LMFDB - Embedded newform 8002.2.a.f.1.4

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:	$1$
Dimension:	$89$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-1.00000 q^{2} -3.26711 q^{3} +1.00000 q^{4} +0.417272 q^{5} +3.26711 q^{6} -4.66212 q^{7} -1.00000 q^{8} +7.67400 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.26711 q^{3} +1.00000 q^{4} +0.417272 q^{5} +3.26711 q^{6} -4.66212 q^{7} -1.00000 q^{8} +7.67400 q^{9} -0.417272 q^{10} +1.62366 q^{11} -3.26711 q^{12} -4.36101 q^{13} +4.66212 q^{14} -1.36327 q^{15} +1.00000 q^{16} +6.32838 q^{17} -7.67400 q^{18} +5.28458 q^{19} +0.417272 q^{20} +15.2317 q^{21} -1.62366 q^{22} -5.24895 q^{23} +3.26711 q^{24} -4.82588 q^{25} +4.36101 q^{26} -15.2705 q^{27} -4.66212 q^{28} -0.958616 q^{29} +1.36327 q^{30} -2.01654 q^{31} -1.00000 q^{32} -5.30467 q^{33} -6.32838 q^{34} -1.94537 q^{35} +7.67400 q^{36} -2.00717 q^{37} -5.28458 q^{38} +14.2479 q^{39} -0.417272 q^{40} +2.19927 q^{41} -15.2317 q^{42} -1.36796 q^{43} +1.62366 q^{44} +3.20215 q^{45} +5.24895 q^{46} -4.48960 q^{47} -3.26711 q^{48} +14.7354 q^{49} +4.82588 q^{50} -20.6755 q^{51} -4.36101 q^{52} +1.46632 q^{53} +15.2705 q^{54} +0.677507 q^{55} +4.66212 q^{56} -17.2653 q^{57} +0.958616 q^{58} +0.871016 q^{59} -1.36327 q^{60} -1.02402 q^{61} +2.01654 q^{62} -35.7771 q^{63} +1.00000 q^{64} -1.81973 q^{65} +5.30467 q^{66} -11.0374 q^{67} +6.32838 q^{68} +17.1489 q^{69} +1.94537 q^{70} +4.38146 q^{71} -7.67400 q^{72} +2.87621 q^{73} +2.00717 q^{74} +15.7667 q^{75} +5.28458 q^{76} -7.56969 q^{77} -14.2479 q^{78} -15.7825 q^{79} +0.417272 q^{80} +26.8683 q^{81} -2.19927 q^{82} +15.7039 q^{83} +15.2317 q^{84} +2.64066 q^{85} +1.36796 q^{86} +3.13190 q^{87} -1.62366 q^{88} +16.0604 q^{89} -3.20215 q^{90} +20.3315 q^{91} -5.24895 q^{92} +6.58825 q^{93} +4.48960 q^{94} +2.20511 q^{95} +3.26711 q^{96} -4.29720 q^{97} -14.7354 q^{98} +12.4600 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) $ 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9} + 18 q^{10} - 26 q^{11} - 12 q^{12} + 2 q^{13} + 27 q^{14} - 21 q^{15} + 89 q^{16} - 60 q^{17} - 95 q^{18} + q^{19} - 18 q^{20} - 6 q^{21} + 26 q^{22} - 45 q^{23} + 12 q^{24} + 107 q^{25} - 2 q^{26} - 45 q^{27} - 27 q^{28} - 18 q^{29} + 21 q^{30} - 38 q^{31} - 89 q^{32} - 29 q^{33} + 60 q^{34} - 47 q^{35} + 95 q^{36} - 15 q^{37} - q^{38} - 38 q^{39} + 18 q^{40} - 50 q^{41} + 6 q^{42} - 15 q^{43} - 26 q^{44} - 35 q^{45} + 45 q^{46} - 121 q^{47} - 12 q^{48} + 132 q^{49} - 107 q^{50} + 6 q^{51} + 2 q^{52} - 46 q^{53} + 45 q^{54} - 37 q^{55} + 27 q^{56} - 42 q^{57} + 18 q^{58} - 34 q^{59} - 21 q^{60} + 41 q^{61} + 38 q^{62} - 131 q^{63} + 89 q^{64} - 57 q^{65} + 29 q^{66} - 11 q^{67} - 60 q^{68} + 15 q^{69} + 47 q^{70} - 66 q^{71} - 95 q^{72} - 47 q^{73} + 15 q^{74} - 46 q^{75} + q^{76} - 106 q^{77} + 38 q^{78} - 51 q^{79} - 18 q^{80} + 113 q^{81} + 50 q^{82} - 141 q^{83} - 6 q^{84} - 7 q^{85} + 15 q^{86} - 110 q^{87} + 26 q^{88} - 30 q^{89} + 35 q^{90} + 37 q^{91} - 45 q^{92} - 44 q^{93} + 121 q^{94} - 98 q^{95} + 12 q^{96} + 3 q^{97} - 132 q^{98} - 71 q^{99}+O(q^{100}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9 + 18 * q^10 - 26 * q^11 - 12 * q^12 + 2 * q^13 + 27 * q^14 - 21 * q^15 + 89 * q^16 - 60 * q^17 - 95 * q^18 + q^19 - 18 * q^20 - 6 * q^21 + 26 * q^22 - 45 * q^23 + 12 * q^24 + 107 * q^25 - 2 * q^26 - 45 * q^27 - 27 * q^28 - 18 * q^29 + 21 * q^30 - 38 * q^31 - 89 * q^32 - 29 * q^33 + 60 * q^34 - 47 * q^35 + 95 * q^36 - 15 * q^37 - q^38 - 38 * q^39 + 18 * q^40 - 50 * q^41 + 6 * q^42 - 15 * q^43 - 26 * q^44 - 35 * q^45 + 45 * q^46 - 121 * q^47 - 12 * q^48 + 132 * q^49 - 107 * q^50 + 6 * q^51 + 2 * q^52 - 46 * q^53 + 45 * q^54 - 37 * q^55 + 27 * q^56 - 42 * q^57 + 18 * q^58 - 34 * q^59 - 21 * q^60 + 41 * q^61 + 38 * q^62 - 131 * q^63 + 89 * q^64 - 57 * q^65 + 29 * q^66 - 11 * q^67 - 60 * q^68 + 15 * q^69 + 47 * q^70 - 66 * q^71 - 95 * q^72 - 47 * q^73 + 15 * q^74 - 46 * q^75 + q^76 - 106 * q^77 + 38 * q^78 - 51 * q^79 - 18 * q^80 + 113 * q^81 + 50 * q^82 - 141 * q^83 - 6 * q^84 - 7 * q^85 + 15 * q^86 - 110 * q^87 + 26 * q^88 - 30 * q^89 + 35 * q^90 + 37 * q^91 - 45 * q^92 - 44 * q^93 + 121 * q^94 - 98 * q^95 + 12 * q^96 + 3 * q^97 - 132 * q^98 - 71 * 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
$3$	−3.26711	−1.88627	−0.943133	−	0.332415i	$-0.892137\pi$
$3$	−3.26711	−1.88627	−0.943133	+	0.332415i	$0.892137\pi$
$4$	1.00000	0.500000
$5$	0.417272	0.186610	0.0933049	−	0.995638i	$-0.470257\pi$
$5$	0.417272	0.186610	0.0933049	+	0.995638i	$0.470257\pi$
$6$	3.26711	1.33379
$7$	−4.66212	−1.76212	−0.881058	−	0.473008i	$-0.843168\pi$
$7$	−4.66212	−1.76212	−0.881058	+	0.473008i	$0.843168\pi$
$8$	−1.00000	−0.353553
$9$	7.67400	2.55800
$10$	−0.417272	−0.131953
$11$	1.62366	0.489551	0.244776	−	0.969580i	$-0.421286\pi$
$11$	1.62366	0.489551	0.244776	+	0.969580i	$0.421286\pi$
$12$	−3.26711	−0.943133
$13$	−4.36101	−1.20953	−0.604763	−	0.796406i	$-0.706732\pi$
$13$	−4.36101	−1.20953	−0.604763	+	0.796406i	$0.706732\pi$
$14$	4.66212	1.24600
$15$	−1.36327	−0.351996
$16$	1.00000	0.250000
$17$	6.32838	1.53486	0.767428	−	0.641135i	$-0.221536\pi$
$17$	6.32838	1.53486	0.767428	+	0.641135i	$0.221536\pi$
$18$	−7.67400	−1.80878
$19$	5.28458	1.21237	0.606183	−	0.795325i	$-0.292700\pi$
$19$	5.28458	1.21237	0.606183	+	0.795325i	$0.292700\pi$
$20$	0.417272	0.0933049
$21$	15.2317	3.32382
$22$	−1.62366	−0.346165
$23$	−5.24895	−1.09448	−0.547241	−	0.836975i	$-0.684322\pi$
$23$	−5.24895	−1.09448	−0.547241	+	0.836975i	$0.684322\pi$
$24$	3.26711	0.666896
$25$	−4.82588	−0.965177
$26$	4.36101	0.855264
$27$	−15.2705	−2.93881
$28$	−4.66212	−0.881058
$29$	−0.958616	−0.178011	−0.0890053	−	0.996031i	$-0.528369\pi$
$29$	−0.958616	−0.178011	−0.0890053	+	0.996031i	$0.528369\pi$
$30$	1.36327	0.248899
$31$	−2.01654	−0.362181	−0.181090	−	0.983466i	$-0.557963\pi$
$31$	−2.01654	−0.362181	−0.181090	+	0.983466i	$0.557963\pi$
$32$	−1.00000	−0.176777
$33$	−5.30467	−0.923424
$34$	−6.32838	−1.08531
$35$	−1.94537	−0.328828
$36$	7.67400	1.27900
$37$	−2.00717	−0.329977	−0.164989	−	0.986295i	$-0.552759\pi$
$37$	−2.00717	−0.329977	−0.164989	+	0.986295i	$0.552759\pi$
$38$	−5.28458	−0.857273
$39$	14.2479	2.28149
$40$	−0.417272	−0.0659765
$41$	2.19927	0.343468	0.171734	−	0.985143i	$-0.445063\pi$
$41$	2.19927	0.343468	0.171734	+	0.985143i	$0.445063\pi$
$42$	−15.2317	−2.35030
$43$	−1.36796	−0.208613	−0.104306	−	0.994545i	$-0.533262\pi$
$43$	−1.36796	−0.208613	−0.104306	+	0.994545i	$0.533262\pi$
$44$	1.62366	0.244776
$45$	3.20215	0.477348
$46$	5.24895	0.773915
$47$	−4.48960	−0.654876	−0.327438	−	0.944873i	$-0.606185\pi$
$47$	−4.48960	−0.654876	−0.327438	+	0.944873i	$0.606185\pi$
$48$	−3.26711	−0.471567
$49$	14.7354	2.10505
$50$	4.82588	0.682483
$51$	−20.6755	−2.89515
$52$	−4.36101	−0.604763
$53$	1.46632	0.201414	0.100707	−	0.994916i	$-0.467889\pi$
$53$	1.46632	0.201414	0.100707	+	0.994916i	$0.467889\pi$
$54$	15.2705	2.07805
$55$	0.677507	0.0913551
$56$	4.66212	0.623002
$57$	−17.2653	−2.28685
$58$	0.958616	0.125872
$59$	0.871016	0.113397	0.0566983	−	0.998391i	$-0.481943\pi$
$59$	0.871016	0.113397	0.0566983	+	0.998391i	$0.481943\pi$
$60$	−1.36327	−0.175998
$61$	−1.02402	−0.131113	−0.0655563	−	0.997849i	$-0.520882\pi$
$61$	−1.02402	−0.131113	−0.0655563	+	0.997849i	$0.520882\pi$
$62$	2.01654	0.256101
$63$	−35.7771	−4.50749
$64$	1.00000	0.125000
$65$	−1.81973	−0.225709
$66$	5.30467	0.652960
$67$	−11.0374	−1.34844	−0.674218	−	0.738532i	$-0.735519\pi$
$67$	−11.0374	−1.34844	−0.674218	+	0.738532i	$0.735519\pi$
$68$	6.32838	0.767428
$69$	17.1489	2.06448
$70$	1.94537	0.232517
$71$	4.38146	0.519984	0.259992	−	0.965611i	$-0.416280\pi$
$71$	4.38146	0.519984	0.259992	+	0.965611i	$0.416280\pi$
$72$	−7.67400	−0.904390
$73$	2.87621	0.336635	0.168318	−	0.985733i	$-0.446167\pi$
$73$	2.87621	0.336635	0.168318	+	0.985733i	$0.446167\pi$
$74$	2.00717	0.233329
$75$	15.7667	1.82058
$76$	5.28458	0.606183
$77$	−7.56969	−0.862646
$78$	−14.2479	−1.61326
$79$	−15.7825	−1.77567	−0.887836	−	0.460160i	$-0.847792\pi$
$79$	−15.7825	−1.77567	−0.887836	+	0.460160i	$0.847792\pi$
$80$	0.417272	0.0466524
$81$	26.8683	2.98537
$82$	−2.19927	−0.242869
$83$	15.7039	1.72373	0.861863	−	0.507141i	$-0.169298\pi$
$83$	15.7039	1.72373	0.861863	+	0.507141i	$0.169298\pi$
$84$	15.2317	1.66191
$85$	2.64066	0.286419
$86$	1.36796	0.147511
$87$	3.13190	0.335775
$88$	−1.62366	−0.173083
$89$	16.0604	1.70240	0.851201	−	0.524839i	$-0.175875\pi$
$89$	16.0604	1.70240	0.851201	+	0.524839i	$0.175875\pi$
$90$	−3.20215	−0.337536
$91$	20.3315	2.13132
$92$	−5.24895	−0.547241
$93$	6.58825	0.683170
$94$	4.48960	0.463067
$95$	2.20511	0.226240
$96$	3.26711	0.333448
$97$	−4.29720	−0.436315	−0.218157	−	0.975914i	$-0.570005\pi$
$97$	−4.29720	−0.436315	−0.218157	+	0.975914i	$0.570005\pi$
$98$	−14.7354	−1.48850
$99$	12.4600	1.25227
$100$	−4.82588	−0.482588
$101$	−10.9415	−1.08872	−0.544361	−	0.838851i	$-0.683228\pi$
$101$	−10.9415	−1.08872	−0.544361	+	0.838851i	$0.683228\pi$
$102$	20.6755	2.04718
$103$	12.6244	1.24392	0.621961	−	0.783048i	$-0.286336\pi$
$103$	12.6244	1.24392	0.621961	+	0.783048i	$0.286336\pi$
$104$	4.36101	0.427632
$105$	6.35575	0.620257
$106$	−1.46632	−0.142421
$107$	5.52068	0.533704	0.266852	−	0.963737i	$-0.414016\pi$
$107$	5.52068	0.533704	0.266852	+	0.963737i	$0.414016\pi$
$108$	−15.2705	−1.46940
$109$	3.81218	0.365140	0.182570	−	0.983193i	$-0.441558\pi$
$109$	3.81218	0.365140	0.182570	+	0.983193i	$0.441558\pi$
$110$	−0.677507	−0.0645978
$111$	6.55766	0.622425
$112$	−4.66212	−0.440529
$113$	3.53387	0.332439	0.166219	−	0.986089i	$-0.446844\pi$
$113$	3.53387	0.332439	0.166219	+	0.986089i	$0.446844\pi$
$114$	17.2653	1.61705
$115$	−2.19024	−0.204241
$116$	−0.958616	−0.0890053
$117$	−33.4664	−3.09397
$118$	−0.871016	−0.0801835
$119$	−29.5037	−2.70460
$120$	1.36327	0.124449
$121$	−8.36373	−0.760339
$122$	1.02402	0.0927106
$123$	−7.18525	−0.647872
$124$	−2.01654	−0.181090
$125$	−4.10007	−0.366721
$126$	35.7771	3.18728
$127$	9.13974	0.811021	0.405510	−	0.914090i	$-0.367094\pi$
$127$	9.13974	0.811021	0.405510	+	0.914090i	$0.367094\pi$
$128$	−1.00000	−0.0883883
$129$	4.46929	0.393499
$130$	1.81973	0.159601
$131$	15.4343	1.34850	0.674250	−	0.738503i	$-0.264467\pi$
$131$	15.4343	1.34850	0.674250	+	0.738503i	$0.264467\pi$
$132$	−5.30467	−0.461712
$133$	−24.6374	−2.13633
$134$	11.0374	0.953488
$135$	−6.37195	−0.548410
$136$	−6.32838	−0.542654
$137$	18.7784	1.60434	0.802172	−	0.597093i	$-0.203678\pi$
$137$	18.7784	1.60434	0.802172	+	0.597093i	$0.203678\pi$
$138$	−17.1489	−1.45981
$139$	−21.5626	−1.82891	−0.914457	−	0.404682i	$-0.867382\pi$
$139$	−21.5626	−1.82891	−0.914457	+	0.404682i	$0.867382\pi$
$140$	−1.94537	−0.164414
$141$	14.6680	1.23527
$142$	−4.38146	−0.367684
$143$	−7.08078	−0.592125
$144$	7.67400	0.639500
$145$	−0.400004	−0.0332185
$146$	−2.87621	−0.238037
$147$	−48.1421	−3.97069
$148$	−2.00717	−0.164989
$149$	14.1746	1.16123	0.580616	−	0.814178i	$-0.302812\pi$
$149$	14.1746	1.16123	0.580616	+	0.814178i	$0.302812\pi$
$150$	−15.7667	−1.28734
$151$	1.68200	0.136879	0.0684396	−	0.997655i	$-0.478198\pi$
$151$	1.68200	0.136879	0.0684396	+	0.997655i	$0.478198\pi$
$152$	−5.28458	−0.428636
$153$	48.5640	3.92617
$154$	7.56969	0.609983
$155$	−0.841445	−0.0675865
$156$	14.2479	1.14074
$157$	19.7012	1.57232	0.786161	−	0.618021i	$-0.212065\pi$
$157$	19.7012	1.57232	0.786161	+	0.618021i	$0.212065\pi$
$158$	15.7825	1.25559
$159$	−4.79062	−0.379921
$160$	−0.417272	−0.0329883
$161$	24.4712	1.92860
$162$	−26.8683	−2.11097
$163$	−10.5939	−0.829777	−0.414889	−	0.909872i	$-0.636179\pi$
$163$	−10.5939	−0.829777	−0.414889	+	0.909872i	$0.636179\pi$
$164$	2.19927	0.171734
$165$	−2.21349	−0.172320
$166$	−15.7039	−1.21886
$167$	−6.15538	−0.476318	−0.238159	−	0.971226i	$-0.576544\pi$
$167$	−6.15538	−0.476318	−0.238159	+	0.971226i	$0.576544\pi$
$168$	−15.2317	−1.17515
$169$	6.01838	0.462952
$170$	−2.64066	−0.202529
$171$	40.5539	3.10124
$172$	−1.36796	−0.104306
$173$	13.3806	1.01731	0.508653	−	0.860972i	$-0.330144\pi$
$173$	13.3806	1.01731	0.508653	+	0.860972i	$0.330144\pi$
$174$	−3.13190	−0.237429
$175$	22.4989	1.70075
$176$	1.62366	0.122388
$177$	−2.84570	−0.213896
$178$	−16.0604	−1.20378
$179$	15.2464	1.13957	0.569783	−	0.821795i	$-0.307027\pi$
$179$	15.2464	1.13957	0.569783	+	0.821795i	$0.307027\pi$
$180$	3.20215	0.238674
$181$	4.05816	0.301641	0.150820	−	0.988561i	$-0.451808\pi$
$181$	4.05816	0.301641	0.150820	+	0.988561i	$0.451808\pi$
$182$	−20.3315	−1.50707
$183$	3.34559	0.247313
$184$	5.24895	0.386958
$185$	−0.837538	−0.0615770
$186$	−6.58825	−0.483074
$187$	10.2751	0.751391
$188$	−4.48960	−0.327438
$189$	71.1928	5.17852
$190$	−2.20511	−0.159976
$191$	−17.0835	−1.23612	−0.618059	−	0.786132i	$-0.712081\pi$
$191$	−17.0835	−1.23612	−0.618059	+	0.786132i	$0.712081\pi$
$192$	−3.26711	−0.235783
$193$	−4.06502	−0.292606	−0.146303	−	0.989240i	$-0.546737\pi$
$193$	−4.06502	−0.292606	−0.146303	+	0.989240i	$0.546737\pi$
$194$	4.29720	0.308521
$195$	5.94525	0.425748
$196$	14.7354	1.05253
$197$	11.7071	0.834097	0.417048	−	0.908884i	$-0.363065\pi$
$197$	11.7071	0.834097	0.417048	+	0.908884i	$0.363065\pi$
$198$	−12.4600	−0.885491
$199$	−17.8174	−1.26304	−0.631521	−	0.775359i	$-0.717569\pi$
$199$	−17.8174	−1.26304	−0.631521	+	0.775359i	$0.717569\pi$
$200$	4.82588	0.341242
$201$	36.0605	2.54351
$202$	10.9415	0.769842
$203$	4.46918	0.313675
$204$	−20.6755	−1.44757
$205$	0.917694	0.0640945
$206$	−12.6244	−0.879586
$207$	−40.2805	−2.79969
$208$	−4.36101	−0.302381
$209$	8.58036	0.593516
$210$	−6.35575	−0.438588
$211$	9.81040	0.675376	0.337688	−	0.941258i	$-0.390355\pi$
$211$	9.81040	0.675376	0.337688	+	0.941258i	$0.390355\pi$
$212$	1.46632	0.100707
$213$	−14.3147	−0.980829
$214$	−5.52068	−0.377386
$215$	−0.570814	−0.0389292
$216$	15.2705	1.03902
$217$	9.40134	0.638205
$218$	−3.81218	−0.258193
$219$	−9.39691	−0.634984
$220$	0.677507	0.0456775
$221$	−27.5981	−1.85645
$222$	−6.55766	−0.440121
$223$	−0.889529	−0.0595673	−0.0297836	−	0.999556i	$-0.509482\pi$
$223$	−0.889529	−0.0595673	−0.0297836	+	0.999556i	$0.509482\pi$
$224$	4.66212	0.311501
$225$	−37.0338	−2.46892
$226$	−3.53387	−0.235070
$227$	−19.3273	−1.28280	−0.641398	−	0.767208i	$-0.721645\pi$
$227$	−19.3273	−1.28280	−0.641398	+	0.767208i	$0.721645\pi$
$228$	−17.2653	−1.14342
$229$	−26.5453	−1.75416	−0.877080	−	0.480344i	$-0.840512\pi$
$229$	−26.5453	−1.75416	−0.877080	+	0.480344i	$0.840512\pi$
$230$	2.19024	0.144420
$231$	24.7310	1.62718
$232$	0.958616	0.0629362
$233$	12.3698	0.810373	0.405187	−	0.914234i	$-0.367207\pi$
$233$	12.3698	0.810373	0.405187	+	0.914234i	$0.367207\pi$
$234$	33.4664	2.18777
$235$	−1.87339	−0.122206
$236$	0.871016	0.0566983
$237$	51.5632	3.34939
$238$	29.5037	1.91244
$239$	−14.4352	−0.933738	−0.466869	−	0.884326i	$-0.654618\pi$
$239$	−14.4352	−0.933738	−0.466869	+	0.884326i	$0.654618\pi$
$240$	−1.36327	−0.0879989
$241$	3.85342	0.248220	0.124110	−	0.992268i	$-0.460392\pi$
$241$	3.85342	0.248220	0.124110	+	0.992268i	$0.460392\pi$
$242$	8.36373	0.537641
$243$	−41.9703	−2.69240
$244$	−1.02402	−0.0655563
$245$	6.14866	0.392823
$246$	7.18525	0.458115
$247$	−23.0461	−1.46639
$248$	2.01654	0.128050
$249$	−51.3063	−3.25141
$250$	4.10007	0.259311
$251$	22.2830	1.40649	0.703247	−	0.710946i	$-0.251733\pi$
$251$	22.2830	1.40649	0.703247	+	0.710946i	$0.251733\pi$
$252$	−35.7771	−2.25375
$253$	−8.52250	−0.535805
$254$	−9.13974	−0.573478
$255$	−8.62731	−0.540263
$256$	1.00000	0.0625000
$257$	4.85910	0.303102	0.151551	−	0.988449i	$-0.451573\pi$
$257$	4.85910	0.303102	0.151551	+	0.988449i	$0.451573\pi$
$258$	−4.46929	−0.278246
$259$	9.35769	0.581458
$260$	−1.81973	−0.112855
$261$	−7.35642	−0.455351
$262$	−15.4343	−0.953533
$263$	−1.02893	−0.0634466	−0.0317233	−	0.999497i	$-0.510100\pi$
$263$	−1.02893	−0.0634466	−0.0317233	+	0.999497i	$0.510100\pi$
$264$	5.30467	0.326480
$265$	0.611854	0.0375859
$266$	24.6374	1.51061
$267$	−52.4712	−3.21118
$268$	−11.0374	−0.674218
$269$	−18.9774	−1.15707	−0.578537	−	0.815656i	$-0.696376\pi$
$269$	−18.9774	−1.15707	−0.578537	+	0.815656i	$0.696376\pi$
$270$	6.37195	0.387784
$271$	29.1428	1.77030	0.885151	−	0.465305i	$-0.154055\pi$
$271$	29.1428	1.77030	0.885151	+	0.465305i	$0.154055\pi$
$272$	6.32838	0.383714
$273$	−66.4254	−4.02025
$274$	−18.7784	−1.13444
$275$	−7.83559	−0.472504
$276$	17.1489	1.03224
$277$	−9.47431	−0.569256	−0.284628	−	0.958638i	$-0.591870\pi$
$277$	−9.47431	−0.569256	−0.284628	+	0.958638i	$0.591870\pi$
$278$	21.5626	1.29324
$279$	−15.4749	−0.926459
$280$	1.94537	0.116258
$281$	8.34072	0.497565	0.248783	−	0.968559i	$-0.419970\pi$
$281$	8.34072	0.497565	0.248783	+	0.968559i	$0.419970\pi$
$282$	−14.6680	−0.873468
$283$	−10.2645	−0.610161	−0.305081	−	0.952327i	$-0.598683\pi$
$283$	−10.2645	−0.610161	−0.305081	+	0.952327i	$0.598683\pi$
$284$	4.38146	0.259992
$285$	−7.20434	−0.426748
$286$	7.08078	0.418696
$287$	−10.2533	−0.605231
$288$	−7.67400	−0.452195
$289$	23.0483	1.35579
$290$	0.400004	0.0234890
$291$	14.0394	0.823006
$292$	2.87621	0.168318
$293$	31.0021	1.81116	0.905582	−	0.424171i	$-0.139435\pi$
$293$	31.0021	1.81116	0.905582	+	0.424171i	$0.139435\pi$
$294$	48.1421	2.80770
$295$	0.363451	0.0211609
$296$	2.00717	0.116665
$297$	−24.7940	−1.43870
$298$	−14.1746	−0.821114
$299$	22.8907	1.32380
$300$	15.7667	0.910290
$301$	6.37762	0.367600
$302$	−1.68200	−0.0967881
$303$	35.7471	2.05362
$304$	5.28458	0.303092
$305$	−0.427296	−0.0244669
$306$	−48.5640	−2.77622
$307$	23.8776	1.36277	0.681385	−	0.731926i	$-0.261378\pi$
$307$	23.8776	1.36277	0.681385	+	0.731926i	$0.261378\pi$
$308$	−7.56969	−0.431323
$309$	−41.2454	−2.34637
$310$	0.841445	0.0477909
$311$	−15.9502	−0.904454	−0.452227	−	0.891903i	$-0.649370\pi$
$311$	−15.9502	−0.904454	−0.452227	+	0.891903i	$0.649370\pi$
$312$	−14.2479	−0.806628
$313$	−32.3642	−1.82933	−0.914667	−	0.404208i	$-0.867547\pi$
$313$	−32.3642	−1.82933	−0.914667	+	0.404208i	$0.867547\pi$
$314$	−19.7012	−1.11180
$315$	−14.9288	−0.841143
$316$	−15.7825	−0.887836
$317$	23.4499	1.31708	0.658539	−	0.752547i	$-0.271175\pi$
$317$	23.4499	1.31708	0.658539	+	0.752547i	$0.271175\pi$
$318$	4.79062	0.268645
$319$	−1.55646	−0.0871453
$320$	0.417272	0.0233262
$321$	−18.0367	−1.00671
$322$	−24.4712	−1.36373
$323$	33.4428	1.86081
$324$	26.8683	1.49268
$325$	21.0457	1.16741
$326$	10.5939	0.586741
$327$	−12.4548	−0.688752
$328$	−2.19927	−0.121434
$329$	20.9311	1.15397
$330$	2.21349	0.121849
$331$	19.2274	1.05683	0.528417	−	0.848985i	$-0.322786\pi$
$331$	19.2274	1.05683	0.528417	+	0.848985i	$0.322786\pi$
$332$	15.7039	0.861863
$333$	−15.4031	−0.844083
$334$	6.15538	0.336808
$335$	−4.60561	−0.251631
$336$	15.2317	0.830955
$337$	30.1848	1.64427	0.822135	−	0.569293i	$-0.192783\pi$
$337$	30.1848	1.64427	0.822135	+	0.569293i	$0.192783\pi$
$338$	−6.01838	−0.327357
$339$	−11.5455	−0.627068
$340$	2.64066	0.143210
$341$	−3.27417	−0.177306
$342$	−40.5539	−2.19290
$343$	−36.0632	−1.94723
$344$	1.36796	0.0737557
$345$	7.15576	0.385253
$346$	−13.3806	−0.719343
$347$	−21.8917	−1.17521	−0.587603	−	0.809149i	$-0.699928\pi$
$347$	−21.8917	−1.17521	−0.587603	+	0.809149i	$0.699928\pi$
$348$	3.13190	0.167888
$349$	−12.8187	−0.686169	−0.343085	−	0.939305i	$-0.611472\pi$
$349$	−12.8187	−0.686169	−0.343085	+	0.939305i	$0.611472\pi$
$350$	−22.4989	−1.20261
$351$	66.5947	3.55456
$352$	−1.62366	−0.0865413
$353$	−18.5139	−0.985393	−0.492697	−	0.870201i	$-0.663989\pi$
$353$	−18.5139	−0.985393	−0.492697	+	0.870201i	$0.663989\pi$
$354$	2.84570	0.151247
$355$	1.82826	0.0970341
$356$	16.0604	0.851201
$357$	96.3917	5.10159
$358$	−15.2464	−0.805796
$359$	25.8167	1.36255	0.681277	−	0.732025i	$-0.261425\pi$
$359$	25.8167	1.36255	0.681277	+	0.732025i	$0.261425\pi$
$360$	−3.20215	−0.168768
$361$	8.92684	0.469833
$362$	−4.05816	−0.213292
$363$	27.3252	1.43420
$364$	20.3315	1.06566
$365$	1.20016	0.0628195
$366$	−3.34559	−0.174877
$367$	23.6047	1.23215	0.616077	−	0.787686i	$-0.288721\pi$
$367$	23.6047	1.23215	0.616077	+	0.787686i	$0.288721\pi$
$368$	−5.24895	−0.273620
$369$	16.8772	0.878592
$370$	0.837538	0.0435415
$371$	−6.83615	−0.354915
$372$	6.58825	0.341585
$373$	−9.05127	−0.468657	−0.234328	−	0.972157i	$-0.575289\pi$
$373$	−9.05127	−0.468657	−0.234328	+	0.972157i	$0.575289\pi$
$374$	−10.2751	−0.531314
$375$	13.3954	0.691734
$376$	4.48960	0.231534
$377$	4.18053	0.215308
$378$	−71.1928	−3.66176
$379$	20.8149	1.06919	0.534595	−	0.845108i	$-0.320464\pi$
$379$	20.8149	1.06919	0.534595	+	0.845108i	$0.320464\pi$
$380$	2.20511	0.113120
$381$	−29.8605	−1.52980
$382$	17.0835	0.874067
$383$	29.4689	1.50579	0.752894	−	0.658142i	$-0.228657\pi$
$383$	29.4689	1.50579	0.752894	+	0.658142i	$0.228657\pi$
$384$	3.26711	0.166724
$385$	−3.15862	−0.160978
$386$	4.06502	0.206904
$387$	−10.4978	−0.533631
$388$	−4.29720	−0.218157
$389$	20.5343	1.04113	0.520564	−	0.853823i	$-0.325722\pi$
$389$	20.5343	1.04113	0.520564	+	0.853823i	$0.325722\pi$
$390$	−5.94525	−0.301049
$391$	−33.2173	−1.67987
$392$	−14.7354	−0.744248
$393$	−50.4255	−2.54363
$394$	−11.7071	−0.589795
$395$	−6.58560	−0.331358
$396$	12.4600	0.626137
$397$	29.3527	1.47317	0.736586	−	0.676344i	$-0.236437\pi$
$397$	29.3527	1.47317	0.736586	+	0.676344i	$0.236437\pi$
$398$	17.8174	0.893105
$399$	80.4930	4.02969
$400$	−4.82588	−0.241294
$401$	10.7990	0.539275	0.269637	−	0.962962i	$-0.413096\pi$
$401$	10.7990	0.539275	0.269637	+	0.962962i	$0.413096\pi$
$402$	−36.0605	−1.79853
$403$	8.79413	0.438067
$404$	−10.9415	−0.544361
$405$	11.2114	0.557099
$406$	−4.46918	−0.221802
$407$	−3.25896	−0.161541
$408$	20.6755	1.02359
$409$	−19.7622	−0.977177	−0.488589	−	0.872514i	$-0.662488\pi$
$409$	−19.7622	−0.977177	−0.488589	+	0.872514i	$0.662488\pi$
$410$	−0.917694	−0.0453217
$411$	−61.3510	−3.02622
$412$	12.6244	0.621961
$413$	−4.06078	−0.199818
$414$	40.2805	1.97968
$415$	6.55280	0.321664
$416$	4.36101	0.213816
$417$	70.4473	3.44982
$418$	−8.58036	−0.419679
$419$	−18.9940	−0.927915	−0.463958	−	0.885857i	$-0.653571\pi$
$419$	−18.9940	−0.927915	−0.463958	+	0.885857i	$0.653571\pi$
$420$	6.35575	0.310129
$421$	−16.5722	−0.807681	−0.403840	−	0.914830i	$-0.632325\pi$
$421$	−16.5722	−0.807681	−0.403840	+	0.914830i	$0.632325\pi$
$422$	−9.81040	−0.477563
$423$	−34.4532	−1.67517
$424$	−1.46632	−0.0712107
$425$	−30.5400	−1.48141
$426$	14.3147	0.693551
$427$	4.77412	0.231036
$428$	5.52068	0.266852
$429$	23.1337	1.11691
$430$	0.570814	0.0275271
$431$	25.5911	1.23268	0.616340	−	0.787480i	$-0.288615\pi$
$431$	25.5911	1.23268	0.616340	+	0.787480i	$0.288615\pi$
$432$	−15.2705	−0.734701
$433$	−23.8275	−1.14508	−0.572538	−	0.819878i	$-0.694041\pi$
$433$	−23.8275	−1.14508	−0.572538	+	0.819878i	$0.694041\pi$
$434$	−9.40134	−0.451279
$435$	1.30686	0.0626590
$436$	3.81218	0.182570
$437$	−27.7385	−1.32691
$438$	9.39691	0.449002
$439$	−32.8923	−1.56986	−0.784932	−	0.619582i	$-0.787302\pi$
$439$	−32.8923	−1.56986	−0.784932	+	0.619582i	$0.787302\pi$
$440$	−0.677507	−0.0322989
$441$	113.079	5.38473
$442$	27.5981	1.31271
$443$	−27.4911	−1.30614	−0.653071	−	0.757297i	$-0.726520\pi$
$443$	−27.4911	−1.30614	−0.653071	+	0.757297i	$0.726520\pi$
$444$	6.55766	0.311213
$445$	6.70157	0.317685
$446$	0.889529	0.0421204
$447$	−46.3101	−2.19039
$448$	−4.66212	−0.220264
$449$	−29.8238	−1.40747	−0.703736	−	0.710461i	$-0.748486\pi$
$449$	−29.8238	−1.40747	−0.703736	+	0.710461i	$0.748486\pi$
$450$	37.0338	1.74579
$451$	3.57086	0.168145
$452$	3.53387	0.166219
$453$	−5.49527	−0.258190
$454$	19.3273	0.907074
$455$	8.48379	0.397726
$456$	17.2653	0.808523
$457$	−11.4932	−0.537627	−0.268814	−	0.963192i	$-0.586632\pi$
$457$	−11.4932	−0.537627	−0.268814	+	0.963192i	$0.586632\pi$
$458$	26.5453	1.24038
$459$	−96.6373	−4.51065
$460$	−2.19024	−0.102120
$461$	−27.9676	−1.30258	−0.651290	−	0.758829i	$-0.725772\pi$
$461$	−27.9676	−1.30258	−0.651290	+	0.758829i	$0.725772\pi$
$462$	−24.7310	−1.15059
$463$	25.0698	1.16509	0.582547	−	0.812797i	$-0.302056\pi$
$463$	25.0698	1.16509	0.582547	+	0.812797i	$0.302056\pi$
$464$	−0.958616	−0.0445026
$465$	2.74909	0.127486
$466$	−12.3698	−0.573021
$467$	−4.41593	−0.204345	−0.102172	−	0.994767i	$-0.532579\pi$
$467$	−4.41593	−0.204345	−0.102172	+	0.994767i	$0.532579\pi$
$468$	−33.4664	−1.54698
$469$	51.4578	2.37610
$470$	1.87339	0.0864129
$471$	−64.3658	−2.96582
$472$	−0.871016	−0.0400918
$473$	−2.22111	−0.102127
$474$	−51.5632	−2.36838
$475$	−25.5028	−1.17015
$476$	−29.5037	−1.35230
$477$	11.2525	0.515218
$478$	14.4352	0.660253
$479$	4.13590	0.188974	0.0944870	−	0.995526i	$-0.469879\pi$
$479$	4.13590	0.188974	0.0944870	+	0.995526i	$0.469879\pi$
$480$	1.36327	0.0622247
$481$	8.75330	0.399116
$482$	−3.85342	−0.175518
$483$	−79.9502	−3.63786
$484$	−8.36373	−0.380170
$485$	−1.79310	−0.0814206
$486$	41.9703	1.90381
$487$	−7.12379	−0.322810	−0.161405	−	0.986888i	$-0.551602\pi$
$487$	−7.12379	−0.322810	−0.161405	+	0.986888i	$0.551602\pi$
$488$	1.02402	0.0463553
$489$	34.6114	1.56518
$490$	−6.14866	−0.277768
$491$	−18.7335	−0.845432	−0.422716	−	0.906262i	$-0.638923\pi$
$491$	−18.7335	−0.845432	−0.422716	+	0.906262i	$0.638923\pi$
$492$	−7.18525	−0.323936
$493$	−6.06648	−0.273221
$494$	23.0461	1.03689
$495$	5.19919	0.233686
$496$	−2.01654	−0.0905452
$497$	−20.4269	−0.916272
$498$	51.3063	2.29909
$499$	−28.1730	−1.26120	−0.630598	−	0.776110i	$-0.717190\pi$
$499$	−28.1730	−1.26120	−0.630598	+	0.776110i	$0.717190\pi$
$500$	−4.10007	−0.183361
$501$	20.1103	0.898462
$502$	−22.2830	−0.994541
$503$	−7.63266	−0.340324	−0.170162	−	0.985416i	$-0.554429\pi$
$503$	−7.63266	−0.340324	−0.170162	+	0.985416i	$0.554429\pi$
$504$	35.7771	1.59364
$505$	−4.56559	−0.203166
$506$	8.52250	0.378871
$507$	−19.6627	−0.873251
$508$	9.13974	0.405510
$509$	−10.1233	−0.448709	−0.224354	−	0.974508i	$-0.572027\pi$
$509$	−10.1233	−0.448709	−0.224354	+	0.974508i	$0.572027\pi$
$510$	8.62731	0.382024
$511$	−13.4093	−0.593191
$512$	−1.00000	−0.0441942
$513$	−80.6981	−3.56291
$514$	−4.85910	−0.214326
$515$	5.26783	0.232128
$516$	4.46929	0.196750
$517$	−7.28958	−0.320596
$518$	−9.35769	−0.411153
$519$	−43.7157	−1.91891
$520$	1.81973	0.0798003
$521$	7.49687	0.328444	0.164222	−	0.986423i	$-0.447489\pi$
$521$	7.49687	0.328444	0.164222	+	0.986423i	$0.447489\pi$
$522$	7.35642	0.321982
$523$	21.9121	0.958149	0.479074	−	0.877774i	$-0.340972\pi$
$523$	21.9121	0.958149	0.479074	+	0.877774i	$0.340972\pi$
$524$	15.4343	0.674250
$525$	−73.5062	−3.20807
$526$	1.02893	0.0448635
$527$	−12.7614	−0.555896
$528$	−5.30467	−0.230856
$529$	4.55147	0.197890
$530$	−0.611854	−0.0265772
$531$	6.68418	0.290069
$532$	−24.6374	−1.06817
$533$	−9.59103	−0.415433
$534$	52.4712	2.27065
$535$	2.30363	0.0995945
$536$	11.0374	0.476744
$537$	−49.8115	−2.14953
$538$	18.9774	0.818175
$539$	23.9252	1.03053
$540$	−6.37195	−0.274205
$541$	7.67261	0.329871	0.164936	−	0.986304i	$-0.447258\pi$
$541$	7.67261	0.329871	0.164936	+	0.986304i	$0.447258\pi$
$542$	−29.1428	−1.25179
$543$	−13.2585	−0.568975
$544$	−6.32838	−0.271327
$545$	1.59071	0.0681387
$546$	66.4254	2.84274
$547$	34.4231	1.47183	0.735913	−	0.677076i	$-0.236753\pi$
$547$	34.4231	1.47183	0.735913	+	0.677076i	$0.236753\pi$
$548$	18.7784	0.802172
$549$	−7.85835	−0.335386
$550$	7.83559	0.334111
$551$	−5.06589	−0.215814
$552$	−17.1489	−0.729905
$553$	73.5800	3.12894
$554$	9.47431	0.402525
$555$	2.73633	0.116151
$556$	−21.5626	−0.914457
$557$	−37.6818	−1.59663	−0.798315	−	0.602240i	$-0.794275\pi$
$557$	−37.6818	−1.59663	−0.798315	+	0.602240i	$0.794275\pi$
$558$	15.4749	0.655105
$559$	5.96570	0.252322
$560$	−1.94537	−0.0822070
$561$	−33.5699	−1.41732
$562$	−8.34072	−0.351832
$563$	−27.2359	−1.14786	−0.573928	−	0.818906i	$-0.694581\pi$
$563$	−27.2359	−1.14786	−0.573928	+	0.818906i	$0.694581\pi$
$564$	14.6680	0.617635
$565$	1.47459	0.0620363
$566$	10.2645	0.431449
$567$	−125.263	−5.26057
$568$	−4.38146	−0.183842
$569$	19.5361	0.818998	0.409499	−	0.912311i	$-0.365704\pi$
$569$	19.5361	0.818998	0.409499	+	0.912311i	$0.365704\pi$
$570$	7.20434	0.301756
$571$	−13.2742	−0.555510	−0.277755	−	0.960652i	$-0.589590\pi$
$571$	−13.2742	−0.555510	−0.277755	+	0.960652i	$0.589590\pi$
$572$	−7.08078	−0.296062
$573$	55.8136	2.33165
$574$	10.2533	0.427963
$575$	25.3308	1.05637
$576$	7.67400	0.319750
$577$	−0.913538	−0.0380311	−0.0190156	−	0.999819i	$-0.506053\pi$
$577$	−0.913538	−0.0380311	−0.0190156	+	0.999819i	$0.506053\pi$
$578$	−23.0483	−0.958685
$579$	13.2809	0.551933
$580$	−0.400004	−0.0166093
$581$	−73.2134	−3.03741
$582$	−14.0394	−0.581953
$583$	2.38080	0.0986027
$584$	−2.87621	−0.119019
$585$	−13.9646	−0.577365
$586$	−31.0021	−1.28069
$587$	−15.9907	−0.660008	−0.330004	−	0.943980i	$-0.607050\pi$
$587$	−15.9907	−0.660008	−0.330004	+	0.943980i	$0.607050\pi$
$588$	−48.1421	−1.98534
$589$	−10.6566	−0.439096
$590$	−0.363451	−0.0149630
$591$	−38.2484	−1.57333
$592$	−2.00717	−0.0824943
$593$	−12.7954	−0.525445	−0.262722	−	0.964871i	$-0.584620\pi$
$593$	−12.7954	−0.525445	−0.262722	+	0.964871i	$0.584620\pi$
$594$	24.7940	1.01731
$595$	−12.3111	−0.504704
$596$	14.1746	0.580616
$597$	58.2114	2.38243
$598$	−22.8907	−0.936071
$599$	17.4172	0.711647	0.355824	−	0.934553i	$-0.384200\pi$
$599$	17.4172	0.711647	0.355824	+	0.934553i	$0.384200\pi$
$600$	−15.7667	−0.643672
$601$	−46.2128	−1.88506	−0.942530	−	0.334120i	$-0.891561\pi$
$601$	−46.2128	−1.88506	−0.942530	+	0.334120i	$0.891561\pi$
$602$	−6.37762	−0.259932
$603$	−84.7012	−3.44930
$604$	1.68200	0.0684396
$605$	−3.48995	−0.141887
$606$	−35.7471	−1.45213
$607$	−19.8269	−0.804748	−0.402374	−	0.915475i	$-0.631815\pi$
$607$	−19.8269	−0.804748	−0.402374	+	0.915475i	$0.631815\pi$
$608$	−5.28458	−0.214318
$609$	−14.6013	−0.591675
$610$	0.427296	0.0173007
$611$	19.5792	0.792089
$612$	48.5640	1.96308
$613$	16.3585	0.660713	0.330356	−	0.943856i	$-0.392831\pi$
$613$	16.3585	0.660713	0.330356	+	0.943856i	$0.392831\pi$
$614$	−23.8776	−0.963623
$615$	−2.99821	−0.120899
$616$	7.56969	0.304992
$617$	−0.972418	−0.0391481	−0.0195740	−	0.999808i	$-0.506231\pi$
$617$	−0.972418	−0.0391481	−0.0195740	+	0.999808i	$0.506231\pi$
$618$	41.2454	1.65913
$619$	−34.8082	−1.39906	−0.699530	−	0.714603i	$-0.746607\pi$
$619$	−34.8082	−1.39906	−0.699530	+	0.714603i	$0.746607\pi$
$620$	−0.841445	−0.0337932
$621$	80.1540	3.21647
$622$	15.9502	0.639546
$623$	−74.8757	−2.99983
$624$	14.2479	0.570372
$625$	22.4186	0.896743
$626$	32.3642	1.29353
$627$	−28.0330	−1.11953
$628$	19.7012	0.786161
$629$	−12.7022	−0.506468
$630$	14.9288	0.594778
$631$	−46.5892	−1.85469	−0.927343	−	0.374212i	$-0.877913\pi$
$631$	−46.5892	−1.85469	−0.927343	+	0.374212i	$0.877913\pi$
$632$	15.7825	0.627795
$633$	−32.0517	−1.27394
$634$	−23.4499	−0.931314
$635$	3.81376	0.151344
$636$	−4.79062	−0.189961
$637$	−64.2610	−2.54611
$638$	1.55646	0.0616210
$639$	33.6234	1.33012
$640$	−0.417272	−0.0164941
$641$	2.47814	0.0978807	0.0489403	−	0.998802i	$-0.484416\pi$
$641$	2.47814	0.0978807	0.0489403	+	0.998802i	$0.484416\pi$
$642$	18.0367	0.711851
$643$	−28.1902	−1.11171	−0.555856	−	0.831278i	$-0.687610\pi$
$643$	−28.1902	−1.11171	−0.555856	+	0.831278i	$0.687610\pi$
$644$	24.4712	0.964302
$645$	1.86491	0.0734308
$646$	−33.4428	−1.31579
$647$	−36.6880	−1.44235	−0.721176	−	0.692752i	$-0.756398\pi$
$647$	−36.6880	−1.44235	−0.721176	+	0.692752i	$0.756398\pi$
$648$	−26.8683	−1.05549
$649$	1.41423	0.0555135
$650$	−21.0457	−0.825481
$651$	−30.7152	−1.20382
$652$	−10.5939	−0.414889
$653$	−3.94524	−0.154389	−0.0771946	−	0.997016i	$-0.524596\pi$
$653$	−3.94524	−0.154389	−0.0771946	+	0.997016i	$0.524596\pi$
$654$	12.4548	0.487021
$655$	6.44030	0.251643
$656$	2.19927	0.0858670
$657$	22.0721	0.861114
$658$	−20.9311	−0.815978
$659$	−15.5118	−0.604253	−0.302127	−	0.953268i	$-0.597697\pi$
$659$	−15.5118	−0.604253	−0.302127	+	0.953268i	$0.597697\pi$
$660$	−2.21349	−0.0861600
$661$	15.0480	0.585299	0.292650	−	0.956220i	$-0.405463\pi$
$661$	15.0480	0.585299	0.292650	+	0.956220i	$0.405463\pi$
$662$	−19.2274	−0.747295
$663$	90.1660	3.50176
$664$	−15.7039	−0.609429
$665$	−10.2805	−0.398660
$666$	15.4031	0.596856
$667$	5.03173	0.194829
$668$	−6.15538	−0.238159
$669$	2.90619	0.112360
$670$	4.60561	0.177930
$671$	−1.66266	−0.0641864
$672$	−15.2317	−0.587574
$673$	33.4865	1.29081	0.645406	−	0.763840i	$-0.276688\pi$
$673$	33.4865	1.29081	0.645406	+	0.763840i	$0.276688\pi$
$674$	−30.1848	−1.16267
$675$	73.6936	2.83647
$676$	6.01838	0.231476
$677$	22.5785	0.867764	0.433882	−	0.900970i	$-0.357144\pi$
$677$	22.5785	0.867764	0.433882	+	0.900970i	$0.357144\pi$
$678$	11.5455	0.443404
$679$	20.0341	0.768837
$680$	−2.64066	−0.101265
$681$	63.1443	2.41969
$682$	3.27417	0.125374
$683$	−43.5000	−1.66448	−0.832241	−	0.554414i	$-0.812942\pi$
$683$	−43.5000	−1.66448	−0.832241	+	0.554414i	$0.812942\pi$
$684$	40.5539	1.55062
$685$	7.83569	0.299386
$686$	36.0632	1.37690
$687$	86.7263	3.30881
$688$	−1.36796	−0.0521532
$689$	−6.39463	−0.243616
$690$	−7.15576	−0.272415
$691$	−19.8511	−0.755170	−0.377585	−	0.925975i	$-0.623245\pi$
$691$	−19.8511	−0.755170	−0.377585	+	0.925975i	$0.623245\pi$
$692$	13.3806	0.508653
$693$	−58.0898	−2.20665
$694$	21.8917	0.830997
$695$	−8.99747	−0.341293
$696$	−3.13190	−0.118714
$697$	13.9178	0.527174
$698$	12.8187	0.485195
$699$	−40.4135	−1.52858
$700$	22.4989	0.850377
$701$	−22.1846	−0.837900	−0.418950	−	0.908009i	$-0.637602\pi$
$701$	−22.1846	−0.837900	−0.418950	+	0.908009i	$0.637602\pi$
$702$	−66.5947	−2.51345
$703$	−10.6071	−0.400054
$704$	1.62366	0.0611939
$705$	6.12056	0.230514
$706$	18.5139	0.696778
$707$	51.0106	1.91845
$708$	−2.84570	−0.106948
$709$	−9.13701	−0.343148	−0.171574	−	0.985171i	$-0.554885\pi$
$709$	−9.13701	−0.343148	−0.171574	+	0.985171i	$0.554885\pi$
$710$	−1.82826	−0.0686135
$711$	−121.115	−4.54217
$712$	−16.0604	−0.601890
$713$	10.5847	0.396400
$714$	−96.3917	−3.60737
$715$	−2.95461	−0.110496
$716$	15.2464	0.569783
$717$	47.1615	1.76128
$718$	−25.8167	−0.963472
$719$	−9.33760	−0.348234	−0.174117	−	0.984725i	$-0.555707\pi$
$719$	−9.33760	−0.348234	−0.174117	+	0.984725i	$0.555707\pi$
$720$	3.20215	0.119337
$721$	−58.8566	−2.19194
$722$	−8.92684	−0.332222
$723$	−12.5895	−0.468210
$724$	4.05816	0.150820
$725$	4.62617	0.171812
$726$	−27.3252	−1.01413
$727$	−34.3631	−1.27446	−0.637229	−	0.770674i	$-0.719919\pi$
$727$	−34.3631	−1.27446	−0.637229	+	0.770674i	$0.719919\pi$
$728$	−20.3315	−0.753537
$729$	56.5166	2.09321
$730$	−1.20016	−0.0444201
$731$	−8.65700	−0.320191
$732$	3.34559	0.123657
$733$	−30.7870	−1.13714	−0.568572	−	0.822633i	$-0.692504\pi$
$733$	−30.7870	−1.13714	−0.568572	+	0.822633i	$0.692504\pi$
$734$	−23.6047	−0.871265
$735$	−20.0883	−0.740970
$736$	5.24895	0.193479
$737$	−17.9210	−0.660129
$738$	−16.8772	−0.621258
$739$	11.3315	0.416837	0.208419	−	0.978040i	$-0.433168\pi$
$739$	11.3315	0.416837	0.208419	+	0.978040i	$0.433168\pi$
$740$	−0.837538	−0.0307885
$741$	75.2942	2.76600
$742$	6.83615	0.250963
$743$	−11.4725	−0.420885	−0.210442	−	0.977606i	$-0.567490\pi$
$743$	−11.4725	−0.420885	−0.210442	+	0.977606i	$0.567490\pi$
$744$	−6.58825	−0.241537
$745$	5.91468	0.216697
$746$	9.05127	0.331390
$747$	120.512	4.40929
$748$	10.2751	0.375696
$749$	−25.7381	−0.940449
$750$	−13.3954	−0.489130
$751$	1.26403	0.0461252	0.0230626	−	0.999734i	$-0.492658\pi$
$751$	1.26403	0.0461252	0.0230626	+	0.999734i	$0.492658\pi$
$752$	−4.48960	−0.163719
$753$	−72.8011	−2.65302
$754$	−4.18053	−0.152246
$755$	0.701851	0.0255430
$756$	71.1928	2.58926
$757$	11.9864	0.435651	0.217826	−	0.975988i	$-0.430104\pi$
$757$	11.9864	0.435651	0.217826	+	0.975988i	$0.430104\pi$
$758$	−20.8149	−0.756032
$759$	27.8439	1.01067
$760$	−2.20511	−0.0799878
$761$	−42.2105	−1.53013	−0.765064	−	0.643954i	$-0.777293\pi$
$761$	−42.2105	−1.53013	−0.765064	+	0.643954i	$0.777293\pi$
$762$	29.8605	1.08173
$763$	−17.7728	−0.643419
$764$	−17.0835	−0.618059
$765$	20.2644	0.732661
$766$	−29.4689	−1.06475
$767$	−3.79851	−0.137156
$768$	−3.26711	−0.117892
$769$	−10.4681	−0.377491	−0.188746	−	0.982026i	$-0.560442\pi$
$769$	−10.4681	−0.377491	−0.188746	+	0.982026i	$0.560442\pi$
$770$	3.15862	0.113829
$771$	−15.8752	−0.571732
$772$	−4.06502	−0.146303
$773$	8.39157	0.301824	0.150912	−	0.988547i	$-0.451779\pi$
$773$	8.39157	0.301824	0.150912	+	0.988547i	$0.451779\pi$
$774$	10.4978	0.377334
$775$	9.73158	0.349569
$776$	4.29720	0.154261
$777$	−30.5726	−1.09679
$778$	−20.5343	−0.736189
$779$	11.6222	0.416409
$780$	5.94525	0.212874
$781$	7.11400	0.254559
$782$	33.2173	1.18785
$783$	14.6385	0.523138
$784$	14.7354	0.526263
$785$	8.22074	0.293411
$786$	50.4255	1.79862
$787$	6.98120	0.248853	0.124427	−	0.992229i	$-0.460291\pi$
$787$	6.98120	0.248853	0.124427	+	0.992229i	$0.460291\pi$
$788$	11.7071	0.417048
$789$	3.36163	0.119677
$790$	6.58560	0.234305
$791$	−16.4753	−0.585796
$792$	−12.4600	−0.442745
$793$	4.46577	0.158584
$794$	−29.3527	−1.04169
$795$	−1.99899	−0.0708970
$796$	−17.8174	−0.631521
$797$	−16.5017	−0.584519	−0.292259	−	0.956339i	$-0.594407\pi$
$797$	−16.5017	−0.584519	−0.292259	+	0.956339i	$0.594407\pi$
$798$	−80.4930	−2.84942
$799$	−28.4119	−1.00514
$800$	4.82588	0.170621
$801$	123.248	4.35475
$802$	−10.7990	−0.381325
$803$	4.66999	0.164800
$804$	36.0605	1.27175
$805$	10.2112	0.359896
$806$	−8.79413	−0.309760
$807$	62.0013	2.18255
$808$	10.9415	0.384921
$809$	−45.3597	−1.59476	−0.797381	−	0.603477i	$-0.793782\pi$
$809$	−45.3597	−1.59476	−0.797381	+	0.603477i	$0.793782\pi$
$810$	−11.2114	−0.393928
$811$	−27.4757	−0.964803	−0.482401	−	0.875950i	$-0.660235\pi$
$811$	−27.4757	−0.964803	−0.482401	+	0.875950i	$0.660235\pi$
$812$	4.46918	0.156838
$813$	−95.2128	−3.33926
$814$	3.25896	0.114227
$815$	−4.42053	−0.154845
$816$	−20.6755	−0.723787
$817$	−7.22913	−0.252915
$818$	19.7622	0.690969
$819$	156.024	5.45193
$820$	0.917694	0.0320473
$821$	−17.0634	−0.595517	−0.297758	−	0.954641i	$-0.596239\pi$
$821$	−17.0634	−0.595517	−0.297758	+	0.954641i	$0.596239\pi$
$822$	61.3510	2.13986
$823$	36.4515	1.27062	0.635309	−	0.772258i	$-0.280873\pi$
$823$	36.4515	1.27062	0.635309	+	0.772258i	$0.280873\pi$
$824$	−12.6244	−0.439793
$825$	25.5997	0.891268
$826$	4.06078	0.141293
$827$	−7.40843	−0.257616	−0.128808	−	0.991670i	$-0.541115\pi$
$827$	−7.40843	−0.257616	−0.128808	+	0.991670i	$0.541115\pi$
$828$	−40.2805	−1.39984
$829$	9.15327	0.317906	0.158953	−	0.987286i	$-0.449188\pi$
$829$	9.15327	0.317906	0.158953	+	0.987286i	$0.449188\pi$
$830$	−6.55280	−0.227451
$831$	30.9536	1.07377
$832$	−4.36101	−0.151191
$833$	93.2509	3.23095
$834$	−70.4473	−2.43939
$835$	−2.56847	−0.0888856
$836$	8.58036	0.296758
$837$	30.7935	1.06438
$838$	18.9940	0.656135
$839$	−17.7434	−0.612569	−0.306285	−	0.951940i	$-0.599086\pi$
$839$	−17.7434	−0.612569	−0.306285	+	0.951940i	$0.599086\pi$
$840$	−6.35575	−0.219294
$841$	−28.0811	−0.968312
$842$	16.5722	0.571116
$843$	−27.2500	−0.938541
$844$	9.81040	0.337688
$845$	2.51130	0.0863914
$846$	34.4532	1.18453
$847$	38.9927	1.33981
$848$	1.46632	0.0503536
$849$	33.5352	1.15093
$850$	30.5400	1.04751
$851$	10.5356	0.361154
$852$	−14.3147	−0.490414
$853$	−2.51210	−0.0860128	−0.0430064	−	0.999075i	$-0.513694\pi$
$853$	−2.51210	−0.0860128	−0.0430064	+	0.999075i	$0.513694\pi$
$854$	−4.77412	−0.163367
$855$	16.9220	0.578721
$856$	−5.52068	−0.188693
$857$	27.4337	0.937116	0.468558	−	0.883433i	$-0.344774\pi$
$857$	27.4337	0.937116	0.468558	+	0.883433i	$0.344774\pi$
$858$	−23.1337	−0.789771
$859$	35.9519	1.22666	0.613332	−	0.789825i	$-0.289829\pi$
$859$	35.9519	1.22666	0.613332	+	0.789825i	$0.289829\pi$
$860$	−0.570814	−0.0194646
$861$	33.4985	1.14163
$862$	−25.5911	−0.871637
$863$	−32.6514	−1.11147	−0.555733	−	0.831361i	$-0.687562\pi$
$863$	−32.6514	−1.11147	−0.555733	+	0.831361i	$0.687562\pi$
$864$	15.2705	0.519512
$865$	5.58334	0.189839
$866$	23.8275	0.809691
$867$	−75.3015	−2.55737
$868$	9.40134	0.319102
$869$	−25.6254	−0.869283
$870$	−1.30686	−0.0443066
$871$	48.1343	1.63097
$872$	−3.81218	−0.129097
$873$	−32.9767	−1.11609
$874$	27.7385	0.938269
$875$	19.1150	0.646205
$876$	−9.39691	−0.317492
$877$	−0.723249	−0.0244224	−0.0122112	−	0.999925i	$-0.503887\pi$
$877$	−0.723249	−0.0244224	−0.0122112	+	0.999925i	$0.503887\pi$
$878$	32.8923	1.11006
$879$	−101.287	−3.41634
$880$	0.677507	0.0228388
$881$	−3.12665	−0.105340	−0.0526698	−	0.998612i	$-0.516773\pi$
$881$	−3.12665	−0.105340	−0.0526698	+	0.998612i	$0.516773\pi$
$882$	−113.079	−3.80758
$883$	28.9590	0.974547	0.487273	−	0.873250i	$-0.337992\pi$
$883$	28.9590	0.974547	0.487273	+	0.873250i	$0.337992\pi$
$884$	−27.5981	−0.928224
$885$	−1.18743	−0.0399151
$886$	27.4911	0.923581
$887$	−11.9116	−0.399953	−0.199976	−	0.979801i	$-0.564087\pi$
$887$	−11.9116	−0.399953	−0.199976	+	0.979801i	$0.564087\pi$
$888$	−6.55766	−0.220061
$889$	−42.6106	−1.42911
$890$	−6.70157	−0.224637
$891$	43.6250	1.46149
$892$	−0.889529	−0.0297836
$893$	−23.7257	−0.793950
$894$	46.3101	1.54884
$895$	6.36188	0.212654
$896$	4.66212	0.155751
$897$	−74.7864	−2.49705
$898$	29.8238	0.995234
$899$	1.93309	0.0644720
$900$	−37.0338	−1.23446
$901$	9.27942	0.309142
$902$	−3.57086	−0.118897
$903$	−20.8364	−0.693391
$904$	−3.53387	−0.117535
$905$	1.69336	0.0562891
$906$	5.49527	0.182568
$907$	36.1433	1.20012	0.600059	−	0.799956i	$-0.295144\pi$
$907$	36.1433	1.20012	0.600059	+	0.799956i	$0.295144\pi$
$908$	−19.3273	−0.641398
$909$	−83.9652	−2.78495
$910$	−8.48379	−0.281235
$911$	27.1321	0.898928	0.449464	−	0.893298i	$-0.351615\pi$
$911$	27.1321	0.898928	0.449464	+	0.893298i	$0.351615\pi$
$912$	−17.2653	−0.571712
$913$	25.4978	0.843852
$914$	11.4932	0.380160
$915$	1.39602	0.0461511
$916$	−26.5453	−0.877080
$917$	−71.9565	−2.37621
$918$	96.6373	3.18951
$919$	−36.7803	−1.21327	−0.606636	−	0.794980i	$-0.707481\pi$
$919$	−36.7803	−1.21327	−0.606636	+	0.794980i	$0.707481\pi$
$920$	2.19024	0.0722101
$921$	−78.0109	−2.57055
$922$	27.9676	0.921063
$923$	−19.1076	−0.628934
$924$	24.7310	0.813590
$925$	9.68639	0.318487
$926$	−25.0698	−0.823846
$927$	96.8800	3.18196
$928$	0.958616	0.0314681
$929$	10.9886	0.360523	0.180261	−	0.983619i	$-0.442306\pi$
$929$	10.9886	0.360523	0.180261	+	0.983619i	$0.442306\pi$
$930$	−2.74909	−0.0901463
$931$	77.8703	2.55210
$932$	12.3698	0.405187
$933$	52.1111	1.70604
$934$	4.41593	0.144494
$935$	4.28752	0.140217
$936$	33.4664	1.09388
$937$	−43.0624	−1.40679	−0.703393	−	0.710801i	$-0.748333\pi$
$937$	−43.0624	−1.40679	−0.703393	+	0.710801i	$0.748333\pi$
$938$	−51.4578	−1.68016
$939$	105.737	3.45061
$940$	−1.87339	−0.0611031
$941$	−6.14408	−0.200291	−0.100146	−	0.994973i	$-0.531931\pi$
$941$	−6.14408	−0.200291	−0.100146	+	0.994973i	$0.531931\pi$
$942$	64.3658	2.09715
$943$	−11.5439	−0.375920
$944$	0.871016	0.0283492
$945$	29.7068	0.966362
$946$	2.22111	0.0722144
$947$	20.9374	0.680374	0.340187	−	0.940358i	$-0.389510\pi$
$947$	20.9374	0.680374	0.340187	+	0.940358i	$0.389510\pi$
$948$	51.5632	1.67470
$949$	−12.5432	−0.407169
$950$	25.5028	0.827420
$951$	−76.6134	−2.48436
$952$	29.5037	0.956219
$953$	16.7765	0.543444	0.271722	−	0.962376i	$-0.412407\pi$
$953$	16.7765	0.543444	0.271722	+	0.962376i	$0.412407\pi$
$954$	−11.2525	−0.364314
$955$	−7.12847	−0.230672
$956$	−14.4352	−0.466869
$957$	5.08514	0.164379
$958$	−4.13590	−0.133625
$959$	−87.5470	−2.82704
$960$	−1.36327	−0.0439995
$961$	−26.9336	−0.868825
$962$	−8.75330	−0.282218
$963$	42.3657	1.36522
$964$	3.85342	0.124110
$965$	−1.69622	−0.0546032
$966$	79.9502	2.57236
$967$	41.1190	1.32230	0.661149	−	0.750254i	$-0.270069\pi$
$967$	41.1190	1.32230	0.661149	+	0.750254i	$0.270069\pi$
$968$	8.36373	0.268821
$969$	−109.261	−3.50998
$970$	1.79310	0.0575731
$971$	40.7940	1.30914	0.654570	−	0.756001i	$-0.272850\pi$
$971$	40.7940	1.30914	0.654570	+	0.756001i	$0.272850\pi$
$972$	−41.9703	−1.34620
$973$	100.527	3.22276
$974$	7.12379	0.228261
$975$	−68.7586	−2.20204
$976$	−1.02402	−0.0327782
$977$	−7.19875	−0.230308	−0.115154	−	0.993348i	$-0.536736\pi$
$977$	−7.19875	−0.230308	−0.115154	+	0.993348i	$0.536736\pi$
$978$	−34.6114	−1.10675
$979$	26.0767	0.833414
$980$	6.14866	0.196412
$981$	29.2546	0.934029
$982$	18.7335	0.597811
$983$	−20.0680	−0.640071	−0.320036	−	0.947406i	$-0.603695\pi$
$983$	−20.0680	−0.640071	−0.320036	+	0.947406i	$0.603695\pi$
$984$	7.18525	0.229057
$985$	4.88505	0.155651
$986$	6.06648	0.193196
$987$	−68.3841	−2.17669
$988$	−23.0461	−0.733194
$989$	7.18038	0.228323
$990$	−5.19919	−0.165241
$991$	−16.4798	−0.523499	−0.261749	−	0.965136i	$-0.584299\pi$
$991$	−16.4798	−0.523499	−0.261749	+	0.965136i	$0.584299\pi$
$992$	2.01654	0.0640251
$993$	−62.8181	−1.99347
$994$	20.4269	0.647902
$995$	−7.43470	−0.235696
$996$	−51.3063	−1.62570
$997$	46.5702	1.47489	0.737446	−	0.675406i	$-0.236032\pi$
$997$	46.5702	1.47489	0.737446	+	0.675406i	$0.236032\pi$
$998$	28.1730	0.891800
$999$	30.6505	0.969739

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.f.1.4	✓	89

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.f.1.4	✓	89	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.26711 q^{3} +1.00000 q^{4} +0.417272 q^{5} +3.26711 q^{6} -4.66212 q^{7} -1.00000 q^{8} +7.67400 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.26711 q^{3} +1.00000 q^{4} +0.417272 q^{5} +3.26711 q^{6} -4.66212 q^{7} -1.00000 q^{8} +7.67400 q^{9} -0.417272 q^{10} +1.62366 q^{11} -3.26711 q^{12} -4.36101 q^{13} +4.66212 q^{14} -1.36327 q^{15} +1.00000 q^{16} +6.32838 q^{17} -7.67400 q^{18} +5.28458 q^{19} +0.417272 q^{20} +15.2317 q^{21} -1.62366 q^{22} -5.24895 q^{23} +3.26711 q^{24} -4.82588 q^{25} +4.36101 q^{26} -15.2705 q^{27} -4.66212 q^{28} -0.958616 q^{29} +1.36327 q^{30} -2.01654 q^{31} -1.00000 q^{32} -5.30467 q^{33} -6.32838 q^{34} -1.94537 q^{35} +7.67400 q^{36} -2.00717 q^{37} -5.28458 q^{38} +14.2479 q^{39} -0.417272 q^{40} +2.19927 q^{41} -15.2317 q^{42} -1.36796 q^{43} +1.62366 q^{44} +3.20215 q^{45} +5.24895 q^{46} -4.48960 q^{47} -3.26711 q^{48} +14.7354 q^{49} +4.82588 q^{50} -20.6755 q^{51} -4.36101 q^{52} +1.46632 q^{53} +15.2705 q^{54} +0.677507 q^{55} +4.66212 q^{56} -17.2653 q^{57} +0.958616 q^{58} +0.871016 q^{59} -1.36327 q^{60} -1.02402 q^{61} +2.01654 q^{62} -35.7771 q^{63} +1.00000 q^{64} -1.81973 q^{65} +5.30467 q^{66} -11.0374 q^{67} +6.32838 q^{68} +17.1489 q^{69} +1.94537 q^{70} +4.38146 q^{71} -7.67400 q^{72} +2.87621 q^{73} +2.00717 q^{74} +15.7667 q^{75} +5.28458 q^{76} -7.56969 q^{77} -14.2479 q^{78} -15.7825 q^{79} +0.417272 q^{80} +26.8683 q^{81} -2.19927 q^{82} +15.7039 q^{83} +15.2317 q^{84} +2.64066 q^{85} +1.36796 q^{86} +3.13190 q^{87} -1.62366 q^{88} +16.0604 q^{89} -3.20215 q^{90} +20.3315 q^{91} -5.24895 q^{92} +6.58825 q^{93} +4.48960 q^{94} +2.20511 q^{95} +3.26711 q^{96} -4.29720 q^{97} -14.7354 q^{98} +12.4600 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9} + 18 q^{10} - 26 q^{11} - 12 q^{12} + 2 q^{13} + 27 q^{14} - 21 q^{15} + 89 q^{16} - 60 q^{17} - 95 q^{18} + q^{19} - 18 q^{20} - 6 q^{21} + 26 q^{22} - 45 q^{23} + 12 q^{24} + 107 q^{25} - 2 q^{26} - 45 q^{27} - 27 q^{28} - 18 q^{29} + 21 q^{30} - 38 q^{31} - 89 q^{32} - 29 q^{33} + 60 q^{34} - 47 q^{35} + 95 q^{36} - 15 q^{37} - q^{38} - 38 q^{39} + 18 q^{40} - 50 q^{41} + 6 q^{42} - 15 q^{43} - 26 q^{44} - 35 q^{45} + 45 q^{46} - 121 q^{47} - 12 q^{48} + 132 q^{49} - 107 q^{50} + 6 q^{51} + 2 q^{52} - 46 q^{53} + 45 q^{54} - 37 q^{55} + 27 q^{56} - 42 q^{57} + 18 q^{58} - 34 q^{59} - 21 q^{60} + 41 q^{61} + 38 q^{62} - 131 q^{63} + 89 q^{64} - 57 q^{65} + 29 q^{66} - 11 q^{67} - 60 q^{68} + 15 q^{69} + 47 q^{70} - 66 q^{71} - 95 q^{72} - 47 q^{73} + 15 q^{74} - 46 q^{75} + q^{76} - 106 q^{77} + 38 q^{78} - 51 q^{79} - 18 q^{80} + 113 q^{81} + 50 q^{82} - 141 q^{83} - 6 q^{84} - 7 q^{85} + 15 q^{86} - 110 q^{87} + 26 q^{88} - 30 q^{89} + 35 q^{90} + 37 q^{91} - 45 q^{92} - 44 q^{93} + 121 q^{94} - 98 q^{95} + 12 q^{96} + 3 q^{97} - 132 q^{98} - 71 q^{99}+O(q^{100}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9 + 18 * q^10 - 26 * q^11 - 12 * q^12 + 2 * q^13 + 27 * q^14 - 21 * q^15 + 89 * q^16 - 60 * q^17 - 95 * q^18 + q^19 - 18 * q^20 - 6 * q^21 + 26 * q^22 - 45 * q^23 + 12 * q^24 + 107 * q^25 - 2 * q^26 - 45 * q^27 - 27 * q^28 - 18 * q^29 + 21 * q^30 - 38 * q^31 - 89 * q^32 - 29 * q^33 + 60 * q^34 - 47 * q^35 + 95 * q^36 - 15 * q^37 - q^38 - 38 * q^39 + 18 * q^40 - 50 * q^41 + 6 * q^42 - 15 * q^43 - 26 * q^44 - 35 * q^45 + 45 * q^46 - 121 * q^47 - 12 * q^48 + 132 * q^49 - 107 * q^50 + 6 * q^51 + 2 * q^52 - 46 * q^53 + 45 * q^54 - 37 * q^55 + 27 * q^56 - 42 * q^57 + 18 * q^58 - 34 * q^59 - 21 * q^60 + 41 * q^61 + 38 * q^62 - 131 * q^63 + 89 * q^64 - 57 * q^65 + 29 * q^66 - 11 * q^67 - 60 * q^68 + 15 * q^69 + 47 * q^70 - 66 * q^71 - 95 * q^72 - 47 * q^73 + 15 * q^74 - 46 * q^75 + q^76 - 106 * q^77 + 38 * q^78 - 51 * q^79 - 18 * q^80 + 113 * q^81 + 50 * q^82 - 141 * q^83 - 6 * q^84 - 7 * q^85 + 15 * q^86 - 110 * q^87 + 26 * q^88 - 30 * q^89 + 35 * q^90 + 37 * q^91 - 45 * q^92 - 44 * q^93 + 121 * q^94 - 98 * q^95 + 12 * q^96 + 3 * q^97 - 132 * q^98 - 71 * q^99

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