LMFDB - Embedded newform 8043.2.a.t.1.5

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:	$0$
Dimension:	$52$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
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)

$f(q)$	$=$	$q-2.45692 q^{2} -1.00000 q^{3} +4.03647 q^{4} -2.79922 q^{5} +2.45692 q^{6} +1.00000 q^{7} -5.00345 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.45692 q^{2} -1.00000 q^{3} +4.03647 q^{4} -2.79922 q^{5} +2.45692 q^{6} +1.00000 q^{7} -5.00345 q^{8} +1.00000 q^{9} +6.87748 q^{10} -0.984367 q^{11} -4.03647 q^{12} +4.11840 q^{13} -2.45692 q^{14} +2.79922 q^{15} +4.22016 q^{16} -1.41398 q^{17} -2.45692 q^{18} +7.64864 q^{19} -11.2990 q^{20} -1.00000 q^{21} +2.41851 q^{22} -3.00054 q^{23} +5.00345 q^{24} +2.83565 q^{25} -10.1186 q^{26} -1.00000 q^{27} +4.03647 q^{28} -2.48107 q^{29} -6.87748 q^{30} -1.69867 q^{31} -0.361700 q^{32} +0.984367 q^{33} +3.47404 q^{34} -2.79922 q^{35} +4.03647 q^{36} -4.85422 q^{37} -18.7921 q^{38} -4.11840 q^{39} +14.0058 q^{40} +2.85338 q^{41} +2.45692 q^{42} +7.04176 q^{43} -3.97337 q^{44} -2.79922 q^{45} +7.37209 q^{46} +9.67674 q^{47} -4.22016 q^{48} +1.00000 q^{49} -6.96697 q^{50} +1.41398 q^{51} +16.6238 q^{52} -2.69129 q^{53} +2.45692 q^{54} +2.75546 q^{55} -5.00345 q^{56} -7.64864 q^{57} +6.09580 q^{58} +8.15845 q^{59} +11.2990 q^{60} -7.43534 q^{61} +4.17350 q^{62} +1.00000 q^{63} -7.55165 q^{64} -11.5283 q^{65} -2.41851 q^{66} +6.56297 q^{67} -5.70748 q^{68} +3.00054 q^{69} +6.87748 q^{70} +14.0094 q^{71} -5.00345 q^{72} +5.55255 q^{73} +11.9264 q^{74} -2.83565 q^{75} +30.8735 q^{76} -0.984367 q^{77} +10.1186 q^{78} +9.46209 q^{79} -11.8132 q^{80} +1.00000 q^{81} -7.01053 q^{82} +12.1889 q^{83} -4.03647 q^{84} +3.95804 q^{85} -17.3011 q^{86} +2.48107 q^{87} +4.92523 q^{88} -8.31439 q^{89} +6.87748 q^{90} +4.11840 q^{91} -12.1116 q^{92} +1.69867 q^{93} -23.7750 q^{94} -21.4103 q^{95} +0.361700 q^{96} -7.65077 q^{97} -2.45692 q^{98} -0.984367 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) $ 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9 - 2 * q^10 + 9 * q^11 - 61 * q^12 + 44 * q^13 + 3 * q^14 + 7 * q^15 + 95 * q^16 - 6 * q^17 + 3 * q^18 + 7 * q^19 - 21 * q^20 - 52 * q^21 + 19 * q^22 - 4 * q^23 - 24 * q^24 + 83 * q^25 - 5 * q^26 - 52 * q^27 + 61 * q^28 + 31 * q^29 + 2 * q^30 + 11 * q^31 + 71 * q^32 - 9 * q^33 + 17 * q^34 - 7 * q^35 + 61 * q^36 + 71 * q^37 - 8 * q^38 - 44 * q^39 + 20 * q^40 - 25 * q^41 - 3 * q^42 + 75 * q^43 + 14 * q^44 - 7 * q^45 + 36 * q^46 - 20 * q^47 - 95 * q^48 + 52 * q^49 + 26 * q^50 + 6 * q^51 + 88 * q^52 + 70 * q^53 - 3 * q^54 + 12 * q^55 + 24 * q^56 - 7 * q^57 + 48 * q^58 - 27 * q^59 + 21 * q^60 + 59 * q^61 - 23 * q^62 + 52 * q^63 + 138 * q^64 + 44 * q^65 - 19 * q^66 + 65 * q^67 - 8 * q^68 + 4 * q^69 - 2 * q^70 - 11 * q^71 + 24 * q^72 + 34 * q^73 + 38 * q^74 - 83 * q^75 + 31 * q^76 + 9 * q^77 + 5 * q^78 + 74 * q^79 - 5 * q^80 + 52 * q^81 + 51 * q^82 - 30 * q^83 - 61 * q^84 + 70 * q^85 + 29 * q^86 - 31 * q^87 + 90 * q^88 - q^89 - 2 * q^90 + 44 * q^91 + 34 * q^92 - 11 * q^93 + 27 * q^94 + 9 * q^95 - 71 * q^96 + 73 * q^97 + 3 * q^98 + 9 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.45692	−1.73731	−0.868654	−	0.495420i	$-0.835014\pi$
$2$	−2.45692	−1.73731	−0.868654	+	0.495420i	$0.835014\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.03647	2.01824
$5$	−2.79922	−1.25185	−0.625925	−	0.779883i	$-0.715279\pi$
$5$	−2.79922	−1.25185	−0.625925	+	0.779883i	$0.715279\pi$
$6$	2.45692	1.00303
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−5.00345	−1.76899
$9$	1.00000	0.333333
$10$	6.87748	2.17485
$11$	−0.984367	−0.296798	−0.148399	−	0.988928i	$-0.547412\pi$
$11$	−0.984367	−0.296798	−0.148399	+	0.988928i	$0.547412\pi$
$12$	−4.03647	−1.16523
$13$	4.11840	1.14224	0.571120	−	0.820867i	$-0.306509\pi$
$13$	4.11840	1.14224	0.571120	+	0.820867i	$0.306509\pi$
$14$	−2.45692	−0.656640
$15$	2.79922	0.722756
$16$	4.22016	1.05504
$17$	−1.41398	−0.342940	−0.171470	−	0.985189i	$-0.554852\pi$
$17$	−1.41398	−0.342940	−0.171470	+	0.985189i	$0.554852\pi$
$18$	−2.45692	−0.579102
$19$	7.64864	1.75472	0.877360	−	0.479833i	$-0.159303\pi$
$19$	7.64864	1.75472	0.877360	+	0.479833i	$0.159303\pi$
$20$	−11.2990	−2.52653
$21$	−1.00000	−0.218218
$22$	2.41851	0.515629
$23$	−3.00054	−0.625655	−0.312828	−	0.949810i	$-0.601276\pi$
$23$	−3.00054	−0.625655	−0.312828	+	0.949810i	$0.601276\pi$
$24$	5.00345	1.02133
$25$	2.83565	0.567130
$26$	−10.1186	−1.98442
$27$	−1.00000	−0.192450
$28$	4.03647	0.762821
$29$	−2.48107	−0.460723	−0.230362	−	0.973105i	$-0.573991\pi$
$29$	−2.48107	−0.460723	−0.230362	+	0.973105i	$0.573991\pi$
$30$	−6.87748	−1.25565
$31$	−1.69867	−0.305090	−0.152545	−	0.988297i	$-0.548747\pi$
$31$	−1.69867	−0.305090	−0.152545	+	0.988297i	$0.548747\pi$
$32$	−0.361700	−0.0639401
$33$	0.984367	0.171356
$34$	3.47404	0.595792
$35$	−2.79922	−0.473155
$36$	4.03647	0.672745
$37$	−4.85422	−0.798029	−0.399014	−	0.916945i	$-0.630648\pi$
$37$	−4.85422	−0.798029	−0.399014	+	0.916945i	$0.630648\pi$
$38$	−18.7921	−3.04849
$39$	−4.11840	−0.659472
$40$	14.0058	2.21451
$41$	2.85338	0.445623	0.222811	−	0.974862i	$-0.428477\pi$
$41$	2.85338	0.445623	0.222811	+	0.974862i	$0.428477\pi$
$42$	2.45692	0.379111
$43$	7.04176	1.07386	0.536929	−	0.843628i	$-0.319584\pi$
$43$	7.04176	1.07386	0.536929	+	0.843628i	$0.319584\pi$
$44$	−3.97337	−0.599008
$45$	−2.79922	−0.417284
$46$	7.37209	1.08696
$47$	9.67674	1.41150	0.705749	−	0.708462i	$-0.250611\pi$
$47$	9.67674	1.41150	0.705749	+	0.708462i	$0.250611\pi$
$48$	−4.22016	−0.609128
$49$	1.00000	0.142857
$50$	−6.96697	−0.985279
$51$	1.41398	0.197997
$52$	16.6238	2.30531
$53$	−2.69129	−0.369678	−0.184839	−	0.982769i	$-0.559176\pi$
$53$	−2.69129	−0.369678	−0.184839	+	0.982769i	$0.559176\pi$
$54$	2.45692	0.334345
$55$	2.75546	0.371546
$56$	−5.00345	−0.668615
$57$	−7.64864	−1.01309
$58$	6.09580	0.800418
$59$	8.15845	1.06214	0.531070	−	0.847328i	$-0.321790\pi$
$59$	8.15845	1.06214	0.531070	+	0.847328i	$0.321790\pi$
$60$	11.2990	1.45869
$61$	−7.43534	−0.951998	−0.475999	−	0.879446i	$-0.657913\pi$
$61$	−7.43534	−0.951998	−0.475999	+	0.879446i	$0.657913\pi$
$62$	4.17350	0.530034
$63$	1.00000	0.125988
$64$	−7.55165	−0.943956
$65$	−11.5283	−1.42991
$66$	−2.41851	−0.297698
$67$	6.56297	0.801794	0.400897	−	0.916123i	$-0.368698\pi$
$67$	6.56297	0.801794	0.400897	+	0.916123i	$0.368698\pi$
$68$	−5.70748	−0.692134
$69$	3.00054	0.361222
$70$	6.87748	0.822016
$71$	14.0094	1.66261	0.831306	−	0.555815i	$-0.187594\pi$
$71$	14.0094	1.66261	0.831306	+	0.555815i	$0.187594\pi$
$72$	−5.00345	−0.589663
$73$	5.55255	0.649877	0.324938	−	0.945735i	$-0.394656\pi$
$73$	5.55255	0.649877	0.324938	+	0.945735i	$0.394656\pi$
$74$	11.9264	1.38642
$75$	−2.83565	−0.327433
$76$	30.8735	3.54144
$77$	−0.984367	−0.112179
$78$	10.1186	1.14571
$79$	9.46209	1.06457	0.532284	−	0.846566i	$-0.321334\pi$
$79$	9.46209	1.06457	0.532284	+	0.846566i	$0.321334\pi$
$80$	−11.8132	−1.32075
$81$	1.00000	0.111111
$82$	−7.01053	−0.774184
$83$	12.1889	1.33791	0.668955	−	0.743303i	$-0.266742\pi$
$83$	12.1889	1.33791	0.668955	+	0.743303i	$0.266742\pi$
$84$	−4.03647	−0.440415
$85$	3.95804	0.429310
$86$	−17.3011	−1.86562
$87$	2.48107	0.265999
$88$	4.92523	0.525032
$89$	−8.31439	−0.881324	−0.440662	−	0.897673i	$-0.645256\pi$
$89$	−8.31439	−0.881324	−0.440662	+	0.897673i	$0.645256\pi$
$90$	6.87748	0.724950
$91$	4.11840	0.431726
$92$	−12.1116	−1.26272
$93$	1.69867	0.176144
$94$	−23.7750	−2.45220
$95$	−21.4103	−2.19665
$96$	0.361700	0.0369158
$97$	−7.65077	−0.776818	−0.388409	−	0.921487i	$-0.626975\pi$
$97$	−7.65077	−0.776818	−0.388409	+	0.921487i	$0.626975\pi$
$98$	−2.45692	−0.248187
$99$	−0.984367	−0.0989326
$100$	11.4460	1.14460
$101$	−17.3784	−1.72922	−0.864610	−	0.502444i	$-0.832434\pi$
$101$	−17.3784	−1.72922	−0.864610	+	0.502444i	$0.832434\pi$
$102$	−3.47404	−0.343981
$103$	7.78687	0.767263	0.383631	−	0.923486i	$-0.374673\pi$
$103$	7.78687	0.767263	0.383631	+	0.923486i	$0.374673\pi$
$104$	−20.6062	−2.02061
$105$	2.79922	0.273176
$106$	6.61230	0.642244
$107$	−6.85574	−0.662770	−0.331385	−	0.943496i	$-0.607516\pi$
$107$	−6.85574	−0.662770	−0.331385	+	0.943496i	$0.607516\pi$
$108$	−4.03647	−0.388410
$109$	−16.2961	−1.56088	−0.780440	−	0.625231i	$-0.785005\pi$
$109$	−16.2961	−1.56088	−0.780440	+	0.625231i	$0.785005\pi$
$110$	−6.76996	−0.645490
$111$	4.85422	0.460742
$112$	4.22016	0.398768
$113$	−2.31882	−0.218136	−0.109068	−	0.994034i	$-0.534787\pi$
$113$	−2.31882	−0.218136	−0.109068	+	0.994034i	$0.534787\pi$
$114$	18.7921	1.76004
$115$	8.39917	0.783227
$116$	−10.0148	−0.929848
$117$	4.11840	0.380746
$118$	−20.0447	−1.84526
$119$	−1.41398	−0.129619
$120$	−14.0058	−1.27855
$121$	−10.0310	−0.911911
$122$	18.2681	1.65391
$123$	−2.85338	−0.257281
$124$	−6.85662	−0.615743
$125$	6.05850	0.541889
$126$	−2.45692	−0.218880
$127$	19.7333	1.75104	0.875522	−	0.483179i	$-0.160518\pi$
$127$	19.7333	1.75104	0.875522	+	0.483179i	$0.160518\pi$
$128$	19.2772	1.70388
$129$	−7.04176	−0.619992
$130$	28.3242	2.48420
$131$	18.2284	1.59262	0.796310	−	0.604889i	$-0.206782\pi$
$131$	18.2284	1.59262	0.796310	+	0.604889i	$0.206782\pi$
$132$	3.97337	0.345837
$133$	7.64864	0.663222
$134$	−16.1247	−1.39296
$135$	2.79922	0.240919
$136$	7.07477	0.606657
$137$	−2.80549	−0.239689	−0.119845	−	0.992793i	$-0.538240\pi$
$137$	−2.80549	−0.239689	−0.119845	+	0.992793i	$0.538240\pi$
$138$	−7.37209	−0.627554
$139$	−12.2230	−1.03674	−0.518371	−	0.855156i	$-0.673461\pi$
$139$	−12.2230	−1.03674	−0.518371	+	0.855156i	$0.673461\pi$
$140$	−11.2990	−0.954938
$141$	−9.67674	−0.814929
$142$	−34.4201	−2.88847
$143$	−4.05402	−0.339014
$144$	4.22016	0.351680
$145$	6.94507	0.576757
$146$	−13.6422	−1.12904
$147$	−1.00000	−0.0824786
$148$	−19.5939	−1.61061
$149$	23.6444	1.93703	0.968513	−	0.248963i	$-0.0800898\pi$
$149$	23.6444	1.93703	0.968513	+	0.248963i	$0.0800898\pi$
$150$	6.96697	0.568851
$151$	−6.36352	−0.517856	−0.258928	−	0.965897i	$-0.583369\pi$
$151$	−6.36352	−0.517856	−0.258928	+	0.965897i	$0.583369\pi$
$152$	−38.2696	−3.10408
$153$	−1.41398	−0.114313
$154$	2.41851	0.194889
$155$	4.75495	0.381927
$156$	−16.6238	−1.33097
$157$	9.84345	0.785593	0.392796	−	0.919625i	$-0.371508\pi$
$157$	9.84345	0.785593	0.392796	+	0.919625i	$0.371508\pi$
$158$	−23.2476	−1.84948
$159$	2.69129	0.213433
$160$	1.01248	0.0800435
$161$	−3.00054	−0.236475
$162$	−2.45692	−0.193034
$163$	−3.61022	−0.282774	−0.141387	−	0.989954i	$-0.545156\pi$
$163$	−3.61022	−0.282774	−0.141387	+	0.989954i	$0.545156\pi$
$164$	11.5176	0.899372
$165$	−2.75546	−0.214512
$166$	−29.9473	−2.32436
$167$	−10.1072	−0.782120	−0.391060	−	0.920365i	$-0.627891\pi$
$167$	−10.1072	−0.782120	−0.391060	+	0.920365i	$0.627891\pi$
$168$	5.00345	0.386025
$169$	3.96123	0.304710
$170$	−9.72460	−0.745843
$171$	7.64864	0.584906
$172$	28.4238	2.16730
$173$	0.0346365	0.00263337	0.00131668	−	0.999999i	$-0.499581\pi$
$173$	0.0346365	0.00263337	0.00131668	+	0.999999i	$0.499581\pi$
$174$	−6.09580	−0.462121
$175$	2.83565	0.214355
$176$	−4.15418	−0.313133
$177$	−8.15845	−0.613227
$178$	20.4278	1.53113
$179$	−15.2196	−1.13756	−0.568782	−	0.822489i	$-0.692585\pi$
$179$	−15.2196	−1.13756	−0.568782	+	0.822489i	$0.692585\pi$
$180$	−11.2990	−0.842177
$181$	9.51027	0.706893	0.353446	−	0.935455i	$-0.385010\pi$
$181$	9.51027	0.706893	0.353446	+	0.935455i	$0.385010\pi$
$182$	−10.1186	−0.750040
$183$	7.43534	0.549636
$184$	15.0131	1.10678
$185$	13.5880	0.999013
$186$	−4.17350	−0.306016
$187$	1.39187	0.101784
$188$	39.0599	2.84874
$189$	−1.00000	−0.0727393
$190$	52.6034	3.81625
$191$	−9.31258	−0.673835	−0.336917	−	0.941534i	$-0.609384\pi$
$191$	−9.31258	−0.673835	−0.336917	+	0.941534i	$0.609384\pi$
$192$	7.55165	0.544993
$193$	−3.50958	−0.252625	−0.126313	−	0.991990i	$-0.540314\pi$
$193$	−3.50958	−0.252625	−0.126313	+	0.991990i	$0.540314\pi$
$194$	18.7974	1.34957
$195$	11.5283	0.825561
$196$	4.03647	0.288319
$197$	18.2121	1.29756	0.648779	−	0.760977i	$-0.275280\pi$
$197$	18.2121	1.29756	0.648779	+	0.760977i	$0.275280\pi$
$198$	2.41851	0.171876
$199$	−22.8073	−1.61676	−0.808382	−	0.588659i	$-0.799656\pi$
$199$	−22.8073	−1.61676	−0.808382	+	0.588659i	$0.799656\pi$
$200$	−14.1880	−1.00325
$201$	−6.56297	−0.462916
$202$	42.6975	3.00418
$203$	−2.48107	−0.174137
$204$	5.70748	0.399604
$205$	−7.98724	−0.557853
$206$	−19.1317	−1.33297
$207$	−3.00054	−0.208552
$208$	17.3803	1.20511
$209$	−7.52907	−0.520797
$210$	−6.87748	−0.474591
$211$	−16.4783	−1.13441	−0.567207	−	0.823575i	$-0.691976\pi$
$211$	−16.4783	−1.13441	−0.567207	+	0.823575i	$0.691976\pi$
$212$	−10.8633	−0.746097
$213$	−14.0094	−0.959909
$214$	16.8440	1.15143
$215$	−19.7114	−1.34431
$216$	5.00345	0.340442
$217$	−1.69867	−0.115313
$218$	40.0382	2.71173
$219$	−5.55255	−0.375206
$220$	11.1223	0.749868
$221$	−5.82333	−0.391720
$222$	−11.9264	−0.800451
$223$	−12.9237	−0.865435	−0.432717	−	0.901530i	$-0.642445\pi$
$223$	−12.9237	−0.865435	−0.432717	+	0.901530i	$0.642445\pi$
$224$	−0.361700	−0.0241671
$225$	2.83565	0.189043
$226$	5.69716	0.378969
$227$	0.269983	0.0179194	0.00895969	−	0.999960i	$-0.497148\pi$
$227$	0.269983	0.0179194	0.00895969	+	0.999960i	$0.497148\pi$
$228$	−30.8735	−2.04465
$229$	−22.0587	−1.45768	−0.728840	−	0.684684i	$-0.759940\pi$
$229$	−22.0587	−1.45768	−0.728840	+	0.684684i	$0.759940\pi$
$230$	−20.6361	−1.36071
$231$	0.984367	0.0647666
$232$	12.4139	0.815014
$233$	−14.9230	−0.977640	−0.488820	−	0.872385i	$-0.662572\pi$
$233$	−14.9230	−0.977640	−0.488820	+	0.872385i	$0.662572\pi$
$234$	−10.1186	−0.661473
$235$	−27.0874	−1.76698
$236$	32.9314	2.14365
$237$	−9.46209	−0.614629
$238$	3.47404	0.225188
$239$	18.5534	1.20012	0.600061	−	0.799954i	$-0.295143\pi$
$239$	18.5534	1.20012	0.600061	+	0.799954i	$0.295143\pi$
$240$	11.8132	0.762537
$241$	2.43794	0.157041	0.0785207	−	0.996912i	$-0.474980\pi$
$241$	2.43794	0.157041	0.0785207	+	0.996912i	$0.474980\pi$
$242$	24.6455	1.58427
$243$	−1.00000	−0.0641500
$244$	−30.0125	−1.92136
$245$	−2.79922	−0.178836
$246$	7.01053	0.446975
$247$	31.5002	2.00431
$248$	8.49920	0.539700
$249$	−12.1889	−0.772442
$250$	−14.8853	−0.941427
$251$	1.35703	0.0856551	0.0428275	−	0.999082i	$-0.486363\pi$
$251$	1.35703	0.0856551	0.0428275	+	0.999082i	$0.486363\pi$
$252$	4.03647	0.254274
$253$	2.95363	0.185693
$254$	−48.4831	−3.04210
$255$	−3.95804	−0.247862
$256$	−32.2594	−2.01621
$257$	−4.66411	−0.290939	−0.145470	−	0.989363i	$-0.546469\pi$
$257$	−4.66411	−0.290939	−0.145470	+	0.989363i	$0.546469\pi$
$258$	17.3011	1.07712
$259$	−4.85422	−0.301627
$260$	−46.5338	−2.88590
$261$	−2.48107	−0.153574
$262$	−44.7857	−2.76687
$263$	9.57669	0.590524	0.295262	−	0.955416i	$-0.404593\pi$
$263$	9.57669	0.590524	0.295262	+	0.955416i	$0.404593\pi$
$264$	−4.92523	−0.303127
$265$	7.53353	0.462781
$266$	−18.7921	−1.15222
$267$	8.31439	0.508832
$268$	26.4912	1.61821
$269$	10.3269	0.629639	0.314820	−	0.949152i	$-0.398056\pi$
$269$	10.3269	0.629639	0.314820	+	0.949152i	$0.398056\pi$
$270$	−6.87748	−0.418550
$271$	−12.6993	−0.771427	−0.385713	−	0.922619i	$-0.626045\pi$
$271$	−12.6993	−0.771427	−0.385713	+	0.922619i	$0.626045\pi$
$272$	−5.96721	−0.361815
$273$	−4.11840	−0.249257
$274$	6.89288	0.416414
$275$	−2.79132	−0.168323
$276$	12.1116	0.729032
$277$	15.4665	0.929294	0.464647	−	0.885496i	$-0.346181\pi$
$277$	15.4665	0.929294	0.464647	+	0.885496i	$0.346181\pi$
$278$	30.0310	1.80114
$279$	−1.69867	−0.101697
$280$	14.0058	0.837006
$281$	21.9734	1.31082	0.655411	−	0.755272i	$-0.272495\pi$
$281$	21.9734	1.31082	0.655411	+	0.755272i	$0.272495\pi$
$282$	23.7750	1.41578
$283$	−22.6149	−1.34431	−0.672157	−	0.740409i	$-0.734632\pi$
$283$	−22.6149	−1.34431	−0.672157	+	0.740409i	$0.734632\pi$
$284$	56.5486	3.35554
$285$	21.4103	1.26823
$286$	9.96041	0.588971
$287$	2.85338	0.168430
$288$	−0.361700	−0.0213134
$289$	−15.0007	−0.882392
$290$	−17.0635	−1.00200
$291$	7.65077	0.448496
$292$	22.4127	1.31160
$293$	−5.75828	−0.336403	−0.168201	−	0.985753i	$-0.553796\pi$
$293$	−5.75828	−0.336403	−0.168201	+	0.985753i	$0.553796\pi$
$294$	2.45692	0.143291
$295$	−22.8373	−1.32964
$296$	24.2879	1.41170
$297$	0.984367	0.0571187
$298$	−58.0925	−3.36521
$299$	−12.3574	−0.714648
$300$	−11.4460	−0.660836
$301$	7.04176	0.405880
$302$	15.6347	0.899674
$303$	17.3784	0.998365
$304$	32.2785	1.85130
$305$	20.8132	1.19176
$306$	3.47404	0.198597
$307$	17.0315	0.972039	0.486020	−	0.873948i	$-0.338448\pi$
$307$	17.0315	0.972039	0.486020	+	0.873948i	$0.338448\pi$
$308$	−3.97337	−0.226404
$309$	−7.78687	−0.442979
$310$	−11.6825	−0.663524
$311$	0.487175	0.0276252	0.0138126	−	0.999905i	$-0.495603\pi$
$311$	0.487175	0.0276252	0.0138126	+	0.999905i	$0.495603\pi$
$312$	20.6062	1.16660
$313$	28.5388	1.61311	0.806553	−	0.591162i	$-0.201331\pi$
$313$	28.5388	1.61311	0.806553	+	0.591162i	$0.201331\pi$
$314$	−24.1846	−1.36482
$315$	−2.79922	−0.157718
$316$	38.1935	2.14855
$317$	−32.6115	−1.83164	−0.915822	−	0.401583i	$-0.868460\pi$
$317$	−32.6115	−1.83164	−0.915822	+	0.401583i	$0.868460\pi$
$318$	−6.61230	−0.370800
$319$	2.44228	0.136742
$320$	21.1388	1.18169
$321$	6.85574	0.382650
$322$	7.37209	0.410831
$323$	−10.8150	−0.601763
$324$	4.03647	0.224248
$325$	11.6783	0.647798
$326$	8.87002	0.491265
$327$	16.2961	0.901174
$328$	−14.2768	−0.788302
$329$	9.67674	0.533496
$330$	6.76996	0.372674
$331$	18.1294	0.996481	0.498241	−	0.867039i	$-0.333980\pi$
$331$	18.1294	0.996481	0.498241	+	0.867039i	$0.333980\pi$
$332$	49.2003	2.70022
$333$	−4.85422	−0.266010
$334$	24.8326	1.35878
$335$	−18.3712	−1.00373
$336$	−4.22016	−0.230229
$337$	29.6809	1.61682	0.808411	−	0.588619i	$-0.200328\pi$
$337$	29.6809	1.61682	0.808411	+	0.588619i	$0.200328\pi$
$338$	−9.73245	−0.529375
$339$	2.31882	0.125941
$340$	15.9765	0.866448
$341$	1.67211	0.0905499
$342$	−18.7921	−1.01616
$343$	1.00000	0.0539949
$344$	−35.2331	−1.89964
$345$	−8.39917	−0.452196
$346$	−0.0850993	−0.00457497
$347$	−16.9246	−0.908558	−0.454279	−	0.890860i	$-0.650103\pi$
$347$	−16.9246	−0.908558	−0.454279	+	0.890860i	$0.650103\pi$
$348$	10.0148	0.536848
$349$	5.92140	0.316965	0.158483	−	0.987362i	$-0.449340\pi$
$349$	5.92140	0.316965	0.158483	+	0.987362i	$0.449340\pi$
$350$	−6.96697	−0.372400
$351$	−4.11840	−0.219824
$352$	0.356045	0.0189773
$353$	5.18274	0.275850	0.137925	−	0.990443i	$-0.455957\pi$
$353$	5.18274	0.275850	0.137925	+	0.990443i	$0.455957\pi$
$354$	20.0447	1.06536
$355$	−39.2155	−2.08134
$356$	−33.5608	−1.77872
$357$	1.41398	0.0748356
$358$	37.3933	1.97630
$359$	−10.4370	−0.550846	−0.275423	−	0.961323i	$-0.588818\pi$
$359$	−10.4370	−0.550846	−0.275423	+	0.961323i	$0.588818\pi$
$360$	14.0058	0.738170
$361$	39.5017	2.07904
$362$	−23.3660	−1.22809
$363$	10.0310	0.526492
$364$	16.6238	0.871325
$365$	−15.5428	−0.813549
$366$	−18.2681	−0.954887
$367$	19.9298	1.04033	0.520163	−	0.854067i	$-0.325871\pi$
$367$	19.9298	1.04033	0.520163	+	0.854067i	$0.325871\pi$
$368$	−12.6627	−0.660091
$369$	2.85338	0.148541
$370$	−33.3848	−1.73559
$371$	−2.69129	−0.139725
$372$	6.85662	0.355499
$373$	2.61334	0.135313	0.0676567	−	0.997709i	$-0.478448\pi$
$373$	2.61334	0.135313	0.0676567	+	0.997709i	$0.478448\pi$
$374$	−3.41972	−0.176830
$375$	−6.05850	−0.312859
$376$	−48.4171	−2.49692
$377$	−10.2180	−0.526256
$378$	2.45692	0.126370
$379$	−31.0034	−1.59254	−0.796269	−	0.604942i	$-0.793196\pi$
$379$	−31.0034	−1.59254	−0.796269	+	0.604942i	$0.793196\pi$
$380$	−86.4219	−4.43335
$381$	−19.7333	−1.01097
$382$	22.8803	1.17066
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	−19.2772	−0.983737
$385$	2.75546	0.140431
$386$	8.62277	0.438887
$387$	7.04176	0.357953
$388$	−30.8821	−1.56780
$389$	12.5464	0.636126	0.318063	−	0.948070i	$-0.396968\pi$
$389$	12.5464	0.636126	0.318063	+	0.948070i	$0.396968\pi$
$390$	−28.3242	−1.43425
$391$	4.24269	0.214562
$392$	−5.00345	−0.252713
$393$	−18.2284	−0.919500
$394$	−44.7457	−2.25426
$395$	−26.4865	−1.33268
$396$	−3.97337	−0.199669
$397$	−16.5278	−0.829506	−0.414753	−	0.909934i	$-0.636132\pi$
$397$	−16.5278	−0.829506	−0.414753	+	0.909934i	$0.636132\pi$
$398$	56.0357	2.80881
$399$	−7.64864	−0.382911
$400$	11.9669	0.598345
$401$	−18.3159	−0.914654	−0.457327	−	0.889299i	$-0.651193\pi$
$401$	−18.3159	−0.914654	−0.457327	+	0.889299i	$0.651193\pi$
$402$	16.1247	0.804228
$403$	−6.99580	−0.348485
$404$	−70.1476	−3.48997
$405$	−2.79922	−0.139095
$406$	6.09580	0.302529
$407$	4.77833	0.236853
$408$	−7.07477	−0.350254
$409$	14.3630	0.710207	0.355103	−	0.934827i	$-0.384446\pi$
$409$	14.3630	0.710207	0.355103	+	0.934827i	$0.384446\pi$
$410$	19.6240	0.969163
$411$	2.80549	0.138385
$412$	31.4315	1.54852
$413$	8.15845	0.401451
$414$	7.37209	0.362318
$415$	−34.1195	−1.67486
$416$	−1.48963	−0.0730349
$417$	12.2230	0.598564
$418$	18.4983	0.904784
$419$	5.32951	0.260363	0.130182	−	0.991490i	$-0.458444\pi$
$419$	5.32951	0.260363	0.130182	+	0.991490i	$0.458444\pi$
$420$	11.2990	0.551334
$421$	9.38637	0.457464	0.228732	−	0.973489i	$-0.426542\pi$
$421$	9.38637	0.457464	0.228732	+	0.973489i	$0.426542\pi$
$422$	40.4860	1.97083
$423$	9.67674	0.470499
$424$	13.4658	0.653955
$425$	−4.00955	−0.194492
$426$	34.4201	1.66766
$427$	−7.43534	−0.359821
$428$	−27.6730	−1.33763
$429$	4.05402	0.195730
$430$	48.4295	2.33548
$431$	15.2493	0.734535	0.367267	−	0.930115i	$-0.380293\pi$
$431$	15.2493	0.734535	0.367267	+	0.930115i	$0.380293\pi$
$432$	−4.22016	−0.203043
$433$	−25.2305	−1.21250	−0.606249	−	0.795275i	$-0.707327\pi$
$433$	−25.2305	−1.21250	−0.606249	+	0.795275i	$0.707327\pi$
$434$	4.17350	0.200334
$435$	−6.94507	−0.332991
$436$	−65.7786	−3.15022
$437$	−22.9500	−1.09785
$438$	13.6422	0.651849
$439$	6.31008	0.301164	0.150582	−	0.988598i	$-0.451885\pi$
$439$	6.31008	0.301164	0.150582	+	0.988598i	$0.451885\pi$
$440$	−13.7868	−0.657261
$441$	1.00000	0.0476190
$442$	14.3075	0.680537
$443$	−34.4740	−1.63791	−0.818956	−	0.573857i	$-0.805447\pi$
$443$	−34.4740	−1.63791	−0.818956	+	0.573857i	$0.805447\pi$
$444$	19.5939	0.929886
$445$	23.2738	1.10329
$446$	31.7525	1.50353
$447$	−23.6444	−1.11834
$448$	−7.55165	−0.356782
$449$	−24.9641	−1.17813	−0.589065	−	0.808086i	$-0.700504\pi$
$449$	−24.9641	−1.17813	−0.589065	+	0.808086i	$0.700504\pi$
$450$	−6.96697	−0.328426
$451$	−2.80877	−0.132260
$452$	−9.35984	−0.440250
$453$	6.36352	0.298984
$454$	−0.663327	−0.0311315
$455$	−11.5283	−0.540456
$456$	38.2696	1.79214
$457$	−28.9746	−1.35538	−0.677688	−	0.735349i	$-0.737018\pi$
$457$	−28.9746	−1.35538	−0.677688	+	0.735349i	$0.737018\pi$
$458$	54.1966	2.53244
$459$	1.41398	0.0659988
$460$	33.9030	1.58074
$461$	30.3866	1.41525	0.707623	−	0.706590i	$-0.249768\pi$
$461$	30.3866	1.41525	0.707623	+	0.706590i	$0.249768\pi$
$462$	−2.41851	−0.112519
$463$	19.5012	0.906299	0.453150	−	0.891434i	$-0.350300\pi$
$463$	19.5012	0.906299	0.453150	+	0.891434i	$0.350300\pi$
$464$	−10.4705	−0.486081
$465$	−4.75495	−0.220505
$466$	36.6647	1.69846
$467$	−13.0969	−0.606051	−0.303025	−	0.952982i	$-0.597997\pi$
$467$	−13.0969	−0.606051	−0.303025	+	0.952982i	$0.597997\pi$
$468$	16.6238	0.768436
$469$	6.56297	0.303050
$470$	66.5516	3.06979
$471$	−9.84345	−0.453562
$472$	−40.8204	−1.87891
$473$	−6.93167	−0.318719
$474$	23.2476	1.06780
$475$	21.6889	0.995154
$476$	−5.70748	−0.261602
$477$	−2.69129	−0.123226
$478$	−45.5844	−2.08498
$479$	−1.59688	−0.0729632	−0.0364816	−	0.999334i	$-0.511615\pi$
$479$	−1.59688	−0.0729632	−0.0364816	+	0.999334i	$0.511615\pi$
$480$	−1.01248	−0.0462131
$481$	−19.9916	−0.911540
$482$	−5.98983	−0.272829
$483$	3.00054	0.136529
$484$	−40.4899	−1.84045
$485$	21.4162	0.972460
$486$	2.45692	0.111448
$487$	3.18973	0.144540	0.0722702	−	0.997385i	$-0.476976\pi$
$487$	3.18973	0.144540	0.0722702	+	0.997385i	$0.476976\pi$
$488$	37.2024	1.68407
$489$	3.61022	0.163260
$490$	6.87748	0.310693
$491$	27.0604	1.22122	0.610609	−	0.791933i	$-0.290925\pi$
$491$	27.0604	1.22122	0.610609	+	0.791933i	$0.290925\pi$
$492$	−11.5176	−0.519253
$493$	3.50818	0.158000
$494$	−77.3935	−3.48210
$495$	2.75546	0.123849
$496$	−7.16865	−0.321882
$497$	14.0094	0.628408
$498$	29.9473	1.34197
$499$	9.49967	0.425264	0.212632	−	0.977132i	$-0.431796\pi$
$499$	9.49967	0.425264	0.212632	+	0.977132i	$0.431796\pi$
$500$	24.4550	1.09366
$501$	10.1072	0.451557
$502$	−3.33412	−0.148809
$503$	−30.7234	−1.36989	−0.684945	−	0.728595i	$-0.740174\pi$
$503$	−30.7234	−1.36989	−0.684945	+	0.728595i	$0.740174\pi$
$504$	−5.00345	−0.222872
$505$	48.6461	2.16472
$506$	−7.25684	−0.322606
$507$	−3.96123	−0.175925
$508$	79.6527	3.53402
$509$	−21.5144	−0.953610	−0.476805	−	0.879009i	$-0.658205\pi$
$509$	−21.5144	−0.953610	−0.476805	+	0.879009i	$0.658205\pi$
$510$	9.72460	0.430613
$511$	5.55255	0.245630
$512$	40.7043	1.79889
$513$	−7.64864	−0.337696
$514$	11.4594	0.505450
$515$	−21.7972	−0.960498
$516$	−28.4238	−1.25129
$517$	−9.52546	−0.418929
$518$	11.9264	0.524018
$519$	−0.0346365	−0.00152037
$520$	57.6815	2.52950
$521$	−23.6374	−1.03557	−0.517787	−	0.855510i	$-0.673244\pi$
$521$	−23.6374	−1.03557	−0.517787	+	0.855510i	$0.673244\pi$
$522$	6.09580	0.266806
$523$	14.2272	0.622112	0.311056	−	0.950392i	$-0.399317\pi$
$523$	14.2272	0.622112	0.311056	+	0.950392i	$0.399317\pi$
$524$	73.5783	3.21428
$525$	−2.83565	−0.123758
$526$	−23.5292	−1.02592
$527$	2.40188	0.104627
$528$	4.15418	0.180788
$529$	−13.9968	−0.608555
$530$	−18.5093	−0.803993
$531$	8.15845	0.354047
$532$	30.8735	1.33854
$533$	11.7514	0.509008
$534$	−20.4278	−0.883998
$535$	19.1908	0.829689
$536$	−32.8375	−1.41836
$537$	15.2196	0.656772
$538$	−25.3723	−1.09388
$539$	−0.984367	−0.0423997
$540$	11.2990	0.486231
$541$	−19.4255	−0.835166	−0.417583	−	0.908639i	$-0.637123\pi$
$541$	−19.4255	−0.835166	−0.417583	+	0.908639i	$0.637123\pi$
$542$	31.2012	1.34020
$543$	−9.51027	−0.408125
$544$	0.511436	0.0219276
$545$	45.6163	1.95399
$546$	10.1186	0.433036
$547$	19.6489	0.840125	0.420063	−	0.907495i	$-0.362008\pi$
$547$	19.6489	0.840125	0.420063	+	0.907495i	$0.362008\pi$
$548$	−11.3243	−0.483750
$549$	−7.43534	−0.317333
$550$	6.85806	0.292429
$551$	−18.9768	−0.808440
$552$	−15.0131	−0.638998
$553$	9.46209	0.402369
$554$	−38.0001	−1.61447
$555$	−13.5880	−0.576780
$556$	−49.3379	−2.09239
$557$	0.547473	0.0231972	0.0115986	−	0.999933i	$-0.496308\pi$
$557$	0.547473	0.0231972	0.0115986	+	0.999933i	$0.496308\pi$
$558$	4.17350	0.176678
$559$	29.0008	1.22660
$560$	−11.8132	−0.499197
$561$	−1.39187	−0.0587649
$562$	−53.9869	−2.27730
$563$	−19.3828	−0.816888	−0.408444	−	0.912783i	$-0.633928\pi$
$563$	−19.3828	−0.816888	−0.408444	+	0.912783i	$0.633928\pi$
$564$	−39.0599	−1.64472
$565$	6.49089	0.273074
$566$	55.5630	2.33549
$567$	1.00000	0.0419961
$568$	−70.0955	−2.94114
$569$	8.21397	0.344347	0.172174	−	0.985067i	$-0.444921\pi$
$569$	8.21397	0.344347	0.172174	+	0.985067i	$0.444921\pi$
$570$	−52.6034	−2.20331
$571$	−16.4006	−0.686343	−0.343172	−	0.939273i	$-0.611501\pi$
$571$	−16.4006	−0.686343	−0.343172	+	0.939273i	$0.611501\pi$
$572$	−16.3639	−0.684210
$573$	9.31258	0.389039
$574$	−7.01053	−0.292614
$575$	−8.50847	−0.354828
$576$	−7.55165	−0.314652
$577$	−26.0604	−1.08491	−0.542453	−	0.840086i	$-0.682504\pi$
$577$	−26.0604	−1.08491	−0.542453	+	0.840086i	$0.682504\pi$
$578$	36.8555	1.53299
$579$	3.50958	0.145853
$580$	28.0336	1.16403
$581$	12.1889	0.505682
$582$	−18.7974	−0.779176
$583$	2.64922	0.109719
$584$	−27.7819	−1.14962
$585$	−11.5283	−0.476638
$586$	14.1477	0.584435
$587$	−16.1899	−0.668230	−0.334115	−	0.942532i	$-0.608437\pi$
$587$	−16.1899	−0.668230	−0.334115	+	0.942532i	$0.608437\pi$
$588$	−4.03647	−0.166461
$589$	−12.9925	−0.535347
$590$	56.1096	2.30999
$591$	−18.2121	−0.749145
$592$	−20.4856	−0.841952
$593$	24.0417	0.987273	0.493637	−	0.869668i	$-0.335667\pi$
$593$	24.0417	0.987273	0.493637	+	0.869668i	$0.335667\pi$
$594$	−2.41851	−0.0992328
$595$	3.95804	0.162264
$596$	95.4400	3.90938
$597$	22.8073	0.933439
$598$	30.3612	1.24156
$599$	12.6353	0.516266	0.258133	−	0.966109i	$-0.416893\pi$
$599$	12.6353	0.516266	0.258133	+	0.966109i	$0.416893\pi$
$600$	14.1880	0.579225
$601$	44.4543	1.81333	0.906664	−	0.421853i	$-0.138620\pi$
$601$	44.4543	1.81333	0.906664	+	0.421853i	$0.138620\pi$
$602$	−17.3011	−0.705138
$603$	6.56297	0.267265
$604$	−25.6862	−1.04515
$605$	28.0791	1.14158
$606$	−42.6975	−1.73447
$607$	37.5003	1.52209	0.761044	−	0.648700i	$-0.224687\pi$
$607$	37.5003	1.52209	0.761044	+	0.648700i	$0.224687\pi$
$608$	−2.76651	−0.112197
$609$	2.48107	0.100538
$610$	−51.1364	−2.07045
$611$	39.8527	1.61227
$612$	−5.70748	−0.230711
$613$	43.4573	1.75523	0.877613	−	0.479370i	$-0.159135\pi$
$613$	43.4573	1.75523	0.877613	+	0.479370i	$0.159135\pi$
$614$	−41.8451	−1.68873
$615$	7.98724	0.322077
$616$	4.92523	0.198443
$617$	−27.3678	−1.10178	−0.550892	−	0.834576i	$-0.685712\pi$
$617$	−27.3678	−1.10178	−0.550892	+	0.834576i	$0.685712\pi$
$618$	19.1317	0.769591
$619$	35.9429	1.44467	0.722334	−	0.691544i	$-0.243069\pi$
$619$	35.9429	1.44467	0.722334	+	0.691544i	$0.243069\pi$
$620$	19.1932	0.770818
$621$	3.00054	0.120407
$622$	−1.19695	−0.0479934
$623$	−8.31439	−0.333109
$624$	−17.3803	−0.695769
$625$	−31.1373	−1.24549
$626$	−70.1175	−2.80246
$627$	7.52907	0.300682
$628$	39.7328	1.58551
$629$	6.86376	0.273676
$630$	6.87748	0.274005
$631$	10.8669	0.432603	0.216301	−	0.976327i	$-0.430601\pi$
$631$	10.8669	0.432603	0.216301	+	0.976327i	$0.430601\pi$
$632$	−47.3431	−1.88321
$633$	16.4783	0.654955
$634$	80.1240	3.18213
$635$	−55.2378	−2.19204
$636$	10.8633	0.430759
$637$	4.11840	0.163177
$638$	−6.00050	−0.237562
$639$	14.0094	0.554204
$640$	−53.9613	−2.13301
$641$	−3.36714	−0.132994	−0.0664971	−	0.997787i	$-0.521182\pi$
$641$	−3.36714	−0.132994	−0.0664971	+	0.997787i	$0.521182\pi$
$642$	−16.8440	−0.664781
$643$	50.4220	1.98845	0.994225	−	0.107311i	$-0.0342241\pi$
$643$	50.4220	1.98845	0.994225	+	0.107311i	$0.0342241\pi$
$644$	−12.1116	−0.477263
$645$	19.7114	0.776137
$646$	26.5717	1.04545
$647$	37.9099	1.49039	0.745195	−	0.666847i	$-0.232356\pi$
$647$	37.9099	1.49039	0.745195	+	0.666847i	$0.232356\pi$
$648$	−5.00345	−0.196554
$649$	−8.03091	−0.315241
$650$	−28.6928	−1.12542
$651$	1.69867	0.0665760
$652$	−14.5725	−0.570705
$653$	19.4852	0.762515	0.381258	−	0.924469i	$-0.375491\pi$
$653$	19.4852	0.762515	0.381258	+	0.924469i	$0.375491\pi$
$654$	−40.0382	−1.56562
$655$	−51.0253	−1.99372
$656$	12.0417	0.470150
$657$	5.55255	0.216626
$658$	−23.7750	−0.926846
$659$	37.0870	1.44471	0.722353	−	0.691525i	$-0.243061\pi$
$659$	37.0870	1.44471	0.722353	+	0.691525i	$0.243061\pi$
$660$	−11.1223	−0.432937
$661$	−24.5936	−0.956580	−0.478290	−	0.878202i	$-0.658743\pi$
$661$	−24.5936	−0.956580	−0.478290	+	0.878202i	$0.658743\pi$
$662$	−44.5425	−1.73119
$663$	5.82333	0.226159
$664$	−60.9868	−2.36675
$665$	−21.4103	−0.830254
$666$	11.9264	0.462140
$667$	7.44454	0.288254
$668$	−40.7975	−1.57850
$669$	12.9237	0.499659
$670$	45.1367	1.74378
$671$	7.31910	0.282551
$672$	0.361700	0.0139529
$673$	−29.5007	−1.13717	−0.568585	−	0.822624i	$-0.692509\pi$
$673$	−29.5007	−1.13717	−0.568585	+	0.822624i	$0.692509\pi$
$674$	−72.9237	−2.80891
$675$	−2.83565	−0.109144
$676$	15.9894	0.614977
$677$	26.6087	1.02266	0.511328	−	0.859386i	$-0.329154\pi$
$677$	26.6087	1.02266	0.511328	+	0.859386i	$0.329154\pi$
$678$	−5.69716	−0.218798
$679$	−7.65077	−0.293610
$680$	−19.8039	−0.759444
$681$	−0.269983	−0.0103458
$682$	−4.10825	−0.157313
$683$	48.8348	1.86861	0.934306	−	0.356471i	$-0.116020\pi$
$683$	48.8348	1.86861	0.934306	+	0.356471i	$0.116020\pi$
$684$	30.8735	1.18048
$685$	7.85320	0.300055
$686$	−2.45692	−0.0938058
$687$	22.0587	0.841592
$688$	29.7173	1.13296
$689$	−11.0838	−0.422260
$690$	20.6361	0.785604
$691$	−38.2014	−1.45325	−0.726624	−	0.687035i	$-0.758912\pi$
$691$	−38.2014	−1.45325	−0.726624	+	0.687035i	$0.758912\pi$
$692$	0.139809	0.00531475
$693$	−0.984367	−0.0373930
$694$	41.5823	1.57844
$695$	34.2149	1.29785
$696$	−12.4139	−0.470548
$697$	−4.03461	−0.152822
$698$	−14.5484	−0.550666
$699$	14.9230	0.564441
$700$	11.4460	0.432619
$701$	44.9529	1.69785	0.848923	−	0.528517i	$-0.177252\pi$
$701$	44.9529	1.69785	0.848923	+	0.528517i	$0.177252\pi$
$702$	10.1186	0.381902
$703$	−37.1282	−1.40032
$704$	7.43359	0.280164
$705$	27.0874	1.02017
$706$	−12.7336	−0.479235
$707$	−17.3784	−0.653583
$708$	−32.9314	−1.23764
$709$	−2.74087	−0.102936	−0.0514678	−	0.998675i	$-0.516390\pi$
$709$	−2.74087	−0.102936	−0.0514678	+	0.998675i	$0.516390\pi$
$710$	96.3494	3.61593
$711$	9.46209	0.354856
$712$	41.6007	1.55905
$713$	5.09692	0.190881
$714$	−3.47404	−0.130013
$715$	11.3481	0.424395
$716$	−61.4333	−2.29587
$717$	−18.5534	−0.692891
$718$	25.6430	0.956988
$719$	−38.4321	−1.43328	−0.716639	−	0.697445i	$-0.754320\pi$
$719$	−38.4321	−1.43328	−0.716639	+	0.697445i	$0.754320\pi$
$720$	−11.8132	−0.440251
$721$	7.78687	0.289998
$722$	−97.0528	−3.61193
$723$	−2.43794	−0.0906679
$724$	38.3879	1.42668
$725$	−7.03545	−0.261290
$726$	−24.6455	−0.914678
$727$	3.18885	0.118268	0.0591339	−	0.998250i	$-0.481166\pi$
$727$	3.18885	0.118268	0.0591339	+	0.998250i	$0.481166\pi$
$728$	−20.6062	−0.763718
$729$	1.00000	0.0370370
$730$	38.1875	1.41338
$731$	−9.95689	−0.368269
$732$	30.0125	1.10930
$733$	−5.90014	−0.217926	−0.108963	−	0.994046i	$-0.534753\pi$
$733$	−5.90014	−0.217926	−0.108963	+	0.994046i	$0.534753\pi$
$734$	−48.9660	−1.80737
$735$	2.79922	0.103251
$736$	1.08529	0.0400045
$737$	−6.46037	−0.237971
$738$	−7.01053	−0.258061
$739$	2.89835	0.106617	0.0533087	−	0.998578i	$-0.483023\pi$
$739$	2.89835	0.106617	0.0533087	+	0.998578i	$0.483023\pi$
$740$	54.8478	2.01624
$741$	−31.5002	−1.15719
$742$	6.61230	0.242745
$743$	−17.0729	−0.626344	−0.313172	−	0.949696i	$-0.601392\pi$
$743$	−17.0729	−0.626344	−0.313172	+	0.949696i	$0.601392\pi$
$744$	−8.49920	−0.311596
$745$	−66.1860	−2.42487
$746$	−6.42077	−0.235081
$747$	12.1889	0.445970
$748$	5.61825	0.205424
$749$	−6.85574	−0.250503
$750$	14.8853	0.543533
$751$	45.9207	1.67567	0.837835	−	0.545924i	$-0.183821\pi$
$751$	45.9207	1.67567	0.837835	+	0.545924i	$0.183821\pi$
$752$	40.8374	1.48919
$753$	−1.35703	−0.0494530
$754$	25.1049	0.914268
$755$	17.8129	0.648278
$756$	−4.03647	−0.146805
$757$	17.1969	0.625033	0.312517	−	0.949912i	$-0.398828\pi$
$757$	17.1969	0.625033	0.312517	+	0.949912i	$0.398828\pi$
$758$	76.1730	2.76673
$759$	−2.95363	−0.107210
$760$	107.125	3.88584
$761$	−28.3587	−1.02800	−0.514002	−	0.857789i	$-0.671837\pi$
$761$	−28.3587	−1.02800	−0.514002	+	0.857789i	$0.671837\pi$
$762$	48.4831	1.75636
$763$	−16.2961	−0.589957
$764$	−37.5900	−1.35996
$765$	3.95804	0.143103
$766$	2.45692	0.0887722
$767$	33.5998	1.21322
$768$	32.2594	1.16406
$769$	23.5743	0.850110	0.425055	−	0.905167i	$-0.360255\pi$
$769$	23.5743	0.850110	0.425055	+	0.905167i	$0.360255\pi$
$770$	−6.76996	−0.243972
$771$	4.66411	0.167974
$772$	−14.1663	−0.509857
$773$	−19.0351	−0.684645	−0.342323	−	0.939582i	$-0.611214\pi$
$773$	−19.0351	−0.684645	−0.342323	+	0.939582i	$0.611214\pi$
$774$	−17.3011	−0.621874
$775$	−4.81683	−0.173026
$776$	38.2803	1.37418
$777$	4.85422	0.174144
$778$	−30.8255	−1.10515
$779$	21.8245	0.781943
$780$	46.5338	1.66618
$781$	−13.7904	−0.493459
$782$	−10.4240	−0.372761
$783$	2.48107	0.0886662
$784$	4.22016	0.150720
$785$	−27.5540	−0.983445
$786$	44.7857	1.59745
$787$	34.4990	1.22976	0.614878	−	0.788622i	$-0.289205\pi$
$787$	34.4990	1.22976	0.614878	+	0.788622i	$0.289205\pi$
$788$	73.5126	2.61878
$789$	−9.57669	−0.340939
$790$	65.0753	2.31528
$791$	−2.31882	−0.0824477
$792$	4.92523	0.175011
$793$	−30.6217	−1.08741
$794$	40.6075	1.44111
$795$	−7.53353	−0.267187
$796$	−92.0608	−3.26301
$797$	29.7462	1.05366	0.526832	−	0.849970i	$-0.323380\pi$
$797$	29.7462	1.05366	0.526832	+	0.849970i	$0.323380\pi$
$798$	18.7921	0.665234
$799$	−13.6827	−0.484059
$800$	−1.02565	−0.0362623
$801$	−8.31439	−0.293775
$802$	45.0008	1.58903
$803$	−5.46574	−0.192882
$804$	−26.4912	−0.934274
$805$	8.39917	0.296032
$806$	17.1881	0.605426
$807$	−10.3269	−0.363522
$808$	86.9522	3.05897
$809$	−32.8674	−1.15555	−0.577777	−	0.816194i	$-0.696080\pi$
$809$	−32.8674	−1.15555	−0.577777	+	0.816194i	$0.696080\pi$
$810$	6.87748	0.241650
$811$	19.1632	0.672910	0.336455	−	0.941700i	$-0.390772\pi$
$811$	19.1632	0.672910	0.336455	+	0.941700i	$0.390772\pi$
$812$	−10.0148	−0.351449
$813$	12.6993	0.445383
$814$	−11.7400	−0.411487
$815$	10.1058	0.353991
$816$	5.96721	0.208894
$817$	53.8599	1.88432
$818$	−35.2889	−1.23385
$819$	4.11840	0.143909
$820$	−32.2403	−1.12588
$821$	34.3594	1.19915	0.599575	−	0.800318i	$-0.295336\pi$
$821$	34.3594	1.19915	0.599575	+	0.800318i	$0.295336\pi$
$822$	−6.89288	−0.240417
$823$	9.59194	0.334354	0.167177	−	0.985927i	$-0.446535\pi$
$823$	9.59194	0.334354	0.167177	+	0.985927i	$0.446535\pi$
$824$	−38.9612	−1.35728
$825$	2.79132	0.0971813
$826$	−20.0447	−0.697444
$827$	10.9137	0.379506	0.189753	−	0.981832i	$-0.439231\pi$
$827$	10.9137	0.379506	0.189753	+	0.981832i	$0.439231\pi$
$828$	−12.1116	−0.420907
$829$	−38.1767	−1.32593	−0.662966	−	0.748650i	$-0.730702\pi$
$829$	−38.1767	−1.32593	−0.662966	+	0.748650i	$0.730702\pi$
$830$	83.8291	2.90975
$831$	−15.4665	−0.536528
$832$	−31.1007	−1.07822
$833$	−1.41398	−0.0489914
$834$	−30.0310	−1.03989
$835$	28.2923	0.979097
$836$	−30.3909	−1.05109
$837$	1.69867	0.0587145
$838$	−13.0942	−0.452331
$839$	−34.0424	−1.17528	−0.587638	−	0.809124i	$-0.699942\pi$
$839$	−34.0424	−1.17528	−0.587638	+	0.809124i	$0.699942\pi$
$840$	−14.0058	−0.483245
$841$	−22.8443	−0.787734
$842$	−23.0616	−0.794755
$843$	−21.9734	−0.756804
$844$	−66.5143	−2.28952
$845$	−11.0884	−0.381452
$846$	−23.7750	−0.817402
$847$	−10.0310	−0.344670
$848$	−11.3577	−0.390025
$849$	22.6149	0.776140
$850$	9.85115	0.337892
$851$	14.5653	0.499291
$852$	−56.5486	−1.93732
$853$	44.3957	1.52008	0.760041	−	0.649876i	$-0.225179\pi$
$853$	44.3957	1.52008	0.760041	+	0.649876i	$0.225179\pi$
$854$	18.2681	0.625120
$855$	−21.4103	−0.732215
$856$	34.3024	1.17243
$857$	3.70560	0.126581	0.0632905	−	0.997995i	$-0.479841\pi$
$857$	3.70560	0.126581	0.0632905	+	0.997995i	$0.479841\pi$
$858$	−9.96041	−0.340043
$859$	14.6259	0.499031	0.249515	−	0.968371i	$-0.419729\pi$
$859$	14.6259	0.499031	0.249515	+	0.968371i	$0.419729\pi$
$860$	−79.5647	−2.71313
$861$	−2.85338	−0.0972429
$862$	−37.4664	−1.27611
$863$	40.5646	1.38083	0.690417	−	0.723412i	$-0.257427\pi$
$863$	40.5646	1.38083	0.690417	+	0.723412i	$0.257427\pi$
$864$	0.361700	0.0123053
$865$	−0.0969554	−0.00329658
$866$	61.9893	2.10648
$867$	15.0007	0.509449
$868$	−6.85662	−0.232729
$869$	−9.31417	−0.315961
$870$	17.0635	0.578507
$871$	27.0289	0.915841
$872$	81.5366	2.76118
$873$	−7.65077	−0.258939
$874$	56.3865	1.90730
$875$	6.05850	0.204815
$876$	−22.4127	−0.757255
$877$	44.0283	1.48673	0.743364	−	0.668887i	$-0.233229\pi$
$877$	44.0283	1.48673	0.743364	+	0.668887i	$0.233229\pi$
$878$	−15.5034	−0.523214
$879$	5.75828	0.194222
$880$	11.6285	0.391996
$881$	−10.4937	−0.353540	−0.176770	−	0.984252i	$-0.556565\pi$
$881$	−10.4937	−0.353540	−0.176770	+	0.984252i	$0.556565\pi$
$882$	−2.45692	−0.0827289
$883$	−18.8625	−0.634775	−0.317388	−	0.948296i	$-0.602806\pi$
$883$	−18.8625	−0.634775	−0.317388	+	0.948296i	$0.602806\pi$
$884$	−23.5057	−0.790582
$885$	22.8373	0.767668
$886$	84.7001	2.84555
$887$	26.3462	0.884619	0.442310	−	0.896862i	$-0.354159\pi$
$887$	26.3462	0.884619	0.442310	+	0.896862i	$0.354159\pi$
$888$	−24.2879	−0.815048
$889$	19.7333	0.661832
$890$	−57.1820	−1.91675
$891$	−0.984367	−0.0329775
$892$	−52.1661	−1.74665
$893$	74.0139	2.47678
$894$	58.0925	1.94290
$895$	42.6029	1.42406
$896$	19.2772	0.644007
$897$	12.3574	0.412602
$898$	61.3349	2.04677
$899$	4.21451	0.140562
$900$	11.4460	0.381534
$901$	3.80543	0.126777
$902$	6.90093	0.229776
$903$	−7.04176	−0.234335
$904$	11.6021	0.385880
$905$	−26.6214	−0.884924
$906$	−15.6347	−0.519427
$907$	−13.5496	−0.449908	−0.224954	−	0.974369i	$-0.572223\pi$
$907$	−13.5496	−0.449908	−0.224954	+	0.974369i	$0.572223\pi$
$908$	1.08978	0.0361656
$909$	−17.3784	−0.576406
$910$	28.3242	0.938938
$911$	−2.70429	−0.0895972	−0.0447986	−	0.998996i	$-0.514265\pi$
$911$	−2.70429	−0.0895972	−0.0447986	+	0.998996i	$0.514265\pi$
$912$	−32.2785	−1.06885
$913$	−11.9984	−0.397088
$914$	71.1885	2.35471
$915$	−20.8132	−0.688062
$916$	−89.0394	−2.94194
$917$	18.2284	0.601954
$918$	−3.47404	−0.114660
$919$	48.6203	1.60384	0.801918	−	0.597434i	$-0.203813\pi$
$919$	48.6203	1.60384	0.801918	+	0.597434i	$0.203813\pi$
$920$	−42.0249	−1.38552
$921$	−17.0315	−0.561207
$922$	−74.6576	−2.45872
$923$	57.6964	1.89910
$924$	3.97337	0.130714
$925$	−13.7649	−0.452586
$926$	−47.9130	−1.57452
$927$	7.78687	0.255754
$928$	0.897403	0.0294587
$929$	20.5529	0.674319	0.337160	−	0.941447i	$-0.390534\pi$
$929$	20.5529	0.674319	0.337160	+	0.941447i	$0.390534\pi$
$930$	11.6825	0.383086
$931$	7.64864	0.250674
$932$	−60.2364	−1.97311
$933$	−0.487175	−0.0159494
$934$	32.1780	1.05290
$935$	−3.89616	−0.127418
$936$	−20.6062	−0.673536
$937$	42.5885	1.39131	0.695653	−	0.718378i	$-0.255115\pi$
$937$	42.5885	1.39131	0.695653	+	0.718378i	$0.255115\pi$
$938$	−16.1247	−0.526491
$939$	−28.5388	−0.931327
$940$	−109.337	−3.56619
$941$	3.51407	0.114556	0.0572778	−	0.998358i	$-0.481758\pi$
$941$	3.51407	0.114556	0.0572778	+	0.998358i	$0.481758\pi$
$942$	24.1846	0.787977
$943$	−8.56167	−0.278806
$944$	34.4300	1.12060
$945$	2.79922	0.0910587
$946$	17.0306	0.553712
$947$	39.5180	1.28416	0.642081	−	0.766637i	$-0.278071\pi$
$947$	39.5180	1.28416	0.642081	+	0.766637i	$0.278071\pi$
$948$	−38.1935	−1.24047
$949$	22.8676	0.742315
$950$	−53.2879	−1.72889
$951$	32.6115	1.05750
$952$	7.07477	0.229295
$953$	26.3266	0.852803	0.426401	−	0.904534i	$-0.359781\pi$
$953$	26.3266	0.852803	0.426401	+	0.904534i	$0.359781\pi$
$954$	6.61230	0.214081
$955$	26.0680	0.843540
$956$	74.8904	2.42213
$957$	−2.44228	−0.0789478
$958$	3.92341	0.126760
$959$	−2.80549	−0.0905941
$960$	−21.1388	−0.682250
$961$	−28.1145	−0.906920
$962$	49.1179	1.58362
$963$	−6.85574	−0.220923
$964$	9.84067	0.316947
$965$	9.82409	0.316249
$966$	−7.37209	−0.237193
$967$	39.9678	1.28528	0.642639	−	0.766169i	$-0.277840\pi$
$967$	39.9678	1.28528	0.642639	+	0.766169i	$0.277840\pi$
$968$	50.1898	1.61316
$969$	10.8150	0.347428
$970$	−52.6180	−1.68946
$971$	−0.846076	−0.0271519	−0.0135759	−	0.999908i	$-0.504321\pi$
$971$	−0.846076	−0.0271519	−0.0135759	+	0.999908i	$0.504321\pi$
$972$	−4.03647	−0.129470
$973$	−12.2230	−0.391852
$974$	−7.83692	−0.251111
$975$	−11.6783	−0.374006
$976$	−31.3783	−1.00440
$977$	37.1676	1.18910	0.594548	−	0.804060i	$-0.297331\pi$
$977$	37.1676	1.18910	0.594548	+	0.804060i	$0.297331\pi$
$978$	−8.87002	−0.283632
$979$	8.18441	0.261575
$980$	−11.2990	−0.360933
$981$	−16.2961	−0.520293
$982$	−66.4852	−2.12163
$983$	−11.1171	−0.354579	−0.177289	−	0.984159i	$-0.556733\pi$
$983$	−11.1171	−0.354579	−0.177289	+	0.984159i	$0.556733\pi$
$984$	14.2768	0.455126
$985$	−50.9797	−1.62435
$986$	−8.61933	−0.274495
$987$	−9.67674	−0.308014
$988$	127.150	4.04517
$989$	−21.1291	−0.671865
$990$	−6.76996	−0.215163
$991$	−14.1924	−0.450838	−0.225419	−	0.974262i	$-0.572375\pi$
$991$	−14.1924	−0.450838	−0.225419	+	0.974262i	$0.572375\pi$
$992$	0.614408	0.0195075
$993$	−18.1294	−0.575319
$994$	−34.4201	−1.09174
$995$	63.8426	2.02395
$996$	−49.2003	−1.55897
$997$	49.1056	1.55519	0.777595	−	0.628766i	$-0.216440\pi$
$997$	49.1056	1.55519	0.777595	+	0.628766i	$0.216440\pi$
$998$	−23.3400	−0.738814
$999$	4.85422	0.153581

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.t.1.5	✓	52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.t.1.5	✓	52	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)

\(f(q)\)	\(=\)	\(q-2.45692 q^{2} -1.00000 q^{3} +4.03647 q^{4} -2.79922 q^{5} +2.45692 q^{6} +1.00000 q^{7} -5.00345 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.45692 q^{2} -1.00000 q^{3} +4.03647 q^{4} -2.79922 q^{5} +2.45692 q^{6} +1.00000 q^{7} -5.00345 q^{8} +1.00000 q^{9} +6.87748 q^{10} -0.984367 q^{11} -4.03647 q^{12} +4.11840 q^{13} -2.45692 q^{14} +2.79922 q^{15} +4.22016 q^{16} -1.41398 q^{17} -2.45692 q^{18} +7.64864 q^{19} -11.2990 q^{20} -1.00000 q^{21} +2.41851 q^{22} -3.00054 q^{23} +5.00345 q^{24} +2.83565 q^{25} -10.1186 q^{26} -1.00000 q^{27} +4.03647 q^{28} -2.48107 q^{29} -6.87748 q^{30} -1.69867 q^{31} -0.361700 q^{32} +0.984367 q^{33} +3.47404 q^{34} -2.79922 q^{35} +4.03647 q^{36} -4.85422 q^{37} -18.7921 q^{38} -4.11840 q^{39} +14.0058 q^{40} +2.85338 q^{41} +2.45692 q^{42} +7.04176 q^{43} -3.97337 q^{44} -2.79922 q^{45} +7.37209 q^{46} +9.67674 q^{47} -4.22016 q^{48} +1.00000 q^{49} -6.96697 q^{50} +1.41398 q^{51} +16.6238 q^{52} -2.69129 q^{53} +2.45692 q^{54} +2.75546 q^{55} -5.00345 q^{56} -7.64864 q^{57} +6.09580 q^{58} +8.15845 q^{59} +11.2990 q^{60} -7.43534 q^{61} +4.17350 q^{62} +1.00000 q^{63} -7.55165 q^{64} -11.5283 q^{65} -2.41851 q^{66} +6.56297 q^{67} -5.70748 q^{68} +3.00054 q^{69} +6.87748 q^{70} +14.0094 q^{71} -5.00345 q^{72} +5.55255 q^{73} +11.9264 q^{74} -2.83565 q^{75} +30.8735 q^{76} -0.984367 q^{77} +10.1186 q^{78} +9.46209 q^{79} -11.8132 q^{80} +1.00000 q^{81} -7.01053 q^{82} +12.1889 q^{83} -4.03647 q^{84} +3.95804 q^{85} -17.3011 q^{86} +2.48107 q^{87} +4.92523 q^{88} -8.31439 q^{89} +6.87748 q^{90} +4.11840 q^{91} -12.1116 q^{92} +1.69867 q^{93} -23.7750 q^{94} -21.4103 q^{95} +0.361700 q^{96} -7.65077 q^{97} -2.45692 q^{98} -0.984367 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9 - 2 * q^10 + 9 * q^11 - 61 * q^12 + 44 * q^13 + 3 * q^14 + 7 * q^15 + 95 * q^16 - 6 * q^17 + 3 * q^18 + 7 * q^19 - 21 * q^20 - 52 * q^21 + 19 * q^22 - 4 * q^23 - 24 * q^24 + 83 * q^25 - 5 * q^26 - 52 * q^27 + 61 * q^28 + 31 * q^29 + 2 * q^30 + 11 * q^31 + 71 * q^32 - 9 * q^33 + 17 * q^34 - 7 * q^35 + 61 * q^36 + 71 * q^37 - 8 * q^38 - 44 * q^39 + 20 * q^40 - 25 * q^41 - 3 * q^42 + 75 * q^43 + 14 * q^44 - 7 * q^45 + 36 * q^46 - 20 * q^47 - 95 * q^48 + 52 * q^49 + 26 * q^50 + 6 * q^51 + 88 * q^52 + 70 * q^53 - 3 * q^54 + 12 * q^55 + 24 * q^56 - 7 * q^57 + 48 * q^58 - 27 * q^59 + 21 * q^60 + 59 * q^61 - 23 * q^62 + 52 * q^63 + 138 * q^64 + 44 * q^65 - 19 * q^66 + 65 * q^67 - 8 * q^68 + 4 * q^69 - 2 * q^70 - 11 * q^71 + 24 * q^72 + 34 * q^73 + 38 * q^74 - 83 * q^75 + 31 * q^76 + 9 * q^77 + 5 * q^78 + 74 * q^79 - 5 * q^80 + 52 * q^81 + 51 * q^82 - 30 * q^83 - 61 * q^84 + 70 * q^85 + 29 * q^86 - 31 * q^87 + 90 * q^88 - q^89 - 2 * q^90 + 44 * q^91 + 34 * q^92 - 11 * q^93 + 27 * q^94 + 9 * q^95 - 71 * q^96 + 73 * q^97 + 3 * q^98 + 9 * q^99

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