LMFDB - Embedded newform 8043.2.a.t.1.49

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			1.49
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)

$f(q)$	$=$	$q+2.68488 q^{2} -1.00000 q^{3} +5.20856 q^{4} +3.44204 q^{5} -2.68488 q^{6} +1.00000 q^{7} +8.61460 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.68488 q^{2} -1.00000 q^{3} +5.20856 q^{4} +3.44204 q^{5} -2.68488 q^{6} +1.00000 q^{7} +8.61460 q^{8} +1.00000 q^{9} +9.24146 q^{10} +3.27164 q^{11} -5.20856 q^{12} +1.33241 q^{13} +2.68488 q^{14} -3.44204 q^{15} +12.7120 q^{16} -7.53643 q^{17} +2.68488 q^{18} -3.13611 q^{19} +17.9281 q^{20} -1.00000 q^{21} +8.78394 q^{22} -0.459345 q^{23} -8.61460 q^{24} +6.84767 q^{25} +3.57736 q^{26} -1.00000 q^{27} +5.20856 q^{28} +6.72310 q^{29} -9.24146 q^{30} -6.53246 q^{31} +16.9010 q^{32} -3.27164 q^{33} -20.2344 q^{34} +3.44204 q^{35} +5.20856 q^{36} -0.880422 q^{37} -8.42006 q^{38} -1.33241 q^{39} +29.6518 q^{40} -0.451821 q^{41} -2.68488 q^{42} +7.55791 q^{43} +17.0405 q^{44} +3.44204 q^{45} -1.23328 q^{46} -0.0684051 q^{47} -12.7120 q^{48} +1.00000 q^{49} +18.3851 q^{50} +7.53643 q^{51} +6.93994 q^{52} +7.55395 q^{53} -2.68488 q^{54} +11.2611 q^{55} +8.61460 q^{56} +3.13611 q^{57} +18.0507 q^{58} +12.2404 q^{59} -17.9281 q^{60} -3.24480 q^{61} -17.5388 q^{62} +1.00000 q^{63} +19.9530 q^{64} +4.58621 q^{65} -8.78394 q^{66} +0.106231 q^{67} -39.2540 q^{68} +0.459345 q^{69} +9.24146 q^{70} -12.0897 q^{71} +8.61460 q^{72} -4.46689 q^{73} -2.36383 q^{74} -6.84767 q^{75} -16.3346 q^{76} +3.27164 q^{77} -3.57736 q^{78} -12.3693 q^{79} +43.7553 q^{80} +1.00000 q^{81} -1.21308 q^{82} -5.02289 q^{83} -5.20856 q^{84} -25.9407 q^{85} +20.2920 q^{86} -6.72310 q^{87} +28.1838 q^{88} +7.39650 q^{89} +9.24146 q^{90} +1.33241 q^{91} -2.39253 q^{92} +6.53246 q^{93} -0.183659 q^{94} -10.7946 q^{95} -16.9010 q^{96} +10.2811 q^{97} +2.68488 q^{98} +3.27164 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.68488	1.89849	0.949247	−	0.314531i	$-0.101847\pi$
$2$	2.68488	1.89849	0.949247	+	0.314531i	$0.101847\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.20856	2.60428
$5$	3.44204	1.53933	0.769664	−	0.638449i	$-0.220424\pi$
$5$	3.44204	1.53933	0.769664	+	0.638449i	$0.220424\pi$
$6$	−2.68488	−1.09610
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	8.61460	3.04572
$9$	1.00000	0.333333
$10$	9.24146	2.92241
$11$	3.27164	0.986435	0.493218	−	0.869906i	$-0.335821\pi$
$11$	3.27164	0.986435	0.493218	+	0.869906i	$0.335821\pi$
$12$	−5.20856	−1.50358
$13$	1.33241	0.369544	0.184772	−	0.982781i	$-0.440845\pi$
$13$	1.33241	0.369544	0.184772	+	0.982781i	$0.440845\pi$
$14$	2.68488	0.717564
$15$	−3.44204	−0.888732
$16$	12.7120	3.17800
$17$	−7.53643	−1.82785	−0.913926	−	0.405880i	$-0.866965\pi$
$17$	−7.53643	−1.82785	−0.913926	+	0.405880i	$0.866965\pi$
$18$	2.68488	0.632832
$19$	−3.13611	−0.719472	−0.359736	−	0.933054i	$-0.617133\pi$
$19$	−3.13611	−0.719472	−0.359736	+	0.933054i	$0.617133\pi$
$20$	17.9281	4.00885
$21$	−1.00000	−0.218218
$22$	8.78394	1.87274
$23$	−0.459345	−0.0957800	−0.0478900	−	0.998853i	$-0.515250\pi$
$23$	−0.459345	−0.0957800	−0.0478900	+	0.998853i	$0.515250\pi$
$24$	−8.61460	−1.75845
$25$	6.84767	1.36953
$26$	3.57736	0.701577
$27$	−1.00000	−0.192450
$28$	5.20856	0.984326
$29$	6.72310	1.24845	0.624224	−	0.781246i	$-0.285415\pi$
$29$	6.72310	1.24845	0.624224	+	0.781246i	$0.285415\pi$
$30$	−9.24146	−1.68725
$31$	−6.53246	−1.17326	−0.586632	−	0.809854i	$-0.699546\pi$
$31$	−6.53246	−1.17326	−0.586632	+	0.809854i	$0.699546\pi$
$32$	16.9010	2.98770
$33$	−3.27164	−0.569519
$34$	−20.2344	−3.47017
$35$	3.44204	0.581812
$36$	5.20856	0.868094
$37$	−0.880422	−0.144741	−0.0723703	−	0.997378i	$-0.523056\pi$
$37$	−0.880422	−0.144741	−0.0723703	+	0.997378i	$0.523056\pi$
$38$	−8.42006	−1.36591
$39$	−1.33241	−0.213356
$40$	29.6518	4.68837
$41$	−0.451821	−0.0705626	−0.0352813	−	0.999377i	$-0.511233\pi$
$41$	−0.451821	−0.0705626	−0.0352813	+	0.999377i	$0.511233\pi$
$42$	−2.68488	−0.414285
$43$	7.55791	1.15257	0.576285	−	0.817249i	$-0.304502\pi$
$43$	7.55791	1.15257	0.576285	+	0.817249i	$0.304502\pi$
$44$	17.0405	2.56896
$45$	3.44204	0.513110
$46$	−1.23328	−0.181838
$47$	−0.0684051	−0.00997792	−0.00498896	−	0.999988i	$-0.501588\pi$
$47$	−0.0684051	−0.00997792	−0.00498896	+	0.999988i	$0.501588\pi$
$48$	−12.7120	−1.83482
$49$	1.00000	0.142857
$50$	18.3851	2.60005
$51$	7.53643	1.05531
$52$	6.93994	0.962397
$53$	7.55395	1.03762	0.518808	−	0.854891i	$-0.326376\pi$
$53$	7.55395	1.03762	0.518808	+	0.854891i	$0.326376\pi$
$54$	−2.68488	−0.365365
$55$	11.2611	1.51845
$56$	8.61460	1.15117
$57$	3.13611	0.415388
$58$	18.0507	2.37017
$59$	12.2404	1.59356	0.796781	−	0.604268i	$-0.206534\pi$
$59$	12.2404	1.59356	0.796781	+	0.604268i	$0.206534\pi$
$60$	−17.9281	−2.31451
$61$	−3.24480	−0.415455	−0.207727	−	0.978187i	$-0.566607\pi$
$61$	−3.24480	−0.415455	−0.207727	+	0.978187i	$0.566607\pi$
$62$	−17.5388	−2.22743
$63$	1.00000	0.125988
$64$	19.9530	2.49413
$65$	4.58621	0.568850
$66$	−8.78394	−1.08123
$67$	0.106231	0.0129781	0.00648907	−	0.999979i	$-0.497934\pi$
$67$	0.106231	0.0129781	0.00648907	+	0.999979i	$0.497934\pi$
$68$	−39.2540	−4.76024
$69$	0.459345	0.0552986
$70$	9.24146	1.10457
$71$	−12.0897	−1.43478	−0.717390	−	0.696672i	$-0.754663\pi$
$71$	−12.0897	−1.43478	−0.717390	+	0.696672i	$0.754663\pi$
$72$	8.61460	1.01524
$73$	−4.46689	−0.522810	−0.261405	−	0.965229i	$-0.584186\pi$
$73$	−4.46689	−0.522810	−0.261405	+	0.965229i	$0.584186\pi$
$74$	−2.36383	−0.274789
$75$	−6.84767	−0.790700
$76$	−16.3346	−1.87371
$77$	3.27164	0.372838
$78$	−3.57736	−0.405056
$79$	−12.3693	−1.39166	−0.695828	−	0.718209i	$-0.744962\pi$
$79$	−12.3693	−1.39166	−0.695828	+	0.718209i	$0.744962\pi$
$80$	43.7553	4.89199
$81$	1.00000	0.111111
$82$	−1.21308	−0.133963
$83$	−5.02289	−0.551333	−0.275667	−	0.961253i	$-0.588899\pi$
$83$	−5.02289	−0.551333	−0.275667	+	0.961253i	$0.588899\pi$
$84$	−5.20856	−0.568301
$85$	−25.9407	−2.81367
$86$	20.2920	2.18815
$87$	−6.72310	−0.720792
$88$	28.1838	3.00441
$89$	7.39650	0.784027	0.392013	−	0.919959i	$-0.371779\pi$
$89$	7.39650	0.784027	0.392013	+	0.919959i	$0.371779\pi$
$90$	9.24146	0.974136
$91$	1.33241	0.139675
$92$	−2.39253	−0.249438
$93$	6.53246	0.677384
$94$	−0.183659	−0.0189430
$95$	−10.7946	−1.10750
$96$	−16.9010	−1.72495
$97$	10.2811	1.04389	0.521945	−	0.852979i	$-0.325207\pi$
$97$	10.2811	1.04389	0.521945	+	0.852979i	$0.325207\pi$
$98$	2.68488	0.271214
$99$	3.27164	0.328812
$100$	35.6665	3.56665
$101$	0.0449292	0.00447062	0.00223531	−	0.999998i	$-0.499288\pi$
$101$	0.0449292	0.00447062	0.00223531	+	0.999998i	$0.499288\pi$
$102$	20.2344	2.00350
$103$	−6.13525	−0.604524	−0.302262	−	0.953225i	$-0.597742\pi$
$103$	−6.13525	−0.604524	−0.302262	+	0.953225i	$0.597742\pi$
$104$	11.4782	1.12553
$105$	−3.44204	−0.335909
$106$	20.2814	1.96991
$107$	8.25094	0.797648	0.398824	−	0.917027i	$-0.369418\pi$
$107$	8.25094	0.797648	0.398824	+	0.917027i	$0.369418\pi$
$108$	−5.20856	−0.501194
$109$	−4.15762	−0.398228	−0.199114	−	0.979976i	$-0.563806\pi$
$109$	−4.15762	−0.398228	−0.199114	+	0.979976i	$0.563806\pi$
$110$	30.2347	2.88277
$111$	0.880422	0.0835660
$112$	12.7120	1.20117
$113$	−16.7174	−1.57264	−0.786320	−	0.617819i	$-0.788016\pi$
$113$	−16.7174	−1.57264	−0.786320	+	0.617819i	$0.788016\pi$
$114$	8.42006	0.788611
$115$	−1.58109	−0.147437
$116$	35.0177	3.25131
$117$	1.33241	0.123181
$118$	32.8639	3.02537
$119$	−7.53643	−0.690863
$120$	−29.6518	−2.70683
$121$	−0.296399	−0.0269454
$122$	−8.71190	−0.788738
$123$	0.451821	0.0407394
$124$	−34.0247	−3.05551
$125$	6.35975	0.568833
$126$	2.68488	0.239188
$127$	5.62925	0.499515	0.249758	−	0.968308i	$-0.419649\pi$
$127$	5.62925	0.499515	0.249758	+	0.968308i	$0.419649\pi$
$128$	19.7695	1.74739
$129$	−7.55791	−0.665436
$130$	12.3134	1.07996
$131$	9.10799	0.795769	0.397885	−	0.917435i	$-0.369744\pi$
$131$	9.10799	0.795769	0.397885	+	0.917435i	$0.369744\pi$
$132$	−17.0405	−1.48319
$133$	−3.13611	−0.271935
$134$	0.285216	0.0246389
$135$	−3.44204	−0.296244
$136$	−64.9233	−5.56713
$137$	11.9924	1.02458	0.512291	−	0.858812i	$-0.328797\pi$
$137$	11.9924	1.02458	0.512291	+	0.858812i	$0.328797\pi$
$138$	1.23328	0.104984
$139$	6.39852	0.542716	0.271358	−	0.962479i	$-0.412527\pi$
$139$	6.39852	0.542716	0.271358	+	0.962479i	$0.412527\pi$
$140$	17.9281	1.51520
$141$	0.0684051	0.00576075
$142$	−32.4593	−2.72392
$143$	4.35916	0.364531
$144$	12.7120	1.05933
$145$	23.1412	1.92177
$146$	−11.9930	−0.992551
$147$	−1.00000	−0.0824786
$148$	−4.58574	−0.376945
$149$	−7.61163	−0.623569	−0.311784	−	0.950153i	$-0.600927\pi$
$149$	−7.61163	−0.623569	−0.311784	+	0.950153i	$0.600927\pi$
$150$	−18.3851	−1.50114
$151$	−0.0758010	−0.00616860	−0.00308430	−	0.999995i	$-0.500982\pi$
$151$	−0.0758010	−0.00616860	−0.00308430	+	0.999995i	$0.500982\pi$
$152$	−27.0163	−2.19131
$153$	−7.53643	−0.609284
$154$	8.78394	0.707830
$155$	−22.4850	−1.80604
$156$	−6.93994	−0.555640
$157$	4.88684	0.390012	0.195006	−	0.980802i	$-0.437527\pi$
$157$	4.88684	0.390012	0.195006	+	0.980802i	$0.437527\pi$
$158$	−33.2101	−2.64205
$159$	−7.55395	−0.599068
$160$	58.1739	4.59905
$161$	−0.459345	−0.0362015
$162$	2.68488	0.210944
$163$	−16.6030	−1.30045	−0.650223	−	0.759744i	$-0.725324\pi$
$163$	−16.6030	−1.30045	−0.650223	+	0.759744i	$0.725324\pi$
$164$	−2.35334	−0.183765
$165$	−11.2611	−0.876677
$166$	−13.4858	−1.04670
$167$	−20.1144	−1.55650	−0.778250	−	0.627955i	$-0.783892\pi$
$167$	−20.1144	−1.55650	−0.778250	+	0.627955i	$0.783892\pi$
$168$	−8.61460	−0.664631
$169$	−11.2247	−0.863437
$170$	−69.6476	−5.34173
$171$	−3.13611	−0.239824
$172$	39.3658	3.00162
$173$	−14.1216	−1.07364	−0.536822	−	0.843695i	$-0.680375\pi$
$173$	−14.1216	−1.07364	−0.536822	+	0.843695i	$0.680375\pi$
$174$	−18.0507	−1.36842
$175$	6.84767	0.517635
$176$	41.5891	3.13489
$177$	−12.2404	−0.920043
$178$	19.8587	1.48847
$179$	−20.8393	−1.55760	−0.778801	−	0.627271i	$-0.784172\pi$
$179$	−20.8393	−1.55760	−0.778801	+	0.627271i	$0.784172\pi$
$180$	17.9281	1.33628
$181$	4.63270	0.344346	0.172173	−	0.985067i	$-0.444921\pi$
$181$	4.63270	0.344346	0.172173	+	0.985067i	$0.444921\pi$
$182$	3.57736	0.265171
$183$	3.24480	0.239863
$184$	−3.95707	−0.291719
$185$	−3.03045	−0.222803
$186$	17.5388	1.28601
$187$	−24.6565	−1.80306
$188$	−0.356293	−0.0259853
$189$	−1.00000	−0.0727393
$190$	−28.9822	−2.10259
$191$	6.00156	0.434258	0.217129	−	0.976143i	$-0.430331\pi$
$191$	6.00156	0.434258	0.217129	+	0.976143i	$0.430331\pi$
$192$	−19.9530	−1.43999
$193$	18.3340	1.31971	0.659855	−	0.751393i	$-0.270618\pi$
$193$	18.3340	1.31971	0.659855	+	0.751393i	$0.270618\pi$
$194$	27.6036	1.98182
$195$	−4.58621	−0.328426
$196$	5.20856	0.372040
$197$	−6.79479	−0.484109	−0.242054	−	0.970263i	$-0.577821\pi$
$197$	−6.79479	−0.484109	−0.242054	+	0.970263i	$0.577821\pi$
$198$	8.78394	0.624247
$199$	19.6988	1.39641	0.698204	−	0.715899i	$-0.253983\pi$
$199$	19.6988	1.39641	0.698204	+	0.715899i	$0.253983\pi$
$200$	58.9899	4.17122
$201$	−0.106231	−0.00749293
$202$	0.120629	0.00848745
$203$	6.72310	0.471869
$204$	39.2540	2.74833
$205$	−1.55519	−0.108619
$206$	−16.4724	−1.14769
$207$	−0.459345	−0.0319267
$208$	16.9376	1.17441
$209$	−10.2602	−0.709713
$210$	−9.24146	−0.637722
$211$	−26.4688	−1.82219	−0.911094	−	0.412198i	$-0.864761\pi$
$211$	−26.4688	−1.82219	−0.911094	+	0.412198i	$0.864761\pi$
$212$	39.3453	2.70224
$213$	12.0897	0.828370
$214$	22.1527	1.51433
$215$	26.0146	1.77418
$216$	−8.61460	−0.586149
$217$	−6.53246	−0.443452
$218$	−11.1627	−0.756034
$219$	4.46689	0.301844
$220$	58.6542	3.95447
$221$	−10.0416	−0.675472
$222$	2.36383	0.158650
$223$	8.77708	0.587756	0.293878	−	0.955843i	$-0.405054\pi$
$223$	8.77708	0.587756	0.293878	+	0.955843i	$0.405054\pi$
$224$	16.9010	1.12924
$225$	6.84767	0.456511
$226$	−44.8842	−2.98565
$227$	−19.8560	−1.31789	−0.658944	−	0.752192i	$-0.728996\pi$
$227$	−19.8560	−1.31789	−0.658944	+	0.752192i	$0.728996\pi$
$228$	16.3346	1.08179
$229$	−13.3012	−0.878966	−0.439483	−	0.898251i	$-0.644838\pi$
$229$	−13.3012	−0.878966	−0.439483	+	0.898251i	$0.644838\pi$
$230$	−4.24502	−0.279908
$231$	−3.27164	−0.215258
$232$	57.9168	3.80242
$233$	−13.8329	−0.906221	−0.453110	−	0.891454i	$-0.649686\pi$
$233$	−13.8329	−0.906221	−0.453110	+	0.891454i	$0.649686\pi$
$234$	3.57736	0.233859
$235$	−0.235454	−0.0153593
$236$	63.7548	4.15008
$237$	12.3693	0.803473
$238$	−20.2344	−1.31160
$239$	15.1025	0.976899	0.488450	−	0.872592i	$-0.337563\pi$
$239$	15.1025	0.976899	0.488450	+	0.872592i	$0.337563\pi$
$240$	−43.7553	−2.82439
$241$	−20.9641	−1.35041	−0.675207	−	0.737628i	$-0.735946\pi$
$241$	−20.9641	−1.35041	−0.675207	+	0.737628i	$0.735946\pi$
$242$	−0.795795	−0.0511557
$243$	−1.00000	−0.0641500
$244$	−16.9008	−1.08196
$245$	3.44204	0.219904
$246$	1.21308	0.0773434
$247$	−4.17858	−0.265877
$248$	−56.2745	−3.57343
$249$	5.02289	0.318313
$250$	17.0751	1.07993
$251$	−6.84691	−0.432173	−0.216086	−	0.976374i	$-0.569329\pi$
$251$	−6.84691	−0.432173	−0.216086	+	0.976374i	$0.569329\pi$
$252$	5.20856	0.328109
$253$	−1.50281	−0.0944808
$254$	15.1138	0.948327
$255$	25.9407	1.62447
$256$	13.1726	0.823285
$257$	13.7835	0.859791	0.429895	−	0.902879i	$-0.358550\pi$
$257$	13.7835	0.859791	0.429895	+	0.902879i	$0.358550\pi$
$258$	−20.2920	−1.26333
$259$	−0.880422	−0.0547068
$260$	23.8876	1.48145
$261$	6.72310	0.416149
$262$	24.4538	1.51076
$263$	10.7404	0.662280	0.331140	−	0.943582i	$-0.392567\pi$
$263$	10.7404	0.662280	0.331140	+	0.943582i	$0.392567\pi$
$264$	−28.1838	−1.73459
$265$	26.0010	1.59723
$266$	−8.42006	−0.516267
$267$	−7.39650	−0.452658
$268$	0.553309	0.0337987
$269$	−2.67432	−0.163056	−0.0815280	−	0.996671i	$-0.525980\pi$
$269$	−2.67432	−0.163056	−0.0815280	+	0.996671i	$0.525980\pi$
$270$	−9.24146	−0.562418
$271$	1.40266	0.0852053	0.0426026	−	0.999092i	$-0.486435\pi$
$271$	1.40266	0.0852053	0.0426026	+	0.999092i	$0.486435\pi$
$272$	−95.8032	−5.80892
$273$	−1.33241	−0.0806411
$274$	32.1982	1.94517
$275$	22.4031	1.35096
$276$	2.39253	0.144013
$277$	7.50122	0.450705	0.225352	−	0.974277i	$-0.427647\pi$
$277$	7.50122	0.450705	0.225352	+	0.974277i	$0.427647\pi$
$278$	17.1792	1.03034
$279$	−6.53246	−0.391088
$280$	29.6518	1.77204
$281$	6.15369	0.367099	0.183549	−	0.983011i	$-0.441241\pi$
$281$	6.15369	0.367099	0.183549	+	0.983011i	$0.441241\pi$
$282$	0.183659	0.0109368
$283$	−23.5799	−1.40168	−0.700839	−	0.713320i	$-0.747191\pi$
$283$	−23.5799	−1.40168	−0.700839	+	0.713320i	$0.747191\pi$
$284$	−62.9698	−3.73657
$285$	10.7946	0.639418
$286$	11.7038	0.692061
$287$	−0.451821	−0.0266702
$288$	16.9010	0.995900
$289$	39.7978	2.34104
$290$	62.1313	3.64847
$291$	−10.2811	−0.602691
$292$	−23.2661	−1.36154
$293$	20.8363	1.21727	0.608636	−	0.793450i	$-0.291717\pi$
$293$	20.8363	1.21727	0.608636	+	0.793450i	$0.291717\pi$
$294$	−2.68488	−0.156585
$295$	42.1319	2.45302
$296$	−7.58448	−0.440839
$297$	−3.27164	−0.189840
$298$	−20.4363	−1.18384
$299$	−0.612036	−0.0353949
$300$	−35.6665	−2.05921
$301$	7.55791	0.435630
$302$	−0.203516	−0.0117111
$303$	−0.0449292	−0.00258111
$304$	−39.8662	−2.28649
$305$	−11.1688	−0.639521
$306$	−20.2344	−1.15672
$307$	−20.3036	−1.15879	−0.579393	−	0.815049i	$-0.696710\pi$
$307$	−20.3036	−1.15879	−0.579393	+	0.815049i	$0.696710\pi$
$308$	17.0405	0.970974
$309$	6.13525	0.349022
$310$	−60.3695	−3.42875
$311$	25.0195	1.41873	0.709363	−	0.704844i	$-0.248983\pi$
$311$	25.0195	1.41873	0.709363	+	0.704844i	$0.248983\pi$
$312$	−11.4782	−0.649824
$313$	−32.1780	−1.81881	−0.909403	−	0.415916i	$-0.863461\pi$
$313$	−32.1780	−1.81881	−0.909403	+	0.415916i	$0.863461\pi$
$314$	13.1206	0.740436
$315$	3.44204	0.193937
$316$	−64.4263	−3.62426
$317$	30.4750	1.71165	0.855823	−	0.517269i	$-0.173051\pi$
$317$	30.4750	1.71165	0.855823	+	0.517269i	$0.173051\pi$
$318$	−20.2814	−1.13733
$319$	21.9955	1.23151
$320$	68.6792	3.83929
$321$	−8.25094	−0.460522
$322$	−1.23328	−0.0687283
$323$	23.6351	1.31509
$324$	5.20856	0.289365
$325$	9.12390	0.506103
$326$	−44.5770	−2.46889
$327$	4.15762	0.229917
$328$	−3.89226	−0.214914
$329$	−0.0684051	−0.00377130
$330$	−30.2347	−1.66437
$331$	29.4419	1.61827	0.809137	−	0.587620i	$-0.199935\pi$
$331$	29.4419	1.61827	0.809137	+	0.587620i	$0.199935\pi$
$332$	−26.1620	−1.43583
$333$	−0.880422	−0.0482468
$334$	−54.0047	−2.95501
$335$	0.365650	0.0199776
$336$	−12.7120	−0.693497
$337$	30.1562	1.64271	0.821356	−	0.570416i	$-0.193218\pi$
$337$	30.1562	1.64271	0.821356	+	0.570416i	$0.193218\pi$
$338$	−30.1369	−1.63923
$339$	16.7174	0.907964
$340$	−135.114	−7.32758
$341$	−21.3718	−1.15735
$342$	−8.42006	−0.455305
$343$	1.00000	0.0539949
$344$	65.1083	3.51041
$345$	1.58109	0.0851228
$346$	−37.9147	−2.03831
$347$	29.3694	1.57663	0.788317	−	0.615269i	$-0.210953\pi$
$347$	29.3694	1.57663	0.788317	+	0.615269i	$0.210953\pi$
$348$	−35.0177	−1.87714
$349$	3.13286	0.167698	0.0838491	−	0.996478i	$-0.473279\pi$
$349$	3.13286	0.167698	0.0838491	+	0.996478i	$0.473279\pi$
$350$	18.3851	0.982727
$351$	−1.33241	−0.0711188
$352$	55.2939	2.94717
$353$	4.62449	0.246137	0.123068	−	0.992398i	$-0.460727\pi$
$353$	4.62449	0.246137	0.123068	+	0.992398i	$0.460727\pi$
$354$	−32.8639	−1.74670
$355$	−41.6132	−2.20860
$356$	38.5251	2.04183
$357$	7.53643	0.398870
$358$	−55.9510	−2.95710
$359$	4.99114	0.263422	0.131711	−	0.991288i	$-0.457953\pi$
$359$	4.99114	0.263422	0.131711	+	0.991288i	$0.457953\pi$
$360$	29.6518	1.56279
$361$	−9.16483	−0.482359
$362$	12.4382	0.653739
$363$	0.296399	0.0155569
$364$	6.93994	0.363752
$365$	−15.3752	−0.804776
$366$	8.71190	0.455378
$367$	0.453879	0.0236923	0.0118462	−	0.999930i	$-0.496229\pi$
$367$	0.453879	0.0236923	0.0118462	+	0.999930i	$0.496229\pi$
$368$	−5.83920	−0.304389
$369$	−0.451821	−0.0235209
$370$	−8.13639	−0.422991
$371$	7.55395	0.392182
$372$	34.0247	1.76410
$373$	15.1654	0.785234	0.392617	−	0.919702i	$-0.371570\pi$
$373$	15.1654	0.785234	0.392617	+	0.919702i	$0.371570\pi$
$374$	−66.1995	−3.42310
$375$	−6.35975	−0.328416
$376$	−0.589283	−0.0303899
$377$	8.95792	0.461356
$378$	−2.68488	−0.138095
$379$	−18.5153	−0.951068	−0.475534	−	0.879697i	$-0.657745\pi$
$379$	−18.5153	−0.951068	−0.475534	+	0.879697i	$0.657745\pi$
$380$	−56.2245	−2.88425
$381$	−5.62925	−0.288395
$382$	16.1135	0.824436
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	−19.7695	−1.00886
$385$	11.2611	0.573920
$386$	49.2245	2.50546
$387$	7.55791	0.384190
$388$	53.5499	2.71859
$389$	9.23294	0.468129	0.234064	−	0.972221i	$-0.424797\pi$
$389$	9.23294	0.468129	0.234064	+	0.972221i	$0.424797\pi$
$390$	−12.3134	−0.623514
$391$	3.46182	0.175072
$392$	8.61460	0.435103
$393$	−9.10799	−0.459438
$394$	−18.2432	−0.919078
$395$	−42.5757	−2.14222
$396$	17.0405	0.856319
$397$	−5.20286	−0.261124	−0.130562	−	0.991440i	$-0.541678\pi$
$397$	−5.20286	−0.261124	−0.130562	+	0.991440i	$0.541678\pi$
$398$	52.8887	2.65107
$399$	3.13611	0.157002
$400$	87.0476	4.35238
$401$	16.8281	0.840353	0.420177	−	0.907442i	$-0.361968\pi$
$401$	16.8281	0.840353	0.420177	+	0.907442i	$0.361968\pi$
$402$	−0.285216	−0.0142253
$403$	−8.70391	−0.433573
$404$	0.234017	0.0116428
$405$	3.44204	0.171037
$406$	18.0507	0.895841
$407$	−2.88042	−0.142777
$408$	64.9233	3.21418
$409$	−12.1402	−0.600293	−0.300147	−	0.953893i	$-0.597036\pi$
$409$	−12.1402	−0.600293	−0.300147	+	0.953893i	$0.597036\pi$
$410$	−4.17549	−0.206213
$411$	−11.9924	−0.591543
$412$	−31.9558	−1.57435
$413$	12.2404	0.602310
$414$	−1.23328	−0.0606126
$415$	−17.2890	−0.848684
$416$	22.5190	1.10409
$417$	−6.39852	−0.313337
$418$	−27.5474	−1.34739
$419$	−13.6163	−0.665199	−0.332599	−	0.943068i	$-0.607926\pi$
$419$	−13.6163	−0.665199	−0.332599	+	0.943068i	$0.607926\pi$
$420$	−17.9281	−0.874802
$421$	0.900135	0.0438699	0.0219349	−	0.999759i	$-0.493017\pi$
$421$	0.900135	0.0438699	0.0219349	+	0.999759i	$0.493017\pi$
$422$	−71.0655	−3.45941
$423$	−0.0684051	−0.00332597
$424$	65.0743	3.16029
$425$	−51.6070	−2.50330
$426$	32.4593	1.57266
$427$	−3.24480	−0.157027
$428$	42.9755	2.07730
$429$	−4.35916	−0.210462
$430$	69.8461	3.36828
$431$	−30.4479	−1.46662	−0.733312	−	0.679892i	$-0.762027\pi$
$431$	−30.4479	−1.46662	−0.733312	+	0.679892i	$0.762027\pi$
$432$	−12.7120	−0.611607
$433$	18.1446	0.871971	0.435986	−	0.899954i	$-0.356400\pi$
$433$	18.1446	0.871971	0.435986	+	0.899954i	$0.356400\pi$
$434$	−17.5388	−0.841891
$435$	−23.1412	−1.10954
$436$	−21.6552	−1.03710
$437$	1.44056	0.0689111
$438$	11.9930	0.573050
$439$	−2.20081	−0.105039	−0.0525196	−	0.998620i	$-0.516725\pi$
$439$	−2.20081	−0.105039	−0.0525196	+	0.998620i	$0.516725\pi$
$440$	97.0100	4.62477
$441$	1.00000	0.0476190
$442$	−26.9605	−1.28238
$443$	−27.7351	−1.31774	−0.658868	−	0.752259i	$-0.728964\pi$
$443$	−27.7351	−1.31774	−0.658868	+	0.752259i	$0.728964\pi$
$444$	4.58574	0.217629
$445$	25.4591	1.20688
$446$	23.5654	1.11585
$447$	7.61163	0.360018
$448$	19.9530	0.942693
$449$	−7.03909	−0.332195	−0.166098	−	0.986109i	$-0.553117\pi$
$449$	−7.03909	−0.332195	−0.166098	+	0.986109i	$0.553117\pi$
$450$	18.3851	0.866684
$451$	−1.47819	−0.0696055
$452$	−87.0736	−4.09560
$453$	0.0758010	0.00356144
$454$	−53.3108	−2.50200
$455$	4.58621	0.215005
$456$	27.0163	1.26515
$457$	18.4268	0.861969	0.430985	−	0.902359i	$-0.358166\pi$
$457$	18.4268	0.861969	0.430985	+	0.902359i	$0.358166\pi$
$458$	−35.7120	−1.66871
$459$	7.53643	0.351770
$460$	−8.23519	−0.383967
$461$	27.8708	1.29807	0.649035	−	0.760758i	$-0.275173\pi$
$461$	27.8708	1.29807	0.649035	+	0.760758i	$0.275173\pi$
$462$	−8.78394	−0.408666
$463$	3.51821	0.163505	0.0817525	−	0.996653i	$-0.473948\pi$
$463$	3.51821	0.163505	0.0817525	+	0.996653i	$0.473948\pi$
$464$	85.4641	3.96757
$465$	22.4850	1.04272
$466$	−37.1395	−1.72046
$467$	7.47876	0.346076	0.173038	−	0.984915i	$-0.444642\pi$
$467$	7.47876	0.346076	0.173038	+	0.984915i	$0.444642\pi$
$468$	6.93994	0.320799
$469$	0.106231	0.00490527
$470$	−0.632164	−0.0291595
$471$	−4.88684	−0.225174
$472$	105.446	4.85354
$473$	24.7267	1.13694
$474$	33.2101	1.52539
$475$	−21.4750	−0.985341
$476$	−39.2540	−1.79920
$477$	7.55395	0.345872
$478$	40.5483	1.85464
$479$	−4.17760	−0.190879	−0.0954397	−	0.995435i	$-0.530426\pi$
$479$	−4.17760	−0.190879	−0.0954397	+	0.995435i	$0.530426\pi$
$480$	−58.1739	−2.65526
$481$	−1.17308	−0.0534880
$482$	−56.2859	−2.56375
$483$	0.459345	0.0209009
$484$	−1.54381	−0.0701734
$485$	35.3881	1.60689
$486$	−2.68488	−0.121788
$487$	−6.44241	−0.291934	−0.145967	−	0.989289i	$-0.546629\pi$
$487$	−6.44241	−0.291934	−0.145967	+	0.989289i	$0.546629\pi$
$488$	−27.9527	−1.26536
$489$	16.6030	0.750813
$490$	9.24146	0.417487
$491$	−2.82697	−0.127579	−0.0637897	−	0.997963i	$-0.520319\pi$
$491$	−2.82697	−0.127579	−0.0637897	+	0.997963i	$0.520319\pi$
$492$	2.35334	0.106097
$493$	−50.6681	−2.28198
$494$	−11.2190	−0.504766
$495$	11.2611	0.506149
$496$	−83.0406	−3.72863
$497$	−12.0897	−0.542296
$498$	13.4858	0.604315
$499$	9.07995	0.406474	0.203237	−	0.979130i	$-0.434854\pi$
$499$	9.07995	0.406474	0.203237	+	0.979130i	$0.434854\pi$
$500$	33.1252	1.48140
$501$	20.1144	0.898646
$502$	−18.3831	−0.820478
$503$	−10.6207	−0.473553	−0.236777	−	0.971564i	$-0.576091\pi$
$503$	−10.6207	−0.473553	−0.236777	+	0.971564i	$0.576091\pi$
$504$	8.61460	0.383725
$505$	0.154648	0.00688176
$506$	−4.03486	−0.179371
$507$	11.2247	0.498506
$508$	29.3203	1.30088
$509$	18.5518	0.822292	0.411146	−	0.911570i	$-0.365129\pi$
$509$	18.5518	0.822292	0.411146	+	0.911570i	$0.365129\pi$
$510$	69.6476	3.08405
$511$	−4.46689	−0.197603
$512$	−4.17226	−0.184390
$513$	3.13611	0.138463
$514$	37.0070	1.63231
$515$	−21.1178	−0.930561
$516$	−39.3658	−1.73298
$517$	−0.223797	−0.00984257
$518$	−2.36383	−0.103861
$519$	14.1216	0.619869
$520$	39.5084	1.73256
$521$	−1.57026	−0.0687945	−0.0343972	−	0.999408i	$-0.510951\pi$
$521$	−1.57026	−0.0687945	−0.0343972	+	0.999408i	$0.510951\pi$
$522$	18.0507	0.790057
$523$	0.310911	0.0135952	0.00679759	−	0.999977i	$-0.497836\pi$
$523$	0.310911	0.0135952	0.00679759	+	0.999977i	$0.497836\pi$
$524$	47.4396	2.07241
$525$	−6.84767	−0.298857
$526$	28.8366	1.25734
$527$	49.2314	2.14455
$528$	−41.5891	−1.80993
$529$	−22.7890	−0.990826
$530$	69.8096	3.03234
$531$	12.2404	0.531187
$532$	−16.3346	−0.708195
$533$	−0.602011	−0.0260760
$534$	−19.8587	−0.859369
$535$	28.4001	1.22784
$536$	0.915134	0.0395278
$537$	20.8393	0.899282
$538$	−7.18021	−0.309561
$539$	3.27164	0.140919
$540$	−17.9281	−0.771503
$541$	−13.8881	−0.597098	−0.298549	−	0.954394i	$-0.596503\pi$
$541$	−13.8881	−0.597098	−0.298549	+	0.954394i	$0.596503\pi$
$542$	3.76596	0.161762
$543$	−4.63270	−0.198808
$544$	−127.373	−5.46108
$545$	−14.3107	−0.613004
$546$	−3.57736	−0.153097
$547$	−22.7344	−0.972052	−0.486026	−	0.873944i	$-0.661554\pi$
$547$	−22.7344	−0.972052	−0.486026	+	0.873944i	$0.661554\pi$
$548$	62.4634	2.66830
$549$	−3.24480	−0.138485
$550$	60.1495	2.56478
$551$	−21.0844	−0.898224
$552$	3.95707	0.168424
$553$	−12.3693	−0.525996
$554$	20.1398	0.855660
$555$	3.03045	0.128636
$556$	33.3271	1.41338
$557$	−42.7361	−1.81079	−0.905393	−	0.424574i	$-0.860424\pi$
$557$	−42.7361	−1.81079	−0.905393	+	0.424574i	$0.860424\pi$
$558$	−17.5388	−0.742478
$559$	10.0702	0.425925
$560$	43.7553	1.84900
$561$	24.6565	1.04100
$562$	16.5219	0.696935
$563$	−18.8503	−0.794444	−0.397222	−	0.917722i	$-0.630026\pi$
$563$	−18.8503	−0.794444	−0.397222	+	0.917722i	$0.630026\pi$
$564$	0.356293	0.0150026
$565$	−57.5420	−2.42081
$566$	−63.3090	−2.66108
$567$	1.00000	0.0419961
$568$	−104.148	−4.36994
$569$	28.7113	1.20364	0.601820	−	0.798632i	$-0.294442\pi$
$569$	28.7113	1.20364	0.601820	+	0.798632i	$0.294442\pi$
$570$	28.9822	1.21393
$571$	−27.6263	−1.15612	−0.578061	−	0.815993i	$-0.696191\pi$
$571$	−27.6263	−1.15612	−0.578061	+	0.815993i	$0.696191\pi$
$572$	22.7050	0.949342
$573$	−6.00156	−0.250719
$574$	−1.21308	−0.0506332
$575$	−3.14544	−0.131174
$576$	19.9530	0.831377
$577$	−38.7045	−1.61129	−0.805644	−	0.592400i	$-0.798180\pi$
$577$	−38.7045	−1.61129	−0.805644	+	0.592400i	$0.798180\pi$
$578$	106.852	4.44446
$579$	−18.3340	−0.761935
$580$	120.532	5.00484
$581$	−5.02289	−0.208384
$582$	−27.6036	−1.14420
$583$	24.7138	1.02354
$584$	−38.4804	−1.59233
$585$	4.58621	0.189617
$586$	55.9430	2.31098
$587$	4.25740	0.175722	0.0878609	−	0.996133i	$-0.471997\pi$
$587$	4.25740	0.175722	0.0878609	+	0.996133i	$0.471997\pi$
$588$	−5.20856	−0.214798
$589$	20.4865	0.844131
$590$	113.119	4.65704
$591$	6.79479	0.279500
$592$	−11.1919	−0.459986
$593$	23.2231	0.953658	0.476829	−	0.878996i	$-0.341786\pi$
$593$	23.2231	0.953658	0.476829	+	0.878996i	$0.341786\pi$
$594$	−8.78394	−0.360409
$595$	−25.9407	−1.06347
$596$	−39.6457	−1.62395
$597$	−19.6988	−0.806216
$598$	−1.64324	−0.0671971
$599$	−31.1228	−1.27164	−0.635821	−	0.771837i	$-0.719338\pi$
$599$	−31.1228	−1.27164	−0.635821	+	0.771837i	$0.719338\pi$
$600$	−58.9899	−2.40825
$601$	2.06608	0.0842770	0.0421385	−	0.999112i	$-0.486583\pi$
$601$	2.06608	0.0842770	0.0421385	+	0.999112i	$0.486583\pi$
$602$	20.2920	0.827042
$603$	0.106231	0.00432604
$604$	−0.394814	−0.0160648
$605$	−1.02022	−0.0414778
$606$	−0.120629	−0.00490023
$607$	36.0249	1.46221	0.731103	−	0.682267i	$-0.239006\pi$
$607$	36.0249	1.46221	0.731103	+	0.682267i	$0.239006\pi$
$608$	−53.0033	−2.14957
$609$	−6.72310	−0.272434
$610$	−29.9867	−1.21413
$611$	−0.0911437	−0.00368728
$612$	−39.2540	−1.58675
$613$	−41.7472	−1.68615	−0.843076	−	0.537794i	$-0.819258\pi$
$613$	−41.7472	−1.68615	−0.843076	+	0.537794i	$0.819258\pi$
$614$	−54.5125	−2.19995
$615$	1.55519	0.0627113
$616$	28.1838	1.13556
$617$	35.2637	1.41966	0.709831	−	0.704372i	$-0.248771\pi$
$617$	35.2637	1.41966	0.709831	+	0.704372i	$0.248771\pi$
$618$	16.4724	0.662617
$619$	−39.3400	−1.58121	−0.790603	−	0.612328i	$-0.790233\pi$
$619$	−39.3400	−1.58121	−0.790603	+	0.612328i	$0.790233\pi$
$620$	−117.115	−4.70343
$621$	0.459345	0.0184329
$622$	67.1743	2.69344
$623$	7.39650	0.296334
$624$	−16.9376	−0.678047
$625$	−12.3478	−0.493912
$626$	−86.3939	−3.45299
$627$	10.2602	0.409753
$628$	25.4534	1.01570
$629$	6.63524	0.264564
$630$	9.24146	0.368189
$631$	3.87666	0.154327	0.0771637	−	0.997018i	$-0.475414\pi$
$631$	3.87666	0.154327	0.0771637	+	0.997018i	$0.475414\pi$
$632$	−106.557	−4.23859
$633$	26.4688	1.05204
$634$	81.8216	3.24955
$635$	19.3761	0.768918
$636$	−39.3453	−1.56014
$637$	1.33241	0.0527920
$638$	59.0553	2.33802
$639$	−12.0897	−0.478260
$640$	68.0475	2.68981
$641$	9.59431	0.378953	0.189476	−	0.981885i	$-0.439321\pi$
$641$	9.59431	0.378953	0.189476	+	0.981885i	$0.439321\pi$
$642$	−22.1527	−0.874299
$643$	19.5945	0.772732	0.386366	−	0.922346i	$-0.373730\pi$
$643$	19.5945	0.772732	0.386366	+	0.922346i	$0.373730\pi$
$644$	−2.39253	−0.0942788
$645$	−26.0146	−1.02433
$646$	63.4572	2.49669
$647$	−50.1636	−1.97213	−0.986067	−	0.166350i	$-0.946802\pi$
$647$	−50.1636	−1.97213	−0.986067	+	0.166350i	$0.946802\pi$
$648$	8.61460	0.338413
$649$	40.0461	1.57195
$650$	24.4965	0.960834
$651$	6.53246	0.256027
$652$	−86.4777	−3.38673
$653$	26.0941	1.02114	0.510570	−	0.859836i	$-0.329434\pi$
$653$	26.0941	1.02114	0.510570	+	0.859836i	$0.329434\pi$
$654$	11.1627	0.436496
$655$	31.3501	1.22495
$656$	−5.74356	−0.224248
$657$	−4.46689	−0.174270
$658$	−0.183659	−0.00715979
$659$	−14.0055	−0.545577	−0.272789	−	0.962074i	$-0.587946\pi$
$659$	−14.0055	−0.545577	−0.272789	+	0.962074i	$0.587946\pi$
$660$	−58.6542	−2.28311
$661$	26.9420	1.04792	0.523961	−	0.851743i	$-0.324454\pi$
$661$	26.9420	1.04792	0.523961	+	0.851743i	$0.324454\pi$
$662$	79.0479	3.07228
$663$	10.0416	0.389984
$664$	−43.2702	−1.67921
$665$	−10.7946	−0.418597
$666$	−2.36383	−0.0915964
$667$	−3.08822	−0.119576
$668$	−104.767	−4.05357
$669$	−8.77708	−0.339341
$670$	0.981726	0.0379274
$671$	−10.6158	−0.409819
$672$	−16.9010	−0.651970
$673$	23.7891	0.917004	0.458502	−	0.888693i	$-0.348386\pi$
$673$	23.7891	0.917004	0.458502	+	0.888693i	$0.348386\pi$
$674$	80.9657	3.11868
$675$	−6.84767	−0.263567
$676$	−58.4645	−2.24863
$677$	1.14643	0.0440608	0.0220304	−	0.999757i	$-0.492987\pi$
$677$	1.14643	0.0440608	0.0220304	+	0.999757i	$0.492987\pi$
$678$	44.8842	1.72377
$679$	10.2811	0.394554
$680$	−223.469	−8.56964
$681$	19.8560	0.760883
$682$	−57.3807	−2.19722
$683$	34.2486	1.31049	0.655243	−	0.755418i	$-0.272566\pi$
$683$	34.2486	1.31049	0.655243	+	0.755418i	$0.272566\pi$
$684$	−16.3346	−0.624570
$685$	41.2785	1.57717
$686$	2.68488	0.102509
$687$	13.3012	0.507471
$688$	96.0762	3.66287
$689$	10.0650	0.383445
$690$	4.24502	0.161605
$691$	−24.2824	−0.923745	−0.461873	−	0.886946i	$-0.652822\pi$
$691$	−24.2824	−0.923745	−0.461873	+	0.886946i	$0.652822\pi$
$692$	−73.5532	−2.79607
$693$	3.27164	0.124279
$694$	78.8533	2.99323
$695$	22.0240	0.835418
$696$	−57.9168	−2.19533
$697$	3.40512	0.128978
$698$	8.41134	0.318374
$699$	13.8329	0.523207
$700$	35.6665	1.34807
$701$	−10.7356	−0.405479	−0.202740	−	0.979233i	$-0.564985\pi$
$701$	−10.7356	−0.405479	−0.202740	+	0.979233i	$0.564985\pi$
$702$	−3.57736	−0.135019
$703$	2.76110	0.104137
$704$	65.2791	2.46030
$705$	0.235454	0.00886769
$706$	12.4162	0.467290
$707$	0.0449292	0.00168974
$708$	−63.7548	−2.39605
$709$	39.5118	1.48390	0.741948	−	0.670458i	$-0.233902\pi$
$709$	39.5118	1.48390	0.741948	+	0.670458i	$0.233902\pi$
$710$	−111.726	−4.19301
$711$	−12.3693	−0.463885
$712$	63.7178	2.38793
$713$	3.00065	0.112375
$714$	20.2344	0.757253
$715$	15.0044	0.561134
$716$	−108.543	−4.05644
$717$	−15.1025	−0.564013
$718$	13.4006	0.500106
$719$	0.207923	0.00775422	0.00387711	−	0.999992i	$-0.498766\pi$
$719$	0.207923	0.00775422	0.00387711	+	0.999992i	$0.498766\pi$
$720$	43.7553	1.63066
$721$	−6.13525	−0.228489
$722$	−24.6064	−0.915757
$723$	20.9641	0.779662
$724$	24.1297	0.896774
$725$	46.0375	1.70979
$726$	0.795795	0.0295347
$727$	33.9130	1.25776	0.628882	−	0.777501i	$-0.283513\pi$
$727$	33.9130	1.25776	0.628882	+	0.777501i	$0.283513\pi$
$728$	11.4782	0.425410
$729$	1.00000	0.0370370
$730$	−41.2806	−1.52786
$731$	−56.9596	−2.10673
$732$	16.9008	0.624670
$733$	−18.1547	−0.670559	−0.335279	−	0.942119i	$-0.608831\pi$
$733$	−18.1547	−0.670559	−0.335279	+	0.942119i	$0.608831\pi$
$734$	1.21861	0.0449797
$735$	−3.44204	−0.126962
$736$	−7.76338	−0.286162
$737$	0.347548	0.0128021
$738$	−1.21308	−0.0446543
$739$	30.7806	1.13228	0.566142	−	0.824308i	$-0.308435\pi$
$739$	30.7806	1.13228	0.566142	+	0.824308i	$0.308435\pi$
$740$	−15.7843	−0.580243
$741$	4.17858	0.153504
$742$	20.2814	0.744555
$743$	41.3493	1.51696	0.758479	−	0.651698i	$-0.225943\pi$
$743$	41.3493	1.51696	0.758479	+	0.651698i	$0.225943\pi$
$744$	56.2745	2.06312
$745$	−26.1996	−0.959878
$746$	40.7172	1.49076
$747$	−5.02289	−0.183778
$748$	−128.425	−4.69567
$749$	8.25094	0.301483
$750$	−17.0751	−0.623496
$751$	−35.6285	−1.30010	−0.650052	−	0.759890i	$-0.725253\pi$
$751$	−35.6285	−1.30010	−0.650052	+	0.759890i	$0.725253\pi$
$752$	−0.869567	−0.0317098
$753$	6.84691	0.249515
$754$	24.0509	0.875883
$755$	−0.260910	−0.00949550
$756$	−5.20856	−0.189434
$757$	−47.4998	−1.72641	−0.863205	−	0.504854i	$-0.831546\pi$
$757$	−47.4998	−1.72641	−0.863205	+	0.504854i	$0.831546\pi$
$758$	−49.7114	−1.80560
$759$	1.50281	0.0545485
$760$	−92.9913	−3.37315
$761$	−2.69606	−0.0977320	−0.0488660	−	0.998805i	$-0.515561\pi$
$761$	−2.69606	−0.0977320	−0.0488660	+	0.998805i	$0.515561\pi$
$762$	−15.1138	−0.547517
$763$	−4.15762	−0.150516
$764$	31.2595	1.13093
$765$	−25.9407	−0.937889
$766$	−2.68488	−0.0970085
$767$	16.3092	0.588891
$768$	−13.1726	−0.475324
$769$	27.1771	0.980031	0.490015	−	0.871714i	$-0.336991\pi$
$769$	27.1771	0.980031	0.490015	+	0.871714i	$0.336991\pi$
$770$	30.2347	1.08958
$771$	−13.7835	−0.496400
$772$	95.4938	3.43690
$773$	−20.4659	−0.736109	−0.368054	−	0.929804i	$-0.619976\pi$
$773$	−20.4659	−0.736109	−0.368054	+	0.929804i	$0.619976\pi$
$774$	20.2920	0.729382
$775$	−44.7321	−1.60682
$776$	88.5678	3.17940
$777$	0.880422	0.0315850
$778$	24.7893	0.888740
$779$	1.41696	0.0507679
$780$	−23.8876	−0.855313
$781$	−39.5530	−1.41532
$782$	9.29456	0.332373
$783$	−6.72310	−0.240264
$784$	12.7120	0.454000
$785$	16.8207	0.600357
$786$	−24.4538	−0.872240
$787$	−22.8095	−0.813071	−0.406536	−	0.913635i	$-0.633263\pi$
$787$	−22.8095	−0.813071	−0.406536	+	0.913635i	$0.633263\pi$
$788$	−35.3911	−1.26076
$789$	−10.7404	−0.382368
$790$	−114.310	−4.06698
$791$	−16.7174	−0.594402
$792$	28.1838	1.00147
$793$	−4.32341	−0.153529
$794$	−13.9690	−0.495743
$795$	−26.0010	−0.922162
$796$	102.602	3.63664
$797$	3.39917	0.120405	0.0602024	−	0.998186i	$-0.480825\pi$
$797$	3.39917	0.120405	0.0602024	+	0.998186i	$0.480825\pi$
$798$	8.42006	0.298067
$799$	0.515531	0.0182382
$800$	115.732	4.09175
$801$	7.39650	0.261342
$802$	45.1813	1.59541
$803$	−14.6140	−0.515718
$804$	−0.553309	−0.0195137
$805$	−1.58109	−0.0557259
$806$	−23.3689	−0.823135
$807$	2.67432	0.0941404
$808$	0.387047	0.0136163
$809$	−26.1310	−0.918718	−0.459359	−	0.888251i	$-0.651921\pi$
$809$	−26.1310	−0.918718	−0.459359	+	0.888251i	$0.651921\pi$
$810$	9.24146	0.324712
$811$	12.0794	0.424165	0.212082	−	0.977252i	$-0.431975\pi$
$811$	12.0794	0.424165	0.212082	+	0.977252i	$0.431975\pi$
$812$	35.0177	1.22888
$813$	−1.40266	−0.0491933
$814$	−7.73358	−0.271062
$815$	−57.1482	−2.00181
$816$	95.8032	3.35378
$817$	−23.7024	−0.829242
$818$	−32.5949	−1.13965
$819$	1.33241	0.0465582
$820$	−8.10030	−0.282875
$821$	−45.0421	−1.57198	−0.785990	−	0.618240i	$-0.787846\pi$
$821$	−45.0421	−1.57198	−0.785990	+	0.618240i	$0.787846\pi$
$822$	−32.1982	−1.12304
$823$	47.9400	1.67108	0.835542	−	0.549426i	$-0.185154\pi$
$823$	47.9400	1.67108	0.835542	+	0.549426i	$0.185154\pi$
$824$	−52.8527	−1.84121
$825$	−22.4031	−0.779975
$826$	32.8639	1.14348
$827$	−42.4287	−1.47539	−0.737694	−	0.675135i	$-0.764085\pi$
$827$	−42.4287	−1.47539	−0.737694	+	0.675135i	$0.764085\pi$
$828$	−2.39253	−0.0831461
$829$	−16.6926	−0.579758	−0.289879	−	0.957063i	$-0.593615\pi$
$829$	−16.6926	−0.579758	−0.289879	+	0.957063i	$0.593615\pi$
$830$	−46.4188	−1.61122
$831$	−7.50122	−0.260214
$832$	26.5856	0.921691
$833$	−7.53643	−0.261122
$834$	−17.1792	−0.594869
$835$	−69.2347	−2.39597
$836$	−53.4409	−1.84829
$837$	6.53246	0.225795
$838$	−36.5580	−1.26288
$839$	26.0780	0.900313	0.450156	−	0.892950i	$-0.351368\pi$
$839$	26.0780	0.900313	0.450156	+	0.892950i	$0.351368\pi$
$840$	−29.6518	−1.02309
$841$	16.2000	0.558622
$842$	2.41675	0.0832867
$843$	−6.15369	−0.211945
$844$	−137.865	−4.74549
$845$	−38.6359	−1.32911
$846$	−0.183659	−0.00631434
$847$	−0.296399	−0.0101844
$848$	96.0259	3.29754
$849$	23.5799	0.809259
$850$	−138.558	−4.75251
$851$	0.404418	0.0138633
$852$	62.9698	2.15731
$853$	2.97843	0.101980	0.0509898	−	0.998699i	$-0.483762\pi$
$853$	2.97843	0.101980	0.0509898	+	0.998699i	$0.483762\pi$
$854$	−8.71190	−0.298115
$855$	−10.7946	−0.369168
$856$	71.0785	2.42941
$857$	16.2127	0.553814	0.276907	−	0.960897i	$-0.410691\pi$
$857$	16.2127	0.553814	0.276907	+	0.960897i	$0.410691\pi$
$858$	−11.7038	−0.399561
$859$	43.5673	1.48650	0.743248	−	0.669016i	$-0.233284\pi$
$859$	43.5673	1.48650	0.743248	+	0.669016i	$0.233284\pi$
$860$	135.499	4.62048
$861$	0.451821	0.0153980
$862$	−81.7489	−2.78438
$863$	8.66342	0.294906	0.147453	−	0.989069i	$-0.452892\pi$
$863$	8.66342	0.294906	0.147453	+	0.989069i	$0.452892\pi$
$864$	−16.9010	−0.574983
$865$	−48.6071	−1.65269
$866$	48.7159	1.65543
$867$	−39.7978	−1.35160
$868$	−34.0247	−1.15487
$869$	−40.4679	−1.37278
$870$	−62.1313	−2.10645
$871$	0.141543	0.00479599
$872$	−35.8163	−1.21289
$873$	10.2811	0.347964
$874$	3.86771	0.130827
$875$	6.35975	0.214999
$876$	23.2661	0.786087
$877$	−21.2168	−0.716440	−0.358220	−	0.933637i	$-0.616616\pi$
$877$	−21.2168	−0.716440	−0.358220	+	0.933637i	$0.616616\pi$
$878$	−5.90891	−0.199416
$879$	−20.8363	−0.702792
$880$	143.151	4.82563
$881$	29.3055	0.987327	0.493663	−	0.869653i	$-0.335657\pi$
$881$	29.3055	0.987327	0.493663	+	0.869653i	$0.335657\pi$
$882$	2.68488	0.0904045
$883$	4.27516	0.143870	0.0719352	−	0.997409i	$-0.477083\pi$
$883$	4.27516	0.143870	0.0719352	+	0.997409i	$0.477083\pi$
$884$	−52.3024	−1.75912
$885$	−42.1319	−1.41625
$886$	−74.4654	−2.50171
$887$	−37.2429	−1.25049	−0.625246	−	0.780427i	$-0.715002\pi$
$887$	−37.2429	−1.25049	−0.625246	+	0.780427i	$0.715002\pi$
$888$	7.58448	0.254519
$889$	5.62925	0.188799
$890$	68.3545	2.29125
$891$	3.27164	0.109604
$892$	45.7160	1.53068
$893$	0.214526	0.00717883
$894$	20.4363	0.683492
$895$	−71.7298	−2.39766
$896$	19.7695	0.660452
$897$	0.612036	0.0204353
$898$	−18.8991	−0.630671
$899$	−43.9183	−1.46476
$900$	35.6665	1.18888
$901$	−56.9298	−1.89661
$902$	−3.96877	−0.132146
$903$	−7.55791	−0.251511
$904$	−144.014	−4.78982
$905$	15.9460	0.530062
$906$	0.203516	0.00676138
$907$	−4.59725	−0.152649	−0.0763246	−	0.997083i	$-0.524319\pi$
$907$	−4.59725	−0.152649	−0.0763246	+	0.997083i	$0.524319\pi$
$908$	−103.421	−3.43215
$909$	0.0449292	0.00149021
$910$	12.3134	0.408186
$911$	−45.3307	−1.50187	−0.750937	−	0.660374i	$-0.770398\pi$
$911$	−45.3307	−1.50187	−0.750937	+	0.660374i	$0.770398\pi$
$912$	39.8662	1.32010
$913$	−16.4331	−0.543855
$914$	49.4737	1.63644
$915$	11.1688	0.369228
$916$	−69.2800	−2.28907
$917$	9.10799	0.300772
$918$	20.2344	0.667834
$919$	19.5320	0.644302	0.322151	−	0.946688i	$-0.395594\pi$
$919$	19.5320	0.644302	0.322151	+	0.946688i	$0.395594\pi$
$920$	−13.6204	−0.449052
$921$	20.3036	0.669025
$922$	74.8296	2.46438
$923$	−16.1084	−0.530214
$924$	−17.0405	−0.560592
$925$	−6.02884	−0.198227
$926$	9.44595	0.310413
$927$	−6.13525	−0.201508
$928$	113.627	3.72999
$929$	44.4753	1.45919	0.729593	−	0.683881i	$-0.239709\pi$
$929$	44.4753	1.45919	0.729593	+	0.683881i	$0.239709\pi$
$930$	60.3695	1.97959
$931$	−3.13611	−0.102782
$932$	−72.0493	−2.36005
$933$	−25.0195	−0.819101
$934$	20.0796	0.657023
$935$	−84.8686	−2.77550
$936$	11.4782	0.375176
$937$	−1.66209	−0.0542980	−0.0271490	−	0.999631i	$-0.508643\pi$
$937$	−1.66209	−0.0542980	−0.0271490	+	0.999631i	$0.508643\pi$
$938$	0.285216	0.00931264
$939$	32.1780	1.05009
$940$	−1.22637	−0.0399999
$941$	58.7235	1.91433	0.957166	−	0.289539i	$-0.0935019\pi$
$941$	58.7235	1.91433	0.957166	+	0.289539i	$0.0935019\pi$
$942$	−13.1206	−0.427491
$943$	0.207542	0.00675849
$944$	155.600	5.06434
$945$	−3.44204	−0.111970
$946$	66.3882	2.15847
$947$	−10.2443	−0.332894	−0.166447	−	0.986050i	$-0.553229\pi$
$947$	−10.2443	−0.332894	−0.166447	+	0.986050i	$0.553229\pi$
$948$	64.4263	2.09247
$949$	−5.95172	−0.193201
$950$	−57.6578	−1.87067
$951$	−30.4750	−0.988219
$952$	−64.9233	−2.10418
$953$	24.9447	0.808037	0.404019	−	0.914751i	$-0.367613\pi$
$953$	24.9447	0.808037	0.404019	+	0.914751i	$0.367613\pi$
$954$	20.2814	0.656636
$955$	20.6576	0.668465
$956$	78.6623	2.54412
$957$	−21.9955	−0.711014
$958$	−11.2163	−0.362384
$959$	11.9924	0.387256
$960$	−68.6792	−2.21661
$961$	11.6730	0.376548
$962$	−3.14958	−0.101547
$963$	8.25094	0.265883
$964$	−109.193	−3.51686
$965$	63.1064	2.03147
$966$	1.23328	0.0396803
$967$	−22.5714	−0.725848	−0.362924	−	0.931819i	$-0.618221\pi$
$967$	−22.5714	−0.725848	−0.362924	+	0.931819i	$0.618221\pi$
$968$	−2.55336	−0.0820681
$969$	−23.6351	−0.759267
$970$	95.0127	3.05067
$971$	−8.15723	−0.261778	−0.130889	−	0.991397i	$-0.541783\pi$
$971$	−8.15723	−0.261778	−0.130889	+	0.991397i	$0.541783\pi$
$972$	−5.20856	−0.167065
$973$	6.39852	0.205127
$974$	−17.2971	−0.554234
$975$	−9.12390	−0.292199
$976$	−41.2480	−1.32032
$977$	−16.4649	−0.526758	−0.263379	−	0.964692i	$-0.584837\pi$
$977$	−16.4649	−0.526758	−0.263379	+	0.964692i	$0.584837\pi$
$978$	44.5770	1.42541
$979$	24.1986	0.773392
$980$	17.9281	0.572692
$981$	−4.15762	−0.132743
$982$	−7.59006	−0.242209
$983$	22.7954	0.727060	0.363530	−	0.931582i	$-0.381571\pi$
$983$	22.7954	0.727060	0.363530	+	0.931582i	$0.381571\pi$
$984$	3.89226	0.124081
$985$	−23.3880	−0.745203
$986$	−136.038	−4.33232
$987$	0.0684051	0.00217736
$988$	−21.7644	−0.692418
$989$	−3.47169	−0.110393
$990$	30.2347	0.960922
$991$	−27.9100	−0.886590	−0.443295	−	0.896376i	$-0.646191\pi$
$991$	−27.9100	−0.886590	−0.443295	+	0.896376i	$0.646191\pi$
$992$	−110.405	−3.50536
$993$	−29.4419	−0.934311
$994$	−32.4593	−1.02955
$995$	67.8040	2.14953
$996$	26.1620	0.828976
$997$	28.0031	0.886867	0.443433	−	0.896307i	$-0.353760\pi$
$997$	28.0031	0.886867	0.443433	+	0.896307i	$0.353760\pi$
$998$	24.3786	0.771690
$999$	0.880422	0.0278553

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.t.1.49	✓	52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.t.1.49	✓	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)

\(f(q)\)	\(=\)	\(q+2.68488 q^{2} -1.00000 q^{3} +5.20856 q^{4} +3.44204 q^{5} -2.68488 q^{6} +1.00000 q^{7} +8.61460 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.68488 q^{2} -1.00000 q^{3} +5.20856 q^{4} +3.44204 q^{5} -2.68488 q^{6} +1.00000 q^{7} +8.61460 q^{8} +1.00000 q^{9} +9.24146 q^{10} +3.27164 q^{11} -5.20856 q^{12} +1.33241 q^{13} +2.68488 q^{14} -3.44204 q^{15} +12.7120 q^{16} -7.53643 q^{17} +2.68488 q^{18} -3.13611 q^{19} +17.9281 q^{20} -1.00000 q^{21} +8.78394 q^{22} -0.459345 q^{23} -8.61460 q^{24} +6.84767 q^{25} +3.57736 q^{26} -1.00000 q^{27} +5.20856 q^{28} +6.72310 q^{29} -9.24146 q^{30} -6.53246 q^{31} +16.9010 q^{32} -3.27164 q^{33} -20.2344 q^{34} +3.44204 q^{35} +5.20856 q^{36} -0.880422 q^{37} -8.42006 q^{38} -1.33241 q^{39} +29.6518 q^{40} -0.451821 q^{41} -2.68488 q^{42} +7.55791 q^{43} +17.0405 q^{44} +3.44204 q^{45} -1.23328 q^{46} -0.0684051 q^{47} -12.7120 q^{48} +1.00000 q^{49} +18.3851 q^{50} +7.53643 q^{51} +6.93994 q^{52} +7.55395 q^{53} -2.68488 q^{54} +11.2611 q^{55} +8.61460 q^{56} +3.13611 q^{57} +18.0507 q^{58} +12.2404 q^{59} -17.9281 q^{60} -3.24480 q^{61} -17.5388 q^{62} +1.00000 q^{63} +19.9530 q^{64} +4.58621 q^{65} -8.78394 q^{66} +0.106231 q^{67} -39.2540 q^{68} +0.459345 q^{69} +9.24146 q^{70} -12.0897 q^{71} +8.61460 q^{72} -4.46689 q^{73} -2.36383 q^{74} -6.84767 q^{75} -16.3346 q^{76} +3.27164 q^{77} -3.57736 q^{78} -12.3693 q^{79} +43.7553 q^{80} +1.00000 q^{81} -1.21308 q^{82} -5.02289 q^{83} -5.20856 q^{84} -25.9407 q^{85} +20.2920 q^{86} -6.72310 q^{87} +28.1838 q^{88} +7.39650 q^{89} +9.24146 q^{90} +1.33241 q^{91} -2.39253 q^{92} +6.53246 q^{93} -0.183659 q^{94} -10.7946 q^{95} -16.9010 q^{96} +10.2811 q^{97} +2.68488 q^{98} +3.27164 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) \) 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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