LMFDB - Embedded newform 8003.2.a.c.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8003,2,Mod(1,8003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8003, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8003 = 53 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8003.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:	$63.9042767376$
Analytic rank:	$0$
Dimension:	$172$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30956 q^{2} -1.91926 q^{3} +3.33407 q^{4} +3.27349 q^{5} +4.43265 q^{6} -1.87972 q^{7} -3.08112 q^{8} +0.683571 q^{9} +O(q^{10})$	\(q-2.30956 q^{2} -1.91926 q^{3} +3.33407 q^{4} +3.27349 q^{5} +4.43265 q^{6} -1.87972 q^{7} -3.08112 q^{8} +0.683571 q^{9} -7.56031 q^{10} -2.21776 q^{11} -6.39896 q^{12} -3.64849 q^{13} +4.34132 q^{14} -6.28268 q^{15} +0.447889 q^{16} +3.59852 q^{17} -1.57875 q^{18} +1.47965 q^{19} +10.9140 q^{20} +3.60767 q^{21} +5.12206 q^{22} +7.78404 q^{23} +5.91348 q^{24} +5.71571 q^{25} +8.42641 q^{26} +4.44584 q^{27} -6.26711 q^{28} +2.80949 q^{29} +14.5102 q^{30} +4.86111 q^{31} +5.12781 q^{32} +4.25647 q^{33} -8.31100 q^{34} -6.15322 q^{35} +2.27907 q^{36} -4.38957 q^{37} -3.41735 q^{38} +7.00241 q^{39} -10.0860 q^{40} +10.2889 q^{41} -8.33213 q^{42} +7.72765 q^{43} -7.39418 q^{44} +2.23766 q^{45} -17.9777 q^{46} +10.9635 q^{47} -0.859617 q^{48} -3.46667 q^{49} -13.2008 q^{50} -6.90651 q^{51} -12.1643 q^{52} -1.00000 q^{53} -10.2679 q^{54} -7.25981 q^{55} +5.79163 q^{56} -2.83985 q^{57} -6.48869 q^{58} +3.58544 q^{59} -20.9469 q^{60} -0.904674 q^{61} -11.2270 q^{62} -1.28492 q^{63} -12.7388 q^{64} -11.9433 q^{65} -9.83057 q^{66} -8.44392 q^{67} +11.9977 q^{68} -14.9396 q^{69} +14.2112 q^{70} -13.6198 q^{71} -2.10616 q^{72} -3.01943 q^{73} +10.1380 q^{74} -10.9699 q^{75} +4.93327 q^{76} +4.16876 q^{77} -16.1725 q^{78} -2.84607 q^{79} +1.46616 q^{80} -10.5834 q^{81} -23.7628 q^{82} -5.76261 q^{83} +12.0282 q^{84} +11.7797 q^{85} -17.8475 q^{86} -5.39215 q^{87} +6.83319 q^{88} +12.5944 q^{89} -5.16801 q^{90} +6.85812 q^{91} +25.9526 q^{92} -9.32974 q^{93} -25.3209 q^{94} +4.84363 q^{95} -9.84162 q^{96} +18.1031 q^{97} +8.00649 q^{98} -1.51600 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 172 q + 8 q^{2} + 25 q^{3} + 188 q^{4} + 27 q^{5} + 10 q^{6} + 31 q^{7} + 21 q^{8} + 179 q^{9}+O(q^{10}) $ 172 * q + 8 * q^2 + 25 * q^3 + 188 * q^4 + 27 * q^5 + 10 * q^6 + 31 * q^7 + 21 * q^8 + 179 * q^9	\( 172 q + 8 q^{2} + 25 q^{3} + 188 q^{4} + 27 q^{5} + 10 q^{6} + 31 q^{7} + 21 q^{8} + 179 q^{9} + 20 q^{10} - 3 q^{11} + 66 q^{12} + 121 q^{13} + 12 q^{14} + 30 q^{15} + 212 q^{16} + 8 q^{17} + 40 q^{18} + 41 q^{19} + 64 q^{20} + 56 q^{21} + 50 q^{22} + 28 q^{23} + 30 q^{24} + 231 q^{25} + 38 q^{26} + 100 q^{27} + 80 q^{28} + 26 q^{29} + 55 q^{30} + 66 q^{31} + 65 q^{32} + 99 q^{33} + 81 q^{34} + 36 q^{35} + 212 q^{36} + 153 q^{37} + q^{38} + 20 q^{39} + 59 q^{40} + 40 q^{41} + 50 q^{42} + 39 q^{43} - 51 q^{44} + 123 q^{45} + 59 q^{46} + 29 q^{47} + 128 q^{48} + 245 q^{49} + 19 q^{50} + 36 q^{51} + 215 q^{52} - 172 q^{53} + 40 q^{54} + 40 q^{55} + 15 q^{56} + 54 q^{57} + 44 q^{58} - 54 q^{60} + 100 q^{61} - 29 q^{62} + 92 q^{63} + 253 q^{64} + 77 q^{65} + 14 q^{66} + 126 q^{67} - 27 q^{68} + 47 q^{69} + 72 q^{70} + 38 q^{71} + 65 q^{72} + 185 q^{73} + 48 q^{74} + 75 q^{75} + 38 q^{76} + 120 q^{77} + 75 q^{78} + 79 q^{79} + 43 q^{80} + 232 q^{81} + 110 q^{82} + 90 q^{83} + 158 q^{84} + 115 q^{85} + 68 q^{86} + 61 q^{87} + 15 q^{88} - 36 q^{89} - 6 q^{90} + 33 q^{91} + 139 q^{92} + 103 q^{93} - 24 q^{94} - 45 q^{95} + 34 q^{96} + 159 q^{97} - 36 q^{98} + 27 q^{99}+O(q^{100}) \) 172 * q + 8 * q^2 + 25 * q^3 + 188 * q^4 + 27 * q^5 + 10 * q^6 + 31 * q^7 + 21 * q^8 + 179 * q^9 + 20 * q^10 - 3 * q^11 + 66 * q^12 + 121 * q^13 + 12 * q^14 + 30 * q^15 + 212 * q^16 + 8 * q^17 + 40 * q^18 + 41 * q^19 + 64 * q^20 + 56 * q^21 + 50 * q^22 + 28 * q^23 + 30 * q^24 + 231 * q^25 + 38 * q^26 + 100 * q^27 + 80 * q^28 + 26 * q^29 + 55 * q^30 + 66 * q^31 + 65 * q^32 + 99 * q^33 + 81 * q^34 + 36 * q^35 + 212 * q^36 + 153 * q^37 + q^38 + 20 * q^39 + 59 * q^40 + 40 * q^41 + 50 * q^42 + 39 * q^43 - 51 * q^44 + 123 * q^45 + 59 * q^46 + 29 * q^47 + 128 * q^48 + 245 * q^49 + 19 * q^50 + 36 * q^51 + 215 * q^52 - 172 * q^53 + 40 * q^54 + 40 * q^55 + 15 * q^56 + 54 * q^57 + 44 * q^58 - 54 * q^60 + 100 * q^61 - 29 * q^62 + 92 * q^63 + 253 * q^64 + 77 * q^65 + 14 * q^66 + 126 * q^67 - 27 * q^68 + 47 * q^69 + 72 * q^70 + 38 * q^71 + 65 * q^72 + 185 * q^73 + 48 * q^74 + 75 * q^75 + 38 * q^76 + 120 * q^77 + 75 * q^78 + 79 * q^79 + 43 * q^80 + 232 * q^81 + 110 * q^82 + 90 * q^83 + 158 * q^84 + 115 * q^85 + 68 * q^86 + 61 * q^87 + 15 * q^88 - 36 * q^89 - 6 * q^90 + 33 * q^91 + 139 * q^92 + 103 * q^93 - 24 * q^94 - 45 * q^95 + 34 * q^96 + 159 * q^97 - 36 * q^98 + 27 * 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 $

$2$	−2.30956	−1.63311	−0.816553	−	0.577270i	$-0.804118\pi$
$2$	−2.30956	−1.63311	−0.816553	+	0.577270i	$0.804118\pi$
$3$	−1.91926	−1.10809	−0.554044	−	0.832488i	$-0.686916\pi$
$3$	−1.91926	−1.10809	−0.554044	+	0.832488i	$0.686916\pi$
$4$	3.33407	1.66704
$5$	3.27349	1.46395	0.731974	−	0.681333i	$-0.238599\pi$
$5$	3.27349	1.46395	0.731974	+	0.681333i	$0.238599\pi$
$6$	4.43265	1.80962
$7$	−1.87972	−0.710466	−0.355233	−	0.934778i	$-0.615598\pi$
$7$	−1.87972	−0.710466	−0.355233	+	0.934778i	$0.615598\pi$
$8$	−3.08112	−1.08934
$9$	0.683571	0.227857
$10$	−7.56031	−2.39078
$11$	−2.21776	−0.668680	−0.334340	−	0.942452i	$-0.608513\pi$
$11$	−2.21776	−0.668680	−0.334340	+	0.942452i	$0.608513\pi$
$12$	−6.39896	−1.84722
$13$	−3.64849	−1.01191	−0.505955	−	0.862560i	$-0.668860\pi$
$13$	−3.64849	−1.01191	−0.505955	+	0.862560i	$0.668860\pi$
$14$	4.34132	1.16027
$15$	−6.28268	−1.62218
$16$	0.447889	0.111972
$17$	3.59852	0.872769	0.436385	−	0.899760i	$-0.356259\pi$
$17$	3.59852	0.872769	0.436385	+	0.899760i	$0.356259\pi$
$18$	−1.57875	−0.372114
$19$	1.47965	0.339456	0.169728	−	0.985491i	$-0.445711\pi$
$19$	1.47965	0.339456	0.169728	+	0.985491i	$0.445711\pi$
$20$	10.9140	2.44045
$21$	3.60767	0.787258
$22$	5.12206	1.09203
$23$	7.78404	1.62309	0.811543	−	0.584293i	$-0.198628\pi$
$23$	7.78404	1.62309	0.811543	+	0.584293i	$0.198628\pi$
$24$	5.91348	1.20708
$25$	5.71571	1.14314
$26$	8.42641	1.65256
$27$	4.44584	0.855602
$28$	−6.26711	−1.18437
$29$	2.80949	0.521709	0.260855	−	0.965378i	$-0.415996\pi$
$29$	2.80949	0.521709	0.260855	+	0.965378i	$0.415996\pi$
$30$	14.5102	2.64919
$31$	4.86111	0.873080	0.436540	−	0.899685i	$-0.356204\pi$
$31$	4.86111	0.873080	0.436540	+	0.899685i	$0.356204\pi$
$32$	5.12781	0.906477
$33$	4.25647	0.740956
$34$	−8.31100	−1.42532
$35$	−6.15322	−1.04008
$36$	2.27907	0.379845
$37$	−4.38957	−0.721641	−0.360820	−	0.932635i	$-0.617503\pi$
$37$	−4.38957	−0.721641	−0.360820	+	0.932635i	$0.617503\pi$
$38$	−3.41735	−0.554368
$39$	7.00241	1.12128
$40$	−10.0860	−1.59474
$41$	10.2889	1.60686	0.803428	−	0.595402i	$-0.203007\pi$
$41$	10.2889	1.60686	0.803428	+	0.595402i	$0.203007\pi$
$42$	−8.33213	−1.28568
$43$	7.72765	1.17846	0.589228	−	0.807967i	$-0.299432\pi$
$43$	7.72765	1.17846	0.589228	+	0.807967i	$0.299432\pi$
$44$	−7.39418	−1.11471
$45$	2.23766	0.333570
$46$	−17.9777	−2.65067
$47$	10.9635	1.59919	0.799596	−	0.600538i	$-0.205047\pi$
$47$	10.9635	1.59919	0.799596	+	0.600538i	$0.205047\pi$
$48$	−0.859617	−0.124075
$49$	−3.46667	−0.495239
$50$	−13.2008	−1.86687
$51$	−6.90651	−0.967104
$52$	−12.1643	−1.68689
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	−10.2679	−1.39729
$55$	−7.25981	−0.978913
$56$	5.79163	0.773939
$57$	−2.83985	−0.376147
$58$	−6.48869	−0.852006
$59$	3.58544	0.466785	0.233392	−	0.972383i	$-0.425017\pi$
$59$	3.58544	0.466785	0.233392	+	0.972383i	$0.425017\pi$
$60$	−20.9469	−2.70423
$61$	−0.904674	−0.115832	−0.0579158	−	0.998321i	$-0.518445\pi$
$61$	−0.904674	−0.115832	−0.0579158	+	0.998321i	$0.518445\pi$
$62$	−11.2270	−1.42583
$63$	−1.28492	−0.161884
$64$	−12.7388	−1.59235
$65$	−11.9433	−1.48138
$66$	−9.83057	−1.21006
$67$	−8.44392	−1.03159	−0.515795	−	0.856712i	$-0.672503\pi$
$67$	−8.44392	−1.03159	−0.515795	+	0.856712i	$0.672503\pi$
$68$	11.9977	1.45494
$69$	−14.9396	−1.79852
$70$	14.2112	1.69857
$71$	−13.6198	−1.61638	−0.808188	−	0.588925i	$-0.799551\pi$
$71$	−13.6198	−1.61638	−0.808188	+	0.588925i	$0.799551\pi$
$72$	−2.10616	−0.248214
$73$	−3.01943	−0.353398	−0.176699	−	0.984265i	$-0.556542\pi$
$73$	−3.01943	−0.353398	−0.176699	+	0.984265i	$0.556542\pi$
$74$	10.1380	1.17852
$75$	−10.9699	−1.26670
$76$	4.93327	0.565885
$77$	4.16876	0.475074
$78$	−16.1725	−1.83117
$79$	−2.84607	−0.320208	−0.160104	−	0.987100i	$-0.551183\pi$
$79$	−2.84607	−0.320208	−0.160104	+	0.987100i	$0.551183\pi$
$80$	1.46616	0.163922
$81$	−10.5834	−1.17594
$82$	−23.7628	−2.62417
$83$	−5.76261	−0.632528	−0.316264	−	0.948671i	$-0.602429\pi$
$83$	−5.76261	−0.632528	−0.316264	+	0.948671i	$0.602429\pi$
$84$	12.0282	1.31239
$85$	11.7797	1.27769
$86$	−17.8475	−1.92454
$87$	−5.39215	−0.578099
$88$	6.83319	0.728420
$89$	12.5944	1.33500	0.667501	−	0.744609i	$-0.267364\pi$
$89$	12.5944	1.33500	0.667501	+	0.744609i	$0.267364\pi$
$90$	−5.16801	−0.544756
$91$	6.85812	0.718927
$92$	25.9526	2.70574
$93$	−9.32974	−0.967449
$94$	−25.3209	−2.61165
$95$	4.84363	0.496946
$96$	−9.84162	−1.00446
$97$	18.1031	1.83809	0.919045	−	0.394153i	$-0.128962\pi$
$97$	18.1031	1.83809	0.919045	+	0.394153i	$0.128962\pi$
$98$	8.00649	0.808777
$99$	−1.51600	−0.152363
$100$	19.0566	1.90566
$101$	3.92619	0.390671	0.195335	−	0.980736i	$-0.437420\pi$
$101$	3.92619	0.390671	0.195335	+	0.980736i	$0.437420\pi$
$102$	15.9510	1.57938
$103$	−13.3680	−1.31718	−0.658592	−	0.752501i	$-0.728848\pi$
$103$	−13.3680	−1.31718	−0.658592	+	0.752501i	$0.728848\pi$
$104$	11.2414	1.10231
$105$	11.8096	1.15250
$106$	2.30956	0.224324
$107$	−1.64347	−0.158880	−0.0794399	−	0.996840i	$-0.525313\pi$
$107$	−1.64347	−0.158880	−0.0794399	+	0.996840i	$0.525313\pi$
$108$	14.8227	1.42632
$109$	4.02805	0.385817	0.192909	−	0.981217i	$-0.438208\pi$
$109$	4.02805	0.385817	0.192909	+	0.981217i	$0.438208\pi$
$110$	16.7670	1.59867
$111$	8.42474	0.799641
$112$	−0.841904	−0.0795525
$113$	7.43035	0.698989	0.349494	−	0.936938i	$-0.386353\pi$
$113$	7.43035	0.698989	0.349494	+	0.936938i	$0.386353\pi$
$114$	6.55880	0.614288
$115$	25.4810	2.37611
$116$	9.36704	0.869708
$117$	−2.49400	−0.230570
$118$	−8.28080	−0.762309
$119$	−6.76419	−0.620073
$120$	19.3577	1.76711
$121$	−6.08153	−0.552867
$122$	2.08940	0.189165
$123$	−19.7471	−1.78054
$124$	16.2073	1.45546
$125$	2.34286	0.209552
$126$	2.96760	0.264375
$127$	−15.2184	−1.35042	−0.675208	−	0.737627i	$-0.735946\pi$
$127$	−15.2184	−1.35042	−0.675208	+	0.737627i	$0.735946\pi$
$128$	19.1653	1.69399
$129$	−14.8314	−1.30583
$130$	27.5837	2.41925
$131$	−3.66294	−0.320032	−0.160016	−	0.987114i	$-0.551155\pi$
$131$	−3.66294	−0.320032	−0.160016	+	0.987114i	$0.551155\pi$
$132$	14.1914	1.23520
$133$	−2.78133	−0.241172
$134$	19.5018	1.68470
$135$	14.5534	1.25256
$136$	−11.0875	−0.950742
$137$	3.47664	0.297029	0.148515	−	0.988910i	$-0.452551\pi$
$137$	3.47664	0.297029	0.148515	+	0.988910i	$0.452551\pi$
$138$	34.5040	2.93717
$139$	−12.9927	−1.10202	−0.551011	−	0.834498i	$-0.685758\pi$
$139$	−12.9927	−1.10202	−0.551011	+	0.834498i	$0.685758\pi$
$140$	−20.5153	−1.73386
$141$	−21.0419	−1.77204
$142$	31.4558	2.63971
$143$	8.09148	0.676644
$144$	0.306164	0.0255137
$145$	9.19682	0.763755
$146$	6.97357	0.577136
$147$	6.65345	0.548767
$148$	−14.6351	−1.20300
$149$	−9.82114	−0.804579	−0.402290	−	0.915512i	$-0.631785\pi$
$149$	−9.82114	−0.804579	−0.402290	+	0.915512i	$0.631785\pi$
$150$	25.3358	2.06866
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	−4.55899	−0.369783
$153$	2.45984	0.198866
$154$	−9.62801	−0.775847
$155$	15.9128	1.27814
$156$	23.3465	1.86922
$157$	−1.49729	−0.119496	−0.0597482	−	0.998213i	$-0.519030\pi$
$157$	−1.49729	−0.119496	−0.0597482	+	0.998213i	$0.519030\pi$
$158$	6.57318	0.522934
$159$	1.91926	0.152207
$160$	16.7858	1.32704
$161$	−14.6318	−1.15315
$162$	24.4431	1.92043
$163$	−13.1307	−1.02848	−0.514238	−	0.857647i	$-0.671925\pi$
$163$	−13.1307	−1.02848	−0.514238	+	0.857647i	$0.671925\pi$
$164$	34.3039	2.67869
$165$	13.9335	1.08472
$166$	13.3091	1.03299
$167$	−15.5508	−1.20336	−0.601680	−	0.798737i	$-0.705502\pi$
$167$	−15.5508	−1.20336	−0.601680	+	0.798737i	$0.705502\pi$
$168$	−11.1157	−0.857591
$169$	0.311484	0.0239603
$170$	−27.2059	−2.08660
$171$	1.01145	0.0773474
$172$	25.7645	1.96453
$173$	4.19065	0.318609	0.159305	−	0.987229i	$-0.449075\pi$
$173$	4.19065	0.318609	0.159305	+	0.987229i	$0.449075\pi$
$174$	12.4535	0.944097
$175$	−10.7439	−0.812163
$176$	−0.993311	−0.0748737
$177$	−6.88141	−0.517238
$178$	−29.0875	−2.18020
$179$	17.5057	1.30844	0.654218	−	0.756306i	$-0.272998\pi$
$179$	17.5057	1.30844	0.654218	+	0.756306i	$0.272998\pi$
$180$	7.46051	0.556074
$181$	19.1505	1.42344	0.711721	−	0.702462i	$-0.247916\pi$
$181$	19.1505	1.42344	0.711721	+	0.702462i	$0.247916\pi$
$182$	−15.8393	−1.17408
$183$	1.73631	0.128352
$184$	−23.9836	−1.76809
$185$	−14.3692	−1.05644
$186$	21.5476	1.57995
$187$	−7.98066	−0.583604
$188$	36.5531	2.66591
$189$	−8.35691	−0.607876
$190$	−11.1867	−0.811565
$191$	11.0182	0.797250	0.398625	−	0.917114i	$-0.369488\pi$
$191$	11.0182	0.797250	0.398625	+	0.917114i	$0.369488\pi$
$192$	24.4490	1.76446
$193$	−1.58339	−0.113975	−0.0569874	−	0.998375i	$-0.518149\pi$
$193$	−1.58339	−0.113975	−0.0569874	+	0.998375i	$0.518149\pi$
$194$	−41.8102	−3.00180
$195$	22.9223	1.64150
$196$	−11.5581	−0.825580
$197$	15.4228	1.09883	0.549414	−	0.835550i	$-0.314851\pi$
$197$	15.4228	1.09883	0.549414	+	0.835550i	$0.314851\pi$
$198$	3.50129	0.248826
$199$	−21.9639	−1.55698	−0.778490	−	0.627657i	$-0.784014\pi$
$199$	−21.9639	−1.55698	−0.778490	+	0.627657i	$0.784014\pi$
$200$	−17.6108	−1.24527
$201$	16.2061	1.14309
$202$	−9.06779	−0.638007
$203$	−5.28104	−0.370656
$204$	−23.0268	−1.61220
$205$	33.6806	2.35235
$206$	30.8741	2.15110
$207$	5.32094	0.369831
$208$	−1.63412	−0.113306
$209$	−3.28152	−0.226988
$210$	−27.2751	−1.88216
$211$	−1.00415	−0.0691283	−0.0345641	−	0.999402i	$-0.511004\pi$
$211$	−1.00415	−0.0691283	−0.0345641	+	0.999402i	$0.511004\pi$
$212$	−3.33407	−0.228985
$213$	26.1400	1.79108
$214$	3.79569	0.259468
$215$	25.2964	1.72520
$216$	−13.6982	−0.932041
$217$	−9.13750	−0.620294
$218$	−9.30303	−0.630081
$219$	5.79509	0.391596
$220$	−24.2047	−1.63188
$221$	−13.1292	−0.883163
$222$	−19.4575	−1.30590
$223$	23.3504	1.56366	0.781830	−	0.623492i	$-0.214287\pi$
$223$	23.3504	1.56366	0.781830	+	0.623492i	$0.214287\pi$
$224$	−9.63882	−0.644021
$225$	3.90709	0.260473
$226$	−17.1609	−1.14152
$227$	8.81313	0.584948	0.292474	−	0.956273i	$-0.405521\pi$
$227$	8.81313	0.584948	0.292474	+	0.956273i	$0.405521\pi$
$228$	−9.46825	−0.627050
$229$	20.4097	1.34871	0.674357	−	0.738406i	$-0.264421\pi$
$229$	20.4097	1.34871	0.674357	+	0.738406i	$0.264421\pi$
$230$	−58.8498	−3.88044
$231$	−8.00095	−0.526424
$232$	−8.65637	−0.568319
$233$	28.9327	1.89544	0.947721	−	0.319101i	$-0.103381\pi$
$233$	28.9327	1.89544	0.947721	+	0.319101i	$0.103381\pi$
$234$	5.76005	0.376546
$235$	35.8889	2.34113
$236$	11.9541	0.778147
$237$	5.46236	0.354819
$238$	15.6223	1.01264
$239$	−8.18888	−0.529695	−0.264847	−	0.964290i	$-0.585322\pi$
$239$	−8.18888	−0.529695	−0.264847	+	0.964290i	$0.585322\pi$
$240$	−2.81394	−0.181639
$241$	−21.8283	−1.40609	−0.703043	−	0.711147i	$-0.748176\pi$
$241$	−21.8283	−1.40609	−0.703043	+	0.711147i	$0.748176\pi$
$242$	14.0457	0.902890
$243$	6.97490	0.447440
$244$	−3.01625	−0.193095
$245$	−11.3481	−0.725003
$246$	45.6071	2.90781
$247$	−5.39850	−0.343499
$248$	−14.9776	−0.951082
$249$	11.0600	0.700896
$250$	−5.41097	−0.342220
$251$	−28.5648	−1.80299	−0.901497	−	0.432785i	$-0.857531\pi$
$251$	−28.5648	−1.80299	−0.901497	+	0.432785i	$0.857531\pi$
$252$	−4.28401	−0.269867
$253$	−17.2632	−1.08533
$254$	35.1479	2.20537
$255$	−22.6083	−1.41579
$256$	−18.7860	−1.17412
$257$	−12.0070	−0.748975	−0.374487	−	0.927232i	$-0.622181\pi$
$257$	−12.0070	−0.748975	−0.374487	+	0.927232i	$0.622181\pi$
$258$	34.2540	2.13256
$259$	8.25114	0.512701
$260$	−39.8198	−2.46952
$261$	1.92048	0.118875
$262$	8.45978	0.522647
$263$	−15.7180	−0.969214	−0.484607	−	0.874732i	$-0.661037\pi$
$263$	−15.7180	−0.969214	−0.484607	+	0.874732i	$0.661037\pi$
$264$	−13.1147	−0.807153
$265$	−3.27349	−0.201089
$266$	6.42365	0.393859
$267$	−24.1719	−1.47930
$268$	−28.1526	−1.71970
$269$	21.7690	1.32728	0.663641	−	0.748052i	$-0.269010\pi$
$269$	21.7690	1.32728	0.663641	+	0.748052i	$0.269010\pi$
$270$	−33.6119	−2.04556
$271$	−29.4138	−1.78676	−0.893380	−	0.449301i	$-0.851673\pi$
$271$	−29.4138	−1.78676	−0.893380	+	0.449301i	$0.851673\pi$
$272$	1.61174	0.0977260
$273$	−13.1625	−0.796633
$274$	−8.02950	−0.485080
$275$	−12.6761	−0.764396
$276$	−49.8098	−2.99820
$277$	10.3320	0.620787	0.310393	−	0.950608i	$-0.399539\pi$
$277$	10.3320	0.620787	0.310393	+	0.950608i	$0.399539\pi$
$278$	30.0073	1.79972
$279$	3.32291	0.198937
$280$	18.9588	1.13301
$281$	20.8017	1.24092	0.620462	−	0.784237i	$-0.286945\pi$
$281$	20.8017	1.24092	0.620462	+	0.784237i	$0.286945\pi$
$282$	48.5974	2.89394
$283$	16.2866	0.968138	0.484069	−	0.875030i	$-0.339158\pi$
$283$	16.2866	0.968138	0.484069	+	0.875030i	$0.339158\pi$
$284$	−45.4095	−2.69456
$285$	−9.29619	−0.550659
$286$	−18.6878	−1.10503
$287$	−19.3402	−1.14162
$288$	3.50522	0.206547
$289$	−4.05066	−0.238274
$290$	−21.2406	−1.24729
$291$	−34.7446	−2.03676
$292$	−10.0670	−0.589127
$293$	−19.9458	−1.16525	−0.582624	−	0.812742i	$-0.697974\pi$
$293$	−19.9458	−1.16525	−0.582624	+	0.812742i	$0.697974\pi$
$294$	−15.3666	−0.896196
$295$	11.7369	0.683348
$296$	13.5248	0.786112
$297$	−9.85981	−0.572124
$298$	22.6825	1.31396
$299$	−28.4000	−1.64242
$300$	−36.5746	−2.11163
$301$	−14.5258	−0.837252
$302$	−2.30956	−0.132900
$303$	−7.53540	−0.432897
$304$	0.662721	0.0380097
$305$	−2.96144	−0.169571
$306$	−5.68115	−0.324770
$307$	−16.9915	−0.969755	−0.484878	−	0.874582i	$-0.661136\pi$
$307$	−16.9915	−0.969755	−0.484878	+	0.874582i	$0.661136\pi$
$308$	13.8989	0.791966
$309$	25.6566	1.45955
$310$	−36.7515	−2.08734
$311$	8.81642	0.499933	0.249967	−	0.968254i	$-0.419580\pi$
$311$	8.81642	0.499933	0.249967	+	0.968254i	$0.419580\pi$
$312$	−21.5753	−1.22146
$313$	16.4759	0.931271	0.465635	−	0.884977i	$-0.345826\pi$
$313$	16.4759	0.931271	0.465635	+	0.884977i	$0.345826\pi$
$314$	3.45807	0.195150
$315$	−4.20616	−0.236990
$316$	−9.48901	−0.533799
$317$	24.3009	1.36487	0.682437	−	0.730945i	$-0.260920\pi$
$317$	24.3009	1.36487	0.682437	+	0.730945i	$0.260920\pi$
$318$	−4.43265	−0.248571
$319$	−6.23078	−0.348857
$320$	−41.7002	−2.33111
$321$	3.15424	0.176053
$322$	33.7930	1.88321
$323$	5.32456	0.296267
$324$	−35.2860	−1.96033
$325$	−20.8537	−1.15676
$326$	30.3262	1.67961
$327$	−7.73089	−0.427519
$328$	−31.7013	−1.75041
$329$	−20.6083	−1.13617
$330$	−32.1802	−1.77146
$331$	−13.8721	−0.762481	−0.381241	−	0.924476i	$-0.624503\pi$
$331$	−13.8721	−0.762481	−0.381241	+	0.924476i	$0.624503\pi$
$332$	−19.2129	−1.05445
$333$	−3.00058	−0.164431
$334$	35.9156	1.96522
$335$	−27.6411	−1.51019
$336$	1.61584	0.0881510
$337$	33.2050	1.80879	0.904396	−	0.426694i	$-0.140322\pi$
$337$	33.2050	1.80879	0.904396	+	0.426694i	$0.140322\pi$
$338$	−0.719391	−0.0391297
$339$	−14.2608	−0.774540
$340$	39.2744	2.12995
$341$	−10.7808	−0.583812
$342$	−2.33600	−0.126316
$343$	19.6744	1.06232
$344$	−23.8098	−1.28374
$345$	−48.9047	−2.63294
$346$	−9.67856	−0.520322
$347$	12.3421	0.662559	0.331280	−	0.943533i	$-0.392520\pi$
$347$	12.3421	0.662559	0.331280	+	0.943533i	$0.392520\pi$
$348$	−17.9778	−0.963712
$349$	−19.5434	−1.04614	−0.523068	−	0.852291i	$-0.675213\pi$
$349$	−19.5434	−1.04614	−0.523068	+	0.852291i	$0.675213\pi$
$350$	24.8137	1.32635
$351$	−16.2206	−0.865791
$352$	−11.3723	−0.606144
$353$	−36.7985	−1.95859	−0.979295	−	0.202441i	$-0.935113\pi$
$353$	−36.7985	−1.95859	−0.979295	+	0.202441i	$0.935113\pi$
$354$	15.8930	0.844705
$355$	−44.5843	−2.36629
$356$	41.9906	2.22550
$357$	12.9823	0.687094
$358$	−40.4305	−2.13682
$359$	23.7785	1.25498	0.627490	−	0.778625i	$-0.284082\pi$
$359$	23.7785	1.25498	0.627490	+	0.778625i	$0.284082\pi$
$360$	−6.89449	−0.363372
$361$	−16.8106	−0.884770
$362$	−44.2291	−2.32463
$363$	11.6721	0.612624
$364$	22.8655	1.19848
$365$	−9.88407	−0.517356
$366$	−4.01011	−0.209612
$367$	29.9281	1.56224	0.781118	−	0.624384i	$-0.214650\pi$
$367$	29.9281	1.56224	0.781118	+	0.624384i	$0.214650\pi$
$368$	3.48639	0.181741
$369$	7.03319	0.366133
$370$	33.1865	1.72529
$371$	1.87972	0.0975900
$372$	−31.1060	−1.61277
$373$	32.5622	1.68601	0.843003	−	0.537909i	$-0.180786\pi$
$373$	32.5622	1.68601	0.843003	+	0.537909i	$0.180786\pi$
$374$	18.4318	0.953087
$375$	−4.49656	−0.232201
$376$	−33.7799	−1.74206
$377$	−10.2504	−0.527922
$378$	19.3008	0.992725
$379$	29.8578	1.53369	0.766846	−	0.641831i	$-0.221825\pi$
$379$	29.8578	1.53369	0.766846	+	0.641831i	$0.221825\pi$
$380$	16.1490	0.828426
$381$	29.2081	1.49638
$382$	−25.4472	−1.30199
$383$	1.66611	0.0851343	0.0425672	−	0.999094i	$-0.486446\pi$
$383$	1.66611	0.0851343	0.0425672	+	0.999094i	$0.486446\pi$
$384$	−36.7833	−1.87709
$385$	13.6464	0.695484
$386$	3.65693	0.186133
$387$	5.28239	0.268519
$388$	60.3570	3.06416
$389$	−5.04024	−0.255550	−0.127775	−	0.991803i	$-0.540784\pi$
$389$	−5.04024	−0.255550	−0.127775	+	0.991803i	$0.540784\pi$
$390$	−52.9404	−2.68074
$391$	28.0110	1.41658
$392$	10.6812	0.539483
$393$	7.03014	0.354624
$394$	−35.6198	−1.79450
$395$	−9.31658	−0.468768
$396$	−5.05444	−0.253995
$397$	27.0756	1.35888	0.679442	−	0.733729i	$-0.262222\pi$
$397$	27.0756	1.35888	0.679442	+	0.733729i	$0.262222\pi$
$398$	50.7270	2.54271
$399$	5.33810	0.267239
$400$	2.56000	0.128000
$401$	−26.7171	−1.33419	−0.667094	−	0.744973i	$-0.732462\pi$
$401$	−26.7171	−1.33419	−0.667094	+	0.744973i	$0.732462\pi$
$402$	−37.4290	−1.86679
$403$	−17.7357	−0.883478
$404$	13.0902	0.651262
$405$	−34.6447	−1.72151
$406$	12.1969	0.605321
$407$	9.73502	0.482547
$408$	21.2798	1.05351
$409$	−18.0560	−0.892810	−0.446405	−	0.894831i	$-0.647296\pi$
$409$	−18.0560	−0.892810	−0.446405	+	0.894831i	$0.647296\pi$
$410$	−77.7873	−3.84164
$411$	−6.67258	−0.329134
$412$	−44.5697	−2.19579
$413$	−6.73961	−0.331635
$414$	−12.2890	−0.603973
$415$	−18.8638	−0.925988
$416$	−18.7088	−0.917273
$417$	24.9363	1.22114
$418$	7.57887	0.370695
$419$	−30.6121	−1.49550	−0.747749	−	0.663981i	$-0.768865\pi$
$419$	−30.6121	−1.49550	−0.747749	+	0.663981i	$0.768865\pi$
$420$	39.3742	1.92126
$421$	−1.17832	−0.0574280	−0.0287140	−	0.999588i	$-0.509141\pi$
$421$	−1.17832	−0.0574280	−0.0287140	+	0.999588i	$0.509141\pi$
$422$	2.31914	0.112894
$423$	7.49433	0.364387
$424$	3.08112	0.149632
$425$	20.5681	0.997699
$426$	−60.3720	−2.92503
$427$	1.70053	0.0822944
$428$	−5.47943	−0.264858
$429$	−15.5297	−0.749780
$430$	−58.4235	−2.81743
$431$	2.85372	0.137459	0.0687294	−	0.997635i	$-0.478105\pi$
$431$	2.85372	0.137459	0.0687294	+	0.997635i	$0.478105\pi$
$432$	1.99124	0.0958037
$433$	33.2602	1.59839	0.799193	−	0.601075i	$-0.205261\pi$
$433$	33.2602	1.59839	0.799193	+	0.601075i	$0.205261\pi$
$434$	21.1036	1.01301
$435$	−17.6511	−0.846307
$436$	13.4298	0.643171
$437$	11.5177	0.550966
$438$	−13.3841	−0.639517
$439$	8.26120	0.394286	0.197143	−	0.980375i	$-0.436834\pi$
$439$	8.26120	0.394286	0.197143	+	0.980375i	$0.436834\pi$
$440$	22.3683	1.06637
$441$	−2.36971	−0.112844
$442$	30.3226	1.44230
$443$	29.1001	1.38259	0.691294	−	0.722574i	$-0.257041\pi$
$443$	29.1001	1.38259	0.691294	+	0.722574i	$0.257041\pi$
$444$	28.0887	1.33303
$445$	41.2276	1.95437
$446$	−53.9292	−2.55362
$447$	18.8494	0.891544
$448$	23.9453	1.13131
$449$	23.5257	1.11025	0.555123	−	0.831768i	$-0.312671\pi$
$449$	23.5257	1.11025	0.555123	+	0.831768i	$0.312671\pi$
$450$	−9.02366	−0.425379
$451$	−22.8183	−1.07447
$452$	24.7733	1.16524
$453$	−1.91926	−0.0901748
$454$	−20.3545	−0.955283
$455$	22.4500	1.05247
$456$	8.74990	0.409752
$457$	−34.0547	−1.59301	−0.796505	−	0.604632i	$-0.793320\pi$
$457$	−34.0547	−1.59301	−0.796505	+	0.604632i	$0.793320\pi$
$458$	−47.1375	−2.20259
$459$	15.9984	0.746743
$460$	84.9553	3.96106
$461$	9.89205	0.460719	0.230359	−	0.973106i	$-0.426010\pi$
$461$	9.89205	0.460719	0.230359	+	0.973106i	$0.426010\pi$
$462$	18.4787	0.859706
$463$	10.1614	0.472238	0.236119	−	0.971724i	$-0.424124\pi$
$463$	10.1614	0.472238	0.236119	+	0.971724i	$0.424124\pi$
$464$	1.25834	0.0584170
$465$	−30.5408	−1.41629
$466$	−66.8217	−3.09546
$467$	37.1208	1.71775	0.858873	−	0.512189i	$-0.171165\pi$
$467$	37.1208	1.71775	0.858873	+	0.512189i	$0.171165\pi$
$468$	−8.31518	−0.384369
$469$	15.8722	0.732909
$470$	−82.8876	−3.82332
$471$	2.87368	0.132412
$472$	−11.0472	−0.508487
$473$	−17.1381	−0.788010
$474$	−12.6157	−0.579456
$475$	8.45727	0.388046
$476$	−22.5523	−1.03368
$477$	−0.683571	−0.0312985
$478$	18.9127	0.865048
$479$	−42.1038	−1.92377	−0.961886	−	0.273452i	$-0.911835\pi$
$479$	−42.1038	−1.92377	−0.961886	+	0.273452i	$0.911835\pi$
$480$	−32.2164	−1.47047
$481$	16.0153	0.730235
$482$	50.4139	2.29629
$483$	28.0822	1.27779
$484$	−20.2763	−0.921648
$485$	59.2602	2.69087
$486$	−16.1090	−0.730717
$487$	22.9855	1.04157	0.520786	−	0.853687i	$-0.325639\pi$
$487$	22.9855	1.04157	0.520786	+	0.853687i	$0.325639\pi$
$488$	2.78741	0.126180
$489$	25.2013	1.13964
$490$	26.2091	1.18401
$491$	25.3716	1.14500	0.572502	−	0.819903i	$-0.305973\pi$
$491$	25.3716	1.14500	0.572502	+	0.819903i	$0.305973\pi$
$492$	−65.8383	−2.96822
$493$	10.1100	0.455332
$494$	12.4682	0.560970
$495$	−4.96259	−0.223052
$496$	2.17724	0.0977608
$497$	25.6014	1.14838
$498$	−25.5436	−1.14464
$499$	10.4864	0.469437	0.234719	−	0.972063i	$-0.424583\pi$
$499$	10.4864	0.469437	0.234719	+	0.972063i	$0.424583\pi$
$500$	7.81126	0.349330
$501$	29.8462	1.33343
$502$	65.9721	2.94448
$503$	−38.2463	−1.70532	−0.852658	−	0.522469i	$-0.825011\pi$
$503$	−38.2463	−1.70532	−0.852658	+	0.522469i	$0.825011\pi$
$504$	3.95899	0.176347
$505$	12.8523	0.571922
$506$	39.8703	1.77245
$507$	−0.597819	−0.0265501
$508$	−50.7393	−2.25119
$509$	−5.55991	−0.246439	−0.123219	−	0.992379i	$-0.539322\pi$
$509$	−5.55991	−0.246439	−0.123219	+	0.992379i	$0.539322\pi$
$510$	52.2154	2.31213
$511$	5.67568	0.251077
$512$	5.05669	0.223476
$513$	6.57830	0.290439
$514$	27.7309	1.22316
$515$	−43.7598	−1.92829
$516$	−49.4489	−2.17687
$517$	−24.3144	−1.06935
$518$	−19.0565	−0.837295
$519$	−8.04296	−0.353047
$520$	36.7987	1.61373
$521$	−14.3266	−0.627659	−0.313829	−	0.949479i	$-0.601612\pi$
$521$	−14.3266	−0.627659	−0.313829	+	0.949479i	$0.601612\pi$
$522$	−4.43548	−0.194135
$523$	37.0173	1.61865	0.809327	−	0.587358i	$-0.199832\pi$
$523$	37.0173	1.61865	0.809327	+	0.587358i	$0.199832\pi$
$524$	−12.2125	−0.533505
$525$	20.6204	0.899947
$526$	36.3017	1.58283
$527$	17.4928	0.761998
$528$	1.90643	0.0829665
$529$	37.5913	1.63441
$530$	7.56031	0.328399
$531$	2.45090	0.106360
$532$	−9.27315	−0.402042
$533$	−37.5390	−1.62599
$534$	55.8266	2.41585
$535$	−5.37986	−0.232592
$536$	26.0167	1.12375
$537$	−33.5980	−1.44986
$538$	−50.2769	−2.16759
$539$	7.68825	0.331156
$540$	48.5220	2.08806
$541$	−8.13669	−0.349824	−0.174912	−	0.984584i	$-0.555964\pi$
$541$	−8.13669	−0.349824	−0.174912	+	0.984584i	$0.555964\pi$
$542$	67.9329	2.91797
$543$	−36.7548	−1.57730
$544$	18.4525	0.791146
$545$	13.1858	0.564816
$546$	30.3997	1.30099
$547$	−37.8013	−1.61627	−0.808133	−	0.589001i	$-0.799522\pi$
$547$	−37.8013	−1.61627	−0.808133	+	0.589001i	$0.799522\pi$
$548$	11.5914	0.495158
$549$	−0.618408	−0.0263930
$550$	29.2762	1.24834
$551$	4.15707	0.177097
$552$	46.0308	1.95920
$553$	5.34981	0.227497
$554$	−23.8623	−1.01381
$555$	27.5783	1.17063
$556$	−43.3184	−1.83711
$557$	32.2145	1.36497	0.682486	−	0.730899i	$-0.260899\pi$
$557$	32.2145	1.36497	0.682486	+	0.730899i	$0.260899\pi$
$558$	−7.67446	−0.324886
$559$	−28.1943	−1.19249
$560$	−2.75596	−0.116461
$561$	15.3170	0.646684
$562$	−48.0427	−2.02656
$563$	13.4393	0.566398	0.283199	−	0.959061i	$-0.408604\pi$
$563$	13.4393	0.566398	0.283199	+	0.959061i	$0.408604\pi$
$564$	−70.1550	−2.95406
$565$	24.3232	1.02328
$566$	−37.6149	−1.58107
$567$	19.8939	0.835464
$568$	41.9643	1.76078
$569$	−27.0179	−1.13265	−0.566325	−	0.824182i	$-0.691635\pi$
$569$	−27.0179	−1.13265	−0.566325	+	0.824182i	$0.691635\pi$
$570$	21.4701	0.899285
$571$	19.8098	0.829015	0.414507	−	0.910046i	$-0.363954\pi$
$571$	19.8098	0.829015	0.414507	+	0.910046i	$0.363954\pi$
$572$	26.9776	1.12799
$573$	−21.1468	−0.883422
$574$	44.6674	1.86438
$575$	44.4913	1.85542
$576$	−8.70785	−0.362827
$577$	15.6485	0.651458	0.325729	−	0.945463i	$-0.394390\pi$
$577$	15.6485	0.651458	0.325729	+	0.945463i	$0.394390\pi$
$578$	9.35524	0.389127
$579$	3.03894	0.126294
$580$	30.6629	1.27321
$581$	10.8321	0.449390
$582$	80.2447	3.32625
$583$	2.21776	0.0918503
$584$	9.30324	0.384971
$585$	−8.16408	−0.337543
$586$	46.0661	1.90297
$587$	16.3338	0.674166	0.337083	−	0.941475i	$-0.390560\pi$
$587$	16.3338	0.674166	0.337083	+	0.941475i	$0.390560\pi$
$588$	22.1831	0.914815
$589$	7.19276	0.296372
$590$	−27.1071	−1.11598
$591$	−29.6004	−1.21760
$592$	−1.96604	−0.0808038
$593$	16.3928	0.673172	0.336586	−	0.941653i	$-0.390728\pi$
$593$	16.3928	0.673172	0.336586	+	0.941653i	$0.390728\pi$
$594$	22.7718	0.934339
$595$	−22.1425	−0.907753
$596$	−32.7444	−1.34126
$597$	42.1545	1.72527
$598$	65.5916	2.68224
$599$	−22.8961	−0.935509	−0.467754	−	0.883858i	$-0.654937\pi$
$599$	−22.8961	−0.935509	−0.467754	+	0.883858i	$0.654937\pi$
$600$	33.7997	1.37987
$601$	23.0659	0.940879	0.470439	−	0.882432i	$-0.344095\pi$
$601$	23.0659	0.940879	0.470439	+	0.882432i	$0.344095\pi$
$602$	33.5482	1.36732
$603$	−5.77202	−0.235055
$604$	3.33407	0.135661
$605$	−19.9078	−0.809367
$606$	17.4035	0.706967
$607$	11.1633	0.453102	0.226551	−	0.973999i	$-0.427255\pi$
$607$	11.1633	0.453102	0.226551	+	0.973999i	$0.427255\pi$
$608$	7.58739	0.307709
$609$	10.1357	0.410720
$610$	6.83962	0.276928
$611$	−40.0003	−1.61824
$612$	8.20129	0.331517
$613$	44.3273	1.79036	0.895182	−	0.445701i	$-0.147046\pi$
$613$	44.3273	1.79036	0.895182	+	0.445701i	$0.147046\pi$
$614$	39.2429	1.58371
$615$	−64.6419	−2.60661
$616$	−12.8444	−0.517518
$617$	35.7813	1.44050	0.720250	−	0.693714i	$-0.244027\pi$
$617$	35.7813	1.44050	0.720250	+	0.693714i	$0.244027\pi$
$618$	−59.2555	−2.38361
$619$	−15.7385	−0.632584	−0.316292	−	0.948662i	$-0.602438\pi$
$619$	−15.7385	−0.632584	−0.316292	+	0.948662i	$0.602438\pi$
$620$	53.0543	2.13071
$621$	34.6066	1.38871
$622$	−20.3621	−0.816444
$623$	−23.6739	−0.948474
$624$	3.13630	0.125553
$625$	−20.9092	−0.836369
$626$	−38.0520	−1.52086
$627$	6.29810	0.251522
$628$	−4.99206	−0.199205
$629$	−15.7960	−0.629826
$630$	9.71438	0.387030
$631$	−38.1728	−1.51964	−0.759818	−	0.650136i	$-0.774712\pi$
$631$	−38.1728	−1.51964	−0.759818	+	0.650136i	$0.774712\pi$
$632$	8.76909	0.348816
$633$	1.92722	0.0766001
$634$	−56.1244	−2.22898
$635$	−49.8173	−1.97694
$636$	6.39896	0.253735
$637$	12.6481	0.501137
$638$	14.3904	0.569720
$639$	−9.31011	−0.368302
$640$	62.7375	2.47992
$641$	1.02145	0.0403449	0.0201725	−	0.999797i	$-0.493578\pi$
$641$	1.02145	0.0403449	0.0201725	+	0.999797i	$0.493578\pi$
$642$	−7.28492	−0.287513
$643$	10.0459	0.396171	0.198085	−	0.980185i	$-0.436528\pi$
$643$	10.0459	0.396171	0.198085	+	0.980185i	$0.436528\pi$
$644$	−48.7834	−1.92234
$645$	−48.5504	−1.91167
$646$	−12.2974	−0.483835
$647$	−47.4236	−1.86442	−0.932208	−	0.361924i	$-0.882120\pi$
$647$	−47.4236	−1.86442	−0.932208	+	0.361924i	$0.882120\pi$
$648$	32.6088	1.28100
$649$	−7.95166	−0.312130
$650$	48.1629	1.88910
$651$	17.5373	0.687339
$652$	−43.7787	−1.71451
$653$	23.8511	0.933366	0.466683	−	0.884425i	$-0.345449\pi$
$653$	23.8511	0.933366	0.466683	+	0.884425i	$0.345449\pi$
$654$	17.8550	0.698184
$655$	−11.9906	−0.468510
$656$	4.60829	0.179923
$657$	−2.06400	−0.0805242
$658$	47.5961	1.85549
$659$	−6.02906	−0.234859	−0.117429	−	0.993081i	$-0.537465\pi$
$659$	−6.02906	−0.234859	−0.117429	+	0.993081i	$0.537465\pi$
$660$	46.4552	1.80827
$661$	−12.4097	−0.482680	−0.241340	−	0.970441i	$-0.577587\pi$
$661$	−12.4097	−0.482680	−0.241340	+	0.970441i	$0.577587\pi$
$662$	32.0385	1.24521
$663$	25.1983	0.978622
$664$	17.7553	0.689038
$665$	−9.10464	−0.353063
$666$	6.93003	0.268533
$667$	21.8692	0.846778
$668$	−51.8476	−2.00605
$669$	−44.8156	−1.73267
$670$	63.8387	2.46631
$671$	2.00635	0.0774543
$672$	18.4994	0.713631
$673$	−3.96652	−0.152898	−0.0764490	−	0.997073i	$-0.524358\pi$
$673$	−3.96652	−0.152898	−0.0764490	+	0.997073i	$0.524358\pi$
$674$	−76.6890	−2.95395
$675$	25.4111	0.978074
$676$	1.03851	0.0399427
$677$	30.6803	1.17914	0.589570	−	0.807717i	$-0.299297\pi$
$677$	30.6803	1.17914	0.589570	+	0.807717i	$0.299297\pi$
$678$	32.9362	1.26491
$679$	−34.0286	−1.30590
$680$	−36.2947	−1.39184
$681$	−16.9147	−0.648174
$682$	24.8989	0.953427
$683$	33.9964	1.30083	0.650417	−	0.759577i	$-0.274594\pi$
$683$	33.9964	1.30083	0.650417	+	0.759577i	$0.274594\pi$
$684$	3.37224	0.128941
$685$	11.3807	0.434835
$686$	−45.4391	−1.73487
$687$	−39.1716	−1.49449
$688$	3.46113	0.131954
$689$	3.64849	0.138996
$690$	112.948	4.29987
$691$	−6.44366	−0.245128	−0.122564	−	0.992461i	$-0.539112\pi$
$691$	−6.44366	−0.245128	−0.122564	+	0.992461i	$0.539112\pi$
$692$	13.9719	0.531133
$693$	2.84964	0.108249
$694$	−28.5049	−1.08203
$695$	−42.5313	−1.61330
$696$	16.6139	0.629747
$697$	37.0248	1.40241
$698$	45.1368	1.70845
$699$	−55.5294	−2.10031
$700$	−35.8209	−1.35390
$701$	23.4318	0.885009	0.442504	−	0.896766i	$-0.354090\pi$
$701$	23.4318	0.885009	0.442504	+	0.896766i	$0.354090\pi$
$702$	37.4625	1.41393
$703$	−6.49505	−0.244965
$704$	28.2516	1.06477
$705$	−68.8802	−2.59418
$706$	84.9885	3.19858
$707$	−7.38013	−0.277558
$708$	−22.9431	−0.862255
$709$	9.46196	0.355351	0.177676	−	0.984089i	$-0.443142\pi$
$709$	9.46196	0.355351	0.177676	+	0.984089i	$0.443142\pi$
$710$	102.970	3.86440
$711$	−1.94549	−0.0729616
$712$	−38.8048	−1.45427
$713$	37.8391	1.41708
$714$	−29.9833	−1.12210
$715$	26.4874	0.990571
$716$	58.3652	2.18121
$717$	15.7166	0.586948
$718$	−54.9178	−2.04952
$719$	−10.6941	−0.398823	−0.199412	−	0.979916i	$-0.563903\pi$
$719$	−10.6941	−0.398823	−0.199412	+	0.979916i	$0.563903\pi$
$720$	1.00222	0.0373506
$721$	25.1279	0.935813
$722$	38.8252	1.44492
$723$	41.8943	1.55807
$724$	63.8490	2.37293
$725$	16.0582	0.596387
$726$	−26.9573	−1.00048
$727$	−44.6229	−1.65497	−0.827485	−	0.561487i	$-0.810229\pi$
$727$	−44.6229	−1.65497	−0.827485	+	0.561487i	$0.810229\pi$
$728$	−21.1307	−0.783156
$729$	18.3637	0.680136
$730$	22.8279	0.844897
$731$	27.8081	1.02852
$732$	5.78897	0.213967
$733$	9.32718	0.344507	0.172254	−	0.985053i	$-0.444895\pi$
$733$	9.32718	0.344507	0.172254	+	0.985053i	$0.444895\pi$
$734$	−69.1208	−2.55130
$735$	21.7800	0.803367
$736$	39.9151	1.47129
$737$	18.7266	0.689804
$738$	−16.2436	−0.597934
$739$	5.90595	0.217254	0.108627	−	0.994083i	$-0.465355\pi$
$739$	5.90595	0.217254	0.108627	+	0.994083i	$0.465355\pi$
$740$	−47.9079	−1.76113
$741$	10.3611	0.380626
$742$	−4.34132	−0.159375
$743$	−34.7745	−1.27575	−0.637876	−	0.770139i	$-0.720187\pi$
$743$	−34.7745	−1.27575	−0.637876	+	0.770139i	$0.720187\pi$
$744$	28.7460	1.05388
$745$	−32.1494	−1.17786
$746$	−75.2043	−2.75343
$747$	−3.93915	−0.144126
$748$	−26.6081	−0.972888
$749$	3.08925	0.112879
$750$	10.3851	0.379210
$751$	49.1285	1.79273	0.896363	−	0.443321i	$-0.146200\pi$
$751$	49.1285	1.79273	0.896363	+	0.443321i	$0.146200\pi$
$752$	4.91044	0.179065
$753$	54.8234	1.99787
$754$	23.6739	0.862153
$755$	3.27349	0.119134
$756$	−27.8625	−1.01335
$757$	−6.20659	−0.225582	−0.112791	−	0.993619i	$-0.535979\pi$
$757$	−6.20659	−0.225582	−0.112791	+	0.993619i	$0.535979\pi$
$758$	−68.9584	−2.50468
$759$	33.1325	1.20263
$760$	−14.9238	−0.541343
$761$	33.8891	1.22848	0.614239	−	0.789120i	$-0.289463\pi$
$761$	33.8891	1.22848	0.614239	+	0.789120i	$0.289463\pi$
$762$	−67.4580	−2.44374
$763$	−7.57159	−0.274110
$764$	36.7355	1.32904
$765$	8.05226	0.291130
$766$	−3.84799	−0.139033
$767$	−13.0815	−0.472344
$768$	36.0552	1.30103
$769$	36.0049	1.29837	0.649186	−	0.760630i	$-0.275110\pi$
$769$	36.0049	1.29837	0.649186	+	0.760630i	$0.275110\pi$
$770$	−31.5171	−1.13580
$771$	23.0446	0.829929
$772$	−5.27913	−0.190000
$773$	4.75472	0.171016	0.0855078	−	0.996338i	$-0.472749\pi$
$773$	4.75472	0.171016	0.0855078	+	0.996338i	$0.472749\pi$
$774$	−12.2000	−0.438520
$775$	27.7847	0.998055
$776$	−55.7778	−2.00230
$777$	−15.8361	−0.568117
$778$	11.6407	0.417341
$779$	15.2240	0.545457
$780$	76.4246	2.73644
$781$	30.2055	1.08084
$782$	−64.6932	−2.31342
$783$	12.4905	0.446375
$784$	−1.55268	−0.0554530
$785$	−4.90134	−0.174936
$786$	−16.2365	−0.579138
$787$	−20.5082	−0.731039	−0.365519	−	0.930804i	$-0.619109\pi$
$787$	−20.5082	−0.731039	−0.365519	+	0.930804i	$0.619109\pi$
$788$	51.4207	1.83178
$789$	30.1670	1.07397
$790$	21.5172	0.765548
$791$	−13.9669	−0.496608
$792$	4.67097	0.165976
$793$	3.30069	0.117211
$794$	−62.5327	−2.21920
$795$	6.28268	0.222824
$796$	−73.2292	−2.59554
$797$	11.7873	0.417527	0.208763	−	0.977966i	$-0.433056\pi$
$797$	11.7873	0.417527	0.208763	+	0.977966i	$0.433056\pi$
$798$	−12.3287	−0.436430
$799$	39.4524	1.39573
$800$	29.3091	1.03623
$801$	8.60915	0.304190
$802$	61.7048	2.17887
$803$	6.69639	0.236310
$804$	54.0323	1.90557
$805$	−47.8969	−1.68815
$806$	40.9617	1.44281
$807$	−41.7805	−1.47074
$808$	−12.0971	−0.425574
$809$	19.2214	0.675787	0.337894	−	0.941184i	$-0.390286\pi$
$809$	19.2214	0.675787	0.337894	+	0.941184i	$0.390286\pi$
$810$	80.0142	2.81141
$811$	27.7705	0.975153	0.487577	−	0.873080i	$-0.337881\pi$
$811$	27.7705	0.975153	0.487577	+	0.873080i	$0.337881\pi$
$812$	−17.6074	−0.617897
$813$	56.4528	1.97989
$814$	−22.4836	−0.788051
$815$	−42.9832	−1.50564
$816$	−3.09335	−0.108289
$817$	11.4343	0.400034
$818$	41.7013	1.45805
$819$	4.68801	0.163812
$820$	112.293	3.92146
$821$	8.56562	0.298942	0.149471	−	0.988766i	$-0.452243\pi$
$821$	8.56562	0.298942	0.149471	+	0.988766i	$0.452243\pi$
$822$	15.4107	0.537511
$823$	−40.6392	−1.41659	−0.708297	−	0.705914i	$-0.750536\pi$
$823$	−40.6392	−1.41659	−0.708297	+	0.705914i	$0.750536\pi$
$824$	41.1882	1.43486
$825$	24.3287	0.847018
$826$	15.5655	0.541594
$827$	7.78134	0.270584	0.135292	−	0.990806i	$-0.456803\pi$
$827$	7.78134	0.270584	0.135292	+	0.990806i	$0.456803\pi$
$828$	17.7404	0.616522
$829$	−10.8179	−0.375721	−0.187861	−	0.982196i	$-0.560155\pi$
$829$	−10.8179	−0.375721	−0.187861	+	0.982196i	$0.560155\pi$
$830$	43.5671	1.51224
$831$	−19.8297	−0.687886
$832$	46.4773	1.61131
$833$	−12.4749	−0.432229
$834$	−57.5920	−1.99425
$835$	−50.9055	−1.76166
$836$	−10.9408	−0.378396
$837$	21.6117	0.747009
$838$	70.7005	2.44231
$839$	17.1284	0.591336	0.295668	−	0.955291i	$-0.404458\pi$
$839$	17.1284	0.591336	0.295668	+	0.955291i	$0.404458\pi$
$840$	−36.3869	−1.25547
$841$	−21.1068	−0.727820
$842$	2.72141	0.0937860
$843$	−39.9239	−1.37505
$844$	−3.34789	−0.115239
$845$	1.01964	0.0350766
$846$	−17.3086	−0.595082
$847$	11.4315	0.392793
$848$	−0.447889	−0.0153806
$849$	−31.2583	−1.07278
$850$	−47.5032	−1.62935
$851$	−34.1686	−1.17128
$852$	87.1527	2.98580
$853$	−11.5163	−0.394312	−0.197156	−	0.980372i	$-0.563170\pi$
$853$	−11.5163	−0.394312	−0.197156	+	0.980372i	$0.563170\pi$
$854$	−3.92748	−0.134395
$855$	3.31096	0.113232
$856$	5.06372	0.173074
$857$	7.93496	0.271053	0.135527	−	0.990774i	$-0.456727\pi$
$857$	7.93496	0.271053	0.135527	+	0.990774i	$0.456727\pi$
$858$	35.8668	1.22447
$859$	−49.3300	−1.68312	−0.841558	−	0.540166i	$-0.818361\pi$
$859$	−49.3300	−1.68312	−0.841558	+	0.540166i	$0.818361\pi$
$860$	84.3398	2.87596
$861$	37.1189	1.26501
$862$	−6.59084	−0.224485
$863$	−31.2389	−1.06338	−0.531692	−	0.846938i	$-0.678443\pi$
$863$	−31.2389	−1.06338	−0.531692	+	0.846938i	$0.678443\pi$
$864$	22.7974	0.775584
$865$	13.7180	0.466427
$866$	−76.8166	−2.61033
$867$	7.77427	0.264028
$868$	−30.4651	−1.03405
$869$	6.31191	0.214117
$870$	40.7663	1.38211
$871$	30.8076	1.04388
$872$	−12.4109	−0.420286
$873$	12.3747	0.418821
$874$	−26.6008	−0.899786
$875$	−4.40391	−0.148879
$876$	19.3212	0.652804
$877$	−7.40098	−0.249913	−0.124957	−	0.992162i	$-0.539879\pi$
$877$	−7.40098	−0.249913	−0.124957	+	0.992162i	$0.539879\pi$
$878$	−19.0797	−0.643910
$879$	38.2813	1.29120
$880$	−3.25159	−0.109611
$881$	−45.5784	−1.53557	−0.767787	−	0.640705i	$-0.778642\pi$
$881$	−45.5784	−1.53557	−0.767787	+	0.640705i	$0.778642\pi$
$882$	5.47300	0.184285
$883$	32.6686	1.09939	0.549693	−	0.835367i	$-0.314745\pi$
$883$	32.6686	1.09939	0.549693	+	0.835367i	$0.314745\pi$
$884$	−43.7736	−1.47226
$885$	−22.5262	−0.757209
$886$	−67.2085	−2.25791
$887$	9.07105	0.304576	0.152288	−	0.988336i	$-0.451336\pi$
$887$	9.07105	0.304576	0.152288	+	0.988336i	$0.451336\pi$
$888$	−25.9576	−0.871081
$889$	28.6063	0.959424
$890$	−95.2175	−3.19170
$891$	23.4716	0.786327
$892$	77.8520	2.60668
$893$	16.2222	0.542855
$894$	−43.5337	−1.45599
$895$	57.3046	1.91548
$896$	−36.0254	−1.20352
$897$	54.5071	1.81994
$898$	−54.3340	−1.81315
$899$	13.6572	0.455494
$900$	13.0265	0.434217
$901$	−3.59852	−0.119884
$902$	52.7003	1.75473
$903$	27.8788	0.927748
$904$	−22.8938	−0.761437
$905$	62.6887	2.08384
$906$	4.43265	0.147265
$907$	31.4090	1.04292	0.521459	−	0.853276i	$-0.325388\pi$
$907$	31.4090	1.04292	0.521459	+	0.853276i	$0.325388\pi$
$908$	29.3836	0.975130
$909$	2.68383	0.0890171
$910$	−51.8496	−1.71880
$911$	−15.9759	−0.529306	−0.264653	−	0.964344i	$-0.585257\pi$
$911$	−15.9759	−0.529306	−0.264653	+	0.964344i	$0.585257\pi$
$912$	−1.27194	−0.0421180
$913$	12.7801	0.422959
$914$	78.6513	2.60155
$915$	5.68378	0.187900
$916$	68.0475	2.24835
$917$	6.88528	0.227372
$918$	−36.9494	−1.21951
$919$	−23.1449	−0.763480	−0.381740	−	0.924270i	$-0.624675\pi$
$919$	−23.1449	−0.763480	−0.381740	+	0.924270i	$0.624675\pi$
$920$	−78.5099	−2.58839
$921$	32.6111	1.07457
$922$	−22.8463	−0.752403
$923$	49.6918	1.63563
$924$	−26.6757	−0.877567
$925$	−25.0895	−0.824938
$926$	−23.4683	−0.771215
$927$	−9.13794	−0.300129
$928$	14.4065	0.472918
$929$	−0.134367	−0.00440844	−0.00220422	−	0.999998i	$-0.500702\pi$
$929$	−0.134367	−0.00440844	−0.00220422	+	0.999998i	$0.500702\pi$
$930$	70.5358	2.31296
$931$	−5.12947	−0.168112
$932$	96.4635	3.15977
$933$	−16.9210	−0.553970
$934$	−85.7327	−2.80526
$935$	−26.1246	−0.854365
$936$	7.68431	0.251170
$937$	−58.4414	−1.90920	−0.954599	−	0.297894i	$-0.903716\pi$
$937$	−58.4414	−1.90920	−0.954599	+	0.297894i	$0.903716\pi$
$938$	−36.6578	−1.19692
$939$	−31.6215	−1.03193
$940$	119.656	3.90275
$941$	58.1044	1.89415	0.947074	−	0.321014i	$-0.104024\pi$
$941$	58.1044	1.89415	0.947074	+	0.321014i	$0.104024\pi$
$942$	−6.63695	−0.216243
$943$	80.0893	2.60806
$944$	1.60588	0.0522670
$945$	−27.3562	−0.889898
$946$	39.5815	1.28690
$947$	40.0725	1.30218	0.651090	−	0.759000i	$-0.274312\pi$
$947$	40.0725	1.30218	0.651090	+	0.759000i	$0.274312\pi$
$948$	18.2119	0.591495
$949$	11.0164	0.357607
$950$	−19.5326	−0.633721
$951$	−46.6398	−1.51240
$952$	20.8413	0.675470
$953$	17.5718	0.569207	0.284603	−	0.958645i	$-0.408138\pi$
$953$	17.5718	0.569207	0.284603	+	0.958645i	$0.408138\pi$
$954$	1.57875	0.0511138
$955$	36.0680	1.16713
$956$	−27.3023	−0.883020
$957$	11.9585	0.386564
$958$	97.2413	3.14172
$959$	−6.53509	−0.211029
$960$	80.0336	2.58307
$961$	−7.36965	−0.237731
$962$	−36.9883	−1.19255
$963$	−1.12343	−0.0362019
$964$	−72.7772	−2.34400
$965$	−5.18320	−0.166853
$966$	−64.8577	−2.08676
$967$	4.12595	0.132682	0.0663408	−	0.997797i	$-0.478868\pi$
$967$	4.12595	0.132682	0.0663408	+	0.997797i	$0.478868\pi$
$968$	18.7379	0.602260
$969$	−10.2192	−0.328289
$970$	−136.865	−4.39447
$971$	12.3135	0.395159	0.197579	−	0.980287i	$-0.436692\pi$
$971$	12.3135	0.395159	0.197579	+	0.980287i	$0.436692\pi$
$972$	23.2548	0.745898
$973$	24.4225	0.782949
$974$	−53.0864	−1.70100
$975$	40.0237	1.28179
$976$	−0.405194	−0.0129699
$977$	−55.6143	−1.77926	−0.889630	−	0.456683i	$-0.849038\pi$
$977$	−55.6143	−1.77926	−0.889630	+	0.456683i	$0.849038\pi$
$978$	−58.2039	−1.86116
$979$	−27.9314	−0.892690
$980$	−37.8354	−1.20861
$981$	2.75346	0.0879111
$982$	−58.5972	−1.86991
$983$	−3.71278	−0.118419	−0.0592096	−	0.998246i	$-0.518858\pi$
$983$	−3.71278	−0.118419	−0.0592096	+	0.998246i	$0.518858\pi$
$984$	60.8432	1.93961
$985$	50.4862	1.60863
$986$	−23.3497	−0.743605
$987$	39.5527	1.25898
$988$	−17.9990	−0.572624
$989$	60.1524	1.91273
$990$	11.4614	0.364268
$991$	33.0253	1.04908	0.524542	−	0.851385i	$-0.324237\pi$
$991$	33.0253	1.04908	0.524542	+	0.851385i	$0.324237\pi$
$992$	24.9268	0.791428
$993$	26.6243	0.844895
$994$	−59.1280	−1.87543
$995$	−71.8985	−2.27934
$996$	36.8747	1.16842
$997$	−16.2510	−0.514674	−0.257337	−	0.966322i	$-0.582845\pi$
$997$	−16.2510	−0.514674	−0.257337	+	0.966322i	$0.582845\pi$
$998$	−24.2191	−0.766641
$999$	−19.5153	−0.617437

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8003.2.a.c.1.19	✓	172

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8003.2.a.c.1.19	✓	172	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:	\( N \)	\(=\)	\( 8003 = 53 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30956 q^{2} -1.91926 q^{3} +3.33407 q^{4} +3.27349 q^{5} +4.43265 q^{6} -1.87972 q^{7} -3.08112 q^{8} +0.683571 q^{9} +O(q^{10})\)	\(q-2.30956 q^{2} -1.91926 q^{3} +3.33407 q^{4} +3.27349 q^{5} +4.43265 q^{6} -1.87972 q^{7} -3.08112 q^{8} +0.683571 q^{9} -7.56031 q^{10} -2.21776 q^{11} -6.39896 q^{12} -3.64849 q^{13} +4.34132 q^{14} -6.28268 q^{15} +0.447889 q^{16} +3.59852 q^{17} -1.57875 q^{18} +1.47965 q^{19} +10.9140 q^{20} +3.60767 q^{21} +5.12206 q^{22} +7.78404 q^{23} +5.91348 q^{24} +5.71571 q^{25} +8.42641 q^{26} +4.44584 q^{27} -6.26711 q^{28} +2.80949 q^{29} +14.5102 q^{30} +4.86111 q^{31} +5.12781 q^{32} +4.25647 q^{33} -8.31100 q^{34} -6.15322 q^{35} +2.27907 q^{36} -4.38957 q^{37} -3.41735 q^{38} +7.00241 q^{39} -10.0860 q^{40} +10.2889 q^{41} -8.33213 q^{42} +7.72765 q^{43} -7.39418 q^{44} +2.23766 q^{45} -17.9777 q^{46} +10.9635 q^{47} -0.859617 q^{48} -3.46667 q^{49} -13.2008 q^{50} -6.90651 q^{51} -12.1643 q^{52} -1.00000 q^{53} -10.2679 q^{54} -7.25981 q^{55} +5.79163 q^{56} -2.83985 q^{57} -6.48869 q^{58} +3.58544 q^{59} -20.9469 q^{60} -0.904674 q^{61} -11.2270 q^{62} -1.28492 q^{63} -12.7388 q^{64} -11.9433 q^{65} -9.83057 q^{66} -8.44392 q^{67} +11.9977 q^{68} -14.9396 q^{69} +14.2112 q^{70} -13.6198 q^{71} -2.10616 q^{72} -3.01943 q^{73} +10.1380 q^{74} -10.9699 q^{75} +4.93327 q^{76} +4.16876 q^{77} -16.1725 q^{78} -2.84607 q^{79} +1.46616 q^{80} -10.5834 q^{81} -23.7628 q^{82} -5.76261 q^{83} +12.0282 q^{84} +11.7797 q^{85} -17.8475 q^{86} -5.39215 q^{87} +6.83319 q^{88} +12.5944 q^{89} -5.16801 q^{90} +6.85812 q^{91} +25.9526 q^{92} -9.32974 q^{93} -25.3209 q^{94} +4.84363 q^{95} -9.84162 q^{96} +18.1031 q^{97} +8.00649 q^{98} -1.51600 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 172 q + 8 q^{2} + 25 q^{3} + 188 q^{4} + 27 q^{5} + 10 q^{6} + 31 q^{7} + 21 q^{8} + 179 q^{9}+O(q^{10}) \) 172 * q + 8 * q^2 + 25 * q^3 + 188 * q^4 + 27 * q^5 + 10 * q^6 + 31 * q^7 + 21 * q^8 + 179 * q^9	\( 172 q + 8 q^{2} + 25 q^{3} + 188 q^{4} + 27 q^{5} + 10 q^{6} + 31 q^{7} + 21 q^{8} + 179 q^{9} + 20 q^{10} - 3 q^{11} + 66 q^{12} + 121 q^{13} + 12 q^{14} + 30 q^{15} + 212 q^{16} + 8 q^{17} + 40 q^{18} + 41 q^{19} + 64 q^{20} + 56 q^{21} + 50 q^{22} + 28 q^{23} + 30 q^{24} + 231 q^{25} + 38 q^{26} + 100 q^{27} + 80 q^{28} + 26 q^{29} + 55 q^{30} + 66 q^{31} + 65 q^{32} + 99 q^{33} + 81 q^{34} + 36 q^{35} + 212 q^{36} + 153 q^{37} + q^{38} + 20 q^{39} + 59 q^{40} + 40 q^{41} + 50 q^{42} + 39 q^{43} - 51 q^{44} + 123 q^{45} + 59 q^{46} + 29 q^{47} + 128 q^{48} + 245 q^{49} + 19 q^{50} + 36 q^{51} + 215 q^{52} - 172 q^{53} + 40 q^{54} + 40 q^{55} + 15 q^{56} + 54 q^{57} + 44 q^{58} - 54 q^{60} + 100 q^{61} - 29 q^{62} + 92 q^{63} + 253 q^{64} + 77 q^{65} + 14 q^{66} + 126 q^{67} - 27 q^{68} + 47 q^{69} + 72 q^{70} + 38 q^{71} + 65 q^{72} + 185 q^{73} + 48 q^{74} + 75 q^{75} + 38 q^{76} + 120 q^{77} + 75 q^{78} + 79 q^{79} + 43 q^{80} + 232 q^{81} + 110 q^{82} + 90 q^{83} + 158 q^{84} + 115 q^{85} + 68 q^{86} + 61 q^{87} + 15 q^{88} - 36 q^{89} - 6 q^{90} + 33 q^{91} + 139 q^{92} + 103 q^{93} - 24 q^{94} - 45 q^{95} + 34 q^{96} + 159 q^{97} - 36 q^{98} + 27 q^{99}+O(q^{100}) \) 172 * q + 8 * q^2 + 25 * q^3 + 188 * q^4 + 27 * q^5 + 10 * q^6 + 31 * q^7 + 21 * q^8 + 179 * q^9 + 20 * q^10 - 3 * q^11 + 66 * q^12 + 121 * q^13 + 12 * q^14 + 30 * q^15 + 212 * q^16 + 8 * q^17 + 40 * q^18 + 41 * q^19 + 64 * q^20 + 56 * q^21 + 50 * q^22 + 28 * q^23 + 30 * q^24 + 231 * q^25 + 38 * q^26 + 100 * q^27 + 80 * q^28 + 26 * q^29 + 55 * q^30 + 66 * q^31 + 65 * q^32 + 99 * q^33 + 81 * q^34 + 36 * q^35 + 212 * q^36 + 153 * q^37 + q^38 + 20 * q^39 + 59 * q^40 + 40 * q^41 + 50 * q^42 + 39 * q^43 - 51 * q^44 + 123 * q^45 + 59 * q^46 + 29 * q^47 + 128 * q^48 + 245 * q^49 + 19 * q^50 + 36 * q^51 + 215 * q^52 - 172 * q^53 + 40 * q^54 + 40 * q^55 + 15 * q^56 + 54 * q^57 + 44 * q^58 - 54 * q^60 + 100 * q^61 - 29 * q^62 + 92 * q^63 + 253 * q^64 + 77 * q^65 + 14 * q^66 + 126 * q^67 - 27 * q^68 + 47 * q^69 + 72 * q^70 + 38 * q^71 + 65 * q^72 + 185 * q^73 + 48 * q^74 + 75 * q^75 + 38 * q^76 + 120 * q^77 + 75 * q^78 + 79 * q^79 + 43 * q^80 + 232 * q^81 + 110 * q^82 + 90 * q^83 + 158 * q^84 + 115 * q^85 + 68 * q^86 + 61 * q^87 + 15 * q^88 - 36 * q^89 - 6 * q^90 + 33 * q^91 + 139 * q^92 + 103 * q^93 - 24 * q^94 - 45 * q^95 + 34 * q^96 + 159 * q^97 - 36 * q^98 + 27 * q^99

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