LMFDB - Embedded newform 8470.2.a.dh.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$67.6332905120$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 16x^{6} + 69x^{4} - 10x^{3} - 70x^{2} + 10x + 5 $ x^8 - 16x^6 + 69x^4 - 10x^3 - 70x^2 + 10*x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.00431$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +2.00431 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00431 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.01724 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.00431 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00431 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.01724 q^{9} -1.00000 q^{10} +2.00431 q^{12} -4.02930 q^{13} -1.00000 q^{14} -2.00431 q^{15} +1.00000 q^{16} +0.300566 q^{17} +1.01724 q^{18} +7.73531 q^{19} -1.00000 q^{20} -2.00431 q^{21} -6.48607 q^{23} +2.00431 q^{24} +1.00000 q^{25} -4.02930 q^{26} -3.97405 q^{27} -1.00000 q^{28} +1.81673 q^{29} -2.00431 q^{30} +7.53480 q^{31} +1.00000 q^{32} +0.300566 q^{34} +1.00000 q^{35} +1.01724 q^{36} +9.11210 q^{37} +7.73531 q^{38} -8.07596 q^{39} -1.00000 q^{40} +9.68330 q^{41} -2.00431 q^{42} +3.42569 q^{43} -1.01724 q^{45} -6.48607 q^{46} -6.21202 q^{47} +2.00431 q^{48} +1.00000 q^{49} +1.00000 q^{50} +0.602426 q^{51} -4.02930 q^{52} +6.39502 q^{53} -3.97405 q^{54} -1.00000 q^{56} +15.5039 q^{57} +1.81673 q^{58} +5.51945 q^{59} -2.00431 q^{60} -3.62844 q^{61} +7.53480 q^{62} -1.01724 q^{63} +1.00000 q^{64} +4.02930 q^{65} +0.505448 q^{67} +0.300566 q^{68} -13.0001 q^{69} +1.00000 q^{70} +8.37915 q^{71} +1.01724 q^{72} +1.40274 q^{73} +9.11210 q^{74} +2.00431 q^{75} +7.73531 q^{76} -8.07596 q^{78} +6.20005 q^{79} -1.00000 q^{80} -11.0169 q^{81} +9.68330 q^{82} +14.5768 q^{83} -2.00431 q^{84} -0.300566 q^{85} +3.42569 q^{86} +3.64127 q^{87} +1.92815 q^{89} -1.01724 q^{90} +4.02930 q^{91} -6.48607 q^{92} +15.1020 q^{93} -6.21202 q^{94} -7.73531 q^{95} +2.00431 q^{96} -16.4944 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} - 8 q^{5} - 8 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 + 8 * q^4 - 8 * q^5 - 8 * q^7 + 8 * q^8 + 8 * q^9	$ 8 q + 8 q^{2} + 8 q^{4} - 8 q^{5} - 8 q^{7} + 8 q^{8} + 8 q^{9} - 8 q^{10} - q^{13} - 8 q^{14} + 8 q^{16} + 6 q^{17} + 8 q^{18} + 5 q^{19} - 8 q^{20} + 10 q^{23} + 8 q^{25} - q^{26} - 8 q^{28} + 3 q^{29} - 8 q^{31} + 8 q^{32} + 6 q^{34} + 8 q^{35} + 8 q^{36} - 6 q^{37} + 5 q^{38} + 35 q^{39} - 8 q^{40} + 11 q^{41} - 5 q^{43} - 8 q^{45} + 10 q^{46} - 15 q^{47} + 8 q^{49} + 8 q^{50} - 6 q^{51} - q^{52} - 16 q^{53} - 8 q^{56} + 38 q^{57} + 3 q^{58} - 9 q^{59} + 32 q^{61} - 8 q^{62} - 8 q^{63} + 8 q^{64} + q^{65} + 33 q^{67} + 6 q^{68} - 22 q^{69} + 8 q^{70} + 11 q^{71} + 8 q^{72} - 34 q^{73} - 6 q^{74} + 5 q^{76} + 35 q^{78} + 31 q^{79} - 8 q^{80} + 20 q^{81} + 11 q^{82} + 50 q^{83} - 6 q^{85} - 5 q^{86} - 12 q^{87} + q^{89} - 8 q^{90} + q^{91} + 10 q^{92} + 26 q^{93} - 15 q^{94} - 5 q^{95} - 4 q^{97} + 8 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 - 8 * q^5 - 8 * q^7 + 8 * q^8 + 8 * q^9 - 8 * q^10 - q^13 - 8 * q^14 + 8 * q^16 + 6 * q^17 + 8 * q^18 + 5 * q^19 - 8 * q^20 + 10 * q^23 + 8 * q^25 - q^26 - 8 * q^28 + 3 * q^29 - 8 * q^31 + 8 * q^32 + 6 * q^34 + 8 * q^35 + 8 * q^36 - 6 * q^37 + 5 * q^38 + 35 * q^39 - 8 * q^40 + 11 * q^41 - 5 * q^43 - 8 * q^45 + 10 * q^46 - 15 * q^47 + 8 * q^49 + 8 * q^50 - 6 * q^51 - q^52 - 16 * q^53 - 8 * q^56 + 38 * q^57 + 3 * q^58 - 9 * q^59 + 32 * q^61 - 8 * q^62 - 8 * q^63 + 8 * q^64 + q^65 + 33 * q^67 + 6 * q^68 - 22 * q^69 + 8 * q^70 + 11 * q^71 + 8 * q^72 - 34 * q^73 - 6 * q^74 + 5 * q^76 + 35 * q^78 + 31 * q^79 - 8 * q^80 + 20 * q^81 + 11 * q^82 + 50 * q^83 - 6 * q^85 - 5 * q^86 - 12 * q^87 + q^89 - 8 * q^90 + q^91 + 10 * q^92 + 26 * q^93 - 15 * q^94 - 5 * q^95 - 4 * q^97 + 8 * 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
$3$	2.00431	1.15719	0.578593	−	0.815616i	$-0.303602\pi$
$3$	2.00431	1.15719	0.578593	+	0.815616i	$0.303602\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.00431	0.818254
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	1.01724	0.339080
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	2.00431	0.578593
$13$	−4.02930	−1.11753	−0.558764	−	0.829327i	$-0.688724\pi$
$13$	−4.02930	−1.11753	−0.558764	+	0.829327i	$0.688724\pi$
$14$	−1.00000	−0.267261
$15$	−2.00431	−0.517509
$16$	1.00000	0.250000
$17$	0.300566	0.0728979	0.0364490	−	0.999336i	$-0.488395\pi$
$17$	0.300566	0.0728979	0.0364490	+	0.999336i	$0.488395\pi$
$18$	1.01724	0.239766
$19$	7.73531	1.77460	0.887301	−	0.461191i	$-0.152578\pi$
$19$	7.73531	1.77460	0.887301	+	0.461191i	$0.152578\pi$
$20$	−1.00000	−0.223607
$21$	−2.00431	−0.437375
$22$	0	0
$23$	−6.48607	−1.35244	−0.676219	−	0.736700i	$-0.736383\pi$
$23$	−6.48607	−1.35244	−0.676219	+	0.736700i	$0.736383\pi$
$24$	2.00431	0.409127
$25$	1.00000	0.200000
$26$	−4.02930	−0.790211
$27$	−3.97405	−0.764807
$28$	−1.00000	−0.188982
$29$	1.81673	0.337358	0.168679	−	0.985671i	$-0.446050\pi$
$29$	1.81673	0.337358	0.168679	+	0.985671i	$0.446050\pi$
$30$	−2.00431	−0.365934
$31$	7.53480	1.35329	0.676645	−	0.736310i	$-0.263433\pi$
$31$	7.53480	1.35329	0.676645	+	0.736310i	$0.263433\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.300566	0.0515466
$35$	1.00000	0.169031
$36$	1.01724	0.169540
$37$	9.11210	1.49802	0.749010	−	0.662559i	$-0.230530\pi$
$37$	9.11210	1.49802	0.749010	+	0.662559i	$0.230530\pi$
$38$	7.73531	1.25483
$39$	−8.07596	−1.29319
$40$	−1.00000	−0.158114
$41$	9.68330	1.51228	0.756139	−	0.654411i	$-0.227083\pi$
$41$	9.68330	1.51228	0.756139	+	0.654411i	$0.227083\pi$
$42$	−2.00431	−0.309271
$43$	3.42569	0.522412	0.261206	−	0.965283i	$-0.415880\pi$
$43$	3.42569	0.522412	0.261206	+	0.965283i	$0.415880\pi$
$44$	0	0
$45$	−1.01724	−0.151641
$46$	−6.48607	−0.956319
$47$	−6.21202	−0.906116	−0.453058	−	0.891481i	$-0.649667\pi$
$47$	−6.21202	−0.906116	−0.453058	+	0.891481i	$0.649667\pi$
$48$	2.00431	0.289297
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0.602426	0.0843565
$52$	−4.02930	−0.558764
$53$	6.39502	0.878423	0.439212	−	0.898384i	$-0.355258\pi$
$53$	6.39502	0.878423	0.439212	+	0.898384i	$0.355258\pi$
$54$	−3.97405	−0.540800
$55$	0	0
$56$	−1.00000	−0.133631
$57$	15.5039	2.05355
$58$	1.81673	0.238548
$59$	5.51945	0.718571	0.359286	−	0.933228i	$-0.383020\pi$
$59$	5.51945	0.718571	0.359286	+	0.933228i	$0.383020\pi$
$60$	−2.00431	−0.258755
$61$	−3.62844	−0.464574	−0.232287	−	0.972647i	$-0.574621\pi$
$61$	−3.62844	−0.464574	−0.232287	+	0.972647i	$0.574621\pi$
$62$	7.53480	0.956920
$63$	−1.01724	−0.128160
$64$	1.00000	0.125000
$65$	4.02930	0.499774
$66$	0	0
$67$	0.505448	0.0617503	0.0308751	−	0.999523i	$-0.490171\pi$
$67$	0.505448	0.0617503	0.0308751	+	0.999523i	$0.490171\pi$
$68$	0.300566	0.0364490
$69$	−13.0001	−1.56502
$70$	1.00000	0.119523
$71$	8.37915	0.994422	0.497211	−	0.867630i	$-0.334357\pi$
$71$	8.37915	0.994422	0.497211	+	0.867630i	$0.334357\pi$
$72$	1.01724	0.119883
$73$	1.40274	0.164178	0.0820892	−	0.996625i	$-0.473841\pi$
$73$	1.40274	0.164178	0.0820892	+	0.996625i	$0.473841\pi$
$74$	9.11210	1.05926
$75$	2.00431	0.231437
$76$	7.73531	0.887301
$77$	0	0
$78$	−8.07596	−0.914422
$79$	6.20005	0.697560	0.348780	−	0.937205i	$-0.386596\pi$
$79$	6.20005	0.697560	0.348780	+	0.937205i	$0.386596\pi$
$80$	−1.00000	−0.111803
$81$	−11.0169	−1.22410
$82$	9.68330	1.06934
$83$	14.5768	1.60001	0.800006	−	0.599992i	$-0.204830\pi$
$83$	14.5768	1.60001	0.800006	+	0.599992i	$0.204830\pi$
$84$	−2.00431	−0.218688
$85$	−0.300566	−0.0326009
$86$	3.42569	0.369401
$87$	3.64127	0.390386
$88$	0	0
$89$	1.92815	0.204384	0.102192	−	0.994765i	$-0.467414\pi$
$89$	1.92815	0.204384	0.102192	+	0.994765i	$0.467414\pi$
$90$	−1.01724	−0.107227
$91$	4.02930	0.422386
$92$	−6.48607	−0.676219
$93$	15.1020	1.56601
$94$	−6.21202	−0.640721
$95$	−7.73531	−0.793626
$96$	2.00431	0.204564
$97$	−16.4944	−1.67475	−0.837374	−	0.546630i	$-0.815910\pi$
$97$	−16.4944	−1.67475	−0.837374	+	0.546630i	$0.815910\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	9.44404	0.939717	0.469859	−	0.882742i	$-0.344305\pi$
$101$	9.44404	0.939717	0.469859	+	0.882742i	$0.344305\pi$
$102$	0.602426	0.0596491
$103$	−3.51983	−0.346820	−0.173410	−	0.984850i	$-0.555479\pi$
$103$	−3.51983	−0.346820	−0.173410	+	0.984850i	$0.555479\pi$
$104$	−4.02930	−0.395106
$105$	2.00431	0.195600
$106$	6.39502	0.621139
$107$	10.4251	1.00783	0.503917	−	0.863752i	$-0.331892\pi$
$107$	10.4251	1.00783	0.503917	+	0.863752i	$0.331892\pi$
$108$	−3.97405	−0.382404
$109$	12.1571	1.16444	0.582221	−	0.813030i	$-0.302184\pi$
$109$	12.1571	1.16444	0.582221	+	0.813030i	$0.302184\pi$
$110$	0	0
$111$	18.2634	1.73349
$112$	−1.00000	−0.0944911
$113$	−6.67216	−0.627664	−0.313832	−	0.949478i	$-0.601613\pi$
$113$	−6.67216	−0.627664	−0.313832	+	0.949478i	$0.601613\pi$
$114$	15.5039	1.45208
$115$	6.48607	0.604829
$116$	1.81673	0.168679
$117$	−4.09877	−0.378932
$118$	5.51945	0.508106
$119$	−0.300566	−0.0275528
$120$	−2.00431	−0.182967
$121$	0	0
$122$	−3.62844	−0.328503
$123$	19.4083	1.74999
$124$	7.53480	0.676645
$125$	−1.00000	−0.0894427
$126$	−1.01724	−0.0906231
$127$	−12.8919	−1.14397	−0.571984	−	0.820265i	$-0.693826\pi$
$127$	−12.8919	−1.14397	−0.571984	+	0.820265i	$0.693826\pi$
$128$	1.00000	0.0883883
$129$	6.86612	0.604528
$130$	4.02930	0.353393
$131$	20.3862	1.78115	0.890577	−	0.454833i	$-0.150301\pi$
$131$	20.3862	1.78115	0.890577	+	0.454833i	$0.150301\pi$
$132$	0	0
$133$	−7.73531	−0.670737
$134$	0.505448	0.0436640
$135$	3.97405	0.342032
$136$	0.300566	0.0257733
$137$	−17.6954	−1.51182	−0.755911	−	0.654674i	$-0.772806\pi$
$137$	−17.6954	−1.51182	−0.755911	+	0.654674i	$0.772806\pi$
$138$	−13.0001	−1.10664
$139$	13.4774	1.14314	0.571570	−	0.820553i	$-0.306335\pi$
$139$	13.4774	1.14314	0.571570	+	0.820553i	$0.306335\pi$
$140$	1.00000	0.0845154
$141$	−12.4508	−1.04855
$142$	8.37915	0.703163
$143$	0	0
$144$	1.01724	0.0847701
$145$	−1.81673	−0.150871
$146$	1.40274	0.116092
$147$	2.00431	0.165312
$148$	9.11210	0.749010
$149$	−1.72417	−0.141249	−0.0706246	−	0.997503i	$-0.522499\pi$
$149$	−1.72417	−0.141249	−0.0706246	+	0.997503i	$0.522499\pi$
$150$	2.00431	0.163651
$151$	−9.14895	−0.744531	−0.372266	−	0.928126i	$-0.621419\pi$
$151$	−9.14895	−0.744531	−0.372266	+	0.928126i	$0.621419\pi$
$152$	7.73531	0.627417
$153$	0.305748	0.0247183
$154$	0	0
$155$	−7.53480	−0.605209
$156$	−8.07596	−0.646594
$157$	−0.876025	−0.0699144	−0.0349572	−	0.999389i	$-0.511129\pi$
$157$	−0.876025	−0.0699144	−0.0349572	+	0.999389i	$0.511129\pi$
$158$	6.20005	0.493250
$159$	12.8176	1.01650
$160$	−1.00000	−0.0790569
$161$	6.48607	0.511174
$162$	−11.0169	−0.865573
$163$	−6.49269	−0.508547	−0.254274	−	0.967132i	$-0.581836\pi$
$163$	−6.49269	−0.508547	−0.254274	+	0.967132i	$0.581836\pi$
$164$	9.68330	0.756139
$165$	0	0
$166$	14.5768	1.13138
$167$	−20.0322	−1.55014	−0.775070	−	0.631875i	$-0.782286\pi$
$167$	−20.0322	−1.55014	−0.775070	+	0.631875i	$0.782286\pi$
$168$	−2.00431	−0.154636
$169$	3.23529	0.248868
$170$	−0.300566	−0.0230524
$171$	7.86868	0.601733
$172$	3.42569	0.261206
$173$	1.52088	0.115630	0.0578151	−	0.998327i	$-0.481587\pi$
$173$	1.52088	0.115630	0.0578151	+	0.998327i	$0.481587\pi$
$174$	3.64127	0.276044
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	11.0627	0.831521
$178$	1.92815	0.144521
$179$	24.1512	1.80515	0.902573	−	0.430537i	$-0.141676\pi$
$179$	24.1512	1.80515	0.902573	+	0.430537i	$0.141676\pi$
$180$	−1.01724	−0.0758207
$181$	16.6623	1.23850	0.619249	−	0.785195i	$-0.287437\pi$
$181$	16.6623	1.23850	0.619249	+	0.785195i	$0.287437\pi$
$182$	4.02930	0.298672
$183$	−7.27250	−0.537598
$184$	−6.48607	−0.478159
$185$	−9.11210	−0.669935
$186$	15.1020	1.10733
$187$	0	0
$188$	−6.21202	−0.453058
$189$	3.97405	0.289070
$190$	−7.73531	−0.561178
$191$	−2.97548	−0.215298	−0.107649	−	0.994189i	$-0.534332\pi$
$191$	−2.97548	−0.215298	−0.107649	+	0.994189i	$0.534332\pi$
$192$	2.00431	0.144648
$193$	−11.5006	−0.827831	−0.413916	−	0.910315i	$-0.635839\pi$
$193$	−11.5006	−0.827831	−0.413916	+	0.910315i	$0.635839\pi$
$194$	−16.4944	−1.18423
$195$	8.07596	0.578331
$196$	1.00000	0.0714286
$197$	16.0597	1.14420	0.572101	−	0.820183i	$-0.306128\pi$
$197$	16.0597	1.14420	0.572101	+	0.820183i	$0.306128\pi$
$198$	0	0
$199$	11.0899	0.786140	0.393070	−	0.919509i	$-0.371413\pi$
$199$	11.0899	0.786140	0.393070	+	0.919509i	$0.371413\pi$
$200$	1.00000	0.0707107
$201$	1.01307	0.0714566
$202$	9.44404	0.664480
$203$	−1.81673	−0.127509
$204$	0.602426	0.0421783
$205$	−9.68330	−0.676311
$206$	−3.51983	−0.245238
$207$	−6.59790	−0.458586
$208$	−4.02930	−0.279382
$209$	0	0
$210$	2.00431	0.138310
$211$	−25.5146	−1.75650	−0.878250	−	0.478203i	$-0.841288\pi$
$211$	−25.5146	−1.75650	−0.878250	+	0.478203i	$0.841288\pi$
$212$	6.39502	0.439212
$213$	16.7944	1.15073
$214$	10.4251	0.712646
$215$	−3.42569	−0.233630
$216$	−3.97405	−0.270400
$217$	−7.53480	−0.511495
$218$	12.1571	0.823385
$219$	2.81152	0.189985
$220$	0	0
$221$	−1.21107	−0.0814655
$222$	18.2634	1.22576
$223$	−23.1843	−1.55254	−0.776268	−	0.630403i	$-0.782890\pi$
$223$	−23.1843	−1.55254	−0.776268	+	0.630403i	$0.782890\pi$
$224$	−1.00000	−0.0668153
$225$	1.01724	0.0678161
$226$	−6.67216	−0.443826
$227$	−4.97362	−0.330111	−0.165055	−	0.986284i	$-0.552780\pi$
$227$	−4.97362	−0.330111	−0.165055	+	0.986284i	$0.552780\pi$
$228$	15.5039	1.02677
$229$	−25.6627	−1.69584	−0.847921	−	0.530123i	$-0.822146\pi$
$229$	−25.6627	−1.69584	−0.847921	+	0.530123i	$0.822146\pi$
$230$	6.48607	0.427679
$231$	0	0
$232$	1.81673	0.119274
$233$	3.81437	0.249888	0.124944	−	0.992164i	$-0.460125\pi$
$233$	3.81437	0.249888	0.124944	+	0.992164i	$0.460125\pi$
$234$	−4.09877	−0.267945
$235$	6.21202	0.405227
$236$	5.51945	0.359286
$237$	12.4268	0.807207
$238$	−0.300566	−0.0194828
$239$	6.58100	0.425689	0.212845	−	0.977086i	$-0.431727\pi$
$239$	6.58100	0.425689	0.212845	+	0.977086i	$0.431727\pi$
$240$	−2.00431	−0.129377
$241$	23.9679	1.54391	0.771955	−	0.635677i	$-0.219279\pi$
$241$	23.9679	1.54391	0.771955	+	0.635677i	$0.219279\pi$
$242$	0	0
$243$	−10.1592	−0.651710
$244$	−3.62844	−0.232287
$245$	−1.00000	−0.0638877
$246$	19.4083	1.23743
$247$	−31.1679	−1.98317
$248$	7.53480	0.478460
$249$	29.2164	1.85151
$250$	−1.00000	−0.0632456
$251$	5.08595	0.321022	0.160511	−	0.987034i	$-0.448686\pi$
$251$	5.08595	0.321022	0.160511	+	0.987034i	$0.448686\pi$
$252$	−1.01724	−0.0640802
$253$	0	0
$254$	−12.8919	−0.808908
$255$	−0.602426	−0.0377254
$256$	1.00000	0.0625000
$257$	11.0968	0.692198	0.346099	−	0.938198i	$-0.387506\pi$
$257$	11.0968	0.692198	0.346099	+	0.938198i	$0.387506\pi$
$258$	6.86612	0.427466
$259$	−9.11210	−0.566198
$260$	4.02930	0.249887
$261$	1.84805	0.114391
$262$	20.3862	1.25947
$263$	−2.64779	−0.163270	−0.0816348	−	0.996662i	$-0.526014\pi$
$263$	−2.64779	−0.163270	−0.0816348	+	0.996662i	$0.526014\pi$
$264$	0	0
$265$	−6.39502	−0.392843
$266$	−7.73531	−0.474282
$267$	3.86460	0.236510
$268$	0.505448	0.0308751
$269$	−3.42199	−0.208642	−0.104321	−	0.994544i	$-0.533267\pi$
$269$	−3.42199	−0.208642	−0.104321	+	0.994544i	$0.533267\pi$
$270$	3.97405	0.241853
$271$	−21.1788	−1.28652	−0.643259	−	0.765649i	$-0.722418\pi$
$271$	−21.1788	−1.28652	−0.643259	+	0.765649i	$0.722418\pi$
$272$	0.300566	0.0182245
$273$	8.07596	0.488779
$274$	−17.6954	−1.06902
$275$	0	0
$276$	−13.0001	−0.782512
$277$	27.3915	1.64580	0.822899	−	0.568188i	$-0.192355\pi$
$277$	27.3915	1.64580	0.822899	+	0.568188i	$0.192355\pi$
$278$	13.4774	0.808322
$279$	7.66470	0.458874
$280$	1.00000	0.0597614
$281$	11.2620	0.671832	0.335916	−	0.941892i	$-0.390954\pi$
$281$	11.2620	0.671832	0.335916	+	0.941892i	$0.390954\pi$
$282$	−12.4508	−0.741433
$283$	−20.2928	−1.20628	−0.603142	−	0.797634i	$-0.706085\pi$
$283$	−20.2928	−1.20628	−0.603142	+	0.797634i	$0.706085\pi$
$284$	8.37915	0.497211
$285$	−15.5039	−0.918373
$286$	0	0
$287$	−9.68330	−0.571587
$288$	1.01724	0.0599415
$289$	−16.9097	−0.994686
$290$	−1.81673	−0.106682
$291$	−33.0597	−1.93800
$292$	1.40274	0.0820892
$293$	−11.9548	−0.698405	−0.349202	−	0.937047i	$-0.613547\pi$
$293$	−11.9548	−0.698405	−0.349202	+	0.937047i	$0.613547\pi$
$294$	2.00431	0.116893
$295$	−5.51945	−0.321355
$296$	9.11210	0.529630
$297$	0	0
$298$	−1.72417	−0.0998783
$299$	26.1343	1.51139
$300$	2.00431	0.115719
$301$	−3.42569	−0.197453
$302$	−9.14895	−0.526463
$303$	18.9287	1.08743
$304$	7.73531	0.443651
$305$	3.62844	0.207764
$306$	0.305748	0.0174785
$307$	−19.6625	−1.12220	−0.561100	−	0.827748i	$-0.689622\pi$
$307$	−19.6625	−1.12220	−0.561100	+	0.827748i	$0.689622\pi$
$308$	0	0
$309$	−7.05482	−0.401335
$310$	−7.53480	−0.427948
$311$	−10.8025	−0.612552	−0.306276	−	0.951943i	$-0.599083\pi$
$311$	−10.8025	−0.612552	−0.306276	+	0.951943i	$0.599083\pi$
$312$	−8.07596	−0.457211
$313$	−14.7449	−0.833431	−0.416715	−	0.909037i	$-0.636819\pi$
$313$	−14.7449	−0.833431	−0.416715	+	0.909037i	$0.636819\pi$
$314$	−0.876025	−0.0494370
$315$	1.01724	0.0573151
$316$	6.20005	0.348780
$317$	17.4668	0.981035	0.490518	−	0.871431i	$-0.336808\pi$
$317$	17.4668	0.981035	0.490518	+	0.871431i	$0.336808\pi$
$318$	12.8176	0.718774
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	20.8951	1.16625
$322$	6.48607	0.361455
$323$	2.32497	0.129365
$324$	−11.0169	−0.612052
$325$	−4.02930	−0.223506
$326$	−6.49269	−0.359597
$327$	24.3666	1.34748
$328$	9.68330	0.534671
$329$	6.21202	0.342480
$330$	0	0
$331$	22.3375	1.22778	0.613890	−	0.789391i	$-0.289604\pi$
$331$	22.3375	1.22778	0.613890	+	0.789391i	$0.289604\pi$
$332$	14.5768	0.800006
$333$	9.26920	0.507949
$334$	−20.0322	−1.09611
$335$	−0.505448	−0.0276156
$336$	−2.00431	−0.109344
$337$	14.2386	0.775624	0.387812	−	0.921739i	$-0.373231\pi$
$337$	14.2386	0.775624	0.387812	+	0.921739i	$0.373231\pi$
$338$	3.23529	0.175976
$339$	−13.3731	−0.726325
$340$	−0.300566	−0.0163005
$341$	0	0
$342$	7.86868	0.425489
$343$	−1.00000	−0.0539949
$344$	3.42569	0.184701
$345$	13.0001	0.699900
$346$	1.52088	0.0817629
$347$	20.6300	1.10748	0.553739	−	0.832690i	$-0.313200\pi$
$347$	20.6300	1.10748	0.553739	+	0.832690i	$0.313200\pi$
$348$	3.64127	0.195193
$349$	−16.0047	−0.856711	−0.428355	−	0.903610i	$-0.640907\pi$
$349$	−16.0047	−0.856711	−0.428355	+	0.903610i	$0.640907\pi$
$350$	−1.00000	−0.0534522
$351$	16.0127	0.854693
$352$	0	0
$353$	−1.46881	−0.0781767	−0.0390883	−	0.999236i	$-0.512445\pi$
$353$	−1.46881	−0.0781767	−0.0390883	+	0.999236i	$0.512445\pi$
$354$	11.0627	0.587974
$355$	−8.37915	−0.444719
$356$	1.92815	0.102192
$357$	−0.602426	−0.0318838
$358$	24.1512	1.27643
$359$	−8.43193	−0.445020	−0.222510	−	0.974930i	$-0.571425\pi$
$359$	−8.43193	−0.445020	−0.222510	+	0.974930i	$0.571425\pi$
$360$	−1.01724	−0.0536133
$361$	40.8350	2.14921
$362$	16.6623	0.875750
$363$	0	0
$364$	4.02930	0.211193
$365$	−1.40274	−0.0734228
$366$	−7.27250	−0.380139
$367$	27.2590	1.42291	0.711454	−	0.702732i	$-0.248037\pi$
$367$	27.2590	1.42291	0.711454	+	0.702732i	$0.248037\pi$
$368$	−6.48607	−0.338110
$369$	9.85025	0.512784
$370$	−9.11210	−0.473715
$371$	−6.39502	−0.332013
$372$	15.1020	0.783004
$373$	14.7417	0.763299	0.381649	−	0.924307i	$-0.375356\pi$
$373$	14.7417	0.763299	0.381649	+	0.924307i	$0.375356\pi$
$374$	0	0
$375$	−2.00431	−0.103502
$376$	−6.21202	−0.320360
$377$	−7.32014	−0.377006
$378$	3.97405	0.204403
$379$	−26.2074	−1.34618	−0.673091	−	0.739560i	$-0.735034\pi$
$379$	−26.2074	−1.34618	−0.673091	+	0.739560i	$0.735034\pi$
$380$	−7.73531	−0.396813
$381$	−25.8392	−1.32378
$382$	−2.97548	−0.152239
$383$	−18.0535	−0.922492	−0.461246	−	0.887272i	$-0.652597\pi$
$383$	−18.0535	−0.922492	−0.461246	+	0.887272i	$0.652597\pi$
$384$	2.00431	0.102282
$385$	0	0
$386$	−11.5006	−0.585365
$387$	3.48475	0.177140
$388$	−16.4944	−0.837374
$389$	−21.4054	−1.08530	−0.542648	−	0.839960i	$-0.682578\pi$
$389$	−21.4054	−1.08530	−0.542648	+	0.839960i	$0.682578\pi$
$390$	8.07596	0.408942
$391$	−1.94949	−0.0985900
$392$	1.00000	0.0505076
$393$	40.8602	2.06113
$394$	16.0597	0.809074
$395$	−6.20005	−0.311959
$396$	0	0
$397$	4.87615	0.244727	0.122364	−	0.992485i	$-0.460953\pi$
$397$	4.87615	0.244727	0.122364	+	0.992485i	$0.460953\pi$
$398$	11.0899	0.555885
$399$	−15.5039	−0.776167
$400$	1.00000	0.0500000
$401$	15.6680	0.782423	0.391212	−	0.920301i	$-0.372056\pi$
$401$	15.6680	0.782423	0.391212	+	0.920301i	$0.372056\pi$
$402$	1.01307	0.0505274
$403$	−30.3600	−1.51234
$404$	9.44404	0.469859
$405$	11.0169	0.547436
$406$	−1.81673	−0.0901626
$407$	0	0
$408$	0.602426	0.0298245
$409$	37.1821	1.83854	0.919268	−	0.393633i	$-0.128782\pi$
$409$	37.1821	1.83854	0.919268	+	0.393633i	$0.128782\pi$
$410$	−9.68330	−0.478224
$411$	−35.4670	−1.74946
$412$	−3.51983	−0.173410
$413$	−5.51945	−0.271594
$414$	−6.59790	−0.324269
$415$	−14.5768	−0.715547
$416$	−4.02930	−0.197553
$417$	27.0129	1.32283
$418$	0	0
$419$	−27.2599	−1.33173	−0.665865	−	0.746072i	$-0.731938\pi$
$419$	−27.2599	−1.33173	−0.665865	+	0.746072i	$0.731938\pi$
$420$	2.00431	0.0978001
$421$	−13.1997	−0.643315	−0.321658	−	0.946856i	$-0.604240\pi$
$421$	−13.1997	−0.643315	−0.321658	+	0.946856i	$0.604240\pi$
$422$	−25.5146	−1.24203
$423$	−6.31912	−0.307246
$424$	6.39502	0.310570
$425$	0.300566	0.0145796
$426$	16.7944	0.813690
$427$	3.62844	0.175592
$428$	10.4251	0.503917
$429$	0	0
$430$	−3.42569	−0.165201
$431$	−17.4534	−0.840699	−0.420350	−	0.907362i	$-0.638093\pi$
$431$	−17.4534	−0.840699	−0.420350	+	0.907362i	$0.638093\pi$
$432$	−3.97405	−0.191202
$433$	−28.2800	−1.35905	−0.679525	−	0.733652i	$-0.737814\pi$
$433$	−28.2800	−1.35905	−0.679525	+	0.733652i	$0.737814\pi$
$434$	−7.53480	−0.361682
$435$	−3.64127	−0.174586
$436$	12.1571	0.582221
$437$	−50.1718	−2.40004
$438$	2.81152	0.134340
$439$	35.7484	1.70618	0.853090	−	0.521764i	$-0.174726\pi$
$439$	35.7484	1.70618	0.853090	+	0.521764i	$0.174726\pi$
$440$	0	0
$441$	1.01724	0.0484401
$442$	−1.21107	−0.0576048
$443$	14.9441	0.710016	0.355008	−	0.934863i	$-0.384478\pi$
$443$	14.9441	0.710016	0.355008	+	0.934863i	$0.384478\pi$
$444$	18.2634	0.866744
$445$	−1.92815	−0.0914031
$446$	−23.1843	−1.09781
$447$	−3.45576	−0.163452
$448$	−1.00000	−0.0472456
$449$	13.9847	0.659981	0.329990	−	0.943984i	$-0.392954\pi$
$449$	13.9847	0.659981	0.329990	+	0.943984i	$0.392954\pi$
$450$	1.01724	0.0479532
$451$	0	0
$452$	−6.67216	−0.313832
$453$	−18.3373	−0.861562
$454$	−4.97362	−0.233423
$455$	−4.02930	−0.188897
$456$	15.5039	0.726038
$457$	10.2714	0.480477	0.240238	−	0.970714i	$-0.422774\pi$
$457$	10.2714	0.480477	0.240238	+	0.970714i	$0.422774\pi$
$458$	−25.6627	−1.19914
$459$	−1.19447	−0.0557529
$460$	6.48607	0.302415
$461$	−17.4058	−0.810671	−0.405335	−	0.914168i	$-0.632845\pi$
$461$	−17.4058	−0.810671	−0.405335	+	0.914168i	$0.632845\pi$
$462$	0	0
$463$	−30.3943	−1.41254	−0.706271	−	0.707942i	$-0.749624\pi$
$463$	−30.3943	−1.41254	−0.706271	+	0.707942i	$0.749624\pi$
$464$	1.81673	0.0843394
$465$	−15.1020	−0.700340
$466$	3.81437	0.176697
$467$	23.8052	1.10158	0.550788	−	0.834646i	$-0.314327\pi$
$467$	23.8052	1.10158	0.550788	+	0.834646i	$0.314327\pi$
$468$	−4.09877	−0.189466
$469$	−0.505448	−0.0233394
$470$	6.21202	0.286539
$471$	−1.75582	−0.0809040
$472$	5.51945	0.254053
$473$	0	0
$474$	12.4268	0.570782
$475$	7.73531	0.354920
$476$	−0.300566	−0.0137764
$477$	6.50528	0.297856
$478$	6.58100	0.301008
$479$	7.79996	0.356389	0.178195	−	0.983995i	$-0.442974\pi$
$479$	7.79996	0.356389	0.178195	+	0.983995i	$0.442974\pi$
$480$	−2.00431	−0.0914836
$481$	−36.7154	−1.67408
$482$	23.9679	1.09171
$483$	13.0001	0.591523
$484$	0	0
$485$	16.4944	0.748970
$486$	−10.1592	−0.460829
$487$	−19.3109	−0.875060	−0.437530	−	0.899204i	$-0.644147\pi$
$487$	−19.3109	−0.875060	−0.437530	+	0.899204i	$0.644147\pi$
$488$	−3.62844	−0.164252
$489$	−13.0133	−0.588484
$490$	−1.00000	−0.0451754
$491$	16.5180	0.745449	0.372724	−	0.927942i	$-0.378424\pi$
$491$	16.5180	0.745449	0.372724	+	0.927942i	$0.378424\pi$
$492$	19.4083	0.874994
$493$	0.546046	0.0245927
$494$	−31.1679	−1.40231
$495$	0	0
$496$	7.53480	0.338322
$497$	−8.37915	−0.375856
$498$	29.2164	1.30922
$499$	11.8692	0.531341	0.265670	−	0.964064i	$-0.414407\pi$
$499$	11.8692	0.531341	0.265670	+	0.964064i	$0.414407\pi$
$500$	−1.00000	−0.0447214
$501$	−40.1507	−1.79380
$502$	5.08595	0.226997
$503$	−6.76151	−0.301481	−0.150741	−	0.988573i	$-0.548166\pi$
$503$	−6.76151	−0.301481	−0.150741	+	0.988573i	$0.548166\pi$
$504$	−1.01724	−0.0453115
$505$	−9.44404	−0.420254
$506$	0	0
$507$	6.48451	0.287987
$508$	−12.8919	−0.571984
$509$	−37.3337	−1.65479	−0.827393	−	0.561624i	$-0.810177\pi$
$509$	−37.3337	−1.65479	−0.827393	+	0.561624i	$0.810177\pi$
$510$	−0.602426	−0.0266759
$511$	−1.40274	−0.0620536
$512$	1.00000	0.0441942
$513$	−30.7405	−1.35723
$514$	11.0968	0.489458
$515$	3.51983	0.155102
$516$	6.86612	0.302264
$517$	0	0
$518$	−9.11210	−0.400363
$519$	3.04831	0.133806
$520$	4.02930	0.176697
$521$	−13.6299	−0.597137	−0.298569	−	0.954388i	$-0.596509\pi$
$521$	−13.6299	−0.597137	−0.298569	+	0.954388i	$0.596509\pi$
$522$	1.84805	0.0808869
$523$	−7.03010	−0.307405	−0.153702	−	0.988117i	$-0.549120\pi$
$523$	−7.03010	−0.307405	−0.153702	+	0.988117i	$0.549120\pi$
$524$	20.3862	0.890577
$525$	−2.00431	−0.0874751
$526$	−2.64779	−0.115449
$527$	2.26470	0.0986520
$528$	0	0
$529$	19.0691	0.829091
$530$	−6.39502	−0.277782
$531$	5.61461	0.243653
$532$	−7.73531	−0.335368
$533$	−39.0170	−1.69001
$534$	3.86460	0.167238
$535$	−10.4251	−0.450717
$536$	0.505448	0.0218320
$537$	48.4064	2.08889
$538$	−3.42199	−0.147532
$539$	0	0
$540$	3.97405	0.171016
$541$	15.3397	0.659507	0.329753	−	0.944067i	$-0.393034\pi$
$541$	15.3397	0.659507	0.329753	+	0.944067i	$0.393034\pi$
$542$	−21.1788	−0.909706
$543$	33.3963	1.43317
$544$	0.300566	0.0128867
$545$	−12.1571	−0.520755
$546$	8.07596	0.345619
$547$	17.6527	0.754775	0.377388	−	0.926055i	$-0.376822\pi$
$547$	17.6527	0.754775	0.377388	+	0.926055i	$0.376822\pi$
$548$	−17.6954	−0.755911
$549$	−3.69100	−0.157528
$550$	0	0
$551$	14.0529	0.598675
$552$	−13.0001	−0.553320
$553$	−6.20005	−0.263653
$554$	27.3915	1.16375
$555$	−18.2634	−0.775239
$556$	13.4774	0.571570
$557$	27.5742	1.16836	0.584178	−	0.811626i	$-0.301417\pi$
$557$	27.5742	1.16836	0.584178	+	0.811626i	$0.301417\pi$
$558$	7.66470	0.324473
$559$	−13.8031	−0.583810
$560$	1.00000	0.0422577
$561$	0	0
$562$	11.2620	0.475057
$563$	−14.3283	−0.603866	−0.301933	−	0.953329i	$-0.597632\pi$
$563$	−14.3283	−0.603866	−0.301933	+	0.953329i	$0.597632\pi$
$564$	−12.4508	−0.524273
$565$	6.67216	0.280700
$566$	−20.2928	−0.852971
$567$	11.0169	0.462668
$568$	8.37915	0.351581
$569$	−29.2285	−1.22532	−0.612662	−	0.790345i	$-0.709901\pi$
$569$	−29.2285	−1.22532	−0.612662	+	0.790345i	$0.709901\pi$
$570$	−15.5039	−0.649388
$571$	−42.5911	−1.78238	−0.891190	−	0.453630i	$-0.850129\pi$
$571$	−42.5911	−1.78238	−0.891190	+	0.453630i	$0.850129\pi$
$572$	0	0
$573$	−5.96378	−0.249140
$574$	−9.68330	−0.404173
$575$	−6.48607	−0.270488
$576$	1.01724	0.0423851
$577$	25.4138	1.05799	0.528994	−	0.848625i	$-0.322569\pi$
$577$	25.4138	1.05799	0.528994	+	0.848625i	$0.322569\pi$
$578$	−16.9097	−0.703349
$579$	−23.0507	−0.957955
$580$	−1.81673	−0.0754354
$581$	−14.5768	−0.604748
$582$	−33.0597	−1.37037
$583$	0	0
$584$	1.40274	0.0580458
$585$	4.09877	0.169463
$586$	−11.9548	−0.493847
$587$	48.0000	1.98117	0.990586	−	0.136892i	$-0.0437115\pi$
$587$	48.0000	1.98117	0.990586	+	0.136892i	$0.0437115\pi$
$588$	2.00431	0.0826562
$589$	58.2840	2.40155
$590$	−5.51945	−0.227232
$591$	32.1884	1.32406
$592$	9.11210	0.374505
$593$	3.34128	0.137210	0.0686050	−	0.997644i	$-0.478145\pi$
$593$	3.34128	0.137210	0.0686050	+	0.997644i	$0.478145\pi$
$594$	0	0
$595$	0.300566	0.0123220
$596$	−1.72417	−0.0706246
$597$	22.2275	0.909711
$598$	26.1343	1.06871
$599$	−43.1060	−1.76126	−0.880632	−	0.473800i	$-0.842882\pi$
$599$	−43.1060	−1.76126	−0.880632	+	0.473800i	$0.842882\pi$
$600$	2.00431	0.0818254
$601$	−17.8432	−0.727840	−0.363920	−	0.931430i	$-0.618562\pi$
$601$	−17.8432	−0.727840	−0.363920	+	0.931430i	$0.618562\pi$
$602$	−3.42569	−0.139621
$603$	0.514162	0.0209383
$604$	−9.14895	−0.372266
$605$	0	0
$606$	18.9287	0.768928
$607$	−11.3238	−0.459619	−0.229809	−	0.973236i	$-0.573810\pi$
$607$	−11.3238	−0.459619	−0.229809	+	0.973236i	$0.573810\pi$
$608$	7.73531	0.313708
$609$	−3.64127	−0.147552
$610$	3.62844	0.146911
$611$	25.0301	1.01261
$612$	0.305748	0.0123591
$613$	−36.2483	−1.46405	−0.732027	−	0.681275i	$-0.761426\pi$
$613$	−36.2483	−1.46405	−0.732027	+	0.681275i	$0.761426\pi$
$614$	−19.6625	−0.793516
$615$	−19.4083	−0.782618
$616$	0	0
$617$	40.1661	1.61702	0.808512	−	0.588479i	$-0.200273\pi$
$617$	40.1661	1.61702	0.808512	+	0.588479i	$0.200273\pi$
$618$	−7.05482	−0.283787
$619$	8.86070	0.356141	0.178071	−	0.984018i	$-0.443014\pi$
$619$	8.86070	0.356141	0.178071	+	0.984018i	$0.443014\pi$
$620$	−7.53480	−0.302605
$621$	25.7760	1.03436
$622$	−10.8025	−0.433139
$623$	−1.92815	−0.0772497
$624$	−8.07596	−0.323297
$625$	1.00000	0.0400000
$626$	−14.7449	−0.589325
$627$	0	0
$628$	−0.876025	−0.0349572
$629$	2.73879	0.109203
$630$	1.01724	0.0405279
$631$	−2.39090	−0.0951802	−0.0475901	−	0.998867i	$-0.515154\pi$
$631$	−2.39090	−0.0951802	−0.0475901	+	0.998867i	$0.515154\pi$
$632$	6.20005	0.246625
$633$	−51.1391	−2.03260
$634$	17.4668	0.693697
$635$	12.8919	0.511598
$636$	12.8176	0.508250
$637$	−4.02930	−0.159647
$638$	0	0
$639$	8.52362	0.337189
$640$	−1.00000	−0.0395285
$641$	−7.27551	−0.287365	−0.143683	−	0.989624i	$-0.545894\pi$
$641$	−7.27551	−0.287365	−0.143683	+	0.989624i	$0.545894\pi$
$642$	20.8951	0.824664
$643$	−9.04508	−0.356703	−0.178352	−	0.983967i	$-0.557076\pi$
$643$	−9.04508	−0.356703	−0.178352	+	0.983967i	$0.557076\pi$
$644$	6.48607	0.255587
$645$	−6.86612	−0.270353
$646$	2.32497	0.0914747
$647$	−31.0318	−1.21999	−0.609993	−	0.792407i	$-0.708828\pi$
$647$	−31.0318	−1.21999	−0.609993	+	0.792407i	$0.708828\pi$
$648$	−11.0169	−0.432786
$649$	0	0
$650$	−4.02930	−0.158042
$651$	−15.1020	−0.591895
$652$	−6.49269	−0.254274
$653$	42.3720	1.65814	0.829072	−	0.559142i	$-0.188870\pi$
$653$	42.3720	1.65814	0.829072	+	0.559142i	$0.188870\pi$
$654$	24.3666	0.952810
$655$	−20.3862	−0.796556
$656$	9.68330	0.378069
$657$	1.42693	0.0556697
$658$	6.21202	0.242170
$659$	25.5490	0.995247	0.497624	−	0.867393i	$-0.334206\pi$
$659$	25.5490	0.995247	0.497624	+	0.867393i	$0.334206\pi$
$660$	0	0
$661$	1.55717	0.0605669	0.0302834	−	0.999541i	$-0.490359\pi$
$661$	1.55717	0.0605669	0.0302834	+	0.999541i	$0.490359\pi$
$662$	22.3375	0.868172
$663$	−2.42736	−0.0942707
$664$	14.5768	0.565690
$665$	7.73531	0.299962
$666$	9.26920	0.359174
$667$	−11.7834	−0.456255
$668$	−20.0322	−0.775070
$669$	−46.4684	−1.79657
$670$	−0.505448	−0.0195271
$671$	0	0
$672$	−2.00431	−0.0773178
$673$	25.0244	0.964620	0.482310	−	0.876001i	$-0.339798\pi$
$673$	25.0244	0.964620	0.482310	+	0.876001i	$0.339798\pi$
$674$	14.2386	0.548449
$675$	−3.97405	−0.152961
$676$	3.23529	0.124434
$677$	23.4532	0.901380	0.450690	−	0.892681i	$-0.351178\pi$
$677$	23.4532	0.901380	0.450690	+	0.892681i	$0.351178\pi$
$678$	−13.3731	−0.513589
$679$	16.4944	0.632995
$680$	−0.300566	−0.0115262
$681$	−9.96865	−0.381999
$682$	0	0
$683$	−36.6815	−1.40358	−0.701790	−	0.712384i	$-0.747616\pi$
$683$	−36.6815	−1.40358	−0.701790	+	0.712384i	$0.747616\pi$
$684$	7.86868	0.300866
$685$	17.6954	0.676107
$686$	−1.00000	−0.0381802
$687$	−51.4360	−1.96240
$688$	3.42569	0.130603
$689$	−25.7675	−0.981662
$690$	13.0001	0.494904
$691$	27.6172	1.05061	0.525303	−	0.850915i	$-0.323952\pi$
$691$	27.6172	1.05061	0.525303	+	0.850915i	$0.323952\pi$
$692$	1.52088	0.0578151
$693$	0	0
$694$	20.6300	0.783106
$695$	−13.4774	−0.511228
$696$	3.64127	0.138022
$697$	2.91047	0.110242
$698$	−16.0047	−0.605786
$699$	7.64517	0.289167
$700$	−1.00000	−0.0377964
$701$	29.9881	1.13263	0.566317	−	0.824188i	$-0.308368\pi$
$701$	29.9881	1.13263	0.566317	+	0.824188i	$0.308368\pi$
$702$	16.0127	0.604359
$703$	70.4849	2.65839
$704$	0	0
$705$	12.4508	0.468924
$706$	−1.46881	−0.0552793
$707$	−9.44404	−0.355180
$708$	11.0627	0.415760
$709$	−5.18512	−0.194731	−0.0973656	−	0.995249i	$-0.531042\pi$
$709$	−5.18512	−0.194731	−0.0973656	+	0.995249i	$0.531042\pi$
$710$	−8.37915	−0.314464
$711$	6.30695	0.236529
$712$	1.92815	0.0722605
$713$	−48.8712	−1.83024
$714$	−0.602426	−0.0225452
$715$	0	0
$716$	24.1512	0.902573
$717$	13.1903	0.492602
$718$	−8.43193	−0.314677
$719$	−16.7828	−0.625892	−0.312946	−	0.949771i	$-0.601316\pi$
$719$	−16.7828	−0.625892	−0.312946	+	0.949771i	$0.601316\pi$
$720$	−1.01724	−0.0379103
$721$	3.51983	0.131085
$722$	40.8350	1.51972
$723$	48.0390	1.78659
$724$	16.6623	0.619249
$725$	1.81673	0.0674715
$726$	0	0
$727$	9.72794	0.360789	0.180395	−	0.983594i	$-0.442263\pi$
$727$	9.72794	0.360789	0.180395	+	0.983594i	$0.442263\pi$
$728$	4.02930	0.149336
$729$	12.6888	0.469954
$730$	−1.40274	−0.0519178
$731$	1.02964	0.0380828
$732$	−7.27250	−0.268799
$733$	10.5873	0.391050	0.195525	−	0.980699i	$-0.437359\pi$
$733$	10.5873	0.391050	0.195525	+	0.980699i	$0.437359\pi$
$734$	27.2590	1.00615
$735$	−2.00431	−0.0739299
$736$	−6.48607	−0.239080
$737$	0	0
$738$	9.85025	0.362593
$739$	−11.6909	−0.430058	−0.215029	−	0.976608i	$-0.568985\pi$
$739$	−11.6909	−0.430058	−0.215029	+	0.976608i	$0.568985\pi$
$740$	−9.11210	−0.334967
$741$	−62.4700	−2.29489
$742$	−6.39502	−0.234768
$743$	2.32433	0.0852713	0.0426356	−	0.999091i	$-0.486425\pi$
$743$	2.32433	0.0852713	0.0426356	+	0.999091i	$0.486425\pi$
$744$	15.1020	0.553667
$745$	1.72417	0.0631686
$746$	14.7417	0.539734
$747$	14.8281	0.542533
$748$	0	0
$749$	−10.4251	−0.380925
$750$	−2.00431	−0.0731869
$751$	−13.6597	−0.498450	−0.249225	−	0.968446i	$-0.580176\pi$
$751$	−13.6597	−0.498450	−0.249225	+	0.968446i	$0.580176\pi$
$752$	−6.21202	−0.226529
$753$	10.1938	0.371483
$754$	−7.32014	−0.266584
$755$	9.14895	0.332965
$756$	3.97405	0.144535
$757$	20.4404	0.742919	0.371460	−	0.928449i	$-0.378857\pi$
$757$	20.4404	0.742919	0.371460	+	0.928449i	$0.378857\pi$
$758$	−26.2074	−0.951894
$759$	0	0
$760$	−7.73531	−0.280589
$761$	5.54838	0.201129	0.100564	−	0.994931i	$-0.467935\pi$
$761$	5.54838	0.201129	0.100564	+	0.994931i	$0.467935\pi$
$762$	−25.8392	−0.936057
$763$	−12.1571	−0.440118
$764$	−2.97548	−0.107649
$765$	−0.305748	−0.0110543
$766$	−18.0535	−0.652300
$767$	−22.2395	−0.803023
$768$	2.00431	0.0723242
$769$	8.18880	0.295296	0.147648	−	0.989040i	$-0.452830\pi$
$769$	8.18880	0.295296	0.147648	+	0.989040i	$0.452830\pi$
$770$	0	0
$771$	22.2413	0.801002
$772$	−11.5006	−0.413916
$773$	10.5045	0.377819	0.188909	−	0.981995i	$-0.439505\pi$
$773$	10.5045	0.377819	0.188909	+	0.981995i	$0.439505\pi$
$774$	3.48475	0.125257
$775$	7.53480	0.270658
$776$	−16.4944	−0.592113
$777$	−18.2634	−0.655197
$778$	−21.4054	−0.767420
$779$	74.9033	2.68369
$780$	8.07596	0.289166
$781$	0	0
$782$	−1.94949	−0.0697137
$783$	−7.21977	−0.258013
$784$	1.00000	0.0357143
$785$	0.876025	0.0312667
$786$	40.8602	1.45744
$787$	11.8432	0.422166	0.211083	−	0.977468i	$-0.432301\pi$
$787$	11.8432	0.422166	0.211083	+	0.977468i	$0.432301\pi$
$788$	16.0597	0.572101
$789$	−5.30697	−0.188933
$790$	−6.20005	−0.220588
$791$	6.67216	0.237235
$792$	0	0
$793$	14.6201	0.519174
$794$	4.87615	0.173048
$795$	−12.8176	−0.454592
$796$	11.0899	0.393070
$797$	−22.3237	−0.790747	−0.395374	−	0.918520i	$-0.629385\pi$
$797$	−22.3237	−0.790747	−0.395374	+	0.918520i	$0.629385\pi$
$798$	−15.5039	−0.548833
$799$	−1.86712	−0.0660540
$800$	1.00000	0.0353553
$801$	1.96139	0.0693025
$802$	15.6680	0.553257
$803$	0	0
$804$	1.01307	0.0357283
$805$	−6.48607	−0.228604
$806$	−30.3600	−1.06938
$807$	−6.85871	−0.241438
$808$	9.44404	0.332240
$809$	−32.0996	−1.12856	−0.564282	−	0.825582i	$-0.690847\pi$
$809$	−32.0996	−1.12856	−0.564282	+	0.825582i	$0.690847\pi$
$810$	11.0169	0.387096
$811$	42.4498	1.49061	0.745306	−	0.666722i	$-0.232303\pi$
$811$	42.4498	1.49061	0.745306	+	0.666722i	$0.232303\pi$
$812$	−1.81673	−0.0637546
$813$	−42.4487	−1.48874
$814$	0	0
$815$	6.49269	0.227429
$816$	0.602426	0.0210891
$817$	26.4987	0.927074
$818$	37.1821	1.30004
$819$	4.09877	0.143223
$820$	−9.68330	−0.338156
$821$	−17.7352	−0.618962	−0.309481	−	0.950906i	$-0.600155\pi$
$821$	−17.7352	−0.618962	−0.309481	+	0.950906i	$0.600155\pi$
$822$	−35.4670	−1.23706
$823$	37.9393	1.32248	0.661241	−	0.750173i	$-0.270030\pi$
$823$	37.9393	1.32248	0.661241	+	0.750173i	$0.270030\pi$
$824$	−3.51983	−0.122619
$825$	0	0
$826$	−5.51945	−0.192046
$827$	40.9256	1.42312	0.711561	−	0.702624i	$-0.247988\pi$
$827$	40.9256	1.42312	0.711561	+	0.702624i	$0.247988\pi$
$828$	−6.59790	−0.229293
$829$	2.81991	0.0979395	0.0489698	−	0.998800i	$-0.484406\pi$
$829$	2.81991	0.0979395	0.0489698	+	0.998800i	$0.484406\pi$
$830$	−14.5768	−0.505968
$831$	54.9010	1.90449
$832$	−4.02930	−0.139691
$833$	0.300566	0.0104140
$834$	27.0129	0.935379
$835$	20.0322	0.693244
$836$	0	0
$837$	−29.9437	−1.03501
$838$	−27.2599	−0.941676
$839$	−55.3938	−1.91241	−0.956204	−	0.292701i	$-0.905446\pi$
$839$	−55.3938	−1.91241	−0.956204	+	0.292701i	$0.905446\pi$
$840$	2.00431	0.0691551
$841$	−25.6995	−0.886190
$842$	−13.1997	−0.454892
$843$	22.5724	0.777435
$844$	−25.5146	−0.878250
$845$	−3.23529	−0.111297
$846$	−6.31912	−0.217256
$847$	0	0
$848$	6.39502	0.219606
$849$	−40.6730	−1.39589
$850$	0.300566	0.0103093
$851$	−59.1017	−2.02598
$852$	16.7944	0.575366
$853$	−49.6786	−1.70096	−0.850481	−	0.526006i	$-0.823689\pi$
$853$	−49.6786	−1.70096	−0.850481	+	0.526006i	$0.823689\pi$
$854$	3.62844	0.124163
$855$	−7.86868	−0.269103
$856$	10.4251	0.356323
$857$	−20.7002	−0.707104	−0.353552	−	0.935415i	$-0.615026\pi$
$857$	−20.7002	−0.707104	−0.353552	+	0.935415i	$0.615026\pi$
$858$	0	0
$859$	−15.2630	−0.520766	−0.260383	−	0.965505i	$-0.583849\pi$
$859$	−15.2630	−0.520766	−0.260383	+	0.965505i	$0.583849\pi$
$860$	−3.42569	−0.116815
$861$	−19.4083	−0.661433
$862$	−17.4534	−0.594464
$863$	8.86338	0.301713	0.150857	−	0.988556i	$-0.451797\pi$
$863$	8.86338	0.301713	0.150857	+	0.988556i	$0.451797\pi$
$864$	−3.97405	−0.135200
$865$	−1.52088	−0.0517114
$866$	−28.2800	−0.960994
$867$	−33.8921	−1.15104
$868$	−7.53480	−0.255748
$869$	0	0
$870$	−3.64127	−0.123451
$871$	−2.03660	−0.0690076
$872$	12.1571	0.411693
$873$	−16.7787	−0.567874
$874$	−50.1718	−1.69709
$875$	1.00000	0.0338062
$876$	2.81152	0.0949925
$877$	16.7377	0.565192	0.282596	−	0.959239i	$-0.408804\pi$
$877$	16.7377	0.565192	0.282596	+	0.959239i	$0.408804\pi$
$878$	35.7484	1.20645
$879$	−23.9610	−0.808185
$880$	0	0
$881$	−4.10239	−0.138213	−0.0691065	−	0.997609i	$-0.522015\pi$
$881$	−4.10239	−0.138213	−0.0691065	+	0.997609i	$0.522015\pi$
$882$	1.01724	0.0342523
$883$	40.7341	1.37081	0.685406	−	0.728161i	$-0.259625\pi$
$883$	40.7341	1.37081	0.685406	+	0.728161i	$0.259625\pi$
$884$	−1.21107	−0.0407327
$885$	−11.0627	−0.371867
$886$	14.9441	0.502057
$887$	−17.7893	−0.597305	−0.298652	−	0.954362i	$-0.596537\pi$
$887$	−17.7893	−0.597305	−0.298652	+	0.954362i	$0.596537\pi$
$888$	18.2634	0.612881
$889$	12.8919	0.432379
$890$	−1.92815	−0.0646318
$891$	0	0
$892$	−23.1843	−0.776268
$893$	−48.0519	−1.60800
$894$	−3.45576	−0.115578
$895$	−24.1512	−0.807286
$896$	−1.00000	−0.0334077
$897$	52.3812	1.74896
$898$	13.9847	0.466677
$899$	13.6887	0.456542
$900$	1.01724	0.0339080
$901$	1.92212	0.0640352
$902$	0	0
$903$	−6.86612	−0.228490
$904$	−6.67216	−0.221913
$905$	−16.6623	−0.553873
$906$	−18.3373	−0.609216
$907$	−3.77888	−0.125476	−0.0627379	−	0.998030i	$-0.519983\pi$
$907$	−3.77888	−0.125476	−0.0627379	+	0.998030i	$0.519983\pi$
$908$	−4.97362	−0.165055
$909$	9.60687	0.318640
$910$	−4.02930	−0.133570
$911$	36.8482	1.22084	0.610418	−	0.792080i	$-0.291002\pi$
$911$	36.8482	1.22084	0.610418	+	0.792080i	$0.291002\pi$
$912$	15.5039	0.513386
$913$	0	0
$914$	10.2714	0.339748
$915$	7.27250	0.240421
$916$	−25.6627	−0.847921
$917$	−20.3862	−0.673213
$918$	−1.19447	−0.0394232
$919$	−56.7946	−1.87348	−0.936740	−	0.350025i	$-0.886173\pi$
$919$	−56.7946	−1.87348	−0.936740	+	0.350025i	$0.886173\pi$
$920$	6.48607	0.213839
$921$	−39.4098	−1.29860
$922$	−17.4058	−0.573231
$923$	−33.7621	−1.11129
$924$	0	0
$925$	9.11210	0.299604
$926$	−30.3943	−0.998818
$927$	−3.58052	−0.117600
$928$	1.81673	0.0596369
$929$	−18.0311	−0.591580	−0.295790	−	0.955253i	$-0.595583\pi$
$929$	−18.0311	−0.591580	−0.295790	+	0.955253i	$0.595583\pi$
$930$	−15.1020	−0.495215
$931$	7.73531	0.253515
$932$	3.81437	0.124944
$933$	−21.6514	−0.708836
$934$	23.8052	0.778931
$935$	0	0
$936$	−4.09877	−0.133973
$937$	−20.7850	−0.679016	−0.339508	−	0.940603i	$-0.610261\pi$
$937$	−20.7850	−0.679016	−0.339508	+	0.940603i	$0.610261\pi$
$938$	−0.505448	−0.0165035
$939$	−29.5533	−0.964435
$940$	6.21202	0.202614
$941$	4.11596	0.134176	0.0670882	−	0.997747i	$-0.478629\pi$
$941$	4.11596	0.134176	0.0670882	+	0.997747i	$0.478629\pi$
$942$	−1.75582	−0.0572078
$943$	−62.8066	−2.04526
$944$	5.51945	0.179643
$945$	−3.97405	−0.129276
$946$	0	0
$947$	−31.0569	−1.00921	−0.504607	−	0.863349i	$-0.668363\pi$
$947$	−31.0569	−1.00921	−0.504607	+	0.863349i	$0.668363\pi$
$948$	12.4268	0.403604
$949$	−5.65207	−0.183474
$950$	7.73531	0.250967
$951$	35.0089	1.13524
$952$	−0.300566	−0.00974140
$953$	17.8888	0.579474	0.289737	−	0.957106i	$-0.406432\pi$
$953$	17.8888	0.579474	0.289737	+	0.957106i	$0.406432\pi$
$954$	6.50528	0.210616
$955$	2.97548	0.0962843
$956$	6.58100	0.212845
$957$	0	0
$958$	7.79996	0.252005
$959$	17.6954	0.571415
$960$	−2.00431	−0.0646887
$961$	25.7731	0.831392
$962$	−36.7154	−1.18375
$963$	10.6049	0.341737
$964$	23.9679	0.771955
$965$	11.5006	0.370217
$966$	13.0001	0.418270
$967$	−59.5313	−1.91440	−0.957199	−	0.289431i	$-0.906534\pi$
$967$	−59.5313	−1.91440	−0.957199	+	0.289431i	$0.906534\pi$
$968$	0	0
$969$	4.65995	0.149699
$970$	16.4944	0.529602
$971$	−43.1529	−1.38484	−0.692421	−	0.721494i	$-0.743456\pi$
$971$	−43.1529	−1.38484	−0.692421	+	0.721494i	$0.743456\pi$
$972$	−10.1592	−0.325855
$973$	−13.4774	−0.432066
$974$	−19.3109	−0.618761
$975$	−8.07596	−0.258638
$976$	−3.62844	−0.116143
$977$	−46.0411	−1.47299	−0.736493	−	0.676445i	$-0.763520\pi$
$977$	−46.0411	−1.47299	−0.736493	+	0.676445i	$0.763520\pi$
$978$	−13.0133	−0.416121
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	12.3667	0.394840
$982$	16.5180	0.527112
$983$	45.1941	1.44147	0.720734	−	0.693212i	$-0.243805\pi$
$983$	45.1941	1.44147	0.720734	+	0.693212i	$0.243805\pi$
$984$	19.4083	0.618714
$985$	−16.0597	−0.511703
$986$	0.546046	0.0173896
$987$	12.4508	0.396313
$988$	−31.1679	−0.991584
$989$	−22.2192	−0.706531
$990$	0	0
$991$	−32.5442	−1.03380	−0.516900	−	0.856046i	$-0.672914\pi$
$991$	−32.5442	−1.03380	−0.516900	+	0.856046i	$0.672914\pi$
$992$	7.53480	0.239230
$993$	44.7712	1.42077
$994$	−8.37915	−0.265770
$995$	−11.0899	−0.351573
$996$	29.2164	0.925756
$997$	−44.8088	−1.41911	−0.709555	−	0.704650i	$-0.751104\pi$
$997$	−44.8088	−1.41911	−0.709555	+	0.704650i	$0.751104\pi$
$998$	11.8692	0.375715
$999$	−36.2120	−1.14570

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.dh.1.7		8
11.5	even	5		770.2.n.k.421.4	✓	16
11.9	even	5		770.2.n.k.631.4	yes	16
11.10	odd	2		8470.2.a.dg.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.k.421.4	✓	16	11.5	even	5
770.2.n.k.631.4	yes	16	11.9	even	5
8470.2.a.dg.1.7		8	11.10	odd	2
8470.2.a.dh.1.7		8	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.00431 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00431 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.01724 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.00431 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00431 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.01724 q^{9} -1.00000 q^{10} +2.00431 q^{12} -4.02930 q^{13} -1.00000 q^{14} -2.00431 q^{15} +1.00000 q^{16} +0.300566 q^{17} +1.01724 q^{18} +7.73531 q^{19} -1.00000 q^{20} -2.00431 q^{21} -6.48607 q^{23} +2.00431 q^{24} +1.00000 q^{25} -4.02930 q^{26} -3.97405 q^{27} -1.00000 q^{28} +1.81673 q^{29} -2.00431 q^{30} +7.53480 q^{31} +1.00000 q^{32} +0.300566 q^{34} +1.00000 q^{35} +1.01724 q^{36} +9.11210 q^{37} +7.73531 q^{38} -8.07596 q^{39} -1.00000 q^{40} +9.68330 q^{41} -2.00431 q^{42} +3.42569 q^{43} -1.01724 q^{45} -6.48607 q^{46} -6.21202 q^{47} +2.00431 q^{48} +1.00000 q^{49} +1.00000 q^{50} +0.602426 q^{51} -4.02930 q^{52} +6.39502 q^{53} -3.97405 q^{54} -1.00000 q^{56} +15.5039 q^{57} +1.81673 q^{58} +5.51945 q^{59} -2.00431 q^{60} -3.62844 q^{61} +7.53480 q^{62} -1.01724 q^{63} +1.00000 q^{64} +4.02930 q^{65} +0.505448 q^{67} +0.300566 q^{68} -13.0001 q^{69} +1.00000 q^{70} +8.37915 q^{71} +1.01724 q^{72} +1.40274 q^{73} +9.11210 q^{74} +2.00431 q^{75} +7.73531 q^{76} -8.07596 q^{78} +6.20005 q^{79} -1.00000 q^{80} -11.0169 q^{81} +9.68330 q^{82} +14.5768 q^{83} -2.00431 q^{84} -0.300566 q^{85} +3.42569 q^{86} +3.64127 q^{87} +1.92815 q^{89} -1.01724 q^{90} +4.02930 q^{91} -6.48607 q^{92} +15.1020 q^{93} -6.21202 q^{94} -7.73531 q^{95} +2.00431 q^{96} -16.4944 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} - 8 q^{5} - 8 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 + 8 * q^4 - 8 * q^5 - 8 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} + 8 q^{4} - 8 q^{5} - 8 q^{7} + 8 q^{8} + 8 q^{9} - 8 q^{10} - q^{13} - 8 q^{14} + 8 q^{16} + 6 q^{17} + 8 q^{18} + 5 q^{19} - 8 q^{20} + 10 q^{23} + 8 q^{25} - q^{26} - 8 q^{28} + 3 q^{29} - 8 q^{31} + 8 q^{32} + 6 q^{34} + 8 q^{35} + 8 q^{36} - 6 q^{37} + 5 q^{38} + 35 q^{39} - 8 q^{40} + 11 q^{41} - 5 q^{43} - 8 q^{45} + 10 q^{46} - 15 q^{47} + 8 q^{49} + 8 q^{50} - 6 q^{51} - q^{52} - 16 q^{53} - 8 q^{56} + 38 q^{57} + 3 q^{58} - 9 q^{59} + 32 q^{61} - 8 q^{62} - 8 q^{63} + 8 q^{64} + q^{65} + 33 q^{67} + 6 q^{68} - 22 q^{69} + 8 q^{70} + 11 q^{71} + 8 q^{72} - 34 q^{73} - 6 q^{74} + 5 q^{76} + 35 q^{78} + 31 q^{79} - 8 q^{80} + 20 q^{81} + 11 q^{82} + 50 q^{83} - 6 q^{85} - 5 q^{86} - 12 q^{87} + q^{89} - 8 q^{90} + q^{91} + 10 q^{92} + 26 q^{93} - 15 q^{94} - 5 q^{95} - 4 q^{97} + 8 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 - 8 * q^5 - 8 * q^7 + 8 * q^8 + 8 * q^9 - 8 * q^10 - q^13 - 8 * q^14 + 8 * q^16 + 6 * q^17 + 8 * q^18 + 5 * q^19 - 8 * q^20 + 10 * q^23 + 8 * q^25 - q^26 - 8 * q^28 + 3 * q^29 - 8 * q^31 + 8 * q^32 + 6 * q^34 + 8 * q^35 + 8 * q^36 - 6 * q^37 + 5 * q^38 + 35 * q^39 - 8 * q^40 + 11 * q^41 - 5 * q^43 - 8 * q^45 + 10 * q^46 - 15 * q^47 + 8 * q^49 + 8 * q^50 - 6 * q^51 - q^52 - 16 * q^53 - 8 * q^56 + 38 * q^57 + 3 * q^58 - 9 * q^59 + 32 * q^61 - 8 * q^62 - 8 * q^63 + 8 * q^64 + q^65 + 33 * q^67 + 6 * q^68 - 22 * q^69 + 8 * q^70 + 11 * q^71 + 8 * q^72 - 34 * q^73 - 6 * q^74 + 5 * q^76 + 35 * q^78 + 31 * q^79 - 8 * q^80 + 20 * q^81 + 11 * q^82 + 50 * q^83 - 6 * q^85 - 5 * q^86 - 12 * q^87 + q^89 - 8 * q^90 + q^91 + 10 * q^92 + 26 * q^93 - 15 * q^94 - 5 * q^95 - 4 * q^97 + 8 * q^98

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