LMFDB - Embedded newform 2006.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2006,2,Mod(1,2006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2006, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2006 = 2 \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2006.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:	$16.0179906455$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.17557$ of defining polynomial
Character	$\chi$	$=$	2006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.07768 q^{3} +1.00000 q^{4} +0.557537 q^{5} +3.07768 q^{6} +3.52015 q^{7} -1.00000 q^{8} +6.47214 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.07768 q^{3} +1.00000 q^{4} +0.557537 q^{5} +3.07768 q^{6} +3.52015 q^{7} -1.00000 q^{8} +6.47214 q^{9} -0.557537 q^{10} -0.442463 q^{11} -3.07768 q^{12} -2.71592 q^{13} -3.52015 q^{14} -1.71592 q^{15} +1.00000 q^{16} +1.00000 q^{17} -6.47214 q^{18} -6.31375 q^{19} +0.557537 q^{20} -10.8339 q^{21} +0.442463 q^{22} +3.08831 q^{23} +3.07768 q^{24} -4.68915 q^{25} +2.71592 q^{26} -10.6861 q^{27} +3.52015 q^{28} -4.34052 q^{29} +1.71592 q^{30} +2.52015 q^{31} -1.00000 q^{32} +1.36176 q^{33} -1.00000 q^{34} +1.96261 q^{35} +6.47214 q^{36} -6.47214 q^{37} +6.31375 q^{38} +8.35875 q^{39} -0.557537 q^{40} +2.90398 q^{41} +10.8339 q^{42} +10.2627 q^{43} -0.442463 q^{44} +3.60845 q^{45} -3.08831 q^{46} +4.17744 q^{47} -3.07768 q^{48} +5.39144 q^{49} +4.68915 q^{50} -3.07768 q^{51} -2.71592 q^{52} -5.25982 q^{53} +10.6861 q^{54} -0.246690 q^{55} -3.52015 q^{56} +19.4317 q^{57} +4.34052 q^{58} +1.00000 q^{59} -1.71592 q^{60} -3.28408 q^{61} -2.52015 q^{62} +22.7829 q^{63} +1.00000 q^{64} -1.51423 q^{65} -1.36176 q^{66} -6.61907 q^{67} +1.00000 q^{68} -9.50483 q^{69} -1.96261 q^{70} -10.8233 q^{71} -6.47214 q^{72} +6.47684 q^{73} +6.47214 q^{74} +14.4317 q^{75} -6.31375 q^{76} -1.55754 q^{77} -8.35875 q^{78} +1.10736 q^{79} +0.557537 q^{80} +13.4721 q^{81} -2.90398 q^{82} +9.14475 q^{83} -10.8339 q^{84} +0.557537 q^{85} -10.2627 q^{86} +13.3587 q^{87} +0.442463 q^{88} -2.72062 q^{89} -3.60845 q^{90} -9.56044 q^{91} +3.08831 q^{92} -7.75621 q^{93} -4.17744 q^{94} -3.52015 q^{95} +3.07768 q^{96} -14.5711 q^{97} -5.39144 q^{98} -2.86368 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} - 4 q^{8} + 8 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 - 4 * q^8 + 8 * q^9	$ 4 q - 4 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} - 4 q^{8} + 8 q^{9} - 2 q^{10} - 2 q^{11} - 14 q^{13} - 2 q^{14} - 10 q^{15} + 4 q^{16} + 4 q^{17} - 8 q^{18} - 4 q^{19} + 2 q^{20} - 10 q^{21} + 2 q^{22} - 6 q^{23} - 4 q^{25} + 14 q^{26} + 2 q^{28} - 14 q^{29} + 10 q^{30} - 2 q^{31} - 4 q^{32} - 10 q^{33} - 4 q^{34} - 4 q^{35} + 8 q^{36} - 8 q^{37} + 4 q^{38} - 10 q^{39} - 2 q^{40} + 10 q^{42} - 14 q^{43} - 2 q^{44} - 16 q^{45} + 6 q^{46} + 8 q^{47} - 12 q^{49} + 4 q^{50} - 14 q^{52} - 2 q^{53} + 14 q^{55} - 2 q^{56} + 20 q^{57} + 14 q^{58} + 4 q^{59} - 10 q^{60} - 10 q^{61} + 2 q^{62} + 24 q^{63} + 4 q^{64} - 2 q^{65} + 10 q^{66} + 10 q^{67} + 4 q^{68} - 30 q^{69} + 4 q^{70} - 16 q^{71} - 8 q^{72} + 10 q^{73} + 8 q^{74} - 4 q^{76} - 6 q^{77} + 10 q^{78} - 26 q^{79} + 2 q^{80} + 36 q^{81} + 18 q^{83} - 10 q^{84} + 2 q^{85} + 14 q^{86} + 10 q^{87} + 2 q^{88} - 16 q^{89} + 16 q^{90} - 2 q^{91} - 6 q^{92} - 10 q^{93} - 8 q^{94} - 2 q^{95} - 16 q^{97} + 12 q^{98} - 24 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 - 4 * q^8 + 8 * q^9 - 2 * q^10 - 2 * q^11 - 14 * q^13 - 2 * q^14 - 10 * q^15 + 4 * q^16 + 4 * q^17 - 8 * q^18 - 4 * q^19 + 2 * q^20 - 10 * q^21 + 2 * q^22 - 6 * q^23 - 4 * q^25 + 14 * q^26 + 2 * q^28 - 14 * q^29 + 10 * q^30 - 2 * q^31 - 4 * q^32 - 10 * q^33 - 4 * q^34 - 4 * q^35 + 8 * q^36 - 8 * q^37 + 4 * q^38 - 10 * q^39 - 2 * q^40 + 10 * q^42 - 14 * q^43 - 2 * q^44 - 16 * q^45 + 6 * q^46 + 8 * q^47 - 12 * q^49 + 4 * q^50 - 14 * q^52 - 2 * q^53 + 14 * q^55 - 2 * q^56 + 20 * q^57 + 14 * q^58 + 4 * q^59 - 10 * q^60 - 10 * q^61 + 2 * q^62 + 24 * q^63 + 4 * q^64 - 2 * q^65 + 10 * q^66 + 10 * q^67 + 4 * q^68 - 30 * q^69 + 4 * q^70 - 16 * q^71 - 8 * q^72 + 10 * q^73 + 8 * q^74 - 4 * q^76 - 6 * q^77 + 10 * q^78 - 26 * q^79 + 2 * q^80 + 36 * q^81 + 18 * q^83 - 10 * q^84 + 2 * q^85 + 14 * q^86 + 10 * q^87 + 2 * q^88 - 16 * q^89 + 16 * q^90 - 2 * q^91 - 6 * q^92 - 10 * q^93 - 8 * q^94 - 2 * q^95 - 16 * q^97 + 12 * q^98 - 24 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−3.07768	−1.77690	−0.888451	−	0.458972i	$-0.848218\pi$
$3$	−3.07768	−1.77690	−0.888451	+	0.458972i	$0.848218\pi$
$4$	1.00000	0.500000
$5$	0.557537	0.249338	0.124669	−	0.992198i	$-0.460213\pi$
$5$	0.557537	0.249338	0.124669	+	0.992198i	$0.460213\pi$
$6$	3.07768	1.25646
$7$	3.52015	1.33049	0.665245	−	0.746625i	$-0.268327\pi$
$7$	3.52015	1.33049	0.665245	+	0.746625i	$0.268327\pi$
$8$	−1.00000	−0.353553
$9$	6.47214	2.15738
$10$	−0.557537	−0.176309
$11$	−0.442463	−0.133408	−0.0667039	−	0.997773i	$-0.521248\pi$
$11$	−0.442463	−0.133408	−0.0667039	+	0.997773i	$0.521248\pi$
$12$	−3.07768	−0.888451
$13$	−2.71592	−0.753261	−0.376630	−	0.926364i	$-0.622917\pi$
$13$	−2.71592	−0.753261	−0.376630	+	0.926364i	$0.622917\pi$
$14$	−3.52015	−0.940799
$15$	−1.71592	−0.443049
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−6.47214	−1.52550
$19$	−6.31375	−1.44847	−0.724237	−	0.689551i	$-0.757808\pi$
$19$	−6.31375	−1.44847	−0.724237	+	0.689551i	$0.757808\pi$
$20$	0.557537	0.124669
$21$	−10.8339	−2.36415
$22$	0.442463	0.0943335
$23$	3.08831	0.643956	0.321978	−	0.946747i	$-0.395652\pi$
$23$	3.08831	0.643956	0.321978	+	0.946747i	$0.395652\pi$
$24$	3.07768	0.628230
$25$	−4.68915	−0.937831
$26$	2.71592	0.532636
$27$	−10.6861	−2.05655
$28$	3.52015	0.665245
$29$	−4.34052	−0.806014	−0.403007	−	0.915197i	$-0.632035\pi$
$29$	−4.34052	−0.806014	−0.403007	+	0.915197i	$0.632035\pi$
$30$	1.71592	0.313283
$31$	2.52015	0.452632	0.226316	−	0.974054i	$-0.427332\pi$
$31$	2.52015	0.452632	0.226316	+	0.974054i	$0.427332\pi$
$32$	−1.00000	−0.176777
$33$	1.36176	0.237052
$34$	−1.00000	−0.171499
$35$	1.96261	0.331742
$36$	6.47214	1.07869
$37$	−6.47214	−1.06401	−0.532006	−	0.846740i	$-0.678562\pi$
$37$	−6.47214	−1.06401	−0.532006	+	0.846740i	$0.678562\pi$
$38$	6.31375	1.02423
$39$	8.35875	1.33847
$40$	−0.557537	−0.0881543
$41$	2.90398	0.453525	0.226763	−	0.973950i	$-0.427186\pi$
$41$	2.90398	0.453525	0.226763	+	0.973950i	$0.427186\pi$
$42$	10.8339	1.67171
$43$	10.2627	1.56505	0.782525	−	0.622619i	$-0.213931\pi$
$43$	10.2627	1.56505	0.782525	+	0.622619i	$0.213931\pi$
$44$	−0.442463	−0.0667039
$45$	3.60845	0.537916
$46$	−3.08831	−0.455346
$47$	4.17744	0.609342	0.304671	−	0.952458i	$-0.401454\pi$
$47$	4.17744	0.609342	0.304671	+	0.952458i	$0.401454\pi$
$48$	−3.07768	−0.444225
$49$	5.39144	0.770205
$50$	4.68915	0.663146
$51$	−3.07768	−0.430962
$52$	−2.71592	−0.376630
$53$	−5.25982	−0.722492	−0.361246	−	0.932471i	$-0.617648\pi$
$53$	−5.25982	−0.722492	−0.361246	+	0.932471i	$0.617648\pi$
$54$	10.6861	1.45420
$55$	−0.246690	−0.0332636
$56$	−3.52015	−0.470399
$57$	19.4317	2.57380
$58$	4.34052	0.569938
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	−1.71592	−0.221524
$61$	−3.28408	−0.420483	−0.210242	−	0.977649i	$-0.567425\pi$
$61$	−3.28408	−0.420483	−0.210242	+	0.977649i	$0.567425\pi$
$62$	−2.52015	−0.320059
$63$	22.7829	2.87037
$64$	1.00000	0.125000
$65$	−1.51423	−0.187817
$66$	−1.36176	−0.167621
$67$	−6.61907	−0.808649	−0.404324	−	0.914616i	$-0.632493\pi$
$67$	−6.61907	−0.808649	−0.404324	+	0.914616i	$0.632493\pi$
$68$	1.00000	0.121268
$69$	−9.50483	−1.14425
$70$	−1.96261	−0.234577
$71$	−10.8233	−1.28449	−0.642243	−	0.766501i	$-0.721996\pi$
$71$	−10.8233	−1.28449	−0.642243	+	0.766501i	$0.721996\pi$
$72$	−6.47214	−0.762749
$73$	6.47684	0.758056	0.379028	−	0.925385i	$-0.376258\pi$
$73$	6.47684	0.758056	0.379028	+	0.925385i	$0.376258\pi$
$74$	6.47214	0.752371
$75$	14.4317	1.66643
$76$	−6.31375	−0.724237
$77$	−1.55754	−0.177498
$78$	−8.35875	−0.946442
$79$	1.10736	0.124587	0.0622936	−	0.998058i	$-0.480158\pi$
$79$	1.10736	0.124587	0.0622936	+	0.998058i	$0.480158\pi$
$80$	0.557537	0.0623345
$81$	13.4721	1.49690
$82$	−2.90398	−0.320691
$83$	9.14475	1.00377	0.501883	−	0.864935i	$-0.332641\pi$
$83$	9.14475	1.00377	0.501883	+	0.864935i	$0.332641\pi$
$84$	−10.8339	−1.18208
$85$	0.557537	0.0604733
$86$	−10.2627	−1.10666
$87$	13.3587	1.43221
$88$	0.442463	0.0471668
$89$	−2.72062	−0.288385	−0.144193	−	0.989550i	$-0.546058\pi$
$89$	−2.72062	−0.288385	−0.144193	+	0.989550i	$0.546058\pi$
$90$	−3.60845	−0.380364
$91$	−9.56044	−1.00221
$92$	3.08831	0.321978
$93$	−7.75621	−0.804282
$94$	−4.17744	−0.430870
$95$	−3.52015	−0.361159
$96$	3.07768	0.314115
$97$	−14.5711	−1.47947	−0.739734	−	0.672900i	$-0.765048\pi$
$97$	−14.5711	−1.47947	−0.739734	+	0.672900i	$0.765048\pi$
$98$	−5.39144	−0.544617
$99$	−2.86368	−0.287811
$100$	−4.68915	−0.468915
$101$	4.18504	0.416427	0.208213	−	0.978083i	$-0.433235\pi$
$101$	4.18504	0.416427	0.208213	+	0.978083i	$0.433235\pi$
$102$	3.07768	0.304736
$103$	−15.3138	−1.50891	−0.754454	−	0.656352i	$-0.772098\pi$
$103$	−15.3138	−1.50891	−0.754454	+	0.656352i	$0.772098\pi$
$104$	2.71592	0.266318
$105$	−6.04029	−0.589472
$106$	5.25982	0.510879
$107$	10.5439	1.01932	0.509659	−	0.860377i	$-0.329772\pi$
$107$	10.5439	1.01932	0.509659	+	0.860377i	$0.329772\pi$
$108$	−10.6861	−1.02827
$109$	−1.53629	−0.147150	−0.0735751	−	0.997290i	$-0.523441\pi$
$109$	−1.53629	−0.147150	−0.0735751	+	0.997290i	$0.523441\pi$
$110$	0.246690	0.0235209
$111$	19.9192	1.89065
$112$	3.52015	0.332623
$113$	6.49047	0.610572	0.305286	−	0.952261i	$-0.401248\pi$
$113$	6.49047	0.610572	0.305286	+	0.952261i	$0.401248\pi$
$114$	−19.4317	−1.81995
$115$	1.72184	0.160563
$116$	−4.34052	−0.403007
$117$	−17.5778	−1.62507
$118$	−1.00000	−0.0920575
$119$	3.52015	0.322691
$120$	1.71592	0.156641
$121$	−10.8042	−0.982202
$122$	3.28408	0.297327
$123$	−8.93752	−0.805869
$124$	2.52015	0.226316
$125$	−5.40206	−0.483175
$126$	−22.7829	−2.02966
$127$	11.0730	0.982568	0.491284	−	0.870999i	$-0.336528\pi$
$127$	11.0730	0.982568	0.491284	+	0.870999i	$0.336528\pi$
$128$	−1.00000	−0.0883883
$129$	−31.5854	−2.78094
$130$	1.51423	0.132806
$131$	6.33801	0.553755	0.276877	−	0.960905i	$-0.410700\pi$
$131$	6.33801	0.553755	0.276877	+	0.960905i	$0.410700\pi$
$132$	1.36176	0.118526
$133$	−22.2253	−1.92718
$134$	6.61907	0.571801
$135$	−5.95791	−0.512775
$136$	−1.00000	−0.0857493
$137$	−6.92522	−0.591662	−0.295831	−	0.955240i	$-0.595596\pi$
$137$	−6.92522	−0.591662	−0.295831	+	0.955240i	$0.595596\pi$
$138$	9.50483	0.809105
$139$	−20.7081	−1.75644	−0.878219	−	0.478259i	$-0.841268\pi$
$139$	−20.7081	−1.75644	−0.878219	+	0.478259i	$0.841268\pi$
$140$	1.96261	0.165871
$141$	−12.8568	−1.08274
$142$	10.8233	0.908269
$143$	1.20170	0.100491
$144$	6.47214	0.539345
$145$	−2.42000	−0.200970
$146$	−6.47684	−0.536027
$147$	−16.5931	−1.36858
$148$	−6.47214	−0.532006
$149$	−12.0160	−0.984392	−0.492196	−	0.870484i	$-0.663806\pi$
$149$	−12.0160	−0.984392	−0.492196	+	0.870484i	$0.663806\pi$
$150$	−14.4317	−1.17835
$151$	−1.73207	−0.140954	−0.0704768	−	0.997513i	$-0.522452\pi$
$151$	−1.73207	−0.140954	−0.0704768	+	0.997513i	$0.522452\pi$
$152$	6.31375	0.512113
$153$	6.47214	0.523241
$154$	1.55754	0.125510
$155$	1.40507	0.112858
$156$	8.35875	0.669235
$157$	−10.7099	−0.854742	−0.427371	−	0.904076i	$-0.640560\pi$
$157$	−10.7099	−0.854742	−0.427371	+	0.904076i	$0.640560\pi$
$158$	−1.10736	−0.0880965
$159$	16.1881	1.28380
$160$	−0.557537	−0.0440771
$161$	10.8713	0.856778
$162$	−13.4721	−1.05847
$163$	−3.52786	−0.276324	−0.138162	−	0.990410i	$-0.544119\pi$
$163$	−3.52786	−0.276324	−0.138162	+	0.990410i	$0.544119\pi$
$164$	2.90398	0.226763
$165$	0.759232	0.0591062
$166$	−9.14475	−0.709770
$167$	4.18806	0.324082	0.162041	−	0.986784i	$-0.448192\pi$
$167$	4.18806	0.324082	0.162041	+	0.986784i	$0.448192\pi$
$168$	10.8339	0.835853
$169$	−5.62377	−0.432598
$170$	−0.557537	−0.0427611
$171$	−40.8635	−3.12491
$172$	10.2627	0.782525
$173$	2.40758	0.183045	0.0915225	−	0.995803i	$-0.470827\pi$
$173$	2.40758	0.183045	0.0915225	+	0.995803i	$0.470827\pi$
$174$	−13.3587	−1.01272
$175$	−16.5065	−1.24777
$176$	−0.442463	−0.0333519
$177$	−3.07768	−0.231333
$178$	2.72062	0.203919
$179$	−13.1749	−0.984740	−0.492370	−	0.870386i	$-0.663869\pi$
$179$	−13.1749	−0.984740	−0.492370	+	0.870386i	$0.663869\pi$
$180$	3.60845	0.268958
$181$	−19.0165	−1.41349	−0.706744	−	0.707469i	$-0.749837\pi$
$181$	−19.0165	−1.41349	−0.706744	+	0.707469i	$0.749837\pi$
$182$	9.56044	0.708667
$183$	10.1074	0.747157
$184$	−3.08831	−0.227673
$185$	−3.60845	−0.265299
$186$	7.75621	0.568713
$187$	−0.442463	−0.0323561
$188$	4.17744	0.304671
$189$	−37.6168	−2.73622
$190$	3.52015	0.255378
$191$	14.0585	1.01724	0.508619	−	0.860991i	$-0.330156\pi$
$191$	14.0585	1.01724	0.508619	+	0.860991i	$0.330156\pi$
$192$	−3.07768	−0.222113
$193$	1.47214	0.105967	0.0529833	−	0.998595i	$-0.483127\pi$
$193$	1.47214	0.105967	0.0529833	+	0.998595i	$0.483127\pi$
$194$	14.5711	1.04614
$195$	4.66031	0.333731
$196$	5.39144	0.385103
$197$	7.91209	0.563713	0.281857	−	0.959457i	$-0.409050\pi$
$197$	7.91209	0.563713	0.281857	+	0.959457i	$0.409050\pi$
$198$	2.86368	0.203513
$199$	−20.3243	−1.44075	−0.720374	−	0.693586i	$-0.756030\pi$
$199$	−20.3243	−1.44075	−0.720374	+	0.693586i	$0.756030\pi$
$200$	4.68915	0.331573
$201$	20.3714	1.43689
$202$	−4.18504	−0.294458
$203$	−15.2793	−1.07239
$204$	−3.07768	−0.215481
$205$	1.61907	0.113081
$206$	15.3138	1.06696
$207$	19.9879	1.38926
$208$	−2.71592	−0.188315
$209$	2.79360	0.193238
$210$	6.04029	0.416820
$211$	21.7667	1.49848	0.749242	−	0.662297i	$-0.230418\pi$
$211$	21.7667	1.49848	0.749242	+	0.662297i	$0.230418\pi$
$212$	−5.25982	−0.361246
$213$	33.3106	2.28241
$214$	−10.5439	−0.720766
$215$	5.72184	0.390226
$216$	10.6861	0.727099
$217$	8.87129	0.602222
$218$	1.53629	0.104051
$219$	−19.9337	−1.34699
$220$	−0.246690	−0.0166318
$221$	−2.71592	−0.182693
$222$	−19.9192	−1.33689
$223$	9.34393	0.625716	0.312858	−	0.949800i	$-0.398714\pi$
$223$	9.34393	0.625716	0.312858	+	0.949800i	$0.398714\pi$
$224$	−3.52015	−0.235200
$225$	−30.3488	−2.02326
$226$	−6.49047	−0.431740
$227$	−16.3677	−1.08636	−0.543181	−	0.839616i	$-0.682780\pi$
$227$	−16.3677	−1.08636	−0.543181	+	0.839616i	$0.682780\pi$
$228$	19.4317	1.28690
$229$	−10.7983	−0.713572	−0.356786	−	0.934186i	$-0.616127\pi$
$229$	−10.7983	−0.713572	−0.356786	+	0.934186i	$0.616127\pi$
$230$	−1.72184	−0.113535
$231$	4.79360	0.315396
$232$	4.34052	0.284969
$233$	14.7994	0.969542	0.484771	−	0.874641i	$-0.338903\pi$
$233$	14.7994	0.969542	0.484771	+	0.874641i	$0.338903\pi$
$234$	17.5778	1.14910
$235$	2.32907	0.151932
$236$	1.00000	0.0650945
$237$	−3.40809	−0.221379
$238$	−3.52015	−0.228177
$239$	−8.56044	−0.553729	−0.276864	−	0.960909i	$-0.589295\pi$
$239$	−8.56044	−0.553729	−0.276864	+	0.960909i	$0.589295\pi$
$240$	−1.71592	−0.110762
$241$	−2.61218	−0.168265	−0.0841327	−	0.996455i	$-0.526812\pi$
$241$	−2.61218	−0.168265	−0.0841327	+	0.996455i	$0.526812\pi$
$242$	10.8042	0.694522
$243$	−9.40456	−0.603303
$244$	−3.28408	−0.210242
$245$	3.00592	0.192041
$246$	8.93752	0.569836
$247$	17.1477	1.09108
$248$	−2.52015	−0.160029
$249$	−28.1446	−1.78359
$250$	5.40206	0.341656
$251$	−0.555741	−0.0350781	−0.0175390	−	0.999846i	$-0.505583\pi$
$251$	−0.555741	−0.0350781	−0.0175390	+	0.999846i	$0.505583\pi$
$252$	22.7829	1.43519
$253$	−1.36646	−0.0859087
$254$	−11.0730	−0.694781
$255$	−1.71592	−0.107455
$256$	1.00000	0.0625000
$257$	−21.7829	−1.35878	−0.679389	−	0.733778i	$-0.737755\pi$
$257$	−21.7829	−1.35878	−0.679389	+	0.733778i	$0.737755\pi$
$258$	31.5854	1.96642
$259$	−22.7829	−1.41566
$260$	−1.51423	−0.0939083
$261$	−28.0924	−1.73888
$262$	−6.33801	−0.391564
$263$	16.6161	1.02459	0.512295	−	0.858810i	$-0.328795\pi$
$263$	16.6161	1.02459	0.512295	+	0.858810i	$0.328795\pi$
$264$	−1.36176	−0.0838107
$265$	−2.93254	−0.180145
$266$	22.2253	1.36272
$267$	8.37321	0.512432
$268$	−6.61907	−0.404324
$269$	−26.6215	−1.62314	−0.811570	−	0.584255i	$-0.801387\pi$
$269$	−26.6215	−1.62314	−0.811570	+	0.584255i	$0.801387\pi$
$270$	5.95791	0.362587
$271$	−24.4394	−1.48459	−0.742295	−	0.670073i	$-0.766263\pi$
$271$	−24.4394	−1.48459	−0.742295	+	0.670073i	$0.766263\pi$
$272$	1.00000	0.0606339
$273$	29.4240	1.78082
$274$	6.92522	0.418368
$275$	2.07478	0.125114
$276$	−9.50483	−0.572123
$277$	−5.13754	−0.308685	−0.154342	−	0.988017i	$-0.549326\pi$
$277$	−5.13754	−0.308685	−0.154342	+	0.988017i	$0.549326\pi$
$278$	20.7081	1.24199
$279$	16.3107	0.976498
$280$	−1.96261	−0.117288
$281$	19.5527	1.16642	0.583209	−	0.812322i	$-0.301797\pi$
$281$	19.5527	1.16642	0.583209	+	0.812322i	$0.301797\pi$
$282$	12.8568	0.765613
$283$	13.2012	0.784729	0.392365	−	0.919810i	$-0.371657\pi$
$283$	13.2012	0.784729	0.392365	+	0.919810i	$0.371657\pi$
$284$	−10.8233	−0.642243
$285$	10.8339	0.641745
$286$	−1.20170	−0.0710578
$287$	10.2224	0.603411
$288$	−6.47214	−0.381374
$289$	1.00000	0.0588235
$290$	2.42000	0.142107
$291$	44.8451	2.62887
$292$	6.47684	0.379028
$293$	−3.37708	−0.197291	−0.0986457	−	0.995123i	$-0.531451\pi$
$293$	−3.37708	−0.197291	−0.0986457	+	0.995123i	$0.531451\pi$
$294$	16.5931	0.967731
$295$	0.557537	0.0324610
$296$	6.47214	0.376185
$297$	4.72822	0.274359
$298$	12.0160	0.696070
$299$	−8.38759	−0.485067
$300$	14.4317	0.833216
$301$	36.1263	2.08228
$302$	1.73207	0.0996693
$303$	−12.8802	−0.739950
$304$	−6.31375	−0.362118
$305$	−1.83099	−0.104842
$306$	−6.47214	−0.369987
$307$	29.7477	1.69779	0.848895	−	0.528562i	$-0.177269\pi$
$307$	29.7477	1.69779	0.848895	+	0.528562i	$0.177269\pi$
$308$	−1.55754	−0.0887489
$309$	47.1309	2.68118
$310$	−1.40507	−0.0798028
$311$	−15.9425	−0.904015	−0.452007	−	0.892014i	$-0.649292\pi$
$311$	−15.9425	−0.904015	−0.452007	+	0.892014i	$0.649292\pi$
$312$	−8.35875	−0.473221
$313$	−19.8522	−1.12211	−0.561057	−	0.827777i	$-0.689605\pi$
$313$	−19.8522	−1.12211	−0.561057	+	0.827777i	$0.689605\pi$
$314$	10.7099	0.604394
$315$	12.7023	0.715693
$316$	1.10736	0.0622936
$317$	10.8539	0.609617	0.304808	−	0.952414i	$-0.401408\pi$
$317$	10.8539	0.609617	0.304808	+	0.952414i	$0.401408\pi$
$318$	−16.1881	−0.907781
$319$	1.92052	0.107529
$320$	0.557537	0.0311672
$321$	−32.4508	−1.81123
$322$	−10.8713	−0.605833
$323$	−6.31375	−0.351307
$324$	13.4721	0.748452
$325$	12.7354	0.706431
$326$	3.52786	0.195390
$327$	4.72822	0.261471
$328$	−2.90398	−0.160345
$329$	14.7052	0.810723
$330$	−0.759232	−0.0417944
$331$	24.8546	1.36613	0.683067	−	0.730356i	$-0.260646\pi$
$331$	24.8546	1.36613	0.683067	+	0.730356i	$0.260646\pi$
$332$	9.14475	0.501883
$333$	−41.8885	−2.29548
$334$	−4.18806	−0.229160
$335$	−3.69038	−0.201627
$336$	−10.8339	−0.591038
$337$	−13.8066	−0.752095	−0.376047	−	0.926600i	$-0.622717\pi$
$337$	−13.8066	−0.752095	−0.376047	+	0.926600i	$0.622717\pi$
$338$	5.62377	0.305893
$339$	−19.9756	−1.08493
$340$	0.557537	0.0302367
$341$	−1.11507	−0.0603846
$342$	40.8635	2.20964
$343$	−5.66239	−0.305740
$344$	−10.2627	−0.553329
$345$	−5.29929	−0.285304
$346$	−2.40758	−0.129432
$347$	−25.7935	−1.38467	−0.692334	−	0.721578i	$-0.743417\pi$
$347$	−25.7935	−1.38467	−0.692334	+	0.721578i	$0.743417\pi$
$348$	13.3587	0.716104
$349$	−12.3168	−0.659302	−0.329651	−	0.944103i	$-0.606931\pi$
$349$	−12.3168	−0.659302	−0.329651	+	0.944103i	$0.606931\pi$
$350$	16.5065	0.882310
$351$	29.0227	1.54912
$352$	0.442463	0.0235834
$353$	−17.2550	−0.918391	−0.459196	−	0.888335i	$-0.651862\pi$
$353$	−17.2550	−0.918391	−0.459196	+	0.888335i	$0.651862\pi$
$354$	3.07768	0.163577
$355$	−6.03437	−0.320271
$356$	−2.72062	−0.144193
$357$	−10.8339	−0.573391
$358$	13.1749	0.696317
$359$	−30.3896	−1.60390	−0.801952	−	0.597389i	$-0.796205\pi$
$359$	−30.3896	−1.60390	−0.801952	+	0.597389i	$0.796205\pi$
$360$	−3.60845	−0.190182
$361$	20.8635	1.09808
$362$	19.0165	0.999487
$363$	33.2520	1.74528
$364$	−9.56044	−0.501103
$365$	3.61107	0.189012
$366$	−10.1074	−0.528320
$367$	−33.5552	−1.75157	−0.875784	−	0.482702i	$-0.839655\pi$
$367$	−33.5552	−1.75157	−0.875784	+	0.482702i	$0.839655\pi$
$368$	3.08831	0.160989
$369$	18.7949	0.978425
$370$	3.60845	0.187594
$371$	−18.5153	−0.961268
$372$	−7.75621	−0.402141
$373$	22.6346	1.17198	0.585988	−	0.810320i	$-0.300707\pi$
$373$	22.6346	1.17198	0.585988	+	0.810320i	$0.300707\pi$
$374$	0.442463	0.0228792
$375$	16.6258	0.858554
$376$	−4.17744	−0.215435
$377$	11.7885	0.607139
$378$	37.6168	1.93480
$379$	−36.3826	−1.86885	−0.934425	−	0.356159i	$-0.884086\pi$
$379$	−36.3826	−1.86885	−0.934425	+	0.356159i	$0.884086\pi$
$380$	−3.52015	−0.180580
$381$	−34.0791	−1.74593
$382$	−14.0585	−0.719296
$383$	5.56970	0.284598	0.142299	−	0.989824i	$-0.454551\pi$
$383$	5.56970	0.284598	0.142299	+	0.989824i	$0.454551\pi$
$384$	3.07768	0.157057
$385$	−0.868383	−0.0442569
$386$	−1.47214	−0.0749297
$387$	66.4217	3.37641
$388$	−14.5711	−0.739734
$389$	36.6787	1.85969	0.929843	−	0.367957i	$-0.119943\pi$
$389$	36.6787	1.85969	0.929843	+	0.367957i	$0.119943\pi$
$390$	−4.66031	−0.235984
$391$	3.08831	0.156182
$392$	−5.39144	−0.272309
$393$	−19.5064	−0.983967
$394$	−7.91209	−0.398605
$395$	0.617391	0.0310643
$396$	−2.86368	−0.143906
$397$	−24.5960	−1.23444	−0.617220	−	0.786791i	$-0.711741\pi$
$397$	−24.5960	−1.23444	−0.617220	+	0.786791i	$0.711741\pi$
$398$	20.3243	1.01876
$399$	68.4025	3.42441
$400$	−4.68915	−0.234458
$401$	1.89606	0.0946846	0.0473423	−	0.998879i	$-0.484925\pi$
$401$	1.89606	0.0946846	0.0473423	+	0.998879i	$0.484925\pi$
$402$	−20.3714	−1.01603
$403$	−6.84452	−0.340950
$404$	4.18504	0.208213
$405$	7.51121	0.373235
$406$	15.2793	0.758297
$407$	2.86368	0.141948
$408$	3.07768	0.152368
$409$	−23.5247	−1.16322	−0.581612	−	0.813467i	$-0.697578\pi$
$409$	−23.5247	−1.16322	−0.581612	+	0.813467i	$0.697578\pi$
$410$	−1.61907	−0.0799604
$411$	21.3136	1.05132
$412$	−15.3138	−0.754454
$413$	3.52015	0.173215
$414$	−19.9879	−0.982353
$415$	5.09853	0.250277
$416$	2.71592	0.133159
$417$	63.7330	3.12102
$418$	−2.79360	−0.136640
$419$	−19.7192	−0.963347	−0.481674	−	0.876351i	$-0.659971\pi$
$419$	−19.7192	−0.963347	−0.481674	+	0.876351i	$0.659971\pi$
$420$	−6.04029	−0.294736
$421$	9.55574	0.465718	0.232859	−	0.972510i	$-0.425192\pi$
$421$	9.55574	0.465718	0.232859	+	0.972510i	$0.425192\pi$
$422$	−21.7667	−1.05959
$423$	27.0369	1.31458
$424$	5.25982	0.255439
$425$	−4.68915	−0.227457
$426$	−33.3106	−1.61390
$427$	−11.5604	−0.559449
$428$	10.5439	0.509659
$429$	−3.69844	−0.178562
$430$	−5.72184	−0.275932
$431$	−14.0155	−0.675104	−0.337552	−	0.941307i	$-0.609599\pi$
$431$	−14.0155	−0.675104	−0.337552	+	0.941307i	$0.609599\pi$
$432$	−10.6861	−0.514137
$433$	31.4590	1.51182	0.755912	−	0.654674i	$-0.227194\pi$
$433$	31.4590	1.51182	0.755912	+	0.654674i	$0.227194\pi$
$434$	−8.87129	−0.425835
$435$	7.44799	0.357104
$436$	−1.53629	−0.0735751
$437$	−19.4988	−0.932754
$438$	19.9337	0.952467
$439$	−19.3123	−0.921724	−0.460862	−	0.887472i	$-0.652460\pi$
$439$	−19.3123	−0.921724	−0.460862	+	0.887472i	$0.652460\pi$
$440$	0.246690	0.0117605
$441$	34.8941	1.66162
$442$	2.71592	0.129183
$443$	17.5095	0.831903	0.415951	−	0.909387i	$-0.363449\pi$
$443$	17.5095	0.831903	0.415951	+	0.909387i	$0.363449\pi$
$444$	19.9192	0.945323
$445$	−1.51685	−0.0719054
$446$	−9.34393	−0.442448
$447$	36.9815	1.74917
$448$	3.52015	0.166311
$449$	−1.79702	−0.0848066	−0.0424033	−	0.999101i	$-0.513501\pi$
$449$	−1.79702	−0.0848066	−0.0424033	+	0.999101i	$0.513501\pi$
$450$	30.3488	1.43066
$451$	−1.28490	−0.0605038
$452$	6.49047	0.305286
$453$	5.33076	0.250461
$454$	16.3677	0.768173
$455$	−5.33030	−0.249888
$456$	−19.4317	−0.909974
$457$	−13.6894	−0.640361	−0.320181	−	0.947357i	$-0.603744\pi$
$457$	−13.6894	−0.640361	−0.320181	+	0.947357i	$0.603744\pi$
$458$	10.7983	0.504572
$459$	−10.6861	−0.498786
$460$	1.72184	0.0802813
$461$	10.5409	0.490938	0.245469	−	0.969404i	$-0.421058\pi$
$461$	10.5409	0.490938	0.245469	+	0.969404i	$0.421058\pi$
$462$	−4.79360	−0.223019
$463$	20.9710	0.974607	0.487303	−	0.873233i	$-0.337981\pi$
$463$	20.9710	0.974607	0.487303	+	0.873233i	$0.337981\pi$
$464$	−4.34052	−0.201504
$465$	−4.32437	−0.200538
$466$	−14.7994	−0.685570
$467$	7.51943	0.347958	0.173979	−	0.984749i	$-0.444338\pi$
$467$	7.51943	0.347958	0.173979	+	0.984749i	$0.444338\pi$
$468$	−17.5778	−0.812535
$469$	−23.3001	−1.07590
$470$	−2.32907	−0.107432
$471$	32.9616	1.51879
$472$	−1.00000	−0.0460287
$473$	−4.54088	−0.208790
$474$	3.40809	0.156539
$475$	29.6061	1.35842
$476$	3.52015	0.161346
$477$	−34.0423	−1.55869
$478$	8.56044	0.391545
$479$	33.5586	1.53333	0.766667	−	0.642045i	$-0.221914\pi$
$479$	33.5586	1.53333	0.766667	+	0.642045i	$0.221914\pi$
$480$	1.71592	0.0783207
$481$	17.5778	0.801479
$482$	2.61218	0.118982
$483$	−33.4584	−1.52241
$484$	−10.8042	−0.491101
$485$	−8.12390	−0.368887
$486$	9.40456	0.426600
$487$	−12.4665	−0.564911	−0.282455	−	0.959280i	$-0.591149\pi$
$487$	−12.4665	−0.564911	−0.282455	+	0.959280i	$0.591149\pi$
$488$	3.28408	0.148663
$489$	10.8576	0.491000
$490$	−3.00592	−0.135794
$491$	22.9912	1.03758	0.518789	−	0.854902i	$-0.326383\pi$
$491$	22.9912	1.03758	0.518789	+	0.854902i	$0.326383\pi$
$492$	−8.93752	−0.402935
$493$	−4.34052	−0.195487
$494$	−17.1477	−0.771509
$495$	−1.59661	−0.0717622
$496$	2.52015	0.113158
$497$	−38.0995	−1.70900
$498$	28.1446	1.26119
$499$	29.4771	1.31958	0.659789	−	0.751451i	$-0.270646\pi$
$499$	29.4771	1.31958	0.659789	+	0.751451i	$0.270646\pi$
$500$	−5.40206	−0.241587
$501$	−12.8895	−0.575861
$502$	0.555741	0.0248040
$503$	−33.2644	−1.48319	−0.741593	−	0.670850i	$-0.765930\pi$
$503$	−33.2644	−1.48319	−0.741593	+	0.670850i	$0.765930\pi$
$504$	−22.7829	−1.01483
$505$	2.33331	0.103831
$506$	1.36646	0.0607467
$507$	17.3082	0.768684
$508$	11.0730	0.491284
$509$	16.8713	0.747807	0.373903	−	0.927468i	$-0.378019\pi$
$509$	16.8713	0.747807	0.373903	+	0.927468i	$0.378019\pi$
$510$	1.71592	0.0759823
$511$	22.7994	1.00859
$512$	−1.00000	−0.0441942
$513$	67.4696	2.97886
$514$	21.7829	0.960801
$515$	−8.53798	−0.376228
$516$	−31.5854	−1.39047
$517$	−1.84836	−0.0812909
$518$	22.7829	1.00102
$519$	−7.40977	−0.325253
$520$	1.51423	0.0664032
$521$	−29.2550	−1.28169	−0.640843	−	0.767672i	$-0.721415\pi$
$521$	−29.2550	−1.28169	−0.640843	+	0.767672i	$0.721415\pi$
$522$	28.0924	1.22957
$523$	6.33730	0.277111	0.138555	−	0.990355i	$-0.455754\pi$
$523$	6.33730	0.277111	0.138555	+	0.990355i	$0.455754\pi$
$524$	6.33801	0.276877
$525$	50.8018	2.21717
$526$	−16.6161	−0.724494
$527$	2.52015	0.109779
$528$	1.36176	0.0592631
$529$	−13.4624	−0.585321
$530$	2.93254	0.127381
$531$	6.47214	0.280867
$532$	−22.2253	−0.963590
$533$	−7.88697	−0.341623
$534$	−8.37321	−0.362344
$535$	5.87861	0.254154
$536$	6.61907	0.285900
$537$	40.5483	1.74979
$538$	26.6215	1.14773
$539$	−2.38551	−0.102751
$540$	−5.95791	−0.256388
$541$	−22.8925	−0.984227	−0.492113	−	0.870531i	$-0.663775\pi$
$541$	−22.8925	−0.984227	−0.492113	+	0.870531i	$0.663775\pi$
$542$	24.4394	1.04976
$543$	58.5269	2.51163
$544$	−1.00000	−0.0428746
$545$	−0.856540	−0.0366901
$546$	−29.4240	−1.25923
$547$	14.0827	0.602131	0.301066	−	0.953603i	$-0.402658\pi$
$547$	14.0827	0.602131	0.301066	+	0.953603i	$0.402658\pi$
$548$	−6.92522	−0.295831
$549$	−21.2550	−0.907142
$550$	−2.07478	−0.0884689
$551$	27.4050	1.16749
$552$	9.50483	0.404552
$553$	3.89806	0.165762
$554$	5.13754	0.218273
$555$	11.1057	0.471410
$556$	−20.7081	−0.878219
$557$	−12.0891	−0.512233	−0.256116	−	0.966646i	$-0.582443\pi$
$557$	−12.0891	−0.512233	−0.256116	+	0.966646i	$0.582443\pi$
$558$	−16.3107	−0.690488
$559$	−27.8727	−1.17889
$560$	1.96261	0.0829354
$561$	1.36176	0.0574937
$562$	−19.5527	−0.824782
$563$	−42.1865	−1.77795	−0.888974	−	0.457958i	$-0.848581\pi$
$563$	−42.1865	−1.77795	−0.888974	+	0.457958i	$0.848581\pi$
$564$	−12.8568	−0.541370
$565$	3.61868	0.152239
$566$	−13.2012	−0.554887
$567$	47.4239	1.99162
$568$	10.8233	0.454135
$569$	−11.9573	−0.501276	−0.250638	−	0.968081i	$-0.580640\pi$
$569$	−11.9573	−0.501276	−0.250638	+	0.968081i	$0.580640\pi$
$570$	−10.8339	−0.453782
$571$	13.7085	0.573684	0.286842	−	0.957978i	$-0.407395\pi$
$571$	13.7085	0.573684	0.286842	+	0.957978i	$0.407395\pi$
$572$	1.20170	0.0502454
$573$	−43.2677	−1.80753
$574$	−10.2224	−0.426676
$575$	−14.4815	−0.603922
$576$	6.47214	0.269672
$577$	−10.6620	−0.443864	−0.221932	−	0.975062i	$-0.571236\pi$
$577$	−10.6620	−0.443864	−0.221932	+	0.975062i	$0.571236\pi$
$578$	−1.00000	−0.0415945
$579$	−4.53077	−0.188292
$580$	−2.42000	−0.100485
$581$	32.1908	1.33550
$582$	−44.8451	−1.85889
$583$	2.32728	0.0963860
$584$	−6.47684	−0.268013
$585$	−9.80027	−0.405191
$586$	3.37708	0.139506
$587$	19.4441	0.802546	0.401273	−	0.915959i	$-0.368568\pi$
$587$	19.4441	0.802546	0.401273	+	0.915959i	$0.368568\pi$
$588$	−16.5931	−0.684289
$589$	−15.9116	−0.655625
$590$	−0.557537	−0.0229534
$591$	−24.3509	−1.00166
$592$	−6.47214	−0.266003
$593$	45.2430	1.85791	0.928955	−	0.370193i	$-0.120709\pi$
$593$	45.2430	1.85791	0.928955	+	0.370193i	$0.120709\pi$
$594$	−4.72822	−0.194001
$595$	1.96261	0.0804592
$596$	−12.0160	−0.492196
$597$	62.5516	2.56007
$598$	8.38759	0.342994
$599$	30.4263	1.24319	0.621593	−	0.783341i	$-0.286486\pi$
$599$	30.4263	1.24319	0.621593	+	0.783341i	$0.286486\pi$
$600$	−14.4317	−0.589173
$601$	−45.4377	−1.85344	−0.926720	−	0.375752i	$-0.877384\pi$
$601$	−45.4377	−1.85344	−0.926720	+	0.375752i	$0.877384\pi$
$602$	−36.1263	−1.47240
$603$	−42.8395	−1.74456
$604$	−1.73207	−0.0704768
$605$	−6.02375	−0.244900
$606$	12.8802	0.523223
$607$	−19.8286	−0.804817	−0.402409	−	0.915460i	$-0.631827\pi$
$607$	−19.8286	−0.804817	−0.402409	+	0.915460i	$0.631827\pi$
$608$	6.31375	0.256056
$609$	47.0247	1.90554
$610$	1.83099	0.0741348
$611$	−11.3456	−0.458993
$612$	6.47214	0.261621
$613$	46.2307	1.86724	0.933621	−	0.358262i	$-0.116631\pi$
$613$	46.2307	1.86724	0.933621	+	0.358262i	$0.116631\pi$
$614$	−29.7477	−1.20052
$615$	−4.98300	−0.200934
$616$	1.55754	0.0627549
$617$	−17.8445	−0.718393	−0.359197	−	0.933262i	$-0.616949\pi$
$617$	−17.8445	−0.718393	−0.359197	+	0.933262i	$0.616949\pi$
$618$	−47.1309	−1.89588
$619$	−11.6186	−0.466990	−0.233495	−	0.972358i	$-0.575016\pi$
$619$	−11.6186	−0.466990	−0.233495	+	0.972358i	$0.575016\pi$
$620$	1.40507	0.0564291
$621$	−33.0020	−1.32433
$622$	15.9425	0.639235
$623$	−9.57698	−0.383694
$624$	8.35875	0.334618
$625$	20.4339	0.817357
$626$	19.8522	0.793455
$627$	−8.59783	−0.343364
$628$	−10.7099	−0.427371
$629$	−6.47214	−0.258061
$630$	−12.7023	−0.506071
$631$	−13.7376	−0.546884	−0.273442	−	0.961889i	$-0.588162\pi$
$631$	−13.7376	−0.546884	−0.273442	+	0.961889i	$0.588162\pi$
$632$	−1.10736	−0.0440483
$633$	−66.9911	−2.66266
$634$	−10.8539	−0.431064
$635$	6.17359	0.244992
$636$	16.1881	0.641898
$637$	−14.6427	−0.580165
$638$	−1.92052	−0.0760342
$639$	−70.0497	−2.77112
$640$	−0.557537	−0.0220386
$641$	11.1078	0.438730	0.219365	−	0.975643i	$-0.429601\pi$
$641$	11.1078	0.438730	0.219365	+	0.975643i	$0.429601\pi$
$642$	32.4508	1.28073
$643$	−0.301908	−0.0119061	−0.00595304	−	0.999982i	$-0.501895\pi$
$643$	−0.301908	−0.0119061	−0.00595304	+	0.999982i	$0.501895\pi$
$644$	10.8713	0.428389
$645$	−17.6100	−0.693394
$646$	6.31375	0.248411
$647$	18.0502	0.709627	0.354813	−	0.934937i	$-0.384544\pi$
$647$	18.0502	0.709627	0.354813	+	0.934937i	$0.384544\pi$
$648$	−13.4721	−0.529235
$649$	−0.442463	−0.0173682
$650$	−12.7354	−0.499522
$651$	−27.3030	−1.07009
$652$	−3.52786	−0.138162
$653$	−24.2884	−0.950479	−0.475239	−	0.879857i	$-0.657639\pi$
$653$	−24.2884	−0.950479	−0.475239	+	0.879857i	$0.657639\pi$
$654$	−4.72822	−0.184888
$655$	3.53367	0.138072
$656$	2.90398	0.113381
$657$	41.9190	1.63541
$658$	−14.7052	−0.573268
$659$	−32.1281	−1.25153	−0.625766	−	0.780011i	$-0.715214\pi$
$659$	−32.1281	−1.25153	−0.625766	+	0.780011i	$0.715214\pi$
$660$	0.759232	0.0295531
$661$	8.25763	0.321184	0.160592	−	0.987021i	$-0.448660\pi$
$661$	8.25763	0.321184	0.160592	+	0.987021i	$0.448660\pi$
$662$	−24.8546	−0.966003
$663$	8.35875	0.324627
$664$	−9.14475	−0.354885
$665$	−12.3914	−0.480519
$666$	41.8885	1.62315
$667$	−13.4048	−0.519038
$668$	4.18806	0.162041
$669$	−28.7577	−1.11184
$670$	3.69038	0.142572
$671$	1.45309	0.0560957
$672$	10.8339	0.417927
$673$	−36.9991	−1.42621	−0.713105	−	0.701058i	$-0.752712\pi$
$673$	−36.9991	−1.42621	−0.713105	+	0.701058i	$0.752712\pi$
$674$	13.8066	0.531811
$675$	50.1089	1.92869
$676$	−5.62377	−0.216299
$677$	−15.3554	−0.590156	−0.295078	−	0.955473i	$-0.595346\pi$
$677$	−15.3554	−0.590156	−0.295078	+	0.955473i	$0.595346\pi$
$678$	19.9756	0.767159
$679$	−51.2923	−1.96842
$680$	−0.557537	−0.0213805
$681$	50.3746	1.93036
$682$	1.11507	0.0426984
$683$	−6.79443	−0.259982	−0.129991	−	0.991515i	$-0.541495\pi$
$683$	−6.79443	−0.259982	−0.129991	+	0.991515i	$0.541495\pi$
$684$	−40.8635	−1.56245
$685$	−3.86106	−0.147524
$686$	5.66239	0.216191
$687$	33.2338	1.26795
$688$	10.2627	0.391263
$689$	14.2853	0.544225
$690$	5.29929	0.201740
$691$	2.84193	0.108112	0.0540561	−	0.998538i	$-0.482785\pi$
$691$	2.84193	0.108112	0.0540561	+	0.998538i	$0.482785\pi$
$692$	2.40758	0.0915225
$693$	−10.0806	−0.382930
$694$	25.7935	0.979108
$695$	−11.5455	−0.437946
$696$	−13.3587	−0.506362
$697$	2.90398	0.109996
$698$	12.3168	0.466197
$699$	−45.5479	−1.72278
$700$	−16.5065	−0.623887
$701$	−31.2542	−1.18045	−0.590227	−	0.807237i	$-0.700962\pi$
$701$	−31.2542	−1.18045	−0.590227	+	0.807237i	$0.700962\pi$
$702$	−29.0227	−1.09539
$703$	40.8635	1.54119
$704$	−0.442463	−0.0166760
$705$	−7.16815	−0.269968
$706$	17.2550	0.649401
$707$	14.7320	0.554052
$708$	−3.07768	−0.115666
$709$	34.3651	1.29061	0.645305	−	0.763925i	$-0.276730\pi$
$709$	34.3651	1.29061	0.645305	+	0.763925i	$0.276730\pi$
$710$	6.03437	0.226466
$711$	7.16696	0.268782
$712$	2.72062	0.101960
$713$	7.78298	0.291475
$714$	10.8339	0.405448
$715$	0.669989	0.0250562
$716$	−13.1749	−0.492370
$717$	26.3463	0.983922
$718$	30.3896	1.13413
$719$	−18.0006	−0.671309	−0.335655	−	0.941985i	$-0.608957\pi$
$719$	−18.0006	−0.671309	−0.335655	+	0.941985i	$0.608957\pi$
$720$	3.60845	0.134479
$721$	−53.9067	−2.00759
$722$	−20.8635	−0.776458
$723$	8.03947	0.298991
$724$	−19.0165	−0.706744
$725$	20.3534	0.755905
$726$	−33.2520	−1.23410
$727$	12.6369	0.468677	0.234338	−	0.972155i	$-0.424708\pi$
$727$	12.6369	0.468677	0.234338	+	0.972155i	$0.424708\pi$
$728$	9.56044	0.354334
$729$	−11.4721	−0.424894
$730$	−3.61107	−0.133652
$731$	10.2627	0.379581
$732$	10.1074	0.373579
$733$	−29.3050	−1.08240	−0.541202	−	0.840893i	$-0.682030\pi$
$733$	−29.3050	−1.08240	−0.541202	+	0.840893i	$0.682030\pi$
$734$	33.5552	1.23855
$735$	−9.25128	−0.341238
$736$	−3.08831	−0.113836
$737$	2.92870	0.107880
$738$	−18.7949	−0.691851
$739$	−31.2060	−1.14793	−0.573966	−	0.818879i	$-0.694596\pi$
$739$	−31.2060	−1.14793	−0.573966	+	0.818879i	$0.694596\pi$
$740$	−3.60845	−0.132649
$741$	−52.7750	−1.93874
$742$	18.5153	0.679719
$743$	26.9039	0.987007	0.493504	−	0.869744i	$-0.335716\pi$
$743$	26.9039	0.987007	0.493504	+	0.869744i	$0.335716\pi$
$744$	7.75621	0.284357
$745$	−6.69938	−0.245446
$746$	−22.6346	−0.828711
$747$	59.1860	2.16550
$748$	−0.442463	−0.0161781
$749$	37.1161	1.35619
$750$	−16.6258	−0.607089
$751$	41.1011	1.49980	0.749900	−	0.661552i	$-0.230102\pi$
$751$	41.1011	1.49980	0.749900	+	0.661552i	$0.230102\pi$
$752$	4.17744	0.152335
$753$	1.71040	0.0623303
$754$	−11.7885	−0.429312
$755$	−0.965691	−0.0351451
$756$	−37.6168	−1.36811
$757$	45.7698	1.66353	0.831766	−	0.555126i	$-0.187330\pi$
$757$	45.7698	1.66353	0.831766	+	0.555126i	$0.187330\pi$
$758$	36.3826	1.32148
$759$	4.20554	0.152651
$760$	3.52015	0.127689
$761$	1.95598	0.0709041	0.0354520	−	0.999371i	$-0.488713\pi$
$761$	1.95598	0.0709041	0.0354520	+	0.999371i	$0.488713\pi$
$762$	34.0791	1.23456
$763$	−5.40798	−0.195782
$764$	14.0585	0.508619
$765$	3.60845	0.130464
$766$	−5.56970	−0.201241
$767$	−2.71592	−0.0980662
$768$	−3.07768	−0.111056
$769$	13.3920	0.482929	0.241465	−	0.970410i	$-0.422372\pi$
$769$	13.3920	0.482929	0.241465	+	0.970410i	$0.422372\pi$
$770$	0.868383	0.0312944
$771$	67.0408	2.41441
$772$	1.47214	0.0529833
$773$	−21.6806	−0.779797	−0.389899	−	0.920858i	$-0.627490\pi$
$773$	−21.6806	−0.779797	−0.389899	+	0.920858i	$0.627490\pi$
$774$	−66.4217	−2.38748
$775$	−11.8174	−0.424492
$776$	14.5711	0.523071
$777$	70.1185	2.51549
$778$	−36.6787	−1.31500
$779$	−18.3350	−0.656919
$780$	4.66031	0.166866
$781$	4.78890	0.171360
$782$	−3.08831	−0.110438
$783$	46.3834	1.65761
$784$	5.39144	0.192551
$785$	−5.97115	−0.213120
$786$	19.5064	0.695770
$787$	12.6181	0.449787	0.224893	−	0.974383i	$-0.427797\pi$
$787$	12.6181	0.449787	0.224893	+	0.974383i	$0.427797\pi$
$788$	7.91209	0.281857
$789$	−51.1390	−1.82060
$790$	−0.617391	−0.0219658
$791$	22.8474	0.812361
$792$	2.86368	0.101757
$793$	8.91930	0.316734
$794$	24.5960	0.872881
$795$	9.02543	0.320099
$796$	−20.3243	−0.720374
$797$	29.1369	1.03208	0.516041	−	0.856564i	$-0.327405\pi$
$797$	29.1369	1.03208	0.516041	+	0.856564i	$0.327405\pi$
$798$	−68.4025	−2.42142
$799$	4.17744	0.147787
$800$	4.68915	0.165787
$801$	−17.6082	−0.622156
$802$	−1.89606	−0.0669522
$803$	−2.86576	−0.101131
$804$	20.3714	0.718444
$805$	6.06114	0.213627
$806$	6.84452	0.241088
$807$	81.9325	2.88416
$808$	−4.18504	−0.147229
$809$	−47.2975	−1.66289	−0.831446	−	0.555606i	$-0.812486\pi$
$809$	−47.2975	−1.66289	−0.831446	+	0.555606i	$0.812486\pi$
$810$	−7.51121	−0.263917
$811$	−8.91215	−0.312948	−0.156474	−	0.987682i	$-0.550013\pi$
$811$	−8.91215	−0.312948	−0.156474	+	0.987682i	$0.550013\pi$
$812$	−15.2793	−0.536197
$813$	75.2169	2.63797
$814$	−2.86368	−0.100372
$815$	−1.96691	−0.0688980
$816$	−3.07768	−0.107740
$817$	−64.7963	−2.26694
$818$	23.5247	0.822523
$819$	−61.8765	−2.16214
$820$	1.61907	0.0565405
$821$	21.9751	0.766937	0.383468	−	0.923554i	$-0.374729\pi$
$821$	21.9751	0.766937	0.383468	+	0.923554i	$0.374729\pi$
$822$	−21.3136	−0.743399
$823$	6.65068	0.231828	0.115914	−	0.993259i	$-0.463020\pi$
$823$	6.65068	0.231828	0.115914	+	0.993259i	$0.463020\pi$
$824$	15.3138	0.533480
$825$	−6.38551	−0.222315
$826$	−3.52015	−0.122482
$827$	19.3423	0.672597	0.336299	−	0.941755i	$-0.390825\pi$
$827$	19.3423	0.672597	0.336299	+	0.941755i	$0.390825\pi$
$828$	19.9879	0.694629
$829$	19.5860	0.680249	0.340125	−	0.940380i	$-0.389531\pi$
$829$	19.5860	0.680249	0.340125	+	0.940380i	$0.389531\pi$
$830$	−5.09853	−0.176973
$831$	15.8117	0.548502
$832$	−2.71592	−0.0941576
$833$	5.39144	0.186802
$834$	−63.7330	−2.20689
$835$	2.33499	0.0808058
$836$	2.79360	0.0966188
$837$	−26.9306	−0.930859
$838$	19.7192	0.681189
$839$	7.32264	0.252806	0.126403	−	0.991979i	$-0.459657\pi$
$839$	7.32264	0.252806	0.126403	+	0.991979i	$0.459657\pi$
$840$	6.04029	0.208410
$841$	−10.1599	−0.350341
$842$	−9.55574	−0.329313
$843$	−60.1771	−2.07261
$844$	21.7667	0.749242
$845$	−3.13546	−0.107863
$846$	−27.0369	−0.929549
$847$	−38.0325	−1.30681
$848$	−5.25982	−0.180623
$849$	−40.6291	−1.39439
$850$	4.68915	0.160837
$851$	−19.9879	−0.685177
$852$	33.3106	1.14120
$853$	−27.1255	−0.928758	−0.464379	−	0.885636i	$-0.653723\pi$
$853$	−27.1255	−0.928758	−0.464379	+	0.885636i	$0.653723\pi$
$854$	11.5604	0.395590
$855$	−22.7829	−0.779158
$856$	−10.5439	−0.360383
$857$	0.0111944	0.000382394 0	0.000191197	−	1.00000i	$-0.499939\pi$
$857$	0.0111944	0.000382394 0	0.000191197		1.00000i	$0.499939\pi$
$858$	3.69844	0.126263
$859$	18.6388	0.635949	0.317974	−	0.948099i	$-0.396997\pi$
$859$	18.6388	0.635949	0.317974	+	0.948099i	$0.396997\pi$
$860$	5.72184	0.195113
$861$	−31.4614	−1.07220
$862$	14.0155	0.477371
$863$	46.5470	1.58448	0.792239	−	0.610211i	$-0.208915\pi$
$863$	46.5470	1.58448	0.792239	+	0.610211i	$0.208915\pi$
$864$	10.6861	0.363550
$865$	1.34231	0.0456401
$866$	−31.4590	−1.06902
$867$	−3.07768	−0.104524
$868$	8.87129	0.301111
$869$	−0.489965	−0.0166209
$870$	−7.44799	−0.252510
$871$	17.9769	0.609123
$872$	1.53629	0.0520255
$873$	−94.3059	−3.19177
$874$	19.4988	0.659556
$875$	−19.0160	−0.642859
$876$	−19.9337	−0.673496
$877$	47.1194	1.59111	0.795555	−	0.605881i	$-0.207179\pi$
$877$	47.1194	1.59111	0.795555	+	0.605881i	$0.207179\pi$
$878$	19.3123	0.651758
$879$	10.3936	0.350567
$880$	−0.246690	−0.00831590
$881$	53.8768	1.81516	0.907578	−	0.419884i	$-0.137929\pi$
$881$	53.8768	1.81516	0.907578	+	0.419884i	$0.137929\pi$
$882$	−34.8941	−1.17495
$883$	−43.0488	−1.44871	−0.724353	−	0.689429i	$-0.757861\pi$
$883$	−43.0488	−1.44871	−0.724353	+	0.689429i	$0.757861\pi$
$884$	−2.71592	−0.0913463
$885$	−1.71592	−0.0576801
$886$	−17.5095	−0.588244
$887$	1.54441	0.0518561	0.0259281	−	0.999664i	$-0.491746\pi$
$887$	1.54441	0.0518561	0.0259281	+	0.999664i	$0.491746\pi$
$888$	−19.9192	−0.668444
$889$	38.9785	1.30730
$890$	1.51685	0.0508448
$891$	−5.96093	−0.199699
$892$	9.34393	0.312858
$893$	−26.3753	−0.882615
$894$	−36.9815	−1.23685
$895$	−7.34550	−0.245533
$896$	−3.52015	−0.117600
$897$	25.8144	0.861916
$898$	1.79702	0.0599673
$899$	−10.9387	−0.364828
$900$	−30.3488	−1.01163
$901$	−5.25982	−0.175230
$902$	1.28490	0.0427826
$903$	−111.185	−3.70002
$904$	−6.49047	−0.215870
$905$	−10.6024	−0.352436
$906$	−5.33076	−0.177102
$907$	−17.8120	−0.591439	−0.295719	−	0.955275i	$-0.595559\pi$
$907$	−17.8120	−0.591439	−0.295719	+	0.955275i	$0.595559\pi$
$908$	−16.3677	−0.543181
$909$	27.0861	0.898391
$910$	5.33030	0.176698
$911$	49.6153	1.64383	0.821914	−	0.569612i	$-0.192906\pi$
$911$	49.6153	1.64383	0.821914	+	0.569612i	$0.192906\pi$
$912$	19.4317	0.643449
$913$	−4.04622	−0.133910
$914$	13.6894	0.452804
$915$	5.63522	0.186295
$916$	−10.7983	−0.356786
$917$	22.3107	0.736765
$918$	10.6861	0.352695
$919$	26.4239	0.871645	0.435823	−	0.900033i	$-0.356458\pi$
$919$	26.4239	0.871645	0.435823	+	0.900033i	$0.356458\pi$
$920$	−1.72184	−0.0567675
$921$	−91.5539	−3.01680
$922$	−10.5409	−0.347145
$923$	29.3952	0.967554
$924$	4.79360	0.157698
$925$	30.3488	0.997864
$926$	−20.9710	−0.689151
$927$	−99.1127	−3.25529
$928$	4.34052	0.142485
$929$	19.3369	0.634424	0.317212	−	0.948355i	$-0.397253\pi$
$929$	19.3369	0.634424	0.317212	+	0.948355i	$0.397253\pi$
$930$	4.32437	0.141802
$931$	−34.0402	−1.11562
$932$	14.7994	0.484771
$933$	49.0659	1.60635
$934$	−7.51943	−0.246043
$935$	−0.246690	−0.00806761
$936$	17.5778	0.574549
$937$	34.6383	1.13158	0.565792	−	0.824548i	$-0.308570\pi$
$937$	34.6383	1.13158	0.565792	+	0.824548i	$0.308570\pi$
$938$	23.3001	0.760776
$939$	61.0989	1.99389
$940$	2.32907	0.0759660
$941$	−5.99337	−0.195378	−0.0976891	−	0.995217i	$-0.531145\pi$
$941$	−5.99337	−0.195378	−0.0976891	+	0.995217i	$0.531145\pi$
$942$	−32.9616	−1.07395
$943$	8.96837	0.292050
$944$	1.00000	0.0325472
$945$	−20.9727	−0.682243
$946$	4.54088	0.147637
$947$	31.0572	1.00922	0.504612	−	0.863346i	$-0.331635\pi$
$947$	31.0572	1.00922	0.504612	+	0.863346i	$0.331635\pi$
$948$	−3.40809	−0.110690
$949$	−17.5906	−0.571014
$950$	−29.6061	−0.960550
$951$	−33.4049	−1.08323
$952$	−3.52015	−0.114089
$953$	42.6110	1.38031	0.690154	−	0.723663i	$-0.257543\pi$
$953$	42.6110	1.38031	0.690154	+	0.723663i	$0.257543\pi$
$954$	34.0423	1.10216
$955$	7.83814	0.253636
$956$	−8.56044	−0.276864
$957$	−5.91076	−0.191068
$958$	−33.5586	−1.08423
$959$	−24.3778	−0.787200
$960$	−1.71592	−0.0553811
$961$	−24.6489	−0.795124
$962$	−17.5778	−0.566731
$963$	68.2415	2.19905
$964$	−2.61218	−0.0841327
$965$	0.820770	0.0264215
$966$	33.4584	1.07651
$967$	−10.2588	−0.329902	−0.164951	−	0.986302i	$-0.552747\pi$
$967$	−10.2588	−0.329902	−0.164951	+	0.986302i	$0.552747\pi$
$968$	10.8042	0.347261
$969$	19.4317	0.624237
$970$	8.12390	0.260843
$971$	11.8100	0.378999	0.189500	−	0.981881i	$-0.439313\pi$
$971$	11.8100	0.378999	0.189500	+	0.981881i	$0.439313\pi$
$972$	−9.40456	−0.301652
$973$	−72.8955	−2.33692
$974$	12.4665	0.399452
$975$	−39.1954	−1.25526
$976$	−3.28408	−0.105121
$977$	−54.9530	−1.75810	−0.879051	−	0.476728i	$-0.841823\pi$
$977$	−54.9530	−1.75810	−0.879051	+	0.476728i	$0.841823\pi$
$978$	−10.8576	−0.347189
$979$	1.20378	0.0384728
$980$	3.00592	0.0960207
$981$	−9.94310	−0.317459
$982$	−22.9912	−0.733679
$983$	−25.6830	−0.819161	−0.409580	−	0.912274i	$-0.634325\pi$
$983$	−25.6830	−0.819161	−0.409580	+	0.912274i	$0.634325\pi$
$984$	8.93752	0.284918
$985$	4.41128	0.140555
$986$	4.34052	0.138230
$987$	−45.2579	−1.44058
$988$	17.1477	0.545539
$989$	31.6944	1.00782
$990$	1.59661	0.0507435
$991$	−3.03896	−0.0965357	−0.0482679	−	0.998834i	$-0.515370\pi$
$991$	−3.03896	−0.0965357	−0.0482679	+	0.998834i	$0.515370\pi$
$992$	−2.52015	−0.0800147
$993$	−76.4947	−2.42749
$994$	38.0995	1.20844
$995$	−11.3315	−0.359233
$996$	−28.1446	−0.891797
$997$	13.0840	0.414375	0.207187	−	0.978301i	$-0.433569\pi$
$997$	13.0840	0.414375	0.207187	+	0.978301i	$0.433569\pi$
$998$	−29.4771	−0.933082
$999$	69.1621	2.18819

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2006.2.a.q.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2006.2.a.q.1.1	✓	4	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:	\( N \)	\(=\)	\( 2006 = 2 \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2006.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.07768 q^{3} +1.00000 q^{4} +0.557537 q^{5} +3.07768 q^{6} +3.52015 q^{7} -1.00000 q^{8} +6.47214 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.07768 q^{3} +1.00000 q^{4} +0.557537 q^{5} +3.07768 q^{6} +3.52015 q^{7} -1.00000 q^{8} +6.47214 q^{9} -0.557537 q^{10} -0.442463 q^{11} -3.07768 q^{12} -2.71592 q^{13} -3.52015 q^{14} -1.71592 q^{15} +1.00000 q^{16} +1.00000 q^{17} -6.47214 q^{18} -6.31375 q^{19} +0.557537 q^{20} -10.8339 q^{21} +0.442463 q^{22} +3.08831 q^{23} +3.07768 q^{24} -4.68915 q^{25} +2.71592 q^{26} -10.6861 q^{27} +3.52015 q^{28} -4.34052 q^{29} +1.71592 q^{30} +2.52015 q^{31} -1.00000 q^{32} +1.36176 q^{33} -1.00000 q^{34} +1.96261 q^{35} +6.47214 q^{36} -6.47214 q^{37} +6.31375 q^{38} +8.35875 q^{39} -0.557537 q^{40} +2.90398 q^{41} +10.8339 q^{42} +10.2627 q^{43} -0.442463 q^{44} +3.60845 q^{45} -3.08831 q^{46} +4.17744 q^{47} -3.07768 q^{48} +5.39144 q^{49} +4.68915 q^{50} -3.07768 q^{51} -2.71592 q^{52} -5.25982 q^{53} +10.6861 q^{54} -0.246690 q^{55} -3.52015 q^{56} +19.4317 q^{57} +4.34052 q^{58} +1.00000 q^{59} -1.71592 q^{60} -3.28408 q^{61} -2.52015 q^{62} +22.7829 q^{63} +1.00000 q^{64} -1.51423 q^{65} -1.36176 q^{66} -6.61907 q^{67} +1.00000 q^{68} -9.50483 q^{69} -1.96261 q^{70} -10.8233 q^{71} -6.47214 q^{72} +6.47684 q^{73} +6.47214 q^{74} +14.4317 q^{75} -6.31375 q^{76} -1.55754 q^{77} -8.35875 q^{78} +1.10736 q^{79} +0.557537 q^{80} +13.4721 q^{81} -2.90398 q^{82} +9.14475 q^{83} -10.8339 q^{84} +0.557537 q^{85} -10.2627 q^{86} +13.3587 q^{87} +0.442463 q^{88} -2.72062 q^{89} -3.60845 q^{90} -9.56044 q^{91} +3.08831 q^{92} -7.75621 q^{93} -4.17744 q^{94} -3.52015 q^{95} +3.07768 q^{96} -14.5711 q^{97} -5.39144 q^{98} -2.86368 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} - 4 q^{8} + 8 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 - 4 * q^8 + 8 * q^9	\( 4 q - 4 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} - 4 q^{8} + 8 q^{9} - 2 q^{10} - 2 q^{11} - 14 q^{13} - 2 q^{14} - 10 q^{15} + 4 q^{16} + 4 q^{17} - 8 q^{18} - 4 q^{19} + 2 q^{20} - 10 q^{21} + 2 q^{22} - 6 q^{23} - 4 q^{25} + 14 q^{26} + 2 q^{28} - 14 q^{29} + 10 q^{30} - 2 q^{31} - 4 q^{32} - 10 q^{33} - 4 q^{34} - 4 q^{35} + 8 q^{36} - 8 q^{37} + 4 q^{38} - 10 q^{39} - 2 q^{40} + 10 q^{42} - 14 q^{43} - 2 q^{44} - 16 q^{45} + 6 q^{46} + 8 q^{47} - 12 q^{49} + 4 q^{50} - 14 q^{52} - 2 q^{53} + 14 q^{55} - 2 q^{56} + 20 q^{57} + 14 q^{58} + 4 q^{59} - 10 q^{60} - 10 q^{61} + 2 q^{62} + 24 q^{63} + 4 q^{64} - 2 q^{65} + 10 q^{66} + 10 q^{67} + 4 q^{68} - 30 q^{69} + 4 q^{70} - 16 q^{71} - 8 q^{72} + 10 q^{73} + 8 q^{74} - 4 q^{76} - 6 q^{77} + 10 q^{78} - 26 q^{79} + 2 q^{80} + 36 q^{81} + 18 q^{83} - 10 q^{84} + 2 q^{85} + 14 q^{86} + 10 q^{87} + 2 q^{88} - 16 q^{89} + 16 q^{90} - 2 q^{91} - 6 q^{92} - 10 q^{93} - 8 q^{94} - 2 q^{95} - 16 q^{97} + 12 q^{98} - 24 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 - 4 * q^8 + 8 * q^9 - 2 * q^10 - 2 * q^11 - 14 * q^13 - 2 * q^14 - 10 * q^15 + 4 * q^16 + 4 * q^17 - 8 * q^18 - 4 * q^19 + 2 * q^20 - 10 * q^21 + 2 * q^22 - 6 * q^23 - 4 * q^25 + 14 * q^26 + 2 * q^28 - 14 * q^29 + 10 * q^30 - 2 * q^31 - 4 * q^32 - 10 * q^33 - 4 * q^34 - 4 * q^35 + 8 * q^36 - 8 * q^37 + 4 * q^38 - 10 * q^39 - 2 * q^40 + 10 * q^42 - 14 * q^43 - 2 * q^44 - 16 * q^45 + 6 * q^46 + 8 * q^47 - 12 * q^49 + 4 * q^50 - 14 * q^52 - 2 * q^53 + 14 * q^55 - 2 * q^56 + 20 * q^57 + 14 * q^58 + 4 * q^59 - 10 * q^60 - 10 * q^61 + 2 * q^62 + 24 * q^63 + 4 * q^64 - 2 * q^65 + 10 * q^66 + 10 * q^67 + 4 * q^68 - 30 * q^69 + 4 * q^70 - 16 * q^71 - 8 * q^72 + 10 * q^73 + 8 * q^74 - 4 * q^76 - 6 * q^77 + 10 * q^78 - 26 * q^79 + 2 * q^80 + 36 * q^81 + 18 * q^83 - 10 * q^84 + 2 * q^85 + 14 * q^86 + 10 * q^87 + 2 * q^88 - 16 * q^89 + 16 * q^90 - 2 * q^91 - 6 * q^92 - 10 * q^93 - 8 * q^94 - 2 * q^95 - 16 * q^97 + 12 * q^98 - 24 * q^99

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