LMFDB - Embedded newform 1339.2.a.e.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1339,2,Mod(1,1339)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1339, 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("1339.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1339 = 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1339.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:	$10.6919688306$
Analytic rank:	$1$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	1339.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.08679 q^{2} -1.93108 q^{3} +2.35469 q^{4} +1.87136 q^{5} -4.02977 q^{6} -0.548741 q^{7} +0.740165 q^{8} +0.729089 q^{9} +O(q^{10})$	\(q+2.08679 q^{2} -1.93108 q^{3} +2.35469 q^{4} +1.87136 q^{5} -4.02977 q^{6} -0.548741 q^{7} +0.740165 q^{8} +0.729089 q^{9} +3.90513 q^{10} -6.22646 q^{11} -4.54711 q^{12} -1.00000 q^{13} -1.14511 q^{14} -3.61376 q^{15} -3.16481 q^{16} -6.48507 q^{17} +1.52145 q^{18} +3.43740 q^{19} +4.40647 q^{20} +1.05967 q^{21} -12.9933 q^{22} -0.591066 q^{23} -1.42932 q^{24} -1.49801 q^{25} -2.08679 q^{26} +4.38532 q^{27} -1.29212 q^{28} -3.31121 q^{29} -7.54115 q^{30} +7.28855 q^{31} -8.08463 q^{32} +12.0238 q^{33} -13.5330 q^{34} -1.02689 q^{35} +1.71678 q^{36} -2.70751 q^{37} +7.17313 q^{38} +1.93108 q^{39} +1.38512 q^{40} -2.81928 q^{41} +2.21130 q^{42} +8.64936 q^{43} -14.6614 q^{44} +1.36439 q^{45} -1.23343 q^{46} -2.97088 q^{47} +6.11152 q^{48} -6.69888 q^{49} -3.12603 q^{50} +12.5232 q^{51} -2.35469 q^{52} -5.03559 q^{53} +9.15125 q^{54} -11.6520 q^{55} -0.406159 q^{56} -6.63791 q^{57} -6.90981 q^{58} -5.16664 q^{59} -8.50928 q^{60} +15.5246 q^{61} +15.2097 q^{62} -0.400081 q^{63} -10.5413 q^{64} -1.87136 q^{65} +25.0912 q^{66} +6.18003 q^{67} -15.2703 q^{68} +1.14140 q^{69} -2.14291 q^{70} -8.76878 q^{71} +0.539646 q^{72} -4.29279 q^{73} -5.65001 q^{74} +2.89279 q^{75} +8.09401 q^{76} +3.41672 q^{77} +4.02977 q^{78} -1.40134 q^{79} -5.92250 q^{80} -10.6557 q^{81} -5.88325 q^{82} +4.60998 q^{83} +2.49519 q^{84} -12.1359 q^{85} +18.0494 q^{86} +6.39423 q^{87} -4.60861 q^{88} -7.36760 q^{89} +2.84719 q^{90} +0.548741 q^{91} -1.39178 q^{92} -14.0748 q^{93} -6.19961 q^{94} +6.43261 q^{95} +15.6121 q^{96} -4.20939 q^{97} -13.9792 q^{98} -4.53964 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9}+O(q^{10}) $ 21 * q - q^2 - 6 * q^3 + 15 * q^4 - 18 * q^5 - 8 * q^6 - 2 * q^7 + 6 * q^8 + 11 * q^9	\( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9} - 10 q^{10} - 5 q^{11} - 22 q^{12} - 21 q^{13} - 24 q^{14} - q^{15} - 5 q^{16} - 18 q^{17} + 10 q^{18} - 6 q^{19} - 30 q^{20} - 17 q^{21} - 8 q^{22} - 14 q^{23} - 13 q^{24} + 11 q^{25} + q^{26} - 33 q^{27} - 2 q^{28} - 36 q^{29} + 2 q^{30} + q^{31} - 9 q^{32} - 11 q^{33} - 25 q^{34} - 37 q^{35} + 3 q^{36} + 5 q^{37} - 13 q^{38} + 6 q^{39} - 2 q^{40} - 32 q^{41} + 7 q^{42} + 2 q^{43} + 4 q^{44} - 32 q^{45} - 5 q^{46} - 12 q^{47} - 18 q^{48} + 5 q^{49} - 2 q^{50} - 27 q^{51} - 15 q^{52} - 49 q^{53} - 31 q^{54} - 9 q^{55} - 53 q^{56} + 14 q^{57} - 2 q^{58} - 34 q^{59} + 25 q^{60} - 27 q^{61} - 12 q^{62} + 41 q^{63} - 70 q^{64} + 18 q^{65} - 15 q^{66} - 12 q^{67} - 20 q^{68} - 70 q^{69} + 70 q^{70} - 26 q^{71} - 14 q^{72} + 23 q^{73} + 20 q^{74} - 9 q^{75} - 36 q^{76} - 62 q^{77} + 8 q^{78} - 7 q^{79} - 41 q^{80} - 3 q^{81} - 33 q^{82} - q^{83} - 48 q^{84} + 10 q^{85} - 23 q^{86} - 7 q^{87} + 9 q^{88} - 16 q^{89} - 30 q^{90} + 2 q^{91} - 27 q^{92} + 2 q^{93} - 37 q^{94} - 35 q^{95} + 55 q^{96} - 30 q^{97} - 43 q^{98} - 42 q^{99}+O(q^{100}) \) 21 * q - q^2 - 6 * q^3 + 15 * q^4 - 18 * q^5 - 8 * q^6 - 2 * q^7 + 6 * q^8 + 11 * q^9 - 10 * q^10 - 5 * q^11 - 22 * q^12 - 21 * q^13 - 24 * q^14 - q^15 - 5 * q^16 - 18 * q^17 + 10 * q^18 - 6 * q^19 - 30 * q^20 - 17 * q^21 - 8 * q^22 - 14 * q^23 - 13 * q^24 + 11 * q^25 + q^26 - 33 * q^27 - 2 * q^28 - 36 * q^29 + 2 * q^30 + q^31 - 9 * q^32 - 11 * q^33 - 25 * q^34 - 37 * q^35 + 3 * q^36 + 5 * q^37 - 13 * q^38 + 6 * q^39 - 2 * q^40 - 32 * q^41 + 7 * q^42 + 2 * q^43 + 4 * q^44 - 32 * q^45 - 5 * q^46 - 12 * q^47 - 18 * q^48 + 5 * q^49 - 2 * q^50 - 27 * q^51 - 15 * q^52 - 49 * q^53 - 31 * q^54 - 9 * q^55 - 53 * q^56 + 14 * q^57 - 2 * q^58 - 34 * q^59 + 25 * q^60 - 27 * q^61 - 12 * q^62 + 41 * q^63 - 70 * q^64 + 18 * q^65 - 15 * q^66 - 12 * q^67 - 20 * q^68 - 70 * q^69 + 70 * q^70 - 26 * q^71 - 14 * q^72 + 23 * q^73 + 20 * q^74 - 9 * q^75 - 36 * q^76 - 62 * q^77 + 8 * q^78 - 7 * q^79 - 41 * q^80 - 3 * q^81 - 33 * q^82 - q^83 - 48 * q^84 + 10 * q^85 - 23 * q^86 - 7 * q^87 + 9 * q^88 - 16 * q^89 - 30 * q^90 + 2 * q^91 - 27 * q^92 + 2 * q^93 - 37 * q^94 - 35 * q^95 + 55 * q^96 - 30 * q^97 - 43 * q^98 - 42 * 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 $

$2$	2.08679	1.47558	0.737792	−	0.675029i	$-0.235869\pi$
$2$	2.08679	1.47558	0.737792	+	0.675029i	$0.235869\pi$
$3$	−1.93108	−1.11491	−0.557456	−	0.830206i	$-0.688223\pi$
$3$	−1.93108	−1.11491	−0.557456	+	0.830206i	$0.688223\pi$
$4$	2.35469	1.17735
$5$	1.87136	0.836898	0.418449	−	0.908240i	$-0.362574\pi$
$5$	1.87136	0.836898	0.418449	+	0.908240i	$0.362574\pi$
$6$	−4.02977	−1.64515
$7$	−0.548741	−0.207405	−0.103702	−	0.994608i	$-0.533069\pi$
$7$	−0.548741	−0.207405	−0.103702	+	0.994608i	$0.533069\pi$
$8$	0.740165	0.261688
$9$	0.729089	0.243030
$10$	3.90513	1.23491
$11$	−6.22646	−1.87735	−0.938675	−	0.344804i	$-0.887945\pi$
$11$	−6.22646	−1.87735	−0.938675	+	0.344804i	$0.887945\pi$
$12$	−4.54711	−1.31264
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−1.14511	−0.306043
$15$	−3.61376	−0.933068
$16$	−3.16481	−0.791203
$17$	−6.48507	−1.57286	−0.786431	−	0.617678i	$-0.788073\pi$
$17$	−6.48507	−1.57286	−0.786431	+	0.617678i	$0.788073\pi$
$18$	1.52145	0.358610
$19$	3.43740	0.788593	0.394297	−	0.918983i	$-0.370988\pi$
$19$	3.43740	0.788593	0.394297	+	0.918983i	$0.370988\pi$
$20$	4.40647	0.985318
$21$	1.05967	0.231238
$22$	−12.9933	−2.77019
$23$	−0.591066	−0.123246	−0.0616229	−	0.998100i	$-0.519628\pi$
$23$	−0.591066	−0.123246	−0.0616229	+	0.998100i	$0.519628\pi$
$24$	−1.42932	−0.291759
$25$	−1.49801	−0.299602
$26$	−2.08679	−0.409253
$27$	4.38532	0.843956
$28$	−1.29212	−0.244187
$29$	−3.31121	−0.614877	−0.307438	−	0.951568i	$-0.599472\pi$
$29$	−3.31121	−0.614877	−0.307438	+	0.951568i	$0.599472\pi$
$30$	−7.54115	−1.37682
$31$	7.28855	1.30906	0.654531	−	0.756035i	$-0.272866\pi$
$31$	7.28855	1.30906	0.654531	+	0.756035i	$0.272866\pi$
$32$	−8.08463	−1.42917
$33$	12.0238	2.09308
$34$	−13.5330	−2.32089
$35$	−1.02689	−0.173577
$36$	1.71678	0.286130
$37$	−2.70751	−0.445113	−0.222556	−	0.974920i	$-0.571440\pi$
$37$	−2.70751	−0.445113	−0.222556	+	0.974920i	$0.571440\pi$
$38$	7.17313	1.16363
$39$	1.93108	0.309221
$40$	1.38512	0.219006
$41$	−2.81928	−0.440298	−0.220149	−	0.975466i	$-0.570654\pi$
$41$	−2.81928	−0.440298	−0.220149	+	0.975466i	$0.570654\pi$
$42$	2.21130	0.341211
$43$	8.64936	1.31901	0.659507	−	0.751698i	$-0.270765\pi$
$43$	8.64936	1.31901	0.659507	+	0.751698i	$0.270765\pi$
$44$	−14.6614	−2.21029
$45$	1.36439	0.203391
$46$	−1.23343	−0.181860
$47$	−2.97088	−0.433348	−0.216674	−	0.976244i	$-0.569521\pi$
$47$	−2.97088	−0.433348	−0.216674	+	0.976244i	$0.569521\pi$
$48$	6.11152	0.882122
$49$	−6.69888	−0.956983
$50$	−3.12603	−0.442088
$51$	12.5232	1.75360
$52$	−2.35469	−0.326537
$53$	−5.03559	−0.691692	−0.345846	−	0.938291i	$-0.612408\pi$
$53$	−5.03559	−0.691692	−0.345846	+	0.938291i	$0.612408\pi$
$54$	9.15125	1.24533
$55$	−11.6520	−1.57115
$56$	−0.406159	−0.0542753
$57$	−6.63791	−0.879212
$58$	−6.90981	−0.907302
$59$	−5.16664	−0.672639	−0.336320	−	0.941748i	$-0.609182\pi$
$59$	−5.16664	−0.672639	−0.336320	+	0.941748i	$0.609182\pi$
$60$	−8.50928	−1.09854
$61$	15.5246	1.98772	0.993861	−	0.110635i	$-0.0352885\pi$
$61$	15.5246	1.98772	0.993861	+	0.110635i	$0.0352885\pi$
$62$	15.2097	1.93163
$63$	−0.400081	−0.0504055
$64$	−10.5413	−1.31766
$65$	−1.87136	−0.232114
$66$	25.0912	3.08851
$67$	6.18003	0.755011	0.377505	−	0.926007i	$-0.376782\pi$
$67$	6.18003	0.755011	0.377505	+	0.926007i	$0.376782\pi$
$68$	−15.2703	−1.85180
$69$	1.14140	0.137408
$70$	−2.14291	−0.256127
$71$	−8.76878	−1.04066	−0.520331	−	0.853964i	$-0.674192\pi$
$71$	−8.76878	−1.04066	−0.520331	+	0.853964i	$0.674192\pi$
$72$	0.539646	0.0635979
$73$	−4.29279	−0.502433	−0.251217	−	0.967931i	$-0.580831\pi$
$73$	−4.29279	−0.502433	−0.251217	+	0.967931i	$0.580831\pi$
$74$	−5.65001	−0.656801
$75$	2.89279	0.334030
$76$	8.09401	0.928447
$77$	3.41672	0.389371
$78$	4.02977	0.456281
$79$	−1.40134	−0.157663	−0.0788314	−	0.996888i	$-0.525119\pi$
$79$	−1.40134	−0.157663	−0.0788314	+	0.996888i	$0.525119\pi$
$80$	−5.92250	−0.662156
$81$	−10.6557	−1.18397
$82$	−5.88325	−0.649696
$83$	4.60998	0.506011	0.253005	−	0.967465i	$-0.418581\pi$
$83$	4.60998	0.506011	0.253005	+	0.967465i	$0.418581\pi$
$84$	2.49519	0.272247
$85$	−12.1359	−1.31632
$86$	18.0494	1.94632
$87$	6.39423	0.685534
$88$	−4.60861	−0.491280
$89$	−7.36760	−0.780964	−0.390482	−	0.920611i	$-0.627692\pi$
$89$	−7.36760	−0.780964	−0.390482	+	0.920611i	$0.627692\pi$
$90$	2.84719	0.300120
$91$	0.548741	0.0575237
$92$	−1.39178	−0.145103
$93$	−14.0748	−1.45949
$94$	−6.19961	−0.639441
$95$	6.43261	0.659972
$96$	15.6121	1.59340
$97$	−4.20939	−0.427398	−0.213699	−	0.976900i	$-0.568551\pi$
$97$	−4.20939	−0.427398	−0.213699	+	0.976900i	$0.568551\pi$
$98$	−13.9792	−1.41211
$99$	−4.53964	−0.456251
$100$	−3.52735	−0.352735
$101$	−13.7863	−1.37179	−0.685896	−	0.727700i	$-0.740589\pi$
$101$	−13.7863	−1.37179	−0.685896	+	0.727700i	$0.740589\pi$
$102$	26.1333	2.58759
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	−0.740165	−0.0725792
$105$	1.98302	0.193523
$106$	−10.5082	−1.02065
$107$	−1.78149	−0.172224	−0.0861118	−	0.996285i	$-0.527444\pi$
$107$	−1.78149	−0.172224	−0.0861118	+	0.996285i	$0.527444\pi$
$108$	10.3261	0.993627
$109$	8.15411	0.781022	0.390511	−	0.920598i	$-0.372298\pi$
$109$	8.15411	0.781022	0.390511	+	0.920598i	$0.372298\pi$
$110$	−24.3152	−2.31836
$111$	5.22844	0.496262
$112$	1.73666	0.164099
$113$	0.282165	0.0265438	0.0132719	−	0.999912i	$-0.495775\pi$
$113$	0.282165	0.0265438	0.0132719	+	0.999912i	$0.495775\pi$
$114$	−13.8519	−1.29735
$115$	−1.10610	−0.103144
$116$	−7.79688	−0.723923
$117$	−0.729089	−0.0674043
$118$	−10.7817	−0.992535
$119$	3.55863	0.326219
$120$	−2.67478	−0.244172
$121$	27.7688	2.52444
$122$	32.3966	2.93305
$123$	5.44427	0.490894
$124$	17.1623	1.54122
$125$	−12.1601	−1.08763
$126$	−0.834885	−0.0743774
$127$	10.9646	0.972952	0.486476	−	0.873694i	$-0.338282\pi$
$127$	10.9646	0.972952	0.486476	+	0.873694i	$0.338282\pi$
$128$	−5.82820	−0.515145
$129$	−16.7026	−1.47059
$130$	−3.90513	−0.342503
$131$	−3.89077	−0.339938	−0.169969	−	0.985449i	$-0.554367\pi$
$131$	−3.89077	−0.339938	−0.169969	+	0.985449i	$0.554367\pi$
$132$	28.3124	2.46428
$133$	−1.88624	−0.163558
$134$	12.8964	1.11408
$135$	8.20652	0.706305
$136$	−4.80003	−0.411599
$137$	7.89377	0.674410	0.337205	−	0.941431i	$-0.390518\pi$
$137$	7.89377	0.674410	0.337205	+	0.941431i	$0.390518\pi$
$138$	2.38186	0.202757
$139$	−3.66825	−0.311137	−0.155568	−	0.987825i	$-0.549721\pi$
$139$	−3.66825	−0.311137	−0.155568	+	0.987825i	$0.549721\pi$
$140$	−2.41801	−0.204360
$141$	5.73703	0.483145
$142$	−18.2986	−1.53558
$143$	6.22646	0.520683
$144$	−2.30743	−0.192286
$145$	−6.19647	−0.514589
$146$	−8.95815	−0.741382
$147$	12.9361	1.06695
$148$	−6.37536	−0.524051
$149$	1.71335	0.140363	0.0701814	−	0.997534i	$-0.477642\pi$
$149$	1.71335	0.140363	0.0701814	+	0.997534i	$0.477642\pi$
$150$	6.03664	0.492889
$151$	13.8555	1.12754	0.563772	−	0.825930i	$-0.309350\pi$
$151$	13.8555	1.12754	0.563772	+	0.825930i	$0.309350\pi$
$152$	2.54424	0.206365
$153$	−4.72819	−0.382252
$154$	7.12997	0.574549
$155$	13.6395	1.09555
$156$	4.54711	0.364060
$157$	−6.61393	−0.527849	−0.263924	−	0.964543i	$-0.585017\pi$
$157$	−6.61393	−0.527849	−0.263924	+	0.964543i	$0.585017\pi$
$158$	−2.92430	−0.232644
$159$	9.72416	0.771176
$160$	−15.1293	−1.19607
$161$	0.324343	0.0255618
$162$	−22.2362	−1.74704
$163$	−1.41517	−0.110845	−0.0554224	−	0.998463i	$-0.517651\pi$
$163$	−1.41517	−0.110845	−0.0554224	+	0.998463i	$0.517651\pi$
$164$	−6.63854	−0.518383
$165$	22.5009	1.75169
$166$	9.62005	0.746661
$167$	11.5571	0.894315	0.447157	−	0.894455i	$-0.352436\pi$
$167$	11.5571	0.894315	0.447157	+	0.894455i	$0.352436\pi$
$168$	0.784328	0.0605122
$169$	1.00000	0.0769231
$170$	−25.3251	−1.94235
$171$	2.50617	0.191651
$172$	20.3666	1.55294
$173$	−5.07598	−0.385919	−0.192960	−	0.981207i	$-0.561809\pi$
$173$	−5.07598	−0.385919	−0.192960	+	0.981207i	$0.561809\pi$
$174$	13.3434	1.01156
$175$	0.822021	0.0621389
$176$	19.7056	1.48536
$177$	9.97722	0.749934
$178$	−15.3746	−1.15238
$179$	11.5212	0.861137	0.430569	−	0.902558i	$-0.358313\pi$
$179$	11.5212	0.861137	0.430569	+	0.902558i	$0.358313\pi$
$180$	3.21271	0.239461
$181$	−15.3955	−1.14434	−0.572168	−	0.820137i	$-0.693897\pi$
$181$	−15.3955	−1.14434	−0.572168	+	0.820137i	$0.693897\pi$
$182$	1.14511	0.0848810
$183$	−29.9793	−2.21614
$184$	−0.437487	−0.0322520
$185$	−5.06673	−0.372514
$186$	−29.3712	−2.15360
$187$	40.3791	2.95281
$188$	−6.99551	−0.510200
$189$	−2.40641	−0.175040
$190$	13.4235	0.973843
$191$	−0.744546	−0.0538734	−0.0269367	−	0.999637i	$-0.508575\pi$
$191$	−0.744546	−0.0538734	−0.0269367	+	0.999637i	$0.508575\pi$
$192$	20.3561	1.46908
$193$	−10.0191	−0.721192	−0.360596	−	0.932722i	$-0.617427\pi$
$193$	−10.0191	−0.721192	−0.360596	+	0.932722i	$0.617427\pi$
$194$	−8.78410	−0.630662
$195$	3.61376	0.258786
$196$	−15.7738	−1.12670
$197$	15.5945	1.11106	0.555532	−	0.831495i	$-0.312515\pi$
$197$	15.5945	1.11106	0.555532	+	0.831495i	$0.312515\pi$
$198$	−9.47328	−0.673237
$199$	16.5420	1.17263	0.586314	−	0.810084i	$-0.300578\pi$
$199$	16.5420	1.17263	0.586314	+	0.810084i	$0.300578\pi$
$200$	−1.10878	−0.0784023
$201$	−11.9342	−0.841771
$202$	−28.7692	−2.02419
$203$	1.81700	0.127528
$204$	29.4883	2.06460
$205$	−5.27589	−0.368484
$206$	−2.08679	−0.145394
$207$	−0.430940	−0.0299524
$208$	3.16481	0.219440
$209$	−21.4028	−1.48046
$210$	4.13814	0.285559
$211$	−25.7841	−1.77505	−0.887526	−	0.460758i	$-0.847578\pi$
$211$	−25.7841	−1.77505	−0.887526	+	0.460758i	$0.847578\pi$
$212$	−11.8573	−0.814361
$213$	16.9333	1.16025
$214$	−3.71760	−0.254130
$215$	16.1861	1.10388
$216$	3.24586	0.220853
$217$	−3.99953	−0.271506
$218$	17.0159	1.15246
$219$	8.28974	0.560169
$220$	−27.4368	−1.84979
$221$	6.48507	0.436233
$222$	10.9107	0.732275
$223$	13.7963	0.923866	0.461933	−	0.886915i	$-0.347156\pi$
$223$	13.7963	0.923866	0.461933	+	0.886915i	$0.347156\pi$
$224$	4.43637	0.296417
$225$	−1.09218	−0.0728122
$226$	0.588819	0.0391676
$227$	−8.63677	−0.573242	−0.286621	−	0.958044i	$-0.592532\pi$
$227$	−8.63677	−0.573242	−0.286621	+	0.958044i	$0.592532\pi$
$228$	−15.6302	−1.03514
$229$	−27.7710	−1.83516	−0.917581	−	0.397549i	$-0.869861\pi$
$229$	−27.7710	−1.83516	−0.917581	+	0.397549i	$0.869861\pi$
$230$	−2.30819	−0.152198
$231$	−6.59797	−0.434115
$232$	−2.45084	−0.160906
$233$	11.6840	0.765446	0.382723	−	0.923863i	$-0.374986\pi$
$233$	11.6840	0.765446	0.382723	+	0.923863i	$0.374986\pi$
$234$	−1.52145	−0.0994606
$235$	−5.55959	−0.362668
$236$	−12.1658	−0.791929
$237$	2.70610	0.175780
$238$	7.42611	0.481363
$239$	−16.3576	−1.05809	−0.529044	−	0.848595i	$-0.677449\pi$
$239$	−16.3576	−1.05809	−0.529044	+	0.848595i	$0.677449\pi$
$240$	11.4369	0.738246
$241$	−23.5396	−1.51632	−0.758159	−	0.652070i	$-0.773901\pi$
$241$	−23.5396	−1.51632	−0.758159	+	0.652070i	$0.773901\pi$
$242$	57.9477	3.72502
$243$	7.42108	0.476063
$244$	36.5556	2.34024
$245$	−12.5360	−0.800897
$246$	11.3611	0.724354
$247$	−3.43740	−0.218716
$248$	5.39473	0.342566
$249$	−8.90226	−0.564158
$250$	−25.3756	−1.60489
$251$	0.787271	0.0496921	0.0248460	−	0.999691i	$-0.492090\pi$
$251$	0.787271	0.0496921	0.0248460	+	0.999691i	$0.492090\pi$
$252$	−0.942067	−0.0593446
$253$	3.68025	0.231376
$254$	22.8808	1.43567
$255$	23.4355	1.46759
$256$	8.92035	0.557522
$257$	8.61920	0.537651	0.268825	−	0.963189i	$-0.413365\pi$
$257$	8.61920	0.537651	0.268825	+	0.963189i	$0.413365\pi$
$258$	−34.8549	−2.16997
$259$	1.48572	0.0923185
$260$	−4.40647	−0.273278
$261$	−2.41417	−0.149433
$262$	−8.11922	−0.501607
$263$	−12.7376	−0.785432	−0.392716	−	0.919660i	$-0.628464\pi$
$263$	−12.7376	−0.785432	−0.392716	+	0.919660i	$0.628464\pi$
$264$	8.89962	0.547734
$265$	−9.42341	−0.578876
$266$	−3.93619	−0.241343
$267$	14.2275	0.870706
$268$	14.5521	0.888908
$269$	29.3050	1.78676	0.893380	−	0.449302i	$-0.148327\pi$
$269$	29.3050	1.78676	0.893380	+	0.449302i	$0.148327\pi$
$270$	17.1253	1.04221
$271$	14.3277	0.870346	0.435173	−	0.900347i	$-0.356687\pi$
$271$	14.3277	0.870346	0.435173	+	0.900347i	$0.356687\pi$
$272$	20.5240	1.24445
$273$	−1.05967	−0.0641339
$274$	16.4726	0.995148
$275$	9.32731	0.562458
$276$	2.68764	0.161777
$277$	6.10566	0.366853	0.183427	−	0.983033i	$-0.441281\pi$
$277$	6.10566	0.366853	0.183427	+	0.983033i	$0.441281\pi$
$278$	−7.65486	−0.459108
$279$	5.31400	0.318141
$280$	−0.760070	−0.0454229
$281$	−4.13541	−0.246698	−0.123349	−	0.992363i	$-0.539363\pi$
$281$	−4.13541	−0.246698	−0.123349	+	0.992363i	$0.539363\pi$
$282$	11.9720	0.712920
$283$	−29.3433	−1.74428	−0.872138	−	0.489260i	$-0.837267\pi$
$283$	−29.3433	−1.74428	−0.872138	+	0.489260i	$0.837267\pi$
$284$	−20.6478	−1.22522
$285$	−12.4219	−0.735811
$286$	12.9933	0.768311
$287$	1.54706	0.0913199
$288$	−5.89441	−0.347331
$289$	25.0562	1.47389
$290$	−12.9307	−0.759319
$291$	8.12868	0.476512
$292$	−10.1082	−0.591537
$293$	−19.9887	−1.16775	−0.583877	−	0.811842i	$-0.698465\pi$
$293$	−19.9887	−1.16775	−0.583877	+	0.811842i	$0.698465\pi$
$294$	26.9949	1.57438
$295$	−9.66864	−0.562930
$296$	−2.00401	−0.116481
$297$	−27.3051	−1.58440
$298$	3.57540	0.207117
$299$	0.591066	0.0341823
$300$	6.81162	0.393269
$301$	−4.74626	−0.273570
$302$	28.9135	1.66379
$303$	26.6226	1.52943
$304$	−10.8787	−0.623937
$305$	29.0521	1.66352
$306$	−9.86675	−0.564044
$307$	−3.01342	−0.171985	−0.0859925	−	0.996296i	$-0.527406\pi$
$307$	−3.01342	−0.171985	−0.0859925	+	0.996296i	$0.527406\pi$
$308$	8.04531	0.458424
$309$	1.93108	0.109856
$310$	28.4628	1.61658
$311$	30.0498	1.70397	0.851985	−	0.523567i	$-0.175399\pi$
$311$	30.0498	1.70397	0.851985	+	0.523567i	$0.175399\pi$
$312$	1.42932	0.0809194
$313$	15.6927	0.887004	0.443502	−	0.896273i	$-0.353736\pi$
$313$	15.6927	0.887004	0.443502	+	0.896273i	$0.353736\pi$
$314$	−13.8019	−0.778885
$315$	−0.748695	−0.0421842
$316$	−3.29972	−0.185623
$317$	21.4699	1.20587	0.602935	−	0.797790i	$-0.293998\pi$
$317$	21.4699	1.20587	0.602935	+	0.797790i	$0.293998\pi$
$318$	20.2923	1.13793
$319$	20.6172	1.15434
$320$	−19.7266	−1.10275
$321$	3.44022	0.192014
$322$	0.676835	0.0377185
$323$	−22.2918	−1.24035
$324$	−25.0909	−1.39394
$325$	1.49801	0.0830947
$326$	−2.95316	−0.163561
$327$	−15.7463	−0.870771
$328$	−2.08673	−0.115221
$329$	1.63025	0.0898784
$330$	46.9547	2.58477
$331$	−6.42972	−0.353409	−0.176705	−	0.984264i	$-0.556544\pi$
$331$	−6.42972	−0.353409	−0.176705	+	0.984264i	$0.556544\pi$
$332$	10.8551	0.595749
$333$	−1.97402	−0.108176
$334$	24.1172	1.31964
$335$	11.5651	0.631867
$336$	−3.35364	−0.182956
$337$	0.333048	0.0181423	0.00907113	−	0.999959i	$-0.497113\pi$
$337$	0.333048	0.0181423	0.00907113	+	0.999959i	$0.497113\pi$
$338$	2.08679	0.113506
$339$	−0.544884	−0.0295940
$340$	−28.5763	−1.54977
$341$	−45.3819	−2.45757
$342$	5.22984	0.282798
$343$	7.51714	0.405888
$344$	6.40195	0.345170
$345$	2.13597	0.114997
$346$	−10.5925	−0.569456
$347$	−32.3486	−1.73656	−0.868282	−	0.496071i	$-0.834776\pi$
$347$	−32.3486	−1.73656	−0.868282	+	0.496071i	$0.834776\pi$
$348$	15.0564	0.807110
$349$	−29.0650	−1.55582	−0.777908	−	0.628378i	$-0.783719\pi$
$349$	−29.0650	−1.55582	−0.777908	+	0.628378i	$0.783719\pi$
$350$	1.71538	0.0916911
$351$	−4.38532	−0.234071
$352$	50.3386	2.68306
$353$	13.8528	0.737313	0.368656	−	0.929566i	$-0.379818\pi$
$353$	13.8528	0.737313	0.368656	+	0.929566i	$0.379818\pi$
$354$	20.8204	1.10659
$355$	−16.4095	−0.870928
$356$	−17.3484	−0.919464
$357$	−6.87201	−0.363705
$358$	24.0424	1.27068
$359$	−18.5239	−0.977655	−0.488827	−	0.872381i	$-0.662575\pi$
$359$	−18.5239	−0.977655	−0.488827	+	0.872381i	$0.662575\pi$
$360$	1.00987	0.0532249
$361$	−7.18429	−0.378121
$362$	−32.1271	−1.68856
$363$	−53.6240	−2.81453
$364$	1.29212	0.0677253
$365$	−8.03336	−0.420485
$366$	−62.5606	−3.27009
$367$	−17.9154	−0.935176	−0.467588	−	0.883946i	$-0.654877\pi$
$367$	−17.9154	−0.935176	−0.467588	+	0.883946i	$0.654877\pi$
$368$	1.87061	0.0975125
$369$	−2.05551	−0.107005
$370$	−10.5732	−0.549675
$371$	2.76324	0.143460
$372$	−33.1418	−1.71832
$373$	2.65542	0.137492	0.0687462	−	0.997634i	$-0.478100\pi$
$373$	2.65542	0.137492	0.0687462	+	0.997634i	$0.478100\pi$
$374$	84.2626	4.35712
$375$	23.4822	1.21262
$376$	−2.19894	−0.113402
$377$	3.31121	0.170536
$378$	−5.02167	−0.258287
$379$	−23.0183	−1.18237	−0.591185	−	0.806536i	$-0.701340\pi$
$379$	−23.0183	−1.18237	−0.591185	+	0.806536i	$0.701340\pi$
$380$	15.1468	0.777015
$381$	−21.1736	−1.08476
$382$	−1.55371	−0.0794947
$383$	−21.2260	−1.08460	−0.542299	−	0.840186i	$-0.682446\pi$
$383$	−21.2260	−1.08460	−0.542299	+	0.840186i	$0.682446\pi$
$384$	11.2548	0.574342
$385$	6.39391	0.325864
$386$	−20.9078	−1.06418
$387$	6.30615	0.320560
$388$	−9.91180	−0.503196
$389$	14.0076	0.710215	0.355107	−	0.934826i	$-0.384444\pi$
$389$	14.0076	0.710215	0.355107	+	0.934826i	$0.384444\pi$
$390$	7.54115	0.381861
$391$	3.83311	0.193849
$392$	−4.95828	−0.250431
$393$	7.51341	0.379001
$394$	32.5425	1.63947
$395$	−2.62241	−0.131948
$396$	−10.6895	−0.537165
$397$	−12.3622	−0.620443	−0.310221	−	0.950664i	$-0.600403\pi$
$397$	−12.3622	−0.620443	−0.310221	+	0.950664i	$0.600403\pi$
$398$	34.5196	1.73031
$399$	3.64249	0.182353
$400$	4.74093	0.237046
$401$	30.7852	1.53734	0.768669	−	0.639647i	$-0.220919\pi$
$401$	30.7852	1.53734	0.768669	+	0.639647i	$0.220919\pi$
$402$	−24.9041	−1.24210
$403$	−7.28855	−0.363069
$404$	−32.4625	−1.61507
$405$	−19.9406	−0.990859
$406$	3.79170	0.188179
$407$	16.8582	0.835632
$408$	9.26926	0.458897
$409$	−27.1408	−1.34203	−0.671014	−	0.741445i	$-0.734141\pi$
$409$	−27.1408	−1.34203	−0.671014	+	0.741445i	$0.734141\pi$
$410$	−11.0097	−0.543729
$411$	−15.2435	−0.751908
$412$	−2.35469	−0.116007
$413$	2.83515	0.139509
$414$	−0.899281	−0.0441972
$415$	8.62693	0.423479
$416$	8.08463	0.396382
$417$	7.08370	0.346890
$418$	−44.6632	−2.18455
$419$	10.4098	0.508552	0.254276	−	0.967132i	$-0.418163\pi$
$419$	10.4098	0.508552	0.254276	+	0.967132i	$0.418163\pi$
$420$	4.66939	0.227843
$421$	7.32329	0.356915	0.178458	−	0.983948i	$-0.442889\pi$
$421$	7.32329	0.356915	0.178458	+	0.983948i	$0.442889\pi$
$422$	−53.8060	−2.61924
$423$	−2.16604	−0.105316
$424$	−3.72717	−0.181007
$425$	9.71472	0.471233
$426$	35.3361	1.71204
$427$	−8.51899	−0.412263
$428$	−4.19487	−0.202767
$429$	−12.0238	−0.580516
$430$	33.7769	1.62887
$431$	9.00402	0.433708	0.216854	−	0.976204i	$-0.430420\pi$
$431$	9.00402	0.433708	0.216854	+	0.976204i	$0.430420\pi$
$432$	−13.8787	−0.667740
$433$	−7.29945	−0.350789	−0.175395	−	0.984498i	$-0.556120\pi$
$433$	−7.29945	−0.350789	−0.175395	+	0.984498i	$0.556120\pi$
$434$	−8.34617	−0.400629
$435$	11.9659	0.573722
$436$	19.2004	0.919533
$437$	−2.03173	−0.0971909
$438$	17.2989	0.826576
$439$	18.5228	0.884046	0.442023	−	0.897004i	$-0.354261\pi$
$439$	18.5228	0.884046	0.442023	+	0.897004i	$0.354261\pi$
$440$	−8.62437	−0.411151
$441$	−4.88408	−0.232575
$442$	13.5330	0.643699
$443$	−26.3332	−1.25113	−0.625564	−	0.780173i	$-0.715131\pi$
$443$	−26.3332	−1.25113	−0.625564	+	0.780173i	$0.715131\pi$
$444$	12.3114	0.584271
$445$	−13.7874	−0.653587
$446$	28.7899	1.36324
$447$	−3.30862	−0.156492
$448$	5.78444	0.273289
$449$	−15.5261	−0.732724	−0.366362	−	0.930472i	$-0.619397\pi$
$449$	−15.5261	−0.732724	−0.366362	+	0.930472i	$0.619397\pi$
$450$	−2.27916	−0.107440
$451$	17.5542	0.826593
$452$	0.664411	0.0312513
$453$	−26.7561	−1.25711
$454$	−18.0231	−0.845867
$455$	1.02689	0.0481415
$456$	−4.91315	−0.230079
$457$	−38.8792	−1.81869	−0.909346	−	0.416040i	$-0.863418\pi$
$457$	−38.8792	−1.81869	−0.909346	+	0.416040i	$0.863418\pi$
$458$	−57.9523	−2.70793
$459$	−28.4391	−1.32743
$460$	−2.60452	−0.121436
$461$	3.68197	0.171486	0.0857431	−	0.996317i	$-0.472674\pi$
$461$	3.68197	0.171486	0.0857431	+	0.996317i	$0.472674\pi$
$462$	−13.7686	−0.640572
$463$	−36.7518	−1.70800	−0.854001	−	0.520272i	$-0.825831\pi$
$463$	−36.7518	−1.70800	−0.854001	+	0.520272i	$0.825831\pi$
$464$	10.4794	0.486493
$465$	−26.3390	−1.22144
$466$	24.3821	1.12948
$467$	−21.0722	−0.975106	−0.487553	−	0.873093i	$-0.662110\pi$
$467$	−21.0722	−0.975106	−0.487553	+	0.873093i	$0.662110\pi$
$468$	−1.71678	−0.0793581
$469$	−3.39124	−0.156593
$470$	−11.6017	−0.535147
$471$	12.7721	0.588505
$472$	−3.82417	−0.176022
$473$	−53.8549	−2.47625
$474$	5.64706	0.259378
$475$	−5.14926	−0.236264
$476$	8.37947	0.384072
$477$	−3.67139	−0.168102
$478$	−34.1349	−1.56130
$479$	−23.7448	−1.08493	−0.542463	−	0.840080i	$-0.682508\pi$
$479$	−23.7448	−1.08493	−0.542463	+	0.840080i	$0.682508\pi$
$480$	29.2159	1.33352
$481$	2.70751	0.123452
$482$	−49.1221	−2.23745
$483$	−0.626333	−0.0284991
$484$	65.3871	2.97214
$485$	−7.87728	−0.357689
$486$	15.4862	0.702470
$487$	33.7088	1.52749	0.763745	−	0.645518i	$-0.223359\pi$
$487$	33.7088	1.52749	0.763745	+	0.645518i	$0.223359\pi$
$488$	11.4908	0.520163
$489$	2.73281	0.123582
$490$	−26.1600	−1.18179
$491$	−38.6471	−1.74412	−0.872060	−	0.489398i	$-0.837217\pi$
$491$	−38.6471	−1.74412	−0.872060	+	0.489398i	$0.837217\pi$
$492$	12.8196	0.577951
$493$	21.4735	0.967116
$494$	−7.17313	−0.322734
$495$	−8.49531	−0.381836
$496$	−23.0669	−1.03573
$497$	4.81179	0.215838
$498$	−18.5771	−0.832461
$499$	−16.5385	−0.740364	−0.370182	−	0.928959i	$-0.620705\pi$
$499$	−16.5385	−0.740364	−0.370182	+	0.928959i	$0.620705\pi$
$500$	−28.6333	−1.28052
$501$	−22.3177	−0.997082
$502$	1.64287	0.0733248
$503$	−28.6960	−1.27949	−0.639747	−	0.768586i	$-0.720961\pi$
$503$	−28.6960	−1.27949	−0.639747	+	0.768586i	$0.720961\pi$
$504$	−0.296126	−0.0131905
$505$	−25.7992	−1.14805
$506$	7.67992	0.341414
$507$	−1.93108	−0.0857625
$508$	25.8183	1.14550
$509$	−9.30825	−0.412581	−0.206291	−	0.978491i	$-0.566139\pi$
$509$	−9.30825	−0.412581	−0.206291	+	0.978491i	$0.566139\pi$
$510$	48.9049	2.16555
$511$	2.35563	0.104207
$512$	30.2713	1.33782
$513$	15.0741	0.665538
$514$	17.9865	0.793349
$515$	−1.87136	−0.0824620
$516$	−39.3296	−1.73139
$517$	18.4981	0.813545
$518$	3.10040	0.136224
$519$	9.80214	0.430266
$520$	−1.38512	−0.0607413
$521$	−11.2376	−0.492330	−0.246165	−	0.969228i	$-0.579171\pi$
$521$	−11.2376	−0.492330	−0.246165	+	0.969228i	$0.579171\pi$
$522$	−5.03786	−0.220501
$523$	39.1410	1.71152	0.855759	−	0.517374i	$-0.173090\pi$
$523$	39.1410	1.71152	0.855759	+	0.517374i	$0.173090\pi$
$524$	−9.16156	−0.400225
$525$	−1.58739	−0.0692794
$526$	−26.5806	−1.15897
$527$	−47.2668	−2.05897
$528$	−38.0532	−1.65605
$529$	−22.6506	−0.984810
$530$	−19.6647	−0.854179
$531$	−3.76694	−0.163471
$532$	−4.44152	−0.192564
$533$	2.81928	0.122117
$534$	29.6897	1.28480
$535$	−3.33382	−0.144134
$536$	4.57424	0.197577
$537$	−22.2485	−0.960092
$538$	61.1535	2.63651
$539$	41.7104	1.79659
$540$	19.3238	0.831564
$541$	40.2473	1.73036	0.865182	−	0.501457i	$-0.167203\pi$
$541$	40.2473	1.73036	0.865182	+	0.501457i	$0.167203\pi$
$542$	29.8989	1.28427
$543$	29.7299	1.27583
$544$	52.4294	2.24789
$545$	15.2593	0.653635
$546$	−2.21130	−0.0946349
$547$	−17.0421	−0.728666	−0.364333	−	0.931269i	$-0.618703\pi$
$547$	−17.0421	−0.728666	−0.364333	+	0.931269i	$0.618703\pi$
$548$	18.5874	0.794013
$549$	11.3188	0.483075
$550$	19.4641	0.829954
$551$	−11.3820	−0.484888
$552$	0.844824	0.0359581
$553$	0.768971	0.0327000
$554$	12.7412	0.541322
$555$	9.78429	0.415320
$556$	−8.63759	−0.366315
$557$	33.6626	1.42633	0.713166	−	0.700995i	$-0.247261\pi$
$557$	33.6626	1.42633	0.713166	+	0.700995i	$0.247261\pi$
$558$	11.0892	0.469443
$559$	−8.64936	−0.365829
$560$	3.24992	0.137334
$561$	−77.9754	−3.29213
$562$	−8.62972	−0.364023
$563$	11.7005	0.493118	0.246559	−	0.969128i	$-0.420700\pi$
$563$	11.7005	0.493118	0.246559	+	0.969128i	$0.420700\pi$
$564$	13.5089	0.568828
$565$	0.528032	0.0222145
$566$	−61.2332	−2.57382
$567$	5.84722	0.245560
$568$	−6.49035	−0.272329
$569$	−37.6764	−1.57948	−0.789738	−	0.613444i	$-0.789783\pi$
$569$	−37.6764	−1.57948	−0.789738	+	0.613444i	$0.789783\pi$
$570$	−25.9219	−1.08575
$571$	0.313565	0.0131223	0.00656115	−	0.999978i	$-0.497912\pi$
$571$	0.313565	0.0131223	0.00656115	+	0.999978i	$0.497912\pi$
$572$	14.6614	0.613024
$573$	1.43778	0.0600641
$574$	3.22838	0.134750
$575$	0.885424	0.0369247
$576$	−7.68554	−0.320231
$577$	−11.0851	−0.461481	−0.230740	−	0.973015i	$-0.574115\pi$
$577$	−11.0851	−0.461481	−0.230740	+	0.973015i	$0.574115\pi$
$578$	52.2870	2.17485
$579$	19.3478	0.804066
$580$	−14.5908	−0.605849
$581$	−2.52968	−0.104949
$582$	16.9628	0.703133
$583$	31.3539	1.29855
$584$	−3.17737	−0.131481
$585$	−1.36439	−0.0564105
$586$	−41.7123	−1.72312
$587$	10.0020	0.412825	0.206413	−	0.978465i	$-0.433821\pi$
$587$	10.0020	0.412825	0.206413	+	0.978465i	$0.433821\pi$
$588$	30.4605	1.25617
$589$	25.0537	1.03232
$590$	−20.1764	−0.830650
$591$	−30.1144	−1.23874
$592$	8.56878	0.352175
$593$	25.3243	1.03994	0.519971	−	0.854184i	$-0.325943\pi$
$593$	25.3243	1.03994	0.519971	+	0.854184i	$0.325943\pi$
$594$	−56.9799	−2.33791
$595$	6.65947	0.273012
$596$	4.03440	0.165256
$597$	−31.9439	−1.30738
$598$	1.23343	0.0504388
$599$	21.5901	0.882149	0.441075	−	0.897470i	$-0.354597\pi$
$599$	21.5901	0.882149	0.441075	+	0.897470i	$0.354597\pi$
$600$	2.14114	0.0874117
$601$	44.1006	1.79890	0.899450	−	0.437025i	$-0.143968\pi$
$601$	44.1006	1.79890	0.899450	+	0.437025i	$0.143968\pi$
$602$	−9.90444	−0.403675
$603$	4.50579	0.183490
$604$	32.6254	1.32751
$605$	51.9655	2.11270
$606$	55.5557	2.25680
$607$	29.1877	1.18469	0.592347	−	0.805683i	$-0.298201\pi$
$607$	29.1877	1.18469	0.592347	+	0.805683i	$0.298201\pi$
$608$	−27.7901	−1.12704
$609$	−3.50878	−0.142183
$610$	60.6257	2.45466
$611$	2.97088	0.120189
$612$	−11.1334	−0.450042
$613$	−25.3348	−1.02326	−0.511631	−	0.859206i	$-0.670958\pi$
$613$	−25.3348	−1.02326	−0.511631	+	0.859206i	$0.670958\pi$
$614$	−6.28838	−0.253778
$615$	10.1882	0.410828
$616$	2.52894	0.101894
$617$	37.1497	1.49559	0.747794	−	0.663930i	$-0.231113\pi$
$617$	37.1497	1.49559	0.747794	+	0.663930i	$0.231113\pi$
$618$	4.02977	0.162101
$619$	−17.1107	−0.687738	−0.343869	−	0.939018i	$-0.611738\pi$
$619$	−17.1107	−0.687738	−0.343869	+	0.939018i	$0.611738\pi$
$620$	32.1168	1.28984
$621$	−2.59202	−0.104014
$622$	62.7077	2.51435
$623$	4.04290	0.161975
$624$	−6.11152	−0.244657
$625$	−15.2659	−0.610636
$626$	32.7474	1.30885
$627$	41.3307	1.65059
$628$	−15.5737	−0.621460
$629$	17.5584	0.700101
$630$	−1.56237	−0.0622463
$631$	26.6201	1.05973	0.529864	−	0.848082i	$-0.322243\pi$
$631$	26.6201	1.05973	0.529864	+	0.848082i	$0.322243\pi$
$632$	−1.03722	−0.0412584
$633$	49.7913	1.97903
$634$	44.8032	1.77936
$635$	20.5187	0.814261
$636$	22.8974	0.907941
$637$	6.69888	0.265419
$638$	43.0237	1.70332
$639$	−6.39322	−0.252912
$640$	−10.9067	−0.431124
$641$	−9.14580	−0.361238	−0.180619	−	0.983553i	$-0.557810\pi$
$641$	−9.14580	−0.361238	−0.180619	+	0.983553i	$0.557810\pi$
$642$	7.17901	0.283333
$643$	−34.1355	−1.34617	−0.673087	−	0.739563i	$-0.735032\pi$
$643$	−34.1355	−1.34617	−0.673087	+	0.739563i	$0.735032\pi$
$644$	0.763726	0.0300950
$645$	−31.2567	−1.23073
$646$	−46.5183	−1.83024
$647$	−1.12626	−0.0442779	−0.0221389	−	0.999755i	$-0.507048\pi$
$647$	−1.12626	−0.0442779	−0.0221389	+	0.999755i	$0.507048\pi$
$648$	−7.88697	−0.309830
$649$	32.1699	1.26278
$650$	3.12603	0.122613
$651$	7.72343	0.302705
$652$	−3.33229	−0.130503
$653$	12.7717	0.499795	0.249897	−	0.968272i	$-0.419603\pi$
$653$	12.7717	0.499795	0.249897	+	0.968272i	$0.419603\pi$
$654$	−32.8592	−1.28490
$655$	−7.28103	−0.284493
$656$	8.92250	0.348365
$657$	−3.12982	−0.122106
$658$	3.40198	0.132623
$659$	−26.4272	−1.02946	−0.514728	−	0.857353i	$-0.672107\pi$
$659$	−26.4272	−1.02946	−0.514728	+	0.857353i	$0.672107\pi$
$660$	52.9827	2.06235
$661$	8.12494	0.316024	0.158012	−	0.987437i	$-0.449492\pi$
$661$	8.12494	0.316024	0.158012	+	0.987437i	$0.449492\pi$
$662$	−13.4175	−0.521485
$663$	−12.5232	−0.486362
$664$	3.41214	0.132417
$665$	−3.52984	−0.136881
$666$	−4.11936	−0.159622
$667$	1.95715	0.0757810
$668$	27.2134	1.05292
$669$	−26.6417	−1.03003
$670$	24.1338	0.932372
$671$	−96.6634	−3.73165
$672$	−8.56700	−0.330479
$673$	6.29301	0.242578	0.121289	−	0.992617i	$-0.461297\pi$
$673$	6.29301	0.242578	0.121289	+	0.992617i	$0.461297\pi$
$674$	0.695000	0.0267704
$675$	−6.56926	−0.252851
$676$	2.35469	0.0905650
$677$	28.3608	1.08999	0.544996	−	0.838439i	$-0.316531\pi$
$677$	28.3608	1.08999	0.544996	+	0.838439i	$0.316531\pi$
$678$	−1.13706	−0.0436685
$679$	2.30986	0.0886444
$680$	−8.98258	−0.344466
$681$	16.6783	0.639115
$682$	−94.7025	−3.62635
$683$	29.1051	1.11367	0.556837	−	0.830622i	$-0.312015\pi$
$683$	29.1051	1.11367	0.556837	+	0.830622i	$0.312015\pi$
$684$	5.90125	0.225640
$685$	14.7721	0.564412
$686$	15.6867	0.598921
$687$	53.6282	2.04604
$688$	−27.3736	−1.04361
$689$	5.03559	0.191841
$690$	4.45732	0.169687
$691$	−36.4207	−1.38551	−0.692754	−	0.721174i	$-0.743603\pi$
$691$	−36.4207	−1.38551	−0.692754	+	0.721174i	$0.743603\pi$
$692$	−11.9524	−0.454360
$693$	2.49109	0.0946287
$694$	−67.5047	−2.56244
$695$	−6.86461	−0.260390
$696$	4.73279	0.179396
$697$	18.2833	0.692528
$698$	−60.6526	−2.29574
$699$	−22.5628	−0.853405
$700$	1.93560	0.0731590
$701$	−30.2284	−1.14171	−0.570856	−	0.821050i	$-0.693389\pi$
$701$	−30.2284	−1.14171	−0.570856	+	0.821050i	$0.693389\pi$
$702$	−9.15125	−0.345392
$703$	−9.30681	−0.351013
$704$	65.6350	2.47371
$705$	10.7360	0.404343
$706$	28.9080	1.08797
$707$	7.56513	0.284516
$708$	23.4933	0.882931
$709$	40.9691	1.53863	0.769314	−	0.638871i	$-0.220598\pi$
$709$	40.9691	1.53863	0.769314	+	0.638871i	$0.220598\pi$
$710$	−34.2433	−1.28513
$711$	−1.02170	−0.0383167
$712$	−5.45324	−0.204369
$713$	−4.30802	−0.161337
$714$	−14.3404	−0.536678
$715$	11.6520	0.435758
$716$	27.1289	1.01386
$717$	31.5880	1.17967
$718$	−38.6555	−1.44261
$719$	38.1444	1.42255	0.711274	−	0.702915i	$-0.248119\pi$
$719$	38.1444	1.42255	0.711274	+	0.702915i	$0.248119\pi$
$720$	−4.31803	−0.160923
$721$	0.548741	0.0204362
$722$	−14.9921	−0.557949
$723$	45.4569	1.69056
$724$	−36.2515	−1.34728
$725$	4.96024	0.184219
$726$	−111.902	−4.15307
$727$	−22.0056	−0.816143	−0.408071	−	0.912950i	$-0.633799\pi$
$727$	−22.0056	−0.816143	−0.408071	+	0.912950i	$0.633799\pi$
$728$	0.406159	0.0150533
$729$	17.6363	0.653198
$730$	−16.7639	−0.620461
$731$	−56.0917	−2.07463
$732$	−70.5921	−2.60916
$733$	−11.7713	−0.434784	−0.217392	−	0.976084i	$-0.569755\pi$
$733$	−11.7713	−0.434784	−0.217392	+	0.976084i	$0.569755\pi$
$734$	−37.3857	−1.37993
$735$	24.2081	0.892930
$736$	4.77855	0.176140
$737$	−38.4797	−1.41742
$738$	−4.28941	−0.157895
$739$	12.3992	0.456114	0.228057	−	0.973648i	$-0.426763\pi$
$739$	12.3992	0.456114	0.228057	+	0.973648i	$0.426763\pi$
$740$	−11.9306	−0.438577
$741$	6.63791	0.243850
$742$	5.76630	0.211687
$743$	7.83651	0.287494	0.143747	−	0.989614i	$-0.454085\pi$
$743$	7.83651	0.287494	0.143747	+	0.989614i	$0.454085\pi$
$744$	−10.4177	−0.381931
$745$	3.20629	0.117469
$746$	5.54130	0.202882
$747$	3.36108	0.122976
$748$	95.0803	3.47648
$749$	0.977579	0.0357200
$750$	49.0025	1.78932
$751$	−27.2601	−0.994735	−0.497368	−	0.867540i	$-0.665700\pi$
$751$	−27.2601	−0.994735	−0.497368	+	0.867540i	$0.665700\pi$
$752$	9.40229	0.342866
$753$	−1.52029	−0.0554023
$754$	6.90981	0.251640
$755$	25.9286	0.943639
$756$	−5.66635	−0.206083
$757$	−7.65895	−0.278369	−0.139185	−	0.990266i	$-0.544448\pi$
$757$	−7.65895	−0.278369	−0.139185	+	0.990266i	$0.544448\pi$
$758$	−48.0343	−1.74469
$759$	−7.10688	−0.257963
$760$	4.76119	0.172707
$761$	−45.1062	−1.63510	−0.817549	−	0.575859i	$-0.804668\pi$
$761$	−45.1062	−1.63510	−0.817549	+	0.575859i	$0.804668\pi$
$762$	−44.1848	−1.60065
$763$	−4.47450	−0.161988
$764$	−1.75317	−0.0634276
$765$	−8.84815	−0.319906
$766$	−44.2942	−1.60041
$767$	5.16664	0.186557
$768$	−17.2260	−0.621588
$769$	−19.3948	−0.699395	−0.349697	−	0.936863i	$-0.613716\pi$
$769$	−19.3948	−0.699395	−0.349697	+	0.936863i	$0.613716\pi$
$770$	13.3427	0.480839
$771$	−16.6444	−0.599434
$772$	−23.5919	−0.849092
$773$	35.6878	1.28360	0.641800	−	0.766872i	$-0.278188\pi$
$773$	35.6878	1.28360	0.641800	+	0.766872i	$0.278188\pi$
$774$	13.1596	0.473012
$775$	−10.9183	−0.392198
$776$	−3.11564	−0.111845
$777$	−2.86906	−0.102927
$778$	29.2310	1.04798
$779$	−9.69099	−0.347216
$780$	8.50928	0.304681
$781$	54.5985	1.95369
$782$	7.99889	0.286040
$783$	−14.5207	−0.518929
$784$	21.2007	0.757168
$785$	−12.3770	−0.441755
$786$	15.6789	0.559248
$787$	43.0761	1.53550	0.767748	−	0.640752i	$-0.221377\pi$
$787$	43.0761	1.53550	0.767748	+	0.640752i	$0.221377\pi$
$788$	36.7203	1.30811
$789$	24.5973	0.875687
$790$	−5.47241	−0.194700
$791$	−0.154835	−0.00550532
$792$	−3.36009	−0.119395
$793$	−15.5246	−0.551295
$794$	−25.7974	−0.915515
$795$	18.1974	0.645395
$796$	38.9512	1.38059
$797$	10.4029	0.368491	0.184246	−	0.982880i	$-0.441016\pi$
$797$	10.4029	0.368491	0.184246	+	0.982880i	$0.441016\pi$
$798$	7.60112	0.269077
$799$	19.2664	0.681596
$800$	12.1109	0.428184
$801$	−5.37163	−0.189797
$802$	64.2422	2.26847
$803$	26.7289	0.943242
$804$	−28.1013	−0.991055
$805$	0.606962	0.0213926
$806$	−15.2097	−0.535738
$807$	−56.5905	−1.99208
$808$	−10.2042	−0.358981
$809$	15.7156	0.552531	0.276266	−	0.961081i	$-0.410903\pi$
$809$	15.7156	0.552531	0.276266	+	0.961081i	$0.410903\pi$
$810$	−41.6119	−1.46209
$811$	−35.6956	−1.25344	−0.626721	−	0.779244i	$-0.715603\pi$
$811$	−35.6956	−1.25344	−0.626721	+	0.779244i	$0.715603\pi$
$812$	4.27847	0.150145
$813$	−27.6680	−0.970359
$814$	35.1796	1.23304
$815$	−2.64829	−0.0927657
$816$	−39.6337	−1.38746
$817$	29.7313	1.04017
$818$	−56.6372	−1.98027
$819$	0.400081	0.0139800
$820$	−12.4231	−0.433833
$821$	−17.9385	−0.626057	−0.313029	−	0.949744i	$-0.601344\pi$
$821$	−17.9385	−0.626057	−0.313029	+	0.949744i	$0.601344\pi$
$822$	−31.8100	−1.10950
$823$	4.44903	0.155083	0.0775417	−	0.996989i	$-0.475293\pi$
$823$	4.44903	0.155083	0.0775417	+	0.996989i	$0.475293\pi$
$824$	−0.740165	−0.0257849
$825$	−18.0118	−0.627092
$826$	5.91636	0.205856
$827$	27.8553	0.968624	0.484312	−	0.874895i	$-0.339070\pi$
$827$	27.8553	0.968624	0.484312	+	0.874895i	$0.339070\pi$
$828$	−1.01473	−0.0352643
$829$	−43.9438	−1.52623	−0.763115	−	0.646263i	$-0.776331\pi$
$829$	−43.9438	−1.52623	−0.763115	+	0.646263i	$0.776331\pi$
$830$	18.0026	0.624879
$831$	−11.7905	−0.409009
$832$	10.5413	0.365454
$833$	43.4428	1.50520
$834$	14.7822	0.511865
$835$	21.6275	0.748450
$836$	−50.3971	−1.74302
$837$	31.9626	1.10479
$838$	21.7231	0.750411
$839$	30.2801	1.04539	0.522693	−	0.852521i	$-0.324927\pi$
$839$	30.2801	1.04539	0.522693	+	0.852521i	$0.324927\pi$
$840$	1.46776	0.0506425
$841$	−18.0359	−0.621926
$842$	15.2822	0.526658
$843$	7.98582	0.275046
$844$	−60.7136	−2.08985
$845$	1.87136	0.0643767
$846$	−4.52006	−0.155403
$847$	−15.2379	−0.523581
$848$	15.9367	0.547269
$849$	56.6643	1.94471
$850$	20.2726	0.695343
$851$	1.60032	0.0548583
$852$	39.8726	1.36601
$853$	−13.1196	−0.449206	−0.224603	−	0.974450i	$-0.572109\pi$
$853$	−13.1196	−0.449206	−0.224603	+	0.974450i	$0.572109\pi$
$854$	−17.7773	−0.608328
$855$	4.68994	0.160393
$856$	−1.31860	−0.0450688
$857$	−51.6448	−1.76415	−0.882077	−	0.471105i	$-0.843855\pi$
$857$	−51.6448	−1.76415	−0.882077	+	0.471105i	$0.843855\pi$
$858$	−25.0912	−0.856600
$859$	47.3475	1.61547	0.807737	−	0.589543i	$-0.200692\pi$
$859$	47.3475	1.61547	0.807737	+	0.589543i	$0.200692\pi$
$860$	38.1132	1.29965
$861$	−2.98750	−0.101814
$862$	18.7895	0.639973
$863$	37.7566	1.28525	0.642624	−	0.766182i	$-0.277846\pi$
$863$	37.7566	1.28525	0.642624	+	0.766182i	$0.277846\pi$
$864$	−35.4537	−1.20616
$865$	−9.49898	−0.322975
$866$	−15.2324	−0.517618
$867$	−48.3856	−1.64326
$868$	−9.41765	−0.319656
$869$	8.72537	0.295988
$870$	24.9703	0.846574
$871$	−6.18003	−0.209402
$872$	6.03539	0.204384
$873$	−3.06902	−0.103870
$874$	−4.23979	−0.143413
$875$	6.67276	0.225580
$876$	19.5198	0.659512
$877$	−48.2952	−1.63081	−0.815407	−	0.578888i	$-0.803487\pi$
$877$	−48.2952	−1.63081	−0.815407	+	0.578888i	$0.803487\pi$
$878$	38.6532	1.30448
$879$	38.5999	1.30194
$880$	36.8763	1.24310
$881$	−30.7455	−1.03584	−0.517921	−	0.855428i	$-0.673294\pi$
$881$	−30.7455	−1.03584	−0.517921	+	0.855428i	$0.673294\pi$
$882$	−10.1920	−0.343184
$883$	−55.7702	−1.87682	−0.938408	−	0.345530i	$-0.887699\pi$
$883$	−55.7702	−1.87682	−0.938408	+	0.345530i	$0.887699\pi$
$884$	15.2703	0.513597
$885$	18.6710	0.627618
$886$	−54.9519	−1.84614
$887$	12.9346	0.434303	0.217151	−	0.976138i	$-0.430323\pi$
$887$	12.9346	0.434303	0.217151	+	0.976138i	$0.430323\pi$
$888$	3.86991	0.129866
$889$	−6.01674	−0.201795
$890$	−28.7715	−0.964421
$891$	66.3473	2.22272
$892$	32.4859	1.08771
$893$	−10.2121	−0.341735
$894$	−6.90439	−0.230917
$895$	21.5604	0.720684
$896$	3.19818	0.106844
$897$	−1.14140	−0.0381102
$898$	−32.3998	−1.08119
$899$	−24.1339	−0.804912
$900$	−2.57175	−0.0857251
$901$	32.6562	1.08794
$902$	36.6318	1.21971
$903$	9.16543	0.305006
$904$	0.208849	0.00694620
$905$	−28.8104	−0.957691
$906$	−55.8344	−1.85498
$907$	49.0583	1.62895	0.814476	−	0.580197i	$-0.197024\pi$
$907$	49.0583	1.62895	0.814476	+	0.580197i	$0.197024\pi$
$908$	−20.3369	−0.674904
$909$	−10.0515	−0.333386
$910$	2.14291	0.0710367
$911$	19.3941	0.642556	0.321278	−	0.946985i	$-0.395888\pi$
$911$	19.3941	0.642556	0.321278	+	0.946985i	$0.395888\pi$
$912$	21.0077	0.695636
$913$	−28.7039	−0.949959
$914$	−81.1327	−2.68363
$915$	−56.1021	−1.85468
$916$	−65.3922	−2.16062
$917$	2.13503	0.0705048
$918$	−59.3465	−1.95873
$919$	15.6744	0.517051	0.258525	−	0.966004i	$-0.416763\pi$
$919$	15.6744	0.517051	0.258525	+	0.966004i	$0.416763\pi$
$920$	−0.818695	−0.0269916
$921$	5.81917	0.191748
$922$	7.68349	0.253042
$923$	8.76878	0.288628
$924$	−15.5362	−0.511103
$925$	4.05589	0.133357
$926$	−76.6933	−2.52030
$927$	−0.729089	−0.0239464
$928$	26.7699	0.878766
$929$	−51.2733	−1.68222	−0.841112	−	0.540861i	$-0.818098\pi$
$929$	−51.2733	−1.68222	−0.841112	+	0.540861i	$0.818098\pi$
$930$	−54.9640	−1.80234
$931$	−23.0267	−0.754671
$932$	27.5123	0.901194
$933$	−58.0288	−1.89978
$934$	−43.9733	−1.43885
$935$	75.5638	2.47120
$936$	−0.539646	−0.0176389
$937$	27.4524	0.896830	0.448415	−	0.893826i	$-0.351989\pi$
$937$	27.4524	0.896830	0.448415	+	0.893826i	$0.351989\pi$
$938$	−7.07680	−0.231066
$939$	−30.3039	−0.988932
$940$	−13.0911	−0.426985
$941$	−49.6328	−1.61798	−0.808992	−	0.587820i	$-0.799986\pi$
$941$	−49.6328	−1.61798	−0.808992	+	0.587820i	$0.799986\pi$
$942$	26.6526	0.868388
$943$	1.66638	0.0542649
$944$	16.3514	0.532194
$945$	−4.50325	−0.146491
$946$	−112.384	−3.65391
$947$	−5.15293	−0.167448	−0.0837238	−	0.996489i	$-0.526681\pi$
$947$	−5.15293	−0.167448	−0.0837238	+	0.996489i	$0.526681\pi$
$948$	6.37203	0.206954
$949$	4.29279	0.139350
$950$	−10.7454	−0.348628
$951$	−41.4602	−1.34444
$952$	2.63397	0.0853675
$953$	30.4562	0.986572	0.493286	−	0.869867i	$-0.335796\pi$
$953$	30.4562	0.986572	0.493286	+	0.869867i	$0.335796\pi$
$954$	−7.66143	−0.248048
$955$	−1.39331	−0.0450865
$956$	−38.5172	−1.24573
$957$	−39.8135	−1.28699
$958$	−49.5504	−1.60090
$959$	−4.33163	−0.139876
$960$	38.0937	1.22947
$961$	22.1230	0.713645
$962$	5.65001	0.182164
$963$	−1.29887	−0.0418554
$964$	−55.4284	−1.78523
$965$	−18.7494	−0.603564
$966$	−1.30703	−0.0420528
$967$	36.9055	1.18680	0.593400	−	0.804908i	$-0.297785\pi$
$967$	36.9055	1.18680	0.593400	+	0.804908i	$0.297785\pi$
$968$	20.5535	0.660616
$969$	43.0473	1.38288
$970$	−16.4382	−0.527799
$971$	28.3606	0.910136	0.455068	−	0.890457i	$-0.349615\pi$
$971$	28.3606	0.910136	0.455068	+	0.890457i	$0.349615\pi$
$972$	17.4744	0.560490
$973$	2.01292	0.0645312
$974$	70.3431	2.25394
$975$	−2.89279	−0.0926433
$976$	−49.1325	−1.57269
$977$	8.19883	0.262304	0.131152	−	0.991362i	$-0.458132\pi$
$977$	8.19883	0.262304	0.131152	+	0.991362i	$0.458132\pi$
$978$	5.70281	0.182356
$979$	45.8741	1.46614
$980$	−29.5185	−0.942933
$981$	5.94507	0.189811
$982$	−80.6484	−2.57360
$983$	−5.48353	−0.174897	−0.0874487	−	0.996169i	$-0.527871\pi$
$983$	−5.48353	−0.174897	−0.0874487	+	0.996169i	$0.527871\pi$
$984$	4.02966	0.128461
$985$	29.1830	0.929847
$986$	44.8106	1.42706
$987$	−3.14814	−0.100207
$988$	−8.09401	−0.257505
$989$	−5.11235	−0.162563
$990$	−17.7279	−0.563430
$991$	49.4131	1.56966	0.784830	−	0.619711i	$-0.212750\pi$
$991$	49.4131	1.56966	0.784830	+	0.619711i	$0.212750\pi$
$992$	−58.9252	−1.87088
$993$	12.4163	0.394020
$994$	10.0412	0.318487
$995$	30.9560	0.981370
$996$	−20.9621	−0.664208
$997$	−2.81664	−0.0892039	−0.0446020	−	0.999005i	$-0.514202\pi$
$997$	−2.81664	−0.0892039	−0.0446020	+	0.999005i	$0.514202\pi$
$998$	−34.5123	−1.09247
$999$	−11.8733	−0.375655

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1339.2.a.e.1.18	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1339.2.a.e.1.18	✓	21	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:	\( N \)	\(=\)	\( 1339 = 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1339.a (trivial)

\(f(q)\)	\(=\)	\(q+2.08679 q^{2} -1.93108 q^{3} +2.35469 q^{4} +1.87136 q^{5} -4.02977 q^{6} -0.548741 q^{7} +0.740165 q^{8} +0.729089 q^{9} +O(q^{10})\)	\(q+2.08679 q^{2} -1.93108 q^{3} +2.35469 q^{4} +1.87136 q^{5} -4.02977 q^{6} -0.548741 q^{7} +0.740165 q^{8} +0.729089 q^{9} +3.90513 q^{10} -6.22646 q^{11} -4.54711 q^{12} -1.00000 q^{13} -1.14511 q^{14} -3.61376 q^{15} -3.16481 q^{16} -6.48507 q^{17} +1.52145 q^{18} +3.43740 q^{19} +4.40647 q^{20} +1.05967 q^{21} -12.9933 q^{22} -0.591066 q^{23} -1.42932 q^{24} -1.49801 q^{25} -2.08679 q^{26} +4.38532 q^{27} -1.29212 q^{28} -3.31121 q^{29} -7.54115 q^{30} +7.28855 q^{31} -8.08463 q^{32} +12.0238 q^{33} -13.5330 q^{34} -1.02689 q^{35} +1.71678 q^{36} -2.70751 q^{37} +7.17313 q^{38} +1.93108 q^{39} +1.38512 q^{40} -2.81928 q^{41} +2.21130 q^{42} +8.64936 q^{43} -14.6614 q^{44} +1.36439 q^{45} -1.23343 q^{46} -2.97088 q^{47} +6.11152 q^{48} -6.69888 q^{49} -3.12603 q^{50} +12.5232 q^{51} -2.35469 q^{52} -5.03559 q^{53} +9.15125 q^{54} -11.6520 q^{55} -0.406159 q^{56} -6.63791 q^{57} -6.90981 q^{58} -5.16664 q^{59} -8.50928 q^{60} +15.5246 q^{61} +15.2097 q^{62} -0.400081 q^{63} -10.5413 q^{64} -1.87136 q^{65} +25.0912 q^{66} +6.18003 q^{67} -15.2703 q^{68} +1.14140 q^{69} -2.14291 q^{70} -8.76878 q^{71} +0.539646 q^{72} -4.29279 q^{73} -5.65001 q^{74} +2.89279 q^{75} +8.09401 q^{76} +3.41672 q^{77} +4.02977 q^{78} -1.40134 q^{79} -5.92250 q^{80} -10.6557 q^{81} -5.88325 q^{82} +4.60998 q^{83} +2.49519 q^{84} -12.1359 q^{85} +18.0494 q^{86} +6.39423 q^{87} -4.60861 q^{88} -7.36760 q^{89} +2.84719 q^{90} +0.548741 q^{91} -1.39178 q^{92} -14.0748 q^{93} -6.19961 q^{94} +6.43261 q^{95} +15.6121 q^{96} -4.20939 q^{97} -13.9792 q^{98} -4.53964 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9}+O(q^{10}) \) 21 * q - q^2 - 6 * q^3 + 15 * q^4 - 18 * q^5 - 8 * q^6 - 2 * q^7 + 6 * q^8 + 11 * q^9	\( 21 q - q^{2} - 6 q^{3} + 15 q^{4} - 18 q^{5} - 8 q^{6} - 2 q^{7} + 6 q^{8} + 11 q^{9} - 10 q^{10} - 5 q^{11} - 22 q^{12} - 21 q^{13} - 24 q^{14} - q^{15} - 5 q^{16} - 18 q^{17} + 10 q^{18} - 6 q^{19} - 30 q^{20} - 17 q^{21} - 8 q^{22} - 14 q^{23} - 13 q^{24} + 11 q^{25} + q^{26} - 33 q^{27} - 2 q^{28} - 36 q^{29} + 2 q^{30} + q^{31} - 9 q^{32} - 11 q^{33} - 25 q^{34} - 37 q^{35} + 3 q^{36} + 5 q^{37} - 13 q^{38} + 6 q^{39} - 2 q^{40} - 32 q^{41} + 7 q^{42} + 2 q^{43} + 4 q^{44} - 32 q^{45} - 5 q^{46} - 12 q^{47} - 18 q^{48} + 5 q^{49} - 2 q^{50} - 27 q^{51} - 15 q^{52} - 49 q^{53} - 31 q^{54} - 9 q^{55} - 53 q^{56} + 14 q^{57} - 2 q^{58} - 34 q^{59} + 25 q^{60} - 27 q^{61} - 12 q^{62} + 41 q^{63} - 70 q^{64} + 18 q^{65} - 15 q^{66} - 12 q^{67} - 20 q^{68} - 70 q^{69} + 70 q^{70} - 26 q^{71} - 14 q^{72} + 23 q^{73} + 20 q^{74} - 9 q^{75} - 36 q^{76} - 62 q^{77} + 8 q^{78} - 7 q^{79} - 41 q^{80} - 3 q^{81} - 33 q^{82} - q^{83} - 48 q^{84} + 10 q^{85} - 23 q^{86} - 7 q^{87} + 9 q^{88} - 16 q^{89} - 30 q^{90} + 2 q^{91} - 27 q^{92} + 2 q^{93} - 37 q^{94} - 35 q^{95} + 55 q^{96} - 30 q^{97} - 43 q^{98} - 42 q^{99}+O(q^{100}) \) 21 * q - q^2 - 6 * q^3 + 15 * q^4 - 18 * q^5 - 8 * q^6 - 2 * q^7 + 6 * q^8 + 11 * q^9 - 10 * q^10 - 5 * q^11 - 22 * q^12 - 21 * q^13 - 24 * q^14 - q^15 - 5 * q^16 - 18 * q^17 + 10 * q^18 - 6 * q^19 - 30 * q^20 - 17 * q^21 - 8 * q^22 - 14 * q^23 - 13 * q^24 + 11 * q^25 + q^26 - 33 * q^27 - 2 * q^28 - 36 * q^29 + 2 * q^30 + q^31 - 9 * q^32 - 11 * q^33 - 25 * q^34 - 37 * q^35 + 3 * q^36 + 5 * q^37 - 13 * q^38 + 6 * q^39 - 2 * q^40 - 32 * q^41 + 7 * q^42 + 2 * q^43 + 4 * q^44 - 32 * q^45 - 5 * q^46 - 12 * q^47 - 18 * q^48 + 5 * q^49 - 2 * q^50 - 27 * q^51 - 15 * q^52 - 49 * q^53 - 31 * q^54 - 9 * q^55 - 53 * q^56 + 14 * q^57 - 2 * q^58 - 34 * q^59 + 25 * q^60 - 27 * q^61 - 12 * q^62 + 41 * q^63 - 70 * q^64 + 18 * q^65 - 15 * q^66 - 12 * q^67 - 20 * q^68 - 70 * q^69 + 70 * q^70 - 26 * q^71 - 14 * q^72 + 23 * q^73 + 20 * q^74 - 9 * q^75 - 36 * q^76 - 62 * q^77 + 8 * q^78 - 7 * q^79 - 41 * q^80 - 3 * q^81 - 33 * q^82 - q^83 - 48 * q^84 + 10 * q^85 - 23 * q^86 - 7 * q^87 + 9 * q^88 - 16 * q^89 - 30 * q^90 + 2 * q^91 - 27 * q^92 + 2 * q^93 - 37 * q^94 - 35 * q^95 + 55 * q^96 - 30 * q^97 - 43 * q^98 - 42 * q^99

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