LMFDB - Embedded newform 8002.2.a.e.1.69

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			1.69
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)

$f(q)$	$=$	$q-1.00000 q^{2} +2.85044 q^{3} +1.00000 q^{4} -0.313038 q^{5} -2.85044 q^{6} -2.28318 q^{7} -1.00000 q^{8} +5.12501 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +2.85044 q^{3} +1.00000 q^{4} -0.313038 q^{5} -2.85044 q^{6} -2.28318 q^{7} -1.00000 q^{8} +5.12501 q^{9} +0.313038 q^{10} -5.09971 q^{11} +2.85044 q^{12} +4.51575 q^{13} +2.28318 q^{14} -0.892296 q^{15} +1.00000 q^{16} -0.126296 q^{17} -5.12501 q^{18} -1.53143 q^{19} -0.313038 q^{20} -6.50806 q^{21} +5.09971 q^{22} +3.91440 q^{23} -2.85044 q^{24} -4.90201 q^{25} -4.51575 q^{26} +6.05722 q^{27} -2.28318 q^{28} -3.65214 q^{29} +0.892296 q^{30} -1.89153 q^{31} -1.00000 q^{32} -14.5364 q^{33} +0.126296 q^{34} +0.714721 q^{35} +5.12501 q^{36} +3.39984 q^{37} +1.53143 q^{38} +12.8719 q^{39} +0.313038 q^{40} +9.79714 q^{41} +6.50806 q^{42} +2.58082 q^{43} -5.09971 q^{44} -1.60432 q^{45} -3.91440 q^{46} -5.41992 q^{47} +2.85044 q^{48} -1.78710 q^{49} +4.90201 q^{50} -0.360001 q^{51} +4.51575 q^{52} +2.81476 q^{53} -6.05722 q^{54} +1.59640 q^{55} +2.28318 q^{56} -4.36525 q^{57} +3.65214 q^{58} +4.87322 q^{59} -0.892296 q^{60} -4.84695 q^{61} +1.89153 q^{62} -11.7013 q^{63} +1.00000 q^{64} -1.41360 q^{65} +14.5364 q^{66} +12.7051 q^{67} -0.126296 q^{68} +11.1578 q^{69} -0.714721 q^{70} +13.0335 q^{71} -5.12501 q^{72} +10.3581 q^{73} -3.39984 q^{74} -13.9729 q^{75} -1.53143 q^{76} +11.6436 q^{77} -12.8719 q^{78} +7.17773 q^{79} -0.313038 q^{80} +1.89070 q^{81} -9.79714 q^{82} +15.1421 q^{83} -6.50806 q^{84} +0.0395356 q^{85} -2.58082 q^{86} -10.4102 q^{87} +5.09971 q^{88} +15.3589 q^{89} +1.60432 q^{90} -10.3103 q^{91} +3.91440 q^{92} -5.39170 q^{93} +5.41992 q^{94} +0.479396 q^{95} -2.85044 q^{96} +0.468441 q^{97} +1.78710 q^{98} -26.1361 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.85044	1.64570	0.822851	−	0.568257i	$-0.192382\pi$
$3$	2.85044	1.64570	0.822851	+	0.568257i	$0.192382\pi$
$4$	1.00000	0.500000
$5$	−0.313038	−0.139995	−0.0699974	−	0.997547i	$-0.522299\pi$
$5$	−0.313038	−0.139995	−0.0699974	+	0.997547i	$0.522299\pi$
$6$	−2.85044	−1.16369
$7$	−2.28318	−0.862960	−0.431480	−	0.902122i	$-0.642008\pi$
$7$	−2.28318	−0.862960	−0.431480	+	0.902122i	$0.642008\pi$
$8$	−1.00000	−0.353553
$9$	5.12501	1.70834
$10$	0.313038	0.0989913
$11$	−5.09971	−1.53762	−0.768811	−	0.639476i	$-0.779151\pi$
$11$	−5.09971	−1.53762	−0.768811	+	0.639476i	$0.779151\pi$
$12$	2.85044	0.822851
$13$	4.51575	1.25244	0.626222	−	0.779645i	$-0.284600\pi$
$13$	4.51575	1.25244	0.626222	+	0.779645i	$0.284600\pi$
$14$	2.28318	0.610205
$15$	−0.892296	−0.230390
$16$	1.00000	0.250000
$17$	−0.126296	−0.0306314	−0.0153157	−	0.999883i	$-0.504875\pi$
$17$	−0.126296	−0.0306314	−0.0153157	+	0.999883i	$0.504875\pi$
$18$	−5.12501	−1.20798
$19$	−1.53143	−0.351334	−0.175667	−	0.984450i	$-0.556208\pi$
$19$	−1.53143	−0.351334	−0.175667	+	0.984450i	$0.556208\pi$
$20$	−0.313038	−0.0699974
$21$	−6.50806	−1.42018
$22$	5.09971	1.08726
$23$	3.91440	0.816208	0.408104	−	0.912935i	$-0.366190\pi$
$23$	3.91440	0.816208	0.408104	+	0.912935i	$0.366190\pi$
$24$	−2.85044	−0.581844
$25$	−4.90201	−0.980401
$26$	−4.51575	−0.885612
$27$	6.05722	1.16571
$28$	−2.28318	−0.431480
$29$	−3.65214	−0.678186	−0.339093	−	0.940753i	$-0.610120\pi$
$29$	−3.65214	−0.678186	−0.339093	+	0.940753i	$0.610120\pi$
$30$	0.892296	0.162910
$31$	−1.89153	−0.339729	−0.169864	−	0.985467i	$-0.554333\pi$
$31$	−1.89153	−0.339729	−0.169864	+	0.985467i	$0.554333\pi$
$32$	−1.00000	−0.176777
$33$	−14.5364	−2.53047
$34$	0.126296	0.0216597
$35$	0.714721	0.120810
$36$	5.12501	0.854168
$37$	3.39984	0.558930	0.279465	−	0.960156i	$-0.409843\pi$
$37$	3.39984	0.558930	0.279465	+	0.960156i	$0.409843\pi$
$38$	1.53143	0.248431
$39$	12.8719	2.06115
$40$	0.313038	0.0494956
$41$	9.79714	1.53006	0.765028	−	0.643997i	$-0.222725\pi$
$41$	9.79714	1.53006	0.765028	+	0.643997i	$0.222725\pi$
$42$	6.50806	1.00422
$43$	2.58082	0.393571	0.196785	−	0.980447i	$-0.436950\pi$
$43$	2.58082	0.393571	0.196785	+	0.980447i	$0.436950\pi$
$44$	−5.09971	−0.768811
$45$	−1.60432	−0.239158
$46$	−3.91440	−0.577146
$47$	−5.41992	−0.790577	−0.395289	−	0.918557i	$-0.629355\pi$
$47$	−5.41992	−0.790577	−0.395289	+	0.918557i	$0.629355\pi$
$48$	2.85044	0.411426
$49$	−1.78710	−0.255300
$50$	4.90201	0.693249
$51$	−0.360001	−0.0504102
$52$	4.51575	0.626222
$53$	2.81476	0.386637	0.193319	−	0.981136i	$-0.438075\pi$
$53$	2.81476	0.386637	0.193319	+	0.981136i	$0.438075\pi$
$54$	−6.05722	−0.824283
$55$	1.59640	0.215259
$56$	2.28318	0.305103
$57$	−4.36525	−0.578192
$58$	3.65214	0.479550
$59$	4.87322	0.634440	0.317220	−	0.948352i	$-0.397251\pi$
$59$	4.87322	0.634440	0.317220	+	0.948352i	$0.397251\pi$
$60$	−0.892296	−0.115195
$61$	−4.84695	−0.620588	−0.310294	−	0.950641i	$-0.600427\pi$
$61$	−4.84695	−0.620588	−0.310294	+	0.950641i	$0.600427\pi$
$62$	1.89153	0.240225
$63$	−11.7013	−1.47423
$64$	1.00000	0.125000
$65$	−1.41360	−0.175336
$66$	14.5364	1.78931
$67$	12.7051	1.55217	0.776087	−	0.630626i	$-0.217202\pi$
$67$	12.7051	1.55217	0.776087	+	0.630626i	$0.217202\pi$
$68$	−0.126296	−0.0153157
$69$	11.1578	1.34324
$70$	−0.714721	−0.0854255
$71$	13.0335	1.54679	0.773395	−	0.633924i	$-0.218557\pi$
$71$	13.0335	1.54679	0.773395	+	0.633924i	$0.218557\pi$
$72$	−5.12501	−0.603988
$73$	10.3581	1.21232	0.606161	−	0.795342i	$-0.292709\pi$
$73$	10.3581	1.21232	0.606161	+	0.795342i	$0.292709\pi$
$74$	−3.39984	−0.395223
$75$	−13.9729	−1.61345
$76$	−1.53143	−0.175667
$77$	11.6436	1.32691
$78$	−12.8719	−1.45745
$79$	7.17773	0.807557	0.403779	−	0.914857i	$-0.367697\pi$
$79$	7.17773	0.807557	0.403779	+	0.914857i	$0.367697\pi$
$80$	−0.313038	−0.0349987
$81$	1.89070	0.210078
$82$	−9.79714	−1.08191
$83$	15.1421	1.66206	0.831029	−	0.556229i	$-0.187752\pi$
$83$	15.1421	1.66206	0.831029	+	0.556229i	$0.187752\pi$
$84$	−6.50806	−0.710088
$85$	0.0395356	0.00428824
$86$	−2.58082	−0.278297
$87$	−10.4102	−1.11609
$88$	5.09971	0.543631
$89$	15.3589	1.62804	0.814022	−	0.580835i	$-0.197274\pi$
$89$	15.3589	1.62804	0.814022	+	0.580835i	$0.197274\pi$
$90$	1.60432	0.169110
$91$	−10.3103	−1.08081
$92$	3.91440	0.408104
$93$	−5.39170	−0.559093
$94$	5.41992	0.559023
$95$	0.479396	0.0491850
$96$	−2.85044	−0.290922
$97$	0.468441	0.0475630	0.0237815	−	0.999717i	$-0.492429\pi$
$97$	0.468441	0.0475630	0.0237815	+	0.999717i	$0.492429\pi$
$98$	1.78710	0.180524
$99$	−26.1361	−2.62678
$100$	−4.90201	−0.490201
$101$	17.2612	1.71755	0.858774	−	0.512354i	$-0.171226\pi$
$101$	17.2612	1.71755	0.858774	+	0.512354i	$0.171226\pi$
$102$	0.360001	0.0356454
$103$	9.69105	0.954888	0.477444	−	0.878662i	$-0.341563\pi$
$103$	9.69105	0.954888	0.477444	+	0.878662i	$0.341563\pi$
$104$	−4.51575	−0.442806
$105$	2.03727	0.198817
$106$	−2.81476	−0.273394
$107$	−7.09182	−0.685592	−0.342796	−	0.939410i	$-0.611374\pi$
$107$	−7.09182	−0.685592	−0.342796	+	0.939410i	$0.611374\pi$
$108$	6.05722	0.582856
$109$	9.91879	0.950048	0.475024	−	0.879973i	$-0.342439\pi$
$109$	9.91879	0.950048	0.475024	+	0.879973i	$0.342439\pi$
$110$	−1.59640	−0.152211
$111$	9.69104	0.919833
$112$	−2.28318	−0.215740
$113$	−2.93470	−0.276074	−0.138037	−	0.990427i	$-0.544079\pi$
$113$	−2.93470	−0.276074	−0.138037	+	0.990427i	$0.544079\pi$
$114$	4.36525	0.408843
$115$	−1.22535	−0.114265
$116$	−3.65214	−0.339093
$117$	23.1433	2.13960
$118$	−4.87322	−0.448617
$119$	0.288357	0.0264337
$120$	0.892296	0.0814551
$121$	15.0071	1.36428
$122$	4.84695	0.438822
$123$	27.9262	2.51802
$124$	−1.89153	−0.169864
$125$	3.09970	0.277246
$126$	11.7013	1.04244
$127$	−11.7557	−1.04315	−0.521575	−	0.853205i	$-0.674655\pi$
$127$	−11.7557	−1.04315	−0.521575	+	0.853205i	$0.674655\pi$
$128$	−1.00000	−0.0883883
$129$	7.35646	0.647701
$130$	1.41360	0.123981
$131$	20.2362	1.76805	0.884023	−	0.467444i	$-0.154825\pi$
$131$	20.2362	1.76805	0.884023	+	0.467444i	$0.154825\pi$
$132$	−14.5364	−1.26523
$133$	3.49653	0.303187
$134$	−12.7051	−1.09755
$135$	−1.89614	−0.163194
$136$	0.126296	0.0108298
$137$	−9.73430	−0.831657	−0.415829	−	0.909443i	$-0.636508\pi$
$137$	−9.73430	−0.831657	−0.415829	+	0.909443i	$0.636508\pi$
$138$	−11.1578	−0.949811
$139$	−20.3125	−1.72288	−0.861442	−	0.507857i	$-0.830438\pi$
$139$	−20.3125	−1.72288	−0.861442	+	0.507857i	$0.830438\pi$
$140$	0.714721	0.0604050
$141$	−15.4492	−1.30106
$142$	−13.0335	−1.09375
$143$	−23.0290	−1.92579
$144$	5.12501	0.427084
$145$	1.14326	0.0949425
$146$	−10.3581	−0.857241
$147$	−5.09402	−0.420147
$148$	3.39984	0.279465
$149$	−10.6168	−0.869765	−0.434882	−	0.900487i	$-0.643210\pi$
$149$	−10.6168	−0.869765	−0.434882	+	0.900487i	$0.643210\pi$
$150$	13.9729	1.14088
$151$	−16.3462	−1.33024	−0.665118	−	0.746739i	$-0.731619\pi$
$151$	−16.3462	−1.33024	−0.665118	+	0.746739i	$0.731619\pi$
$152$	1.53143	0.124215
$153$	−0.647271	−0.0523287
$154$	−11.6436	−0.938264
$155$	0.592121	0.0475603
$156$	12.8719	1.03058
$157$	−17.6267	−1.40676	−0.703380	−	0.710814i	$-0.748327\pi$
$157$	−17.6267	−1.40676	−0.703380	+	0.710814i	$0.748327\pi$
$158$	−7.17773	−0.571029
$159$	8.02331	0.636289
$160$	0.313038	0.0247478
$161$	−8.93726	−0.704355
$162$	−1.89070	−0.148547
$163$	18.3326	1.43592	0.717959	−	0.696085i	$-0.245076\pi$
$163$	18.3326	1.43592	0.717959	+	0.696085i	$0.245076\pi$
$164$	9.79714	0.765028
$165$	4.55045	0.354252
$166$	−15.1421	−1.17525
$167$	16.3929	1.26852	0.634262	−	0.773118i	$-0.281304\pi$
$167$	16.3929	1.26852	0.634262	+	0.773118i	$0.281304\pi$
$168$	6.50806	0.502108
$169$	7.39201	0.568616
$170$	−0.0395356	−0.00303224
$171$	−7.84860	−0.600197
$172$	2.58082	0.196785
$173$	9.30822	0.707691	0.353846	−	0.935304i	$-0.384874\pi$
$173$	9.30822	0.707691	0.353846	+	0.935304i	$0.384874\pi$
$174$	10.4102	0.789196
$175$	11.1922	0.846047
$176$	−5.09971	−0.384405
$177$	13.8908	1.04410
$178$	−15.3589	−1.15120
$179$	4.65786	0.348145	0.174073	−	0.984733i	$-0.444307\pi$
$179$	4.65786	0.348145	0.174073	+	0.984733i	$0.444307\pi$
$180$	−1.60432	−0.119579
$181$	−23.5063	−1.74721	−0.873606	−	0.486634i	$-0.838224\pi$
$181$	−23.5063	−1.74721	−0.873606	+	0.486634i	$0.838224\pi$
$182$	10.3103	0.764248
$183$	−13.8159	−1.02130
$184$	−3.91440	−0.288573
$185$	−1.06428	−0.0782473
$186$	5.39170	0.395338
$187$	0.644076	0.0470995
$188$	−5.41992	−0.395289
$189$	−13.8297	−1.00596
$190$	−0.479396	−0.0347790
$191$	8.38597	0.606788	0.303394	−	0.952865i	$-0.401880\pi$
$191$	8.38597	0.606788	0.303394	+	0.952865i	$0.401880\pi$
$192$	2.85044	0.205713
$193$	3.91843	0.282055	0.141028	−	0.990006i	$-0.454959\pi$
$193$	3.91843	0.282055	0.141028	+	0.990006i	$0.454959\pi$
$194$	−0.468441	−0.0336321
$195$	−4.02939	−0.288550
$196$	−1.78710	−0.127650
$197$	−19.7703	−1.40858	−0.704288	−	0.709914i	$-0.748734\pi$
$197$	−19.7703	−1.40858	−0.704288	+	0.709914i	$0.748734\pi$
$198$	26.1361	1.85741
$199$	9.79565	0.694395	0.347198	−	0.937792i	$-0.387133\pi$
$199$	9.79565	0.694395	0.347198	+	0.937792i	$0.387133\pi$
$200$	4.90201	0.346624
$201$	36.2151	2.55442
$202$	−17.2612	−1.21449
$203$	8.33849	0.585247
$204$	−0.360001	−0.0252051
$205$	−3.06688	−0.214200
$206$	−9.69105	−0.675208
$207$	20.0613	1.39436
$208$	4.51575	0.313111
$209$	7.80985	0.540219
$210$	−2.03727	−0.140585
$211$	−0.528876	−0.0364093	−0.0182047	−	0.999834i	$-0.505795\pi$
$211$	−0.528876	−0.0364093	−0.0182047	+	0.999834i	$0.505795\pi$
$212$	2.81476	0.193319
$213$	37.1512	2.54556
$214$	7.09182	0.484787
$215$	−0.807893	−0.0550979
$216$	−6.05722	−0.412141
$217$	4.31870	0.293173
$218$	−9.91879	−0.671785
$219$	29.5251	1.99512
$220$	1.59640	0.107630
$221$	−0.570324	−0.0383641
$222$	−9.69104	−0.650420
$223$	−9.28276	−0.621620	−0.310810	−	0.950472i	$-0.600600\pi$
$223$	−9.28276	−0.621620	−0.310810	+	0.950472i	$0.600600\pi$
$224$	2.28318	0.152551
$225$	−25.1228	−1.67486
$226$	2.93470	0.195214
$227$	−7.14966	−0.474540	−0.237270	−	0.971444i	$-0.576253\pi$
$227$	−7.14966	−0.474540	−0.237270	+	0.971444i	$0.576253\pi$
$228$	−4.36525	−0.289096
$229$	6.52394	0.431114	0.215557	−	0.976491i	$-0.430843\pi$
$229$	6.52394	0.431114	0.215557	+	0.976491i	$0.430843\pi$
$230$	1.22535	0.0807975
$231$	33.1893	2.18369
$232$	3.65214	0.239775
$233$	18.2621	1.19639	0.598196	−	0.801350i	$-0.295884\pi$
$233$	18.2621	1.19639	0.598196	+	0.801350i	$0.295884\pi$
$234$	−23.1433	−1.51292
$235$	1.69664	0.110677
$236$	4.87322	0.317220
$237$	20.4597	1.32900
$238$	−0.288357	−0.0186914
$239$	19.0323	1.23110	0.615549	−	0.788098i	$-0.288934\pi$
$239$	19.0323	1.23110	0.615549	+	0.788098i	$0.288934\pi$
$240$	−0.892296	−0.0575975
$241$	−29.4304	−1.89578	−0.947890	−	0.318598i	$-0.896788\pi$
$241$	−29.4304	−1.89578	−0.947890	+	0.318598i	$0.896788\pi$
$242$	−15.0071	−0.964691
$243$	−12.7823	−0.819986
$244$	−4.84695	−0.310294
$245$	0.559429	0.0357406
$246$	−27.9262	−1.78051
$247$	−6.91556	−0.440026
$248$	1.89153	0.120112
$249$	43.1616	2.73525
$250$	−3.09970	−0.196042
$251$	0.246912	0.0155850	0.00779248	−	0.999970i	$-0.497520\pi$
$251$	0.246912	0.0155850	0.00779248	+	0.999970i	$0.497520\pi$
$252$	−11.7013	−0.737113
$253$	−19.9623	−1.25502
$254$	11.7557	0.737618
$255$	0.112694	0.00705716
$256$	1.00000	0.0625000
$257$	−23.2977	−1.45327	−0.726634	−	0.687024i	$-0.758917\pi$
$257$	−23.2977	−1.45327	−0.726634	+	0.687024i	$0.758917\pi$
$258$	−7.35646	−0.457993
$259$	−7.76244	−0.482334
$260$	−1.41360	−0.0876678
$261$	−18.7173	−1.15857
$262$	−20.2362	−1.25020
$263$	−22.5527	−1.39066	−0.695329	−	0.718692i	$-0.744741\pi$
$263$	−22.5527	−1.39066	−0.695329	+	0.718692i	$0.744741\pi$
$264$	14.5364	0.894655
$265$	−0.881127	−0.0541272
$266$	−3.49653	−0.214386
$267$	43.7797	2.67928
$268$	12.7051	0.776087
$269$	8.40284	0.512330	0.256165	−	0.966633i	$-0.417541\pi$
$269$	8.40284	0.512330	0.256165	+	0.966633i	$0.417541\pi$
$270$	1.89614	0.115395
$271$	7.05797	0.428741	0.214370	−	0.976752i	$-0.431230\pi$
$271$	7.05797	0.428741	0.214370	+	0.976752i	$0.431230\pi$
$272$	−0.126296	−0.00765785
$273$	−29.3888	−1.77869
$274$	9.73430	0.588070
$275$	24.9988	1.50749
$276$	11.1578	0.671618
$277$	20.2293	1.21546	0.607730	−	0.794144i	$-0.292080\pi$
$277$	20.2293	1.21546	0.607730	+	0.794144i	$0.292080\pi$
$278$	20.3125	1.21826
$279$	−9.69411	−0.580371
$280$	−0.714721	−0.0427128
$281$	4.89710	0.292137	0.146068	−	0.989275i	$-0.453338\pi$
$281$	4.89710	0.292137	0.146068	+	0.989275i	$0.453338\pi$
$282$	15.4492	0.919985
$283$	−4.95342	−0.294451	−0.147225	−	0.989103i	$-0.547034\pi$
$283$	−4.95342	−0.294451	−0.147225	+	0.989103i	$0.547034\pi$
$284$	13.0335	0.773395
$285$	1.36649	0.0809438
$286$	23.0290	1.36174
$287$	−22.3686	−1.32038
$288$	−5.12501	−0.301994
$289$	−16.9840	−0.999062
$290$	−1.14326	−0.0671345
$291$	1.33526	0.0782746
$292$	10.3581	0.606161
$293$	24.0835	1.40698	0.703488	−	0.710707i	$-0.251625\pi$
$293$	24.0835	1.40698	0.703488	+	0.710707i	$0.251625\pi$
$294$	5.09402	0.297089
$295$	−1.52550	−0.0888183
$296$	−3.39984	−0.197612
$297$	−30.8901	−1.79242
$298$	10.6168	0.615017
$299$	17.6764	1.02225
$300$	−13.9729	−0.806725
$301$	−5.89246	−0.339636
$302$	16.3462	0.940618
$303$	49.2019	2.82657
$304$	−1.53143	−0.0878335
$305$	1.51728	0.0868791
$306$	0.647271	0.0370020
$307$	18.5857	1.06074	0.530371	−	0.847766i	$-0.322053\pi$
$307$	18.5857	1.06074	0.530371	+	0.847766i	$0.322053\pi$
$308$	11.6436	0.663453
$309$	27.6238	1.57146
$310$	−0.592121	−0.0336302
$311$	1.49213	0.0846107	0.0423054	−	0.999105i	$-0.486530\pi$
$311$	1.49213	0.0846107	0.0423054	+	0.999105i	$0.486530\pi$
$312$	−12.8719	−0.728727
$313$	−21.4703	−1.21358	−0.606788	−	0.794864i	$-0.707542\pi$
$313$	−21.4703	−1.21358	−0.606788	+	0.794864i	$0.707542\pi$
$314$	17.6267	0.994730
$315$	3.66295	0.206384
$316$	7.17773	0.403779
$317$	0.219981	0.0123554	0.00617769	−	0.999981i	$-0.498034\pi$
$317$	0.219981	0.0123554	0.00617769	+	0.999981i	$0.498034\pi$
$318$	−8.02331	−0.449925
$319$	18.6249	1.04279
$320$	−0.313038	−0.0174994
$321$	−20.2148	−1.12828
$322$	8.93726	0.498054
$323$	0.193414	0.0107619
$324$	1.89070	0.105039
$325$	−22.1362	−1.22790
$326$	−18.3326	−1.01535
$327$	28.2729	1.56350
$328$	−9.79714	−0.540957
$329$	12.3747	0.682237
$330$	−4.55045	−0.250494
$331$	33.6739	1.85089	0.925443	−	0.378886i	$-0.123693\pi$
$331$	33.6739	1.85089	0.925443	+	0.378886i	$0.123693\pi$
$332$	15.1421	0.831029
$333$	17.4242	0.954841
$334$	−16.3929	−0.896982
$335$	−3.97718	−0.217296
$336$	−6.50806	−0.355044
$337$	8.05332	0.438692	0.219346	−	0.975647i	$-0.429608\pi$
$337$	8.05332	0.438692	0.219346	+	0.975647i	$0.429608\pi$
$338$	−7.39201	−0.402073
$339$	−8.36520	−0.454335
$340$	0.0395356	0.00214412
$341$	9.64626	0.522375
$342$	7.84860	0.424403
$343$	20.0625	1.08327
$344$	−2.58082	−0.139148
$345$	−3.49280	−0.188046
$346$	−9.30822	−0.500413
$347$	16.1313	0.865975	0.432987	−	0.901400i	$-0.357459\pi$
$347$	16.1313	0.865975	0.432987	+	0.901400i	$0.357459\pi$
$348$	−10.4102	−0.558046
$349$	27.8134	1.48882	0.744408	−	0.667725i	$-0.232732\pi$
$349$	27.8134	1.48882	0.744408	+	0.667725i	$0.232732\pi$
$350$	−11.1922	−0.598246
$351$	27.3529	1.45999
$352$	5.09971	0.271816
$353$	−21.9528	−1.16843	−0.584214	−	0.811599i	$-0.698597\pi$
$353$	−21.9528	−1.16843	−0.584214	+	0.811599i	$0.698597\pi$
$354$	−13.8908	−0.738289
$355$	−4.07998	−0.216543
$356$	15.3589	0.814022
$357$	0.821945	0.0435020
$358$	−4.65786	−0.246176
$359$	16.6779	0.880228	0.440114	−	0.897942i	$-0.354938\pi$
$359$	16.6779	0.880228	0.440114	+	0.897942i	$0.354938\pi$
$360$	1.60432	0.0845552
$361$	−16.6547	−0.876564
$362$	23.5063	1.23547
$363$	42.7768	2.24520
$364$	−10.3103	−0.540405
$365$	−3.24247	−0.169719
$366$	13.8159	0.722170
$367$	−3.50595	−0.183009	−0.0915046	−	0.995805i	$-0.529168\pi$
$367$	−3.50595	−0.183009	−0.0915046	+	0.995805i	$0.529168\pi$
$368$	3.91440	0.204052
$369$	50.2105	2.61385
$370$	1.06428	0.0553292
$371$	−6.42660	−0.333652
$372$	−5.39170	−0.279546
$373$	−25.6247	−1.32679	−0.663397	−	0.748268i	$-0.730886\pi$
$373$	−25.6247	−1.32679	−0.663397	+	0.748268i	$0.730886\pi$
$374$	−0.644076	−0.0333044
$375$	8.83552	0.456264
$376$	5.41992	0.279511
$377$	−16.4922	−0.849390
$378$	13.8297	0.711323
$379$	−3.14771	−0.161687	−0.0808434	−	0.996727i	$-0.525761\pi$
$379$	−3.14771	−0.161687	−0.0808434	+	0.996727i	$0.525761\pi$
$380$	0.479396	0.0245925
$381$	−33.5089	−1.71671
$382$	−8.38597	−0.429064
$383$	−12.0515	−0.615804	−0.307902	−	0.951418i	$-0.599627\pi$
$383$	−12.0515	−0.615804	−0.307902	+	0.951418i	$0.599627\pi$
$384$	−2.85044	−0.145461
$385$	−3.64487	−0.185760
$386$	−3.91843	−0.199443
$387$	13.2267	0.672352
$388$	0.468441	0.0237815
$389$	−17.0974	−0.866872	−0.433436	−	0.901184i	$-0.642699\pi$
$389$	−17.0974	−0.866872	−0.433436	+	0.901184i	$0.642699\pi$
$390$	4.02939	0.204036
$391$	−0.494374	−0.0250016
$392$	1.78710	0.0902621
$393$	57.6821	2.90968
$394$	19.7703	0.996014
$395$	−2.24690	−0.113054
$396$	−26.1361	−1.31339
$397$	−22.4612	−1.12729	−0.563647	−	0.826016i	$-0.690602\pi$
$397$	−22.4612	−1.12729	−0.563647	+	0.826016i	$0.690602\pi$
$398$	−9.79565	−0.491012
$399$	9.96664	0.498956
$400$	−4.90201	−0.245100
$401$	5.74712	0.286997	0.143499	−	0.989650i	$-0.454165\pi$
$401$	5.74712	0.286997	0.143499	+	0.989650i	$0.454165\pi$
$402$	−36.2151	−1.80625
$403$	−8.54168	−0.425492
$404$	17.2612	0.858774
$405$	−0.591861	−0.0294098
$406$	−8.33849	−0.413832
$407$	−17.3382	−0.859423
$408$	0.360001	0.0178227
$409$	−2.50970	−0.124097	−0.0620483	−	0.998073i	$-0.519763\pi$
$409$	−2.50970	−0.124097	−0.0620483	+	0.998073i	$0.519763\pi$
$410$	3.06688	0.151462
$411$	−27.7470	−1.36866
$412$	9.69105	0.477444
$413$	−11.1264	−0.547496
$414$	−20.0613	−0.985960
$415$	−4.74004	−0.232679
$416$	−4.51575	−0.221403
$417$	−57.8996	−2.83535
$418$	−7.80985	−0.381992
$419$	15.1892	0.742039	0.371020	−	0.928625i	$-0.379008\pi$
$419$	15.1892	0.742039	0.371020	+	0.928625i	$0.379008\pi$
$420$	2.03727	0.0994086
$421$	−18.0094	−0.877727	−0.438863	−	0.898554i	$-0.644619\pi$
$421$	−18.0094	−0.877727	−0.438863	+	0.898554i	$0.644619\pi$
$422$	0.528876	0.0257453
$423$	−27.7772	−1.35057
$424$	−2.81476	−0.136697
$425$	0.619106	0.0300311
$426$	−37.1512	−1.79998
$427$	11.0664	0.535543
$428$	−7.09182	−0.342796
$429$	−65.6429	−3.16927
$430$	0.807893	0.0389601
$431$	34.4904	1.66134	0.830672	−	0.556762i	$-0.187956\pi$
$431$	34.4904	1.66134	0.830672	+	0.556762i	$0.187956\pi$
$432$	6.05722	0.291428
$433$	26.3045	1.26412	0.632058	−	0.774921i	$-0.282211\pi$
$433$	26.3045	1.26412	0.632058	+	0.774921i	$0.282211\pi$
$434$	−4.31870	−0.207304
$435$	3.25879	0.156247
$436$	9.91879	0.475024
$437$	−5.99462	−0.286762
$438$	−29.5251	−1.41076
$439$	28.0203	1.33734	0.668668	−	0.743561i	$-0.266865\pi$
$439$	28.0203	1.33734	0.668668	+	0.743561i	$0.266865\pi$
$440$	−1.59640	−0.0761056
$441$	−9.15890	−0.436138
$442$	0.570324	0.0271275
$443$	21.6295	1.02765	0.513824	−	0.857896i	$-0.328228\pi$
$443$	21.6295	1.02765	0.513824	+	0.857896i	$0.328228\pi$
$444$	9.69104	0.459916
$445$	−4.80793	−0.227918
$446$	9.28276	0.439552
$447$	−30.2627	−1.43137
$448$	−2.28318	−0.107870
$449$	−36.1825	−1.70756	−0.853778	−	0.520637i	$-0.825695\pi$
$449$	−36.1825	−1.70756	−0.853778	+	0.520637i	$0.825695\pi$
$450$	25.1228	1.18430
$451$	−49.9626	−2.35265
$452$	−2.93470	−0.138037
$453$	−46.5939	−2.18917
$454$	7.14966	0.335550
$455$	3.22750	0.151308
$456$	4.36525	0.204422
$457$	27.9605	1.30794	0.653968	−	0.756522i	$-0.273103\pi$
$457$	27.9605	1.30794	0.653968	+	0.756522i	$0.273103\pi$
$458$	−6.52394	−0.304843
$459$	−0.765005	−0.0357074
$460$	−1.22535	−0.0571324
$461$	11.6428	0.542259	0.271130	−	0.962543i	$-0.412603\pi$
$461$	11.6428	0.542259	0.271130	+	0.962543i	$0.412603\pi$
$462$	−33.1893	−1.54410
$463$	16.3305	0.758941	0.379470	−	0.925204i	$-0.376106\pi$
$463$	16.3305	0.758941	0.379470	+	0.925204i	$0.376106\pi$
$464$	−3.65214	−0.169546
$465$	1.68781	0.0782701
$466$	−18.2621	−0.845977
$467$	22.0980	1.02257	0.511287	−	0.859410i	$-0.329169\pi$
$467$	22.0980	1.02257	0.511287	+	0.859410i	$0.329169\pi$
$468$	23.1433	1.06980
$469$	−29.0080	−1.33946
$470$	−1.69664	−0.0782603
$471$	−50.2437	−2.31511
$472$	−4.87322	−0.224308
$473$	−13.1614	−0.605163
$474$	−20.4597	−0.939744
$475$	7.50708	0.344449
$476$	0.288357	0.0132168
$477$	14.4257	0.660506
$478$	−19.0323	−0.870518
$479$	36.5487	1.66995	0.834977	−	0.550284i	$-0.185481\pi$
$479$	36.5487	1.66995	0.834977	+	0.550284i	$0.185481\pi$
$480$	0.892296	0.0407275
$481$	15.3528	0.700029
$482$	29.4304	1.34052
$483$	−25.4751	−1.15916
$484$	15.0071	0.682140
$485$	−0.146640	−0.00665858
$486$	12.7823	0.579818
$487$	24.1704	1.09527	0.547633	−	0.836719i	$-0.315529\pi$
$487$	24.1704	1.09527	0.547633	+	0.836719i	$0.315529\pi$
$488$	4.84695	0.219411
$489$	52.2559	2.36309
$490$	−0.559429	−0.0252724
$491$	−29.9975	−1.35377	−0.676884	−	0.736090i	$-0.736670\pi$
$491$	−29.9975	−1.35377	−0.676884	+	0.736090i	$0.736670\pi$
$492$	27.9262	1.25901
$493$	0.461253	0.0207738
$494$	6.91556	0.311146
$495$	8.18158	0.367735
$496$	−1.89153	−0.0849322
$497$	−29.7578	−1.33482
$498$	−43.1616	−1.93412
$499$	34.6845	1.55269	0.776345	−	0.630308i	$-0.217072\pi$
$499$	34.6845	1.55269	0.776345	+	0.630308i	$0.217072\pi$
$500$	3.09970	0.138623
$501$	46.7271	2.08761
$502$	−0.246912	−0.0110202
$503$	−10.5901	−0.472191	−0.236095	−	0.971730i	$-0.575868\pi$
$503$	−10.5901	−0.472191	−0.236095	+	0.971730i	$0.575868\pi$
$504$	11.7013	0.521218
$505$	−5.40339	−0.240448
$506$	19.9623	0.887432
$507$	21.0705	0.935774
$508$	−11.7557	−0.521575
$509$	2.43681	0.108010	0.0540048	−	0.998541i	$-0.482801\pi$
$509$	2.43681	0.108010	0.0540048	+	0.998541i	$0.482801\pi$
$510$	−0.112694	−0.00499017
$511$	−23.6494	−1.04619
$512$	−1.00000	−0.0441942
$513$	−9.27620	−0.409554
$514$	23.2977	1.02762
$515$	−3.03367	−0.133679
$516$	7.35646	0.323850
$517$	27.6401	1.21561
$518$	7.76244	0.341062
$519$	26.5325	1.16465
$520$	1.41360	0.0619905
$521$	−31.3164	−1.37199	−0.685997	−	0.727604i	$-0.740634\pi$
$521$	−31.3164	−1.37199	−0.685997	+	0.727604i	$0.740634\pi$
$522$	18.7173	0.819232
$523$	0.0933031	0.00407986	0.00203993	−	0.999998i	$-0.499351\pi$
$523$	0.0933031	0.00407986	0.00203993	+	0.999998i	$0.499351\pi$
$524$	20.2362	0.884023
$525$	31.9026	1.39234
$526$	22.5527	0.983344
$527$	0.238894	0.0104064
$528$	−14.5364	−0.632617
$529$	−7.67751	−0.333805
$530$	0.881127	0.0382737
$531$	24.9753	1.08384
$532$	3.49653	0.151594
$533$	44.2415	1.91631
$534$	−43.7797	−1.89453
$535$	2.22001	0.0959793
$536$	−12.7051	−0.548777
$537$	13.2770	0.572943
$538$	−8.40284	−0.362272
$539$	9.11369	0.392554
$540$	−1.89614	−0.0815968
$541$	−29.6014	−1.27266	−0.636331	−	0.771416i	$-0.719549\pi$
$541$	−29.6014	−1.27266	−0.636331	+	0.771416i	$0.719549\pi$
$542$	−7.05797	−0.303166
$543$	−67.0034	−2.87539
$544$	0.126296	0.00541492
$545$	−3.10496	−0.133002
$546$	29.3888	1.25772
$547$	−34.5062	−1.47538	−0.737689	−	0.675140i	$-0.764083\pi$
$547$	−34.5062	−1.47538	−0.737689	+	0.675140i	$0.764083\pi$
$548$	−9.73430	−0.415829
$549$	−24.8407	−1.06017
$550$	−24.9988	−1.06595
$551$	5.59300	0.238270
$552$	−11.1578	−0.474905
$553$	−16.3880	−0.696890
$554$	−20.2293	−0.859460
$555$	−3.03366	−0.128772
$556$	−20.3125	−0.861442
$557$	4.91952	0.208447	0.104223	−	0.994554i	$-0.466764\pi$
$557$	4.91952	0.208447	0.104223	+	0.994554i	$0.466764\pi$
$558$	9.69411	0.410385
$559$	11.6543	0.492926
$560$	0.714721	0.0302025
$561$	1.83590	0.0775117
$562$	−4.89710	−0.206572
$563$	1.47055	0.0619763	0.0309881	−	0.999520i	$-0.490135\pi$
$563$	1.47055	0.0619763	0.0309881	+	0.999520i	$0.490135\pi$
$564$	−15.4492	−0.650528
$565$	0.918674	0.0386489
$566$	4.95342	0.208208
$567$	−4.31681	−0.181289
$568$	−13.0335	−0.546873
$569$	2.90985	0.121987	0.0609937	−	0.998138i	$-0.480573\pi$
$569$	2.90985	0.121987	0.0609937	+	0.998138i	$0.480573\pi$
$570$	−1.36649	−0.0572359
$571$	14.5735	0.609883	0.304942	−	0.952371i	$-0.401363\pi$
$571$	14.5735	0.609883	0.304942	+	0.952371i	$0.401363\pi$
$572$	−23.0290	−0.962893
$573$	23.9037	0.998592
$574$	22.3686	0.933648
$575$	−19.1884	−0.800211
$576$	5.12501	0.213542
$577$	−34.6853	−1.44397	−0.721983	−	0.691911i	$-0.756769\pi$
$577$	−34.6853	−1.44397	−0.721983	+	0.691911i	$0.756769\pi$
$578$	16.9840	0.706443
$579$	11.1693	0.464179
$580$	1.14326	0.0474712
$581$	−34.5720	−1.43429
$582$	−1.33526	−0.0553485
$583$	−14.3545	−0.594501
$584$	−10.3581	−0.428621
$585$	−7.24472	−0.299532
$586$	−24.0835	−0.994882
$587$	−24.6833	−1.01879	−0.509395	−	0.860533i	$-0.670131\pi$
$587$	−24.6833	−1.01879	−0.509395	+	0.860533i	$0.670131\pi$
$588$	−5.09402	−0.210074
$589$	2.89675	0.119358
$590$	1.52550	0.0628040
$591$	−56.3541	−2.31810
$592$	3.39984	0.139733
$593$	34.4904	1.41635	0.708175	−	0.706037i	$-0.249519\pi$
$593$	34.4904	1.41635	0.708175	+	0.706037i	$0.249519\pi$
$594$	30.8901	1.26743
$595$	−0.0902668	−0.00370058
$596$	−10.6168	−0.434882
$597$	27.9219	1.14277
$598$	−17.6764	−0.722843
$599$	45.6328	1.86451	0.932253	−	0.361806i	$-0.117840\pi$
$599$	45.6328	1.86451	0.932253	+	0.361806i	$0.117840\pi$
$600$	13.9729	0.570440
$601$	−1.34248	−0.0547608	−0.0273804	−	0.999625i	$-0.508717\pi$
$601$	−1.34248	−0.0547608	−0.0273804	+	0.999625i	$0.508717\pi$
$602$	5.89246	0.240159
$603$	65.1137	2.65164
$604$	−16.3462	−0.665118
$605$	−4.69778	−0.190992
$606$	−49.2019	−1.99869
$607$	16.2039	0.657697	0.328848	−	0.944383i	$-0.393339\pi$
$607$	16.2039	0.657697	0.328848	+	0.944383i	$0.393339\pi$
$608$	1.53143	0.0621077
$609$	23.7684	0.963143
$610$	−1.51728	−0.0614328
$611$	−24.4750	−0.990154
$612$	−0.647271	−0.0261644
$613$	8.82589	0.356474	0.178237	−	0.983988i	$-0.442961\pi$
$613$	8.82589	0.356474	0.178237	+	0.983988i	$0.442961\pi$
$614$	−18.5857	−0.750058
$615$	−8.74195	−0.352509
$616$	−11.6436	−0.469132
$617$	−42.6776	−1.71813	−0.859067	−	0.511862i	$-0.828956\pi$
$617$	−42.6776	−1.71813	−0.859067	+	0.511862i	$0.828956\pi$
$618$	−27.6238	−1.11119
$619$	−41.4758	−1.66705	−0.833526	−	0.552481i	$-0.813681\pi$
$619$	−41.4758	−1.66705	−0.833526	+	0.552481i	$0.813681\pi$
$620$	0.592121	0.0237801
$621$	23.7103	0.951463
$622$	−1.49213	−0.0598288
$623$	−35.0672	−1.40494
$624$	12.8719	0.515288
$625$	23.5397	0.941588
$626$	21.4703	0.858127
$627$	22.2615	0.889040
$628$	−17.6267	−0.703380
$629$	−0.429388	−0.0171208
$630$	−3.66295	−0.145936
$631$	26.0385	1.03657	0.518287	−	0.855206i	$-0.326570\pi$
$631$	26.0385	1.03657	0.518287	+	0.855206i	$0.326570\pi$
$632$	−7.17773	−0.285515
$633$	−1.50753	−0.0599190
$634$	−0.219981	−0.00873657
$635$	3.67998	0.146036
$636$	8.02331	0.318145
$637$	−8.07009	−0.319749
$638$	−18.6249	−0.737366
$639$	66.7968	2.64244
$640$	0.313038	0.0123739
$641$	15.4863	0.611672	0.305836	−	0.952084i	$-0.401064\pi$
$641$	15.4863	0.611672	0.305836	+	0.952084i	$0.401064\pi$
$642$	20.2148	0.797815
$643$	−35.6985	−1.40781	−0.703905	−	0.710294i	$-0.748562\pi$
$643$	−35.6985	−1.40781	−0.703905	+	0.710294i	$0.748562\pi$
$644$	−8.93726	−0.352177
$645$	−2.30285	−0.0906747
$646$	−0.193414	−0.00760978
$647$	−17.3586	−0.682438	−0.341219	−	0.939984i	$-0.610840\pi$
$647$	−17.3586	−0.682438	−0.341219	+	0.939984i	$0.610840\pi$
$648$	−1.89070	−0.0742737
$649$	−24.8520	−0.975528
$650$	22.1362	0.868255
$651$	12.3102	0.482475
$652$	18.3326	0.717959
$653$	8.27603	0.323866	0.161933	−	0.986802i	$-0.448227\pi$
$653$	8.27603	0.323866	0.161933	+	0.986802i	$0.448227\pi$
$654$	−28.2729	−1.10556
$655$	−6.33470	−0.247517
$656$	9.79714	0.382514
$657$	53.0853	2.07105
$658$	−12.3747	−0.482414
$659$	35.5468	1.38471	0.692354	−	0.721558i	$-0.256574\pi$
$659$	35.5468	1.38471	0.692354	+	0.721558i	$0.256574\pi$
$660$	4.55045	0.177126
$661$	48.1785	1.87393	0.936963	−	0.349429i	$-0.113624\pi$
$661$	48.1785	1.87393	0.936963	+	0.349429i	$0.113624\pi$
$662$	−33.6739	−1.30877
$663$	−1.62567	−0.0631359
$664$	−15.1421	−0.587626
$665$	−1.09455	−0.0424447
$666$	−17.4242	−0.675174
$667$	−14.2959	−0.553540
$668$	16.3929	0.634262
$669$	−26.4600	−1.02300
$670$	3.97718	0.153652
$671$	24.7180	0.954229
$672$	6.50806	0.251054
$673$	−24.7850	−0.955392	−0.477696	−	0.878525i	$-0.658528\pi$
$673$	−24.7850	−0.955392	−0.477696	+	0.878525i	$0.658528\pi$
$674$	−8.05332	−0.310202
$675$	−29.6925	−1.14287
$676$	7.39201	0.284308
$677$	−5.54626	−0.213160	−0.106580	−	0.994304i	$-0.533990\pi$
$677$	−5.54626	−0.213160	−0.106580	+	0.994304i	$0.533990\pi$
$678$	8.36520	0.321264
$679$	−1.06954	−0.0410450
$680$	−0.0395356	−0.00151612
$681$	−20.3797	−0.780951
$682$	−9.64626	−0.369375
$683$	−26.1287	−0.999787	−0.499893	−	0.866087i	$-0.666627\pi$
$683$	−26.1287	−0.999787	−0.499893	+	0.866087i	$0.666627\pi$
$684$	−7.84860	−0.300099
$685$	3.04720	0.116428
$686$	−20.0625	−0.765990
$687$	18.5961	0.709485
$688$	2.58082	0.0983927
$689$	12.7108	0.484241
$690$	3.49280	0.132969
$691$	14.0736	0.535386	0.267693	−	0.963504i	$-0.413739\pi$
$691$	14.0736	0.535386	0.267693	+	0.963504i	$0.413739\pi$
$692$	9.30822	0.353846
$693$	59.6733	2.26680
$694$	−16.1313	−0.612337
$695$	6.35858	0.241195
$696$	10.4102	0.394598
$697$	−1.23734	−0.0468678
$698$	−27.8134	−1.05275
$699$	52.0551	1.96891
$700$	11.1922	0.423024
$701$	−8.52882	−0.322129	−0.161064	−	0.986944i	$-0.551493\pi$
$701$	−8.52882	−0.322129	−0.161064	+	0.986944i	$0.551493\pi$
$702$	−27.3529	−1.03237
$703$	−5.20662	−0.196371
$704$	−5.09971	−0.192203
$705$	4.83618	0.182141
$706$	21.9528	0.826204
$707$	−39.4103	−1.48218
$708$	13.8908	0.522049
$709$	26.6170	0.999623	0.499811	−	0.866134i	$-0.333403\pi$
$709$	26.6170	0.999623	0.499811	+	0.866134i	$0.333403\pi$
$710$	4.07998	0.153119
$711$	36.7859	1.37958
$712$	−15.3589	−0.575600
$713$	−7.40420	−0.277289
$714$	−0.821945	−0.0307605
$715$	7.20896	0.269600
$716$	4.65786	0.174073
$717$	54.2505	2.02602
$718$	−16.6779	−0.622415
$719$	−34.5514	−1.28855	−0.644275	−	0.764794i	$-0.722841\pi$
$719$	−34.5514	−1.28855	−0.644275	+	0.764794i	$0.722841\pi$
$720$	−1.60432	−0.0597896
$721$	−22.1264	−0.824030
$722$	16.6547	0.619825
$723$	−83.8896	−3.11989
$724$	−23.5063	−0.873606
$725$	17.9028	0.664894
$726$	−42.7768	−1.58759
$727$	9.78483	0.362899	0.181450	−	0.983400i	$-0.441921\pi$
$727$	9.78483	0.362899	0.181450	+	0.983400i	$0.441921\pi$
$728$	10.3103	0.382124
$729$	−42.1073	−1.55953
$730$	3.24247	0.120009
$731$	−0.325948	−0.0120556
$732$	−13.8159	−0.510652
$733$	−11.4147	−0.421611	−0.210806	−	0.977528i	$-0.567609\pi$
$733$	−11.4147	−0.421611	−0.210806	+	0.977528i	$0.567609\pi$
$734$	3.50595	0.129407
$735$	1.59462	0.0588184
$736$	−3.91440	−0.144287
$737$	−64.7923	−2.38666
$738$	−50.2105	−1.84827
$739$	−21.5087	−0.791211	−0.395606	−	0.918420i	$-0.629465\pi$
$739$	−21.5087	−0.791211	−0.395606	+	0.918420i	$0.629465\pi$
$740$	−1.06428	−0.0391237
$741$	−19.7124	−0.724153
$742$	6.42660	0.235928
$743$	−15.7827	−0.579013	−0.289506	−	0.957176i	$-0.593491\pi$
$743$	−15.7827	−0.579013	−0.289506	+	0.957176i	$0.593491\pi$
$744$	5.39170	0.197669
$745$	3.32347	0.121763
$746$	25.6247	0.938185
$747$	77.6033	2.83935
$748$	0.644076	0.0235497
$749$	16.1919	0.591639
$750$	−8.83552	−0.322628
$751$	−34.9217	−1.27431	−0.637155	−	0.770736i	$-0.719889\pi$
$751$	−34.9217	−1.27431	−0.637155	+	0.770736i	$0.719889\pi$
$752$	−5.41992	−0.197644
$753$	0.703809	0.0256482
$754$	16.4922	0.600609
$755$	5.11698	0.186226
$756$	−13.8297	−0.502981
$757$	−34.1293	−1.24045	−0.620225	−	0.784424i	$-0.712959\pi$
$757$	−34.1293	−1.24045	−0.620225	+	0.784424i	$0.712959\pi$
$758$	3.14771	0.114330
$759$	−56.9013	−2.06539
$760$	−0.479396	−0.0173895
$761$	46.3356	1.67966	0.839832	−	0.542846i	$-0.182653\pi$
$761$	46.3356	1.67966	0.839832	+	0.542846i	$0.182653\pi$
$762$	33.5089	1.21390
$763$	−22.6464	−0.819854
$764$	8.38597	0.303394
$765$	0.202620	0.00732575
$766$	12.0515	0.435439
$767$	22.0063	0.794600
$768$	2.85044	0.102856
$769$	−30.2435	−1.09061	−0.545304	−	0.838238i	$-0.683586\pi$
$769$	−30.2435	−1.09061	−0.545304	+	0.838238i	$0.683586\pi$
$770$	3.64487	0.131352
$771$	−66.4086	−2.39165
$772$	3.91843	0.141028
$773$	−5.74404	−0.206599	−0.103299	−	0.994650i	$-0.532940\pi$
$773$	−5.74404	−0.206599	−0.103299	+	0.994650i	$0.532940\pi$
$774$	−13.2267	−0.475424
$775$	9.27230	0.333071
$776$	−0.468441	−0.0168161
$777$	−22.1264	−0.793779
$778$	17.0974	0.612971
$779$	−15.0036	−0.537561
$780$	−4.02939	−0.144275
$781$	−66.4671	−2.37838
$782$	0.494374	0.0176788
$783$	−22.1218	−0.790569
$784$	−1.78710	−0.0638249
$785$	5.51781	0.196939
$786$	−57.6821	−2.05745
$787$	−33.0042	−1.17647	−0.588237	−	0.808689i	$-0.700178\pi$
$787$	−33.0042	−1.17647	−0.588237	+	0.808689i	$0.700178\pi$
$788$	−19.7703	−0.704288
$789$	−64.2851	−2.28861
$790$	2.24690	0.0799411
$791$	6.70045	0.238241
$792$	26.1361	0.928705
$793$	−21.8876	−0.777252
$794$	22.4612	0.797117
$795$	−2.51160	−0.0890772
$796$	9.79565	0.347198
$797$	−29.0801	−1.03007	−0.515035	−	0.857169i	$-0.672221\pi$
$797$	−29.0801	−1.03007	−0.515035	+	0.857169i	$0.672221\pi$
$798$	−9.96664	−0.352815
$799$	0.684517	0.0242165
$800$	4.90201	0.173312
$801$	78.7147	2.78125
$802$	−5.74712	−0.202938
$803$	−52.8233	−1.86409
$804$	36.2151	1.27721
$805$	2.79770	0.0986060
$806$	8.54168	0.300868
$807$	23.9518	0.843143
$808$	−17.2612	−0.607245
$809$	12.8104	0.450388	0.225194	−	0.974314i	$-0.427698\pi$
$809$	12.8104	0.450388	0.225194	+	0.974314i	$0.427698\pi$
$810$	0.591861	0.0207959
$811$	−32.2734	−1.13327	−0.566636	−	0.823968i	$-0.691755\pi$
$811$	−32.2734	−1.13327	−0.566636	+	0.823968i	$0.691755\pi$
$812$	8.33849	0.292624
$813$	20.1183	0.705580
$814$	17.3382	0.607704
$815$	−5.73879	−0.201021
$816$	−0.360001	−0.0126025
$817$	−3.95234	−0.138275
$818$	2.50970	0.0877496
$819$	−52.8402	−1.84639
$820$	−3.06688	−0.107100
$821$	−2.79108	−0.0974092	−0.0487046	−	0.998813i	$-0.515509\pi$
$821$	−2.79108	−0.0974092	−0.0487046	+	0.998813i	$0.515509\pi$
$822$	27.7470	0.967789
$823$	0.0566805	0.00197576	0.000987879	−	1.00000i	$-0.499686\pi$
$823$	0.0566805	0.00197576	0.000987879		1.00000i	$0.499686\pi$
$824$	−9.69105	−0.337604
$825$	71.2577	2.48087
$826$	11.1264	0.387138
$827$	−20.9898	−0.729888	−0.364944	−	0.931030i	$-0.618912\pi$
$827$	−20.9898	−0.729888	−0.364944	+	0.931030i	$0.618912\pi$
$828$	20.0613	0.697179
$829$	32.6478	1.13390	0.566952	−	0.823751i	$-0.308123\pi$
$829$	32.6478	1.13390	0.566952	+	0.823751i	$0.308123\pi$
$830$	4.74004	0.164529
$831$	57.6624	2.00029
$832$	4.51575	0.156556
$833$	0.225704	0.00782019
$834$	57.8996	2.00490
$835$	−5.13161	−0.177587
$836$	7.80985	0.270109
$837$	−11.4574	−0.396026
$838$	−15.1892	−0.524701
$839$	43.5629	1.50396	0.751980	−	0.659186i	$-0.229099\pi$
$839$	43.5629	1.50396	0.751980	+	0.659186i	$0.229099\pi$
$840$	−2.03727	−0.0702925
$841$	−15.6619	−0.540064
$842$	18.0094	0.620647
$843$	13.9589	0.480770
$844$	−0.528876	−0.0182047
$845$	−2.31398	−0.0796034
$846$	27.7772	0.954999
$847$	−34.2638	−1.17732
$848$	2.81476	0.0966593
$849$	−14.1194	−0.484578
$850$	−0.619106	−0.0212352
$851$	13.3083	0.456203
$852$	37.1512	1.27278
$853$	−42.3883	−1.45135	−0.725674	−	0.688039i	$-0.758472\pi$
$853$	−42.3883	−1.45135	−0.725674	+	0.688039i	$0.758472\pi$
$854$	−11.0664	−0.378686
$855$	2.45691	0.0840245
$856$	7.09182	0.242393
$857$	7.82112	0.267164	0.133582	−	0.991038i	$-0.457352\pi$
$857$	7.82112	0.267164	0.133582	+	0.991038i	$0.457352\pi$
$858$	65.6429	2.24101
$859$	7.26388	0.247840	0.123920	−	0.992292i	$-0.460453\pi$
$859$	7.26388	0.247840	0.123920	+	0.992292i	$0.460453\pi$
$860$	−0.807893	−0.0275489
$861$	−63.7604	−2.17295
$862$	−34.4904	−1.17475
$863$	−24.2946	−0.826999	−0.413500	−	0.910504i	$-0.635694\pi$
$863$	−24.2946	−0.826999	−0.413500	+	0.910504i	$0.635694\pi$
$864$	−6.05722	−0.206071
$865$	−2.91383	−0.0990731
$866$	−26.3045	−0.893864
$867$	−48.4120	−1.64416
$868$	4.31870	0.146586
$869$	−36.6043	−1.24172
$870$	−3.25879	−0.110483
$871$	57.3731	1.94401
$872$	−9.91879	−0.335893
$873$	2.40077	0.0812536
$874$	5.99462	0.202771
$875$	−7.07718	−0.239252
$876$	29.5251	0.997561
$877$	35.0514	1.18360	0.591801	−	0.806084i	$-0.298417\pi$
$877$	35.0514	1.18360	0.591801	+	0.806084i	$0.298417\pi$
$878$	−28.0203	−0.945639
$879$	68.6487	2.31546
$880$	1.59640	0.0538148
$881$	−4.45994	−0.150259	−0.0751297	−	0.997174i	$-0.523937\pi$
$881$	−4.45994	−0.150259	−0.0751297	+	0.997174i	$0.523937\pi$
$882$	9.15890	0.308396
$883$	32.0004	1.07690	0.538450	−	0.842657i	$-0.319010\pi$
$883$	32.0004	1.07690	0.538450	+	0.842657i	$0.319010\pi$
$884$	−0.570324	−0.0191821
$885$	−4.34836	−0.146168
$886$	−21.6295	−0.726657
$887$	24.3683	0.818206	0.409103	−	0.912488i	$-0.365842\pi$
$887$	24.3683	0.818206	0.409103	+	0.912488i	$0.365842\pi$
$888$	−9.69104	−0.325210
$889$	26.8404	0.900197
$890$	4.80793	0.161162
$891$	−9.64203	−0.323020
$892$	−9.28276	−0.310810
$893$	8.30024	0.277757
$894$	30.2627	1.01213
$895$	−1.45809	−0.0487385
$896$	2.28318	0.0762756
$897$	50.3856	1.68233
$898$	36.1825	1.20742
$899$	6.90814	0.230399
$900$	−25.1228	−0.837428
$901$	−0.355494	−0.0118432
$902$	49.9626	1.66357
$903$	−16.7961	−0.558940
$904$	2.93470	0.0976068
$905$	7.35837	0.244601
$906$	46.5939	1.54798
$907$	1.24427	0.0413152	0.0206576	−	0.999787i	$-0.493424\pi$
$907$	1.24427	0.0413152	0.0206576	+	0.999787i	$0.493424\pi$
$908$	−7.14966	−0.237270
$909$	88.4636	2.93415
$910$	−3.22750	−0.106991
$911$	7.51471	0.248973	0.124487	−	0.992221i	$-0.460272\pi$
$911$	7.51471	0.248973	0.124487	+	0.992221i	$0.460272\pi$
$912$	−4.36525	−0.144548
$913$	−77.2202	−2.55562
$914$	−27.9605	−0.924850
$915$	4.32491	0.142977
$916$	6.52394	0.215557
$917$	−46.2029	−1.52575
$918$	0.765005	0.0252489
$919$	32.6393	1.07667	0.538336	−	0.842730i	$-0.319053\pi$
$919$	32.6393	1.07667	0.538336	+	0.842730i	$0.319053\pi$
$920$	1.22535	0.0403987
$921$	52.9774	1.74567
$922$	−11.6428	−0.383435
$923$	58.8560	1.93727
$924$	33.1893	1.09185
$925$	−16.6660	−0.547976
$926$	−16.3305	−0.536652
$927$	49.6668	1.63127
$928$	3.65214	0.119887
$929$	12.5117	0.410494	0.205247	−	0.978710i	$-0.434200\pi$
$929$	12.5117	0.410494	0.205247	+	0.978710i	$0.434200\pi$
$930$	−1.68781	−0.0553453
$931$	2.73682	0.0896955
$932$	18.2621	0.598196
$933$	4.25322	0.139244
$934$	−22.0980	−0.723068
$935$	−0.201620	−0.00659368
$936$	−23.1433	−0.756462
$937$	16.5641	0.541126	0.270563	−	0.962702i	$-0.412790\pi$
$937$	16.5641	0.541126	0.270563	+	0.962702i	$0.412790\pi$
$938$	29.0080	0.947145
$939$	−61.1999	−1.99718
$940$	1.69664	0.0553384
$941$	13.3971	0.436734	0.218367	−	0.975867i	$-0.429927\pi$
$941$	13.3971	0.436734	0.218367	+	0.975867i	$0.429927\pi$
$942$	50.2437	1.63703
$943$	38.3499	1.24884
$944$	4.87322	0.158610
$945$	4.32922	0.140830
$946$	13.1614	0.427915
$947$	12.6312	0.410457	0.205229	−	0.978714i	$-0.434206\pi$
$947$	12.6312	0.410457	0.205229	+	0.978714i	$0.434206\pi$
$948$	20.4597	0.664500
$949$	46.7745	1.51837
$950$	−7.50708	−0.243562
$951$	0.627044	0.0203333
$952$	−0.288357	−0.00934571
$953$	−27.5551	−0.892597	−0.446299	−	0.894884i	$-0.647258\pi$
$953$	−27.5551	−0.892597	−0.446299	+	0.894884i	$0.647258\pi$
$954$	−14.4257	−0.467048
$955$	−2.62513	−0.0849471
$956$	19.0323	0.615549
$957$	53.0891	1.71613
$958$	−36.5487	−1.18084
$959$	22.2251	0.717687
$960$	−0.892296	−0.0287987
$961$	−27.4221	−0.884584
$962$	−15.3528	−0.494995
$963$	−36.3456	−1.17122
$964$	−29.4304	−0.947890
$965$	−1.22662	−0.0394862
$966$	25.4751	0.819649
$967$	−54.2019	−1.74302	−0.871509	−	0.490380i	$-0.836858\pi$
$967$	−54.2019	−1.74302	−0.871509	+	0.490380i	$0.836858\pi$
$968$	−15.0071	−0.482346
$969$	0.551316	0.0177108
$970$	0.146640	0.00470832
$971$	−17.7978	−0.571159	−0.285580	−	0.958355i	$-0.592186\pi$
$971$	−17.7978	−0.571159	−0.285580	+	0.958355i	$0.592186\pi$
$972$	−12.7823	−0.409993
$973$	46.3770	1.48678
$974$	−24.1704	−0.774470
$975$	−63.0981	−2.02076
$976$	−4.84695	−0.155147
$977$	13.4925	0.431662	0.215831	−	0.976431i	$-0.430754\pi$
$977$	13.4925	0.431662	0.215831	+	0.976431i	$0.430754\pi$
$978$	−52.2559	−1.67096
$979$	−78.3261	−2.50331
$980$	0.559429	0.0178703
$981$	50.8339	1.62300
$982$	29.9975	0.957259
$983$	49.6062	1.58219	0.791095	−	0.611693i	$-0.209511\pi$
$983$	49.6062	1.58219	0.791095	+	0.611693i	$0.209511\pi$
$984$	−27.9262	−0.890254
$985$	6.18886	0.197193
$986$	−0.461253	−0.0146893
$987$	35.2732	1.12276
$988$	−6.91556	−0.220013
$989$	10.1023	0.321236
$990$	−8.18158	−0.260028
$991$	−14.5456	−0.462056	−0.231028	−	0.972947i	$-0.574209\pi$
$991$	−14.5456	−0.462056	−0.231028	+	0.972947i	$0.574209\pi$
$992$	1.89153	0.0600562
$993$	95.9855	3.04601
$994$	29.7578	0.943860
$995$	−3.06641	−0.0972118
$996$	43.1616	1.36763
$997$	35.7749	1.13300	0.566501	−	0.824061i	$-0.308297\pi$
$997$	35.7749	1.13300	0.566501	+	0.824061i	$0.308297\pi$
$998$	−34.6845	−1.09792
$999$	20.5936	0.651551

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.e.1.69	✓	77

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.e.1.69	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.85044 q^{3} +1.00000 q^{4} -0.313038 q^{5} -2.85044 q^{6} -2.28318 q^{7} -1.00000 q^{8} +5.12501 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +2.85044 q^{3} +1.00000 q^{4} -0.313038 q^{5} -2.85044 q^{6} -2.28318 q^{7} -1.00000 q^{8} +5.12501 q^{9} +0.313038 q^{10} -5.09971 q^{11} +2.85044 q^{12} +4.51575 q^{13} +2.28318 q^{14} -0.892296 q^{15} +1.00000 q^{16} -0.126296 q^{17} -5.12501 q^{18} -1.53143 q^{19} -0.313038 q^{20} -6.50806 q^{21} +5.09971 q^{22} +3.91440 q^{23} -2.85044 q^{24} -4.90201 q^{25} -4.51575 q^{26} +6.05722 q^{27} -2.28318 q^{28} -3.65214 q^{29} +0.892296 q^{30} -1.89153 q^{31} -1.00000 q^{32} -14.5364 q^{33} +0.126296 q^{34} +0.714721 q^{35} +5.12501 q^{36} +3.39984 q^{37} +1.53143 q^{38} +12.8719 q^{39} +0.313038 q^{40} +9.79714 q^{41} +6.50806 q^{42} +2.58082 q^{43} -5.09971 q^{44} -1.60432 q^{45} -3.91440 q^{46} -5.41992 q^{47} +2.85044 q^{48} -1.78710 q^{49} +4.90201 q^{50} -0.360001 q^{51} +4.51575 q^{52} +2.81476 q^{53} -6.05722 q^{54} +1.59640 q^{55} +2.28318 q^{56} -4.36525 q^{57} +3.65214 q^{58} +4.87322 q^{59} -0.892296 q^{60} -4.84695 q^{61} +1.89153 q^{62} -11.7013 q^{63} +1.00000 q^{64} -1.41360 q^{65} +14.5364 q^{66} +12.7051 q^{67} -0.126296 q^{68} +11.1578 q^{69} -0.714721 q^{70} +13.0335 q^{71} -5.12501 q^{72} +10.3581 q^{73} -3.39984 q^{74} -13.9729 q^{75} -1.53143 q^{76} +11.6436 q^{77} -12.8719 q^{78} +7.17773 q^{79} -0.313038 q^{80} +1.89070 q^{81} -9.79714 q^{82} +15.1421 q^{83} -6.50806 q^{84} +0.0395356 q^{85} -2.58082 q^{86} -10.4102 q^{87} +5.09971 q^{88} +15.3589 q^{89} +1.60432 q^{90} -10.3103 q^{91} +3.91440 q^{92} -5.39170 q^{93} +5.41992 q^{94} +0.479396 q^{95} -2.85044 q^{96} +0.468441 q^{97} +1.78710 q^{98} -26.1361 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) 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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