LMFDB - Embedded newform 8470.2.a.br.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, 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("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.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:	$67.6332905120$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} +1.00000 q^{10} -0.732051 q^{12} -5.46410 q^{13} +1.00000 q^{14} +0.732051 q^{15} +1.00000 q^{16} -3.46410 q^{17} +2.46410 q^{18} -3.26795 q^{19} -1.00000 q^{20} +0.732051 q^{21} +2.19615 q^{23} +0.732051 q^{24} +1.00000 q^{25} +5.46410 q^{26} +4.00000 q^{27} -1.00000 q^{28} +1.26795 q^{29} -0.732051 q^{30} +2.00000 q^{31} -1.00000 q^{32} +3.46410 q^{34} +1.00000 q^{35} -2.46410 q^{36} -2.73205 q^{37} +3.26795 q^{38} +4.00000 q^{39} +1.00000 q^{40} -8.19615 q^{41} -0.732051 q^{42} -2.00000 q^{43} +2.46410 q^{45} -2.19615 q^{46} -6.92820 q^{47} -0.732051 q^{48} +1.00000 q^{49} -1.00000 q^{50} +2.53590 q^{51} -5.46410 q^{52} -10.7321 q^{53} -4.00000 q^{54} +1.00000 q^{56} +2.39230 q^{57} -1.26795 q^{58} -6.92820 q^{59} +0.732051 q^{60} -8.92820 q^{61} -2.00000 q^{62} +2.46410 q^{63} +1.00000 q^{64} +5.46410 q^{65} -4.00000 q^{67} -3.46410 q^{68} -1.60770 q^{69} -1.00000 q^{70} +2.53590 q^{71} +2.46410 q^{72} -6.39230 q^{73} +2.73205 q^{74} -0.732051 q^{75} -3.26795 q^{76} -4.00000 q^{78} +1.80385 q^{79} -1.00000 q^{80} +4.46410 q^{81} +8.19615 q^{82} -4.39230 q^{83} +0.732051 q^{84} +3.46410 q^{85} +2.00000 q^{86} -0.928203 q^{87} -3.46410 q^{89} -2.46410 q^{90} +5.46410 q^{91} +2.19615 q^{92} -1.46410 q^{93} +6.92820 q^{94} +3.26795 q^{95} +0.732051 q^{96} -16.5885 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} - 4 q^{13} + 2 q^{14} - 2 q^{15} + 2 q^{16} - 2 q^{18} - 10 q^{19} - 2 q^{20} - 2 q^{21} - 6 q^{23} - 2 q^{24} + 2 q^{25} + 4 q^{26} + 8 q^{27} - 2 q^{28} + 6 q^{29} + 2 q^{30} + 4 q^{31} - 2 q^{32} + 2 q^{35} + 2 q^{36} - 2 q^{37} + 10 q^{38} + 8 q^{39} + 2 q^{40} - 6 q^{41} + 2 q^{42} - 4 q^{43} - 2 q^{45} + 6 q^{46} + 2 q^{48} + 2 q^{49} - 2 q^{50} + 12 q^{51} - 4 q^{52} - 18 q^{53} - 8 q^{54} + 2 q^{56} - 16 q^{57} - 6 q^{58} - 2 q^{60} - 4 q^{61} - 4 q^{62} - 2 q^{63} + 2 q^{64} + 4 q^{65} - 8 q^{67} - 24 q^{69} - 2 q^{70} + 12 q^{71} - 2 q^{72} + 8 q^{73} + 2 q^{74} + 2 q^{75} - 10 q^{76} - 8 q^{78} + 14 q^{79} - 2 q^{80} + 2 q^{81} + 6 q^{82} + 12 q^{83} - 2 q^{84} + 4 q^{86} + 12 q^{87} + 2 q^{90} + 4 q^{91} - 6 q^{92} + 4 q^{93} + 10 q^{95} - 2 q^{96} - 2 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^12 - 4 * q^13 + 2 * q^14 - 2 * q^15 + 2 * q^16 - 2 * q^18 - 10 * q^19 - 2 * q^20 - 2 * q^21 - 6 * q^23 - 2 * q^24 + 2 * q^25 + 4 * q^26 + 8 * q^27 - 2 * q^28 + 6 * q^29 + 2 * q^30 + 4 * q^31 - 2 * q^32 + 2 * q^35 + 2 * q^36 - 2 * q^37 + 10 * q^38 + 8 * q^39 + 2 * q^40 - 6 * q^41 + 2 * q^42 - 4 * q^43 - 2 * q^45 + 6 * q^46 + 2 * q^48 + 2 * q^49 - 2 * q^50 + 12 * q^51 - 4 * q^52 - 18 * q^53 - 8 * q^54 + 2 * q^56 - 16 * q^57 - 6 * q^58 - 2 * q^60 - 4 * q^61 - 4 * q^62 - 2 * q^63 + 2 * q^64 + 4 * q^65 - 8 * q^67 - 24 * q^69 - 2 * q^70 + 12 * q^71 - 2 * q^72 + 8 * q^73 + 2 * q^74 + 2 * q^75 - 10 * q^76 - 8 * q^78 + 14 * q^79 - 2 * q^80 + 2 * q^81 + 6 * q^82 + 12 * q^83 - 2 * q^84 + 4 * q^86 + 12 * q^87 + 2 * q^90 + 4 * q^91 - 6 * q^92 + 4 * q^93 + 10 * q^95 - 2 * q^96 - 2 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−0.732051	−0.422650	−0.211325	−	0.977416i	$-0.567778\pi$
$3$	−0.732051	−0.422650	−0.211325	+	0.977416i	$0.567778\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.732051	0.298858
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	−2.46410	−0.821367
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	−0.732051	−0.211325
$13$	−5.46410	−1.51547	−0.757735	−	0.652563i	$-0.773694\pi$
$13$	−5.46410	−1.51547	−0.757735	+	0.652563i	$0.773694\pi$
$14$	1.00000	0.267261
$15$	0.732051	0.189015
$16$	1.00000	0.250000
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	2.46410	0.580794
$19$	−3.26795	−0.749719	−0.374859	−	0.927082i	$-0.622309\pi$
$19$	−3.26795	−0.749719	−0.374859	+	0.927082i	$0.622309\pi$
$20$	−1.00000	−0.223607
$21$	0.732051	0.159747
$22$	0	0
$23$	2.19615	0.457929	0.228965	−	0.973435i	$-0.426466\pi$
$23$	2.19615	0.457929	0.228965	+	0.973435i	$0.426466\pi$
$24$	0.732051	0.149429
$25$	1.00000	0.200000
$26$	5.46410	1.07160
$27$	4.00000	0.769800
$28$	−1.00000	−0.188982
$29$	1.26795	0.235452	0.117726	−	0.993046i	$-0.462440\pi$
$29$	1.26795	0.235452	0.117726	+	0.993046i	$0.462440\pi$
$30$	−0.732051	−0.133654
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	3.46410	0.594089
$35$	1.00000	0.169031
$36$	−2.46410	−0.410684
$37$	−2.73205	−0.449146	−0.224573	−	0.974457i	$-0.572099\pi$
$37$	−2.73205	−0.449146	−0.224573	+	0.974457i	$0.572099\pi$
$38$	3.26795	0.530131
$39$	4.00000	0.640513
$40$	1.00000	0.158114
$41$	−8.19615	−1.28002	−0.640012	−	0.768365i	$-0.721071\pi$
$41$	−8.19615	−1.28002	−0.640012	+	0.768365i	$0.721071\pi$
$42$	−0.732051	−0.112958
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	2.46410	0.367327
$46$	−2.19615	−0.323805
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	−0.732051	−0.105662
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	2.53590	0.355097
$52$	−5.46410	−0.757735
$53$	−10.7321	−1.47416	−0.737080	−	0.675805i	$-0.763796\pi$
$53$	−10.7321	−1.47416	−0.737080	+	0.675805i	$0.763796\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	1.00000	0.133631
$57$	2.39230	0.316869
$58$	−1.26795	−0.166490
$59$	−6.92820	−0.901975	−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820	−0.901975	−0.450988	+	0.892530i	$0.648928\pi$
$60$	0.732051	0.0945074
$61$	−8.92820	−1.14314	−0.571570	−	0.820554i	$-0.693665\pi$
$61$	−8.92820	−1.14314	−0.571570	+	0.820554i	$0.693665\pi$
$62$	−2.00000	−0.254000
$63$	2.46410	0.310448
$64$	1.00000	0.125000
$65$	5.46410	0.677738
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−3.46410	−0.420084
$69$	−1.60770	−0.193544
$70$	−1.00000	−0.119523
$71$	2.53590	0.300956	0.150478	−	0.988613i	$-0.451919\pi$
$71$	2.53590	0.300956	0.150478	+	0.988613i	$0.451919\pi$
$72$	2.46410	0.290397
$73$	−6.39230	−0.748163	−0.374081	−	0.927396i	$-0.622042\pi$
$73$	−6.39230	−0.748163	−0.374081	+	0.927396i	$0.622042\pi$
$74$	2.73205	0.317594
$75$	−0.732051	−0.0845299
$76$	−3.26795	−0.374859
$77$	0	0
$78$	−4.00000	−0.452911
$79$	1.80385	0.202949	0.101474	−	0.994838i	$-0.467644\pi$
$79$	1.80385	0.202949	0.101474	+	0.994838i	$0.467644\pi$
$80$	−1.00000	−0.111803
$81$	4.46410	0.496011
$82$	8.19615	0.905114
$83$	−4.39230	−0.482118	−0.241059	−	0.970510i	$-0.577495\pi$
$83$	−4.39230	−0.482118	−0.241059	+	0.970510i	$0.577495\pi$
$84$	0.732051	0.0798733
$85$	3.46410	0.375735
$86$	2.00000	0.215666
$87$	−0.928203	−0.0995138
$88$	0	0
$89$	−3.46410	−0.367194	−0.183597	−	0.983002i	$-0.558774\pi$
$89$	−3.46410	−0.367194	−0.183597	+	0.983002i	$0.558774\pi$
$90$	−2.46410	−0.259739
$91$	5.46410	0.572793
$92$	2.19615	0.228965
$93$	−1.46410	−0.151820
$94$	6.92820	0.714590
$95$	3.26795	0.335285
$96$	0.732051	0.0747146
$97$	−16.5885	−1.68430	−0.842151	−	0.539241i	$-0.818711\pi$
$97$	−16.5885	−1.68430	−0.842151	+	0.539241i	$0.818711\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−19.8564	−1.97579	−0.987893	−	0.155136i	$-0.950418\pi$
$101$	−19.8564	−1.97579	−0.987893	+	0.155136i	$0.950418\pi$
$102$	−2.53590	−0.251091
$103$	−8.39230	−0.826918	−0.413459	−	0.910523i	$-0.635680\pi$
$103$	−8.39230	−0.826918	−0.413459	+	0.910523i	$0.635680\pi$
$104$	5.46410	0.535799
$105$	−0.732051	−0.0714408
$106$	10.7321	1.04239
$107$	19.8564	1.91959	0.959796	−	0.280700i	$-0.0905665\pi$
$107$	19.8564	1.91959	0.959796	+	0.280700i	$0.0905665\pi$
$108$	4.00000	0.384900
$109$	−1.66025	−0.159023	−0.0795117	−	0.996834i	$-0.525336\pi$
$109$	−1.66025	−0.159023	−0.0795117	+	0.996834i	$0.525336\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	−1.00000	−0.0944911
$113$	−19.8564	−1.86793	−0.933967	−	0.357360i	$-0.883677\pi$
$113$	−19.8564	−1.86793	−0.933967	+	0.357360i	$0.883677\pi$
$114$	−2.39230	−0.224060
$115$	−2.19615	−0.204792
$116$	1.26795	0.117726
$117$	13.4641	1.24476
$118$	6.92820	0.637793
$119$	3.46410	0.317554
$120$	−0.732051	−0.0668268
$121$	0	0
$122$	8.92820	0.808322
$123$	6.00000	0.541002
$124$	2.00000	0.179605
$125$	−1.00000	−0.0894427
$126$	−2.46410	−0.219520
$127$	−14.9282	−1.32466	−0.662332	−	0.749211i	$-0.730433\pi$
$127$	−14.9282	−1.32466	−0.662332	+	0.749211i	$0.730433\pi$
$128$	−1.00000	−0.0883883
$129$	1.46410	0.128907
$130$	−5.46410	−0.479233
$131$	11.6603	1.01876	0.509381	−	0.860541i	$-0.329875\pi$
$131$	11.6603	1.01876	0.509381	+	0.860541i	$0.329875\pi$
$132$	0	0
$133$	3.26795	0.283367
$134$	4.00000	0.345547
$135$	−4.00000	−0.344265
$136$	3.46410	0.297044
$137$	12.9282	1.10453	0.552265	−	0.833668i	$-0.313763\pi$
$137$	12.9282	1.10453	0.552265	+	0.833668i	$0.313763\pi$
$138$	1.60770	0.136856
$139$	3.66025	0.310459	0.155229	−	0.987878i	$-0.450388\pi$
$139$	3.66025	0.310459	0.155229	+	0.987878i	$0.450388\pi$
$140$	1.00000	0.0845154
$141$	5.07180	0.427122
$142$	−2.53590	−0.212808
$143$	0	0
$144$	−2.46410	−0.205342
$145$	−1.26795	−0.105297
$146$	6.39230	0.529031
$147$	−0.732051	−0.0603785
$148$	−2.73205	−0.224573
$149$	10.7321	0.879204	0.439602	−	0.898193i	$-0.355120\pi$
$149$	10.7321	0.879204	0.439602	+	0.898193i	$0.355120\pi$
$150$	0.732051	0.0597717
$151$	13.1244	1.06804	0.534022	−	0.845470i	$-0.320680\pi$
$151$	13.1244	1.06804	0.534022	+	0.845470i	$0.320680\pi$
$152$	3.26795	0.265066
$153$	8.53590	0.690086
$154$	0	0
$155$	−2.00000	−0.160644
$156$	4.00000	0.320256
$157$	11.4641	0.914935	0.457467	−	0.889226i	$-0.348757\pi$
$157$	11.4641	0.914935	0.457467	+	0.889226i	$0.348757\pi$
$158$	−1.80385	−0.143506
$159$	7.85641	0.623054
$160$	1.00000	0.0790569
$161$	−2.19615	−0.173081
$162$	−4.46410	−0.350733
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−8.19615	−0.640012
$165$	0	0
$166$	4.39230	0.340909
$167$	13.8564	1.07224	0.536120	−	0.844141i	$-0.319889\pi$
$167$	13.8564	1.07224	0.536120	+	0.844141i	$0.319889\pi$
$168$	−0.732051	−0.0564789
$169$	16.8564	1.29665
$170$	−3.46410	−0.265684
$171$	8.05256	0.615795
$172$	−2.00000	−0.152499
$173$	12.9282	0.982913	0.491457	−	0.870902i	$-0.336465\pi$
$173$	12.9282	0.982913	0.491457	+	0.870902i	$0.336465\pi$
$174$	0.928203	0.0703669
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	5.07180	0.381220
$178$	3.46410	0.259645
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	2.46410	0.183663
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	−5.46410	−0.405026
$183$	6.53590	0.483148
$184$	−2.19615	−0.161903
$185$	2.73205	0.200864
$186$	1.46410	0.107353
$187$	0	0
$188$	−6.92820	−0.505291
$189$	−4.00000	−0.290957
$190$	−3.26795	−0.237082
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	−0.732051	−0.0528312
$193$	8.39230	0.604091	0.302046	−	0.953293i	$-0.402330\pi$
$193$	8.39230	0.604091	0.302046	+	0.953293i	$0.402330\pi$
$194$	16.5885	1.19098
$195$	−4.00000	−0.286446
$196$	1.00000	0.0714286
$197$	−24.2487	−1.72765	−0.863825	−	0.503793i	$-0.831938\pi$
$197$	−24.2487	−1.72765	−0.863825	+	0.503793i	$0.831938\pi$
$198$	0	0
$199$	−10.9282	−0.774680	−0.387340	−	0.921937i	$-0.626606\pi$
$199$	−10.9282	−0.774680	−0.387340	+	0.921937i	$0.626606\pi$
$200$	−1.00000	−0.0707107
$201$	2.92820	0.206540
$202$	19.8564	1.39709
$203$	−1.26795	−0.0889926
$204$	2.53590	0.177548
$205$	8.19615	0.572444
$206$	8.39230	0.584720
$207$	−5.41154	−0.376128
$208$	−5.46410	−0.378867
$209$	0	0
$210$	0.732051	0.0505163
$211$	−13.0718	−0.899900	−0.449950	−	0.893054i	$-0.648558\pi$
$211$	−13.0718	−0.899900	−0.449950	+	0.893054i	$0.648558\pi$
$212$	−10.7321	−0.737080
$213$	−1.85641	−0.127199
$214$	−19.8564	−1.35736
$215$	2.00000	0.136399
$216$	−4.00000	−0.272166
$217$	−2.00000	−0.135769
$218$	1.66025	0.112447
$219$	4.67949	0.316211
$220$	0	0
$221$	18.9282	1.27325
$222$	−2.00000	−0.134231
$223$	−18.5359	−1.24126	−0.620628	−	0.784105i	$-0.713122\pi$
$223$	−18.5359	−1.24126	−0.620628	+	0.784105i	$0.713122\pi$
$224$	1.00000	0.0668153
$225$	−2.46410	−0.164273
$226$	19.8564	1.32083
$227$	6.92820	0.459841	0.229920	−	0.973209i	$-0.426153\pi$
$227$	6.92820	0.459841	0.229920	+	0.973209i	$0.426153\pi$
$228$	2.39230	0.158434
$229$	3.60770	0.238403	0.119202	−	0.992870i	$-0.461967\pi$
$229$	3.60770	0.238403	0.119202	+	0.992870i	$0.461967\pi$
$230$	2.19615	0.144810
$231$	0	0
$232$	−1.26795	−0.0832449
$233$	−19.8564	−1.30084	−0.650418	−	0.759576i	$-0.725406\pi$
$233$	−19.8564	−1.30084	−0.650418	+	0.759576i	$0.725406\pi$
$234$	−13.4641	−0.880176
$235$	6.92820	0.451946
$236$	−6.92820	−0.450988
$237$	−1.32051	−0.0857762
$238$	−3.46410	−0.224544
$239$	−4.73205	−0.306091	−0.153045	−	0.988219i	$-0.548908\pi$
$239$	−4.73205	−0.306091	−0.153045	+	0.988219i	$0.548908\pi$
$240$	0.732051	0.0472537
$241$	−6.73205	−0.433650	−0.216825	−	0.976211i	$-0.569570\pi$
$241$	−6.73205	−0.433650	−0.216825	+	0.976211i	$0.569570\pi$
$242$	0	0
$243$	−15.2679	−0.979439
$244$	−8.92820	−0.571570
$245$	−1.00000	−0.0638877
$246$	−6.00000	−0.382546
$247$	17.8564	1.13618
$248$	−2.00000	−0.127000
$249$	3.21539	0.203767
$250$	1.00000	0.0632456
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	2.46410	0.155224
$253$	0	0
$254$	14.9282	0.936679
$255$	−2.53590	−0.158804
$256$	1.00000	0.0625000
$257$	−6.33975	−0.395462	−0.197731	−	0.980256i	$-0.563357\pi$
$257$	−6.33975	−0.395462	−0.197731	+	0.980256i	$0.563357\pi$
$258$	−1.46410	−0.0911510
$259$	2.73205	0.169761
$260$	5.46410	0.338869
$261$	−3.12436	−0.193393
$262$	−11.6603	−0.720373
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	10.7321	0.659265
$266$	−3.26795	−0.200371
$267$	2.53590	0.155194
$268$	−4.00000	−0.244339
$269$	7.60770	0.463849	0.231925	−	0.972734i	$-0.425498\pi$
$269$	7.60770	0.463849	0.231925	+	0.972734i	$0.425498\pi$
$270$	4.00000	0.243432
$271$	20.3923	1.23874	0.619372	−	0.785098i	$-0.287387\pi$
$271$	20.3923	1.23874	0.619372	+	0.785098i	$0.287387\pi$
$272$	−3.46410	−0.210042
$273$	−4.00000	−0.242091
$274$	−12.9282	−0.781021
$275$	0	0
$276$	−1.60770	−0.0967719
$277$	−20.9282	−1.25745	−0.628727	−	0.777626i	$-0.716424\pi$
$277$	−20.9282	−1.25745	−0.628727	+	0.777626i	$0.716424\pi$
$278$	−3.66025	−0.219527
$279$	−4.92820	−0.295044
$280$	−1.00000	−0.0597614
$281$	−1.60770	−0.0959071	−0.0479535	−	0.998850i	$-0.515270\pi$
$281$	−1.60770	−0.0959071	−0.0479535	+	0.998850i	$0.515270\pi$
$282$	−5.07180	−0.302021
$283$	−23.7128	−1.40958	−0.704790	−	0.709416i	$-0.748959\pi$
$283$	−23.7128	−1.40958	−0.704790	+	0.709416i	$0.748959\pi$
$284$	2.53590	0.150478
$285$	−2.39230	−0.141708
$286$	0	0
$287$	8.19615	0.483804
$288$	2.46410	0.145199
$289$	−5.00000	−0.294118
$290$	1.26795	0.0744565
$291$	12.1436	0.711870
$292$	−6.39230	−0.374081
$293$	−14.5359	−0.849196	−0.424598	−	0.905382i	$-0.639585\pi$
$293$	−14.5359	−0.849196	−0.424598	+	0.905382i	$0.639585\pi$
$294$	0.732051	0.0426941
$295$	6.92820	0.403376
$296$	2.73205	0.158797
$297$	0	0
$298$	−10.7321	−0.621691
$299$	−12.0000	−0.693978
$300$	−0.732051	−0.0422650
$301$	2.00000	0.115278
$302$	−13.1244	−0.755222
$303$	14.5359	0.835066
$304$	−3.26795	−0.187430
$305$	8.92820	0.511227
$306$	−8.53590	−0.487965
$307$	−3.60770	−0.205902	−0.102951	−	0.994686i	$-0.532828\pi$
$307$	−3.60770	−0.205902	−0.102951	+	0.994686i	$0.532828\pi$
$308$	0	0
$309$	6.14359	0.349497
$310$	2.00000	0.113592
$311$	11.0718	0.627824	0.313912	−	0.949452i	$-0.398360\pi$
$311$	11.0718	0.627824	0.313912	+	0.949452i	$0.398360\pi$
$312$	−4.00000	−0.226455
$313$	11.8038	0.667193	0.333596	−	0.942716i	$-0.391738\pi$
$313$	11.8038	0.667193	0.333596	+	0.942716i	$0.391738\pi$
$314$	−11.4641	−0.646957
$315$	−2.46410	−0.138836
$316$	1.80385	0.101474
$317$	−0.588457	−0.0330511	−0.0165255	−	0.999863i	$-0.505260\pi$
$317$	−0.588457	−0.0330511	−0.0165255	+	0.999863i	$0.505260\pi$
$318$	−7.85641	−0.440565
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−14.5359	−0.811315
$322$	2.19615	0.122387
$323$	11.3205	0.629890
$324$	4.46410	0.248006
$325$	−5.46410	−0.303094
$326$	4.00000	0.221540
$327$	1.21539	0.0672112
$328$	8.19615	0.452557
$329$	6.92820	0.381964
$330$	0	0
$331$	22.7846	1.25236	0.626178	−	0.779680i	$-0.284618\pi$
$331$	22.7846	1.25236	0.626178	+	0.779680i	$0.284618\pi$
$332$	−4.39230	−0.241059
$333$	6.73205	0.368914
$334$	−13.8564	−0.758189
$335$	4.00000	0.218543
$336$	0.732051	0.0399366
$337$	18.7846	1.02326	0.511631	−	0.859205i	$-0.329041\pi$
$337$	18.7846	1.02326	0.511631	+	0.859205i	$0.329041\pi$
$338$	−16.8564	−0.916868
$339$	14.5359	0.789482
$340$	3.46410	0.187867
$341$	0	0
$342$	−8.05256	−0.435433
$343$	−1.00000	−0.0539949
$344$	2.00000	0.107833
$345$	1.60770	0.0865554
$346$	−12.9282	−0.695025
$347$	36.9282	1.98241	0.991205	−	0.132336i	$-0.0422478\pi$
$347$	36.9282	1.98241	0.991205	+	0.132336i	$0.0422478\pi$
$348$	−0.928203	−0.0497569
$349$	26.3923	1.41275	0.706374	−	0.707839i	$-0.250330\pi$
$349$	26.3923	1.41275	0.706374	+	0.707839i	$0.250330\pi$
$350$	1.00000	0.0534522
$351$	−21.8564	−1.16661
$352$	0	0
$353$	3.80385	0.202458	0.101229	−	0.994863i	$-0.467722\pi$
$353$	3.80385	0.202458	0.101229	+	0.994863i	$0.467722\pi$
$354$	−5.07180	−0.269563
$355$	−2.53590	−0.134592
$356$	−3.46410	−0.183597
$357$	−2.53590	−0.134214
$358$	6.00000	0.317110
$359$	−4.05256	−0.213886	−0.106943	−	0.994265i	$-0.534106\pi$
$359$	−4.05256	−0.213886	−0.106943	+	0.994265i	$0.534106\pi$
$360$	−2.46410	−0.129870
$361$	−8.32051	−0.437921
$362$	−14.0000	−0.735824
$363$	0	0
$364$	5.46410	0.286397
$365$	6.39230	0.334589
$366$	−6.53590	−0.341637
$367$	−8.39230	−0.438075	−0.219037	−	0.975716i	$-0.570292\pi$
$367$	−8.39230	−0.438075	−0.219037	+	0.975716i	$0.570292\pi$
$368$	2.19615	0.114482
$369$	20.1962	1.05137
$370$	−2.73205	−0.142033
$371$	10.7321	0.557180
$372$	−1.46410	−0.0759101
$373$	2.39230	0.123869	0.0619344	−	0.998080i	$-0.480273\pi$
$373$	2.39230	0.123869	0.0619344	+	0.998080i	$0.480273\pi$
$374$	0	0
$375$	0.732051	0.0378029
$376$	6.92820	0.357295
$377$	−6.92820	−0.356821
$378$	4.00000	0.205738
$379$	33.8564	1.73909	0.869543	−	0.493857i	$-0.164413\pi$
$379$	33.8564	1.73909	0.869543	+	0.493857i	$0.164413\pi$
$380$	3.26795	0.167642
$381$	10.9282	0.559869
$382$	12.0000	0.613973
$383$	−26.5359	−1.35592	−0.677961	−	0.735098i	$-0.737136\pi$
$383$	−26.5359	−1.35592	−0.677961	+	0.735098i	$0.737136\pi$
$384$	0.732051	0.0373573
$385$	0	0
$386$	−8.39230	−0.427157
$387$	4.92820	0.250515
$388$	−16.5885	−0.842151
$389$	−22.3923	−1.13533	−0.567667	−	0.823258i	$-0.692154\pi$
$389$	−22.3923	−1.13533	−0.567667	+	0.823258i	$0.692154\pi$
$390$	4.00000	0.202548
$391$	−7.60770	−0.384738
$392$	−1.00000	−0.0505076
$393$	−8.53590	−0.430579
$394$	24.2487	1.22163
$395$	−1.80385	−0.0907614
$396$	0	0
$397$	9.60770	0.482196	0.241098	−	0.970501i	$-0.422492\pi$
$397$	9.60770	0.482196	0.241098	+	0.970501i	$0.422492\pi$
$398$	10.9282	0.547781
$399$	−2.39230	−0.119765
$400$	1.00000	0.0500000
$401$	9.46410	0.472615	0.236307	−	0.971678i	$-0.424063\pi$
$401$	9.46410	0.472615	0.236307	+	0.971678i	$0.424063\pi$
$402$	−2.92820	−0.146046
$403$	−10.9282	−0.544373
$404$	−19.8564	−0.987893
$405$	−4.46410	−0.221823
$406$	1.26795	0.0629273
$407$	0	0
$408$	−2.53590	−0.125546
$409$	−35.1244	−1.73679	−0.868394	−	0.495875i	$-0.834847\pi$
$409$	−35.1244	−1.73679	−0.868394	+	0.495875i	$0.834847\pi$
$410$	−8.19615	−0.404779
$411$	−9.46410	−0.466830
$412$	−8.39230	−0.413459
$413$	6.92820	0.340915
$414$	5.41154	0.265963
$415$	4.39230	0.215610
$416$	5.46410	0.267900
$417$	−2.67949	−0.131215
$418$	0	0
$419$	17.0718	0.834012	0.417006	−	0.908904i	$-0.363079\pi$
$419$	17.0718	0.834012	0.417006	+	0.908904i	$0.363079\pi$
$420$	−0.732051	−0.0357204
$421$	−8.14359	−0.396894	−0.198447	−	0.980112i	$-0.563590\pi$
$421$	−8.14359	−0.396894	−0.198447	+	0.980112i	$0.563590\pi$
$422$	13.0718	0.636325
$423$	17.0718	0.830059
$424$	10.7321	0.521194
$425$	−3.46410	−0.168034
$426$	1.85641	0.0899432
$427$	8.92820	0.432066
$428$	19.8564	0.959796
$429$	0	0
$430$	−2.00000	−0.0964486
$431$	21.1244	1.01752	0.508762	−	0.860907i	$-0.330103\pi$
$431$	21.1244	1.01752	0.508762	+	0.860907i	$0.330103\pi$
$432$	4.00000	0.192450
$433$	−7.80385	−0.375029	−0.187514	−	0.982262i	$-0.560043\pi$
$433$	−7.80385	−0.375029	−0.187514	+	0.982262i	$0.560043\pi$
$434$	2.00000	0.0960031
$435$	0.928203	0.0445039
$436$	−1.66025	−0.0795117
$437$	−7.17691	−0.343318
$438$	−4.67949	−0.223595
$439$	22.2487	1.06187	0.530937	−	0.847412i	$-0.321840\pi$
$439$	22.2487	1.06187	0.530937	+	0.847412i	$0.321840\pi$
$440$	0	0
$441$	−2.46410	−0.117338
$442$	−18.9282	−0.900323
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	2.00000	0.0949158
$445$	3.46410	0.164214
$446$	18.5359	0.877700
$447$	−7.85641	−0.371595
$448$	−1.00000	−0.0472456
$449$	19.6077	0.925344	0.462672	−	0.886529i	$-0.346891\pi$
$449$	19.6077	0.925344	0.462672	+	0.886529i	$0.346891\pi$
$450$	2.46410	0.116159
$451$	0	0
$452$	−19.8564	−0.933967
$453$	−9.60770	−0.451409
$454$	−6.92820	−0.325157
$455$	−5.46410	−0.256161
$456$	−2.39230	−0.112030
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	−3.60770	−0.168577
$459$	−13.8564	−0.646762
$460$	−2.19615	−0.102396
$461$	−1.60770	−0.0748778	−0.0374389	−	0.999299i	$-0.511920\pi$
$461$	−1.60770	−0.0748778	−0.0374389	+	0.999299i	$0.511920\pi$
$462$	0	0
$463$	−17.5167	−0.814068	−0.407034	−	0.913413i	$-0.633437\pi$
$463$	−17.5167	−0.814068	−0.407034	+	0.913413i	$0.633437\pi$
$464$	1.26795	0.0588631
$465$	1.46410	0.0678961
$466$	19.8564	0.919830
$467$	4.05256	0.187530	0.0937650	−	0.995594i	$-0.470110\pi$
$467$	4.05256	0.187530	0.0937650	+	0.995594i	$0.470110\pi$
$468$	13.4641	0.622378
$469$	4.00000	0.184703
$470$	−6.92820	−0.319574
$471$	−8.39230	−0.386697
$472$	6.92820	0.318896
$473$	0	0
$474$	1.32051	0.0606529
$475$	−3.26795	−0.149944
$476$	3.46410	0.158777
$477$	26.4449	1.21083
$478$	4.73205	0.216439
$479$	−8.78461	−0.401379	−0.200690	−	0.979655i	$-0.564318\pi$
$479$	−8.78461	−0.401379	−0.200690	+	0.979655i	$0.564318\pi$
$480$	−0.732051	−0.0334134
$481$	14.9282	0.680667
$482$	6.73205	0.306637
$483$	1.60770	0.0731527
$484$	0	0
$485$	16.5885	0.753243
$486$	15.2679	0.692568
$487$	−6.19615	−0.280774	−0.140387	−	0.990097i	$-0.544835\pi$
$487$	−6.19615	−0.280774	−0.140387	+	0.990097i	$0.544835\pi$
$488$	8.92820	0.404161
$489$	2.92820	0.132418
$490$	1.00000	0.0451754
$491$	−27.7128	−1.25066	−0.625331	−	0.780360i	$-0.715036\pi$
$491$	−27.7128	−1.25066	−0.625331	+	0.780360i	$0.715036\pi$
$492$	6.00000	0.270501
$493$	−4.39230	−0.197819
$494$	−17.8564	−0.803398
$495$	0	0
$496$	2.00000	0.0898027
$497$	−2.53590	−0.113751
$498$	−3.21539	−0.144085
$499$	39.8564	1.78422	0.892109	−	0.451820i	$-0.149225\pi$
$499$	39.8564	1.78422	0.892109	+	0.451820i	$0.149225\pi$
$500$	−1.00000	−0.0447214
$501$	−10.1436	−0.453182
$502$	−12.0000	−0.535586
$503$	32.7846	1.46179	0.730897	−	0.682488i	$-0.239102\pi$
$503$	32.7846	1.46179	0.730897	+	0.682488i	$0.239102\pi$
$504$	−2.46410	−0.109760
$505$	19.8564	0.883598
$506$	0	0
$507$	−12.3397	−0.548027
$508$	−14.9282	−0.662332
$509$	24.9282	1.10492	0.552462	−	0.833538i	$-0.313689\pi$
$509$	24.9282	1.10492	0.552462	+	0.833538i	$0.313689\pi$
$510$	2.53590	0.112291
$511$	6.39230	0.282779
$512$	−1.00000	−0.0441942
$513$	−13.0718	−0.577134
$514$	6.33975	0.279634
$515$	8.39230	0.369809
$516$	1.46410	0.0644535
$517$	0	0
$518$	−2.73205	−0.120039
$519$	−9.46410	−0.415428
$520$	−5.46410	−0.239617
$521$	−10.3923	−0.455295	−0.227648	−	0.973744i	$-0.573103\pi$
$521$	−10.3923	−0.455295	−0.227648	+	0.973744i	$0.573103\pi$
$522$	3.12436	0.136749
$523$	−33.1769	−1.45073	−0.725363	−	0.688367i	$-0.758328\pi$
$523$	−33.1769	−1.45073	−0.725363	+	0.688367i	$0.758328\pi$
$524$	11.6603	0.509381
$525$	0.732051	0.0319493
$526$	−24.0000	−1.04645
$527$	−6.92820	−0.301797
$528$	0	0
$529$	−18.1769	−0.790301
$530$	−10.7321	−0.466170
$531$	17.0718	0.740853
$532$	3.26795	0.141684
$533$	44.7846	1.93984
$534$	−2.53590	−0.109739
$535$	−19.8564	−0.858467
$536$	4.00000	0.172774
$537$	4.39230	0.189542
$538$	−7.60770	−0.327991
$539$	0	0
$540$	−4.00000	−0.172133
$541$	17.2679	0.742407	0.371204	−	0.928552i	$-0.378945\pi$
$541$	17.2679	0.742407	0.371204	+	0.928552i	$0.378945\pi$
$542$	−20.3923	−0.875924
$543$	−10.2487	−0.439814
$544$	3.46410	0.148522
$545$	1.66025	0.0711175
$546$	4.00000	0.171184
$547$	12.7846	0.546630	0.273315	−	0.961925i	$-0.411880\pi$
$547$	12.7846	0.546630	0.273315	+	0.961925i	$0.411880\pi$
$548$	12.9282	0.552265
$549$	22.0000	0.938937
$550$	0	0
$551$	−4.14359	−0.176523
$552$	1.60770	0.0684280
$553$	−1.80385	−0.0767074
$554$	20.9282	0.889154
$555$	−2.00000	−0.0848953
$556$	3.66025	0.155229
$557$	46.3923	1.96571	0.982853	−	0.184393i	$-0.0590320\pi$
$557$	46.3923	1.96571	0.982853	+	0.184393i	$0.0590320\pi$
$558$	4.92820	0.208627
$559$	10.9282	0.462214
$560$	1.00000	0.0422577
$561$	0	0
$562$	1.60770	0.0678165
$563$	−18.9282	−0.797729	−0.398864	−	0.917010i	$-0.630596\pi$
$563$	−18.9282	−0.797729	−0.398864	+	0.917010i	$0.630596\pi$
$564$	5.07180	0.213561
$565$	19.8564	0.835365
$566$	23.7128	0.996724
$567$	−4.46410	−0.187475
$568$	−2.53590	−0.106404
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	2.39230	0.100203
$571$	−24.3923	−1.02079	−0.510393	−	0.859941i	$-0.670500\pi$
$571$	−24.3923	−1.02079	−0.510393	+	0.859941i	$0.670500\pi$
$572$	0	0
$573$	8.78461	0.366982
$574$	−8.19615	−0.342101
$575$	2.19615	0.0915859
$576$	−2.46410	−0.102671
$577$	−3.41154	−0.142024	−0.0710122	−	0.997475i	$-0.522623\pi$
$577$	−3.41154	−0.142024	−0.0710122	+	0.997475i	$0.522623\pi$
$578$	5.00000	0.207973
$579$	−6.14359	−0.255319
$580$	−1.26795	−0.0526487
$581$	4.39230	0.182224
$582$	−12.1436	−0.503368
$583$	0	0
$584$	6.39230	0.264515
$585$	−13.4641	−0.556672
$586$	14.5359	0.600472
$587$	−42.5885	−1.75781	−0.878907	−	0.476993i	$-0.841727\pi$
$587$	−42.5885	−1.75781	−0.878907	+	0.476993i	$0.841727\pi$
$588$	−0.732051	−0.0301893
$589$	−6.53590	−0.269307
$590$	−6.92820	−0.285230
$591$	17.7513	0.730190
$592$	−2.73205	−0.112287
$593$	48.2487	1.98134	0.990669	−	0.136293i	$-0.0435189\pi$
$593$	48.2487	1.98134	0.990669	+	0.136293i	$0.0435189\pi$
$594$	0	0
$595$	−3.46410	−0.142014
$596$	10.7321	0.439602
$597$	8.00000	0.327418
$598$	12.0000	0.490716
$599$	−25.1769	−1.02870	−0.514350	−	0.857580i	$-0.671967\pi$
$599$	−25.1769	−1.02870	−0.514350	+	0.857580i	$0.671967\pi$
$600$	0.732051	0.0298858
$601$	−39.5167	−1.61192	−0.805959	−	0.591971i	$-0.798350\pi$
$601$	−39.5167	−1.61192	−0.805959	+	0.591971i	$0.798350\pi$
$602$	−2.00000	−0.0815139
$603$	9.85641	0.401384
$604$	13.1244	0.534022
$605$	0	0
$606$	−14.5359	−0.590481
$607$	−7.07180	−0.287035	−0.143518	−	0.989648i	$-0.545841\pi$
$607$	−7.07180	−0.287035	−0.143518	+	0.989648i	$0.545841\pi$
$608$	3.26795	0.132533
$609$	0.928203	0.0376127
$610$	−8.92820	−0.361492
$611$	37.8564	1.53151
$612$	8.53590	0.345043
$613$	−27.1769	−1.09767	−0.548833	−	0.835932i	$-0.684928\pi$
$613$	−27.1769	−1.09767	−0.548833	+	0.835932i	$0.684928\pi$
$614$	3.60770	0.145595
$615$	−6.00000	−0.241943
$616$	0	0
$617$	−17.3205	−0.697297	−0.348649	−	0.937253i	$-0.613359\pi$
$617$	−17.3205	−0.697297	−0.348649	+	0.937253i	$0.613359\pi$
$618$	−6.14359	−0.247132
$619$	−12.7846	−0.513857	−0.256928	−	0.966430i	$-0.582710\pi$
$619$	−12.7846	−0.513857	−0.256928	+	0.966430i	$0.582710\pi$
$620$	−2.00000	−0.0803219
$621$	8.78461	0.352514
$622$	−11.0718	−0.443939
$623$	3.46410	0.138786
$624$	4.00000	0.160128
$625$	1.00000	0.0400000
$626$	−11.8038	−0.471777
$627$	0	0
$628$	11.4641	0.457467
$629$	9.46410	0.377358
$630$	2.46410	0.0981722
$631$	−33.0718	−1.31657	−0.658284	−	0.752770i	$-0.728717\pi$
$631$	−33.0718	−1.31657	−0.658284	+	0.752770i	$0.728717\pi$
$632$	−1.80385	−0.0717532
$633$	9.56922	0.380342
$634$	0.588457	0.0233706
$635$	14.9282	0.592408
$636$	7.85641	0.311527
$637$	−5.46410	−0.216496
$638$	0	0
$639$	−6.24871	−0.247195
$640$	1.00000	0.0395285
$641$	47.5692	1.87887	0.939436	−	0.342725i	$-0.111350\pi$
$641$	47.5692	1.87887	0.939436	+	0.342725i	$0.111350\pi$
$642$	14.5359	0.573686
$643$	36.7321	1.44857	0.724285	−	0.689500i	$-0.242170\pi$
$643$	36.7321	1.44857	0.724285	+	0.689500i	$0.242170\pi$
$644$	−2.19615	−0.0865405
$645$	−1.46410	−0.0576489
$646$	−11.3205	−0.445399
$647$	−33.4641	−1.31561	−0.657805	−	0.753188i	$-0.728515\pi$
$647$	−33.4641	−1.31561	−0.657805	+	0.753188i	$0.728515\pi$
$648$	−4.46410	−0.175366
$649$	0	0
$650$	5.46410	0.214320
$651$	1.46410	0.0573827
$652$	−4.00000	−0.156652
$653$	−5.66025	−0.221503	−0.110751	−	0.993848i	$-0.535326\pi$
$653$	−5.66025	−0.221503	−0.110751	+	0.993848i	$0.535326\pi$
$654$	−1.21539	−0.0475255
$655$	−11.6603	−0.455604
$656$	−8.19615	−0.320006
$657$	15.7513	0.614516
$658$	−6.92820	−0.270089
$659$	18.2487	0.710869	0.355434	−	0.934701i	$-0.384333\pi$
$659$	18.2487	0.710869	0.355434	+	0.934701i	$0.384333\pi$
$660$	0	0
$661$	−46.0000	−1.78919	−0.894596	−	0.446875i	$-0.852537\pi$
$661$	−46.0000	−1.78919	−0.894596	+	0.446875i	$0.852537\pi$
$662$	−22.7846	−0.885549
$663$	−13.8564	−0.538138
$664$	4.39230	0.170454
$665$	−3.26795	−0.126726
$666$	−6.73205	−0.260862
$667$	2.78461	0.107821
$668$	13.8564	0.536120
$669$	13.5692	0.524616
$670$	−4.00000	−0.154533
$671$	0	0
$672$	−0.732051	−0.0282395
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	−18.7846	−0.723556
$675$	4.00000	0.153960
$676$	16.8564	0.648323
$677$	31.8564	1.22434	0.612171	−	0.790726i	$-0.290297\pi$
$677$	31.8564	1.22434	0.612171	+	0.790726i	$0.290297\pi$
$678$	−14.5359	−0.558248
$679$	16.5885	0.636607
$680$	−3.46410	−0.132842
$681$	−5.07180	−0.194352
$682$	0	0
$683$	−42.2487	−1.61660	−0.808301	−	0.588769i	$-0.799613\pi$
$683$	−42.2487	−1.61660	−0.808301	+	0.588769i	$0.799613\pi$
$684$	8.05256	0.307897
$685$	−12.9282	−0.493961
$686$	1.00000	0.0381802
$687$	−2.64102	−0.100761
$688$	−2.00000	−0.0762493
$689$	58.6410	2.23404
$690$	−1.60770	−0.0612039
$691$	−6.53590	−0.248637	−0.124319	−	0.992242i	$-0.539675\pi$
$691$	−6.53590	−0.248637	−0.124319	+	0.992242i	$0.539675\pi$
$692$	12.9282	0.491457
$693$	0	0
$694$	−36.9282	−1.40178
$695$	−3.66025	−0.138841
$696$	0.928203	0.0351835
$697$	28.3923	1.07544
$698$	−26.3923	−0.998963
$699$	14.5359	0.549798
$700$	−1.00000	−0.0377964
$701$	23.9090	0.903029	0.451515	−	0.892264i	$-0.350884\pi$
$701$	23.9090	0.903029	0.451515	+	0.892264i	$0.350884\pi$
$702$	21.8564	0.824917
$703$	8.92820	0.336734
$704$	0	0
$705$	−5.07180	−0.191015
$706$	−3.80385	−0.143160
$707$	19.8564	0.746777
$708$	5.07180	0.190610
$709$	−9.32051	−0.350039	−0.175020	−	0.984565i	$-0.555999\pi$
$709$	−9.32051	−0.350039	−0.175020	+	0.984565i	$0.555999\pi$
$710$	2.53590	0.0951706
$711$	−4.44486	−0.166695
$712$	3.46410	0.129823
$713$	4.39230	0.164493
$714$	2.53590	0.0949036
$715$	0	0
$716$	−6.00000	−0.224231
$717$	3.46410	0.129369
$718$	4.05256	0.151240
$719$	27.7128	1.03351	0.516757	−	0.856132i	$-0.327139\pi$
$719$	27.7128	1.03351	0.516757	+	0.856132i	$0.327139\pi$
$720$	2.46410	0.0918316
$721$	8.39230	0.312546
$722$	8.32051	0.309657
$723$	4.92820	0.183282
$724$	14.0000	0.520306
$725$	1.26795	0.0470905
$726$	0	0
$727$	−4.00000	−0.148352	−0.0741759	−	0.997245i	$-0.523633\pi$
$727$	−4.00000	−0.148352	−0.0741759	+	0.997245i	$0.523633\pi$
$728$	−5.46410	−0.202513
$729$	−2.21539	−0.0820515
$730$	−6.39230	−0.236590
$731$	6.92820	0.256249
$732$	6.53590	0.241574
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	8.39230	0.309766
$735$	0.732051	0.0270021
$736$	−2.19615	−0.0809513
$737$	0	0
$738$	−20.1962	−0.743431
$739$	43.7128	1.60800	0.804001	−	0.594628i	$-0.202701\pi$
$739$	43.7128	1.60800	0.804001	+	0.594628i	$0.202701\pi$
$740$	2.73205	0.100432
$741$	−13.0718	−0.480204
$742$	−10.7321	−0.393986
$743$	−8.78461	−0.322276	−0.161138	−	0.986932i	$-0.551516\pi$
$743$	−8.78461	−0.322276	−0.161138	+	0.986932i	$0.551516\pi$
$744$	1.46410	0.0536766
$745$	−10.7321	−0.393192
$746$	−2.39230	−0.0875885
$747$	10.8231	0.395996
$748$	0	0
$749$	−19.8564	−0.725537
$750$	−0.732051	−0.0267307
$751$	−32.3923	−1.18201	−0.591006	−	0.806667i	$-0.701269\pi$
$751$	−32.3923	−1.18201	−0.591006	+	0.806667i	$0.701269\pi$
$752$	−6.92820	−0.252646
$753$	−8.78461	−0.320129
$754$	6.92820	0.252310
$755$	−13.1244	−0.477644
$756$	−4.00000	−0.145479
$757$	29.3731	1.06758	0.533791	−	0.845616i	$-0.320767\pi$
$757$	29.3731	1.06758	0.533791	+	0.845616i	$0.320767\pi$
$758$	−33.8564	−1.22972
$759$	0	0
$760$	−3.26795	−0.118541
$761$	29.6603	1.07518	0.537592	−	0.843205i	$-0.319334\pi$
$761$	29.6603	1.07518	0.537592	+	0.843205i	$0.319334\pi$
$762$	−10.9282	−0.395887
$763$	1.66025	0.0601052
$764$	−12.0000	−0.434145
$765$	−8.53590	−0.308616
$766$	26.5359	0.958781
$767$	37.8564	1.36692
$768$	−0.732051	−0.0264156
$769$	7.80385	0.281414	0.140707	−	0.990051i	$-0.455062\pi$
$769$	7.80385	0.281414	0.140707	+	0.990051i	$0.455062\pi$
$770$	0	0
$771$	4.64102	0.167142
$772$	8.39230	0.302046
$773$	12.9282	0.464995	0.232498	−	0.972597i	$-0.425310\pi$
$773$	12.9282	0.464995	0.232498	+	0.972597i	$0.425310\pi$
$774$	−4.92820	−0.177141
$775$	2.00000	0.0718421
$776$	16.5885	0.595491
$777$	−2.00000	−0.0717496
$778$	22.3923	0.802803
$779$	26.7846	0.959658
$780$	−4.00000	−0.143223
$781$	0	0
$782$	7.60770	0.272051
$783$	5.07180	0.181251
$784$	1.00000	0.0357143
$785$	−11.4641	−0.409171
$786$	8.53590	0.304465
$787$	−45.8564	−1.63460	−0.817302	−	0.576209i	$-0.804531\pi$
$787$	−45.8564	−1.63460	−0.817302	+	0.576209i	$0.804531\pi$
$788$	−24.2487	−0.863825
$789$	−17.5692	−0.625481
$790$	1.80385	0.0641780
$791$	19.8564	0.706013
$792$	0	0
$793$	48.7846	1.73239
$794$	−9.60770	−0.340964
$795$	−7.85641	−0.278638
$796$	−10.9282	−0.387340
$797$	−46.3923	−1.64330	−0.821650	−	0.569993i	$-0.806946\pi$
$797$	−46.3923	−1.64330	−0.821650	+	0.569993i	$0.806946\pi$
$798$	2.39230	0.0846867
$799$	24.0000	0.849059
$800$	−1.00000	−0.0353553
$801$	8.53590	0.301601
$802$	−9.46410	−0.334189
$803$	0	0
$804$	2.92820	0.103270
$805$	2.19615	0.0774042
$806$	10.9282	0.384930
$807$	−5.56922	−0.196046
$808$	19.8564	0.698546
$809$	−31.1769	−1.09612	−0.548061	−	0.836438i	$-0.684634\pi$
$809$	−31.1769	−1.09612	−0.548061	+	0.836438i	$0.684634\pi$
$810$	4.46410	0.156853
$811$	18.8756	0.662814	0.331407	−	0.943488i	$-0.392477\pi$
$811$	18.8756	0.662814	0.331407	+	0.943488i	$0.392477\pi$
$812$	−1.26795	−0.0444963
$813$	−14.9282	−0.523555
$814$	0	0
$815$	4.00000	0.140114
$816$	2.53590	0.0887742
$817$	6.53590	0.228662
$818$	35.1244	1.22809
$819$	−13.4641	−0.470474
$820$	8.19615	0.286222
$821$	47.9090	1.67203	0.836017	−	0.548703i	$-0.184878\pi$
$821$	47.9090	1.67203	0.836017	+	0.548703i	$0.184878\pi$
$822$	9.46410	0.330098
$823$	−25.1244	−0.875780	−0.437890	−	0.899029i	$-0.644274\pi$
$823$	−25.1244	−0.875780	−0.437890	+	0.899029i	$0.644274\pi$
$824$	8.39230	0.292360
$825$	0	0
$826$	−6.92820	−0.241063
$827$	−1.85641	−0.0645536	−0.0322768	−	0.999479i	$-0.510276\pi$
$827$	−1.85641	−0.0645536	−0.0322768	+	0.999479i	$0.510276\pi$
$828$	−5.41154	−0.188064
$829$	−46.2487	−1.60628	−0.803142	−	0.595788i	$-0.796840\pi$
$829$	−46.2487	−1.60628	−0.803142	+	0.595788i	$0.796840\pi$
$830$	−4.39230	−0.152459
$831$	15.3205	0.531463
$832$	−5.46410	−0.189434
$833$	−3.46410	−0.120024
$834$	2.67949	0.0927832
$835$	−13.8564	−0.479521
$836$	0	0
$837$	8.00000	0.276520
$838$	−17.0718	−0.589735
$839$	−4.14359	−0.143053	−0.0715264	−	0.997439i	$-0.522787\pi$
$839$	−4.14359	−0.143053	−0.0715264	+	0.997439i	$0.522787\pi$
$840$	0.732051	0.0252582
$841$	−27.3923	−0.944562
$842$	8.14359	0.280647
$843$	1.17691	0.0405351
$844$	−13.0718	−0.449950
$845$	−16.8564	−0.579878
$846$	−17.0718	−0.586940
$847$	0	0
$848$	−10.7321	−0.368540
$849$	17.3590	0.595759
$850$	3.46410	0.118818
$851$	−6.00000	−0.205677
$852$	−1.85641	−0.0635994
$853$	13.2154	0.452486	0.226243	−	0.974071i	$-0.427356\pi$
$853$	13.2154	0.452486	0.226243	+	0.974071i	$0.427356\pi$
$854$	−8.92820	−0.305517
$855$	−8.05256	−0.275392
$856$	−19.8564	−0.678678
$857$	20.5359	0.701493	0.350746	−	0.936470i	$-0.385928\pi$
$857$	20.5359	0.701493	0.350746	+	0.936470i	$0.385928\pi$
$858$	0	0
$859$	28.7846	0.982118	0.491059	−	0.871126i	$-0.336610\pi$
$859$	28.7846	0.982118	0.491059	+	0.871126i	$0.336610\pi$
$860$	2.00000	0.0681994
$861$	−6.00000	−0.204479
$862$	−21.1244	−0.719498
$863$	37.5167	1.27708	0.638541	−	0.769588i	$-0.279538\pi$
$863$	37.5167	1.27708	0.638541	+	0.769588i	$0.279538\pi$
$864$	−4.00000	−0.136083
$865$	−12.9282	−0.439572
$866$	7.80385	0.265186
$867$	3.66025	0.124309
$868$	−2.00000	−0.0678844
$869$	0	0
$870$	−0.928203	−0.0314690
$871$	21.8564	0.740576
$872$	1.66025	0.0562233
$873$	40.8756	1.38343
$874$	7.17691	0.242763
$875$	1.00000	0.0338062
$876$	4.67949	0.158105
$877$	−13.3205	−0.449802	−0.224901	−	0.974382i	$-0.572206\pi$
$877$	−13.3205	−0.449802	−0.224901	+	0.974382i	$0.572206\pi$
$878$	−22.2487	−0.750858
$879$	10.6410	0.358913
$880$	0	0
$881$	24.9282	0.839853	0.419926	−	0.907558i	$-0.362056\pi$
$881$	24.9282	0.839853	0.419926	+	0.907558i	$0.362056\pi$
$882$	2.46410	0.0829706
$883$	−56.3923	−1.89775	−0.948876	−	0.315649i	$-0.897778\pi$
$883$	−56.3923	−1.89775	−0.948876	+	0.315649i	$0.897778\pi$
$884$	18.9282	0.636624
$885$	−5.07180	−0.170487
$886$	0	0
$887$	13.8564	0.465253	0.232626	−	0.972566i	$-0.425268\pi$
$887$	13.8564	0.465253	0.232626	+	0.972566i	$0.425268\pi$
$888$	−2.00000	−0.0671156
$889$	14.9282	0.500676
$890$	−3.46410	−0.116117
$891$	0	0
$892$	−18.5359	−0.620628
$893$	22.6410	0.757653
$894$	7.85641	0.262758
$895$	6.00000	0.200558
$896$	1.00000	0.0334077
$897$	8.78461	0.293310
$898$	−19.6077	−0.654317
$899$	2.53590	0.0845769
$900$	−2.46410	−0.0821367
$901$	37.1769	1.23854
$902$	0	0
$903$	−1.46410	−0.0487223
$904$	19.8564	0.660414
$905$	−14.0000	−0.465376
$906$	9.60770	0.319194
$907$	59.0333	1.96017	0.980085	−	0.198580i	$-0.0636331\pi$
$907$	59.0333	1.96017	0.980085	+	0.198580i	$0.0636331\pi$
$908$	6.92820	0.229920
$909$	48.9282	1.62285
$910$	5.46410	0.181133
$911$	2.53590	0.0840181	0.0420090	−	0.999117i	$-0.486624\pi$
$911$	2.53590	0.0840181	0.0420090	+	0.999117i	$0.486624\pi$
$912$	2.39230	0.0792171
$913$	0	0
$914$	2.00000	0.0661541
$915$	−6.53590	−0.216070
$916$	3.60770	0.119202
$917$	−11.6603	−0.385056
$918$	13.8564	0.457330
$919$	−47.3731	−1.56269	−0.781347	−	0.624097i	$-0.785467\pi$
$919$	−47.3731	−1.56269	−0.781347	+	0.624097i	$0.785467\pi$
$920$	2.19615	0.0724050
$921$	2.64102	0.0870244
$922$	1.60770	0.0529466
$923$	−13.8564	−0.456089
$924$	0	0
$925$	−2.73205	−0.0898293
$926$	17.5167	0.575633
$927$	20.6795	0.679204
$928$	−1.26795	−0.0416225
$929$	−46.3923	−1.52208	−0.761041	−	0.648704i	$-0.775311\pi$
$929$	−46.3923	−1.52208	−0.761041	+	0.648704i	$0.775311\pi$
$930$	−1.46410	−0.0480098
$931$	−3.26795	−0.107103
$932$	−19.8564	−0.650418
$933$	−8.10512	−0.265350
$934$	−4.05256	−0.132604
$935$	0	0
$936$	−13.4641	−0.440088
$937$	42.7846	1.39771	0.698856	−	0.715262i	$-0.253693\pi$
$937$	42.7846	1.39771	0.698856	+	0.715262i	$0.253693\pi$
$938$	−4.00000	−0.130605
$939$	−8.64102	−0.281989
$940$	6.92820	0.225973
$941$	−26.7846	−0.873153	−0.436577	−	0.899667i	$-0.643809\pi$
$941$	−26.7846	−0.873153	−0.436577	+	0.899667i	$0.643809\pi$
$942$	8.39230	0.273436
$943$	−18.0000	−0.586161
$944$	−6.92820	−0.225494
$945$	4.00000	0.130120
$946$	0	0
$947$	12.6795	0.412028	0.206014	−	0.978549i	$-0.433951\pi$
$947$	12.6795	0.412028	0.206014	+	0.978549i	$0.433951\pi$
$948$	−1.32051	−0.0428881
$949$	34.9282	1.13382
$950$	3.26795	0.106026
$951$	0.430781	0.0139690
$952$	−3.46410	−0.112272
$953$	−33.4641	−1.08401	−0.542004	−	0.840376i	$-0.682334\pi$
$953$	−33.4641	−1.08401	−0.542004	+	0.840376i	$0.682334\pi$
$954$	−26.4449	−0.856184
$955$	12.0000	0.388311
$956$	−4.73205	−0.153045
$957$	0	0
$958$	8.78461	0.283818
$959$	−12.9282	−0.417473
$960$	0.732051	0.0236268
$961$	−27.0000	−0.870968
$962$	−14.9282	−0.481305
$963$	−48.9282	−1.57669
$964$	−6.73205	−0.216825
$965$	−8.39230	−0.270158
$966$	−1.60770	−0.0517267
$967$	−13.0718	−0.420361	−0.210180	−	0.977663i	$-0.567405\pi$
$967$	−13.0718	−0.420361	−0.210180	+	0.977663i	$0.567405\pi$
$968$	0	0
$969$	−8.28719	−0.266223
$970$	−16.5885	−0.532623
$971$	29.0718	0.932958	0.466479	−	0.884532i	$-0.345522\pi$
$971$	29.0718	0.932958	0.466479	+	0.884532i	$0.345522\pi$
$972$	−15.2679	−0.489720
$973$	−3.66025	−0.117342
$974$	6.19615	0.198538
$975$	4.00000	0.128103
$976$	−8.92820	−0.285785
$977$	21.7128	0.694654	0.347327	−	0.937744i	$-0.387089\pi$
$977$	21.7128	0.694654	0.347327	+	0.937744i	$0.387089\pi$
$978$	−2.92820	−0.0936336
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	4.09103	0.130617
$982$	27.7128	0.884351
$983$	25.1769	0.803019	0.401509	−	0.915855i	$-0.368486\pi$
$983$	25.1769	0.803019	0.401509	+	0.915855i	$0.368486\pi$
$984$	−6.00000	−0.191273
$985$	24.2487	0.772628
$986$	4.39230	0.139879
$987$	−5.07180	−0.161437
$988$	17.8564	0.568088
$989$	−4.39230	−0.139667
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	−2.00000	−0.0635001
$993$	−16.6795	−0.529308
$994$	2.53590	0.0804338
$995$	10.9282	0.346447
$996$	3.21539	0.101884
$997$	37.7128	1.19438	0.597188	−	0.802101i	$-0.296284\pi$
$997$	37.7128	1.19438	0.597188	+	0.802101i	$0.296284\pi$
$998$	−39.8564	−1.26163
$999$	−10.9282	−0.345753

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.br.1.1		2
11.10	odd	2		770.2.a.j.1.1	✓	2
33.32	even	2		6930.2.a.bv.1.2		2
44.43	even	2		6160.2.a.t.1.2		2
55.32	even	4		3850.2.c.x.1849.4		4
55.43	even	4		3850.2.c.x.1849.1		4
55.54	odd	2		3850.2.a.bd.1.2		2
77.76	even	2		5390.2.a.bs.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.j.1.1	✓	2	11.10	odd	2
3850.2.a.bd.1.2		2	55.54	odd	2
3850.2.c.x.1849.1		4	55.43	even	4
3850.2.c.x.1849.4		4	55.32	even	4
5390.2.a.bs.1.2		2	77.76	even	2
6160.2.a.t.1.2		2	44.43	even	2
6930.2.a.bv.1.2		2	33.32	even	2
8470.2.a.br.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.732051 q^{3} +1.00000 q^{4} -1.00000 q^{5} +0.732051 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.46410 q^{9} +1.00000 q^{10} -0.732051 q^{12} -5.46410 q^{13} +1.00000 q^{14} +0.732051 q^{15} +1.00000 q^{16} -3.46410 q^{17} +2.46410 q^{18} -3.26795 q^{19} -1.00000 q^{20} +0.732051 q^{21} +2.19615 q^{23} +0.732051 q^{24} +1.00000 q^{25} +5.46410 q^{26} +4.00000 q^{27} -1.00000 q^{28} +1.26795 q^{29} -0.732051 q^{30} +2.00000 q^{31} -1.00000 q^{32} +3.46410 q^{34} +1.00000 q^{35} -2.46410 q^{36} -2.73205 q^{37} +3.26795 q^{38} +4.00000 q^{39} +1.00000 q^{40} -8.19615 q^{41} -0.732051 q^{42} -2.00000 q^{43} +2.46410 q^{45} -2.19615 q^{46} -6.92820 q^{47} -0.732051 q^{48} +1.00000 q^{49} -1.00000 q^{50} +2.53590 q^{51} -5.46410 q^{52} -10.7321 q^{53} -4.00000 q^{54} +1.00000 q^{56} +2.39230 q^{57} -1.26795 q^{58} -6.92820 q^{59} +0.732051 q^{60} -8.92820 q^{61} -2.00000 q^{62} +2.46410 q^{63} +1.00000 q^{64} +5.46410 q^{65} -4.00000 q^{67} -3.46410 q^{68} -1.60770 q^{69} -1.00000 q^{70} +2.53590 q^{71} +2.46410 q^{72} -6.39230 q^{73} +2.73205 q^{74} -0.732051 q^{75} -3.26795 q^{76} -4.00000 q^{78} +1.80385 q^{79} -1.00000 q^{80} +4.46410 q^{81} +8.19615 q^{82} -4.39230 q^{83} +0.732051 q^{84} +3.46410 q^{85} +2.00000 q^{86} -0.928203 q^{87} -3.46410 q^{89} -2.46410 q^{90} +5.46410 q^{91} +2.19615 q^{92} -1.46410 q^{93} +6.92820 q^{94} +3.26795 q^{95} +0.732051 q^{96} -16.5885 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} - 4 q^{13} + 2 q^{14} - 2 q^{15} + 2 q^{16} - 2 q^{18} - 10 q^{19} - 2 q^{20} - 2 q^{21} - 6 q^{23} - 2 q^{24} + 2 q^{25} + 4 q^{26} + 8 q^{27} - 2 q^{28} + 6 q^{29} + 2 q^{30} + 4 q^{31} - 2 q^{32} + 2 q^{35} + 2 q^{36} - 2 q^{37} + 10 q^{38} + 8 q^{39} + 2 q^{40} - 6 q^{41} + 2 q^{42} - 4 q^{43} - 2 q^{45} + 6 q^{46} + 2 q^{48} + 2 q^{49} - 2 q^{50} + 12 q^{51} - 4 q^{52} - 18 q^{53} - 8 q^{54} + 2 q^{56} - 16 q^{57} - 6 q^{58} - 2 q^{60} - 4 q^{61} - 4 q^{62} - 2 q^{63} + 2 q^{64} + 4 q^{65} - 8 q^{67} - 24 q^{69} - 2 q^{70} + 12 q^{71} - 2 q^{72} + 8 q^{73} + 2 q^{74} + 2 q^{75} - 10 q^{76} - 8 q^{78} + 14 q^{79} - 2 q^{80} + 2 q^{81} + 6 q^{82} + 12 q^{83} - 2 q^{84} + 4 q^{86} + 12 q^{87} + 2 q^{90} + 4 q^{91} - 6 q^{92} + 4 q^{93} + 10 q^{95} - 2 q^{96} - 2 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^12 - 4 * q^13 + 2 * q^14 - 2 * q^15 + 2 * q^16 - 2 * q^18 - 10 * q^19 - 2 * q^20 - 2 * q^21 - 6 * q^23 - 2 * q^24 + 2 * q^25 + 4 * q^26 + 8 * q^27 - 2 * q^28 + 6 * q^29 + 2 * q^30 + 4 * q^31 - 2 * q^32 + 2 * q^35 + 2 * q^36 - 2 * q^37 + 10 * q^38 + 8 * q^39 + 2 * q^40 - 6 * q^41 + 2 * q^42 - 4 * q^43 - 2 * q^45 + 6 * q^46 + 2 * q^48 + 2 * q^49 - 2 * q^50 + 12 * q^51 - 4 * q^52 - 18 * q^53 - 8 * q^54 + 2 * q^56 - 16 * q^57 - 6 * q^58 - 2 * q^60 - 4 * q^61 - 4 * q^62 - 2 * q^63 + 2 * q^64 + 4 * q^65 - 8 * q^67 - 24 * q^69 - 2 * q^70 + 12 * q^71 - 2 * q^72 + 8 * q^73 + 2 * q^74 + 2 * q^75 - 10 * q^76 - 8 * q^78 + 14 * q^79 - 2 * q^80 + 2 * q^81 + 6 * q^82 + 12 * q^83 - 2 * q^84 + 4 * q^86 + 12 * q^87 + 2 * q^90 + 4 * q^91 - 6 * q^92 + 4 * q^93 + 10 * q^95 - 2 * q^96 - 2 * q^97 - 2 * q^98

Introduction

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