LMFDB - Embedded newform 6038.2.a.d.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6038,2,Mod(1,6038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6038, 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("6038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6038 = 2 \cdot 3019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6038.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:	$48.2136727404$
Analytic rank:	$0$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	6038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.18407 q^{3} +1.00000 q^{4} +2.82880 q^{5} +2.18407 q^{6} -4.34635 q^{7} -1.00000 q^{8} +1.77018 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.18407 q^{3} +1.00000 q^{4} +2.82880 q^{5} +2.18407 q^{6} -4.34635 q^{7} -1.00000 q^{8} +1.77018 q^{9} -2.82880 q^{10} -5.10940 q^{11} -2.18407 q^{12} +1.49187 q^{13} +4.34635 q^{14} -6.17832 q^{15} +1.00000 q^{16} +2.74241 q^{17} -1.77018 q^{18} +7.11472 q^{19} +2.82880 q^{20} +9.49275 q^{21} +5.10940 q^{22} -2.22934 q^{23} +2.18407 q^{24} +3.00212 q^{25} -1.49187 q^{26} +2.68601 q^{27} -4.34635 q^{28} -5.22351 q^{29} +6.17832 q^{30} +1.28246 q^{31} -1.00000 q^{32} +11.1593 q^{33} -2.74241 q^{34} -12.2950 q^{35} +1.77018 q^{36} +10.8635 q^{37} -7.11472 q^{38} -3.25835 q^{39} -2.82880 q^{40} -0.414107 q^{41} -9.49275 q^{42} -7.40257 q^{43} -5.10940 q^{44} +5.00750 q^{45} +2.22934 q^{46} -4.54934 q^{47} -2.18407 q^{48} +11.8907 q^{49} -3.00212 q^{50} -5.98964 q^{51} +1.49187 q^{52} -8.58927 q^{53} -2.68601 q^{54} -14.4535 q^{55} +4.34635 q^{56} -15.5391 q^{57} +5.22351 q^{58} -10.7218 q^{59} -6.17832 q^{60} -4.49619 q^{61} -1.28246 q^{62} -7.69383 q^{63} +1.00000 q^{64} +4.22019 q^{65} -11.1593 q^{66} +2.37942 q^{67} +2.74241 q^{68} +4.86905 q^{69} +12.2950 q^{70} -5.81418 q^{71} -1.77018 q^{72} -7.44172 q^{73} -10.8635 q^{74} -6.55686 q^{75} +7.11472 q^{76} +22.2072 q^{77} +3.25835 q^{78} -7.80847 q^{79} +2.82880 q^{80} -11.1770 q^{81} +0.414107 q^{82} -8.14913 q^{83} +9.49275 q^{84} +7.75775 q^{85} +7.40257 q^{86} +11.4085 q^{87} +5.10940 q^{88} -7.46661 q^{89} -5.00750 q^{90} -6.48417 q^{91} -2.22934 q^{92} -2.80098 q^{93} +4.54934 q^{94} +20.1261 q^{95} +2.18407 q^{96} +14.7776 q^{97} -11.8907 q^{98} -9.04458 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) $ 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9} - 18 q^{10} - 4 q^{11} + 8 q^{12} + 40 q^{13} - 32 q^{14} + 6 q^{15} + 69 q^{16} + 6 q^{17} - 79 q^{18} + 13 q^{19} + 18 q^{20} + 23 q^{21} + 4 q^{22} + 11 q^{23} - 8 q^{24} + 111 q^{25} - 40 q^{26} + 32 q^{27} + 32 q^{28} + 23 q^{29} - 6 q^{30} + 30 q^{31} - 69 q^{32} + 37 q^{33} - 6 q^{34} - 12 q^{35} + 79 q^{36} + 81 q^{37} - 13 q^{38} - 8 q^{39} - 18 q^{40} - 13 q^{41} - 23 q^{42} + 42 q^{43} - 4 q^{44} + 89 q^{45} - 11 q^{46} + 35 q^{47} + 8 q^{48} + 115 q^{49} - 111 q^{50} - 21 q^{51} + 40 q^{52} + 41 q^{53} - 32 q^{54} + 28 q^{55} - 32 q^{56} + 39 q^{57} - 23 q^{58} - 24 q^{59} + 6 q^{60} + 69 q^{61} - 30 q^{62} + 79 q^{63} + 69 q^{64} + 15 q^{65} - 37 q^{66} + 87 q^{67} + 6 q^{68} + 51 q^{69} + 12 q^{70} - 22 q^{71} - 79 q^{72} + 104 q^{73} - 81 q^{74} + 39 q^{75} + 13 q^{76} + 47 q^{77} + 8 q^{78} + 16 q^{79} + 18 q^{80} + 105 q^{81} + 13 q^{82} + 30 q^{83} + 23 q^{84} + 74 q^{85} - 42 q^{86} + 47 q^{87} + 4 q^{88} - 4 q^{89} - 89 q^{90} + 70 q^{91} + 11 q^{92} + 104 q^{93} - 35 q^{94} - 36 q^{95} - 8 q^{96} + 158 q^{97} - 115 q^{98} + 34 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9 - 18 * q^10 - 4 * q^11 + 8 * q^12 + 40 * q^13 - 32 * q^14 + 6 * q^15 + 69 * q^16 + 6 * q^17 - 79 * q^18 + 13 * q^19 + 18 * q^20 + 23 * q^21 + 4 * q^22 + 11 * q^23 - 8 * q^24 + 111 * q^25 - 40 * q^26 + 32 * q^27 + 32 * q^28 + 23 * q^29 - 6 * q^30 + 30 * q^31 - 69 * q^32 + 37 * q^33 - 6 * q^34 - 12 * q^35 + 79 * q^36 + 81 * q^37 - 13 * q^38 - 8 * q^39 - 18 * q^40 - 13 * q^41 - 23 * q^42 + 42 * q^43 - 4 * q^44 + 89 * q^45 - 11 * q^46 + 35 * q^47 + 8 * q^48 + 115 * q^49 - 111 * q^50 - 21 * q^51 + 40 * q^52 + 41 * q^53 - 32 * q^54 + 28 * q^55 - 32 * q^56 + 39 * q^57 - 23 * q^58 - 24 * q^59 + 6 * q^60 + 69 * q^61 - 30 * q^62 + 79 * q^63 + 69 * q^64 + 15 * q^65 - 37 * q^66 + 87 * q^67 + 6 * q^68 + 51 * q^69 + 12 * q^70 - 22 * q^71 - 79 * q^72 + 104 * q^73 - 81 * q^74 + 39 * q^75 + 13 * q^76 + 47 * q^77 + 8 * q^78 + 16 * q^79 + 18 * q^80 + 105 * q^81 + 13 * q^82 + 30 * q^83 + 23 * q^84 + 74 * q^85 - 42 * q^86 + 47 * q^87 + 4 * q^88 - 4 * q^89 - 89 * q^90 + 70 * q^91 + 11 * q^92 + 104 * q^93 - 35 * q^94 - 36 * q^95 - 8 * q^96 + 158 * q^97 - 115 * q^98 + 34 * 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
$3$	−2.18407	−1.26098	−0.630488	−	0.776199i	$-0.717145\pi$
$3$	−2.18407	−1.26098	−0.630488	+	0.776199i	$0.717145\pi$
$4$	1.00000	0.500000
$5$	2.82880	1.26508	0.632539	−	0.774528i	$-0.282013\pi$
$5$	2.82880	1.26508	0.632539	+	0.774528i	$0.282013\pi$
$6$	2.18407	0.891645
$7$	−4.34635	−1.64276	−0.821382	−	0.570378i	$-0.806797\pi$
$7$	−4.34635	−1.64276	−0.821382	+	0.570378i	$0.806797\pi$
$8$	−1.00000	−0.353553
$9$	1.77018	0.590061
$10$	−2.82880	−0.894546
$11$	−5.10940	−1.54054	−0.770272	−	0.637716i	$-0.779879\pi$
$11$	−5.10940	−1.54054	−0.770272	+	0.637716i	$0.779879\pi$
$12$	−2.18407	−0.630488
$13$	1.49187	0.413769	0.206885	−	0.978365i	$-0.433668\pi$
$13$	1.49187	0.413769	0.206885	+	0.978365i	$0.433668\pi$
$14$	4.34635	1.16161
$15$	−6.17832	−1.59523
$16$	1.00000	0.250000
$17$	2.74241	0.665133	0.332567	−	0.943080i	$-0.392085\pi$
$17$	2.74241	0.665133	0.332567	+	0.943080i	$0.392085\pi$
$18$	−1.77018	−0.417236
$19$	7.11472	1.63223	0.816114	−	0.577890i	$-0.196124\pi$
$19$	7.11472	1.63223	0.816114	+	0.577890i	$0.196124\pi$
$20$	2.82880	0.632539
$21$	9.49275	2.07149
$22$	5.10940	1.08933
$23$	−2.22934	−0.464850	−0.232425	−	0.972614i	$-0.574666\pi$
$23$	−2.22934	−0.464850	−0.232425	+	0.972614i	$0.574666\pi$
$24$	2.18407	0.445822
$25$	3.00212	0.600425
$26$	−1.49187	−0.292579
$27$	2.68601	0.516923
$28$	−4.34635	−0.821382
$29$	−5.22351	−0.969981	−0.484991	−	0.874519i	$-0.661177\pi$
$29$	−5.22351	−0.969981	−0.484991	+	0.874519i	$0.661177\pi$
$30$	6.17832	1.12800
$31$	1.28246	0.230336	0.115168	−	0.993346i	$-0.463259\pi$
$31$	1.28246	0.230336	0.115168	+	0.993346i	$0.463259\pi$
$32$	−1.00000	−0.176777
$33$	11.1593	1.94259
$34$	−2.74241	−0.470320
$35$	−12.2950	−2.07823
$36$	1.77018	0.295031
$37$	10.8635	1.78596	0.892978	−	0.450101i	$-0.148612\pi$
$37$	10.8635	1.78596	0.892978	+	0.450101i	$0.148612\pi$
$38$	−7.11472	−1.15416
$39$	−3.25835	−0.521753
$40$	−2.82880	−0.447273
$41$	−0.414107	−0.0646726	−0.0323363	−	0.999477i	$-0.510295\pi$
$41$	−0.414107	−0.0646726	−0.0323363	+	0.999477i	$0.510295\pi$
$42$	−9.49275	−1.46476
$43$	−7.40257	−1.12888	−0.564440	−	0.825474i	$-0.690908\pi$
$43$	−7.40257	−1.12888	−0.564440	+	0.825474i	$0.690908\pi$
$44$	−5.10940	−0.770272
$45$	5.00750	0.746474
$46$	2.22934	0.328698
$47$	−4.54934	−0.663589	−0.331794	−	0.943352i	$-0.607654\pi$
$47$	−4.54934	−0.663589	−0.331794	+	0.943352i	$0.607654\pi$
$48$	−2.18407	−0.315244
$49$	11.8907	1.69868
$50$	−3.00212	−0.424564
$51$	−5.98964	−0.838717
$52$	1.49187	0.206885
$53$	−8.58927	−1.17983	−0.589914	−	0.807466i	$-0.700838\pi$
$53$	−8.58927	−1.17983	−0.589914	+	0.807466i	$0.700838\pi$
$54$	−2.68601	−0.365520
$55$	−14.4535	−1.94891
$56$	4.34635	0.580805
$57$	−15.5391	−2.05820
$58$	5.22351	0.685880
$59$	−10.7218	−1.39586	−0.697928	−	0.716167i	$-0.745895\pi$
$59$	−10.7218	−1.39586	−0.697928	+	0.716167i	$0.745895\pi$
$60$	−6.17832	−0.797617
$61$	−4.49619	−0.575679	−0.287839	−	0.957679i	$-0.592937\pi$
$61$	−4.49619	−0.575679	−0.287839	+	0.957679i	$0.592937\pi$
$62$	−1.28246	−0.162872
$63$	−7.69383	−0.969331
$64$	1.00000	0.125000
$65$	4.22019	0.523451
$66$	−11.1593	−1.37362
$67$	2.37942	0.290693	0.145346	−	0.989381i	$-0.453570\pi$
$67$	2.37942	0.290693	0.145346	+	0.989381i	$0.453570\pi$
$68$	2.74241	0.332567
$69$	4.86905	0.586165
$70$	12.2950	1.46953
$71$	−5.81418	−0.690016	−0.345008	−	0.938600i	$-0.612124\pi$
$71$	−5.81418	−0.690016	−0.345008	+	0.938600i	$0.612124\pi$
$72$	−1.77018	−0.208618
$73$	−7.44172	−0.870987	−0.435494	−	0.900192i	$-0.643426\pi$
$73$	−7.44172	−0.870987	−0.435494	+	0.900192i	$0.643426\pi$
$74$	−10.8635	−1.26286
$75$	−6.55686	−0.757121
$76$	7.11472	0.816114
$77$	22.2072	2.53075
$78$	3.25835	0.368935
$79$	−7.80847	−0.878521	−0.439261	−	0.898360i	$-0.644760\pi$
$79$	−7.80847	−0.878521	−0.439261	+	0.898360i	$0.644760\pi$
$80$	2.82880	0.316270
$81$	−11.1770	−1.24189
$82$	0.414107	0.0457304
$83$	−8.14913	−0.894483	−0.447242	−	0.894413i	$-0.647594\pi$
$83$	−8.14913	−0.894483	−0.447242	+	0.894413i	$0.647594\pi$
$84$	9.49275	1.03574
$85$	7.75775	0.841446
$86$	7.40257	0.798239
$87$	11.4085	1.22312
$88$	5.10940	0.544664
$89$	−7.46661	−0.791459	−0.395729	−	0.918367i	$-0.629508\pi$
$89$	−7.46661	−0.791459	−0.395729	+	0.918367i	$0.629508\pi$
$90$	−5.00750	−0.527837
$91$	−6.48417	−0.679725
$92$	−2.22934	−0.232425
$93$	−2.80098	−0.290448
$94$	4.54934	0.469228
$95$	20.1261	2.06490
$96$	2.18407	0.222911
$97$	14.7776	1.50043	0.750217	−	0.661192i	$-0.229949\pi$
$97$	14.7776	1.50043	0.750217	+	0.661192i	$0.229949\pi$
$98$	−11.8907	−1.20114
$99$	−9.04458	−0.909015
$100$	3.00212	0.300212
$101$	13.3525	1.32862	0.664309	−	0.747458i	$-0.268726\pi$
$101$	13.3525	1.32862	0.664309	+	0.747458i	$0.268726\pi$
$102$	5.98964	0.593063
$103$	19.5702	1.92830	0.964152	−	0.265350i	$-0.0854874\pi$
$103$	19.5702	1.92830	0.964152	+	0.265350i	$0.0854874\pi$
$104$	−1.49187	−0.146289
$105$	26.8531	2.62059
$106$	8.58927	0.834264
$107$	9.09767	0.879505	0.439753	−	0.898119i	$-0.355066\pi$
$107$	9.09767	0.879505	0.439753	+	0.898119i	$0.355066\pi$
$108$	2.68601	0.258462
$109$	−9.92787	−0.950917	−0.475459	−	0.879738i	$-0.657718\pi$
$109$	−9.92787	−0.950917	−0.475459	+	0.879738i	$0.657718\pi$
$110$	14.4535	1.37809
$111$	−23.7268	−2.25205
$112$	−4.34635	−0.410691
$113$	−6.70293	−0.630559	−0.315280	−	0.948999i	$-0.602098\pi$
$113$	−6.70293	−0.630559	−0.315280	+	0.948999i	$0.602098\pi$
$114$	15.5391	1.45537
$115$	−6.30637	−0.588072
$116$	−5.22351	−0.484991
$117$	2.64088	0.244149
$118$	10.7218	0.987020
$119$	−11.9195	−1.09266
$120$	6.17832	0.564001
$121$	15.1060	1.37327
$122$	4.49619	0.407066
$123$	0.904440	0.0815506
$124$	1.28246	0.115168
$125$	−5.65160	−0.505494
$126$	7.69383	0.685421
$127$	−1.73515	−0.153970	−0.0769848	−	0.997032i	$-0.524529\pi$
$127$	−1.73515	−0.153970	−0.0769848	+	0.997032i	$0.524529\pi$
$128$	−1.00000	−0.0883883
$129$	16.1678	1.42349
$130$	−4.22019	−0.370135
$131$	−0.522119	−0.0456177	−0.0228089	−	0.999740i	$-0.507261\pi$
$131$	−0.522119	−0.0456177	−0.0228089	+	0.999740i	$0.507261\pi$
$132$	11.1593	0.971294
$133$	−30.9230	−2.68137
$134$	−2.37942	−0.205551
$135$	7.59820	0.653949
$136$	−2.74241	−0.235160
$137$	15.2827	1.30569	0.652843	−	0.757494i	$-0.273576\pi$
$137$	15.2827	1.30569	0.652843	+	0.757494i	$0.273576\pi$
$138$	−4.86905	−0.414481
$139$	2.73521	0.231997	0.115999	−	0.993249i	$-0.462993\pi$
$139$	2.73521	0.231997	0.115999	+	0.993249i	$0.462993\pi$
$140$	−12.2950	−1.03911
$141$	9.93609	0.836770
$142$	5.81418	0.487915
$143$	−7.62254	−0.637429
$144$	1.77018	0.147515
$145$	−14.7763	−1.22710
$146$	7.44172	0.615881
$147$	−25.9702	−2.14199
$148$	10.8635	0.892978
$149$	−7.41392	−0.607372	−0.303686	−	0.952772i	$-0.598217\pi$
$149$	−7.41392	−0.607372	−0.303686	+	0.952772i	$0.598217\pi$
$150$	6.55686	0.535366
$151$	−12.4648	−1.01437	−0.507185	−	0.861837i	$-0.669314\pi$
$151$	−12.4648	−1.01437	−0.507185	+	0.861837i	$0.669314\pi$
$152$	−7.11472	−0.577080
$153$	4.85458	0.392469
$154$	−22.2072	−1.78951
$155$	3.62782	0.291393
$156$	−3.25835	−0.260877
$157$	13.4050	1.06984	0.534919	−	0.844903i	$-0.320342\pi$
$157$	13.4050	1.06984	0.534919	+	0.844903i	$0.320342\pi$
$158$	7.80847	0.621208
$159$	18.7596	1.48773
$160$	−2.82880	−0.223636
$161$	9.68949	0.763639
$162$	11.1770	0.878148
$163$	15.3216	1.20008	0.600041	−	0.799969i	$-0.295151\pi$
$163$	15.3216	1.20008	0.600041	+	0.799969i	$0.295151\pi$
$164$	−0.414107	−0.0323363
$165$	31.5675	2.45753
$166$	8.14913	0.632495
$167$	−3.97244	−0.307397	−0.153698	−	0.988118i	$-0.549118\pi$
$167$	−3.97244	−0.307397	−0.153698	+	0.988118i	$0.549118\pi$
$168$	−9.49275	−0.732381
$169$	−10.7743	−0.828795
$170$	−7.75775	−0.594992
$171$	12.5944	0.963115
$172$	−7.40257	−0.564440
$173$	10.2111	0.776335	0.388168	−	0.921589i	$-0.373108\pi$
$173$	10.2111	0.776335	0.388168	+	0.921589i	$0.373108\pi$
$174$	−11.4085	−0.864879
$175$	−13.0483	−0.986357
$176$	−5.10940	−0.385136
$177$	23.4172	1.76014
$178$	7.46661	0.559646
$179$	23.7902	1.77816	0.889081	−	0.457751i	$-0.151345\pi$
$179$	23.7902	1.77816	0.889081	+	0.457751i	$0.151345\pi$
$180$	5.00750	0.373237
$181$	−3.34887	−0.248919	−0.124460	−	0.992225i	$-0.539720\pi$
$181$	−3.34887	−0.248919	−0.124460	+	0.992225i	$0.539720\pi$
$182$	6.48417	0.480638
$183$	9.82003	0.725917
$184$	2.22934	0.164349
$185$	30.7308	2.25937
$186$	2.80098	0.205378
$187$	−14.0121	−1.02467
$188$	−4.54934	−0.331794
$189$	−11.6743	−0.849183
$190$	−20.1261	−1.46010
$191$	−6.48250	−0.469057	−0.234529	−	0.972109i	$-0.575355\pi$
$191$	−6.48250	−0.469057	−0.234529	+	0.972109i	$0.575355\pi$
$192$	−2.18407	−0.157622
$193$	11.6415	0.837976	0.418988	−	0.907992i	$-0.362385\pi$
$193$	11.6415	0.837976	0.418988	+	0.907992i	$0.362385\pi$
$194$	−14.7776	−1.06097
$195$	−9.21722	−0.660059
$196$	11.8907	0.849338
$197$	0.921097	0.0656255	0.0328127	−	0.999462i	$-0.489554\pi$
$197$	0.921097	0.0656255	0.0328127	+	0.999462i	$0.489554\pi$
$198$	9.04458	0.642770
$199$	0.0200824	0.00142361	0.000711803	−	1.00000i	$-0.499773\pi$
$199$	0.0200824	0.00142361	0.000711803		1.00000i	$0.499773\pi$
$200$	−3.00212	−0.212282
$201$	−5.19683	−0.366556
$202$	−13.3525	−0.939475
$203$	22.7032	1.59345
$204$	−5.98964	−0.419359
$205$	−1.17143	−0.0818160
$206$	−19.5702	−1.36352
$207$	−3.94634	−0.274290
$208$	1.49187	0.103442
$209$	−36.3520	−2.51452
$210$	−26.8531	−1.85304
$211$	6.72557	0.463008	0.231504	−	0.972834i	$-0.425635\pi$
$211$	6.72557	0.463008	0.231504	+	0.972834i	$0.425635\pi$
$212$	−8.58927	−0.589914
$213$	12.6986	0.870094
$214$	−9.09767	−0.621904
$215$	−20.9404	−1.42812
$216$	−2.68601	−0.182760
$217$	−5.57400	−0.378388
$218$	9.92787	0.672400
$219$	16.2533	1.09829
$220$	−14.4535	−0.974454
$221$	4.09131	0.275212
$222$	23.7268	1.59244
$223$	9.86289	0.660468	0.330234	−	0.943899i	$-0.392872\pi$
$223$	9.86289	0.660468	0.330234	+	0.943899i	$0.392872\pi$
$224$	4.34635	0.290402
$225$	5.31431	0.354287
$226$	6.70293	0.445873
$227$	0.132543	0.00879722	0.00439861	−	0.999990i	$-0.498600\pi$
$227$	0.132543	0.00879722	0.00439861	+	0.999990i	$0.498600\pi$
$228$	−15.5391	−1.02910
$229$	−22.9051	−1.51361	−0.756804	−	0.653641i	$-0.773240\pi$
$229$	−22.9051	−1.51361	−0.756804	+	0.653641i	$0.773240\pi$
$230$	6.30637	0.415830
$231$	−48.5023	−3.19122
$232$	5.22351	0.342940
$233$	−22.8287	−1.49556	−0.747778	−	0.663949i	$-0.768879\pi$
$233$	−22.8287	−1.49556	−0.747778	+	0.663949i	$0.768879\pi$
$234$	−2.64088	−0.172639
$235$	−12.8692	−0.839492
$236$	−10.7218	−0.697928
$237$	17.0543	1.10779
$238$	11.9195	0.772625
$239$	18.6625	1.20718	0.603590	−	0.797295i	$-0.293737\pi$
$239$	18.6625	1.20718	0.603590	+	0.797295i	$0.293737\pi$
$240$	−6.17832	−0.398809
$241$	7.26302	0.467853	0.233926	−	0.972254i	$-0.424843\pi$
$241$	7.26302	0.467853	0.233926	+	0.972254i	$0.424843\pi$
$242$	−15.1060	−0.971051
$243$	16.3534	1.04907
$244$	−4.49619	−0.287839
$245$	33.6365	2.14896
$246$	−0.904440	−0.0576650
$247$	10.6142	0.675366
$248$	−1.28246	−0.0814361
$249$	17.7983	1.12792
$250$	5.65160	0.357438
$251$	−11.5358	−0.728131	−0.364065	−	0.931373i	$-0.618612\pi$
$251$	−11.5358	−0.728131	−0.364065	+	0.931373i	$0.618612\pi$
$252$	−7.69383	−0.484666
$253$	11.3906	0.716121
$254$	1.73515	0.108873
$255$	−16.9435	−1.06104
$256$	1.00000	0.0625000
$257$	12.1212	0.756099	0.378050	−	0.925785i	$-0.376595\pi$
$257$	12.1212	0.756099	0.378050	+	0.925785i	$0.376595\pi$
$258$	−16.1678	−1.00656
$259$	−47.2167	−2.93390
$260$	4.22019	0.261725
$261$	−9.24657	−0.572348
$262$	0.522119	0.0322566
$263$	16.5165	1.01845	0.509227	−	0.860632i	$-0.329931\pi$
$263$	16.5165	1.01845	0.509227	+	0.860632i	$0.329931\pi$
$264$	−11.1593	−0.686809
$265$	−24.2974	−1.49257
$266$	30.9230	1.89601
$267$	16.3076	0.998010
$268$	2.37942	0.145346
$269$	4.38345	0.267264	0.133632	−	0.991031i	$-0.457336\pi$
$269$	4.38345	0.267264	0.133632	+	0.991031i	$0.457336\pi$
$270$	−7.59820	−0.462412
$271$	9.31481	0.565834	0.282917	−	0.959144i	$-0.408698\pi$
$271$	9.31481	0.565834	0.282917	+	0.959144i	$0.408698\pi$
$272$	2.74241	0.166283
$273$	14.1619	0.857117
$274$	−15.2827	−0.923259
$275$	−15.3391	−0.924980
$276$	4.86905	0.293082
$277$	0.879332	0.0528339	0.0264170	−	0.999651i	$-0.491590\pi$
$277$	0.879332	0.0528339	0.0264170	+	0.999651i	$0.491590\pi$
$278$	−2.73521	−0.164047
$279$	2.27018	0.135912
$280$	12.2950	0.734764
$281$	1.68828	0.100714	0.0503572	−	0.998731i	$-0.483964\pi$
$281$	1.68828	0.100714	0.0503572	+	0.998731i	$0.483964\pi$
$282$	−9.93609	−0.591685
$283$	4.85693	0.288715	0.144357	−	0.989526i	$-0.453889\pi$
$283$	4.85693	0.288715	0.144357	+	0.989526i	$0.453889\pi$
$284$	−5.81418	−0.345008
$285$	−43.9570	−2.60379
$286$	7.62254	0.450731
$287$	1.79985	0.106242
$288$	−1.77018	−0.104309
$289$	−9.47916	−0.557598
$290$	14.7763	0.867693
$291$	−32.2753	−1.89201
$292$	−7.44172	−0.435494
$293$	−4.05063	−0.236640	−0.118320	−	0.992975i	$-0.537751\pi$
$293$	−4.05063	−0.236640	−0.118320	+	0.992975i	$0.537751\pi$
$294$	25.9702	1.51461
$295$	−30.3298	−1.76587
$296$	−10.8635	−0.631431
$297$	−13.7239	−0.796343
$298$	7.41392	0.429477
$299$	−3.32588	−0.192341
$300$	−6.55686	−0.378561
$301$	32.1741	1.85449
$302$	12.4648	0.717268
$303$	−29.1628	−1.67536
$304$	7.11472	0.408057
$305$	−12.7188	−0.728279
$306$	−4.85458	−0.277518
$307$	−16.3041	−0.930523	−0.465261	−	0.885173i	$-0.654040\pi$
$307$	−16.3041	−0.930523	−0.465261	+	0.885173i	$0.654040\pi$
$308$	22.2072	1.26537
$309$	−42.7427	−2.43155
$310$	−3.62782	−0.206046
$311$	15.1011	0.856307	0.428153	−	0.903706i	$-0.359164\pi$
$311$	15.1011	0.856307	0.428153	+	0.903706i	$0.359164\pi$
$312$	3.25835	0.184468
$313$	6.45421	0.364813	0.182407	−	0.983223i	$-0.441611\pi$
$313$	6.45421	0.364813	0.182407	+	0.983223i	$0.441611\pi$
$314$	−13.4050	−0.756490
$315$	−21.7643	−1.22628
$316$	−7.80847	−0.439261
$317$	13.0052	0.730445	0.365223	−	0.930920i	$-0.380993\pi$
$317$	13.0052	0.730445	0.365223	+	0.930920i	$0.380993\pi$
$318$	−18.7596	−1.05199
$319$	26.6890	1.49430
$320$	2.82880	0.158135
$321$	−19.8700	−1.10904
$322$	−9.68949	−0.539974
$323$	19.5115	1.08565
$324$	−11.1770	−0.620945
$325$	4.47877	0.248437
$326$	−15.3216	−0.848586
$327$	21.6832	1.19908
$328$	0.414107	0.0228652
$329$	19.7730	1.09012
$330$	−31.5675	−1.73773
$331$	−3.44060	−0.189112	−0.0945562	−	0.995520i	$-0.530143\pi$
$331$	−3.44060	−0.189112	−0.0945562	+	0.995520i	$0.530143\pi$
$332$	−8.14913	−0.447242
$333$	19.2305	1.05382
$334$	3.97244	0.217362
$335$	6.73091	0.367749
$336$	9.49275	0.517872
$337$	26.3414	1.43491	0.717454	−	0.696606i	$-0.245307\pi$
$337$	26.3414	1.43491	0.717454	+	0.696606i	$0.245307\pi$
$338$	10.7743	0.586047
$339$	14.6397	0.795120
$340$	7.75775	0.420723
$341$	−6.55259	−0.354843
$342$	−12.5944	−0.681025
$343$	−21.2568	−1.14776
$344$	7.40257	0.399120
$345$	13.7736	0.741545
$346$	−10.2111	−0.548952
$347$	8.40867	0.451401	0.225701	−	0.974197i	$-0.427533\pi$
$347$	8.40867	0.451401	0.225701	+	0.974197i	$0.427533\pi$
$348$	11.4085	0.611562
$349$	18.3086	0.980037	0.490019	−	0.871712i	$-0.336990\pi$
$349$	18.3086	0.980037	0.490019	+	0.871712i	$0.336990\pi$
$350$	13.0483	0.697459
$351$	4.00717	0.213887
$352$	5.10940	0.272332
$353$	−23.1379	−1.23150	−0.615752	−	0.787940i	$-0.711148\pi$
$353$	−23.1379	−1.23150	−0.615752	+	0.787940i	$0.711148\pi$
$354$	−23.4172	−1.24461
$355$	−16.4472	−0.872925
$356$	−7.46661	−0.395729
$357$	26.0330	1.37781
$358$	−23.7902	−1.25735
$359$	29.5307	1.55857	0.779286	−	0.626669i	$-0.215582\pi$
$359$	29.5307	1.55857	0.779286	+	0.626669i	$0.215582\pi$
$360$	−5.00750	−0.263918
$361$	31.6192	1.66417
$362$	3.34887	0.176013
$363$	−32.9927	−1.73167
$364$	−6.48417	−0.339863
$365$	−21.0511	−1.10187
$366$	−9.82003	−0.513301
$367$	−22.4523	−1.17200	−0.586000	−	0.810311i	$-0.699298\pi$
$367$	−22.4523	−1.17200	−0.586000	+	0.810311i	$0.699298\pi$
$368$	−2.22934	−0.116212
$369$	−0.733045	−0.0381608
$370$	−30.7308	−1.59762
$371$	37.3319	1.93818
$372$	−2.80098	−0.145224
$373$	10.6018	0.548940	0.274470	−	0.961596i	$-0.411498\pi$
$373$	10.6018	0.548940	0.274470	+	0.961596i	$0.411498\pi$
$374$	14.0121	0.724549
$375$	12.3435	0.637416
$376$	4.54934	0.234614
$377$	−7.79277	−0.401348
$378$	11.6743	0.600463
$379$	19.0951	0.980847	0.490424	−	0.871484i	$-0.336842\pi$
$379$	19.0951	0.980847	0.490424	+	0.871484i	$0.336842\pi$
$380$	20.1261	1.03245
$381$	3.78970	0.194152
$382$	6.48250	0.331674
$383$	35.7385	1.82615	0.913075	−	0.407791i	$-0.133701\pi$
$383$	35.7385	1.82615	0.913075	+	0.407791i	$0.133701\pi$
$384$	2.18407	0.111456
$385$	62.8199	3.20160
$386$	−11.6415	−0.592539
$387$	−13.1039	−0.666109
$388$	14.7776	0.750217
$389$	−14.2907	−0.724567	−0.362284	−	0.932068i	$-0.618003\pi$
$389$	−14.2907	−0.724567	−0.362284	+	0.932068i	$0.618003\pi$
$390$	9.21722	0.466732
$391$	−6.11378	−0.309187
$392$	−11.8907	−0.600572
$393$	1.14035	0.0575229
$394$	−0.921097	−0.0464042
$395$	−22.0886	−1.11140
$396$	−9.04458	−0.454507
$397$	6.04623	0.303451	0.151726	−	0.988423i	$-0.451517\pi$
$397$	6.04623	0.303451	0.151726	+	0.988423i	$0.451517\pi$
$398$	−0.0200824	−0.00100664
$399$	67.5382	3.38114
$400$	3.00212	0.150106
$401$	30.1295	1.50460	0.752299	−	0.658822i	$-0.228945\pi$
$401$	30.1295	1.50460	0.752299	+	0.658822i	$0.228945\pi$
$402$	5.19683	0.259195
$403$	1.91325	0.0953060
$404$	13.3525	0.664309
$405$	−31.6175	−1.57109
$406$	−22.7032	−1.12674
$407$	−55.5062	−2.75134
$408$	5.98964	0.296531
$409$	1.12173	0.0554660	0.0277330	−	0.999615i	$-0.491171\pi$
$409$	1.12173	0.0554660	0.0277330	+	0.999615i	$0.491171\pi$
$410$	1.17143	0.0578526
$411$	−33.3785	−1.64644
$412$	19.5702	0.964152
$413$	46.6006	2.29306
$414$	3.94634	0.193952
$415$	−23.0523	−1.13159
$416$	−1.49187	−0.0731447
$417$	−5.97390	−0.292543
$418$	36.3520	1.77803
$419$	−20.4075	−0.996969	−0.498485	−	0.866899i	$-0.666110\pi$
$419$	−20.4075	−0.996969	−0.498485	+	0.866899i	$0.666110\pi$
$420$	26.8531	1.31030
$421$	−17.1077	−0.833780	−0.416890	−	0.908957i	$-0.636880\pi$
$421$	−17.1077	−0.833780	−0.416890	+	0.908957i	$0.636880\pi$
$422$	−6.72557	−0.327396
$423$	−8.05316	−0.391558
$424$	8.58927	0.417132
$425$	8.23307	0.399363
$426$	−12.6986	−0.615249
$427$	19.5420	0.945705
$428$	9.09767	0.439753
$429$	16.6482	0.803783
$430$	20.9404	1.00984
$431$	17.1688	0.826993	0.413497	−	0.910506i	$-0.364307\pi$
$431$	17.1688	0.826993	0.413497	+	0.910506i	$0.364307\pi$
$432$	2.68601	0.129231
$433$	−13.9661	−0.671170	−0.335585	−	0.942010i	$-0.608934\pi$
$433$	−13.9661	−0.671170	−0.335585	+	0.942010i	$0.608934\pi$
$434$	5.57400	0.267561
$435$	32.2725	1.54735
$436$	−9.92787	−0.475459
$437$	−15.8611	−0.758741
$438$	−16.2533	−0.776611
$439$	3.45086	0.164701	0.0823504	−	0.996603i	$-0.473757\pi$
$439$	3.45086	0.164701	0.0823504	+	0.996603i	$0.473757\pi$
$440$	14.4535	0.689043
$441$	21.0488	1.00232
$442$	−4.09131	−0.194604
$443$	19.1235	0.908583	0.454291	−	0.890853i	$-0.349893\pi$
$443$	19.1235	0.908583	0.454291	+	0.890853i	$0.349893\pi$
$444$	−23.7268	−1.12602
$445$	−21.1216	−1.00126
$446$	−9.86289	−0.467021
$447$	16.1926	0.765881
$448$	−4.34635	−0.205346
$449$	35.1296	1.65787	0.828935	−	0.559345i	$-0.188947\pi$
$449$	35.1296	1.65787	0.828935	+	0.559345i	$0.188947\pi$
$450$	−5.31431	−0.250519
$451$	2.11584	0.0996310
$452$	−6.70293	−0.315280
$453$	27.2240	1.27910
$454$	−0.132543	−0.00622057
$455$	−18.3424	−0.859906
$456$	15.5391	0.727684
$457$	10.9999	0.514556	0.257278	−	0.966337i	$-0.417174\pi$
$457$	10.9999	0.514556	0.257278	+	0.966337i	$0.417174\pi$
$458$	22.9051	1.07028
$459$	7.36616	0.343823
$460$	−6.30637	−0.294036
$461$	10.0345	0.467354	0.233677	−	0.972314i	$-0.424924\pi$
$461$	10.0345	0.467354	0.233677	+	0.972314i	$0.424924\pi$
$462$	48.5023	2.25653
$463$	13.4426	0.624732	0.312366	−	0.949962i	$-0.398879\pi$
$463$	13.4426	0.624732	0.312366	+	0.949962i	$0.398879\pi$
$464$	−5.22351	−0.242495
$465$	−7.92343	−0.367440
$466$	22.8287	1.05752
$467$	−9.04106	−0.418371	−0.209185	−	0.977876i	$-0.567081\pi$
$467$	−9.04106	−0.418371	−0.209185	+	0.977876i	$0.567081\pi$
$468$	2.64088	0.122075
$469$	−10.3418	−0.477539
$470$	12.8692	0.593611
$471$	−29.2776	−1.34904
$472$	10.7218	0.493510
$473$	37.8227	1.73909
$474$	−17.0543	−0.783329
$475$	21.3593	0.980031
$476$	−11.9195	−0.546329
$477$	−15.2046	−0.696170
$478$	−18.6625	−0.853605
$479$	15.8773	0.725452	0.362726	−	0.931896i	$-0.381846\pi$
$479$	15.8773	0.725452	0.362726	+	0.931896i	$0.381846\pi$
$480$	6.17832	0.282000
$481$	16.2069	0.738973
$482$	−7.26302	−0.330822
$483$	−21.1626	−0.962930
$484$	15.1060	0.686637
$485$	41.8028	1.89817
$486$	−16.3534	−0.741804
$487$	9.60849	0.435402	0.217701	−	0.976015i	$-0.430144\pi$
$487$	9.60849	0.435402	0.217701	+	0.976015i	$0.430144\pi$
$488$	4.49619	0.203533
$489$	−33.4636	−1.51327
$490$	−33.6365	−1.51954
$491$	−38.7351	−1.74809	−0.874046	−	0.485843i	$-0.838513\pi$
$491$	−38.7351	−1.74809	−0.874046	+	0.485843i	$0.838513\pi$
$492$	0.904440	0.0407753
$493$	−14.3250	−0.645167
$494$	−10.6142	−0.477556
$495$	−25.5853	−1.14998
$496$	1.28246	0.0575840
$497$	25.2704	1.13353
$498$	−17.7983	−0.797561
$499$	−6.73412	−0.301460	−0.150730	−	0.988575i	$-0.548162\pi$
$499$	−6.73412	−0.301460	−0.150730	+	0.988575i	$0.548162\pi$
$500$	−5.65160	−0.252747
$501$	8.67611	0.387620
$502$	11.5358	0.514866
$503$	14.3236	0.638656	0.319328	−	0.947644i	$-0.396543\pi$
$503$	14.3236	0.638656	0.319328	+	0.947644i	$0.396543\pi$
$504$	7.69383	0.342710
$505$	37.7715	1.68081
$506$	−11.3906	−0.506374
$507$	23.5320	1.04509
$508$	−1.73515	−0.0769848
$509$	20.6022	0.913175	0.456588	−	0.889678i	$-0.349071\pi$
$509$	20.6022	0.913175	0.456588	+	0.889678i	$0.349071\pi$
$510$	16.9435	0.750271
$511$	32.3443	1.43083
$512$	−1.00000	−0.0441942
$513$	19.1102	0.843737
$514$	−12.1212	−0.534643
$515$	55.3601	2.43946
$516$	16.1678	0.711746
$517$	23.2444	1.02229
$518$	47.2167	2.07458
$519$	−22.3018	−0.978940
$520$	−4.22019	−0.185068
$521$	34.1603	1.49659	0.748295	−	0.663366i	$-0.230873\pi$
$521$	34.1603	1.49659	0.748295	+	0.663366i	$0.230873\pi$
$522$	9.24657	0.404711
$523$	−36.7285	−1.60602	−0.803012	−	0.595963i	$-0.796770\pi$
$523$	−36.7285	−1.60602	−0.803012	+	0.595963i	$0.796770\pi$
$524$	−0.522119	−0.0228089
$525$	28.4984	1.24377
$526$	−16.5165	−0.720156
$527$	3.51703	0.153204
$528$	11.1593	0.485647
$529$	−18.0300	−0.783915
$530$	24.2974	1.05541
$531$	−18.9795	−0.823641
$532$	−30.9230	−1.34068
$533$	−0.617792	−0.0267595
$534$	−16.3076	−0.705700
$535$	25.7355	1.11264
$536$	−2.37942	−0.102775
$537$	−51.9595	−2.24222
$538$	−4.38345	−0.188984
$539$	−60.7545	−2.61688
$540$	7.59820	0.326974
$541$	8.12954	0.349516	0.174758	−	0.984611i	$-0.444086\pi$
$541$	8.12954	0.349516	0.174758	+	0.984611i	$0.444086\pi$
$542$	−9.31481	−0.400105
$543$	7.31418	0.313882
$544$	−2.74241	−0.117580
$545$	−28.0840	−1.20299
$546$	−14.1619	−0.606073
$547$	38.3751	1.64080	0.820400	−	0.571789i	$-0.193751\pi$
$547$	38.3751	1.64080	0.820400	+	0.571789i	$0.193751\pi$
$548$	15.2827	0.652843
$549$	−7.95909	−0.339686
$550$	15.3391	0.654060
$551$	−37.1638	−1.58323
$552$	−4.86905	−0.207240
$553$	33.9383	1.44320
$554$	−0.879332	−0.0373592
$555$	−67.1184	−2.84902
$556$	2.73521	0.115999
$557$	−17.9252	−0.759514	−0.379757	−	0.925086i	$-0.623992\pi$
$557$	−17.9252	−0.759514	−0.379757	+	0.925086i	$0.623992\pi$
$558$	−2.27018	−0.0961046
$559$	−11.0436	−0.467096
$560$	−12.2950	−0.519557
$561$	30.6035	1.29208
$562$	−1.68828	−0.0712159
$563$	14.1089	0.594621	0.297310	−	0.954781i	$-0.403910\pi$
$563$	14.1089	0.594621	0.297310	+	0.954781i	$0.403910\pi$
$564$	9.93609	0.418385
$565$	−18.9613	−0.797707
$566$	−4.85693	−0.204152
$567$	48.5791	2.04013
$568$	5.81418	0.243957
$569$	−24.1611	−1.01289	−0.506443	−	0.862274i	$-0.669040\pi$
$569$	−24.1611	−1.01289	−0.506443	+	0.862274i	$0.669040\pi$
$570$	43.9570	1.84116
$571$	18.5291	0.775421	0.387710	−	0.921781i	$-0.373266\pi$
$571$	18.5291	0.775421	0.387710	+	0.921781i	$0.373266\pi$
$572$	−7.62254	−0.318715
$573$	14.1583	0.591470
$574$	−1.79985	−0.0751244
$575$	−6.69276	−0.279107
$576$	1.77018	0.0737576
$577$	13.6541	0.568427	0.284214	−	0.958761i	$-0.408267\pi$
$577$	13.6541	0.568427	0.284214	+	0.958761i	$0.408267\pi$
$578$	9.47916	0.394281
$579$	−25.4260	−1.05667
$580$	−14.7763	−0.613551
$581$	35.4189	1.46943
$582$	32.2753	1.33785
$583$	43.8861	1.81757
$584$	7.44172	0.307940
$585$	7.47052	0.308868
$586$	4.05063	0.167330
$587$	−27.8440	−1.14925	−0.574623	−	0.818418i	$-0.694851\pi$
$587$	−27.8440	−1.14925	−0.574623	+	0.818418i	$0.694851\pi$
$588$	−25.9702	−1.07099
$589$	9.12433	0.375961
$590$	30.3298	1.24866
$591$	−2.01175	−0.0827521
$592$	10.8635	0.446489
$593$	7.58643	0.311537	0.155769	−	0.987794i	$-0.450215\pi$
$593$	7.58643	0.311537	0.155769	+	0.987794i	$0.450215\pi$
$594$	13.7239	0.563099
$595$	−33.7179	−1.38230
$596$	−7.41392	−0.303686
$597$	−0.0438615	−0.00179513
$598$	3.32588	0.136005
$599$	14.0630	0.574597	0.287299	−	0.957841i	$-0.407243\pi$
$599$	14.0630	0.574597	0.287299	+	0.957841i	$0.407243\pi$
$600$	6.55686	0.267683
$601$	−3.07032	−0.125241	−0.0626205	−	0.998037i	$-0.519946\pi$
$601$	−3.07032	−0.125241	−0.0626205	+	0.998037i	$0.519946\pi$
$602$	−32.1741	−1.31132
$603$	4.21201	0.171526
$604$	−12.4648	−0.507185
$605$	42.7319	1.73730
$606$	29.1628	1.18466
$607$	22.7619	0.923878	0.461939	−	0.886912i	$-0.347154\pi$
$607$	22.7619	0.923878	0.461939	+	0.886912i	$0.347154\pi$
$608$	−7.11472	−0.288540
$609$	−49.5854	−2.00930
$610$	12.7188	0.514971
$611$	−6.78700	−0.274573
$612$	4.85458	0.196235
$613$	8.10257	0.327260	0.163630	−	0.986522i	$-0.447680\pi$
$613$	8.10257	0.327260	0.163630	+	0.986522i	$0.447680\pi$
$614$	16.3041	0.657979
$615$	2.55848	0.103168
$616$	−22.2072	−0.894755
$617$	35.7925	1.44095	0.720475	−	0.693480i	$-0.243924\pi$
$617$	35.7925	1.44095	0.720475	+	0.693480i	$0.243924\pi$
$618$	42.7427	1.71936
$619$	−13.4528	−0.540713	−0.270356	−	0.962760i	$-0.587142\pi$
$619$	−13.4528	−0.540713	−0.270356	+	0.962760i	$0.587142\pi$
$620$	3.62782	0.145697
$621$	−5.98804	−0.240292
$622$	−15.1011	−0.605500
$623$	32.4525	1.30018
$624$	−3.25835	−0.130438
$625$	−30.9979	−1.23991
$626$	−6.45421	−0.257962
$627$	79.3954	3.17075
$628$	13.4050	0.534919
$629$	29.7923	1.18790
$630$	21.7643	0.867111
$631$	−21.2022	−0.844046	−0.422023	−	0.906585i	$-0.638680\pi$
$631$	−21.2022	−0.844046	−0.422023	+	0.906585i	$0.638680\pi$
$632$	7.80847	0.310604
$633$	−14.6892	−0.583842
$634$	−13.0052	−0.516503
$635$	−4.90840	−0.194784
$636$	18.7596	0.743867
$637$	17.7394	0.702859
$638$	−26.6890	−1.05663
$639$	−10.2922	−0.407152
$640$	−2.82880	−0.111818
$641$	−21.4045	−0.845428	−0.422714	−	0.906263i	$-0.638923\pi$
$641$	−21.4045	−0.845428	−0.422714	+	0.906263i	$0.638923\pi$
$642$	19.8700	0.784206
$643$	−24.3335	−0.959620	−0.479810	−	0.877372i	$-0.659294\pi$
$643$	−24.3335	−0.959620	−0.479810	+	0.877372i	$0.659294\pi$
$644$	9.68949	0.381819
$645$	45.7354	1.80083
$646$	−19.5115	−0.767670
$647$	−31.3275	−1.23161	−0.615805	−	0.787899i	$-0.711169\pi$
$647$	−31.3275	−1.23161	−0.615805	+	0.787899i	$0.711169\pi$
$648$	11.1770	0.439074
$649$	54.7819	2.15038
$650$	−4.47877	−0.175672
$651$	12.1740	0.477138
$652$	15.3216	0.600041
$653$	−2.22028	−0.0868861	−0.0434431	−	0.999056i	$-0.513833\pi$
$653$	−2.22028	−0.0868861	−0.0434431	+	0.999056i	$0.513833\pi$
$654$	−21.6832	−0.847881
$655$	−1.47697	−0.0577100
$656$	−0.414107	−0.0161682
$657$	−13.1732	−0.513936
$658$	−19.7730	−0.770831
$659$	−50.2676	−1.95815	−0.979074	−	0.203506i	$-0.934766\pi$
$659$	−50.2676	−1.95815	−0.979074	+	0.203506i	$0.934766\pi$
$660$	31.5675	1.22876
$661$	43.8455	1.70539	0.852697	−	0.522407i	$-0.174966\pi$
$661$	43.8455	1.70539	0.852697	+	0.522407i	$0.174966\pi$
$662$	3.44060	0.133723
$663$	−8.93574	−0.347035
$664$	8.14913	0.316248
$665$	−87.4752	−3.39214
$666$	−19.2305	−0.745165
$667$	11.6450	0.450896
$668$	−3.97244	−0.153698
$669$	−21.5413	−0.832834
$670$	−6.73091	−0.260038
$671$	22.9729	0.886858
$672$	−9.49275	−0.366191
$673$	6.99426	0.269609	0.134805	−	0.990872i	$-0.456959\pi$
$673$	6.99426	0.269609	0.134805	+	0.990872i	$0.456959\pi$
$674$	−26.3414	−1.01463
$675$	8.06374	0.310374
$676$	−10.7743	−0.414398
$677$	−9.07362	−0.348728	−0.174364	−	0.984681i	$-0.555787\pi$
$677$	−9.07362	−0.348728	−0.174364	+	0.984681i	$0.555787\pi$
$678$	−14.6397	−0.562235
$679$	−64.2284	−2.46486
$680$	−7.75775	−0.297496
$681$	−0.289485	−0.0110931
$682$	6.55259	0.250912
$683$	−33.4617	−1.28038	−0.640189	−	0.768217i	$-0.721144\pi$
$683$	−33.4617	−1.28038	−0.640189	+	0.768217i	$0.721144\pi$
$684$	12.5944	0.481557
$685$	43.2316	1.65180
$686$	21.2568	0.811588
$687$	50.0264	1.90862
$688$	−7.40257	−0.282220
$689$	−12.8140	−0.488176
$690$	−13.7736	−0.524351
$691$	9.10451	0.346352	0.173176	−	0.984891i	$-0.444597\pi$
$691$	9.10451	0.346352	0.173176	+	0.984891i	$0.444597\pi$
$692$	10.2111	0.388168
$693$	39.3109	1.49330
$694$	−8.40867	−0.319189
$695$	7.73736	0.293495
$696$	−11.4085	−0.432439
$697$	−1.13565	−0.0430159
$698$	−18.3086	−0.692991
$699$	49.8595	1.88586
$700$	−13.0483	−0.493178
$701$	19.3967	0.732602	0.366301	−	0.930496i	$-0.380624\pi$
$701$	19.3967	0.732602	0.366301	+	0.930496i	$0.380624\pi$
$702$	−4.00717	−0.151241
$703$	77.2911	2.91509
$704$	−5.10940	−0.192568
$705$	28.1072	1.05858
$706$	23.1379	0.870805
$707$	−58.0344	−2.18261
$708$	23.4172	0.880071
$709$	24.1209	0.905879	0.452940	−	0.891541i	$-0.350375\pi$
$709$	24.1209	0.905879	0.452940	+	0.891541i	$0.350375\pi$
$710$	16.4472	0.617251
$711$	−13.8224	−0.518381
$712$	7.46661	0.279823
$713$	−2.85904	−0.107072
$714$	−26.0330	−0.974262
$715$	−21.5627	−0.806398
$716$	23.7902	0.889081
$717$	−40.7604	−1.52222
$718$	−29.5307	−1.10208
$719$	26.5957	0.991853	0.495927	−	0.868364i	$-0.334829\pi$
$719$	26.5957	0.991853	0.495927	+	0.868364i	$0.334829\pi$
$720$	5.00750	0.186618
$721$	−85.0587	−3.16775
$722$	−31.6192	−1.17675
$723$	−15.8630	−0.589951
$724$	−3.34887	−0.124460
$725$	−15.6816	−0.582401
$726$	32.9927	1.22447
$727$	43.0074	1.59506	0.797529	−	0.603281i	$-0.206140\pi$
$727$	43.0074	1.59506	0.797529	+	0.603281i	$0.206140\pi$
$728$	6.48417	0.240319
$729$	−2.18598	−0.0809624
$730$	21.0511	0.779138
$731$	−20.3009	−0.750856
$732$	9.82003	0.362959
$733$	−34.0821	−1.25885	−0.629426	−	0.777060i	$-0.716710\pi$
$733$	−34.0821	−1.25885	−0.629426	+	0.777060i	$0.716710\pi$
$734$	22.4523	0.828729
$735$	−73.4647	−2.70979
$736$	2.22934	0.0821746
$737$	−12.1574	−0.447824
$738$	0.733045	0.0269838
$739$	11.7941	0.433854	0.216927	−	0.976188i	$-0.430397\pi$
$739$	11.7941	0.433854	0.216927	+	0.976188i	$0.430397\pi$
$740$	30.7308	1.12969
$741$	−23.1822	−0.851620
$742$	−37.3319	−1.37050
$743$	−43.3867	−1.59170	−0.795852	−	0.605491i	$-0.792977\pi$
$743$	−43.3867	−1.59170	−0.795852	+	0.605491i	$0.792977\pi$
$744$	2.80098	0.102689
$745$	−20.9725	−0.768373
$746$	−10.6018	−0.388159
$747$	−14.4255	−0.527800
$748$	−14.0121	−0.512333
$749$	−39.5416	−1.44482
$750$	−12.3435	−0.450721
$751$	31.7249	1.15766	0.578830	−	0.815448i	$-0.303510\pi$
$751$	31.7249	1.15766	0.578830	+	0.815448i	$0.303510\pi$
$752$	−4.54934	−0.165897
$753$	25.1950	0.918156
$754$	7.79277	0.283796
$755$	−35.2604	−1.28326
$756$	−11.6743	−0.424592
$757$	39.3660	1.43078	0.715390	−	0.698725i	$-0.246249\pi$
$757$	39.3660	1.43078	0.715390	+	0.698725i	$0.246249\pi$
$758$	−19.0951	−0.693564
$759$	−24.8779	−0.903012
$760$	−20.1261	−0.730052
$761$	−43.6809	−1.58343	−0.791716	−	0.610889i	$-0.790812\pi$
$761$	−43.6809	−1.58343	−0.791716	+	0.610889i	$0.790812\pi$
$762$	−3.78970	−0.137286
$763$	43.1500	1.56213
$764$	−6.48250	−0.234529
$765$	13.7326	0.496505
$766$	−35.7385	−1.29128
$767$	−15.9955	−0.577563
$768$	−2.18407	−0.0788110
$769$	5.57345	0.200984	0.100492	−	0.994938i	$-0.467958\pi$
$769$	5.57345	0.200984	0.100492	+	0.994938i	$0.467958\pi$
$770$	−62.8199	−2.26387
$771$	−26.4736	−0.953423
$772$	11.6415	0.418988
$773$	6.06530	0.218154	0.109077	−	0.994033i	$-0.465211\pi$
$773$	6.06530	0.218154	0.109077	+	0.994033i	$0.465211\pi$
$774$	13.1039	0.471010
$775$	3.85010	0.138300
$776$	−14.7776	−0.530483
$777$	103.125	3.69958
$778$	14.2907	0.512346
$779$	−2.94625	−0.105561
$780$	−9.21722	−0.330029
$781$	29.7070	1.06300
$782$	6.11378	0.218628
$783$	−14.0304	−0.501406
$784$	11.8907	0.424669
$785$	37.9202	1.35343
$786$	−1.14035	−0.0406748
$787$	4.38651	0.156362	0.0781810	−	0.996939i	$-0.475089\pi$
$787$	4.38651	0.156362	0.0781810	+	0.996939i	$0.475089\pi$
$788$	0.921097	0.0328127
$789$	−36.0734	−1.28425
$790$	22.0886	0.785878
$791$	29.1333	1.03586
$792$	9.04458	0.321385
$793$	−6.70772	−0.238198
$794$	−6.04623	−0.214573
$795$	53.0672	1.88210
$796$	0.0200824	0.000711803 0
$797$	−34.4159	−1.21907	−0.609537	−	0.792757i	$-0.708645\pi$
$797$	−34.4159	−1.21907	−0.609537	+	0.792757i	$0.708645\pi$
$798$	−67.5382	−2.39083
$799$	−12.4762	−0.441375
$800$	−3.00212	−0.106141
$801$	−13.2173	−0.467009
$802$	−30.1295	−1.06391
$803$	38.0227	1.34179
$804$	−5.19683	−0.183278
$805$	27.4097	0.966063
$806$	−1.91325	−0.0673915
$807$	−9.57379	−0.337013
$808$	−13.3525	−0.469738
$809$	−3.63860	−0.127926	−0.0639632	−	0.997952i	$-0.520374\pi$
$809$	−3.63860	−0.127926	−0.0639632	+	0.997952i	$0.520374\pi$
$810$	31.6175	1.11093
$811$	50.0537	1.75762	0.878812	−	0.477169i	$-0.158337\pi$
$811$	50.0537	1.75762	0.878812	+	0.477169i	$0.158337\pi$
$812$	22.7032	0.796725
$813$	−20.3442	−0.713504
$814$	55.5062	1.94549
$815$	43.3418	1.51820
$816$	−5.98964	−0.209679
$817$	−52.6672	−1.84259
$818$	−1.12173	−0.0392204
$819$	−11.4782	−0.401079
$820$	−1.17143	−0.0409080
$821$	50.6145	1.76646	0.883228	−	0.468943i	$-0.155365\pi$
$821$	50.6145	1.76646	0.883228	+	0.468943i	$0.155365\pi$
$822$	33.3785	1.16421
$823$	43.3453	1.51092	0.755460	−	0.655195i	$-0.227413\pi$
$823$	43.3453	1.51092	0.755460	+	0.655195i	$0.227413\pi$
$824$	−19.5702	−0.681759
$825$	33.5017	1.16638
$826$	−46.6006	−1.62144
$827$	44.3039	1.54060	0.770299	−	0.637683i	$-0.220107\pi$
$827$	44.3039	1.54060	0.770299	+	0.637683i	$0.220107\pi$
$828$	−3.94634	−0.137145
$829$	48.7507	1.69318	0.846591	−	0.532245i	$-0.178651\pi$
$829$	48.7507	1.69318	0.846591	+	0.532245i	$0.178651\pi$
$830$	23.0523	0.800156
$831$	−1.92053	−0.0666223
$832$	1.49187	0.0517211
$833$	32.6093	1.12985
$834$	5.97390	0.206859
$835$	−11.2373	−0.388881
$836$	−36.3520	−1.25726
$837$	3.44470	0.119066
$838$	20.4075	0.704964
$839$	−11.9239	−0.411659	−0.205829	−	0.978588i	$-0.565989\pi$
$839$	−11.9239	−0.411659	−0.205829	+	0.978588i	$0.565989\pi$
$840$	−26.8531	−0.926520
$841$	−1.71495	−0.0591363
$842$	17.1077	0.589572
$843$	−3.68733	−0.126999
$844$	6.72557	0.231504
$845$	−30.4785	−1.04849
$846$	8.05316	0.276873
$847$	−65.6559	−2.25596
$848$	−8.58927	−0.294957
$849$	−10.6079	−0.364062
$850$	−8.23307	−0.282392
$851$	−24.2185	−0.830201
$852$	12.6986	0.435047
$853$	−49.4356	−1.69264	−0.846321	−	0.532673i	$-0.821188\pi$
$853$	−49.4356	−1.69264	−0.846321	+	0.532673i	$0.821188\pi$
$854$	−19.5420	−0.668714
$855$	35.6270	1.21842
$856$	−9.09767	−0.310952
$857$	−18.6562	−0.637284	−0.318642	−	0.947875i	$-0.603227\pi$
$857$	−18.6562	−0.637284	−0.318642	+	0.947875i	$0.603227\pi$
$858$	−16.6482	−0.568360
$859$	10.4619	0.356956	0.178478	−	0.983944i	$-0.442883\pi$
$859$	10.4619	0.356956	0.178478	+	0.983944i	$0.442883\pi$
$860$	−20.9404	−0.714062
$861$	−3.93101	−0.133968
$862$	−17.1688	−0.584772
$863$	−46.3749	−1.57862	−0.789310	−	0.613995i	$-0.789562\pi$
$863$	−46.3749	−1.57862	−0.789310	+	0.613995i	$0.789562\pi$
$864$	−2.68601	−0.0913800
$865$	28.8852	0.982125
$866$	13.9661	0.474589
$867$	20.7032	0.703117
$868$	−5.57400	−0.189194
$869$	39.8966	1.35340
$870$	−32.2725	−1.09414
$871$	3.54978	0.120280
$872$	9.92787	0.336200
$873$	26.1590	0.885347
$874$	15.8611	0.536511
$875$	24.5638	0.830408
$876$	16.2533	0.549147
$877$	4.49488	0.151781	0.0758906	−	0.997116i	$-0.475820\pi$
$877$	4.49488	0.151781	0.0758906	+	0.997116i	$0.475820\pi$
$878$	−3.45086	−0.116461
$879$	8.84689	0.298398
$880$	−14.4535	−0.487227
$881$	17.0399	0.574090	0.287045	−	0.957917i	$-0.407327\pi$
$881$	17.0399	0.574090	0.287045	+	0.957917i	$0.407327\pi$
$882$	−21.0488	−0.708749
$883$	−21.6595	−0.728900	−0.364450	−	0.931223i	$-0.618743\pi$
$883$	−21.6595	−0.728900	−0.364450	+	0.931223i	$0.618743\pi$
$884$	4.09131	0.137606
$885$	66.2426	2.22672
$886$	−19.1235	−0.642465
$887$	−30.4582	−1.02268	−0.511342	−	0.859377i	$-0.670852\pi$
$887$	−30.4582	−1.02268	−0.511342	+	0.859377i	$0.670852\pi$
$888$	23.7268	0.796219
$889$	7.54156	0.252936
$890$	21.1216	0.707996
$891$	57.1078	1.91318
$892$	9.86289	0.330234
$893$	−32.3672	−1.08313
$894$	−16.1926	−0.541560
$895$	67.2977	2.24951
$896$	4.34635	0.145201
$897$	7.26397	0.242537
$898$	−35.1296	−1.17229
$899$	−6.69893	−0.223422
$900$	5.31431	0.177144
$901$	−23.5553	−0.784742
$902$	−2.11584	−0.0704497
$903$	−70.2707	−2.33846
$904$	6.70293	0.222936
$905$	−9.47329	−0.314903
$906$	−27.2240	−0.904458
$907$	41.8454	1.38945	0.694727	−	0.719274i	$-0.255525\pi$
$907$	41.8454	1.38945	0.694727	+	0.719274i	$0.255525\pi$
$908$	0.132543	0.00439861
$909$	23.6363	0.783966
$910$	18.3424	0.608045
$911$	13.5321	0.448339	0.224169	−	0.974550i	$-0.428033\pi$
$911$	13.5321	0.448339	0.224169	+	0.974550i	$0.428033\pi$
$912$	−15.5391	−0.514550
$913$	41.6372	1.37799
$914$	−10.9999	−0.363846
$915$	27.7789	0.918343
$916$	−22.9051	−0.756804
$917$	2.26931	0.0749392
$918$	−7.36616	−0.243119
$919$	−49.1927	−1.62272	−0.811358	−	0.584550i	$-0.801271\pi$
$919$	−49.1927	−1.62272	−0.811358	+	0.584550i	$0.801271\pi$
$920$	6.30637	0.207915
$921$	35.6093	1.17337
$922$	−10.0345	−0.330469
$923$	−8.67397	−0.285507
$924$	−48.5023	−1.59561
$925$	32.6137	1.07233
$926$	−13.4426	−0.441752
$927$	34.6428	1.13782
$928$	5.22351	0.171470
$929$	8.04957	0.264098	0.132049	−	0.991243i	$-0.457844\pi$
$929$	8.04957	0.264098	0.132049	+	0.991243i	$0.457844\pi$
$930$	7.92343	0.259819
$931$	84.5992	2.77263
$932$	−22.8287	−0.747778
$933$	−32.9820	−1.07978
$934$	9.04106	0.295833
$935$	−39.6375	−1.29628
$936$	−2.64088	−0.0863197
$937$	−27.6837	−0.904386	−0.452193	−	0.891920i	$-0.649358\pi$
$937$	−27.6837	−0.904386	−0.452193	+	0.891920i	$0.649358\pi$
$938$	10.3418	0.337671
$939$	−14.0965	−0.460021
$940$	−12.8692	−0.419746
$941$	−52.3497	−1.70655	−0.853276	−	0.521459i	$-0.825388\pi$
$941$	−52.3497	−1.70655	−0.853276	+	0.521459i	$0.825388\pi$
$942$	29.2776	0.953916
$943$	0.923186	0.0300631
$944$	−10.7218	−0.348964
$945$	−33.0244	−1.07428
$946$	−37.8227	−1.22972
$947$	18.0499	0.586542	0.293271	−	0.956029i	$-0.405256\pi$
$947$	18.0499	0.586542	0.293271	+	0.956029i	$0.405256\pi$
$948$	17.0543	0.553897
$949$	−11.1020	−0.360388
$950$	−21.3593	−0.692986
$951$	−28.4043	−0.921074
$952$	11.9195	0.386313
$953$	30.7720	0.996803	0.498402	−	0.866946i	$-0.333920\pi$
$953$	30.7720	0.996803	0.498402	+	0.866946i	$0.333920\pi$
$954$	15.2046	0.492267
$955$	−18.3377	−0.593395
$956$	18.6625	0.603590
$957$	−58.2908	−1.88427
$958$	−15.8773	−0.512972
$959$	−66.4237	−2.14493
$960$	−6.17832	−0.199404
$961$	−29.3553	−0.946945
$962$	−16.2069	−0.522533
$963$	16.1045	0.518962
$964$	7.26302	0.233926
$965$	32.9316	1.06011
$966$	21.1626	0.680895
$967$	−37.4705	−1.20497	−0.602485	−	0.798130i	$-0.705823\pi$
$967$	−37.4705	−1.20497	−0.602485	+	0.798130i	$0.705823\pi$
$968$	−15.1060	−0.485525
$969$	−42.6146	−1.36898
$970$	−41.8028	−1.34221
$971$	43.4410	1.39409	0.697045	−	0.717028i	$-0.254498\pi$
$971$	43.4410	1.39409	0.697045	+	0.717028i	$0.254498\pi$
$972$	16.3534	0.524535
$973$	−11.8882	−0.381117
$974$	−9.60849	−0.307876
$975$	−9.78196	−0.313273
$976$	−4.49619	−0.143920
$977$	−34.6678	−1.10912	−0.554560	−	0.832143i	$-0.687114\pi$
$977$	−34.6678	−1.10912	−0.554560	+	0.832143i	$0.687114\pi$
$978$	33.4636	1.07005
$979$	38.1499	1.21928
$980$	33.6365	1.07448
$981$	−17.5741	−0.561099
$982$	38.7351	1.23609
$983$	61.1234	1.94954	0.974768	−	0.223222i	$-0.0716575\pi$
$983$	61.1234	1.94954	0.974768	+	0.223222i	$0.0716575\pi$
$984$	−0.904440	−0.0288325
$985$	2.60560	0.0830214
$986$	14.3250	0.456202
$987$	−43.1857	−1.37462
$988$	10.6142	0.337683
$989$	16.5028	0.524760
$990$	25.5853	0.813155
$991$	8.69122	0.276086	0.138043	−	0.990426i	$-0.455919\pi$
$991$	8.69122	0.276086	0.138043	+	0.990426i	$0.455919\pi$
$992$	−1.28246	−0.0407181
$993$	7.51453	0.238466
$994$	−25.2704	−0.801529
$995$	0.0568092	0.00180097
$996$	17.7983	0.563961
$997$	8.46841	0.268197	0.134099	−	0.990968i	$-0.457186\pi$
$997$	8.46841	0.268197	0.134099	+	0.990968i	$0.457186\pi$
$998$	6.73412	0.213165
$999$	29.1796	0.923202

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.d.1.13	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.d.1.13	✓	69	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:	\( N \)	\(=\)	\( 6038 = 2 \cdot 3019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.18407 q^{3} +1.00000 q^{4} +2.82880 q^{5} +2.18407 q^{6} -4.34635 q^{7} -1.00000 q^{8} +1.77018 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.18407 q^{3} +1.00000 q^{4} +2.82880 q^{5} +2.18407 q^{6} -4.34635 q^{7} -1.00000 q^{8} +1.77018 q^{9} -2.82880 q^{10} -5.10940 q^{11} -2.18407 q^{12} +1.49187 q^{13} +4.34635 q^{14} -6.17832 q^{15} +1.00000 q^{16} +2.74241 q^{17} -1.77018 q^{18} +7.11472 q^{19} +2.82880 q^{20} +9.49275 q^{21} +5.10940 q^{22} -2.22934 q^{23} +2.18407 q^{24} +3.00212 q^{25} -1.49187 q^{26} +2.68601 q^{27} -4.34635 q^{28} -5.22351 q^{29} +6.17832 q^{30} +1.28246 q^{31} -1.00000 q^{32} +11.1593 q^{33} -2.74241 q^{34} -12.2950 q^{35} +1.77018 q^{36} +10.8635 q^{37} -7.11472 q^{38} -3.25835 q^{39} -2.82880 q^{40} -0.414107 q^{41} -9.49275 q^{42} -7.40257 q^{43} -5.10940 q^{44} +5.00750 q^{45} +2.22934 q^{46} -4.54934 q^{47} -2.18407 q^{48} +11.8907 q^{49} -3.00212 q^{50} -5.98964 q^{51} +1.49187 q^{52} -8.58927 q^{53} -2.68601 q^{54} -14.4535 q^{55} +4.34635 q^{56} -15.5391 q^{57} +5.22351 q^{58} -10.7218 q^{59} -6.17832 q^{60} -4.49619 q^{61} -1.28246 q^{62} -7.69383 q^{63} +1.00000 q^{64} +4.22019 q^{65} -11.1593 q^{66} +2.37942 q^{67} +2.74241 q^{68} +4.86905 q^{69} +12.2950 q^{70} -5.81418 q^{71} -1.77018 q^{72} -7.44172 q^{73} -10.8635 q^{74} -6.55686 q^{75} +7.11472 q^{76} +22.2072 q^{77} +3.25835 q^{78} -7.80847 q^{79} +2.82880 q^{80} -11.1770 q^{81} +0.414107 q^{82} -8.14913 q^{83} +9.49275 q^{84} +7.75775 q^{85} +7.40257 q^{86} +11.4085 q^{87} +5.10940 q^{88} -7.46661 q^{89} -5.00750 q^{90} -6.48417 q^{91} -2.22934 q^{92} -2.80098 q^{93} +4.54934 q^{94} +20.1261 q^{95} +2.18407 q^{96} +14.7776 q^{97} -11.8907 q^{98} -9.04458 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9} - 18 q^{10} - 4 q^{11} + 8 q^{12} + 40 q^{13} - 32 q^{14} + 6 q^{15} + 69 q^{16} + 6 q^{17} - 79 q^{18} + 13 q^{19} + 18 q^{20} + 23 q^{21} + 4 q^{22} + 11 q^{23} - 8 q^{24} + 111 q^{25} - 40 q^{26} + 32 q^{27} + 32 q^{28} + 23 q^{29} - 6 q^{30} + 30 q^{31} - 69 q^{32} + 37 q^{33} - 6 q^{34} - 12 q^{35} + 79 q^{36} + 81 q^{37} - 13 q^{38} - 8 q^{39} - 18 q^{40} - 13 q^{41} - 23 q^{42} + 42 q^{43} - 4 q^{44} + 89 q^{45} - 11 q^{46} + 35 q^{47} + 8 q^{48} + 115 q^{49} - 111 q^{50} - 21 q^{51} + 40 q^{52} + 41 q^{53} - 32 q^{54} + 28 q^{55} - 32 q^{56} + 39 q^{57} - 23 q^{58} - 24 q^{59} + 6 q^{60} + 69 q^{61} - 30 q^{62} + 79 q^{63} + 69 q^{64} + 15 q^{65} - 37 q^{66} + 87 q^{67} + 6 q^{68} + 51 q^{69} + 12 q^{70} - 22 q^{71} - 79 q^{72} + 104 q^{73} - 81 q^{74} + 39 q^{75} + 13 q^{76} + 47 q^{77} + 8 q^{78} + 16 q^{79} + 18 q^{80} + 105 q^{81} + 13 q^{82} + 30 q^{83} + 23 q^{84} + 74 q^{85} - 42 q^{86} + 47 q^{87} + 4 q^{88} - 4 q^{89} - 89 q^{90} + 70 q^{91} + 11 q^{92} + 104 q^{93} - 35 q^{94} - 36 q^{95} - 8 q^{96} + 158 q^{97} - 115 q^{98} + 34 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9 - 18 * q^10 - 4 * q^11 + 8 * q^12 + 40 * q^13 - 32 * q^14 + 6 * q^15 + 69 * q^16 + 6 * q^17 - 79 * q^18 + 13 * q^19 + 18 * q^20 + 23 * q^21 + 4 * q^22 + 11 * q^23 - 8 * q^24 + 111 * q^25 - 40 * q^26 + 32 * q^27 + 32 * q^28 + 23 * q^29 - 6 * q^30 + 30 * q^31 - 69 * q^32 + 37 * q^33 - 6 * q^34 - 12 * q^35 + 79 * q^36 + 81 * q^37 - 13 * q^38 - 8 * q^39 - 18 * q^40 - 13 * q^41 - 23 * q^42 + 42 * q^43 - 4 * q^44 + 89 * q^45 - 11 * q^46 + 35 * q^47 + 8 * q^48 + 115 * q^49 - 111 * q^50 - 21 * q^51 + 40 * q^52 + 41 * q^53 - 32 * q^54 + 28 * q^55 - 32 * q^56 + 39 * q^57 - 23 * q^58 - 24 * q^59 + 6 * q^60 + 69 * q^61 - 30 * q^62 + 79 * q^63 + 69 * q^64 + 15 * q^65 - 37 * q^66 + 87 * q^67 + 6 * q^68 + 51 * q^69 + 12 * q^70 - 22 * q^71 - 79 * q^72 + 104 * q^73 - 81 * q^74 + 39 * q^75 + 13 * q^76 + 47 * q^77 + 8 * q^78 + 16 * q^79 + 18 * q^80 + 105 * q^81 + 13 * q^82 + 30 * q^83 + 23 * q^84 + 74 * q^85 - 42 * q^86 + 47 * q^87 + 4 * q^88 - 4 * q^89 - 89 * q^90 + 70 * q^91 + 11 * q^92 + 104 * q^93 - 35 * q^94 - 36 * q^95 - 8 * q^96 + 158 * q^97 - 115 * q^98 + 34 * q^99

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