LMFDB - Embedded newform 8049.2.a.d.1.20

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:	$0$
Dimension:	$129$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q-2.16635 q^{2} +1.00000 q^{3} +2.69306 q^{4} +3.84293 q^{5} -2.16635 q^{6} +1.73846 q^{7} -1.50141 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.16635 q^{2} +1.00000 q^{3} +2.69306 q^{4} +3.84293 q^{5} -2.16635 q^{6} +1.73846 q^{7} -1.50141 q^{8} +1.00000 q^{9} -8.32512 q^{10} +4.75687 q^{11} +2.69306 q^{12} -6.74726 q^{13} -3.76611 q^{14} +3.84293 q^{15} -2.13355 q^{16} +2.34530 q^{17} -2.16635 q^{18} +3.67271 q^{19} +10.3492 q^{20} +1.73846 q^{21} -10.3050 q^{22} +3.41586 q^{23} -1.50141 q^{24} +9.76811 q^{25} +14.6169 q^{26} +1.00000 q^{27} +4.68178 q^{28} -0.923971 q^{29} -8.32512 q^{30} -3.30410 q^{31} +7.62482 q^{32} +4.75687 q^{33} -5.08073 q^{34} +6.68078 q^{35} +2.69306 q^{36} +3.54033 q^{37} -7.95636 q^{38} -6.74726 q^{39} -5.76980 q^{40} -2.53932 q^{41} -3.76611 q^{42} +0.448524 q^{43} +12.8105 q^{44} +3.84293 q^{45} -7.39993 q^{46} +8.56763 q^{47} -2.13355 q^{48} -3.97776 q^{49} -21.1611 q^{50} +2.34530 q^{51} -18.1708 q^{52} +5.17121 q^{53} -2.16635 q^{54} +18.2803 q^{55} -2.61014 q^{56} +3.67271 q^{57} +2.00164 q^{58} +2.63076 q^{59} +10.3492 q^{60} -1.15859 q^{61} +7.15783 q^{62} +1.73846 q^{63} -12.2509 q^{64} -25.9292 q^{65} -10.3050 q^{66} -2.76278 q^{67} +6.31603 q^{68} +3.41586 q^{69} -14.4729 q^{70} -3.18424 q^{71} -1.50141 q^{72} -0.274760 q^{73} -7.66957 q^{74} +9.76811 q^{75} +9.89082 q^{76} +8.26963 q^{77} +14.6169 q^{78} -9.96891 q^{79} -8.19908 q^{80} +1.00000 q^{81} +5.50104 q^{82} -4.99069 q^{83} +4.68178 q^{84} +9.01282 q^{85} -0.971658 q^{86} -0.923971 q^{87} -7.14201 q^{88} +8.72509 q^{89} -8.32512 q^{90} -11.7298 q^{91} +9.19911 q^{92} -3.30410 q^{93} -18.5605 q^{94} +14.1140 q^{95} +7.62482 q^{96} -5.02525 q^{97} +8.61720 q^{98} +4.75687 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 129 q + 8 q^{2} + 129 q^{3} + 158 q^{4} + 11 q^{5} + 8 q^{6} + 40 q^{7} + 18 q^{8} + 129 q^{9}+O(q^{10}) $ 129 * q + 8 * q^2 + 129 * q^3 + 158 * q^4 + 11 * q^5 + 8 * q^6 + 40 * q^7 + 18 * q^8 + 129 * q^9	\( 129 q + 8 q^{2} + 129 q^{3} + 158 q^{4} + 11 q^{5} + 8 q^{6} + 40 q^{7} + 18 q^{8} + 129 q^{9} + 20 q^{10} + 48 q^{11} + 158 q^{12} + 77 q^{13} + 13 q^{14} + 11 q^{15} + 212 q^{16} + 9 q^{17} + 8 q^{18} + 68 q^{19} + 19 q^{20} + 40 q^{21} + 45 q^{22} + 64 q^{23} + 18 q^{24} + 188 q^{25} + 19 q^{26} + 129 q^{27} + 69 q^{28} + 23 q^{29} + 20 q^{30} + 133 q^{31} + 24 q^{32} + 48 q^{33} + 63 q^{34} + 26 q^{35} + 158 q^{36} + 147 q^{37} + 9 q^{38} + 77 q^{39} + 58 q^{40} + 21 q^{41} + 13 q^{42} + 76 q^{43} + 110 q^{44} + 11 q^{45} + 48 q^{46} + 85 q^{47} + 212 q^{48} + 213 q^{49} + 17 q^{50} + 9 q^{51} + 139 q^{52} + 30 q^{53} + 8 q^{54} + 103 q^{55} + 19 q^{56} + 68 q^{57} + 94 q^{58} + 64 q^{59} + 19 q^{60} + 110 q^{61} - 10 q^{62} + 40 q^{63} + 288 q^{64} - 8 q^{65} + 45 q^{66} + 118 q^{67} - 15 q^{68} + 64 q^{69} + 75 q^{70} + 154 q^{71} + 18 q^{72} + 137 q^{73} + 28 q^{74} + 188 q^{75} + 156 q^{76} + 17 q^{77} + 19 q^{78} + 157 q^{79} + 2 q^{80} + 129 q^{81} + 72 q^{82} + 39 q^{83} + 69 q^{84} + 127 q^{85} + 54 q^{86} + 23 q^{87} + 97 q^{88} + 31 q^{89} + 20 q^{90} + 137 q^{91} + 82 q^{92} + 133 q^{93} + 40 q^{94} + 68 q^{95} + 24 q^{96} + 170 q^{97} - 21 q^{98} + 48 q^{99}+O(q^{100}) \) 129 * q + 8 * q^2 + 129 * q^3 + 158 * q^4 + 11 * q^5 + 8 * q^6 + 40 * q^7 + 18 * q^8 + 129 * q^9 + 20 * q^10 + 48 * q^11 + 158 * q^12 + 77 * q^13 + 13 * q^14 + 11 * q^15 + 212 * q^16 + 9 * q^17 + 8 * q^18 + 68 * q^19 + 19 * q^20 + 40 * q^21 + 45 * q^22 + 64 * q^23 + 18 * q^24 + 188 * q^25 + 19 * q^26 + 129 * q^27 + 69 * q^28 + 23 * q^29 + 20 * q^30 + 133 * q^31 + 24 * q^32 + 48 * q^33 + 63 * q^34 + 26 * q^35 + 158 * q^36 + 147 * q^37 + 9 * q^38 + 77 * q^39 + 58 * q^40 + 21 * q^41 + 13 * q^42 + 76 * q^43 + 110 * q^44 + 11 * q^45 + 48 * q^46 + 85 * q^47 + 212 * q^48 + 213 * q^49 + 17 * q^50 + 9 * q^51 + 139 * q^52 + 30 * q^53 + 8 * q^54 + 103 * q^55 + 19 * q^56 + 68 * q^57 + 94 * q^58 + 64 * q^59 + 19 * q^60 + 110 * q^61 - 10 * q^62 + 40 * q^63 + 288 * q^64 - 8 * q^65 + 45 * q^66 + 118 * q^67 - 15 * q^68 + 64 * q^69 + 75 * q^70 + 154 * q^71 + 18 * q^72 + 137 * q^73 + 28 * q^74 + 188 * q^75 + 156 * q^76 + 17 * q^77 + 19 * q^78 + 157 * q^79 + 2 * q^80 + 129 * q^81 + 72 * q^82 + 39 * q^83 + 69 * q^84 + 127 * q^85 + 54 * q^86 + 23 * q^87 + 97 * q^88 + 31 * q^89 + 20 * q^90 + 137 * q^91 + 82 * q^92 + 133 * q^93 + 40 * q^94 + 68 * q^95 + 24 * q^96 + 170 * q^97 - 21 * q^98 + 48 * 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 $

$2$	−2.16635	−1.53184	−0.765919	−	0.642937i	$-0.777716\pi$
$2$	−2.16635	−1.53184	−0.765919	+	0.642937i	$0.777716\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.69306	1.34653
$5$	3.84293	1.71861	0.859305	−	0.511463i	$-0.170896\pi$
$5$	3.84293	1.71861	0.859305	+	0.511463i	$0.170896\pi$
$6$	−2.16635	−0.884407
$7$	1.73846	0.657076	0.328538	−	0.944491i	$-0.393444\pi$
$7$	1.73846	0.657076	0.328538	+	0.944491i	$0.393444\pi$
$8$	−1.50141	−0.530828
$9$	1.00000	0.333333
$10$	−8.32512	−2.63263
$11$	4.75687	1.43425	0.717126	−	0.696944i	$-0.245457\pi$
$11$	4.75687	1.43425	0.717126	+	0.696944i	$0.245457\pi$
$12$	2.69306	0.777419
$13$	−6.74726	−1.87135	−0.935677	−	0.352858i	$-0.885210\pi$
$13$	−6.74726	−1.87135	−0.935677	+	0.352858i	$0.885210\pi$
$14$	−3.76611	−1.00653
$15$	3.84293	0.992240
$16$	−2.13355	−0.533387
$17$	2.34530	0.568819	0.284409	−	0.958703i	$-0.408203\pi$
$17$	2.34530	0.568819	0.284409	+	0.958703i	$0.408203\pi$
$18$	−2.16635	−0.510613
$19$	3.67271	0.842577	0.421288	−	0.906927i	$-0.361578\pi$
$19$	3.67271	0.842577	0.421288	+	0.906927i	$0.361578\pi$
$20$	10.3492	2.31416
$21$	1.73846	0.379363
$22$	−10.3050	−2.19704
$23$	3.41586	0.712256	0.356128	−	0.934437i	$-0.384097\pi$
$23$	3.41586	0.712256	0.356128	+	0.934437i	$0.384097\pi$
$24$	−1.50141	−0.306473
$25$	9.76811	1.95362
$26$	14.6169	2.86661
$27$	1.00000	0.192450
$28$	4.68178	0.884772
$29$	−0.923971	−0.171577	−0.0857886	−	0.996313i	$-0.527341\pi$
$29$	−0.923971	−0.171577	−0.0857886	+	0.996313i	$0.527341\pi$
$30$	−8.32512	−1.51995
$31$	−3.30410	−0.593434	−0.296717	−	0.954965i	$-0.595892\pi$
$31$	−3.30410	−0.593434	−0.296717	+	0.954965i	$0.595892\pi$
$32$	7.62482	1.34789
$33$	4.75687	0.828065
$34$	−5.08073	−0.871338
$35$	6.68078	1.12926
$36$	2.69306	0.448843
$37$	3.54033	0.582026	0.291013	−	0.956719i	$-0.406008\pi$
$37$	3.54033	0.582026	0.291013	+	0.956719i	$0.406008\pi$
$38$	−7.95636	−1.29069
$39$	−6.74726	−1.08043
$40$	−5.76980	−0.912286
$41$	−2.53932	−0.396575	−0.198287	−	0.980144i	$-0.563538\pi$
$41$	−2.53932	−0.396575	−0.198287	+	0.980144i	$0.563538\pi$
$42$	−3.76611	−0.581123
$43$	0.448524	0.0683993	0.0341996	−	0.999415i	$-0.489112\pi$
$43$	0.448524	0.0683993	0.0341996	+	0.999415i	$0.489112\pi$
$44$	12.8105	1.93126
$45$	3.84293	0.572870
$46$	−7.39993	−1.09106
$47$	8.56763	1.24972	0.624859	−	0.780738i	$-0.285156\pi$
$47$	8.56763	1.24972	0.624859	+	0.780738i	$0.285156\pi$
$48$	−2.13355	−0.307951
$49$	−3.97776	−0.568251
$50$	−21.1611	−2.99263
$51$	2.34530	0.328408
$52$	−18.1708	−2.51983
$53$	5.17121	0.710320	0.355160	−	0.934806i	$-0.384426\pi$
$53$	5.17121	0.710320	0.355160	+	0.934806i	$0.384426\pi$
$54$	−2.16635	−0.294802
$55$	18.2803	2.46492
$56$	−2.61014	−0.348794
$57$	3.67271	0.486462
$58$	2.00164	0.262829
$59$	2.63076	0.342496	0.171248	−	0.985228i	$-0.445220\pi$
$59$	2.63076	0.342496	0.171248	+	0.985228i	$0.445220\pi$
$60$	10.3492	1.33608
$61$	−1.15859	−0.148342	−0.0741710	−	0.997246i	$-0.523631\pi$
$61$	−1.15859	−0.148342	−0.0741710	+	0.997246i	$0.523631\pi$
$62$	7.15783	0.909046
$63$	1.73846	0.219025
$64$	−12.2509	−1.53136
$65$	−25.9292	−3.21613
$66$	−10.3050	−1.26846
$67$	−2.76278	−0.337527	−0.168763	−	0.985657i	$-0.553977\pi$
$67$	−2.76278	−0.337527	−0.168763	+	0.985657i	$0.553977\pi$
$68$	6.31603	0.765931
$69$	3.41586	0.411221
$70$	−14.4729	−1.72984
$71$	−3.18424	−0.377899	−0.188950	−	0.981987i	$-0.560508\pi$
$71$	−3.18424	−0.377899	−0.188950	+	0.981987i	$0.560508\pi$
$72$	−1.50141	−0.176943
$73$	−0.274760	−0.0321582	−0.0160791	−	0.999871i	$-0.505118\pi$
$73$	−0.274760	−0.0321582	−0.0160791	+	0.999871i	$0.505118\pi$
$74$	−7.66957	−0.891570
$75$	9.76811	1.12792
$76$	9.89082	1.13455
$77$	8.26963	0.942412
$78$	14.6169	1.65504
$79$	−9.96891	−1.12159	−0.560795	−	0.827955i	$-0.689505\pi$
$79$	−9.96891	−1.12159	−0.560795	+	0.827955i	$0.689505\pi$
$80$	−8.19908	−0.916685
$81$	1.00000	0.111111
$82$	5.50104	0.607488
$83$	−4.99069	−0.547799	−0.273900	−	0.961758i	$-0.588314\pi$
$83$	−4.99069	−0.547799	−0.273900	+	0.961758i	$0.588314\pi$
$84$	4.68178	0.510824
$85$	9.01282	0.977578
$86$	−0.971658	−0.104777
$87$	−0.923971	−0.0990601
$88$	−7.14201	−0.761340
$89$	8.72509	0.924858	0.462429	−	0.886656i	$-0.346978\pi$
$89$	8.72509	0.924858	0.462429	+	0.886656i	$0.346978\pi$
$90$	−8.32512	−0.877545
$91$	−11.7298	−1.22962
$92$	9.19911	0.959073
$93$	−3.30410	−0.342619
$94$	−18.5605	−1.91437
$95$	14.1140	1.44806
$96$	7.62482	0.778205
$97$	−5.02525	−0.510237	−0.255118	−	0.966910i	$-0.582114\pi$
$97$	−5.02525	−0.510237	−0.255118	+	0.966910i	$0.582114\pi$
$98$	8.61720	0.870469
$99$	4.75687	0.478084
$100$	26.3061	2.63061
$101$	−7.74643	−0.770799	−0.385400	−	0.922750i	$-0.625936\pi$
$101$	−7.74643	−0.770799	−0.385400	+	0.922750i	$0.625936\pi$
$102$	−5.08073	−0.503067
$103$	4.40114	0.433657	0.216828	−	0.976210i	$-0.430429\pi$
$103$	4.40114	0.433657	0.216828	+	0.976210i	$0.430429\pi$
$104$	10.1304	0.993366
$105$	6.68078	0.651977
$106$	−11.2026	−1.08810
$107$	−3.85381	−0.372562	−0.186281	−	0.982497i	$-0.559643\pi$
$107$	−3.85381	−0.372562	−0.186281	+	0.982497i	$0.559643\pi$
$108$	2.69306	0.259140
$109$	0.714470	0.0684338	0.0342169	−	0.999414i	$-0.489106\pi$
$109$	0.714470	0.0684338	0.0342169	+	0.999414i	$0.489106\pi$
$110$	−39.6015	−3.77586
$111$	3.54033	0.336033
$112$	−3.70909	−0.350476
$113$	10.1624	0.956001	0.478001	−	0.878360i	$-0.341362\pi$
$113$	10.1624	0.956001	0.478001	+	0.878360i	$0.341362\pi$
$114$	−7.95636	−0.745181
$115$	13.1269	1.22409
$116$	−2.48831	−0.231034
$117$	−6.74726	−0.623784
$118$	−5.69914	−0.524649
$119$	4.07721	0.373757
$120$	−5.76980	−0.526708
$121$	11.6278	1.05708
$122$	2.50990	0.227236
$123$	−2.53932	−0.228962
$124$	−8.89815	−0.799077
$125$	18.3235	1.63890
$126$	−3.76611	−0.335512
$127$	18.8629	1.67381	0.836907	−	0.547345i	$-0.184362\pi$
$127$	18.8629	1.67381	0.836907	+	0.547345i	$0.184362\pi$
$128$	11.2901	0.997912
$129$	0.448524	0.0394903
$130$	56.1717	4.92659
$131$	1.02945	0.0899433	0.0449716	−	0.998988i	$-0.485680\pi$
$131$	1.02945	0.0899433	0.0449716	+	0.998988i	$0.485680\pi$
$132$	12.8105	1.11501
$133$	6.38485	0.553637
$134$	5.98513	0.517037
$135$	3.84293	0.330747
$136$	−3.52125	−0.301945
$137$	13.7852	1.17775	0.588876	−	0.808223i	$-0.299571\pi$
$137$	13.7852	1.17775	0.588876	+	0.808223i	$0.299571\pi$
$138$	−7.39993	−0.629924
$139$	−19.6766	−1.66894	−0.834472	−	0.551050i	$-0.814227\pi$
$139$	−19.6766	−1.66894	−0.834472	+	0.551050i	$0.814227\pi$
$140$	17.9917	1.52058
$141$	8.56763	0.721525
$142$	6.89816	0.578881
$143$	−32.0959	−2.68399
$144$	−2.13355	−0.177796
$145$	−3.55076	−0.294874
$146$	0.595226	0.0492612
$147$	−3.97776	−0.328080
$148$	9.53431	0.783715
$149$	18.8854	1.54715	0.773575	−	0.633704i	$-0.218466\pi$
$149$	18.8854	1.54715	0.773575	+	0.633704i	$0.218466\pi$
$150$	−21.1611	−1.72780
$151$	−6.94496	−0.565173	−0.282586	−	0.959242i	$-0.591192\pi$
$151$	−6.94496	−0.565173	−0.282586	+	0.959242i	$0.591192\pi$
$152$	−5.51423	−0.447263
$153$	2.34530	0.189606
$154$	−17.9149	−1.44362
$155$	−12.6974	−1.01988
$156$	−18.1708	−1.45483
$157$	8.32010	0.664016	0.332008	−	0.943277i	$-0.392274\pi$
$157$	8.32010	0.664016	0.332008	+	0.943277i	$0.392274\pi$
$158$	21.5961	1.71810
$159$	5.17121	0.410103
$160$	29.3017	2.31650
$161$	5.93833	0.468006
$162$	−2.16635	−0.170204
$163$	6.44902	0.505126	0.252563	−	0.967580i	$-0.418727\pi$
$163$	6.44902	0.505126	0.252563	+	0.967580i	$0.418727\pi$
$164$	−6.83853	−0.533999
$165$	18.2803	1.42312
$166$	10.8116	0.839140
$167$	−15.4789	−1.19779	−0.598897	−	0.800826i	$-0.704394\pi$
$167$	−15.4789	−1.19779	−0.598897	+	0.800826i	$0.704394\pi$
$168$	−2.61014	−0.201376
$169$	32.5255	2.50196
$170$	−19.5249	−1.49749
$171$	3.67271	0.280859
$172$	1.20790	0.0921016
$173$	−18.8877	−1.43601	−0.718004	−	0.696039i	$-0.754944\pi$
$173$	−18.8877	−1.43601	−0.718004	+	0.696039i	$0.754944\pi$
$174$	2.00164	0.151744
$175$	16.9815	1.28368
$176$	−10.1490	−0.765012
$177$	2.63076	0.197740
$178$	−18.9016	−1.41673
$179$	2.11099	0.157782	0.0788912	−	0.996883i	$-0.474862\pi$
$179$	2.11099	0.157782	0.0788912	+	0.996883i	$0.474862\pi$
$180$	10.3492	0.771387
$181$	16.0191	1.19069	0.595346	−	0.803469i	$-0.297015\pi$
$181$	16.0191	1.19069	0.595346	+	0.803469i	$0.297015\pi$
$182$	25.4109	1.88358
$183$	−1.15859	−0.0856453
$184$	−5.12859	−0.378085
$185$	13.6052	1.00028
$186$	7.15783	0.524838
$187$	11.1563	0.815829
$188$	23.0731	1.68278
$189$	1.73846	0.126454
$190$	−30.5757	−2.21820
$191$	−3.38524	−0.244947	−0.122474	−	0.992472i	$-0.539083\pi$
$191$	−3.38524	−0.244947	−0.122474	+	0.992472i	$0.539083\pi$
$192$	−12.2509	−0.884134
$193$	9.36064	0.673794	0.336897	−	0.941542i	$-0.390623\pi$
$193$	9.36064	0.673794	0.336897	+	0.941542i	$0.390623\pi$
$194$	10.8864	0.781601
$195$	−25.9292	−1.85683
$196$	−10.7123	−0.765167
$197$	28.0358	1.99747	0.998734	−	0.0503110i	$-0.0160212\pi$
$197$	28.0358	1.99747	0.998734	+	0.0503110i	$0.0160212\pi$
$198$	−10.3050	−0.732347
$199$	−1.81714	−0.128814	−0.0644068	−	0.997924i	$-0.520516\pi$
$199$	−1.81714	−0.128814	−0.0644068	+	0.997924i	$0.520516\pi$
$200$	−14.6659	−1.03704
$201$	−2.76278	−0.194871
$202$	16.7815	1.18074
$203$	−1.60629	−0.112739
$204$	6.31603	0.442211
$205$	−9.75841	−0.681557
$206$	−9.53439	−0.664292
$207$	3.41586	0.237419
$208$	14.3956	0.998156
$209$	17.4706	1.20847
$210$	−14.4729	−0.998724
$211$	16.9521	1.16703	0.583516	−	0.812102i	$-0.301677\pi$
$211$	16.9521	1.16703	0.583516	+	0.812102i	$0.301677\pi$
$212$	13.9264	0.956467
$213$	−3.18424	−0.218180
$214$	8.34868	0.570704
$215$	1.72365	0.117552
$216$	−1.50141	−0.102158
$217$	−5.74405	−0.389932
$218$	−1.54779	−0.104830
$219$	−0.274760	−0.0185666
$220$	49.2300	3.31909
$221$	−15.8243	−1.06446
$222$	−7.66957	−0.514748
$223$	−20.6423	−1.38231	−0.691154	−	0.722708i	$-0.742897\pi$
$223$	−20.6423	−1.38231	−0.691154	+	0.722708i	$0.742897\pi$
$224$	13.2555	0.885667
$225$	9.76811	0.651207
$226$	−22.0154	−1.46444
$227$	−25.0272	−1.66112	−0.830558	−	0.556933i	$-0.811978\pi$
$227$	−25.0272	−1.66112	−0.830558	+	0.556933i	$0.811978\pi$
$228$	9.89082	0.655035
$229$	12.0453	0.795977	0.397988	−	0.917390i	$-0.369708\pi$
$229$	12.0453	0.795977	0.397988	+	0.917390i	$0.369708\pi$
$230$	−28.4374	−1.87511
$231$	8.26963	0.544102
$232$	1.38726	0.0910779
$233$	−23.9643	−1.56995	−0.784977	−	0.619525i	$-0.787325\pi$
$233$	−23.9643	−1.56995	−0.784977	+	0.619525i	$0.787325\pi$
$234$	14.6169	0.955537
$235$	32.9248	2.14778
$236$	7.08480	0.461181
$237$	−9.96891	−0.647551
$238$	−8.83265	−0.572536
$239$	−4.26160	−0.275660	−0.137830	−	0.990456i	$-0.544013\pi$
$239$	−4.26160	−0.275660	−0.137830	+	0.990456i	$0.544013\pi$
$240$	−8.19908	−0.529248
$241$	10.7017	0.689356	0.344678	−	0.938721i	$-0.387988\pi$
$241$	10.7017	0.689356	0.344678	+	0.938721i	$0.387988\pi$
$242$	−25.1900	−1.61927
$243$	1.00000	0.0641500
$244$	−3.12015	−0.199747
$245$	−15.2862	−0.976602
$246$	5.50104	0.350734
$247$	−24.7807	−1.57676
$248$	4.96080	0.315011
$249$	−4.99069	−0.316272
$250$	−39.6950	−2.51053
$251$	−14.6224	−0.922961	−0.461480	−	0.887150i	$-0.652682\pi$
$251$	−14.6224	−0.922961	−0.461480	+	0.887150i	$0.652682\pi$
$252$	4.68178	0.294924
$253$	16.2488	1.02155
$254$	−40.8636	−2.56401
$255$	9.01282	0.564405
$256$	0.0435867	0.00272417
$257$	−22.4523	−1.40054	−0.700269	−	0.713879i	$-0.746937\pi$
$257$	−22.4523	−1.40054	−0.700269	+	0.713879i	$0.746937\pi$
$258$	−0.971658	−0.0604928
$259$	6.15471	0.382435
$260$	−69.8290	−4.33061
$261$	−0.923971	−0.0571924
$262$	−2.23014	−0.137779
$263$	4.46385	0.275253	0.137626	−	0.990484i	$-0.456053\pi$
$263$	4.46385	0.275253	0.137626	+	0.990484i	$0.456053\pi$
$264$	−7.14201	−0.439560
$265$	19.8726	1.22076
$266$	−13.8318	−0.848083
$267$	8.72509	0.533967
$268$	−7.44032	−0.454490
$269$	−4.17777	−0.254723	−0.127362	−	0.991856i	$-0.540651\pi$
$269$	−4.17777	−0.254723	−0.127362	+	0.991856i	$0.540651\pi$
$270$	−8.32512	−0.506651
$271$	22.9215	1.39238	0.696191	−	0.717856i	$-0.254877\pi$
$271$	22.9215	1.39238	0.696191	+	0.717856i	$0.254877\pi$
$272$	−5.00381	−0.303401
$273$	−11.7298	−0.709922
$274$	−29.8636	−1.80413
$275$	46.4656	2.80198
$276$	9.19911	0.553721
$277$	26.5217	1.59353	0.796767	−	0.604287i	$-0.206542\pi$
$277$	26.5217	1.59353	0.796767	+	0.604287i	$0.206542\pi$
$278$	42.6263	2.55655
$279$	−3.30410	−0.197811
$280$	−10.0306	−0.599441
$281$	15.0889	0.900125	0.450063	−	0.892997i	$-0.351402\pi$
$281$	15.0889	0.900125	0.450063	+	0.892997i	$0.351402\pi$
$282$	−18.5605	−1.10526
$283$	−0.507477	−0.0301664	−0.0150832	−	0.999886i	$-0.504801\pi$
$283$	−0.507477	−0.0301664	−0.0150832	+	0.999886i	$0.504801\pi$
$284$	−8.57534	−0.508853
$285$	14.1140	0.836038
$286$	69.5308	4.11144
$287$	−4.41450	−0.260580
$288$	7.62482	0.449297
$289$	−11.4996	−0.676445
$290$	7.69217	0.451700
$291$	−5.02525	−0.294585
$292$	−0.739946	−0.0433020
$293$	−28.3650	−1.65710	−0.828550	−	0.559916i	$-0.810834\pi$
$293$	−28.3650	−1.65710	−0.828550	+	0.559916i	$0.810834\pi$
$294$	8.61720	0.502565
$295$	10.1098	0.588617
$296$	−5.31547	−0.308955
$297$	4.75687	0.276022
$298$	−40.9123	−2.36999
$299$	−23.0477	−1.33288
$300$	26.3061	1.51878
$301$	0.779741	0.0449435
$302$	15.0452	0.865753
$303$	−7.74643	−0.445021
$304$	−7.83590	−0.449420
$305$	−4.45237	−0.254942
$306$	−5.08073	−0.290446
$307$	−17.4748	−0.997340	−0.498670	−	0.866792i	$-0.666178\pi$
$307$	−17.4748	−0.997340	−0.498670	+	0.866792i	$0.666178\pi$
$308$	22.2706	1.26899
$309$	4.40114	0.250372
$310$	27.5070	1.56230
$311$	19.4449	1.10262	0.551309	−	0.834301i	$-0.314128\pi$
$311$	19.4449	1.10262	0.551309	+	0.834301i	$0.314128\pi$
$312$	10.1304	0.573520
$313$	−9.25254	−0.522984	−0.261492	−	0.965206i	$-0.584215\pi$
$313$	−9.25254	−0.522984	−0.261492	+	0.965206i	$0.584215\pi$
$314$	−18.0242	−1.01717
$315$	6.68078	0.376419
$316$	−26.8469	−1.51025
$317$	−15.8160	−0.888316	−0.444158	−	0.895949i	$-0.646497\pi$
$317$	−15.8160	−0.888316	−0.444158	+	0.895949i	$0.646497\pi$
$318$	−11.2026	−0.628212
$319$	−4.39522	−0.246085
$320$	−47.0794	−2.63182
$321$	−3.85381	−0.215099
$322$	−12.8645	−0.716910
$323$	8.61360	0.479273
$324$	2.69306	0.149614
$325$	−65.9080	−3.65592
$326$	−13.9708	−0.773771
$327$	0.714470	0.0395103
$328$	3.81255	0.210513
$329$	14.8945	0.821160
$330$	−39.6015	−2.17999
$331$	−2.46290	−0.135373	−0.0676866	−	0.997707i	$-0.521562\pi$
$331$	−2.46290	−0.135373	−0.0676866	+	0.997707i	$0.521562\pi$
$332$	−13.4402	−0.737628
$333$	3.54033	0.194009
$334$	33.5327	1.83483
$335$	−10.6172	−0.580077
$336$	−3.70909	−0.202347
$337$	13.1314	0.715312	0.357656	−	0.933853i	$-0.383576\pi$
$337$	13.1314	0.715312	0.357656	+	0.933853i	$0.383576\pi$
$338$	−70.4616	−3.83260
$339$	10.1624	0.551947
$340$	24.2721	1.31634
$341$	−15.7172	−0.851134
$342$	−7.95636	−0.430231
$343$	−19.0844	−1.03046
$344$	−0.673417	−0.0363082
$345$	13.1269	0.706728
$346$	40.9174	2.19973
$347$	24.2474	1.30167	0.650835	−	0.759219i	$-0.274419\pi$
$347$	24.2474	1.30167	0.650835	+	0.759219i	$0.274419\pi$
$348$	−2.48831	−0.133387
$349$	22.5659	1.20792	0.603962	−	0.797013i	$-0.293588\pi$
$349$	22.5659	1.20792	0.603962	+	0.797013i	$0.293588\pi$
$350$	−36.7877	−1.96639
$351$	−6.74726	−0.360142
$352$	36.2703	1.93321
$353$	−19.3636	−1.03062	−0.515311	−	0.857003i	$-0.672324\pi$
$353$	−19.3636	−1.03062	−0.515311	+	0.857003i	$0.672324\pi$
$354$	−5.69914	−0.302906
$355$	−12.2368	−0.649462
$356$	23.4972	1.24535
$357$	4.07721	0.215789
$358$	−4.57313	−0.241697
$359$	−12.7669	−0.673810	−0.336905	−	0.941539i	$-0.609380\pi$
$359$	−12.7669	−0.673810	−0.336905	+	0.941539i	$0.609380\pi$
$360$	−5.76980	−0.304095
$361$	−5.51123	−0.290064
$362$	−34.7030	−1.82395
$363$	11.6278	0.610304
$364$	−31.5892	−1.65572
$365$	−1.05588	−0.0552675
$366$	2.50990	0.131195
$367$	−20.8296	−1.08730	−0.543648	−	0.839314i	$-0.682957\pi$
$367$	−20.8296	−1.08730	−0.543648	+	0.839314i	$0.682957\pi$
$368$	−7.28790	−0.379908
$369$	−2.53932	−0.132192
$370$	−29.4736	−1.53226
$371$	8.98993	0.466734
$372$	−8.89815	−0.461347
$373$	−35.5139	−1.83884	−0.919420	−	0.393278i	$-0.871341\pi$
$373$	−35.5139	−1.83884	−0.919420	+	0.393278i	$0.871341\pi$
$374$	−24.1684	−1.24972
$375$	18.3235	0.946221
$376$	−12.8635	−0.663385
$377$	6.23428	0.321082
$378$	−3.76611	−0.193708
$379$	15.7018	0.806549	0.403274	−	0.915079i	$-0.367872\pi$
$379$	15.7018	0.806549	0.403274	+	0.915079i	$0.367872\pi$
$380$	38.0097	1.94986
$381$	18.8629	0.966377
$382$	7.33360	0.375220
$383$	−3.07552	−0.157152	−0.0785759	−	0.996908i	$-0.525037\pi$
$383$	−3.07552	−0.157152	−0.0785759	+	0.996908i	$0.525037\pi$
$384$	11.2901	0.576145
$385$	31.7796	1.61964
$386$	−20.2784	−1.03214
$387$	0.448524	0.0227998
$388$	−13.5333	−0.687049
$389$	−0.309286	−0.0156814	−0.00784070	−	0.999969i	$-0.502496\pi$
$389$	−0.309286	−0.0156814	−0.00784070	+	0.999969i	$0.502496\pi$
$390$	56.1717	2.84437
$391$	8.01121	0.405144
$392$	5.97223	0.301643
$393$	1.02945	0.0519288
$394$	−60.7352	−3.05980
$395$	−38.3098	−1.92758
$396$	12.8105	0.643754
$397$	−22.2613	−1.11726	−0.558631	−	0.829417i	$-0.688673\pi$
$397$	−22.2613	−1.11726	−0.558631	+	0.829417i	$0.688673\pi$
$398$	3.93655	0.197322
$399$	6.38485	0.319642
$400$	−20.8407	−1.04204
$401$	20.1247	1.00498	0.502489	−	0.864584i	$-0.332418\pi$
$401$	20.1247	1.00498	0.502489	+	0.864584i	$0.332418\pi$
$402$	5.98513	0.298511
$403$	22.2936	1.11053
$404$	−20.8616	−1.03790
$405$	3.84293	0.190957
$406$	3.47978	0.172698
$407$	16.8409	0.834772
$408$	−3.52125	−0.174328
$409$	20.8247	1.02972	0.514858	−	0.857275i	$-0.327845\pi$
$409$	20.8247	1.02972	0.514858	+	0.857275i	$0.327845\pi$
$410$	21.1401	1.04404
$411$	13.7852	0.679976
$412$	11.8525	0.583932
$413$	4.57347	0.225046
$414$	−7.39993	−0.363687
$415$	−19.1789	−0.941453
$416$	−51.4467	−2.52238
$417$	−19.6766	−0.963566
$418$	−37.8474	−1.85118
$419$	29.5626	1.44423	0.722114	−	0.691774i	$-0.243171\pi$
$419$	29.5626	1.44423	0.722114	+	0.691774i	$0.243171\pi$
$420$	17.9917	0.877907
$421$	25.2047	1.22840	0.614201	−	0.789149i	$-0.289478\pi$
$421$	25.2047	1.22840	0.614201	+	0.789149i	$0.289478\pi$
$422$	−36.7242	−1.78771
$423$	8.56763	0.416573
$424$	−7.76409	−0.377057
$425$	22.9091	1.11126
$426$	6.89816	0.334217
$427$	−2.01416	−0.0974720
$428$	−10.3785	−0.501665
$429$	−32.0959	−1.54960
$430$	−3.73401	−0.180070
$431$	3.89778	0.187750	0.0938748	−	0.995584i	$-0.470075\pi$
$431$	3.89778	0.187750	0.0938748	+	0.995584i	$0.470075\pi$
$432$	−2.13355	−0.102650
$433$	−9.96158	−0.478723	−0.239362	−	0.970931i	$-0.576938\pi$
$433$	−9.96158	−0.478723	−0.239362	+	0.970931i	$0.576938\pi$
$434$	12.4436	0.597312
$435$	−3.55076	−0.170246
$436$	1.92411	0.0921482
$437$	12.5454	0.600130
$438$	0.595226	0.0284410
$439$	25.4167	1.21307	0.606537	−	0.795055i	$-0.292558\pi$
$439$	25.4167	1.21307	0.606537	+	0.795055i	$0.292558\pi$
$440$	−27.4462	−1.30845
$441$	−3.97776	−0.189417
$442$	34.2810	1.63058
$443$	31.8954	1.51540	0.757699	−	0.652605i	$-0.226324\pi$
$443$	31.8954	1.51540	0.757699	+	0.652605i	$0.226324\pi$
$444$	9.53431	0.452478
$445$	33.5299	1.58947
$446$	44.7183	2.11747
$447$	18.8854	0.893248
$448$	−21.2977	−1.00622
$449$	5.26569	0.248503	0.124252	−	0.992251i	$-0.460347\pi$
$449$	5.26569	0.248503	0.124252	+	0.992251i	$0.460347\pi$
$450$	−21.1611	−0.997544
$451$	−12.0792	−0.568788
$452$	27.3680	1.28728
$453$	−6.94496	−0.326303
$454$	54.2176	2.54456
$455$	−45.0770	−2.11324
$456$	−5.51423	−0.258227
$457$	−14.6248	−0.684118	−0.342059	−	0.939679i	$-0.611124\pi$
$457$	−14.6248	−0.684118	−0.342059	+	0.939679i	$0.611124\pi$
$458$	−26.0943	−1.21931
$459$	2.34530	0.109469
$460$	35.3515	1.64827
$461$	−28.8774	−1.34496	−0.672478	−	0.740117i	$-0.734770\pi$
$461$	−28.8774	−1.34496	−0.672478	+	0.740117i	$0.734770\pi$
$462$	−17.9149	−0.833477
$463$	−31.0325	−1.44220	−0.721101	−	0.692830i	$-0.756363\pi$
$463$	−31.0325	−1.44220	−0.721101	+	0.692830i	$0.756363\pi$
$464$	1.97134	0.0915171
$465$	−12.6974	−0.588829
$466$	51.9150	2.40492
$467$	−12.2632	−0.567472	−0.283736	−	0.958902i	$-0.591574\pi$
$467$	−12.2632	−0.567472	−0.283736	+	0.958902i	$0.591574\pi$
$468$	−18.1708	−0.839944
$469$	−4.80298	−0.221781
$470$	−71.3266	−3.29005
$471$	8.32010	0.383370
$472$	−3.94985	−0.181806
$473$	2.13357	0.0981017
$474$	21.5961	0.991943
$475$	35.8754	1.64608
$476$	10.9802	0.503275
$477$	5.17121	0.236773
$478$	9.23211	0.422267
$479$	20.0104	0.914299	0.457150	−	0.889390i	$-0.348870\pi$
$479$	20.0104	0.914299	0.457150	+	0.889390i	$0.348870\pi$
$480$	29.3017	1.33743
$481$	−23.8875	−1.08918
$482$	−23.1835	−1.05598
$483$	5.93833	0.270203
$484$	31.3145	1.42339
$485$	−19.3117	−0.876898
$486$	−2.16635	−0.0982675
$487$	−12.2455	−0.554897	−0.277449	−	0.960740i	$-0.589489\pi$
$487$	−12.2455	−0.554897	−0.277449	+	0.960740i	$0.589489\pi$
$488$	1.73951	0.0787441
$489$	6.44902	0.291635
$490$	33.1153	1.49600
$491$	7.52576	0.339633	0.169816	−	0.985476i	$-0.445683\pi$
$491$	7.52576	0.339633	0.169816	+	0.985476i	$0.445683\pi$
$492$	−6.83853	−0.308305
$493$	−2.16699	−0.0975963
$494$	53.6836	2.41534
$495$	18.2803	0.821640
$496$	7.04947	0.316530
$497$	−5.53567	−0.248309
$498$	10.8116	0.484478
$499$	−8.88916	−0.397933	−0.198967	−	0.980006i	$-0.563759\pi$
$499$	−8.88916	−0.397933	−0.198967	+	0.980006i	$0.563759\pi$
$500$	49.3462	2.20683
$501$	−15.4789	−0.691546
$502$	31.6773	1.41383
$503$	28.6931	1.27936	0.639681	−	0.768641i	$-0.279067\pi$
$503$	28.6931	1.27936	0.639681	+	0.768641i	$0.279067\pi$
$504$	−2.61014	−0.116265
$505$	−29.7690	−1.32470
$506$	−35.2005	−1.56486
$507$	32.5255	1.44451
$508$	50.7990	2.25384
$509$	−44.5438	−1.97437	−0.987185	−	0.159578i	$-0.948987\pi$
$509$	−44.5438	−1.97437	−0.987185	+	0.159578i	$0.948987\pi$
$510$	−19.5249	−0.864577
$511$	−0.477660	−0.0211304
$512$	−22.6746	−1.00208
$513$	3.67271	0.162154
$514$	48.6396	2.14540
$515$	16.9133	0.745287
$516$	1.20790	0.0531749
$517$	40.7552	1.79241
$518$	−13.3332	−0.585829
$519$	−18.8877	−0.829080
$520$	38.9304	1.70721
$521$	−13.9927	−0.613030	−0.306515	−	0.951866i	$-0.599163\pi$
$521$	−13.9927	−0.613030	−0.306515	+	0.951866i	$0.599163\pi$
$522$	2.00164	0.0876095
$523$	−21.9748	−0.960891	−0.480446	−	0.877025i	$-0.659525\pi$
$523$	−21.9748	−0.960891	−0.480446	+	0.877025i	$0.659525\pi$
$524$	2.77236	0.121111
$525$	16.9815	0.741132
$526$	−9.67024	−0.421643
$527$	−7.74911	−0.337557
$528$	−10.1490	−0.441680
$529$	−11.3319	−0.492692
$530$	−43.0509	−1.87001
$531$	2.63076	0.114165
$532$	17.1948	0.745489
$533$	17.1334	0.742131
$534$	−18.9016	−0.817951
$535$	−14.8099	−0.640288
$536$	4.14805	0.179169
$537$	2.11099	0.0910957
$538$	9.05050	0.390195
$539$	−18.9217	−0.815015
$540$	10.3492	0.445360
$541$	−14.7489	−0.634103	−0.317051	−	0.948408i	$-0.602693\pi$
$541$	−14.7489	−0.634103	−0.317051	+	0.948408i	$0.602693\pi$
$542$	−49.6559	−2.13291
$543$	16.0191	0.687447
$544$	17.8825	0.766706
$545$	2.74566	0.117611
$546$	25.4109	1.08749
$547$	−38.7263	−1.65582	−0.827908	−	0.560863i	$-0.810469\pi$
$547$	−38.7263	−1.65582	−0.827908	+	0.560863i	$0.810469\pi$
$548$	37.1245	1.58588
$549$	−1.15859	−0.0494474
$550$	−100.661	−4.29219
$551$	−3.39348	−0.144567
$552$	−5.12859	−0.218287
$553$	−17.3306	−0.736970
$554$	−57.4552	−2.44104
$555$	13.6052	0.577509
$556$	−52.9902	−2.24728
$557$	−4.25012	−0.180083	−0.0900416	−	0.995938i	$-0.528700\pi$
$557$	−4.25012	−0.180083	−0.0900416	+	0.995938i	$0.528700\pi$
$558$	7.15783	0.303015
$559$	−3.02631	−0.127999
$560$	−14.2538	−0.602332
$561$	11.1563	0.471019
$562$	−32.6877	−1.37885
$563$	−22.4343	−0.945493	−0.472747	−	0.881198i	$-0.656737\pi$
$563$	−22.4343	−0.945493	−0.472747	+	0.881198i	$0.656737\pi$
$564$	23.0731	0.971555
$565$	39.0535	1.64299
$566$	1.09937	0.0462100
$567$	1.73846	0.0730085
$568$	4.78084	0.200599
$569$	−14.7103	−0.616689	−0.308345	−	0.951275i	$-0.599775\pi$
$569$	−14.7103	−0.616689	−0.308345	+	0.951275i	$0.599775\pi$
$570$	−30.5757	−1.28068
$571$	−12.3908	−0.518540	−0.259270	−	0.965805i	$-0.583482\pi$
$571$	−12.3908	−0.518540	−0.259270	+	0.965805i	$0.583482\pi$
$572$	−86.4361	−3.61407
$573$	−3.38524	−0.141420
$574$	9.56334	0.399166
$575$	33.3665	1.39148
$576$	−12.2509	−0.510455
$577$	11.9386	0.497010	0.248505	−	0.968631i	$-0.420061\pi$
$577$	11.9386	0.497010	0.248505	+	0.968631i	$0.420061\pi$
$578$	24.9121	1.03621
$579$	9.36064	0.389015
$580$	−9.56240	−0.397057
$581$	−8.67611	−0.359946
$582$	10.8864	0.451257
$583$	24.5988	1.01878
$584$	0.412527	0.0170705
$585$	−25.9292	−1.07204
$586$	61.4484	2.53841
$587$	−20.1587	−0.832040	−0.416020	−	0.909355i	$-0.636575\pi$
$587$	−20.1587	−0.832040	−0.416020	+	0.909355i	$0.636575\pi$
$588$	−10.7123	−0.441769
$589$	−12.1350	−0.500014
$590$	−21.9014	−0.901667
$591$	28.0358	1.15324
$592$	−7.55346	−0.310445
$593$	−15.0871	−0.619551	−0.309776	−	0.950810i	$-0.600254\pi$
$593$	−15.0871	−0.619551	−0.309776	+	0.950810i	$0.600254\pi$
$594$	−10.3050	−0.422821
$595$	15.6684	0.642343
$596$	50.8595	2.08328
$597$	−1.81714	−0.0743706
$598$	49.9293	2.04176
$599$	30.7163	1.25503	0.627517	−	0.778603i	$-0.284071\pi$
$599$	30.7163	1.25503	0.627517	+	0.778603i	$0.284071\pi$
$600$	−14.6659	−0.598733
$601$	−32.5031	−1.32583	−0.662915	−	0.748694i	$-0.730681\pi$
$601$	−32.5031	−1.32583	−0.662915	+	0.748694i	$0.730681\pi$
$602$	−1.68919	−0.0688462
$603$	−2.76278	−0.112509
$604$	−18.7032	−0.761022
$605$	44.6850	1.81670
$606$	16.7815	0.681700
$607$	−41.5835	−1.68782	−0.843911	−	0.536483i	$-0.819752\pi$
$607$	−41.5835	−1.68782	−0.843911	+	0.536483i	$0.819752\pi$
$608$	28.0037	1.13570
$609$	−1.60629	−0.0650900
$610$	9.64539	0.390530
$611$	−57.8081	−2.33866
$612$	6.31603	0.255310
$613$	4.41545	0.178338	0.0891692	−	0.996016i	$-0.471579\pi$
$613$	4.41545	0.178338	0.0891692	+	0.996016i	$0.471579\pi$
$614$	37.8565	1.52776
$615$	−9.75841	−0.393497
$616$	−12.4161	−0.500259
$617$	8.47693	0.341268	0.170634	−	0.985334i	$-0.445418\pi$
$617$	8.47693	0.341268	0.170634	+	0.985334i	$0.445418\pi$
$618$	−9.53439	−0.383529
$619$	16.3518	0.657233	0.328616	−	0.944464i	$-0.393418\pi$
$619$	16.3518	0.657233	0.328616	+	0.944464i	$0.393418\pi$
$620$	−34.1949	−1.37330
$621$	3.41586	0.137074
$622$	−42.1244	−1.68903
$623$	15.1682	0.607702
$624$	14.3956	0.576286
$625$	21.5754	0.863014
$626$	20.0442	0.801127
$627$	17.4706	0.697709
$628$	22.4065	0.894118
$629$	8.30312	0.331067
$630$	−14.4729	−0.576614
$631$	5.68619	0.226364	0.113182	−	0.993574i	$-0.463896\pi$
$631$	5.68619	0.226364	0.113182	+	0.993574i	$0.463896\pi$
$632$	14.9674	0.595371
$633$	16.9521	0.673786
$634$	34.2630	1.36076
$635$	72.4889	2.87663
$636$	13.9264	0.552216
$637$	26.8390	1.06340
$638$	9.52156	0.376962
$639$	−3.18424	−0.125966
$640$	43.3870	1.71502
$641$	−12.4893	−0.493296	−0.246648	−	0.969105i	$-0.579329\pi$
$641$	−12.4893	−0.493296	−0.246648	+	0.969105i	$0.579329\pi$
$642$	8.34868	0.329496
$643$	−20.8221	−0.821142	−0.410571	−	0.911829i	$-0.634671\pi$
$643$	−20.8221	−0.821142	−0.410571	+	0.911829i	$0.634671\pi$
$644$	15.9923	0.630184
$645$	1.72365	0.0678685
$646$	−18.6600	−0.734169
$647$	−25.6081	−1.00676	−0.503379	−	0.864065i	$-0.667910\pi$
$647$	−25.6081	−1.00676	−0.503379	+	0.864065i	$0.667910\pi$
$648$	−1.50141	−0.0589808
$649$	12.5142	0.491225
$650$	142.779	5.60027
$651$	−5.74405	−0.225127
$652$	17.3676	0.680167
$653$	2.13441	0.0835258	0.0417629	−	0.999128i	$-0.486703\pi$
$653$	2.13441	0.0835258	0.0417629	+	0.999128i	$0.486703\pi$
$654$	−1.54779	−0.0605234
$655$	3.95609	0.154577
$656$	5.41776	0.211528
$657$	−0.274760	−0.0107194
$658$	−32.2666	−1.25788
$659$	−16.7462	−0.652341	−0.326171	−	0.945311i	$-0.605758\pi$
$659$	−16.7462	−0.652341	−0.326171	+	0.945311i	$0.605758\pi$
$660$	49.2300	1.91628
$661$	4.42341	0.172051	0.0860254	−	0.996293i	$-0.472583\pi$
$661$	4.42341	0.172051	0.0860254	+	0.996293i	$0.472583\pi$
$662$	5.33549	0.207370
$663$	−15.8243	−0.614567
$664$	7.49306	0.290787
$665$	24.5365	0.951486
$666$	−7.66957	−0.297190
$667$	−3.15615	−0.122207
$668$	−41.6856	−1.61286
$669$	−20.6423	−0.798076
$670$	23.0004	0.888585
$671$	−5.51126	−0.212760
$672$	13.2555	0.511340
$673$	1.61455	0.0622365	0.0311182	−	0.999516i	$-0.490093\pi$
$673$	1.61455	0.0622365	0.0311182	+	0.999516i	$0.490093\pi$
$674$	−28.4471	−1.09574
$675$	9.76811	0.375975
$676$	87.5932	3.36897
$677$	−0.677398	−0.0260345	−0.0130173	−	0.999915i	$-0.504144\pi$
$677$	−0.677398	−0.0260345	−0.0130173	+	0.999915i	$0.504144\pi$
$678$	−22.0154	−0.845494
$679$	−8.73620	−0.335264
$680$	−13.5319	−0.518925
$681$	−25.0272	−0.959045
$682$	34.0489	1.30380
$683$	−4.12918	−0.157999	−0.0789994	−	0.996875i	$-0.525173\pi$
$683$	−4.12918	−0.157999	−0.0789994	+	0.996875i	$0.525173\pi$
$684$	9.89082	0.378185
$685$	52.9757	2.02410
$686$	41.3434	1.57850
$687$	12.0453	0.459557
$688$	−0.956948	−0.0364833
$689$	−34.8915	−1.32926
$690$	−28.4374	−1.08259
$691$	−25.2479	−0.960476	−0.480238	−	0.877138i	$-0.659450\pi$
$691$	−25.2479	−0.960476	−0.480238	+	0.877138i	$0.659450\pi$
$692$	−50.8658	−1.93363
$693$	8.26963	0.314137
$694$	−52.5284	−1.99395
$695$	−75.6157	−2.86827
$696$	1.38726	0.0525839
$697$	−5.95546	−0.225579
$698$	−48.8856	−1.85035
$699$	−23.9643	−0.906413
$700$	45.7321	1.72851
$701$	35.7393	1.34986	0.674928	−	0.737883i	$-0.264175\pi$
$701$	35.7393	1.34986	0.674928	+	0.737883i	$0.264175\pi$
$702$	14.6169	0.551680
$703$	13.0026	0.490402
$704$	−58.2761	−2.19636
$705$	32.9248	1.24002
$706$	41.9484	1.57875
$707$	−13.4669	−0.506474
$708$	7.08480	0.266263
$709$	11.4501	0.430017	0.215009	−	0.976612i	$-0.431022\pi$
$709$	11.4501	0.430017	0.215009	+	0.976612i	$0.431022\pi$
$710$	26.5092	0.994871
$711$	−9.96891	−0.373863
$712$	−13.0999	−0.490940
$713$	−11.2863	−0.422677
$714$	−8.83265	−0.330554
$715$	−123.342	−4.61274
$716$	5.68501	0.212459
$717$	−4.26160	−0.159152
$718$	27.6575	1.03217
$719$	21.6446	0.807207	0.403603	−	0.914934i	$-0.367758\pi$
$719$	21.6446	0.807207	0.403603	+	0.914934i	$0.367758\pi$
$720$	−8.19908	−0.305562
$721$	7.65120	0.284946
$722$	11.9392	0.444332
$723$	10.7017	0.398000
$724$	43.1405	1.60330
$725$	−9.02545	−0.335197
$726$	−25.1900	−0.934887
$727$	17.2905	0.641268	0.320634	−	0.947203i	$-0.396104\pi$
$727$	17.2905	0.641268	0.320634	+	0.947203i	$0.396104\pi$
$728$	17.6113	0.652717
$729$	1.00000	0.0370370
$730$	2.28741	0.0846609
$731$	1.05192	0.0389068
$732$	−3.12015	−0.115324
$733$	−6.76932	−0.250031	−0.125015	−	0.992155i	$-0.539898\pi$
$733$	−6.76932	−0.250031	−0.125015	+	0.992155i	$0.539898\pi$
$734$	45.1241	1.66556
$735$	−15.2862	−0.563841
$736$	26.0453	0.960043
$737$	−13.1422	−0.484098
$738$	5.50104	0.202496
$739$	3.09807	0.113965	0.0569823	−	0.998375i	$-0.481852\pi$
$739$	3.09807	0.113965	0.0569823	+	0.998375i	$0.481852\pi$
$740$	36.6397	1.34690
$741$	−24.7807	−0.910342
$742$	−19.4753	−0.714961
$743$	−28.1746	−1.03363	−0.516813	−	0.856098i	$-0.672882\pi$
$743$	−28.1746	−1.03363	−0.516813	+	0.856098i	$0.672882\pi$
$744$	4.96080	0.181872
$745$	72.5752	2.65895
$746$	76.9354	2.81681
$747$	−4.99069	−0.182600
$748$	30.0446	1.09854
$749$	−6.69969	−0.244801
$750$	−39.6950	−1.44946
$751$	48.3056	1.76270	0.881348	−	0.472468i	$-0.156637\pi$
$751$	48.3056	1.76270	0.881348	+	0.472468i	$0.156637\pi$
$752$	−18.2795	−0.666584
$753$	−14.6224	−0.532872
$754$	−13.5056	−0.491845
$755$	−26.6890	−0.971311
$756$	4.68178	0.170275
$757$	18.3876	0.668309	0.334154	−	0.942518i	$-0.391549\pi$
$757$	18.3876	0.668309	0.334154	+	0.942518i	$0.391549\pi$
$758$	−34.0156	−1.23550
$759$	16.2488	0.589794
$760$	−21.1908	−0.768671
$761$	36.8271	1.33498	0.667491	−	0.744618i	$-0.267368\pi$
$761$	36.8271	1.33498	0.667491	+	0.744618i	$0.267368\pi$
$762$	−40.8636	−1.48033
$763$	1.24208	0.0449662
$764$	−9.11665	−0.329829
$765$	9.01282	0.325859
$766$	6.66264	0.240731
$767$	−17.7504	−0.640931
$768$	0.0435867	0.00157280
$769$	−39.5384	−1.42579	−0.712895	−	0.701271i	$-0.752616\pi$
$769$	−39.5384	−1.42579	−0.712895	+	0.701271i	$0.752616\pi$
$770$	−68.8457	−2.48103
$771$	−22.4523	−0.808601
$772$	25.2088	0.907284
$773$	15.9248	0.572775	0.286387	−	0.958114i	$-0.407546\pi$
$773$	15.9248	0.572775	0.286387	+	0.958114i	$0.407546\pi$
$774$	−0.971658	−0.0349255
$775$	−32.2748	−1.15935
$776$	7.54495	0.270848
$777$	6.15471	0.220799
$778$	0.670020	0.0240214
$779$	−9.32616	−0.334145
$780$	−69.8290	−2.50028
$781$	−15.1470	−0.542003
$782$	−17.3551	−0.620616
$783$	−0.923971	−0.0330200
$784$	8.48674	0.303098
$785$	31.9736	1.14118
$786$	−2.23014	−0.0795465
$787$	16.8337	0.600056	0.300028	−	0.953930i	$-0.403004\pi$
$787$	16.8337	0.600056	0.300028	+	0.953930i	$0.403004\pi$
$788$	75.5020	2.68965
$789$	4.46385	0.158917
$790$	82.9924	2.95274
$791$	17.6670	0.628165
$792$	−7.14201	−0.253780
$793$	7.81730	0.277600
$794$	48.2256	1.71146
$795$	19.8726	0.704808
$796$	−4.89366	−0.173451
$797$	16.6902	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$797$	16.6902	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$798$	−13.8318	−0.489641
$799$	20.0937	0.710863
$800$	74.4801	2.63327
$801$	8.72509	0.308286
$802$	−43.5970	−1.53946
$803$	−1.30700	−0.0461230
$804$	−7.44032	−0.262400
$805$	22.8206	0.804320
$806$	−48.2958	−1.70115
$807$	−4.17777	−0.147064
$808$	11.6306	0.409161
$809$	−12.0692	−0.424332	−0.212166	−	0.977234i	$-0.568052\pi$
$809$	−12.0692	−0.424332	−0.212166	+	0.977234i	$0.568052\pi$
$810$	−8.32512	−0.292515
$811$	48.0294	1.68654	0.843270	−	0.537491i	$-0.180628\pi$
$811$	48.0294	1.68654	0.843270	+	0.537491i	$0.180628\pi$
$812$	−4.32583	−0.151807
$813$	22.9215	0.803892
$814$	−36.4832	−1.27874
$815$	24.7831	0.868115
$816$	−5.00381	−0.175168
$817$	1.64730	0.0576316
$818$	−45.1136	−1.57736
$819$	−11.7298	−0.409874
$820$	−26.2800	−0.917737
$821$	8.55777	0.298668	0.149334	−	0.988787i	$-0.452287\pi$
$821$	8.55777	0.298668	0.149334	+	0.988787i	$0.452287\pi$
$822$	−29.8636	−1.04161
$823$	23.9597	0.835181	0.417590	−	0.908635i	$-0.362875\pi$
$823$	23.9597	0.835181	0.417590	+	0.908635i	$0.362875\pi$
$824$	−6.60790	−0.230197
$825$	46.4656	1.61773
$826$	−9.90773	−0.344734
$827$	52.6284	1.83007	0.915035	−	0.403374i	$-0.132163\pi$
$827$	52.6284	1.83007	0.915035	+	0.403374i	$0.132163\pi$
$828$	9.19911	0.319691
$829$	6.91553	0.240186	0.120093	−	0.992763i	$-0.461681\pi$
$829$	6.91553	0.240186	0.120093	+	0.992763i	$0.461681\pi$
$830$	41.5481	1.44215
$831$	26.5217	0.920027
$832$	82.6601	2.86572
$833$	−9.32903	−0.323232
$834$	42.6263	1.47603
$835$	−59.4843	−2.05854
$836$	47.0494	1.62724
$837$	−3.30410	−0.114206
$838$	−64.0429	−2.21232
$839$	−3.37052	−0.116363	−0.0581816	−	0.998306i	$-0.518530\pi$
$839$	−3.37052	−0.116363	−0.0581816	+	0.998306i	$0.518530\pi$
$840$	−10.0306	−0.346088
$841$	−28.1463	−0.970561
$842$	−54.6022	−1.88172
$843$	15.0889	0.519688
$844$	45.6531	1.57144
$845$	124.993	4.29990
$846$	−18.5605	−0.638122
$847$	20.2146	0.694580
$848$	−11.0330	−0.378876
$849$	−0.507477	−0.0174166
$850$	−49.6291	−1.70226
$851$	12.0932	0.414551
$852$	−8.57534	−0.293786
$853$	7.46342	0.255543	0.127771	−	0.991804i	$-0.459218\pi$
$853$	7.46342	0.255543	0.127771	+	0.991804i	$0.459218\pi$
$854$	4.36337	0.149311
$855$	14.1140	0.482687
$856$	5.78613	0.197766
$857$	45.1883	1.54360	0.771802	−	0.635863i	$-0.219356\pi$
$857$	45.1883	1.54360	0.771802	+	0.635863i	$0.219356\pi$
$858$	69.5308	2.37374
$859$	2.84258	0.0969875	0.0484938	−	0.998823i	$-0.484558\pi$
$859$	2.84258	0.0969875	0.0484938	+	0.998823i	$0.484558\pi$
$860$	4.64188	0.158287
$861$	−4.41450	−0.150446
$862$	−8.44395	−0.287602
$863$	31.5546	1.07413	0.537066	−	0.843540i	$-0.319533\pi$
$863$	31.5546	1.07413	0.537066	+	0.843540i	$0.319533\pi$
$864$	7.62482	0.259402
$865$	−72.5843	−2.46794
$866$	21.5802	0.733326
$867$	−11.4996	−0.390546
$868$	−15.4691	−0.525054
$869$	−47.4209	−1.60864
$870$	7.69217	0.260789
$871$	18.6412	0.631632
$872$	−1.07271	−0.0363266
$873$	−5.02525	−0.170079
$874$	−27.1778	−0.919302
$875$	31.8547	1.07688
$876$	−0.739946	−0.0250004
$877$	5.74259	0.193913	0.0969567	−	0.995289i	$-0.469089\pi$
$877$	5.74259	0.193913	0.0969567	+	0.995289i	$0.469089\pi$
$878$	−55.0614	−1.85823
$879$	−28.3650	−0.956727
$880$	−39.0020	−1.31476
$881$	49.5935	1.67085	0.835424	−	0.549606i	$-0.185222\pi$
$881$	49.5935	1.67085	0.835424	+	0.549606i	$0.185222\pi$
$882$	8.61720	0.290156
$883$	−18.7690	−0.631628	−0.315814	−	0.948821i	$-0.602278\pi$
$883$	−18.7690	−0.631628	−0.315814	+	0.948821i	$0.602278\pi$
$884$	−42.6159	−1.43333
$885$	10.1098	0.339838
$886$	−69.0965	−2.32134
$887$	−6.47276	−0.217334	−0.108667	−	0.994078i	$-0.534658\pi$
$887$	−6.47276	−0.217334	−0.108667	+	0.994078i	$0.534658\pi$
$888$	−5.31547	−0.178376
$889$	32.7924	1.09982
$890$	−72.6374	−2.43481
$891$	4.75687	0.159361
$892$	−55.5908	−1.86132
$893$	31.4664	1.05298
$894$	−40.9123	−1.36831
$895$	8.11237	0.271166
$896$	19.6274	0.655704
$897$	−23.0477	−0.769540
$898$	−11.4073	−0.380667
$899$	3.05290	0.101820
$900$	26.3061	0.876870
$901$	12.1280	0.404043
$902$	26.1678	0.871291
$903$	0.779741	0.0259482
$904$	−15.2579	−0.507472
$905$	61.5604	2.04634
$906$	15.0452	0.499843
$907$	38.9596	1.29363	0.646817	−	0.762645i	$-0.276100\pi$
$907$	38.9596	1.29363	0.646817	+	0.762645i	$0.276100\pi$
$908$	−67.3998	−2.23674
$909$	−7.74643	−0.256933
$910$	97.6523	3.23714
$911$	17.9970	0.596266	0.298133	−	0.954524i	$-0.403636\pi$
$911$	17.9970	0.596266	0.298133	+	0.954524i	$0.403636\pi$
$912$	−7.83590	−0.259473
$913$	−23.7401	−0.785682
$914$	31.6823	1.04796
$915$	−4.45237	−0.147191
$916$	32.4387	1.07181
$917$	1.78965	0.0590996
$918$	−5.08073	−0.167689
$919$	34.0478	1.12313	0.561566	−	0.827432i	$-0.310199\pi$
$919$	34.0478	1.12313	0.561566	+	0.827432i	$0.310199\pi$
$920$	−19.7088	−0.649781
$921$	−17.4748	−0.575815
$922$	62.5585	2.06025
$923$	21.4849	0.707183
$924$	22.2706	0.732650
$925$	34.5823	1.13706
$926$	67.2271	2.20922
$927$	4.40114	0.144552
$928$	−7.04512	−0.231267
$929$	−20.5317	−0.673623	−0.336811	−	0.941572i	$-0.609348\pi$
$929$	−20.5317	−0.673623	−0.336811	+	0.941572i	$0.609348\pi$
$930$	27.5070	0.901992
$931$	−14.6091	−0.478795
$932$	−64.5373	−2.11399
$933$	19.4449	0.636597
$934$	26.5663	0.869275
$935$	42.8728	1.40209
$936$	10.1304	0.331122
$937$	−15.1438	−0.494725	−0.247363	−	0.968923i	$-0.579564\pi$
$937$	−15.1438	−0.494725	−0.247363	+	0.968923i	$0.579564\pi$
$938$	10.4049	0.339733
$939$	−9.25254	−0.301945
$940$	88.6685	2.89205
$941$	−49.5619	−1.61567	−0.807835	−	0.589408i	$-0.799361\pi$
$941$	−49.5619	−1.61567	−0.807835	+	0.589408i	$0.799361\pi$
$942$	−18.0242	−0.587261
$943$	−8.67394	−0.282462
$944$	−5.61286	−0.182683
$945$	6.68078	0.217326
$946$	−4.62206	−0.150276
$947$	−2.81943	−0.0916193	−0.0458096	−	0.998950i	$-0.514587\pi$
$947$	−2.81943	−0.0916193	−0.0458096	+	0.998950i	$0.514587\pi$
$948$	−26.8469	−0.871946
$949$	1.85388	0.0601794
$950$	−77.7185	−2.52152
$951$	−15.8160	−0.512869
$952$	−6.12155	−0.198401
$953$	25.5994	0.829246	0.414623	−	0.909993i	$-0.363913\pi$
$953$	25.5994	0.829246	0.414623	+	0.909993i	$0.363913\pi$
$954$	−11.2026	−0.362698
$955$	−13.0092	−0.420969
$956$	−11.4767	−0.371184
$957$	−4.39522	−0.142077
$958$	−43.3495	−1.40056
$959$	23.9651	0.773873
$960$	−47.0794	−1.51948
$961$	−20.0829	−0.647836
$962$	51.7486	1.66844
$963$	−3.85381	−0.124187
$964$	28.8202	0.928238
$965$	35.9723	1.15799
$966$	−12.8645	−0.413908
$967$	−18.9630	−0.609808	−0.304904	−	0.952383i	$-0.598624\pi$
$967$	−18.9630	−0.609808	−0.304904	+	0.952383i	$0.598624\pi$
$968$	−17.4581	−0.561126
$969$	8.61360	0.276709
$970$	41.8358	1.34327
$971$	−39.5105	−1.26795	−0.633976	−	0.773352i	$-0.718578\pi$
$971$	−39.5105	−1.26795	−0.633976	+	0.773352i	$0.718578\pi$
$972$	2.69306	0.0863799
$973$	−34.2069	−1.09662
$974$	26.5280	0.850013
$975$	−65.9080	−2.11074
$976$	2.47191	0.0791238
$977$	−47.1158	−1.50737	−0.753684	−	0.657236i	$-0.771725\pi$
$977$	−47.1158	−1.50737	−0.753684	+	0.657236i	$0.771725\pi$
$978$	−13.9708	−0.446737
$979$	41.5042	1.32648
$980$	−41.1667	−1.31502
$981$	0.714470	0.0228113
$982$	−16.3034	−0.520263
$983$	56.1821	1.79193	0.895966	−	0.444124i	$-0.146485\pi$
$983$	56.1821	1.79193	0.895966	+	0.444124i	$0.146485\pi$
$984$	3.81255	0.121540
$985$	107.740	3.43287
$986$	4.69445	0.149502
$987$	14.8945	0.474097
$988$	−66.7359	−2.12315
$989$	1.53209	0.0487177
$990$	−39.6015	−1.25862
$991$	42.6562	1.35502	0.677509	−	0.735514i	$-0.263059\pi$
$991$	42.6562	1.35502	0.677509	+	0.735514i	$0.263059\pi$
$992$	−25.1932	−0.799885
$993$	−2.46290	−0.0781577
$994$	11.9922	0.380369
$995$	−6.98314	−0.221380
$996$	−13.4402	−0.425870
$997$	45.7971	1.45041	0.725204	−	0.688534i	$-0.241745\pi$
$997$	45.7971	1.45041	0.725204	+	0.688534i	$0.241745\pi$
$998$	19.2570	0.609570
$999$	3.54033	0.112011

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8049.2.a.d.1.20	✓	129

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8049.2.a.d.1.20	✓	129	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)

\(f(q)\)	\(=\)	\(q-2.16635 q^{2} +1.00000 q^{3} +2.69306 q^{4} +3.84293 q^{5} -2.16635 q^{6} +1.73846 q^{7} -1.50141 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.16635 q^{2} +1.00000 q^{3} +2.69306 q^{4} +3.84293 q^{5} -2.16635 q^{6} +1.73846 q^{7} -1.50141 q^{8} +1.00000 q^{9} -8.32512 q^{10} +4.75687 q^{11} +2.69306 q^{12} -6.74726 q^{13} -3.76611 q^{14} +3.84293 q^{15} -2.13355 q^{16} +2.34530 q^{17} -2.16635 q^{18} +3.67271 q^{19} +10.3492 q^{20} +1.73846 q^{21} -10.3050 q^{22} +3.41586 q^{23} -1.50141 q^{24} +9.76811 q^{25} +14.6169 q^{26} +1.00000 q^{27} +4.68178 q^{28} -0.923971 q^{29} -8.32512 q^{30} -3.30410 q^{31} +7.62482 q^{32} +4.75687 q^{33} -5.08073 q^{34} +6.68078 q^{35} +2.69306 q^{36} +3.54033 q^{37} -7.95636 q^{38} -6.74726 q^{39} -5.76980 q^{40} -2.53932 q^{41} -3.76611 q^{42} +0.448524 q^{43} +12.8105 q^{44} +3.84293 q^{45} -7.39993 q^{46} +8.56763 q^{47} -2.13355 q^{48} -3.97776 q^{49} -21.1611 q^{50} +2.34530 q^{51} -18.1708 q^{52} +5.17121 q^{53} -2.16635 q^{54} +18.2803 q^{55} -2.61014 q^{56} +3.67271 q^{57} +2.00164 q^{58} +2.63076 q^{59} +10.3492 q^{60} -1.15859 q^{61} +7.15783 q^{62} +1.73846 q^{63} -12.2509 q^{64} -25.9292 q^{65} -10.3050 q^{66} -2.76278 q^{67} +6.31603 q^{68} +3.41586 q^{69} -14.4729 q^{70} -3.18424 q^{71} -1.50141 q^{72} -0.274760 q^{73} -7.66957 q^{74} +9.76811 q^{75} +9.89082 q^{76} +8.26963 q^{77} +14.6169 q^{78} -9.96891 q^{79} -8.19908 q^{80} +1.00000 q^{81} +5.50104 q^{82} -4.99069 q^{83} +4.68178 q^{84} +9.01282 q^{85} -0.971658 q^{86} -0.923971 q^{87} -7.14201 q^{88} +8.72509 q^{89} -8.32512 q^{90} -11.7298 q^{91} +9.19911 q^{92} -3.30410 q^{93} -18.5605 q^{94} +14.1140 q^{95} +7.62482 q^{96} -5.02525 q^{97} +8.61720 q^{98} +4.75687 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 129 q + 8 q^{2} + 129 q^{3} + 158 q^{4} + 11 q^{5} + 8 q^{6} + 40 q^{7} + 18 q^{8} + 129 q^{9}+O(q^{10}) \) 129 * q + 8 * q^2 + 129 * q^3 + 158 * q^4 + 11 * q^5 + 8 * q^6 + 40 * q^7 + 18 * q^8 + 129 * q^9	\( 129 q + 8 q^{2} + 129 q^{3} + 158 q^{4} + 11 q^{5} + 8 q^{6} + 40 q^{7} + 18 q^{8} + 129 q^{9} + 20 q^{10} + 48 q^{11} + 158 q^{12} + 77 q^{13} + 13 q^{14} + 11 q^{15} + 212 q^{16} + 9 q^{17} + 8 q^{18} + 68 q^{19} + 19 q^{20} + 40 q^{21} + 45 q^{22} + 64 q^{23} + 18 q^{24} + 188 q^{25} + 19 q^{26} + 129 q^{27} + 69 q^{28} + 23 q^{29} + 20 q^{30} + 133 q^{31} + 24 q^{32} + 48 q^{33} + 63 q^{34} + 26 q^{35} + 158 q^{36} + 147 q^{37} + 9 q^{38} + 77 q^{39} + 58 q^{40} + 21 q^{41} + 13 q^{42} + 76 q^{43} + 110 q^{44} + 11 q^{45} + 48 q^{46} + 85 q^{47} + 212 q^{48} + 213 q^{49} + 17 q^{50} + 9 q^{51} + 139 q^{52} + 30 q^{53} + 8 q^{54} + 103 q^{55} + 19 q^{56} + 68 q^{57} + 94 q^{58} + 64 q^{59} + 19 q^{60} + 110 q^{61} - 10 q^{62} + 40 q^{63} + 288 q^{64} - 8 q^{65} + 45 q^{66} + 118 q^{67} - 15 q^{68} + 64 q^{69} + 75 q^{70} + 154 q^{71} + 18 q^{72} + 137 q^{73} + 28 q^{74} + 188 q^{75} + 156 q^{76} + 17 q^{77} + 19 q^{78} + 157 q^{79} + 2 q^{80} + 129 q^{81} + 72 q^{82} + 39 q^{83} + 69 q^{84} + 127 q^{85} + 54 q^{86} + 23 q^{87} + 97 q^{88} + 31 q^{89} + 20 q^{90} + 137 q^{91} + 82 q^{92} + 133 q^{93} + 40 q^{94} + 68 q^{95} + 24 q^{96} + 170 q^{97} - 21 q^{98} + 48 q^{99}+O(q^{100}) \) 129 * q + 8 * q^2 + 129 * q^3 + 158 * q^4 + 11 * q^5 + 8 * q^6 + 40 * q^7 + 18 * q^8 + 129 * q^9 + 20 * q^10 + 48 * q^11 + 158 * q^12 + 77 * q^13 + 13 * q^14 + 11 * q^15 + 212 * q^16 + 9 * q^17 + 8 * q^18 + 68 * q^19 + 19 * q^20 + 40 * q^21 + 45 * q^22 + 64 * q^23 + 18 * q^24 + 188 * q^25 + 19 * q^26 + 129 * q^27 + 69 * q^28 + 23 * q^29 + 20 * q^30 + 133 * q^31 + 24 * q^32 + 48 * q^33 + 63 * q^34 + 26 * q^35 + 158 * q^36 + 147 * q^37 + 9 * q^38 + 77 * q^39 + 58 * q^40 + 21 * q^41 + 13 * q^42 + 76 * q^43 + 110 * q^44 + 11 * q^45 + 48 * q^46 + 85 * q^47 + 212 * q^48 + 213 * q^49 + 17 * q^50 + 9 * q^51 + 139 * q^52 + 30 * q^53 + 8 * q^54 + 103 * q^55 + 19 * q^56 + 68 * q^57 + 94 * q^58 + 64 * q^59 + 19 * q^60 + 110 * q^61 - 10 * q^62 + 40 * q^63 + 288 * q^64 - 8 * q^65 + 45 * q^66 + 118 * q^67 - 15 * q^68 + 64 * q^69 + 75 * q^70 + 154 * q^71 + 18 * q^72 + 137 * q^73 + 28 * q^74 + 188 * q^75 + 156 * q^76 + 17 * q^77 + 19 * q^78 + 157 * q^79 + 2 * q^80 + 129 * q^81 + 72 * q^82 + 39 * q^83 + 69 * q^84 + 127 * q^85 + 54 * q^86 + 23 * q^87 + 97 * q^88 + 31 * q^89 + 20 * q^90 + 137 * q^91 + 82 * q^92 + 133 * q^93 + 40 * q^94 + 68 * q^95 + 24 * q^96 + 170 * q^97 - 21 * q^98 + 48 * q^99

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