LMFDB - Embedded newform 8043.2.a.n.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$1$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.36925 q^{2} +1.00000 q^{3} -0.125168 q^{4} +0.412992 q^{5} -1.36925 q^{6} +1.00000 q^{7} +2.90988 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.36925 q^{2} +1.00000 q^{3} -0.125168 q^{4} +0.412992 q^{5} -1.36925 q^{6} +1.00000 q^{7} +2.90988 q^{8} +1.00000 q^{9} -0.565488 q^{10} +0.145400 q^{11} -0.125168 q^{12} -2.28507 q^{13} -1.36925 q^{14} +0.412992 q^{15} -3.73400 q^{16} -4.89406 q^{17} -1.36925 q^{18} -1.37831 q^{19} -0.0516933 q^{20} +1.00000 q^{21} -0.199088 q^{22} +4.31146 q^{23} +2.90988 q^{24} -4.82944 q^{25} +3.12883 q^{26} +1.00000 q^{27} -0.125168 q^{28} +5.58084 q^{29} -0.565488 q^{30} -3.31254 q^{31} -0.706993 q^{32} +0.145400 q^{33} +6.70117 q^{34} +0.412992 q^{35} -0.125168 q^{36} +0.515734 q^{37} +1.88725 q^{38} -2.28507 q^{39} +1.20176 q^{40} +9.41395 q^{41} -1.36925 q^{42} +1.65213 q^{43} -0.0181994 q^{44} +0.412992 q^{45} -5.90344 q^{46} +2.22219 q^{47} -3.73400 q^{48} +1.00000 q^{49} +6.61268 q^{50} -4.89406 q^{51} +0.286017 q^{52} -9.08135 q^{53} -1.36925 q^{54} +0.0600490 q^{55} +2.90988 q^{56} -1.37831 q^{57} -7.64154 q^{58} -14.1739 q^{59} -0.0516933 q^{60} +9.87878 q^{61} +4.53567 q^{62} +1.00000 q^{63} +8.43604 q^{64} -0.943718 q^{65} -0.199088 q^{66} -1.93721 q^{67} +0.612579 q^{68} +4.31146 q^{69} -0.565488 q^{70} +8.65792 q^{71} +2.90988 q^{72} -11.7271 q^{73} -0.706167 q^{74} -4.82944 q^{75} +0.172520 q^{76} +0.145400 q^{77} +3.12883 q^{78} +0.650114 q^{79} -1.54211 q^{80} +1.00000 q^{81} -12.8900 q^{82} -11.7108 q^{83} -0.125168 q^{84} -2.02121 q^{85} -2.26217 q^{86} +5.58084 q^{87} +0.423096 q^{88} +9.73143 q^{89} -0.565488 q^{90} -2.28507 q^{91} -0.539655 q^{92} -3.31254 q^{93} -3.04272 q^{94} -0.569233 q^{95} -0.706993 q^{96} -11.9101 q^{97} -1.36925 q^{98} +0.145400 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9}+O(q^{10}) $ 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9	\( 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9} - 14 q^{10} - 29 q^{11} + 33 q^{12} - 40 q^{13} - 9 q^{14} - 27 q^{15} + 23 q^{16} - 46 q^{17} - 9 q^{18} + 17 q^{19} - 53 q^{20} + 40 q^{21} - 15 q^{22} - 62 q^{23} - 18 q^{24} + 35 q^{25} - 29 q^{26} + 40 q^{27} + 33 q^{28} - 47 q^{29} - 14 q^{30} + q^{31} - 45 q^{32} - 29 q^{33} + 9 q^{34} - 27 q^{35} + 33 q^{36} - 49 q^{37} - 52 q^{38} - 40 q^{39} - 12 q^{40} - 49 q^{41} - 9 q^{42} - 45 q^{43} - 68 q^{44} - 27 q^{45} - 36 q^{46} - 88 q^{47} + 23 q^{48} + 40 q^{49} - 32 q^{50} - 46 q^{51} - 34 q^{52} - 96 q^{53} - 9 q^{54} + 26 q^{55} - 18 q^{56} + 17 q^{57} + 12 q^{58} - 45 q^{59} - 53 q^{60} - 23 q^{61} - 15 q^{62} + 40 q^{63} + 50 q^{64} - 32 q^{65} - 15 q^{66} - 31 q^{67} - 76 q^{68} - 62 q^{69} - 14 q^{70} - 89 q^{71} - 18 q^{72} + 4 q^{73} + 10 q^{74} + 35 q^{75} + 51 q^{76} - 29 q^{77} - 29 q^{78} - 44 q^{79} - 53 q^{80} + 40 q^{81} - 15 q^{82} - 60 q^{83} + 33 q^{84} - 24 q^{85} - 9 q^{86} - 47 q^{87} - 64 q^{88} - 39 q^{89} - 14 q^{90} - 40 q^{91} - 80 q^{92} + q^{93} + 51 q^{94} - 71 q^{95} - 45 q^{96} - 21 q^{97} - 9 q^{98} - 29 q^{99}+O(q^{100}) \) 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9 - 14 * q^10 - 29 * q^11 + 33 * q^12 - 40 * q^13 - 9 * q^14 - 27 * q^15 + 23 * q^16 - 46 * q^17 - 9 * q^18 + 17 * q^19 - 53 * q^20 + 40 * q^21 - 15 * q^22 - 62 * q^23 - 18 * q^24 + 35 * q^25 - 29 * q^26 + 40 * q^27 + 33 * q^28 - 47 * q^29 - 14 * q^30 + q^31 - 45 * q^32 - 29 * q^33 + 9 * q^34 - 27 * q^35 + 33 * q^36 - 49 * q^37 - 52 * q^38 - 40 * q^39 - 12 * q^40 - 49 * q^41 - 9 * q^42 - 45 * q^43 - 68 * q^44 - 27 * q^45 - 36 * q^46 - 88 * q^47 + 23 * q^48 + 40 * q^49 - 32 * q^50 - 46 * q^51 - 34 * q^52 - 96 * q^53 - 9 * q^54 + 26 * q^55 - 18 * q^56 + 17 * q^57 + 12 * q^58 - 45 * q^59 - 53 * q^60 - 23 * q^61 - 15 * q^62 + 40 * q^63 + 50 * q^64 - 32 * q^65 - 15 * q^66 - 31 * q^67 - 76 * q^68 - 62 * q^69 - 14 * q^70 - 89 * q^71 - 18 * q^72 + 4 * q^73 + 10 * q^74 + 35 * q^75 + 51 * q^76 - 29 * q^77 - 29 * q^78 - 44 * q^79 - 53 * q^80 + 40 * q^81 - 15 * q^82 - 60 * q^83 + 33 * q^84 - 24 * q^85 - 9 * q^86 - 47 * q^87 - 64 * q^88 - 39 * q^89 - 14 * q^90 - 40 * q^91 - 80 * q^92 + q^93 + 51 * q^94 - 71 * q^95 - 45 * q^96 - 21 * q^97 - 9 * q^98 - 29 * 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 $

$2$	−1.36925	−0.968203	−0.484101	−	0.875012i	$-0.660853\pi$
$2$	−1.36925	−0.968203	−0.484101	+	0.875012i	$0.660853\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.125168	−0.0625839
$5$	0.412992	0.184696	0.0923479	−	0.995727i	$-0.470563\pi$
$5$	0.412992	0.184696	0.0923479	+	0.995727i	$0.470563\pi$
$6$	−1.36925	−0.558992
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.90988	1.02880
$9$	1.00000	0.333333
$10$	−0.565488	−0.178823
$11$	0.145400	0.0438397	0.0219199	−	0.999760i	$-0.493022\pi$
$11$	0.145400	0.0438397	0.0219199	+	0.999760i	$0.493022\pi$
$12$	−0.125168	−0.0361328
$13$	−2.28507	−0.633765	−0.316883	−	0.948465i	$-0.602636\pi$
$13$	−2.28507	−0.633765	−0.316883	+	0.948465i	$0.602636\pi$
$14$	−1.36925	−0.365946
$15$	0.412992	0.106634
$16$	−3.73400	−0.933499
$17$	−4.89406	−1.18698	−0.593492	−	0.804840i	$-0.702251\pi$
$17$	−4.89406	−1.18698	−0.593492	+	0.804840i	$0.702251\pi$
$18$	−1.36925	−0.322734
$19$	−1.37831	−0.316207	−0.158103	−	0.987423i	$-0.550538\pi$
$19$	−1.37831	−0.316207	−0.158103	+	0.987423i	$0.550538\pi$
$20$	−0.0516933	−0.0115590
$21$	1.00000	0.218218
$22$	−0.199088	−0.0424457
$23$	4.31146	0.899001	0.449500	−	0.893280i	$-0.351602\pi$
$23$	4.31146	0.899001	0.449500	+	0.893280i	$0.351602\pi$
$24$	2.90988	0.593976
$25$	−4.82944	−0.965887
$26$	3.12883	0.613613
$27$	1.00000	0.192450
$28$	−0.125168	−0.0236545
$29$	5.58084	1.03634	0.518168	−	0.855279i	$-0.326614\pi$
$29$	5.58084	1.03634	0.518168	+	0.855279i	$0.326614\pi$
$30$	−0.565488	−0.103243
$31$	−3.31254	−0.594949	−0.297475	−	0.954730i	$-0.596144\pi$
$31$	−3.31254	−0.594949	−0.297475	+	0.954730i	$0.596144\pi$
$32$	−0.706993	−0.124980
$33$	0.145400	0.0253109
$34$	6.70117	1.14924
$35$	0.412992	0.0698084
$36$	−0.125168	−0.0208613
$37$	0.515734	0.0847862	0.0423931	−	0.999101i	$-0.486502\pi$
$37$	0.515734	0.0847862	0.0423931	+	0.999101i	$0.486502\pi$
$38$	1.88725	0.306152
$39$	−2.28507	−0.365905
$40$	1.20176	0.190014
$41$	9.41395	1.47021	0.735106	−	0.677952i	$-0.237132\pi$
$41$	9.41395	1.47021	0.735106	+	0.677952i	$0.237132\pi$
$42$	−1.36925	−0.211279
$43$	1.65213	0.251947	0.125974	−	0.992034i	$-0.459795\pi$
$43$	1.65213	0.251947	0.125974	+	0.992034i	$0.459795\pi$
$44$	−0.0181994	−0.00274366
$45$	0.412992	0.0615653
$46$	−5.90344	−0.870415
$47$	2.22219	0.324140	0.162070	−	0.986779i	$-0.448183\pi$
$47$	2.22219	0.324140	0.162070	+	0.986779i	$0.448183\pi$
$48$	−3.73400	−0.538956
$49$	1.00000	0.142857
$50$	6.61268	0.935175
$51$	−4.89406	−0.685306
$52$	0.286017	0.0396635
$53$	−9.08135	−1.24742	−0.623710	−	0.781656i	$-0.714375\pi$
$53$	−9.08135	−1.24742	−0.623710	+	0.781656i	$0.714375\pi$
$54$	−1.36925	−0.186331
$55$	0.0600490	0.00809701
$56$	2.90988	0.388848
$57$	−1.37831	−0.182562
$58$	−7.64154	−1.00338
$59$	−14.1739	−1.84528	−0.922640	−	0.385661i	$-0.873973\pi$
$59$	−14.1739	−1.84528	−0.922640	+	0.385661i	$0.873973\pi$
$60$	−0.0516933	−0.00667358
$61$	9.87878	1.26485	0.632424	−	0.774622i	$-0.282060\pi$
$61$	9.87878	1.26485	0.632424	+	0.774622i	$0.282060\pi$
$62$	4.53567	0.576031
$63$	1.00000	0.125988
$64$	8.43604	1.05451
$65$	−0.943718	−0.117054
$66$	−0.199088	−0.0245061
$67$	−1.93721	−0.236667	−0.118334	−	0.992974i	$-0.537755\pi$
$67$	−1.93721	−0.236667	−0.118334	+	0.992974i	$0.537755\pi$
$68$	0.612579	0.0742861
$69$	4.31146	0.519038
$70$	−0.565488	−0.0675887
$71$	8.65792	1.02751	0.513753	−	0.857938i	$-0.328255\pi$
$71$	8.65792	1.02751	0.513753	+	0.857938i	$0.328255\pi$
$72$	2.90988	0.342932
$73$	−11.7271	−1.37255	−0.686277	−	0.727341i	$-0.740756\pi$
$73$	−11.7271	−1.37255	−0.686277	+	0.727341i	$0.740756\pi$
$74$	−0.706167	−0.0820902
$75$	−4.82944	−0.557655
$76$	0.172520	0.0197894
$77$	0.145400	0.0165699
$78$	3.12883	0.354270
$79$	0.650114	0.0731435	0.0365717	−	0.999331i	$-0.488356\pi$
$79$	0.650114	0.0731435	0.0365717	+	0.999331i	$0.488356\pi$
$80$	−1.54211	−0.172413
$81$	1.00000	0.111111
$82$	−12.8900	−1.42346
$83$	−11.7108	−1.28543	−0.642715	−	0.766106i	$-0.722192\pi$
$83$	−11.7108	−1.28543	−0.642715	+	0.766106i	$0.722192\pi$
$84$	−0.125168	−0.0136569
$85$	−2.02121	−0.219231
$86$	−2.26217	−0.243936
$87$	5.58084	0.598329
$88$	0.423096	0.0451021
$89$	9.73143	1.03153	0.515765	−	0.856730i	$-0.327508\pi$
$89$	9.73143	1.03153	0.515765	+	0.856730i	$0.327508\pi$
$90$	−0.565488	−0.0596076
$91$	−2.28507	−0.239541
$92$	−0.539655	−0.0562629
$93$	−3.31254	−0.343494
$94$	−3.04272	−0.313833
$95$	−0.569233	−0.0584020
$96$	−0.706993	−0.0721572
$97$	−11.9101	−1.20929	−0.604644	−	0.796496i	$-0.706685\pi$
$97$	−11.9101	−1.20929	−0.604644	+	0.796496i	$0.706685\pi$
$98$	−1.36925	−0.138315
$99$	0.145400	0.0146132
$100$	0.604490	0.0604490
$101$	−0.822242	−0.0818162	−0.0409081	−	0.999163i	$-0.513025\pi$
$101$	−0.822242	−0.0818162	−0.0409081	+	0.999163i	$0.513025\pi$
$102$	6.70117	0.663515
$103$	0.468884	0.0462005	0.0231002	−	0.999733i	$-0.492646\pi$
$103$	0.468884	0.0462005	0.0231002	+	0.999733i	$0.492646\pi$
$104$	−6.64928	−0.652015
$105$	0.412992	0.0403039
$106$	12.4346	1.20775
$107$	−6.50083	−0.628459	−0.314229	−	0.949347i	$-0.601746\pi$
$107$	−6.50083	−0.628459	−0.314229	+	0.949347i	$0.601746\pi$
$108$	−0.125168	−0.0120443
$109$	11.5333	1.10469	0.552346	−	0.833615i	$-0.313733\pi$
$109$	11.5333	1.10469	0.552346	+	0.833615i	$0.313733\pi$
$110$	−0.0822219	−0.00783955
$111$	0.515734	0.0489513
$112$	−3.73400	−0.352830
$113$	−16.3922	−1.54205	−0.771023	−	0.636808i	$-0.780254\pi$
$113$	−16.3922	−1.54205	−0.771023	+	0.636808i	$0.780254\pi$
$114$	1.88725	0.176757
$115$	1.78060	0.166042
$116$	−0.698541	−0.0648579
$117$	−2.28507	−0.211255
$118$	19.4075	1.78661
$119$	−4.89406	−0.448638
$120$	1.20176	0.109705
$121$	−10.9789	−0.998078
$122$	−13.5265	−1.22463
$123$	9.41395	0.848827
$124$	0.414623	0.0372342
$125$	−4.05948	−0.363091
$126$	−1.36925	−0.121982
$127$	−11.1021	−0.985152	−0.492576	−	0.870269i	$-0.663945\pi$
$127$	−11.1021	−0.985152	−0.492576	+	0.870269i	$0.663945\pi$
$128$	−10.1370	−0.895995
$129$	1.65213	0.145462
$130$	1.29218	0.113332
$131$	−2.85694	−0.249612	−0.124806	−	0.992181i	$-0.539831\pi$
$131$	−2.85694	−0.249612	−0.124806	+	0.992181i	$0.539831\pi$
$132$	−0.0181994	−0.00158405
$133$	−1.37831	−0.119515
$134$	2.65251	0.229142
$135$	0.412992	0.0355447
$136$	−14.2411	−1.22117
$137$	−20.1444	−1.72106	−0.860528	−	0.509403i	$-0.829866\pi$
$137$	−20.1444	−1.72106	−0.860528	+	0.509403i	$0.829866\pi$
$138$	−5.90344	−0.502534
$139$	−7.99544	−0.678164	−0.339082	−	0.940757i	$-0.610116\pi$
$139$	−7.99544	−0.678164	−0.339082	+	0.940757i	$0.610116\pi$
$140$	−0.0516933	−0.00436888
$141$	2.22219	0.187142
$142$	−11.8548	−0.994834
$143$	−0.332249	−0.0277841
$144$	−3.73400	−0.311166
$145$	2.30484	0.191407
$146$	16.0573	1.32891
$147$	1.00000	0.0824786
$148$	−0.0645533	−0.00530625
$149$	−7.66228	−0.627718	−0.313859	−	0.949470i	$-0.601622\pi$
$149$	−7.66228	−0.627718	−0.313859	+	0.949470i	$0.601622\pi$
$150$	6.61268	0.539923
$151$	2.98212	0.242681	0.121341	−	0.992611i	$-0.461281\pi$
$151$	2.98212	0.242681	0.121341	+	0.992611i	$0.461281\pi$
$152$	−4.01072	−0.325312
$153$	−4.89406	−0.395661
$154$	−0.199088	−0.0160430
$155$	−1.36805	−0.109885
$156$	0.286017	0.0228997
$157$	20.4457	1.63174	0.815871	−	0.578233i	$-0.196258\pi$
$157$	20.4457	1.63174	0.815871	+	0.578233i	$0.196258\pi$
$158$	−0.890165	−0.0708177
$159$	−9.08135	−0.720198
$160$	−0.291983	−0.0230833
$161$	4.31146	0.339790
$162$	−1.36925	−0.107578
$163$	23.4289	1.83509	0.917545	−	0.397631i	$-0.130168\pi$
$163$	23.4289	1.83509	0.917545	+	0.397631i	$0.130168\pi$
$164$	−1.17832	−0.0920115
$165$	0.0600490	0.00467481
$166$	16.0350	1.24456
$167$	14.0112	1.08422	0.542110	−	0.840308i	$-0.317626\pi$
$167$	14.0112	1.08422	0.542110	+	0.840308i	$0.317626\pi$
$168$	2.90988	0.224502
$169$	−7.77844	−0.598342
$170$	2.76753	0.212260
$171$	−1.37831	−0.105402
$172$	−0.206793	−0.0157678
$173$	−3.62809	−0.275838	−0.137919	−	0.990443i	$-0.544041\pi$
$173$	−3.62809	−0.275838	−0.137919	+	0.990443i	$0.544041\pi$
$174$	−7.64154	−0.579303
$175$	−4.82944	−0.365071
$176$	−0.542923	−0.0409244
$177$	−14.1739	−1.06537
$178$	−13.3247	−0.998729
$179$	6.03010	0.450711	0.225356	−	0.974277i	$-0.427646\pi$
$179$	6.03010	0.450711	0.225356	+	0.974277i	$0.427646\pi$
$180$	−0.0516933	−0.00385299
$181$	−13.5026	−1.00364	−0.501822	−	0.864971i	$-0.667337\pi$
$181$	−13.5026	−1.00364	−0.501822	+	0.864971i	$0.667337\pi$
$182$	3.12883	0.231924
$183$	9.87878	0.730261
$184$	12.5458	0.924889
$185$	0.212994	0.0156597
$186$	4.53567	0.332572
$187$	−0.711596	−0.0520371
$188$	−0.278147	−0.0202859
$189$	1.00000	0.0727393
$190$	0.779419	0.0565450
$191$	3.61688	0.261708	0.130854	−	0.991402i	$-0.458228\pi$
$191$	3.61688	0.261708	0.130854	+	0.991402i	$0.458228\pi$
$192$	8.43604	0.608819
$193$	−9.14902	−0.658561	−0.329280	−	0.944232i	$-0.606806\pi$
$193$	−9.14902	−0.658561	−0.329280	+	0.944232i	$0.606806\pi$
$194$	16.3079	1.17084
$195$	−0.943718	−0.0675810
$196$	−0.125168	−0.00894055
$197$	13.5427	0.964874	0.482437	−	0.875931i	$-0.339752\pi$
$197$	13.5427	0.964874	0.482437	+	0.875931i	$0.339752\pi$
$198$	−0.199088	−0.0141486
$199$	8.53934	0.605338	0.302669	−	0.953096i	$-0.402122\pi$
$199$	8.53934	0.605338	0.302669	+	0.953096i	$0.402122\pi$
$200$	−14.0531	−0.993702
$201$	−1.93721	−0.136640
$202$	1.12585	0.0792146
$203$	5.58084	0.391698
$204$	0.612579	0.0428891
$205$	3.88789	0.271542
$206$	−0.642017	−0.0447314
$207$	4.31146	0.299667
$208$	8.53246	0.591619
$209$	−0.200407	−0.0138624
$210$	−0.565488	−0.0390224
$211$	−8.86701	−0.610430	−0.305215	−	0.952283i	$-0.598728\pi$
$211$	−8.86701	−0.610430	−0.305215	+	0.952283i	$0.598728\pi$
$212$	1.13669	0.0780683
$213$	8.65792	0.593231
$214$	8.90122	0.608475
$215$	0.682316	0.0465336
$216$	2.90988	0.197992
$217$	−3.31254	−0.224870
$218$	−15.7919	−1.06957
$219$	−11.7271	−0.792444
$220$	−0.00751620	−0.000506742 0
$221$	11.1833	0.752269
$222$	−0.706167	−0.0473948
$223$	−18.8464	−1.26205	−0.631025	−	0.775762i	$-0.717365\pi$
$223$	−18.8464	−1.26205	−0.631025	+	0.775762i	$0.717365\pi$
$224$	−0.706993	−0.0472380
$225$	−4.82944	−0.321962
$226$	22.4449	1.49301
$227$	−6.61140	−0.438814	−0.219407	−	0.975633i	$-0.570412\pi$
$227$	−6.61140	−0.438814	−0.219407	+	0.975633i	$0.570412\pi$
$228$	0.172520	0.0114254
$229$	22.4426	1.48305	0.741523	−	0.670928i	$-0.234104\pi$
$229$	22.4426	1.48305	0.741523	+	0.670928i	$0.234104\pi$
$230$	−2.43808	−0.160762
$231$	0.145400	0.00956661
$232$	16.2395	1.06618
$233$	−20.6774	−1.35462	−0.677311	−	0.735697i	$-0.736855\pi$
$233$	−20.6774	−1.35462	−0.677311	+	0.735697i	$0.736855\pi$
$234$	3.12883	0.204538
$235$	0.917748	0.0598673
$236$	1.77411	0.115485
$237$	0.650114	0.0422294
$238$	6.70117	0.434372
$239$	−15.9820	−1.03379	−0.516895	−	0.856049i	$-0.672912\pi$
$239$	−15.9820	−1.03379	−0.516895	+	0.856049i	$0.672912\pi$
$240$	−1.54211	−0.0995429
$241$	5.92275	0.381518	0.190759	−	0.981637i	$-0.438905\pi$
$241$	5.92275	0.381518	0.190759	+	0.981637i	$0.438905\pi$
$242$	15.0327	0.966342
$243$	1.00000	0.0641500
$244$	−1.23650	−0.0791591
$245$	0.412992	0.0263851
$246$	−12.8900	−0.821837
$247$	3.14954	0.200401
$248$	−9.63907	−0.612082
$249$	−11.7108	−0.742143
$250$	5.55843	0.351546
$251$	−6.51150	−0.411002	−0.205501	−	0.978657i	$-0.565882\pi$
$251$	−6.51150	−0.411002	−0.205501	+	0.978657i	$0.565882\pi$
$252$	−0.125168	−0.00788482
$253$	0.626885	0.0394119
$254$	15.2015	0.953827
$255$	−2.02121	−0.126573
$256$	−2.99201	−0.187001
$257$	18.6449	1.16304	0.581520	−	0.813532i	$-0.302458\pi$
$257$	18.6449	1.16304	0.581520	+	0.813532i	$0.302458\pi$
$258$	−2.26217	−0.140836
$259$	0.515734	0.0320462
$260$	0.118123	0.00732568
$261$	5.58084	0.345445
$262$	3.91186	0.241675
$263$	−6.04012	−0.372450	−0.186225	−	0.982507i	$-0.559625\pi$
$263$	−6.04012	−0.372450	−0.186225	+	0.982507i	$0.559625\pi$
$264$	0.423096	0.0260397
$265$	−3.75053	−0.230393
$266$	1.88725	0.115715
$267$	9.73143	0.595554
$268$	0.242476	0.0148116
$269$	13.2256	0.806379	0.403190	−	0.915116i	$-0.367901\pi$
$269$	13.2256	0.806379	0.403190	+	0.915116i	$0.367901\pi$
$270$	−0.565488	−0.0344145
$271$	15.1057	0.917603	0.458802	−	0.888539i	$-0.348279\pi$
$271$	15.1057	0.917603	0.458802	+	0.888539i	$0.348279\pi$
$272$	18.2744	1.10805
$273$	−2.28507	−0.138299
$274$	27.5827	1.66633
$275$	−0.702200	−0.0423442
$276$	−0.539655	−0.0324834
$277$	19.9944	1.20135	0.600674	−	0.799494i	$-0.294899\pi$
$277$	19.9944	1.20135	0.600674	+	0.799494i	$0.294899\pi$
$278$	10.9477	0.656600
$279$	−3.31254	−0.198316
$280$	1.20176	0.0718187
$281$	−23.4048	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$281$	−23.4048	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$282$	−3.04272	−0.181192
$283$	10.3362	0.614421	0.307210	−	0.951642i	$-0.400604\pi$
$283$	10.3362	0.614421	0.307210	+	0.951642i	$0.400604\pi$
$284$	−1.08369	−0.0643053
$285$	−0.569233	−0.0337184
$286$	0.454931	0.0269006
$287$	9.41395	0.555688
$288$	−0.706993	−0.0416600
$289$	6.95185	0.408932
$290$	−3.15590	−0.185321
$291$	−11.9101	−0.698183
$292$	1.46785	0.0858997
$293$	−15.8653	−0.926859	−0.463430	−	0.886134i	$-0.653381\pi$
$293$	−15.8653	−0.926859	−0.463430	+	0.886134i	$0.653381\pi$
$294$	−1.36925	−0.0798560
$295$	−5.85370	−0.340816
$296$	1.50072	0.0872277
$297$	0.145400	0.00843696
$298$	10.4915	0.607758
$299$	−9.85199	−0.569755
$300$	0.604490	0.0349002
$301$	1.65213	0.0952271
$302$	−4.08325	−0.234965
$303$	−0.822242	−0.0472366
$304$	5.14662	0.295179
$305$	4.07986	0.233612
$306$	6.70117	0.383080
$307$	−8.50378	−0.485336	−0.242668	−	0.970109i	$-0.578023\pi$
$307$	−8.50378	−0.485336	−0.242668	+	0.970109i	$0.578023\pi$
$308$	−0.0181994	−0.00103701
$309$	0.468884	0.0266739
$310$	1.87320	0.106391
$311$	21.8829	1.24086	0.620431	−	0.784261i	$-0.286958\pi$
$311$	21.8829	1.24086	0.620431	+	0.784261i	$0.286958\pi$
$312$	−6.64928	−0.376441
$313$	−3.25130	−0.183775	−0.0918873	−	0.995769i	$-0.529290\pi$
$313$	−3.25130	−0.183775	−0.0918873	+	0.995769i	$0.529290\pi$
$314$	−27.9951	−1.57986
$315$	0.412992	0.0232695
$316$	−0.0813732	−0.00457760
$317$	−26.8352	−1.50721	−0.753607	−	0.657325i	$-0.771688\pi$
$317$	−26.8352	−1.50721	−0.753607	+	0.657325i	$0.771688\pi$
$318$	12.4346	0.697298
$319$	0.811453	0.0454327
$320$	3.48402	0.194763
$321$	−6.50083	−0.362841
$322$	−5.90344	−0.328986
$323$	6.74555	0.375332
$324$	−0.125168	−0.00695376
$325$	11.0356	0.612146
$326$	−32.0799	−1.77674
$327$	11.5333	0.637794
$328$	27.3934	1.51255
$329$	2.22219	0.122513
$330$	−0.0822219	−0.00452616
$331$	−16.3229	−0.897189	−0.448595	−	0.893735i	$-0.648075\pi$
$331$	−16.3229	−0.897189	−0.448595	+	0.893735i	$0.648075\pi$
$332$	1.46582	0.0804471
$333$	0.515734	0.0282621
$334$	−19.1848	−1.04974
$335$	−0.800051	−0.0437115
$336$	−3.73400	−0.203706
$337$	−10.6223	−0.578633	−0.289316	−	0.957234i	$-0.593428\pi$
$337$	−10.6223	−0.578633	−0.289316	+	0.957234i	$0.593428\pi$
$338$	10.6506	0.579316
$339$	−16.3922	−0.890300
$340$	0.252990	0.0137203
$341$	−0.481643	−0.0260824
$342$	1.88725	0.102051
$343$	1.00000	0.0539949
$344$	4.80749	0.259202
$345$	1.78060	0.0958642
$346$	4.96774	0.267067
$347$	−31.7312	−1.70342	−0.851711	−	0.524012i	$-0.824435\pi$
$347$	−31.7312	−1.70342	−0.851711	+	0.524012i	$0.824435\pi$
$348$	−0.698541	−0.0374457
$349$	20.8317	1.11510	0.557548	−	0.830145i	$-0.311742\pi$
$349$	20.8317	1.11510	0.557548	+	0.830145i	$0.311742\pi$
$350$	6.61268	0.353463
$351$	−2.28507	−0.121968
$352$	−0.102797	−0.00547908
$353$	18.8478	1.00316	0.501582	−	0.865110i	$-0.332751\pi$
$353$	18.8478	1.00316	0.501582	+	0.865110i	$0.332751\pi$
$354$	19.4075	1.03150
$355$	3.57566	0.189776
$356$	−1.21806	−0.0645571
$357$	−4.89406	−0.259021
$358$	−8.25669	−0.436380
$359$	−24.4492	−1.29038	−0.645188	−	0.764023i	$-0.723221\pi$
$359$	−24.4492	−1.29038	−0.645188	+	0.764023i	$0.723221\pi$
$360$	1.20176	0.0633381
$361$	−17.1003	−0.900013
$362$	18.4884	0.971730
$363$	−10.9789	−0.576241
$364$	0.286017	0.0149914
$365$	−4.84320	−0.253505
$366$	−13.5265	−0.707040
$367$	−10.8434	−0.566020	−0.283010	−	0.959117i	$-0.591333\pi$
$367$	−10.8434	−0.566020	−0.283010	+	0.959117i	$0.591333\pi$
$368$	−16.0990	−0.839217
$369$	9.41395	0.490071
$370$	−0.291641	−0.0151617
$371$	−9.08135	−0.471480
$372$	0.414623	0.0214972
$373$	−32.1217	−1.66320	−0.831600	−	0.555375i	$-0.812575\pi$
$373$	−32.1217	−1.66320	−0.831600	+	0.555375i	$0.812575\pi$
$374$	0.974350	0.0503824
$375$	−4.05948	−0.209631
$376$	6.46630	0.333474
$377$	−12.7526	−0.656794
$378$	−1.36925	−0.0704264
$379$	−13.4103	−0.688841	−0.344421	−	0.938815i	$-0.611925\pi$
$379$	−13.4103	−0.688841	−0.344421	+	0.938815i	$0.611925\pi$
$380$	0.0712495	0.00365502
$381$	−11.1021	−0.568778
$382$	−4.95240	−0.253387
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	−10.1370	−0.517303
$385$	0.0600490	0.00306038
$386$	12.5272	0.637620
$387$	1.65213	0.0839824
$388$	1.49076	0.0756819
$389$	25.5258	1.29421	0.647105	−	0.762401i	$-0.275980\pi$
$389$	25.5258	1.29421	0.647105	+	0.762401i	$0.275980\pi$
$390$	1.29218	0.0654321
$391$	−21.1005	−1.06710
$392$	2.90988	0.146971
$393$	−2.85694	−0.144114
$394$	−18.5432	−0.934194
$395$	0.268492	0.0135093
$396$	−0.0181994	−0.000914553 0
$397$	6.85885	0.344236	0.172118	−	0.985076i	$-0.444939\pi$
$397$	6.85885	0.344236	0.172118	+	0.985076i	$0.444939\pi$
$398$	−11.6925	−0.586090
$399$	−1.37831	−0.0690019
$400$	18.0331	0.901655
$401$	−13.8024	−0.689260	−0.344630	−	0.938739i	$-0.611996\pi$
$401$	−13.8024	−0.689260	−0.344630	+	0.938739i	$0.611996\pi$
$402$	2.65251	0.132295
$403$	7.56939	0.377058
$404$	0.102918	0.00512037
$405$	0.412992	0.0205218
$406$	−7.64154	−0.379243
$407$	0.0749877	0.00371700
$408$	−14.2411	−0.705040
$409$	27.5600	1.36275	0.681377	−	0.731932i	$-0.261381\pi$
$409$	27.5600	1.36275	0.681377	+	0.731932i	$0.261381\pi$
$410$	−5.32347	−0.262908
$411$	−20.1444	−0.993652
$412$	−0.0586891	−0.00289140
$413$	−14.1739	−0.697451
$414$	−5.90344	−0.290138
$415$	−4.83648	−0.237413
$416$	1.61553	0.0792079
$417$	−7.99544	−0.391538
$418$	0.274406	0.0134216
$419$	−6.85174	−0.334730	−0.167365	−	0.985895i	$-0.553526\pi$
$419$	−6.85174	−0.334730	−0.167365	+	0.985895i	$0.553526\pi$
$420$	−0.0516933	−0.00252237
$421$	−14.3873	−0.701195	−0.350598	−	0.936526i	$-0.614022\pi$
$421$	−14.3873	−0.701195	−0.350598	+	0.936526i	$0.614022\pi$
$422$	12.1411	0.591020
$423$	2.22219	0.108047
$424$	−26.4256	−1.28334
$425$	23.6356	1.14649
$426$	−11.8548	−0.574368
$427$	9.87878	0.478068
$428$	0.813694	0.0393314
$429$	−0.332249	−0.0160412
$430$	−0.934258	−0.0450539
$431$	−2.81868	−0.135771	−0.0678854	−	0.997693i	$-0.521625\pi$
$431$	−2.81868	−0.135771	−0.0678854	+	0.997693i	$0.521625\pi$
$432$	−3.73400	−0.179652
$433$	1.19424	0.0573917	0.0286959	−	0.999588i	$-0.490865\pi$
$433$	1.19424	0.0573917	0.0286959	+	0.999588i	$0.490865\pi$
$434$	4.53567	0.217719
$435$	2.30484	0.110509
$436$	−1.44360	−0.0691359
$437$	−5.94253	−0.284270
$438$	16.0573	0.767246
$439$	−1.12766	−0.0538201	−0.0269100	−	0.999638i	$-0.508567\pi$
$439$	−1.12766	−0.0538201	−0.0269100	+	0.999638i	$0.508567\pi$
$440$	0.174735	0.00833018
$441$	1.00000	0.0476190
$442$	−15.3127	−0.728349
$443$	−6.60499	−0.313812	−0.156906	−	0.987614i	$-0.550152\pi$
$443$	−6.60499	−0.313812	−0.156906	+	0.987614i	$0.550152\pi$
$444$	−0.0645533	−0.00306356
$445$	4.01900	0.190519
$446$	25.8054	1.22192
$447$	−7.66228	−0.362413
$448$	8.43604	0.398566
$449$	40.7154	1.92148	0.960740	−	0.277452i	$-0.0894899\pi$
$449$	40.7154	1.92148	0.960740	+	0.277452i	$0.0894899\pi$
$450$	6.61268	0.311725
$451$	1.36879	0.0644537
$452$	2.05177	0.0965071
$453$	2.98212	0.140112
$454$	9.05263	0.424861
$455$	−0.943718	−0.0442422
$456$	−4.01072	−0.187819
$457$	24.9855	1.16877	0.584387	−	0.811475i	$-0.301335\pi$
$457$	24.9855	1.16877	0.584387	+	0.811475i	$0.301335\pi$
$458$	−30.7294	−1.43589
$459$	−4.89406	−0.228435
$460$	−0.222873	−0.0103915
$461$	−12.7032	−0.591647	−0.295824	−	0.955243i	$-0.595594\pi$
$461$	−12.7032	−0.591647	−0.295824	+	0.955243i	$0.595594\pi$
$462$	−0.199088	−0.00926242
$463$	29.4329	1.36786	0.683932	−	0.729546i	$-0.260269\pi$
$463$	29.4329	1.36786	0.683932	+	0.729546i	$0.260269\pi$
$464$	−20.8388	−0.967419
$465$	−1.36805	−0.0634419
$466$	28.3124	1.31155
$467$	4.74051	0.219365	0.109682	−	0.993967i	$-0.465017\pi$
$467$	4.74051	0.219365	0.109682	+	0.993967i	$0.465017\pi$
$468$	0.286017	0.0132212
$469$	−1.93721	−0.0894518
$470$	−1.25662	−0.0579636
$471$	20.4457	0.942087
$472$	−41.2442	−1.89842
$473$	0.240219	0.0110453
$474$	−0.890165	−0.0408866
$475$	6.65647	0.305420
$476$	0.612579	0.0280775
$477$	−9.08135	−0.415807
$478$	21.8833	1.00092
$479$	−3.73999	−0.170884	−0.0854422	−	0.996343i	$-0.527230\pi$
$479$	−3.73999	−0.170884	−0.0854422	+	0.996343i	$0.527230\pi$
$480$	−0.291983	−0.0133271
$481$	−1.17849	−0.0537345
$482$	−8.10969	−0.369386
$483$	4.31146	0.196178
$484$	1.37420	0.0624636
$485$	−4.91878	−0.223351
$486$	−1.36925	−0.0621102
$487$	−32.4260	−1.46936	−0.734682	−	0.678412i	$-0.762669\pi$
$487$	−32.4260	−1.46936	−0.734682	+	0.678412i	$0.762669\pi$
$488$	28.7460	1.30127
$489$	23.4289	1.05949
$490$	−0.565488	−0.0255461
$491$	−26.9629	−1.21682	−0.608409	−	0.793624i	$-0.708192\pi$
$491$	−26.9629	−1.21682	−0.608409	+	0.793624i	$0.708192\pi$
$492$	−1.17832	−0.0531229
$493$	−27.3130	−1.23011
$494$	−4.31250	−0.194029
$495$	0.0600490	0.00269900
$496$	12.3690	0.555385
$497$	8.65792	0.388361
$498$	16.0350	0.718545
$499$	−31.4530	−1.40803	−0.704014	−	0.710186i	$-0.748611\pi$
$499$	−31.4530	−1.40803	−0.704014	+	0.710186i	$0.748611\pi$
$500$	0.508116	0.0227236
$501$	14.0112	0.625974
$502$	8.91584	0.397933
$503$	−35.4395	−1.58017	−0.790084	−	0.612999i	$-0.789963\pi$
$503$	−35.4395	−1.58017	−0.790084	+	0.612999i	$0.789963\pi$
$504$	2.90988	0.129616
$505$	−0.339580	−0.0151111
$506$	−0.858360	−0.0381587
$507$	−7.77844	−0.345453
$508$	1.38962	0.0616546
$509$	−30.9451	−1.37162	−0.685809	−	0.727782i	$-0.740551\pi$
$509$	−30.9451	−1.37162	−0.685809	+	0.727782i	$0.740551\pi$
$510$	2.76753	0.122548
$511$	−11.7271	−0.518776
$512$	24.3708	1.07705
$513$	−1.37831	−0.0608540
$514$	−25.5295	−1.12606
$515$	0.193645	0.00853303
$516$	−0.206793	−0.00910356
$517$	0.323106	0.0142102
$518$	−0.706167	−0.0310272
$519$	−3.62809	−0.159255
$520$	−2.74610	−0.120424
$521$	−25.7298	−1.12724	−0.563622	−	0.826033i	$-0.690593\pi$
$521$	−25.7298	−1.12724	−0.563622	+	0.826033i	$0.690593\pi$
$522$	−7.64154	−0.334461
$523$	−33.9434	−1.48424	−0.742120	−	0.670267i	$-0.766180\pi$
$523$	−33.9434	−1.48424	−0.742120	+	0.670267i	$0.766180\pi$
$524$	0.357597	0.0156217
$525$	−4.82944	−0.210774
$526$	8.27041	0.360607
$527$	16.2118	0.706195
$528$	−0.542923	−0.0236277
$529$	−4.41135	−0.191798
$530$	5.13539	0.223067
$531$	−14.1739	−0.615094
$532$	0.172520	0.00747970
$533$	−21.5116	−0.931769
$534$	−13.3247	−0.576617
$535$	−2.68479	−0.116074
$536$	−5.63703	−0.243483
$537$	6.03010	0.260218
$538$	−18.1091	−0.780739
$539$	0.145400	0.00626282
$540$	−0.0516933	−0.00222453
$541$	−30.1584	−1.29661	−0.648306	−	0.761380i	$-0.724522\pi$
$541$	−30.1584	−1.29661	−0.648306	+	0.761380i	$0.724522\pi$
$542$	−20.6833	−0.888426
$543$	−13.5026	−0.579454
$544$	3.46007	0.148349
$545$	4.76317	0.204032
$546$	3.12883	0.133901
$547$	29.8793	1.27755	0.638773	−	0.769395i	$-0.279442\pi$
$547$	29.8793	1.27755	0.638773	+	0.769395i	$0.279442\pi$
$548$	2.52143	0.107710
$549$	9.87878	0.421616
$550$	0.961484	0.0409978
$551$	−7.69214	−0.327696
$552$	12.5458	0.533985
$553$	0.650114	0.0276456
$554$	−27.3772	−1.16315
$555$	0.212994	0.00904111
$556$	1.00077	0.0424421
$557$	−46.6023	−1.97460	−0.987301	−	0.158860i	$-0.949218\pi$
$557$	−46.6023	−1.97460	−0.987301	+	0.158860i	$0.949218\pi$
$558$	4.53567	0.192010
$559$	−3.77523	−0.159675
$560$	−1.54211	−0.0651661
$561$	−0.711596	−0.0300436
$562$	32.0469	1.35182
$563$	−40.8118	−1.72001	−0.860006	−	0.510283i	$-0.829541\pi$
$563$	−40.8118	−1.72001	−0.860006	+	0.510283i	$0.829541\pi$
$564$	−0.278147	−0.0117121
$565$	−6.76984	−0.284809
$566$	−14.1527	−0.594884
$567$	1.00000	0.0419961
$568$	25.1935	1.05710
$569$	−31.2184	−1.30874	−0.654371	−	0.756173i	$-0.727067\pi$
$569$	−31.2184	−1.30874	−0.654371	+	0.756173i	$0.727067\pi$
$570$	0.779419	0.0326463
$571$	−8.14583	−0.340892	−0.170446	−	0.985367i	$-0.554521\pi$
$571$	−8.14583	−0.340892	−0.170446	+	0.985367i	$0.554521\pi$
$572$	0.0415869	0.00173884
$573$	3.61688	0.151097
$574$	−12.8900	−0.538018
$575$	−20.8219	−0.868333
$576$	8.43604	0.351502
$577$	31.8764	1.32703	0.663515	−	0.748163i	$-0.269064\pi$
$577$	31.8764	1.32703	0.663515	+	0.748163i	$0.269064\pi$
$578$	−9.51878	−0.395929
$579$	−9.14902	−0.380220
$580$	−0.288492	−0.0119790
$581$	−11.7108	−0.485847
$582$	16.3079	0.675983
$583$	−1.32043	−0.0546865
$584$	−34.1244	−1.41208
$585$	−0.943718	−0.0390179
$586$	21.7235	0.897388
$587$	18.0047	0.743135	0.371567	−	0.928406i	$-0.378820\pi$
$587$	18.0047	0.743135	0.371567	+	0.928406i	$0.378820\pi$
$588$	−0.125168	−0.00516183
$589$	4.56571	0.188127
$590$	8.01515	0.329979
$591$	13.5427	0.557071
$592$	−1.92575	−0.0791479
$593$	−6.34775	−0.260671	−0.130335	−	0.991470i	$-0.541605\pi$
$593$	−6.34775	−0.260671	−0.130335	+	0.991470i	$0.541605\pi$
$594$	−0.199088	−0.00816868
$595$	−2.02121	−0.0828615
$596$	0.959070	0.0392850
$597$	8.53934	0.349492
$598$	13.4898	0.551639
$599$	−17.9813	−0.734697	−0.367348	−	0.930083i	$-0.619734\pi$
$599$	−17.9813	−0.734697	−0.367348	+	0.930083i	$0.619734\pi$
$600$	−14.0531	−0.573714
$601$	16.2860	0.664320	0.332160	−	0.943223i	$-0.392223\pi$
$601$	16.2860	0.664320	0.332160	+	0.943223i	$0.392223\pi$
$602$	−2.26217	−0.0921991
$603$	−1.93721	−0.0788891
$604$	−0.373265	−0.0151879
$605$	−4.53418	−0.184341
$606$	1.12585	0.0457346
$607$	9.32117	0.378335	0.189167	−	0.981945i	$-0.439421\pi$
$607$	9.32117	0.378335	0.189167	+	0.981945i	$0.439421\pi$
$608$	0.974457	0.0395195
$609$	5.58084	0.226147
$610$	−5.58633	−0.226184
$611$	−5.07787	−0.205429
$612$	0.612579	0.0247620
$613$	16.4466	0.664274	0.332137	−	0.943231i	$-0.392230\pi$
$613$	16.4466	0.664274	0.332137	+	0.943231i	$0.392230\pi$
$614$	11.6438	0.469904
$615$	3.88789	0.156775
$616$	0.423096	0.0170470
$617$	8.72458	0.351238	0.175619	−	0.984458i	$-0.443807\pi$
$617$	8.72458	0.351238	0.175619	+	0.984458i	$0.443807\pi$
$618$	−0.642017	−0.0258257
$619$	−22.8399	−0.918014	−0.459007	−	0.888433i	$-0.651795\pi$
$619$	−22.8399	−0.918014	−0.459007	+	0.888433i	$0.651795\pi$
$620$	0.171236	0.00687700
$621$	4.31146	0.173013
$622$	−29.9630	−1.20141
$623$	9.73143	0.389881
$624$	8.53246	0.341572
$625$	22.4707	0.898826
$626$	4.45183	0.177931
$627$	−0.200407	−0.00800347
$628$	−2.55914	−0.102121
$629$	−2.52404	−0.100640
$630$	−0.565488	−0.0225296
$631$	−29.8510	−1.18835	−0.594174	−	0.804337i	$-0.702521\pi$
$631$	−29.8510	−1.18835	−0.594174	+	0.804337i	$0.702521\pi$
$632$	1.89175	0.0752498
$633$	−8.86701	−0.352432
$634$	36.7440	1.45929
$635$	−4.58508	−0.181953
$636$	1.13669	0.0450728
$637$	−2.28507	−0.0905379
$638$	−1.11108	−0.0439880
$639$	8.65792	0.342502
$640$	−4.18651	−0.165486
$641$	41.2872	1.63075	0.815373	−	0.578935i	$-0.196532\pi$
$641$	41.2872	1.63075	0.815373	+	0.578935i	$0.196532\pi$
$642$	8.90122	0.351303
$643$	−9.43023	−0.371892	−0.185946	−	0.982560i	$-0.559535\pi$
$643$	−9.43023	−0.371892	−0.185946	+	0.982560i	$0.559535\pi$
$644$	−0.539655	−0.0212654
$645$	0.682316	0.0268662
$646$	−9.23631	−0.363398
$647$	4.18668	0.164595	0.0822977	−	0.996608i	$-0.473774\pi$
$647$	4.18668	0.164595	0.0822977	+	0.996608i	$0.473774\pi$
$648$	2.90988	0.114311
$649$	−2.06088	−0.0808966
$650$	−15.1105	−0.592681
$651$	−3.31254	−0.129829
$652$	−2.93254	−0.114847
$653$	11.6864	0.457323	0.228661	−	0.973506i	$-0.426565\pi$
$653$	11.6864	0.457323	0.228661	+	0.973506i	$0.426565\pi$
$654$	−15.7919	−0.617514
$655$	−1.17990	−0.0461023
$656$	−35.1517	−1.37244
$657$	−11.7271	−0.457518
$658$	−3.04272	−0.118618
$659$	0.306255	0.0119300	0.00596500	−	0.999982i	$-0.498101\pi$
$659$	0.306255	0.0119300	0.00596500	+	0.999982i	$0.498101\pi$
$660$	−0.00751620	−0.000292568 0
$661$	28.1920	1.09654	0.548271	−	0.836301i	$-0.315286\pi$
$661$	28.1920	1.09654	0.548271	+	0.836301i	$0.315286\pi$
$662$	22.3501	0.868661
$663$	11.1833	0.434323
$664$	−34.0770	−1.32244
$665$	−0.569233	−0.0220739
$666$	−0.706167	−0.0273634
$667$	24.0615	0.931666
$668$	−1.75375	−0.0678546
$669$	−18.8464	−0.728645
$670$	1.09547	0.0423215
$671$	1.43637	0.0554506
$672$	−0.706993	−0.0272728
$673$	36.1286	1.39266	0.696328	−	0.717724i	$-0.254816\pi$
$673$	36.1286	1.39266	0.696328	+	0.717724i	$0.254816\pi$
$674$	14.5445	0.560234
$675$	−4.82944	−0.185885
$676$	0.973610	0.0374465
$677$	−13.7074	−0.526819	−0.263410	−	0.964684i	$-0.584847\pi$
$677$	−13.7074	−0.526819	−0.263410	+	0.964684i	$0.584847\pi$
$678$	22.4449	0.861991
$679$	−11.9101	−0.457068
$680$	−5.88147	−0.225544
$681$	−6.61140	−0.253349
$682$	0.659487	0.0252531
$683$	−42.8250	−1.63865	−0.819327	−	0.573327i	$-0.805652\pi$
$683$	−42.8250	−1.63865	−0.819327	+	0.573327i	$0.805652\pi$
$684$	0.172520	0.00659648
$685$	−8.31950	−0.317872
$686$	−1.36925	−0.0522780
$687$	22.4426	0.856237
$688$	−6.16904	−0.235192
$689$	20.7516	0.790571
$690$	−2.43808	−0.0928159
$691$	30.7012	1.16793	0.583964	−	0.811779i	$-0.301501\pi$
$691$	30.7012	1.16793	0.583964	+	0.811779i	$0.301501\pi$
$692$	0.454119	0.0172630
$693$	0.145400	0.00552329
$694$	43.4478	1.64926
$695$	−3.30206	−0.125254
$696$	16.2395	0.615558
$697$	−46.0725	−1.74512
$698$	−28.5237	−1.07964
$699$	−20.6774	−0.782091
$700$	0.604490	0.0228476
$701$	−2.68126	−0.101270	−0.0506350	−	0.998717i	$-0.516125\pi$
$701$	−2.68126	−0.101270	−0.0506350	+	0.998717i	$0.516125\pi$
$702$	3.12883	0.118090
$703$	−0.710843	−0.0268100
$704$	1.22660	0.0462292
$705$	0.917748	0.0345644
$706$	−25.8072	−0.971267
$707$	−0.822242	−0.0309236
$708$	1.77411	0.0666752
$709$	32.7460	1.22980	0.614901	−	0.788604i	$-0.289196\pi$
$709$	32.7460	1.22980	0.614901	+	0.788604i	$0.289196\pi$
$710$	−4.89595	−0.183742
$711$	0.650114	0.0243812
$712$	28.3172	1.06123
$713$	−14.2819	−0.534860
$714$	6.70117	0.250785
$715$	−0.137216	−0.00513160
$716$	−0.754774	−0.0282072
$717$	−15.9820	−0.596859
$718$	33.4769	1.24935
$719$	41.6635	1.55379	0.776893	−	0.629632i	$-0.216794\pi$
$719$	41.6635	1.55379	0.776893	+	0.629632i	$0.216794\pi$
$720$	−1.54211	−0.0574711
$721$	0.468884	0.0174621
$722$	23.4144	0.871395
$723$	5.92275	0.220269
$724$	1.69009	0.0628119
$725$	−26.9523	−1.00098
$726$	15.0327	0.557918
$727$	8.46691	0.314020	0.157010	−	0.987597i	$-0.449814\pi$
$727$	8.46691	0.314020	0.157010	+	0.987597i	$0.449814\pi$
$728$	−6.64928	−0.246439
$729$	1.00000	0.0370370
$730$	6.63153	0.245444
$731$	−8.08562	−0.299057
$732$	−1.23650	−0.0457025
$733$	−14.4348	−0.533160	−0.266580	−	0.963813i	$-0.585894\pi$
$733$	−14.4348	−0.533160	−0.266580	+	0.963813i	$0.585894\pi$
$734$	14.8472	0.548022
$735$	0.412992	0.0152335
$736$	−3.04817	−0.112357
$737$	−0.281670	−0.0103754
$738$	−12.8900	−0.474488
$739$	28.1267	1.03466	0.517329	−	0.855787i	$-0.326927\pi$
$739$	28.1267	1.03466	0.517329	+	0.855787i	$0.326927\pi$
$740$	−0.0266600	−0.000980042 0
$741$	3.14954	0.115701
$742$	12.4346	0.456488
$743$	−34.8134	−1.27718	−0.638589	−	0.769548i	$-0.720482\pi$
$743$	−34.8134	−1.27718	−0.638589	+	0.769548i	$0.720482\pi$
$744$	−9.63907	−0.353385
$745$	−3.16446	−0.115937
$746$	43.9825	1.61031
$747$	−11.7108	−0.428476
$748$	0.0890689	0.00325668
$749$	−6.50083	−0.237535
$750$	5.55843	0.202965
$751$	53.2275	1.94230	0.971150	−	0.238471i	$-0.0766463\pi$
$751$	53.2275	1.94230	0.971150	+	0.238471i	$0.0766463\pi$
$752$	−8.29766	−0.302584
$753$	−6.51150	−0.237292
$754$	17.4615	0.635909
$755$	1.23159	0.0448222
$756$	−0.125168	−0.00455231
$757$	4.14058	0.150492	0.0752460	−	0.997165i	$-0.476026\pi$
$757$	4.14058	0.150492	0.0752460	+	0.997165i	$0.476026\pi$
$758$	18.3620	0.666938
$759$	0.626885	0.0227545
$760$	−1.65640	−0.0600838
$761$	−30.4927	−1.10536	−0.552679	−	0.833394i	$-0.686395\pi$
$761$	−30.4927	−1.10536	−0.552679	+	0.833394i	$0.686395\pi$
$762$	15.2015	0.550692
$763$	11.5333	0.417534
$764$	−0.452717	−0.0163787
$765$	−2.02121	−0.0730770
$766$	1.36925	0.0494728
$767$	32.3883	1.16947
$768$	−2.99201	−0.107965
$769$	−3.33772	−0.120361	−0.0601807	−	0.998188i	$-0.519168\pi$
$769$	−3.33772	−0.120361	−0.0601807	+	0.998188i	$0.519168\pi$
$770$	−0.0822219	−0.00296307
$771$	18.6449	0.671481
$772$	1.14516	0.0412153
$773$	−29.4600	−1.05960	−0.529802	−	0.848121i	$-0.677734\pi$
$773$	−29.4600	−1.05960	−0.529802	+	0.848121i	$0.677734\pi$
$774$	−2.26217	−0.0813119
$775$	15.9977	0.574654
$776$	−34.6569	−1.24411
$777$	0.515734	0.0185019
$778$	−34.9511	−1.25306
$779$	−12.9754	−0.464891
$780$	0.118123	0.00422948
$781$	1.25886	0.0450456
$782$	28.8918	1.03317
$783$	5.58084	0.199443
$784$	−3.73400	−0.133357
$785$	8.44391	0.301376
$786$	3.91186	0.139531
$787$	50.5016	1.80019	0.900094	−	0.435696i	$-0.143498\pi$
$787$	50.5016	1.80019	0.900094	+	0.435696i	$0.143498\pi$
$788$	−1.69510	−0.0603856
$789$	−6.04012	−0.215034
$790$	−0.367631	−0.0130797
$791$	−16.3922	−0.582838
$792$	0.423096	0.0150340
$793$	−22.5737	−0.801617
$794$	−9.39145	−0.333290
$795$	−3.75053	−0.133018
$796$	−1.06885	−0.0378844
$797$	29.8592	1.05767	0.528833	−	0.848726i	$-0.322630\pi$
$797$	29.8592	1.05767	0.528833	+	0.848726i	$0.322630\pi$
$798$	1.88725	0.0668079
$799$	−10.8755	−0.384749
$800$	3.41438	0.120717
$801$	9.73143	0.343843
$802$	18.8989	0.667344
$803$	−1.70512	−0.0601723
$804$	0.242476	0.00855146
$805$	1.78060	0.0627578
$806$	−10.3643	−0.365069
$807$	13.2256	0.465563
$808$	−2.39262	−0.0841722
$809$	10.8987	0.383177	0.191589	−	0.981475i	$-0.438636\pi$
$809$	10.8987	0.383177	0.191589	+	0.981475i	$0.438636\pi$
$810$	−0.565488	−0.0198692
$811$	24.4896	0.859945	0.429973	−	0.902842i	$-0.358523\pi$
$811$	24.4896	0.859945	0.429973	+	0.902842i	$0.358523\pi$
$812$	−0.698541	−0.0245140
$813$	15.1057	0.529779
$814$	−0.102677	−0.00359881
$815$	9.67594	0.338933
$816$	18.2744	0.639733
$817$	−2.27715	−0.0796673
$818$	−37.7364	−1.31942
$819$	−2.28507	−0.0798469
$820$	−0.486638	−0.0169941
$821$	21.9369	0.765603	0.382802	−	0.923831i	$-0.374959\pi$
$821$	21.9369	0.765603	0.382802	+	0.923831i	$0.374959\pi$
$822$	27.5827	0.962057
$823$	−35.8916	−1.25110	−0.625551	−	0.780184i	$-0.715126\pi$
$823$	−35.8916	−1.25110	−0.625551	+	0.780184i	$0.715126\pi$
$824$	1.36439	0.0475309
$825$	−0.702200	−0.0244475
$826$	19.4075	0.675273
$827$	24.8139	0.862863	0.431432	−	0.902146i	$-0.358009\pi$
$827$	24.8139	0.862863	0.431432	+	0.902146i	$0.358009\pi$
$828$	−0.539655	−0.0187543
$829$	−14.2578	−0.495193	−0.247596	−	0.968863i	$-0.579641\pi$
$829$	−14.2578	−0.495193	−0.247596	+	0.968863i	$0.579641\pi$
$830$	6.62232	0.229864
$831$	19.9944	0.693598
$832$	−19.2770	−0.668309
$833$	−4.89406	−0.169569
$834$	10.9477	0.379088
$835$	5.78652	0.200251
$836$	0.0250844	0.000867563 0
$837$	−3.31254	−0.114498
$838$	9.38172	0.324086
$839$	−31.6152	−1.09148	−0.545740	−	0.837955i	$-0.683751\pi$
$839$	−31.6152	−1.09148	−0.545740	+	0.837955i	$0.683751\pi$
$840$	1.20176	0.0414645
$841$	2.14576	0.0739918
$842$	19.6998	0.678899
$843$	−23.4048	−0.806104
$844$	1.10986	0.0382031
$845$	−3.21244	−0.110511
$846$	−3.04272	−0.104611
$847$	−10.9789	−0.377238
$848$	33.9098	1.16447
$849$	10.3362	0.354736
$850$	−32.3629	−1.11004
$851$	2.22357	0.0762229
$852$	−1.08369	−0.0371267
$853$	14.7443	0.504835	0.252417	−	0.967618i	$-0.418774\pi$
$853$	14.7443	0.504835	0.252417	+	0.967618i	$0.418774\pi$
$854$	−13.5265	−0.462866
$855$	−0.569233	−0.0194673
$856$	−18.9166	−0.646556
$857$	12.8218	0.437983	0.218992	−	0.975727i	$-0.429723\pi$
$857$	12.8218	0.437983	0.218992	+	0.975727i	$0.429723\pi$
$858$	0.454931	0.0155311
$859$	−15.8237	−0.539896	−0.269948	−	0.962875i	$-0.587007\pi$
$859$	−15.8237	−0.539896	−0.269948	+	0.962875i	$0.587007\pi$
$860$	−0.0854039	−0.00291225
$861$	9.41395	0.320827
$862$	3.85946	0.131454
$863$	−3.86528	−0.131576	−0.0657878	−	0.997834i	$-0.520956\pi$
$863$	−3.86528	−0.131576	−0.0657878	+	0.997834i	$0.520956\pi$
$864$	−0.706993	−0.0240524
$865$	−1.49837	−0.0509462
$866$	−1.63521	−0.0555668
$867$	6.95185	0.236097
$868$	0.414623	0.0140732
$869$	0.0945265	0.00320659
$870$	−3.15590	−0.106995
$871$	4.42666	0.149992
$872$	33.5605	1.13650
$873$	−11.9101	−0.403096
$874$	8.13679	0.275231
$875$	−4.05948	−0.137236
$876$	1.46785	0.0495942
$877$	16.1021	0.543730	0.271865	−	0.962335i	$-0.412360\pi$
$877$	16.1021	0.543730	0.271865	+	0.962335i	$0.412360\pi$
$878$	1.54404	0.0521087
$879$	−15.8653	−0.535123
$880$	−0.224223	−0.00755856
$881$	6.03404	0.203292	0.101646	−	0.994821i	$-0.467589\pi$
$881$	6.03404	0.203292	0.101646	+	0.994821i	$0.467589\pi$
$882$	−1.36925	−0.0461049
$883$	−42.0374	−1.41467	−0.707335	−	0.706878i	$-0.750103\pi$
$883$	−42.0374	−1.41467	−0.707335	+	0.706878i	$0.750103\pi$
$884$	−1.39979	−0.0470799
$885$	−5.85370	−0.196770
$886$	9.04385	0.303834
$887$	19.7093	0.661774	0.330887	−	0.943670i	$-0.392652\pi$
$887$	19.7093	0.661774	0.330887	+	0.943670i	$0.392652\pi$
$888$	1.50072	0.0503610
$889$	−11.1021	−0.372353
$890$	−5.50300	−0.184461
$891$	0.145400	0.00487108
$892$	2.35896	0.0789840
$893$	−3.06287	−0.102495
$894$	10.4915	0.350890
$895$	2.49039	0.0832444
$896$	−10.1370	−0.338654
$897$	−9.85199	−0.328948
$898$	−55.7494	−1.86038
$899$	−18.4867	−0.616567
$900$	0.604490	0.0201497
$901$	44.4447	1.48067
$902$	−1.87421	−0.0624042
$903$	1.65213	0.0549794
$904$	−47.6992	−1.58645
$905$	−5.57649	−0.185369
$906$	−4.08325	−0.135657
$907$	8.29815	0.275536	0.137768	−	0.990465i	$-0.456007\pi$
$907$	8.29815	0.275536	0.137768	+	0.990465i	$0.456007\pi$
$908$	0.827534	0.0274627
$909$	−0.822242	−0.0272721
$910$	1.29218	0.0428354
$911$	−15.8283	−0.524417	−0.262208	−	0.965011i	$-0.584451\pi$
$911$	−15.8283	−0.524417	−0.262208	+	0.965011i	$0.584451\pi$
$912$	5.14662	0.170421
$913$	−1.70275	−0.0563529
$914$	−34.2113	−1.13161
$915$	4.07986	0.134876
$916$	−2.80908	−0.0928147
$917$	−2.85694	−0.0943446
$918$	6.70117	0.221172
$919$	−38.5614	−1.27202	−0.636012	−	0.771679i	$-0.719417\pi$
$919$	−38.5614	−1.27202	−0.636012	+	0.771679i	$0.719417\pi$
$920$	5.18132	0.170823
$921$	−8.50378	−0.280209
$922$	17.3938	0.572834
$923$	−19.7840	−0.651198
$924$	−0.0181994	−0.000598715 0
$925$	−2.49071	−0.0818939
$926$	−40.3009	−1.32437
$927$	0.468884	0.0154002
$928$	−3.94561	−0.129521
$929$	−3.16362	−0.103795	−0.0518975	−	0.998652i	$-0.516527\pi$
$929$	−3.16362	−0.103795	−0.0518975	+	0.998652i	$0.516527\pi$
$930$	1.87320	0.0614246
$931$	−1.37831	−0.0451724
$932$	2.58814	0.0847775
$933$	21.8829	0.716413
$934$	−6.49092	−0.212389
$935$	−0.293884	−0.00961103
$936$	−6.64928	−0.217338
$937$	−19.3443	−0.631952	−0.315976	−	0.948767i	$-0.602332\pi$
$937$	−19.3443	−0.631952	−0.315976	+	0.948767i	$0.602332\pi$
$938$	2.65251	0.0866075
$939$	−3.25130	−0.106102
$940$	−0.114872	−0.00374672
$941$	−13.3692	−0.435822	−0.217911	−	0.975969i	$-0.569924\pi$
$941$	−13.3692	−0.435822	−0.217911	+	0.975969i	$0.569924\pi$
$942$	−27.9951	−0.912131
$943$	40.5878	1.32172
$944$	52.9252	1.72257
$945$	0.412992	0.0134346
$946$	−0.328919	−0.0106941
$947$	−38.1483	−1.23965	−0.619827	−	0.784739i	$-0.712797\pi$
$947$	−38.1483	−1.23965	−0.619827	+	0.784739i	$0.712797\pi$
$948$	−0.0813732	−0.00264288
$949$	26.7973	0.869876
$950$	−9.11435	−0.295708
$951$	−26.8352	−0.870191
$952$	−14.2411	−0.461557
$953$	−34.1839	−1.10733	−0.553663	−	0.832741i	$-0.686770\pi$
$953$	−34.1839	−1.10733	−0.553663	+	0.832741i	$0.686770\pi$
$954$	12.4346	0.402585
$955$	1.49374	0.0483364
$956$	2.00043	0.0646985
$957$	0.811453	0.0262306
$958$	5.12096	0.165451
$959$	−20.1444	−0.650498
$960$	3.48402	0.112446
$961$	−20.0271	−0.646036
$962$	1.61364	0.0520259
$963$	−6.50083	−0.209486
$964$	−0.741337	−0.0238768
$965$	−3.77847	−0.121633
$966$	−5.90344	−0.189940
$967$	−24.8385	−0.798752	−0.399376	−	0.916787i	$-0.630773\pi$
$967$	−24.8385	−0.798752	−0.399376	+	0.916787i	$0.630773\pi$
$968$	−31.9471	−1.02682
$969$	6.74555	0.216698
$970$	6.73502	0.216249
$971$	−23.1269	−0.742178	−0.371089	−	0.928597i	$-0.621016\pi$
$971$	−23.1269	−0.742178	−0.371089	+	0.928597i	$0.621016\pi$
$972$	−0.125168	−0.00401476
$973$	−7.99544	−0.256322
$974$	44.3992	1.42264
$975$	11.0356	0.353423
$976$	−36.8873	−1.18074
$977$	−8.07371	−0.258301	−0.129150	−	0.991625i	$-0.541225\pi$
$977$	−8.07371	−0.258301	−0.129150	+	0.991625i	$0.541225\pi$
$978$	−32.0799	−1.02580
$979$	1.41495	0.0452220
$980$	−0.0516933	−0.00165128
$981$	11.5333	0.368231
$982$	36.9188	1.17813
$983$	−8.96126	−0.285820	−0.142910	−	0.989736i	$-0.545646\pi$
$983$	−8.96126	−0.285820	−0.142910	+	0.989736i	$0.545646\pi$
$984$	27.3934	0.873270
$985$	5.59301	0.178208
$986$	37.3982	1.19100
$987$	2.22219	0.0707331
$988$	−0.394221	−0.0125419
$989$	7.12308	0.226501
$990$	−0.0822219	−0.00261318
$991$	42.3954	1.34674	0.673368	−	0.739308i	$-0.264847\pi$
$991$	42.3954	1.34674	0.673368	+	0.739308i	$0.264847\pi$
$992$	2.34194	0.0743567
$993$	−16.3229	−0.517992
$994$	−11.8548	−0.376012
$995$	3.52668	0.111803
$996$	1.46582	0.0464462
$997$	22.6607	0.717672	0.358836	−	0.933401i	$-0.383174\pi$
$997$	22.6607	0.717672	0.358836	+	0.933401i	$0.383174\pi$
$998$	43.0668	1.36326
$999$	0.515734	0.0163171

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.n.1.14	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.n.1.14	✓	40	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q-1.36925 q^{2} +1.00000 q^{3} -0.125168 q^{4} +0.412992 q^{5} -1.36925 q^{6} +1.00000 q^{7} +2.90988 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.36925 q^{2} +1.00000 q^{3} -0.125168 q^{4} +0.412992 q^{5} -1.36925 q^{6} +1.00000 q^{7} +2.90988 q^{8} +1.00000 q^{9} -0.565488 q^{10} +0.145400 q^{11} -0.125168 q^{12} -2.28507 q^{13} -1.36925 q^{14} +0.412992 q^{15} -3.73400 q^{16} -4.89406 q^{17} -1.36925 q^{18} -1.37831 q^{19} -0.0516933 q^{20} +1.00000 q^{21} -0.199088 q^{22} +4.31146 q^{23} +2.90988 q^{24} -4.82944 q^{25} +3.12883 q^{26} +1.00000 q^{27} -0.125168 q^{28} +5.58084 q^{29} -0.565488 q^{30} -3.31254 q^{31} -0.706993 q^{32} +0.145400 q^{33} +6.70117 q^{34} +0.412992 q^{35} -0.125168 q^{36} +0.515734 q^{37} +1.88725 q^{38} -2.28507 q^{39} +1.20176 q^{40} +9.41395 q^{41} -1.36925 q^{42} +1.65213 q^{43} -0.0181994 q^{44} +0.412992 q^{45} -5.90344 q^{46} +2.22219 q^{47} -3.73400 q^{48} +1.00000 q^{49} +6.61268 q^{50} -4.89406 q^{51} +0.286017 q^{52} -9.08135 q^{53} -1.36925 q^{54} +0.0600490 q^{55} +2.90988 q^{56} -1.37831 q^{57} -7.64154 q^{58} -14.1739 q^{59} -0.0516933 q^{60} +9.87878 q^{61} +4.53567 q^{62} +1.00000 q^{63} +8.43604 q^{64} -0.943718 q^{65} -0.199088 q^{66} -1.93721 q^{67} +0.612579 q^{68} +4.31146 q^{69} -0.565488 q^{70} +8.65792 q^{71} +2.90988 q^{72} -11.7271 q^{73} -0.706167 q^{74} -4.82944 q^{75} +0.172520 q^{76} +0.145400 q^{77} +3.12883 q^{78} +0.650114 q^{79} -1.54211 q^{80} +1.00000 q^{81} -12.8900 q^{82} -11.7108 q^{83} -0.125168 q^{84} -2.02121 q^{85} -2.26217 q^{86} +5.58084 q^{87} +0.423096 q^{88} +9.73143 q^{89} -0.565488 q^{90} -2.28507 q^{91} -0.539655 q^{92} -3.31254 q^{93} -3.04272 q^{94} -0.569233 q^{95} -0.706993 q^{96} -11.9101 q^{97} -1.36925 q^{98} +0.145400 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9}+O(q^{10}) \) 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9	\( 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9} - 14 q^{10} - 29 q^{11} + 33 q^{12} - 40 q^{13} - 9 q^{14} - 27 q^{15} + 23 q^{16} - 46 q^{17} - 9 q^{18} + 17 q^{19} - 53 q^{20} + 40 q^{21} - 15 q^{22} - 62 q^{23} - 18 q^{24} + 35 q^{25} - 29 q^{26} + 40 q^{27} + 33 q^{28} - 47 q^{29} - 14 q^{30} + q^{31} - 45 q^{32} - 29 q^{33} + 9 q^{34} - 27 q^{35} + 33 q^{36} - 49 q^{37} - 52 q^{38} - 40 q^{39} - 12 q^{40} - 49 q^{41} - 9 q^{42} - 45 q^{43} - 68 q^{44} - 27 q^{45} - 36 q^{46} - 88 q^{47} + 23 q^{48} + 40 q^{49} - 32 q^{50} - 46 q^{51} - 34 q^{52} - 96 q^{53} - 9 q^{54} + 26 q^{55} - 18 q^{56} + 17 q^{57} + 12 q^{58} - 45 q^{59} - 53 q^{60} - 23 q^{61} - 15 q^{62} + 40 q^{63} + 50 q^{64} - 32 q^{65} - 15 q^{66} - 31 q^{67} - 76 q^{68} - 62 q^{69} - 14 q^{70} - 89 q^{71} - 18 q^{72} + 4 q^{73} + 10 q^{74} + 35 q^{75} + 51 q^{76} - 29 q^{77} - 29 q^{78} - 44 q^{79} - 53 q^{80} + 40 q^{81} - 15 q^{82} - 60 q^{83} + 33 q^{84} - 24 q^{85} - 9 q^{86} - 47 q^{87} - 64 q^{88} - 39 q^{89} - 14 q^{90} - 40 q^{91} - 80 q^{92} + q^{93} + 51 q^{94} - 71 q^{95} - 45 q^{96} - 21 q^{97} - 9 q^{98} - 29 q^{99}+O(q^{100}) \) 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9 - 14 * q^10 - 29 * q^11 + 33 * q^12 - 40 * q^13 - 9 * q^14 - 27 * q^15 + 23 * q^16 - 46 * q^17 - 9 * q^18 + 17 * q^19 - 53 * q^20 + 40 * q^21 - 15 * q^22 - 62 * q^23 - 18 * q^24 + 35 * q^25 - 29 * q^26 + 40 * q^27 + 33 * q^28 - 47 * q^29 - 14 * q^30 + q^31 - 45 * q^32 - 29 * q^33 + 9 * q^34 - 27 * q^35 + 33 * q^36 - 49 * q^37 - 52 * q^38 - 40 * q^39 - 12 * q^40 - 49 * q^41 - 9 * q^42 - 45 * q^43 - 68 * q^44 - 27 * q^45 - 36 * q^46 - 88 * q^47 + 23 * q^48 + 40 * q^49 - 32 * q^50 - 46 * q^51 - 34 * q^52 - 96 * q^53 - 9 * q^54 + 26 * q^55 - 18 * q^56 + 17 * q^57 + 12 * q^58 - 45 * q^59 - 53 * q^60 - 23 * q^61 - 15 * q^62 + 40 * q^63 + 50 * q^64 - 32 * q^65 - 15 * q^66 - 31 * q^67 - 76 * q^68 - 62 * q^69 - 14 * q^70 - 89 * q^71 - 18 * q^72 + 4 * q^73 + 10 * q^74 + 35 * q^75 + 51 * q^76 - 29 * q^77 - 29 * q^78 - 44 * q^79 - 53 * q^80 + 40 * q^81 - 15 * q^82 - 60 * q^83 + 33 * q^84 - 24 * q^85 - 9 * q^86 - 47 * q^87 - 64 * q^88 - 39 * q^89 - 14 * q^90 - 40 * q^91 - 80 * q^92 + q^93 + 51 * q^94 - 71 * q^95 - 45 * q^96 - 21 * q^97 - 9 * q^98 - 29 * q^99

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