LMFDB - Embedded newform 8470.2.a.bo.1.1

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:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.82843 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.82843 q^{6} +1.00000 q^{7} -1.00000 q^{8} +5.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.82843 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.82843 q^{6} +1.00000 q^{7} -1.00000 q^{8} +5.00000 q^{9} +1.00000 q^{10} -2.82843 q^{12} -2.00000 q^{13} -1.00000 q^{14} +2.82843 q^{15} +1.00000 q^{16} +3.65685 q^{17} -5.00000 q^{18} -2.82843 q^{19} -1.00000 q^{20} -2.82843 q^{21} +2.82843 q^{23} +2.82843 q^{24} +1.00000 q^{25} +2.00000 q^{26} -5.65685 q^{27} +1.00000 q^{28} -8.82843 q^{29} -2.82843 q^{30} +5.65685 q^{31} -1.00000 q^{32} -3.65685 q^{34} -1.00000 q^{35} +5.00000 q^{36} +8.82843 q^{37} +2.82843 q^{38} +5.65685 q^{39} +1.00000 q^{40} -4.82843 q^{41} +2.82843 q^{42} -1.65685 q^{43} -5.00000 q^{45} -2.82843 q^{46} -8.00000 q^{47} -2.82843 q^{48} +1.00000 q^{49} -1.00000 q^{50} -10.3431 q^{51} -2.00000 q^{52} +6.48528 q^{53} +5.65685 q^{54} -1.00000 q^{56} +8.00000 q^{57} +8.82843 q^{58} -8.00000 q^{59} +2.82843 q^{60} -13.3137 q^{61} -5.65685 q^{62} +5.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -7.31371 q^{67} +3.65685 q^{68} -8.00000 q^{69} +1.00000 q^{70} +5.65685 q^{71} -5.00000 q^{72} +3.65685 q^{73} -8.82843 q^{74} -2.82843 q^{75} -2.82843 q^{76} -5.65685 q^{78} +10.8284 q^{79} -1.00000 q^{80} +1.00000 q^{81} +4.82843 q^{82} +13.6569 q^{83} -2.82843 q^{84} -3.65685 q^{85} +1.65685 q^{86} +24.9706 q^{87} -0.343146 q^{89} +5.00000 q^{90} -2.00000 q^{91} +2.82843 q^{92} -16.0000 q^{93} +8.00000 q^{94} +2.82843 q^{95} +2.82843 q^{96} +4.82843 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 10 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8 + 10 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 10 q^{9} + 2 q^{10} - 4 q^{13} - 2 q^{14} + 2 q^{16} - 4 q^{17} - 10 q^{18} - 2 q^{20} + 2 q^{25} + 4 q^{26} + 2 q^{28} - 12 q^{29} - 2 q^{32} + 4 q^{34} - 2 q^{35} + 10 q^{36} + 12 q^{37} + 2 q^{40} - 4 q^{41} + 8 q^{43} - 10 q^{45} - 16 q^{47} + 2 q^{49} - 2 q^{50} - 32 q^{51} - 4 q^{52} - 4 q^{53} - 2 q^{56} + 16 q^{57} + 12 q^{58} - 16 q^{59} - 4 q^{61} + 10 q^{63} + 2 q^{64} + 4 q^{65} + 8 q^{67} - 4 q^{68} - 16 q^{69} + 2 q^{70} - 10 q^{72} - 4 q^{73} - 12 q^{74} + 16 q^{79} - 2 q^{80} + 2 q^{81} + 4 q^{82} + 16 q^{83} + 4 q^{85} - 8 q^{86} + 16 q^{87} - 12 q^{89} + 10 q^{90} - 4 q^{91} - 32 q^{93} + 16 q^{94} + 4 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8 + 10 * q^9 + 2 * q^10 - 4 * q^13 - 2 * q^14 + 2 * q^16 - 4 * q^17 - 10 * q^18 - 2 * q^20 + 2 * q^25 + 4 * q^26 + 2 * q^28 - 12 * q^29 - 2 * q^32 + 4 * q^34 - 2 * q^35 + 10 * q^36 + 12 * q^37 + 2 * q^40 - 4 * q^41 + 8 * q^43 - 10 * q^45 - 16 * q^47 + 2 * q^49 - 2 * q^50 - 32 * q^51 - 4 * q^52 - 4 * q^53 - 2 * q^56 + 16 * q^57 + 12 * q^58 - 16 * q^59 - 4 * q^61 + 10 * q^63 + 2 * q^64 + 4 * q^65 + 8 * q^67 - 4 * q^68 - 16 * q^69 + 2 * q^70 - 10 * q^72 - 4 * q^73 - 12 * q^74 + 16 * q^79 - 2 * q^80 + 2 * q^81 + 4 * q^82 + 16 * q^83 + 4 * q^85 - 8 * q^86 + 16 * q^87 - 12 * q^89 + 10 * q^90 - 4 * q^91 - 32 * q^93 + 16 * q^94 + 4 * 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
$3$	−2.82843	−1.63299	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	−2.82843	−1.63299	−0.816497	+	0.577350i	$0.804087\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.82843	1.15470
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	5.00000	1.66667
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	−2.82843	−0.816497
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−1.00000	−0.267261
$15$	2.82843	0.730297
$16$	1.00000	0.250000
$17$	3.65685	0.886917	0.443459	−	0.896295i	$-0.353751\pi$
$17$	3.65685	0.886917	0.443459	+	0.896295i	$0.353751\pi$
$18$	−5.00000	−1.17851
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	−1.00000	−0.223607
$21$	−2.82843	−0.617213
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	2.82843	0.577350
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	−5.65685	−1.08866
$28$	1.00000	0.188982
$29$	−8.82843	−1.63940	−0.819699	−	0.572795i	$-0.805859\pi$
$29$	−8.82843	−1.63940	−0.819699	+	0.572795i	$0.805859\pi$
$30$	−2.82843	−0.516398
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.65685	−0.627145
$35$	−1.00000	−0.169031
$36$	5.00000	0.833333
$37$	8.82843	1.45138	0.725692	−	0.688019i	$-0.241520\pi$
$37$	8.82843	1.45138	0.725692	+	0.688019i	$0.241520\pi$
$38$	2.82843	0.458831
$39$	5.65685	0.905822
$40$	1.00000	0.158114
$41$	−4.82843	−0.754074	−0.377037	−	0.926198i	$-0.623057\pi$
$41$	−4.82843	−0.754074	−0.377037	+	0.926198i	$0.623057\pi$
$42$	2.82843	0.436436
$43$	−1.65685	−0.252668	−0.126334	−	0.991988i	$-0.540321\pi$
$43$	−1.65685	−0.252668	−0.126334	+	0.991988i	$0.540321\pi$
$44$	0	0
$45$	−5.00000	−0.745356
$46$	−2.82843	−0.417029
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	−2.82843	−0.408248
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−10.3431	−1.44833
$52$	−2.00000	−0.277350
$53$	6.48528	0.890822	0.445411	−	0.895326i	$-0.353058\pi$
$53$	6.48528	0.890822	0.445411	+	0.895326i	$0.353058\pi$
$54$	5.65685	0.769800
$55$	0	0
$56$	−1.00000	−0.133631
$57$	8.00000	1.05963
$58$	8.82843	1.15923
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	2.82843	0.365148
$61$	−13.3137	−1.70465	−0.852323	−	0.523016i	$-0.824807\pi$
$61$	−13.3137	−1.70465	−0.852323	+	0.523016i	$0.824807\pi$
$62$	−5.65685	−0.718421
$63$	5.00000	0.629941
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	−7.31371	−0.893512	−0.446756	−	0.894656i	$-0.647421\pi$
$67$	−7.31371	−0.893512	−0.446756	+	0.894656i	$0.647421\pi$
$68$	3.65685	0.443459
$69$	−8.00000	−0.963087
$70$	1.00000	0.119523
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	−5.00000	−0.589256
$73$	3.65685	0.428002	0.214001	−	0.976833i	$-0.431350\pi$
$73$	3.65685	0.428002	0.214001	+	0.976833i	$0.431350\pi$
$74$	−8.82843	−1.02628
$75$	−2.82843	−0.326599
$76$	−2.82843	−0.324443
$77$	0	0
$78$	−5.65685	−0.640513
$79$	10.8284	1.21829	0.609147	−	0.793058i	$-0.291512\pi$
$79$	10.8284	1.21829	0.609147	+	0.793058i	$0.291512\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	4.82843	0.533211
$83$	13.6569	1.49903	0.749517	−	0.661985i	$-0.230286\pi$
$83$	13.6569	1.49903	0.749517	+	0.661985i	$0.230286\pi$
$84$	−2.82843	−0.308607
$85$	−3.65685	−0.396642
$86$	1.65685	0.178663
$87$	24.9706	2.67713
$88$	0	0
$89$	−0.343146	−0.0363734	−0.0181867	−	0.999835i	$-0.505789\pi$
$89$	−0.343146	−0.0363734	−0.0181867	+	0.999835i	$0.505789\pi$
$90$	5.00000	0.527046
$91$	−2.00000	−0.209657
$92$	2.82843	0.294884
$93$	−16.0000	−1.65912
$94$	8.00000	0.825137
$95$	2.82843	0.290191
$96$	2.82843	0.288675
$97$	4.82843	0.490252	0.245126	−	0.969491i	$-0.421171\pi$
$97$	4.82843	0.490252	0.245126	+	0.969491i	$0.421171\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	10.3431	1.02412
$103$	5.65685	0.557386	0.278693	−	0.960380i	$-0.410099\pi$
$103$	5.65685	0.557386	0.278693	+	0.960380i	$0.410099\pi$
$104$	2.00000	0.196116
$105$	2.82843	0.276026
$106$	−6.48528	−0.629906
$107$	−9.65685	−0.933563	−0.466782	−	0.884373i	$-0.654587\pi$
$107$	−9.65685	−0.933563	−0.466782	+	0.884373i	$0.654587\pi$
$108$	−5.65685	−0.544331
$109$	2.48528	0.238047	0.119023	−	0.992891i	$-0.462024\pi$
$109$	2.48528	0.238047	0.119023	+	0.992891i	$0.462024\pi$
$110$	0	0
$111$	−24.9706	−2.37010
$112$	1.00000	0.0944911
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−8.00000	−0.749269
$115$	−2.82843	−0.263752
$116$	−8.82843	−0.819699
$117$	−10.0000	−0.924500
$118$	8.00000	0.736460
$119$	3.65685	0.335223
$120$	−2.82843	−0.258199
$121$	0	0
$122$	13.3137	1.20537
$123$	13.6569	1.23140
$124$	5.65685	0.508001
$125$	−1.00000	−0.0894427
$126$	−5.00000	−0.445435
$127$	3.31371	0.294044	0.147022	−	0.989133i	$-0.453031\pi$
$127$	3.31371	0.294044	0.147022	+	0.989133i	$0.453031\pi$
$128$	−1.00000	−0.0883883
$129$	4.68629	0.412605
$130$	−2.00000	−0.175412
$131$	−13.1716	−1.15081	−0.575403	−	0.817870i	$-0.695155\pi$
$131$	−13.1716	−1.15081	−0.575403	+	0.817870i	$0.695155\pi$
$132$	0	0
$133$	−2.82843	−0.245256
$134$	7.31371	0.631808
$135$	5.65685	0.486864
$136$	−3.65685	−0.313573
$137$	−17.3137	−1.47921	−0.739605	−	0.673041i	$-0.764988\pi$
$137$	−17.3137	−1.47921	−0.739605	+	0.673041i	$0.764988\pi$
$138$	8.00000	0.681005
$139$	−13.1716	−1.11720	−0.558599	−	0.829438i	$-0.688661\pi$
$139$	−13.1716	−1.11720	−0.558599	+	0.829438i	$0.688661\pi$
$140$	−1.00000	−0.0845154
$141$	22.6274	1.90557
$142$	−5.65685	−0.474713
$143$	0	0
$144$	5.00000	0.416667
$145$	8.82843	0.733161
$146$	−3.65685	−0.302643
$147$	−2.82843	−0.233285
$148$	8.82843	0.725692
$149$	18.4853	1.51437	0.757187	−	0.653199i	$-0.226573\pi$
$149$	18.4853	1.51437	0.757187	+	0.653199i	$0.226573\pi$
$150$	2.82843	0.230940
$151$	−5.17157	−0.420857	−0.210428	−	0.977609i	$-0.567486\pi$
$151$	−5.17157	−0.420857	−0.210428	+	0.977609i	$0.567486\pi$
$152$	2.82843	0.229416
$153$	18.2843	1.47820
$154$	0	0
$155$	−5.65685	−0.454369
$156$	5.65685	0.452911
$157$	4.34315	0.346621	0.173310	−	0.984867i	$-0.444554\pi$
$157$	4.34315	0.346621	0.173310	+	0.984867i	$0.444554\pi$
$158$	−10.8284	−0.861463
$159$	−18.3431	−1.45471
$160$	1.00000	0.0790569
$161$	2.82843	0.222911
$162$	−1.00000	−0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	−4.82843	−0.377037
$165$	0	0
$166$	−13.6569	−1.05998
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	2.82843	0.218218
$169$	−9.00000	−0.692308
$170$	3.65685	0.280468
$171$	−14.1421	−1.08148
$172$	−1.65685	−0.126334
$173$	−21.3137	−1.62045	−0.810226	−	0.586118i	$-0.800655\pi$
$173$	−21.3137	−1.62045	−0.810226	+	0.586118i	$0.800655\pi$
$174$	−24.9706	−1.89301
$175$	1.00000	0.0755929
$176$	0	0
$177$	22.6274	1.70078
$178$	0.343146	0.0257199
$179$	17.6569	1.31974	0.659868	−	0.751382i	$-0.270612\pi$
$179$	17.6569	1.31974	0.659868	+	0.751382i	$0.270612\pi$
$180$	−5.00000	−0.372678
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	2.00000	0.148250
$183$	37.6569	2.78367
$184$	−2.82843	−0.208514
$185$	−8.82843	−0.649079
$186$	16.0000	1.17318
$187$	0	0
$188$	−8.00000	−0.583460
$189$	−5.65685	−0.411476
$190$	−2.82843	−0.205196
$191$	−3.31371	−0.239772	−0.119886	−	0.992788i	$-0.538253\pi$
$191$	−3.31371	−0.239772	−0.119886	+	0.992788i	$0.538253\pi$
$192$	−2.82843	−0.204124
$193$	1.31371	0.0945628	0.0472814	−	0.998882i	$-0.484944\pi$
$193$	1.31371	0.0945628	0.0472814	+	0.998882i	$0.484944\pi$
$194$	−4.82843	−0.346661
$195$	−5.65685	−0.405096
$196$	1.00000	0.0714286
$197$	23.6569	1.68548	0.842741	−	0.538320i	$-0.180941\pi$
$197$	23.6569	1.68548	0.842741	+	0.538320i	$0.180941\pi$
$198$	0	0
$199$	11.3137	0.802008	0.401004	−	0.916076i	$-0.368661\pi$
$199$	11.3137	0.802008	0.401004	+	0.916076i	$0.368661\pi$
$200$	−1.00000	−0.0707107
$201$	20.6863	1.45910
$202$	10.0000	0.703598
$203$	−8.82843	−0.619634
$204$	−10.3431	−0.724165
$205$	4.82843	0.337232
$206$	−5.65685	−0.394132
$207$	14.1421	0.982946
$208$	−2.00000	−0.138675
$209$	0	0
$210$	−2.82843	−0.195180
$211$	7.31371	0.503496	0.251748	−	0.967793i	$-0.418995\pi$
$211$	7.31371	0.503496	0.251748	+	0.967793i	$0.418995\pi$
$212$	6.48528	0.445411
$213$	−16.0000	−1.09630
$214$	9.65685	0.660129
$215$	1.65685	0.112997
$216$	5.65685	0.384900
$217$	5.65685	0.384012
$218$	−2.48528	−0.168324
$219$	−10.3431	−0.698925
$220$	0	0
$221$	−7.31371	−0.491973
$222$	24.9706	1.67591
$223$	28.2843	1.89405	0.947027	−	0.321153i	$-0.104070\pi$
$223$	28.2843	1.89405	0.947027	+	0.321153i	$0.104070\pi$
$224$	−1.00000	−0.0668153
$225$	5.00000	0.333333
$226$	−2.00000	−0.133038
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	8.00000	0.529813
$229$	21.3137	1.40845	0.704225	−	0.709977i	$-0.251295\pi$
$229$	21.3137	1.40845	0.704225	+	0.709977i	$0.251295\pi$
$230$	2.82843	0.186501
$231$	0	0
$232$	8.82843	0.579615
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	10.0000	0.653720
$235$	8.00000	0.521862
$236$	−8.00000	−0.520756
$237$	−30.6274	−1.98946
$238$	−3.65685	−0.237039
$239$	24.4853	1.58382	0.791911	−	0.610637i	$-0.209087\pi$
$239$	24.4853	1.58382	0.791911	+	0.610637i	$0.209087\pi$
$240$	2.82843	0.182574
$241$	3.17157	0.204299	0.102149	−	0.994769i	$-0.467428\pi$
$241$	3.17157	0.204299	0.102149	+	0.994769i	$0.467428\pi$
$242$	0	0
$243$	14.1421	0.907218
$244$	−13.3137	−0.852323
$245$	−1.00000	−0.0638877
$246$	−13.6569	−0.870729
$247$	5.65685	0.359937
$248$	−5.65685	−0.359211
$249$	−38.6274	−2.44791
$250$	1.00000	0.0632456
$251$	11.3137	0.714115	0.357057	−	0.934082i	$-0.383780\pi$
$251$	11.3137	0.714115	0.357057	+	0.934082i	$0.383780\pi$
$252$	5.00000	0.314970
$253$	0	0
$254$	−3.31371	−0.207921
$255$	10.3431	0.647713
$256$	1.00000	0.0625000
$257$	28.8284	1.79827	0.899134	−	0.437674i	$-0.144197\pi$
$257$	28.8284	1.79827	0.899134	+	0.437674i	$0.144197\pi$
$258$	−4.68629	−0.291756
$259$	8.82843	0.548572
$260$	2.00000	0.124035
$261$	−44.1421	−2.73233
$262$	13.1716	0.813742
$263$	−30.6274	−1.88857	−0.944284	−	0.329133i	$-0.893244\pi$
$263$	−30.6274	−1.88857	−0.944284	+	0.329133i	$0.893244\pi$
$264$	0	0
$265$	−6.48528	−0.398388
$266$	2.82843	0.173422
$267$	0.970563	0.0593975
$268$	−7.31371	−0.446756
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	−5.65685	−0.344265
$271$	16.9706	1.03089	0.515444	−	0.856923i	$-0.327627\pi$
$271$	16.9706	1.03089	0.515444	+	0.856923i	$0.327627\pi$
$272$	3.65685	0.221729
$273$	5.65685	0.342368
$274$	17.3137	1.04596
$275$	0	0
$276$	−8.00000	−0.481543
$277$	−9.31371	−0.559607	−0.279803	−	0.960057i	$-0.590269\pi$
$277$	−9.31371	−0.559607	−0.279803	+	0.960057i	$0.590269\pi$
$278$	13.1716	0.789978
$279$	28.2843	1.69334
$280$	1.00000	0.0597614
$281$	−26.9706	−1.60893	−0.804464	−	0.594001i	$-0.797548\pi$
$281$	−26.9706	−1.60893	−0.804464	+	0.594001i	$0.797548\pi$
$282$	−22.6274	−1.34744
$283$	−22.6274	−1.34506	−0.672530	−	0.740070i	$-0.734792\pi$
$283$	−22.6274	−1.34506	−0.672530	+	0.740070i	$0.734792\pi$
$284$	5.65685	0.335673
$285$	−8.00000	−0.473879
$286$	0	0
$287$	−4.82843	−0.285013
$288$	−5.00000	−0.294628
$289$	−3.62742	−0.213377
$290$	−8.82843	−0.518423
$291$	−13.6569	−0.800579
$292$	3.65685	0.214001
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	2.82843	0.164957
$295$	8.00000	0.465778
$296$	−8.82843	−0.513142
$297$	0	0
$298$	−18.4853	−1.07082
$299$	−5.65685	−0.327144
$300$	−2.82843	−0.163299
$301$	−1.65685	−0.0954995
$302$	5.17157	0.297591
$303$	28.2843	1.62489
$304$	−2.82843	−0.162221
$305$	13.3137	0.762341
$306$	−18.2843	−1.04524
$307$	−24.9706	−1.42515	−0.712573	−	0.701598i	$-0.752470\pi$
$307$	−24.9706	−1.42515	−0.712573	+	0.701598i	$0.752470\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	5.65685	0.321288
$311$	−32.9706	−1.86959	−0.934795	−	0.355189i	$-0.884417\pi$
$311$	−32.9706	−1.86959	−0.934795	+	0.355189i	$0.884417\pi$
$312$	−5.65685	−0.320256
$313$	−3.17157	−0.179268	−0.0896339	−	0.995975i	$-0.528570\pi$
$313$	−3.17157	−0.179268	−0.0896339	+	0.995975i	$0.528570\pi$
$314$	−4.34315	−0.245098
$315$	−5.00000	−0.281718
$316$	10.8284	0.609147
$317$	8.82843	0.495854	0.247927	−	0.968779i	$-0.420251\pi$
$317$	8.82843	0.495854	0.247927	+	0.968779i	$0.420251\pi$
$318$	18.3431	1.02863
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	27.3137	1.52450
$322$	−2.82843	−0.157622
$323$	−10.3431	−0.575508
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	−12.0000	−0.664619
$327$	−7.02944	−0.388729
$328$	4.82843	0.266605
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−4.97056	−0.273207	−0.136603	−	0.990626i	$-0.543619\pi$
$331$	−4.97056	−0.273207	−0.136603	+	0.990626i	$0.543619\pi$
$332$	13.6569	0.749517
$333$	44.1421	2.41897
$334$	−16.0000	−0.875481
$335$	7.31371	0.399591
$336$	−2.82843	−0.154303
$337$	−6.68629	−0.364226	−0.182113	−	0.983278i	$-0.558294\pi$
$337$	−6.68629	−0.364226	−0.182113	+	0.983278i	$0.558294\pi$
$338$	9.00000	0.489535
$339$	−5.65685	−0.307238
$340$	−3.65685	−0.198321
$341$	0	0
$342$	14.1421	0.764719
$343$	1.00000	0.0539949
$344$	1.65685	0.0893316
$345$	8.00000	0.430706
$346$	21.3137	1.14583
$347$	4.97056	0.266834	0.133417	−	0.991060i	$-0.457405\pi$
$347$	4.97056	0.266834	0.133417	+	0.991060i	$0.457405\pi$
$348$	24.9706	1.33856
$349$	6.97056	0.373126	0.186563	−	0.982443i	$-0.440265\pi$
$349$	6.97056	0.373126	0.186563	+	0.982443i	$0.440265\pi$
$350$	−1.00000	−0.0534522
$351$	11.3137	0.603881
$352$	0	0
$353$	−19.1716	−1.02040	−0.510200	−	0.860056i	$-0.670429\pi$
$353$	−19.1716	−1.02040	−0.510200	+	0.860056i	$0.670429\pi$
$354$	−22.6274	−1.20263
$355$	−5.65685	−0.300235
$356$	−0.343146	−0.0181867
$357$	−10.3431	−0.547417
$358$	−17.6569	−0.933194
$359$	17.4558	0.921284	0.460642	−	0.887586i	$-0.347619\pi$
$359$	17.4558	0.921284	0.460642	+	0.887586i	$0.347619\pi$
$360$	5.00000	0.263523
$361$	−11.0000	−0.578947
$362$	14.0000	0.735824
$363$	0	0
$364$	−2.00000	−0.104828
$365$	−3.65685	−0.191408
$366$	−37.6569	−1.96836
$367$	−28.2843	−1.47643	−0.738213	−	0.674567i	$-0.764330\pi$
$367$	−28.2843	−1.47643	−0.738213	+	0.674567i	$0.764330\pi$
$368$	2.82843	0.147442
$369$	−24.1421	−1.25679
$370$	8.82843	0.458968
$371$	6.48528	0.336699
$372$	−16.0000	−0.829561
$373$	7.65685	0.396457	0.198228	−	0.980156i	$-0.436481\pi$
$373$	7.65685	0.396457	0.198228	+	0.980156i	$0.436481\pi$
$374$	0	0
$375$	2.82843	0.146059
$376$	8.00000	0.412568
$377$	17.6569	0.909374
$378$	5.65685	0.290957
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	2.82843	0.145095
$381$	−9.37258	−0.480172
$382$	3.31371	0.169544
$383$	−18.3431	−0.937291	−0.468645	−	0.883386i	$-0.655258\pi$
$383$	−18.3431	−0.937291	−0.468645	+	0.883386i	$0.655258\pi$
$384$	2.82843	0.144338
$385$	0	0
$386$	−1.31371	−0.0668660
$387$	−8.28427	−0.421113
$388$	4.82843	0.245126
$389$	−2.97056	−0.150614	−0.0753068	−	0.997160i	$-0.523994\pi$
$389$	−2.97056	−0.150614	−0.0753068	+	0.997160i	$0.523994\pi$
$390$	5.65685	0.286446
$391$	10.3431	0.523075
$392$	−1.00000	−0.0505076
$393$	37.2548	1.87926
$394$	−23.6569	−1.19182
$395$	−10.8284	−0.544837
$396$	0	0
$397$	10.9706	0.550597	0.275298	−	0.961359i	$-0.411223\pi$
$397$	10.9706	0.550597	0.275298	+	0.961359i	$0.411223\pi$
$398$	−11.3137	−0.567105
$399$	8.00000	0.400501
$400$	1.00000	0.0500000
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	−20.6863	−1.03174
$403$	−11.3137	−0.563576
$404$	−10.0000	−0.497519
$405$	−1.00000	−0.0496904
$406$	8.82843	0.438147
$407$	0	0
$408$	10.3431	0.512062
$409$	−16.1421	−0.798177	−0.399089	−	0.916912i	$-0.630673\pi$
$409$	−16.1421	−0.798177	−0.399089	+	0.916912i	$0.630673\pi$
$410$	−4.82843	−0.238459
$411$	48.9706	2.41554
$412$	5.65685	0.278693
$413$	−8.00000	−0.393654
$414$	−14.1421	−0.695048
$415$	−13.6569	−0.670389
$416$	2.00000	0.0980581
$417$	37.2548	1.82438
$418$	0	0
$419$	14.6274	0.714596	0.357298	−	0.933990i	$-0.383698\pi$
$419$	14.6274	0.714596	0.357298	+	0.933990i	$0.383698\pi$
$420$	2.82843	0.138013
$421$	−16.6274	−0.810371	−0.405185	−	0.914235i	$-0.632793\pi$
$421$	−16.6274	−0.810371	−0.405185	+	0.914235i	$0.632793\pi$
$422$	−7.31371	−0.356026
$423$	−40.0000	−1.94487
$424$	−6.48528	−0.314953
$425$	3.65685	0.177383
$426$	16.0000	0.775203
$427$	−13.3137	−0.644296
$428$	−9.65685	−0.466782
$429$	0	0
$430$	−1.65685	−0.0799006
$431$	−19.7990	−0.953684	−0.476842	−	0.878989i	$-0.658219\pi$
$431$	−19.7990	−0.953684	−0.476842	+	0.878989i	$0.658219\pi$
$432$	−5.65685	−0.272166
$433$	15.1716	0.729099	0.364550	−	0.931184i	$-0.381223\pi$
$433$	15.1716	0.729099	0.364550	+	0.931184i	$0.381223\pi$
$434$	−5.65685	−0.271538
$435$	−24.9706	−1.19725
$436$	2.48528	0.119023
$437$	−8.00000	−0.382692
$438$	10.3431	0.494215
$439$	8.97056	0.428142	0.214071	−	0.976818i	$-0.431328\pi$
$439$	8.97056	0.428142	0.214071	+	0.976818i	$0.431328\pi$
$440$	0	0
$441$	5.00000	0.238095
$442$	7.31371	0.347878
$443$	−31.3137	−1.48776	−0.743880	−	0.668314i	$-0.767016\pi$
$443$	−31.3137	−1.48776	−0.743880	+	0.668314i	$0.767016\pi$
$444$	−24.9706	−1.18505
$445$	0.343146	0.0162667
$446$	−28.2843	−1.33930
$447$	−52.2843	−2.47296
$448$	1.00000	0.0472456
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	−5.00000	−0.235702
$451$	0	0
$452$	2.00000	0.0940721
$453$	14.6274	0.687256
$454$	−24.0000	−1.12638
$455$	2.00000	0.0937614
$456$	−8.00000	−0.374634
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	−21.3137	−0.995924
$459$	−20.6863	−0.965553
$460$	−2.82843	−0.131876
$461$	0.343146	0.0159819	0.00799095	−	0.999968i	$-0.497456\pi$
$461$	0.343146	0.0159819	0.00799095	+	0.999968i	$0.497456\pi$
$462$	0	0
$463$	41.4558	1.92662	0.963308	−	0.268398i	$-0.0864941\pi$
$463$	41.4558	1.92662	0.963308	+	0.268398i	$0.0864941\pi$
$464$	−8.82843	−0.409849
$465$	16.0000	0.741982
$466$	−6.00000	−0.277945
$467$	−27.7990	−1.28638	−0.643192	−	0.765705i	$-0.722390\pi$
$467$	−27.7990	−1.28638	−0.643192	+	0.765705i	$0.722390\pi$
$468$	−10.0000	−0.462250
$469$	−7.31371	−0.337716
$470$	−8.00000	−0.369012
$471$	−12.2843	−0.566029
$472$	8.00000	0.368230
$473$	0	0
$474$	30.6274	1.40676
$475$	−2.82843	−0.129777
$476$	3.65685	0.167612
$477$	32.4264	1.48470
$478$	−24.4853	−1.11993
$479$	−12.6863	−0.579651	−0.289826	−	0.957079i	$-0.593597\pi$
$479$	−12.6863	−0.579651	−0.289826	+	0.957079i	$0.593597\pi$
$480$	−2.82843	−0.129099
$481$	−17.6569	−0.805083
$482$	−3.17157	−0.144461
$483$	−8.00000	−0.364013
$484$	0	0
$485$	−4.82843	−0.219248
$486$	−14.1421	−0.641500
$487$	36.7696	1.66619	0.833094	−	0.553132i	$-0.186567\pi$
$487$	36.7696	1.66619	0.833094	+	0.553132i	$0.186567\pi$
$488$	13.3137	0.602683
$489$	−33.9411	−1.53487
$490$	1.00000	0.0451754
$491$	31.3137	1.41317	0.706584	−	0.707629i	$-0.250235\pi$
$491$	31.3137	1.41317	0.706584	+	0.707629i	$0.250235\pi$
$492$	13.6569	0.615699
$493$	−32.2843	−1.45401
$494$	−5.65685	−0.254514
$495$	0	0
$496$	5.65685	0.254000
$497$	5.65685	0.253745
$498$	38.6274	1.73094
$499$	−4.97056	−0.222513	−0.111256	−	0.993792i	$-0.535488\pi$
$499$	−4.97056	−0.222513	−0.111256	+	0.993792i	$0.535488\pi$
$500$	−1.00000	−0.0447214
$501$	−45.2548	−2.02184
$502$	−11.3137	−0.504956
$503$	−19.3137	−0.861156	−0.430578	−	0.902553i	$-0.641690\pi$
$503$	−19.3137	−0.861156	−0.430578	+	0.902553i	$0.641690\pi$
$504$	−5.00000	−0.222718
$505$	10.0000	0.444994
$506$	0	0
$507$	25.4558	1.13053
$508$	3.31371	0.147022
$509$	−33.3137	−1.47660	−0.738302	−	0.674470i	$-0.764372\pi$
$509$	−33.3137	−1.47660	−0.738302	+	0.674470i	$0.764372\pi$
$510$	−10.3431	−0.458002
$511$	3.65685	0.161770
$512$	−1.00000	−0.0441942
$513$	16.0000	0.706417
$514$	−28.8284	−1.27157
$515$	−5.65685	−0.249271
$516$	4.68629	0.206302
$517$	0	0
$518$	−8.82843	−0.387899
$519$	60.2843	2.64619
$520$	−2.00000	−0.0877058
$521$	−37.5980	−1.64720	−0.823599	−	0.567173i	$-0.808037\pi$
$521$	−37.5980	−1.64720	−0.823599	+	0.567173i	$0.808037\pi$
$522$	44.1421	1.93205
$523$	21.6569	0.946988	0.473494	−	0.880797i	$-0.342993\pi$
$523$	21.6569	0.946988	0.473494	+	0.880797i	$0.342993\pi$
$524$	−13.1716	−0.575403
$525$	−2.82843	−0.123443
$526$	30.6274	1.33542
$527$	20.6863	0.901109
$528$	0	0
$529$	−15.0000	−0.652174
$530$	6.48528	0.281703
$531$	−40.0000	−1.73585
$532$	−2.82843	−0.122628
$533$	9.65685	0.418285
$534$	−0.970563	−0.0420004
$535$	9.65685	0.417502
$536$	7.31371	0.315904
$537$	−49.9411	−2.15512
$538$	−2.00000	−0.0862261
$539$	0	0
$540$	5.65685	0.243432
$541$	−29.1127	−1.25165	−0.625826	−	0.779962i	$-0.715238\pi$
$541$	−29.1127	−1.25165	−0.625826	+	0.779962i	$0.715238\pi$
$542$	−16.9706	−0.728948
$543$	39.5980	1.69931
$544$	−3.65685	−0.156786
$545$	−2.48528	−0.106458
$546$	−5.65685	−0.242091
$547$	−0.686292	−0.0293437	−0.0146719	−	0.999892i	$-0.504670\pi$
$547$	−0.686292	−0.0293437	−0.0146719	+	0.999892i	$0.504670\pi$
$548$	−17.3137	−0.739605
$549$	−66.5685	−2.84108
$550$	0	0
$551$	24.9706	1.06378
$552$	8.00000	0.340503
$553$	10.8284	0.460472
$554$	9.31371	0.395702
$555$	24.9706	1.05994
$556$	−13.1716	−0.558599
$557$	−6.97056	−0.295352	−0.147676	−	0.989036i	$-0.547179\pi$
$557$	−6.97056	−0.295352	−0.147676	+	0.989036i	$0.547179\pi$
$558$	−28.2843	−1.19737
$559$	3.31371	0.140155
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	26.9706	1.13768
$563$	−22.6274	−0.953632	−0.476816	−	0.879003i	$-0.658209\pi$
$563$	−22.6274	−0.953632	−0.476816	+	0.879003i	$0.658209\pi$
$564$	22.6274	0.952786
$565$	−2.00000	−0.0841406
$566$	22.6274	0.951101
$567$	1.00000	0.0419961
$568$	−5.65685	−0.237356
$569$	28.6274	1.20012	0.600062	−	0.799954i	$-0.295143\pi$
$569$	28.6274	1.20012	0.600062	+	0.799954i	$0.295143\pi$
$570$	8.00000	0.335083
$571$	−43.5980	−1.82452	−0.912259	−	0.409613i	$-0.865664\pi$
$571$	−43.5980	−1.82452	−0.912259	+	0.409613i	$0.865664\pi$
$572$	0	0
$573$	9.37258	0.391545
$574$	4.82843	0.201535
$575$	2.82843	0.117954
$576$	5.00000	0.208333
$577$	−37.1127	−1.54502	−0.772511	−	0.635001i	$-0.780999\pi$
$577$	−37.1127	−1.54502	−0.772511	+	0.635001i	$0.780999\pi$
$578$	3.62742	0.150881
$579$	−3.71573	−0.154420
$580$	8.82843	0.366580
$581$	13.6569	0.566582
$582$	13.6569	0.566095
$583$	0	0
$584$	−3.65685	−0.151322
$585$	10.0000	0.413449
$586$	2.00000	0.0826192
$587$	−9.85786	−0.406878	−0.203439	−	0.979088i	$-0.565212\pi$
$587$	−9.85786	−0.406878	−0.203439	+	0.979088i	$0.565212\pi$
$588$	−2.82843	−0.116642
$589$	−16.0000	−0.659269
$590$	−8.00000	−0.329355
$591$	−66.9117	−2.75238
$592$	8.82843	0.362846
$593$	−31.6569	−1.29999	−0.649996	−	0.759938i	$-0.725229\pi$
$593$	−31.6569	−1.29999	−0.649996	+	0.759938i	$0.725229\pi$
$594$	0	0
$595$	−3.65685	−0.149916
$596$	18.4853	0.757187
$597$	−32.0000	−1.30967
$598$	5.65685	0.231326
$599$	10.3431	0.422609	0.211305	−	0.977420i	$-0.432229\pi$
$599$	10.3431	0.422609	0.211305	+	0.977420i	$0.432229\pi$
$600$	2.82843	0.115470
$601$	−7.17157	−0.292535	−0.146267	−	0.989245i	$-0.546726\pi$
$601$	−7.17157	−0.292535	−0.146267	+	0.989245i	$0.546726\pi$
$602$	1.65685	0.0675283
$603$	−36.5685	−1.48919
$604$	−5.17157	−0.210428
$605$	0	0
$606$	−28.2843	−1.14897
$607$	−18.3431	−0.744525	−0.372263	−	0.928127i	$-0.621418\pi$
$607$	−18.3431	−0.744525	−0.372263	+	0.928127i	$0.621418\pi$
$608$	2.82843	0.114708
$609$	24.9706	1.01186
$610$	−13.3137	−0.539056
$611$	16.0000	0.647291
$612$	18.2843	0.739098
$613$	10.9706	0.443097	0.221548	−	0.975149i	$-0.428889\pi$
$613$	10.9706	0.443097	0.221548	+	0.975149i	$0.428889\pi$
$614$	24.9706	1.00773
$615$	−13.6569	−0.550698
$616$	0	0
$617$	−43.6569	−1.75756	−0.878779	−	0.477228i	$-0.841642\pi$
$617$	−43.6569	−1.75756	−0.878779	+	0.477228i	$0.841642\pi$
$618$	16.0000	0.643614
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	−5.65685	−0.227185
$621$	−16.0000	−0.642058
$622$	32.9706	1.32200
$623$	−0.343146	−0.0137478
$624$	5.65685	0.226455
$625$	1.00000	0.0400000
$626$	3.17157	0.126762
$627$	0	0
$628$	4.34315	0.173310
$629$	32.2843	1.28726
$630$	5.00000	0.199205
$631$	22.6274	0.900783	0.450392	−	0.892831i	$-0.351284\pi$
$631$	22.6274	0.900783	0.450392	+	0.892831i	$0.351284\pi$
$632$	−10.8284	−0.430732
$633$	−20.6863	−0.822206
$634$	−8.82843	−0.350622
$635$	−3.31371	−0.131501
$636$	−18.3431	−0.727353
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	28.2843	1.11891
$640$	1.00000	0.0395285
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−27.3137	−1.07799
$643$	−14.1421	−0.557711	−0.278856	−	0.960333i	$-0.589955\pi$
$643$	−14.1421	−0.557711	−0.278856	+	0.960333i	$0.589955\pi$
$644$	2.82843	0.111456
$645$	−4.68629	−0.184523
$646$	10.3431	0.406946
$647$	−44.2843	−1.74099	−0.870497	−	0.492173i	$-0.836203\pi$
$647$	−44.2843	−1.74099	−0.870497	+	0.492173i	$0.836203\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	2.00000	0.0784465
$651$	−16.0000	−0.627089
$652$	12.0000	0.469956
$653$	−43.4558	−1.70056	−0.850279	−	0.526332i	$-0.823567\pi$
$653$	−43.4558	−1.70056	−0.850279	+	0.526332i	$0.823567\pi$
$654$	7.02944	0.274873
$655$	13.1716	0.514656
$656$	−4.82843	−0.188518
$657$	18.2843	0.713337
$658$	8.00000	0.311872
$659$	−1.65685	−0.0645419	−0.0322709	−	0.999479i	$-0.510274\pi$
$659$	−1.65685	−0.0645419	−0.0322709	+	0.999479i	$0.510274\pi$
$660$	0	0
$661$	−6.00000	−0.233373	−0.116686	−	0.993169i	$-0.537227\pi$
$661$	−6.00000	−0.233373	−0.116686	+	0.993169i	$0.537227\pi$
$662$	4.97056	0.193186
$663$	20.6863	0.803389
$664$	−13.6569	−0.529989
$665$	2.82843	0.109682
$666$	−44.1421	−1.71047
$667$	−24.9706	−0.966864
$668$	16.0000	0.619059
$669$	−80.0000	−3.09298
$670$	−7.31371	−0.282553
$671$	0	0
$672$	2.82843	0.109109
$673$	−8.62742	−0.332562	−0.166281	−	0.986078i	$-0.553176\pi$
$673$	−8.62742	−0.332562	−0.166281	+	0.986078i	$0.553176\pi$
$674$	6.68629	0.257546
$675$	−5.65685	−0.217732
$676$	−9.00000	−0.346154
$677$	14.0000	0.538064	0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000	0.538064	0.269032	+	0.963131i	$0.413296\pi$
$678$	5.65685	0.217250
$679$	4.82843	0.185298
$680$	3.65685	0.140234
$681$	−67.8823	−2.60125
$682$	0	0
$683$	33.6569	1.28784	0.643922	−	0.765091i	$-0.277306\pi$
$683$	33.6569	1.28784	0.643922	+	0.765091i	$0.277306\pi$
$684$	−14.1421	−0.540738
$685$	17.3137	0.661523
$686$	−1.00000	−0.0381802
$687$	−60.2843	−2.29999
$688$	−1.65685	−0.0631670
$689$	−12.9706	−0.494139
$690$	−8.00000	−0.304555
$691$	−32.9706	−1.25426	−0.627130	−	0.778915i	$-0.715770\pi$
$691$	−32.9706	−1.25426	−0.627130	+	0.778915i	$0.715770\pi$
$692$	−21.3137	−0.810226
$693$	0	0
$694$	−4.97056	−0.188680
$695$	13.1716	0.499626
$696$	−24.9706	−0.946507
$697$	−17.6569	−0.668801
$698$	−6.97056	−0.263840
$699$	−16.9706	−0.641886
$700$	1.00000	0.0377964
$701$	−35.1716	−1.32841	−0.664206	−	0.747550i	$-0.731230\pi$
$701$	−35.1716	−1.32841	−0.664206	+	0.747550i	$0.731230\pi$
$702$	−11.3137	−0.427008
$703$	−24.9706	−0.941783
$704$	0	0
$705$	−22.6274	−0.852198
$706$	19.1716	0.721532
$707$	−10.0000	−0.376089
$708$	22.6274	0.850390
$709$	−30.2843	−1.13735	−0.568675	−	0.822562i	$-0.692544\pi$
$709$	−30.2843	−1.13735	−0.568675	+	0.822562i	$0.692544\pi$
$710$	5.65685	0.212298
$711$	54.1421	2.03049
$712$	0.343146	0.0128599
$713$	16.0000	0.599205
$714$	10.3431	0.387083
$715$	0	0
$716$	17.6569	0.659868
$717$	−69.2548	−2.58637
$718$	−17.4558	−0.651446
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	−5.00000	−0.186339
$721$	5.65685	0.210672
$722$	11.0000	0.409378
$723$	−8.97056	−0.333619
$724$	−14.0000	−0.520306
$725$	−8.82843	−0.327880
$726$	0	0
$727$	27.3137	1.01301	0.506505	−	0.862237i	$-0.330937\pi$
$727$	27.3137	1.01301	0.506505	+	0.862237i	$0.330937\pi$
$728$	2.00000	0.0741249
$729$	−43.0000	−1.59259
$730$	3.65685	0.135346
$731$	−6.05887	−0.224096
$732$	37.6569	1.39184
$733$	46.0000	1.69905	0.849524	−	0.527549i	$-0.176889\pi$
$733$	46.0000	1.69905	0.849524	+	0.527549i	$0.176889\pi$
$734$	28.2843	1.04399
$735$	2.82843	0.104328
$736$	−2.82843	−0.104257
$737$	0	0
$738$	24.1421	0.888684
$739$	49.2548	1.81187	0.905934	−	0.423419i	$-0.139170\pi$
$739$	49.2548	1.81187	0.905934	+	0.423419i	$0.139170\pi$
$740$	−8.82843	−0.324539
$741$	−16.0000	−0.587775
$742$	−6.48528	−0.238082
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	16.0000	0.586588
$745$	−18.4853	−0.677248
$746$	−7.65685	−0.280337
$747$	68.2843	2.49839
$748$	0	0
$749$	−9.65685	−0.352854
$750$	−2.82843	−0.103280
$751$	−39.5980	−1.44495	−0.722475	−	0.691397i	$-0.756996\pi$
$751$	−39.5980	−1.44495	−0.722475	+	0.691397i	$0.756996\pi$
$752$	−8.00000	−0.291730
$753$	−32.0000	−1.16614
$754$	−17.6569	−0.643025
$755$	5.17157	0.188213
$756$	−5.65685	−0.205738
$757$	37.1127	1.34888	0.674442	−	0.738328i	$-0.264384\pi$
$757$	37.1127	1.34888	0.674442	+	0.738328i	$0.264384\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	−2.82843	−0.102598
$761$	−43.4558	−1.57527	−0.787637	−	0.616140i	$-0.788695\pi$
$761$	−43.4558	−1.57527	−0.787637	+	0.616140i	$0.788695\pi$
$762$	9.37258	0.339533
$763$	2.48528	0.0899732
$764$	−3.31371	−0.119886
$765$	−18.2843	−0.661069
$766$	18.3431	0.662765
$767$	16.0000	0.577727
$768$	−2.82843	−0.102062
$769$	−2.48528	−0.0896215	−0.0448108	−	0.998995i	$-0.514268\pi$
$769$	−2.48528	−0.0896215	−0.0448108	+	0.998995i	$0.514268\pi$
$770$	0	0
$771$	−81.5391	−2.93656
$772$	1.31371	0.0472814
$773$	−33.3137	−1.19821	−0.599105	−	0.800670i	$-0.704477\pi$
$773$	−33.3137	−1.19821	−0.599105	+	0.800670i	$0.704477\pi$
$774$	8.28427	0.297772
$775$	5.65685	0.203200
$776$	−4.82843	−0.173330
$777$	−24.9706	−0.895814
$778$	2.97056	0.106500
$779$	13.6569	0.489308
$780$	−5.65685	−0.202548
$781$	0	0
$782$	−10.3431	−0.369870
$783$	49.9411	1.78475
$784$	1.00000	0.0357143
$785$	−4.34315	−0.155014
$786$	−37.2548	−1.32884
$787$	14.6274	0.521411	0.260706	−	0.965418i	$-0.416045\pi$
$787$	14.6274	0.521411	0.260706	+	0.965418i	$0.416045\pi$
$788$	23.6569	0.842741
$789$	86.6274	3.08402
$790$	10.8284	0.385258
$791$	2.00000	0.0711118
$792$	0	0
$793$	26.6274	0.945567
$794$	−10.9706	−0.389331
$795$	18.3431	0.650564
$796$	11.3137	0.401004
$797$	−50.2843	−1.78116	−0.890580	−	0.454826i	$-0.849701\pi$
$797$	−50.2843	−1.78116	−0.890580	+	0.454826i	$0.849701\pi$
$798$	−8.00000	−0.283197
$799$	−29.2548	−1.03496
$800$	−1.00000	−0.0353553
$801$	−1.71573	−0.0606223
$802$	−10.0000	−0.353112
$803$	0	0
$804$	20.6863	0.729549
$805$	−2.82843	−0.0996890
$806$	11.3137	0.398508
$807$	−5.65685	−0.199131
$808$	10.0000	0.351799
$809$	34.2843	1.20537	0.602685	−	0.797979i	$-0.294097\pi$
$809$	34.2843	1.20537	0.602685	+	0.797979i	$0.294097\pi$
$810$	1.00000	0.0351364
$811$	6.14214	0.215680	0.107840	−	0.994168i	$-0.465607\pi$
$811$	6.14214	0.215680	0.107840	+	0.994168i	$0.465607\pi$
$812$	−8.82843	−0.309817
$813$	−48.0000	−1.68343
$814$	0	0
$815$	−12.0000	−0.420342
$816$	−10.3431	−0.362083
$817$	4.68629	0.163953
$818$	16.1421	0.564397
$819$	−10.0000	−0.349428
$820$	4.82843	0.168616
$821$	−9.79899	−0.341987	−0.170994	−	0.985272i	$-0.554698\pi$
$821$	−9.79899	−0.341987	−0.170994	+	0.985272i	$0.554698\pi$
$822$	−48.9706	−1.70804
$823$	26.8284	0.935180	0.467590	−	0.883945i	$-0.345122\pi$
$823$	26.8284	0.935180	0.467590	+	0.883945i	$0.345122\pi$
$824$	−5.65685	−0.197066
$825$	0	0
$826$	8.00000	0.278356
$827$	−41.2548	−1.43457	−0.717286	−	0.696779i	$-0.754616\pi$
$827$	−41.2548	−1.43457	−0.717286	+	0.696779i	$0.754616\pi$
$828$	14.1421	0.491473
$829$	−51.2548	−1.78015	−0.890077	−	0.455810i	$-0.849350\pi$
$829$	−51.2548	−1.78015	−0.890077	+	0.455810i	$0.849350\pi$
$830$	13.6569	0.474036
$831$	26.3431	0.913834
$832$	−2.00000	−0.0693375
$833$	3.65685	0.126702
$834$	−37.2548	−1.29003
$835$	−16.0000	−0.553703
$836$	0	0
$837$	−32.0000	−1.10608
$838$	−14.6274	−0.505296
$839$	−45.6569	−1.57625	−0.788125	−	0.615515i	$-0.788948\pi$
$839$	−45.6569	−1.57625	−0.788125	+	0.615515i	$0.788948\pi$
$840$	−2.82843	−0.0975900
$841$	48.9411	1.68763
$842$	16.6274	0.573019
$843$	76.2843	2.62737
$844$	7.31371	0.251748
$845$	9.00000	0.309609
$846$	40.0000	1.37523
$847$	0	0
$848$	6.48528	0.222705
$849$	64.0000	2.19647
$850$	−3.65685	−0.125429
$851$	24.9706	0.855980
$852$	−16.0000	−0.548151
$853$	2.68629	0.0919769	0.0459884	−	0.998942i	$-0.485356\pi$
$853$	2.68629	0.0919769	0.0459884	+	0.998942i	$0.485356\pi$
$854$	13.3137	0.455586
$855$	14.1421	0.483651
$856$	9.65685	0.330064
$857$	−26.9706	−0.921297	−0.460648	−	0.887583i	$-0.652383\pi$
$857$	−26.9706	−0.921297	−0.460648	+	0.887583i	$0.652383\pi$
$858$	0	0
$859$	27.3137	0.931932	0.465966	−	0.884803i	$-0.345707\pi$
$859$	27.3137	0.931932	0.465966	+	0.884803i	$0.345707\pi$
$860$	1.65685	0.0564983
$861$	13.6569	0.465424
$862$	19.7990	0.674356
$863$	36.7696	1.25165	0.625825	−	0.779963i	$-0.284762\pi$
$863$	36.7696	1.25165	0.625825	+	0.779963i	$0.284762\pi$
$864$	5.65685	0.192450
$865$	21.3137	0.724688
$866$	−15.1716	−0.515551
$867$	10.2599	0.348444
$868$	5.65685	0.192006
$869$	0	0
$870$	24.9706	0.846581
$871$	14.6274	0.495631
$872$	−2.48528	−0.0841622
$873$	24.1421	0.817087
$874$	8.00000	0.270604
$875$	−1.00000	−0.0338062
$876$	−10.3431	−0.349463
$877$	−30.9706	−1.04580	−0.522901	−	0.852394i	$-0.675150\pi$
$877$	−30.9706	−1.04580	−0.522901	+	0.852394i	$0.675150\pi$
$878$	−8.97056	−0.302742
$879$	5.65685	0.190801
$880$	0	0
$881$	−25.3137	−0.852841	−0.426420	−	0.904525i	$-0.640226\pi$
$881$	−25.3137	−0.852841	−0.426420	+	0.904525i	$0.640226\pi$
$882$	−5.00000	−0.168359
$883$	30.3431	1.02113	0.510564	−	0.859840i	$-0.329437\pi$
$883$	30.3431	1.02113	0.510564	+	0.859840i	$0.329437\pi$
$884$	−7.31371	−0.245987
$885$	−22.6274	−0.760612
$886$	31.3137	1.05200
$887$	−22.6274	−0.759754	−0.379877	−	0.925037i	$-0.624034\pi$
$887$	−22.6274	−0.759754	−0.379877	+	0.925037i	$0.624034\pi$
$888$	24.9706	0.837957
$889$	3.31371	0.111138
$890$	−0.343146	−0.0115023
$891$	0	0
$892$	28.2843	0.947027
$893$	22.6274	0.757198
$894$	52.2843	1.74865
$895$	−17.6569	−0.590204
$896$	−1.00000	−0.0334077
$897$	16.0000	0.534224
$898$	6.00000	0.200223
$899$	−49.9411	−1.66563
$900$	5.00000	0.166667
$901$	23.7157	0.790085
$902$	0	0
$903$	4.68629	0.155950
$904$	−2.00000	−0.0665190
$905$	14.0000	0.465376
$906$	−14.6274	−0.485963
$907$	6.34315	0.210621	0.105310	−	0.994439i	$-0.466416\pi$
$907$	6.34315	0.210621	0.105310	+	0.994439i	$0.466416\pi$
$908$	24.0000	0.796468
$909$	−50.0000	−1.65840
$910$	−2.00000	−0.0662994
$911$	−10.3431	−0.342684	−0.171342	−	0.985212i	$-0.554810\pi$
$911$	−10.3431	−0.342684	−0.171342	+	0.985212i	$0.554810\pi$
$912$	8.00000	0.264906
$913$	0	0
$914$	−22.0000	−0.727695
$915$	−37.6569	−1.24490
$916$	21.3137	0.704225
$917$	−13.1716	−0.434964
$918$	20.6863	0.682749
$919$	52.7696	1.74071	0.870353	−	0.492428i	$-0.163890\pi$
$919$	52.7696	1.74071	0.870353	+	0.492428i	$0.163890\pi$
$920$	2.82843	0.0932505
$921$	70.6274	2.32725
$922$	−0.343146	−0.0113009
$923$	−11.3137	−0.372395
$924$	0	0
$925$	8.82843	0.290277
$926$	−41.4558	−1.36232
$927$	28.2843	0.928977
$928$	8.82843	0.289807
$929$	−5.02944	−0.165010	−0.0825052	−	0.996591i	$-0.526292\pi$
$929$	−5.02944	−0.165010	−0.0825052	+	0.996591i	$0.526292\pi$
$930$	−16.0000	−0.524661
$931$	−2.82843	−0.0926980
$932$	6.00000	0.196537
$933$	93.2548	3.05303
$934$	27.7990	0.909611
$935$	0	0
$936$	10.0000	0.326860
$937$	20.6274	0.673868	0.336934	−	0.941528i	$-0.390610\pi$
$937$	20.6274	0.673868	0.336934	+	0.941528i	$0.390610\pi$
$938$	7.31371	0.238801
$939$	8.97056	0.292743
$940$	8.00000	0.260931
$941$	9.31371	0.303618	0.151809	−	0.988410i	$-0.451490\pi$
$941$	9.31371	0.303618	0.151809	+	0.988410i	$0.451490\pi$
$942$	12.2843	0.400243
$943$	−13.6569	−0.444728
$944$	−8.00000	−0.260378
$945$	5.65685	0.184017
$946$	0	0
$947$	−43.5980	−1.41674	−0.708372	−	0.705839i	$-0.750570\pi$
$947$	−43.5980	−1.41674	−0.708372	+	0.705839i	$0.750570\pi$
$948$	−30.6274	−0.994732
$949$	−7.31371	−0.237413
$950$	2.82843	0.0917663
$951$	−24.9706	−0.809726
$952$	−3.65685	−0.118519
$953$	−47.2548	−1.53073	−0.765367	−	0.643594i	$-0.777443\pi$
$953$	−47.2548	−1.53073	−0.765367	+	0.643594i	$0.777443\pi$
$954$	−32.4264	−1.04984
$955$	3.31371	0.107229
$956$	24.4853	0.791911
$957$	0	0
$958$	12.6863	0.409875
$959$	−17.3137	−0.559089
$960$	2.82843	0.0912871
$961$	1.00000	0.0322581
$962$	17.6569	0.569280
$963$	−48.2843	−1.55594
$964$	3.17157	0.102149
$965$	−1.31371	−0.0422898
$966$	8.00000	0.257396
$967$	−36.6863	−1.17975	−0.589876	−	0.807494i	$-0.700823\pi$
$967$	−36.6863	−1.17975	−0.589876	+	0.807494i	$0.700823\pi$
$968$	0	0
$969$	29.2548	0.939801
$970$	4.82843	0.155031
$971$	25.9411	0.832490	0.416245	−	0.909252i	$-0.363346\pi$
$971$	25.9411	0.832490	0.416245	+	0.909252i	$0.363346\pi$
$972$	14.1421	0.453609
$973$	−13.1716	−0.422261
$974$	−36.7696	−1.17817
$975$	5.65685	0.181164
$976$	−13.3137	−0.426161
$977$	24.6274	0.787901	0.393950	−	0.919132i	$-0.371108\pi$
$977$	24.6274	0.787901	0.393950	+	0.919132i	$0.371108\pi$
$978$	33.9411	1.08532
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	12.4264	0.396745
$982$	−31.3137	−0.999261
$983$	−39.5980	−1.26298	−0.631490	−	0.775384i	$-0.717556\pi$
$983$	−39.5980	−1.26298	−0.631490	+	0.775384i	$0.717556\pi$
$984$	−13.6569	−0.435365
$985$	−23.6569	−0.753770
$986$	32.2843	1.02814
$987$	22.6274	0.720239
$988$	5.65685	0.179969
$989$	−4.68629	−0.149015
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	−5.65685	−0.179605
$993$	14.0589	0.446145
$994$	−5.65685	−0.179425
$995$	−11.3137	−0.358669
$996$	−38.6274	−1.22396
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	4.97056	0.157340
$999$	−49.9411	−1.58007

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.bo.1.1		2
11.10	odd	2		770.2.a.i.1.1	✓	2
33.32	even	2		6930.2.a.br.1.2		2
44.43	even	2		6160.2.a.ba.1.2		2
55.32	even	4		3850.2.c.u.1849.4		4
55.43	even	4		3850.2.c.u.1849.1		4
55.54	odd	2		3850.2.a.bi.1.2		2
77.76	even	2		5390.2.a.bt.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.i.1.1	✓	2	11.10	odd	2
3850.2.a.bi.1.2		2	55.54	odd	2
3850.2.c.u.1849.1		4	55.43	even	4
3850.2.c.u.1849.4		4	55.32	even	4
5390.2.a.bt.1.2		2	77.76	even	2
6160.2.a.ba.1.2		2	44.43	even	2
6930.2.a.br.1.2		2	33.32	even	2
8470.2.a.bo.1.1		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.82843 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.82843 q^{6} +1.00000 q^{7} -1.00000 q^{8} +5.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.82843 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.82843 q^{6} +1.00000 q^{7} -1.00000 q^{8} +5.00000 q^{9} +1.00000 q^{10} -2.82843 q^{12} -2.00000 q^{13} -1.00000 q^{14} +2.82843 q^{15} +1.00000 q^{16} +3.65685 q^{17} -5.00000 q^{18} -2.82843 q^{19} -1.00000 q^{20} -2.82843 q^{21} +2.82843 q^{23} +2.82843 q^{24} +1.00000 q^{25} +2.00000 q^{26} -5.65685 q^{27} +1.00000 q^{28} -8.82843 q^{29} -2.82843 q^{30} +5.65685 q^{31} -1.00000 q^{32} -3.65685 q^{34} -1.00000 q^{35} +5.00000 q^{36} +8.82843 q^{37} +2.82843 q^{38} +5.65685 q^{39} +1.00000 q^{40} -4.82843 q^{41} +2.82843 q^{42} -1.65685 q^{43} -5.00000 q^{45} -2.82843 q^{46} -8.00000 q^{47} -2.82843 q^{48} +1.00000 q^{49} -1.00000 q^{50} -10.3431 q^{51} -2.00000 q^{52} +6.48528 q^{53} +5.65685 q^{54} -1.00000 q^{56} +8.00000 q^{57} +8.82843 q^{58} -8.00000 q^{59} +2.82843 q^{60} -13.3137 q^{61} -5.65685 q^{62} +5.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -7.31371 q^{67} +3.65685 q^{68} -8.00000 q^{69} +1.00000 q^{70} +5.65685 q^{71} -5.00000 q^{72} +3.65685 q^{73} -8.82843 q^{74} -2.82843 q^{75} -2.82843 q^{76} -5.65685 q^{78} +10.8284 q^{79} -1.00000 q^{80} +1.00000 q^{81} +4.82843 q^{82} +13.6569 q^{83} -2.82843 q^{84} -3.65685 q^{85} +1.65685 q^{86} +24.9706 q^{87} -0.343146 q^{89} +5.00000 q^{90} -2.00000 q^{91} +2.82843 q^{92} -16.0000 q^{93} +8.00000 q^{94} +2.82843 q^{95} +2.82843 q^{96} +4.82843 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 10 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8 + 10 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} - 2 q^{8} + 10 q^{9} + 2 q^{10} - 4 q^{13} - 2 q^{14} + 2 q^{16} - 4 q^{17} - 10 q^{18} - 2 q^{20} + 2 q^{25} + 4 q^{26} + 2 q^{28} - 12 q^{29} - 2 q^{32} + 4 q^{34} - 2 q^{35} + 10 q^{36} + 12 q^{37} + 2 q^{40} - 4 q^{41} + 8 q^{43} - 10 q^{45} - 16 q^{47} + 2 q^{49} - 2 q^{50} - 32 q^{51} - 4 q^{52} - 4 q^{53} - 2 q^{56} + 16 q^{57} + 12 q^{58} - 16 q^{59} - 4 q^{61} + 10 q^{63} + 2 q^{64} + 4 q^{65} + 8 q^{67} - 4 q^{68} - 16 q^{69} + 2 q^{70} - 10 q^{72} - 4 q^{73} - 12 q^{74} + 16 q^{79} - 2 q^{80} + 2 q^{81} + 4 q^{82} + 16 q^{83} + 4 q^{85} - 8 q^{86} + 16 q^{87} - 12 q^{89} + 10 q^{90} - 4 q^{91} - 32 q^{93} + 16 q^{94} + 4 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 - 2 * q^8 + 10 * q^9 + 2 * q^10 - 4 * q^13 - 2 * q^14 + 2 * q^16 - 4 * q^17 - 10 * q^18 - 2 * q^20 + 2 * q^25 + 4 * q^26 + 2 * q^28 - 12 * q^29 - 2 * q^32 + 4 * q^34 - 2 * q^35 + 10 * q^36 + 12 * q^37 + 2 * q^40 - 4 * q^41 + 8 * q^43 - 10 * q^45 - 16 * q^47 + 2 * q^49 - 2 * q^50 - 32 * q^51 - 4 * q^52 - 4 * q^53 - 2 * q^56 + 16 * q^57 + 12 * q^58 - 16 * q^59 - 4 * q^61 + 10 * q^63 + 2 * q^64 + 4 * q^65 + 8 * q^67 - 4 * q^68 - 16 * q^69 + 2 * q^70 - 10 * q^72 - 4 * q^73 - 12 * q^74 + 16 * q^79 - 2 * q^80 + 2 * q^81 + 4 * q^82 + 16 * q^83 + 4 * q^85 - 8 * q^86 + 16 * q^87 - 12 * q^89 + 10 * q^90 - 4 * q^91 - 32 * q^93 + 16 * q^94 + 4 * q^97 - 2 * q^98

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