LMFDB - Embedded newform 8023.2.a.e.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8023,2,Mod(1,8023)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8023, 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("8023.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8023 = 71 \cdot 113 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8023.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:	$64.0639775417$
Analytic rank:	$0$
Dimension:	$172$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8023.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30435 q^{2} -0.933250 q^{3} +3.31004 q^{4} -0.282102 q^{5} +2.15054 q^{6} +1.67948 q^{7} -3.01880 q^{8} -2.12904 q^{9} +O(q^{10})$	\(q-2.30435 q^{2} -0.933250 q^{3} +3.31004 q^{4} -0.282102 q^{5} +2.15054 q^{6} +1.67948 q^{7} -3.01880 q^{8} -2.12904 q^{9} +0.650063 q^{10} +1.69294 q^{11} -3.08910 q^{12} -6.05529 q^{13} -3.87013 q^{14} +0.263272 q^{15} +0.336302 q^{16} +4.76313 q^{17} +4.90607 q^{18} +1.16089 q^{19} -0.933771 q^{20} -1.56738 q^{21} -3.90113 q^{22} -8.59381 q^{23} +2.81730 q^{24} -4.92042 q^{25} +13.9535 q^{26} +4.78668 q^{27} +5.55917 q^{28} +4.29516 q^{29} -0.606671 q^{30} +5.56183 q^{31} +5.26265 q^{32} -1.57994 q^{33} -10.9759 q^{34} -0.473786 q^{35} -7.04723 q^{36} +10.2663 q^{37} -2.67509 q^{38} +5.65110 q^{39} +0.851611 q^{40} -4.88233 q^{41} +3.61179 q^{42} +0.321977 q^{43} +5.60371 q^{44} +0.600608 q^{45} +19.8032 q^{46} -10.4764 q^{47} -0.313854 q^{48} -4.17933 q^{49} +11.3384 q^{50} -4.44519 q^{51} -20.0433 q^{52} -7.71618 q^{53} -11.0302 q^{54} -0.477583 q^{55} -5.07003 q^{56} -1.08340 q^{57} -9.89757 q^{58} -10.2440 q^{59} +0.871441 q^{60} -6.63068 q^{61} -12.8164 q^{62} -3.57570 q^{63} -12.7996 q^{64} +1.70821 q^{65} +3.64073 q^{66} -1.76278 q^{67} +15.7662 q^{68} +8.02017 q^{69} +1.09177 q^{70} -1.00000 q^{71} +6.42717 q^{72} +10.0400 q^{73} -23.6572 q^{74} +4.59198 q^{75} +3.84258 q^{76} +2.84327 q^{77} -13.0221 q^{78} -1.89175 q^{79} -0.0948716 q^{80} +1.91997 q^{81} +11.2506 q^{82} +14.0005 q^{83} -5.18809 q^{84} -1.34369 q^{85} -0.741949 q^{86} -4.00846 q^{87} -5.11066 q^{88} +10.2649 q^{89} -1.38401 q^{90} -10.1698 q^{91} -28.4459 q^{92} -5.19057 q^{93} +24.1414 q^{94} -0.327489 q^{95} -4.91136 q^{96} -0.0308119 q^{97} +9.63066 q^{98} -3.60435 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 172 q + 24 q^{2} + 18 q^{3} + 180 q^{4} + 28 q^{5} + 16 q^{6} + 4 q^{7} + 72 q^{8} + 198 q^{9}+O(q^{10}) $ 172 * q + 24 * q^2 + 18 * q^3 + 180 * q^4 + 28 * q^5 + 16 * q^6 + 4 * q^7 + 72 * q^8 + 198 * q^9	\( 172 q + 24 q^{2} + 18 q^{3} + 180 q^{4} + 28 q^{5} + 16 q^{6} + 4 q^{7} + 72 q^{8} + 198 q^{9} + 14 q^{10} + 20 q^{11} + 54 q^{12} + 36 q^{13} + 26 q^{14} + 32 q^{15} + 196 q^{16} + 123 q^{17} + 74 q^{18} + 20 q^{19} + 70 q^{20} + 37 q^{21} + 11 q^{22} + 22 q^{23} + 62 q^{24} + 210 q^{25} + 50 q^{26} + 69 q^{27} + 42 q^{28} + 58 q^{29} + 36 q^{30} + 10 q^{31} + 168 q^{32} + 124 q^{33} + 5 q^{34} + 59 q^{35} + 192 q^{36} + 40 q^{37} + 58 q^{38} + 15 q^{39} + 7 q^{40} + 155 q^{41} - 6 q^{42} + 19 q^{43} + 22 q^{44} + 76 q^{45} + q^{46} + 71 q^{47} + 144 q^{48} + 206 q^{49} + 126 q^{50} + 33 q^{51} + 71 q^{52} + 101 q^{53} + 92 q^{54} - 2 q^{55} + 57 q^{56} + 114 q^{57} + 4 q^{58} + 71 q^{59} + 38 q^{60} + 50 q^{61} + 86 q^{62} + 14 q^{63} + 240 q^{64} + 143 q^{65} + 21 q^{66} + 8 q^{67} + 192 q^{68} + 41 q^{69} - 12 q^{70} - 172 q^{71} + 156 q^{72} + 128 q^{73} + 30 q^{74} + 72 q^{75} + 74 q^{76} + 127 q^{77} + 107 q^{78} + 2 q^{79} + 50 q^{80} + 236 q^{81} + 42 q^{82} + 140 q^{83} + 71 q^{84} + 55 q^{85} + 46 q^{86} + 100 q^{87} - 31 q^{88} + 215 q^{89} - 7 q^{90} + 22 q^{91} - 15 q^{92} + 60 q^{93} + 5 q^{94} + 74 q^{95} + 182 q^{96} + 120 q^{97} + 164 q^{98} + 60 q^{99}+O(q^{100}) \) 172 * q + 24 * q^2 + 18 * q^3 + 180 * q^4 + 28 * q^5 + 16 * q^6 + 4 * q^7 + 72 * q^8 + 198 * q^9 + 14 * q^10 + 20 * q^11 + 54 * q^12 + 36 * q^13 + 26 * q^14 + 32 * q^15 + 196 * q^16 + 123 * q^17 + 74 * q^18 + 20 * q^19 + 70 * q^20 + 37 * q^21 + 11 * q^22 + 22 * q^23 + 62 * q^24 + 210 * q^25 + 50 * q^26 + 69 * q^27 + 42 * q^28 + 58 * q^29 + 36 * q^30 + 10 * q^31 + 168 * q^32 + 124 * q^33 + 5 * q^34 + 59 * q^35 + 192 * q^36 + 40 * q^37 + 58 * q^38 + 15 * q^39 + 7 * q^40 + 155 * q^41 - 6 * q^42 + 19 * q^43 + 22 * q^44 + 76 * q^45 + q^46 + 71 * q^47 + 144 * q^48 + 206 * q^49 + 126 * q^50 + 33 * q^51 + 71 * q^52 + 101 * q^53 + 92 * q^54 - 2 * q^55 + 57 * q^56 + 114 * q^57 + 4 * q^58 + 71 * q^59 + 38 * q^60 + 50 * q^61 + 86 * q^62 + 14 * q^63 + 240 * q^64 + 143 * q^65 + 21 * q^66 + 8 * q^67 + 192 * q^68 + 41 * q^69 - 12 * q^70 - 172 * q^71 + 156 * q^72 + 128 * q^73 + 30 * q^74 + 72 * q^75 + 74 * q^76 + 127 * q^77 + 107 * q^78 + 2 * q^79 + 50 * q^80 + 236 * q^81 + 42 * q^82 + 140 * q^83 + 71 * q^84 + 55 * q^85 + 46 * q^86 + 100 * q^87 - 31 * q^88 + 215 * q^89 - 7 * q^90 + 22 * q^91 - 15 * q^92 + 60 * q^93 + 5 * q^94 + 74 * q^95 + 182 * q^96 + 120 * q^97 + 164 * q^98 + 60 * 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 $

$2$	−2.30435	−1.62942	−0.814712	−	0.579866i	$-0.803105\pi$
$2$	−2.30435	−1.62942	−0.814712	+	0.579866i	$0.803105\pi$
$3$	−0.933250	−0.538812	−0.269406	−	0.963027i	$-0.586827\pi$
$3$	−0.933250	−0.538812	−0.269406	+	0.963027i	$0.586827\pi$
$4$	3.31004	1.65502
$5$	−0.282102	−0.126160	−0.0630800	−	0.998008i	$-0.520092\pi$
$5$	−0.282102	−0.126160	−0.0630800	+	0.998008i	$0.520092\pi$
$6$	2.15054	0.877953
$7$	1.67948	0.634786	0.317393	−	0.948294i	$-0.397193\pi$
$7$	1.67948	0.634786	0.317393	+	0.948294i	$0.397193\pi$
$8$	−3.01880	−1.06731
$9$	−2.12904	−0.709682
$10$	0.650063	0.205568
$11$	1.69294	0.510441	0.255220	−	0.966883i	$-0.417852\pi$
$11$	1.69294	0.510441	0.255220	+	0.966883i	$0.417852\pi$
$12$	−3.08910	−0.891746
$13$	−6.05529	−1.67944	−0.839718	−	0.543023i	$-0.817280\pi$
$13$	−6.05529	−1.67944	−0.839718	+	0.543023i	$0.817280\pi$
$14$	−3.87013	−1.03433
$15$	0.263272	0.0679765
$16$	0.336302	0.0840756
$17$	4.76313	1.15523	0.577614	−	0.816310i	$-0.303984\pi$
$17$	4.76313	1.15523	0.577614	+	0.816310i	$0.303984\pi$
$18$	4.90607	1.15637
$19$	1.16089	0.266326	0.133163	−	0.991094i	$-0.457487\pi$
$19$	1.16089	0.266326	0.133163	+	0.991094i	$0.457487\pi$
$20$	−0.933771	−0.208798
$21$	−1.56738	−0.342030
$22$	−3.90113	−0.831725
$23$	−8.59381	−1.79193	−0.895967	−	0.444121i	$-0.853516\pi$
$23$	−8.59381	−1.79193	−0.895967	+	0.444121i	$0.853516\pi$
$24$	2.81730	0.575078
$25$	−4.92042	−0.984084
$26$	13.9535	2.73651
$27$	4.78668	0.921197
$28$	5.55917	1.05058
$29$	4.29516	0.797591	0.398796	−	0.917040i	$-0.369428\pi$
$29$	4.29516	0.797591	0.398796	+	0.917040i	$0.369428\pi$
$30$	−0.606671	−0.110763
$31$	5.56183	0.998933	0.499467	−	0.866333i	$-0.333529\pi$
$31$	5.56183	0.998933	0.499467	+	0.866333i	$0.333529\pi$
$32$	5.26265	0.930314
$33$	−1.57994	−0.275032
$34$	−10.9759	−1.88236
$35$	−0.473786	−0.0800845
$36$	−7.04723	−1.17454
$37$	10.2663	1.68777	0.843885	−	0.536525i	$-0.180263\pi$
$37$	10.2663	1.68777	0.843885	+	0.536525i	$0.180263\pi$
$38$	−2.67509	−0.433957
$39$	5.65110	0.904900
$40$	0.851611	0.134652
$41$	−4.88233	−0.762492	−0.381246	−	0.924474i	$-0.624505\pi$
$41$	−4.88233	−0.762492	−0.381246	+	0.924474i	$0.624505\pi$
$42$	3.61179	0.557312
$43$	0.321977	0.0491011	0.0245505	−	0.999699i	$-0.492185\pi$
$43$	0.321977	0.0491011	0.0245505	+	0.999699i	$0.492185\pi$
$44$	5.60371	0.844791
$45$	0.600608	0.0895334
$46$	19.8032	2.91982
$47$	−10.4764	−1.52814	−0.764072	−	0.645131i	$-0.776803\pi$
$47$	−10.4764	−1.52814	−0.764072	+	0.645131i	$0.776803\pi$
$48$	−0.313854	−0.0453009
$49$	−4.17933	−0.597047
$50$	11.3384	1.60349
$51$	−4.44519	−0.622451
$52$	−20.0433	−2.77950
$53$	−7.71618	−1.05990	−0.529950	−	0.848029i	$-0.677789\pi$
$53$	−7.71618	−1.05990	−0.529950	+	0.848029i	$0.677789\pi$
$54$	−11.0302	−1.50102
$55$	−0.477583	−0.0643972
$56$	−5.07003	−0.677512
$57$	−1.08340	−0.143499
$58$	−9.89757	−1.29961
$59$	−10.2440	−1.33366	−0.666829	−	0.745211i	$-0.732349\pi$
$59$	−10.2440	−1.33366	−0.666829	+	0.745211i	$0.732349\pi$
$60$	0.871441	0.112503
$61$	−6.63068	−0.848972	−0.424486	−	0.905434i	$-0.639545\pi$
$61$	−6.63068	−0.848972	−0.424486	+	0.905434i	$0.639545\pi$
$62$	−12.8164	−1.62769
$63$	−3.57570	−0.450496
$64$	−12.7996	−1.59995
$65$	1.70821	0.211878
$66$	3.64073	0.448143
$67$	−1.76278	−0.215358	−0.107679	−	0.994186i	$-0.534342\pi$
$67$	−1.76278	−0.215358	−0.107679	+	0.994186i	$0.534342\pi$
$68$	15.7662	1.91193
$69$	8.02017	0.965515
$70$	1.09177	0.130492
$71$	−1.00000	−0.118678
$71$	−1.00000	−0.118678
$72$	6.42717	0.757449
$73$	10.0400	1.17509	0.587545	−	0.809192i	$-0.300095\pi$
$73$	10.0400	1.17509	0.587545	+	0.809192i	$0.300095\pi$
$74$	−23.6572	−2.75009
$75$	4.59198	0.530236
$76$	3.84258	0.440775
$77$	2.84327	0.324021
$78$	−13.0221	−1.47447
$79$	−1.89175	−0.212839	−0.106419	−	0.994321i	$-0.533939\pi$
$79$	−1.89175	−0.212839	−0.106419	+	0.994321i	$0.533939\pi$
$80$	−0.0948716	−0.0106070
$81$	1.91997	0.213330
$82$	11.2506	1.24242
$83$	14.0005	1.53676	0.768379	−	0.639995i	$-0.221064\pi$
$83$	14.0005	1.53676	0.768379	+	0.639995i	$0.221064\pi$
$84$	−5.18809	−0.566067
$85$	−1.34369	−0.145744
$86$	−0.741949	−0.0800065
$87$	−4.00846	−0.429752
$88$	−5.11066	−0.544798
$89$	10.2649	1.08808	0.544041	−	0.839059i	$-0.316894\pi$
$89$	10.2649	1.08808	0.544041	+	0.839059i	$0.316894\pi$
$90$	−1.38401	−0.145888
$91$	−10.1698	−1.06608
$92$	−28.4459	−2.96569
$93$	−5.19057	−0.538237
$94$	24.1414	2.48999
$95$	−0.327489	−0.0335996
$96$	−4.91136	−0.501264
$97$	−0.0308119	−0.00312848	−0.00156424	−	0.999999i	$-0.500498\pi$
$97$	−0.0308119	−0.00312848	−0.00156424	+	0.999999i	$0.500498\pi$
$98$	9.63066	0.972843
$99$	−3.60435	−0.362251
$100$	−16.2868	−1.62868
$101$	8.59596	0.855330	0.427665	−	0.903937i	$-0.359336\pi$
$101$	8.59596	0.855330	0.427665	+	0.903937i	$0.359336\pi$
$102$	10.2433	1.01424
$103$	−4.83108	−0.476021	−0.238010	−	0.971263i	$-0.576495\pi$
$103$	−4.83108	−0.476021	−0.238010	+	0.971263i	$0.576495\pi$
$104$	18.2797	1.79248
$105$	0.442161	0.0431505
$106$	17.7808	1.72702
$107$	12.5740	1.21557	0.607787	−	0.794100i	$-0.292057\pi$
$107$	12.5740	1.21557	0.607787	+	0.794100i	$0.292057\pi$
$108$	15.8441	1.52460
$109$	−11.7610	−1.12650	−0.563250	−	0.826287i	$-0.690449\pi$
$109$	−11.7610	−1.12650	−0.563250	+	0.826287i	$0.690449\pi$
$110$	1.10052	0.104930
$111$	−9.58102	−0.909390
$112$	0.564815	0.0533700
$113$	1.00000	0.0940721
$113$	1.00000	0.0940721
$114$	2.49653	0.233821
$115$	2.42433	0.226070
$116$	14.2172	1.32003
$117$	12.8920	1.19186
$118$	23.6058	2.17309
$119$	7.99960	0.733322
$120$	−0.794766	−0.0725519
$121$	−8.13395	−0.739450
$122$	15.2794	1.38334
$123$	4.55643	0.410840
$124$	18.4099	1.65326
$125$	2.79857	0.250312
$126$	8.23967	0.734048
$127$	−7.16975	−0.636212	−0.318106	−	0.948055i	$-0.603047\pi$
$127$	−7.16975	−0.636212	−0.318106	+	0.948055i	$0.603047\pi$
$128$	18.9695	1.67668
$129$	−0.300485	−0.0264562
$130$	−3.93632	−0.345238
$131$	−16.3542	−1.42887	−0.714435	−	0.699702i	$-0.753316\pi$
$131$	−16.3542	−1.42887	−0.714435	+	0.699702i	$0.753316\pi$
$132$	−5.22966	−0.455184
$133$	1.94969	0.169060
$134$	4.06207	0.350910
$135$	−1.35033	−0.116218
$136$	−14.3789	−1.23298
$137$	22.5413	1.92584	0.962918	−	0.269793i	$-0.0869552\pi$
$137$	22.5413	1.92584	0.962918	+	0.269793i	$0.0869552\pi$
$138$	−18.4813	−1.57323
$139$	17.7590	1.50630	0.753148	−	0.657851i	$-0.228534\pi$
$139$	17.7590	1.50630	0.753148	+	0.657851i	$0.228534\pi$
$140$	−1.56825	−0.132542
$141$	9.77712	0.823382
$142$	2.30435	0.193377
$143$	−10.2513	−0.857253
$144$	−0.716003	−0.0596669
$145$	−1.21167	−0.100624
$146$	−23.1356	−1.91472
$147$	3.90036	0.321696
$148$	33.9819	2.79330
$149$	6.29608	0.515795	0.257898	−	0.966172i	$-0.416970\pi$
$149$	6.29608	0.515795	0.257898	+	0.966172i	$0.416970\pi$
$150$	−10.5815	−0.863979
$151$	−20.9951	−1.70855	−0.854277	−	0.519819i	$-0.826000\pi$
$151$	−20.9951	−1.70855	−0.854277	+	0.519819i	$0.826000\pi$
$152$	−3.50449	−0.284251
$153$	−10.1409	−0.819844
$154$	−6.55190	−0.527967
$155$	−1.56900	−0.126025
$156$	18.7054	1.49763
$157$	−4.02742	−0.321423	−0.160712	−	0.987001i	$-0.551379\pi$
$157$	−4.02742	−0.321423	−0.160712	+	0.987001i	$0.551379\pi$
$158$	4.35926	0.346804
$159$	7.20113	0.571086
$160$	−1.48460	−0.117368
$161$	−14.4332	−1.13749
$162$	−4.42428	−0.347604
$163$	7.31710	0.573119	0.286560	−	0.958062i	$-0.407488\pi$
$163$	7.31710	0.573119	0.286560	+	0.958062i	$0.407488\pi$
$164$	−16.1607	−1.26194
$165$	0.445704	0.0346980
$166$	−32.2622	−2.50403
$167$	0.229136	0.0177310	0.00886552	−	0.999961i	$-0.497178\pi$
$167$	0.229136	0.0177310	0.00886552	+	0.999961i	$0.497178\pi$
$168$	4.73161	0.365052
$169$	23.6666	1.82051
$170$	3.09633	0.237478
$171$	−2.47158	−0.189006
$172$	1.06576	0.0812633
$173$	−13.8002	−1.04921	−0.524606	−	0.851345i	$-0.675787\pi$
$173$	−13.8002	−1.04921	−0.524606	+	0.851345i	$0.675787\pi$
$174$	9.23690	0.700248
$175$	−8.26377	−0.624682
$176$	0.569340	0.0429156
$177$	9.56023	0.718591
$178$	−23.6540	−1.77295
$179$	13.2779	0.992438	0.496219	−	0.868197i	$-0.334721\pi$
$179$	13.2779	0.992438	0.496219	+	0.868197i	$0.334721\pi$
$180$	1.98804	0.148180
$181$	−21.9217	−1.62943	−0.814713	−	0.579865i	$-0.803105\pi$
$181$	−21.9217	−1.62943	−0.814713	+	0.579865i	$0.803105\pi$
$182$	23.4347	1.73710
$183$	6.18808	0.457436
$184$	25.9430	1.91255
$185$	−2.89615	−0.212929
$186$	11.9609	0.877017
$187$	8.06369	0.589676
$188$	−34.6774	−2.52911
$189$	8.03916	0.584763
$190$	0.754649	0.0547480
$191$	4.31530	0.312244	0.156122	−	0.987738i	$-0.450101\pi$
$191$	4.31530	0.312244	0.156122	+	0.987738i	$0.450101\pi$
$192$	11.9452	0.862073
$193$	4.85886	0.349748	0.174874	−	0.984591i	$-0.444048\pi$
$193$	4.85886	0.349748	0.174874	+	0.984591i	$0.444048\pi$
$194$	0.0710016	0.00509762
$195$	−1.59419	−0.114162
$196$	−13.8338	−0.988126
$197$	−27.2118	−1.93876	−0.969381	−	0.245561i	$-0.921028\pi$
$197$	−27.2118	−1.93876	−0.969381	+	0.245561i	$0.921028\pi$
$198$	8.30569	0.590260
$199$	23.3270	1.65361	0.826804	−	0.562490i	$-0.190157\pi$
$199$	23.3270	1.65361	0.826804	+	0.562490i	$0.190157\pi$
$200$	14.8538	1.05032
$201$	1.64512	0.116038
$202$	−19.8081	−1.39369
$203$	7.21366	0.506299
$204$	−14.7138	−1.03017
$205$	1.37732	0.0961959
$206$	11.1325	0.775639
$207$	18.2966	1.27170
$208$	−2.03641	−0.141200
$209$	1.96531	0.135943
$210$	−1.01890	−0.0703105
$211$	−3.69292	−0.254231	−0.127115	−	0.991888i	$-0.540572\pi$
$211$	−3.69292	−0.254231	−0.127115	+	0.991888i	$0.540572\pi$
$212$	−25.5409	−1.75416
$213$	0.933250	0.0639452
$214$	−28.9749	−1.98069
$215$	−0.0908305	−0.00619459
$216$	−14.4500	−0.983201
$217$	9.34100	0.634108
$218$	27.1015	1.83554
$219$	−9.36980	−0.633152
$220$	−1.58082	−0.106579
$221$	−28.8421	−1.94013
$222$	22.0781	1.48178
$223$	26.8341	1.79694	0.898472	−	0.439031i	$-0.144678\pi$
$223$	26.8341	1.79694	0.898472	+	0.439031i	$0.144678\pi$
$224$	8.83854	0.590550
$225$	10.4758	0.698386
$226$	−2.30435	−0.153283
$227$	4.58504	0.304319	0.152160	−	0.988356i	$-0.451377\pi$
$227$	4.58504	0.304319	0.152160	+	0.988356i	$0.451377\pi$
$228$	−3.58609	−0.237495
$229$	−20.6376	−1.36377	−0.681885	−	0.731459i	$-0.738840\pi$
$229$	−20.6376	−1.36377	−0.681885	+	0.731459i	$0.738840\pi$
$230$	−5.58652	−0.368364
$231$	−2.65348	−0.174586
$232$	−12.9662	−0.851276
$233$	18.2844	1.19785	0.598926	−	0.800804i	$-0.295594\pi$
$233$	18.2844	1.19785	0.598926	+	0.800804i	$0.295594\pi$
$234$	−29.7077	−1.94205
$235$	2.95542	0.192791
$236$	−33.9081	−2.20723
$237$	1.76548	0.114680
$238$	−18.4339	−1.19489
$239$	−4.88767	−0.316157	−0.158078	−	0.987427i	$-0.550530\pi$
$239$	−4.88767	−0.316157	−0.158078	+	0.987427i	$0.550530\pi$
$240$	0.0885389	0.00571516
$241$	−18.5637	−1.19579	−0.597897	−	0.801573i	$-0.703997\pi$
$241$	−18.5637	−1.19579	−0.597897	+	0.801573i	$0.703997\pi$
$242$	18.7435	1.20488
$243$	−16.1518	−1.03614
$244$	−21.9479	−1.40507
$245$	1.17900	0.0753235
$246$	−10.4996	−0.669432
$247$	−7.02950	−0.447277
$248$	−16.7901	−1.06617
$249$	−13.0660	−0.828024
$250$	−6.44890	−0.407864
$251$	21.0295	1.32737	0.663684	−	0.748013i	$-0.268992\pi$
$251$	21.0295	1.32737	0.663684	+	0.748013i	$0.268992\pi$
$252$	−11.8357	−0.745580
$253$	−14.5488	−0.914676
$254$	16.5216	1.03666
$255$	1.25400	0.0785284
$256$	−18.1133	−1.13208
$257$	1.63524	0.102004	0.0510018	−	0.998699i	$-0.483759\pi$
$257$	1.63524	0.102004	0.0510018	+	0.998699i	$0.483759\pi$
$258$	0.692424	0.0431084
$259$	17.2421	1.07137
$260$	5.65426	0.350662
$261$	−9.14459	−0.566036
$262$	37.6858	2.32823
$263$	2.89001	0.178206	0.0891030	−	0.996022i	$-0.471600\pi$
$263$	2.89001	0.178206	0.0891030	+	0.996022i	$0.471600\pi$
$264$	4.76952	0.293544
$265$	2.17675	0.133717
$266$	−4.49277	−0.275470
$267$	−9.57975	−0.586271
$268$	−5.83489	−0.356422
$269$	20.7332	1.26413	0.632064	−	0.774916i	$-0.282208\pi$
$269$	20.7332	1.26413	0.632064	+	0.774916i	$0.282208\pi$
$270$	3.11164	0.189369
$271$	−6.78151	−0.411948	−0.205974	−	0.978558i	$-0.566036\pi$
$271$	−6.78151	−0.411948	−0.205974	+	0.978558i	$0.566036\pi$
$272$	1.60185	0.0971265
$273$	9.49094	0.574418
$274$	−51.9432	−3.13800
$275$	−8.32998	−0.502317
$276$	26.5471	1.59795
$277$	2.51629	0.151189	0.0755945	−	0.997139i	$-0.475915\pi$
$277$	2.51629	0.151189	0.0755945	+	0.997139i	$0.475915\pi$
$278$	−40.9230	−2.45440
$279$	−11.8414	−0.708925
$280$	1.43027	0.0854749
$281$	−11.3193	−0.675250	−0.337625	−	0.941281i	$-0.609624\pi$
$281$	−11.3193	−0.675250	−0.337625	+	0.941281i	$0.609624\pi$
$282$	−22.5299	−1.34164
$283$	8.02695	0.477152	0.238576	−	0.971124i	$-0.423319\pi$
$283$	8.02695	0.477152	0.238576	+	0.971124i	$0.423319\pi$
$284$	−3.31004	−0.196415
$285$	0.305629	0.0181039
$286$	23.6225	1.39683
$287$	−8.19980	−0.484019
$288$	−11.2044	−0.660226
$289$	5.68738	0.334552
$290$	2.79213	0.163959
$291$	0.0287552	0.00168566
$292$	33.2327	1.94480
$293$	−5.08891	−0.297297	−0.148649	−	0.988890i	$-0.547492\pi$
$293$	−5.08891	−0.297297	−0.148649	+	0.988890i	$0.547492\pi$
$294$	−8.98781	−0.524180
$295$	2.88986	0.168254
$296$	−30.9919	−1.80137
$297$	8.10357	0.470217
$298$	−14.5084	−0.840449
$299$	52.0380	3.00944
$300$	15.1997	0.877552
$301$	0.540756	0.0311686
$302$	48.3800	2.78396
$303$	−8.02218	−0.460862
$304$	0.390409	0.0223915
$305$	1.87053	0.107106
$306$	23.3682	1.33587
$307$	−14.4285	−0.823480	−0.411740	−	0.911301i	$-0.635079\pi$
$307$	−14.4285	−0.823480	−0.411740	+	0.911301i	$0.635079\pi$
$308$	9.41134	0.536261
$309$	4.50861	0.256486
$310$	3.61554	0.205349
$311$	34.6462	1.96461	0.982304	−	0.187295i	$-0.0599719\pi$
$311$	34.6462	1.96461	0.982304	+	0.187295i	$0.0599719\pi$
$312$	−17.0596	−0.965808
$313$	−34.2714	−1.93714	−0.968568	−	0.248750i	$-0.919980\pi$
$313$	−34.2714	−1.93714	−0.968568	+	0.248750i	$0.919980\pi$
$314$	9.28060	0.523735
$315$	1.00871	0.0568345
$316$	−6.26178	−0.352253
$317$	−28.0560	−1.57578	−0.787892	−	0.615814i	$-0.788827\pi$
$317$	−28.0560	−1.57578	−0.787892	+	0.615814i	$0.788827\pi$
$318$	−16.5939	−0.930542
$319$	7.27146	0.407123
$320$	3.61080	0.201850
$321$	−11.7347	−0.654966
$322$	33.2591	1.85346
$323$	5.52945	0.307667
$324$	6.35517	0.353065
$325$	29.7946	1.65271
$326$	−16.8612	−0.933855
$327$	10.9759	0.606971
$328$	14.7388	0.813814
$329$	−17.5950	−0.970044
$330$	−1.02706	−0.0565377
$331$	11.6338	0.639450	0.319725	−	0.947510i	$-0.396410\pi$
$331$	11.6338	0.639450	0.319725	+	0.947510i	$0.396410\pi$
$332$	46.3424	2.54337
$333$	−21.8574	−1.19778
$334$	−0.528009	−0.0288914
$335$	0.497285	0.0271696
$336$	−0.527113	−0.0287564
$337$	18.9296	1.03116	0.515579	−	0.856842i	$-0.327577\pi$
$337$	18.9296	1.03116	0.515579	+	0.856842i	$0.327577\pi$
$338$	−54.5361	−2.96638
$339$	−0.933250	−0.0506872
$340$	−4.44767	−0.241209
$341$	9.41584	0.509897
$342$	5.69539	0.307971
$343$	−18.7755	−1.01378
$344$	−0.971986	−0.0524060
$345$	−2.26251	−0.121809
$346$	31.8006	1.70961
$347$	29.6956	1.59414	0.797071	−	0.603885i	$-0.206382\pi$
$347$	29.6956	1.59414	0.797071	+	0.603885i	$0.206382\pi$
$348$	−13.2682	−0.711249
$349$	−19.0151	−1.01786	−0.508928	−	0.860809i	$-0.669958\pi$
$349$	−19.0151	−1.01786	−0.508928	+	0.860809i	$0.669958\pi$
$350$	19.0426	1.01787
$351$	−28.9847	−1.54709
$352$	8.90935	0.474870
$353$	−19.2688	−1.02557	−0.512786	−	0.858516i	$-0.671387\pi$
$353$	−19.2688	−1.02557	−0.512786	+	0.858516i	$0.671387\pi$
$354$	−22.0301	−1.17089
$355$	0.282102	0.0149724
$356$	33.9774	1.80080
$357$	−7.46562	−0.395123
$358$	−30.5970	−1.61710
$359$	14.7118	0.776461	0.388230	−	0.921562i	$-0.373086\pi$
$359$	14.7118	0.776461	0.388230	+	0.921562i	$0.373086\pi$
$360$	−1.81312	−0.0955598
$361$	−17.6523	−0.929071
$362$	50.5153	2.65502
$363$	7.59101	0.398425
$364$	−33.6624	−1.76439
$365$	−2.83230	−0.148249
$366$	−14.2595	−0.745358
$367$	1.22598	0.0639958	0.0319979	−	0.999488i	$-0.489813\pi$
$367$	1.22598	0.0639958	0.0319979	+	0.999488i	$0.489813\pi$
$368$	−2.89012	−0.150658
$369$	10.3947	0.541126
$370$	6.67374	0.346951
$371$	−12.9592	−0.672809
$372$	−17.1810	−0.890795
$373$	15.0902	0.781341	0.390671	−	0.920531i	$-0.372243\pi$
$373$	15.0902	0.781341	0.390671	+	0.920531i	$0.372243\pi$
$374$	−18.5816	−0.960832
$375$	−2.61177	−0.134871
$376$	31.6263	1.63100
$377$	−26.0085	−1.33950
$378$	−18.5251	−0.952826
$379$	−4.56246	−0.234358	−0.117179	−	0.993111i	$-0.537385\pi$
$379$	−4.56246	−0.234358	−0.117179	+	0.993111i	$0.537385\pi$
$380$	−1.08400	−0.0556081
$381$	6.69116	0.342799
$382$	−9.94398	−0.508778
$383$	−3.72927	−0.190557	−0.0952785	−	0.995451i	$-0.530374\pi$
$383$	−3.72927	−0.190557	−0.0952785	+	0.995451i	$0.530374\pi$
$384$	−17.7033	−0.903417
$385$	−0.802093	−0.0408784
$386$	−11.1965	−0.569888
$387$	−0.685504	−0.0348461
$388$	−0.101989	−0.00517770
$389$	15.8620	0.804237	0.402119	−	0.915588i	$-0.368274\pi$
$389$	15.8620	0.804237	0.402119	+	0.915588i	$0.368274\pi$
$390$	3.67357	0.186019
$391$	−40.9334	−2.07009
$392$	12.6166	0.637234
$393$	15.2625	0.769892
$394$	62.7057	3.15907
$395$	0.533667	0.0268517
$396$	−11.9305	−0.599533
$397$	21.2059	1.06429	0.532147	−	0.846652i	$-0.321385\pi$
$397$	21.2059	1.06429	0.532147	+	0.846652i	$0.321385\pi$
$398$	−53.7537	−2.69443
$399$	−1.81955	−0.0910913
$400$	−1.65475	−0.0827374
$401$	6.34469	0.316839	0.158419	−	0.987372i	$-0.449360\pi$
$401$	6.34469	0.316839	0.158419	+	0.987372i	$0.449360\pi$
$402$	−3.79093	−0.189074
$403$	−33.6785	−1.67764
$404$	28.4530	1.41559
$405$	−0.541627	−0.0269137
$406$	−16.6228	−0.824976
$407$	17.3802	0.861507
$408$	13.4191	0.664347
$409$	−31.8733	−1.57603	−0.788016	−	0.615655i	$-0.788892\pi$
$409$	−31.8733	−1.57603	−0.788016	+	0.615655i	$0.788892\pi$
$410$	−3.17382	−0.156744
$411$	−21.0367	−1.03766
$412$	−15.9911	−0.787825
$413$	−17.2047	−0.846586
$414$	−42.1618	−2.07214
$415$	−3.94958	−0.193877
$416$	−31.8669	−1.56240
$417$	−16.5736	−0.811611
$418$	−4.52877	−0.221509
$419$	−22.7297	−1.11042	−0.555208	−	0.831711i	$-0.687361\pi$
$419$	−22.7297	−1.11042	−0.555208	+	0.831711i	$0.687361\pi$
$420$	1.46357	0.0714150
$421$	−8.30315	−0.404671	−0.202335	−	0.979316i	$-0.564853\pi$
$421$	−8.30315	−0.404671	−0.202335	+	0.979316i	$0.564853\pi$
$422$	8.50979	0.414250
$423$	22.3048	1.08450
$424$	23.2936	1.13124
$425$	−23.4366	−1.13684
$426$	−2.15054	−0.104194
$427$	−11.1361	−0.538915
$428$	41.6205	2.01180
$429$	9.56698	0.461898
$430$	0.209306	0.0100936
$431$	8.68742	0.418458	0.209229	−	0.977867i	$-0.432905\pi$
$431$	8.68742	0.418458	0.209229	+	0.977867i	$0.432905\pi$
$432$	1.60977	0.0774502
$433$	32.9652	1.58421	0.792103	−	0.610387i	$-0.208986\pi$
$433$	32.9652	1.58421	0.792103	+	0.610387i	$0.208986\pi$
$434$	−21.5250	−1.03323
$435$	1.13080	0.0542175
$436$	−38.9294	−1.86438
$437$	−9.97643	−0.477238
$438$	21.5913	1.03167
$439$	−31.8836	−1.52172	−0.760861	−	0.648915i	$-0.775223\pi$
$439$	−31.8836	−1.52172	−0.760861	+	0.648915i	$0.775223\pi$
$440$	1.44173	0.0687317
$441$	8.89798	0.423714
$442$	66.4625	3.16130
$443$	10.7667	0.511543	0.255772	−	0.966737i	$-0.417670\pi$
$443$	10.7667	0.511543	0.255772	+	0.966737i	$0.417670\pi$
$444$	−31.7136	−1.50506
$445$	−2.89576	−0.137272
$446$	−61.8352	−2.92798
$447$	−5.87582	−0.277917
$448$	−21.4967	−1.01563
$449$	4.43244	0.209180	0.104590	−	0.994515i	$-0.466647\pi$
$449$	4.43244	0.209180	0.104590	+	0.994515i	$0.466647\pi$
$450$	−24.1399	−1.13797
$451$	−8.26550	−0.389207
$452$	3.31004	0.155691
$453$	19.5936	0.920589
$454$	−10.5655	−0.495865
$455$	2.86892	0.134497
$456$	3.27056	0.153158
$457$	−37.1151	−1.73617	−0.868084	−	0.496417i	$-0.834649\pi$
$457$	−37.1151	−1.73617	−0.868084	+	0.496417i	$0.834649\pi$
$458$	47.5563	2.22216
$459$	22.7996	1.06419
$460$	8.02465	0.374151
$461$	8.33726	0.388305	0.194152	−	0.980971i	$-0.437804\pi$
$461$	8.33726	0.388305	0.194152	+	0.980971i	$0.437804\pi$
$462$	6.11455	0.284475
$463$	−7.16146	−0.332821	−0.166411	−	0.986057i	$-0.553218\pi$
$463$	−7.16146	−0.332821	−0.166411	+	0.986057i	$0.553218\pi$
$464$	1.44447	0.0670580
$465$	1.46427	0.0679040
$466$	−42.1338	−1.95181
$467$	8.72226	0.403618	0.201809	−	0.979425i	$-0.435318\pi$
$467$	8.72226	0.403618	0.201809	+	0.979425i	$0.435318\pi$
$468$	42.6731	1.97256
$469$	−2.96057	−0.136706
$470$	−6.81034	−0.314138
$471$	3.75859	0.173187
$472$	30.9247	1.42342
$473$	0.545089	0.0250632
$474$	−4.06828	−0.186862
$475$	−5.71204	−0.262087
$476$	26.4790	1.21366
$477$	16.4281	0.752191
$478$	11.2629	0.515153
$479$	−10.7178	−0.489710	−0.244855	−	0.969560i	$-0.578740\pi$
$479$	−10.7178	−0.489710	−0.244855	+	0.969560i	$0.578740\pi$
$480$	1.38551	0.0632395
$481$	−62.1654	−2.83450
$482$	42.7773	1.94845
$483$	13.4698	0.612895
$484$	−26.9237	−1.22381
$485$	0.00869212	0.000394689 0
$486$	37.2196	1.68831
$487$	35.1612	1.59331	0.796653	−	0.604438i	$-0.206602\pi$
$487$	35.1612	1.59331	0.796653	+	0.604438i	$0.206602\pi$
$488$	20.0167	0.906115
$489$	−6.82868	−0.308804
$490$	−2.71683	−0.122734
$491$	−18.3707	−0.829058	−0.414529	−	0.910036i	$-0.636054\pi$
$491$	−18.3707	−0.829058	−0.414529	+	0.910036i	$0.636054\pi$
$492$	15.0820	0.679949
$493$	20.4584	0.921400
$494$	16.1985	0.728803
$495$	1.01679	0.0457015
$496$	1.87045	0.0839859
$497$	−1.67948	−0.0753352
$498$	30.1087	1.34920
$499$	14.8324	0.663989	0.331995	−	0.943281i	$-0.392278\pi$
$499$	14.8324	0.663989	0.331995	+	0.943281i	$0.392278\pi$
$500$	9.26340	0.414272
$501$	−0.213841	−0.00955370
$502$	−48.4593	−2.16285
$503$	−28.1042	−1.25310	−0.626552	−	0.779379i	$-0.715535\pi$
$503$	−28.1042	−1.25310	−0.626552	+	0.779379i	$0.715535\pi$
$504$	10.7943	0.480818
$505$	−2.42494	−0.107908
$506$	33.5256	1.49040
$507$	−22.0868	−0.980910
$508$	−23.7322	−1.05294
$509$	24.5891	1.08989	0.544945	−	0.838472i	$-0.316550\pi$
$509$	24.5891	1.08989	0.544945	+	0.838472i	$0.316550\pi$
$510$	−2.88965	−0.127956
$511$	16.8620	0.745930
$512$	3.80030	0.167951
$513$	5.55679	0.245338
$514$	−3.76818	−0.166207
$515$	1.36286	0.0600548
$516$	−0.994619	−0.0437857
$517$	−17.7360	−0.780027
$518$	−39.7319	−1.74572
$519$	12.8791	0.565328
$520$	−5.15676	−0.226139
$521$	17.8367	0.781439	0.390719	−	0.920510i	$-0.372226\pi$
$521$	17.8367	0.781439	0.390719	+	0.920510i	$0.372226\pi$
$522$	21.0724	0.922312
$523$	31.3073	1.36897	0.684486	−	0.729026i	$-0.260027\pi$
$523$	31.3073	1.36897	0.684486	+	0.729026i	$0.260027\pi$
$524$	−54.1330	−2.36481
$525$	7.71216	0.336586
$526$	−6.65961	−0.290373
$527$	26.4917	1.15400
$528$	−0.531336	−0.0231235
$529$	50.8536	2.21103
$530$	−5.01601	−0.217881
$531$	21.8100	0.946472
$532$	6.45356	0.279797
$533$	29.5639	1.28056
$534$	22.0751	0.955284
$535$	−3.54715	−0.153357
$536$	5.32149	0.229854
$537$	−12.3916	−0.534737
$538$	−47.7767	−2.05980
$539$	−7.07536	−0.304757
$540$	−4.46966	−0.192344
$541$	−0.361817	−0.0155557	−0.00777786	−	0.999970i	$-0.502476\pi$
$541$	−0.361817	−0.0155557	−0.00777786	+	0.999970i	$0.502476\pi$
$542$	15.6270	0.671237
$543$	20.4584	0.877954
$544$	25.0667	1.07472
$545$	3.31780	0.142119
$546$	−21.8705	−0.935970
$547$	−8.19034	−0.350193	−0.175097	−	0.984551i	$-0.556024\pi$
$547$	−8.19034	−0.350193	−0.175097	+	0.984551i	$0.556024\pi$
$548$	74.6128	3.18730
$549$	14.1170	0.602500
$550$	19.1952	0.818487
$551$	4.98619	0.212419
$552$	−24.2113	−1.03050
$553$	−3.17717	−0.135107
$554$	−5.79842	−0.246351
$555$	2.70283	0.114729
$556$	58.7830	2.49295
$557$	37.5300	1.59020	0.795099	−	0.606479i	$-0.207419\pi$
$557$	37.5300	1.59020	0.795099	+	0.606479i	$0.207419\pi$
$558$	27.2867	1.15514
$559$	−1.94967	−0.0824621
$560$	−0.159335	−0.00673315
$561$	−7.52544	−0.317724
$562$	26.0836	1.10027
$563$	2.60028	0.109589	0.0547944	−	0.998498i	$-0.482550\pi$
$563$	2.60028	0.109589	0.0547944	+	0.998498i	$0.482550\pi$
$564$	32.3627	1.36272
$565$	−0.282102	−0.0118681
$566$	−18.4969	−0.777483
$567$	3.22455	0.135419
$568$	3.01880	0.126666
$569$	−2.32748	−0.0975729	−0.0487864	−	0.998809i	$-0.515535\pi$
$569$	−2.32748	−0.0975729	−0.0487864	+	0.998809i	$0.515535\pi$
$570$	−0.704276	−0.0294989
$571$	−21.5421	−0.901509	−0.450755	−	0.892648i	$-0.648845\pi$
$571$	−21.5421	−0.901509	−0.450755	+	0.892648i	$0.648845\pi$
$572$	−33.9321	−1.41877
$573$	−4.02725	−0.168241
$574$	18.8952	0.788672
$575$	42.2851	1.76341
$576$	27.2509	1.13546
$577$	40.3839	1.68120	0.840601	−	0.541654i	$-0.182202\pi$
$577$	40.3839	1.68120	0.840601	+	0.541654i	$0.182202\pi$
$578$	−13.1057	−0.545127
$579$	−4.53453	−0.188449
$580$	−4.01070	−0.166535
$581$	23.5137	0.975512
$582$	−0.0662622	−0.00274666
$583$	−13.0630	−0.541016
$584$	−30.3087	−1.25418
$585$	−3.63686	−0.150366
$586$	11.7266	0.484423
$587$	24.6613	1.01788	0.508940	−	0.860802i	$-0.330037\pi$
$587$	24.6613	1.01788	0.508940	+	0.860802i	$0.330037\pi$
$588$	12.9104	0.532414
$589$	6.45665	0.266041
$590$	−6.65926	−0.274157
$591$	25.3954	1.04463
$592$	3.45258	0.141900
$593$	19.4069	0.796948	0.398474	−	0.917180i	$-0.369540\pi$
$593$	19.4069	0.796948	0.398474	+	0.917180i	$0.369540\pi$
$594$	−18.6735	−0.766182
$595$	−2.25671	−0.0925159
$596$	20.8403	0.853652
$597$	−21.7699	−0.890984
$598$	−119.914	−4.90365
$599$	2.83497	0.115834	0.0579169	−	0.998321i	$-0.481554\pi$
$599$	2.83497	0.115834	0.0579169	+	0.998321i	$0.481554\pi$
$600$	−13.8623	−0.565925
$601$	5.26525	0.214774	0.107387	−	0.994217i	$-0.465752\pi$
$601$	5.26525	0.214774	0.107387	+	0.994217i	$0.465752\pi$
$602$	−1.24609	−0.0507869
$603$	3.75304	0.152836
$604$	−69.4946	−2.82769
$605$	2.29461	0.0932890
$606$	18.4859	0.750939
$607$	−15.4741	−0.628076	−0.314038	−	0.949410i	$-0.601682\pi$
$607$	−15.4741	−0.628076	−0.314038	+	0.949410i	$0.601682\pi$
$608$	6.10933	0.247766
$609$	−6.73214	−0.272800
$610$	−4.31036	−0.174522
$611$	63.4378	2.56642
$612$	−33.5669	−1.35686
$613$	2.94571	0.118976	0.0594881	−	0.998229i	$-0.481053\pi$
$613$	2.94571	0.118976	0.0594881	+	0.998229i	$0.481053\pi$
$614$	33.2484	1.34180
$615$	−1.28538	−0.0518315
$616$	−8.58327	−0.345830
$617$	−27.3304	−1.10028	−0.550139	−	0.835073i	$-0.685425\pi$
$617$	−27.3304	−1.10028	−0.550139	+	0.835073i	$0.685425\pi$
$618$	−10.3894	−0.417924
$619$	−18.5520	−0.745669	−0.372835	−	0.927898i	$-0.621614\pi$
$619$	−18.5520	−0.745669	−0.372835	+	0.927898i	$0.621614\pi$
$620$	−5.19347	−0.208575
$621$	−41.1358	−1.65072
$622$	−79.8372	−3.20118
$623$	17.2398	0.690698
$624$	1.90048	0.0760800
$625$	23.8126	0.952504
$626$	78.9735	3.15641
$627$	−1.83413	−0.0732480
$628$	−13.3309	−0.531963
$629$	48.8997	1.94976
$630$	−2.32443	−0.0926075
$631$	−32.9303	−1.31094	−0.655468	−	0.755223i	$-0.727528\pi$
$631$	−32.9303	−1.31094	−0.655468	+	0.755223i	$0.727528\pi$
$632$	5.71082	0.227164
$633$	3.44642	0.136983
$634$	64.6510	2.56762
$635$	2.02260	0.0802645
$636$	23.8360	0.945161
$637$	25.3071	1.00270
$638$	−16.7560	−0.663376
$639$	2.12904	0.0842237
$640$	−5.35134	−0.211530
$641$	38.2505	1.51081	0.755403	−	0.655261i	$-0.227441\pi$
$641$	38.2505	1.51081	0.755403	+	0.655261i	$0.227441\pi$
$642$	27.0408	1.06722
$643$	9.77910	0.385650	0.192825	−	0.981233i	$-0.438235\pi$
$643$	9.77910	0.385650	0.192825	+	0.981233i	$0.438235\pi$
$644$	−47.7744	−1.88258
$645$	0.0847676	0.00333772
$646$	−12.7418	−0.501319
$647$	−23.8066	−0.935934	−0.467967	−	0.883746i	$-0.655013\pi$
$647$	−23.8066	−0.935934	−0.467967	+	0.883746i	$0.655013\pi$
$648$	−5.79600	−0.227688
$649$	−17.3425	−0.680753
$650$	−68.6572	−2.69296
$651$	−8.71749	−0.341665
$652$	24.2199	0.948525
$653$	28.2657	1.10612	0.553061	−	0.833140i	$-0.313459\pi$
$653$	28.2657	1.10612	0.553061	+	0.833140i	$0.313459\pi$
$654$	−25.2925	−0.989013
$655$	4.61355	0.180266
$656$	−1.64194	−0.0641069
$657$	−21.3755	−0.833939
$658$	40.5451	1.58061
$659$	37.5997	1.46467	0.732337	−	0.680942i	$-0.238429\pi$
$659$	37.5997	1.46467	0.732337	+	0.680942i	$0.238429\pi$
$660$	1.47530	0.0574259
$661$	10.2176	0.397419	0.198710	−	0.980058i	$-0.436325\pi$
$661$	10.2176	0.397419	0.198710	+	0.980058i	$0.436325\pi$
$662$	−26.8083	−1.04193
$663$	26.9169	1.04537
$664$	−42.2648	−1.64019
$665$	−0.550012	−0.0213286
$666$	50.3672	1.95169
$667$	−36.9118	−1.42923
$668$	0.758449	0.0293453
$669$	−25.0429	−0.968215
$670$	−1.14592	−0.0442708
$671$	−11.2254	−0.433350
$672$	−8.24856	−0.318195
$673$	−2.73103	−0.105274	−0.0526368	−	0.998614i	$-0.516763\pi$
$673$	−2.73103	−0.105274	−0.0526368	+	0.998614i	$0.516763\pi$
$674$	−43.6204	−1.68019
$675$	−23.5525	−0.906535
$676$	78.3374	3.01298
$677$	9.62394	0.369878	0.184939	−	0.982750i	$-0.440791\pi$
$677$	9.62394	0.369878	0.184939	+	0.982750i	$0.440791\pi$
$678$	2.15054	0.0825909
$679$	−0.0517482	−0.00198591
$680$	4.05633	0.155553
$681$	−4.27898	−0.163971
$682$	−21.6974	−0.830838
$683$	32.3114	1.23636	0.618181	−	0.786036i	$-0.287870\pi$
$683$	32.3114	1.23636	0.618181	+	0.786036i	$0.287870\pi$
$684$	−8.18103	−0.312810
$685$	−6.35896	−0.242964
$686$	43.2654	1.65188
$687$	19.2600	0.734816
$688$	0.108282	0.00412820
$689$	46.7237	1.78003
$690$	5.21362	0.198479
$691$	−44.2690	−1.68407	−0.842036	−	0.539421i	$-0.818643\pi$
$691$	−44.2690	−1.68407	−0.842036	+	0.539421i	$0.818643\pi$
$692$	−45.6794	−1.73647
$693$	−6.05345	−0.229951
$694$	−68.4291	−2.59753
$695$	−5.00985	−0.190034
$696$	12.1007	0.458678
$697$	−23.2552	−0.880852
$698$	43.8175	1.65852
$699$	−17.0639	−0.645417
$700$	−27.3534	−1.03386
$701$	14.0162	0.529383	0.264692	−	0.964333i	$-0.414730\pi$
$701$	14.0162	0.529383	0.264692	+	0.964333i	$0.414730\pi$
$702$	66.7911	2.52087
$703$	11.9180	0.449496
$704$	−21.6690	−0.816680
$705$	−2.75815	−0.103878
$706$	44.4020	1.67109
$707$	14.4368	0.542951
$708$	31.6448	1.18928
$709$	−25.0562	−0.941004	−0.470502	−	0.882399i	$-0.655927\pi$
$709$	−25.0562	−0.941004	−0.470502	+	0.882399i	$0.655927\pi$
$710$	−0.650063	−0.0243964
$711$	4.02762	0.151048
$712$	−30.9878	−1.16132
$713$	−47.7973	−1.79002
$714$	17.2034	0.643822
$715$	2.89190	0.108151
$716$	43.9505	1.64251
$717$	4.56141	0.170349
$718$	−33.9013	−1.26518
$719$	38.8510	1.44890	0.724448	−	0.689329i	$-0.242095\pi$
$719$	38.8510	1.44890	0.724448	+	0.689329i	$0.242095\pi$
$720$	0.201986	0.00752757
$721$	−8.11373	−0.302171
$722$	40.6772	1.51385
$723$	17.3246	0.644308
$724$	−72.5617	−2.69673
$725$	−21.1340	−0.784897
$726$	−17.4924	−0.649202
$727$	−39.0714	−1.44908	−0.724539	−	0.689234i	$-0.757947\pi$
$727$	−39.0714	−1.44908	−0.724539	+	0.689234i	$0.757947\pi$
$728$	30.7005	1.13784
$729$	9.31381	0.344956
$730$	6.52661	0.241561
$731$	1.53362	0.0567229
$732$	20.4828	0.757067
$733$	17.0698	0.630489	0.315244	−	0.949011i	$-0.397914\pi$
$733$	17.0698	0.630489	0.315244	+	0.949011i	$0.397914\pi$
$734$	−2.82510	−0.104276
$735$	−1.10030	−0.0405852
$736$	−45.2262	−1.66706
$737$	−2.98429	−0.109928
$738$	−23.9531	−0.881724
$739$	50.8622	1.87099	0.935497	−	0.353334i	$-0.114952\pi$
$739$	50.8622	1.87099	0.935497	+	0.353334i	$0.114952\pi$
$740$	−9.58637	−0.352402
$741$	6.56028	0.240998
$742$	29.8626	1.09629
$743$	0.149025	0.00546718	0.00273359	−	0.999996i	$-0.499130\pi$
$743$	0.149025	0.00546718	0.00273359	+	0.999996i	$0.499130\pi$
$744$	15.6693	0.574465
$745$	−1.77614	−0.0650727
$746$	−34.7732	−1.27314
$747$	−29.8078	−1.09061
$748$	26.6912	0.975926
$749$	21.1178	0.771629
$750$	6.01843	0.219762
$751$	33.5099	1.22279	0.611397	−	0.791324i	$-0.290608\pi$
$751$	33.5099	1.22279	0.611397	+	0.791324i	$0.290608\pi$
$752$	−3.52325	−0.128480
$753$	−19.6257	−0.715202
$754$	59.9327	2.18262
$755$	5.92275	0.215551
$756$	26.6100	0.967795
$757$	43.5835	1.58407	0.792034	−	0.610477i	$-0.209022\pi$
$757$	43.5835	1.58407	0.792034	+	0.610477i	$0.209022\pi$
$758$	10.5135	0.381869
$759$	13.5777	0.492839
$760$	0.988624	0.0358612
$761$	49.5560	1.79640	0.898202	−	0.439582i	$-0.144873\pi$
$761$	49.5560	1.79640	0.898202	+	0.439582i	$0.144873\pi$
$762$	−15.4188	−0.558564
$763$	−19.7524	−0.715085
$764$	14.2838	0.516771
$765$	2.86077	0.103432
$766$	8.59356	0.310498
$767$	62.0305	2.23979
$768$	16.9042	0.609977
$769$	6.07870	0.219204	0.109602	−	0.993976i	$-0.465042\pi$
$769$	6.07870	0.219204	0.109602	+	0.993976i	$0.465042\pi$
$770$	1.84830	0.0666083
$771$	−1.52609	−0.0549608
$772$	16.0830	0.578841
$773$	−16.3560	−0.588285	−0.294143	−	0.955762i	$-0.595034\pi$
$773$	−16.3560	−0.588285	−0.294143	+	0.955762i	$0.595034\pi$
$774$	1.57964	0.0567791
$775$	−27.3665	−0.983034
$776$	0.0930152	0.00333905
$777$	−16.0912	−0.577268
$778$	−36.5517	−1.31044
$779$	−5.66783	−0.203071
$780$	−5.27683	−0.188941
$781$	−1.69294	−0.0605782
$782$	94.3250	3.37306
$783$	20.5596	0.734739
$784$	−1.40552	−0.0501971
$785$	1.13615	0.0405508
$786$	−35.1702	−1.25448
$787$	27.2569	0.971604	0.485802	−	0.874069i	$-0.338528\pi$
$787$	27.2569	0.971604	0.485802	+	0.874069i	$0.338528\pi$
$788$	−90.0723	−3.20869
$789$	−2.69711	−0.0960195
$790$	−1.22976	−0.0437528
$791$	1.67948	0.0597156
$792$	10.8808	0.386633
$793$	40.1507	1.42579
$794$	−48.8659	−1.73419
$795$	−2.03145	−0.0720482
$796$	77.2134	2.73676
$797$	−28.0186	−0.992469	−0.496235	−	0.868188i	$-0.665284\pi$
$797$	−28.0186	−0.992469	−0.496235	+	0.868188i	$0.665284\pi$
$798$	4.19288	0.148426
$799$	−49.9006	−1.76536
$800$	−25.8944	−0.915506
$801$	−21.8545	−0.772191
$802$	−14.6204	−0.516264
$803$	16.9971	0.599814
$804$	5.44541	0.192045
$805$	4.07163	0.143506
$806$	77.6071	2.73359
$807$	−19.3493	−0.681127
$808$	−25.9495	−0.912901
$809$	−22.9733	−0.807697	−0.403848	−	0.914826i	$-0.632328\pi$
$809$	−22.9733	−0.807697	−0.403848	+	0.914826i	$0.632328\pi$
$810$	1.24810	0.0438538
$811$	41.2509	1.44852	0.724258	−	0.689529i	$-0.242182\pi$
$811$	41.2509	1.44852	0.724258	+	0.689529i	$0.242182\pi$
$812$	23.8775	0.837937
$813$	6.32885	0.221962
$814$	−40.0502	−1.40376
$815$	−2.06417	−0.0723047
$816$	−1.49493	−0.0523329
$817$	0.373779	0.0130769
$818$	73.4473	2.56802
$819$	21.6519	0.756579
$820$	4.55898	0.159206
$821$	28.8512	1.00691	0.503456	−	0.864021i	$-0.332061\pi$
$821$	28.8512	1.00691	0.503456	+	0.864021i	$0.332061\pi$
$822$	48.4760	1.69079
$823$	−44.7482	−1.55983	−0.779913	−	0.625888i	$-0.784737\pi$
$823$	−44.7482	−1.55983	−0.779913	+	0.625888i	$0.784737\pi$
$824$	14.5841	0.508061
$825$	7.77395	0.270654
$826$	39.6456	1.37945
$827$	32.6904	1.13676	0.568379	−	0.822767i	$-0.307571\pi$
$827$	32.6904	1.13676	0.568379	+	0.822767i	$0.307571\pi$
$828$	60.5626	2.10469
$829$	49.4868	1.71875	0.859373	−	0.511349i	$-0.170854\pi$
$829$	49.4868	1.71875	0.859373	+	0.511349i	$0.170854\pi$
$830$	9.10123	0.315908
$831$	−2.34832	−0.0814625
$832$	77.5054	2.68701
$833$	−19.9067	−0.689726
$834$	38.1913	1.32246
$835$	−0.0646397	−0.00223695
$836$	6.50527	0.224989
$837$	26.6227	0.920214
$838$	52.3772	1.80934
$839$	14.7069	0.507738	0.253869	−	0.967239i	$-0.418297\pi$
$839$	14.7069	0.507738	0.253869	+	0.967239i	$0.418297\pi$
$840$	−1.33480	−0.0460549
$841$	−10.5516	−0.363848
$842$	19.1334	0.659380
$843$	10.5637	0.363833
$844$	−12.2237	−0.420758
$845$	−6.67639	−0.229675
$846$	−51.3981	−1.76710
$847$	−13.6608	−0.469392
$848$	−2.59497	−0.0891116
$849$	−7.49114	−0.257095
$850$	54.0062	1.85240
$851$	−88.2266	−3.02437
$852$	3.08910	0.105831
$853$	−15.0265	−0.514499	−0.257249	−	0.966345i	$-0.582816\pi$
$853$	−15.0265	−0.514499	−0.257249	+	0.966345i	$0.582816\pi$
$854$	25.6616	0.878121
$855$	0.697238	0.0238450
$856$	−37.9584	−1.29739
$857$	57.6049	1.96775	0.983873	−	0.178869i	$-0.0572438\pi$
$857$	57.6049	1.96775	0.983873	+	0.178869i	$0.0572438\pi$
$858$	−22.0457	−0.752628
$859$	−30.2630	−1.03256	−0.516280	−	0.856420i	$-0.672684\pi$
$859$	−30.2630	−1.03256	−0.516280	+	0.856420i	$0.672684\pi$
$860$	−0.300653	−0.0102522
$861$	7.65246	0.260795
$862$	−20.0189	−0.681846
$863$	25.0262	0.851902	0.425951	−	0.904746i	$-0.359940\pi$
$863$	25.0262	0.851902	0.425951	+	0.904746i	$0.359940\pi$
$864$	25.1906	0.857002
$865$	3.89308	0.132369
$866$	−75.9635	−2.58134
$867$	−5.30775	−0.180261
$868$	30.9191	1.04946
$869$	−3.20262	−0.108642
$870$	−2.60575	−0.0883432
$871$	10.6742	0.361680
$872$	35.5041	1.20232
$873$	0.0656000	0.00222022
$874$	22.9892	0.777622
$875$	4.70016	0.158894
$876$	−31.0144	−1.04788
$877$	50.9856	1.72166	0.860831	−	0.508891i	$-0.169944\pi$
$877$	50.9856	1.72166	0.860831	+	0.508891i	$0.169944\pi$
$878$	73.4711	2.47953
$879$	4.74922	0.160187
$880$	−0.160612	−0.00541423
$881$	30.4100	1.02454	0.512270	−	0.858824i	$-0.328805\pi$
$881$	30.4100	1.02454	0.512270	+	0.858824i	$0.328805\pi$
$882$	−20.5041	−0.690409
$883$	35.2748	1.18709	0.593546	−	0.804800i	$-0.297727\pi$
$883$	35.2748	1.18709	0.593546	+	0.804800i	$0.297727\pi$
$884$	−95.4687	−3.21096
$885$	−2.69696	−0.0906574
$886$	−24.8104	−0.833521
$887$	−20.2765	−0.680820	−0.340410	−	0.940277i	$-0.610566\pi$
$887$	−20.2765	−0.680820	−0.340410	+	0.940277i	$0.610566\pi$
$888$	28.9232	0.970600
$889$	−12.0415	−0.403858
$890$	6.67286	0.223675
$891$	3.25039	0.108892
$892$	88.8220	2.97398
$893$	−12.1619	−0.406984
$894$	13.5400	0.452844
$895$	−3.74573	−0.125206
$896$	31.8590	1.06433
$897$	−48.5645	−1.62152
$898$	−10.2139	−0.340842
$899$	23.8889	0.796741
$900$	34.6753	1.15584
$901$	−36.7532	−1.22443
$902$	19.0466	0.634183
$903$	−0.504660	−0.0167940
$904$	−3.01880	−0.100404
$905$	6.18415	0.205568
$906$	−45.1506	−1.50003
$907$	23.6416	0.785007	0.392503	−	0.919751i	$-0.371609\pi$
$907$	23.6416	0.785007	0.392503	+	0.919751i	$0.371609\pi$
$908$	15.1767	0.503655
$909$	−18.3012	−0.607012
$910$	−6.61099	−0.219152
$911$	−16.6181	−0.550581	−0.275290	−	0.961361i	$-0.588774\pi$
$911$	−16.6181	−0.550581	−0.275290	+	0.961361i	$0.588774\pi$
$912$	−0.364349	−0.0120648
$913$	23.7021	0.784424
$914$	85.5262	2.82895
$915$	−1.74567	−0.0577102
$916$	−68.3113	−2.25707
$917$	−27.4666	−0.907026
$918$	−52.5383	−1.73402
$919$	−13.1595	−0.434092	−0.217046	−	0.976161i	$-0.569642\pi$
$919$	−13.1595	−0.434092	−0.217046	+	0.976161i	$0.569642\pi$
$920$	−7.31859	−0.241287
$921$	13.4654	0.443701
$922$	−19.2120	−0.632713
$923$	6.05529	0.199312
$924$	−8.78313	−0.288944
$925$	−50.5145	−1.66091
$926$	16.5025	0.542307
$927$	10.2856	0.337823
$928$	22.6039	0.742010
$929$	16.3583	0.536698	0.268349	−	0.963322i	$-0.413522\pi$
$929$	16.3583	0.536698	0.268349	+	0.963322i	$0.413522\pi$
$930$	−3.37420	−0.110644
$931$	−4.85173	−0.159009
$932$	60.5223	1.98247
$933$	−32.3336	−1.05855
$934$	−20.0992	−0.657665
$935$	−2.27479	−0.0743935
$936$	−38.9184	−1.27209
$937$	51.3273	1.67679	0.838394	−	0.545064i	$-0.183495\pi$
$937$	51.3273	1.67679	0.838394	+	0.545064i	$0.183495\pi$
$938$	6.82219	0.222752
$939$	31.9838	1.04375
$940$	9.78258	0.319073
$941$	2.47871	0.0808038	0.0404019	−	0.999184i	$-0.487136\pi$
$941$	2.47871	0.0808038	0.0404019	+	0.999184i	$0.487136\pi$
$942$	−8.66112	−0.282195
$943$	41.9578	1.36633
$944$	−3.44509	−0.112128
$945$	−2.26786	−0.0737736
$946$	−1.25608	−0.0408386
$947$	−20.4056	−0.663094	−0.331547	−	0.943439i	$-0.607571\pi$
$947$	−20.4056	−0.663094	−0.331547	+	0.943439i	$0.607571\pi$
$948$	5.84380	0.189798
$949$	−60.7949	−1.97349
$950$	13.1626	0.427050
$951$	26.1833	0.849051
$952$	−24.1492	−0.782681
$953$	−18.0509	−0.584727	−0.292364	−	0.956307i	$-0.594442\pi$
$953$	−18.0509	−0.584727	−0.292364	+	0.956307i	$0.594442\pi$
$954$	−37.8561	−1.22564
$955$	−1.21736	−0.0393927
$956$	−16.1784	−0.523246
$957$	−6.78608	−0.219363
$958$	24.6977	0.797946
$959$	37.8578	1.22249
$960$	−3.36978	−0.108759
$961$	−0.0660929	−0.00213203
$962$	143.251	4.61860
$963$	−26.7706	−0.862671
$964$	−61.4467	−1.97906
$965$	−1.37069	−0.0441242
$966$	−31.0391	−0.998666
$967$	−43.2484	−1.39078	−0.695388	−	0.718635i	$-0.744767\pi$
$967$	−43.2484	−1.39078	−0.695388	+	0.718635i	$0.744767\pi$
$968$	24.5548	0.789221
$969$	−5.16036	−0.165775
$970$	−0.0200297	−0.000643115 0
$971$	6.90014	0.221436	0.110718	−	0.993852i	$-0.464685\pi$
$971$	6.90014	0.221436	0.110718	+	0.993852i	$0.464685\pi$
$972$	−53.4633	−1.71484
$973$	29.8259	0.956175
$974$	−81.0238	−2.59617
$975$	−27.8058	−0.890498
$976$	−2.22991	−0.0713778
$977$	32.1927	1.02994	0.514968	−	0.857210i	$-0.327804\pi$
$977$	32.1927	1.02994	0.514968	+	0.857210i	$0.327804\pi$
$978$	15.7357	0.503172
$979$	17.3779	0.555401
$980$	3.90254	0.124662
$981$	25.0397	0.799456
$982$	42.3326	1.35089
$983$	26.4189	0.842633	0.421316	−	0.906914i	$-0.361568\pi$
$983$	26.4189	0.842633	0.421316	+	0.906914i	$0.361568\pi$
$984$	−13.7550	−0.438493
$985$	7.67652	0.244594
$986$	−47.1434	−1.50135
$987$	16.4205	0.522671
$988$	−23.2680	−0.740253
$989$	−2.76701	−0.0879858
$990$	−2.34305	−0.0744671
$991$	9.93574	0.315619	0.157810	−	0.987470i	$-0.449557\pi$
$991$	9.93574	0.315619	0.157810	+	0.987470i	$0.449557\pi$
$992$	29.2699	0.929321
$993$	−10.8572	−0.344543
$994$	3.87013	0.122753
$995$	−6.58060	−0.208619
$996$	−43.2490	−1.37040
$997$	42.8627	1.35747	0.678737	−	0.734381i	$-0.262528\pi$
$997$	42.8627	1.35747	0.678737	+	0.734381i	$0.262528\pi$
$998$	−34.1791	−1.08192
$999$	49.1415	1.55477

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8023.2.a.e.1.20	✓	172

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8023.2.a.e.1.20	✓	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:	\( N \)	\(=\)	\( 8023 = 71 \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8023.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30435 q^{2} -0.933250 q^{3} +3.31004 q^{4} -0.282102 q^{5} +2.15054 q^{6} +1.67948 q^{7} -3.01880 q^{8} -2.12904 q^{9} +O(q^{10})\)	\(q-2.30435 q^{2} -0.933250 q^{3} +3.31004 q^{4} -0.282102 q^{5} +2.15054 q^{6} +1.67948 q^{7} -3.01880 q^{8} -2.12904 q^{9} +0.650063 q^{10} +1.69294 q^{11} -3.08910 q^{12} -6.05529 q^{13} -3.87013 q^{14} +0.263272 q^{15} +0.336302 q^{16} +4.76313 q^{17} +4.90607 q^{18} +1.16089 q^{19} -0.933771 q^{20} -1.56738 q^{21} -3.90113 q^{22} -8.59381 q^{23} +2.81730 q^{24} -4.92042 q^{25} +13.9535 q^{26} +4.78668 q^{27} +5.55917 q^{28} +4.29516 q^{29} -0.606671 q^{30} +5.56183 q^{31} +5.26265 q^{32} -1.57994 q^{33} -10.9759 q^{34} -0.473786 q^{35} -7.04723 q^{36} +10.2663 q^{37} -2.67509 q^{38} +5.65110 q^{39} +0.851611 q^{40} -4.88233 q^{41} +3.61179 q^{42} +0.321977 q^{43} +5.60371 q^{44} +0.600608 q^{45} +19.8032 q^{46} -10.4764 q^{47} -0.313854 q^{48} -4.17933 q^{49} +11.3384 q^{50} -4.44519 q^{51} -20.0433 q^{52} -7.71618 q^{53} -11.0302 q^{54} -0.477583 q^{55} -5.07003 q^{56} -1.08340 q^{57} -9.89757 q^{58} -10.2440 q^{59} +0.871441 q^{60} -6.63068 q^{61} -12.8164 q^{62} -3.57570 q^{63} -12.7996 q^{64} +1.70821 q^{65} +3.64073 q^{66} -1.76278 q^{67} +15.7662 q^{68} +8.02017 q^{69} +1.09177 q^{70} -1.00000 q^{71} +6.42717 q^{72} +10.0400 q^{73} -23.6572 q^{74} +4.59198 q^{75} +3.84258 q^{76} +2.84327 q^{77} -13.0221 q^{78} -1.89175 q^{79} -0.0948716 q^{80} +1.91997 q^{81} +11.2506 q^{82} +14.0005 q^{83} -5.18809 q^{84} -1.34369 q^{85} -0.741949 q^{86} -4.00846 q^{87} -5.11066 q^{88} +10.2649 q^{89} -1.38401 q^{90} -10.1698 q^{91} -28.4459 q^{92} -5.19057 q^{93} +24.1414 q^{94} -0.327489 q^{95} -4.91136 q^{96} -0.0308119 q^{97} +9.63066 q^{98} -3.60435 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 172 q + 24 q^{2} + 18 q^{3} + 180 q^{4} + 28 q^{5} + 16 q^{6} + 4 q^{7} + 72 q^{8} + 198 q^{9}+O(q^{10}) \) 172 * q + 24 * q^2 + 18 * q^3 + 180 * q^4 + 28 * q^5 + 16 * q^6 + 4 * q^7 + 72 * q^8 + 198 * q^9	\( 172 q + 24 q^{2} + 18 q^{3} + 180 q^{4} + 28 q^{5} + 16 q^{6} + 4 q^{7} + 72 q^{8} + 198 q^{9} + 14 q^{10} + 20 q^{11} + 54 q^{12} + 36 q^{13} + 26 q^{14} + 32 q^{15} + 196 q^{16} + 123 q^{17} + 74 q^{18} + 20 q^{19} + 70 q^{20} + 37 q^{21} + 11 q^{22} + 22 q^{23} + 62 q^{24} + 210 q^{25} + 50 q^{26} + 69 q^{27} + 42 q^{28} + 58 q^{29} + 36 q^{30} + 10 q^{31} + 168 q^{32} + 124 q^{33} + 5 q^{34} + 59 q^{35} + 192 q^{36} + 40 q^{37} + 58 q^{38} + 15 q^{39} + 7 q^{40} + 155 q^{41} - 6 q^{42} + 19 q^{43} + 22 q^{44} + 76 q^{45} + q^{46} + 71 q^{47} + 144 q^{48} + 206 q^{49} + 126 q^{50} + 33 q^{51} + 71 q^{52} + 101 q^{53} + 92 q^{54} - 2 q^{55} + 57 q^{56} + 114 q^{57} + 4 q^{58} + 71 q^{59} + 38 q^{60} + 50 q^{61} + 86 q^{62} + 14 q^{63} + 240 q^{64} + 143 q^{65} + 21 q^{66} + 8 q^{67} + 192 q^{68} + 41 q^{69} - 12 q^{70} - 172 q^{71} + 156 q^{72} + 128 q^{73} + 30 q^{74} + 72 q^{75} + 74 q^{76} + 127 q^{77} + 107 q^{78} + 2 q^{79} + 50 q^{80} + 236 q^{81} + 42 q^{82} + 140 q^{83} + 71 q^{84} + 55 q^{85} + 46 q^{86} + 100 q^{87} - 31 q^{88} + 215 q^{89} - 7 q^{90} + 22 q^{91} - 15 q^{92} + 60 q^{93} + 5 q^{94} + 74 q^{95} + 182 q^{96} + 120 q^{97} + 164 q^{98} + 60 q^{99}+O(q^{100}) \) 172 * q + 24 * q^2 + 18 * q^3 + 180 * q^4 + 28 * q^5 + 16 * q^6 + 4 * q^7 + 72 * q^8 + 198 * q^9 + 14 * q^10 + 20 * q^11 + 54 * q^12 + 36 * q^13 + 26 * q^14 + 32 * q^15 + 196 * q^16 + 123 * q^17 + 74 * q^18 + 20 * q^19 + 70 * q^20 + 37 * q^21 + 11 * q^22 + 22 * q^23 + 62 * q^24 + 210 * q^25 + 50 * q^26 + 69 * q^27 + 42 * q^28 + 58 * q^29 + 36 * q^30 + 10 * q^31 + 168 * q^32 + 124 * q^33 + 5 * q^34 + 59 * q^35 + 192 * q^36 + 40 * q^37 + 58 * q^38 + 15 * q^39 + 7 * q^40 + 155 * q^41 - 6 * q^42 + 19 * q^43 + 22 * q^44 + 76 * q^45 + q^46 + 71 * q^47 + 144 * q^48 + 206 * q^49 + 126 * q^50 + 33 * q^51 + 71 * q^52 + 101 * q^53 + 92 * q^54 - 2 * q^55 + 57 * q^56 + 114 * q^57 + 4 * q^58 + 71 * q^59 + 38 * q^60 + 50 * q^61 + 86 * q^62 + 14 * q^63 + 240 * q^64 + 143 * q^65 + 21 * q^66 + 8 * q^67 + 192 * q^68 + 41 * q^69 - 12 * q^70 - 172 * q^71 + 156 * q^72 + 128 * q^73 + 30 * q^74 + 72 * q^75 + 74 * q^76 + 127 * q^77 + 107 * q^78 + 2 * q^79 + 50 * q^80 + 236 * q^81 + 42 * q^82 + 140 * q^83 + 71 * q^84 + 55 * q^85 + 46 * q^86 + 100 * q^87 - 31 * q^88 + 215 * q^89 - 7 * q^90 + 22 * q^91 - 15 * q^92 + 60 * q^93 + 5 * q^94 + 74 * q^95 + 182 * q^96 + 120 * q^97 + 164 * q^98 + 60 * q^99

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