LMFDB - Embedded newform 8042.2.a.a.1.59

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			1.59
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +2.02561 q^{3} +1.00000 q^{4} -1.40347 q^{5} +2.02561 q^{6} +3.13899 q^{7} +1.00000 q^{8} +1.10308 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.02561 q^{3} +1.00000 q^{4} -1.40347 q^{5} +2.02561 q^{6} +3.13899 q^{7} +1.00000 q^{8} +1.10308 q^{9} -1.40347 q^{10} -3.31834 q^{11} +2.02561 q^{12} -3.31414 q^{13} +3.13899 q^{14} -2.84289 q^{15} +1.00000 q^{16} -1.17811 q^{17} +1.10308 q^{18} -2.36567 q^{19} -1.40347 q^{20} +6.35835 q^{21} -3.31834 q^{22} -6.99532 q^{23} +2.02561 q^{24} -3.03026 q^{25} -3.31414 q^{26} -3.84241 q^{27} +3.13899 q^{28} +1.67203 q^{29} -2.84289 q^{30} -0.292298 q^{31} +1.00000 q^{32} -6.72165 q^{33} -1.17811 q^{34} -4.40549 q^{35} +1.10308 q^{36} -5.04873 q^{37} -2.36567 q^{38} -6.71314 q^{39} -1.40347 q^{40} -2.57518 q^{41} +6.35835 q^{42} -8.23043 q^{43} -3.31834 q^{44} -1.54815 q^{45} -6.99532 q^{46} +10.4302 q^{47} +2.02561 q^{48} +2.85323 q^{49} -3.03026 q^{50} -2.38638 q^{51} -3.31414 q^{52} +9.58256 q^{53} -3.84241 q^{54} +4.65720 q^{55} +3.13899 q^{56} -4.79191 q^{57} +1.67203 q^{58} -10.2680 q^{59} -2.84289 q^{60} -7.11974 q^{61} -0.292298 q^{62} +3.46256 q^{63} +1.00000 q^{64} +4.65131 q^{65} -6.72165 q^{66} -3.65974 q^{67} -1.17811 q^{68} -14.1698 q^{69} -4.40549 q^{70} +11.7494 q^{71} +1.10308 q^{72} +3.72322 q^{73} -5.04873 q^{74} -6.13811 q^{75} -2.36567 q^{76} -10.4162 q^{77} -6.71314 q^{78} -8.97847 q^{79} -1.40347 q^{80} -11.0925 q^{81} -2.57518 q^{82} +6.40575 q^{83} +6.35835 q^{84} +1.65345 q^{85} -8.23043 q^{86} +3.38687 q^{87} -3.31834 q^{88} -12.1959 q^{89} -1.54815 q^{90} -10.4030 q^{91} -6.99532 q^{92} -0.592082 q^{93} +10.4302 q^{94} +3.32016 q^{95} +2.02561 q^{96} +11.5168 q^{97} +2.85323 q^{98} -3.66040 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	2.02561	1.16948	0.584742	−	0.811219i	$-0.301196\pi$
$3$	2.02561	1.16948	0.584742	+	0.811219i	$0.301196\pi$
$4$	1.00000	0.500000
$5$	−1.40347	−0.627653	−0.313826	−	0.949480i	$-0.601611\pi$
$5$	−1.40347	−0.627653	−0.313826	+	0.949480i	$0.601611\pi$
$6$	2.02561	0.826951
$7$	3.13899	1.18643	0.593213	−	0.805046i	$-0.297859\pi$
$7$	3.13899	1.18643	0.593213	+	0.805046i	$0.297859\pi$
$8$	1.00000	0.353553
$9$	1.10308	0.367694
$10$	−1.40347	−0.443818
$11$	−3.31834	−1.00052	−0.500258	−	0.865876i	$-0.666762\pi$
$11$	−3.31834	−1.00052	−0.500258	+	0.865876i	$0.666762\pi$
$12$	2.02561	0.584742
$13$	−3.31414	−0.919176	−0.459588	−	0.888132i	$-0.652003\pi$
$13$	−3.31414	−0.919176	−0.459588	+	0.888132i	$0.652003\pi$
$14$	3.13899	0.838929
$15$	−2.84289	−0.734030
$16$	1.00000	0.250000
$17$	−1.17811	−0.285733	−0.142867	−	0.989742i	$-0.545632\pi$
$17$	−1.17811	−0.285733	−0.142867	+	0.989742i	$0.545632\pi$
$18$	1.10308	0.259999
$19$	−2.36567	−0.542722	−0.271361	−	0.962478i	$-0.587474\pi$
$19$	−2.36567	−0.542722	−0.271361	+	0.962478i	$0.587474\pi$
$20$	−1.40347	−0.313826
$21$	6.35835	1.38751
$22$	−3.31834	−0.707472
$23$	−6.99532	−1.45863	−0.729313	−	0.684180i	$-0.760160\pi$
$23$	−6.99532	−1.45863	−0.729313	+	0.684180i	$0.760160\pi$
$24$	2.02561	0.413475
$25$	−3.03026	−0.606052
$26$	−3.31414	−0.649956
$27$	−3.84241	−0.739472
$28$	3.13899	0.593213
$29$	1.67203	0.310488	0.155244	−	0.987876i	$-0.450384\pi$
$29$	1.67203	0.310488	0.155244	+	0.987876i	$0.450384\pi$
$30$	−2.84289	−0.519038
$31$	−0.292298	−0.0524983	−0.0262492	−	0.999655i	$-0.508356\pi$
$31$	−0.292298	−0.0524983	−0.0262492	+	0.999655i	$0.508356\pi$
$32$	1.00000	0.176777
$33$	−6.72165	−1.17009
$34$	−1.17811	−0.202044
$35$	−4.40549	−0.744663
$36$	1.10308	0.183847
$37$	−5.04873	−0.830006	−0.415003	−	0.909820i	$-0.636219\pi$
$37$	−5.04873	−0.830006	−0.415003	+	0.909820i	$0.636219\pi$
$38$	−2.36567	−0.383762
$39$	−6.71314	−1.07496
$40$	−1.40347	−0.221909
$41$	−2.57518	−0.402175	−0.201088	−	0.979573i	$-0.564448\pi$
$41$	−2.57518	−0.402175	−0.201088	+	0.979573i	$0.564448\pi$
$42$	6.35835	0.981115
$43$	−8.23043	−1.25513	−0.627565	−	0.778564i	$-0.715948\pi$
$43$	−8.23043	−1.25513	−0.627565	+	0.778564i	$0.715948\pi$
$44$	−3.31834	−0.500258
$45$	−1.54815	−0.230784
$46$	−6.99532	−1.03140
$47$	10.4302	1.52140	0.760698	−	0.649106i	$-0.224857\pi$
$47$	10.4302	1.52140	0.760698	+	0.649106i	$0.224857\pi$
$48$	2.02561	0.292371
$49$	2.85323	0.407604
$50$	−3.03026	−0.428543
$51$	−2.38638	−0.334161
$52$	−3.31414	−0.459588
$53$	9.58256	1.31627	0.658133	−	0.752902i	$-0.271346\pi$
$53$	9.58256	1.31627	0.658133	+	0.752902i	$0.271346\pi$
$54$	−3.84241	−0.522886
$55$	4.65720	0.627977
$56$	3.13899	0.419465
$57$	−4.79191	−0.634705
$58$	1.67203	0.219548
$59$	−10.2680	−1.33678	−0.668392	−	0.743809i	$-0.733017\pi$
$59$	−10.2680	−1.33678	−0.668392	+	0.743809i	$0.733017\pi$
$60$	−2.84289	−0.367015
$61$	−7.11974	−0.911589	−0.455795	−	0.890085i	$-0.650645\pi$
$61$	−7.11974	−0.911589	−0.455795	+	0.890085i	$0.650645\pi$
$62$	−0.292298	−0.0371219
$63$	3.46256	0.436242
$64$	1.00000	0.125000
$65$	4.65131	0.576924
$66$	−6.72165	−0.827378
$67$	−3.65974	−0.447108	−0.223554	−	0.974692i	$-0.571766\pi$
$67$	−3.65974	−0.447108	−0.223554	+	0.974692i	$0.571766\pi$
$68$	−1.17811	−0.142867
$69$	−14.1698	−1.70584
$70$	−4.40549	−0.526556
$71$	11.7494	1.39440	0.697199	−	0.716878i	$-0.254430\pi$
$71$	11.7494	1.39440	0.697199	+	0.716878i	$0.254430\pi$
$72$	1.10308	0.130000
$73$	3.72322	0.435771	0.217885	−	0.975974i	$-0.430084\pi$
$73$	3.72322	0.435771	0.217885	+	0.975974i	$0.430084\pi$
$74$	−5.04873	−0.586903
$75$	−6.13811	−0.708768
$76$	−2.36567	−0.271361
$77$	−10.4162	−1.18704
$78$	−6.71314	−0.760113
$79$	−8.97847	−1.01016	−0.505079	−	0.863073i	$-0.668536\pi$
$79$	−8.97847	−1.01016	−0.505079	+	0.863073i	$0.668536\pi$
$80$	−1.40347	−0.156913
$81$	−11.0925	−1.23250
$82$	−2.57518	−0.284381
$83$	6.40575	0.703123	0.351561	−	0.936165i	$-0.385651\pi$
$83$	6.40575	0.703123	0.351561	+	0.936165i	$0.385651\pi$
$84$	6.35835	0.693753
$85$	1.65345	0.179341
$86$	−8.23043	−0.887510
$87$	3.38687	0.363111
$88$	−3.31834	−0.353736
$89$	−12.1959	−1.29276	−0.646380	−	0.763016i	$-0.723718\pi$
$89$	−12.1959	−1.29276	−0.646380	+	0.763016i	$0.723718\pi$
$90$	−1.54815	−0.163189
$91$	−10.4030	−1.09053
$92$	−6.99532	−0.729313
$93$	−0.592082	−0.0613960
$94$	10.4302	1.07579
$95$	3.32016	0.340641
$96$	2.02561	0.206738
$97$	11.5168	1.16935	0.584675	−	0.811268i	$-0.301222\pi$
$97$	11.5168	1.16935	0.584675	+	0.811268i	$0.301222\pi$
$98$	2.85323	0.288220
$99$	−3.66040	−0.367884
$100$	−3.03026	−0.303026
$101$	−18.0044	−1.79151	−0.895753	−	0.444553i	$-0.853363\pi$
$101$	−18.0044	−1.79151	−0.895753	+	0.444553i	$0.853363\pi$
$102$	−2.38638	−0.236287
$103$	−3.15847	−0.311214	−0.155607	−	0.987819i	$-0.549733\pi$
$103$	−3.15847	−0.311214	−0.155607	+	0.987819i	$0.549733\pi$
$104$	−3.31414	−0.324978
$105$	−8.92378	−0.870872
$106$	9.58256	0.930741
$107$	11.6403	1.12531	0.562653	−	0.826693i	$-0.309781\pi$
$107$	11.6403	1.12531	0.562653	+	0.826693i	$0.309781\pi$
$108$	−3.84241	−0.369736
$109$	16.3869	1.56958	0.784789	−	0.619763i	$-0.212771\pi$
$109$	16.3869	1.56958	0.784789	+	0.619763i	$0.212771\pi$
$110$	4.65720	0.444047
$111$	−10.2267	−0.970679
$112$	3.13899	0.296606
$113$	−11.4477	−1.07691	−0.538455	−	0.842654i	$-0.680992\pi$
$113$	−11.4477	−1.07691	−0.538455	+	0.842654i	$0.680992\pi$
$114$	−4.79191	−0.448804
$115$	9.81776	0.915511
$116$	1.67203	0.155244
$117$	−3.65577	−0.337976
$118$	−10.2680	−0.945249
$119$	−3.69807	−0.339001
$120$	−2.84289	−0.259519
$121$	0.0113668	0.00103335
$122$	−7.11974	−0.644591
$123$	−5.21630	−0.470338
$124$	−0.292298	−0.0262492
$125$	11.2703	1.00804
$126$	3.46256	0.308469
$127$	10.9598	0.972523	0.486262	−	0.873813i	$-0.338360\pi$
$127$	10.9598	0.972523	0.486262	+	0.873813i	$0.338360\pi$
$128$	1.00000	0.0883883
$129$	−16.6716	−1.46785
$130$	4.65131	0.407947
$131$	4.36120	0.381039	0.190520	−	0.981683i	$-0.438983\pi$
$131$	4.36120	0.381039	0.190520	+	0.981683i	$0.438983\pi$
$132$	−6.72165	−0.585044
$133$	−7.42580	−0.643898
$134$	−3.65974	−0.316153
$135$	5.39272	0.464132
$136$	−1.17811	−0.101022
$137$	−1.78357	−0.152381	−0.0761905	−	0.997093i	$-0.524276\pi$
$137$	−1.78357	−0.152381	−0.0761905	+	0.997093i	$0.524276\pi$
$138$	−14.1698	−1.20621
$139$	−7.45251	−0.632114	−0.316057	−	0.948740i	$-0.602359\pi$
$139$	−7.45251	−0.632114	−0.316057	+	0.948740i	$0.602359\pi$
$140$	−4.40549	−0.372332
$141$	21.1274	1.77925
$142$	11.7494	0.985988
$143$	10.9974	0.919651
$144$	1.10308	0.0919236
$145$	−2.34665	−0.194879
$146$	3.72322	0.308136
$147$	5.77952	0.476687
$148$	−5.04873	−0.415003
$149$	−4.88481	−0.400179	−0.200090	−	0.979778i	$-0.564123\pi$
$149$	−4.88481	−0.400179	−0.200090	+	0.979778i	$0.564123\pi$
$150$	−6.13811	−0.501175
$151$	19.3351	1.57347	0.786735	−	0.617291i	$-0.211770\pi$
$151$	19.3351	1.57347	0.786735	+	0.617291i	$0.211770\pi$
$152$	−2.36567	−0.191881
$153$	−1.29955	−0.105063
$154$	−10.4162	−0.839363
$155$	0.410233	0.0329507
$156$	−6.71314	−0.537481
$157$	14.1094	1.12605	0.563026	−	0.826439i	$-0.309637\pi$
$157$	14.1094	1.12605	0.563026	+	0.826439i	$0.309637\pi$
$158$	−8.97847	−0.714289
$159$	19.4105	1.53935
$160$	−1.40347	−0.110954
$161$	−21.9582	−1.73055
$162$	−11.0925	−0.871506
$163$	15.6471	1.22557	0.612787	−	0.790248i	$-0.290048\pi$
$163$	15.6471	1.22557	0.612787	+	0.790248i	$0.290048\pi$
$164$	−2.57518	−0.201088
$165$	9.43366	0.734410
$166$	6.40575	0.497183
$167$	4.79395	0.370967	0.185483	−	0.982647i	$-0.440615\pi$
$167$	4.79395	0.370967	0.185483	+	0.982647i	$0.440615\pi$
$168$	6.35835	0.490557
$169$	−2.01649	−0.155115
$170$	1.65345	0.126813
$171$	−2.60953	−0.199556
$172$	−8.23043	−0.627565
$173$	−1.82840	−0.139011	−0.0695055	−	0.997582i	$-0.522142\pi$
$173$	−1.82840	−0.139011	−0.0695055	+	0.997582i	$0.522142\pi$
$174$	3.38687	0.256758
$175$	−9.51194	−0.719035
$176$	−3.31834	−0.250129
$177$	−20.7990	−1.56335
$178$	−12.1959	−0.914120
$179$	8.51227	0.636237	0.318118	−	0.948051i	$-0.396949\pi$
$179$	8.51227	0.636237	0.318118	+	0.948051i	$0.396949\pi$
$180$	−1.54815	−0.115392
$181$	2.50544	0.186228	0.0931138	−	0.995655i	$-0.470318\pi$
$181$	2.50544	0.186228	0.0931138	+	0.995655i	$0.470318\pi$
$182$	−10.4030	−0.771124
$183$	−14.4218	−1.06609
$184$	−6.99532	−0.515702
$185$	7.08576	0.520955
$186$	−0.592082	−0.0434135
$187$	3.90936	0.285881
$188$	10.4302	0.760698
$189$	−12.0613	−0.877328
$190$	3.32016	0.240869
$191$	−8.95790	−0.648171	−0.324085	−	0.946028i	$-0.605056\pi$
$191$	−8.95790	−0.648171	−0.324085	+	0.946028i	$0.605056\pi$
$192$	2.02561	0.146186
$193$	−19.6719	−1.41601	−0.708006	−	0.706206i	$-0.750405\pi$
$193$	−19.6719	−1.41601	−0.708006	+	0.706206i	$0.750405\pi$
$194$	11.5168	0.826855
$195$	9.42172	0.674703
$196$	2.85323	0.203802
$197$	15.5963	1.11119	0.555596	−	0.831452i	$-0.312490\pi$
$197$	15.5963	1.11119	0.555596	+	0.831452i	$0.312490\pi$
$198$	−3.66040	−0.260133
$199$	−13.4087	−0.950515	−0.475257	−	0.879847i	$-0.657645\pi$
$199$	−13.4087	−0.950515	−0.475257	+	0.879847i	$0.657645\pi$
$200$	−3.03026	−0.214272
$201$	−7.41319	−0.522886
$202$	−18.0044	−1.26679
$203$	5.24847	0.368371
$204$	−2.38638	−0.167080
$205$	3.61420	0.252426
$206$	−3.15847	−0.220061
$207$	−7.71642	−0.536328
$208$	−3.31414	−0.229794
$209$	7.85009	0.543002
$210$	−8.92378	−0.615799
$211$	−26.9085	−1.85246	−0.926230	−	0.376960i	$-0.876970\pi$
$211$	−26.9085	−1.85246	−0.926230	+	0.376960i	$0.876970\pi$
$212$	9.58256	0.658133
$213$	23.7997	1.63073
$214$	11.6403	0.795712
$215$	11.5512	0.787785
$216$	−3.84241	−0.261443
$217$	−0.917521	−0.0622854
$218$	16.3869	1.10986
$219$	7.54179	0.509627
$220$	4.65720	0.313989
$221$	3.90441	0.262639
$222$	−10.2267	−0.686373
$223$	10.1417	0.679141	0.339570	−	0.940581i	$-0.389718\pi$
$223$	10.1417	0.679141	0.339570	+	0.940581i	$0.389718\pi$
$224$	3.13899	0.209732
$225$	−3.34263	−0.222842
$226$	−11.4477	−0.761490
$227$	−15.7277	−1.04388	−0.521941	−	0.852982i	$-0.674792\pi$
$227$	−15.7277	−1.04388	−0.521941	+	0.852982i	$0.674792\pi$
$228$	−4.79191	−0.317352
$229$	7.52891	0.497524	0.248762	−	0.968565i	$-0.419976\pi$
$229$	7.52891	0.497524	0.248762	+	0.968565i	$0.419976\pi$
$230$	9.81776	0.647364
$231$	−21.0992	−1.38822
$232$	1.67203	0.109774
$233$	−11.8437	−0.775906	−0.387953	−	0.921679i	$-0.626818\pi$
$233$	−11.8437	−0.775906	−0.387953	+	0.921679i	$0.626818\pi$
$234$	−3.65577	−0.238985
$235$	−14.6385	−0.954909
$236$	−10.2680	−0.668392
$237$	−18.1869	−1.18136
$238$	−3.69807	−0.239710
$239$	−20.8890	−1.35120	−0.675599	−	0.737269i	$-0.736115\pi$
$239$	−20.8890	−1.35120	−0.675599	+	0.737269i	$0.736115\pi$
$240$	−2.84289	−0.183508
$241$	23.4202	1.50863	0.754313	−	0.656515i	$-0.227970\pi$
$241$	23.4202	1.50863	0.754313	+	0.656515i	$0.227970\pi$
$242$	0.0113668	0.000730687 0
$243$	−10.9417	−0.701912
$244$	−7.11974	−0.455795
$245$	−4.00444	−0.255834
$246$	−5.21630	−0.332579
$247$	7.84015	0.498857
$248$	−0.292298	−0.0185610
$249$	12.9755	0.822291
$250$	11.2703	0.712794
$251$	11.6532	0.735544	0.367772	−	0.929916i	$-0.380121\pi$
$251$	11.6532	0.735544	0.367772	+	0.929916i	$0.380121\pi$
$252$	3.46256	0.218121
$253$	23.2129	1.45938
$254$	10.9598	0.687678
$255$	3.34923	0.209737
$256$	1.00000	0.0625000
$257$	17.2869	1.07833	0.539164	−	0.842201i	$-0.318740\pi$
$257$	17.2869	1.07833	0.539164	+	0.842201i	$0.318740\pi$
$258$	−16.6716	−1.03793
$259$	−15.8479	−0.984739
$260$	4.65131	0.288462
$261$	1.84439	0.114165
$262$	4.36120	0.269436
$263$	−20.4887	−1.26339	−0.631695	−	0.775217i	$-0.717640\pi$
$263$	−20.4887	−1.26339	−0.631695	+	0.775217i	$0.717640\pi$
$264$	−6.72165	−0.413689
$265$	−13.4489	−0.826158
$266$	−7.42580	−0.455305
$267$	−24.7041	−1.51186
$268$	−3.65974	−0.223554
$269$	17.9353	1.09354	0.546769	−	0.837284i	$-0.315858\pi$
$269$	17.9353	1.09354	0.546769	+	0.837284i	$0.315858\pi$
$270$	5.39272	0.328191
$271$	−25.9394	−1.57571	−0.787853	−	0.615864i	$-0.788807\pi$
$271$	−25.9394	−1.57571	−0.787853	+	0.615864i	$0.788807\pi$
$272$	−1.17811	−0.0714333
$273$	−21.0724	−1.27536
$274$	−1.78357	−0.107750
$275$	10.0554	0.606365
$276$	−14.1698	−0.852920
$277$	3.56266	0.214060	0.107030	−	0.994256i	$-0.465866\pi$
$277$	3.56266	0.214060	0.107030	+	0.994256i	$0.465866\pi$
$278$	−7.45251	−0.446972
$279$	−0.322429	−0.0193033
$280$	−4.40549	−0.263278
$281$	−2.40111	−0.143238	−0.0716190	−	0.997432i	$-0.522817\pi$
$281$	−2.40111	−0.143238	−0.0716190	+	0.997432i	$0.522817\pi$
$282$	21.1274	1.25812
$283$	24.9558	1.48347	0.741734	−	0.670694i	$-0.234003\pi$
$283$	24.9558	1.48347	0.741734	+	0.670694i	$0.234003\pi$
$284$	11.7494	0.697199
$285$	6.72533	0.398374
$286$	10.9974	0.650292
$287$	−8.08345	−0.477151
$288$	1.10308	0.0649998
$289$	−15.6121	−0.918356
$290$	−2.34665	−0.137800
$291$	23.3284	1.36754
$292$	3.72322	0.217885
$293$	26.6301	1.55575	0.777875	−	0.628420i	$-0.216298\pi$
$293$	26.6301	1.55575	0.777875	+	0.628420i	$0.216298\pi$
$294$	5.77952	0.337069
$295$	14.4109	0.839036
$296$	−5.04873	−0.293451
$297$	12.7504	0.739854
$298$	−4.88481	−0.282969
$299$	23.1835	1.34073
$300$	−6.13811	−0.354384
$301$	−25.8352	−1.48912
$302$	19.3351	1.11261
$303$	−36.4698	−2.09514
$304$	−2.36567	−0.135680
$305$	9.99237	0.572162
$306$	−1.29955	−0.0742904
$307$	−2.48205	−0.141658	−0.0708291	−	0.997488i	$-0.522565\pi$
$307$	−2.48205	−0.141658	−0.0708291	+	0.997488i	$0.522565\pi$
$308$	−10.4162	−0.593519
$309$	−6.39782	−0.363959
$310$	0.410233	0.0232997
$311$	−2.30513	−0.130712	−0.0653561	−	0.997862i	$-0.520818\pi$
$311$	−2.30513	−0.130712	−0.0653561	+	0.997862i	$0.520818\pi$
$312$	−6.71314	−0.380057
$313$	5.09640	0.288065	0.144033	−	0.989573i	$-0.453993\pi$
$313$	5.09640	0.288065	0.144033	+	0.989573i	$0.453993\pi$
$314$	14.1094	0.796240
$315$	−4.85962	−0.273808
$316$	−8.97847	−0.505079
$317$	3.14856	0.176841	0.0884204	−	0.996083i	$-0.471818\pi$
$317$	3.14856	0.176841	0.0884204	+	0.996083i	$0.471818\pi$
$318$	19.4105	1.08849
$319$	−5.54836	−0.310648
$320$	−1.40347	−0.0784566
$321$	23.5786	1.31603
$322$	−21.9582	−1.22368
$323$	2.78701	0.155074
$324$	−11.0925	−0.616248
$325$	10.0427	0.557069
$326$	15.6471	0.866612
$327$	33.1934	1.83560
$328$	−2.57518	−0.142190
$329$	32.7401	1.80502
$330$	9.43366	0.519306
$331$	25.9353	1.42553	0.712767	−	0.701400i	$-0.247441\pi$
$331$	25.9353	1.42553	0.712767	+	0.701400i	$0.247441\pi$
$332$	6.40575	0.351561
$333$	−5.56916	−0.305188
$334$	4.79395	0.262313
$335$	5.13635	0.280629
$336$	6.35835	0.346876
$337$	−5.36067	−0.292014	−0.146007	−	0.989284i	$-0.546642\pi$
$337$	−5.36067	−0.292014	−0.146007	+	0.989284i	$0.546642\pi$
$338$	−2.01649	−0.109683
$339$	−23.1886	−1.25943
$340$	1.65345	0.0896707
$341$	0.969945	0.0525255
$342$	−2.60953	−0.141107
$343$	−13.0167	−0.702833
$344$	−8.23043	−0.443755
$345$	19.8869	1.07068
$346$	−1.82840	−0.0982956
$347$	20.0869	1.07832	0.539161	−	0.842203i	$-0.318741\pi$
$347$	20.0869	1.07832	0.539161	+	0.842203i	$0.318741\pi$
$348$	3.38687	0.181555
$349$	−25.1034	−1.34375	−0.671877	−	0.740663i	$-0.734512\pi$
$349$	−25.1034	−1.34375	−0.671877	+	0.740663i	$0.734512\pi$
$350$	−9.51194	−0.508435
$351$	12.7343	0.679705
$352$	−3.31834	−0.176868
$353$	23.5854	1.25532	0.627661	−	0.778487i	$-0.284012\pi$
$353$	23.5854	1.25532	0.627661	+	0.778487i	$0.284012\pi$
$354$	−20.7990	−1.10545
$355$	−16.4900	−0.875198
$356$	−12.1959	−0.646380
$357$	−7.49083	−0.396457
$358$	8.51227	0.449887
$359$	14.2407	0.751595	0.375798	−	0.926702i	$-0.377369\pi$
$359$	14.2407	0.751595	0.375798	+	0.926702i	$0.377369\pi$
$360$	−1.54815	−0.0815946
$361$	−13.4036	−0.705453
$362$	2.50544	0.131683
$363$	0.0230247	0.00120848
$364$	−10.4030	−0.545267
$365$	−5.22545	−0.273513
$366$	−14.4218	−0.753839
$367$	−15.7133	−0.820228	−0.410114	−	0.912034i	$-0.634511\pi$
$367$	−15.7133	−0.820228	−0.410114	+	0.912034i	$0.634511\pi$
$368$	−6.99532	−0.364657
$369$	−2.84063	−0.147878
$370$	7.08576	0.368371
$371$	30.0795	1.56165
$372$	−0.592082	−0.0306980
$373$	−5.06110	−0.262054	−0.131027	−	0.991379i	$-0.541827\pi$
$373$	−5.06110	−0.262054	−0.131027	+	0.991379i	$0.541827\pi$
$374$	3.90936	0.202148
$375$	22.8291	1.17889
$376$	10.4302	0.537895
$377$	−5.54133	−0.285393
$378$	−12.0613	−0.620364
$379$	14.2646	0.732723	0.366362	−	0.930473i	$-0.380603\pi$
$379$	14.2646	0.732723	0.366362	+	0.930473i	$0.380603\pi$
$380$	3.32016	0.170320
$381$	22.2002	1.13735
$382$	−8.95790	−0.458326
$383$	−26.8851	−1.37376	−0.686882	−	0.726769i	$-0.741021\pi$
$383$	−26.8851	−1.37376	−0.686882	+	0.726769i	$0.741021\pi$
$384$	2.02561	0.103369
$385$	14.6189	0.745048
$386$	−19.6719	−1.00127
$387$	−9.07885	−0.461504
$388$	11.5168	0.584675
$389$	−19.5166	−0.989533	−0.494767	−	0.869026i	$-0.664746\pi$
$389$	−19.5166	−0.989533	−0.494767	+	0.869026i	$0.664746\pi$
$390$	9.42172	0.477087
$391$	8.24125	0.416778
$392$	2.85323	0.144110
$393$	8.83407	0.445620
$394$	15.5963	0.785732
$395$	12.6011	0.634028
$396$	−3.66040	−0.183942
$397$	−12.3196	−0.618304	−0.309152	−	0.951013i	$-0.600045\pi$
$397$	−12.3196	−0.618304	−0.309152	+	0.951013i	$0.600045\pi$
$398$	−13.4087	−0.672115
$399$	−15.0417	−0.753029
$400$	−3.03026	−0.151513
$401$	−37.6213	−1.87872	−0.939359	−	0.342935i	$-0.888579\pi$
$401$	−37.6213	−1.87872	−0.939359	+	0.342935i	$0.888579\pi$
$402$	−7.41319	−0.369736
$403$	0.968717	0.0482552
$404$	−18.0044	−0.895753
$405$	15.5680	0.773579
$406$	5.24847	0.260477
$407$	16.7534	0.830434
$408$	−2.38638	−0.118144
$409$	−31.4047	−1.55286	−0.776431	−	0.630202i	$-0.782972\pi$
$409$	−31.4047	−1.55286	−0.776431	+	0.630202i	$0.782972\pi$
$410$	3.61420	0.178492
$411$	−3.61282	−0.178207
$412$	−3.15847	−0.155607
$413$	−32.2312	−1.58599
$414$	−7.71642	−0.379242
$415$	−8.99031	−0.441317
$416$	−3.31414	−0.162489
$417$	−15.0959	−0.739247
$418$	7.85009	0.383960
$419$	9.80992	0.479246	0.239623	−	0.970866i	$-0.422976\pi$
$419$	9.80992	0.479246	0.239623	+	0.970866i	$0.422976\pi$
$420$	−8.92378	−0.435436
$421$	5.24081	0.255422	0.127711	−	0.991811i	$-0.459237\pi$
$421$	5.24081	0.255422	0.127711	+	0.991811i	$0.459237\pi$
$422$	−26.9085	−1.30989
$423$	11.5053	0.559409
$424$	9.58256	0.465370
$425$	3.56997	0.173169
$426$	23.7997	1.15310
$427$	−22.3488	−1.08153
$428$	11.6403	0.562653
$429$	22.2765	1.07552
$430$	11.5512	0.557048
$431$	8.97057	0.432097	0.216048	−	0.976383i	$-0.430683\pi$
$431$	8.97057	0.432097	0.216048	+	0.976383i	$0.430683\pi$
$432$	−3.84241	−0.184868
$433$	−30.5789	−1.46953	−0.734764	−	0.678322i	$-0.762707\pi$
$433$	−30.5789	−1.46953	−0.734764	+	0.678322i	$0.762707\pi$
$434$	−0.917521	−0.0440424
$435$	−4.75339	−0.227908
$436$	16.3869	0.784789
$437$	16.5486	0.791628
$438$	7.54179	0.360361
$439$	−28.2743	−1.34946	−0.674728	−	0.738066i	$-0.735739\pi$
$439$	−28.2743	−1.34946	−0.674728	+	0.738066i	$0.735739\pi$
$440$	4.65720	0.222023
$441$	3.14735	0.149874
$442$	3.90441	0.185714
$443$	29.7383	1.41291	0.706455	−	0.707758i	$-0.250293\pi$
$443$	29.7383	1.41291	0.706455	+	0.707758i	$0.250293\pi$
$444$	−10.2267	−0.485339
$445$	17.1166	0.811405
$446$	10.1417	0.480225
$447$	−9.89470	−0.468003
$448$	3.13899	0.148303
$449$	−21.0145	−0.991738	−0.495869	−	0.868397i	$-0.665150\pi$
$449$	−21.0145	−0.991738	−0.495869	+	0.868397i	$0.665150\pi$
$450$	−3.34263	−0.157573
$451$	8.54531	0.402383
$452$	−11.4477	−0.538455
$453$	39.1653	1.84015
$454$	−15.7277	−0.738136
$455$	14.6004	0.684477
$456$	−4.79191	−0.224402
$457$	32.7526	1.53210	0.766051	−	0.642779i	$-0.222219\pi$
$457$	32.7526	1.53210	0.766051	+	0.642779i	$0.222219\pi$
$458$	7.52891	0.351803
$459$	4.52677	0.211292
$460$	9.81776	0.457755
$461$	9.69798	0.451680	0.225840	−	0.974164i	$-0.427487\pi$
$461$	9.69798	0.451680	0.225840	+	0.974164i	$0.427487\pi$
$462$	−21.0992	−0.981622
$463$	−33.4658	−1.55529	−0.777645	−	0.628704i	$-0.783586\pi$
$463$	−33.4658	−1.55529	−0.777645	+	0.628704i	$0.783586\pi$
$464$	1.67203	0.0776220
$465$	0.830971	0.0385354
$466$	−11.8437	−0.548648
$467$	−26.4721	−1.22498	−0.612491	−	0.790478i	$-0.709832\pi$
$467$	−26.4721	−1.22498	−0.612491	+	0.790478i	$0.709832\pi$
$468$	−3.65577	−0.168988
$469$	−11.4879	−0.530460
$470$	−14.6385	−0.675222
$471$	28.5801	1.31690
$472$	−10.2680	−0.472624
$473$	27.3114	1.25578
$474$	−18.1869	−0.835350
$475$	7.16859	0.328917
$476$	−3.69807	−0.169501
$477$	10.5704	0.483984
$478$	−20.8890	−0.955442
$479$	−21.2265	−0.969865	−0.484932	−	0.874552i	$-0.661156\pi$
$479$	−21.2265	−0.969865	−0.484932	+	0.874552i	$0.661156\pi$
$480$	−2.84289	−0.129759
$481$	16.7322	0.762921
$482$	23.4202	1.06676
$483$	−44.4787	−2.02385
$484$	0.0113668	0.000516674 0
$485$	−16.1635	−0.733946
$486$	−10.9417	−0.496327
$487$	−37.2100	−1.68614	−0.843072	−	0.537800i	$-0.819256\pi$
$487$	−37.2100	−1.68614	−0.843072	+	0.537800i	$0.819256\pi$
$488$	−7.11974	−0.322296
$489$	31.6949	1.43329
$490$	−4.00444	−0.180902
$491$	−7.91452	−0.357177	−0.178589	−	0.983924i	$-0.557153\pi$
$491$	−7.91452	−0.357177	−0.178589	+	0.983924i	$0.557153\pi$
$492$	−5.21630	−0.235169
$493$	−1.96983	−0.0887167
$494$	7.84015	0.352745
$495$	5.13728	0.230904
$496$	−0.292298	−0.0131246
$497$	36.8812	1.65435
$498$	12.9755	0.581448
$499$	12.9964	0.581798	0.290899	−	0.956754i	$-0.406046\pi$
$499$	12.9964	0.581798	0.290899	+	0.956754i	$0.406046\pi$
$500$	11.2703	0.504022
$501$	9.71066	0.433840
$502$	11.6532	0.520108
$503$	−14.4361	−0.643676	−0.321838	−	0.946795i	$-0.604301\pi$
$503$	−14.4361	−0.643676	−0.321838	+	0.946795i	$0.604301\pi$
$504$	3.46256	0.154235
$505$	25.2687	1.12444
$506$	23.2129	1.03194
$507$	−4.08462	−0.181404
$508$	10.9598	0.486262
$509$	22.7790	1.00966	0.504830	−	0.863219i	$-0.331555\pi$
$509$	22.7790	1.00966	0.504830	+	0.863219i	$0.331555\pi$
$510$	3.34923	0.148306
$511$	11.6871	0.517009
$512$	1.00000	0.0441942
$513$	9.08986	0.401327
$514$	17.2869	0.762492
$515$	4.43284	0.195334
$516$	−16.6716	−0.733927
$517$	−34.6108	−1.52218
$518$	−15.8479	−0.696316
$519$	−3.70363	−0.162571
$520$	4.65131	0.203973
$521$	−41.2809	−1.80855	−0.904275	−	0.426950i	$-0.859588\pi$
$521$	−41.2809	−1.80855	−0.904275	+	0.426950i	$0.859588\pi$
$522$	1.84439	0.0807266
$523$	9.22828	0.403524	0.201762	−	0.979435i	$-0.435333\pi$
$523$	9.22828	0.403524	0.201762	+	0.979435i	$0.435333\pi$
$524$	4.36120	0.190520
$525$	−19.2675	−0.840901
$526$	−20.4887	−0.893351
$527$	0.344359	0.0150005
$528$	−6.72165	−0.292522
$529$	25.9346	1.12759
$530$	−13.4489	−0.584182
$531$	−11.3265	−0.491528
$532$	−7.42580	−0.321949
$533$	8.53449	0.369670
$534$	−24.7041	−1.06905
$535$	−16.3368	−0.706302
$536$	−3.65974	−0.158077
$537$	17.2425	0.744069
$538$	17.9353	0.773248
$539$	−9.46798	−0.407815
$540$	5.39272	0.232066
$541$	12.2409	0.526277	0.263139	−	0.964758i	$-0.415242\pi$
$541$	12.2409	0.526277	0.263139	+	0.964758i	$0.415242\pi$
$542$	−25.9394	−1.11419
$543$	5.07503	0.217790
$544$	−1.17811	−0.0505110
$545$	−22.9986	−0.985150
$546$	−21.0724	−0.901818
$547$	−27.0379	−1.15606	−0.578029	−	0.816016i	$-0.696178\pi$
$547$	−27.0379	−1.15606	−0.578029	+	0.816016i	$0.696178\pi$
$548$	−1.78357	−0.0761905
$549$	−7.85366	−0.335186
$550$	10.0554	0.428765
$551$	−3.95546	−0.168508
$552$	−14.1698	−0.603106
$553$	−28.1833	−1.19848
$554$	3.56266	0.151363
$555$	14.3530	0.609249
$556$	−7.45251	−0.316057
$557$	21.8575	0.926131	0.463066	−	0.886324i	$-0.346749\pi$
$557$	21.8575	0.926131	0.463066	+	0.886324i	$0.346749\pi$
$558$	−0.322429	−0.0136495
$559$	27.2768	1.15369
$560$	−4.40549	−0.186166
$561$	7.91883	0.334333
$562$	−2.40111	−0.101285
$563$	−32.5808	−1.37312	−0.686559	−	0.727074i	$-0.740880\pi$
$563$	−32.5808	−1.37312	−0.686559	+	0.727074i	$0.740880\pi$
$564$	21.1274	0.889625
$565$	16.0666	0.675926
$566$	24.9558	1.04897
$567$	−34.8191	−1.46226
$568$	11.7494	0.492994
$569$	−2.12686	−0.0891626	−0.0445813	−	0.999006i	$-0.514195\pi$
$569$	−2.12686	−0.0891626	−0.0445813	+	0.999006i	$0.514195\pi$
$570$	6.72533	0.281693
$571$	11.7787	0.492923	0.246461	−	0.969153i	$-0.420732\pi$
$571$	11.7787	0.492923	0.246461	+	0.969153i	$0.420732\pi$
$572$	10.9974	0.459826
$573$	−18.1452	−0.758026
$574$	−8.08345	−0.337397
$575$	21.1976	0.884003
$576$	1.10308	0.0459618
$577$	23.7674	0.989450	0.494725	−	0.869050i	$-0.335269\pi$
$577$	23.7674	0.989450	0.494725	+	0.869050i	$0.335269\pi$
$578$	−15.6121	−0.649376
$579$	−39.8475	−1.65600
$580$	−2.34665	−0.0974393
$581$	20.1076	0.834202
$582$	23.3284	0.966994
$583$	−31.7982	−1.31695
$584$	3.72322	0.154068
$585$	5.13078	0.212132
$586$	26.6301	1.10008
$587$	32.1082	1.32525	0.662624	−	0.748952i	$-0.269443\pi$
$587$	32.1082	1.32525	0.662624	+	0.748952i	$0.269443\pi$
$588$	5.77952	0.238344
$589$	0.691481	0.0284920
$590$	14.4109	0.593288
$591$	31.5920	1.29952
$592$	−5.04873	−0.207501
$593$	24.5936	1.00994	0.504969	−	0.863138i	$-0.331504\pi$
$593$	24.5936	1.00994	0.504969	+	0.863138i	$0.331504\pi$
$594$	12.7504	0.523156
$595$	5.19014	0.212775
$596$	−4.88481	−0.200090
$597$	−27.1607	−1.11161
$598$	23.1835	0.948043
$599$	5.49597	0.224559	0.112280	−	0.993677i	$-0.464185\pi$
$599$	5.49597	0.224559	0.112280	+	0.993677i	$0.464185\pi$
$600$	−6.13811	−0.250587
$601$	−14.9590	−0.610191	−0.305095	−	0.952322i	$-0.598688\pi$
$601$	−14.9590	−0.610191	−0.305095	+	0.952322i	$0.598688\pi$
$602$	−25.8352	−1.05296
$603$	−4.03699	−0.164399
$604$	19.3351	0.786735
$605$	−0.0159530	−0.000648584 0
$606$	−36.4698	−1.48149
$607$	−18.1977	−0.738623	−0.369312	−	0.929306i	$-0.620406\pi$
$607$	−18.1977	−0.738623	−0.369312	+	0.929306i	$0.620406\pi$
$608$	−2.36567	−0.0959405
$609$	10.6313	0.430804
$610$	9.99237	0.404579
$611$	−34.5670	−1.39843
$612$	−1.29955	−0.0525313
$613$	−20.3102	−0.820322	−0.410161	−	0.912013i	$-0.634527\pi$
$613$	−20.3102	−0.820322	−0.410161	+	0.912013i	$0.634527\pi$
$614$	−2.48205	−0.100168
$615$	7.32094	0.295209
$616$	−10.4162	−0.419681
$617$	8.25776	0.332445	0.166222	−	0.986088i	$-0.446843\pi$
$617$	8.25776	0.332445	0.166222	+	0.986088i	$0.446843\pi$
$618$	−6.39782	−0.257358
$619$	−7.82306	−0.314435	−0.157218	−	0.987564i	$-0.550252\pi$
$619$	−7.82306	−0.314435	−0.157218	+	0.987564i	$0.550252\pi$
$620$	0.410233	0.0164754
$621$	26.8789	1.07861
$622$	−2.30513	−0.0924274
$623$	−38.2827	−1.53376
$624$	−6.71314	−0.268741
$625$	−0.666230	−0.0266492
$626$	5.09640	0.203693
$627$	15.9012	0.635032
$628$	14.1094	0.563026
$629$	5.94795	0.237160
$630$	−4.85962	−0.193612
$631$	38.3415	1.52635	0.763175	−	0.646191i	$-0.223639\pi$
$631$	38.3415	1.52635	0.763175	+	0.646191i	$0.223639\pi$
$632$	−8.97847	−0.357144
$633$	−54.5061	−2.16642
$634$	3.14856	0.125045
$635$	−15.3818	−0.610407
$636$	19.4105	0.769676
$637$	−9.45600	−0.374660
$638$	−5.54836	−0.219661
$639$	12.9606	0.512712
$640$	−1.40347	−0.0554772
$641$	48.6835	1.92288	0.961442	−	0.275009i	$-0.0886810\pi$
$641$	48.6835	1.92288	0.961442	+	0.275009i	$0.0886810\pi$
$642$	23.5786	0.930572
$643$	−33.4492	−1.31911	−0.659554	−	0.751657i	$-0.729255\pi$
$643$	−33.4492	−1.31911	−0.659554	+	0.751657i	$0.729255\pi$
$644$	−21.9582	−0.865275
$645$	23.3982	0.921303
$646$	2.78701	0.109654
$647$	−39.5909	−1.55648	−0.778240	−	0.627967i	$-0.783887\pi$
$647$	−39.5909	−1.55648	−0.778240	+	0.627967i	$0.783887\pi$
$648$	−11.0925	−0.435753
$649$	34.0728	1.33747
$650$	10.0427	0.393907
$651$	−1.85854	−0.0728418
$652$	15.6471	0.612787
$653$	−29.7959	−1.16600	−0.583002	−	0.812471i	$-0.698122\pi$
$653$	−29.7959	−1.16600	−0.583002	+	0.812471i	$0.698122\pi$
$654$	33.1934	1.29796
$655$	−6.12083	−0.239160
$656$	−2.57518	−0.100544
$657$	4.10703	0.160230
$658$	32.7401	1.27634
$659$	−28.8985	−1.12573	−0.562863	−	0.826550i	$-0.690300\pi$
$659$	−28.8985	−1.12573	−0.562863	+	0.826550i	$0.690300\pi$
$660$	9.43366	0.367205
$661$	−0.199112	−0.00774454	−0.00387227	−	0.999993i	$-0.501233\pi$
$661$	−0.199112	−0.00774454	−0.00387227	+	0.999993i	$0.501233\pi$
$662$	25.9353	1.00801
$663$	7.90881	0.307153
$664$	6.40575	0.248591
$665$	10.4219	0.404145
$666$	−5.56916	−0.215801
$667$	−11.6964	−0.452886
$668$	4.79395	0.185483
$669$	20.5432	0.794245
$670$	5.13635	0.198434
$671$	23.6257	0.912060
$672$	6.35835	0.245279
$673$	−4.99642	−0.192598	−0.0962989	−	0.995352i	$-0.530700\pi$
$673$	−4.99642	−0.192598	−0.0962989	+	0.995352i	$0.530700\pi$
$674$	−5.36067	−0.206485
$675$	11.6435	0.448158
$676$	−2.01649	−0.0775574
$677$	0.864166	0.0332126	0.0166063	−	0.999862i	$-0.494714\pi$
$677$	0.864166	0.0332126	0.0166063	+	0.999862i	$0.494714\pi$
$678$	−23.1886	−0.890551
$679$	36.1509	1.38735
$680$	1.65345	0.0634067
$681$	−31.8581	−1.22080
$682$	0.969945	0.0371411
$683$	20.3379	0.778209	0.389105	−	0.921194i	$-0.372784\pi$
$683$	20.3379	0.778209	0.389105	+	0.921194i	$0.372784\pi$
$684$	−2.60953	−0.0997778
$685$	2.50320	0.0956423
$686$	−13.0167	−0.496978
$687$	15.2506	0.581847
$688$	−8.23043	−0.313782
$689$	−31.7579	−1.20988
$690$	19.8869	0.757082
$691$	−20.1124	−0.765112	−0.382556	−	0.923932i	$-0.624956\pi$
$691$	−20.1124	−0.765112	−0.382556	+	0.923932i	$0.624956\pi$
$692$	−1.82840	−0.0695055
$693$	−11.4899	−0.436467
$694$	20.0869	0.762489
$695$	10.4594	0.396748
$696$	3.38687	0.128379
$697$	3.03384	0.114915
$698$	−25.1034	−0.950178
$699$	−23.9907	−0.907410
$700$	−9.51194	−0.359518
$701$	21.1180	0.797615	0.398808	−	0.917035i	$-0.369424\pi$
$701$	21.1180	0.797615	0.398808	+	0.917035i	$0.369424\pi$
$702$	12.7343	0.480624
$703$	11.9436	0.450462
$704$	−3.31834	−0.125065
$705$	−29.6518	−1.11675
$706$	23.5854	0.887647
$707$	−56.5156	−2.12549
$708$	−20.7990	−0.781674
$709$	−24.4050	−0.916550	−0.458275	−	0.888810i	$-0.651533\pi$
$709$	−24.4050	−0.916550	−0.458275	+	0.888810i	$0.651533\pi$
$710$	−16.4900	−0.618858
$711$	−9.90400	−0.371429
$712$	−12.1959	−0.457060
$713$	2.04472	0.0765755
$714$	−7.49083	−0.280337
$715$	−15.4346	−0.577222
$716$	8.51227	0.318118
$717$	−42.3130	−1.58021
$718$	14.2407	0.531458
$719$	−3.69374	−0.137753	−0.0688767	−	0.997625i	$-0.521942\pi$
$719$	−3.69374	−0.137753	−0.0688767	+	0.997625i	$0.521942\pi$
$720$	−1.54815	−0.0576961
$721$	−9.91440	−0.369232
$722$	−13.4036	−0.498831
$723$	47.4401	1.76432
$724$	2.50544	0.0931138
$725$	−5.06668	−0.188172
$726$	0.0230247	0.000854527 0
$727$	−13.5920	−0.504101	−0.252051	−	0.967714i	$-0.581105\pi$
$727$	−13.5920	−0.504101	−0.252051	+	0.967714i	$0.581105\pi$
$728$	−10.4030	−0.385562
$729$	11.1137	0.411619
$730$	−5.22545	−0.193403
$731$	9.69634	0.358632
$732$	−14.4218	−0.533045
$733$	−11.9200	−0.440274	−0.220137	−	0.975469i	$-0.570650\pi$
$733$	−11.9200	−0.440274	−0.220137	+	0.975469i	$0.570650\pi$
$734$	−15.7133	−0.579989
$735$	−8.11141	−0.299194
$736$	−6.99532	−0.257851
$737$	12.1442	0.447339
$738$	−2.84063	−0.104565
$739$	−17.5149	−0.644295	−0.322147	−	0.946690i	$-0.604405\pi$
$739$	−17.5149	−0.644295	−0.322147	+	0.946690i	$0.604405\pi$
$740$	7.08576	0.260478
$741$	15.8811	0.583405
$742$	30.0795	1.10425
$743$	7.66521	0.281209	0.140605	−	0.990066i	$-0.455095\pi$
$743$	7.66521	0.281209	0.140605	+	0.990066i	$0.455095\pi$
$744$	−0.592082	−0.0217068
$745$	6.85571	0.251174
$746$	−5.06110	−0.185300
$747$	7.06607	0.258534
$748$	3.90936	0.142940
$749$	36.5386	1.33509
$750$	22.8291	0.833602
$751$	−50.9569	−1.85944	−0.929721	−	0.368264i	$-0.879952\pi$
$751$	−50.9569	−1.85944	−0.929721	+	0.368264i	$0.879952\pi$
$752$	10.4302	0.380349
$753$	23.6048	0.860207
$754$	−5.54133	−0.201803
$755$	−27.1363	−0.987592
$756$	−12.0613	−0.438664
$757$	35.6644	1.29625	0.648123	−	0.761536i	$-0.275554\pi$
$757$	35.6644	1.29625	0.648123	+	0.761536i	$0.275554\pi$
$758$	14.2646	0.518113
$759$	47.0201	1.70672
$760$	3.32016	0.120435
$761$	−18.2924	−0.663099	−0.331550	−	0.943438i	$-0.607571\pi$
$761$	−18.2924	−0.663099	−0.331550	+	0.943438i	$0.607571\pi$
$762$	22.2002	0.804228
$763$	51.4382	1.86219
$764$	−8.95790	−0.324085
$765$	1.82389	0.0659428
$766$	−26.8851	−0.971398
$767$	34.0297	1.22874
$768$	2.02561	0.0730928
$769$	−6.81498	−0.245754	−0.122877	−	0.992422i	$-0.539212\pi$
$769$	−6.81498	−0.245754	−0.122877	+	0.992422i	$0.539212\pi$
$770$	14.6189	0.526828
$771$	35.0165	1.26109
$772$	−19.6719	−0.708006
$773$	46.0860	1.65760	0.828799	−	0.559546i	$-0.189024\pi$
$773$	46.0860	1.65760	0.828799	+	0.559546i	$0.189024\pi$
$774$	−9.07885	−0.326333
$775$	0.885740	0.0318167
$776$	11.5168	0.413428
$777$	−32.1016	−1.15164
$778$	−19.5166	−0.699706
$779$	6.09202	0.218269
$780$	9.42172	0.337352
$781$	−38.9885	−1.39512
$782$	8.24125	0.294707
$783$	−6.42462	−0.229597
$784$	2.85323	0.101901
$785$	−19.8022	−0.706770
$786$	8.83407	0.315101
$787$	29.2515	1.04270	0.521352	−	0.853342i	$-0.325428\pi$
$787$	29.2515	1.04270	0.521352	+	0.853342i	$0.325428\pi$
$788$	15.5963	0.555596
$789$	−41.5021	−1.47751
$790$	12.6011	0.448326
$791$	−35.9342	−1.27767
$792$	−3.66040	−0.130067
$793$	23.5958	0.837911
$794$	−12.3196	−0.437207
$795$	−27.2421	−0.966179
$796$	−13.4087	−0.475257
$797$	39.0393	1.38284	0.691422	−	0.722451i	$-0.256985\pi$
$797$	39.0393	1.38284	0.691422	+	0.722451i	$0.256985\pi$
$798$	−15.0417	−0.532472
$799$	−12.2879	−0.434714
$800$	−3.03026	−0.107136
$801$	−13.4531	−0.475341
$802$	−37.6213	−1.32845
$803$	−12.3549	−0.435996
$804$	−7.41319	−0.261443
$805$	30.8178	1.08618
$806$	0.968717	0.0341216
$807$	36.3300	1.27888
$808$	−18.0044	−0.633393
$809$	51.7104	1.81804	0.909020	−	0.416752i	$-0.136832\pi$
$809$	51.7104	1.81804	0.909020	+	0.416752i	$0.136832\pi$
$810$	15.5680	0.547003
$811$	−10.1422	−0.356142	−0.178071	−	0.984018i	$-0.556986\pi$
$811$	−10.1422	−0.356142	−0.178071	+	0.984018i	$0.556986\pi$
$812$	5.24847	0.184185
$813$	−52.5430	−1.84276
$814$	16.7534	0.587206
$815$	−21.9603	−0.769235
$816$	−2.38638	−0.0835402
$817$	19.4705	0.681186
$818$	−31.4047	−1.09804
$819$	−11.4754	−0.400983
$820$	3.61420	0.126213
$821$	−11.3839	−0.397300	−0.198650	−	0.980070i	$-0.563656\pi$
$821$	−11.3839	−0.397300	−0.198650	+	0.980070i	$0.563656\pi$
$822$	−3.61282	−0.126011
$823$	−20.0501	−0.698903	−0.349451	−	0.936954i	$-0.613632\pi$
$823$	−20.0501	−0.698903	−0.349451	+	0.936954i	$0.613632\pi$
$824$	−3.15847	−0.110031
$825$	20.3683	0.709134
$826$	−32.2312	−1.12147
$827$	10.3120	0.358584	0.179292	−	0.983796i	$-0.442619\pi$
$827$	10.3120	0.358584	0.179292	+	0.983796i	$0.442619\pi$
$828$	−7.71642	−0.268164
$829$	−25.4095	−0.882509	−0.441254	−	0.897382i	$-0.645466\pi$
$829$	−25.4095	−0.882509	−0.441254	+	0.897382i	$0.645466\pi$
$830$	−8.99031	−0.312058
$831$	7.21655	0.250339
$832$	−3.31414	−0.114897
$833$	−3.36142	−0.116466
$834$	−15.0959	−0.522727
$835$	−6.72819	−0.232838
$836$	7.85009	0.271501
$837$	1.12313	0.0388210
$838$	9.80992	0.338878
$839$	−6.84006	−0.236145	−0.118073	−	0.993005i	$-0.537672\pi$
$839$	−6.84006	−0.236145	−0.118073	+	0.993005i	$0.537672\pi$
$840$	−8.92378	−0.307900
$841$	−26.2043	−0.903597
$842$	5.24081	0.180610
$843$	−4.86370	−0.167515
$844$	−26.9085	−0.926230
$845$	2.83009	0.0973582
$846$	11.5053	0.395562
$847$	0.0356803	0.00122599
$848$	9.58256	0.329066
$849$	50.5507	1.73489
$850$	3.56997	0.122449
$851$	35.3175	1.21067
$852$	23.7997	0.815363
$853$	−5.93470	−0.203200	−0.101600	−	0.994825i	$-0.532396\pi$
$853$	−5.93470	−0.203200	−0.101600	+	0.994825i	$0.532396\pi$
$854$	−22.3488	−0.764759
$855$	3.66241	0.125252
$856$	11.6403	0.397856
$857$	−29.3586	−1.00287	−0.501435	−	0.865195i	$-0.667194\pi$
$857$	−29.3586	−1.00287	−0.501435	+	0.865195i	$0.667194\pi$
$858$	22.2765	0.760506
$859$	−23.4911	−0.801506	−0.400753	−	0.916186i	$-0.631252\pi$
$859$	−23.4911	−0.801506	−0.400753	+	0.916186i	$0.631252\pi$
$860$	11.5512	0.393893
$861$	−16.3739	−0.558020
$862$	8.97057	0.305539
$863$	13.5820	0.462336	0.231168	−	0.972914i	$-0.425745\pi$
$863$	13.5820	0.462336	0.231168	+	0.972914i	$0.425745\pi$
$864$	−3.84241	−0.130721
$865$	2.56612	0.0872506
$866$	−30.5789	−1.03911
$867$	−31.6239	−1.07400
$868$	−0.917521	−0.0311427
$869$	29.7936	1.01068
$870$	−4.75339	−0.161155
$871$	12.1289	0.410971
$872$	16.3869	0.554930
$873$	12.7039	0.429963
$874$	16.5486	0.559765
$875$	35.3772	1.19597
$876$	7.54179	0.254813
$877$	−37.7745	−1.27555	−0.637777	−	0.770221i	$-0.720146\pi$
$877$	−37.7745	−1.27555	−0.637777	+	0.770221i	$0.720146\pi$
$878$	−28.2743	−0.954210
$879$	53.9422	1.81942
$880$	4.65720	0.156994
$881$	35.2441	1.18740	0.593702	−	0.804685i	$-0.297666\pi$
$881$	35.2441	1.18740	0.593702	+	0.804685i	$0.297666\pi$
$882$	3.14735	0.105977
$883$	−44.4056	−1.49437	−0.747183	−	0.664618i	$-0.768594\pi$
$883$	−44.4056	−1.49437	−0.747183	+	0.664618i	$0.768594\pi$
$884$	3.90441	0.131320
$885$	29.1909	0.981240
$886$	29.7383	0.999079
$887$	52.5358	1.76398	0.881990	−	0.471267i	$-0.156203\pi$
$887$	52.5358	1.76398	0.881990	+	0.471267i	$0.156203\pi$
$888$	−10.2267	−0.343187
$889$	34.4026	1.15383
$890$	17.1166	0.573750
$891$	36.8085	1.23313
$892$	10.1417	0.339570
$893$	−24.6743	−0.825695
$894$	−9.89470	−0.330928
$895$	−11.9468	−0.399336
$896$	3.13899	0.104866
$897$	46.9606	1.56797
$898$	−21.0145	−0.701264
$899$	−0.488731	−0.0163001
$900$	−3.34263	−0.111421
$901$	−11.2893	−0.376101
$902$	8.54531	0.284528
$903$	−52.3320	−1.74150
$904$	−11.4477	−0.380745
$905$	−3.51631	−0.116886
$906$	39.1653	1.30118
$907$	−37.7435	−1.25325	−0.626627	−	0.779320i	$-0.715565\pi$
$907$	−37.7435	−1.25325	−0.626627	+	0.779320i	$0.715565\pi$
$908$	−15.7277	−0.521941
$909$	−19.8604	−0.658726
$910$	14.6004	0.483998
$911$	43.4422	1.43930	0.719652	−	0.694335i	$-0.244302\pi$
$911$	43.4422	1.43930	0.719652	+	0.694335i	$0.244302\pi$
$912$	−4.79191	−0.158676
$913$	−21.2564	−0.703486
$914$	32.7526	1.08336
$915$	20.2406	0.669134
$916$	7.52891	0.248762
$917$	13.6897	0.452075
$918$	4.52677	0.149406
$919$	45.8006	1.51082	0.755411	−	0.655251i	$-0.227437\pi$
$919$	45.8006	1.51082	0.755411	+	0.655251i	$0.227437\pi$
$920$	9.81776	0.323682
$921$	−5.02767	−0.165667
$922$	9.69798	0.319386
$923$	−38.9391	−1.28170
$924$	−21.0992	−0.694111
$925$	15.2990	0.503026
$926$	−33.4658	−1.09976
$927$	−3.48406	−0.114431
$928$	1.67203	0.0548870
$929$	−53.2790	−1.74803	−0.874013	−	0.485902i	$-0.838491\pi$
$929$	−53.2790	−1.74803	−0.874013	+	0.485902i	$0.838491\pi$
$930$	0.830971	0.0272486
$931$	−6.74980	−0.221216
$932$	−11.8437	−0.387953
$933$	−4.66929	−0.152866
$934$	−26.4721	−0.866193
$935$	−5.48669	−0.179434
$936$	−3.65577	−0.119493
$937$	−4.28688	−0.140046	−0.0700231	−	0.997545i	$-0.522307\pi$
$937$	−4.28688	−0.140046	−0.0700231	+	0.997545i	$0.522307\pi$
$938$	−11.4879	−0.375092
$939$	10.3233	0.336888
$940$	−14.6385	−0.477454
$941$	44.5893	1.45357	0.726785	−	0.686865i	$-0.241013\pi$
$941$	44.5893	1.45357	0.726785	+	0.686865i	$0.241013\pi$
$942$	28.5801	0.931190
$943$	18.0142	0.586623
$944$	−10.2680	−0.334196
$945$	16.9277	0.550657
$946$	27.3114	0.887969
$947$	−30.7864	−1.00042	−0.500211	−	0.865904i	$-0.666744\pi$
$947$	−30.7864	−1.00042	−0.500211	+	0.865904i	$0.666744\pi$
$948$	−18.1869	−0.590682
$949$	−12.3393	−0.400550
$950$	7.16859	0.232580
$951$	6.37775	0.206813
$952$	−3.69807	−0.119855
$953$	53.8007	1.74277	0.871387	−	0.490596i	$-0.163221\pi$
$953$	53.8007	1.74277	0.871387	+	0.490596i	$0.163221\pi$
$954$	10.5704	0.342228
$955$	12.5722	0.406826
$956$	−20.8890	−0.675599
$957$	−11.2388	−0.363298
$958$	−21.2265	−0.685798
$959$	−5.59861	−0.180789
$960$	−2.84289	−0.0917538
$961$	−30.9146	−0.997244
$962$	16.7322	0.539467
$963$	12.8402	0.413769
$964$	23.4202	0.754313
$965$	27.6090	0.888764
$966$	−44.4787	−1.43108
$967$	35.0765	1.12799	0.563993	−	0.825780i	$-0.309265\pi$
$967$	35.0765	1.12799	0.563993	+	0.825780i	$0.309265\pi$
$968$	0.0113668	0.000365344 0
$969$	5.64539	0.181356
$970$	−16.1635	−0.518978
$971$	21.0641	0.675980	0.337990	−	0.941150i	$-0.390253\pi$
$971$	21.0641	0.675980	0.337990	+	0.941150i	$0.390253\pi$
$972$	−10.9417	−0.350956
$973$	−23.3933	−0.749955
$974$	−37.2100	−1.19228
$975$	20.3426	0.651483
$976$	−7.11974	−0.227897
$977$	−1.26695	−0.0405333	−0.0202666	−	0.999795i	$-0.506452\pi$
$977$	−1.26695	−0.0405333	−0.0202666	+	0.999795i	$0.506452\pi$
$978$	31.6949	1.01349
$979$	40.4700	1.29343
$980$	−4.00444	−0.127917
$981$	18.0761	0.577125
$982$	−7.91452	−0.252562
$983$	14.7648	0.470925	0.235463	−	0.971883i	$-0.424340\pi$
$983$	14.7648	0.470925	0.235463	+	0.971883i	$0.424340\pi$
$984$	−5.21630	−0.166289
$985$	−21.8890	−0.697443
$986$	−1.96983	−0.0627322
$987$	66.3187	2.11095
$988$	7.84015	0.249428
$989$	57.5745	1.83076
$990$	5.13728	0.163273
$991$	24.6407	0.782738	0.391369	−	0.920234i	$-0.372002\pi$
$991$	24.6407	0.782738	0.391369	+	0.920234i	$0.372002\pi$
$992$	−0.292298	−0.00928048
$993$	52.5348	1.66714
$994$	36.8812	1.16980
$995$	18.8187	0.596593
$996$	12.9755	0.411146
$997$	−13.6608	−0.432641	−0.216320	−	0.976322i	$-0.569406\pi$
$997$	−13.6608	−0.432641	−0.216320	+	0.976322i	$0.569406\pi$
$998$	12.9964	0.411393
$999$	19.3993	0.613766

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.59	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.59	✓	67	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.02561 q^{3} +1.00000 q^{4} -1.40347 q^{5} +2.02561 q^{6} +3.13899 q^{7} +1.00000 q^{8} +1.10308 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.02561 q^{3} +1.00000 q^{4} -1.40347 q^{5} +2.02561 q^{6} +3.13899 q^{7} +1.00000 q^{8} +1.10308 q^{9} -1.40347 q^{10} -3.31834 q^{11} +2.02561 q^{12} -3.31414 q^{13} +3.13899 q^{14} -2.84289 q^{15} +1.00000 q^{16} -1.17811 q^{17} +1.10308 q^{18} -2.36567 q^{19} -1.40347 q^{20} +6.35835 q^{21} -3.31834 q^{22} -6.99532 q^{23} +2.02561 q^{24} -3.03026 q^{25} -3.31414 q^{26} -3.84241 q^{27} +3.13899 q^{28} +1.67203 q^{29} -2.84289 q^{30} -0.292298 q^{31} +1.00000 q^{32} -6.72165 q^{33} -1.17811 q^{34} -4.40549 q^{35} +1.10308 q^{36} -5.04873 q^{37} -2.36567 q^{38} -6.71314 q^{39} -1.40347 q^{40} -2.57518 q^{41} +6.35835 q^{42} -8.23043 q^{43} -3.31834 q^{44} -1.54815 q^{45} -6.99532 q^{46} +10.4302 q^{47} +2.02561 q^{48} +2.85323 q^{49} -3.03026 q^{50} -2.38638 q^{51} -3.31414 q^{52} +9.58256 q^{53} -3.84241 q^{54} +4.65720 q^{55} +3.13899 q^{56} -4.79191 q^{57} +1.67203 q^{58} -10.2680 q^{59} -2.84289 q^{60} -7.11974 q^{61} -0.292298 q^{62} +3.46256 q^{63} +1.00000 q^{64} +4.65131 q^{65} -6.72165 q^{66} -3.65974 q^{67} -1.17811 q^{68} -14.1698 q^{69} -4.40549 q^{70} +11.7494 q^{71} +1.10308 q^{72} +3.72322 q^{73} -5.04873 q^{74} -6.13811 q^{75} -2.36567 q^{76} -10.4162 q^{77} -6.71314 q^{78} -8.97847 q^{79} -1.40347 q^{80} -11.0925 q^{81} -2.57518 q^{82} +6.40575 q^{83} +6.35835 q^{84} +1.65345 q^{85} -8.23043 q^{86} +3.38687 q^{87} -3.31834 q^{88} -12.1959 q^{89} -1.54815 q^{90} -10.4030 q^{91} -6.99532 q^{92} -0.592082 q^{93} +10.4302 q^{94} +3.32016 q^{95} +2.02561 q^{96} +11.5168 q^{97} +2.85323 q^{98} -3.66040 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

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