LMFDB - Embedded newform 8038.2.a.a.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, 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("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.1837531447$
Analytic rank:	$1$
Dimension:	$75$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.53842 q^{3} +1.00000 q^{4} +2.78579 q^{5} -2.53842 q^{6} -2.33236 q^{7} +1.00000 q^{8} +3.44358 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.53842 q^{3} +1.00000 q^{4} +2.78579 q^{5} -2.53842 q^{6} -2.33236 q^{7} +1.00000 q^{8} +3.44358 q^{9} +2.78579 q^{10} +1.65786 q^{11} -2.53842 q^{12} +6.48185 q^{13} -2.33236 q^{14} -7.07152 q^{15} +1.00000 q^{16} +6.65733 q^{17} +3.44358 q^{18} -7.49137 q^{19} +2.78579 q^{20} +5.92052 q^{21} +1.65786 q^{22} -9.04726 q^{23} -2.53842 q^{24} +2.76065 q^{25} +6.48185 q^{26} -1.12599 q^{27} -2.33236 q^{28} -7.64788 q^{29} -7.07152 q^{30} -0.149330 q^{31} +1.00000 q^{32} -4.20834 q^{33} +6.65733 q^{34} -6.49748 q^{35} +3.44358 q^{36} -10.5394 q^{37} -7.49137 q^{38} -16.4537 q^{39} +2.78579 q^{40} -8.50425 q^{41} +5.92052 q^{42} -6.66822 q^{43} +1.65786 q^{44} +9.59310 q^{45} -9.04726 q^{46} -3.20501 q^{47} -2.53842 q^{48} -1.56008 q^{49} +2.76065 q^{50} -16.8991 q^{51} +6.48185 q^{52} -3.05885 q^{53} -1.12599 q^{54} +4.61845 q^{55} -2.33236 q^{56} +19.0163 q^{57} -7.64788 q^{58} +14.4686 q^{59} -7.07152 q^{60} -2.29357 q^{61} -0.149330 q^{62} -8.03168 q^{63} +1.00000 q^{64} +18.0571 q^{65} -4.20834 q^{66} +6.09738 q^{67} +6.65733 q^{68} +22.9658 q^{69} -6.49748 q^{70} -9.88290 q^{71} +3.44358 q^{72} -6.79109 q^{73} -10.5394 q^{74} -7.00768 q^{75} -7.49137 q^{76} -3.86673 q^{77} -16.4537 q^{78} +5.45629 q^{79} +2.78579 q^{80} -7.47250 q^{81} -8.50425 q^{82} -2.48393 q^{83} +5.92052 q^{84} +18.5460 q^{85} -6.66822 q^{86} +19.4135 q^{87} +1.65786 q^{88} +9.82202 q^{89} +9.59310 q^{90} -15.1180 q^{91} -9.04726 q^{92} +0.379063 q^{93} -3.20501 q^{94} -20.8694 q^{95} -2.53842 q^{96} +12.0378 q^{97} -1.56008 q^{98} +5.70897 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9}+O(q^{10}) $ 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9} - 29 q^{10} - 30 q^{11} - 30 q^{12} - 23 q^{13} - 31 q^{14} - 22 q^{15} + 75 q^{16} - 48 q^{17} + 55 q^{18} - 58 q^{19} - 29 q^{20} - 5 q^{21} - 30 q^{22} - 79 q^{23} - 30 q^{24} + 42 q^{25} - 23 q^{26} - 108 q^{27} - 31 q^{28} - 39 q^{29} - 22 q^{30} - 95 q^{31} + 75 q^{32} - 44 q^{33} - 48 q^{34} - 60 q^{35} + 55 q^{36} - 16 q^{37} - 58 q^{38} - 57 q^{39} - 29 q^{40} - 85 q^{41} - 5 q^{42} - 55 q^{43} - 30 q^{44} - 48 q^{45} - 79 q^{46} - 88 q^{47} - 30 q^{48} + 26 q^{49} + 42 q^{50} - 39 q^{51} - 23 q^{52} - 79 q^{53} - 108 q^{54} - 78 q^{55} - 31 q^{56} - 40 q^{57} - 39 q^{58} - 74 q^{59} - 22 q^{60} - 21 q^{61} - 95 q^{62} - 97 q^{63} + 75 q^{64} - 63 q^{65} - 44 q^{66} - 59 q^{67} - 48 q^{68} + 3 q^{69} - 60 q^{70} - 72 q^{71} + 55 q^{72} - 91 q^{73} - 16 q^{74} - 95 q^{75} - 58 q^{76} - 60 q^{77} - 57 q^{78} - 64 q^{79} - 29 q^{80} + 47 q^{81} - 85 q^{82} - 105 q^{83} - 5 q^{84} + 14 q^{85} - 55 q^{86} - 75 q^{87} - 30 q^{88} - 78 q^{89} - 48 q^{90} - 89 q^{91} - 79 q^{92} + 27 q^{93} - 88 q^{94} - 53 q^{95} - 30 q^{96} - 79 q^{97} + 26 q^{98} - 57 q^{99}+O(q^{100}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9 - 29 * q^10 - 30 * q^11 - 30 * q^12 - 23 * q^13 - 31 * q^14 - 22 * q^15 + 75 * q^16 - 48 * q^17 + 55 * q^18 - 58 * q^19 - 29 * q^20 - 5 * q^21 - 30 * q^22 - 79 * q^23 - 30 * q^24 + 42 * q^25 - 23 * q^26 - 108 * q^27 - 31 * q^28 - 39 * q^29 - 22 * q^30 - 95 * q^31 + 75 * q^32 - 44 * q^33 - 48 * q^34 - 60 * q^35 + 55 * q^36 - 16 * q^37 - 58 * q^38 - 57 * q^39 - 29 * q^40 - 85 * q^41 - 5 * q^42 - 55 * q^43 - 30 * q^44 - 48 * q^45 - 79 * q^46 - 88 * q^47 - 30 * q^48 + 26 * q^49 + 42 * q^50 - 39 * q^51 - 23 * q^52 - 79 * q^53 - 108 * q^54 - 78 * q^55 - 31 * q^56 - 40 * q^57 - 39 * q^58 - 74 * q^59 - 22 * q^60 - 21 * q^61 - 95 * q^62 - 97 * q^63 + 75 * q^64 - 63 * q^65 - 44 * q^66 - 59 * q^67 - 48 * q^68 + 3 * q^69 - 60 * q^70 - 72 * q^71 + 55 * q^72 - 91 * q^73 - 16 * q^74 - 95 * q^75 - 58 * q^76 - 60 * q^77 - 57 * q^78 - 64 * q^79 - 29 * q^80 + 47 * q^81 - 85 * q^82 - 105 * q^83 - 5 * q^84 + 14 * q^85 - 55 * q^86 - 75 * q^87 - 30 * q^88 - 78 * q^89 - 48 * q^90 - 89 * q^91 - 79 * q^92 + 27 * q^93 - 88 * q^94 - 53 * q^95 - 30 * q^96 - 79 * q^97 + 26 * q^98 - 57 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.53842	−1.46556	−0.732779	−	0.680467i	$-0.761777\pi$
$3$	−2.53842	−1.46556	−0.732779	+	0.680467i	$0.761777\pi$
$4$	1.00000	0.500000
$5$	2.78579	1.24584	0.622922	−	0.782284i	$-0.285945\pi$
$5$	2.78579	1.24584	0.622922	+	0.782284i	$0.285945\pi$
$6$	−2.53842	−1.03631
$7$	−2.33236	−0.881550	−0.440775	−	0.897618i	$-0.645296\pi$
$7$	−2.33236	−0.881550	−0.440775	+	0.897618i	$0.645296\pi$
$8$	1.00000	0.353553
$9$	3.44358	1.14786
$10$	2.78579	0.880945
$11$	1.65786	0.499863	0.249932	−	0.968263i	$-0.419592\pi$
$11$	1.65786	0.499863	0.249932	+	0.968263i	$0.419592\pi$
$12$	−2.53842	−0.732779
$13$	6.48185	1.79774	0.898871	−	0.438214i	$-0.144389\pi$
$13$	6.48185	1.79774	0.898871	+	0.438214i	$0.144389\pi$
$14$	−2.33236	−0.623350
$15$	−7.07152	−1.82586
$16$	1.00000	0.250000
$17$	6.65733	1.61464	0.807320	−	0.590114i	$-0.200917\pi$
$17$	6.65733	1.61464	0.807320	+	0.590114i	$0.200917\pi$
$18$	3.44358	0.811660
$19$	−7.49137	−1.71864	−0.859320	−	0.511439i	$-0.829113\pi$
$19$	−7.49137	−1.71864	−0.859320	+	0.511439i	$0.829113\pi$
$20$	2.78579	0.622922
$21$	5.92052	1.29196
$22$	1.65786	0.353457
$23$	−9.04726	−1.88648	−0.943242	−	0.332106i	$-0.892241\pi$
$23$	−9.04726	−1.88648	−0.943242	+	0.332106i	$0.892241\pi$
$24$	−2.53842	−0.518153
$25$	2.76065	0.552129
$26$	6.48185	1.27120
$27$	−1.12599	−0.216698
$28$	−2.33236	−0.440775
$29$	−7.64788	−1.42018	−0.710088	−	0.704113i	$-0.751345\pi$
$29$	−7.64788	−1.42018	−0.710088	+	0.704113i	$0.751345\pi$
$30$	−7.07152	−1.29108
$31$	−0.149330	−0.0268205	−0.0134103	−	0.999910i	$-0.504269\pi$
$31$	−0.149330	−0.0268205	−0.0134103	+	0.999910i	$0.504269\pi$
$32$	1.00000	0.176777
$33$	−4.20834	−0.732579
$34$	6.65733	1.14172
$35$	−6.49748	−1.09827
$36$	3.44358	0.573930
$37$	−10.5394	−1.73267	−0.866335	−	0.499464i	$-0.833530\pi$
$37$	−10.5394	−1.73267	−0.866335	+	0.499464i	$0.833530\pi$
$38$	−7.49137	−1.21526
$39$	−16.4537	−2.63469
$40$	2.78579	0.440473
$41$	−8.50425	−1.32814	−0.664071	−	0.747670i	$-0.731173\pi$
$41$	−8.50425	−1.32814	−0.664071	+	0.747670i	$0.731173\pi$
$42$	5.92052	0.913556
$43$	−6.66822	−1.01689	−0.508447	−	0.861093i	$-0.669780\pi$
$43$	−6.66822	−1.01689	−0.508447	+	0.861093i	$0.669780\pi$
$44$	1.65786	0.249932
$45$	9.59310	1.43006
$46$	−9.04726	−1.33395
$47$	−3.20501	−0.467498	−0.233749	−	0.972297i	$-0.575099\pi$
$47$	−3.20501	−0.467498	−0.233749	+	0.972297i	$0.575099\pi$
$48$	−2.53842	−0.366389
$49$	−1.56008	−0.222869
$50$	2.76065	0.390414
$51$	−16.8991	−2.36635
$52$	6.48185	0.898871
$53$	−3.05885	−0.420166	−0.210083	−	0.977684i	$-0.567373\pi$
$53$	−3.05885	−0.420166	−0.210083	+	0.977684i	$0.567373\pi$
$54$	−1.12599	−0.153228
$55$	4.61845	0.622752
$56$	−2.33236	−0.311675
$57$	19.0163	2.51877
$58$	−7.64788	−1.00422
$59$	14.4686	1.88365	0.941825	−	0.336103i	$-0.109109\pi$
$59$	14.4686	1.88365	0.941825	+	0.336103i	$0.109109\pi$
$60$	−7.07152	−0.912929
$61$	−2.29357	−0.293662	−0.146831	−	0.989162i	$-0.546907\pi$
$61$	−2.29357	−0.293662	−0.146831	+	0.989162i	$0.546907\pi$
$62$	−0.149330	−0.0189650
$63$	−8.03168	−1.01190
$64$	1.00000	0.125000
$65$	18.0571	2.23971
$66$	−4.20834	−0.518011
$67$	6.09738	0.744914	0.372457	−	0.928049i	$-0.378515\pi$
$67$	6.09738	0.744914	0.372457	+	0.928049i	$0.378515\pi$
$68$	6.65733	0.807320
$69$	22.9658	2.76475
$70$	−6.49748	−0.776598
$71$	−9.88290	−1.17288	−0.586442	−	0.809991i	$-0.699472\pi$
$71$	−9.88290	−1.17288	−0.586442	+	0.809991i	$0.699472\pi$
$72$	3.44358	0.405830
$73$	−6.79109	−0.794836	−0.397418	−	0.917638i	$-0.630094\pi$
$73$	−6.79109	−0.794836	−0.397418	+	0.917638i	$0.630094\pi$
$74$	−10.5394	−1.22518
$75$	−7.00768	−0.809177
$76$	−7.49137	−0.859320
$77$	−3.86673	−0.440655
$78$	−16.4537	−1.86301
$79$	5.45629	0.613881	0.306941	−	0.951729i	$-0.400695\pi$
$79$	5.45629	0.613881	0.306941	+	0.951729i	$0.400695\pi$
$80$	2.78579	0.311461
$81$	−7.47250	−0.830277
$82$	−8.50425	−0.939138
$83$	−2.48393	−0.272647	−0.136324	−	0.990664i	$-0.543529\pi$
$83$	−2.48393	−0.272647	−0.136324	+	0.990664i	$0.543529\pi$
$84$	5.92052	0.645982
$85$	18.5460	2.01159
$86$	−6.66822	−0.719053
$87$	19.4135	2.08135
$88$	1.65786	0.176728
$89$	9.82202	1.04113	0.520566	−	0.853822i	$-0.325721\pi$
$89$	9.82202	1.04113	0.520566	+	0.853822i	$0.325721\pi$
$90$	9.59310	1.01120
$91$	−15.1180	−1.58480
$92$	−9.04726	−0.943242
$93$	0.379063	0.0393070
$94$	−3.20501	−0.330571
$95$	−20.8694	−2.14116
$96$	−2.53842	−0.259076
$97$	12.0378	1.22226	0.611128	−	0.791532i	$-0.290716\pi$
$97$	12.0378	1.22226	0.611128	+	0.791532i	$0.290716\pi$
$98$	−1.56008	−0.157592
$99$	5.70897	0.573773
$100$	2.76065	0.276065
$101$	9.08793	0.904282	0.452141	−	0.891946i	$-0.350660\pi$
$101$	9.08793	0.904282	0.452141	+	0.891946i	$0.350660\pi$
$102$	−16.8991	−1.67326
$103$	0.874604	0.0861773	0.0430887	−	0.999071i	$-0.486280\pi$
$103$	0.874604	0.0861773	0.0430887	+	0.999071i	$0.486280\pi$
$104$	6.48185	0.635598
$105$	16.4933	1.60959
$106$	−3.05885	−0.297102
$107$	−6.91687	−0.668680	−0.334340	−	0.942453i	$-0.608513\pi$
$107$	−6.91687	−0.668680	−0.334340	+	0.942453i	$0.608513\pi$
$108$	−1.12599	−0.108349
$109$	−1.53898	−0.147407	−0.0737036	−	0.997280i	$-0.523482\pi$
$109$	−1.53898	−0.147407	−0.0737036	+	0.997280i	$0.523482\pi$
$110$	4.61845	0.440352
$111$	26.7535	2.53933
$112$	−2.33236	−0.220388
$113$	6.92378	0.651334	0.325667	−	0.945484i	$-0.394411\pi$
$113$	6.92378	0.651334	0.325667	+	0.945484i	$0.394411\pi$
$114$	19.0163	1.78104
$115$	−25.2038	−2.35027
$116$	−7.64788	−0.710088
$117$	22.3208	2.06356
$118$	14.4686	1.33194
$119$	−15.5273	−1.42339
$120$	−7.07152	−0.645538
$121$	−8.25150	−0.750137
$122$	−2.29357	−0.207650
$123$	21.5874	1.94647
$124$	−0.149330	−0.0134103
$125$	−6.23838	−0.557978
$126$	−8.03168	−0.715519
$127$	−6.96111	−0.617699	−0.308849	−	0.951111i	$-0.599944\pi$
$127$	−6.96111	−0.617699	−0.308849	+	0.951111i	$0.599944\pi$
$128$	1.00000	0.0883883
$129$	16.9268	1.49032
$130$	18.0571	1.58371
$131$	−2.78077	−0.242957	−0.121479	−	0.992594i	$-0.538764\pi$
$131$	−2.78077	−0.242957	−0.121479	+	0.992594i	$0.538764\pi$
$132$	−4.20834	−0.366289
$133$	17.4726	1.51507
$134$	6.09738	0.526734
$135$	−3.13679	−0.269971
$136$	6.65733	0.570862
$137$	2.58445	0.220804	0.110402	−	0.993887i	$-0.464786\pi$
$137$	2.58445	0.220804	0.110402	+	0.993887i	$0.464786\pi$
$138$	22.9658	1.95497
$139$	15.2398	1.29262	0.646311	−	0.763074i	$-0.276311\pi$
$139$	15.2398	1.29262	0.646311	+	0.763074i	$0.276311\pi$
$140$	−6.49748	−0.549137
$141$	8.13566	0.685146
$142$	−9.88290	−0.829354
$143$	10.7460	0.898625
$144$	3.44358	0.286965
$145$	−21.3054	−1.76932
$146$	−6.79109	−0.562034
$147$	3.96015	0.326627
$148$	−10.5394	−0.866335
$149$	−4.16882	−0.341523	−0.170761	−	0.985312i	$-0.554623\pi$
$149$	−4.16882	−0.341523	−0.170761	+	0.985312i	$0.554623\pi$
$150$	−7.00768	−0.572175
$151$	−10.2353	−0.832935	−0.416467	−	0.909151i	$-0.636732\pi$
$151$	−10.2353	−0.832935	−0.416467	+	0.909151i	$0.636732\pi$
$152$	−7.49137	−0.607631
$153$	22.9251	1.85338
$154$	−3.86673	−0.311590
$155$	−0.416003	−0.0334142
$156$	−16.4537	−1.31735
$157$	−12.0053	−0.958126	−0.479063	−	0.877781i	$-0.659023\pi$
$157$	−12.0053	−0.958126	−0.479063	+	0.877781i	$0.659023\pi$
$158$	5.45629	0.434079
$159$	7.76466	0.615777
$160$	2.78579	0.220236
$161$	21.1015	1.66303
$162$	−7.47250	−0.587095
$163$	−5.17732	−0.405519	−0.202759	−	0.979229i	$-0.564991\pi$
$163$	−5.17732	−0.405519	−0.202759	+	0.979229i	$0.564991\pi$
$164$	−8.50425	−0.664071
$165$	−11.7236	−0.912679
$166$	−2.48393	−0.192791
$167$	13.9075	1.07620	0.538098	−	0.842882i	$-0.319143\pi$
$167$	13.9075	1.07620	0.538098	+	0.842882i	$0.319143\pi$
$168$	5.92052	0.456778
$169$	29.0144	2.23187
$170$	18.5460	1.42241
$171$	−25.7972	−1.97276
$172$	−6.66822	−0.508447
$173$	−14.1119	−1.07291	−0.536454	−	0.843930i	$-0.680236\pi$
$173$	−14.1119	−1.07291	−0.536454	+	0.843930i	$0.680236\pi$
$174$	19.4135	1.47174
$175$	−6.43883	−0.486730
$176$	1.65786	0.124966
$177$	−36.7274	−2.76060
$178$	9.82202	0.736191
$179$	−19.6463	−1.46843	−0.734217	−	0.678915i	$-0.762451\pi$
$179$	−19.6463	−1.46843	−0.734217	+	0.678915i	$0.762451\pi$
$180$	9.59310	0.715028
$181$	3.48901	0.259336	0.129668	−	0.991557i	$-0.458609\pi$
$181$	3.48901	0.259336	0.129668	+	0.991557i	$0.458609\pi$
$182$	−15.1180	−1.12062
$183$	5.82206	0.430379
$184$	−9.04726	−0.666973
$185$	−29.3606	−2.15864
$186$	0.379063	0.0277943
$187$	11.0369	0.807100
$188$	−3.20501	−0.233749
$189$	2.62623	0.191030
$190$	−20.8694	−1.51403
$191$	−24.7184	−1.78856	−0.894282	−	0.447504i	$-0.852313\pi$
$191$	−24.7184	−1.78856	−0.894282	+	0.447504i	$0.852313\pi$
$192$	−2.53842	−0.183195
$193$	−15.9019	−1.14465	−0.572323	−	0.820029i	$-0.693957\pi$
$193$	−15.9019	−1.14465	−0.572323	+	0.820029i	$0.693957\pi$
$194$	12.0378	0.864265
$195$	−45.8365	−3.28242
$196$	−1.56008	−0.111434
$197$	17.3682	1.23743	0.618717	−	0.785614i	$-0.287653\pi$
$197$	17.3682	1.23743	0.618717	+	0.785614i	$0.287653\pi$
$198$	5.70897	0.405719
$199$	13.9870	0.991509	0.495754	−	0.868463i	$-0.334892\pi$
$199$	13.9870	0.991509	0.495754	+	0.868463i	$0.334892\pi$
$200$	2.76065	0.195207
$201$	−15.4777	−1.09171
$202$	9.08793	0.639424
$203$	17.8376	1.25196
$204$	−16.8991	−1.18317
$205$	−23.6911	−1.65466
$206$	0.874604	0.0609366
$207$	−31.1550	−2.16542
$208$	6.48185	0.449435
$209$	−12.4196	−0.859085
$210$	16.4933	1.13815
$211$	−21.6379	−1.48961	−0.744806	−	0.667281i	$-0.767458\pi$
$211$	−21.6379	−1.48961	−0.744806	+	0.667281i	$0.767458\pi$
$212$	−3.05885	−0.210083
$213$	25.0870	1.71893
$214$	−6.91687	−0.472828
$215$	−18.5763	−1.26689
$216$	−1.12599	−0.0766141
$217$	0.348292	0.0236436
$218$	−1.53898	−0.104233
$219$	17.2386	1.16488
$220$	4.61845	0.311376
$221$	43.1518	2.90271
$222$	26.7535	1.79558
$223$	11.6587	0.780723	0.390361	−	0.920662i	$-0.372350\pi$
$223$	11.6587	0.780723	0.390361	+	0.920662i	$0.372350\pi$
$224$	−2.33236	−0.155838
$225$	9.50650	0.633767
$226$	6.92378	0.460563
$227$	−13.3567	−0.886516	−0.443258	−	0.896394i	$-0.646177\pi$
$227$	−13.3567	−0.886516	−0.443258	+	0.896394i	$0.646177\pi$
$228$	19.0163	1.25938
$229$	−24.9308	−1.64747	−0.823735	−	0.566975i	$-0.808114\pi$
$229$	−24.9308	−1.64747	−0.823735	+	0.566975i	$0.808114\pi$
$230$	−25.2038	−1.66189
$231$	9.81539	0.645805
$232$	−7.64788	−0.502108
$233$	14.3173	0.937960	0.468980	−	0.883209i	$-0.344622\pi$
$233$	14.3173	0.937960	0.468980	+	0.883209i	$0.344622\pi$
$234$	22.3208	1.45915
$235$	−8.92849	−0.582430
$236$	14.4686	0.941825
$237$	−13.8504	−0.899678
$238$	−15.5273	−1.00649
$239$	−17.9043	−1.15813	−0.579065	−	0.815281i	$-0.696582\pi$
$239$	−17.9043	−1.15813	−0.579065	+	0.815281i	$0.696582\pi$
$240$	−7.07152	−0.456464
$241$	−12.9863	−0.836523	−0.418261	−	0.908327i	$-0.637360\pi$
$241$	−12.9863	−0.836523	−0.418261	+	0.908327i	$0.637360\pi$
$242$	−8.25150	−0.530427
$243$	22.3463	1.43352
$244$	−2.29357	−0.146831
$245$	−4.34607	−0.277660
$246$	21.5874	1.37636
$247$	−48.5580	−3.08967
$248$	−0.149330	−0.00948248
$249$	6.30527	0.399580
$250$	−6.23838	−0.394550
$251$	10.3679	0.654418	0.327209	−	0.944952i	$-0.393892\pi$
$251$	10.3679	0.654418	0.327209	+	0.944952i	$0.393892\pi$
$252$	−8.03168	−0.505948
$253$	−14.9991	−0.942984
$254$	−6.96111	−0.436779
$255$	−47.0774	−2.94810
$256$	1.00000	0.0625000
$257$	−25.7968	−1.60916	−0.804579	−	0.593846i	$-0.797609\pi$
$257$	−25.7968	−1.60916	−0.804579	+	0.593846i	$0.797609\pi$
$258$	16.9268	1.05381
$259$	24.5818	1.52744
$260$	18.0571	1.11985
$261$	−26.3361	−1.63016
$262$	−2.78077	−0.171797
$263$	−5.52634	−0.340769	−0.170384	−	0.985378i	$-0.554501\pi$
$263$	−5.52634	−0.340769	−0.170384	+	0.985378i	$0.554501\pi$
$264$	−4.20834	−0.259006
$265$	−8.52133	−0.523461
$266$	17.4726	1.07131
$267$	−24.9324	−1.52584
$268$	6.09738	0.372457
$269$	−1.68187	−0.102546	−0.0512728	−	0.998685i	$-0.516328\pi$
$269$	−1.68187	−0.102546	−0.0512728	+	0.998685i	$0.516328\pi$
$270$	−3.13679	−0.190899
$271$	−0.697683	−0.0423812	−0.0211906	−	0.999775i	$-0.506746\pi$
$271$	−0.697683	−0.0423812	−0.0211906	+	0.999775i	$0.506746\pi$
$272$	6.65733	0.403660
$273$	38.3759	2.32262
$274$	2.58445	0.156132
$275$	4.57676	0.275989
$276$	22.9658	1.38238
$277$	7.25026	0.435626	0.217813	−	0.975991i	$-0.430108\pi$
$277$	7.25026	0.435626	0.217813	+	0.975991i	$0.430108\pi$
$278$	15.2398	0.914021
$279$	−0.514231	−0.0307862
$280$	−6.49748	−0.388299
$281$	−7.00415	−0.417832	−0.208916	−	0.977934i	$-0.566994\pi$
$281$	−7.00415	−0.417832	−0.208916	+	0.977934i	$0.566994\pi$
$282$	8.13566	0.484471
$283$	1.66364	0.0988932	0.0494466	−	0.998777i	$-0.484254\pi$
$283$	1.66364	0.0988932	0.0494466	+	0.998777i	$0.484254\pi$
$284$	−9.88290	−0.586442
$285$	52.9754	3.13799
$286$	10.7460	0.635424
$287$	19.8350	1.17082
$288$	3.44358	0.202915
$289$	27.3201	1.60706
$290$	−21.3054	−1.25110
$291$	−30.5571	−1.79129
$292$	−6.79109	−0.397418
$293$	27.6045	1.61267	0.806337	−	0.591457i	$-0.201447\pi$
$293$	27.6045	1.61267	0.806337	+	0.591457i	$0.201447\pi$
$294$	3.96015	0.230960
$295$	40.3065	2.34674
$296$	−10.5394	−0.612591
$297$	−1.86674	−0.108319
$298$	−4.16882	−0.241493
$299$	−58.6430	−3.39141
$300$	−7.00768	−0.404589
$301$	15.5527	0.896444
$302$	−10.2353	−0.588974
$303$	−23.0690	−1.32528
$304$	−7.49137	−0.429660
$305$	−6.38942	−0.365857
$306$	22.9251	1.31054
$307$	34.2077	1.95234	0.976169	−	0.217011i	$-0.0696309\pi$
$307$	34.2077	1.95234	0.976169	+	0.217011i	$0.0696309\pi$
$308$	−3.86673	−0.220327
$309$	−2.22011	−0.126298
$310$	−0.416003	−0.0236274
$311$	12.5159	0.709713	0.354857	−	0.934921i	$-0.384530\pi$
$311$	12.5159	0.709713	0.354857	+	0.934921i	$0.384530\pi$
$312$	−16.4537	−0.931505
$313$	−30.1496	−1.70416	−0.852078	−	0.523416i	$-0.824658\pi$
$313$	−30.1496	−1.70416	−0.852078	+	0.523416i	$0.824658\pi$
$314$	−12.0053	−0.677497
$315$	−22.3746	−1.26067
$316$	5.45629	0.306941
$317$	−8.73279	−0.490482	−0.245241	−	0.969462i	$-0.578867\pi$
$317$	−8.73279	−0.490482	−0.245241	+	0.969462i	$0.578867\pi$
$318$	7.76466	0.435420
$319$	−12.6791	−0.709894
$320$	2.78579	0.155731
$321$	17.5579	0.979989
$322$	21.1015	1.17594
$323$	−49.8726	−2.77498
$324$	−7.47250	−0.415139
$325$	17.8941	0.992585
$326$	−5.17732	−0.286745
$327$	3.90657	0.216034
$328$	−8.50425	−0.469569
$329$	7.47524	0.412123
$330$	−11.7236	−0.645362
$331$	−10.4379	−0.573717	−0.286858	−	0.957973i	$-0.592611\pi$
$331$	−10.4379	−0.573717	−0.286858	+	0.957973i	$0.592611\pi$
$332$	−2.48393	−0.136324
$333$	−36.2933	−1.98886
$334$	13.9075	0.760985
$335$	16.9860	0.928047
$336$	5.92052	0.322991
$337$	10.4767	0.570702	0.285351	−	0.958423i	$-0.407890\pi$
$337$	10.4767	0.570702	0.285351	+	0.958423i	$0.407890\pi$
$338$	29.0144	1.57817
$339$	−17.5755	−0.954568
$340$	18.5460	1.00580
$341$	−0.247569	−0.0134066
$342$	−25.7972	−1.39495
$343$	19.9652	1.07802
$344$	−6.66822	−0.359527
$345$	63.9778	3.44445
$346$	−14.1119	−0.758660
$347$	−5.55275	−0.298087	−0.149043	−	0.988831i	$-0.547619\pi$
$347$	−5.55275	−0.298087	−0.149043	+	0.988831i	$0.547619\pi$
$348$	19.4135	1.04067
$349$	25.7503	1.37838	0.689191	−	0.724579i	$-0.257966\pi$
$349$	25.7503	1.37838	0.689191	+	0.724579i	$0.257966\pi$
$350$	−6.43883	−0.344170
$351$	−7.29852	−0.389566
$352$	1.65786	0.0883642
$353$	6.96541	0.370731	0.185366	−	0.982670i	$-0.440653\pi$
$353$	6.96541	0.370731	0.185366	+	0.982670i	$0.440653\pi$
$354$	−36.7274	−1.95204
$355$	−27.5317	−1.46123
$356$	9.82202	0.520566
$357$	39.4149	2.08606
$358$	−19.6463	−1.03834
$359$	33.8592	1.78702	0.893510	−	0.449044i	$-0.148235\pi$
$359$	33.8592	1.78702	0.893510	+	0.449044i	$0.148235\pi$
$360$	9.59310	0.505601
$361$	37.1207	1.95372
$362$	3.48901	0.183378
$363$	20.9458	1.09937
$364$	−15.1180	−0.792400
$365$	−18.9186	−0.990243
$366$	5.82206	0.304324
$367$	−21.2171	−1.10752	−0.553761	−	0.832676i	$-0.686808\pi$
$367$	−21.2171	−1.10752	−0.553761	+	0.832676i	$0.686808\pi$
$368$	−9.04726	−0.471621
$369$	−29.2851	−1.52452
$370$	−29.3606	−1.52639
$371$	7.13436	0.370397
$372$	0.379063	0.0196535
$373$	−3.39962	−0.176026	−0.0880129	−	0.996119i	$-0.528052\pi$
$373$	−3.39962	−0.176026	−0.0880129	+	0.996119i	$0.528052\pi$
$374$	11.0369	0.570706
$375$	15.8356	0.817749
$376$	−3.20501	−0.165286
$377$	−49.5724	−2.55311
$378$	2.62623	0.135078
$379$	−2.93410	−0.150715	−0.0753573	−	0.997157i	$-0.524010\pi$
$379$	−2.93410	−0.150715	−0.0753573	+	0.997157i	$0.524010\pi$
$380$	−20.8694	−1.07058
$381$	17.6702	0.905273
$382$	−24.7184	−1.26471
$383$	−0.522717	−0.0267096	−0.0133548	−	0.999911i	$-0.504251\pi$
$383$	−0.522717	−0.0267096	−0.0133548	+	0.999911i	$0.504251\pi$
$384$	−2.53842	−0.129538
$385$	−10.7719	−0.548987
$386$	−15.9019	−0.809386
$387$	−22.9626	−1.16725
$388$	12.0378	0.611128
$389$	4.43961	0.225097	0.112549	−	0.993646i	$-0.464099\pi$
$389$	4.43961	0.225097	0.112549	+	0.993646i	$0.464099\pi$
$390$	−45.8365	−2.32102
$391$	−60.2306	−3.04599
$392$	−1.56008	−0.0787961
$393$	7.05877	0.356068
$394$	17.3682	0.874999
$395$	15.2001	0.764800
$396$	5.70897	0.286887
$397$	−7.72706	−0.387810	−0.193905	−	0.981020i	$-0.562115\pi$
$397$	−7.72706	−0.387810	−0.193905	+	0.981020i	$0.562115\pi$
$398$	13.9870	0.701102
$399$	−44.3528	−2.22042
$400$	2.76065	0.138032
$401$	29.2893	1.46264	0.731320	−	0.682035i	$-0.238905\pi$
$401$	29.2893	1.46264	0.731320	+	0.682035i	$0.238905\pi$
$402$	−15.4777	−0.771959
$403$	−0.967936	−0.0482163
$404$	9.08793	0.452141
$405$	−20.8168	−1.03440
$406$	17.8376	0.885267
$407$	−17.4729	−0.866098
$408$	−16.8991	−0.836631
$409$	−9.29022	−0.459372	−0.229686	−	0.973265i	$-0.573770\pi$
$409$	−9.29022	−0.459372	−0.229686	+	0.973265i	$0.573770\pi$
$410$	−23.6911	−1.17002
$411$	−6.56042	−0.323602
$412$	0.874604	0.0430887
$413$	−33.7460	−1.66053
$414$	−31.1550	−1.53118
$415$	−6.91973	−0.339676
$416$	6.48185	0.317799
$417$	−38.6850	−1.89441
$418$	−12.4196	−0.607465
$419$	−9.43895	−0.461123	−0.230561	−	0.973058i	$-0.574056\pi$
$419$	−9.43895	−0.461123	−0.230561	+	0.973058i	$0.574056\pi$
$420$	16.4933	0.804793
$421$	−13.6953	−0.667468	−0.333734	−	0.942667i	$-0.608309\pi$
$421$	−13.6953	−0.667468	−0.333734	+	0.942667i	$0.608309\pi$
$422$	−21.6379	−1.05331
$423$	−11.0367	−0.536623
$424$	−3.05885	−0.148551
$425$	18.3785	0.891490
$426$	25.0870	1.21547
$427$	5.34945	0.258878
$428$	−6.91687	−0.334340
$429$	−27.2779	−1.31699
$430$	−18.5763	−0.895828
$431$	24.3705	1.17388	0.586942	−	0.809629i	$-0.300332\pi$
$431$	24.3705	1.17388	0.586942	+	0.809629i	$0.300332\pi$
$432$	−1.12599	−0.0541744
$433$	15.3883	0.739515	0.369758	−	0.929128i	$-0.379441\pi$
$433$	15.3883	0.739515	0.369758	+	0.929128i	$0.379441\pi$
$434$	0.348292	0.0167186
$435$	54.0821	2.59304
$436$	−1.53898	−0.0737036
$437$	67.7764	3.24219
$438$	17.2386	0.823694
$439$	−4.82832	−0.230443	−0.115221	−	0.993340i	$-0.536758\pi$
$439$	−4.82832	−0.230443	−0.115221	+	0.993340i	$0.536758\pi$
$440$	4.61845	0.220176
$441$	−5.37227	−0.255822
$442$	43.1518	2.05252
$443$	40.9451	1.94536	0.972681	−	0.232146i	$-0.0745748\pi$
$443$	40.9451	1.94536	0.972681	+	0.232146i	$0.0745748\pi$
$444$	26.7535	1.26966
$445$	27.3621	1.29709
$446$	11.6587	0.552054
$447$	10.5822	0.500521
$448$	−2.33236	−0.110194
$449$	−21.9733	−1.03698	−0.518492	−	0.855083i	$-0.673506\pi$
$449$	−21.9733	−1.03698	−0.518492	+	0.855083i	$0.673506\pi$
$450$	9.50650	0.448141
$451$	−14.0989	−0.663889
$452$	6.92378	0.325667
$453$	25.9814	1.22071
$454$	−13.3567	−0.626861
$455$	−42.1157	−1.97441
$456$	19.0163	0.890518
$457$	−12.1240	−0.567135	−0.283567	−	0.958952i	$-0.591518\pi$
$457$	−12.1240	−0.567135	−0.283567	+	0.958952i	$0.591518\pi$
$458$	−24.9308	−1.16494
$459$	−7.49611	−0.349889
$460$	−25.2038	−1.17513
$461$	37.2463	1.73473	0.867366	−	0.497671i	$-0.165811\pi$
$461$	37.2463	1.73473	0.867366	+	0.497671i	$0.165811\pi$
$462$	9.81539	0.456653
$463$	14.4253	0.670401	0.335201	−	0.942147i	$-0.391196\pi$
$463$	14.4253	0.670401	0.335201	+	0.942147i	$0.391196\pi$
$464$	−7.64788	−0.355044
$465$	1.05599	0.0489704
$466$	14.3173	0.663238
$467$	2.71364	0.125572	0.0627861	−	0.998027i	$-0.480001\pi$
$467$	2.71364	0.125572	0.0627861	+	0.998027i	$0.480001\pi$
$468$	22.3208	1.03178
$469$	−14.2213	−0.656679
$470$	−8.92849	−0.411840
$471$	30.4745	1.40419
$472$	14.4686	0.665971
$473$	−11.0550	−0.508308
$474$	−13.8504	−0.636169
$475$	−20.6810	−0.948911
$476$	−15.5273	−0.711693
$477$	−10.5334	−0.482292
$478$	−17.9043	−0.818922
$479$	1.36125	0.0621970	0.0310985	−	0.999516i	$-0.490099\pi$
$479$	1.36125	0.0621970	0.0310985	+	0.999516i	$0.490099\pi$
$480$	−7.07152	−0.322769
$481$	−68.3149	−3.11489
$482$	−12.9863	−0.591511
$483$	−53.5645	−2.43727
$484$	−8.25150	−0.375068
$485$	33.5349	1.52274
$486$	22.3463	1.01365
$487$	−22.3546	−1.01298	−0.506492	−	0.862245i	$-0.669058\pi$
$487$	−22.3546	−1.01298	−0.506492	+	0.862245i	$0.669058\pi$
$488$	−2.29357	−0.103825
$489$	13.1422	0.594311
$490$	−4.34607	−0.196335
$491$	−17.7968	−0.803157	−0.401578	−	0.915825i	$-0.631538\pi$
$491$	−17.7968	−0.803157	−0.401578	+	0.915825i	$0.631538\pi$
$492$	21.5874	0.973234
$493$	−50.9145	−2.29307
$494$	−48.5580	−2.18473
$495$	15.9040	0.714832
$496$	−0.149330	−0.00670513
$497$	23.0505	1.03396
$498$	6.30527	0.282546
$499$	−21.8916	−0.980004	−0.490002	−	0.871721i	$-0.663004\pi$
$499$	−21.8916	−0.980004	−0.490002	+	0.871721i	$0.663004\pi$
$500$	−6.23838	−0.278989
$501$	−35.3031	−1.57723
$502$	10.3679	0.462743
$503$	35.7204	1.59270	0.796348	−	0.604839i	$-0.206762\pi$
$503$	35.7204	1.59270	0.796348	+	0.604839i	$0.206762\pi$
$504$	−8.03168	−0.357759
$505$	25.3171	1.12660
$506$	−14.9991	−0.666791
$507$	−73.6507	−3.27094
$508$	−6.96111	−0.308849
$509$	−18.0444	−0.799803	−0.399902	−	0.916558i	$-0.630956\pi$
$509$	−18.0444	−0.799803	−0.399902	+	0.916558i	$0.630956\pi$
$510$	−47.0774	−2.08462
$511$	15.8393	0.700688
$512$	1.00000	0.0441942
$513$	8.43524	0.372425
$514$	−25.7968	−1.13785
$515$	2.43647	0.107364
$516$	16.9268	0.745159
$517$	−5.31345	−0.233685
$518$	24.5818	1.08006
$519$	35.8219	1.57241
$520$	18.0571	0.791856
$521$	18.3005	0.801760	0.400880	−	0.916131i	$-0.368705\pi$
$521$	18.3005	0.801760	0.400880	+	0.916131i	$0.368705\pi$
$522$	−26.3361	−1.15270
$523$	−7.83035	−0.342397	−0.171199	−	0.985237i	$-0.554764\pi$
$523$	−7.83035	−0.342397	−0.171199	+	0.985237i	$0.554764\pi$
$524$	−2.78077	−0.121479
$525$	16.3445	0.713330
$526$	−5.52634	−0.240960
$527$	−0.994141	−0.0433055
$528$	−4.20834	−0.183145
$529$	58.8529	2.55882
$530$	−8.52133	−0.370143
$531$	49.8238	2.16217
$532$	17.4726	0.757534
$533$	−55.1233	−2.38765
$534$	−24.9324	−1.07893
$535$	−19.2690	−0.833071
$536$	6.09738	0.263367
$537$	49.8706	2.15208
$538$	−1.68187	−0.0725107
$539$	−2.58640	−0.111404
$540$	−3.13679	−0.134986
$541$	−32.7933	−1.40990	−0.704948	−	0.709259i	$-0.749030\pi$
$541$	−32.7933	−1.40990	−0.704948	+	0.709259i	$0.749030\pi$
$542$	−0.697683	−0.0299681
$543$	−8.85658	−0.380072
$544$	6.65733	0.285431
$545$	−4.28727	−0.183646
$546$	38.3759	1.64234
$547$	−31.8921	−1.36361	−0.681803	−	0.731536i	$-0.738804\pi$
$547$	−31.8921	−1.36361	−0.681803	+	0.731536i	$0.738804\pi$
$548$	2.58445	0.110402
$549$	−7.89811	−0.337083
$550$	4.57676	0.195154
$551$	57.2931	2.44077
$552$	22.9658	0.977487
$553$	−12.7261	−0.541167
$554$	7.25026	0.308034
$555$	74.5297	3.16361
$556$	15.2398	0.646311
$557$	19.9963	0.847271	0.423636	−	0.905833i	$-0.360754\pi$
$557$	19.9963	0.847271	0.423636	+	0.905833i	$0.360754\pi$
$558$	−0.514231	−0.0217691
$559$	−43.2224	−1.82811
$560$	−6.49748	−0.274569
$561$	−28.0163	−1.18285
$562$	−7.00415	−0.295452
$563$	−34.5311	−1.45531	−0.727657	−	0.685941i	$-0.759391\pi$
$563$	−34.5311	−1.45531	−0.727657	+	0.685941i	$0.759391\pi$
$564$	8.13566	0.342573
$565$	19.2882	0.811462
$566$	1.66364	0.0699280
$567$	17.4286	0.731931
$568$	−9.88290	−0.414677
$569$	7.29304	0.305740	0.152870	−	0.988246i	$-0.451148\pi$
$569$	7.29304	0.305740	0.152870	+	0.988246i	$0.451148\pi$
$570$	52.9754	2.21889
$571$	−2.94309	−0.123164	−0.0615822	−	0.998102i	$-0.519615\pi$
$571$	−2.94309	−0.123164	−0.0615822	+	0.998102i	$0.519615\pi$
$572$	10.7460	0.449313
$573$	62.7458	2.62124
$574$	19.8350	0.827897
$575$	−24.9763	−1.04158
$576$	3.44358	0.143483
$577$	23.5195	0.979131	0.489565	−	0.871967i	$-0.337155\pi$
$577$	23.5195	0.979131	0.489565	+	0.871967i	$0.337155\pi$
$578$	27.3201	1.13637
$579$	40.3658	1.67754
$580$	−21.3054	−0.884659
$581$	5.79344	0.240352
$582$	−30.5571	−1.26663
$583$	−5.07115	−0.210026
$584$	−6.79109	−0.281017
$585$	62.1811	2.57087
$586$	27.6045	1.14033
$587$	−12.3131	−0.508218	−0.254109	−	0.967176i	$-0.581782\pi$
$587$	−12.3131	−0.508218	−0.254109	+	0.967176i	$0.581782\pi$
$588$	3.96015	0.163314
$589$	1.11869	0.0460948
$590$	40.3065	1.65939
$591$	−44.0879	−1.81353
$592$	−10.5394	−0.433167
$593$	−40.7490	−1.67336	−0.836680	−	0.547692i	$-0.815507\pi$
$593$	−40.7490	−1.67336	−0.836680	+	0.547692i	$0.815507\pi$
$594$	−1.86674	−0.0765932
$595$	−43.2559	−1.77332
$596$	−4.16882	−0.170761
$597$	−35.5048	−1.45311
$598$	−58.6430	−2.39809
$599$	21.9352	0.896247	0.448124	−	0.893972i	$-0.352092\pi$
$599$	21.9352	0.896247	0.448124	+	0.893972i	$0.352092\pi$
$600$	−7.00768	−0.286087
$601$	0.318391	0.0129875	0.00649373	−	0.999979i	$-0.497933\pi$
$601$	0.318391	0.0129875	0.00649373	+	0.999979i	$0.497933\pi$
$602$	15.5527	0.633881
$603$	20.9968	0.855057
$604$	−10.2353	−0.416467
$605$	−22.9870	−0.934554
$606$	−23.0690	−0.937113
$607$	14.0939	0.572052	0.286026	−	0.958222i	$-0.407666\pi$
$607$	14.0939	0.572052	0.286026	+	0.958222i	$0.407666\pi$
$608$	−7.49137	−0.303815
$609$	−45.2794	−1.83481
$610$	−6.38942	−0.258700
$611$	−20.7744	−0.840441
$612$	22.9251	0.926691
$613$	−25.5110	−1.03038	−0.515191	−	0.857076i	$-0.672279\pi$
$613$	−25.5110	−1.03038	−0.515191	+	0.857076i	$0.672279\pi$
$614$	34.2077	1.38051
$615$	60.1380	2.42500
$616$	−3.86673	−0.155795
$617$	−21.5151	−0.866164	−0.433082	−	0.901354i	$-0.642574\pi$
$617$	−21.5151	−0.866164	−0.433082	+	0.901354i	$0.642574\pi$
$618$	−2.22011	−0.0893061
$619$	−30.5377	−1.22741	−0.613706	−	0.789534i	$-0.710322\pi$
$619$	−30.5377	−1.22741	−0.613706	+	0.789534i	$0.710322\pi$
$620$	−0.416003	−0.0167071
$621$	10.1872	0.408796
$622$	12.5159	0.501843
$623$	−22.9085	−0.917810
$624$	−16.4537	−0.658674
$625$	−31.1821	−1.24728
$626$	−30.1496	−1.20502
$627$	31.5263	1.25904
$628$	−12.0053	−0.479063
$629$	−70.1644	−2.79764
$630$	−22.3746	−0.891425
$631$	−15.7183	−0.625735	−0.312868	−	0.949797i	$-0.601290\pi$
$631$	−15.7183	−0.625735	−0.312868	+	0.949797i	$0.601290\pi$
$632$	5.45629	0.217040
$633$	54.9260	2.18311
$634$	−8.73279	−0.346823
$635$	−19.3922	−0.769557
$636$	7.76466	0.307889
$637$	−10.1122	−0.400661
$638$	−12.6791	−0.501971
$639$	−34.0326	−1.34631
$640$	2.78579	0.110118
$641$	−0.220484	−0.00870858	−0.00435429	−	0.999991i	$-0.501386\pi$
$641$	−0.220484	−0.00870858	−0.00435429	+	0.999991i	$0.501386\pi$
$642$	17.5579	0.692957
$643$	15.9992	0.630946	0.315473	−	0.948935i	$-0.397837\pi$
$643$	15.9992	0.630946	0.315473	+	0.948935i	$0.397837\pi$
$644$	21.1015	0.831515
$645$	47.1545	1.85670
$646$	−49.8726	−1.96221
$647$	0.708068	0.0278370	0.0139185	−	0.999903i	$-0.495569\pi$
$647$	0.708068	0.0278370	0.0139185	+	0.999903i	$0.495569\pi$
$648$	−7.47250	−0.293547
$649$	23.9869	0.941568
$650$	17.8941	0.701864
$651$	−0.884113	−0.0346511
$652$	−5.17732	−0.202759
$653$	−17.2750	−0.676021	−0.338011	−	0.941142i	$-0.609754\pi$
$653$	−17.2750	−0.676021	−0.338011	+	0.941142i	$0.609754\pi$
$654$	3.90657	0.152759
$655$	−7.74665	−0.302687
$656$	−8.50425	−0.332035
$657$	−23.3856	−0.912361
$658$	7.47524	0.291415
$659$	−0.820731	−0.0319711	−0.0159856	−	0.999872i	$-0.505089\pi$
$659$	−0.820731	−0.0319711	−0.0159856	+	0.999872i	$0.505089\pi$
$660$	−11.7236	−0.456340
$661$	21.4948	0.836050	0.418025	−	0.908436i	$-0.362722\pi$
$661$	21.4948	0.836050	0.418025	+	0.908436i	$0.362722\pi$
$662$	−10.4379	−0.405679
$663$	−109.537	−4.25408
$664$	−2.48393	−0.0963953
$665$	48.6751	1.88754
$666$	−36.2933	−1.40634
$667$	69.1923	2.67914
$668$	13.9075	0.538098
$669$	−29.5946	−1.14419
$670$	16.9860	0.656228
$671$	−3.80242	−0.146791
$672$	5.92052	0.228389
$673$	−29.2929	−1.12916	−0.564578	−	0.825380i	$-0.690961\pi$
$673$	−29.2929	−1.12916	−0.564578	+	0.825380i	$0.690961\pi$
$674$	10.4767	0.403547
$675$	−3.10847	−0.119645
$676$	29.0144	1.11594
$677$	−43.9928	−1.69078	−0.845390	−	0.534149i	$-0.820632\pi$
$677$	−43.9928	−1.69078	−0.845390	+	0.534149i	$0.820632\pi$
$678$	−17.5755	−0.674982
$679$	−28.0766	−1.07748
$680$	18.5460	0.711205
$681$	33.9049	1.29924
$682$	−0.247569	−0.00947989
$683$	−35.8641	−1.37230	−0.686151	−	0.727459i	$-0.740701\pi$
$683$	−35.8641	−1.37230	−0.686151	+	0.727459i	$0.740701\pi$
$684$	−25.7972	−0.986379
$685$	7.19974	0.275088
$686$	19.9652	0.762276
$687$	63.2847	2.41446
$688$	−6.66822	−0.254224
$689$	−19.8270	−0.755350
$690$	63.9778	2.43560
$691$	−2.56221	−0.0974711	−0.0487355	−	0.998812i	$-0.515519\pi$
$691$	−2.56221	−0.0974711	−0.0487355	+	0.998812i	$0.515519\pi$
$692$	−14.1119	−0.536454
$693$	−13.3154	−0.505810
$694$	−5.55275	−0.210779
$695$	42.4549	1.61041
$696$	19.4135	0.735868
$697$	−56.6156	−2.14447
$698$	25.7503	0.974664
$699$	−36.3434	−1.37464
$700$	−6.43883	−0.243365
$701$	−1.36282	−0.0514731	−0.0257365	−	0.999669i	$-0.508193\pi$
$701$	−1.36282	−0.0514731	−0.0257365	+	0.999669i	$0.508193\pi$
$702$	−7.29852	−0.275465
$703$	78.9547	2.97783
$704$	1.65786	0.0624829
$705$	22.6643	0.853585
$706$	6.96541	0.262147
$707$	−21.1963	−0.797170
$708$	−36.7274	−1.38030
$709$	37.4108	1.40499	0.702496	−	0.711688i	$-0.252069\pi$
$709$	37.4108	1.40499	0.702496	+	0.711688i	$0.252069\pi$
$710$	−27.5317	−1.03325
$711$	18.7892	0.704650
$712$	9.82202	0.368096
$713$	1.35103	0.0505965
$714$	39.4149	1.47506
$715$	29.9361	1.11955
$716$	−19.6463	−0.734217
$717$	45.4485	1.69731
$718$	33.8592	1.26361
$719$	34.7499	1.29595	0.647976	−	0.761660i	$-0.275616\pi$
$719$	34.7499	1.29595	0.647976	+	0.761660i	$0.275616\pi$
$720$	9.59310	0.357514
$721$	−2.03990	−0.0759697
$722$	37.1207	1.38149
$723$	32.9648	1.22597
$724$	3.48901	0.129668
$725$	−21.1131	−0.784120
$726$	20.9458	0.777371
$727$	−15.5393	−0.576320	−0.288160	−	0.957582i	$-0.593044\pi$
$727$	−15.5393	−0.576320	−0.288160	+	0.957582i	$0.593044\pi$
$728$	−15.1180	−0.560311
$729$	−34.3069	−1.27062
$730$	−18.9186	−0.700207
$731$	−44.3926	−1.64192
$732$	5.82206	0.215189
$733$	4.67096	0.172526	0.0862629	−	0.996272i	$-0.472507\pi$
$733$	4.67096	0.172526	0.0862629	+	0.996272i	$0.472507\pi$
$734$	−21.2171	−0.783136
$735$	11.0321	0.406927
$736$	−9.04726	−0.333486
$737$	10.1086	0.372355
$738$	−29.2851	−1.07800
$739$	−9.04679	−0.332792	−0.166396	−	0.986059i	$-0.553213\pi$
$739$	−9.04679	−0.332792	−0.166396	+	0.986059i	$0.553213\pi$
$740$	−29.3606	−1.07932
$741$	123.261	4.52809
$742$	7.13436	0.261910
$743$	10.8526	0.398143	0.199072	−	0.979985i	$-0.436207\pi$
$743$	10.8526	0.398143	0.199072	+	0.979985i	$0.436207\pi$
$744$	0.379063	0.0138971
$745$	−11.6135	−0.425484
$746$	−3.39962	−0.124469
$747$	−8.55363	−0.312961
$748$	11.0369	0.403550
$749$	16.1327	0.589475
$750$	15.8356	0.578236
$751$	27.8417	1.01596	0.507979	−	0.861369i	$-0.330393\pi$
$751$	27.8417	1.01596	0.507979	+	0.861369i	$0.330393\pi$
$752$	−3.20501	−0.116875
$753$	−26.3182	−0.959087
$754$	−49.5724	−1.80532
$755$	−28.5134	−1.03771
$756$	2.62623	0.0955149
$757$	−1.37800	−0.0500843	−0.0250421	−	0.999686i	$-0.507972\pi$
$757$	−1.37800	−0.0500843	−0.0250421	+	0.999686i	$0.507972\pi$
$758$	−2.93410	−0.106571
$759$	38.0740	1.38200
$760$	−20.8694	−0.757014
$761$	42.1545	1.52810	0.764049	−	0.645158i	$-0.223208\pi$
$761$	42.1545	1.52810	0.764049	+	0.645158i	$0.223208\pi$
$762$	17.6702	0.640125
$763$	3.58945	0.129947
$764$	−24.7184	−0.894282
$765$	63.8645	2.30902
$766$	−0.522717	−0.0188865
$767$	93.7832	3.38632
$768$	−2.53842	−0.0915974
$769$	−43.4512	−1.56689	−0.783445	−	0.621461i	$-0.786540\pi$
$769$	−43.4512	−1.56689	−0.783445	+	0.621461i	$0.786540\pi$
$770$	−10.7719	−0.388193
$771$	65.4831	2.35831
$772$	−15.9019	−0.572323
$773$	43.3124	1.55784	0.778919	−	0.627125i	$-0.215768\pi$
$773$	43.3124	1.55784	0.778919	+	0.627125i	$0.215768\pi$
$774$	−22.9626	−0.825372
$775$	−0.412248	−0.0148084
$776$	12.0378	0.432133
$777$	−62.3988	−2.23855
$778$	4.43961	0.159168
$779$	63.7085	2.28260
$780$	−45.8365	−1.64121
$781$	−16.3845	−0.586282
$782$	−60.2306	−2.15384
$783$	8.61146	0.307748
$784$	−1.56008	−0.0557172
$785$	−33.4442	−1.19368
$786$	7.05877	0.251778
$787$	−30.0497	−1.07116	−0.535579	−	0.844485i	$-0.679907\pi$
$787$	−30.0497	−1.07116	−0.535579	+	0.844485i	$0.679907\pi$
$788$	17.3682	0.618717
$789$	14.0282	0.499416
$790$	15.2001	0.540796
$791$	−16.1488	−0.574184
$792$	5.70897	0.202859
$793$	−14.8666	−0.527928
$794$	−7.72706	−0.274223
$795$	21.6307	0.767163
$796$	13.9870	0.495754
$797$	−40.3542	−1.42942	−0.714710	−	0.699421i	$-0.753441\pi$
$797$	−40.3542	−1.42942	−0.714710	+	0.699421i	$0.753441\pi$
$798$	−44.3528	−1.57007
$799$	−21.3368	−0.754842
$800$	2.76065	0.0976035
$801$	33.8229	1.19507
$802$	29.2893	1.03424
$803$	−11.2587	−0.397310
$804$	−15.4777	−0.545857
$805$	58.7844	2.07188
$806$	−0.967936	−0.0340941
$807$	4.26930	0.150287
$808$	9.08793	0.319712
$809$	2.43502	0.0856107	0.0428053	−	0.999083i	$-0.486370\pi$
$809$	2.43502	0.0856107	0.0428053	+	0.999083i	$0.486370\pi$
$810$	−20.8168	−0.731429
$811$	−16.1894	−0.568488	−0.284244	−	0.958752i	$-0.591742\pi$
$811$	−16.1894	−0.568488	−0.284244	+	0.958752i	$0.591742\pi$
$812$	17.8376	0.625978
$813$	1.77101	0.0621122
$814$	−17.4729	−0.612424
$815$	−14.4229	−0.505213
$816$	−16.8991	−0.591587
$817$	49.9542	1.74767
$818$	−9.29022	−0.324825
$819$	−52.0601	−1.81913
$820$	−23.6911	−0.827329
$821$	−17.4759	−0.609913	−0.304956	−	0.952366i	$-0.598642\pi$
$821$	−17.4759	−0.609913	−0.304956	+	0.952366i	$0.598642\pi$
$822$	−6.56042	−0.228821
$823$	−0.0627882	−0.00218866	−0.00109433	−	0.999999i	$-0.500348\pi$
$823$	−0.0627882	−0.00218866	−0.00109433	+	0.999999i	$0.500348\pi$
$824$	0.874604	0.0304683
$825$	−11.6177	−0.404478
$826$	−33.7460	−1.17417
$827$	50.2732	1.74817	0.874084	−	0.485774i	$-0.161462\pi$
$827$	50.2732	1.74817	0.874084	+	0.485774i	$0.161462\pi$
$828$	−31.1550	−1.08271
$829$	39.6863	1.37836	0.689180	−	0.724590i	$-0.257971\pi$
$829$	39.6863	1.37836	0.689180	+	0.724590i	$0.257971\pi$
$830$	−6.91973	−0.240187
$831$	−18.4042	−0.638435
$832$	6.48185	0.224718
$833$	−10.3860	−0.359853
$834$	−38.6850	−1.33955
$835$	38.7435	1.34077
$836$	−12.4196	−0.429542
$837$	0.168145	0.00581194
$838$	−9.43895	−0.326063
$839$	13.6659	0.471798	0.235899	−	0.971778i	$-0.424197\pi$
$839$	13.6659	0.471798	0.235899	+	0.971778i	$0.424197\pi$
$840$	16.4933	0.569074
$841$	29.4900	1.01690
$842$	−13.6953	−0.471971
$843$	17.7795	0.612358
$844$	−21.6379	−0.744806
$845$	80.8281	2.78057
$846$	−11.0367	−0.379450
$847$	19.2455	0.661283
$848$	−3.05885	−0.105041
$849$	−4.22302	−0.144934
$850$	18.3785	0.630378
$851$	95.3529	3.26865
$852$	25.0870	0.859465
$853$	22.9096	0.784409	0.392204	−	0.919878i	$-0.371713\pi$
$853$	22.9096	0.784409	0.392204	+	0.919878i	$0.371713\pi$
$854$	5.34945	0.183054
$855$	−71.8655	−2.45775
$856$	−6.91687	−0.236414
$857$	−23.3829	−0.798743	−0.399372	−	0.916789i	$-0.630772\pi$
$857$	−23.3829	−0.798743	−0.399372	+	0.916789i	$0.630772\pi$
$858$	−27.2779	−0.931251
$859$	−5.82953	−0.198901	−0.0994506	−	0.995043i	$-0.531709\pi$
$859$	−5.82953	−0.198901	−0.0994506	+	0.995043i	$0.531709\pi$
$860$	−18.5763	−0.633446
$861$	−50.3496	−1.71591
$862$	24.3705	0.830062
$863$	28.6109	0.973928	0.486964	−	0.873422i	$-0.338104\pi$
$863$	28.6109	0.973928	0.486964	+	0.873422i	$0.338104\pi$
$864$	−1.12599	−0.0383071
$865$	−39.3128	−1.33668
$866$	15.3883	0.522916
$867$	−69.3498	−2.35524
$868$	0.348292	0.0118218
$869$	9.04577	0.306857
$870$	54.0821	1.83355
$871$	39.5223	1.33916
$872$	−1.53898	−0.0521163
$873$	41.4532	1.40298
$874$	67.7764	2.29257
$875$	14.5502	0.491885
$876$	17.2386	0.582439
$877$	−50.4859	−1.70479	−0.852393	−	0.522901i	$-0.824850\pi$
$877$	−50.4859	−1.70479	−0.852393	+	0.522901i	$0.824850\pi$
$878$	−4.82832	−0.162948
$879$	−70.0719	−2.36347
$880$	4.61845	0.155688
$881$	−18.9441	−0.638244	−0.319122	−	0.947714i	$-0.603388\pi$
$881$	−18.9441	−0.638244	−0.319122	+	0.947714i	$0.603388\pi$
$882$	−5.37227	−0.180894
$883$	18.2185	0.613102	0.306551	−	0.951854i	$-0.400825\pi$
$883$	18.2185	0.613102	0.306551	+	0.951854i	$0.400825\pi$
$884$	43.1518	1.45135
$885$	−102.315	−3.43928
$886$	40.9451	1.37558
$887$	18.7842	0.630711	0.315356	−	0.948974i	$-0.397876\pi$
$887$	18.7842	0.630711	0.315356	+	0.948974i	$0.397876\pi$
$888$	26.7535	0.897788
$889$	16.2358	0.544533
$890$	27.3621	0.917180
$891$	−12.3883	−0.415025
$892$	11.6587	0.390361
$893$	24.0099	0.803461
$894$	10.5822	0.353922
$895$	−54.7306	−1.82944
$896$	−2.33236	−0.0779188
$897$	148.861	4.97031
$898$	−21.9733	−0.733258
$899$	1.14206	0.0380898
$900$	9.50650	0.316883
$901$	−20.3638	−0.678417
$902$	−14.0989	−0.469441
$903$	−39.4793	−1.31379
$904$	6.92378	0.230282
$905$	9.71967	0.323093
$906$	25.9814	0.863175
$907$	−38.3803	−1.27440	−0.637199	−	0.770699i	$-0.719907\pi$
$907$	−38.3803	−1.27440	−0.637199	+	0.770699i	$0.719907\pi$
$908$	−13.3567	−0.443258
$909$	31.2950	1.03799
$910$	−42.1157	−1.39612
$911$	51.8219	1.71694	0.858468	−	0.512868i	$-0.171417\pi$
$911$	51.8219	1.71694	0.858468	+	0.512868i	$0.171417\pi$
$912$	19.0163	0.629691
$913$	−4.11801	−0.136286
$914$	−12.1240	−0.401025
$915$	16.2190	0.536185
$916$	−24.9308	−0.823735
$917$	6.48577	0.214179
$918$	−7.49611	−0.247409
$919$	−27.7400	−0.915057	−0.457529	−	0.889195i	$-0.651265\pi$
$919$	−27.7400	−0.915057	−0.457529	+	0.889195i	$0.651265\pi$
$920$	−25.2038	−0.830945
$921$	−86.8336	−2.86126
$922$	37.2463	1.22664
$923$	−64.0595	−2.10854
$924$	9.81539	0.322903
$925$	−29.0956	−0.956657
$926$	14.4253	0.474045
$927$	3.01177	0.0989195
$928$	−7.64788	−0.251054
$929$	1.51097	0.0495733	0.0247867	−	0.999693i	$-0.492109\pi$
$929$	1.51097	0.0495733	0.0247867	+	0.999693i	$0.492109\pi$
$930$	1.05599	0.0346273
$931$	11.6872	0.383031
$932$	14.3173	0.468980
$933$	−31.7707	−1.04013
$934$	2.71364	0.0887930
$935$	30.7466	1.00552
$936$	22.3208	0.729577
$937$	−52.3833	−1.71129	−0.855644	−	0.517566i	$-0.826838\pi$
$937$	−52.3833	−1.71129	−0.855644	+	0.517566i	$0.826838\pi$
$938$	−14.2213	−0.464342
$939$	76.5323	2.49754
$940$	−8.92849	−0.291215
$941$	−3.16320	−0.103117	−0.0515587	−	0.998670i	$-0.516419\pi$
$941$	−3.16320	−0.103117	−0.0515587	+	0.998670i	$0.516419\pi$
$942$	30.4745	0.992912
$943$	76.9402	2.50552
$944$	14.4686	0.470913
$945$	7.31612	0.237993
$946$	−11.0550	−0.359428
$947$	−39.9153	−1.29707	−0.648536	−	0.761184i	$-0.724619\pi$
$947$	−39.9153	−1.29707	−0.648536	+	0.761184i	$0.724619\pi$
$948$	−13.8504	−0.449839
$949$	−44.0188	−1.42891
$950$	−20.6810	−0.670981
$951$	22.1675	0.718830
$952$	−15.5273	−0.503243
$953$	32.1845	1.04256	0.521279	−	0.853386i	$-0.325455\pi$
$953$	32.1845	1.04256	0.521279	+	0.853386i	$0.325455\pi$
$954$	−10.5334	−0.341032
$955$	−68.8605	−2.22827
$956$	−17.9043	−0.579065
$957$	32.1849	1.04039
$958$	1.36125	0.0439799
$959$	−6.02787	−0.194650
$960$	−7.07152	−0.228232
$961$	−30.9777	−0.999281
$962$	−68.3149	−2.20256
$963$	−23.8188	−0.767551
$964$	−12.9863	−0.418261
$965$	−44.2995	−1.42605
$966$	−53.5645	−1.72341
$967$	−8.66875	−0.278768	−0.139384	−	0.990238i	$-0.544512\pi$
$967$	−8.66875	−0.278768	−0.139384	+	0.990238i	$0.544512\pi$
$968$	−8.25150	−0.265213
$969$	126.598	4.06690
$970$	33.5349	1.07674
$971$	37.6680	1.20882	0.604412	−	0.796672i	$-0.293408\pi$
$971$	37.6680	1.20882	0.604412	+	0.796672i	$0.293408\pi$
$972$	22.3463	0.716759
$973$	−35.5447	−1.13951
$974$	−22.3546	−0.716288
$975$	−45.4227	−1.45469
$976$	−2.29357	−0.0734155
$977$	−24.1749	−0.773423	−0.386711	−	0.922201i	$-0.626389\pi$
$977$	−24.1749	−0.773423	−0.386711	+	0.922201i	$0.626389\pi$
$978$	13.1422	0.420242
$979$	16.2835	0.520424
$980$	−4.34607	−0.138830
$981$	−5.29959	−0.169203
$982$	−17.7968	−0.567917
$983$	−0.0700196	−0.00223328	−0.00111664	−	0.999999i	$-0.500355\pi$
$983$	−0.0700196	−0.00223328	−0.00111664	+	0.999999i	$0.500355\pi$
$984$	21.5874	0.688180
$985$	48.3843	1.54165
$986$	−50.9145	−1.62145
$987$	−18.9753	−0.603991
$988$	−48.5580	−1.54483
$989$	60.3292	1.91836
$990$	15.9040	0.505463
$991$	30.7852	0.977923	0.488962	−	0.872305i	$-0.337376\pi$
$991$	30.7852	0.977923	0.488962	+	0.872305i	$0.337376\pi$
$992$	−0.149330	−0.00474124
$993$	26.4957	0.840815
$994$	23.0505	0.731118
$995$	38.9648	1.23527
$996$	6.30527	0.199790
$997$	44.9656	1.42407	0.712037	−	0.702142i	$-0.247773\pi$
$997$	44.9656	1.42407	0.712037	+	0.702142i	$0.247773\pi$
$998$	−21.8916	−0.692967
$999$	11.8673	0.375465

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.a.1.13	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.a.1.13	✓	75	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.53842 q^{3} +1.00000 q^{4} +2.78579 q^{5} -2.53842 q^{6} -2.33236 q^{7} +1.00000 q^{8} +3.44358 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.53842 q^{3} +1.00000 q^{4} +2.78579 q^{5} -2.53842 q^{6} -2.33236 q^{7} +1.00000 q^{8} +3.44358 q^{9} +2.78579 q^{10} +1.65786 q^{11} -2.53842 q^{12} +6.48185 q^{13} -2.33236 q^{14} -7.07152 q^{15} +1.00000 q^{16} +6.65733 q^{17} +3.44358 q^{18} -7.49137 q^{19} +2.78579 q^{20} +5.92052 q^{21} +1.65786 q^{22} -9.04726 q^{23} -2.53842 q^{24} +2.76065 q^{25} +6.48185 q^{26} -1.12599 q^{27} -2.33236 q^{28} -7.64788 q^{29} -7.07152 q^{30} -0.149330 q^{31} +1.00000 q^{32} -4.20834 q^{33} +6.65733 q^{34} -6.49748 q^{35} +3.44358 q^{36} -10.5394 q^{37} -7.49137 q^{38} -16.4537 q^{39} +2.78579 q^{40} -8.50425 q^{41} +5.92052 q^{42} -6.66822 q^{43} +1.65786 q^{44} +9.59310 q^{45} -9.04726 q^{46} -3.20501 q^{47} -2.53842 q^{48} -1.56008 q^{49} +2.76065 q^{50} -16.8991 q^{51} +6.48185 q^{52} -3.05885 q^{53} -1.12599 q^{54} +4.61845 q^{55} -2.33236 q^{56} +19.0163 q^{57} -7.64788 q^{58} +14.4686 q^{59} -7.07152 q^{60} -2.29357 q^{61} -0.149330 q^{62} -8.03168 q^{63} +1.00000 q^{64} +18.0571 q^{65} -4.20834 q^{66} +6.09738 q^{67} +6.65733 q^{68} +22.9658 q^{69} -6.49748 q^{70} -9.88290 q^{71} +3.44358 q^{72} -6.79109 q^{73} -10.5394 q^{74} -7.00768 q^{75} -7.49137 q^{76} -3.86673 q^{77} -16.4537 q^{78} +5.45629 q^{79} +2.78579 q^{80} -7.47250 q^{81} -8.50425 q^{82} -2.48393 q^{83} +5.92052 q^{84} +18.5460 q^{85} -6.66822 q^{86} +19.4135 q^{87} +1.65786 q^{88} +9.82202 q^{89} +9.59310 q^{90} -15.1180 q^{91} -9.04726 q^{92} +0.379063 q^{93} -3.20501 q^{94} -20.8694 q^{95} -2.53842 q^{96} +12.0378 q^{97} -1.56008 q^{98} +5.70897 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9}+O(q^{10}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9} - 29 q^{10} - 30 q^{11} - 30 q^{12} - 23 q^{13} - 31 q^{14} - 22 q^{15} + 75 q^{16} - 48 q^{17} + 55 q^{18} - 58 q^{19} - 29 q^{20} - 5 q^{21} - 30 q^{22} - 79 q^{23} - 30 q^{24} + 42 q^{25} - 23 q^{26} - 108 q^{27} - 31 q^{28} - 39 q^{29} - 22 q^{30} - 95 q^{31} + 75 q^{32} - 44 q^{33} - 48 q^{34} - 60 q^{35} + 55 q^{36} - 16 q^{37} - 58 q^{38} - 57 q^{39} - 29 q^{40} - 85 q^{41} - 5 q^{42} - 55 q^{43} - 30 q^{44} - 48 q^{45} - 79 q^{46} - 88 q^{47} - 30 q^{48} + 26 q^{49} + 42 q^{50} - 39 q^{51} - 23 q^{52} - 79 q^{53} - 108 q^{54} - 78 q^{55} - 31 q^{56} - 40 q^{57} - 39 q^{58} - 74 q^{59} - 22 q^{60} - 21 q^{61} - 95 q^{62} - 97 q^{63} + 75 q^{64} - 63 q^{65} - 44 q^{66} - 59 q^{67} - 48 q^{68} + 3 q^{69} - 60 q^{70} - 72 q^{71} + 55 q^{72} - 91 q^{73} - 16 q^{74} - 95 q^{75} - 58 q^{76} - 60 q^{77} - 57 q^{78} - 64 q^{79} - 29 q^{80} + 47 q^{81} - 85 q^{82} - 105 q^{83} - 5 q^{84} + 14 q^{85} - 55 q^{86} - 75 q^{87} - 30 q^{88} - 78 q^{89} - 48 q^{90} - 89 q^{91} - 79 q^{92} + 27 q^{93} - 88 q^{94} - 53 q^{95} - 30 q^{96} - 79 q^{97} + 26 q^{98} - 57 q^{99}+O(q^{100}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9 - 29 * q^10 - 30 * q^11 - 30 * q^12 - 23 * q^13 - 31 * q^14 - 22 * q^15 + 75 * q^16 - 48 * q^17 + 55 * q^18 - 58 * q^19 - 29 * q^20 - 5 * q^21 - 30 * q^22 - 79 * q^23 - 30 * q^24 + 42 * q^25 - 23 * q^26 - 108 * q^27 - 31 * q^28 - 39 * q^29 - 22 * q^30 - 95 * q^31 + 75 * q^32 - 44 * q^33 - 48 * q^34 - 60 * q^35 + 55 * q^36 - 16 * q^37 - 58 * q^38 - 57 * q^39 - 29 * q^40 - 85 * q^41 - 5 * q^42 - 55 * q^43 - 30 * q^44 - 48 * q^45 - 79 * q^46 - 88 * q^47 - 30 * q^48 + 26 * q^49 + 42 * q^50 - 39 * q^51 - 23 * q^52 - 79 * q^53 - 108 * q^54 - 78 * q^55 - 31 * q^56 - 40 * q^57 - 39 * q^58 - 74 * q^59 - 22 * q^60 - 21 * q^61 - 95 * q^62 - 97 * q^63 + 75 * q^64 - 63 * q^65 - 44 * q^66 - 59 * q^67 - 48 * q^68 + 3 * q^69 - 60 * q^70 - 72 * q^71 + 55 * q^72 - 91 * q^73 - 16 * q^74 - 95 * q^75 - 58 * q^76 - 60 * q^77 - 57 * q^78 - 64 * q^79 - 29 * q^80 + 47 * q^81 - 85 * q^82 - 105 * q^83 - 5 * q^84 + 14 * q^85 - 55 * q^86 - 75 * q^87 - 30 * q^88 - 78 * q^89 - 48 * q^90 - 89 * q^91 - 79 * q^92 + 27 * q^93 - 88 * q^94 - 53 * q^95 - 30 * q^96 - 79 * q^97 + 26 * q^98 - 57 * q^99

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