LMFDB - Embedded newform 8042.2.a.c.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, 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("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.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:	$64.2156933055$
Analytic rank:	$0$
Dimension:	$86$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.10336 q^{3} +1.00000 q^{4} -2.86600 q^{5} +2.10336 q^{6} +3.42965 q^{7} -1.00000 q^{8} +1.42413 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.10336 q^{3} +1.00000 q^{4} -2.86600 q^{5} +2.10336 q^{6} +3.42965 q^{7} -1.00000 q^{8} +1.42413 q^{9} +2.86600 q^{10} -1.51191 q^{11} -2.10336 q^{12} -1.17483 q^{13} -3.42965 q^{14} +6.02824 q^{15} +1.00000 q^{16} -7.28415 q^{17} -1.42413 q^{18} -3.37924 q^{19} -2.86600 q^{20} -7.21380 q^{21} +1.51191 q^{22} -2.46602 q^{23} +2.10336 q^{24} +3.21397 q^{25} +1.17483 q^{26} +3.31463 q^{27} +3.42965 q^{28} +4.84908 q^{29} -6.02824 q^{30} -8.53296 q^{31} -1.00000 q^{32} +3.18010 q^{33} +7.28415 q^{34} -9.82939 q^{35} +1.42413 q^{36} -3.32091 q^{37} +3.37924 q^{38} +2.47109 q^{39} +2.86600 q^{40} -9.68874 q^{41} +7.21380 q^{42} -1.96662 q^{43} -1.51191 q^{44} -4.08155 q^{45} +2.46602 q^{46} -0.645154 q^{47} -2.10336 q^{48} +4.76251 q^{49} -3.21397 q^{50} +15.3212 q^{51} -1.17483 q^{52} -9.65253 q^{53} -3.31463 q^{54} +4.33315 q^{55} -3.42965 q^{56} +7.10777 q^{57} -4.84908 q^{58} -0.822649 q^{59} +6.02824 q^{60} +10.3443 q^{61} +8.53296 q^{62} +4.88426 q^{63} +1.00000 q^{64} +3.36706 q^{65} -3.18010 q^{66} -5.13750 q^{67} -7.28415 q^{68} +5.18693 q^{69} +9.82939 q^{70} -10.7203 q^{71} -1.42413 q^{72} +6.12864 q^{73} +3.32091 q^{74} -6.76013 q^{75} -3.37924 q^{76} -5.18534 q^{77} -2.47109 q^{78} +15.3907 q^{79} -2.86600 q^{80} -11.2442 q^{81} +9.68874 q^{82} -6.42446 q^{83} -7.21380 q^{84} +20.8764 q^{85} +1.96662 q^{86} -10.1994 q^{87} +1.51191 q^{88} -12.7659 q^{89} +4.08155 q^{90} -4.02925 q^{91} -2.46602 q^{92} +17.9479 q^{93} +0.645154 q^{94} +9.68491 q^{95} +2.10336 q^{96} +3.11750 q^{97} -4.76251 q^{98} -2.15316 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) $ 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9 + 4 * q^10 + 13 * q^11 + 12 * q^12 + 45 * q^13 - 35 * q^14 + 17 * q^15 + 86 * q^16 + 5 * q^17 - 72 * q^18 + 47 * q^19 - 4 * q^20 + 15 * q^21 - 13 * q^22 + 6 * q^23 - 12 * q^24 + 112 * q^25 - 45 * q^26 + 51 * q^27 + 35 * q^28 - 14 * q^29 - 17 * q^30 + 24 * q^31 - 86 * q^32 + 43 * q^33 - 5 * q^34 + 42 * q^35 + 72 * q^36 + 61 * q^37 - 47 * q^38 + 20 * q^39 + 4 * q^40 - 16 * q^41 - 15 * q^42 + 72 * q^43 + 13 * q^44 + 6 * q^45 - 6 * q^46 + 11 * q^47 + 12 * q^48 + 89 * q^49 - 112 * q^50 + 56 * q^51 + 45 * q^52 - 7 * q^53 - 51 * q^54 + 48 * q^55 - 35 * q^56 + 65 * q^57 + 14 * q^58 + 24 * q^59 + 17 * q^60 + 31 * q^61 - 24 * q^62 + 98 * q^63 + 86 * q^64 - 9 * q^65 - 43 * q^66 + 157 * q^67 + 5 * q^68 + q^69 - 42 * q^70 - 11 * q^71 - 72 * q^72 + 74 * q^73 - 61 * q^74 + 76 * q^75 + 47 * q^76 - 13 * q^77 - 20 * q^78 + 57 * q^79 - 4 * q^80 + 34 * q^81 + 16 * q^82 + 65 * q^83 + 15 * q^84 + 102 * q^85 - 72 * q^86 + 49 * q^87 - 13 * q^88 - 34 * q^89 - 6 * q^90 + 91 * q^91 + 6 * q^92 + 57 * q^93 - 11 * q^94 - 13 * q^95 - 12 * q^96 + 64 * q^97 - 89 * q^98 + 56 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.10336	−1.21438	−0.607188	−	0.794558i	$-0.707703\pi$
$3$	−2.10336	−1.21438	−0.607188	+	0.794558i	$0.707703\pi$
$4$	1.00000	0.500000
$5$	−2.86600	−1.28172	−0.640858	−	0.767660i	$-0.721421\pi$
$5$	−2.86600	−1.28172	−0.640858	+	0.767660i	$0.721421\pi$
$6$	2.10336	0.858694
$7$	3.42965	1.29629	0.648143	−	0.761518i	$-0.275546\pi$
$7$	3.42965	1.29629	0.648143	+	0.761518i	$0.275546\pi$
$8$	−1.00000	−0.353553
$9$	1.42413	0.474709
$10$	2.86600	0.906309
$11$	−1.51191	−0.455859	−0.227930	−	0.973678i	$-0.573196\pi$
$11$	−1.51191	−0.455859	−0.227930	+	0.973678i	$0.573196\pi$
$12$	−2.10336	−0.607188
$13$	−1.17483	−0.325839	−0.162919	−	0.986639i	$-0.552091\pi$
$13$	−1.17483	−0.325839	−0.162919	+	0.986639i	$0.552091\pi$
$14$	−3.42965	−0.916613
$15$	6.02824	1.55648
$16$	1.00000	0.250000
$17$	−7.28415	−1.76667	−0.883333	−	0.468745i	$-0.844706\pi$
$17$	−7.28415	−1.76667	−0.883333	+	0.468745i	$0.844706\pi$
$18$	−1.42413	−0.335670
$19$	−3.37924	−0.775251	−0.387626	−	0.921817i	$-0.626705\pi$
$19$	−3.37924	−0.775251	−0.387626	+	0.921817i	$0.626705\pi$
$20$	−2.86600	−0.640858
$21$	−7.21380	−1.57418
$22$	1.51191	0.322341
$23$	−2.46602	−0.514200	−0.257100	−	0.966385i	$-0.582767\pi$
$23$	−2.46602	−0.514200	−0.257100	+	0.966385i	$0.582767\pi$
$24$	2.10336	0.429347
$25$	3.21397	0.642793
$26$	1.17483	0.230403
$27$	3.31463	0.637900
$28$	3.42965	0.648143
$29$	4.84908	0.900451	0.450226	−	0.892915i	$-0.351344\pi$
$29$	4.84908	0.900451	0.450226	+	0.892915i	$0.351344\pi$
$30$	−6.02824	−1.10060
$31$	−8.53296	−1.53257	−0.766283	−	0.642503i	$-0.777896\pi$
$31$	−8.53296	−1.53257	−0.766283	+	0.642503i	$0.777896\pi$
$32$	−1.00000	−0.176777
$33$	3.18010	0.553585
$34$	7.28415	1.24922
$35$	−9.82939	−1.66147
$36$	1.42413	0.237355
$37$	−3.32091	−0.545954	−0.272977	−	0.962021i	$-0.588008\pi$
$37$	−3.32091	−0.545954	−0.272977	+	0.962021i	$0.588008\pi$
$38$	3.37924	0.548185
$39$	2.47109	0.395691
$40$	2.86600	0.453155
$41$	−9.68874	−1.51313	−0.756563	−	0.653921i	$-0.773123\pi$
$41$	−9.68874	−1.51313	−0.756563	+	0.653921i	$0.773123\pi$
$42$	7.21380	1.11311
$43$	−1.96662	−0.299907	−0.149954	−	0.988693i	$-0.547912\pi$
$43$	−1.96662	−0.299907	−0.149954	+	0.988693i	$0.547912\pi$
$44$	−1.51191	−0.227930
$45$	−4.08155	−0.608442
$46$	2.46602	0.363595
$47$	−0.645154	−0.0941054	−0.0470527	−	0.998892i	$-0.514983\pi$
$47$	−0.645154	−0.0941054	−0.0470527	+	0.998892i	$0.514983\pi$
$48$	−2.10336	−0.303594
$49$	4.76251	0.680359
$50$	−3.21397	−0.454524
$51$	15.3212	2.14540
$52$	−1.17483	−0.162919
$53$	−9.65253	−1.32588	−0.662938	−	0.748674i	$-0.730691\pi$
$53$	−9.65253	−1.32588	−0.662938	+	0.748674i	$0.730691\pi$
$54$	−3.31463	−0.451064
$55$	4.33315	0.584282
$56$	−3.42965	−0.458307
$57$	7.10777	0.941447
$58$	−4.84908	−0.636715
$59$	−0.822649	−0.107100	−0.0535499	−	0.998565i	$-0.517054\pi$
$59$	−0.822649	−0.107100	−0.0535499	+	0.998565i	$0.517054\pi$
$60$	6.02824	0.778242
$61$	10.3443	1.32445	0.662223	−	0.749307i	$-0.269613\pi$
$61$	10.3443	1.32445	0.662223	+	0.749307i	$0.269613\pi$
$62$	8.53296	1.08369
$63$	4.88426	0.615359
$64$	1.00000	0.125000
$65$	3.36706	0.417632
$66$	−3.18010	−0.391443
$67$	−5.13750	−0.627645	−0.313823	−	0.949482i	$-0.601610\pi$
$67$	−5.13750	−0.627645	−0.313823	+	0.949482i	$0.601610\pi$
$68$	−7.28415	−0.883333
$69$	5.18693	0.624433
$70$	9.82939	1.17484
$71$	−10.7203	−1.27226	−0.636132	−	0.771581i	$-0.719466\pi$
$71$	−10.7203	−1.27226	−0.636132	+	0.771581i	$0.719466\pi$
$72$	−1.42413	−0.167835
$73$	6.12864	0.717302	0.358651	−	0.933472i	$-0.383237\pi$
$73$	6.12864	0.717302	0.358651	+	0.933472i	$0.383237\pi$
$74$	3.32091	0.386048
$75$	−6.76013	−0.780593
$76$	−3.37924	−0.387626
$77$	−5.18534	−0.590924
$78$	−2.47109	−0.279795
$79$	15.3907	1.73159	0.865794	−	0.500400i	$-0.166814\pi$
$79$	15.3907	1.73159	0.865794	+	0.500400i	$0.166814\pi$
$80$	−2.86600	−0.320429
$81$	−11.2442	−1.24936
$82$	9.68874	1.06994
$83$	−6.42446	−0.705176	−0.352588	−	0.935779i	$-0.614698\pi$
$83$	−6.42446	−0.705176	−0.352588	+	0.935779i	$0.614698\pi$
$84$	−7.21380	−0.787090
$85$	20.8764	2.26436
$86$	1.96662	0.212066
$87$	−10.1994	−1.09349
$88$	1.51191	0.161171
$89$	−12.7659	−1.35318	−0.676591	−	0.736359i	$-0.736544\pi$
$89$	−12.7659	−1.35318	−0.676591	+	0.736359i	$0.736544\pi$
$90$	4.08155	0.430234
$91$	−4.02925	−0.422380
$92$	−2.46602	−0.257100
$93$	17.9479	1.86111
$94$	0.645154	0.0665425
$95$	9.68491	0.993651
$96$	2.10336	0.214673
$97$	3.11750	0.316534	0.158267	−	0.987396i	$-0.449409\pi$
$97$	3.11750	0.316534	0.158267	+	0.987396i	$0.449409\pi$
$98$	−4.76251	−0.481086
$99$	−2.15316	−0.216401
$100$	3.21397	0.321397
$101$	−12.3350	−1.22737	−0.613687	−	0.789549i	$-0.710314\pi$
$101$	−12.3350	−1.22737	−0.613687	+	0.789549i	$0.710314\pi$
$102$	−15.3212	−1.51703
$103$	−7.97045	−0.785352	−0.392676	−	0.919677i	$-0.628451\pi$
$103$	−7.97045	−0.785352	−0.392676	+	0.919677i	$0.628451\pi$
$104$	1.17483	0.115201
$105$	20.6748	2.01765
$106$	9.65253	0.937537
$107$	0.791050	0.0764737	0.0382369	−	0.999269i	$-0.487826\pi$
$107$	0.791050	0.0764737	0.0382369	+	0.999269i	$0.487826\pi$
$108$	3.31463	0.318950
$109$	−1.22575	−0.117406	−0.0587029	−	0.998275i	$-0.518696\pi$
$109$	−1.22575	−0.117406	−0.0587029	+	0.998275i	$0.518696\pi$
$110$	−4.33315	−0.413150
$111$	6.98507	0.662994
$112$	3.42965	0.324072
$113$	12.5726	1.18273	0.591363	−	0.806405i	$-0.298590\pi$
$113$	12.5726	1.18273	0.591363	+	0.806405i	$0.298590\pi$
$114$	−7.10777	−0.665703
$115$	7.06761	0.659058
$116$	4.84908	0.450226
$117$	−1.67311	−0.154679
$118$	0.822649	0.0757310
$119$	−24.9821	−2.29011
$120$	−6.02824	−0.550300
$121$	−8.71412	−0.792192
$122$	−10.3443	−0.936525
$123$	20.3789	1.83750
$124$	−8.53296	−0.766283
$125$	5.11877	0.457837
$126$	−4.88426	−0.435125
$127$	3.10468	0.275496	0.137748	−	0.990467i	$-0.456014\pi$
$127$	3.10468	0.275496	0.137748	+	0.990467i	$0.456014\pi$
$128$	−1.00000	−0.0883883
$129$	4.13652	0.364200
$130$	−3.36706	−0.295311
$131$	0.692149	0.0604734	0.0302367	−	0.999543i	$-0.490374\pi$
$131$	0.692149	0.0604734	0.0302367	+	0.999543i	$0.490374\pi$
$132$	3.18010	0.276792
$133$	−11.5896	−1.00495
$134$	5.13750	0.443812
$135$	−9.49973	−0.817606
$136$	7.28415	0.624611
$137$	5.61252	0.479510	0.239755	−	0.970833i	$-0.422933\pi$
$137$	5.61252	0.479510	0.239755	+	0.970833i	$0.422933\pi$
$138$	−5.18693	−0.441540
$139$	−3.42114	−0.290177	−0.145089	−	0.989419i	$-0.546347\pi$
$139$	−3.42114	−0.290177	−0.145089	+	0.989419i	$0.546347\pi$
$140$	−9.82939	−0.830735
$141$	1.35699	0.114279
$142$	10.7203	0.899626
$143$	1.77624	0.148537
$144$	1.42413	0.118677
$145$	−13.8975	−1.15412
$146$	−6.12864	−0.507209
$147$	−10.0173	−0.826212
$148$	−3.32091	−0.272977
$149$	17.9357	1.46935	0.734677	−	0.678417i	$-0.237334\pi$
$149$	17.9357	1.46935	0.734677	+	0.678417i	$0.237334\pi$
$150$	6.76013	0.551963
$151$	−13.4578	−1.09518	−0.547591	−	0.836746i	$-0.684455\pi$
$151$	−13.4578	−1.09518	−0.547591	+	0.836746i	$0.684455\pi$
$152$	3.37924	0.274093
$153$	−10.3736	−0.838653
$154$	5.18534	0.417847
$155$	24.4555	1.96431
$156$	2.47109	0.197845
$157$	6.87183	0.548432	0.274216	−	0.961668i	$-0.411582\pi$
$157$	6.87183	0.548432	0.274216	+	0.961668i	$0.411582\pi$
$158$	−15.3907	−1.22442
$159$	20.3028	1.61011
$160$	2.86600	0.226577
$161$	−8.45758	−0.666551
$162$	11.2442	0.883431
$163$	−24.4552	−1.91548	−0.957740	−	0.287635i	$-0.907131\pi$
$163$	−24.4552	−1.91548	−0.957740	+	0.287635i	$0.907131\pi$
$164$	−9.68874	−0.756563
$165$	−9.11418	−0.709538
$166$	6.42446	0.498635
$167$	−13.8409	−1.07104	−0.535522	−	0.844521i	$-0.679885\pi$
$167$	−13.8409	−1.07104	−0.535522	+	0.844521i	$0.679885\pi$
$168$	7.21380	0.556557
$169$	−11.6198	−0.893829
$170$	−20.8764	−1.60115
$171$	−4.81247	−0.368019
$172$	−1.96662	−0.149954
$173$	−7.18295	−0.546110	−0.273055	−	0.961998i	$-0.588034\pi$
$173$	−7.18295	−0.546110	−0.273055	+	0.961998i	$0.588034\pi$
$174$	10.1994	0.773212
$175$	11.0228	0.833245
$176$	−1.51191	−0.113965
$177$	1.73033	0.130059
$178$	12.7659	0.956845
$179$	−2.51685	−0.188118	−0.0940591	−	0.995567i	$-0.529984\pi$
$179$	−2.51685	−0.188118	−0.0940591	+	0.995567i	$0.529984\pi$
$180$	−4.08155	−0.304221
$181$	8.28994	0.616186	0.308093	−	0.951356i	$-0.400309\pi$
$181$	8.28994	0.616186	0.308093	+	0.951356i	$0.400309\pi$
$182$	4.02925	0.298668
$183$	−21.7577	−1.60838
$184$	2.46602	0.181797
$185$	9.51773	0.699758
$186$	−17.9479	−1.31600
$187$	11.0130	0.805351
$188$	−0.645154	−0.0470527
$189$	11.3680	0.826902
$190$	−9.68491	−0.702617
$191$	−12.2518	−0.886512	−0.443256	−	0.896395i	$-0.646177\pi$
$191$	−12.2518	−0.886512	−0.443256	+	0.896395i	$0.646177\pi$
$192$	−2.10336	−0.151797
$193$	4.61562	0.332239	0.166120	−	0.986106i	$-0.446876\pi$
$193$	4.61562	0.332239	0.166120	+	0.986106i	$0.446876\pi$
$194$	−3.11750	−0.223824
$195$	−7.08214	−0.507163
$196$	4.76251	0.340180
$197$	−12.3012	−0.876426	−0.438213	−	0.898871i	$-0.644388\pi$
$197$	−12.3012	−0.876426	−0.438213	+	0.898871i	$0.644388\pi$
$198$	2.15316	0.153018
$199$	−18.0187	−1.27731	−0.638655	−	0.769493i	$-0.720509\pi$
$199$	−18.0187	−1.27731	−0.638655	+	0.769493i	$0.720509\pi$
$200$	−3.21397	−0.227262
$201$	10.8060	0.762197
$202$	12.3350	0.867885
$203$	16.6307	1.16724
$204$	15.3212	1.07270
$205$	27.7679	1.93940
$206$	7.97045	0.555328
$207$	−3.51193	−0.244096
$208$	−1.17483	−0.0814596
$209$	5.10912	0.353405
$210$	−20.6748	−1.42669
$211$	−27.1505	−1.86911	−0.934557	−	0.355813i	$-0.884204\pi$
$211$	−27.1505	−1.86911	−0.934557	+	0.355813i	$0.884204\pi$
$212$	−9.65253	−0.662938
$213$	22.5486	1.54501
$214$	−0.791050	−0.0540751
$215$	5.63634	0.384396
$216$	−3.31463	−0.225532
$217$	−29.2651	−1.98664
$218$	1.22575	0.0830184
$219$	−12.8907	−0.871075
$220$	4.33315	0.292141
$221$	8.55762	0.575648
$222$	−6.98507	−0.468807
$223$	−2.51162	−0.168190	−0.0840952	−	0.996458i	$-0.526800\pi$
$223$	−2.51162	−0.168190	−0.0840952	+	0.996458i	$0.526800\pi$
$224$	−3.42965	−0.229153
$225$	4.57710	0.305140
$226$	−12.5726	−0.836314
$227$	22.5312	1.49545	0.747723	−	0.664011i	$-0.231147\pi$
$227$	22.5312	1.49545	0.747723	+	0.664011i	$0.231147\pi$
$228$	7.10777	0.470723
$229$	−18.3882	−1.21512	−0.607562	−	0.794273i	$-0.707852\pi$
$229$	−18.3882	−1.21512	−0.607562	+	0.794273i	$0.707852\pi$
$230$	−7.06761	−0.466025
$231$	10.9066	0.717604
$232$	−4.84908	−0.318358
$233$	−8.43449	−0.552561	−0.276281	−	0.961077i	$-0.589102\pi$
$233$	−8.43449	−0.552561	−0.276281	+	0.961077i	$0.589102\pi$
$234$	1.67311	0.109374
$235$	1.84901	0.120616
$236$	−0.822649	−0.0535499
$237$	−32.3722	−2.10280
$238$	24.9821	1.61935
$239$	19.8057	1.28112	0.640561	−	0.767907i	$-0.278702\pi$
$239$	19.8057	1.28112	0.640561	+	0.767907i	$0.278702\pi$
$240$	6.02824	0.389121
$241$	−28.3492	−1.82613	−0.913066	−	0.407811i	$-0.866292\pi$
$241$	−28.3492	−1.82613	−0.913066	+	0.407811i	$0.866292\pi$
$242$	8.71412	0.560165
$243$	13.7068	0.879293
$244$	10.3443	0.662223
$245$	−13.6494	−0.872026
$246$	−20.3789	−1.29931
$247$	3.97003	0.252607
$248$	8.53296	0.541844
$249$	13.5130	0.856349
$250$	−5.11877	−0.323740
$251$	−21.0654	−1.32964	−0.664818	−	0.747006i	$-0.731491\pi$
$251$	−21.0654	−1.32964	−0.664818	+	0.747006i	$0.731491\pi$
$252$	4.88426	0.307680
$253$	3.72841	0.234403
$254$	−3.10468	−0.194805
$255$	−43.9106	−2.74979
$256$	1.00000	0.0625000
$257$	−6.93510	−0.432600	−0.216300	−	0.976327i	$-0.569399\pi$
$257$	−6.93510	−0.432600	−0.216300	+	0.976327i	$0.569399\pi$
$258$	−4.13652	−0.257528
$259$	−11.3896	−0.707713
$260$	3.36706	0.208816
$261$	6.90571	0.427453
$262$	−0.692149	−0.0427611
$263$	−30.6371	−1.88916	−0.944581	−	0.328278i	$-0.893532\pi$
$263$	−30.6371	−1.88916	−0.944581	+	0.328278i	$0.893532\pi$
$264$	−3.18010	−0.195722
$265$	27.6642	1.69940
$266$	11.5896	0.710605
$267$	26.8513	1.64327
$268$	−5.13750	−0.313823
$269$	−5.70907	−0.348088	−0.174044	−	0.984738i	$-0.555684\pi$
$269$	−5.70907	−0.348088	−0.174044	+	0.984738i	$0.555684\pi$
$270$	9.49973	0.578135
$271$	31.4036	1.90763	0.953817	−	0.300388i	$-0.0971161\pi$
$271$	31.4036	1.90763	0.953817	+	0.300388i	$0.0971161\pi$
$272$	−7.28415	−0.441667
$273$	8.47497	0.512928
$274$	−5.61252	−0.339065
$275$	−4.85924	−0.293023
$276$	5.18693	0.312216
$277$	4.12553	0.247879	0.123940	−	0.992290i	$-0.460447\pi$
$277$	4.12553	0.247879	0.123940	+	0.992290i	$0.460447\pi$
$278$	3.42114	0.205186
$279$	−12.1520	−0.727523
$280$	9.82939	0.587418
$281$	−23.9809	−1.43058	−0.715290	−	0.698828i	$-0.753705\pi$
$281$	−23.9809	−1.43058	−0.715290	+	0.698828i	$0.753705\pi$
$282$	−1.35699	−0.0808077
$283$	−14.1399	−0.840531	−0.420265	−	0.907401i	$-0.638063\pi$
$283$	−14.1399	−0.840531	−0.420265	+	0.907401i	$0.638063\pi$
$284$	−10.7203	−0.636132
$285$	−20.3709	−1.20667
$286$	−1.77624	−0.105031
$287$	−33.2290	−1.96145
$288$	−1.42413	−0.0839176
$289$	36.0589	2.12111
$290$	13.8975	0.816087
$291$	−6.55723	−0.384392
$292$	6.12864	0.358651
$293$	8.61165	0.503098	0.251549	−	0.967845i	$-0.419060\pi$
$293$	8.61165	0.503098	0.251549	+	0.967845i	$0.419060\pi$
$294$	10.0173	0.584220
$295$	2.35771	0.137271
$296$	3.32091	0.193024
$297$	−5.01143	−0.290793
$298$	−17.9357	−1.03899
$299$	2.89715	0.167546
$300$	−6.76013	−0.390297
$301$	−6.74483	−0.388766
$302$	13.4578	0.774411
$303$	25.9449	1.49049
$304$	−3.37924	−0.193813
$305$	−29.6466	−1.69756
$306$	10.3736	0.593017
$307$	−10.2691	−0.586091	−0.293045	−	0.956099i	$-0.594669\pi$
$307$	−10.2691	−0.586091	−0.293045	+	0.956099i	$0.594669\pi$
$308$	−5.18534	−0.295462
$309$	16.7647	0.953713
$310$	−24.4555	−1.38898
$311$	−3.65819	−0.207437	−0.103719	−	0.994607i	$-0.533074\pi$
$311$	−3.65819	−0.207437	−0.103719	+	0.994607i	$0.533074\pi$
$312$	−2.47109	−0.139898
$313$	−13.5030	−0.763235	−0.381617	−	0.924320i	$-0.624633\pi$
$313$	−13.5030	−0.763235	−0.381617	+	0.924320i	$0.624633\pi$
$314$	−6.87183	−0.387800
$315$	−13.9983	−0.788715
$316$	15.3907	0.865794
$317$	0.0257917	0.00144860	0.000724302	−	1.00000i	$-0.499769\pi$
$317$	0.0257917	0.00144860	0.000724302		1.00000i	$0.499769\pi$
$318$	−20.3028	−1.13852
$319$	−7.33139	−0.410479
$320$	−2.86600	−0.160214
$321$	−1.66386	−0.0928679
$322$	8.45758	0.471323
$323$	24.6149	1.36961
$324$	−11.2442	−0.624680
$325$	−3.77586	−0.209447
$326$	24.4552	1.35445
$327$	2.57820	0.142575
$328$	9.68874	0.534971
$329$	−2.21265	−0.121988
$330$	9.11418	0.501719
$331$	14.0491	0.772210	0.386105	−	0.922455i	$-0.373820\pi$
$331$	14.0491	0.772210	0.386105	+	0.922455i	$0.373820\pi$
$332$	−6.42446	−0.352588
$333$	−4.72940	−0.259170
$334$	13.8409	0.757343
$335$	14.7241	0.804462
$336$	−7.21380	−0.393545
$337$	24.2966	1.32352	0.661761	−	0.749715i	$-0.269809\pi$
$337$	24.2966	1.32352	0.661761	+	0.749715i	$0.269809\pi$
$338$	11.6198	0.632033
$339$	−26.4446	−1.43628
$340$	20.8764	1.13218
$341$	12.9011	0.698634
$342$	4.81247	0.260229
$343$	−7.67380	−0.414346
$344$	1.96662	0.106033
$345$	−14.8657	−0.800345
$346$	7.18295	0.386158
$347$	0.948564	0.0509216	0.0254608	−	0.999676i	$-0.491895\pi$
$347$	0.948564	0.0509216	0.0254608	+	0.999676i	$0.491895\pi$
$348$	−10.1994	−0.546743
$349$	−10.6983	−0.572665	−0.286333	−	0.958130i	$-0.592436\pi$
$349$	−10.6983	−0.572665	−0.286333	+	0.958130i	$0.592436\pi$
$350$	−11.0228	−0.589193
$351$	−3.89412	−0.207853
$352$	1.51191	0.0805853
$353$	−10.1166	−0.538453	−0.269227	−	0.963077i	$-0.586768\pi$
$353$	−10.1166	−0.538453	−0.269227	+	0.963077i	$0.586768\pi$
$354$	−1.73033	−0.0919659
$355$	30.7243	1.63068
$356$	−12.7659	−0.676591
$357$	52.5464	2.78105
$358$	2.51685	0.133020
$359$	0.328326	0.0173284	0.00866420	−	0.999962i	$-0.497242\pi$
$359$	0.328326	0.0173284	0.00866420	+	0.999962i	$0.497242\pi$
$360$	4.08155	0.215117
$361$	−7.58073	−0.398986
$362$	−8.28994	−0.435710
$363$	18.3289	0.962019
$364$	−4.02925	−0.211190
$365$	−17.5647	−0.919377
$366$	21.7577	1.13729
$367$	15.4184	0.804835	0.402417	−	0.915456i	$-0.368170\pi$
$367$	15.4184	0.804835	0.402417	+	0.915456i	$0.368170\pi$
$368$	−2.46602	−0.128550
$369$	−13.7980	−0.718295
$370$	−9.51773	−0.494803
$371$	−33.1048	−1.71872
$372$	17.9479	0.930555
$373$	26.7105	1.38302	0.691509	−	0.722368i	$-0.256946\pi$
$373$	26.7105	1.38302	0.691509	+	0.722368i	$0.256946\pi$
$374$	−11.0130	−0.569469
$375$	−10.7666	−0.555986
$376$	0.645154	0.0332713
$377$	−5.69683	−0.293402
$378$	−11.3680	−0.584708
$379$	30.8165	1.58294	0.791469	−	0.611209i	$-0.209317\pi$
$379$	30.8165	1.58294	0.791469	+	0.611209i	$0.209317\pi$
$380$	9.68491	0.496826
$381$	−6.53027	−0.334556
$382$	12.2518	0.626858
$383$	36.1044	1.84485	0.922425	−	0.386177i	$-0.126205\pi$
$383$	36.1044	1.84485	0.922425	+	0.386177i	$0.126205\pi$
$384$	2.10336	0.107337
$385$	14.8612	0.757397
$386$	−4.61562	−0.234929
$387$	−2.80072	−0.142369
$388$	3.11750	0.158267
$389$	30.8209	1.56268	0.781342	−	0.624103i	$-0.214536\pi$
$389$	30.8209	1.56268	0.781342	+	0.624103i	$0.214536\pi$
$390$	7.08214	0.358618
$391$	17.9629	0.908421
$392$	−4.76251	−0.240543
$393$	−1.45584	−0.0734374
$394$	12.3012	0.619727
$395$	−44.1098	−2.21940
$396$	−2.15316	−0.108200
$397$	14.9668	0.751161	0.375581	−	0.926790i	$-0.377443\pi$
$397$	14.9668	0.751161	0.375581	+	0.926790i	$0.377443\pi$
$398$	18.0187	0.903195
$399$	24.3772	1.22038
$400$	3.21397	0.160698
$401$	−4.58532	−0.228980	−0.114490	−	0.993424i	$-0.536523\pi$
$401$	−4.58532	−0.228980	−0.114490	+	0.993424i	$0.536523\pi$
$402$	−10.8060	−0.538955
$403$	10.0248	0.499369
$404$	−12.3350	−0.613687
$405$	32.2260	1.60132
$406$	−16.6307	−0.825365
$407$	5.02093	0.248878
$408$	−15.3212	−0.758513
$409$	17.5640	0.868482	0.434241	−	0.900797i	$-0.357017\pi$
$409$	17.5640	0.868482	0.434241	+	0.900797i	$0.357017\pi$
$410$	−27.7679	−1.37136
$411$	−11.8052	−0.582305
$412$	−7.97045	−0.392676
$413$	−2.82140	−0.138832
$414$	3.51193	0.172602
$415$	18.4125	0.903835
$416$	1.17483	0.0576007
$417$	7.19589	0.352384
$418$	−5.10912	−0.249895
$419$	35.6335	1.74081	0.870405	−	0.492336i	$-0.163857\pi$
$419$	35.6335	1.74081	0.870405	+	0.492336i	$0.163857\pi$
$420$	20.6748	1.00882
$421$	37.4786	1.82660	0.913298	−	0.407291i	$-0.133527\pi$
$421$	37.4786	1.82660	0.913298	+	0.407291i	$0.133527\pi$
$422$	27.1505	1.32166
$423$	−0.918782	−0.0446727
$424$	9.65253	0.468768
$425$	−23.4110	−1.13560
$426$	−22.5486	−1.09248
$427$	35.4772	1.71686
$428$	0.791050	0.0382369
$429$	−3.73607	−0.180379
$430$	−5.63634	−0.271809
$431$	31.3578	1.51045	0.755226	−	0.655464i	$-0.227527\pi$
$431$	31.3578	1.51045	0.755226	+	0.655464i	$0.227527\pi$
$432$	3.31463	0.159475
$433$	21.2769	1.02250	0.511250	−	0.859432i	$-0.329182\pi$
$433$	21.2769	1.02250	0.511250	+	0.859432i	$0.329182\pi$
$434$	29.2651	1.40477
$435$	29.2314	1.40154
$436$	−1.22575	−0.0587029
$437$	8.33327	0.398634
$438$	12.8907	0.615943
$439$	−2.41774	−0.115393	−0.0576963	−	0.998334i	$-0.518376\pi$
$439$	−2.41774	−0.115393	−0.0576963	+	0.998334i	$0.518376\pi$
$440$	−4.33315	−0.206575
$441$	6.78243	0.322973
$442$	−8.55762	−0.407045
$443$	29.6406	1.40827	0.704133	−	0.710068i	$-0.251336\pi$
$443$	29.6406	1.40827	0.704133	+	0.710068i	$0.251336\pi$
$444$	6.98507	0.331497
$445$	36.5871	1.73439
$446$	2.51162	0.118929
$447$	−37.7254	−1.78435
$448$	3.42965	0.162036
$449$	−17.6851	−0.834611	−0.417305	−	0.908766i	$-0.637025\pi$
$449$	−17.6851	−0.834611	−0.417305	+	0.908766i	$0.637025\pi$
$450$	−4.57710	−0.215767
$451$	14.6485	0.689773
$452$	12.5726	0.591363
$453$	28.3067	1.32996
$454$	−22.5312	−1.05744
$455$	11.5478	0.541371
$456$	−7.10777	−0.332852
$457$	0.459735	0.0215055	0.0107527	−	0.999942i	$-0.496577\pi$
$457$	0.459735	0.0215055	0.0107527	+	0.999942i	$0.496577\pi$
$458$	18.3882	0.859222
$459$	−24.1443	−1.12696
$460$	7.06761	0.329529
$461$	7.88790	0.367376	0.183688	−	0.982985i	$-0.441196\pi$
$461$	7.88790	0.367376	0.183688	+	0.982985i	$0.441196\pi$
$462$	−10.9066	−0.507423
$463$	−37.5488	−1.74504	−0.872521	−	0.488576i	$-0.837517\pi$
$463$	−37.5488	−1.74504	−0.872521	+	0.488576i	$0.837517\pi$
$464$	4.84908	0.225113
$465$	−51.4387	−2.38541
$466$	8.43449	0.390720
$467$	−41.4338	−1.91733	−0.958664	−	0.284542i	$-0.908158\pi$
$467$	−41.4338	−1.91733	−0.958664	+	0.284542i	$0.908158\pi$
$468$	−1.67311	−0.0773393
$469$	−17.6198	−0.813608
$470$	−1.84901	−0.0852886
$471$	−14.4539	−0.666003
$472$	0.822649	0.0378655
$473$	2.97336	0.136715
$474$	32.3722	1.48690
$475$	−10.8608	−0.498326
$476$	−24.9821	−1.14505
$477$	−13.7464	−0.629406
$478$	−19.8057	−0.905890
$479$	−27.2202	−1.24372	−0.621860	−	0.783128i	$-0.713623\pi$
$479$	−27.2202	−1.24372	−0.621860	+	0.783128i	$0.713623\pi$
$480$	−6.02824	−0.275150
$481$	3.90150	0.177893
$482$	28.3492	1.29127
$483$	17.7894	0.809444
$484$	−8.71412	−0.396096
$485$	−8.93477	−0.405707
$486$	−13.7068	−0.621754
$487$	7.85504	0.355946	0.177973	−	0.984035i	$-0.443046\pi$
$487$	7.85504	0.355946	0.177973	+	0.984035i	$0.443046\pi$
$488$	−10.3443	−0.468262
$489$	51.4381	2.32611
$490$	13.6494	0.616616
$491$	28.6606	1.29343	0.646717	−	0.762730i	$-0.276141\pi$
$491$	28.6606	1.29343	0.646717	+	0.762730i	$0.276141\pi$
$492$	20.3789	0.918752
$493$	−35.3214	−1.59080
$494$	−3.97003	−0.178620
$495$	6.17096	0.277364
$496$	−8.53296	−0.383141
$497$	−36.7668	−1.64922
$498$	−13.5130	−0.605530
$499$	35.3208	1.58118	0.790589	−	0.612347i	$-0.209775\pi$
$499$	35.3208	1.58118	0.790589	+	0.612347i	$0.209775\pi$
$500$	5.11877	0.228918
$501$	29.1125	1.30065
$502$	21.0654	0.940194
$503$	−23.4898	−1.04736	−0.523679	−	0.851915i	$-0.675441\pi$
$503$	−23.4898	−1.04736	−0.523679	+	0.851915i	$0.675441\pi$
$504$	−4.88426	−0.217562
$505$	35.3520	1.57314
$506$	−3.72841	−0.165748
$507$	24.4406	1.08544
$508$	3.10468	0.137748
$509$	13.6674	0.605797	0.302898	−	0.953023i	$-0.402046\pi$
$509$	13.6674	0.605797	0.302898	+	0.953023i	$0.402046\pi$
$510$	43.9106	1.94439
$511$	21.0191	0.929830
$512$	−1.00000	−0.0441942
$513$	−11.2009	−0.494533
$514$	6.93510	0.305894
$515$	22.8433	1.00660
$516$	4.13652	0.182100
$517$	0.975417	0.0428988
$518$	11.3896	0.500429
$519$	15.1083	0.663183
$520$	−3.36706	−0.147655
$521$	−20.5598	−0.900740	−0.450370	−	0.892842i	$-0.648708\pi$
$521$	−20.5598	−0.900740	−0.450370	+	0.892842i	$0.648708\pi$
$522$	−6.90571	−0.302255
$523$	23.1195	1.01094	0.505472	−	0.862843i	$-0.331318\pi$
$523$	23.1195	1.01094	0.505472	+	0.862843i	$0.331318\pi$
$524$	0.692149	0.0302367
$525$	−23.1849	−1.01187
$526$	30.6371	1.33584
$527$	62.1554	2.70753
$528$	3.18010	0.138396
$529$	−16.9188	−0.735598
$530$	−27.6642	−1.20165
$531$	−1.17156	−0.0508413
$532$	−11.5896	−0.502474
$533$	11.3826	0.493035
$534$	−26.8513	−1.16197
$535$	−2.26715	−0.0980175
$536$	5.13750	0.221906
$537$	5.29384	0.228446
$538$	5.70907	0.246136
$539$	−7.20051	−0.310148
$540$	−9.49973	−0.408803
$541$	−28.1794	−1.21153	−0.605764	−	0.795644i	$-0.707133\pi$
$541$	−28.1794	−1.21153	−0.605764	+	0.795644i	$0.707133\pi$
$542$	−31.4036	−1.34890
$543$	−17.4367	−0.748282
$544$	7.28415	0.312305
$545$	3.51301	0.150481
$546$	−8.47497	−0.362695
$547$	39.8372	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$547$	39.8372	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$548$	5.61252	0.239755
$549$	14.7315	0.628727
$550$	4.85924	0.207199
$551$	−16.3862	−0.698076
$552$	−5.18693	−0.220770
$553$	52.7847	2.24464
$554$	−4.12553	−0.175277
$555$	−20.0192	−0.849769
$556$	−3.42114	−0.145089
$557$	27.5288	1.16643	0.583216	−	0.812317i	$-0.301794\pi$
$557$	27.5288	1.16643	0.583216	+	0.812317i	$0.301794\pi$
$558$	12.1520	0.514437
$559$	2.31044	0.0977213
$560$	−9.82939	−0.415368
$561$	−23.1643	−0.977999
$562$	23.9809	1.01157
$563$	−23.5606	−0.992962	−0.496481	−	0.868047i	$-0.665375\pi$
$563$	−23.5606	−0.992962	−0.496481	+	0.868047i	$0.665375\pi$
$564$	1.35699	0.0571397
$565$	−36.0330	−1.51592
$566$	14.1399	0.594345
$567$	−38.5638	−1.61953
$568$	10.7203	0.449813
$569$	−41.1421	−1.72476	−0.862382	−	0.506257i	$-0.831029\pi$
$569$	−41.1421	−1.72476	−0.862382	+	0.506257i	$0.831029\pi$
$570$	20.3709	0.853242
$571$	−8.88139	−0.371675	−0.185837	−	0.982581i	$-0.559500\pi$
$571$	−8.88139	−0.371675	−0.185837	+	0.982581i	$0.559500\pi$
$572$	1.77624	0.0742683
$573$	25.7700	1.07656
$574$	33.2290	1.38695
$575$	−7.92570	−0.330525
$576$	1.42413	0.0593387
$577$	23.0948	0.961450	0.480725	−	0.876871i	$-0.340373\pi$
$577$	23.0948	0.961450	0.480725	+	0.876871i	$0.340373\pi$
$578$	−36.0589	−1.49985
$579$	−9.70831	−0.403463
$580$	−13.8975	−0.577061
$581$	−22.0337	−0.914110
$582$	6.55723	0.271806
$583$	14.5938	0.604413
$584$	−6.12864	−0.253605
$585$	4.79512	0.198254
$586$	−8.61165	−0.355744
$587$	−3.56091	−0.146974	−0.0734872	−	0.997296i	$-0.523413\pi$
$587$	−3.56091	−0.146974	−0.0734872	+	0.997296i	$0.523413\pi$
$588$	−10.0173	−0.413106
$589$	28.8349	1.18812
$590$	−2.35771	−0.0970655
$591$	25.8739	1.06431
$592$	−3.32091	−0.136489
$593$	−9.58874	−0.393762	−0.196881	−	0.980427i	$-0.563081\pi$
$593$	−9.58874	−0.393762	−0.196881	+	0.980427i	$0.563081\pi$
$594$	5.01143	0.205622
$595$	71.5988	2.93526
$596$	17.9357	0.734677
$597$	37.8998	1.55114
$598$	−2.89715	−0.118473
$599$	26.7794	1.09418	0.547088	−	0.837075i	$-0.315736\pi$
$599$	26.7794	1.09418	0.547088	+	0.837075i	$0.315736\pi$
$600$	6.76013	0.275981
$601$	21.0157	0.857249	0.428624	−	0.903483i	$-0.358998\pi$
$601$	21.0157	0.857249	0.428624	+	0.903483i	$0.358998\pi$
$602$	6.74483	0.274899
$603$	−7.31645	−0.297949
$604$	−13.4578	−0.547591
$605$	24.9747	1.01536
$606$	−25.9449	−1.05394
$607$	−32.5724	−1.32207	−0.661037	−	0.750353i	$-0.729883\pi$
$607$	−32.5724	−1.32207	−0.661037	+	0.750353i	$0.729883\pi$
$608$	3.37924	0.137046
$609$	−34.9803	−1.41747
$610$	29.6466	1.20036
$611$	0.757945	0.0306632
$612$	−10.3736	−0.419327
$613$	7.23880	0.292372	0.146186	−	0.989257i	$-0.453300\pi$
$613$	7.23880	0.292372	0.146186	+	0.989257i	$0.453300\pi$
$614$	10.2691	0.414429
$615$	−58.4060	−2.35516
$616$	5.18534	0.208923
$617$	14.0526	0.565737	0.282868	−	0.959159i	$-0.408714\pi$
$617$	14.0526	0.565737	0.282868	+	0.959159i	$0.408714\pi$
$618$	−16.7647	−0.674377
$619$	−2.24759	−0.0903381	−0.0451691	−	0.998979i	$-0.514383\pi$
$619$	−2.24759	−0.0903381	−0.0451691	+	0.998979i	$0.514383\pi$
$620$	24.4555	0.982156
$621$	−8.17393	−0.328009
$622$	3.65819	0.146680
$623$	−43.7826	−1.75411
$624$	2.47109	0.0989226
$625$	−30.7403	−1.22961
$626$	13.5030	0.539689
$627$	−10.7463	−0.429167
$628$	6.87183	0.274216
$629$	24.1900	0.964519
$630$	13.9983	0.557706
$631$	9.18148	0.365509	0.182754	−	0.983159i	$-0.441499\pi$
$631$	9.18148	0.365509	0.182754	+	0.983159i	$0.441499\pi$
$632$	−15.3907	−0.612209
$633$	57.1072	2.26981
$634$	−0.0257917	−0.00102432
$635$	−8.89803	−0.353108
$636$	20.3028	0.805057
$637$	−5.59513	−0.221687
$638$	7.33139	0.290252
$639$	−15.2671	−0.603955
$640$	2.86600	0.113289
$641$	−4.87165	−0.192419	−0.0962093	−	0.995361i	$-0.530672\pi$
$641$	−4.87165	−0.192419	−0.0962093	+	0.995361i	$0.530672\pi$
$642$	1.66386	0.0656675
$643$	11.1134	0.438269	0.219134	−	0.975695i	$-0.429677\pi$
$643$	11.1134	0.438269	0.219134	+	0.975695i	$0.429677\pi$
$644$	−8.45758	−0.333275
$645$	−11.8553	−0.466801
$646$	−24.6149	−0.968461
$647$	−4.83049	−0.189906	−0.0949530	−	0.995482i	$-0.530270\pi$
$647$	−4.83049	−0.189906	−0.0949530	+	0.995482i	$0.530270\pi$
$648$	11.2442	0.441716
$649$	1.24377	0.0488224
$650$	3.77586	0.148101
$651$	61.5551	2.41253
$652$	−24.4552	−0.957740
$653$	−1.00841	−0.0394620	−0.0197310	−	0.999805i	$-0.506281\pi$
$653$	−1.00841	−0.0394620	−0.0197310	+	0.999805i	$0.506281\pi$
$654$	−2.57820	−0.100816
$655$	−1.98370	−0.0775096
$656$	−9.68874	−0.378282
$657$	8.72796	0.340510
$658$	2.21265	0.0862582
$659$	−40.6090	−1.58190	−0.790951	−	0.611879i	$-0.790414\pi$
$659$	−40.6090	−1.58190	−0.790951	+	0.611879i	$0.790414\pi$
$660$	−9.11418	−0.354769
$661$	−33.6944	−1.31056	−0.655280	−	0.755386i	$-0.727449\pi$
$661$	−33.6944	−1.31056	−0.655280	+	0.755386i	$0.727449\pi$
$662$	−14.0491	−0.546035
$663$	−17.9998	−0.699053
$664$	6.42446	0.249317
$665$	33.2159	1.28806
$666$	4.72940	0.183261
$667$	−11.9579	−0.463012
$668$	−13.8409	−0.535522
$669$	5.28284	0.204246
$670$	−14.7241	−0.568841
$671$	−15.6396	−0.603761
$672$	7.21380	0.278278
$673$	18.7103	0.721231	0.360615	−	0.932715i	$-0.382567\pi$
$673$	18.7103	0.721231	0.360615	+	0.932715i	$0.382567\pi$
$674$	−24.2966	−0.935871
$675$	10.6531	0.410038
$676$	−11.6198	−0.446915
$677$	−35.0072	−1.34544	−0.672719	−	0.739898i	$-0.734874\pi$
$677$	−35.0072	−1.34544	−0.672719	+	0.739898i	$0.734874\pi$
$678$	26.4446	1.01560
$679$	10.6919	0.410319
$680$	−20.8764	−0.800573
$681$	−47.3912	−1.81603
$682$	−12.9011	−0.494009
$683$	31.3377	1.19911	0.599553	−	0.800335i	$-0.295345\pi$
$683$	31.3377	1.19911	0.599553	+	0.800335i	$0.295345\pi$
$684$	−4.81247	−0.184010
$685$	−16.0855	−0.614595
$686$	7.67380	0.292987
$687$	38.6769	1.47562
$688$	−1.96662	−0.0749768
$689$	11.3401	0.432022
$690$	14.8657	0.565929
$691$	−12.4762	−0.474618	−0.237309	−	0.971434i	$-0.576265\pi$
$691$	−12.4762	−0.474618	−0.237309	+	0.971434i	$0.576265\pi$
$692$	−7.18295	−0.273055
$693$	−7.38459	−0.280517
$694$	−0.948564	−0.0360070
$695$	9.80500	0.371925
$696$	10.1994	0.386606
$697$	70.5742	2.67319
$698$	10.6983	0.404935
$699$	17.7408	0.671017
$700$	11.0228	0.416622
$701$	22.1317	0.835905	0.417952	−	0.908469i	$-0.362748\pi$
$701$	22.1317	0.835905	0.417952	+	0.908469i	$0.362748\pi$
$702$	3.89412	0.146974
$703$	11.2222	0.423252
$704$	−1.51191	−0.0569824
$705$	−3.88914	−0.146474
$706$	10.1166	0.380744
$707$	−42.3046	−1.59103
$708$	1.73033	0.0650297
$709$	35.7153	1.34132	0.670658	−	0.741767i	$-0.266012\pi$
$709$	35.7153	1.34132	0.670658	+	0.741767i	$0.266012\pi$
$710$	−30.7243	−1.15306
$711$	21.9183	0.822002
$712$	12.7659	0.478422
$713$	21.0424	0.788046
$714$	−52.5464	−1.96650
$715$	−5.09070	−0.190381
$716$	−2.51685	−0.0940591
$717$	−41.6585	−1.55576
$718$	−0.328326	−0.0122530
$719$	28.8487	1.07587	0.537937	−	0.842985i	$-0.319204\pi$
$719$	28.8487	1.07587	0.537937	+	0.842985i	$0.319204\pi$
$720$	−4.08155	−0.152111
$721$	−27.3359	−1.01804
$722$	7.58073	0.282125
$723$	59.6286	2.21761
$724$	8.28994	0.308093
$725$	15.5848	0.578804
$726$	−18.3289	−0.680251
$727$	−30.3028	−1.12387	−0.561935	−	0.827182i	$-0.689943\pi$
$727$	−30.3028	−1.12387	−0.561935	+	0.827182i	$0.689943\pi$
$728$	4.02925	0.149334
$729$	4.90233	0.181568
$730$	17.5647	0.650098
$731$	14.3252	0.529836
$732$	−21.7577	−0.804188
$733$	22.3506	0.825538	0.412769	−	0.910836i	$-0.364562\pi$
$733$	22.3506	0.825538	0.412769	+	0.910836i	$0.364562\pi$
$734$	−15.4184	−0.569104
$735$	28.7096	1.05897
$736$	2.46602	0.0908986
$737$	7.76745	0.286118
$738$	13.7980	0.507912
$739$	−40.2470	−1.48051	−0.740255	−	0.672327i	$-0.765295\pi$
$739$	−40.2470	−1.48051	−0.740255	+	0.672327i	$0.765295\pi$
$740$	9.51773	0.349879
$741$	−8.35040	−0.306760
$742$	33.1048	1.21532
$743$	−12.5623	−0.460867	−0.230433	−	0.973088i	$-0.574014\pi$
$743$	−12.5623	−0.460867	−0.230433	+	0.973088i	$0.574014\pi$
$744$	−17.9479	−0.658002
$745$	−51.4039	−1.88329
$746$	−26.7105	−0.977941
$747$	−9.14925	−0.334754
$748$	11.0130	0.402676
$749$	2.71303	0.0991319
$750$	10.7666	0.393142
$751$	7.38580	0.269512	0.134756	−	0.990879i	$-0.456975\pi$
$751$	7.38580	0.269512	0.134756	+	0.990879i	$0.456975\pi$
$752$	−0.645154	−0.0235263
$753$	44.3081	1.61468
$754$	5.69683	0.207466
$755$	38.5702	1.40371
$756$	11.3680	0.413451
$757$	−24.3461	−0.884874	−0.442437	−	0.896799i	$-0.645886\pi$
$757$	−24.3461	−0.884874	−0.442437	+	0.896799i	$0.645886\pi$
$758$	−30.8165	−1.11931
$759$	−7.84219	−0.284653
$760$	−9.68491	−0.351309
$761$	−10.2975	−0.373285	−0.186643	−	0.982428i	$-0.559761\pi$
$761$	−10.2975	−0.373285	−0.186643	+	0.982428i	$0.559761\pi$
$762$	6.53027	0.236567
$763$	−4.20390	−0.152191
$764$	−12.2518	−0.443256
$765$	29.7307	1.07491
$766$	−36.1044	−1.30451
$767$	0.966471	0.0348972
$768$	−2.10336	−0.0758985
$769$	−30.8671	−1.11310	−0.556548	−	0.830815i	$-0.687875\pi$
$769$	−30.8671	−1.11310	−0.556548	+	0.830815i	$0.687875\pi$
$770$	−14.8612	−0.535560
$771$	14.5870	0.525339
$772$	4.61562	0.166120
$773$	−9.07233	−0.326309	−0.163154	−	0.986601i	$-0.552167\pi$
$773$	−9.07233	−0.326309	−0.163154	+	0.986601i	$0.552167\pi$
$774$	2.80072	0.100670
$775$	−27.4247	−0.985123
$776$	−3.11750	−0.111912
$777$	23.9564	0.859430
$778$	−30.8209	−1.10498
$779$	32.7406	1.17305
$780$	−7.08214	−0.253581
$781$	16.2081	0.579973
$782$	−17.9629	−0.642350
$783$	16.0729	0.574398
$784$	4.76251	0.170090
$785$	−19.6947	−0.702933
$786$	1.45584	0.0519281
$787$	8.12236	0.289531	0.144765	−	0.989466i	$-0.453757\pi$
$787$	8.12236	0.289531	0.144765	+	0.989466i	$0.453757\pi$
$788$	−12.3012	−0.438213
$789$	64.4408	2.29415
$790$	44.1098	1.56936
$791$	43.1195	1.53315
$792$	2.15316	0.0765092
$793$	−12.1527	−0.431556
$794$	−14.9668	−0.531151
$795$	−58.1877	−2.06371
$796$	−18.0187	−0.638655
$797$	−17.1401	−0.607132	−0.303566	−	0.952810i	$-0.598177\pi$
$797$	−17.1401	−0.607132	−0.303566	+	0.952810i	$0.598177\pi$
$798$	−24.3772	−0.862942
$799$	4.69940	0.166253
$800$	−3.21397	−0.113631
$801$	−18.1803	−0.642369
$802$	4.58532	0.161913
$803$	−9.26597	−0.326989
$804$	10.8060	0.381099
$805$	24.2395	0.854328
$806$	−10.0248	−0.353107
$807$	12.0082	0.422710
$808$	12.3350	0.433943
$809$	26.6469	0.936854	0.468427	−	0.883502i	$-0.344821\pi$
$809$	26.6469	0.936854	0.468427	+	0.883502i	$0.344821\pi$
$810$	−32.2260	−1.13231
$811$	−8.06325	−0.283139	−0.141570	−	0.989928i	$-0.545215\pi$
$811$	−8.06325	−0.283139	−0.141570	+	0.989928i	$0.545215\pi$
$812$	16.6307	0.583621
$813$	−66.0532	−2.31659
$814$	−5.02093	−0.175984
$815$	70.0887	2.45510
$816$	15.3212	0.536349
$817$	6.64569	0.232503
$818$	−17.5640	−0.614110
$819$	−5.73817	−0.200508
$820$	27.7679	0.969698
$821$	29.0669	1.01444	0.507221	−	0.861816i	$-0.330673\pi$
$821$	29.0669	1.01444	0.507221	+	0.861816i	$0.330673\pi$
$822$	11.8052	0.411752
$823$	33.1970	1.15718	0.578588	−	0.815620i	$-0.303604\pi$
$823$	33.1970	1.15718	0.578588	+	0.815620i	$0.303604\pi$
$824$	7.97045	0.277664
$825$	10.2207	0.355841
$826$	2.82140	0.0981691
$827$	50.0174	1.73928	0.869638	−	0.493689i	$-0.164352\pi$
$827$	50.0174	1.73928	0.869638	+	0.493689i	$0.164352\pi$
$828$	−3.51193	−0.122048
$829$	−45.3776	−1.57603	−0.788015	−	0.615655i	$-0.788891\pi$
$829$	−45.3776	−1.57603	−0.788015	+	0.615655i	$0.788891\pi$
$830$	−18.4125	−0.639108
$831$	−8.67749	−0.301019
$832$	−1.17483	−0.0407298
$833$	−34.6909	−1.20197
$834$	−7.19589	−0.249173
$835$	39.6682	1.37277
$836$	5.10912	0.176703
$837$	−28.2836	−0.977624
$838$	−35.6335	−1.23094
$839$	1.03059	0.0355799	0.0177900	−	0.999842i	$-0.494337\pi$
$839$	1.03059	0.0355799	0.0177900	+	0.999842i	$0.494337\pi$
$840$	−20.6748	−0.713347
$841$	−5.48644	−0.189188
$842$	−37.4786	−1.29160
$843$	50.4405	1.73726
$844$	−27.1505	−0.934557
$845$	33.3023	1.14563
$846$	0.918782	0.0315884
$847$	−29.8864	−1.02691
$848$	−9.65253	−0.331469
$849$	29.7414	1.02072
$850$	23.4110	0.802992
$851$	8.18942	0.280730
$852$	22.5486	0.772503
$853$	13.8081	0.472781	0.236390	−	0.971658i	$-0.424036\pi$
$853$	13.8081	0.472781	0.236390	+	0.971658i	$0.424036\pi$
$854$	−35.4772	−1.21400
$855$	13.7926	0.471696
$856$	−0.791050	−0.0270375
$857$	51.1022	1.74562	0.872809	−	0.488062i	$-0.162296\pi$
$857$	51.1022	1.74562	0.872809	+	0.488062i	$0.162296\pi$
$858$	3.73607	0.127547
$859$	−8.73015	−0.297869	−0.148934	−	0.988847i	$-0.547584\pi$
$859$	−8.73015	−0.297869	−0.148934	+	0.988847i	$0.547584\pi$
$860$	5.63634	0.192198
$861$	69.8926	2.38193
$862$	−31.3578	−1.06805
$863$	23.4756	0.799118	0.399559	−	0.916707i	$-0.369163\pi$
$863$	23.4756	0.799118	0.399559	+	0.916707i	$0.369163\pi$
$864$	−3.31463	−0.112766
$865$	20.5864	0.699957
$866$	−21.2769	−0.723017
$867$	−75.8449	−2.57583
$868$	−29.2651	−0.993322
$869$	−23.2694	−0.789361
$870$	−29.2314	−0.991037
$871$	6.03567	0.204511
$872$	1.22575	0.0415092
$873$	4.43972	0.150262
$874$	−8.33327	−0.281877
$875$	17.5556	0.593488
$876$	−12.8907	−0.435538
$877$	18.8682	0.637133	0.318567	−	0.947900i	$-0.396799\pi$
$877$	18.8682	0.637133	0.318567	+	0.947900i	$0.396799\pi$
$878$	2.41774	0.0815949
$879$	−18.1134	−0.610950
$880$	4.33315	0.146070
$881$	−28.8640	−0.972452	−0.486226	−	0.873833i	$-0.661627\pi$
$881$	−28.8640	−0.972452	−0.486226	+	0.873833i	$0.661627\pi$
$882$	−6.78243	−0.228376
$883$	−43.3633	−1.45929	−0.729645	−	0.683826i	$-0.760315\pi$
$883$	−43.3633	−1.45929	−0.729645	+	0.683826i	$0.760315\pi$
$884$	8.55762	0.287824
$885$	−4.95912	−0.166699
$886$	−29.6406	−0.995794
$887$	12.0826	0.405695	0.202848	−	0.979210i	$-0.434980\pi$
$887$	12.0826	0.405695	0.202848	+	0.979210i	$0.434980\pi$
$888$	−6.98507	−0.234404
$889$	10.6480	0.357122
$890$	−36.5871	−1.22640
$891$	17.0003	0.569532
$892$	−2.51162	−0.0840952
$893$	2.18013	0.0729553
$894$	37.7254	1.26172
$895$	7.21330	0.241114
$896$	−3.42965	−0.114577
$897$	−6.09374	−0.203464
$898$	17.6851	0.590159
$899$	−41.3770	−1.38000
$900$	4.57710	0.152570
$901$	70.3105	2.34238
$902$	−14.6485	−0.487743
$903$	14.1868	0.472108
$904$	−12.5726	−0.418157
$905$	−23.7590	−0.789775
$906$	−28.3067	−0.940426
$907$	−18.5127	−0.614703	−0.307351	−	0.951596i	$-0.599443\pi$
$907$	−18.5127	−0.614703	−0.307351	+	0.951596i	$0.599443\pi$
$908$	22.5312	0.747723
$909$	−17.5666	−0.582646
$910$	−11.5478	−0.382807
$911$	20.7011	0.685857	0.342929	−	0.939361i	$-0.388581\pi$
$911$	20.7011	0.685857	0.342929	+	0.939361i	$0.388581\pi$
$912$	7.10777	0.235362
$913$	9.71323	0.321461
$914$	−0.459735	−0.0152067
$915$	62.3576	2.06148
$916$	−18.3882	−0.607562
$917$	2.37383	0.0783908
$918$	24.1443	0.796879
$919$	−18.9549	−0.625265	−0.312633	−	0.949874i	$-0.601211\pi$
$919$	−18.9549	−0.625265	−0.312633	+	0.949874i	$0.601211\pi$
$920$	−7.06761	−0.233012
$921$	21.5997	0.711734
$922$	−7.88790	−0.259774
$923$	12.5945	0.414552
$924$	10.9066	0.358802
$925$	−10.6733	−0.350936
$926$	37.5488	1.23393
$927$	−11.3509	−0.372814
$928$	−4.84908	−0.159179
$929$	−16.5427	−0.542749	−0.271375	−	0.962474i	$-0.587478\pi$
$929$	−16.5427	−0.542749	−0.271375	+	0.962474i	$0.587478\pi$
$930$	51.4387	1.68674
$931$	−16.0937	−0.527449
$932$	−8.43449	−0.276281
$933$	7.69450	0.251907
$934$	41.4338	1.35576
$935$	−31.5633	−1.03223
$936$	1.67311	0.0546872
$937$	−18.9277	−0.618340	−0.309170	−	0.951007i	$-0.600051\pi$
$937$	−18.9277	−0.618340	−0.309170	+	0.951007i	$0.600051\pi$
$938$	17.6198	0.575308
$939$	28.4017	0.926854
$940$	1.84901	0.0603081
$941$	33.7968	1.10175	0.550873	−	0.834589i	$-0.314295\pi$
$941$	33.7968	1.10175	0.550873	+	0.834589i	$0.314295\pi$
$942$	14.4539	0.470935
$943$	23.8926	0.778050
$944$	−0.822649	−0.0267749
$945$	−32.5808	−1.05985
$946$	−2.97336	−0.0966724
$947$	45.4634	1.47736	0.738681	−	0.674055i	$-0.235449\pi$
$947$	45.4634	1.47736	0.738681	+	0.674055i	$0.235449\pi$
$948$	−32.3722	−1.05140
$949$	−7.20009	−0.233725
$950$	10.8608	0.352370
$951$	−0.0542492	−0.00175915
$952$	24.9821	0.809675
$953$	−44.7905	−1.45091	−0.725453	−	0.688271i	$-0.758370\pi$
$953$	−44.7905	−1.45091	−0.725453	+	0.688271i	$0.758370\pi$
$954$	13.7464	0.445057
$955$	35.1138	1.13626
$956$	19.8057	0.640561
$957$	15.4206	0.498476
$958$	27.2202	0.879443
$959$	19.2490	0.621582
$960$	6.02824	0.194561
$961$	41.8115	1.34876
$962$	−3.90150	−0.125789
$963$	1.12656	0.0363028
$964$	−28.3492	−0.913066
$965$	−13.2284	−0.425836
$966$	−17.7894	−0.572363
$967$	60.4996	1.94554	0.972768	−	0.231782i	$-0.0744556\pi$
$967$	60.4996	1.94554	0.972768	+	0.231782i	$0.0744556\pi$
$968$	8.71412	0.280082
$969$	−51.7741	−1.66322
$970$	8.93477	0.286878
$971$	−11.0649	−0.355091	−0.177545	−	0.984113i	$-0.556816\pi$
$971$	−11.0649	−0.355091	−0.177545	+	0.984113i	$0.556816\pi$
$972$	13.7068	0.439647
$973$	−11.7333	−0.376153
$974$	−7.85504	−0.251692
$975$	7.94199	0.254347
$976$	10.3443	0.331111
$977$	−36.3948	−1.16437	−0.582186	−	0.813056i	$-0.697802\pi$
$977$	−36.3948	−1.16437	−0.582186	+	0.813056i	$0.697802\pi$
$978$	−51.4381	−1.64481
$979$	19.3009	0.616861
$980$	−13.6494	−0.436013
$981$	−1.74563	−0.0557336
$982$	−28.6606	−0.914596
$983$	32.5565	1.03839	0.519196	−	0.854655i	$-0.326231\pi$
$983$	32.5565	1.03839	0.519196	+	0.854655i	$0.326231\pi$
$984$	−20.3789	−0.649656
$985$	35.2553	1.12333
$986$	35.3214	1.12486
$987$	4.65401	0.148139
$988$	3.97003	0.126303
$989$	4.84973	0.154212
$990$	−6.17096	−0.196126
$991$	17.7721	0.564550	0.282275	−	0.959334i	$-0.408911\pi$
$991$	17.7721	0.564550	0.282275	+	0.959334i	$0.408911\pi$
$992$	8.53296	0.270922
$993$	−29.5504	−0.937754
$994$	36.7668	1.16617
$995$	51.6416	1.63715
$996$	13.5130	0.428174
$997$	−24.1251	−0.764049	−0.382024	−	0.924152i	$-0.624773\pi$
$997$	−24.1251	−0.764049	−0.382024	+	0.924152i	$0.624773\pi$
$998$	−35.3208	−1.11806
$999$	−11.0076	−0.348264

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.c.1.16	✓	86

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.c.1.16	✓	86	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.10336 q^{3} +1.00000 q^{4} -2.86600 q^{5} +2.10336 q^{6} +3.42965 q^{7} -1.00000 q^{8} +1.42413 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.10336 q^{3} +1.00000 q^{4} -2.86600 q^{5} +2.10336 q^{6} +3.42965 q^{7} -1.00000 q^{8} +1.42413 q^{9} +2.86600 q^{10} -1.51191 q^{11} -2.10336 q^{12} -1.17483 q^{13} -3.42965 q^{14} +6.02824 q^{15} +1.00000 q^{16} -7.28415 q^{17} -1.42413 q^{18} -3.37924 q^{19} -2.86600 q^{20} -7.21380 q^{21} +1.51191 q^{22} -2.46602 q^{23} +2.10336 q^{24} +3.21397 q^{25} +1.17483 q^{26} +3.31463 q^{27} +3.42965 q^{28} +4.84908 q^{29} -6.02824 q^{30} -8.53296 q^{31} -1.00000 q^{32} +3.18010 q^{33} +7.28415 q^{34} -9.82939 q^{35} +1.42413 q^{36} -3.32091 q^{37} +3.37924 q^{38} +2.47109 q^{39} +2.86600 q^{40} -9.68874 q^{41} +7.21380 q^{42} -1.96662 q^{43} -1.51191 q^{44} -4.08155 q^{45} +2.46602 q^{46} -0.645154 q^{47} -2.10336 q^{48} +4.76251 q^{49} -3.21397 q^{50} +15.3212 q^{51} -1.17483 q^{52} -9.65253 q^{53} -3.31463 q^{54} +4.33315 q^{55} -3.42965 q^{56} +7.10777 q^{57} -4.84908 q^{58} -0.822649 q^{59} +6.02824 q^{60} +10.3443 q^{61} +8.53296 q^{62} +4.88426 q^{63} +1.00000 q^{64} +3.36706 q^{65} -3.18010 q^{66} -5.13750 q^{67} -7.28415 q^{68} +5.18693 q^{69} +9.82939 q^{70} -10.7203 q^{71} -1.42413 q^{72} +6.12864 q^{73} +3.32091 q^{74} -6.76013 q^{75} -3.37924 q^{76} -5.18534 q^{77} -2.47109 q^{78} +15.3907 q^{79} -2.86600 q^{80} -11.2442 q^{81} +9.68874 q^{82} -6.42446 q^{83} -7.21380 q^{84} +20.8764 q^{85} +1.96662 q^{86} -10.1994 q^{87} +1.51191 q^{88} -12.7659 q^{89} +4.08155 q^{90} -4.02925 q^{91} -2.46602 q^{92} +17.9479 q^{93} +0.645154 q^{94} +9.68491 q^{95} +2.10336 q^{96} +3.11750 q^{97} -4.76251 q^{98} -2.15316 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9 + 4 * q^10 + 13 * q^11 + 12 * q^12 + 45 * q^13 - 35 * q^14 + 17 * q^15 + 86 * q^16 + 5 * q^17 - 72 * q^18 + 47 * q^19 - 4 * q^20 + 15 * q^21 - 13 * q^22 + 6 * q^23 - 12 * q^24 + 112 * q^25 - 45 * q^26 + 51 * q^27 + 35 * q^28 - 14 * q^29 - 17 * q^30 + 24 * q^31 - 86 * q^32 + 43 * q^33 - 5 * q^34 + 42 * q^35 + 72 * q^36 + 61 * q^37 - 47 * q^38 + 20 * q^39 + 4 * q^40 - 16 * q^41 - 15 * q^42 + 72 * q^43 + 13 * q^44 + 6 * q^45 - 6 * q^46 + 11 * q^47 + 12 * q^48 + 89 * q^49 - 112 * q^50 + 56 * q^51 + 45 * q^52 - 7 * q^53 - 51 * q^54 + 48 * q^55 - 35 * q^56 + 65 * q^57 + 14 * q^58 + 24 * q^59 + 17 * q^60 + 31 * q^61 - 24 * q^62 + 98 * q^63 + 86 * q^64 - 9 * q^65 - 43 * q^66 + 157 * q^67 + 5 * q^68 + q^69 - 42 * q^70 - 11 * q^71 - 72 * q^72 + 74 * q^73 - 61 * q^74 + 76 * q^75 + 47 * q^76 - 13 * q^77 - 20 * q^78 + 57 * q^79 - 4 * q^80 + 34 * q^81 + 16 * q^82 + 65 * q^83 + 15 * q^84 + 102 * q^85 - 72 * q^86 + 49 * q^87 - 13 * q^88 - 34 * q^89 - 6 * q^90 + 91 * q^91 + 6 * q^92 + 57 * q^93 - 11 * q^94 - 13 * q^95 - 12 * q^96 + 64 * q^97 - 89 * q^98 + 56 * q^99

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