LMFDB - Embedded newform 8007.2.a.i.1.37

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, 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("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$0$
Dimension:	$63$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.37
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.845709 q^{2} +1.00000 q^{3} -1.28478 q^{4} -0.0676094 q^{5} +0.845709 q^{6} -2.72063 q^{7} -2.77796 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.845709 q^{2} +1.00000 q^{3} -1.28478 q^{4} -0.0676094 q^{5} +0.845709 q^{6} -2.72063 q^{7} -2.77796 q^{8} +1.00000 q^{9} -0.0571779 q^{10} +3.60018 q^{11} -1.28478 q^{12} +4.54708 q^{13} -2.30086 q^{14} -0.0676094 q^{15} +0.220204 q^{16} +1.00000 q^{17} +0.845709 q^{18} -7.93383 q^{19} +0.0868630 q^{20} -2.72063 q^{21} +3.04470 q^{22} +8.56382 q^{23} -2.77796 q^{24} -4.99543 q^{25} +3.84550 q^{26} +1.00000 q^{27} +3.49540 q^{28} -2.86974 q^{29} -0.0571779 q^{30} -1.58439 q^{31} +5.74216 q^{32} +3.60018 q^{33} +0.845709 q^{34} +0.183940 q^{35} -1.28478 q^{36} +11.7925 q^{37} -6.70971 q^{38} +4.54708 q^{39} +0.187817 q^{40} -3.37357 q^{41} -2.30086 q^{42} -4.88999 q^{43} -4.62543 q^{44} -0.0676094 q^{45} +7.24250 q^{46} -4.53952 q^{47} +0.220204 q^{48} +0.401817 q^{49} -4.22468 q^{50} +1.00000 q^{51} -5.84198 q^{52} -12.6175 q^{53} +0.845709 q^{54} -0.243406 q^{55} +7.55781 q^{56} -7.93383 q^{57} -2.42697 q^{58} +6.87792 q^{59} +0.0868630 q^{60} +7.55888 q^{61} -1.33993 q^{62} -2.72063 q^{63} +4.41578 q^{64} -0.307425 q^{65} +3.04470 q^{66} +11.4867 q^{67} -1.28478 q^{68} +8.56382 q^{69} +0.155560 q^{70} -1.26257 q^{71} -2.77796 q^{72} -12.1528 q^{73} +9.97303 q^{74} -4.99543 q^{75} +10.1932 q^{76} -9.79475 q^{77} +3.84550 q^{78} +5.41143 q^{79} -0.0148879 q^{80} +1.00000 q^{81} -2.85305 q^{82} +1.96361 q^{83} +3.49540 q^{84} -0.0676094 q^{85} -4.13551 q^{86} -2.86974 q^{87} -10.0012 q^{88} -5.81324 q^{89} -0.0571779 q^{90} -12.3709 q^{91} -11.0026 q^{92} -1.58439 q^{93} -3.83912 q^{94} +0.536401 q^{95} +5.74216 q^{96} +14.0780 q^{97} +0.339820 q^{98} +3.60018 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 63 q + 10 q^{2} + 63 q^{3} + 70 q^{4} + 19 q^{5} + 10 q^{6} + 11 q^{7} + 27 q^{8} + 63 q^{9}+O(q^{10}) $ 63 * q + 10 * q^2 + 63 * q^3 + 70 * q^4 + 19 * q^5 + 10 * q^6 + 11 * q^7 + 27 * q^8 + 63 * q^9	\( 63 q + 10 q^{2} + 63 q^{3} + 70 q^{4} + 19 q^{5} + 10 q^{6} + 11 q^{7} + 27 q^{8} + 63 q^{9} + 4 q^{10} + 23 q^{11} + 70 q^{12} + 10 q^{13} + 18 q^{14} + 19 q^{15} + 72 q^{16} + 63 q^{17} + 10 q^{18} + 6 q^{19} + 48 q^{20} + 11 q^{21} + 21 q^{22} + 44 q^{23} + 27 q^{24} + 110 q^{25} + 41 q^{26} + 63 q^{27} + 26 q^{28} + 35 q^{29} + 4 q^{30} + q^{31} + 54 q^{32} + 23 q^{33} + 10 q^{34} + 47 q^{35} + 70 q^{36} + 40 q^{37} + 38 q^{38} + 10 q^{39} - 10 q^{40} + 35 q^{41} + 18 q^{42} + 27 q^{43} + 46 q^{44} + 19 q^{45} + 8 q^{46} + 29 q^{47} + 72 q^{48} + 114 q^{49} + 27 q^{50} + 63 q^{51} - q^{52} + 75 q^{53} + 10 q^{54} + 5 q^{55} + 24 q^{56} + 6 q^{57} + 41 q^{58} + 105 q^{59} + 48 q^{60} + 5 q^{61} + 22 q^{62} + 11 q^{63} + 61 q^{64} + 49 q^{65} + 21 q^{66} + 4 q^{67} + 70 q^{68} + 44 q^{69} - 16 q^{70} + 16 q^{71} + 27 q^{72} + 39 q^{73} + 54 q^{74} + 110 q^{75} + 6 q^{76} + 88 q^{77} + 41 q^{78} + 16 q^{79} + 102 q^{80} + 63 q^{81} - 29 q^{82} + 73 q^{83} + 26 q^{84} + 19 q^{85} + 46 q^{86} + 35 q^{87} + 18 q^{88} + 88 q^{89} + 4 q^{90} - 15 q^{91} + 110 q^{92} + q^{93} - 8 q^{94} + 28 q^{95} + 54 q^{96} + 70 q^{97} + 33 q^{98} + 23 q^{99}+O(q^{100}) \) 63 * q + 10 * q^2 + 63 * q^3 + 70 * q^4 + 19 * q^5 + 10 * q^6 + 11 * q^7 + 27 * q^8 + 63 * q^9 + 4 * q^10 + 23 * q^11 + 70 * q^12 + 10 * q^13 + 18 * q^14 + 19 * q^15 + 72 * q^16 + 63 * q^17 + 10 * q^18 + 6 * q^19 + 48 * q^20 + 11 * q^21 + 21 * q^22 + 44 * q^23 + 27 * q^24 + 110 * q^25 + 41 * q^26 + 63 * q^27 + 26 * q^28 + 35 * q^29 + 4 * q^30 + q^31 + 54 * q^32 + 23 * q^33 + 10 * q^34 + 47 * q^35 + 70 * q^36 + 40 * q^37 + 38 * q^38 + 10 * q^39 - 10 * q^40 + 35 * q^41 + 18 * q^42 + 27 * q^43 + 46 * q^44 + 19 * q^45 + 8 * q^46 + 29 * q^47 + 72 * q^48 + 114 * q^49 + 27 * q^50 + 63 * q^51 - q^52 + 75 * q^53 + 10 * q^54 + 5 * q^55 + 24 * q^56 + 6 * q^57 + 41 * q^58 + 105 * q^59 + 48 * q^60 + 5 * q^61 + 22 * q^62 + 11 * q^63 + 61 * q^64 + 49 * q^65 + 21 * q^66 + 4 * q^67 + 70 * q^68 + 44 * q^69 - 16 * q^70 + 16 * q^71 + 27 * q^72 + 39 * q^73 + 54 * q^74 + 110 * q^75 + 6 * q^76 + 88 * q^77 + 41 * q^78 + 16 * q^79 + 102 * q^80 + 63 * q^81 - 29 * q^82 + 73 * q^83 + 26 * q^84 + 19 * q^85 + 46 * q^86 + 35 * q^87 + 18 * q^88 + 88 * q^89 + 4 * q^90 - 15 * q^91 + 110 * q^92 + q^93 - 8 * q^94 + 28 * q^95 + 54 * q^96 + 70 * q^97 + 33 * q^98 + 23 * 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 $

$2$	0.845709	0.598006	0.299003	−	0.954252i	$-0.403346\pi$
$2$	0.845709	0.598006	0.299003	+	0.954252i	$0.403346\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.28478	−0.642388
$5$	−0.0676094	−0.0302358	−0.0151179	−	0.999886i	$-0.504812\pi$
$5$	−0.0676094	−0.0302358	−0.0151179	+	0.999886i	$0.504812\pi$
$6$	0.845709	0.345259
$7$	−2.72063	−1.02830	−0.514150	−	0.857700i	$-0.671893\pi$
$7$	−2.72063	−1.02830	−0.514150	+	0.857700i	$0.671893\pi$
$8$	−2.77796	−0.982159
$9$	1.00000	0.333333
$10$	−0.0571779	−0.0180812
$11$	3.60018	1.08549	0.542747	−	0.839896i	$-0.317384\pi$
$11$	3.60018	1.08549	0.542747	+	0.839896i	$0.317384\pi$
$12$	−1.28478	−0.370883
$13$	4.54708	1.26113	0.630566	−	0.776135i	$-0.282823\pi$
$13$	4.54708	1.26113	0.630566	+	0.776135i	$0.282823\pi$
$14$	−2.30086	−0.614930
$15$	−0.0676094	−0.0174567
$16$	0.220204	0.0550510
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0.845709	0.199335
$19$	−7.93383	−1.82014	−0.910072	−	0.414450i	$-0.863974\pi$
$19$	−7.93383	−1.82014	−0.910072	+	0.414450i	$0.863974\pi$
$20$	0.0868630	0.0194232
$21$	−2.72063	−0.593690
$22$	3.04470	0.649133
$23$	8.56382	1.78568	0.892840	−	0.450374i	$-0.148709\pi$
$23$	8.56382	1.78568	0.892840	+	0.450374i	$0.148709\pi$
$24$	−2.77796	−0.567050
$25$	−4.99543	−0.999086
$26$	3.84550	0.754166
$27$	1.00000	0.192450
$28$	3.49540	0.660568
$29$	−2.86974	−0.532898	−0.266449	−	0.963849i	$-0.585850\pi$
$29$	−2.86974	−0.532898	−0.266449	+	0.963849i	$0.585850\pi$
$30$	−0.0571779	−0.0104392
$31$	−1.58439	−0.284564	−0.142282	−	0.989826i	$-0.545444\pi$
$31$	−1.58439	−0.284564	−0.142282	+	0.989826i	$0.545444\pi$
$32$	5.74216	1.01508
$33$	3.60018	0.626711
$34$	0.845709	0.145038
$35$	0.183940	0.0310915
$36$	−1.28478	−0.214129
$37$	11.7925	1.93868	0.969339	−	0.245729i	$-0.0790273\pi$
$37$	11.7925	1.93868	0.969339	+	0.245729i	$0.0790273\pi$
$38$	−6.70971	−1.08846
$39$	4.54708	0.728115
$40$	0.187817	0.0296964
$41$	−3.37357	−0.526862	−0.263431	−	0.964678i	$-0.584854\pi$
$41$	−3.37357	−0.526862	−0.263431	+	0.964678i	$0.584854\pi$
$42$	−2.30086	−0.355030
$43$	−4.88999	−0.745717	−0.372858	−	0.927888i	$-0.621622\pi$
$43$	−4.88999	−0.745717	−0.372858	+	0.927888i	$0.621622\pi$
$44$	−4.62543	−0.697309
$45$	−0.0676094	−0.0100786
$46$	7.24250	1.06785
$47$	−4.53952	−0.662158	−0.331079	−	0.943603i	$-0.607413\pi$
$47$	−4.53952	−0.662158	−0.331079	+	0.943603i	$0.607413\pi$
$48$	0.220204	0.0317837
$49$	0.401817	0.0574024
$50$	−4.22468	−0.597460
$51$	1.00000	0.140028
$52$	−5.84198	−0.810137
$53$	−12.6175	−1.73315	−0.866573	−	0.499051i	$-0.833682\pi$
$53$	−12.6175	−1.73315	−0.866573	+	0.499051i	$0.833682\pi$
$54$	0.845709	0.115086
$55$	−0.243406	−0.0328208
$56$	7.55781	1.00995
$57$	−7.93383	−1.05086
$58$	−2.42697	−0.318676
$59$	6.87792	0.895429	0.447715	−	0.894177i	$-0.352238\pi$
$59$	6.87792	0.895429	0.447715	+	0.894177i	$0.352238\pi$
$60$	0.0868630	0.0112140
$61$	7.55888	0.967815	0.483907	−	0.875119i	$-0.339217\pi$
$61$	7.55888	0.967815	0.483907	+	0.875119i	$0.339217\pi$
$62$	−1.33993	−0.170171
$63$	−2.72063	−0.342767
$64$	4.41578	0.551973
$65$	−0.307425	−0.0381314
$66$	3.04470	0.374777
$67$	11.4867	1.40333	0.701663	−	0.712509i	$-0.252441\pi$
$67$	11.4867	1.40333	0.701663	+	0.712509i	$0.252441\pi$
$68$	−1.28478	−0.155802
$69$	8.56382	1.03096
$70$	0.155560	0.0185929
$71$	−1.26257	−0.149839	−0.0749195	−	0.997190i	$-0.523870\pi$
$71$	−1.26257	−0.149839	−0.0749195	+	0.997190i	$0.523870\pi$
$72$	−2.77796	−0.327386
$73$	−12.1528	−1.42237	−0.711187	−	0.703003i	$-0.751842\pi$
$73$	−12.1528	−1.42237	−0.711187	+	0.703003i	$0.751842\pi$
$74$	9.97303	1.15934
$75$	−4.99543	−0.576822
$76$	10.1932	1.16924
$77$	−9.79475	−1.11621
$78$	3.84550	0.435418
$79$	5.41143	0.608833	0.304417	−	0.952539i	$-0.401539\pi$
$79$	5.41143	0.608833	0.304417	+	0.952539i	$0.401539\pi$
$80$	−0.0148879	−0.00166451
$81$	1.00000	0.111111
$82$	−2.85305	−0.315067
$83$	1.96361	0.215534	0.107767	−	0.994176i	$-0.465630\pi$
$83$	1.96361	0.215534	0.107767	+	0.994176i	$0.465630\pi$
$84$	3.49540	0.381379
$85$	−0.0676094	−0.00733327
$86$	−4.13551	−0.445943
$87$	−2.86974	−0.307669
$88$	−10.0012	−1.06613
$89$	−5.81324	−0.616203	−0.308101	−	0.951354i	$-0.599694\pi$
$89$	−5.81324	−0.616203	−0.308101	+	0.951354i	$0.599694\pi$
$90$	−0.0571779	−0.00602708
$91$	−12.3709	−1.29682
$92$	−11.0026	−1.14710
$93$	−1.58439	−0.164293
$94$	−3.83912	−0.395974
$95$	0.536401	0.0550336
$96$	5.74216	0.586057
$97$	14.0780	1.42940	0.714700	−	0.699431i	$-0.246563\pi$
$97$	14.0780	1.42940	0.714700	+	0.699431i	$0.246563\pi$
$98$	0.339820	0.0343270
$99$	3.60018	0.361832
$100$	6.41801	0.641801
$101$	11.3329	1.12766	0.563832	−	0.825890i	$-0.309327\pi$
$101$	11.3329	1.12766	0.563832	+	0.825890i	$0.309327\pi$
$102$	0.845709	0.0837377
$103$	18.2566	1.79888	0.899441	−	0.437043i	$-0.143974\pi$
$103$	18.2566	1.79888	0.899441	+	0.437043i	$0.143974\pi$
$104$	−12.6316	−1.23863
$105$	0.183940	0.0179507
$106$	−10.6707	−1.03643
$107$	−8.86696	−0.857201	−0.428600	−	0.903494i	$-0.640993\pi$
$107$	−8.86696	−0.857201	−0.428600	+	0.903494i	$0.640993\pi$
$108$	−1.28478	−0.123628
$109$	16.6977	1.59935	0.799676	−	0.600432i	$-0.205005\pi$
$109$	16.6977	1.59935	0.799676	+	0.600432i	$0.205005\pi$
$110$	−0.205851	−0.0196271
$111$	11.7925	1.11930
$112$	−0.599094	−0.0566090
$113$	10.2571	0.964906	0.482453	−	0.875922i	$-0.339746\pi$
$113$	10.2571	0.964906	0.482453	+	0.875922i	$0.339746\pi$
$114$	−6.70971	−0.628422
$115$	−0.578995	−0.0539916
$116$	3.68698	0.342327
$117$	4.54708	0.420378
$118$	5.81672	0.535472
$119$	−2.72063	−0.249400
$120$	0.187817	0.0171452
$121$	1.96129	0.178299
$122$	6.39261	0.578760
$123$	−3.37357	−0.304184
$124$	2.03558	0.182801
$125$	0.675785	0.0604440
$126$	−2.30086	−0.204977
$127$	−14.3299	−1.27158	−0.635788	−	0.771864i	$-0.719325\pi$
$127$	−14.3299	−1.27158	−0.635788	+	0.771864i	$0.719325\pi$
$128$	−7.74985	−0.684996
$129$	−4.88999	−0.430540
$130$	−0.259992	−0.0228028
$131$	12.8088	1.11911	0.559557	−	0.828792i	$-0.310971\pi$
$131$	12.8088	1.11911	0.559557	+	0.828792i	$0.310971\pi$
$132$	−4.62543	−0.402592
$133$	21.5850	1.87166
$134$	9.71441	0.839198
$135$	−0.0676094	−0.00581889
$136$	−2.77796	−0.238208
$137$	1.11420	0.0951924	0.0475962	−	0.998867i	$-0.484844\pi$
$137$	1.11420	0.0951924	0.0475962	+	0.998867i	$0.484844\pi$
$138$	7.24250	0.616523
$139$	1.44881	0.122886	0.0614431	−	0.998111i	$-0.480430\pi$
$139$	1.44881	0.122886	0.0614431	+	0.998111i	$0.480430\pi$
$140$	−0.236322	−0.0199728
$141$	−4.53952	−0.382297
$142$	−1.06776	−0.0896047
$143$	16.3703	1.36895
$144$	0.220204	0.0183503
$145$	0.194022	0.0161126
$146$	−10.2777	−0.850589
$147$	0.401817	0.0331413
$148$	−15.1507	−1.24538
$149$	3.36721	0.275853	0.137926	−	0.990442i	$-0.455956\pi$
$149$	3.36721	0.275853	0.137926	+	0.990442i	$0.455956\pi$
$150$	−4.22468	−0.344944
$151$	11.4667	0.933145	0.466572	−	0.884483i	$-0.345489\pi$
$151$	11.4667	0.933145	0.466572	+	0.884483i	$0.345489\pi$
$152$	22.0399	1.78767
$153$	1.00000	0.0808452
$154$	−8.28350	−0.667504
$155$	0.107119	0.00860404
$156$	−5.84198	−0.467733
$157$	1.00000	0.0798087
$157$	1.00000	0.0798087
$158$	4.57649	0.364086
$159$	−12.6175	−1.00063
$160$	−0.388224	−0.0306918
$161$	−23.2990	−1.83622
$162$	0.845709	0.0664452
$163$	21.6489	1.69567	0.847835	−	0.530260i	$-0.177906\pi$
$163$	21.6489	1.69567	0.847835	+	0.530260i	$0.177906\pi$
$164$	4.33428	0.338450
$165$	−0.243406	−0.0189491
$166$	1.66064	0.128891
$167$	12.7447	0.986213	0.493107	−	0.869969i	$-0.335861\pi$
$167$	12.7447	0.986213	0.493107	+	0.869969i	$0.335861\pi$
$168$	7.55781	0.583098
$169$	7.67593	0.590456
$170$	−0.0571779	−0.00438534
$171$	−7.93383	−0.606715
$172$	6.28255	0.479040
$173$	12.8480	0.976812	0.488406	−	0.872617i	$-0.337579\pi$
$173$	12.8480	0.976812	0.488406	+	0.872617i	$0.337579\pi$
$174$	−2.42697	−0.183988
$175$	13.5907	1.02736
$176$	0.792774	0.0597576
$177$	6.87792	0.516976
$178$	−4.91631	−0.368493
$179$	−0.332088	−0.0248214	−0.0124107	−	0.999923i	$-0.503951\pi$
$179$	−0.332088	−0.0248214	−0.0124107	+	0.999923i	$0.503951\pi$
$180$	0.0868630	0.00647438
$181$	−5.12215	−0.380726	−0.190363	−	0.981714i	$-0.560967\pi$
$181$	−5.12215	−0.380726	−0.190363	+	0.981714i	$0.560967\pi$
$182$	−10.4622	−0.775509
$183$	7.55888	0.558768
$184$	−23.7900	−1.75382
$185$	−0.797285	−0.0586175
$186$	−1.33993	−0.0982485
$187$	3.60018	0.263271
$188$	5.83227	0.425362
$189$	−2.72063	−0.197897
$190$	0.453639	0.0329104
$191$	−10.6002	−0.767003	−0.383502	−	0.923540i	$-0.625282\pi$
$191$	−10.6002	−0.767003	−0.383502	+	0.923540i	$0.625282\pi$
$192$	4.41578	0.318682
$193$	−1.42635	−0.102671	−0.0513353	−	0.998681i	$-0.516348\pi$
$193$	−1.42635	−0.102671	−0.0513353	+	0.998681i	$0.516348\pi$
$194$	11.9059	0.854791
$195$	−0.307425	−0.0220152
$196$	−0.516245	−0.0368746
$197$	15.5119	1.10517	0.552587	−	0.833455i	$-0.313641\pi$
$197$	15.5119	1.10517	0.552587	+	0.833455i	$0.313641\pi$
$198$	3.04470	0.216378
$199$	9.44215	0.669337	0.334668	−	0.942336i	$-0.391376\pi$
$199$	9.44215	0.669337	0.334668	+	0.942336i	$0.391376\pi$
$200$	13.8771	0.981261
$201$	11.4867	0.810210
$202$	9.58431	0.674350
$203$	7.80750	0.547979
$204$	−1.28478	−0.0899524
$205$	0.228085	0.0159301
$206$	15.4398	1.07574
$207$	8.56382	0.595227
$208$	1.00129	0.0694267
$209$	−28.5632	−1.97576
$210$	0.155560	0.0107346
$211$	19.1781	1.32028	0.660138	−	0.751144i	$-0.270498\pi$
$211$	19.1781	1.32028	0.660138	+	0.751144i	$0.270498\pi$
$212$	16.2107	1.11335
$213$	−1.26257	−0.0865096
$214$	−7.49886	−0.512612
$215$	0.330609	0.0225474
$216$	−2.77796	−0.189017
$217$	4.31053	0.292618
$218$	14.1214	0.956423
$219$	−12.1528	−0.821208
$220$	0.312722	0.0210837
$221$	4.54708	0.305870
$222$	9.97303	0.669346
$223$	−18.5209	−1.24025	−0.620126	−	0.784502i	$-0.712919\pi$
$223$	−18.5209	−1.24025	−0.620126	+	0.784502i	$0.712919\pi$
$224$	−15.6223	−1.04381
$225$	−4.99543	−0.333029
$226$	8.67451	0.577020
$227$	2.20573	0.146399	0.0731996	−	0.997317i	$-0.476679\pi$
$227$	2.20573	0.146399	0.0731996	+	0.997317i	$0.476679\pi$
$228$	10.1932	0.675061
$229$	−15.1431	−1.00069	−0.500344	−	0.865827i	$-0.666793\pi$
$229$	−15.1431	−1.00069	−0.500344	+	0.865827i	$0.666793\pi$
$230$	−0.489661	−0.0322873
$231$	−9.79475	−0.644447
$232$	7.97205	0.523390
$233$	−14.2834	−0.935737	−0.467868	−	0.883798i	$-0.654978\pi$
$233$	−14.2834	−0.935737	−0.467868	+	0.883798i	$0.654978\pi$
$234$	3.84550	0.251389
$235$	0.306914	0.0200209
$236$	−8.83659	−0.575213
$237$	5.41143	0.351510
$238$	−2.30086	−0.149143
$239$	6.57780	0.425483	0.212741	−	0.977109i	$-0.431761\pi$
$239$	6.57780	0.425483	0.212741	+	0.977109i	$0.431761\pi$
$240$	−0.0148879	−0.000961008 0
$241$	3.13257	0.201786	0.100893	−	0.994897i	$-0.467830\pi$
$241$	3.13257	0.201786	0.100893	+	0.994897i	$0.467830\pi$
$242$	1.65868	0.106624
$243$	1.00000	0.0641500
$244$	−9.71147	−0.621713
$245$	−0.0271666	−0.00173561
$246$	−2.85305	−0.181904
$247$	−36.0757	−2.29544
$248$	4.40137	0.279487
$249$	1.96361	0.124439
$250$	0.571517	0.0361459
$251$	−21.1655	−1.33595	−0.667977	−	0.744182i	$-0.732840\pi$
$251$	−21.1655	−1.33595	−0.667977	+	0.744182i	$0.732840\pi$
$252$	3.49540	0.220189
$253$	30.8313	1.93835
$254$	−12.1189	−0.760410
$255$	−0.0676094	−0.00423386
$256$	−15.3857	−0.961605
$257$	−9.59364	−0.598435	−0.299217	−	0.954185i	$-0.596726\pi$
$257$	−9.59364	−0.598435	−0.299217	+	0.954185i	$0.596726\pi$
$258$	−4.13551	−0.257466
$259$	−32.0830	−1.99354
$260$	0.394973	0.0244952
$261$	−2.86974	−0.177633
$262$	10.8326	0.669237
$263$	0.710142	0.0437892	0.0218946	−	0.999760i	$-0.493030\pi$
$263$	0.710142	0.0437892	0.0218946	+	0.999760i	$0.493030\pi$
$264$	−10.0012	−0.615529
$265$	0.853061	0.0524031
$266$	18.2546	1.11926
$267$	−5.81324	−0.355765
$268$	−14.7579	−0.901480
$269$	32.0767	1.95575	0.977876	−	0.209184i	$-0.0670808\pi$
$269$	32.0767	1.95575	0.977876	+	0.209184i	$0.0670808\pi$
$270$	−0.0571779	−0.00347973
$271$	19.8130	1.20355	0.601776	−	0.798665i	$-0.294460\pi$
$271$	19.8130	1.20355	0.601776	+	0.798665i	$0.294460\pi$
$272$	0.220204	0.0133518
$273$	−12.3709	−0.748722
$274$	0.942288	0.0569257
$275$	−17.9844	−1.08450
$276$	−11.0026	−0.662279
$277$	14.3279	0.860881	0.430440	−	0.902619i	$-0.358358\pi$
$277$	14.3279	0.860881	0.430440	+	0.902619i	$0.358358\pi$
$278$	1.22527	0.0734868
$279$	−1.58439	−0.0948548
$280$	−0.510979	−0.0305368
$281$	14.8436	0.885496	0.442748	−	0.896646i	$-0.354004\pi$
$281$	14.8436	0.885496	0.442748	+	0.896646i	$0.354004\pi$
$282$	−3.83912	−0.228616
$283$	18.3056	1.08816	0.544078	−	0.839035i	$-0.316880\pi$
$283$	18.3056	1.08816	0.544078	+	0.839035i	$0.316880\pi$
$284$	1.62211	0.0962548
$285$	0.536401	0.0317737
$286$	13.8445	0.818643
$287$	9.17822	0.541773
$288$	5.74216	0.338360
$289$	1.00000	0.0588235
$290$	0.164086	0.00963545
$291$	14.0780	0.825265
$292$	15.6136	0.913716
$293$	23.2137	1.35616	0.678079	−	0.734989i	$-0.262813\pi$
$293$	23.2137	1.35616	0.678079	+	0.734989i	$0.262813\pi$
$294$	0.339820	0.0198187
$295$	−0.465012	−0.0270741
$296$	−32.7592	−1.90409
$297$	3.60018	0.208904
$298$	2.84768	0.164962
$299$	38.9404	2.25198
$300$	6.41801	0.370544
$301$	13.3038	0.766821
$302$	9.69747	0.558026
$303$	11.3329	0.651057
$304$	−1.74706	−0.100201
$305$	−0.511051	−0.0292627
$306$	0.845709	0.0483460
$307$	−30.5090	−1.74124	−0.870622	−	0.491953i	$-0.836283\pi$
$307$	−30.5090	−1.74124	−0.870622	+	0.491953i	$0.836283\pi$
$308$	12.5841	0.717043
$309$	18.2566	1.03858
$310$	0.0905919	0.00514527
$311$	22.3925	1.26976	0.634882	−	0.772609i	$-0.281049\pi$
$311$	22.3925	1.26976	0.634882	+	0.772609i	$0.281049\pi$
$312$	−12.6316	−0.715125
$313$	−17.1031	−0.966725	−0.483362	−	0.875420i	$-0.660585\pi$
$313$	−17.1031	−0.966725	−0.483362	+	0.875420i	$0.660585\pi$
$314$	0.845709	0.0477261
$315$	0.183940	0.0103638
$316$	−6.95248	−0.391107
$317$	−15.9722	−0.897089	−0.448544	−	0.893761i	$-0.648057\pi$
$317$	−15.9722	−0.897089	−0.448544	+	0.893761i	$0.648057\pi$
$318$	−10.6707	−0.598384
$319$	−10.3316	−0.578458
$320$	−0.298549	−0.0166894
$321$	−8.86696	−0.494905
$322$	−19.7041	−1.09807
$323$	−7.93383	−0.441450
$324$	−1.28478	−0.0713765
$325$	−22.7146	−1.25998
$326$	18.3086	1.01402
$327$	16.6977	0.923386
$328$	9.37164	0.517462
$329$	12.3504	0.680897
$330$	−0.205851	−0.0113317
$331$	−1.00861	−0.0554380	−0.0277190	−	0.999616i	$-0.508824\pi$
$331$	−1.00861	−0.0554380	−0.0277190	+	0.999616i	$0.508824\pi$
$332$	−2.52280	−0.138457
$333$	11.7925	0.646226
$334$	10.7783	0.589762
$335$	−0.776610	−0.0424307
$336$	−0.599094	−0.0326832
$337$	15.9942	0.871260	0.435630	−	0.900126i	$-0.356526\pi$
$337$	15.9942	0.871260	0.435630	+	0.900126i	$0.356526\pi$
$338$	6.49160	0.353096
$339$	10.2571	0.557089
$340$	0.0868630	0.00471081
$341$	−5.70408	−0.308893
$342$	−6.70971	−0.362819
$343$	17.9512	0.969274
$344$	13.5842	0.732412
$345$	−0.578995	−0.0311720
$346$	10.8656	0.584140
$347$	15.6542	0.840360	0.420180	−	0.907441i	$-0.361967\pi$
$347$	15.6542	0.840360	0.420180	+	0.907441i	$0.361967\pi$
$348$	3.68698	0.197643
$349$	11.1913	0.599059	0.299529	−	0.954087i	$-0.403170\pi$
$349$	11.1913	0.599059	0.299529	+	0.954087i	$0.403170\pi$
$350$	11.4938	0.614368
$351$	4.54708	0.242705
$352$	20.6728	1.10186
$353$	3.63531	0.193488	0.0967440	−	0.995309i	$-0.469157\pi$
$353$	3.63531	0.193488	0.0967440	+	0.995309i	$0.469157\pi$
$354$	5.81672	0.309155
$355$	0.0853613	0.00453051
$356$	7.46872	0.395841
$357$	−2.72063	−0.143991
$358$	−0.280850	−0.0148434
$359$	−27.5112	−1.45199	−0.725994	−	0.687701i	$-0.758620\pi$
$359$	−27.5112	−1.45199	−0.725994	+	0.687701i	$0.758620\pi$
$360$	0.187817	0.00989880
$361$	43.9456	2.31293
$362$	−4.33185	−0.227677
$363$	1.96129	0.102941
$364$	15.8939	0.833064
$365$	0.821641	0.0430067
$366$	6.39261	0.334147
$367$	−26.0936	−1.36207	−0.681037	−	0.732249i	$-0.738471\pi$
$367$	−26.0936	−1.36207	−0.681037	+	0.732249i	$0.738471\pi$
$368$	1.88579	0.0983036
$369$	−3.37357	−0.175621
$370$	−0.674271	−0.0350537
$371$	34.3275	1.78219
$372$	2.03558	0.105540
$373$	15.2289	0.788523	0.394261	−	0.918998i	$-0.371000\pi$
$373$	15.2289	0.788523	0.394261	+	0.918998i	$0.371000\pi$
$374$	3.04470	0.157438
$375$	0.675785	0.0348974
$376$	12.6106	0.650344
$377$	−13.0490	−0.672055
$378$	−2.30086	−0.118343
$379$	18.8058	0.965990	0.482995	−	0.875623i	$-0.339549\pi$
$379$	18.8058	0.965990	0.482995	+	0.875623i	$0.339549\pi$
$380$	−0.689156	−0.0353529
$381$	−14.3299	−0.734145
$382$	−8.96468	−0.458673
$383$	17.9410	0.916742	0.458371	−	0.888761i	$-0.348433\pi$
$383$	17.9410	0.916742	0.458371	+	0.888761i	$0.348433\pi$
$384$	−7.74985	−0.395483
$385$	0.662217	0.0337497
$386$	−1.20627	−0.0613977
$387$	−4.88999	−0.248572
$388$	−18.0870	−0.918230
$389$	−6.67710	−0.338542	−0.169271	−	0.985570i	$-0.554141\pi$
$389$	−6.67710	−0.338542	−0.169271	+	0.985570i	$0.554141\pi$
$390$	−0.259992	−0.0131652
$391$	8.56382	0.433091
$392$	−1.11623	−0.0563782
$393$	12.8088	0.646121
$394$	13.1185	0.660901
$395$	−0.365863	−0.0184086
$396$	−4.62543	−0.232436
$397$	8.38618	0.420890	0.210445	−	0.977606i	$-0.432509\pi$
$397$	8.38618	0.420890	0.210445	+	0.977606i	$0.432509\pi$
$398$	7.98531	0.400268
$399$	21.5850	1.08060
$400$	−1.10001	−0.0550007
$401$	21.9559	1.09642	0.548212	−	0.836340i	$-0.315309\pi$
$401$	21.9559	1.09642	0.548212	+	0.836340i	$0.315309\pi$
$402$	9.71441	0.484511
$403$	−7.20433	−0.358873
$404$	−14.5602	−0.724398
$405$	−0.0676094	−0.00335954
$406$	6.60288	0.327695
$407$	42.4552	2.10442
$408$	−2.77796	−0.137530
$409$	20.0643	0.992115	0.496058	−	0.868290i	$-0.334780\pi$
$409$	20.0643	0.992115	0.496058	+	0.868290i	$0.334780\pi$
$410$	0.192893	0.00952632
$411$	1.11420	0.0549594
$412$	−23.4557	−1.15558
$413$	−18.7123	−0.920770
$414$	7.24250	0.355949
$415$	−0.132759	−0.00651686
$416$	26.1100	1.28015
$417$	1.44881	0.0709484
$418$	−24.1561	−1.18152
$419$	−18.1461	−0.886493	−0.443246	−	0.896400i	$-0.646173\pi$
$419$	−18.1461	−0.886493	−0.443246	+	0.896400i	$0.646173\pi$
$420$	−0.236322	−0.0115313
$421$	25.4690	1.24128	0.620642	−	0.784094i	$-0.286872\pi$
$421$	25.4690	1.24128	0.620642	+	0.784094i	$0.286872\pi$
$422$	16.2191	0.789534
$423$	−4.53952	−0.220719
$424$	35.0509	1.70222
$425$	−4.99543	−0.242314
$426$	−1.06776	−0.0517333
$427$	−20.5649	−0.995205
$428$	11.3921	0.550656
$429$	16.3703	0.790365
$430$	0.279599	0.0134835
$431$	−3.84856	−0.185378	−0.0926892	−	0.995695i	$-0.529546\pi$
$431$	−3.84856	−0.185378	−0.0926892	+	0.995695i	$0.529546\pi$
$432$	0.220204	0.0105946
$433$	2.89189	0.138975	0.0694877	−	0.997583i	$-0.477864\pi$
$433$	2.89189	0.138975	0.0694877	+	0.997583i	$0.477864\pi$
$434$	3.64545	0.174987
$435$	0.194022	0.00930263
$436$	−21.4528	−1.02741
$437$	−67.9439	−3.25020
$438$	−10.2777	−0.491088
$439$	−19.8298	−0.946425	−0.473212	−	0.880948i	$-0.656906\pi$
$439$	−19.8298	−0.946425	−0.473212	+	0.880948i	$0.656906\pi$
$440$	0.676173	0.0322353
$441$	0.401817	0.0191341
$442$	3.84550	0.182912
$443$	−32.2679	−1.53309	−0.766547	−	0.642188i	$-0.778027\pi$
$443$	−32.2679	−1.53309	−0.766547	+	0.642188i	$0.778027\pi$
$444$	−15.1507	−0.719023
$445$	0.393030	0.0186314
$446$	−15.6633	−0.741679
$447$	3.36721	0.159264
$448$	−12.0137	−0.567594
$449$	−16.1139	−0.760462	−0.380231	−	0.924892i	$-0.624155\pi$
$449$	−16.1139	−0.760462	−0.380231	+	0.924892i	$0.624155\pi$
$450$	−4.22468	−0.199153
$451$	−12.1454	−0.571906
$452$	−13.1781	−0.619844
$453$	11.4667	0.538751
$454$	1.86540	0.0875477
$455$	0.836390	0.0392106
$456$	22.0399	1.03211
$457$	−11.9301	−0.558065	−0.279033	−	0.960282i	$-0.590014\pi$
$457$	−11.9301	−0.558065	−0.279033	+	0.960282i	$0.590014\pi$
$458$	−12.8067	−0.598417
$459$	1.00000	0.0466760
$460$	0.743879	0.0346835
$461$	−22.5631	−1.05087	−0.525433	−	0.850835i	$-0.676097\pi$
$461$	−22.5631	−1.05087	−0.525433	+	0.850835i	$0.676097\pi$
$462$	−8.28350	−0.385383
$463$	23.4874	1.09155	0.545777	−	0.837931i	$-0.316235\pi$
$463$	23.4874	1.09155	0.545777	+	0.837931i	$0.316235\pi$
$464$	−0.631929	−0.0293366
$465$	0.107119	0.00496755
$466$	−12.0796	−0.559576
$467$	4.46272	0.206510	0.103255	−	0.994655i	$-0.467074\pi$
$467$	4.46272	0.206510	0.103255	+	0.994655i	$0.467074\pi$
$468$	−5.84198	−0.270046
$469$	−31.2511	−1.44304
$470$	0.259560	0.0119726
$471$	1.00000	0.0460776
$472$	−19.1066	−0.879454
$473$	−17.6048	−0.809472
$474$	4.57649	0.210205
$475$	39.6329	1.81848
$476$	3.49540	0.160211
$477$	−12.6175	−0.577715
$478$	5.56291	0.254441
$479$	−23.7000	−1.08288	−0.541441	−	0.840739i	$-0.682121\pi$
$479$	−23.7000	−1.08288	−0.541441	+	0.840739i	$0.682121\pi$
$480$	−0.388224	−0.0177199
$481$	53.6215	2.44493
$482$	2.64924	0.120670
$483$	−23.2990	−1.06014
$484$	−2.51982	−0.114537
$485$	−0.951803	−0.0432191
$486$	0.845709	0.0383621
$487$	−42.7524	−1.93730	−0.968648	−	0.248436i	$-0.920084\pi$
$487$	−42.7524	−1.93730	−0.968648	+	0.248436i	$0.920084\pi$
$488$	−20.9983	−0.950548
$489$	21.6489	0.978996
$490$	−0.0229750	−0.00103791
$491$	36.0110	1.62515	0.812577	−	0.582853i	$-0.198064\pi$
$491$	36.0110	1.62515	0.812577	+	0.582853i	$0.198064\pi$
$492$	4.33428	0.195404
$493$	−2.86974	−0.129247
$494$	−30.5096	−1.37269
$495$	−0.243406	−0.0109403
$496$	−0.348889	−0.0156656
$497$	3.43497	0.154080
$498$	1.66064	0.0744152
$499$	−3.52160	−0.157649	−0.0788243	−	0.996889i	$-0.525117\pi$
$499$	−3.52160	−0.157649	−0.0788243	+	0.996889i	$0.525117\pi$
$500$	−0.868233	−0.0388285
$501$	12.7447	0.569390
$502$	−17.8999	−0.798910
$503$	−31.7215	−1.41439	−0.707195	−	0.707018i	$-0.750040\pi$
$503$	−31.7215	−1.41439	−0.707195	+	0.707018i	$0.750040\pi$
$504$	7.55781	0.336652
$505$	−0.766209	−0.0340959
$506$	26.0743	1.15914
$507$	7.67593	0.340900
$508$	18.4108	0.816845
$509$	17.1133	0.758533	0.379267	−	0.925287i	$-0.376176\pi$
$509$	17.1133	0.758533	0.379267	+	0.925287i	$0.376176\pi$
$510$	−0.0571779	−0.00253188
$511$	33.0632	1.46263
$512$	2.48789	0.109950
$513$	−7.93383	−0.350287
$514$	−8.11343	−0.357868
$515$	−1.23432	−0.0543907
$516$	6.28255	0.276574
$517$	−16.3431	−0.718769
$518$	−27.1329	−1.19215
$519$	12.8480	0.563963
$520$	0.854017	0.0374511
$521$	4.93923	0.216391	0.108196	−	0.994130i	$-0.465493\pi$
$521$	4.93923	0.216391	0.108196	+	0.994130i	$0.465493\pi$
$522$	−2.42697	−0.106225
$523$	−36.1733	−1.58175	−0.790874	−	0.611978i	$-0.790374\pi$
$523$	−36.1733	−1.58175	−0.790874	+	0.611978i	$0.790374\pi$
$524$	−16.4565	−0.718906
$525$	13.5907	0.593147
$526$	0.600573	0.0261862
$527$	−1.58439	−0.0690170
$528$	0.792774	0.0345011
$529$	50.3390	2.18865
$530$	0.721441	0.0313374
$531$	6.87792	0.298476
$532$	−27.7319	−1.20233
$533$	−15.3399	−0.664443
$534$	−4.91631	−0.212750
$535$	0.599490	0.0259182
$536$	−31.9097	−1.37829
$537$	−0.332088	−0.0143307
$538$	27.1276	1.16955
$539$	1.44661	0.0623100
$540$	0.0868630	0.00373799
$541$	32.8568	1.41263	0.706313	−	0.707900i	$-0.250357\pi$
$541$	32.8568	1.41263	0.706313	+	0.707900i	$0.250357\pi$
$542$	16.7560	0.719732
$543$	−5.12215	−0.219812
$544$	5.74216	0.246193
$545$	−1.12892	−0.0483578
$546$	−10.4622	−0.447740
$547$	14.8499	0.634937	0.317468	−	0.948269i	$-0.397167\pi$
$547$	14.8499	0.634937	0.317468	+	0.948269i	$0.397167\pi$
$548$	−1.43150	−0.0611505
$549$	7.55888	0.322605
$550$	−15.2096	−0.648539
$551$	22.7680	0.969951
$552$	−23.7900	−1.01257
$553$	−14.7225	−0.626064
$554$	12.1172	0.514812
$555$	−0.797285	−0.0338429
$556$	−1.86139	−0.0789407
$557$	−22.5229	−0.954326	−0.477163	−	0.878815i	$-0.658335\pi$
$557$	−22.5229	−0.954326	−0.477163	+	0.878815i	$0.658335\pi$
$558$	−1.33993	−0.0567238
$559$	−22.2352	−0.940448
$560$	0.0405044	0.00171162
$561$	3.60018	0.152000
$562$	12.5534	0.529532
$563$	14.5796	0.614455	0.307228	−	0.951636i	$-0.400599\pi$
$563$	14.5796	0.614455	0.307228	+	0.951636i	$0.400599\pi$
$564$	5.83227	0.245583
$565$	−0.693476	−0.0291747
$566$	15.4812	0.650724
$567$	−2.72063	−0.114256
$568$	3.50736	0.147166
$569$	−35.8263	−1.50192	−0.750959	−	0.660349i	$-0.770408\pi$
$569$	−35.8263	−1.50192	−0.750959	+	0.660349i	$0.770408\pi$
$570$	0.453639	0.0190009
$571$	14.7827	0.618636	0.309318	−	0.950959i	$-0.399899\pi$
$571$	14.7827	0.618636	0.309318	+	0.950959i	$0.399899\pi$
$572$	−21.0322	−0.879399
$573$	−10.6002	−0.442830
$574$	7.76210	0.323984
$575$	−42.7800	−1.78405
$576$	4.41578	0.183991
$577$	−27.3255	−1.13758	−0.568788	−	0.822484i	$-0.692588\pi$
$577$	−27.3255	−1.13758	−0.568788	+	0.822484i	$0.692588\pi$
$578$	0.845709	0.0351768
$579$	−1.42635	−0.0592769
$580$	−0.249274	−0.0103506
$581$	−5.34226	−0.221634
$582$	11.9059	0.493514
$583$	−45.4252	−1.88132
$584$	33.7600	1.39700
$585$	−0.307425	−0.0127105
$586$	19.6320	0.810991
$587$	−12.2837	−0.507001	−0.253501	−	0.967335i	$-0.581582\pi$
$587$	−12.2837	−0.507001	−0.253501	+	0.967335i	$0.581582\pi$
$588$	−0.516245	−0.0212896
$589$	12.5703	0.517948
$590$	−0.393265	−0.0161905
$591$	15.5119	0.638072
$592$	2.59676	0.106726
$593$	−19.7061	−0.809232	−0.404616	−	0.914487i	$-0.632595\pi$
$593$	−19.7061	−0.809232	−0.404616	+	0.914487i	$0.632595\pi$
$594$	3.04470	0.124926
$595$	0.183940	0.00754081
$596$	−4.32611	−0.177205
$597$	9.44215	0.386442
$598$	32.9322	1.34670
$599$	−20.9251	−0.854976	−0.427488	−	0.904021i	$-0.640601\pi$
$599$	−20.9251	−0.854976	−0.427488	+	0.904021i	$0.640601\pi$
$600$	13.8771	0.566531
$601$	−5.16273	−0.210592	−0.105296	−	0.994441i	$-0.533579\pi$
$601$	−5.16273	−0.210592	−0.105296	+	0.994441i	$0.533579\pi$
$602$	11.2512	0.458564
$603$	11.4867	0.467775
$604$	−14.7321	−0.599441
$605$	−0.132601	−0.00539101
$606$	9.58431	0.389336
$607$	−29.7349	−1.20690	−0.603451	−	0.797400i	$-0.706208\pi$
$607$	−29.7349	−1.20690	−0.603451	+	0.797400i	$0.706208\pi$
$608$	−45.5573	−1.84759
$609$	7.80750	0.316376
$610$	−0.432200	−0.0174993
$611$	−20.6416	−0.835069
$612$	−1.28478	−0.0519340
$613$	2.84644	0.114967	0.0574834	−	0.998346i	$-0.481692\pi$
$613$	2.84644	0.114967	0.0574834	+	0.998346i	$0.481692\pi$
$614$	−25.8018	−1.04127
$615$	0.228085	0.00919726
$616$	27.2095	1.09630
$617$	18.4915	0.744439	0.372220	−	0.928145i	$-0.378597\pi$
$617$	18.4915	0.744439	0.372220	+	0.928145i	$0.378597\pi$
$618$	15.4398	0.621080
$619$	−20.6235	−0.828927	−0.414463	−	0.910066i	$-0.636031\pi$
$619$	−20.6235	−0.828927	−0.414463	+	0.910066i	$0.636031\pi$
$620$	−0.137625	−0.00552714
$621$	8.56382	0.343654
$622$	18.9376	0.759327
$623$	15.8157	0.633642
$624$	1.00129	0.0400835
$625$	24.9315	0.997258
$626$	−14.4642	−0.578108
$627$	−28.5632	−1.14070
$628$	−1.28478	−0.0512682
$629$	11.7925	0.470198
$630$	0.155560	0.00619765
$631$	−31.5894	−1.25755	−0.628776	−	0.777586i	$-0.716444\pi$
$631$	−31.5894	−1.25755	−0.628776	+	0.777586i	$0.716444\pi$
$632$	−15.0328	−0.597971
$633$	19.1781	0.762262
$634$	−13.5078	−0.536465
$635$	0.968838	0.0384472
$636$	16.2107	0.642794
$637$	1.82709	0.0723920
$638$	−8.73752	−0.345922
$639$	−1.26257	−0.0499463
$640$	0.523963	0.0207114
$641$	49.8370	1.96844	0.984222	−	0.176936i	$-0.0566186\pi$
$641$	49.8370	1.96844	0.984222	+	0.176936i	$0.0566186\pi$
$642$	−7.49886	−0.295956
$643$	8.12348	0.320359	0.160179	−	0.987088i	$-0.448793\pi$
$643$	8.12348	0.320359	0.160179	+	0.987088i	$0.448793\pi$
$644$	29.9340	1.17956
$645$	0.330609	0.0130177
$646$	−6.70971	−0.263990
$647$	36.3668	1.42973	0.714863	−	0.699264i	$-0.246489\pi$
$647$	36.3668	1.42973	0.714863	+	0.699264i	$0.246489\pi$
$648$	−2.77796	−0.109129
$649$	24.7617	0.971984
$650$	−19.2099	−0.753476
$651$	4.31053	0.168943
$652$	−27.8140	−1.08928
$653$	11.7569	0.460084	0.230042	−	0.973181i	$-0.426114\pi$
$653$	11.7569	0.460084	0.230042	+	0.973181i	$0.426114\pi$
$654$	14.1214	0.552191
$655$	−0.865998	−0.0338374
$656$	−0.742873	−0.0290043
$657$	−12.1528	−0.474125
$658$	10.4448	0.407181
$659$	−43.0595	−1.67736	−0.838679	−	0.544625i	$-0.816672\pi$
$659$	−43.0595	−1.67736	−0.838679	+	0.544625i	$0.816672\pi$
$660$	0.312722	0.0121727
$661$	−34.7731	−1.35252	−0.676259	−	0.736664i	$-0.736400\pi$
$661$	−34.7731	−1.35252	−0.676259	+	0.736664i	$0.736400\pi$
$662$	−0.852987	−0.0331523
$663$	4.54708	0.176594
$664$	−5.45484	−0.211689
$665$	−1.45935	−0.0565911
$666$	9.97303	0.386447
$667$	−24.5760	−0.951585
$668$	−16.3741	−0.633532
$669$	−18.5209	−0.716060
$670$	−0.656786	−0.0253738
$671$	27.2133	1.05056
$672$	−15.6223	−0.602642
$673$	−10.8498	−0.418230	−0.209115	−	0.977891i	$-0.567058\pi$
$673$	−10.8498	−0.418230	−0.209115	+	0.977891i	$0.567058\pi$
$674$	13.5264	0.521019
$675$	−4.99543	−0.192274
$676$	−9.86185	−0.379302
$677$	−28.1123	−1.08044	−0.540221	−	0.841523i	$-0.681659\pi$
$677$	−28.1123	−1.08044	−0.540221	+	0.841523i	$0.681659\pi$
$678$	8.67451	0.333143
$679$	−38.3009	−1.46985
$680$	0.187817	0.00720243
$681$	2.20573	0.0845236
$682$	−4.82399	−0.184720
$683$	30.4014	1.16328	0.581639	−	0.813447i	$-0.302412\pi$
$683$	30.4014	1.16328	0.581639	+	0.813447i	$0.302412\pi$
$684$	10.1932	0.389746
$685$	−0.0753303	−0.00287822
$686$	15.1815	0.579632
$687$	−15.1431	−0.577747
$688$	−1.07680	−0.0410525
$689$	−57.3727	−2.18573
$690$	−0.489661	−0.0186411
$691$	−16.0667	−0.611207	−0.305604	−	0.952159i	$-0.598858\pi$
$691$	−16.0667	−0.611207	−0.305604	+	0.952159i	$0.598858\pi$
$692$	−16.5068	−0.627492
$693$	−9.79475	−0.372072
$694$	13.2389	0.502541
$695$	−0.0979530	−0.00371557
$696$	7.97205	0.302180
$697$	−3.37357	−0.127783
$698$	9.46461	0.358241
$699$	−14.2834	−0.540248
$700$	−17.4610	−0.659964
$701$	10.6335	0.401621	0.200811	−	0.979630i	$-0.435642\pi$
$701$	10.6335	0.401621	0.200811	+	0.979630i	$0.435642\pi$
$702$	3.84550	0.145139
$703$	−93.5597	−3.52867
$704$	15.8976	0.599164
$705$	0.306914	0.0115591
$706$	3.07441	0.115707
$707$	−30.8325	−1.15958
$708$	−8.83659	−0.332099
$709$	−15.6598	−0.588117	−0.294059	−	0.955787i	$-0.595006\pi$
$709$	−15.6598	−0.588117	−0.294059	+	0.955787i	$0.595006\pi$
$710$	0.0721908	0.00270927
$711$	5.41143	0.202944
$712$	16.1490	0.605209
$713$	−13.5684	−0.508141
$714$	−2.30086	−0.0861075
$715$	−1.10679	−0.0413914
$716$	0.426659	0.0159450
$717$	6.57780	0.245653
$718$	−23.2665	−0.868298
$719$	−21.7412	−0.810810	−0.405405	−	0.914137i	$-0.632870\pi$
$719$	−21.7412	−0.810810	−0.405405	+	0.914137i	$0.632870\pi$
$720$	−0.0148879	−0.000554838 0
$721$	−49.6695	−1.84979
$722$	37.1652	1.38314
$723$	3.13257	0.116501
$724$	6.58082	0.244574
$725$	14.3356	0.532411
$726$	1.65868	0.0615593
$727$	34.3241	1.27301	0.636504	−	0.771273i	$-0.280380\pi$
$727$	34.3241	1.27301	0.636504	+	0.771273i	$0.280380\pi$
$728$	34.3659	1.27369
$729$	1.00000	0.0370370
$730$	0.694869	0.0257183
$731$	−4.88999	−0.180863
$732$	−9.71147	−0.358946
$733$	10.3988	0.384089	0.192045	−	0.981386i	$-0.438488\pi$
$733$	10.3988	0.384089	0.192045	+	0.981386i	$0.438488\pi$
$734$	−22.0676	−0.814529
$735$	−0.0271666	−0.00100205
$736$	49.1748	1.81261
$737$	41.3542	1.52330
$738$	−2.85305	−0.105022
$739$	16.9984	0.625297	0.312648	−	0.949869i	$-0.398784\pi$
$739$	16.9984	0.625297	0.312648	+	0.949869i	$0.398784\pi$
$740$	1.02433	0.0376552
$741$	−36.0757	−1.32528
$742$	29.0311	1.06576
$743$	−9.93833	−0.364602	−0.182301	−	0.983243i	$-0.558355\pi$
$743$	−9.93833	−0.364602	−0.182301	+	0.983243i	$0.558355\pi$
$744$	4.40137	0.161362
$745$	−0.227655	−0.00834064
$746$	12.8792	0.471542
$747$	1.96361	0.0718448
$748$	−4.62543	−0.169122
$749$	24.1237	0.881460
$750$	0.571517	0.0208689
$751$	−22.2959	−0.813587	−0.406794	−	0.913520i	$-0.633353\pi$
$751$	−22.2959	−0.813587	−0.406794	+	0.913520i	$0.633353\pi$
$752$	−0.999622	−0.0364525
$753$	−21.1655	−0.771314
$754$	−11.0356	−0.401893
$755$	−0.775255	−0.0282144
$756$	3.49540	0.127126
$757$	17.5436	0.637634	0.318817	−	0.947816i	$-0.396714\pi$
$757$	17.5436	0.637634	0.318817	+	0.947816i	$0.396714\pi$
$758$	15.9042	0.577668
$759$	30.8313	1.11910
$760$	−1.49010	−0.0540517
$761$	22.6549	0.821238	0.410619	−	0.911807i	$-0.365313\pi$
$761$	22.6549	0.821238	0.410619	+	0.911807i	$0.365313\pi$
$762$	−12.1189	−0.439023
$763$	−45.4283	−1.64461
$764$	13.6189	0.492714
$765$	−0.0676094	−0.00244442
$766$	15.1729	0.548218
$767$	31.2745	1.12926
$768$	−15.3857	−0.555183
$769$	−33.2206	−1.19797	−0.598983	−	0.800762i	$-0.704428\pi$
$769$	−33.2206	−1.19797	−0.598983	+	0.800762i	$0.704428\pi$
$770$	0.560043	0.0201825
$771$	−9.59364	−0.345507
$772$	1.83254	0.0659544
$773$	35.9824	1.29420	0.647099	−	0.762406i	$-0.275982\pi$
$773$	35.9824	1.29420	0.647099	+	0.762406i	$0.275982\pi$
$774$	−4.13551	−0.148648
$775$	7.91469	0.284304
$776$	−39.1081	−1.40390
$777$	−32.0830	−1.15097
$778$	−5.64688	−0.202451
$779$	26.7653	0.958966
$780$	0.394973	0.0141423
$781$	−4.54546	−0.162649
$782$	7.24250	0.258991
$783$	−2.86974	−0.102556
$784$	0.0884817	0.00316006
$785$	−0.0676094	−0.00241308
$786$	10.8326	0.386384
$787$	−2.20781	−0.0787000	−0.0393500	−	0.999225i	$-0.512529\pi$
$787$	−2.20781	−0.0787000	−0.0393500	+	0.999225i	$0.512529\pi$
$788$	−19.9293	−0.709951
$789$	0.710142	0.0252817
$790$	−0.309414	−0.0110085
$791$	−27.9057	−0.992213
$792$	−10.0012	−0.355376
$793$	34.3708	1.22054
$794$	7.09227	0.251695
$795$	0.853061	0.0302549
$796$	−12.1311	−0.429974
$797$	−16.8741	−0.597712	−0.298856	−	0.954298i	$-0.596605\pi$
$797$	−16.8741	−0.597712	−0.298856	+	0.954298i	$0.596605\pi$
$798$	18.2546	0.646206
$799$	−4.53952	−0.160597
$800$	−28.6845	−1.01415
$801$	−5.81324	−0.205401
$802$	18.5683	0.655668
$803$	−43.7521	−1.54398
$804$	−14.7579	−0.520470
$805$	1.57523	0.0555196
$806$	−6.09277	−0.214609
$807$	32.0767	1.12915
$808$	−31.4823	−1.10754
$809$	3.11208	0.109415	0.0547074	−	0.998502i	$-0.482577\pi$
$809$	3.11208	0.109415	0.0547074	+	0.998502i	$0.482577\pi$
$810$	−0.0571779	−0.00200903
$811$	−7.59672	−0.266757	−0.133378	−	0.991065i	$-0.542583\pi$
$811$	−7.59672	−0.266757	−0.133378	+	0.991065i	$0.542583\pi$
$812$	−10.0309	−0.352016
$813$	19.8130	0.694871
$814$	35.9047	1.25846
$815$	−1.46367	−0.0512700
$816$	0.220204	0.00770869
$817$	38.7963	1.35731
$818$	16.9686	0.593291
$819$	−12.3709	−0.432275
$820$	−0.293038	−0.0102333
$821$	15.0471	0.525149	0.262574	−	0.964912i	$-0.415428\pi$
$821$	15.0471	0.525149	0.262574	+	0.964912i	$0.415428\pi$
$822$	0.942288	0.0328661
$823$	−49.5353	−1.72669	−0.863347	−	0.504611i	$-0.831636\pi$
$823$	−49.5353	−1.72669	−0.863347	+	0.504611i	$0.831636\pi$
$824$	−50.7163	−1.76679
$825$	−17.9844	−0.626138
$826$	−15.8251	−0.550627
$827$	−50.9287	−1.77097	−0.885483	−	0.464672i	$-0.846172\pi$
$827$	−50.9287	−1.77097	−0.885483	+	0.464672i	$0.846172\pi$
$828$	−11.0026	−0.382367
$829$	26.1306	0.907552	0.453776	−	0.891116i	$-0.350077\pi$
$829$	26.1306	0.907552	0.453776	+	0.891116i	$0.350077\pi$
$830$	−0.112275	−0.00389713
$831$	14.3279	0.497030
$832$	20.0789	0.696111
$833$	0.401817	0.0139221
$834$	1.22527	0.0424276
$835$	−0.861660	−0.0298190
$836$	36.6973	1.26920
$837$	−1.58439	−0.0547644
$838$	−15.3463	−0.530128
$839$	54.6356	1.88623	0.943116	−	0.332465i	$-0.107880\pi$
$839$	54.6356	1.88623	0.943116	+	0.332465i	$0.107880\pi$
$840$	−0.510979	−0.0176304
$841$	−20.7646	−0.716020
$842$	21.5394	0.742296
$843$	14.8436	0.511242
$844$	−24.6396	−0.848130
$845$	−0.518965	−0.0178529
$846$	−3.83912	−0.131991
$847$	−5.33593	−0.183345
$848$	−2.77842	−0.0954114
$849$	18.3056	0.628247
$850$	−4.22468	−0.144905
$851$	100.989	3.46186
$852$	1.62211	0.0555727
$853$	−35.0704	−1.20079	−0.600393	−	0.799705i	$-0.704989\pi$
$853$	−35.0704	−1.20079	−0.600393	+	0.799705i	$0.704989\pi$
$854$	−17.3919	−0.595139
$855$	0.536401	0.0183445
$856$	24.6321	0.841907
$857$	−12.9917	−0.443790	−0.221895	−	0.975071i	$-0.571224\pi$
$857$	−12.9917	−0.443790	−0.221895	+	0.975071i	$0.571224\pi$
$858$	13.8445	0.472644
$859$	−4.60616	−0.157160	−0.0785801	−	0.996908i	$-0.525039\pi$
$859$	−4.60616	−0.157160	−0.0785801	+	0.996908i	$0.525039\pi$
$860$	−0.424759	−0.0144842
$861$	9.17822	0.312793
$862$	−3.25476	−0.110858
$863$	−42.3752	−1.44247	−0.721235	−	0.692691i	$-0.756425\pi$
$863$	−42.3752	−1.44247	−0.721235	+	0.692691i	$0.756425\pi$
$864$	5.74216	0.195352
$865$	−0.868642	−0.0295347
$866$	2.44570	0.0831081
$867$	1.00000	0.0339618
$868$	−5.53807	−0.187974
$869$	19.4821	0.660885
$870$	0.164086	0.00556303
$871$	52.2310	1.76978
$872$	−46.3857	−1.57082
$873$	14.0780	0.476467
$874$	−57.4607	−1.94364
$875$	−1.83856	−0.0621547
$876$	15.6136	0.527534
$877$	9.14107	0.308672	0.154336	−	0.988018i	$-0.450676\pi$
$877$	9.14107	0.308672	0.154336	+	0.988018i	$0.450676\pi$
$878$	−16.7702	−0.565968
$879$	23.2137	0.782978
$880$	−0.0535990	−0.00180682
$881$	20.0387	0.675123	0.337561	−	0.941304i	$-0.390398\pi$
$881$	20.0387	0.675123	0.337561	+	0.941304i	$0.390398\pi$
$882$	0.339820	0.0114423
$883$	−19.7565	−0.664858	−0.332429	−	0.943128i	$-0.607868\pi$
$883$	−19.7565	−0.664858	−0.332429	+	0.943128i	$0.607868\pi$
$884$	−5.84198	−0.196487
$885$	−0.465012	−0.0156312
$886$	−27.2893	−0.916800
$887$	51.3233	1.72327	0.861633	−	0.507532i	$-0.169442\pi$
$887$	51.3233	1.72327	0.861633	+	0.507532i	$0.169442\pi$
$888$	−32.7592	−1.09933
$889$	38.9864	1.30756
$890$	0.332389	0.0111417
$891$	3.60018	0.120611
$892$	23.7952	0.796723
$893$	36.0158	1.20522
$894$	2.84768	0.0952407
$895$	0.0224523	0.000750497 0
$896$	21.0845	0.704382
$897$	38.9404	1.30018
$898$	−13.6277	−0.454761
$899$	4.54679	0.151644
$900$	6.41801	0.213934
$901$	−12.6175	−0.420349
$902$	−10.2715	−0.342004
$903$	13.3038	0.442724
$904$	−28.4938	−0.947691
$905$	0.346305	0.0115116
$906$	9.69747	0.322177
$907$	9.51449	0.315923	0.157962	−	0.987445i	$-0.449508\pi$
$907$	9.51449	0.315923	0.157962	+	0.987445i	$0.449508\pi$
$908$	−2.83387	−0.0940451
$909$	11.3329	0.375888
$910$	0.707342	0.0234482
$911$	40.2448	1.33337	0.666684	−	0.745340i	$-0.267713\pi$
$911$	40.2448	1.33337	0.666684	+	0.745340i	$0.267713\pi$
$912$	−1.74706	−0.0578510
$913$	7.06935	0.233961
$914$	−10.0894	−0.333727
$915$	−0.511051	−0.0168948
$916$	19.4556	0.642830
$917$	−34.8481	−1.15079
$918$	0.845709	0.0279126
$919$	10.8992	0.359532	0.179766	−	0.983709i	$-0.442466\pi$
$919$	10.8992	0.359532	0.179766	+	0.983709i	$0.442466\pi$
$920$	1.60843	0.0530283
$921$	−30.5090	−1.00531
$922$	−19.0818	−0.628425
$923$	−5.74099	−0.188967
$924$	12.5841	0.413985
$925$	−58.9087	−1.93690
$926$	19.8635	0.652756
$927$	18.2566	0.599627
$928$	−16.4785	−0.540934
$929$	−6.16555	−0.202285	−0.101143	−	0.994872i	$-0.532250\pi$
$929$	−6.16555	−0.202285	−0.101143	+	0.994872i	$0.532250\pi$
$930$	0.0905919	0.00297062
$931$	−3.18794	−0.104481
$932$	18.3510	0.601106
$933$	22.3925	0.733099
$934$	3.77416	0.123494
$935$	−0.243406	−0.00796023
$936$	−12.6316	−0.412878
$937$	26.4229	0.863199	0.431599	−	0.902065i	$-0.357949\pi$
$937$	26.4229	0.863199	0.431599	+	0.902065i	$0.357949\pi$
$938$	−26.4293	−0.862947
$939$	−17.1031	−0.558139
$940$	−0.394317	−0.0128612
$941$	−18.6631	−0.608401	−0.304201	−	0.952608i	$-0.598389\pi$
$941$	−18.6631	−0.608401	−0.304201	+	0.952608i	$0.598389\pi$
$942$	0.845709	0.0275547
$943$	−28.8906	−0.940808
$944$	1.51455	0.0492943
$945$	0.183940	0.00598357
$946$	−14.8886	−0.484069
$947$	36.0058	1.17003	0.585016	−	0.811022i	$-0.301088\pi$
$947$	36.0058	1.17003	0.585016	+	0.811022i	$0.301088\pi$
$948$	−6.95248	−0.225806
$949$	−55.2596	−1.79380
$950$	33.5179	1.08746
$951$	−15.9722	−0.517934
$952$	7.55781	0.244950
$953$	20.7084	0.670810	0.335405	−	0.942074i	$-0.391127\pi$
$953$	20.7084	0.670810	0.335405	+	0.942074i	$0.391127\pi$
$954$	−10.6707	−0.345477
$955$	0.716673	0.0231910
$956$	−8.45101	−0.273325
$957$	−10.3316	−0.333973
$958$	−20.0433	−0.647570
$959$	−3.03132	−0.0978864
$960$	−0.298549	−0.00963561
$961$	−28.4897	−0.919023
$962$	45.3482	1.46208
$963$	−8.86696	−0.285734
$964$	−4.02465	−0.129625
$965$	0.0964344	0.00310433
$966$	−19.7041	−0.633971
$967$	−60.5510	−1.94719	−0.973594	−	0.228288i	$-0.926687\pi$
$967$	−60.5510	−1.94719	−0.973594	+	0.228288i	$0.926687\pi$
$968$	−5.44838	−0.175118
$969$	−7.93383	−0.254871
$970$	−0.804948	−0.0258453
$971$	−14.5629	−0.467347	−0.233673	−	0.972315i	$-0.575075\pi$
$971$	−14.5629	−0.467347	−0.233673	+	0.972315i	$0.575075\pi$
$972$	−1.28478	−0.0412092
$973$	−3.94167	−0.126364
$974$	−36.1561	−1.15852
$975$	−22.7146	−0.727450
$976$	1.66450	0.0532792
$977$	−3.35587	−0.107364	−0.0536819	−	0.998558i	$-0.517096\pi$
$977$	−3.35587	−0.107364	−0.0536819	+	0.998558i	$0.517096\pi$
$978$	18.3086	0.585446
$979$	−20.9287	−0.668885
$980$	0.0349030	0.00111493
$981$	16.6977	0.533117
$982$	30.4548	0.971853
$983$	−7.26365	−0.231675	−0.115837	−	0.993268i	$-0.536955\pi$
$983$	−7.26365	−0.231675	−0.115837	+	0.993268i	$0.536955\pi$
$984$	9.37164	0.298757
$985$	−1.04875	−0.0334159
$986$	−2.42697	−0.0772904
$987$	12.3504	0.393116
$988$	46.3493	1.47457
$989$	−41.8770	−1.33161
$990$	−0.205851	−0.00654236
$991$	25.3317	0.804688	0.402344	−	0.915489i	$-0.368196\pi$
$991$	25.3317	0.804688	0.402344	+	0.915489i	$0.368196\pi$
$992$	−9.09780	−0.288855
$993$	−1.00861	−0.0320071
$994$	2.90499	0.0921405
$995$	−0.638378	−0.0202380
$996$	−2.52280	−0.0799380
$997$	−3.88720	−0.123109	−0.0615544	−	0.998104i	$-0.519606\pi$
$997$	−3.88720	−0.123109	−0.0615544	+	0.998104i	$0.519606\pi$
$998$	−2.97825	−0.0942749
$999$	11.7925	0.373099

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.i.1.37	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.i.1.37	✓	63	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q+0.845709 q^{2} +1.00000 q^{3} -1.28478 q^{4} -0.0676094 q^{5} +0.845709 q^{6} -2.72063 q^{7} -2.77796 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.845709 q^{2} +1.00000 q^{3} -1.28478 q^{4} -0.0676094 q^{5} +0.845709 q^{6} -2.72063 q^{7} -2.77796 q^{8} +1.00000 q^{9} -0.0571779 q^{10} +3.60018 q^{11} -1.28478 q^{12} +4.54708 q^{13} -2.30086 q^{14} -0.0676094 q^{15} +0.220204 q^{16} +1.00000 q^{17} +0.845709 q^{18} -7.93383 q^{19} +0.0868630 q^{20} -2.72063 q^{21} +3.04470 q^{22} +8.56382 q^{23} -2.77796 q^{24} -4.99543 q^{25} +3.84550 q^{26} +1.00000 q^{27} +3.49540 q^{28} -2.86974 q^{29} -0.0571779 q^{30} -1.58439 q^{31} +5.74216 q^{32} +3.60018 q^{33} +0.845709 q^{34} +0.183940 q^{35} -1.28478 q^{36} +11.7925 q^{37} -6.70971 q^{38} +4.54708 q^{39} +0.187817 q^{40} -3.37357 q^{41} -2.30086 q^{42} -4.88999 q^{43} -4.62543 q^{44} -0.0676094 q^{45} +7.24250 q^{46} -4.53952 q^{47} +0.220204 q^{48} +0.401817 q^{49} -4.22468 q^{50} +1.00000 q^{51} -5.84198 q^{52} -12.6175 q^{53} +0.845709 q^{54} -0.243406 q^{55} +7.55781 q^{56} -7.93383 q^{57} -2.42697 q^{58} +6.87792 q^{59} +0.0868630 q^{60} +7.55888 q^{61} -1.33993 q^{62} -2.72063 q^{63} +4.41578 q^{64} -0.307425 q^{65} +3.04470 q^{66} +11.4867 q^{67} -1.28478 q^{68} +8.56382 q^{69} +0.155560 q^{70} -1.26257 q^{71} -2.77796 q^{72} -12.1528 q^{73} +9.97303 q^{74} -4.99543 q^{75} +10.1932 q^{76} -9.79475 q^{77} +3.84550 q^{78} +5.41143 q^{79} -0.0148879 q^{80} +1.00000 q^{81} -2.85305 q^{82} +1.96361 q^{83} +3.49540 q^{84} -0.0676094 q^{85} -4.13551 q^{86} -2.86974 q^{87} -10.0012 q^{88} -5.81324 q^{89} -0.0571779 q^{90} -12.3709 q^{91} -11.0026 q^{92} -1.58439 q^{93} -3.83912 q^{94} +0.536401 q^{95} +5.74216 q^{96} +14.0780 q^{97} +0.339820 q^{98} +3.60018 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q + 10 q^{2} + 63 q^{3} + 70 q^{4} + 19 q^{5} + 10 q^{6} + 11 q^{7} + 27 q^{8} + 63 q^{9}+O(q^{10}) \) 63 * q + 10 * q^2 + 63 * q^3 + 70 * q^4 + 19 * q^5 + 10 * q^6 + 11 * q^7 + 27 * q^8 + 63 * q^9	\( 63 q + 10 q^{2} + 63 q^{3} + 70 q^{4} + 19 q^{5} + 10 q^{6} + 11 q^{7} + 27 q^{8} + 63 q^{9} + 4 q^{10} + 23 q^{11} + 70 q^{12} + 10 q^{13} + 18 q^{14} + 19 q^{15} + 72 q^{16} + 63 q^{17} + 10 q^{18} + 6 q^{19} + 48 q^{20} + 11 q^{21} + 21 q^{22} + 44 q^{23} + 27 q^{24} + 110 q^{25} + 41 q^{26} + 63 q^{27} + 26 q^{28} + 35 q^{29} + 4 q^{30} + q^{31} + 54 q^{32} + 23 q^{33} + 10 q^{34} + 47 q^{35} + 70 q^{36} + 40 q^{37} + 38 q^{38} + 10 q^{39} - 10 q^{40} + 35 q^{41} + 18 q^{42} + 27 q^{43} + 46 q^{44} + 19 q^{45} + 8 q^{46} + 29 q^{47} + 72 q^{48} + 114 q^{49} + 27 q^{50} + 63 q^{51} - q^{52} + 75 q^{53} + 10 q^{54} + 5 q^{55} + 24 q^{56} + 6 q^{57} + 41 q^{58} + 105 q^{59} + 48 q^{60} + 5 q^{61} + 22 q^{62} + 11 q^{63} + 61 q^{64} + 49 q^{65} + 21 q^{66} + 4 q^{67} + 70 q^{68} + 44 q^{69} - 16 q^{70} + 16 q^{71} + 27 q^{72} + 39 q^{73} + 54 q^{74} + 110 q^{75} + 6 q^{76} + 88 q^{77} + 41 q^{78} + 16 q^{79} + 102 q^{80} + 63 q^{81} - 29 q^{82} + 73 q^{83} + 26 q^{84} + 19 q^{85} + 46 q^{86} + 35 q^{87} + 18 q^{88} + 88 q^{89} + 4 q^{90} - 15 q^{91} + 110 q^{92} + q^{93} - 8 q^{94} + 28 q^{95} + 54 q^{96} + 70 q^{97} + 33 q^{98} + 23 q^{99}+O(q^{100}) \) 63 * q + 10 * q^2 + 63 * q^3 + 70 * q^4 + 19 * q^5 + 10 * q^6 + 11 * q^7 + 27 * q^8 + 63 * q^9 + 4 * q^10 + 23 * q^11 + 70 * q^12 + 10 * q^13 + 18 * q^14 + 19 * q^15 + 72 * q^16 + 63 * q^17 + 10 * q^18 + 6 * q^19 + 48 * q^20 + 11 * q^21 + 21 * q^22 + 44 * q^23 + 27 * q^24 + 110 * q^25 + 41 * q^26 + 63 * q^27 + 26 * q^28 + 35 * q^29 + 4 * q^30 + q^31 + 54 * q^32 + 23 * q^33 + 10 * q^34 + 47 * q^35 + 70 * q^36 + 40 * q^37 + 38 * q^38 + 10 * q^39 - 10 * q^40 + 35 * q^41 + 18 * q^42 + 27 * q^43 + 46 * q^44 + 19 * q^45 + 8 * q^46 + 29 * q^47 + 72 * q^48 + 114 * q^49 + 27 * q^50 + 63 * q^51 - q^52 + 75 * q^53 + 10 * q^54 + 5 * q^55 + 24 * q^56 + 6 * q^57 + 41 * q^58 + 105 * q^59 + 48 * q^60 + 5 * q^61 + 22 * q^62 + 11 * q^63 + 61 * q^64 + 49 * q^65 + 21 * q^66 + 4 * q^67 + 70 * q^68 + 44 * q^69 - 16 * q^70 + 16 * q^71 + 27 * q^72 + 39 * q^73 + 54 * q^74 + 110 * q^75 + 6 * q^76 + 88 * q^77 + 41 * q^78 + 16 * q^79 + 102 * q^80 + 63 * q^81 - 29 * q^82 + 73 * q^83 + 26 * q^84 + 19 * q^85 + 46 * q^86 + 35 * q^87 + 18 * q^88 + 88 * q^89 + 4 * q^90 - 15 * q^91 + 110 * q^92 + q^93 - 8 * q^94 + 28 * q^95 + 54 * q^96 + 70 * q^97 + 33 * q^98 + 23 * q^99

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