LMFDB - Embedded newform 8002.2.a.e.1.51

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.8962916974$
Analytic rank:	$0$
Dimension:	$77$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.51
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.19273 q^{3} +1.00000 q^{4} -0.879538 q^{5} -1.19273 q^{6} -2.85585 q^{7} -1.00000 q^{8} -1.57740 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.19273 q^{3} +1.00000 q^{4} -0.879538 q^{5} -1.19273 q^{6} -2.85585 q^{7} -1.00000 q^{8} -1.57740 q^{9} +0.879538 q^{10} -0.846358 q^{11} +1.19273 q^{12} -5.02997 q^{13} +2.85585 q^{14} -1.04905 q^{15} +1.00000 q^{16} -5.13586 q^{17} +1.57740 q^{18} -2.55760 q^{19} -0.879538 q^{20} -3.40625 q^{21} +0.846358 q^{22} -3.35499 q^{23} -1.19273 q^{24} -4.22641 q^{25} +5.02997 q^{26} -5.45959 q^{27} -2.85585 q^{28} +3.60120 q^{29} +1.04905 q^{30} -3.68141 q^{31} -1.00000 q^{32} -1.00947 q^{33} +5.13586 q^{34} +2.51183 q^{35} -1.57740 q^{36} -1.05497 q^{37} +2.55760 q^{38} -5.99938 q^{39} +0.879538 q^{40} -8.11812 q^{41} +3.40625 q^{42} +8.66462 q^{43} -0.846358 q^{44} +1.38739 q^{45} +3.35499 q^{46} +1.38679 q^{47} +1.19273 q^{48} +1.15588 q^{49} +4.22641 q^{50} -6.12568 q^{51} -5.02997 q^{52} +3.10581 q^{53} +5.45959 q^{54} +0.744404 q^{55} +2.85585 q^{56} -3.05052 q^{57} -3.60120 q^{58} -0.466311 q^{59} -1.04905 q^{60} -4.78790 q^{61} +3.68141 q^{62} +4.50483 q^{63} +1.00000 q^{64} +4.42405 q^{65} +1.00947 q^{66} -13.3782 q^{67} -5.13586 q^{68} -4.00158 q^{69} -2.51183 q^{70} -4.10978 q^{71} +1.57740 q^{72} +11.6853 q^{73} +1.05497 q^{74} -5.04096 q^{75} -2.55760 q^{76} +2.41707 q^{77} +5.99938 q^{78} +16.3415 q^{79} -0.879538 q^{80} -1.77959 q^{81} +8.11812 q^{82} -10.7732 q^{83} -3.40625 q^{84} +4.51718 q^{85} -8.66462 q^{86} +4.29525 q^{87} +0.846358 q^{88} +16.6430 q^{89} -1.38739 q^{90} +14.3648 q^{91} -3.35499 q^{92} -4.39092 q^{93} -1.38679 q^{94} +2.24951 q^{95} -1.19273 q^{96} +14.0127 q^{97} -1.15588 q^{98} +1.33505 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) $ 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.19273	0.688621	0.344311	−	0.938856i	$-0.388113\pi$
$3$	1.19273	0.688621	0.344311	+	0.938856i	$0.388113\pi$
$4$	1.00000	0.500000
$5$	−0.879538	−0.393341	−0.196671	−	0.980470i	$-0.563013\pi$
$5$	−0.879538	−0.393341	−0.196671	+	0.980470i	$0.563013\pi$
$6$	−1.19273	−0.486929
$7$	−2.85585	−1.07941	−0.539705	−	0.841854i	$-0.681464\pi$
$7$	−2.85585	−1.07941	−0.539705	+	0.841854i	$0.681464\pi$
$8$	−1.00000	−0.353553
$9$	−1.57740	−0.525801
$10$	0.879538	0.278134
$11$	−0.846358	−0.255187	−0.127593	−	0.991827i	$-0.540725\pi$
$11$	−0.846358	−0.255187	−0.127593	+	0.991827i	$0.540725\pi$
$12$	1.19273	0.344311
$13$	−5.02997	−1.39506	−0.697531	−	0.716554i	$-0.745718\pi$
$13$	−5.02997	−1.39506	−0.697531	+	0.716554i	$0.745718\pi$
$14$	2.85585	0.763258
$15$	−1.04905	−0.270863
$16$	1.00000	0.250000
$17$	−5.13586	−1.24563	−0.622814	−	0.782370i	$-0.714011\pi$
$17$	−5.13586	−1.24563	−0.622814	+	0.782370i	$0.714011\pi$
$18$	1.57740	0.371797
$19$	−2.55760	−0.586754	−0.293377	−	0.955997i	$-0.594779\pi$
$19$	−2.55760	−0.586754	−0.293377	+	0.955997i	$0.594779\pi$
$20$	−0.879538	−0.196671
$21$	−3.40625	−0.743305
$22$	0.846358	0.180444
$23$	−3.35499	−0.699563	−0.349781	−	0.936831i	$-0.613744\pi$
$23$	−3.35499	−0.699563	−0.349781	+	0.936831i	$0.613744\pi$
$24$	−1.19273	−0.243464
$25$	−4.22641	−0.845283
$26$	5.02997	0.986458
$27$	−5.45959	−1.05070
$28$	−2.85585	−0.539705
$29$	3.60120	0.668726	0.334363	−	0.942444i	$-0.391479\pi$
$29$	3.60120	0.668726	0.334363	+	0.942444i	$0.391479\pi$
$30$	1.04905	0.191529
$31$	−3.68141	−0.661201	−0.330600	−	0.943771i	$-0.607251\pi$
$31$	−3.68141	−0.661201	−0.330600	+	0.943771i	$0.607251\pi$
$32$	−1.00000	−0.176777
$33$	−1.00947	−0.175727
$34$	5.13586	0.880792
$35$	2.51183	0.424577
$36$	−1.57740	−0.262900
$37$	−1.05497	−0.173436	−0.0867180	−	0.996233i	$-0.527638\pi$
$37$	−1.05497	−0.173436	−0.0867180	+	0.996233i	$0.527638\pi$
$38$	2.55760	0.414898
$39$	−5.99938	−0.960670
$40$	0.879538	0.139067
$41$	−8.11812	−1.26784	−0.633919	−	0.773400i	$-0.718555\pi$
$41$	−8.11812	−1.26784	−0.633919	+	0.773400i	$0.718555\pi$
$42$	3.40625	0.525596
$43$	8.66462	1.32134	0.660671	−	0.750676i	$-0.270272\pi$
$43$	8.66462	1.32134	0.660671	+	0.750676i	$0.270272\pi$
$44$	−0.846358	−0.127593
$45$	1.38739	0.206819
$46$	3.35499	0.494666
$47$	1.38679	0.202284	0.101142	−	0.994872i	$-0.467750\pi$
$47$	1.38679	0.202284	0.101142	+	0.994872i	$0.467750\pi$
$48$	1.19273	0.172155
$49$	1.15588	0.165126
$50$	4.22641	0.597705
$51$	−6.12568	−0.857766
$52$	−5.02997	−0.697531
$53$	3.10581	0.426615	0.213308	−	0.976985i	$-0.431576\pi$
$53$	3.10581	0.426615	0.213308	+	0.976985i	$0.431576\pi$
$54$	5.45959	0.742956
$55$	0.744404	0.100375
$56$	2.85585	0.381629
$57$	−3.05052	−0.404051
$58$	−3.60120	−0.472861
$59$	−0.466311	−0.0607085	−0.0303543	−	0.999539i	$-0.509664\pi$
$59$	−0.466311	−0.0607085	−0.0303543	+	0.999539i	$0.509664\pi$
$60$	−1.04905	−0.135432
$61$	−4.78790	−0.613028	−0.306514	−	0.951866i	$-0.599163\pi$
$61$	−4.78790	−0.613028	−0.306514	+	0.951866i	$0.599163\pi$
$62$	3.68141	0.467539
$63$	4.50483	0.567555
$64$	1.00000	0.125000
$65$	4.42405	0.548736
$66$	1.00947	0.124258
$67$	−13.3782	−1.63441	−0.817205	−	0.576347i	$-0.804478\pi$
$67$	−13.3782	−1.63441	−0.817205	+	0.576347i	$0.804478\pi$
$68$	−5.13586	−0.622814
$69$	−4.00158	−0.481734
$70$	−2.51183	−0.300221
$71$	−4.10978	−0.487742	−0.243871	−	0.969808i	$-0.578417\pi$
$71$	−4.10978	−0.487742	−0.243871	+	0.969808i	$0.578417\pi$
$72$	1.57740	0.185899
$73$	11.6853	1.36766	0.683829	−	0.729642i	$-0.260313\pi$
$73$	11.6853	1.36766	0.683829	+	0.729642i	$0.260313\pi$
$74$	1.05497	0.122638
$75$	−5.04096	−0.582080
$76$	−2.55760	−0.293377
$77$	2.41707	0.275451
$78$	5.99938	0.679296
$79$	16.3415	1.83857	0.919283	−	0.393598i	$-0.128770\pi$
$79$	16.3415	1.83857	0.919283	+	0.393598i	$0.128770\pi$
$80$	−0.879538	−0.0983353
$81$	−1.77959	−0.197733
$82$	8.11812	0.896497
$83$	−10.7732	−1.18252	−0.591258	−	0.806482i	$-0.701369\pi$
$83$	−10.7732	−1.18252	−0.591258	+	0.806482i	$0.701369\pi$
$84$	−3.40625	−0.371652
$85$	4.51718	0.489957
$86$	−8.66462	−0.934329
$87$	4.29525	0.460499
$88$	0.846358	0.0902221
$89$	16.6430	1.76415	0.882075	−	0.471110i	$-0.156146\pi$
$89$	16.6430	1.76415	0.882075	+	0.471110i	$0.156146\pi$
$90$	−1.38739	−0.146243
$91$	14.3648	1.50585
$92$	−3.35499	−0.349781
$93$	−4.39092	−0.455317
$94$	−1.38679	−0.143036
$95$	2.24951	0.230794
$96$	−1.19273	−0.121732
$97$	14.0127	1.42278	0.711389	−	0.702798i	$-0.248066\pi$
$97$	14.0127	1.42278	0.711389	+	0.702798i	$0.248066\pi$
$98$	−1.15588	−0.116762
$99$	1.33505	0.134177
$100$	−4.22641	−0.422641
$101$	−14.7832	−1.47099	−0.735493	−	0.677532i	$-0.763049\pi$
$101$	−14.7832	−1.47099	−0.735493	+	0.677532i	$0.763049\pi$
$102$	6.12568	0.606532
$103$	2.59589	0.255780	0.127890	−	0.991788i	$-0.459180\pi$
$103$	2.59589	0.255780	0.127890	+	0.991788i	$0.459180\pi$
$104$	5.02997	0.493229
$105$	2.99593	0.292372
$106$	−3.10581	−0.301662
$107$	5.59785	0.541164	0.270582	−	0.962697i	$-0.412784\pi$
$107$	5.59785	0.541164	0.270582	+	0.962697i	$0.412784\pi$
$108$	−5.45959	−0.525349
$109$	6.95007	0.665696	0.332848	−	0.942981i	$-0.391990\pi$
$109$	6.95007	0.665696	0.332848	+	0.942981i	$0.391990\pi$
$110$	−0.744404	−0.0709761
$111$	−1.25829	−0.119432
$112$	−2.85585	−0.269853
$113$	−6.50303	−0.611753	−0.305877	−	0.952071i	$-0.598950\pi$
$113$	−6.50303	−0.611753	−0.305877	+	0.952071i	$0.598950\pi$
$114$	3.05052	0.285707
$115$	2.95084	0.275167
$116$	3.60120	0.334363
$117$	7.93429	0.733525
$118$	0.466311	0.0429274
$119$	14.6672	1.34454
$120$	1.04905	0.0957646
$121$	−10.2837	−0.934880
$122$	4.78790	0.433476
$123$	−9.68270	−0.873060
$124$	−3.68141	−0.330600
$125$	8.11498	0.725826
$126$	−4.50483	−0.401322
$127$	−2.33773	−0.207440	−0.103720	−	0.994607i	$-0.533075\pi$
$127$	−2.33773	−0.207440	−0.103720	+	0.994607i	$0.533075\pi$
$128$	−1.00000	−0.0883883
$129$	10.3345	0.909904
$130$	−4.42405	−0.388015
$131$	12.0840	1.05578	0.527890	−	0.849313i	$-0.322983\pi$
$131$	12.0840	1.05578	0.527890	+	0.849313i	$0.322983\pi$
$132$	−1.00947	−0.0878634
$133$	7.30412	0.633348
$134$	13.3782	1.15570
$135$	4.80192	0.413283
$136$	5.13586	0.440396
$137$	3.08422	0.263503	0.131751	−	0.991283i	$-0.457940\pi$
$137$	3.08422	0.263503	0.131751	+	0.991283i	$0.457940\pi$
$138$	4.00158	0.340637
$139$	−15.4434	−1.30989	−0.654945	−	0.755676i	$-0.727308\pi$
$139$	−15.4434	−1.30989	−0.654945	+	0.755676i	$0.727308\pi$
$140$	2.51183	0.212288
$141$	1.65406	0.139297
$142$	4.10978	0.344885
$143$	4.25716	0.356001
$144$	−1.57740	−0.131450
$145$	−3.16739	−0.263038
$146$	−11.6853	−0.967081
$147$	1.37865	0.113709
$148$	−1.05497	−0.0867180
$149$	−24.1752	−1.98051	−0.990255	−	0.139263i	$-0.955527\pi$
$149$	−24.1752	−1.98051	−0.990255	+	0.139263i	$0.955527\pi$
$150$	5.04096	0.411592
$151$	−12.0404	−0.979835	−0.489918	−	0.871769i	$-0.662973\pi$
$151$	−12.0404	−0.979835	−0.489918	+	0.871769i	$0.662973\pi$
$152$	2.55760	0.207449
$153$	8.10131	0.654952
$154$	−2.41707	−0.194773
$155$	3.23794	0.260078
$156$	−5.99938	−0.480335
$157$	2.62683	0.209644	0.104822	−	0.994491i	$-0.466573\pi$
$157$	2.62683	0.209644	0.104822	+	0.994491i	$0.466573\pi$
$158$	−16.3415	−1.30006
$159$	3.70438	0.293776
$160$	0.879538	0.0695336
$161$	9.58134	0.755115
$162$	1.77959	0.139818
$163$	1.43604	0.112480	0.0562398	−	0.998417i	$-0.482089\pi$
$163$	1.43604	0.112480	0.0562398	+	0.998417i	$0.482089\pi$
$164$	−8.11812	−0.633919
$165$	0.887871	0.0691206
$166$	10.7732	0.836165
$167$	−8.06728	−0.624265	−0.312132	−	0.950039i	$-0.601043\pi$
$167$	−8.06728	−0.624265	−0.312132	+	0.950039i	$0.601043\pi$
$168$	3.40625	0.262798
$169$	12.3006	0.946201
$170$	−4.51718	−0.346452
$171$	4.03436	0.308516
$172$	8.66462	0.660671
$173$	4.91211	0.373461	0.186730	−	0.982411i	$-0.440211\pi$
$173$	4.91211	0.373461	0.186730	+	0.982411i	$0.440211\pi$
$174$	−4.29525	−0.325622
$175$	12.0700	0.912407
$176$	−0.846358	−0.0637966
$177$	−0.556182	−0.0418052
$178$	−16.6430	−1.24744
$179$	−6.00742	−0.449016	−0.224508	−	0.974472i	$-0.572077\pi$
$179$	−6.00742	−0.449016	−0.224508	+	0.974472i	$0.572077\pi$
$180$	1.38739	0.103410
$181$	−16.4150	−1.22012	−0.610060	−	0.792355i	$-0.708855\pi$
$181$	−16.4150	−1.22012	−0.610060	+	0.792355i	$0.708855\pi$
$182$	−14.3648	−1.06479
$183$	−5.71066	−0.422144
$184$	3.35499	0.247333
$185$	0.927886	0.0682195
$186$	4.39092	0.321958
$187$	4.34677	0.317868
$188$	1.38679	0.101142
$189$	15.5918	1.13413
$190$	−2.24951	−0.163196
$191$	−1.55968	−0.112854	−0.0564271	−	0.998407i	$-0.517971\pi$
$191$	−1.55968	−0.112854	−0.0564271	+	0.998407i	$0.517971\pi$
$192$	1.19273	0.0860777
$193$	12.7563	0.918222	0.459111	−	0.888379i	$-0.348168\pi$
$193$	12.7563	0.918222	0.459111	+	0.888379i	$0.348168\pi$
$194$	−14.0127	−1.00606
$195$	5.27668	0.377871
$196$	1.15588	0.0825631
$197$	19.1347	1.36329	0.681645	−	0.731683i	$-0.261265\pi$
$197$	19.1347	1.36329	0.681645	+	0.731683i	$0.261265\pi$
$198$	−1.33505	−0.0948777
$199$	5.87108	0.416190	0.208095	−	0.978109i	$-0.433274\pi$
$199$	5.87108	0.416190	0.208095	+	0.978109i	$0.433274\pi$
$200$	4.22641	0.298853
$201$	−15.9566	−1.12549
$202$	14.7832	1.04014
$203$	−10.2845	−0.721829
$204$	−6.12568	−0.428883
$205$	7.14020	0.498693
$206$	−2.59589	−0.180864
$207$	5.29216	0.367831
$208$	−5.02997	−0.348766
$209$	2.16465	0.149732
$210$	−2.99593	−0.206739
$211$	−5.69678	−0.392183	−0.196091	−	0.980586i	$-0.562825\pi$
$211$	−5.69678	−0.392183	−0.196091	+	0.980586i	$0.562825\pi$
$212$	3.10581	0.213308
$213$	−4.90185	−0.335869
$214$	−5.59785	−0.382661
$215$	−7.62086	−0.519738
$216$	5.45959	0.371478
$217$	10.5136	0.713707
$218$	−6.95007	−0.470718
$219$	13.9373	0.941799
$220$	0.744404	0.0501877
$221$	25.8332	1.73773
$222$	1.25829	0.0844509
$223$	−5.98253	−0.400620	−0.200310	−	0.979733i	$-0.564195\pi$
$223$	−5.98253	−0.400620	−0.200310	+	0.979733i	$0.564195\pi$
$224$	2.85585	0.190815
$225$	6.66675	0.444450
$226$	6.50303	0.432575
$227$	27.3608	1.81600	0.907999	−	0.418972i	$-0.137609\pi$
$227$	27.3608	1.81600	0.907999	+	0.418972i	$0.137609\pi$
$228$	−3.05052	−0.202026
$229$	−18.5008	−1.22257	−0.611284	−	0.791411i	$-0.709347\pi$
$229$	−18.5008	−1.22257	−0.611284	+	0.791411i	$0.709347\pi$
$230$	−2.95084	−0.194572
$231$	2.88291	0.189681
$232$	−3.60120	−0.236430
$233$	−24.7780	−1.62326	−0.811629	−	0.584174i	$-0.801419\pi$
$233$	−24.7780	−1.62326	−0.811629	+	0.584174i	$0.801419\pi$
$234$	−7.93429	−0.518681
$235$	−1.21973	−0.0795666
$236$	−0.466311	−0.0303543
$237$	19.4910	1.26608
$238$	−14.6672	−0.950736
$239$	−16.9634	−1.09727	−0.548634	−	0.836062i	$-0.684852\pi$
$239$	−16.9634	−1.09727	−0.548634	+	0.836062i	$0.684852\pi$
$240$	−1.04905	−0.0677158
$241$	19.8361	1.27775	0.638877	−	0.769309i	$-0.279399\pi$
$241$	19.8361	1.27775	0.638877	+	0.769309i	$0.279399\pi$
$242$	10.2837	0.661060
$243$	14.2562	0.914536
$244$	−4.78790	−0.306514
$245$	−1.01664	−0.0649510
$246$	9.68270	0.617347
$247$	12.8647	0.818558
$248$	3.68141	0.233770
$249$	−12.8495	−0.814306
$250$	−8.11498	−0.513236
$251$	−6.04403	−0.381496	−0.190748	−	0.981639i	$-0.561091\pi$
$251$	−6.04403	−0.381496	−0.190748	+	0.981639i	$0.561091\pi$
$252$	4.50483	0.283777
$253$	2.83952	0.178519
$254$	2.33773	0.146682
$255$	5.38776	0.337395
$256$	1.00000	0.0625000
$257$	−15.8085	−0.986104	−0.493052	−	0.870000i	$-0.664119\pi$
$257$	−15.8085	−0.986104	−0.493052	+	0.870000i	$0.664119\pi$
$258$	−10.3345	−0.643399
$259$	3.01284	0.187208
$260$	4.42405	0.274368
$261$	−5.68054	−0.351617
$262$	−12.0840	−0.746549
$263$	18.4639	1.13853	0.569265	−	0.822154i	$-0.307228\pi$
$263$	18.4639	1.13853	0.569265	+	0.822154i	$0.307228\pi$
$264$	1.00947	0.0621288
$265$	−2.73167	−0.167805
$266$	−7.30412	−0.447845
$267$	19.8505	1.21483
$268$	−13.3782	−0.817205
$269$	10.3624	0.631809	0.315905	−	0.948791i	$-0.397692\pi$
$269$	10.3624	0.631809	0.315905	+	0.948791i	$0.397692\pi$
$270$	−4.80192	−0.292235
$271$	5.49331	0.333695	0.166848	−	0.985983i	$-0.446641\pi$
$271$	5.49331	0.333695	0.166848	+	0.985983i	$0.446641\pi$
$272$	−5.13586	−0.311407
$273$	17.1333	1.03696
$274$	−3.08422	−0.186324
$275$	3.57706	0.215705
$276$	−4.00158	−0.240867
$277$	21.8693	1.31400	0.657001	−	0.753890i	$-0.271825\pi$
$277$	21.8693	1.31400	0.657001	+	0.753890i	$0.271825\pi$
$278$	15.4434	0.926233
$279$	5.80706	0.347660
$280$	−2.51183	−0.150111
$281$	24.3768	1.45420	0.727100	−	0.686532i	$-0.240868\pi$
$281$	24.3768	1.45420	0.727100	+	0.686532i	$0.240868\pi$
$282$	−1.65406	−0.0984979
$283$	−27.6683	−1.64471	−0.822355	−	0.568975i	$-0.807340\pi$
$283$	−27.6683	−1.64471	−0.822355	+	0.568975i	$0.807340\pi$
$284$	−4.10978	−0.243871
$285$	2.68305	0.158930
$286$	−4.25716	−0.251731
$287$	23.1841	1.36852
$288$	1.57740	0.0929493
$289$	9.37703	0.551590
$290$	3.16739	0.185996
$291$	16.7134	0.979755
$292$	11.6853	0.683829
$293$	31.4378	1.83662	0.918309	−	0.395865i	$-0.129555\pi$
$293$	31.4378	1.83662	0.918309	+	0.395865i	$0.129555\pi$
$294$	−1.37865	−0.0804047
$295$	0.410138	0.0238792
$296$	1.05497	0.0613189
$297$	4.62077	0.268124
$298$	24.1752	1.40043
$299$	16.8755	0.975934
$300$	−5.04096	−0.291040
$301$	−24.7448	−1.42627
$302$	12.0404	0.692848
$303$	−17.6324	−1.01295
$304$	−2.55760	−0.146688
$305$	4.21114	0.241129
$306$	−8.10131	−0.463121
$307$	7.26068	0.414389	0.207194	−	0.978300i	$-0.433567\pi$
$307$	7.26068	0.414389	0.207194	+	0.978300i	$0.433567\pi$
$308$	2.41707	0.137725
$309$	3.09618	0.176136
$310$	−3.23794	−0.183903
$311$	−33.6177	−1.90629	−0.953143	−	0.302521i	$-0.902172\pi$
$311$	−33.6177	−1.90629	−0.953143	+	0.302521i	$0.902172\pi$
$312$	5.99938	0.339648
$313$	−24.6944	−1.39581	−0.697905	−	0.716191i	$-0.745884\pi$
$313$	−24.6944	−1.39581	−0.697905	+	0.716191i	$0.745884\pi$
$314$	−2.62683	−0.148241
$315$	−3.96217	−0.223243
$316$	16.3415	0.919283
$317$	−9.27158	−0.520744	−0.260372	−	0.965508i	$-0.583845\pi$
$317$	−9.27158	−0.520744	−0.260372	+	0.965508i	$0.583845\pi$
$318$	−3.70438	−0.207731
$319$	−3.04790	−0.170650
$320$	−0.879538	−0.0491677
$321$	6.67670	0.372657
$322$	−9.58134	−0.533947
$323$	13.1355	0.730877
$324$	−1.77959	−0.0988664
$325$	21.2587	1.17922
$326$	−1.43604	−0.0795351
$327$	8.28953	0.458412
$328$	8.11812	0.448248
$329$	−3.96046	−0.218347
$330$	−0.887871	−0.0488757
$331$	−9.59128	−0.527185	−0.263592	−	0.964634i	$-0.584907\pi$
$331$	−9.59128	−0.527185	−0.263592	+	0.964634i	$0.584907\pi$
$332$	−10.7732	−0.591258
$333$	1.66411	0.0911927
$334$	8.06728	0.441422
$335$	11.7667	0.642881
$336$	−3.40625	−0.185826
$337$	−2.65982	−0.144890	−0.0724449	−	0.997372i	$-0.523080\pi$
$337$	−2.65982	−0.144890	−0.0724449	+	0.997372i	$0.523080\pi$
$338$	−12.3006	−0.669065
$339$	−7.75634	−0.421266
$340$	4.51718	0.244979
$341$	3.11579	0.168729
$342$	−4.03436	−0.218153
$343$	16.6899	0.901171
$344$	−8.66462	−0.467165
$345$	3.51954	0.189486
$346$	−4.91211	−0.264077
$347$	−24.1241	−1.29505	−0.647524	−	0.762045i	$-0.724196\pi$
$347$	−24.1241	−1.29505	−0.647524	+	0.762045i	$0.724196\pi$
$348$	4.29525	0.230249
$349$	−13.0268	−0.697310	−0.348655	−	0.937251i	$-0.613362\pi$
$349$	−13.0268	−0.697310	−0.348655	+	0.937251i	$0.613362\pi$
$350$	−12.0700	−0.645169
$351$	27.4616	1.46579
$352$	0.846358	0.0451110
$353$	16.5059	0.878518	0.439259	−	0.898360i	$-0.355241\pi$
$353$	16.5059	0.878518	0.439259	+	0.898360i	$0.355241\pi$
$354$	0.556182	0.0295607
$355$	3.61471	0.191849
$356$	16.6430	0.882075
$357$	17.4940	0.925882
$358$	6.00742	0.317502
$359$	−31.9398	−1.68572	−0.842858	−	0.538136i	$-0.819129\pi$
$359$	−31.9398	−1.68572	−0.842858	+	0.538136i	$0.819129\pi$
$360$	−1.38739	−0.0731216
$361$	−12.4587	−0.655720
$362$	16.4150	0.862756
$363$	−12.2656	−0.643778
$364$	14.3648	0.752923
$365$	−10.2776	−0.537957
$366$	5.71066	0.298501
$367$	−8.18891	−0.427458	−0.213729	−	0.976893i	$-0.568561\pi$
$367$	−8.18891	−0.427458	−0.213729	+	0.976893i	$0.568561\pi$
$368$	−3.35499	−0.174891
$369$	12.8055	0.666630
$370$	−0.927886	−0.0482385
$371$	−8.86972	−0.460493
$372$	−4.39092	−0.227658
$373$	−12.8455	−0.665113	−0.332556	−	0.943083i	$-0.607911\pi$
$373$	−12.8455	−0.665113	−0.332556	+	0.943083i	$0.607911\pi$
$374$	−4.34677	−0.224766
$375$	9.67896	0.499819
$376$	−1.38679	−0.0715182
$377$	−18.1139	−0.932915
$378$	−15.5918	−0.801955
$379$	−5.50411	−0.282727	−0.141364	−	0.989958i	$-0.545149\pi$
$379$	−5.50411	−0.282727	−0.141364	+	0.989958i	$0.545149\pi$
$380$	2.24951	0.115397
$381$	−2.78827	−0.142847
$382$	1.55968	0.0798000
$383$	−4.60091	−0.235096	−0.117548	−	0.993067i	$-0.537503\pi$
$383$	−4.60091	−0.235096	−0.117548	+	0.993067i	$0.537503\pi$
$384$	−1.19273	−0.0608661
$385$	−2.12591	−0.108346
$386$	−12.7563	−0.649281
$387$	−13.6676	−0.694762
$388$	14.0127	0.711389
$389$	29.4172	1.49151	0.745755	−	0.666221i	$-0.232089\pi$
$389$	29.4172	1.49151	0.745755	+	0.666221i	$0.232089\pi$
$390$	−5.27668	−0.267195
$391$	17.2307	0.871395
$392$	−1.15588	−0.0583809
$393$	14.4129	0.727032
$394$	−19.1347	−0.963991
$395$	−14.3730	−0.723184
$396$	1.33505	0.0670886
$397$	−28.1632	−1.41347	−0.706735	−	0.707479i	$-0.749832\pi$
$397$	−28.1632	−1.41347	−0.706735	+	0.707479i	$0.749832\pi$
$398$	−5.87108	−0.294291
$399$	8.71183	0.436137
$400$	−4.22641	−0.211321
$401$	23.5894	1.17800	0.588999	−	0.808133i	$-0.299522\pi$
$401$	23.5894	1.17800	0.588999	+	0.808133i	$0.299522\pi$
$402$	15.9566	0.795842
$403$	18.5174	0.922416
$404$	−14.7832	−0.735493
$405$	1.56522	0.0777765
$406$	10.2845	0.510411
$407$	0.892882	0.0442585
$408$	6.12568	0.303266
$409$	−20.5473	−1.01600	−0.508000	−	0.861357i	$-0.669615\pi$
$409$	−20.5473	−1.01600	−0.508000	+	0.861357i	$0.669615\pi$
$410$	−7.14020	−0.352629
$411$	3.67863	0.181453
$412$	2.59589	0.127890
$413$	1.33171	0.0655294
$414$	−5.29216	−0.260096
$415$	9.47547	0.465132
$416$	5.02997	0.246615
$417$	−18.4197	−0.902019
$418$	−2.16465	−0.105876
$419$	10.1840	0.497522	0.248761	−	0.968565i	$-0.419977\pi$
$419$	10.1840	0.497522	0.248761	+	0.968565i	$0.419977\pi$
$420$	2.99593	0.146186
$421$	21.2809	1.03717	0.518584	−	0.855027i	$-0.326459\pi$
$421$	21.2809	1.03717	0.518584	+	0.855027i	$0.326459\pi$
$422$	5.69678	0.277315
$423$	−2.18752	−0.106361
$424$	−3.10581	−0.150831
$425$	21.7063	1.05291
$426$	4.90185	0.237495
$427$	13.6735	0.661709
$428$	5.59785	0.270582
$429$	5.07763	0.245150
$430$	7.62086	0.367510
$431$	−26.8272	−1.29222	−0.646110	−	0.763244i	$-0.723605\pi$
$431$	−26.8272	−1.29222	−0.646110	+	0.763244i	$0.723605\pi$
$432$	−5.45959	−0.262675
$433$	−11.0028	−0.528763	−0.264381	−	0.964418i	$-0.585168\pi$
$433$	−11.0028	−0.528763	−0.264381	+	0.964418i	$0.585168\pi$
$434$	−10.5136	−0.504667
$435$	−3.77783	−0.181133
$436$	6.95007	0.332848
$437$	8.58071	0.410471
$438$	−13.9373	−0.665952
$439$	−4.05311	−0.193445	−0.0967223	−	0.995311i	$-0.530836\pi$
$439$	−4.05311	−0.193445	−0.0967223	+	0.995311i	$0.530836\pi$
$440$	−0.744404	−0.0354881
$441$	−1.82329	−0.0868235
$442$	−25.8332	−1.22876
$443$	33.7660	1.60427	0.802136	−	0.597142i	$-0.203697\pi$
$443$	33.7660	1.60427	0.802136	+	0.597142i	$0.203697\pi$
$444$	−1.25829	−0.0597158
$445$	−14.6381	−0.693913
$446$	5.98253	0.283281
$447$	−28.8344	−1.36382
$448$	−2.85585	−0.134926
$449$	−0.476417	−0.0224835	−0.0112417	−	0.999937i	$-0.503578\pi$
$449$	−0.476417	−0.0224835	−0.0112417	+	0.999937i	$0.503578\pi$
$450$	−6.66675	−0.314274
$451$	6.87084	0.323535
$452$	−6.50303	−0.305877
$453$	−14.3609	−0.674735
$454$	−27.3608	−1.28410
$455$	−12.6344	−0.592311
$456$	3.05052	0.142854
$457$	−17.6693	−0.826537	−0.413268	−	0.910609i	$-0.635613\pi$
$457$	−17.6693	−0.826537	−0.413268	+	0.910609i	$0.635613\pi$
$458$	18.5008	0.864486
$459$	28.0397	1.30878
$460$	2.95084	0.137583
$461$	−13.0416	−0.607406	−0.303703	−	0.952767i	$-0.598223\pi$
$461$	−13.0416	−0.607406	−0.303703	+	0.952767i	$0.598223\pi$
$462$	−2.88291	−0.134125
$463$	6.62965	0.308106	0.154053	−	0.988063i	$-0.450767\pi$
$463$	6.62965	0.308106	0.154053	+	0.988063i	$0.450767\pi$
$464$	3.60120	0.167181
$465$	3.86198	0.179095
$466$	24.7780	1.14782
$467$	30.7789	1.42428	0.712138	−	0.702040i	$-0.247727\pi$
$467$	30.7789	1.42428	0.712138	+	0.702040i	$0.247727\pi$
$468$	7.93429	0.366763
$469$	38.2062	1.76420
$470$	1.21973	0.0562621
$471$	3.13309	0.144365
$472$	0.466311	0.0214637
$473$	−7.33337	−0.337189
$474$	−19.4910	−0.895250
$475$	10.8095	0.495973
$476$	14.6672	0.672272
$477$	−4.89910	−0.224315
$478$	16.9634	0.775886
$479$	35.5807	1.62572	0.812861	−	0.582458i	$-0.197909\pi$
$479$	35.5807	1.62572	0.812861	+	0.582458i	$0.197909\pi$
$480$	1.04905	0.0478823
$481$	5.30647	0.241954
$482$	−19.8361	−0.903508
$483$	11.4279	0.519988
$484$	−10.2837	−0.467440
$485$	−12.3247	−0.559637
$486$	−14.2562	−0.646675
$487$	−30.2807	−1.37215	−0.686075	−	0.727531i	$-0.740668\pi$
$487$	−30.2807	−1.37215	−0.686075	+	0.727531i	$0.740668\pi$
$488$	4.78790	0.216738
$489$	1.71281	0.0774558
$490$	1.01664	0.0459273
$491$	−7.83648	−0.353655	−0.176828	−	0.984242i	$-0.556584\pi$
$491$	−7.83648	−0.353655	−0.176828	+	0.984242i	$0.556584\pi$
$492$	−9.68270	−0.436530
$493$	−18.4952	−0.832984
$494$	−12.8647	−0.578808
$495$	−1.17422	−0.0527775
$496$	−3.68141	−0.165300
$497$	11.7369	0.526473
$498$	12.8495	0.575801
$499$	−23.5213	−1.05296	−0.526479	−	0.850188i	$-0.676488\pi$
$499$	−23.5213	−1.05296	−0.526479	+	0.850188i	$0.676488\pi$
$500$	8.11498	0.362913
$501$	−9.62206	−0.429882
$502$	6.04403	0.269758
$503$	9.49545	0.423381	0.211691	−	0.977337i	$-0.432103\pi$
$503$	9.49545	0.423381	0.211691	+	0.977337i	$0.432103\pi$
$504$	−4.50483	−0.200661
$505$	13.0024	0.578600
$506$	−2.83952	−0.126232
$507$	14.6713	0.651574
$508$	−2.33773	−0.103720
$509$	6.65640	0.295040	0.147520	−	0.989059i	$-0.452871\pi$
$509$	6.65640	0.295040	0.147520	+	0.989059i	$0.452871\pi$
$510$	−5.38776	−0.238574
$511$	−33.3714	−1.47626
$512$	−1.00000	−0.0441942
$513$	13.9635	0.616501
$514$	15.8085	0.697281
$515$	−2.28318	−0.100609
$516$	10.3345	0.454952
$517$	−1.17372	−0.0516201
$518$	−3.01284	−0.132376
$519$	5.85881	0.257173
$520$	−4.42405	−0.194007
$521$	−4.49037	−0.196727	−0.0983633	−	0.995151i	$-0.531361\pi$
$521$	−4.49037	−0.196727	−0.0983633	+	0.995151i	$0.531361\pi$
$522$	5.68054	0.248630
$523$	0.503028	0.0219959	0.0109979	−	0.999940i	$-0.496499\pi$
$523$	0.503028	0.0219959	0.0109979	+	0.999940i	$0.496499\pi$
$524$	12.0840	0.527890
$525$	14.3962	0.628303
$526$	−18.4639	−0.805062
$527$	18.9072	0.823610
$528$	−1.00947	−0.0439317
$529$	−11.7441	−0.510612
$530$	2.73167	0.118656
$531$	0.735560	0.0319206
$532$	7.30412	0.316674
$533$	40.8339	1.76871
$534$	−19.8505	−0.859015
$535$	−4.92352	−0.212862
$536$	13.3782	0.577851
$537$	−7.16521	−0.309202
$538$	−10.3624	−0.446757
$539$	−0.978292	−0.0421380
$540$	4.80192	0.206642
$541$	−16.0275	−0.689074	−0.344537	−	0.938773i	$-0.611964\pi$
$541$	−16.0275	−0.689074	−0.344537	+	0.938773i	$0.611964\pi$
$542$	−5.49331	−0.235958
$543$	−19.5787	−0.840201
$544$	5.13586	0.220198
$545$	−6.11285	−0.261846
$546$	−17.1333	−0.733239
$547$	14.9542	0.639397	0.319699	−	0.947519i	$-0.396418\pi$
$547$	14.9542	0.639397	0.319699	+	0.947519i	$0.396418\pi$
$548$	3.08422	0.131751
$549$	7.55245	0.322331
$550$	−3.57706	−0.152526
$551$	−9.21043	−0.392377
$552$	4.00158	0.170319
$553$	−46.6690	−1.98457
$554$	−21.8693	−0.929139
$555$	1.10671	0.0469774
$556$	−15.4434	−0.654945
$557$	−13.9886	−0.592716	−0.296358	−	0.955077i	$-0.595772\pi$
$557$	−13.9886	−0.592716	−0.296358	+	0.955077i	$0.595772\pi$
$558$	−5.80706	−0.245833
$559$	−43.5828	−1.84335
$560$	2.51183	0.106144
$561$	5.18451	0.218890
$562$	−24.3768	−1.02827
$563$	−0.948070	−0.0399564	−0.0199782	−	0.999800i	$-0.506360\pi$
$563$	−0.948070	−0.0399564	−0.0199782	+	0.999800i	$0.506360\pi$
$564$	1.65406	0.0696485
$565$	5.71966	0.240628
$566$	27.6683	1.16299
$567$	5.08226	0.213435
$568$	4.10978	0.172443
$569$	−30.9897	−1.29916	−0.649578	−	0.760295i	$-0.725054\pi$
$569$	−30.9897	−1.29916	−0.649578	+	0.760295i	$0.725054\pi$
$570$	−2.68305	−0.112380
$571$	−29.5904	−1.23832	−0.619159	−	0.785266i	$-0.712526\pi$
$571$	−29.5904	−1.23832	−0.619159	+	0.785266i	$0.712526\pi$
$572$	4.25716	0.178001
$573$	−1.86027	−0.0777139
$574$	−23.1841	−0.967687
$575$	14.1796	0.591328
$576$	−1.57740	−0.0657251
$577$	−29.0532	−1.20950	−0.604749	−	0.796416i	$-0.706727\pi$
$577$	−29.0532	−1.20950	−0.604749	+	0.796416i	$0.706727\pi$
$578$	−9.37703	−0.390033
$579$	15.2148	0.632307
$580$	−3.16739	−0.131519
$581$	30.7667	1.27642
$582$	−16.7134	−0.692791
$583$	−2.62862	−0.108866
$584$	−11.6853	−0.483540
$585$	−6.97851	−0.288526
$586$	−31.4378	−1.29868
$587$	13.6434	0.563122	0.281561	−	0.959543i	$-0.409148\pi$
$587$	13.6434	0.563122	0.281561	+	0.959543i	$0.409148\pi$
$588$	1.37865	0.0568547
$589$	9.41557	0.387962
$590$	−0.410138	−0.0168851
$591$	22.8224	0.938790
$592$	−1.05497	−0.0433590
$593$	10.4939	0.430931	0.215466	−	0.976511i	$-0.430873\pi$
$593$	10.4939	0.430931	0.215466	+	0.976511i	$0.430873\pi$
$594$	−4.62077	−0.189592
$595$	−12.9004	−0.528865
$596$	−24.1752	−0.990255
$597$	7.00260	0.286597
$598$	−16.8755	−0.690090
$599$	20.5624	0.840156	0.420078	−	0.907488i	$-0.362003\pi$
$599$	20.5624	0.840156	0.420078	+	0.907488i	$0.362003\pi$
$600$	5.04096	0.205796
$601$	−15.5029	−0.632375	−0.316188	−	0.948697i	$-0.602403\pi$
$601$	−15.5029	−0.632375	−0.316188	+	0.948697i	$0.602403\pi$
$602$	24.7448	1.00852
$603$	21.1029	0.859375
$604$	−12.0404	−0.489918
$605$	9.04488	0.367727
$606$	17.6324	0.716266
$607$	45.3696	1.84149	0.920747	−	0.390159i	$-0.127580\pi$
$607$	45.3696	1.84149	0.920747	+	0.390159i	$0.127580\pi$
$608$	2.55760	0.103724
$609$	−12.2666	−0.497067
$610$	−4.21114	−0.170504
$611$	−6.97551	−0.282199
$612$	8.10131	0.327476
$613$	14.5504	0.587684	0.293842	−	0.955854i	$-0.405066\pi$
$613$	14.5504	0.587684	0.293842	+	0.955854i	$0.405066\pi$
$614$	−7.26068	−0.293017
$615$	8.51630	0.343411
$616$	−2.41707	−0.0973866
$617$	−41.6269	−1.67583	−0.837917	−	0.545797i	$-0.816227\pi$
$617$	−41.6269	−1.67583	−0.837917	+	0.545797i	$0.816227\pi$
$618$	−3.09618	−0.124547
$619$	−28.9738	−1.16455	−0.582277	−	0.812990i	$-0.697838\pi$
$619$	−28.9738	−1.16455	−0.582277	+	0.812990i	$0.697838\pi$
$620$	3.23794	0.130039
$621$	18.3169	0.735030
$622$	33.6177	1.34795
$623$	−47.5298	−1.90424
$624$	−5.99938	−0.240167
$625$	13.9946	0.559785
$626$	24.6944	0.986986
$627$	2.58183	0.103108
$628$	2.62683	0.104822
$629$	5.41817	0.216037
$630$	3.96217	0.157856
$631$	−19.2173	−0.765028	−0.382514	−	0.923950i	$-0.624942\pi$
$631$	−19.2173	−0.765028	−0.382514	+	0.923950i	$0.624942\pi$
$632$	−16.3415	−0.650031
$633$	−6.79471	−0.270065
$634$	9.27158	0.368221
$635$	2.05612	0.0815946
$636$	3.70438	0.146888
$637$	−5.81406	−0.230362
$638$	3.04790	0.120668
$639$	6.48278	0.256455
$640$	0.879538	0.0347668
$641$	−21.5134	−0.849730	−0.424865	−	0.905257i	$-0.639678\pi$
$641$	−21.5134	−0.849730	−0.424865	+	0.905257i	$0.639678\pi$
$642$	−6.67670	−0.263508
$643$	46.0605	1.81645	0.908225	−	0.418483i	$-0.137438\pi$
$643$	46.0605	1.81645	0.908225	+	0.418483i	$0.137438\pi$
$644$	9.58134	0.377558
$645$	−9.08960	−0.357903
$646$	−13.1355	−0.516808
$647$	36.6002	1.43890	0.719451	−	0.694544i	$-0.244394\pi$
$647$	36.6002	1.43890	0.719451	+	0.694544i	$0.244394\pi$
$648$	1.77959	0.0699091
$649$	0.394666	0.0154920
$650$	−21.2587	−0.833836
$651$	12.5398	0.491474
$652$	1.43604	0.0562398
$653$	−15.6819	−0.613681	−0.306840	−	0.951761i	$-0.599272\pi$
$653$	−15.6819	−0.613681	−0.306840	+	0.951761i	$0.599272\pi$
$654$	−8.28953	−0.324146
$655$	−10.6283	−0.415282
$656$	−8.11812	−0.316959
$657$	−18.4324	−0.719116
$658$	3.96046	0.154395
$659$	29.4633	1.14773	0.573863	−	0.818951i	$-0.305444\pi$
$659$	29.4633	1.14773	0.573863	+	0.818951i	$0.305444\pi$
$660$	0.887871	0.0345603
$661$	32.3916	1.25989	0.629943	−	0.776641i	$-0.283078\pi$
$661$	32.3916	1.25989	0.629943	+	0.776641i	$0.283078\pi$
$662$	9.59128	0.372776
$663$	30.8120	1.19664
$664$	10.7732	0.418083
$665$	−6.42425	−0.249122
$666$	−1.66411	−0.0644830
$667$	−12.0820	−0.467816
$668$	−8.06728	−0.312132
$669$	−7.13553	−0.275876
$670$	−11.7667	−0.454586
$671$	4.05228	0.156437
$672$	3.40625	0.131399
$673$	−26.0659	−1.00477	−0.502383	−	0.864645i	$-0.667543\pi$
$673$	−26.0659	−1.00477	−0.502383	+	0.864645i	$0.667543\pi$
$674$	2.65982	0.102453
$675$	23.0745	0.888137
$676$	12.3006	0.473100
$677$	−3.23708	−0.124411	−0.0622055	−	0.998063i	$-0.519813\pi$
$677$	−3.23708	−0.124411	−0.0622055	+	0.998063i	$0.519813\pi$
$678$	7.75634	0.297880
$679$	−40.0183	−1.53576
$680$	−4.51718	−0.173226
$681$	32.6339	1.25054
$682$	−3.11579	−0.119310
$683$	−31.6503	−1.21107	−0.605533	−	0.795820i	$-0.707040\pi$
$683$	−31.6503	−1.21107	−0.605533	+	0.795820i	$0.707040\pi$
$684$	4.03436	0.154258
$685$	−2.71269	−0.103646
$686$	−16.6899	−0.637224
$687$	−22.0664	−0.841886
$688$	8.66462	0.330335
$689$	−15.6221	−0.595155
$690$	−3.51954	−0.133987
$691$	11.0632	0.420863	0.210431	−	0.977609i	$-0.432513\pi$
$691$	11.0632	0.420863	0.210431	+	0.977609i	$0.432513\pi$
$692$	4.91211	0.186730
$693$	−3.81270	−0.144832
$694$	24.1241	0.915738
$695$	13.5830	0.515234
$696$	−4.29525	−0.162811
$697$	41.6935	1.57925
$698$	13.0268	0.493073
$699$	−29.5533	−1.11781
$700$	12.0700	0.456203
$701$	−3.85913	−0.145757	−0.0728787	−	0.997341i	$-0.523219\pi$
$701$	−3.85913	−0.145757	−0.0728787	+	0.997341i	$0.523219\pi$
$702$	−27.4616	−1.03647
$703$	2.69819	0.101764
$704$	−0.846358	−0.0318983
$705$	−1.45481	−0.0547913
$706$	−16.5059	−0.621206
$707$	42.2187	1.58780
$708$	−0.556182	−0.0209026
$709$	2.30173	0.0864433	0.0432216	−	0.999066i	$-0.486238\pi$
$709$	2.30173	0.0864433	0.0432216	+	0.999066i	$0.486238\pi$
$710$	−3.61471	−0.135658
$711$	−25.7772	−0.966719
$712$	−16.6430	−0.623721
$713$	12.3511	0.462551
$714$	−17.4940	−0.654697
$715$	−3.74433	−0.140030
$716$	−6.00742	−0.224508
$717$	−20.2327	−0.755602
$718$	31.9398	1.19198
$719$	−36.3976	−1.35740	−0.678701	−	0.734415i	$-0.737457\pi$
$719$	−36.3976	−1.35740	−0.678701	+	0.734415i	$0.737457\pi$
$720$	1.38739	0.0517048
$721$	−7.41346	−0.276092
$722$	12.4587	0.463664
$723$	23.6590	0.879888
$724$	−16.4150	−0.610060
$725$	−15.2202	−0.565262
$726$	12.2656	0.455220
$727$	11.6931	0.433674	0.216837	−	0.976208i	$-0.430426\pi$
$727$	11.6931	0.433674	0.216837	+	0.976208i	$0.430426\pi$
$728$	−14.3648	−0.532397
$729$	22.3425	0.827502
$730$	10.2776	0.380393
$731$	−44.5002	−1.64590
$732$	−5.71066	−0.211072
$733$	35.4224	1.30836	0.654178	−	0.756340i	$-0.273015\pi$
$733$	35.4224	1.30836	0.654178	+	0.756340i	$0.273015\pi$
$734$	8.18891	0.302258
$735$	−1.21258	−0.0447266
$736$	3.35499	0.123666
$737$	11.3228	0.417080
$738$	−12.8055	−0.471379
$739$	−47.6231	−1.75185	−0.875923	−	0.482451i	$-0.839747\pi$
$739$	−47.6231	−1.75185	−0.875923	+	0.482451i	$0.839747\pi$
$740$	0.927886	0.0341098
$741$	15.3440	0.563677
$742$	8.86972	0.325618
$743$	−23.2372	−0.852489	−0.426244	−	0.904608i	$-0.640164\pi$
$743$	−23.2372	−0.852489	−0.426244	+	0.904608i	$0.640164\pi$
$744$	4.39092	0.160979
$745$	21.2630	0.779017
$746$	12.8455	0.470306
$747$	16.9937	0.621768
$748$	4.34677	0.158934
$749$	−15.9866	−0.584138
$750$	−9.67896	−0.353425
$751$	−17.9495	−0.654987	−0.327494	−	0.944853i	$-0.606204\pi$
$751$	−17.9495	−0.654987	−0.327494	+	0.944853i	$0.606204\pi$
$752$	1.38679	0.0505710
$753$	−7.20888	−0.262706
$754$	18.1139	0.659670
$755$	10.5900	0.385410
$756$	15.5918	0.567067
$757$	14.7212	0.535049	0.267525	−	0.963551i	$-0.413794\pi$
$757$	14.7212	0.535049	0.267525	+	0.963551i	$0.413794\pi$
$758$	5.50411	0.199918
$759$	3.38677	0.122932
$760$	−2.24951	−0.0815982
$761$	13.2085	0.478810	0.239405	−	0.970920i	$-0.423048\pi$
$761$	13.2085	0.478810	0.239405	+	0.970920i	$0.423048\pi$
$762$	2.78827	0.101008
$763$	−19.8484	−0.718559
$764$	−1.55968	−0.0564271
$765$	−7.12541	−0.257620
$766$	4.60091	0.166238
$767$	2.34553	0.0846922
$768$	1.19273	0.0430388
$769$	32.3689	1.16725	0.583626	−	0.812023i	$-0.301633\pi$
$769$	32.3689	1.16725	0.583626	+	0.812023i	$0.301633\pi$
$770$	2.12591	0.0766124
$771$	−18.8552	−0.679052
$772$	12.7563	0.459111
$773$	40.5491	1.45845	0.729224	−	0.684275i	$-0.239881\pi$
$773$	40.5491	1.45845	0.729224	+	0.684275i	$0.239881\pi$
$774$	13.6676	0.491271
$775$	15.5592	0.558901
$776$	−14.0127	−0.503028
$777$	3.59349	0.128916
$778$	−29.4172	−1.05466
$779$	20.7629	0.743908
$780$	5.27668	0.188936
$781$	3.47835	0.124465
$782$	−17.2307	−0.616170
$783$	−19.6611	−0.702629
$784$	1.15588	0.0412816
$785$	−2.31040	−0.0824616
$786$	−14.4129	−0.514090
$787$	18.2595	0.650881	0.325441	−	0.945562i	$-0.394487\pi$
$787$	18.2595	0.650881	0.325441	+	0.945562i	$0.394487\pi$
$788$	19.1347	0.681645
$789$	22.0223	0.784016
$790$	14.3730	0.511368
$791$	18.5717	0.660333
$792$	−1.33505	−0.0474388
$793$	24.0830	0.855213
$794$	28.1632	0.999474
$795$	−3.25814	−0.115554
$796$	5.87108	0.208095
$797$	24.9977	0.885464	0.442732	−	0.896654i	$-0.354009\pi$
$797$	24.9977	0.885464	0.442732	+	0.896654i	$0.354009\pi$
$798$	−8.71183	−0.308395
$799$	−7.12235	−0.251971
$800$	4.22641	0.149426
$801$	−26.2526	−0.927591
$802$	−23.5894	−0.832971
$803$	−9.88993	−0.349008
$804$	−15.9566	−0.562745
$805$	−8.42715	−0.297018
$806$	−18.5174	−0.652247
$807$	12.3596	0.435077
$808$	14.7832	0.520072
$809$	−17.4311	−0.612847	−0.306423	−	0.951895i	$-0.599132\pi$
$809$	−17.4311	−0.612847	−0.306423	+	0.951895i	$0.599132\pi$
$810$	−1.56522	−0.0549963
$811$	−10.9414	−0.384205	−0.192102	−	0.981375i	$-0.561531\pi$
$811$	−10.9414	−0.384205	−0.192102	+	0.981375i	$0.561531\pi$
$812$	−10.2845	−0.360915
$813$	6.55202	0.229790
$814$	−0.892882	−0.0312955
$815$	−1.26305	−0.0442429
$816$	−6.12568	−0.214442
$817$	−22.1606	−0.775302
$818$	20.5473	0.718420
$819$	−22.6591	−0.791775
$820$	7.14020	0.249346
$821$	8.96973	0.313046	0.156523	−	0.987674i	$-0.449972\pi$
$821$	8.96973	0.313046	0.156523	+	0.987674i	$0.449972\pi$
$822$	−3.67863	−0.128307
$823$	−40.1711	−1.40028	−0.700138	−	0.714008i	$-0.746878\pi$
$823$	−40.1711	−1.40028	−0.700138	+	0.714008i	$0.746878\pi$
$824$	−2.59589	−0.0904320
$825$	4.26645	0.148539
$826$	−1.33171	−0.0463363
$827$	43.8976	1.52647	0.763234	−	0.646122i	$-0.223610\pi$
$827$	43.8976	1.52647	0.763234	+	0.646122i	$0.223610\pi$
$828$	5.29216	0.183915
$829$	20.1595	0.700168	0.350084	−	0.936718i	$-0.386153\pi$
$829$	20.1595	0.700168	0.350084	+	0.936718i	$0.386153\pi$
$830$	−9.47547	−0.328898
$831$	26.0842	0.904849
$832$	−5.02997	−0.174383
$833$	−5.93645	−0.205686
$834$	18.4197	0.637824
$835$	7.09548	0.245549
$836$	2.16465	0.0748658
$837$	20.0990	0.694723
$838$	−10.1840	−0.351802
$839$	−11.6408	−0.401884	−0.200942	−	0.979603i	$-0.564400\pi$
$839$	−11.6408	−0.401884	−0.200942	+	0.979603i	$0.564400\pi$
$840$	−2.99593	−0.103369
$841$	−16.0314	−0.552806
$842$	−21.2809	−0.733388
$843$	29.0749	1.00139
$844$	−5.69678	−0.196091
$845$	−10.8189	−0.372180
$846$	2.18752	0.0752086
$847$	29.3687	1.00912
$848$	3.10581	0.106654
$849$	−33.0007	−1.13258
$850$	−21.7063	−0.744518
$851$	3.53941	0.121329
$852$	−4.90185	−0.167935
$853$	−10.5333	−0.360652	−0.180326	−	0.983607i	$-0.557715\pi$
$853$	−10.5333	−0.360652	−0.180326	+	0.983607i	$0.557715\pi$
$854$	−13.6735	−0.467899
$855$	−3.54838	−0.121352
$856$	−5.59785	−0.191330
$857$	−29.7850	−1.01744	−0.508718	−	0.860933i	$-0.669880\pi$
$857$	−29.7850	−1.01744	−0.508718	+	0.860933i	$0.669880\pi$
$858$	−5.07763	−0.173347
$859$	−30.0626	−1.02572	−0.512861	−	0.858472i	$-0.671414\pi$
$859$	−30.0626	−1.02572	−0.512861	+	0.858472i	$0.671414\pi$
$860$	−7.62086	−0.259869
$861$	27.6524	0.942390
$862$	26.8272	0.913738
$863$	29.8927	1.01756	0.508780	−	0.860897i	$-0.330097\pi$
$863$	29.8927	1.01756	0.508780	+	0.860897i	$0.330097\pi$
$864$	5.45959	0.185739
$865$	−4.32039	−0.146898
$866$	11.0028	0.373892
$867$	11.1842	0.379837
$868$	10.5136	0.356853
$869$	−13.8308	−0.469177
$870$	3.77783	0.128081
$871$	67.2921	2.28011
$872$	−6.95007	−0.235359
$873$	−22.1037	−0.748098
$874$	−8.58071	−0.290247
$875$	−23.1752	−0.783464
$876$	13.9373	0.470899
$877$	27.2310	0.919526	0.459763	−	0.888042i	$-0.347934\pi$
$877$	27.2310	0.919526	0.459763	+	0.888042i	$0.347934\pi$
$878$	4.05311	0.136786
$879$	37.4967	1.26473
$880$	0.744404	0.0250939
$881$	−48.2162	−1.62444	−0.812222	−	0.583349i	$-0.801742\pi$
$881$	−48.2162	−1.62444	−0.812222	+	0.583349i	$0.801742\pi$
$882$	1.82329	0.0613935
$883$	19.9012	0.669727	0.334864	−	0.942267i	$-0.391310\pi$
$883$	19.9012	0.669727	0.334864	+	0.942267i	$0.391310\pi$
$884$	25.8332	0.868865
$885$	0.489183	0.0164437
$886$	−33.7660	−1.13439
$887$	37.8589	1.27118	0.635588	−	0.772028i	$-0.280758\pi$
$887$	37.8589	1.27118	0.635588	+	0.772028i	$0.280758\pi$
$888$	1.25829	0.0422255
$889$	6.67620	0.223912
$890$	14.6381	0.490670
$891$	1.50617	0.0504587
$892$	−5.98253	−0.200310
$893$	−3.54685	−0.118691
$894$	28.8344	0.964368
$895$	5.28375	0.176616
$896$	2.85585	0.0954073
$897$	20.1278	0.672049
$898$	0.476417	0.0158982
$899$	−13.2575	−0.442162
$900$	6.66675	0.222225
$901$	−15.9510	−0.531404
$902$	−6.87084	−0.228774
$903$	−29.5138	−0.982159
$904$	6.50303	0.216287
$905$	14.4377	0.479924
$906$	14.3609	0.477110
$907$	33.1077	1.09932	0.549662	−	0.835387i	$-0.314757\pi$
$907$	33.1077	1.09932	0.549662	+	0.835387i	$0.314757\pi$
$908$	27.3608	0.907999
$909$	23.3191	0.773446
$910$	12.6344	0.418827
$911$	−5.46211	−0.180968	−0.0904839	−	0.995898i	$-0.528841\pi$
$911$	−5.46211	−0.180968	−0.0904839	+	0.995898i	$0.528841\pi$
$912$	−3.05052	−0.101013
$913$	9.11801	0.301762
$914$	17.6693	0.584450
$915$	5.02274	0.166047
$916$	−18.5008	−0.611284
$917$	−34.5100	−1.13962
$918$	−28.0397	−0.925447
$919$	40.8886	1.34879	0.674395	−	0.738371i	$-0.264405\pi$
$919$	40.8886	1.34879	0.674395	+	0.738371i	$0.264405\pi$
$920$	−2.95084	−0.0972862
$921$	8.66001	0.285357
$922$	13.0416	0.429501
$923$	20.6721	0.680430
$924$	2.88291	0.0948407
$925$	4.45874	0.146602
$926$	−6.62965	−0.217864
$927$	−4.09476	−0.134489
$928$	−3.60120	−0.118215
$929$	23.9245	0.784939	0.392470	−	0.919765i	$-0.371621\pi$
$929$	23.9245	0.784939	0.392470	+	0.919765i	$0.371621\pi$
$930$	−3.86198	−0.126639
$931$	−2.95629	−0.0968884
$932$	−24.7780	−0.811629
$933$	−40.0967	−1.31271
$934$	−30.7789	−1.00711
$935$	−3.82315	−0.125030
$936$	−7.93429	−0.259340
$937$	29.9758	0.979268	0.489634	−	0.871928i	$-0.337130\pi$
$937$	29.9758	0.979268	0.489634	+	0.871928i	$0.337130\pi$
$938$	−38.2062	−1.24748
$939$	−29.4537	−0.961184
$940$	−1.21973	−0.0397833
$941$	46.5376	1.51708	0.758542	−	0.651624i	$-0.225912\pi$
$941$	46.5376	1.51708	0.758542	+	0.651624i	$0.225912\pi$
$942$	−3.13309	−0.102082
$943$	27.2362	0.886932
$944$	−0.466311	−0.0151771
$945$	−13.7136	−0.446102
$946$	7.33337	0.238428
$947$	−13.3102	−0.432522	−0.216261	−	0.976336i	$-0.569386\pi$
$947$	−13.3102	−0.432522	−0.216261	+	0.976336i	$0.569386\pi$
$948$	19.4910	0.633038
$949$	−58.7766	−1.90797
$950$	−10.8095	−0.350706
$951$	−11.0585	−0.358595
$952$	−14.6672	−0.475368
$953$	15.8393	0.513085	0.256542	−	0.966533i	$-0.417417\pi$
$953$	15.8393	0.513085	0.256542	+	0.966533i	$0.417417\pi$
$954$	4.89910	0.158614
$955$	1.37180	0.0443903
$956$	−16.9634	−0.548634
$957$	−3.63532	−0.117513
$958$	−35.5807	−1.14956
$959$	−8.80807	−0.284427
$960$	−1.04905	−0.0338579
$961$	−17.4472	−0.562814
$962$	−5.30647	−0.171087
$963$	−8.83006	−0.284545
$964$	19.8361	0.638877
$965$	−11.2197	−0.361174
$966$	−11.4279	−0.367687
$967$	−51.1450	−1.64471	−0.822357	−	0.568972i	$-0.807341\pi$
$967$	−51.1450	−1.64471	−0.822357	+	0.568972i	$0.807341\pi$
$968$	10.2837	0.330530
$969$	15.6670	0.503298
$970$	12.3247	0.395723
$971$	−18.6528	−0.598596	−0.299298	−	0.954160i	$-0.596752\pi$
$971$	−18.6528	−0.598596	−0.299298	+	0.954160i	$0.596752\pi$
$972$	14.2562	0.457268
$973$	44.1040	1.41391
$974$	30.2807	0.970257
$975$	25.3559	0.812038
$976$	−4.78790	−0.153257
$977$	−20.7296	−0.663198	−0.331599	−	0.943420i	$-0.607588\pi$
$977$	−20.7296	−0.663198	−0.331599	+	0.943420i	$0.607588\pi$
$978$	−1.71281	−0.0547695
$979$	−14.0859	−0.450187
$980$	−1.01664	−0.0324755
$981$	−10.9631	−0.350023
$982$	7.83648	0.250072
$983$	−19.3227	−0.616298	−0.308149	−	0.951338i	$-0.599710\pi$
$983$	−19.3227	−0.616298	−0.308149	+	0.951338i	$0.599710\pi$
$984$	9.68270	0.308673
$985$	−16.8297	−0.536238
$986$	18.4952	0.589009
$987$	−4.72375	−0.150359
$988$	12.8647	0.409279
$989$	−29.0697	−0.924361
$990$	1.17422	0.0373193
$991$	25.3675	0.805824	0.402912	−	0.915239i	$-0.367998\pi$
$991$	25.3675	0.805824	0.402912	+	0.915239i	$0.367998\pi$
$992$	3.68141	0.116885
$993$	−11.4398	−0.363031
$994$	−11.7369	−0.372273
$995$	−5.16384	−0.163705
$996$	−12.8495	−0.407153
$997$	−0.980226	−0.0310441	−0.0155220	−	0.999880i	$-0.504941\pi$
$997$	−0.980226	−0.0310441	−0.0155220	+	0.999880i	$0.504941\pi$
$998$	23.5213	0.744554
$999$	5.75970	0.182229

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.e.1.51	✓	77

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.e.1.51	✓	77	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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.19273 q^{3} +1.00000 q^{4} -0.879538 q^{5} -1.19273 q^{6} -2.85585 q^{7} -1.00000 q^{8} -1.57740 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.19273 q^{3} +1.00000 q^{4} -0.879538 q^{5} -1.19273 q^{6} -2.85585 q^{7} -1.00000 q^{8} -1.57740 q^{9} +0.879538 q^{10} -0.846358 q^{11} +1.19273 q^{12} -5.02997 q^{13} +2.85585 q^{14} -1.04905 q^{15} +1.00000 q^{16} -5.13586 q^{17} +1.57740 q^{18} -2.55760 q^{19} -0.879538 q^{20} -3.40625 q^{21} +0.846358 q^{22} -3.35499 q^{23} -1.19273 q^{24} -4.22641 q^{25} +5.02997 q^{26} -5.45959 q^{27} -2.85585 q^{28} +3.60120 q^{29} +1.04905 q^{30} -3.68141 q^{31} -1.00000 q^{32} -1.00947 q^{33} +5.13586 q^{34} +2.51183 q^{35} -1.57740 q^{36} -1.05497 q^{37} +2.55760 q^{38} -5.99938 q^{39} +0.879538 q^{40} -8.11812 q^{41} +3.40625 q^{42} +8.66462 q^{43} -0.846358 q^{44} +1.38739 q^{45} +3.35499 q^{46} +1.38679 q^{47} +1.19273 q^{48} +1.15588 q^{49} +4.22641 q^{50} -6.12568 q^{51} -5.02997 q^{52} +3.10581 q^{53} +5.45959 q^{54} +0.744404 q^{55} +2.85585 q^{56} -3.05052 q^{57} -3.60120 q^{58} -0.466311 q^{59} -1.04905 q^{60} -4.78790 q^{61} +3.68141 q^{62} +4.50483 q^{63} +1.00000 q^{64} +4.42405 q^{65} +1.00947 q^{66} -13.3782 q^{67} -5.13586 q^{68} -4.00158 q^{69} -2.51183 q^{70} -4.10978 q^{71} +1.57740 q^{72} +11.6853 q^{73} +1.05497 q^{74} -5.04096 q^{75} -2.55760 q^{76} +2.41707 q^{77} +5.99938 q^{78} +16.3415 q^{79} -0.879538 q^{80} -1.77959 q^{81} +8.11812 q^{82} -10.7732 q^{83} -3.40625 q^{84} +4.51718 q^{85} -8.66462 q^{86} +4.29525 q^{87} +0.846358 q^{88} +16.6430 q^{89} -1.38739 q^{90} +14.3648 q^{91} -3.35499 q^{92} -4.39092 q^{93} -1.38679 q^{94} +2.24951 q^{95} -1.19273 q^{96} +14.0127 q^{97} -1.15588 q^{98} +1.33505 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * q^99

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