LMFDB - Embedded newform 8042.2.a.a.1.38

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, 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("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2156933055$
Analytic rank:	$1$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.38
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.286464 q^{3} +1.00000 q^{4} -1.60771 q^{5} +0.286464 q^{6} +2.79262 q^{7} +1.00000 q^{8} -2.91794 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.286464 q^{3} +1.00000 q^{4} -1.60771 q^{5} +0.286464 q^{6} +2.79262 q^{7} +1.00000 q^{8} -2.91794 q^{9} -1.60771 q^{10} -2.80813 q^{11} +0.286464 q^{12} -3.84050 q^{13} +2.79262 q^{14} -0.460552 q^{15} +1.00000 q^{16} +6.31179 q^{17} -2.91794 q^{18} +5.93298 q^{19} -1.60771 q^{20} +0.799983 q^{21} -2.80813 q^{22} +4.43946 q^{23} +0.286464 q^{24} -2.41525 q^{25} -3.84050 q^{26} -1.69527 q^{27} +2.79262 q^{28} -8.35758 q^{29} -0.460552 q^{30} -2.93002 q^{31} +1.00000 q^{32} -0.804428 q^{33} +6.31179 q^{34} -4.48973 q^{35} -2.91794 q^{36} -6.71476 q^{37} +5.93298 q^{38} -1.10016 q^{39} -1.60771 q^{40} -7.16886 q^{41} +0.799983 q^{42} +2.58569 q^{43} -2.80813 q^{44} +4.69121 q^{45} +4.43946 q^{46} -11.5697 q^{47} +0.286464 q^{48} +0.798700 q^{49} -2.41525 q^{50} +1.80810 q^{51} -3.84050 q^{52} +8.04707 q^{53} -1.69527 q^{54} +4.51468 q^{55} +2.79262 q^{56} +1.69958 q^{57} -8.35758 q^{58} -1.69126 q^{59} -0.460552 q^{60} +10.5297 q^{61} -2.93002 q^{62} -8.14868 q^{63} +1.00000 q^{64} +6.17443 q^{65} -0.804428 q^{66} +2.34885 q^{67} +6.31179 q^{68} +1.27174 q^{69} -4.48973 q^{70} -1.74046 q^{71} -2.91794 q^{72} -2.97642 q^{73} -6.71476 q^{74} -0.691883 q^{75} +5.93298 q^{76} -7.84204 q^{77} -1.10016 q^{78} -11.1018 q^{79} -1.60771 q^{80} +8.26818 q^{81} -7.16886 q^{82} +6.34135 q^{83} +0.799983 q^{84} -10.1475 q^{85} +2.58569 q^{86} -2.39414 q^{87} -2.80813 q^{88} +1.19753 q^{89} +4.69121 q^{90} -10.7250 q^{91} +4.43946 q^{92} -0.839343 q^{93} -11.5697 q^{94} -9.53853 q^{95} +0.286464 q^{96} +1.60006 q^{97} +0.798700 q^{98} +8.19396 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * 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
$3$	0.286464	0.165390	0.0826949	−	0.996575i	$-0.473647\pi$
$3$	0.286464	0.165390	0.0826949	+	0.996575i	$0.473647\pi$
$4$	1.00000	0.500000
$5$	−1.60771	−0.718992	−0.359496	−	0.933147i	$-0.617051\pi$
$5$	−1.60771	−0.718992	−0.359496	+	0.933147i	$0.617051\pi$
$6$	0.286464	0.116948
$7$	2.79262	1.05551	0.527755	−	0.849397i	$-0.323034\pi$
$7$	2.79262	1.05551	0.527755	+	0.849397i	$0.323034\pi$
$8$	1.00000	0.353553
$9$	−2.91794	−0.972646
$10$	−1.60771	−0.508404
$11$	−2.80813	−0.846684	−0.423342	−	0.905970i	$-0.639143\pi$
$11$	−2.80813	−0.846684	−0.423342	+	0.905970i	$0.639143\pi$
$12$	0.286464	0.0826949
$13$	−3.84050	−1.06516	−0.532582	−	0.846379i	$-0.678778\pi$
$13$	−3.84050	−1.06516	−0.532582	+	0.846379i	$0.678778\pi$
$14$	2.79262	0.746358
$15$	−0.460552	−0.118914
$16$	1.00000	0.250000
$17$	6.31179	1.53083	0.765417	−	0.643535i	$-0.222533\pi$
$17$	6.31179	1.53083	0.765417	+	0.643535i	$0.222533\pi$
$18$	−2.91794	−0.687765
$19$	5.93298	1.36112	0.680559	−	0.732693i	$-0.261737\pi$
$19$	5.93298	1.36112	0.680559	+	0.732693i	$0.261737\pi$
$20$	−1.60771	−0.359496
$21$	0.799983	0.174571
$22$	−2.80813	−0.598696
$23$	4.43946	0.925690	0.462845	−	0.886439i	$-0.346829\pi$
$23$	4.43946	0.925690	0.462845	+	0.886439i	$0.346829\pi$
$24$	0.286464	0.0584741
$25$	−2.41525	−0.483051
$26$	−3.84050	−0.753184
$27$	−1.69527	−0.326256
$28$	2.79262	0.527755
$29$	−8.35758	−1.55196	−0.775982	−	0.630755i	$-0.782745\pi$
$29$	−8.35758	−1.55196	−0.775982	+	0.630755i	$0.782745\pi$
$30$	−0.460552	−0.0840848
$31$	−2.93002	−0.526247	−0.263123	−	0.964762i	$-0.584753\pi$
$31$	−2.93002	−0.526247	−0.263123	+	0.964762i	$0.584753\pi$
$32$	1.00000	0.176777
$33$	−0.804428	−0.140033
$34$	6.31179	1.08246
$35$	−4.48973	−0.758902
$36$	−2.91794	−0.486323
$37$	−6.71476	−1.10390	−0.551950	−	0.833877i	$-0.686116\pi$
$37$	−6.71476	−1.10390	−0.551950	+	0.833877i	$0.686116\pi$
$38$	5.93298	0.962456
$39$	−1.10016	−0.176167
$40$	−1.60771	−0.254202
$41$	−7.16886	−1.11959	−0.559794	−	0.828632i	$-0.689120\pi$
$41$	−7.16886	−1.11959	−0.559794	+	0.828632i	$0.689120\pi$
$42$	0.799983	0.123440
$43$	2.58569	0.394314	0.197157	−	0.980372i	$-0.436829\pi$
$43$	2.58569	0.394314	0.197157	+	0.980372i	$0.436829\pi$
$44$	−2.80813	−0.423342
$45$	4.69121	0.699325
$46$	4.43946	0.654562
$47$	−11.5697	−1.68761	−0.843807	−	0.536647i	$-0.819691\pi$
$47$	−11.5697	−1.68761	−0.843807	+	0.536647i	$0.819691\pi$
$48$	0.286464	0.0413475
$49$	0.798700	0.114100
$50$	−2.41525	−0.341569
$51$	1.80810	0.253184
$52$	−3.84050	−0.532582
$53$	8.04707	1.10535	0.552675	−	0.833397i	$-0.313607\pi$
$53$	8.04707	1.10535	0.552675	+	0.833397i	$0.313607\pi$
$54$	−1.69527	−0.230698
$55$	4.51468	0.608759
$56$	2.79262	0.373179
$57$	1.69958	0.225115
$58$	−8.35758	−1.09740
$59$	−1.69126	−0.220184	−0.110092	−	0.993921i	$-0.535115\pi$
$59$	−1.69126	−0.220184	−0.110092	+	0.993921i	$0.535115\pi$
$60$	−0.460552	−0.0594570
$61$	10.5297	1.34819	0.674096	−	0.738644i	$-0.264533\pi$
$61$	10.5297	1.34819	0.674096	+	0.738644i	$0.264533\pi$
$62$	−2.93002	−0.372113
$63$	−8.14868	−1.02664
$64$	1.00000	0.125000
$65$	6.17443	0.765844
$66$	−0.804428	−0.0990182
$67$	2.34885	0.286958	0.143479	−	0.989653i	$-0.454171\pi$
$67$	2.34885	0.286958	0.143479	+	0.989653i	$0.454171\pi$
$68$	6.31179	0.765417
$69$	1.27174	0.153100
$70$	−4.48973	−0.536625
$71$	−1.74046	−0.206555	−0.103277	−	0.994653i	$-0.532933\pi$
$71$	−1.74046	−0.206555	−0.103277	+	0.994653i	$0.532933\pi$
$72$	−2.91794	−0.343882
$73$	−2.97642	−0.348363	−0.174182	−	0.984714i	$-0.555728\pi$
$73$	−2.97642	−0.348363	−0.174182	+	0.984714i	$0.555728\pi$
$74$	−6.71476	−0.780575
$75$	−0.691883	−0.0798917
$76$	5.93298	0.680559
$77$	−7.84204	−0.893683
$78$	−1.10016	−0.124569
$79$	−11.1018	−1.24905	−0.624526	−	0.781004i	$-0.714708\pi$
$79$	−11.1018	−1.24905	−0.624526	+	0.781004i	$0.714708\pi$
$80$	−1.60771	−0.179748
$81$	8.26818	0.918687
$82$	−7.16886	−0.791668
$83$	6.34135	0.696054	0.348027	−	0.937485i	$-0.386852\pi$
$83$	6.34135	0.696054	0.348027	+	0.937485i	$0.386852\pi$
$84$	0.799983	0.0872853
$85$	−10.1475	−1.10066
$86$	2.58569	0.278822
$87$	−2.39414	−0.256679
$88$	−2.80813	−0.299348
$89$	1.19753	0.126938	0.0634691	−	0.997984i	$-0.479784\pi$
$89$	1.19753	0.126938	0.0634691	+	0.997984i	$0.479784\pi$
$90$	4.69121	0.494497
$91$	−10.7250	−1.12429
$92$	4.43946	0.462845
$93$	−0.839343	−0.0870358
$94$	−11.5697	−1.19332
$95$	−9.53853	−0.978633
$96$	0.286464	0.0292371
$97$	1.60006	0.162461	0.0812307	−	0.996695i	$-0.474115\pi$
$97$	1.60006	0.162461	0.0812307	+	0.996695i	$0.474115\pi$
$98$	0.798700	0.0806809
$99$	8.19396	0.823524
$100$	−2.41525	−0.241525
$101$	−7.26364	−0.722760	−0.361380	−	0.932419i	$-0.617694\pi$
$101$	−7.26364	−0.722760	−0.361380	+	0.932419i	$0.617694\pi$
$102$	1.80810	0.179028
$103$	−15.1757	−1.49530	−0.747651	−	0.664092i	$-0.768818\pi$
$103$	−15.1757	−1.49530	−0.747651	+	0.664092i	$0.768818\pi$
$104$	−3.84050	−0.376592
$105$	−1.28614	−0.125515
$106$	8.04707	0.781601
$107$	11.7207	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$107$	11.7207	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$108$	−1.69527	−0.163128
$109$	−3.49552	−0.334810	−0.167405	−	0.985888i	$-0.553539\pi$
$109$	−3.49552	−0.334810	−0.167405	+	0.985888i	$0.553539\pi$
$110$	4.51468	0.430458
$111$	−1.92353	−0.182574
$112$	2.79262	0.263877
$113$	−0.995047	−0.0936061	−0.0468031	−	0.998904i	$-0.514903\pi$
$113$	−0.995047	−0.0936061	−0.0468031	+	0.998904i	$0.514903\pi$
$114$	1.69958	0.159180
$115$	−7.13738	−0.665564
$116$	−8.35758	−0.775982
$117$	11.2063	1.03603
$118$	−1.69126	−0.155694
$119$	17.6264	1.61581
$120$	−0.460552	−0.0420424
$121$	−3.11439	−0.283126
$122$	10.5297	0.953316
$123$	−2.05362	−0.185168
$124$	−2.93002	−0.263123
$125$	11.9216	1.06630
$126$	−8.14868	−0.725942
$127$	−8.03249	−0.712768	−0.356384	−	0.934340i	$-0.615990\pi$
$127$	−8.03249	−0.712768	−0.356384	+	0.934340i	$0.615990\pi$
$128$	1.00000	0.0883883
$129$	0.740706	0.0652155
$130$	6.17443	0.541533
$131$	−16.1838	−1.41399	−0.706994	−	0.707219i	$-0.749949\pi$
$131$	−16.1838	−1.41399	−0.706994	+	0.707219i	$0.749949\pi$
$132$	−0.804428	−0.0700165
$133$	16.5685	1.43667
$134$	2.34885	0.202910
$135$	2.72552	0.234575
$136$	6.31179	0.541231
$137$	0.0658877	0.00562917	0.00281458	−	0.999996i	$-0.499104\pi$
$137$	0.0658877	0.00562917	0.00281458	+	0.999996i	$0.499104\pi$
$138$	1.27174	0.108258
$139$	4.24219	0.359818	0.179909	−	0.983683i	$-0.442420\pi$
$139$	4.24219	0.359818	0.179909	+	0.983683i	$0.442420\pi$
$140$	−4.48973	−0.379451
$141$	−3.31430	−0.279114
$142$	−1.74046	−0.146056
$143$	10.7846	0.901857
$144$	−2.91794	−0.243162
$145$	13.4366	1.11585
$146$	−2.97642	−0.246330
$147$	0.228798	0.0188710
$148$	−6.71476	−0.551950
$149$	−5.53405	−0.453367	−0.226684	−	0.973968i	$-0.572788\pi$
$149$	−5.53405	−0.453367	−0.226684	+	0.973968i	$0.572788\pi$
$150$	−0.691883	−0.0564920
$151$	2.42829	0.197612	0.0988059	−	0.995107i	$-0.468498\pi$
$151$	2.42829	0.197612	0.0988059	+	0.995107i	$0.468498\pi$
$152$	5.93298	0.481228
$153$	−18.4174	−1.48896
$154$	−7.84204	−0.631929
$155$	4.71063	0.378367
$156$	−1.10016	−0.0880836
$157$	−21.9082	−1.74846	−0.874232	−	0.485509i	$-0.838634\pi$
$157$	−21.9082	−1.74846	−0.874232	+	0.485509i	$0.838634\pi$
$158$	−11.1018	−0.883214
$159$	2.30519	0.182814
$160$	−1.60771	−0.127101
$161$	12.3977	0.977075
$162$	8.26818	0.649610
$163$	−10.4069	−0.815134	−0.407567	−	0.913175i	$-0.633623\pi$
$163$	−10.4069	−0.815134	−0.407567	+	0.913175i	$0.633623\pi$
$164$	−7.16886	−0.559794
$165$	1.29329	0.100683
$166$	6.34135	0.492184
$167$	−9.63580	−0.745641	−0.372820	−	0.927904i	$-0.621609\pi$
$167$	−9.63580	−0.745641	−0.372820	+	0.927904i	$0.621609\pi$
$168$	0.799983	0.0617200
$169$	1.74946	0.134574
$170$	−10.1475	−0.778282
$171$	−17.3121	−1.32389
$172$	2.58569	0.197157
$173$	7.39950	0.562573	0.281287	−	0.959624i	$-0.409239\pi$
$173$	7.39950	0.562573	0.281287	+	0.959624i	$0.409239\pi$
$174$	−2.39414	−0.181500
$175$	−6.74488	−0.509865
$176$	−2.80813	−0.211671
$177$	−0.484486	−0.0364162
$178$	1.19753	0.0897589
$179$	−7.54748	−0.564125	−0.282062	−	0.959396i	$-0.591019\pi$
$179$	−7.54748	−0.564125	−0.282062	+	0.959396i	$0.591019\pi$
$180$	4.69121	0.349662
$181$	−11.8909	−0.883841	−0.441921	−	0.897054i	$-0.645703\pi$
$181$	−11.8909	−0.883841	−0.441921	+	0.897054i	$0.645703\pi$
$182$	−10.7250	−0.794993
$183$	3.01638	0.222977
$184$	4.43946	0.327281
$185$	10.7954	0.793694
$186$	−0.839343	−0.0615436
$187$	−17.7243	−1.29613
$188$	−11.5697	−0.843807
$189$	−4.73425	−0.344366
$190$	−9.53853	−0.691998
$191$	3.78170	0.273634	0.136817	−	0.990596i	$-0.456313\pi$
$191$	3.78170	0.273634	0.136817	+	0.990596i	$0.456313\pi$
$192$	0.286464	0.0206737
$193$	−5.42342	−0.390386	−0.195193	−	0.980765i	$-0.562533\pi$
$193$	−5.42342	−0.390386	−0.195193	+	0.980765i	$0.562533\pi$
$194$	1.60006	0.114878
$195$	1.76875	0.126663
$196$	0.798700	0.0570500
$197$	12.5608	0.894917	0.447458	−	0.894305i	$-0.352329\pi$
$197$	12.5608	0.894917	0.447458	+	0.894305i	$0.352329\pi$
$198$	8.19396	0.582320
$199$	−7.03375	−0.498610	−0.249305	−	0.968425i	$-0.580202\pi$
$199$	−7.03375	−0.498610	−0.249305	+	0.968425i	$0.580202\pi$
$200$	−2.41525	−0.170784
$201$	0.672860	0.0474599
$202$	−7.26364	−0.511068
$203$	−23.3395	−1.63811
$204$	1.80810	0.126592
$205$	11.5255	0.804974
$206$	−15.1757	−1.05734
$207$	−12.9541	−0.900369
$208$	−3.84050	−0.266291
$209$	−16.6606	−1.15244
$210$	−1.28614	−0.0887523
$211$	−17.7471	−1.22176	−0.610881	−	0.791722i	$-0.709185\pi$
$211$	−17.7471	−1.22176	−0.610881	+	0.791722i	$0.709185\pi$
$212$	8.04707	0.552675
$213$	−0.498579	−0.0341621
$214$	11.7207	0.801212
$215$	−4.15705	−0.283508
$216$	−1.69527	−0.115349
$217$	−8.18241	−0.555458
$218$	−3.49552	−0.236746
$219$	−0.852635	−0.0576157
$220$	4.51468	0.304379
$221$	−24.2404	−1.63059
$222$	−1.92353	−0.129099
$223$	−3.88124	−0.259907	−0.129953	−	0.991520i	$-0.541483\pi$
$223$	−3.88124	−0.259907	−0.129953	+	0.991520i	$0.541483\pi$
$224$	2.79262	0.186589
$225$	7.04757	0.469838
$226$	−0.995047	−0.0661895
$227$	6.43434	0.427062	0.213531	−	0.976936i	$-0.431503\pi$
$227$	6.43434	0.427062	0.213531	+	0.976936i	$0.431503\pi$
$228$	1.69958	0.112558
$229$	−16.8246	−1.11180	−0.555900	−	0.831249i	$-0.687626\pi$
$229$	−16.8246	−1.11180	−0.555900	+	0.831249i	$0.687626\pi$
$230$	−7.13738	−0.470625
$231$	−2.24646	−0.147806
$232$	−8.35758	−0.548702
$233$	7.65063	0.501209	0.250605	−	0.968090i	$-0.419371\pi$
$233$	7.65063	0.501209	0.250605	+	0.968090i	$0.419371\pi$
$234$	11.2063	0.732582
$235$	18.6008	1.21338
$236$	−1.69126	−0.110092
$237$	−3.18027	−0.206581
$238$	17.6264	1.14255
$239$	−21.2919	−1.37726	−0.688630	−	0.725112i	$-0.741788\pi$
$239$	−21.2919	−1.37726	−0.688630	+	0.725112i	$0.741788\pi$
$240$	−0.460552	−0.0297285
$241$	−8.87105	−0.571435	−0.285717	−	0.958314i	$-0.592232\pi$
$241$	−8.87105	−0.571435	−0.285717	+	0.958314i	$0.592232\pi$
$242$	−3.11439	−0.200200
$243$	7.45435	0.478197
$244$	10.5297	0.674096
$245$	−1.28408	−0.0820369
$246$	−2.05362	−0.130934
$247$	−22.7856	−1.44981
$248$	−2.93002	−0.186056
$249$	1.81657	0.115120
$250$	11.9216	0.753989
$251$	−6.71100	−0.423594	−0.211797	−	0.977314i	$-0.567932\pi$
$251$	−6.71100	−0.423594	−0.211797	+	0.977314i	$0.567932\pi$
$252$	−8.14868	−0.513319
$253$	−12.4666	−0.783767
$254$	−8.03249	−0.504003
$255$	−2.90690	−0.182037
$256$	1.00000	0.0625000
$257$	20.3098	1.26689	0.633446	−	0.773787i	$-0.281640\pi$
$257$	20.3098	1.26689	0.633446	+	0.773787i	$0.281640\pi$
$258$	0.740706	0.0461143
$259$	−18.7517	−1.16518
$260$	6.17443	0.382922
$261$	24.3869	1.50951
$262$	−16.1838	−0.999841
$263$	8.47407	0.522534	0.261267	−	0.965267i	$-0.415860\pi$
$263$	8.47407	0.522534	0.261267	+	0.965267i	$0.415860\pi$
$264$	−0.804428	−0.0495091
$265$	−12.9374	−0.794738
$266$	16.5685	1.01588
$267$	0.343049	0.0209943
$268$	2.34885	0.143479
$269$	3.19895	0.195044	0.0975218	−	0.995233i	$-0.468908\pi$
$269$	3.19895	0.195044	0.0975218	+	0.995233i	$0.468908\pi$
$270$	2.72552	0.165870
$271$	−11.5013	−0.698655	−0.349327	−	0.937001i	$-0.613590\pi$
$271$	−11.5013	−0.698655	−0.349327	+	0.937001i	$0.613590\pi$
$272$	6.31179	0.382708
$273$	−3.07233	−0.185946
$274$	0.0658877	0.00398042
$275$	6.78236	0.408992
$276$	1.27174	0.0765499
$277$	−7.52682	−0.452243	−0.226121	−	0.974099i	$-0.572605\pi$
$277$	−7.52682	−0.452243	−0.226121	+	0.974099i	$0.572605\pi$
$278$	4.24219	0.254430
$279$	8.54961	0.511852
$280$	−4.48973	−0.268313
$281$	24.0854	1.43681	0.718407	−	0.695623i	$-0.244872\pi$
$281$	24.0854	1.43681	0.718407	+	0.695623i	$0.244872\pi$
$282$	−3.31430	−0.197363
$283$	0.0878452	0.00522186	0.00261093	−	0.999997i	$-0.499169\pi$
$283$	0.0878452	0.00522186	0.00261093	+	0.999997i	$0.499169\pi$
$284$	−1.74046	−0.103277
$285$	−2.73244	−0.161856
$286$	10.7846	0.637709
$287$	−20.0199	−1.18174
$288$	−2.91794	−0.171941
$289$	22.8387	1.34345
$290$	13.4366	0.789025
$291$	0.458359	0.0268695
$292$	−2.97642	−0.174182
$293$	−16.8336	−0.983427	−0.491714	−	0.870757i	$-0.663629\pi$
$293$	−16.8336	−0.983427	−0.491714	+	0.870757i	$0.663629\pi$
$294$	0.228798	0.0133438
$295$	2.71907	0.158310
$296$	−6.71476	−0.390287
$297$	4.76056	0.276235
$298$	−5.53405	−0.320579
$299$	−17.0497	−0.986012
$300$	−0.691883	−0.0399459
$301$	7.22083	0.416202
$302$	2.42829	0.139733
$303$	−2.08077	−0.119537
$304$	5.93298	0.340280
$305$	−16.9288	−0.969339
$306$	−18.4174	−1.05285
$307$	11.9965	0.684674	0.342337	−	0.939577i	$-0.388782\pi$
$307$	11.9965	0.684674	0.342337	+	0.939577i	$0.388782\pi$
$308$	−7.84204	−0.446842
$309$	−4.34727	−0.247308
$310$	4.71063	0.267546
$311$	−22.5755	−1.28014	−0.640070	−	0.768317i	$-0.721094\pi$
$311$	−22.5755	−1.28014	−0.640070	+	0.768317i	$0.721094\pi$
$312$	−1.10016	−0.0622845
$313$	−5.62510	−0.317950	−0.158975	−	0.987283i	$-0.550819\pi$
$313$	−5.62510	−0.317950	−0.158975	+	0.987283i	$0.550819\pi$
$314$	−21.9082	−1.23635
$315$	13.1007	0.738144
$316$	−11.1018	−0.624526
$317$	−4.58584	−0.257566	−0.128783	−	0.991673i	$-0.541107\pi$
$317$	−4.58584	−0.257566	−0.128783	+	0.991673i	$0.541107\pi$
$318$	2.30519	0.129269
$319$	23.4692	1.31402
$320$	−1.60771	−0.0898740
$321$	3.35756	0.187401
$322$	12.3977	0.690896
$323$	37.4477	2.08365
$324$	8.26818	0.459343
$325$	9.27579	0.514528
$326$	−10.4069	−0.576387
$327$	−1.00134	−0.0553742
$328$	−7.16886	−0.395834
$329$	−32.3097	−1.78129
$330$	1.29329	0.0711933
$331$	6.06897	0.333581	0.166790	−	0.985992i	$-0.446660\pi$
$331$	6.06897	0.333581	0.166790	+	0.985992i	$0.446660\pi$
$332$	6.34135	0.348027
$333$	19.5933	1.07370
$334$	−9.63580	−0.527247
$335$	−3.77628	−0.206320
$336$	0.799983	0.0436426
$337$	7.93629	0.432318	0.216159	−	0.976358i	$-0.430647\pi$
$337$	7.93629	0.432318	0.216159	+	0.976358i	$0.430647\pi$
$338$	1.74946	0.0951579
$339$	−0.285045	−0.0154815
$340$	−10.1475	−0.550328
$341$	8.22788	0.445565
$342$	−17.3121	−0.936129
$343$	−17.3178	−0.935076
$344$	2.58569	0.139411
$345$	−2.04460	−0.110077
$346$	7.39950	0.397799
$347$	−19.3327	−1.03783	−0.518917	−	0.854825i	$-0.673665\pi$
$347$	−19.3327	−1.03783	−0.518917	+	0.854825i	$0.673665\pi$
$348$	−2.39414	−0.128340
$349$	21.0920	1.12903	0.564514	−	0.825424i	$-0.309064\pi$
$349$	21.0920	1.12903	0.564514	+	0.825424i	$0.309064\pi$
$350$	−6.74488	−0.360529
$351$	6.51070	0.347516
$352$	−2.80813	−0.149674
$353$	16.9829	0.903906	0.451953	−	0.892042i	$-0.350727\pi$
$353$	16.9829	0.903906	0.451953	+	0.892042i	$0.350727\pi$
$354$	−0.484486	−0.0257501
$355$	2.79816	0.148511
$356$	1.19753	0.0634691
$357$	5.04932	0.267238
$358$	−7.54748	−0.398896
$359$	9.51825	0.502354	0.251177	−	0.967941i	$-0.419182\pi$
$359$	9.51825	0.502354	0.251177	+	0.967941i	$0.419182\pi$
$360$	4.69121	0.247249
$361$	16.2002	0.852644
$362$	−11.8909	−0.624970
$363$	−0.892158	−0.0468261
$364$	−10.7250	−0.562145
$365$	4.78523	0.250470
$366$	3.01638	0.157669
$367$	19.4346	1.01448	0.507239	−	0.861806i	$-0.330666\pi$
$367$	19.4346	1.01448	0.507239	+	0.861806i	$0.330666\pi$
$368$	4.43946	0.231423
$369$	20.9183	1.08896
$370$	10.7954	0.561227
$371$	22.4724	1.16671
$372$	−0.839343	−0.0435179
$373$	9.28432	0.480724	0.240362	−	0.970683i	$-0.422734\pi$
$373$	9.28432	0.480724	0.240362	+	0.970683i	$0.422734\pi$
$374$	−17.7243	−0.916504
$375$	3.41511	0.176355
$376$	−11.5697	−0.596661
$377$	32.0973	1.65310
$378$	−4.73425	−0.243503
$379$	−3.66971	−0.188500	−0.0942501	−	0.995549i	$-0.530045\pi$
$379$	−3.66971	−0.188500	−0.0942501	+	0.995549i	$0.530045\pi$
$380$	−9.53853	−0.489316
$381$	−2.30102	−0.117885
$382$	3.78170	0.193489
$383$	−23.4873	−1.20014	−0.600072	−	0.799946i	$-0.704861\pi$
$383$	−23.4873	−1.20014	−0.600072	+	0.799946i	$0.704861\pi$
$384$	0.286464	0.0146185
$385$	12.6078	0.642551
$386$	−5.42342	−0.276045
$387$	−7.54488	−0.383528
$388$	1.60006	0.0812307
$389$	9.54823	0.484114	0.242057	−	0.970262i	$-0.422178\pi$
$389$	9.54823	0.484114	0.242057	+	0.970262i	$0.422178\pi$
$390$	1.76875	0.0895641
$391$	28.0209	1.41708
$392$	0.798700	0.0403404
$393$	−4.63608	−0.233859
$394$	12.5608	0.632802
$395$	17.8486	0.898059
$396$	8.19396	0.411762
$397$	−20.1192	−1.00975	−0.504877	−	0.863191i	$-0.668462\pi$
$397$	−20.1192	−1.00975	−0.504877	+	0.863191i	$0.668462\pi$
$398$	−7.03375	−0.352570
$399$	4.74628	0.237611
$400$	−2.41525	−0.120763
$401$	−32.6230	−1.62912	−0.814558	−	0.580082i	$-0.803021\pi$
$401$	−32.6230	−1.62912	−0.814558	+	0.580082i	$0.803021\pi$
$402$	0.672860	0.0335592
$403$	11.2527	0.560539
$404$	−7.26364	−0.361380
$405$	−13.2929	−0.660528
$406$	−23.3395	−1.15832
$407$	18.8559	0.934654
$408$	1.80810	0.0895141
$409$	−22.8967	−1.13217	−0.566084	−	0.824347i	$-0.691542\pi$
$409$	−22.8967	−1.13217	−0.566084	+	0.824347i	$0.691542\pi$
$410$	11.5255	0.569203
$411$	0.0188744	0.000931007 0
$412$	−15.1757	−0.747651
$413$	−4.72305	−0.232406
$414$	−12.9541	−0.636657
$415$	−10.1951	−0.500457
$416$	−3.84050	−0.188296
$417$	1.21523	0.0595102
$418$	−16.6606	−0.814896
$419$	31.1934	1.52390	0.761949	−	0.647636i	$-0.224242\pi$
$419$	31.1934	1.52390	0.761949	+	0.647636i	$0.224242\pi$
$420$	−1.28614	−0.0627574
$421$	16.3398	0.796351	0.398176	−	0.917309i	$-0.369643\pi$
$421$	16.3398	0.796351	0.398176	+	0.917309i	$0.369643\pi$
$422$	−17.7471	−0.863916
$423$	33.7597	1.64145
$424$	8.04707	0.390800
$425$	−15.2446	−0.739470
$426$	−0.498579	−0.0241562
$427$	29.4054	1.42303
$428$	11.7207	0.566542
$429$	3.08941	0.149158
$430$	−4.15705	−0.200471
$431$	−25.7091	−1.23837	−0.619183	−	0.785247i	$-0.712536\pi$
$431$	−25.7091	−1.23837	−0.619183	+	0.785247i	$0.712536\pi$
$432$	−1.69527	−0.0815639
$433$	31.8605	1.53112	0.765560	−	0.643364i	$-0.222462\pi$
$433$	31.8605	1.53112	0.765560	+	0.643364i	$0.222462\pi$
$434$	−8.18241	−0.392768
$435$	3.84910	0.184550
$436$	−3.49552	−0.167405
$437$	26.3392	1.25997
$438$	−0.852635	−0.0407405
$439$	−5.77546	−0.275648	−0.137824	−	0.990457i	$-0.544011\pi$
$439$	−5.77546	−0.275648	−0.137824	+	0.990457i	$0.544011\pi$
$440$	4.51468	0.215229
$441$	−2.33056	−0.110979
$442$	−24.2404	−1.15300
$443$	5.79476	0.275317	0.137659	−	0.990480i	$-0.456042\pi$
$443$	5.79476	0.275317	0.137659	+	0.990480i	$0.456042\pi$
$444$	−1.92353	−0.0912869
$445$	−1.92529	−0.0912675
$446$	−3.88124	−0.183782
$447$	−1.58530	−0.0749824
$448$	2.79262	0.131939
$449$	17.4960	0.825685	0.412843	−	0.910802i	$-0.364536\pi$
$449$	17.4960	0.825685	0.412843	+	0.910802i	$0.364536\pi$
$450$	7.04757	0.332225
$451$	20.1311	0.947937
$452$	−0.995047	−0.0468031
$453$	0.695618	0.0326830
$454$	6.43434	0.301979
$455$	17.2428	0.808355
$456$	1.69958	0.0795902
$457$	8.82881	0.412994	0.206497	−	0.978447i	$-0.433794\pi$
$457$	8.82881	0.412994	0.206497	+	0.978447i	$0.433794\pi$
$458$	−16.8246	−0.786162
$459$	−10.7002	−0.499443
$460$	−7.13738	−0.332782
$461$	11.2409	0.523541	0.261771	−	0.965130i	$-0.415694\pi$
$461$	11.2409	0.523541	0.261771	+	0.965130i	$0.415694\pi$
$462$	−2.24646	−0.104515
$463$	18.6472	0.866609	0.433305	−	0.901248i	$-0.357347\pi$
$463$	18.6472	0.866609	0.433305	+	0.901248i	$0.357347\pi$
$464$	−8.35758	−0.387991
$465$	1.34942	0.0625780
$466$	7.65063	0.354408
$467$	−29.6285	−1.37105	−0.685523	−	0.728051i	$-0.740426\pi$
$467$	−29.6285	−1.37105	−0.685523	+	0.728051i	$0.740426\pi$
$468$	11.2063	0.518014
$469$	6.55944	0.302887
$470$	18.6008	0.857989
$471$	−6.27590	−0.289178
$472$	−1.69126	−0.0778468
$473$	−7.26096	−0.333859
$474$	−3.18027	−0.146075
$475$	−14.3297	−0.657490
$476$	17.6264	0.807904
$477$	−23.4809	−1.07511
$478$	−21.2919	−0.973871
$479$	9.42038	0.430428	0.215214	−	0.976567i	$-0.430955\pi$
$479$	9.42038	0.430428	0.215214	+	0.976567i	$0.430955\pi$
$480$	−0.460552	−0.0210212
$481$	25.7880	1.17583
$482$	−8.87105	−0.404065
$483$	3.55149	0.161598
$484$	−3.11439	−0.141563
$485$	−2.57244	−0.116808
$486$	7.45435	0.338136
$487$	17.6041	0.797717	0.398859	−	0.917012i	$-0.369406\pi$
$487$	17.6041	0.797717	0.398859	+	0.917012i	$0.369406\pi$
$488$	10.5297	0.476658
$489$	−2.98121	−0.134815
$490$	−1.28408	−0.0580089
$491$	−15.7270	−0.709748	−0.354874	−	0.934914i	$-0.615476\pi$
$491$	−15.7270	−0.709748	−0.354874	+	0.934914i	$0.615476\pi$
$492$	−2.05362	−0.0925842
$493$	−52.7513	−2.37580
$494$	−22.7856	−1.02517
$495$	−13.1735	−0.592107
$496$	−2.93002	−0.131562
$497$	−4.86044	−0.218021
$498$	1.81657	0.0814023
$499$	−28.0601	−1.25614	−0.628072	−	0.778155i	$-0.716156\pi$
$499$	−28.0601	−1.25614	−0.628072	+	0.778155i	$0.716156\pi$
$500$	11.9216	0.533151
$501$	−2.76031	−0.123321
$502$	−6.71100	−0.299527
$503$	37.4399	1.66936	0.834680	−	0.550735i	$-0.185652\pi$
$503$	37.4399	1.66936	0.834680	+	0.550735i	$0.185652\pi$
$504$	−8.14868	−0.362971
$505$	11.6779	0.519658
$506$	−12.4666	−0.554207
$507$	0.501156	0.0222571
$508$	−8.03249	−0.356384
$509$	−12.2242	−0.541826	−0.270913	−	0.962604i	$-0.587326\pi$
$509$	−12.2242	−0.541826	−0.270913	+	0.962604i	$0.587326\pi$
$510$	−2.90690	−0.128720
$511$	−8.31198	−0.367700
$512$	1.00000	0.0441942
$513$	−10.0580	−0.444073
$514$	20.3098	0.895828
$515$	24.3981	1.07511
$516$	0.740706	0.0326078
$517$	32.4893	1.42888
$518$	−18.7517	−0.823904
$519$	2.11969	0.0930439
$520$	6.17443	0.270767
$521$	21.2537	0.931140	0.465570	−	0.885011i	$-0.345849\pi$
$521$	21.2537	0.931140	0.465570	+	0.885011i	$0.345849\pi$
$522$	24.3869	1.06739
$523$	39.0820	1.70894	0.854469	−	0.519502i	$-0.173883\pi$
$523$	39.0820	1.70894	0.854469	+	0.519502i	$0.173883\pi$
$524$	−16.1838	−0.706994
$525$	−1.93216	−0.0843265
$526$	8.47407	0.369487
$527$	−18.4936	−0.805596
$528$	−0.804428	−0.0350082
$529$	−3.29124	−0.143097
$530$	−12.9374	−0.561964
$531$	4.93501	0.214161
$532$	16.5685	0.718337
$533$	27.5320	1.19254
$534$	0.343049	0.0148452
$535$	−18.8436	−0.814678
$536$	2.34885	0.101455
$537$	−2.16208	−0.0933005
$538$	3.19895	0.137917
$539$	−2.24286	−0.0966066
$540$	2.72552	0.117288
$541$	12.8113	0.550802	0.275401	−	0.961329i	$-0.411189\pi$
$541$	12.8113	0.550802	0.275401	+	0.961329i	$0.411189\pi$
$542$	−11.5013	−0.494024
$543$	−3.40630	−0.146178
$544$	6.31179	0.270616
$545$	5.61980	0.240726
$546$	−3.07233	−0.131484
$547$	6.94580	0.296981	0.148490	−	0.988914i	$-0.452559\pi$
$547$	6.94580	0.296981	0.148490	+	0.988914i	$0.452559\pi$
$548$	0.0658877	0.00281458
$549$	−30.7251	−1.31131
$550$	6.78236	0.289201
$551$	−49.5854	−2.11241
$552$	1.27174	0.0541289
$553$	−31.0031	−1.31839
$554$	−7.52682	−0.319784
$555$	3.09249	0.131269
$556$	4.24219	0.179909
$557$	13.3623	0.566180	0.283090	−	0.959093i	$-0.408641\pi$
$557$	13.3623	0.566180	0.283090	+	0.959093i	$0.408641\pi$
$558$	8.54961	0.361934
$559$	−9.93034	−0.420009
$560$	−4.48973	−0.189726
$561$	−5.07738	−0.214367
$562$	24.0854	1.01598
$563$	−18.9292	−0.797771	−0.398885	−	0.917001i	$-0.630603\pi$
$563$	−18.9292	−0.797771	−0.398885	+	0.917001i	$0.630603\pi$
$564$	−3.31430	−0.139557
$565$	1.59975	0.0673020
$566$	0.0878452	0.00369241
$567$	23.0898	0.969683
$568$	−1.74046	−0.0730281
$569$	−19.5325	−0.818845	−0.409423	−	0.912345i	$-0.634270\pi$
$569$	−19.5325	−0.818845	−0.409423	+	0.912345i	$0.634270\pi$
$570$	−2.73244	−0.114449
$571$	44.6786	1.86974	0.934870	−	0.354991i	$-0.115516\pi$
$571$	44.6786	1.86974	0.934870	+	0.354991i	$0.115516\pi$
$572$	10.7846	0.450929
$573$	1.08332	0.0452563
$574$	−20.0199	−0.835613
$575$	−10.7224	−0.447156
$576$	−2.91794	−0.121581
$577$	−27.1585	−1.13062	−0.565312	−	0.824877i	$-0.691244\pi$
$577$	−27.1585	−1.13062	−0.565312	+	0.824877i	$0.691244\pi$
$578$	22.8387	0.949963
$579$	−1.55361	−0.0645659
$580$	13.4366	0.557925
$581$	17.7090	0.734691
$582$	0.458359	0.0189996
$583$	−22.5973	−0.935883
$584$	−2.97642	−0.123165
$585$	−18.0166	−0.744895
$586$	−16.8336	−0.695388
$587$	14.7993	0.610830	0.305415	−	0.952219i	$-0.401205\pi$
$587$	14.7993	0.610830	0.305415	+	0.952219i	$0.401205\pi$
$588$	0.228798	0.00943549
$589$	−17.3837	−0.716284
$590$	2.71907	0.111942
$591$	3.59820	0.148010
$592$	−6.71476	−0.275975
$593$	−20.4590	−0.840150	−0.420075	−	0.907489i	$-0.637996\pi$
$593$	−20.4590	−0.840150	−0.420075	+	0.907489i	$0.637996\pi$
$594$	4.76056	0.195328
$595$	−28.3382	−1.16175
$596$	−5.53405	−0.226684
$597$	−2.01491	−0.0824649
$598$	−17.0497	−0.697216
$599$	13.5140	0.552166	0.276083	−	0.961134i	$-0.410963\pi$
$599$	13.5140	0.552166	0.276083	+	0.961134i	$0.410963\pi$
$600$	−0.691883	−0.0282460
$601$	−41.8395	−1.70667	−0.853335	−	0.521364i	$-0.825423\pi$
$601$	−41.8395	−1.70667	−0.853335	+	0.521364i	$0.825423\pi$
$602$	7.22083	0.294299
$603$	−6.85380	−0.279108
$604$	2.42829	0.0988059
$605$	5.00704	0.203565
$606$	−2.08077	−0.0845255
$607$	14.8850	0.604162	0.302081	−	0.953282i	$-0.402319\pi$
$607$	14.8850	0.604162	0.302081	+	0.953282i	$0.402319\pi$
$608$	5.93298	0.240614
$609$	−6.68592	−0.270927
$610$	−16.9288	−0.685426
$611$	44.4334	1.79758
$612$	−18.4174	−0.744480
$613$	−14.6883	−0.593256	−0.296628	−	0.954993i	$-0.595862\pi$
$613$	−14.6883	−0.593256	−0.296628	+	0.954993i	$0.595862\pi$
$614$	11.9965	0.484138
$615$	3.30163	0.133135
$616$	−7.84204	−0.315965
$617$	45.4360	1.82918	0.914592	−	0.404378i	$-0.132512\pi$
$617$	45.4360	1.82918	0.914592	+	0.404378i	$0.132512\pi$
$618$	−4.34727	−0.174873
$619$	−43.5996	−1.75242	−0.876208	−	0.481933i	$-0.839935\pi$
$619$	−43.5996	−1.75242	−0.876208	+	0.481933i	$0.839935\pi$
$620$	4.71063	0.189183
$621$	−7.52609	−0.302012
$622$	−22.5755	−0.905195
$623$	3.34425	0.133984
$624$	−1.10016	−0.0440418
$625$	−7.09027	−0.283611
$626$	−5.62510	−0.224824
$627$	−4.77265	−0.190601
$628$	−21.9082	−0.874232
$629$	−42.3821	−1.68989
$630$	13.1007	0.521946
$631$	−20.6577	−0.822370	−0.411185	−	0.911552i	$-0.634885\pi$
$631$	−20.6577	−0.822370	−0.411185	+	0.911552i	$0.634885\pi$
$632$	−11.1018	−0.441607
$633$	−5.08390	−0.202067
$634$	−4.58584	−0.182127
$635$	12.9139	0.512474
$636$	2.30519	0.0914069
$637$	−3.06741	−0.121535
$638$	23.4692	0.929155
$639$	5.07856	0.200905
$640$	−1.60771	−0.0635505
$641$	−1.92238	−0.0759295	−0.0379647	−	0.999279i	$-0.512087\pi$
$641$	−1.92238	−0.0759295	−0.0379647	+	0.999279i	$0.512087\pi$
$642$	3.35756	0.132512
$643$	17.3563	0.684467	0.342234	−	0.939615i	$-0.388817\pi$
$643$	17.3563	0.684467	0.342234	+	0.939615i	$0.388817\pi$
$644$	12.3977	0.488537
$645$	−1.19084	−0.0468894
$646$	37.4477	1.47336
$647$	−9.30951	−0.365995	−0.182997	−	0.983113i	$-0.558580\pi$
$647$	−9.30951	−0.365995	−0.182997	+	0.983113i	$0.558580\pi$
$648$	8.26818	0.324805
$649$	4.74930	0.186426
$650$	9.27579	0.363826
$651$	−2.34396	−0.0918671
$652$	−10.4069	−0.407567
$653$	−0.0177535	−0.000694749 0	−0.000347375	−	1.00000i	$-0.500111\pi$
$653$	−0.0177535	−0.000694749 0	−0.000347375		1.00000i	$0.500111\pi$
$654$	−1.00134	−0.0391555
$655$	26.0190	1.01665
$656$	−7.16886	−0.279897
$657$	8.68500	0.338834
$658$	−32.3097	−1.25956
$659$	−21.6251	−0.842393	−0.421197	−	0.906969i	$-0.638390\pi$
$659$	−21.6251	−0.842393	−0.421197	+	0.906969i	$0.638390\pi$
$660$	1.29329	0.0503413
$661$	−21.5438	−0.837956	−0.418978	−	0.907996i	$-0.637612\pi$
$661$	−21.5438	−0.837956	−0.418978	+	0.907996i	$0.637612\pi$
$662$	6.06897	0.235877
$663$	−6.94400	−0.269683
$664$	6.34135	0.246092
$665$	−26.6375	−1.03296
$666$	19.5933	0.759223
$667$	−37.1031	−1.43664
$668$	−9.63580	−0.372820
$669$	−1.11183	−0.0429859
$670$	−3.77628	−0.145890
$671$	−29.5689	−1.14149
$672$	0.799983	0.0308600
$673$	−11.8832	−0.458062	−0.229031	−	0.973419i	$-0.573556\pi$
$673$	−11.8832	−0.458062	−0.229031	+	0.973419i	$0.573556\pi$
$674$	7.93629	0.305695
$675$	4.09452	0.157598
$676$	1.74946	0.0672868
$677$	50.5615	1.94324	0.971619	−	0.236552i	$-0.0760173\pi$
$677$	50.5615	1.94324	0.971619	+	0.236552i	$0.0760173\pi$
$678$	−0.285045	−0.0109471
$679$	4.46835	0.171479
$680$	−10.1475	−0.389141
$681$	1.84321	0.0706318
$682$	8.22788	0.315062
$683$	17.2570	0.660320	0.330160	−	0.943925i	$-0.392897\pi$
$683$	17.2570	0.660320	0.330160	+	0.943925i	$0.392897\pi$
$684$	−17.3121	−0.661943
$685$	−0.105929	−0.00404732
$686$	−17.3178	−0.661198
$687$	−4.81963	−0.183880
$688$	2.58569	0.0985785
$689$	−30.9048	−1.17738
$690$	−2.04460	−0.0778365
$691$	26.7862	1.01899	0.509497	−	0.860472i	$-0.329831\pi$
$691$	26.7862	1.01899	0.509497	+	0.860472i	$0.329831\pi$
$692$	7.39950	0.281287
$693$	22.8826	0.869237
$694$	−19.3327	−0.733860
$695$	−6.82023	−0.258706
$696$	−2.39414	−0.0907498
$697$	−45.2483	−1.71390
$698$	21.0920	0.798343
$699$	2.19163	0.0828949
$700$	−6.74488	−0.254932
$701$	26.2778	0.992499	0.496250	−	0.868180i	$-0.334710\pi$
$701$	26.2778	0.992499	0.496250	+	0.868180i	$0.334710\pi$
$702$	6.51070	0.245731
$703$	−39.8385	−1.50254
$704$	−2.80813	−0.105836
$705$	5.32844	0.200681
$706$	16.9829	0.639158
$707$	−20.2846	−0.762880
$708$	−0.484486	−0.0182081
$709$	5.37322	0.201795	0.100898	−	0.994897i	$-0.467829\pi$
$709$	5.37322	0.201795	0.100898	+	0.994897i	$0.467829\pi$
$710$	2.79816	0.105013
$711$	32.3944	1.21489
$712$	1.19753	0.0448794
$713$	−13.0077	−0.487141
$714$	5.04932	0.188966
$715$	−17.3386	−0.648428
$716$	−7.54748	−0.282062
$717$	−6.09936	−0.227785
$718$	9.51825	0.355218
$719$	−32.4214	−1.20911	−0.604556	−	0.796562i	$-0.706650\pi$
$719$	−32.4214	−1.20911	−0.604556	+	0.796562i	$0.706650\pi$
$720$	4.69121	0.174831
$721$	−42.3798	−1.57831
$722$	16.2002	0.602910
$723$	−2.54123	−0.0945095
$724$	−11.8909	−0.441921
$725$	20.1857	0.749678
$726$	−0.892158	−0.0331111
$727$	−7.25408	−0.269039	−0.134520	−	0.990911i	$-0.542949\pi$
$727$	−7.25408	−0.269039	−0.134520	+	0.990911i	$0.542949\pi$
$728$	−10.7250	−0.397497
$729$	−22.6691	−0.839598
$730$	4.78523	0.177109
$731$	16.3203	0.603629
$732$	3.01638	0.111489
$733$	−5.11130	−0.188790	−0.0943950	−	0.995535i	$-0.530092\pi$
$733$	−5.11130	−0.188790	−0.0943950	+	0.995535i	$0.530092\pi$
$734$	19.4346	0.717344
$735$	−0.367842	−0.0135681
$736$	4.43946	0.163640
$737$	−6.59589	−0.242963
$738$	20.9183	0.770013
$739$	−4.90316	−0.180365	−0.0901827	−	0.995925i	$-0.528745\pi$
$739$	−4.90316	−0.180365	−0.0901827	+	0.995925i	$0.528745\pi$
$740$	10.7954	0.396847
$741$	−6.52725	−0.239785
$742$	22.4724	0.824987
$743$	−21.1578	−0.776203	−0.388101	−	0.921617i	$-0.626869\pi$
$743$	−21.1578	−0.776203	−0.388101	+	0.921617i	$0.626869\pi$
$744$	−0.839343	−0.0307718
$745$	8.89718	0.325967
$746$	9.28432	0.339923
$747$	−18.5037	−0.677014
$748$	−17.7243	−0.648066
$749$	32.7315	1.19598
$750$	3.41511	0.124702
$751$	12.0423	0.439431	0.219716	−	0.975564i	$-0.429487\pi$
$751$	12.0423	0.439431	0.219716	+	0.975564i	$0.429487\pi$
$752$	−11.5697	−0.421903
$753$	−1.92246	−0.0700582
$754$	32.0973	1.16892
$755$	−3.90400	−0.142081
$756$	−4.73425	−0.172183
$757$	13.1224	0.476940	0.238470	−	0.971150i	$-0.423354\pi$
$757$	13.1224	0.476940	0.238470	+	0.971150i	$0.423354\pi$
$758$	−3.66971	−0.133290
$759$	−3.57122	−0.129627
$760$	−9.53853	−0.345999
$761$	15.6891	0.568728	0.284364	−	0.958716i	$-0.408218\pi$
$761$	15.6891	0.568728	0.284364	+	0.958716i	$0.408218\pi$
$762$	−2.30102	−0.0833570
$763$	−9.76164	−0.353395
$764$	3.78170	0.136817
$765$	29.6099	1.07055
$766$	−23.4873	−0.848630
$767$	6.49531	0.234532
$768$	0.286464	0.0103369
$769$	−31.6328	−1.14071	−0.570355	−	0.821399i	$-0.693194\pi$
$769$	−31.6328	−1.14071	−0.570355	+	0.821399i	$0.693194\pi$
$770$	12.6078	0.454352
$771$	5.81802	0.209531
$772$	−5.42342	−0.195193
$773$	−13.6798	−0.492028	−0.246014	−	0.969266i	$-0.579121\pi$
$773$	−13.6798	−0.492028	−0.246014	+	0.969266i	$0.579121\pi$
$774$	−7.54488	−0.271195
$775$	7.07674	0.254204
$776$	1.60006	0.0574388
$777$	−5.37169	−0.192708
$778$	9.54823	0.342321
$779$	−42.5327	−1.52389
$780$	1.76875	0.0633314
$781$	4.88745	0.174887
$782$	28.0209	1.00203
$783$	14.1684	0.506337
$784$	0.798700	0.0285250
$785$	35.2221	1.25713
$786$	−4.63608	−0.165364
$787$	37.7398	1.34528	0.672640	−	0.739970i	$-0.265160\pi$
$787$	37.7398	1.34528	0.672640	+	0.739970i	$0.265160\pi$
$788$	12.5608	0.447458
$789$	2.42751	0.0864218
$790$	17.8486	0.635023
$791$	−2.77878	−0.0988022
$792$	8.19396	0.291160
$793$	−40.4394	−1.43605
$794$	−20.1192	−0.714005
$795$	−3.70609	−0.131442
$796$	−7.03375	−0.249305
$797$	27.3024	0.967100	0.483550	−	0.875317i	$-0.339347\pi$
$797$	27.3024	0.967100	0.483550	+	0.875317i	$0.339347\pi$
$798$	4.74628	0.168016
$799$	−73.0254	−2.58345
$800$	−2.41525	−0.0853922
$801$	−3.49433	−0.123466
$802$	−32.6230	−1.15196
$803$	8.35817	0.294954
$804$	0.672860	0.0237299
$805$	−19.9319	−0.702509
$806$	11.2527	0.396361
$807$	0.916384	0.0322582
$808$	−7.26364	−0.255534
$809$	−43.9995	−1.54694	−0.773470	−	0.633833i	$-0.781481\pi$
$809$	−43.9995	−1.54694	−0.773470	+	0.633833i	$0.781481\pi$
$810$	−13.2929	−0.467064
$811$	9.13564	0.320796	0.160398	−	0.987052i	$-0.448722\pi$
$811$	9.13564	0.320796	0.160398	+	0.987052i	$0.448722\pi$
$812$	−23.3395	−0.819057
$813$	−3.29471	−0.115550
$814$	18.8559	0.660900
$815$	16.7314	0.586075
$816$	1.80810	0.0632961
$817$	15.3408	0.536708
$818$	−22.8967	−0.800564
$819$	31.2950	1.09354
$820$	11.5255	0.402487
$821$	27.3798	0.955560	0.477780	−	0.878479i	$-0.341442\pi$
$821$	27.3798	0.955560	0.477780	+	0.878479i	$0.341442\pi$
$822$	0.0188744	0.000658321 0
$823$	−20.4981	−0.714519	−0.357259	−	0.934005i	$-0.616289\pi$
$823$	−20.4981	−0.714519	−0.357259	+	0.934005i	$0.616289\pi$
$824$	−15.1757	−0.528669
$825$	1.94290	0.0676430
$826$	−4.72305	−0.164336
$827$	19.8664	0.690823	0.345411	−	0.938451i	$-0.387739\pi$
$827$	19.8664	0.690823	0.345411	+	0.938451i	$0.387739\pi$
$828$	−12.9541	−0.450185
$829$	−6.51306	−0.226208	−0.113104	−	0.993583i	$-0.536079\pi$
$829$	−6.51306	−0.226208	−0.113104	+	0.993583i	$0.536079\pi$
$830$	−10.1951	−0.353877
$831$	−2.15616	−0.0747964
$832$	−3.84050	−0.133145
$833$	5.04122	0.174668
$834$	1.21523	0.0420801
$835$	15.4916	0.536109
$836$	−16.6606	−0.576219
$837$	4.96718	0.171691
$838$	31.1934	1.07756
$839$	−2.36991	−0.0818185	−0.0409093	−	0.999163i	$-0.513025\pi$
$839$	−2.36991	−0.0818185	−0.0409093	+	0.999163i	$0.513025\pi$
$840$	−1.28614	−0.0443762
$841$	40.8492	1.40859
$842$	16.3398	0.563105
$843$	6.89958	0.237634
$844$	−17.7471	−0.610881
$845$	−2.81263	−0.0967573
$846$	33.7597	1.16068
$847$	−8.69728	−0.298842
$848$	8.04707	0.276338
$849$	0.0251645	0.000863642 0
$850$	−15.2446	−0.522885
$851$	−29.8099	−1.02187
$852$	−0.498579	−0.0170810
$853$	39.1139	1.33923	0.669617	−	0.742707i	$-0.266458\pi$
$853$	39.1139	1.33923	0.669617	+	0.742707i	$0.266458\pi$
$854$	29.4054	1.00623
$855$	27.8329	0.951864
$856$	11.7207	0.400606
$857$	37.2785	1.27341	0.636704	−	0.771108i	$-0.280297\pi$
$857$	37.2785	1.27341	0.636704	+	0.771108i	$0.280297\pi$
$858$	3.08941	0.105471
$859$	22.8456	0.779481	0.389740	−	0.920925i	$-0.372565\pi$
$859$	22.8456	0.779481	0.389740	+	0.920925i	$0.372565\pi$
$860$	−4.15705	−0.141754
$861$	−5.73496	−0.195447
$862$	−25.7091	−0.875657
$863$	−4.92656	−0.167702	−0.0838510	−	0.996478i	$-0.526722\pi$
$863$	−4.92656	−0.167702	−0.0838510	+	0.996478i	$0.526722\pi$
$864$	−1.69527	−0.0576744
$865$	−11.8963	−0.404485
$866$	31.8605	1.08267
$867$	6.54244	0.222193
$868$	−8.18241	−0.277729
$869$	31.1754	1.05755
$870$	3.84910	0.130497
$871$	−9.02077	−0.305657
$872$	−3.49552	−0.118373
$873$	−4.66887	−0.158017
$874$	26.3392	0.890937
$875$	33.2925	1.12549
$876$	−0.852635	−0.0288079
$877$	30.9851	1.04629	0.523147	−	0.852243i	$-0.324758\pi$
$877$	30.9851	1.04629	0.523147	+	0.852243i	$0.324758\pi$
$878$	−5.77546	−0.194912
$879$	−4.82220	−0.162649
$880$	4.51468	0.152190
$881$	−5.07226	−0.170889	−0.0854443	−	0.996343i	$-0.527231\pi$
$881$	−5.07226	−0.170889	−0.0854443	+	0.996343i	$0.527231\pi$
$882$	−2.33056	−0.0784739
$883$	46.0993	1.55136	0.775682	−	0.631123i	$-0.217406\pi$
$883$	46.0993	1.55136	0.775682	+	0.631123i	$0.217406\pi$
$884$	−24.2404	−0.815294
$885$	0.778915	0.0261829
$886$	5.79476	0.194679
$887$	39.5767	1.32885	0.664427	−	0.747353i	$-0.268676\pi$
$887$	39.5767	1.32885	0.664427	+	0.747353i	$0.268676\pi$
$888$	−1.92353	−0.0645496
$889$	−22.4316	−0.752333
$890$	−1.92529	−0.0645359
$891$	−23.2182	−0.777838
$892$	−3.88124	−0.129953
$893$	−68.6428	−2.29704
$894$	−1.58530	−0.0530205
$895$	12.1342	0.405601
$896$	2.79262	0.0932947
$897$	−4.88413	−0.163076
$898$	17.4960	0.583848
$899$	24.4879	0.816716
$900$	7.04757	0.234919
$901$	50.7914	1.69211
$902$	20.1311	0.670293
$903$	2.06851	0.0688356
$904$	−0.995047	−0.0330948
$905$	19.1171	0.635475
$906$	0.695618	0.0231104
$907$	−4.73449	−0.157206	−0.0786031	−	0.996906i	$-0.525046\pi$
$907$	−4.73449	−0.157206	−0.0786031	+	0.996906i	$0.525046\pi$
$908$	6.43434	0.213531
$909$	21.1949	0.702989
$910$	17.2428	0.571594
$911$	−49.9281	−1.65419	−0.827096	−	0.562061i	$-0.810009\pi$
$911$	−49.9281	−1.65419	−0.827096	+	0.562061i	$0.810009\pi$
$912$	1.69958	0.0562788
$913$	−17.8074	−0.589338
$914$	8.82881	0.292031
$915$	−4.84948	−0.160319
$916$	−16.8246	−0.555900
$917$	−45.1952	−1.49248
$918$	−10.7002	−0.353159
$919$	−30.1088	−0.993198	−0.496599	−	0.867980i	$-0.665418\pi$
$919$	−30.1088	−0.993198	−0.496599	+	0.867980i	$0.665418\pi$
$920$	−7.13738	−0.235312
$921$	3.43655	0.113238
$922$	11.2409	0.370199
$923$	6.68425	0.220015
$924$	−2.24646	−0.0739030
$925$	16.2179	0.533240
$926$	18.6472	0.612785
$927$	44.2817	1.45440
$928$	−8.35758	−0.274351
$929$	25.8719	0.848830	0.424415	−	0.905468i	$-0.360480\pi$
$929$	25.8719	0.848830	0.424415	+	0.905468i	$0.360480\pi$
$930$	1.34942	0.0442494
$931$	4.73867	0.155304
$932$	7.65063	0.250605
$933$	−6.46706	−0.211722
$934$	−29.6285	−0.969475
$935$	28.4957	0.931908
$936$	11.2063	0.366291
$937$	−7.03842	−0.229935	−0.114968	−	0.993369i	$-0.536676\pi$
$937$	−7.03842	−0.229935	−0.114968	+	0.993369i	$0.536676\pi$
$938$	6.55944	0.214173
$939$	−1.61139	−0.0525856
$940$	18.6008	0.606690
$941$	−7.44005	−0.242539	−0.121269	−	0.992620i	$-0.538696\pi$
$941$	−7.44005	−0.242539	−0.121269	+	0.992620i	$0.538696\pi$
$942$	−6.27590	−0.204480
$943$	−31.8258	−1.03639
$944$	−1.69126	−0.0550460
$945$	7.61132	0.247596
$946$	−7.26096	−0.236074
$947$	31.5154	1.02411	0.512057	−	0.858952i	$-0.328884\pi$
$947$	31.5154	1.02411	0.512057	+	0.858952i	$0.328884\pi$
$948$	−3.18027	−0.103290
$949$	11.4309	0.371064
$950$	−14.3297	−0.464915
$951$	−1.31368	−0.0425989
$952$	17.6264	0.571275
$953$	1.92388	0.0623205	0.0311603	−	0.999514i	$-0.490080\pi$
$953$	1.92388	0.0623205	0.0311603	+	0.999514i	$0.490080\pi$
$954$	−23.4809	−0.760221
$955$	−6.07989	−0.196741
$956$	−21.2919	−0.688630
$957$	6.72308	0.217326
$958$	9.42038	0.304359
$959$	0.183999	0.00594164
$960$	−0.460552	−0.0148642
$961$	−22.4150	−0.723065
$962$	25.7880	0.831440
$963$	−34.2003	−1.10209
$964$	−8.87105	−0.285717
$965$	8.71931	0.280685
$966$	3.55149	0.114267
$967$	−21.5556	−0.693181	−0.346590	−	0.938017i	$-0.612661\pi$
$967$	−21.5556	−0.693181	−0.346590	+	0.938017i	$0.612661\pi$
$968$	−3.11439	−0.100100
$969$	10.7274	0.344614
$970$	−2.57244	−0.0825960
$971$	37.5090	1.20372	0.601861	−	0.798601i	$-0.294426\pi$
$971$	37.5090	1.20372	0.601861	+	0.798601i	$0.294426\pi$
$972$	7.45435	0.239099
$973$	11.8468	0.379791
$974$	17.6041	0.564071
$975$	2.65718	0.0850978
$976$	10.5297	0.337048
$977$	26.2000	0.838211	0.419106	−	0.907937i	$-0.362344\pi$
$977$	26.2000	0.838211	0.419106	+	0.907937i	$0.362344\pi$
$978$	−2.98121	−0.0953286
$979$	−3.36283	−0.107477
$980$	−1.28408	−0.0410185
$981$	10.1997	0.325652
$982$	−15.7270	−0.501867
$983$	9.89385	0.315565	0.157782	−	0.987474i	$-0.449566\pi$
$983$	9.89385	0.315565	0.157782	+	0.987474i	$0.449566\pi$
$984$	−2.05362	−0.0654669
$985$	−20.1941	−0.643438
$986$	−52.7513	−1.67994
$987$	−9.25555	−0.294608
$988$	−22.7856	−0.724907
$989$	11.4791	0.365013
$990$	−13.1735	−0.418683
$991$	32.1186	1.02028	0.510141	−	0.860091i	$-0.329593\pi$
$991$	32.1186	1.02028	0.510141	+	0.860091i	$0.329593\pi$
$992$	−2.93002	−0.0930281
$993$	1.73854	0.0551709
$994$	−4.86044	−0.154164
$995$	11.3083	0.358496
$996$	1.81657	0.0575601
$997$	1.92760	0.0610477	0.0305239	−	0.999534i	$-0.490282\pi$
$997$	1.92760	0.0610477	0.0305239	+	0.999534i	$0.490282\pi$
$998$	−28.0601	−0.888228
$999$	11.3834	0.360153

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.38	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.38	✓	67	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.286464 q^{3} +1.00000 q^{4} -1.60771 q^{5} +0.286464 q^{6} +2.79262 q^{7} +1.00000 q^{8} -2.91794 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.286464 q^{3} +1.00000 q^{4} -1.60771 q^{5} +0.286464 q^{6} +2.79262 q^{7} +1.00000 q^{8} -2.91794 q^{9} -1.60771 q^{10} -2.80813 q^{11} +0.286464 q^{12} -3.84050 q^{13} +2.79262 q^{14} -0.460552 q^{15} +1.00000 q^{16} +6.31179 q^{17} -2.91794 q^{18} +5.93298 q^{19} -1.60771 q^{20} +0.799983 q^{21} -2.80813 q^{22} +4.43946 q^{23} +0.286464 q^{24} -2.41525 q^{25} -3.84050 q^{26} -1.69527 q^{27} +2.79262 q^{28} -8.35758 q^{29} -0.460552 q^{30} -2.93002 q^{31} +1.00000 q^{32} -0.804428 q^{33} +6.31179 q^{34} -4.48973 q^{35} -2.91794 q^{36} -6.71476 q^{37} +5.93298 q^{38} -1.10016 q^{39} -1.60771 q^{40} -7.16886 q^{41} +0.799983 q^{42} +2.58569 q^{43} -2.80813 q^{44} +4.69121 q^{45} +4.43946 q^{46} -11.5697 q^{47} +0.286464 q^{48} +0.798700 q^{49} -2.41525 q^{50} +1.80810 q^{51} -3.84050 q^{52} +8.04707 q^{53} -1.69527 q^{54} +4.51468 q^{55} +2.79262 q^{56} +1.69958 q^{57} -8.35758 q^{58} -1.69126 q^{59} -0.460552 q^{60} +10.5297 q^{61} -2.93002 q^{62} -8.14868 q^{63} +1.00000 q^{64} +6.17443 q^{65} -0.804428 q^{66} +2.34885 q^{67} +6.31179 q^{68} +1.27174 q^{69} -4.48973 q^{70} -1.74046 q^{71} -2.91794 q^{72} -2.97642 q^{73} -6.71476 q^{74} -0.691883 q^{75} +5.93298 q^{76} -7.84204 q^{77} -1.10016 q^{78} -11.1018 q^{79} -1.60771 q^{80} +8.26818 q^{81} -7.16886 q^{82} +6.34135 q^{83} +0.799983 q^{84} -10.1475 q^{85} +2.58569 q^{86} -2.39414 q^{87} -2.80813 q^{88} +1.19753 q^{89} +4.69121 q^{90} -10.7250 q^{91} +4.43946 q^{92} -0.839343 q^{93} -11.5697 q^{94} -9.53853 q^{95} +0.286464 q^{96} +1.60006 q^{97} +0.798700 q^{98} +8.19396 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

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