LMFDB - Embedded newform 8470.2.a.cy.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:	$1$
Dimension:	$6$
Coefficient field:	6.6.13298000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 3x^{3} + 26x^{2} + 13x - 1 $ x^6 - x^5 - 10x^4 + 3x^3 + 26x^2 + 13x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24667$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -0.418877 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.418877 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.82454 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.418877 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.418877 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.82454 q^{9} -1.00000 q^{10} -0.418877 q^{12} +5.30073 q^{13} +1.00000 q^{14} -0.418877 q^{15} +1.00000 q^{16} -1.47618 q^{17} +2.82454 q^{18} -0.952490 q^{19} +1.00000 q^{20} +0.418877 q^{21} +2.37872 q^{23} +0.418877 q^{24} +1.00000 q^{25} -5.30073 q^{26} +2.43977 q^{27} -1.00000 q^{28} -5.97768 q^{29} +0.418877 q^{30} +2.34330 q^{31} -1.00000 q^{32} +1.47618 q^{34} -1.00000 q^{35} -2.82454 q^{36} +0.0224026 q^{37} +0.952490 q^{38} -2.22035 q^{39} -1.00000 q^{40} +1.53186 q^{41} -0.418877 q^{42} -3.83101 q^{43} -2.82454 q^{45} -2.37872 q^{46} -1.50443 q^{47} -0.418877 q^{48} +1.00000 q^{49} -1.00000 q^{50} +0.618340 q^{51} +5.30073 q^{52} -11.3487 q^{53} -2.43977 q^{54} +1.00000 q^{56} +0.398977 q^{57} +5.97768 q^{58} +0.768866 q^{59} -0.418877 q^{60} -8.06011 q^{61} -2.34330 q^{62} +2.82454 q^{63} +1.00000 q^{64} +5.30073 q^{65} +12.1841 q^{67} -1.47618 q^{68} -0.996391 q^{69} +1.00000 q^{70} -11.9348 q^{71} +2.82454 q^{72} -7.28500 q^{73} -0.0224026 q^{74} -0.418877 q^{75} -0.952490 q^{76} +2.22035 q^{78} +12.8107 q^{79} +1.00000 q^{80} +7.45166 q^{81} -1.53186 q^{82} -6.13526 q^{83} +0.418877 q^{84} -1.47618 q^{85} +3.83101 q^{86} +2.50392 q^{87} +13.4596 q^{89} +2.82454 q^{90} -5.30073 q^{91} +2.37872 q^{92} -0.981556 q^{93} +1.50443 q^{94} -0.952490 q^{95} +0.418877 q^{96} +18.4862 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} - 6 q^{8} + 15 q^{9}+O(q^{10}) $ 6 * q - 6 * q^2 + q^3 + 6 * q^4 + 6 * q^5 - q^6 - 6 * q^7 - 6 * q^8 + 15 * q^9	$ 6 q - 6 q^{2} + q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} - 6 q^{8} + 15 q^{9} - 6 q^{10} + q^{12} - 2 q^{13} + 6 q^{14} + q^{15} + 6 q^{16} - 7 q^{17} - 15 q^{18} - 11 q^{19} + 6 q^{20} - q^{21} - 6 q^{23} - q^{24} + 6 q^{25} + 2 q^{26} + 4 q^{27} - 6 q^{28} - 2 q^{29} - q^{30} - 6 q^{32} + 7 q^{34} - 6 q^{35} + 15 q^{36} - 14 q^{37} + 11 q^{38} - 20 q^{39} - 6 q^{40} - 13 q^{41} + q^{42} - 19 q^{43} + 15 q^{45} + 6 q^{46} + 22 q^{47} + q^{48} + 6 q^{49} - 6 q^{50} + 14 q^{51} - 2 q^{52} - 10 q^{53} - 4 q^{54} + 6 q^{56} - 32 q^{57} + 2 q^{58} - 7 q^{59} + q^{60} - 22 q^{61} - 15 q^{63} + 6 q^{64} - 2 q^{65} + 5 q^{67} - 7 q^{68} - 36 q^{69} + 6 q^{70} + 8 q^{71} - 15 q^{72} - 13 q^{73} + 14 q^{74} + q^{75} - 11 q^{76} + 20 q^{78} + 16 q^{79} + 6 q^{80} + 18 q^{81} + 13 q^{82} + 5 q^{83} - q^{84} - 7 q^{85} + 19 q^{86} - 14 q^{87} + q^{89} - 15 q^{90} + 2 q^{91} - 6 q^{92} - 42 q^{93} - 22 q^{94} - 11 q^{95} - q^{96} - 3 q^{97} - 6 q^{98}+O(q^{100}) $ 6 * q - 6 * q^2 + q^3 + 6 * q^4 + 6 * q^5 - q^6 - 6 * q^7 - 6 * q^8 + 15 * q^9 - 6 * q^10 + q^12 - 2 * q^13 + 6 * q^14 + q^15 + 6 * q^16 - 7 * q^17 - 15 * q^18 - 11 * q^19 + 6 * q^20 - q^21 - 6 * q^23 - q^24 + 6 * q^25 + 2 * q^26 + 4 * q^27 - 6 * q^28 - 2 * q^29 - q^30 - 6 * q^32 + 7 * q^34 - 6 * q^35 + 15 * q^36 - 14 * q^37 + 11 * q^38 - 20 * q^39 - 6 * q^40 - 13 * q^41 + q^42 - 19 * q^43 + 15 * q^45 + 6 * q^46 + 22 * q^47 + q^48 + 6 * q^49 - 6 * q^50 + 14 * q^51 - 2 * q^52 - 10 * q^53 - 4 * q^54 + 6 * q^56 - 32 * q^57 + 2 * q^58 - 7 * q^59 + q^60 - 22 * q^61 - 15 * q^63 + 6 * q^64 - 2 * q^65 + 5 * q^67 - 7 * q^68 - 36 * q^69 + 6 * q^70 + 8 * q^71 - 15 * q^72 - 13 * q^73 + 14 * q^74 + q^75 - 11 * q^76 + 20 * q^78 + 16 * q^79 + 6 * q^80 + 18 * q^81 + 13 * q^82 + 5 * q^83 - q^84 - 7 * q^85 + 19 * q^86 - 14 * q^87 + q^89 - 15 * q^90 + 2 * q^91 - 6 * q^92 - 42 * q^93 - 22 * q^94 - 11 * q^95 - q^96 - 3 * 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
$3$	−0.418877	−0.241839	−0.120919	−	0.992662i	$-0.538584\pi$
$3$	−0.418877	−0.241839	−0.120919	+	0.992662i	$0.538584\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.418877	0.171006
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	−2.82454	−0.941514
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	−0.418877	−0.120919
$13$	5.30073	1.47016	0.735078	−	0.677982i	$-0.237145\pi$
$13$	5.30073	1.47016	0.735078	+	0.677982i	$0.237145\pi$
$14$	1.00000	0.267261
$15$	−0.418877	−0.108154
$16$	1.00000	0.250000
$17$	−1.47618	−0.358027	−0.179014	−	0.983847i	$-0.557291\pi$
$17$	−1.47618	−0.358027	−0.179014	+	0.983847i	$0.557291\pi$
$18$	2.82454	0.665751
$19$	−0.952490	−0.218516	−0.109258	−	0.994013i	$-0.534848\pi$
$19$	−0.952490	−0.218516	−0.109258	+	0.994013i	$0.534848\pi$
$20$	1.00000	0.223607
$21$	0.418877	0.0914065
$22$	0	0
$23$	2.37872	0.495997	0.247999	−	0.968760i	$-0.420227\pi$
$23$	2.37872	0.495997	0.247999	+	0.968760i	$0.420227\pi$
$24$	0.418877	0.0855030
$25$	1.00000	0.200000
$26$	−5.30073	−1.03956
$27$	2.43977	0.469534
$28$	−1.00000	−0.188982
$29$	−5.97768	−1.11003	−0.555014	−	0.831841i	$-0.687287\pi$
$29$	−5.97768	−1.11003	−0.555014	+	0.831841i	$0.687287\pi$
$30$	0.418877	0.0764762
$31$	2.34330	0.420869	0.210435	−	0.977608i	$-0.432512\pi$
$31$	2.34330	0.420869	0.210435	+	0.977608i	$0.432512\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.47618	0.253163
$35$	−1.00000	−0.169031
$36$	−2.82454	−0.470757
$37$	0.0224026	0.00368297	0.00184148	−	0.999998i	$-0.499414\pi$
$37$	0.0224026	0.00368297	0.00184148	+	0.999998i	$0.499414\pi$
$38$	0.952490	0.154514
$39$	−2.22035	−0.355541
$40$	−1.00000	−0.158114
$41$	1.53186	0.239236	0.119618	−	0.992820i	$-0.461833\pi$
$41$	1.53186	0.239236	0.119618	+	0.992820i	$0.461833\pi$
$42$	−0.418877	−0.0646342
$43$	−3.83101	−0.584223	−0.292112	−	0.956384i	$-0.594358\pi$
$43$	−3.83101	−0.584223	−0.292112	+	0.956384i	$0.594358\pi$
$44$	0	0
$45$	−2.82454	−0.421058
$46$	−2.37872	−0.350723
$47$	−1.50443	−0.219443	−0.109722	−	0.993962i	$-0.534996\pi$
$47$	−1.50443	−0.219443	−0.109722	+	0.993962i	$0.534996\pi$
$48$	−0.418877	−0.0604597
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	0.618340	0.0865849
$52$	5.30073	0.735078
$53$	−11.3487	−1.55887	−0.779434	−	0.626484i	$-0.784493\pi$
$53$	−11.3487	−1.55887	−0.779434	+	0.626484i	$0.784493\pi$
$54$	−2.43977	−0.332010
$55$	0	0
$56$	1.00000	0.133631
$57$	0.398977	0.0528457
$58$	5.97768	0.784908
$59$	0.768866	0.100098	0.0500489	−	0.998747i	$-0.484062\pi$
$59$	0.768866	0.100098	0.0500489	+	0.998747i	$0.484062\pi$
$60$	−0.418877	−0.0540768
$61$	−8.06011	−1.03199	−0.515996	−	0.856591i	$-0.672578\pi$
$61$	−8.06011	−1.03199	−0.515996	+	0.856591i	$0.672578\pi$
$62$	−2.34330	−0.297600
$63$	2.82454	0.355859
$64$	1.00000	0.125000
$65$	5.30073	0.657474
$66$	0	0
$67$	12.1841	1.48852	0.744260	−	0.667890i	$-0.232802\pi$
$67$	12.1841	1.48852	0.744260	+	0.667890i	$0.232802\pi$
$68$	−1.47618	−0.179014
$69$	−0.996391	−0.119951
$70$	1.00000	0.119523
$71$	−11.9348	−1.41640	−0.708199	−	0.706013i	$-0.750492\pi$
$71$	−11.9348	−1.41640	−0.708199	+	0.706013i	$0.750492\pi$
$72$	2.82454	0.332875
$73$	−7.28500	−0.852644	−0.426322	−	0.904571i	$-0.640191\pi$
$73$	−7.28500	−0.852644	−0.426322	+	0.904571i	$0.640191\pi$
$74$	−0.0224026	−0.00260425
$75$	−0.418877	−0.0483678
$76$	−0.952490	−0.109258
$77$	0	0
$78$	2.22035	0.251406
$79$	12.8107	1.44131	0.720657	−	0.693292i	$-0.243840\pi$
$79$	12.8107	1.44131	0.720657	+	0.693292i	$0.243840\pi$
$80$	1.00000	0.111803
$81$	7.45166	0.827962
$82$	−1.53186	−0.169166
$83$	−6.13526	−0.673433	−0.336716	−	0.941606i	$-0.609316\pi$
$83$	−6.13526	−0.673433	−0.336716	+	0.941606i	$0.609316\pi$
$84$	0.418877	0.0457033
$85$	−1.47618	−0.160115
$86$	3.83101	0.413108
$87$	2.50392	0.268448
$88$	0	0
$89$	13.4596	1.42671	0.713357	−	0.700801i	$-0.247174\pi$
$89$	13.4596	1.42671	0.713357	+	0.700801i	$0.247174\pi$
$90$	2.82454	0.297733
$91$	−5.30073	−0.555667
$92$	2.37872	0.247999
$93$	−0.981556	−0.101783
$94$	1.50443	0.155170
$95$	−0.952490	−0.0977235
$96$	0.418877	0.0427515
$97$	18.4862	1.87699	0.938495	−	0.345292i	$-0.112220\pi$
$97$	18.4862	1.87699	0.938495	+	0.345292i	$0.112220\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−8.92131	−0.887704	−0.443852	−	0.896100i	$-0.646388\pi$
$101$	−8.92131	−0.887704	−0.443852	+	0.896100i	$0.646388\pi$
$102$	−0.618340	−0.0612248
$103$	−12.3386	−1.21576	−0.607881	−	0.794028i	$-0.707980\pi$
$103$	−12.3386	−1.21576	−0.607881	+	0.794028i	$0.707980\pi$
$104$	−5.30073	−0.519779
$105$	0.418877	0.0408782
$106$	11.3487	1.10229
$107$	−9.65719	−0.933596	−0.466798	−	0.884364i	$-0.654593\pi$
$107$	−9.65719	−0.933596	−0.466798	+	0.884364i	$0.654593\pi$
$108$	2.43977	0.234767
$109$	5.91826	0.566867	0.283433	−	0.958992i	$-0.408526\pi$
$109$	5.91826	0.566867	0.283433	+	0.958992i	$0.408526\pi$
$110$	0	0
$111$	−0.00938394	−0.000890684 0
$112$	−1.00000	−0.0944911
$113$	12.5957	1.18490	0.592452	−	0.805606i	$-0.298160\pi$
$113$	12.5957	1.18490	0.592452	+	0.805606i	$0.298160\pi$
$114$	−0.398977	−0.0373676
$115$	2.37872	0.221817
$116$	−5.97768	−0.555014
$117$	−14.9721	−1.38417
$118$	−0.768866	−0.0707799
$119$	1.47618	0.135322
$120$	0.418877	0.0382381
$121$	0	0
$122$	8.06011	0.729729
$123$	−0.641661	−0.0578566
$124$	2.34330	0.210435
$125$	1.00000	0.0894427
$126$	−2.82454	−0.251630
$127$	2.54206	0.225572	0.112786	−	0.993619i	$-0.464023\pi$
$127$	2.54206	0.225572	0.112786	+	0.993619i	$0.464023\pi$
$128$	−1.00000	−0.0883883
$129$	1.60472	0.141288
$130$	−5.30073	−0.464904
$131$	−7.80495	−0.681921	−0.340961	−	0.940078i	$-0.610752\pi$
$131$	−7.80495	−0.681921	−0.340961	+	0.940078i	$0.610752\pi$
$132$	0	0
$133$	0.952490	0.0825914
$134$	−12.1841	−1.05254
$135$	2.43977	0.209982
$136$	1.47618	0.126582
$137$	19.7473	1.68713	0.843564	−	0.537029i	$-0.180453\pi$
$137$	19.7473	1.68713	0.843564	+	0.537029i	$0.180453\pi$
$138$	0.996391	0.0848184
$139$	−13.2995	−1.12805	−0.564023	−	0.825759i	$-0.690747\pi$
$139$	−13.2995	−1.12805	−0.564023	+	0.825759i	$0.690747\pi$
$140$	−1.00000	−0.0845154
$141$	0.630170	0.0530699
$142$	11.9348	1.00155
$143$	0	0
$144$	−2.82454	−0.235378
$145$	−5.97768	−0.496420
$146$	7.28500	0.602911
$147$	−0.418877	−0.0345484
$148$	0.0224026	0.00184148
$149$	−3.95398	−0.323923	−0.161961	−	0.986797i	$-0.551782\pi$
$149$	−3.95398	−0.323923	−0.161961	+	0.986797i	$0.551782\pi$
$150$	0.418877	0.0342012
$151$	9.41563	0.766233	0.383117	−	0.923700i	$-0.374851\pi$
$151$	9.41563	0.766233	0.383117	+	0.923700i	$0.374851\pi$
$152$	0.952490	0.0772572
$153$	4.16954	0.337088
$154$	0	0
$155$	2.34330	0.188218
$156$	−2.22035	−0.177771
$157$	−8.62372	−0.688248	−0.344124	−	0.938924i	$-0.611824\pi$
$157$	−8.62372	−0.688248	−0.344124	+	0.938924i	$0.611824\pi$
$158$	−12.8107	−1.01916
$159$	4.75372	0.376995
$160$	−1.00000	−0.0790569
$161$	−2.37872	−0.187469
$162$	−7.45166	−0.585458
$163$	−10.9905	−0.860842	−0.430421	−	0.902628i	$-0.641635\pi$
$163$	−10.9905	−0.860842	−0.430421	+	0.902628i	$0.641635\pi$
$164$	1.53186	0.119618
$165$	0	0
$166$	6.13526	0.476189
$167$	−4.26261	−0.329851	−0.164925	−	0.986306i	$-0.552738\pi$
$167$	−4.26261	−0.329851	−0.164925	+	0.986306i	$0.552738\pi$
$168$	−0.418877	−0.0323171
$169$	15.0977	1.16136
$170$	1.47618	0.113218
$171$	2.69035	0.205736
$172$	−3.83101	−0.292112
$173$	−21.8753	−1.66315	−0.831575	−	0.555413i	$-0.812560\pi$
$173$	−21.8753	−1.66315	−0.831575	+	0.555413i	$0.812560\pi$
$174$	−2.50392	−0.189821
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−0.322061	−0.0242076
$178$	−13.4596	−1.00884
$179$	12.0148	0.898028	0.449014	−	0.893525i	$-0.351775\pi$
$179$	12.0148	0.898028	0.449014	+	0.893525i	$0.351775\pi$
$180$	−2.82454	−0.210529
$181$	−21.0621	−1.56554	−0.782768	−	0.622314i	$-0.786193\pi$
$181$	−21.0621	−1.56554	−0.782768	+	0.622314i	$0.786193\pi$
$182$	5.30073	0.392916
$183$	3.37620	0.249576
$184$	−2.37872	−0.175361
$185$	0.0224026	0.00164707
$186$	0.981556	0.0719711
$187$	0	0
$188$	−1.50443	−0.109722
$189$	−2.43977	−0.177467
$190$	0.952490	0.0691009
$191$	−7.60935	−0.550593	−0.275296	−	0.961359i	$-0.588776\pi$
$191$	−7.60935	−0.550593	−0.275296	+	0.961359i	$0.588776\pi$
$192$	−0.418877	−0.0302299
$193$	−20.6268	−1.48475	−0.742375	−	0.669985i	$-0.766301\pi$
$193$	−20.6268	−1.48475	−0.742375	+	0.669985i	$0.766301\pi$
$194$	−18.4862	−1.32723
$195$	−2.22035	−0.159003
$196$	1.00000	0.0714286
$197$	10.1376	0.722273	0.361136	−	0.932513i	$-0.382389\pi$
$197$	10.1376	0.722273	0.361136	+	0.932513i	$0.382389\pi$
$198$	0	0
$199$	22.6636	1.60658	0.803290	−	0.595588i	$-0.203081\pi$
$199$	22.6636	1.60658	0.803290	+	0.595588i	$0.203081\pi$
$200$	−1.00000	−0.0707107
$201$	−5.10363	−0.359982
$202$	8.92131	0.627702
$203$	5.97768	0.419551
$204$	0.618340	0.0432925
$205$	1.53186	0.106990
$206$	12.3386	0.859673
$207$	−6.71879	−0.466988
$208$	5.30073	0.367539
$209$	0	0
$210$	−0.418877	−0.0289053
$211$	−13.2771	−0.914033	−0.457016	−	0.889458i	$-0.651082\pi$
$211$	−13.2771	−0.914033	−0.457016	+	0.889458i	$0.651082\pi$
$212$	−11.3487	−0.779434
$213$	4.99921	0.342540
$214$	9.65719	0.660152
$215$	−3.83101	−0.261272
$216$	−2.43977	−0.166005
$217$	−2.34330	−0.159074
$218$	−5.91826	−0.400835
$219$	3.05152	0.206203
$220$	0	0
$221$	−7.82485	−0.526356
$222$	0.00938394	0.000629809 0
$223$	−10.6151	−0.710841	−0.355420	−	0.934707i	$-0.615662\pi$
$223$	−10.6151	−0.710841	−0.355420	+	0.934707i	$0.615662\pi$
$224$	1.00000	0.0668153
$225$	−2.82454	−0.188303
$226$	−12.5957	−0.837853
$227$	24.3926	1.61899	0.809497	−	0.587124i	$-0.199740\pi$
$227$	24.3926	1.61899	0.809497	+	0.587124i	$0.199740\pi$
$228$	0.398977	0.0264229
$229$	11.2804	0.745433	0.372716	−	0.927945i	$-0.378426\pi$
$229$	11.2804	0.745433	0.372716	+	0.927945i	$0.378426\pi$
$230$	−2.37872	−0.156848
$231$	0	0
$232$	5.97768	0.392454
$233$	−9.18604	−0.601798	−0.300899	−	0.953656i	$-0.597287\pi$
$233$	−9.18604	−0.601798	−0.300899	+	0.953656i	$0.597287\pi$
$234$	14.9721	0.978758
$235$	−1.50443	−0.0981379
$236$	0.768866	0.0500489
$237$	−5.36610	−0.348566
$238$	−1.47618	−0.0956868
$239$	3.05287	0.197474	0.0987368	−	0.995114i	$-0.468520\pi$
$239$	3.05287	0.197474	0.0987368	+	0.995114i	$0.468520\pi$
$240$	−0.418877	−0.0270384
$241$	5.27801	0.339987	0.169993	−	0.985445i	$-0.445625\pi$
$241$	5.27801	0.339987	0.169993	+	0.985445i	$0.445625\pi$
$242$	0	0
$243$	−10.4406	−0.669767
$244$	−8.06011	−0.515996
$245$	1.00000	0.0638877
$246$	0.641661	0.0409108
$247$	−5.04889	−0.321253
$248$	−2.34330	−0.148800
$249$	2.56992	0.162862
$250$	−1.00000	−0.0632456
$251$	−2.11801	−0.133688	−0.0668438	−	0.997763i	$-0.521293\pi$
$251$	−2.11801	−0.133688	−0.0668438	+	0.997763i	$0.521293\pi$
$252$	2.82454	0.177929
$253$	0	0
$254$	−2.54206	−0.159503
$255$	0.618340	0.0387219
$256$	1.00000	0.0625000
$257$	−26.8253	−1.67331	−0.836657	−	0.547727i	$-0.815493\pi$
$257$	−26.8253	−1.67331	−0.836657	+	0.547727i	$0.815493\pi$
$258$	−1.60472	−0.0999056
$259$	−0.0224026	−0.00139203
$260$	5.30073	0.328737
$261$	16.8842	1.04511
$262$	7.80495	0.482191
$263$	8.68185	0.535346	0.267673	−	0.963510i	$-0.413745\pi$
$263$	8.68185	0.535346	0.267673	+	0.963510i	$0.413745\pi$
$264$	0	0
$265$	−11.3487	−0.697147
$266$	−0.952490	−0.0584009
$267$	−5.63791	−0.345035
$268$	12.1841	0.744260
$269$	−11.4629	−0.698903	−0.349452	−	0.936954i	$-0.613632\pi$
$269$	−11.4629	−0.698903	−0.349452	+	0.936954i	$0.613632\pi$
$270$	−2.43977	−0.148480
$271$	−17.1909	−1.04427	−0.522137	−	0.852862i	$-0.674865\pi$
$271$	−17.1909	−1.04427	−0.522137	+	0.852862i	$0.674865\pi$
$272$	−1.47618	−0.0895068
$273$	2.22035	0.134382
$274$	−19.7473	−1.19298
$275$	0	0
$276$	−0.996391	−0.0599757
$277$	−15.2055	−0.913610	−0.456805	−	0.889567i	$-0.651006\pi$
$277$	−15.2055	−0.913610	−0.456805	+	0.889567i	$0.651006\pi$
$278$	13.2995	0.797649
$279$	−6.61875	−0.396254
$280$	1.00000	0.0597614
$281$	2.39475	0.142859	0.0714293	−	0.997446i	$-0.477244\pi$
$281$	2.39475	0.142859	0.0714293	+	0.997446i	$0.477244\pi$
$282$	−0.630170	−0.0375261
$283$	12.0220	0.714632	0.357316	−	0.933983i	$-0.383692\pi$
$283$	12.0220	0.714632	0.357316	+	0.933983i	$0.383692\pi$
$284$	−11.9348	−0.708199
$285$	0.398977	0.0236333
$286$	0	0
$287$	−1.53186	−0.0904228
$288$	2.82454	0.166438
$289$	−14.8209	−0.871816
$290$	5.97768	0.351022
$291$	−7.74345	−0.453929
$292$	−7.28500	−0.426322
$293$	−11.2424	−0.656790	−0.328395	−	0.944540i	$-0.606508\pi$
$293$	−11.2424	−0.656790	−0.328395	+	0.944540i	$0.606508\pi$
$294$	0.418877	0.0244294
$295$	0.768866	0.0447651
$296$	−0.0224026	−0.00130213
$297$	0	0
$298$	3.95398	0.229048
$299$	12.6089	0.729194
$300$	−0.418877	−0.0241839
$301$	3.83101	0.220816
$302$	−9.41563	−0.541809
$303$	3.73694	0.214681
$304$	−0.952490	−0.0546291
$305$	−8.06011	−0.461521
$306$	−4.16954	−0.238357
$307$	−25.1111	−1.43317	−0.716584	−	0.697501i	$-0.754295\pi$
$307$	−25.1111	−1.43317	−0.716584	+	0.697501i	$0.754295\pi$
$308$	0	0
$309$	5.16837	0.294018
$310$	−2.34330	−0.133091
$311$	−2.62754	−0.148994	−0.0744971	−	0.997221i	$-0.523735\pi$
$311$	−2.62754	−0.148994	−0.0744971	+	0.997221i	$0.523735\pi$
$312$	2.22035	0.125703
$313$	3.90115	0.220506	0.110253	−	0.993904i	$-0.464834\pi$
$313$	3.90115	0.220506	0.110253	+	0.993904i	$0.464834\pi$
$314$	8.62372	0.486665
$315$	2.82454	0.159145
$316$	12.8107	0.720657
$317$	−27.5825	−1.54919	−0.774594	−	0.632459i	$-0.782046\pi$
$317$	−27.5825	−1.54919	−0.774594	+	0.632459i	$0.782046\pi$
$318$	−4.75372	−0.266576
$319$	0	0
$320$	1.00000	0.0559017
$321$	4.04518	0.225780
$322$	2.37872	0.132561
$323$	1.40605	0.0782348
$324$	7.45166	0.413981
$325$	5.30073	0.294031
$326$	10.9905	0.608708
$327$	−2.47903	−0.137090
$328$	−1.53186	−0.0845828
$329$	1.50443	0.0829417
$330$	0	0
$331$	33.4595	1.83910	0.919551	−	0.392972i	$-0.128553\pi$
$331$	33.4595	1.83910	0.919551	+	0.392972i	$0.128553\pi$
$332$	−6.13526	−0.336716
$333$	−0.0632771	−0.00346756
$334$	4.26261	0.233240
$335$	12.1841	0.665687
$336$	0.418877	0.0228516
$337$	−15.9275	−0.867627	−0.433813	−	0.901003i	$-0.642832\pi$
$337$	−15.9275	−0.867627	−0.433813	+	0.901003i	$0.642832\pi$
$338$	−15.0977	−0.821206
$339$	−5.27605	−0.286556
$340$	−1.47618	−0.0800573
$341$	0	0
$342$	−2.69035	−0.145477
$343$	−1.00000	−0.0539949
$344$	3.83101	0.206554
$345$	−0.996391	−0.0536439
$346$	21.8753	1.17602
$347$	−19.0799	−1.02426	−0.512132	−	0.858907i	$-0.671144\pi$
$347$	−19.0799	−1.02426	−0.512132	+	0.858907i	$0.671144\pi$
$348$	2.50392	0.134224
$349$	17.3451	0.928460	0.464230	−	0.885715i	$-0.346331\pi$
$349$	17.3451	0.928460	0.464230	+	0.885715i	$0.346331\pi$
$350$	1.00000	0.0534522
$351$	12.9325	0.690288
$352$	0	0
$353$	−23.2547	−1.23772	−0.618861	−	0.785501i	$-0.712405\pi$
$353$	−23.2547	−1.23772	−0.618861	+	0.785501i	$0.712405\pi$
$354$	0.322061	0.0171173
$355$	−11.9348	−0.633433
$356$	13.4596	0.713357
$357$	−0.618340	−0.0327260
$358$	−12.0148	−0.635002
$359$	0.200151	0.0105636	0.00528179	−	0.999986i	$-0.498319\pi$
$359$	0.200151	0.0105636	0.00528179	+	0.999986i	$0.498319\pi$
$360$	2.82454	0.148866
$361$	−18.0928	−0.952251
$362$	21.0621	1.10700
$363$	0	0
$364$	−5.30073	−0.277834
$365$	−7.28500	−0.381314
$366$	−3.37620	−0.176477
$367$	13.0368	0.680516	0.340258	−	0.940332i	$-0.389486\pi$
$367$	13.0368	0.680516	0.340258	+	0.940332i	$0.389486\pi$
$368$	2.37872	0.123999
$369$	−4.32680	−0.225244
$370$	−0.0224026	−0.00116466
$371$	11.3487	0.589197
$372$	−0.981556	−0.0508913
$373$	−16.1910	−0.838336	−0.419168	−	0.907909i	$-0.637678\pi$
$373$	−16.1910	−0.838336	−0.419168	+	0.907909i	$0.637678\pi$
$374$	0	0
$375$	−0.418877	−0.0216307
$376$	1.50443	0.0775848
$377$	−31.6861	−1.63192
$378$	2.43977	0.125488
$379$	38.0576	1.95489	0.977443	−	0.211198i	$-0.0677366\pi$
$379$	38.0576	1.95489	0.977443	+	0.211198i	$0.0677366\pi$
$380$	−0.952490	−0.0488617
$381$	−1.06481	−0.0545520
$382$	7.60935	0.389328
$383$	−3.26994	−0.167086	−0.0835431	−	0.996504i	$-0.526624\pi$
$383$	−3.26994	−0.167086	−0.0835431	+	0.996504i	$0.526624\pi$
$384$	0.418877	0.0213757
$385$	0	0
$386$	20.6268	1.04988
$387$	10.8208	0.550054
$388$	18.4862	0.938495
$389$	−30.0096	−1.52155	−0.760775	−	0.649016i	$-0.775181\pi$
$389$	−30.0096	−1.52155	−0.760775	+	0.649016i	$0.775181\pi$
$390$	2.22035	0.112432
$391$	−3.51143	−0.177580
$392$	−1.00000	−0.0505076
$393$	3.26931	0.164915
$394$	−10.1376	−0.510724
$395$	12.8107	0.644575
$396$	0	0
$397$	15.5047	0.778158	0.389079	−	0.921204i	$-0.372793\pi$
$397$	15.5047	0.778158	0.389079	+	0.921204i	$0.372793\pi$
$398$	−22.6636	−1.13602
$399$	−0.398977	−0.0199738
$400$	1.00000	0.0500000
$401$	−5.70472	−0.284880	−0.142440	−	0.989803i	$-0.545495\pi$
$401$	−5.70472	−0.284880	−0.142440	+	0.989803i	$0.545495\pi$
$402$	5.10363	0.254546
$403$	12.4212	0.618744
$404$	−8.92131	−0.443852
$405$	7.45166	0.370276
$406$	−5.97768	−0.296667
$407$	0	0
$408$	−0.618340	−0.0306124
$409$	−1.28797	−0.0636860	−0.0318430	−	0.999493i	$-0.510138\pi$
$409$	−1.28797	−0.0636860	−0.0318430	+	0.999493i	$0.510138\pi$
$410$	−1.53186	−0.0756532
$411$	−8.27171	−0.408013
$412$	−12.3386	−0.607881
$413$	−0.768866	−0.0378334
$414$	6.71879	0.330211
$415$	−6.13526	−0.301168
$416$	−5.30073	−0.259889
$417$	5.57084	0.272805
$418$	0	0
$419$	36.2889	1.77283	0.886414	−	0.462893i	$-0.153189\pi$
$419$	36.2889	1.77283	0.886414	+	0.462893i	$0.153189\pi$
$420$	0.418877	0.0204391
$421$	−13.4042	−0.653279	−0.326639	−	0.945149i	$-0.605916\pi$
$421$	−13.4042	−0.653279	−0.326639	+	0.945149i	$0.605916\pi$
$422$	13.2771	0.646319
$423$	4.24931	0.206609
$424$	11.3487	0.551143
$425$	−1.47618	−0.0716054
$426$	−4.99921	−0.242213
$427$	8.06011	0.390056
$428$	−9.65719	−0.466798
$429$	0	0
$430$	3.83101	0.184748
$431$	−2.38684	−0.114970	−0.0574851	−	0.998346i	$-0.518308\pi$
$431$	−2.38684	−0.114970	−0.0574851	+	0.998346i	$0.518308\pi$
$432$	2.43977	0.117383
$433$	−22.6678	−1.08935	−0.544673	−	0.838648i	$-0.683346\pi$
$433$	−22.6678	−1.08935	−0.544673	+	0.838648i	$0.683346\pi$
$434$	2.34330	0.112482
$435$	2.50392	0.120054
$436$	5.91826	0.283433
$437$	−2.26571	−0.108383
$438$	−3.05152	−0.145807
$439$	−17.7876	−0.848958	−0.424479	−	0.905438i	$-0.639543\pi$
$439$	−17.7876	−0.848958	−0.424479	+	0.905438i	$0.639543\pi$
$440$	0	0
$441$	−2.82454	−0.134502
$442$	7.82485	0.372190
$443$	−24.0914	−1.14462	−0.572309	−	0.820038i	$-0.693952\pi$
$443$	−24.0914	−1.14462	−0.572309	+	0.820038i	$0.693952\pi$
$444$	−0.00938394	−0.000445342 0
$445$	13.4596	0.638046
$446$	10.6151	0.502640
$447$	1.65623	0.0783371
$448$	−1.00000	−0.0472456
$449$	−6.06955	−0.286440	−0.143220	−	0.989691i	$-0.545746\pi$
$449$	−6.06955	−0.286440	−0.143220	+	0.989691i	$0.545746\pi$
$450$	2.82454	0.133150
$451$	0	0
$452$	12.5957	0.592452
$453$	−3.94399	−0.185305
$454$	−24.3926	−1.14480
$455$	−5.30073	−0.248502
$456$	−0.398977	−0.0186838
$457$	−4.92065	−0.230178	−0.115089	−	0.993355i	$-0.536715\pi$
$457$	−4.92065	−0.230178	−0.115089	+	0.993355i	$0.536715\pi$
$458$	−11.2804	−0.527101
$459$	−3.60155	−0.168106
$460$	2.37872	0.110908
$461$	33.6918	1.56918	0.784591	−	0.620013i	$-0.212873\pi$
$461$	33.6918	1.56918	0.784591	+	0.620013i	$0.212873\pi$
$462$	0	0
$463$	2.26163	0.105107	0.0525533	−	0.998618i	$-0.483264\pi$
$463$	2.26163	0.105107	0.0525533	+	0.998618i	$0.483264\pi$
$464$	−5.97768	−0.277507
$465$	−0.981556	−0.0455185
$466$	9.18604	0.425535
$467$	13.3924	0.619727	0.309863	−	0.950781i	$-0.399717\pi$
$467$	13.3924	0.619727	0.309863	+	0.950781i	$0.399717\pi$
$468$	−14.9721	−0.692087
$469$	−12.1841	−0.562608
$470$	1.50443	0.0693940
$471$	3.61228	0.166445
$472$	−0.768866	−0.0353899
$473$	0	0
$474$	5.36610	0.246473
$475$	−0.952490	−0.0437033
$476$	1.47618	0.0676608
$477$	32.0550	1.46770
$478$	−3.05287	−0.139635
$479$	−16.6637	−0.761385	−0.380692	−	0.924702i	$-0.624314\pi$
$479$	−16.6637	−0.761385	−0.380692	+	0.924702i	$0.624314\pi$
$480$	0.418877	0.0191190
$481$	0.118750	0.00541454
$482$	−5.27801	−0.240407
$483$	0.996391	0.0453374
$484$	0	0
$485$	18.4862	0.839416
$486$	10.4406	0.473597
$487$	−12.6959	−0.575308	−0.287654	−	0.957734i	$-0.592875\pi$
$487$	−12.6959	−0.575308	−0.287654	+	0.957734i	$0.592875\pi$
$488$	8.06011	0.364864
$489$	4.60367	0.208185
$490$	−1.00000	−0.0451754
$491$	−15.4392	−0.696761	−0.348380	−	0.937353i	$-0.613268\pi$
$491$	−15.4392	−0.696761	−0.348380	+	0.937353i	$0.613268\pi$
$492$	−0.641661	−0.0289283
$493$	8.82416	0.397420
$494$	5.04889	0.227160
$495$	0	0
$496$	2.34330	0.105217
$497$	11.9348	0.535348
$498$	−2.56992	−0.115161
$499$	−5.96950	−0.267231	−0.133616	−	0.991033i	$-0.542659\pi$
$499$	−5.96950	−0.267231	−0.133616	+	0.991033i	$0.542659\pi$
$500$	1.00000	0.0447214
$501$	1.78551	0.0797707
$502$	2.11801	0.0945314
$503$	1.00487	0.0448052	0.0224026	−	0.999749i	$-0.492868\pi$
$503$	1.00487	0.0448052	0.0224026	+	0.999749i	$0.492868\pi$
$504$	−2.82454	−0.125815
$505$	−8.92131	−0.396993
$506$	0	0
$507$	−6.32408	−0.280862
$508$	2.54206	0.112786
$509$	19.2453	0.853034	0.426517	−	0.904480i	$-0.359740\pi$
$509$	19.2453	0.853034	0.426517	+	0.904480i	$0.359740\pi$
$510$	−0.618340	−0.0273806
$511$	7.28500	0.322269
$512$	−1.00000	−0.0441942
$513$	−2.32386	−0.102601
$514$	26.8253	1.18321
$515$	−12.3386	−0.543705
$516$	1.60472	0.0706439
$517$	0	0
$518$	0.0224026	0.000984314 0
$519$	9.16307	0.402214
$520$	−5.30073	−0.232452
$521$	4.02400	0.176295	0.0881473	−	0.996107i	$-0.471905\pi$
$521$	4.02400	0.176295	0.0881473	+	0.996107i	$0.471905\pi$
$522$	−16.8842	−0.739002
$523$	1.07196	0.0468734	0.0234367	−	0.999725i	$-0.492539\pi$
$523$	1.07196	0.0468734	0.0234367	+	0.999725i	$0.492539\pi$
$524$	−7.80495	−0.340961
$525$	0.418877	0.0182813
$526$	−8.68185	−0.378547
$527$	−3.45914	−0.150683
$528$	0	0
$529$	−17.3417	−0.753987
$530$	11.3487	0.492957
$531$	−2.17170	−0.0942436
$532$	0.952490	0.0412957
$533$	8.11997	0.351715
$534$	5.63791	0.243976
$535$	−9.65719	−0.417517
$536$	−12.1841	−0.526272
$537$	−5.03273	−0.217178
$538$	11.4629	0.494199
$539$	0	0
$540$	2.43977	0.104991
$541$	31.8711	1.37025	0.685123	−	0.728428i	$-0.259749\pi$
$541$	31.8711	1.37025	0.685123	+	0.728428i	$0.259749\pi$
$542$	17.1909	0.738413
$543$	8.82245	0.378607
$544$	1.47618	0.0632909
$545$	5.91826	0.253511
$546$	−2.22035	−0.0950224
$547$	7.60840	0.325312	0.162656	−	0.986683i	$-0.447994\pi$
$547$	7.60840	0.325312	0.162656	+	0.986683i	$0.447994\pi$
$548$	19.7473	0.843564
$549$	22.7661	0.971635
$550$	0	0
$551$	5.69369	0.242559
$552$	0.996391	0.0424092
$553$	−12.8107	−0.544765
$554$	15.2055	0.646020
$555$	−0.00938394	−0.000398326 0
$556$	−13.2995	−0.564023
$557$	−25.1526	−1.06575	−0.532875	−	0.846194i	$-0.678888\pi$
$557$	−25.1526	−1.06575	−0.532875	+	0.846194i	$0.678888\pi$
$558$	6.61875	0.280194
$559$	−20.3071	−0.858900
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−2.39475	−0.101016
$563$	−36.3670	−1.53268	−0.766342	−	0.642432i	$-0.777925\pi$
$563$	−36.3670	−1.53268	−0.766342	+	0.642432i	$0.777925\pi$
$564$	0.630170	0.0265349
$565$	12.5957	0.529905
$566$	−12.0220	−0.505321
$567$	−7.45166	−0.312940
$568$	11.9348	0.500773
$569$	−23.7370	−0.995109	−0.497554	−	0.867433i	$-0.665769\pi$
$569$	−23.7370	−0.995109	−0.497554	+	0.867433i	$0.665769\pi$
$570$	−0.398977	−0.0167113
$571$	42.7475	1.78893	0.894464	−	0.447139i	$-0.147557\pi$
$571$	42.7475	1.78893	0.894464	+	0.447139i	$0.147557\pi$
$572$	0	0
$573$	3.18738	0.133155
$574$	1.53186	0.0639386
$575$	2.37872	0.0991994
$576$	−2.82454	−0.117689
$577$	25.6758	1.06890	0.534449	−	0.845201i	$-0.320519\pi$
$577$	25.6758	1.06890	0.534449	+	0.845201i	$0.320519\pi$
$578$	14.8209	0.616467
$579$	8.64010	0.359070
$580$	−5.97768	−0.248210
$581$	6.13526	0.254534
$582$	7.74345	0.320976
$583$	0	0
$584$	7.28500	0.301455
$585$	−14.9721	−0.619021
$586$	11.2424	0.464421
$587$	−37.8416	−1.56189	−0.780945	−	0.624600i	$-0.785262\pi$
$587$	−37.8416	−1.56189	−0.780945	+	0.624600i	$0.785262\pi$
$588$	−0.418877	−0.0172742
$589$	−2.23197	−0.0919668
$590$	−0.768866	−0.0316537
$591$	−4.24640	−0.174674
$592$	0.0224026	0.000920741 0
$593$	−8.15354	−0.334826	−0.167413	−	0.985887i	$-0.553541\pi$
$593$	−8.15354	−0.334826	−0.167413	+	0.985887i	$0.553541\pi$
$594$	0	0
$595$	1.47618	0.0605177
$596$	−3.95398	−0.161961
$597$	−9.49326	−0.388533
$598$	−12.6089	−0.515618
$599$	−10.0438	−0.410378	−0.205189	−	0.978722i	$-0.565781\pi$
$599$	−10.0438	−0.410378	−0.205189	+	0.978722i	$0.565781\pi$
$600$	0.418877	0.0171006
$601$	−19.4237	−0.792307	−0.396154	−	0.918184i	$-0.629655\pi$
$601$	−19.4237	−0.792307	−0.396154	+	0.918184i	$0.629655\pi$
$602$	−3.83101	−0.156140
$603$	−34.4144	−1.40146
$604$	9.41563	0.383117
$605$	0	0
$606$	−3.73694	−0.151803
$607$	−14.8851	−0.604169	−0.302084	−	0.953281i	$-0.597682\pi$
$607$	−14.8851	−0.604169	−0.302084	+	0.953281i	$0.597682\pi$
$608$	0.952490	0.0386286
$609$	−2.50392	−0.101464
$610$	8.06011	0.326345
$611$	−7.97455	−0.322616
$612$	4.16954	0.168544
$613$	18.1069	0.731332	0.365666	−	0.930746i	$-0.380841\pi$
$613$	18.1069	0.731332	0.365666	+	0.930746i	$0.380841\pi$
$614$	25.1111	1.01340
$615$	−0.641661	−0.0258743
$616$	0	0
$617$	−33.3189	−1.34137	−0.670684	−	0.741743i	$-0.733999\pi$
$617$	−33.3189	−1.34137	−0.670684	+	0.741743i	$0.733999\pi$
$618$	−5.16837	−0.207902
$619$	−13.0672	−0.525214	−0.262607	−	0.964903i	$-0.584582\pi$
$619$	−13.0672	−0.525214	−0.262607	+	0.964903i	$0.584582\pi$
$620$	2.34330	0.0941092
$621$	5.80352	0.232887
$622$	2.62754	0.105355
$623$	−13.4596	−0.539247
$624$	−2.22035	−0.0888853
$625$	1.00000	0.0400000
$626$	−3.90115	−0.155921
$627$	0	0
$628$	−8.62372	−0.344124
$629$	−0.0330704	−0.00131860
$630$	−2.82454	−0.112532
$631$	39.3161	1.56515	0.782575	−	0.622557i	$-0.213906\pi$
$631$	39.3161	1.56515	0.782575	+	0.622557i	$0.213906\pi$
$632$	−12.8107	−0.509581
$633$	5.56147	0.221049
$634$	27.5825	1.09544
$635$	2.54206	0.100879
$636$	4.75372	0.188497
$637$	5.30073	0.210022
$638$	0	0
$639$	33.7103	1.33356
$640$	−1.00000	−0.0395285
$641$	33.0127	1.30392	0.651962	−	0.758251i	$-0.273946\pi$
$641$	33.0127	1.30392	0.651962	+	0.758251i	$0.273946\pi$
$642$	−4.04518	−0.159650
$643$	0.128880	0.00508254	0.00254127	−	0.999997i	$-0.499191\pi$
$643$	0.128880	0.00508254	0.00254127	+	0.999997i	$0.499191\pi$
$644$	−2.37872	−0.0937347
$645$	1.60472	0.0631858
$646$	−1.40605	−0.0553203
$647$	31.9843	1.25743	0.628716	−	0.777635i	$-0.283581\pi$
$647$	31.9843	1.25743	0.628716	+	0.777635i	$0.283581\pi$
$648$	−7.45166	−0.292729
$649$	0	0
$650$	−5.30073	−0.207912
$651$	0.981556	0.0384702
$652$	−10.9905	−0.430421
$653$	−30.0198	−1.17477	−0.587384	−	0.809309i	$-0.699842\pi$
$653$	−30.0198	−1.17477	−0.587384	+	0.809309i	$0.699842\pi$
$654$	2.47903	0.0969376
$655$	−7.80495	−0.304965
$656$	1.53186	0.0598091
$657$	20.5768	0.802776
$658$	−1.50443	−0.0586486
$659$	19.3178	0.752515	0.376258	−	0.926515i	$-0.377211\pi$
$659$	19.3178	0.752515	0.376258	+	0.926515i	$0.377211\pi$
$660$	0	0
$661$	−0.389498	−0.0151497	−0.00757487	−	0.999971i	$-0.502411\pi$
$661$	−0.389498	−0.0151497	−0.00757487	+	0.999971i	$0.502411\pi$
$662$	−33.4595	−1.30044
$663$	3.27765	0.127293
$664$	6.13526	0.238094
$665$	0.952490	0.0369360
$666$	0.0632771	0.00245194
$667$	−14.2192	−0.550571
$668$	−4.26261	−0.164925
$669$	4.44643	0.171909
$670$	−12.1841	−0.470712
$671$	0	0
$672$	−0.418877	−0.0161585
$673$	−31.7397	−1.22347	−0.611737	−	0.791061i	$-0.709529\pi$
$673$	−31.7397	−1.22347	−0.611737	+	0.791061i	$0.709529\pi$
$674$	15.9275	0.613505
$675$	2.43977	0.0939067
$676$	15.0977	0.580681
$677$	−42.1706	−1.62075	−0.810374	−	0.585913i	$-0.800736\pi$
$677$	−42.1706	−1.62075	−0.810374	+	0.585913i	$0.800736\pi$
$678$	5.27605	0.202625
$679$	−18.4862	−0.709436
$680$	1.47618	0.0566091
$681$	−10.2175	−0.391536
$682$	0	0
$683$	−22.4212	−0.857923	−0.428961	−	0.903323i	$-0.641120\pi$
$683$	−22.4212	−0.857923	−0.428961	+	0.903323i	$0.641120\pi$
$684$	2.69035	0.102868
$685$	19.7473	0.754507
$686$	1.00000	0.0381802
$687$	−4.72512	−0.180275
$688$	−3.83101	−0.146056
$689$	−60.1565	−2.29178
$690$	0.996391	0.0379320
$691$	−21.3409	−0.811844	−0.405922	−	0.913908i	$-0.633050\pi$
$691$	−21.3409	−0.811844	−0.405922	+	0.913908i	$0.633050\pi$
$692$	−21.8753	−0.831575
$693$	0	0
$694$	19.0799	0.724264
$695$	−13.2995	−0.504478
$696$	−2.50392	−0.0949107
$697$	−2.26131	−0.0856531
$698$	−17.3451	−0.656520
$699$	3.84782	0.145538
$700$	−1.00000	−0.0377964
$701$	22.4894	0.849412	0.424706	−	0.905331i	$-0.360377\pi$
$701$	22.4894	0.849412	0.424706	+	0.905331i	$0.360377\pi$
$702$	−12.9325	−0.488107
$703$	−0.0213383	−0.000804788 0
$704$	0	0
$705$	0.630170	0.0237336
$706$	23.2547	0.875201
$707$	8.92131	0.335521
$708$	−0.322061	−0.0121038
$709$	−38.6442	−1.45131	−0.725656	−	0.688058i	$-0.758464\pi$
$709$	−38.6442	−1.45131	−0.725656	+	0.688058i	$0.758464\pi$
$710$	11.9348	0.447905
$711$	−36.1843	−1.35702
$712$	−13.4596	−0.504419
$713$	5.57405	0.208750
$714$	0.618340	0.0231408
$715$	0	0
$716$	12.0148	0.449014
$717$	−1.27878	−0.0477568
$718$	−0.200151	−0.00746958
$719$	17.1230	0.638580	0.319290	−	0.947657i	$-0.396556\pi$
$719$	17.1230	0.638580	0.319290	+	0.947657i	$0.396556\pi$
$720$	−2.82454	−0.105264
$721$	12.3386	0.459514
$722$	18.0928	0.673343
$723$	−2.21084	−0.0822220
$724$	−21.0621	−0.782768
$725$	−5.97768	−0.222006
$726$	0	0
$727$	−21.9967	−0.815811	−0.407906	−	0.913024i	$-0.633741\pi$
$727$	−21.9967	−0.815811	−0.407906	+	0.913024i	$0.633741\pi$
$728$	5.30073	0.196458
$729$	−17.9816	−0.665987
$730$	7.28500	0.269630
$731$	5.65527	0.209168
$732$	3.37620	0.124788
$733$	−13.0302	−0.481283	−0.240641	−	0.970614i	$-0.577358\pi$
$733$	−13.0302	−0.481283	−0.240641	+	0.970614i	$0.577358\pi$
$734$	−13.0368	−0.481198
$735$	−0.418877	−0.0154505
$736$	−2.37872	−0.0876807
$737$	0	0
$738$	4.32680	0.159272
$739$	17.3424	0.637949	0.318974	−	0.947763i	$-0.396662\pi$
$739$	17.3424	0.637949	0.318974	+	0.947763i	$0.396662\pi$
$740$	0.0224026	0.000823536 0
$741$	2.11487	0.0776915
$742$	−11.3487	−0.416625
$743$	27.3421	1.00308	0.501542	−	0.865133i	$-0.332766\pi$
$743$	27.3421	1.00308	0.501542	+	0.865133i	$0.332766\pi$
$744$	0.981556	0.0359856
$745$	−3.95398	−0.144863
$746$	16.1910	0.592793
$747$	17.3293	0.634046
$748$	0	0
$749$	9.65719	0.352866
$750$	0.418877	0.0152952
$751$	−37.6779	−1.37488	−0.687442	−	0.726239i	$-0.741267\pi$
$751$	−37.6779	−1.37488	−0.687442	+	0.726239i	$0.741267\pi$
$752$	−1.50443	−0.0548608
$753$	0.887186	0.0323309
$754$	31.6861	1.15394
$755$	9.41563	0.342670
$756$	−2.43977	−0.0887335
$757$	−8.49638	−0.308806	−0.154403	−	0.988008i	$-0.549345\pi$
$757$	−8.49638	−0.308806	−0.154403	+	0.988008i	$0.549345\pi$
$758$	−38.0576	−1.38231
$759$	0	0
$760$	0.952490	0.0345505
$761$	−14.7525	−0.534777	−0.267388	−	0.963589i	$-0.586161\pi$
$761$	−14.7525	−0.534777	−0.267388	+	0.963589i	$0.586161\pi$
$762$	1.06481	0.0385741
$763$	−5.91826	−0.214256
$764$	−7.60935	−0.275296
$765$	4.16954	0.150750
$766$	3.26994	0.118148
$767$	4.07555	0.147160
$768$	−0.418877	−0.0151149
$769$	−39.2157	−1.41415	−0.707077	−	0.707136i	$-0.749987\pi$
$769$	−39.2157	−1.41415	−0.707077	+	0.707136i	$0.749987\pi$
$770$	0	0
$771$	11.2365	0.404672
$772$	−20.6268	−0.742375
$773$	1.26797	0.0456058	0.0228029	−	0.999740i	$-0.492741\pi$
$773$	1.26797	0.0456058	0.0228029	+	0.999740i	$0.492741\pi$
$774$	−10.8208	−0.388947
$775$	2.34330	0.0841739
$776$	−18.4862	−0.663616
$777$	0.00938394	0.000336647 0
$778$	30.0096	1.07590
$779$	−1.45908	−0.0522770
$780$	−2.22035	−0.0795014
$781$	0	0
$782$	3.51143	0.125568
$783$	−14.5842	−0.521195
$784$	1.00000	0.0357143
$785$	−8.62372	−0.307794
$786$	−3.26931	−0.116613
$787$	−24.7449	−0.882060	−0.441030	−	0.897492i	$-0.645387\pi$
$787$	−24.7449	−0.882060	−0.441030	+	0.897492i	$0.645387\pi$
$788$	10.1376	0.361136
$789$	−3.63663	−0.129467
$790$	−12.8107	−0.455783
$791$	−12.5957	−0.447851
$792$	0	0
$793$	−42.7245	−1.51719
$794$	−15.5047	−0.550241
$795$	4.75372	0.168597
$796$	22.6636	0.803290
$797$	−24.3368	−0.862054	−0.431027	−	0.902339i	$-0.641849\pi$
$797$	−24.3368	−0.862054	−0.431027	+	0.902339i	$0.641849\pi$
$798$	0.398977	0.0141236
$799$	2.22081	0.0785666
$800$	−1.00000	−0.0353553
$801$	−38.0172	−1.34327
$802$	5.70472	0.201441
$803$	0	0
$804$	−5.10363	−0.179991
$805$	−2.37872	−0.0838388
$806$	−12.4212	−0.437518
$807$	4.80153	0.169022
$808$	8.92131	0.313851
$809$	41.5305	1.46013	0.730067	−	0.683376i	$-0.239489\pi$
$809$	41.5305	1.46013	0.730067	+	0.683376i	$0.239489\pi$
$810$	−7.45166	−0.261825
$811$	−44.5784	−1.56536	−0.782680	−	0.622424i	$-0.786148\pi$
$811$	−44.5784	−1.56536	−0.782680	+	0.622424i	$0.786148\pi$
$812$	5.97768	0.209776
$813$	7.20089	0.252546
$814$	0	0
$815$	−10.9905	−0.384980
$816$	0.618340	0.0216462
$817$	3.64900	0.127662
$818$	1.28797	0.0450328
$819$	14.9721	0.523168
$820$	1.53186	0.0534949
$821$	18.0614	0.630348	0.315174	−	0.949034i	$-0.397937\pi$
$821$	18.0614	0.630348	0.315174	+	0.949034i	$0.397937\pi$
$822$	8.27171	0.288509
$823$	55.8537	1.94694	0.973468	−	0.228823i	$-0.0734877\pi$
$823$	55.8537	1.94694	0.973468	+	0.228823i	$0.0734877\pi$
$824$	12.3386	0.429836
$825$	0	0
$826$	0.768866	0.0267523
$827$	48.4371	1.68432	0.842161	−	0.539227i	$-0.181283\pi$
$827$	48.4371	1.68432	0.842161	+	0.539227i	$0.181283\pi$
$828$	−6.71879	−0.233494
$829$	15.6816	0.544643	0.272321	−	0.962206i	$-0.412209\pi$
$829$	15.6816	0.544643	0.272321	+	0.962206i	$0.412209\pi$
$830$	6.13526	0.212958
$831$	6.36924	0.220946
$832$	5.30073	0.183770
$833$	−1.47618	−0.0511467
$834$	−5.57084	−0.192903
$835$	−4.26261	−0.147514
$836$	0	0
$837$	5.71711	0.197612
$838$	−36.2889	−1.25358
$839$	−22.7942	−0.786945	−0.393472	−	0.919336i	$-0.628726\pi$
$839$	−22.7942	−0.786945	−0.393472	+	0.919336i	$0.628726\pi$
$840$	−0.418877	−0.0144526
$841$	6.73269	0.232162
$842$	13.4042	0.461938
$843$	−1.00310	−0.0345488
$844$	−13.2771	−0.457016
$845$	15.0977	0.519377
$846$	−4.24931	−0.146094
$847$	0	0
$848$	−11.3487	−0.389717
$849$	−5.03573	−0.172826
$850$	1.47618	0.0506327
$851$	0.0532895	0.00182674
$852$	4.99921	0.171270
$853$	−13.4043	−0.458953	−0.229477	−	0.973314i	$-0.573701\pi$
$853$	−13.4043	−0.458953	−0.229477	+	0.973314i	$0.573701\pi$
$854$	−8.06011	−0.275811
$855$	2.69035	0.0920080
$856$	9.65719	0.330076
$857$	−6.20533	−0.211970	−0.105985	−	0.994368i	$-0.533800\pi$
$857$	−6.20533	−0.211970	−0.105985	+	0.994368i	$0.533800\pi$
$858$	0	0
$859$	−34.2020	−1.16696	−0.583479	−	0.812128i	$-0.698309\pi$
$859$	−34.2020	−1.16696	−0.583479	+	0.812128i	$0.698309\pi$
$860$	−3.83101	−0.130636
$861$	0.641661	0.0218678
$862$	2.38684	0.0812962
$863$	−12.9621	−0.441233	−0.220617	−	0.975361i	$-0.570807\pi$
$863$	−12.9621	−0.441233	−0.220617	+	0.975361i	$0.570807\pi$
$864$	−2.43977	−0.0830026
$865$	−21.8753	−0.743783
$866$	22.6678	0.770284
$867$	6.20813	0.210839
$868$	−2.34330	−0.0795368
$869$	0	0
$870$	−2.50392	−0.0848907
$871$	64.5844	2.18836
$872$	−5.91826	−0.200418
$873$	−52.2151	−1.76721
$874$	2.26571	0.0766387
$875$	−1.00000	−0.0338062
$876$	3.05152	0.103101
$877$	−6.05485	−0.204458	−0.102229	−	0.994761i	$-0.532597\pi$
$877$	−6.05485	−0.204458	−0.102229	+	0.994761i	$0.532597\pi$
$878$	17.7876	0.600304
$879$	4.70920	0.158837
$880$	0	0
$881$	−19.9937	−0.673605	−0.336803	−	0.941575i	$-0.609346\pi$
$881$	−19.9937	−0.673605	−0.336803	+	0.941575i	$0.609346\pi$
$882$	2.82454	0.0951073
$883$	−7.40727	−0.249275	−0.124637	−	0.992202i	$-0.539777\pi$
$883$	−7.40727	−0.249275	−0.124637	+	0.992202i	$0.539777\pi$
$884$	−7.82485	−0.263178
$885$	−0.322061	−0.0108260
$886$	24.0914	0.809367
$887$	−57.8294	−1.94172	−0.970860	−	0.239647i	$-0.922968\pi$
$887$	−57.8294	−1.94172	−0.970860	+	0.239647i	$0.922968\pi$
$888$	0.00938394	0.000314904 0
$889$	−2.54206	−0.0852580
$890$	−13.4596	−0.451166
$891$	0	0
$892$	−10.6151	−0.355420
$893$	1.43295	0.0479519
$894$	−1.65623	−0.0553927
$895$	12.0148	0.401610
$896$	1.00000	0.0334077
$897$	−5.28160	−0.176347
$898$	6.06955	0.202543
$899$	−14.0075	−0.467177
$900$	−2.82454	−0.0941514
$901$	16.7528	0.558117
$902$	0	0
$903$	−1.60472	−0.0534018
$904$	−12.5957	−0.418927
$905$	−21.0621	−0.700129
$906$	3.94399	0.131030
$907$	−15.8050	−0.524795	−0.262398	−	0.964960i	$-0.584513\pi$
$907$	−15.8050	−0.524795	−0.262398	+	0.964960i	$0.584513\pi$
$908$	24.3926	0.809497
$909$	25.1986	0.835786
$910$	5.30073	0.175717
$911$	−36.0360	−1.19393	−0.596963	−	0.802269i	$-0.703626\pi$
$911$	−36.0360	−1.19393	−0.596963	+	0.802269i	$0.703626\pi$
$912$	0.398977	0.0132114
$913$	0	0
$914$	4.92065	0.162761
$915$	3.37620	0.111614
$916$	11.2804	0.372716
$917$	7.80495	0.257742
$918$	3.60155	0.118869
$919$	40.4661	1.33485	0.667427	−	0.744675i	$-0.267396\pi$
$919$	40.4661	1.33485	0.667427	+	0.744675i	$0.267396\pi$
$920$	−2.37872	−0.0784240
$921$	10.5185	0.346596
$922$	−33.6918	−1.10958
$923$	−63.2630	−2.08233
$924$	0	0
$925$	0.0224026	0.000736593 0
$926$	−2.26163	−0.0743216
$927$	34.8510	1.14466
$928$	5.97768	0.196227
$929$	29.6298	0.972122	0.486061	−	0.873925i	$-0.338433\pi$
$929$	29.6298	0.972122	0.486061	+	0.873925i	$0.338433\pi$
$930$	0.981556	0.0321865
$931$	−0.952490	−0.0312166
$932$	−9.18604	−0.300899
$933$	1.10062	0.0360326
$934$	−13.3924	−0.438213
$935$	0	0
$936$	14.9721	0.489379
$937$	8.28639	0.270705	0.135352	−	0.990798i	$-0.456783\pi$
$937$	8.28639	0.270705	0.135352	+	0.990798i	$0.456783\pi$
$938$	12.1841	0.397824
$939$	−1.63410	−0.0533269
$940$	−1.50443	−0.0490690
$941$	−30.8591	−1.00598	−0.502988	−	0.864293i	$-0.667766\pi$
$941$	−30.8591	−1.00598	−0.502988	+	0.864293i	$0.667766\pi$
$942$	−3.61228	−0.117694
$943$	3.64386	0.118661
$944$	0.768866	0.0250245
$945$	−2.43977	−0.0793657
$946$	0	0
$947$	12.2671	0.398627	0.199313	−	0.979936i	$-0.436129\pi$
$947$	12.2671	0.398627	0.199313	+	0.979936i	$0.436129\pi$
$948$	−5.36610	−0.174283
$949$	−38.6158	−1.25352
$950$	0.952490	0.0309029
$951$	11.5537	0.374654
$952$	−1.47618	−0.0478434
$953$	7.17969	0.232573	0.116286	−	0.993216i	$-0.462901\pi$
$953$	7.17969	0.232573	0.116286	+	0.993216i	$0.462901\pi$
$954$	−32.0550	−1.03782
$955$	−7.60935	−0.246233
$956$	3.05287	0.0987368
$957$	0	0
$958$	16.6637	0.538380
$959$	−19.7473	−0.637674
$960$	−0.418877	−0.0135192
$961$	−25.5089	−0.822869
$962$	−0.118750	−0.00382866
$963$	27.2771	0.878994
$964$	5.27801	0.169993
$965$	−20.6268	−0.664000
$966$	−0.996391	−0.0320584
$967$	−5.07937	−0.163341	−0.0816707	−	0.996659i	$-0.526026\pi$
$967$	−5.07937	−0.163341	−0.0816707	+	0.996659i	$0.526026\pi$
$968$	0	0
$969$	−0.588963	−0.0189202
$970$	−18.4862	−0.593557
$971$	−40.3117	−1.29366	−0.646832	−	0.762633i	$-0.723906\pi$
$971$	−40.3117	−1.29366	−0.646832	+	0.762633i	$0.723906\pi$
$972$	−10.4406	−0.334884
$973$	13.2995	0.426361
$974$	12.6959	0.406804
$975$	−2.22035	−0.0711082
$976$	−8.06011	−0.257998
$977$	−28.1223	−0.899711	−0.449855	−	0.893101i	$-0.648524\pi$
$977$	−28.1223	−0.899711	−0.449855	+	0.893101i	$0.648524\pi$
$978$	−4.60367	−0.147209
$979$	0	0
$980$	1.00000	0.0319438
$981$	−16.7164	−0.533713
$982$	15.4392	0.492684
$983$	23.1675	0.738927	0.369464	−	0.929245i	$-0.379541\pi$
$983$	23.1675	0.738927	0.369464	+	0.929245i	$0.379541\pi$
$984$	0.641661	0.0204554
$985$	10.1376	0.323010
$986$	−8.82416	−0.281019
$987$	−0.630170	−0.0200585
$988$	−5.04889	−0.160627
$989$	−9.11289	−0.289773
$990$	0	0
$991$	−33.5898	−1.06701	−0.533507	−	0.845795i	$-0.679126\pi$
$991$	−33.5898	−1.06701	−0.533507	+	0.845795i	$0.679126\pi$
$992$	−2.34330	−0.0743999
$993$	−14.0154	−0.444766
$994$	−11.9348	−0.378548
$995$	22.6636	0.718484
$996$	2.56992	0.0814311
$997$	22.6238	0.716502	0.358251	−	0.933625i	$-0.383373\pi$
$997$	22.6238	0.716502	0.358251	+	0.933625i	$0.383373\pi$
$998$	5.96950	0.188961
$999$	0.0546572	0.00172928

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.cy.1.3		6
11.7	odd	10		770.2.n.i.71.2	✓	12
11.8	odd	10		770.2.n.i.141.2	yes	12
11.10	odd	2		8470.2.a.de.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.i.71.2	✓	12	11.7	odd	10
770.2.n.i.141.2	yes	12	11.8	odd	10
8470.2.a.cy.1.3		6	1.1	even	1	trivial
8470.2.a.de.1.3		6	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.418877 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.418877 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.82454 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.418877 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.418877 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.82454 q^{9} -1.00000 q^{10} -0.418877 q^{12} +5.30073 q^{13} +1.00000 q^{14} -0.418877 q^{15} +1.00000 q^{16} -1.47618 q^{17} +2.82454 q^{18} -0.952490 q^{19} +1.00000 q^{20} +0.418877 q^{21} +2.37872 q^{23} +0.418877 q^{24} +1.00000 q^{25} -5.30073 q^{26} +2.43977 q^{27} -1.00000 q^{28} -5.97768 q^{29} +0.418877 q^{30} +2.34330 q^{31} -1.00000 q^{32} +1.47618 q^{34} -1.00000 q^{35} -2.82454 q^{36} +0.0224026 q^{37} +0.952490 q^{38} -2.22035 q^{39} -1.00000 q^{40} +1.53186 q^{41} -0.418877 q^{42} -3.83101 q^{43} -2.82454 q^{45} -2.37872 q^{46} -1.50443 q^{47} -0.418877 q^{48} +1.00000 q^{49} -1.00000 q^{50} +0.618340 q^{51} +5.30073 q^{52} -11.3487 q^{53} -2.43977 q^{54} +1.00000 q^{56} +0.398977 q^{57} +5.97768 q^{58} +0.768866 q^{59} -0.418877 q^{60} -8.06011 q^{61} -2.34330 q^{62} +2.82454 q^{63} +1.00000 q^{64} +5.30073 q^{65} +12.1841 q^{67} -1.47618 q^{68} -0.996391 q^{69} +1.00000 q^{70} -11.9348 q^{71} +2.82454 q^{72} -7.28500 q^{73} -0.0224026 q^{74} -0.418877 q^{75} -0.952490 q^{76} +2.22035 q^{78} +12.8107 q^{79} +1.00000 q^{80} +7.45166 q^{81} -1.53186 q^{82} -6.13526 q^{83} +0.418877 q^{84} -1.47618 q^{85} +3.83101 q^{86} +2.50392 q^{87} +13.4596 q^{89} +2.82454 q^{90} -5.30073 q^{91} +2.37872 q^{92} -0.981556 q^{93} +1.50443 q^{94} -0.952490 q^{95} +0.418877 q^{96} +18.4862 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} - 6 q^{8} + 15 q^{9}+O(q^{10}) \) 6 * q - 6 * q^2 + q^3 + 6 * q^4 + 6 * q^5 - q^6 - 6 * q^7 - 6 * q^8 + 15 * q^9	\( 6 q - 6 q^{2} + q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} - 6 q^{8} + 15 q^{9} - 6 q^{10} + q^{12} - 2 q^{13} + 6 q^{14} + q^{15} + 6 q^{16} - 7 q^{17} - 15 q^{18} - 11 q^{19} + 6 q^{20} - q^{21} - 6 q^{23} - q^{24} + 6 q^{25} + 2 q^{26} + 4 q^{27} - 6 q^{28} - 2 q^{29} - q^{30} - 6 q^{32} + 7 q^{34} - 6 q^{35} + 15 q^{36} - 14 q^{37} + 11 q^{38} - 20 q^{39} - 6 q^{40} - 13 q^{41} + q^{42} - 19 q^{43} + 15 q^{45} + 6 q^{46} + 22 q^{47} + q^{48} + 6 q^{49} - 6 q^{50} + 14 q^{51} - 2 q^{52} - 10 q^{53} - 4 q^{54} + 6 q^{56} - 32 q^{57} + 2 q^{58} - 7 q^{59} + q^{60} - 22 q^{61} - 15 q^{63} + 6 q^{64} - 2 q^{65} + 5 q^{67} - 7 q^{68} - 36 q^{69} + 6 q^{70} + 8 q^{71} - 15 q^{72} - 13 q^{73} + 14 q^{74} + q^{75} - 11 q^{76} + 20 q^{78} + 16 q^{79} + 6 q^{80} + 18 q^{81} + 13 q^{82} + 5 q^{83} - q^{84} - 7 q^{85} + 19 q^{86} - 14 q^{87} + q^{89} - 15 q^{90} + 2 q^{91} - 6 q^{92} - 42 q^{93} - 22 q^{94} - 11 q^{95} - q^{96} - 3 q^{97} - 6 q^{98}+O(q^{100}) \) 6 * q - 6 * q^2 + q^3 + 6 * q^4 + 6 * q^5 - q^6 - 6 * q^7 - 6 * q^8 + 15 * q^9 - 6 * q^10 + q^12 - 2 * q^13 + 6 * q^14 + q^15 + 6 * q^16 - 7 * q^17 - 15 * q^18 - 11 * q^19 + 6 * q^20 - q^21 - 6 * q^23 - q^24 + 6 * q^25 + 2 * q^26 + 4 * q^27 - 6 * q^28 - 2 * q^29 - q^30 - 6 * q^32 + 7 * q^34 - 6 * q^35 + 15 * q^36 - 14 * q^37 + 11 * q^38 - 20 * q^39 - 6 * q^40 - 13 * q^41 + q^42 - 19 * q^43 + 15 * q^45 + 6 * q^46 + 22 * q^47 + q^48 + 6 * q^49 - 6 * q^50 + 14 * q^51 - 2 * q^52 - 10 * q^53 - 4 * q^54 + 6 * q^56 - 32 * q^57 + 2 * q^58 - 7 * q^59 + q^60 - 22 * q^61 - 15 * q^63 + 6 * q^64 - 2 * q^65 + 5 * q^67 - 7 * q^68 - 36 * q^69 + 6 * q^70 + 8 * q^71 - 15 * q^72 - 13 * q^73 + 14 * q^74 + q^75 - 11 * q^76 + 20 * q^78 + 16 * q^79 + 6 * q^80 + 18 * q^81 + 13 * q^82 + 5 * q^83 - q^84 - 7 * q^85 + 19 * q^86 - 14 * q^87 + q^89 - 15 * q^90 + 2 * q^91 - 6 * q^92 - 42 * q^93 - 22 * q^94 - 11 * q^95 - q^96 - 3 * q^97 - 6 * q^98

Introduction

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