LMFDB - Embedded newform 8015.2.a.n.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8015,2,Mod(1,8015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8015, 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("8015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8015 = 5 \cdot 7 \cdot 229 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8015.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:	$64.0000972201$
Analytic rank:	$0$
Dimension:	$68$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.36878 q^{2} +0.541505 q^{3} -0.126449 q^{4} -1.00000 q^{5} -0.741199 q^{6} +1.00000 q^{7} +2.91064 q^{8} -2.70677 q^{9} +O(q^{10})$	\(q-1.36878 q^{2} +0.541505 q^{3} -0.126449 q^{4} -1.00000 q^{5} -0.741199 q^{6} +1.00000 q^{7} +2.91064 q^{8} -2.70677 q^{9} +1.36878 q^{10} -4.26507 q^{11} -0.0684729 q^{12} -2.32288 q^{13} -1.36878 q^{14} -0.541505 q^{15} -3.73111 q^{16} +1.53014 q^{17} +3.70497 q^{18} +1.28761 q^{19} +0.126449 q^{20} +0.541505 q^{21} +5.83793 q^{22} +5.27188 q^{23} +1.57612 q^{24} +1.00000 q^{25} +3.17951 q^{26} -3.09025 q^{27} -0.126449 q^{28} +1.01560 q^{29} +0.741199 q^{30} -0.101572 q^{31} -0.714209 q^{32} -2.30956 q^{33} -2.09442 q^{34} -1.00000 q^{35} +0.342269 q^{36} +0.801495 q^{37} -1.76246 q^{38} -1.25785 q^{39} -2.91064 q^{40} -2.15676 q^{41} -0.741199 q^{42} -5.96226 q^{43} +0.539315 q^{44} +2.70677 q^{45} -7.21603 q^{46} -6.73318 q^{47} -2.02042 q^{48} +1.00000 q^{49} -1.36878 q^{50} +0.828577 q^{51} +0.293727 q^{52} +2.69434 q^{53} +4.22986 q^{54} +4.26507 q^{55} +2.91064 q^{56} +0.697249 q^{57} -1.39012 q^{58} -9.37360 q^{59} +0.0684729 q^{60} +12.4660 q^{61} +0.139029 q^{62} -2.70677 q^{63} +8.43982 q^{64} +2.32288 q^{65} +3.16127 q^{66} -10.5194 q^{67} -0.193485 q^{68} +2.85475 q^{69} +1.36878 q^{70} -4.66998 q^{71} -7.87843 q^{72} -4.26934 q^{73} -1.09707 q^{74} +0.541505 q^{75} -0.162818 q^{76} -4.26507 q^{77} +1.72172 q^{78} -5.86428 q^{79} +3.73111 q^{80} +6.44693 q^{81} +2.95212 q^{82} -14.9508 q^{83} -0.0684729 q^{84} -1.53014 q^{85} +8.16101 q^{86} +0.549950 q^{87} -12.4141 q^{88} +4.87539 q^{89} -3.70497 q^{90} -2.32288 q^{91} -0.666626 q^{92} -0.0550015 q^{93} +9.21623 q^{94} -1.28761 q^{95} -0.386748 q^{96} -3.28835 q^{97} -1.36878 q^{98} +11.5446 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 68 q + 9 q^{2} + 83 q^{4} - 68 q^{5} + 5 q^{6} + 68 q^{7} + 30 q^{8} + 86 q^{9}+O(q^{10}) $ 68 * q + 9 * q^2 + 83 * q^4 - 68 * q^5 + 5 * q^6 + 68 * q^7 + 30 * q^8 + 86 * q^9	\( 68 q + 9 q^{2} + 83 q^{4} - 68 q^{5} + 5 q^{6} + 68 q^{7} + 30 q^{8} + 86 q^{9} - 9 q^{10} + 5 q^{11} + 9 q^{12} + 15 q^{13} + 9 q^{14} + 109 q^{16} + 7 q^{17} + 39 q^{18} + 20 q^{19} - 83 q^{20} + 56 q^{22} + 36 q^{23} + q^{24} + 68 q^{25} + q^{26} + 12 q^{27} + 83 q^{28} - 16 q^{29} - 5 q^{30} + 31 q^{31} + 79 q^{32} + 45 q^{33} + 31 q^{34} - 68 q^{35} + 114 q^{36} + 72 q^{37} + 8 q^{38} + 47 q^{39} - 30 q^{40} + 6 q^{41} + 5 q^{42} + 75 q^{43} + 15 q^{44} - 86 q^{45} + 29 q^{46} - 10 q^{47} + 44 q^{48} + 68 q^{49} + 9 q^{50} + 23 q^{51} + 37 q^{52} + 41 q^{53} + 4 q^{54} - 5 q^{55} + 30 q^{56} + 55 q^{57} + 66 q^{58} - 5 q^{59} - 9 q^{60} - 2 q^{61} + 3 q^{62} + 86 q^{63} + 162 q^{64} - 15 q^{65} - 23 q^{66} + 92 q^{67} + 35 q^{68} - 25 q^{69} - 9 q^{70} - 2 q^{71} + 128 q^{72} + 80 q^{73} + 18 q^{74} + 71 q^{76} + 5 q^{77} + 20 q^{78} + 100 q^{79} - 109 q^{80} + 140 q^{81} + 36 q^{82} - 60 q^{83} + 9 q^{84} - 7 q^{85} - 27 q^{86} + 24 q^{87} + 175 q^{88} + 19 q^{89} - 39 q^{90} + 15 q^{91} + 75 q^{92} + 37 q^{93} + 11 q^{94} - 20 q^{95} + 15 q^{96} + 96 q^{97} + 9 q^{98} + 3 q^{99}+O(q^{100}) \) 68 * q + 9 * q^2 + 83 * q^4 - 68 * q^5 + 5 * q^6 + 68 * q^7 + 30 * q^8 + 86 * q^9 - 9 * q^10 + 5 * q^11 + 9 * q^12 + 15 * q^13 + 9 * q^14 + 109 * q^16 + 7 * q^17 + 39 * q^18 + 20 * q^19 - 83 * q^20 + 56 * q^22 + 36 * q^23 + q^24 + 68 * q^25 + q^26 + 12 * q^27 + 83 * q^28 - 16 * q^29 - 5 * q^30 + 31 * q^31 + 79 * q^32 + 45 * q^33 + 31 * q^34 - 68 * q^35 + 114 * q^36 + 72 * q^37 + 8 * q^38 + 47 * q^39 - 30 * q^40 + 6 * q^41 + 5 * q^42 + 75 * q^43 + 15 * q^44 - 86 * q^45 + 29 * q^46 - 10 * q^47 + 44 * q^48 + 68 * q^49 + 9 * q^50 + 23 * q^51 + 37 * q^52 + 41 * q^53 + 4 * q^54 - 5 * q^55 + 30 * q^56 + 55 * q^57 + 66 * q^58 - 5 * q^59 - 9 * q^60 - 2 * q^61 + 3 * q^62 + 86 * q^63 + 162 * q^64 - 15 * q^65 - 23 * q^66 + 92 * q^67 + 35 * q^68 - 25 * q^69 - 9 * q^70 - 2 * q^71 + 128 * q^72 + 80 * q^73 + 18 * q^74 + 71 * q^76 + 5 * q^77 + 20 * q^78 + 100 * q^79 - 109 * q^80 + 140 * q^81 + 36 * q^82 - 60 * q^83 + 9 * q^84 - 7 * q^85 - 27 * q^86 + 24 * q^87 + 175 * q^88 + 19 * q^89 - 39 * q^90 + 15 * q^91 + 75 * q^92 + 37 * q^93 + 11 * q^94 - 20 * q^95 + 15 * q^96 + 96 * q^97 + 9 * q^98 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.36878	−0.967872	−0.483936	−	0.875104i	$-0.660793\pi$
$2$	−1.36878	−0.967872	−0.483936	+	0.875104i	$0.660793\pi$
$3$	0.541505	0.312638	0.156319	−	0.987707i	$-0.450037\pi$
$3$	0.541505	0.312638	0.156319	+	0.987707i	$0.450037\pi$
$4$	−0.126449	−0.0632246
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.741199	−0.302593
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.91064	1.02906
$9$	−2.70677	−0.902257
$10$	1.36878	0.432845
$11$	−4.26507	−1.28597	−0.642984	−	0.765880i	$-0.722304\pi$
$11$	−4.26507	−1.28597	−0.642984	+	0.765880i	$0.722304\pi$
$12$	−0.0684729	−0.0197664
$13$	−2.32288	−0.644252	−0.322126	−	0.946697i	$-0.604398\pi$
$13$	−2.32288	−0.644252	−0.322126	+	0.946697i	$0.604398\pi$
$14$	−1.36878	−0.365821
$15$	−0.541505	−0.139816
$16$	−3.73111	−0.932778
$17$	1.53014	0.371113	0.185556	−	0.982634i	$-0.440591\pi$
$17$	1.53014	0.371113	0.185556	+	0.982634i	$0.440591\pi$
$18$	3.70497	0.873269
$19$	1.28761	0.295399	0.147699	−	0.989032i	$-0.452813\pi$
$19$	1.28761	0.295399	0.147699	+	0.989032i	$0.452813\pi$
$20$	0.126449	0.0282749
$21$	0.541505	0.118166
$22$	5.83793	1.24465
$23$	5.27188	1.09926	0.549632	−	0.835407i	$-0.314768\pi$
$23$	5.27188	1.09926	0.549632	+	0.835407i	$0.314768\pi$
$24$	1.57612	0.321725
$25$	1.00000	0.200000
$26$	3.17951	0.623554
$27$	−3.09025	−0.594718
$28$	−0.126449	−0.0238967
$29$	1.01560	0.188591	0.0942957	−	0.995544i	$-0.469940\pi$
$29$	1.01560	0.188591	0.0942957	+	0.995544i	$0.469940\pi$
$30$	0.741199	0.135324
$31$	−0.101572	−0.0182428	−0.00912140	−	0.999958i	$-0.502903\pi$
$31$	−0.101572	−0.0182428	−0.00912140	+	0.999958i	$0.502903\pi$
$32$	−0.714209	−0.126256
$33$	−2.30956	−0.402042
$34$	−2.09442	−0.359189
$35$	−1.00000	−0.169031
$36$	0.342269	0.0570449
$37$	0.801495	0.131765	0.0658825	−	0.997827i	$-0.479014\pi$
$37$	0.801495	0.131765	0.0658825	+	0.997827i	$0.479014\pi$
$38$	−1.76246	−0.285908
$39$	−1.25785	−0.201418
$40$	−2.91064	−0.460212
$41$	−2.15676	−0.336829	−0.168414	−	0.985716i	$-0.553865\pi$
$41$	−2.15676	−0.336829	−0.168414	+	0.985716i	$0.553865\pi$
$42$	−0.741199	−0.114370
$43$	−5.96226	−0.909236	−0.454618	−	0.890686i	$-0.650224\pi$
$43$	−5.96226	−0.909236	−0.454618	+	0.890686i	$0.650224\pi$
$44$	0.539315	0.0813048
$45$	2.70677	0.403502
$46$	−7.21603	−1.06395
$47$	−6.73318	−0.982136	−0.491068	−	0.871121i	$-0.663393\pi$
$47$	−6.73318	−0.982136	−0.491068	+	0.871121i	$0.663393\pi$
$48$	−2.02042	−0.291622
$49$	1.00000	0.142857
$50$	−1.36878	−0.193574
$51$	0.828577	0.116024
$52$	0.293727	0.0407326
$53$	2.69434	0.370096	0.185048	−	0.982730i	$-0.440756\pi$
$53$	2.69434	0.370096	0.185048	+	0.982730i	$0.440756\pi$
$54$	4.22986	0.575611
$55$	4.26507	0.575102
$56$	2.91064	0.388950
$57$	0.697249	0.0923529
$58$	−1.39012	−0.182532
$59$	−9.37360	−1.22034	−0.610170	−	0.792271i	$-0.708899\pi$
$59$	−9.37360	−1.22034	−0.610170	+	0.792271i	$0.708899\pi$
$60$	0.0684729	0.00883981
$61$	12.4660	1.59611	0.798053	−	0.602587i	$-0.205863\pi$
$61$	12.4660	1.59611	0.798053	+	0.602587i	$0.205863\pi$
$62$	0.139029	0.0176567
$63$	−2.70677	−0.341021
$64$	8.43982	1.05498
$65$	2.32288	0.288118
$66$	3.16127	0.389125
$67$	−10.5194	−1.28515	−0.642574	−	0.766224i	$-0.722133\pi$
$67$	−10.5194	−1.28515	−0.642574	+	0.766224i	$0.722133\pi$
$68$	−0.193485	−0.0234635
$69$	2.85475	0.343672
$70$	1.36878	0.163600
$71$	−4.66998	−0.554225	−0.277113	−	0.960837i	$-0.589378\pi$
$71$	−4.66998	−0.554225	−0.277113	+	0.960837i	$0.589378\pi$
$72$	−7.87843	−0.928482
$73$	−4.26934	−0.499688	−0.249844	−	0.968286i	$-0.580379\pi$
$73$	−4.26934	−0.499688	−0.249844	+	0.968286i	$0.580379\pi$
$74$	−1.09707	−0.127532
$75$	0.541505	0.0625276
$76$	−0.162818	−0.0186765
$77$	−4.26507	−0.486050
$78$	1.72172	0.194947
$79$	−5.86428	−0.659784	−0.329892	−	0.944019i	$-0.607012\pi$
$79$	−5.86428	−0.659784	−0.329892	+	0.944019i	$0.607012\pi$
$80$	3.73111	0.417151
$81$	6.44693	0.716326
$82$	2.95212	0.326007
$83$	−14.9508	−1.64106	−0.820531	−	0.571602i	$-0.806322\pi$
$83$	−14.9508	−1.64106	−0.820531	+	0.571602i	$0.806322\pi$
$84$	−0.0684729	−0.00747100
$85$	−1.53014	−0.165967
$86$	8.16101	0.880024
$87$	0.549950	0.0589608
$88$	−12.4141	−1.32334
$89$	4.87539	0.516790	0.258395	−	0.966039i	$-0.416806\pi$
$89$	4.87539	0.516790	0.258395	+	0.966039i	$0.416806\pi$
$90$	−3.70497	−0.390538
$91$	−2.32288	−0.243505
$92$	−0.666626	−0.0695005
$93$	−0.0550015	−0.00570339
$94$	9.21623	0.950581
$95$	−1.28761	−0.132106
$96$	−0.386748	−0.0394723
$97$	−3.28835	−0.333881	−0.166941	−	0.985967i	$-0.553389\pi$
$97$	−3.28835	−0.333881	−0.166941	+	0.985967i	$0.553389\pi$
$98$	−1.36878	−0.138267
$99$	11.5446	1.16027
$100$	−0.126449	−0.0126449
$101$	13.1169	1.30518	0.652590	−	0.757711i	$-0.273682\pi$
$101$	13.1169	1.30518	0.652590	+	0.757711i	$0.273682\pi$
$102$	−1.13414	−0.112296
$103$	7.70871	0.759561	0.379781	−	0.925077i	$-0.375999\pi$
$103$	7.70871	0.759561	0.379781	+	0.925077i	$0.375999\pi$
$104$	−6.76107	−0.662977
$105$	−0.541505	−0.0528455
$106$	−3.68795	−0.358205
$107$	−16.8849	−1.63232	−0.816160	−	0.577825i	$-0.803902\pi$
$107$	−16.8849	−1.63232	−0.816160	+	0.577825i	$0.803902\pi$
$108$	0.390759	0.0376008
$109$	−7.55878	−0.723999	−0.362000	−	0.932178i	$-0.617906\pi$
$109$	−7.55878	−0.723999	−0.362000	+	0.932178i	$0.617906\pi$
$110$	−5.83793	−0.556625
$111$	0.434014	0.0411947
$112$	−3.73111	−0.352557
$113$	6.02563	0.566844	0.283422	−	0.958995i	$-0.408530\pi$
$113$	6.02563	0.566844	0.283422	+	0.958995i	$0.408530\pi$
$114$	−0.954379	−0.0893858
$115$	−5.27188	−0.491606
$116$	−0.128421	−0.0119236
$117$	6.28752	0.581282
$118$	12.8304	1.18113
$119$	1.53014	0.140267
$120$	−1.57612	−0.143880
$121$	7.19084	0.653713
$122$	−17.0632	−1.54483
$123$	−1.16789	−0.105305
$124$	0.0128437	0.00115339
$125$	−1.00000	−0.0894427
$126$	3.70497	0.330065
$127$	−13.5654	−1.20373	−0.601867	−	0.798597i	$-0.705576\pi$
$127$	−13.5654	−1.20373	−0.601867	+	0.798597i	$0.705576\pi$
$128$	−10.1238	−0.894827
$129$	−3.22859	−0.284262
$130$	−3.17951	−0.278862
$131$	−11.0303	−0.963721	−0.481860	−	0.876248i	$-0.660039\pi$
$131$	−11.0303	−0.963721	−0.481860	+	0.876248i	$0.660039\pi$
$132$	0.292042	0.0254190
$133$	1.28761	0.111650
$134$	14.3987	1.24386
$135$	3.09025	0.265966
$136$	4.45367	0.381899
$137$	19.0012	1.62338	0.811692	−	0.584086i	$-0.198547\pi$
$137$	19.0012	1.62338	0.811692	+	0.584086i	$0.198547\pi$
$138$	−3.90752	−0.332630
$139$	−15.5694	−1.32058	−0.660288	−	0.751012i	$-0.729566\pi$
$139$	−15.5694	−1.32058	−0.660288	+	0.751012i	$0.729566\pi$
$140$	0.126449	0.0106869
$141$	−3.64605	−0.307053
$142$	6.39217	0.536419
$143$	9.90727	0.828488
$144$	10.0993	0.841606
$145$	−1.01560	−0.0843406
$146$	5.84377	0.483634
$147$	0.541505	0.0446626
$148$	−0.101348	−0.00833079
$149$	21.5122	1.76235	0.881175	−	0.472790i	$-0.156753\pi$
$149$	21.5122	1.76235	0.881175	+	0.472790i	$0.156753\pi$
$150$	−0.741199	−0.0605187
$151$	11.2293	0.913828	0.456914	−	0.889511i	$-0.348955\pi$
$151$	11.2293	0.913828	0.456914	+	0.889511i	$0.348955\pi$
$152$	3.74777	0.303985
$153$	−4.14173	−0.334839
$154$	5.83793	0.470434
$155$	0.101572	0.00815843
$156$	0.159055	0.0127346
$157$	23.0016	1.83573	0.917863	−	0.396898i	$-0.129913\pi$
$157$	23.0016	1.83573	0.917863	+	0.396898i	$0.129913\pi$
$158$	8.02690	0.638586
$159$	1.45900	0.115706
$160$	0.714209	0.0564632
$161$	5.27188	0.415483
$162$	−8.82442	−0.693312
$163$	23.5366	1.84353	0.921766	−	0.387746i	$-0.126746\pi$
$163$	23.5366	1.84353	0.921766	+	0.387746i	$0.126746\pi$
$164$	0.272720	0.0212959
$165$	2.30956	0.179799
$166$	20.4643	1.58834
$167$	−16.5512	−1.28077	−0.640383	−	0.768055i	$-0.721225\pi$
$167$	−16.5512	−1.28077	−0.640383	+	0.768055i	$0.721225\pi$
$168$	1.57612	0.121601
$169$	−7.60421	−0.584939
$170$	2.09442	0.160634
$171$	−3.48528	−0.266526
$172$	0.753923	0.0574861
$173$	−10.6514	−0.809808	−0.404904	−	0.914359i	$-0.632695\pi$
$173$	−10.6514	−0.809808	−0.404904	+	0.914359i	$0.632695\pi$
$174$	−0.752759	−0.0570665
$175$	1.00000	0.0755929
$176$	15.9135	1.19952
$177$	−5.07585	−0.381524
$178$	−6.67332	−0.500186
$179$	13.3822	1.00023	0.500116	−	0.865958i	$-0.333291\pi$
$179$	13.3822	1.00023	0.500116	+	0.865958i	$0.333291\pi$
$180$	−0.342269	−0.0255112
$181$	−17.1647	−1.27584	−0.637921	−	0.770102i	$-0.720205\pi$
$181$	−17.1647	−1.27584	−0.637921	+	0.770102i	$0.720205\pi$
$182$	3.17951	0.235681
$183$	6.75040	0.499004
$184$	15.3445	1.13121
$185$	−0.801495	−0.0589271
$186$	0.0752848	0.00552015
$187$	−6.52614	−0.477239
$188$	0.851406	0.0620951
$189$	−3.09025	−0.224782
$190$	1.76246	0.127862
$191$	3.53488	0.255775	0.127888	−	0.991789i	$-0.459180\pi$
$191$	3.53488	0.255775	0.127888	+	0.991789i	$0.459180\pi$
$192$	4.57020	0.329826
$193$	−10.8665	−0.782186	−0.391093	−	0.920351i	$-0.627903\pi$
$193$	−10.8665	−0.782186	−0.391093	+	0.920351i	$0.627903\pi$
$194$	4.50101	0.323154
$195$	1.25785	0.0900768
$196$	−0.126449	−0.00903209
$197$	14.2620	1.01613	0.508063	−	0.861320i	$-0.330362\pi$
$197$	14.2620	1.01613	0.508063	+	0.861320i	$0.330362\pi$
$198$	−15.8020	−1.12300
$199$	20.5658	1.45787	0.728934	−	0.684584i	$-0.240016\pi$
$199$	20.5658	1.45787	0.728934	+	0.684584i	$0.240016\pi$
$200$	2.91064	0.205813
$201$	−5.69630	−0.401786
$202$	−17.9541	−1.26325
$203$	1.01560	0.0712808
$204$	−0.104773	−0.00733557
$205$	2.15676	0.150634
$206$	−10.5515	−0.735158
$207$	−14.2698	−0.991819
$208$	8.66694	0.600944
$209$	−5.49177	−0.379873
$210$	0.741199	0.0511476
$211$	14.8989	1.02568	0.512840	−	0.858484i	$-0.328593\pi$
$211$	14.8989	1.02568	0.512840	+	0.858484i	$0.328593\pi$
$212$	−0.340697	−0.0233992
$213$	−2.52882	−0.173272
$214$	23.1116	1.57988
$215$	5.96226	0.406623
$216$	−8.99458	−0.612003
$217$	−0.101572	−0.00689513
$218$	10.3463	0.700738
$219$	−2.31187	−0.156222
$220$	−0.539315	−0.0363606
$221$	−3.55433	−0.239090
$222$	−0.594068	−0.0398712
$223$	21.6929	1.45267	0.726333	−	0.687343i	$-0.241223\pi$
$223$	21.6929	1.45267	0.726333	+	0.687343i	$0.241223\pi$
$224$	−0.714209	−0.0477201
$225$	−2.70677	−0.180451
$226$	−8.24775	−0.548632
$227$	21.2693	1.41169	0.705846	−	0.708365i	$-0.250567\pi$
$227$	21.2693	1.41169	0.705846	+	0.708365i	$0.250567\pi$
$228$	−0.0881666	−0.00583898
$229$	−1.00000	−0.0660819
$229$	−1.00000	−0.0660819
$230$	7.21603	0.475811
$231$	−2.30956	−0.151958
$232$	2.95603	0.194073
$233$	−5.48811	−0.359538	−0.179769	−	0.983709i	$-0.557535\pi$
$233$	−5.48811	−0.359538	−0.179769	+	0.983709i	$0.557535\pi$
$234$	−8.60622	−0.562606
$235$	6.73318	0.439224
$236$	1.18529	0.0771555
$237$	−3.17554	−0.206273
$238$	−2.09442	−0.135761
$239$	4.63995	0.300133	0.150067	−	0.988676i	$-0.452051\pi$
$239$	4.63995	0.300133	0.150067	+	0.988676i	$0.452051\pi$
$240$	2.02042	0.130417
$241$	10.0024	0.644310	0.322155	−	0.946687i	$-0.395593\pi$
$241$	10.0024	0.644310	0.322155	+	0.946687i	$0.395593\pi$
$242$	−9.84266	−0.632710
$243$	12.7618	0.818669
$244$	−1.57632	−0.100913
$245$	−1.00000	−0.0638877
$246$	1.59859	0.101922
$247$	−2.99098	−0.190311
$248$	−0.295638	−0.0187730
$249$	−8.09593	−0.513058
$250$	1.36878	0.0865691
$251$	−14.5796	−0.920255	−0.460127	−	0.887853i	$-0.652196\pi$
$251$	−14.5796	−0.920255	−0.460127	+	0.887853i	$0.652196\pi$
$252$	0.342269	0.0215609
$253$	−22.4850	−1.41362
$254$	18.5680	1.16506
$255$	−0.828577	−0.0518875
$256$	−3.02239	−0.188900
$257$	−0.0444559	−0.00277308	−0.00138654	−	0.999999i	$-0.500441\pi$
$257$	−0.0444559	−0.00277308	−0.00138654	+	0.999999i	$0.500441\pi$
$258$	4.41922	0.275129
$259$	0.801495	0.0498025
$260$	−0.293727	−0.0182162
$261$	−2.74899	−0.170158
$262$	15.0980	0.932758
$263$	1.14922	0.0708640	0.0354320	−	0.999372i	$-0.488719\pi$
$263$	1.14922	0.0708640	0.0354320	+	0.999372i	$0.488719\pi$
$264$	−6.72228	−0.413728
$265$	−2.69434	−0.165512
$266$	−1.76246	−0.108063
$267$	2.64005	0.161568
$268$	1.33017	0.0812529
$269$	2.60752	0.158983	0.0794916	−	0.996836i	$-0.474670\pi$
$269$	2.60752	0.158983	0.0794916	+	0.996836i	$0.474670\pi$
$270$	−4.22986	−0.257421
$271$	22.8294	1.38679	0.693393	−	0.720559i	$-0.256115\pi$
$271$	22.8294	1.38679	0.693393	+	0.720559i	$0.256115\pi$
$272$	−5.70911	−0.346166
$273$	−1.25785	−0.0761288
$274$	−26.0084	−1.57123
$275$	−4.26507	−0.257194
$276$	−0.360981	−0.0217285
$277$	−10.3010	−0.618929	−0.309465	−	0.950911i	$-0.600150\pi$
$277$	−10.3010	−0.618929	−0.309465	+	0.950911i	$0.600150\pi$
$278$	21.3110	1.27815
$279$	0.274931	0.0164597
$280$	−2.91064	−0.173944
$281$	−25.9572	−1.54848	−0.774238	−	0.632895i	$-0.781867\pi$
$281$	−25.9572	−1.54848	−0.774238	+	0.632895i	$0.781867\pi$
$282$	4.99063	0.297188
$283$	6.22847	0.370244	0.185122	−	0.982716i	$-0.440732\pi$
$283$	6.22847	0.370244	0.185122	+	0.982716i	$0.440732\pi$
$284$	0.590516	0.0350407
$285$	−0.697249	−0.0413015
$286$	−13.5608	−0.801870
$287$	−2.15676	−0.127309
$288$	1.93320	0.113915
$289$	−14.6587	−0.862275
$290$	1.39012	0.0816309
$291$	−1.78066	−0.104384
$292$	0.539855	0.0315926
$293$	4.25445	0.248548	0.124274	−	0.992248i	$-0.460340\pi$
$293$	4.25445	0.248548	0.124274	+	0.992248i	$0.460340\pi$
$294$	−0.741199	−0.0432276
$295$	9.37360	0.545752
$296$	2.33286	0.135595
$297$	13.1801	0.764788
$298$	−29.4455	−1.70573
$299$	−12.2460	−0.708203
$300$	−0.0684729	−0.00395328
$301$	−5.96226	−0.343659
$302$	−15.3704	−0.884468
$303$	7.10287	0.408049
$304$	−4.80423	−0.275542
$305$	−12.4660	−0.713801
$306$	5.66911	0.324081
$307$	13.0292	0.743616	0.371808	−	0.928310i	$-0.378738\pi$
$307$	13.0292	0.743616	0.371808	+	0.928310i	$0.378738\pi$
$308$	0.539315	0.0307303
$309$	4.17430	0.237468
$310$	−0.139029	−0.00789631
$311$	−14.0732	−0.798016	−0.399008	−	0.916947i	$-0.630645\pi$
$311$	−14.0732	−0.798016	−0.399008	+	0.916947i	$0.630645\pi$
$312$	−3.66115	−0.207272
$313$	−10.3566	−0.585389	−0.292695	−	0.956206i	$-0.594552\pi$
$313$	−10.3566	−0.585389	−0.292695	+	0.956206i	$0.594552\pi$
$314$	−31.4840	−1.77675
$315$	2.70677	0.152509
$316$	0.741534	0.0417146
$317$	−26.7893	−1.50464	−0.752319	−	0.658799i	$-0.771065\pi$
$317$	−26.7893	−1.50464	−0.752319	+	0.658799i	$0.771065\pi$
$318$	−1.99704	−0.111989
$319$	−4.33159	−0.242522
$320$	−8.43982	−0.471800
$321$	−9.14323	−0.510326
$322$	−7.21603	−0.402134
$323$	1.97023	0.109626
$324$	−0.815210	−0.0452894
$325$	−2.32288	−0.128850
$326$	−32.2164	−1.78430
$327$	−4.09311	−0.226350
$328$	−6.27753	−0.346619
$329$	−6.73318	−0.371212
$330$	−3.16127	−0.174022
$331$	−17.0324	−0.936187	−0.468094	−	0.883679i	$-0.655059\pi$
$331$	−17.0324	−0.936187	−0.468094	+	0.883679i	$0.655059\pi$
$332$	1.89052	0.103756
$333$	−2.16947	−0.118886
$334$	22.6548	1.23962
$335$	10.5194	0.574735
$336$	−2.02042	−0.110223
$337$	4.31116	0.234844	0.117422	−	0.993082i	$-0.462537\pi$
$337$	4.31116	0.234844	0.117422	+	0.993082i	$0.462537\pi$
$338$	10.4085	0.566146
$339$	3.26291	0.177217
$340$	0.193485	0.0104932
$341$	0.433210	0.0234597
$342$	4.77057	0.257963
$343$	1.00000	0.0539949
$344$	−17.3540	−0.935663
$345$	−2.85475	−0.153695
$346$	14.5793	0.783790
$347$	−11.4785	−0.616198	−0.308099	−	0.951354i	$-0.599693\pi$
$347$	−11.4785	−0.616198	−0.308099	+	0.951354i	$0.599693\pi$
$348$	−0.0695408	−0.00372778
$349$	−18.9212	−1.01283	−0.506414	−	0.862291i	$-0.669029\pi$
$349$	−18.9212	−1.01283	−0.506414	+	0.862291i	$0.669029\pi$
$350$	−1.36878	−0.0731642
$351$	7.17828	0.383148
$352$	3.04615	0.162361
$353$	−0.0344964	−0.00183606	−0.000918030	−	1.00000i	$-0.500292\pi$
$353$	−0.0344964	−0.00183606	−0.000918030		1.00000i	$0.500292\pi$
$354$	6.94771	0.369267
$355$	4.66998	0.247857
$356$	−0.616489	−0.0326739
$357$	0.828577	0.0438529
$358$	−18.3173	−0.968097
$359$	11.5295	0.608506	0.304253	−	0.952591i	$-0.401593\pi$
$359$	11.5295	0.608506	0.304253	+	0.952591i	$0.401593\pi$
$360$	7.87843	0.415230
$361$	−17.3421	−0.912740
$362$	23.4947	1.23485
$363$	3.89388	0.204375
$364$	0.293727	0.0153955
$365$	4.26934	0.223467
$366$	−9.23979	−0.482971
$367$	2.37224	0.123830	0.0619150	−	0.998081i	$-0.480279\pi$
$367$	2.37224	0.123830	0.0619150	+	0.998081i	$0.480279\pi$
$368$	−19.6700	−1.02537
$369$	5.83785	0.303906
$370$	1.09707	0.0570339
$371$	2.69434	0.139883
$372$	0.00695490	0.000360595 0
$373$	34.3278	1.77742	0.888712	−	0.458465i	$-0.151601\pi$
$373$	34.3278	1.77742	0.888712	+	0.458465i	$0.151601\pi$
$374$	8.93284	0.461906
$375$	−0.541505	−0.0279632
$376$	−19.5978	−1.01068
$377$	−2.35911	−0.121500
$378$	4.22986	0.217560
$379$	19.5652	1.00499	0.502497	−	0.864579i	$-0.332415\pi$
$379$	19.5652	1.00499	0.502497	+	0.864579i	$0.332415\pi$
$380$	0.162818	0.00835238
$381$	−7.34572	−0.376333
$382$	−4.83847	−0.247558
$383$	30.0222	1.53406	0.767030	−	0.641611i	$-0.221734\pi$
$383$	30.0222	1.53406	0.767030	+	0.641611i	$0.221734\pi$
$384$	−5.48209	−0.279757
$385$	4.26507	0.217368
$386$	14.8738	0.757055
$387$	16.1385	0.820365
$388$	0.415809	0.0211095
$389$	31.2724	1.58557	0.792786	−	0.609500i	$-0.208630\pi$
$389$	31.2724	1.58557	0.792786	+	0.609500i	$0.208630\pi$
$390$	−1.72172	−0.0871827
$391$	8.06670	0.407951
$392$	2.91064	0.147009
$393$	−5.97296	−0.301296
$394$	−19.5215	−0.983479
$395$	5.86428	0.295064
$396$	−1.45980	−0.0733579
$397$	12.2253	0.613570	0.306785	−	0.951779i	$-0.400747\pi$
$397$	12.2253	0.613570	0.306785	+	0.951779i	$0.400747\pi$
$398$	−28.1499	−1.41103
$399$	0.697249	0.0349061
$400$	−3.73111	−0.186556
$401$	29.2903	1.46269	0.731345	−	0.682008i	$-0.238893\pi$
$401$	29.2903	1.46269	0.731345	+	0.682008i	$0.238893\pi$
$402$	7.79696	0.388877
$403$	0.235939	0.0117530
$404$	−1.65862	−0.0825195
$405$	−6.44693	−0.320351
$406$	−1.39012	−0.0689907
$407$	−3.41843	−0.169446
$408$	2.41168	0.119396
$409$	25.0985	1.24104	0.620521	−	0.784190i	$-0.286921\pi$
$409$	25.0985	1.24104	0.620521	+	0.784190i	$0.286921\pi$
$410$	−2.95212	−0.145795
$411$	10.2893	0.507531
$412$	−0.974760	−0.0480230
$413$	−9.37360	−0.461245
$414$	19.5322	0.959953
$415$	14.9508	0.733905
$416$	1.65903	0.0813404
$417$	−8.43089	−0.412862
$418$	7.51700	0.367669
$419$	−19.9645	−0.975329	−0.487665	−	0.873031i	$-0.662151\pi$
$419$	−19.9645	−0.975329	−0.487665	+	0.873031i	$0.662151\pi$
$420$	0.0684729	0.00334113
$421$	25.8958	1.26209	0.631043	−	0.775748i	$-0.282627\pi$
$421$	25.8958	1.26209	0.631043	+	0.775748i	$0.282627\pi$
$422$	−20.3932	−0.992727
$423$	18.2252	0.886139
$424$	7.84224	0.380853
$425$	1.53014	0.0742225
$426$	3.46139	0.167705
$427$	12.4660	0.603272
$428$	2.13508	0.103203
$429$	5.36484	0.259017
$430$	−8.16101	−0.393559
$431$	−22.0080	−1.06009	−0.530043	−	0.847971i	$-0.677824\pi$
$431$	−22.0080	−1.06009	−0.530043	+	0.847971i	$0.677824\pi$
$432$	11.5301	0.554740
$433$	13.8189	0.664092	0.332046	−	0.943263i	$-0.392261\pi$
$433$	13.8189	0.664092	0.332046	+	0.943263i	$0.392261\pi$
$434$	0.139029	0.00667360
$435$	−0.549950	−0.0263681
$436$	0.955801	0.0457746
$437$	6.78815	0.324721
$438$	3.16443	0.151202
$439$	17.3587	0.828484	0.414242	−	0.910167i	$-0.364047\pi$
$439$	17.3587	0.828484	0.414242	+	0.910167i	$0.364047\pi$
$440$	12.4141	0.591817
$441$	−2.70677	−0.128894
$442$	4.86509	0.231409
$443$	−16.3856	−0.778504	−0.389252	−	0.921131i	$-0.627267\pi$
$443$	−16.3856	−0.778504	−0.389252	+	0.921131i	$0.627267\pi$
$444$	−0.0548807	−0.00260452
$445$	−4.87539	−0.231116
$446$	−29.6928	−1.40599
$447$	11.6490	0.550978
$448$	8.43982	0.398744
$449$	36.0231	1.70003	0.850017	−	0.526755i	$-0.176592\pi$
$449$	36.0231	1.70003	0.850017	+	0.526755i	$0.176592\pi$
$450$	3.70497	0.174654
$451$	9.19872	0.433151
$452$	−0.761936	−0.0358385
$453$	6.08072	0.285697
$454$	−29.1129	−1.36634
$455$	2.32288	0.108899
$456$	2.02944	0.0950371
$457$	1.22007	0.0570726	0.0285363	−	0.999593i	$-0.490915\pi$
$457$	1.22007	0.0570726	0.0285363	+	0.999593i	$0.490915\pi$
$458$	1.36878	0.0639588
$459$	−4.72850	−0.220707
$460$	0.666626	0.0310816
$461$	−24.3588	−1.13450	−0.567252	−	0.823544i	$-0.691993\pi$
$461$	−24.3588	−1.13450	−0.567252	+	0.823544i	$0.691993\pi$
$462$	3.16127	0.147076
$463$	−3.87040	−0.179873	−0.0899363	−	0.995948i	$-0.528666\pi$
$463$	−3.87040	−0.179873	−0.0899363	+	0.995948i	$0.528666\pi$
$464$	−3.78930	−0.175914
$465$	0.0550015	0.00255063
$466$	7.51200	0.347986
$467$	−41.4667	−1.91885	−0.959425	−	0.281965i	$-0.909014\pi$
$467$	−41.4667	−1.91885	−0.959425	+	0.281965i	$0.909014\pi$
$468$	−0.795052	−0.0367513
$469$	−10.5194	−0.485740
$470$	−9.21623	−0.425113
$471$	12.4555	0.573918
$472$	−27.2831	−1.25581
$473$	25.4295	1.16925
$474$	4.34660	0.199646
$475$	1.28761	0.0590798
$476$	−0.193485	−0.00886835
$477$	−7.29296	−0.333922
$478$	−6.35105	−0.290490
$479$	24.8064	1.13343	0.566716	−	0.823913i	$-0.308214\pi$
$479$	24.8064	1.13343	0.566716	+	0.823913i	$0.308214\pi$
$480$	0.386748	0.0176525
$481$	−1.86178	−0.0848899
$482$	−13.6910	−0.623609
$483$	2.85475	0.129896
$484$	−0.909276	−0.0413307
$485$	3.28835	0.149316
$486$	−17.4680	−0.792366
$487$	38.4492	1.74230	0.871150	−	0.491018i	$-0.163375\pi$
$487$	38.4492	1.74230	0.871150	+	0.491018i	$0.163375\pi$
$488$	36.2840	1.64250
$489$	12.7452	0.576358
$490$	1.36878	0.0618350
$491$	−37.0674	−1.67283	−0.836415	−	0.548096i	$-0.815353\pi$
$491$	−37.0674	−1.67283	−0.836415	+	0.548096i	$0.815353\pi$
$492$	0.147679	0.00665790
$493$	1.55400	0.0699887
$494$	4.09398	0.184197
$495$	−11.5446	−0.518890
$496$	0.378975	0.0170165
$497$	−4.66998	−0.209477
$498$	11.0815	0.496575
$499$	10.9761	0.491358	0.245679	−	0.969351i	$-0.420989\pi$
$499$	10.9761	0.491358	0.245679	+	0.969351i	$0.420989\pi$
$500$	0.126449	0.00565498
$501$	−8.96253	−0.400416
$502$	19.9562	0.890688
$503$	23.0341	1.02704	0.513519	−	0.858078i	$-0.328342\pi$
$503$	23.0341	1.02704	0.513519	+	0.858078i	$0.328342\pi$
$504$	−7.87843	−0.350933
$505$	−13.1169	−0.583694
$506$	30.7769	1.36820
$507$	−4.11771	−0.182874
$508$	1.71533	0.0761056
$509$	20.3807	0.903359	0.451679	−	0.892180i	$-0.350825\pi$
$509$	20.3807	0.903359	0.451679	+	0.892180i	$0.350825\pi$
$510$	1.13414	0.0502204
$511$	−4.26934	−0.188864
$512$	24.3846	1.07766
$513$	−3.97904	−0.175679
$514$	0.0608502	0.00268399
$515$	−7.70871	−0.339686
$516$	0.408253	0.0179723
$517$	28.7175	1.26299
$518$	−1.09707	−0.0482024
$519$	−5.76776	−0.253177
$520$	6.76107	0.296493
$521$	−3.46576	−0.151838	−0.0759189	−	0.997114i	$-0.524189\pi$
$521$	−3.46576	−0.151838	−0.0759189	+	0.997114i	$0.524189\pi$
$522$	3.76275	0.164691
$523$	4.37900	0.191480	0.0957401	−	0.995406i	$-0.469478\pi$
$523$	4.37900	0.191480	0.0957401	+	0.995406i	$0.469478\pi$
$524$	1.39477	0.0609309
$525$	0.541505	0.0236332
$526$	−1.57303	−0.0685872
$527$	−0.155418	−0.00677014
$528$	8.61722	0.375016
$529$	4.79276	0.208381
$530$	3.68795	0.160194
$531$	25.3722	1.10106
$532$	−0.162818	−0.00705905
$533$	5.00990	0.217003
$534$	−3.61364	−0.156377
$535$	16.8849	0.729996
$536$	−30.6181	−1.32250
$537$	7.24653	0.312711
$538$	−3.56911	−0.153875
$539$	−4.26507	−0.183710
$540$	−0.390759	−0.0168156
$541$	31.5380	1.35592	0.677962	−	0.735097i	$-0.262863\pi$
$541$	31.5380	1.35592	0.677962	+	0.735097i	$0.262863\pi$
$542$	−31.2484	−1.34223
$543$	−9.29477	−0.398877
$544$	−1.09284	−0.0468550
$545$	7.55878	0.323782
$546$	1.72172	0.0736829
$547$	−24.3141	−1.03960	−0.519798	−	0.854289i	$-0.673993\pi$
$547$	−24.3141	−1.03960	−0.519798	+	0.854289i	$0.673993\pi$
$548$	−2.40269	−0.102638
$549$	−33.7426	−1.44010
$550$	5.83793	0.248930
$551$	1.30770	0.0557097
$552$	8.30914	0.353660
$553$	−5.86428	−0.249375
$554$	14.0998	0.599044
$555$	−0.434014	−0.0184228
$556$	1.96874	0.0834930
$557$	1.09186	0.0462634	0.0231317	−	0.999732i	$-0.492636\pi$
$557$	1.09186	0.0462634	0.0231317	+	0.999732i	$0.492636\pi$
$558$	−0.376320	−0.0159309
$559$	13.8496	0.585778
$560$	3.73111	0.157668
$561$	−3.53394	−0.149203
$562$	35.5296	1.49873
$563$	−7.74289	−0.326324	−0.163162	−	0.986599i	$-0.552169\pi$
$563$	−7.74289	−0.326324	−0.163162	+	0.986599i	$0.552169\pi$
$564$	0.461040	0.0194133
$565$	−6.02563	−0.253500
$566$	−8.52539	−0.358349
$567$	6.44693	0.270746
$568$	−13.5926	−0.570334
$569$	0.807802	0.0338648	0.0169324	−	0.999857i	$-0.494610\pi$
$569$	0.807802	0.0338648	0.0169324	+	0.999857i	$0.494610\pi$
$570$	0.954379	0.0399745
$571$	32.9270	1.37795	0.688977	−	0.724784i	$-0.258060\pi$
$571$	32.9270	1.37795	0.688977	+	0.724784i	$0.258060\pi$
$572$	−1.25277	−0.0523808
$573$	1.91416	0.0799650
$574$	2.95212	0.123219
$575$	5.27188	0.219853
$576$	−22.8447	−0.951861
$577$	23.9113	0.995442	0.497721	−	0.867337i	$-0.334170\pi$
$577$	23.9113	0.995442	0.497721	+	0.867337i	$0.334170\pi$
$578$	20.0645	0.834572
$579$	−5.88425	−0.244541
$580$	0.128421	0.00533240
$581$	−14.9508	−0.620263
$582$	2.43732	0.101030
$583$	−11.4915	−0.475931
$584$	−12.4265	−0.514212
$585$	−6.28752	−0.259957
$586$	−5.82340	−0.240562
$587$	47.3837	1.95573	0.977867	−	0.209230i	$-0.0670957\pi$
$587$	47.3837	1.95573	0.977867	+	0.209230i	$0.0670957\pi$
$588$	−0.0684729	−0.00282377
$589$	−0.130785	−0.00538890
$590$	−12.8304	−0.528218
$591$	7.72294	0.317679
$592$	−2.99047	−0.122907
$593$	42.0762	1.72786	0.863931	−	0.503611i	$-0.167995\pi$
$593$	42.0762	1.72786	0.863931	+	0.503611i	$0.167995\pi$
$594$	−18.0406	−0.740217
$595$	−1.53014	−0.0627295
$596$	−2.72021	−0.111424
$597$	11.1365	0.455785
$598$	16.7620	0.685450
$599$	20.6840	0.845123	0.422562	−	0.906334i	$-0.361131\pi$
$599$	20.6840	0.845123	0.422562	+	0.906334i	$0.361131\pi$
$600$	1.57612	0.0643450
$601$	12.5213	0.510753	0.255377	−	0.966842i	$-0.417801\pi$
$601$	12.5213	0.510753	0.255377	+	0.966842i	$0.417801\pi$
$602$	8.16101	0.332618
$603$	28.4736	1.15953
$604$	−1.41994	−0.0577764
$605$	−7.19084	−0.292349
$606$	−9.72224	−0.394939
$607$	−27.8454	−1.13021	−0.565105	−	0.825019i	$-0.691164\pi$
$607$	−27.8454	−1.13021	−0.565105	+	0.825019i	$0.691164\pi$
$608$	−0.919626	−0.0372957
$609$	0.549950	0.0222851
$610$	17.0632	0.690867
$611$	15.6404	0.632743
$612$	0.523719	0.0211701
$613$	42.6910	1.72427	0.862137	−	0.506675i	$-0.169125\pi$
$613$	42.6910	1.72427	0.862137	+	0.506675i	$0.169125\pi$
$614$	−17.8341	−0.719724
$615$	1.16789	0.0470940
$616$	−12.4141	−0.500177
$617$	−42.1432	−1.69662	−0.848312	−	0.529497i	$-0.822381\pi$
$617$	−42.1432	−1.69662	−0.848312	+	0.529497i	$0.822381\pi$
$618$	−5.71369	−0.229838
$619$	6.94060	0.278966	0.139483	−	0.990224i	$-0.455456\pi$
$619$	6.94060	0.278966	0.139483	+	0.990224i	$0.455456\pi$
$620$	−0.0128437	−0.000515813 0
$621$	−16.2914	−0.653752
$622$	19.2630	0.772377
$623$	4.87539	0.195328
$624$	4.69319	0.187878
$625$	1.00000	0.0400000
$626$	14.1759	0.566582
$627$	−2.97382	−0.118763
$628$	−2.90853	−0.116063
$629$	1.22640	0.0488997
$630$	−3.70497	−0.147609
$631$	−9.70888	−0.386504	−0.193252	−	0.981149i	$-0.561904\pi$
$631$	−9.70888	−0.386504	−0.193252	+	0.981149i	$0.561904\pi$
$632$	−17.0688	−0.678960
$633$	8.06781	0.320667
$634$	36.6686	1.45630
$635$	13.5654	0.538326
$636$	−0.184489	−0.00731547
$637$	−2.32288	−0.0920361
$638$	5.92898	0.234731
$639$	12.6406	0.500054
$640$	10.1238	0.400179
$641$	−36.4042	−1.43788	−0.718940	−	0.695072i	$-0.755372\pi$
$641$	−36.4042	−1.43788	−0.718940	+	0.695072i	$0.755372\pi$
$642$	12.5150	0.493930
$643$	−9.98850	−0.393908	−0.196954	−	0.980413i	$-0.563105\pi$
$643$	−9.98850	−0.393908	−0.196954	+	0.980413i	$0.563105\pi$
$644$	−0.666626	−0.0262687
$645$	3.22859	0.127126
$646$	−2.69680	−0.106104
$647$	2.42266	0.0952444	0.0476222	−	0.998865i	$-0.484836\pi$
$647$	2.42266	0.0952444	0.0476222	+	0.998865i	$0.484836\pi$
$648$	18.7647	0.737146
$649$	39.9791	1.56932
$650$	3.17951	0.124711
$651$	−0.0550015	−0.00215568
$652$	−2.97619	−0.116557
$653$	13.0698	0.511461	0.255731	−	0.966748i	$-0.417684\pi$
$653$	13.0698	0.511461	0.255731	+	0.966748i	$0.417684\pi$
$654$	5.60256	0.219077
$655$	11.0303	0.430989
$656$	8.04710	0.314186
$657$	11.5561	0.450848
$658$	9.21623	0.359286
$659$	−22.8313	−0.889383	−0.444691	−	0.895684i	$-0.646687\pi$
$659$	−22.8313	−0.889383	−0.444691	+	0.895684i	$0.646687\pi$
$660$	−0.292042	−0.0113677
$661$	2.63451	0.102470	0.0512352	−	0.998687i	$-0.483684\pi$
$661$	2.63451	0.102470	0.0512352	+	0.998687i	$0.483684\pi$
$662$	23.3136	0.906109
$663$	−1.92469	−0.0747487
$664$	−43.5163	−1.68876
$665$	−1.28761	−0.0499315
$666$	2.96951	0.115066
$667$	5.35410	0.207312
$668$	2.09288	0.0809760
$669$	11.7468	0.454158
$670$	−14.3987	−0.556270
$671$	−53.1684	−2.05254
$672$	−0.386748	−0.0149191
$673$	4.12754	0.159105	0.0795526	−	0.996831i	$-0.474651\pi$
$673$	4.12754	0.159105	0.0795526	+	0.996831i	$0.474651\pi$
$674$	−5.90102	−0.227299
$675$	−3.09025	−0.118944
$676$	0.961546	0.0369825
$677$	0.134340	0.00516309	0.00258155	−	0.999997i	$-0.499178\pi$
$677$	0.134340	0.00516309	0.00258155	+	0.999997i	$0.499178\pi$
$678$	−4.46619	−0.171523
$679$	−3.28835	−0.126195
$680$	−4.45367	−0.170790
$681$	11.5174	0.441349
$682$	−0.592968	−0.0227059
$683$	−11.9219	−0.456177	−0.228089	−	0.973640i	$-0.573248\pi$
$683$	−11.9219	−0.456177	−0.228089	+	0.973640i	$0.573248\pi$
$684$	0.440711	0.0168510
$685$	−19.0012	−0.725999
$686$	−1.36878	−0.0522602
$687$	−0.541505	−0.0206597
$688$	22.2459	0.848116
$689$	−6.25864	−0.238435
$690$	3.90752	0.148757
$691$	−9.69838	−0.368944	−0.184472	−	0.982838i	$-0.559057\pi$
$691$	−9.69838	−0.368944	−0.184472	+	0.982838i	$0.559057\pi$
$692$	1.34686	0.0511998
$693$	11.5446	0.438542
$694$	15.7115	0.596400
$695$	15.5694	0.590580
$696$	1.60070	0.0606745
$697$	−3.30013	−0.125001
$698$	25.8989	0.980287
$699$	−2.97184	−0.112405
$700$	−0.126449	−0.00477933
$701$	−23.6706	−0.894026	−0.447013	−	0.894528i	$-0.647512\pi$
$701$	−23.6706	−0.894026	−0.447013	+	0.894528i	$0.647512\pi$
$702$	−9.82547	−0.370838
$703$	1.03202	0.0389232
$704$	−35.9964	−1.35667
$705$	3.64605	0.137318
$706$	0.0472179	0.00177707
$707$	13.1169	0.493312
$708$	0.641838	0.0241217
$709$	14.1926	0.533014	0.266507	−	0.963833i	$-0.414130\pi$
$709$	14.1926	0.533014	0.266507	+	0.963833i	$0.414130\pi$
$710$	−6.39217	−0.239894
$711$	15.8733	0.595295
$712$	14.1905	0.531811
$713$	−0.535474	−0.0200537
$714$	−1.13414	−0.0424440
$715$	−9.90727	−0.370511
$716$	−1.69217	−0.0632393
$717$	2.51255	0.0938330
$718$	−15.7814	−0.588955
$719$	46.3161	1.72730	0.863649	−	0.504094i	$-0.168173\pi$
$719$	46.3161	1.72730	0.863649	+	0.504094i	$0.168173\pi$
$720$	−10.0993	−0.376378
$721$	7.70871	0.287087
$722$	23.7374	0.883415
$723$	5.41634	0.201436
$724$	2.17046	0.0806647
$725$	1.01560	0.0377183
$726$	−5.32985	−0.197809
$727$	44.5465	1.65214	0.826070	−	0.563568i	$-0.190572\pi$
$727$	44.5465	1.65214	0.826070	+	0.563568i	$0.190572\pi$
$728$	−6.76107	−0.250582
$729$	−12.4302	−0.460379
$730$	−5.84377	−0.216288
$731$	−9.12308	−0.337429
$732$	−0.853582	−0.0315493
$733$	−21.0807	−0.778632	−0.389316	−	0.921104i	$-0.627289\pi$
$733$	−21.0807	−0.778632	−0.389316	+	0.921104i	$0.627289\pi$
$734$	−3.24707	−0.119851
$735$	−0.541505	−0.0199737
$736$	−3.76523	−0.138788
$737$	44.8659	1.65266
$738$	−7.99071	−0.294142
$739$	3.96915	0.146008	0.0730038	−	0.997332i	$-0.476741\pi$
$739$	3.96915	0.146008	0.0730038	+	0.997332i	$0.476741\pi$
$740$	0.101348	0.00372564
$741$	−1.61963	−0.0594986
$742$	−3.68795	−0.135389
$743$	−10.1235	−0.371396	−0.185698	−	0.982607i	$-0.559455\pi$
$743$	−10.1235	−0.371396	−0.185698	+	0.982607i	$0.559455\pi$
$744$	−0.160089	−0.00586916
$745$	−21.5122	−0.788147
$746$	−46.9871	−1.72032
$747$	40.4684	1.48066
$748$	0.825226	0.0301732
$749$	−16.8849	−0.616959
$750$	0.741199	0.0270648
$751$	15.5471	0.567321	0.283661	−	0.958925i	$-0.408451\pi$
$751$	15.5471	0.567321	0.283661	+	0.958925i	$0.408451\pi$
$752$	25.1223	0.916115
$753$	−7.89491	−0.287707
$754$	3.22910	0.117597
$755$	−11.2293	−0.408676
$756$	0.390759	0.0142118
$757$	12.7621	0.463846	0.231923	−	0.972734i	$-0.425498\pi$
$757$	12.7621	0.463846	0.231923	+	0.972734i	$0.425498\pi$
$758$	−26.7803	−0.972706
$759$	−12.1757	−0.441951
$760$	−3.74777	−0.135946
$761$	−26.7019	−0.967944	−0.483972	−	0.875083i	$-0.660806\pi$
$761$	−26.7019	−0.967944	−0.483972	+	0.875083i	$0.660806\pi$
$762$	10.0547	0.364242
$763$	−7.55878	−0.273646
$764$	−0.446983	−0.0161713
$765$	4.14173	0.149745
$766$	−41.0936	−1.48477
$767$	21.7738	0.786207
$768$	−1.63664	−0.0590572
$769$	48.4859	1.74844	0.874222	−	0.485526i	$-0.161372\pi$
$769$	48.4859	1.74844	0.874222	+	0.485526i	$0.161372\pi$
$770$	−5.83793	−0.210385
$771$	−0.0240731	−0.000866971 0
$772$	1.37406	0.0494534
$773$	51.7915	1.86281	0.931405	−	0.363984i	$-0.118584\pi$
$773$	51.7915	1.86281	0.931405	+	0.363984i	$0.118584\pi$
$774$	−22.0900	−0.794008
$775$	−0.101572	−0.00364856
$776$	−9.57118	−0.343585
$777$	0.434014	0.0155701
$778$	−42.8049	−1.53463
$779$	−2.77707	−0.0994988
$780$	−0.159055	−0.00569507
$781$	19.9178	0.712716
$782$	−11.0415	−0.394844
$783$	−3.13844	−0.112159
$784$	−3.73111	−0.133254
$785$	−23.0016	−0.820961
$786$	8.17564	0.291616
$787$	−33.0923	−1.17961	−0.589807	−	0.807544i	$-0.700796\pi$
$787$	−33.0923	−1.17961	−0.589807	+	0.807544i	$0.700796\pi$
$788$	−1.80342	−0.0642441
$789$	0.622309	0.0221548
$790$	−8.02690	−0.285584
$791$	6.02563	0.214247
$792$	33.6021	1.19400
$793$	−28.9571	−1.02830
$794$	−16.7337	−0.593857
$795$	−1.45900	−0.0517453
$796$	−2.60052	−0.0921732
$797$	−32.5458	−1.15283	−0.576416	−	0.817156i	$-0.695549\pi$
$797$	−32.5458	−1.15283	−0.576416	+	0.817156i	$0.695549\pi$
$798$	−0.954379	−0.0337846
$799$	−10.3027	−0.364483
$800$	−0.714209	−0.0252511
$801$	−13.1966	−0.466278
$802$	−40.0919	−1.41570
$803$	18.2090	0.642583
$804$	0.720292	0.0254028
$805$	−5.27188	−0.185810
$806$	−0.322948	−0.0113754
$807$	1.41198	0.0497042
$808$	38.1785	1.34312
$809$	15.5182	0.545589	0.272795	−	0.962072i	$-0.412052\pi$
$809$	15.5182	0.545589	0.272795	+	0.962072i	$0.412052\pi$
$810$	8.82442	0.310058
$811$	31.9020	1.12023	0.560115	−	0.828415i	$-0.310757\pi$
$811$	31.9020	1.12023	0.560115	+	0.828415i	$0.310757\pi$
$812$	−0.128421	−0.00450670
$813$	12.3622	0.433562
$814$	4.67908	0.164002
$815$	−23.5366	−0.824453
$816$	−3.09151	−0.108225
$817$	−7.67709	−0.268587
$818$	−34.3543	−1.20117
$819$	6.28752	0.219704
$820$	−0.272720	−0.00952380
$821$	−10.8250	−0.377796	−0.188898	−	0.981997i	$-0.560492\pi$
$821$	−10.8250	−0.377796	−0.188898	+	0.981997i	$0.560492\pi$
$822$	−14.0837	−0.491225
$823$	19.2973	0.672661	0.336330	−	0.941744i	$-0.390814\pi$
$823$	19.2973	0.672661	0.336330	+	0.941744i	$0.390814\pi$
$824$	22.4372	0.781638
$825$	−2.30956	−0.0804085
$826$	12.8304	0.446426
$827$	2.72131	0.0946294	0.0473147	−	0.998880i	$-0.484934\pi$
$827$	2.72131	0.0946294	0.0473147	+	0.998880i	$0.484934\pi$
$828$	1.80440	0.0627074
$829$	−2.99251	−0.103934	−0.0519670	−	0.998649i	$-0.516549\pi$
$829$	−2.99251	−0.103934	−0.0519670	+	0.998649i	$0.516549\pi$
$830$	−20.4643	−0.710326
$831$	−5.57806	−0.193501
$832$	−19.6047	−0.679672
$833$	1.53014	0.0530161
$834$	11.5400	0.399598
$835$	16.5512	0.572776
$836$	0.694430	0.0240173
$837$	0.313881	0.0108493
$838$	27.3269	0.943994
$839$	−47.2142	−1.63001	−0.815007	−	0.579451i	$-0.803267\pi$
$839$	−47.2142	−1.63001	−0.815007	+	0.579451i	$0.803267\pi$
$840$	−1.57612	−0.0543814
$841$	−27.9686	−0.964433
$842$	−35.4456	−1.22154
$843$	−14.0559	−0.484112
$844$	−1.88395	−0.0648483
$845$	7.60421	0.261593
$846$	−24.9462	−0.857669
$847$	7.19084	0.247080
$848$	−10.0529	−0.345217
$849$	3.37275	0.115752
$850$	−2.09442	−0.0718379
$851$	4.22539	0.144844
$852$	0.319767	0.0109550
$853$	6.71791	0.230017	0.115008	−	0.993365i	$-0.463311\pi$
$853$	6.71791	0.230017	0.115008	+	0.993365i	$0.463311\pi$
$854$	−17.0632	−0.583889
$855$	3.48528	0.119194
$856$	−49.1457	−1.67976
$857$	17.5317	0.598872	0.299436	−	0.954116i	$-0.403202\pi$
$857$	17.5317	0.598872	0.299436	+	0.954116i	$0.403202\pi$
$858$	−7.34326	−0.250695
$859$	−27.6551	−0.943581	−0.471790	−	0.881711i	$-0.656392\pi$
$859$	−27.6551	−0.943581	−0.471790	+	0.881711i	$0.656392\pi$
$860$	−0.753923	−0.0257086
$861$	−1.16789	−0.0398017
$862$	30.1240	1.02603
$863$	44.7374	1.52288	0.761440	−	0.648236i	$-0.224493\pi$
$863$	44.7374	1.52288	0.761440	+	0.648236i	$0.224493\pi$
$864$	2.20708	0.0750864
$865$	10.6514	0.362157
$866$	−18.9149	−0.642755
$867$	−7.93775	−0.269580
$868$	0.0128437	0.000435942 0
$869$	25.0116	0.848460
$870$	0.752759	0.0255209
$871$	24.4353	0.827959
$872$	−22.0008	−0.745042
$873$	8.90081	0.301247
$874$	−9.29147	−0.314289
$875$	−1.00000	−0.0338062
$876$	0.292334	0.00987705
$877$	16.0848	0.543146	0.271573	−	0.962418i	$-0.412456\pi$
$877$	16.0848	0.543146	0.271573	+	0.962418i	$0.412456\pi$
$878$	−23.7601	−0.801866
$879$	2.30381	0.0777055
$880$	−15.9135	−0.536443
$881$	−56.7866	−1.91319	−0.956594	−	0.291425i	$-0.905871\pi$
$881$	−56.7866	−1.91319	−0.956594	+	0.291425i	$0.905871\pi$
$882$	3.70497	0.124753
$883$	16.9155	0.569253	0.284626	−	0.958639i	$-0.408130\pi$
$883$	16.9155	0.569253	0.284626	+	0.958639i	$0.408130\pi$
$884$	0.449443	0.0151164
$885$	5.07585	0.170623
$886$	22.4283	0.753492
$887$	−37.4077	−1.25603	−0.628015	−	0.778202i	$-0.716132\pi$
$887$	−37.4077	−1.25603	−0.628015	+	0.778202i	$0.716132\pi$
$888$	1.26326	0.0423921
$889$	−13.5654	−0.454968
$890$	6.67332	0.223690
$891$	−27.4966	−0.921172
$892$	−2.74305	−0.0918442
$893$	−8.66974	−0.290122
$894$	−15.9449	−0.533276
$895$	−13.3822	−0.447318
$896$	−10.1238	−0.338213
$897$	−6.63126	−0.221411
$898$	−49.3076	−1.64541
$899$	−0.103156	−0.00344044
$900$	0.342269	0.0114090
$901$	4.12271	0.137347
$902$	−12.5910	−0.419234
$903$	−3.22859	−0.107441
$904$	17.5384	0.583319
$905$	17.1647	0.570574
$906$	−8.32316	−0.276518
$907$	−11.9845	−0.397939	−0.198969	−	0.980006i	$-0.563759\pi$
$907$	−11.9845	−0.397939	−0.198969	+	0.980006i	$0.563759\pi$
$908$	−2.68949	−0.0892537
$909$	−35.5045	−1.17761
$910$	−3.17951	−0.105400
$911$	39.6318	1.31306	0.656530	−	0.754300i	$-0.272024\pi$
$911$	39.6318	1.31306	0.656530	+	0.754300i	$0.272024\pi$
$912$	−2.60151	−0.0861448
$913$	63.7662	2.11035
$914$	−1.67001	−0.0552389
$915$	−6.75040	−0.223161
$916$	0.126449	0.00417800
$917$	−11.0303	−0.364252
$918$	6.47226	0.213616
$919$	28.7161	0.947256	0.473628	−	0.880725i	$-0.342944\pi$
$919$	28.7161	0.947256	0.473628	+	0.880725i	$0.342944\pi$
$920$	−15.3445	−0.505894
$921$	7.05537	0.232483
$922$	33.3418	1.09805
$923$	10.8478	0.357061
$924$	0.292042	0.00960747
$925$	0.801495	0.0263530
$926$	5.29771	0.174094
$927$	−20.8657	−0.685320
$928$	−0.725348	−0.0238107
$929$	2.32011	0.0761205	0.0380602	−	0.999275i	$-0.487882\pi$
$929$	2.32011	0.0761205	0.0380602	+	0.999275i	$0.487882\pi$
$930$	−0.0752848	−0.00246869
$931$	1.28761	0.0421998
$932$	0.693967	0.0227316
$933$	−7.62069	−0.249490
$934$	56.7587	1.85720
$935$	6.52614	0.213428
$936$	18.3007	0.598176
$937$	−0.981393	−0.0320607	−0.0160304	−	0.999872i	$-0.505103\pi$
$937$	−0.981393	−0.0320607	−0.0160304	+	0.999872i	$0.505103\pi$
$938$	14.3987	0.470134
$939$	−5.60815	−0.183015
$940$	−0.851406	−0.0277698
$941$	21.5921	0.703881	0.351941	−	0.936022i	$-0.385522\pi$
$941$	21.5921	0.703881	0.351941	+	0.936022i	$0.385522\pi$
$942$	−17.0488	−0.555479
$943$	−11.3702	−0.370264
$944$	34.9740	1.13831
$945$	3.09025	0.100526
$946$	−34.8073	−1.13168
$947$	−5.03825	−0.163721	−0.0818606	−	0.996644i	$-0.526086\pi$
$947$	−5.03825	−0.163721	−0.0818606	+	0.996644i	$0.526086\pi$
$948$	0.401544	0.0130416
$949$	9.91719	0.321925
$950$	−1.76246	−0.0571816
$951$	−14.5065	−0.470407
$952$	4.45367	0.144344
$953$	38.6171	1.25093	0.625466	−	0.780252i	$-0.284909\pi$
$953$	38.6171	1.25093	0.625466	+	0.780252i	$0.284909\pi$
$954$	9.98244	0.323193
$955$	−3.53488	−0.114386
$956$	−0.586718	−0.0189758
$957$	−2.34558	−0.0758217
$958$	−33.9544	−1.09702
$959$	19.0012	0.613581
$960$	−4.57020	−0.147503
$961$	−30.9897	−0.999667
$962$	2.54836	0.0821625
$963$	45.7035	1.47277
$964$	−1.26479	−0.0407362
$965$	10.8665	0.349804
$966$	−3.90752	−0.125722
$967$	−3.38874	−0.108974	−0.0544872	−	0.998514i	$-0.517352\pi$
$967$	−3.38874	−0.108974	−0.0544872	+	0.998514i	$0.517352\pi$
$968$	20.9299	0.672713
$969$	1.06689	0.0342733
$970$	−4.50101	−0.144519
$971$	23.8778	0.766274	0.383137	−	0.923692i	$-0.374844\pi$
$971$	23.8778	0.766274	0.383137	+	0.923692i	$0.374844\pi$
$972$	−1.61372	−0.0517600
$973$	−15.5694	−0.499131
$974$	−52.6284	−1.68632
$975$	−1.25785	−0.0402836
$976$	−46.5120	−1.48881
$977$	1.83886	0.0588305	0.0294152	−	0.999567i	$-0.490635\pi$
$977$	1.83886	0.0588305	0.0294152	+	0.999567i	$0.490635\pi$
$978$	−17.4453	−0.557841
$979$	−20.7939	−0.664575
$980$	0.126449	0.00403927
$981$	20.4599	0.653234
$982$	50.7371	1.61909
$983$	13.4874	0.430180	0.215090	−	0.976594i	$-0.430995\pi$
$983$	13.4874	0.430180	0.215090	+	0.976594i	$0.430995\pi$
$984$	−3.39931	−0.108366
$985$	−14.2620	−0.454425
$986$	−2.12708	−0.0677400
$987$	−3.64605	−0.116055
$988$	0.378207	0.0120324
$989$	−31.4323	−0.999491
$990$	15.8020	0.502219
$991$	16.4896	0.523811	0.261905	−	0.965094i	$-0.415649\pi$
$991$	16.4896	0.523811	0.261905	+	0.965094i	$0.415649\pi$
$992$	0.0725434	0.00230325
$993$	−9.22315	−0.292688
$994$	6.39217	0.202747
$995$	−20.5658	−0.651979
$996$	1.02372	0.0324379
$997$	12.3344	0.390633	0.195317	−	0.980740i	$-0.437427\pi$
$997$	12.3344	0.390633	0.195317	+	0.980740i	$0.437427\pi$
$998$	−15.0239	−0.475572
$999$	−2.47682	−0.0783630

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8015.2.a.n.1.20	✓	68

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8015.2.a.n.1.20	✓	68	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:	\( N \)	\(=\)	\( 8015 = 5 \cdot 7 \cdot 229 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8015.a (trivial)

\(f(q)\)	\(=\)	\(q-1.36878 q^{2} +0.541505 q^{3} -0.126449 q^{4} -1.00000 q^{5} -0.741199 q^{6} +1.00000 q^{7} +2.91064 q^{8} -2.70677 q^{9} +O(q^{10})\)	\(q-1.36878 q^{2} +0.541505 q^{3} -0.126449 q^{4} -1.00000 q^{5} -0.741199 q^{6} +1.00000 q^{7} +2.91064 q^{8} -2.70677 q^{9} +1.36878 q^{10} -4.26507 q^{11} -0.0684729 q^{12} -2.32288 q^{13} -1.36878 q^{14} -0.541505 q^{15} -3.73111 q^{16} +1.53014 q^{17} +3.70497 q^{18} +1.28761 q^{19} +0.126449 q^{20} +0.541505 q^{21} +5.83793 q^{22} +5.27188 q^{23} +1.57612 q^{24} +1.00000 q^{25} +3.17951 q^{26} -3.09025 q^{27} -0.126449 q^{28} +1.01560 q^{29} +0.741199 q^{30} -0.101572 q^{31} -0.714209 q^{32} -2.30956 q^{33} -2.09442 q^{34} -1.00000 q^{35} +0.342269 q^{36} +0.801495 q^{37} -1.76246 q^{38} -1.25785 q^{39} -2.91064 q^{40} -2.15676 q^{41} -0.741199 q^{42} -5.96226 q^{43} +0.539315 q^{44} +2.70677 q^{45} -7.21603 q^{46} -6.73318 q^{47} -2.02042 q^{48} +1.00000 q^{49} -1.36878 q^{50} +0.828577 q^{51} +0.293727 q^{52} +2.69434 q^{53} +4.22986 q^{54} +4.26507 q^{55} +2.91064 q^{56} +0.697249 q^{57} -1.39012 q^{58} -9.37360 q^{59} +0.0684729 q^{60} +12.4660 q^{61} +0.139029 q^{62} -2.70677 q^{63} +8.43982 q^{64} +2.32288 q^{65} +3.16127 q^{66} -10.5194 q^{67} -0.193485 q^{68} +2.85475 q^{69} +1.36878 q^{70} -4.66998 q^{71} -7.87843 q^{72} -4.26934 q^{73} -1.09707 q^{74} +0.541505 q^{75} -0.162818 q^{76} -4.26507 q^{77} +1.72172 q^{78} -5.86428 q^{79} +3.73111 q^{80} +6.44693 q^{81} +2.95212 q^{82} -14.9508 q^{83} -0.0684729 q^{84} -1.53014 q^{85} +8.16101 q^{86} +0.549950 q^{87} -12.4141 q^{88} +4.87539 q^{89} -3.70497 q^{90} -2.32288 q^{91} -0.666626 q^{92} -0.0550015 q^{93} +9.21623 q^{94} -1.28761 q^{95} -0.386748 q^{96} -3.28835 q^{97} -1.36878 q^{98} +11.5446 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q + 9 q^{2} + 83 q^{4} - 68 q^{5} + 5 q^{6} + 68 q^{7} + 30 q^{8} + 86 q^{9}+O(q^{10}) \) 68 * q + 9 * q^2 + 83 * q^4 - 68 * q^5 + 5 * q^6 + 68 * q^7 + 30 * q^8 + 86 * q^9	\( 68 q + 9 q^{2} + 83 q^{4} - 68 q^{5} + 5 q^{6} + 68 q^{7} + 30 q^{8} + 86 q^{9} - 9 q^{10} + 5 q^{11} + 9 q^{12} + 15 q^{13} + 9 q^{14} + 109 q^{16} + 7 q^{17} + 39 q^{18} + 20 q^{19} - 83 q^{20} + 56 q^{22} + 36 q^{23} + q^{24} + 68 q^{25} + q^{26} + 12 q^{27} + 83 q^{28} - 16 q^{29} - 5 q^{30} + 31 q^{31} + 79 q^{32} + 45 q^{33} + 31 q^{34} - 68 q^{35} + 114 q^{36} + 72 q^{37} + 8 q^{38} + 47 q^{39} - 30 q^{40} + 6 q^{41} + 5 q^{42} + 75 q^{43} + 15 q^{44} - 86 q^{45} + 29 q^{46} - 10 q^{47} + 44 q^{48} + 68 q^{49} + 9 q^{50} + 23 q^{51} + 37 q^{52} + 41 q^{53} + 4 q^{54} - 5 q^{55} + 30 q^{56} + 55 q^{57} + 66 q^{58} - 5 q^{59} - 9 q^{60} - 2 q^{61} + 3 q^{62} + 86 q^{63} + 162 q^{64} - 15 q^{65} - 23 q^{66} + 92 q^{67} + 35 q^{68} - 25 q^{69} - 9 q^{70} - 2 q^{71} + 128 q^{72} + 80 q^{73} + 18 q^{74} + 71 q^{76} + 5 q^{77} + 20 q^{78} + 100 q^{79} - 109 q^{80} + 140 q^{81} + 36 q^{82} - 60 q^{83} + 9 q^{84} - 7 q^{85} - 27 q^{86} + 24 q^{87} + 175 q^{88} + 19 q^{89} - 39 q^{90} + 15 q^{91} + 75 q^{92} + 37 q^{93} + 11 q^{94} - 20 q^{95} + 15 q^{96} + 96 q^{97} + 9 q^{98} + 3 q^{99}+O(q^{100}) \) 68 * q + 9 * q^2 + 83 * q^4 - 68 * q^5 + 5 * q^6 + 68 * q^7 + 30 * q^8 + 86 * q^9 - 9 * q^10 + 5 * q^11 + 9 * q^12 + 15 * q^13 + 9 * q^14 + 109 * q^16 + 7 * q^17 + 39 * q^18 + 20 * q^19 - 83 * q^20 + 56 * q^22 + 36 * q^23 + q^24 + 68 * q^25 + q^26 + 12 * q^27 + 83 * q^28 - 16 * q^29 - 5 * q^30 + 31 * q^31 + 79 * q^32 + 45 * q^33 + 31 * q^34 - 68 * q^35 + 114 * q^36 + 72 * q^37 + 8 * q^38 + 47 * q^39 - 30 * q^40 + 6 * q^41 + 5 * q^42 + 75 * q^43 + 15 * q^44 - 86 * q^45 + 29 * q^46 - 10 * q^47 + 44 * q^48 + 68 * q^49 + 9 * q^50 + 23 * q^51 + 37 * q^52 + 41 * q^53 + 4 * q^54 - 5 * q^55 + 30 * q^56 + 55 * q^57 + 66 * q^58 - 5 * q^59 - 9 * q^60 - 2 * q^61 + 3 * q^62 + 86 * q^63 + 162 * q^64 - 15 * q^65 - 23 * q^66 + 92 * q^67 + 35 * q^68 - 25 * q^69 - 9 * q^70 - 2 * q^71 + 128 * q^72 + 80 * q^73 + 18 * q^74 + 71 * q^76 + 5 * q^77 + 20 * q^78 + 100 * q^79 - 109 * q^80 + 140 * q^81 + 36 * q^82 - 60 * q^83 + 9 * q^84 - 7 * q^85 - 27 * q^86 + 24 * q^87 + 175 * q^88 + 19 * q^89 - 39 * q^90 + 15 * q^91 + 75 * q^92 + 37 * q^93 + 11 * q^94 - 20 * q^95 + 15 * q^96 + 96 * q^97 + 9 * q^98 + 3 * q^99

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