LMFDB - Embedded newform 8002.2.a.d.1.38

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			1.38
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)

$f(q)$	$=$	$q+1.00000 q^{2} -0.147869 q^{3} +1.00000 q^{4} +3.03338 q^{5} -0.147869 q^{6} +1.52845 q^{7} +1.00000 q^{8} -2.97813 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.147869 q^{3} +1.00000 q^{4} +3.03338 q^{5} -0.147869 q^{6} +1.52845 q^{7} +1.00000 q^{8} -2.97813 q^{9} +3.03338 q^{10} +1.68897 q^{11} -0.147869 q^{12} -5.00206 q^{13} +1.52845 q^{14} -0.448543 q^{15} +1.00000 q^{16} -5.10938 q^{17} -2.97813 q^{18} -5.98503 q^{19} +3.03338 q^{20} -0.226011 q^{21} +1.68897 q^{22} -6.33776 q^{23} -0.147869 q^{24} +4.20139 q^{25} -5.00206 q^{26} +0.883981 q^{27} +1.52845 q^{28} +4.38843 q^{29} -0.448543 q^{30} -3.33959 q^{31} +1.00000 q^{32} -0.249746 q^{33} -5.10938 q^{34} +4.63637 q^{35} -2.97813 q^{36} -6.79581 q^{37} -5.98503 q^{38} +0.739650 q^{39} +3.03338 q^{40} +6.96861 q^{41} -0.226011 q^{42} +5.66100 q^{43} +1.68897 q^{44} -9.03381 q^{45} -6.33776 q^{46} -6.11889 q^{47} -0.147869 q^{48} -4.66384 q^{49} +4.20139 q^{50} +0.755519 q^{51} -5.00206 q^{52} +1.39823 q^{53} +0.883981 q^{54} +5.12328 q^{55} +1.52845 q^{56} +0.885001 q^{57} +4.38843 q^{58} -9.64608 q^{59} -0.448543 q^{60} -7.15860 q^{61} -3.33959 q^{62} -4.55193 q^{63} +1.00000 q^{64} -15.1731 q^{65} -0.249746 q^{66} -1.61887 q^{67} -5.10938 q^{68} +0.937159 q^{69} +4.63637 q^{70} +3.73307 q^{71} -2.97813 q^{72} -10.8633 q^{73} -6.79581 q^{74} -0.621255 q^{75} -5.98503 q^{76} +2.58150 q^{77} +0.739650 q^{78} +10.2411 q^{79} +3.03338 q^{80} +8.80369 q^{81} +6.96861 q^{82} -12.1182 q^{83} -0.226011 q^{84} -15.4987 q^{85} +5.66100 q^{86} -0.648913 q^{87} +1.68897 q^{88} +9.49062 q^{89} -9.03381 q^{90} -7.64540 q^{91} -6.33776 q^{92} +0.493821 q^{93} -6.11889 q^{94} -18.1549 q^{95} -0.147869 q^{96} -2.09853 q^{97} -4.66384 q^{98} -5.02997 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.147869	−0.0853722	−0.0426861	−	0.999089i	$-0.513592\pi$
$3$	−0.147869	−0.0853722	−0.0426861	+	0.999089i	$0.513592\pi$
$4$	1.00000	0.500000
$5$	3.03338	1.35657	0.678284	−	0.734800i	$-0.262724\pi$
$5$	3.03338	1.35657	0.678284	+	0.734800i	$0.262724\pi$
$6$	−0.147869	−0.0603673
$7$	1.52845	0.577700	0.288850	−	0.957374i	$-0.406727\pi$
$7$	1.52845	0.577700	0.288850	+	0.957374i	$0.406727\pi$
$8$	1.00000	0.353553
$9$	−2.97813	−0.992712
$10$	3.03338	0.959239
$11$	1.68897	0.509243	0.254621	−	0.967041i	$-0.418049\pi$
$11$	1.68897	0.509243	0.254621	+	0.967041i	$0.418049\pi$
$12$	−0.147869	−0.0426861
$13$	−5.00206	−1.38732	−0.693661	−	0.720302i	$-0.744003\pi$
$13$	−5.00206	−1.38732	−0.693661	+	0.720302i	$0.744003\pi$
$14$	1.52845	0.408496
$15$	−0.448543	−0.115813
$16$	1.00000	0.250000
$17$	−5.10938	−1.23921	−0.619603	−	0.784915i	$-0.712707\pi$
$17$	−5.10938	−1.23921	−0.619603	+	0.784915i	$0.712707\pi$
$18$	−2.97813	−0.701953
$19$	−5.98503	−1.37306	−0.686530	−	0.727101i	$-0.740867\pi$
$19$	−5.98503	−1.37306	−0.686530	+	0.727101i	$0.740867\pi$
$20$	3.03338	0.678284
$21$	−0.226011	−0.0493196
$22$	1.68897	0.360089
$23$	−6.33776	−1.32151	−0.660757	−	0.750599i	$-0.729765\pi$
$23$	−6.33776	−1.32151	−0.660757	+	0.750599i	$0.729765\pi$
$24$	−0.147869	−0.0301836
$25$	4.20139	0.840277
$26$	−5.00206	−0.980984
$27$	0.883981	0.170122
$28$	1.52845	0.288850
$29$	4.38843	0.814911	0.407456	−	0.913225i	$-0.366416\pi$
$29$	4.38843	0.814911	0.407456	+	0.913225i	$0.366416\pi$
$30$	−0.448543	−0.0818923
$31$	−3.33959	−0.599807	−0.299904	−	0.953969i	$-0.596955\pi$
$31$	−3.33959	−0.599807	−0.299904	+	0.953969i	$0.596955\pi$
$32$	1.00000	0.176777
$33$	−0.249746	−0.0434752
$34$	−5.10938	−0.876251
$35$	4.63637	0.783690
$36$	−2.97813	−0.496356
$37$	−6.79581	−1.11723	−0.558613	−	0.829429i	$-0.688666\pi$
$37$	−6.79581	−1.11723	−0.558613	+	0.829429i	$0.688666\pi$
$38$	−5.98503	−0.970900
$39$	0.739650	0.118439
$40$	3.03338	0.479619
$41$	6.96861	1.08831	0.544157	−	0.838984i	$-0.316850\pi$
$41$	6.96861	1.08831	0.544157	+	0.838984i	$0.316850\pi$
$42$	−0.226011	−0.0348742
$43$	5.66100	0.863295	0.431647	−	0.902042i	$-0.357932\pi$
$43$	5.66100	0.863295	0.431647	+	0.902042i	$0.357932\pi$
$44$	1.68897	0.254621
$45$	−9.03381	−1.34668
$46$	−6.33776	−0.934452
$47$	−6.11889	−0.892532	−0.446266	−	0.894900i	$-0.647247\pi$
$47$	−6.11889	−0.892532	−0.446266	+	0.894900i	$0.647247\pi$
$48$	−0.147869	−0.0213431
$49$	−4.66384	−0.666262
$50$	4.20139	0.594166
$51$	0.755519	0.105794
$52$	−5.00206	−0.693661
$53$	1.39823	0.192062	0.0960310	−	0.995378i	$-0.469385\pi$
$53$	1.39823	0.192062	0.0960310	+	0.995378i	$0.469385\pi$
$54$	0.883981	0.120295
$55$	5.12328	0.690822
$56$	1.52845	0.204248
$57$	0.885001	0.117221
$58$	4.38843	0.576229
$59$	−9.64608	−1.25581	−0.627906	−	0.778289i	$-0.716088\pi$
$59$	−9.64608	−1.25581	−0.627906	+	0.778289i	$0.716088\pi$
$60$	−0.448543	−0.0579066
$61$	−7.15860	−0.916564	−0.458282	−	0.888807i	$-0.651535\pi$
$61$	−7.15860	−0.916564	−0.458282	+	0.888807i	$0.651535\pi$
$62$	−3.33959	−0.424128
$63$	−4.55193	−0.573490
$64$	1.00000	0.125000
$65$	−15.1731	−1.88200
$66$	−0.249746	−0.0307416
$67$	−1.61887	−0.197776	−0.0988880	−	0.995099i	$-0.531529\pi$
$67$	−1.61887	−0.197776	−0.0988880	+	0.995099i	$0.531529\pi$
$68$	−5.10938	−0.619603
$69$	0.937159	0.112821
$70$	4.63637	0.554152
$71$	3.73307	0.443034	0.221517	−	0.975157i	$-0.428899\pi$
$71$	3.73307	0.443034	0.221517	+	0.975157i	$0.428899\pi$
$72$	−2.97813	−0.350977
$73$	−10.8633	−1.27145	−0.635725	−	0.771916i	$-0.719299\pi$
$73$	−10.8633	−1.27145	−0.635725	+	0.771916i	$0.719299\pi$
$74$	−6.79581	−0.789997
$75$	−0.621255	−0.0717363
$76$	−5.98503	−0.686530
$77$	2.58150	0.294190
$78$	0.739650	0.0837488
$79$	10.2411	1.15222	0.576109	−	0.817373i	$-0.304570\pi$
$79$	10.2411	1.15222	0.576109	+	0.817373i	$0.304570\pi$
$80$	3.03338	0.339142
$81$	8.80369	0.978188
$82$	6.96861	0.769554
$83$	−12.1182	−1.33015	−0.665075	−	0.746777i	$-0.731600\pi$
$83$	−12.1182	−1.33015	−0.665075	+	0.746777i	$0.731600\pi$
$84$	−0.226011	−0.0246598
$85$	−15.4987	−1.68107
$86$	5.66100	0.610442
$87$	−0.648913	−0.0695708
$88$	1.68897	0.180044
$89$	9.49062	1.00600	0.503002	−	0.864285i	$-0.332229\pi$
$89$	9.49062	1.00600	0.503002	+	0.864285i	$0.332229\pi$
$90$	−9.03381	−0.952247
$91$	−7.64540	−0.801456
$92$	−6.33776	−0.660757
$93$	0.493821	0.0512069
$94$	−6.11889	−0.631116
$95$	−18.1549	−1.86265
$96$	−0.147869	−0.0150918
$97$	−2.09853	−0.213073	−0.106537	−	0.994309i	$-0.533976\pi$
$97$	−2.09853	−0.213073	−0.106537	+	0.994309i	$0.533976\pi$
$98$	−4.66384	−0.471119
$99$	−5.02997	−0.505531
$100$	4.20139	0.420139
$101$	10.0033	0.995365	0.497682	−	0.867359i	$-0.334185\pi$
$101$	10.0033	0.995365	0.497682	+	0.867359i	$0.334185\pi$
$102$	0.755519	0.0748075
$103$	−8.51838	−0.839341	−0.419670	−	0.907677i	$-0.637854\pi$
$103$	−8.51838	−0.839341	−0.419670	+	0.907677i	$0.637854\pi$
$104$	−5.00206	−0.490492
$105$	−0.685576	−0.0669054
$106$	1.39823	0.135808
$107$	7.42703	0.717998	0.358999	−	0.933338i	$-0.383118\pi$
$107$	7.42703	0.717998	0.358999	+	0.933338i	$0.383118\pi$
$108$	0.883981	0.0850611
$109$	1.76613	0.169164	0.0845822	−	0.996417i	$-0.473044\pi$
$109$	1.76613	0.169164	0.0845822	+	0.996417i	$0.473044\pi$
$110$	5.12328	0.488485
$111$	1.00489	0.0953800
$112$	1.52845	0.144425
$113$	1.05804	0.0995317	0.0497659	−	0.998761i	$-0.484152\pi$
$113$	1.05804	0.0995317	0.0497659	+	0.998761i	$0.484152\pi$
$114$	0.885001	0.0828879
$115$	−19.2248	−1.79272
$116$	4.38843	0.407456
$117$	14.8968	1.37721
$118$	−9.64608	−0.887994
$119$	−7.80944	−0.715890
$120$	−0.448543	−0.0409462
$121$	−8.14739	−0.740672
$122$	−7.15860	−0.648109
$123$	−1.03044	−0.0929118
$124$	−3.33959	−0.299904
$125$	−2.42250	−0.216675
$126$	−4.55193	−0.405519
$127$	−8.45190	−0.749984	−0.374992	−	0.927028i	$-0.622355\pi$
$127$	−8.45190	−0.749984	−0.374992	+	0.927028i	$0.622355\pi$
$128$	1.00000	0.0883883
$129$	−0.837087	−0.0737014
$130$	−15.1731	−1.33077
$131$	0.624251	0.0545411	0.0272705	−	0.999628i	$-0.491318\pi$
$131$	0.624251	0.0545411	0.0272705	+	0.999628i	$0.491318\pi$
$132$	−0.249746	−0.0217376
$133$	−9.14783	−0.793217
$134$	−1.61887	−0.139849
$135$	2.68145	0.230782
$136$	−5.10938	−0.438126
$137$	−1.41833	−0.121176	−0.0605880	−	0.998163i	$-0.519298\pi$
$137$	−1.41833	−0.121176	−0.0605880	+	0.998163i	$0.519298\pi$
$138$	0.937159	0.0797763
$139$	2.72561	0.231183	0.115592	−	0.993297i	$-0.463124\pi$
$139$	2.72561	0.231183	0.115592	+	0.993297i	$0.463124\pi$
$140$	4.63637	0.391845
$141$	0.904795	0.0761975
$142$	3.73307	0.313272
$143$	−8.44831	−0.706483
$144$	−2.97813	−0.248178
$145$	13.3118	1.10548
$146$	−10.8633	−0.899051
$147$	0.689637	0.0568803
$148$	−6.79581	−0.558613
$149$	4.62853	0.379184	0.189592	−	0.981863i	$-0.439284\pi$
$149$	4.62853	0.379184	0.189592	+	0.981863i	$0.439284\pi$
$150$	−0.621255	−0.0507252
$151$	9.27158	0.754511	0.377255	−	0.926109i	$-0.376868\pi$
$151$	9.27158	0.754511	0.377255	+	0.926109i	$0.376868\pi$
$152$	−5.98503	−0.485450
$153$	15.2164	1.23017
$154$	2.58150	0.208024
$155$	−10.1302	−0.813680
$156$	0.739650	0.0592194
$157$	−8.05630	−0.642962	−0.321481	−	0.946916i	$-0.604181\pi$
$157$	−8.05630	−0.642962	−0.321481	+	0.946916i	$0.604181\pi$
$158$	10.2411	0.814741
$159$	−0.206755	−0.0163968
$160$	3.03338	0.239810
$161$	−9.68696	−0.763440
$162$	8.80369	0.691683
$163$	0.800201	0.0626766	0.0313383	−	0.999509i	$-0.490023\pi$
$163$	0.800201	0.0626766	0.0313383	+	0.999509i	$0.490023\pi$
$164$	6.96861	0.544157
$165$	−0.757574	−0.0589770
$166$	−12.1182	−0.940557
$167$	1.23448	0.0955268	0.0477634	−	0.998859i	$-0.484791\pi$
$167$	1.23448	0.0955268	0.0477634	+	0.998859i	$0.484791\pi$
$168$	−0.226011	−0.0174371
$169$	12.0206	0.924660
$170$	−15.4987	−1.18869
$171$	17.8242	1.36305
$172$	5.66100	0.431647
$173$	−15.1897	−1.15485	−0.577425	−	0.816443i	$-0.695943\pi$
$173$	−15.1897	−1.15485	−0.577425	+	0.816443i	$0.695943\pi$
$174$	−0.648913	−0.0491940
$175$	6.42161	0.485428
$176$	1.68897	0.127311
$177$	1.42636	0.107212
$178$	9.49062	0.711352
$179$	15.3799	1.14954	0.574772	−	0.818313i	$-0.305091\pi$
$179$	15.3799	1.14954	0.574772	+	0.818313i	$0.305091\pi$
$180$	−9.03381	−0.673340
$181$	−6.74037	−0.501008	−0.250504	−	0.968116i	$-0.580596\pi$
$181$	−6.74037	−0.501008	−0.250504	+	0.968116i	$0.580596\pi$
$182$	−7.64540	−0.566715
$183$	1.05853	0.0782491
$184$	−6.33776	−0.467226
$185$	−20.6143	−1.51559
$186$	0.493821	0.0362087
$187$	−8.62957	−0.631057
$188$	−6.11889	−0.446266
$189$	1.35112	0.0982797
$190$	−18.1549	−1.31709
$191$	13.8382	1.00130	0.500649	−	0.865650i	$-0.333095\pi$
$191$	13.8382	1.00130	0.500649	+	0.865650i	$0.333095\pi$
$192$	−0.147869	−0.0106715
$193$	13.4609	0.968936	0.484468	−	0.874809i	$-0.339013\pi$
$193$	13.4609	0.968936	0.484468	+	0.874809i	$0.339013\pi$
$194$	−2.09853	−0.150665
$195$	2.24364	0.160670
$196$	−4.66384	−0.333131
$197$	3.18525	0.226940	0.113470	−	0.993541i	$-0.463803\pi$
$197$	3.18525	0.226940	0.113470	+	0.993541i	$0.463803\pi$
$198$	−5.02997	−0.357464
$199$	−2.33119	−0.165253	−0.0826267	−	0.996581i	$-0.526331\pi$
$199$	−2.33119	−0.165253	−0.0826267	+	0.996581i	$0.526331\pi$
$200$	4.20139	0.297083
$201$	0.239380	0.0168846
$202$	10.0033	0.703829
$203$	6.70750	0.470775
$204$	0.755519	0.0528969
$205$	21.1384	1.47637
$206$	−8.51838	−0.593503
$207$	18.8747	1.31188
$208$	−5.00206	−0.346830
$209$	−10.1085	−0.699221
$210$	−0.685576	−0.0473092
$211$	−0.379580	−0.0261313	−0.0130657	−	0.999915i	$-0.504159\pi$
$211$	−0.379580	−0.0261313	−0.0130657	+	0.999915i	$0.504159\pi$
$212$	1.39823	0.0960310
$213$	−0.552005	−0.0378228
$214$	7.42703	0.507701
$215$	17.1720	1.17112
$216$	0.883981	0.0601473
$217$	−5.10440	−0.346509
$218$	1.76613	0.119617
$219$	1.60634	0.108547
$220$	5.12328	0.345411
$221$	25.5574	1.71918
$222$	1.00489	0.0674438
$223$	−16.6576	−1.11548	−0.557738	−	0.830017i	$-0.688331\pi$
$223$	−16.6576	−1.11548	−0.557738	+	0.830017i	$0.688331\pi$
$224$	1.52845	0.102124
$225$	−12.5123	−0.834153
$226$	1.05804	0.0703796
$227$	−2.80593	−0.186236	−0.0931180	−	0.995655i	$-0.529683\pi$
$227$	−2.80593	−0.186236	−0.0931180	+	0.995655i	$0.529683\pi$
$228$	0.885001	0.0586106
$229$	26.1669	1.72916	0.864579	−	0.502498i	$-0.167585\pi$
$229$	26.1669	1.72916	0.864579	+	0.502498i	$0.167585\pi$
$230$	−19.2248	−1.26765
$231$	−0.381725	−0.0251156
$232$	4.38843	0.288115
$233$	13.7130	0.898370	0.449185	−	0.893439i	$-0.351714\pi$
$233$	13.7130	0.898370	0.449185	+	0.893439i	$0.351714\pi$
$234$	14.8968	0.973834
$235$	−18.5609	−1.21078
$236$	−9.64608	−0.627906
$237$	−1.51435	−0.0983674
$238$	−7.80944	−0.506211
$239$	1.26814	0.0820290	0.0410145	−	0.999159i	$-0.486941\pi$
$239$	1.26814	0.0820290	0.0410145	+	0.999159i	$0.486941\pi$
$240$	−0.448543	−0.0289533
$241$	9.21934	0.593870	0.296935	−	0.954898i	$-0.404036\pi$
$241$	9.21934	0.593870	0.296935	+	0.954898i	$0.404036\pi$
$242$	−8.14739	−0.523734
$243$	−3.95374	−0.253632
$244$	−7.15860	−0.458282
$245$	−14.1472	−0.903830
$246$	−1.03044	−0.0656985
$247$	29.9375	1.90488
$248$	−3.33959	−0.212064
$249$	1.79191	0.113558
$250$	−2.42250	−0.153212
$251$	16.1939	1.02215	0.511076	−	0.859535i	$-0.329247\pi$
$251$	16.1939	1.02215	0.511076	+	0.859535i	$0.329247\pi$
$252$	−4.55193	−0.286745
$253$	−10.7043	−0.672972
$254$	−8.45190	−0.530319
$255$	2.29178	0.143517
$256$	1.00000	0.0625000
$257$	−5.61661	−0.350354	−0.175177	−	0.984537i	$-0.556050\pi$
$257$	−5.61661	−0.350354	−0.175177	+	0.984537i	$0.556050\pi$
$258$	−0.837087	−0.0521148
$259$	−10.3871	−0.645421
$260$	−15.1731	−0.940998
$261$	−13.0693	−0.808972
$262$	0.624251	0.0385664
$263$	21.0817	1.29995	0.649976	−	0.759955i	$-0.274779\pi$
$263$	21.0817	1.29995	0.649976	+	0.759955i	$0.274779\pi$
$264$	−0.249746	−0.0153708
$265$	4.24137	0.260545
$266$	−9.14783	−0.560889
$267$	−1.40337	−0.0858848
$268$	−1.61887	−0.0988880
$269$	0.807967	0.0492626	0.0246313	−	0.999697i	$-0.492159\pi$
$269$	0.807967	0.0492626	0.0246313	+	0.999697i	$0.492159\pi$
$270$	2.68145	0.163188
$271$	4.12981	0.250868	0.125434	−	0.992102i	$-0.459968\pi$
$271$	4.12981	0.250868	0.125434	+	0.992102i	$0.459968\pi$
$272$	−5.10938	−0.309802
$273$	1.13052	0.0684221
$274$	−1.41833	−0.0856844
$275$	7.09600	0.427905
$276$	0.937159	0.0564103
$277$	8.53105	0.512581	0.256291	−	0.966600i	$-0.417500\pi$
$277$	8.53105	0.512581	0.256291	+	0.966600i	$0.417500\pi$
$278$	2.72561	0.163471
$279$	9.94574	0.595436
$280$	4.63637	0.277076
$281$	−23.6939	−1.41346	−0.706729	−	0.707484i	$-0.749830\pi$
$281$	−23.6939	−1.41346	−0.706729	+	0.707484i	$0.749830\pi$
$282$	0.904795	0.0538797
$283$	−9.57763	−0.569331	−0.284666	−	0.958627i	$-0.591883\pi$
$283$	−9.57763	−0.569331	−0.284666	+	0.958627i	$0.591883\pi$
$284$	3.73307	0.221517
$285$	2.68454	0.159019
$286$	−8.44831	−0.499559
$287$	10.6512	0.628719
$288$	−2.97813	−0.175488
$289$	9.10576	0.535633
$290$	13.3118	0.781694
$291$	0.310307	0.0181905
$292$	−10.8633	−0.635725
$293$	16.2454	0.949064	0.474532	−	0.880238i	$-0.342617\pi$
$293$	16.2454	0.949064	0.474532	+	0.880238i	$0.342617\pi$
$294$	0.689637	0.0402204
$295$	−29.2602	−1.70360
$296$	−6.79581	−0.394999
$297$	1.49301	0.0866335
$298$	4.62853	0.268123
$299$	31.7019	1.83337
$300$	−0.621255	−0.0358682
$301$	8.65257	0.498726
$302$	9.27158	0.533520
$303$	−1.47918	−0.0849765
$304$	−5.98503	−0.343265
$305$	−21.7147	−1.24338
$306$	15.2164	0.869865
$307$	−14.0419	−0.801416	−0.400708	−	0.916206i	$-0.631236\pi$
$307$	−14.0419	−0.801416	−0.400708	+	0.916206i	$0.631236\pi$
$308$	2.58150	0.147095
$309$	1.25960	0.0716564
$310$	−10.1302	−0.575358
$311$	16.2367	0.920700	0.460350	−	0.887737i	$-0.347724\pi$
$311$	16.2367	0.920700	0.460350	+	0.887737i	$0.347724\pi$
$312$	0.739650	0.0418744
$313$	−2.82674	−0.159777	−0.0798884	−	0.996804i	$-0.525456\pi$
$313$	−2.82674	−0.159777	−0.0798884	+	0.996804i	$0.525456\pi$
$314$	−8.05630	−0.454643
$315$	−13.8077	−0.777978
$316$	10.2411	0.576109
$317$	25.8605	1.45247	0.726236	−	0.687446i	$-0.241268\pi$
$317$	25.8605	1.45247	0.726236	+	0.687446i	$0.241268\pi$
$318$	−0.206755	−0.0115943
$319$	7.41191	0.414988
$320$	3.03338	0.169571
$321$	−1.09823	−0.0612971
$322$	−9.68696	−0.539833
$323$	30.5798	1.70151
$324$	8.80369	0.489094
$325$	−21.0156	−1.16573
$326$	0.800201	0.0443190
$327$	−0.261156	−0.0144419
$328$	6.96861	0.384777
$329$	−9.35243	−0.515616
$330$	−0.757574	−0.0417031
$331$	−15.0544	−0.827466	−0.413733	−	0.910398i	$-0.635775\pi$
$331$	−15.0544	−0.827466	−0.413733	+	0.910398i	$0.635775\pi$
$332$	−12.1182	−0.665075
$333$	20.2389	1.10908
$334$	1.23448	0.0675476
$335$	−4.91063	−0.268296
$336$	−0.226011	−0.0123299
$337$	−8.67759	−0.472699	−0.236349	−	0.971668i	$-0.575951\pi$
$337$	−8.67759	−0.472699	−0.236349	+	0.971668i	$0.575951\pi$
$338$	12.0206	0.653833
$339$	−0.156451	−0.00849725
$340$	−15.4987	−0.840534
$341$	−5.64045	−0.305448
$342$	17.8242	0.963824
$343$	−17.8276	−0.962600
$344$	5.66100	0.305221
$345$	2.84276	0.153049
$346$	−15.1897	−0.816603
$347$	5.40134	0.289959	0.144980	−	0.989435i	$-0.453688\pi$
$347$	5.40134	0.289959	0.144980	+	0.989435i	$0.453688\pi$
$348$	−0.648913	−0.0347854
$349$	9.64764	0.516426	0.258213	−	0.966088i	$-0.416866\pi$
$349$	9.64764	0.516426	0.258213	+	0.966088i	$0.416866\pi$
$350$	6.42161	0.343250
$351$	−4.42172	−0.236014
$352$	1.68897	0.0900222
$353$	4.80876	0.255945	0.127972	−	0.991778i	$-0.459153\pi$
$353$	4.80876	0.255945	0.127972	+	0.991778i	$0.459153\pi$
$354$	1.42636	0.0758100
$355$	11.3238	0.601005
$356$	9.49062	0.503002
$357$	1.15477	0.0611171
$358$	15.3799	0.812851
$359$	7.81401	0.412408	0.206204	−	0.978509i	$-0.433889\pi$
$359$	7.81401	0.412408	0.206204	+	0.978509i	$0.433889\pi$
$360$	−9.03381	−0.476124
$361$	16.8206	0.885294
$362$	−6.74037	−0.354266
$363$	1.20475	0.0632328
$364$	−7.64540	−0.400728
$365$	−32.9524	−1.72481
$366$	1.05853	0.0553305
$367$	28.2687	1.47562	0.737808	−	0.675010i	$-0.235861\pi$
$367$	28.2687	1.47562	0.737808	+	0.675010i	$0.235861\pi$
$368$	−6.33776	−0.330379
$369$	−20.7535	−1.08038
$370$	−20.6143	−1.07169
$371$	2.13713	0.110954
$372$	0.493821	0.0256035
$373$	−16.5536	−0.857113	−0.428556	−	0.903515i	$-0.640978\pi$
$373$	−16.5536	−0.857113	−0.428556	+	0.903515i	$0.640978\pi$
$374$	−8.62957	−0.446225
$375$	0.358213	0.0184980
$376$	−6.11889	−0.315558
$377$	−21.9512	−1.13054
$378$	1.35112	0.0694942
$379$	−20.9324	−1.07523	−0.537613	−	0.843192i	$-0.680674\pi$
$379$	−20.9324	−1.07523	−0.537613	+	0.843192i	$0.680674\pi$
$380$	−18.1549	−0.931325
$381$	1.24977	0.0640278
$382$	13.8382	0.708024
$383$	−28.0183	−1.43167	−0.715835	−	0.698270i	$-0.753954\pi$
$383$	−28.0183	−1.43167	−0.715835	+	0.698270i	$0.753954\pi$
$384$	−0.147869	−0.00754591
$385$	7.83068	0.399088
$386$	13.4609	0.685142
$387$	−16.8592	−0.857003
$388$	−2.09853	−0.106537
$389$	−4.15683	−0.210760	−0.105380	−	0.994432i	$-0.533606\pi$
$389$	−4.15683	−0.210760	−0.105380	+	0.994432i	$0.533606\pi$
$390$	2.24364	0.113611
$391$	32.3820	1.63763
$392$	−4.66384	−0.235559
$393$	−0.0923074	−0.00465629
$394$	3.18525	0.160471
$395$	31.0652	1.56306
$396$	−5.02997	−0.252766
$397$	−32.5354	−1.63291	−0.816453	−	0.577412i	$-0.804063\pi$
$397$	−32.5354	−1.63291	−0.816453	+	0.577412i	$0.804063\pi$
$398$	−2.33119	−0.116852
$399$	1.35268	0.0677188
$400$	4.20139	0.210069
$401$	−28.8647	−1.44143	−0.720717	−	0.693230i	$-0.756187\pi$
$401$	−28.8647	−1.44143	−0.720717	+	0.693230i	$0.756187\pi$
$402$	0.239380	0.0119392
$403$	16.7048	0.832126
$404$	10.0033	0.497682
$405$	26.7049	1.32698
$406$	6.70750	0.332888
$407$	−11.4779	−0.568939
$408$	0.755519	0.0374038
$409$	−15.2958	−0.756331	−0.378165	−	0.925738i	$-0.623445\pi$
$409$	−15.2958	−0.756331	−0.378165	+	0.925738i	$0.623445\pi$
$410$	21.1384	1.04395
$411$	0.209727	0.0103451
$412$	−8.51838	−0.419670
$413$	−14.7436	−0.725484
$414$	18.8747	0.927641
$415$	−36.7592	−1.80444
$416$	−5.00206	−0.245246
$417$	−0.403033	−0.0197366
$418$	−10.1085	−0.494424
$419$	3.42456	0.167301	0.0836505	−	0.996495i	$-0.473342\pi$
$419$	3.42456	0.167301	0.0836505	+	0.996495i	$0.473342\pi$
$420$	−0.685576	−0.0334527
$421$	8.46044	0.412336	0.206168	−	0.978517i	$-0.433901\pi$
$421$	8.46044	0.412336	0.206168	+	0.978517i	$0.433901\pi$
$422$	−0.379580	−0.0184777
$423$	18.2229	0.886027
$424$	1.39823	0.0679042
$425$	−21.4665	−1.04128
$426$	−0.552005	−0.0267447
$427$	−10.9416	−0.529500
$428$	7.42703	0.358999
$429$	1.24924	0.0603140
$430$	17.1720	0.828106
$431$	−30.2577	−1.45746	−0.728731	−	0.684800i	$-0.759890\pi$
$431$	−30.2577	−1.45746	−0.728731	+	0.684800i	$0.759890\pi$
$432$	0.883981	0.0425306
$433$	−23.7655	−1.14210	−0.571048	−	0.820917i	$-0.693463\pi$
$433$	−23.7655	−1.14210	−0.571048	+	0.820917i	$0.693463\pi$
$434$	−5.10440	−0.245019
$435$	−1.96840	−0.0943775
$436$	1.76613	0.0845822
$437$	37.9317	1.81452
$438$	1.60634	0.0767540
$439$	−34.6870	−1.65552	−0.827761	−	0.561081i	$-0.810386\pi$
$439$	−34.6870	−1.65552	−0.827761	+	0.561081i	$0.810386\pi$
$440$	5.12328	0.244243
$441$	13.8895	0.661406
$442$	25.5574	1.21564
$443$	21.4850	1.02078	0.510392	−	0.859942i	$-0.329500\pi$
$443$	21.4850	1.02078	0.510392	+	0.859942i	$0.329500\pi$
$444$	1.00489	0.0476900
$445$	28.7886	1.36471
$446$	−16.6576	−0.788761
$447$	−0.684416	−0.0323718
$448$	1.52845	0.0722126
$449$	−4.67975	−0.220851	−0.110425	−	0.993884i	$-0.535221\pi$
$449$	−4.67975	−0.220851	−0.110425	+	0.993884i	$0.535221\pi$
$450$	−12.5123	−0.589835
$451$	11.7697	0.554216
$452$	1.05804	0.0497659
$453$	−1.37098	−0.0644143
$454$	−2.80593	−0.131689
$455$	−23.1914	−1.08723
$456$	0.885001	0.0414440
$457$	33.3988	1.56233	0.781164	−	0.624326i	$-0.214626\pi$
$457$	33.3988	1.56233	0.781164	+	0.624326i	$0.214626\pi$
$458$	26.1669	1.22270
$459$	−4.51660	−0.210817
$460$	−19.2248	−0.896362
$461$	−9.61956	−0.448028	−0.224014	−	0.974586i	$-0.571916\pi$
$461$	−9.61956	−0.448028	−0.224014	+	0.974586i	$0.571916\pi$
$462$	−0.381725	−0.0177594
$463$	−15.4685	−0.718882	−0.359441	−	0.933168i	$-0.617033\pi$
$463$	−15.4685	−0.718882	−0.359441	+	0.933168i	$0.617033\pi$
$464$	4.38843	0.203728
$465$	1.49795	0.0694657
$466$	13.7130	0.635244
$467$	7.25229	0.335596	0.167798	−	0.985821i	$-0.446334\pi$
$467$	7.25229	0.335596	0.167798	+	0.985821i	$0.446334\pi$
$468$	14.8968	0.688605
$469$	−2.47436	−0.114255
$470$	−18.5609	−0.856151
$471$	1.19128	0.0548911
$472$	−9.64608	−0.443997
$473$	9.56124	0.439627
$474$	−1.51435	−0.0695563
$475$	−25.1454	−1.15375
$476$	−7.80944	−0.357945
$477$	−4.16412	−0.190662
$478$	1.26814	0.0580033
$479$	−19.6130	−0.896139	−0.448069	−	0.893999i	$-0.647888\pi$
$479$	−19.6130	−0.896139	−0.448069	+	0.893999i	$0.647888\pi$
$480$	−0.448543	−0.0204731
$481$	33.9931	1.54995
$482$	9.21934	0.419929
$483$	1.43240	0.0651766
$484$	−8.14739	−0.370336
$485$	−6.36562	−0.289048
$486$	−3.95374	−0.179345
$487$	−18.8027	−0.852032	−0.426016	−	0.904716i	$-0.640083\pi$
$487$	−18.8027	−0.852032	−0.426016	+	0.904716i	$0.640083\pi$
$488$	−7.15860	−0.324054
$489$	−0.118325	−0.00535084
$490$	−14.1472	−0.639104
$491$	−3.07141	−0.138611	−0.0693053	−	0.997595i	$-0.522078\pi$
$491$	−3.07141	−0.138611	−0.0693053	+	0.997595i	$0.522078\pi$
$492$	−1.03044	−0.0464559
$493$	−22.4222	−1.00984
$494$	29.9375	1.34695
$495$	−15.2578	−0.685787
$496$	−3.33959	−0.149952
$497$	5.70582	0.255941
$498$	1.79191	0.0802975
$499$	3.41472	0.152864	0.0764319	−	0.997075i	$-0.475647\pi$
$499$	3.41472	0.152864	0.0764319	+	0.997075i	$0.475647\pi$
$500$	−2.42250	−0.108338
$501$	−0.182541	−0.00815533
$502$	16.1939	0.722771
$503$	−30.5782	−1.36341	−0.681706	−	0.731626i	$-0.738762\pi$
$503$	−30.5782	−1.36341	−0.681706	+	0.731626i	$0.738762\pi$
$504$	−4.55193	−0.202759
$505$	30.3438	1.35028
$506$	−10.7043	−0.475863
$507$	−1.77747	−0.0789403
$508$	−8.45190	−0.374992
$509$	−6.16955	−0.273461	−0.136730	−	0.990608i	$-0.543659\pi$
$509$	−6.16955	−0.273461	−0.136730	+	0.990608i	$0.543659\pi$
$510$	2.29178	0.101482
$511$	−16.6040	−0.734517
$512$	1.00000	0.0441942
$513$	−5.29065	−0.233588
$514$	−5.61661	−0.247738
$515$	−25.8395	−1.13862
$516$	−0.837087	−0.0368507
$517$	−10.3346	−0.454515
$518$	−10.3871	−0.456382
$519$	2.24609	0.0985922
$520$	−15.1731	−0.665386
$521$	−20.4688	−0.896755	−0.448378	−	0.893844i	$-0.647998\pi$
$521$	−20.4688	−0.896755	−0.448378	+	0.893844i	$0.647998\pi$
$522$	−13.0693	−0.572029
$523$	−29.3226	−1.28219	−0.641094	−	0.767463i	$-0.721519\pi$
$523$	−29.3226	−1.28219	−0.641094	+	0.767463i	$0.721519\pi$
$524$	0.624251	0.0272705
$525$	−0.949558	−0.0414421
$526$	21.0817	0.919205
$527$	17.0632	0.743285
$528$	−0.249746	−0.0108688
$529$	17.1672	0.746402
$530$	4.24137	0.184233
$531$	28.7273	1.24666
$532$	−9.14783	−0.396609
$533$	−34.8574	−1.50984
$534$	−1.40337	−0.0607297
$535$	22.5290	0.974013
$536$	−1.61887	−0.0699243
$537$	−2.27420	−0.0981392
$538$	0.807967	0.0348339
$539$	−7.87706	−0.339289
$540$	2.68145	0.115391
$541$	−0.227063	−0.00976220	−0.00488110	−	0.999988i	$-0.501554\pi$
$541$	−0.227063	−0.00976220	−0.00488110	+	0.999988i	$0.501554\pi$
$542$	4.12981	0.177391
$543$	0.996692	0.0427721
$544$	−5.10938	−0.219063
$545$	5.35734	0.229483
$546$	1.13052	0.0483817
$547$	−29.6671	−1.26847	−0.634237	−	0.773139i	$-0.718686\pi$
$547$	−29.6671	−1.26847	−0.634237	+	0.773139i	$0.718686\pi$
$548$	−1.41833	−0.0605880
$549$	21.3193	0.909884
$550$	7.09600	0.302574
$551$	−26.2649	−1.11892
$552$	0.937159	0.0398881
$553$	15.6531	0.665637
$554$	8.53105	0.362450
$555$	3.04821	0.129389
$556$	2.72561	0.115592
$557$	−16.8169	−0.712556	−0.356278	−	0.934380i	$-0.615954\pi$
$557$	−16.8169	−0.712556	−0.356278	+	0.934380i	$0.615954\pi$
$558$	9.94574	0.421037
$559$	−28.3167	−1.19767
$560$	4.63637	0.195922
$561$	1.27605	0.0538747
$562$	−23.6939	−0.999466
$563$	−1.02718	−0.0432904	−0.0216452	−	0.999766i	$-0.506890\pi$
$563$	−1.02718	−0.0432904	−0.0216452	+	0.999766i	$0.506890\pi$
$564$	0.904795	0.0380987
$565$	3.20943	0.135022
$566$	−9.57763	−0.402578
$567$	13.4560	0.565100
$568$	3.73307	0.156636
$569$	18.3874	0.770840	0.385420	−	0.922741i	$-0.374057\pi$
$569$	18.3874	0.770840	0.385420	+	0.922741i	$0.374057\pi$
$570$	2.68454	0.112443
$571$	20.1941	0.845098	0.422549	−	0.906340i	$-0.361135\pi$
$571$	20.1941	0.845098	0.422549	+	0.906340i	$0.361135\pi$
$572$	−8.44831	−0.353242
$573$	−2.04624	−0.0854830
$574$	10.6512	0.444572
$575$	−26.6274	−1.11044
$576$	−2.97813	−0.124089
$577$	17.0197	0.708540	0.354270	−	0.935143i	$-0.384729\pi$
$577$	17.0197	0.708540	0.354270	+	0.935143i	$0.384729\pi$
$578$	9.10576	0.378750
$579$	−1.99045	−0.0827203
$580$	13.3118	0.552741
$581$	−18.5221	−0.768428
$582$	0.310307	0.0128626
$583$	2.36157	0.0978061
$584$	−10.8633	−0.449525
$585$	45.1876	1.86828
$586$	16.2454	0.671090
$587$	7.26573	0.299889	0.149944	−	0.988694i	$-0.452091\pi$
$587$	7.26573	0.299889	0.149944	+	0.988694i	$0.452091\pi$
$588$	0.689637	0.0284401
$589$	19.9875	0.823572
$590$	−29.2602	−1.20462
$591$	−0.471000	−0.0193743
$592$	−6.79581	−0.279306
$593$	−9.19197	−0.377469	−0.188735	−	0.982028i	$-0.560439\pi$
$593$	−9.19197	−0.377469	−0.188735	+	0.982028i	$0.560439\pi$
$594$	1.49301	0.0612591
$595$	−23.6890	−0.971154
$596$	4.62853	0.189592
$597$	0.344710	0.0141081
$598$	31.7019	1.29639
$599$	47.1001	1.92446	0.962229	−	0.272243i	$-0.0877653\pi$
$599$	47.1001	1.92446	0.962229	+	0.272243i	$0.0877653\pi$
$600$	−0.621255	−0.0253626
$601$	37.6612	1.53623	0.768116	−	0.640311i	$-0.221195\pi$
$601$	37.6612	1.53623	0.768116	+	0.640311i	$0.221195\pi$
$602$	8.65257	0.352652
$603$	4.82120	0.196334
$604$	9.27158	0.377255
$605$	−24.7141	−1.00477
$606$	−1.47918	−0.0600875
$607$	27.3911	1.11177	0.555884	−	0.831260i	$-0.312380\pi$
$607$	27.3911	1.11177	0.555884	+	0.831260i	$0.312380\pi$
$608$	−5.98503	−0.242725
$609$	−0.991832	−0.0401911
$610$	−21.7147	−0.879204
$611$	30.6071	1.23823
$612$	15.2164	0.615087
$613$	8.57158	0.346203	0.173101	−	0.984904i	$-0.444621\pi$
$613$	8.57158	0.346203	0.173101	+	0.984904i	$0.444621\pi$
$614$	−14.0419	−0.566687
$615$	−3.12572	−0.126041
$616$	2.58150	0.104012
$617$	−34.5169	−1.38960	−0.694799	−	0.719204i	$-0.744507\pi$
$617$	−34.5169	−1.38960	−0.694799	+	0.719204i	$0.744507\pi$
$618$	1.25960	0.0506687
$619$	6.53061	0.262487	0.131244	−	0.991350i	$-0.458103\pi$
$619$	6.53061	0.262487	0.131244	+	0.991350i	$0.458103\pi$
$620$	−10.1302	−0.406840
$621$	−5.60246	−0.224819
$622$	16.2367	0.651033
$623$	14.5060	0.581169
$624$	0.739650	0.0296097
$625$	−28.3553	−1.13421
$626$	−2.82674	−0.112979
$627$	1.49474	0.0596940
$628$	−8.05630	−0.321481
$629$	34.7224	1.38447
$630$	−13.8077	−0.550114
$631$	18.0857	0.719982	0.359991	−	0.932956i	$-0.382780\pi$
$631$	18.0857	0.719982	0.359991	+	0.932956i	$0.382780\pi$
$632$	10.2411	0.407370
$633$	0.0561281	0.00223089
$634$	25.8605	1.02705
$635$	−25.6378	−1.01740
$636$	−0.206755	−0.00819838
$637$	23.3288	0.924320
$638$	7.41191	0.293440
$639$	−11.1176	−0.439805
$640$	3.03338	0.119905
$641$	44.7350	1.76693	0.883464	−	0.468500i	$-0.155205\pi$
$641$	44.7350	1.76693	0.883464	+	0.468500i	$0.155205\pi$
$642$	−1.09823	−0.0433436
$643$	−48.5385	−1.91417	−0.957085	−	0.289807i	$-0.906409\pi$
$643$	−48.5385	−1.91417	−0.957085	+	0.289807i	$0.906409\pi$
$644$	−9.68696	−0.381720
$645$	−2.53920	−0.0999810
$646$	30.5798	1.20315
$647$	22.1434	0.870547	0.435273	−	0.900298i	$-0.356652\pi$
$647$	22.1434	0.870547	0.435273	+	0.900298i	$0.356652\pi$
$648$	8.80369	0.345842
$649$	−16.2919	−0.639513
$650$	−21.0156	−0.824299
$651$	0.754782	0.0295822
$652$	0.800201	0.0313383
$653$	11.1423	0.436033	0.218017	−	0.975945i	$-0.430041\pi$
$653$	11.1423	0.436033	0.218017	+	0.975945i	$0.430041\pi$
$654$	−0.261156	−0.0102120
$655$	1.89359	0.0739887
$656$	6.96861	0.272078
$657$	32.3523	1.26218
$658$	−9.35243	−0.364596
$659$	41.1353	1.60240	0.801201	−	0.598395i	$-0.204195\pi$
$659$	41.1353	1.60240	0.801201	+	0.598395i	$0.204195\pi$
$660$	−0.757574	−0.0294885
$661$	−2.04701	−0.0796195	−0.0398098	−	0.999207i	$-0.512675\pi$
$661$	−2.04701	−0.0796195	−0.0398098	+	0.999207i	$0.512675\pi$
$662$	−15.0544	−0.585106
$663$	−3.77915	−0.146770
$664$	−12.1182	−0.470279
$665$	−27.7488	−1.07605
$666$	20.2389	0.784240
$667$	−27.8128	−1.07692
$668$	1.23448	0.0477634
$669$	2.46315	0.0952307
$670$	−4.91063	−0.189714
$671$	−12.0906	−0.466754
$672$	−0.226011	−0.00871855
$673$	−27.2595	−1.05078	−0.525388	−	0.850863i	$-0.676080\pi$
$673$	−27.2595	−1.05078	−0.525388	+	0.850863i	$0.676080\pi$
$674$	−8.67759	−0.334248
$675$	3.71395	0.142950
$676$	12.0206	0.462330
$677$	41.2211	1.58426	0.792128	−	0.610355i	$-0.208973\pi$
$677$	41.2211	1.58426	0.792128	+	0.610355i	$0.208973\pi$
$678$	−0.156451	−0.00600846
$679$	−3.20750	−0.123092
$680$	−15.4987	−0.594347
$681$	0.414910	0.0158994
$682$	−5.64045	−0.215984
$683$	17.3757	0.664863	0.332431	−	0.943127i	$-0.392131\pi$
$683$	17.3757	0.664863	0.332431	+	0.943127i	$0.392131\pi$
$684$	17.8242	0.681526
$685$	−4.30233	−0.164384
$686$	−17.8276	−0.680661
$687$	−3.86927	−0.147622
$688$	5.66100	0.215824
$689$	−6.99404	−0.266452
$690$	2.84276	0.108222
$691$	10.2344	0.389334	0.194667	−	0.980869i	$-0.437637\pi$
$691$	10.2344	0.389334	0.194667	+	0.980869i	$0.437637\pi$
$692$	−15.1897	−0.577425
$693$	−7.68807	−0.292045
$694$	5.40134	0.205032
$695$	8.26780	0.313616
$696$	−0.648913	−0.0245970
$697$	−35.6053	−1.34865
$698$	9.64764	0.365168
$699$	−2.02773	−0.0766959
$700$	6.42161	0.242714
$701$	−38.8472	−1.46724	−0.733620	−	0.679560i	$-0.762171\pi$
$701$	−38.8472	−1.46724	−0.733620	+	0.679560i	$0.762171\pi$
$702$	−4.42172	−0.166887
$703$	40.6732	1.53402
$704$	1.68897	0.0636553
$705$	2.74459	0.103367
$706$	4.80876	0.180980
$707$	15.2895	0.575023
$708$	1.42636	0.0536058
$709$	−34.1430	−1.28227	−0.641134	−	0.767429i	$-0.721536\pi$
$709$	−34.1430	−1.28227	−0.641134	+	0.767429i	$0.721536\pi$
$710$	11.3238	0.424975
$711$	−30.4995	−1.14382
$712$	9.49062	0.355676
$713$	21.1655	0.792654
$714$	1.15477	0.0432163
$715$	−25.6269	−0.958392
$716$	15.3799	0.574772
$717$	−0.187518	−0.00700300
$718$	7.81401	0.291616
$719$	6.46607	0.241144	0.120572	−	0.992705i	$-0.461527\pi$
$719$	6.46607	0.241144	0.120572	+	0.992705i	$0.461527\pi$
$720$	−9.03381	−0.336670
$721$	−13.0199	−0.484887
$722$	16.8206	0.625998
$723$	−1.36325	−0.0507000
$724$	−6.74037	−0.250504
$725$	18.4375	0.684751
$726$	1.20475	0.0447124
$727$	35.7727	1.32674	0.663368	−	0.748294i	$-0.269127\pi$
$727$	35.7727	1.32674	0.663368	+	0.748294i	$0.269127\pi$
$728$	−7.64540	−0.283357
$729$	−25.8264	−0.956535
$730$	−32.9524	−1.21962
$731$	−28.9242	−1.06980
$732$	1.05853	0.0391246
$733$	−14.4914	−0.535251	−0.267625	−	0.963523i	$-0.586239\pi$
$733$	−14.4914	−0.535251	−0.267625	+	0.963523i	$0.586239\pi$
$734$	28.2687	1.04342
$735$	2.09193	0.0771620
$736$	−6.33776	−0.233613
$737$	−2.73421	−0.100716
$738$	−20.7535	−0.763945
$739$	−21.7969	−0.801811	−0.400906	−	0.916119i	$-0.631304\pi$
$739$	−21.7969	−0.801811	−0.400906	+	0.916119i	$0.631304\pi$
$740$	−20.6143	−0.757796
$741$	−4.42683	−0.162623
$742$	2.13713	0.0784565
$743$	−10.4369	−0.382894	−0.191447	−	0.981503i	$-0.561318\pi$
$743$	−10.4369	−0.382894	−0.191447	+	0.981503i	$0.561318\pi$
$744$	0.493821	0.0181044
$745$	14.0401	0.514389
$746$	−16.5536	−0.606070
$747$	36.0897	1.32045
$748$	−8.62957	−0.315528
$749$	11.3519	0.414788
$750$	0.358213	0.0130801
$751$	−18.7364	−0.683701	−0.341850	−	0.939754i	$-0.611054\pi$
$751$	−18.7364	−0.683701	−0.341850	+	0.939754i	$0.611054\pi$
$752$	−6.11889	−0.223133
$753$	−2.39458	−0.0872635
$754$	−21.9512	−0.799415
$755$	28.1242	1.02355
$756$	1.35112	0.0491398
$757$	21.0788	0.766123	0.383062	−	0.923723i	$-0.374870\pi$
$757$	21.0788	0.766123	0.383062	+	0.923723i	$0.374870\pi$
$758$	−20.9324	−0.760299
$759$	1.58283	0.0574531
$760$	−18.1549	−0.658546
$761$	−28.4475	−1.03122	−0.515611	−	0.856823i	$-0.672435\pi$
$761$	−28.4475	−1.03122	−0.515611	+	0.856823i	$0.672435\pi$
$762$	1.24977	0.0452745
$763$	2.69944	0.0977264
$764$	13.8382	0.500649
$765$	46.1572	1.66882
$766$	−28.0183	−1.01234
$767$	48.2503	1.74222
$768$	−0.147869	−0.00533576
$769$	−43.1949	−1.55765	−0.778823	−	0.627243i	$-0.784183\pi$
$769$	−43.1949	−1.55765	−0.778823	+	0.627243i	$0.784183\pi$
$770$	7.83068	0.282198
$771$	0.830522	0.0299105
$772$	13.4609	0.484468
$773$	−47.2274	−1.69865	−0.849326	−	0.527869i	$-0.822991\pi$
$773$	−47.2274	−1.69865	−0.849326	+	0.527869i	$0.822991\pi$
$774$	−16.8592	−0.605992
$775$	−14.0309	−0.504004
$776$	−2.09853	−0.0753327
$777$	1.53593	0.0551011
$778$	−4.15683	−0.149030
$779$	−41.7073	−1.49432
$780$	2.24364	0.0803351
$781$	6.30503	0.225612
$782$	32.3820	1.15798
$783$	3.87929	0.138635
$784$	−4.66384	−0.166566
$785$	−24.4378	−0.872222
$786$	−0.0923074	−0.00329250
$787$	−8.85534	−0.315659	−0.157829	−	0.987466i	$-0.550450\pi$
$787$	−8.85534	−0.315659	−0.157829	+	0.987466i	$0.550450\pi$
$788$	3.18525	0.113470
$789$	−3.11733	−0.110980
$790$	31.0652	1.10525
$791$	1.61716	0.0574995
$792$	−5.02997	−0.178732
$793$	35.8077	1.27157
$794$	−32.5354	−1.15464
$795$	−0.627167	−0.0222433
$796$	−2.33119	−0.0826267
$797$	−1.06590	−0.0377559	−0.0188780	−	0.999822i	$-0.506009\pi$
$797$	−1.06590	−0.0377559	−0.0188780	+	0.999822i	$0.506009\pi$
$798$	1.35268	0.0478844
$799$	31.2637	1.10603
$800$	4.20139	0.148541
$801$	−28.2643	−0.998671
$802$	−28.8647	−1.01925
$803$	−18.3477	−0.647476
$804$	0.239380	0.00844229
$805$	−29.3842	−1.03566
$806$	16.7048	0.588402
$807$	−0.119473	−0.00420566
$808$	10.0033	0.351915
$809$	18.2352	0.641115	0.320558	−	0.947229i	$-0.396130\pi$
$809$	18.2352	0.641115	0.320558	+	0.947229i	$0.396130\pi$
$810$	26.7049	0.938315
$811$	−7.91392	−0.277895	−0.138948	−	0.990300i	$-0.544372\pi$
$811$	−7.91392	−0.277895	−0.138948	+	0.990300i	$0.544372\pi$
$812$	6.70750	0.235387
$813$	−0.610671	−0.0214172
$814$	−11.4779	−0.402300
$815$	2.42731	0.0850250
$816$	0.755519	0.0264485
$817$	−33.8813	−1.18536
$818$	−15.2958	−0.534807
$819$	22.7690	0.795615
$820$	21.1384	0.738186
$821$	6.72205	0.234601	0.117301	−	0.993096i	$-0.462576\pi$
$821$	6.72205	0.234601	0.117301	+	0.993096i	$0.462576\pi$
$822$	0.209727	0.00731507
$823$	0.772747	0.0269363	0.0134681	−	0.999909i	$-0.495713\pi$
$823$	0.772747	0.0269363	0.0134681	+	0.999909i	$0.495713\pi$
$824$	−8.51838	−0.296752
$825$	−1.04928	−0.0365312
$826$	−14.7436	−0.512994
$827$	28.5459	0.992637	0.496318	−	0.868141i	$-0.334685\pi$
$827$	28.5459	0.992637	0.496318	+	0.868141i	$0.334685\pi$
$828$	18.8747	0.655942
$829$	35.2376	1.22385	0.611926	−	0.790915i	$-0.290395\pi$
$829$	35.2376	1.22385	0.611926	+	0.790915i	$0.290395\pi$
$830$	−36.7592	−1.27593
$831$	−1.26148	−0.0437602
$832$	−5.00206	−0.173415
$833$	23.8293	0.825637
$834$	−0.403033	−0.0139559
$835$	3.74464	0.129589
$836$	−10.1085	−0.349610
$837$	−2.95213	−0.102041
$838$	3.42456	0.118300
$839$	−44.0237	−1.51987	−0.759933	−	0.650002i	$-0.774768\pi$
$839$	−44.0237	−1.51987	−0.759933	+	0.650002i	$0.774768\pi$
$840$	−0.685576	−0.0236546
$841$	−9.74167	−0.335920
$842$	8.46044	0.291566
$843$	3.50359	0.120670
$844$	−0.379580	−0.0130657
$845$	36.4630	1.25436
$846$	18.2229	0.626516
$847$	−12.4529	−0.427886
$848$	1.39823	0.0480155
$849$	1.41624	0.0486051
$850$	−21.4665	−0.736294
$851$	43.0703	1.47643
$852$	−0.552005	−0.0189114
$853$	1.07146	0.0366861	0.0183430	−	0.999832i	$-0.494161\pi$
$853$	1.07146	0.0366861	0.0183430	+	0.999832i	$0.494161\pi$
$854$	−10.9416	−0.374413
$855$	54.0676	1.84907
$856$	7.42703	0.253851
$857$	22.5612	0.770676	0.385338	−	0.922775i	$-0.374085\pi$
$857$	22.5612	0.770676	0.385338	+	0.922775i	$0.374085\pi$
$858$	1.24924	0.0426485
$859$	30.3902	1.03690	0.518450	−	0.855108i	$-0.326509\pi$
$859$	30.3902	1.03690	0.518450	+	0.855108i	$0.326509\pi$
$860$	17.1720	0.585559
$861$	−1.57498	−0.0536752
$862$	−30.2577	−1.03058
$863$	−12.1716	−0.414326	−0.207163	−	0.978306i	$-0.566423\pi$
$863$	−12.1716	−0.414326	−0.207163	+	0.978306i	$0.566423\pi$
$864$	0.883981	0.0300736
$865$	−46.0761	−1.56663
$866$	−23.7655	−0.807583
$867$	−1.34646	−0.0457282
$868$	−5.10440	−0.173255
$869$	17.2969	0.586758
$870$	−1.96840	−0.0667350
$871$	8.09766	0.274379
$872$	1.76613	0.0598087
$873$	6.24969	0.211520
$874$	37.9317	1.28306
$875$	−3.70267	−0.125173
$876$	1.60634	0.0542733
$877$	−48.0641	−1.62301	−0.811505	−	0.584345i	$-0.801351\pi$
$877$	−48.0641	−1.62301	−0.811505	+	0.584345i	$0.801351\pi$
$878$	−34.6870	−1.17063
$879$	−2.40219	−0.0810237
$880$	5.12328	0.172706
$881$	−40.8597	−1.37660	−0.688299	−	0.725427i	$-0.741642\pi$
$881$	−40.8597	−1.37660	−0.688299	+	0.725427i	$0.741642\pi$
$882$	13.8895	0.467685
$883$	1.06944	0.0359894	0.0179947	−	0.999838i	$-0.494272\pi$
$883$	1.06944	0.0359894	0.0179947	+	0.999838i	$0.494272\pi$
$884$	25.5574	0.859589
$885$	4.32668	0.145440
$886$	21.4850	0.721804
$887$	18.2230	0.611867	0.305934	−	0.952053i	$-0.401031\pi$
$887$	18.2230	0.611867	0.305934	+	0.952053i	$0.401031\pi$
$888$	1.00489	0.0337219
$889$	−12.9183	−0.433266
$890$	28.7886	0.964997
$891$	14.8691	0.498135
$892$	−16.6576	−0.557738
$893$	36.6218	1.22550
$894$	−0.684416	−0.0228903
$895$	46.6529	1.55944
$896$	1.52845	0.0510620
$897$	−4.68772	−0.156519
$898$	−4.67975	−0.156165
$899$	−14.6555	−0.488790
$900$	−12.5123	−0.417076
$901$	−7.14410	−0.238004
$902$	11.7697	0.391890
$903$	−1.27945	−0.0425773
$904$	1.05804	0.0351898
$905$	−20.4461	−0.679651
$906$	−1.37098	−0.0455478
$907$	11.0637	0.367363	0.183681	−	0.982986i	$-0.441199\pi$
$907$	11.0637	0.367363	0.183681	+	0.982986i	$0.441199\pi$
$908$	−2.80593	−0.0931180
$909$	−29.7912	−0.988110
$910$	−23.1914	−0.768788
$911$	11.7477	0.389218	0.194609	−	0.980881i	$-0.437656\pi$
$911$	11.7477	0.389218	0.194609	+	0.980881i	$0.437656\pi$
$912$	0.885001	0.0293053
$913$	−20.4673	−0.677369
$914$	33.3988	1.10473
$915$	3.21094	0.106150
$916$	26.1669	0.864579
$917$	0.954138	0.0315084
$918$	−4.51660	−0.149070
$919$	44.6166	1.47177	0.735884	−	0.677108i	$-0.236767\pi$
$919$	44.6166	1.47177	0.735884	+	0.677108i	$0.236767\pi$
$920$	−19.2248	−0.633824
$921$	2.07637	0.0684187
$922$	−9.61956	−0.316803
$923$	−18.6730	−0.614630
$924$	−0.381725	−0.0125578
$925$	−28.5518	−0.938779
$926$	−15.4685	−0.508326
$927$	25.3689	0.833223
$928$	4.38843	0.144057
$929$	−37.8601	−1.24215	−0.621074	−	0.783752i	$-0.713303\pi$
$929$	−37.8601	−1.24215	−0.621074	+	0.783752i	$0.713303\pi$
$930$	1.49795	0.0491196
$931$	27.9132	0.914818
$932$	13.7130	0.449185
$933$	−2.40091	−0.0786022
$934$	7.25229	0.237302
$935$	−26.1768	−0.856072
$936$	14.8968	0.486917
$937$	46.7158	1.52614	0.763070	−	0.646316i	$-0.223691\pi$
$937$	46.7158	1.52614	0.763070	+	0.646316i	$0.223691\pi$
$938$	−2.47436	−0.0807906
$939$	0.417988	0.0136405
$940$	−18.5609	−0.605390
$941$	38.0183	1.23936	0.619680	−	0.784854i	$-0.287262\pi$
$941$	38.0183	1.23936	0.619680	+	0.784854i	$0.287262\pi$
$942$	1.19128	0.0388139
$943$	−44.1654	−1.43822
$944$	−9.64608	−0.313953
$945$	4.09847	0.133323
$946$	9.56124	0.310863
$947$	25.4462	0.826890	0.413445	−	0.910529i	$-0.364325\pi$
$947$	25.4462	0.826890	0.413445	+	0.910529i	$0.364325\pi$
$948$	−1.51435	−0.0491837
$949$	54.3387	1.76391
$950$	−25.1454	−0.815825
$951$	−3.82397	−0.124001
$952$	−7.80944	−0.253105
$953$	−28.8205	−0.933588	−0.466794	−	0.884366i	$-0.654591\pi$
$953$	−28.8205	−0.933588	−0.466794	+	0.884366i	$0.654591\pi$
$954$	−4.16412	−0.134818
$955$	41.9765	1.35833
$956$	1.26814	0.0410145
$957$	−1.09599	−0.0354284
$958$	−19.6130	−0.633666
$959$	−2.16785	−0.0700034
$960$	−0.448543	−0.0144767
$961$	−19.8472	−0.640231
$962$	33.9931	1.09598
$963$	−22.1187	−0.712765
$964$	9.21934	0.296935
$965$	40.8320	1.31443
$966$	1.43240	0.0460868
$967$	−40.6246	−1.30640	−0.653199	−	0.757187i	$-0.726573\pi$
$967$	−40.6246	−1.30640	−0.653199	+	0.757187i	$0.726573\pi$
$968$	−8.14739	−0.261867
$969$	−4.52181	−0.145261
$970$	−6.36562	−0.204388
$971$	38.4231	1.23306	0.616528	−	0.787333i	$-0.288539\pi$
$971$	38.4231	1.23306	0.616528	+	0.787333i	$0.288539\pi$
$972$	−3.95374	−0.126816
$973$	4.16596	0.133555
$974$	−18.8027	−0.602478
$975$	3.10755	0.0995213
$976$	−7.15860	−0.229141
$977$	−25.4492	−0.814191	−0.407095	−	0.913386i	$-0.633458\pi$
$977$	−25.4492	−0.814191	−0.407095	+	0.913386i	$0.633458\pi$
$978$	−0.118325	−0.00378361
$979$	16.0293	0.512300
$980$	−14.1472	−0.451915
$981$	−5.25977	−0.167932
$982$	−3.07141	−0.0980125
$983$	20.2639	0.646320	0.323160	−	0.946344i	$-0.395255\pi$
$983$	20.2639	0.646320	0.323160	+	0.946344i	$0.395255\pi$
$984$	−1.03044	−0.0328493
$985$	9.66207	0.307859
$986$	−22.4222	−0.714067
$987$	1.38294	0.0440193
$988$	29.9375	0.952438
$989$	−35.8781	−1.14086
$990$	−15.2578	−0.484925
$991$	−3.16385	−0.100503	−0.0502515	−	0.998737i	$-0.516002\pi$
$991$	−3.16385	−0.100503	−0.0502515	+	0.998737i	$0.516002\pi$
$992$	−3.33959	−0.106032
$993$	2.22608	0.0706426
$994$	5.70582	0.180977
$995$	−7.07137	−0.224178
$996$	1.79191	0.0567789
$997$	−49.9682	−1.58251	−0.791254	−	0.611488i	$-0.790571\pi$
$997$	−49.9682	−1.58251	−0.791254	+	0.611488i	$0.790571\pi$
$998$	3.41472	0.108091
$999$	−6.00737	−0.190065

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.d.1.38	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.d.1.38	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.147869 q^{3} +1.00000 q^{4} +3.03338 q^{5} -0.147869 q^{6} +1.52845 q^{7} +1.00000 q^{8} -2.97813 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.147869 q^{3} +1.00000 q^{4} +3.03338 q^{5} -0.147869 q^{6} +1.52845 q^{7} +1.00000 q^{8} -2.97813 q^{9} +3.03338 q^{10} +1.68897 q^{11} -0.147869 q^{12} -5.00206 q^{13} +1.52845 q^{14} -0.448543 q^{15} +1.00000 q^{16} -5.10938 q^{17} -2.97813 q^{18} -5.98503 q^{19} +3.03338 q^{20} -0.226011 q^{21} +1.68897 q^{22} -6.33776 q^{23} -0.147869 q^{24} +4.20139 q^{25} -5.00206 q^{26} +0.883981 q^{27} +1.52845 q^{28} +4.38843 q^{29} -0.448543 q^{30} -3.33959 q^{31} +1.00000 q^{32} -0.249746 q^{33} -5.10938 q^{34} +4.63637 q^{35} -2.97813 q^{36} -6.79581 q^{37} -5.98503 q^{38} +0.739650 q^{39} +3.03338 q^{40} +6.96861 q^{41} -0.226011 q^{42} +5.66100 q^{43} +1.68897 q^{44} -9.03381 q^{45} -6.33776 q^{46} -6.11889 q^{47} -0.147869 q^{48} -4.66384 q^{49} +4.20139 q^{50} +0.755519 q^{51} -5.00206 q^{52} +1.39823 q^{53} +0.883981 q^{54} +5.12328 q^{55} +1.52845 q^{56} +0.885001 q^{57} +4.38843 q^{58} -9.64608 q^{59} -0.448543 q^{60} -7.15860 q^{61} -3.33959 q^{62} -4.55193 q^{63} +1.00000 q^{64} -15.1731 q^{65} -0.249746 q^{66} -1.61887 q^{67} -5.10938 q^{68} +0.937159 q^{69} +4.63637 q^{70} +3.73307 q^{71} -2.97813 q^{72} -10.8633 q^{73} -6.79581 q^{74} -0.621255 q^{75} -5.98503 q^{76} +2.58150 q^{77} +0.739650 q^{78} +10.2411 q^{79} +3.03338 q^{80} +8.80369 q^{81} +6.96861 q^{82} -12.1182 q^{83} -0.226011 q^{84} -15.4987 q^{85} +5.66100 q^{86} -0.648913 q^{87} +1.68897 q^{88} +9.49062 q^{89} -9.03381 q^{90} -7.64540 q^{91} -6.33776 q^{92} +0.493821 q^{93} -6.11889 q^{94} -18.1549 q^{95} -0.147869 q^{96} -2.09853 q^{97} -4.66384 q^{98} -5.02997 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 25 q^{3} + 69 q^{4} - 33 q^{5} - 25 q^{6} - 19 q^{7} + 69 q^{8} + 54 q^{9}+O(q^{10}) \) 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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