LMFDB - Embedded newform 869.2.a.g.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [869,2,Mod(1,869)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(869, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("869.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 869 = 11 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	869.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:	$6.93899993565$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 4 x^{17} - 22 x^{16} + 106 x^{15} + 154 x^{14} - 1097 x^{13} - 124 x^{12} + 5565 x^{11} + \cdots - 53 $ x^18 - 4x^17 - 22x^16 + 106x^15 + 154x^14 - 1097x^13 - 124x^12 + 5565x^11 - 3154x^10 - 14068x^9 + 14526x^8 + 15037x^7 - 24233x^6 - 928x^5 + 13949x^4 - 6248x^3 + 156x^2 + 373*x - 53
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Root			$2.46820$ of defining polynomial
Character	$\chi$	$=$	869.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.46820 q^{2} +0.912264 q^{3} +4.09201 q^{4} -1.28776 q^{5} +2.25165 q^{6} +2.61736 q^{7} +5.16351 q^{8} -2.16777 q^{9} +O(q^{10})$	\(q+2.46820 q^{2} +0.912264 q^{3} +4.09201 q^{4} -1.28776 q^{5} +2.25165 q^{6} +2.61736 q^{7} +5.16351 q^{8} -2.16777 q^{9} -3.17845 q^{10} +1.00000 q^{11} +3.73300 q^{12} +5.21419 q^{13} +6.46018 q^{14} -1.17478 q^{15} +4.56055 q^{16} -3.11539 q^{17} -5.35050 q^{18} -2.33058 q^{19} -5.26953 q^{20} +2.38773 q^{21} +2.46820 q^{22} +0.942834 q^{23} +4.71048 q^{24} -3.34168 q^{25} +12.8697 q^{26} -4.71437 q^{27} +10.7103 q^{28} -2.31509 q^{29} -2.89958 q^{30} +1.36058 q^{31} +0.929330 q^{32} +0.912264 q^{33} -7.68941 q^{34} -3.37053 q^{35} -8.87056 q^{36} +8.05603 q^{37} -5.75233 q^{38} +4.75672 q^{39} -6.64935 q^{40} -0.503115 q^{41} +5.89339 q^{42} -1.83695 q^{43} +4.09201 q^{44} +2.79157 q^{45} +2.32710 q^{46} +5.10159 q^{47} +4.16042 q^{48} -0.149406 q^{49} -8.24793 q^{50} -2.84206 q^{51} +21.3365 q^{52} +0.0506846 q^{53} -11.6360 q^{54} -1.28776 q^{55} +13.5148 q^{56} -2.12610 q^{57} -5.71410 q^{58} -2.31495 q^{59} -4.80720 q^{60} -8.30400 q^{61} +3.35819 q^{62} -5.67386 q^{63} -6.82732 q^{64} -6.71462 q^{65} +2.25165 q^{66} -3.35748 q^{67} -12.7482 q^{68} +0.860114 q^{69} -8.31915 q^{70} -12.4315 q^{71} -11.1933 q^{72} +5.26652 q^{73} +19.8839 q^{74} -3.04849 q^{75} -9.53675 q^{76} +2.61736 q^{77} +11.7405 q^{78} -1.00000 q^{79} -5.87289 q^{80} +2.20257 q^{81} -1.24179 q^{82} -11.4118 q^{83} +9.77061 q^{84} +4.01187 q^{85} -4.53395 q^{86} -2.11197 q^{87} +5.16351 q^{88} +2.24769 q^{89} +6.89016 q^{90} +13.6474 q^{91} +3.85809 q^{92} +1.24121 q^{93} +12.5917 q^{94} +3.00122 q^{95} +0.847794 q^{96} -0.550198 q^{97} -0.368764 q^{98} -2.16777 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 4 q^{2} + 8 q^{3} + 24 q^{4} - 3 q^{5} + 2 q^{6} + 10 q^{7} - 6 q^{8} + 18 q^{9}+O(q^{10}) $ 18 * q + 4 * q^2 + 8 * q^3 + 24 * q^4 - 3 * q^5 + 2 * q^6 + 10 * q^7 - 6 * q^8 + 18 * q^9	\( 18 q + 4 q^{2} + 8 q^{3} + 24 q^{4} - 3 q^{5} + 2 q^{6} + 10 q^{7} - 6 q^{8} + 18 q^{9} + 7 q^{10} + 18 q^{11} + 5 q^{12} + 4 q^{13} - 7 q^{14} + 4 q^{15} + 32 q^{16} + 19 q^{17} + 9 q^{18} + 47 q^{19} - 5 q^{20} + 7 q^{21} + 4 q^{22} - q^{23} + 43 q^{24} + 23 q^{25} - q^{26} - q^{27} + 20 q^{28} + 5 q^{29} + 7 q^{30} + 7 q^{31} - 23 q^{32} + 8 q^{33} + 18 q^{34} + 15 q^{35} + 24 q^{36} - 8 q^{37} - 6 q^{38} + 26 q^{39} - 6 q^{40} + 37 q^{41} - 47 q^{42} + 22 q^{43} + 24 q^{44} - 9 q^{45} + 36 q^{46} - 26 q^{47} - 34 q^{48} + 20 q^{49} - 32 q^{50} - q^{51} + 53 q^{52} - 19 q^{53} + 16 q^{54} - 3 q^{55} - 52 q^{56} - 6 q^{57} + 29 q^{58} + 20 q^{59} + 8 q^{60} + 47 q^{61} - 11 q^{62} + 8 q^{63} + 4 q^{64} - q^{65} + 2 q^{66} + 16 q^{67} + 52 q^{68} - q^{69} - 69 q^{70} - 8 q^{71} - 24 q^{72} + 20 q^{73} + 26 q^{74} - 18 q^{75} + 54 q^{76} + 10 q^{77} - 29 q^{78} - 18 q^{79} - 7 q^{80} + 2 q^{81} - 2 q^{82} + 15 q^{83} - 92 q^{84} + 28 q^{85} - 27 q^{86} - 15 q^{87} - 6 q^{88} - 21 q^{89} + 27 q^{90} + 60 q^{91} - 62 q^{92} - 27 q^{93} + 20 q^{94} - 11 q^{95} + 84 q^{96} - 17 q^{97} - 57 q^{98} + 18 q^{99}+O(q^{100}) \) 18 * q + 4 * q^2 + 8 * q^3 + 24 * q^4 - 3 * q^5 + 2 * q^6 + 10 * q^7 - 6 * q^8 + 18 * q^9 + 7 * q^10 + 18 * q^11 + 5 * q^12 + 4 * q^13 - 7 * q^14 + 4 * q^15 + 32 * q^16 + 19 * q^17 + 9 * q^18 + 47 * q^19 - 5 * q^20 + 7 * q^21 + 4 * q^22 - q^23 + 43 * q^24 + 23 * q^25 - q^26 - q^27 + 20 * q^28 + 5 * q^29 + 7 * q^30 + 7 * q^31 - 23 * q^32 + 8 * q^33 + 18 * q^34 + 15 * q^35 + 24 * q^36 - 8 * q^37 - 6 * q^38 + 26 * q^39 - 6 * q^40 + 37 * q^41 - 47 * q^42 + 22 * q^43 + 24 * q^44 - 9 * q^45 + 36 * q^46 - 26 * q^47 - 34 * q^48 + 20 * q^49 - 32 * q^50 - q^51 + 53 * q^52 - 19 * q^53 + 16 * q^54 - 3 * q^55 - 52 * q^56 - 6 * q^57 + 29 * q^58 + 20 * q^59 + 8 * q^60 + 47 * q^61 - 11 * q^62 + 8 * q^63 + 4 * q^64 - q^65 + 2 * q^66 + 16 * q^67 + 52 * q^68 - q^69 - 69 * q^70 - 8 * q^71 - 24 * q^72 + 20 * q^73 + 26 * q^74 - 18 * q^75 + 54 * q^76 + 10 * q^77 - 29 * q^78 - 18 * q^79 - 7 * q^80 + 2 * q^81 - 2 * q^82 + 15 * q^83 - 92 * q^84 + 28 * q^85 - 27 * q^86 - 15 * q^87 - 6 * q^88 - 21 * q^89 + 27 * q^90 + 60 * q^91 - 62 * q^92 - 27 * q^93 + 20 * q^94 - 11 * q^95 + 84 * q^96 - 17 * q^97 - 57 * q^98 + 18 * 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 $

$2$	2.46820	1.74528	0.872641	−	0.488363i	$-0.162406\pi$
$2$	2.46820	1.74528	0.872641	+	0.488363i	$0.162406\pi$
$3$	0.912264	0.526696	0.263348	−	0.964701i	$-0.415173\pi$
$3$	0.912264	0.526696	0.263348	+	0.964701i	$0.415173\pi$
$4$	4.09201	2.04601
$5$	−1.28776	−0.575903	−0.287952	−	0.957645i	$-0.592974\pi$
$5$	−1.28776	−0.575903	−0.287952	+	0.957645i	$0.592974\pi$
$6$	2.25165	0.919232
$7$	2.61736	0.989271	0.494635	−	0.869101i	$-0.335302\pi$
$7$	2.61736	0.989271	0.494635	+	0.869101i	$0.335302\pi$
$8$	5.16351	1.82558
$9$	−2.16777	−0.722592
$10$	−3.17845	−1.00511
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	3.73300	1.07762
$13$	5.21419	1.44616	0.723078	−	0.690766i	$-0.242727\pi$
$13$	5.21419	1.44616	0.723078	+	0.690766i	$0.242727\pi$
$14$	6.46018	1.72656
$15$	−1.17478	−0.303326
$16$	4.56055	1.14014
$17$	−3.11539	−0.755594	−0.377797	−	0.925889i	$-0.623318\pi$
$17$	−3.11539	−0.755594	−0.377797	+	0.925889i	$0.623318\pi$
$18$	−5.35050	−1.26113
$19$	−2.33058	−0.534671	−0.267335	−	0.963604i	$-0.586143\pi$
$19$	−2.33058	−0.534671	−0.267335	+	0.963604i	$0.586143\pi$
$20$	−5.26953	−1.17830
$21$	2.38773	0.521045
$22$	2.46820	0.526222
$23$	0.942834	0.196595	0.0982973	−	0.995157i	$-0.468660\pi$
$23$	0.942834	0.196595	0.0982973	+	0.995157i	$0.468660\pi$
$24$	4.71048	0.961523
$25$	−3.34168	−0.668335
$26$	12.8697	2.52395
$27$	−4.71437	−0.907282
$28$	10.7103	2.02405
$29$	−2.31509	−0.429901	−0.214950	−	0.976625i	$-0.568959\pi$
$29$	−2.31509	−0.429901	−0.214950	+	0.976625i	$0.568959\pi$
$30$	−2.89958	−0.529389
$31$	1.36058	0.244368	0.122184	−	0.992507i	$-0.461010\pi$
$31$	1.36058	0.244368	0.122184	+	0.992507i	$0.461010\pi$
$32$	0.929330	0.164284
$33$	0.912264	0.158805
$34$	−7.68941	−1.31872
$35$	−3.37053	−0.569724
$36$	−8.87056	−1.47843
$37$	8.05603	1.32440	0.662202	−	0.749326i	$-0.269622\pi$
$37$	8.05603	1.32440	0.662202	+	0.749326i	$0.269622\pi$
$38$	−5.75233	−0.933151
$39$	4.75672	0.761684
$40$	−6.64935	−1.05136
$41$	−0.503115	−0.0785734	−0.0392867	−	0.999228i	$-0.512509\pi$
$41$	−0.503115	−0.0785734	−0.0392867	+	0.999228i	$0.512509\pi$
$42$	5.89339	0.909369
$43$	−1.83695	−0.280132	−0.140066	−	0.990142i	$-0.544731\pi$
$43$	−1.83695	−0.280132	−0.140066	+	0.990142i	$0.544731\pi$
$44$	4.09201	0.616894
$45$	2.79157	0.416143
$46$	2.32710	0.343113
$47$	5.10159	0.744143	0.372072	−	0.928204i	$-0.378648\pi$
$47$	5.10159	0.744143	0.372072	+	0.928204i	$0.378648\pi$
$48$	4.16042	0.600505
$49$	−0.149406	−0.0213437
$50$	−8.24793	−1.16643
$51$	−2.84206	−0.397968
$52$	21.3365	2.95885
$53$	0.0506846	0.00696206	0.00348103	−	0.999994i	$-0.498892\pi$
$53$	0.0506846	0.00696206	0.00348103	+	0.999994i	$0.498892\pi$
$54$	−11.6360	−1.58346
$55$	−1.28776	−0.173641
$56$	13.5148	1.80599
$57$	−2.12610	−0.281609
$58$	−5.71410	−0.750298
$59$	−2.31495	−0.301381	−0.150690	−	0.988581i	$-0.548150\pi$
$59$	−2.31495	−0.301381	−0.150690	+	0.988581i	$0.548150\pi$
$60$	−4.80720	−0.620607
$61$	−8.30400	−1.06322	−0.531609	−	0.846990i	$-0.678412\pi$
$61$	−8.30400	−1.06322	−0.531609	+	0.846990i	$0.678412\pi$
$62$	3.35819	0.426491
$63$	−5.67386	−0.714839
$64$	−6.82732	−0.853416
$65$	−6.71462	−0.832846
$66$	2.25165	0.277159
$67$	−3.35748	−0.410181	−0.205091	−	0.978743i	$-0.565749\pi$
$67$	−3.35748	−0.410181	−0.205091	+	0.978743i	$0.565749\pi$
$68$	−12.7482	−1.54595
$69$	0.860114	0.103546
$70$	−8.31915	−0.994329
$71$	−12.4315	−1.47535	−0.737675	−	0.675155i	$-0.764077\pi$
$71$	−12.4315	−1.47535	−0.737675	+	0.675155i	$0.764077\pi$
$72$	−11.1933	−1.31915
$73$	5.26652	0.616400	0.308200	−	0.951322i	$-0.400273\pi$
$73$	5.26652	0.616400	0.308200	+	0.951322i	$0.400273\pi$
$74$	19.8839	2.31146
$75$	−3.04849	−0.352009
$76$	−9.53675	−1.09394
$77$	2.61736	0.298276
$78$	11.7405	1.32935
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	−5.87289	−0.656609
$81$	2.20257	0.244730
$82$	−1.24179	−0.137133
$83$	−11.4118	−1.25261	−0.626303	−	0.779580i	$-0.715433\pi$
$83$	−11.4118	−1.25261	−0.626303	+	0.779580i	$0.715433\pi$
$84$	9.77061	1.06606
$85$	4.01187	0.435149
$86$	−4.53395	−0.488909
$87$	−2.11197	−0.226427
$88$	5.16351	0.550432
$89$	2.24769	0.238255	0.119127	−	0.992879i	$-0.461990\pi$
$89$	2.24769	0.238255	0.119127	+	0.992879i	$0.461990\pi$
$90$	6.89016	0.726286
$91$	13.6474	1.43064
$92$	3.85809	0.402234
$93$	1.24121	0.128708
$94$	12.5917	1.29874
$95$	3.00122	0.307919
$96$	0.847794	0.0865276
$97$	−0.550198	−0.0558641	−0.0279321	−	0.999610i	$-0.508892\pi$
$97$	−0.550198	−0.0558641	−0.0279321	+	0.999610i	$0.508892\pi$
$98$	−0.368764	−0.0372508
$99$	−2.16777	−0.217870
$100$	−13.6742	−1.36742
$101$	3.03228	0.301723	0.150862	−	0.988555i	$-0.451795\pi$
$101$	3.03228	0.301723	0.150862	+	0.988555i	$0.451795\pi$
$102$	−7.01477	−0.694566
$103$	13.5816	1.33823	0.669115	−	0.743159i	$-0.266673\pi$
$103$	13.5816	1.33823	0.669115	+	0.743159i	$0.266673\pi$
$104$	26.9235	2.64007
$105$	−3.07482	−0.300071
$106$	0.125100	0.0121508
$107$	−10.7911	−1.04321	−0.521607	−	0.853186i	$-0.674667\pi$
$107$	−10.7911	−1.04321	−0.521607	+	0.853186i	$0.674667\pi$
$108$	−19.2913	−1.85630
$109$	−0.844508	−0.0808892	−0.0404446	−	0.999182i	$-0.512877\pi$
$109$	−0.844508	−0.0808892	−0.0404446	+	0.999182i	$0.512877\pi$
$110$	−3.17845	−0.303053
$111$	7.34923	0.697558
$112$	11.9366	1.12790
$113$	4.12100	0.387671	0.193835	−	0.981034i	$-0.437907\pi$
$113$	4.12100	0.387671	0.193835	+	0.981034i	$0.437907\pi$
$114$	−5.24764	−0.491487
$115$	−1.21414	−0.113219
$116$	−9.47336	−0.879580
$117$	−11.3032	−1.04498
$118$	−5.71376	−0.525994
$119$	−8.15412	−0.747487
$120$	−6.06597	−0.553744
$121$	1.00000	0.0909091
$122$	−20.4959	−1.85562
$123$	−0.458974	−0.0413843
$124$	5.56753	0.499979
$125$	10.7421	0.960800
$126$	−14.0042	−1.24759
$127$	18.1473	1.61032	0.805158	−	0.593060i	$-0.202080\pi$
$127$	18.1473	1.61032	0.805158	+	0.593060i	$0.202080\pi$
$128$	−18.7099	−1.65373
$129$	−1.67578	−0.147544
$130$	−16.5730	−1.45355
$131$	6.13039	0.535615	0.267808	−	0.963472i	$-0.413701\pi$
$131$	6.13039	0.535615	0.267808	+	0.963472i	$0.413701\pi$
$132$	3.73300	0.324916
$133$	−6.09997	−0.528934
$134$	−8.28693	−0.715882
$135$	6.07098	0.522506
$136$	−16.0864	−1.37939
$137$	−6.89473	−0.589057	−0.294528	−	0.955643i	$-0.595163\pi$
$137$	−6.89473	−0.589057	−0.294528	+	0.955643i	$0.595163\pi$
$138$	2.12293	0.180716
$139$	−15.0798	−1.27905	−0.639525	−	0.768770i	$-0.720869\pi$
$139$	−15.0798	−1.27905	−0.639525	+	0.768770i	$0.720869\pi$
$140$	−13.7923	−1.16566
$141$	4.65399	0.391937
$142$	−30.6835	−2.57490
$143$	5.21419	0.436033
$144$	−9.88624	−0.823854
$145$	2.98127	0.247581
$146$	12.9988	1.07579
$147$	−0.136298	−0.0112416
$148$	32.9654	2.70974
$149$	3.07301	0.251751	0.125875	−	0.992046i	$-0.459826\pi$
$149$	3.07301	0.251751	0.125875	+	0.992046i	$0.459826\pi$
$150$	−7.52429	−0.614355
$151$	−13.9528	−1.13546	−0.567732	−	0.823213i	$-0.692179\pi$
$151$	−13.9528	−1.13546	−0.567732	+	0.823213i	$0.692179\pi$
$152$	−12.0340	−0.976082
$153$	6.75347	0.545986
$154$	6.46018	0.520576
$155$	−1.75210	−0.140732
$156$	19.4646	1.55841
$157$	16.8428	1.34420	0.672102	−	0.740459i	$-0.265392\pi$
$157$	16.8428	1.34420	0.672102	+	0.740459i	$0.265392\pi$
$158$	−2.46820	−0.196359
$159$	0.0462377	0.00366689
$160$	−1.19675	−0.0946116
$161$	2.46774	0.194485
$162$	5.43639	0.427123
$163$	14.0376	1.09951	0.549753	−	0.835327i	$-0.314722\pi$
$163$	14.0376	1.09951	0.549753	+	0.835327i	$0.314722\pi$
$164$	−2.05875	−0.160762
$165$	−1.17478	−0.0914562
$166$	−28.1666	−2.18615
$167$	5.71366	0.442136	0.221068	−	0.975258i	$-0.429046\pi$
$167$	5.71366	0.442136	0.221068	+	0.975258i	$0.429046\pi$
$168$	12.3290	0.951207
$169$	14.1878	1.09137
$170$	9.90211	0.759457
$171$	5.05217	0.386349
$172$	−7.51681	−0.573151
$173$	12.5264	0.952365	0.476183	−	0.879346i	$-0.342020\pi$
$173$	12.5264	0.952365	0.476183	+	0.879346i	$0.342020\pi$
$174$	−5.21276	−0.395179
$175$	−8.74639	−0.661165
$176$	4.56055	0.343764
$177$	−2.11184	−0.158736
$178$	5.54776	0.415822
$179$	−18.7530	−1.40166	−0.700832	−	0.713326i	$-0.747188\pi$
$179$	−18.7530	−1.40166	−0.700832	+	0.713326i	$0.747188\pi$
$180$	11.4231	0.851431
$181$	25.0465	1.86169	0.930847	−	0.365410i	$-0.119071\pi$
$181$	25.0465	1.86169	0.930847	+	0.365410i	$0.119071\pi$
$182$	33.6846	2.49687
$183$	−7.57544	−0.559993
$184$	4.86833	0.358898
$185$	−10.3742	−0.762728
$186$	3.06356	0.224631
$187$	−3.11539	−0.227820
$188$	20.8758	1.52252
$189$	−12.3392	−0.897547
$190$	7.40761	0.537405
$191$	−6.72633	−0.486700	−0.243350	−	0.969939i	$-0.578246\pi$
$191$	−6.72633	−0.486700	−0.243350	+	0.969939i	$0.578246\pi$
$192$	−6.22832	−0.449490
$193$	2.02743	0.145938	0.0729688	−	0.997334i	$-0.476753\pi$
$193$	2.02743	0.145938	0.0729688	+	0.997334i	$0.476753\pi$
$194$	−1.35800	−0.0974986
$195$	−6.12550	−0.438657
$196$	−0.611371	−0.0436694
$197$	11.2700	0.802952	0.401476	−	0.915870i	$-0.368497\pi$
$197$	11.2700	0.802952	0.401476	+	0.915870i	$0.368497\pi$
$198$	−5.35050	−0.380244
$199$	21.8699	1.55032	0.775158	−	0.631767i	$-0.217670\pi$
$199$	21.8699	1.55032	0.775158	+	0.631767i	$0.217670\pi$
$200$	−17.2548	−1.22010
$201$	−3.06291	−0.216041
$202$	7.48428	0.526592
$203$	−6.05942	−0.425288
$204$	−11.6297	−0.814245
$205$	0.647891	0.0452507
$206$	33.5220	2.33559
$207$	−2.04385	−0.142058
$208$	23.7796	1.64882
$209$	−2.33058	−0.161209
$210$	−7.58926	−0.523709
$211$	9.97087	0.686423	0.343211	−	0.939258i	$-0.388485\pi$
$211$	9.97087	0.686423	0.343211	+	0.939258i	$0.388485\pi$
$212$	0.207402	0.0142444
$213$	−11.3408	−0.777061
$214$	−26.6346	−1.82070
$215$	2.36554	0.161329
$216$	−24.3427	−1.65631
$217$	3.56114	0.241746
$218$	−2.08442	−0.141174
$219$	4.80446	0.324655
$220$	−5.26953	−0.355271
$221$	−16.2443	−1.09271
$222$	18.1394	1.21743
$223$	24.4073	1.63443	0.817217	−	0.576330i	$-0.195516\pi$
$223$	24.4073	1.63443	0.817217	+	0.576330i	$0.195516\pi$
$224$	2.43239	0.162521
$225$	7.24400	0.482934
$226$	10.1714	0.676594
$227$	−5.16357	−0.342718	−0.171359	−	0.985209i	$-0.554816\pi$
$227$	−5.16357	−0.342718	−0.171359	+	0.985209i	$0.554816\pi$
$228$	−8.70003	−0.576174
$229$	−20.5566	−1.35842	−0.679211	−	0.733943i	$-0.737678\pi$
$229$	−20.5566	−1.35842	−0.679211	+	0.733943i	$0.737678\pi$
$230$	−2.99675	−0.197600
$231$	2.38773	0.157101
$232$	−11.9540	−0.784816
$233$	−13.2180	−0.865939	−0.432969	−	0.901409i	$-0.642534\pi$
$233$	−13.2180	−0.865939	−0.432969	+	0.901409i	$0.642534\pi$
$234$	−27.8985	−1.82378
$235$	−6.56961	−0.428554
$236$	−9.47281	−0.616627
$237$	−0.912264	−0.0592579
$238$	−20.1260	−1.30457
$239$	−14.2435	−0.921333	−0.460667	−	0.887573i	$-0.652390\pi$
$239$	−14.2435	−0.921333	−0.460667	+	0.887573i	$0.652390\pi$
$240$	−5.35762	−0.345833
$241$	0.361949	0.0233152	0.0116576	−	0.999932i	$-0.496289\pi$
$241$	0.361949	0.0233152	0.0116576	+	0.999932i	$0.496289\pi$
$242$	2.46820	0.158662
$243$	16.1524	1.03618
$244$	−33.9801	−2.17535
$245$	0.192399	0.0122919
$246$	−1.13284	−0.0722272
$247$	−12.1521	−0.773218
$248$	7.02539	0.446113
$249$	−10.4106	−0.659742
$250$	26.5136	1.67687
$251$	−10.6885	−0.674650	−0.337325	−	0.941388i	$-0.609522\pi$
$251$	−10.6885	−0.674650	−0.337325	+	0.941388i	$0.609522\pi$
$252$	−23.2175	−1.46256
$253$	0.942834	0.0592755
$254$	44.7913	2.81046
$255$	3.65989	0.229191
$256$	−32.5251	−2.03282
$257$	−1.50047	−0.0935966	−0.0467983	−	0.998904i	$-0.514902\pi$
$257$	−1.50047	−0.0935966	−0.0467983	+	0.998904i	$0.514902\pi$
$258$	−4.13616	−0.257506
$259$	21.0856	1.31019
$260$	−27.4763	−1.70401
$261$	5.01859	0.310643
$262$	15.1310	0.934799
$263$	27.8152	1.71516	0.857580	−	0.514351i	$-0.171967\pi$
$263$	27.8152	1.71516	0.857580	+	0.514351i	$0.171967\pi$
$264$	4.71048	0.289910
$265$	−0.0652695	−0.00400948
$266$	−15.0559	−0.923139
$267$	2.05049	0.125488
$268$	−13.7389	−0.839234
$269$	−15.1259	−0.922244	−0.461122	−	0.887337i	$-0.652553\pi$
$269$	−15.1259	−0.922244	−0.461122	+	0.887337i	$0.652553\pi$
$270$	14.9844	0.911921
$271$	10.5103	0.638456	0.319228	−	0.947678i	$-0.396576\pi$
$271$	10.5103	0.638456	0.319228	+	0.947678i	$0.396576\pi$
$272$	−14.2079	−0.861481
$273$	12.4501	0.753512
$274$	−17.0176	−1.02807
$275$	−3.34168	−0.201511
$276$	3.51960	0.211855
$277$	−7.24774	−0.435474	−0.217737	−	0.976007i	$-0.569868\pi$
$277$	−7.24774	−0.435474	−0.217737	+	0.976007i	$0.569868\pi$
$278$	−37.2199	−2.23230
$279$	−2.94944	−0.176578
$280$	−17.4038	−1.04007
$281$	28.7038	1.71232	0.856161	−	0.516709i	$-0.172843\pi$
$281$	28.7038	1.71232	0.856161	+	0.516709i	$0.172843\pi$
$282$	11.4870	0.684040
$283$	29.5010	1.75365	0.876825	−	0.480810i	$-0.159657\pi$
$283$	29.5010	1.75365	0.876825	+	0.480810i	$0.159657\pi$
$284$	−50.8700	−3.01858
$285$	2.73790	0.162179
$286$	12.8697	0.760999
$287$	−1.31684	−0.0777303
$288$	−2.01458	−0.118710
$289$	−7.29433	−0.429078
$290$	7.35838	0.432099
$291$	−0.501925	−0.0294234
$292$	21.5507	1.26116
$293$	17.5390	1.02464	0.512320	−	0.858795i	$-0.328786\pi$
$293$	17.5390	1.02464	0.512320	+	0.858795i	$0.328786\pi$
$294$	−0.336410	−0.0196198
$295$	2.98110	0.173566
$296$	41.5974	2.41780
$297$	−4.71437	−0.273556
$298$	7.58480	0.439376
$299$	4.91612	0.284306
$300$	−12.4745	−0.720214
$301$	−4.80796	−0.277126
$302$	−34.4384	−1.98171
$303$	2.76624	0.158916
$304$	−10.6287	−0.609598
$305$	10.6936	0.612311
$306$	16.6689	0.952899
$307$	16.8003	0.958846	0.479423	−	0.877584i	$-0.340846\pi$
$307$	16.8003	0.958846	0.479423	+	0.877584i	$0.340846\pi$
$308$	10.7103	0.610275
$309$	12.3900	0.704840
$310$	−4.32455	−0.245618
$311$	−9.20449	−0.521939	−0.260969	−	0.965347i	$-0.584042\pi$
$311$	−9.20449	−0.521939	−0.260969	+	0.965347i	$0.584042\pi$
$312$	24.5614	1.39051
$313$	−1.21853	−0.0688756	−0.0344378	−	0.999407i	$-0.510964\pi$
$313$	−1.21853	−0.0688756	−0.0344378	+	0.999407i	$0.510964\pi$
$314$	41.5715	2.34601
$315$	7.30656	0.411678
$316$	−4.09201	−0.230194
$317$	9.57240	0.537640	0.268820	−	0.963190i	$-0.413366\pi$
$317$	9.57240	0.537640	0.268820	+	0.963190i	$0.413366\pi$
$318$	0.114124	0.00639975
$319$	−2.31509	−0.129620
$320$	8.79195	0.491485
$321$	−9.84433	−0.549457
$322$	6.09088	0.339431
$323$	7.26066	0.403994
$324$	9.01296	0.500720
$325$	−17.4241	−0.966518
$326$	34.6475	1.91895
$327$	−0.770414	−0.0426040
$328$	−2.59784	−0.143442
$329$	13.3527	0.736159
$330$	−2.89958	−0.159617
$331$	−14.6522	−0.805358	−0.402679	−	0.915341i	$-0.631921\pi$
$331$	−14.6522	−0.805358	−0.402679	+	0.915341i	$0.631921\pi$
$332$	−46.6972	−2.56284
$333$	−17.4637	−0.957003
$334$	14.1025	0.771652
$335$	4.32362	0.236225
$336$	10.8893	0.594062
$337$	−23.0814	−1.25732	−0.628661	−	0.777680i	$-0.716397\pi$
$337$	−23.0814	−1.25732	−0.628661	+	0.777680i	$0.716397\pi$
$338$	35.0183	1.90474
$339$	3.75944	0.204184
$340$	16.4166	0.890318
$341$	1.36058	0.0736798
$342$	12.4698	0.674287
$343$	−18.7126	−1.01039
$344$	−9.48509	−0.511402
$345$	−1.10762	−0.0596322
$346$	30.9177	1.66214
$347$	4.10900	0.220583	0.110291	−	0.993899i	$-0.464822\pi$
$347$	4.10900	0.220583	0.110291	+	0.993899i	$0.464822\pi$
$348$	−8.64221	−0.463271
$349$	9.81669	0.525476	0.262738	−	0.964867i	$-0.415375\pi$
$349$	9.81669	0.525476	0.262738	+	0.964867i	$0.415375\pi$
$350$	−21.5878	−1.15392
$351$	−24.5816	−1.31207
$352$	0.929330	0.0495335
$353$	35.1950	1.87324	0.936620	−	0.350348i	$-0.113937\pi$
$353$	35.1950	1.87324	0.936620	+	0.350348i	$0.113937\pi$
$354$	−5.21246	−0.277039
$355$	16.0088	0.849659
$356$	9.19759	0.487471
$357$	−7.43871	−0.393698
$358$	−46.2862	−2.44630
$359$	17.8027	0.939588	0.469794	−	0.882776i	$-0.344328\pi$
$359$	17.8027	0.939588	0.469794	+	0.882776i	$0.344328\pi$
$360$	14.4143	0.759701
$361$	−13.5684	−0.714127
$362$	61.8198	3.24918
$363$	0.912264	0.0478814
$364$	55.8455	2.92710
$365$	−6.78201	−0.354987
$366$	−18.6977	−0.977345
$367$	−6.29659	−0.328679	−0.164340	−	0.986404i	$-0.552549\pi$
$367$	−6.29659	−0.328679	−0.164340	+	0.986404i	$0.552549\pi$
$368$	4.29984	0.224145
$369$	1.09064	0.0567765
$370$	−25.6057	−1.33118
$371$	0.132660	0.00688736
$372$	5.07906	0.263337
$373$	11.7946	0.610699	0.305349	−	0.952240i	$-0.401227\pi$
$373$	11.7946	0.610699	0.305349	+	0.952240i	$0.401227\pi$
$374$	−7.68941	−0.397610
$375$	9.79960	0.506049
$376$	26.3421	1.35849
$377$	−12.0713	−0.621704
$378$	−30.4557	−1.56647
$379$	−1.18627	−0.0609347	−0.0304673	−	0.999536i	$-0.509700\pi$
$379$	−1.18627	−0.0609347	−0.0304673	+	0.999536i	$0.509700\pi$
$380$	12.2810	0.630004
$381$	16.5552	0.848147
$382$	−16.6019	−0.849428
$383$	−10.6465	−0.544011	−0.272006	−	0.962296i	$-0.587687\pi$
$383$	−10.6465	−0.544011	−0.272006	+	0.962296i	$0.587687\pi$
$384$	−17.0683	−0.871015
$385$	−3.37053	−0.171778
$386$	5.00411	0.254702
$387$	3.98209	0.202421
$388$	−2.25142	−0.114298
$389$	−28.6247	−1.45133	−0.725665	−	0.688048i	$-0.758468\pi$
$389$	−28.6247	−1.45133	−0.725665	+	0.688048i	$0.758468\pi$
$390$	−15.1190	−0.765579
$391$	−2.93730	−0.148546
$392$	−0.771459	−0.0389646
$393$	5.59254	0.282106
$394$	27.8165	1.40138
$395$	1.28776	0.0647942
$396$	−8.87056	−0.445763
$397$	−37.3820	−1.87615	−0.938075	−	0.346431i	$-0.887393\pi$
$397$	−37.3820	−1.87615	−0.938075	+	0.346431i	$0.887393\pi$
$398$	53.9793	2.70574
$399$	−5.56478	−0.278587
$400$	−15.2399	−0.761994
$401$	−6.75597	−0.337377	−0.168688	−	0.985669i	$-0.553953\pi$
$401$	−6.75597	−0.337377	−0.168688	+	0.985669i	$0.553953\pi$
$402$	−7.55987	−0.377052
$403$	7.09435	0.353395
$404$	12.4081	0.617328
$405$	−2.83638	−0.140941
$406$	−14.9559	−0.742247
$407$	8.05603	0.399323
$408$	−14.6750	−0.726521
$409$	−38.0630	−1.88209	−0.941046	−	0.338278i	$-0.890156\pi$
$409$	−38.0630	−1.88209	−0.941046	+	0.338278i	$0.890156\pi$
$410$	1.59912	0.0789751
$411$	−6.28981	−0.310254
$412$	55.5759	2.73803
$413$	−6.05907	−0.298147
$414$	−5.04464	−0.247930
$415$	14.6956	0.721380
$416$	4.84570	0.237580
$417$	−13.7567	−0.673670
$418$	−5.75233	−0.281356
$419$	8.72390	0.426190	0.213095	−	0.977031i	$-0.431646\pi$
$419$	8.72390	0.426190	0.213095	+	0.977031i	$0.431646\pi$
$420$	−12.5822	−0.613948
$421$	−26.1326	−1.27362	−0.636812	−	0.771019i	$-0.719747\pi$
$421$	−26.1326	−1.27362	−0.636812	+	0.771019i	$0.719747\pi$
$422$	24.6101	1.19800
$423$	−11.0591	−0.537712
$424$	0.261710	0.0127098
$425$	10.4106	0.504990
$426$	−27.9914	−1.35619
$427$	−21.7346	−1.05181
$428$	−44.1573	−2.13442
$429$	4.75672	0.229656
$430$	5.83864	0.281564
$431$	−26.7028	−1.28623	−0.643113	−	0.765771i	$-0.722358\pi$
$431$	−26.7028	−1.28623	−0.643113	+	0.765771i	$0.722358\pi$
$432$	−21.5001	−1.03443
$433$	33.0847	1.58995	0.794973	−	0.606644i	$-0.207485\pi$
$433$	33.0847	1.58995	0.794973	+	0.606644i	$0.207485\pi$
$434$	8.78962	0.421915
$435$	2.71971	0.130400
$436$	−3.45574	−0.165500
$437$	−2.19735	−0.105113
$438$	11.8584	0.566614
$439$	22.0205	1.05098	0.525490	−	0.850800i	$-0.323882\pi$
$439$	22.0205	1.05098	0.525490	+	0.850800i	$0.323882\pi$
$440$	−6.64935	−0.316996
$441$	0.323878	0.0154228
$442$	−40.0941	−1.90708
$443$	11.3741	0.540398	0.270199	−	0.962804i	$-0.412910\pi$
$443$	11.3741	0.540398	0.270199	+	0.962804i	$0.412910\pi$
$444$	30.0731	1.42721
$445$	−2.89449	−0.137212
$446$	60.2421	2.85255
$447$	2.80339	0.132596
$448$	−17.8696	−0.844259
$449$	14.9818	0.707034	0.353517	−	0.935428i	$-0.384986\pi$
$449$	14.9818	0.707034	0.353517	+	0.935428i	$0.384986\pi$
$450$	17.8797	0.842855
$451$	−0.503115	−0.0236908
$452$	16.8632	0.793177
$453$	−12.7287	−0.598044
$454$	−12.7447	−0.598139
$455$	−17.5746	−0.823910
$456$	−10.9781	−0.514098
$457$	−25.9069	−1.21187	−0.605937	−	0.795512i	$-0.707202\pi$
$457$	−25.9069	−1.21187	−0.605937	+	0.795512i	$0.707202\pi$
$458$	−50.7379	−2.37083
$459$	14.6871	0.685536
$460$	−4.96829	−0.231648
$461$	−13.9392	−0.649215	−0.324608	−	0.945849i	$-0.605232\pi$
$461$	−13.9392	−0.649215	−0.324608	+	0.945849i	$0.605232\pi$
$462$	5.89339	0.274185
$463$	1.97027	0.0915663	0.0457832	−	0.998951i	$-0.485422\pi$
$463$	1.97027	0.0915663	0.0457832	+	0.998951i	$0.485422\pi$
$464$	−10.5581	−0.490146
$465$	−1.59838	−0.0741232
$466$	−32.6246	−1.51131
$467$	18.9187	0.875455	0.437727	−	0.899108i	$-0.355783\pi$
$467$	18.9187	0.875455	0.437727	+	0.899108i	$0.355783\pi$
$468$	−46.2528	−2.13804
$469$	−8.78775	−0.405780
$470$	−16.2151	−0.747948
$471$	15.3651	0.707986
$472$	−11.9533	−0.550194
$473$	−1.83695	−0.0844629
$474$	−2.25165	−0.103422
$475$	7.78803	0.357340
$476$	−33.3668	−1.52936
$477$	−0.109873	−0.00503073
$478$	−35.1557	−1.60799
$479$	−13.3848	−0.611567	−0.305783	−	0.952101i	$-0.598918\pi$
$479$	−13.3848	−0.611567	−0.305783	+	0.952101i	$0.598918\pi$
$480$	−1.09175	−0.0498315
$481$	42.0057	1.91529
$482$	0.893362	0.0406915
$483$	2.25123	0.102435
$484$	4.09201	0.186001
$485$	0.708522	0.0321723
$486$	39.8675	1.80843
$487$	27.4130	1.24220	0.621101	−	0.783731i	$-0.286686\pi$
$487$	27.4130	1.24220	0.621101	+	0.783731i	$0.286686\pi$
$488$	−42.8778	−1.94099
$489$	12.8059	0.579105
$490$	0.474879	0.0214528
$491$	38.0469	1.71703	0.858516	−	0.512786i	$-0.171387\pi$
$491$	38.0469	1.71703	0.858516	+	0.512786i	$0.171387\pi$
$492$	−1.87813	−0.0846725
$493$	7.21240	0.324830
$494$	−29.9937	−1.34948
$495$	2.79157	0.125472
$496$	6.20501	0.278613
$497$	−32.5378	−1.45952
$498$	−25.6953	−1.15144
$499$	−16.0780	−0.719748	−0.359874	−	0.933001i	$-0.617180\pi$
$499$	−16.0780	−0.719748	−0.359874	+	0.933001i	$0.617180\pi$
$500$	43.9567	1.96580
$501$	5.21237	0.232871
$502$	−26.3813	−1.17745
$503$	17.3435	0.773307	0.386653	−	0.922225i	$-0.373631\pi$
$503$	17.3435	0.773307	0.386653	+	0.922225i	$0.373631\pi$
$504$	−29.2970	−1.30499
$505$	−3.90485	−0.173763
$506$	2.32710	0.103452
$507$	12.9430	0.574819
$508$	74.2592	3.29472
$509$	−13.6993	−0.607210	−0.303605	−	0.952798i	$-0.598190\pi$
$509$	−13.6993	−0.607210	−0.303605	+	0.952798i	$0.598190\pi$
$510$	9.03334	0.400003
$511$	13.7844	0.609786
$512$	−42.8586	−1.89410
$513$	10.9872	0.485097
$514$	−3.70346	−0.163352
$515$	−17.4898	−0.770691
$516$	−6.85731	−0.301876
$517$	5.10159	0.224368
$518$	52.0434	2.28666
$519$	11.4274	0.501607
$520$	−34.6710	−1.52042
$521$	24.1242	1.05690	0.528450	−	0.848965i	$-0.322774\pi$
$521$	24.1242	1.05690	0.528450	+	0.848965i	$0.322774\pi$
$522$	12.3869	0.542159
$523$	−1.45233	−0.0635058	−0.0317529	−	0.999496i	$-0.510109\pi$
$523$	−1.45233	−0.0635058	−0.0317529	+	0.999496i	$0.510109\pi$
$524$	25.0857	1.09587
$525$	−7.97901	−0.348233
$526$	68.6535	2.99344
$527$	−4.23875	−0.184643
$528$	4.16042	0.181059
$529$	−22.1111	−0.961351
$530$	−0.161098	−0.00699766
$531$	5.01829	0.217775
$532$	−24.9611	−1.08220
$533$	−2.62334	−0.113629
$534$	5.06102	0.219012
$535$	13.8963	0.600791
$536$	−17.3364	−0.748818
$537$	−17.1077	−0.738251
$538$	−37.3338	−1.60957
$539$	−0.149406	−0.00643537
$540$	24.8425	1.06905
$541$	−10.2955	−0.442636	−0.221318	−	0.975202i	$-0.571036\pi$
$541$	−10.2955	−0.442636	−0.221318	+	0.975202i	$0.571036\pi$
$542$	25.9416	1.11429
$543$	22.8490	0.980546
$544$	−2.89523	−0.124132
$545$	1.08752	0.0465844
$546$	30.7292	1.31509
$547$	39.7842	1.70105	0.850525	−	0.525935i	$-0.176284\pi$
$547$	39.7842	1.70105	0.850525	+	0.525935i	$0.176284\pi$
$548$	−28.2133	−1.20521
$549$	18.0012	0.768273
$550$	−8.24793	−0.351693
$551$	5.39548	0.229855
$552$	4.44120	0.189030
$553$	−2.61736	−0.111302
$554$	−17.8889	−0.760025
$555$	−9.46403	−0.401726
$556$	−61.7067	−2.61695
$557$	−8.93344	−0.378522	−0.189261	−	0.981927i	$-0.560609\pi$
$557$	−8.93344	−0.378522	−0.189261	+	0.981927i	$0.560609\pi$
$558$	−7.27981	−0.308179
$559$	−9.57819	−0.405114
$560$	−15.3715	−0.649564
$561$	−2.84206	−0.119992
$562$	70.8466	2.98848
$563$	21.7880	0.918254	0.459127	−	0.888371i	$-0.348162\pi$
$563$	21.7880	0.918254	0.459127	+	0.888371i	$0.348162\pi$
$564$	19.0442	0.801906
$565$	−5.30685	−0.223261
$566$	72.8143	3.06061
$567$	5.76493	0.242104
$568$	−64.1903	−2.69337
$569$	8.26754	0.346593	0.173297	−	0.984870i	$-0.444558\pi$
$569$	8.26754	0.346593	0.173297	+	0.984870i	$0.444558\pi$
$570$	6.75770	0.283049
$571$	35.2320	1.47441	0.737206	−	0.675668i	$-0.236145\pi$
$571$	35.2320	1.47441	0.737206	+	0.675668i	$0.236145\pi$
$572$	21.3365	0.892126
$573$	−6.13619	−0.256343
$574$	−3.25021	−0.135661
$575$	−3.15065	−0.131391
$576$	14.8001	0.616671
$577$	−28.0349	−1.16711	−0.583553	−	0.812075i	$-0.698338\pi$
$577$	−28.0349	−1.16711	−0.583553	+	0.812075i	$0.698338\pi$
$578$	−18.0039	−0.748862
$579$	1.84955	0.0768647
$580$	12.1994	0.506553
$581$	−29.8688	−1.23917
$582$	−1.23885	−0.0513521
$583$	0.0506846	0.00209914
$584$	27.1937	1.12528
$585$	14.5558	0.601808
$586$	43.2898	1.78829
$587$	23.0598	0.951779	0.475890	−	0.879505i	$-0.342126\pi$
$587$	23.0598	0.951779	0.475890	+	0.879505i	$0.342126\pi$
$588$	−0.557732	−0.0230005
$589$	−3.17095	−0.130657
$590$	7.35794	0.302922
$591$	10.2812	0.422911
$592$	36.7399	1.51000
$593$	−20.2690	−0.832346	−0.416173	−	0.909285i	$-0.636629\pi$
$593$	−20.2690	−0.832346	−0.416173	+	0.909285i	$0.636629\pi$
$594$	−11.6360	−0.477432
$595$	10.5005	0.430480
$596$	12.5748	0.515084
$597$	19.9511	0.816545
$598$	12.1340	0.496195
$599$	−39.7935	−1.62592	−0.812960	−	0.582320i	$-0.802145\pi$
$599$	−39.7935	−1.62592	−0.812960	+	0.582320i	$0.802145\pi$
$600$	−15.7409	−0.642620
$601$	21.2248	0.865776	0.432888	−	0.901448i	$-0.357495\pi$
$601$	21.2248	0.865776	0.432888	+	0.901448i	$0.357495\pi$
$602$	−11.8670	−0.483663
$603$	7.27826	0.296394
$604$	−57.0951	−2.32317
$605$	−1.28776	−0.0523548
$606$	6.82764	0.277354
$607$	−41.4108	−1.68081	−0.840407	−	0.541956i	$-0.817684\pi$
$607$	−41.4108	−1.68081	−0.840407	+	0.541956i	$0.817684\pi$
$608$	−2.16587	−0.0878378
$609$	−5.52779	−0.223997
$610$	26.3938	1.06865
$611$	26.6007	1.07615
$612$	27.6353	1.11709
$613$	0.552181	0.0223024	0.0111512	−	0.999938i	$-0.496450\pi$
$613$	0.552181	0.0223024	0.0111512	+	0.999938i	$0.496450\pi$
$614$	41.4666	1.67346
$615$	0.591047	0.0238333
$616$	13.5148	0.544526
$617$	−46.4632	−1.87054	−0.935268	−	0.353940i	$-0.884842\pi$
$617$	−46.4632	−1.87054	−0.935268	+	0.353940i	$0.884842\pi$
$618$	30.5809	1.23014
$619$	−3.37540	−0.135669	−0.0678345	−	0.997697i	$-0.521609\pi$
$619$	−3.37540	−0.135669	−0.0678345	+	0.997697i	$0.521609\pi$
$620$	−7.16964	−0.287939
$621$	−4.44487	−0.178367
$622$	−22.7185	−0.910930
$623$	5.88303	0.235699
$624$	21.6932	0.868425
$625$	2.87519	0.115008
$626$	−3.00759	−0.120207
$627$	−2.12610	−0.0849083
$628$	68.9210	2.75025
$629$	−25.0977	−1.00071
$630$	18.0341	0.718494
$631$	0.706167	0.0281121	0.0140560	−	0.999901i	$-0.495526\pi$
$631$	0.706167	0.0281121	0.0140560	+	0.999901i	$0.495526\pi$
$632$	−5.16351	−0.205393
$633$	9.09606	0.361536
$634$	23.6266	0.938333
$635$	−23.3694	−0.927387
$636$	0.189205	0.00750248
$637$	−0.779031	−0.0308663
$638$	−5.71410	−0.226223
$639$	26.9488	1.06608
$640$	24.0938	0.952391
$641$	−24.4745	−0.966684	−0.483342	−	0.875432i	$-0.660577\pi$
$641$	−24.4745	−0.966684	−0.483342	+	0.875432i	$0.660577\pi$
$642$	−24.2978	−0.958957
$643$	−36.2341	−1.42893	−0.714466	−	0.699670i	$-0.753330\pi$
$643$	−36.2341	−1.42893	−0.714466	+	0.699670i	$0.753330\pi$
$644$	10.0980	0.397918
$645$	2.15800	0.0849712
$646$	17.9208	0.705083
$647$	−37.5534	−1.47638	−0.738188	−	0.674595i	$-0.764318\pi$
$647$	−37.5534	−1.47638	−0.738188	+	0.674595i	$0.764318\pi$
$648$	11.3730	0.446774
$649$	−2.31495	−0.0908697
$650$	−43.0063	−1.68684
$651$	3.24870	0.127327
$652$	57.4419	2.24960
$653$	−14.4128	−0.564018	−0.282009	−	0.959412i	$-0.591001\pi$
$653$	−14.4128	−0.564018	−0.282009	+	0.959412i	$0.591001\pi$
$654$	−1.90154	−0.0743560
$655$	−7.89447	−0.308462
$656$	−2.29448	−0.0895844
$657$	−11.4166	−0.445405
$658$	32.9572	1.28480
$659$	22.0689	0.859684	0.429842	−	0.902904i	$-0.358569\pi$
$659$	22.0689	0.859684	0.429842	+	0.902904i	$0.358569\pi$
$660$	−4.80720	−0.187120
$661$	47.2652	1.83840	0.919202	−	0.393786i	$-0.128835\pi$
$661$	47.2652	1.83840	0.919202	+	0.393786i	$0.128835\pi$
$662$	−36.1646	−1.40558
$663$	−14.8190	−0.575524
$664$	−58.9248	−2.28673
$665$	7.85529	0.304615
$666$	−43.1038	−1.67024
$667$	−2.18274	−0.0845161
$668$	23.3804	0.904614
$669$	22.2659	0.860850
$670$	10.6716	0.412279
$671$	−8.30400	−0.320572
$672$	2.21899	0.0855992
$673$	−40.4856	−1.56061	−0.780303	−	0.625402i	$-0.784935\pi$
$673$	−40.4856	−1.56061	−0.780303	+	0.625402i	$0.784935\pi$
$674$	−56.9694	−2.19438
$675$	15.7539	0.606368
$676$	58.0566	2.23295
$677$	8.38062	0.322093	0.161047	−	0.986947i	$-0.448513\pi$
$677$	8.38062	0.322093	0.161047	+	0.986947i	$0.448513\pi$
$678$	9.27904	0.356359
$679$	−1.44007	−0.0552647
$680$	20.7154	0.794398
$681$	−4.71053	−0.180508
$682$	3.35819	0.128592
$683$	−6.73352	−0.257651	−0.128825	−	0.991667i	$-0.541121\pi$
$683$	−6.73352	−0.257651	−0.128825	+	0.991667i	$0.541121\pi$
$684$	20.6735	0.790472
$685$	8.87875	0.339240
$686$	−46.1864	−1.76341
$687$	−18.7531	−0.715475
$688$	−8.37748	−0.319388
$689$	0.264279	0.0100682
$690$	−2.73383	−0.104075
$691$	39.2806	1.49430	0.747152	−	0.664654i	$-0.231421\pi$
$691$	39.2806	1.49430	0.747152	+	0.664654i	$0.231421\pi$
$692$	51.2582	1.94855
$693$	−5.67386	−0.215532
$694$	10.1418	0.384979
$695$	19.4191	0.736609
$696$	−10.9052	−0.413359
$697$	1.56740	0.0593696
$698$	24.2296	0.917103
$699$	−12.0583	−0.456086
$700$	−35.7903	−1.35275
$701$	−34.6027	−1.30692	−0.653462	−	0.756959i	$-0.726684\pi$
$701$	−34.6027	−1.30692	−0.653462	+	0.756959i	$0.726684\pi$
$702$	−60.6724	−2.28993
$703$	−18.7752	−0.708120
$704$	−6.82732	−0.257314
$705$	−5.99322	−0.225718
$706$	86.8683	3.26933
$707$	7.93659	0.298486
$708$	−8.64170	−0.324775
$709$	28.5446	1.07202	0.536008	−	0.844213i	$-0.319932\pi$
$709$	28.5446	1.07202	0.536008	+	0.844213i	$0.319932\pi$
$710$	39.5130	1.48289
$711$	2.16777	0.0812979
$712$	11.6060	0.434953
$713$	1.28281	0.0480414
$714$	−18.3602	−0.687114
$715$	−6.71462	−0.251113
$716$	−76.7375	−2.86782
$717$	−12.9938	−0.485262
$718$	43.9405	1.63985
$719$	−32.6441	−1.21742	−0.608710	−	0.793392i	$-0.708313\pi$
$719$	−32.6441	−1.21742	−0.608710	+	0.793392i	$0.708313\pi$
$720$	12.7311	0.474460
$721$	35.5479	1.32387
$722$	−33.4896	−1.24635
$723$	0.330193	0.0122800
$724$	102.491	3.80904
$725$	7.73627	0.287318
$726$	2.25165	0.0835666
$727$	34.9400	1.29585	0.647926	−	0.761704i	$-0.275637\pi$
$727$	34.9400	1.29585	0.647926	+	0.761704i	$0.275637\pi$
$728$	70.4687	2.61174
$729$	8.12758	0.301021
$730$	−16.7394	−0.619551
$731$	5.72281	0.211666
$732$	−30.9988	−1.14575
$733$	20.7139	0.765084	0.382542	−	0.923938i	$-0.375049\pi$
$733$	20.7139	0.765084	0.382542	+	0.923938i	$0.375049\pi$
$734$	−15.5412	−0.573638
$735$	0.175518	0.00647410
$736$	0.876204	0.0322973
$737$	−3.35748	−0.123674
$738$	2.69192	0.0990909
$739$	−20.1460	−0.741082	−0.370541	−	0.928816i	$-0.620828\pi$
$739$	−20.1460	−0.741082	−0.370541	+	0.928816i	$0.620828\pi$
$740$	−42.4515	−1.56055
$741$	−11.0859	−0.407250
$742$	0.327432	0.0120204
$743$	−20.0265	−0.734701	−0.367351	−	0.930082i	$-0.619735\pi$
$743$	−20.0265	−0.734701	−0.367351	+	0.930082i	$0.619735\pi$
$744$	6.40901	0.234966
$745$	−3.95729	−0.144984
$746$	29.1113	1.06584
$747$	24.7382	0.905122
$748$	−12.7482	−0.466121
$749$	−28.2442	−1.03202
$750$	24.1874	0.883198
$751$	−26.8635	−0.980265	−0.490132	−	0.871648i	$-0.663052\pi$
$751$	−26.8635	−0.980265	−0.490132	+	0.871648i	$0.663052\pi$
$752$	23.2660	0.848425
$753$	−9.75069	−0.355335
$754$	−29.7944	−1.08505
$755$	17.9679	0.653918
$756$	−50.4923	−1.83639
$757$	35.3421	1.28453	0.642265	−	0.766483i	$-0.277995\pi$
$757$	35.3421	1.28453	0.642265	+	0.766483i	$0.277995\pi$
$758$	−2.92796	−0.106348
$759$	0.860114	0.0312201
$760$	15.4968	0.562129
$761$	−25.4928	−0.924114	−0.462057	−	0.886850i	$-0.652888\pi$
$761$	−25.4928	−0.924114	−0.462057	+	0.886850i	$0.652888\pi$
$762$	40.8615	1.48025
$763$	−2.21039	−0.0800213
$764$	−27.5242	−0.995792
$765$	−8.69684	−0.314435
$766$	−26.2777	−0.949453
$767$	−12.0706	−0.435844
$768$	−29.6714	−1.07068
$769$	13.8736	0.500294	0.250147	−	0.968208i	$-0.419521\pi$
$769$	13.8736	0.500294	0.250147	+	0.968208i	$0.419521\pi$
$770$	−8.31915	−0.299801
$771$	−1.36882	−0.0492969
$772$	8.29627	0.298589
$773$	37.0888	1.33399	0.666995	−	0.745062i	$-0.267580\pi$
$773$	37.0888	1.33399	0.666995	+	0.745062i	$0.267580\pi$
$774$	9.82859	0.353281
$775$	−4.54663	−0.163320
$776$	−2.84095	−0.101984
$777$	19.2356	0.690073
$778$	−70.6515	−2.53298
$779$	1.17255	0.0420109
$780$	−25.0657	−0.897494
$781$	−12.4315	−0.444835
$782$	−7.24984	−0.259254
$783$	10.9142	0.390041
$784$	−0.681373	−0.0243348
$785$	−21.6895	−0.774131
$786$	13.8035	0.492355
$787$	0.617525	0.0220124	0.0110062	−	0.999939i	$-0.496497\pi$
$787$	0.617525	0.0220124	0.0110062	+	0.999939i	$0.496497\pi$
$788$	46.1169	1.64285
$789$	25.3748	0.903367
$790$	3.17845	0.113084
$791$	10.7861	0.383511
$792$	−11.1933	−0.397738
$793$	−43.2986	−1.53758
$794$	−92.2664	−3.27441
$795$	−0.0595430	−0.00211177
$796$	89.4920	3.17196
$797$	31.5120	1.11621	0.558106	−	0.829770i	$-0.311528\pi$
$797$	31.5120	1.11621	0.558106	+	0.829770i	$0.311528\pi$
$798$	−13.7350	−0.486213
$799$	−15.8934	−0.562270
$800$	−3.10552	−0.109797
$801$	−4.87249	−0.172161
$802$	−16.6751	−0.588818
$803$	5.26652	0.185851
$804$	−12.5335	−0.442021
$805$	−3.17785	−0.112005
$806$	17.5103	0.616773
$807$	−13.7988	−0.485742
$808$	15.6572	0.550819
$809$	−37.8308	−1.33006	−0.665030	−	0.746817i	$-0.731581\pi$
$809$	−37.8308	−1.33006	−0.665030	+	0.746817i	$0.731581\pi$
$810$	−7.00076	−0.245982
$811$	−18.1751	−0.638213	−0.319107	−	0.947719i	$-0.603383\pi$
$811$	−18.1751	−0.638213	−0.319107	+	0.947719i	$0.603383\pi$
$812$	−24.7952	−0.870142
$813$	9.58818	0.336272
$814$	19.8839	0.696930
$815$	−18.0770	−0.633209
$816$	−12.9614	−0.453738
$817$	4.28114	0.149778
$818$	−93.9470	−3.28478
$819$	−29.5846	−1.03377
$820$	2.65118	0.0925832
$821$	5.48810	0.191536	0.0957680	−	0.995404i	$-0.469469\pi$
$821$	5.48810	0.191536	0.0957680	+	0.995404i	$0.469469\pi$
$822$	−15.5245	−0.541480
$823$	23.6476	0.824305	0.412152	−	0.911115i	$-0.364777\pi$
$823$	23.6476	0.824305	0.412152	+	0.911115i	$0.364777\pi$
$824$	70.1285	2.44304
$825$	−3.04849	−0.106135
$826$	−14.9550	−0.520351
$827$	−4.85783	−0.168923	−0.0844616	−	0.996427i	$-0.526917\pi$
$827$	−4.85783	−0.168923	−0.0844616	+	0.996427i	$0.526917\pi$
$828$	−8.36347	−0.290651
$829$	−51.9083	−1.80285	−0.901425	−	0.432935i	$-0.857478\pi$
$829$	−51.9083	−1.80285	−0.901425	+	0.432935i	$0.857478\pi$
$830$	36.2717	1.25901
$831$	−6.61185	−0.229363
$832$	−35.5990	−1.23417
$833$	0.465458	0.0161272
$834$	−33.9544	−1.17574
$835$	−7.35782	−0.254628
$836$	−9.53675	−0.329835
$837$	−6.41430	−0.221711
$838$	21.5323	0.743822
$839$	−39.7272	−1.37154	−0.685768	−	0.727821i	$-0.740533\pi$
$839$	−39.7272	−1.37154	−0.685768	+	0.727821i	$0.740533\pi$
$840$	−15.8768	−0.547803
$841$	−23.6404	−0.815185
$842$	−64.5005	−2.22283
$843$	26.1854	0.901873
$844$	40.8009	1.40443
$845$	−18.2704	−0.628523
$846$	−27.2961	−0.938458
$847$	2.61736	0.0899337
$848$	0.231150	0.00793771
$849$	26.9127	0.923640
$850$	25.6955	0.881350
$851$	7.59550	0.260371
$852$	−46.4068	−1.58987
$853$	16.0671	0.550127	0.275064	−	0.961426i	$-0.411301\pi$
$853$	16.0671	0.550127	0.275064	+	0.961426i	$0.411301\pi$
$854$	−53.6453	−1.83571
$855$	−6.50597	−0.222499
$856$	−55.7199	−1.90447
$857$	−25.5581	−0.873047	−0.436523	−	0.899693i	$-0.643790\pi$
$857$	−25.5581	−0.873047	−0.436523	+	0.899693i	$0.643790\pi$
$858$	11.7405	0.400815
$859$	22.5070	0.767929	0.383964	−	0.923348i	$-0.374559\pi$
$859$	22.5070	0.767929	0.383964	+	0.923348i	$0.374559\pi$
$860$	9.67984	0.330080
$861$	−1.20130	−0.0409402
$862$	−65.9077	−2.24483
$863$	−42.7846	−1.45641	−0.728203	−	0.685361i	$-0.759644\pi$
$863$	−42.7846	−1.45641	−0.728203	+	0.685361i	$0.759644\pi$
$864$	−4.38121	−0.149052
$865$	−16.1310	−0.548470
$866$	81.6596	2.77490
$867$	−6.65435	−0.225994
$868$	14.5723	0.494614
$869$	−1.00000	−0.0339227
$870$	6.71278	0.227585
$871$	−17.5065	−0.593187
$872$	−4.36063	−0.147669
$873$	1.19270	0.0403669
$874$	−5.42349	−0.183452
$875$	28.1159	0.950491
$876$	19.6599	0.664246
$877$	−44.0367	−1.48701	−0.743507	−	0.668728i	$-0.766839\pi$
$877$	−44.0367	−1.48701	−0.743507	+	0.668728i	$0.766839\pi$
$878$	54.3509	1.83425
$879$	16.0002	0.539674
$880$	−5.87289	−0.197975
$881$	−10.6596	−0.359132	−0.179566	−	0.983746i	$-0.557469\pi$
$881$	−10.6596	−0.359132	−0.179566	+	0.983746i	$0.557469\pi$
$882$	0.799397	0.0269171
$883$	7.96620	0.268084	0.134042	−	0.990976i	$-0.457204\pi$
$883$	7.96620	0.268084	0.134042	+	0.990976i	$0.457204\pi$
$884$	−66.4717	−2.23569
$885$	2.71955	0.0914166
$886$	28.0735	0.943147
$887$	12.5506	0.421408	0.210704	−	0.977550i	$-0.432424\pi$
$887$	12.5506	0.421408	0.210704	+	0.977550i	$0.432424\pi$
$888$	37.9478	1.27344
$889$	47.4982	1.59304
$890$	−7.14417	−0.239473
$891$	2.20257	0.0737890
$892$	99.8751	3.34406
$893$	−11.8896	−0.397872
$894$	6.91934	0.231417
$895$	24.1493	0.807223
$896$	−48.9705	−1.63599
$897$	4.48480	0.149743
$898$	36.9780	1.23397
$899$	−3.14987	−0.105054
$900$	29.6426	0.988085
$901$	−0.157902	−0.00526049
$902$	−1.24179	−0.0413470
$903$	−4.38612	−0.145961
$904$	21.2788	0.707722
$905$	−32.2539	−1.07216
$906$	−31.4169	−1.04376
$907$	−15.7829	−0.524063	−0.262032	−	0.965059i	$-0.584392\pi$
$907$	−15.7829	−0.524063	−0.262032	+	0.965059i	$0.584392\pi$
$908$	−21.1294	−0.701203
$909$	−6.57330	−0.218023
$910$	−43.3777	−1.43796
$911$	2.65756	0.0880488	0.0440244	−	0.999030i	$-0.485982\pi$
$911$	2.65756	0.0880488	0.0440244	+	0.999030i	$0.485982\pi$
$912$	−9.69618	−0.321073
$913$	−11.4118	−0.377675
$914$	−63.9435	−2.11506
$915$	9.75534	0.322502
$916$	−84.1181	−2.77934
$917$	16.0455	0.529868
$918$	36.2508	1.19645
$919$	−26.4858	−0.873684	−0.436842	−	0.899538i	$-0.643903\pi$
$919$	−26.4858	−0.873684	−0.436842	+	0.899538i	$0.643903\pi$
$920$	−6.26924	−0.206691
$921$	15.3263	0.505020
$922$	−34.4048	−1.13306
$923$	−64.8204	−2.13359
$924$	9.77061	0.321429
$925$	−26.9207	−0.885146
$926$	4.86303	0.159809
$927$	−29.4417	−0.966994
$928$	−2.15148	−0.0706258
$929$	−39.5785	−1.29853	−0.649265	−	0.760562i	$-0.724923\pi$
$929$	−39.5785	−1.29853	−0.649265	+	0.760562i	$0.724923\pi$
$930$	−3.94513	−0.129366
$931$	0.348202	0.0114119
$932$	−54.0881	−1.77172
$933$	−8.39692	−0.274903
$934$	46.6952	1.52791
$935$	4.01187	0.131202
$936$	−58.3641	−1.90769
$937$	40.2619	1.31530	0.657650	−	0.753324i	$-0.271551\pi$
$937$	40.2619	1.31530	0.657650	+	0.753324i	$0.271551\pi$
$938$	−21.6899	−0.708201
$939$	−1.11162	−0.0362765
$940$	−26.8830	−0.876825
$941$	−15.2303	−0.496494	−0.248247	−	0.968697i	$-0.579854\pi$
$941$	−15.2303	−0.496494	−0.248247	+	0.968697i	$0.579854\pi$
$942$	37.9241	1.23563
$943$	−0.474354	−0.0154471
$944$	−10.5574	−0.343615
$945$	15.8900	0.516900
$946$	−4.53395	−0.147411
$947$	−34.7732	−1.12998	−0.564988	−	0.825099i	$-0.691119\pi$
$947$	−34.7732	−1.12998	−0.564988	+	0.825099i	$0.691119\pi$
$948$	−3.73300	−0.121242
$949$	27.4606	0.891410
$950$	19.2224	0.623658
$951$	8.73256	0.283173
$952$	−42.1039	−1.36459
$953$	32.3507	1.04794	0.523971	−	0.851736i	$-0.324450\pi$
$953$	32.3507	1.04794	0.523971	+	0.851736i	$0.324450\pi$
$954$	−0.271188	−0.00878004
$955$	8.66189	0.280292
$956$	−58.2845	−1.88505
$957$	−2.11197	−0.0682703
$958$	−33.0364	−1.06736
$959$	−18.0460	−0.582737
$960$	8.02058	0.258863
$961$	−29.1488	−0.940284
$962$	103.678	3.34273
$963$	23.3927	0.753818
$964$	1.48110	0.0477030
$965$	−2.61084	−0.0840460
$966$	5.55649	0.178777
$967$	23.1798	0.745413	0.372707	−	0.927949i	$-0.378430\pi$
$967$	23.1798	0.745413	0.372707	+	0.927949i	$0.378430\pi$
$968$	5.16351	0.165961
$969$	6.62364	0.212782
$970$	1.74877	0.0561498
$971$	−13.0420	−0.418538	−0.209269	−	0.977858i	$-0.567108\pi$
$971$	−13.0420	−0.418538	−0.209269	+	0.977858i	$0.567108\pi$
$972$	66.0960	2.12003
$973$	−39.4693	−1.26533
$974$	67.6608	2.16799
$975$	−15.8954	−0.509061
$976$	−37.8708	−1.21221
$977$	1.22071	0.0390540	0.0195270	−	0.999809i	$-0.493784\pi$
$977$	1.22071	0.0390540	0.0195270	+	0.999809i	$0.493784\pi$
$978$	31.6077	1.01070
$979$	2.24769	0.0718366
$980$	0.787299	0.0251493
$981$	1.83070	0.0584499
$982$	93.9074	2.99670
$983$	−11.3056	−0.360592	−0.180296	−	0.983612i	$-0.557705\pi$
$983$	−11.3056	−0.360592	−0.180296	+	0.983612i	$0.557705\pi$
$984$	−2.36991	−0.0755501
$985$	−14.5130	−0.462423
$986$	17.8017	0.566920
$987$	12.1812	0.387732
$988$	−49.7264	−1.58201
$989$	−1.73194	−0.0550724
$990$	6.89016	0.218984
$991$	−36.8559	−1.17077	−0.585383	−	0.810757i	$-0.699056\pi$
$991$	−36.8559	−1.17077	−0.585383	+	0.810757i	$0.699056\pi$
$992$	1.26443	0.0401458
$993$	−13.3667	−0.424179
$994$	−80.3099	−2.54728
$995$	−28.1632	−0.892833
$996$	−42.6001	−1.34984
$997$	−61.4850	−1.94725	−0.973625	−	0.228152i	$-0.926732\pi$
$997$	−61.4850	−1.94725	−0.973625	+	0.228152i	$0.926732\pi$
$998$	−39.6836	−1.25616
$999$	−37.9791	−1.20161

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	869.2.a.g.1.17	✓	18
3.2	odd	2		7821.2.a.n.1.2		18
11.10	odd	2		9559.2.a.k.1.2		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
869.2.a.g.1.17	✓	18	1.1	even	1	trivial
7821.2.a.n.1.2		18	3.2	odd	2
9559.2.a.k.1.2		18	11.10	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:	\( N \)	\(=\)	\( 869 = 11 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	869.a (trivial)

\(f(q)\)	\(=\)	\(q+2.46820 q^{2} +0.912264 q^{3} +4.09201 q^{4} -1.28776 q^{5} +2.25165 q^{6} +2.61736 q^{7} +5.16351 q^{8} -2.16777 q^{9} +O(q^{10})\)	\(q+2.46820 q^{2} +0.912264 q^{3} +4.09201 q^{4} -1.28776 q^{5} +2.25165 q^{6} +2.61736 q^{7} +5.16351 q^{8} -2.16777 q^{9} -3.17845 q^{10} +1.00000 q^{11} +3.73300 q^{12} +5.21419 q^{13} +6.46018 q^{14} -1.17478 q^{15} +4.56055 q^{16} -3.11539 q^{17} -5.35050 q^{18} -2.33058 q^{19} -5.26953 q^{20} +2.38773 q^{21} +2.46820 q^{22} +0.942834 q^{23} +4.71048 q^{24} -3.34168 q^{25} +12.8697 q^{26} -4.71437 q^{27} +10.7103 q^{28} -2.31509 q^{29} -2.89958 q^{30} +1.36058 q^{31} +0.929330 q^{32} +0.912264 q^{33} -7.68941 q^{34} -3.37053 q^{35} -8.87056 q^{36} +8.05603 q^{37} -5.75233 q^{38} +4.75672 q^{39} -6.64935 q^{40} -0.503115 q^{41} +5.89339 q^{42} -1.83695 q^{43} +4.09201 q^{44} +2.79157 q^{45} +2.32710 q^{46} +5.10159 q^{47} +4.16042 q^{48} -0.149406 q^{49} -8.24793 q^{50} -2.84206 q^{51} +21.3365 q^{52} +0.0506846 q^{53} -11.6360 q^{54} -1.28776 q^{55} +13.5148 q^{56} -2.12610 q^{57} -5.71410 q^{58} -2.31495 q^{59} -4.80720 q^{60} -8.30400 q^{61} +3.35819 q^{62} -5.67386 q^{63} -6.82732 q^{64} -6.71462 q^{65} +2.25165 q^{66} -3.35748 q^{67} -12.7482 q^{68} +0.860114 q^{69} -8.31915 q^{70} -12.4315 q^{71} -11.1933 q^{72} +5.26652 q^{73} +19.8839 q^{74} -3.04849 q^{75} -9.53675 q^{76} +2.61736 q^{77} +11.7405 q^{78} -1.00000 q^{79} -5.87289 q^{80} +2.20257 q^{81} -1.24179 q^{82} -11.4118 q^{83} +9.77061 q^{84} +4.01187 q^{85} -4.53395 q^{86} -2.11197 q^{87} +5.16351 q^{88} +2.24769 q^{89} +6.89016 q^{90} +13.6474 q^{91} +3.85809 q^{92} +1.24121 q^{93} +12.5917 q^{94} +3.00122 q^{95} +0.847794 q^{96} -0.550198 q^{97} -0.368764 q^{98} -2.16777 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 4 q^{2} + 8 q^{3} + 24 q^{4} - 3 q^{5} + 2 q^{6} + 10 q^{7} - 6 q^{8} + 18 q^{9}+O(q^{10}) \) 18 * q + 4 * q^2 + 8 * q^3 + 24 * q^4 - 3 * q^5 + 2 * q^6 + 10 * q^7 - 6 * q^8 + 18 * q^9	\( 18 q + 4 q^{2} + 8 q^{3} + 24 q^{4} - 3 q^{5} + 2 q^{6} + 10 q^{7} - 6 q^{8} + 18 q^{9} + 7 q^{10} + 18 q^{11} + 5 q^{12} + 4 q^{13} - 7 q^{14} + 4 q^{15} + 32 q^{16} + 19 q^{17} + 9 q^{18} + 47 q^{19} - 5 q^{20} + 7 q^{21} + 4 q^{22} - q^{23} + 43 q^{24} + 23 q^{25} - q^{26} - q^{27} + 20 q^{28} + 5 q^{29} + 7 q^{30} + 7 q^{31} - 23 q^{32} + 8 q^{33} + 18 q^{34} + 15 q^{35} + 24 q^{36} - 8 q^{37} - 6 q^{38} + 26 q^{39} - 6 q^{40} + 37 q^{41} - 47 q^{42} + 22 q^{43} + 24 q^{44} - 9 q^{45} + 36 q^{46} - 26 q^{47} - 34 q^{48} + 20 q^{49} - 32 q^{50} - q^{51} + 53 q^{52} - 19 q^{53} + 16 q^{54} - 3 q^{55} - 52 q^{56} - 6 q^{57} + 29 q^{58} + 20 q^{59} + 8 q^{60} + 47 q^{61} - 11 q^{62} + 8 q^{63} + 4 q^{64} - q^{65} + 2 q^{66} + 16 q^{67} + 52 q^{68} - q^{69} - 69 q^{70} - 8 q^{71} - 24 q^{72} + 20 q^{73} + 26 q^{74} - 18 q^{75} + 54 q^{76} + 10 q^{77} - 29 q^{78} - 18 q^{79} - 7 q^{80} + 2 q^{81} - 2 q^{82} + 15 q^{83} - 92 q^{84} + 28 q^{85} - 27 q^{86} - 15 q^{87} - 6 q^{88} - 21 q^{89} + 27 q^{90} + 60 q^{91} - 62 q^{92} - 27 q^{93} + 20 q^{94} - 11 q^{95} + 84 q^{96} - 17 q^{97} - 57 q^{98} + 18 q^{99}+O(q^{100}) \) 18 * q + 4 * q^2 + 8 * q^3 + 24 * q^4 - 3 * q^5 + 2 * q^6 + 10 * q^7 - 6 * q^8 + 18 * q^9 + 7 * q^10 + 18 * q^11 + 5 * q^12 + 4 * q^13 - 7 * q^14 + 4 * q^15 + 32 * q^16 + 19 * q^17 + 9 * q^18 + 47 * q^19 - 5 * q^20 + 7 * q^21 + 4 * q^22 - q^23 + 43 * q^24 + 23 * q^25 - q^26 - q^27 + 20 * q^28 + 5 * q^29 + 7 * q^30 + 7 * q^31 - 23 * q^32 + 8 * q^33 + 18 * q^34 + 15 * q^35 + 24 * q^36 - 8 * q^37 - 6 * q^38 + 26 * q^39 - 6 * q^40 + 37 * q^41 - 47 * q^42 + 22 * q^43 + 24 * q^44 - 9 * q^45 + 36 * q^46 - 26 * q^47 - 34 * q^48 + 20 * q^49 - 32 * q^50 - q^51 + 53 * q^52 - 19 * q^53 + 16 * q^54 - 3 * q^55 - 52 * q^56 - 6 * q^57 + 29 * q^58 + 20 * q^59 + 8 * q^60 + 47 * q^61 - 11 * q^62 + 8 * q^63 + 4 * q^64 - q^65 + 2 * q^66 + 16 * q^67 + 52 * q^68 - q^69 - 69 * q^70 - 8 * q^71 - 24 * q^72 + 20 * q^73 + 26 * q^74 - 18 * q^75 + 54 * q^76 + 10 * q^77 - 29 * q^78 - 18 * q^79 - 7 * q^80 + 2 * q^81 - 2 * q^82 + 15 * q^83 - 92 * q^84 + 28 * q^85 - 27 * q^86 - 15 * q^87 - 6 * q^88 - 21 * q^89 + 27 * q^90 + 60 * q^91 - 62 * q^92 - 27 * q^93 + 20 * q^94 - 11 * q^95 + 84 * q^96 - 17 * q^97 - 57 * q^98 + 18 * q^99

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