LMFDB - Embedded newform 8043.2.a.o.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$1$
Dimension:	$41$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.86200 q^{2} -1.00000 q^{3} +1.46705 q^{4} +0.268615 q^{5} +1.86200 q^{6} +1.00000 q^{7} +0.992360 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.86200 q^{2} -1.00000 q^{3} +1.46705 q^{4} +0.268615 q^{5} +1.86200 q^{6} +1.00000 q^{7} +0.992360 q^{8} +1.00000 q^{9} -0.500162 q^{10} +1.06642 q^{11} -1.46705 q^{12} +1.44076 q^{13} -1.86200 q^{14} -0.268615 q^{15} -4.78187 q^{16} +7.39371 q^{17} -1.86200 q^{18} +7.57668 q^{19} +0.394071 q^{20} -1.00000 q^{21} -1.98567 q^{22} +0.546917 q^{23} -0.992360 q^{24} -4.92785 q^{25} -2.68270 q^{26} -1.00000 q^{27} +1.46705 q^{28} -2.04541 q^{29} +0.500162 q^{30} -7.36408 q^{31} +6.91912 q^{32} -1.06642 q^{33} -13.7671 q^{34} +0.268615 q^{35} +1.46705 q^{36} +5.86348 q^{37} -14.1078 q^{38} -1.44076 q^{39} +0.266563 q^{40} -9.45102 q^{41} +1.86200 q^{42} +6.63122 q^{43} +1.56449 q^{44} +0.268615 q^{45} -1.01836 q^{46} -7.64458 q^{47} +4.78187 q^{48} +1.00000 q^{49} +9.17565 q^{50} -7.39371 q^{51} +2.11366 q^{52} -6.35378 q^{53} +1.86200 q^{54} +0.286457 q^{55} +0.992360 q^{56} -7.57668 q^{57} +3.80855 q^{58} -3.71663 q^{59} -0.394071 q^{60} -11.5103 q^{61} +13.7119 q^{62} +1.00000 q^{63} -3.31967 q^{64} +0.387011 q^{65} +1.98567 q^{66} -12.2502 q^{67} +10.8469 q^{68} -0.546917 q^{69} -0.500162 q^{70} -9.50162 q^{71} +0.992360 q^{72} +2.61402 q^{73} -10.9178 q^{74} +4.92785 q^{75} +11.1153 q^{76} +1.06642 q^{77} +2.68270 q^{78} -11.2378 q^{79} -1.28448 q^{80} +1.00000 q^{81} +17.5978 q^{82} +7.62977 q^{83} -1.46705 q^{84} +1.98607 q^{85} -12.3473 q^{86} +2.04541 q^{87} +1.05827 q^{88} -10.6360 q^{89} -0.500162 q^{90} +1.44076 q^{91} +0.802353 q^{92} +7.36408 q^{93} +14.2342 q^{94} +2.03521 q^{95} -6.91912 q^{96} -16.5486 q^{97} -1.86200 q^{98} +1.06642 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 41 q - 4 q^{2} - 41 q^{3} + 34 q^{4} + 5 q^{5} + 4 q^{6} + 41 q^{7} - 15 q^{8} + 41 q^{9}+O(q^{10}) $ 41 * q - 4 * q^2 - 41 * q^3 + 34 * q^4 + 5 * q^5 + 4 * q^6 + 41 * q^7 - 15 * q^8 + 41 * q^9	\( 41 q - 4 q^{2} - 41 q^{3} + 34 q^{4} + 5 q^{5} + 4 q^{6} + 41 q^{7} - 15 q^{8} + 41 q^{9} - 18 q^{10} - 17 q^{11} - 34 q^{12} - 38 q^{13} - 4 q^{14} - 5 q^{15} + 16 q^{16} + 4 q^{17} - 4 q^{18} - 15 q^{19} + 23 q^{20} - 41 q^{21} - 31 q^{22} - 8 q^{23} + 15 q^{24} + 16 q^{25} + 15 q^{26} - 41 q^{27} + 34 q^{28} - 27 q^{29} + 18 q^{30} - 23 q^{31} - 24 q^{32} + 17 q^{33} - 9 q^{34} + 5 q^{35} + 34 q^{36} - 81 q^{37} + 38 q^{39} - 52 q^{40} + 29 q^{41} + 4 q^{42} - 47 q^{43} - 20 q^{44} + 5 q^{45} - 40 q^{46} + 26 q^{47} - 16 q^{48} + 41 q^{49} - 23 q^{50} - 4 q^{51} - 64 q^{52} - 66 q^{53} + 4 q^{54} - 14 q^{55} - 15 q^{56} + 15 q^{57} - 50 q^{58} + 41 q^{59} - 23 q^{60} - 47 q^{61} + 5 q^{62} + 41 q^{63} - 37 q^{64} - 24 q^{65} + 31 q^{66} - 73 q^{67} + 10 q^{68} + 8 q^{69} - 18 q^{70} + q^{71} - 15 q^{72} - 14 q^{73} + 18 q^{74} - 16 q^{75} - 37 q^{76} - 17 q^{77} - 15 q^{78} - 80 q^{79} + 63 q^{80} + 41 q^{81} - 31 q^{82} + 20 q^{83} - 34 q^{84} - 82 q^{85} - 39 q^{86} + 27 q^{87} - 132 q^{88} + 17 q^{89} - 18 q^{90} - 38 q^{91} - 26 q^{92} + 23 q^{93} - 9 q^{94} - q^{95} + 24 q^{96} - 73 q^{97} - 4 q^{98} - 17 q^{99}+O(q^{100}) \) 41 * q - 4 * q^2 - 41 * q^3 + 34 * q^4 + 5 * q^5 + 4 * q^6 + 41 * q^7 - 15 * q^8 + 41 * q^9 - 18 * q^10 - 17 * q^11 - 34 * q^12 - 38 * q^13 - 4 * q^14 - 5 * q^15 + 16 * q^16 + 4 * q^17 - 4 * q^18 - 15 * q^19 + 23 * q^20 - 41 * q^21 - 31 * q^22 - 8 * q^23 + 15 * q^24 + 16 * q^25 + 15 * q^26 - 41 * q^27 + 34 * q^28 - 27 * q^29 + 18 * q^30 - 23 * q^31 - 24 * q^32 + 17 * q^33 - 9 * q^34 + 5 * q^35 + 34 * q^36 - 81 * q^37 + 38 * q^39 - 52 * q^40 + 29 * q^41 + 4 * q^42 - 47 * q^43 - 20 * q^44 + 5 * q^45 - 40 * q^46 + 26 * q^47 - 16 * q^48 + 41 * q^49 - 23 * q^50 - 4 * q^51 - 64 * q^52 - 66 * q^53 + 4 * q^54 - 14 * q^55 - 15 * q^56 + 15 * q^57 - 50 * q^58 + 41 * q^59 - 23 * q^60 - 47 * q^61 + 5 * q^62 + 41 * q^63 - 37 * q^64 - 24 * q^65 + 31 * q^66 - 73 * q^67 + 10 * q^68 + 8 * q^69 - 18 * q^70 + q^71 - 15 * q^72 - 14 * q^73 + 18 * q^74 - 16 * q^75 - 37 * q^76 - 17 * q^77 - 15 * q^78 - 80 * q^79 + 63 * q^80 + 41 * q^81 - 31 * q^82 + 20 * q^83 - 34 * q^84 - 82 * q^85 - 39 * q^86 + 27 * q^87 - 132 * q^88 + 17 * q^89 - 18 * q^90 - 38 * q^91 - 26 * q^92 + 23 * q^93 - 9 * q^94 - q^95 + 24 * q^96 - 73 * q^97 - 4 * q^98 - 17 * 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 $

$2$	−1.86200	−1.31663	−0.658317	−	0.752741i	$-0.728731\pi$
$2$	−1.86200	−1.31663	−0.658317	+	0.752741i	$0.728731\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.46705	0.733523
$5$	0.268615	0.120128	0.0600642	−	0.998195i	$-0.480869\pi$
$5$	0.268615	0.120128	0.0600642	+	0.998195i	$0.480869\pi$
$6$	1.86200	0.760159
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.992360	0.350852
$9$	1.00000	0.333333
$10$	−0.500162	−0.158165
$11$	1.06642	0.321538	0.160769	−	0.986992i	$-0.448603\pi$
$11$	1.06642	0.321538	0.160769	+	0.986992i	$0.448603\pi$
$12$	−1.46705	−0.423500
$13$	1.44076	0.399595	0.199798	−	0.979837i	$-0.435972\pi$
$13$	1.44076	0.399595	0.199798	+	0.979837i	$0.435972\pi$
$14$	−1.86200	−0.497641
$15$	−0.268615	−0.0693562
$16$	−4.78187	−1.19547
$17$	7.39371	1.79324	0.896619	−	0.442803i	$-0.146016\pi$
$17$	7.39371	1.79324	0.896619	+	0.442803i	$0.146016\pi$
$18$	−1.86200	−0.438878
$19$	7.57668	1.73821	0.869105	−	0.494627i	$-0.164695\pi$
$19$	7.57668	1.73821	0.869105	+	0.494627i	$0.164695\pi$
$20$	0.394071	0.0881170
$21$	−1.00000	−0.218218
$22$	−1.98567	−0.423347
$23$	0.546917	0.114040	0.0570201	−	0.998373i	$-0.481840\pi$
$23$	0.546917	0.114040	0.0570201	+	0.998373i	$0.481840\pi$
$24$	−0.992360	−0.202565
$25$	−4.92785	−0.985569
$26$	−2.68270	−0.526120
$27$	−1.00000	−0.192450
$28$	1.46705	0.277246
$29$	−2.04541	−0.379823	−0.189912	−	0.981801i	$-0.560820\pi$
$29$	−2.04541	−0.379823	−0.189912	+	0.981801i	$0.560820\pi$
$30$	0.500162	0.0913167
$31$	−7.36408	−1.32263	−0.661314	−	0.750109i	$-0.730001\pi$
$31$	−7.36408	−1.32263	−0.661314	+	0.750109i	$0.730001\pi$
$32$	6.91912	1.22314
$33$	−1.06642	−0.185640
$34$	−13.7671	−2.36104
$35$	0.268615	0.0454043
$36$	1.46705	0.244508
$37$	5.86348	0.963951	0.481975	−	0.876185i	$-0.339919\pi$
$37$	5.86348	0.963951	0.481975	+	0.876185i	$0.339919\pi$
$38$	−14.1078	−2.28859
$39$	−1.44076	−0.230706
$40$	0.266563	0.0421473
$41$	−9.45102	−1.47600	−0.738001	−	0.674800i	$-0.764230\pi$
$41$	−9.45102	−1.47600	−0.738001	+	0.674800i	$0.764230\pi$
$42$	1.86200	0.287313
$43$	6.63122	1.01125	0.505626	−	0.862753i	$-0.331262\pi$
$43$	6.63122	1.01125	0.505626	+	0.862753i	$0.331262\pi$
$44$	1.56449	0.235855
$45$	0.268615	0.0400428
$46$	−1.01836	−0.150149
$47$	−7.64458	−1.11508	−0.557538	−	0.830151i	$-0.688254\pi$
$47$	−7.64458	−1.11508	−0.557538	+	0.830151i	$0.688254\pi$
$48$	4.78187	0.690203
$49$	1.00000	0.142857
$50$	9.17565	1.29763
$51$	−7.39371	−1.03533
$52$	2.11366	0.293112
$53$	−6.35378	−0.872759	−0.436379	−	0.899763i	$-0.643739\pi$
$53$	−6.35378	−0.872759	−0.436379	+	0.899763i	$0.643739\pi$
$54$	1.86200	0.253386
$55$	0.286457	0.0386258
$56$	0.992360	0.132610
$57$	−7.57668	−1.00356
$58$	3.80855	0.500088
$59$	−3.71663	−0.483864	−0.241932	−	0.970293i	$-0.577781\pi$
$59$	−3.71663	−0.483864	−0.241932	+	0.970293i	$0.577781\pi$
$60$	−0.394071	−0.0508744
$61$	−11.5103	−1.47374	−0.736870	−	0.676034i	$-0.763697\pi$
$61$	−11.5103	−1.47374	−0.736870	+	0.676034i	$0.763697\pi$
$62$	13.7119	1.74142
$63$	1.00000	0.125988
$64$	−3.31967	−0.414959
$65$	0.387011	0.0480028
$66$	1.98567	0.244420
$67$	−12.2502	−1.49661	−0.748303	−	0.663358i	$-0.769131\pi$
$67$	−12.2502	−1.49661	−0.748303	+	0.663358i	$0.769131\pi$
$68$	10.8469	1.31538
$69$	−0.546917	−0.0658411
$70$	−0.500162	−0.0597808
$71$	−9.50162	−1.12764	−0.563818	−	0.825899i	$-0.690668\pi$
$71$	−9.50162	−1.12764	−0.563818	+	0.825899i	$0.690668\pi$
$72$	0.992360	0.116951
$73$	2.61402	0.305948	0.152974	−	0.988230i	$-0.451115\pi$
$73$	2.61402	0.305948	0.152974	+	0.988230i	$0.451115\pi$
$74$	−10.9178	−1.26917
$75$	4.92785	0.569019
$76$	11.1153	1.27502
$77$	1.06642	0.121530
$78$	2.68270	0.303756
$79$	−11.2378	−1.26435	−0.632175	−	0.774826i	$-0.717838\pi$
$79$	−11.2378	−1.26435	−0.632175	+	0.774826i	$0.717838\pi$
$80$	−1.28448	−0.143610
$81$	1.00000	0.111111
$82$	17.5978	1.94335
$83$	7.62977	0.837476	0.418738	−	0.908107i	$-0.362473\pi$
$83$	7.62977	0.837476	0.418738	+	0.908107i	$0.362473\pi$
$84$	−1.46705	−0.160068
$85$	1.98607	0.215419
$86$	−12.3473	−1.33145
$87$	2.04541	0.219291
$88$	1.05827	0.112812
$89$	−10.6360	−1.12742	−0.563708	−	0.825974i	$-0.690626\pi$
$89$	−10.6360	−1.12742	−0.563708	+	0.825974i	$0.690626\pi$
$90$	−0.500162	−0.0527217
$91$	1.44076	0.151033
$92$	0.802353	0.0836511
$93$	7.36408	0.763620
$94$	14.2342	1.46815
$95$	2.03521	0.208809
$96$	−6.91912	−0.706180
$97$	−16.5486	−1.68026	−0.840128	−	0.542388i	$-0.817520\pi$
$97$	−16.5486	−1.68026	−0.840128	+	0.542388i	$0.817520\pi$
$98$	−1.86200	−0.188090
$99$	1.06642	0.107179
$100$	−7.22938	−0.722938
$101$	5.00592	0.498108	0.249054	−	0.968490i	$-0.419880\pi$
$101$	5.00592	0.498108	0.249054	+	0.968490i	$0.419880\pi$
$102$	13.7671	1.36315
$103$	−5.25835	−0.518121	−0.259060	−	0.965861i	$-0.583413\pi$
$103$	−5.25835	−0.518121	−0.259060	+	0.965861i	$0.583413\pi$
$104$	1.42975	0.140199
$105$	−0.268615	−0.0262142
$106$	11.8307	1.14910
$107$	0.651473	0.0629802	0.0314901	−	0.999504i	$-0.489975\pi$
$107$	0.651473	0.0629802	0.0314901	+	0.999504i	$0.489975\pi$
$108$	−1.46705	−0.141167
$109$	1.16865	0.111936	0.0559682	−	0.998433i	$-0.482175\pi$
$109$	1.16865	0.111936	0.0559682	+	0.998433i	$0.482175\pi$
$110$	−0.533383	−0.0508560
$111$	−5.86348	−0.556537
$112$	−4.78187	−0.451844
$113$	−0.143115	−0.0134631	−0.00673155	−	0.999977i	$-0.502143\pi$
$113$	−0.143115	−0.0134631	−0.00673155	+	0.999977i	$0.502143\pi$
$114$	14.1078	1.32132
$115$	0.146910	0.0136995
$116$	−3.00071	−0.278609
$117$	1.44076	0.133198
$118$	6.92037	0.637072
$119$	7.39371	0.677780
$120$	−0.266563	−0.0243338
$121$	−9.86275	−0.896614
$122$	21.4322	1.94038
$123$	9.45102	0.852170
$124$	−10.8035	−0.970179
$125$	−2.66677	−0.238523
$126$	−1.86200	−0.165880
$127$	−3.88739	−0.344950	−0.172475	−	0.985014i	$-0.555176\pi$
$127$	−3.88739	−0.344950	−0.172475	+	0.985014i	$0.555176\pi$
$128$	−7.65701	−0.676790
$129$	−6.63122	−0.583846
$130$	−0.720614	−0.0632020
$131$	2.30994	0.201820	0.100910	−	0.994896i	$-0.467825\pi$
$131$	2.30994	0.201820	0.100910	+	0.994896i	$0.467825\pi$
$132$	−1.56449	−0.136171
$133$	7.57668	0.656982
$134$	22.8100	1.97048
$135$	−0.268615	−0.0231187
$136$	7.33722	0.629162
$137$	−5.99003	−0.511763	−0.255881	−	0.966708i	$-0.582366\pi$
$137$	−5.99003	−0.511763	−0.255881	+	0.966708i	$0.582366\pi$
$138$	1.01836	0.0866886
$139$	−16.1279	−1.36795	−0.683975	−	0.729506i	$-0.739750\pi$
$139$	−16.1279	−1.36795	−0.683975	+	0.729506i	$0.739750\pi$
$140$	0.394071	0.0333051
$141$	7.64458	0.643790
$142$	17.6920	1.48468
$143$	1.53646	0.128485
$144$	−4.78187	−0.398489
$145$	−0.549429	−0.0456276
$146$	−4.86731	−0.402821
$147$	−1.00000	−0.0824786
$148$	8.60200	0.707080
$149$	16.1603	1.32391	0.661953	−	0.749546i	$-0.269728\pi$
$149$	16.1603	1.32391	0.661953	+	0.749546i	$0.269728\pi$
$150$	−9.17565	−0.749189
$151$	−15.8289	−1.28813	−0.644067	−	0.764969i	$-0.722754\pi$
$151$	−15.8289	−1.28813	−0.644067	+	0.764969i	$0.722754\pi$
$152$	7.51880	0.609855
$153$	7.39371	0.597746
$154$	−1.98567	−0.160010
$155$	−1.97811	−0.158885
$156$	−2.11366	−0.169228
$157$	−5.99005	−0.478058	−0.239029	−	0.971012i	$-0.576829\pi$
$157$	−5.99005	−0.478058	−0.239029	+	0.971012i	$0.576829\pi$
$158$	20.9248	1.66469
$159$	6.35378	0.503887
$160$	1.85858	0.146934
$161$	0.546917	0.0431031
$162$	−1.86200	−0.146293
$163$	−4.37629	−0.342777	−0.171389	−	0.985204i	$-0.554825\pi$
$163$	−4.37629	−0.342777	−0.171389	+	0.985204i	$0.554825\pi$
$164$	−13.8651	−1.08268
$165$	−0.286457	−0.0223006
$166$	−14.2066	−1.10265
$167$	18.1049	1.40100	0.700500	−	0.713652i	$-0.252960\pi$
$167$	18.1049	1.40100	0.700500	+	0.713652i	$0.252960\pi$
$168$	−0.992360	−0.0765622
$169$	−10.9242	−0.840324
$170$	−3.69805	−0.283628
$171$	7.57668	0.579403
$172$	9.72830	0.741776
$173$	−16.1668	−1.22914	−0.614571	−	0.788861i	$-0.710671\pi$
$173$	−16.1668	−1.22914	−0.614571	+	0.788861i	$0.710671\pi$
$174$	−3.80855	−0.288726
$175$	−4.92785	−0.372510
$176$	−5.09948	−0.384387
$177$	3.71663	0.279359
$178$	19.8043	1.48439
$179$	−4.44660	−0.332355	−0.166177	−	0.986096i	$-0.553142\pi$
$179$	−4.44660	−0.332355	−0.166177	+	0.986096i	$0.553142\pi$
$180$	0.394071	0.0293723
$181$	−3.51904	−0.261568	−0.130784	−	0.991411i	$-0.541749\pi$
$181$	−3.51904	−0.261568	−0.130784	+	0.991411i	$0.541749\pi$
$182$	−2.68270	−0.198855
$183$	11.5103	0.850865
$184$	0.542739	0.0400112
$185$	1.57502	0.115798
$186$	−13.7119	−1.00541
$187$	7.88480	0.576593
$188$	−11.2150	−0.817935
$189$	−1.00000	−0.0727393
$190$	−3.78957	−0.274924
$191$	9.45095	0.683847	0.341923	−	0.939728i	$-0.388922\pi$
$191$	9.45095	0.683847	0.341923	+	0.939728i	$0.388922\pi$
$192$	3.31967	0.239577
$193$	−3.31980	−0.238964	−0.119482	−	0.992836i	$-0.538123\pi$
$193$	−3.31980	−0.238964	−0.119482	+	0.992836i	$0.538123\pi$
$194$	30.8135	2.21228
$195$	−0.387011	−0.0277144
$196$	1.46705	0.104789
$197$	−4.93397	−0.351531	−0.175765	−	0.984432i	$-0.556240\pi$
$197$	−4.93397	−0.351531	−0.175765	+	0.984432i	$0.556240\pi$
$198$	−1.98567	−0.141116
$199$	0.435100	0.0308434	0.0154217	−	0.999881i	$-0.495091\pi$
$199$	0.435100	0.0308434	0.0154217	+	0.999881i	$0.495091\pi$
$200$	−4.89020	−0.345789
$201$	12.2502	0.864065
$202$	−9.32103	−0.655825
$203$	−2.04541	−0.143560
$204$	−10.8469	−0.759436
$205$	−2.53869	−0.177310
$206$	9.79105	0.682175
$207$	0.546917	0.0380134
$208$	−6.88953	−0.477703
$209$	8.07992	0.558900
$210$	0.500162	0.0345145
$211$	−21.5916	−1.48642	−0.743212	−	0.669056i	$-0.766699\pi$
$211$	−21.5916	−1.48642	−0.743212	+	0.669056i	$0.766699\pi$
$212$	−9.32129	−0.640189
$213$	9.50162	0.651040
$214$	−1.21304	−0.0829219
$215$	1.78125	0.121480
$216$	−0.992360	−0.0675215
$217$	−7.36408	−0.499907
$218$	−2.17603	−0.147379
$219$	−2.61402	−0.176639
$220$	0.420245	0.0283329
$221$	10.6526	0.716569
$222$	10.9178	0.732755
$223$	−13.9096	−0.931453	−0.465727	−	0.884929i	$-0.654207\pi$
$223$	−13.9096	−0.931453	−0.465727	+	0.884929i	$0.654207\pi$
$224$	6.91912	0.462303
$225$	−4.92785	−0.328523
$226$	0.266480	0.0177260
$227$	−15.6192	−1.03668	−0.518341	−	0.855174i	$-0.673450\pi$
$227$	−15.6192	−1.03668	−0.518341	+	0.855174i	$0.673450\pi$
$228$	−11.1153	−0.736132
$229$	21.4522	1.41760	0.708800	−	0.705410i	$-0.249237\pi$
$229$	21.4522	1.41760	0.708800	+	0.705410i	$0.249237\pi$
$230$	−0.273547	−0.0180372
$231$	−1.06642	−0.0701652
$232$	−2.02978	−0.133262
$233$	26.6293	1.74454	0.872272	−	0.489021i	$-0.162646\pi$
$233$	26.6293	1.74454	0.872272	+	0.489021i	$0.162646\pi$
$234$	−2.68270	−0.175373
$235$	−2.05345	−0.133952
$236$	−5.45247	−0.354926
$237$	11.2378	0.729973
$238$	−13.7671	−0.892388
$239$	13.3419	0.863015	0.431507	−	0.902109i	$-0.357982\pi$
$239$	13.3419	0.863015	0.431507	+	0.902109i	$0.357982\pi$
$240$	1.28448	0.0829131
$241$	−24.2841	−1.56428	−0.782139	−	0.623104i	$-0.785871\pi$
$241$	−24.2841	−1.56428	−0.782139	+	0.623104i	$0.785871\pi$
$242$	18.3644	1.18051
$243$	−1.00000	−0.0641500
$244$	−16.8861	−1.08102
$245$	0.268615	0.0171612
$246$	−17.5978	−1.12200
$247$	10.9162	0.694580
$248$	−7.30782	−0.464047
$249$	−7.62977	−0.483517
$250$	4.96553	0.314048
$251$	12.3630	0.780345	0.390173	−	0.920742i	$-0.372415\pi$
$251$	12.3630	0.780345	0.390173	+	0.920742i	$0.372415\pi$
$252$	1.46705	0.0924152
$253$	0.583243	0.0366682
$254$	7.23831	0.454172
$255$	−1.98607	−0.124372
$256$	20.8967	1.30604
$257$	−1.21481	−0.0757779	−0.0378889	−	0.999282i	$-0.512063\pi$
$257$	−1.21481	−0.0757779	−0.0378889	+	0.999282i	$0.512063\pi$
$258$	12.3473	0.768711
$259$	5.86348	0.364339
$260$	0.567763	0.0352111
$261$	−2.04541	−0.126608
$262$	−4.30111	−0.265723
$263$	−7.79396	−0.480596	−0.240298	−	0.970699i	$-0.577245\pi$
$263$	−7.79396	−0.480596	−0.240298	+	0.970699i	$0.577245\pi$
$264$	−1.05827	−0.0651321
$265$	−1.70672	−0.104843
$266$	−14.1078	−0.865004
$267$	10.6360	0.650914
$268$	−17.9717	−1.09779
$269$	−3.66251	−0.223307	−0.111654	−	0.993747i	$-0.535615\pi$
$269$	−3.66251	−0.223307	−0.111654	+	0.993747i	$0.535615\pi$
$270$	0.500162	0.0304389
$271$	15.2391	0.925710	0.462855	−	0.886434i	$-0.346825\pi$
$271$	15.2391	0.925710	0.462855	+	0.886434i	$0.346825\pi$
$272$	−35.3557	−2.14376
$273$	−1.44076	−0.0871988
$274$	11.1534	0.673804
$275$	−5.25515	−0.316897
$276$	−0.802353	−0.0482960
$277$	−25.8639	−1.55401	−0.777006	−	0.629493i	$-0.783263\pi$
$277$	−25.8639	−1.55401	−0.777006	+	0.629493i	$0.783263\pi$
$278$	30.0301	1.80109
$279$	−7.36408	−0.440876
$280$	0.266563	0.0159302
$281$	−1.04481	−0.0623282	−0.0311641	−	0.999514i	$-0.509921\pi$
$281$	−1.04481	−0.0623282	−0.0311641	+	0.999514i	$0.509921\pi$
$282$	−14.2342	−0.847635
$283$	23.5522	1.40004	0.700018	−	0.714126i	$-0.253175\pi$
$283$	23.5522	1.40004	0.700018	+	0.714126i	$0.253175\pi$
$284$	−13.9393	−0.827147
$285$	−2.03521	−0.120556
$286$	−2.86088	−0.169167
$287$	−9.45102	−0.557876
$288$	6.91912	0.407713
$289$	37.6670	2.21570
$290$	1.02304	0.0600748
$291$	16.5486	0.970096
$292$	3.83489	0.224420
$293$	1.55605	0.0909053	0.0454526	−	0.998966i	$-0.485527\pi$
$293$	1.55605	0.0909053	0.0454526	+	0.998966i	$0.485527\pi$
$294$	1.86200	0.108594
$295$	−0.998345	−0.0581259
$296$	5.81869	0.338204
$297$	−1.06642	−0.0618799
$298$	−30.0905	−1.74310
$299$	0.787977	0.0455699
$300$	7.22938	0.417388
$301$	6.63122	0.382217
$302$	29.4734	1.69600
$303$	−5.00592	−0.287583
$304$	−36.2307	−2.07797
$305$	−3.09184	−0.177038
$306$	−13.7671	−0.787012
$307$	−17.2544	−0.984761	−0.492380	−	0.870380i	$-0.663873\pi$
$307$	−17.2544	−0.984761	−0.492380	+	0.870380i	$0.663873\pi$
$308$	1.56449	0.0891449
$309$	5.25835	0.299137
$310$	3.68324	0.209194
$311$	−1.70755	−0.0968265	−0.0484132	−	0.998827i	$-0.515416\pi$
$311$	−1.70755	−0.0968265	−0.0484132	+	0.998827i	$0.515416\pi$
$312$	−1.42975	−0.0809438
$313$	0.0257596	0.00145602	0.000728008	−	1.00000i	$-0.499768\pi$
$313$	0.0257596	0.00145602	0.000728008		1.00000i	$0.499768\pi$
$314$	11.1535	0.629427
$315$	0.268615	0.0151348
$316$	−16.4864	−0.927430
$317$	32.8550	1.84532	0.922660	−	0.385615i	$-0.126011\pi$
$317$	32.8550	1.84532	0.922660	+	0.385615i	$0.126011\pi$
$318$	−11.8307	−0.663435
$319$	−2.18126	−0.122127
$320$	−0.891716	−0.0498484
$321$	−0.651473	−0.0363617
$322$	−1.01836	−0.0567510
$323$	56.0198	3.11703
$324$	1.46705	0.0815026
$325$	−7.09985	−0.393829
$326$	8.14865	0.451312
$327$	−1.16865	−0.0646265
$328$	−9.37882	−0.517858
$329$	−7.64458	−0.421459
$330$	0.533383	0.0293617
$331$	20.9966	1.15408	0.577039	−	0.816717i	$-0.304208\pi$
$331$	20.9966	1.15408	0.577039	+	0.816717i	$0.304208\pi$
$332$	11.1932	0.614308
$333$	5.86348	0.321317
$334$	−33.7114	−1.84460
$335$	−3.29060	−0.179785
$336$	4.78187	0.260872
$337$	−34.9567	−1.90421	−0.952107	−	0.305764i	$-0.901088\pi$
$337$	−34.9567	−1.90421	−0.952107	+	0.305764i	$0.901088\pi$
$338$	20.3409	1.10640
$339$	0.143115	0.00777292
$340$	2.91365	0.158015
$341$	−7.85320	−0.425275
$342$	−14.1078	−0.762862
$343$	1.00000	0.0539949
$344$	6.58055	0.354800
$345$	−0.146910	−0.00790939
$346$	30.1027	1.61833
$347$	0.678920	0.0364463	0.0182232	−	0.999834i	$-0.494199\pi$
$347$	0.678920	0.0364463	0.0182232	+	0.999834i	$0.494199\pi$
$348$	3.00071	0.160855
$349$	−10.9777	−0.587625	−0.293812	−	0.955863i	$-0.594924\pi$
$349$	−10.9777	−0.587625	−0.293812	+	0.955863i	$0.594924\pi$
$350$	9.17565	0.490459
$351$	−1.44076	−0.0769021
$352$	7.37868	0.393285
$353$	−8.95985	−0.476885	−0.238442	−	0.971157i	$-0.576637\pi$
$353$	−8.95985	−0.476885	−0.238442	+	0.971157i	$0.576637\pi$
$354$	−6.92037	−0.367813
$355$	−2.55228	−0.135461
$356$	−15.6035	−0.826985
$357$	−7.39371	−0.391317
$358$	8.27958	0.437589
$359$	15.2419	0.804435	0.402218	−	0.915544i	$-0.368239\pi$
$359$	15.2419	0.804435	0.402218	+	0.915544i	$0.368239\pi$
$360$	0.266563	0.0140491
$361$	38.4061	2.02137
$362$	6.55245	0.344389
$363$	9.86275	0.517660
$364$	2.11366	0.110786
$365$	0.702166	0.0367531
$366$	−21.4322	−1.12028
$367$	18.6203	0.971971	0.485986	−	0.873967i	$-0.338461\pi$
$367$	18.6203	0.971971	0.485986	+	0.873967i	$0.338461\pi$
$368$	−2.61529	−0.136331
$369$	−9.45102	−0.492001
$370$	−2.93269	−0.152463
$371$	−6.35378	−0.329872
$372$	10.8035	0.560133
$373$	19.1726	0.992718	0.496359	−	0.868117i	$-0.334670\pi$
$373$	19.1726	0.992718	0.496359	+	0.868117i	$0.334670\pi$
$374$	−14.6815	−0.759162
$375$	2.66677	0.137712
$376$	−7.58617	−0.391227
$377$	−2.94695	−0.151775
$378$	1.86200	0.0957710
$379$	9.25475	0.475384	0.237692	−	0.971341i	$-0.423609\pi$
$379$	9.25475	0.475384	0.237692	+	0.971341i	$0.423609\pi$
$380$	2.98575	0.153166
$381$	3.88739	0.199157
$382$	−17.5977	−0.900375
$383$	1.00000	0.0510976
$383$	1.00000	0.0510976
$384$	7.65701	0.390745
$385$	0.286457	0.0145992
$386$	6.18147	0.314628
$387$	6.63122	0.337084
$388$	−24.2776	−1.23251
$389$	18.8938	0.957952	0.478976	−	0.877828i	$-0.341008\pi$
$389$	18.8938	0.957952	0.478976	+	0.877828i	$0.341008\pi$
$390$	0.720614	0.0364897
$391$	4.04375	0.204501
$392$	0.992360	0.0501217
$393$	−2.30994	−0.116521
$394$	9.18706	0.462837
$395$	−3.01864	−0.151884
$396$	1.56449	0.0786184
$397$	−18.2181	−0.914342	−0.457171	−	0.889379i	$-0.651137\pi$
$397$	−18.2181	−0.914342	−0.457171	+	0.889379i	$0.651137\pi$
$398$	−0.810156	−0.0406095
$399$	−7.57668	−0.379309
$400$	23.5643	1.17822
$401$	16.3293	0.815445	0.407723	−	0.913106i	$-0.366323\pi$
$401$	16.3293	0.815445	0.407723	+	0.913106i	$0.366323\pi$
$402$	−22.8100	−1.13766
$403$	−10.6099	−0.528516
$404$	7.34392	0.365374
$405$	0.268615	0.0133476
$406$	3.80855	0.189015
$407$	6.25293	0.309946
$408$	−7.33722	−0.363247
$409$	10.5701	0.522659	0.261329	−	0.965250i	$-0.415839\pi$
$409$	10.5701	0.522659	0.261329	+	0.965250i	$0.415839\pi$
$410$	4.72704	0.233452
$411$	5.99003	0.295466
$412$	−7.71424	−0.380053
$413$	−3.71663	−0.182883
$414$	−1.01836	−0.0500497
$415$	2.04947	0.100605
$416$	9.96880	0.488761
$417$	16.1279	0.789786
$418$	−15.0448	−0.735866
$419$	32.8029	1.60252	0.801262	−	0.598313i	$-0.204162\pi$
$419$	32.8029	1.60252	0.801262	+	0.598313i	$0.204162\pi$
$420$	−0.394071	−0.0192287
$421$	−1.65340	−0.0805819	−0.0402910	−	0.999188i	$-0.512828\pi$
$421$	−1.65340	−0.0805819	−0.0402910	+	0.999188i	$0.512828\pi$
$422$	40.2035	1.95708
$423$	−7.64458	−0.371692
$424$	−6.30524	−0.306209
$425$	−36.4351	−1.76736
$426$	−17.6920	−0.857182
$427$	−11.5103	−0.557022
$428$	0.955741	0.0461975
$429$	−1.53646	−0.0741808
$430$	−3.31668	−0.159945
$431$	−38.5275	−1.85580	−0.927902	−	0.372824i	$-0.878390\pi$
$431$	−38.5275	−1.85580	−0.927902	+	0.372824i	$0.878390\pi$
$432$	4.78187	0.230068
$433$	−20.4418	−0.982371	−0.491185	−	0.871055i	$-0.663436\pi$
$433$	−20.4418	−0.982371	−0.491185	+	0.871055i	$0.663436\pi$
$434$	13.7119	0.658194
$435$	0.549429	0.0263431
$436$	1.71446	0.0821080
$437$	4.14382	0.198226
$438$	4.86731	0.232569
$439$	−20.6963	−0.987779	−0.493890	−	0.869525i	$-0.664425\pi$
$439$	−20.6963	−0.987779	−0.493890	+	0.869525i	$0.664425\pi$
$440$	0.284268	0.0135520
$441$	1.00000	0.0476190
$442$	−19.8351	−0.943459
$443$	0.0536249	0.00254780	0.00127390	−	0.999999i	$-0.499595\pi$
$443$	0.0536249	0.00254780	0.00127390	+	0.999999i	$0.499595\pi$
$444$	−8.60200	−0.408233
$445$	−2.85700	−0.135435
$446$	25.8996	1.22638
$447$	−16.1603	−0.764357
$448$	−3.31967	−0.156840
$449$	17.6183	0.831459	0.415729	−	0.909488i	$-0.363526\pi$
$449$	17.6183	0.831459	0.415729	+	0.909488i	$0.363526\pi$
$450$	9.17565	0.432544
$451$	−10.0788	−0.474590
$452$	−0.209956	−0.00987549
$453$	15.8289	0.743705
$454$	29.0830	1.36493
$455$	0.387011	0.0181433
$456$	−7.51880	−0.352100
$457$	0.237045	0.0110885	0.00554425	−	0.999985i	$-0.498235\pi$
$457$	0.237045	0.0110885	0.00554425	+	0.999985i	$0.498235\pi$
$458$	−39.9440	−1.86646
$459$	−7.39371	−0.345109
$460$	0.215524	0.0100489
$461$	1.07154	0.0499068	0.0249534	−	0.999689i	$-0.492056\pi$
$461$	1.07154	0.0499068	0.0249534	+	0.999689i	$0.492056\pi$
$462$	1.98567	0.0923819
$463$	8.36369	0.388693	0.194347	−	0.980933i	$-0.437741\pi$
$463$	8.36369	0.388693	0.194347	+	0.980933i	$0.437741\pi$
$464$	9.78088	0.454066
$465$	1.97811	0.0917325
$466$	−49.5838	−2.29692
$467$	22.7486	1.05268	0.526340	−	0.850274i	$-0.323564\pi$
$467$	22.7486	1.05268	0.526340	+	0.850274i	$0.323564\pi$
$468$	2.11366	0.0977041
$469$	−12.2502	−0.565664
$470$	3.82353	0.176366
$471$	5.99005	0.276007
$472$	−3.68824	−0.169765
$473$	7.07166	0.325155
$474$	−20.9248	−0.961107
$475$	−37.3367	−1.71313
$476$	10.8469	0.497168
$477$	−6.35378	−0.290920
$478$	−24.8426	−1.13627
$479$	−1.26965	−0.0580116	−0.0290058	−	0.999579i	$-0.509234\pi$
$479$	−1.26965	−0.0580116	−0.0290058	+	0.999579i	$0.509234\pi$
$480$	−1.85858	−0.0848323
$481$	8.44788	0.385190
$482$	45.2170	2.05958
$483$	−0.546917	−0.0248856
$484$	−14.4691	−0.657687
$485$	−4.44521	−0.201847
$486$	1.86200	0.0844621
$487$	7.88643	0.357368	0.178684	−	0.983906i	$-0.442816\pi$
$487$	7.88643	0.357368	0.178684	+	0.983906i	$0.442816\pi$
$488$	−11.4223	−0.517065
$489$	4.37629	0.197903
$490$	−0.500162	−0.0225950
$491$	−37.7489	−1.70358	−0.851791	−	0.523881i	$-0.824484\pi$
$491$	−37.7489	−1.70358	−0.851791	+	0.523881i	$0.824484\pi$
$492$	13.8651	0.625087
$493$	−15.1232	−0.681113
$494$	−20.3259	−0.914508
$495$	0.286457	0.0128753
$496$	35.2141	1.58116
$497$	−9.50162	−0.426206
$498$	14.2066	0.636615
$499$	32.4746	1.45376	0.726881	−	0.686763i	$-0.240969\pi$
$499$	32.4746	1.45376	0.726881	+	0.686763i	$0.240969\pi$
$500$	−3.91228	−0.174962
$501$	−18.1049	−0.808868
$502$	−23.0199	−1.02743
$503$	−21.8548	−0.974459	−0.487230	−	0.873274i	$-0.661993\pi$
$503$	−21.8548	−0.974459	−0.487230	+	0.873274i	$0.661993\pi$
$504$	0.992360	0.0442032
$505$	1.34467	0.0598369
$506$	−1.08600	−0.0482785
$507$	10.9242	0.485161
$508$	−5.70297	−0.253029
$509$	−2.01516	−0.0893203	−0.0446602	−	0.999002i	$-0.514220\pi$
$509$	−2.01516	−0.0893203	−0.0446602	+	0.999002i	$0.514220\pi$
$510$	3.69805	0.163753
$511$	2.61402	0.115637
$512$	−23.5957	−1.04279
$513$	−7.57668	−0.334519
$514$	2.26198	0.0997717
$515$	−1.41247	−0.0622410
$516$	−9.72830	−0.428265
$517$	−8.15233	−0.358539
$518$	−10.9178	−0.479701
$519$	16.1668	0.709646
$520$	0.384054	0.0168419
$521$	−9.69716	−0.424840	−0.212420	−	0.977178i	$-0.568135\pi$
$521$	−9.69716	−0.424840	−0.212420	+	0.977178i	$0.568135\pi$
$522$	3.80855	0.166696
$523$	0.906382	0.0396333	0.0198167	−	0.999804i	$-0.493692\pi$
$523$	0.906382	0.0396333	0.0198167	+	0.999804i	$0.493692\pi$
$524$	3.38879	0.148040
$525$	4.92785	0.215069
$526$	14.5124	0.632769
$527$	−54.4479	−2.37179
$528$	5.09948	0.221926
$529$	−22.7009	−0.986995
$530$	3.17792	0.138040
$531$	−3.71663	−0.161288
$532$	11.1153	0.481911
$533$	−13.6167	−0.589803
$534$	−19.8043	−0.857014
$535$	0.174996	0.00756572
$536$	−12.1566	−0.525087
$537$	4.44660	0.191885
$538$	6.81960	0.294014
$539$	1.06642	0.0459339
$540$	−0.394071	−0.0169581
$541$	6.40653	0.275438	0.137719	−	0.990471i	$-0.456023\pi$
$541$	6.40653	0.275438	0.137719	+	0.990471i	$0.456023\pi$
$542$	−28.3752	−1.21882
$543$	3.51904	0.151016
$544$	51.1580	2.19338
$545$	0.313918	0.0134468
$546$	2.68270	0.114809
$547$	11.7879	0.504013	0.252006	−	0.967726i	$-0.418910\pi$
$547$	11.7879	0.504013	0.252006	+	0.967726i	$0.418910\pi$
$548$	−8.78765	−0.375390
$549$	−11.5103	−0.491247
$550$	9.78509	0.417238
$551$	−15.4974	−0.660212
$552$	−0.542739	−0.0231005
$553$	−11.2378	−0.477879
$554$	48.1587	2.04607
$555$	−1.57502	−0.0668560
$556$	−23.6604	−1.00342
$557$	−27.3437	−1.15859	−0.579294	−	0.815118i	$-0.696672\pi$
$557$	−27.3437	−1.15859	−0.579294	+	0.815118i	$0.696672\pi$
$558$	13.7119	0.580472
$559$	9.55400	0.404091
$560$	−1.28448	−0.0542793
$561$	−7.88480	−0.332896
$562$	1.94544	0.0820634
$563$	5.34869	0.225420	0.112710	−	0.993628i	$-0.464047\pi$
$563$	5.34869	0.225420	0.112710	+	0.993628i	$0.464047\pi$
$564$	11.2150	0.472235
$565$	−0.0384428	−0.00161730
$566$	−43.8543	−1.84333
$567$	1.00000	0.0419961
$568$	−9.42903	−0.395633
$569$	−36.5178	−1.53091	−0.765453	−	0.643492i	$-0.777485\pi$
$569$	−36.5178	−1.53091	−0.765453	+	0.643492i	$0.777485\pi$
$570$	3.78957	0.158728
$571$	−6.18621	−0.258885	−0.129443	−	0.991587i	$-0.541319\pi$
$571$	−6.18621	−0.258885	−0.129443	+	0.991587i	$0.541319\pi$
$572$	2.25405	0.0942466
$573$	−9.45095	−0.394819
$574$	17.5978	0.734518
$575$	−2.69512	−0.112394
$576$	−3.31967	−0.138320
$577$	6.40641	0.266702	0.133351	−	0.991069i	$-0.457426\pi$
$577$	6.40641	0.266702	0.133351	+	0.991069i	$0.457426\pi$
$578$	−70.1359	−2.91727
$579$	3.31980	0.137966
$580$	−0.806037	−0.0334689
$581$	7.62977	0.316536
$582$	−30.8135	−1.27726
$583$	−6.77579	−0.280625
$584$	2.59405	0.107343
$585$	0.387011	0.0160009
$586$	−2.89736	−0.119689
$587$	10.5043	0.433560	0.216780	−	0.976221i	$-0.430445\pi$
$587$	10.5043	0.433560	0.216780	+	0.976221i	$0.430445\pi$
$588$	−1.46705	−0.0605000
$589$	−55.7953	−2.29901
$590$	1.85892	0.0765305
$591$	4.93397	0.202957
$592$	−28.0384	−1.15237
$593$	−19.1249	−0.785365	−0.392682	−	0.919674i	$-0.628453\pi$
$593$	−19.1249	−0.785365	−0.392682	+	0.919674i	$0.628453\pi$
$594$	1.98567	0.0814732
$595$	1.98607	0.0814207
$596$	23.7079	0.971115
$597$	−0.435100	−0.0178074
$598$	−1.46721	−0.0599988
$599$	−26.7655	−1.09361	−0.546804	−	0.837260i	$-0.684156\pi$
$599$	−26.7655	−1.09361	−0.546804	+	0.837260i	$0.684156\pi$
$600$	4.89020	0.199641
$601$	29.7011	1.21153	0.605767	−	0.795642i	$-0.292866\pi$
$601$	29.7011	1.21153	0.605767	+	0.795642i	$0.292866\pi$
$602$	−12.3473	−0.503240
$603$	−12.2502	−0.498868
$604$	−23.2217	−0.944877
$605$	−2.64929	−0.107709
$606$	9.32103	0.378641
$607$	−27.4421	−1.11384	−0.556920	−	0.830566i	$-0.688017\pi$
$607$	−27.4421	−1.11384	−0.556920	+	0.830566i	$0.688017\pi$
$608$	52.4240	2.12607
$609$	2.04541	0.0828842
$610$	5.75701	0.233094
$611$	−11.0140	−0.445579
$612$	10.8469	0.438461
$613$	34.2505	1.38336	0.691682	−	0.722202i	$-0.256870\pi$
$613$	34.2505	1.38336	0.691682	+	0.722202i	$0.256870\pi$
$614$	32.1277	1.29657
$615$	2.53869	0.102370
$616$	1.05827	0.0426390
$617$	1.09788	0.0441989	0.0220994	−	0.999756i	$-0.492965\pi$
$617$	1.09788	0.0441989	0.0220994	+	0.999756i	$0.492965\pi$
$618$	−9.79105	−0.393854
$619$	20.0197	0.804658	0.402329	−	0.915495i	$-0.368201\pi$
$619$	20.0197	0.804658	0.402329	+	0.915495i	$0.368201\pi$
$620$	−2.90198	−0.116546
$621$	−0.546917	−0.0219470
$622$	3.17947	0.127485
$623$	−10.6360	−0.426123
$624$	6.88953	0.275802
$625$	23.9229	0.956916
$626$	−0.0479643	−0.00191704
$627$	−8.07992	−0.322681
$628$	−8.78768	−0.350667
$629$	43.3529	1.72859
$630$	−0.500162	−0.0199269
$631$	−1.96011	−0.0780309	−0.0390154	−	0.999239i	$-0.512422\pi$
$631$	−1.96011	−0.0780309	−0.0390154	+	0.999239i	$0.512422\pi$
$632$	−11.1519	−0.443600
$633$	21.5916	0.858188
$634$	−61.1760	−2.42961
$635$	−1.04421	−0.0414383
$636$	9.32129	0.369613
$637$	1.44076	0.0570850
$638$	4.06152	0.160797
$639$	−9.50162	−0.375878
$640$	−2.05679	−0.0813018
$641$	−25.5883	−1.01068	−0.505338	−	0.862922i	$-0.668632\pi$
$641$	−25.5883	−1.01068	−0.505338	+	0.862922i	$0.668632\pi$
$642$	1.21304	0.0478750
$643$	−7.75371	−0.305776	−0.152888	−	0.988243i	$-0.548857\pi$
$643$	−7.75371	−0.305776	−0.152888	+	0.988243i	$0.548857\pi$
$644$	0.802353	0.0316171
$645$	−1.78125	−0.0701366
$646$	−104.309	−4.10398
$647$	−28.8722	−1.13508	−0.567542	−	0.823344i	$-0.692106\pi$
$647$	−28.8722	−1.13508	−0.567542	+	0.823344i	$0.692106\pi$
$648$	0.992360	0.0389836
$649$	−3.96349	−0.155580
$650$	13.2199	0.518528
$651$	7.36408	0.288621
$652$	−6.42022	−0.251435
$653$	−20.2859	−0.793847	−0.396924	−	0.917852i	$-0.629922\pi$
$653$	−20.2859	−0.793847	−0.396924	+	0.917852i	$0.629922\pi$
$654$	2.17603	0.0850894
$655$	0.620485	0.0242444
$656$	45.1935	1.76451
$657$	2.61402	0.101983
$658$	14.2342	0.554907
$659$	28.7676	1.12063	0.560314	−	0.828280i	$-0.310680\pi$
$659$	28.7676	1.12063	0.560314	+	0.828280i	$0.310680\pi$
$660$	−0.420245	−0.0163580
$661$	−43.0546	−1.67463	−0.837314	−	0.546722i	$-0.815875\pi$
$661$	−43.0546	−1.67463	−0.837314	+	0.546722i	$0.815875\pi$
$662$	−39.0957	−1.51950
$663$	−10.6526	−0.413711
$664$	7.57148	0.293830
$665$	2.03521	0.0789222
$666$	−10.9178	−0.423057
$667$	−1.11867	−0.0433151
$668$	26.5608	1.02767
$669$	13.9096	0.537775
$670$	6.12711	0.236711
$671$	−12.2748	−0.473863
$672$	−6.91912	−0.266911
$673$	34.0839	1.31384	0.656918	−	0.753962i	$-0.271860\pi$
$673$	34.0839	1.31384	0.656918	+	0.753962i	$0.271860\pi$
$674$	65.0895	2.50715
$675$	4.92785	0.189673
$676$	−16.0263	−0.616397
$677$	49.9864	1.92113	0.960567	−	0.278047i	$-0.0896871\pi$
$677$	49.9864	1.92113	0.960567	+	0.278047i	$0.0896871\pi$
$678$	−0.266480	−0.0102341
$679$	−16.5486	−0.635077
$680$	1.97089	0.0755802
$681$	15.6192	0.598529
$682$	14.6227	0.559931
$683$	−4.69123	−0.179505	−0.0897525	−	0.995964i	$-0.528608\pi$
$683$	−4.69123	−0.179505	−0.0897525	+	0.995964i	$0.528608\pi$
$684$	11.1153	0.425006
$685$	−1.60901	−0.0614773
$686$	−1.86200	−0.0710915
$687$	−21.4522	−0.818451
$688$	−31.7096	−1.20892
$689$	−9.15428	−0.348750
$690$	0.273547	0.0104138
$691$	−3.48923	−0.132737	−0.0663683	−	0.997795i	$-0.521141\pi$
$691$	−3.48923	−0.132737	−0.0663683	+	0.997795i	$0.521141\pi$
$692$	−23.7175	−0.901605
$693$	1.06642	0.0405099
$694$	−1.26415	−0.0479864
$695$	−4.33220	−0.164330
$696$	2.02978	0.0769387
$697$	−69.8781	−2.64682
$698$	20.4406	0.773686
$699$	−26.6293	−1.00721
$700$	−7.22938	−0.273245
$701$	−31.1270	−1.17565	−0.587825	−	0.808988i	$-0.700016\pi$
$701$	−31.1270	−1.17565	−0.587825	+	0.808988i	$0.700016\pi$
$702$	2.68270	0.101252
$703$	44.4258	1.67555
$704$	−3.54016	−0.133425
$705$	2.05345	0.0773375
$706$	16.6832	0.627882
$707$	5.00592	0.188267
$708$	5.45247	0.204916
$709$	−12.9759	−0.487321	−0.243661	−	0.969861i	$-0.578348\pi$
$709$	−12.9759	−0.487321	−0.243661	+	0.969861i	$0.578348\pi$
$710$	4.75235	0.178353
$711$	−11.2378	−0.421450
$712$	−10.5548	−0.395556
$713$	−4.02754	−0.150833
$714$	13.7671	0.515221
$715$	0.412716	0.0154347
$716$	−6.52337	−0.243790
$717$	−13.3419	−0.498262
$718$	−28.3804	−1.05915
$719$	22.7660	0.849029	0.424514	−	0.905421i	$-0.360445\pi$
$719$	22.7660	0.849029	0.424514	+	0.905421i	$0.360445\pi$
$720$	−1.28448	−0.0478699
$721$	−5.25835	−0.195831
$722$	−71.5122	−2.66141
$723$	24.2841	0.903136
$724$	−5.16259	−0.191866
$725$	10.0795	0.374342
$726$	−18.3644	−0.681569
$727$	−37.2635	−1.38203	−0.691014	−	0.722842i	$-0.742836\pi$
$727$	−37.2635	−1.38203	−0.691014	+	0.722842i	$0.742836\pi$
$728$	1.42975	0.0529902
$729$	1.00000	0.0370370
$730$	−1.30743	−0.0483903
$731$	49.0293	1.81341
$732$	16.8861	0.624129
$733$	−17.1813	−0.634607	−0.317304	−	0.948324i	$-0.602777\pi$
$733$	−17.1813	−0.634607	−0.317304	+	0.948324i	$0.602777\pi$
$734$	−34.6710	−1.27973
$735$	−0.268615	−0.00990803
$736$	3.78419	0.139487
$737$	−13.0639	−0.481215
$738$	17.5978	0.647784
$739$	−14.3472	−0.527772	−0.263886	−	0.964554i	$-0.585004\pi$
$739$	−14.3472	−0.527772	−0.263886	+	0.964554i	$0.585004\pi$
$740$	2.31063	0.0849405
$741$	−10.9162	−0.401016
$742$	11.8307	0.434320
$743$	−6.43084	−0.235925	−0.117962	−	0.993018i	$-0.537636\pi$
$743$	−6.43084	−0.235925	−0.117962	+	0.993018i	$0.537636\pi$
$744$	7.30782	0.267918
$745$	4.34091	0.159039
$746$	−35.6993	−1.30704
$747$	7.62977	0.279159
$748$	11.5674	0.422945
$749$	0.651473	0.0238043
$750$	−4.96553	−0.181316
$751$	15.5000	0.565604	0.282802	−	0.959178i	$-0.408736\pi$
$751$	15.5000	0.565604	0.282802	+	0.959178i	$0.408736\pi$
$752$	36.5554	1.33304
$753$	−12.3630	−0.450533
$754$	5.48722	0.199833
$755$	−4.25188	−0.154742
$756$	−1.46705	−0.0533560
$757$	−33.3619	−1.21256	−0.606280	−	0.795251i	$-0.707339\pi$
$757$	−33.3619	−1.21256	−0.606280	+	0.795251i	$0.707339\pi$
$758$	−17.2323	−0.625907
$759$	−0.583243	−0.0211704
$760$	2.01966	0.0732609
$761$	30.9380	1.12150	0.560752	−	0.827984i	$-0.310512\pi$
$761$	30.9380	1.12150	0.560752	+	0.827984i	$0.310512\pi$
$762$	−7.23831	−0.262216
$763$	1.16865	0.0423080
$764$	13.8650	0.501617
$765$	1.98607	0.0718063
$766$	−1.86200	−0.0672768
$767$	−5.35478	−0.193350
$768$	−20.8967	−0.754045
$769$	−36.4924	−1.31595	−0.657974	−	0.753040i	$-0.728586\pi$
$769$	−36.4924	−1.31595	−0.657974	+	0.753040i	$0.728586\pi$
$770$	−0.533383	−0.0192218
$771$	1.21481	0.0437504
$772$	−4.87030	−0.175286
$773$	2.19783	0.0790505	0.0395252	−	0.999219i	$-0.487415\pi$
$773$	2.19783	0.0790505	0.0395252	+	0.999219i	$0.487415\pi$
$774$	−12.3473	−0.443816
$775$	36.2891	1.30354
$776$	−16.4222	−0.589522
$777$	−5.86348	−0.210351
$778$	−35.1802	−1.26127
$779$	−71.6074	−2.56560
$780$	−0.567763	−0.0203292
$781$	−10.1327	−0.362577
$782$	−7.52946	−0.269253
$783$	2.04541	0.0730970
$784$	−4.78187	−0.170781
$785$	−1.60902	−0.0574284
$786$	4.30111	0.153415
$787$	11.2406	0.400685	0.200343	−	0.979726i	$-0.435795\pi$
$787$	11.2406	0.400685	0.200343	+	0.979726i	$0.435795\pi$
$788$	−7.23837	−0.257856
$789$	7.79396	0.277472
$790$	5.62072	0.199976
$791$	−0.143115	−0.00508857
$792$	1.05827	0.0376040
$793$	−16.5836	−0.588900
$794$	33.9222	1.20385
$795$	1.70672	0.0605312
$796$	0.638312	0.0226244
$797$	−1.47115	−0.0521107	−0.0260553	−	0.999661i	$-0.508295\pi$
$797$	−1.47115	−0.0521107	−0.0260553	+	0.999661i	$0.508295\pi$
$798$	14.1078	0.499410
$799$	−56.5218	−1.99960
$800$	−34.0964	−1.20549
$801$	−10.6360	−0.375805
$802$	−30.4051	−1.07364
$803$	2.78764	0.0983737
$804$	17.9717	0.633812
$805$	0.146910	0.00517791
$806$	19.7556	0.695862
$807$	3.66251	0.128926
$808$	4.96767	0.174762
$809$	31.6126	1.11144	0.555720	−	0.831369i	$-0.312443\pi$
$809$	31.6126	1.11144	0.555720	+	0.831369i	$0.312443\pi$
$810$	−0.500162	−0.0175739
$811$	19.1973	0.674108	0.337054	−	0.941485i	$-0.390570\pi$
$811$	19.1973	0.674108	0.337054	+	0.941485i	$0.390570\pi$
$812$	−3.00071	−0.105304
$813$	−15.2391	−0.534459
$814$	−11.6430	−0.408086
$815$	−1.17554	−0.0411773
$816$	35.3557	1.23770
$817$	50.2426	1.75777
$818$	−19.6816	−0.688150
$819$	1.44076	0.0503443
$820$	−3.72438	−0.130061
$821$	20.5326	0.716594	0.358297	−	0.933608i	$-0.383358\pi$
$821$	20.5326	0.716594	0.358297	+	0.933608i	$0.383358\pi$
$822$	−11.1534	−0.389021
$823$	29.4845	1.02776	0.513882	−	0.857861i	$-0.328207\pi$
$823$	29.4845	1.02776	0.513882	+	0.857861i	$0.328207\pi$
$824$	−5.21817	−0.181784
$825$	5.25515	0.182961
$826$	6.92037	0.240790
$827$	24.8544	0.864272	0.432136	−	0.901808i	$-0.357760\pi$
$827$	24.8544	0.864272	0.432136	+	0.901808i	$0.357760\pi$
$828$	0.802353	0.0278837
$829$	−11.7088	−0.406664	−0.203332	−	0.979110i	$-0.565177\pi$
$829$	−11.7088	−0.406664	−0.203332	+	0.979110i	$0.565177\pi$
$830$	−3.81612	−0.132460
$831$	25.8639	0.897210
$832$	−4.78285	−0.165816
$833$	7.39371	0.256177
$834$	−30.0301	−1.03986
$835$	4.86326	0.168300
$836$	11.8536	0.409966
$837$	7.36408	0.254540
$838$	−61.0789	−2.10994
$839$	30.4157	1.05006	0.525032	−	0.851082i	$-0.324053\pi$
$839$	30.4157	1.05006	0.525032	+	0.851082i	$0.324053\pi$
$840$	−0.266563	−0.00919730
$841$	−24.8163	−0.855734
$842$	3.07864	0.106097
$843$	1.04481	0.0359852
$844$	−31.6758	−1.09033
$845$	−2.93441	−0.100947
$846$	14.2342	0.489382
$847$	−9.86275	−0.338888
$848$	30.3829	1.04335
$849$	−23.5522	−0.808311
$850$	67.8421	2.32697
$851$	3.20684	0.109929
$852$	13.9393	0.477553
$853$	−36.2220	−1.24022	−0.620108	−	0.784516i	$-0.712911\pi$
$853$	−36.2220	−1.24022	−0.620108	+	0.784516i	$0.712911\pi$
$854$	21.4322	0.733393
$855$	2.03521	0.0696029
$856$	0.646495	0.0220968
$857$	15.3235	0.523439	0.261720	−	0.965144i	$-0.415710\pi$
$857$	15.3235	0.523439	0.261720	+	0.965144i	$0.415710\pi$
$858$	2.86088	0.0976689
$859$	6.54411	0.223282	0.111641	−	0.993749i	$-0.464389\pi$
$859$	6.54411	0.223282	0.111641	+	0.993749i	$0.464389\pi$
$860$	2.61317	0.0891085
$861$	9.45102	0.322090
$862$	71.7382	2.44341
$863$	−51.1352	−1.74066	−0.870331	−	0.492468i	$-0.836095\pi$
$863$	−51.1352	−1.74066	−0.870331	+	0.492468i	$0.836095\pi$
$864$	−6.91912	−0.235393
$865$	−4.34267	−0.147655
$866$	38.0627	1.29342
$867$	−37.6670	−1.27924
$868$	−10.8035	−0.366693
$869$	−11.9842	−0.406536
$870$	−1.02304	−0.0346842
$871$	−17.6497	−0.598036
$872$	1.15972	0.0392731
$873$	−16.5486	−0.560085
$874$	−7.71579	−0.260991
$875$	−2.66677	−0.0901534
$876$	−3.83489	−0.129569
$877$	−6.89360	−0.232780	−0.116390	−	0.993204i	$-0.537132\pi$
$877$	−6.89360	−0.232780	−0.116390	+	0.993204i	$0.537132\pi$
$878$	38.5365	1.30054
$879$	−1.55605	−0.0524842
$880$	−1.36980	−0.0461759
$881$	16.0172	0.539634	0.269817	−	0.962912i	$-0.413037\pi$
$881$	16.0172	0.539634	0.269817	+	0.962912i	$0.413037\pi$
$882$	−1.86200	−0.0626968
$883$	−41.6638	−1.40210	−0.701049	−	0.713113i	$-0.747285\pi$
$883$	−41.6638	−1.40210	−0.701049	+	0.713113i	$0.747285\pi$
$884$	15.6278	0.525620
$885$	0.998345	0.0335590
$886$	−0.0998496	−0.00335451
$887$	58.1756	1.95335	0.976673	−	0.214730i	$-0.0688872\pi$
$887$	58.1756	1.95335	0.976673	+	0.214730i	$0.0688872\pi$
$888$	−5.81869	−0.195262
$889$	−3.88739	−0.130379
$890$	5.31973	0.178318
$891$	1.06642	0.0357264
$892$	−20.4060	−0.683243
$893$	−57.9206	−1.93824
$894$	30.0905	1.00638
$895$	−1.19443	−0.0399253
$896$	−7.65701	−0.255803
$897$	−0.787977	−0.0263098
$898$	−32.8053	−1.09473
$899$	15.0626	0.502365
$900$	−7.22938	−0.240979
$901$	−46.9780	−1.56506
$902$	18.7666	0.624861
$903$	−6.63122	−0.220673
$904$	−0.142021	−0.00472356
$905$	−0.945268	−0.0314218
$906$	−29.4734	−0.979187
$907$	0.485050	0.0161058	0.00805292	−	0.999968i	$-0.497437\pi$
$907$	0.485050	0.0161058	0.00805292	+	0.999968i	$0.497437\pi$
$908$	−22.9141	−0.760431
$909$	5.00592	0.166036
$910$	−0.720614	−0.0238881
$911$	51.5077	1.70653	0.853263	−	0.521481i	$-0.174620\pi$
$911$	51.5077	1.70653	0.853263	+	0.521481i	$0.174620\pi$
$912$	36.2307	1.19972
$913$	8.13654	0.269280
$914$	−0.441378	−0.0145995
$915$	3.09184	0.102213
$916$	31.4713	1.03984
$917$	2.30994	0.0762809
$918$	13.7671	0.454382
$919$	7.32341	0.241577	0.120789	−	0.992678i	$-0.461458\pi$
$919$	7.32341	0.241577	0.120789	+	0.992678i	$0.461458\pi$
$920$	0.145788	0.00480649
$921$	17.2544	0.568552
$922$	−1.99522	−0.0657089
$923$	−13.6896	−0.450598
$924$	−1.56449	−0.0514678
$925$	−28.8943	−0.950040
$926$	−15.5732	−0.511767
$927$	−5.25835	−0.172707
$928$	−14.1524	−0.464576
$929$	−33.9887	−1.11513	−0.557566	−	0.830132i	$-0.688265\pi$
$929$	−33.9887	−1.11513	−0.557566	+	0.830132i	$0.688265\pi$
$930$	−3.68324	−0.120778
$931$	7.57668	0.248316
$932$	39.0664	1.27966
$933$	1.70755	0.0559028
$934$	−42.3579	−1.38599
$935$	2.11798	0.0692653
$936$	1.42975	0.0467329
$937$	−29.7134	−0.970694	−0.485347	−	0.874322i	$-0.661307\pi$
$937$	−29.7134	−0.970694	−0.485347	+	0.874322i	$0.661307\pi$
$938$	22.8100	0.744772
$939$	−0.0257596	−0.000840631 0
$940$	−3.01251	−0.0982573
$941$	15.1038	0.492370	0.246185	−	0.969223i	$-0.420823\pi$
$941$	15.1038	0.492370	0.246185	+	0.969223i	$0.420823\pi$
$942$	−11.1535	−0.363400
$943$	−5.16893	−0.168323
$944$	17.7724	0.578444
$945$	−0.268615	−0.00873806
$946$	−13.1674	−0.428110
$947$	44.6487	1.45089	0.725443	−	0.688282i	$-0.241635\pi$
$947$	44.6487	1.45089	0.725443	+	0.688282i	$0.241635\pi$
$948$	16.4864	0.535452
$949$	3.76618	0.122255
$950$	69.5210	2.25556
$951$	−32.8550	−1.06540
$952$	7.33722	0.237801
$953$	34.4991	1.11753	0.558767	−	0.829325i	$-0.311275\pi$
$953$	34.4991	1.11753	0.558767	+	0.829325i	$0.311275\pi$
$954$	11.8307	0.383034
$955$	2.53867	0.0821495
$956$	19.5732	0.633041
$957$	2.18126	0.0705103
$958$	2.36408	0.0763800
$959$	−5.99003	−0.193428
$960$	0.891716	0.0287800
$961$	23.2297	0.749346
$962$	−15.7300	−0.507154
$963$	0.651473	0.0209934
$964$	−35.6259	−1.14743
$965$	−0.891749	−0.0287064
$966$	1.01836	0.0327652
$967$	38.8771	1.25020	0.625102	−	0.780543i	$-0.285058\pi$
$967$	38.8771	1.25020	0.625102	+	0.780543i	$0.285058\pi$
$968$	−9.78740	−0.314579
$969$	−56.0198	−1.79962
$970$	8.27699	0.265758
$971$	0.674669	0.0216512	0.0108256	−	0.999941i	$-0.496554\pi$
$971$	0.674669	0.0216512	0.0108256	+	0.999941i	$0.496554\pi$
$972$	−1.46705	−0.0470555
$973$	−16.1279	−0.517036
$974$	−14.6845	−0.470523
$975$	7.09985	0.227377
$976$	55.0407	1.76181
$977$	−41.8285	−1.33821	−0.669107	−	0.743166i	$-0.733323\pi$
$977$	−41.8285	−1.33821	−0.669107	+	0.743166i	$0.733323\pi$
$978$	−8.14865	−0.260565
$979$	−11.3425	−0.362506
$980$	0.394071	0.0125881
$981$	1.16865	0.0373121
$982$	70.2884	2.24299
$983$	−11.7710	−0.375436	−0.187718	−	0.982223i	$-0.560109\pi$
$983$	−11.7710	−0.375436	−0.187718	+	0.982223i	$0.560109\pi$
$984$	9.37882	0.298986
$985$	−1.32534	−0.0422289
$986$	28.1593	0.896776
$987$	7.64458	0.243330
$988$	16.0146	0.509491
$989$	3.62673	0.115323
$990$	−0.533383	−0.0169520
$991$	−51.8749	−1.64786	−0.823931	−	0.566691i	$-0.808223\pi$
$991$	−51.8749	−1.64786	−0.823931	+	0.566691i	$0.808223\pi$
$992$	−50.9530	−1.61776
$993$	−20.9966	−0.666307
$994$	17.6920	0.561157
$995$	0.116875	0.00370517
$996$	−11.1932	−0.354671
$997$	0.717182	0.0227134	0.0113567	−	0.999936i	$-0.496385\pi$
$997$	0.717182	0.0227134	0.0113567	+	0.999936i	$0.496385\pi$
$998$	−60.4677	−1.91407
$999$	−5.86348	−0.185512

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.o.1.9	✓	41

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.o.1.9	✓	41	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q-1.86200 q^{2} -1.00000 q^{3} +1.46705 q^{4} +0.268615 q^{5} +1.86200 q^{6} +1.00000 q^{7} +0.992360 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.86200 q^{2} -1.00000 q^{3} +1.46705 q^{4} +0.268615 q^{5} +1.86200 q^{6} +1.00000 q^{7} +0.992360 q^{8} +1.00000 q^{9} -0.500162 q^{10} +1.06642 q^{11} -1.46705 q^{12} +1.44076 q^{13} -1.86200 q^{14} -0.268615 q^{15} -4.78187 q^{16} +7.39371 q^{17} -1.86200 q^{18} +7.57668 q^{19} +0.394071 q^{20} -1.00000 q^{21} -1.98567 q^{22} +0.546917 q^{23} -0.992360 q^{24} -4.92785 q^{25} -2.68270 q^{26} -1.00000 q^{27} +1.46705 q^{28} -2.04541 q^{29} +0.500162 q^{30} -7.36408 q^{31} +6.91912 q^{32} -1.06642 q^{33} -13.7671 q^{34} +0.268615 q^{35} +1.46705 q^{36} +5.86348 q^{37} -14.1078 q^{38} -1.44076 q^{39} +0.266563 q^{40} -9.45102 q^{41} +1.86200 q^{42} +6.63122 q^{43} +1.56449 q^{44} +0.268615 q^{45} -1.01836 q^{46} -7.64458 q^{47} +4.78187 q^{48} +1.00000 q^{49} +9.17565 q^{50} -7.39371 q^{51} +2.11366 q^{52} -6.35378 q^{53} +1.86200 q^{54} +0.286457 q^{55} +0.992360 q^{56} -7.57668 q^{57} +3.80855 q^{58} -3.71663 q^{59} -0.394071 q^{60} -11.5103 q^{61} +13.7119 q^{62} +1.00000 q^{63} -3.31967 q^{64} +0.387011 q^{65} +1.98567 q^{66} -12.2502 q^{67} +10.8469 q^{68} -0.546917 q^{69} -0.500162 q^{70} -9.50162 q^{71} +0.992360 q^{72} +2.61402 q^{73} -10.9178 q^{74} +4.92785 q^{75} +11.1153 q^{76} +1.06642 q^{77} +2.68270 q^{78} -11.2378 q^{79} -1.28448 q^{80} +1.00000 q^{81} +17.5978 q^{82} +7.62977 q^{83} -1.46705 q^{84} +1.98607 q^{85} -12.3473 q^{86} +2.04541 q^{87} +1.05827 q^{88} -10.6360 q^{89} -0.500162 q^{90} +1.44076 q^{91} +0.802353 q^{92} +7.36408 q^{93} +14.2342 q^{94} +2.03521 q^{95} -6.91912 q^{96} -16.5486 q^{97} -1.86200 q^{98} +1.06642 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 41 q - 4 q^{2} - 41 q^{3} + 34 q^{4} + 5 q^{5} + 4 q^{6} + 41 q^{7} - 15 q^{8} + 41 q^{9}+O(q^{10}) \) 41 * q - 4 * q^2 - 41 * q^3 + 34 * q^4 + 5 * q^5 + 4 * q^6 + 41 * q^7 - 15 * q^8 + 41 * q^9	\( 41 q - 4 q^{2} - 41 q^{3} + 34 q^{4} + 5 q^{5} + 4 q^{6} + 41 q^{7} - 15 q^{8} + 41 q^{9} - 18 q^{10} - 17 q^{11} - 34 q^{12} - 38 q^{13} - 4 q^{14} - 5 q^{15} + 16 q^{16} + 4 q^{17} - 4 q^{18} - 15 q^{19} + 23 q^{20} - 41 q^{21} - 31 q^{22} - 8 q^{23} + 15 q^{24} + 16 q^{25} + 15 q^{26} - 41 q^{27} + 34 q^{28} - 27 q^{29} + 18 q^{30} - 23 q^{31} - 24 q^{32} + 17 q^{33} - 9 q^{34} + 5 q^{35} + 34 q^{36} - 81 q^{37} + 38 q^{39} - 52 q^{40} + 29 q^{41} + 4 q^{42} - 47 q^{43} - 20 q^{44} + 5 q^{45} - 40 q^{46} + 26 q^{47} - 16 q^{48} + 41 q^{49} - 23 q^{50} - 4 q^{51} - 64 q^{52} - 66 q^{53} + 4 q^{54} - 14 q^{55} - 15 q^{56} + 15 q^{57} - 50 q^{58} + 41 q^{59} - 23 q^{60} - 47 q^{61} + 5 q^{62} + 41 q^{63} - 37 q^{64} - 24 q^{65} + 31 q^{66} - 73 q^{67} + 10 q^{68} + 8 q^{69} - 18 q^{70} + q^{71} - 15 q^{72} - 14 q^{73} + 18 q^{74} - 16 q^{75} - 37 q^{76} - 17 q^{77} - 15 q^{78} - 80 q^{79} + 63 q^{80} + 41 q^{81} - 31 q^{82} + 20 q^{83} - 34 q^{84} - 82 q^{85} - 39 q^{86} + 27 q^{87} - 132 q^{88} + 17 q^{89} - 18 q^{90} - 38 q^{91} - 26 q^{92} + 23 q^{93} - 9 q^{94} - q^{95} + 24 q^{96} - 73 q^{97} - 4 q^{98} - 17 q^{99}+O(q^{100}) \) 41 * q - 4 * q^2 - 41 * q^3 + 34 * q^4 + 5 * q^5 + 4 * q^6 + 41 * q^7 - 15 * q^8 + 41 * q^9 - 18 * q^10 - 17 * q^11 - 34 * q^12 - 38 * q^13 - 4 * q^14 - 5 * q^15 + 16 * q^16 + 4 * q^17 - 4 * q^18 - 15 * q^19 + 23 * q^20 - 41 * q^21 - 31 * q^22 - 8 * q^23 + 15 * q^24 + 16 * q^25 + 15 * q^26 - 41 * q^27 + 34 * q^28 - 27 * q^29 + 18 * q^30 - 23 * q^31 - 24 * q^32 + 17 * q^33 - 9 * q^34 + 5 * q^35 + 34 * q^36 - 81 * q^37 + 38 * q^39 - 52 * q^40 + 29 * q^41 + 4 * q^42 - 47 * q^43 - 20 * q^44 + 5 * q^45 - 40 * q^46 + 26 * q^47 - 16 * q^48 + 41 * q^49 - 23 * q^50 - 4 * q^51 - 64 * q^52 - 66 * q^53 + 4 * q^54 - 14 * q^55 - 15 * q^56 + 15 * q^57 - 50 * q^58 + 41 * q^59 - 23 * q^60 - 47 * q^61 + 5 * q^62 + 41 * q^63 - 37 * q^64 - 24 * q^65 + 31 * q^66 - 73 * q^67 + 10 * q^68 + 8 * q^69 - 18 * q^70 + q^71 - 15 * q^72 - 14 * q^73 + 18 * q^74 - 16 * q^75 - 37 * q^76 - 17 * q^77 - 15 * q^78 - 80 * q^79 + 63 * q^80 + 41 * q^81 - 31 * q^82 + 20 * q^83 - 34 * q^84 - 82 * q^85 - 39 * q^86 + 27 * q^87 - 132 * q^88 + 17 * q^89 - 18 * q^90 - 38 * q^91 - 26 * q^92 + 23 * q^93 - 9 * q^94 - q^95 + 24 * q^96 - 73 * q^97 - 4 * q^98 - 17 * q^99

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