LMFDB - Embedded newform 8002.2.a.e.1.21

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			1.21
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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.56339 q^{3} +1.00000 q^{4} +2.57083 q^{5} +1.56339 q^{6} +0.482155 q^{7} -1.00000 q^{8} -0.555797 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.56339 q^{3} +1.00000 q^{4} +2.57083 q^{5} +1.56339 q^{6} +0.482155 q^{7} -1.00000 q^{8} -0.555797 q^{9} -2.57083 q^{10} -1.83398 q^{11} -1.56339 q^{12} +2.81070 q^{13} -0.482155 q^{14} -4.01923 q^{15} +1.00000 q^{16} +3.01322 q^{17} +0.555797 q^{18} +3.88411 q^{19} +2.57083 q^{20} -0.753798 q^{21} +1.83398 q^{22} +7.79637 q^{23} +1.56339 q^{24} +1.60918 q^{25} -2.81070 q^{26} +5.55911 q^{27} +0.482155 q^{28} +8.66328 q^{29} +4.01923 q^{30} +6.90859 q^{31} -1.00000 q^{32} +2.86723 q^{33} -3.01322 q^{34} +1.23954 q^{35} -0.555797 q^{36} +6.65062 q^{37} -3.88411 q^{38} -4.39423 q^{39} -2.57083 q^{40} -11.5818 q^{41} +0.753798 q^{42} +6.28175 q^{43} -1.83398 q^{44} -1.42886 q^{45} -7.79637 q^{46} -4.21061 q^{47} -1.56339 q^{48} -6.76753 q^{49} -1.60918 q^{50} -4.71086 q^{51} +2.81070 q^{52} +0.124695 q^{53} -5.55911 q^{54} -4.71485 q^{55} -0.482155 q^{56} -6.07240 q^{57} -8.66328 q^{58} +12.7703 q^{59} -4.01923 q^{60} +9.66903 q^{61} -6.90859 q^{62} -0.267980 q^{63} +1.00000 q^{64} +7.22584 q^{65} -2.86723 q^{66} -14.1323 q^{67} +3.01322 q^{68} -12.1888 q^{69} -1.23954 q^{70} +12.5032 q^{71} +0.555797 q^{72} -7.95519 q^{73} -6.65062 q^{74} -2.51578 q^{75} +3.88411 q^{76} -0.884261 q^{77} +4.39423 q^{78} -7.39974 q^{79} +2.57083 q^{80} -7.02370 q^{81} +11.5818 q^{82} -4.97925 q^{83} -0.753798 q^{84} +7.74649 q^{85} -6.28175 q^{86} -13.5441 q^{87} +1.83398 q^{88} -7.00892 q^{89} +1.42886 q^{90} +1.35519 q^{91} +7.79637 q^{92} -10.8008 q^{93} +4.21061 q^{94} +9.98540 q^{95} +1.56339 q^{96} +3.02340 q^{97} +6.76753 q^{98} +1.01932 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.56339	−0.902626	−0.451313	−	0.892366i	$-0.649044\pi$
$3$	−1.56339	−0.902626	−0.451313	+	0.892366i	$0.649044\pi$
$4$	1.00000	0.500000
$5$	2.57083	1.14971	0.574856	−	0.818255i	$-0.305058\pi$
$5$	2.57083	1.14971	0.574856	+	0.818255i	$0.305058\pi$
$6$	1.56339	0.638253
$7$	0.482155	0.182237	0.0911187	−	0.995840i	$-0.470956\pi$
$7$	0.482155	0.182237	0.0911187	+	0.995840i	$0.470956\pi$
$8$	−1.00000	−0.353553
$9$	−0.555797	−0.185266
$10$	−2.57083	−0.812969
$11$	−1.83398	−0.552965	−0.276483	−	0.961019i	$-0.589169\pi$
$11$	−1.83398	−0.552965	−0.276483	+	0.961019i	$0.589169\pi$
$12$	−1.56339	−0.451313
$13$	2.81070	0.779548	0.389774	−	0.920911i	$-0.372553\pi$
$13$	2.81070	0.779548	0.389774	+	0.920911i	$0.372553\pi$
$14$	−0.482155	−0.128861
$15$	−4.01923	−1.03776
$16$	1.00000	0.250000
$17$	3.01322	0.730814	0.365407	−	0.930848i	$-0.380930\pi$
$17$	3.01322	0.730814	0.365407	+	0.930848i	$0.380930\pi$
$18$	0.555797	0.131003
$19$	3.88411	0.891076	0.445538	−	0.895263i	$-0.353012\pi$
$19$	3.88411	0.891076	0.445538	+	0.895263i	$0.353012\pi$
$20$	2.57083	0.574856
$21$	−0.753798	−0.164492
$22$	1.83398	0.391005
$23$	7.79637	1.62566	0.812828	−	0.582504i	$-0.197927\pi$
$23$	7.79637	1.62566	0.812828	+	0.582504i	$0.197927\pi$
$24$	1.56339	0.319127
$25$	1.60918	0.321836
$26$	−2.81070	−0.551223
$27$	5.55911	1.06985
$28$	0.482155	0.0911187
$29$	8.66328	1.60873	0.804366	−	0.594135i	$-0.202505\pi$
$29$	8.66328	1.60873	0.804366	+	0.594135i	$0.202505\pi$
$30$	4.01923	0.733807
$31$	6.90859	1.24082	0.620409	−	0.784278i	$-0.286966\pi$
$31$	6.90859	1.24082	0.620409	+	0.784278i	$0.286966\pi$
$32$	−1.00000	−0.176777
$33$	2.86723	0.499121
$34$	−3.01322	−0.516764
$35$	1.23954	0.209520
$36$	−0.555797	−0.0926329
$37$	6.65062	1.09336	0.546678	−	0.837343i	$-0.315892\pi$
$37$	6.65062	1.09336	0.546678	+	0.837343i	$0.315892\pi$
$38$	−3.88411	−0.630086
$39$	−4.39423	−0.703640
$40$	−2.57083	−0.406484
$41$	−11.5818	−1.80877	−0.904386	−	0.426716i	$-0.859671\pi$
$41$	−11.5818	−1.80877	−0.904386	+	0.426716i	$0.859671\pi$
$42$	0.753798	0.116314
$43$	6.28175	0.957957	0.478979	−	0.877827i	$-0.341007\pi$
$43$	6.28175	0.957957	0.478979	+	0.877827i	$0.341007\pi$
$44$	−1.83398	−0.276483
$45$	−1.42886	−0.213002
$46$	−7.79637	−1.14951
$47$	−4.21061	−0.614180	−0.307090	−	0.951680i	$-0.599355\pi$
$47$	−4.21061	−0.614180	−0.307090	+	0.951680i	$0.599355\pi$
$48$	−1.56339	−0.225657
$49$	−6.76753	−0.966790
$50$	−1.60918	−0.227572
$51$	−4.71086	−0.659652
$52$	2.81070	0.389774
$53$	0.124695	0.0171281	0.00856406	−	0.999963i	$-0.497274\pi$
$53$	0.124695	0.0171281	0.00856406	+	0.999963i	$0.497274\pi$
$54$	−5.55911	−0.756500
$55$	−4.71485	−0.635750
$56$	−0.482155	−0.0644306
$57$	−6.07240	−0.804309
$58$	−8.66328	−1.13754
$59$	12.7703	1.66255	0.831274	−	0.555862i	$-0.187612\pi$
$59$	12.7703	1.66255	0.831274	+	0.555862i	$0.187612\pi$
$60$	−4.01923	−0.518880
$61$	9.66903	1.23799	0.618996	−	0.785394i	$-0.287540\pi$
$61$	9.66903	1.23799	0.618996	+	0.785394i	$0.287540\pi$
$62$	−6.90859	−0.877391
$63$	−0.267980	−0.0337624
$64$	1.00000	0.125000
$65$	7.22584	0.896255
$66$	−2.86723	−0.352932
$67$	−14.1323	−1.72654	−0.863269	−	0.504745i	$-0.831587\pi$
$67$	−14.1323	−1.72654	−0.863269	+	0.504745i	$0.831587\pi$
$68$	3.01322	0.365407
$69$	−12.1888	−1.46736
$70$	−1.23954	−0.148153
$71$	12.5032	1.48386	0.741931	−	0.670477i	$-0.233910\pi$
$71$	12.5032	1.48386	0.741931	+	0.670477i	$0.233910\pi$
$72$	0.555797	0.0655014
$73$	−7.95519	−0.931085	−0.465542	−	0.885026i	$-0.654141\pi$
$73$	−7.95519	−0.931085	−0.465542	+	0.885026i	$0.654141\pi$
$74$	−6.65062	−0.773119
$75$	−2.51578	−0.290497
$76$	3.88411	0.445538
$77$	−0.884261	−0.100771
$78$	4.39423	0.497549
$79$	−7.39974	−0.832536	−0.416268	−	0.909242i	$-0.636662\pi$
$79$	−7.39974	−0.832536	−0.416268	+	0.909242i	$0.636662\pi$
$80$	2.57083	0.287428
$81$	−7.02370	−0.780411
$82$	11.5818	1.27899
$83$	−4.97925	−0.546544	−0.273272	−	0.961937i	$-0.588106\pi$
$83$	−4.97925	−0.546544	−0.273272	+	0.961937i	$0.588106\pi$
$84$	−0.753798	−0.0822461
$85$	7.74649	0.840225
$86$	−6.28175	−0.677378
$87$	−13.5441	−1.45208
$88$	1.83398	0.195503
$89$	−7.00892	−0.742944	−0.371472	−	0.928444i	$-0.621147\pi$
$89$	−7.00892	−0.742944	−0.371472	+	0.928444i	$0.621147\pi$
$90$	1.42886	0.150615
$91$	1.35519	0.142063
$92$	7.79637	0.812828
$93$	−10.8008	−1.12000
$94$	4.21061	0.434291
$95$	9.98540	1.02448
$96$	1.56339	0.159563
$97$	3.02340	0.306980	0.153490	−	0.988150i	$-0.450949\pi$
$97$	3.02340	0.306980	0.153490	+	0.988150i	$0.450949\pi$
$98$	6.76753	0.683623
$99$	1.01932	0.102446
$100$	1.60918	0.160918
$101$	4.65196	0.462888	0.231444	−	0.972848i	$-0.425655\pi$
$101$	4.65196	0.462888	0.231444	+	0.972848i	$0.425655\pi$
$102$	4.71086	0.466444
$103$	−15.1305	−1.49085	−0.745426	−	0.666588i	$-0.767754\pi$
$103$	−15.1305	−1.49085	−0.745426	+	0.666588i	$0.767754\pi$
$104$	−2.81070	−0.275612
$105$	−1.93789	−0.189119
$106$	−0.124695	−0.0121114
$107$	−7.19364	−0.695435	−0.347717	−	0.937599i	$-0.613043\pi$
$107$	−7.19364	−0.695435	−0.347717	+	0.937599i	$0.613043\pi$
$108$	5.55911	0.534926
$109$	3.54498	0.339547	0.169774	−	0.985483i	$-0.445696\pi$
$109$	3.54498	0.339547	0.169774	+	0.985483i	$0.445696\pi$
$110$	4.71485	0.449543
$111$	−10.3975	−0.986891
$112$	0.482155	0.0455593
$113$	−11.5327	−1.08491	−0.542454	−	0.840086i	$-0.682505\pi$
$113$	−11.5327	−1.08491	−0.542454	+	0.840086i	$0.682505\pi$
$114$	6.07240	0.568732
$115$	20.0432	1.86903
$116$	8.66328	0.804366
$117$	−1.56218	−0.144424
$118$	−12.7703	−1.17560
$119$	1.45284	0.133182
$120$	4.01923	0.366903
$121$	−7.63653	−0.694230
$122$	−9.66903	−0.875393
$123$	18.1069	1.63264
$124$	6.90859	0.620409
$125$	−8.71723	−0.779693
$126$	0.267980	0.0238736
$127$	−9.77415	−0.867315	−0.433658	−	0.901078i	$-0.642777\pi$
$127$	−9.77415	−0.867315	−0.433658	+	0.901078i	$0.642777\pi$
$128$	−1.00000	−0.0883883
$129$	−9.82085	−0.864677
$130$	−7.22584	−0.633748
$131$	0.856748	0.0748544	0.0374272	−	0.999299i	$-0.488084\pi$
$131$	0.856748	0.0748544	0.0374272	+	0.999299i	$0.488084\pi$
$132$	2.86723	0.249560
$133$	1.87274	0.162387
$134$	14.1323	1.22085
$135$	14.2916	1.23002
$136$	−3.01322	−0.258382
$137$	14.1980	1.21302	0.606510	−	0.795076i	$-0.292569\pi$
$137$	14.1980	1.21302	0.606510	+	0.795076i	$0.292569\pi$
$138$	12.1888	1.03758
$139$	14.3374	1.21608	0.608040	−	0.793907i	$-0.291956\pi$
$139$	14.3374	1.21608	0.608040	+	0.793907i	$0.291956\pi$
$140$	1.23954	0.104760
$141$	6.58284	0.554375
$142$	−12.5032	−1.04925
$143$	−5.15476	−0.431063
$144$	−0.555797	−0.0463165
$145$	22.2718	1.84958
$146$	7.95519	0.658376
$147$	10.5803	0.872650
$148$	6.65062	0.546678
$149$	11.9062	0.975393	0.487696	−	0.873013i	$-0.337837\pi$
$149$	11.9062	0.975393	0.487696	+	0.873013i	$0.337837\pi$
$150$	2.51578	0.205413
$151$	−7.99865	−0.650921	−0.325460	−	0.945556i	$-0.605519\pi$
$151$	−7.99865	−0.650921	−0.325460	+	0.945556i	$0.605519\pi$
$152$	−3.88411	−0.315043
$153$	−1.67474	−0.135395
$154$	0.884261	0.0712558
$155$	17.7608	1.42658
$156$	−4.39423	−0.351820
$157$	17.4152	1.38988	0.694940	−	0.719067i	$-0.255431\pi$
$157$	17.4152	1.38988	0.694940	+	0.719067i	$0.255431\pi$
$158$	7.39974	0.588692
$159$	−0.194947	−0.0154603
$160$	−2.57083	−0.203242
$161$	3.75906	0.296255
$162$	7.02370	0.551834
$163$	20.4398	1.60097	0.800484	−	0.599355i	$-0.204576\pi$
$163$	20.4398	1.60097	0.800484	+	0.599355i	$0.204576\pi$
$164$	−11.5818	−0.904386
$165$	7.37117	0.573845
$166$	4.97925	0.386465
$167$	18.8288	1.45701	0.728506	−	0.685039i	$-0.240215\pi$
$167$	18.8288	1.45701	0.728506	+	0.685039i	$0.240215\pi$
$168$	0.753798	0.0581568
$169$	−5.09997	−0.392305
$170$	−7.74649	−0.594129
$171$	−2.15878	−0.165086
$172$	6.28175	0.478979
$173$	12.6001	0.957970	0.478985	−	0.877823i	$-0.341005\pi$
$173$	12.6001	0.957970	0.478985	+	0.877823i	$0.341005\pi$
$174$	13.5441	1.02678
$175$	0.775873	0.0586505
$176$	−1.83398	−0.138241
$177$	−19.9650	−1.50066
$178$	7.00892	0.525341
$179$	0.426075	0.0318463	0.0159232	−	0.999873i	$-0.494931\pi$
$179$	0.426075	0.0318463	0.0159232	+	0.999873i	$0.494931\pi$
$180$	−1.42886	−0.106501
$181$	12.8128	0.952368	0.476184	−	0.879346i	$-0.342020\pi$
$181$	12.8128	0.952368	0.476184	+	0.879346i	$0.342020\pi$
$182$	−1.35519	−0.100454
$183$	−15.1165	−1.11744
$184$	−7.79637	−0.574756
$185$	17.0976	1.25704
$186$	10.8008	0.791956
$187$	−5.52618	−0.404115
$188$	−4.21061	−0.307090
$189$	2.68035	0.194967
$190$	−9.98540	−0.724417
$191$	−19.4401	−1.40664	−0.703319	−	0.710875i	$-0.748299\pi$
$191$	−19.4401	−1.40664	−0.703319	+	0.710875i	$0.748299\pi$
$192$	−1.56339	−0.112828
$193$	−20.4603	−1.47277	−0.736383	−	0.676565i	$-0.763468\pi$
$193$	−20.4603	−1.47277	−0.736383	+	0.676565i	$0.763468\pi$
$194$	−3.02340	−0.217067
$195$	−11.2968	−0.808983
$196$	−6.76753	−0.483395
$197$	−14.8602	−1.05875	−0.529373	−	0.848389i	$-0.677573\pi$
$197$	−14.8602	−1.05875	−0.529373	+	0.848389i	$0.677573\pi$
$198$	−1.01932	−0.0724399
$199$	22.9014	1.62343	0.811717	−	0.584051i	$-0.198533\pi$
$199$	22.9014	1.62343	0.811717	+	0.584051i	$0.198533\pi$
$200$	−1.60918	−0.113786
$201$	22.0944	1.55842
$202$	−4.65196	−0.327311
$203$	4.17704	0.293171
$204$	−4.71086	−0.329826
$205$	−29.7748	−2.07957
$206$	15.1305	1.05419
$207$	−4.33320	−0.301178
$208$	2.81070	0.194887
$209$	−7.12337	−0.492734
$210$	1.93789	0.133727
$211$	26.2928	1.81007	0.905034	−	0.425338i	$-0.139845\pi$
$211$	26.2928	1.81007	0.905034	+	0.425338i	$0.139845\pi$
$212$	0.124695	0.00856406
$213$	−19.5475	−1.33937
$214$	7.19364	0.491747
$215$	16.1493	1.10137
$216$	−5.55911	−0.378250
$217$	3.33101	0.226124
$218$	−3.54498	−0.240096
$219$	12.4371	0.840422
$220$	−4.71485	−0.317875
$221$	8.46927	0.569704
$222$	10.3975	0.697838
$223$	−5.09847	−0.341419	−0.170709	−	0.985321i	$-0.554606\pi$
$223$	−5.09847	−0.341419	−0.170709	+	0.985321i	$0.554606\pi$
$224$	−0.482155	−0.0322153
$225$	−0.894377	−0.0596252
$226$	11.5327	0.767145
$227$	−9.23300	−0.612816	−0.306408	−	0.951900i	$-0.599127\pi$
$227$	−9.23300	−0.612816	−0.306408	+	0.951900i	$0.599127\pi$
$228$	−6.07240	−0.402154
$229$	−24.7043	−1.63251	−0.816253	−	0.577695i	$-0.803952\pi$
$229$	−24.7043	−1.63251	−0.816253	+	0.577695i	$0.803952\pi$
$230$	−20.0432	−1.32161
$231$	1.38245	0.0909585
$232$	−8.66328	−0.568772
$233$	−6.50005	−0.425832	−0.212916	−	0.977070i	$-0.568296\pi$
$233$	−6.50005	−0.425832	−0.212916	+	0.977070i	$0.568296\pi$
$234$	1.56218	0.102123
$235$	−10.8248	−0.706130
$236$	12.7703	0.831274
$237$	11.5687	0.751469
$238$	−1.45284	−0.0941736
$239$	−5.18780	−0.335571	−0.167786	−	0.985824i	$-0.553662\pi$
$239$	−5.18780	−0.335571	−0.167786	+	0.985824i	$0.553662\pi$
$240$	−4.01923	−0.259440
$241$	−19.9305	−1.28384	−0.641918	−	0.766774i	$-0.721861\pi$
$241$	−19.9305	−1.28384	−0.641918	+	0.766774i	$0.721861\pi$
$242$	7.63653	0.490895
$243$	−5.69653	−0.365433
$244$	9.66903	0.618996
$245$	−17.3982	−1.11153
$246$	−18.1069	−1.15445
$247$	10.9171	0.694636
$248$	−6.90859	−0.438696
$249$	7.78454	0.493325
$250$	8.71723	0.551326
$251$	8.86713	0.559688	0.279844	−	0.960045i	$-0.409717\pi$
$251$	8.86713	0.559688	0.279844	+	0.960045i	$0.409717\pi$
$252$	−0.267980	−0.0168812
$253$	−14.2984	−0.898930
$254$	9.77415	0.613285
$255$	−12.1108	−0.758409
$256$	1.00000	0.0625000
$257$	20.8020	1.29759	0.648796	−	0.760962i	$-0.275273\pi$
$257$	20.8020	1.29759	0.648796	+	0.760962i	$0.275273\pi$
$258$	9.82085	0.611419
$259$	3.20663	0.199250
$260$	7.22584	0.448127
$261$	−4.81503	−0.298043
$262$	−0.856748	−0.0529300
$263$	17.1358	1.05664	0.528319	−	0.849046i	$-0.322823\pi$
$263$	17.1358	1.05664	0.528319	+	0.849046i	$0.322823\pi$
$264$	−2.86723	−0.176466
$265$	0.320569	0.0196924
$266$	−1.87274	−0.114825
$267$	10.9577	0.670601
$268$	−14.1323	−0.863269
$269$	−26.7135	−1.62875	−0.814377	−	0.580337i	$-0.802921\pi$
$269$	−26.7135	−1.62875	−0.814377	+	0.580337i	$0.802921\pi$
$270$	−14.2916	−0.869756
$271$	−14.3826	−0.873679	−0.436840	−	0.899539i	$-0.643902\pi$
$271$	−14.3826	−0.873679	−0.436840	+	0.899539i	$0.643902\pi$
$272$	3.01322	0.182704
$273$	−2.11870	−0.128230
$274$	−14.1980	−0.857735
$275$	−2.95120	−0.177964
$276$	−12.1888	−0.733680
$277$	3.28907	0.197621	0.0988106	−	0.995106i	$-0.468496\pi$
$277$	3.28907	0.197621	0.0988106	+	0.995106i	$0.468496\pi$
$278$	−14.3374	−0.859898
$279$	−3.83977	−0.229881
$280$	−1.23954	−0.0740766
$281$	19.5666	1.16724	0.583622	−	0.812026i	$-0.301635\pi$
$281$	19.5666	1.16724	0.583622	+	0.812026i	$0.301635\pi$
$282$	−6.58284	−0.392003
$283$	−7.65265	−0.454903	−0.227452	−	0.973789i	$-0.573039\pi$
$283$	−7.65265	−0.454903	−0.227452	+	0.973789i	$0.573039\pi$
$284$	12.5032	0.741931
$285$	−15.6111	−0.924723
$286$	5.15476	0.304807
$287$	−5.58422	−0.329626
$288$	0.555797	0.0327507
$289$	−7.92048	−0.465911
$290$	−22.2718	−1.30785
$291$	−4.72677	−0.277088
$292$	−7.95519	−0.465542
$293$	4.82001	0.281588	0.140794	−	0.990039i	$-0.455034\pi$
$293$	4.82001	0.281588	0.140794	+	0.990039i	$0.455034\pi$
$294$	−10.5803	−0.617056
$295$	32.8302	1.91145
$296$	−6.65062	−0.386560
$297$	−10.1953	−0.591591
$298$	−11.9062	−0.689707
$299$	21.9132	1.26728
$300$	−2.51578	−0.145249
$301$	3.02877	0.174576
$302$	7.99865	0.460271
$303$	−7.27286	−0.417815
$304$	3.88411	0.222769
$305$	24.8575	1.42333
$306$	1.67474	0.0957386
$307$	18.4937	1.05549	0.527745	−	0.849403i	$-0.323038\pi$
$307$	18.4937	1.05549	0.527745	+	0.849403i	$0.323038\pi$
$308$	−0.884261	−0.0503854
$309$	23.6549	1.34568
$310$	−17.7608	−1.00875
$311$	−28.8408	−1.63541	−0.817705	−	0.575638i	$-0.804754\pi$
$311$	−28.8408	−1.63541	−0.817705	+	0.575638i	$0.804754\pi$
$312$	4.39423	0.248774
$313$	−9.11885	−0.515428	−0.257714	−	0.966221i	$-0.582969\pi$
$313$	−9.11885	−0.515428	−0.257714	+	0.966221i	$0.582969\pi$
$314$	−17.4152	−0.982794
$315$	−0.688933	−0.0388170
$316$	−7.39974	−0.416268
$317$	−6.34888	−0.356588	−0.178294	−	0.983977i	$-0.557058\pi$
$317$	−6.34888	−0.356588	−0.178294	+	0.983977i	$0.557058\pi$
$318$	0.194947	0.0109321
$319$	−15.8883	−0.889572
$320$	2.57083	0.143714
$321$	11.2465	0.627718
$322$	−3.75906	−0.209484
$323$	11.7037	0.651211
$324$	−7.02370	−0.390205
$325$	4.52292	0.250886
$326$	−20.4398	−1.13205
$327$	−5.54220	−0.306484
$328$	11.5818	0.639497
$329$	−2.03017	−0.111927
$330$	−7.37117	−0.405769
$331$	−30.7917	−1.69246	−0.846232	−	0.532814i	$-0.821135\pi$
$331$	−30.7917	−1.69246	−0.846232	+	0.532814i	$0.821135\pi$
$332$	−4.97925	−0.273272
$333$	−3.69640	−0.202561
$334$	−18.8288	−1.03026
$335$	−36.3318	−1.98502
$336$	−0.753798	−0.0411231
$337$	−29.5257	−1.60837	−0.804185	−	0.594380i	$-0.797398\pi$
$337$	−29.5257	−1.60837	−0.804185	+	0.594380i	$0.797398\pi$
$338$	5.09997	0.277402
$339$	18.0302	0.979266
$340$	7.74649	0.420113
$341$	−12.6702	−0.686129
$342$	2.15878	0.116733
$343$	−6.63808	−0.358423
$344$	−6.28175	−0.338689
$345$	−31.3354	−1.68704
$346$	−12.6001	−0.677387
$347$	5.32202	0.285701	0.142850	−	0.989744i	$-0.454373\pi$
$347$	5.32202	0.285701	0.142850	+	0.989744i	$0.454373\pi$
$348$	−13.5441	−0.726042
$349$	7.18624	0.384671	0.192335	−	0.981329i	$-0.438394\pi$
$349$	7.18624	0.384671	0.192335	+	0.981329i	$0.438394\pi$
$350$	−0.775873	−0.0414722
$351$	15.6250	0.834001
$352$	1.83398	0.0977513
$353$	−15.5249	−0.826306	−0.413153	−	0.910662i	$-0.635573\pi$
$353$	−15.5249	−0.826306	−0.413153	+	0.910662i	$0.635573\pi$
$354$	19.9650	1.06113
$355$	32.1437	1.70601
$356$	−7.00892	−0.371472
$357$	−2.27136	−0.120213
$358$	−0.426075	−0.0225187
$359$	30.3154	1.59999	0.799994	−	0.600008i	$-0.204836\pi$
$359$	30.3154	1.59999	0.799994	+	0.600008i	$0.204836\pi$
$360$	1.42886	0.0753076
$361$	−3.91368	−0.205983
$362$	−12.8128	−0.673426
$363$	11.9389	0.626630
$364$	1.35519	0.0710314
$365$	−20.4515	−1.07048
$366$	15.1165	0.790153
$367$	33.6321	1.75558	0.877791	−	0.479044i	$-0.159017\pi$
$367$	33.6321	1.75558	0.877791	+	0.479044i	$0.159017\pi$
$368$	7.79637	0.406414
$369$	6.43713	0.335104
$370$	−17.0976	−0.888864
$371$	0.0601221	0.00312138
$372$	−10.8008	−0.559998
$373$	16.3879	0.848536	0.424268	−	0.905537i	$-0.360531\pi$
$373$	16.3879	0.848536	0.424268	+	0.905537i	$0.360531\pi$
$374$	5.52618	0.285752
$375$	13.6285	0.703771
$376$	4.21061	0.217146
$377$	24.3499	1.25408
$378$	−2.68035	−0.137863
$379$	36.7281	1.88660	0.943298	−	0.331948i	$-0.107706\pi$
$379$	36.7281	1.88660	0.943298	+	0.331948i	$0.107706\pi$
$380$	9.98540	0.512240
$381$	15.2808	0.782862
$382$	19.4401	0.994643
$383$	14.2378	0.727518	0.363759	−	0.931493i	$-0.381493\pi$
$383$	14.2378	0.727518	0.363759	+	0.931493i	$0.381493\pi$
$384$	1.56339	0.0797816
$385$	−2.27329	−0.115857
$386$	20.4603	1.04140
$387$	−3.49138	−0.177477
$388$	3.02340	0.153490
$389$	9.42029	0.477628	0.238814	−	0.971065i	$-0.423241\pi$
$389$	9.42029	0.477628	0.238814	+	0.971065i	$0.423241\pi$
$390$	11.2968	0.572037
$391$	23.4922	1.18805
$392$	6.76753	0.341812
$393$	−1.33943	−0.0675655
$394$	14.8602	0.748646
$395$	−19.0235	−0.957176
$396$	1.01932	0.0512228
$397$	−26.1917	−1.31452	−0.657262	−	0.753662i	$-0.728285\pi$
$397$	−26.1917	−1.31452	−0.657262	+	0.753662i	$0.728285\pi$
$398$	−22.9014	−1.14794
$399$	−2.92784	−0.146575
$400$	1.60918	0.0804589
$401$	30.3863	1.51742	0.758709	−	0.651429i	$-0.225830\pi$
$401$	30.3863	1.51742	0.758709	+	0.651429i	$0.225830\pi$
$402$	−22.0944	−1.10197
$403$	19.4180	0.967277
$404$	4.65196	0.231444
$405$	−18.0567	−0.897247
$406$	−4.17704	−0.207303
$407$	−12.1971	−0.604587
$408$	4.71086	0.233222
$409$	−8.62823	−0.426639	−0.213319	−	0.976983i	$-0.568427\pi$
$409$	−8.62823	−0.426639	−0.213319	+	0.976983i	$0.568427\pi$
$410$	29.7748	1.47047
$411$	−22.1971	−1.09490
$412$	−15.1305	−0.745426
$413$	6.15725	0.302978
$414$	4.33320	0.212965
$415$	−12.8008	−0.628368
$416$	−2.81070	−0.137806
$417$	−22.4150	−1.09767
$418$	7.12337	0.348416
$419$	−28.3564	−1.38530	−0.692651	−	0.721273i	$-0.743557\pi$
$419$	−28.3564	−1.38530	−0.692651	+	0.721273i	$0.743557\pi$
$420$	−1.93789	−0.0945593
$421$	−21.2358	−1.03497	−0.517484	−	0.855693i	$-0.673131\pi$
$421$	−21.2358	−1.03497	−0.517484	+	0.855693i	$0.673131\pi$
$422$	−26.2928	−1.27991
$423$	2.34025	0.113787
$424$	−0.124695	−0.00605571
$425$	4.84882	0.235202
$426$	19.5475	0.947079
$427$	4.66197	0.225608
$428$	−7.19364	−0.347717
$429$	8.05892	0.389088
$430$	−16.1493	−0.778789
$431$	13.0626	0.629205	0.314603	−	0.949224i	$-0.398129\pi$
$431$	13.0626	0.629205	0.314603	+	0.949224i	$0.398129\pi$
$432$	5.55911	0.267463
$433$	27.3065	1.31227	0.656133	−	0.754646i	$-0.272191\pi$
$433$	27.3065	1.31227	0.656133	+	0.754646i	$0.272191\pi$
$434$	−3.33101	−0.159893
$435$	−34.8197	−1.66948
$436$	3.54498	0.169774
$437$	30.2820	1.44858
$438$	−12.4371	−0.594268
$439$	8.17965	0.390393	0.195197	−	0.980764i	$-0.437466\pi$
$439$	8.17965	0.390393	0.195197	+	0.980764i	$0.437466\pi$
$440$	4.71485	0.224772
$441$	3.76137	0.179113
$442$	−8.46927	−0.402842
$443$	16.5928	0.788350	0.394175	−	0.919035i	$-0.371030\pi$
$443$	16.5928	0.788350	0.394175	+	0.919035i	$0.371030\pi$
$444$	−10.3975	−0.493446
$445$	−18.0188	−0.854171
$446$	5.09847	0.241420
$447$	−18.6141	−0.880415
$448$	0.482155	0.0227797
$449$	16.2280	0.765845	0.382922	−	0.923781i	$-0.374918\pi$
$449$	16.2280	0.765845	0.382922	+	0.923781i	$0.374918\pi$
$450$	0.894377	0.0421614
$451$	21.2407	1.00019
$452$	−11.5327	−0.542454
$453$	12.5050	0.587538
$454$	9.23300	0.433326
$455$	3.48397	0.163331
$456$	6.07240	0.284366
$457$	−24.3462	−1.13887	−0.569433	−	0.822038i	$-0.692837\pi$
$457$	−24.3462	−1.13887	−0.569433	+	0.822038i	$0.692837\pi$
$458$	24.7043	1.15436
$459$	16.7509	0.781863
$460$	20.0432	0.934517
$461$	−28.8708	−1.34465	−0.672323	−	0.740257i	$-0.734704\pi$
$461$	−28.8708	−1.34465	−0.672323	+	0.740257i	$0.734704\pi$
$462$	−1.38245	−0.0643173
$463$	24.1740	1.12346	0.561730	−	0.827320i	$-0.310136\pi$
$463$	24.1740	1.12346	0.561730	+	0.827320i	$0.310136\pi$
$464$	8.66328	0.402183
$465$	−27.7672	−1.28767
$466$	6.50005	0.301109
$467$	13.8050	0.638818	0.319409	−	0.947617i	$-0.396516\pi$
$467$	13.8050	0.638818	0.319409	+	0.947617i	$0.396516\pi$
$468$	−1.56218	−0.0722118
$469$	−6.81396	−0.314640
$470$	10.8248	0.499309
$471$	−27.2268	−1.25454
$472$	−12.7703	−0.587800
$473$	−11.5206	−0.529717
$474$	−11.5687	−0.531369
$475$	6.25023	0.286780
$476$	1.45284	0.0665908
$477$	−0.0693050	−0.00317326
$478$	5.18780	0.237285
$479$	24.0629	1.09946	0.549732	−	0.835341i	$-0.314730\pi$
$479$	24.0629	1.09946	0.549732	+	0.835341i	$0.314730\pi$
$480$	4.01923	0.183452
$481$	18.6929	0.852323
$482$	19.9305	0.907809
$483$	−5.87689	−0.267408
$484$	−7.63653	−0.347115
$485$	7.77265	0.352938
$486$	5.69653	0.258400
$487$	12.7521	0.577855	0.288927	−	0.957351i	$-0.406701\pi$
$487$	12.7521	0.577855	0.288927	+	0.957351i	$0.406701\pi$
$488$	−9.66903	−0.437696
$489$	−31.9554	−1.44508
$490$	17.3982	0.785969
$491$	−24.9257	−1.12488	−0.562441	−	0.826837i	$-0.690138\pi$
$491$	−24.9257	−1.12488	−0.562441	+	0.826837i	$0.690138\pi$
$492$	18.1069	0.816322
$493$	26.1044	1.17568
$494$	−10.9171	−0.491182
$495$	2.62050	0.117783
$496$	6.90859	0.310205
$497$	6.02850	0.270415
$498$	−7.78454	−0.348833
$499$	6.55961	0.293649	0.146824	−	0.989163i	$-0.453095\pi$
$499$	6.55961	0.293649	0.146824	+	0.989163i	$0.453095\pi$
$500$	−8.71723	−0.389847
$501$	−29.4368	−1.31514
$502$	−8.86713	−0.395760
$503$	3.35422	0.149557	0.0747787	−	0.997200i	$-0.476175\pi$
$503$	3.35422	0.149557	0.0747787	+	0.997200i	$0.476175\pi$
$504$	0.267980	0.0119368
$505$	11.9594	0.532187
$506$	14.2984	0.635640
$507$	7.97327	0.354105
$508$	−9.77415	−0.433658
$509$	−9.08249	−0.402574	−0.201287	−	0.979532i	$-0.564512\pi$
$509$	−9.08249	−0.402574	−0.201287	+	0.979532i	$0.564512\pi$
$510$	12.1108	0.536276
$511$	−3.83563	−0.169678
$512$	−1.00000	−0.0441942
$513$	21.5922	0.953320
$514$	−20.8020	−0.917536
$515$	−38.8980	−1.71405
$516$	−9.82085	−0.432339
$517$	7.72216	0.339620
$518$	−3.20663	−0.140891
$519$	−19.6990	−0.864689
$520$	−7.22584	−0.316874
$521$	−14.7494	−0.646184	−0.323092	−	0.946368i	$-0.604722\pi$
$521$	−14.7494	−0.646184	−0.323092	+	0.946368i	$0.604722\pi$
$522$	4.81503	0.210748
$523$	13.3464	0.583598	0.291799	−	0.956480i	$-0.405746\pi$
$523$	13.3464	0.583598	0.291799	+	0.956480i	$0.405746\pi$
$524$	0.856748	0.0374272
$525$	−1.21300	−0.0529395
$526$	−17.1358	−0.747156
$527$	20.8171	0.906808
$528$	2.86723	0.124780
$529$	37.7834	1.64275
$530$	−0.320569	−0.0139246
$531$	−7.09769	−0.308013
$532$	1.87274	0.0811937
$533$	−32.5529	−1.41002
$534$	−10.9577	−0.474186
$535$	−18.4936	−0.799549
$536$	14.1323	0.610423
$537$	−0.666123	−0.0287453
$538$	26.7135	1.15170
$539$	12.4115	0.534601
$540$	14.2916	0.615010
$541$	−22.2126	−0.954995	−0.477498	−	0.878633i	$-0.658456\pi$
$541$	−22.2126	−0.954995	−0.477498	+	0.878633i	$0.658456\pi$
$542$	14.3826	0.617784
$543$	−20.0315	−0.859632
$544$	−3.01322	−0.129191
$545$	9.11354	0.390381
$546$	2.11870	0.0906720
$547$	3.51595	0.150331	0.0751655	−	0.997171i	$-0.476051\pi$
$547$	3.51595	0.150331	0.0751655	+	0.997171i	$0.476051\pi$
$548$	14.1980	0.606510
$549$	−5.37402	−0.229358
$550$	2.95120	0.125839
$551$	33.6492	1.43350
$552$	12.1888	0.518790
$553$	−3.56782	−0.151719
$554$	−3.28907	−0.139739
$555$	−26.7303	−1.13464
$556$	14.3374	0.608040
$557$	−3.35666	−0.142226	−0.0711132	−	0.997468i	$-0.522655\pi$
$557$	−3.35666	−0.142226	−0.0711132	+	0.997468i	$0.522655\pi$
$558$	3.83977	0.162551
$559$	17.6561	0.746773
$560$	1.23954	0.0523801
$561$	8.63961	0.364764
$562$	−19.5666	−0.825366
$563$	7.54703	0.318070	0.159035	−	0.987273i	$-0.449162\pi$
$563$	7.54703	0.318070	0.159035	+	0.987273i	$0.449162\pi$
$564$	6.58284	0.277188
$565$	−29.6487	−1.24733
$566$	7.65265	0.321665
$567$	−3.38651	−0.142220
$568$	−12.5032	−0.524624
$569$	−34.7307	−1.45599	−0.727993	−	0.685585i	$-0.759546\pi$
$569$	−34.7307	−1.45599	−0.727993	+	0.685585i	$0.759546\pi$
$570$	15.6111	0.653878
$571$	13.5722	0.567977	0.283988	−	0.958828i	$-0.408342\pi$
$571$	13.5722	0.567977	0.283988	+	0.958828i	$0.408342\pi$
$572$	−5.15476	−0.215531
$573$	30.3926	1.26967
$574$	5.58422	0.233081
$575$	12.5458	0.523194
$576$	−0.555797	−0.0231582
$577$	22.8088	0.949543	0.474771	−	0.880109i	$-0.342531\pi$
$577$	22.8088	0.949543	0.474771	+	0.880109i	$0.342531\pi$
$578$	7.92048	0.329449
$579$	31.9875	1.32936
$580$	22.2718	0.924788
$581$	−2.40077	−0.0996007
$582$	4.72677	0.195931
$583$	−0.228687	−0.00947125
$584$	7.95519	0.329188
$585$	−4.01610	−0.166045
$586$	−4.82001	−0.199113
$587$	43.6212	1.80044	0.900221	−	0.435434i	$-0.143405\pi$
$587$	43.6212	1.80044	0.900221	+	0.435434i	$0.143405\pi$
$588$	10.5803	0.436325
$589$	26.8337	1.10566
$590$	−32.8302	−1.35160
$591$	23.2324	0.955652
$592$	6.65062	0.273339
$593$	−1.74748	−0.0717605	−0.0358802	−	0.999356i	$-0.511423\pi$
$593$	−1.74748	−0.0717605	−0.0358802	+	0.999356i	$0.511423\pi$
$594$	10.1953	0.418318
$595$	3.73501	0.153120
$596$	11.9062	0.487696
$597$	−35.8038	−1.46535
$598$	−21.9132	−0.896099
$599$	29.0213	1.18578	0.592889	−	0.805284i	$-0.297987\pi$
$599$	29.0213	1.18578	0.592889	+	0.805284i	$0.297987\pi$
$600$	2.51578	0.102706
$601$	26.1920	1.06839	0.534197	−	0.845360i	$-0.320614\pi$
$601$	26.1920	1.06839	0.534197	+	0.845360i	$0.320614\pi$
$602$	−3.02877	−0.123444
$603$	7.85471	0.319868
$604$	−7.99865	−0.325460
$605$	−19.6322	−0.798164
$606$	7.27286	0.295440
$607$	−28.5212	−1.15764	−0.578820	−	0.815456i	$-0.696486\pi$
$607$	−28.5212	−1.15764	−0.578820	+	0.815456i	$0.696486\pi$
$608$	−3.88411	−0.157521
$609$	−6.53037	−0.264624
$610$	−24.8575	−1.00645
$611$	−11.8348	−0.478783
$612$	−1.67474	−0.0676974
$613$	10.4111	0.420501	0.210250	−	0.977648i	$-0.432572\pi$
$613$	10.4111	0.420501	0.210250	+	0.977648i	$0.432572\pi$
$614$	−18.4937	−0.746344
$615$	46.5498	1.87707
$616$	0.884261	0.0356279
$617$	37.7077	1.51806	0.759028	−	0.651058i	$-0.225675\pi$
$617$	37.7077	1.51806	0.759028	+	0.651058i	$0.225675\pi$
$618$	−23.6549	−0.951541
$619$	−27.9668	−1.12408	−0.562041	−	0.827110i	$-0.689984\pi$
$619$	−27.9668	−1.12408	−0.562041	+	0.827110i	$0.689984\pi$
$620$	17.7608	0.713292
$621$	43.3409	1.73921
$622$	28.8408	1.15641
$623$	−3.37938	−0.135392
$624$	−4.39423	−0.175910
$625$	−30.4564	−1.21826
$626$	9.11885	0.364462
$627$	11.1366	0.444755
$628$	17.4152	0.694940
$629$	20.0398	0.799040
$630$	0.688933	0.0274477
$631$	0.359647	0.0143173	0.00715866	−	0.999974i	$-0.497721\pi$
$631$	0.359647	0.0143173	0.00715866	+	0.999974i	$0.497721\pi$
$632$	7.39974	0.294346
$633$	−41.1060	−1.63382
$634$	6.34888	0.252146
$635$	−25.1277	−0.997162
$636$	−0.194947	−0.00773015
$637$	−19.0215	−0.753659
$638$	15.8883	0.629022
$639$	−6.94927	−0.274909
$640$	−2.57083	−0.101621
$641$	−0.603991	−0.0238562	−0.0119281	−	0.999929i	$-0.503797\pi$
$641$	−0.603991	−0.0238562	−0.0119281	+	0.999929i	$0.503797\pi$
$642$	−11.2465	−0.443864
$643$	23.6777	0.933755	0.466878	−	0.884322i	$-0.345379\pi$
$643$	23.6777	0.933755	0.466878	+	0.884322i	$0.345379\pi$
$644$	3.75906	0.148128
$645$	−25.2478	−0.994129
$646$	−11.7037	−0.460476
$647$	−20.2367	−0.795587	−0.397793	−	0.917475i	$-0.630224\pi$
$647$	−20.2367	−0.795587	−0.397793	+	0.917475i	$0.630224\pi$
$648$	7.02370	0.275917
$649$	−23.4204	−0.919331
$650$	−4.52292	−0.177403
$651$	−5.20768	−0.204105
$652$	20.4398	0.800484
$653$	−40.2859	−1.57651	−0.788254	−	0.615350i	$-0.789015\pi$
$653$	−40.2859	−1.57651	−0.788254	+	0.615350i	$0.789015\pi$
$654$	5.54220	0.216717
$655$	2.20255	0.0860609
$656$	−11.5818	−0.452193
$657$	4.42148	0.172498
$658$	2.03017	0.0791441
$659$	46.7650	1.82170	0.910852	−	0.412733i	$-0.135426\pi$
$659$	46.7650	1.82170	0.910852	+	0.412733i	$0.135426\pi$
$660$	7.37117	0.286922
$661$	3.72126	0.144740	0.0723700	−	0.997378i	$-0.476944\pi$
$661$	3.72126	0.144740	0.0723700	+	0.997378i	$0.476944\pi$
$662$	30.7917	1.19675
$663$	−13.2408	−0.514230
$664$	4.97925	0.193232
$665$	4.81451	0.186699
$666$	3.69640	0.143233
$667$	67.5421	2.61524
$668$	18.8288	0.728506
$669$	7.97092	0.308174
$670$	36.3318	1.40362
$671$	−17.7328	−0.684566
$672$	0.753798	0.0290784
$673$	−48.9769	−1.88792	−0.943959	−	0.330061i	$-0.892931\pi$
$673$	−48.9769	−1.88792	−0.943959	+	0.330061i	$0.892931\pi$
$674$	29.5257	1.13729
$675$	8.94561	0.344317
$676$	−5.09997	−0.196153
$677$	39.5601	1.52042	0.760209	−	0.649679i	$-0.225097\pi$
$677$	39.5601	1.52042	0.760209	+	0.649679i	$0.225097\pi$
$678$	−18.0302	−0.692446
$679$	1.45775	0.0559432
$680$	−7.74649	−0.297064
$681$	14.4348	0.553144
$682$	12.6702	0.485167
$683$	−20.1698	−0.771774	−0.385887	−	0.922546i	$-0.626105\pi$
$683$	−20.1698	−0.771774	−0.385887	+	0.922546i	$0.626105\pi$
$684$	−2.15878	−0.0825430
$685$	36.5008	1.39462
$686$	6.63808	0.253443
$687$	38.6225	1.47354
$688$	6.28175	0.239489
$689$	0.350479	0.0133522
$690$	31.3354	1.19292
$691$	−9.63277	−0.366448	−0.183224	−	0.983071i	$-0.558653\pi$
$691$	−9.63277	−0.366448	−0.183224	+	0.983071i	$0.558653\pi$
$692$	12.6001	0.478985
$693$	0.491470	0.0186694
$694$	−5.32202	−0.202021
$695$	36.8590	1.39814
$696$	13.5441	0.513389
$697$	−34.8985	−1.32188
$698$	−7.18624	−0.272003
$699$	10.1621	0.384368
$700$	0.775873	0.0293252
$701$	21.4472	0.810050	0.405025	−	0.914306i	$-0.367263\pi$
$701$	21.4472	0.810050	0.405025	+	0.914306i	$0.367263\pi$
$702$	−15.6250	−0.589728
$703$	25.8317	0.974263
$704$	−1.83398	−0.0691206
$705$	16.9234	0.637372
$706$	15.5249	0.584287
$707$	2.24297	0.0843554
$708$	−19.9650	−0.750330
$709$	0.938031	0.0352285	0.0176143	−	0.999845i	$-0.494393\pi$
$709$	0.938031	0.0352285	0.0176143	+	0.999845i	$0.494393\pi$
$710$	−32.1437	−1.20633
$711$	4.11276	0.154240
$712$	7.00892	0.262670
$713$	53.8619	2.01714
$714$	2.27136	0.0850036
$715$	−13.2520	−0.495597
$716$	0.426075	0.0159232
$717$	8.11059	0.302895
$718$	−30.3154	−1.13136
$719$	−20.2287	−0.754404	−0.377202	−	0.926131i	$-0.623114\pi$
$719$	−20.2287	−0.754404	−0.377202	+	0.926131i	$0.623114\pi$
$720$	−1.42886	−0.0532505
$721$	−7.29524	−0.271689
$722$	3.91368	0.145652
$723$	31.1592	1.15882
$724$	12.8128	0.476184
$725$	13.9408	0.517747
$726$	−11.9389	−0.443094
$727$	−15.4574	−0.573282	−0.286641	−	0.958038i	$-0.592539\pi$
$727$	−15.4574	−0.573282	−0.286641	+	0.958038i	$0.592539\pi$
$728$	−1.35519	−0.0502268
$729$	29.9770	1.11026
$730$	20.4515	0.756943
$731$	18.9283	0.700089
$732$	−15.1165	−0.558722
$733$	22.3192	0.824377	0.412189	−	0.911098i	$-0.364764\pi$
$733$	22.3192	0.824377	0.412189	+	0.911098i	$0.364764\pi$
$734$	−33.6321	−1.24138
$735$	27.2002	1.00330
$736$	−7.79637	−0.287378
$737$	25.9184	0.954715
$738$	−6.43713	−0.236954
$739$	−7.09386	−0.260952	−0.130476	−	0.991451i	$-0.541651\pi$
$739$	−7.09386	−0.260952	−0.130476	+	0.991451i	$0.541651\pi$
$740$	17.0976	0.628521
$741$	−17.0677	−0.626997
$742$	−0.0601221	−0.00220715
$743$	8.31167	0.304926	0.152463	−	0.988309i	$-0.451280\pi$
$743$	8.31167	0.304926	0.152463	+	0.988309i	$0.451280\pi$
$744$	10.8008	0.395978
$745$	30.6088	1.12142
$746$	−16.3879	−0.600005
$747$	2.76746	0.101256
$748$	−5.52618	−0.202057
$749$	−3.46845	−0.126734
$750$	−13.6285	−0.497642
$751$	5.84066	0.213129	0.106564	−	0.994306i	$-0.466015\pi$
$751$	5.84066	0.213129	0.106564	+	0.994306i	$0.466015\pi$
$752$	−4.21061	−0.153545
$753$	−13.8628	−0.505190
$754$	−24.3499	−0.886770
$755$	−20.5632	−0.748371
$756$	2.68035	0.0974835
$757$	24.6551	0.896105	0.448053	−	0.894007i	$-0.352118\pi$
$757$	24.6551	0.896105	0.448053	+	0.894007i	$0.352118\pi$
$758$	−36.7281	−1.33402
$759$	22.3540	0.811398
$760$	−9.98540	−0.362208
$761$	−21.2573	−0.770577	−0.385288	−	0.922796i	$-0.625898\pi$
$761$	−21.2573	−0.770577	−0.385288	+	0.922796i	$0.625898\pi$
$762$	−15.2808	−0.553567
$763$	1.70923	0.0618782
$764$	−19.4401	−0.703319
$765$	−4.30548	−0.155665
$766$	−14.2378	−0.514433
$767$	35.8934	1.29604
$768$	−1.56339	−0.0564141
$769$	24.6780	0.889910	0.444955	−	0.895553i	$-0.353220\pi$
$769$	24.6780	0.889910	0.444955	+	0.895553i	$0.353220\pi$
$770$	2.27329	0.0819236
$771$	−32.5217	−1.17124
$772$	−20.4603	−0.736383
$773$	24.2113	0.870819	0.435410	−	0.900232i	$-0.356604\pi$
$773$	24.2113	0.870819	0.435410	+	0.900232i	$0.356604\pi$
$774$	3.49138	0.125495
$775$	11.1172	0.399340
$776$	−3.02340	−0.108534
$777$	−5.01323	−0.179848
$778$	−9.42029	−0.337734
$779$	−44.9850	−1.61175
$780$	−11.2968	−0.404492
$781$	−22.9307	−0.820523
$782$	−23.4922	−0.840079
$783$	48.1602	1.72110
$784$	−6.76753	−0.241697
$785$	44.7714	1.59796
$786$	1.33943	0.0477760
$787$	−6.80747	−0.242660	−0.121330	−	0.992612i	$-0.538716\pi$
$787$	−6.80747	−0.242660	−0.121330	+	0.992612i	$0.538716\pi$
$788$	−14.8602	−0.529373
$789$	−26.7900	−0.953749
$790$	19.0235	0.676826
$791$	−5.56056	−0.197711
$792$	−1.01932	−0.0362200
$793$	27.1767	0.965074
$794$	26.1917	0.929509
$795$	−0.501176	−0.0177749
$796$	22.9014	0.811717
$797$	−19.4447	−0.688765	−0.344383	−	0.938829i	$-0.611912\pi$
$797$	−19.4447	−0.688765	−0.344383	+	0.938829i	$0.611912\pi$
$798$	2.92784	0.103644
$799$	−12.6875	−0.448852
$800$	−1.60918	−0.0568931
$801$	3.89554	0.137642
$802$	−30.3863	−1.07298
$803$	14.5896	0.514857
$804$	22.0944	0.779209
$805$	9.66390	0.340608
$806$	−19.4180	−0.683968
$807$	41.7638	1.47016
$808$	−4.65196	−0.163656
$809$	−15.0885	−0.530483	−0.265241	−	0.964182i	$-0.585452\pi$
$809$	−15.0885	−0.530483	−0.265241	+	0.964182i	$0.585452\pi$
$810$	18.0567	0.634449
$811$	−34.9365	−1.22679	−0.613393	−	0.789778i	$-0.710196\pi$
$811$	−34.9365	−1.22679	−0.613393	+	0.789778i	$0.710196\pi$
$812$	4.17704	0.146585
$813$	22.4856	0.788606
$814$	12.1971	0.427508
$815$	52.5473	1.84065
$816$	−4.71086	−0.164913
$817$	24.3990	0.853613
$818$	8.62823	0.301679
$819$	−0.753212	−0.0263194
$820$	−29.7748	−1.03978
$821$	48.8191	1.70380	0.851900	−	0.523705i	$-0.175450\pi$
$821$	48.8191	1.70380	0.851900	+	0.523705i	$0.175450\pi$
$822$	22.1971	0.774214
$823$	4.42711	0.154319	0.0771597	−	0.997019i	$-0.475415\pi$
$823$	4.42711	0.154319	0.0771597	+	0.997019i	$0.475415\pi$
$824$	15.1305	0.527096
$825$	4.61389	0.160635
$826$	−6.15725	−0.214238
$827$	26.3041	0.914682	0.457341	−	0.889292i	$-0.348802\pi$
$827$	26.3041	0.914682	0.457341	+	0.889292i	$0.348802\pi$
$828$	−4.33320	−0.150589
$829$	−9.93193	−0.344950	−0.172475	−	0.985014i	$-0.555176\pi$
$829$	−9.93193	−0.344950	−0.172475	+	0.985014i	$0.555176\pi$
$830$	12.8008	0.444323
$831$	−5.14212	−0.178378
$832$	2.81070	0.0974435
$833$	−20.3921	−0.706543
$834$	22.4150	0.776166
$835$	48.4056	1.67514
$836$	−7.12337	−0.246367
$837$	38.4056	1.32749
$838$	28.3564	0.979557
$839$	−49.8945	−1.72255	−0.861275	−	0.508139i	$-0.830334\pi$
$839$	−49.8945	−1.72255	−0.861275	+	0.508139i	$0.830334\pi$
$840$	1.93789	0.0668635
$841$	46.0525	1.58802
$842$	21.2358	0.731833
$843$	−30.5903	−1.05358
$844$	26.2928	0.905034
$845$	−13.1112	−0.451038
$846$	−2.34025	−0.0804593
$847$	−3.68199	−0.126515
$848$	0.124695	0.00428203
$849$	11.9641	0.410607
$850$	−4.84882	−0.166313
$851$	51.8507	1.77742
$852$	−19.5475	−0.669686
$853$	27.5719	0.944042	0.472021	−	0.881587i	$-0.343525\pi$
$853$	27.5719	0.944042	0.472021	+	0.881587i	$0.343525\pi$
$854$	−4.66197	−0.159529
$855$	−5.54986	−0.189801
$856$	7.19364	0.245873
$857$	−5.53048	−0.188918	−0.0944588	−	0.995529i	$-0.530112\pi$
$857$	−5.53048	−0.188918	−0.0944588	+	0.995529i	$0.530112\pi$
$858$	−8.05892	−0.275127
$859$	−20.1704	−0.688204	−0.344102	−	0.938932i	$-0.611817\pi$
$859$	−20.1704	−0.688204	−0.344102	+	0.938932i	$0.611817\pi$
$860$	16.1493	0.550687
$861$	8.73033	0.297529
$862$	−13.0626	−0.444915
$863$	−48.5919	−1.65409	−0.827044	−	0.562138i	$-0.809979\pi$
$863$	−48.5919	−1.65409	−0.827044	+	0.562138i	$0.809979\pi$
$864$	−5.55911	−0.189125
$865$	32.3928	1.10139
$866$	−27.3065	−0.927912
$867$	12.3828	0.420543
$868$	3.33101	0.113062
$869$	13.5710	0.460363
$870$	34.8197	1.18050
$871$	−39.7217	−1.34592
$872$	−3.54498	−0.120048
$873$	−1.68040	−0.0568728
$874$	−30.2820	−1.02430
$875$	−4.20306	−0.142089
$876$	12.4371	0.420211
$877$	−14.6068	−0.493235	−0.246618	−	0.969113i	$-0.579319\pi$
$877$	−14.6068	−0.493235	−0.246618	+	0.969113i	$0.579319\pi$
$878$	−8.17965	−0.276050
$879$	−7.53557	−0.254169
$880$	−4.71485	−0.158938
$881$	−7.76894	−0.261742	−0.130871	−	0.991399i	$-0.541777\pi$
$881$	−7.76894	−0.261742	−0.130871	+	0.991399i	$0.541777\pi$
$882$	−3.76137	−0.126652
$883$	38.1786	1.28481	0.642405	−	0.766365i	$-0.277937\pi$
$883$	38.1786	1.28481	0.642405	+	0.766365i	$0.277937\pi$
$884$	8.46927	0.284852
$885$	−51.3266	−1.72533
$886$	−16.5928	−0.557447
$887$	−1.40045	−0.0470225	−0.0235112	−	0.999724i	$-0.507485\pi$
$887$	−1.40045	−0.0470225	−0.0235112	+	0.999724i	$0.507485\pi$
$888$	10.3975	0.348919
$889$	−4.71265	−0.158057
$890$	18.0188	0.603990
$891$	12.8813	0.431540
$892$	−5.09847	−0.170709
$893$	−16.3545	−0.547282
$894$	18.6141	0.622548
$895$	1.09537	0.0366141
$896$	−0.482155	−0.0161077
$897$	−34.2590	−1.14388
$898$	−16.2280	−0.541534
$899$	59.8510	1.99614
$900$	−0.894377	−0.0298126
$901$	0.375733	0.0125175
$902$	−21.2407	−0.707239
$903$	−4.73517	−0.157577
$904$	11.5327	0.383573
$905$	32.9396	1.09495
$906$	−12.5050	−0.415452
$907$	−29.5541	−0.981329	−0.490665	−	0.871349i	$-0.663246\pi$
$907$	−29.5541	−0.981329	−0.490665	+	0.871349i	$0.663246\pi$
$908$	−9.23300	−0.306408
$909$	−2.58555	−0.0857573
$910$	−3.48397	−0.115493
$911$	7.85558	0.260267	0.130134	−	0.991496i	$-0.458459\pi$
$911$	7.85558	0.260267	0.130134	+	0.991496i	$0.458459\pi$
$912$	−6.07240	−0.201077
$913$	9.13184	0.302220
$914$	24.3462	0.805300
$915$	−38.8620	−1.28474
$916$	−24.7043	−0.816253
$917$	0.413085	0.0136413
$918$	−16.7509	−0.552861
$919$	1.95358	0.0644428	0.0322214	−	0.999481i	$-0.489742\pi$
$919$	1.95358	0.0644428	0.0322214	+	0.999481i	$0.489742\pi$
$920$	−20.0432	−0.660803
$921$	−28.9129	−0.952713
$922$	28.8708	0.950809
$923$	35.1428	1.15674
$924$	1.38245	0.0454792
$925$	10.7020	0.351881
$926$	−24.1740	−0.794407
$927$	8.40949	0.276204
$928$	−8.66328	−0.284386
$929$	−51.8454	−1.70099	−0.850497	−	0.525980i	$-0.823699\pi$
$929$	−51.8454	−1.70099	−0.850497	+	0.525980i	$0.823699\pi$
$930$	27.7672	0.910521
$931$	−26.2858	−0.861483
$932$	−6.50005	−0.212916
$933$	45.0895	1.47616
$934$	−13.8050	−0.451712
$935$	−14.2069	−0.464615
$936$	1.56218	0.0510614
$937$	−28.4593	−0.929724	−0.464862	−	0.885383i	$-0.653896\pi$
$937$	−28.4593	−0.929724	−0.464862	+	0.885383i	$0.653896\pi$
$938$	6.81396	0.222484
$939$	14.2564	0.465239
$940$	−10.8248	−0.353065
$941$	13.1630	0.429101	0.214550	−	0.976713i	$-0.431171\pi$
$941$	13.1630	0.429101	0.214550	+	0.976713i	$0.431171\pi$
$942$	27.2268	0.887096
$943$	−90.2959	−2.94044
$944$	12.7703	0.415637
$945$	6.89074	0.224156
$946$	11.5206	0.374566
$947$	42.5173	1.38163	0.690814	−	0.723033i	$-0.257252\pi$
$947$	42.5173	1.38163	0.690814	+	0.723033i	$0.257252\pi$
$948$	11.5687	0.375734
$949$	−22.3597	−0.725825
$950$	−6.25023	−0.202784
$951$	9.92580	0.321866
$952$	−1.45284	−0.0470868
$953$	−6.78189	−0.219687	−0.109844	−	0.993949i	$-0.535035\pi$
$953$	−6.78189	−0.219687	−0.109844	+	0.993949i	$0.535035\pi$
$954$	0.0693050	0.00224383
$955$	−49.9773	−1.61723
$956$	−5.18780	−0.167786
$957$	24.8396	0.802951
$958$	−24.0629	−0.777438
$959$	6.84565	0.221058
$960$	−4.01923	−0.129720
$961$	16.7286	0.539631
$962$	−18.6929	−0.602683
$963$	3.99820	0.128840
$964$	−19.9305	−0.641918
$965$	−52.6000	−1.69325
$966$	5.87689	0.189086
$967$	5.89217	0.189479	0.0947397	−	0.995502i	$-0.469798\pi$
$967$	5.89217	0.189479	0.0947397	+	0.995502i	$0.469798\pi$
$968$	7.63653	0.245447
$969$	−18.2975	−0.587800
$970$	−7.77265	−0.249565
$971$	−39.9452	−1.28190	−0.640952	−	0.767581i	$-0.721460\pi$
$971$	−39.9452	−1.28190	−0.640952	+	0.767581i	$0.721460\pi$
$972$	−5.69653	−0.182716
$973$	6.91283	0.221615
$974$	−12.7521	−0.408605
$975$	−7.07110	−0.226457
$976$	9.66903	0.309498
$977$	35.3159	1.12986	0.564929	−	0.825140i	$-0.308904\pi$
$977$	35.3159	1.12986	0.564929	+	0.825140i	$0.308904\pi$
$978$	31.9554	1.02182
$979$	12.8542	0.410822
$980$	−17.3982	−0.555764
$981$	−1.97029	−0.0629065
$982$	24.9257	0.795412
$983$	30.8331	0.983422	0.491711	−	0.870758i	$-0.336372\pi$
$983$	30.8331	0.983422	0.491711	+	0.870758i	$0.336372\pi$
$984$	−18.1069	−0.577227
$985$	−38.2031	−1.21725
$986$	−26.1044	−0.831334
$987$	3.17395	0.101028
$988$	10.9171	0.347318
$989$	48.9748	1.55731
$990$	−2.62050	−0.0832850
$991$	−50.5702	−1.60642	−0.803208	−	0.595699i	$-0.796875\pi$
$991$	−50.5702	−1.60642	−0.803208	+	0.595699i	$0.796875\pi$
$992$	−6.90859	−0.219348
$993$	48.1396	1.52766
$994$	−6.02850	−0.191212
$995$	58.8755	1.86648
$996$	7.78454	0.246662
$997$	−55.4530	−1.75622	−0.878108	−	0.478463i	$-0.841194\pi$
$997$	−55.4530	−1.75622	−0.878108	+	0.478463i	$0.841194\pi$
$998$	−6.55961	−0.207641
$999$	36.9716	1.16973

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.e.1.21	✓	77

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.e.1.21	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.56339 q^{3} +1.00000 q^{4} +2.57083 q^{5} +1.56339 q^{6} +0.482155 q^{7} -1.00000 q^{8} -0.555797 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.56339 q^{3} +1.00000 q^{4} +2.57083 q^{5} +1.56339 q^{6} +0.482155 q^{7} -1.00000 q^{8} -0.555797 q^{9} -2.57083 q^{10} -1.83398 q^{11} -1.56339 q^{12} +2.81070 q^{13} -0.482155 q^{14} -4.01923 q^{15} +1.00000 q^{16} +3.01322 q^{17} +0.555797 q^{18} +3.88411 q^{19} +2.57083 q^{20} -0.753798 q^{21} +1.83398 q^{22} +7.79637 q^{23} +1.56339 q^{24} +1.60918 q^{25} -2.81070 q^{26} +5.55911 q^{27} +0.482155 q^{28} +8.66328 q^{29} +4.01923 q^{30} +6.90859 q^{31} -1.00000 q^{32} +2.86723 q^{33} -3.01322 q^{34} +1.23954 q^{35} -0.555797 q^{36} +6.65062 q^{37} -3.88411 q^{38} -4.39423 q^{39} -2.57083 q^{40} -11.5818 q^{41} +0.753798 q^{42} +6.28175 q^{43} -1.83398 q^{44} -1.42886 q^{45} -7.79637 q^{46} -4.21061 q^{47} -1.56339 q^{48} -6.76753 q^{49} -1.60918 q^{50} -4.71086 q^{51} +2.81070 q^{52} +0.124695 q^{53} -5.55911 q^{54} -4.71485 q^{55} -0.482155 q^{56} -6.07240 q^{57} -8.66328 q^{58} +12.7703 q^{59} -4.01923 q^{60} +9.66903 q^{61} -6.90859 q^{62} -0.267980 q^{63} +1.00000 q^{64} +7.22584 q^{65} -2.86723 q^{66} -14.1323 q^{67} +3.01322 q^{68} -12.1888 q^{69} -1.23954 q^{70} +12.5032 q^{71} +0.555797 q^{72} -7.95519 q^{73} -6.65062 q^{74} -2.51578 q^{75} +3.88411 q^{76} -0.884261 q^{77} +4.39423 q^{78} -7.39974 q^{79} +2.57083 q^{80} -7.02370 q^{81} +11.5818 q^{82} -4.97925 q^{83} -0.753798 q^{84} +7.74649 q^{85} -6.28175 q^{86} -13.5441 q^{87} +1.83398 q^{88} -7.00892 q^{89} +1.42886 q^{90} +1.35519 q^{91} +7.79637 q^{92} -10.8008 q^{93} +4.21061 q^{94} +9.98540 q^{95} +1.56339 q^{96} +3.02340 q^{97} +6.76753 q^{98} +1.01932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) 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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