LMFDB - Embedded newform 8002.2.a.f.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.8962916974$
Analytic rank:	$1$
Dimension:	$89$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.19996 q^{3} +1.00000 q^{4} -2.22300 q^{5} +2.19996 q^{6} +1.97846 q^{7} -1.00000 q^{8} +1.83980 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.19996 q^{3} +1.00000 q^{4} -2.22300 q^{5} +2.19996 q^{6} +1.97846 q^{7} -1.00000 q^{8} +1.83980 q^{9} +2.22300 q^{10} -2.62642 q^{11} -2.19996 q^{12} -0.806727 q^{13} -1.97846 q^{14} +4.89050 q^{15} +1.00000 q^{16} -5.61120 q^{17} -1.83980 q^{18} -2.10990 q^{19} -2.22300 q^{20} -4.35253 q^{21} +2.62642 q^{22} +0.710150 q^{23} +2.19996 q^{24} -0.0582708 q^{25} +0.806727 q^{26} +2.55238 q^{27} +1.97846 q^{28} -0.956772 q^{29} -4.89050 q^{30} +9.64235 q^{31} -1.00000 q^{32} +5.77801 q^{33} +5.61120 q^{34} -4.39812 q^{35} +1.83980 q^{36} +5.08736 q^{37} +2.10990 q^{38} +1.77476 q^{39} +2.22300 q^{40} -9.75506 q^{41} +4.35253 q^{42} +12.8752 q^{43} -2.62642 q^{44} -4.08988 q^{45} -0.710150 q^{46} +1.40781 q^{47} -2.19996 q^{48} -3.08569 q^{49} +0.0582708 q^{50} +12.3444 q^{51} -0.806727 q^{52} +4.57788 q^{53} -2.55238 q^{54} +5.83854 q^{55} -1.97846 q^{56} +4.64169 q^{57} +0.956772 q^{58} -6.21383 q^{59} +4.89050 q^{60} -15.2147 q^{61} -9.64235 q^{62} +3.63998 q^{63} +1.00000 q^{64} +1.79335 q^{65} -5.77801 q^{66} +11.7275 q^{67} -5.61120 q^{68} -1.56230 q^{69} +4.39812 q^{70} +3.35656 q^{71} -1.83980 q^{72} +3.90662 q^{73} -5.08736 q^{74} +0.128193 q^{75} -2.10990 q^{76} -5.19628 q^{77} -1.77476 q^{78} -6.45058 q^{79} -2.22300 q^{80} -11.1345 q^{81} +9.75506 q^{82} +6.24555 q^{83} -4.35253 q^{84} +12.4737 q^{85} -12.8752 q^{86} +2.10486 q^{87} +2.62642 q^{88} +9.16497 q^{89} +4.08988 q^{90} -1.59608 q^{91} +0.710150 q^{92} -21.2127 q^{93} -1.40781 q^{94} +4.69031 q^{95} +2.19996 q^{96} +4.38886 q^{97} +3.08569 q^{98} -4.83210 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) $ 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9} + 18 q^{10} - 26 q^{11} - 12 q^{12} + 2 q^{13} + 27 q^{14} - 21 q^{15} + 89 q^{16} - 60 q^{17} - 95 q^{18} + q^{19} - 18 q^{20} - 6 q^{21} + 26 q^{22} - 45 q^{23} + 12 q^{24} + 107 q^{25} - 2 q^{26} - 45 q^{27} - 27 q^{28} - 18 q^{29} + 21 q^{30} - 38 q^{31} - 89 q^{32} - 29 q^{33} + 60 q^{34} - 47 q^{35} + 95 q^{36} - 15 q^{37} - q^{38} - 38 q^{39} + 18 q^{40} - 50 q^{41} + 6 q^{42} - 15 q^{43} - 26 q^{44} - 35 q^{45} + 45 q^{46} - 121 q^{47} - 12 q^{48} + 132 q^{49} - 107 q^{50} + 6 q^{51} + 2 q^{52} - 46 q^{53} + 45 q^{54} - 37 q^{55} + 27 q^{56} - 42 q^{57} + 18 q^{58} - 34 q^{59} - 21 q^{60} + 41 q^{61} + 38 q^{62} - 131 q^{63} + 89 q^{64} - 57 q^{65} + 29 q^{66} - 11 q^{67} - 60 q^{68} + 15 q^{69} + 47 q^{70} - 66 q^{71} - 95 q^{72} - 47 q^{73} + 15 q^{74} - 46 q^{75} + q^{76} - 106 q^{77} + 38 q^{78} - 51 q^{79} - 18 q^{80} + 113 q^{81} + 50 q^{82} - 141 q^{83} - 6 q^{84} - 7 q^{85} + 15 q^{86} - 110 q^{87} + 26 q^{88} - 30 q^{89} + 35 q^{90} + 37 q^{91} - 45 q^{92} - 44 q^{93} + 121 q^{94} - 98 q^{95} + 12 q^{96} + 3 q^{97} - 132 q^{98} - 71 q^{99}+O(q^{100}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9 + 18 * q^10 - 26 * q^11 - 12 * q^12 + 2 * q^13 + 27 * q^14 - 21 * q^15 + 89 * q^16 - 60 * q^17 - 95 * q^18 + q^19 - 18 * q^20 - 6 * q^21 + 26 * q^22 - 45 * q^23 + 12 * q^24 + 107 * q^25 - 2 * q^26 - 45 * q^27 - 27 * q^28 - 18 * q^29 + 21 * q^30 - 38 * q^31 - 89 * q^32 - 29 * q^33 + 60 * q^34 - 47 * q^35 + 95 * q^36 - 15 * q^37 - q^38 - 38 * q^39 + 18 * q^40 - 50 * q^41 + 6 * q^42 - 15 * q^43 - 26 * q^44 - 35 * q^45 + 45 * q^46 - 121 * q^47 - 12 * q^48 + 132 * q^49 - 107 * q^50 + 6 * q^51 + 2 * q^52 - 46 * q^53 + 45 * q^54 - 37 * q^55 + 27 * q^56 - 42 * q^57 + 18 * q^58 - 34 * q^59 - 21 * q^60 + 41 * q^61 + 38 * q^62 - 131 * q^63 + 89 * q^64 - 57 * q^65 + 29 * q^66 - 11 * q^67 - 60 * q^68 + 15 * q^69 + 47 * q^70 - 66 * q^71 - 95 * q^72 - 47 * q^73 + 15 * q^74 - 46 * q^75 + q^76 - 106 * q^77 + 38 * q^78 - 51 * q^79 - 18 * q^80 + 113 * q^81 + 50 * q^82 - 141 * q^83 - 6 * q^84 - 7 * q^85 + 15 * q^86 - 110 * q^87 + 26 * q^88 - 30 * q^89 + 35 * q^90 + 37 * q^91 - 45 * q^92 - 44 * q^93 + 121 * q^94 - 98 * q^95 + 12 * q^96 + 3 * q^97 - 132 * q^98 - 71 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.19996	−1.27014	−0.635072	−	0.772453i	$-0.719030\pi$
$3$	−2.19996	−1.27014	−0.635072	+	0.772453i	$0.719030\pi$
$4$	1.00000	0.500000
$5$	−2.22300	−0.994156	−0.497078	−	0.867706i	$-0.665594\pi$
$5$	−2.22300	−0.994156	−0.497078	+	0.867706i	$0.665594\pi$
$6$	2.19996	0.898128
$7$	1.97846	0.747788	0.373894	−	0.927471i	$-0.378022\pi$
$7$	1.97846	0.747788	0.373894	+	0.927471i	$0.378022\pi$
$8$	−1.00000	−0.353553
$9$	1.83980	0.613268
$10$	2.22300	0.702974
$11$	−2.62642	−0.791896	−0.395948	−	0.918273i	$-0.629584\pi$
$11$	−2.62642	−0.791896	−0.395948	+	0.918273i	$0.629584\pi$
$12$	−2.19996	−0.635072
$13$	−0.806727	−0.223746	−0.111873	−	0.993723i	$-0.535685\pi$
$13$	−0.806727	−0.223746	−0.111873	+	0.993723i	$0.535685\pi$
$14$	−1.97846	−0.528766
$15$	4.89050	1.26272
$16$	1.00000	0.250000
$17$	−5.61120	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$17$	−5.61120	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$18$	−1.83980	−0.433646
$19$	−2.10990	−0.484044	−0.242022	−	0.970271i	$-0.577811\pi$
$19$	−2.10990	−0.484044	−0.242022	+	0.970271i	$0.577811\pi$
$20$	−2.22300	−0.497078
$21$	−4.35253	−0.949799
$22$	2.62642	0.559955
$23$	0.710150	0.148076	0.0740382	−	0.997255i	$-0.476411\pi$
$23$	0.710150	0.148076	0.0740382	+	0.997255i	$0.476411\pi$
$24$	2.19996	0.449064
$25$	−0.0582708	−0.0116542
$26$	0.806727	0.158212
$27$	2.55238	0.491206
$28$	1.97846	0.373894
$29$	−0.956772	−0.177668	−0.0888341	−	0.996046i	$-0.528314\pi$
$29$	−0.956772	−0.177668	−0.0888341	+	0.996046i	$0.528314\pi$
$30$	−4.89050	−0.892879
$31$	9.64235	1.73182	0.865909	−	0.500202i	$-0.166741\pi$
$31$	9.64235	1.73182	0.865909	+	0.500202i	$0.166741\pi$
$32$	−1.00000	−0.176777
$33$	5.77801	1.00582
$34$	5.61120	0.962313
$35$	−4.39812	−0.743418
$36$	1.83980	0.306634
$37$	5.08736	0.836356	0.418178	−	0.908365i	$-0.362669\pi$
$37$	5.08736	0.836356	0.418178	+	0.908365i	$0.362669\pi$
$38$	2.10990	0.342271
$39$	1.77476	0.284190
$40$	2.22300	0.351487
$41$	−9.75506	−1.52348	−0.761742	−	0.647880i	$-0.775656\pi$
$41$	−9.75506	−1.52348	−0.761742	+	0.647880i	$0.775656\pi$
$42$	4.35253	0.671609
$43$	12.8752	1.96345	0.981724	−	0.190308i	$-0.0609487\pi$
$43$	12.8752	1.96345	0.981724	+	0.190308i	$0.0609487\pi$
$44$	−2.62642	−0.395948
$45$	−4.08988	−0.609684
$46$	−0.710150	−0.104706
$47$	1.40781	0.205350	0.102675	−	0.994715i	$-0.467260\pi$
$47$	1.40781	0.205350	0.102675	+	0.994715i	$0.467260\pi$
$48$	−2.19996	−0.317536
$49$	−3.08569	−0.440813
$50$	0.0582708	0.00824074
$51$	12.3444	1.72856
$52$	−0.806727	−0.111873
$53$	4.57788	0.628820	0.314410	−	0.949287i	$-0.398193\pi$
$53$	4.57788	0.628820	0.314410	+	0.949287i	$0.398193\pi$
$54$	−2.55238	−0.347335
$55$	5.83854	0.787268
$56$	−1.97846	−0.264383
$57$	4.64169	0.614806
$58$	0.956772	0.125630
$59$	−6.21383	−0.808971	−0.404486	−	0.914544i	$-0.632549\pi$
$59$	−6.21383	−0.808971	−0.404486	+	0.914544i	$0.632549\pi$
$60$	4.89050	0.631361
$61$	−15.2147	−1.94804	−0.974022	−	0.226454i	$-0.927287\pi$
$61$	−15.2147	−1.94804	−0.974022	+	0.226454i	$0.927287\pi$
$62$	−9.64235	−1.22458
$63$	3.63998	0.458594
$64$	1.00000	0.125000
$65$	1.79335	0.222438
$66$	−5.77801	−0.711224
$67$	11.7275	1.43274	0.716371	−	0.697720i	$-0.245802\pi$
$67$	11.7275	1.43274	0.716371	+	0.697720i	$0.245802\pi$
$68$	−5.61120	−0.680458
$69$	−1.56230	−0.188079
$70$	4.39812	0.525676
$71$	3.35656	0.398351	0.199175	−	0.979964i	$-0.436174\pi$
$71$	3.35656	0.398351	0.199175	+	0.979964i	$0.436174\pi$
$72$	−1.83980	−0.216823
$73$	3.90662	0.457235	0.228617	−	0.973516i	$-0.426580\pi$
$73$	3.90662	0.457235	0.228617	+	0.973516i	$0.426580\pi$
$74$	−5.08736	−0.591393
$75$	0.128193	0.0148025
$76$	−2.10990	−0.242022
$77$	−5.19628	−0.592171
$78$	−1.77476	−0.200952
$79$	−6.45058	−0.725747	−0.362873	−	0.931838i	$-0.618204\pi$
$79$	−6.45058	−0.725747	−0.362873	+	0.931838i	$0.618204\pi$
$80$	−2.22300	−0.248539
$81$	−11.1345	−1.23717
$82$	9.75506	1.07727
$83$	6.24555	0.685538	0.342769	−	0.939420i	$-0.388635\pi$
$83$	6.24555	0.685538	0.342769	+	0.939420i	$0.388635\pi$
$84$	−4.35253	−0.474900
$85$	12.4737	1.35296
$86$	−12.8752	−1.38837
$87$	2.10486	0.225664
$88$	2.62642	0.279978
$89$	9.16497	0.971485	0.485742	−	0.874102i	$-0.338549\pi$
$89$	9.16497	0.971485	0.485742	+	0.874102i	$0.338549\pi$
$90$	4.08988	0.431112
$91$	−1.59608	−0.167315
$92$	0.710150	0.0740382
$93$	−21.2127	−2.19966
$94$	−1.40781	−0.145204
$95$	4.69031	0.481215
$96$	2.19996	0.224532
$97$	4.38886	0.445621	0.222810	−	0.974862i	$-0.428477\pi$
$97$	4.38886	0.445621	0.222810	+	0.974862i	$0.428477\pi$
$98$	3.08569	0.311702
$99$	−4.83210	−0.485645
$100$	−0.0582708	−0.00582708
$101$	2.71647	0.270299	0.135150	−	0.990825i	$-0.456849\pi$
$101$	2.71647	0.270299	0.135150	+	0.990825i	$0.456849\pi$
$102$	−12.3444	−1.22228
$103$	0.909821	0.0896473	0.0448237	−	0.998995i	$-0.485727\pi$
$103$	0.909821	0.0896473	0.0448237	+	0.998995i	$0.485727\pi$
$104$	0.806727	0.0791061
$105$	9.67567	0.944248
$106$	−4.57788	−0.444643
$107$	−20.1568	−1.94863	−0.974316	−	0.225186i	$-0.927701\pi$
$107$	−20.1568	−1.94863	−0.974316	+	0.225186i	$0.927701\pi$
$108$	2.55238	0.245603
$109$	7.95041	0.761511	0.380756	−	0.924676i	$-0.375664\pi$
$109$	7.95041	0.761511	0.380756	+	0.924676i	$0.375664\pi$
$110$	−5.83854	−0.556683
$111$	−11.1920	−1.06229
$112$	1.97846	0.186947
$113$	−8.55336	−0.804632	−0.402316	−	0.915501i	$-0.631795\pi$
$113$	−8.55336	−0.804632	−0.402316	+	0.915501i	$0.631795\pi$
$114$	−4.64169	−0.434734
$115$	−1.57866	−0.147211
$116$	−0.956772	−0.0888341
$117$	−1.48422	−0.137216
$118$	6.21383	0.572029
$119$	−11.1015	−1.01768
$120$	−4.89050	−0.446440
$121$	−4.10190	−0.372900
$122$	15.2147	1.37748
$123$	21.4607	1.93505
$124$	9.64235	0.865909
$125$	11.2445	1.00574
$126$	−3.63998	−0.324275
$127$	−7.58025	−0.672639	−0.336319	−	0.941748i	$-0.609182\pi$
$127$	−7.58025	−0.672639	−0.336319	+	0.941748i	$0.609182\pi$
$128$	−1.00000	−0.0883883
$129$	−28.3249	−2.49386
$130$	−1.79335	−0.157288
$131$	5.19286	0.453702	0.226851	−	0.973929i	$-0.427157\pi$
$131$	5.19286	0.453702	0.226851	+	0.973929i	$0.427157\pi$
$132$	5.77801	0.502911
$133$	−4.17435	−0.361962
$134$	−11.7275	−1.01310
$135$	−5.67394	−0.488335
$136$	5.61120	0.481156
$137$	6.37061	0.544278	0.272139	−	0.962258i	$-0.412269\pi$
$137$	6.37061	0.544278	0.272139	+	0.962258i	$0.412269\pi$
$138$	1.56230	0.132992
$139$	1.02203	0.0866872	0.0433436	−	0.999060i	$-0.486199\pi$
$139$	1.02203	0.0866872	0.0433436	+	0.999060i	$0.486199\pi$
$140$	−4.39812	−0.371709
$141$	−3.09711	−0.260824
$142$	−3.35656	−0.281676
$143$	2.11881	0.177184
$144$	1.83980	0.153317
$145$	2.12690	0.176630
$146$	−3.90662	−0.323314
$147$	6.78838	0.559896
$148$	5.08736	0.418178
$149$	23.5381	1.92832	0.964160	−	0.265323i	$-0.0854785\pi$
$149$	23.5381	1.92832	0.964160	+	0.265323i	$0.0854785\pi$
$150$	−0.128193	−0.0104669
$151$	23.8870	1.94390	0.971948	−	0.235197i	$-0.0755736\pi$
$151$	23.8870	1.94390	0.971948	+	0.235197i	$0.0755736\pi$
$152$	2.10990	0.171135
$153$	−10.3235	−0.834606
$154$	5.19628	0.418728
$155$	−21.4349	−1.72170
$156$	1.77476	0.142095
$157$	2.85329	0.227718	0.113859	−	0.993497i	$-0.463679\pi$
$157$	2.85329	0.227718	0.113859	+	0.993497i	$0.463679\pi$
$158$	6.45058	0.513180
$159$	−10.0711	−0.798692
$160$	2.22300	0.175744
$161$	1.40500	0.110730
$162$	11.1345	0.874812
$163$	10.6734	0.836004	0.418002	−	0.908446i	$-0.362730\pi$
$163$	10.6734	0.836004	0.418002	+	0.908446i	$0.362730\pi$
$164$	−9.75506	−0.761742
$165$	−12.8445	−0.999945
$166$	−6.24555	−0.484749
$167$	−10.9974	−0.851002	−0.425501	−	0.904958i	$-0.639902\pi$
$167$	−10.9974	−0.851002	−0.425501	+	0.904958i	$0.639902\pi$
$168$	4.35253	0.335805
$169$	−12.3492	−0.949938
$170$	−12.4737	−0.956689
$171$	−3.88180	−0.296849
$172$	12.8752	0.981724
$173$	−8.67305	−0.659399	−0.329700	−	0.944086i	$-0.606947\pi$
$173$	−8.67305	−0.659399	−0.329700	+	0.944086i	$0.606947\pi$
$174$	−2.10486	−0.159569
$175$	−0.115287	−0.00871485
$176$	−2.62642	−0.197974
$177$	13.6701	1.02751
$178$	−9.16497	−0.686943
$179$	17.6774	1.32127	0.660637	−	0.750706i	$-0.270286\pi$
$179$	17.6774	1.32127	0.660637	+	0.750706i	$0.270286\pi$
$180$	−4.08988	−0.304842
$181$	8.99579	0.668652	0.334326	−	0.942458i	$-0.391491\pi$
$181$	8.99579	0.668652	0.334326	+	0.942458i	$0.391491\pi$
$182$	1.59608	0.118309
$183$	33.4717	2.47430
$184$	−0.710150	−0.0523529
$185$	−11.3092	−0.831468
$186$	21.2127	1.55539
$187$	14.7374	1.07770
$188$	1.40781	0.102675
$189$	5.04979	0.367318
$190$	−4.69031	−0.340271
$191$	11.3933	0.824389	0.412194	−	0.911096i	$-0.364762\pi$
$191$	11.3933	0.824389	0.412194	+	0.911096i	$0.364762\pi$
$192$	−2.19996	−0.158768
$193$	−0.604308	−0.0434990	−0.0217495	−	0.999763i	$-0.506924\pi$
$193$	−0.604308	−0.0434990	−0.0217495	+	0.999763i	$0.506924\pi$
$194$	−4.38886	−0.315102
$195$	−3.94530	−0.282529
$196$	−3.08569	−0.220406
$197$	−4.16639	−0.296843	−0.148422	−	0.988924i	$-0.547419\pi$
$197$	−4.16639	−0.296843	−0.148422	+	0.988924i	$0.547419\pi$
$198$	4.83210	0.343403
$199$	−1.02270	−0.0724974	−0.0362487	−	0.999343i	$-0.511541\pi$
$199$	−1.02270	−0.0724974	−0.0362487	+	0.999343i	$0.511541\pi$
$200$	0.0582708	0.00412037
$201$	−25.8000	−1.81979
$202$	−2.71647	−0.191130
$203$	−1.89294	−0.132858
$204$	12.3444	0.864280
$205$	21.6855	1.51458
$206$	−0.909821	−0.0633902
$207$	1.30654	0.0908105
$208$	−0.806727	−0.0559365
$209$	5.54149	0.383313
$210$	−9.67567	−0.667684
$211$	22.6782	1.56123	0.780617	−	0.625009i	$-0.214905\pi$
$211$	22.6782	1.56123	0.780617	+	0.625009i	$0.214905\pi$
$212$	4.57788	0.314410
$213$	−7.38429	−0.505963
$214$	20.1568	1.37789
$215$	−28.6216	−1.95197
$216$	−2.55238	−0.173668
$217$	19.0770	1.29503
$218$	−7.95041	−0.538470
$219$	−8.59438	−0.580755
$220$	5.83854	0.393634
$221$	4.52671	0.304499
$222$	11.1920	0.751155
$223$	−22.0906	−1.47930	−0.739649	−	0.672992i	$-0.765009\pi$
$223$	−22.0906	−1.47930	−0.739649	+	0.672992i	$0.765009\pi$
$224$	−1.97846	−0.132192
$225$	−0.107207	−0.00714712
$226$	8.55336	0.568961
$227$	−1.18679	−0.0787700	−0.0393850	−	0.999224i	$-0.512540\pi$
$227$	−1.18679	−0.0787700	−0.0393850	+	0.999224i	$0.512540\pi$
$228$	4.64169	0.307403
$229$	19.8722	1.31319	0.656597	−	0.754241i	$-0.271995\pi$
$229$	19.8722	1.31319	0.656597	+	0.754241i	$0.271995\pi$
$230$	1.57866	0.104094
$231$	11.4316	0.752142
$232$	0.956772	0.0628152
$233$	19.5714	1.28217	0.641083	−	0.767472i	$-0.278486\pi$
$233$	19.5714	1.28217	0.641083	+	0.767472i	$0.278486\pi$
$234$	1.48422	0.0970265
$235$	−3.12955	−0.204149
$236$	−6.21383	−0.404486
$237$	14.1910	0.921803
$238$	11.1015	0.719606
$239$	18.8636	1.22019	0.610093	−	0.792330i	$-0.291132\pi$
$239$	18.8636	1.22019	0.610093	+	0.792330i	$0.291132\pi$
$240$	4.89050	0.315680
$241$	8.08984	0.521112	0.260556	−	0.965459i	$-0.416094\pi$
$241$	8.08984	0.521112	0.260556	+	0.965459i	$0.416094\pi$
$242$	4.10190	0.263680
$243$	16.8383	1.08018
$244$	−15.2147	−0.974022
$245$	6.85949	0.438237
$246$	−21.4607	−1.36828
$247$	1.70211	0.108303
$248$	−9.64235	−0.612290
$249$	−13.7399	−0.870733
$250$	−11.2445	−0.711167
$251$	−13.2800	−0.838226	−0.419113	−	0.907934i	$-0.637659\pi$
$251$	−13.2800	−0.838226	−0.419113	+	0.907934i	$0.637659\pi$
$252$	3.63998	0.229297
$253$	−1.86515	−0.117261
$254$	7.58025	0.475627
$255$	−27.4416	−1.71846
$256$	1.00000	0.0625000
$257$	−13.0763	−0.815675	−0.407837	−	0.913055i	$-0.633717\pi$
$257$	−13.0763	−0.815675	−0.407837	+	0.913055i	$0.633717\pi$
$258$	28.3249	1.76343
$259$	10.0651	0.625417
$260$	1.79335	0.111219
$261$	−1.76027	−0.108958
$262$	−5.19286	−0.320816
$263$	−15.2040	−0.937517	−0.468758	−	0.883326i	$-0.655299\pi$
$263$	−15.2040	−0.937517	−0.468758	+	0.883326i	$0.655299\pi$
$264$	−5.77801	−0.355612
$265$	−10.1766	−0.625145
$266$	4.17435	0.255946
$267$	−20.1625	−1.23393
$268$	11.7275	0.716371
$269$	14.4841	0.883111	0.441555	−	0.897234i	$-0.354427\pi$
$269$	14.4841	0.883111	0.441555	+	0.897234i	$0.354427\pi$
$270$	5.67394	0.345305
$271$	3.62007	0.219903	0.109952	−	0.993937i	$-0.464930\pi$
$271$	3.62007	0.219903	0.109952	+	0.993937i	$0.464930\pi$
$272$	−5.61120	−0.340229
$273$	3.51130	0.212514
$274$	−6.37061	−0.384863
$275$	0.153044	0.00922889
$276$	−1.56230	−0.0940393
$277$	−18.6610	−1.12123	−0.560616	−	0.828076i	$-0.689436\pi$
$277$	−18.6610	−1.12123	−0.560616	+	0.828076i	$0.689436\pi$
$278$	−1.02203	−0.0612971
$279$	17.7400	1.06207
$280$	4.39812	0.262838
$281$	1.27539	0.0760832	0.0380416	−	0.999276i	$-0.487888\pi$
$281$	1.27539	0.0760832	0.0380416	+	0.999276i	$0.487888\pi$
$282$	3.09711	0.184430
$283$	−20.0065	−1.18926	−0.594632	−	0.803998i	$-0.702702\pi$
$283$	−20.0065	−1.18926	−0.594632	+	0.803998i	$0.702702\pi$
$284$	3.35656	0.199175
$285$	−10.3185	−0.611213
$286$	−2.11881	−0.125288
$287$	−19.3000	−1.13924
$288$	−1.83980	−0.108411
$289$	14.4856	0.852092
$290$	−2.12690	−0.124896
$291$	−9.65529	−0.566003
$292$	3.90662	0.228617
$293$	21.1465	1.23539	0.617696	−	0.786417i	$-0.288066\pi$
$293$	21.1465	1.23539	0.617696	+	0.786417i	$0.288066\pi$
$294$	−6.78838	−0.395906
$295$	13.8133	0.804243
$296$	−5.08736	−0.295697
$297$	−6.70363	−0.388984
$298$	−23.5381	−1.36353
$299$	−0.572897	−0.0331315
$300$	0.128193	0.00740124
$301$	25.4731	1.46824
$302$	−23.8870	−1.37454
$303$	−5.97612	−0.343319
$304$	−2.10990	−0.121011
$305$	33.8223	1.93666
$306$	10.3235	0.590156
$307$	−9.90494	−0.565305	−0.282652	−	0.959222i	$-0.591214\pi$
$307$	−9.90494	−0.565305	−0.282652	+	0.959222i	$0.591214\pi$
$308$	−5.19628	−0.296085
$309$	−2.00157	−0.113865
$310$	21.4349	1.21742
$311$	6.53121	0.370351	0.185176	−	0.982705i	$-0.440715\pi$
$311$	6.53121	0.370351	0.185176	+	0.982705i	$0.440715\pi$
$312$	−1.77476	−0.100476
$313$	−31.1250	−1.75929	−0.879644	−	0.475632i	$-0.842219\pi$
$313$	−31.1250	−1.75929	−0.879644	+	0.475632i	$0.842219\pi$
$314$	−2.85329	−0.161021
$315$	−8.09168	−0.455914
$316$	−6.45058	−0.362873
$317$	−6.51677	−0.366018	−0.183009	−	0.983111i	$-0.558584\pi$
$317$	−6.51677	−0.366018	−0.183009	+	0.983111i	$0.558584\pi$
$318$	10.0711	0.564761
$319$	2.51289	0.140695
$320$	−2.22300	−0.124269
$321$	44.3441	2.47504
$322$	−1.40500	−0.0782978
$323$	11.8391	0.658743
$324$	−11.1345	−0.618585
$325$	0.0470087	0.00260757
$326$	−10.6734	−0.591144
$327$	−17.4906	−0.967230
$328$	9.75506	0.538633
$329$	2.78529	0.153558
$330$	12.8445	0.707068
$331$	−12.0113	−0.660202	−0.330101	−	0.943946i	$-0.607083\pi$
$331$	−12.0113	−0.660202	−0.330101	+	0.943946i	$0.607083\pi$
$332$	6.24555	0.342769
$333$	9.35973	0.512910
$334$	10.9974	0.601750
$335$	−26.0702	−1.42437
$336$	−4.35253	−0.237450
$337$	−23.7848	−1.29564	−0.647821	−	0.761792i	$-0.724320\pi$
$337$	−23.7848	−1.29564	−0.647821	+	0.761792i	$0.724320\pi$
$338$	12.3492	0.671707
$339$	18.8170	1.02200
$340$	12.4737	0.676481
$341$	−25.3249	−1.37142
$342$	3.88180	0.209904
$343$	−19.9541	−1.07742
$344$	−12.8752	−0.694184
$345$	3.47299	0.186979
$346$	8.67305	0.466266
$347$	3.10810	0.166851	0.0834257	−	0.996514i	$-0.473414\pi$
$347$	3.10810	0.166851	0.0834257	+	0.996514i	$0.473414\pi$
$348$	2.10486	0.112832
$349$	9.07318	0.485676	0.242838	−	0.970067i	$-0.421922\pi$
$349$	9.07318	0.485676	0.242838	+	0.970067i	$0.421922\pi$
$350$	0.115287	0.00616233
$351$	−2.05908	−0.109905
$352$	2.62642	0.139989
$353$	1.35163	0.0719402	0.0359701	−	0.999353i	$-0.488548\pi$
$353$	1.35163	0.0719402	0.0359701	+	0.999353i	$0.488548\pi$
$354$	−13.6701	−0.726560
$355$	−7.46164	−0.396023
$356$	9.16497	0.485742
$357$	24.4229	1.29260
$358$	−17.6774	−0.934282
$359$	−16.8330	−0.888409	−0.444205	−	0.895925i	$-0.646514\pi$
$359$	−16.8330	−0.888409	−0.444205	+	0.895925i	$0.646514\pi$
$360$	4.08988	0.215556
$361$	−14.5483	−0.765701
$362$	−8.99579	−0.472808
$363$	9.02400	0.473637
$364$	−1.59608	−0.0836573
$365$	−8.68441	−0.454563
$366$	−33.4717	−1.74959
$367$	−18.2940	−0.954938	−0.477469	−	0.878649i	$-0.658446\pi$
$367$	−18.2940	−0.954938	−0.477469	+	0.878649i	$0.658446\pi$
$368$	0.710150	0.0370191
$369$	−17.9474	−0.934304
$370$	11.3092	0.587937
$371$	9.05715	0.470224
$372$	−21.2127	−1.09983
$373$	−31.4381	−1.62781	−0.813903	−	0.581001i	$-0.802661\pi$
$373$	−31.4381	−1.62781	−0.813903	+	0.581001i	$0.802661\pi$
$374$	−14.7374	−0.762052
$375$	−24.7375	−1.27744
$376$	−1.40781	−0.0726020
$377$	0.771854	0.0397525
$378$	−5.04979	−0.259733
$379$	1.11141	0.0570895	0.0285448	−	0.999593i	$-0.490913\pi$
$379$	1.11141	0.0570895	0.0285448	+	0.999593i	$0.490913\pi$
$380$	4.69031	0.240608
$381$	16.6762	0.854349
$382$	−11.3933	−0.582931
$383$	10.6124	0.542268	0.271134	−	0.962542i	$-0.412601\pi$
$383$	10.6124	0.542268	0.271134	+	0.962542i	$0.412601\pi$
$384$	2.19996	0.112266
$385$	11.5513	0.588710
$386$	0.604308	0.0307585
$387$	23.6878	1.20412
$388$	4.38886	0.222810
$389$	38.6013	1.95716	0.978582	−	0.205859i	$-0.0659987\pi$
$389$	38.6013	1.95716	0.978582	+	0.205859i	$0.0659987\pi$
$390$	3.94530	0.199778
$391$	−3.98479	−0.201520
$392$	3.08569	0.155851
$393$	−11.4241	−0.576267
$394$	4.16639	0.209900
$395$	14.3396	0.721505
$396$	−4.83210	−0.242822
$397$	−6.52169	−0.327314	−0.163657	−	0.986517i	$-0.552329\pi$
$397$	−6.52169	−0.327314	−0.163657	+	0.986517i	$0.552329\pi$
$398$	1.02270	0.0512634
$399$	9.18339	0.459745
$400$	−0.0582708	−0.00291354
$401$	−11.7778	−0.588156	−0.294078	−	0.955781i	$-0.595012\pi$
$401$	−11.7778	−0.588156	−0.294078	+	0.955781i	$0.595012\pi$
$402$	25.8000	1.28679
$403$	−7.77875	−0.387487
$404$	2.71647	0.135150
$405$	24.7521	1.22994
$406$	1.89294	0.0939449
$407$	−13.3615	−0.662307
$408$	−12.3444	−0.611138
$409$	5.73549	0.283602	0.141801	−	0.989895i	$-0.454711\pi$
$409$	5.73549	0.283602	0.141801	+	0.989895i	$0.454711\pi$
$410$	−21.6855	−1.07097
$411$	−14.0151	−0.691312
$412$	0.909821	0.0448237
$413$	−12.2938	−0.604939
$414$	−1.30654	−0.0642127
$415$	−13.8839	−0.681532
$416$	0.806727	0.0395531
$417$	−2.24841	−0.110105
$418$	−5.54149	−0.271043
$419$	−20.8212	−1.01718	−0.508592	−	0.861008i	$-0.669834\pi$
$419$	−20.8212	−1.01718	−0.508592	+	0.861008i	$0.669834\pi$
$420$	9.67567	0.472124
$421$	10.2287	0.498518	0.249259	−	0.968437i	$-0.419813\pi$
$421$	10.2287	0.498518	0.249259	+	0.968437i	$0.419813\pi$
$422$	−22.6782	−1.10396
$423$	2.59009	0.125934
$424$	−4.57788	−0.222321
$425$	0.326969	0.0158603
$426$	7.38429	0.357770
$427$	−30.1017	−1.45672
$428$	−20.1568	−0.974316
$429$	−4.66128	−0.225049
$430$	28.6216	1.38025
$431$	−13.8459	−0.666936	−0.333468	−	0.942761i	$-0.608219\pi$
$431$	−13.8459	−0.666936	−0.333468	+	0.942761i	$0.608219\pi$
$432$	2.55238	0.122801
$433$	−4.28394	−0.205873	−0.102936	−	0.994688i	$-0.532824\pi$
$433$	−4.28394	−0.205873	−0.102936	+	0.994688i	$0.532824\pi$
$434$	−19.0770	−0.915726
$435$	−4.67910	−0.224345
$436$	7.95041	0.380756
$437$	−1.49834	−0.0716755
$438$	8.59438	0.410655
$439$	−16.5345	−0.789146	−0.394573	−	0.918865i	$-0.629108\pi$
$439$	−16.5345	−0.789146	−0.394573	+	0.918865i	$0.629108\pi$
$440$	−5.83854	−0.278341
$441$	−5.67706	−0.270336
$442$	−4.52671	−0.215314
$443$	13.2088	0.627571	0.313786	−	0.949494i	$-0.398403\pi$
$443$	13.2088	0.627571	0.313786	+	0.949494i	$0.398403\pi$
$444$	−11.1920	−0.531147
$445$	−20.3737	−0.965807
$446$	22.0906	1.04602
$447$	−51.7829	−2.44924
$448$	1.97846	0.0934735
$449$	−29.3972	−1.38734	−0.693671	−	0.720292i	$-0.744008\pi$
$449$	−29.3972	−1.38734	−0.693671	+	0.720292i	$0.744008\pi$
$450$	0.107207	0.00505378
$451$	25.6209	1.20644
$452$	−8.55336	−0.402316
$453$	−52.5503	−2.46903
$454$	1.18679	0.0556988
$455$	3.54808	0.166337
$456$	−4.64169	−0.217367
$457$	8.38001	0.392001	0.196000	−	0.980604i	$-0.437205\pi$
$457$	8.38001	0.392001	0.196000	+	0.980604i	$0.437205\pi$
$458$	−19.8722	−0.928569
$459$	−14.3219	−0.668490
$460$	−1.57866	−0.0736055
$461$	2.42680	0.113027	0.0565137	−	0.998402i	$-0.482002\pi$
$461$	2.42680	0.113027	0.0565137	+	0.998402i	$0.482002\pi$
$462$	−11.4316	−0.531845
$463$	28.4538	1.32236	0.661181	−	0.750227i	$-0.270056\pi$
$463$	28.4538	1.32236	0.661181	+	0.750227i	$0.270056\pi$
$464$	−0.956772	−0.0444170
$465$	47.1559	2.18680
$466$	−19.5714	−0.906628
$467$	−14.7273	−0.681497	−0.340748	−	0.940155i	$-0.610680\pi$
$467$	−14.7273	−0.681497	−0.340748	+	0.940155i	$0.610680\pi$
$468$	−1.48422	−0.0686081
$469$	23.2024	1.07139
$470$	3.12955	0.144355
$471$	−6.27712	−0.289234
$472$	6.21383	0.286015
$473$	−33.8157	−1.55485
$474$	−14.1910	−0.651813
$475$	0.122946	0.00564113
$476$	−11.1015	−0.508838
$477$	8.42240	0.385635
$478$	−18.8636	−0.862802
$479$	20.6686	0.944372	0.472186	−	0.881499i	$-0.343465\pi$
$479$	20.6686	0.944372	0.472186	+	0.881499i	$0.343465\pi$
$480$	−4.89050	−0.223220
$481$	−4.10411	−0.187131
$482$	−8.08984	−0.368482
$483$	−3.09095	−0.140643
$484$	−4.10190	−0.186450
$485$	−9.75643	−0.443017
$486$	−16.8383	−0.763802
$487$	22.6031	1.02424	0.512121	−	0.858913i	$-0.328860\pi$
$487$	22.6031	1.02424	0.512121	+	0.858913i	$0.328860\pi$
$488$	15.2147	0.688738
$489$	−23.4810	−1.06185
$490$	−6.85949	−0.309880
$491$	−10.0217	−0.452273	−0.226137	−	0.974096i	$-0.572610\pi$
$491$	−10.0217	−0.452273	−0.226137	+	0.974096i	$0.572610\pi$
$492$	21.4607	0.967523
$493$	5.36864	0.241791
$494$	−1.70211	−0.0765817
$495$	10.7418	0.482806
$496$	9.64235	0.432954
$497$	6.64083	0.297882
$498$	13.7399	0.615701
$499$	−27.9862	−1.25283	−0.626417	−	0.779488i	$-0.715479\pi$
$499$	−27.9862	−1.25283	−0.626417	+	0.779488i	$0.715479\pi$
$500$	11.2445	0.502871
$501$	24.1937	1.08090
$502$	13.2800	0.592715
$503$	−15.2696	−0.680838	−0.340419	−	0.940274i	$-0.610569\pi$
$503$	−15.2696	−0.680838	−0.340419	+	0.940274i	$0.610569\pi$
$504$	−3.63998	−0.162138
$505$	−6.03872	−0.268719
$506$	1.86515	0.0829162
$507$	27.1677	1.20656
$508$	−7.58025	−0.336319
$509$	−6.92973	−0.307155	−0.153577	−	0.988137i	$-0.549079\pi$
$509$	−6.92973	−0.307155	−0.153577	+	0.988137i	$0.549079\pi$
$510$	27.4416	1.21513
$511$	7.72909	0.341915
$512$	−1.00000	−0.0441942
$513$	−5.38527	−0.237765
$514$	13.0763	0.576769
$515$	−2.02253	−0.0891234
$516$	−28.3249	−1.24693
$517$	−3.69749	−0.162616
$518$	−10.0651	−0.442237
$519$	19.0803	0.837533
$520$	−1.79335	−0.0786438
$521$	−3.15905	−0.138401	−0.0692003	−	0.997603i	$-0.522045\pi$
$521$	−3.15905	−0.138401	−0.0692003	+	0.997603i	$0.522045\pi$
$522$	1.76027	0.0770451
$523$	0.945916	0.0413620	0.0206810	−	0.999786i	$-0.493417\pi$
$523$	0.945916	0.0413620	0.0206810	+	0.999786i	$0.493417\pi$
$524$	5.19286	0.226851
$525$	0.253625	0.0110691
$526$	15.2040	0.662925
$527$	−54.1052	−2.35686
$528$	5.77801	0.251456
$529$	−22.4957	−0.978073
$530$	10.1766	0.442044
$531$	−11.4322	−0.496116
$532$	−4.17435	−0.180981
$533$	7.86967	0.340873
$534$	20.1625	0.872518
$535$	44.8086	1.93724
$536$	−11.7275	−0.506551
$537$	−38.8896	−1.67821
$538$	−14.4841	−0.624453
$539$	8.10433	0.349078
$540$	−5.67394	−0.244168
$541$	29.8030	1.28133	0.640666	−	0.767820i	$-0.278659\pi$
$541$	29.8030	1.28133	0.640666	+	0.767820i	$0.278659\pi$
$542$	−3.62007	−0.155495
$543$	−19.7903	−0.849285
$544$	5.61120	0.240578
$545$	−17.6738	−0.757061
$546$	−3.51130	−0.150270
$547$	−28.9364	−1.23723	−0.618616	−	0.785693i	$-0.712306\pi$
$547$	−28.9364	−1.23723	−0.618616	+	0.785693i	$0.712306\pi$
$548$	6.37061	0.272139
$549$	−27.9921	−1.19467
$550$	−0.153044	−0.00652581
$551$	2.01869	0.0859992
$552$	1.56230	0.0664958
$553$	−12.7622	−0.542705
$554$	18.6610	0.792831
$555$	24.8797	1.05609
$556$	1.02203	0.0433436
$557$	28.7223	1.21700	0.608502	−	0.793552i	$-0.291771\pi$
$557$	28.7223	1.21700	0.608502	+	0.793552i	$0.291771\pi$
$558$	−17.7400	−0.750995
$559$	−10.3868	−0.439314
$560$	−4.39812	−0.185854
$561$	−32.4216	−1.36884
$562$	−1.27539	−0.0537989
$563$	−13.1265	−0.553215	−0.276607	−	0.960983i	$-0.589210\pi$
$563$	−13.1265	−0.553215	−0.276607	+	0.960983i	$0.589210\pi$
$564$	−3.09711	−0.130412
$565$	19.0141	0.799930
$566$	20.0065	0.840937
$567$	−22.0292	−0.925141
$568$	−3.35656	−0.140838
$569$	−18.1326	−0.760158	−0.380079	−	0.924954i	$-0.624103\pi$
$569$	−18.1326	−0.760158	−0.380079	+	0.924954i	$0.624103\pi$
$570$	10.3185	0.432193
$571$	3.58847	0.150173	0.0750864	−	0.997177i	$-0.476077\pi$
$571$	3.58847	0.150173	0.0750864	+	0.997177i	$0.476077\pi$
$572$	2.11881	0.0885918
$573$	−25.0647	−1.04709
$574$	19.3000	0.805567
$575$	−0.0413810	−0.00172571
$576$	1.83980	0.0766585
$577$	35.5456	1.47978	0.739891	−	0.672727i	$-0.234877\pi$
$577$	35.5456	1.47978	0.739891	+	0.672727i	$0.234877\pi$
$578$	−14.4856	−0.602520
$579$	1.32945	0.0552501
$580$	2.12690	0.0883149
$581$	12.3566	0.512637
$582$	9.65529	0.400225
$583$	−12.0234	−0.497960
$584$	−3.90662	−0.161657
$585$	3.29942	0.136414
$586$	−21.1465	−0.873555
$587$	18.4703	0.762350	0.381175	−	0.924503i	$-0.375520\pi$
$587$	18.4703	0.762350	0.381175	+	0.924503i	$0.375520\pi$
$588$	6.78838	0.279948
$589$	−20.3444	−0.838276
$590$	−13.8133	−0.568686
$591$	9.16588	0.377034
$592$	5.08736	0.209089
$593$	−35.9845	−1.47771	−0.738853	−	0.673867i	$-0.764632\pi$
$593$	−35.9845	−1.47771	−0.738853	+	0.673867i	$0.764632\pi$
$594$	6.70363	0.275053
$595$	24.6787	1.01173
$596$	23.5381	0.964160
$597$	2.24990	0.0920822
$598$	0.572897	0.0234275
$599$	26.1928	1.07021	0.535104	−	0.844786i	$-0.320273\pi$
$599$	26.1928	1.07021	0.535104	+	0.844786i	$0.320273\pi$
$600$	−0.128193	−0.00523347
$601$	−46.3252	−1.88964	−0.944822	−	0.327584i	$-0.893765\pi$
$601$	−46.3252	−1.88964	−0.944822	+	0.327584i	$0.893765\pi$
$602$	−25.4731	−1.03821
$603$	21.5763	0.878654
$604$	23.8870	0.971948
$605$	9.11853	0.370721
$606$	5.97612	0.242763
$607$	17.7590	0.720815	0.360408	−	0.932795i	$-0.382638\pi$
$607$	17.7590	0.720815	0.360408	+	0.932795i	$0.382638\pi$
$608$	2.10990	0.0855677
$609$	4.16438	0.168749
$610$	−33.8223	−1.36942
$611$	−1.13572	−0.0459461
$612$	−10.3235	−0.417303
$613$	−31.9519	−1.29053	−0.645263	−	0.763961i	$-0.723252\pi$
$613$	−31.9519	−1.29053	−0.645263	+	0.763961i	$0.723252\pi$
$614$	9.90494	0.399731
$615$	−47.7071	−1.92374
$616$	5.19628	0.209364
$617$	−16.4416	−0.661912	−0.330956	−	0.943646i	$-0.607371\pi$
$617$	−16.4416	−0.661912	−0.330956	+	0.943646i	$0.607371\pi$
$618$	2.00157	0.0805148
$619$	23.2184	0.933228	0.466614	−	0.884461i	$-0.345474\pi$
$619$	23.2184	0.933228	0.466614	+	0.884461i	$0.345474\pi$
$620$	−21.4349	−0.860848
$621$	1.81257	0.0727360
$622$	−6.53121	−0.261878
$623$	18.1325	0.726465
$624$	1.77476	0.0710474
$625$	−24.7053	−0.988210
$626$	31.1250	1.24400
$627$	−12.1910	−0.486863
$628$	2.85329	0.113859
$629$	−28.5462	−1.13821
$630$	8.09168	0.322380
$631$	−13.5272	−0.538510	−0.269255	−	0.963069i	$-0.586777\pi$
$631$	−13.5272	−0.538510	−0.269255	+	0.963069i	$0.586777\pi$
$632$	6.45058	0.256590
$633$	−49.8911	−1.98299
$634$	6.51677	0.258814
$635$	16.8509	0.668708
$636$	−10.0711	−0.399346
$637$	2.48931	0.0986301
$638$	−2.51289	−0.0994862
$639$	6.17542	0.244296
$640$	2.22300	0.0878718
$641$	−5.74946	−0.227090	−0.113545	−	0.993533i	$-0.536221\pi$
$641$	−5.74946	−0.227090	−0.113545	+	0.993533i	$0.536221\pi$
$642$	−44.3441	−1.75012
$643$	−18.7655	−0.740040	−0.370020	−	0.929024i	$-0.620649\pi$
$643$	−18.7655	−0.740040	−0.370020	+	0.929024i	$0.620649\pi$
$644$	1.40500	0.0553649
$645$	62.9662	2.47929
$646$	−11.8391	−0.465802
$647$	−7.78877	−0.306208	−0.153104	−	0.988210i	$-0.548927\pi$
$647$	−7.78877	−0.306208	−0.153104	+	0.988210i	$0.548927\pi$
$648$	11.1345	0.437406
$649$	16.3201	0.640621
$650$	−0.0470087	−0.00184383
$651$	−41.9686	−1.64488
$652$	10.6734	0.418002
$653$	−0.103169	−0.00403730	−0.00201865	−	0.999998i	$-0.500643\pi$
$653$	−0.103169	−0.00403730	−0.00201865	+	0.999998i	$0.500643\pi$
$654$	17.4906	0.683935
$655$	−11.5437	−0.451050
$656$	−9.75506	−0.380871
$657$	7.18741	0.280407
$658$	−2.78529	−0.108582
$659$	−30.5347	−1.18946	−0.594732	−	0.803924i	$-0.702742\pi$
$659$	−30.5347	−1.18946	−0.594732	+	0.803924i	$0.702742\pi$
$660$	−12.8445	−0.499972
$661$	−0.991743	−0.0385743	−0.0192872	−	0.999814i	$-0.506140\pi$
$661$	−0.991743	−0.0385743	−0.0192872	+	0.999814i	$0.506140\pi$
$662$	12.0113	0.466833
$663$	−9.95856	−0.386758
$664$	−6.24555	−0.242374
$665$	9.27959	0.359847
$666$	−9.35973	−0.362682
$667$	−0.679452	−0.0263085
$668$	−10.9974	−0.425501
$669$	48.5984	1.87892
$670$	26.0702	1.00718
$671$	39.9603	1.54265
$672$	4.35253	0.167902
$673$	33.9450	1.30848	0.654242	−	0.756285i	$-0.272988\pi$
$673$	33.9450	1.30848	0.654242	+	0.756285i	$0.272988\pi$
$674$	23.7848	0.916158
$675$	−0.148729	−0.00572459
$676$	−12.3492	−0.474969
$677$	18.3036	0.703466	0.351733	−	0.936100i	$-0.385593\pi$
$677$	18.3036	0.703466	0.351733	+	0.936100i	$0.385593\pi$
$678$	−18.8170	−0.722663
$679$	8.68318	0.333230
$680$	−12.4737	−0.478345
$681$	2.61089	0.100049
$682$	25.3249	0.969740
$683$	−38.6382	−1.47845	−0.739224	−	0.673460i	$-0.764807\pi$
$683$	−38.6382	−1.47845	−0.739224	+	0.673460i	$0.764807\pi$
$684$	−3.88180	−0.148424
$685$	−14.1619	−0.541097
$686$	19.9541	0.761853
$687$	−43.7180	−1.66795
$688$	12.8752	0.490862
$689$	−3.69310	−0.140696
$690$	−3.47299	−0.132214
$691$	−4.77117	−0.181504	−0.0907519	−	0.995874i	$-0.528927\pi$
$691$	−4.77117	−0.181504	−0.0907519	+	0.995874i	$0.528927\pi$
$692$	−8.67305	−0.329700
$693$	−9.56013	−0.363159
$694$	−3.10810	−0.117982
$695$	−2.27197	−0.0861806
$696$	−2.10486	−0.0797844
$697$	54.7376	2.07333
$698$	−9.07318	−0.343425
$699$	−43.0562	−1.62854
$700$	−0.115287	−0.00435742
$701$	−15.8975	−0.600442	−0.300221	−	0.953870i	$-0.597060\pi$
$701$	−15.8975	−0.600442	−0.300221	+	0.953870i	$0.597060\pi$
$702$	2.05908	0.0777148
$703$	−10.7338	−0.404833
$704$	−2.62642	−0.0989870
$705$	6.88487	0.259299
$706$	−1.35163	−0.0508694
$707$	5.37444	0.202126
$708$	13.6701	0.513755
$709$	29.2113	1.09705	0.548527	−	0.836133i	$-0.315189\pi$
$709$	29.2113	1.09705	0.548527	+	0.836133i	$0.315189\pi$
$710$	7.46164	0.280030
$711$	−11.8678	−0.445077
$712$	−9.16497	−0.343472
$713$	6.84751	0.256441
$714$	−24.4229	−0.914004
$715$	−4.71011	−0.176148
$716$	17.6774	0.660637
$717$	−41.4991	−1.54981
$718$	16.8330	0.628200
$719$	6.14979	0.229348	0.114674	−	0.993403i	$-0.463418\pi$
$719$	6.14979	0.229348	0.114674	+	0.993403i	$0.463418\pi$
$720$	−4.08988	−0.152421
$721$	1.80005	0.0670372
$722$	14.5483	0.541433
$723$	−17.7973	−0.661888
$724$	8.99579	0.334326
$725$	0.0557519	0.00207057
$726$	−9.02400	−0.334912
$727$	10.6275	0.394154	0.197077	−	0.980388i	$-0.436855\pi$
$727$	10.6275	0.394154	0.197077	+	0.980388i	$0.436855\pi$
$728$	1.59608	0.0591546
$729$	−3.63998	−0.134814
$730$	8.68441	0.321424
$731$	−72.2453	−2.67209
$732$	33.4717	1.23715
$733$	1.32181	0.0488223	0.0244111	−	0.999702i	$-0.492229\pi$
$733$	1.32181	0.0488223	0.0244111	+	0.999702i	$0.492229\pi$
$734$	18.2940	0.675243
$735$	−15.0906	−0.556624
$736$	−0.710150	−0.0261765
$737$	−30.8014	−1.13458
$738$	17.9474	0.660653
$739$	−43.2336	−1.59037	−0.795187	−	0.606364i	$-0.792627\pi$
$739$	−43.2336	−1.59037	−0.795187	+	0.606364i	$0.792627\pi$
$740$	−11.3092	−0.415734
$741$	−3.74457	−0.137560
$742$	−9.05715	−0.332499
$743$	−13.7583	−0.504744	−0.252372	−	0.967630i	$-0.581211\pi$
$743$	−13.7583	−0.504744	−0.252372	+	0.967630i	$0.581211\pi$
$744$	21.2127	0.777697
$745$	−52.3253	−1.91705
$746$	31.4381	1.15103
$747$	11.4906	0.420419
$748$	14.7374	0.538852
$749$	−39.8795	−1.45716
$750$	24.7375	0.903285
$751$	27.6029	1.00724	0.503622	−	0.863924i	$-0.332000\pi$
$751$	27.6029	1.00724	0.503622	+	0.863924i	$0.332000\pi$
$752$	1.40781	0.0513374
$753$	29.2154	1.06467
$754$	−0.771854	−0.0281093
$755$	−53.1008	−1.93253
$756$	5.04979	0.183659
$757$	38.3761	1.39480	0.697401	−	0.716681i	$-0.254340\pi$
$757$	38.3761	1.39480	0.697401	+	0.716681i	$0.254340\pi$
$758$	−1.11141	−0.0403684
$759$	4.10325	0.148939
$760$	−4.69031	−0.170135
$761$	−20.8661	−0.756396	−0.378198	−	0.925725i	$-0.623456\pi$
$761$	−20.8661	−0.756396	−0.378198	+	0.925725i	$0.623456\pi$
$762$	−16.6762	−0.604116
$763$	15.7296	0.569449
$764$	11.3933	0.412194
$765$	22.9492	0.829728
$766$	−10.6124	−0.383441
$767$	5.01286	0.181004
$768$	−2.19996	−0.0793840
$769$	−23.8148	−0.858785	−0.429393	−	0.903118i	$-0.641272\pi$
$769$	−23.8148	−0.858785	−0.429393	+	0.903118i	$0.641272\pi$
$770$	−11.5513	−0.416281
$771$	28.7672	1.03602
$772$	−0.604308	−0.0217495
$773$	−50.5678	−1.81880	−0.909399	−	0.415925i	$-0.863458\pi$
$773$	−50.5678	−1.81880	−0.909399	+	0.415925i	$0.863458\pi$
$774$	−23.6878	−0.851441
$775$	−0.561868	−0.0201829
$776$	−4.38886	−0.157551
$777$	−22.1429	−0.794370
$778$	−38.6013	−1.38392
$779$	20.5822	0.737434
$780$	−3.94530	−0.141264
$781$	−8.81575	−0.315452
$782$	3.98479	0.142496
$783$	−2.44205	−0.0872716
$784$	−3.08569	−0.110203
$785$	−6.34287	−0.226387
$786$	11.4241	0.407482
$787$	12.9591	0.461944	0.230972	−	0.972960i	$-0.425809\pi$
$787$	12.9591	0.461944	0.230972	+	0.972960i	$0.425809\pi$
$788$	−4.16639	−0.148422
$789$	33.4481	1.19078
$790$	−14.3396	−0.510181
$791$	−16.9225	−0.601695
$792$	4.83210	0.171701
$793$	12.2741	0.435867
$794$	6.52169	0.231446
$795$	22.3881	0.794025
$796$	−1.02270	−0.0362487
$797$	−11.5649	−0.409651	−0.204825	−	0.978799i	$-0.565663\pi$
$797$	−11.5649	−0.409651	−0.204825	+	0.978799i	$0.565663\pi$
$798$	−9.18339	−0.325089
$799$	−7.89948	−0.279463
$800$	0.0582708	0.00206018
$801$	16.8617	0.595780
$802$	11.7778	0.415889
$803$	−10.2604	−0.362083
$804$	−25.8000	−0.909894
$805$	−3.12332	−0.110083
$806$	7.77875	0.273995
$807$	−31.8643	−1.12168
$808$	−2.71647	−0.0955652
$809$	−39.4704	−1.38771	−0.693854	−	0.720116i	$-0.744088\pi$
$809$	−39.4704	−1.38771	−0.693854	+	0.720116i	$0.744088\pi$
$810$	−24.7521	−0.869699
$811$	−48.2957	−1.69589	−0.847946	−	0.530083i	$-0.822161\pi$
$811$	−48.2957	−1.69589	−0.847946	+	0.530083i	$0.822161\pi$
$812$	−1.89294	−0.0664291
$813$	−7.96398	−0.279309
$814$	13.3615	0.468322
$815$	−23.7269	−0.831119
$816$	12.3444	0.432140
$817$	−27.1654	−0.950396
$818$	−5.73549	−0.200537
$819$	−2.93647	−0.102609
$820$	21.6855	0.757290
$821$	47.7459	1.66634	0.833172	−	0.553014i	$-0.186522\pi$
$821$	47.7459	1.66634	0.833172	+	0.553014i	$0.186522\pi$
$822$	14.0151	0.488831
$823$	−7.81992	−0.272585	−0.136293	−	0.990669i	$-0.543519\pi$
$823$	−7.81992	−0.272585	−0.136293	+	0.990669i	$0.543519\pi$
$824$	−0.909821	−0.0316951
$825$	−0.336690	−0.0117220
$826$	12.2938	0.427757
$827$	−13.0265	−0.452977	−0.226488	−	0.974014i	$-0.572724\pi$
$827$	−13.0265	−0.452977	−0.226488	+	0.974014i	$0.572724\pi$
$828$	1.30654	0.0454053
$829$	−7.87005	−0.273338	−0.136669	−	0.990617i	$-0.543640\pi$
$829$	−7.87005	−0.273338	−0.136669	+	0.990617i	$0.543640\pi$
$830$	13.8839	0.481916
$831$	41.0534	1.42413
$832$	−0.806727	−0.0279682
$833$	17.3144	0.599909
$834$	2.24841	0.0778562
$835$	24.4472	0.846029
$836$	5.54149	0.191656
$837$	24.6109	0.850679
$838$	20.8212	0.719257
$839$	25.2236	0.870815	0.435408	−	0.900233i	$-0.356604\pi$
$839$	25.2236	0.870815	0.435408	+	0.900233i	$0.356604\pi$
$840$	−9.67567	−0.333842
$841$	−28.0846	−0.968434
$842$	−10.2287	−0.352505
$843$	−2.80579	−0.0966367
$844$	22.6782	0.780617
$845$	27.4523	0.944386
$846$	−2.59009	−0.0890490
$847$	−8.11546	−0.278850
$848$	4.57788	0.157205
$849$	44.0135	1.51054
$850$	−0.326969	−0.0112150
$851$	3.61278	0.123845
$852$	−7.38429	−0.252982
$853$	−3.02551	−0.103591	−0.0517957	−	0.998658i	$-0.516494\pi$
$853$	−3.02551	−0.103591	−0.0517957	+	0.998658i	$0.516494\pi$
$854$	30.1017	1.03006
$855$	8.62924	0.295114
$856$	20.1568	0.688945
$857$	3.60624	0.123187	0.0615935	−	0.998101i	$-0.480382\pi$
$857$	3.60624	0.123187	0.0615935	+	0.998101i	$0.480382\pi$
$858$	4.66128	0.159134
$859$	35.2823	1.20382	0.601908	−	0.798566i	$-0.294407\pi$
$859$	35.2823	1.20382	0.601908	+	0.798566i	$0.294407\pi$
$860$	−28.6216	−0.975987
$861$	42.4592	1.44700
$862$	13.8459	0.471595
$863$	20.0310	0.681864	0.340932	−	0.940088i	$-0.389257\pi$
$863$	20.0310	0.681864	0.340932	+	0.940088i	$0.389257\pi$
$864$	−2.55238	−0.0868338
$865$	19.2802	0.655546
$866$	4.28394	0.145574
$867$	−31.8676	−1.08228
$868$	19.0770	0.647516
$869$	16.9419	0.574716
$870$	4.67910	0.158636
$871$	−9.46089	−0.320570
$872$	−7.95041	−0.269235
$873$	8.07463	0.273285
$874$	1.49834	0.0506823
$875$	22.2469	0.752082
$876$	−8.59438	−0.290377
$877$	−12.0724	−0.407654	−0.203827	−	0.979007i	$-0.565338\pi$
$877$	−12.0724	−0.407654	−0.203827	+	0.979007i	$0.565338\pi$
$878$	16.5345	0.558011
$879$	−46.5214	−1.56913
$880$	5.83854	0.196817
$881$	−21.2157	−0.714774	−0.357387	−	0.933956i	$-0.616332\pi$
$881$	−21.2157	−0.714774	−0.357387	+	0.933956i	$0.616332\pi$
$882$	5.67706	0.191157
$883$	−41.2618	−1.38857	−0.694285	−	0.719700i	$-0.744279\pi$
$883$	−41.2618	−1.38857	−0.694285	+	0.719700i	$0.744279\pi$
$884$	4.52671	0.152250
$885$	−30.3887	−1.02151
$886$	−13.2088	−0.443760
$887$	11.2847	0.378902	0.189451	−	0.981890i	$-0.439329\pi$
$887$	11.2847	0.378902	0.189451	+	0.981890i	$0.439329\pi$
$888$	11.1920	0.375577
$889$	−14.9972	−0.502991
$890$	20.3737	0.682929
$891$	29.2440	0.979711
$892$	−22.0906	−0.739649
$893$	−2.97033	−0.0993983
$894$	51.7829	1.73188
$895$	−39.2969	−1.31355
$896$	−1.97846	−0.0660958
$897$	1.26035	0.0420818
$898$	29.3972	0.980999
$899$	−9.22553	−0.307689
$900$	−0.107207	−0.00357356
$901$	−25.6874	−0.855771
$902$	−25.6209	−0.853083
$903$	−56.0396	−1.86488
$904$	8.55336	0.284481
$905$	−19.9976	−0.664744
$906$	52.5503	1.74587
$907$	−3.69524	−0.122698	−0.0613491	−	0.998116i	$-0.519540\pi$
$907$	−3.69524	−0.122698	−0.0613491	+	0.998116i	$0.519540\pi$
$908$	−1.18679	−0.0393850
$909$	4.99778	0.165766
$910$	−3.54808	−0.117618
$911$	−51.7983	−1.71615	−0.858077	−	0.513521i	$-0.828341\pi$
$911$	−51.7983	−1.71615	−0.858077	+	0.513521i	$0.828341\pi$
$912$	4.64169	0.153702
$913$	−16.4035	−0.542875
$914$	−8.38001	−0.277186
$915$	−74.4075	−2.45984
$916$	19.8722	0.656597
$917$	10.2739	0.339273
$918$	14.3219	0.472694
$919$	−24.1718	−0.797355	−0.398678	−	0.917091i	$-0.630531\pi$
$919$	−24.1718	−0.797355	−0.398678	+	0.917091i	$0.630531\pi$
$920$	1.57866	0.0520470
$921$	21.7904	0.718019
$922$	−2.42680	−0.0799224
$923$	−2.70783	−0.0891293
$924$	11.4316	0.376071
$925$	−0.296444	−0.00974703
$926$	−28.4538	−0.935051
$927$	1.67389	0.0549778
$928$	0.956772	0.0314076
$929$	20.3521	0.667730	0.333865	−	0.942621i	$-0.391647\pi$
$929$	20.3521	0.667730	0.333865	+	0.942621i	$0.391647\pi$
$930$	−47.1559	−1.54630
$931$	6.51050	0.213373
$932$	19.5714	0.641083
$933$	−14.3684	−0.470400
$934$	14.7273	0.481891
$935$	−32.7612	−1.07141
$936$	1.48422	0.0485132
$937$	−10.0279	−0.327597	−0.163798	−	0.986494i	$-0.552375\pi$
$937$	−10.0279	−0.327597	−0.163798	+	0.986494i	$0.552375\pi$
$938$	−23.2024	−0.757585
$939$	68.4736	2.23455
$940$	−3.12955	−0.102075
$941$	10.3107	0.336120	0.168060	−	0.985777i	$-0.446250\pi$
$941$	10.3107	0.336120	0.168060	+	0.985777i	$0.446250\pi$
$942$	6.27712	0.204520
$943$	−6.92755	−0.225592
$944$	−6.21383	−0.202243
$945$	−11.2257	−0.365171
$946$	33.8157	1.09944
$947$	−5.51848	−0.179326	−0.0896632	−	0.995972i	$-0.528579\pi$
$947$	−5.51848	−0.179326	−0.0896632	+	0.995972i	$0.528579\pi$
$948$	14.1910	0.460902
$949$	−3.15157	−0.102304
$950$	−0.122946	−0.00398888
$951$	14.3366	0.464896
$952$	11.1015	0.359803
$953$	−36.2520	−1.17432	−0.587159	−	0.809472i	$-0.699754\pi$
$953$	−36.2520	−1.17432	−0.587159	+	0.809472i	$0.699754\pi$
$954$	−8.42240	−0.272685
$955$	−25.3273	−0.819571
$956$	18.8636	0.610093
$957$	−5.52824	−0.178703
$958$	−20.6686	−0.667772
$959$	12.6040	0.407004
$960$	4.89050	0.157840
$961$	61.9749	1.99919
$962$	4.10411	0.132322
$963$	−37.0846	−1.19503
$964$	8.08984	0.260556
$965$	1.34338	0.0432448
$966$	3.09095	0.0994495
$967$	−40.6093	−1.30591	−0.652953	−	0.757399i	$-0.726470\pi$
$967$	−40.6093	−1.30591	−0.652953	+	0.757399i	$0.726470\pi$
$968$	4.10190	0.131840
$969$	−26.0454	−0.836700
$970$	9.75643	0.313260
$971$	−38.1666	−1.22483	−0.612413	−	0.790538i	$-0.709801\pi$
$971$	−38.1666	−1.22483	−0.612413	+	0.790538i	$0.709801\pi$
$972$	16.8383	0.540090
$973$	2.02204	0.0648237
$974$	−22.6031	−0.724249
$975$	−0.103417	−0.00331199
$976$	−15.2147	−0.487011
$977$	8.85419	0.283271	0.141635	−	0.989919i	$-0.454764\pi$
$977$	8.85419	0.283271	0.141635	+	0.989919i	$0.454764\pi$
$978$	23.4810	0.750839
$979$	−24.0711	−0.769315
$980$	6.85949	0.219118
$981$	14.6272	0.467010
$982$	10.0217	0.319806
$983$	22.7843	0.726706	0.363353	−	0.931651i	$-0.381632\pi$
$983$	22.7843	0.726706	0.363353	+	0.931651i	$0.381632\pi$
$984$	−21.4607	−0.684142
$985$	9.26189	0.295108
$986$	−5.36864	−0.170972
$987$	−6.12751	−0.195041
$988$	1.70211	0.0541514
$989$	9.14332	0.290741
$990$	−10.7418	−0.341396
$991$	61.6043	1.95692	0.978462	−	0.206427i	$-0.0661836\pi$
$991$	61.6043	1.95692	0.978462	+	0.206427i	$0.0661836\pi$
$992$	−9.64235	−0.306145
$993$	26.4244	0.838552
$994$	−6.64083	−0.210634
$995$	2.27347	0.0720737
$996$	−13.7399	−0.435367
$997$	−6.08411	−0.192686	−0.0963429	−	0.995348i	$-0.530715\pi$
$997$	−6.08411	−0.192686	−0.0963429	+	0.995348i	$0.530715\pi$
$998$	27.9862	0.885888
$999$	12.9849	0.410823

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.f.1.18	✓	89

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.f.1.18	✓	89	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.19996 q^{3} +1.00000 q^{4} -2.22300 q^{5} +2.19996 q^{6} +1.97846 q^{7} -1.00000 q^{8} +1.83980 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.19996 q^{3} +1.00000 q^{4} -2.22300 q^{5} +2.19996 q^{6} +1.97846 q^{7} -1.00000 q^{8} +1.83980 q^{9} +2.22300 q^{10} -2.62642 q^{11} -2.19996 q^{12} -0.806727 q^{13} -1.97846 q^{14} +4.89050 q^{15} +1.00000 q^{16} -5.61120 q^{17} -1.83980 q^{18} -2.10990 q^{19} -2.22300 q^{20} -4.35253 q^{21} +2.62642 q^{22} +0.710150 q^{23} +2.19996 q^{24} -0.0582708 q^{25} +0.806727 q^{26} +2.55238 q^{27} +1.97846 q^{28} -0.956772 q^{29} -4.89050 q^{30} +9.64235 q^{31} -1.00000 q^{32} +5.77801 q^{33} +5.61120 q^{34} -4.39812 q^{35} +1.83980 q^{36} +5.08736 q^{37} +2.10990 q^{38} +1.77476 q^{39} +2.22300 q^{40} -9.75506 q^{41} +4.35253 q^{42} +12.8752 q^{43} -2.62642 q^{44} -4.08988 q^{45} -0.710150 q^{46} +1.40781 q^{47} -2.19996 q^{48} -3.08569 q^{49} +0.0582708 q^{50} +12.3444 q^{51} -0.806727 q^{52} +4.57788 q^{53} -2.55238 q^{54} +5.83854 q^{55} -1.97846 q^{56} +4.64169 q^{57} +0.956772 q^{58} -6.21383 q^{59} +4.89050 q^{60} -15.2147 q^{61} -9.64235 q^{62} +3.63998 q^{63} +1.00000 q^{64} +1.79335 q^{65} -5.77801 q^{66} +11.7275 q^{67} -5.61120 q^{68} -1.56230 q^{69} +4.39812 q^{70} +3.35656 q^{71} -1.83980 q^{72} +3.90662 q^{73} -5.08736 q^{74} +0.128193 q^{75} -2.10990 q^{76} -5.19628 q^{77} -1.77476 q^{78} -6.45058 q^{79} -2.22300 q^{80} -11.1345 q^{81} +9.75506 q^{82} +6.24555 q^{83} -4.35253 q^{84} +12.4737 q^{85} -12.8752 q^{86} +2.10486 q^{87} +2.62642 q^{88} +9.16497 q^{89} +4.08988 q^{90} -1.59608 q^{91} +0.710150 q^{92} -21.2127 q^{93} -1.40781 q^{94} +4.69031 q^{95} +2.19996 q^{96} +4.38886 q^{97} +3.08569 q^{98} -4.83210 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9}+O(q^{10}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9	\( 89 q - 89 q^{2} - 12 q^{3} + 89 q^{4} - 18 q^{5} + 12 q^{6} - 27 q^{7} - 89 q^{8} + 95 q^{9} + 18 q^{10} - 26 q^{11} - 12 q^{12} + 2 q^{13} + 27 q^{14} - 21 q^{15} + 89 q^{16} - 60 q^{17} - 95 q^{18} + q^{19} - 18 q^{20} - 6 q^{21} + 26 q^{22} - 45 q^{23} + 12 q^{24} + 107 q^{25} - 2 q^{26} - 45 q^{27} - 27 q^{28} - 18 q^{29} + 21 q^{30} - 38 q^{31} - 89 q^{32} - 29 q^{33} + 60 q^{34} - 47 q^{35} + 95 q^{36} - 15 q^{37} - q^{38} - 38 q^{39} + 18 q^{40} - 50 q^{41} + 6 q^{42} - 15 q^{43} - 26 q^{44} - 35 q^{45} + 45 q^{46} - 121 q^{47} - 12 q^{48} + 132 q^{49} - 107 q^{50} + 6 q^{51} + 2 q^{52} - 46 q^{53} + 45 q^{54} - 37 q^{55} + 27 q^{56} - 42 q^{57} + 18 q^{58} - 34 q^{59} - 21 q^{60} + 41 q^{61} + 38 q^{62} - 131 q^{63} + 89 q^{64} - 57 q^{65} + 29 q^{66} - 11 q^{67} - 60 q^{68} + 15 q^{69} + 47 q^{70} - 66 q^{71} - 95 q^{72} - 47 q^{73} + 15 q^{74} - 46 q^{75} + q^{76} - 106 q^{77} + 38 q^{78} - 51 q^{79} - 18 q^{80} + 113 q^{81} + 50 q^{82} - 141 q^{83} - 6 q^{84} - 7 q^{85} + 15 q^{86} - 110 q^{87} + 26 q^{88} - 30 q^{89} + 35 q^{90} + 37 q^{91} - 45 q^{92} - 44 q^{93} + 121 q^{94} - 98 q^{95} + 12 q^{96} + 3 q^{97} - 132 q^{98} - 71 q^{99}+O(q^{100}) \) 89 * q - 89 * q^2 - 12 * q^3 + 89 * q^4 - 18 * q^5 + 12 * q^6 - 27 * q^7 - 89 * q^8 + 95 * q^9 + 18 * q^10 - 26 * q^11 - 12 * q^12 + 2 * q^13 + 27 * q^14 - 21 * q^15 + 89 * q^16 - 60 * q^17 - 95 * q^18 + q^19 - 18 * q^20 - 6 * q^21 + 26 * q^22 - 45 * q^23 + 12 * q^24 + 107 * q^25 - 2 * q^26 - 45 * q^27 - 27 * q^28 - 18 * q^29 + 21 * q^30 - 38 * q^31 - 89 * q^32 - 29 * q^33 + 60 * q^34 - 47 * q^35 + 95 * q^36 - 15 * q^37 - q^38 - 38 * q^39 + 18 * q^40 - 50 * q^41 + 6 * q^42 - 15 * q^43 - 26 * q^44 - 35 * q^45 + 45 * q^46 - 121 * q^47 - 12 * q^48 + 132 * q^49 - 107 * q^50 + 6 * q^51 + 2 * q^52 - 46 * q^53 + 45 * q^54 - 37 * q^55 + 27 * q^56 - 42 * q^57 + 18 * q^58 - 34 * q^59 - 21 * q^60 + 41 * q^61 + 38 * q^62 - 131 * q^63 + 89 * q^64 - 57 * q^65 + 29 * q^66 - 11 * q^67 - 60 * q^68 + 15 * q^69 + 47 * q^70 - 66 * q^71 - 95 * q^72 - 47 * q^73 + 15 * q^74 - 46 * q^75 + q^76 - 106 * q^77 + 38 * q^78 - 51 * q^79 - 18 * q^80 + 113 * q^81 + 50 * q^82 - 141 * q^83 - 6 * q^84 - 7 * q^85 + 15 * q^86 - 110 * q^87 + 26 * q^88 - 30 * q^89 + 35 * q^90 + 37 * q^91 - 45 * q^92 - 44 * q^93 + 121 * q^94 - 98 * q^95 + 12 * q^96 + 3 * q^97 - 132 * q^98 - 71 * q^99

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