LMFDB - Embedded newform 1045.2.a.j.1.2

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			1.2
Root			$-1.98926$ 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)

$f(q)$	$=$	$q-1.53772 q^{2} -2.83656 q^{3} +0.364584 q^{4} -1.00000 q^{5} +4.36184 q^{6} -2.01727 q^{7} +2.51481 q^{8} +5.04608 q^{9} +O(q^{10})$	\(q-1.53772 q^{2} -2.83656 q^{3} +0.364584 q^{4} -1.00000 q^{5} +4.36184 q^{6} -2.01727 q^{7} +2.51481 q^{8} +5.04608 q^{9} +1.53772 q^{10} -1.00000 q^{11} -1.03417 q^{12} -4.70522 q^{13} +3.10200 q^{14} +2.83656 q^{15} -4.59625 q^{16} -5.74124 q^{17} -7.75947 q^{18} +1.00000 q^{19} -0.364584 q^{20} +5.72211 q^{21} +1.53772 q^{22} -4.84191 q^{23} -7.13342 q^{24} +1.00000 q^{25} +7.23531 q^{26} -5.80384 q^{27} -0.735466 q^{28} -9.25093 q^{29} -4.36184 q^{30} -5.18786 q^{31} +2.03812 q^{32} +2.83656 q^{33} +8.82842 q^{34} +2.01727 q^{35} +1.83972 q^{36} -4.73659 q^{37} -1.53772 q^{38} +13.3466 q^{39} -2.51481 q^{40} -5.93607 q^{41} -8.79901 q^{42} -10.2731 q^{43} -0.364584 q^{44} -5.04608 q^{45} +7.44550 q^{46} +9.94533 q^{47} +13.0375 q^{48} -2.93062 q^{49} -1.53772 q^{50} +16.2854 q^{51} -1.71545 q^{52} -2.07121 q^{53} +8.92469 q^{54} +1.00000 q^{55} -5.07306 q^{56} -2.83656 q^{57} +14.2254 q^{58} +10.0737 q^{59} +1.03417 q^{60} +13.8810 q^{61} +7.97748 q^{62} -10.1793 q^{63} +6.05844 q^{64} +4.70522 q^{65} -4.36184 q^{66} +4.81791 q^{67} -2.09317 q^{68} +13.7344 q^{69} -3.10200 q^{70} -4.53572 q^{71} +12.6900 q^{72} -15.2563 q^{73} +7.28355 q^{74} -2.83656 q^{75} +0.364584 q^{76} +2.01727 q^{77} -20.5234 q^{78} -2.34481 q^{79} +4.59625 q^{80} +1.32471 q^{81} +9.12801 q^{82} +13.1004 q^{83} +2.08619 q^{84} +5.74124 q^{85} +15.7971 q^{86} +26.2408 q^{87} -2.51481 q^{88} -11.8462 q^{89} +7.75947 q^{90} +9.49170 q^{91} -1.76528 q^{92} +14.7157 q^{93} -15.2931 q^{94} -1.00000 q^{95} -5.78125 q^{96} +1.34789 q^{97} +4.50647 q^{98} -5.04608 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.53772	−1.08733	−0.543666	−	0.839301i	$-0.682964\pi$
$2$	−1.53772	−1.08733	−0.543666	+	0.839301i	$0.682964\pi$
$3$	−2.83656	−1.63769	−0.818845	−	0.574015i	$-0.805385\pi$
$3$	−2.83656	−1.63769	−0.818845	+	0.574015i	$0.805385\pi$
$4$	0.364584	0.182292
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	4.36184	1.78071
$7$	−2.01727	−0.762457	−0.381228	−	0.924481i	$-0.624499\pi$
$7$	−2.01727	−0.762457	−0.381228	+	0.924481i	$0.624499\pi$
$8$	2.51481	0.889120
$9$	5.04608	1.68203
$10$	1.53772	0.486270
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.03417	−0.298538
$13$	−4.70522	−1.30499	−0.652497	−	0.757792i	$-0.726278\pi$
$13$	−4.70522	−1.30499	−0.652497	+	0.757792i	$0.726278\pi$
$14$	3.10200	0.829044
$15$	2.83656	0.732397
$16$	−4.59625	−1.14906
$17$	−5.74124	−1.39246	−0.696228	−	0.717821i	$-0.745140\pi$
$17$	−5.74124	−1.39246	−0.696228	+	0.717821i	$0.745140\pi$
$18$	−7.75947	−1.82892
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.364584	−0.0815235
$21$	5.72211	1.24867
$22$	1.53772	0.327843
$23$	−4.84191	−1.00961	−0.504804	−	0.863234i	$-0.668435\pi$
$23$	−4.84191	−1.00961	−0.504804	+	0.863234i	$0.668435\pi$
$24$	−7.13342	−1.45610
$25$	1.00000	0.200000
$26$	7.23531	1.41896
$27$	−5.80384	−1.11695
$28$	−0.735466	−0.138990
$29$	−9.25093	−1.71786	−0.858928	−	0.512097i	$-0.828869\pi$
$29$	−9.25093	−1.71786	−0.858928	+	0.512097i	$0.828869\pi$
$30$	−4.36184	−0.796359
$31$	−5.18786	−0.931768	−0.465884	−	0.884846i	$-0.654264\pi$
$31$	−5.18786	−0.931768	−0.465884	+	0.884846i	$0.654264\pi$
$32$	2.03812	0.360292
$33$	2.83656	0.493782
$34$	8.82842	1.51406
$35$	2.01727	0.340981
$36$	1.83972	0.306621
$37$	−4.73659	−0.778690	−0.389345	−	0.921092i	$-0.627299\pi$
$37$	−4.73659	−0.778690	−0.389345	+	0.921092i	$0.627299\pi$
$38$	−1.53772	−0.249451
$39$	13.3466	2.13717
$40$	−2.51481	−0.397627
$41$	−5.93607	−0.927058	−0.463529	−	0.886082i	$-0.653417\pi$
$41$	−5.93607	−0.927058	−0.463529	+	0.886082i	$0.653417\pi$
$42$	−8.79901	−1.35772
$43$	−10.2731	−1.56663	−0.783313	−	0.621627i	$-0.786472\pi$
$43$	−10.2731	−1.56663	−0.783313	+	0.621627i	$0.786472\pi$
$44$	−0.364584	−0.0549632
$45$	−5.04608	−0.752226
$46$	7.44550	1.09778
$47$	9.94533	1.45068	0.725338	−	0.688393i	$-0.241683\pi$
$47$	9.94533	1.45068	0.725338	+	0.688393i	$0.241683\pi$
$48$	13.0375	1.88181
$49$	−2.93062	−0.418660
$50$	−1.53772	−0.217467
$51$	16.2854	2.28041
$52$	−1.71545	−0.237890
$53$	−2.07121	−0.284503	−0.142251	−	0.989831i	$-0.545434\pi$
$53$	−2.07121	−0.284503	−0.142251	+	0.989831i	$0.545434\pi$
$54$	8.92469	1.21450
$55$	1.00000	0.134840
$56$	−5.07306	−0.677916
$57$	−2.83656	−0.375712
$58$	14.2254	1.86788
$59$	10.0737	1.31149	0.655743	−	0.754985i	$-0.272356\pi$
$59$	10.0737	1.31149	0.655743	+	0.754985i	$0.272356\pi$
$60$	1.03417	0.133510
$61$	13.8810	1.77728	0.888639	−	0.458607i	$-0.151651\pi$
$61$	13.8810	1.77728	0.888639	+	0.458607i	$0.151651\pi$
$62$	7.97748	1.01314
$63$	−10.1793	−1.28247
$64$	6.05844	0.757305
$65$	4.70522	0.583611
$66$	−4.36184	−0.536905
$67$	4.81791	0.588601	0.294300	−	0.955713i	$-0.404913\pi$
$67$	4.81791	0.588601	0.294300	+	0.955713i	$0.404913\pi$
$68$	−2.09317	−0.253834
$69$	13.7344	1.65342
$70$	−3.10200	−0.370760
$71$	−4.53572	−0.538291	−0.269146	−	0.963100i	$-0.586741\pi$
$71$	−4.53572	−0.538291	−0.269146	+	0.963100i	$0.586741\pi$
$72$	12.6900	1.49553
$73$	−15.2563	−1.78562	−0.892809	−	0.450435i	$-0.851269\pi$
$73$	−15.2563	−1.78562	−0.892809	+	0.450435i	$0.851269\pi$
$74$	7.28355	0.846695
$75$	−2.83656	−0.327538
$76$	0.364584	0.0418207
$77$	2.01727	0.229889
$78$	−20.5234	−2.32382
$79$	−2.34481	−0.263812	−0.131906	−	0.991262i	$-0.542110\pi$
$79$	−2.34481	−0.263812	−0.131906	+	0.991262i	$0.542110\pi$
$80$	4.59625	0.513876
$81$	1.32471	0.147190
$82$	9.12801	1.00802
$83$	13.1004	1.43795	0.718977	−	0.695034i	$-0.244611\pi$
$83$	13.1004	1.43795	0.718977	+	0.695034i	$0.244611\pi$
$84$	2.08619	0.227622
$85$	5.74124	0.622725
$86$	15.7971	1.70344
$87$	26.2408	2.81331
$88$	−2.51481	−0.268080
$89$	−11.8462	−1.25569	−0.627847	−	0.778336i	$-0.716064\pi$
$89$	−11.8462	−1.25569	−0.627847	+	0.778336i	$0.716064\pi$
$90$	7.75947	0.817920
$91$	9.49170	0.995001
$92$	−1.76528	−0.184044
$93$	14.7157	1.52595
$94$	−15.2931	−1.57737
$95$	−1.00000	−0.102598
$96$	−5.78125	−0.590046
$97$	1.34789	0.136858	0.0684290	−	0.997656i	$-0.478201\pi$
$97$	1.34789	0.136858	0.0684290	+	0.997656i	$0.478201\pi$
$98$	4.50647	0.455222
$99$	−5.04608	−0.507150
$100$	0.364584	0.0364584
$101$	12.5088	1.24467	0.622334	−	0.782752i	$-0.286184\pi$
$101$	12.5088	1.24467	0.622334	+	0.782752i	$0.286184\pi$
$102$	−25.0424	−2.47956
$103$	−15.5211	−1.52934	−0.764668	−	0.644425i	$-0.777097\pi$
$103$	−15.5211	−1.52934	−0.764668	+	0.644425i	$0.777097\pi$
$104$	−11.8327	−1.16030
$105$	−5.72211	−0.558421
$106$	3.18494	0.309349
$107$	8.02816	0.776111	0.388056	−	0.921636i	$-0.373147\pi$
$107$	8.02816	0.776111	0.388056	+	0.921636i	$0.373147\pi$
$108$	−2.11599	−0.203611
$109$	−17.4014	−1.66675	−0.833375	−	0.552709i	$-0.813594\pi$
$109$	−17.4014	−1.66675	−0.833375	+	0.552709i	$0.813594\pi$
$110$	−1.53772	−0.146616
$111$	13.4356	1.27525
$112$	9.27188	0.876110
$113$	−4.14618	−0.390040	−0.195020	−	0.980799i	$-0.562477\pi$
$113$	−4.14618	−0.390040	−0.195020	+	0.980799i	$0.562477\pi$
$114$	4.36184	0.408524
$115$	4.84191	0.451510
$116$	−3.37275	−0.313152
$117$	−23.7429	−2.19503
$118$	−15.4905	−1.42602
$119$	11.5816	1.06169
$120$	7.13342	0.651189
$121$	1.00000	0.0909091
$122$	−21.3451	−1.93249
$123$	16.8380	1.51823
$124$	−1.89141	−0.169854
$125$	−1.00000	−0.0894427
$126$	15.6529	1.39448
$127$	3.54518	0.314584	0.157292	−	0.987552i	$-0.449724\pi$
$127$	3.54518	0.314584	0.157292	+	0.987552i	$0.449724\pi$
$128$	−13.3924	−1.18373
$129$	29.1402	2.56565
$130$	−7.23531	−0.634579
$131$	6.40923	0.559977	0.279989	−	0.960003i	$-0.409669\pi$
$131$	6.40923	0.559977	0.279989	+	0.960003i	$0.409669\pi$
$132$	1.03417	0.0900126
$133$	−2.01727	−0.174920
$134$	−7.40859	−0.640005
$135$	5.80384	0.499515
$136$	−14.4381	−1.23806
$137$	16.5172	1.41116	0.705579	−	0.708631i	$-0.250687\pi$
$137$	16.5172	1.41116	0.705579	+	0.708631i	$0.250687\pi$
$138$	−21.1196	−1.79782
$139$	−1.34728	−0.114275	−0.0571376	−	0.998366i	$-0.518197\pi$
$139$	−1.34728	−0.114275	−0.0571376	+	0.998366i	$0.518197\pi$
$140$	0.735466	0.0621582
$141$	−28.2106	−2.37576
$142$	6.97467	0.585301
$143$	4.70522	0.393470
$144$	−23.1930	−1.93275
$145$	9.25093	0.768248
$146$	23.4600	1.94156
$147$	8.31288	0.685635
$148$	−1.72689	−0.141949
$149$	10.1150	0.828651	0.414326	−	0.910129i	$-0.364017\pi$
$149$	10.1150	0.828651	0.414326	+	0.910129i	$0.364017\pi$
$150$	4.36184	0.356143
$151$	7.86645	0.640163	0.320081	−	0.947390i	$-0.396290\pi$
$151$	7.86645	0.640163	0.320081	+	0.947390i	$0.396290\pi$
$152$	2.51481	0.203978
$153$	−28.9708	−2.34215
$154$	−3.10200	−0.249966
$155$	5.18786	0.416699
$156$	4.86598	0.389590
$157$	4.66142	0.372022	0.186011	−	0.982548i	$-0.440444\pi$
$157$	4.66142	0.372022	0.186011	+	0.982548i	$0.440444\pi$
$158$	3.60566	0.286851
$159$	5.87512	0.465927
$160$	−2.03812	−0.161127
$161$	9.76744	0.769782
$162$	−2.03703	−0.160044
$163$	−12.6308	−0.989319	−0.494659	−	0.869087i	$-0.664707\pi$
$163$	−12.6308	−0.989319	−0.494659	+	0.869087i	$0.664707\pi$
$164$	−2.16420	−0.168995
$165$	−2.83656	−0.220826
$166$	−20.1447	−1.56353
$167$	−4.40813	−0.341112	−0.170556	−	0.985348i	$-0.554556\pi$
$167$	−4.40813	−0.341112	−0.170556	+	0.985348i	$0.554556\pi$
$168$	14.3900	1.11022
$169$	9.13909	0.703007
$170$	−8.82842	−0.677109
$171$	5.04608	0.385884
$172$	−3.74540	−0.285584
$173$	17.2090	1.30837	0.654186	−	0.756333i	$-0.273011\pi$
$173$	17.2090	1.30837	0.654186	+	0.756333i	$0.273011\pi$
$174$	−40.3511	−3.05901
$175$	−2.01727	−0.152491
$176$	4.59625	0.346455
$177$	−28.5747	−2.14781
$178$	18.2161	1.36536
$179$	4.64276	0.347016	0.173508	−	0.984832i	$-0.444490\pi$
$179$	4.64276	0.347016	0.173508	+	0.984832i	$0.444490\pi$
$180$	−1.83972	−0.137125
$181$	−4.88640	−0.363203	−0.181602	−	0.983372i	$-0.558128\pi$
$181$	−4.88640	−0.363203	−0.181602	+	0.983372i	$0.558128\pi$
$182$	−14.5956	−1.08190
$183$	−39.3743	−2.91063
$184$	−12.1765	−0.897663
$185$	4.73659	0.348241
$186$	−22.6286	−1.65921
$187$	5.74124	0.419841
$188$	3.62591	0.264447
$189$	11.7079	0.851626
$190$	1.53772	0.111558
$191$	−7.26536	−0.525703	−0.262851	−	0.964836i	$-0.584663\pi$
$191$	−7.26536	−0.525703	−0.262851	+	0.964836i	$0.584663\pi$
$192$	−17.1851	−1.24023
$193$	−12.3608	−0.889749	−0.444875	−	0.895593i	$-0.646752\pi$
$193$	−12.3608	−0.889749	−0.444875	+	0.895593i	$0.646752\pi$
$194$	−2.07268	−0.148810
$195$	−13.3466	−0.955773
$196$	−1.06846	−0.0763184
$197$	−20.7580	−1.47895	−0.739473	−	0.673187i	$-0.764925\pi$
$197$	−20.7580	−1.47895	−0.739473	+	0.673187i	$0.764925\pi$
$198$	7.75947	0.551441
$199$	4.14644	0.293933	0.146967	−	0.989141i	$-0.453049\pi$
$199$	4.14644	0.293933	0.146967	+	0.989141i	$0.453049\pi$
$200$	2.51481	0.177824
$201$	−13.6663	−0.963945
$202$	−19.2350	−1.35337
$203$	18.6616	1.30979
$204$	5.93740	0.415701
$205$	5.93607	0.414593
$206$	23.8670	1.66290
$207$	−24.4327	−1.69819
$208$	21.6264	1.49952
$209$	−1.00000	−0.0691714
$210$	8.79901	0.607190
$211$	−26.2648	−1.80814	−0.904071	−	0.427383i	$-0.859436\pi$
$211$	−26.2648	−1.80814	−0.904071	+	0.427383i	$0.859436\pi$
$212$	−0.755131	−0.0518626
$213$	12.8659	0.881554
$214$	−12.3451	−0.843891
$215$	10.2731	0.700617
$216$	−14.5956	−0.993103
$217$	10.4653	0.710433
$218$	26.7584	1.81231
$219$	43.2755	2.92429
$220$	0.364584	0.0245803
$221$	27.0138	1.81714
$222$	−20.6602	−1.38662
$223$	−11.9068	−0.797339	−0.398669	−	0.917095i	$-0.630528\pi$
$223$	−11.9068	−0.797339	−0.398669	+	0.917095i	$0.630528\pi$
$224$	−4.11144	−0.274707
$225$	5.04608	0.336406
$226$	6.37567	0.424103
$227$	11.7626	0.780712	0.390356	−	0.920664i	$-0.372352\pi$
$227$	11.7626	0.780712	0.390356	+	0.920664i	$0.372352\pi$
$228$	−1.03417	−0.0684893
$229$	6.72710	0.444539	0.222270	−	0.974985i	$-0.428654\pi$
$229$	6.72710	0.444539	0.222270	+	0.974985i	$0.428654\pi$
$230$	−7.44550	−0.490942
$231$	−5.72211	−0.376487
$232$	−23.2644	−1.52738
$233$	−16.1436	−1.05760	−0.528800	−	0.848746i	$-0.677358\pi$
$233$	−16.1436	−1.05760	−0.528800	+	0.848746i	$0.677358\pi$
$234$	36.5100	2.38673
$235$	−9.94533	−0.648762
$236$	3.67272	0.239073
$237$	6.65120	0.432042
$238$	−17.8093	−1.15441
$239$	−5.67826	−0.367296	−0.183648	−	0.982992i	$-0.558791\pi$
$239$	−5.67826	−0.367296	−0.183648	+	0.982992i	$0.558791\pi$
$240$	−13.0375	−0.841570
$241$	−23.3578	−1.50461	−0.752303	−	0.658817i	$-0.771057\pi$
$241$	−23.3578	−1.50461	−0.752303	+	0.658817i	$0.771057\pi$
$242$	−1.53772	−0.0988484
$243$	13.6539	0.875899
$244$	5.06079	0.323984
$245$	2.93062	0.187230
$246$	−25.8922	−1.65082
$247$	−4.70522	−0.299386
$248$	−13.0465	−0.828454
$249$	−37.1601	−2.35492
$250$	1.53772	0.0972540
$251$	12.2543	0.773482	0.386741	−	0.922188i	$-0.373601\pi$
$251$	12.2543	0.773482	0.386741	+	0.922188i	$0.373601\pi$
$252$	−3.71122	−0.233785
$253$	4.84191	0.304408
$254$	−5.45150	−0.342058
$255$	−16.2854	−1.01983
$256$	8.47693	0.529808
$257$	27.1675	1.69466	0.847331	−	0.531065i	$-0.178208\pi$
$257$	27.1675	1.69466	0.847331	+	0.531065i	$0.178208\pi$
$258$	−44.8094	−2.78971
$259$	9.55498	0.593717
$260$	1.71545	0.106388
$261$	−46.6810	−2.88948
$262$	−9.85561	−0.608882
$263$	23.0435	1.42092	0.710461	−	0.703736i	$-0.248486\pi$
$263$	23.0435	1.42092	0.710461	+	0.703736i	$0.248486\pi$
$264$	7.13342	0.439032
$265$	2.07121	0.127233
$266$	3.10200	0.190196
$267$	33.6025	2.05644
$268$	1.75653	0.107297
$269$	−16.4904	−1.00544	−0.502718	−	0.864450i	$-0.667667\pi$
$269$	−16.4904	−1.00544	−0.502718	+	0.864450i	$0.667667\pi$
$270$	−8.92469	−0.543139
$271$	−24.1144	−1.46485	−0.732424	−	0.680849i	$-0.761611\pi$
$271$	−24.1144	−1.46485	−0.732424	+	0.680849i	$0.761611\pi$
$272$	26.3882	1.60002
$273$	−26.9238	−1.62950
$274$	−25.3988	−1.53440
$275$	−1.00000	−0.0603023
$276$	5.00734	0.301406
$277$	0.954174	0.0573307	0.0286654	−	0.999589i	$-0.490874\pi$
$277$	0.954174	0.0573307	0.0286654	+	0.999589i	$0.490874\pi$
$278$	2.07175	0.124255
$279$	−26.1784	−1.56726
$280$	5.07306	0.303173
$281$	−28.2870	−1.68746	−0.843730	−	0.536767i	$-0.819645\pi$
$281$	−28.2870	−1.68746	−0.843730	+	0.536767i	$0.819645\pi$
$282$	43.3799	2.58324
$283$	25.3951	1.50958	0.754791	−	0.655965i	$-0.227738\pi$
$283$	25.3951	1.50958	0.754791	+	0.655965i	$0.227738\pi$
$284$	−1.65365	−0.0981262
$285$	2.83656	0.168023
$286$	−7.23531	−0.427833
$287$	11.9747	0.706842
$288$	10.2845	0.606021
$289$	15.9618	0.938932
$290$	−14.2254	−0.835341
$291$	−3.82339	−0.224131
$292$	−5.56222	−0.325504
$293$	−25.5231	−1.49108	−0.745538	−	0.666463i	$-0.767807\pi$
$293$	−25.5231	−1.49108	−0.745538	+	0.666463i	$0.767807\pi$
$294$	−12.7829	−0.745513
$295$	−10.0737	−0.586514
$296$	−11.9116	−0.692349
$297$	5.80384	0.336773
$298$	−15.5540	−0.901019
$299$	22.7822	1.31753
$300$	−1.03417	−0.0597076
$301$	20.7235	1.19449
$302$	−12.0964	−0.696070
$303$	−35.4819	−2.03838
$304$	−4.59625	−0.263613
$305$	−13.8810	−0.794823
$306$	44.5490	2.54669
$307$	11.8158	0.674366	0.337183	−	0.941439i	$-0.390526\pi$
$307$	11.8158	0.674366	0.337183	+	0.941439i	$0.390526\pi$
$308$	0.735466	0.0419070
$309$	44.0264	2.50458
$310$	−7.97748	−0.453091
$311$	26.0291	1.47598	0.737988	−	0.674813i	$-0.235776\pi$
$311$	26.0291	1.47598	0.737988	+	0.674813i	$0.235776\pi$
$312$	33.5643	1.90020
$313$	23.7167	1.34055	0.670273	−	0.742115i	$-0.266177\pi$
$313$	23.7167	1.34055	0.670273	+	0.742115i	$0.266177\pi$
$314$	−7.16797	−0.404512
$315$	10.1793	0.573540
$316$	−0.854882	−0.0480908
$317$	16.0845	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$317$	16.0845	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$318$	−9.03429	−0.506618
$319$	9.25093	0.517953
$320$	−6.05844	−0.338677
$321$	−22.7724	−1.27103
$322$	−15.0196	−0.837009
$323$	−5.74124	−0.319451
$324$	0.482968	0.0268315
$325$	−4.70522	−0.260999
$326$	19.4226	1.07572
$327$	49.3601	2.72962
$328$	−14.9281	−0.824266
$329$	−20.0624	−1.10608
$330$	4.36184	0.240111
$331$	14.5415	0.799276	0.399638	−	0.916673i	$-0.369136\pi$
$331$	14.5415	0.799276	0.399638	+	0.916673i	$0.369136\pi$
$332$	4.77620	0.262128
$333$	−23.9012	−1.30978
$334$	6.77848	0.370902
$335$	−4.81791	−0.263230
$336$	−26.3003	−1.43480
$337$	24.9558	1.35943	0.679714	−	0.733477i	$-0.262104\pi$
$337$	24.9558	1.35943	0.679714	+	0.733477i	$0.262104\pi$
$338$	−14.0534	−0.764402
$339$	11.7609	0.638764
$340$	2.09317	0.113518
$341$	5.18786	0.280939
$342$	−7.75947	−0.419584
$343$	20.0327	1.08167
$344$	−25.8348	−1.39292
$345$	−13.7344	−0.739434
$346$	−26.4626	−1.42264
$347$	−11.1305	−0.597517	−0.298759	−	0.954329i	$-0.596573\pi$
$347$	−11.1305	−0.597517	−0.298759	+	0.954329i	$0.596573\pi$
$348$	9.56700	0.512845
$349$	−11.3457	−0.607319	−0.303660	−	0.952781i	$-0.598209\pi$
$349$	−11.3457	−0.607319	−0.303660	+	0.952781i	$0.598209\pi$
$350$	3.10200	0.165809
$351$	27.3084	1.45761
$352$	−2.03812	−0.108632
$353$	24.3514	1.29609	0.648046	−	0.761601i	$-0.275586\pi$
$353$	24.3514	1.29609	0.648046	+	0.761601i	$0.275586\pi$
$354$	43.9399	2.33538
$355$	4.53572	0.240731
$356$	−4.31894	−0.228903
$357$	−32.8520	−1.73871
$358$	−7.13926	−0.377322
$359$	−6.00208	−0.316777	−0.158389	−	0.987377i	$-0.550630\pi$
$359$	−6.00208	−0.316777	−0.158389	+	0.987377i	$0.550630\pi$
$360$	−12.6900	−0.668819
$361$	1.00000	0.0526316
$362$	7.51392	0.394923
$363$	−2.83656	−0.148881
$364$	3.46053	0.181381
$365$	15.2563	0.798553
$366$	60.5466	3.16482
$367$	−5.31680	−0.277535	−0.138767	−	0.990325i	$-0.544314\pi$
$367$	−5.31680	−0.277535	−0.138767	+	0.990325i	$0.544314\pi$
$368$	22.2546	1.16010
$369$	−29.9539	−1.55934
$370$	−7.28355	−0.378654
$371$	4.17819	0.216921
$372$	5.36511	0.278168
$373$	−18.7216	−0.969370	−0.484685	−	0.874689i	$-0.661066\pi$
$373$	−18.7216	−0.969370	−0.484685	+	0.874689i	$0.661066\pi$
$374$	−8.82842	−0.456507
$375$	2.83656	0.146479
$376$	25.0106	1.28983
$377$	43.5277	2.24179
$378$	−18.0035	−0.926001
$379$	15.7282	0.807901	0.403951	−	0.914781i	$-0.367637\pi$
$379$	15.7282	0.807901	0.403951	+	0.914781i	$0.367637\pi$
$380$	−0.364584	−0.0187028
$381$	−10.0561	−0.515191
$382$	11.1721	0.571614
$383$	3.42938	0.175233	0.0876166	−	0.996154i	$-0.472075\pi$
$383$	3.42938	0.175233	0.0876166	+	0.996154i	$0.472075\pi$
$384$	37.9884	1.93859
$385$	−2.01727	−0.102810
$386$	19.0074	0.967453
$387$	−51.8387	−2.63511
$388$	0.491421	0.0249481
$389$	−20.4234	−1.03551	−0.517754	−	0.855530i	$-0.673232\pi$
$389$	−20.4234	−1.03551	−0.517754	+	0.855530i	$0.673232\pi$
$390$	20.5234	1.03924
$391$	27.7986	1.40583
$392$	−7.36995	−0.372239
$393$	−18.1802	−0.917069
$394$	31.9200	1.60811
$395$	2.34481	0.117980
$396$	−1.83972	−0.0924496
$397$	−3.35903	−0.168585	−0.0842926	−	0.996441i	$-0.526863\pi$
$397$	−3.35903	−0.168585	−0.0842926	+	0.996441i	$0.526863\pi$
$398$	−6.37607	−0.319603
$399$	5.72211	0.286464
$400$	−4.59625	−0.229812
$401$	−0.619983	−0.0309605	−0.0154802	−	0.999880i	$-0.504928\pi$
$401$	−0.619983	−0.0309605	−0.0154802	+	0.999880i	$0.504928\pi$
$402$	21.0149	1.04813
$403$	24.4100	1.21595
$404$	4.56050	0.226893
$405$	−1.32471	−0.0658253
$406$	−28.6964	−1.42418
$407$	4.73659	0.234784
$408$	40.9547	2.02756
$409$	−5.50461	−0.272185	−0.136093	−	0.990696i	$-0.543454\pi$
$409$	−5.50461	−0.272185	−0.136093	+	0.990696i	$0.543454\pi$
$410$	−9.12801	−0.450800
$411$	−46.8520	−2.31104
$412$	−5.65873	−0.278786
$413$	−20.3214	−0.999951
$414$	37.5706	1.84650
$415$	−13.1004	−0.643073
$416$	−9.58980	−0.470178
$417$	3.82166	0.187147
$418$	1.53772	0.0752124
$419$	15.8476	0.774207	0.387104	−	0.922036i	$-0.373476\pi$
$419$	15.8476	0.774207	0.387104	+	0.922036i	$0.373476\pi$
$420$	−2.08619	−0.101796
$421$	−15.1166	−0.736740	−0.368370	−	0.929679i	$-0.620084\pi$
$421$	−15.1166	−0.736740	−0.368370	+	0.929679i	$0.620084\pi$
$422$	40.3879	1.96605
$423$	50.1850	2.44008
$424$	−5.20870	−0.252957
$425$	−5.74124	−0.278491
$426$	−19.7841	−0.958542
$427$	−28.0017	−1.35510
$428$	2.92694	0.141479
$429$	−13.3466	−0.644382
$430$	−15.7971	−0.761804
$431$	−13.9715	−0.672983	−0.336492	−	0.941686i	$-0.609240\pi$
$431$	−13.9715	−0.672983	−0.336492	+	0.941686i	$0.609240\pi$
$432$	26.6759	1.28344
$433$	25.8282	1.24123	0.620613	−	0.784117i	$-0.286884\pi$
$433$	25.8282	1.24123	0.620613	+	0.784117i	$0.286884\pi$
$434$	−16.0927	−0.772477
$435$	−26.2408	−1.25815
$436$	−6.34427	−0.303835
$437$	−4.84191	−0.231620
$438$	−66.5457	−3.17968
$439$	4.96718	0.237071	0.118535	−	0.992950i	$-0.462180\pi$
$439$	4.96718	0.237071	0.118535	+	0.992950i	$0.462180\pi$
$440$	2.51481	0.119889
$441$	−14.7881	−0.704197
$442$	−41.5397	−1.97584
$443$	−38.4761	−1.82806	−0.914028	−	0.405651i	$-0.867045\pi$
$443$	−38.4761	−1.82806	−0.914028	+	0.405651i	$0.867045\pi$
$444$	4.89842	0.232469
$445$	11.8462	0.561564
$446$	18.3093	0.866973
$447$	−28.6918	−1.35707
$448$	−12.2215	−0.577412
$449$	−24.0945	−1.13709	−0.568544	−	0.822653i	$-0.692493\pi$
$449$	−24.0945	−1.13709	−0.568544	+	0.822653i	$0.692493\pi$
$450$	−7.75947	−0.365785
$451$	5.93607	0.279519
$452$	−1.51163	−0.0711012
$453$	−22.3137	−1.04839
$454$	−18.0876	−0.848893
$455$	−9.49170	−0.444978
$456$	−7.13342	−0.334053
$457$	7.51214	0.351403	0.175702	−	0.984443i	$-0.443781\pi$
$457$	7.51214	0.351403	0.175702	+	0.984443i	$0.443781\pi$
$458$	−10.3444	−0.483362
$459$	33.3213	1.55530
$460$	1.76528	0.0823068
$461$	−0.669728	−0.0311924	−0.0155962	−	0.999878i	$-0.504965\pi$
$461$	−0.669728	−0.0311924	−0.0155962	+	0.999878i	$0.504965\pi$
$462$	8.79901	0.409367
$463$	2.37086	0.110183	0.0550917	−	0.998481i	$-0.482455\pi$
$463$	2.37086	0.110183	0.0550917	+	0.998481i	$0.482455\pi$
$464$	42.5196	1.97392
$465$	−14.7157	−0.682424
$466$	24.8243	1.14996
$467$	3.85813	0.178533	0.0892664	−	0.996008i	$-0.471548\pi$
$467$	3.85813	0.178533	0.0892664	+	0.996008i	$0.471548\pi$
$468$	−8.65630	−0.400138
$469$	−9.71902	−0.448783
$470$	15.2931	0.705420
$471$	−13.2224	−0.609257
$472$	25.3335	1.16607
$473$	10.2731	0.472356
$474$	−10.2277	−0.469773
$475$	1.00000	0.0458831
$476$	4.22248	0.193537
$477$	−10.4515	−0.478541
$478$	8.73158	0.399373
$479$	−28.0062	−1.27964	−0.639818	−	0.768526i	$-0.720990\pi$
$479$	−28.0062	−1.27964	−0.639818	+	0.768526i	$0.720990\pi$
$480$	5.78125	0.263877
$481$	22.2867	1.01619
$482$	35.9177	1.63601
$483$	−27.7060	−1.26066
$484$	0.364584	0.0165720
$485$	−1.34789	−0.0612047
$486$	−20.9959	−0.952393
$487$	2.97414	0.134771	0.0673856	−	0.997727i	$-0.478534\pi$
$487$	2.97414	0.134771	0.0673856	+	0.997727i	$0.478534\pi$
$488$	34.9081	1.58021
$489$	35.8280	1.62020
$490$	−4.50647	−0.203582
$491$	−8.67268	−0.391392	−0.195696	−	0.980665i	$-0.562697\pi$
$491$	−8.67268	−0.391392	−0.195696	+	0.980665i	$0.562697\pi$
$492$	6.13888	0.276762
$493$	53.1118	2.39204
$494$	7.23531	0.325532
$495$	5.04608	0.226805
$496$	23.8447	1.07066
$497$	9.14978	0.410424
$498$	57.1418	2.56058
$499$	7.12060	0.318762	0.159381	−	0.987217i	$-0.449050\pi$
$499$	7.12060	0.318762	0.159381	+	0.987217i	$0.449050\pi$
$500$	−0.364584	−0.0163047
$501$	12.5039	0.558635
$502$	−18.8436	−0.841032
$503$	−13.8153	−0.615993	−0.307997	−	0.951387i	$-0.599659\pi$
$503$	−13.8153	−0.615993	−0.307997	+	0.951387i	$0.599659\pi$
$504$	−25.5991	−1.14027
$505$	−12.5088	−0.556633
$506$	−7.44550	−0.330993
$507$	−25.9236	−1.15131
$508$	1.29252	0.0573462
$509$	−2.06694	−0.0916155	−0.0458078	−	0.998950i	$-0.514586\pi$
$509$	−2.06694	−0.0916155	−0.0458078	+	0.998950i	$0.514586\pi$
$510$	25.0424	1.10889
$511$	30.7762	1.36146
$512$	13.7497	0.607657
$513$	−5.80384	−0.256246
$514$	−41.7760	−1.84266
$515$	15.5211	0.683940
$516$	10.6241	0.467698
$517$	−9.94533	−0.437395
$518$	−14.6929	−0.645568
$519$	−48.8143	−2.14271
$520$	11.8327	0.518900
$521$	2.18954	0.0959255	0.0479627	−	0.998849i	$-0.484727\pi$
$521$	2.18954	0.0959255	0.0479627	+	0.998849i	$0.484727\pi$
$522$	71.7823	3.14183
$523$	−30.5980	−1.33796	−0.668978	−	0.743283i	$-0.733268\pi$
$523$	−30.5980	−1.33796	−0.668978	+	0.743283i	$0.733268\pi$
$524$	2.33671	0.102079
$525$	5.72211	0.249734
$526$	−35.4344	−1.54502
$527$	29.7848	1.29744
$528$	−13.0375	−0.567386
$529$	0.444081	0.0193079
$530$	−3.18494	−0.138345
$531$	50.8328	2.20595
$532$	−0.735466	−0.0318865
$533$	27.9305	1.20980
$534$	−51.6712	−2.23603
$535$	−8.02816	−0.347087
$536$	12.1161	0.523337
$537$	−13.1695	−0.568305
$538$	25.3576	1.09324
$539$	2.93062	0.126231
$540$	2.11599	0.0910577
$541$	9.73661	0.418610	0.209305	−	0.977850i	$-0.432880\pi$
$541$	9.73661	0.418610	0.209305	+	0.977850i	$0.432880\pi$
$542$	37.0813	1.59278
$543$	13.8606	0.594814
$544$	−11.7013	−0.501690
$545$	17.4014	0.745393
$546$	41.4013	1.77181
$547$	−17.4994	−0.748220	−0.374110	−	0.927384i	$-0.622052\pi$
$547$	−17.4994	−0.748220	−0.374110	+	0.927384i	$0.622052\pi$
$548$	6.02191	0.257243
$549$	70.0446	2.98943
$550$	1.53772	0.0655686
$551$	−9.25093	−0.394103
$552$	34.5394	1.47009
$553$	4.73012	0.201145
$554$	−1.46725	−0.0623376
$555$	−13.4356	−0.570310
$556$	−0.491199	−0.0208315
$557$	24.7800	1.04996	0.524980	−	0.851114i	$-0.324073\pi$
$557$	24.7800	1.04996	0.524980	+	0.851114i	$0.324073\pi$
$558$	40.2550	1.70413
$559$	48.3370	2.04444
$560$	−9.27188	−0.391808
$561$	−16.2854	−0.687569
$562$	43.4975	1.83483
$563$	7.87016	0.331688	0.165844	−	0.986152i	$-0.446965\pi$
$563$	7.87016	0.331688	0.165844	+	0.986152i	$0.446965\pi$
$564$	−10.2851	−0.433082
$565$	4.14618	0.174431
$566$	−39.0506	−1.64142
$567$	−2.67230	−0.112226
$568$	−11.4065	−0.478606
$569$	31.5275	1.32170	0.660850	−	0.750518i	$-0.270196\pi$
$569$	31.5275	1.32170	0.660850	+	0.750518i	$0.270196\pi$
$570$	−4.36184	−0.182697
$571$	−29.9140	−1.25186	−0.625932	−	0.779878i	$-0.715281\pi$
$571$	−29.9140	−1.25186	−0.625932	+	0.779878i	$0.715281\pi$
$572$	1.71545	0.0717265
$573$	20.6086	0.860938
$574$	−18.4137	−0.768572
$575$	−4.84191	−0.201922
$576$	30.5714	1.27381
$577$	−14.6760	−0.610970	−0.305485	−	0.952197i	$-0.598819\pi$
$577$	−14.6760	−0.610970	−0.305485	+	0.952197i	$0.598819\pi$
$578$	−24.5448	−1.02093
$579$	35.0621	1.45713
$580$	3.37275	0.140046
$581$	−26.4270	−1.09638
$582$	5.87930	0.243705
$583$	2.07121	0.0857808
$584$	−38.3668	−1.58763
$585$	23.7429	0.981649
$586$	39.2474	1.62130
$587$	−29.5301	−1.21884	−0.609419	−	0.792848i	$-0.708597\pi$
$587$	−29.5301	−1.21884	−0.609419	+	0.792848i	$0.708597\pi$
$588$	3.03075	0.124986
$589$	−5.18786	−0.213762
$590$	15.4905	0.637736
$591$	58.8813	2.42205
$592$	21.7705	0.894763
$593$	−6.17556	−0.253600	−0.126800	−	0.991928i	$-0.540471\pi$
$593$	−6.17556	−0.253600	−0.126800	+	0.991928i	$0.540471\pi$
$594$	−8.92469	−0.366184
$595$	−11.5816	−0.474801
$596$	3.68776	0.151057
$597$	−11.7616	−0.481372
$598$	−35.0327	−1.43259
$599$	−31.0482	−1.26860	−0.634298	−	0.773088i	$-0.718711\pi$
$599$	−31.0482	−1.26860	−0.634298	+	0.773088i	$0.718711\pi$
$600$	−7.13342	−0.291221
$601$	−13.7230	−0.559771	−0.279885	−	0.960033i	$-0.590296\pi$
$601$	−13.7230	−0.559771	−0.279885	+	0.960033i	$0.590296\pi$
$602$	−31.8670	−1.29880
$603$	24.3116	0.990043
$604$	2.86798	0.116697
$605$	−1.00000	−0.0406558
$606$	54.5612	2.21640
$607$	−25.5372	−1.03652	−0.518261	−	0.855223i	$-0.673420\pi$
$607$	−25.5372	−1.03652	−0.518261	+	0.855223i	$0.673420\pi$
$608$	2.03812	0.0826566
$609$	−52.9349	−2.14503
$610$	21.3451	0.864237
$611$	−46.7950	−1.89312
$612$	−10.5623	−0.426955
$613$	−29.2502	−1.18141	−0.590703	−	0.806889i	$-0.701149\pi$
$613$	−29.2502	−1.18141	−0.590703	+	0.806889i	$0.701149\pi$
$614$	−18.1695	−0.733260
$615$	−16.8380	−0.678975
$616$	5.07306	0.204399
$617$	−1.86862	−0.0752276	−0.0376138	−	0.999292i	$-0.511976\pi$
$617$	−1.86862	−0.0752276	−0.0376138	+	0.999292i	$0.511976\pi$
$618$	−67.7004	−2.72331
$619$	22.3005	0.896332	0.448166	−	0.893950i	$-0.352077\pi$
$619$	22.3005	0.896332	0.448166	+	0.893950i	$0.352077\pi$
$620$	1.89141	0.0759610
$621$	28.1017	1.12768
$622$	−40.0255	−1.60488
$623$	23.8970	0.957413
$624$	−61.3445	−2.45574
$625$	1.00000	0.0400000
$626$	−36.4696	−1.45762
$627$	2.83656	0.113281
$628$	1.69948	0.0678167
$629$	27.1939	1.08429
$630$	−15.6529	−0.623628
$631$	−29.8888	−1.18985	−0.594927	−	0.803780i	$-0.702819\pi$
$631$	−29.8888	−1.18985	−0.594927	+	0.803780i	$0.702819\pi$
$632$	−5.89676	−0.234561
$633$	74.5017	2.96117
$634$	−24.7334	−0.982290
$635$	−3.54518	−0.140686
$636$	2.14198	0.0849348
$637$	13.7892	0.546348
$638$	−14.2254	−0.563187
$639$	−22.8876	−0.905421
$640$	13.3924	0.529382
$641$	30.0650	1.18750	0.593748	−	0.804651i	$-0.297648\pi$
$641$	30.0650	1.18750	0.593748	+	0.804651i	$0.297648\pi$
$642$	35.0175	1.38203
$643$	−12.1532	−0.479274	−0.239637	−	0.970863i	$-0.577028\pi$
$643$	−12.1532	−0.479274	−0.239637	+	0.970863i	$0.577028\pi$
$644$	3.56106	0.140325
$645$	−29.1402	−1.14739
$646$	8.82842	0.347350
$647$	−19.6336	−0.771876	−0.385938	−	0.922525i	$-0.626122\pi$
$647$	−19.6336	−0.771876	−0.385938	+	0.922525i	$0.626122\pi$
$648$	3.33139	0.130869
$649$	−10.0737	−0.395428
$650$	7.23531	0.283792
$651$	−29.6855	−1.16347
$652$	−4.60498	−0.180345
$653$	20.2846	0.793798	0.396899	−	0.917862i	$-0.370086\pi$
$653$	20.2846	0.793798	0.396899	+	0.917862i	$0.370086\pi$
$654$	−75.9020	−2.96800
$655$	−6.40923	−0.250429
$656$	27.2836	1.06525
$657$	−76.9847	−3.00346
$658$	30.8504	1.20267
$659$	−14.2342	−0.554485	−0.277242	−	0.960800i	$-0.589421\pi$
$659$	−14.2342	−0.554485	−0.277242	+	0.960800i	$0.589421\pi$
$660$	−1.03417	−0.0402549
$661$	−16.6865	−0.649029	−0.324514	−	0.945881i	$-0.605201\pi$
$661$	−16.6865	−0.649029	−0.324514	+	0.945881i	$0.605201\pi$
$662$	−22.3608	−0.869078
$663$	−76.6263	−2.97592
$664$	32.9450	1.27851
$665$	2.01727	0.0782264
$666$	36.7534	1.42416
$667$	44.7922	1.73436
$668$	−1.60714	−0.0621820
$669$	33.7744	1.30579
$670$	7.40859	0.286219
$671$	−13.8810	−0.535870
$672$	11.6623	0.449885
$673$	−16.4361	−0.633564	−0.316782	−	0.948498i	$-0.602602\pi$
$673$	−16.4361	−0.633564	−0.316782	+	0.948498i	$0.602602\pi$
$674$	−38.3750	−1.47815
$675$	−5.80384	−0.223390
$676$	3.33197	0.128153
$677$	1.59022	0.0611170	0.0305585	−	0.999533i	$-0.490271\pi$
$677$	1.59022	0.0611170	0.0305585	+	0.999533i	$0.490271\pi$
$678$	−18.0850	−0.694549
$679$	−2.71907	−0.104348
$680$	14.4381	0.553677
$681$	−33.3654	−1.27856
$682$	−7.97748	−0.305474
$683$	43.9225	1.68065	0.840324	−	0.542084i	$-0.182365\pi$
$683$	43.9225	1.68065	0.840324	+	0.542084i	$0.182365\pi$
$684$	1.83972	0.0703436
$685$	−16.5172	−0.631089
$686$	−30.8048	−1.17613
$687$	−19.0818	−0.728017
$688$	47.2175	1.80015
$689$	9.74550	0.371274
$690$	21.1196	0.804011
$691$	19.8971	0.756922	0.378461	−	0.925617i	$-0.376453\pi$
$691$	19.8971	0.756922	0.378461	+	0.925617i	$0.376453\pi$
$692$	6.27412	0.238506
$693$	10.1793	0.386680
$694$	17.1156	0.649700
$695$	1.34728	0.0511054
$696$	65.9908	2.50137
$697$	34.0804	1.29089
$698$	17.4465	0.660358
$699$	45.7922	1.73202
$700$	−0.735466	−0.0277980
$701$	−3.01626	−0.113923	−0.0569613	−	0.998376i	$-0.518141\pi$
$701$	−3.01626	−0.113923	−0.0569613	+	0.998376i	$0.518141\pi$
$702$	−41.9926	−1.58491
$703$	−4.73659	−0.178644
$704$	−6.05844	−0.228336
$705$	28.2106	1.06247
$706$	−37.4456	−1.40928
$707$	−25.2336	−0.949006
$708$	−10.4179	−0.391528
$709$	−19.3868	−0.728085	−0.364042	−	0.931382i	$-0.618604\pi$
$709$	−19.3868	−0.728085	−0.364042	+	0.931382i	$0.618604\pi$
$710$	−6.97467	−0.261755
$711$	−11.8321	−0.443739
$712$	−29.7910	−1.11646
$713$	25.1192	0.940720
$714$	50.5172	1.89056
$715$	−4.70522	−0.175965
$716$	1.69268	0.0632583
$717$	16.1067	0.601517
$718$	9.22952	0.344442
$719$	11.6979	0.436260	0.218130	−	0.975920i	$-0.430004\pi$
$719$	11.6979	0.436260	0.218130	+	0.975920i	$0.430004\pi$
$720$	23.1930	0.864354
$721$	31.3102	1.16605
$722$	−1.53772	−0.0572280
$723$	66.2558	2.46408
$724$	−1.78151	−0.0662091
$725$	−9.25093	−0.343571
$726$	4.36184	0.161883
$727$	−30.9023	−1.14610	−0.573052	−	0.819519i	$-0.694241\pi$
$727$	−30.9023	−1.14610	−0.573052	+	0.819519i	$0.694241\pi$
$728$	23.8699	0.884676
$729$	−42.7043	−1.58164
$730$	−23.4600	−0.868293
$731$	58.9801	2.18146
$732$	−14.3552	−0.530585
$733$	−29.7115	−1.09742	−0.548710	−	0.836012i	$-0.684881\pi$
$733$	−29.7115	−1.09742	−0.548710	+	0.836012i	$0.684881\pi$
$734$	8.17576	0.301773
$735$	−8.31288	−0.306625
$736$	−9.86839	−0.363754
$737$	−4.81791	−0.177470
$738$	46.0607	1.69552
$739$	28.8684	1.06194	0.530970	−	0.847390i	$-0.321828\pi$
$739$	28.8684	1.06194	0.530970	+	0.847390i	$0.321828\pi$
$740$	1.72689	0.0634816
$741$	13.3466	0.490301
$742$	−6.42489	−0.235865
$743$	26.2048	0.961361	0.480680	−	0.876896i	$-0.340390\pi$
$743$	26.2048	0.961361	0.480680	+	0.876896i	$0.340390\pi$
$744$	37.0072	1.35675
$745$	−10.1150	−0.370584
$746$	28.7886	1.05403
$747$	66.1056	2.41868
$748$	2.09317	0.0765337
$749$	−16.1950	−0.591751
$750$	−4.36184	−0.159272
$751$	−19.7513	−0.720735	−0.360367	−	0.932810i	$-0.617349\pi$
$751$	−19.7513	−0.720735	−0.360367	+	0.932810i	$0.617349\pi$
$752$	−45.7112	−1.66692
$753$	−34.7599	−1.26672
$754$	−66.9334	−2.43757
$755$	−7.86645	−0.286289
$756$	4.26853	0.155245
$757$	−19.8301	−0.720737	−0.360368	−	0.932810i	$-0.617349\pi$
$757$	−19.8301	−0.720737	−0.360368	+	0.932810i	$0.617349\pi$
$758$	−24.1855	−0.878457
$759$	−13.7344	−0.498526
$760$	−2.51481	−0.0912218
$761$	16.8734	0.611661	0.305831	−	0.952086i	$-0.401066\pi$
$761$	16.8734	0.611661	0.305831	+	0.952086i	$0.401066\pi$
$762$	15.4635	0.560184
$763$	35.1033	1.27082
$764$	−2.64884	−0.0958315
$765$	28.9708	1.04744
$766$	−5.27343	−0.190537
$767$	−47.3990	−1.71148
$768$	−24.0453	−0.867661
$769$	16.4390	0.592804	0.296402	−	0.955063i	$-0.404213\pi$
$769$	16.4390	0.592804	0.296402	+	0.955063i	$0.404213\pi$
$770$	3.10200	0.111788
$771$	−77.0623	−2.77533
$772$	−4.50655	−0.162194
$773$	−31.5015	−1.13303	−0.566516	−	0.824051i	$-0.691709\pi$
$773$	−31.5015	−1.13303	−0.566516	+	0.824051i	$0.691709\pi$
$774$	79.7135	2.86524
$775$	−5.18786	−0.186354
$776$	3.38970	0.121683
$777$	−27.1033	−0.972325
$778$	31.4055	1.12594
$779$	−5.93607	−0.212682
$780$	−4.86598	−0.174230
$781$	4.53572	0.162301
$782$	−42.7464	−1.52861
$783$	53.6910	1.91876
$784$	13.4698	0.481066
$785$	−4.66142	−0.166373
$786$	27.9560	0.997159
$787$	−15.4972	−0.552415	−0.276207	−	0.961098i	$-0.589078\pi$
$787$	−15.4972	−0.552415	−0.276207	+	0.961098i	$0.589078\pi$
$788$	−7.56804	−0.269600
$789$	−65.3643	−2.32703
$790$	−3.60566	−0.128284
$791$	8.36397	0.297389
$792$	−12.6900	−0.450918
$793$	−65.3131	−2.31934
$794$	5.16526	0.183308
$795$	−5.87512	−0.208369
$796$	1.51173	0.0535818
$797$	3.03614	0.107546	0.0537729	−	0.998553i	$-0.482875\pi$
$797$	3.03614	0.107546	0.0537729	+	0.998553i	$0.482875\pi$
$798$	−8.79901	−0.311482
$799$	−57.0986	−2.02000
$800$	2.03812	0.0720584
$801$	−59.7769	−2.11211
$802$	0.953360	0.0336643
$803$	15.2563	0.538384
$804$	−4.98251	−0.175720
$805$	−9.76744	−0.344257
$806$	−37.5358	−1.32214
$807$	46.7760	1.64659
$808$	31.4572	1.10666
$809$	−47.1638	−1.65819	−0.829095	−	0.559108i	$-0.811144\pi$
$809$	−47.1638	−1.65819	−0.829095	+	0.559108i	$0.811144\pi$
$810$	2.03703	0.0715740
$811$	−5.89815	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$811$	−5.89815	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$812$	6.80374	0.238765
$813$	68.4021	2.39897
$814$	−7.28355	−0.255288
$815$	12.6308	0.442437
$816$	−74.8516	−2.62033
$817$	−10.2731	−0.359409
$818$	8.46455	0.295956
$819$	47.8959	1.67362
$820$	2.16420	0.0755771
$821$	5.79242	0.202157	0.101078	−	0.994878i	$-0.467771\pi$
$821$	5.79242	0.202157	0.101078	+	0.994878i	$0.467771\pi$
$822$	72.0453	2.51287
$823$	−29.5146	−1.02881	−0.514407	−	0.857546i	$-0.671988\pi$
$823$	−29.5146	−1.02881	−0.514407	+	0.857546i	$0.671988\pi$
$824$	−39.0325	−1.35976
$825$	2.83656	0.0987564
$826$	31.2486	1.08728
$827$	26.7950	0.931754	0.465877	−	0.884849i	$-0.345739\pi$
$827$	26.7950	0.931754	0.465877	+	0.884849i	$0.345739\pi$
$828$	−8.90777	−0.309566
$829$	−27.7792	−0.964813	−0.482407	−	0.875947i	$-0.660237\pi$
$829$	−27.7792	−0.964813	−0.482407	+	0.875947i	$0.660237\pi$
$830$	20.1447	0.699234
$831$	−2.70657	−0.0938900
$832$	−28.5063	−0.988277
$833$	16.8254	0.582965
$834$	−5.87664	−0.203491
$835$	4.40813	0.152550
$836$	−0.364584	−0.0126094
$837$	30.1095	1.04074
$838$	−24.3692	−0.841821
$839$	−26.6561	−0.920269	−0.460135	−	0.887849i	$-0.652199\pi$
$839$	−26.6561	−0.920269	−0.460135	+	0.887849i	$0.652199\pi$
$840$	−14.3900	−0.496504
$841$	56.5798	1.95103
$842$	23.2452	0.801081
$843$	80.2378	2.76354
$844$	−9.57573	−0.329610
$845$	−9.13909	−0.314394
$846$	−77.1705	−2.65318
$847$	−2.01727	−0.0693143
$848$	9.51979	0.326911
$849$	−72.0348	−2.47223
$850$	8.82842	0.302812
$851$	22.9341	0.786171
$852$	4.69069	0.160700
$853$	23.7885	0.814504	0.407252	−	0.913316i	$-0.366487\pi$
$853$	23.7885	0.814504	0.407252	+	0.913316i	$0.366487\pi$
$854$	43.0588	1.47344
$855$	−5.04608	−0.172572
$856$	20.1893	0.690056
$857$	−47.7351	−1.63060	−0.815299	−	0.579040i	$-0.803428\pi$
$857$	−47.7351	−1.63060	−0.815299	+	0.579040i	$0.803428\pi$
$858$	20.5234	0.700658
$859$	−5.86761	−0.200200	−0.100100	−	0.994977i	$-0.531916\pi$
$859$	−5.86761	−0.200200	−0.100100	+	0.994977i	$0.531916\pi$
$860$	3.74540	0.127717
$861$	−33.9669	−1.15759
$862$	21.4842	0.731756
$863$	22.9280	0.780478	0.390239	−	0.920714i	$-0.372392\pi$
$863$	22.9280	0.780478	0.390239	+	0.920714i	$0.372392\pi$
$864$	−11.8289	−0.402428
$865$	−17.2090	−0.585122
$866$	−39.7166	−1.34963
$867$	−45.2767	−1.53768
$868$	3.81549	0.129506
$869$	2.34481	0.0795423
$870$	40.3511	1.36803
$871$	−22.6693	−0.768120
$872$	−43.7612	−1.48194
$873$	6.80159	0.230199
$874$	7.44550	0.251848
$875$	2.01727	0.0681962
$876$	15.7776	0.533075
$877$	55.0317	1.85829	0.929145	−	0.369716i	$-0.120545\pi$
$877$	55.0317	1.85829	0.929145	+	0.369716i	$0.120545\pi$
$878$	−7.63813	−0.257775
$879$	72.3979	2.44192
$880$	−4.59625	−0.154939
$881$	38.7321	1.30492	0.652459	−	0.757824i	$-0.273737\pi$
$881$	38.7321	1.30492	0.652459	+	0.757824i	$0.273737\pi$
$882$	22.7400	0.765696
$883$	−0.517241	−0.0174066	−0.00870328	−	0.999962i	$-0.502770\pi$
$883$	−0.517241	−0.0174066	−0.00870328	+	0.999962i	$0.502770\pi$
$884$	9.84881	0.331251
$885$	28.5747	0.960528
$886$	59.1655	1.98771
$887$	−52.7106	−1.76985	−0.884924	−	0.465736i	$-0.845790\pi$
$887$	−52.7106	−1.76985	−0.884924	+	0.465736i	$0.845790\pi$
$888$	33.7881	1.13385
$889$	−7.15160	−0.239857
$890$	−18.2161	−0.610607
$891$	−1.32471	−0.0443794
$892$	−4.34104	−0.145349
$893$	9.94533	0.332808
$894$	44.1199	1.47559
$895$	−4.64276	−0.155190
$896$	27.0161	0.902546
$897$	−64.6232	−2.15771
$898$	37.0505	1.23639
$899$	47.9926	1.60064
$900$	1.83972	0.0613241
$901$	11.8913	0.396157
$902$	−9.12801	−0.303930
$903$	−58.7836	−1.95620
$904$	−10.4269	−0.346792
$905$	4.88640	0.162429
$906$	34.3122	1.13995
$907$	−9.12788	−0.303086	−0.151543	−	0.988451i	$-0.548424\pi$
$907$	−9.12788	−0.303086	−0.151543	+	0.988451i	$0.548424\pi$
$908$	4.28846	0.142318
$909$	63.1203	2.09357
$910$	14.5956	0.483839
$911$	3.25455	0.107828	0.0539141	−	0.998546i	$-0.482830\pi$
$911$	3.25455	0.107828	0.0539141	+	0.998546i	$0.482830\pi$
$912$	13.0375	0.431716
$913$	−13.1004	−0.433559
$914$	−11.5516	−0.382092
$915$	39.3743	1.30167
$916$	2.45259	0.0810360
$917$	−12.9292	−0.426959
$918$	−51.2388	−1.69113
$919$	20.0976	0.662957	0.331479	−	0.943463i	$-0.392453\pi$
$919$	20.0976	0.662957	0.331479	+	0.943463i	$0.392453\pi$
$920$	12.1765	0.401447
$921$	−33.5164	−1.10440
$922$	1.02985	0.0339165
$923$	21.3416	0.702466
$924$	−2.08619	−0.0686307
$925$	−4.73659	−0.155738
$926$	−3.64572	−0.119806
$927$	−78.3206	−2.57238
$928$	−18.8545	−0.618929
$929$	11.0406	0.362231	0.181116	−	0.983462i	$-0.442029\pi$
$929$	11.0406	0.362231	0.181116	+	0.983462i	$0.442029\pi$
$930$	22.6286	0.742022
$931$	−2.93062	−0.0960471
$932$	−5.88569	−0.192792
$933$	−73.8333	−2.41719
$934$	−5.93272	−0.194124
$935$	−5.74124	−0.187759
$936$	−59.7090	−1.95165
$937$	−46.7942	−1.52870	−0.764349	−	0.644802i	$-0.776940\pi$
$937$	−46.7942	−1.52870	−0.764349	+	0.644802i	$0.776940\pi$
$938$	14.9451	0.487976
$939$	−67.2738	−2.19540
$940$	−3.62591	−0.118264
$941$	−25.4873	−0.830861	−0.415430	−	0.909625i	$-0.636369\pi$
$941$	−25.4873	−0.830861	−0.415430	+	0.909625i	$0.636369\pi$
$942$	20.3324	0.662465
$943$	28.7419	0.935965
$944$	−46.3012	−1.50698
$945$	−11.7079	−0.380859
$946$	−15.7971	−0.513608
$947$	−5.99767	−0.194898	−0.0974490	−	0.995241i	$-0.531068\pi$
$947$	−5.99767	−0.194898	−0.0974490	+	0.995241i	$0.531068\pi$
$948$	2.42492	0.0787579
$949$	71.7844	2.33022
$950$	−1.53772	−0.0498902
$951$	−45.6246	−1.47948
$952$	29.1256	0.943968
$953$	−45.6410	−1.47846	−0.739228	−	0.673455i	$-0.764810\pi$
$953$	−45.6410	−1.47846	−0.739228	+	0.673455i	$0.764810\pi$
$954$	16.0715	0.520334
$955$	7.26536	0.235101
$956$	−2.07021	−0.0669552
$957$	−26.2408	−0.848246
$958$	43.0657	1.39139
$959$	−33.3196	−1.07595
$960$	17.1851	0.554648
$961$	−4.08608	−0.131809
$962$	−34.2707	−1.10493
$963$	40.5107	1.30544
$964$	−8.51588	−0.274278
$965$	12.3608	0.397908
$966$	42.6040	1.37076
$967$	24.9342	0.801828	0.400914	−	0.916116i	$-0.368693\pi$
$967$	24.9342	0.801828	0.400914	+	0.916116i	$0.368693\pi$
$968$	2.51481	0.0808291
$969$	16.2854	0.523162
$970$	2.07268	0.0665499
$971$	−30.9950	−0.994678	−0.497339	−	0.867556i	$-0.665689\pi$
$971$	−30.9950	−0.994678	−0.497339	+	0.867556i	$0.665689\pi$
$972$	4.97800	0.159670
$973$	2.71784	0.0871299
$974$	−4.57340	−0.146541
$975$	13.3466	0.427435
$976$	−63.8004	−2.04220
$977$	−53.5055	−1.71179	−0.855896	−	0.517148i	$-0.826994\pi$
$977$	−53.5055	−1.71179	−0.855896	+	0.517148i	$0.826994\pi$
$978$	−55.0934	−1.76169
$979$	11.8462	0.378606
$980$	1.06846	0.0341306
$981$	−87.8088	−2.80352
$982$	13.3362	0.425574
$983$	−6.15000	−0.196154	−0.0980772	−	0.995179i	$-0.531269\pi$
$983$	−6.15000	−0.196154	−0.0980772	+	0.995179i	$0.531269\pi$
$984$	42.3445	1.34989
$985$	20.7580	0.661404
$986$	−81.6712	−2.60094
$987$	56.9083	1.81141
$988$	−1.71545	−0.0545757
$989$	49.7412	1.58168
$990$	−7.75947	−0.246612
$991$	39.7330	1.26216	0.631081	−	0.775717i	$-0.282612\pi$
$991$	39.7330	1.26216	0.631081	+	0.775717i	$0.282612\pi$
$992$	−10.5735	−0.335708
$993$	−41.2480	−1.30897
$994$	−14.0698	−0.446267
$995$	−4.14644	−0.131451
$996$	−13.5480	−0.429284
$997$	−45.8109	−1.45085	−0.725423	−	0.688304i	$-0.758356\pi$
$997$	−45.8109	−1.45085	−0.725423	+	0.688304i	$0.758356\pi$
$998$	−10.9495	−0.346600
$999$	27.4904	0.869758

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.j.1.2	✓	9	1.1	even	1	trivial
5225.2.a.q.1.8		9	5.4	even	2
9405.2.a.bi.1.8		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)

\(f(q)\)	\(=\)	\(q-1.53772 q^{2} -2.83656 q^{3} +0.364584 q^{4} -1.00000 q^{5} +4.36184 q^{6} -2.01727 q^{7} +2.51481 q^{8} +5.04608 q^{9} +O(q^{10})\)	\(q-1.53772 q^{2} -2.83656 q^{3} +0.364584 q^{4} -1.00000 q^{5} +4.36184 q^{6} -2.01727 q^{7} +2.51481 q^{8} +5.04608 q^{9} +1.53772 q^{10} -1.00000 q^{11} -1.03417 q^{12} -4.70522 q^{13} +3.10200 q^{14} +2.83656 q^{15} -4.59625 q^{16} -5.74124 q^{17} -7.75947 q^{18} +1.00000 q^{19} -0.364584 q^{20} +5.72211 q^{21} +1.53772 q^{22} -4.84191 q^{23} -7.13342 q^{24} +1.00000 q^{25} +7.23531 q^{26} -5.80384 q^{27} -0.735466 q^{28} -9.25093 q^{29} -4.36184 q^{30} -5.18786 q^{31} +2.03812 q^{32} +2.83656 q^{33} +8.82842 q^{34} +2.01727 q^{35} +1.83972 q^{36} -4.73659 q^{37} -1.53772 q^{38} +13.3466 q^{39} -2.51481 q^{40} -5.93607 q^{41} -8.79901 q^{42} -10.2731 q^{43} -0.364584 q^{44} -5.04608 q^{45} +7.44550 q^{46} +9.94533 q^{47} +13.0375 q^{48} -2.93062 q^{49} -1.53772 q^{50} +16.2854 q^{51} -1.71545 q^{52} -2.07121 q^{53} +8.92469 q^{54} +1.00000 q^{55} -5.07306 q^{56} -2.83656 q^{57} +14.2254 q^{58} +10.0737 q^{59} +1.03417 q^{60} +13.8810 q^{61} +7.97748 q^{62} -10.1793 q^{63} +6.05844 q^{64} +4.70522 q^{65} -4.36184 q^{66} +4.81791 q^{67} -2.09317 q^{68} +13.7344 q^{69} -3.10200 q^{70} -4.53572 q^{71} +12.6900 q^{72} -15.2563 q^{73} +7.28355 q^{74} -2.83656 q^{75} +0.364584 q^{76} +2.01727 q^{77} -20.5234 q^{78} -2.34481 q^{79} +4.59625 q^{80} +1.32471 q^{81} +9.12801 q^{82} +13.1004 q^{83} +2.08619 q^{84} +5.74124 q^{85} +15.7971 q^{86} +26.2408 q^{87} -2.51481 q^{88} -11.8462 q^{89} +7.75947 q^{90} +9.49170 q^{91} -1.76528 q^{92} +14.7157 q^{93} -15.2931 q^{94} -1.00000 q^{95} -5.78125 q^{96} +1.34789 q^{97} +4.50647 q^{98} -5.04608 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 3 q^{2} + 3 q^{3} + 9 q^{4} - 9 q^{5} + 6 q^{6} - 9 q^{7} + 15 q^{8} + 16 q^{9}+O(q^{10}) \) 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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