LMFDB - Embedded newform 8470.2.a.df.1.2

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			1.2
Root			$2.40765$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.34292 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.34292 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.19656 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.34292 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.34292 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.19656 q^{9} +1.00000 q^{10} -1.34292 q^{12} +4.31157 q^{13} +1.00000 q^{14} -1.34292 q^{15} +1.00000 q^{16} -2.63565 q^{17} -1.19656 q^{18} +1.56877 q^{19} +1.00000 q^{20} -1.34292 q^{21} +5.45794 q^{23} -1.34292 q^{24} +1.00000 q^{25} +4.31157 q^{26} +5.63565 q^{27} +1.00000 q^{28} +2.83920 q^{29} -1.34292 q^{30} +1.14637 q^{31} +1.00000 q^{32} -2.63565 q^{34} +1.00000 q^{35} -1.19656 q^{36} -0.839198 q^{37} +1.56877 q^{38} -5.79011 q^{39} +1.00000 q^{40} +3.47854 q^{41} -1.34292 q^{42} -2.01075 q^{43} -1.19656 q^{45} +5.45794 q^{46} -8.39442 q^{47} -1.34292 q^{48} +1.00000 q^{49} +1.00000 q^{50} +3.53948 q^{51} +4.31157 q^{52} +1.40604 q^{53} +5.63565 q^{54} +1.00000 q^{56} -2.10674 q^{57} +2.83920 q^{58} +2.32408 q^{59} -1.34292 q^{60} -5.24387 q^{61} +1.14637 q^{62} -1.19656 q^{63} +1.00000 q^{64} +4.31157 q^{65} -1.60636 q^{67} -2.63565 q^{68} -7.32959 q^{69} +1.00000 q^{70} -13.8057 q^{71} -1.19656 q^{72} -1.14830 q^{73} -0.839198 q^{74} -1.34292 q^{75} +1.56877 q^{76} -5.79011 q^{78} +9.65408 q^{79} +1.00000 q^{80} -3.97858 q^{81} +3.47854 q^{82} -2.51880 q^{83} -1.34292 q^{84} -2.63565 q^{85} -2.01075 q^{86} -3.81282 q^{87} +2.19326 q^{89} -1.19656 q^{90} +4.31157 q^{91} +5.45794 q^{92} -1.53948 q^{93} -8.39442 q^{94} +1.56877 q^{95} -1.34292 q^{96} -7.56372 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.34292	−0.775337	−0.387669	−	0.921799i	$-0.626719\pi$
$3$	−1.34292	−0.775337	−0.387669	+	0.921799i	$0.626719\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.34292	−0.548246
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	−1.19656	−0.398853
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	−1.34292	−0.387669
$13$	4.31157	1.19582	0.597908	−	0.801565i	$-0.295999\pi$
$13$	4.31157	1.19582	0.597908	+	0.801565i	$0.295999\pi$
$14$	1.00000	0.267261
$15$	−1.34292	−0.346741
$16$	1.00000	0.250000
$17$	−2.63565	−0.639240	−0.319620	−	0.947546i	$-0.603555\pi$
$17$	−2.63565	−0.639240	−0.319620	+	0.947546i	$0.603555\pi$
$18$	−1.19656	−0.282031
$19$	1.56877	0.359901	0.179951	−	0.983676i	$-0.442406\pi$
$19$	1.56877	0.359901	0.179951	+	0.983676i	$0.442406\pi$
$20$	1.00000	0.223607
$21$	−1.34292	−0.293050
$22$	0	0
$23$	5.45794	1.13806	0.569030	−	0.822317i	$-0.307319\pi$
$23$	5.45794	1.13806	0.569030	+	0.822317i	$0.307319\pi$
$24$	−1.34292	−0.274123
$25$	1.00000	0.200000
$26$	4.31157	0.845569
$27$	5.63565	1.08458
$28$	1.00000	0.188982
$29$	2.83920	0.527226	0.263613	−	0.964629i	$-0.415086\pi$
$29$	2.83920	0.527226	0.263613	+	0.964629i	$0.415086\pi$
$30$	−1.34292	−0.245183
$31$	1.14637	0.205893	0.102947	−	0.994687i	$-0.467173\pi$
$31$	1.14637	0.205893	0.102947	+	0.994687i	$0.467173\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.63565	−0.452011
$35$	1.00000	0.169031
$36$	−1.19656	−0.199426
$37$	−0.839198	−0.137963	−0.0689816	−	0.997618i	$-0.521975\pi$
$37$	−0.839198	−0.137963	−0.0689816	+	0.997618i	$0.521975\pi$
$38$	1.56877	0.254489
$39$	−5.79011	−0.927160
$40$	1.00000	0.158114
$41$	3.47854	0.543256	0.271628	−	0.962402i	$-0.412438\pi$
$41$	3.47854	0.543256	0.271628	+	0.962402i	$0.412438\pi$
$42$	−1.34292	−0.207218
$43$	−2.01075	−0.306637	−0.153318	−	0.988177i	$-0.548996\pi$
$43$	−2.01075	−0.306637	−0.153318	+	0.988177i	$0.548996\pi$
$44$	0	0
$45$	−1.19656	−0.178372
$46$	5.45794	0.804729
$47$	−8.39442	−1.22445	−0.612226	−	0.790683i	$-0.709726\pi$
$47$	−8.39442	−1.22445	−0.612226	+	0.790683i	$0.709726\pi$
$48$	−1.34292	−0.193834
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	3.53948	0.495626
$52$	4.31157	0.597908
$53$	1.40604	0.193134	0.0965672	−	0.995326i	$-0.469214\pi$
$53$	1.40604	0.193134	0.0965672	+	0.995326i	$0.469214\pi$
$54$	5.63565	0.766915
$55$	0	0
$56$	1.00000	0.133631
$57$	−2.10674	−0.279045
$58$	2.83920	0.372805
$59$	2.32408	0.302569	0.151285	−	0.988490i	$-0.451659\pi$
$59$	2.32408	0.302569	0.151285	+	0.988490i	$0.451659\pi$
$60$	−1.34292	−0.173371
$61$	−5.24387	−0.671409	−0.335704	−	0.941967i	$-0.608974\pi$
$61$	−5.24387	−0.671409	−0.335704	+	0.941967i	$0.608974\pi$
$62$	1.14637	0.145589
$63$	−1.19656	−0.150752
$64$	1.00000	0.125000
$65$	4.31157	0.534785
$66$	0	0
$67$	−1.60636	−0.196248	−0.0981241	−	0.995174i	$-0.531284\pi$
$67$	−1.60636	−0.196248	−0.0981241	+	0.995174i	$0.531284\pi$
$68$	−2.63565	−0.319620
$69$	−7.32959	−0.882379
$70$	1.00000	0.119523
$71$	−13.8057	−1.63844	−0.819220	−	0.573480i	$-0.805593\pi$
$71$	−13.8057	−1.63844	−0.819220	+	0.573480i	$0.805593\pi$
$72$	−1.19656	−0.141016
$73$	−1.14830	−0.134398	−0.0671990	−	0.997740i	$-0.521406\pi$
$73$	−1.14830	−0.134398	−0.0671990	+	0.997740i	$0.521406\pi$
$74$	−0.839198	−0.0975548
$75$	−1.34292	−0.155067
$76$	1.56877	0.179951
$77$	0	0
$78$	−5.79011	−0.655601
$79$	9.65408	1.08617	0.543084	−	0.839678i	$-0.317256\pi$
$79$	9.65408	1.08617	0.543084	+	0.839678i	$0.317256\pi$
$80$	1.00000	0.111803
$81$	−3.97858	−0.442064
$82$	3.47854	0.384140
$83$	−2.51880	−0.276475	−0.138237	−	0.990399i	$-0.544144\pi$
$83$	−2.51880	−0.276475	−0.138237	+	0.990399i	$0.544144\pi$
$84$	−1.34292	−0.146525
$85$	−2.63565	−0.285877
$86$	−2.01075	−0.216825
$87$	−3.81282	−0.408778
$88$	0	0
$89$	2.19326	0.232485	0.116242	−	0.993221i	$-0.462915\pi$
$89$	2.19326	0.232485	0.116242	+	0.993221i	$0.462915\pi$
$90$	−1.19656	−0.126128
$91$	4.31157	0.451976
$92$	5.45794	0.569030
$93$	−1.53948	−0.159637
$94$	−8.39442	−0.865818
$95$	1.56877	0.160953
$96$	−1.34292	−0.137062
$97$	−7.56372	−0.767979	−0.383990	−	0.923337i	$-0.625450\pi$
$97$	−7.56372	−0.767979	−0.383990	+	0.923337i	$0.625450\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	17.6449	1.75574	0.877868	−	0.478902i	$-0.158965\pi$
$101$	17.6449	1.75574	0.877868	+	0.478902i	$0.158965\pi$
$102$	3.53948	0.350461
$103$	10.1528	1.00039	0.500194	−	0.865913i	$-0.333262\pi$
$103$	10.1528	1.00039	0.500194	+	0.865913i	$0.333262\pi$
$104$	4.31157	0.422785
$105$	−1.34292	−0.131056
$106$	1.40604	0.136567
$107$	15.4232	1.49102	0.745508	−	0.666497i	$-0.232207\pi$
$107$	15.4232	1.49102	0.745508	+	0.666497i	$0.232207\pi$
$108$	5.63565	0.542291
$109$	8.49656	0.813823	0.406911	−	0.913468i	$-0.366606\pi$
$109$	8.49656	0.813823	0.406911	+	0.913468i	$0.366606\pi$
$110$	0	0
$111$	1.12698	0.106968
$112$	1.00000	0.0944911
$113$	2.72299	0.256158	0.128079	−	0.991764i	$-0.459119\pi$
$113$	2.72299	0.256158	0.128079	+	0.991764i	$0.459119\pi$
$114$	−2.10674	−0.197314
$115$	5.45794	0.508956
$116$	2.83920	0.263613
$117$	−5.15905	−0.476954
$118$	2.32408	0.213949
$119$	−2.63565	−0.241610
$120$	−1.34292	−0.122592
$121$	0	0
$122$	−5.24387	−0.474758
$123$	−4.67141	−0.421207
$124$	1.14637	0.102947
$125$	1.00000	0.0894427
$126$	−1.19656	−0.106598
$127$	−9.21928	−0.818079	−0.409040	−	0.912517i	$-0.634136\pi$
$127$	−9.21928	−0.818079	−0.409040	+	0.912517i	$0.634136\pi$
$128$	1.00000	0.0883883
$129$	2.70028	0.237747
$130$	4.31157	0.378150
$131$	17.5357	1.53210	0.766052	−	0.642779i	$-0.222219\pi$
$131$	17.5357	1.53210	0.766052	+	0.642779i	$0.222219\pi$
$132$	0	0
$133$	1.56877	0.136030
$134$	−1.60636	−0.138768
$135$	5.63565	0.485040
$136$	−2.63565	−0.226005
$137$	9.12927	0.779966	0.389983	−	0.920822i	$-0.372481\pi$
$137$	9.12927	0.779966	0.389983	+	0.920822i	$0.372481\pi$
$138$	−7.32959	−0.623936
$139$	0.657844	0.0557976	0.0278988	−	0.999611i	$-0.491118\pi$
$139$	0.657844	0.0557976	0.0278988	+	0.999611i	$0.491118\pi$
$140$	1.00000	0.0845154
$141$	11.2731	0.949363
$142$	−13.8057	−1.15855
$143$	0	0
$144$	−1.19656	−0.0997131
$145$	2.83920	0.235783
$146$	−1.14830	−0.0950337
$147$	−1.34292	−0.110762
$148$	−0.839198	−0.0689816
$149$	−20.5982	−1.68747	−0.843737	−	0.536757i	$-0.819649\pi$
$149$	−20.5982	−1.68747	−0.843737	+	0.536757i	$0.819649\pi$
$150$	−1.34292	−0.109649
$151$	−4.29219	−0.349293	−0.174647	−	0.984631i	$-0.555878\pi$
$151$	−4.29219	−0.349293	−0.174647	+	0.984631i	$0.555878\pi$
$152$	1.56877	0.127244
$153$	3.15371	0.254962
$154$	0	0
$155$	1.14637	0.0920783
$156$	−5.79011	−0.463580
$157$	10.9039	0.870226	0.435113	−	0.900376i	$-0.356708\pi$
$157$	10.9039	0.870226	0.435113	+	0.900376i	$0.356708\pi$
$158$	9.65408	0.768037
$159$	−1.88820	−0.149744
$160$	1.00000	0.0790569
$161$	5.45794	0.430146
$162$	−3.97858	−0.312587
$163$	12.3525	0.967526	0.483763	−	0.875199i	$-0.339270\pi$
$163$	12.3525	0.967526	0.483763	+	0.875199i	$0.339270\pi$
$164$	3.47854	0.271628
$165$	0	0
$166$	−2.51880	−0.195497
$167$	−6.93334	−0.536518	−0.268259	−	0.963347i	$-0.586448\pi$
$167$	−6.93334	−0.536518	−0.268259	+	0.963347i	$0.586448\pi$
$168$	−1.34292	−0.103609
$169$	5.58967	0.429975
$170$	−2.63565	−0.202145
$171$	−1.87713	−0.143548
$172$	−2.01075	−0.153318
$173$	−11.5843	−0.880741	−0.440370	−	0.897816i	$-0.645153\pi$
$173$	−11.5843	−0.880741	−0.440370	+	0.897816i	$0.645153\pi$
$174$	−3.81282	−0.289049
$175$	1.00000	0.0755929
$176$	0	0
$177$	−3.12106	−0.234593
$178$	2.19326	0.164392
$179$	20.4825	1.53093	0.765466	−	0.643476i	$-0.222508\pi$
$179$	20.4825	1.53093	0.765466	+	0.643476i	$0.222508\pi$
$180$	−1.19656	−0.0891861
$181$	−8.01175	−0.595509	−0.297754	−	0.954642i	$-0.596238\pi$
$181$	−8.01175	−0.595509	−0.297754	+	0.954642i	$0.596238\pi$
$182$	4.31157	0.319595
$183$	7.04211	0.520568
$184$	5.45794	0.402365
$185$	−0.839198	−0.0616990
$186$	−1.53948	−0.112880
$187$	0	0
$188$	−8.39442	−0.612226
$189$	5.63565	0.409934
$190$	1.56877	0.113811
$191$	7.72371	0.558868	0.279434	−	0.960165i	$-0.409853\pi$
$191$	7.72371	0.558868	0.279434	+	0.960165i	$0.409853\pi$
$192$	−1.34292	−0.0969171
$193$	−16.3855	−1.17945	−0.589727	−	0.807603i	$-0.700765\pi$
$193$	−16.3855	−1.17945	−0.589727	+	0.807603i	$0.700765\pi$
$194$	−7.56372	−0.543043
$195$	−5.79011	−0.414639
$196$	1.00000	0.0714286
$197$	5.51971	0.393263	0.196631	−	0.980477i	$-0.437000\pi$
$197$	5.51971	0.393263	0.196631	+	0.980477i	$0.437000\pi$
$198$	0	0
$199$	4.27443	0.303006	0.151503	−	0.988457i	$-0.451589\pi$
$199$	4.27443	0.303006	0.151503	+	0.988457i	$0.451589\pi$
$200$	1.00000	0.0707107
$201$	2.15722	0.152159
$202$	17.6449	1.24149
$203$	2.83920	0.199273
$204$	3.53948	0.247813
$205$	3.47854	0.242952
$206$	10.1528	0.707381
$207$	−6.53074	−0.453918
$208$	4.31157	0.298954
$209$	0	0
$210$	−1.34292	−0.0926705
$211$	−19.6106	−1.35005	−0.675026	−	0.737794i	$-0.735868\pi$
$211$	−19.6106	−1.35005	−0.675026	+	0.737794i	$0.735868\pi$
$212$	1.40604	0.0965672
$213$	18.5400	1.27034
$214$	15.4232	1.05431
$215$	−2.01075	−0.137132
$216$	5.63565	0.383458
$217$	1.14637	0.0778204
$218$	8.49656	0.575459
$219$	1.54207	0.104204
$220$	0	0
$221$	−11.3638	−0.764413
$222$	1.12698	0.0756378
$223$	4.01943	0.269161	0.134580	−	0.990903i	$-0.457031\pi$
$223$	4.01943	0.269161	0.134580	+	0.990903i	$0.457031\pi$
$224$	1.00000	0.0668153
$225$	−1.19656	−0.0797705
$226$	2.72299	0.181131
$227$	20.4787	1.35922	0.679610	−	0.733574i	$-0.262149\pi$
$227$	20.4787	1.35922	0.679610	+	0.733574i	$0.262149\pi$
$228$	−2.10674	−0.139522
$229$	8.74872	0.578132	0.289066	−	0.957309i	$-0.406655\pi$
$229$	8.74872	0.578132	0.289066	+	0.957309i	$0.406655\pi$
$230$	5.45794	0.359886
$231$	0	0
$232$	2.83920	0.186402
$233$	14.1462	0.926748	0.463374	−	0.886163i	$-0.346639\pi$
$233$	14.1462	0.926748	0.463374	+	0.886163i	$0.346639\pi$
$234$	−5.15905	−0.337257
$235$	−8.39442	−0.547591
$236$	2.32408	0.151285
$237$	−12.9647	−0.842147
$238$	−2.63565	−0.170844
$239$	2.12134	0.137218	0.0686091	−	0.997644i	$-0.478144\pi$
$239$	2.12134	0.137218	0.0686091	+	0.997644i	$0.478144\pi$
$240$	−1.34292	−0.0866853
$241$	17.4357	1.12313	0.561566	−	0.827432i	$-0.310199\pi$
$241$	17.4357	1.12313	0.561566	+	0.827432i	$0.310199\pi$
$242$	0	0
$243$	−11.5640	−0.741833
$244$	−5.24387	−0.335704
$245$	1.00000	0.0638877
$246$	−4.67141	−0.297838
$247$	6.76388	0.430376
$248$	1.14637	0.0727943
$249$	3.38256	0.214361
$250$	1.00000	0.0632456
$251$	0.936840	0.0591328	0.0295664	−	0.999563i	$-0.490587\pi$
$251$	0.936840	0.0591328	0.0295664	+	0.999563i	$0.490587\pi$
$252$	−1.19656	−0.0753760
$253$	0	0
$254$	−9.21928	−0.578469
$255$	3.53948	0.221651
$256$	1.00000	0.0625000
$257$	13.7242	0.856095	0.428047	−	0.903756i	$-0.359202\pi$
$257$	13.7242	0.856095	0.428047	+	0.903756i	$0.359202\pi$
$258$	2.70028	0.168112
$259$	−0.839198	−0.0521452
$260$	4.31157	0.267392
$261$	−3.39726	−0.210285
$262$	17.5357	1.08336
$263$	−18.9341	−1.16753	−0.583763	−	0.811924i	$-0.698420\pi$
$263$	−18.9341	−1.16753	−0.583763	+	0.811924i	$0.698420\pi$
$264$	0	0
$265$	1.40604	0.0863723
$266$	1.56877	0.0961877
$267$	−2.94538	−0.180254
$268$	−1.60636	−0.0981241
$269$	12.9639	0.790422	0.395211	−	0.918590i	$-0.370672\pi$
$269$	12.9639	0.790422	0.395211	+	0.918590i	$0.370672\pi$
$270$	5.63565	0.342975
$271$	23.5085	1.42804	0.714020	−	0.700125i	$-0.246873\pi$
$271$	23.5085	1.42804	0.714020	+	0.700125i	$0.246873\pi$
$272$	−2.63565	−0.159810
$273$	−5.79011	−0.350434
$274$	9.12927	0.551519
$275$	0	0
$276$	−7.32959	−0.441190
$277$	2.44560	0.146942	0.0734710	−	0.997297i	$-0.476592\pi$
$277$	2.44560	0.146942	0.0734710	+	0.997297i	$0.476592\pi$
$278$	0.657844	0.0394549
$279$	−1.37169	−0.0821211
$280$	1.00000	0.0597614
$281$	−18.0116	−1.07448	−0.537242	−	0.843428i	$-0.680534\pi$
$281$	−18.0116	−1.07448	−0.537242	+	0.843428i	$0.680534\pi$
$282$	11.2731	0.671301
$283$	−9.78930	−0.581914	−0.290957	−	0.956736i	$-0.593974\pi$
$283$	−9.78930	−0.581914	−0.290957	+	0.956736i	$0.593974\pi$
$284$	−13.8057	−0.819220
$285$	−2.10674	−0.124793
$286$	0	0
$287$	3.47854	0.205332
$288$	−1.19656	−0.0705078
$289$	−10.0533	−0.591372
$290$	2.83920	0.166723
$291$	10.1575	0.595443
$292$	−1.14830	−0.0671990
$293$	−8.44836	−0.493558	−0.246779	−	0.969072i	$-0.579372\pi$
$293$	−8.44836	−0.493558	−0.246779	+	0.969072i	$0.579372\pi$
$294$	−1.34292	−0.0783209
$295$	2.32408	0.135313
$296$	−0.839198	−0.0487774
$297$	0	0
$298$	−20.5982	−1.19322
$299$	23.5323	1.36091
$300$	−1.34292	−0.0775337
$301$	−2.01075	−0.115898
$302$	−4.29219	−0.246988
$303$	−23.6958	−1.36129
$304$	1.56877	0.0899753
$305$	−5.24387	−0.300263
$306$	3.15371	0.180286
$307$	−9.34689	−0.533455	−0.266728	−	0.963772i	$-0.585942\pi$
$307$	−9.34689	−0.533455	−0.266728	+	0.963772i	$0.585942\pi$
$308$	0	0
$309$	−13.6345	−0.775638
$310$	1.14637	0.0651092
$311$	−27.6729	−1.56919	−0.784593	−	0.620011i	$-0.787128\pi$
$311$	−27.6729	−1.56919	−0.784593	+	0.620011i	$0.787128\pi$
$312$	−5.79011	−0.327801
$313$	5.34524	0.302131	0.151065	−	0.988524i	$-0.451730\pi$
$313$	5.34524	0.302131	0.151065	+	0.988524i	$0.451730\pi$
$314$	10.9039	0.615343
$315$	−1.19656	−0.0674184
$316$	9.65408	0.543084
$317$	7.13917	0.400976	0.200488	−	0.979696i	$-0.435747\pi$
$317$	7.13917	0.400976	0.200488	+	0.979696i	$0.435747\pi$
$318$	−1.88820	−0.105885
$319$	0	0
$320$	1.00000	0.0559017
$321$	−20.7122	−1.15604
$322$	5.45794	0.304159
$323$	−4.13474	−0.230063
$324$	−3.97858	−0.221032
$325$	4.31157	0.239163
$326$	12.3525	0.684144
$327$	−11.4102	−0.630987
$328$	3.47854	0.192070
$329$	−8.39442	−0.462799
$330$	0	0
$331$	5.83207	0.320559	0.160280	−	0.987072i	$-0.448760\pi$
$331$	5.83207	0.320559	0.160280	+	0.987072i	$0.448760\pi$
$332$	−2.51880	−0.138237
$333$	1.00415	0.0550270
$334$	−6.93334	−0.379375
$335$	−1.60636	−0.0877649
$336$	−1.34292	−0.0732625
$337$	22.2848	1.21393	0.606964	−	0.794729i	$-0.292387\pi$
$337$	22.2848	1.21393	0.606964	+	0.794729i	$0.292387\pi$
$338$	5.58967	0.304038
$339$	−3.65677	−0.198609
$340$	−2.63565	−0.142938
$341$	0	0
$342$	−1.87713	−0.101503
$343$	1.00000	0.0539949
$344$	−2.01075	−0.108412
$345$	−7.32959	−0.394612
$346$	−11.5843	−0.622778
$347$	16.6253	0.892495	0.446248	−	0.894910i	$-0.352760\pi$
$347$	16.6253	0.892495	0.446248	+	0.894910i	$0.352760\pi$
$348$	−3.81282	−0.204389
$349$	3.23934	0.173398	0.0866989	−	0.996235i	$-0.472368\pi$
$349$	3.23934	0.173398	0.0866989	+	0.996235i	$0.472368\pi$
$350$	1.00000	0.0534522
$351$	24.2985	1.29696
$352$	0	0
$353$	−24.6580	−1.31241	−0.656205	−	0.754582i	$-0.727839\pi$
$353$	−24.6580	−1.31241	−0.656205	+	0.754582i	$0.727839\pi$
$354$	−3.12106	−0.165882
$355$	−13.8057	−0.732732
$356$	2.19326	0.116242
$357$	3.53948	0.187329
$358$	20.4825	1.08253
$359$	20.7493	1.09511	0.547554	−	0.836771i	$-0.315559\pi$
$359$	20.7493	1.09511	0.547554	+	0.836771i	$0.315559\pi$
$360$	−1.19656	−0.0630641
$361$	−16.5390	−0.870471
$362$	−8.01175	−0.421088
$363$	0	0
$364$	4.31157	0.225988
$365$	−1.14830	−0.0601046
$366$	7.04211	0.368097
$367$	2.12160	0.110746	0.0553732	−	0.998466i	$-0.482365\pi$
$367$	2.12160	0.110746	0.0553732	+	0.998466i	$0.482365\pi$
$368$	5.45794	0.284515
$369$	−4.16227	−0.216679
$370$	−0.839198	−0.0436278
$371$	1.40604	0.0729979
$372$	−1.53948	−0.0798184
$373$	−7.19996	−0.372800	−0.186400	−	0.982474i	$-0.559682\pi$
$373$	−7.19996	−0.372800	−0.186400	+	0.982474i	$0.559682\pi$
$374$	0	0
$375$	−1.34292	−0.0693482
$376$	−8.39442	−0.432909
$377$	12.2414	0.630465
$378$	5.63565	0.289867
$379$	−23.1300	−1.18811	−0.594055	−	0.804424i	$-0.702474\pi$
$379$	−23.1300	−1.18811	−0.594055	+	0.804424i	$0.702474\pi$
$380$	1.56877	0.0804764
$381$	12.3808	0.634287
$382$	7.72371	0.395180
$383$	31.4479	1.60691	0.803455	−	0.595365i	$-0.202993\pi$
$383$	31.4479	1.60691	0.803455	+	0.595365i	$0.202993\pi$
$384$	−1.34292	−0.0685308
$385$	0	0
$386$	−16.3855	−0.834000
$387$	2.40598	0.122303
$388$	−7.56372	−0.383990
$389$	17.5618	0.890420	0.445210	−	0.895426i	$-0.353129\pi$
$389$	17.5618	0.890420	0.445210	+	0.895426i	$0.353129\pi$
$390$	−5.79011	−0.293194
$391$	−14.3852	−0.727493
$392$	1.00000	0.0505076
$393$	−23.5491	−1.18790
$394$	5.51971	0.278079
$395$	9.65408	0.485749
$396$	0	0
$397$	32.7839	1.64538	0.822689	−	0.568492i	$-0.192473\pi$
$397$	32.7839	1.64538	0.822689	+	0.568492i	$0.192473\pi$
$398$	4.27443	0.214258
$399$	−2.10674	−0.105469
$400$	1.00000	0.0500000
$401$	9.25146	0.461996	0.230998	−	0.972954i	$-0.425801\pi$
$401$	9.25146	0.461996	0.230998	+	0.972954i	$0.425801\pi$
$402$	2.15722	0.107592
$403$	4.94264	0.246210
$404$	17.6449	0.877868
$405$	−3.97858	−0.197697
$406$	2.83920	0.140907
$407$	0	0
$408$	3.53948	0.175230
$409$	−0.0174947	−0.000865057 0	−0.000432529	−	1.00000i	$-0.500138\pi$
$409$	−0.0174947	−0.000865057 0	−0.000432529		1.00000i	$0.500138\pi$
$410$	3.47854	0.171793
$411$	−12.2599	−0.604737
$412$	10.1528	0.500194
$413$	2.32408	0.114360
$414$	−6.53074	−0.320968
$415$	−2.51880	−0.123643
$416$	4.31157	0.211392
$417$	−0.883434	−0.0432620
$418$	0	0
$419$	32.4579	1.58567	0.792837	−	0.609433i	$-0.208603\pi$
$419$	32.4579	1.58567	0.792837	+	0.609433i	$0.208603\pi$
$420$	−1.34292	−0.0655279
$421$	9.15329	0.446104	0.223052	−	0.974807i	$-0.428398\pi$
$421$	9.15329	0.446104	0.223052	+	0.974807i	$0.428398\pi$
$422$	−19.6106	−0.954631
$423$	10.0444	0.488376
$424$	1.40604	0.0682833
$425$	−2.63565	−0.127848
$426$	18.5400	0.898268
$427$	−5.24387	−0.253769
$428$	15.4232	0.745508
$429$	0	0
$430$	−2.01075	−0.0969670
$431$	−26.0024	−1.25249	−0.626247	−	0.779625i	$-0.715410\pi$
$431$	−26.0024	−1.25249	−0.626247	+	0.779625i	$0.715410\pi$
$432$	5.63565	0.271146
$433$	23.6006	1.13417	0.567085	−	0.823659i	$-0.308071\pi$
$433$	23.6006	1.13417	0.567085	+	0.823659i	$0.308071\pi$
$434$	1.14637	0.0550273
$435$	−3.81282	−0.182811
$436$	8.49656	0.406911
$437$	8.56227	0.409589
$438$	1.54207	0.0736832
$439$	31.7894	1.51722	0.758612	−	0.651543i	$-0.225878\pi$
$439$	31.7894	1.51722	0.758612	+	0.651543i	$0.225878\pi$
$440$	0	0
$441$	−1.19656	−0.0569789
$442$	−11.3638	−0.540522
$443$	36.1071	1.71550	0.857750	−	0.514067i	$-0.171862\pi$
$443$	36.1071	1.71550	0.857750	+	0.514067i	$0.171862\pi$
$444$	1.12698	0.0534840
$445$	2.19326	0.103970
$446$	4.01943	0.190326
$447$	27.6619	1.30836
$448$	1.00000	0.0472456
$449$	−4.52269	−0.213439	−0.106719	−	0.994289i	$-0.534035\pi$
$449$	−4.52269	−0.213439	−0.106719	+	0.994289i	$0.534035\pi$
$450$	−1.19656	−0.0564063
$451$	0	0
$452$	2.72299	0.128079
$453$	5.76408	0.270820
$454$	20.4787	0.961114
$455$	4.31157	0.202130
$456$	−2.10674	−0.0986572
$457$	−15.1593	−0.709120	−0.354560	−	0.935033i	$-0.615369\pi$
$457$	−15.1593	−0.709120	−0.354560	+	0.935033i	$0.615369\pi$
$458$	8.74872	0.408801
$459$	−14.8536	−0.693308
$460$	5.45794	0.254478
$461$	−27.6126	−1.28605	−0.643024	−	0.765846i	$-0.722320\pi$
$461$	−27.6126	−1.28605	−0.643024	+	0.765846i	$0.722320\pi$
$462$	0	0
$463$	13.3557	0.620692	0.310346	−	0.950624i	$-0.399555\pi$
$463$	13.3557	0.620692	0.310346	+	0.950624i	$0.399555\pi$
$464$	2.83920	0.131806
$465$	−1.53948	−0.0713917
$466$	14.1462	0.655310
$467$	26.3287	1.21835	0.609173	−	0.793038i	$-0.291502\pi$
$467$	26.3287	1.21835	0.609173	+	0.793038i	$0.291502\pi$
$468$	−5.15905	−0.238477
$469$	−1.60636	−0.0741749
$470$	−8.39442	−0.387206
$471$	−14.6431	−0.674719
$472$	2.32408	0.106974
$473$	0	0
$474$	−12.9647	−0.595488
$475$	1.56877	0.0719803
$476$	−2.63565	−0.120805
$477$	−1.68241	−0.0770322
$478$	2.12134	0.0970279
$479$	8.18258	0.373872	0.186936	−	0.982372i	$-0.440144\pi$
$479$	8.18258	0.373872	0.186936	+	0.982372i	$0.440144\pi$
$480$	−1.34292	−0.0612958
$481$	−3.61826	−0.164979
$482$	17.4357	0.794174
$483$	−7.32959	−0.333508
$484$	0	0
$485$	−7.56372	−0.343451
$486$	−11.5640	−0.524555
$487$	−9.09993	−0.412357	−0.206179	−	0.978514i	$-0.566103\pi$
$487$	−9.09993	−0.412357	−0.206179	+	0.978514i	$0.566103\pi$
$488$	−5.24387	−0.237379
$489$	−16.5885	−0.750159
$490$	1.00000	0.0451754
$491$	−3.80239	−0.171600	−0.0857998	−	0.996312i	$-0.527345\pi$
$491$	−3.80239	−0.171600	−0.0857998	+	0.996312i	$0.527345\pi$
$492$	−4.67141	−0.210603
$493$	−7.48314	−0.337024
$494$	6.76388	0.304321
$495$	0	0
$496$	1.14637	0.0514733
$497$	−13.8057	−0.619272
$498$	3.38256	0.151576
$499$	−22.9856	−1.02898	−0.514488	−	0.857497i	$-0.672018\pi$
$499$	−22.9856	−1.02898	−0.514488	+	0.857497i	$0.672018\pi$
$500$	1.00000	0.0447214
$501$	9.31094	0.415982
$502$	0.936840	0.0418132
$503$	13.4287	0.598756	0.299378	−	0.954135i	$-0.403221\pi$
$503$	13.4287	0.598756	0.299378	+	0.954135i	$0.403221\pi$
$504$	−1.19656	−0.0532989
$505$	17.6449	0.785189
$506$	0	0
$507$	−7.50650	−0.333375
$508$	−9.21928	−0.409040
$509$	36.9453	1.63757	0.818785	−	0.574100i	$-0.194648\pi$
$509$	36.9453	1.63757	0.818785	+	0.574100i	$0.194648\pi$
$510$	3.53948	0.156731
$511$	−1.14830	−0.0507977
$512$	1.00000	0.0441942
$513$	8.84106	0.390342
$514$	13.7242	0.605350
$515$	10.1528	0.447387
$516$	2.70028	0.118873
$517$	0	0
$518$	−0.839198	−0.0368722
$519$	15.5569	0.682871
$520$	4.31157	0.189075
$521$	12.8367	0.562388	0.281194	−	0.959651i	$-0.409270\pi$
$521$	12.8367	0.562388	0.281194	+	0.959651i	$0.409270\pi$
$522$	−3.39726	−0.148694
$523$	−8.66972	−0.379100	−0.189550	−	0.981871i	$-0.560703\pi$
$523$	−8.66972	−0.379100	−0.189550	+	0.981871i	$0.560703\pi$
$524$	17.5357	0.766052
$525$	−1.34292	−0.0586100
$526$	−18.9341	−0.825565
$527$	−3.02142	−0.131615
$528$	0	0
$529$	6.78911	0.295179
$530$	1.40604	0.0610745
$531$	−2.78090	−0.120681
$532$	1.56877	0.0680149
$533$	14.9980	0.649634
$534$	−2.94538	−0.127459
$535$	15.4232	0.666803
$536$	−1.60636	−0.0693842
$537$	−27.5064	−1.18699
$538$	12.9639	0.558913
$539$	0	0
$540$	5.63565	0.242520
$541$	−38.6634	−1.66227	−0.831136	−	0.556070i	$-0.812309\pi$
$541$	−38.6634	−1.66227	−0.831136	+	0.556070i	$0.812309\pi$
$542$	23.5085	1.00978
$543$	10.7592	0.461720
$544$	−2.63565	−0.113003
$545$	8.49656	0.363953
$546$	−5.79011	−0.247794
$547$	−6.20768	−0.265421	−0.132711	−	0.991155i	$-0.542368\pi$
$547$	−6.20768	−0.265421	−0.132711	+	0.991155i	$0.542368\pi$
$548$	9.12927	0.389983
$549$	6.27459	0.267793
$550$	0	0
$551$	4.45406	0.189749
$552$	−7.32959	−0.311968
$553$	9.65408	0.410533
$554$	2.44560	0.103904
$555$	1.12698	0.0478376
$556$	0.657844	0.0278988
$557$	−36.9739	−1.56663	−0.783317	−	0.621623i	$-0.786474\pi$
$557$	−36.9739	−1.56663	−0.783317	+	0.621623i	$0.786474\pi$
$558$	−1.37169	−0.0580684
$559$	−8.66950	−0.366681
$560$	1.00000	0.0422577
$561$	0	0
$562$	−18.0116	−0.759776
$563$	26.0483	1.09780	0.548901	−	0.835887i	$-0.315046\pi$
$563$	26.0483	1.09780	0.548901	+	0.835887i	$0.315046\pi$
$564$	11.2731	0.474681
$565$	2.72299	0.114557
$566$	−9.78930	−0.411475
$567$	−3.97858	−0.167085
$568$	−13.8057	−0.579276
$569$	−18.6278	−0.780920	−0.390460	−	0.920620i	$-0.627684\pi$
$569$	−18.6278	−0.780920	−0.390460	+	0.920620i	$0.627684\pi$
$570$	−2.10674	−0.0882417
$571$	34.4156	1.44025	0.720123	−	0.693846i	$-0.244085\pi$
$571$	34.4156	1.44025	0.720123	+	0.693846i	$0.244085\pi$
$572$	0	0
$573$	−10.3724	−0.433311
$574$	3.47854	0.145191
$575$	5.45794	0.227612
$576$	−1.19656	−0.0498566
$577$	24.5630	1.02257	0.511285	−	0.859411i	$-0.329170\pi$
$577$	24.5630	1.02257	0.511285	+	0.859411i	$0.329170\pi$
$578$	−10.0533	−0.418163
$579$	22.0045	0.914474
$580$	2.83920	0.117891
$581$	−2.51880	−0.104498
$582$	10.1575	0.421042
$583$	0	0
$584$	−1.14830	−0.0475169
$585$	−5.15905	−0.213300
$586$	−8.44836	−0.348999
$587$	18.3415	0.757037	0.378518	−	0.925594i	$-0.376434\pi$
$587$	18.3415	0.757037	0.378518	+	0.925594i	$0.376434\pi$
$588$	−1.34292	−0.0553812
$589$	1.79839	0.0741013
$590$	2.32408	0.0956809
$591$	−7.41254	−0.304911
$592$	−0.839198	−0.0344908
$593$	−10.9987	−0.451664	−0.225832	−	0.974166i	$-0.572510\pi$
$593$	−10.9987	−0.451664	−0.225832	+	0.974166i	$0.572510\pi$
$594$	0	0
$595$	−2.63565	−0.108051
$596$	−20.5982	−0.843737
$597$	−5.74023	−0.234932
$598$	23.5323	0.962308
$599$	4.76034	0.194502	0.0972511	−	0.995260i	$-0.468995\pi$
$599$	4.76034	0.194502	0.0972511	+	0.995260i	$0.468995\pi$
$600$	−1.34292	−0.0548246
$601$	−16.5137	−0.673608	−0.336804	−	0.941575i	$-0.609346\pi$
$601$	−16.5137	−0.673608	−0.336804	+	0.941575i	$0.609346\pi$
$602$	−2.01075	−0.0819521
$603$	1.92210	0.0782741
$604$	−4.29219	−0.174647
$605$	0	0
$606$	−23.6958	−0.962576
$607$	3.14655	0.127714	0.0638572	−	0.997959i	$-0.479660\pi$
$607$	3.14655	0.127714	0.0638572	+	0.997959i	$0.479660\pi$
$608$	1.56877	0.0636222
$609$	−3.81282	−0.154503
$610$	−5.24387	−0.212318
$611$	−36.1932	−1.46422
$612$	3.15371	0.127481
$613$	−45.9924	−1.85761	−0.928807	−	0.370564i	$-0.879164\pi$
$613$	−45.9924	−1.85761	−0.928807	+	0.370564i	$0.879164\pi$
$614$	−9.34689	−0.377210
$615$	−4.67141	−0.188369
$616$	0	0
$617$	45.5056	1.83198	0.915992	−	0.401196i	$-0.131405\pi$
$617$	45.5056	1.83198	0.915992	+	0.401196i	$0.131405\pi$
$618$	−13.6345	−0.548459
$619$	−22.5870	−0.907849	−0.453925	−	0.891040i	$-0.649976\pi$
$619$	−22.5870	−0.907849	−0.453925	+	0.891040i	$0.649976\pi$
$620$	1.14637	0.0460391
$621$	30.7591	1.23432
$622$	−27.6729	−1.10958
$623$	2.19326	0.0878710
$624$	−5.79011	−0.231790
$625$	1.00000	0.0400000
$626$	5.34524	0.213639
$627$	0	0
$628$	10.9039	0.435113
$629$	2.21184	0.0881916
$630$	−1.19656	−0.0476720
$631$	14.8009	0.589213	0.294606	−	0.955619i	$-0.404811\pi$
$631$	14.8009	0.589213	0.294606	+	0.955619i	$0.404811\pi$
$632$	9.65408	0.384019
$633$	26.3356	1.04675
$634$	7.13917	0.283533
$635$	−9.21928	−0.365856
$636$	−1.88820	−0.0748721
$637$	4.31157	0.170831
$638$	0	0
$639$	16.5194	0.653496
$640$	1.00000	0.0395285
$641$	−34.5333	−1.36398	−0.681992	−	0.731360i	$-0.738886\pi$
$641$	−34.5333	−1.36398	−0.681992	+	0.731360i	$0.738886\pi$
$642$	−20.7122	−0.817444
$643$	−16.2144	−0.639434	−0.319717	−	0.947513i	$-0.603588\pi$
$643$	−16.2144	−0.639434	−0.319717	+	0.947513i	$0.603588\pi$
$644$	5.45794	0.215073
$645$	2.70028	0.106324
$646$	−4.13474	−0.162679
$647$	−43.3962	−1.70608	−0.853041	−	0.521844i	$-0.825244\pi$
$647$	−43.3962	−1.70608	−0.853041	+	0.521844i	$0.825244\pi$
$648$	−3.97858	−0.156293
$649$	0	0
$650$	4.31157	0.169114
$651$	−1.53948	−0.0603370
$652$	12.3525	0.483763
$653$	7.75135	0.303334	0.151667	−	0.988432i	$-0.451536\pi$
$653$	7.75135	0.303334	0.151667	+	0.988432i	$0.451536\pi$
$654$	−11.4102	−0.446175
$655$	17.5357	0.685178
$656$	3.47854	0.135814
$657$	1.37400	0.0536050
$658$	−8.39442	−0.327248
$659$	6.80255	0.264990	0.132495	−	0.991184i	$-0.457701\pi$
$659$	6.80255	0.264990	0.132495	+	0.991184i	$0.457701\pi$
$660$	0	0
$661$	−17.3537	−0.674980	−0.337490	−	0.941329i	$-0.609578\pi$
$661$	−17.3537	−0.674980	−0.337490	+	0.941329i	$0.609578\pi$
$662$	5.83207	0.226670
$663$	15.2607	0.592678
$664$	−2.51880	−0.0977486
$665$	1.56877	0.0608344
$666$	1.00415	0.0389100
$667$	15.4962	0.600014
$668$	−6.93334	−0.268259
$669$	−5.39779	−0.208690
$670$	−1.60636	−0.0620592
$671$	0	0
$672$	−1.34292	−0.0518044
$673$	−47.8170	−1.84321	−0.921605	−	0.388130i	$-0.873121\pi$
$673$	−47.8170	−1.84321	−0.921605	+	0.388130i	$0.873121\pi$
$674$	22.2848	0.858377
$675$	5.63565	0.216916
$676$	5.58967	0.214987
$677$	−7.12176	−0.273711	−0.136856	−	0.990591i	$-0.543700\pi$
$677$	−7.12176	−0.273711	−0.136856	+	0.990591i	$0.543700\pi$
$678$	−3.65677	−0.140437
$679$	−7.56372	−0.290269
$680$	−2.63565	−0.101073
$681$	−27.5013	−1.05385
$682$	0	0
$683$	17.8302	0.682252	0.341126	−	0.940018i	$-0.389192\pi$
$683$	17.8302	0.682252	0.341126	+	0.940018i	$0.389192\pi$
$684$	−1.87713	−0.0717738
$685$	9.12927	0.348812
$686$	1.00000	0.0381802
$687$	−11.7489	−0.448247
$688$	−2.01075	−0.0766591
$689$	6.06225	0.230953
$690$	−7.32959	−0.279033
$691$	−15.7399	−0.598772	−0.299386	−	0.954132i	$-0.596782\pi$
$691$	−15.7399	−0.598772	−0.299386	+	0.954132i	$0.596782\pi$
$692$	−11.5843	−0.440370
$693$	0	0
$694$	16.6253	0.631089
$695$	0.657844	0.0249535
$696$	−3.81282	−0.144525
$697$	−9.16822	−0.347271
$698$	3.23934	0.122611
$699$	−18.9973	−0.718542
$700$	1.00000	0.0377964
$701$	−19.1512	−0.723329	−0.361665	−	0.932308i	$-0.617791\pi$
$701$	−19.1512	−0.723329	−0.361665	+	0.932308i	$0.617791\pi$
$702$	24.2985	0.917089
$703$	−1.31651	−0.0496532
$704$	0	0
$705$	11.2731	0.424568
$706$	−24.6580	−0.928014
$707$	17.6449	0.663606
$708$	−3.12106	−0.117297
$709$	6.08007	0.228342	0.114171	−	0.993461i	$-0.463579\pi$
$709$	6.08007	0.228342	0.114171	+	0.993461i	$0.463579\pi$
$710$	−13.8057	−0.518120
$711$	−11.5517	−0.433221
$712$	2.19326	0.0821958
$713$	6.25679	0.234319
$714$	3.53948	0.132462
$715$	0	0
$716$	20.4825	0.765466
$717$	−2.84880	−0.106390
$718$	20.7493	0.774358
$719$	27.3905	1.02149	0.510746	−	0.859732i	$-0.329369\pi$
$719$	27.3905	1.02149	0.510746	+	0.859732i	$0.329369\pi$
$720$	−1.19656	−0.0445931
$721$	10.1528	0.378111
$722$	−16.5390	−0.615516
$723$	−23.4148	−0.870805
$724$	−8.01175	−0.297754
$725$	2.83920	0.105445
$726$	0	0
$727$	28.3641	1.05197	0.525984	−	0.850495i	$-0.323697\pi$
$727$	28.3641	1.05197	0.525984	+	0.850495i	$0.323697\pi$
$728$	4.31157	0.159798
$729$	27.4653	1.01724
$730$	−1.14830	−0.0425004
$731$	5.29964	0.196014
$732$	7.04211	0.260284
$733$	−48.3933	−1.78745	−0.893724	−	0.448618i	$-0.851916\pi$
$733$	−48.3933	−1.78745	−0.893724	+	0.448618i	$0.851916\pi$
$734$	2.12160	0.0783096
$735$	−1.34292	−0.0495345
$736$	5.45794	0.201182
$737$	0	0
$738$	−4.16227	−0.153215
$739$	6.07467	0.223460	0.111730	−	0.993739i	$-0.464361\pi$
$739$	6.07467	0.223460	0.111730	+	0.993739i	$0.464361\pi$
$740$	−0.839198	−0.0308495
$741$	−9.08337	−0.333686
$742$	1.40604	0.0516173
$743$	32.1862	1.18080	0.590398	−	0.807112i	$-0.298971\pi$
$743$	32.1862	1.18080	0.590398	+	0.807112i	$0.298971\pi$
$744$	−1.53948	−0.0564401
$745$	−20.5982	−0.754661
$746$	−7.19996	−0.263609
$747$	3.01389	0.110273
$748$	0	0
$749$	15.4232	0.563551
$750$	−1.34292	−0.0490366
$751$	−24.5881	−0.897234	−0.448617	−	0.893724i	$-0.648083\pi$
$751$	−24.5881	−0.897234	−0.448617	+	0.893724i	$0.648083\pi$
$752$	−8.39442	−0.306113
$753$	−1.25810	−0.0458479
$754$	12.2414	0.445806
$755$	−4.29219	−0.156209
$756$	5.63565	0.204967
$757$	−25.5181	−0.927470	−0.463735	−	0.885974i	$-0.653491\pi$
$757$	−25.5181	−0.927470	−0.463735	+	0.885974i	$0.653491\pi$
$758$	−23.1300	−0.840121
$759$	0	0
$760$	1.56877	0.0569054
$761$	−13.0436	−0.472830	−0.236415	−	0.971652i	$-0.575972\pi$
$761$	−13.0436	−0.472830	−0.236415	+	0.971652i	$0.575972\pi$
$762$	12.3808	0.448509
$763$	8.49656	0.307596
$764$	7.72371	0.279434
$765$	3.15371	0.114023
$766$	31.4479	1.13626
$767$	10.0204	0.361817
$768$	−1.34292	−0.0484586
$769$	−48.0699	−1.73344	−0.866722	−	0.498792i	$-0.833777\pi$
$769$	−48.0699	−1.73344	−0.866722	+	0.498792i	$0.833777\pi$
$770$	0	0
$771$	−18.4306	−0.663762
$772$	−16.3855	−0.589727
$773$	−40.5127	−1.45714	−0.728570	−	0.684972i	$-0.759815\pi$
$773$	−40.5127	−1.45714	−0.728570	+	0.684972i	$0.759815\pi$
$774$	2.40598	0.0864811
$775$	1.14637	0.0411787
$776$	−7.56372	−0.271522
$777$	1.12698	0.0404301
$778$	17.5618	0.629622
$779$	5.45704	0.195519
$780$	−5.79011	−0.207319
$781$	0	0
$782$	−14.3852	−0.514415
$783$	16.0007	0.571820
$784$	1.00000	0.0357143
$785$	10.9039	0.389177
$786$	−23.5491	−0.839970
$787$	36.3703	1.29646	0.648231	−	0.761444i	$-0.275509\pi$
$787$	36.3703	1.29646	0.648231	+	0.761444i	$0.275509\pi$
$788$	5.51971	0.196631
$789$	25.4270	0.905226
$790$	9.65408	0.343477
$791$	2.72299	0.0968185
$792$	0	0
$793$	−22.6093	−0.802881
$794$	32.7839	1.16346
$795$	−1.88820	−0.0669677
$796$	4.27443	0.151503
$797$	30.7299	1.08851	0.544254	−	0.838920i	$-0.316813\pi$
$797$	30.7299	1.08851	0.544254	+	0.838920i	$0.316813\pi$
$798$	−2.10674	−0.0745779
$799$	22.1248	0.782719
$800$	1.00000	0.0353553
$801$	−2.62436	−0.0927272
$802$	9.25146	0.326680
$803$	0	0
$804$	2.15722	0.0760793
$805$	5.45794	0.192367
$806$	4.94264	0.174097
$807$	−17.4095	−0.612843
$808$	17.6449	0.620747
$809$	−45.6988	−1.60668	−0.803342	−	0.595517i	$-0.796947\pi$
$809$	−45.6988	−1.60668	−0.803342	+	0.595517i	$0.796947\pi$
$810$	−3.97858	−0.139793
$811$	−9.43169	−0.331191	−0.165596	−	0.986194i	$-0.552955\pi$
$811$	−9.43169	−0.331191	−0.165596	+	0.986194i	$0.552955\pi$
$812$	2.83920	0.0996363
$813$	−31.5701	−1.10721
$814$	0	0
$815$	12.3525	0.432691
$816$	3.53948	0.123907
$817$	−3.15441	−0.110359
$818$	−0.0174947	−0.000611688 0
$819$	−5.15905	−0.180272
$820$	3.47854	0.121476
$821$	−12.2170	−0.426376	−0.213188	−	0.977011i	$-0.568385\pi$
$821$	−12.2170	−0.426376	−0.213188	+	0.977011i	$0.568385\pi$
$822$	−12.2599	−0.427613
$823$	8.03161	0.279964	0.139982	−	0.990154i	$-0.455295\pi$
$823$	8.03161	0.279964	0.139982	+	0.990154i	$0.455295\pi$
$824$	10.1528	0.353691
$825$	0	0
$826$	2.32408	0.0808651
$827$	9.90168	0.344315	0.172158	−	0.985069i	$-0.444926\pi$
$827$	9.90168	0.344315	0.172158	+	0.985069i	$0.444926\pi$
$828$	−6.53074	−0.226959
$829$	−20.2243	−0.702420	−0.351210	−	0.936297i	$-0.614230\pi$
$829$	−20.2243	−0.702420	−0.351210	+	0.936297i	$0.614230\pi$
$830$	−2.51880	−0.0874290
$831$	−3.28426	−0.113930
$832$	4.31157	0.149477
$833$	−2.63565	−0.0913200
$834$	−0.883434	−0.0305908
$835$	−6.93334	−0.239938
$836$	0	0
$837$	6.46052	0.223308
$838$	32.4579	1.12124
$839$	35.7528	1.23432	0.617162	−	0.786836i	$-0.288282\pi$
$839$	35.7528	1.23432	0.617162	+	0.786836i	$0.288282\pi$
$840$	−1.34292	−0.0463352
$841$	−20.9390	−0.722033
$842$	9.15329	0.315443
$843$	24.1883	0.833088
$844$	−19.6106	−0.675026
$845$	5.58967	0.192291
$846$	10.0444	0.345334
$847$	0	0
$848$	1.40604	0.0482836
$849$	13.1463	0.451179
$850$	−2.63565	−0.0904022
$851$	−4.58029	−0.157010
$852$	18.5400	0.635171
$853$	23.0500	0.789218	0.394609	−	0.918849i	$-0.370880\pi$
$853$	23.0500	0.789218	0.394609	+	0.918849i	$0.370880\pi$
$854$	−5.24387	−0.179442
$855$	−1.87713	−0.0641964
$856$	15.4232	0.527154
$857$	27.3702	0.934948	0.467474	−	0.884007i	$-0.345164\pi$
$857$	27.3702	0.934948	0.467474	+	0.884007i	$0.345164\pi$
$858$	0	0
$859$	43.4968	1.48409	0.742045	−	0.670350i	$-0.233856\pi$
$859$	43.4968	1.48409	0.742045	+	0.670350i	$0.233856\pi$
$860$	−2.01075	−0.0685660
$861$	−4.67141	−0.159201
$862$	−26.0024	−0.885647
$863$	−17.9594	−0.611345	−0.305672	−	0.952137i	$-0.598881\pi$
$863$	−17.9594	−0.611345	−0.305672	+	0.952137i	$0.598881\pi$
$864$	5.63565	0.191729
$865$	−11.5843	−0.393879
$866$	23.6006	0.801980
$867$	13.5008	0.458513
$868$	1.14637	0.0389102
$869$	0	0
$870$	−3.81282	−0.129267
$871$	−6.92595	−0.234677
$872$	8.49656	0.287730
$873$	9.05042	0.306310
$874$	8.56227	0.289623
$875$	1.00000	0.0338062
$876$	1.54207	0.0521019
$877$	32.8608	1.10963	0.554816	−	0.831973i	$-0.312789\pi$
$877$	32.8608	1.10963	0.554816	+	0.831973i	$0.312789\pi$
$878$	31.7894	1.07284
$879$	11.3455	0.382674
$880$	0	0
$881$	−24.9270	−0.839812	−0.419906	−	0.907568i	$-0.637937\pi$
$881$	−24.9270	−0.839812	−0.419906	+	0.907568i	$0.637937\pi$
$882$	−1.19656	−0.0402902
$883$	−18.1957	−0.612334	−0.306167	−	0.951978i	$-0.599047\pi$
$883$	−18.1957	−0.612334	−0.306167	+	0.951978i	$0.599047\pi$
$884$	−11.3638	−0.382207
$885$	−3.12106	−0.104913
$886$	36.1071	1.21304
$887$	−38.9219	−1.30687	−0.653434	−	0.756983i	$-0.726672\pi$
$887$	−38.9219	−1.30687	−0.653434	+	0.756983i	$0.726672\pi$
$888$	1.12698	0.0378189
$889$	−9.21928	−0.309205
$890$	2.19326	0.0735182
$891$	0	0
$892$	4.01943	0.134580
$893$	−13.1689	−0.440682
$894$	27.6619	0.925151
$895$	20.4825	0.684654
$896$	1.00000	0.0334077
$897$	−31.6021	−1.05516
$898$	−4.52269	−0.150924
$899$	3.25476	0.108552
$900$	−1.19656	−0.0398853
$901$	−3.70583	−0.123459
$902$	0	0
$903$	2.70028	0.0898598
$904$	2.72299	0.0905655
$905$	−8.01175	−0.266320
$906$	5.76408	0.191499
$907$	−47.4377	−1.57514	−0.787572	−	0.616223i	$-0.788662\pi$
$907$	−47.4377	−1.57514	−0.787572	+	0.616223i	$0.788662\pi$
$908$	20.4787	0.679610
$909$	−21.1132	−0.700280
$910$	4.31157	0.142927
$911$	−10.2118	−0.338333	−0.169167	−	0.985587i	$-0.554108\pi$
$911$	−10.2118	−0.338333	−0.169167	+	0.985587i	$0.554108\pi$
$912$	−2.10674	−0.0697612
$913$	0	0
$914$	−15.1593	−0.501423
$915$	7.04211	0.232805
$916$	8.74872	0.289066
$917$	17.5357	0.579081
$918$	−14.8536	−0.490243
$919$	−33.7533	−1.11342	−0.556710	−	0.830707i	$-0.687936\pi$
$919$	−33.7533	−1.11342	−0.556710	+	0.830707i	$0.687936\pi$
$920$	5.45794	0.179943
$921$	12.5522	0.413608
$922$	−27.6126	−0.909373
$923$	−59.5244	−1.95927
$924$	0	0
$925$	−0.839198	−0.0275927
$926$	13.3557	0.438895
$927$	−12.1484	−0.399007
$928$	2.83920	0.0932012
$929$	41.2273	1.35262	0.676312	−	0.736615i	$-0.263577\pi$
$929$	41.2273	1.35262	0.676312	+	0.736615i	$0.263577\pi$
$930$	−1.53948	−0.0504816
$931$	1.56877	0.0514145
$932$	14.1462	0.463374
$933$	37.1626	1.21665
$934$	26.3287	0.861500
$935$	0	0
$936$	−5.15905	−0.168629
$937$	−32.2808	−1.05457	−0.527283	−	0.849690i	$-0.676789\pi$
$937$	−32.2808	−1.05457	−0.527283	+	0.849690i	$0.676789\pi$
$938$	−1.60636	−0.0524496
$939$	−7.17825	−0.234253
$940$	−8.39442	−0.273796
$941$	−47.3632	−1.54400	−0.771998	−	0.635625i	$-0.780743\pi$
$941$	−47.3632	−1.54400	−0.771998	+	0.635625i	$0.780743\pi$
$942$	−14.6431	−0.477098
$943$	18.9857	0.618258
$944$	2.32408	0.0756424
$945$	5.63565	0.183328
$946$	0	0
$947$	25.4907	0.828335	0.414167	−	0.910201i	$-0.364073\pi$
$947$	25.4907	0.828335	0.414167	+	0.910201i	$0.364073\pi$
$948$	−12.9647	−0.421073
$949$	−4.95097	−0.160715
$950$	1.56877	0.0508977
$951$	−9.58736	−0.310891
$952$	−2.63565	−0.0854220
$953$	−39.8135	−1.28969	−0.644843	−	0.764315i	$-0.723077\pi$
$953$	−39.8135	−1.28969	−0.644843	+	0.764315i	$0.723077\pi$
$954$	−1.68241	−0.0544700
$955$	7.72371	0.249934
$956$	2.12134	0.0686091
$957$	0	0
$958$	8.18258	0.264367
$959$	9.12927	0.294800
$960$	−1.34292	−0.0433427
$961$	−29.6858	−0.957608
$962$	−3.61826	−0.116657
$963$	−18.4547	−0.594695
$964$	17.4357	0.561566
$965$	−16.3855	−0.527468
$966$	−7.32959	−0.235826
$967$	29.9522	0.963198	0.481599	−	0.876392i	$-0.340056\pi$
$967$	29.9522	0.963198	0.481599	+	0.876392i	$0.340056\pi$
$968$	0	0
$969$	5.55264	0.178377
$970$	−7.56372	−0.242856
$971$	8.45499	0.271333	0.135667	−	0.990755i	$-0.456682\pi$
$971$	8.45499	0.271333	0.135667	+	0.990755i	$0.456682\pi$
$972$	−11.5640	−0.370917
$973$	0.657844	0.0210895
$974$	−9.09993	−0.291581
$975$	−5.79011	−0.185432
$976$	−5.24387	−0.167852
$977$	−4.10946	−0.131473	−0.0657366	−	0.997837i	$-0.520940\pi$
$977$	−4.10946	−0.131473	−0.0657366	+	0.997837i	$0.520940\pi$
$978$	−16.5885	−0.530442
$979$	0	0
$980$	1.00000	0.0319438
$981$	−10.1666	−0.324595
$982$	−3.80239	−0.121339
$983$	20.8447	0.664842	0.332421	−	0.943131i	$-0.392135\pi$
$983$	20.8447	0.664842	0.332421	+	0.943131i	$0.392135\pi$
$984$	−4.67141	−0.148919
$985$	5.51971	0.175873
$986$	−7.48314	−0.238312
$987$	11.2731	0.358825
$988$	6.76388	0.215188
$989$	−10.9746	−0.348971
$990$	0	0
$991$	29.5690	0.939290	0.469645	−	0.882855i	$-0.344382\pi$
$991$	29.5690	0.939290	0.469645	+	0.882855i	$0.344382\pi$
$992$	1.14637	0.0363971
$993$	−7.83202	−0.248542
$994$	−13.8057	−0.437891
$995$	4.27443	0.135508
$996$	3.38256	0.107181
$997$	57.0730	1.80752	0.903760	−	0.428040i	$-0.140796\pi$
$997$	57.0730	1.80752	0.903760	+	0.428040i	$0.140796\pi$
$998$	−22.9856	−0.727596
$999$	−4.72943	−0.149632

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.df.1.2	yes	6
11.10	odd	2		8470.2.a.cz.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.cz.1.1	✓	6	11.10	odd	2
8470.2.a.df.1.2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.34292 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.34292 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.19656 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.34292 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.34292 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.19656 q^{9} +1.00000 q^{10} -1.34292 q^{12} +4.31157 q^{13} +1.00000 q^{14} -1.34292 q^{15} +1.00000 q^{16} -2.63565 q^{17} -1.19656 q^{18} +1.56877 q^{19} +1.00000 q^{20} -1.34292 q^{21} +5.45794 q^{23} -1.34292 q^{24} +1.00000 q^{25} +4.31157 q^{26} +5.63565 q^{27} +1.00000 q^{28} +2.83920 q^{29} -1.34292 q^{30} +1.14637 q^{31} +1.00000 q^{32} -2.63565 q^{34} +1.00000 q^{35} -1.19656 q^{36} -0.839198 q^{37} +1.56877 q^{38} -5.79011 q^{39} +1.00000 q^{40} +3.47854 q^{41} -1.34292 q^{42} -2.01075 q^{43} -1.19656 q^{45} +5.45794 q^{46} -8.39442 q^{47} -1.34292 q^{48} +1.00000 q^{49} +1.00000 q^{50} +3.53948 q^{51} +4.31157 q^{52} +1.40604 q^{53} +5.63565 q^{54} +1.00000 q^{56} -2.10674 q^{57} +2.83920 q^{58} +2.32408 q^{59} -1.34292 q^{60} -5.24387 q^{61} +1.14637 q^{62} -1.19656 q^{63} +1.00000 q^{64} +4.31157 q^{65} -1.60636 q^{67} -2.63565 q^{68} -7.32959 q^{69} +1.00000 q^{70} -13.8057 q^{71} -1.19656 q^{72} -1.14830 q^{73} -0.839198 q^{74} -1.34292 q^{75} +1.56877 q^{76} -5.79011 q^{78} +9.65408 q^{79} +1.00000 q^{80} -3.97858 q^{81} +3.47854 q^{82} -2.51880 q^{83} -1.34292 q^{84} -2.63565 q^{85} -2.01075 q^{86} -3.81282 q^{87} +2.19326 q^{89} -1.19656 q^{90} +4.31157 q^{91} +5.45794 q^{92} -1.53948 q^{93} -8.39442 q^{94} +1.56877 q^{95} -1.34292 q^{96} -7.56372 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 4 q^{3} + 6 q^{4} + 6 q^{5} + 4 q^{6} + 6 q^{7} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) 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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