LMFDB - Embedded newform 8470.2.a.bi.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:	$1$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} +2.46410 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +0.535898 q^{17} +2.00000 q^{18} +7.92820 q^{19} +1.00000 q^{20} +1.00000 q^{21} +2.46410 q^{23} +1.00000 q^{24} +1.00000 q^{25} -2.46410 q^{26} +5.00000 q^{27} -1.00000 q^{28} -6.92820 q^{29} +1.00000 q^{30} -7.46410 q^{31} -1.00000 q^{32} -0.535898 q^{34} -1.00000 q^{35} -2.00000 q^{36} +4.92820 q^{37} -7.92820 q^{38} -2.46410 q^{39} -1.00000 q^{40} -12.3923 q^{41} -1.00000 q^{42} -8.39230 q^{43} -2.00000 q^{45} -2.46410 q^{46} -7.46410 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -0.535898 q^{51} +2.46410 q^{52} -0.535898 q^{53} -5.00000 q^{54} +1.00000 q^{56} -7.92820 q^{57} +6.92820 q^{58} -3.92820 q^{59} -1.00000 q^{60} +8.92820 q^{61} +7.46410 q^{62} +2.00000 q^{63} +1.00000 q^{64} +2.46410 q^{65} -14.3923 q^{67} +0.535898 q^{68} -2.46410 q^{69} +1.00000 q^{70} +8.00000 q^{71} +2.00000 q^{72} +7.46410 q^{73} -4.92820 q^{74} -1.00000 q^{75} +7.92820 q^{76} +2.46410 q^{78} +14.4641 q^{79} +1.00000 q^{80} +1.00000 q^{81} +12.3923 q^{82} +1.00000 q^{83} +1.00000 q^{84} +0.535898 q^{85} +8.39230 q^{86} +6.92820 q^{87} -10.5359 q^{89} +2.00000 q^{90} -2.46410 q^{91} +2.46410 q^{92} +7.46410 q^{93} +7.46410 q^{94} +7.92820 q^{95} +1.00000 q^{96} -14.9282 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} - 4 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 - 4 * 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} - 4 q^{9} - 2 q^{10} - 2 q^{12} - 2 q^{13} + 2 q^{14} - 2 q^{15} + 2 q^{16} + 8 q^{17} + 4 q^{18} + 2 q^{19} + 2 q^{20} + 2 q^{21} - 2 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{26} + 10 q^{27} - 2 q^{28} + 2 q^{30} - 8 q^{31} - 2 q^{32} - 8 q^{34} - 2 q^{35} - 4 q^{36} - 4 q^{37} - 2 q^{38} + 2 q^{39} - 2 q^{40} - 4 q^{41} - 2 q^{42} + 4 q^{43} - 4 q^{45} + 2 q^{46} - 8 q^{47} - 2 q^{48} + 2 q^{49} - 2 q^{50} - 8 q^{51} - 2 q^{52} - 8 q^{53} - 10 q^{54} + 2 q^{56} - 2 q^{57} + 6 q^{59} - 2 q^{60} + 4 q^{61} + 8 q^{62} + 4 q^{63} + 2 q^{64} - 2 q^{65} - 8 q^{67} + 8 q^{68} + 2 q^{69} + 2 q^{70} + 16 q^{71} + 4 q^{72} + 8 q^{73} + 4 q^{74} - 2 q^{75} + 2 q^{76} - 2 q^{78} + 22 q^{79} + 2 q^{80} + 2 q^{81} + 4 q^{82} + 2 q^{83} + 2 q^{84} + 8 q^{85} - 4 q^{86} - 28 q^{89} + 4 q^{90} + 2 q^{91} - 2 q^{92} + 8 q^{93} + 8 q^{94} + 2 q^{95} + 2 q^{96} - 16 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 - 4 * q^9 - 2 * q^10 - 2 * q^12 - 2 * q^13 + 2 * q^14 - 2 * q^15 + 2 * q^16 + 8 * q^17 + 4 * q^18 + 2 * q^19 + 2 * q^20 + 2 * q^21 - 2 * q^23 + 2 * q^24 + 2 * q^25 + 2 * q^26 + 10 * q^27 - 2 * q^28 + 2 * q^30 - 8 * q^31 - 2 * q^32 - 8 * q^34 - 2 * q^35 - 4 * q^36 - 4 * q^37 - 2 * q^38 + 2 * q^39 - 2 * q^40 - 4 * q^41 - 2 * q^42 + 4 * q^43 - 4 * q^45 + 2 * q^46 - 8 * q^47 - 2 * q^48 + 2 * q^49 - 2 * q^50 - 8 * q^51 - 2 * q^52 - 8 * q^53 - 10 * q^54 + 2 * q^56 - 2 * q^57 + 6 * q^59 - 2 * q^60 + 4 * q^61 + 8 * q^62 + 4 * q^63 + 2 * q^64 - 2 * q^65 - 8 * q^67 + 8 * q^68 + 2 * q^69 + 2 * q^70 + 16 * q^71 + 4 * q^72 + 8 * q^73 + 4 * q^74 - 2 * q^75 + 2 * q^76 - 2 * q^78 + 22 * q^79 + 2 * q^80 + 2 * q^81 + 4 * q^82 + 2 * q^83 + 2 * q^84 + 8 * q^85 - 4 * q^86 - 28 * q^89 + 4 * q^90 + 2 * q^91 - 2 * q^92 + 8 * q^93 + 8 * q^94 + 2 * q^95 + 2 * q^96 - 16 * 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
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	−2.00000	−0.666667
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	−1.00000	−0.288675
$13$	2.46410	0.683419	0.341709	−	0.939806i	$-0.388994\pi$
$13$	2.46410	0.683419	0.341709	+	0.939806i	$0.388994\pi$
$14$	1.00000	0.267261
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	0.535898	0.129974	0.0649872	−	0.997886i	$-0.479299\pi$
$17$	0.535898	0.129974	0.0649872	+	0.997886i	$0.479299\pi$
$18$	2.00000	0.471405
$19$	7.92820	1.81885	0.909427	−	0.415863i	$-0.136520\pi$
$19$	7.92820	1.81885	0.909427	+	0.415863i	$0.136520\pi$
$20$	1.00000	0.223607
$21$	1.00000	0.218218
$22$	0	0
$23$	2.46410	0.513801	0.256900	−	0.966438i	$-0.417299\pi$
$23$	2.46410	0.513801	0.256900	+	0.966438i	$0.417299\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−2.46410	−0.483250
$27$	5.00000	0.962250
$28$	−1.00000	−0.188982
$29$	−6.92820	−1.28654	−0.643268	−	0.765641i	$-0.722422\pi$
$29$	−6.92820	−1.28654	−0.643268	+	0.765641i	$0.722422\pi$
$30$	1.00000	0.182574
$31$	−7.46410	−1.34059	−0.670296	−	0.742094i	$-0.733833\pi$
$31$	−7.46410	−1.34059	−0.670296	+	0.742094i	$0.733833\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−0.535898	−0.0919058
$35$	−1.00000	−0.169031
$36$	−2.00000	−0.333333
$37$	4.92820	0.810192	0.405096	−	0.914274i	$-0.367238\pi$
$37$	4.92820	0.810192	0.405096	+	0.914274i	$0.367238\pi$
$38$	−7.92820	−1.28612
$39$	−2.46410	−0.394572
$40$	−1.00000	−0.158114
$41$	−12.3923	−1.93535	−0.967676	−	0.252195i	$-0.918848\pi$
$41$	−12.3923	−1.93535	−0.967676	+	0.252195i	$0.918848\pi$
$42$	−1.00000	−0.154303
$43$	−8.39230	−1.27981	−0.639907	−	0.768452i	$-0.721027\pi$
$43$	−8.39230	−1.27981	−0.639907	+	0.768452i	$0.721027\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	−2.46410	−0.363312
$47$	−7.46410	−1.08875	−0.544376	−	0.838842i	$-0.683233\pi$
$47$	−7.46410	−1.08875	−0.544376	+	0.838842i	$0.683233\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−0.535898	−0.0750408
$52$	2.46410	0.341709
$53$	−0.535898	−0.0736113	−0.0368057	−	0.999322i	$-0.511718\pi$
$53$	−0.535898	−0.0736113	−0.0368057	+	0.999322i	$0.511718\pi$
$54$	−5.00000	−0.680414
$55$	0	0
$56$	1.00000	0.133631
$57$	−7.92820	−1.05012
$58$	6.92820	0.909718
$59$	−3.92820	−0.511409	−0.255704	−	0.966755i	$-0.582307\pi$
$59$	−3.92820	−0.511409	−0.255704	+	0.966755i	$0.582307\pi$
$60$	−1.00000	−0.129099
$61$	8.92820	1.14314	0.571570	−	0.820554i	$-0.306335\pi$
$61$	8.92820	1.14314	0.571570	+	0.820554i	$0.306335\pi$
$62$	7.46410	0.947942
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	2.46410	0.305634
$66$	0	0
$67$	−14.3923	−1.75830	−0.879150	−	0.476545i	$-0.841889\pi$
$67$	−14.3923	−1.75830	−0.879150	+	0.476545i	$0.841889\pi$
$68$	0.535898	0.0649872
$69$	−2.46410	−0.296643
$70$	1.00000	0.119523
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	2.00000	0.235702
$73$	7.46410	0.873607	0.436804	−	0.899557i	$-0.356111\pi$
$73$	7.46410	0.873607	0.436804	+	0.899557i	$0.356111\pi$
$74$	−4.92820	−0.572892
$75$	−1.00000	−0.115470
$76$	7.92820	0.909427
$77$	0	0
$78$	2.46410	0.279005
$79$	14.4641	1.62734	0.813669	−	0.581328i	$-0.197467\pi$
$79$	14.4641	1.62734	0.813669	+	0.581328i	$0.197467\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	12.3923	1.36850
$83$	1.00000	0.109764	0.0548821	−	0.998493i	$-0.482522\pi$
$83$	1.00000	0.109764	0.0548821	+	0.998493i	$0.482522\pi$
$84$	1.00000	0.109109
$85$	0.535898	0.0581263
$86$	8.39230	0.904966
$87$	6.92820	0.742781
$88$	0	0
$89$	−10.5359	−1.11680	−0.558401	−	0.829571i	$-0.688585\pi$
$89$	−10.5359	−1.11680	−0.558401	+	0.829571i	$0.688585\pi$
$90$	2.00000	0.210819
$91$	−2.46410	−0.258308
$92$	2.46410	0.256900
$93$	7.46410	0.773991
$94$	7.46410	0.769863
$95$	7.92820	0.813416
$96$	1.00000	0.102062
$97$	−14.9282	−1.51573	−0.757865	−	0.652412i	$-0.773757\pi$
$97$	−14.9282	−1.51573	−0.757865	+	0.652412i	$0.773757\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	18.4641	1.83725	0.918623	−	0.395134i	$-0.129302\pi$
$101$	18.4641	1.83725	0.918623	+	0.395134i	$0.129302\pi$
$102$	0.535898	0.0530618
$103$	5.46410	0.538394	0.269197	−	0.963085i	$-0.413242\pi$
$103$	5.46410	0.538394	0.269197	+	0.963085i	$0.413242\pi$
$104$	−2.46410	−0.241625
$105$	1.00000	0.0975900
$106$	0.535898	0.0520511
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	5.00000	0.481125
$109$	2.39230	0.229141	0.114571	−	0.993415i	$-0.463451\pi$
$109$	2.39230	0.229141	0.114571	+	0.993415i	$0.463451\pi$
$110$	0	0
$111$	−4.92820	−0.467764
$112$	−1.00000	−0.0944911
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	7.92820	0.742544
$115$	2.46410	0.229779
$116$	−6.92820	−0.643268
$117$	−4.92820	−0.455613
$118$	3.92820	0.361620
$119$	−0.535898	−0.0491257
$120$	1.00000	0.0912871
$121$	0	0
$122$	−8.92820	−0.808322
$123$	12.3923	1.11738
$124$	−7.46410	−0.670296
$125$	1.00000	0.0894427
$126$	−2.00000	−0.178174
$127$	7.39230	0.655961	0.327980	−	0.944684i	$-0.393632\pi$
$127$	7.39230	0.655961	0.327980	+	0.944684i	$0.393632\pi$
$128$	−1.00000	−0.0883883
$129$	8.39230	0.738901
$130$	−2.46410	−0.216116
$131$	−16.8564	−1.47275	−0.736376	−	0.676573i	$-0.763464\pi$
$131$	−16.8564	−1.47275	−0.736376	+	0.676573i	$0.763464\pi$
$132$	0	0
$133$	−7.92820	−0.687462
$134$	14.3923	1.24331
$135$	5.00000	0.430331
$136$	−0.535898	−0.0459529
$137$	−5.92820	−0.506481	−0.253240	−	0.967403i	$-0.581496\pi$
$137$	−5.92820	−0.506481	−0.253240	+	0.967403i	$0.581496\pi$
$138$	2.46410	0.209758
$139$	10.8564	0.920828	0.460414	−	0.887704i	$-0.347701\pi$
$139$	10.8564	0.920828	0.460414	+	0.887704i	$0.347701\pi$
$140$	−1.00000	−0.0845154
$141$	7.46410	0.628591
$142$	−8.00000	−0.671345
$143$	0	0
$144$	−2.00000	−0.166667
$145$	−6.92820	−0.575356
$146$	−7.46410	−0.617733
$147$	−1.00000	−0.0824786
$148$	4.92820	0.405096
$149$	16.9282	1.38681	0.693406	−	0.720547i	$-0.256109\pi$
$149$	16.9282	1.38681	0.693406	+	0.720547i	$0.256109\pi$
$150$	1.00000	0.0816497
$151$	12.4641	1.01431	0.507157	−	0.861854i	$-0.330696\pi$
$151$	12.4641	1.01431	0.507157	+	0.861854i	$0.330696\pi$
$152$	−7.92820	−0.643062
$153$	−1.07180	−0.0866496
$154$	0	0
$155$	−7.46410	−0.599531
$156$	−2.46410	−0.197286
$157$	14.3205	1.14290	0.571450	−	0.820637i	$-0.306381\pi$
$157$	14.3205	1.14290	0.571450	+	0.820637i	$0.306381\pi$
$158$	−14.4641	−1.15070
$159$	0.535898	0.0424995
$160$	−1.00000	−0.0790569
$161$	−2.46410	−0.194198
$162$	−1.00000	−0.0785674
$163$	−22.0000	−1.72317	−0.861586	−	0.507611i	$-0.830529\pi$
$163$	−22.0000	−1.72317	−0.861586	+	0.507611i	$0.830529\pi$
$164$	−12.3923	−0.967676
$165$	0	0
$166$	−1.00000	−0.0776151
$167$	−16.3923	−1.26847	−0.634237	−	0.773138i	$-0.718686\pi$
$167$	−16.3923	−1.26847	−0.634237	+	0.773138i	$0.718686\pi$
$168$	−1.00000	−0.0771517
$169$	−6.92820	−0.532939
$170$	−0.535898	−0.0411015
$171$	−15.8564	−1.21257
$172$	−8.39230	−0.639907
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	−6.92820	−0.525226
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	3.92820	0.295262
$178$	10.5359	0.789699
$179$	12.3923	0.926244	0.463122	−	0.886294i	$-0.346729\pi$
$179$	12.3923	0.926244	0.463122	+	0.886294i	$0.346729\pi$
$180$	−2.00000	−0.149071
$181$	3.53590	0.262821	0.131411	−	0.991328i	$-0.458049\pi$
$181$	3.53590	0.262821	0.131411	+	0.991328i	$0.458049\pi$
$182$	2.46410	0.182651
$183$	−8.92820	−0.659992
$184$	−2.46410	−0.181656
$185$	4.92820	0.362329
$186$	−7.46410	−0.547294
$187$	0	0
$188$	−7.46410	−0.544376
$189$	−5.00000	−0.363696
$190$	−7.92820	−0.575172
$191$	−17.3923	−1.25846	−0.629232	−	0.777218i	$-0.716630\pi$
$191$	−17.3923	−1.25846	−0.629232	+	0.777218i	$0.716630\pi$
$192$	−1.00000	−0.0721688
$193$	−19.0000	−1.36765	−0.683825	−	0.729646i	$-0.739685\pi$
$193$	−19.0000	−1.36765	−0.683825	+	0.729646i	$0.739685\pi$
$194$	14.9282	1.07178
$195$	−2.46410	−0.176458
$196$	1.00000	0.0714286
$197$	−4.00000	−0.284988	−0.142494	−	0.989796i	$-0.545512\pi$
$197$	−4.00000	−0.284988	−0.142494	+	0.989796i	$0.545512\pi$
$198$	0	0
$199$	1.85641	0.131597	0.0657986	−	0.997833i	$-0.479041\pi$
$199$	1.85641	0.131597	0.0657986	+	0.997833i	$0.479041\pi$
$200$	−1.00000	−0.0707107
$201$	14.3923	1.01515
$202$	−18.4641	−1.29913
$203$	6.92820	0.486265
$204$	−0.535898	−0.0375204
$205$	−12.3923	−0.865516
$206$	−5.46410	−0.380702
$207$	−4.92820	−0.342534
$208$	2.46410	0.170855
$209$	0	0
$210$	−1.00000	−0.0690066
$211$	24.0000	1.65223	0.826114	−	0.563503i	$-0.190547\pi$
$211$	24.0000	1.65223	0.826114	+	0.563503i	$0.190547\pi$
$212$	−0.535898	−0.0368057
$213$	−8.00000	−0.548151
$214$	−6.00000	−0.410152
$215$	−8.39230	−0.572350
$216$	−5.00000	−0.340207
$217$	7.46410	0.506696
$218$	−2.39230	−0.162027
$219$	−7.46410	−0.504377
$220$	0	0
$221$	1.32051	0.0888270
$222$	4.92820	0.330759
$223$	3.46410	0.231973	0.115987	−	0.993251i	$-0.462997\pi$
$223$	3.46410	0.231973	0.115987	+	0.993251i	$0.462997\pi$
$224$	1.00000	0.0668153
$225$	−2.00000	−0.133333
$226$	−1.00000	−0.0665190
$227$	16.0000	1.06196	0.530979	−	0.847385i	$-0.321824\pi$
$227$	16.0000	1.06196	0.530979	+	0.847385i	$0.321824\pi$
$228$	−7.92820	−0.525058
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	−2.46410	−0.162478
$231$	0	0
$232$	6.92820	0.454859
$233$	−8.85641	−0.580202	−0.290101	−	0.956996i	$-0.593689\pi$
$233$	−8.85641	−0.580202	−0.290101	+	0.956996i	$0.593689\pi$
$234$	4.92820	0.322167
$235$	−7.46410	−0.486904
$236$	−3.92820	−0.255704
$237$	−14.4641	−0.939544
$238$	0.535898	0.0347371
$239$	5.53590	0.358087	0.179044	−	0.983841i	$-0.442700\pi$
$239$	5.53590	0.358087	0.179044	+	0.983841i	$0.442700\pi$
$240$	−1.00000	−0.0645497
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	8.92820	0.571570
$245$	1.00000	0.0638877
$246$	−12.3923	−0.790104
$247$	19.5359	1.24304
$248$	7.46410	0.473971
$249$	−1.00000	−0.0633724
$250$	−1.00000	−0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	−7.39230	−0.463834
$255$	−0.535898	−0.0335593
$256$	1.00000	0.0625000
$257$	−18.3923	−1.14728	−0.573640	−	0.819107i	$-0.694469\pi$
$257$	−18.3923	−1.14728	−0.573640	+	0.819107i	$0.694469\pi$
$258$	−8.39230	−0.522482
$259$	−4.92820	−0.306224
$260$	2.46410	0.152817
$261$	13.8564	0.857690
$262$	16.8564	1.04139
$263$	−9.53590	−0.588009	−0.294004	−	0.955804i	$-0.594988\pi$
$263$	−9.53590	−0.588009	−0.294004	+	0.955804i	$0.594988\pi$
$264$	0	0
$265$	−0.535898	−0.0329200
$266$	7.92820	0.486109
$267$	10.5359	0.644787
$268$	−14.3923	−0.879150
$269$	24.3205	1.48285	0.741424	−	0.671037i	$-0.234151\pi$
$269$	24.3205	1.48285	0.741424	+	0.671037i	$0.234151\pi$
$270$	−5.00000	−0.304290
$271$	−2.14359	−0.130214	−0.0651070	−	0.997878i	$-0.520739\pi$
$271$	−2.14359	−0.130214	−0.0651070	+	0.997878i	$0.520739\pi$
$272$	0.535898	0.0324936
$273$	2.46410	0.149134
$274$	5.92820	0.358136
$275$	0	0
$276$	−2.46410	−0.148321
$277$	7.32051	0.439847	0.219923	−	0.975517i	$-0.429419\pi$
$277$	7.32051	0.439847	0.219923	+	0.975517i	$0.429419\pi$
$278$	−10.8564	−0.651124
$279$	14.9282	0.893728
$280$	1.00000	0.0597614
$281$	−17.0000	−1.01413	−0.507067	−	0.861906i	$-0.669271\pi$
$281$	−17.0000	−1.01413	−0.507067	+	0.861906i	$0.669271\pi$
$282$	−7.46410	−0.444481
$283$	−27.7846	−1.65162	−0.825812	−	0.563946i	$-0.809283\pi$
$283$	−27.7846	−1.65162	−0.825812	+	0.563946i	$0.809283\pi$
$284$	8.00000	0.474713
$285$	−7.92820	−0.469626
$286$	0	0
$287$	12.3923	0.731495
$288$	2.00000	0.117851
$289$	−16.7128	−0.983107
$290$	6.92820	0.406838
$291$	14.9282	0.875107
$292$	7.46410	0.436804
$293$	13.5359	0.790776	0.395388	−	0.918514i	$-0.370610\pi$
$293$	13.5359	0.790776	0.395388	+	0.918514i	$0.370610\pi$
$294$	1.00000	0.0583212
$295$	−3.92820	−0.228709
$296$	−4.92820	−0.286446
$297$	0	0
$298$	−16.9282	−0.980624
$299$	6.07180	0.351141
$300$	−1.00000	−0.0577350
$301$	8.39230	0.483724
$302$	−12.4641	−0.717228
$303$	−18.4641	−1.06073
$304$	7.92820	0.454714
$305$	8.92820	0.511227
$306$	1.07180	0.0612705
$307$	1.07180	0.0611707	0.0305853	−	0.999532i	$-0.490263\pi$
$307$	1.07180	0.0611707	0.0305853	+	0.999532i	$0.490263\pi$
$308$	0	0
$309$	−5.46410	−0.310842
$310$	7.46410	0.423932
$311$	−11.3205	−0.641927	−0.320964	−	0.947092i	$-0.604007\pi$
$311$	−11.3205	−0.641927	−0.320964	+	0.947092i	$0.604007\pi$
$312$	2.46410	0.139502
$313$	−26.7846	−1.51396	−0.756978	−	0.653441i	$-0.773325\pi$
$313$	−26.7846	−1.51396	−0.756978	+	0.653441i	$0.773325\pi$
$314$	−14.3205	−0.808153
$315$	2.00000	0.112687
$316$	14.4641	0.813669
$317$	−1.60770	−0.0902972	−0.0451486	−	0.998980i	$-0.514376\pi$
$317$	−1.60770	−0.0902972	−0.0451486	+	0.998980i	$0.514376\pi$
$318$	−0.535898	−0.0300517
$319$	0	0
$320$	1.00000	0.0559017
$321$	−6.00000	−0.334887
$322$	2.46410	0.137319
$323$	4.24871	0.236405
$324$	1.00000	0.0555556
$325$	2.46410	0.136684
$326$	22.0000	1.21847
$327$	−2.39230	−0.132295
$328$	12.3923	0.684251
$329$	7.46410	0.411509
$330$	0	0
$331$	−5.46410	−0.300334	−0.150167	−	0.988661i	$-0.547981\pi$
$331$	−5.46410	−0.300334	−0.150167	+	0.988661i	$0.547981\pi$
$332$	1.00000	0.0548821
$333$	−9.85641	−0.540128
$334$	16.3923	0.896947
$335$	−14.3923	−0.786336
$336$	1.00000	0.0545545
$337$	9.78461	0.533002	0.266501	−	0.963835i	$-0.414132\pi$
$337$	9.78461	0.533002	0.266501	+	0.963835i	$0.414132\pi$
$338$	6.92820	0.376845
$339$	−1.00000	−0.0543125
$340$	0.535898	0.0290632
$341$	0	0
$342$	15.8564	0.857416
$343$	−1.00000	−0.0539949
$344$	8.39230	0.452483
$345$	−2.46410	−0.132663
$346$	18.0000	0.967686
$347$	12.9282	0.694022	0.347011	−	0.937861i	$-0.387197\pi$
$347$	12.9282	0.694022	0.347011	+	0.937861i	$0.387197\pi$
$348$	6.92820	0.371391
$349$	8.46410	0.453073	0.226536	−	0.974003i	$-0.427260\pi$
$349$	8.46410	0.453073	0.226536	+	0.974003i	$0.427260\pi$
$350$	1.00000	0.0534522
$351$	12.3205	0.657620
$352$	0	0
$353$	−27.4641	−1.46177	−0.730883	−	0.682502i	$-0.760892\pi$
$353$	−27.4641	−1.46177	−0.730883	+	0.682502i	$0.760892\pi$
$354$	−3.92820	−0.208782
$355$	8.00000	0.424596
$356$	−10.5359	−0.558401
$357$	0.535898	0.0283628
$358$	−12.3923	−0.654954
$359$	−35.7128	−1.88485	−0.942425	−	0.334417i	$-0.891460\pi$
$359$	−35.7128	−1.88485	−0.942425	+	0.334417i	$0.891460\pi$
$360$	2.00000	0.105409
$361$	43.8564	2.30823
$362$	−3.53590	−0.185843
$363$	0	0
$364$	−2.46410	−0.129154
$365$	7.46410	0.390689
$366$	8.92820	0.466685
$367$	−14.5359	−0.758768	−0.379384	−	0.925239i	$-0.623864\pi$
$367$	−14.5359	−0.758768	−0.379384	+	0.925239i	$0.623864\pi$
$368$	2.46410	0.128450
$369$	24.7846	1.29024
$370$	−4.92820	−0.256205
$371$	0.535898	0.0278225
$372$	7.46410	0.386996
$373$	−11.0718	−0.573276	−0.286638	−	0.958039i	$-0.592538\pi$
$373$	−11.0718	−0.573276	−0.286638	+	0.958039i	$0.592538\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	7.46410	0.384932
$377$	−17.0718	−0.879242
$378$	5.00000	0.257172
$379$	−3.85641	−0.198090	−0.0990451	−	0.995083i	$-0.531579\pi$
$379$	−3.85641	−0.198090	−0.0990451	+	0.995083i	$0.531579\pi$
$380$	7.92820	0.406708
$381$	−7.39230	−0.378719
$382$	17.3923	0.889868
$383$	−14.5359	−0.742750	−0.371375	−	0.928483i	$-0.621114\pi$
$383$	−14.5359	−0.742750	−0.371375	+	0.928483i	$0.621114\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	19.0000	0.967075
$387$	16.7846	0.853210
$388$	−14.9282	−0.757865
$389$	21.3205	1.08099	0.540496	−	0.841346i	$-0.318236\pi$
$389$	21.3205	1.08099	0.540496	+	0.841346i	$0.318236\pi$
$390$	2.46410	0.124775
$391$	1.32051	0.0667810
$392$	−1.00000	−0.0505076
$393$	16.8564	0.850293
$394$	4.00000	0.201517
$395$	14.4641	0.727768
$396$	0	0
$397$	−24.9282	−1.25111	−0.625555	−	0.780180i	$-0.715128\pi$
$397$	−24.9282	−1.25111	−0.625555	+	0.780180i	$0.715128\pi$
$398$	−1.85641	−0.0930532
$399$	7.92820	0.396907
$400$	1.00000	0.0500000
$401$	31.8564	1.59083	0.795417	−	0.606063i	$-0.207252\pi$
$401$	31.8564	1.59083	0.795417	+	0.606063i	$0.207252\pi$
$402$	−14.3923	−0.717823
$403$	−18.3923	−0.916186
$404$	18.4641	0.918623
$405$	1.00000	0.0496904
$406$	−6.92820	−0.343841
$407$	0	0
$408$	0.535898	0.0265309
$409$	−23.7128	−1.17252	−0.586262	−	0.810122i	$-0.699401\pi$
$409$	−23.7128	−1.17252	−0.586262	+	0.810122i	$0.699401\pi$
$410$	12.3923	0.612012
$411$	5.92820	0.292417
$412$	5.46410	0.269197
$413$	3.92820	0.193294
$414$	4.92820	0.242208
$415$	1.00000	0.0490881
$416$	−2.46410	−0.120813
$417$	−10.8564	−0.531641
$418$	0	0
$419$	−16.7128	−0.816474	−0.408237	−	0.912876i	$-0.633856\pi$
$419$	−16.7128	−0.816474	−0.408237	+	0.912876i	$0.633856\pi$
$420$	1.00000	0.0487950
$421$	−21.8564	−1.06522	−0.532608	−	0.846362i	$-0.678788\pi$
$421$	−21.8564	−1.06522	−0.532608	+	0.846362i	$0.678788\pi$
$422$	−24.0000	−1.16830
$423$	14.9282	0.725834
$424$	0.535898	0.0260255
$425$	0.535898	0.0259949
$426$	8.00000	0.387601
$427$	−8.92820	−0.432066
$428$	6.00000	0.290021
$429$	0	0
$430$	8.39230	0.404713
$431$	−24.3205	−1.17148	−0.585739	−	0.810500i	$-0.699196\pi$
$431$	−24.3205	−1.17148	−0.585739	+	0.810500i	$0.699196\pi$
$432$	5.00000	0.240563
$433$	−26.2487	−1.26143	−0.630716	−	0.776014i	$-0.717239\pi$
$433$	−26.2487	−1.26143	−0.630716	+	0.776014i	$0.717239\pi$
$434$	−7.46410	−0.358288
$435$	6.92820	0.332182
$436$	2.39230	0.114571
$437$	19.5359	0.934529
$438$	7.46410	0.356649
$439$	29.4641	1.40624	0.703122	−	0.711069i	$-0.251789\pi$
$439$	29.4641	1.40624	0.703122	+	0.711069i	$0.251789\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	−1.32051	−0.0628102
$443$	39.7128	1.88681	0.943406	−	0.331639i	$-0.107602\pi$
$443$	39.7128	1.88681	0.943406	+	0.331639i	$0.107602\pi$
$444$	−4.92820	−0.233882
$445$	−10.5359	−0.499449
$446$	−3.46410	−0.164030
$447$	−16.9282	−0.800677
$448$	−1.00000	−0.0472456
$449$	−21.9282	−1.03486	−0.517428	−	0.855727i	$-0.673110\pi$
$449$	−21.9282	−1.03486	−0.517428	+	0.855727i	$0.673110\pi$
$450$	2.00000	0.0942809
$451$	0	0
$452$	1.00000	0.0470360
$453$	−12.4641	−0.585615
$454$	−16.0000	−0.750917
$455$	−2.46410	−0.115519
$456$	7.92820	0.371272
$457$	−29.6410	−1.38655	−0.693274	−	0.720674i	$-0.743832\pi$
$457$	−29.6410	−1.38655	−0.693274	+	0.720674i	$0.743832\pi$
$458$	18.0000	0.841085
$459$	2.67949	0.125068
$460$	2.46410	0.114889
$461$	−39.8564	−1.85630	−0.928149	−	0.372209i	$-0.878600\pi$
$461$	−39.8564	−1.85630	−0.928149	+	0.372209i	$0.878600\pi$
$462$	0	0
$463$	−1.53590	−0.0713793	−0.0356896	−	0.999363i	$-0.511363\pi$
$463$	−1.53590	−0.0713793	−0.0356896	+	0.999363i	$0.511363\pi$
$464$	−6.92820	−0.321634
$465$	7.46410	0.346139
$466$	8.85641	0.410265
$467$	−2.07180	−0.0958713	−0.0479357	−	0.998850i	$-0.515264\pi$
$467$	−2.07180	−0.0958713	−0.0479357	+	0.998850i	$0.515264\pi$
$468$	−4.92820	−0.227806
$469$	14.3923	0.664575
$470$	7.46410	0.344293
$471$	−14.3205	−0.659854
$472$	3.92820	0.180810
$473$	0	0
$474$	14.4641	0.664358
$475$	7.92820	0.363771
$476$	−0.535898	−0.0245629
$477$	1.07180	0.0490742
$478$	−5.53590	−0.253206
$479$	9.46410	0.432426	0.216213	−	0.976346i	$-0.430629\pi$
$479$	9.46410	0.432426	0.216213	+	0.976346i	$0.430629\pi$
$480$	1.00000	0.0456435
$481$	12.1436	0.553700
$482$	−4.00000	−0.182195
$483$	2.46410	0.112121
$484$	0	0
$485$	−14.9282	−0.677855
$486$	16.0000	0.725775
$487$	−32.3205	−1.46458	−0.732291	−	0.680992i	$-0.761549\pi$
$487$	−32.3205	−1.46458	−0.732291	+	0.680992i	$0.761549\pi$
$488$	−8.92820	−0.404161
$489$	22.0000	0.994874
$490$	−1.00000	−0.0451754
$491$	−16.5359	−0.746255	−0.373127	−	0.927780i	$-0.621715\pi$
$491$	−16.5359	−0.746255	−0.373127	+	0.927780i	$0.621715\pi$
$492$	12.3923	0.558688
$493$	−3.71281	−0.167217
$494$	−19.5359	−0.878962
$495$	0	0
$496$	−7.46410	−0.335148
$497$	−8.00000	−0.358849
$498$	1.00000	0.0448111
$499$	4.92820	0.220617	0.110308	−	0.993897i	$-0.464816\pi$
$499$	4.92820	0.220617	0.110308	+	0.993897i	$0.464816\pi$
$500$	1.00000	0.0447214
$501$	16.3923	0.732354
$502$	0	0
$503$	18.2487	0.813670	0.406835	−	0.913502i	$-0.366632\pi$
$503$	18.2487	0.813670	0.406835	+	0.913502i	$0.366632\pi$
$504$	−2.00000	−0.0890871
$505$	18.4641	0.821642
$506$	0	0
$507$	6.92820	0.307692
$508$	7.39230	0.327980
$509$	8.32051	0.368800	0.184400	−	0.982851i	$-0.440966\pi$
$509$	8.32051	0.368800	0.184400	+	0.982851i	$0.440966\pi$
$510$	0.535898	0.0237300
$511$	−7.46410	−0.330192
$512$	−1.00000	−0.0441942
$513$	39.6410	1.75019
$514$	18.3923	0.811250
$515$	5.46410	0.240777
$516$	8.39230	0.369451
$517$	0	0
$518$	4.92820	0.216533
$519$	18.0000	0.790112
$520$	−2.46410	−0.108058
$521$	16.0000	0.700973	0.350486	−	0.936568i	$-0.386016\pi$
$521$	16.0000	0.700973	0.350486	+	0.936568i	$0.386016\pi$
$522$	−13.8564	−0.606478
$523$	−9.92820	−0.434130	−0.217065	−	0.976157i	$-0.569648\pi$
$523$	−9.92820	−0.434130	−0.217065	+	0.976157i	$0.569648\pi$
$524$	−16.8564	−0.736376
$525$	1.00000	0.0436436
$526$	9.53590	0.415785
$527$	−4.00000	−0.174243
$528$	0	0
$529$	−16.9282	−0.736009
$530$	0.535898	0.0232779
$531$	7.85641	0.340939
$532$	−7.92820	−0.343731
$533$	−30.5359	−1.32266
$534$	−10.5359	−0.455933
$535$	6.00000	0.259403
$536$	14.3923	0.621653
$537$	−12.3923	−0.534767
$538$	−24.3205	−1.04853
$539$	0	0
$540$	5.00000	0.215166
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	2.14359	0.0920752
$543$	−3.53590	−0.151740
$544$	−0.535898	−0.0229765
$545$	2.39230	0.102475
$546$	−2.46410	−0.105454
$547$	41.4641	1.77288	0.886438	−	0.462846i	$-0.153172\pi$
$547$	41.4641	1.77288	0.886438	+	0.462846i	$0.153172\pi$
$548$	−5.92820	−0.253240
$549$	−17.8564	−0.762093
$550$	0	0
$551$	−54.9282	−2.34002
$552$	2.46410	0.104879
$553$	−14.4641	−0.615076
$554$	−7.32051	−0.311019
$555$	−4.92820	−0.209191
$556$	10.8564	0.460414
$557$	13.3205	0.564408	0.282204	−	0.959354i	$-0.408934\pi$
$557$	13.3205	0.564408	0.282204	+	0.959354i	$0.408934\pi$
$558$	−14.9282	−0.631961
$559$	−20.6795	−0.874649
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	17.0000	0.717102
$563$	12.7128	0.535781	0.267891	−	0.963449i	$-0.413673\pi$
$563$	12.7128	0.535781	0.267891	+	0.963449i	$0.413673\pi$
$564$	7.46410	0.314295
$565$	1.00000	0.0420703
$566$	27.7846	1.16787
$567$	−1.00000	−0.0419961
$568$	−8.00000	−0.335673
$569$	−22.7128	−0.952171	−0.476085	−	0.879399i	$-0.657945\pi$
$569$	−22.7128	−0.952171	−0.476085	+	0.879399i	$0.657945\pi$
$570$	7.92820	0.332076
$571$	25.8564	1.08206	0.541028	−	0.841004i	$-0.318035\pi$
$571$	25.8564	1.08206	0.541028	+	0.841004i	$0.318035\pi$
$572$	0	0
$573$	17.3923	0.726574
$574$	−12.3923	−0.517245
$575$	2.46410	0.102760
$576$	−2.00000	−0.0833333
$577$	0.679492	0.0282876	0.0141438	−	0.999900i	$-0.495498\pi$
$577$	0.679492	0.0282876	0.0141438	+	0.999900i	$0.495498\pi$
$578$	16.7128	0.695161
$579$	19.0000	0.789613
$580$	−6.92820	−0.287678
$581$	−1.00000	−0.0414870
$582$	−14.9282	−0.618794
$583$	0	0
$584$	−7.46410	−0.308867
$585$	−4.92820	−0.203756
$586$	−13.5359	−0.559163
$587$	−1.14359	−0.0472012	−0.0236006	−	0.999721i	$-0.507513\pi$
$587$	−1.14359	−0.0472012	−0.0236006	+	0.999721i	$0.507513\pi$
$588$	−1.00000	−0.0412393
$589$	−59.1769	−2.43834
$590$	3.92820	0.161722
$591$	4.00000	0.164538
$592$	4.92820	0.202548
$593$	−28.0000	−1.14982	−0.574911	−	0.818216i	$-0.694963\pi$
$593$	−28.0000	−1.14982	−0.574911	+	0.818216i	$0.694963\pi$
$594$	0	0
$595$	−0.535898	−0.0219697
$596$	16.9282	0.693406
$597$	−1.85641	−0.0759777
$598$	−6.07180	−0.248294
$599$	−4.60770	−0.188265	−0.0941327	−	0.995560i	$-0.530008\pi$
$599$	−4.60770	−0.188265	−0.0941327	+	0.995560i	$0.530008\pi$
$600$	1.00000	0.0408248
$601$	−37.3205	−1.52234	−0.761168	−	0.648555i	$-0.775374\pi$
$601$	−37.3205	−1.52234	−0.761168	+	0.648555i	$0.775374\pi$
$602$	−8.39230	−0.342045
$603$	28.7846	1.17220
$604$	12.4641	0.507157
$605$	0	0
$606$	18.4641	0.750053
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	−7.92820	−0.321531
$609$	−6.92820	−0.280745
$610$	−8.92820	−0.361492
$611$	−18.3923	−0.744073
$612$	−1.07180	−0.0433248
$613$	−33.4641	−1.35160	−0.675801	−	0.737084i	$-0.736202\pi$
$613$	−33.4641	−1.35160	−0.675801	+	0.737084i	$0.736202\pi$
$614$	−1.07180	−0.0432542
$615$	12.3923	0.499706
$616$	0	0
$617$	−42.7846	−1.72244	−0.861222	−	0.508229i	$-0.830300\pi$
$617$	−42.7846	−1.72244	−0.861222	+	0.508229i	$0.830300\pi$
$618$	5.46410	0.219798
$619$	19.9282	0.800982	0.400491	−	0.916301i	$-0.368840\pi$
$619$	19.9282	0.800982	0.400491	+	0.916301i	$0.368840\pi$
$620$	−7.46410	−0.299766
$621$	12.3205	0.494405
$622$	11.3205	0.453911
$623$	10.5359	0.422112
$624$	−2.46410	−0.0986430
$625$	1.00000	0.0400000
$626$	26.7846	1.07053
$627$	0	0
$628$	14.3205	0.571450
$629$	2.64102	0.105304
$630$	−2.00000	−0.0796819
$631$	25.0718	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$631$	25.0718	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$632$	−14.4641	−0.575351
$633$	−24.0000	−0.953914
$634$	1.60770	0.0638497
$635$	7.39230	0.293355
$636$	0.535898	0.0212498
$637$	2.46410	0.0976313
$638$	0	0
$639$	−16.0000	−0.632950
$640$	−1.00000	−0.0395285
$641$	−7.14359	−0.282155	−0.141077	−	0.989999i	$-0.545057\pi$
$641$	−7.14359	−0.282155	−0.141077	+	0.989999i	$0.545057\pi$
$642$	6.00000	0.236801
$643$	41.5692	1.63933	0.819665	−	0.572843i	$-0.194160\pi$
$643$	41.5692	1.63933	0.819665	+	0.572843i	$0.194160\pi$
$644$	−2.46410	−0.0970992
$645$	8.39230	0.330447
$646$	−4.24871	−0.167163
$647$	23.3205	0.916824	0.458412	−	0.888740i	$-0.348418\pi$
$647$	23.3205	0.916824	0.458412	+	0.888740i	$0.348418\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	−2.46410	−0.0966500
$651$	−7.46410	−0.292541
$652$	−22.0000	−0.861586
$653$	−18.5359	−0.725366	−0.362683	−	0.931913i	$-0.618139\pi$
$653$	−18.5359	−0.725366	−0.362683	+	0.931913i	$0.618139\pi$
$654$	2.39230	0.0935465
$655$	−16.8564	−0.658634
$656$	−12.3923	−0.483838
$657$	−14.9282	−0.582405
$658$	−7.46410	−0.290981
$659$	−5.32051	−0.207258	−0.103629	−	0.994616i	$-0.533045\pi$
$659$	−5.32051	−0.207258	−0.103629	+	0.994616i	$0.533045\pi$
$660$	0	0
$661$	−21.3923	−0.832064	−0.416032	−	0.909350i	$-0.636580\pi$
$661$	−21.3923	−0.832064	−0.416032	+	0.909350i	$0.636580\pi$
$662$	5.46410	0.212368
$663$	−1.32051	−0.0512843
$664$	−1.00000	−0.0388075
$665$	−7.92820	−0.307443
$666$	9.85641	0.381928
$667$	−17.0718	−0.661023
$668$	−16.3923	−0.634237
$669$	−3.46410	−0.133930
$670$	14.3923	0.556023
$671$	0	0
$672$	−1.00000	−0.0385758
$673$	−11.1436	−0.429554	−0.214777	−	0.976663i	$-0.568902\pi$
$673$	−11.1436	−0.429554	−0.214777	+	0.976663i	$0.568902\pi$
$674$	−9.78461	−0.376889
$675$	5.00000	0.192450
$676$	−6.92820	−0.266469
$677$	12.6077	0.484553	0.242277	−	0.970207i	$-0.422106\pi$
$677$	12.6077	0.484553	0.242277	+	0.970207i	$0.422106\pi$
$678$	1.00000	0.0384048
$679$	14.9282	0.572892
$680$	−0.535898	−0.0205508
$681$	−16.0000	−0.613121
$682$	0	0
$683$	8.78461	0.336134	0.168067	−	0.985776i	$-0.446248\pi$
$683$	8.78461	0.336134	0.168067	+	0.985776i	$0.446248\pi$
$684$	−15.8564	−0.606285
$685$	−5.92820	−0.226505
$686$	1.00000	0.0381802
$687$	18.0000	0.686743
$688$	−8.39230	−0.319954
$689$	−1.32051	−0.0503074
$690$	2.46410	0.0938067
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	−12.9282	−0.490748
$695$	10.8564	0.411807
$696$	−6.92820	−0.262613
$697$	−6.64102	−0.251546
$698$	−8.46410	−0.320371
$699$	8.85641	0.334980
$700$	−1.00000	−0.0377964
$701$	−9.60770	−0.362878	−0.181439	−	0.983402i	$-0.558075\pi$
$701$	−9.60770	−0.362878	−0.181439	+	0.983402i	$0.558075\pi$
$702$	−12.3205	−0.465008
$703$	39.0718	1.47362
$704$	0	0
$705$	7.46410	0.281114
$706$	27.4641	1.03363
$707$	−18.4641	−0.694414
$708$	3.92820	0.147631
$709$	−7.85641	−0.295054	−0.147527	−	0.989058i	$-0.547131\pi$
$709$	−7.85641	−0.295054	−0.147527	+	0.989058i	$0.547131\pi$
$710$	−8.00000	−0.300235
$711$	−28.9282	−1.08489
$712$	10.5359	0.394849
$713$	−18.3923	−0.688797
$714$	−0.535898	−0.0200555
$715$	0	0
$716$	12.3923	0.463122
$717$	−5.53590	−0.206742
$718$	35.7128	1.33279
$719$	−15.7128	−0.585989	−0.292995	−	0.956114i	$-0.594652\pi$
$719$	−15.7128	−0.585989	−0.292995	+	0.956114i	$0.594652\pi$
$720$	−2.00000	−0.0745356
$721$	−5.46410	−0.203494
$722$	−43.8564	−1.63217
$723$	−4.00000	−0.148762
$724$	3.53590	0.131411
$725$	−6.92820	−0.257307
$726$	0	0
$727$	20.7846	0.770859	0.385429	−	0.922737i	$-0.374053\pi$
$727$	20.7846	0.770859	0.385429	+	0.922737i	$0.374053\pi$
$728$	2.46410	0.0913257
$729$	13.0000	0.481481
$730$	−7.46410	−0.276259
$731$	−4.49742	−0.166343
$732$	−8.92820	−0.329996
$733$	−14.1769	−0.523636	−0.261818	−	0.965117i	$-0.584322\pi$
$733$	−14.1769	−0.523636	−0.261818	+	0.965117i	$0.584322\pi$
$734$	14.5359	0.536530
$735$	−1.00000	−0.0368856
$736$	−2.46410	−0.0908280
$737$	0	0
$738$	−24.7846	−0.912334
$739$	−28.2487	−1.03915	−0.519573	−	0.854426i	$-0.673909\pi$
$739$	−28.2487	−1.03915	−0.519573	+	0.854426i	$0.673909\pi$
$740$	4.92820	0.181164
$741$	−19.5359	−0.717669
$742$	−0.535898	−0.0196734
$743$	−46.9282	−1.72163	−0.860814	−	0.508919i	$-0.830045\pi$
$743$	−46.9282	−1.72163	−0.860814	+	0.508919i	$0.830045\pi$
$744$	−7.46410	−0.273647
$745$	16.9282	0.620201
$746$	11.0718	0.405367
$747$	−2.00000	−0.0731762
$748$	0	0
$749$	−6.00000	−0.219235
$750$	1.00000	0.0365148
$751$	36.1769	1.32011	0.660057	−	0.751215i	$-0.270532\pi$
$751$	36.1769	1.32011	0.660057	+	0.751215i	$0.270532\pi$
$752$	−7.46410	−0.272188
$753$	0	0
$754$	17.0718	0.621718
$755$	12.4641	0.453615
$756$	−5.00000	−0.181848
$757$	7.32051	0.266068	0.133034	−	0.991111i	$-0.457528\pi$
$757$	7.32051	0.266068	0.133034	+	0.991111i	$0.457528\pi$
$758$	3.85641	0.140071
$759$	0	0
$760$	−7.92820	−0.287586
$761$	0.248711	0.00901578	0.00450789	−	0.999990i	$-0.498565\pi$
$761$	0.248711	0.00901578	0.00450789	+	0.999990i	$0.498565\pi$
$762$	7.39230	0.267795
$763$	−2.39230	−0.0866073
$764$	−17.3923	−0.629232
$765$	−1.07180	−0.0387509
$766$	14.5359	0.525203
$767$	−9.67949	−0.349506
$768$	−1.00000	−0.0360844
$769$	−18.7846	−0.677390	−0.338695	−	0.940896i	$-0.609986\pi$
$769$	−18.7846	−0.677390	−0.338695	+	0.940896i	$0.609986\pi$
$770$	0	0
$771$	18.3923	0.662383
$772$	−19.0000	−0.683825
$773$	−4.46410	−0.160563	−0.0802813	−	0.996772i	$-0.525582\pi$
$773$	−4.46410	−0.160563	−0.0802813	+	0.996772i	$0.525582\pi$
$774$	−16.7846	−0.603310
$775$	−7.46410	−0.268118
$776$	14.9282	0.535891
$777$	4.92820	0.176798
$778$	−21.3205	−0.764377
$779$	−98.2487	−3.52013
$780$	−2.46410	−0.0882290
$781$	0	0
$782$	−1.32051	−0.0472213
$783$	−34.6410	−1.23797
$784$	1.00000	0.0357143
$785$	14.3205	0.511121
$786$	−16.8564	−0.601248
$787$	48.4974	1.72875	0.864373	−	0.502851i	$-0.167715\pi$
$787$	48.4974	1.72875	0.864373	+	0.502851i	$0.167715\pi$
$788$	−4.00000	−0.142494
$789$	9.53590	0.339487
$790$	−14.4641	−0.514610
$791$	−1.00000	−0.0355559
$792$	0	0
$793$	22.0000	0.781243
$794$	24.9282	0.884669
$795$	0.535898	0.0190064
$796$	1.85641	0.0657986
$797$	−0.607695	−0.0215257	−0.0107628	−	0.999942i	$-0.503426\pi$
$797$	−0.607695	−0.0215257	−0.0107628	+	0.999942i	$0.503426\pi$
$798$	−7.92820	−0.280655
$799$	−4.00000	−0.141510
$800$	−1.00000	−0.0353553
$801$	21.0718	0.744535
$802$	−31.8564	−1.12489
$803$	0	0
$804$	14.3923	0.507577
$805$	−2.46410	−0.0868482
$806$	18.3923	0.647841
$807$	−24.3205	−0.856122
$808$	−18.4641	−0.649565
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	−1.00000	−0.0351364
$811$	19.7128	0.692210	0.346105	−	0.938196i	$-0.387504\pi$
$811$	19.7128	0.692210	0.346105	+	0.938196i	$0.387504\pi$
$812$	6.92820	0.243132
$813$	2.14359	0.0751791
$814$	0	0
$815$	−22.0000	−0.770626
$816$	−0.535898	−0.0187602
$817$	−66.5359	−2.32780
$818$	23.7128	0.829099
$819$	4.92820	0.172205
$820$	−12.3923	−0.432758
$821$	−23.0718	−0.805211	−0.402606	−	0.915374i	$-0.631895\pi$
$821$	−23.0718	−0.805211	−0.402606	+	0.915374i	$0.631895\pi$
$822$	−5.92820	−0.206770
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	−5.46410	−0.190351
$825$	0	0
$826$	−3.92820	−0.136680
$827$	−2.39230	−0.0831886	−0.0415943	−	0.999135i	$-0.513244\pi$
$827$	−2.39230	−0.0831886	−0.0415943	+	0.999135i	$0.513244\pi$
$828$	−4.92820	−0.171267
$829$	−0.607695	−0.0211061	−0.0105531	−	0.999944i	$-0.503359\pi$
$829$	−0.607695	−0.0211061	−0.0105531	+	0.999944i	$0.503359\pi$
$830$	−1.00000	−0.0347105
$831$	−7.32051	−0.253946
$832$	2.46410	0.0854274
$833$	0.535898	0.0185678
$834$	10.8564	0.375927
$835$	−16.3923	−0.567279
$836$	0	0
$837$	−37.3205	−1.28999
$838$	16.7128	0.577335
$839$	−42.2487	−1.45859	−0.729294	−	0.684201i	$-0.760151\pi$
$839$	−42.2487	−1.45859	−0.729294	+	0.684201i	$0.760151\pi$
$840$	−1.00000	−0.0345033
$841$	19.0000	0.655172
$842$	21.8564	0.753222
$843$	17.0000	0.585511
$844$	24.0000	0.826114
$845$	−6.92820	−0.238337
$846$	−14.9282	−0.513242
$847$	0	0
$848$	−0.535898	−0.0184028
$849$	27.7846	0.953565
$850$	−0.535898	−0.0183812
$851$	12.1436	0.416277
$852$	−8.00000	−0.274075
$853$	−16.4641	−0.563720	−0.281860	−	0.959456i	$-0.590951\pi$
$853$	−16.4641	−0.563720	−0.281860	+	0.959456i	$0.590951\pi$
$854$	8.92820	0.305517
$855$	−15.8564	−0.542278
$856$	−6.00000	−0.205076
$857$	6.92820	0.236663	0.118331	−	0.992974i	$-0.462245\pi$
$857$	6.92820	0.236663	0.118331	+	0.992974i	$0.462245\pi$
$858$	0	0
$859$	42.6410	1.45489	0.727446	−	0.686165i	$-0.240707\pi$
$859$	42.6410	1.45489	0.727446	+	0.686165i	$0.240707\pi$
$860$	−8.39230	−0.286175
$861$	−12.3923	−0.422329
$862$	24.3205	0.828360
$863$	−40.7846	−1.38832	−0.694162	−	0.719819i	$-0.744225\pi$
$863$	−40.7846	−1.38832	−0.694162	+	0.719819i	$0.744225\pi$
$864$	−5.00000	−0.170103
$865$	−18.0000	−0.612018
$866$	26.2487	0.891968
$867$	16.7128	0.567597
$868$	7.46410	0.253348
$869$	0	0
$870$	−6.92820	−0.234888
$871$	−35.4641	−1.20166
$872$	−2.39230	−0.0810137
$873$	29.8564	1.01049
$874$	−19.5359	−0.660812
$875$	−1.00000	−0.0338062
$876$	−7.46410	−0.252189
$877$	−12.3923	−0.418458	−0.209229	−	0.977867i	$-0.567095\pi$
$877$	−12.3923	−0.418458	−0.209229	+	0.977867i	$0.567095\pi$
$878$	−29.4641	−0.994365
$879$	−13.5359	−0.456555
$880$	0	0
$881$	49.7128	1.67487	0.837434	−	0.546539i	$-0.184055\pi$
$881$	49.7128	1.67487	0.837434	+	0.546539i	$0.184055\pi$
$882$	2.00000	0.0673435
$883$	6.78461	0.228320	0.114160	−	0.993462i	$-0.463582\pi$
$883$	6.78461	0.228320	0.114160	+	0.993462i	$0.463582\pi$
$884$	1.32051	0.0444135
$885$	3.92820	0.132045
$886$	−39.7128	−1.33418
$887$	33.4641	1.12361	0.561807	−	0.827268i	$-0.310106\pi$
$887$	33.4641	1.12361	0.561807	+	0.827268i	$0.310106\pi$
$888$	4.92820	0.165380
$889$	−7.39230	−0.247930
$890$	10.5359	0.353164
$891$	0	0
$892$	3.46410	0.115987
$893$	−59.1769	−1.98028
$894$	16.9282	0.566164
$895$	12.3923	0.414229
$896$	1.00000	0.0334077
$897$	−6.07180	−0.202731
$898$	21.9282	0.731754
$899$	51.7128	1.72472
$900$	−2.00000	−0.0666667
$901$	−0.287187	−0.00956759
$902$	0	0
$903$	−8.39230	−0.279278
$904$	−1.00000	−0.0332595
$905$	3.53590	0.117537
$906$	12.4641	0.414092
$907$	26.6410	0.884600	0.442300	−	0.896867i	$-0.354163\pi$
$907$	26.6410	0.884600	0.442300	+	0.896867i	$0.354163\pi$
$908$	16.0000	0.530979
$909$	−36.9282	−1.22483
$910$	2.46410	0.0816842
$911$	−11.3923	−0.377444	−0.188722	−	0.982031i	$-0.560434\pi$
$911$	−11.3923	−0.377444	−0.188722	+	0.982031i	$0.560434\pi$
$912$	−7.92820	−0.262529
$913$	0	0
$914$	29.6410	0.980438
$915$	−8.92820	−0.295157
$916$	−18.0000	−0.594737
$917$	16.8564	0.556648
$918$	−2.67949	−0.0884364
$919$	51.7128	1.70585	0.852924	−	0.522035i	$-0.174827\pi$
$919$	51.7128	1.70585	0.852924	+	0.522035i	$0.174827\pi$
$920$	−2.46410	−0.0812390
$921$	−1.07180	−0.0353169
$922$	39.8564	1.31260
$923$	19.7128	0.648855
$924$	0	0
$925$	4.92820	0.162038
$926$	1.53590	0.0504728
$927$	−10.9282	−0.358929
$928$	6.92820	0.227429
$929$	1.85641	0.0609067	0.0304534	−	0.999536i	$-0.490305\pi$
$929$	1.85641	0.0609067	0.0304534	+	0.999536i	$0.490305\pi$
$930$	−7.46410	−0.244758
$931$	7.92820	0.259836
$932$	−8.85641	−0.290101
$933$	11.3205	0.370617
$934$	2.07180	0.0677913
$935$	0	0
$936$	4.92820	0.161083
$937$	−33.8564	−1.10604	−0.553020	−	0.833168i	$-0.686525\pi$
$937$	−33.8564	−1.10604	−0.553020	+	0.833168i	$0.686525\pi$
$938$	−14.3923	−0.469925
$939$	26.7846	0.874083
$940$	−7.46410	−0.243452
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	14.3205	0.466587
$943$	−30.5359	−0.994386
$944$	−3.92820	−0.127852
$945$	−5.00000	−0.162650
$946$	0	0
$947$	50.3923	1.63753	0.818765	−	0.574129i	$-0.194659\pi$
$947$	50.3923	1.63753	0.818765	+	0.574129i	$0.194659\pi$
$948$	−14.4641	−0.469772
$949$	18.3923	0.597039
$950$	−7.92820	−0.257225
$951$	1.60770	0.0521331
$952$	0.535898	0.0173686
$953$	50.5692	1.63810	0.819049	−	0.573724i	$-0.194502\pi$
$953$	50.5692	1.63810	0.819049	+	0.573724i	$0.194502\pi$
$954$	−1.07180	−0.0347007
$955$	−17.3923	−0.562802
$956$	5.53590	0.179044
$957$	0	0
$958$	−9.46410	−0.305771
$959$	5.92820	0.191432
$960$	−1.00000	−0.0322749
$961$	24.7128	0.797188
$962$	−12.1436	−0.391525
$963$	−12.0000	−0.386695
$964$	4.00000	0.128831
$965$	−19.0000	−0.611632
$966$	−2.46410	−0.0792812
$967$	43.7128	1.40571	0.702855	−	0.711333i	$-0.251908\pi$
$967$	43.7128	1.40571	0.702855	+	0.711333i	$0.251908\pi$
$968$	0	0
$969$	−4.24871	−0.136488
$970$	14.9282	0.479316
$971$	−42.7128	−1.37072	−0.685360	−	0.728205i	$-0.740355\pi$
$971$	−42.7128	−1.37072	−0.685360	+	0.728205i	$0.740355\pi$
$972$	−16.0000	−0.513200
$973$	−10.8564	−0.348040
$974$	32.3205	1.03562
$975$	−2.46410	−0.0789144
$976$	8.92820	0.285785
$977$	−8.85641	−0.283342	−0.141671	−	0.989914i	$-0.545247\pi$
$977$	−8.85641	−0.283342	−0.141671	+	0.989914i	$0.545247\pi$
$978$	−22.0000	−0.703482
$979$	0	0
$980$	1.00000	0.0319438
$981$	−4.78461	−0.152761
$982$	16.5359	0.527682
$983$	−26.7846	−0.854296	−0.427148	−	0.904182i	$-0.640482\pi$
$983$	−26.7846	−0.854296	−0.427148	+	0.904182i	$0.640482\pi$
$984$	−12.3923	−0.395052
$985$	−4.00000	−0.127451
$986$	3.71281	0.118240
$987$	−7.46410	−0.237585
$988$	19.5359	0.621520
$989$	−20.6795	−0.657570
$990$	0	0
$991$	−33.1051	−1.05162	−0.525809	−	0.850602i	$-0.676237\pi$
$991$	−33.1051	−1.05162	−0.525809	+	0.850602i	$0.676237\pi$
$992$	7.46410	0.236985
$993$	5.46410	0.173398
$994$	8.00000	0.253745
$995$	1.85641	0.0588520
$996$	−1.00000	−0.0316862
$997$	−4.32051	−0.136832	−0.0684159	−	0.997657i	$-0.521794\pi$
$997$	−4.32051	−0.136832	−0.0684159	+	0.997657i	$0.521794\pi$
$998$	−4.92820	−0.156000
$999$	24.6410	0.779607

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.bi.1.2	✓	2	1.1	even	1	trivial
8470.2.a.bw.1.1	yes	2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} +2.46410 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +0.535898 q^{17} +2.00000 q^{18} +7.92820 q^{19} +1.00000 q^{20} +1.00000 q^{21} +2.46410 q^{23} +1.00000 q^{24} +1.00000 q^{25} -2.46410 q^{26} +5.00000 q^{27} -1.00000 q^{28} -6.92820 q^{29} +1.00000 q^{30} -7.46410 q^{31} -1.00000 q^{32} -0.535898 q^{34} -1.00000 q^{35} -2.00000 q^{36} +4.92820 q^{37} -7.92820 q^{38} -2.46410 q^{39} -1.00000 q^{40} -12.3923 q^{41} -1.00000 q^{42} -8.39230 q^{43} -2.00000 q^{45} -2.46410 q^{46} -7.46410 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -0.535898 q^{51} +2.46410 q^{52} -0.535898 q^{53} -5.00000 q^{54} +1.00000 q^{56} -7.92820 q^{57} +6.92820 q^{58} -3.92820 q^{59} -1.00000 q^{60} +8.92820 q^{61} +7.46410 q^{62} +2.00000 q^{63} +1.00000 q^{64} +2.46410 q^{65} -14.3923 q^{67} +0.535898 q^{68} -2.46410 q^{69} +1.00000 q^{70} +8.00000 q^{71} +2.00000 q^{72} +7.46410 q^{73} -4.92820 q^{74} -1.00000 q^{75} +7.92820 q^{76} +2.46410 q^{78} +14.4641 q^{79} +1.00000 q^{80} +1.00000 q^{81} +12.3923 q^{82} +1.00000 q^{83} +1.00000 q^{84} +0.535898 q^{85} +8.39230 q^{86} +6.92820 q^{87} -10.5359 q^{89} +2.00000 q^{90} -2.46410 q^{91} +2.46410 q^{92} +7.46410 q^{93} +7.46410 q^{94} +7.92820 q^{95} +1.00000 q^{96} -14.9282 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} - 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^7 - 2 * q^8 - 4 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{7} - 2 q^{8} - 4 q^{9} - 2 q^{10} - 2 q^{12} - 2 q^{13} + 2 q^{14} - 2 q^{15} + 2 q^{16} + 8 q^{17} + 4 q^{18} + 2 q^{19} + 2 q^{20} + 2 q^{21} - 2 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{26} + 10 q^{27} - 2 q^{28} + 2 q^{30} - 8 q^{31} - 2 q^{32} - 8 q^{34} - 2 q^{35} - 4 q^{36} - 4 q^{37} - 2 q^{38} + 2 q^{39} - 2 q^{40} - 4 q^{41} - 2 q^{42} + 4 q^{43} - 4 q^{45} + 2 q^{46} - 8 q^{47} - 2 q^{48} + 2 q^{49} - 2 q^{50} - 8 q^{51} - 2 q^{52} - 8 q^{53} - 10 q^{54} + 2 q^{56} - 2 q^{57} + 6 q^{59} - 2 q^{60} + 4 q^{61} + 8 q^{62} + 4 q^{63} + 2 q^{64} - 2 q^{65} - 8 q^{67} + 8 q^{68} + 2 q^{69} + 2 q^{70} + 16 q^{71} + 4 q^{72} + 8 q^{73} + 4 q^{74} - 2 q^{75} + 2 q^{76} - 2 q^{78} + 22 q^{79} + 2 q^{80} + 2 q^{81} + 4 q^{82} + 2 q^{83} + 2 q^{84} + 8 q^{85} - 4 q^{86} - 28 q^{89} + 4 q^{90} + 2 q^{91} - 2 q^{92} + 8 q^{93} + 8 q^{94} + 2 q^{95} + 2 q^{96} - 16 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 - 2 * q^7 - 2 * q^8 - 4 * q^9 - 2 * q^10 - 2 * q^12 - 2 * q^13 + 2 * q^14 - 2 * q^15 + 2 * q^16 + 8 * q^17 + 4 * q^18 + 2 * q^19 + 2 * q^20 + 2 * q^21 - 2 * q^23 + 2 * q^24 + 2 * q^25 + 2 * q^26 + 10 * q^27 - 2 * q^28 + 2 * q^30 - 8 * q^31 - 2 * q^32 - 8 * q^34 - 2 * q^35 - 4 * q^36 - 4 * q^37 - 2 * q^38 + 2 * q^39 - 2 * q^40 - 4 * q^41 - 2 * q^42 + 4 * q^43 - 4 * q^45 + 2 * q^46 - 8 * q^47 - 2 * q^48 + 2 * q^49 - 2 * q^50 - 8 * q^51 - 2 * q^52 - 8 * q^53 - 10 * q^54 + 2 * q^56 - 2 * q^57 + 6 * q^59 - 2 * q^60 + 4 * q^61 + 8 * q^62 + 4 * q^63 + 2 * q^64 - 2 * q^65 - 8 * q^67 + 8 * q^68 + 2 * q^69 + 2 * q^70 + 16 * q^71 + 4 * q^72 + 8 * q^73 + 4 * q^74 - 2 * q^75 + 2 * q^76 - 2 * q^78 + 22 * q^79 + 2 * q^80 + 2 * q^81 + 4 * q^82 + 2 * q^83 + 2 * q^84 + 8 * q^85 - 4 * q^86 - 28 * q^89 + 4 * q^90 + 2 * q^91 - 2 * q^92 + 8 * q^93 + 8 * q^94 + 2 * q^95 + 2 * q^96 - 16 * q^97 - 2 * q^98

Introduction

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