LMFDB - Embedded newform 8007.2.a.f.1.39

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.39
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.82604 q^{2} -1.00000 q^{3} +1.33443 q^{4} -0.167627 q^{5} -1.82604 q^{6} +3.32340 q^{7} -1.21535 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.82604 q^{2} -1.00000 q^{3} +1.33443 q^{4} -0.167627 q^{5} -1.82604 q^{6} +3.32340 q^{7} -1.21535 q^{8} +1.00000 q^{9} -0.306095 q^{10} -0.790492 q^{11} -1.33443 q^{12} -2.24590 q^{13} +6.06866 q^{14} +0.167627 q^{15} -4.88815 q^{16} -1.00000 q^{17} +1.82604 q^{18} +7.55949 q^{19} -0.223688 q^{20} -3.32340 q^{21} -1.44347 q^{22} +0.417165 q^{23} +1.21535 q^{24} -4.97190 q^{25} -4.10112 q^{26} -1.00000 q^{27} +4.43485 q^{28} -2.98837 q^{29} +0.306095 q^{30} -8.95148 q^{31} -6.49528 q^{32} +0.790492 q^{33} -1.82604 q^{34} -0.557092 q^{35} +1.33443 q^{36} -1.22329 q^{37} +13.8039 q^{38} +2.24590 q^{39} +0.203726 q^{40} -9.37300 q^{41} -6.06866 q^{42} -2.86920 q^{43} -1.05486 q^{44} -0.167627 q^{45} +0.761762 q^{46} +4.19886 q^{47} +4.88815 q^{48} +4.04495 q^{49} -9.07891 q^{50} +1.00000 q^{51} -2.99701 q^{52} +3.33576 q^{53} -1.82604 q^{54} +0.132508 q^{55} -4.03910 q^{56} -7.55949 q^{57} -5.45688 q^{58} +5.24784 q^{59} +0.223688 q^{60} -4.29098 q^{61} -16.3458 q^{62} +3.32340 q^{63} -2.08435 q^{64} +0.376475 q^{65} +1.44347 q^{66} -2.10530 q^{67} -1.33443 q^{68} -0.417165 q^{69} -1.01727 q^{70} +3.75354 q^{71} -1.21535 q^{72} +12.0241 q^{73} -2.23378 q^{74} +4.97190 q^{75} +10.0876 q^{76} -2.62712 q^{77} +4.10112 q^{78} -10.6687 q^{79} +0.819389 q^{80} +1.00000 q^{81} -17.1155 q^{82} -6.26459 q^{83} -4.43485 q^{84} +0.167627 q^{85} -5.23929 q^{86} +2.98837 q^{87} +0.960726 q^{88} +5.46315 q^{89} -0.306095 q^{90} -7.46402 q^{91} +0.556680 q^{92} +8.95148 q^{93} +7.66730 q^{94} -1.26718 q^{95} +6.49528 q^{96} +1.46789 q^{97} +7.38626 q^{98} -0.790492 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) $ 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * 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 $

$2$	1.82604	1.29121	0.645604	−	0.763673i	$-0.276606\pi$
$2$	1.82604	1.29121	0.645604	+	0.763673i	$0.276606\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.33443	0.667217
$5$	−0.167627	−0.0749653	−0.0374826	−	0.999297i	$-0.511934\pi$
$5$	−0.167627	−0.0749653	−0.0374826	+	0.999297i	$0.511934\pi$
$6$	−1.82604	−0.745479
$7$	3.32340	1.25613	0.628063	−	0.778163i	$-0.283848\pi$
$7$	3.32340	1.25613	0.628063	+	0.778163i	$0.283848\pi$
$8$	−1.21535	−0.429692
$9$	1.00000	0.333333
$10$	−0.306095	−0.0967957
$11$	−0.790492	−0.238342	−0.119171	−	0.992874i	$-0.538024\pi$
$11$	−0.790492	−0.238342	−0.119171	+	0.992874i	$0.538024\pi$
$12$	−1.33443	−0.385218
$13$	−2.24590	−0.622901	−0.311451	−	0.950262i	$-0.600815\pi$
$13$	−2.24590	−0.622901	−0.311451	+	0.950262i	$0.600815\pi$
$14$	6.06866	1.62192
$15$	0.167627	0.0432812
$16$	−4.88815	−1.22204
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.82604	0.430403
$19$	7.55949	1.73426	0.867132	−	0.498078i	$-0.165961\pi$
$19$	7.55949	1.73426	0.867132	+	0.498078i	$0.165961\pi$
$20$	−0.223688	−0.0500181
$21$	−3.32340	−0.725224
$22$	−1.44347	−0.307749
$23$	0.417165	0.0869850	0.0434925	−	0.999054i	$-0.486152\pi$
$23$	0.417165	0.0869850	0.0434925	+	0.999054i	$0.486152\pi$
$24$	1.21535	0.248083
$25$	−4.97190	−0.994380
$26$	−4.10112	−0.804295
$27$	−1.00000	−0.192450
$28$	4.43485	0.838108
$29$	−2.98837	−0.554925	−0.277463	−	0.960736i	$-0.589494\pi$
$29$	−2.98837	−0.554925	−0.277463	+	0.960736i	$0.589494\pi$
$30$	0.306095	0.0558850
$31$	−8.95148	−1.60773	−0.803867	−	0.594809i	$-0.797228\pi$
$31$	−8.95148	−1.60773	−0.803867	+	0.594809i	$0.797228\pi$
$32$	−6.49528	−1.14821
$33$	0.790492	0.137607
$34$	−1.82604	−0.313164
$35$	−0.557092	−0.0941658
$36$	1.33443	0.222406
$37$	−1.22329	−0.201107	−0.100554	−	0.994932i	$-0.532061\pi$
$37$	−1.22329	−0.201107	−0.100554	+	0.994932i	$0.532061\pi$
$38$	13.8039	2.23930
$39$	2.24590	0.359632
$40$	0.203726	0.0322120
$41$	−9.37300	−1.46382	−0.731908	−	0.681404i	$-0.761370\pi$
$41$	−9.37300	−1.46382	−0.731908	+	0.681404i	$0.761370\pi$
$42$	−6.06866	−0.936415
$43$	−2.86920	−0.437549	−0.218775	−	0.975775i	$-0.570206\pi$
$43$	−2.86920	−0.437549	−0.218775	+	0.975775i	$0.570206\pi$
$44$	−1.05486	−0.159026
$45$	−0.167627	−0.0249884
$46$	0.761762	0.112316
$47$	4.19886	0.612467	0.306233	−	0.951956i	$-0.400931\pi$
$47$	4.19886	0.612467	0.306233	+	0.951956i	$0.400931\pi$
$48$	4.88815	0.705544
$49$	4.04495	0.577851
$50$	−9.07891	−1.28395
$51$	1.00000	0.140028
$52$	−2.99701	−0.415610
$53$	3.33576	0.458202	0.229101	−	0.973403i	$-0.426421\pi$
$53$	3.33576	0.458202	0.229101	+	0.973403i	$0.426421\pi$
$54$	−1.82604	−0.248493
$55$	0.132508	0.0178674
$56$	−4.03910	−0.539747
$57$	−7.55949	−1.00128
$58$	−5.45688	−0.716524
$59$	5.24784	0.683210	0.341605	−	0.939844i	$-0.389029\pi$
$59$	5.24784	0.683210	0.341605	+	0.939844i	$0.389029\pi$
$60$	0.223688	0.0288780
$61$	−4.29098	−0.549404	−0.274702	−	0.961529i	$-0.588579\pi$
$61$	−4.29098	−0.549404	−0.274702	+	0.961529i	$0.588579\pi$
$62$	−16.3458	−2.07592
$63$	3.32340	0.418708
$64$	−2.08435	−0.260543
$65$	0.376475	0.0466960
$66$	1.44347	0.177679
$67$	−2.10530	−0.257203	−0.128602	−	0.991696i	$-0.541049\pi$
$67$	−2.10530	−0.257203	−0.128602	+	0.991696i	$0.541049\pi$
$68$	−1.33443	−0.161824
$69$	−0.417165	−0.0502208
$70$	−1.01727	−0.121588
$71$	3.75354	0.445463	0.222731	−	0.974880i	$-0.428503\pi$
$71$	3.75354	0.445463	0.222731	+	0.974880i	$0.428503\pi$
$72$	−1.21535	−0.143231
$73$	12.0241	1.40732	0.703658	−	0.710539i	$-0.251549\pi$
$73$	12.0241	1.40732	0.703658	+	0.710539i	$0.251549\pi$
$74$	−2.23378	−0.259671
$75$	4.97190	0.574106
$76$	10.0876	1.15713
$77$	−2.62712	−0.299388
$78$	4.10112	0.464360
$79$	−10.6687	−1.20032	−0.600159	−	0.799880i	$-0.704896\pi$
$79$	−10.6687	−1.20032	−0.600159	+	0.799880i	$0.704896\pi$
$80$	0.819389	0.0916105
$81$	1.00000	0.111111
$82$	−17.1155	−1.89009
$83$	−6.26459	−0.687628	−0.343814	−	0.939038i	$-0.611719\pi$
$83$	−6.26459	−0.687628	−0.343814	+	0.939038i	$0.611719\pi$
$84$	−4.43485	−0.483882
$85$	0.167627	0.0181817
$86$	−5.23929	−0.564967
$87$	2.98837	0.320386
$88$	0.960726	0.102414
$89$	5.46315	0.579093	0.289547	−	0.957164i	$-0.406495\pi$
$89$	5.46315	0.579093	0.289547	+	0.957164i	$0.406495\pi$
$90$	−0.306095	−0.0322652
$91$	−7.46402	−0.782442
$92$	0.556680	0.0580379
$93$	8.95148	0.928225
$94$	7.66730	0.790822
$95$	−1.26718	−0.130010
$96$	6.49528	0.662921
$97$	1.46789	0.149042	0.0745209	−	0.997219i	$-0.476257\pi$
$97$	1.46789	0.149042	0.0745209	+	0.997219i	$0.476257\pi$
$98$	7.38626	0.746125
$99$	−0.790492	−0.0794474
$100$	−6.63467	−0.663467
$101$	−11.5861	−1.15286	−0.576432	−	0.817145i	$-0.695555\pi$
$101$	−11.5861	−1.15286	−0.576432	+	0.817145i	$0.695555\pi$
$102$	1.82604	0.180805
$103$	−17.6860	−1.74266	−0.871328	−	0.490701i	$-0.836741\pi$
$103$	−17.6860	−1.74266	−0.871328	+	0.490701i	$0.836741\pi$
$104$	2.72956	0.267656
$105$	0.557092	0.0543666
$106$	6.09125	0.591635
$107$	−12.8920	−1.24632	−0.623160	−	0.782095i	$-0.714151\pi$
$107$	−12.8920	−1.24632	−0.623160	+	0.782095i	$0.714151\pi$
$108$	−1.33443	−0.128406
$109$	8.10349	0.776173	0.388087	−	0.921623i	$-0.373136\pi$
$109$	8.10349	0.776173	0.388087	+	0.921623i	$0.373136\pi$
$110$	0.241966	0.0230705
$111$	1.22329	0.116109
$112$	−16.2453	−1.53503
$113$	18.4938	1.73975	0.869873	−	0.493276i	$-0.164201\pi$
$113$	18.4938	1.73975	0.869873	+	0.493276i	$0.164201\pi$
$114$	−13.8039	−1.29286
$115$	−0.0699284	−0.00652085
$116$	−3.98778	−0.370256
$117$	−2.24590	−0.207634
$118$	9.58278	0.882166
$119$	−3.32340	−0.304655
$120$	−0.203726	−0.0185976
$121$	−10.3751	−0.943193
$122$	−7.83552	−0.709395
$123$	9.37300	0.845135
$124$	−11.9452	−1.07271
$125$	1.67156	0.149509
$126$	6.06866	0.540639
$127$	3.10262	0.275313	0.137656	−	0.990480i	$-0.456043\pi$
$127$	3.10262	0.275313	0.137656	+	0.990480i	$0.456043\pi$
$128$	9.18444	0.811798
$129$	2.86920	0.252619
$130$	0.687460	0.0602942
$131$	−13.0950	−1.14412	−0.572058	−	0.820213i	$-0.693855\pi$
$131$	−13.0950	−1.14412	−0.572058	+	0.820213i	$0.693855\pi$
$132$	1.05486	0.0918137
$133$	25.1232	2.17845
$134$	−3.84437	−0.332103
$135$	0.167627	0.0144271
$136$	1.21535	0.104216
$137$	3.91930	0.334848	0.167424	−	0.985885i	$-0.446455\pi$
$137$	3.91930	0.334848	0.167424	+	0.985885i	$0.446455\pi$
$138$	−0.761762	−0.0648455
$139$	−16.4286	−1.39345	−0.696727	−	0.717336i	$-0.745361\pi$
$139$	−16.4286	−1.39345	−0.696727	+	0.717336i	$0.745361\pi$
$140$	−0.743403	−0.0628290
$141$	−4.19886	−0.353608
$142$	6.85412	0.575185
$143$	1.77537	0.148464
$144$	−4.88815	−0.407346
$145$	0.500932	0.0416001
$146$	21.9565	1.81714
$147$	−4.04495	−0.333622
$148$	−1.63240	−0.134182
$149$	−21.0121	−1.72138	−0.860691	−	0.509128i	$-0.829968\pi$
$149$	−21.0121	−1.72138	−0.860691	+	0.509128i	$0.829968\pi$
$150$	9.07891	0.741290
$151$	−11.2749	−0.917542	−0.458771	−	0.888555i	$-0.651710\pi$
$151$	−11.2749	−0.917542	−0.458771	+	0.888555i	$0.651710\pi$
$152$	−9.18744	−0.745200
$153$	−1.00000	−0.0808452
$154$	−4.79723	−0.386572
$155$	1.50051	0.120524
$156$	2.99701	0.239953
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	−19.4815	−1.54986
$159$	−3.33576	−0.264543
$160$	1.08879	0.0860761
$161$	1.38641	0.109264
$162$	1.82604	0.143468
$163$	0.929838	0.0728305	0.0364153	−	0.999337i	$-0.488406\pi$
$163$	0.929838	0.0728305	0.0364153	+	0.999337i	$0.488406\pi$
$164$	−12.5076	−0.976683
$165$	−0.132508	−0.0103157
$166$	−11.4394	−0.887871
$167$	6.00705	0.464839	0.232420	−	0.972616i	$-0.425336\pi$
$167$	6.00705	0.464839	0.232420	+	0.972616i	$0.425336\pi$
$168$	4.03910	0.311623
$169$	−7.95592	−0.611994
$170$	0.306095	0.0234764
$171$	7.55949	0.578088
$172$	−3.82876	−0.291940
$173$	13.1729	1.00151	0.500757	−	0.865588i	$-0.333055\pi$
$173$	13.1729	1.00151	0.500757	+	0.865588i	$0.333055\pi$
$174$	5.45688	0.413685
$175$	−16.5236	−1.24907
$176$	3.86405	0.291263
$177$	−5.24784	−0.394452
$178$	9.97595	0.747729
$179$	3.81256	0.284964	0.142482	−	0.989797i	$-0.454492\pi$
$179$	3.81256	0.284964	0.142482	+	0.989797i	$0.454492\pi$
$180$	−0.223688	−0.0166727
$181$	−3.28420	−0.244112	−0.122056	−	0.992523i	$-0.538949\pi$
$181$	−3.28420	−0.244112	−0.122056	+	0.992523i	$0.538949\pi$
$182$	−13.6296	−1.01030
$183$	4.29098	0.317199
$184$	−0.507003	−0.0373767
$185$	0.205057	0.0150761
$186$	16.3458	1.19853
$187$	0.790492	0.0578065
$188$	5.60310	0.408648
$189$	−3.32340	−0.241741
$190$	−2.31392	−0.167869
$191$	16.1713	1.17011	0.585056	−	0.810993i	$-0.301072\pi$
$191$	16.1713	1.17011	0.585056	+	0.810993i	$0.301072\pi$
$192$	2.08435	0.150425
$193$	15.8666	1.14210	0.571050	−	0.820915i	$-0.306536\pi$
$193$	15.8666	1.14210	0.571050	+	0.820915i	$0.306536\pi$
$194$	2.68043	0.192444
$195$	−0.376475	−0.0269599
$196$	5.39773	0.385552
$197$	15.0792	1.07435	0.537173	−	0.843472i	$-0.319492\pi$
$197$	15.0792	1.07435	0.537173	+	0.843472i	$0.319492\pi$
$198$	−1.44347	−0.102583
$199$	−17.3826	−1.23222	−0.616109	−	0.787661i	$-0.711292\pi$
$199$	−17.3826	−1.23222	−0.616109	+	0.787661i	$0.711292\pi$
$200$	6.04261	0.427277
$201$	2.10530	0.148496
$202$	−21.1568	−1.48859
$203$	−9.93152	−0.697056
$204$	1.33443	0.0934291
$205$	1.57117	0.109735
$206$	−32.2955	−2.25013
$207$	0.417165	0.0289950
$208$	10.9783	0.761210
$209$	−5.97571	−0.413349
$210$	1.01727	0.0701986
$211$	−2.15143	−0.148111	−0.0740553	−	0.997254i	$-0.523594\pi$
$211$	−2.15143	−0.148111	−0.0740553	+	0.997254i	$0.523594\pi$
$212$	4.45136	0.305720
$213$	−3.75354	−0.257188
$214$	−23.5414	−1.60926
$215$	0.480957	0.0328010
$216$	1.21535	0.0826942
$217$	−29.7493	−2.01951
$218$	14.7973	1.00220
$219$	−12.0241	−0.812514
$220$	0.176823	0.0119214
$221$	2.24590	0.151076
$222$	2.23378	0.149921
$223$	−6.40349	−0.428809	−0.214405	−	0.976745i	$-0.568781\pi$
$223$	−6.40349	−0.428809	−0.214405	+	0.976745i	$0.568781\pi$
$224$	−21.5864	−1.44230
$225$	−4.97190	−0.331460
$226$	33.7704	2.24637
$227$	−10.0649	−0.668032	−0.334016	−	0.942567i	$-0.608404\pi$
$227$	−10.0649	−0.668032	−0.334016	+	0.942567i	$0.608404\pi$
$228$	−10.0876	−0.668070
$229$	2.39902	0.158532	0.0792658	−	0.996854i	$-0.474742\pi$
$229$	2.39902	0.158532	0.0792658	+	0.996854i	$0.474742\pi$
$230$	−0.127692	−0.00841978
$231$	2.62712	0.172852
$232$	3.63192	0.238447
$233$	−18.1937	−1.19191	−0.595953	−	0.803019i	$-0.703226\pi$
$233$	−18.1937	−1.19191	−0.595953	+	0.803019i	$0.703226\pi$
$234$	−4.10112	−0.268098
$235$	−0.703844	−0.0459137
$236$	7.00289	0.455849
$237$	10.6687	0.693004
$238$	−6.06866	−0.393373
$239$	−7.50963	−0.485758	−0.242879	−	0.970057i	$-0.578092\pi$
$239$	−7.50963	−0.485758	−0.242879	+	0.970057i	$0.578092\pi$
$240$	−0.819389	−0.0528913
$241$	−15.4902	−0.997809	−0.498904	−	0.866657i	$-0.666264\pi$
$241$	−15.4902	−0.997809	−0.498904	+	0.866657i	$0.666264\pi$
$242$	−18.9454	−1.21786
$243$	−1.00000	−0.0641500
$244$	−5.72603	−0.366572
$245$	−0.678045	−0.0433187
$246$	17.1155	1.09124
$247$	−16.9779	−1.08028
$248$	10.8792	0.690830
$249$	6.26459	0.397002
$250$	3.05235	0.193047
$251$	−4.47396	−0.282394	−0.141197	−	0.989982i	$-0.545095\pi$
$251$	−4.47396	−0.282394	−0.141197	+	0.989982i	$0.545095\pi$
$252$	4.43485	0.279369
$253$	−0.329766	−0.0207322
$254$	5.66551	0.355486
$255$	−0.167627	−0.0104972
$256$	20.9399	1.30874
$257$	12.9682	0.808933	0.404467	−	0.914553i	$-0.367457\pi$
$257$	12.9682	0.808933	0.404467	+	0.914553i	$0.367457\pi$
$258$	5.23929	0.326184
$259$	−4.06547	−0.252616
$260$	0.502381	0.0311564
$261$	−2.98837	−0.184975
$262$	−23.9121	−1.47729
$263$	−6.98264	−0.430568	−0.215284	−	0.976551i	$-0.569068\pi$
$263$	−6.98264	−0.430568	−0.215284	+	0.976551i	$0.569068\pi$
$264$	−0.960726	−0.0591286
$265$	−0.559166	−0.0343493
$266$	45.8760	2.81284
$267$	−5.46315	−0.334340
$268$	−2.80938	−0.171610
$269$	−16.6172	−1.01317	−0.506583	−	0.862191i	$-0.669092\pi$
$269$	−16.6172	−1.01317	−0.506583	+	0.862191i	$0.669092\pi$
$270$	0.306095	0.0186283
$271$	−4.42732	−0.268940	−0.134470	−	0.990918i	$-0.542933\pi$
$271$	−4.42732	−0.268940	−0.134470	+	0.990918i	$0.542933\pi$
$272$	4.88815	0.296388
$273$	7.46402	0.451743
$274$	7.15680	0.432358
$275$	3.93025	0.237003
$276$	−0.556680	−0.0335082
$277$	7.02563	0.422129	0.211065	−	0.977472i	$-0.432307\pi$
$277$	7.02563	0.422129	0.211065	+	0.977472i	$0.432307\pi$
$278$	−29.9993	−1.79924
$279$	−8.95148	−0.535911
$280$	0.677063	0.0404623
$281$	10.4169	0.621419	0.310709	−	0.950505i	$-0.399433\pi$
$281$	10.4169	0.621419	0.310709	+	0.950505i	$0.399433\pi$
$282$	−7.66730	−0.456581
$283$	−11.8295	−0.703193	−0.351596	−	0.936152i	$-0.614361\pi$
$283$	−11.8295	−0.703193	−0.351596	+	0.936152i	$0.614361\pi$
$284$	5.00885	0.297220
$285$	1.26718	0.0750611
$286$	3.24190	0.191698
$287$	−31.1502	−1.83874
$288$	−6.49528	−0.382738
$289$	1.00000	0.0588235
$290$	0.914724	0.0537144
$291$	−1.46789	−0.0860494
$292$	16.0454	0.938985
$293$	1.93768	0.113201	0.0566003	−	0.998397i	$-0.481974\pi$
$293$	1.93768	0.113201	0.0566003	+	0.998397i	$0.481974\pi$
$294$	−7.38626	−0.430776
$295$	−0.879682	−0.0512170
$296$	1.48673	0.0864141
$297$	0.790492	0.0458690
$298$	−38.3691	−2.22266
$299$	−0.936913	−0.0541831
$300$	6.63467	0.383053
$301$	−9.53549	−0.549616
$302$	−20.5885	−1.18474
$303$	11.5861	0.665606
$304$	−36.9519	−2.11934
$305$	0.719287	0.0411862
$306$	−1.82604	−0.104388
$307$	−15.8996	−0.907436	−0.453718	−	0.891145i	$-0.649903\pi$
$307$	−15.8996	−0.907436	−0.453718	+	0.891145i	$0.649903\pi$
$308$	−3.50571	−0.199757
$309$	17.6860	1.00612
$310$	2.74000	0.155622
$311$	−10.4760	−0.594042	−0.297021	−	0.954871i	$-0.595993\pi$
$311$	−10.4760	−0.594042	−0.297021	+	0.954871i	$0.595993\pi$
$312$	−2.72956	−0.154531
$313$	−18.9373	−1.07040	−0.535201	−	0.844725i	$-0.679764\pi$
$313$	−18.9373	−1.07040	−0.535201	+	0.844725i	$0.679764\pi$
$314$	−1.82604	−0.103050
$315$	−0.557092	−0.0313886
$316$	−14.2366	−0.800873
$317$	−10.4507	−0.586970	−0.293485	−	0.955964i	$-0.594815\pi$
$317$	−10.4507	−0.586970	−0.293485	+	0.955964i	$0.594815\pi$
$318$	−6.09125	−0.341580
$319$	2.36228	0.132262
$320$	0.349394	0.0195317
$321$	12.8920	0.719563
$322$	2.53164	0.141083
$323$	−7.55949	−0.420621
$324$	1.33443	0.0741352
$325$	11.1664	0.619401
$326$	1.69792	0.0940393
$327$	−8.10349	−0.448124
$328$	11.3915	0.628990
$329$	13.9545	0.769335
$330$	−0.241966	−0.0133198
$331$	−15.2077	−0.835889	−0.417945	−	0.908473i	$-0.637249\pi$
$331$	−15.2077	−0.835889	−0.417945	+	0.908473i	$0.637249\pi$
$332$	−8.35968	−0.458797
$333$	−1.22329	−0.0670357
$334$	10.9691	0.600204
$335$	0.352906	0.0192813
$336$	16.2453	0.886252
$337$	−33.4952	−1.82460	−0.912301	−	0.409521i	$-0.865696\pi$
$337$	−33.4952	−1.82460	−0.912301	+	0.409521i	$0.865696\pi$
$338$	−14.5279	−0.790211
$339$	−18.4938	−1.00444
$340$	0.223688	0.0121312
$341$	7.07607	0.383191
$342$	13.8039	0.746432
$343$	−9.82078	−0.530272
$344$	3.48709	0.188011
$345$	0.0699284	0.00376482
$346$	24.0542	1.29316
$347$	1.36704	0.0733868	0.0366934	−	0.999327i	$-0.488318\pi$
$347$	1.36704	0.0733868	0.0366934	+	0.999327i	$0.488318\pi$
$348$	3.98778	0.213767
$349$	−1.63934	−0.0877518	−0.0438759	−	0.999037i	$-0.513971\pi$
$349$	−1.63934	−0.0877518	−0.0438759	+	0.999037i	$0.513971\pi$
$350$	−30.1728	−1.61280
$351$	2.24590	0.119877
$352$	5.13446	0.273668
$353$	21.5686	1.14798	0.573990	−	0.818862i	$-0.305395\pi$
$353$	21.5686	1.14798	0.573990	+	0.818862i	$0.305395\pi$
$354$	−9.58278	−0.509319
$355$	−0.629196	−0.0333942
$356$	7.29022	0.386381
$357$	3.32340	0.175893
$358$	6.96190	0.367948
$359$	29.8507	1.57546	0.787730	−	0.616020i	$-0.211256\pi$
$359$	29.8507	1.57546	0.787730	+	0.616020i	$0.211256\pi$
$360$	0.203726	0.0107373
$361$	38.1458	2.00767
$362$	−5.99708	−0.315200
$363$	10.3751	0.544553
$364$	−9.96025	−0.522059
$365$	−2.01557	−0.105500
$366$	7.83552	0.409569
$367$	−10.7930	−0.563388	−0.281694	−	0.959504i	$-0.590896\pi$
$367$	−10.7930	−0.563388	−0.281694	+	0.959504i	$0.590896\pi$
$368$	−2.03917	−0.106299
$369$	−9.37300	−0.487939
$370$	0.374442	0.0194663
$371$	11.0861	0.575560
$372$	11.9452	0.619328
$373$	−2.07987	−0.107692	−0.0538458	−	0.998549i	$-0.517148\pi$
$373$	−2.07987	−0.107692	−0.0538458	+	0.998549i	$0.517148\pi$
$374$	1.44347	0.0746402
$375$	−1.67156	−0.0863192
$376$	−5.10310	−0.263172
$377$	6.71158	0.345664
$378$	−6.06866	−0.312138
$379$	31.6435	1.62542	0.812708	−	0.582671i	$-0.197992\pi$
$379$	31.6435	1.62542	0.812708	+	0.582671i	$0.197992\pi$
$380$	−1.69096	−0.0867446
$381$	−3.10262	−0.158952
$382$	29.5295	1.51086
$383$	14.9047	0.761592	0.380796	−	0.924659i	$-0.375650\pi$
$383$	14.9047	0.761592	0.380796	+	0.924659i	$0.375650\pi$
$384$	−9.18444	−0.468692
$385$	0.440377	0.0224437
$386$	28.9731	1.47469
$387$	−2.86920	−0.145850
$388$	1.95881	0.0994433
$389$	−17.3086	−0.877583	−0.438792	−	0.898589i	$-0.644593\pi$
$389$	−17.3086	−0.877583	−0.438792	+	0.898589i	$0.644593\pi$
$390$	−0.687460	−0.0348109
$391$	−0.417165	−0.0210970
$392$	−4.91605	−0.248298
$393$	13.0950	0.660556
$394$	27.5352	1.38720
$395$	1.78836	0.0899822
$396$	−1.05486	−0.0530087
$397$	30.4420	1.52784	0.763921	−	0.645310i	$-0.223272\pi$
$397$	30.4420	1.52784	0.763921	+	0.645310i	$0.223272\pi$
$398$	−31.7413	−1.59105
$399$	−25.1232	−1.25773
$400$	24.3034	1.21517
$401$	−31.4646	−1.57127	−0.785633	−	0.618693i	$-0.787663\pi$
$401$	−31.4646	−1.57127	−0.785633	+	0.618693i	$0.787663\pi$
$402$	3.84437	0.191740
$403$	20.1042	1.00146
$404$	−15.4609	−0.769210
$405$	−0.167627	−0.00832948
$406$	−18.1354	−0.900044
$407$	0.966999	0.0479324
$408$	−1.21535	−0.0601689
$409$	−25.1796	−1.24505	−0.622527	−	0.782599i	$-0.713894\pi$
$409$	−25.1796	−1.24505	−0.622527	+	0.782599i	$0.713894\pi$
$410$	2.86903	0.141691
$411$	−3.91930	−0.193325
$412$	−23.6008	−1.16273
$413$	17.4406	0.858198
$414$	0.761762	0.0374386
$415$	1.05012	0.0515482
$416$	14.5878	0.715224
$417$	16.4286	0.804511
$418$	−10.9119	−0.533719
$419$	39.4605	1.92777	0.963886	−	0.266315i	$-0.0858060\pi$
$419$	39.4605	1.92777	0.963886	+	0.266315i	$0.0858060\pi$
$420$	0.743403	0.0362743
$421$	3.15855	0.153938	0.0769692	−	0.997033i	$-0.475476\pi$
$421$	3.15855	0.153938	0.0769692	+	0.997033i	$0.475476\pi$
$422$	−3.92861	−0.191242
$423$	4.19886	0.204156
$424$	−4.05413	−0.196886
$425$	4.97190	0.241173
$426$	−6.85412	−0.332083
$427$	−14.2606	−0.690120
$428$	−17.2036	−0.831565
$429$	−1.77537	−0.0857156
$430$	0.878248	0.0423529
$431$	8.80135	0.423946	0.211973	−	0.977276i	$-0.432011\pi$
$431$	8.80135	0.423946	0.211973	+	0.977276i	$0.432011\pi$
$432$	4.88815	0.235181
$433$	12.8301	0.616573	0.308287	−	0.951293i	$-0.400244\pi$
$433$	12.8301	0.616573	0.308287	+	0.951293i	$0.400244\pi$
$434$	−54.3235	−2.60761
$435$	−0.500932	−0.0240179
$436$	10.8136	0.517876
$437$	3.15356	0.150855
$438$	−21.9565	−1.04912
$439$	−21.6115	−1.03146	−0.515731	−	0.856750i	$-0.672480\pi$
$439$	−21.6115	−1.03146	−0.515731	+	0.856750i	$0.672480\pi$
$440$	−0.161044	−0.00767748
$441$	4.04495	0.192617
$442$	4.10112	0.195070
$443$	−38.4004	−1.82446	−0.912230	−	0.409679i	$-0.865641\pi$
$443$	−38.4004	−1.82446	−0.912230	+	0.409679i	$0.865641\pi$
$444$	1.63240	0.0774701
$445$	−0.915774	−0.0434119
$446$	−11.6930	−0.553682
$447$	21.0121	0.993840
$448$	−6.92711	−0.327275
$449$	22.9394	1.08258	0.541289	−	0.840837i	$-0.317936\pi$
$449$	22.9394	1.08258	0.541289	+	0.840837i	$0.317936\pi$
$450$	−9.07891	−0.427984
$451$	7.40928	0.348889
$452$	24.6787	1.16079
$453$	11.2749	0.529743
$454$	−18.3790	−0.862568
$455$	1.25118	0.0586560
$456$	9.18744	0.430241
$457$	−29.3028	−1.37073	−0.685364	−	0.728201i	$-0.740357\pi$
$457$	−29.3028	−1.37073	−0.685364	+	0.728201i	$0.740357\pi$
$458$	4.38071	0.204697
$459$	1.00000	0.0466760
$460$	−0.0933148	−0.00435082
$461$	36.5452	1.70208	0.851040	−	0.525100i	$-0.175972\pi$
$461$	36.5452	1.70208	0.851040	+	0.525100i	$0.175972\pi$
$462$	4.79723	0.223187
$463$	26.5273	1.23283	0.616413	−	0.787423i	$-0.288585\pi$
$463$	26.5273	1.23283	0.616413	+	0.787423i	$0.288585\pi$
$464$	14.6076	0.678140
$465$	−1.50051	−0.0695847
$466$	−33.2224	−1.53900
$467$	16.4222	0.759927	0.379964	−	0.925001i	$-0.375937\pi$
$467$	16.4222	0.759927	0.379964	+	0.925001i	$0.375937\pi$
$468$	−2.99701	−0.138537
$469$	−6.99674	−0.323079
$470$	−1.28525	−0.0592842
$471$	1.00000	0.0460776
$472$	−6.37797	−0.293570
$473$	2.26808	0.104286
$474$	19.4815	0.894813
$475$	−37.5850	−1.72452
$476$	−4.43485	−0.203271
$477$	3.33576	0.152734
$478$	−13.7129	−0.627214
$479$	27.1235	1.23930	0.619652	−	0.784877i	$-0.287274\pi$
$479$	27.1235	1.23930	0.619652	+	0.784877i	$0.287274\pi$
$480$	−1.08879	−0.0496961
$481$	2.74739	0.125270
$482$	−28.2857	−1.28838
$483$	−1.38641	−0.0630836
$484$	−13.8449	−0.629314
$485$	−0.246059	−0.0111730
$486$	−1.82604	−0.0828310
$487$	−27.7682	−1.25830	−0.629149	−	0.777285i	$-0.716597\pi$
$487$	−27.7682	−1.25830	−0.629149	+	0.777285i	$0.716597\pi$
$488$	5.21506	0.236075
$489$	−0.929838	−0.0420487
$490$	−1.23814	−0.0559335
$491$	−0.640190	−0.0288914	−0.0144457	−	0.999896i	$-0.504598\pi$
$491$	−0.640190	−0.0288914	−0.0144457	+	0.999896i	$0.504598\pi$
$492$	12.5076	0.563888
$493$	2.98837	0.134589
$494$	−31.0023	−1.39486
$495$	0.132508	0.00595580
$496$	43.7562	1.96471
$497$	12.4745	0.559557
$498$	11.4394	0.512612
$499$	−16.2768	−0.728650	−0.364325	−	0.931272i	$-0.618700\pi$
$499$	−16.2768	−0.728650	−0.364325	+	0.931272i	$0.618700\pi$
$500$	2.23059	0.0997551
$501$	−6.00705	−0.268375
$502$	−8.16965	−0.364629
$503$	11.2630	0.502191	0.251095	−	0.967962i	$-0.419209\pi$
$503$	11.2630	0.502191	0.251095	+	0.967962i	$0.419209\pi$
$504$	−4.03910	−0.179916
$505$	1.94215	0.0864247
$506$	−0.602167	−0.0267696
$507$	7.95592	0.353335
$508$	4.14024	0.183693
$509$	−6.48398	−0.287397	−0.143699	−	0.989621i	$-0.545900\pi$
$509$	−6.48398	−0.287397	−0.143699	+	0.989621i	$0.545900\pi$
$510$	−0.306095	−0.0135541
$511$	39.9609	1.76776
$512$	19.8683	0.878061
$513$	−7.55949	−0.333759
$514$	23.6805	1.04450
$515$	2.96466	0.130639
$516$	3.82876	0.168552
$517$	−3.31917	−0.145977
$518$	−7.42372	−0.326180
$519$	−13.1729	−0.578225
$520$	−0.457550	−0.0200649
$521$	−35.6849	−1.56338	−0.781691	−	0.623666i	$-0.785643\pi$
$521$	−35.6849	−1.56338	−0.781691	+	0.623666i	$0.785643\pi$
$522$	−5.45688	−0.238841
$523$	9.54047	0.417176	0.208588	−	0.978004i	$-0.433113\pi$
$523$	9.54047	0.417176	0.208588	+	0.978004i	$0.433113\pi$
$524$	−17.4744	−0.763374
$525$	16.5236	0.721149
$526$	−12.7506	−0.555953
$527$	8.95148	0.389933
$528$	−3.86405	−0.168161
$529$	−22.8260	−0.992434
$530$	−1.02106	−0.0443520
$531$	5.24784	0.227737
$532$	33.5252	1.45350
$533$	21.0508	0.911813
$534$	−9.97595	−0.431702
$535$	2.16106	0.0934307
$536$	2.55868	0.110518
$537$	−3.81256	−0.164524
$538$	−30.3437	−1.30821
$539$	−3.19750	−0.137726
$540$	0.223688	0.00962599
$541$	9.65625	0.415155	0.207577	−	0.978219i	$-0.433442\pi$
$541$	9.65625	0.415155	0.207577	+	0.978219i	$0.433442\pi$
$542$	−8.08447	−0.347258
$543$	3.28420	0.140938
$544$	6.49528	0.278483
$545$	−1.35837	−0.0581860
$546$	13.6296	0.583294
$547$	37.9903	1.62435	0.812173	−	0.583416i	$-0.198284\pi$
$547$	37.9903	1.62435	0.812173	+	0.583416i	$0.198284\pi$
$548$	5.23004	0.223416
$549$	−4.29098	−0.183135
$550$	7.17680	0.306020
$551$	−22.5905	−0.962388
$552$	0.507003	0.0215795
$553$	−35.4562	−1.50775
$554$	12.8291	0.545057
$555$	−0.205057	−0.00870417
$556$	−21.9229	−0.929737
$557$	−45.7237	−1.93738	−0.968688	−	0.248282i	$-0.920134\pi$
$557$	−45.7237	−1.93738	−0.968688	+	0.248282i	$0.920134\pi$
$558$	−16.3458	−0.691973
$559$	6.44395	0.272550
$560$	2.72315	0.115074
$561$	−0.790492	−0.0333746
$562$	19.0217	0.802381
$563$	19.1959	0.809010	0.404505	−	0.914536i	$-0.367444\pi$
$563$	19.1959	0.809010	0.404505	+	0.914536i	$0.367444\pi$
$564$	−5.60310	−0.235933
$565$	−3.10006	−0.130421
$566$	−21.6012	−0.907968
$567$	3.32340	0.139569
$568$	−4.56187	−0.191412
$569$	46.7144	1.95837	0.979184	−	0.202972i	$-0.0650602\pi$
$569$	46.7144	1.95837	0.979184	+	0.202972i	$0.0650602\pi$
$570$	2.31392	0.0969195
$571$	−10.9140	−0.456736	−0.228368	−	0.973575i	$-0.573339\pi$
$571$	−10.9140	−0.456736	−0.228368	+	0.973575i	$0.573339\pi$
$572$	2.36911	0.0990575
$573$	−16.1713	−0.675565
$574$	−56.8815	−2.37419
$575$	−2.07410	−0.0864961
$576$	−2.08435	−0.0868478
$577$	−13.3027	−0.553800	−0.276900	−	0.960899i	$-0.589307\pi$
$577$	−13.3027	−0.553800	−0.276900	+	0.960899i	$0.589307\pi$
$578$	1.82604	0.0759534
$579$	−15.8666	−0.659392
$580$	0.668461	0.0277563
$581$	−20.8197	−0.863747
$582$	−2.68043	−0.111108
$583$	−2.63689	−0.109209
$584$	−14.6135	−0.604712
$585$	0.376475	0.0155653
$586$	3.53829	0.146166
$587$	20.2978	0.837779	0.418889	−	0.908037i	$-0.362420\pi$
$587$	20.2978	0.837779	0.418889	+	0.908037i	$0.362420\pi$
$588$	−5.39773	−0.222598
$589$	−67.6686	−2.78824
$590$	−1.60634	−0.0661318
$591$	−15.0792	−0.620274
$592$	5.97962	0.245761
$593$	18.9889	0.779782	0.389891	−	0.920861i	$-0.372513\pi$
$593$	18.9889	0.779782	0.389891	+	0.920861i	$0.372513\pi$
$594$	1.44347	0.0592264
$595$	0.557092	0.0228386
$596$	−28.0393	−1.14853
$597$	17.3826	0.711422
$598$	−1.71084	−0.0699616
$599$	29.0150	1.18552	0.592761	−	0.805379i	$-0.298038\pi$
$599$	29.0150	1.18552	0.592761	+	0.805379i	$0.298038\pi$
$600$	−6.04261	−0.246689
$601$	40.1606	1.63818	0.819092	−	0.573661i	$-0.194477\pi$
$601$	40.1606	1.63818	0.819092	+	0.573661i	$0.194477\pi$
$602$	−17.4122	−0.709669
$603$	−2.10530	−0.0857344
$604$	−15.0457	−0.612199
$605$	1.73916	0.0707067
$606$	21.1568	0.859436
$607$	24.9201	1.01148	0.505739	−	0.862687i	$-0.331220\pi$
$607$	24.9201	1.01148	0.505739	+	0.862687i	$0.331220\pi$
$608$	−49.1009	−1.99131
$609$	9.93152	0.402445
$610$	1.31345	0.0531800
$611$	−9.43024	−0.381507
$612$	−1.33443	−0.0539413
$613$	18.3843	0.742537	0.371268	−	0.928526i	$-0.378923\pi$
$613$	18.3843	0.742537	0.371268	+	0.928526i	$0.378923\pi$
$614$	−29.0333	−1.17169
$615$	−1.57117	−0.0633557
$616$	3.19287	0.128645
$617$	−24.2759	−0.977312	−0.488656	−	0.872476i	$-0.662513\pi$
$617$	−24.2759	−0.977312	−0.488656	+	0.872476i	$0.662513\pi$
$618$	32.2955	1.29911
$619$	−13.4919	−0.542285	−0.271143	−	0.962539i	$-0.587402\pi$
$619$	−13.4919	−0.542285	−0.271143	+	0.962539i	$0.587402\pi$
$620$	2.00234	0.0804158
$621$	−0.417165	−0.0167403
$622$	−19.1297	−0.767031
$623$	18.1562	0.727413
$624$	−10.9783	−0.439485
$625$	24.5793	0.983172
$626$	−34.5804	−1.38211
$627$	5.97571	0.238647
$628$	−1.33443	−0.0532497
$629$	1.22329	0.0487757
$630$	−1.01727	−0.0405292
$631$	2.20017	0.0875873	0.0437937	−	0.999041i	$-0.486056\pi$
$631$	2.20017	0.0875873	0.0437937	+	0.999041i	$0.486056\pi$
$632$	12.9662	0.515767
$633$	2.15143	0.0855117
$634$	−19.0834	−0.757900
$635$	−0.520084	−0.0206389
$636$	−4.45136	−0.176508
$637$	−9.08458	−0.359944
$638$	4.31362	0.170778
$639$	3.75354	0.148488
$640$	−1.53956	−0.0608566
$641$	−24.6827	−0.974908	−0.487454	−	0.873149i	$-0.662074\pi$
$641$	−24.6827	−0.974908	−0.487454	+	0.873149i	$0.662074\pi$
$642$	23.5414	0.929105
$643$	12.4848	0.492352	0.246176	−	0.969225i	$-0.420826\pi$
$643$	12.4848	0.492352	0.246176	+	0.969225i	$0.420826\pi$
$644$	1.85007	0.0729028
$645$	−0.480957	−0.0189377
$646$	−13.8039	−0.543109
$647$	−25.6869	−1.00985	−0.504927	−	0.863162i	$-0.668481\pi$
$647$	−25.6869	−1.00985	−0.504927	+	0.863162i	$0.668481\pi$
$648$	−1.21535	−0.0477435
$649$	−4.14837	−0.162838
$650$	20.3903	0.799775
$651$	29.7493	1.16597
$652$	1.24081	0.0485938
$653$	4.38199	0.171480	0.0857402	−	0.996318i	$-0.472674\pi$
$653$	4.38199	0.171480	0.0857402	+	0.996318i	$0.472674\pi$
$654$	−14.7973	−0.578621
$655$	2.19508	0.0857690
$656$	45.8166	1.78884
$657$	12.0241	0.469105
$658$	25.4815	0.993371
$659$	29.0589	1.13197	0.565986	−	0.824415i	$-0.308496\pi$
$659$	29.0589	1.13197	0.565986	+	0.824415i	$0.308496\pi$
$660$	−0.176823	−0.00688284
$661$	−7.59455	−0.295394	−0.147697	−	0.989033i	$-0.547186\pi$
$661$	−7.59455	−0.295394	−0.147697	+	0.989033i	$0.547186\pi$
$662$	−27.7699	−1.07931
$663$	−2.24590	−0.0872237
$664$	7.61369	0.295468
$665$	−4.21133	−0.163308
$666$	−2.23378	−0.0865571
$667$	−1.24664	−0.0482702
$668$	8.01601	0.310149
$669$	6.40349	0.247573
$670$	0.644421	0.0248962
$671$	3.39199	0.130946
$672$	21.5864	0.832712
$673$	−49.2062	−1.89676	−0.948380	−	0.317135i	$-0.897279\pi$
$673$	−49.2062	−1.89676	−0.948380	+	0.317135i	$0.897279\pi$
$674$	−61.1637	−2.35594
$675$	4.97190	0.191369
$676$	−10.6166	−0.408333
$677$	33.6580	1.29358	0.646791	−	0.762667i	$-0.276110\pi$
$677$	33.6580	1.29358	0.646791	+	0.762667i	$0.276110\pi$
$678$	−33.7704	−1.29694
$679$	4.87839	0.187215
$680$	−0.203726	−0.00781255
$681$	10.0649	0.385688
$682$	12.9212	0.494779
$683$	−15.4901	−0.592714	−0.296357	−	0.955077i	$-0.595772\pi$
$683$	−15.4901	−0.592714	−0.296357	+	0.955077i	$0.595772\pi$
$684$	10.0876	0.385710
$685$	−0.656982	−0.0251020
$686$	−17.9332	−0.684692
$687$	−2.39902	−0.0915283
$688$	14.0251	0.534702
$689$	−7.49180	−0.285415
$690$	0.127692	0.00486116
$691$	50.9101	1.93671	0.968356	−	0.249571i	$-0.0802896\pi$
$691$	50.9101	1.93671	0.968356	+	0.249571i	$0.0802896\pi$
$692$	17.5783	0.668227
$693$	−2.62712	−0.0997959
$694$	2.49628	0.0947576
$695$	2.75388	0.104461
$696$	−3.63192	−0.137667
$697$	9.37300	0.355028
$698$	−2.99350	−0.113306
$699$	18.1937	0.688148
$700$	−22.0496	−0.833398
$701$	6.08252	0.229733	0.114867	−	0.993381i	$-0.463356\pi$
$701$	6.08252	0.229733	0.114867	+	0.993381i	$0.463356\pi$
$702$	4.10112	0.154787
$703$	−9.24742	−0.348773
$704$	1.64766	0.0620985
$705$	0.703844	0.0265083
$706$	39.3852	1.48228
$707$	−38.5053	−1.44814
$708$	−7.00289	−0.263185
$709$	15.5988	0.585825	0.292912	−	0.956139i	$-0.405376\pi$
$709$	15.5988	0.585825	0.292912	+	0.956139i	$0.405376\pi$
$710$	−1.14894	−0.0431189
$711$	−10.6687	−0.400106
$712$	−6.63966	−0.248832
$713$	−3.73425	−0.139849
$714$	6.06866	0.227114
$715$	−0.297600	−0.0111296
$716$	5.08761	0.190133
$717$	7.50963	0.280452
$718$	54.5087	2.03425
$719$	−6.88361	−0.256715	−0.128358	−	0.991728i	$-0.540971\pi$
$719$	−6.88361	−0.256715	−0.128358	+	0.991728i	$0.540971\pi$
$720$	0.819389	0.0305368
$721$	−58.7777	−2.18899
$722$	69.6559	2.59233
$723$	15.4902	0.576085
$724$	−4.38254	−0.162876
$725$	14.8579	0.551807
$726$	18.9454	0.703131
$727$	33.4014	1.23879	0.619394	−	0.785080i	$-0.287378\pi$
$727$	33.4014	1.23879	0.619394	+	0.785080i	$0.287378\pi$
$728$	9.07142	0.336209
$729$	1.00000	0.0370370
$730$	−3.68052	−0.136222
$731$	2.86920	0.106121
$732$	5.72603	0.211640
$733$	46.3072	1.71040	0.855198	−	0.518302i	$-0.173436\pi$
$733$	46.3072	1.71040	0.855198	+	0.518302i	$0.173436\pi$
$734$	−19.7084	−0.727450
$735$	0.678045	0.0250101
$736$	−2.70960	−0.0998773
$737$	1.66422	0.0613024
$738$	−17.1155	−0.630030
$739$	−16.4868	−0.606478	−0.303239	−	0.952915i	$-0.598068\pi$
$739$	−16.4868	−0.606478	−0.303239	+	0.952915i	$0.598068\pi$
$740$	0.273634	0.0100590
$741$	16.9779	0.623698
$742$	20.2436	0.743167
$743$	−42.1914	−1.54785	−0.773926	−	0.633276i	$-0.781710\pi$
$743$	−42.1914	−1.54785	−0.773926	+	0.633276i	$0.781710\pi$
$744$	−10.8792	−0.398851
$745$	3.52221	0.129044
$746$	−3.79793	−0.139052
$747$	−6.26459	−0.229209
$748$	1.05486	0.0385695
$749$	−42.8453	−1.56553
$750$	−3.05235	−0.111456
$751$	27.8285	1.01548	0.507738	−	0.861512i	$-0.330482\pi$
$751$	27.8285	1.01548	0.507738	+	0.861512i	$0.330482\pi$
$752$	−20.5247	−0.748458
$753$	4.47396	0.163040
$754$	12.2556	0.446324
$755$	1.88999	0.0687838
$756$	−4.43485	−0.161294
$757$	−11.3376	−0.412072	−0.206036	−	0.978544i	$-0.566056\pi$
$757$	−11.3376	−0.412072	−0.206036	+	0.978544i	$0.566056\pi$
$758$	57.7823	2.09875
$759$	0.329766	0.0119697
$760$	1.54007	0.0558641
$761$	−14.8333	−0.537706	−0.268853	−	0.963181i	$-0.586645\pi$
$761$	−14.8333	−0.537706	−0.268853	+	0.963181i	$0.586645\pi$
$762$	−5.66551	−0.205240
$763$	26.9311	0.974971
$764$	21.5795	0.780719
$765$	0.167627	0.00606058
$766$	27.2165	0.983374
$767$	−11.7861	−0.425573
$768$	−20.9399	−0.755603
$769$	−45.3793	−1.63642	−0.818210	−	0.574920i	$-0.805033\pi$
$769$	−45.3793	−1.63642	−0.818210	+	0.574920i	$0.805033\pi$
$770$	0.804147	0.0289795
$771$	−12.9682	−0.467038
$772$	21.1729	0.762029
$773$	−42.4650	−1.52736	−0.763680	−	0.645595i	$-0.776609\pi$
$773$	−42.4650	−1.52736	−0.763680	+	0.645595i	$0.776609\pi$
$774$	−5.23929	−0.188322
$775$	44.5059	1.59870
$776$	−1.78401	−0.0640421
$777$	4.06547	0.145848
$778$	−31.6063	−1.13314
$779$	−70.8550	−2.53864
$780$	−0.502381	−0.0179881
$781$	−2.96714	−0.106173
$782$	−0.761762	−0.0272406
$783$	2.98837	0.106795
$784$	−19.7724	−0.706156
$785$	0.167627	0.00598288
$786$	23.9121	0.852915
$787$	35.9496	1.28146	0.640732	−	0.767765i	$-0.278631\pi$
$787$	35.9496	1.28146	0.640732	+	0.767765i	$0.278631\pi$
$788$	20.1222	0.716822
$789$	6.98264	0.248589
$790$	3.26563	0.116186
$791$	61.4621	2.18534
$792$	0.960726	0.0341379
$793$	9.63713	0.342225
$794$	55.5885	1.97276
$795$	0.559166	0.0198316
$796$	−23.1959	−0.822157
$797$	−23.1830	−0.821183	−0.410592	−	0.911819i	$-0.634678\pi$
$797$	−23.1830	−0.821183	−0.410592	+	0.911819i	$0.634678\pi$
$798$	−45.8760	−1.62399
$799$	−4.19886	−0.148545
$800$	32.2939	1.14176
$801$	5.46315	0.193031
$802$	−57.4557	−2.02883
$803$	−9.50496	−0.335423
$804$	2.80938	0.0990792
$805$	−0.232400	−0.00819101
$806$	36.7111	1.29309
$807$	16.6172	0.584952
$808$	14.0812	0.495376
$809$	7.71907	0.271388	0.135694	−	0.990751i	$-0.456674\pi$
$809$	7.71907	0.271388	0.135694	+	0.990751i	$0.456674\pi$
$810$	−0.306095	−0.0107551
$811$	−29.6601	−1.04151	−0.520754	−	0.853707i	$-0.674349\pi$
$811$	−29.6601	−1.04151	−0.520754	+	0.853707i	$0.674349\pi$
$812$	−13.2530	−0.465088
$813$	4.42732	0.155273
$814$	1.76578	0.0618906
$815$	−0.155866	−0.00545976
$816$	−4.88815	−0.171120
$817$	−21.6897	−0.758826
$818$	−45.9791	−1.60762
$819$	−7.46402	−0.260814
$820$	2.09662	0.0732173
$821$	−41.8880	−1.46190	−0.730951	−	0.682430i	$-0.760923\pi$
$821$	−41.8880	−1.46190	−0.730951	+	0.682430i	$0.760923\pi$
$822$	−7.15680	−0.249622
$823$	−49.4767	−1.72465	−0.862325	−	0.506356i	$-0.830992\pi$
$823$	−49.4767	−1.72465	−0.862325	+	0.506356i	$0.830992\pi$
$824$	21.4948	0.748805
$825$	−3.93025	−0.136834
$826$	31.8474	1.10811
$827$	31.8902	1.10893	0.554466	−	0.832207i	$-0.312923\pi$
$827$	31.8902	1.10893	0.554466	+	0.832207i	$0.312923\pi$
$828$	0.556680	0.0193460
$829$	10.8768	0.377765	0.188883	−	0.982000i	$-0.439513\pi$
$829$	10.8768	0.377765	0.188883	+	0.982000i	$0.439513\pi$
$830$	1.91756	0.0665595
$831$	−7.02563	−0.243717
$832$	4.68124	0.162293
$833$	−4.04495	−0.140149
$834$	29.9993	1.03879
$835$	−1.00695	−0.0348468
$836$	−7.97419	−0.275793
$837$	8.95148	0.309408
$838$	72.0566	2.48915
$839$	−54.5822	−1.88439	−0.942193	−	0.335071i	$-0.891240\pi$
$839$	−54.5822	−1.88439	−0.942193	+	0.335071i	$0.891240\pi$
$840$	−0.677063	−0.0233609
$841$	−20.0697	−0.692058
$842$	5.76766	0.198767
$843$	−10.4169	−0.358776
$844$	−2.87094	−0.0988219
$845$	1.33363	0.0458783
$846$	7.66730	0.263607
$847$	−34.4806	−1.18477
$848$	−16.3057	−0.559941
$849$	11.8295	0.405988
$850$	9.07891	0.311404
$851$	−0.510313	−0.0174933
$852$	−5.00885	−0.171600
$853$	−8.97469	−0.307288	−0.153644	−	0.988126i	$-0.549101\pi$
$853$	−8.97469	−0.307288	−0.153644	+	0.988126i	$0.549101\pi$
$854$	−26.0405	−0.891089
$855$	−1.26718	−0.0433365
$856$	15.6684	0.535533
$857$	21.7054	0.741442	0.370721	−	0.928744i	$-0.379111\pi$
$857$	21.7054	0.741442	0.370721	+	0.928744i	$0.379111\pi$
$858$	−3.24190	−0.110677
$859$	51.6995	1.76396	0.881981	−	0.471284i	$-0.156209\pi$
$859$	51.6995	1.76396	0.881981	+	0.471284i	$0.156209\pi$
$860$	0.641805	0.0218854
$861$	31.1502	1.06159
$862$	16.0716	0.547402
$863$	41.7475	1.42110	0.710551	−	0.703646i	$-0.248446\pi$
$863$	41.7475	1.42110	0.710551	+	0.703646i	$0.248446\pi$
$864$	6.49528	0.220974
$865$	−2.20813	−0.0750788
$866$	23.4282	0.796124
$867$	−1.00000	−0.0339618
$868$	−39.6985	−1.34745
$869$	8.43350	0.286087
$870$	−0.914724	−0.0310120
$871$	4.72830	0.160212
$872$	−9.84859	−0.333515
$873$	1.46789	0.0496806
$874$	5.75853	0.194785
$875$	5.55527	0.187802
$876$	−16.0454	−0.542123
$877$	41.0439	1.38595	0.692977	−	0.720959i	$-0.256299\pi$
$877$	41.0439	1.38595	0.692977	+	0.720959i	$0.256299\pi$
$878$	−39.4636	−1.33183
$879$	−1.93768	−0.0653564
$880$	−0.647720	−0.0218346
$881$	−8.84113	−0.297865	−0.148933	−	0.988847i	$-0.547584\pi$
$881$	−8.84113	−0.297865	−0.148933	+	0.988847i	$0.547584\pi$
$882$	7.38626	0.248708
$883$	0.749367	0.0252182	0.0126091	−	0.999921i	$-0.495986\pi$
$883$	0.749367	0.0252182	0.0126091	+	0.999921i	$0.495986\pi$
$884$	2.99701	0.100800
$885$	0.879682	0.0295702
$886$	−70.1209	−2.35576
$887$	0.920209	0.0308976	0.0154488	−	0.999881i	$-0.495082\pi$
$887$	0.920209	0.0308976	0.0154488	+	0.999881i	$0.495082\pi$
$888$	−1.48673	−0.0498912
$889$	10.3112	0.345827
$890$	−1.67224	−0.0560537
$891$	−0.790492	−0.0264825
$892$	−8.54503	−0.286109
$893$	31.7412	1.06218
$894$	38.3691	1.28325
$895$	−0.639089	−0.0213624
$896$	30.5235	1.01972
$897$	0.936913	0.0312826
$898$	41.8884	1.39783
$899$	26.7503	0.892172
$900$	−6.63467	−0.221156
$901$	−3.33576	−0.111130
$902$	13.5297	0.450488
$903$	9.53549	0.317321
$904$	−22.4764	−0.747555
$905$	0.550521	0.0183000
$906$	20.5885	0.684008
$907$	−49.8197	−1.65424	−0.827118	−	0.562029i	$-0.810021\pi$
$907$	−49.8197	−1.65424	−0.827118	+	0.562029i	$0.810021\pi$
$908$	−13.4310	−0.445722
$909$	−11.5861	−0.384288
$910$	2.28470	0.0757371
$911$	−17.5975	−0.583030	−0.291515	−	0.956566i	$-0.594159\pi$
$911$	−17.5975	−0.583030	−0.291515	+	0.956566i	$0.594159\pi$
$912$	36.9519	1.22360
$913$	4.95211	0.163891
$914$	−53.5082	−1.76989
$915$	−0.719287	−0.0237789
$916$	3.20133	0.105775
$917$	−43.5199	−1.43715
$918$	1.82604	0.0602684
$919$	12.1033	0.399251	0.199626	−	0.979872i	$-0.436027\pi$
$919$	12.1033	0.399251	0.199626	+	0.979872i	$0.436027\pi$
$920$	0.0849876	0.00280196
$921$	15.8996	0.523909
$922$	66.7332	2.19774
$923$	−8.43008	−0.277479
$924$	3.50571	0.115330
$925$	6.08206	0.199977
$926$	48.4399	1.59183
$927$	−17.6860	−0.580885
$928$	19.4103	0.637173
$929$	−52.3519	−1.71761	−0.858805	−	0.512302i	$-0.828793\pi$
$929$	−52.3519	−1.71761	−0.858805	+	0.512302i	$0.828793\pi$
$930$	−2.74000	−0.0898483
$931$	30.5778	1.00215
$932$	−24.2783	−0.795261
$933$	10.4760	0.342970
$934$	29.9876	0.981224
$935$	−0.132508	−0.00433348
$936$	2.72956	0.0892186
$937$	38.4380	1.25571	0.627857	−	0.778329i	$-0.283932\pi$
$937$	38.4380	1.25571	0.627857	+	0.778329i	$0.283932\pi$
$938$	−12.7763	−0.417162
$939$	18.9373	0.617997
$940$	−0.939234	−0.0306344
$941$	50.9150	1.65978	0.829891	−	0.557926i	$-0.188403\pi$
$941$	50.9150	1.65978	0.829891	+	0.557926i	$0.188403\pi$
$942$	1.82604	0.0594957
$943$	−3.91009	−0.127330
$944$	−25.6522	−0.834909
$945$	0.557092	0.0181222
$946$	4.14161	0.134655
$947$	−18.0649	−0.587031	−0.293516	−	0.955954i	$-0.594825\pi$
$947$	−18.0649	−0.587031	−0.293516	+	0.955954i	$0.594825\pi$
$948$	14.2366	0.462384
$949$	−27.0050	−0.876619
$950$	−68.6319	−2.22671
$951$	10.4507	0.338887
$952$	4.03910	0.130908
$953$	31.9241	1.03412	0.517061	−	0.855949i	$-0.327026\pi$
$953$	31.9241	1.03412	0.517061	+	0.855949i	$0.327026\pi$
$954$	6.09125	0.197212
$955$	−2.71075	−0.0877178
$956$	−10.0211	−0.324106
$957$	−2.36228	−0.0763616
$958$	49.5286	1.60020
$959$	13.0254	0.420611
$960$	−0.349394	−0.0112766
$961$	49.1290	1.58481
$962$	5.01684	0.161750
$963$	−12.8920	−0.415440
$964$	−20.6706	−0.665755
$965$	−2.65967	−0.0856179
$966$	−2.53164	−0.0814541
$967$	58.4931	1.88101	0.940506	−	0.339778i	$-0.110352\pi$
$967$	58.4931	1.88101	0.940506	+	0.339778i	$0.110352\pi$
$968$	12.6094	0.405282
$969$	7.55949	0.242846
$970$	−0.449314	−0.0144266
$971$	46.3161	1.48635	0.743177	−	0.669095i	$-0.233318\pi$
$971$	46.3161	1.48635	0.743177	+	0.669095i	$0.233318\pi$
$972$	−1.33443	−0.0428020
$973$	−54.5987	−1.75035
$974$	−50.7060	−1.62472
$975$	−11.1664	−0.357611
$976$	20.9750	0.671393
$977$	−3.33760	−0.106779	−0.0533897	−	0.998574i	$-0.517003\pi$
$977$	−3.33760	−0.106779	−0.0533897	+	0.998574i	$0.517003\pi$
$978$	−1.69792	−0.0542936
$979$	−4.31858	−0.138022
$980$	−0.904807	−0.0289030
$981$	8.10349	0.258724
$982$	−1.16901	−0.0373047
$983$	−11.3562	−0.362208	−0.181104	−	0.983464i	$-0.557967\pi$
$983$	−11.3562	−0.362208	−0.181104	+	0.983464i	$0.557967\pi$
$984$	−11.3915	−0.363147
$985$	−2.52768	−0.0805387
$986$	5.45688	0.173783
$987$	−13.9545	−0.444176
$988$	−22.6559	−0.720779
$989$	−1.19693	−0.0380602
$990$	0.241966	0.00769017
$991$	45.9817	1.46066	0.730328	−	0.683097i	$-0.239367\pi$
$991$	45.9817	1.46066	0.730328	+	0.683097i	$0.239367\pi$
$992$	58.1423	1.84602
$993$	15.2077	0.482601
$994$	22.7790	0.722504
$995$	2.91380	0.0923736
$996$	8.35968	0.264887
$997$	−18.8695	−0.597604	−0.298802	−	0.954315i	$-0.596587\pi$
$997$	−18.8695	−0.597604	−0.298802	+	0.954315i	$0.596587\pi$
$998$	−29.7222	−0.940838
$999$	1.22329	0.0387031

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.39	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.39	✓	48	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q+1.82604 q^{2} -1.00000 q^{3} +1.33443 q^{4} -0.167627 q^{5} -1.82604 q^{6} +3.32340 q^{7} -1.21535 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.82604 q^{2} -1.00000 q^{3} +1.33443 q^{4} -0.167627 q^{5} -1.82604 q^{6} +3.32340 q^{7} -1.21535 q^{8} +1.00000 q^{9} -0.306095 q^{10} -0.790492 q^{11} -1.33443 q^{12} -2.24590 q^{13} +6.06866 q^{14} +0.167627 q^{15} -4.88815 q^{16} -1.00000 q^{17} +1.82604 q^{18} +7.55949 q^{19} -0.223688 q^{20} -3.32340 q^{21} -1.44347 q^{22} +0.417165 q^{23} +1.21535 q^{24} -4.97190 q^{25} -4.10112 q^{26} -1.00000 q^{27} +4.43485 q^{28} -2.98837 q^{29} +0.306095 q^{30} -8.95148 q^{31} -6.49528 q^{32} +0.790492 q^{33} -1.82604 q^{34} -0.557092 q^{35} +1.33443 q^{36} -1.22329 q^{37} +13.8039 q^{38} +2.24590 q^{39} +0.203726 q^{40} -9.37300 q^{41} -6.06866 q^{42} -2.86920 q^{43} -1.05486 q^{44} -0.167627 q^{45} +0.761762 q^{46} +4.19886 q^{47} +4.88815 q^{48} +4.04495 q^{49} -9.07891 q^{50} +1.00000 q^{51} -2.99701 q^{52} +3.33576 q^{53} -1.82604 q^{54} +0.132508 q^{55} -4.03910 q^{56} -7.55949 q^{57} -5.45688 q^{58} +5.24784 q^{59} +0.223688 q^{60} -4.29098 q^{61} -16.3458 q^{62} +3.32340 q^{63} -2.08435 q^{64} +0.376475 q^{65} +1.44347 q^{66} -2.10530 q^{67} -1.33443 q^{68} -0.417165 q^{69} -1.01727 q^{70} +3.75354 q^{71} -1.21535 q^{72} +12.0241 q^{73} -2.23378 q^{74} +4.97190 q^{75} +10.0876 q^{76} -2.62712 q^{77} +4.10112 q^{78} -10.6687 q^{79} +0.819389 q^{80} +1.00000 q^{81} -17.1155 q^{82} -6.26459 q^{83} -4.43485 q^{84} +0.167627 q^{85} -5.23929 q^{86} +2.98837 q^{87} +0.960726 q^{88} +5.46315 q^{89} -0.306095 q^{90} -7.46402 q^{91} +0.556680 q^{92} +8.95148 q^{93} +7.66730 q^{94} -1.26718 q^{95} +6.49528 q^{96} +1.46789 q^{97} +7.38626 q^{98} -0.790492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * q^99

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