LMFDB - Embedded newform 8002.2.a.d.1.37

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:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.37
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)

$f(q)$	$=$	$q+1.00000 q^{2} -0.320920 q^{3} +1.00000 q^{4} -0.577212 q^{5} -0.320920 q^{6} +1.17502 q^{7} +1.00000 q^{8} -2.89701 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.320920 q^{3} +1.00000 q^{4} -0.577212 q^{5} -0.320920 q^{6} +1.17502 q^{7} +1.00000 q^{8} -2.89701 q^{9} -0.577212 q^{10} +2.26550 q^{11} -0.320920 q^{12} -3.59296 q^{13} +1.17502 q^{14} +0.185239 q^{15} +1.00000 q^{16} -6.50500 q^{17} -2.89701 q^{18} +7.90845 q^{19} -0.577212 q^{20} -0.377086 q^{21} +2.26550 q^{22} -2.05633 q^{23} -0.320920 q^{24} -4.66683 q^{25} -3.59296 q^{26} +1.89247 q^{27} +1.17502 q^{28} +9.22327 q^{29} +0.185239 q^{30} +1.06916 q^{31} +1.00000 q^{32} -0.727043 q^{33} -6.50500 q^{34} -0.678235 q^{35} -2.89701 q^{36} -4.93329 q^{37} +7.90845 q^{38} +1.15305 q^{39} -0.577212 q^{40} +4.68274 q^{41} -0.377086 q^{42} -10.2765 q^{43} +2.26550 q^{44} +1.67219 q^{45} -2.05633 q^{46} +7.93369 q^{47} -0.320920 q^{48} -5.61933 q^{49} -4.66683 q^{50} +2.08758 q^{51} -3.59296 q^{52} +7.88708 q^{53} +1.89247 q^{54} -1.30767 q^{55} +1.17502 q^{56} -2.53798 q^{57} +9.22327 q^{58} -10.0055 q^{59} +0.185239 q^{60} -1.49477 q^{61} +1.06916 q^{62} -3.40404 q^{63} +1.00000 q^{64} +2.07390 q^{65} -0.727043 q^{66} -15.3392 q^{67} -6.50500 q^{68} +0.659918 q^{69} -0.678235 q^{70} -13.6360 q^{71} -2.89701 q^{72} +10.3370 q^{73} -4.93329 q^{74} +1.49768 q^{75} +7.90845 q^{76} +2.66200 q^{77} +1.15305 q^{78} -2.94742 q^{79} -0.577212 q^{80} +8.08370 q^{81} +4.68274 q^{82} +10.6520 q^{83} -0.377086 q^{84} +3.75477 q^{85} -10.2765 q^{86} -2.95993 q^{87} +2.26550 q^{88} -5.27348 q^{89} +1.67219 q^{90} -4.22179 q^{91} -2.05633 q^{92} -0.343114 q^{93} +7.93369 q^{94} -4.56486 q^{95} -0.320920 q^{96} -6.86024 q^{97} -5.61933 q^{98} -6.56317 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−0.320920	−0.185283	−0.0926416	−	0.995700i	$-0.529531\pi$
$3$	−0.320920	−0.185283	−0.0926416	+	0.995700i	$0.529531\pi$
$4$	1.00000	0.500000
$5$	−0.577212	−0.258137	−0.129069	−	0.991636i	$-0.541199\pi$
$5$	−0.577212	−0.258137	−0.129069	+	0.991636i	$0.541199\pi$
$6$	−0.320920	−0.131015
$7$	1.17502	0.444115	0.222057	−	0.975034i	$-0.428723\pi$
$7$	1.17502	0.444115	0.222057	+	0.975034i	$0.428723\pi$
$8$	1.00000	0.353553
$9$	−2.89701	−0.965670
$10$	−0.577212	−0.182531
$11$	2.26550	0.683073	0.341537	−	0.939868i	$-0.389053\pi$
$11$	2.26550	0.683073	0.341537	+	0.939868i	$0.389053\pi$
$12$	−0.320920	−0.0926416
$13$	−3.59296	−0.996506	−0.498253	−	0.867032i	$-0.666025\pi$
$13$	−3.59296	−0.996506	−0.498253	+	0.867032i	$0.666025\pi$
$14$	1.17502	0.314037
$15$	0.185239	0.0478285
$16$	1.00000	0.250000
$17$	−6.50500	−1.57769	−0.788847	−	0.614589i	$-0.789322\pi$
$17$	−6.50500	−1.57769	−0.788847	+	0.614589i	$0.789322\pi$
$18$	−2.89701	−0.682832
$19$	7.90845	1.81432	0.907162	−	0.420782i	$-0.138244\pi$
$19$	7.90845	1.81432	0.907162	+	0.420782i	$0.138244\pi$
$20$	−0.577212	−0.129069
$21$	−0.377086	−0.0822870
$22$	2.26550	0.483006
$23$	−2.05633	−0.428775	−0.214388	−	0.976749i	$-0.568776\pi$
$23$	−2.05633	−0.428775	−0.214388	+	0.976749i	$0.568776\pi$
$24$	−0.320920	−0.0655075
$25$	−4.66683	−0.933365
$26$	−3.59296	−0.704636
$27$	1.89247	0.364206
$28$	1.17502	0.222057
$29$	9.22327	1.71272	0.856359	−	0.516381i	$-0.172721\pi$
$29$	9.22327	1.71272	0.856359	+	0.516381i	$0.172721\pi$
$30$	0.185239	0.0338198
$31$	1.06916	0.192027	0.0960134	−	0.995380i	$-0.469391\pi$
$31$	1.06916	0.192027	0.0960134	+	0.995380i	$0.469391\pi$
$32$	1.00000	0.176777
$33$	−0.727043	−0.126562
$34$	−6.50500	−1.11560
$35$	−0.678235	−0.114643
$36$	−2.89701	−0.482835
$37$	−4.93329	−0.811027	−0.405514	−	0.914089i	$-0.632907\pi$
$37$	−4.93329	−0.811027	−0.405514	+	0.914089i	$0.632907\pi$
$38$	7.90845	1.28292
$39$	1.15305	0.184636
$40$	−0.577212	−0.0912653
$41$	4.68274	0.731321	0.365661	−	0.930748i	$-0.380843\pi$
$41$	4.68274	0.731321	0.365661	+	0.930748i	$0.380843\pi$
$42$	−0.377086	−0.0581857
$43$	−10.2765	−1.56715	−0.783576	−	0.621296i	$-0.786606\pi$
$43$	−10.2765	−1.56715	−0.783576	+	0.621296i	$0.786606\pi$
$44$	2.26550	0.341537
$45$	1.67219	0.249275
$46$	−2.05633	−0.303190
$47$	7.93369	1.15725	0.578624	−	0.815595i	$-0.303590\pi$
$47$	7.93369	1.15725	0.578624	+	0.815595i	$0.303590\pi$
$48$	−0.320920	−0.0463208
$49$	−5.61933	−0.802762
$50$	−4.66683	−0.659989
$51$	2.08758	0.292320
$52$	−3.59296	−0.498253
$53$	7.88708	1.08337	0.541687	−	0.840580i	$-0.317786\pi$
$53$	7.88708	1.08337	0.541687	+	0.840580i	$0.317786\pi$
$54$	1.89247	0.257532
$55$	−1.30767	−0.176327
$56$	1.17502	0.157018
$57$	−2.53798	−0.336164
$58$	9.22327	1.21107
$59$	−10.0055	−1.30260	−0.651302	−	0.758819i	$-0.725777\pi$
$59$	−10.0055	−1.30260	−0.651302	+	0.758819i	$0.725777\pi$
$60$	0.185239	0.0239142
$61$	−1.49477	−0.191386	−0.0956931	−	0.995411i	$-0.530507\pi$
$61$	−1.49477	−0.191386	−0.0956931	+	0.995411i	$0.530507\pi$
$62$	1.06916	0.135783
$63$	−3.40404	−0.428869
$64$	1.00000	0.125000
$65$	2.07390	0.257235
$66$	−0.727043	−0.0894928
$67$	−15.3392	−1.87398	−0.936991	−	0.349353i	$-0.886401\pi$
$67$	−15.3392	−1.87398	−0.936991	+	0.349353i	$0.886401\pi$
$68$	−6.50500	−0.788847
$69$	0.659918	0.0794448
$70$	−0.678235	−0.0810646
$71$	−13.6360	−1.61830	−0.809149	−	0.587603i	$-0.800072\pi$
$71$	−13.6360	−1.61830	−0.809149	+	0.587603i	$0.800072\pi$
$72$	−2.89701	−0.341416
$73$	10.3370	1.20985	0.604926	−	0.796282i	$-0.293203\pi$
$73$	10.3370	1.20985	0.604926	+	0.796282i	$0.293203\pi$
$74$	−4.93329	−0.573483
$75$	1.49768	0.172937
$76$	7.90845	0.907162
$77$	2.66200	0.303363
$78$	1.15305	0.130557
$79$	−2.94742	−0.331610	−0.165805	−	0.986159i	$-0.553022\pi$
$79$	−2.94742	−0.331610	−0.165805	+	0.986159i	$0.553022\pi$
$80$	−0.577212	−0.0645343
$81$	8.08370	0.898189
$82$	4.68274	0.517122
$83$	10.6520	1.16921	0.584604	−	0.811318i	$-0.301250\pi$
$83$	10.6520	1.16921	0.584604	+	0.811318i	$0.301250\pi$
$84$	−0.377086	−0.0411435
$85$	3.75477	0.407262
$86$	−10.2765	−1.10814
$87$	−2.95993	−0.317338
$88$	2.26550	0.241503
$89$	−5.27348	−0.558988	−0.279494	−	0.960147i	$-0.590167\pi$
$89$	−5.27348	−0.558988	−0.279494	+	0.960147i	$0.590167\pi$
$90$	1.67219	0.176264
$91$	−4.22179	−0.442563
$92$	−2.05633	−0.214388
$93$	−0.343114	−0.0355793
$94$	7.93369	0.818298
$95$	−4.56486	−0.468345
$96$	−0.320920	−0.0327537
$97$	−6.86024	−0.696552	−0.348276	−	0.937392i	$-0.613233\pi$
$97$	−6.86024	−0.696552	−0.348276	+	0.937392i	$0.613233\pi$
$98$	−5.61933	−0.567638
$99$	−6.56317	−0.659624
$100$	−4.66683	−0.466683
$101$	−12.8857	−1.28217	−0.641085	−	0.767470i	$-0.721516\pi$
$101$	−12.8857	−1.28217	−0.641085	+	0.767470i	$0.721516\pi$
$102$	2.08758	0.206702
$103$	−9.04064	−0.890801	−0.445400	−	0.895331i	$-0.646939\pi$
$103$	−9.04064	−0.890801	−0.445400	+	0.895331i	$0.646939\pi$
$104$	−3.59296	−0.352318
$105$	0.217659	0.0212413
$106$	7.88708	0.766061
$107$	1.15547	0.111703	0.0558516	−	0.998439i	$-0.482213\pi$
$107$	1.15547	0.111703	0.0558516	+	0.998439i	$0.482213\pi$
$108$	1.89247	0.182103
$109$	−2.11438	−0.202521	−0.101260	−	0.994860i	$-0.532287\pi$
$109$	−2.11438	−0.202521	−0.101260	+	0.994860i	$0.532287\pi$
$110$	−1.30767	−0.124682
$111$	1.58319	0.150270
$112$	1.17502	0.111029
$113$	−3.74136	−0.351958	−0.175979	−	0.984394i	$-0.556309\pi$
$113$	−3.74136	−0.351958	−0.175979	+	0.984394i	$0.556309\pi$
$114$	−2.53798	−0.237704
$115$	1.18694	0.110683
$116$	9.22327	0.856359
$117$	10.4088	0.962297
$118$	−10.0055	−0.921080
$119$	−7.64349	−0.700678
$120$	0.185239	0.0169099
$121$	−5.86752	−0.533411
$122$	−1.49477	−0.135330
$123$	−1.50278	−0.135501
$124$	1.06916	0.0960134
$125$	5.57981	0.499074
$126$	−3.40404	−0.303256
$127$	−10.2430	−0.908916	−0.454458	−	0.890768i	$-0.650167\pi$
$127$	−10.2430	−0.908916	−0.454458	+	0.890768i	$0.650167\pi$
$128$	1.00000	0.0883883
$129$	3.29793	0.290367
$130$	2.07390	0.181893
$131$	5.54978	0.484886	0.242443	−	0.970166i	$-0.422051\pi$
$131$	5.54978	0.484886	0.242443	+	0.970166i	$0.422051\pi$
$132$	−0.727043	−0.0632810
$133$	9.29257	0.805768
$134$	−15.3392	−1.32511
$135$	−1.09236	−0.0940150
$136$	−6.50500	−0.557799
$137$	−6.95871	−0.594523	−0.297261	−	0.954796i	$-0.596073\pi$
$137$	−6.95871	−0.594523	−0.297261	+	0.954796i	$0.596073\pi$
$138$	0.659918	0.0561760
$139$	−9.81928	−0.832861	−0.416430	−	0.909168i	$-0.636719\pi$
$139$	−9.81928	−0.832861	−0.416430	+	0.909168i	$0.636719\pi$
$140$	−0.678235	−0.0573213
$141$	−2.54608	−0.214418
$142$	−13.6360	−1.14431
$143$	−8.13983	−0.680687
$144$	−2.89701	−0.241418
$145$	−5.32379	−0.442116
$146$	10.3370	0.855494
$147$	1.80336	0.148738
$148$	−4.93329	−0.405514
$149$	10.4459	0.855758	0.427879	−	0.903836i	$-0.359261\pi$
$149$	10.4459	0.855758	0.427879	+	0.903836i	$0.359261\pi$
$150$	1.49768	0.122285
$151$	−24.5097	−1.99457	−0.997286	−	0.0736222i	$-0.976544\pi$
$151$	−24.5097	−1.99457	−0.997286	+	0.0736222i	$0.976544\pi$
$152$	7.90845	0.641460
$153$	18.8451	1.52353
$154$	2.66200	0.214510
$155$	−0.617132	−0.0495692
$156$	1.15305	0.0923179
$157$	6.34175	0.506126	0.253063	−	0.967450i	$-0.418562\pi$
$157$	6.34175	0.506126	0.253063	+	0.967450i	$0.418562\pi$
$158$	−2.94742	−0.234484
$159$	−2.53112	−0.200731
$160$	−0.577212	−0.0456327
$161$	−2.41623	−0.190425
$162$	8.08370	0.635116
$163$	10.2935	0.806248	0.403124	−	0.915145i	$-0.367924\pi$
$163$	10.2935	0.806248	0.403124	+	0.915145i	$0.367924\pi$
$164$	4.68274	0.365661
$165$	0.419658	0.0326704
$166$	10.6520	0.826756
$167$	−11.2117	−0.867584	−0.433792	−	0.901013i	$-0.642825\pi$
$167$	−11.2117	−0.867584	−0.433792	+	0.901013i	$0.642825\pi$
$168$	−0.377086	−0.0290929
$169$	−0.0906726	−0.00697482
$170$	3.75477	0.287978
$171$	−22.9109	−1.75204
$172$	−10.2765	−0.783576
$173$	−13.9442	−1.06016	−0.530080	−	0.847948i	$-0.677838\pi$
$173$	−13.9442	−1.06016	−0.530080	+	0.847948i	$0.677838\pi$
$174$	−2.95993	−0.224392
$175$	−5.48360	−0.414521
$176$	2.26550	0.170768
$177$	3.21096	0.241351
$178$	−5.27348	−0.395264
$179$	10.4516	0.781190	0.390595	−	0.920563i	$-0.372269\pi$
$179$	10.4516	0.781190	0.390595	+	0.920563i	$0.372269\pi$
$180$	1.67219	0.124638
$181$	−2.12544	−0.157983	−0.0789915	−	0.996875i	$-0.525170\pi$
$181$	−2.12544	−0.157983	−0.0789915	+	0.996875i	$0.525170\pi$
$182$	−4.22179	−0.312940
$183$	0.479702	0.0354606
$184$	−2.05633	−0.151595
$185$	2.84755	0.209356
$186$	−0.343114	−0.0251584
$187$	−14.7371	−1.07768
$188$	7.93369	0.578624
$189$	2.22368	0.161749
$190$	−4.56486	−0.331170
$191$	−15.6364	−1.13141	−0.565706	−	0.824607i	$-0.691396\pi$
$191$	−15.6364	−1.13141	−0.565706	+	0.824607i	$0.691396\pi$
$192$	−0.320920	−0.0231604
$193$	−0.142439	−0.0102530	−0.00512649	−	0.999987i	$-0.501632\pi$
$193$	−0.142439	−0.0102530	−0.00512649	+	0.999987i	$0.501632\pi$
$194$	−6.86024	−0.492536
$195$	−0.665555	−0.0476614
$196$	−5.61933	−0.401381
$197$	−5.48547	−0.390823	−0.195412	−	0.980721i	$-0.562604\pi$
$197$	−5.48547	−0.390823	−0.195412	+	0.980721i	$0.562604\pi$
$198$	−6.56317	−0.466424
$199$	−0.953994	−0.0676268	−0.0338134	−	0.999428i	$-0.510765\pi$
$199$	−0.953994	−0.0676268	−0.0338134	+	0.999428i	$0.510765\pi$
$200$	−4.66683	−0.329994
$201$	4.92265	0.347217
$202$	−12.8857	−0.906632
$203$	10.8375	0.760644
$204$	2.08758	0.146160
$205$	−2.70294	−0.188781
$206$	−9.04064	−0.629891
$207$	5.95722	0.414055
$208$	−3.59296	−0.249127
$209$	17.9166	1.23932
$210$	0.217659	0.0150199
$211$	12.5773	0.865854	0.432927	−	0.901429i	$-0.357481\pi$
$211$	12.5773	0.865854	0.432927	+	0.901429i	$0.357481\pi$
$212$	7.88708	0.541687
$213$	4.37607	0.299843
$214$	1.15547	0.0789861
$215$	5.93172	0.404540
$216$	1.89247	0.128766
$217$	1.25628	0.0852819
$218$	−2.11438	−0.143204
$219$	−3.31734	−0.224165
$220$	−1.30767	−0.0881633
$221$	23.3722	1.57218
$222$	1.58319	0.106257
$223$	12.5610	0.841144	0.420572	−	0.907259i	$-0.361829\pi$
$223$	12.5610	0.841144	0.420572	+	0.907259i	$0.361829\pi$
$224$	1.17502	0.0785092
$225$	13.5198	0.901323
$226$	−3.74136	−0.248872
$227$	−1.43749	−0.0954096	−0.0477048	−	0.998861i	$-0.515191\pi$
$227$	−1.43749	−0.0954096	−0.0477048	+	0.998861i	$0.515191\pi$
$228$	−2.53798	−0.168082
$229$	1.96151	0.129620	0.0648101	−	0.997898i	$-0.479356\pi$
$229$	1.96151	0.129620	0.0648101	+	0.997898i	$0.479356\pi$
$230$	1.18694	0.0782646
$231$	−0.854289	−0.0562081
$232$	9.22327	0.605537
$233$	−19.0396	−1.24732	−0.623662	−	0.781694i	$-0.714356\pi$
$233$	−19.0396	−1.24732	−0.623662	+	0.781694i	$0.714356\pi$
$234$	10.4088	0.680446
$235$	−4.57942	−0.298729
$236$	−10.0055	−0.651302
$237$	0.945884	0.0614418
$238$	−7.64349	−0.495454
$239$	0.663987	0.0429498	0.0214749	−	0.999769i	$-0.493164\pi$
$239$	0.663987	0.0429498	0.0214749	+	0.999769i	$0.493164\pi$
$240$	0.185239	0.0119571
$241$	12.0809	0.778198	0.389099	−	0.921196i	$-0.372786\pi$
$241$	12.0809	0.778198	0.389099	+	0.921196i	$0.372786\pi$
$242$	−5.86752	−0.377178
$243$	−8.27162	−0.530625
$244$	−1.49477	−0.0956931
$245$	3.24355	0.207223
$246$	−1.50278	−0.0958140
$247$	−28.4147	−1.80799
$248$	1.06916	0.0678917
$249$	−3.41844	−0.216635
$250$	5.57981	0.352898
$251$	−9.87004	−0.622991	−0.311496	−	0.950248i	$-0.600830\pi$
$251$	−9.87004	−0.622991	−0.311496	+	0.950248i	$0.600830\pi$
$252$	−3.40404	−0.214434
$253$	−4.65862	−0.292885
$254$	−10.2430	−0.642701
$255$	−1.20498	−0.0754587
$256$	1.00000	0.0625000
$257$	16.9814	1.05927	0.529636	−	0.848225i	$-0.322328\pi$
$257$	16.9814	1.05927	0.529636	+	0.848225i	$0.322328\pi$
$258$	3.29793	0.205320
$259$	−5.79670	−0.360189
$260$	2.07390	0.128618
$261$	−26.7199	−1.65392
$262$	5.54978	0.342866
$263$	−28.0511	−1.72971	−0.864854	−	0.502024i	$-0.832589\pi$
$263$	−28.0511	−1.72971	−0.864854	+	0.502024i	$0.832589\pi$
$264$	−0.727043	−0.0447464
$265$	−4.55252	−0.279659
$266$	9.29257	0.569764
$267$	1.69237	0.103571
$268$	−15.3392	−0.936991
$269$	10.3385	0.630350	0.315175	−	0.949034i	$-0.397937\pi$
$269$	10.3385	0.630350	0.315175	+	0.949034i	$0.397937\pi$
$270$	−1.09236	−0.0664787
$271$	−0.914784	−0.0555692	−0.0277846	−	0.999614i	$-0.508845\pi$
$271$	−0.914784	−0.0555692	−0.0277846	+	0.999614i	$0.508845\pi$
$272$	−6.50500	−0.394424
$273$	1.35485	0.0819995
$274$	−6.95871	−0.420391
$275$	−10.5727	−0.637557
$276$	0.659918	0.0397224
$277$	9.18601	0.551934	0.275967	−	0.961167i	$-0.411002\pi$
$277$	9.18601	0.551934	0.275967	+	0.961167i	$0.411002\pi$
$278$	−9.81928	−0.588921
$279$	−3.09737	−0.185434
$280$	−0.678235	−0.0405323
$281$	9.81157	0.585309	0.292655	−	0.956218i	$-0.405461\pi$
$281$	9.81157	0.585309	0.292655	+	0.956218i	$0.405461\pi$
$282$	−2.54608	−0.151617
$283$	21.3863	1.27129	0.635643	−	0.771983i	$-0.280735\pi$
$283$	21.3863	1.27129	0.635643	+	0.771983i	$0.280735\pi$
$284$	−13.6360	−0.809149
$285$	1.46495	0.0867763
$286$	−8.13983	−0.481318
$287$	5.50230	0.324791
$288$	−2.89701	−0.170708
$289$	25.3150	1.48912
$290$	−5.32379	−0.312623
$291$	2.20159	0.129059
$292$	10.3370	0.604926
$293$	−30.1353	−1.76052	−0.880262	−	0.474489i	$-0.842633\pi$
$293$	−30.1353	−1.76052	−0.880262	+	0.474489i	$0.842633\pi$
$294$	1.80336	0.105174
$295$	5.77530	0.336251
$296$	−4.93329	−0.286741
$297$	4.28738	0.248779
$298$	10.4459	0.605113
$299$	7.38831	0.427277
$300$	1.49768	0.0864684
$301$	−12.0751	−0.695995
$302$	−24.5097	−1.41038
$303$	4.13526	0.237565
$304$	7.90845	0.453581
$305$	0.862802	0.0494039
$306$	18.8451	1.07730
$307$	−14.1139	−0.805524	−0.402762	−	0.915305i	$-0.631950\pi$
$307$	−14.1139	−0.805524	−0.402762	+	0.915305i	$0.631950\pi$
$308$	2.66200	0.151682
$309$	2.90132	0.165050
$310$	−0.617132	−0.0350508
$311$	−9.07469	−0.514578	−0.257289	−	0.966334i	$-0.582829\pi$
$311$	−9.07469	−0.514578	−0.257289	+	0.966334i	$0.582829\pi$
$312$	1.15305	0.0652786
$313$	−11.1055	−0.627720	−0.313860	−	0.949469i	$-0.601622\pi$
$313$	−11.1055	−0.627720	−0.313860	+	0.949469i	$0.601622\pi$
$314$	6.34175	0.357885
$315$	1.96485	0.110707
$316$	−2.94742	−0.165805
$317$	17.5616	0.986355	0.493178	−	0.869929i	$-0.335835\pi$
$317$	17.5616	0.986355	0.493178	+	0.869929i	$0.335835\pi$
$318$	−2.53112	−0.141938
$319$	20.8953	1.16991
$320$	−0.577212	−0.0322672
$321$	−0.370812	−0.0206967
$322$	−2.41623	−0.134651
$323$	−51.4445	−2.86245
$324$	8.08370	0.449095
$325$	16.7677	0.930104
$326$	10.2935	0.570104
$327$	0.678545	0.0375236
$328$	4.68274	0.258561
$329$	9.32223	0.513951
$330$	0.419658	0.0231014
$331$	25.9603	1.42691	0.713455	−	0.700702i	$-0.247130\pi$
$331$	25.9603	1.42691	0.713455	+	0.700702i	$0.247130\pi$
$332$	10.6520	0.584604
$333$	14.2918	0.783185
$334$	−11.2117	−0.613475
$335$	8.85398	0.483745
$336$	−0.377086	−0.0205718
$337$	−9.99780	−0.544615	−0.272307	−	0.962210i	$-0.587787\pi$
$337$	−9.99780	−0.544615	−0.272307	+	0.962210i	$0.587787\pi$
$338$	−0.0906726	−0.00493194
$339$	1.20068	0.0652118
$340$	3.75477	0.203631
$341$	2.42218	0.131168
$342$	−22.9109	−1.23888
$343$	−14.8279	−0.800634
$344$	−10.2765	−0.554072
$345$	−0.380913	−0.0205077
$346$	−13.9442	−0.749646
$347$	8.92467	0.479102	0.239551	−	0.970884i	$-0.423000\pi$
$347$	8.92467	0.479102	0.239551	+	0.970884i	$0.423000\pi$
$348$	−2.95993	−0.158669
$349$	−5.32879	−0.285243	−0.142622	−	0.989777i	$-0.545553\pi$
$349$	−5.32879	−0.285243	−0.142622	+	0.989777i	$0.545553\pi$
$350$	−5.48360	−0.293111
$351$	−6.79955	−0.362933
$352$	2.26550	0.120751
$353$	−30.6804	−1.63295	−0.816475	−	0.577381i	$-0.804075\pi$
$353$	−30.6804	−1.63295	−0.816475	+	0.577381i	$0.804075\pi$
$354$	3.21096	0.170661
$355$	7.87088	0.417743
$356$	−5.27348	−0.279494
$357$	2.45295	0.129824
$358$	10.4516	0.552385
$359$	−16.0218	−0.845600	−0.422800	−	0.906223i	$-0.638953\pi$
$359$	−16.0218	−0.845600	−0.422800	+	0.906223i	$0.638953\pi$
$360$	1.67219	0.0881322
$361$	43.5436	2.29177
$362$	−2.12544	−0.111711
$363$	1.88300	0.0988320
$364$	−4.22179	−0.221282
$365$	−5.96663	−0.312308
$366$	0.479702	0.0250744
$367$	10.5595	0.551199	0.275600	−	0.961273i	$-0.411124\pi$
$367$	10.5595	0.551199	0.275600	+	0.961273i	$0.411124\pi$
$368$	−2.05633	−0.107194
$369$	−13.5659	−0.706215
$370$	2.84755	0.148037
$371$	9.26746	0.481143
$372$	−0.343114	−0.0177897
$373$	29.8566	1.54592	0.772959	−	0.634456i	$-0.218776\pi$
$373$	29.8566	1.54592	0.772959	+	0.634456i	$0.218776\pi$
$374$	−14.7371	−0.762036
$375$	−1.79067	−0.0924699
$376$	7.93369	0.409149
$377$	−33.1388	−1.70673
$378$	2.22368	0.114374
$379$	14.3024	0.734665	0.367333	−	0.930090i	$-0.380271\pi$
$379$	14.3024	0.734665	0.367333	+	0.930090i	$0.380271\pi$
$380$	−4.56486	−0.234172
$381$	3.28717	0.168407
$382$	−15.6364	−0.800029
$383$	−16.6469	−0.850619	−0.425310	−	0.905048i	$-0.639835\pi$
$383$	−16.6469	−0.850619	−0.425310	+	0.905048i	$0.639835\pi$
$384$	−0.320920	−0.0163769
$385$	−1.53654	−0.0783093
$386$	−0.142439	−0.00724995
$387$	29.7711	1.51335
$388$	−6.86024	−0.348276
$389$	−18.3463	−0.930194	−0.465097	−	0.885260i	$-0.653980\pi$
$389$	−18.3463	−0.930194	−0.465097	+	0.885260i	$0.653980\pi$
$390$	−0.665555	−0.0337017
$391$	13.3765	0.676476
$392$	−5.61933	−0.283819
$393$	−1.78103	−0.0898413
$394$	−5.48547	−0.276354
$395$	1.70129	0.0856010
$396$	−6.56317	−0.329812
$397$	10.4848	0.526216	0.263108	−	0.964766i	$-0.415252\pi$
$397$	10.4848	0.526216	0.263108	+	0.964766i	$0.415252\pi$
$398$	−0.953994	−0.0478194
$399$	−2.98217	−0.149295
$400$	−4.66683	−0.233341
$401$	17.0943	0.853650	0.426825	−	0.904334i	$-0.359632\pi$
$401$	17.0943	0.853650	0.426825	+	0.904334i	$0.359632\pi$
$402$	4.92265	0.245520
$403$	−3.84144	−0.191356
$404$	−12.8857	−0.641085
$405$	−4.66601	−0.231856
$406$	10.8375	0.537856
$407$	−11.1764	−0.553991
$408$	2.08758	0.103351
$409$	−21.7528	−1.07561	−0.537803	−	0.843070i	$-0.680746\pi$
$409$	−21.7528	−1.07561	−0.537803	+	0.843070i	$0.680746\pi$
$410$	−2.70294	−0.133489
$411$	2.23319	0.110155
$412$	−9.04064	−0.445400
$413$	−11.7566	−0.578506
$414$	5.95722	0.292781
$415$	−6.14847	−0.301816
$416$	−3.59296	−0.176159
$417$	3.15120	0.154315
$418$	17.9166	0.876329
$419$	−32.3706	−1.58141	−0.790704	−	0.612198i	$-0.790285\pi$
$419$	−32.3706	−1.58141	−0.790704	+	0.612198i	$0.790285\pi$
$420$	0.217659	0.0106207
$421$	−1.97430	−0.0962213	−0.0481107	−	0.998842i	$-0.515320\pi$
$421$	−1.97430	−0.0962213	−0.0481107	+	0.998842i	$0.515320\pi$
$422$	12.5773	0.612252
$423$	−22.9840	−1.11752
$424$	7.88708	0.383031
$425$	30.3577	1.47256
$426$	4.37607	0.212021
$427$	−1.75639	−0.0849975
$428$	1.15547	0.0558516
$429$	2.61223	0.126120
$430$	5.93172	0.286053
$431$	22.7710	1.09684	0.548420	−	0.836203i	$-0.315229\pi$
$431$	22.7710	1.09684	0.548420	+	0.836203i	$0.315229\pi$
$432$	1.89247	0.0910514
$433$	14.6734	0.705159	0.352579	−	0.935782i	$-0.385305\pi$
$433$	14.6734	0.705159	0.352579	+	0.935782i	$0.385305\pi$
$434$	1.25628	0.0603034
$435$	1.70851	0.0819167
$436$	−2.11438	−0.101260
$437$	−16.2624	−0.777937
$438$	−3.31734	−0.158509
$439$	8.72841	0.416584	0.208292	−	0.978067i	$-0.433210\pi$
$439$	8.72841	0.416584	0.208292	+	0.978067i	$0.433210\pi$
$440$	−1.30767	−0.0623409
$441$	16.2793	0.775203
$442$	23.3722	1.11170
$443$	33.7038	1.60132	0.800658	−	0.599121i	$-0.204483\pi$
$443$	33.7038	1.60132	0.800658	+	0.599121i	$0.204483\pi$
$444$	1.58319	0.0751348
$445$	3.04392	0.144296
$446$	12.5610	0.594779
$447$	−3.35228	−0.158558
$448$	1.17502	0.0555144
$449$	−21.5697	−1.01794	−0.508970	−	0.860784i	$-0.669973\pi$
$449$	−21.5697	−1.01794	−0.508970	+	0.860784i	$0.669973\pi$
$450$	13.5198	0.637332
$451$	10.6087	0.499546
$452$	−3.74136	−0.175979
$453$	7.86565	0.369561
$454$	−1.43749	−0.0674648
$455$	2.43687	0.114242
$456$	−2.53798	−0.118852
$457$	−13.6228	−0.637247	−0.318623	−	0.947881i	$-0.603221\pi$
$457$	−13.6228	−0.637247	−0.318623	+	0.947881i	$0.603221\pi$
$458$	1.96151	0.0916553
$459$	−12.3105	−0.574605
$460$	1.18694	0.0553414
$461$	0.695552	0.0323951	0.0161975	−	0.999869i	$-0.494844\pi$
$461$	0.695552	0.0323951	0.0161975	+	0.999869i	$0.494844\pi$
$462$	−0.854289	−0.0397451
$463$	−14.5576	−0.676550	−0.338275	−	0.941047i	$-0.609843\pi$
$463$	−14.5576	−0.676550	−0.338275	+	0.941047i	$0.609843\pi$
$464$	9.22327	0.428179
$465$	0.198050	0.00918435
$466$	−19.0396	−0.881991
$467$	−37.9116	−1.75434	−0.877171	−	0.480179i	$-0.840572\pi$
$467$	−37.9116	−1.75434	−0.877171	+	0.480179i	$0.840572\pi$
$468$	10.4088	0.481148
$469$	−18.0238	−0.832264
$470$	−4.57942	−0.211233
$471$	−2.03519	−0.0937767
$472$	−10.0055	−0.460540
$473$	−23.2814	−1.07048
$474$	0.945884	0.0434459
$475$	−36.9074	−1.69343
$476$	−7.64349	−0.350339
$477$	−22.8490	−1.04618
$478$	0.663987	0.0303701
$479$	16.1073	0.735964	0.367982	−	0.929833i	$-0.380049\pi$
$479$	16.1073	0.735964	0.367982	+	0.929833i	$0.380049\pi$
$480$	0.185239	0.00845496
$481$	17.7251	0.808194
$482$	12.0809	0.550269
$483$	0.775416	0.0352826
$484$	−5.86752	−0.266705
$485$	3.95982	0.179806
$486$	−8.27162	−0.375208
$487$	−17.9176	−0.811924	−0.405962	−	0.913890i	$-0.633063\pi$
$487$	−17.9176	−0.811924	−0.405962	+	0.913890i	$0.633063\pi$
$488$	−1.49477	−0.0676652
$489$	−3.30338	−0.149384
$490$	3.24355	0.146529
$491$	23.0916	1.04211	0.521054	−	0.853524i	$-0.325539\pi$
$491$	23.0916	1.04211	0.521054	+	0.853524i	$0.325539\pi$
$492$	−1.50278	−0.0677507
$493$	−59.9974	−2.70215
$494$	−28.4147	−1.27844
$495$	3.78834	0.170273
$496$	1.06916	0.0480067
$497$	−16.0226	−0.718711
$498$	−3.41844	−0.153184
$499$	15.1955	0.680243	0.340122	−	0.940381i	$-0.389532\pi$
$499$	15.1955	0.680243	0.340122	+	0.940381i	$0.389532\pi$
$500$	5.57981	0.249537
$501$	3.59804	0.160749
$502$	−9.87004	−0.440522
$503$	−30.3484	−1.35317	−0.676584	−	0.736365i	$-0.736541\pi$
$503$	−30.3484	−1.35317	−0.676584	+	0.736365i	$0.736541\pi$
$504$	−3.40404	−0.151628
$505$	7.43776	0.330976
$506$	−4.65862	−0.207101
$507$	0.0290986	0.00129232
$508$	−10.2430	−0.454458
$509$	−23.7464	−1.05254	−0.526270	−	0.850317i	$-0.676410\pi$
$509$	−23.7464	−1.05254	−0.526270	+	0.850317i	$0.676410\pi$
$510$	−1.20498	−0.0533574
$511$	12.1461	0.537313
$512$	1.00000	0.0441942
$513$	14.9665	0.660787
$514$	16.9814	0.749019
$515$	5.21837	0.229949
$516$	3.29793	0.145183
$517$	17.9738	0.790485
$518$	−5.79670	−0.254692
$519$	4.47498	0.196430
$520$	2.07390	0.0909465
$521$	−33.5797	−1.47115	−0.735577	−	0.677441i	$-0.763089\pi$
$521$	−33.5797	−1.47115	−0.735577	+	0.677441i	$0.763089\pi$
$522$	−26.7199	−1.16950
$523$	10.6291	0.464779	0.232390	−	0.972623i	$-0.425346\pi$
$523$	10.6291	0.464779	0.232390	+	0.972623i	$0.425346\pi$
$524$	5.54978	0.242443
$525$	1.75980	0.0768038
$526$	−28.0511	−1.22309
$527$	−6.95488	−0.302959
$528$	−0.727043	−0.0316405
$529$	−18.7715	−0.816152
$530$	−4.55252	−0.197749
$531$	28.9860	1.25789
$532$	9.29257	0.402884
$533$	−16.8249	−0.728766
$534$	1.69237	0.0732358
$535$	−0.666950	−0.0288348
$536$	−15.3392	−0.662553
$537$	−3.35413	−0.144741
$538$	10.3385	0.445725
$539$	−12.7306	−0.548345
$540$	−1.09236	−0.0470075
$541$	12.1083	0.520575	0.260288	−	0.965531i	$-0.416183\pi$
$541$	12.1083	0.520575	0.260288	+	0.965531i	$0.416183\pi$
$542$	−0.914784	−0.0392933
$543$	0.682097	0.0292716
$544$	−6.50500	−0.278900
$545$	1.22044	0.0522781
$546$	1.35485	0.0579824
$547$	19.3525	0.827451	0.413726	−	0.910402i	$-0.364227\pi$
$547$	19.3525	0.827451	0.413726	+	0.910402i	$0.364227\pi$
$548$	−6.95871	−0.297261
$549$	4.33038	0.184816
$550$	−10.5727	−0.450821
$551$	72.9418	3.10742
$552$	0.659918	0.0280880
$553$	−3.46327	−0.147273
$554$	9.18601	0.390276
$555$	−0.913837	−0.0387902
$556$	−9.81928	−0.416430
$557$	−28.4402	−1.20505	−0.602526	−	0.798099i	$-0.705839\pi$
$557$	−28.4402	−1.20505	−0.602526	+	0.798099i	$0.705839\pi$
$558$	−3.09737	−0.131122
$559$	36.9230	1.56168
$560$	−0.678235	−0.0286607
$561$	4.72942	0.199676
$562$	9.81157	0.413876
$563$	−22.2268	−0.936749	−0.468374	−	0.883530i	$-0.655160\pi$
$563$	−22.2268	−0.936749	−0.468374	+	0.883530i	$0.655160\pi$
$564$	−2.54608	−0.107209
$565$	2.15956	0.0908534
$566$	21.3863	0.898935
$567$	9.49849	0.398899
$568$	−13.6360	−0.572155
$569$	23.3940	0.980729	0.490364	−	0.871517i	$-0.336864\pi$
$569$	23.3940	0.980729	0.490364	+	0.871517i	$0.336864\pi$
$570$	1.46495	0.0613601
$571$	23.5160	0.984113	0.492056	−	0.870563i	$-0.336245\pi$
$571$	23.5160	0.984113	0.492056	+	0.870563i	$0.336245\pi$
$572$	−8.13983	−0.340344
$573$	5.01804	0.209632
$574$	5.50230	0.229662
$575$	9.59655	0.400204
$576$	−2.89701	−0.120709
$577$	26.1311	1.08785	0.543925	−	0.839134i	$-0.316938\pi$
$577$	26.1311	1.08785	0.543925	+	0.839134i	$0.316938\pi$
$578$	25.3150	1.05297
$579$	0.0457115	0.00189970
$580$	−5.32379	−0.221058
$581$	12.5163	0.519263
$582$	2.20159	0.0912587
$583$	17.8682	0.740024
$584$	10.3370	0.427747
$585$	−6.00811	−0.248405
$586$	−30.1353	−1.24488
$587$	14.0428	0.579610	0.289805	−	0.957086i	$-0.406410\pi$
$587$	14.0428	0.579610	0.289805	+	0.957086i	$0.406410\pi$
$588$	1.80336	0.0743691
$589$	8.45540	0.348399
$590$	5.77530	0.237765
$591$	1.76040	0.0724130
$592$	−4.93329	−0.202757
$593$	−26.8924	−1.10434	−0.552170	−	0.833732i	$-0.686200\pi$
$593$	−26.8924	−1.10434	−0.552170	+	0.833732i	$0.686200\pi$
$594$	4.28738	0.175913
$595$	4.41192	0.180871
$596$	10.4459	0.427879
$597$	0.306156	0.0125301
$598$	7.38831	0.302131
$599$	−4.28453	−0.175061	−0.0875306	−	0.996162i	$-0.527898\pi$
$599$	−4.28453	−0.175061	−0.0875306	+	0.996162i	$0.527898\pi$
$600$	1.49768	0.0611424
$601$	−8.56703	−0.349456	−0.174728	−	0.984617i	$-0.555905\pi$
$601$	−8.56703	−0.349456	−0.174728	+	0.984617i	$0.555905\pi$
$602$	−12.0751	−0.492143
$603$	44.4378	1.80965
$604$	−24.5097	−0.997286
$605$	3.38681	0.137693
$606$	4.13526	0.167984
$607$	−4.95971	−0.201308	−0.100654	−	0.994921i	$-0.532094\pi$
$607$	−4.95971	−0.201308	−0.100654	+	0.994921i	$0.532094\pi$
$608$	7.90845	0.320730
$609$	−3.47797	−0.140934
$610$	0.862802	0.0349338
$611$	−28.5054	−1.15320
$612$	18.8451	0.761766
$613$	45.9782	1.85704	0.928520	−	0.371282i	$-0.121082\pi$
$613$	45.9782	1.85704	0.928520	+	0.371282i	$0.121082\pi$
$614$	−14.1139	−0.569592
$615$	0.867426	0.0349780
$616$	2.66200	0.107255
$617$	39.4625	1.58870	0.794350	−	0.607460i	$-0.207812\pi$
$617$	39.4625	1.58870	0.794350	+	0.607460i	$0.207812\pi$
$618$	2.90132	0.116708
$619$	−25.0207	−1.00567	−0.502834	−	0.864383i	$-0.667709\pi$
$619$	−25.0207	−1.00567	−0.502834	+	0.864383i	$0.667709\pi$
$620$	−0.617132	−0.0247846
$621$	−3.89154	−0.156162
$622$	−9.07469	−0.363862
$623$	−6.19644	−0.248255
$624$	1.15305	0.0461590
$625$	20.1134	0.804536
$626$	−11.1055	−0.443865
$627$	−5.74979	−0.229624
$628$	6.34175	0.253063
$629$	32.0910	1.27955
$630$	1.96485	0.0782816
$631$	−8.55131	−0.340422	−0.170211	−	0.985408i	$-0.554445\pi$
$631$	−8.55131	−0.340422	−0.170211	+	0.985408i	$0.554445\pi$
$632$	−2.94742	−0.117242
$633$	−4.03629	−0.160428
$634$	17.5616	0.697458
$635$	5.91237	0.234625
$636$	−2.53112	−0.100365
$637$	20.1900	0.799957
$638$	20.8953	0.827253
$639$	39.5037	1.56274
$640$	−0.577212	−0.0228163
$641$	−13.9271	−0.550086	−0.275043	−	0.961432i	$-0.588692\pi$
$641$	−13.9271	−0.550086	−0.275043	+	0.961432i	$0.588692\pi$
$642$	−0.370812	−0.0146348
$643$	20.2737	0.799515	0.399757	−	0.916621i	$-0.369094\pi$
$643$	20.2737	0.799515	0.399757	+	0.916621i	$0.369094\pi$
$644$	−2.41623	−0.0952127
$645$	−1.90361	−0.0749545
$646$	−51.4445	−2.02406
$647$	−37.1894	−1.46206	−0.731032	−	0.682343i	$-0.760961\pi$
$647$	−37.1894	−1.46206	−0.731032	+	0.682343i	$0.760961\pi$
$648$	8.08370	0.317558
$649$	−22.6674	−0.889774
$650$	16.7677	0.657683
$651$	−0.403166	−0.0158013
$652$	10.2935	0.403124
$653$	−29.1127	−1.13927	−0.569634	−	0.821898i	$-0.692915\pi$
$653$	−29.1127	−1.13927	−0.569634	+	0.821898i	$0.692915\pi$
$654$	0.678545	0.0265332
$655$	−3.20340	−0.125167
$656$	4.68274	0.182830
$657$	−29.9463	−1.16832
$658$	9.32223	0.363418
$659$	22.3613	0.871073	0.435536	−	0.900171i	$-0.356559\pi$
$659$	22.3613	0.871073	0.435536	+	0.900171i	$0.356559\pi$
$660$	0.419658	0.0163352
$661$	−22.8121	−0.887290	−0.443645	−	0.896203i	$-0.646315\pi$
$661$	−22.8121	−0.887290	−0.443645	+	0.896203i	$0.646315\pi$
$662$	25.9603	1.00898
$663$	−7.50059	−0.291299
$664$	10.6520	0.413378
$665$	−5.36379	−0.207999
$666$	14.2918	0.553795
$667$	−18.9661	−0.734371
$668$	−11.2117	−0.433792
$669$	−4.03106	−0.155850
$670$	8.85398	0.342059
$671$	−3.38641	−0.130731
$672$	−0.377086	−0.0145464
$673$	−25.6011	−0.986850	−0.493425	−	0.869788i	$-0.664255\pi$
$673$	−25.6011	−0.986850	−0.493425	+	0.869788i	$0.664255\pi$
$674$	−9.99780	−0.385101
$675$	−8.83182	−0.339937
$676$	−0.0906726	−0.00348741
$677$	34.2603	1.31673	0.658364	−	0.752699i	$-0.271249\pi$
$677$	34.2603	1.31673	0.658364	+	0.752699i	$0.271249\pi$
$678$	1.20068	0.0461117
$679$	−8.06090	−0.309349
$680$	3.75477	0.143989
$681$	0.461319	0.0176778
$682$	2.42218	0.0927500
$683$	25.0524	0.958603	0.479301	−	0.877650i	$-0.340890\pi$
$683$	25.0524	0.958603	0.479301	+	0.877650i	$0.340890\pi$
$684$	−22.9109	−0.876019
$685$	4.01665	0.153468
$686$	−14.8279	−0.566133
$687$	−0.629487	−0.0240164
$688$	−10.2765	−0.391788
$689$	−28.3379	−1.07959
$690$	−0.380913	−0.0145011
$691$	−12.3308	−0.469084	−0.234542	−	0.972106i	$-0.575359\pi$
$691$	−12.3308	−0.469084	−0.234542	+	0.972106i	$0.575359\pi$
$692$	−13.9442	−0.530080
$693$	−7.71184	−0.292949
$694$	8.92467	0.338776
$695$	5.66781	0.214992
$696$	−2.95993	−0.112196
$697$	−30.4612	−1.15380
$698$	−5.32879	−0.201698
$699$	6.11017	0.231108
$700$	−5.48360	−0.207261
$701$	−17.5052	−0.661161	−0.330580	−	0.943778i	$-0.607244\pi$
$701$	−17.5052	−0.661161	−0.330580	+	0.943778i	$0.607244\pi$
$702$	−6.79955	−0.256632
$703$	−39.0147	−1.47147
$704$	2.26550	0.0853842
$705$	1.46963	0.0553494
$706$	−30.6804	−1.15467
$707$	−15.1409	−0.569431
$708$	3.21096	0.120675
$709$	12.6909	0.476616	0.238308	−	0.971190i	$-0.423407\pi$
$709$	12.6909	0.476616	0.238308	+	0.971190i	$0.423407\pi$
$710$	7.87088	0.295389
$711$	8.53870	0.320226
$712$	−5.27348	−0.197632
$713$	−2.19855	−0.0823363
$714$	2.45295	0.0917993
$715$	4.69841	0.175711
$716$	10.4516	0.390595
$717$	−0.213087	−0.00795787
$718$	−16.0218	−0.597929
$719$	28.0933	1.04770	0.523852	−	0.851809i	$-0.324495\pi$
$719$	28.0933	1.04770	0.523852	+	0.851809i	$0.324495\pi$
$720$	1.67219	0.0623189
$721$	−10.6229	−0.395618
$722$	43.5436	1.62053
$723$	−3.87699	−0.144187
$724$	−2.12544	−0.0789915
$725$	−43.0434	−1.59859
$726$	1.88300	0.0698848
$727$	17.9355	0.665192	0.332596	−	0.943069i	$-0.392076\pi$
$727$	17.9355	0.665192	0.332596	+	0.943069i	$0.392076\pi$
$728$	−4.22179	−0.156470
$729$	−21.5966	−0.799873
$730$	−5.96663	−0.220835
$731$	66.8486	2.47249
$732$	0.479702	0.0177303
$733$	−42.6348	−1.57475	−0.787375	−	0.616474i	$-0.788561\pi$
$733$	−42.6348	−1.57475	−0.787375	+	0.616474i	$0.788561\pi$
$734$	10.5595	0.389757
$735$	−1.04092	−0.0383949
$736$	−2.05633	−0.0757975
$737$	−34.7509	−1.28007
$738$	−13.5659	−0.499370
$739$	9.60825	0.353445	0.176723	−	0.984261i	$-0.443450\pi$
$739$	9.60825	0.353445	0.176723	+	0.984261i	$0.443450\pi$
$740$	2.84755	0.104678
$741$	9.11885	0.334989
$742$	9.26746	0.340219
$743$	−27.2262	−0.998833	−0.499417	−	0.866362i	$-0.666452\pi$
$743$	−27.2262	−0.998833	−0.499417	+	0.866362i	$0.666452\pi$
$744$	−0.343114	−0.0125792
$745$	−6.02948	−0.220903
$746$	29.8566	1.09313
$747$	−30.8590	−1.12907
$748$	−14.7371	−0.538840
$749$	1.35769	0.0496091
$750$	−1.79067	−0.0653861
$751$	47.5126	1.73376	0.866880	−	0.498517i	$-0.166122\pi$
$751$	47.5126	1.73376	0.866880	+	0.498517i	$0.166122\pi$
$752$	7.93369	0.289312
$753$	3.16749	0.115430
$754$	−33.1388	−1.20684
$755$	14.1473	0.514873
$756$	2.22368	0.0808746
$757$	3.20174	0.116369	0.0581847	−	0.998306i	$-0.481469\pi$
$757$	3.20174	0.116369	0.0581847	+	0.998306i	$0.481469\pi$
$758$	14.3024	0.519487
$759$	1.49504	0.0542666
$760$	−4.56486	−0.165585
$761$	−4.70879	−0.170694	−0.0853468	−	0.996351i	$-0.527200\pi$
$761$	−4.70879	−0.170694	−0.0853468	+	0.996351i	$0.527200\pi$
$762$	3.28717	0.119082
$763$	−2.48443	−0.0899424
$764$	−15.6364	−0.565706
$765$	−10.8776	−0.393280
$766$	−16.6469	−0.601479
$767$	35.9493	1.29805
$768$	−0.320920	−0.0115802
$769$	31.4670	1.13473	0.567364	−	0.823467i	$-0.307963\pi$
$769$	31.4670	1.13473	0.567364	+	0.823467i	$0.307963\pi$
$770$	−1.53654	−0.0553731
$771$	−5.44968	−0.196265
$772$	−0.142439	−0.00512649
$773$	−32.8029	−1.17984	−0.589919	−	0.807463i	$-0.700840\pi$
$773$	−32.8029	−1.17984	−0.589919	+	0.807463i	$0.700840\pi$
$774$	29.7711	1.07010
$775$	−4.98958	−0.179231
$776$	−6.86024	−0.246268
$777$	1.86028	0.0667370
$778$	−18.3463	−0.657746
$779$	37.0332	1.32685
$780$	−0.665555	−0.0238307
$781$	−30.8924	−1.10542
$782$	13.3765	0.478341
$783$	17.4547	0.623781
$784$	−5.61933	−0.200690
$785$	−3.66053	−0.130650
$786$	−1.78103	−0.0635274
$787$	55.6568	1.98395	0.991976	−	0.126426i	$-0.0403507\pi$
$787$	55.6568	1.98395	0.991976	+	0.126426i	$0.0403507\pi$
$788$	−5.48547	−0.195412
$789$	9.00217	0.320486
$790$	1.70129	0.0605290
$791$	−4.39617	−0.156310
$792$	−6.56317	−0.233212
$793$	5.37065	0.190718
$794$	10.4848	0.372091
$795$	1.46099	0.0518161
$796$	−0.953994	−0.0338134
$797$	32.7094	1.15863	0.579313	−	0.815105i	$-0.303321\pi$
$797$	32.7094	1.15863	0.579313	+	0.815105i	$0.303321\pi$
$798$	−2.98217	−0.105568
$799$	−51.6087	−1.82578
$800$	−4.66683	−0.164997
$801$	15.2773	0.539798
$802$	17.0943	0.603622
$803$	23.4184	0.826417
$804$	4.92265	0.173609
$805$	1.39468	0.0491559
$806$	−3.84144	−0.135309
$807$	−3.31783	−0.116793
$808$	−12.8857	−0.453316
$809$	52.9003	1.85988	0.929938	−	0.367716i	$-0.119860\pi$
$809$	52.9003	1.85988	0.929938	+	0.367716i	$0.119860\pi$
$810$	−4.66601	−0.163947
$811$	−17.7865	−0.624568	−0.312284	−	0.949989i	$-0.601094\pi$
$811$	−17.7865	−0.624568	−0.312284	+	0.949989i	$0.601094\pi$
$812$	10.8375	0.380322
$813$	0.293572	0.0102960
$814$	−11.1764	−0.391731
$815$	−5.94153	−0.208123
$816$	2.08758	0.0730800
$817$	−81.2712	−2.84332
$818$	−21.7528	−0.760569
$819$	12.2306	0.427370
$820$	−2.70294	−0.0943906
$821$	28.9782	1.01134	0.505672	−	0.862726i	$-0.331245\pi$
$821$	28.9782	1.01134	0.505672	+	0.862726i	$0.331245\pi$
$822$	2.23319	0.0778914
$823$	−40.3900	−1.40791	−0.703953	−	0.710246i	$-0.748584\pi$
$823$	−40.3900	−1.40791	−0.703953	+	0.710246i	$0.748584\pi$
$824$	−9.04064	−0.314946
$825$	3.39298	0.118129
$826$	−11.7566	−0.409066
$827$	−28.4224	−0.988344	−0.494172	−	0.869364i	$-0.664529\pi$
$827$	−28.4224	−0.988344	−0.494172	+	0.869364i	$0.664529\pi$
$828$	5.95722	0.207028
$829$	14.4854	0.503097	0.251549	−	0.967845i	$-0.419060\pi$
$829$	14.4854	0.503097	0.251549	+	0.967845i	$0.419060\pi$
$830$	−6.14847	−0.213416
$831$	−2.94797	−0.102264
$832$	−3.59296	−0.124563
$833$	36.5538	1.26651
$834$	3.15120	0.109117
$835$	6.47151	0.223956
$836$	17.9166	0.619658
$837$	2.02335	0.0699372
$838$	−32.3706	−1.11822
$839$	18.7534	0.647438	0.323719	−	0.946153i	$-0.395067\pi$
$839$	18.7534	0.647438	0.323719	+	0.946153i	$0.395067\pi$
$840$	0.217659	0.00750995
$841$	56.0687	1.93340
$842$	−1.97430	−0.0680388
$843$	−3.14873	−0.108448
$844$	12.5773	0.432927
$845$	0.0523374	0.00180046
$846$	−22.9840	−0.790206
$847$	−6.89444	−0.236896
$848$	7.88708	0.270844
$849$	−6.86330	−0.235548
$850$	30.3577	1.04126
$851$	10.1445	0.347748
$852$	4.37607	0.149922
$853$	34.0091	1.16445	0.582224	−	0.813029i	$-0.302183\pi$
$853$	34.0091	1.16445	0.582224	+	0.813029i	$0.302183\pi$
$854$	−1.75639	−0.0601023
$855$	13.2244	0.452266
$856$	1.15547	0.0394930
$857$	−30.4720	−1.04090	−0.520452	−	0.853891i	$-0.674237\pi$
$857$	−30.4720	−1.04090	−0.520452	+	0.853891i	$0.674237\pi$
$858$	2.61223	0.0891802
$859$	−32.7574	−1.11767	−0.558834	−	0.829279i	$-0.688751\pi$
$859$	−32.7574	−1.11767	−0.558834	+	0.829279i	$0.688751\pi$
$860$	5.93172	0.202270
$861$	−1.76580	−0.0601782
$862$	22.7710	0.775583
$863$	−6.84310	−0.232942	−0.116471	−	0.993194i	$-0.537158\pi$
$863$	−6.84310	−0.232942	−0.116471	+	0.993194i	$0.537158\pi$
$864$	1.89247	0.0643830
$865$	8.04878	0.273667
$866$	14.6734	0.498622
$867$	−8.12409	−0.275909
$868$	1.25628	0.0426410
$869$	−6.67737	−0.226514
$870$	1.70851	0.0579238
$871$	55.1131	1.86744
$872$	−2.11438	−0.0716018
$873$	19.8742	0.672639
$874$	−16.2624	−0.550084
$875$	6.55638	0.221646
$876$	−3.31734	−0.112082
$877$	45.7322	1.54427	0.772133	−	0.635461i	$-0.219190\pi$
$877$	45.7322	1.54427	0.772133	+	0.635461i	$0.219190\pi$
$878$	8.72841	0.294569
$879$	9.67102	0.326195
$880$	−1.30767	−0.0440817
$881$	−31.9784	−1.07738	−0.538690	−	0.842504i	$-0.681080\pi$
$881$	−31.9784	−1.07738	−0.538690	+	0.842504i	$0.681080\pi$
$882$	16.2793	0.548151
$883$	−57.9659	−1.95071	−0.975354	−	0.220646i	$-0.929184\pi$
$883$	−57.9659	−1.95071	−0.975354	+	0.220646i	$0.929184\pi$
$884$	23.3722	0.786091
$885$	−1.85341	−0.0623016
$886$	33.7038	1.13230
$887$	28.7602	0.965672	0.482836	−	0.875711i	$-0.339607\pi$
$887$	28.7602	0.965672	0.482836	+	0.875711i	$0.339607\pi$
$888$	1.58319	0.0531284
$889$	−12.0357	−0.403663
$890$	3.04392	0.102032
$891$	18.3136	0.613529
$892$	12.5610	0.420572
$893$	62.7432	2.09962
$894$	−3.35228	−0.112117
$895$	−6.03280	−0.201654
$896$	1.17502	0.0392546
$897$	−2.37106	−0.0791673
$898$	−21.5697	−0.719792
$899$	9.86114	0.328888
$900$	13.5198	0.450661
$901$	−51.3055	−1.70923
$902$	10.6087	0.353232
$903$	3.87513	0.128956
$904$	−3.74136	−0.124436
$905$	1.22683	0.0407813
$906$	7.86565	0.261319
$907$	−28.3684	−0.941956	−0.470978	−	0.882145i	$-0.656099\pi$
$907$	−28.3684	−0.941956	−0.470978	+	0.882145i	$0.656099\pi$
$908$	−1.43749	−0.0477048
$909$	37.3299	1.23815
$910$	2.43687	0.0807814
$911$	−6.92452	−0.229420	−0.114710	−	0.993399i	$-0.536594\pi$
$911$	−6.92452	−0.229420	−0.114710	+	0.993399i	$0.536594\pi$
$912$	−2.53798	−0.0840409
$913$	24.1321	0.798655
$914$	−13.6228	−0.450601
$915$	−0.276890	−0.00915371
$916$	1.96151	0.0648101
$917$	6.52109	0.215345
$918$	−12.3105	−0.406307
$919$	39.1721	1.29217	0.646084	−	0.763266i	$-0.276406\pi$
$919$	39.1721	1.29217	0.646084	+	0.763266i	$0.276406\pi$
$920$	1.18694	0.0391323
$921$	4.52944	0.149250
$922$	0.695552	0.0229068
$923$	48.9936	1.61264
$924$	−0.854289	−0.0281040
$925$	23.0228	0.756985
$926$	−14.5576	−0.478393
$927$	26.1908	0.860220
$928$	9.22327	0.302769
$929$	−49.2184	−1.61480	−0.807402	−	0.590002i	$-0.799127\pi$
$929$	−49.2184	−1.61480	−0.807402	+	0.590002i	$0.799127\pi$
$930$	0.198050	0.00649431
$931$	−44.4402	−1.45647
$932$	−19.0396	−0.623662
$933$	2.91225	0.0953427
$934$	−37.9116	−1.24051
$935$	8.50642	0.278190
$936$	10.4088	0.340223
$937$	7.45945	0.243690	0.121845	−	0.992549i	$-0.461119\pi$
$937$	7.45945	0.243690	0.121845	+	0.992549i	$0.461119\pi$
$938$	−18.0238	−0.588499
$939$	3.56397	0.116306
$940$	−4.57942	−0.149364
$941$	54.1979	1.76680	0.883401	−	0.468618i	$-0.155248\pi$
$941$	54.1979	1.76680	0.883401	+	0.468618i	$0.155248\pi$
$942$	−2.03519	−0.0663101
$943$	−9.62928	−0.313572
$944$	−10.0055	−0.325651
$945$	−1.28354	−0.0417535
$946$	−23.2814	−0.756943
$947$	−14.8059	−0.481126	−0.240563	−	0.970634i	$-0.577332\pi$
$947$	−14.8059	−0.481126	−0.240563	+	0.970634i	$0.577332\pi$
$948$	0.945884	0.0307209
$949$	−37.1403	−1.20562
$950$	−36.9074	−1.19743
$951$	−5.63585	−0.182755
$952$	−7.64349	−0.247727
$953$	−23.1500	−0.749903	−0.374951	−	0.927045i	$-0.622341\pi$
$953$	−23.1500	−0.749903	−0.374951	+	0.927045i	$0.622341\pi$
$954$	−22.8490	−0.739762
$955$	9.02554	0.292060
$956$	0.663987	0.0214749
$957$	−6.70571	−0.216765
$958$	16.1073	0.520405
$959$	−8.17661	−0.264036
$960$	0.185239	0.00597856
$961$	−29.8569	−0.963126
$962$	17.7251	0.571480
$963$	−3.34740	−0.107868
$964$	12.0809	0.389099
$965$	0.0822176	0.00264668
$966$	0.775416	0.0249486
$967$	−9.18932	−0.295509	−0.147754	−	0.989024i	$-0.547205\pi$
$967$	−9.18932	−0.295509	−0.147754	+	0.989024i	$0.547205\pi$
$968$	−5.86752	−0.188589
$969$	16.5096	0.530363
$970$	3.95982	0.127142
$971$	41.5856	1.33455	0.667273	−	0.744814i	$-0.267462\pi$
$971$	41.5856	1.33455	0.667273	+	0.744814i	$0.267462\pi$
$972$	−8.27162	−0.265312
$973$	−11.5378	−0.369886
$974$	−17.9176	−0.574117
$975$	−5.38109	−0.172333
$976$	−1.49477	−0.0478465
$977$	25.3519	0.811079	0.405540	−	0.914077i	$-0.367084\pi$
$977$	25.3519	0.811079	0.405540	+	0.914077i	$0.367084\pi$
$978$	−3.30338	−0.105631
$979$	−11.9471	−0.381830
$980$	3.24355	0.103611
$981$	6.12537	0.195568
$982$	23.0916	0.736881
$983$	−23.4072	−0.746573	−0.373286	−	0.927716i	$-0.621769\pi$
$983$	−23.4072	−0.746573	−0.373286	+	0.927716i	$0.621769\pi$
$984$	−1.50278	−0.0479070
$985$	3.16628	0.100886
$986$	−59.9974	−1.91071
$987$	−2.99169	−0.0952264
$988$	−28.4147	−0.903993
$989$	21.1319	0.671956
$990$	3.78834	0.120401
$991$	9.09638	0.288956	0.144478	−	0.989508i	$-0.453850\pi$
$991$	9.09638	0.288956	0.144478	+	0.989508i	$0.453850\pi$
$992$	1.06916	0.0339458
$993$	−8.33119	−0.264382
$994$	−16.0226	−0.508205
$995$	0.550657	0.0174570
$996$	−3.41844	−0.108317
$997$	48.8908	1.54839	0.774193	−	0.632950i	$-0.218156\pi$
$997$	48.8908	1.54839	0.774193	+	0.632950i	$0.218156\pi$
$998$	15.1955	0.481004
$999$	−9.33609	−0.295381

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.37	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.37	✓	69	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.320920 q^{3} +1.00000 q^{4} -0.577212 q^{5} -0.320920 q^{6} +1.17502 q^{7} +1.00000 q^{8} -2.89701 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.320920 q^{3} +1.00000 q^{4} -0.577212 q^{5} -0.320920 q^{6} +1.17502 q^{7} +1.00000 q^{8} -2.89701 q^{9} -0.577212 q^{10} +2.26550 q^{11} -0.320920 q^{12} -3.59296 q^{13} +1.17502 q^{14} +0.185239 q^{15} +1.00000 q^{16} -6.50500 q^{17} -2.89701 q^{18} +7.90845 q^{19} -0.577212 q^{20} -0.377086 q^{21} +2.26550 q^{22} -2.05633 q^{23} -0.320920 q^{24} -4.66683 q^{25} -3.59296 q^{26} +1.89247 q^{27} +1.17502 q^{28} +9.22327 q^{29} +0.185239 q^{30} +1.06916 q^{31} +1.00000 q^{32} -0.727043 q^{33} -6.50500 q^{34} -0.678235 q^{35} -2.89701 q^{36} -4.93329 q^{37} +7.90845 q^{38} +1.15305 q^{39} -0.577212 q^{40} +4.68274 q^{41} -0.377086 q^{42} -10.2765 q^{43} +2.26550 q^{44} +1.67219 q^{45} -2.05633 q^{46} +7.93369 q^{47} -0.320920 q^{48} -5.61933 q^{49} -4.66683 q^{50} +2.08758 q^{51} -3.59296 q^{52} +7.88708 q^{53} +1.89247 q^{54} -1.30767 q^{55} +1.17502 q^{56} -2.53798 q^{57} +9.22327 q^{58} -10.0055 q^{59} +0.185239 q^{60} -1.49477 q^{61} +1.06916 q^{62} -3.40404 q^{63} +1.00000 q^{64} +2.07390 q^{65} -0.727043 q^{66} -15.3392 q^{67} -6.50500 q^{68} +0.659918 q^{69} -0.678235 q^{70} -13.6360 q^{71} -2.89701 q^{72} +10.3370 q^{73} -4.93329 q^{74} +1.49768 q^{75} +7.90845 q^{76} +2.66200 q^{77} +1.15305 q^{78} -2.94742 q^{79} -0.577212 q^{80} +8.08370 q^{81} +4.68274 q^{82} +10.6520 q^{83} -0.377086 q^{84} +3.75477 q^{85} -10.2765 q^{86} -2.95993 q^{87} +2.26550 q^{88} -5.27348 q^{89} +1.67219 q^{90} -4.22179 q^{91} -2.05633 q^{92} -0.343114 q^{93} +7.93369 q^{94} -4.56486 q^{95} -0.320920 q^{96} -6.86024 q^{97} -5.61933 q^{98} -6.56317 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9} - 33 q^{10} - 30 q^{11} - 25 q^{12} - 58 q^{13} - 19 q^{14} + 2 q^{15} + 69 q^{16} - 80 q^{17} + 54 q^{18} - 40 q^{19} - 33 q^{20} - 32 q^{21} - 30 q^{22} - 45 q^{23} - 25 q^{24} + 42 q^{25} - 58 q^{26} - 76 q^{27} - 19 q^{28} - 44 q^{29} + 2 q^{30} - 12 q^{31} + 69 q^{32} - 41 q^{33} - 80 q^{34} - 49 q^{35} + 54 q^{36} - 47 q^{37} - 40 q^{38} - 14 q^{39} - 33 q^{40} - 94 q^{41} - 32 q^{42} - 10 q^{43} - 30 q^{44} - 89 q^{45} - 45 q^{46} - 85 q^{47} - 25 q^{48} + 32 q^{49} + 42 q^{50} - 10 q^{51} - 58 q^{52} - 41 q^{53} - 76 q^{54} - 27 q^{55} - 19 q^{56} - 72 q^{57} - 44 q^{58} - 75 q^{59} + 2 q^{60} - 98 q^{61} - 12 q^{62} - 61 q^{63} + 69 q^{64} - 47 q^{65} - 41 q^{66} - 22 q^{67} - 80 q^{68} - 74 q^{69} - 49 q^{70} - 22 q^{71} + 54 q^{72} - 129 q^{73} - 47 q^{74} - 106 q^{75} - 40 q^{76} - 108 q^{77} - 14 q^{78} + 21 q^{79} - 33 q^{80} + 13 q^{81} - 94 q^{82} - 111 q^{83} - 32 q^{84} - 67 q^{85} - 10 q^{86} - 38 q^{87} - 30 q^{88} - 112 q^{89} - 89 q^{90} - 55 q^{91} - 45 q^{92} - 90 q^{93} - 85 q^{94} - 38 q^{95} - 25 q^{96} - 98 q^{97} + 32 q^{98} - 51 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 25 * q^3 + 69 * q^4 - 33 * q^5 - 25 * q^6 - 19 * q^7 + 69 * q^8 + 54 * q^9 - 33 * q^10 - 30 * q^11 - 25 * q^12 - 58 * q^13 - 19 * q^14 + 2 * q^15 + 69 * q^16 - 80 * q^17 + 54 * q^18 - 40 * q^19 - 33 * q^20 - 32 * q^21 - 30 * q^22 - 45 * q^23 - 25 * q^24 + 42 * q^25 - 58 * q^26 - 76 * q^27 - 19 * q^28 - 44 * q^29 + 2 * q^30 - 12 * q^31 + 69 * q^32 - 41 * q^33 - 80 * q^34 - 49 * q^35 + 54 * q^36 - 47 * q^37 - 40 * q^38 - 14 * q^39 - 33 * q^40 - 94 * q^41 - 32 * q^42 - 10 * q^43 - 30 * q^44 - 89 * q^45 - 45 * q^46 - 85 * q^47 - 25 * q^48 + 32 * q^49 + 42 * q^50 - 10 * q^51 - 58 * q^52 - 41 * q^53 - 76 * q^54 - 27 * q^55 - 19 * q^56 - 72 * q^57 - 44 * q^58 - 75 * q^59 + 2 * q^60 - 98 * q^61 - 12 * q^62 - 61 * q^63 + 69 * q^64 - 47 * q^65 - 41 * q^66 - 22 * q^67 - 80 * q^68 - 74 * q^69 - 49 * q^70 - 22 * q^71 + 54 * q^72 - 129 * q^73 - 47 * q^74 - 106 * q^75 - 40 * q^76 - 108 * q^77 - 14 * q^78 + 21 * q^79 - 33 * q^80 + 13 * q^81 - 94 * q^82 - 111 * q^83 - 32 * q^84 - 67 * q^85 - 10 * q^86 - 38 * q^87 - 30 * q^88 - 112 * q^89 - 89 * q^90 - 55 * q^91 - 45 * q^92 - 90 * q^93 - 85 * q^94 - 38 * q^95 - 25 * q^96 - 98 * q^97 + 32 * q^98 - 51 * q^99

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