LMFDB - Embedded newform 8049.2.a.b.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8049,2,Mod(1,8049)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8049, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8049.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8049 = 3 \cdot 2683 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8049.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:	$64.2715885869$
Analytic rank:	$1$
Dimension:	$104$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.98457 q^{2} -1.00000 q^{3} +1.93852 q^{4} +3.95408 q^{5} +1.98457 q^{6} -0.848310 q^{7} +0.122004 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.98457 q^{2} -1.00000 q^{3} +1.93852 q^{4} +3.95408 q^{5} +1.98457 q^{6} -0.848310 q^{7} +0.122004 q^{8} +1.00000 q^{9} -7.84715 q^{10} +1.74172 q^{11} -1.93852 q^{12} +0.886882 q^{13} +1.68353 q^{14} -3.95408 q^{15} -4.11917 q^{16} +5.15720 q^{17} -1.98457 q^{18} -3.00318 q^{19} +7.66507 q^{20} +0.848310 q^{21} -3.45657 q^{22} -4.08559 q^{23} -0.122004 q^{24} +10.6347 q^{25} -1.76008 q^{26} -1.00000 q^{27} -1.64447 q^{28} -1.27813 q^{29} +7.84715 q^{30} -2.82090 q^{31} +7.93079 q^{32} -1.74172 q^{33} -10.2348 q^{34} -3.35428 q^{35} +1.93852 q^{36} -9.06131 q^{37} +5.96003 q^{38} -0.886882 q^{39} +0.482413 q^{40} +8.97216 q^{41} -1.68353 q^{42} -4.69424 q^{43} +3.37636 q^{44} +3.95408 q^{45} +8.10815 q^{46} -6.64103 q^{47} +4.11917 q^{48} -6.28037 q^{49} -21.1053 q^{50} -5.15720 q^{51} +1.71924 q^{52} +0.864805 q^{53} +1.98457 q^{54} +6.88689 q^{55} -0.103497 q^{56} +3.00318 q^{57} +2.53653 q^{58} -1.74937 q^{59} -7.66507 q^{60} -8.20595 q^{61} +5.59827 q^{62} -0.848310 q^{63} -7.50086 q^{64} +3.50680 q^{65} +3.45657 q^{66} -3.76807 q^{67} +9.99735 q^{68} +4.08559 q^{69} +6.65681 q^{70} +5.73429 q^{71} +0.122004 q^{72} -4.42082 q^{73} +17.9828 q^{74} -10.6347 q^{75} -5.82174 q^{76} -1.47752 q^{77} +1.76008 q^{78} -13.2554 q^{79} -16.2875 q^{80} +1.00000 q^{81} -17.8059 q^{82} +4.32535 q^{83} +1.64447 q^{84} +20.3919 q^{85} +9.31605 q^{86} +1.27813 q^{87} +0.212497 q^{88} -4.48033 q^{89} -7.84715 q^{90} -0.752351 q^{91} -7.92001 q^{92} +2.82090 q^{93} +13.1796 q^{94} -11.8748 q^{95} -7.93079 q^{96} +8.06233 q^{97} +12.4638 q^{98} +1.74172 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 104 q - 9 q^{2} - 104 q^{3} + 87 q^{4} - 15 q^{5} + 9 q^{6} - 10 q^{7} - 27 q^{8} + 104 q^{9}+O(q^{10}) $ 104 * q - 9 * q^2 - 104 * q^3 + 87 * q^4 - 15 * q^5 + 9 * q^6 - 10 * q^7 - 27 * q^8 + 104 * q^9	\( 104 q - 9 q^{2} - 104 q^{3} + 87 q^{4} - 15 q^{5} + 9 q^{6} - 10 q^{7} - 27 q^{8} + 104 q^{9} + 8 q^{10} - 52 q^{11} - 87 q^{12} + 35 q^{13} - 23 q^{14} + 15 q^{15} + 53 q^{16} - 19 q^{17} - 9 q^{18} - 22 q^{19} - 35 q^{20} + 10 q^{21} - q^{22} - 70 q^{23} + 27 q^{24} + 79 q^{25} - 39 q^{26} - 104 q^{27} - 9 q^{28} - 37 q^{29} - 8 q^{30} - 47 q^{31} - 53 q^{32} + 52 q^{33} - 17 q^{34} - 54 q^{35} + 87 q^{36} + 65 q^{37} - 33 q^{38} - 35 q^{39} + 14 q^{40} - 47 q^{41} + 23 q^{42} - 30 q^{43} - 122 q^{44} - 15 q^{45} - 6 q^{46} - 101 q^{47} - 53 q^{48} + 78 q^{49} - 64 q^{50} + 19 q^{51} + 41 q^{52} - 48 q^{53} + 9 q^{54} - 29 q^{55} - 71 q^{56} + 22 q^{57} - 2 q^{58} - 86 q^{59} + 35 q^{60} + 34 q^{61} - 36 q^{62} - 10 q^{63} - 15 q^{64} - 64 q^{65} + q^{66} - 38 q^{67} - 33 q^{68} + 70 q^{69} - 29 q^{70} - 176 q^{71} - 27 q^{72} + 69 q^{73} - 86 q^{74} - 79 q^{75} - 54 q^{76} - 45 q^{77} + 39 q^{78} - 101 q^{79} - 76 q^{80} + 104 q^{81} + 38 q^{82} - 67 q^{83} + 9 q^{84} + 3 q^{85} - 90 q^{86} + 37 q^{87} + 7 q^{88} - 91 q^{89} + 8 q^{90} - 47 q^{91} - 136 q^{92} + 47 q^{93} - 20 q^{94} - 130 q^{95} + 53 q^{96} + 86 q^{97} - 44 q^{98} - 52 q^{99}+O(q^{100}) \) 104 * q - 9 * q^2 - 104 * q^3 + 87 * q^4 - 15 * q^5 + 9 * q^6 - 10 * q^7 - 27 * q^8 + 104 * q^9 + 8 * q^10 - 52 * q^11 - 87 * q^12 + 35 * q^13 - 23 * q^14 + 15 * q^15 + 53 * q^16 - 19 * q^17 - 9 * q^18 - 22 * q^19 - 35 * q^20 + 10 * q^21 - q^22 - 70 * q^23 + 27 * q^24 + 79 * q^25 - 39 * q^26 - 104 * q^27 - 9 * q^28 - 37 * q^29 - 8 * q^30 - 47 * q^31 - 53 * q^32 + 52 * q^33 - 17 * q^34 - 54 * q^35 + 87 * q^36 + 65 * q^37 - 33 * q^38 - 35 * q^39 + 14 * q^40 - 47 * q^41 + 23 * q^42 - 30 * q^43 - 122 * q^44 - 15 * q^45 - 6 * q^46 - 101 * q^47 - 53 * q^48 + 78 * q^49 - 64 * q^50 + 19 * q^51 + 41 * q^52 - 48 * q^53 + 9 * q^54 - 29 * q^55 - 71 * q^56 + 22 * q^57 - 2 * q^58 - 86 * q^59 + 35 * q^60 + 34 * q^61 - 36 * q^62 - 10 * q^63 - 15 * q^64 - 64 * q^65 + q^66 - 38 * q^67 - 33 * q^68 + 70 * q^69 - 29 * q^70 - 176 * q^71 - 27 * q^72 + 69 * q^73 - 86 * q^74 - 79 * q^75 - 54 * q^76 - 45 * q^77 + 39 * q^78 - 101 * q^79 - 76 * q^80 + 104 * q^81 + 38 * q^82 - 67 * q^83 + 9 * q^84 + 3 * q^85 - 90 * q^86 + 37 * q^87 + 7 * q^88 - 91 * q^89 + 8 * q^90 - 47 * q^91 - 136 * q^92 + 47 * q^93 - 20 * q^94 - 130 * q^95 + 53 * q^96 + 86 * q^97 - 44 * q^98 - 52 * 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 $

$2$	−1.98457	−1.40330	−0.701652	−	0.712520i	$-0.747554\pi$
$2$	−1.98457	−1.40330	−0.701652	+	0.712520i	$0.747554\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.93852	0.969262
$5$	3.95408	1.76832	0.884158	−	0.467188i	$-0.154733\pi$
$5$	3.95408	1.76832	0.884158	+	0.467188i	$0.154733\pi$
$6$	1.98457	0.810198
$7$	−0.848310	−0.320631	−0.160316	−	0.987066i	$-0.551251\pi$
$7$	−0.848310	−0.320631	−0.160316	+	0.987066i	$0.551251\pi$
$8$	0.122004	0.0431349
$9$	1.00000	0.333333
$10$	−7.84715	−2.48149
$11$	1.74172	0.525148	0.262574	−	0.964912i	$-0.415429\pi$
$11$	1.74172	0.525148	0.262574	+	0.964912i	$0.415429\pi$
$12$	−1.93852	−0.559604
$13$	0.886882	0.245977	0.122988	−	0.992408i	$-0.460752\pi$
$13$	0.886882	0.245977	0.122988	+	0.992408i	$0.460752\pi$
$14$	1.68353	0.449943
$15$	−3.95408	−1.02094
$16$	−4.11917	−1.02979
$17$	5.15720	1.25080	0.625402	−	0.780303i	$-0.284935\pi$
$17$	5.15720	1.25080	0.625402	+	0.780303i	$0.284935\pi$
$18$	−1.98457	−0.467768
$19$	−3.00318	−0.688978	−0.344489	−	0.938790i	$-0.611948\pi$
$19$	−3.00318	−0.688978	−0.344489	+	0.938790i	$0.611948\pi$
$20$	7.66507	1.71396
$21$	0.848310	0.185116
$22$	−3.45657	−0.736942
$23$	−4.08559	−0.851905	−0.425952	−	0.904746i	$-0.640061\pi$
$23$	−4.08559	−0.851905	−0.425952	+	0.904746i	$0.640061\pi$
$24$	−0.122004	−0.0249040
$25$	10.6347	2.12694
$26$	−1.76008	−0.345180
$27$	−1.00000	−0.192450
$28$	−1.64447	−0.310776
$29$	−1.27813	−0.237342	−0.118671	−	0.992934i	$-0.537863\pi$
$29$	−1.27813	−0.237342	−0.118671	+	0.992934i	$0.537863\pi$
$30$	7.84715	1.43269
$31$	−2.82090	−0.506648	−0.253324	−	0.967382i	$-0.581524\pi$
$31$	−2.82090	−0.506648	−0.253324	+	0.967382i	$0.581524\pi$
$32$	7.93079	1.40198
$33$	−1.74172	−0.303194
$34$	−10.2348	−1.75526
$35$	−3.35428	−0.566977
$36$	1.93852	0.323087
$37$	−9.06131	−1.48967	−0.744835	−	0.667249i	$-0.767472\pi$
$37$	−9.06131	−1.48967	−0.744835	+	0.667249i	$0.767472\pi$
$38$	5.96003	0.966845
$39$	−0.886882	−0.142015
$40$	0.482413	0.0762762
$41$	8.97216	1.40122	0.700608	−	0.713546i	$-0.252912\pi$
$41$	8.97216	1.40122	0.700608	+	0.713546i	$0.252912\pi$
$42$	−1.68353	−0.259775
$43$	−4.69424	−0.715864	−0.357932	−	0.933748i	$-0.616518\pi$
$43$	−4.69424	−0.715864	−0.357932	+	0.933748i	$0.616518\pi$
$44$	3.37636	0.509006
$45$	3.95408	0.589439
$46$	8.10815	1.19548
$47$	−6.64103	−0.968693	−0.484347	−	0.874876i	$-0.660943\pi$
$47$	−6.64103	−0.968693	−0.484347	+	0.874876i	$0.660943\pi$
$48$	4.11917	0.594551
$49$	−6.28037	−0.897196
$50$	−21.1053	−2.98475
$51$	−5.15720	−0.722152
$52$	1.71924	0.238416
$53$	0.864805	0.118790	0.0593951	−	0.998235i	$-0.481083\pi$
$53$	0.864805	0.118790	0.0593951	+	0.998235i	$0.481083\pi$
$54$	1.98457	0.270066
$55$	6.88689	0.928628
$56$	−0.103497	−0.0138304
$57$	3.00318	0.397782
$58$	2.53653	0.333063
$59$	−1.74937	−0.227749	−0.113874	−	0.993495i	$-0.536326\pi$
$59$	−1.74937	−0.227749	−0.113874	+	0.993495i	$0.536326\pi$
$60$	−7.66507	−0.989556
$61$	−8.20595	−1.05066	−0.525332	−	0.850897i	$-0.676059\pi$
$61$	−8.20595	−1.05066	−0.525332	+	0.850897i	$0.676059\pi$
$62$	5.59827	0.710981
$63$	−0.848310	−0.106877
$64$	−7.50086	−0.937608
$65$	3.50680	0.434965
$66$	3.45657	0.425474
$67$	−3.76807	−0.460343	−0.230172	−	0.973150i	$-0.573929\pi$
$67$	−3.76807	−0.460343	−0.230172	+	0.973150i	$0.573929\pi$
$68$	9.99735	1.21236
$69$	4.08559	0.491847
$70$	6.65681	0.795641
$71$	5.73429	0.680535	0.340268	−	0.940329i	$-0.389482\pi$
$71$	5.73429	0.680535	0.340268	+	0.940329i	$0.389482\pi$
$72$	0.122004	0.0143783
$73$	−4.42082	−0.517417	−0.258709	−	0.965955i	$-0.583297\pi$
$73$	−4.42082	−0.517417	−0.258709	+	0.965955i	$0.583297\pi$
$74$	17.9828	2.09046
$75$	−10.6347	−1.22799
$76$	−5.82174	−0.667800
$77$	−1.47752	−0.168379
$78$	1.76008	0.199290
$79$	−13.2554	−1.49135	−0.745676	−	0.666308i	$-0.767873\pi$
$79$	−13.2554	−1.49135	−0.745676	+	0.666308i	$0.767873\pi$
$80$	−16.2875	−1.82100
$81$	1.00000	0.111111
$82$	−17.8059	−1.96633
$83$	4.32535	0.474769	0.237384	−	0.971416i	$-0.423710\pi$
$83$	4.32535	0.474769	0.237384	+	0.971416i	$0.423710\pi$
$84$	1.64447	0.179426
$85$	20.3919	2.21182
$86$	9.31605	1.00458
$87$	1.27813	0.137029
$88$	0.212497	0.0226522
$89$	−4.48033	−0.474914	−0.237457	−	0.971398i	$-0.576314\pi$
$89$	−4.48033	−0.474914	−0.237457	+	0.971398i	$0.576314\pi$
$90$	−7.84715	−0.827162
$91$	−0.752351	−0.0788679
$92$	−7.92001	−0.825719
$93$	2.82090	0.292513
$94$	13.1796	1.35937
$95$	−11.8748	−1.21833
$96$	−7.93079	−0.809432
$97$	8.06233	0.818605	0.409303	−	0.912399i	$-0.365772\pi$
$97$	8.06233	0.818605	0.409303	+	0.912399i	$0.365772\pi$
$98$	12.4638	1.25904
$99$	1.74172	0.175049
$100$	20.6156	2.06156
$101$	−3.86203	−0.384286	−0.192143	−	0.981367i	$-0.561544\pi$
$101$	−3.86203	−0.384286	−0.192143	+	0.981367i	$0.561544\pi$
$102$	10.2348	1.01340
$103$	−15.7204	−1.54898	−0.774489	−	0.632588i	$-0.781993\pi$
$103$	−15.7204	−1.54898	−0.774489	+	0.632588i	$0.781993\pi$
$104$	0.108203	0.0106102
$105$	3.35428	0.327344
$106$	−1.71627	−0.166699
$107$	−15.4933	−1.49780	−0.748899	−	0.662684i	$-0.769417\pi$
$107$	−15.4933	−1.49780	−0.748899	+	0.662684i	$0.769417\pi$
$108$	−1.93852	−0.186535
$109$	−6.31203	−0.604583	−0.302292	−	0.953216i	$-0.597752\pi$
$109$	−6.31203	−0.604583	−0.302292	+	0.953216i	$0.597752\pi$
$110$	−13.6675	−1.30315
$111$	9.06131	0.860061
$112$	3.49434	0.330184
$113$	5.64307	0.530855	0.265428	−	0.964131i	$-0.414487\pi$
$113$	5.64307	0.530855	0.265428	+	0.964131i	$0.414487\pi$
$114$	−5.96003	−0.558208
$115$	−16.1547	−1.50644
$116$	−2.47768	−0.230047
$117$	0.886882	0.0819923
$118$	3.47176	0.319601
$119$	−4.37490	−0.401047
$120$	−0.482413	−0.0440381
$121$	−7.96641	−0.724219
$122$	16.2853	1.47440
$123$	−8.97216	−0.808992
$124$	−5.46837	−0.491074
$125$	22.2801	1.99279
$126$	1.68353	0.149981
$127$	−9.09257	−0.806835	−0.403418	−	0.915016i	$-0.632178\pi$
$127$	−9.09257	−0.806835	−0.403418	+	0.915016i	$0.632178\pi$
$128$	−0.975571	−0.0862291
$129$	4.69424	0.413304
$130$	−6.95950	−0.610388
$131$	−18.4896	−1.61544	−0.807722	−	0.589564i	$-0.799300\pi$
$131$	−18.4896	−1.61544	−0.807722	+	0.589564i	$0.799300\pi$
$132$	−3.37636	−0.293875
$133$	2.54763	0.220908
$134$	7.47801	0.646001
$135$	−3.95408	−0.340313
$136$	0.629199	0.0539534
$137$	−2.04250	−0.174503	−0.0872514	−	0.996186i	$-0.527808\pi$
$137$	−2.04250	−0.174503	−0.0872514	+	0.996186i	$0.527808\pi$
$138$	−8.10815	−0.690211
$139$	−11.0963	−0.941179	−0.470589	−	0.882352i	$-0.655959\pi$
$139$	−11.0963	−0.941179	−0.470589	+	0.882352i	$0.655959\pi$
$140$	−6.50236	−0.549549
$141$	6.64103	0.559275
$142$	−11.3801	−0.954998
$143$	1.54470	0.129174
$144$	−4.11917	−0.343264
$145$	−5.05381	−0.419696
$146$	8.77343	0.726094
$147$	6.28037	0.517996
$148$	−17.5656	−1.44388
$149$	−4.80127	−0.393335	−0.196668	−	0.980470i	$-0.563012\pi$
$149$	−4.80127	−0.393335	−0.196668	+	0.980470i	$0.563012\pi$
$150$	21.1053	1.72324
$151$	−4.71325	−0.383559	−0.191779	−	0.981438i	$-0.561426\pi$
$151$	−4.71325	−0.383559	−0.191779	+	0.981438i	$0.561426\pi$
$152$	−0.366401	−0.0297190
$153$	5.15720	0.416935
$154$	2.93224	0.236287
$155$	−11.1540	−0.895914
$156$	−1.71924	−0.137650
$157$	−7.90039	−0.630520	−0.315260	−	0.949005i	$-0.602092\pi$
$157$	−7.90039	−0.630520	−0.315260	+	0.949005i	$0.602092\pi$
$158$	26.3063	2.09282
$159$	−0.864805	−0.0685835
$160$	31.3589	2.47914
$161$	3.46585	0.273147
$162$	−1.98457	−0.155923
$163$	19.2102	1.50466	0.752329	−	0.658788i	$-0.228931\pi$
$163$	19.2102	1.50466	0.752329	+	0.658788i	$0.228931\pi$
$164$	17.3927	1.35815
$165$	−6.88689	−0.536144
$166$	−8.58396	−0.666245
$167$	−10.8244	−0.837617	−0.418808	−	0.908075i	$-0.637552\pi$
$167$	−10.8244	−0.837617	−0.418808	+	0.908075i	$0.637552\pi$
$168$	0.103497	0.00798499
$169$	−12.2134	−0.939495
$170$	−40.4693	−3.10385
$171$	−3.00318	−0.229659
$172$	−9.09989	−0.693860
$173$	−12.3031	−0.935384	−0.467692	−	0.883892i	$-0.654914\pi$
$173$	−12.3031	−0.935384	−0.467692	+	0.883892i	$0.654914\pi$
$174$	−2.53653	−0.192294
$175$	−9.02154	−0.681964
$176$	−7.17444	−0.540794
$177$	1.74937	0.131491
$178$	8.89153	0.666449
$179$	−3.70939	−0.277253	−0.138626	−	0.990345i	$-0.544269\pi$
$179$	−3.70939	−0.277253	−0.138626	+	0.990345i	$0.544269\pi$
$180$	7.66507	0.571321
$181$	16.5977	1.23370	0.616851	−	0.787080i	$-0.288408\pi$
$181$	16.5977	1.23370	0.616851	+	0.787080i	$0.288408\pi$
$182$	1.49310	0.110676
$183$	8.20595	0.606602
$184$	−0.498459	−0.0367469
$185$	−35.8291	−2.63421
$186$	−5.59827	−0.410485
$187$	8.98239	0.656857
$188$	−12.8738	−0.938918
$189$	0.848310	0.0617055
$190$	23.5664	1.70969
$191$	−6.58032	−0.476135	−0.238067	−	0.971249i	$-0.576514\pi$
$191$	−6.58032	−0.476135	−0.238067	+	0.971249i	$0.576514\pi$
$192$	7.50086	0.541328
$193$	−0.689777	−0.0496512	−0.0248256	−	0.999692i	$-0.507903\pi$
$193$	−0.689777	−0.0496512	−0.0248256	+	0.999692i	$0.507903\pi$
$194$	−16.0003	−1.14875
$195$	−3.50680	−0.251127
$196$	−12.1746	−0.869618
$197$	6.43942	0.458790	0.229395	−	0.973333i	$-0.426325\pi$
$197$	6.43942	0.458790	0.229395	+	0.973333i	$0.426325\pi$
$198$	−3.45657	−0.245647
$199$	21.0426	1.49167	0.745836	−	0.666129i	$-0.232050\pi$
$199$	21.0426	1.49167	0.745836	+	0.666129i	$0.232050\pi$
$200$	1.29748	0.0917456
$201$	3.76807	0.265779
$202$	7.66448	0.539271
$203$	1.08425	0.0760992
$204$	−9.99735	−0.699955
$205$	35.4766	2.47779
$206$	31.1983	2.17369
$207$	−4.08559	−0.283968
$208$	−3.65322	−0.253305
$209$	−5.23070	−0.361815
$210$	−6.65681	−0.459364
$211$	−20.2854	−1.39651	−0.698253	−	0.715851i	$-0.746039\pi$
$211$	−20.2854	−1.39651	−0.698253	+	0.715851i	$0.746039\pi$
$212$	1.67645	0.115139
$213$	−5.73429	−0.392907
$214$	30.7477	2.10187
$215$	−18.5614	−1.26587
$216$	−0.122004	−0.00830132
$217$	2.39299	0.162447
$218$	12.5267	0.848414
$219$	4.42082	0.298731
$220$	13.3504	0.900084
$221$	4.57383	0.307669
$222$	−17.9828	−1.20693
$223$	20.7548	1.38985	0.694924	−	0.719083i	$-0.255438\pi$
$223$	20.7548	1.38985	0.694924	+	0.719083i	$0.255438\pi$
$224$	−6.72777	−0.449518
$225$	10.6347	0.708981
$226$	−11.1991	−0.744951
$227$	11.1757	0.741757	0.370878	−	0.928681i	$-0.379057\pi$
$227$	11.1757	0.741757	0.370878	+	0.928681i	$0.379057\pi$
$228$	5.82174	0.385554
$229$	−11.2445	−0.743059	−0.371529	−	0.928421i	$-0.621166\pi$
$229$	−11.2445	−0.743059	−0.371529	+	0.928421i	$0.621166\pi$
$230$	32.0602	2.11399
$231$	1.47752	0.0972136
$232$	−0.155937	−0.0102377
$233$	−21.2010	−1.38892	−0.694461	−	0.719531i	$-0.744357\pi$
$233$	−21.2010	−1.38892	−0.694461	+	0.719531i	$0.744357\pi$
$234$	−1.76008	−0.115060
$235$	−26.2591	−1.71296
$236$	−3.39120	−0.220748
$237$	13.2554	0.861033
$238$	8.68231	0.562790
$239$	−11.1279	−0.719803	−0.359901	−	0.932990i	$-0.617190\pi$
$239$	−11.1279	−0.719803	−0.359901	+	0.932990i	$0.617190\pi$
$240$	16.2875	1.05136
$241$	−2.55675	−0.164695	−0.0823473	−	0.996604i	$-0.526242\pi$
$241$	−2.55675	−0.164695	−0.0823473	+	0.996604i	$0.526242\pi$
$242$	15.8099	1.01630
$243$	−1.00000	−0.0641500
$244$	−15.9074	−1.01837
$245$	−24.8331	−1.58653
$246$	17.8059	1.13526
$247$	−2.66347	−0.169473
$248$	−0.344161	−0.0218542
$249$	−4.32535	−0.274108
$250$	−44.2164	−2.79649
$251$	−0.784701	−0.0495299	−0.0247650	−	0.999693i	$-0.507884\pi$
$251$	−0.784701	−0.0495299	−0.0247650	+	0.999693i	$0.507884\pi$
$252$	−1.64447	−0.103592
$253$	−7.11595	−0.447376
$254$	18.0449	1.13224
$255$	−20.3919	−1.27699
$256$	16.9378	1.05861
$257$	4.28330	0.267185	0.133592	−	0.991036i	$-0.457349\pi$
$257$	4.28330	0.267185	0.133592	+	0.991036i	$0.457349\pi$
$258$	−9.31605	−0.579992
$259$	7.68680	0.477634
$260$	6.79802	0.421595
$261$	−1.27813	−0.0791140
$262$	36.6939	2.26696
$263$	31.2445	1.92662	0.963310	−	0.268392i	$-0.0864921\pi$
$263$	31.2445	1.92662	0.963310	+	0.268392i	$0.0864921\pi$
$264$	−0.212497	−0.0130783
$265$	3.41951	0.210059
$266$	−5.05596	−0.310001
$267$	4.48033	0.274192
$268$	−7.30449	−0.446193
$269$	−7.73573	−0.471656	−0.235828	−	0.971795i	$-0.575780\pi$
$269$	−7.73573	−0.471656	−0.235828	+	0.971795i	$0.575780\pi$
$270$	7.84715	0.477562
$271$	13.3939	0.813621	0.406810	−	0.913513i	$-0.366641\pi$
$271$	13.3939	0.813621	0.406810	+	0.913513i	$0.366641\pi$
$272$	−21.2434	−1.28807
$273$	0.752351	0.0455344
$274$	4.05349	0.244880
$275$	18.5227	1.11696
$276$	7.92001	0.476729
$277$	−2.27322	−0.136585	−0.0682923	−	0.997665i	$-0.521755\pi$
$277$	−2.27322	−0.136585	−0.0682923	+	0.997665i	$0.521755\pi$
$278$	22.0215	1.32076
$279$	−2.82090	−0.168883
$280$	−0.409236	−0.0244565
$281$	23.2792	1.38872	0.694360	−	0.719628i	$-0.255688\pi$
$281$	23.2792	1.38872	0.694360	+	0.719628i	$0.255688\pi$
$282$	−13.1796	−0.784833
$283$	2.74644	0.163259	0.0816296	−	0.996663i	$-0.473988\pi$
$283$	2.74644	0.163259	0.0816296	+	0.996663i	$0.473988\pi$
$284$	11.1161	0.659617
$285$	11.8748	0.703404
$286$	−3.06557	−0.181271
$287$	−7.61117	−0.449273
$288$	7.93079	0.467326
$289$	9.59669	0.564511
$290$	10.0296	0.588961
$291$	−8.06233	−0.472622
$292$	−8.56986	−0.501513
$293$	14.0102	0.818487	0.409243	−	0.912425i	$-0.365793\pi$
$293$	14.0102	0.818487	0.409243	+	0.912425i	$0.365793\pi$
$294$	−12.4638	−0.726906
$295$	−6.91715	−0.402732
$296$	−1.10552	−0.0642568
$297$	−1.74172	−0.101065
$298$	9.52846	0.551969
$299$	−3.62344	−0.209549
$300$	−20.6156	−1.19024
$301$	3.98217	0.229528
$302$	9.35378	0.538249
$303$	3.86203	0.221868
$304$	12.3706	0.709505
$305$	−32.4470	−1.85791
$306$	−10.2348	−0.585086
$307$	16.7112	0.953761	0.476880	−	0.878968i	$-0.341767\pi$
$307$	16.7112	0.953761	0.476880	+	0.878968i	$0.341767\pi$
$308$	−2.86420	−0.163203
$309$	15.7204	0.894303
$310$	22.1360	1.25724
$311$	−25.9783	−1.47310	−0.736548	−	0.676385i	$-0.763546\pi$
$311$	−25.9783	−1.47310	−0.736548	+	0.676385i	$0.763546\pi$
$312$	−0.108203	−0.00612580
$313$	13.7853	0.779193	0.389597	−	0.920986i	$-0.372614\pi$
$313$	13.7853	0.779193	0.389597	+	0.920986i	$0.372614\pi$
$314$	15.6789	0.884811
$315$	−3.35428	−0.188992
$316$	−25.6960	−1.44551
$317$	9.31023	0.522915	0.261457	−	0.965215i	$-0.415797\pi$
$317$	9.31023	0.522915	0.261457	+	0.965215i	$0.415797\pi$
$318$	1.71627	0.0962435
$319$	−2.22614	−0.124640
$320$	−29.6590	−1.65799
$321$	15.4933	0.864754
$322$	−6.87822	−0.383308
$323$	−15.4880	−0.861776
$324$	1.93852	0.107696
$325$	9.43174	0.523179
$326$	−38.1240	−2.11149
$327$	6.31203	0.349056
$328$	1.09464	0.0604414
$329$	5.63365	0.310593
$330$	13.6675	0.752372
$331$	32.8606	1.80618	0.903092	−	0.429447i	$-0.141292\pi$
$331$	32.8606	1.80618	0.903092	+	0.429447i	$0.141292\pi$
$332$	8.38479	0.460175
$333$	−9.06131	−0.496557
$334$	21.4818	1.17543
$335$	−14.8992	−0.814032
$336$	−3.49434	−0.190632
$337$	6.24516	0.340195	0.170098	−	0.985427i	$-0.445592\pi$
$337$	6.24516	0.340195	0.170098	+	0.985427i	$0.445592\pi$
$338$	24.2384	1.31840
$339$	−5.64307	−0.306489
$340$	39.5303	2.14383
$341$	−4.91321	−0.266065
$342$	5.96003	0.322282
$343$	11.2659	0.608300
$344$	−0.572716	−0.0308788
$345$	16.1547	0.869742
$346$	24.4163	1.31263
$347$	31.8916	1.71203	0.856016	−	0.516950i	$-0.172933\pi$
$347$	31.8916	1.71203	0.856016	+	0.516950i	$0.172933\pi$
$348$	2.47768	0.132817
$349$	−8.25997	−0.442146	−0.221073	−	0.975257i	$-0.570956\pi$
$349$	−8.25997	−0.442146	−0.221073	+	0.975257i	$0.570956\pi$
$350$	17.9039	0.957003
$351$	−0.886882	−0.0473383
$352$	13.8132	0.736246
$353$	−22.6286	−1.20440	−0.602200	−	0.798345i	$-0.705709\pi$
$353$	−22.6286	−1.20440	−0.602200	+	0.798345i	$0.705709\pi$
$354$	−3.47176	−0.184522
$355$	22.6738	1.20340
$356$	−8.68522	−0.460316
$357$	4.37490	0.231544
$358$	7.36155	0.389070
$359$	−4.68760	−0.247402	−0.123701	−	0.992320i	$-0.539476\pi$
$359$	−4.68760	−0.247402	−0.123701	+	0.992320i	$0.539476\pi$
$360$	0.482413	0.0254254
$361$	−9.98088	−0.525310
$362$	−32.9394	−1.73126
$363$	7.96641	0.418128
$364$	−1.45845	−0.0764436
$365$	−17.4802	−0.914958
$366$	−16.2853	−0.851246
$367$	−27.8948	−1.45610	−0.728048	−	0.685526i	$-0.759572\pi$
$367$	−27.8948	−1.45610	−0.728048	+	0.685526i	$0.759572\pi$
$368$	16.8293	0.877286
$369$	8.97216	0.467072
$370$	71.1054	3.69659
$371$	−0.733623	−0.0380878
$372$	5.46837	0.283522
$373$	3.44737	0.178498	0.0892489	−	0.996009i	$-0.471553\pi$
$373$	3.44737	0.178498	0.0892489	+	0.996009i	$0.471553\pi$
$374$	−17.8262	−0.921771
$375$	−22.2801	−1.15054
$376$	−0.810232	−0.0417845
$377$	−1.13355	−0.0583807
$378$	−1.68353	−0.0865916
$379$	25.3726	1.30330	0.651652	−	0.758518i	$-0.274076\pi$
$379$	25.3726	1.30330	0.651652	+	0.758518i	$0.274076\pi$
$380$	−23.0196	−1.18088
$381$	9.09257	0.465827
$382$	13.0591	0.668162
$383$	−5.93977	−0.303508	−0.151754	−	0.988418i	$-0.548492\pi$
$383$	−5.93977	−0.303508	−0.151754	+	0.988418i	$0.548492\pi$
$384$	0.975571	0.0497844
$385$	−5.84222	−0.297747
$386$	1.36891	0.0696758
$387$	−4.69424	−0.238621
$388$	15.6290	0.793443
$389$	6.09951	0.309258	0.154629	−	0.987973i	$-0.450582\pi$
$389$	6.09951	0.309258	0.154629	+	0.987973i	$0.450582\pi$
$390$	6.95950	0.352408
$391$	−21.0702	−1.06557
$392$	−0.766230	−0.0387005
$393$	18.4896	0.932677
$394$	−12.7795	−0.643822
$395$	−52.4130	−2.63718
$396$	3.37636	0.169669
$397$	−8.75822	−0.439563	−0.219781	−	0.975549i	$-0.570534\pi$
$397$	−8.75822	−0.439563	−0.219781	+	0.975549i	$0.570534\pi$
$398$	−41.7606	−2.09327
$399$	−2.54763	−0.127541
$400$	−43.8062	−2.19031
$401$	6.52103	0.325645	0.162822	−	0.986655i	$-0.447940\pi$
$401$	6.52103	0.325645	0.162822	+	0.986655i	$0.447940\pi$
$402$	−7.47801	−0.372969
$403$	−2.50180	−0.124624
$404$	−7.48664	−0.372474
$405$	3.95408	0.196480
$406$	−2.15177	−0.106790
$407$	−15.7823	−0.782297
$408$	−0.629199	−0.0311500
$409$	10.1629	0.502521	0.251261	−	0.967919i	$-0.419155\pi$
$409$	10.1629	0.502521	0.251261	+	0.967919i	$0.419155\pi$
$410$	−70.4058	−3.47710
$411$	2.04250	0.100749
$412$	−30.4744	−1.50137
$413$	1.48401	0.0730234
$414$	8.10815	0.398494
$415$	17.1028	0.839541
$416$	7.03367	0.344854
$417$	11.0963	0.543390
$418$	10.3807	0.507737
$419$	9.51380	0.464780	0.232390	−	0.972623i	$-0.425346\pi$
$419$	9.51380	0.464780	0.232390	+	0.972623i	$0.425346\pi$
$420$	6.50236	0.317283
$421$	32.3603	1.57714	0.788572	−	0.614943i	$-0.210821\pi$
$421$	32.3603	1.57714	0.788572	+	0.614943i	$0.210821\pi$
$422$	40.2579	1.95972
$423$	−6.64103	−0.322898
$424$	0.105510	0.00512401
$425$	54.8453	2.66039
$426$	11.3801	0.551368
$427$	6.96119	0.336876
$428$	−30.0342	−1.45176
$429$	−1.54470	−0.0745788
$430$	36.8364	1.77641
$431$	−5.41376	−0.260772	−0.130386	−	0.991463i	$-0.541622\pi$
$431$	−5.41376	−0.260772	−0.130386	+	0.991463i	$0.541622\pi$
$432$	4.11917	0.198184
$433$	−15.6455	−0.751874	−0.375937	−	0.926645i	$-0.622679\pi$
$433$	−15.6455	−0.751874	−0.375937	+	0.926645i	$0.622679\pi$
$434$	−4.74907	−0.227963
$435$	5.05381	0.242311
$436$	−12.2360	−0.585999
$437$	12.2698	0.586943
$438$	−8.77343	−0.419210
$439$	27.4859	1.31183	0.655914	−	0.754835i	$-0.272283\pi$
$439$	27.4859	1.31183	0.655914	+	0.754835i	$0.272283\pi$
$440$	0.840228	0.0400563
$441$	−6.28037	−0.299065
$442$	−9.07709	−0.431753
$443$	−17.4922	−0.831081	−0.415541	−	0.909575i	$-0.636408\pi$
$443$	−17.4922	−0.831081	−0.415541	+	0.909575i	$0.636408\pi$
$444$	17.5656	0.833625
$445$	−17.7156	−0.839798
$446$	−41.1895	−1.95038
$447$	4.80127	0.227092
$448$	6.36306	0.300626
$449$	25.2136	1.18990	0.594951	−	0.803762i	$-0.297172\pi$
$449$	25.2136	1.18990	0.594951	+	0.803762i	$0.297172\pi$
$450$	−21.1053	−0.994916
$451$	15.6270	0.735846
$452$	10.9392	0.514538
$453$	4.71325	0.221448
$454$	−22.1790	−1.04091
$455$	−2.97485	−0.139463
$456$	0.366401	0.0171583
$457$	18.6095	0.870517	0.435258	−	0.900306i	$-0.356657\pi$
$457$	18.6095	0.870517	0.435258	+	0.900306i	$0.356657\pi$
$458$	22.3155	1.04274
$459$	−5.15720	−0.240717
$460$	−31.3163	−1.46013
$461$	35.4103	1.64922	0.824610	−	0.565701i	$-0.191394\pi$
$461$	35.4103	1.64922	0.824610	+	0.565701i	$0.191394\pi$
$462$	−2.93224	−0.136420
$463$	18.1550	0.843732	0.421866	−	0.906658i	$-0.361375\pi$
$463$	18.1550	0.843732	0.421866	+	0.906658i	$0.361375\pi$
$464$	5.26482	0.244413
$465$	11.1540	0.517256
$466$	42.0748	1.94908
$467$	12.8079	0.592679	0.296340	−	0.955083i	$-0.404234\pi$
$467$	12.8079	0.592679	0.296340	+	0.955083i	$0.404234\pi$
$468$	1.71924	0.0794720
$469$	3.19649	0.147600
$470$	52.1131	2.40380
$471$	7.90039	0.364031
$472$	−0.213431	−0.00982394
$473$	−8.17604	−0.375935
$474$	−26.3063	−1.20829
$475$	−31.9380	−1.46542
$476$	−8.48085	−0.388719
$477$	0.864805	0.0395967
$478$	22.0841	1.01010
$479$	35.5005	1.62206	0.811030	−	0.585004i	$-0.198907\pi$
$479$	35.5005	1.62206	0.811030	+	0.585004i	$0.198907\pi$
$480$	−31.3589	−1.43133
$481$	−8.03631	−0.366424
$482$	5.07405	0.231117
$483$	−3.46585	−0.157702
$484$	−15.4431	−0.701958
$485$	31.8791	1.44755
$486$	1.98457	0.0900220
$487$	−17.1704	−0.778067	−0.389033	−	0.921224i	$-0.627191\pi$
$487$	−17.1704	−0.778067	−0.389033	+	0.921224i	$0.627191\pi$
$488$	−1.00116	−0.0453204
$489$	−19.2102	−0.868714
$490$	49.2830	2.22638
$491$	−35.2966	−1.59291	−0.796457	−	0.604696i	$-0.793295\pi$
$491$	−35.2966	−1.59291	−0.796457	+	0.604696i	$0.793295\pi$
$492$	−17.3927	−0.784125
$493$	−6.59155	−0.296868
$494$	5.28585	0.237822
$495$	6.88689	0.309543
$496$	11.6198	0.521743
$497$	−4.86446	−0.218201
$498$	8.58396	0.384657
$499$	−20.5266	−0.918896	−0.459448	−	0.888205i	$-0.651953\pi$
$499$	−20.5266	−0.918896	−0.459448	+	0.888205i	$0.651953\pi$
$500$	43.1905	1.93154
$501$	10.8244	0.483598
$502$	1.55730	0.0695055
$503$	−12.3221	−0.549417	−0.274708	−	0.961528i	$-0.588581\pi$
$503$	−12.3221	−0.549417	−0.274708	+	0.961528i	$0.588581\pi$
$504$	−0.103497	−0.00461013
$505$	−15.2708	−0.679540
$506$	14.1221	0.627805
$507$	12.2134	0.542418
$508$	−17.6262	−0.782035
$509$	−22.6967	−1.00601	−0.503007	−	0.864282i	$-0.667773\pi$
$509$	−22.6967	−1.00601	−0.503007	+	0.864282i	$0.667773\pi$
$510$	40.4693	1.79201
$511$	3.75022	0.165900
$512$	−31.6632	−1.39933
$513$	3.00318	0.132594
$514$	−8.50051	−0.374941
$515$	−62.1597	−2.73908
$516$	9.09989	0.400600
$517$	−11.5668	−0.508708
$518$	−15.2550	−0.670266
$519$	12.3031	0.540044
$520$	0.427844	0.0187622
$521$	−33.5267	−1.46883	−0.734415	−	0.678700i	$-0.762544\pi$
$521$	−33.5267	−1.46883	−0.734415	+	0.678700i	$0.762544\pi$
$522$	2.53653	0.111021
$523$	−11.3161	−0.494819	−0.247410	−	0.968911i	$-0.579579\pi$
$523$	−11.3161	−0.494819	−0.247410	+	0.968911i	$0.579579\pi$
$524$	−35.8425	−1.56579
$525$	9.02154	0.393732
$526$	−62.0070	−2.70363
$527$	−14.5479	−0.633717
$528$	7.17444	0.312228
$529$	−6.30795	−0.274259
$530$	−6.78625	−0.294776
$531$	−1.74937	−0.0759163
$532$	4.93865	0.214117
$533$	7.95725	0.344667
$534$	−8.89153	−0.384774
$535$	−61.2619	−2.64858
$536$	−0.459720	−0.0198569
$537$	3.70939	0.160072
$538$	15.3521	0.661876
$539$	−10.9386	−0.471161
$540$	−7.66507	−0.329852
$541$	19.2040	0.825643	0.412821	−	0.910812i	$-0.364543\pi$
$541$	19.2040	0.825643	0.412821	+	0.910812i	$0.364543\pi$
$542$	−26.5811	−1.14176
$543$	−16.5977	−0.712278
$544$	40.9006	1.75360
$545$	−24.9583	−1.06909
$546$	−1.49310	−0.0638986
$547$	13.8264	0.591175	0.295587	−	0.955316i	$-0.404485\pi$
$547$	13.8264	0.591175	0.295587	+	0.955316i	$0.404485\pi$
$548$	−3.95944	−0.169139
$549$	−8.20595	−0.350222
$550$	−36.7596	−1.56743
$551$	3.83845	0.163523
$552$	0.498459	0.0212158
$553$	11.2447	0.478174
$554$	4.51137	0.191670
$555$	35.8291	1.52086
$556$	−21.5105	−0.912249
$557$	−0.317524	−0.0134539	−0.00672697	−	0.999977i	$-0.502141\pi$
$557$	−0.317524	−0.0134539	−0.00672697	+	0.999977i	$0.502141\pi$
$558$	5.59827	0.236994
$559$	−4.16324	−0.176086
$560$	13.8169	0.583869
$561$	−8.98239	−0.379237
$562$	−46.1992	−1.94880
$563$	35.5734	1.49924	0.749619	−	0.661869i	$-0.230237\pi$
$563$	35.5734	1.49924	0.749619	+	0.661869i	$0.230237\pi$
$564$	12.8738	0.542084
$565$	22.3131	0.938720
$566$	−5.45052	−0.229102
$567$	−0.848310	−0.0356257
$568$	0.699607	0.0293549
$569$	25.2167	1.05714	0.528569	−	0.848890i	$-0.322729\pi$
$569$	25.2167	1.05714	0.528569	+	0.848890i	$0.322729\pi$
$570$	−23.5664	−0.987089
$571$	−6.26022	−0.261982	−0.130991	−	0.991384i	$-0.541816\pi$
$571$	−6.26022	−0.261982	−0.130991	+	0.991384i	$0.541816\pi$
$572$	2.99444	0.125204
$573$	6.58032	0.274897
$574$	15.1049	0.630467
$575$	−43.4491	−1.81195
$576$	−7.50086	−0.312536
$577$	3.66730	0.152672	0.0763358	−	0.997082i	$-0.475678\pi$
$577$	3.66730	0.152672	0.0763358	+	0.997082i	$0.475678\pi$
$578$	−19.0453	−0.792180
$579$	0.689777	0.0286661
$580$	−9.79692	−0.406795
$581$	−3.66924	−0.152226
$582$	16.0003	0.663232
$583$	1.50625	0.0623824
$584$	−0.539357	−0.0223188
$585$	3.50680	0.144988
$586$	−27.8043	−1.14859
$587$	9.08227	0.374866	0.187433	−	0.982277i	$-0.439983\pi$
$587$	9.08227	0.374866	0.187433	+	0.982277i	$0.439983\pi$
$588$	12.1746	0.502074
$589$	8.47167	0.349069
$590$	13.7276	0.565156
$591$	−6.43942	−0.264882
$592$	37.3251	1.53405
$593$	24.8119	1.01890	0.509451	−	0.860500i	$-0.329849\pi$
$593$	24.8119	1.01890	0.509451	+	0.860500i	$0.329849\pi$
$594$	3.45657	0.141825
$595$	−17.2987	−0.709178
$596$	−9.30737	−0.381245
$597$	−21.0426	−0.861217
$598$	7.19097	0.294061
$599$	−35.7004	−1.45868	−0.729339	−	0.684153i	$-0.760172\pi$
$599$	−35.7004	−1.45868	−0.729339	+	0.684153i	$0.760172\pi$
$600$	−1.29748	−0.0529693
$601$	16.8634	0.687874	0.343937	−	0.938993i	$-0.388239\pi$
$601$	16.8634	0.687874	0.343937	+	0.938993i	$0.388239\pi$
$602$	−7.90290	−0.322098
$603$	−3.76807	−0.153448
$604$	−9.13674	−0.371769
$605$	−31.4998	−1.28065
$606$	−7.66448	−0.311348
$607$	−25.4929	−1.03473	−0.517363	−	0.855766i	$-0.673086\pi$
$607$	−25.4929	−1.03473	−0.517363	+	0.855766i	$0.673086\pi$
$608$	−23.8176	−0.965932
$609$	−1.08425	−0.0439359
$610$	64.3933	2.60721
$611$	−5.88981	−0.238276
$612$	9.99735	0.404119
$613$	−40.6226	−1.64073	−0.820366	−	0.571839i	$-0.806230\pi$
$613$	−40.6226	−1.64073	−0.820366	+	0.571839i	$0.806230\pi$
$614$	−33.1646	−1.33842
$615$	−35.4766	−1.43055
$616$	−0.180263	−0.00726301
$617$	−0.134832	−0.00542814	−0.00271407	−	0.999996i	$-0.500864\pi$
$617$	−0.134832	−0.00542814	−0.00271407	+	0.999996i	$0.500864\pi$
$618$	−31.1983	−1.25498
$619$	24.3711	0.979556	0.489778	−	0.871847i	$-0.337078\pi$
$619$	24.3711	0.979556	0.489778	+	0.871847i	$0.337078\pi$
$620$	−21.6224	−0.868375
$621$	4.08559	0.163949
$622$	51.5559	2.06720
$623$	3.80071	0.152272
$624$	3.65322	0.146246
$625$	34.9236	1.39694
$626$	−27.3580	−1.09345
$627$	5.23070	0.208894
$628$	−15.3151	−0.611139
$629$	−46.7310	−1.86329
$630$	6.65681	0.265214
$631$	−9.03563	−0.359703	−0.179851	−	0.983694i	$-0.557562\pi$
$631$	−9.03563	−0.359703	−0.179851	+	0.983694i	$0.557562\pi$
$632$	−1.61722	−0.0643294
$633$	20.2854	0.806274
$634$	−18.4768	−0.733808
$635$	−35.9527	−1.42674
$636$	−1.67645	−0.0664754
$637$	−5.56995	−0.220689
$638$	4.41793	0.174907
$639$	5.73429	0.226845
$640$	−3.85748	−0.152480
$641$	−4.67743	−0.184747	−0.0923736	−	0.995724i	$-0.529445\pi$
$641$	−4.67743	−0.184747	−0.0923736	+	0.995724i	$0.529445\pi$
$642$	−30.7477	−1.21351
$643$	36.6098	1.44375	0.721874	−	0.692024i	$-0.243281\pi$
$643$	36.6098	1.44375	0.721874	+	0.692024i	$0.243281\pi$
$644$	6.71863	0.264751
$645$	18.5614	0.730853
$646$	30.7371	1.20933
$647$	−35.7366	−1.40495	−0.702476	−	0.711707i	$-0.747922\pi$
$647$	−35.7366	−1.40495	−0.702476	+	0.711707i	$0.747922\pi$
$648$	0.122004	0.00479277
$649$	−3.04692	−0.119602
$650$	−18.7180	−0.734179
$651$	−2.39299	−0.0937888
$652$	37.2394	1.45841
$653$	22.0350	0.862295	0.431147	−	0.902282i	$-0.358109\pi$
$653$	22.0350	0.862295	0.431147	+	0.902282i	$0.358109\pi$
$654$	−12.5267	−0.489832
$655$	−73.1093	−2.85662
$656$	−36.9579	−1.44296
$657$	−4.42082	−0.172472
$658$	−11.1804	−0.435857
$659$	12.6755	0.493768	0.246884	−	0.969045i	$-0.420593\pi$
$659$	12.6755	0.493768	0.246884	+	0.969045i	$0.420593\pi$
$660$	−13.3504	−0.519664
$661$	43.2975	1.68408	0.842039	−	0.539417i	$-0.181355\pi$
$661$	43.2975	1.68408	0.842039	+	0.539417i	$0.181355\pi$
$662$	−65.2143	−2.53462
$663$	−4.57383	−0.177633
$664$	0.527710	0.0204791
$665$	10.0735	0.390635
$666$	17.9828	0.696820
$667$	5.22190	0.202193
$668$	−20.9833	−0.811870
$669$	−20.7548	−0.802429
$670$	29.5686	1.14233
$671$	−14.2925	−0.551755
$672$	6.72777	0.259529
$673$	−6.51420	−0.251104	−0.125552	−	0.992087i	$-0.540070\pi$
$673$	−6.51420	−0.251104	−0.125552	+	0.992087i	$0.540070\pi$
$674$	−12.3940	−0.477398
$675$	−10.6347	−0.409330
$676$	−23.6760	−0.910617
$677$	10.5039	0.403696	0.201848	−	0.979417i	$-0.435305\pi$
$677$	10.5039	0.403696	0.201848	+	0.979417i	$0.435305\pi$
$678$	11.1991	0.430098
$679$	−6.83935	−0.262470
$680$	2.48790	0.0954066
$681$	−11.1757	−0.428254
$682$	9.75061	0.373370
$683$	−1.59916	−0.0611903	−0.0305951	−	0.999532i	$-0.509740\pi$
$683$	−1.59916	−0.0611903	−0.0305951	+	0.999532i	$0.509740\pi$
$684$	−5.82174	−0.222600
$685$	−8.07621	−0.308576
$686$	−22.3579	−0.853630
$687$	11.2445	0.429005
$688$	19.3364	0.737192
$689$	0.766981	0.0292196
$690$	−32.0602	−1.22051
$691$	8.34011	0.317273	0.158636	−	0.987337i	$-0.449290\pi$
$691$	8.34011	0.317273	0.158636	+	0.987337i	$0.449290\pi$
$692$	−23.8498	−0.906632
$693$	−1.47752	−0.0561263
$694$	−63.2912	−2.40250
$695$	−43.8758	−1.66430
$696$	0.155937	0.00591076
$697$	46.2712	1.75265
$698$	16.3925	0.620465
$699$	21.2010	0.801894
$700$	−17.4885	−0.661002
$701$	10.8714	0.410607	0.205304	−	0.978698i	$-0.434182\pi$
$701$	10.8714	0.410607	0.205304	+	0.978698i	$0.434182\pi$
$702$	1.76008	0.0664300
$703$	27.2128	1.02635
$704$	−13.0644	−0.492383
$705$	26.2591	0.988976
$706$	44.9081	1.69014
$707$	3.27620	0.123214
$708$	3.39120	0.127449
$709$	−10.2651	−0.385513	−0.192756	−	0.981247i	$-0.561743\pi$
$709$	−10.2651	−0.385513	−0.192756	+	0.981247i	$0.561743\pi$
$710$	−44.9978	−1.68874
$711$	−13.2554	−0.497118
$712$	−0.546618	−0.0204854
$713$	11.5250	0.431616
$714$	−8.68231	−0.324927
$715$	6.10786	0.228421
$716$	−7.19074	−0.268731
$717$	11.1279	0.415578
$718$	9.30288	0.347180
$719$	20.6174	0.768897	0.384449	−	0.923146i	$-0.374392\pi$
$719$	20.6174	0.768897	0.384449	+	0.923146i	$0.374392\pi$
$720$	−16.2875	−0.607000
$721$	13.3358	0.496650
$722$	19.8078	0.737169
$723$	2.55675	0.0950865
$724$	32.1751	1.19578
$725$	−13.5925	−0.504813
$726$	−15.8099	−0.586761
$727$	33.3554	1.23708	0.618541	−	0.785753i	$-0.287724\pi$
$727$	33.3554	1.23708	0.618541	+	0.785753i	$0.287724\pi$
$728$	−0.0917899	−0.00340196
$729$	1.00000	0.0370370
$730$	34.6908	1.28396
$731$	−24.2091	−0.895406
$732$	15.9074	0.587956
$733$	−29.4223	−1.08674	−0.543368	−	0.839494i	$-0.682851\pi$
$733$	−29.4223	−1.08674	−0.543368	+	0.839494i	$0.682851\pi$
$734$	55.3592	2.04334
$735$	24.8331	0.915981
$736$	−32.4019	−1.19435
$737$	−6.56292	−0.241748
$738$	−17.8059	−0.655444
$739$	−15.3829	−0.565868	−0.282934	−	0.959139i	$-0.591308\pi$
$739$	−15.3829	−0.565868	−0.282934	+	0.959139i	$0.591308\pi$
$740$	−69.4556	−2.55324
$741$	2.66347	0.0978451
$742$	1.45593	0.0534488
$743$	0.931987	0.0341913	0.0170956	−	0.999854i	$-0.494558\pi$
$743$	0.931987	0.0341913	0.0170956	+	0.999854i	$0.494558\pi$
$744$	0.344161	0.0126175
$745$	−18.9846	−0.695541
$746$	−6.84154	−0.250487
$747$	4.32535	0.158256
$748$	17.4126	0.636667
$749$	13.1432	0.480241
$750$	44.2164	1.61456
$751$	−53.7749	−1.96228	−0.981138	−	0.193310i	$-0.938078\pi$
$751$	−53.7749	−1.96228	−0.981138	+	0.193310i	$0.938078\pi$
$752$	27.3555	0.997554
$753$	0.784701	0.0285961
$754$	2.24961	0.0819258
$755$	−18.6365	−0.678253
$756$	1.64447	0.0598088
$757$	−35.8676	−1.30363	−0.651815	−	0.758378i	$-0.725992\pi$
$757$	−35.8676	−1.30363	−0.651815	+	0.758378i	$0.725992\pi$
$758$	−50.3538	−1.82893
$759$	7.11595	0.258293
$760$	−1.44878	−0.0525526
$761$	−54.5859	−1.97874	−0.989369	−	0.145425i	$-0.953545\pi$
$761$	−54.5859	−1.97874	−0.989369	+	0.145425i	$0.953545\pi$
$762$	−18.0449	−0.653696
$763$	5.35456	0.193848
$764$	−12.7561	−0.461499
$765$	20.3919	0.737272
$766$	11.7879	0.425914
$767$	−1.55149	−0.0560210
$768$	−16.9378	−0.611191
$769$	−30.6360	−1.10476	−0.552382	−	0.833591i	$-0.686281\pi$
$769$	−30.6360	−1.10476	−0.552382	+	0.833591i	$0.686281\pi$
$770$	11.5943	0.417830
$771$	−4.28330	−0.154259
$772$	−1.33715	−0.0481250
$773$	−13.6296	−0.490224	−0.245112	−	0.969495i	$-0.578825\pi$
$773$	−13.6296	−0.490224	−0.245112	+	0.969495i	$0.578825\pi$
$774$	9.31605	0.334858
$775$	−29.9994	−1.07761
$776$	0.983636	0.0353105
$777$	−7.68680	−0.275762
$778$	−12.1049	−0.433983
$779$	−26.9450	−0.965407
$780$	−6.79802	−0.243408
$781$	9.98753	0.357382
$782$	41.8153	1.49531
$783$	1.27813	0.0456765
$784$	25.8699	0.923926
$785$	−31.2387	−1.11496
$786$	−36.6939	−1.30883
$787$	37.2140	1.32654	0.663269	−	0.748381i	$-0.269169\pi$
$787$	37.2140	1.32654	0.663269	+	0.748381i	$0.269169\pi$
$788$	12.4830	0.444688
$789$	−31.2445	−1.11233
$790$	104.017	3.70077
$791$	−4.78707	−0.170209
$792$	0.212497	0.00755074
$793$	−7.27772	−0.258439
$794$	17.3813	0.616840
$795$	−3.41951	−0.121277
$796$	40.7916	1.44582
$797$	−15.6514	−0.554401	−0.277200	−	0.960812i	$-0.589407\pi$
$797$	−15.6514	−0.554401	−0.277200	+	0.960812i	$0.589407\pi$
$798$	5.05596	0.178979
$799$	−34.2491	−1.21165
$800$	84.3416	2.98193
$801$	−4.48033	−0.158305
$802$	−12.9414	−0.456978
$803$	−7.69982	−0.271721
$804$	7.30449	0.257610
$805$	13.7042	0.483011
$806$	4.96501	0.174885
$807$	7.73573	0.272310
$808$	−0.471183	−0.0165762
$809$	8.65885	0.304429	0.152215	−	0.988347i	$-0.451360\pi$
$809$	8.65885	0.304429	0.152215	+	0.988347i	$0.451360\pi$
$810$	−7.84715	−0.275721
$811$	−52.6392	−1.84841	−0.924206	−	0.381895i	$-0.875271\pi$
$811$	−52.6392	−1.84841	−0.924206	+	0.381895i	$0.875271\pi$
$812$	2.10184	0.0737601
$813$	−13.3939	−0.469744
$814$	31.3210	1.09780
$815$	75.9585	2.66071
$816$	21.2434	0.743667
$817$	14.0977	0.493215
$818$	−20.1689	−0.705190
$819$	−0.752351	−0.0262893
$820$	68.7722	2.40163
$821$	6.87113	0.239804	0.119902	−	0.992786i	$-0.461742\pi$
$821$	6.87113	0.239804	0.119902	+	0.992786i	$0.461742\pi$
$822$	−4.05349	−0.141382
$823$	−46.9401	−1.63623	−0.818114	−	0.575056i	$-0.804980\pi$
$823$	−46.9401	−1.63623	−0.818114	+	0.575056i	$0.804980\pi$
$824$	−1.91795	−0.0668151
$825$	−18.5227	−0.644877
$826$	−2.94513	−0.102474
$827$	20.8625	0.725461	0.362730	−	0.931894i	$-0.381845\pi$
$827$	20.8625	0.725461	0.362730	+	0.931894i	$0.381845\pi$
$828$	−7.92001	−0.275240
$829$	46.0073	1.59790	0.798949	−	0.601399i	$-0.205390\pi$
$829$	46.0073	1.59790	0.798949	+	0.601399i	$0.205390\pi$
$830$	−33.9416	−1.17813
$831$	2.27322	0.0788571
$832$	−6.65238	−0.230630
$833$	−32.3891	−1.12222
$834$	−22.0215	−0.762541
$835$	−42.8005	−1.48117
$836$	−10.1398	−0.350694
$837$	2.82090	0.0975044
$838$	−18.8808	−0.652227
$839$	−38.1303	−1.31640	−0.658201	−	0.752842i	$-0.728682\pi$
$839$	−38.1303	−1.31640	−0.658201	+	0.752842i	$0.728682\pi$
$840$	0.409236	0.0141200
$841$	−27.3664	−0.943669
$842$	−64.2213	−2.21321
$843$	−23.2792	−0.801778
$844$	−39.3238	−1.35358
$845$	−48.2929	−1.66133
$846$	13.1796	0.453124
$847$	6.75799	0.232207
$848$	−3.56228	−0.122329
$849$	−2.74644	−0.0942577
$850$	−108.844	−3.73333
$851$	37.0208	1.26906
$852$	−11.1161	−0.380830
$853$	−8.39100	−0.287302	−0.143651	−	0.989628i	$-0.545884\pi$
$853$	−8.39100	−0.287302	−0.143651	+	0.989628i	$0.545884\pi$
$854$	−13.8150	−0.472739
$855$	−11.8748	−0.406110
$856$	−1.89025	−0.0646075
$857$	20.3512	0.695183	0.347591	−	0.937646i	$-0.387000\pi$
$857$	20.3512	0.695183	0.347591	+	0.937646i	$0.387000\pi$
$858$	3.06557	0.104657
$859$	−52.7318	−1.79919	−0.899593	−	0.436730i	$-0.856137\pi$
$859$	−52.7318	−1.79919	−0.899593	+	0.436730i	$0.856137\pi$
$860$	−35.9816	−1.22696
$861$	7.61117	0.259388
$862$	10.7440	0.365942
$863$	35.3625	1.20375	0.601877	−	0.798589i	$-0.294420\pi$
$863$	35.3625	1.20375	0.601877	+	0.798589i	$0.294420\pi$
$864$	−7.93079	−0.269811
$865$	−48.6472	−1.65405
$866$	31.0496	1.05511
$867$	−9.59669	−0.325921
$868$	4.63888	0.157454
$869$	−23.0872	−0.783181
$870$	−10.0296	−0.340037
$871$	−3.34184	−0.113234
$872$	−0.770093	−0.0260787
$873$	8.06233	0.272868
$874$	−24.3503	−0.823660
$875$	−18.9004	−0.638951
$876$	8.56986	0.289549
$877$	−11.0581	−0.373406	−0.186703	−	0.982416i	$-0.559780\pi$
$877$	−11.0581	−0.373406	−0.186703	+	0.982416i	$0.559780\pi$
$878$	−54.5477	−1.84089
$879$	−14.0102	−0.472554
$880$	−28.3683	−0.956295
$881$	0.879266	0.0296232	0.0148116	−	0.999890i	$-0.495285\pi$
$881$	0.879266	0.0296232	0.0148116	+	0.999890i	$0.495285\pi$
$882$	12.4638	0.419679
$883$	25.8022	0.868314	0.434157	−	0.900837i	$-0.357046\pi$
$883$	25.8022	0.868314	0.434157	+	0.900837i	$0.357046\pi$
$884$	8.86647	0.298212
$885$	6.91715	0.232518
$886$	34.7146	1.16626
$887$	−26.5488	−0.891423	−0.445712	−	0.895177i	$-0.647049\pi$
$887$	−26.5488	−0.891423	−0.445712	+	0.895177i	$0.647049\pi$
$888$	1.10552	0.0370987
$889$	7.71332	0.258696
$890$	35.1578	1.17849
$891$	1.74172	0.0583498
$892$	40.2338	1.34713
$893$	19.9442	0.667408
$894$	−9.52846	−0.318679
$895$	−14.6672	−0.490271
$896$	0.827587	0.0276477
$897$	3.62344	0.120983
$898$	−50.0381	−1.66979
$899$	3.60546	0.120249
$900$	20.6156	0.687188
$901$	4.45997	0.148583
$902$	−31.0129	−1.03262
$903$	−3.98217	−0.132518
$904$	0.688477	0.0228984
$905$	65.6288	2.18157
$906$	−9.35378	−0.310758
$907$	45.7161	1.51798	0.758989	−	0.651104i	$-0.225694\pi$
$907$	45.7161	1.51798	0.758989	+	0.651104i	$0.225694\pi$
$908$	21.6643	0.718957
$909$	−3.86203	−0.128095
$910$	5.90381	0.195709
$911$	−18.1460	−0.601203	−0.300602	−	0.953750i	$-0.597187\pi$
$911$	−18.1460	−0.601203	−0.300602	+	0.953750i	$0.597187\pi$
$912$	−12.3706	−0.409633
$913$	7.53354	0.249324
$914$	−36.9319	−1.22160
$915$	32.4470	1.07266
$916$	−21.7978	−0.720218
$917$	15.6849	0.517962
$918$	10.2348	0.337800
$919$	−17.4896	−0.576928	−0.288464	−	0.957491i	$-0.593145\pi$
$919$	−17.4896	−0.576928	−0.288464	+	0.957491i	$0.593145\pi$
$920$	−1.97094	−0.0649801
$921$	−16.7112	−0.550654
$922$	−70.2742	−2.31436
$923$	5.08564	0.167396
$924$	2.86420	0.0942254
$925$	−96.3644	−3.16844
$926$	−36.0298	−1.18401
$927$	−15.7204	−0.516326
$928$	−10.1365	−0.332748
$929$	−34.4514	−1.13031	−0.565157	−	0.824983i	$-0.691184\pi$
$929$	−34.4514	−1.13031	−0.565157	+	0.824983i	$0.691184\pi$
$930$	−22.1360	−0.725867
$931$	18.8611	0.618148
$932$	−41.0986	−1.34623
$933$	25.9783	0.850493
$934$	−25.4182	−0.831709
$935$	35.5171	1.16153
$936$	0.108203	0.00353673
$937$	−17.5024	−0.571777	−0.285889	−	0.958263i	$-0.592289\pi$
$937$	−17.5024	−0.571777	−0.285889	+	0.958263i	$0.592289\pi$
$938$	−6.34367	−0.207128
$939$	−13.7853	−0.449868
$940$	−50.9039	−1.66030
$941$	−16.1718	−0.527184	−0.263592	−	0.964634i	$-0.584907\pi$
$941$	−16.1718	−0.527184	−0.263592	+	0.964634i	$0.584907\pi$
$942$	−15.6789	−0.510846
$943$	−36.6566	−1.19370
$944$	7.20597	0.234534
$945$	3.35428	0.109115
$946$	16.2259	0.527551
$947$	16.1232	0.523932	0.261966	−	0.965077i	$-0.415629\pi$
$947$	16.1232	0.523932	0.261966	+	0.965077i	$0.415629\pi$
$948$	25.6960	0.834566
$949$	−3.92074	−0.127273
$950$	63.3833	2.05642
$951$	−9.31023	−0.301905
$952$	−0.533756	−0.0172991
$953$	8.08704	0.261965	0.130983	−	0.991385i	$-0.458187\pi$
$953$	8.08704	0.261965	0.130983	+	0.991385i	$0.458187\pi$
$954$	−1.71627	−0.0555662
$955$	−26.0191	−0.841957
$956$	−21.5717	−0.697678
$957$	2.22614	0.0719608
$958$	−70.4533	−2.27624
$959$	1.73268	0.0559510
$960$	29.6590	0.957240
$961$	−23.0425	−0.743308
$962$	15.9486	0.514205
$963$	−15.4933	−0.499266
$964$	−4.95632	−0.159632
$965$	−2.72743	−0.0877991
$966$	6.87822	0.221303
$967$	−44.2152	−1.42186	−0.710932	−	0.703261i	$-0.751727\pi$
$967$	−44.2152	−1.42186	−0.710932	+	0.703261i	$0.751727\pi$
$968$	−0.971935	−0.0312392
$969$	15.4880	0.497547
$970$	−63.2663	−2.03136
$971$	−28.0655	−0.900666	−0.450333	−	0.892861i	$-0.648695\pi$
$971$	−28.0655	−0.900666	−0.450333	+	0.892861i	$0.648695\pi$
$972$	−1.93852	−0.0621782
$973$	9.41314	0.301771
$974$	34.0760	1.09186
$975$	−9.43174	−0.302057
$976$	33.8017	1.08197
$977$	49.2419	1.57539	0.787695	−	0.616066i	$-0.211274\pi$
$977$	49.2419	1.57539	0.787695	+	0.616066i	$0.211274\pi$
$978$	38.1240	1.21907
$979$	−7.80348	−0.249400
$980$	−48.1395	−1.53776
$981$	−6.31203	−0.201528
$982$	70.0486	2.23534
$983$	−0.725294	−0.0231333	−0.0115666	−	0.999933i	$-0.503682\pi$
$983$	−0.725294	−0.0231333	−0.0115666	+	0.999933i	$0.503682\pi$
$984$	−1.09464	−0.0348958
$985$	25.4620	0.811286
$986$	13.0814	0.416597
$987$	−5.63365	−0.179321
$988$	−5.16320	−0.164263
$989$	19.1787	0.609848
$990$	−13.6675	−0.434382
$991$	42.4285	1.34778	0.673892	−	0.738829i	$-0.264621\pi$
$991$	42.4285	1.34778	0.673892	+	0.738829i	$0.264621\pi$
$992$	−22.3719	−0.710309
$993$	−32.8606	−1.04280
$994$	9.65387	0.306202
$995$	83.2041	2.63775
$996$	−8.38479	−0.265682
$997$	1.82548	0.0578135	0.0289068	−	0.999582i	$-0.490797\pi$
$997$	1.82548	0.0578135	0.0289068	+	0.999582i	$0.490797\pi$
$998$	40.7365	1.28949
$999$	9.06131	0.286687

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8049.2.a.b.1.19	✓	104

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8049.2.a.b.1.19	✓	104	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:	\( N \)	\(=\)	\( 8049 = 3 \cdot 2683 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8049.a (trivial)

\(f(q)\)	\(=\)	\(q-1.98457 q^{2} -1.00000 q^{3} +1.93852 q^{4} +3.95408 q^{5} +1.98457 q^{6} -0.848310 q^{7} +0.122004 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.98457 q^{2} -1.00000 q^{3} +1.93852 q^{4} +3.95408 q^{5} +1.98457 q^{6} -0.848310 q^{7} +0.122004 q^{8} +1.00000 q^{9} -7.84715 q^{10} +1.74172 q^{11} -1.93852 q^{12} +0.886882 q^{13} +1.68353 q^{14} -3.95408 q^{15} -4.11917 q^{16} +5.15720 q^{17} -1.98457 q^{18} -3.00318 q^{19} +7.66507 q^{20} +0.848310 q^{21} -3.45657 q^{22} -4.08559 q^{23} -0.122004 q^{24} +10.6347 q^{25} -1.76008 q^{26} -1.00000 q^{27} -1.64447 q^{28} -1.27813 q^{29} +7.84715 q^{30} -2.82090 q^{31} +7.93079 q^{32} -1.74172 q^{33} -10.2348 q^{34} -3.35428 q^{35} +1.93852 q^{36} -9.06131 q^{37} +5.96003 q^{38} -0.886882 q^{39} +0.482413 q^{40} +8.97216 q^{41} -1.68353 q^{42} -4.69424 q^{43} +3.37636 q^{44} +3.95408 q^{45} +8.10815 q^{46} -6.64103 q^{47} +4.11917 q^{48} -6.28037 q^{49} -21.1053 q^{50} -5.15720 q^{51} +1.71924 q^{52} +0.864805 q^{53} +1.98457 q^{54} +6.88689 q^{55} -0.103497 q^{56} +3.00318 q^{57} +2.53653 q^{58} -1.74937 q^{59} -7.66507 q^{60} -8.20595 q^{61} +5.59827 q^{62} -0.848310 q^{63} -7.50086 q^{64} +3.50680 q^{65} +3.45657 q^{66} -3.76807 q^{67} +9.99735 q^{68} +4.08559 q^{69} +6.65681 q^{70} +5.73429 q^{71} +0.122004 q^{72} -4.42082 q^{73} +17.9828 q^{74} -10.6347 q^{75} -5.82174 q^{76} -1.47752 q^{77} +1.76008 q^{78} -13.2554 q^{79} -16.2875 q^{80} +1.00000 q^{81} -17.8059 q^{82} +4.32535 q^{83} +1.64447 q^{84} +20.3919 q^{85} +9.31605 q^{86} +1.27813 q^{87} +0.212497 q^{88} -4.48033 q^{89} -7.84715 q^{90} -0.752351 q^{91} -7.92001 q^{92} +2.82090 q^{93} +13.1796 q^{94} -11.8748 q^{95} -7.93079 q^{96} +8.06233 q^{97} +12.4638 q^{98} +1.74172 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 9 q^{2} - 104 q^{3} + 87 q^{4} - 15 q^{5} + 9 q^{6} - 10 q^{7} - 27 q^{8} + 104 q^{9}+O(q^{10}) \) 104 * q - 9 * q^2 - 104 * q^3 + 87 * q^4 - 15 * q^5 + 9 * q^6 - 10 * q^7 - 27 * q^8 + 104 * q^9	\( 104 q - 9 q^{2} - 104 q^{3} + 87 q^{4} - 15 q^{5} + 9 q^{6} - 10 q^{7} - 27 q^{8} + 104 q^{9} + 8 q^{10} - 52 q^{11} - 87 q^{12} + 35 q^{13} - 23 q^{14} + 15 q^{15} + 53 q^{16} - 19 q^{17} - 9 q^{18} - 22 q^{19} - 35 q^{20} + 10 q^{21} - q^{22} - 70 q^{23} + 27 q^{24} + 79 q^{25} - 39 q^{26} - 104 q^{27} - 9 q^{28} - 37 q^{29} - 8 q^{30} - 47 q^{31} - 53 q^{32} + 52 q^{33} - 17 q^{34} - 54 q^{35} + 87 q^{36} + 65 q^{37} - 33 q^{38} - 35 q^{39} + 14 q^{40} - 47 q^{41} + 23 q^{42} - 30 q^{43} - 122 q^{44} - 15 q^{45} - 6 q^{46} - 101 q^{47} - 53 q^{48} + 78 q^{49} - 64 q^{50} + 19 q^{51} + 41 q^{52} - 48 q^{53} + 9 q^{54} - 29 q^{55} - 71 q^{56} + 22 q^{57} - 2 q^{58} - 86 q^{59} + 35 q^{60} + 34 q^{61} - 36 q^{62} - 10 q^{63} - 15 q^{64} - 64 q^{65} + q^{66} - 38 q^{67} - 33 q^{68} + 70 q^{69} - 29 q^{70} - 176 q^{71} - 27 q^{72} + 69 q^{73} - 86 q^{74} - 79 q^{75} - 54 q^{76} - 45 q^{77} + 39 q^{78} - 101 q^{79} - 76 q^{80} + 104 q^{81} + 38 q^{82} - 67 q^{83} + 9 q^{84} + 3 q^{85} - 90 q^{86} + 37 q^{87} + 7 q^{88} - 91 q^{89} + 8 q^{90} - 47 q^{91} - 136 q^{92} + 47 q^{93} - 20 q^{94} - 130 q^{95} + 53 q^{96} + 86 q^{97} - 44 q^{98} - 52 q^{99}+O(q^{100}) \) 104 * q - 9 * q^2 - 104 * q^3 + 87 * q^4 - 15 * q^5 + 9 * q^6 - 10 * q^7 - 27 * q^8 + 104 * q^9 + 8 * q^10 - 52 * q^11 - 87 * q^12 + 35 * q^13 - 23 * q^14 + 15 * q^15 + 53 * q^16 - 19 * q^17 - 9 * q^18 - 22 * q^19 - 35 * q^20 + 10 * q^21 - q^22 - 70 * q^23 + 27 * q^24 + 79 * q^25 - 39 * q^26 - 104 * q^27 - 9 * q^28 - 37 * q^29 - 8 * q^30 - 47 * q^31 - 53 * q^32 + 52 * q^33 - 17 * q^34 - 54 * q^35 + 87 * q^36 + 65 * q^37 - 33 * q^38 - 35 * q^39 + 14 * q^40 - 47 * q^41 + 23 * q^42 - 30 * q^43 - 122 * q^44 - 15 * q^45 - 6 * q^46 - 101 * q^47 - 53 * q^48 + 78 * q^49 - 64 * q^50 + 19 * q^51 + 41 * q^52 - 48 * q^53 + 9 * q^54 - 29 * q^55 - 71 * q^56 + 22 * q^57 - 2 * q^58 - 86 * q^59 + 35 * q^60 + 34 * q^61 - 36 * q^62 - 10 * q^63 - 15 * q^64 - 64 * q^65 + q^66 - 38 * q^67 - 33 * q^68 + 70 * q^69 - 29 * q^70 - 176 * q^71 - 27 * q^72 + 69 * q^73 - 86 * q^74 - 79 * q^75 - 54 * q^76 - 45 * q^77 + 39 * q^78 - 101 * q^79 - 76 * q^80 + 104 * q^81 + 38 * q^82 - 67 * q^83 + 9 * q^84 + 3 * q^85 - 90 * q^86 + 37 * q^87 + 7 * q^88 - 91 * q^89 + 8 * q^90 - 47 * q^91 - 136 * q^92 + 47 * q^93 - 20 * q^94 - 130 * q^95 + 53 * q^96 + 86 * q^97 - 44 * q^98 - 52 * q^99

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