LMFDB - Embedded newform 8470.2.a.dh.1.8

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			1.8
Root			$3.18761$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +3.18761 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.18761 q^{6} -1.00000 q^{7} +1.00000 q^{8} +7.16083 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +3.18761 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.18761 q^{6} -1.00000 q^{7} +1.00000 q^{8} +7.16083 q^{9} -1.00000 q^{10} +3.18761 q^{12} +5.69102 q^{13} -1.00000 q^{14} -3.18761 q^{15} +1.00000 q^{16} -0.749022 q^{17} +7.16083 q^{18} +6.78967 q^{19} -1.00000 q^{20} -3.18761 q^{21} +3.94010 q^{23} +3.18761 q^{24} +1.00000 q^{25} +5.69102 q^{26} +13.2631 q^{27} -1.00000 q^{28} -5.73837 q^{29} -3.18761 q^{30} -7.10952 q^{31} +1.00000 q^{32} -0.749022 q^{34} +1.00000 q^{35} +7.16083 q^{36} -11.6050 q^{37} +6.78967 q^{38} +18.1407 q^{39} -1.00000 q^{40} +11.4458 q^{41} -3.18761 q^{42} +2.33345 q^{43} -7.16083 q^{45} +3.94010 q^{46} -5.00025 q^{47} +3.18761 q^{48} +1.00000 q^{49} +1.00000 q^{50} -2.38759 q^{51} +5.69102 q^{52} -10.7671 q^{53} +13.2631 q^{54} -1.00000 q^{56} +21.6428 q^{57} -5.73837 q^{58} +5.22904 q^{59} -3.18761 q^{60} +7.68398 q^{61} -7.10952 q^{62} -7.16083 q^{63} +1.00000 q^{64} -5.69102 q^{65} -7.39587 q^{67} -0.749022 q^{68} +12.5595 q^{69} +1.00000 q^{70} -0.575191 q^{71} +7.16083 q^{72} -12.5704 q^{73} -11.6050 q^{74} +3.18761 q^{75} +6.78967 q^{76} +18.1407 q^{78} +6.83481 q^{79} -1.00000 q^{80} +20.7950 q^{81} +11.4458 q^{82} +10.5252 q^{83} -3.18761 q^{84} +0.749022 q^{85} +2.33345 q^{86} -18.2916 q^{87} +4.80467 q^{89} -7.16083 q^{90} -5.69102 q^{91} +3.94010 q^{92} -22.6624 q^{93} -5.00025 q^{94} -6.78967 q^{95} +3.18761 q^{96} +16.2717 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.18761	1.84036	0.920182	−	0.391490i	$-0.128040\pi$
$3$	3.18761	1.84036	0.920182	+	0.391490i	$0.128040\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	3.18761	1.30133
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	7.16083	2.38694
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	3.18761	0.920182
$13$	5.69102	1.57841	0.789203	−	0.614133i	$-0.210494\pi$
$13$	5.69102	1.57841	0.789203	+	0.614133i	$0.210494\pi$
$14$	−1.00000	−0.267261
$15$	−3.18761	−0.823036
$16$	1.00000	0.250000
$17$	−0.749022	−0.181664	−0.0908322	−	0.995866i	$-0.528953\pi$
$17$	−0.749022	−0.181664	−0.0908322	+	0.995866i	$0.528953\pi$
$18$	7.16083	1.68782
$19$	6.78967	1.55766	0.778828	−	0.627237i	$-0.215814\pi$
$19$	6.78967	1.55766	0.778828	+	0.627237i	$0.215814\pi$
$20$	−1.00000	−0.223607
$21$	−3.18761	−0.695592
$22$	0	0
$23$	3.94010	0.821567	0.410783	−	0.911733i	$-0.365255\pi$
$23$	3.94010	0.821567	0.410783	+	0.911733i	$0.365255\pi$
$24$	3.18761	0.650667
$25$	1.00000	0.200000
$26$	5.69102	1.11610
$27$	13.2631	2.55248
$28$	−1.00000	−0.188982
$29$	−5.73837	−1.06559	−0.532794	−	0.846245i	$-0.678858\pi$
$29$	−5.73837	−1.06559	−0.532794	+	0.846245i	$0.678858\pi$
$30$	−3.18761	−0.581974
$31$	−7.10952	−1.27691	−0.638454	−	0.769660i	$-0.720426\pi$
$31$	−7.10952	−1.27691	−0.638454	+	0.769660i	$0.720426\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−0.749022	−0.128456
$35$	1.00000	0.169031
$36$	7.16083	1.19347
$37$	−11.6050	−1.90785	−0.953923	−	0.300052i	$-0.902996\pi$
$37$	−11.6050	−1.90785	−0.953923	+	0.300052i	$0.902996\pi$
$38$	6.78967	1.10143
$39$	18.1407	2.90484
$40$	−1.00000	−0.158114
$41$	11.4458	1.78754	0.893768	−	0.448530i	$-0.148052\pi$
$41$	11.4458	1.78754	0.893768	+	0.448530i	$0.148052\pi$
$42$	−3.18761	−0.491858
$43$	2.33345	0.355847	0.177924	−	0.984044i	$-0.443062\pi$
$43$	2.33345	0.355847	0.177924	+	0.984044i	$0.443062\pi$
$44$	0	0
$45$	−7.16083	−1.06747
$46$	3.94010	0.580936
$47$	−5.00025	−0.729361	−0.364680	−	0.931133i	$-0.618822\pi$
$47$	−5.00025	−0.729361	−0.364680	+	0.931133i	$0.618822\pi$
$48$	3.18761	0.460091
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−2.38759	−0.334329
$52$	5.69102	0.789203
$53$	−10.7671	−1.47897	−0.739487	−	0.673171i	$-0.764932\pi$
$53$	−10.7671	−1.47897	−0.739487	+	0.673171i	$0.764932\pi$
$54$	13.2631	1.80488
$55$	0	0
$56$	−1.00000	−0.133631
$57$	21.6428	2.86666
$58$	−5.73837	−0.753484
$59$	5.22904	0.680763	0.340381	−	0.940287i	$-0.389444\pi$
$59$	5.22904	0.680763	0.340381	+	0.940287i	$0.389444\pi$
$60$	−3.18761	−0.411518
$61$	7.68398	0.983833	0.491917	−	0.870642i	$-0.336296\pi$
$61$	7.68398	0.983833	0.491917	+	0.870642i	$0.336296\pi$
$62$	−7.10952	−0.902911
$63$	−7.16083	−0.902179
$64$	1.00000	0.125000
$65$	−5.69102	−0.705884
$66$	0	0
$67$	−7.39587	−0.903549	−0.451775	−	0.892132i	$-0.649209\pi$
$67$	−7.39587	−0.903549	−0.451775	+	0.892132i	$0.649209\pi$
$68$	−0.749022	−0.0908322
$69$	12.5595	1.51198
$70$	1.00000	0.119523
$71$	−0.575191	−0.0682626	−0.0341313	−	0.999417i	$-0.510866\pi$
$71$	−0.575191	−0.0682626	−0.0341313	+	0.999417i	$0.510866\pi$
$72$	7.16083	0.843912
$73$	−12.5704	−1.47125	−0.735627	−	0.677387i	$-0.763112\pi$
$73$	−12.5704	−1.47125	−0.735627	+	0.677387i	$0.763112\pi$
$74$	−11.6050	−1.34905
$75$	3.18761	0.368073
$76$	6.78967	0.778828
$77$	0	0
$78$	18.1407	2.05403
$79$	6.83481	0.768976	0.384488	−	0.923130i	$-0.374378\pi$
$79$	6.83481	0.768976	0.384488	+	0.923130i	$0.374378\pi$
$80$	−1.00000	−0.111803
$81$	20.7950	2.31055
$82$	11.4458	1.26398
$83$	10.5252	1.15530	0.577648	−	0.816286i	$-0.303971\pi$
$83$	10.5252	1.15530	0.577648	+	0.816286i	$0.303971\pi$
$84$	−3.18761	−0.347796
$85$	0.749022	0.0812428
$86$	2.33345	0.251622
$87$	−18.2916	−1.96107
$88$	0	0
$89$	4.80467	0.509294	0.254647	−	0.967034i	$-0.418041\pi$
$89$	4.80467	0.509294	0.254647	+	0.967034i	$0.418041\pi$
$90$	−7.16083	−0.754817
$91$	−5.69102	−0.596581
$92$	3.94010	0.410783
$93$	−22.6624	−2.34998
$94$	−5.00025	−0.515736
$95$	−6.78967	−0.696605
$96$	3.18761	0.325334
$97$	16.2717	1.65214	0.826071	−	0.563566i	$-0.190571\pi$
$97$	16.2717	1.65214	0.826071	+	0.563566i	$0.190571\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−6.38488	−0.635319	−0.317659	−	0.948205i	$-0.602897\pi$
$101$	−6.38488	−0.635319	−0.317659	+	0.948205i	$0.602897\pi$
$102$	−2.38759	−0.236406
$103$	2.32870	0.229454	0.114727	−	0.993397i	$-0.463401\pi$
$103$	2.32870	0.229454	0.114727	+	0.993397i	$0.463401\pi$
$104$	5.69102	0.558051
$105$	3.18761	0.311078
$106$	−10.7671	−1.04579
$107$	1.88876	0.182593	0.0912967	−	0.995824i	$-0.470899\pi$
$107$	1.88876	0.182593	0.0912967	+	0.995824i	$0.470899\pi$
$108$	13.2631	1.27624
$109$	7.76554	0.743804	0.371902	−	0.928272i	$-0.378706\pi$
$109$	7.76554	0.743804	0.371902	+	0.928272i	$0.378706\pi$
$110$	0	0
$111$	−36.9921	−3.51113
$112$	−1.00000	−0.0944911
$113$	7.75283	0.729325	0.364663	−	0.931140i	$-0.381184\pi$
$113$	7.75283	0.729325	0.364663	+	0.931140i	$0.381184\pi$
$114$	21.6428	2.02703
$115$	−3.94010	−0.367416
$116$	−5.73837	−0.532794
$117$	40.7524	3.76756
$118$	5.22904	0.481372
$119$	0.749022	0.0686627
$120$	−3.18761	−0.290987
$121$	0	0
$122$	7.68398	0.695675
$123$	36.4847	3.28972
$124$	−7.10952	−0.638454
$125$	−1.00000	−0.0894427
$126$	−7.16083	−0.637937
$127$	−15.5292	−1.37799	−0.688996	−	0.724765i	$-0.741948\pi$
$127$	−15.5292	−1.37799	−0.688996	+	0.724765i	$0.741948\pi$
$128$	1.00000	0.0883883
$129$	7.43810	0.654889
$130$	−5.69102	−0.499136
$131$	0.957959	0.0836973	0.0418486	−	0.999124i	$-0.486675\pi$
$131$	0.957959	0.0836973	0.0418486	+	0.999124i	$0.486675\pi$
$132$	0	0
$133$	−6.78967	−0.588739
$134$	−7.39587	−0.638906
$135$	−13.2631	−1.14150
$136$	−0.749022	−0.0642281
$137$	−4.66176	−0.398281	−0.199140	−	0.979971i	$-0.563815\pi$
$137$	−4.66176	−0.398281	−0.199140	+	0.979971i	$0.563815\pi$
$138$	12.5595	1.06913
$139$	4.46428	0.378655	0.189328	−	0.981914i	$-0.439369\pi$
$139$	4.46428	0.378655	0.189328	+	0.981914i	$0.439369\pi$
$140$	1.00000	0.0845154
$141$	−15.9388	−1.34229
$142$	−0.575191	−0.0482690
$143$	0	0
$144$	7.16083	0.596736
$145$	5.73837	0.476545
$146$	−12.5704	−1.04033
$147$	3.18761	0.262909
$148$	−11.6050	−0.953923
$149$	15.6247	1.28003	0.640013	−	0.768364i	$-0.278929\pi$
$149$	15.6247	1.28003	0.640013	+	0.768364i	$0.278929\pi$
$150$	3.18761	0.260267
$151$	−17.7850	−1.44732	−0.723660	−	0.690157i	$-0.757542\pi$
$151$	−17.7850	−1.44732	−0.723660	+	0.690157i	$0.757542\pi$
$152$	6.78967	0.550715
$153$	−5.36361	−0.433623
$154$	0	0
$155$	7.10952	0.571051
$156$	18.1407	1.45242
$157$	−7.37542	−0.588623	−0.294311	−	0.955710i	$-0.595090\pi$
$157$	−7.37542	−0.588623	−0.294311	+	0.955710i	$0.595090\pi$
$158$	6.83481	0.543748
$159$	−34.3212	−2.72185
$160$	−1.00000	−0.0790569
$161$	−3.94010	−0.310523
$162$	20.7950	1.63381
$163$	−10.4614	−0.819402	−0.409701	−	0.912220i	$-0.634367\pi$
$163$	−10.4614	−0.819402	−0.409701	+	0.912220i	$0.634367\pi$
$164$	11.4458	0.893768
$165$	0	0
$166$	10.5252	0.816918
$167$	−13.8928	−1.07506	−0.537528	−	0.843246i	$-0.680642\pi$
$167$	−13.8928	−1.07506	−0.537528	+	0.843246i	$0.680642\pi$
$168$	−3.18761	−0.245929
$169$	19.3877	1.49136
$170$	0.749022	0.0574473
$171$	48.6196	3.71804
$172$	2.33345	0.177924
$173$	9.79571	0.744754	0.372377	−	0.928082i	$-0.378543\pi$
$173$	9.79571	0.744754	0.372377	+	0.928082i	$0.378543\pi$
$174$	−18.2916	−1.38669
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	16.6681	1.25285
$178$	4.80467	0.360126
$179$	0.676043	0.0505298	0.0252649	−	0.999681i	$-0.491957\pi$
$179$	0.676043	0.0505298	0.0252649	+	0.999681i	$0.491957\pi$
$180$	−7.16083	−0.533737
$181$	−5.50028	−0.408833	−0.204416	−	0.978884i	$-0.565530\pi$
$181$	−5.50028	−0.408833	−0.204416	+	0.978884i	$0.565530\pi$
$182$	−5.69102	−0.421847
$183$	24.4935	1.81061
$184$	3.94010	0.290468
$185$	11.6050	0.853215
$186$	−22.6624	−1.66168
$187$	0	0
$188$	−5.00025	−0.364680
$189$	−13.2631	−0.964747
$190$	−6.78967	−0.492574
$191$	−12.5963	−0.911434	−0.455717	−	0.890125i	$-0.650617\pi$
$191$	−12.5963	−0.911434	−0.455717	+	0.890125i	$0.650617\pi$
$192$	3.18761	0.230046
$193$	−2.47246	−0.177971	−0.0889857	−	0.996033i	$-0.528363\pi$
$193$	−2.47246	−0.177971	−0.0889857	+	0.996033i	$0.528363\pi$
$194$	16.2717	1.16824
$195$	−18.1407	−1.29908
$196$	1.00000	0.0714286
$197$	8.91129	0.634903	0.317451	−	0.948275i	$-0.397173\pi$
$197$	8.91129	0.634903	0.317451	+	0.948275i	$0.397173\pi$
$198$	0	0
$199$	26.4542	1.87529	0.937644	−	0.347598i	$-0.113002\pi$
$199$	26.4542	1.87529	0.937644	+	0.347598i	$0.113002\pi$
$200$	1.00000	0.0707107
$201$	−23.5751	−1.66286
$202$	−6.38488	−0.449238
$203$	5.73837	0.402754
$204$	−2.38759	−0.167164
$205$	−11.4458	−0.799410
$206$	2.32870	0.162248
$207$	28.2144	1.96103
$208$	5.69102	0.394601
$209$	0	0
$210$	3.18761	0.219966
$211$	−8.29206	−0.570849	−0.285424	−	0.958401i	$-0.592135\pi$
$211$	−8.29206	−0.570849	−0.285424	+	0.958401i	$0.592135\pi$
$212$	−10.7671	−0.739487
$213$	−1.83348	−0.125628
$214$	1.88876	0.129113
$215$	−2.33345	−0.159140
$216$	13.2631	0.902438
$217$	7.10952	0.482626
$218$	7.76554	0.525949
$219$	−40.0695	−2.70764
$220$	0	0
$221$	−4.26270	−0.286740
$222$	−36.9921	−2.48275
$223$	0.390611	0.0261573	0.0130786	−	0.999914i	$-0.495837\pi$
$223$	0.390611	0.0261573	0.0130786	+	0.999914i	$0.495837\pi$
$224$	−1.00000	−0.0668153
$225$	7.16083	0.477388
$226$	7.75283	0.515711
$227$	−4.91748	−0.326385	−0.163192	−	0.986594i	$-0.552179\pi$
$227$	−4.91748	−0.326385	−0.163192	+	0.986594i	$0.552179\pi$
$228$	21.6428	1.43333
$229$	−9.18360	−0.606869	−0.303435	−	0.952852i	$-0.598133\pi$
$229$	−9.18360	−0.606869	−0.303435	+	0.952852i	$0.598133\pi$
$230$	−3.94010	−0.259802
$231$	0	0
$232$	−5.73837	−0.376742
$233$	−20.1599	−1.32072	−0.660359	−	0.750950i	$-0.729596\pi$
$233$	−20.1599	−1.32072	−0.660359	+	0.750950i	$0.729596\pi$
$234$	40.7524	2.66407
$235$	5.00025	0.326180
$236$	5.22904	0.340381
$237$	21.7867	1.41520
$238$	0.749022	0.0485519
$239$	9.77852	0.632520	0.316260	−	0.948673i	$-0.397573\pi$
$239$	9.77852	0.632520	0.316260	+	0.948673i	$0.397573\pi$
$240$	−3.18761	−0.205759
$241$	−12.3008	−0.792366	−0.396183	−	0.918172i	$-0.629665\pi$
$241$	−12.3008	−0.792366	−0.396183	+	0.918172i	$0.629665\pi$
$242$	0	0
$243$	26.4969	1.69978
$244$	7.68398	0.491917
$245$	−1.00000	−0.0638877
$246$	36.4847	2.32618
$247$	38.6401	2.45861
$248$	−7.10952	−0.451455
$249$	33.5503	2.12617
$250$	−1.00000	−0.0632456
$251$	−20.8441	−1.31567	−0.657833	−	0.753164i	$-0.728527\pi$
$251$	−20.8441	−1.31567	−0.657833	+	0.753164i	$0.728527\pi$
$252$	−7.16083	−0.451090
$253$	0	0
$254$	−15.5292	−0.974388
$255$	2.38759	0.149516
$256$	1.00000	0.0625000
$257$	7.52911	0.469653	0.234827	−	0.972037i	$-0.424548\pi$
$257$	7.52911	0.469653	0.234827	+	0.972037i	$0.424548\pi$
$258$	7.43810	0.463076
$259$	11.6050	0.721098
$260$	−5.69102	−0.352942
$261$	−41.0914	−2.54350
$262$	0.957959	0.0591829
$263$	3.27610	0.202013	0.101007	−	0.994886i	$-0.467794\pi$
$263$	3.27610	0.202013	0.101007	+	0.994886i	$0.467794\pi$
$264$	0	0
$265$	10.7671	0.661417
$266$	−6.78967	−0.416301
$267$	15.3154	0.937287
$268$	−7.39587	−0.451775
$269$	1.12272	0.0684535	0.0342267	−	0.999414i	$-0.489103\pi$
$269$	1.12272	0.0684535	0.0342267	+	0.999414i	$0.489103\pi$
$270$	−13.2631	−0.807165
$271$	9.82214	0.596652	0.298326	−	0.954464i	$-0.403572\pi$
$271$	9.82214	0.596652	0.298326	+	0.954464i	$0.403572\pi$
$272$	−0.749022	−0.0454161
$273$	−18.1407	−1.09793
$274$	−4.66176	−0.281627
$275$	0	0
$276$	12.5595	0.755991
$277$	−0.891218	−0.0535481	−0.0267741	−	0.999642i	$-0.508523\pi$
$277$	−0.891218	−0.0535481	−0.0267741	+	0.999642i	$0.508523\pi$
$278$	4.46428	0.267750
$279$	−50.9101	−3.04791
$280$	1.00000	0.0597614
$281$	17.3571	1.03544	0.517720	−	0.855550i	$-0.326781\pi$
$281$	17.3571	1.03544	0.517720	+	0.855550i	$0.326781\pi$
$282$	−15.9388	−0.949142
$283$	−2.98931	−0.177696	−0.0888481	−	0.996045i	$-0.528319\pi$
$283$	−2.98931	−0.177696	−0.0888481	+	0.996045i	$0.528319\pi$
$284$	−0.575191	−0.0341313
$285$	−21.6428	−1.28201
$286$	0	0
$287$	−11.4458	−0.675625
$288$	7.16083	0.421956
$289$	−16.4390	−0.966998
$290$	5.73837	0.336968
$291$	51.8678	3.04054
$292$	−12.5704	−0.735627
$293$	17.5887	1.02754	0.513771	−	0.857927i	$-0.328248\pi$
$293$	17.5887	1.02754	0.513771	+	0.857927i	$0.328248\pi$
$294$	3.18761	0.185905
$295$	−5.22904	−0.304446
$296$	−11.6050	−0.674525
$297$	0	0
$298$	15.6247	0.905115
$299$	22.4232	1.29677
$300$	3.18761	0.184036
$301$	−2.33345	−0.134498
$302$	−17.7850	−1.02341
$303$	−20.3525	−1.16922
$304$	6.78967	0.389414
$305$	−7.68398	−0.439984
$306$	−5.36361	−0.306617
$307$	1.28791	0.0735047	0.0367524	−	0.999324i	$-0.488299\pi$
$307$	1.28791	0.0735047	0.0367524	+	0.999324i	$0.488299\pi$
$308$	0	0
$309$	7.42298	0.422279
$310$	7.10952	0.403794
$311$	−0.0496330	−0.00281443	−0.00140721	−	0.999999i	$-0.500448\pi$
$311$	−0.0496330	−0.00281443	−0.00140721	+	0.999999i	$0.500448\pi$
$312$	18.1407	1.02702
$313$	−27.6778	−1.56444	−0.782220	−	0.623003i	$-0.785913\pi$
$313$	−27.6778	−1.56444	−0.782220	+	0.623003i	$0.785913\pi$
$314$	−7.37542	−0.416219
$315$	7.16083	0.403467
$316$	6.83481	0.384488
$317$	9.23290	0.518571	0.259286	−	0.965801i	$-0.416513\pi$
$317$	9.23290	0.518571	0.259286	+	0.965801i	$0.416513\pi$
$318$	−34.3212	−1.92464
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	6.02062	0.336038
$322$	−3.94010	−0.219573
$323$	−5.08561	−0.282971
$324$	20.7950	1.15528
$325$	5.69102	0.315681
$326$	−10.4614	−0.579405
$327$	24.7535	1.36887
$328$	11.4458	0.631989
$329$	5.00025	0.275673
$330$	0	0
$331$	16.5715	0.910851	0.455425	−	0.890274i	$-0.349487\pi$
$331$	16.5715	0.910851	0.455425	+	0.890274i	$0.349487\pi$
$332$	10.5252	0.577648
$333$	−83.1012	−4.55392
$334$	−13.8928	−0.760180
$335$	7.39587	0.404080
$336$	−3.18761	−0.173898
$337$	−6.46713	−0.352287	−0.176144	−	0.984364i	$-0.556362\pi$
$337$	−6.46713	−0.352287	−0.176144	+	0.984364i	$0.556362\pi$
$338$	19.3877	1.05455
$339$	24.7130	1.34222
$340$	0.749022	0.0406214
$341$	0	0
$342$	48.6196	2.62905
$343$	−1.00000	−0.0539949
$344$	2.33345	0.125811
$345$	−12.5595	−0.676179
$346$	9.79571	0.526621
$347$	−29.5567	−1.58669	−0.793344	−	0.608774i	$-0.791662\pi$
$347$	−29.5567	−1.58669	−0.793344	+	0.608774i	$0.791662\pi$
$348$	−18.2916	−0.980535
$349$	−25.4079	−1.36005	−0.680027	−	0.733187i	$-0.738032\pi$
$349$	−25.4079	−1.36005	−0.680027	+	0.733187i	$0.738032\pi$
$350$	−1.00000	−0.0534522
$351$	75.4804	4.02885
$352$	0	0
$353$	13.7939	0.734174	0.367087	−	0.930187i	$-0.380355\pi$
$353$	13.7939	0.734174	0.367087	+	0.930187i	$0.380355\pi$
$354$	16.6681	0.885900
$355$	0.575191	0.0305280
$356$	4.80467	0.254647
$357$	2.38759	0.126364
$358$	0.676043	0.0357300
$359$	22.7783	1.20219	0.601096	−	0.799177i	$-0.294731\pi$
$359$	22.7783	1.20219	0.601096	+	0.799177i	$0.294731\pi$
$360$	−7.16083	−0.377409
$361$	27.0996	1.42629
$362$	−5.50028	−0.289089
$363$	0	0
$364$	−5.69102	−0.298291
$365$	12.5704	0.657965
$366$	24.4935	1.28030
$367$	−9.71895	−0.507325	−0.253662	−	0.967293i	$-0.581635\pi$
$367$	−9.71895	−0.507325	−0.253662	+	0.967293i	$0.581635\pi$
$368$	3.94010	0.205392
$369$	81.9615	4.26674
$370$	11.6050	0.603314
$371$	10.7671	0.558999
$372$	−22.6624	−1.17499
$373$	−6.44329	−0.333621	−0.166811	−	0.985989i	$-0.553347\pi$
$373$	−6.44329	−0.333621	−0.166811	+	0.985989i	$0.553347\pi$
$374$	0	0
$375$	−3.18761	−0.164607
$376$	−5.00025	−0.257868
$377$	−32.6572	−1.68193
$378$	−13.2631	−0.682179
$379$	−28.8120	−1.47997	−0.739987	−	0.672621i	$-0.765169\pi$
$379$	−28.8120	−1.47997	−0.739987	+	0.672621i	$0.765169\pi$
$380$	−6.78967	−0.348303
$381$	−49.5009	−2.53601
$382$	−12.5963	−0.644481
$383$	19.2162	0.981903	0.490952	−	0.871187i	$-0.336649\pi$
$383$	19.2162	0.981903	0.490952	+	0.871187i	$0.336649\pi$
$384$	3.18761	0.162667
$385$	0	0
$386$	−2.47246	−0.125845
$387$	16.7094	0.849387
$388$	16.2717	0.826071
$389$	−1.16875	−0.0592582	−0.0296291	−	0.999561i	$-0.509433\pi$
$389$	−1.16875	−0.0592582	−0.0296291	+	0.999561i	$0.509433\pi$
$390$	−18.1407	−0.918592
$391$	−2.95122	−0.149249
$392$	1.00000	0.0505076
$393$	3.05360	0.154034
$394$	8.91129	0.448944
$395$	−6.83481	−0.343896
$396$	0	0
$397$	−34.1835	−1.71562	−0.857811	−	0.513966i	$-0.828176\pi$
$397$	−34.1835	−1.71562	−0.857811	+	0.513966i	$0.828176\pi$
$398$	26.4542	1.32603
$399$	−21.6428	−1.08349
$400$	1.00000	0.0500000
$401$	24.0445	1.20072	0.600362	−	0.799728i	$-0.295023\pi$
$401$	24.0445	1.20072	0.600362	+	0.799728i	$0.295023\pi$
$402$	−23.5751	−1.17582
$403$	−40.4605	−2.01548
$404$	−6.38488	−0.317659
$405$	−20.7950	−1.03331
$406$	5.73837	0.284790
$407$	0	0
$408$	−2.38759	−0.118203
$409$	−5.46092	−0.270025	−0.135013	−	0.990844i	$-0.543107\pi$
$409$	−5.46092	−0.270025	−0.135013	+	0.990844i	$0.543107\pi$
$410$	−11.4458	−0.565268
$411$	−14.8598	−0.732982
$412$	2.32870	0.114727
$413$	−5.22904	−0.257304
$414$	28.2144	1.38666
$415$	−10.5252	−0.516664
$416$	5.69102	0.279025
$417$	14.2304	0.696864
$418$	0	0
$419$	−37.5281	−1.83337	−0.916684	−	0.399613i	$-0.869145\pi$
$419$	−37.5281	−1.83337	−0.916684	+	0.399613i	$0.869145\pi$
$420$	3.18761	0.155539
$421$	9.35357	0.455865	0.227933	−	0.973677i	$-0.426803\pi$
$421$	9.35357	0.455865	0.227933	+	0.973677i	$0.426803\pi$
$422$	−8.29206	−0.403651
$423$	−35.8059	−1.74094
$424$	−10.7671	−0.522896
$425$	−0.749022	−0.0363329
$426$	−1.83348	−0.0888325
$427$	−7.68398	−0.371854
$428$	1.88876	0.0912967
$429$	0	0
$430$	−2.33345	−0.112529
$431$	−5.10019	−0.245668	−0.122834	−	0.992427i	$-0.539198\pi$
$431$	−5.10019	−0.245668	−0.122834	+	0.992427i	$0.539198\pi$
$432$	13.2631	0.638120
$433$	−15.0790	−0.724649	−0.362325	−	0.932052i	$-0.618017\pi$
$433$	−15.0790	−0.724649	−0.362325	+	0.932052i	$0.618017\pi$
$434$	7.10952	0.341268
$435$	18.2916	0.877017
$436$	7.76554	0.371902
$437$	26.7520	1.27972
$438$	−40.0695	−1.91459
$439$	−8.35774	−0.398893	−0.199447	−	0.979909i	$-0.563914\pi$
$439$	−8.35774	−0.398893	−0.199447	+	0.979909i	$0.563914\pi$
$440$	0	0
$441$	7.16083	0.340992
$442$	−4.26270	−0.202756
$443$	6.68524	0.317626	0.158813	−	0.987309i	$-0.449233\pi$
$443$	6.68524	0.317626	0.158813	+	0.987309i	$0.449233\pi$
$444$	−36.9921	−1.75557
$445$	−4.80467	−0.227763
$446$	0.390611	0.0184960
$447$	49.8054	2.35572
$448$	−1.00000	−0.0472456
$449$	−7.97205	−0.376224	−0.188112	−	0.982148i	$-0.560237\pi$
$449$	−7.97205	−0.376224	−0.188112	+	0.982148i	$0.560237\pi$
$450$	7.16083	0.337565
$451$	0	0
$452$	7.75283	0.364663
$453$	−56.6915	−2.66360
$454$	−4.91748	−0.230789
$455$	5.69102	0.266799
$456$	21.6428	1.01352
$457$	−2.34209	−0.109558	−0.0547791	−	0.998498i	$-0.517445\pi$
$457$	−2.34209	−0.109558	−0.0547791	+	0.998498i	$0.517445\pi$
$458$	−9.18360	−0.429121
$459$	−9.93433	−0.463695
$460$	−3.94010	−0.183708
$461$	−0.312762	−0.0145668	−0.00728340	−	0.999973i	$-0.502318\pi$
$461$	−0.312762	−0.0145668	−0.00728340	+	0.999973i	$0.502318\pi$
$462$	0	0
$463$	15.8769	0.737863	0.368931	−	0.929457i	$-0.379724\pi$
$463$	15.8769	0.737863	0.368931	+	0.929457i	$0.379724\pi$
$464$	−5.73837	−0.266397
$465$	22.6624	1.05094
$466$	−20.1599	−0.933888
$467$	−0.774386	−0.0358343	−0.0179172	−	0.999839i	$-0.505704\pi$
$467$	−0.774386	−0.0358343	−0.0179172	+	0.999839i	$0.505704\pi$
$468$	40.7524	1.88378
$469$	7.39587	0.341510
$470$	5.00025	0.230644
$471$	−23.5099	−1.08328
$472$	5.22904	0.240686
$473$	0	0
$474$	21.7867	1.00069
$475$	6.78967	0.311531
$476$	0.749022	0.0343313
$477$	−77.1013	−3.53022
$478$	9.77852	0.447259
$479$	30.3787	1.38804	0.694019	−	0.719956i	$-0.255838\pi$
$479$	30.3787	1.38804	0.694019	+	0.719956i	$0.255838\pi$
$480$	−3.18761	−0.145494
$481$	−66.0441	−3.01135
$482$	−12.3008	−0.560287
$483$	−12.5595	−0.571476
$484$	0	0
$485$	−16.2717	−0.738861
$486$	26.4969	1.20192
$487$	9.43710	0.427636	0.213818	−	0.976874i	$-0.431410\pi$
$487$	9.43710	0.427636	0.213818	+	0.976874i	$0.431410\pi$
$488$	7.68398	0.347838
$489$	−33.3469	−1.50800
$490$	−1.00000	−0.0451754
$491$	28.7129	1.29579	0.647897	−	0.761728i	$-0.275649\pi$
$491$	28.7129	1.29579	0.647897	+	0.761728i	$0.275649\pi$
$492$	36.4847	1.64486
$493$	4.29816	0.193579
$494$	38.6401	1.73850
$495$	0	0
$496$	−7.10952	−0.319227
$497$	0.575191	0.0258008
$498$	33.5503	1.50343
$499$	27.6602	1.23824	0.619120	−	0.785297i	$-0.287490\pi$
$499$	27.6602	1.23824	0.619120	+	0.785297i	$0.287490\pi$
$500$	−1.00000	−0.0447214
$501$	−44.2848	−1.97850
$502$	−20.8441	−0.930316
$503$	−1.65413	−0.0737539	−0.0368770	−	0.999320i	$-0.511741\pi$
$503$	−1.65413	−0.0737539	−0.0368770	+	0.999320i	$0.511741\pi$
$504$	−7.16083	−0.318969
$505$	6.38488	0.284123
$506$	0	0
$507$	61.8004	2.74465
$508$	−15.5292	−0.688996
$509$	11.0335	0.489050	0.244525	−	0.969643i	$-0.421368\pi$
$509$	11.0335	0.489050	0.244525	+	0.969643i	$0.421368\pi$
$510$	2.38759	0.105724
$511$	12.5704	0.556082
$512$	1.00000	0.0441942
$513$	90.0519	3.97589
$514$	7.52911	0.332095
$515$	−2.32870	−0.102615
$516$	7.43810	0.327444
$517$	0	0
$518$	11.6050	0.509893
$519$	31.2249	1.37062
$520$	−5.69102	−0.249568
$521$	19.2485	0.843294	0.421647	−	0.906760i	$-0.361452\pi$
$521$	19.2485	0.843294	0.421647	+	0.906760i	$0.361452\pi$
$522$	−41.0914	−1.79852
$523$	4.09540	0.179079	0.0895396	−	0.995983i	$-0.471460\pi$
$523$	4.09540	0.179079	0.0895396	+	0.995983i	$0.471460\pi$
$524$	0.957959	0.0418486
$525$	−3.18761	−0.139118
$526$	3.27610	0.142845
$527$	5.32519	0.231969
$528$	0	0
$529$	−7.47564	−0.325028
$530$	10.7671	0.467692
$531$	37.4442	1.62494
$532$	−6.78967	−0.294369
$533$	65.1383	2.82146
$534$	15.3154	0.662762
$535$	−1.88876	−0.0816582
$536$	−7.39587	−0.319453
$537$	2.15496	0.0929933
$538$	1.12272	0.0484039
$539$	0	0
$540$	−13.2631	−0.570752
$541$	−12.0631	−0.518632	−0.259316	−	0.965793i	$-0.583497\pi$
$541$	−12.0631	−0.518632	−0.259316	+	0.965793i	$0.583497\pi$
$542$	9.82214	0.421897
$543$	−17.5327	−0.752402
$544$	−0.749022	−0.0321140
$545$	−7.76554	−0.332639
$546$	−18.1407	−0.776352
$547$	−5.69776	−0.243619	−0.121809	−	0.992554i	$-0.538870\pi$
$547$	−5.69776	−0.243619	−0.121809	+	0.992554i	$0.538870\pi$
$548$	−4.66176	−0.199140
$549$	55.0237	2.34835
$550$	0	0
$551$	−38.9616	−1.65982
$552$	12.5595	0.534567
$553$	−6.83481	−0.290645
$554$	−0.891218	−0.0378642
$555$	36.9921	1.57023
$556$	4.46428	0.189328
$557$	8.87789	0.376168	0.188084	−	0.982153i	$-0.439772\pi$
$557$	8.87789	0.376168	0.188084	+	0.982153i	$0.439772\pi$
$558$	−50.9101	−2.15520
$559$	13.2797	0.561671
$560$	1.00000	0.0422577
$561$	0	0
$562$	17.3571	0.732167
$563$	20.1280	0.848294	0.424147	−	0.905593i	$-0.360574\pi$
$563$	20.1280	0.848294	0.424147	+	0.905593i	$0.360574\pi$
$564$	−15.9388	−0.671145
$565$	−7.75283	−0.326164
$566$	−2.98931	−0.125650
$567$	−20.7950	−0.873306
$568$	−0.575191	−0.0241345
$569$	24.8357	1.04117	0.520583	−	0.853811i	$-0.325715\pi$
$569$	24.8357	1.04117	0.520583	+	0.853811i	$0.325715\pi$
$570$	−21.6428	−0.906516
$571$	−28.6955	−1.20087	−0.600435	−	0.799674i	$-0.705006\pi$
$571$	−28.6955	−1.20087	−0.600435	+	0.799674i	$0.705006\pi$
$572$	0	0
$573$	−40.1520	−1.67737
$574$	−11.4458	−0.477739
$575$	3.94010	0.164313
$576$	7.16083	0.298368
$577$	−15.3853	−0.640499	−0.320249	−	0.947333i	$-0.603767\pi$
$577$	−15.3853	−0.640499	−0.320249	+	0.947333i	$0.603767\pi$
$578$	−16.4390	−0.683771
$579$	−7.88122	−0.327532
$580$	5.73837	0.238273
$581$	−10.5252	−0.436661
$582$	51.8678	2.14999
$583$	0	0
$584$	−12.5704	−0.520167
$585$	−40.7524	−1.68491
$586$	17.5887	0.726582
$587$	−36.1775	−1.49321	−0.746603	−	0.665270i	$-0.768317\pi$
$587$	−36.1775	−1.49321	−0.746603	+	0.665270i	$0.768317\pi$
$588$	3.18761	0.131455
$589$	−48.2713	−1.98898
$590$	−5.22904	−0.215276
$591$	28.4057	1.16845
$592$	−11.6050	−0.476961
$593$	22.6239	0.929052	0.464526	−	0.885560i	$-0.346225\pi$
$593$	22.6239	0.929052	0.464526	+	0.885560i	$0.346225\pi$
$594$	0	0
$595$	−0.749022	−0.0307069
$596$	15.6247	0.640013
$597$	84.3255	3.45121
$598$	22.4232	0.916952
$599$	2.93462	0.119906	0.0599528	−	0.998201i	$-0.480905\pi$
$599$	2.93462	0.119906	0.0599528	+	0.998201i	$0.480905\pi$
$600$	3.18761	0.130133
$601$	−33.3078	−1.35865	−0.679327	−	0.733836i	$-0.737728\pi$
$601$	−33.3078	−1.35865	−0.679327	+	0.733836i	$0.737728\pi$
$602$	−2.33345	−0.0951041
$603$	−52.9606	−2.15672
$604$	−17.7850	−0.723660
$605$	0	0
$606$	−20.3525	−0.826762
$607$	27.9310	1.13369	0.566843	−	0.823826i	$-0.308165\pi$
$607$	27.9310	1.13369	0.566843	+	0.823826i	$0.308165\pi$
$608$	6.78967	0.275357
$609$	18.2916	0.741215
$610$	−7.68398	−0.311115
$611$	−28.4565	−1.15123
$612$	−5.36361	−0.216811
$613$	13.2976	0.537087	0.268543	−	0.963268i	$-0.413458\pi$
$613$	13.2976	0.537087	0.268543	+	0.963268i	$0.413458\pi$
$614$	1.28791	0.0519757
$615$	−36.4847	−1.47121
$616$	0	0
$617$	−17.0581	−0.686735	−0.343368	−	0.939201i	$-0.611568\pi$
$617$	−17.0581	−0.686735	−0.343368	+	0.939201i	$0.611568\pi$
$618$	7.42298	0.298596
$619$	−29.0446	−1.16740	−0.583700	−	0.811969i	$-0.698396\pi$
$619$	−29.0446	−1.16740	−0.583700	+	0.811969i	$0.698396\pi$
$620$	7.10952	0.285525
$621$	52.2578	2.09703
$622$	−0.0496330	−0.00199010
$623$	−4.80467	−0.192495
$624$	18.1407	0.726210
$625$	1.00000	0.0400000
$626$	−27.6778	−1.10623
$627$	0	0
$628$	−7.37542	−0.294311
$629$	8.69238	0.346588
$630$	7.16083	0.285294
$631$	32.0364	1.27535	0.637675	−	0.770305i	$-0.279896\pi$
$631$	32.0364	1.27535	0.637675	+	0.770305i	$0.279896\pi$
$632$	6.83481	0.271874
$633$	−26.4318	−1.05057
$634$	9.23290	0.366685
$635$	15.5292	0.616257
$636$	−34.3212	−1.36093
$637$	5.69102	0.225486
$638$	0	0
$639$	−4.11884	−0.162939
$640$	−1.00000	−0.0395285
$641$	7.75431	0.306277	0.153138	−	0.988205i	$-0.451062\pi$
$641$	7.75431	0.306277	0.153138	+	0.988205i	$0.451062\pi$
$642$	6.02062	0.237615
$643$	24.2104	0.954763	0.477382	−	0.878696i	$-0.341586\pi$
$643$	24.2104	0.954763	0.477382	+	0.878696i	$0.341586\pi$
$644$	−3.94010	−0.155262
$645$	−7.43810	−0.292875
$646$	−5.08561	−0.200091
$647$	−0.861530	−0.0338702	−0.0169351	−	0.999857i	$-0.505391\pi$
$647$	−0.861530	−0.0338702	−0.0169351	+	0.999857i	$0.505391\pi$
$648$	20.7950	0.816903
$649$	0	0
$650$	5.69102	0.223220
$651$	22.6624	0.888208
$652$	−10.4614	−0.409701
$653$	−21.7225	−0.850068	−0.425034	−	0.905177i	$-0.639738\pi$
$653$	−21.7225	−0.850068	−0.425034	+	0.905177i	$0.639738\pi$
$654$	24.7535	0.967938
$655$	−0.957959	−0.0374306
$656$	11.4458	0.446884
$657$	−90.0144	−3.51180
$658$	5.00025	0.194930
$659$	−35.0148	−1.36398	−0.681991	−	0.731360i	$-0.738886\pi$
$659$	−35.0148	−1.36398	−0.681991	+	0.731360i	$0.738886\pi$
$660$	0	0
$661$	21.6301	0.841315	0.420657	−	0.907220i	$-0.361799\pi$
$661$	21.6301	0.841315	0.420657	+	0.907220i	$0.361799\pi$
$662$	16.5715	0.644069
$663$	−13.5878	−0.527706
$664$	10.5252	0.408459
$665$	6.78967	0.263292
$666$	−83.1012	−3.22011
$667$	−22.6097	−0.875452
$668$	−13.8928	−0.537528
$669$	1.24511	0.0481389
$670$	7.39587	0.285727
$671$	0	0
$672$	−3.18761	−0.122965
$673$	−34.4741	−1.32888	−0.664440	−	0.747342i	$-0.731330\pi$
$673$	−34.4741	−1.32888	−0.664440	+	0.747342i	$0.731330\pi$
$674$	−6.46713	−0.249105
$675$	13.2631	0.510496
$676$	19.3877	0.745682
$677$	−13.5315	−0.520058	−0.260029	−	0.965601i	$-0.583732\pi$
$677$	−13.5315	−0.520058	−0.260029	+	0.965601i	$0.583732\pi$
$678$	24.7130	0.949096
$679$	−16.2717	−0.624451
$680$	0.749022	0.0287237
$681$	−15.6750	−0.600667
$682$	0	0
$683$	19.6188	0.750694	0.375347	−	0.926884i	$-0.377524\pi$
$683$	19.6188	0.750694	0.375347	+	0.926884i	$0.377524\pi$
$684$	48.6196	1.85902
$685$	4.66176	0.178117
$686$	−1.00000	−0.0381802
$687$	−29.2737	−1.11686
$688$	2.33345	0.0889618
$689$	−61.2757	−2.33442
$690$	−12.5595	−0.478131
$691$	−27.4372	−1.04376	−0.521880	−	0.853019i	$-0.674769\pi$
$691$	−27.4372	−1.04376	−0.521880	+	0.853019i	$0.674769\pi$
$692$	9.79571	0.372377
$693$	0	0
$694$	−29.5567	−1.12196
$695$	−4.46428	−0.169340
$696$	−18.2916	−0.693343
$697$	−8.57316	−0.324732
$698$	−25.4079	−0.961703
$699$	−64.2617	−2.43060
$700$	−1.00000	−0.0377964
$701$	−32.2446	−1.21786	−0.608930	−	0.793224i	$-0.708401\pi$
$701$	−32.2446	−1.21786	−0.608930	+	0.793224i	$0.708401\pi$
$702$	75.4804	2.84883
$703$	−78.7939	−2.97177
$704$	0	0
$705$	15.9388	0.600290
$706$	13.7939	0.519139
$707$	6.38488	0.240128
$708$	16.6681	0.626426
$709$	20.8485	0.782982	0.391491	−	0.920182i	$-0.371959\pi$
$709$	20.8485	0.782982	0.391491	+	0.920182i	$0.371959\pi$
$710$	0.575191	0.0215865
$711$	48.9429	1.83550
$712$	4.80467	0.180063
$713$	−28.0122	−1.04907
$714$	2.38759	0.0893531
$715$	0	0
$716$	0.676043	0.0252649
$717$	31.1701	1.16407
$718$	22.7783	0.850078
$719$	−40.3188	−1.50364	−0.751818	−	0.659370i	$-0.770823\pi$
$719$	−40.3188	−1.50364	−0.751818	+	0.659370i	$0.770823\pi$
$720$	−7.16083	−0.266868
$721$	−2.32870	−0.0867254
$722$	27.0996	1.00854
$723$	−39.2102	−1.45824
$724$	−5.50028	−0.204416
$725$	−5.73837	−0.213118
$726$	0	0
$727$	39.1899	1.45347	0.726736	−	0.686917i	$-0.241036\pi$
$727$	39.1899	1.45347	0.726736	+	0.686917i	$0.241036\pi$
$728$	−5.69102	−0.210923
$729$	22.0768	0.817660
$730$	12.5704	0.465251
$731$	−1.74780	−0.0646448
$732$	24.4935	0.905306
$733$	−24.4077	−0.901518	−0.450759	−	0.892646i	$-0.648847\pi$
$733$	−24.4077	−0.901518	−0.450759	+	0.892646i	$0.648847\pi$
$734$	−9.71895	−0.358733
$735$	−3.18761	−0.117577
$736$	3.94010	0.145234
$737$	0	0
$738$	81.9615	3.01704
$739$	−41.5299	−1.52770	−0.763851	−	0.645392i	$-0.776694\pi$
$739$	−41.5299	−1.52770	−0.763851	+	0.645392i	$0.776694\pi$
$740$	11.6050	0.426607
$741$	123.170	4.52475
$742$	10.7671	0.395272
$743$	−22.6025	−0.829207	−0.414603	−	0.910002i	$-0.636080\pi$
$743$	−22.6025	−0.829207	−0.414603	+	0.910002i	$0.636080\pi$
$744$	−22.6624	−0.830842
$745$	−15.6247	−0.572445
$746$	−6.44329	−0.235906
$747$	75.3695	2.75763
$748$	0	0
$749$	−1.88876	−0.0690138
$750$	−3.18761	−0.116395
$751$	−22.7802	−0.831262	−0.415631	−	0.909533i	$-0.636439\pi$
$751$	−22.7802	−0.831262	−0.415631	+	0.909533i	$0.636439\pi$
$752$	−5.00025	−0.182340
$753$	−66.4426	−2.42130
$754$	−32.6572	−1.18930
$755$	17.7850	0.647261
$756$	−13.2631	−0.482373
$757$	6.77323	0.246177	0.123089	−	0.992396i	$-0.460720\pi$
$757$	6.77323	0.246177	0.123089	+	0.992396i	$0.460720\pi$
$758$	−28.8120	−1.04650
$759$	0	0
$760$	−6.78967	−0.246287
$761$	7.24515	0.262637	0.131318	−	0.991340i	$-0.458079\pi$
$761$	7.24515	0.262637	0.131318	+	0.991340i	$0.458079\pi$
$762$	−49.5009	−1.79323
$763$	−7.76554	−0.281132
$764$	−12.5963	−0.455717
$765$	5.36361	0.193922
$766$	19.2162	0.694310
$767$	29.7586	1.07452
$768$	3.18761	0.115023
$769$	−1.45679	−0.0525333	−0.0262667	−	0.999655i	$-0.508362\pi$
$769$	−1.45679	−0.0525333	−0.0262667	+	0.999655i	$0.508362\pi$
$770$	0	0
$771$	23.9998	0.864333
$772$	−2.47246	−0.0889857
$773$	−25.4505	−0.915390	−0.457695	−	0.889109i	$-0.651325\pi$
$773$	−25.4505	−0.915390	−0.457695	+	0.889109i	$0.651325\pi$
$774$	16.7094	0.600607
$775$	−7.10952	−0.255382
$776$	16.2717	0.584121
$777$	36.9921	1.32708
$778$	−1.16875	−0.0419018
$779$	77.7133	2.78437
$780$	−18.1407	−0.649542
$781$	0	0
$782$	−2.95122	−0.105535
$783$	−76.1084	−2.71989
$784$	1.00000	0.0357143
$785$	7.37542	0.263240
$786$	3.05360	0.108918
$787$	18.9942	0.677069	0.338535	−	0.940954i	$-0.390069\pi$
$787$	18.9942	0.677069	0.338535	+	0.940954i	$0.390069\pi$
$788$	8.91129	0.317451
$789$	10.4429	0.371778
$790$	−6.83481	−0.243171
$791$	−7.75283	−0.275659
$792$	0	0
$793$	43.7297	1.55289
$794$	−34.1835	−1.21313
$795$	34.3212	1.21725
$796$	26.4542	0.937644
$797$	−44.9953	−1.59381	−0.796907	−	0.604102i	$-0.793532\pi$
$797$	−44.9953	−1.59381	−0.796907	+	0.604102i	$0.793532\pi$
$798$	−21.6428	−0.766146
$799$	3.74529	0.132499
$800$	1.00000	0.0353553
$801$	34.4054	1.21566
$802$	24.0445	0.849041
$803$	0	0
$804$	−23.5751	−0.831430
$805$	3.94010	0.138870
$806$	−40.4605	−1.42516
$807$	3.57879	0.125979
$808$	−6.38488	−0.224619
$809$	−31.2609	−1.09907	−0.549537	−	0.835469i	$-0.685196\pi$
$809$	−31.2609	−1.09907	−0.549537	+	0.835469i	$0.685196\pi$
$810$	−20.7950	−0.730661
$811$	27.5975	0.969078	0.484539	−	0.874770i	$-0.338987\pi$
$811$	27.5975	0.969078	0.484539	+	0.874770i	$0.338987\pi$
$812$	5.73837	0.201377
$813$	31.3091	1.09806
$814$	0	0
$815$	10.4614	0.366448
$816$	−2.38759	−0.0835822
$817$	15.8433	0.554288
$818$	−5.46092	−0.190937
$819$	−40.7524	−1.42400
$820$	−11.4458	−0.399705
$821$	−21.3493	−0.745096	−0.372548	−	0.928013i	$-0.621516\pi$
$821$	−21.3493	−0.745096	−0.372548	+	0.928013i	$0.621516\pi$
$822$	−14.8598	−0.518296
$823$	−52.8518	−1.84230	−0.921149	−	0.389210i	$-0.872748\pi$
$823$	−52.8518	−1.84230	−0.921149	+	0.389210i	$0.872748\pi$
$824$	2.32870	0.0811242
$825$	0	0
$826$	−5.22904	−0.181941
$827$	17.8294	0.619990	0.309995	−	0.950738i	$-0.399673\pi$
$827$	17.8294	0.619990	0.309995	+	0.950738i	$0.399673\pi$
$828$	28.2144	0.980517
$829$	−46.2177	−1.60521	−0.802603	−	0.596513i	$-0.796552\pi$
$829$	−46.2177	−1.60521	−0.802603	+	0.596513i	$0.796552\pi$
$830$	−10.5252	−0.365337
$831$	−2.84085	−0.0985481
$832$	5.69102	0.197301
$833$	−0.749022	−0.0259521
$834$	14.2304	0.492757
$835$	13.8928	0.480780
$836$	0	0
$837$	−94.2942	−3.25928
$838$	−37.5281	−1.29639
$839$	−19.4145	−0.670263	−0.335131	−	0.942171i	$-0.608781\pi$
$839$	−19.4145	−0.670263	−0.335131	+	0.942171i	$0.608781\pi$
$840$	3.18761	0.109983
$841$	3.92884	0.135477
$842$	9.35357	0.322345
$843$	55.3277	1.90559
$844$	−8.29206	−0.285424
$845$	−19.3877	−0.666958
$846$	−35.8059	−1.23103
$847$	0	0
$848$	−10.7671	−0.369743
$849$	−9.52875	−0.327026
$850$	−0.749022	−0.0256912
$851$	−45.7247	−1.56742
$852$	−1.83348	−0.0628141
$853$	−23.3656	−0.800021	−0.400011	−	0.916510i	$-0.630994\pi$
$853$	−23.3656	−0.800021	−0.400011	+	0.916510i	$0.630994\pi$
$854$	−7.68398	−0.262940
$855$	−48.6196	−1.66276
$856$	1.88876	0.0645565
$857$	−32.8025	−1.12051	−0.560256	−	0.828320i	$-0.689297\pi$
$857$	−32.8025	−1.12051	−0.560256	+	0.828320i	$0.689297\pi$
$858$	0	0
$859$	−5.80773	−0.198157	−0.0990786	−	0.995080i	$-0.531590\pi$
$859$	−5.80773	−0.198157	−0.0990786	+	0.995080i	$0.531590\pi$
$860$	−2.33345	−0.0795698
$861$	−36.4847	−1.24340
$862$	−5.10019	−0.173713
$863$	−44.4137	−1.51186	−0.755931	−	0.654652i	$-0.772815\pi$
$863$	−44.4137	−1.51186	−0.755931	+	0.654652i	$0.772815\pi$
$864$	13.2631	0.451219
$865$	−9.79571	−0.333064
$866$	−15.0790	−0.512405
$867$	−52.4009	−1.77963
$868$	7.10952	0.241313
$869$	0	0
$870$	18.2916	0.620145
$871$	−42.0901	−1.42617
$872$	7.76554	0.262974
$873$	116.519	3.94357
$874$	26.7520	0.904898
$875$	1.00000	0.0338062
$876$	−40.0695	−1.35382
$877$	−17.3464	−0.585746	−0.292873	−	0.956151i	$-0.594611\pi$
$877$	−17.3464	−0.585746	−0.292873	+	0.956151i	$0.594611\pi$
$878$	−8.35774	−0.282060
$879$	56.0658	1.89105
$880$	0	0
$881$	2.06132	0.0694478	0.0347239	−	0.999397i	$-0.488945\pi$
$881$	2.06132	0.0694478	0.0347239	+	0.999397i	$0.488945\pi$
$882$	7.16083	0.241118
$883$	−46.5639	−1.56700	−0.783501	−	0.621391i	$-0.786568\pi$
$883$	−46.5639	−1.56700	−0.783501	+	0.621391i	$0.786568\pi$
$884$	−4.26270	−0.143370
$885$	−16.6681	−0.560292
$886$	6.68524	0.224595
$887$	27.9025	0.936873	0.468437	−	0.883497i	$-0.344817\pi$
$887$	27.9025	0.936873	0.468437	+	0.883497i	$0.344817\pi$
$888$	−36.9921	−1.24137
$889$	15.5292	0.520832
$890$	−4.80467	−0.161053
$891$	0	0
$892$	0.390611	0.0130786
$893$	−33.9500	−1.13609
$894$	49.8054	1.66574
$895$	−0.676043	−0.0225976
$896$	−1.00000	−0.0334077
$897$	71.4762	2.38652
$898$	−7.97205	−0.266031
$899$	40.7971	1.36066
$900$	7.16083	0.238694
$901$	8.06478	0.268677
$902$	0	0
$903$	−7.43810	−0.247525
$904$	7.75283	0.257855
$905$	5.50028	0.182836
$906$	−56.6915	−1.88345
$907$	−7.30550	−0.242575	−0.121288	−	0.992617i	$-0.538702\pi$
$907$	−7.30550	−0.242575	−0.121288	+	0.992617i	$0.538702\pi$
$908$	−4.91748	−0.163192
$909$	−45.7210	−1.51647
$910$	5.69102	0.188656
$911$	−28.9807	−0.960174	−0.480087	−	0.877221i	$-0.659395\pi$
$911$	−28.9807	−0.960174	−0.480087	+	0.877221i	$0.659395\pi$
$912$	21.6428	0.716664
$913$	0	0
$914$	−2.34209	−0.0774693
$915$	−24.4935	−0.809730
$916$	−9.18360	−0.303435
$917$	−0.957959	−0.0316346
$918$	−9.93433	−0.327882
$919$	5.06468	0.167068	0.0835342	−	0.996505i	$-0.473379\pi$
$919$	5.06468	0.167068	0.0835342	+	0.996505i	$0.473379\pi$
$920$	−3.94010	−0.129901
$921$	4.10534	0.135276
$922$	−0.312762	−0.0103003
$923$	−3.27342	−0.107746
$924$	0	0
$925$	−11.6050	−0.381569
$926$	15.8769	0.521748
$927$	16.6754	0.547693
$928$	−5.73837	−0.188371
$929$	−23.6262	−0.775151	−0.387576	−	0.921838i	$-0.626687\pi$
$929$	−23.6262	−0.775151	−0.387576	+	0.921838i	$0.626687\pi$
$930$	22.6624	0.743128
$931$	6.78967	0.222522
$932$	−20.1599	−0.660359
$933$	−0.158210	−0.00517957
$934$	−0.774386	−0.0253387
$935$	0	0
$936$	40.7524	1.33203
$937$	−39.5066	−1.29063	−0.645313	−	0.763919i	$-0.723273\pi$
$937$	−39.5066	−1.29063	−0.645313	+	0.763919i	$0.723273\pi$
$938$	7.39587	0.241484
$939$	−88.2258	−2.87914
$940$	5.00025	0.163090
$941$	14.2738	0.465313	0.232656	−	0.972559i	$-0.425258\pi$
$941$	14.2738	0.465313	0.232656	+	0.972559i	$0.425258\pi$
$942$	−23.5099	−0.765995
$943$	45.0976	1.46858
$944$	5.22904	0.170191
$945$	13.2631	0.431448
$946$	0	0
$947$	15.8228	0.514171	0.257085	−	0.966389i	$-0.417238\pi$
$947$	15.8228	0.514171	0.257085	+	0.966389i	$0.417238\pi$
$948$	21.7867	0.707598
$949$	−71.5384	−2.32223
$950$	6.78967	0.220286
$951$	29.4308	0.954360
$952$	0.749022	0.0242759
$953$	31.9949	1.03642	0.518208	−	0.855255i	$-0.326599\pi$
$953$	31.9949	1.03642	0.518208	+	0.855255i	$0.326599\pi$
$954$	−77.1013	−2.49625
$955$	12.5963	0.407606
$956$	9.77852	0.316260
$957$	0	0
$958$	30.3787	0.981491
$959$	4.66176	0.150536
$960$	−3.18761	−0.102880
$961$	19.5453	0.630495
$962$	−66.0441	−2.12935
$963$	13.5251	0.435840
$964$	−12.3008	−0.396183
$965$	2.47246	0.0795912
$966$	−12.5595	−0.404094
$967$	−24.1383	−0.776237	−0.388118	−	0.921610i	$-0.626875\pi$
$967$	−24.1383	−0.776237	−0.388118	+	0.921610i	$0.626875\pi$
$968$	0	0
$969$	−16.2109	−0.520770
$970$	−16.2717	−0.522453
$971$	−24.3861	−0.782587	−0.391293	−	0.920266i	$-0.627972\pi$
$971$	−24.3861	−0.782587	−0.391293	+	0.920266i	$0.627972\pi$
$972$	26.4969	0.849889
$973$	−4.46428	−0.143118
$974$	9.43710	0.302384
$975$	18.1407	0.580968
$976$	7.68398	0.245958
$977$	54.4580	1.74227	0.871133	−	0.491046i	$-0.163385\pi$
$977$	54.4580	1.74227	0.871133	+	0.491046i	$0.163385\pi$
$978$	−33.3469	−1.06632
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	55.6077	1.77542
$982$	28.7129	0.916264
$983$	−52.4013	−1.67134	−0.835670	−	0.549231i	$-0.814921\pi$
$983$	−52.4013	−1.67134	−0.835670	+	0.549231i	$0.814921\pi$
$984$	36.4847	1.16309
$985$	−8.91129	−0.283937
$986$	4.29816	0.136881
$987$	15.9388	0.507338
$988$	38.6401	1.22931
$989$	9.19400	0.292352
$990$	0	0
$991$	13.7527	0.436870	0.218435	−	0.975851i	$-0.429905\pi$
$991$	13.7527	0.436870	0.218435	+	0.975851i	$0.429905\pi$
$992$	−7.10952	−0.225728
$993$	52.8233	1.67630
$994$	0.575191	0.0182440
$995$	−26.4542	−0.838654
$996$	33.5503	1.06308
$997$	38.7130	1.22605	0.613026	−	0.790062i	$-0.289952\pi$
$997$	38.7130	1.22605	0.613026	+	0.790062i	$0.289952\pi$
$998$	27.6602	0.875568
$999$	−153.918	−4.86974

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.dh.1.8		8
11.3	even	5		770.2.n.k.141.1	yes	16
11.4	even	5		770.2.n.k.71.1	✓	16
11.10	odd	2		8470.2.a.dg.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.k.71.1	✓	16	11.4	even	5
770.2.n.k.141.1	yes	16	11.3	even	5
8470.2.a.dg.1.8		8	11.10	odd	2
8470.2.a.dh.1.8		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +3.18761 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.18761 q^{6} -1.00000 q^{7} +1.00000 q^{8} +7.16083 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +3.18761 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.18761 q^{6} -1.00000 q^{7} +1.00000 q^{8} +7.16083 q^{9} -1.00000 q^{10} +3.18761 q^{12} +5.69102 q^{13} -1.00000 q^{14} -3.18761 q^{15} +1.00000 q^{16} -0.749022 q^{17} +7.16083 q^{18} +6.78967 q^{19} -1.00000 q^{20} -3.18761 q^{21} +3.94010 q^{23} +3.18761 q^{24} +1.00000 q^{25} +5.69102 q^{26} +13.2631 q^{27} -1.00000 q^{28} -5.73837 q^{29} -3.18761 q^{30} -7.10952 q^{31} +1.00000 q^{32} -0.749022 q^{34} +1.00000 q^{35} +7.16083 q^{36} -11.6050 q^{37} +6.78967 q^{38} +18.1407 q^{39} -1.00000 q^{40} +11.4458 q^{41} -3.18761 q^{42} +2.33345 q^{43} -7.16083 q^{45} +3.94010 q^{46} -5.00025 q^{47} +3.18761 q^{48} +1.00000 q^{49} +1.00000 q^{50} -2.38759 q^{51} +5.69102 q^{52} -10.7671 q^{53} +13.2631 q^{54} -1.00000 q^{56} +21.6428 q^{57} -5.73837 q^{58} +5.22904 q^{59} -3.18761 q^{60} +7.68398 q^{61} -7.10952 q^{62} -7.16083 q^{63} +1.00000 q^{64} -5.69102 q^{65} -7.39587 q^{67} -0.749022 q^{68} +12.5595 q^{69} +1.00000 q^{70} -0.575191 q^{71} +7.16083 q^{72} -12.5704 q^{73} -11.6050 q^{74} +3.18761 q^{75} +6.78967 q^{76} +18.1407 q^{78} +6.83481 q^{79} -1.00000 q^{80} +20.7950 q^{81} +11.4458 q^{82} +10.5252 q^{83} -3.18761 q^{84} +0.749022 q^{85} +2.33345 q^{86} -18.2916 q^{87} +4.80467 q^{89} -7.16083 q^{90} -5.69102 q^{91} +3.94010 q^{92} -22.6624 q^{93} -5.00025 q^{94} -6.78967 q^{95} +3.18761 q^{96} +16.2717 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} - 8 q^{5} - 8 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 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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