LMFDB - Embedded newform 8039.2.a.b.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8039,2,Mod(1,8039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8039, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8039 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8039.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:	$64.1917381849$
Analytic rank:	$0$
Dimension:	$391$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.62255 q^{2} +0.822293 q^{3} +4.87774 q^{4} -3.51986 q^{5} -2.15650 q^{6} -0.523743 q^{7} -7.54702 q^{8} -2.32383 q^{9} +O(q^{10})$	\(q-2.62255 q^{2} +0.822293 q^{3} +4.87774 q^{4} -3.51986 q^{5} -2.15650 q^{6} -0.523743 q^{7} -7.54702 q^{8} -2.32383 q^{9} +9.23098 q^{10} -0.382984 q^{11} +4.01093 q^{12} -2.17167 q^{13} +1.37354 q^{14} -2.89435 q^{15} +10.0369 q^{16} -5.99530 q^{17} +6.09436 q^{18} +1.37774 q^{19} -17.1690 q^{20} -0.430670 q^{21} +1.00439 q^{22} -2.27351 q^{23} -6.20586 q^{24} +7.38938 q^{25} +5.69531 q^{26} -4.37775 q^{27} -2.55468 q^{28} +5.25315 q^{29} +7.59057 q^{30} -1.57191 q^{31} -11.2282 q^{32} -0.314925 q^{33} +15.7230 q^{34} +1.84350 q^{35} -11.3351 q^{36} +2.34450 q^{37} -3.61320 q^{38} -1.78575 q^{39} +26.5644 q^{40} -0.763235 q^{41} +1.12945 q^{42} -9.99682 q^{43} -1.86810 q^{44} +8.17956 q^{45} +5.96238 q^{46} -8.20958 q^{47} +8.25327 q^{48} -6.72569 q^{49} -19.3790 q^{50} -4.92989 q^{51} -10.5929 q^{52} -3.32732 q^{53} +11.4808 q^{54} +1.34805 q^{55} +3.95270 q^{56} +1.13291 q^{57} -13.7766 q^{58} -4.62075 q^{59} -14.1179 q^{60} +5.88594 q^{61} +4.12240 q^{62} +1.21709 q^{63} +9.37267 q^{64} +7.64398 q^{65} +0.825906 q^{66} +12.5836 q^{67} -29.2436 q^{68} -1.86949 q^{69} -4.83466 q^{70} -11.6905 q^{71} +17.5380 q^{72} -13.9911 q^{73} -6.14857 q^{74} +6.07624 q^{75} +6.72028 q^{76} +0.200585 q^{77} +4.68321 q^{78} -7.46107 q^{79} -35.3285 q^{80} +3.37171 q^{81} +2.00162 q^{82} -14.0000 q^{83} -2.10070 q^{84} +21.1026 q^{85} +26.2171 q^{86} +4.31962 q^{87} +2.89039 q^{88} -10.0144 q^{89} -21.4513 q^{90} +1.13740 q^{91} -11.0896 q^{92} -1.29257 q^{93} +21.5300 q^{94} -4.84946 q^{95} -9.23287 q^{96} -10.9092 q^{97} +17.6384 q^{98} +0.889993 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9}+O(q^{10}) $ 391 * q + 14 * q^2 + 12 * q^3 + 446 * q^4 + 22 * q^5 + 40 * q^6 + 63 * q^7 + 36 * q^8 + 501 * q^9	\( 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9} + 40 q^{10} + 57 q^{11} + 20 q^{12} + 83 q^{13} + 21 q^{14} + 60 q^{15} + 548 q^{16} + 59 q^{17} + 54 q^{18} + 131 q^{19} + 35 q^{20} + 121 q^{21} + 89 q^{22} + 34 q^{23} + 110 q^{24} + 609 q^{25} + 54 q^{26} + 27 q^{27} + 182 q^{28} + 102 q^{29} + 92 q^{30} + 88 q^{31} + 76 q^{32} + 131 q^{33} + 128 q^{34} + 31 q^{35} + 654 q^{36} + 135 q^{37} + 23 q^{38} + 96 q^{39} + 113 q^{40} + 128 q^{41} + 45 q^{42} + 140 q^{43} + 151 q^{44} + 77 q^{45} + 245 q^{46} + 22 q^{47} + 25 q^{48} + 712 q^{49} + 53 q^{50} + 102 q^{51} + 174 q^{52} + 54 q^{53} + 131 q^{54} + 101 q^{55} + 43 q^{56} + 226 q^{57} + 109 q^{58} + 40 q^{59} + 123 q^{60} + 249 q^{61} + 28 q^{62} + 139 q^{63} + 730 q^{64} + 227 q^{65} + 55 q^{66} + 169 q^{67} + 48 q^{68} + 89 q^{69} + 98 q^{70} + 66 q^{71} + 120 q^{72} + 324 q^{73} + 60 q^{74} + 19 q^{75} + 356 q^{76} + 83 q^{77} - 11 q^{78} + 195 q^{79} + 26 q^{80} + 807 q^{81} + 49 q^{82} + 74 q^{83} + 252 q^{84} + 373 q^{85} + 100 q^{86} + 43 q^{87} + 211 q^{88} + 207 q^{89} + 10 q^{90} + 189 q^{91} + 30 q^{92} + 172 q^{93} + 130 q^{94} + 43 q^{95} + 203 q^{96} + 254 q^{97} + 26 q^{98} + 273 q^{99}+O(q^{100}) \) 391 * q + 14 * q^2 + 12 * q^3 + 446 * q^4 + 22 * q^5 + 40 * q^6 + 63 * q^7 + 36 * q^8 + 501 * q^9 + 40 * q^10 + 57 * q^11 + 20 * q^12 + 83 * q^13 + 21 * q^14 + 60 * q^15 + 548 * q^16 + 59 * q^17 + 54 * q^18 + 131 * q^19 + 35 * q^20 + 121 * q^21 + 89 * q^22 + 34 * q^23 + 110 * q^24 + 609 * q^25 + 54 * q^26 + 27 * q^27 + 182 * q^28 + 102 * q^29 + 92 * q^30 + 88 * q^31 + 76 * q^32 + 131 * q^33 + 128 * q^34 + 31 * q^35 + 654 * q^36 + 135 * q^37 + 23 * q^38 + 96 * q^39 + 113 * q^40 + 128 * q^41 + 45 * q^42 + 140 * q^43 + 151 * q^44 + 77 * q^45 + 245 * q^46 + 22 * q^47 + 25 * q^48 + 712 * q^49 + 53 * q^50 + 102 * q^51 + 174 * q^52 + 54 * q^53 + 131 * q^54 + 101 * q^55 + 43 * q^56 + 226 * q^57 + 109 * q^58 + 40 * q^59 + 123 * q^60 + 249 * q^61 + 28 * q^62 + 139 * q^63 + 730 * q^64 + 227 * q^65 + 55 * q^66 + 169 * q^67 + 48 * q^68 + 89 * q^69 + 98 * q^70 + 66 * q^71 + 120 * q^72 + 324 * q^73 + 60 * q^74 + 19 * q^75 + 356 * q^76 + 83 * q^77 - 11 * q^78 + 195 * q^79 + 26 * q^80 + 807 * q^81 + 49 * q^82 + 74 * q^83 + 252 * q^84 + 373 * q^85 + 100 * q^86 + 43 * q^87 + 211 * q^88 + 207 * q^89 + 10 * q^90 + 189 * q^91 + 30 * q^92 + 172 * q^93 + 130 * q^94 + 43 * q^95 + 203 * q^96 + 254 * q^97 + 26 * q^98 + 273 * 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 $

$2$	−2.62255	−1.85442	−0.927210	−	0.374542i	$-0.877800\pi$
$2$	−2.62255	−1.85442	−0.927210	+	0.374542i	$0.877800\pi$
$3$	0.822293	0.474751	0.237375	−	0.971418i	$-0.423713\pi$
$3$	0.822293	0.474751	0.237375	+	0.971418i	$0.423713\pi$
$4$	4.87774	2.43887
$5$	−3.51986	−1.57413	−0.787064	−	0.616872i	$-0.788400\pi$
$5$	−3.51986	−1.57413	−0.787064	+	0.616872i	$0.788400\pi$
$6$	−2.15650	−0.880387
$7$	−0.523743	−0.197956	−0.0989781	−	0.995090i	$-0.531557\pi$
$7$	−0.523743	−0.197956	−0.0989781	+	0.995090i	$0.531557\pi$
$8$	−7.54702	−2.66827
$9$	−2.32383	−0.774612
$10$	9.23098	2.91909
$11$	−0.382984	−0.115474	−0.0577371	−	0.998332i	$-0.518389\pi$
$11$	−0.382984	−0.115474	−0.0577371	+	0.998332i	$0.518389\pi$
$12$	4.01093	1.15786
$13$	−2.17167	−0.602314	−0.301157	−	0.953575i	$-0.597373\pi$
$13$	−2.17167	−0.602314	−0.301157	+	0.953575i	$0.597373\pi$
$14$	1.37354	0.367094
$15$	−2.89435	−0.747318
$16$	10.0369	2.50923
$17$	−5.99530	−1.45407	−0.727037	−	0.686598i	$-0.759103\pi$
$17$	−5.99530	−1.45407	−0.727037	+	0.686598i	$0.759103\pi$
$18$	6.09436	1.43646
$19$	1.37774	0.316076	0.158038	−	0.987433i	$-0.449483\pi$
$19$	1.37774	0.316076	0.158038	+	0.987433i	$0.449483\pi$
$20$	−17.1690	−3.83910
$21$	−0.430670	−0.0939799
$22$	1.00439	0.214138
$23$	−2.27351	−0.474059	−0.237030	−	0.971502i	$-0.576174\pi$
$23$	−2.27351	−0.474059	−0.237030	+	0.971502i	$0.576174\pi$
$24$	−6.20586	−1.26677
$25$	7.38938	1.47788
$26$	5.69531	1.11694
$27$	−4.37775	−0.842498
$28$	−2.55468	−0.482790
$29$	5.25315	0.975485	0.487742	−	0.872988i	$-0.337821\pi$
$29$	5.25315	0.975485	0.487742	+	0.872988i	$0.337821\pi$
$30$	7.59057	1.38584
$31$	−1.57191	−0.282323	−0.141162	−	0.989987i	$-0.545084\pi$
$31$	−1.57191	−0.282323	−0.141162	+	0.989987i	$0.545084\pi$
$32$	−11.2282	−1.98489
$33$	−0.314925	−0.0548215
$34$	15.7230	2.69646
$35$	1.84350	0.311608
$36$	−11.3351	−1.88918
$37$	2.34450	0.385434	0.192717	−	0.981254i	$-0.438270\pi$
$37$	2.34450	0.385434	0.192717	+	0.981254i	$0.438270\pi$
$38$	−3.61320	−0.586138
$39$	−1.78575	−0.285949
$40$	26.5644	4.20020
$41$	−0.763235	−0.119197	−0.0595986	−	0.998222i	$-0.518982\pi$
$41$	−0.763235	−0.119197	−0.0595986	+	0.998222i	$0.518982\pi$
$42$	1.12945	0.174278
$43$	−9.99682	−1.52450	−0.762250	−	0.647282i	$-0.775905\pi$
$43$	−9.99682	−1.52450	−0.762250	+	0.647282i	$0.775905\pi$
$44$	−1.86810	−0.281627
$45$	8.17956	1.21934
$46$	5.96238	0.879105
$47$	−8.20958	−1.19749	−0.598745	−	0.800940i	$-0.704334\pi$
$47$	−8.20958	−1.19749	−0.598745	+	0.800940i	$0.704334\pi$
$48$	8.25327	1.19126
$49$	−6.72569	−0.960813
$50$	−19.3790	−2.74060
$51$	−4.92989	−0.690323
$52$	−10.5929	−1.46897
$53$	−3.32732	−0.457042	−0.228521	−	0.973539i	$-0.573389\pi$
$53$	−3.32732	−0.457042	−0.228521	+	0.973539i	$0.573389\pi$
$54$	11.4808	1.56235
$55$	1.34805	0.181771
$56$	3.95270	0.528201
$57$	1.13291	0.150057
$58$	−13.7766	−1.80896
$59$	−4.62075	−0.601570	−0.300785	−	0.953692i	$-0.597249\pi$
$59$	−4.62075	−0.601570	−0.300785	+	0.953692i	$0.597249\pi$
$60$	−14.1179	−1.82261
$61$	5.88594	0.753617	0.376809	−	0.926291i	$-0.377021\pi$
$61$	5.88594	0.753617	0.376809	+	0.926291i	$0.377021\pi$
$62$	4.12240	0.523546
$63$	1.21709	0.153339
$64$	9.37267	1.17158
$65$	7.64398	0.948119
$66$	0.825906	0.101662
$67$	12.5836	1.53733	0.768664	−	0.639653i	$-0.220922\pi$
$67$	12.5836	1.53733	0.768664	+	0.639653i	$0.220922\pi$
$68$	−29.2436	−3.54630
$69$	−1.86949	−0.225060
$70$	−4.83466	−0.577852
$71$	−11.6905	−1.38741	−0.693705	−	0.720259i	$-0.744023\pi$
$71$	−11.6905	−1.38741	−0.693705	+	0.720259i	$0.744023\pi$
$72$	17.5380	2.06688
$73$	−13.9911	−1.63754	−0.818770	−	0.574122i	$-0.805343\pi$
$73$	−13.9911	−1.63754	−0.818770	+	0.574122i	$0.805343\pi$
$74$	−6.14857	−0.714756
$75$	6.07624	0.701623
$76$	6.72028	0.770869
$77$	0.200585	0.0228588
$78$	4.68321	0.530270
$79$	−7.46107	−0.839435	−0.419718	−	0.907655i	$-0.637871\pi$
$79$	−7.46107	−0.839435	−0.419718	+	0.907655i	$0.637871\pi$
$80$	−35.3285	−3.94984
$81$	3.37171	0.374635
$82$	2.00162	0.221042
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	−2.10070	−0.229205
$85$	21.1026	2.28890
$86$	26.2171	2.82706
$87$	4.31962	0.463112
$88$	2.89039	0.308117
$89$	−10.0144	−1.06153	−0.530764	−	0.847520i	$-0.678095\pi$
$89$	−10.0144	−1.06153	−0.530764	+	0.847520i	$0.678095\pi$
$90$	−21.4513	−2.26116
$91$	1.13740	0.119232
$92$	−11.0896	−1.15617
$93$	−1.29257	−0.134033
$94$	21.5300	2.22065
$95$	−4.84946	−0.497544
$96$	−9.23287	−0.942326
$97$	−10.9092	−1.10766	−0.553829	−	0.832631i	$-0.686834\pi$
$97$	−10.9092	−1.10766	−0.553829	+	0.832631i	$0.686834\pi$
$98$	17.6384	1.78175
$99$	0.889993	0.0894476
$100$	36.0435	3.60435
$101$	11.5142	1.14570	0.572851	−	0.819660i	$-0.305837\pi$
$101$	11.5142	1.14570	0.572851	+	0.819660i	$0.305837\pi$
$102$	12.9289	1.28015
$103$	−0.0871809	−0.00859019	−0.00429510	−	0.999991i	$-0.501367\pi$
$103$	−0.0871809	−0.00859019	−0.00429510	+	0.999991i	$0.501367\pi$
$104$	16.3897	1.60714
$105$	1.51590	0.147936
$106$	8.72604	0.847548
$107$	−17.1301	−1.65603	−0.828013	−	0.560710i	$-0.810528\pi$
$107$	−17.1301	−1.65603	−0.828013	+	0.560710i	$0.810528\pi$
$108$	−21.3535	−2.05475
$109$	9.32981	0.893634	0.446817	−	0.894625i	$-0.352558\pi$
$109$	9.32981	0.893634	0.446817	+	0.894625i	$0.352558\pi$
$110$	−3.53532	−0.337080
$111$	1.92787	0.182985
$112$	−5.25676	−0.496717
$113$	−14.1422	−1.33039	−0.665193	−	0.746671i	$-0.731651\pi$
$113$	−14.1422	−1.33039	−0.665193	+	0.746671i	$0.731651\pi$
$114$	−2.97110	−0.278269
$115$	8.00242	0.746229
$116$	25.6235	2.37908
$117$	5.04661	0.466559
$118$	12.1181	1.11556
$119$	3.14000	0.287843
$120$	21.8437	1.99405
$121$	−10.8533	−0.986666
$122$	−15.4361	−1.39752
$123$	−0.627602	−0.0565890
$124$	−7.66737	−0.688550
$125$	−8.41028	−0.752239
$126$	−3.19188	−0.284355
$127$	−11.0386	−0.979513	−0.489757	−	0.871859i	$-0.662914\pi$
$127$	−11.0386	−0.979513	−0.489757	+	0.871859i	$0.662914\pi$
$128$	−2.12385	−0.187724
$129$	−8.22031	−0.723758
$130$	−20.0467	−1.75821
$131$	−16.1790	−1.41357	−0.706783	−	0.707430i	$-0.749854\pi$
$131$	−16.1790	−1.41357	−0.706783	+	0.707430i	$0.749854\pi$
$132$	−1.53613	−0.133703
$133$	−0.721583	−0.0625692
$134$	−33.0010	−2.85085
$135$	15.4090	1.32620
$136$	45.2466	3.87987
$137$	16.4439	1.40489	0.702447	−	0.711737i	$-0.252091\pi$
$137$	16.4439	1.40489	0.702447	+	0.711737i	$0.252091\pi$
$138$	4.90282	0.417356
$139$	17.9818	1.52519	0.762596	−	0.646875i	$-0.223924\pi$
$139$	17.9818	1.52519	0.762596	+	0.646875i	$0.223924\pi$
$140$	8.99212	0.759973
$141$	−6.75067	−0.568509
$142$	30.6589	2.57284
$143$	0.831717	0.0695517
$144$	−23.3241	−1.94368
$145$	−18.4903	−1.53554
$146$	36.6924	3.03668
$147$	−5.53049	−0.456147
$148$	11.4359	0.940024
$149$	−11.4661	−0.939342	−0.469671	−	0.882841i	$-0.655628\pi$
$149$	−11.4661	−0.939342	−0.469671	+	0.882841i	$0.655628\pi$
$150$	−15.9352	−1.30110
$151$	−18.2814	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$151$	−18.2814	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$152$	−10.3979	−0.843377
$153$	13.9321	1.12634
$154$	−0.526044	−0.0423899
$155$	5.53289	0.444413
$156$	−8.71044	−0.697393
$157$	−17.7461	−1.41629	−0.708145	−	0.706067i	$-0.750468\pi$
$157$	−17.7461	−1.41629	−0.708145	+	0.706067i	$0.750468\pi$
$158$	19.5670	1.55667
$159$	−2.73603	−0.216981
$160$	39.5217	3.12446
$161$	1.19073	0.0938430
$162$	−8.84247	−0.694730
$163$	−15.9807	−1.25171	−0.625854	−	0.779940i	$-0.715249\pi$
$163$	−15.9807	−1.25171	−0.625854	+	0.779940i	$0.715249\pi$
$164$	−3.72286	−0.290707
$165$	1.10849	0.0862959
$166$	36.7157	2.84969
$167$	−8.80682	−0.681492	−0.340746	−	0.940155i	$-0.610680\pi$
$167$	−8.80682	−0.681492	−0.340746	+	0.940155i	$0.610680\pi$
$168$	3.25027	0.250764
$169$	−8.28383	−0.637218
$170$	−55.3425	−4.24458
$171$	−3.20165	−0.244836
$172$	−48.7619	−3.71806
$173$	−20.4313	−1.55336	−0.776682	−	0.629893i	$-0.783099\pi$
$173$	−20.4313	−1.55336	−0.776682	+	0.629893i	$0.783099\pi$
$174$	−11.3284	−0.858804
$175$	−3.87014	−0.292555
$176$	−3.84398	−0.289751
$177$	−3.79961	−0.285596
$178$	26.2633	1.96852
$179$	−0.181565	−0.0135708	−0.00678540	−	0.999977i	$-0.502160\pi$
$179$	−0.181565	−0.0135708	−0.00678540	+	0.999977i	$0.502160\pi$
$180$	39.8978	2.97381
$181$	25.6401	1.90581	0.952906	−	0.303267i	$-0.0980775\pi$
$181$	25.6401	1.90581	0.952906	+	0.303267i	$0.0980775\pi$
$182$	−2.98288	−0.221106
$183$	4.83996	0.357780
$184$	17.1582	1.26492
$185$	−8.25231	−0.606722
$186$	3.38982	0.248554
$187$	2.29611	0.167908
$188$	−40.0442	−2.92052
$189$	2.29282	0.166778
$190$	12.7179	0.922655
$191$	0.103784	0.00750954	0.00375477	−	0.999993i	$-0.498805\pi$
$191$	0.103784	0.00750954	0.00375477	+	0.999993i	$0.498805\pi$
$192$	7.70708	0.556211
$193$	23.2609	1.67436	0.837178	−	0.546930i	$-0.184204\pi$
$193$	23.2609	1.67436	0.837178	+	0.546930i	$0.184204\pi$
$194$	28.6098	2.05406
$195$	6.28559	0.450120
$196$	−32.8062	−2.34330
$197$	17.7307	1.26326	0.631631	−	0.775270i	$-0.282386\pi$
$197$	17.7307	1.26326	0.631631	+	0.775270i	$0.282386\pi$
$198$	−2.33405	−0.165873
$199$	6.06319	0.429808	0.214904	−	0.976635i	$-0.431056\pi$
$199$	6.06319	0.429808	0.214904	+	0.976635i	$0.431056\pi$
$200$	−55.7678	−3.94338
$201$	10.3474	0.729848
$202$	−30.1964	−2.12461
$203$	−2.75130	−0.193103
$204$	−24.0468	−1.68361
$205$	2.68648	0.187632
$206$	0.228636	0.0159298
$207$	5.28326	0.367212
$208$	−21.7969	−1.51134
$209$	−0.527654	−0.0364986
$210$	−3.97551	−0.274336
$211$	−22.5618	−1.55322	−0.776608	−	0.629984i	$-0.783061\pi$
$211$	−22.5618	−1.55322	−0.776608	+	0.629984i	$0.783061\pi$
$212$	−16.2298	−1.11467
$213$	−9.61303	−0.658674
$214$	44.9244	3.07097
$215$	35.1874	2.39976
$216$	33.0390	2.24802
$217$	0.823276	0.0558876
$218$	−24.4679	−1.65717
$219$	−11.5048	−0.777423
$220$	6.57544	0.443316
$221$	13.0198	0.875809
$222$	−5.05592	−0.339331
$223$	5.29525	0.354596	0.177298	−	0.984157i	$-0.443264\pi$
$223$	5.29525	0.354596	0.177298	+	0.984157i	$0.443264\pi$
$224$	5.88069	0.392920
$225$	−17.1717	−1.14478
$226$	37.0886	2.46709
$227$	8.79665	0.583854	0.291927	−	0.956441i	$-0.405704\pi$
$227$	8.79665	0.583854	0.291927	+	0.956441i	$0.405704\pi$
$228$	5.52604	0.365971
$229$	−8.38714	−0.554238	−0.277119	−	0.960836i	$-0.589380\pi$
$229$	−8.38714	−0.554238	−0.277119	+	0.960836i	$0.589380\pi$
$230$	−20.9867	−1.38382
$231$	0.164940	0.0108522
$232$	−39.6456	−2.60286
$233$	0.397755	0.0260578	0.0130289	−	0.999915i	$-0.495853\pi$
$233$	0.397755	0.0260578	0.0130289	+	0.999915i	$0.495853\pi$
$234$	−13.2350	−0.865197
$235$	28.8965	1.88500
$236$	−22.5388	−1.46715
$237$	−6.13518	−0.398523
$238$	−8.23478	−0.533782
$239$	−6.68916	−0.432686	−0.216343	−	0.976317i	$-0.569413\pi$
$239$	−6.68916	−0.432686	−0.216343	+	0.976317i	$0.569413\pi$
$240$	−29.0503	−1.87519
$241$	30.3913	1.95767	0.978837	−	0.204641i	$-0.0656026\pi$
$241$	30.3913	1.95767	0.978837	+	0.204641i	$0.0656026\pi$
$242$	28.4633	1.82969
$243$	15.9058	1.02036
$244$	28.7101	1.83798
$245$	23.6735	1.51244
$246$	1.64592	0.104940
$247$	−2.99201	−0.190377
$248$	11.8632	0.753316
$249$	−11.5121	−0.729550
$250$	22.0564	1.39497
$251$	−29.9361	−1.88955	−0.944774	−	0.327722i	$-0.893719\pi$
$251$	−29.9361	−1.88955	−0.944774	+	0.327722i	$0.893719\pi$
$252$	5.93666	0.373975
$253$	0.870718	0.0547416
$254$	28.9491	1.81643
$255$	17.3525	1.08666
$256$	−13.1755	−0.823466
$257$	28.3947	1.77122	0.885608	−	0.464434i	$-0.153742\pi$
$257$	28.3947	1.77122	0.885608	+	0.464434i	$0.153742\pi$
$258$	21.5581	1.34215
$259$	−1.22792	−0.0762990
$260$	37.2854	2.31234
$261$	−12.2074	−0.755622
$262$	42.4302	2.62135
$263$	−14.9395	−0.921207	−0.460604	−	0.887606i	$-0.652367\pi$
$263$	−14.9395	−0.921207	−0.460604	+	0.887606i	$0.652367\pi$
$264$	2.37675	0.146279
$265$	11.7117	0.719442
$266$	1.89239	0.116030
$267$	−8.23479	−0.503961
$268$	61.3794	3.74935
$269$	20.5169	1.25094	0.625469	−	0.780249i	$-0.284908\pi$
$269$	20.5169	1.25094	0.625469	+	0.780249i	$0.284908\pi$
$270$	−40.4109	−2.45933
$271$	6.48542	0.393961	0.196980	−	0.980407i	$-0.436886\pi$
$271$	6.48542	0.393961	0.196980	+	0.980407i	$0.436886\pi$
$272$	−60.1743	−3.64860
$273$	0.935275	0.0566054
$274$	−43.1247	−2.60526
$275$	−2.83002	−0.170657
$276$	−9.11889	−0.548893
$277$	2.30029	0.138211	0.0691056	−	0.997609i	$-0.477985\pi$
$277$	2.30029	0.138211	0.0691056	+	0.997609i	$0.477985\pi$
$278$	−47.1580	−2.82835
$279$	3.65286	0.218691
$280$	−13.9129	−0.831456
$281$	0.810838	0.0483705	0.0241853	−	0.999707i	$-0.492301\pi$
$281$	0.810838	0.0483705	0.0241853	+	0.999707i	$0.492301\pi$
$282$	17.7039	1.05425
$283$	12.8234	0.762275	0.381137	−	0.924518i	$-0.375533\pi$
$283$	12.8234	0.762275	0.381137	+	0.924518i	$0.375533\pi$
$284$	−57.0234	−3.38372
$285$	−3.98767	−0.236209
$286$	−2.18122	−0.128978
$287$	0.399739	0.0235958
$288$	26.0925	1.53752
$289$	18.9436	1.11433
$290$	48.4917	2.84753
$291$	−8.97052	−0.525861
$292$	−68.2452	−3.99375
$293$	−11.0999	−0.648464	−0.324232	−	0.945978i	$-0.605106\pi$
$293$	−11.0999	−0.648464	−0.324232	+	0.945978i	$0.605106\pi$
$294$	14.5040	0.845888
$295$	16.2644	0.946948
$296$	−17.6940	−1.02844
$297$	1.67661	0.0972868
$298$	30.0705	1.74193
$299$	4.93732	0.285532
$300$	29.6383	1.71117
$301$	5.23576	0.301784
$302$	47.9437	2.75885
$303$	9.46801	0.543923
$304$	13.8283	0.793107
$305$	−20.7177	−1.18629
$306$	−36.5375	−2.08871
$307$	10.0267	0.572253	0.286127	−	0.958192i	$-0.407632\pi$
$307$	10.0267	0.572253	0.286127	+	0.958192i	$0.407632\pi$
$308$	0.978404	0.0557498
$309$	−0.0716882	−0.00407820
$310$	−14.5103	−0.824128
$311$	1.78127	0.101007	0.0505033	−	0.998724i	$-0.483917\pi$
$311$	1.78127	0.101007	0.0505033	+	0.998724i	$0.483917\pi$
$312$	13.4771	0.762990
$313$	19.2068	1.08563	0.542817	−	0.839851i	$-0.317358\pi$
$313$	19.2068	1.08563	0.542817	+	0.839851i	$0.317358\pi$
$314$	46.5399	2.62640
$315$	−4.28399	−0.241375
$316$	−36.3932	−2.04728
$317$	12.9186	0.725579	0.362790	−	0.931871i	$-0.381824\pi$
$317$	12.9186	0.725579	0.362790	+	0.931871i	$0.381824\pi$
$318$	7.17536	0.402374
$319$	−2.01187	−0.112643
$320$	−32.9905	−1.84422
$321$	−14.0859	−0.786199
$322$	−3.12275	−0.174024
$323$	−8.25999	−0.459598
$324$	16.4464	0.913686
$325$	−16.0473	−0.890146
$326$	41.9102	2.32119
$327$	7.67183	0.424253
$328$	5.76015	0.318051
$329$	4.29971	0.237051
$330$	−2.90707	−0.160029
$331$	−4.10708	−0.225746	−0.112873	−	0.993609i	$-0.536005\pi$
$331$	−4.10708	−0.225746	−0.112873	+	0.993609i	$0.536005\pi$
$332$	−68.2885	−3.74782
$333$	−5.44824	−0.298562
$334$	23.0963	1.26377
$335$	−44.2924	−2.41995
$336$	−4.32259	−0.235817
$337$	−13.8455	−0.754214	−0.377107	−	0.926170i	$-0.623081\pi$
$337$	−13.8455	−0.754214	−0.377107	+	0.926170i	$0.623081\pi$
$338$	21.7247	1.18167
$339$	−11.6290	−0.631602
$340$	102.933	5.58233
$341$	0.602017	0.0326010
$342$	8.39647	0.454029
$343$	7.18873	0.388155
$344$	75.4462	4.06778
$345$	6.58033	0.354273
$346$	53.5821	2.88059
$347$	14.5007	0.778437	0.389218	−	0.921146i	$-0.372745\pi$
$347$	14.5007	0.778437	0.389218	+	0.921146i	$0.372745\pi$
$348$	21.0700	1.12947
$349$	28.5486	1.52817	0.764085	−	0.645115i	$-0.223191\pi$
$349$	28.5486	1.52817	0.764085	+	0.645115i	$0.223191\pi$
$350$	10.1496	0.542519
$351$	9.50704	0.507449
$352$	4.30023	0.229203
$353$	−18.9414	−1.00815	−0.504073	−	0.863661i	$-0.668166\pi$
$353$	−18.9414	−1.00815	−0.504073	+	0.863661i	$0.668166\pi$
$354$	9.96464	0.529615
$355$	41.1489	2.18396
$356$	−48.8478	−2.58893
$357$	2.58200	0.136654
$358$	0.476162	0.0251660
$359$	−3.68599	−0.194539	−0.0972696	−	0.995258i	$-0.531011\pi$
$359$	−3.68599	−0.194539	−0.0972696	+	0.995258i	$0.531011\pi$
$360$	−61.7313	−3.25353
$361$	−17.1018	−0.900096
$362$	−67.2422	−3.53417
$363$	−8.92461	−0.468420
$364$	5.54794	0.290791
$365$	49.2468	2.57769
$366$	−12.6930	−0.663475
$367$	−34.9856	−1.82624	−0.913118	−	0.407696i	$-0.866332\pi$
$367$	−34.9856	−1.82624	−0.913118	+	0.407696i	$0.866332\pi$
$368$	−22.8190	−1.18952
$369$	1.77363	0.0923316
$370$	21.6421	1.12512
$371$	1.74266	0.0904743
$372$	−6.30482	−0.326890
$373$	−5.69293	−0.294769	−0.147384	−	0.989079i	$-0.547085\pi$
$373$	−5.69293	−0.294769	−0.147384	+	0.989079i	$0.547085\pi$
$374$	−6.02164	−0.311372
$375$	−6.91571	−0.357126
$376$	61.9578	3.19523
$377$	−11.4081	−0.587548
$378$	−6.01301	−0.309276
$379$	28.2988	1.45361	0.726807	−	0.686842i	$-0.241003\pi$
$379$	28.2988	1.45361	0.726807	+	0.686842i	$0.241003\pi$
$380$	−23.6544	−1.21345
$381$	−9.07692	−0.465025
$382$	−0.272178	−0.0139258
$383$	38.0774	1.94566	0.972831	−	0.231516i	$-0.0743684\pi$
$383$	38.0774	1.94566	0.972831	+	0.231516i	$0.0743684\pi$
$384$	−1.74643	−0.0891220
$385$	−0.706031	−0.0359827
$386$	−61.0028	−3.10496
$387$	23.2310	1.18090
$388$	−53.2121	−2.70144
$389$	−9.84963	−0.499396	−0.249698	−	0.968324i	$-0.580331\pi$
$389$	−9.84963	−0.499396	−0.249698	+	0.968324i	$0.580331\pi$
$390$	−16.4842	−0.834712
$391$	13.6304	0.689317
$392$	50.7589	2.56371
$393$	−13.3039	−0.671092
$394$	−46.4996	−2.34262
$395$	26.2619	1.32138
$396$	4.34116	0.218151
$397$	5.45203	0.273630	0.136815	−	0.990597i	$-0.456313\pi$
$397$	5.45203	0.273630	0.136815	+	0.990597i	$0.456313\pi$
$398$	−15.9010	−0.797044
$399$	−0.593353	−0.0297048
$400$	74.1665	3.70833
$401$	26.4416	1.32043	0.660215	−	0.751076i	$-0.270465\pi$
$401$	26.4416	1.32043	0.660215	+	0.751076i	$0.270465\pi$
$402$	−27.1365	−1.35344
$403$	3.41367	0.170047
$404$	56.1631	2.79422
$405$	−11.8679	−0.589723
$406$	7.21540	0.358094
$407$	−0.897908	−0.0445076
$408$	37.2060	1.84197
$409$	−30.0948	−1.48809	−0.744047	−	0.668127i	$-0.767096\pi$
$409$	−30.0948	−1.48809	−0.744047	+	0.668127i	$0.767096\pi$
$410$	−7.04541	−0.347948
$411$	13.5217	0.666974
$412$	−0.425246	−0.0209504
$413$	2.42008	0.119085
$414$	−13.8556	−0.680965
$415$	49.2780	2.41896
$416$	24.3840	1.19552
$417$	14.7863	0.724086
$418$	1.38380	0.0676838
$419$	15.8739	0.775489	0.387744	−	0.921767i	$-0.373254\pi$
$419$	15.8739	0.775489	0.387744	+	0.921767i	$0.373254\pi$
$420$	7.39415	0.360798
$421$	−21.4122	−1.04357	−0.521783	−	0.853078i	$-0.674733\pi$
$421$	−21.4122	−1.04357	−0.521783	+	0.853078i	$0.674733\pi$
$422$	59.1692	2.88031
$423$	19.0777	0.927589
$424$	25.1113	1.21951
$425$	−44.3016	−2.14894
$426$	25.2106	1.22146
$427$	−3.08272	−0.149183
$428$	−83.5560	−4.03883
$429$	0.683915	0.0330197
$430$	−92.2804	−4.45016
$431$	−13.4937	−0.649967	−0.324984	−	0.945720i	$-0.605359\pi$
$431$	−13.4937	−0.649967	−0.324984	+	0.945720i	$0.605359\pi$
$432$	−43.9391	−2.11402
$433$	2.08015	0.0999656	0.0499828	−	0.998750i	$-0.484083\pi$
$433$	2.08015	0.0999656	0.0499828	+	0.998750i	$0.484083\pi$
$434$	−2.15908	−0.103639
$435$	−15.2045	−0.728998
$436$	45.5084	2.17946
$437$	−3.13231	−0.149839
$438$	30.1719	1.44167
$439$	−31.9605	−1.52539	−0.762696	−	0.646757i	$-0.776125\pi$
$439$	−31.9605	−1.52539	−0.762696	+	0.646757i	$0.776125\pi$
$440$	−10.1738	−0.485015
$441$	15.6294	0.744257
$442$	−34.1451	−1.62412
$443$	41.0759	1.95158	0.975788	−	0.218718i	$-0.0701875\pi$
$443$	41.0759	1.95158	0.975788	+	0.218718i	$0.0701875\pi$
$444$	9.40364	0.446277
$445$	35.2493	1.67098
$446$	−13.8870	−0.657570
$447$	−9.42852	−0.445954
$448$	−4.90887	−0.231922
$449$	−32.2721	−1.52301	−0.761507	−	0.648156i	$-0.775540\pi$
$449$	−32.2721	−1.52301	−0.761507	+	0.648156i	$0.775540\pi$
$450$	45.0336	2.12290
$451$	0.292307	0.0137642
$452$	−68.9821	−3.24464
$453$	−15.0326	−0.706295
$454$	−23.0696	−1.08271
$455$	−4.00348	−0.187686
$456$	−8.55008	−0.400394
$457$	−12.6146	−0.590085	−0.295043	−	0.955484i	$-0.595334\pi$
$457$	−12.6146	−0.590085	−0.295043	+	0.955484i	$0.595334\pi$
$458$	21.9956	1.02779
$459$	26.2459	1.22506
$460$	39.0338	1.81996
$461$	−27.4043	−1.27634	−0.638172	−	0.769894i	$-0.720309\pi$
$461$	−27.4043	−1.27634	−0.638172	+	0.769894i	$0.720309\pi$
$462$	−0.432562	−0.0201246
$463$	−29.0288	−1.34908	−0.674542	−	0.738236i	$-0.735659\pi$
$463$	−29.0288	−1.34908	−0.674542	+	0.738236i	$0.735659\pi$
$464$	52.7253	2.44771
$465$	4.54966	0.210985
$466$	−1.04313	−0.0483221
$467$	−29.4816	−1.36425	−0.682123	−	0.731237i	$-0.738943\pi$
$467$	−29.4816	−1.36425	−0.682123	+	0.731237i	$0.738943\pi$
$468$	24.6161	1.13788
$469$	−6.59056	−0.304324
$470$	−75.7824	−3.49558
$471$	−14.5925	−0.672385
$472$	34.8729	1.60515
$473$	3.82863	0.176040
$474$	16.0898	0.739028
$475$	10.1807	0.467121
$476$	15.3161	0.702012
$477$	7.73213	0.354030
$478$	17.5426	0.802381
$479$	19.5691	0.894135	0.447067	−	0.894500i	$-0.352468\pi$
$479$	19.5691	0.894135	0.447067	+	0.894500i	$0.352468\pi$
$480$	32.4984	1.48334
$481$	−5.09149	−0.232152
$482$	−79.7025	−3.63035
$483$	0.979131	0.0445520
$484$	−52.9397	−2.40635
$485$	38.3987	1.74359
$486$	−41.7136	−1.89217
$487$	−11.4019	−0.516671	−0.258336	−	0.966055i	$-0.583174\pi$
$487$	−11.4019	−0.516671	−0.258336	+	0.966055i	$0.583174\pi$
$488$	−44.4213	−2.01086
$489$	−13.1408	−0.594249
$490$	−62.0848	−2.80470
$491$	1.42906	0.0644928	0.0322464	−	0.999480i	$-0.489734\pi$
$491$	1.42906	0.0644928	0.0322464	+	0.999480i	$0.489734\pi$
$492$	−3.06128	−0.138013
$493$	−31.4942	−1.41843
$494$	7.84668	0.353039
$495$	−3.13265	−0.140802
$496$	−15.7771	−0.708413
$497$	6.12283	0.274646
$498$	30.1910	1.35289
$499$	38.8933	1.74110	0.870552	−	0.492077i	$-0.163762\pi$
$499$	38.8933	1.74110	0.870552	+	0.492077i	$0.163762\pi$
$500$	−41.0232	−1.83461
$501$	−7.24178	−0.323539
$502$	78.5087	3.50402
$503$	−37.2755	−1.66203	−0.831015	−	0.556249i	$-0.812240\pi$
$503$	−37.2755	−1.66203	−0.831015	+	0.556249i	$0.812240\pi$
$504$	−9.18541	−0.409151
$505$	−40.5282	−1.80348
$506$	−2.28350	−0.101514
$507$	−6.81174	−0.302520
$508$	−53.8432	−2.38891
$509$	0.407681	0.0180701	0.00903507	−	0.999959i	$-0.497124\pi$
$509$	0.407681	0.0180701	0.00903507	+	0.999959i	$0.497124\pi$
$510$	−45.5077	−2.01512
$511$	7.32776	0.324161
$512$	38.8009	1.71478
$513$	−6.03142	−0.266294
$514$	−74.4665	−3.28458
$515$	0.306864	0.0135221
$516$	−40.0966	−1.76515
$517$	3.14414	0.138279
$518$	3.22027	0.141490
$519$	−16.8005	−0.737461
$520$	−57.6892	−2.52984
$521$	−43.8897	−1.92284	−0.961422	−	0.275078i	$-0.911296\pi$
$521$	−43.8897	−1.92284	−0.961422	+	0.275078i	$0.911296\pi$
$522$	32.0146	1.40124
$523$	−30.3119	−1.32545	−0.662724	−	0.748864i	$-0.730600\pi$
$523$	−30.3119	−1.32545	−0.662724	+	0.748864i	$0.730600\pi$
$524$	−78.9171	−3.44751
$525$	−3.18238	−0.138891
$526$	39.1794	1.70830
$527$	9.42407	0.410519
$528$	−3.16088	−0.137559
$529$	−17.8312	−0.775268
$530$	−30.7144	−1.33415
$531$	10.7379	0.465983
$532$	−3.51970	−0.152598
$533$	1.65750	0.0717942
$534$	21.5961	0.934555
$535$	60.2953	2.60679
$536$	−94.9684	−4.10201
$537$	−0.149300	−0.00644275
$538$	−53.8065	−2.31976
$539$	2.57584	0.110949
$540$	75.1614	3.23443
$541$	36.3931	1.56466	0.782331	−	0.622863i	$-0.214030\pi$
$541$	36.3931	1.56466	0.782331	+	0.622863i	$0.214030\pi$
$542$	−17.0083	−0.730569
$543$	21.0836	0.904785
$544$	67.3165	2.88617
$545$	−32.8396	−1.40669
$546$	−2.45280	−0.104970
$547$	9.82954	0.420281	0.210140	−	0.977671i	$-0.432608\pi$
$547$	9.82954	0.420281	0.210140	+	0.977671i	$0.432608\pi$
$548$	80.2089	3.42635
$549$	−13.6779	−0.583761
$550$	7.42185	0.316469
$551$	7.23749	0.308327
$552$	14.1091	0.600522
$553$	3.90768	0.166171
$554$	−6.03263	−0.256302
$555$	−6.78581	−0.288042
$556$	87.7104	3.71975
$557$	−9.52805	−0.403717	−0.201858	−	0.979415i	$-0.564698\pi$
$557$	−9.52805	−0.403717	−0.201858	+	0.979415i	$0.564698\pi$
$558$	−9.57978	−0.405545
$559$	21.7098	0.918228
$560$	18.5030	0.781896
$561$	1.88807	0.0797145
$562$	−2.12646	−0.0896993
$563$	36.9316	1.55648	0.778240	−	0.627967i	$-0.216113\pi$
$563$	36.9316	1.55648	0.778240	+	0.627967i	$0.216113\pi$
$564$	−32.9281	−1.38652
$565$	49.7785	2.09420
$566$	−33.6301	−1.41358
$567$	−1.76591	−0.0741613
$568$	88.2286	3.70199
$569$	17.0519	0.714854	0.357427	−	0.933941i	$-0.383654\pi$
$569$	17.0519	0.714854	0.357427	+	0.933941i	$0.383654\pi$
$570$	10.4579	0.438031
$571$	14.6566	0.613361	0.306680	−	0.951813i	$-0.400782\pi$
$571$	14.6566	0.613361	0.306680	+	0.951813i	$0.400782\pi$
$572$	4.05690	0.169628
$573$	0.0853407	0.00356516
$574$	−1.04833	−0.0437566
$575$	−16.7998	−0.700601
$576$	−21.7805	−0.907523
$577$	−13.2859	−0.553101	−0.276550	−	0.960999i	$-0.589191\pi$
$577$	−13.2859	−0.553101	−0.276550	+	0.960999i	$0.589191\pi$
$578$	−49.6806	−2.06644
$579$	19.1273	0.794902
$580$	−90.1910	−3.74498
$581$	7.33240	0.304199
$582$	23.5256	0.975168
$583$	1.27431	0.0527765
$584$	105.591	4.36940
$585$	−17.7633	−0.734424
$586$	29.1100	1.20252
$587$	−15.6148	−0.644494	−0.322247	−	0.946656i	$-0.604438\pi$
$587$	−15.6148	−0.644494	−0.322247	+	0.946656i	$0.604438\pi$
$588$	−26.9763	−1.11248
$589$	−2.16569	−0.0892356
$590$	−42.6540	−1.75604
$591$	14.5798	0.599734
$592$	23.5316	0.967141
$593$	−2.09231	−0.0859208	−0.0429604	−	0.999077i	$-0.513679\pi$
$593$	−2.09231	−0.0859208	−0.0429604	+	0.999077i	$0.513679\pi$
$594$	−4.39699	−0.180411
$595$	−11.0523	−0.453102
$596$	−55.9289	−2.29094
$597$	4.98571	0.204052
$598$	−12.9483	−0.529497
$599$	14.4140	0.588940	0.294470	−	0.955661i	$-0.404857\pi$
$599$	14.4140	0.588940	0.294470	+	0.955661i	$0.404857\pi$
$600$	−45.8575	−1.87212
$601$	15.0145	0.612453	0.306226	−	0.951959i	$-0.400934\pi$
$601$	15.0145	0.612453	0.306226	+	0.951959i	$0.400934\pi$
$602$	−13.7310	−0.559635
$603$	−29.2421	−1.19083
$604$	−89.1718	−3.62835
$605$	38.2021	1.55314
$606$	−24.8303	−1.00866
$607$	−9.05126	−0.367379	−0.183690	−	0.982984i	$-0.558804\pi$
$607$	−9.05126	−0.367379	−0.183690	+	0.982984i	$0.558804\pi$
$608$	−15.4696	−0.627375
$609$	−2.26237	−0.0916759
$610$	54.3330	2.19988
$611$	17.8285	0.721265
$612$	67.9572	2.74701
$613$	13.7519	0.555432	0.277716	−	0.960663i	$-0.410423\pi$
$613$	13.7519	0.555432	0.277716	+	0.960663i	$0.410423\pi$
$614$	−26.2954	−1.06120
$615$	2.20907	0.0890783
$616$	−1.51382	−0.0609936
$617$	8.33208	0.335437	0.167718	−	0.985835i	$-0.446360\pi$
$617$	8.33208	0.335437	0.167718	+	0.985835i	$0.446360\pi$
$618$	0.188006	0.00756270
$619$	−12.2117	−0.490831	−0.245416	−	0.969418i	$-0.578924\pi$
$619$	−12.2117	−0.490831	−0.245416	+	0.969418i	$0.578924\pi$
$620$	26.9880	1.08387
$621$	9.95285	0.399394
$622$	−4.67147	−0.187309
$623$	5.24499	0.210136
$624$	−17.9234	−0.717511
$625$	−7.34393	−0.293757
$626$	−50.3708	−2.01322
$627$	−0.433886	−0.0173278
$628$	−86.5608	−3.45415
$629$	−14.0560	−0.560449
$630$	11.2350	0.447611
$631$	−42.2605	−1.68236	−0.841182	−	0.540751i	$-0.818140\pi$
$631$	−42.2605	−1.68236	−0.841182	+	0.540751i	$0.818140\pi$
$632$	56.3088	2.23984
$633$	−18.5524	−0.737390
$634$	−33.8795	−1.34553
$635$	38.8541	1.54188
$636$	−13.3456	−0.529189
$637$	14.6060	0.578711
$638$	5.27623	0.208888
$639$	27.1668	1.07470
$640$	7.47565	0.295501
$641$	−11.7503	−0.464110	−0.232055	−	0.972703i	$-0.574545\pi$
$641$	−11.7503	−0.464110	−0.232055	+	0.972703i	$0.574545\pi$
$642$	36.9410	1.45794
$643$	5.29636	0.208868	0.104434	−	0.994532i	$-0.466697\pi$
$643$	5.29636	0.208868	0.104434	+	0.994532i	$0.466697\pi$
$644$	5.80809	0.228871
$645$	28.9343	1.13929
$646$	21.6622	0.852288
$647$	19.3799	0.761901	0.380950	−	0.924595i	$-0.375597\pi$
$647$	19.3799	0.761901	0.380950	+	0.924595i	$0.375597\pi$
$648$	−25.4464	−0.999628
$649$	1.76967	0.0694658
$650$	42.0848	1.65070
$651$	0.676974	0.0265327
$652$	−77.9499	−3.05276
$653$	−0.404616	−0.0158338	−0.00791692	−	0.999969i	$-0.502520\pi$
$653$	−0.404616	−0.0158338	−0.00791692	+	0.999969i	$0.502520\pi$
$654$	−20.1197	−0.786744
$655$	56.9478	2.22513
$656$	−7.66051	−0.299093
$657$	32.5131	1.26846
$658$	−11.2762	−0.439591
$659$	4.05852	0.158098	0.0790488	−	0.996871i	$-0.474812\pi$
$659$	4.05852	0.158098	0.0790488	+	0.996871i	$0.474812\pi$
$660$	5.40694	0.210465
$661$	−15.3101	−0.595495	−0.297748	−	0.954645i	$-0.596235\pi$
$661$	−15.3101	−0.595495	−0.297748	+	0.954645i	$0.596235\pi$
$662$	10.7710	0.418627
$663$	10.7061	0.415791
$664$	105.658	4.10034
$665$	2.53987	0.0984919
$666$	14.2882	0.553658
$667$	−11.9431	−0.462438
$668$	−42.9574	−1.66207
$669$	4.35424	0.168345
$670$	116.159	4.48760
$671$	−2.25422	−0.0870233
$672$	4.83565	0.186539
$673$	23.1065	0.890691	0.445345	−	0.895359i	$-0.353081\pi$
$673$	23.1065	0.890691	0.445345	+	0.895359i	$0.353081\pi$
$674$	36.3105	1.39863
$675$	−32.3489	−1.24511
$676$	−40.4064	−1.55409
$677$	−28.6067	−1.09945	−0.549723	−	0.835347i	$-0.685267\pi$
$677$	−28.6067	−1.09945	−0.549723	+	0.835347i	$0.685267\pi$
$678$	30.4977	1.17126
$679$	5.71360	0.219268
$680$	−159.262	−6.10741
$681$	7.23342	0.277185
$682$	−1.57882	−0.0604560
$683$	−16.3228	−0.624575	−0.312287	−	0.949988i	$-0.601095\pi$
$683$	−16.3228	−0.624575	−0.312287	+	0.949988i	$0.601095\pi$
$684$	−15.6168	−0.597124
$685$	−57.8800	−2.21148
$686$	−18.8528	−0.719803
$687$	−6.89668	−0.263125
$688$	−100.337	−3.82532
$689$	7.22585	0.275283
$690$	−17.2572	−0.656971
$691$	3.50045	0.133164	0.0665818	−	0.997781i	$-0.478791\pi$
$691$	3.50045	0.133164	0.0665818	+	0.997781i	$0.478791\pi$
$692$	−99.6588	−3.78846
$693$	−0.466127	−0.0177067
$694$	−38.0287	−1.44355
$695$	−63.2932	−2.40085
$696$	−32.6003	−1.23571
$697$	4.57582	0.173322
$698$	−74.8700	−2.83387
$699$	0.327071	0.0123710
$700$	−18.8775	−0.713504
$701$	20.7125	0.782299	0.391149	−	0.920327i	$-0.372078\pi$
$701$	20.7125	0.782299	0.391149	+	0.920327i	$0.372078\pi$
$702$	−24.9327	−0.941023
$703$	3.23012	0.121826
$704$	−3.58959	−0.135288
$705$	23.7614	0.894906
$706$	49.6746	1.86953
$707$	−6.03046	−0.226799
$708$	−18.5335	−0.696532
$709$	−25.5253	−0.958625	−0.479312	−	0.877644i	$-0.659114\pi$
$709$	−25.5253	−0.958625	−0.479312	+	0.877644i	$0.659114\pi$
$710$	−107.915	−4.04998
$711$	17.3383	0.650236
$712$	75.5791	2.83245
$713$	3.57375	0.133838
$714$	−6.77140	−0.253413
$715$	−2.92752	−0.109483
$716$	−0.885628	−0.0330975
$717$	−5.50044	−0.205418
$718$	9.66668	0.360757
$719$	15.0901	0.562765	0.281383	−	0.959596i	$-0.409207\pi$
$719$	15.0901	0.562765	0.281383	+	0.959596i	$0.409207\pi$
$720$	82.0975	3.05959
$721$	0.0456604	0.00170048
$722$	44.8503	1.66916
$723$	24.9905	0.929408
$724$	125.066	4.64803
$725$	38.8175	1.44165
$726$	23.4052	0.868648
$727$	−36.4981	−1.35364	−0.676820	−	0.736149i	$-0.736642\pi$
$727$	−36.4981	−1.35364	−0.676820	+	0.736149i	$0.736642\pi$
$728$	−8.58397	−0.318143
$729$	2.96407	0.109780
$730$	−129.152	−4.78013
$731$	59.9339	2.21674
$732$	23.6081	0.872581
$733$	−34.5797	−1.27723	−0.638615	−	0.769527i	$-0.720492\pi$
$733$	−34.5797	−1.27723	−0.638615	+	0.769527i	$0.720492\pi$
$734$	91.7514	3.38661
$735$	19.4665	0.718033
$736$	25.5274	0.940953
$737$	−4.81931	−0.177522
$738$	−4.65143	−0.171221
$739$	43.2679	1.59163	0.795817	−	0.605537i	$-0.207041\pi$
$739$	43.2679	1.59163	0.795817	+	0.605537i	$0.207041\pi$
$740$	−40.2527	−1.47972
$741$	−2.46031	−0.0903817
$742$	−4.57020	−0.167777
$743$	5.10456	0.187268	0.0936341	−	0.995607i	$-0.470152\pi$
$743$	5.10456	0.187268	0.0936341	+	0.995607i	$0.470152\pi$
$744$	9.75504	0.357637
$745$	40.3591	1.47864
$746$	14.9300	0.546625
$747$	32.5337	1.19035
$748$	11.1998	0.409506
$749$	8.97175	0.327820
$750$	18.1368	0.662261
$751$	−20.9708	−0.765235	−0.382617	−	0.923907i	$-0.624977\pi$
$751$	−20.9708	−0.765235	−0.382617	+	0.923907i	$0.624977\pi$
$752$	−82.3987	−3.00477
$753$	−24.6162	−0.897065
$754$	29.9183	1.08956
$755$	64.3478	2.34185
$756$	11.1838	0.406750
$757$	16.0986	0.585114	0.292557	−	0.956248i	$-0.405494\pi$
$757$	16.0986	0.585114	0.292557	+	0.956248i	$0.405494\pi$
$758$	−74.2150	−2.69561
$759$	0.715985	0.0259886
$760$	36.5990	1.32758
$761$	51.0608	1.85095	0.925477	−	0.378805i	$-0.123665\pi$
$761$	51.0608	1.85095	0.925477	+	0.378805i	$0.123665\pi$
$762$	23.8046	0.862351
$763$	−4.88642	−0.176900
$764$	0.506231	0.0183148
$765$	−49.0389	−1.77301
$766$	−99.8596	−3.60807
$767$	10.0348	0.362334
$768$	−10.8341	−0.390941
$769$	2.44775	0.0882680	0.0441340	−	0.999026i	$-0.485947\pi$
$769$	2.44775	0.0882680	0.0441340	+	0.999026i	$0.485947\pi$
$770$	1.85160	0.0667270
$771$	23.3488	0.840886
$772$	113.461	4.08354
$773$	−26.3228	−0.946764	−0.473382	−	0.880857i	$-0.656967\pi$
$773$	−26.3228	−0.946764	−0.473382	+	0.880857i	$0.656967\pi$
$774$	−60.9242	−2.18988
$775$	−11.6154	−0.417239
$776$	82.3316	2.95553
$777$	−1.00971	−0.0362230
$778$	25.8311	0.926090
$779$	−1.05154	−0.0376754
$780$	30.6595	1.09779
$781$	4.47729	0.160210
$782$	−35.7463	−1.27828
$783$	−22.9970	−0.821844
$784$	−67.5052	−2.41090
$785$	62.4636	2.22942
$786$	34.8900	1.24449
$787$	−2.09824	−0.0747943	−0.0373972	−	0.999300i	$-0.511907\pi$
$787$	−2.09824	−0.0747943	−0.0373972	+	0.999300i	$0.511907\pi$
$788$	86.4859	3.08093
$789$	−12.2846	−0.437344
$790$	−68.8730	−2.45039
$791$	7.40688	0.263358
$792$	−6.71679	−0.238671
$793$	−12.7823	−0.453914
$794$	−14.2982	−0.507424
$795$	9.63042	0.341556
$796$	29.5747	1.04825
$797$	26.6161	0.942790	0.471395	−	0.881922i	$-0.343751\pi$
$797$	26.6161	0.942790	0.471395	+	0.881922i	$0.343751\pi$
$798$	1.55609	0.0550852
$799$	49.2189	1.74124
$800$	−82.9695	−2.93342
$801$	23.2719	0.822271
$802$	−69.3443	−2.44863
$803$	5.35839	0.189093
$804$	50.4719	1.78001
$805$	−4.19121	−0.147721
$806$	−8.95251	−0.315339
$807$	16.8709	0.593884
$808$	−86.8976	−3.05705
$809$	−39.1844	−1.37765	−0.688825	−	0.724928i	$-0.741873\pi$
$809$	−39.1844	−1.37765	−0.688825	+	0.724928i	$0.741873\pi$
$810$	31.1242	1.09359
$811$	12.0104	0.421741	0.210870	−	0.977514i	$-0.432370\pi$
$811$	12.0104	0.421741	0.210870	+	0.977514i	$0.432370\pi$
$812$	−13.4201	−0.470954
$813$	5.33291	0.187033
$814$	2.35480	0.0825359
$815$	56.2499	1.97035
$816$	−49.4809	−1.73218
$817$	−13.7731	−0.481858
$818$	78.9251	2.75955
$819$	−2.64313	−0.0923583
$820$	13.1039	0.457610
$821$	21.3833	0.746283	0.373141	−	0.927775i	$-0.378281\pi$
$821$	21.3833	0.746283	0.373141	+	0.927775i	$0.378281\pi$
$822$	−35.4612	−1.23685
$823$	−14.3462	−0.500078	−0.250039	−	0.968236i	$-0.580443\pi$
$823$	−14.3462	−0.500078	−0.250039	+	0.968236i	$0.580443\pi$
$824$	0.657956	0.0229210
$825$	−2.32710	−0.0810193
$826$	−6.34678	−0.220833
$827$	−20.1558	−0.700884	−0.350442	−	0.936584i	$-0.613969\pi$
$827$	−20.1558	−0.700884	−0.350442	+	0.936584i	$0.613969\pi$
$828$	25.7704	0.895583
$829$	36.2162	1.25784	0.628920	−	0.777470i	$-0.283497\pi$
$829$	36.2162	1.25784	0.628920	+	0.777470i	$0.283497\pi$
$830$	−129.234	−4.48577
$831$	1.89151	0.0656159
$832$	−20.3544	−0.705662
$833$	40.3226	1.39709
$834$	−38.7777	−1.34276
$835$	30.9987	1.07276
$836$	−2.57376	−0.0890155
$837$	6.88143	0.237857
$838$	−41.6299	−1.43808
$839$	33.4735	1.15563	0.577817	−	0.816166i	$-0.303905\pi$
$839$	33.4735	1.15563	0.577817	+	0.816166i	$0.303905\pi$
$840$	−11.4405	−0.394734
$841$	−1.40445	−0.0484294
$842$	56.1544	1.93521
$843$	0.666746	0.0229640
$844$	−110.051	−3.78809
$845$	29.1579	1.00306
$846$	−50.0321	−1.72014
$847$	5.68435	0.195317
$848$	−33.3960	−1.14682
$849$	10.5446	0.361890
$850$	116.183	3.98504
$851$	−5.33025	−0.182718
$852$	−46.8899	−1.60642
$853$	−28.1398	−0.963490	−0.481745	−	0.876311i	$-0.659997\pi$
$853$	−28.1398	−0.963490	−0.481745	+	0.876311i	$0.659997\pi$
$854$	8.08457	0.276648
$855$	11.2693	0.385403
$856$	129.281	4.41873
$857$	−13.6265	−0.465473	−0.232737	−	0.972540i	$-0.574768\pi$
$857$	−13.6265	−0.465473	−0.232737	+	0.972540i	$0.574768\pi$
$858$	−1.79360	−0.0612324
$859$	−28.8454	−0.984194	−0.492097	−	0.870540i	$-0.663769\pi$
$859$	−28.8454	−0.984194	−0.492097	+	0.870540i	$0.663769\pi$
$860$	171.635	5.85270
$861$	0.328702	0.0112021
$862$	35.3878	1.20531
$863$	39.6610	1.35008	0.675039	−	0.737782i	$-0.264127\pi$
$863$	39.6610	1.35008	0.675039	+	0.737782i	$0.264127\pi$
$864$	49.1543	1.67226
$865$	71.9153	2.44519
$866$	−5.45529	−0.185378
$867$	15.5772	0.529030
$868$	4.01573	0.136303
$869$	2.85747	0.0969331
$870$	39.8744	1.35187
$871$	−27.3274	−0.925954
$872$	−70.4122	−2.38446
$873$	25.3511	0.858004
$874$	8.21463	0.277864
$875$	4.40483	0.148910
$876$	−56.1175	−1.89604
$877$	7.32920	0.247489	0.123745	−	0.992314i	$-0.460510\pi$
$877$	7.32920	0.247489	0.123745	+	0.992314i	$0.460510\pi$
$878$	83.8179	2.82872
$879$	−9.12737	−0.307859
$880$	13.5302	0.456105
$881$	21.0611	0.709567	0.354783	−	0.934949i	$-0.384555\pi$
$881$	21.0611	0.709567	0.354783	+	0.934949i	$0.384555\pi$
$882$	−40.9888	−1.38017
$883$	51.2514	1.72475	0.862373	−	0.506273i	$-0.168977\pi$
$883$	51.2514	1.72475	0.862373	+	0.506273i	$0.168977\pi$
$884$	63.5074	2.13599
$885$	13.3741	0.449564
$886$	−107.723	−3.61904
$887$	15.4332	0.518195	0.259097	−	0.965851i	$-0.416575\pi$
$887$	15.4332	0.518195	0.259097	+	0.965851i	$0.416575\pi$
$888$	−14.5496	−0.488254
$889$	5.78136	0.193901
$890$	−92.4430	−3.09870
$891$	−1.29131	−0.0432606
$892$	25.8289	0.864814
$893$	−11.3107	−0.378498
$894$	24.7267	0.826985
$895$	0.639082	0.0213622
$896$	1.11235	0.0371611
$897$	4.05992	0.135557
$898$	84.6351	2.82431
$899$	−8.25747	−0.275402
$900$	−83.7592	−2.79197
$901$	19.9483	0.664573
$902$	−0.766588	−0.0255246
$903$	4.30533	0.143272
$904$	106.731	3.54983
$905$	−90.2493	−2.99999
$906$	39.4238	1.30977
$907$	58.3730	1.93824	0.969121	−	0.246586i	$-0.0793086\pi$
$907$	58.3730	1.93824	0.969121	+	0.246586i	$0.0793086\pi$
$908$	42.9078	1.42395
$909$	−26.7570	−0.887474
$910$	10.4993	0.348049
$911$	16.8422	0.558008	0.279004	−	0.960290i	$-0.409996\pi$
$911$	16.8422	0.558008	0.279004	+	0.960290i	$0.409996\pi$
$912$	11.3709	0.376528
$913$	5.36178	0.177449
$914$	33.0823	1.09427
$915$	−17.0360	−0.563192
$916$	−40.9103	−1.35171
$917$	8.47364	0.279824
$918$	−68.8311	−2.27177
$919$	−56.2110	−1.85423	−0.927115	−	0.374777i	$-0.877719\pi$
$919$	−56.2110	−1.85423	−0.927115	+	0.374777i	$0.877719\pi$
$920$	−60.3944	−1.99114
$921$	8.24487	0.271678
$922$	71.8689	2.36688
$923$	25.3880	0.835656
$924$	0.804535	0.0264672
$925$	17.3244	0.569624
$926$	76.1294	2.50177
$927$	0.202594	0.00665406
$928$	−58.9834	−1.93623
$929$	4.83658	0.158683	0.0793416	−	0.996847i	$-0.474718\pi$
$929$	4.83658	0.158683	0.0793416	+	0.996847i	$0.474718\pi$
$930$	−11.9317	−0.391255
$931$	−9.26628	−0.303690
$932$	1.94015	0.0635517
$933$	1.46473	0.0479530
$934$	77.3169	2.52989
$935$	−8.08196	−0.264309
$936$	−38.0869	−1.24491
$937$	−21.9495	−0.717058	−0.358529	−	0.933519i	$-0.616722\pi$
$937$	−21.9495	−0.717058	−0.358529	+	0.933519i	$0.616722\pi$
$938$	17.2840	0.564344
$939$	15.7936	0.515406
$940$	140.950	4.59728
$941$	44.1240	1.43840	0.719201	−	0.694803i	$-0.244508\pi$
$941$	44.1240	1.43840	0.719201	+	0.694803i	$0.244508\pi$
$942$	38.2694	1.24688
$943$	1.73522	0.0565065
$944$	−46.3780	−1.50948
$945$	−8.07038	−0.262529
$946$	−10.0407	−0.326453
$947$	−13.3277	−0.433091	−0.216546	−	0.976272i	$-0.569479\pi$
$947$	−13.3277	−0.433091	−0.216546	+	0.976272i	$0.569479\pi$
$948$	−29.9258	−0.971946
$949$	30.3842	0.986313
$950$	−26.6993	−0.866239
$951$	10.6228	0.344469
$952$	−23.6976	−0.768044
$953$	52.6505	1.70552	0.852758	−	0.522307i	$-0.174928\pi$
$953$	52.6505	1.70552	0.852758	+	0.522307i	$0.174928\pi$
$954$	−20.2779	−0.656520
$955$	−0.365304	−0.0118210
$956$	−32.6280	−1.05527
$957$	−1.65435	−0.0534775
$958$	−51.3208	−1.65810
$959$	−8.61235	−0.278107
$960$	−27.1278	−0.875546
$961$	−28.5291	−0.920294
$962$	13.3527	0.430508
$963$	39.8074	1.28278
$964$	148.241	4.77452
$965$	−81.8750	−2.63565
$966$	−2.56782	−0.0826182
$967$	−10.2313	−0.329017	−0.164509	−	0.986376i	$-0.552604\pi$
$967$	−10.2313	−0.329017	−0.164509	+	0.986376i	$0.552604\pi$
$968$	81.9102	2.63269
$969$	−6.79213	−0.218195
$970$	−100.702	−3.23335
$971$	12.6983	0.407509	0.203754	−	0.979022i	$-0.434686\pi$
$971$	12.6983	0.407509	0.203754	+	0.979022i	$0.434686\pi$
$972$	77.5844	2.48852
$973$	−9.41782	−0.301921
$974$	29.9021	0.958125
$975$	−13.1956	−0.422597
$976$	59.0766	1.89100
$977$	4.61502	0.147647	0.0738237	−	0.997271i	$-0.476480\pi$
$977$	4.61502	0.147647	0.0738237	+	0.997271i	$0.476480\pi$
$978$	34.4624	1.10199
$979$	3.83537	0.122579
$980$	115.473	3.68865
$981$	−21.6809	−0.692219
$982$	−3.74779	−0.119597
$983$	2.57444	0.0821118	0.0410559	−	0.999157i	$-0.486928\pi$
$983$	2.57444	0.0821118	0.0410559	+	0.999157i	$0.486928\pi$
$984$	4.73652	0.150995
$985$	−62.4096	−1.98853
$986$	82.5950	2.63036
$987$	3.53562	0.112540
$988$	−14.5943	−0.464305
$989$	22.7278	0.722703
$990$	8.21550	0.261106
$991$	18.7305	0.594995	0.297497	−	0.954723i	$-0.403848\pi$
$991$	18.7305	0.594995	0.297497	+	0.954723i	$0.403848\pi$
$992$	17.6497	0.560379
$993$	−3.37722	−0.107173
$994$	−16.0574	−0.509310
$995$	−21.3415	−0.676573
$996$	−56.1531	−1.77928
$997$	−47.0016	−1.48856	−0.744278	−	0.667870i	$-0.767206\pi$
$997$	−47.0016	−1.48856	−0.744278	+	0.667870i	$0.767206\pi$
$998$	−101.999	−3.22874
$999$	−10.2636	−0.324727

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8039.2.a.b.1.18	✓	391

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8039.2.a.b.1.18	✓	391	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:	\( N \)	\(=\)	\( 8039 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8039.a (trivial)

\(f(q)\)	\(=\)	\(q-2.62255 q^{2} +0.822293 q^{3} +4.87774 q^{4} -3.51986 q^{5} -2.15650 q^{6} -0.523743 q^{7} -7.54702 q^{8} -2.32383 q^{9} +O(q^{10})\)	\(q-2.62255 q^{2} +0.822293 q^{3} +4.87774 q^{4} -3.51986 q^{5} -2.15650 q^{6} -0.523743 q^{7} -7.54702 q^{8} -2.32383 q^{9} +9.23098 q^{10} -0.382984 q^{11} +4.01093 q^{12} -2.17167 q^{13} +1.37354 q^{14} -2.89435 q^{15} +10.0369 q^{16} -5.99530 q^{17} +6.09436 q^{18} +1.37774 q^{19} -17.1690 q^{20} -0.430670 q^{21} +1.00439 q^{22} -2.27351 q^{23} -6.20586 q^{24} +7.38938 q^{25} +5.69531 q^{26} -4.37775 q^{27} -2.55468 q^{28} +5.25315 q^{29} +7.59057 q^{30} -1.57191 q^{31} -11.2282 q^{32} -0.314925 q^{33} +15.7230 q^{34} +1.84350 q^{35} -11.3351 q^{36} +2.34450 q^{37} -3.61320 q^{38} -1.78575 q^{39} +26.5644 q^{40} -0.763235 q^{41} +1.12945 q^{42} -9.99682 q^{43} -1.86810 q^{44} +8.17956 q^{45} +5.96238 q^{46} -8.20958 q^{47} +8.25327 q^{48} -6.72569 q^{49} -19.3790 q^{50} -4.92989 q^{51} -10.5929 q^{52} -3.32732 q^{53} +11.4808 q^{54} +1.34805 q^{55} +3.95270 q^{56} +1.13291 q^{57} -13.7766 q^{58} -4.62075 q^{59} -14.1179 q^{60} +5.88594 q^{61} +4.12240 q^{62} +1.21709 q^{63} +9.37267 q^{64} +7.64398 q^{65} +0.825906 q^{66} +12.5836 q^{67} -29.2436 q^{68} -1.86949 q^{69} -4.83466 q^{70} -11.6905 q^{71} +17.5380 q^{72} -13.9911 q^{73} -6.14857 q^{74} +6.07624 q^{75} +6.72028 q^{76} +0.200585 q^{77} +4.68321 q^{78} -7.46107 q^{79} -35.3285 q^{80} +3.37171 q^{81} +2.00162 q^{82} -14.0000 q^{83} -2.10070 q^{84} +21.1026 q^{85} +26.2171 q^{86} +4.31962 q^{87} +2.89039 q^{88} -10.0144 q^{89} -21.4513 q^{90} +1.13740 q^{91} -11.0896 q^{92} -1.29257 q^{93} +21.5300 q^{94} -4.84946 q^{95} -9.23287 q^{96} -10.9092 q^{97} +17.6384 q^{98} +0.889993 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9}+O(q^{10}) \) 391 * q + 14 * q^2 + 12 * q^3 + 446 * q^4 + 22 * q^5 + 40 * q^6 + 63 * q^7 + 36 * q^8 + 501 * q^9	\( 391 q + 14 q^{2} + 12 q^{3} + 446 q^{4} + 22 q^{5} + 40 q^{6} + 63 q^{7} + 36 q^{8} + 501 q^{9} + 40 q^{10} + 57 q^{11} + 20 q^{12} + 83 q^{13} + 21 q^{14} + 60 q^{15} + 548 q^{16} + 59 q^{17} + 54 q^{18} + 131 q^{19} + 35 q^{20} + 121 q^{21} + 89 q^{22} + 34 q^{23} + 110 q^{24} + 609 q^{25} + 54 q^{26} + 27 q^{27} + 182 q^{28} + 102 q^{29} + 92 q^{30} + 88 q^{31} + 76 q^{32} + 131 q^{33} + 128 q^{34} + 31 q^{35} + 654 q^{36} + 135 q^{37} + 23 q^{38} + 96 q^{39} + 113 q^{40} + 128 q^{41} + 45 q^{42} + 140 q^{43} + 151 q^{44} + 77 q^{45} + 245 q^{46} + 22 q^{47} + 25 q^{48} + 712 q^{49} + 53 q^{50} + 102 q^{51} + 174 q^{52} + 54 q^{53} + 131 q^{54} + 101 q^{55} + 43 q^{56} + 226 q^{57} + 109 q^{58} + 40 q^{59} + 123 q^{60} + 249 q^{61} + 28 q^{62} + 139 q^{63} + 730 q^{64} + 227 q^{65} + 55 q^{66} + 169 q^{67} + 48 q^{68} + 89 q^{69} + 98 q^{70} + 66 q^{71} + 120 q^{72} + 324 q^{73} + 60 q^{74} + 19 q^{75} + 356 q^{76} + 83 q^{77} - 11 q^{78} + 195 q^{79} + 26 q^{80} + 807 q^{81} + 49 q^{82} + 74 q^{83} + 252 q^{84} + 373 q^{85} + 100 q^{86} + 43 q^{87} + 211 q^{88} + 207 q^{89} + 10 q^{90} + 189 q^{91} + 30 q^{92} + 172 q^{93} + 130 q^{94} + 43 q^{95} + 203 q^{96} + 254 q^{97} + 26 q^{98} + 273 q^{99}+O(q^{100}) \) 391 * q + 14 * q^2 + 12 * q^3 + 446 * q^4 + 22 * q^5 + 40 * q^6 + 63 * q^7 + 36 * q^8 + 501 * q^9 + 40 * q^10 + 57 * q^11 + 20 * q^12 + 83 * q^13 + 21 * q^14 + 60 * q^15 + 548 * q^16 + 59 * q^17 + 54 * q^18 + 131 * q^19 + 35 * q^20 + 121 * q^21 + 89 * q^22 + 34 * q^23 + 110 * q^24 + 609 * q^25 + 54 * q^26 + 27 * q^27 + 182 * q^28 + 102 * q^29 + 92 * q^30 + 88 * q^31 + 76 * q^32 + 131 * q^33 + 128 * q^34 + 31 * q^35 + 654 * q^36 + 135 * q^37 + 23 * q^38 + 96 * q^39 + 113 * q^40 + 128 * q^41 + 45 * q^42 + 140 * q^43 + 151 * q^44 + 77 * q^45 + 245 * q^46 + 22 * q^47 + 25 * q^48 + 712 * q^49 + 53 * q^50 + 102 * q^51 + 174 * q^52 + 54 * q^53 + 131 * q^54 + 101 * q^55 + 43 * q^56 + 226 * q^57 + 109 * q^58 + 40 * q^59 + 123 * q^60 + 249 * q^61 + 28 * q^62 + 139 * q^63 + 730 * q^64 + 227 * q^65 + 55 * q^66 + 169 * q^67 + 48 * q^68 + 89 * q^69 + 98 * q^70 + 66 * q^71 + 120 * q^72 + 324 * q^73 + 60 * q^74 + 19 * q^75 + 356 * q^76 + 83 * q^77 - 11 * q^78 + 195 * q^79 + 26 * q^80 + 807 * q^81 + 49 * q^82 + 74 * q^83 + 252 * q^84 + 373 * q^85 + 100 * q^86 + 43 * q^87 + 211 * q^88 + 207 * q^89 + 10 * q^90 + 189 * q^91 + 30 * q^92 + 172 * q^93 + 130 * q^94 + 43 * q^95 + 203 * q^96 + 254 * q^97 + 26 * q^98 + 273 * q^99

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