LMFDB - Embedded newform 2001.2.a.n.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2001,2,Mod(1,2001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2001, 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("2001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2001 = 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2001.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:	$15.9780654445$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} - 28 x^{14} + 27 x^{13} + 316 x^{12} - 295 x^{11} - 1835 x^{10} + 1665 x^{9} + \cdots - 192 $ x^16 - x^15 - 28x^14 + 27x^13 + 316x^12 - 295x^11 - 1835x^10 + 1665x^9 + 5778x^8 - 5124x^7 - 9405x^6 + 8288x^5 + 6405x^4 - 6032x^3 - 400x^2 + 1088x - 192
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-1.53756$ of defining polynomial
Character	$\chi$	$=$	2001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.53756 q^{2} +1.00000 q^{3} +0.364091 q^{4} -2.97145 q^{5} +1.53756 q^{6} +1.52213 q^{7} -2.51531 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.53756 q^{2} +1.00000 q^{3} +0.364091 q^{4} -2.97145 q^{5} +1.53756 q^{6} +1.52213 q^{7} -2.51531 q^{8} +1.00000 q^{9} -4.56878 q^{10} +0.657982 q^{11} +0.364091 q^{12} +3.12537 q^{13} +2.34037 q^{14} -2.97145 q^{15} -4.59562 q^{16} +1.99186 q^{17} +1.53756 q^{18} +6.50919 q^{19} -1.08188 q^{20} +1.52213 q^{21} +1.01169 q^{22} -1.00000 q^{23} -2.51531 q^{24} +3.82949 q^{25} +4.80545 q^{26} +1.00000 q^{27} +0.554195 q^{28} -1.00000 q^{29} -4.56878 q^{30} +6.58309 q^{31} -2.03543 q^{32} +0.657982 q^{33} +3.06261 q^{34} -4.52294 q^{35} +0.364091 q^{36} +7.52960 q^{37} +10.0083 q^{38} +3.12537 q^{39} +7.47410 q^{40} +8.68681 q^{41} +2.34037 q^{42} -3.14842 q^{43} +0.239565 q^{44} -2.97145 q^{45} -1.53756 q^{46} +1.88957 q^{47} -4.59562 q^{48} -4.68311 q^{49} +5.88807 q^{50} +1.99186 q^{51} +1.13792 q^{52} -2.78935 q^{53} +1.53756 q^{54} -1.95516 q^{55} -3.82864 q^{56} +6.50919 q^{57} -1.53756 q^{58} +6.26938 q^{59} -1.08188 q^{60} +1.61173 q^{61} +10.1219 q^{62} +1.52213 q^{63} +6.06165 q^{64} -9.28687 q^{65} +1.01169 q^{66} +5.54031 q^{67} +0.725219 q^{68} -1.00000 q^{69} -6.95429 q^{70} -9.71587 q^{71} -2.51531 q^{72} +4.17202 q^{73} +11.5772 q^{74} +3.82949 q^{75} +2.36994 q^{76} +1.00154 q^{77} +4.80545 q^{78} +9.99227 q^{79} +13.6556 q^{80} +1.00000 q^{81} +13.3565 q^{82} +3.81394 q^{83} +0.554195 q^{84} -5.91871 q^{85} -4.84088 q^{86} -1.00000 q^{87} -1.65503 q^{88} -0.899101 q^{89} -4.56878 q^{90} +4.75723 q^{91} -0.364091 q^{92} +6.58309 q^{93} +2.90533 q^{94} -19.3417 q^{95} -2.03543 q^{96} -3.81354 q^{97} -7.20056 q^{98} +0.657982 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - q^{2} + 16 q^{3} + 25 q^{4} + 3 q^{5} - q^{6} + 13 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q - q^2 + 16 * q^3 + 25 * q^4 + 3 * q^5 - q^6 + 13 * q^7 + 16 * q^9	\( 16 q - q^{2} + 16 q^{3} + 25 q^{4} + 3 q^{5} - q^{6} + 13 q^{7} + 16 q^{9} + 11 q^{10} + 8 q^{11} + 25 q^{12} + 19 q^{13} + 16 q^{14} + 3 q^{15} + 31 q^{16} - 4 q^{17} - q^{18} + 19 q^{19} + 16 q^{20} + 13 q^{21} + 6 q^{22} - 16 q^{23} + 23 q^{25} - 15 q^{26} + 16 q^{27} + 18 q^{28} - 16 q^{29} + 11 q^{30} + 24 q^{31} - 21 q^{32} + 8 q^{33} - 9 q^{34} - 13 q^{35} + 25 q^{36} + 26 q^{37} + 19 q^{39} - 22 q^{40} - 15 q^{41} + 16 q^{42} + 33 q^{43} + 6 q^{44} + 3 q^{45} + q^{46} + 13 q^{47} + 31 q^{48} + 41 q^{49} + 13 q^{50} - 4 q^{51} - 26 q^{52} + 5 q^{53} - q^{54} + 9 q^{55} + 40 q^{56} + 19 q^{57} + q^{58} + 2 q^{59} + 16 q^{60} + 29 q^{61} - 32 q^{62} + 13 q^{63} + 28 q^{64} + 18 q^{65} + 6 q^{66} + 32 q^{67} - 26 q^{68} - 16 q^{69} + 18 q^{70} + 29 q^{71} + 19 q^{73} - 16 q^{74} + 23 q^{75} + 64 q^{76} - 21 q^{77} - 15 q^{78} + 56 q^{79} + 16 q^{81} + 14 q^{82} + 5 q^{83} + 18 q^{84} + 16 q^{85} - 20 q^{86} - 16 q^{87} + q^{88} + 7 q^{89} + 11 q^{90} - 6 q^{91} - 25 q^{92} + 24 q^{93} - 11 q^{94} + 39 q^{95} - 21 q^{96} + 35 q^{97} - 109 q^{98} + 8 q^{99}+O(q^{100}) \) 16 * q - q^2 + 16 * q^3 + 25 * q^4 + 3 * q^5 - q^6 + 13 * q^7 + 16 * q^9 + 11 * q^10 + 8 * q^11 + 25 * q^12 + 19 * q^13 + 16 * q^14 + 3 * q^15 + 31 * q^16 - 4 * q^17 - q^18 + 19 * q^19 + 16 * q^20 + 13 * q^21 + 6 * q^22 - 16 * q^23 + 23 * q^25 - 15 * q^26 + 16 * q^27 + 18 * q^28 - 16 * q^29 + 11 * q^30 + 24 * q^31 - 21 * q^32 + 8 * q^33 - 9 * q^34 - 13 * q^35 + 25 * q^36 + 26 * q^37 + 19 * q^39 - 22 * q^40 - 15 * q^41 + 16 * q^42 + 33 * q^43 + 6 * q^44 + 3 * q^45 + q^46 + 13 * q^47 + 31 * q^48 + 41 * q^49 + 13 * q^50 - 4 * q^51 - 26 * q^52 + 5 * q^53 - q^54 + 9 * q^55 + 40 * q^56 + 19 * q^57 + q^58 + 2 * q^59 + 16 * q^60 + 29 * q^61 - 32 * q^62 + 13 * q^63 + 28 * q^64 + 18 * q^65 + 6 * q^66 + 32 * q^67 - 26 * q^68 - 16 * q^69 + 18 * q^70 + 29 * q^71 + 19 * q^73 - 16 * q^74 + 23 * q^75 + 64 * q^76 - 21 * q^77 - 15 * q^78 + 56 * q^79 + 16 * q^81 + 14 * q^82 + 5 * q^83 + 18 * q^84 + 16 * q^85 - 20 * q^86 - 16 * q^87 + q^88 + 7 * q^89 + 11 * q^90 - 6 * q^91 - 25 * q^92 + 24 * q^93 - 11 * q^94 + 39 * q^95 - 21 * q^96 + 35 * q^97 - 109 * q^98 + 8 * 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 $

$2$	1.53756	1.08722	0.543610	−	0.839338i	$-0.317057\pi$
$2$	1.53756	1.08722	0.543610	+	0.839338i	$0.317057\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.364091	0.182046
$5$	−2.97145	−1.32887	−0.664435	−	0.747346i	$-0.731328\pi$
$5$	−2.97145	−1.32887	−0.664435	+	0.747346i	$0.731328\pi$
$6$	1.53756	0.627706
$7$	1.52213	0.575313	0.287656	−	0.957734i	$-0.407124\pi$
$7$	1.52213	0.575313	0.287656	+	0.957734i	$0.407124\pi$
$8$	−2.51531	−0.889296
$9$	1.00000	0.333333
$10$	−4.56878	−1.44477
$11$	0.657982	0.198389	0.0991945	−	0.995068i	$-0.468373\pi$
$11$	0.657982	0.198389	0.0991945	+	0.995068i	$0.468373\pi$
$12$	0.364091	0.105104
$13$	3.12537	0.866822	0.433411	−	0.901196i	$-0.357310\pi$
$13$	3.12537	0.866822	0.433411	+	0.901196i	$0.357310\pi$
$14$	2.34037	0.625491
$15$	−2.97145	−0.767224
$16$	−4.59562	−1.14890
$17$	1.99186	0.483097	0.241549	−	0.970389i	$-0.422345\pi$
$17$	1.99186	0.483097	0.241549	+	0.970389i	$0.422345\pi$
$18$	1.53756	0.362406
$19$	6.50919	1.49331	0.746656	−	0.665211i	$-0.231658\pi$
$19$	6.50919	1.49331	0.746656	+	0.665211i	$0.231658\pi$
$20$	−1.08188	−0.241915
$21$	1.52213	0.332157
$22$	1.01169	0.215692
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−2.51531	−0.513435
$25$	3.82949	0.765898
$26$	4.80545	0.942426
$27$	1.00000	0.192450
$28$	0.554195	0.104733
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−4.56878	−0.834141
$31$	6.58309	1.18236	0.591179	−	0.806540i	$-0.298663\pi$
$31$	6.58309	1.18236	0.591179	+	0.806540i	$0.298663\pi$
$32$	−2.03543	−0.359816
$33$	0.657982	0.114540
$34$	3.06261	0.525233
$35$	−4.52294	−0.764516
$36$	0.364091	0.0606818
$37$	7.52960	1.23786	0.618929	−	0.785447i	$-0.287567\pi$
$37$	7.52960	1.23786	0.618929	+	0.785447i	$0.287567\pi$
$38$	10.0083	1.62356
$39$	3.12537	0.500460
$40$	7.47410	1.18176
$41$	8.68681	1.35665	0.678326	−	0.734761i	$-0.262706\pi$
$41$	8.68681	1.35665	0.678326	+	0.734761i	$0.262706\pi$
$42$	2.34037	0.361127
$43$	−3.14842	−0.480129	−0.240064	−	0.970757i	$-0.577169\pi$
$43$	−3.14842	−0.480129	−0.240064	+	0.970757i	$0.577169\pi$
$44$	0.239565	0.0361158
$45$	−2.97145	−0.442957
$46$	−1.53756	−0.226701
$47$	1.88957	0.275622	0.137811	−	0.990459i	$-0.455993\pi$
$47$	1.88957	0.275622	0.137811	+	0.990459i	$0.455993\pi$
$48$	−4.59562	−0.663321
$49$	−4.68311	−0.669016
$50$	5.88807	0.832699
$51$	1.99186	0.278916
$52$	1.13792	0.157801
$53$	−2.78935	−0.383146	−0.191573	−	0.981478i	$-0.561359\pi$
$53$	−2.78935	−0.383146	−0.191573	+	0.981478i	$0.561359\pi$
$54$	1.53756	0.209235
$55$	−1.95516	−0.263633
$56$	−3.82864	−0.511623
$57$	6.50919	0.862164
$58$	−1.53756	−0.201892
$59$	6.26938	0.816204	0.408102	−	0.912936i	$-0.366191\pi$
$59$	6.26938	0.816204	0.408102	+	0.912936i	$0.366191\pi$
$60$	−1.08188	−0.139670
$61$	1.61173	0.206361	0.103180	−	0.994663i	$-0.467098\pi$
$61$	1.61173	0.206361	0.103180	+	0.994663i	$0.467098\pi$
$62$	10.1219	1.28548
$63$	1.52213	0.191771
$64$	6.06165	0.757706
$65$	−9.28687	−1.15189
$66$	1.01169	0.124530
$67$	5.54031	0.676857	0.338428	−	0.940992i	$-0.390105\pi$
$67$	5.54031	0.676857	0.338428	+	0.940992i	$0.390105\pi$
$68$	0.725219	0.0879457
$69$	−1.00000	−0.120386
$70$	−6.95429	−0.831196
$71$	−9.71587	−1.15306	−0.576531	−	0.817076i	$-0.695594\pi$
$71$	−9.71587	−1.15306	−0.576531	+	0.817076i	$0.695594\pi$
$72$	−2.51531	−0.296432
$73$	4.17202	0.488298	0.244149	−	0.969738i	$-0.421491\pi$
$73$	4.17202	0.488298	0.244149	+	0.969738i	$0.421491\pi$
$74$	11.5772	1.34582
$75$	3.82949	0.442191
$76$	2.36994	0.271851
$77$	1.00154	0.114136
$78$	4.80545	0.544110
$79$	9.99227	1.12422	0.562109	−	0.827063i	$-0.309990\pi$
$79$	9.99227	1.12422	0.562109	+	0.827063i	$0.309990\pi$
$80$	13.6556	1.52675
$81$	1.00000	0.111111
$82$	13.3565	1.47498
$83$	3.81394	0.418635	0.209317	−	0.977848i	$-0.432876\pi$
$83$	3.81394	0.418635	0.209317	+	0.977848i	$0.432876\pi$
$84$	0.554195	0.0604677
$85$	−5.91871	−0.641974
$86$	−4.84088	−0.522005
$87$	−1.00000	−0.107211
$88$	−1.65503	−0.176427
$89$	−0.899101	−0.0953045	−0.0476523	−	0.998864i	$-0.515174\pi$
$89$	−0.899101	−0.0953045	−0.0476523	+	0.998864i	$0.515174\pi$
$90$	−4.56878	−0.481591
$91$	4.75723	0.498694
$92$	−0.364091	−0.0379591
$93$	6.58309	0.682635
$94$	2.90533	0.299661
$95$	−19.3417	−1.98442
$96$	−2.03543	−0.207740
$97$	−3.81354	−0.387206	−0.193603	−	0.981080i	$-0.562017\pi$
$97$	−3.81354	−0.387206	−0.193603	+	0.981080i	$0.562017\pi$
$98$	−7.20056	−0.727366
$99$	0.657982	0.0661297
$100$	1.39428	0.139428
$101$	−9.37952	−0.933297	−0.466649	−	0.884443i	$-0.654539\pi$
$101$	−9.37952	−0.933297	−0.466649	+	0.884443i	$0.654539\pi$
$102$	3.06261	0.303243
$103$	1.04005	0.102479	0.0512395	−	0.998686i	$-0.483683\pi$
$103$	1.04005	0.102479	0.0512395	+	0.998686i	$0.483683\pi$
$104$	−7.86127	−0.770861
$105$	−4.52294	−0.441394
$106$	−4.28879	−0.416564
$107$	−9.51332	−0.919687	−0.459843	−	0.888000i	$-0.652094\pi$
$107$	−9.51332	−0.919687	−0.459843	+	0.888000i	$0.652094\pi$
$108$	0.364091	0.0350347
$109$	11.0318	1.05666	0.528329	−	0.849040i	$-0.322819\pi$
$109$	11.0318	1.05666	0.528329	+	0.849040i	$0.322819\pi$
$110$	−3.00617	−0.286627
$111$	7.52960	0.714678
$112$	−6.99515	−0.660979
$113$	−16.5846	−1.56015	−0.780076	−	0.625685i	$-0.784819\pi$
$113$	−16.5846	−1.56015	−0.780076	+	0.625685i	$0.784819\pi$
$114$	10.0083	0.937361
$115$	2.97145	0.277089
$116$	−0.364091	−0.0338050
$117$	3.12537	0.288941
$118$	9.63955	0.887392
$119$	3.03188	0.277932
$120$	7.47410	0.682289
$121$	−10.5671	−0.960642
$122$	2.47813	0.224360
$123$	8.68681	0.783264
$124$	2.39685	0.215243
$125$	3.47811	0.311092
$126$	2.34037	0.208497
$127$	−7.11884	−0.631695	−0.315847	−	0.948810i	$-0.602289\pi$
$127$	−7.11884	−0.631695	−0.315847	+	0.948810i	$0.602289\pi$
$128$	13.3910	1.18361
$129$	−3.14842	−0.277203
$130$	−14.2791	−1.25236
$131$	−2.82852	−0.247129	−0.123564	−	0.992337i	$-0.539433\pi$
$131$	−2.82852	−0.247129	−0.123564	+	0.992337i	$0.539433\pi$
$132$	0.239565	0.0208515
$133$	9.90786	0.859121
$134$	8.51856	0.735892
$135$	−2.97145	−0.255741
$136$	−5.01014	−0.429616
$137$	−2.57599	−0.220082	−0.110041	−	0.993927i	$-0.535098\pi$
$137$	−2.57599	−0.220082	−0.110041	+	0.993927i	$0.535098\pi$
$138$	−1.53756	−0.130886
$139$	−15.0031	−1.27255	−0.636273	−	0.771464i	$-0.719525\pi$
$139$	−15.0031	−1.27255	−0.636273	+	0.771464i	$0.719525\pi$
$140$	−1.64676	−0.139177
$141$	1.88957	0.159130
$142$	−14.9387	−1.25363
$143$	2.05644	0.171968
$144$	−4.59562	−0.382968
$145$	2.97145	0.246765
$146$	6.41474	0.530887
$147$	−4.68311	−0.386256
$148$	2.74146	0.225347
$149$	19.2475	1.57682	0.788409	−	0.615151i	$-0.210905\pi$
$149$	19.2475	1.57682	0.788409	+	0.615151i	$0.210905\pi$
$150$	5.88807	0.480759
$151$	17.3611	1.41283	0.706415	−	0.707798i	$-0.250311\pi$
$151$	17.3611	1.41283	0.706415	+	0.707798i	$0.250311\pi$
$152$	−16.3726	−1.32800
$153$	1.99186	0.161032
$154$	1.53992	0.124091
$155$	−19.5613	−1.57120
$156$	1.13792	0.0911065
$157$	−7.87752	−0.628694	−0.314347	−	0.949308i	$-0.601786\pi$
$157$	−7.87752	−0.628694	−0.314347	+	0.949308i	$0.601786\pi$
$158$	15.3637	1.22227
$159$	−2.78935	−0.221210
$160$	6.04815	0.478149
$161$	−1.52213	−0.119961
$162$	1.53756	0.120802
$163$	−15.2023	−1.19074	−0.595370	−	0.803452i	$-0.702994\pi$
$163$	−15.2023	−1.19074	−0.595370	+	0.803452i	$0.702994\pi$
$164$	3.16279	0.246973
$165$	−1.95516	−0.152209
$166$	5.86417	0.455148
$167$	10.9193	0.844961	0.422480	−	0.906372i	$-0.361160\pi$
$167$	10.9193	0.844961	0.422480	+	0.906372i	$0.361160\pi$
$168$	−3.82864	−0.295386
$169$	−3.23205	−0.248620
$170$	−9.10037	−0.697966
$171$	6.50919	0.497770
$172$	−1.14631	−0.0874053
$173$	1.77155	0.134688	0.0673441	−	0.997730i	$-0.478547\pi$
$173$	1.77155	0.134688	0.0673441	+	0.997730i	$0.478547\pi$
$174$	−1.53756	−0.116562
$175$	5.82899	0.440631
$176$	−3.02384	−0.227930
$177$	6.26938	0.471235
$178$	−1.38242	−0.103617
$179$	−14.9962	−1.12087	−0.560433	−	0.828200i	$-0.689365\pi$
$179$	−14.9962	−1.12087	−0.560433	+	0.828200i	$0.689365\pi$
$180$	−1.08188	−0.0806383
$181$	25.1518	1.86952	0.934759	−	0.355281i	$-0.115615\pi$
$181$	25.1518	1.86952	0.934759	+	0.355281i	$0.115615\pi$
$182$	7.31453	0.542189
$183$	1.61173	0.119143
$184$	2.51531	0.185431
$185$	−22.3738	−1.64495
$186$	10.1219	0.742174
$187$	1.31061	0.0958412
$188$	0.687975	0.0501757
$189$	1.52213	0.110719
$190$	−29.7390	−2.15750
$191$	8.60952	0.622963	0.311482	−	0.950252i	$-0.399175\pi$
$191$	8.60952	0.622963	0.311482	+	0.950252i	$0.399175\pi$
$192$	6.06165	0.437462
$193$	−1.52670	−0.109894	−0.0549472	−	0.998489i	$-0.517499\pi$
$193$	−1.52670	−0.109894	−0.0549472	+	0.998489i	$0.517499\pi$
$194$	−5.86354	−0.420978
$195$	−9.28687	−0.665047
$196$	−1.70508	−0.121791
$197$	11.9516	0.851513	0.425757	−	0.904838i	$-0.360008\pi$
$197$	11.9516	0.851513	0.425757	+	0.904838i	$0.360008\pi$
$198$	1.01169	0.0718975
$199$	−20.2312	−1.43415	−0.717076	−	0.696995i	$-0.754520\pi$
$199$	−20.2312	−1.43415	−0.717076	+	0.696995i	$0.754520\pi$
$200$	−9.63234	−0.681110
$201$	5.54031	0.390784
$202$	−14.4216	−1.01470
$203$	−1.52213	−0.106833
$204$	0.725219	0.0507755
$205$	−25.8124	−1.80282
$206$	1.59914	0.111417
$207$	−1.00000	−0.0695048
$208$	−14.3630	−0.995896
$209$	4.28293	0.296257
$210$	−6.95429	−0.479892
$211$	−26.2493	−1.80708	−0.903539	−	0.428505i	$-0.859040\pi$
$211$	−26.2493	−1.80708	−0.903539	+	0.428505i	$0.859040\pi$
$212$	−1.01558	−0.0697501
$213$	−9.71587	−0.665720
$214$	−14.6273	−0.999901
$215$	9.35534	0.638029
$216$	−2.51531	−0.171145
$217$	10.0204	0.680226
$218$	16.9621	1.14882
$219$	4.17202	0.281919
$220$	−0.711855	−0.0479933
$221$	6.22530	0.418759
$222$	11.5772	0.777012
$223$	−13.3354	−0.893005	−0.446502	−	0.894782i	$-0.647331\pi$
$223$	−13.3354	−0.893005	−0.446502	+	0.894782i	$0.647331\pi$
$224$	−3.09819	−0.207006
$225$	3.82949	0.255299
$226$	−25.4999	−1.69623
$227$	−13.9162	−0.923648	−0.461824	−	0.886972i	$-0.652805\pi$
$227$	−13.9162	−0.923648	−0.461824	+	0.886972i	$0.652805\pi$
$228$	2.36994	0.156953
$229$	2.24048	0.148055	0.0740274	−	0.997256i	$-0.476415\pi$
$229$	2.24048	0.148055	0.0740274	+	0.997256i	$0.476415\pi$
$230$	4.56878	0.301256
$231$	1.00154	0.0658963
$232$	2.51531	0.165138
$233$	−16.1306	−1.05675	−0.528376	−	0.849011i	$-0.677199\pi$
$233$	−16.1306	−1.05675	−0.528376	+	0.849011i	$0.677199\pi$
$234$	4.80545	0.314142
$235$	−5.61475	−0.366266
$236$	2.28263	0.148586
$237$	9.99227	0.649068
$238$	4.66170	0.302173
$239$	25.9374	1.67775	0.838874	−	0.544325i	$-0.183214\pi$
$239$	25.9374	1.67775	0.838874	+	0.544325i	$0.183214\pi$
$240$	13.6556	0.881467
$241$	17.9123	1.15384	0.576918	−	0.816802i	$-0.304255\pi$
$241$	17.9123	1.15384	0.576918	+	0.816802i	$0.304255\pi$
$242$	−16.2475	−1.04443
$243$	1.00000	0.0641500
$244$	0.586817	0.0375671
$245$	13.9156	0.889035
$246$	13.3565	0.851579
$247$	20.3436	1.29443
$248$	−16.5585	−1.05147
$249$	3.81394	0.241699
$250$	5.34780	0.338225
$251$	−23.8459	−1.50514	−0.752569	−	0.658513i	$-0.771186\pi$
$251$	−23.8459	−1.50514	−0.752569	+	0.658513i	$0.771186\pi$
$252$	0.554195	0.0349110
$253$	−0.657982	−0.0413670
$254$	−10.9456	−0.686791
$255$	−5.91871	−0.370644
$256$	8.46617	0.529136
$257$	12.0783	0.753425	0.376713	−	0.926330i	$-0.377054\pi$
$257$	12.0783	0.753425	0.376713	+	0.926330i	$0.377054\pi$
$258$	−4.84088	−0.301380
$259$	11.4611	0.712156
$260$	−3.38127	−0.209697
$261$	−1.00000	−0.0618984
$262$	−4.34902	−0.268683
$263$	10.2132	0.629772	0.314886	−	0.949130i	$-0.398034\pi$
$263$	10.2132	0.629772	0.314886	+	0.949130i	$0.398034\pi$
$264$	−1.65503	−0.101860
$265$	8.28839	0.509152
$266$	15.2339	0.934052
$267$	−0.899101	−0.0550241
$268$	2.01718	0.123219
$269$	−16.3462	−0.996648	−0.498324	−	0.866991i	$-0.666051\pi$
$269$	−16.3462	−0.996648	−0.498324	+	0.866991i	$0.666051\pi$
$270$	−4.56878	−0.278047
$271$	−3.90467	−0.237192	−0.118596	−	0.992943i	$-0.537839\pi$
$271$	−3.90467	−0.237192	−0.118596	+	0.992943i	$0.537839\pi$
$272$	−9.15383	−0.555033
$273$	4.75723	0.287921
$274$	−3.96074	−0.239277
$275$	2.51973	0.151946
$276$	−0.364091	−0.0219157
$277$	−22.0571	−1.32529	−0.662643	−	0.748936i	$-0.730565\pi$
$277$	−22.0571	−1.32529	−0.662643	+	0.748936i	$0.730565\pi$
$278$	−23.0682	−1.38354
$279$	6.58309	0.394120
$280$	11.3766	0.679881
$281$	0.0586985	0.00350166	0.00175083	−	0.999998i	$-0.499443\pi$
$281$	0.0586985	0.00350166	0.00175083	+	0.999998i	$0.499443\pi$
$282$	2.90533	0.173010
$283$	11.6006	0.689582	0.344791	−	0.938679i	$-0.387950\pi$
$283$	11.6006	0.689582	0.344791	+	0.938679i	$0.387950\pi$
$284$	−3.53746	−0.209910
$285$	−19.3417	−1.14570
$286$	3.16190	0.186967
$287$	13.2225	0.780499
$288$	−2.03543	−0.119939
$289$	−13.0325	−0.766617
$290$	4.56878	0.268288
$291$	−3.81354	−0.223554
$292$	1.51900	0.0888925
$293$	−16.3619	−0.955873	−0.477936	−	0.878394i	$-0.658615\pi$
$293$	−16.3619	−0.955873	−0.477936	+	0.878394i	$0.658615\pi$
$294$	−7.20056	−0.419945
$295$	−18.6291	−1.08463
$296$	−18.9393	−1.10082
$297$	0.657982	0.0381800
$298$	29.5942	1.71435
$299$	−3.12537	−0.180745
$300$	1.39428	0.0804989
$301$	−4.79231	−0.276224
$302$	26.6938	1.53606
$303$	−9.37952	−0.538839
$304$	−29.9138	−1.71567
$305$	−4.78917	−0.274227
$306$	3.06261	0.175078
$307$	−6.09940	−0.348111	−0.174056	−	0.984736i	$-0.555687\pi$
$307$	−6.09940	−0.348111	−0.174056	+	0.984736i	$0.555687\pi$
$308$	0.364651	0.0207779
$309$	1.04005	0.0591663
$310$	−30.0767	−1.70824
$311$	−11.4644	−0.650084	−0.325042	−	0.945700i	$-0.605378\pi$
$311$	−11.4644	−0.650084	−0.325042	+	0.945700i	$0.605378\pi$
$312$	−7.86127	−0.445057
$313$	19.2263	1.08674	0.543368	−	0.839495i	$-0.317149\pi$
$313$	19.2263	1.08674	0.543368	+	0.839495i	$0.317149\pi$
$314$	−12.1122	−0.683528
$315$	−4.52294	−0.254839
$316$	3.63810	0.204659
$317$	9.21199	0.517397	0.258699	−	0.965958i	$-0.416706\pi$
$317$	9.21199	0.517397	0.258699	+	0.965958i	$0.416706\pi$
$318$	−4.28879	−0.240503
$319$	−0.657982	−0.0368399
$320$	−18.0119	−1.00689
$321$	−9.51332	−0.530982
$322$	−2.34037	−0.130424
$323$	12.9654	0.721414
$324$	0.364091	0.0202273
$325$	11.9686	0.663897
$326$	−23.3745	−1.29459
$327$	11.0318	0.610061
$328$	−21.8500	−1.20647
$329$	2.87618	0.158569
$330$	−3.00617	−0.165484
$331$	−6.85715	−0.376903	−0.188451	−	0.982083i	$-0.560347\pi$
$331$	−6.85715	−0.376903	−0.188451	+	0.982083i	$0.560347\pi$
$332$	1.38862	0.0762105
$333$	7.52960	0.412620
$334$	16.7891	0.918657
$335$	−16.4627	−0.899455
$336$	−6.99515	−0.381617
$337$	12.0008	0.653725	0.326863	−	0.945072i	$-0.394009\pi$
$337$	12.0008	0.653725	0.326863	+	0.945072i	$0.394009\pi$
$338$	−4.96948	−0.270304
$339$	−16.5846	−0.900754
$340$	−2.15495	−0.116868
$341$	4.33156	0.234567
$342$	10.0083	0.541185
$343$	−17.7833	−0.960206
$344$	7.91924	0.426977
$345$	2.97145	0.159977
$346$	2.72386	0.146436
$347$	1.31039	0.0703454	0.0351727	−	0.999381i	$-0.488802\pi$
$347$	1.31039	0.0703454	0.0351727	+	0.999381i	$0.488802\pi$
$348$	−0.364091	−0.0195173
$349$	24.3350	1.30262	0.651310	−	0.758811i	$-0.274220\pi$
$349$	24.3350	1.30262	0.651310	+	0.758811i	$0.274220\pi$
$350$	8.96243	0.479062
$351$	3.12537	0.166820
$352$	−1.33927	−0.0713835
$353$	12.1064	0.644361	0.322181	−	0.946678i	$-0.395584\pi$
$353$	12.1064	0.644361	0.322181	+	0.946678i	$0.395584\pi$
$354$	9.63955	0.512336
$355$	28.8702	1.53227
$356$	−0.327355	−0.0173498
$357$	3.03188	0.160464
$358$	−23.0575	−1.21863
$359$	−0.568054	−0.0299808	−0.0149904	−	0.999888i	$-0.504772\pi$
$359$	−0.568054	−0.0299808	−0.0149904	+	0.999888i	$0.504772\pi$
$360$	7.47410	0.393920
$361$	23.3696	1.22998
$362$	38.6724	2.03258
$363$	−10.5671	−0.554627
$364$	1.73207	0.0907849
$365$	−12.3969	−0.648885
$366$	2.47813	0.129534
$367$	12.8719	0.671908	0.335954	−	0.941878i	$-0.390941\pi$
$367$	12.8719	0.671908	0.335954	+	0.941878i	$0.390941\pi$
$368$	4.59562	0.239563
$369$	8.68681	0.452217
$370$	−34.4011	−1.78843
$371$	−4.24576	−0.220429
$372$	2.39685	0.124271
$373$	−26.8087	−1.38810	−0.694050	−	0.719927i	$-0.744175\pi$
$373$	−26.8087	−1.38810	−0.694050	+	0.719927i	$0.744175\pi$
$374$	2.01514	0.104200
$375$	3.47811	0.179609
$376$	−4.75285	−0.245109
$377$	−3.12537	−0.160965
$378$	2.34037	0.120376
$379$	23.5826	1.21136	0.605678	−	0.795710i	$-0.292902\pi$
$379$	23.5826	1.21136	0.605678	+	0.795710i	$0.292902\pi$
$380$	−7.04214	−0.361254
$381$	−7.11884	−0.364709
$382$	13.2377	0.677298
$383$	4.55238	0.232616	0.116308	−	0.993213i	$-0.462894\pi$
$383$	4.55238	0.232616	0.116308	+	0.993213i	$0.462894\pi$
$384$	13.3910	0.683357
$385$	−2.97601	−0.151672
$386$	−2.34740	−0.119479
$387$	−3.14842	−0.160043
$388$	−1.38848	−0.0704892
$389$	28.4426	1.44210	0.721048	−	0.692885i	$-0.243661\pi$
$389$	28.4426	1.44210	0.721048	+	0.692885i	$0.243661\pi$
$390$	−14.2791	−0.723051
$391$	−1.99186	−0.100733
$392$	11.7795	0.594953
$393$	−2.82852	−0.142680
$394$	18.3762	0.925782
$395$	−29.6915	−1.49394
$396$	0.239565	0.0120386
$397$	−20.0063	−1.00409	−0.502043	−	0.864843i	$-0.667418\pi$
$397$	−20.0063	−1.00409	−0.502043	+	0.864843i	$0.667418\pi$
$398$	−31.1067	−1.55924
$399$	9.90786	0.496013
$400$	−17.5989	−0.879944
$401$	1.05576	0.0527220	0.0263610	−	0.999652i	$-0.491608\pi$
$401$	1.05576	0.0527220	0.0263610	+	0.999652i	$0.491608\pi$
$402$	8.51856	0.424867
$403$	20.5746	1.02489
$404$	−3.41500	−0.169903
$405$	−2.97145	−0.147652
$406$	−2.34037	−0.116151
$407$	4.95434	0.245578
$408$	−5.01014	−0.248039
$409$	11.8403	0.585463	0.292732	−	0.956195i	$-0.405436\pi$
$409$	11.8403	0.585463	0.292732	+	0.956195i	$0.405436\pi$
$410$	−39.6881	−1.96006
$411$	−2.57599	−0.127064
$412$	0.378672	0.0186558
$413$	9.54284	0.469572
$414$	−1.53756	−0.0755670
$415$	−11.3329	−0.556311
$416$	−6.36146	−0.311896
$417$	−15.0031	−0.734705
$418$	6.58526	0.322096
$419$	36.2135	1.76915	0.884573	−	0.466401i	$-0.154450\pi$
$419$	36.2135	1.76915	0.884573	+	0.466401i	$0.154450\pi$
$420$	−1.64676	−0.0803537
$421$	30.5979	1.49125	0.745625	−	0.666366i	$-0.232151\pi$
$421$	30.5979	1.49125	0.745625	+	0.666366i	$0.232151\pi$
$422$	−40.3599	−1.96469
$423$	1.88957	0.0918740
$424$	7.01607	0.340730
$425$	7.62781	0.370003
$426$	−14.9387	−0.723784
$427$	2.45327	0.118722
$428$	−3.46371	−0.167425
$429$	2.05644	0.0992858
$430$	14.3844	0.693678
$431$	−1.70015	−0.0818935	−0.0409467	−	0.999161i	$-0.513037\pi$
$431$	−1.70015	−0.0818935	−0.0409467	+	0.999161i	$0.513037\pi$
$432$	−4.59562	−0.221107
$433$	24.1484	1.16050	0.580248	−	0.814440i	$-0.302956\pi$
$433$	24.1484	1.16050	0.580248	+	0.814440i	$0.302956\pi$
$434$	15.4069	0.739554
$435$	2.97145	0.142470
$436$	4.01659	0.192360
$437$	−6.50919	−0.311377
$438$	6.41474	0.306508
$439$	6.50808	0.310614	0.155307	−	0.987866i	$-0.450363\pi$
$439$	6.50808	0.310614	0.155307	+	0.987866i	$0.450363\pi$
$440$	4.91782	0.234448
$441$	−4.68311	−0.223005
$442$	9.57178	0.455283
$443$	25.4823	1.21070	0.605349	−	0.795960i	$-0.293033\pi$
$443$	25.4823	1.21070	0.605349	+	0.795960i	$0.293033\pi$
$444$	2.74146	0.130104
$445$	2.67163	0.126647
$446$	−20.5040	−0.970892
$447$	19.2475	0.910377
$448$	9.22664	0.435918
$449$	−21.8497	−1.03115	−0.515575	−	0.856845i	$-0.672422\pi$
$449$	−21.8497	−1.03115	−0.515575	+	0.856845i	$0.672422\pi$
$450$	5.88807	0.277566
$451$	5.71577	0.269145
$452$	−6.03832	−0.284019
$453$	17.3611	0.815698
$454$	−21.3969	−1.00421
$455$	−14.1359	−0.662699
$456$	−16.3726	−0.766718
$457$	24.1438	1.12940	0.564700	−	0.825296i	$-0.308992\pi$
$457$	24.1438	1.12940	0.564700	+	0.825296i	$0.308992\pi$
$458$	3.44487	0.160968
$459$	1.99186	0.0929721
$460$	1.08188	0.0504428
$461$	−23.3197	−1.08611	−0.543054	−	0.839698i	$-0.682732\pi$
$461$	−23.3197	−1.08611	−0.543054	+	0.839698i	$0.682732\pi$
$462$	1.53992	0.0716437
$463$	24.5951	1.14303	0.571516	−	0.820591i	$-0.306356\pi$
$463$	24.5951	1.14303	0.571516	+	0.820591i	$0.306356\pi$
$464$	4.59562	0.213346
$465$	−19.5613	−0.907134
$466$	−24.8018	−1.14892
$467$	−6.13480	−0.283885	−0.141942	−	0.989875i	$-0.545335\pi$
$467$	−6.13480	−0.283885	−0.141942	+	0.989875i	$0.545335\pi$
$468$	1.13792	0.0526004
$469$	8.43310	0.389404
$470$	−8.63302	−0.398211
$471$	−7.87752	−0.362977
$472$	−15.7694	−0.725847
$473$	−2.07160	−0.0952523
$474$	15.3637	0.705679
$475$	24.9269	1.14372
$476$	1.10388	0.0505963
$477$	−2.78935	−0.127715
$478$	39.8803	1.82408
$479$	−13.6698	−0.624588	−0.312294	−	0.949986i	$-0.601097\pi$
$479$	−13.6698	−0.624588	−0.312294	+	0.949986i	$0.601097\pi$
$480$	6.04815	0.276059
$481$	23.5328	1.07300
$482$	27.5413	1.25447
$483$	−1.52213	−0.0692595
$484$	−3.84737	−0.174881
$485$	11.3317	0.514547
$486$	1.53756	0.0697451
$487$	−3.77831	−0.171212	−0.0856058	−	0.996329i	$-0.527283\pi$
$487$	−3.77831	−0.171212	−0.0856058	+	0.996329i	$0.527283\pi$
$488$	−4.05400	−0.183516
$489$	−15.2023	−0.687474
$490$	21.3961	0.966576
$491$	−5.21934	−0.235545	−0.117773	−	0.993041i	$-0.537575\pi$
$491$	−5.21934	−0.235545	−0.117773	+	0.993041i	$0.537575\pi$
$492$	3.16279	0.142590
$493$	−1.99186	−0.0897089
$494$	31.2796	1.40733
$495$	−1.95516	−0.0878778
$496$	−30.2534	−1.35842
$497$	−14.7888	−0.663370
$498$	5.86417	0.262780
$499$	−16.8243	−0.753157	−0.376579	−	0.926385i	$-0.622900\pi$
$499$	−16.8243	−0.753157	−0.376579	+	0.926385i	$0.622900\pi$
$500$	1.26635	0.0566329
$501$	10.9193	0.487838
$502$	−36.6645	−1.63642
$503$	−17.5168	−0.781036	−0.390518	−	0.920595i	$-0.627704\pi$
$503$	−17.5168	−0.781036	−0.390518	+	0.920595i	$0.627704\pi$
$504$	−3.82864	−0.170541
$505$	27.8707	1.24023
$506$	−1.01169	−0.0449750
$507$	−3.23205	−0.143541
$508$	−2.59191	−0.114997
$509$	−39.8215	−1.76506	−0.882530	−	0.470257i	$-0.844161\pi$
$509$	−39.8215	−1.76506	−0.882530	+	0.470257i	$0.844161\pi$
$510$	−9.10037	−0.402971
$511$	6.35038	0.280924
$512$	−13.7648	−0.608322
$513$	6.50919	0.287388
$514$	18.5712	0.819138
$515$	−3.09045	−0.136181
$516$	−1.14631	−0.0504635
$517$	1.24330	0.0546804
$518$	17.6221	0.774269
$519$	1.77155	0.0777623
$520$	23.3593	1.02437
$521$	20.5118	0.898637	0.449319	−	0.893372i	$-0.351667\pi$
$521$	20.5118	0.898637	0.449319	+	0.893372i	$0.351667\pi$
$522$	−1.53756	−0.0672972
$523$	−0.217705	−0.00951957	−0.00475978	−	0.999989i	$-0.501515\pi$
$523$	−0.217705	−0.00951957	−0.00475978	+	0.999989i	$0.501515\pi$
$524$	−1.02984	−0.0449887
$525$	5.82899	0.254398
$526$	15.7034	0.684700
$527$	13.1126	0.571194
$528$	−3.02384	−0.131596
$529$	1.00000	0.0434783
$530$	12.7439	0.553560
$531$	6.26938	0.272068
$532$	3.60736	0.156399
$533$	27.1495	1.17598
$534$	−1.38242	−0.0598232
$535$	28.2683	1.22215
$536$	−13.9356	−0.601926
$537$	−14.9962	−0.647132
$538$	−25.1333	−1.08357
$539$	−3.08140	−0.132725
$540$	−1.08188	−0.0465566
$541$	−19.2734	−0.828630	−0.414315	−	0.910134i	$-0.635979\pi$
$541$	−19.2734	−0.828630	−0.414315	+	0.910134i	$0.635979\pi$
$542$	−6.00366	−0.257879
$543$	25.1518	1.07937
$544$	−4.05428	−0.173826
$545$	−32.7805	−1.40416
$546$	7.31453	0.313033
$547$	26.3839	1.12809	0.564047	−	0.825743i	$-0.309244\pi$
$547$	26.3839	1.12809	0.564047	+	0.825743i	$0.309244\pi$
$548$	−0.937895	−0.0400649
$549$	1.61173	0.0687870
$550$	3.87424	0.165198
$551$	−6.50919	−0.277301
$552$	2.51531	0.107059
$553$	15.2096	0.646777
$554$	−33.9142	−1.44088
$555$	−22.3738	−0.949715
$556$	−5.46250	−0.231661
$557$	6.89107	0.291984	0.145992	−	0.989286i	$-0.453363\pi$
$557$	6.89107	0.291984	0.145992	+	0.989286i	$0.453363\pi$
$558$	10.1219	0.428494
$559$	−9.83997	−0.416186
$560$	20.7857	0.878356
$561$	1.31061	0.0553339
$562$	0.0902525	0.00380707
$563$	−32.6691	−1.37684	−0.688418	−	0.725314i	$-0.741695\pi$
$563$	−32.6691	−1.37684	−0.688418	+	0.725314i	$0.741695\pi$
$564$	0.687975	0.0289690
$565$	49.2803	2.07324
$566$	17.8366	0.749727
$567$	1.52213	0.0639236
$568$	24.4384	1.02541
$569$	13.8851	0.582095	0.291048	−	0.956709i	$-0.405996\pi$
$569$	13.8851	0.582095	0.291048	+	0.956709i	$0.405996\pi$
$570$	−29.7390	−1.24563
$571$	−1.56148	−0.0653459	−0.0326729	−	0.999466i	$-0.510402\pi$
$571$	−1.56148	−0.0653459	−0.0326729	+	0.999466i	$0.510402\pi$
$572$	0.748731	0.0313060
$573$	8.60952	0.359668
$574$	20.3304	0.848574
$575$	−3.82949	−0.159701
$576$	6.06165	0.252569
$577$	−32.2938	−1.34441	−0.672205	−	0.740365i	$-0.734653\pi$
$577$	−32.2938	−1.34441	−0.672205	+	0.740365i	$0.734653\pi$
$578$	−20.0382	−0.833481
$579$	−1.52670	−0.0634476
$580$	1.08188	0.0449225
$581$	5.80533	0.240846
$582$	−5.86354	−0.243052
$583$	−1.83534	−0.0760120
$584$	−10.4939	−0.434242
$585$	−9.28687	−0.383965
$586$	−25.1574	−1.03924
$587$	−0.644254	−0.0265912	−0.0132956	−	0.999912i	$-0.504232\pi$
$587$	−0.644254	−0.0265912	−0.0132956	+	0.999912i	$0.504232\pi$
$588$	−1.70508	−0.0703162
$589$	42.8506	1.76563
$590$	−28.6434	−1.17923
$591$	11.9516	0.491621
$592$	−34.6032	−1.42218
$593$	−42.7893	−1.75715	−0.878573	−	0.477609i	$-0.841504\pi$
$593$	−42.7893	−1.75715	−0.878573	+	0.477609i	$0.841504\pi$
$594$	1.01169	0.0415100
$595$	−9.00906	−0.369336
$596$	7.00785	0.287053
$597$	−20.2312	−0.828009
$598$	−4.80545	−0.196509
$599$	−27.8106	−1.13631	−0.568155	−	0.822922i	$-0.692343\pi$
$599$	−27.8106	−1.13631	−0.568155	+	0.822922i	$0.692343\pi$
$600$	−9.63234	−0.393239
$601$	−0.0213529	−0.000871004 0	−0.000435502	−	1.00000i	$-0.500139\pi$
$601$	−0.0213529	−0.000871004 0	−0.000435502		1.00000i	$0.500139\pi$
$602$	−7.36846	−0.300316
$603$	5.54031	0.225619
$604$	6.32104	0.257199
$605$	31.3994	1.27657
$606$	−14.4216	−0.585836
$607$	−3.95786	−0.160645	−0.0803223	−	0.996769i	$-0.525595\pi$
$607$	−3.95786	−0.160645	−0.0803223	+	0.996769i	$0.525595\pi$
$608$	−13.2490	−0.537317
$609$	−1.52213	−0.0616800
$610$	−7.36364	−0.298145
$611$	5.90560	0.238915
$612$	0.725219	0.0293152
$613$	−3.77872	−0.152621	−0.0763105	−	0.997084i	$-0.524314\pi$
$613$	−3.77872	−0.152621	−0.0763105	+	0.997084i	$0.524314\pi$
$614$	−9.37819	−0.378473
$615$	−25.8124	−1.04086
$616$	−2.51917	−0.101500
$617$	−29.3301	−1.18079	−0.590393	−	0.807116i	$-0.701027\pi$
$617$	−29.3301	−1.18079	−0.590393	+	0.807116i	$0.701027\pi$
$618$	1.59914	0.0643267
$619$	−5.53617	−0.222517	−0.111259	−	0.993791i	$-0.535488\pi$
$619$	−5.53617	−0.222517	−0.111259	+	0.993791i	$0.535488\pi$
$620$	−7.12210	−0.286030
$621$	−1.00000	−0.0401286
$622$	−17.6271	−0.706783
$623$	−1.36855	−0.0548299
$624$	−14.3630	−0.574981
$625$	−29.4825	−1.17930
$626$	29.5616	1.18152
$627$	4.28293	0.171044
$628$	−2.86813	−0.114451
$629$	14.9979	0.598006
$630$	−6.95429	−0.277065
$631$	−31.4015	−1.25007	−0.625037	−	0.780595i	$-0.714916\pi$
$631$	−31.4015	−1.25007	−0.625037	+	0.780595i	$0.714916\pi$
$632$	−25.1336	−0.999762
$633$	−26.2493	−1.04332
$634$	14.1640	0.562524
$635$	21.1532	0.839441
$636$	−1.01558	−0.0402702
$637$	−14.6365	−0.579917
$638$	−1.01169	−0.0400531
$639$	−9.71587	−0.384354
$640$	−39.7906	−1.57286
$641$	−9.00627	−0.355726	−0.177863	−	0.984055i	$-0.556918\pi$
$641$	−9.00627	−0.355726	−0.177863	+	0.984055i	$0.556918\pi$
$642$	−14.6273	−0.577293
$643$	−49.6816	−1.95925	−0.979626	−	0.200829i	$-0.935636\pi$
$643$	−49.6816	−1.95925	−0.979626	+	0.200829i	$0.935636\pi$
$644$	−0.554195	−0.0218384
$645$	9.35534	0.368366
$646$	19.9351	0.784336
$647$	−24.7647	−0.973601	−0.486801	−	0.873513i	$-0.661836\pi$
$647$	−24.7647	−0.973601	−0.486801	+	0.873513i	$0.661836\pi$
$648$	−2.51531	−0.0988106
$649$	4.12514	0.161926
$650$	18.4024	0.721802
$651$	10.0204	0.392729
$652$	−5.53504	−0.216769
$653$	−41.6418	−1.62957	−0.814785	−	0.579764i	$-0.803145\pi$
$653$	−41.6418	−1.62957	−0.814785	+	0.579764i	$0.803145\pi$
$654$	16.9621	0.663270
$655$	8.40478	0.328402
$656$	−39.9213	−1.55866
$657$	4.17202	0.162766
$658$	4.42229	0.172399
$659$	31.4482	1.22505	0.612524	−	0.790452i	$-0.290154\pi$
$659$	31.4482	1.22505	0.612524	+	0.790452i	$0.290154\pi$
$660$	−0.711855	−0.0277089
$661$	−27.1590	−1.05636	−0.528181	−	0.849132i	$-0.677126\pi$
$661$	−27.1590	−1.05636	−0.528181	+	0.849132i	$0.677126\pi$
$662$	−10.5433	−0.409776
$663$	6.22530	0.241771
$664$	−9.59324	−0.372290
$665$	−29.4407	−1.14166
$666$	11.5772	0.448608
$667$	1.00000	0.0387202
$668$	3.97562	0.153821
$669$	−13.3354	−0.515577
$670$	−25.3124	−0.977905
$671$	1.06049	0.0409398
$672$	−3.09819	−0.119515
$673$	29.7953	1.14852	0.574262	−	0.818672i	$-0.305289\pi$
$673$	29.7953	1.14852	0.574262	+	0.818672i	$0.305289\pi$
$674$	18.4520	0.710743
$675$	3.82949	0.147397
$676$	−1.17676	−0.0452601
$677$	35.8770	1.37887	0.689433	−	0.724349i	$-0.257860\pi$
$677$	35.8770	1.37887	0.689433	+	0.724349i	$0.257860\pi$
$678$	−25.4999	−0.979317
$679$	−5.80472	−0.222765
$680$	14.8874	0.570905
$681$	−13.9162	−0.533268
$682$	6.66003	0.255026
$683$	0.183601	0.00702530	0.00351265	−	0.999994i	$-0.498882\pi$
$683$	0.183601	0.00702530	0.00351265	+	0.999994i	$0.498882\pi$
$684$	2.36994	0.0906169
$685$	7.65441	0.292460
$686$	−27.3428	−1.04395
$687$	2.24048	0.0854795
$688$	14.4689	0.551622
$689$	−8.71775	−0.332120
$690$	4.56878	0.173930
$691$	−8.73544	−0.332312	−0.166156	−	0.986099i	$-0.553136\pi$
$691$	−8.73544	−0.332312	−0.166156	+	0.986099i	$0.553136\pi$
$692$	0.645005	0.0245194
$693$	1.00154	0.0380452
$694$	2.01480	0.0764808
$695$	44.5809	1.69105
$696$	2.51531	0.0953425
$697$	17.3029	0.655395
$698$	37.4165	1.41623
$699$	−16.1306	−0.610116
$700$	2.12228	0.0802148
$701$	−39.2626	−1.48293	−0.741464	−	0.670992i	$-0.765868\pi$
$701$	−39.2626	−1.48293	−0.741464	+	0.670992i	$0.765868\pi$
$702$	4.80545	0.181370
$703$	49.0116	1.84851
$704$	3.98846	0.150321
$705$	−5.61475	−0.211464
$706$	18.6144	0.700562
$707$	−14.2769	−0.536937
$708$	2.28263	0.0857863
$709$	28.7413	1.07940	0.539701	−	0.841857i	$-0.318537\pi$
$709$	28.7413	1.07940	0.539701	+	0.841857i	$0.318537\pi$
$710$	44.3896	1.66591
$711$	9.99227	0.374739
$712$	2.26152	0.0847539
$713$	−6.58309	−0.246539
$714$	4.66170	0.174460
$715$	−6.11059	−0.228523
$716$	−5.45997	−0.204049
$717$	25.9374	0.968649
$718$	−0.873417	−0.0325956
$719$	9.64982	0.359877	0.179939	−	0.983678i	$-0.442410\pi$
$719$	9.64982	0.359877	0.179939	+	0.983678i	$0.442410\pi$
$720$	13.6556	0.508915
$721$	1.58309	0.0589574
$722$	35.9321	1.33726
$723$	17.9123	0.666167
$724$	9.15755	0.340338
$725$	−3.82949	−0.142224
$726$	−16.2475	−0.603001
$727$	0.623800	0.0231355	0.0115677	−	0.999933i	$-0.496318\pi$
$727$	0.623800	0.0231355	0.0115677	+	0.999933i	$0.496318\pi$
$728$	−11.9659	−0.443486
$729$	1.00000	0.0370370
$730$	−19.0610	−0.705481
$731$	−6.27121	−0.231949
$732$	0.586817	0.0216894
$733$	14.3861	0.531362	0.265681	−	0.964061i	$-0.414403\pi$
$733$	14.3861	0.531362	0.265681	+	0.964061i	$0.414403\pi$
$734$	19.7913	0.730512
$735$	13.9156	0.513285
$736$	2.03543	0.0750268
$737$	3.64543	0.134281
$738$	13.3565	0.491660
$739$	−3.95277	−0.145405	−0.0727025	−	0.997354i	$-0.523162\pi$
$739$	−3.95277	−0.145405	−0.0727025	+	0.997354i	$0.523162\pi$
$740$	−8.14610	−0.299457
$741$	20.3436	0.747342
$742$	−6.52811	−0.239654
$743$	17.5497	0.643837	0.321919	−	0.946767i	$-0.395672\pi$
$743$	17.5497	0.643837	0.321919	+	0.946767i	$0.395672\pi$
$744$	−16.5585	−0.607065
$745$	−57.1930	−2.09539
$746$	−41.2199	−1.50917
$747$	3.81394	0.139545
$748$	0.477181	0.0174475
$749$	−14.4805	−0.529107
$750$	5.34780	0.195274
$751$	−36.8189	−1.34354	−0.671770	−	0.740760i	$-0.734465\pi$
$751$	−36.8189	−1.34354	−0.671770	+	0.740760i	$0.734465\pi$
$752$	−8.68374	−0.316663
$753$	−23.8459	−0.868992
$754$	−4.80545	−0.175004
$755$	−51.5877	−1.87747
$756$	0.554195	0.0201559
$757$	−33.2754	−1.20942	−0.604708	−	0.796447i	$-0.706710\pi$
$757$	−33.2754	−1.20942	−0.604708	+	0.796447i	$0.706710\pi$
$758$	36.2596	1.31701
$759$	−0.657982	−0.0238832
$760$	48.6504	1.76473
$761$	37.2086	1.34881	0.674405	−	0.738361i	$-0.264400\pi$
$761$	37.2086	1.34881	0.674405	+	0.738361i	$0.264400\pi$
$762$	−10.9456	−0.396519
$763$	16.7919	0.607908
$764$	3.13465	0.113408
$765$	−5.91871	−0.213991
$766$	6.99956	0.252904
$767$	19.5941	0.707503
$768$	8.46617	0.305497
$769$	54.5305	1.96642	0.983209	−	0.182481i	$-0.0584129\pi$
$769$	54.5305	1.96642	0.983209	+	0.182481i	$0.0584129\pi$
$770$	−4.57580	−0.164900
$771$	12.0783	0.434990
$772$	−0.555859	−0.0200058
$773$	−25.4337	−0.914787	−0.457393	−	0.889264i	$-0.651217\pi$
$773$	−25.4337	−0.914787	−0.457393	+	0.889264i	$0.651217\pi$
$774$	−4.84088	−0.174002
$775$	25.2099	0.905566
$776$	9.59223	0.344341
$777$	11.4611	0.411163
$778$	43.7322	1.56788
$779$	56.5441	2.02590
$780$	−3.38127	−0.121069
$781$	−6.39286	−0.228755
$782$	−3.06261	−0.109519
$783$	−1.00000	−0.0357371
$784$	21.5218	0.768635
$785$	23.4076	0.835453
$786$	−4.34902	−0.155124
$787$	−47.1336	−1.68013	−0.840066	−	0.542484i	$-0.817484\pi$
$787$	−47.1336	−1.68013	−0.840066	+	0.542484i	$0.817484\pi$
$788$	4.35146	0.155014
$789$	10.2132	0.363599
$790$	−45.6524	−1.62424
$791$	−25.2440	−0.897574
$792$	−1.65503	−0.0588088
$793$	5.03726	0.178878
$794$	−30.7608	−1.09166
$795$	8.28839	0.293959
$796$	−7.36600	−0.261081
$797$	33.2908	1.17922	0.589611	−	0.807687i	$-0.299281\pi$
$797$	33.2908	1.17922	0.589611	+	0.807687i	$0.299281\pi$
$798$	15.2339	0.539275
$799$	3.76376	0.133152
$800$	−7.79464	−0.275582
$801$	−0.899101	−0.0317682
$802$	1.62329	0.0573204
$803$	2.74512	0.0968730
$804$	2.01718	0.0711404
$805$	4.52294	0.159413
$806$	31.6347	1.11429
$807$	−16.3462	−0.575415
$808$	23.5924	0.829977
$809$	−8.67983	−0.305166	−0.152583	−	0.988291i	$-0.548759\pi$
$809$	−8.67983	−0.305166	−0.152583	+	0.988291i	$0.548759\pi$
$810$	−4.56878	−0.160530
$811$	−25.8833	−0.908887	−0.454444	−	0.890776i	$-0.650162\pi$
$811$	−25.8833	−0.908887	−0.454444	+	0.890776i	$0.650162\pi$
$812$	−0.554195	−0.0194484
$813$	−3.90467	−0.136943
$814$	7.61760	0.266997
$815$	45.1729	1.58234
$816$	−9.15383	−0.320448
$817$	−20.4936	−0.716982
$818$	18.2051	0.636527
$819$	4.75723	0.166231
$820$	−9.39806	−0.328195
$821$	15.6591	0.546507	0.273253	−	0.961942i	$-0.411900\pi$
$821$	15.6591	0.546507	0.273253	+	0.961942i	$0.411900\pi$
$822$	−3.96074	−0.138147
$823$	46.7430	1.62936	0.814679	−	0.579912i	$-0.196913\pi$
$823$	46.7430	1.62936	0.814679	+	0.579912i	$0.196913\pi$
$824$	−2.61604	−0.0911341
$825$	2.51973	0.0877259
$826$	14.6727	0.510528
$827$	−32.5350	−1.13135	−0.565676	−	0.824627i	$-0.691385\pi$
$827$	−32.5350	−1.13135	−0.565676	+	0.824627i	$0.691385\pi$
$828$	−0.364091	−0.0126530
$829$	11.9491	0.415008	0.207504	−	0.978234i	$-0.433466\pi$
$829$	11.9491	0.415008	0.207504	+	0.978234i	$0.433466\pi$
$830$	−17.4250	−0.604832
$831$	−22.0571	−0.765154
$832$	18.9449	0.656797
$833$	−9.32810	−0.323200
$834$	−23.0682	−0.798785
$835$	−32.4461	−1.12284
$836$	1.55938	0.0539322
$837$	6.58309	0.227545
$838$	55.6805	1.92345
$839$	−17.8539	−0.616384	−0.308192	−	0.951324i	$-0.599724\pi$
$839$	−17.8539	−0.616384	−0.308192	+	0.951324i	$0.599724\pi$
$840$	11.3766	0.392529
$841$	1.00000	0.0344828
$842$	47.0461	1.62132
$843$	0.0586985	0.00202168
$844$	−9.55715	−0.328971
$845$	9.60387	0.330383
$846$	2.90533	0.0998872
$847$	−16.0845	−0.552669
$848$	12.8188	0.440199
$849$	11.6006	0.398131
$850$	11.7282	0.402274
$851$	−7.52960	−0.258111
$852$	−3.53746	−0.121191
$853$	22.5605	0.772457	0.386228	−	0.922403i	$-0.373778\pi$
$853$	22.5605	0.772457	0.386228	+	0.922403i	$0.373778\pi$
$854$	3.77205	0.129077
$855$	−19.3417	−0.661473
$856$	23.9289	0.817874
$857$	22.9279	0.783204	0.391602	−	0.920135i	$-0.371921\pi$
$857$	22.9279	0.783204	0.391602	+	0.920135i	$0.371921\pi$
$858$	3.16190	0.107945
$859$	−2.33959	−0.0798257	−0.0399129	−	0.999203i	$-0.512708\pi$
$859$	−2.33959	−0.0798257	−0.0399129	+	0.999203i	$0.512708\pi$
$860$	3.40620	0.116150
$861$	13.2225	0.450621
$862$	−2.61409	−0.0890362
$863$	33.2368	1.13140	0.565698	−	0.824613i	$-0.308607\pi$
$863$	33.2368	1.13140	0.565698	+	0.824613i	$0.308607\pi$
$864$	−2.03543	−0.0692466
$865$	−5.26406	−0.178983
$866$	37.1296	1.26171
$867$	−13.0325	−0.442607
$868$	3.64832	0.123832
$869$	6.57473	0.223033
$870$	4.56878	0.154896
$871$	17.3155	0.586715
$872$	−27.7484	−0.939681
$873$	−3.81354	−0.129069
$874$	−10.0083	−0.338535
$875$	5.29415	0.178975
$876$	1.51900	0.0513221
$877$	19.6244	0.662669	0.331335	−	0.943513i	$-0.392501\pi$
$877$	19.6244	0.662669	0.331335	+	0.943513i	$0.392501\pi$
$878$	10.0066	0.337705
$879$	−16.3619	−0.551873
$880$	8.98516	0.302890
$881$	−47.7909	−1.61012	−0.805058	−	0.593196i	$-0.797866\pi$
$881$	−47.7909	−1.61012	−0.805058	+	0.593196i	$0.797866\pi$
$882$	−7.20056	−0.242455
$883$	9.68739	0.326007	0.163003	−	0.986626i	$-0.447882\pi$
$883$	9.68739	0.326007	0.163003	+	0.986626i	$0.447882\pi$
$884$	2.26658	0.0762333
$885$	−18.6291	−0.626211
$886$	39.1805	1.31629
$887$	−10.9256	−0.366845	−0.183422	−	0.983034i	$-0.558718\pi$
$887$	−10.9256	−0.366845	−0.183422	+	0.983034i	$0.558718\pi$
$888$	−18.9393	−0.635560
$889$	−10.8358	−0.363422
$890$	4.10779	0.137693
$891$	0.657982	0.0220432
$892$	−4.85530	−0.162568
$893$	12.2996	0.411589
$894$	29.5942	0.989779
$895$	44.5603	1.48949
$896$	20.3829	0.680945
$897$	−3.12537	−0.104353
$898$	−33.5952	−1.12109
$899$	−6.58309	−0.219559
$900$	1.39428	0.0464761
$901$	−5.55599	−0.185097
$902$	8.78834	0.292620
$903$	−4.79231	−0.159478
$904$	41.7155	1.38744
$905$	−74.7372	−2.48435
$906$	26.6938	0.886842
$907$	−31.7832	−1.05534	−0.527672	−	0.849448i	$-0.676935\pi$
$907$	−31.7832	−1.05534	−0.527672	+	0.849448i	$0.676935\pi$
$908$	−5.06675	−0.168146
$909$	−9.37952	−0.311099
$910$	−21.7347	−0.720499
$911$	6.69493	0.221813	0.110907	−	0.993831i	$-0.464625\pi$
$911$	6.69493	0.221813	0.110907	+	0.993831i	$0.464625\pi$
$912$	−29.9138	−0.990544
$913$	2.50951	0.0830525
$914$	37.1226	1.22791
$915$	−4.78917	−0.158325
$916$	0.815737	0.0269527
$917$	−4.30538	−0.142176
$918$	3.06261	0.101081
$919$	46.2250	1.52482	0.762411	−	0.647093i	$-0.224015\pi$
$919$	46.2250	1.52482	0.762411	+	0.647093i	$0.224015\pi$
$920$	−7.47410	−0.246414
$921$	−6.09940	−0.200982
$922$	−35.8555	−1.18084
$923$	−30.3657	−0.999499
$924$	0.364651	0.0119961
$925$	28.8345	0.948073
$926$	37.8164	1.24273
$927$	1.04005	0.0341597
$928$	2.03543	0.0668161
$929$	−14.9133	−0.489290	−0.244645	−	0.969613i	$-0.578671\pi$
$929$	−14.9133	−0.489290	−0.244645	+	0.969613i	$0.578671\pi$
$930$	−30.0767	−0.986253
$931$	−30.4833	−0.999048
$932$	−5.87301	−0.192377
$933$	−11.4644	−0.375326
$934$	−9.43263	−0.308645
$935$	−3.89440	−0.127361
$936$	−7.86127	−0.256954
$937$	−6.70593	−0.219073	−0.109537	−	0.993983i	$-0.534937\pi$
$937$	−6.70593	−0.219073	−0.109537	+	0.993983i	$0.534937\pi$
$938$	12.9664	0.423368
$939$	19.2263	0.627427
$940$	−2.04428	−0.0666771
$941$	−33.3052	−1.08572	−0.542858	−	0.839824i	$-0.682658\pi$
$941$	−33.3052	−1.08572	−0.542858	+	0.839824i	$0.682658\pi$
$942$	−12.1122	−0.394635
$943$	−8.68681	−0.282882
$944$	−28.8117	−0.937741
$945$	−4.52294	−0.147131
$946$	−3.18521	−0.103560
$947$	27.4034	0.890492	0.445246	−	0.895408i	$-0.353116\pi$
$947$	27.4034	0.890492	0.445246	+	0.895408i	$0.353116\pi$
$948$	3.63810	0.118160
$949$	13.0391	0.423268
$950$	38.3266	1.24348
$951$	9.21199	0.298719
$952$	−7.62611	−0.247164
$953$	−1.09506	−0.0354723	−0.0177362	−	0.999843i	$-0.505646\pi$
$953$	−1.09506	−0.0354723	−0.0177362	+	0.999843i	$0.505646\pi$
$954$	−4.28879	−0.138855
$955$	−25.5827	−0.827838
$956$	9.44356	0.305427
$957$	−0.657982	−0.0212695
$958$	−21.0181	−0.679064
$959$	−3.92100	−0.126616
$960$	−18.0119	−0.581330
$961$	12.3371	0.397972
$962$	36.1831	1.16659
$963$	−9.51332	−0.306562
$964$	6.52173	0.210051
$965$	4.53651	0.146035
$966$	−2.34037	−0.0753002
$967$	21.9241	0.705030	0.352515	−	0.935806i	$-0.385327\pi$
$967$	21.9241	0.705030	0.352515	+	0.935806i	$0.385327\pi$
$968$	26.5794	0.854295
$969$	12.9654	0.416509
$970$	17.4232	0.559425
$971$	−0.0746247	−0.00239482	−0.00119741	−	0.999999i	$-0.500381\pi$
$971$	−0.0746247	−0.00239482	−0.00119741	+	0.999999i	$0.500381\pi$
$972$	0.364091	0.0116782
$973$	−22.8367	−0.732112
$974$	−5.80938	−0.186145
$975$	11.9686	0.383301
$976$	−7.40690	−0.237089
$977$	−58.4019	−1.86844	−0.934221	−	0.356694i	$-0.883904\pi$
$977$	−58.4019	−1.86844	−0.934221	+	0.356694i	$0.883904\pi$
$978$	−23.3745	−0.747434
$979$	−0.591592	−0.0189074
$980$	5.06655	0.161845
$981$	11.0318	0.352219
$982$	−8.02504	−0.256089
$983$	25.8365	0.824056	0.412028	−	0.911171i	$-0.364821\pi$
$983$	25.8365	0.824056	0.412028	+	0.911171i	$0.364821\pi$
$984$	−21.8500	−0.696553
$985$	−35.5134	−1.13155
$986$	−3.06261	−0.0975332
$987$	2.87618	0.0915497
$988$	7.40694	0.235646
$989$	3.14842	0.100114
$990$	−3.00617	−0.0955424
$991$	−19.3585	−0.614944	−0.307472	−	0.951557i	$-0.599483\pi$
$991$	−19.3585	−0.614944	−0.307472	+	0.951557i	$0.599483\pi$
$992$	−13.3994	−0.425431
$993$	−6.85715	−0.217605
$994$	−22.7387	−0.721229
$995$	60.1160	1.90580
$996$	1.38862	0.0440002
$997$	−30.5596	−0.967833	−0.483917	−	0.875114i	$-0.660786\pi$
$997$	−30.5596	−0.967833	−0.483917	+	0.875114i	$0.660786\pi$
$998$	−25.8683	−0.818847
$999$	7.52960	0.238226

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2001.2.a.n.1.12	✓	16
3.2	odd	2		6003.2.a.r.1.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.n.1.12	✓	16	1.1	even	1	trivial
6003.2.a.r.1.5		16	3.2	odd	2

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:	\( N \)	\(=\)	\( 2001 = 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2001.a (trivial)

\(f(q)\)	\(=\)	\(q+1.53756 q^{2} +1.00000 q^{3} +0.364091 q^{4} -2.97145 q^{5} +1.53756 q^{6} +1.52213 q^{7} -2.51531 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.53756 q^{2} +1.00000 q^{3} +0.364091 q^{4} -2.97145 q^{5} +1.53756 q^{6} +1.52213 q^{7} -2.51531 q^{8} +1.00000 q^{9} -4.56878 q^{10} +0.657982 q^{11} +0.364091 q^{12} +3.12537 q^{13} +2.34037 q^{14} -2.97145 q^{15} -4.59562 q^{16} +1.99186 q^{17} +1.53756 q^{18} +6.50919 q^{19} -1.08188 q^{20} +1.52213 q^{21} +1.01169 q^{22} -1.00000 q^{23} -2.51531 q^{24} +3.82949 q^{25} +4.80545 q^{26} +1.00000 q^{27} +0.554195 q^{28} -1.00000 q^{29} -4.56878 q^{30} +6.58309 q^{31} -2.03543 q^{32} +0.657982 q^{33} +3.06261 q^{34} -4.52294 q^{35} +0.364091 q^{36} +7.52960 q^{37} +10.0083 q^{38} +3.12537 q^{39} +7.47410 q^{40} +8.68681 q^{41} +2.34037 q^{42} -3.14842 q^{43} +0.239565 q^{44} -2.97145 q^{45} -1.53756 q^{46} +1.88957 q^{47} -4.59562 q^{48} -4.68311 q^{49} +5.88807 q^{50} +1.99186 q^{51} +1.13792 q^{52} -2.78935 q^{53} +1.53756 q^{54} -1.95516 q^{55} -3.82864 q^{56} +6.50919 q^{57} -1.53756 q^{58} +6.26938 q^{59} -1.08188 q^{60} +1.61173 q^{61} +10.1219 q^{62} +1.52213 q^{63} +6.06165 q^{64} -9.28687 q^{65} +1.01169 q^{66} +5.54031 q^{67} +0.725219 q^{68} -1.00000 q^{69} -6.95429 q^{70} -9.71587 q^{71} -2.51531 q^{72} +4.17202 q^{73} +11.5772 q^{74} +3.82949 q^{75} +2.36994 q^{76} +1.00154 q^{77} +4.80545 q^{78} +9.99227 q^{79} +13.6556 q^{80} +1.00000 q^{81} +13.3565 q^{82} +3.81394 q^{83} +0.554195 q^{84} -5.91871 q^{85} -4.84088 q^{86} -1.00000 q^{87} -1.65503 q^{88} -0.899101 q^{89} -4.56878 q^{90} +4.75723 q^{91} -0.364091 q^{92} +6.58309 q^{93} +2.90533 q^{94} -19.3417 q^{95} -2.03543 q^{96} -3.81354 q^{97} -7.20056 q^{98} +0.657982 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - q^{2} + 16 q^{3} + 25 q^{4} + 3 q^{5} - q^{6} + 13 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q - q^2 + 16 * q^3 + 25 * q^4 + 3 * q^5 - q^6 + 13 * q^7 + 16 * q^9	\( 16 q - q^{2} + 16 q^{3} + 25 q^{4} + 3 q^{5} - q^{6} + 13 q^{7} + 16 q^{9} + 11 q^{10} + 8 q^{11} + 25 q^{12} + 19 q^{13} + 16 q^{14} + 3 q^{15} + 31 q^{16} - 4 q^{17} - q^{18} + 19 q^{19} + 16 q^{20} + 13 q^{21} + 6 q^{22} - 16 q^{23} + 23 q^{25} - 15 q^{26} + 16 q^{27} + 18 q^{28} - 16 q^{29} + 11 q^{30} + 24 q^{31} - 21 q^{32} + 8 q^{33} - 9 q^{34} - 13 q^{35} + 25 q^{36} + 26 q^{37} + 19 q^{39} - 22 q^{40} - 15 q^{41} + 16 q^{42} + 33 q^{43} + 6 q^{44} + 3 q^{45} + q^{46} + 13 q^{47} + 31 q^{48} + 41 q^{49} + 13 q^{50} - 4 q^{51} - 26 q^{52} + 5 q^{53} - q^{54} + 9 q^{55} + 40 q^{56} + 19 q^{57} + q^{58} + 2 q^{59} + 16 q^{60} + 29 q^{61} - 32 q^{62} + 13 q^{63} + 28 q^{64} + 18 q^{65} + 6 q^{66} + 32 q^{67} - 26 q^{68} - 16 q^{69} + 18 q^{70} + 29 q^{71} + 19 q^{73} - 16 q^{74} + 23 q^{75} + 64 q^{76} - 21 q^{77} - 15 q^{78} + 56 q^{79} + 16 q^{81} + 14 q^{82} + 5 q^{83} + 18 q^{84} + 16 q^{85} - 20 q^{86} - 16 q^{87} + q^{88} + 7 q^{89} + 11 q^{90} - 6 q^{91} - 25 q^{92} + 24 q^{93} - 11 q^{94} + 39 q^{95} - 21 q^{96} + 35 q^{97} - 109 q^{98} + 8 q^{99}+O(q^{100}) \) 16 * q - q^2 + 16 * q^3 + 25 * q^4 + 3 * q^5 - q^6 + 13 * q^7 + 16 * q^9 + 11 * q^10 + 8 * q^11 + 25 * q^12 + 19 * q^13 + 16 * q^14 + 3 * q^15 + 31 * q^16 - 4 * q^17 - q^18 + 19 * q^19 + 16 * q^20 + 13 * q^21 + 6 * q^22 - 16 * q^23 + 23 * q^25 - 15 * q^26 + 16 * q^27 + 18 * q^28 - 16 * q^29 + 11 * q^30 + 24 * q^31 - 21 * q^32 + 8 * q^33 - 9 * q^34 - 13 * q^35 + 25 * q^36 + 26 * q^37 + 19 * q^39 - 22 * q^40 - 15 * q^41 + 16 * q^42 + 33 * q^43 + 6 * q^44 + 3 * q^45 + q^46 + 13 * q^47 + 31 * q^48 + 41 * q^49 + 13 * q^50 - 4 * q^51 - 26 * q^52 + 5 * q^53 - q^54 + 9 * q^55 + 40 * q^56 + 19 * q^57 + q^58 + 2 * q^59 + 16 * q^60 + 29 * q^61 - 32 * q^62 + 13 * q^63 + 28 * q^64 + 18 * q^65 + 6 * q^66 + 32 * q^67 - 26 * q^68 - 16 * q^69 + 18 * q^70 + 29 * q^71 + 19 * q^73 - 16 * q^74 + 23 * q^75 + 64 * q^76 - 21 * q^77 - 15 * q^78 + 56 * q^79 + 16 * q^81 + 14 * q^82 + 5 * q^83 + 18 * q^84 + 16 * q^85 - 20 * q^86 - 16 * q^87 + q^88 + 7 * q^89 + 11 * q^90 - 6 * q^91 - 25 * q^92 + 24 * q^93 - 11 * q^94 + 39 * q^95 - 21 * q^96 + 35 * q^97 - 109 * q^98 + 8 * q^99

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