LMFDB - Embedded newform 8015.2.a.n.1.19

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			1.19
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)

$f(q)$	$=$	$q-1.37224 q^{2} +1.58108 q^{3} -0.116968 q^{4} -1.00000 q^{5} -2.16961 q^{6} +1.00000 q^{7} +2.90498 q^{8} -0.500189 q^{9} +O(q^{10})$	\(q-1.37224 q^{2} +1.58108 q^{3} -0.116968 q^{4} -1.00000 q^{5} -2.16961 q^{6} +1.00000 q^{7} +2.90498 q^{8} -0.500189 q^{9} +1.37224 q^{10} +1.81164 q^{11} -0.184935 q^{12} +1.94632 q^{13} -1.37224 q^{14} -1.58108 q^{15} -3.75238 q^{16} +7.31807 q^{17} +0.686378 q^{18} -6.35160 q^{19} +0.116968 q^{20} +1.58108 q^{21} -2.48600 q^{22} -0.698834 q^{23} +4.59300 q^{24} +1.00000 q^{25} -2.67081 q^{26} -5.53408 q^{27} -0.116968 q^{28} +4.09354 q^{29} +2.16961 q^{30} -4.08330 q^{31} -0.660802 q^{32} +2.86435 q^{33} -10.0421 q^{34} -1.00000 q^{35} +0.0585059 q^{36} -5.67222 q^{37} +8.71590 q^{38} +3.07729 q^{39} -2.90498 q^{40} +1.02976 q^{41} -2.16961 q^{42} +0.232334 q^{43} -0.211903 q^{44} +0.500189 q^{45} +0.958966 q^{46} +0.646523 q^{47} -5.93281 q^{48} +1.00000 q^{49} -1.37224 q^{50} +11.5705 q^{51} -0.227656 q^{52} +9.34547 q^{53} +7.59406 q^{54} -1.81164 q^{55} +2.90498 q^{56} -10.0424 q^{57} -5.61730 q^{58} +7.36958 q^{59} +0.184935 q^{60} +7.98253 q^{61} +5.60326 q^{62} -0.500189 q^{63} +8.41154 q^{64} -1.94632 q^{65} -3.93056 q^{66} +9.98438 q^{67} -0.855977 q^{68} -1.10491 q^{69} +1.37224 q^{70} +6.02018 q^{71} -1.45304 q^{72} +2.55398 q^{73} +7.78363 q^{74} +1.58108 q^{75} +0.742931 q^{76} +1.81164 q^{77} -4.22276 q^{78} +3.79056 q^{79} +3.75238 q^{80} -7.24924 q^{81} -1.41308 q^{82} -10.2491 q^{83} -0.184935 q^{84} -7.31807 q^{85} -0.318818 q^{86} +6.47220 q^{87} +5.26278 q^{88} +8.04567 q^{89} -0.686378 q^{90} +1.94632 q^{91} +0.0817409 q^{92} -6.45603 q^{93} -0.887182 q^{94} +6.35160 q^{95} -1.04478 q^{96} -8.16205 q^{97} -1.37224 q^{98} -0.906162 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.37224	−0.970318	−0.485159	−	0.874426i	$-0.661238\pi$
$2$	−1.37224	−0.970318	−0.485159	+	0.874426i	$0.661238\pi$
$3$	1.58108	0.912836	0.456418	−	0.889765i	$-0.349132\pi$
$3$	1.58108	0.912836	0.456418	+	0.889765i	$0.349132\pi$
$4$	−0.116968	−0.0584838
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.16961	−0.885741
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.90498	1.02707
$9$	−0.500189	−0.166730
$10$	1.37224	0.433939
$11$	1.81164	0.546230	0.273115	−	0.961981i	$-0.411946\pi$
$11$	1.81164	0.546230	0.273115	+	0.961981i	$0.411946\pi$
$12$	−0.184935	−0.0533861
$13$	1.94632	0.539812	0.269906	−	0.962887i	$-0.413007\pi$
$13$	1.94632	0.539812	0.269906	+	0.962887i	$0.413007\pi$
$14$	−1.37224	−0.366746
$15$	−1.58108	−0.408233
$16$	−3.75238	−0.938096
$17$	7.31807	1.77489	0.887447	−	0.460910i	$-0.152477\pi$
$17$	7.31807	1.77489	0.887447	+	0.460910i	$0.152477\pi$
$18$	0.686378	0.161781
$19$	−6.35160	−1.45716	−0.728579	−	0.684962i	$-0.759819\pi$
$19$	−6.35160	−1.45716	−0.728579	+	0.684962i	$0.759819\pi$
$20$	0.116968	0.0261547
$21$	1.58108	0.345020
$22$	−2.48600	−0.530016
$23$	−0.698834	−0.145717	−0.0728585	−	0.997342i	$-0.523212\pi$
$23$	−0.698834	−0.145717	−0.0728585	+	0.997342i	$0.523212\pi$
$24$	4.59300	0.937543
$25$	1.00000	0.200000
$26$	−2.67081	−0.523789
$27$	−5.53408	−1.06503
$28$	−0.116968	−0.0221048
$29$	4.09354	0.760150	0.380075	−	0.924956i	$-0.375898\pi$
$29$	4.09354	0.760150	0.380075	+	0.924956i	$0.375898\pi$
$30$	2.16961	0.396116
$31$	−4.08330	−0.733383	−0.366692	−	0.930343i	$-0.619510\pi$
$31$	−4.08330	−0.733383	−0.366692	+	0.930343i	$0.619510\pi$
$32$	−0.660802	−0.116814
$33$	2.86435	0.498619
$34$	−10.0421	−1.72221
$35$	−1.00000	−0.169031
$36$	0.0585059	0.00975098
$37$	−5.67222	−0.932508	−0.466254	−	0.884651i	$-0.654397\pi$
$37$	−5.67222	−0.932508	−0.466254	+	0.884651i	$0.654397\pi$
$38$	8.71590	1.41391
$39$	3.07729	0.492760
$40$	−2.90498	−0.459318
$41$	1.02976	0.160822	0.0804110	−	0.996762i	$-0.474377\pi$
$41$	1.02976	0.160822	0.0804110	+	0.996762i	$0.474377\pi$
$42$	−2.16961	−0.334779
$43$	0.232334	0.0354307	0.0177153	−	0.999843i	$-0.494361\pi$
$43$	0.232334	0.0354307	0.0177153	+	0.999843i	$0.494361\pi$
$44$	−0.211903	−0.0319456
$45$	0.500189	0.0745638
$46$	0.958966	0.141392
$47$	0.646523	0.0943050	0.0471525	−	0.998888i	$-0.484985\pi$
$47$	0.646523	0.0943050	0.0471525	+	0.998888i	$0.484985\pi$
$48$	−5.93281	−0.856328
$49$	1.00000	0.142857
$50$	−1.37224	−0.194064
$51$	11.5705	1.62019
$52$	−0.227656	−0.0315702
$53$	9.34547	1.28370	0.641850	−	0.766831i	$-0.278167\pi$
$53$	9.34547	1.28370	0.641850	+	0.766831i	$0.278167\pi$
$54$	7.59406	1.03342
$55$	−1.81164	−0.244281
$56$	2.90498	0.388194
$57$	−10.0424	−1.33015
$58$	−5.61730	−0.737587
$59$	7.36958	0.959437	0.479718	−	0.877422i	$-0.340739\pi$
$59$	7.36958	0.959437	0.479718	+	0.877422i	$0.340739\pi$
$60$	0.184935	0.0238750
$61$	7.98253	1.02206	0.511029	−	0.859563i	$-0.329264\pi$
$61$	7.98253	1.02206	0.511029	+	0.859563i	$0.329264\pi$
$62$	5.60326	0.711615
$63$	−0.500189	−0.0630179
$64$	8.41154	1.05144
$65$	−1.94632	−0.241411
$66$	−3.93056	−0.483818
$67$	9.98438	1.21979	0.609893	−	0.792484i	$-0.291212\pi$
$67$	9.98438	1.21979	0.609893	+	0.792484i	$0.291212\pi$
$68$	−0.855977	−0.103802
$69$	−1.10491	−0.133016
$70$	1.37224	0.164014
$71$	6.02018	0.714464	0.357232	−	0.934016i	$-0.383720\pi$
$71$	6.02018	0.714464	0.357232	+	0.934016i	$0.383720\pi$
$72$	−1.45304	−0.171242
$73$	2.55398	0.298921	0.149460	−	0.988768i	$-0.452246\pi$
$73$	2.55398	0.298921	0.149460	+	0.988768i	$0.452246\pi$
$74$	7.78363	0.904829
$75$	1.58108	0.182567
$76$	0.742931	0.0852200
$77$	1.81164	0.206455
$78$	−4.22276	−0.478134
$79$	3.79056	0.426471	0.213236	−	0.977001i	$-0.431600\pi$
$79$	3.79056	0.426471	0.213236	+	0.977001i	$0.431600\pi$
$80$	3.75238	0.419529
$81$	−7.24924	−0.805471
$82$	−1.41308	−0.156048
$83$	−10.2491	−1.12499	−0.562494	−	0.826801i	$-0.690158\pi$
$83$	−10.2491	−1.12499	−0.562494	+	0.826801i	$0.690158\pi$
$84$	−0.184935	−0.0201780
$85$	−7.31807	−0.793756
$86$	−0.318818	−0.0343790
$87$	6.47220	0.693893
$88$	5.26278	0.561014
$89$	8.04567	0.852839	0.426419	−	0.904526i	$-0.359775\pi$
$89$	8.04567	0.852839	0.426419	+	0.904526i	$0.359775\pi$
$90$	−0.686378	−0.0723506
$91$	1.94632	0.204030
$92$	0.0817409	0.00852208
$93$	−6.45603	−0.669459
$94$	−0.887182	−0.0915058
$95$	6.35160	0.651661
$96$	−1.04478	−0.106632
$97$	−8.16205	−0.828731	−0.414365	−	0.910111i	$-0.635996\pi$
$97$	−8.16205	−0.828731	−0.414365	+	0.910111i	$0.635996\pi$
$98$	−1.37224	−0.138617
$99$	−0.906162	−0.0910728
$100$	−0.116968	−0.0116968
$101$	4.19695	0.417612	0.208806	−	0.977957i	$-0.433042\pi$
$101$	4.19695	0.417612	0.208806	+	0.977957i	$0.433042\pi$
$102$	−15.8774	−1.57210
$103$	−14.9951	−1.47752	−0.738758	−	0.673971i	$-0.764587\pi$
$103$	−14.9951	−1.47752	−0.738758	+	0.673971i	$0.764587\pi$
$104$	5.65402	0.554422
$105$	−1.58108	−0.154298
$106$	−12.8242	−1.24560
$107$	−16.9815	−1.64166	−0.820832	−	0.571170i	$-0.806490\pi$
$107$	−16.9815	−1.64166	−0.820832	+	0.571170i	$0.806490\pi$
$108$	0.647307	0.0622871
$109$	15.0760	1.44402	0.722010	−	0.691882i	$-0.243218\pi$
$109$	15.0760	1.44402	0.722010	+	0.691882i	$0.243218\pi$
$110$	2.48600	0.237031
$111$	−8.96823	−0.851227
$112$	−3.75238	−0.354567
$113$	17.9510	1.68869	0.844345	−	0.535800i	$-0.179990\pi$
$113$	17.9510	1.68869	0.844345	+	0.535800i	$0.179990\pi$
$114$	13.7805	1.29066
$115$	0.698834	0.0651666
$116$	−0.478811	−0.0444564
$117$	−0.973529	−0.0900027
$118$	−10.1128	−0.930959
$119$	7.31807	0.670847
$120$	−4.59300	−0.419282
$121$	−7.71796	−0.701633
$122$	−10.9539	−0.991721
$123$	1.62814	0.146804
$124$	0.477614	0.0428910
$125$	−1.00000	−0.0894427
$126$	0.686378	0.0611474
$127$	8.45295	0.750078	0.375039	−	0.927009i	$-0.377629\pi$
$127$	8.45295	0.750078	0.375039	+	0.927009i	$0.377629\pi$
$128$	−10.2210	−0.903419
$129$	0.367339	0.0323424
$130$	2.67081	0.234246
$131$	6.72759	0.587792	0.293896	−	0.955837i	$-0.405048\pi$
$131$	6.72759	0.587792	0.293896	+	0.955837i	$0.405048\pi$
$132$	−0.335035	−0.0291611
$133$	−6.35160	−0.550754
$134$	−13.7009	−1.18358
$135$	5.53408	0.476297
$136$	21.2589	1.82293
$137$	−5.92107	−0.505872	−0.252936	−	0.967483i	$-0.581396\pi$
$137$	−5.92107	−0.505872	−0.252936	+	0.967483i	$0.581396\pi$
$138$	1.51620	0.129068
$139$	−7.62831	−0.647025	−0.323512	−	0.946224i	$-0.604864\pi$
$139$	−7.62831	−0.647025	−0.323512	+	0.946224i	$0.604864\pi$
$140$	0.116968	0.00988556
$141$	1.02220	0.0860851
$142$	−8.26112	−0.693257
$143$	3.52603	0.294862
$144$	1.87690	0.156408
$145$	−4.09354	−0.339950
$146$	−3.50466	−0.290048
$147$	1.58108	0.130405
$148$	0.663466	0.0545366
$149$	16.1230	1.32085	0.660425	−	0.750892i	$-0.270376\pi$
$149$	16.1230	1.32085	0.660425	+	0.750892i	$0.270376\pi$
$150$	−2.16961	−0.177148
$151$	−17.5486	−1.42809	−0.714044	−	0.700100i	$-0.753139\pi$
$151$	−17.5486	−1.42809	−0.714044	+	0.700100i	$0.753139\pi$
$152$	−18.4513	−1.49660
$153$	−3.66042	−0.295927
$154$	−2.48600	−0.200327
$155$	4.08330	0.327979
$156$	−0.359943	−0.0288185
$157$	−12.2270	−0.975824	−0.487912	−	0.872893i	$-0.662241\pi$
$157$	−12.2270	−0.975824	−0.487912	+	0.872893i	$0.662241\pi$
$158$	−5.20154	−0.413812
$159$	14.7759	1.17181
$160$	0.660802	0.0522410
$161$	−0.698834	−0.0550759
$162$	9.94767	0.781563
$163$	−4.34906	−0.340644	−0.170322	−	0.985388i	$-0.554481\pi$
$163$	−4.34906	−0.340644	−0.170322	+	0.985388i	$0.554481\pi$
$164$	−0.120449	−0.00940547
$165$	−2.86435	−0.222989
$166$	14.0642	1.09160
$167$	−10.4032	−0.805022	−0.402511	−	0.915415i	$-0.631862\pi$
$167$	−10.4032	−0.805022	−0.402511	+	0.915415i	$0.631862\pi$
$168$	4.59300	0.354358
$169$	−9.21184	−0.708603
$170$	10.0421	0.770196
$171$	3.17700	0.242951
$172$	−0.0271756	−0.00207212
$173$	13.2348	1.00622	0.503112	−	0.864221i	$-0.332188\pi$
$173$	13.2348	1.00622	0.503112	+	0.864221i	$0.332188\pi$
$174$	−8.88139	−0.673297
$175$	1.00000	0.0755929
$176$	−6.79797	−0.512416
$177$	11.6519	0.875809
$178$	−11.0406	−0.827525
$179$	−15.4347	−1.15364	−0.576821	−	0.816871i	$-0.695707\pi$
$179$	−15.4347	−1.15364	−0.576821	+	0.816871i	$0.695707\pi$
$180$	−0.0585059	−0.00436077
$181$	14.8518	1.10393	0.551963	−	0.833869i	$-0.313879\pi$
$181$	14.8518	1.10393	0.551963	+	0.833869i	$0.313879\pi$
$182$	−2.67081	−0.197974
$183$	12.6210	0.932972
$184$	−2.03010	−0.149661
$185$	5.67222	0.417030
$186$	8.85920	0.649588
$187$	13.2577	0.969500
$188$	−0.0756222	−0.00551531
$189$	−5.53408	−0.402545
$190$	−8.71590	−0.632318
$191$	−13.0782	−0.946306	−0.473153	−	0.880980i	$-0.656884\pi$
$191$	−13.0782	−0.946306	−0.473153	+	0.880980i	$0.656884\pi$
$192$	13.2993	0.959795
$193$	21.4936	1.54715	0.773573	−	0.633708i	$-0.218468\pi$
$193$	21.4936	1.54715	0.773573	+	0.633708i	$0.218468\pi$
$194$	11.2003	0.804132
$195$	−3.07729	−0.220369
$196$	−0.116968	−0.00835482
$197$	5.19039	0.369800	0.184900	−	0.982757i	$-0.440804\pi$
$197$	5.19039	0.369800	0.184900	+	0.982757i	$0.440804\pi$
$198$	1.24347	0.0883695
$199$	−21.5625	−1.52852	−0.764261	−	0.644907i	$-0.776896\pi$
$199$	−21.5625	−1.52852	−0.764261	+	0.644907i	$0.776896\pi$
$200$	2.90498	0.205413
$201$	15.7861	1.11347
$202$	−5.75921	−0.405216
$203$	4.09354	0.287310
$204$	−1.35337	−0.0947546
$205$	−1.02976	−0.0719218
$206$	20.5769	1.43366
$207$	0.349549	0.0242954
$208$	−7.30334	−0.506396
$209$	−11.5068	−0.795943
$210$	2.16961	0.149718
$211$	−2.01836	−0.138949	−0.0694747	−	0.997584i	$-0.522132\pi$
$211$	−2.01836	−0.138949	−0.0694747	+	0.997584i	$0.522132\pi$
$212$	−1.09312	−0.0750755
$213$	9.51839	0.652189
$214$	23.3026	1.59293
$215$	−0.232334	−0.0158451
$216$	−16.0764	−1.09386
$217$	−4.08330	−0.277193
$218$	−20.6879	−1.40116
$219$	4.03804	0.272866
$220$	0.211903	0.0142865
$221$	14.2433	0.958109
$222$	12.3065	0.825961
$223$	0.207287	0.0138809	0.00694047	−	0.999976i	$-0.497791\pi$
$223$	0.207287	0.0138809	0.00694047	+	0.999976i	$0.497791\pi$
$224$	−0.660802	−0.0441517
$225$	−0.500189	−0.0333459
$226$	−24.6330	−1.63857
$227$	−5.51669	−0.366156	−0.183078	−	0.983098i	$-0.558606\pi$
$227$	−5.51669	−0.366156	−0.183078	+	0.983098i	$0.558606\pi$
$228$	1.17463	0.0777920
$229$	−1.00000	−0.0660819
$229$	−1.00000	−0.0660819
$230$	−0.958966	−0.0632323
$231$	2.86435	0.188460
$232$	11.8916	0.780724
$233$	27.0971	1.77519	0.887594	−	0.460626i	$-0.152375\pi$
$233$	27.0971	1.77519	0.887594	+	0.460626i	$0.152375\pi$
$234$	1.33591	0.0873312
$235$	−0.646523	−0.0421745
$236$	−0.862001	−0.0561115
$237$	5.99317	0.389298
$238$	−10.0421	−0.650934
$239$	−7.81618	−0.505586	−0.252793	−	0.967520i	$-0.581349\pi$
$239$	−7.81618	−0.505586	−0.252793	+	0.967520i	$0.581349\pi$
$240$	5.93281	0.382962
$241$	−3.17490	−0.204514	−0.102257	−	0.994758i	$-0.532606\pi$
$241$	−3.17490	−0.204514	−0.102257	+	0.994758i	$0.532606\pi$
$242$	10.5909	0.680807
$243$	5.14060	0.329770
$244$	−0.933696	−0.0597738
$245$	−1.00000	−0.0638877
$246$	−2.23419	−0.142447
$247$	−12.3623	−0.786591
$248$	−11.8619	−0.753233
$249$	−16.2047	−1.02693
$250$	1.37224	0.0867878
$251$	12.4149	0.783619	0.391810	−	0.920046i	$-0.371849\pi$
$251$	12.4149	0.783619	0.391810	+	0.920046i	$0.371849\pi$
$252$	0.0585059	0.00368552
$253$	−1.26604	−0.0795950
$254$	−11.5995	−0.727814
$255$	−11.5705	−0.724570
$256$	−2.79743	−0.174839
$257$	−27.2662	−1.70082	−0.850410	−	0.526120i	$-0.823646\pi$
$257$	−27.2662	−1.70082	−0.850410	+	0.526120i	$0.823646\pi$
$258$	−0.504076	−0.0313824
$259$	−5.67222	−0.352455
$260$	0.227656	0.0141186
$261$	−2.04754	−0.126740
$262$	−9.23184	−0.570345
$263$	−9.02067	−0.556238	−0.278119	−	0.960547i	$-0.589711\pi$
$263$	−9.02067	−0.556238	−0.278119	+	0.960547i	$0.589711\pi$
$264$	8.32086	0.512114
$265$	−9.34547	−0.574088
$266$	8.71590	0.534406
$267$	12.7208	0.778502
$268$	−1.16785	−0.0713377
$269$	22.1937	1.35317	0.676586	−	0.736363i	$-0.263459\pi$
$269$	22.1937	1.35317	0.676586	+	0.736363i	$0.263459\pi$
$270$	−7.59406	−0.462160
$271$	27.2588	1.65586	0.827928	−	0.560834i	$-0.189519\pi$
$271$	27.2588	1.65586	0.827928	+	0.560834i	$0.189519\pi$
$272$	−27.4602	−1.66502
$273$	3.07729	0.186246
$274$	8.12511	0.490856
$275$	1.81164	0.109246
$276$	0.129239	0.00777926
$277$	11.2954	0.678674	0.339337	−	0.940665i	$-0.389797\pi$
$277$	11.2954	0.678674	0.339337	+	0.940665i	$0.389797\pi$
$278$	10.4678	0.627819
$279$	2.04242	0.122277
$280$	−2.90498	−0.173606
$281$	9.29222	0.554327	0.277164	−	0.960823i	$-0.410606\pi$
$281$	9.29222	0.554327	0.277164	+	0.960823i	$0.410606\pi$
$282$	−1.40270	−0.0835299
$283$	24.8921	1.47968	0.739840	−	0.672783i	$-0.234901\pi$
$283$	24.8921	1.47968	0.739840	+	0.672783i	$0.234901\pi$
$284$	−0.704166	−0.0417846
$285$	10.0424	0.594860
$286$	−4.83855	−0.286109
$287$	1.02976	0.0607850
$288$	0.330526	0.0194764
$289$	36.5542	2.15025
$290$	5.61730	0.329859
$291$	−12.9048	−0.756496
$292$	−0.298733	−0.0174820
$293$	−15.9484	−0.931715	−0.465858	−	0.884860i	$-0.654254\pi$
$293$	−15.9484	−0.931715	−0.465858	+	0.884860i	$0.654254\pi$
$294$	−2.16961	−0.126534
$295$	−7.36958	−0.429073
$296$	−16.4777	−0.957747
$297$	−10.0258	−0.581753
$298$	−22.1246	−1.28164
$299$	−1.36016	−0.0786598
$300$	−0.184935	−0.0106772
$301$	0.232334	0.0133915
$302$	24.0809	1.38570
$303$	6.63571	0.381212
$304$	23.8336	1.36695
$305$	−7.98253	−0.457078
$306$	5.02296	0.287144
$307$	6.33924	0.361800	0.180900	−	0.983502i	$-0.442099\pi$
$307$	6.33924	0.361800	0.180900	+	0.983502i	$0.442099\pi$
$308$	−0.211903	−0.0120743
$309$	−23.7085	−1.34873
$310$	−5.60326	−0.318244
$311$	18.6709	1.05873	0.529364	−	0.848395i	$-0.322431\pi$
$311$	18.6709	1.05873	0.529364	+	0.848395i	$0.322431\pi$
$312$	8.93946	0.506097
$313$	6.53890	0.369600	0.184800	−	0.982776i	$-0.440836\pi$
$313$	6.53890	0.369600	0.184800	+	0.982776i	$0.440836\pi$
$314$	16.7784	0.946860
$315$	0.500189	0.0281825
$316$	−0.443372	−0.0249416
$317$	13.0076	0.730581	0.365290	−	0.930894i	$-0.380970\pi$
$317$	13.0076	0.730581	0.365290	+	0.930894i	$0.380970\pi$
$318$	−20.2761	−1.13703
$319$	7.41601	0.415217
$320$	−8.41154	−0.470220
$321$	−26.8491	−1.49857
$322$	0.958966	0.0534411
$323$	−46.4815	−2.58630
$324$	0.847926	0.0471070
$325$	1.94632	0.107962
$326$	5.96793	0.330533
$327$	23.8364	1.31815
$328$	2.99144	0.165175
$329$	0.646523	0.0356440
$330$	3.93056	0.216370
$331$	11.0355	0.606568	0.303284	−	0.952900i	$-0.401917\pi$
$331$	11.0355	0.606568	0.303284	+	0.952900i	$0.401917\pi$
$332$	1.19881	0.0657935
$333$	2.83718	0.155477
$334$	14.2756	0.781127
$335$	−9.98438	−0.545505
$336$	−5.93281	−0.323662
$337$	8.50237	0.463154	0.231577	−	0.972817i	$-0.425612\pi$
$337$	8.50237	0.463154	0.231577	+	0.972817i	$0.425612\pi$
$338$	12.6408	0.687570
$339$	28.3820	1.54150
$340$	0.855977	0.0464219
$341$	−7.39748	−0.400596
$342$	−4.35960	−0.235740
$343$	1.00000	0.0539949
$344$	0.674927	0.0363896
$345$	1.10491	0.0594865
$346$	−18.1613	−0.976357
$347$	13.1930	0.708236	0.354118	−	0.935201i	$-0.384781\pi$
$347$	13.1930	0.708236	0.354118	+	0.935201i	$0.384781\pi$
$348$	−0.757037	−0.0405815
$349$	30.3949	1.62700	0.813501	−	0.581564i	$-0.197559\pi$
$349$	30.3949	1.62700	0.813501	+	0.581564i	$0.197559\pi$
$350$	−1.37224	−0.0733491
$351$	−10.7711	−0.574918
$352$	−1.19714	−0.0638075
$353$	35.7438	1.90245	0.951224	−	0.308501i	$-0.0998272\pi$
$353$	35.7438	1.90245	0.951224	+	0.308501i	$0.0998272\pi$
$354$	−15.9891	−0.849813
$355$	−6.02018	−0.319518
$356$	−0.941081	−0.0498772
$357$	11.5705	0.612373
$358$	21.1800	1.11940
$359$	−7.46232	−0.393846	−0.196923	−	0.980419i	$-0.563095\pi$
$359$	−7.46232	−0.393846	−0.196923	+	0.980419i	$0.563095\pi$
$360$	1.45304	0.0765819
$361$	21.3429	1.12331
$362$	−20.3802	−1.07116
$363$	−12.2027	−0.640476
$364$	−0.227656	−0.0119324
$365$	−2.55398	−0.133681
$366$	−17.3190	−0.905279
$367$	−18.0355	−0.941443	−0.470721	−	0.882282i	$-0.656006\pi$
$367$	−18.0355	−0.941443	−0.470721	+	0.882282i	$0.656006\pi$
$368$	2.62229	0.136697
$369$	−0.515076	−0.0268138
$370$	−7.78363	−0.404652
$371$	9.34547	0.485193
$372$	0.755145	0.0391525
$373$	20.9668	1.08562	0.542810	−	0.839856i	$-0.317360\pi$
$373$	20.9668	1.08562	0.542810	+	0.839856i	$0.317360\pi$
$374$	−18.1927	−0.940723
$375$	−1.58108	−0.0816466
$376$	1.87814	0.0968574
$377$	7.96733	0.410338
$378$	7.59406	0.390596
$379$	0.494521	0.0254018	0.0127009	−	0.999919i	$-0.495957\pi$
$379$	0.494521	0.0254018	0.0127009	+	0.999919i	$0.495957\pi$
$380$	−0.742931	−0.0381116
$381$	13.3648	0.684699
$382$	17.9464	0.918217
$383$	−0.587134	−0.0300012	−0.0150006	−	0.999887i	$-0.504775\pi$
$383$	−0.587134	−0.0300012	−0.0150006	+	0.999887i	$0.504775\pi$
$384$	−16.1602	−0.824674
$385$	−1.81164	−0.0923297
$386$	−29.4943	−1.50122
$387$	−0.116211	−0.00590735
$388$	0.954695	0.0484673
$389$	−16.4484	−0.833968	−0.416984	−	0.908914i	$-0.636913\pi$
$389$	−16.4484	−0.833968	−0.416984	+	0.908914i	$0.636913\pi$
$390$	4.22276	0.213828
$391$	−5.11412	−0.258632
$392$	2.90498	0.146724
$393$	10.6369	0.536558
$394$	−7.12244	−0.358823
$395$	−3.79056	−0.190724
$396$	0.105992	0.00532628
$397$	−13.4037	−0.672713	−0.336357	−	0.941735i	$-0.609195\pi$
$397$	−13.4037	−0.672713	−0.336357	+	0.941735i	$0.609195\pi$
$398$	29.5888	1.48315
$399$	−10.0424	−0.502748
$400$	−3.75238	−0.187619
$401$	4.79173	0.239288	0.119644	−	0.992817i	$-0.461825\pi$
$401$	4.79173	0.239288	0.119644	+	0.992817i	$0.461825\pi$
$402$	−21.6623	−1.08041
$403$	−7.94742	−0.395889
$404$	−0.490907	−0.0244235
$405$	7.24924	0.360218
$406$	−5.61730	−0.278782
$407$	−10.2760	−0.509364
$408$	33.6119	1.66404
$409$	23.8432	1.17897	0.589485	−	0.807780i	$-0.299331\pi$
$409$	23.8432	1.17897	0.589485	+	0.807780i	$0.299331\pi$
$410$	1.41308	0.0697870
$411$	−9.36169	−0.461778
$412$	1.75394	0.0864107
$413$	7.36958	0.362633
$414$	−0.479664	−0.0235742
$415$	10.2491	0.503110
$416$	−1.28613	−0.0630579
$417$	−12.0610	−0.590628
$418$	15.7901	0.772317
$419$	2.24734	0.109790	0.0548948	−	0.998492i	$-0.482518\pi$
$419$	2.24734	0.109790	0.0548948	+	0.998492i	$0.482518\pi$
$420$	0.184935	0.00902390
$421$	32.0667	1.56284	0.781419	−	0.624007i	$-0.214496\pi$
$421$	32.0667	1.56284	0.781419	+	0.624007i	$0.214496\pi$
$422$	2.76966	0.134825
$423$	−0.323384	−0.0157235
$424$	27.1484	1.31844
$425$	7.31807	0.354979
$426$	−13.0615	−0.632831
$427$	7.98253	0.386302
$428$	1.98628	0.0960106
$429$	5.57493	0.269160
$430$	0.318818	0.0153748
$431$	26.4339	1.27328	0.636638	−	0.771162i	$-0.280324\pi$
$431$	26.4339	1.27328	0.636638	+	0.771162i	$0.280324\pi$
$432$	20.7660	0.999103
$433$	2.54494	0.122302	0.0611509	−	0.998129i	$-0.480523\pi$
$433$	2.54494	0.122302	0.0611509	+	0.998129i	$0.480523\pi$
$434$	5.60326	0.268965
$435$	−6.47220	−0.310318
$436$	−1.76340	−0.0844518
$437$	4.43872	0.212333
$438$	−5.54115	−0.264766
$439$	−16.9554	−0.809235	−0.404618	−	0.914486i	$-0.632595\pi$
$439$	−16.9554	−0.809235	−0.404618	+	0.914486i	$0.632595\pi$
$440$	−5.26278	−0.250893
$441$	−0.500189	−0.0238185
$442$	−19.5452	−0.929670
$443$	−14.3876	−0.683578	−0.341789	−	0.939777i	$-0.611033\pi$
$443$	−14.3876	−0.683578	−0.341789	+	0.939777i	$0.611033\pi$
$444$	1.04899	0.0497830
$445$	−8.04567	−0.381401
$446$	−0.284446	−0.0134689
$447$	25.4918	1.20572
$448$	8.41154	0.397408
$449$	38.8024	1.83120	0.915599	−	0.402092i	$-0.131717\pi$
$449$	38.8024	1.83120	0.915599	+	0.402092i	$0.131717\pi$
$450$	0.686378	0.0323562
$451$	1.86556	0.0878458
$452$	−2.09969	−0.0987609
$453$	−27.7458	−1.30361
$454$	7.57020	0.355287
$455$	−1.94632	−0.0912449
$456$	−29.1729	−1.36615
$457$	40.4831	1.89372	0.946860	−	0.321645i	$-0.104236\pi$
$457$	40.4831	1.89372	0.946860	+	0.321645i	$0.104236\pi$
$458$	1.37224	0.0641204
$459$	−40.4988	−1.89032
$460$	−0.0817409	−0.00381119
$461$	0.225209	0.0104890	0.00524452	−	0.999986i	$-0.498331\pi$
$461$	0.225209	0.0104890	0.00524452	+	0.999986i	$0.498331\pi$
$462$	−3.93056	−0.182866
$463$	39.3839	1.83032	0.915162	−	0.403086i	$-0.132062\pi$
$463$	39.3839	1.83032	0.915162	+	0.403086i	$0.132062\pi$
$464$	−15.3605	−0.713094
$465$	6.45603	0.299391
$466$	−37.1836	−1.72250
$467$	−17.2539	−0.798416	−0.399208	−	0.916860i	$-0.630715\pi$
$467$	−17.2539	−0.798416	−0.399208	+	0.916860i	$0.630715\pi$
$468$	0.113871	0.00526370
$469$	9.98438	0.461036
$470$	0.887182	0.0409227
$471$	−19.3319	−0.890768
$472$	21.4085	0.985405
$473$	0.420906	0.0193533
$474$	−8.22405	−0.377743
$475$	−6.35160	−0.291432
$476$	−0.855977	−0.0392336
$477$	−4.67450	−0.214031
$478$	10.7256	0.490579
$479$	6.52789	0.298267	0.149134	−	0.988817i	$-0.452352\pi$
$479$	6.52789	0.298267	0.149134	+	0.988817i	$0.452352\pi$
$480$	1.04478	0.0476875
$481$	−11.0400	−0.503379
$482$	4.35672	0.198443
$483$	−1.10491	−0.0502752
$484$	0.902751	0.0410341
$485$	8.16205	0.370620
$486$	−7.05412	−0.319981
$487$	−29.3827	−1.33146	−0.665728	−	0.746194i	$-0.731879\pi$
$487$	−29.3827	−1.33146	−0.665728	+	0.746194i	$0.731879\pi$
$488$	23.1891	1.04972
$489$	−6.87620	−0.310953
$490$	1.37224	0.0619913
$491$	−40.3779	−1.82223	−0.911114	−	0.412155i	$-0.864776\pi$
$491$	−40.3779	−1.82223	−0.911114	+	0.412155i	$0.864776\pi$
$492$	−0.190439	−0.00858566
$493$	29.9568	1.34919
$494$	16.9639	0.763244
$495$	0.906162	0.0407290
$496$	15.3221	0.687984
$497$	6.02018	0.270042
$498$	22.2366	0.996448
$499$	17.9094	0.801733	0.400867	−	0.916136i	$-0.368709\pi$
$499$	17.9094	0.801733	0.400867	+	0.916136i	$0.368709\pi$
$500$	0.116968	0.00523095
$501$	−16.4482	−0.734853
$502$	−17.0361	−0.760360
$503$	24.6021	1.09695	0.548477	−	0.836165i	$-0.315208\pi$
$503$	24.6021	1.09695	0.548477	+	0.836165i	$0.315208\pi$
$504$	−1.45304	−0.0647235
$505$	−4.19695	−0.186762
$506$	1.73730	0.0772324
$507$	−14.5646	−0.646838
$508$	−0.988721	−0.0438674
$509$	33.9250	1.50370	0.751850	−	0.659334i	$-0.229162\pi$
$509$	33.9250	1.50370	0.751850	+	0.659334i	$0.229162\pi$
$510$	15.8774	0.703063
$511$	2.55398	0.112981
$512$	24.2808	1.07307
$513$	35.1502	1.55192
$514$	37.4157	1.65034
$515$	14.9951	0.660765
$516$	−0.0429667	−0.00189150
$517$	1.17127	0.0515122
$518$	7.78363	0.341993
$519$	20.9253	0.918518
$520$	−5.65402	−0.247945
$521$	−2.35676	−0.103252	−0.0516258	−	0.998666i	$-0.516440\pi$
$521$	−2.35676	−0.103252	−0.0516258	+	0.998666i	$0.516440\pi$
$522$	2.80971	0.122978
$523$	27.7565	1.21371	0.606853	−	0.794814i	$-0.292432\pi$
$523$	27.7565	1.21371	0.606853	+	0.794814i	$0.292432\pi$
$524$	−0.786909	−0.0343763
$525$	1.58108	0.0690039
$526$	12.3785	0.539728
$527$	−29.8819	−1.30168
$528$	−10.7481	−0.467752
$529$	−22.5116	−0.978767
$530$	12.8242	0.557047
$531$	−3.68618	−0.159967
$532$	0.742931	0.0322101
$533$	2.00425	0.0868137
$534$	−17.4560	−0.755395
$535$	16.9815	0.734174
$536$	29.0044	1.25280
$537$	−24.4034	−1.05309
$538$	−30.4550	−1.31301
$539$	1.81164	0.0780328
$540$	−0.647307	−0.0278557
$541$	−17.6093	−0.757084	−0.378542	−	0.925584i	$-0.623574\pi$
$541$	−17.6093	−0.757084	−0.378542	+	0.925584i	$0.623574\pi$
$542$	−37.4056	−1.60671
$543$	23.4819	1.00770
$544$	−4.83580	−0.207333
$545$	−15.0760	−0.645786
$546$	−4.22276	−0.180718
$547$	−9.24591	−0.395327	−0.197663	−	0.980270i	$-0.563335\pi$
$547$	−9.24591	−0.395327	−0.197663	+	0.980270i	$0.563335\pi$
$548$	0.692573	0.0295853
$549$	−3.99277	−0.170407
$550$	−2.48600	−0.106003
$551$	−26.0005	−1.10766
$552$	−3.20975	−0.136616
$553$	3.79056	0.161191
$554$	−15.4999	−0.658529
$555$	8.96823	0.380680
$556$	0.892264	0.0378404
$557$	−10.1287	−0.429167	−0.214584	−	0.976706i	$-0.568839\pi$
$557$	−10.1287	−0.429167	−0.214584	+	0.976706i	$0.568839\pi$
$558$	−2.80269	−0.118647
$559$	0.452197	0.0191259
$560$	3.75238	0.158567
$561$	20.9615	0.884995
$562$	−12.7511	−0.537874
$563$	−7.57414	−0.319212	−0.159606	−	0.987181i	$-0.551022\pi$
$563$	−7.57414	−0.319212	−0.159606	+	0.987181i	$0.551022\pi$
$564$	−0.119565	−0.00503458
$565$	−17.9510	−0.755205
$566$	−34.1578	−1.43576
$567$	−7.24924	−0.304440
$568$	17.4885	0.733802
$569$	−46.5954	−1.95338	−0.976690	−	0.214657i	$-0.931137\pi$
$569$	−46.5954	−1.95338	−0.976690	+	0.214657i	$0.931137\pi$
$570$	−13.7805	−0.577203
$571$	25.3283	1.05996	0.529978	−	0.848012i	$-0.322200\pi$
$571$	25.3283	1.05996	0.529978	+	0.848012i	$0.322200\pi$
$572$	−0.412431	−0.0172446
$573$	−20.6777	−0.863822
$574$	−1.41308	−0.0589808
$575$	−0.698834	−0.0291434
$576$	−4.20736	−0.175307
$577$	−34.6623	−1.44301	−0.721506	−	0.692408i	$-0.756550\pi$
$577$	−34.6623	−1.44301	−0.721506	+	0.692408i	$0.756550\pi$
$578$	−50.1610	−2.08642
$579$	33.9831	1.41229
$580$	0.478811	0.0198815
$581$	−10.2491	−0.425205
$582$	17.7085	0.734041
$583$	16.9306	0.701195
$584$	7.41926	0.307011
$585$	0.973529	0.0402504
$586$	21.8850	0.904060
$587$	5.62898	0.232333	0.116166	−	0.993230i	$-0.462939\pi$
$587$	5.62898	0.232333	0.116166	+	0.993230i	$0.462939\pi$
$588$	−0.184935	−0.00762659
$589$	25.9355	1.06865
$590$	10.1128	0.416337
$591$	8.20641	0.337567
$592$	21.2844	0.874782
$593$	5.87412	0.241221	0.120611	−	0.992700i	$-0.461515\pi$
$593$	5.87412	0.241221	0.120611	+	0.992700i	$0.461515\pi$
$594$	13.7577	0.564485
$595$	−7.31807	−0.300012
$596$	−1.88587	−0.0772482
$597$	−34.0919	−1.39529
$598$	1.86645	0.0763250
$599$	−22.3529	−0.913314	−0.456657	−	0.889643i	$-0.650953\pi$
$599$	−22.3529	−0.913314	−0.456657	+	0.889643i	$0.650953\pi$
$600$	4.59300	0.187509
$601$	29.6537	1.20960	0.604800	−	0.796378i	$-0.293253\pi$
$601$	29.6537	1.20960	0.604800	+	0.796378i	$0.293253\pi$
$602$	−0.318818	−0.0129940
$603$	−4.99408	−0.203375
$604$	2.05262	0.0835200
$605$	7.71796	0.313780
$606$	−9.10576	−0.369896
$607$	29.0302	1.17830	0.589151	−	0.808023i	$-0.299463\pi$
$607$	29.0302	1.17830	0.589151	+	0.808023i	$0.299463\pi$
$608$	4.19715	0.170217
$609$	6.47220	0.262267
$610$	10.9539	0.443511
$611$	1.25834	0.0509070
$612$	0.428150	0.0173069
$613$	26.4657	1.06894	0.534470	−	0.845187i	$-0.320511\pi$
$613$	26.4657	1.06894	0.534470	+	0.845187i	$0.320511\pi$
$614$	−8.69894	−0.351061
$615$	−1.62814	−0.0656528
$616$	5.26278	0.212043
$617$	41.7693	1.68157	0.840785	−	0.541369i	$-0.182094\pi$
$617$	41.7693	1.68157	0.840785	+	0.541369i	$0.182094\pi$
$618$	32.5337	1.30870
$619$	13.8667	0.557349	0.278674	−	0.960386i	$-0.410105\pi$
$619$	13.8667	0.557349	0.278674	+	0.960386i	$0.410105\pi$
$620$	−0.477614	−0.0191814
$621$	3.86740	0.155193
$622$	−25.6208	−1.02730
$623$	8.04567	0.322343
$624$	−11.5472	−0.462256
$625$	1.00000	0.0400000
$626$	−8.97292	−0.358630
$627$	−18.1932	−0.726566
$628$	1.43017	0.0570699
$629$	−41.5097	−1.65510
$630$	−0.686378	−0.0273459
$631$	−9.67751	−0.385255	−0.192628	−	0.981272i	$-0.561701\pi$
$631$	−9.67751	−0.385255	−0.192628	+	0.981272i	$0.561701\pi$
$632$	11.0115	0.438014
$633$	−3.19118	−0.126838
$634$	−17.8495	−0.708896
$635$	−8.45295	−0.335445
$636$	−1.72830	−0.0685317
$637$	1.94632	0.0771160
$638$	−10.1765	−0.402892
$639$	−3.01123	−0.119122
$640$	10.2210	0.404021
$641$	−39.6656	−1.56670	−0.783348	−	0.621583i	$-0.786490\pi$
$641$	−39.6656	−1.56670	−0.783348	+	0.621583i	$0.786490\pi$
$642$	36.8433	1.45409
$643$	35.9795	1.41889	0.709446	−	0.704760i	$-0.248945\pi$
$643$	35.9795	1.41889	0.709446	+	0.704760i	$0.248945\pi$
$644$	0.0817409	0.00322104
$645$	−0.367339	−0.0144640
$646$	63.7836	2.50953
$647$	−11.0803	−0.435611	−0.217805	−	0.975992i	$-0.569890\pi$
$647$	−11.0803	−0.435611	−0.217805	+	0.975992i	$0.569890\pi$
$648$	−21.0589	−0.827272
$649$	13.3510	0.524073
$650$	−2.67081	−0.104758
$651$	−6.45603	−0.253032
$652$	0.508698	0.0199222
$653$	−17.0214	−0.666097	−0.333049	−	0.942910i	$-0.608077\pi$
$653$	−17.0214	−0.666097	−0.333049	+	0.942910i	$0.608077\pi$
$654$	−32.7091	−1.27903
$655$	−6.72759	−0.262869
$656$	−3.86407	−0.150866
$657$	−1.27747	−0.0498390
$658$	−0.887182	−0.0345860
$659$	42.1403	1.64155	0.820777	−	0.571249i	$-0.193541\pi$
$659$	42.1403	1.64155	0.820777	+	0.571249i	$0.193541\pi$
$660$	0.335035	0.0130412
$661$	−11.2280	−0.436717	−0.218359	−	0.975869i	$-0.570070\pi$
$661$	−11.2280	−0.436717	−0.218359	+	0.975869i	$0.570070\pi$
$662$	−15.1434	−0.588564
$663$	22.5198	0.874597
$664$	−29.7735	−1.15544
$665$	6.35160	0.246305
$666$	−3.89329	−0.150862
$667$	−2.86070	−0.110767
$668$	1.21683	0.0470807
$669$	0.327737	0.0126710
$670$	13.7009	0.529313
$671$	14.4615	0.558279
$672$	−1.04478	−0.0403033
$673$	−13.7426	−0.529737	−0.264868	−	0.964285i	$-0.585329\pi$
$673$	−13.7426	−0.529737	−0.264868	+	0.964285i	$0.585329\pi$
$674$	−11.6673	−0.449406
$675$	−5.53408	−0.213007
$676$	1.07749	0.0414417
$677$	−29.3444	−1.12780	−0.563899	−	0.825844i	$-0.690699\pi$
$677$	−29.3444	−1.12780	−0.563899	+	0.825844i	$0.690699\pi$
$678$	−38.9468	−1.49574
$679$	−8.16205	−0.313231
$680$	−21.2589	−0.815240
$681$	−8.72232	−0.334240
$682$	10.1511	0.388705
$683$	6.89166	0.263702	0.131851	−	0.991270i	$-0.457908\pi$
$683$	6.89166	0.263702	0.131851	+	0.991270i	$0.457908\pi$
$684$	−0.371606	−0.0142087
$685$	5.92107	0.226233
$686$	−1.37224	−0.0523922
$687$	−1.58108	−0.0603219
$688$	−0.871808	−0.0332374
$689$	18.1893	0.692956
$690$	−1.51620	−0.0577208
$691$	−51.9286	−1.97546	−0.987728	−	0.156185i	$-0.950080\pi$
$691$	−51.9286	−1.97546	−0.987728	+	0.156185i	$0.950080\pi$
$692$	−1.54804	−0.0588478
$693$	−0.906162	−0.0344223
$694$	−18.1039	−0.687214
$695$	7.62831	0.289358
$696$	18.8016	0.712673
$697$	7.53588	0.285442
$698$	−41.7090	−1.57871
$699$	42.8426	1.62046
$700$	−0.116968	−0.00442096
$701$	14.0607	0.531064	0.265532	−	0.964102i	$-0.414452\pi$
$701$	14.0607	0.531064	0.265532	+	0.964102i	$0.414452\pi$
$702$	14.7805	0.557853
$703$	36.0277	1.35881
$704$	15.2387	0.574330
$705$	−1.02220	−0.0384984
$706$	−49.0489	−1.84598
$707$	4.19695	0.157843
$708$	−1.36289	−0.0512206
$709$	6.26641	0.235340	0.117670	−	0.993053i	$-0.462457\pi$
$709$	6.26641	0.235340	0.117670	+	0.993053i	$0.462457\pi$
$710$	8.26112	0.310034
$711$	−1.89600	−0.0711054
$712$	23.3725	0.875921
$713$	2.85355	0.106866
$714$	−15.8774	−0.594196
$715$	−3.52603	−0.131866
$716$	1.80536	0.0674693
$717$	−12.3580	−0.461518
$718$	10.2401	0.382156
$719$	−30.9495	−1.15422	−0.577112	−	0.816665i	$-0.695820\pi$
$719$	−30.9495	−1.15422	−0.577112	+	0.816665i	$0.695820\pi$
$720$	−1.87690	−0.0699480
$721$	−14.9951	−0.558448
$722$	−29.2874	−1.08997
$723$	−5.01978	−0.186687
$724$	−1.73718	−0.0645617
$725$	4.09354	0.152030
$726$	16.7450	0.621465
$727$	−38.3414	−1.42201	−0.711003	−	0.703189i	$-0.751759\pi$
$727$	−38.3414	−1.42201	−0.711003	+	0.703189i	$0.751759\pi$
$728$	5.65402	0.209552
$729$	29.8754	1.10650
$730$	3.50466	0.129713
$731$	1.70024	0.0628856
$732$	−1.47625	−0.0545637
$733$	−21.1383	−0.780762	−0.390381	−	0.920653i	$-0.627657\pi$
$733$	−21.1383	−0.780762	−0.390381	+	0.920653i	$0.627657\pi$
$734$	24.7489	0.913499
$735$	−1.58108	−0.0583190
$736$	0.461791	0.0170219
$737$	18.0881	0.666284
$738$	0.706807	0.0260179
$739$	43.4828	1.59954	0.799771	−	0.600305i	$-0.204954\pi$
$739$	43.4828	1.59954	0.799771	+	0.600305i	$0.204954\pi$
$740$	−0.663466	−0.0243895
$741$	−19.5457	−0.718029
$742$	−12.8242	−0.470791
$743$	19.8246	0.727296	0.363648	−	0.931536i	$-0.381531\pi$
$743$	19.8246	0.727296	0.363648	+	0.931536i	$0.381531\pi$
$744$	−18.7546	−0.687578
$745$	−16.1230	−0.590702
$746$	−28.7714	−1.05340
$747$	5.12650	0.187569
$748$	−1.55072	−0.0567000
$749$	−16.9815	−0.620490
$750$	2.16961	0.0792231
$751$	14.8399	0.541515	0.270758	−	0.962648i	$-0.412726\pi$
$751$	14.8399	0.541515	0.270758	+	0.962648i	$0.412726\pi$
$752$	−2.42600	−0.0884672
$753$	19.6289	0.715316
$754$	−10.9331	−0.398159
$755$	17.5486	0.638661
$756$	0.647307	0.0235423
$757$	10.6728	0.387909	0.193955	−	0.981010i	$-0.437869\pi$
$757$	10.6728	0.387909	0.193955	+	0.981010i	$0.437869\pi$
$758$	−0.678599	−0.0246478
$759$	−2.00170	−0.0726572
$760$	18.4513	0.669298
$761$	−30.0541	−1.08946	−0.544730	−	0.838611i	$-0.683368\pi$
$761$	−30.0541	−1.08946	−0.544730	+	0.838611i	$0.683368\pi$
$762$	−18.3396	−0.664375
$763$	15.0760	0.545789
$764$	1.52972	0.0553435
$765$	3.66042	0.132343
$766$	0.805687	0.0291107
$767$	14.3436	0.517916
$768$	−4.42296	−0.159600
$769$	−24.7815	−0.893643	−0.446822	−	0.894623i	$-0.647444\pi$
$769$	−24.7815	−0.893643	−0.446822	+	0.894623i	$0.647444\pi$
$770$	2.48600	0.0895891
$771$	−43.1101	−1.55257
$772$	−2.51406	−0.0904829
$773$	5.84140	0.210100	0.105050	−	0.994467i	$-0.466500\pi$
$773$	5.84140	0.210100	0.105050	+	0.994467i	$0.466500\pi$
$774$	0.159469	0.00573200
$775$	−4.08330	−0.146677
$776$	−23.7106	−0.851161
$777$	−8.96823	−0.321734
$778$	22.5711	0.809213
$779$	−6.54065	−0.234343
$780$	0.359943	0.0128880
$781$	10.9064	0.390262
$782$	7.01778	0.250955
$783$	−22.6539	−0.809586
$784$	−3.75238	−0.134014
$785$	12.2270	0.436402
$786$	−14.5963	−0.520632
$787$	45.2532	1.61310	0.806551	−	0.591164i	$-0.201331\pi$
$787$	45.2532	1.61310	0.806551	+	0.591164i	$0.201331\pi$
$788$	−0.607107	−0.0216273
$789$	−14.2624	−0.507755
$790$	5.20154	0.185062
$791$	17.9510	0.638265
$792$	−2.63238	−0.0935377
$793$	15.5366	0.551719
$794$	18.3931	0.652746
$795$	−14.7759	−0.524048
$796$	2.52211	0.0893937
$797$	−37.2037	−1.31782	−0.658912	−	0.752220i	$-0.728983\pi$
$797$	−37.2037	−1.31782	−0.658912	+	0.752220i	$0.728983\pi$
$798$	13.7805	0.487825
$799$	4.73130	0.167381
$800$	−0.660802	−0.0233629
$801$	−4.02435	−0.142194
$802$	−6.57539	−0.232185
$803$	4.62689	0.163279
$804$	−1.84646	−0.0651196
$805$	0.698834	0.0246307
$806$	10.9057	0.384138
$807$	35.0900	1.23523
$808$	12.1921	0.428915
$809$	−19.5437	−0.687120	−0.343560	−	0.939131i	$-0.611633\pi$
$809$	−19.5437	−0.687120	−0.343560	+	0.939131i	$0.611633\pi$
$810$	−9.94767	−0.349526
$811$	−10.3168	−0.362272	−0.181136	−	0.983458i	$-0.557977\pi$
$811$	−10.3168	−0.362272	−0.181136	+	0.983458i	$0.557977\pi$
$812$	−0.478811	−0.0168030
$813$	43.0984	1.51153
$814$	14.1011	0.494244
$815$	4.34906	0.152341
$816$	−43.4168	−1.51989
$817$	−1.47570	−0.0516281
$818$	−32.7185	−1.14397
$819$	−0.973529	−0.0340178
$820$	0.120449	0.00420626
$821$	45.1104	1.57436	0.787181	−	0.616722i	$-0.211540\pi$
$821$	45.1104	1.57436	0.787181	+	0.616722i	$0.211540\pi$
$822$	12.8464	0.448071
$823$	22.9143	0.798743	0.399371	−	0.916789i	$-0.369228\pi$
$823$	22.9143	0.798743	0.399371	+	0.916789i	$0.369228\pi$
$824$	−43.5606	−1.51751
$825$	2.86435	0.0997237
$826$	−10.1128	−0.351869
$827$	−48.1039	−1.67274	−0.836369	−	0.548167i	$-0.815326\pi$
$827$	−48.1039	−1.67274	−0.836369	+	0.548167i	$0.815326\pi$
$828$	−0.0408859	−0.00142088
$829$	−30.4808	−1.05864	−0.529320	−	0.848422i	$-0.677553\pi$
$829$	−30.4808	−1.05864	−0.529320	+	0.848422i	$0.677553\pi$
$830$	−14.0642	−0.488176
$831$	17.8589	0.619518
$832$	16.3716	0.567582
$833$	7.31807	0.253556
$834$	16.5505	0.573096
$835$	10.4032	0.360017
$836$	1.34592	0.0465497
$837$	22.5973	0.781078
$838$	−3.08388	−0.106531
$839$	4.31890	0.149105	0.0745525	−	0.997217i	$-0.476247\pi$
$839$	4.31890	0.149105	0.0745525	+	0.997217i	$0.476247\pi$
$840$	−4.59300	−0.158474
$841$	−12.2430	−0.422171
$842$	−44.0031	−1.51645
$843$	14.6917	0.506010
$844$	0.236082	0.00812628
$845$	9.21184	0.316897
$846$	0.443759	0.0152567
$847$	−7.71796	−0.265192
$848$	−35.0678	−1.20423
$849$	39.3563	1.35071
$850$	−10.0421	−0.344442
$851$	3.96394	0.135882
$852$	−1.11334	−0.0381425
$853$	−26.1643	−0.895848	−0.447924	−	0.894072i	$-0.647836\pi$
$853$	−26.1643	−0.895848	−0.447924	+	0.894072i	$0.647836\pi$
$854$	−10.9539	−0.374835
$855$	−3.17700	−0.108651
$856$	−49.3309	−1.68610
$857$	−23.7204	−0.810272	−0.405136	−	0.914256i	$-0.632776\pi$
$857$	−23.7204	−0.810272	−0.405136	+	0.914256i	$0.632776\pi$
$858$	−7.65013	−0.261171
$859$	42.3158	1.44380	0.721898	−	0.692000i	$-0.243270\pi$
$859$	42.3158	1.44380	0.721898	+	0.692000i	$0.243270\pi$
$860$	0.0271756	0.000926679 0
$861$	1.62814	0.0554868
$862$	−36.2736	−1.23548
$863$	−41.7360	−1.42071	−0.710355	−	0.703844i	$-0.751466\pi$
$863$	−41.7360	−1.42071	−0.710355	+	0.703844i	$0.751466\pi$
$864$	3.65693	0.124411
$865$	−13.2348	−0.449997
$866$	−3.49225	−0.118672
$867$	57.7951	1.96282
$868$	0.477614	0.0162113
$869$	6.86712	0.232951
$870$	8.88139	0.301107
$871$	19.4328	0.658456
$872$	43.7955	1.48310
$873$	4.08257	0.138174
$874$	−6.09097	−0.206030
$875$	−1.00000	−0.0338062
$876$	−0.472320	−0.0159582
$877$	−31.6414	−1.06845	−0.534227	−	0.845341i	$-0.679397\pi$
$877$	−31.6414	−1.06845	−0.534227	+	0.845341i	$0.679397\pi$
$878$	23.2668	0.785215
$879$	−25.2157	−0.850504
$880$	6.79797	0.229159
$881$	23.3271	0.785908	0.392954	−	0.919558i	$-0.371453\pi$
$881$	23.3271	0.785908	0.392954	+	0.919558i	$0.371453\pi$
$882$	0.686378	0.0231115
$883$	−40.5238	−1.36373	−0.681867	−	0.731476i	$-0.738832\pi$
$883$	−40.5238	−1.36373	−0.681867	+	0.731476i	$0.738832\pi$
$884$	−1.66601	−0.0560338
$885$	−11.6519	−0.391674
$886$	19.7433	0.663288
$887$	45.5707	1.53011	0.765057	−	0.643963i	$-0.222711\pi$
$887$	45.5707	1.53011	0.765057	+	0.643963i	$0.222711\pi$
$888$	−26.0525	−0.874266
$889$	8.45295	0.283503
$890$	11.0406	0.370080
$891$	−13.1330	−0.439973
$892$	−0.0242458	−0.000811810 0
$893$	−4.10646	−0.137417
$894$	−34.9807	−1.16993
$895$	15.4347	0.515924
$896$	−10.2210	−0.341460
$897$	−2.15051	−0.0718035
$898$	−53.2461	−1.77684
$899$	−16.7152	−0.557482
$900$	0.0585059	0.00195020
$901$	68.3908	2.27843
$902$	−2.55999	−0.0852383
$903$	0.367339	0.0122243
$904$	52.1473	1.73439
$905$	−14.8518	−0.493690
$906$	38.0738	1.26492
$907$	−10.9504	−0.363604	−0.181802	−	0.983335i	$-0.558193\pi$
$907$	−10.9504	−0.363604	−0.181802	+	0.983335i	$0.558193\pi$
$908$	0.645274	0.0214142
$909$	−2.09927	−0.0696284
$910$	2.67081	0.0885365
$911$	43.4486	1.43952	0.719758	−	0.694226i	$-0.244253\pi$
$911$	43.4486	1.43952	0.719758	+	0.694226i	$0.244253\pi$
$912$	37.6829	1.24780
$913$	−18.5677	−0.614502
$914$	−55.5524	−1.83751
$915$	−12.6210	−0.417238
$916$	0.116968	0.00386472
$917$	6.72759	0.222165
$918$	55.5739	1.83421
$919$	−57.3964	−1.89333	−0.946666	−	0.322217i	$-0.895572\pi$
$919$	−57.3964	−1.89333	−0.946666	+	0.322217i	$0.895572\pi$
$920$	2.03010	0.0669304
$921$	10.0228	0.330264
$922$	−0.309040	−0.0101777
$923$	11.7172	0.385677
$924$	−0.335035	−0.0110219
$925$	−5.67222	−0.186502
$926$	−54.0440	−1.77600
$927$	7.50041	0.246346
$928$	−2.70502	−0.0887966
$929$	25.9841	0.852510	0.426255	−	0.904603i	$-0.359833\pi$
$929$	25.9841	0.852510	0.426255	+	0.904603i	$0.359833\pi$
$930$	−8.85920	−0.290504
$931$	−6.35160	−0.208165
$932$	−3.16948	−0.103820
$933$	29.5201	0.966445
$934$	23.6764	0.774717
$935$	−13.2577	−0.433573
$936$	−2.82808	−0.0924387
$937$	33.5126	1.09481	0.547404	−	0.836869i	$-0.315616\pi$
$937$	33.5126	1.09481	0.547404	+	0.836869i	$0.315616\pi$
$938$	−13.7009	−0.447351
$939$	10.3385	0.337385
$940$	0.0756222	0.00246652
$941$	34.7325	1.13225	0.566124	−	0.824320i	$-0.308442\pi$
$941$	34.7325	1.13225	0.566124	+	0.824320i	$0.308442\pi$
$942$	26.5280	0.864328
$943$	−0.719634	−0.0234345
$944$	−27.6535	−0.900044
$945$	5.53408	0.180023
$946$	−0.577583	−0.0187788
$947$	30.8764	1.00335	0.501674	−	0.865057i	$-0.332718\pi$
$947$	30.8764	1.00335	0.501674	+	0.865057i	$0.332718\pi$
$948$	−0.701006	−0.0227676
$949$	4.97086	0.161361
$950$	8.71590	0.282781
$951$	20.5661	0.666901
$952$	21.2589	0.689003
$953$	36.1779	1.17192	0.585958	−	0.810342i	$-0.300718\pi$
$953$	36.1779	1.17192	0.585958	+	0.810342i	$0.300718\pi$
$954$	6.41452	0.207678
$955$	13.0782	0.423201
$956$	0.914239	0.0295686
$957$	11.7253	0.379025
$958$	−8.95781	−0.289414
$959$	−5.92107	−0.191201
$960$	−13.2993	−0.429234
$961$	−14.3266	−0.462149
$962$	15.1494	0.488438
$963$	8.49396	0.273714
$964$	0.371361	0.0119607
$965$	−21.4936	−0.691904
$966$	1.51620	0.0487830
$967$	31.6161	1.01671	0.508353	−	0.861149i	$-0.330254\pi$
$967$	31.6161	1.01671	0.508353	+	0.861149i	$0.330254\pi$
$968$	−22.4205	−0.720623
$969$	−73.4909	−2.36087
$970$	−11.2003	−0.359619
$971$	−44.7818	−1.43712	−0.718558	−	0.695467i	$-0.755198\pi$
$971$	−44.7818	−1.43712	−0.718558	+	0.695467i	$0.755198\pi$
$972$	−0.601283	−0.0192862
$973$	−7.62831	−0.244552
$974$	40.3200	1.29194
$975$	3.07729	0.0985520
$976$	−29.9535	−0.958788
$977$	−40.2010	−1.28614	−0.643072	−	0.765806i	$-0.722340\pi$
$977$	−40.2010	−1.28614	−0.643072	+	0.765806i	$0.722340\pi$
$978$	9.43577	0.301723
$979$	14.5758	0.465846
$980$	0.116968	0.00373639
$981$	−7.54086	−0.240761
$982$	55.4080	1.76814
$983$	15.7088	0.501034	0.250517	−	0.968112i	$-0.419399\pi$
$983$	15.7088	0.501034	0.250517	+	0.968112i	$0.419399\pi$
$984$	4.72971	0.150777
$985$	−5.19039	−0.165380
$986$	−41.1078	−1.30914
$987$	1.02220	0.0325371
$988$	1.44598	0.0460028
$989$	−0.162363	−0.00516285
$990$	−1.24347	−0.0395200
$991$	−25.5144	−0.810493	−0.405247	−	0.914207i	$-0.632814\pi$
$991$	−25.5144	−0.810493	−0.405247	+	0.914207i	$0.632814\pi$
$992$	2.69826	0.0856698
$993$	17.4481	0.553698
$994$	−8.26112	−0.262027
$995$	21.5625	0.683576
$996$	1.89542	0.0600587
$997$	−3.77305	−0.119494	−0.0597468	−	0.998214i	$-0.519029\pi$
$997$	−3.77305	−0.119494	−0.0597468	+	0.998214i	$0.519029\pi$
$998$	−24.5759	−0.777936
$999$	31.3905	0.993152

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8015.2.a.n.1.19	✓	68

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8015.2.a.n.1.19	✓	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)

\(f(q)\)	\(=\)	\(q-1.37224 q^{2} +1.58108 q^{3} -0.116968 q^{4} -1.00000 q^{5} -2.16961 q^{6} +1.00000 q^{7} +2.90498 q^{8} -0.500189 q^{9} +O(q^{10})\)	\(q-1.37224 q^{2} +1.58108 q^{3} -0.116968 q^{4} -1.00000 q^{5} -2.16961 q^{6} +1.00000 q^{7} +2.90498 q^{8} -0.500189 q^{9} +1.37224 q^{10} +1.81164 q^{11} -0.184935 q^{12} +1.94632 q^{13} -1.37224 q^{14} -1.58108 q^{15} -3.75238 q^{16} +7.31807 q^{17} +0.686378 q^{18} -6.35160 q^{19} +0.116968 q^{20} +1.58108 q^{21} -2.48600 q^{22} -0.698834 q^{23} +4.59300 q^{24} +1.00000 q^{25} -2.67081 q^{26} -5.53408 q^{27} -0.116968 q^{28} +4.09354 q^{29} +2.16961 q^{30} -4.08330 q^{31} -0.660802 q^{32} +2.86435 q^{33} -10.0421 q^{34} -1.00000 q^{35} +0.0585059 q^{36} -5.67222 q^{37} +8.71590 q^{38} +3.07729 q^{39} -2.90498 q^{40} +1.02976 q^{41} -2.16961 q^{42} +0.232334 q^{43} -0.211903 q^{44} +0.500189 q^{45} +0.958966 q^{46} +0.646523 q^{47} -5.93281 q^{48} +1.00000 q^{49} -1.37224 q^{50} +11.5705 q^{51} -0.227656 q^{52} +9.34547 q^{53} +7.59406 q^{54} -1.81164 q^{55} +2.90498 q^{56} -10.0424 q^{57} -5.61730 q^{58} +7.36958 q^{59} +0.184935 q^{60} +7.98253 q^{61} +5.60326 q^{62} -0.500189 q^{63} +8.41154 q^{64} -1.94632 q^{65} -3.93056 q^{66} +9.98438 q^{67} -0.855977 q^{68} -1.10491 q^{69} +1.37224 q^{70} +6.02018 q^{71} -1.45304 q^{72} +2.55398 q^{73} +7.78363 q^{74} +1.58108 q^{75} +0.742931 q^{76} +1.81164 q^{77} -4.22276 q^{78} +3.79056 q^{79} +3.75238 q^{80} -7.24924 q^{81} -1.41308 q^{82} -10.2491 q^{83} -0.184935 q^{84} -7.31807 q^{85} -0.318818 q^{86} +6.47220 q^{87} +5.26278 q^{88} +8.04567 q^{89} -0.686378 q^{90} +1.94632 q^{91} +0.0817409 q^{92} -6.45603 q^{93} -0.887182 q^{94} +6.35160 q^{95} -1.04478 q^{96} -8.16205 q^{97} -1.37224 q^{98} -0.906162 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q + 9 q^{2} + 83 q^{4} - 68 q^{5} + 5 q^{6} + 68 q^{7} + 30 q^{8} + 86 q^{9}+O(q^{10}) \) 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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