LMFDB - Embedded newform 1045.2.a.j.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, 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("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1045.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:	$8.34436701122$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 11x^{7} + 11x^{6} + 34x^{5} - 20x^{4} - 36x^{3} + 13x^{2} + 10x - 2 $ x^9 - 2x^8 - 11x^7 + 11x^6 + 34x^5 - 20x^4 - 36x^3 + 13x^2 + 10x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.58924$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64409 q^{2} +0.185485 q^{3} +4.99120 q^{4} -1.00000 q^{5} +0.490440 q^{6} +1.87571 q^{7} +7.90900 q^{8} -2.96560 q^{9} +O(q^{10})$	\(q+2.64409 q^{2} +0.185485 q^{3} +4.99120 q^{4} -1.00000 q^{5} +0.490440 q^{6} +1.87571 q^{7} +7.90900 q^{8} -2.96560 q^{9} -2.64409 q^{10} -1.00000 q^{11} +0.925795 q^{12} +2.60200 q^{13} +4.95955 q^{14} -0.185485 q^{15} +10.9297 q^{16} +4.32709 q^{17} -7.84130 q^{18} +1.00000 q^{19} -4.99120 q^{20} +0.347917 q^{21} -2.64409 q^{22} +5.43061 q^{23} +1.46700 q^{24} +1.00000 q^{25} +6.87992 q^{26} -1.10653 q^{27} +9.36207 q^{28} -3.56987 q^{29} -0.490440 q^{30} -3.56972 q^{31} +13.0811 q^{32} -0.185485 q^{33} +11.4412 q^{34} -1.87571 q^{35} -14.8019 q^{36} -6.04251 q^{37} +2.64409 q^{38} +0.482633 q^{39} -7.90900 q^{40} -5.41389 q^{41} +0.919924 q^{42} -4.97152 q^{43} -4.99120 q^{44} +2.96560 q^{45} +14.3590 q^{46} -6.12841 q^{47} +2.02730 q^{48} -3.48170 q^{49} +2.64409 q^{50} +0.802611 q^{51} +12.9871 q^{52} +7.62653 q^{53} -2.92576 q^{54} +1.00000 q^{55} +14.8350 q^{56} +0.185485 q^{57} -9.43905 q^{58} -5.51407 q^{59} -0.925795 q^{60} -12.0388 q^{61} -9.43864 q^{62} -5.56261 q^{63} +12.7281 q^{64} -2.60200 q^{65} -0.490440 q^{66} -2.94394 q^{67} +21.5974 q^{68} +1.00730 q^{69} -4.95955 q^{70} -12.5978 q^{71} -23.4549 q^{72} +14.2715 q^{73} -15.9769 q^{74} +0.185485 q^{75} +4.99120 q^{76} -1.87571 q^{77} +1.27612 q^{78} -10.8162 q^{79} -10.9297 q^{80} +8.69154 q^{81} -14.3148 q^{82} +15.9110 q^{83} +1.73653 q^{84} -4.32709 q^{85} -13.1451 q^{86} -0.662158 q^{87} -7.90900 q^{88} +1.97457 q^{89} +7.84130 q^{90} +4.88061 q^{91} +27.1053 q^{92} -0.662130 q^{93} -16.2041 q^{94} -1.00000 q^{95} +2.42635 q^{96} -8.59555 q^{97} -9.20591 q^{98} +2.96560 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 3 q^{2} + 3 q^{3} + 9 q^{4} - 9 q^{5} + 6 q^{6} - 9 q^{7} + 15 q^{8} + 16 q^{9}+O(q^{10}) $ 9 * q + 3 * q^2 + 3 * q^3 + 9 * q^4 - 9 * q^5 + 6 * q^6 - 9 * q^7 + 15 * q^8 + 16 * q^9	\( 9 q + 3 q^{2} + 3 q^{3} + 9 q^{4} - 9 q^{5} + 6 q^{6} - 9 q^{7} + 15 q^{8} + 16 q^{9} - 3 q^{10} - 9 q^{11} + 13 q^{12} - 3 q^{13} + 4 q^{14} - 3 q^{15} + 17 q^{16} + 5 q^{17} + 17 q^{18} + 9 q^{19} - 9 q^{20} - q^{21} - 3 q^{22} + 4 q^{23} + 21 q^{24} + 9 q^{25} + 20 q^{26} + 24 q^{27} - 24 q^{28} + 3 q^{29} - 6 q^{30} - q^{31} + 38 q^{32} - 3 q^{33} + 28 q^{34} + 9 q^{35} + 17 q^{36} + 5 q^{37} + 3 q^{38} - 15 q^{40} - 5 q^{41} - 43 q^{42} - 11 q^{43} - 9 q^{44} - 16 q^{45} + 2 q^{46} + 30 q^{47} + 54 q^{48} + 12 q^{49} + 3 q^{50} + 40 q^{51} - 3 q^{52} - q^{53} + 65 q^{54} + 9 q^{55} - 16 q^{56} + 3 q^{57} - 15 q^{58} + 59 q^{59} - 13 q^{60} - 21 q^{61} - 10 q^{62} - 12 q^{63} + 19 q^{64} + 3 q^{65} - 6 q^{66} - 2 q^{67} - 9 q^{68} - 22 q^{69} - 4 q^{70} + 34 q^{71} + 32 q^{72} - 34 q^{73} - 21 q^{74} + 3 q^{75} + 9 q^{76} + 9 q^{77} - 65 q^{78} - 13 q^{79} - 17 q^{80} + 57 q^{81} + 10 q^{82} + 51 q^{83} - 95 q^{84} - 5 q^{85} - 14 q^{86} + 8 q^{87} - 15 q^{88} + 8 q^{89} - 17 q^{90} + 62 q^{91} + 57 q^{92} - 18 q^{93} + 2 q^{94} - 9 q^{95} + 81 q^{96} - 8 q^{97} - 20 q^{98} - 16 q^{99}+O(q^{100}) \) 9 * q + 3 * q^2 + 3 * q^3 + 9 * q^4 - 9 * q^5 + 6 * q^6 - 9 * q^7 + 15 * q^8 + 16 * q^9 - 3 * q^10 - 9 * q^11 + 13 * q^12 - 3 * q^13 + 4 * q^14 - 3 * q^15 + 17 * q^16 + 5 * q^17 + 17 * q^18 + 9 * q^19 - 9 * q^20 - q^21 - 3 * q^22 + 4 * q^23 + 21 * q^24 + 9 * q^25 + 20 * q^26 + 24 * q^27 - 24 * q^28 + 3 * q^29 - 6 * q^30 - q^31 + 38 * q^32 - 3 * q^33 + 28 * q^34 + 9 * q^35 + 17 * q^36 + 5 * q^37 + 3 * q^38 - 15 * q^40 - 5 * q^41 - 43 * q^42 - 11 * q^43 - 9 * q^44 - 16 * q^45 + 2 * q^46 + 30 * q^47 + 54 * q^48 + 12 * q^49 + 3 * q^50 + 40 * q^51 - 3 * q^52 - q^53 + 65 * q^54 + 9 * q^55 - 16 * q^56 + 3 * q^57 - 15 * q^58 + 59 * q^59 - 13 * q^60 - 21 * q^61 - 10 * q^62 - 12 * q^63 + 19 * q^64 + 3 * q^65 - 6 * q^66 - 2 * q^67 - 9 * q^68 - 22 * q^69 - 4 * q^70 + 34 * q^71 + 32 * q^72 - 34 * q^73 - 21 * q^74 + 3 * q^75 + 9 * q^76 + 9 * q^77 - 65 * q^78 - 13 * q^79 - 17 * q^80 + 57 * q^81 + 10 * q^82 + 51 * q^83 - 95 * q^84 - 5 * q^85 - 14 * q^86 + 8 * q^87 - 15 * q^88 + 8 * q^89 - 17 * q^90 + 62 * q^91 + 57 * q^92 - 18 * q^93 + 2 * q^94 - 9 * q^95 + 81 * q^96 - 8 * q^97 - 20 * q^98 - 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.64409	1.86965	0.934826	−	0.355105i	$-0.115555\pi$
$2$	2.64409	1.86965	0.934826	+	0.355105i	$0.115555\pi$
$3$	0.185485	0.107090	0.0535450	−	0.998565i	$-0.482948\pi$
$3$	0.185485	0.107090	0.0535450	+	0.998565i	$0.482948\pi$
$4$	4.99120	2.49560
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.490440	0.200221
$7$	1.87571	0.708953	0.354477	−	0.935065i	$-0.384659\pi$
$7$	1.87571	0.708953	0.354477	+	0.935065i	$0.384659\pi$
$8$	7.90900	2.79625
$9$	−2.96560	−0.988532
$10$	−2.64409	−0.836134
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0.925795	0.267254
$13$	2.60200	0.721665	0.360833	−	0.932631i	$-0.382493\pi$
$13$	2.60200	0.721665	0.360833	+	0.932631i	$0.382493\pi$
$14$	4.95955	1.32550
$15$	−0.185485	−0.0478921
$16$	10.9297	2.73242
$17$	4.32709	1.04947	0.524736	−	0.851265i	$-0.324164\pi$
$17$	4.32709	1.04947	0.524736	+	0.851265i	$0.324164\pi$
$18$	−7.84130	−1.84821
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−4.99120	−1.11607
$21$	0.347917	0.0759218
$22$	−2.64409	−0.563721
$23$	5.43061	1.13236	0.566180	−	0.824281i	$-0.308421\pi$
$23$	5.43061	1.13236	0.566180	+	0.824281i	$0.308421\pi$
$24$	1.46700	0.299451
$25$	1.00000	0.200000
$26$	6.87992	1.34926
$27$	−1.10653	−0.212952
$28$	9.36207	1.76926
$29$	−3.56987	−0.662908	−0.331454	−	0.943471i	$-0.607539\pi$
$29$	−3.56987	−0.662908	−0.331454	+	0.943471i	$0.607539\pi$
$30$	−0.490440	−0.0895416
$31$	−3.56972	−0.641140	−0.320570	−	0.947225i	$-0.603874\pi$
$31$	−3.56972	−0.641140	−0.320570	+	0.947225i	$0.603874\pi$
$32$	13.0811	2.31243
$33$	−0.185485	−0.0322889
$34$	11.4412	1.96215
$35$	−1.87571	−0.317054
$36$	−14.8019	−2.46698
$37$	−6.04251	−0.993383	−0.496692	−	0.867927i	$-0.665452\pi$
$37$	−6.04251	−0.993383	−0.496692	+	0.867927i	$0.665452\pi$
$38$	2.64409	0.428928
$39$	0.482633	0.0772832
$40$	−7.90900	−1.25052
$41$	−5.41389	−0.845507	−0.422753	−	0.906245i	$-0.638936\pi$
$41$	−5.41389	−0.845507	−0.422753	+	0.906245i	$0.638936\pi$
$42$	0.919924	0.141947
$43$	−4.97152	−0.758150	−0.379075	−	0.925366i	$-0.623758\pi$
$43$	−4.97152	−0.758150	−0.379075	+	0.925366i	$0.623758\pi$
$44$	−4.99120	−0.752452
$45$	2.96560	0.442085
$46$	14.3590	2.11712
$47$	−6.12841	−0.893921	−0.446961	−	0.894554i	$-0.647494\pi$
$47$	−6.12841	−0.893921	−0.446961	+	0.894554i	$0.647494\pi$
$48$	2.02730	0.292615
$49$	−3.48170	−0.497385
$50$	2.64409	0.373931
$51$	0.802611	0.112388
$52$	12.9871	1.80099
$53$	7.62653	1.04758	0.523792	−	0.851846i	$-0.324517\pi$
$53$	7.62653	1.04758	0.523792	+	0.851846i	$0.324517\pi$
$54$	−2.92576	−0.398146
$55$	1.00000	0.134840
$56$	14.8350	1.98241
$57$	0.185485	0.0245681
$58$	−9.43905	−1.23941
$59$	−5.51407	−0.717871	−0.358935	−	0.933362i	$-0.616860\pi$
$59$	−5.51407	−0.717871	−0.358935	+	0.933362i	$0.616860\pi$
$60$	−0.925795	−0.119520
$61$	−12.0388	−1.54141	−0.770707	−	0.637190i	$-0.780097\pi$
$61$	−12.0388	−1.54141	−0.770707	+	0.637190i	$0.780097\pi$
$62$	−9.43864	−1.19871
$63$	−5.56261	−0.700823
$64$	12.7281	1.59102
$65$	−2.60200	−0.322739
$66$	−0.490440	−0.0603689
$67$	−2.94394	−0.359660	−0.179830	−	0.983698i	$-0.557555\pi$
$67$	−2.94394	−0.359660	−0.179830	+	0.983698i	$0.557555\pi$
$68$	21.5974	2.61906
$69$	1.00730	0.121265
$70$	−4.95955	−0.592780
$71$	−12.5978	−1.49509	−0.747544	−	0.664212i	$-0.768767\pi$
$71$	−12.5978	−1.49509	−0.747544	+	0.664212i	$0.768767\pi$
$72$	−23.4549	−2.76419
$73$	14.2715	1.67036	0.835179	−	0.549979i	$-0.185364\pi$
$73$	14.2715	1.67036	0.835179	+	0.549979i	$0.185364\pi$
$74$	−15.9769	−1.85728
$75$	0.185485	0.0214180
$76$	4.99120	0.572530
$77$	−1.87571	−0.213757
$78$	1.27612	0.144493
$79$	−10.8162	−1.21692	−0.608460	−	0.793585i	$-0.708212\pi$
$79$	−10.8162	−1.21692	−0.608460	+	0.793585i	$0.708212\pi$
$80$	−10.9297	−1.22198
$81$	8.69154	0.965727
$82$	−14.3148	−1.58080
$83$	15.9110	1.74646	0.873229	−	0.487311i	$-0.162022\pi$
$83$	15.9110	1.74646	0.873229	+	0.487311i	$0.162022\pi$
$84$	1.73653	0.189471
$85$	−4.32709	−0.469338
$86$	−13.1451	−1.41748
$87$	−0.662158	−0.0709908
$88$	−7.90900	−0.843103
$89$	1.97457	0.209304	0.104652	−	0.994509i	$-0.466627\pi$
$89$	1.97457	0.209304	0.104652	+	0.994509i	$0.466627\pi$
$90$	7.84130	0.826545
$91$	4.88061	0.511627
$92$	27.1053	2.82592
$93$	−0.662130	−0.0686597
$94$	−16.2041	−1.67132
$95$	−1.00000	−0.102598
$96$	2.42635	0.247638
$97$	−8.59555	−0.872746	−0.436373	−	0.899766i	$-0.643737\pi$
$97$	−8.59555	−0.872746	−0.436373	+	0.899766i	$0.643737\pi$
$98$	−9.20591	−0.929938
$99$	2.96560	0.298054
$100$	4.99120	0.499120
$101$	11.1255	1.10703	0.553516	−	0.832838i	$-0.313286\pi$
$101$	11.1255	1.10703	0.553516	+	0.832838i	$0.313286\pi$
$102$	2.12217	0.210127
$103$	8.14311	0.802364	0.401182	−	0.915998i	$-0.368599\pi$
$103$	8.14311	0.802364	0.401182	+	0.915998i	$0.368599\pi$
$104$	20.5792	2.01796
$105$	−0.347917	−0.0339533
$106$	20.1652	1.95862
$107$	10.6362	1.02824	0.514119	−	0.857719i	$-0.328119\pi$
$107$	10.6362	1.02824	0.514119	+	0.857719i	$0.328119\pi$
$108$	−5.52292	−0.531443
$109$	16.1573	1.54759	0.773796	−	0.633435i	$-0.218355\pi$
$109$	16.1573	1.54759	0.773796	+	0.633435i	$0.218355\pi$
$110$	2.64409	0.252104
$111$	−1.12080	−0.106381
$112$	20.5010	1.93716
$113$	−13.0612	−1.22869	−0.614347	−	0.789036i	$-0.710581\pi$
$113$	−13.0612	−1.22869	−0.614347	+	0.789036i	$0.710581\pi$
$114$	0.490440	0.0459339
$115$	−5.43061	−0.506407
$116$	−17.8179	−1.65435
$117$	−7.71648	−0.713389
$118$	−14.5797	−1.34217
$119$	8.11638	0.744027
$120$	−1.46700	−0.133919
$121$	1.00000	0.0909091
$122$	−31.8317	−2.88191
$123$	−1.00420	−0.0905453
$124$	−17.8172	−1.60003
$125$	−1.00000	−0.0894427
$126$	−14.7080	−1.31030
$127$	−1.37497	−0.122009	−0.0610046	−	0.998137i	$-0.519430\pi$
$127$	−1.37497	−0.122009	−0.0610046	+	0.998137i	$0.519430\pi$
$128$	7.49213	0.662217
$129$	−0.922144	−0.0811903
$130$	−6.87992	−0.603409
$131$	17.0648	1.49096	0.745478	−	0.666530i	$-0.232221\pi$
$131$	17.0648	1.49096	0.745478	+	0.666530i	$0.232221\pi$
$132$	−0.925795	−0.0805801
$133$	1.87571	0.162645
$134$	−7.78405	−0.672439
$135$	1.10653	0.0952350
$136$	34.2229	2.93459
$137$	−16.4696	−1.40709	−0.703547	−	0.710649i	$-0.748401\pi$
$137$	−16.4696	−1.40709	−0.703547	+	0.710649i	$0.748401\pi$
$138$	2.66339	0.226723
$139$	−14.2843	−1.21158	−0.605788	−	0.795626i	$-0.707142\pi$
$139$	−14.2843	−1.21158	−0.605788	+	0.795626i	$0.707142\pi$
$140$	−9.36207	−0.791239
$141$	−1.13673	−0.0957300
$142$	−33.3098	−2.79530
$143$	−2.60200	−0.217590
$144$	−32.4131	−2.70109
$145$	3.56987	0.296461
$146$	37.7352	3.12299
$147$	−0.645804	−0.0532650
$148$	−30.1594	−2.47909
$149$	3.95012	0.323607	0.161803	−	0.986823i	$-0.448269\pi$
$149$	3.95012	0.323607	0.161803	+	0.986823i	$0.448269\pi$
$150$	0.490440	0.0400442
$151$	8.43269	0.686242	0.343121	−	0.939291i	$-0.388516\pi$
$151$	8.43269	0.686242	0.343121	+	0.939291i	$0.388516\pi$
$152$	7.90900	0.641505
$153$	−12.8324	−1.03744
$154$	−4.95955	−0.399652
$155$	3.56972	0.286726
$156$	2.40892	0.192868
$157$	−6.58825	−0.525800	−0.262900	−	0.964823i	$-0.584679\pi$
$157$	−6.58825	−0.525800	−0.262900	+	0.964823i	$0.584679\pi$
$158$	−28.5990	−2.27522
$159$	1.41461	0.112186
$160$	−13.0811	−1.03415
$161$	10.1863	0.802791
$162$	22.9812	1.80557
$163$	−5.28336	−0.413825	−0.206912	−	0.978360i	$-0.566341\pi$
$163$	−5.28336	−0.413825	−0.206912	+	0.978360i	$0.566341\pi$
$164$	−27.0218	−2.11005
$165$	0.185485	0.0144400
$166$	42.0700	3.26527
$167$	14.1738	1.09680	0.548399	−	0.836217i	$-0.315238\pi$
$167$	14.1738	1.09680	0.548399	+	0.836217i	$0.315238\pi$
$168$	2.75168	0.212297
$169$	−6.22959	−0.479199
$170$	−11.4412	−0.877500
$171$	−2.96560	−0.226785
$172$	−24.8139	−1.89204
$173$	25.7188	1.95536	0.977681	−	0.210096i	$-0.0673775\pi$
$173$	25.7188	1.95536	0.977681	+	0.210096i	$0.0673775\pi$
$174$	−1.75081	−0.132728
$175$	1.87571	0.141791
$176$	−10.9297	−0.823857
$177$	−1.02278	−0.0768768
$178$	5.22094	0.391326
$179$	−18.5258	−1.38468	−0.692340	−	0.721571i	$-0.743420\pi$
$179$	−18.5258	−1.38468	−0.692340	+	0.721571i	$0.743420\pi$
$180$	14.8019	1.10327
$181$	4.78848	0.355925	0.177962	−	0.984037i	$-0.443049\pi$
$181$	4.78848	0.355925	0.177962	+	0.984037i	$0.443049\pi$
$182$	12.9048	0.956565
$183$	−2.23303	−0.165070
$184$	42.9507	3.16637
$185$	6.04251	0.444255
$186$	−1.75073	−0.128370
$187$	−4.32709	−0.316428
$188$	−30.5882	−2.23087
$189$	−2.07553	−0.150973
$190$	−2.64409	−0.191822
$191$	0.948263	0.0686139	0.0343070	−	0.999411i	$-0.489078\pi$
$191$	0.948263	0.0686139	0.0343070	+	0.999411i	$0.489078\pi$
$192$	2.36088	0.170382
$193$	−10.7442	−0.773385	−0.386692	−	0.922209i	$-0.626382\pi$
$193$	−10.7442	−0.773385	−0.386692	+	0.922209i	$0.626382\pi$
$194$	−22.7274	−1.63173
$195$	−0.482633	−0.0345621
$196$	−17.3779	−1.24128
$197$	11.5164	0.820507	0.410253	−	0.911972i	$-0.365440\pi$
$197$	11.5164	0.820507	0.410253	+	0.911972i	$0.365440\pi$
$198$	7.84130	0.557257
$199$	2.52922	0.179292	0.0896459	−	0.995974i	$-0.471426\pi$
$199$	2.52922	0.179292	0.0896459	+	0.995974i	$0.471426\pi$
$200$	7.90900	0.559251
$201$	−0.546059	−0.0385160
$202$	29.4169	2.06977
$203$	−6.69605	−0.469971
$204$	4.00599	0.280476
$205$	5.41389	0.378122
$206$	21.5311	1.50014
$207$	−16.1050	−1.11937
$208$	28.4391	1.97190
$209$	−1.00000	−0.0691714
$210$	−0.919924	−0.0634808
$211$	−8.26595	−0.569051	−0.284526	−	0.958668i	$-0.591836\pi$
$211$	−8.26595	−0.569051	−0.284526	+	0.958668i	$0.591836\pi$
$212$	38.0655	2.61435
$213$	−2.33671	−0.160109
$214$	28.1230	1.92245
$215$	4.97152	0.339055
$216$	−8.75155	−0.595468
$217$	−6.69577	−0.454538
$218$	42.7214	2.89346
$219$	2.64716	0.178879
$220$	4.99120	0.336507
$221$	11.2591	0.757368
$222$	−2.96349	−0.198896
$223$	−1.98558	−0.132964	−0.0664822	−	0.997788i	$-0.521178\pi$
$223$	−1.98558	−0.132964	−0.0664822	+	0.997788i	$0.521178\pi$
$224$	24.5364	1.63940
$225$	−2.96560	−0.197706
$226$	−34.5350	−2.29723
$227$	9.94028	0.659759	0.329880	−	0.944023i	$-0.392992\pi$
$227$	9.94028	0.659759	0.329880	+	0.944023i	$0.392992\pi$
$228$	0.925795	0.0613123
$229$	14.4931	0.957732	0.478866	−	0.877888i	$-0.341048\pi$
$229$	14.4931	0.957732	0.478866	+	0.877888i	$0.341048\pi$
$230$	−14.3590	−0.946806
$231$	−0.347917	−0.0228913
$232$	−28.2341	−1.85366
$233$	14.3928	0.942905	0.471453	−	0.881891i	$-0.343730\pi$
$233$	14.3928	0.942905	0.471453	+	0.881891i	$0.343730\pi$
$234$	−20.4031	−1.33379
$235$	6.12841	0.399774
$236$	−27.5218	−1.79152
$237$	−2.00625	−0.130320
$238$	21.4604	1.39107
$239$	1.23081	0.0796146	0.0398073	−	0.999207i	$-0.487326\pi$
$239$	1.23081	0.0796146	0.0398073	+	0.999207i	$0.487326\pi$
$240$	−2.02730	−0.130862
$241$	−14.8196	−0.954615	−0.477308	−	0.878736i	$-0.658387\pi$
$241$	−14.8196	−0.954615	−0.477308	+	0.878736i	$0.658387\pi$
$242$	2.64409	0.169968
$243$	4.93174	0.316372
$244$	−60.0882	−3.84676
$245$	3.48170	0.222437
$246$	−2.65518	−0.169288
$247$	2.60200	0.165561
$248$	−28.2329	−1.79279
$249$	2.95125	0.187028
$250$	−2.64409	−0.167227
$251$	11.8145	0.745724	0.372862	−	0.927887i	$-0.378377\pi$
$251$	11.8145	0.745724	0.372862	+	0.927887i	$0.378377\pi$
$252$	−27.7641	−1.74897
$253$	−5.43061	−0.341420
$254$	−3.63555	−0.228115
$255$	−0.802611	−0.0502614
$256$	−5.64639	−0.352899
$257$	6.28684	0.392162	0.196081	−	0.980588i	$-0.437178\pi$
$257$	6.28684	0.392162	0.196081	+	0.980588i	$0.437178\pi$
$258$	−2.43823	−0.151798
$259$	−11.3340	−0.704262
$260$	−12.9871	−0.805427
$261$	10.5868	0.655306
$262$	45.1208	2.78757
$263$	−3.29644	−0.203267	−0.101634	−	0.994822i	$-0.532407\pi$
$263$	−3.29644	−0.203267	−0.101634	+	0.994822i	$0.532407\pi$
$264$	−1.46700	−0.0902879
$265$	−7.62653	−0.468494
$266$	4.95955	0.304090
$267$	0.366254	0.0224144
$268$	−14.6938	−0.897568
$269$	11.2424	0.685460	0.342730	−	0.939434i	$-0.388648\pi$
$269$	11.2424	0.685460	0.342730	+	0.939434i	$0.388648\pi$
$270$	2.92576	0.178056
$271$	10.6614	0.647632	0.323816	−	0.946120i	$-0.395034\pi$
$271$	10.6614	0.647632	0.323816	+	0.946120i	$0.395034\pi$
$272$	47.2937	2.86760
$273$	0.905282	0.0547902
$274$	−43.5471	−2.63078
$275$	−1.00000	−0.0603023
$276$	5.02763	0.302628
$277$	−25.2333	−1.51612	−0.758061	−	0.652183i	$-0.773853\pi$
$277$	−25.2333	−1.51612	−0.758061	+	0.652183i	$0.773853\pi$
$278$	−37.7689	−2.26523
$279$	10.5863	0.633787
$280$	−14.8350	−0.886563
$281$	−10.4638	−0.624220	−0.312110	−	0.950046i	$-0.601036\pi$
$281$	−10.4638	−0.624220	−0.312110	+	0.950046i	$0.601036\pi$
$282$	−3.00562	−0.178982
$283$	−14.9555	−0.889014	−0.444507	−	0.895775i	$-0.646621\pi$
$283$	−14.9555	−0.889014	−0.444507	+	0.895775i	$0.646621\pi$
$284$	−62.8784	−3.73114
$285$	−0.185485	−0.0109872
$286$	−6.87992	−0.406818
$287$	−10.1549	−0.599425
$288$	−38.7932	−2.28591
$289$	1.72367	0.101392
$290$	9.43905	0.554280
$291$	−1.59435	−0.0934624
$292$	71.2322	4.16855
$293$	−0.461787	−0.0269779	−0.0134890	−	0.999909i	$-0.504294\pi$
$293$	−0.461787	−0.0269779	−0.0134890	+	0.999909i	$0.504294\pi$
$294$	−1.70756	−0.0995870
$295$	5.51407	0.321042
$296$	−47.7903	−2.77775
$297$	1.10653	0.0642074
$298$	10.4445	0.605032
$299$	14.1305	0.817186
$300$	0.925795	0.0534508
$301$	−9.32515	−0.537493
$302$	22.2968	1.28303
$303$	2.06362	0.118552
$304$	10.9297	0.626861
$305$	12.0388	0.689341
$306$	−33.9300	−1.93965
$307$	14.1600	0.808156	0.404078	−	0.914725i	$-0.367592\pi$
$307$	14.1600	0.808156	0.404078	+	0.914725i	$0.367592\pi$
$308$	−9.36207	−0.533453
$309$	1.51043	0.0859252
$310$	9.43864	0.536079
$311$	7.46064	0.423054	0.211527	−	0.977372i	$-0.432156\pi$
$311$	7.46064	0.423054	0.211527	+	0.977372i	$0.432156\pi$
$312$	3.81715	0.216103
$313$	6.20034	0.350464	0.175232	−	0.984527i	$-0.443932\pi$
$313$	6.20034	0.350464	0.175232	+	0.984527i	$0.443932\pi$
$314$	−17.4199	−0.983063
$315$	5.56261	0.313417
$316$	−53.9859	−3.03695
$317$	5.39294	0.302898	0.151449	−	0.988465i	$-0.451606\pi$
$317$	5.39294	0.302898	0.151449	+	0.988465i	$0.451606\pi$
$318$	3.74035	0.209748
$319$	3.56987	0.199874
$320$	−12.7281	−0.711524
$321$	1.97286	0.110114
$322$	26.9334	1.50094
$323$	4.32709	0.240765
$324$	43.3812	2.41007
$325$	2.60200	0.144333
$326$	−13.9697	−0.773708
$327$	2.99695	0.165732
$328$	−42.8184	−2.36425
$329$	−11.4952	−0.633748
$330$	0.490440	0.0269978
$331$	31.8456	1.75039	0.875195	−	0.483770i	$-0.160733\pi$
$331$	31.8456	1.75039	0.875195	+	0.483770i	$0.160733\pi$
$332$	79.4149	4.35846
$333$	17.9197	0.981991
$334$	37.4767	2.05063
$335$	2.94394	0.160845
$336$	3.80263	0.207451
$337$	20.6584	1.12533	0.562667	−	0.826684i	$-0.309775\pi$
$337$	20.6584	1.12533	0.562667	+	0.826684i	$0.309775\pi$
$338$	−16.4716	−0.895936
$339$	−2.42266	−0.131581
$340$	−21.5974	−1.17128
$341$	3.56972	0.193311
$342$	−7.84130	−0.424009
$343$	−19.6607	−1.06158
$344$	−39.3198	−2.11998
$345$	−1.00730	−0.0542311
$346$	68.0027	3.65585
$347$	−17.0159	−0.913461	−0.456730	−	0.889605i	$-0.650980\pi$
$347$	−17.0159	−0.913461	−0.456730	+	0.889605i	$0.650980\pi$
$348$	−3.30497	−0.177165
$349$	7.20959	0.385920	0.192960	−	0.981207i	$-0.438191\pi$
$349$	7.20959	0.385920	0.192960	+	0.981207i	$0.438191\pi$
$350$	4.95955	0.265099
$351$	−2.87919	−0.153680
$352$	−13.0811	−0.697224
$353$	31.5229	1.67780	0.838898	−	0.544289i	$-0.183201\pi$
$353$	31.5229	1.67780	0.838898	+	0.544289i	$0.183201\pi$
$354$	−2.70432	−0.143733
$355$	12.5978	0.668624
$356$	9.85549	0.522340
$357$	1.50547	0.0796778
$358$	−48.9838	−2.58887
$359$	23.4006	1.23504	0.617518	−	0.786556i	$-0.288138\pi$
$359$	23.4006	1.23504	0.617518	+	0.786556i	$0.288138\pi$
$360$	23.4549	1.23618
$361$	1.00000	0.0526316
$362$	12.6612	0.665456
$363$	0.185485	0.00973546
$364$	24.3601	1.27682
$365$	−14.2715	−0.747007
$366$	−5.90432	−0.308624
$367$	11.4989	0.600236	0.300118	−	0.953902i	$-0.402974\pi$
$367$	11.4989	0.600236	0.300118	+	0.953902i	$0.402974\pi$
$368$	59.3549	3.09409
$369$	16.0554	0.835810
$370$	15.9769	0.830602
$371$	14.3052	0.742688
$372$	−3.30482	−0.171347
$373$	−29.5747	−1.53132	−0.765659	−	0.643246i	$-0.777587\pi$
$373$	−29.5747	−1.53132	−0.765659	+	0.643246i	$0.777587\pi$
$374$	−11.4412	−0.591610
$375$	−0.185485	−0.00957842
$376$	−48.4697	−2.49963
$377$	−9.28881	−0.478398
$378$	−5.48790	−0.282267
$379$	9.83336	0.505106	0.252553	−	0.967583i	$-0.418730\pi$
$379$	9.83336	0.505106	0.252553	+	0.967583i	$0.418730\pi$
$380$	−4.99120	−0.256043
$381$	−0.255037	−0.0130660
$382$	2.50729	0.128284
$383$	16.0204	0.818602	0.409301	−	0.912399i	$-0.365773\pi$
$383$	16.0204	0.818602	0.409301	+	0.912399i	$0.365773\pi$
$384$	1.38968	0.0709169
$385$	1.87571	0.0955952
$386$	−28.4086	−1.44596
$387$	14.7435	0.749455
$388$	−42.9021	−2.17803
$389$	−14.9919	−0.760117	−0.380059	−	0.924962i	$-0.624096\pi$
$389$	−14.9919	−0.760117	−0.380059	+	0.924962i	$0.624096\pi$
$390$	−1.27612	−0.0646191
$391$	23.4987	1.18838
$392$	−27.5367	−1.39082
$393$	3.16527	0.159667
$394$	30.4503	1.53406
$395$	10.8162	0.544223
$396$	14.8019	0.743823
$397$	−28.4247	−1.42660	−0.713298	−	0.700861i	$-0.752800\pi$
$397$	−28.4247	−1.42660	−0.713298	+	0.700861i	$0.752800\pi$
$398$	6.68749	0.335213
$399$	0.347917	0.0174177
$400$	10.9297	0.546485
$401$	−18.0673	−0.902237	−0.451118	−	0.892464i	$-0.648975\pi$
$401$	−18.0673	−0.902237	−0.451118	+	0.892464i	$0.648975\pi$
$402$	−1.44383	−0.0720115
$403$	−9.28841	−0.462688
$404$	55.5298	2.76271
$405$	−8.69154	−0.431886
$406$	−17.7050	−0.878682
$407$	6.04251	0.299516
$408$	6.34785	0.314266
$409$	−14.0939	−0.696899	−0.348449	−	0.937328i	$-0.613292\pi$
$409$	−14.0939	−0.696899	−0.348449	+	0.937328i	$0.613292\pi$
$410$	14.3148	0.706957
$411$	−3.05487	−0.150686
$412$	40.6439	2.00238
$413$	−10.3428	−0.508937
$414$	−42.5830	−2.09284
$415$	−15.9110	−0.781039
$416$	34.0370	1.66880
$417$	−2.64952	−0.129748
$418$	−2.64409	−0.129327
$419$	22.2007	1.08457	0.542286	−	0.840194i	$-0.317559\pi$
$419$	22.2007	1.08457	0.542286	+	0.840194i	$0.317559\pi$
$420$	−1.73653	−0.0847338
$421$	33.6904	1.64197	0.820984	−	0.570951i	$-0.193425\pi$
$421$	33.6904	1.64197	0.820984	+	0.570951i	$0.193425\pi$
$422$	−21.8559	−1.06393
$423$	18.1744	0.883669
$424$	60.3182	2.92931
$425$	4.32709	0.209894
$426$	−6.17848	−0.299348
$427$	−22.5814	−1.09279
$428$	53.0874	2.56607
$429$	−0.482633	−0.0233017
$430$	13.1451	0.633915
$431$	−19.4736	−0.938010	−0.469005	−	0.883196i	$-0.655387\pi$
$431$	−19.4736	−0.938010	−0.469005	+	0.883196i	$0.655387\pi$
$432$	−12.0940	−0.581875
$433$	5.93659	0.285294	0.142647	−	0.989774i	$-0.454439\pi$
$433$	5.93659	0.285294	0.142647	+	0.989774i	$0.454439\pi$
$434$	−17.7042	−0.849828
$435$	0.662158	0.0317481
$436$	80.6445	3.86217
$437$	5.43061	0.259781
$438$	6.99933	0.334441
$439$	−8.79894	−0.419951	−0.209975	−	0.977707i	$-0.567338\pi$
$439$	−8.79894	−0.419951	−0.209975	+	0.977707i	$0.567338\pi$
$440$	7.90900	0.377047
$441$	10.3253	0.491681
$442$	29.7700	1.41602
$443$	−8.44848	−0.401399	−0.200700	−	0.979653i	$-0.564322\pi$
$443$	−8.44848	−0.401399	−0.200700	+	0.979653i	$0.564322\pi$
$444$	−5.59413	−0.265486
$445$	−1.97457	−0.0936037
$446$	−5.25005	−0.248597
$447$	0.732690	0.0346550
$448$	23.8743	1.12796
$449$	−13.1022	−0.618332	−0.309166	−	0.951008i	$-0.600050\pi$
$449$	−13.1022	−0.618332	−0.309166	+	0.951008i	$0.600050\pi$
$450$	−7.84130	−0.369642
$451$	5.41389	0.254930
$452$	−65.1911	−3.06633
$453$	1.56414	0.0734897
$454$	26.2830	1.23352
$455$	−4.88061	−0.228807
$456$	1.46700	0.0686988
$457$	12.5771	0.588334	0.294167	−	0.955754i	$-0.404958\pi$
$457$	12.5771	0.588334	0.294167	+	0.955754i	$0.404958\pi$
$458$	38.3211	1.79063
$459$	−4.78805	−0.223487
$460$	−27.1053	−1.26379
$461$	−39.2043	−1.82592	−0.912962	−	0.408045i	$-0.866211\pi$
$461$	−39.2043	−1.82592	−0.912962	+	0.408045i	$0.866211\pi$
$462$	−0.919924	−0.0427988
$463$	7.83358	0.364057	0.182029	−	0.983293i	$-0.441734\pi$
$463$	7.83358	0.364057	0.182029	+	0.983293i	$0.441734\pi$
$464$	−39.0176	−1.81135
$465$	0.662130	0.0307055
$466$	38.0559	1.76291
$467$	−8.02211	−0.371219	−0.185610	−	0.982624i	$-0.559426\pi$
$467$	−8.02211	−0.371219	−0.185610	+	0.982624i	$0.559426\pi$
$468$	−38.5145	−1.78033
$469$	−5.52200	−0.254982
$470$	16.2041	0.747438
$471$	−1.22202	−0.0563079
$472$	−43.6108	−2.00735
$473$	4.97152	0.228591
$474$	−5.30470	−0.243653
$475$	1.00000	0.0458831
$476$	40.5105	1.85679
$477$	−22.6172	−1.03557
$478$	3.25437	0.148852
$479$	−20.5202	−0.937594	−0.468797	−	0.883306i	$-0.655312\pi$
$479$	−20.5202	−0.937594	−0.468797	+	0.883306i	$0.655312\pi$
$480$	−2.42635	−0.110747
$481$	−15.7226	−0.716890
$482$	−39.1844	−1.78480
$483$	1.88940	0.0859709
$484$	4.99120	0.226873
$485$	8.59555	0.390304
$486$	13.0400	0.591505
$487$	13.4865	0.611134	0.305567	−	0.952171i	$-0.401154\pi$
$487$	13.4865	0.611134	0.305567	+	0.952171i	$0.401154\pi$
$488$	−95.2151	−4.31019
$489$	−0.979985	−0.0443165
$490$	9.20591	0.415881
$491$	40.1691	1.81281	0.906404	−	0.422412i	$-0.138816\pi$
$491$	40.1691	1.81281	0.906404	+	0.422412i	$0.138816\pi$
$492$	−5.01215	−0.225965
$493$	−15.4471	−0.695704
$494$	6.87992	0.309542
$495$	−2.96560	−0.133294
$496$	−39.0159	−1.75187
$497$	−23.6299	−1.05995
$498$	7.80338	0.349678
$499$	39.8970	1.78603	0.893017	−	0.450024i	$-0.148584\pi$
$499$	39.8970	1.78603	0.893017	+	0.450024i	$0.148584\pi$
$500$	−4.99120	−0.223213
$501$	2.62902	0.117456
$502$	31.2385	1.39424
$503$	−20.4729	−0.912844	−0.456422	−	0.889764i	$-0.650869\pi$
$503$	−20.4729	−0.912844	−0.456422	+	0.889764i	$0.650869\pi$
$504$	−43.9947	−1.95968
$505$	−11.1255	−0.495080
$506$	−14.3590	−0.638336
$507$	−1.15550	−0.0513174
$508$	−6.86277	−0.304486
$509$	−23.1998	−1.02831	−0.514157	−	0.857696i	$-0.671895\pi$
$509$	−23.1998	−1.02831	−0.514157	+	0.857696i	$0.671895\pi$
$510$	−2.12217	−0.0939714
$511$	26.7693	1.18421
$512$	−29.9138	−1.32202
$513$	−1.10653	−0.0488545
$514$	16.6229	0.733207
$515$	−8.14311	−0.358828
$516$	−4.60261	−0.202619
$517$	6.12841	0.269527
$518$	−29.9682	−1.31673
$519$	4.77045	0.209400
$520$	−20.5792	−0.902459
$521$	7.43019	0.325523	0.162761	−	0.986665i	$-0.447960\pi$
$521$	7.43019	0.325523	0.162761	+	0.986665i	$0.447960\pi$
$522$	27.9924	1.22519
$523$	−18.6998	−0.817683	−0.408842	−	0.912605i	$-0.634067\pi$
$523$	−18.6998	−0.817683	−0.408842	+	0.912605i	$0.634067\pi$
$524$	85.1738	3.72083
$525$	0.347917	0.0151844
$526$	−8.71607	−0.380039
$527$	−15.4465	−0.672858
$528$	−2.02730	−0.0882268
$529$	6.49154	0.282241
$530$	−20.1652	−0.875921
$531$	16.3525	0.709638
$532$	9.36207	0.405897
$533$	−14.0869	−0.610173
$534$	0.968408	0.0419071
$535$	−10.6362	−0.459842
$536$	−23.2837	−1.00570
$537$	−3.43626	−0.148285
$538$	29.7258	1.28157
$539$	3.48170	0.149967
$540$	5.52292	0.237669
$541$	−25.5055	−1.09657	−0.548285	−	0.836292i	$-0.684719\pi$
$541$	−25.5055	−1.09657	−0.548285	+	0.836292i	$0.684719\pi$
$542$	28.1896	1.21085
$543$	0.888193	0.0381160
$544$	56.6029	2.42683
$545$	−16.1573	−0.692104
$546$	2.39365	0.102439
$547$	−34.4453	−1.47277	−0.736387	−	0.676560i	$-0.763470\pi$
$547$	−34.4453	−1.47277	−0.736387	+	0.676560i	$0.763470\pi$
$548$	−82.2031	−3.51154
$549$	35.7023	1.52374
$550$	−2.64409	−0.112744
$551$	−3.56987	−0.152082
$552$	7.96673	0.339087
$553$	−20.2881	−0.862739
$554$	−66.7191	−2.83462
$555$	1.12080	0.0475752
$556$	−71.2957	−3.02361
$557$	−3.25423	−0.137886	−0.0689431	−	0.997621i	$-0.521963\pi$
$557$	−3.25423	−0.137886	−0.0689431	+	0.997621i	$0.521963\pi$
$558$	27.9912	1.18496
$559$	−12.9359	−0.547131
$560$	−20.5010	−0.866325
$561$	−0.802611	−0.0338863
$562$	−27.6673	−1.16707
$563$	11.8981	0.501443	0.250722	−	0.968059i	$-0.419332\pi$
$563$	11.8981	0.501443	0.250722	+	0.968059i	$0.419332\pi$
$564$	−5.67366	−0.238904
$565$	13.0612	0.549489
$566$	−39.5437	−1.66215
$567$	16.3028	0.684655
$568$	−99.6364	−4.18065
$569$	−43.1186	−1.80763	−0.903814	−	0.427926i	$-0.859244\pi$
$569$	−43.1186	−1.80763	−0.903814	+	0.427926i	$0.859244\pi$
$570$	−0.490440	−0.0205423
$571$	44.5572	1.86466	0.932331	−	0.361606i	$-0.117772\pi$
$571$	44.5572	1.86466	0.932331	+	0.361606i	$0.117772\pi$
$572$	−12.9871	−0.543019
$573$	0.175889	0.00734786
$574$	−26.8505	−1.12072
$575$	5.43061	0.226472
$576$	−37.7465	−1.57277
$577$	−13.0109	−0.541651	−0.270825	−	0.962628i	$-0.587297\pi$
$577$	−13.0109	−0.541651	−0.270825	+	0.962628i	$0.587297\pi$
$578$	4.55753	0.189568
$579$	−1.99289	−0.0828218
$580$	17.8179	0.739850
$581$	29.8445	1.23816
$582$	−4.21560	−0.174742
$583$	−7.62653	−0.315858
$584$	112.874	4.67075
$585$	7.71648	0.319037
$586$	−1.22101	−0.0504393
$587$	22.0197	0.908851	0.454426	−	0.890785i	$-0.349845\pi$
$587$	22.0197	0.908851	0.454426	+	0.890785i	$0.349845\pi$
$588$	−3.22334	−0.132928
$589$	−3.56972	−0.147088
$590$	14.5797	0.600236
$591$	2.13612	0.0878681
$592$	−66.0428	−2.71434
$593$	30.3538	1.24648	0.623240	−	0.782031i	$-0.285816\pi$
$593$	30.3538	1.24648	0.623240	+	0.782031i	$0.285816\pi$
$594$	2.92576	0.120046
$595$	−8.11638	−0.332739
$596$	19.7159	0.807593
$597$	0.469134	0.0192004
$598$	37.3622	1.52785
$599$	22.0406	0.900554	0.450277	−	0.892889i	$-0.351325\pi$
$599$	22.0406	0.900554	0.450277	+	0.892889i	$0.351325\pi$
$600$	1.46700	0.0598902
$601$	40.3514	1.64597	0.822984	−	0.568065i	$-0.192308\pi$
$601$	40.3514	1.64597	0.822984	+	0.568065i	$0.192308\pi$
$602$	−24.6565	−1.00492
$603$	8.73055	0.355535
$604$	42.0892	1.71259
$605$	−1.00000	−0.0406558
$606$	5.45640	0.221651
$607$	32.2446	1.30877	0.654384	−	0.756162i	$-0.272928\pi$
$607$	32.2446	1.30877	0.654384	+	0.756162i	$0.272928\pi$
$608$	13.0811	0.530508
$609$	−1.24202	−0.0503292
$610$	31.8317	1.28883
$611$	−15.9461	−0.645112
$612$	−64.0490	−2.58903
$613$	23.7206	0.958067	0.479034	−	0.877797i	$-0.340987\pi$
$613$	23.7206	0.958067	0.479034	+	0.877797i	$0.340987\pi$
$614$	37.4404	1.51097
$615$	1.00420	0.0404931
$616$	−14.8350	−0.597720
$617$	−44.6024	−1.79563	−0.897813	−	0.440378i	$-0.854844\pi$
$617$	−44.6024	−1.79563	−0.897813	+	0.440378i	$0.854844\pi$
$618$	3.99370	0.160650
$619$	−41.5788	−1.67119	−0.835596	−	0.549344i	$-0.814878\pi$
$619$	−41.5788	−1.67119	−0.835596	+	0.549344i	$0.814878\pi$
$620$	17.8172	0.715555
$621$	−6.00914	−0.241138
$622$	19.7266	0.790965
$623$	3.70373	0.148387
$624$	5.27503	0.211170
$625$	1.00000	0.0400000
$626$	16.3942	0.655246
$627$	−0.185485	−0.00740757
$628$	−32.8833	−1.31219
$629$	−26.1465	−1.04253
$630$	14.7080	0.585982
$631$	−13.0707	−0.520337	−0.260168	−	0.965563i	$-0.583778\pi$
$631$	−13.0707	−0.520337	−0.260168	+	0.965563i	$0.583778\pi$
$632$	−85.5455	−3.40282
$633$	−1.53321	−0.0609397
$634$	14.2594	0.566314
$635$	1.37497	0.0545641
$636$	7.06060	0.279971
$637$	−9.05938	−0.358946
$638$	9.43905	0.373696
$639$	37.3601	1.47794
$640$	−7.49213	−0.296153
$641$	−11.4907	−0.453857	−0.226928	−	0.973911i	$-0.572868\pi$
$641$	−11.4907	−0.453857	−0.226928	+	0.973911i	$0.572868\pi$
$642$	5.21641	0.205875
$643$	16.2149	0.639453	0.319726	−	0.947510i	$-0.396409\pi$
$643$	16.2149	0.639453	0.319726	+	0.947510i	$0.396409\pi$
$644$	50.8418	2.00345
$645$	0.922144	0.0363094
$646$	11.4412	0.450148
$647$	−8.28602	−0.325757	−0.162878	−	0.986646i	$-0.552078\pi$
$647$	−8.28602	−0.325757	−0.162878	+	0.986646i	$0.552078\pi$
$648$	68.7414	2.70042
$649$	5.51407	0.216446
$650$	6.87992	0.269853
$651$	−1.24197	−0.0486765
$652$	−26.3703	−1.03274
$653$	−39.2515	−1.53603	−0.768014	−	0.640432i	$-0.778755\pi$
$653$	−39.2515	−1.53603	−0.768014	+	0.640432i	$0.778755\pi$
$654$	7.92420	0.309861
$655$	−17.0648	−0.666776
$656$	−59.1721	−2.31028
$657$	−42.3236	−1.65120
$658$	−30.3942	−1.18489
$659$	40.4764	1.57674	0.788369	−	0.615203i	$-0.210926\pi$
$659$	40.4764	1.57674	0.788369	+	0.615203i	$0.210926\pi$
$660$	0.925795	0.0360365
$661$	1.62911	0.0633649	0.0316824	−	0.999498i	$-0.489913\pi$
$661$	1.62911	0.0633649	0.0316824	+	0.999498i	$0.489913\pi$
$662$	84.2025	3.27262
$663$	2.08840	0.0811065
$664$	125.840	4.88354
$665$	−1.87571	−0.0727371
$666$	47.3811	1.83598
$667$	−19.3866	−0.750651
$668$	70.7441	2.73717
$669$	−0.368296	−0.0142392
$670$	7.78405	0.300724
$671$	12.0388	0.464754
$672$	4.55113	0.175564
$673$	2.41503	0.0930925	0.0465462	−	0.998916i	$-0.485179\pi$
$673$	2.41503	0.0930925	0.0465462	+	0.998916i	$0.485179\pi$
$674$	54.6226	2.10398
$675$	−1.10653	−0.0425904
$676$	−31.0931	−1.19589
$677$	−33.9619	−1.30526	−0.652630	−	0.757676i	$-0.726335\pi$
$677$	−33.9619	−1.30526	−0.652630	+	0.757676i	$0.726335\pi$
$678$	−6.40573	−0.246011
$679$	−16.1228	−0.618736
$680$	−34.2229	−1.31239
$681$	1.84378	0.0706537
$682$	9.43864	0.361424
$683$	30.3370	1.16081	0.580406	−	0.814327i	$-0.302894\pi$
$683$	30.3370	1.16081	0.580406	+	0.814327i	$0.302894\pi$
$684$	−14.8019	−0.565964
$685$	16.4696	0.629271
$686$	−51.9845	−1.98478
$687$	2.68826	0.102564
$688$	−54.3372	−2.07159
$689$	19.8442	0.756005
$690$	−2.66339	−0.101393
$691$	47.1518	1.79374	0.896870	−	0.442294i	$-0.145835\pi$
$691$	47.1518	1.79374	0.896870	+	0.442294i	$0.145835\pi$
$692$	128.368	4.87980
$693$	5.56261	0.211306
$694$	−44.9915	−1.70785
$695$	14.2843	0.541833
$696$	−5.23701	−0.198508
$697$	−23.4263	−0.887336
$698$	19.0628	0.721537
$699$	2.66966	0.100976
$700$	9.36207	0.353853
$701$	−8.75344	−0.330613	−0.165306	−	0.986242i	$-0.552861\pi$
$701$	−8.75344	−0.330613	−0.165306	+	0.986242i	$0.552861\pi$
$702$	−7.61284	−0.287328
$703$	−6.04251	−0.227898
$704$	−12.7281	−0.479709
$705$	1.13673	0.0428118
$706$	83.3494	3.13689
$707$	20.8683	0.784834
$708$	−5.10490	−0.191854
$709$	−7.93205	−0.297895	−0.148947	−	0.988845i	$-0.547588\pi$
$709$	−7.93205	−0.297895	−0.148947	+	0.988845i	$0.547588\pi$
$710$	33.3098	1.25009
$711$	32.0765	1.20296
$712$	15.6169	0.585268
$713$	−19.3857	−0.726002
$714$	3.98059	0.148970
$715$	2.60200	0.0973093
$716$	−92.4658	−3.45561
$717$	0.228297	0.00852593
$718$	61.8733	2.30909
$719$	48.3802	1.80428	0.902138	−	0.431448i	$-0.141997\pi$
$719$	48.3802	1.80428	0.902138	+	0.431448i	$0.141997\pi$
$720$	32.4131	1.20796
$721$	15.2741	0.568839
$722$	2.64409	0.0984028
$723$	−2.74882	−0.102230
$724$	23.9003	0.888247
$725$	−3.56987	−0.132582
$726$	0.490440	0.0182019
$727$	3.50010	0.129812	0.0649058	−	0.997891i	$-0.479325\pi$
$727$	3.50010	0.129812	0.0649058	+	0.997891i	$0.479325\pi$
$728$	38.6008	1.43064
$729$	−25.1599	−0.931846
$730$	−37.7352	−1.39664
$731$	−21.5122	−0.795657
$732$	−11.1455	−0.411949
$733$	41.7547	1.54225	0.771123	−	0.636686i	$-0.219695\pi$
$733$	41.7547	1.54225	0.771123	+	0.636686i	$0.219695\pi$
$734$	30.4040	1.12223
$735$	0.645804	0.0238208
$736$	71.0382	2.61850
$737$	2.94394	0.108442
$738$	42.4519	1.56268
$739$	4.29195	0.157882	0.0789409	−	0.996879i	$-0.474846\pi$
$739$	4.29195	0.157882	0.0789409	+	0.996879i	$0.474846\pi$
$740$	30.1594	1.10868
$741$	0.482633	0.0177300
$742$	37.8242	1.38857
$743$	−1.67118	−0.0613097	−0.0306549	−	0.999530i	$-0.509759\pi$
$743$	−1.67118	−0.0613097	−0.0306549	+	0.999530i	$0.509759\pi$
$744$	−5.23679	−0.191990
$745$	−3.95012	−0.144721
$746$	−78.1980	−2.86303
$747$	−47.1855	−1.72643
$748$	−21.5974	−0.789678
$749$	19.9504	0.728973
$750$	−0.490440	−0.0179083
$751$	−51.8305	−1.89132	−0.945660	−	0.325156i	$-0.894583\pi$
$751$	−51.8305	−1.89132	−0.945660	+	0.325156i	$0.894583\pi$
$752$	−66.9817	−2.44257
$753$	2.19141	0.0798596
$754$	−24.5604	−0.894438
$755$	−8.43269	−0.306897
$756$	−10.3594	−0.376768
$757$	−33.1222	−1.20385	−0.601924	−	0.798553i	$-0.705599\pi$
$757$	−33.1222	−1.20385	−0.601924	+	0.798553i	$0.705599\pi$
$758$	26.0003	0.944372
$759$	−1.00730	−0.0365626
$760$	−7.90900	−0.286890
$761$	8.54710	0.309832	0.154916	−	0.987928i	$-0.450489\pi$
$761$	8.54710	0.309832	0.154916	+	0.987928i	$0.450489\pi$
$762$	−0.674341	−0.0244288
$763$	30.3065	1.09717
$764$	4.73297	0.171233
$765$	12.8324	0.463956
$766$	42.3593	1.53050
$767$	−14.3476	−0.518063
$768$	−1.04732	−0.0377920
$769$	4.98659	0.179821	0.0899106	−	0.995950i	$-0.471342\pi$
$769$	4.98659	0.179821	0.0899106	+	0.995950i	$0.471342\pi$
$770$	4.95955	0.178730
$771$	1.16612	0.0419966
$772$	−53.6265	−1.93006
$773$	27.3064	0.982142	0.491071	−	0.871120i	$-0.336606\pi$
$773$	27.3064	0.982142	0.491071	+	0.871120i	$0.336606\pi$
$774$	38.9832	1.40122
$775$	−3.56972	−0.128228
$776$	−67.9823	−2.44042
$777$	−2.10230	−0.0754195
$778$	−39.6398	−1.42116
$779$	−5.41389	−0.193973
$780$	−2.40892	−0.0862532
$781$	12.5978	0.450786
$782$	62.1327	2.22186
$783$	3.95017	0.141168
$784$	−38.0539	−1.35907
$785$	6.58825	0.235145
$786$	8.36924	0.298521
$787$	−7.28649	−0.259735	−0.129868	−	0.991531i	$-0.541455\pi$
$787$	−7.28649	−0.259735	−0.129868	+	0.991531i	$0.541455\pi$
$788$	57.4805	2.04766
$789$	−0.611441	−0.0217679
$790$	28.5990	1.01751
$791$	−24.4991	−0.871087
$792$	23.4549	0.833434
$793$	−31.3251	−1.11239
$794$	−75.1575	−2.66724
$795$	−1.41461	−0.0501710
$796$	12.6239	0.447441
$797$	−22.0769	−0.782005	−0.391003	−	0.920390i	$-0.627872\pi$
$797$	−22.0769	−0.782005	−0.391003	+	0.920390i	$0.627872\pi$
$798$	0.919924	0.0325650
$799$	−26.5182	−0.938146
$800$	13.0811	0.462486
$801$	−5.85578	−0.206904
$802$	−47.7715	−1.68687
$803$	−14.2715	−0.503632
$804$	−2.72549	−0.0961206
$805$	−10.1863	−0.359019
$806$	−24.5594	−0.865067
$807$	2.08530	0.0734059
$808$	87.9919	3.09554
$809$	8.65470	0.304283	0.152142	−	0.988359i	$-0.451383\pi$
$809$	8.65470	0.304283	0.152142	+	0.988359i	$0.451383\pi$
$810$	−22.9812	−0.807477
$811$	−19.0546	−0.669096	−0.334548	−	0.942379i	$-0.608584\pi$
$811$	−19.0546	−0.669096	−0.334548	+	0.942379i	$0.608584\pi$
$812$	−33.4214	−1.17286
$813$	1.97753	0.0693549
$814$	15.9769	0.559992
$815$	5.28336	0.185068
$816$	8.77229	0.307092
$817$	−4.97152	−0.173931
$818$	−37.2655	−1.30296
$819$	−14.4739	−0.505760
$820$	27.0218	0.943642
$821$	46.0336	1.60658	0.803292	−	0.595585i	$-0.203080\pi$
$821$	46.0336	1.60658	0.803292	+	0.595585i	$0.203080\pi$
$822$	−8.07735	−0.281730
$823$	18.2090	0.634725	0.317362	−	0.948304i	$-0.397203\pi$
$823$	18.2090	0.634725	0.317362	+	0.948304i	$0.397203\pi$
$824$	64.4039	2.24361
$825$	−0.185485	−0.00645777
$826$	−27.3473	−0.951535
$827$	33.9574	1.18082	0.590408	−	0.807105i	$-0.298967\pi$
$827$	33.9574	1.18082	0.590408	+	0.807105i	$0.298967\pi$
$828$	−80.3833	−2.79351
$829$	−29.6730	−1.03059	−0.515294	−	0.857014i	$-0.672317\pi$
$829$	−29.6730	−1.03059	−0.515294	+	0.857014i	$0.672317\pi$
$830$	−42.0700	−1.46027
$831$	−4.68041	−0.162362
$832$	33.1186	1.14818
$833$	−15.0656	−0.521992
$834$	−7.00557	−0.242583
$835$	−14.1738	−0.490503
$836$	−4.99120	−0.172624
$837$	3.95000	0.136532
$838$	58.7005	2.02777
$839$	55.6503	1.92126	0.960631	−	0.277828i	$-0.0896146\pi$
$839$	55.6503	1.92126	0.960631	+	0.277828i	$0.0896146\pi$
$840$	−2.75168	−0.0949420
$841$	−16.2560	−0.560553
$842$	89.0803	3.06991
$843$	−1.94089	−0.0668477
$844$	−41.2570	−1.42013
$845$	6.22959	0.214304
$846$	48.0547	1.65215
$847$	1.87571	0.0644503
$848$	83.3556	2.86244
$849$	−2.77403	−0.0952045
$850$	11.4412	0.392430
$851$	−32.8146	−1.12487
$852$	−11.6630	−0.399568
$853$	−54.8948	−1.87956	−0.939781	−	0.341777i	$-0.888971\pi$
$853$	−54.8948	−1.87956	−0.939781	+	0.341777i	$0.888971\pi$
$854$	−59.7072	−2.04314
$855$	2.96560	0.101421
$856$	84.1216	2.87522
$857$	−13.4196	−0.458406	−0.229203	−	0.973379i	$-0.573612\pi$
$857$	−13.4196	−0.458406	−0.229203	+	0.973379i	$0.573612\pi$
$858$	−1.27612	−0.0435662
$859$	−14.6846	−0.501031	−0.250516	−	0.968113i	$-0.580600\pi$
$859$	−14.6846	−0.501031	−0.250516	+	0.968113i	$0.580600\pi$
$860$	24.8139	0.846146
$861$	−1.88359	−0.0641924
$862$	−51.4899	−1.75375
$863$	27.4014	0.932755	0.466378	−	0.884586i	$-0.345559\pi$
$863$	27.4014	0.932755	0.466378	+	0.884586i	$0.345559\pi$
$864$	−14.4746	−0.492436
$865$	−25.7188	−0.874464
$866$	15.6969	0.533402
$867$	0.319715	0.0108581
$868$	−33.4199	−1.13435
$869$	10.8162	0.366915
$870$	1.75081	0.0593579
$871$	−7.66015	−0.259554
$872$	127.788	4.32746
$873$	25.4909	0.862737
$874$	14.3590	0.485701
$875$	−1.87571	−0.0634107
$876$	13.2125	0.446410
$877$	−10.0321	−0.338761	−0.169381	−	0.985551i	$-0.554177\pi$
$877$	−10.0321	−0.338761	−0.169381	+	0.985551i	$0.554177\pi$
$878$	−23.2652	−0.785162
$879$	−0.0856548	−0.00288906
$880$	10.9297	0.368440
$881$	23.9718	0.807632	0.403816	−	0.914840i	$-0.367684\pi$
$881$	23.9718	0.807632	0.403816	+	0.914840i	$0.367684\pi$
$882$	27.3010	0.919273
$883$	18.4667	0.621455	0.310728	−	0.950499i	$-0.399427\pi$
$883$	18.4667	0.621455	0.310728	+	0.950499i	$0.399427\pi$
$884$	56.1964	1.89009
$885$	1.02278	0.0343804
$886$	−22.3385	−0.750477
$887$	−21.6040	−0.725393	−0.362696	−	0.931907i	$-0.618144\pi$
$887$	−21.6040	−0.725393	−0.362696	+	0.931907i	$0.618144\pi$
$888$	−8.86439	−0.297470
$889$	−2.57906	−0.0864988
$890$	−5.22094	−0.175006
$891$	−8.69154	−0.291178
$892$	−9.91044	−0.331826
$893$	−6.12841	−0.205080
$894$	1.93730	0.0647929
$895$	18.5258	0.619248
$896$	14.0531	0.469481
$897$	2.62099	0.0875124
$898$	−34.6434	−1.15607
$899$	12.7434	0.425017
$900$	−14.8019	−0.493396
$901$	33.0006	1.09941
$902$	14.3148	0.476630
$903$	−1.72968	−0.0575601
$904$	−103.301	−3.43574
$905$	−4.78848	−0.159174
$906$	4.13572	0.137400
$907$	−23.3508	−0.775352	−0.387676	−	0.921796i	$-0.626722\pi$
$907$	−23.3508	−0.775352	−0.387676	+	0.921796i	$0.626722\pi$
$908$	49.6139	1.64650
$909$	−32.9938	−1.09434
$910$	−12.9048	−0.427789
$911$	−32.6590	−1.08204	−0.541021	−	0.841009i	$-0.681962\pi$
$911$	−32.6590	−1.08204	−0.541021	+	0.841009i	$0.681962\pi$
$912$	2.02730	0.0671306
$913$	−15.9110	−0.526577
$914$	33.2551	1.09998
$915$	2.23303	0.0738216
$916$	72.3381	2.39012
$917$	32.0087	1.05702
$918$	−12.6600	−0.417843
$919$	22.6838	0.748268	0.374134	−	0.927375i	$-0.377940\pi$
$919$	22.6838	0.748268	0.374134	+	0.927375i	$0.377940\pi$
$920$	−42.9507	−1.41604
$921$	2.62648	0.0865454
$922$	−103.659	−3.41384
$923$	−32.7796	−1.07895
$924$	−1.73653	−0.0571275
$925$	−6.04251	−0.198677
$926$	20.7127	0.680661
$927$	−24.1492	−0.793162
$928$	−46.6977	−1.53293
$929$	31.6815	1.03944	0.519719	−	0.854337i	$-0.326037\pi$
$929$	31.6815	1.03944	0.519719	+	0.854337i	$0.326037\pi$
$930$	1.75073	0.0574087
$931$	−3.48170	−0.114108
$932$	71.8375	2.35312
$933$	1.38384	0.0453049
$934$	−21.2112	−0.694051
$935$	4.32709	0.141511
$936$	−61.0297	−1.99482
$937$	−25.8773	−0.845376	−0.422688	−	0.906275i	$-0.638913\pi$
$937$	−25.8773	−0.845376	−0.422688	+	0.906275i	$0.638913\pi$
$938$	−14.6007	−0.476728
$939$	1.15007	0.0375312
$940$	30.5882	0.997676
$941$	−25.1909	−0.821200	−0.410600	−	0.911816i	$-0.634681\pi$
$941$	−25.1909	−0.821200	−0.410600	+	0.911816i	$0.634681\pi$
$942$	−3.23114	−0.105276
$943$	−29.4007	−0.957419
$944$	−60.2671	−1.96153
$945$	2.07553	0.0675172
$946$	13.1451	0.427385
$947$	49.5480	1.61010	0.805048	−	0.593210i	$-0.202140\pi$
$947$	49.5480	1.61010	0.805048	+	0.593210i	$0.202140\pi$
$948$	−10.0136	−0.325227
$949$	37.1346	1.20544
$950$	2.64409	0.0857855
$951$	1.00031	0.0324373
$952$	64.1924	2.08049
$953$	57.2694	1.85514	0.927568	−	0.373654i	$-0.121895\pi$
$953$	57.2694	1.85514	0.927568	+	0.373654i	$0.121895\pi$
$954$	−59.8018	−1.93616
$955$	−0.948263	−0.0306851
$956$	6.14323	0.198686
$957$	0.662158	0.0214045
$958$	−54.2573	−1.75298
$959$	−30.8923	−0.997563
$960$	−2.36088	−0.0761971
$961$	−18.2571	−0.588940
$962$	−41.5720	−1.34034
$963$	−31.5426	−1.01645
$964$	−73.9677	−2.38234
$965$	10.7442	0.345868
$966$	4.99575	0.160736
$967$	−20.0776	−0.645651	−0.322825	−	0.946459i	$-0.604633\pi$
$967$	−20.0776	−0.645651	−0.322825	+	0.946459i	$0.604633\pi$
$968$	7.90900	0.254205
$969$	0.802611	0.0257836
$970$	22.7274	0.729733
$971$	−5.31169	−0.170460	−0.0852302	−	0.996361i	$-0.527163\pi$
$971$	−5.31169	−0.170460	−0.0852302	+	0.996361i	$0.527163\pi$
$972$	24.6153	0.789537
$973$	−26.7932	−0.858951
$974$	35.6596	1.14261
$975$	0.482633	0.0154566
$976$	−131.581	−4.21180
$977$	34.6931	1.10993	0.554965	−	0.831874i	$-0.312732\pi$
$977$	34.6931	1.10993	0.554965	+	0.831874i	$0.312732\pi$
$978$	−2.59117	−0.0828564
$979$	−1.97457	−0.0631076
$980$	17.3779	0.555115
$981$	−47.9161	−1.52984
$982$	106.211	3.38932
$983$	−54.3522	−1.73357	−0.866783	−	0.498685i	$-0.833816\pi$
$983$	−54.3522	−1.73357	−0.866783	+	0.498685i	$0.833816\pi$
$984$	−7.94219	−0.253188
$985$	−11.5164	−0.366942
$986$	−40.8436	−1.30072
$987$	−2.13218	−0.0678681
$988$	12.9871	0.413175
$989$	−26.9984	−0.858499
$990$	−7.84130	−0.249213
$991$	4.05776	0.128899	0.0644495	−	0.997921i	$-0.479471\pi$
$991$	4.05776	0.128899	0.0644495	+	0.997921i	$0.479471\pi$
$992$	−46.6957	−1.48259
$993$	5.90688	0.187449
$994$	−62.4797	−1.98173
$995$	−2.52922	−0.0801817
$996$	14.7303	0.466748
$997$	35.6905	1.13033	0.565165	−	0.824978i	$-0.308813\pi$
$997$	35.6905	1.13033	0.565165	+	0.824978i	$0.308813\pi$
$998$	105.491	3.33926
$999$	6.68623	0.211543

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.2.a.j.1.8	✓	9
3.2	odd	2		9405.2.a.bi.1.2		9
5.4	even	2		5225.2.a.q.1.2		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.j.1.8	✓	9	1.1	even	1	trivial
5225.2.a.q.1.2		9	5.4	even	2
9405.2.a.bi.1.2		9	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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q+2.64409 q^{2} +0.185485 q^{3} +4.99120 q^{4} -1.00000 q^{5} +0.490440 q^{6} +1.87571 q^{7} +7.90900 q^{8} -2.96560 q^{9} +O(q^{10})\)	\(q+2.64409 q^{2} +0.185485 q^{3} +4.99120 q^{4} -1.00000 q^{5} +0.490440 q^{6} +1.87571 q^{7} +7.90900 q^{8} -2.96560 q^{9} -2.64409 q^{10} -1.00000 q^{11} +0.925795 q^{12} +2.60200 q^{13} +4.95955 q^{14} -0.185485 q^{15} +10.9297 q^{16} +4.32709 q^{17} -7.84130 q^{18} +1.00000 q^{19} -4.99120 q^{20} +0.347917 q^{21} -2.64409 q^{22} +5.43061 q^{23} +1.46700 q^{24} +1.00000 q^{25} +6.87992 q^{26} -1.10653 q^{27} +9.36207 q^{28} -3.56987 q^{29} -0.490440 q^{30} -3.56972 q^{31} +13.0811 q^{32} -0.185485 q^{33} +11.4412 q^{34} -1.87571 q^{35} -14.8019 q^{36} -6.04251 q^{37} +2.64409 q^{38} +0.482633 q^{39} -7.90900 q^{40} -5.41389 q^{41} +0.919924 q^{42} -4.97152 q^{43} -4.99120 q^{44} +2.96560 q^{45} +14.3590 q^{46} -6.12841 q^{47} +2.02730 q^{48} -3.48170 q^{49} +2.64409 q^{50} +0.802611 q^{51} +12.9871 q^{52} +7.62653 q^{53} -2.92576 q^{54} +1.00000 q^{55} +14.8350 q^{56} +0.185485 q^{57} -9.43905 q^{58} -5.51407 q^{59} -0.925795 q^{60} -12.0388 q^{61} -9.43864 q^{62} -5.56261 q^{63} +12.7281 q^{64} -2.60200 q^{65} -0.490440 q^{66} -2.94394 q^{67} +21.5974 q^{68} +1.00730 q^{69} -4.95955 q^{70} -12.5978 q^{71} -23.4549 q^{72} +14.2715 q^{73} -15.9769 q^{74} +0.185485 q^{75} +4.99120 q^{76} -1.87571 q^{77} +1.27612 q^{78} -10.8162 q^{79} -10.9297 q^{80} +8.69154 q^{81} -14.3148 q^{82} +15.9110 q^{83} +1.73653 q^{84} -4.32709 q^{85} -13.1451 q^{86} -0.662158 q^{87} -7.90900 q^{88} +1.97457 q^{89} +7.84130 q^{90} +4.88061 q^{91} +27.1053 q^{92} -0.662130 q^{93} -16.2041 q^{94} -1.00000 q^{95} +2.42635 q^{96} -8.59555 q^{97} -9.20591 q^{98} +2.96560 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 3 q^{2} + 3 q^{3} + 9 q^{4} - 9 q^{5} + 6 q^{6} - 9 q^{7} + 15 q^{8} + 16 q^{9}+O(q^{10}) \) 9 * q + 3 * q^2 + 3 * q^3 + 9 * q^4 - 9 * q^5 + 6 * q^6 - 9 * q^7 + 15 * q^8 + 16 * q^9	\( 9 q + 3 q^{2} + 3 q^{3} + 9 q^{4} - 9 q^{5} + 6 q^{6} - 9 q^{7} + 15 q^{8} + 16 q^{9} - 3 q^{10} - 9 q^{11} + 13 q^{12} - 3 q^{13} + 4 q^{14} - 3 q^{15} + 17 q^{16} + 5 q^{17} + 17 q^{18} + 9 q^{19} - 9 q^{20} - q^{21} - 3 q^{22} + 4 q^{23} + 21 q^{24} + 9 q^{25} + 20 q^{26} + 24 q^{27} - 24 q^{28} + 3 q^{29} - 6 q^{30} - q^{31} + 38 q^{32} - 3 q^{33} + 28 q^{34} + 9 q^{35} + 17 q^{36} + 5 q^{37} + 3 q^{38} - 15 q^{40} - 5 q^{41} - 43 q^{42} - 11 q^{43} - 9 q^{44} - 16 q^{45} + 2 q^{46} + 30 q^{47} + 54 q^{48} + 12 q^{49} + 3 q^{50} + 40 q^{51} - 3 q^{52} - q^{53} + 65 q^{54} + 9 q^{55} - 16 q^{56} + 3 q^{57} - 15 q^{58} + 59 q^{59} - 13 q^{60} - 21 q^{61} - 10 q^{62} - 12 q^{63} + 19 q^{64} + 3 q^{65} - 6 q^{66} - 2 q^{67} - 9 q^{68} - 22 q^{69} - 4 q^{70} + 34 q^{71} + 32 q^{72} - 34 q^{73} - 21 q^{74} + 3 q^{75} + 9 q^{76} + 9 q^{77} - 65 q^{78} - 13 q^{79} - 17 q^{80} + 57 q^{81} + 10 q^{82} + 51 q^{83} - 95 q^{84} - 5 q^{85} - 14 q^{86} + 8 q^{87} - 15 q^{88} + 8 q^{89} - 17 q^{90} + 62 q^{91} + 57 q^{92} - 18 q^{93} + 2 q^{94} - 9 q^{95} + 81 q^{96} - 8 q^{97} - 20 q^{98} - 16 q^{99}+O(q^{100}) \) 9 * q + 3 * q^2 + 3 * q^3 + 9 * q^4 - 9 * q^5 + 6 * q^6 - 9 * q^7 + 15 * q^8 + 16 * q^9 - 3 * q^10 - 9 * q^11 + 13 * q^12 - 3 * q^13 + 4 * q^14 - 3 * q^15 + 17 * q^16 + 5 * q^17 + 17 * q^18 + 9 * q^19 - 9 * q^20 - q^21 - 3 * q^22 + 4 * q^23 + 21 * q^24 + 9 * q^25 + 20 * q^26 + 24 * q^27 - 24 * q^28 + 3 * q^29 - 6 * q^30 - q^31 + 38 * q^32 - 3 * q^33 + 28 * q^34 + 9 * q^35 + 17 * q^36 + 5 * q^37 + 3 * q^38 - 15 * q^40 - 5 * q^41 - 43 * q^42 - 11 * q^43 - 9 * q^44 - 16 * q^45 + 2 * q^46 + 30 * q^47 + 54 * q^48 + 12 * q^49 + 3 * q^50 + 40 * q^51 - 3 * q^52 - q^53 + 65 * q^54 + 9 * q^55 - 16 * q^56 + 3 * q^57 - 15 * q^58 + 59 * q^59 - 13 * q^60 - 21 * q^61 - 10 * q^62 - 12 * q^63 + 19 * q^64 + 3 * q^65 - 6 * q^66 - 2 * q^67 - 9 * q^68 - 22 * q^69 - 4 * q^70 + 34 * q^71 + 32 * q^72 - 34 * q^73 - 21 * q^74 + 3 * q^75 + 9 * q^76 + 9 * q^77 - 65 * q^78 - 13 * q^79 - 17 * q^80 + 57 * q^81 + 10 * q^82 + 51 * q^83 - 95 * q^84 - 5 * q^85 - 14 * q^86 + 8 * q^87 - 15 * q^88 + 8 * q^89 - 17 * q^90 + 62 * q^91 + 57 * q^92 - 18 * q^93 + 2 * q^94 - 9 * q^95 + 81 * q^96 - 8 * q^97 - 20 * q^98 - 16 * q^99

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