LMFDB - Embedded newform 8470.2.a.ce.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:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.73205 q^{6} -1.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.73205 q^{6} -1.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} +1.00000 q^{10} +2.73205 q^{12} +1.46410 q^{13} -1.00000 q^{14} +2.73205 q^{15} +1.00000 q^{16} -3.46410 q^{17} +4.46410 q^{18} +2.73205 q^{19} +1.00000 q^{20} -2.73205 q^{21} +1.26795 q^{23} +2.73205 q^{24} +1.00000 q^{25} +1.46410 q^{26} +4.00000 q^{27} -1.00000 q^{28} +4.73205 q^{29} +2.73205 q^{30} +8.92820 q^{31} +1.00000 q^{32} -3.46410 q^{34} -1.00000 q^{35} +4.46410 q^{36} +3.26795 q^{37} +2.73205 q^{38} +4.00000 q^{39} +1.00000 q^{40} +4.73205 q^{41} -2.73205 q^{42} +4.92820 q^{43} +4.46410 q^{45} +1.26795 q^{46} +2.73205 q^{48} +1.00000 q^{49} +1.00000 q^{50} -9.46410 q^{51} +1.46410 q^{52} -4.73205 q^{53} +4.00000 q^{54} -1.00000 q^{56} +7.46410 q^{57} +4.73205 q^{58} -13.8564 q^{59} +2.73205 q^{60} -2.00000 q^{61} +8.92820 q^{62} -4.46410 q^{63} +1.00000 q^{64} +1.46410 q^{65} -10.9282 q^{67} -3.46410 q^{68} +3.46410 q^{69} -1.00000 q^{70} +9.46410 q^{71} +4.46410 q^{72} -11.4641 q^{73} +3.26795 q^{74} +2.73205 q^{75} +2.73205 q^{76} +4.00000 q^{78} -6.73205 q^{79} +1.00000 q^{80} -2.46410 q^{81} +4.73205 q^{82} +4.39230 q^{83} -2.73205 q^{84} -3.46410 q^{85} +4.92820 q^{86} +12.9282 q^{87} -15.4641 q^{89} +4.46410 q^{90} -1.46410 q^{91} +1.26795 q^{92} +24.3923 q^{93} +2.73205 q^{95} +2.73205 q^{96} +5.80385 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} - 4 q^{13} - 2 q^{14} + 2 q^{15} + 2 q^{16} + 2 q^{18} + 2 q^{19} + 2 q^{20} - 2 q^{21} + 6 q^{23} + 2 q^{24} + 2 q^{25} - 4 q^{26} + 8 q^{27} - 2 q^{28} + 6 q^{29} + 2 q^{30} + 4 q^{31} + 2 q^{32} - 2 q^{35} + 2 q^{36} + 10 q^{37} + 2 q^{38} + 8 q^{39} + 2 q^{40} + 6 q^{41} - 2 q^{42} - 4 q^{43} + 2 q^{45} + 6 q^{46} + 2 q^{48} + 2 q^{49} + 2 q^{50} - 12 q^{51} - 4 q^{52} - 6 q^{53} + 8 q^{54} - 2 q^{56} + 8 q^{57} + 6 q^{58} + 2 q^{60} - 4 q^{61} + 4 q^{62} - 2 q^{63} + 2 q^{64} - 4 q^{65} - 8 q^{67} - 2 q^{70} + 12 q^{71} + 2 q^{72} - 16 q^{73} + 10 q^{74} + 2 q^{75} + 2 q^{76} + 8 q^{78} - 10 q^{79} + 2 q^{80} + 2 q^{81} + 6 q^{82} - 12 q^{83} - 2 q^{84} - 4 q^{86} + 12 q^{87} - 24 q^{89} + 2 q^{90} + 4 q^{91} + 6 q^{92} + 28 q^{93} + 2 q^{95} + 2 q^{96} + 22 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^12 - 4 * q^13 - 2 * q^14 + 2 * q^15 + 2 * q^16 + 2 * q^18 + 2 * q^19 + 2 * q^20 - 2 * q^21 + 6 * q^23 + 2 * q^24 + 2 * q^25 - 4 * q^26 + 8 * q^27 - 2 * q^28 + 6 * q^29 + 2 * q^30 + 4 * q^31 + 2 * q^32 - 2 * q^35 + 2 * q^36 + 10 * q^37 + 2 * q^38 + 8 * q^39 + 2 * q^40 + 6 * q^41 - 2 * q^42 - 4 * q^43 + 2 * q^45 + 6 * q^46 + 2 * q^48 + 2 * q^49 + 2 * q^50 - 12 * q^51 - 4 * q^52 - 6 * q^53 + 8 * q^54 - 2 * q^56 + 8 * q^57 + 6 * q^58 + 2 * q^60 - 4 * q^61 + 4 * q^62 - 2 * q^63 + 2 * q^64 - 4 * q^65 - 8 * q^67 - 2 * q^70 + 12 * q^71 + 2 * q^72 - 16 * q^73 + 10 * q^74 + 2 * q^75 + 2 * q^76 + 8 * q^78 - 10 * q^79 + 2 * q^80 + 2 * q^81 + 6 * q^82 - 12 * q^83 - 2 * q^84 - 4 * q^86 + 12 * q^87 - 24 * q^89 + 2 * q^90 + 4 * q^91 + 6 * q^92 + 28 * q^93 + 2 * q^95 + 2 * q^96 + 22 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	2.73205	1.57735	0.788675	−	0.614810i	$-0.210767\pi$
$3$	2.73205	1.57735	0.788675	+	0.614810i	$0.210767\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	2.73205	1.11536
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	4.46410	1.48803
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	2.73205	0.788675
$13$	1.46410	0.406069	0.203034	−	0.979172i	$-0.434920\pi$
$13$	1.46410	0.406069	0.203034	+	0.979172i	$0.434920\pi$
$14$	−1.00000	−0.267261
$15$	2.73205	0.705412
$16$	1.00000	0.250000
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	4.46410	1.05220
$19$	2.73205	0.626775	0.313388	−	0.949625i	$-0.398536\pi$
$19$	2.73205	0.626775	0.313388	+	0.949625i	$0.398536\pi$
$20$	1.00000	0.223607
$21$	−2.73205	−0.596182
$22$	0	0
$23$	1.26795	0.264386	0.132193	−	0.991224i	$-0.457798\pi$
$23$	1.26795	0.264386	0.132193	+	0.991224i	$0.457798\pi$
$24$	2.73205	0.557678
$25$	1.00000	0.200000
$26$	1.46410	0.287134
$27$	4.00000	0.769800
$28$	−1.00000	−0.188982
$29$	4.73205	0.878720	0.439360	−	0.898311i	$-0.355205\pi$
$29$	4.73205	0.878720	0.439360	+	0.898311i	$0.355205\pi$
$30$	2.73205	0.498802
$31$	8.92820	1.60355	0.801776	−	0.597624i	$-0.203889\pi$
$31$	8.92820	1.60355	0.801776	+	0.597624i	$0.203889\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.46410	−0.594089
$35$	−1.00000	−0.169031
$36$	4.46410	0.744017
$37$	3.26795	0.537248	0.268624	−	0.963245i	$-0.413431\pi$
$37$	3.26795	0.537248	0.268624	+	0.963245i	$0.413431\pi$
$38$	2.73205	0.443197
$39$	4.00000	0.640513
$40$	1.00000	0.158114
$41$	4.73205	0.739022	0.369511	−	0.929226i	$-0.379525\pi$
$41$	4.73205	0.739022	0.369511	+	0.929226i	$0.379525\pi$
$42$	−2.73205	−0.421565
$43$	4.92820	0.751544	0.375772	−	0.926712i	$-0.377378\pi$
$43$	4.92820	0.751544	0.375772	+	0.926712i	$0.377378\pi$
$44$	0	0
$45$	4.46410	0.665469
$46$	1.26795	0.186949
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	2.73205	0.394338
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−9.46410	−1.32524
$52$	1.46410	0.203034
$53$	−4.73205	−0.649997	−0.324999	−	0.945715i	$-0.605364\pi$
$53$	−4.73205	−0.649997	−0.324999	+	0.945715i	$0.605364\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	−1.00000	−0.133631
$57$	7.46410	0.988644
$58$	4.73205	0.621349
$59$	−13.8564	−1.80395	−0.901975	−	0.431788i	$-0.857883\pi$
$59$	−13.8564	−1.80395	−0.901975	+	0.431788i	$0.857883\pi$
$60$	2.73205	0.352706
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	8.92820	1.13388
$63$	−4.46410	−0.562424
$64$	1.00000	0.125000
$65$	1.46410	0.181599
$66$	0	0
$67$	−10.9282	−1.33509	−0.667546	−	0.744568i	$-0.732655\pi$
$67$	−10.9282	−1.33509	−0.667546	+	0.744568i	$0.732655\pi$
$68$	−3.46410	−0.420084
$69$	3.46410	0.417029
$70$	−1.00000	−0.119523
$71$	9.46410	1.12318	0.561591	−	0.827415i	$-0.310189\pi$
$71$	9.46410	1.12318	0.561591	+	0.827415i	$0.310189\pi$
$72$	4.46410	0.526099
$73$	−11.4641	−1.34177	−0.670886	−	0.741561i	$-0.734086\pi$
$73$	−11.4641	−1.34177	−0.670886	+	0.741561i	$0.734086\pi$
$74$	3.26795	0.379891
$75$	2.73205	0.315470
$76$	2.73205	0.313388
$77$	0	0
$78$	4.00000	0.452911
$79$	−6.73205	−0.757415	−0.378707	−	0.925516i	$-0.623631\pi$
$79$	−6.73205	−0.757415	−0.378707	+	0.925516i	$0.623631\pi$
$80$	1.00000	0.111803
$81$	−2.46410	−0.273789
$82$	4.73205	0.522568
$83$	4.39230	0.482118	0.241059	−	0.970510i	$-0.422505\pi$
$83$	4.39230	0.482118	0.241059	+	0.970510i	$0.422505\pi$
$84$	−2.73205	−0.298091
$85$	−3.46410	−0.375735
$86$	4.92820	0.531422
$87$	12.9282	1.38605
$88$	0	0
$89$	−15.4641	−1.63919	−0.819596	−	0.572942i	$-0.805802\pi$
$89$	−15.4641	−1.63919	−0.819596	+	0.572942i	$0.805802\pi$
$90$	4.46410	0.470558
$91$	−1.46410	−0.153480
$92$	1.26795	0.132193
$93$	24.3923	2.52936
$94$	0	0
$95$	2.73205	0.280302
$96$	2.73205	0.278839
$97$	5.80385	0.589291	0.294646	−	0.955607i	$-0.404798\pi$
$97$	5.80385	0.589291	0.294646	+	0.955607i	$0.404798\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−9.46410	−0.937086
$103$	10.5359	1.03813	0.519066	−	0.854734i	$-0.326280\pi$
$103$	10.5359	1.03813	0.519066	+	0.854734i	$0.326280\pi$
$104$	1.46410	0.143567
$105$	−2.73205	−0.266621
$106$	−4.73205	−0.459617
$107$	−0.928203	−0.0897328	−0.0448664	−	0.998993i	$-0.514286\pi$
$107$	−0.928203	−0.0897328	−0.0448664	+	0.998993i	$0.514286\pi$
$108$	4.00000	0.384900
$109$	1.80385	0.172777	0.0863886	−	0.996262i	$-0.472467\pi$
$109$	1.80385	0.172777	0.0863886	+	0.996262i	$0.472467\pi$
$110$	0	0
$111$	8.92820	0.847428
$112$	−1.00000	−0.0944911
$113$	12.9282	1.21618	0.608092	−	0.793867i	$-0.291935\pi$
$113$	12.9282	1.21618	0.608092	+	0.793867i	$0.291935\pi$
$114$	7.46410	0.699077
$115$	1.26795	0.118237
$116$	4.73205	0.439360
$117$	6.53590	0.604244
$118$	−13.8564	−1.27559
$119$	3.46410	0.317554
$120$	2.73205	0.249401
$121$	0	0
$122$	−2.00000	−0.181071
$123$	12.9282	1.16570
$124$	8.92820	0.801776
$125$	1.00000	0.0894427
$126$	−4.46410	−0.397694
$127$	−9.85641	−0.874615	−0.437307	−	0.899312i	$-0.644068\pi$
$127$	−9.85641	−0.874615	−0.437307	+	0.899312i	$0.644068\pi$
$128$	1.00000	0.0883883
$129$	13.4641	1.18545
$130$	1.46410	0.128410
$131$	17.6603	1.54298	0.771492	−	0.636239i	$-0.219511\pi$
$131$	17.6603	1.54298	0.771492	+	0.636239i	$0.219511\pi$
$132$	0	0
$133$	−2.73205	−0.236899
$134$	−10.9282	−0.944053
$135$	4.00000	0.344265
$136$	−3.46410	−0.297044
$137$	7.85641	0.671218	0.335609	−	0.942001i	$-0.391058\pi$
$137$	7.85641	0.671218	0.335609	+	0.942001i	$0.391058\pi$
$138$	3.46410	0.294884
$139$	−4.19615	−0.355913	−0.177957	−	0.984038i	$-0.556949\pi$
$139$	−4.19615	−0.355913	−0.177957	+	0.984038i	$0.556949\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	9.46410	0.794210
$143$	0	0
$144$	4.46410	0.372008
$145$	4.73205	0.392975
$146$	−11.4641	−0.948776
$147$	2.73205	0.225336
$148$	3.26795	0.268624
$149$	7.26795	0.595414	0.297707	−	0.954657i	$-0.403778\pi$
$149$	7.26795	0.595414	0.297707	+	0.954657i	$0.403778\pi$
$150$	2.73205	0.223071
$151$	7.80385	0.635068	0.317534	−	0.948247i	$-0.397145\pi$
$151$	7.80385	0.635068	0.317534	+	0.948247i	$0.397145\pi$
$152$	2.73205	0.221599
$153$	−15.4641	−1.25020
$154$	0	0
$155$	8.92820	0.717131
$156$	4.00000	0.320256
$157$	6.39230	0.510161	0.255081	−	0.966920i	$-0.417898\pi$
$157$	6.39230	0.510161	0.255081	+	0.966920i	$0.417898\pi$
$158$	−6.73205	−0.535573
$159$	−12.9282	−1.02527
$160$	1.00000	0.0790569
$161$	−1.26795	−0.0999284
$162$	−2.46410	−0.193598
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	4.73205	0.369511
$165$	0	0
$166$	4.39230	0.340909
$167$	13.8564	1.07224	0.536120	−	0.844141i	$-0.319889\pi$
$167$	13.8564	1.07224	0.536120	+	0.844141i	$0.319889\pi$
$168$	−2.73205	−0.210782
$169$	−10.8564	−0.835108
$170$	−3.46410	−0.265684
$171$	12.1962	0.932663
$172$	4.92820	0.375772
$173$	−0.928203	−0.0705700	−0.0352850	−	0.999377i	$-0.511234\pi$
$173$	−0.928203	−0.0705700	−0.0352850	+	0.999377i	$0.511234\pi$
$174$	12.9282	0.980085
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−37.8564	−2.84546
$178$	−15.4641	−1.15908
$179$	−19.8564	−1.48414	−0.742069	−	0.670324i	$-0.766155\pi$
$179$	−19.8564	−1.48414	−0.742069	+	0.670324i	$0.766155\pi$
$180$	4.46410	0.332734
$181$	−11.8564	−0.881280	−0.440640	−	0.897684i	$-0.645248\pi$
$181$	−11.8564	−0.881280	−0.440640	+	0.897684i	$0.645248\pi$
$182$	−1.46410	−0.108526
$183$	−5.46410	−0.403918
$184$	1.26795	0.0934745
$185$	3.26795	0.240264
$186$	24.3923	1.78853
$187$	0	0
$188$	0	0
$189$	−4.00000	−0.290957
$190$	2.73205	0.198204
$191$	6.92820	0.501307	0.250654	−	0.968077i	$-0.419354\pi$
$191$	6.92820	0.501307	0.250654	+	0.968077i	$0.419354\pi$
$192$	2.73205	0.197169
$193$	25.4641	1.83295	0.916473	−	0.400096i	$-0.131023\pi$
$193$	25.4641	1.83295	0.916473	+	0.400096i	$0.131023\pi$
$194$	5.80385	0.416692
$195$	4.00000	0.286446
$196$	1.00000	0.0714286
$197$	22.3923	1.59539	0.797693	−	0.603064i	$-0.206054\pi$
$197$	22.3923	1.59539	0.797693	+	0.603064i	$0.206054\pi$
$198$	0	0
$199$	−24.7846	−1.75693	−0.878467	−	0.477803i	$-0.841433\pi$
$199$	−24.7846	−1.75693	−0.878467	+	0.477803i	$0.841433\pi$
$200$	1.00000	0.0707107
$201$	−29.8564	−2.10591
$202$	6.00000	0.422159
$203$	−4.73205	−0.332125
$204$	−9.46410	−0.662620
$205$	4.73205	0.330501
$206$	10.5359	0.734071
$207$	5.66025	0.393415
$208$	1.46410	0.101517
$209$	0	0
$210$	−2.73205	−0.188529
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	−4.73205	−0.324999
$213$	25.8564	1.77165
$214$	−0.928203	−0.0634507
$215$	4.92820	0.336101
$216$	4.00000	0.272166
$217$	−8.92820	−0.606086
$218$	1.80385	0.122172
$219$	−31.3205	−2.11644
$220$	0	0
$221$	−5.07180	−0.341166
$222$	8.92820	0.599222
$223$	−8.39230	−0.561990	−0.280995	−	0.959709i	$-0.590664\pi$
$223$	−8.39230	−0.561990	−0.280995	+	0.959709i	$0.590664\pi$
$224$	−1.00000	−0.0668153
$225$	4.46410	0.297607
$226$	12.9282	0.859971
$227$	−13.8564	−0.919682	−0.459841	−	0.888001i	$-0.652094\pi$
$227$	−13.8564	−0.919682	−0.459841	+	0.888001i	$0.652094\pi$
$228$	7.46410	0.494322
$229$	−13.4641	−0.889733	−0.444866	−	0.895597i	$-0.646749\pi$
$229$	−13.4641	−0.889733	−0.444866	+	0.895597i	$0.646749\pi$
$230$	1.26795	0.0836061
$231$	0	0
$232$	4.73205	0.310674
$233$	−7.85641	−0.514690	−0.257345	−	0.966320i	$-0.582848\pi$
$233$	−7.85641	−0.514690	−0.257345	+	0.966320i	$0.582848\pi$
$234$	6.53590	0.427265
$235$	0	0
$236$	−13.8564	−0.901975
$237$	−18.3923	−1.19471
$238$	3.46410	0.224544
$239$	3.80385	0.246050	0.123025	−	0.992404i	$-0.460740\pi$
$239$	3.80385	0.246050	0.123025	+	0.992404i	$0.460740\pi$
$240$	2.73205	0.176353
$241$	18.1962	1.17212	0.586059	−	0.810269i	$-0.300679\pi$
$241$	18.1962	1.17212	0.586059	+	0.810269i	$0.300679\pi$
$242$	0	0
$243$	−18.7321	−1.20166
$244$	−2.00000	−0.128037
$245$	1.00000	0.0638877
$246$	12.9282	0.824272
$247$	4.00000	0.254514
$248$	8.92820	0.566941
$249$	12.0000	0.760469
$250$	1.00000	0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	−4.46410	−0.281212
$253$	0	0
$254$	−9.85641	−0.618446
$255$	−9.46410	−0.592665
$256$	1.00000	0.0625000
$257$	22.9808	1.43350	0.716750	−	0.697330i	$-0.245629\pi$
$257$	22.9808	1.43350	0.716750	+	0.697330i	$0.245629\pi$
$258$	13.4641	0.838238
$259$	−3.26795	−0.203060
$260$	1.46410	0.0907997
$261$	21.1244	1.30756
$262$	17.6603	1.09105
$263$	−18.9282	−1.16716	−0.583582	−	0.812055i	$-0.698349\pi$
$263$	−18.9282	−1.16716	−0.583582	+	0.812055i	$0.698349\pi$
$264$	0	0
$265$	−4.73205	−0.290688
$266$	−2.73205	−0.167513
$267$	−42.2487	−2.58558
$268$	−10.9282	−0.667546
$269$	4.39230	0.267804	0.133902	−	0.990995i	$-0.457249\pi$
$269$	4.39230	0.267804	0.133902	+	0.990995i	$0.457249\pi$
$270$	4.00000	0.243432
$271$	−24.3923	−1.48173	−0.740863	−	0.671656i	$-0.765583\pi$
$271$	−24.3923	−1.48173	−0.740863	+	0.671656i	$0.765583\pi$
$272$	−3.46410	−0.210042
$273$	−4.00000	−0.242091
$274$	7.85641	0.474623
$275$	0	0
$276$	3.46410	0.208514
$277$	−27.8564	−1.67373	−0.836865	−	0.547410i	$-0.815614\pi$
$277$	−27.8564	−1.67373	−0.836865	+	0.547410i	$0.815614\pi$
$278$	−4.19615	−0.251668
$279$	39.8564	2.38614
$280$	−1.00000	−0.0597614
$281$	10.3923	0.619953	0.309976	−	0.950744i	$-0.399679\pi$
$281$	10.3923	0.619953	0.309976	+	0.950744i	$0.399679\pi$
$282$	0	0
$283$	10.9282	0.649614	0.324807	−	0.945780i	$-0.394701\pi$
$283$	10.9282	0.649614	0.324807	+	0.945780i	$0.394701\pi$
$284$	9.46410	0.561591
$285$	7.46410	0.442135
$286$	0	0
$287$	−4.73205	−0.279324
$288$	4.46410	0.263050
$289$	−5.00000	−0.294118
$290$	4.73205	0.277876
$291$	15.8564	0.929519
$292$	−11.4641	−0.670886
$293$	2.53590	0.148149	0.0740744	−	0.997253i	$-0.476400\pi$
$293$	2.53590	0.148149	0.0740744	+	0.997253i	$0.476400\pi$
$294$	2.73205	0.159336
$295$	−13.8564	−0.806751
$296$	3.26795	0.189946
$297$	0	0
$298$	7.26795	0.421021
$299$	1.85641	0.107359
$300$	2.73205	0.157735
$301$	−4.92820	−0.284057
$302$	7.80385	0.449061
$303$	16.3923	0.941713
$304$	2.73205	0.156694
$305$	−2.00000	−0.114520
$306$	−15.4641	−0.884024
$307$	−31.3205	−1.78756	−0.893778	−	0.448510i	$-0.851955\pi$
$307$	−31.3205	−1.78756	−0.893778	+	0.448510i	$0.851955\pi$
$308$	0	0
$309$	28.7846	1.63750
$310$	8.92820	0.507088
$311$	7.85641	0.445496	0.222748	−	0.974876i	$-0.428497\pi$
$311$	7.85641	0.445496	0.222748	+	0.974876i	$0.428497\pi$
$312$	4.00000	0.226455
$313$	27.2679	1.54128	0.770638	−	0.637273i	$-0.219938\pi$
$313$	27.2679	1.54128	0.770638	+	0.637273i	$0.219938\pi$
$314$	6.39230	0.360739
$315$	−4.46410	−0.251524
$316$	−6.73205	−0.378707
$317$	−32.4449	−1.82229	−0.911143	−	0.412091i	$-0.864798\pi$
$317$	−32.4449	−1.82229	−0.911143	+	0.412091i	$0.864798\pi$
$318$	−12.9282	−0.724978
$319$	0	0
$320$	1.00000	0.0559017
$321$	−2.53590	−0.141540
$322$	−1.26795	−0.0706600
$323$	−9.46410	−0.526597
$324$	−2.46410	−0.136895
$325$	1.46410	0.0812137
$326$	8.00000	0.443079
$327$	4.92820	0.272530
$328$	4.73205	0.261284
$329$	0	0
$330$	0	0
$331$	−4.92820	−0.270879	−0.135439	−	0.990786i	$-0.543245\pi$
$331$	−4.92820	−0.270879	−0.135439	+	0.990786i	$0.543245\pi$
$332$	4.39230	0.241059
$333$	14.5885	0.799443
$334$	13.8564	0.758189
$335$	−10.9282	−0.597072
$336$	−2.73205	−0.149046
$337$	30.7846	1.67694	0.838472	−	0.544944i	$-0.183449\pi$
$337$	30.7846	1.67694	0.838472	+	0.544944i	$0.183449\pi$
$338$	−10.8564	−0.590511
$339$	35.3205	1.91835
$340$	−3.46410	−0.187867
$341$	0	0
$342$	12.1962	0.659492
$343$	−1.00000	−0.0539949
$344$	4.92820	0.265711
$345$	3.46410	0.186501
$346$	−0.928203	−0.0499005
$347$	−7.85641	−0.421754	−0.210877	−	0.977513i	$-0.567632\pi$
$347$	−7.85641	−0.421754	−0.210877	+	0.977513i	$0.567632\pi$
$348$	12.9282	0.693024
$349$	−30.3923	−1.62686	−0.813431	−	0.581661i	$-0.802403\pi$
$349$	−30.3923	−1.62686	−0.813431	+	0.581661i	$0.802403\pi$
$350$	−1.00000	−0.0534522
$351$	5.85641	0.312592
$352$	0	0
$353$	−32.4449	−1.72687	−0.863433	−	0.504464i	$-0.831690\pi$
$353$	−32.4449	−1.72687	−0.863433	+	0.504464i	$0.831690\pi$
$354$	−37.8564	−2.01205
$355$	9.46410	0.502302
$356$	−15.4641	−0.819596
$357$	9.46410	0.500893
$358$	−19.8564	−1.04944
$359$	1.26795	0.0669198	0.0334599	−	0.999440i	$-0.489347\pi$
$359$	1.26795	0.0669198	0.0334599	+	0.999440i	$0.489347\pi$
$360$	4.46410	0.235279
$361$	−11.5359	−0.607153
$362$	−11.8564	−0.623159
$363$	0	0
$364$	−1.46410	−0.0767398
$365$	−11.4641	−0.600059
$366$	−5.46410	−0.285613
$367$	29.4641	1.53801	0.769007	−	0.639241i	$-0.220751\pi$
$367$	29.4641	1.53801	0.769007	+	0.639241i	$0.220751\pi$
$368$	1.26795	0.0660964
$369$	21.1244	1.09969
$370$	3.26795	0.169893
$371$	4.73205	0.245676
$372$	24.3923	1.26468
$373$	−30.3923	−1.57365	−0.786827	−	0.617174i	$-0.788278\pi$
$373$	−30.3923	−1.57365	−0.786827	+	0.617174i	$0.788278\pi$
$374$	0	0
$375$	2.73205	0.141082
$376$	0	0
$377$	6.92820	0.356821
$378$	−4.00000	−0.205738
$379$	23.7128	1.21805	0.609023	−	0.793153i	$-0.291562\pi$
$379$	23.7128	1.21805	0.609023	+	0.793153i	$0.291562\pi$
$380$	2.73205	0.140151
$381$	−26.9282	−1.37957
$382$	6.92820	0.354478
$383$	16.3923	0.837608	0.418804	−	0.908077i	$-0.362449\pi$
$383$	16.3923	0.837608	0.418804	+	0.908077i	$0.362449\pi$
$384$	2.73205	0.139419
$385$	0	0
$386$	25.4641	1.29609
$387$	22.0000	1.11832
$388$	5.80385	0.294646
$389$	−24.2487	−1.22946	−0.614729	−	0.788738i	$-0.710735\pi$
$389$	−24.2487	−1.22946	−0.614729	+	0.788738i	$0.710735\pi$
$390$	4.00000	0.202548
$391$	−4.39230	−0.222128
$392$	1.00000	0.0505076
$393$	48.2487	2.43383
$394$	22.3923	1.12811
$395$	−6.73205	−0.338726
$396$	0	0
$397$	−38.3923	−1.92685	−0.963427	−	0.267970i	$-0.913647\pi$
$397$	−38.3923	−1.92685	−0.963427	+	0.267970i	$0.913647\pi$
$398$	−24.7846	−1.24234
$399$	−7.46410	−0.373672
$400$	1.00000	0.0500000
$401$	−25.1769	−1.25728	−0.628638	−	0.777698i	$-0.716387\pi$
$401$	−25.1769	−1.25728	−0.628638	+	0.777698i	$0.716387\pi$
$402$	−29.8564	−1.48910
$403$	13.0718	0.651153
$404$	6.00000	0.298511
$405$	−2.46410	−0.122442
$406$	−4.73205	−0.234848
$407$	0	0
$408$	−9.46410	−0.468543
$409$	−22.1962	−1.09753	−0.548765	−	0.835977i	$-0.684902\pi$
$409$	−22.1962	−1.09753	−0.548765	+	0.835977i	$0.684902\pi$
$410$	4.73205	0.233699
$411$	21.4641	1.05875
$412$	10.5359	0.519066
$413$	13.8564	0.681829
$414$	5.66025	0.278186
$415$	4.39230	0.215610
$416$	1.46410	0.0717835
$417$	−11.4641	−0.561399
$418$	0	0
$419$	32.7846	1.60163	0.800816	−	0.598910i	$-0.204399\pi$
$419$	32.7846	1.60163	0.800816	+	0.598910i	$0.204399\pi$
$420$	−2.73205	−0.133310
$421$	10.7846	0.525610	0.262805	−	0.964849i	$-0.415352\pi$
$421$	10.7846	0.525610	0.262805	+	0.964849i	$0.415352\pi$
$422$	−8.00000	−0.389434
$423$	0	0
$424$	−4.73205	−0.229809
$425$	−3.46410	−0.168034
$426$	25.8564	1.25275
$427$	2.00000	0.0967868
$428$	−0.928203	−0.0448664
$429$	0	0
$430$	4.92820	0.237659
$431$	−18.3397	−0.883394	−0.441697	−	0.897164i	$-0.645623\pi$
$431$	−18.3397	−0.883394	−0.441697	+	0.897164i	$0.645623\pi$
$432$	4.00000	0.192450
$433$	0.732051	0.0351801	0.0175901	−	0.999845i	$-0.494401\pi$
$433$	0.732051	0.0351801	0.0175901	+	0.999845i	$0.494401\pi$
$434$	−8.92820	−0.428567
$435$	12.9282	0.619860
$436$	1.80385	0.0863886
$437$	3.46410	0.165710
$438$	−31.3205	−1.49655
$439$	−36.3923	−1.73691	−0.868455	−	0.495768i	$-0.834887\pi$
$439$	−36.3923	−1.73691	−0.868455	+	0.495768i	$0.834887\pi$
$440$	0	0
$441$	4.46410	0.212576
$442$	−5.07180	−0.241241
$443$	20.7846	0.987507	0.493753	−	0.869602i	$-0.335625\pi$
$443$	20.7846	0.987507	0.493753	+	0.869602i	$0.335625\pi$
$444$	8.92820	0.423714
$445$	−15.4641	−0.733069
$446$	−8.39230	−0.397387
$447$	19.8564	0.939176
$448$	−1.00000	−0.0472456
$449$	−35.3205	−1.66688	−0.833439	−	0.552612i	$-0.813631\pi$
$449$	−35.3205	−1.66688	−0.833439	+	0.552612i	$0.813631\pi$
$450$	4.46410	0.210440
$451$	0	0
$452$	12.9282	0.608092
$453$	21.3205	1.00172
$454$	−13.8564	−0.650313
$455$	−1.46410	−0.0686381
$456$	7.46410	0.349539
$457$	34.0000	1.59045	0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000	1.59045	0.795226	+	0.606313i	$0.207352\pi$
$458$	−13.4641	−0.629136
$459$	−13.8564	−0.646762
$460$	1.26795	0.0591184
$461$	−29.3205	−1.36559	−0.682796	−	0.730609i	$-0.739236\pi$
$461$	−29.3205	−1.36559	−0.682796	+	0.730609i	$0.739236\pi$
$462$	0	0
$463$	12.9808	0.603267	0.301634	−	0.953424i	$-0.402468\pi$
$463$	12.9808	0.603267	0.301634	+	0.953424i	$0.402468\pi$
$464$	4.73205	0.219680
$465$	24.3923	1.13117
$466$	−7.85641	−0.363941
$467$	17.6603	0.817219	0.408610	−	0.912709i	$-0.366014\pi$
$467$	17.6603	0.817219	0.408610	+	0.912709i	$0.366014\pi$
$468$	6.53590	0.302122
$469$	10.9282	0.504618
$470$	0	0
$471$	17.4641	0.804703
$472$	−13.8564	−0.637793
$473$	0	0
$474$	−18.3923	−0.844787
$475$	2.73205	0.125355
$476$	3.46410	0.158777
$477$	−21.1244	−0.967218
$478$	3.80385	0.173984
$479$	−13.8564	−0.633115	−0.316558	−	0.948573i	$-0.602527\pi$
$479$	−13.8564	−0.633115	−0.316558	+	0.948573i	$0.602527\pi$
$480$	2.73205	0.124700
$481$	4.78461	0.218159
$482$	18.1962	0.828812
$483$	−3.46410	−0.157622
$484$	0	0
$485$	5.80385	0.263539
$486$	−18.7321	−0.849703
$487$	4.87564	0.220937	0.110468	−	0.993880i	$-0.464765\pi$
$487$	4.87564	0.220937	0.110468	+	0.993880i	$0.464765\pi$
$488$	−2.00000	−0.0905357
$489$	21.8564	0.988381
$490$	1.00000	0.0451754
$491$	−42.9282	−1.93732	−0.968661	−	0.248385i	$-0.920100\pi$
$491$	−42.9282	−1.93732	−0.968661	+	0.248385i	$0.920100\pi$
$492$	12.9282	0.582848
$493$	−16.3923	−0.738272
$494$	4.00000	0.179969
$495$	0	0
$496$	8.92820	0.400888
$497$	−9.46410	−0.424523
$498$	12.0000	0.537733
$499$	2.00000	0.0895323	0.0447661	−	0.998997i	$-0.485746\pi$
$499$	2.00000	0.0895323	0.0447661	+	0.998997i	$0.485746\pi$
$500$	1.00000	0.0447214
$501$	37.8564	1.69130
$502$	0	0
$503$	18.9282	0.843967	0.421983	−	0.906604i	$-0.361334\pi$
$503$	18.9282	0.843967	0.421983	+	0.906604i	$0.361334\pi$
$504$	−4.46410	−0.198847
$505$	6.00000	0.266996
$506$	0	0
$507$	−29.6603	−1.31726
$508$	−9.85641	−0.437307
$509$	2.78461	0.123426	0.0617128	−	0.998094i	$-0.480344\pi$
$509$	2.78461	0.123426	0.0617128	+	0.998094i	$0.480344\pi$
$510$	−9.46410	−0.419077
$511$	11.4641	0.507142
$512$	1.00000	0.0441942
$513$	10.9282	0.482492
$514$	22.9808	1.01364
$515$	10.5359	0.464267
$516$	13.4641	0.592724
$517$	0	0
$518$	−3.26795	−0.143585
$519$	−2.53590	−0.111314
$520$	1.46410	0.0642051
$521$	10.3923	0.455295	0.227648	−	0.973744i	$-0.426897\pi$
$521$	10.3923	0.455295	0.227648	+	0.973744i	$0.426897\pi$
$522$	21.1244	0.924588
$523$	−19.3205	−0.844827	−0.422413	−	0.906403i	$-0.638817\pi$
$523$	−19.3205	−0.844827	−0.422413	+	0.906403i	$0.638817\pi$
$524$	17.6603	0.771492
$525$	−2.73205	−0.119236
$526$	−18.9282	−0.825309
$527$	−30.9282	−1.34725
$528$	0	0
$529$	−21.3923	−0.930100
$530$	−4.73205	−0.205547
$531$	−61.8564	−2.68434
$532$	−2.73205	−0.118449
$533$	6.92820	0.300094
$534$	−42.2487	−1.82828
$535$	−0.928203	−0.0401297
$536$	−10.9282	−0.472026
$537$	−54.2487	−2.34100
$538$	4.39230	0.189366
$539$	0	0
$540$	4.00000	0.172133
$541$	−17.1244	−0.736234	−0.368117	−	0.929780i	$-0.619997\pi$
$541$	−17.1244	−0.736234	−0.368117	+	0.929780i	$0.619997\pi$
$542$	−24.3923	−1.04774
$543$	−32.3923	−1.39009
$544$	−3.46410	−0.148522
$545$	1.80385	0.0772683
$546$	−4.00000	−0.171184
$547$	−4.78461	−0.204575	−0.102288	−	0.994755i	$-0.532616\pi$
$547$	−4.78461	−0.204575	−0.102288	+	0.994755i	$0.532616\pi$
$548$	7.85641	0.335609
$549$	−8.92820	−0.381046
$550$	0	0
$551$	12.9282	0.550760
$552$	3.46410	0.147442
$553$	6.73205	0.286276
$554$	−27.8564	−1.18351
$555$	8.92820	0.378981
$556$	−4.19615	−0.177957
$557$	12.2487	0.518995	0.259497	−	0.965744i	$-0.416443\pi$
$557$	12.2487	0.518995	0.259497	+	0.965744i	$0.416443\pi$
$558$	39.8564	1.68726
$559$	7.21539	0.305178
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	10.3923	0.438373
$563$	20.7846	0.875967	0.437983	−	0.898983i	$-0.355693\pi$
$563$	20.7846	0.875967	0.437983	+	0.898983i	$0.355693\pi$
$564$	0	0
$565$	12.9282	0.543894
$566$	10.9282	0.459347
$567$	2.46410	0.103483
$568$	9.46410	0.397105
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	7.46410	0.312637
$571$	20.3923	0.853391	0.426696	−	0.904395i	$-0.359678\pi$
$571$	20.3923	0.853391	0.426696	+	0.904395i	$0.359678\pi$
$572$	0	0
$573$	18.9282	0.790737
$574$	−4.73205	−0.197512
$575$	1.26795	0.0528771
$576$	4.46410	0.186004
$577$	8.33975	0.347188	0.173594	−	0.984817i	$-0.444462\pi$
$577$	8.33975	0.347188	0.173594	+	0.984817i	$0.444462\pi$
$578$	−5.00000	−0.207973
$579$	69.5692	2.89120
$580$	4.73205	0.196488
$581$	−4.39230	−0.182224
$582$	15.8564	0.657269
$583$	0	0
$584$	−11.4641	−0.474388
$585$	6.53590	0.270226
$586$	2.53590	0.104757
$587$	−28.9808	−1.19616	−0.598082	−	0.801435i	$-0.704070\pi$
$587$	−28.9808	−1.19616	−0.598082	+	0.801435i	$0.704070\pi$
$588$	2.73205	0.112668
$589$	24.3923	1.00507
$590$	−13.8564	−0.570459
$591$	61.1769	2.51648
$592$	3.26795	0.134312
$593$	−32.5359	−1.33609	−0.668045	−	0.744121i	$-0.732868\pi$
$593$	−32.5359	−1.33609	−0.668045	+	0.744121i	$0.732868\pi$
$594$	0	0
$595$	3.46410	0.142014
$596$	7.26795	0.297707
$597$	−67.7128	−2.77130
$598$	1.85641	0.0759141
$599$	−32.1051	−1.31178	−0.655890	−	0.754857i	$-0.727706\pi$
$599$	−32.1051	−1.31178	−0.655890	+	0.754857i	$0.727706\pi$
$600$	2.73205	0.111536
$601$	−28.4449	−1.16029	−0.580145	−	0.814513i	$-0.697004\pi$
$601$	−28.4449	−1.16029	−0.580145	+	0.814513i	$0.697004\pi$
$602$	−4.92820	−0.200859
$603$	−48.7846	−1.98666
$604$	7.80385	0.317534
$605$	0	0
$606$	16.3923	0.665892
$607$	−34.7846	−1.41186	−0.705932	−	0.708280i	$-0.749472\pi$
$607$	−34.7846	−1.41186	−0.705932	+	0.708280i	$0.749472\pi$
$608$	2.73205	0.110799
$609$	−12.9282	−0.523877
$610$	−2.00000	−0.0809776
$611$	0	0
$612$	−15.4641	−0.625099
$613$	−27.1769	−1.09767	−0.548833	−	0.835932i	$-0.684928\pi$
$613$	−27.1769	−1.09767	−0.548833	+	0.835932i	$0.684928\pi$
$614$	−31.3205	−1.26399
$615$	12.9282	0.521315
$616$	0	0
$617$	8.53590	0.343642	0.171821	−	0.985128i	$-0.445035\pi$
$617$	8.53590	0.343642	0.171821	+	0.985128i	$0.445035\pi$
$618$	28.7846	1.15789
$619$	−10.9282	−0.439242	−0.219621	−	0.975585i	$-0.570482\pi$
$619$	−10.9282	−0.439242	−0.219621	+	0.975585i	$0.570482\pi$
$620$	8.92820	0.358565
$621$	5.07180	0.203524
$622$	7.85641	0.315013
$623$	15.4641	0.619556
$624$	4.00000	0.160128
$625$	1.00000	0.0400000
$626$	27.2679	1.08985
$627$	0	0
$628$	6.39230	0.255081
$629$	−11.3205	−0.451378
$630$	−4.46410	−0.177854
$631$	−12.7846	−0.508947	−0.254474	−	0.967080i	$-0.581902\pi$
$631$	−12.7846	−0.508947	−0.254474	+	0.967080i	$0.581902\pi$
$632$	−6.73205	−0.267787
$633$	−21.8564	−0.868714
$634$	−32.4449	−1.28855
$635$	−9.85641	−0.391140
$636$	−12.9282	−0.512637
$637$	1.46410	0.0580098
$638$	0	0
$639$	42.2487	1.67133
$640$	1.00000	0.0395285
$641$	19.8564	0.784281	0.392140	−	0.919905i	$-0.371735\pi$
$641$	19.8564	0.784281	0.392140	+	0.919905i	$0.371735\pi$
$642$	−2.53590	−0.100084
$643$	−14.7321	−0.580975	−0.290488	−	0.956879i	$-0.593818\pi$
$643$	−14.7321	−0.580975	−0.290488	+	0.956879i	$0.593818\pi$
$644$	−1.26795	−0.0499642
$645$	13.4641	0.530148
$646$	−9.46410	−0.372360
$647$	37.1769	1.46158	0.730788	−	0.682605i	$-0.239153\pi$
$647$	37.1769	1.46158	0.730788	+	0.682605i	$0.239153\pi$
$648$	−2.46410	−0.0967991
$649$	0	0
$650$	1.46410	0.0574268
$651$	−24.3923	−0.956010
$652$	8.00000	0.313304
$653$	−9.80385	−0.383654	−0.191827	−	0.981429i	$-0.561441\pi$
$653$	−9.80385	−0.383654	−0.191827	+	0.981429i	$0.561441\pi$
$654$	4.92820	0.192708
$655$	17.6603	0.690043
$656$	4.73205	0.184756
$657$	−51.1769	−1.99660
$658$	0	0
$659$	−6.24871	−0.243415	−0.121708	−	0.992566i	$-0.538837\pi$
$659$	−6.24871	−0.243415	−0.121708	+	0.992566i	$0.538837\pi$
$660$	0	0
$661$	3.85641	0.149997	0.0749984	−	0.997184i	$-0.476105\pi$
$661$	3.85641	0.149997	0.0749984	+	0.997184i	$0.476105\pi$
$662$	−4.92820	−0.191540
$663$	−13.8564	−0.538138
$664$	4.39230	0.170454
$665$	−2.73205	−0.105944
$666$	14.5885	0.565291
$667$	6.00000	0.232321
$668$	13.8564	0.536120
$669$	−22.9282	−0.886456
$670$	−10.9282	−0.422193
$671$	0	0
$672$	−2.73205	−0.105391
$673$	20.1436	0.776478	0.388239	−	0.921559i	$-0.373083\pi$
$673$	20.1436	0.776478	0.388239	+	0.921559i	$0.373083\pi$
$674$	30.7846	1.18578
$675$	4.00000	0.153960
$676$	−10.8564	−0.417554
$677$	−19.8564	−0.763144	−0.381572	−	0.924339i	$-0.624617\pi$
$677$	−19.8564	−0.763144	−0.381572	+	0.924339i	$0.624617\pi$
$678$	35.3205	1.35648
$679$	−5.80385	−0.222731
$680$	−3.46410	−0.132842
$681$	−37.8564	−1.45066
$682$	0	0
$683$	42.2487	1.61660	0.808301	−	0.588769i	$-0.200387\pi$
$683$	42.2487	1.61660	0.808301	+	0.588769i	$0.200387\pi$
$684$	12.1962	0.466332
$685$	7.85641	0.300178
$686$	−1.00000	−0.0381802
$687$	−36.7846	−1.40342
$688$	4.92820	0.187886
$689$	−6.92820	−0.263944
$690$	3.46410	0.131876
$691$	31.3205	1.19149	0.595744	−	0.803174i	$-0.296857\pi$
$691$	31.3205	1.19149	0.595744	+	0.803174i	$0.296857\pi$
$692$	−0.928203	−0.0352850
$693$	0	0
$694$	−7.85641	−0.298225
$695$	−4.19615	−0.159169
$696$	12.9282	0.490042
$697$	−16.3923	−0.620903
$698$	−30.3923	−1.15037
$699$	−21.4641	−0.811847
$700$	−1.00000	−0.0377964
$701$	−14.1962	−0.536181	−0.268091	−	0.963394i	$-0.586393\pi$
$701$	−14.1962	−0.536181	−0.268091	+	0.963394i	$0.586393\pi$
$702$	5.85641	0.221036
$703$	8.92820	0.336734
$704$	0	0
$705$	0	0
$706$	−32.4449	−1.22108
$707$	−6.00000	−0.225653
$708$	−37.8564	−1.42273
$709$	−2.39230	−0.0898449	−0.0449224	−	0.998990i	$-0.514304\pi$
$709$	−2.39230	−0.0898449	−0.0449224	+	0.998990i	$0.514304\pi$
$710$	9.46410	0.355181
$711$	−30.0526	−1.12706
$712$	−15.4641	−0.579542
$713$	11.3205	0.423956
$714$	9.46410	0.354185
$715$	0	0
$716$	−19.8564	−0.742069
$717$	10.3923	0.388108
$718$	1.26795	0.0473194
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	4.46410	0.166367
$721$	−10.5359	−0.392377
$722$	−11.5359	−0.429322
$723$	49.7128	1.84884
$724$	−11.8564	−0.440640
$725$	4.73205	0.175744
$726$	0	0
$727$	−38.6410	−1.43312	−0.716558	−	0.697528i	$-0.754283\pi$
$727$	−38.6410	−1.43312	−0.716558	+	0.697528i	$0.754283\pi$
$728$	−1.46410	−0.0542632
$729$	−43.7846	−1.62165
$730$	−11.4641	−0.424305
$731$	−17.0718	−0.631423
$732$	−5.46410	−0.201959
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	29.4641	1.08754
$735$	2.73205	0.100773
$736$	1.26795	0.0467372
$737$	0	0
$738$	21.1244	0.777598
$739$	−18.1436	−0.667423	−0.333711	−	0.942675i	$-0.608301\pi$
$739$	−18.1436	−0.667423	−0.333711	+	0.942675i	$0.608301\pi$
$740$	3.26795	0.120132
$741$	10.9282	0.401458
$742$	4.73205	0.173719
$743$	−3.71281	−0.136210	−0.0681049	−	0.997678i	$-0.521695\pi$
$743$	−3.71281	−0.136210	−0.0681049	+	0.997678i	$0.521695\pi$
$744$	24.3923	0.894265
$745$	7.26795	0.266277
$746$	−30.3923	−1.11274
$747$	19.6077	0.717408
$748$	0	0
$749$	0.928203	0.0339158
$750$	2.73205	0.0997604
$751$	2.24871	0.0820566	0.0410283	−	0.999158i	$-0.486937\pi$
$751$	2.24871	0.0820566	0.0410283	+	0.999158i	$0.486937\pi$
$752$	0	0
$753$	0	0
$754$	6.92820	0.252310
$755$	7.80385	0.284011
$756$	−4.00000	−0.145479
$757$	23.3731	0.849509	0.424754	−	0.905309i	$-0.360360\pi$
$757$	23.3731	0.849509	0.424754	+	0.905309i	$0.360360\pi$
$758$	23.7128	0.861288
$759$	0	0
$760$	2.73205	0.0991019
$761$	−19.2679	−0.698463	−0.349231	−	0.937037i	$-0.613557\pi$
$761$	−19.2679	−0.698463	−0.349231	+	0.937037i	$0.613557\pi$
$762$	−26.9282	−0.975506
$763$	−1.80385	−0.0653037
$764$	6.92820	0.250654
$765$	−15.4641	−0.559106
$766$	16.3923	0.592278
$767$	−20.2872	−0.732528
$768$	2.73205	0.0985844
$769$	24.4449	0.881504	0.440752	−	0.897629i	$-0.354712\pi$
$769$	24.4449	0.881504	0.440752	+	0.897629i	$0.354712\pi$
$770$	0	0
$771$	62.7846	2.26113
$772$	25.4641	0.916473
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	22.0000	0.790774
$775$	8.92820	0.320711
$776$	5.80385	0.208346
$777$	−8.92820	−0.320298
$778$	−24.2487	−0.869358
$779$	12.9282	0.463201
$780$	4.00000	0.143223
$781$	0	0
$782$	−4.39230	−0.157069
$783$	18.9282	0.676439
$784$	1.00000	0.0357143
$785$	6.39230	0.228151
$786$	48.2487	1.72097
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	22.3923	0.797693
$789$	−51.7128	−1.84102
$790$	−6.73205	−0.239516
$791$	−12.9282	−0.459674
$792$	0	0
$793$	−2.92820	−0.103984
$794$	−38.3923	−1.36249
$795$	−12.9282	−0.458516
$796$	−24.7846	−0.878467
$797$	6.67949	0.236600	0.118300	−	0.992978i	$-0.462256\pi$
$797$	6.67949	0.236600	0.118300	+	0.992978i	$0.462256\pi$
$798$	−7.46410	−0.264226
$799$	0	0
$800$	1.00000	0.0353553
$801$	−69.0333	−2.43917
$802$	−25.1769	−0.889028
$803$	0	0
$804$	−29.8564	−1.05295
$805$	−1.26795	−0.0446893
$806$	13.0718	0.460434
$807$	12.0000	0.422420
$808$	6.00000	0.211079
$809$	−5.32051	−0.187059	−0.0935296	−	0.995617i	$-0.529815\pi$
$809$	−5.32051	−0.187059	−0.0935296	+	0.995617i	$0.529815\pi$
$810$	−2.46410	−0.0865797
$811$	4.58846	0.161123	0.0805613	−	0.996750i	$-0.474329\pi$
$811$	4.58846	0.161123	0.0805613	+	0.996750i	$0.474329\pi$
$812$	−4.73205	−0.166062
$813$	−66.6410	−2.33720
$814$	0	0
$815$	8.00000	0.280228
$816$	−9.46410	−0.331310
$817$	13.4641	0.471049
$818$	−22.1962	−0.776070
$819$	−6.53590	−0.228383
$820$	4.73205	0.165250
$821$	−52.0526	−1.81665	−0.908323	−	0.418269i	$-0.862637\pi$
$821$	−52.0526	−1.81665	−0.908323	+	0.418269i	$0.862637\pi$
$822$	21.4641	0.748647
$823$	−24.1962	−0.843425	−0.421712	−	0.906730i	$-0.638571\pi$
$823$	−24.1962	−0.843425	−0.421712	+	0.906730i	$0.638571\pi$
$824$	10.5359	0.367035
$825$	0	0
$826$	13.8564	0.482126
$827$	−53.5692	−1.86278	−0.931392	−	0.364017i	$-0.881405\pi$
$827$	−53.5692	−1.86278	−0.931392	+	0.364017i	$0.881405\pi$
$828$	5.66025	0.196707
$829$	40.1051	1.39291	0.696454	−	0.717601i	$-0.254760\pi$
$829$	40.1051	1.39291	0.696454	+	0.717601i	$0.254760\pi$
$830$	4.39230	0.152459
$831$	−76.1051	−2.64006
$832$	1.46410	0.0507586
$833$	−3.46410	−0.120024
$834$	−11.4641	−0.396969
$835$	13.8564	0.479521
$836$	0	0
$837$	35.7128	1.23442
$838$	32.7846	1.13253
$839$	−38.7846	−1.33899	−0.669497	−	0.742815i	$-0.733490\pi$
$839$	−38.7846	−1.33899	−0.669497	+	0.742815i	$0.733490\pi$
$840$	−2.73205	−0.0942647
$841$	−6.60770	−0.227852
$842$	10.7846	0.371662
$843$	28.3923	0.977883
$844$	−8.00000	−0.275371
$845$	−10.8564	−0.373472
$846$	0	0
$847$	0	0
$848$	−4.73205	−0.162499
$849$	29.8564	1.02467
$850$	−3.46410	−0.118818
$851$	4.14359	0.142041
$852$	25.8564	0.885826
$853$	−20.9282	−0.716568	−0.358284	−	0.933613i	$-0.616638\pi$
$853$	−20.9282	−0.716568	−0.358284	+	0.933613i	$0.616638\pi$
$854$	2.00000	0.0684386
$855$	12.1962	0.417100
$856$	−0.928203	−0.0317253
$857$	19.1769	0.655071	0.327535	−	0.944839i	$-0.393782\pi$
$857$	19.1769	0.655071	0.327535	+	0.944839i	$0.393782\pi$
$858$	0	0
$859$	40.7846	1.39155	0.695776	−	0.718258i	$-0.255060\pi$
$859$	40.7846	1.39155	0.695776	+	0.718258i	$0.255060\pi$
$860$	4.92820	0.168050
$861$	−12.9282	−0.440592
$862$	−18.3397	−0.624654
$863$	24.5885	0.837001	0.418500	−	0.908217i	$-0.362556\pi$
$863$	24.5885	0.837001	0.418500	+	0.908217i	$0.362556\pi$
$864$	4.00000	0.136083
$865$	−0.928203	−0.0315599
$866$	0.732051	0.0248761
$867$	−13.6603	−0.463927
$868$	−8.92820	−0.303043
$869$	0	0
$870$	12.9282	0.438307
$871$	−16.0000	−0.542139
$872$	1.80385	0.0610860
$873$	25.9090	0.876886
$874$	3.46410	0.117175
$875$	−1.00000	−0.0338062
$876$	−31.3205	−1.05822
$877$	23.1769	0.782629	0.391314	−	0.920257i	$-0.372021\pi$
$877$	23.1769	0.782629	0.391314	+	0.920257i	$0.372021\pi$
$878$	−36.3923	−1.22818
$879$	6.92820	0.233682
$880$	0	0
$881$	24.9282	0.839853	0.419926	−	0.907558i	$-0.362056\pi$
$881$	24.9282	0.839853	0.419926	+	0.907558i	$0.362056\pi$
$882$	4.46410	0.150314
$883$	−41.1769	−1.38571	−0.692857	−	0.721075i	$-0.743648\pi$
$883$	−41.1769	−1.38571	−0.692857	+	0.721075i	$0.743648\pi$
$884$	−5.07180	−0.170583
$885$	−37.8564	−1.27253
$886$	20.7846	0.698273
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	8.92820	0.299611
$889$	9.85641	0.330573
$890$	−15.4641	−0.518358
$891$	0	0
$892$	−8.39230	−0.280995
$893$	0	0
$894$	19.8564	0.664098
$895$	−19.8564	−0.663726
$896$	−1.00000	−0.0334077
$897$	5.07180	0.169342
$898$	−35.3205	−1.17866
$899$	42.2487	1.40907
$900$	4.46410	0.148803
$901$	16.3923	0.546107
$902$	0	0
$903$	−13.4641	−0.448057
$904$	12.9282	0.429986
$905$	−11.8564	−0.394120
$906$	21.3205	0.708326
$907$	−15.3205	−0.508709	−0.254355	−	0.967111i	$-0.581863\pi$
$907$	−15.3205	−0.508709	−0.254355	+	0.967111i	$0.581863\pi$
$908$	−13.8564	−0.459841
$909$	26.7846	0.888389
$910$	−1.46410	−0.0485345
$911$	14.5359	0.481596	0.240798	−	0.970575i	$-0.422591\pi$
$911$	14.5359	0.481596	0.240798	+	0.970575i	$0.422591\pi$
$912$	7.46410	0.247161
$913$	0	0
$914$	34.0000	1.12462
$915$	−5.46410	−0.180638
$916$	−13.4641	−0.444866
$917$	−17.6603	−0.583193
$918$	−13.8564	−0.457330
$919$	−24.9808	−0.824039	−0.412020	−	0.911175i	$-0.635176\pi$
$919$	−24.9808	−0.824039	−0.412020	+	0.911175i	$0.635176\pi$
$920$	1.26795	0.0418030
$921$	−85.5692	−2.81960
$922$	−29.3205	−0.965620
$923$	13.8564	0.456089
$924$	0	0
$925$	3.26795	0.107450
$926$	12.9808	0.426574
$927$	47.0333	1.54478
$928$	4.73205	0.155337
$929$	26.1051	0.856481	0.428241	−	0.903665i	$-0.359134\pi$
$929$	26.1051	0.856481	0.428241	+	0.903665i	$0.359134\pi$
$930$	24.3923	0.799855
$931$	2.73205	0.0895393
$932$	−7.85641	−0.257345
$933$	21.4641	0.702703
$934$	17.6603	0.577861
$935$	0	0
$936$	6.53590	0.213633
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	10.9282	0.356818
$939$	74.4974	2.43113
$940$	0	0
$941$	35.5692	1.15952	0.579762	−	0.814786i	$-0.303146\pi$
$941$	35.5692	1.15952	0.579762	+	0.814786i	$0.303146\pi$
$942$	17.4641	0.569011
$943$	6.00000	0.195387
$944$	−13.8564	−0.450988
$945$	−4.00000	−0.130120
$946$	0	0
$947$	−25.1769	−0.818140	−0.409070	−	0.912503i	$-0.634147\pi$
$947$	−25.1769	−0.818140	−0.409070	+	0.912503i	$0.634147\pi$
$948$	−18.3923	−0.597354
$949$	−16.7846	−0.544851
$950$	2.73205	0.0886394
$951$	−88.6410	−2.87438
$952$	3.46410	0.112272
$953$	11.3205	0.366707	0.183354	−	0.983047i	$-0.441305\pi$
$953$	11.3205	0.366707	0.183354	+	0.983047i	$0.441305\pi$
$954$	−21.1244	−0.683926
$955$	6.92820	0.224191
$956$	3.80385	0.123025
$957$	0	0
$958$	−13.8564	−0.447680
$959$	−7.85641	−0.253697
$960$	2.73205	0.0881766
$961$	48.7128	1.57138
$962$	4.78461	0.154262
$963$	−4.14359	−0.133525
$964$	18.1962	0.586059
$965$	25.4641	0.819718
$966$	−3.46410	−0.111456
$967$	26.1436	0.840721	0.420361	−	0.907357i	$-0.361904\pi$
$967$	26.1436	0.840721	0.420361	+	0.907357i	$0.361904\pi$
$968$	0	0
$969$	−25.8564	−0.830627
$970$	5.80385	0.186350
$971$	49.8564	1.59997	0.799984	−	0.600021i	$-0.204841\pi$
$971$	49.8564	1.59997	0.799984	+	0.600021i	$0.204841\pi$
$972$	−18.7321	−0.600831
$973$	4.19615	0.134522
$974$	4.87564	0.156226
$975$	4.00000	0.128103
$976$	−2.00000	−0.0640184
$977$	−0.928203	−0.0296959	−0.0148479	−	0.999890i	$-0.504726\pi$
$977$	−0.928203	−0.0296959	−0.0148479	+	0.999890i	$0.504726\pi$
$978$	21.8564	0.698891
$979$	0	0
$980$	1.00000	0.0319438
$981$	8.05256	0.257098
$982$	−42.9282	−1.36989
$983$	−7.60770	−0.242648	−0.121324	−	0.992613i	$-0.538714\pi$
$983$	−7.60770	−0.242648	−0.121324	+	0.992613i	$0.538714\pi$
$984$	12.9282	0.412136
$985$	22.3923	0.713478
$986$	−16.3923	−0.522037
$987$	0	0
$988$	4.00000	0.127257
$989$	6.24871	0.198697
$990$	0	0
$991$	−22.9282	−0.728338	−0.364169	−	0.931333i	$-0.618647\pi$
$991$	−22.9282	−0.728338	−0.364169	+	0.931333i	$0.618647\pi$
$992$	8.92820	0.283471
$993$	−13.4641	−0.427270
$994$	−9.46410	−0.300183
$995$	−24.7846	−0.785725
$996$	12.0000	0.380235
$997$	23.8564	0.755540	0.377770	−	0.925899i	$-0.376691\pi$
$997$	23.8564	0.755540	0.377770	+	0.925899i	$0.376691\pi$
$998$	2.00000	0.0633089
$999$	13.0718	0.413573

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.ce.1.2		2
11.10	odd	2		770.2.a.h.1.2	✓	2
33.32	even	2		6930.2.a.ca.1.1		2
44.43	even	2		6160.2.a.v.1.1		2
55.32	even	4		3850.2.c.s.1849.1		4
55.43	even	4		3850.2.c.s.1849.4		4
55.54	odd	2		3850.2.a.bm.1.1		2
77.76	even	2		5390.2.a.bk.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.h.1.2	✓	2	11.10	odd	2
3850.2.a.bm.1.1		2	55.54	odd	2
3850.2.c.s.1849.1		4	55.32	even	4
3850.2.c.s.1849.4		4	55.43	even	4
5390.2.a.bk.1.1		2	77.76	even	2
6160.2.a.v.1.1		2	44.43	even	2
6930.2.a.ca.1.1		2	33.32	even	2
8470.2.a.ce.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.73205 q^{6} -1.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.73205 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.73205 q^{6} -1.00000 q^{7} +1.00000 q^{8} +4.46410 q^{9} +1.00000 q^{10} +2.73205 q^{12} +1.46410 q^{13} -1.00000 q^{14} +2.73205 q^{15} +1.00000 q^{16} -3.46410 q^{17} +4.46410 q^{18} +2.73205 q^{19} +1.00000 q^{20} -2.73205 q^{21} +1.26795 q^{23} +2.73205 q^{24} +1.00000 q^{25} +1.46410 q^{26} +4.00000 q^{27} -1.00000 q^{28} +4.73205 q^{29} +2.73205 q^{30} +8.92820 q^{31} +1.00000 q^{32} -3.46410 q^{34} -1.00000 q^{35} +4.46410 q^{36} +3.26795 q^{37} +2.73205 q^{38} +4.00000 q^{39} +1.00000 q^{40} +4.73205 q^{41} -2.73205 q^{42} +4.92820 q^{43} +4.46410 q^{45} +1.26795 q^{46} +2.73205 q^{48} +1.00000 q^{49} +1.00000 q^{50} -9.46410 q^{51} +1.46410 q^{52} -4.73205 q^{53} +4.00000 q^{54} -1.00000 q^{56} +7.46410 q^{57} +4.73205 q^{58} -13.8564 q^{59} +2.73205 q^{60} -2.00000 q^{61} +8.92820 q^{62} -4.46410 q^{63} +1.00000 q^{64} +1.46410 q^{65} -10.9282 q^{67} -3.46410 q^{68} +3.46410 q^{69} -1.00000 q^{70} +9.46410 q^{71} +4.46410 q^{72} -11.4641 q^{73} +3.26795 q^{74} +2.73205 q^{75} +2.73205 q^{76} +4.00000 q^{78} -6.73205 q^{79} +1.00000 q^{80} -2.46410 q^{81} +4.73205 q^{82} +4.39230 q^{83} -2.73205 q^{84} -3.46410 q^{85} +4.92820 q^{86} +12.9282 q^{87} -15.4641 q^{89} +4.46410 q^{90} -1.46410 q^{91} +1.26795 q^{92} +24.3923 q^{93} +2.73205 q^{95} +2.73205 q^{96} +5.80385 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{12} - 4 q^{13} - 2 q^{14} + 2 q^{15} + 2 q^{16} + 2 q^{18} + 2 q^{19} + 2 q^{20} - 2 q^{21} + 6 q^{23} + 2 q^{24} + 2 q^{25} - 4 q^{26} + 8 q^{27} - 2 q^{28} + 6 q^{29} + 2 q^{30} + 4 q^{31} + 2 q^{32} - 2 q^{35} + 2 q^{36} + 10 q^{37} + 2 q^{38} + 8 q^{39} + 2 q^{40} + 6 q^{41} - 2 q^{42} - 4 q^{43} + 2 q^{45} + 6 q^{46} + 2 q^{48} + 2 q^{49} + 2 q^{50} - 12 q^{51} - 4 q^{52} - 6 q^{53} + 8 q^{54} - 2 q^{56} + 8 q^{57} + 6 q^{58} + 2 q^{60} - 4 q^{61} + 4 q^{62} - 2 q^{63} + 2 q^{64} - 4 q^{65} - 8 q^{67} - 2 q^{70} + 12 q^{71} + 2 q^{72} - 16 q^{73} + 10 q^{74} + 2 q^{75} + 2 q^{76} + 8 q^{78} - 10 q^{79} + 2 q^{80} + 2 q^{81} + 6 q^{82} - 12 q^{83} - 2 q^{84} - 4 q^{86} + 12 q^{87} - 24 q^{89} + 2 q^{90} + 4 q^{91} + 6 q^{92} + 28 q^{93} + 2 q^{95} + 2 q^{96} + 22 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^12 - 4 * q^13 - 2 * q^14 + 2 * q^15 + 2 * q^16 + 2 * q^18 + 2 * q^19 + 2 * q^20 - 2 * q^21 + 6 * q^23 + 2 * q^24 + 2 * q^25 - 4 * q^26 + 8 * q^27 - 2 * q^28 + 6 * q^29 + 2 * q^30 + 4 * q^31 + 2 * q^32 - 2 * q^35 + 2 * q^36 + 10 * q^37 + 2 * q^38 + 8 * q^39 + 2 * q^40 + 6 * q^41 - 2 * q^42 - 4 * q^43 + 2 * q^45 + 6 * q^46 + 2 * q^48 + 2 * q^49 + 2 * q^50 - 12 * q^51 - 4 * q^52 - 6 * q^53 + 8 * q^54 - 2 * q^56 + 8 * q^57 + 6 * q^58 + 2 * q^60 - 4 * q^61 + 4 * q^62 - 2 * q^63 + 2 * q^64 - 4 * q^65 - 8 * q^67 - 2 * q^70 + 12 * q^71 + 2 * q^72 - 16 * q^73 + 10 * q^74 + 2 * q^75 + 2 * q^76 + 8 * q^78 - 10 * q^79 + 2 * q^80 + 2 * q^81 + 6 * q^82 - 12 * q^83 - 2 * q^84 - 4 * q^86 + 12 * q^87 - 24 * q^89 + 2 * q^90 + 4 * q^91 + 6 * q^92 + 28 * q^93 + 2 * q^95 + 2 * q^96 + 22 * q^97 + 2 * q^98

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