LMFDB - Embedded newform 8470.2.a.df.1.3

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:	$6$
Coefficient field:	6.6.10784448.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 11x^{4} - 4x^{3} + 31x^{2} + 22x - 2 $ x^6 - 11x^4 - 4x^3 + 31x^2 + 22x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.84763$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +0.529317 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.529317 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.71982 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.529317 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.529317 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.71982 q^{9} +1.00000 q^{10} +0.529317 q^{12} -4.81243 q^{13} +1.00000 q^{14} +0.529317 q^{15} +1.00000 q^{16} +6.02760 q^{17} -2.71982 q^{18} +3.59370 q^{19} +1.00000 q^{20} +0.529317 q^{21} -7.06157 q^{23} +0.529317 q^{24} +1.00000 q^{25} -4.81243 q^{26} -3.02760 q^{27} +1.00000 q^{28} +0.353515 q^{29} +0.529317 q^{30} -2.24914 q^{31} +1.00000 q^{32} +6.02760 q^{34} +1.00000 q^{35} -2.71982 q^{36} +1.64649 q^{37} +3.59370 q^{38} -2.54730 q^{39} +1.00000 q^{40} +9.35973 q^{41} +0.529317 q^{42} +9.13818 q^{43} -2.71982 q^{45} -7.06157 q^{46} +10.7634 q^{47} +0.529317 q^{48} +1.00000 q^{49} +1.00000 q^{50} +3.19051 q^{51} -4.81243 q^{52} +4.64885 q^{53} -3.02760 q^{54} +1.00000 q^{56} +1.90221 q^{57} +0.353515 q^{58} +2.78483 q^{59} +0.529317 q^{60} -14.6733 q^{61} -2.24914 q^{62} -2.71982 q^{63} +1.00000 q^{64} -4.81243 q^{65} +9.43079 q^{67} +6.02760 q^{68} -3.73781 q^{69} +1.00000 q^{70} +12.2276 q^{71} -2.71982 q^{72} +5.95077 q^{73} +1.64649 q^{74} +0.529317 q^{75} +3.59370 q^{76} -2.54730 q^{78} +8.04631 q^{79} +1.00000 q^{80} +6.55691 q^{81} +9.35973 q^{82} -16.6025 q^{83} +0.529317 q^{84} +6.02760 q^{85} +9.13818 q^{86} +0.187121 q^{87} +16.0923 q^{89} -2.71982 q^{90} -4.81243 q^{91} -7.06157 q^{92} -1.19051 q^{93} +10.7634 q^{94} +3.59370 q^{95} +0.529317 q^{96} -1.23078 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 4 q^{3} + 6 q^{4} + 6 q^{5} + 4 q^{6} + 6 q^{7} + 6 q^{8} + 2 q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 + 4 * q^3 + 6 * q^4 + 6 * q^5 + 4 * q^6 + 6 * q^7 + 6 * q^8 + 2 * q^9	$ 6 q + 6 q^{2} + 4 q^{3} + 6 q^{4} + 6 q^{5} + 4 q^{6} + 6 q^{7} + 6 q^{8} + 2 q^{9} + 6 q^{10} + 4 q^{12} + 6 q^{14} + 4 q^{15} + 6 q^{16} + 2 q^{17} + 2 q^{18} + 6 q^{20} + 4 q^{21} + 4 q^{23} + 4 q^{24} + 6 q^{25} + 16 q^{27} + 6 q^{28} + 8 q^{29} + 4 q^{30} + 4 q^{31} + 6 q^{32} + 2 q^{34} + 6 q^{35} + 2 q^{36} + 4 q^{37} + 6 q^{40} + 12 q^{41} + 4 q^{42} - 6 q^{43} + 2 q^{45} + 4 q^{46} + 16 q^{47} + 4 q^{48} + 6 q^{49} + 6 q^{50} + 12 q^{53} + 16 q^{54} + 6 q^{56} + 8 q^{57} + 8 q^{58} + 22 q^{59} + 4 q^{60} - 4 q^{61} + 4 q^{62} + 2 q^{63} + 6 q^{64} + 20 q^{67} + 2 q^{68} + 12 q^{69} + 6 q^{70} + 14 q^{71} + 2 q^{72} + 18 q^{73} + 4 q^{74} + 4 q^{75} + 32 q^{79} + 6 q^{80} + 6 q^{81} + 12 q^{82} - 16 q^{83} + 4 q^{84} + 2 q^{85} - 6 q^{86} - 4 q^{87} + 4 q^{89} + 2 q^{90} + 4 q^{92} + 12 q^{93} + 16 q^{94} + 4 q^{96} + 4 q^{97} + 6 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 + 4 * q^3 + 6 * q^4 + 6 * q^5 + 4 * q^6 + 6 * q^7 + 6 * q^8 + 2 * q^9 + 6 * q^10 + 4 * q^12 + 6 * q^14 + 4 * q^15 + 6 * q^16 + 2 * q^17 + 2 * q^18 + 6 * q^20 + 4 * q^21 + 4 * q^23 + 4 * q^24 + 6 * q^25 + 16 * q^27 + 6 * q^28 + 8 * q^29 + 4 * q^30 + 4 * q^31 + 6 * q^32 + 2 * q^34 + 6 * q^35 + 2 * q^36 + 4 * q^37 + 6 * q^40 + 12 * q^41 + 4 * q^42 - 6 * q^43 + 2 * q^45 + 4 * q^46 + 16 * q^47 + 4 * q^48 + 6 * q^49 + 6 * q^50 + 12 * q^53 + 16 * q^54 + 6 * q^56 + 8 * q^57 + 8 * q^58 + 22 * q^59 + 4 * q^60 - 4 * q^61 + 4 * q^62 + 2 * q^63 + 6 * q^64 + 20 * q^67 + 2 * q^68 + 12 * q^69 + 6 * q^70 + 14 * q^71 + 2 * q^72 + 18 * q^73 + 4 * q^74 + 4 * q^75 + 32 * q^79 + 6 * q^80 + 6 * q^81 + 12 * q^82 - 16 * q^83 + 4 * q^84 + 2 * q^85 - 6 * q^86 - 4 * q^87 + 4 * q^89 + 2 * q^90 + 4 * q^92 + 12 * q^93 + 16 * q^94 + 4 * q^96 + 4 * q^97 + 6 * 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
$3$	0.529317	0.305601	0.152801	−	0.988257i	$-0.451171\pi$
$3$	0.529317	0.305601	0.152801	+	0.988257i	$0.451171\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.529317	0.216093
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	−2.71982	−0.906608
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	0.529317	0.152801
$13$	−4.81243	−1.33473	−0.667364	−	0.744732i	$-0.732577\pi$
$13$	−4.81243	−1.33473	−0.667364	+	0.744732i	$0.732577\pi$
$14$	1.00000	0.267261
$15$	0.529317	0.136669
$16$	1.00000	0.250000
$17$	6.02760	1.46191	0.730954	−	0.682427i	$-0.239076\pi$
$17$	6.02760	1.46191	0.730954	+	0.682427i	$0.239076\pi$
$18$	−2.71982	−0.641069
$19$	3.59370	0.824452	0.412226	−	0.911082i	$-0.364751\pi$
$19$	3.59370	0.824452	0.412226	+	0.911082i	$0.364751\pi$
$20$	1.00000	0.223607
$21$	0.529317	0.115506
$22$	0	0
$23$	−7.06157	−1.47244	−0.736220	−	0.676743i	$-0.763391\pi$
$23$	−7.06157	−1.47244	−0.736220	+	0.676743i	$0.763391\pi$
$24$	0.529317	0.108046
$25$	1.00000	0.200000
$26$	−4.81243	−0.943795
$27$	−3.02760	−0.582661
$28$	1.00000	0.188982
$29$	0.353515	0.0656461	0.0328230	−	0.999461i	$-0.489550\pi$
$29$	0.353515	0.0656461	0.0328230	+	0.999461i	$0.489550\pi$
$30$	0.529317	0.0966395
$31$	−2.24914	−0.403958	−0.201979	−	0.979390i	$-0.564737\pi$
$31$	−2.24914	−0.403958	−0.201979	+	0.979390i	$0.564737\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.02760	1.03372
$35$	1.00000	0.169031
$36$	−2.71982	−0.453304
$37$	1.64649	0.270680	0.135340	−	0.990799i	$-0.456787\pi$
$37$	1.64649	0.270680	0.135340	+	0.990799i	$0.456787\pi$
$38$	3.59370	0.582976
$39$	−2.54730	−0.407894
$40$	1.00000	0.158114
$41$	9.35973	1.46174	0.730872	−	0.682515i	$-0.239114\pi$
$41$	9.35973	1.46174	0.730872	+	0.682515i	$0.239114\pi$
$42$	0.529317	0.0816753
$43$	9.13818	1.39356	0.696780	−	0.717285i	$-0.254615\pi$
$43$	9.13818	1.39356	0.696780	+	0.717285i	$0.254615\pi$
$44$	0	0
$45$	−2.71982	−0.405447
$46$	−7.06157	−1.04117
$47$	10.7634	1.57000	0.785002	−	0.619493i	$-0.212662\pi$
$47$	10.7634	1.57000	0.785002	+	0.619493i	$0.212662\pi$
$48$	0.529317	0.0764003
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	3.19051	0.446760
$52$	−4.81243	−0.667364
$53$	4.64885	0.638569	0.319285	−	0.947659i	$-0.396557\pi$
$53$	4.64885	0.638569	0.319285	+	0.947659i	$0.396557\pi$
$54$	−3.02760	−0.412004
$55$	0	0
$56$	1.00000	0.133631
$57$	1.90221	0.251954
$58$	0.353515	0.0464188
$59$	2.78483	0.362554	0.181277	−	0.983432i	$-0.441977\pi$
$59$	2.78483	0.362554	0.181277	+	0.983432i	$0.441977\pi$
$60$	0.529317	0.0683345
$61$	−14.6733	−1.87872	−0.939359	−	0.342935i	$-0.888579\pi$
$61$	−14.6733	−1.87872	−0.939359	+	0.342935i	$0.888579\pi$
$62$	−2.24914	−0.285641
$63$	−2.71982	−0.342666
$64$	1.00000	0.125000
$65$	−4.81243	−0.596908
$66$	0	0
$67$	9.43079	1.15215	0.576077	−	0.817395i	$-0.304583\pi$
$67$	9.43079	1.15215	0.576077	+	0.817395i	$0.304583\pi$
$68$	6.02760	0.730954
$69$	−3.73781	−0.449979
$70$	1.00000	0.119523
$71$	12.2276	1.45115	0.725575	−	0.688143i	$-0.241574\pi$
$71$	12.2276	1.45115	0.725575	+	0.688143i	$0.241574\pi$
$72$	−2.71982	−0.320534
$73$	5.95077	0.696486	0.348243	−	0.937404i	$-0.386778\pi$
$73$	5.95077	0.696486	0.348243	+	0.937404i	$0.386778\pi$
$74$	1.64649	0.191400
$75$	0.529317	0.0611202
$76$	3.59370	0.412226
$77$	0	0
$78$	−2.54730	−0.288425
$79$	8.04631	0.905280	0.452640	−	0.891693i	$-0.350482\pi$
$79$	8.04631	0.905280	0.452640	+	0.891693i	$0.350482\pi$
$80$	1.00000	0.111803
$81$	6.55691	0.728546
$82$	9.35973	1.03361
$83$	−16.6025	−1.82236	−0.911180	−	0.412008i	$-0.864828\pi$
$83$	−16.6025	−1.82236	−0.911180	+	0.412008i	$0.864828\pi$
$84$	0.529317	0.0577532
$85$	6.02760	0.653785
$86$	9.13818	0.985396
$87$	0.187121	0.0200615
$88$	0	0
$89$	16.0923	1.70579	0.852893	−	0.522086i	$-0.174846\pi$
$89$	16.0923	1.70579	0.852893	+	0.522086i	$0.174846\pi$
$90$	−2.71982	−0.286695
$91$	−4.81243	−0.504480
$92$	−7.06157	−0.736220
$93$	−1.19051	−0.123450
$94$	10.7634	1.11016
$95$	3.59370	0.368706
$96$	0.529317	0.0540231
$97$	−1.23078	−0.124967	−0.0624835	−	0.998046i	$-0.519902\pi$
$97$	−1.23078	−0.124967	−0.0624835	+	0.998046i	$0.519902\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−10.8741	−1.08201	−0.541007	−	0.841018i	$-0.681957\pi$
$101$	−10.8741	−1.08201	−0.541007	+	0.841018i	$0.681957\pi$
$102$	3.19051	0.315907
$103$	3.42100	0.337081	0.168541	−	0.985695i	$-0.446095\pi$
$103$	3.42100	0.337081	0.168541	+	0.985695i	$0.446095\pi$
$104$	−4.81243	−0.471897
$105$	0.529317	0.0516560
$106$	4.64885	0.451537
$107$	−9.35136	−0.904030	−0.452015	−	0.892010i	$-0.649295\pi$
$107$	−9.35136	−0.904030	−0.452015	+	0.892010i	$0.649295\pi$
$108$	−3.02760	−0.291331
$109$	19.9100	1.90703	0.953514	−	0.301348i	$-0.0974366\pi$
$109$	19.9100	1.90703	0.953514	+	0.301348i	$0.0974366\pi$
$110$	0	0
$111$	0.871512	0.0827202
$112$	1.00000	0.0944911
$113$	4.10257	0.385938	0.192969	−	0.981205i	$-0.438188\pi$
$113$	4.10257	0.385938	0.192969	+	0.981205i	$0.438188\pi$
$114$	1.90221	0.178158
$115$	−7.06157	−0.658495
$116$	0.353515	0.0328230
$117$	13.0890	1.21007
$118$	2.78483	0.256364
$119$	6.02760	0.552549
$120$	0.529317	0.0483198
$121$	0	0
$122$	−14.6733	−1.32845
$123$	4.95426	0.446710
$124$	−2.24914	−0.201979
$125$	1.00000	0.0894427
$126$	−2.71982	−0.242301
$127$	0.926321	0.0821977	0.0410989	−	0.999155i	$-0.486914\pi$
$127$	0.926321	0.0821977	0.0410989	+	0.999155i	$0.486914\pi$
$128$	1.00000	0.0883883
$129$	4.83699	0.425873
$130$	−4.81243	−0.422078
$131$	−13.3755	−1.16862	−0.584310	−	0.811530i	$-0.698635\pi$
$131$	−13.3755	−1.16862	−0.584310	+	0.811530i	$0.698635\pi$
$132$	0	0
$133$	3.59370	0.311614
$134$	9.43079	0.814697
$135$	−3.02760	−0.260574
$136$	6.02760	0.516862
$137$	19.8175	1.69312	0.846560	−	0.532294i	$-0.178670\pi$
$137$	19.8175	1.69312	0.846560	+	0.532294i	$0.178670\pi$
$138$	−3.73781	−0.318183
$139$	18.7006	1.58616	0.793081	−	0.609117i	$-0.208476\pi$
$139$	18.7006	1.58616	0.793081	+	0.609117i	$0.208476\pi$
$140$	1.00000	0.0845154
$141$	5.69725	0.479795
$142$	12.2276	1.02612
$143$	0	0
$144$	−2.71982	−0.226652
$145$	0.353515	0.0293578
$146$	5.95077	0.492490
$147$	0.529317	0.0436573
$148$	1.64649	0.135340
$149$	−19.6448	−1.60937	−0.804684	−	0.593704i	$-0.797665\pi$
$149$	−19.6448	−1.60937	−0.804684	+	0.593704i	$0.797665\pi$
$150$	0.529317	0.0432185
$151$	1.69178	0.137675	0.0688374	−	0.997628i	$-0.478071\pi$
$151$	1.69178	0.137675	0.0688374	+	0.997628i	$0.478071\pi$
$152$	3.59370	0.291488
$153$	−16.3940	−1.32538
$154$	0	0
$155$	−2.24914	−0.180655
$156$	−2.54730	−0.203947
$157$	−13.6152	−1.08661	−0.543305	−	0.839536i	$-0.682827\pi$
$157$	−13.6152	−1.08661	−0.543305	+	0.839536i	$0.682827\pi$
$158$	8.04631	0.640130
$159$	2.46072	0.195147
$160$	1.00000	0.0790569
$161$	−7.06157	−0.556530
$162$	6.55691	0.515160
$163$	14.0694	1.10200	0.551001	−	0.834505i	$-0.314246\pi$
$163$	14.0694	1.10200	0.551001	+	0.834505i	$0.314246\pi$
$164$	9.35973	0.730872
$165$	0	0
$166$	−16.6025	−1.28860
$167$	17.5422	1.35745	0.678726	−	0.734391i	$-0.262532\pi$
$167$	17.5422	1.35745	0.678726	+	0.734391i	$0.262532\pi$
$168$	0.529317	0.0408377
$169$	10.1595	0.781498
$170$	6.02760	0.462296
$171$	−9.77424	−0.747455
$172$	9.13818	0.696780
$173$	−14.8544	−1.12936	−0.564681	−	0.825310i	$-0.691001\pi$
$173$	−14.8544	−1.12936	−0.564681	+	0.825310i	$0.691001\pi$
$174$	0.187121	0.0141856
$175$	1.00000	0.0755929
$176$	0	0
$177$	1.47406	0.110797
$178$	16.0923	1.20617
$179$	−13.8768	−1.03720	−0.518601	−	0.855016i	$-0.673547\pi$
$179$	−13.8768	−1.03720	−0.518601	+	0.855016i	$0.673547\pi$
$180$	−2.71982	−0.202724
$181$	16.3546	1.21563	0.607815	−	0.794078i	$-0.292046\pi$
$181$	16.3546	1.21563	0.607815	+	0.794078i	$0.292046\pi$
$182$	−4.81243	−0.356721
$183$	−7.76680	−0.574138
$184$	−7.06157	−0.520586
$185$	1.64649	0.121052
$186$	−1.19051	−0.0872922
$187$	0	0
$188$	10.7634	0.785002
$189$	−3.02760	−0.220225
$190$	3.59370	0.260715
$191$	−16.1410	−1.16792	−0.583961	−	0.811782i	$-0.698498\pi$
$191$	−16.1410	−1.16792	−0.583961	+	0.811782i	$0.698498\pi$
$192$	0.529317	0.0382001
$193$	3.72307	0.267992	0.133996	−	0.990982i	$-0.457219\pi$
$193$	3.72307	0.267992	0.133996	+	0.990982i	$0.457219\pi$
$194$	−1.23078	−0.0883650
$195$	−2.54730	−0.182416
$196$	1.00000	0.0714286
$197$	−2.01016	−0.143218	−0.0716091	−	0.997433i	$-0.522813\pi$
$197$	−2.01016	−0.143218	−0.0716091	+	0.997433i	$0.522813\pi$
$198$	0	0
$199$	−9.97363	−0.707012	−0.353506	−	0.935432i	$-0.615011\pi$
$199$	−9.97363	−0.707012	−0.353506	+	0.935432i	$0.615011\pi$
$200$	1.00000	0.0707107
$201$	4.99188	0.352100
$202$	−10.8741	−0.765099
$203$	0.353515	0.0248119
$204$	3.19051	0.223380
$205$	9.35973	0.653712
$206$	3.42100	0.238352
$207$	19.2062	1.33492
$208$	−4.81243	−0.333682
$209$	0	0
$210$	0.529317	0.0365263
$211$	−10.8629	−0.747832	−0.373916	−	0.927463i	$-0.621985\pi$
$211$	−10.8629	−0.747832	−0.373916	+	0.927463i	$0.621985\pi$
$212$	4.64885	0.319285
$213$	6.47228	0.443473
$214$	−9.35136	−0.639246
$215$	9.13818	0.623219
$216$	−3.02760	−0.206002
$217$	−2.24914	−0.152682
$218$	19.9100	1.34847
$219$	3.14984	0.212847
$220$	0	0
$221$	−29.0074	−1.95125
$222$	0.871512	0.0584920
$223$	−12.2446	−0.819956	−0.409978	−	0.912095i	$-0.634464\pi$
$223$	−12.2446	−0.819956	−0.409978	+	0.912095i	$0.634464\pi$
$224$	1.00000	0.0668153
$225$	−2.71982	−0.181322
$226$	4.10257	0.272899
$227$	21.0077	1.39433	0.697164	−	0.716912i	$-0.254445\pi$
$227$	21.0077	1.39433	0.697164	+	0.716912i	$0.254445\pi$
$228$	1.90221	0.125977
$229$	−4.92949	−0.325750	−0.162875	−	0.986647i	$-0.552077\pi$
$229$	−4.92949	−0.325750	−0.162875	+	0.986647i	$0.552077\pi$
$230$	−7.06157	−0.465626
$231$	0	0
$232$	0.353515	0.0232094
$233$	−20.1605	−1.32076	−0.660380	−	0.750932i	$-0.729605\pi$
$233$	−20.1605	−1.32076	−0.660380	+	0.750932i	$0.729605\pi$
$234$	13.0890	0.855652
$235$	10.7634	0.702128
$236$	2.78483	0.181277
$237$	4.25904	0.276655
$238$	6.02760	0.390711
$239$	−9.53194	−0.616570	−0.308285	−	0.951294i	$-0.599755\pi$
$239$	−9.53194	−0.616570	−0.308285	+	0.951294i	$0.599755\pi$
$240$	0.529317	0.0341672
$241$	−6.73710	−0.433975	−0.216987	−	0.976174i	$-0.569623\pi$
$241$	−6.73710	−0.433975	−0.216987	+	0.976174i	$0.569623\pi$
$242$	0	0
$243$	12.5535	0.805306
$244$	−14.6733	−0.939359
$245$	1.00000	0.0638877
$246$	4.95426	0.315872
$247$	−17.2944	−1.10042
$248$	−2.24914	−0.142821
$249$	−8.78798	−0.556915
$250$	1.00000	0.0632456
$251$	8.94574	0.564650	0.282325	−	0.959319i	$-0.408894\pi$
$251$	8.94574	0.564650	0.282325	+	0.959319i	$0.408894\pi$
$252$	−2.71982	−0.171333
$253$	0	0
$254$	0.926321	0.0581226
$255$	3.19051	0.199797
$256$	1.00000	0.0625000
$257$	16.9351	1.05639	0.528193	−	0.849125i	$-0.322870\pi$
$257$	16.9351	1.05639	0.528193	+	0.849125i	$0.322870\pi$
$258$	4.83699	0.301138
$259$	1.64649	0.102308
$260$	−4.81243	−0.298454
$261$	−0.961498	−0.0595152
$262$	−13.3755	−0.826340
$263$	16.0270	0.988266	0.494133	−	0.869386i	$-0.335486\pi$
$263$	16.0270	0.988266	0.494133	+	0.869386i	$0.335486\pi$
$264$	0	0
$265$	4.64885	0.285577
$266$	3.59370	0.220344
$267$	8.51795	0.521290
$268$	9.43079	0.576077
$269$	9.38110	0.571975	0.285988	−	0.958233i	$-0.407678\pi$
$269$	9.38110	0.571975	0.285988	+	0.958233i	$0.407678\pi$
$270$	−3.02760	−0.184254
$271$	−15.8105	−0.960418	−0.480209	−	0.877154i	$-0.659439\pi$
$271$	−15.8105	−0.960418	−0.480209	+	0.877154i	$0.659439\pi$
$272$	6.02760	0.365477
$273$	−2.54730	−0.154170
$274$	19.8175	1.19722
$275$	0	0
$276$	−3.73781	−0.224989
$277$	−24.2012	−1.45411	−0.727054	−	0.686580i	$-0.759111\pi$
$277$	−24.2012	−1.45411	−0.727054	+	0.686580i	$0.759111\pi$
$278$	18.7006	1.12159
$279$	6.11727	0.366231
$280$	1.00000	0.0597614
$281$	16.8649	1.00608	0.503038	−	0.864264i	$-0.332215\pi$
$281$	16.8649	1.00608	0.503038	+	0.864264i	$0.332215\pi$
$282$	5.69725	0.339266
$283$	−3.49996	−0.208051	−0.104025	−	0.994575i	$-0.533172\pi$
$283$	−3.49996	−0.208051	−0.104025	+	0.994575i	$0.533172\pi$
$284$	12.2276	0.725575
$285$	1.90221	0.112677
$286$	0	0
$287$	9.35973	0.552487
$288$	−2.71982	−0.160267
$289$	19.3319	1.13717
$290$	0.353515	0.0207591
$291$	−0.651473	−0.0381900
$292$	5.95077	0.348243
$293$	10.6822	0.624060	0.312030	−	0.950072i	$-0.398991\pi$
$293$	10.6822	0.624060	0.312030	+	0.950072i	$0.398991\pi$
$294$	0.529317	0.0308704
$295$	2.78483	0.162139
$296$	1.64649	0.0957000
$297$	0	0
$298$	−19.6448	−1.13799
$299$	33.9833	1.96531
$300$	0.529317	0.0305601
$301$	9.13818	0.526716
$302$	1.69178	0.0973508
$303$	−5.75584	−0.330664
$304$	3.59370	0.206113
$305$	−14.6733	−0.840188
$306$	−16.3940	−0.937183
$307$	−17.4466	−0.995731	−0.497865	−	0.867254i	$-0.665883\pi$
$307$	−17.4466	−0.995731	−0.497865	+	0.867254i	$0.665883\pi$
$308$	0	0
$309$	1.81079	0.103012
$310$	−2.24914	−0.127743
$311$	9.50939	0.539228	0.269614	−	0.962968i	$-0.413104\pi$
$311$	9.50939	0.539228	0.269614	+	0.962968i	$0.413104\pi$
$312$	−2.54730	−0.144212
$313$	29.1754	1.64909	0.824545	−	0.565796i	$-0.191431\pi$
$313$	29.1754	1.64909	0.824545	+	0.565796i	$0.191431\pi$
$314$	−13.6152	−0.768349
$315$	−2.71982	−0.153245
$316$	8.04631	0.452640
$317$	−28.7999	−1.61756	−0.808782	−	0.588109i	$-0.799873\pi$
$317$	−28.7999	−1.61756	−0.808782	+	0.588109i	$0.799873\pi$
$318$	2.46072	0.137990
$319$	0	0
$320$	1.00000	0.0559017
$321$	−4.94983	−0.276273
$322$	−7.06157	−0.393526
$323$	21.6614	1.20527
$324$	6.55691	0.364273
$325$	−4.81243	−0.266946
$326$	14.0694	0.779233
$327$	10.5387	0.582790
$328$	9.35973	0.516804
$329$	10.7634	0.593406
$330$	0	0
$331$	17.9933	0.989003	0.494502	−	0.869177i	$-0.335351\pi$
$331$	17.9933	0.989003	0.494502	+	0.869177i	$0.335351\pi$
$332$	−16.6025	−0.911180
$333$	−4.47815	−0.245401
$334$	17.5422	0.959864
$335$	9.43079	0.515259
$336$	0.529317	0.0288766
$337$	23.2472	1.26635	0.633177	−	0.774007i	$-0.281751\pi$
$337$	23.2472	1.26635	0.633177	+	0.774007i	$0.281751\pi$
$338$	10.1595	0.552602
$339$	2.17156	0.117943
$340$	6.02760	0.326892
$341$	0	0
$342$	−9.77424	−0.528531
$343$	1.00000	0.0539949
$344$	9.13818	0.492698
$345$	−3.73781	−0.201237
$346$	−14.8544	−0.798579
$347$	−34.5288	−1.85361	−0.926803	−	0.375548i	$-0.877454\pi$
$347$	−34.5288	−1.85361	−0.926803	+	0.375548i	$0.877454\pi$
$348$	0.187121	0.0100308
$349$	12.3015	0.658483	0.329242	−	0.944246i	$-0.393207\pi$
$349$	12.3015	0.658483	0.329242	+	0.944246i	$0.393207\pi$
$350$	1.00000	0.0534522
$351$	14.5701	0.777694
$352$	0	0
$353$	−4.73131	−0.251822	−0.125911	−	0.992042i	$-0.540185\pi$
$353$	−4.73131	−0.251822	−0.125911	+	0.992042i	$0.540185\pi$
$354$	1.47406	0.0783453
$355$	12.2276	0.648974
$356$	16.0923	0.852893
$357$	3.19051	0.168860
$358$	−13.8768	−0.733413
$359$	1.31338	0.0693178	0.0346589	−	0.999399i	$-0.488966\pi$
$359$	1.31338	0.0693178	0.0346589	+	0.999399i	$0.488966\pi$
$360$	−2.71982	−0.143347
$361$	−6.08529	−0.320278
$362$	16.3546	0.859581
$363$	0	0
$364$	−4.81243	−0.252240
$365$	5.95077	0.311478
$366$	−7.76680	−0.405977
$367$	−5.39464	−0.281598	−0.140799	−	0.990038i	$-0.544967\pi$
$367$	−5.39464	−0.281598	−0.140799	+	0.990038i	$0.544967\pi$
$368$	−7.06157	−0.368110
$369$	−25.4568	−1.32523
$370$	1.64649	0.0855967
$371$	4.64885	0.241356
$372$	−1.19051	−0.0617249
$373$	−13.7265	−0.710731	−0.355366	−	0.934727i	$-0.615644\pi$
$373$	−13.7265	−0.710731	−0.355366	+	0.934727i	$0.615644\pi$
$374$	0	0
$375$	0.529317	0.0273338
$376$	10.7634	0.555081
$377$	−1.70127	−0.0876196
$378$	−3.02760	−0.155723
$379$	−12.4209	−0.638019	−0.319009	−	0.947752i	$-0.603350\pi$
$379$	−12.4209	−0.638019	−0.319009	+	0.947752i	$0.603350\pi$
$380$	3.59370	0.184353
$381$	0.490317	0.0251197
$382$	−16.1410	−0.825846
$383$	−6.58507	−0.336481	−0.168241	−	0.985746i	$-0.553809\pi$
$383$	−6.58507	−0.336481	−0.168241	+	0.985746i	$0.553809\pi$
$384$	0.529317	0.0270116
$385$	0	0
$386$	3.72307	0.189499
$387$	−24.8543	−1.26341
$388$	−1.23078	−0.0624835
$389$	−23.8620	−1.20985	−0.604925	−	0.796282i	$-0.706797\pi$
$389$	−23.8620	−1.20985	−0.604925	+	0.796282i	$0.706797\pi$
$390$	−2.54730	−0.128987
$391$	−42.5643	−2.15257
$392$	1.00000	0.0505076
$393$	−7.07986	−0.357132
$394$	−2.01016	−0.101271
$395$	8.04631	0.404854
$396$	0	0
$397$	−14.1058	−0.707952	−0.353976	−	0.935254i	$-0.615171\pi$
$397$	−14.1058	−0.707952	−0.353976	+	0.935254i	$0.615171\pi$
$398$	−9.97363	−0.499933
$399$	1.90221	0.0952295
$400$	1.00000	0.0500000
$401$	−1.50759	−0.0752855	−0.0376428	−	0.999291i	$-0.511985\pi$
$401$	−1.50759	−0.0752855	−0.0376428	+	0.999291i	$0.511985\pi$
$402$	4.99188	0.248972
$403$	10.8238	0.539173
$404$	−10.8741	−0.541007
$405$	6.55691	0.325816
$406$	0.353515	0.0175446
$407$	0	0
$408$	3.19051	0.157954
$409$	−4.42801	−0.218951	−0.109476	−	0.993989i	$-0.534917\pi$
$409$	−4.42801	−0.218951	−0.109476	+	0.993989i	$0.534917\pi$
$410$	9.35973	0.462244
$411$	10.4897	0.517419
$412$	3.42100	0.168541
$413$	2.78483	0.137033
$414$	19.2062	0.943934
$415$	−16.6025	−0.814984
$416$	−4.81243	−0.235949
$417$	9.89852	0.484733
$418$	0	0
$419$	−16.3250	−0.797529	−0.398765	−	0.917053i	$-0.630561\pi$
$419$	−16.3250	−0.797529	−0.398765	+	0.917053i	$0.630561\pi$
$420$	0.529317	0.0258280
$421$	−3.12002	−0.152061	−0.0760303	−	0.997106i	$-0.524225\pi$
$421$	−3.12002	−0.152061	−0.0760303	+	0.997106i	$0.524225\pi$
$422$	−10.8629	−0.528797
$423$	−29.2746	−1.42338
$424$	4.64885	0.225768
$425$	6.02760	0.292381
$426$	6.47228	0.313583
$427$	−14.6733	−0.710089
$428$	−9.35136	−0.452015
$429$	0	0
$430$	9.13818	0.440682
$431$	−4.93779	−0.237845	−0.118923	−	0.992904i	$-0.537944\pi$
$431$	−4.93779	−0.237845	−0.118923	+	0.992904i	$0.537944\pi$
$432$	−3.02760	−0.145665
$433$	31.6620	1.52158	0.760790	−	0.648998i	$-0.224812\pi$
$433$	31.6620	1.52158	0.760790	+	0.648998i	$0.224812\pi$
$434$	−2.24914	−0.107962
$435$	0.187121	0.00897178
$436$	19.9100	0.953514
$437$	−25.3772	−1.21396
$438$	3.14984	0.150505
$439$	3.57840	0.170788	0.0853938	−	0.996347i	$-0.472785\pi$
$439$	3.57840	0.170788	0.0853938	+	0.996347i	$0.472785\pi$
$440$	0	0
$441$	−2.71982	−0.129515
$442$	−29.0074	−1.37974
$443$	11.2916	0.536482	0.268241	−	0.963352i	$-0.413558\pi$
$443$	11.2916	0.536482	0.268241	+	0.963352i	$0.413558\pi$
$444$	0.871512	0.0413601
$445$	16.0923	0.762850
$446$	−12.2446	−0.579797
$447$	−10.3983	−0.491824
$448$	1.00000	0.0472456
$449$	−9.27050	−0.437502	−0.218751	−	0.975781i	$-0.570198\pi$
$449$	−9.27050	−0.437502	−0.218751	+	0.975781i	$0.570198\pi$
$450$	−2.71982	−0.128214
$451$	0	0
$452$	4.10257	0.192969
$453$	0.895485	0.0420736
$454$	21.0077	0.985938
$455$	−4.81243	−0.225610
$456$	1.90221	0.0890790
$457$	−41.7715	−1.95399	−0.976993	−	0.213271i	$-0.931588\pi$
$457$	−41.7715	−1.95399	−0.976993	+	0.213271i	$0.931588\pi$
$458$	−4.92949	−0.230340
$459$	−18.2491	−0.851797
$460$	−7.06157	−0.329247
$461$	14.6501	0.682325	0.341162	−	0.940004i	$-0.389179\pi$
$461$	14.6501	0.682325	0.341162	+	0.940004i	$0.389179\pi$
$462$	0	0
$463$	−28.0291	−1.30262	−0.651312	−	0.758810i	$-0.725781\pi$
$463$	−28.0291	−1.30262	−0.651312	+	0.758810i	$0.725781\pi$
$464$	0.353515	0.0164115
$465$	−1.19051	−0.0552085
$466$	−20.1605	−0.933918
$467$	13.2230	0.611887	0.305943	−	0.952050i	$-0.401028\pi$
$467$	13.2230	0.611887	0.305943	+	0.952050i	$0.401028\pi$
$468$	13.0890	0.605037
$469$	9.43079	0.435474
$470$	10.7634	0.496479
$471$	−7.20674	−0.332069
$472$	2.78483	0.128182
$473$	0	0
$474$	4.25904	0.195624
$475$	3.59370	0.164890
$476$	6.02760	0.276274
$477$	−12.6441	−0.578932
$478$	−9.53194	−0.435981
$479$	−18.6878	−0.853866	−0.426933	−	0.904283i	$-0.640406\pi$
$479$	−18.6878	−0.853866	−0.426933	+	0.904283i	$0.640406\pi$
$480$	0.529317	0.0241599
$481$	−7.92359	−0.361285
$482$	−6.73710	−0.306866
$483$	−3.73781	−0.170076
$484$	0	0
$485$	−1.23078	−0.0558869
$486$	12.5535	0.569437
$487$	−11.4532	−0.518992	−0.259496	−	0.965744i	$-0.583556\pi$
$487$	−11.4532	−0.518992	−0.259496	+	0.965744i	$0.583556\pi$
$488$	−14.6733	−0.664227
$489$	7.44717	0.336773
$490$	1.00000	0.0451754
$491$	32.7871	1.47966	0.739831	−	0.672793i	$-0.234905\pi$
$491$	32.7871	1.47966	0.739831	+	0.672793i	$0.234905\pi$
$492$	4.95426	0.223355
$493$	2.13085	0.0959684
$494$	−17.2944	−0.778114
$495$	0	0
$496$	−2.24914	−0.100989
$497$	12.2276	0.548483
$498$	−8.78798	−0.393799
$499$	−31.0534	−1.39014	−0.695071	−	0.718941i	$-0.744627\pi$
$499$	−31.0534	−1.39014	−0.695071	+	0.718941i	$0.744627\pi$
$500$	1.00000	0.0447214
$501$	9.28535	0.414839
$502$	8.94574	0.399268
$503$	−25.5004	−1.13701	−0.568503	−	0.822681i	$-0.692477\pi$
$503$	−25.5004	−1.13701	−0.568503	+	0.822681i	$0.692477\pi$
$504$	−2.71982	−0.121151
$505$	−10.8741	−0.483891
$506$	0	0
$507$	5.37758	0.238827
$508$	0.926321	0.0410989
$509$	−27.3509	−1.21231	−0.606153	−	0.795348i	$-0.707288\pi$
$509$	−27.3509	−1.21231	−0.606153	+	0.795348i	$0.707288\pi$
$510$	3.19051	0.141278
$511$	5.95077	0.263247
$512$	1.00000	0.0441942
$513$	−10.8803	−0.480377
$514$	16.9351	0.746977
$515$	3.42100	0.150747
$516$	4.83699	0.212937
$517$	0	0
$518$	1.64649	0.0723424
$519$	−7.86269	−0.345134
$520$	−4.81243	−0.211039
$521$	23.0349	1.00918	0.504589	−	0.863359i	$-0.331644\pi$
$521$	23.0349	1.00918	0.504589	+	0.863359i	$0.331644\pi$
$522$	−0.961498	−0.0420836
$523$	43.6113	1.90699	0.953494	−	0.301413i	$-0.0974583\pi$
$523$	43.6113	1.90699	0.953494	+	0.301413i	$0.0974583\pi$
$524$	−13.3755	−0.584310
$525$	0.529317	0.0231013
$526$	16.0270	0.698810
$527$	−13.5569	−0.590548
$528$	0	0
$529$	26.8658	1.16808
$530$	4.64885	0.201933
$531$	−7.57425	−0.328694
$532$	3.59370	0.155807
$533$	−45.0430	−1.95103
$534$	8.51795	0.368608
$535$	−9.35136	−0.404295
$536$	9.43079	0.407348
$537$	−7.34523	−0.316970
$538$	9.38110	0.404448
$539$	0	0
$540$	−3.02760	−0.130287
$541$	28.6593	1.23216	0.616079	−	0.787684i	$-0.288720\pi$
$541$	28.6593	1.23216	0.616079	+	0.787684i	$0.288720\pi$
$542$	−15.8105	−0.679118
$543$	8.65678	0.371498
$544$	6.02760	0.258431
$545$	19.9100	0.852849
$546$	−2.54730	−0.109014
$547$	−0.317491	−0.0135749	−0.00678747	−	0.999977i	$-0.502161\pi$
$547$	−0.317491	−0.0135749	−0.00678747	+	0.999977i	$0.502161\pi$
$548$	19.8175	0.846560
$549$	39.9087	1.70326
$550$	0	0
$551$	1.27043	0.0541221
$552$	−3.73781	−0.159092
$553$	8.04631	0.342164
$554$	−24.2012	−1.02821
$555$	0.871512	0.0369936
$556$	18.7006	0.793081
$557$	2.72599	0.115504	0.0577519	−	0.998331i	$-0.481607\pi$
$557$	2.72599	0.115504	0.0577519	+	0.998331i	$0.481607\pi$
$558$	6.11727	0.258965
$559$	−43.9769	−1.86002
$560$	1.00000	0.0422577
$561$	0	0
$562$	16.8649	0.711404
$563$	−7.36860	−0.310550	−0.155275	−	0.987871i	$-0.549626\pi$
$563$	−7.36860	−0.310550	−0.155275	+	0.987871i	$0.549626\pi$
$564$	5.69725	0.239898
$565$	4.10257	0.172597
$566$	−3.49996	−0.147114
$567$	6.55691	0.275365
$568$	12.2276	0.513059
$569$	12.7784	0.535700	0.267850	−	0.963461i	$-0.413687\pi$
$569$	12.7784	0.535700	0.267850	+	0.963461i	$0.413687\pi$
$570$	1.90221	0.0796747
$571$	−13.5141	−0.565546	−0.282773	−	0.959187i	$-0.591254\pi$
$571$	−13.5141	−0.565546	−0.282773	+	0.959187i	$0.591254\pi$
$572$	0	0
$573$	−8.54370	−0.356918
$574$	9.35973	0.390667
$575$	−7.06157	−0.294488
$576$	−2.71982	−0.113326
$577$	−4.80939	−0.200218	−0.100109	−	0.994976i	$-0.531919\pi$
$577$	−4.80939	−0.200218	−0.100109	+	0.994976i	$0.531919\pi$
$578$	19.3319	0.804102
$579$	1.97068	0.0818987
$580$	0.353515	0.0146789
$581$	−16.6025	−0.688788
$582$	−0.651473	−0.0270044
$583$	0	0
$584$	5.95077	0.246245
$585$	13.0890	0.541162
$586$	10.6822	0.441277
$587$	16.1586	0.666936	0.333468	−	0.942761i	$-0.391781\pi$
$587$	16.1586	0.666936	0.333468	+	0.942761i	$0.391781\pi$
$588$	0.529317	0.0218286
$589$	−8.08275	−0.333044
$590$	2.78483	0.114650
$591$	−1.06401	−0.0437676
$592$	1.64649	0.0676701
$593$	−30.4646	−1.25103	−0.625516	−	0.780211i	$-0.715111\pi$
$593$	−30.4646	−1.25103	−0.625516	+	0.780211i	$0.715111\pi$
$594$	0	0
$595$	6.02760	0.247107
$596$	−19.6448	−0.804684
$597$	−5.27921	−0.216064
$598$	33.9833	1.38968
$599$	30.4537	1.24430	0.622152	−	0.782896i	$-0.286258\pi$
$599$	30.4537	1.24430	0.622152	+	0.782896i	$0.286258\pi$
$600$	0.529317	0.0216093
$601$	17.0612	0.695940	0.347970	−	0.937506i	$-0.386871\pi$
$601$	17.0612	0.695940	0.347970	+	0.937506i	$0.386871\pi$
$602$	9.13818	0.372445
$603$	−25.6501	−1.04455
$604$	1.69178	0.0688374
$605$	0	0
$606$	−5.75584	−0.233815
$607$	−15.7032	−0.637374	−0.318687	−	0.947860i	$-0.603242\pi$
$607$	−15.7032	−0.637374	−0.318687	+	0.947860i	$0.603242\pi$
$608$	3.59370	0.145744
$609$	0.187121	0.00758254
$610$	−14.6733	−0.594103
$611$	−51.7982	−2.09553
$612$	−16.3940	−0.662688
$613$	17.2732	0.697657	0.348829	−	0.937186i	$-0.386579\pi$
$613$	17.2732	0.697657	0.348829	+	0.937186i	$0.386579\pi$
$614$	−17.4466	−0.704088
$615$	4.95426	0.199775
$616$	0	0
$617$	−11.2271	−0.451986	−0.225993	−	0.974129i	$-0.572563\pi$
$617$	−11.2271	−0.451986	−0.225993	+	0.974129i	$0.572563\pi$
$618$	1.81079	0.0728408
$619$	18.3288	0.736696	0.368348	−	0.929688i	$-0.379924\pi$
$619$	18.3288	0.736696	0.368348	+	0.929688i	$0.379924\pi$
$620$	−2.24914	−0.0903277
$621$	21.3796	0.857933
$622$	9.50939	0.381292
$623$	16.0923	0.644726
$624$	−2.54730	−0.101974
$625$	1.00000	0.0400000
$626$	29.1754	1.16608
$627$	0	0
$628$	−13.6152	−0.543305
$629$	9.92435	0.395710
$630$	−2.71982	−0.108360
$631$	−35.0446	−1.39510	−0.697552	−	0.716534i	$-0.745727\pi$
$631$	−35.0446	−1.39510	−0.697552	+	0.716534i	$0.745727\pi$
$632$	8.04631	0.320065
$633$	−5.74991	−0.228538
$634$	−28.7999	−1.14379
$635$	0.926321	0.0367599
$636$	2.46072	0.0975737
$637$	−4.81243	−0.190675
$638$	0	0
$639$	−33.2569	−1.31562
$640$	1.00000	0.0395285
$641$	20.3744	0.804739	0.402370	−	0.915477i	$-0.368187\pi$
$641$	20.3744	0.804739	0.402370	+	0.915477i	$0.368187\pi$
$642$	−4.94983	−0.195354
$643$	−1.34168	−0.0529108	−0.0264554	−	0.999650i	$-0.508422\pi$
$643$	−1.34168	−0.0529108	−0.0264554	+	0.999650i	$0.508422\pi$
$644$	−7.06157	−0.278265
$645$	4.83699	0.190456
$646$	21.6614	0.852257
$647$	21.1028	0.829637	0.414818	−	0.909904i	$-0.363845\pi$
$647$	21.1028	0.829637	0.414818	+	0.909904i	$0.363845\pi$
$648$	6.55691	0.257580
$649$	0	0
$650$	−4.81243	−0.188759
$651$	−1.19051	−0.0466597
$652$	14.0694	0.551001
$653$	48.3564	1.89233	0.946166	−	0.323681i	$-0.104921\pi$
$653$	48.3564	1.89233	0.946166	+	0.323681i	$0.104921\pi$
$654$	10.5387	0.412095
$655$	−13.3755	−0.522623
$656$	9.35973	0.365436
$657$	−16.1851	−0.631439
$658$	10.7634	0.419601
$659$	15.9811	0.622536	0.311268	−	0.950322i	$-0.399246\pi$
$659$	15.9811	0.622536	0.311268	+	0.950322i	$0.399246\pi$
$660$	0	0
$661$	31.1212	1.21048	0.605238	−	0.796045i	$-0.293078\pi$
$661$	31.1212	1.21048	0.605238	+	0.796045i	$0.293078\pi$
$662$	17.9933	0.699331
$663$	−15.3541	−0.596303
$664$	−16.6025	−0.644302
$665$	3.59370	0.139358
$666$	−4.47815	−0.173525
$667$	−2.49637	−0.0966598
$668$	17.5422	0.678726
$669$	−6.48125	−0.250580
$670$	9.43079	0.364343
$671$	0	0
$672$	0.529317	0.0204188
$673$	38.4561	1.48237	0.741186	−	0.671300i	$-0.234264\pi$
$673$	38.4561	1.48237	0.741186	+	0.671300i	$0.234264\pi$
$674$	23.2472	0.895448
$675$	−3.02760	−0.116532
$676$	10.1595	0.390749
$677$	−3.05095	−0.117257	−0.0586287	−	0.998280i	$-0.518673\pi$
$677$	−3.05095	−0.117257	−0.0586287	+	0.998280i	$0.518673\pi$
$678$	2.17156	0.0833983
$679$	−1.23078	−0.0472331
$680$	6.02760	0.231148
$681$	11.1197	0.426108
$682$	0	0
$683$	−29.6414	−1.13420	−0.567099	−	0.823650i	$-0.691934\pi$
$683$	−29.6414	−1.13420	−0.567099	+	0.823650i	$0.691934\pi$
$684$	−9.77424	−0.373728
$685$	19.8175	0.757186
$686$	1.00000	0.0381802
$687$	−2.60926	−0.0995495
$688$	9.13818	0.348390
$689$	−22.3723	−0.852316
$690$	−3.73781	−0.142296
$691$	−29.4999	−1.12223	−0.561114	−	0.827738i	$-0.689627\pi$
$691$	−29.4999	−1.12223	−0.561114	+	0.827738i	$0.689627\pi$
$692$	−14.8544	−0.564681
$693$	0	0
$694$	−34.5288	−1.31070
$695$	18.7006	0.709353
$696$	0.187121	0.00709281
$697$	56.4167	2.13693
$698$	12.3015	0.465618
$699$	−10.6713	−0.403625
$700$	1.00000	0.0377964
$701$	−1.49369	−0.0564158	−0.0282079	−	0.999602i	$-0.508980\pi$
$701$	−1.49369	−0.0564158	−0.0282079	+	0.999602i	$0.508980\pi$
$702$	14.5701	0.549913
$703$	5.91698	0.223163
$704$	0	0
$705$	5.69725	0.214571
$706$	−4.73131	−0.178065
$707$	−10.8741	−0.408962
$708$	1.47406	0.0553985
$709$	−36.0120	−1.35246	−0.676229	−	0.736691i	$-0.736387\pi$
$709$	−36.0120	−1.35246	−0.676229	+	0.736691i	$0.736387\pi$
$710$	12.2276	0.458894
$711$	−21.8845	−0.820734
$712$	16.0923	0.603086
$713$	15.8825	0.594803
$714$	3.19051	0.119402
$715$	0	0
$716$	−13.8768	−0.518601
$717$	−5.04541	−0.188424
$718$	1.31338	0.0490151
$719$	14.9481	0.557472	0.278736	−	0.960368i	$-0.410085\pi$
$719$	14.9481	0.557472	0.278736	+	0.960368i	$0.410085\pi$
$720$	−2.71982	−0.101362
$721$	3.42100	0.127405
$722$	−6.08529	−0.226471
$723$	−3.56606	−0.132623
$724$	16.3546	0.607815
$725$	0.353515	0.0131292
$726$	0	0
$727$	21.3542	0.791984	0.395992	−	0.918254i	$-0.370401\pi$
$727$	21.3542	0.791984	0.395992	+	0.918254i	$0.370401\pi$
$728$	−4.81243	−0.178360
$729$	−13.0260	−0.482444
$730$	5.95077	0.220248
$731$	55.0813	2.03726
$732$	−7.76680	−0.287069
$733$	−26.1929	−0.967455	−0.483728	−	0.875219i	$-0.660717\pi$
$733$	−26.1929	−0.967455	−0.483728	+	0.875219i	$0.660717\pi$
$734$	−5.39464	−0.199120
$735$	0.529317	0.0195241
$736$	−7.06157	−0.260293
$737$	0	0
$738$	−25.4568	−0.937078
$739$	0.973255	0.0358018	0.0179009	−	0.999840i	$-0.494302\pi$
$739$	0.973255	0.0358018	0.0179009	+	0.999840i	$0.494302\pi$
$740$	1.64649	0.0605260
$741$	−9.15424	−0.336289
$742$	4.64885	0.170665
$743$	−47.9841	−1.76037	−0.880183	−	0.474635i	$-0.842580\pi$
$743$	−47.9841	−1.76037	−0.880183	+	0.474635i	$0.842580\pi$
$744$	−1.19051	−0.0436461
$745$	−19.6448	−0.719731
$746$	−13.7265	−0.502563
$747$	45.1559	1.65217
$748$	0	0
$749$	−9.35136	−0.341691
$750$	0.529317	0.0193279
$751$	12.1660	0.443944	0.221972	−	0.975053i	$-0.428751\pi$
$751$	12.1660	0.443944	0.221972	+	0.975053i	$0.428751\pi$
$752$	10.7634	0.392501
$753$	4.73513	0.172558
$754$	−1.70127	−0.0619564
$755$	1.69178	0.0615700
$756$	−3.02760	−0.110113
$757$	33.8459	1.23015	0.615076	−	0.788468i	$-0.289125\pi$
$757$	33.8459	1.23015	0.615076	+	0.788468i	$0.289125\pi$
$758$	−12.4209	−0.451147
$759$	0	0
$760$	3.59370	0.130357
$761$	−14.6460	−0.530917	−0.265459	−	0.964122i	$-0.585523\pi$
$761$	−14.6460	−0.530917	−0.265459	+	0.964122i	$0.585523\pi$
$762$	0.490317	0.0177623
$763$	19.9100	0.720789
$764$	−16.1410	−0.583961
$765$	−16.3940	−0.592726
$766$	−6.58507	−0.237928
$767$	−13.4018	−0.483911
$768$	0.529317	0.0191001
$769$	−15.5100	−0.559304	−0.279652	−	0.960101i	$-0.590219\pi$
$769$	−15.5100	−0.559304	−0.279652	+	0.960101i	$0.590219\pi$
$770$	0	0
$771$	8.96405	0.322832
$772$	3.72307	0.133996
$773$	40.5521	1.45856	0.729278	−	0.684217i	$-0.239856\pi$
$773$	40.5521	1.45856	0.729278	+	0.684217i	$0.239856\pi$
$774$	−24.8543	−0.893368
$775$	−2.24914	−0.0807915
$776$	−1.23078	−0.0441825
$777$	0.871512	0.0312653
$778$	−23.8620	−0.855493
$779$	33.6361	1.20514
$780$	−2.54730	−0.0912079
$781$	0	0
$782$	−42.5643	−1.52210
$783$	−1.07030	−0.0382494
$784$	1.00000	0.0357143
$785$	−13.6152	−0.485947
$786$	−7.07986	−0.252530
$787$	28.5706	1.01843	0.509216	−	0.860639i	$-0.329935\pi$
$787$	28.5706	1.01843	0.509216	+	0.860639i	$0.329935\pi$
$788$	−2.01016	−0.0716091
$789$	8.48335	0.302015
$790$	8.04631	0.286275
$791$	4.10257	0.145871
$792$	0	0
$793$	70.6140	2.50758
$794$	−14.1058	−0.500598
$795$	2.46072	0.0872726
$796$	−9.97363	−0.353506
$797$	−16.0691	−0.569196	−0.284598	−	0.958647i	$-0.591860\pi$
$797$	−16.0691	−0.569196	−0.284598	+	0.958647i	$0.591860\pi$
$798$	1.90221	0.0673374
$799$	64.8775	2.29520
$800$	1.00000	0.0353553
$801$	−43.7683	−1.54648
$802$	−1.50759	−0.0532349
$803$	0	0
$804$	4.99188	0.176050
$805$	−7.06157	−0.248888
$806$	10.8238	0.381253
$807$	4.96557	0.174796
$808$	−10.8741	−0.382549
$809$	30.7932	1.08263	0.541315	−	0.840820i	$-0.317927\pi$
$809$	30.7932	1.08263	0.541315	+	0.840820i	$0.317927\pi$
$810$	6.55691	0.230386
$811$	21.9746	0.771632	0.385816	−	0.922576i	$-0.373920\pi$
$811$	21.9746	0.771632	0.385816	+	0.922576i	$0.373920\pi$
$812$	0.353515	0.0124059
$813$	−8.36875	−0.293505
$814$	0	0
$815$	14.0694	0.492830
$816$	3.19051	0.111690
$817$	32.8399	1.14892
$818$	−4.42801	−0.154822
$819$	13.0890	0.457365
$820$	9.35973	0.326856
$821$	−45.6665	−1.59377	−0.796885	−	0.604130i	$-0.793521\pi$
$821$	−45.6665	−1.59377	−0.796885	+	0.604130i	$0.793521\pi$
$822$	10.4897	0.365871
$823$	2.44948	0.0853834	0.0426917	−	0.999088i	$-0.486407\pi$
$823$	2.44948	0.0853834	0.0426917	+	0.999088i	$0.486407\pi$
$824$	3.42100	0.119176
$825$	0	0
$826$	2.78483	0.0968967
$827$	−31.9237	−1.11009	−0.555047	−	0.831819i	$-0.687300\pi$
$827$	−31.9237	−1.11009	−0.555047	+	0.831819i	$0.687300\pi$
$828$	19.2062	0.667462
$829$	17.0171	0.591029	0.295515	−	0.955338i	$-0.404509\pi$
$829$	17.0171	0.591029	0.295515	+	0.955338i	$0.404509\pi$
$830$	−16.6025	−0.576281
$831$	−12.8101	−0.444377
$832$	−4.81243	−0.166841
$833$	6.02760	0.208844
$834$	9.89852	0.342758
$835$	17.5422	0.607071
$836$	0	0
$837$	6.80949	0.235370
$838$	−16.3250	−0.563938
$839$	46.6872	1.61182	0.805911	−	0.592037i	$-0.201676\pi$
$839$	46.6872	1.61182	0.805911	+	0.592037i	$0.201676\pi$
$840$	0.529317	0.0182632
$841$	−28.8750	−0.995691
$842$	−3.12002	−0.107523
$843$	8.92688	0.307458
$844$	−10.8629	−0.373916
$845$	10.1595	0.349496
$846$	−29.2746	−1.00648
$847$	0	0
$848$	4.64885	0.159642
$849$	−1.85258	−0.0635805
$850$	6.02760	0.206745
$851$	−11.6268	−0.398561
$852$	6.47228	0.221737
$853$	−13.8541	−0.474357	−0.237178	−	0.971466i	$-0.576223\pi$
$853$	−13.8541	−0.474357	−0.237178	+	0.971466i	$0.576223\pi$
$854$	−14.6733	−0.502108
$855$	−9.77424	−0.334272
$856$	−9.35136	−0.319623
$857$	−30.0551	−1.02666	−0.513331	−	0.858191i	$-0.671589\pi$
$857$	−30.0551	−1.02666	−0.513331	+	0.858191i	$0.671589\pi$
$858$	0	0
$859$	44.4219	1.51566	0.757828	−	0.652454i	$-0.226260\pi$
$859$	44.4219	1.51566	0.757828	+	0.652454i	$0.226260\pi$
$860$	9.13818	0.311610
$861$	4.95426	0.168841
$862$	−4.93779	−0.168182
$863$	16.6118	0.565472	0.282736	−	0.959198i	$-0.408758\pi$
$863$	16.6118	0.565472	0.282736	+	0.959198i	$0.408758\pi$
$864$	−3.02760	−0.103001
$865$	−14.8544	−0.505066
$866$	31.6620	1.07592
$867$	10.2327	0.347521
$868$	−2.24914	−0.0763408
$869$	0	0
$870$	0.187121	0.00634401
$871$	−45.3850	−1.53781
$872$	19.9100	0.674236
$873$	3.34751	0.113296
$874$	−25.3772	−0.858396
$875$	1.00000	0.0338062
$876$	3.14984	0.106423
$877$	30.4224	1.02729	0.513646	−	0.858002i	$-0.328295\pi$
$877$	30.4224	1.02729	0.513646	+	0.858002i	$0.328295\pi$
$878$	3.57840	0.120765
$879$	5.65426	0.190714
$880$	0	0
$881$	−28.2470	−0.951666	−0.475833	−	0.879536i	$-0.657853\pi$
$881$	−28.2470	−0.951666	−0.475833	+	0.879536i	$0.657853\pi$
$882$	−2.71982	−0.0915812
$883$	4.84763	0.163136	0.0815679	−	0.996668i	$-0.474007\pi$
$883$	4.84763	0.163136	0.0815679	+	0.996668i	$0.474007\pi$
$884$	−29.0074	−0.975624
$885$	1.47406	0.0495499
$886$	11.2916	0.379350
$887$	−25.4879	−0.855800	−0.427900	−	0.903826i	$-0.640746\pi$
$887$	−25.4879	−0.855800	−0.427900	+	0.903826i	$0.640746\pi$
$888$	0.871512	0.0292460
$889$	0.926321	0.0310678
$890$	16.0923	0.539417
$891$	0	0
$892$	−12.2446	−0.409978
$893$	38.6805	1.29439
$894$	−10.3983	−0.347772
$895$	−13.8768	−0.463851
$896$	1.00000	0.0334077
$897$	17.9879	0.600599
$898$	−9.27050	−0.309360
$899$	−0.795105	−0.0265182
$900$	−2.71982	−0.0906608
$901$	28.0214	0.933529
$902$	0	0
$903$	4.83699	0.160965
$904$	4.10257	0.136450
$905$	16.3546	0.543647
$906$	0.895485	0.0297505
$907$	16.2939	0.541030	0.270515	−	0.962716i	$-0.412806\pi$
$907$	16.2939	0.541030	0.270515	+	0.962716i	$0.412806\pi$
$908$	21.0077	0.697164
$909$	29.5756	0.980962
$910$	−4.81243	−0.159530
$911$	−15.5268	−0.514427	−0.257213	−	0.966355i	$-0.582804\pi$
$911$	−15.5268	−0.514427	−0.257213	+	0.966355i	$0.582804\pi$
$912$	1.90221	0.0629884
$913$	0	0
$914$	−41.7715	−1.38168
$915$	−7.76680	−0.256762
$916$	−4.92949	−0.162875
$917$	−13.3755	−0.441697
$918$	−18.2491	−0.602311
$919$	14.5051	0.478478	0.239239	−	0.970961i	$-0.423102\pi$
$919$	14.5051	0.478478	0.239239	+	0.970961i	$0.423102\pi$
$920$	−7.06157	−0.232813
$921$	−9.23478	−0.304296
$922$	14.6501	0.482476
$923$	−58.8445	−1.93689
$924$	0	0
$925$	1.64649	0.0541361
$926$	−28.0291	−0.921095
$927$	−9.30452	−0.305601
$928$	0.353515	0.0116047
$929$	−25.0735	−0.822636	−0.411318	−	0.911492i	$-0.634931\pi$
$929$	−25.0735	−0.822636	−0.411318	+	0.911492i	$0.634931\pi$
$930$	−1.19051	−0.0390383
$931$	3.59370	0.117779
$932$	−20.1605	−0.660380
$933$	5.03348	0.164789
$934$	13.2230	0.432669
$935$	0	0
$936$	13.0890	0.427826
$937$	−36.1117	−1.17972	−0.589859	−	0.807506i	$-0.700817\pi$
$937$	−36.1117	−1.17972	−0.589859	+	0.807506i	$0.700817\pi$
$938$	9.43079	0.307926
$939$	15.4430	0.503964
$940$	10.7634	0.351064
$941$	25.8134	0.841494	0.420747	−	0.907178i	$-0.361768\pi$
$941$	25.8134	0.841494	0.420747	+	0.907178i	$0.361768\pi$
$942$	−7.20674	−0.234808
$943$	−66.0944	−2.15233
$944$	2.78483	0.0906385
$945$	−3.02760	−0.0984878
$946$	0	0
$947$	21.6527	0.703617	0.351809	−	0.936072i	$-0.385567\pi$
$947$	21.6527	0.703617	0.351809	+	0.936072i	$0.385567\pi$
$948$	4.25904	0.138327
$949$	−28.6377	−0.929618
$950$	3.59370	0.116595
$951$	−15.2443	−0.494329
$952$	6.02760	0.195356
$953$	−50.6680	−1.64130	−0.820649	−	0.571432i	$-0.806388\pi$
$953$	−50.6680	−1.64130	−0.820649	+	0.571432i	$0.806388\pi$
$954$	−12.6441	−0.409367
$955$	−16.1410	−0.522311
$956$	−9.53194	−0.308285
$957$	0	0
$958$	−18.6878	−0.603774
$959$	19.8175	0.639939
$960$	0.529317	0.0170836
$961$	−25.9414	−0.836818
$962$	−7.92359	−0.255467
$963$	25.4341	0.819601
$964$	−6.73710	−0.216987
$965$	3.72307	0.119850
$966$	−3.73781	−0.120262
$967$	38.1065	1.22542	0.612711	−	0.790307i	$-0.290079\pi$
$967$	38.1065	1.22542	0.612711	+	0.790307i	$0.290079\pi$
$968$	0	0
$969$	11.4657	0.368333
$970$	−1.23078	−0.0395180
$971$	14.7852	0.474481	0.237241	−	0.971451i	$-0.423757\pi$
$971$	14.7852	0.474481	0.237241	+	0.971451i	$0.423757\pi$
$972$	12.5535	0.402653
$973$	18.7006	0.599513
$974$	−11.4532	−0.366983
$975$	−2.54730	−0.0815788
$976$	−14.6733	−0.469679
$977$	−44.2971	−1.41719	−0.708595	−	0.705616i	$-0.750671\pi$
$977$	−44.2971	−1.41719	−0.708595	+	0.705616i	$0.750671\pi$
$978$	7.44717	0.238134
$979$	0	0
$980$	1.00000	0.0319438
$981$	−54.1516	−1.72893
$982$	32.7871	1.04628
$983$	50.9868	1.62623	0.813113	−	0.582106i	$-0.197771\pi$
$983$	50.9868	1.62623	0.813113	+	0.582106i	$0.197771\pi$
$984$	4.95426	0.157936
$985$	−2.01016	−0.0640491
$986$	2.13085	0.0678599
$987$	5.69725	0.181346
$988$	−17.2944	−0.550210
$989$	−64.5299	−2.05193
$990$	0	0
$991$	32.2027	1.02295	0.511477	−	0.859297i	$-0.329099\pi$
$991$	32.2027	1.02295	0.511477	+	0.859297i	$0.329099\pi$
$992$	−2.24914	−0.0714103
$993$	9.52417	0.302240
$994$	12.2276	0.387836
$995$	−9.97363	−0.316185
$996$	−8.78798	−0.278458
$997$	−14.4477	−0.457562	−0.228781	−	0.973478i	$-0.573474\pi$
$997$	−14.4477	−0.457562	−0.228781	+	0.973478i	$0.573474\pi$
$998$	−31.0534	−0.982979
$999$	−4.98489	−0.157715

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.df.1.3	yes	6
11.10	odd	2		8470.2.a.cz.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.cz.1.4	✓	6	11.10	odd	2
8470.2.a.df.1.3	yes	6	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.529317 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.529317 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.71982 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.529317 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.529317 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.71982 q^{9} +1.00000 q^{10} +0.529317 q^{12} -4.81243 q^{13} +1.00000 q^{14} +0.529317 q^{15} +1.00000 q^{16} +6.02760 q^{17} -2.71982 q^{18} +3.59370 q^{19} +1.00000 q^{20} +0.529317 q^{21} -7.06157 q^{23} +0.529317 q^{24} +1.00000 q^{25} -4.81243 q^{26} -3.02760 q^{27} +1.00000 q^{28} +0.353515 q^{29} +0.529317 q^{30} -2.24914 q^{31} +1.00000 q^{32} +6.02760 q^{34} +1.00000 q^{35} -2.71982 q^{36} +1.64649 q^{37} +3.59370 q^{38} -2.54730 q^{39} +1.00000 q^{40} +9.35973 q^{41} +0.529317 q^{42} +9.13818 q^{43} -2.71982 q^{45} -7.06157 q^{46} +10.7634 q^{47} +0.529317 q^{48} +1.00000 q^{49} +1.00000 q^{50} +3.19051 q^{51} -4.81243 q^{52} +4.64885 q^{53} -3.02760 q^{54} +1.00000 q^{56} +1.90221 q^{57} +0.353515 q^{58} +2.78483 q^{59} +0.529317 q^{60} -14.6733 q^{61} -2.24914 q^{62} -2.71982 q^{63} +1.00000 q^{64} -4.81243 q^{65} +9.43079 q^{67} +6.02760 q^{68} -3.73781 q^{69} +1.00000 q^{70} +12.2276 q^{71} -2.71982 q^{72} +5.95077 q^{73} +1.64649 q^{74} +0.529317 q^{75} +3.59370 q^{76} -2.54730 q^{78} +8.04631 q^{79} +1.00000 q^{80} +6.55691 q^{81} +9.35973 q^{82} -16.6025 q^{83} +0.529317 q^{84} +6.02760 q^{85} +9.13818 q^{86} +0.187121 q^{87} +16.0923 q^{89} -2.71982 q^{90} -4.81243 q^{91} -7.06157 q^{92} -1.19051 q^{93} +10.7634 q^{94} +3.59370 q^{95} +0.529317 q^{96} -1.23078 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 4 q^{3} + 6 q^{4} + 6 q^{5} + 4 q^{6} + 6 q^{7} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 + 4 * q^3 + 6 * q^4 + 6 * q^5 + 4 * q^6 + 6 * q^7 + 6 * q^8 + 2 * q^9	\( 6 q + 6 q^{2} + 4 q^{3} + 6 q^{4} + 6 q^{5} + 4 q^{6} + 6 q^{7} + 6 q^{8} + 2 q^{9} + 6 q^{10} + 4 q^{12} + 6 q^{14} + 4 q^{15} + 6 q^{16} + 2 q^{17} + 2 q^{18} + 6 q^{20} + 4 q^{21} + 4 q^{23} + 4 q^{24} + 6 q^{25} + 16 q^{27} + 6 q^{28} + 8 q^{29} + 4 q^{30} + 4 q^{31} + 6 q^{32} + 2 q^{34} + 6 q^{35} + 2 q^{36} + 4 q^{37} + 6 q^{40} + 12 q^{41} + 4 q^{42} - 6 q^{43} + 2 q^{45} + 4 q^{46} + 16 q^{47} + 4 q^{48} + 6 q^{49} + 6 q^{50} + 12 q^{53} + 16 q^{54} + 6 q^{56} + 8 q^{57} + 8 q^{58} + 22 q^{59} + 4 q^{60} - 4 q^{61} + 4 q^{62} + 2 q^{63} + 6 q^{64} + 20 q^{67} + 2 q^{68} + 12 q^{69} + 6 q^{70} + 14 q^{71} + 2 q^{72} + 18 q^{73} + 4 q^{74} + 4 q^{75} + 32 q^{79} + 6 q^{80} + 6 q^{81} + 12 q^{82} - 16 q^{83} + 4 q^{84} + 2 q^{85} - 6 q^{86} - 4 q^{87} + 4 q^{89} + 2 q^{90} + 4 q^{92} + 12 q^{93} + 16 q^{94} + 4 q^{96} + 4 q^{97} + 6 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 + 4 * q^3 + 6 * q^4 + 6 * q^5 + 4 * q^6 + 6 * q^7 + 6 * q^8 + 2 * q^9 + 6 * q^10 + 4 * q^12 + 6 * q^14 + 4 * q^15 + 6 * q^16 + 2 * q^17 + 2 * q^18 + 6 * q^20 + 4 * q^21 + 4 * q^23 + 4 * q^24 + 6 * q^25 + 16 * q^27 + 6 * q^28 + 8 * q^29 + 4 * q^30 + 4 * q^31 + 6 * q^32 + 2 * q^34 + 6 * q^35 + 2 * q^36 + 4 * q^37 + 6 * q^40 + 12 * q^41 + 4 * q^42 - 6 * q^43 + 2 * q^45 + 4 * q^46 + 16 * q^47 + 4 * q^48 + 6 * q^49 + 6 * q^50 + 12 * q^53 + 16 * q^54 + 6 * q^56 + 8 * q^57 + 8 * q^58 + 22 * q^59 + 4 * q^60 - 4 * q^61 + 4 * q^62 + 2 * q^63 + 6 * q^64 + 20 * q^67 + 2 * q^68 + 12 * q^69 + 6 * q^70 + 14 * q^71 + 2 * q^72 + 18 * q^73 + 4 * q^74 + 4 * q^75 + 32 * q^79 + 6 * q^80 + 6 * q^81 + 12 * q^82 - 16 * q^83 + 4 * q^84 + 2 * q^85 - 6 * q^86 - 4 * q^87 + 4 * q^89 + 2 * q^90 + 4 * q^92 + 12 * q^93 + 16 * q^94 + 4 * q^96 + 4 * q^97 + 6 * q^98

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