LMFDB - Embedded newform 82.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [82,2,Mod(1,82)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(82, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("82.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 82 = 2 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	82.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:	$0.654773296574$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	82.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -2.82843 q^{5} +1.41421 q^{6} -3.41421 q^{7} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -2.82843 q^{5} +1.41421 q^{6} -3.41421 q^{7} +1.00000 q^{8} -1.00000 q^{9} -2.82843 q^{10} +4.24264 q^{11} +1.41421 q^{12} -3.41421 q^{14} -4.00000 q^{15} +1.00000 q^{16} +7.65685 q^{17} -1.00000 q^{18} -5.41421 q^{19} -2.82843 q^{20} -4.82843 q^{21} +4.24264 q^{22} +1.17157 q^{23} +1.41421 q^{24} +3.00000 q^{25} -5.65685 q^{27} -3.41421 q^{28} -1.65685 q^{29} -4.00000 q^{30} -1.17157 q^{31} +1.00000 q^{32} +6.00000 q^{33} +7.65685 q^{34} +9.65685 q^{35} -1.00000 q^{36} +8.48528 q^{37} -5.41421 q^{38} -2.82843 q^{40} -1.00000 q^{41} -4.82843 q^{42} -1.65685 q^{43} +4.24264 q^{44} +2.82843 q^{45} +1.17157 q^{46} -9.07107 q^{47} +1.41421 q^{48} +4.65685 q^{49} +3.00000 q^{50} +10.8284 q^{51} +12.0000 q^{53} -5.65685 q^{54} -12.0000 q^{55} -3.41421 q^{56} -7.65685 q^{57} -1.65685 q^{58} -1.17157 q^{59} -4.00000 q^{60} +6.00000 q^{61} -1.17157 q^{62} +3.41421 q^{63} +1.00000 q^{64} +6.00000 q^{66} -8.24264 q^{67} +7.65685 q^{68} +1.65685 q^{69} +9.65685 q^{70} -0.585786 q^{71} -1.00000 q^{72} -13.6569 q^{73} +8.48528 q^{74} +4.24264 q^{75} -5.41421 q^{76} -14.4853 q^{77} -10.2426 q^{79} -2.82843 q^{80} -5.00000 q^{81} -1.00000 q^{82} +17.6569 q^{83} -4.82843 q^{84} -21.6569 q^{85} -1.65685 q^{86} -2.34315 q^{87} +4.24264 q^{88} -11.6569 q^{89} +2.82843 q^{90} +1.17157 q^{92} -1.65685 q^{93} -9.07107 q^{94} +15.3137 q^{95} +1.41421 q^{96} +3.65685 q^{97} +4.65685 q^{98} -4.24264 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 4 q^{7} + 2 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^7 + 2 * q^8 - 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{4} - 4 q^{7} + 2 q^{8} - 2 q^{9} - 4 q^{14} - 8 q^{15} + 2 q^{16} + 4 q^{17} - 2 q^{18} - 8 q^{19} - 4 q^{21} + 8 q^{23} + 6 q^{25} - 4 q^{28} + 8 q^{29} - 8 q^{30} - 8 q^{31} + 2 q^{32} + 12 q^{33} + 4 q^{34} + 8 q^{35} - 2 q^{36} - 8 q^{38} - 2 q^{41} - 4 q^{42} + 8 q^{43} + 8 q^{46} - 4 q^{47} - 2 q^{49} + 6 q^{50} + 16 q^{51} + 24 q^{53} - 24 q^{55} - 4 q^{56} - 4 q^{57} + 8 q^{58} - 8 q^{59} - 8 q^{60} + 12 q^{61} - 8 q^{62} + 4 q^{63} + 2 q^{64} + 12 q^{66} - 8 q^{67} + 4 q^{68} - 8 q^{69} + 8 q^{70} - 4 q^{71} - 2 q^{72} - 16 q^{73} - 8 q^{76} - 12 q^{77} - 12 q^{79} - 10 q^{81} - 2 q^{82} + 24 q^{83} - 4 q^{84} - 32 q^{85} + 8 q^{86} - 16 q^{87} - 12 q^{89} + 8 q^{92} + 8 q^{93} - 4 q^{94} + 8 q^{95} - 4 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^7 + 2 * q^8 - 2 * q^9 - 4 * q^14 - 8 * q^15 + 2 * q^16 + 4 * q^17 - 2 * q^18 - 8 * q^19 - 4 * q^21 + 8 * q^23 + 6 * q^25 - 4 * q^28 + 8 * q^29 - 8 * q^30 - 8 * q^31 + 2 * q^32 + 12 * q^33 + 4 * q^34 + 8 * q^35 - 2 * q^36 - 8 * q^38 - 2 * q^41 - 4 * q^42 + 8 * q^43 + 8 * q^46 - 4 * q^47 - 2 * q^49 + 6 * q^50 + 16 * q^51 + 24 * q^53 - 24 * q^55 - 4 * q^56 - 4 * q^57 + 8 * q^58 - 8 * q^59 - 8 * q^60 + 12 * q^61 - 8 * q^62 + 4 * q^63 + 2 * q^64 + 12 * q^66 - 8 * q^67 + 4 * q^68 - 8 * q^69 + 8 * q^70 - 4 * q^71 - 2 * q^72 - 16 * q^73 - 8 * q^76 - 12 * q^77 - 12 * q^79 - 10 * q^81 - 2 * q^82 + 24 * q^83 - 4 * q^84 - 32 * q^85 + 8 * q^86 - 16 * q^87 - 12 * q^89 + 8 * q^92 + 8 * q^93 - 4 * q^94 + 8 * q^95 - 4 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	1.00000	0.500000
$5$	−2.82843	−1.26491	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	1.41421	0.577350
$7$	−3.41421	−1.29045	−0.645226	−	0.763992i	$-0.723237\pi$
$7$	−3.41421	−1.29045	−0.645226	+	0.763992i	$0.723237\pi$
$8$	1.00000	0.353553
$9$	−1.00000	−0.333333
$10$	−2.82843	−0.894427
$11$	4.24264	1.27920	0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264	1.27920	0.639602	+	0.768706i	$0.279099\pi$
$12$	1.41421	0.408248
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−3.41421	−0.912487
$15$	−4.00000	−1.03280
$16$	1.00000	0.250000
$17$	7.65685	1.85706	0.928530	−	0.371257i	$-0.121073\pi$
$17$	7.65685	1.85706	0.928530	+	0.371257i	$0.121073\pi$
$18$	−1.00000	−0.235702
$19$	−5.41421	−1.24211	−0.621053	−	0.783769i	$-0.713295\pi$
$19$	−5.41421	−1.24211	−0.621053	+	0.783769i	$0.713295\pi$
$20$	−2.82843	−0.632456
$21$	−4.82843	−1.05365
$22$	4.24264	0.904534
$23$	1.17157	0.244290	0.122145	−	0.992512i	$-0.461023\pi$
$23$	1.17157	0.244290	0.122145	+	0.992512i	$0.461023\pi$
$24$	1.41421	0.288675
$25$	3.00000	0.600000
$26$	0	0
$27$	−5.65685	−1.08866
$28$	−3.41421	−0.645226
$29$	−1.65685	−0.307670	−0.153835	−	0.988097i	$-0.549162\pi$
$29$	−1.65685	−0.307670	−0.153835	+	0.988097i	$0.549162\pi$
$30$	−4.00000	−0.730297
$31$	−1.17157	−0.210421	−0.105210	−	0.994450i	$-0.533552\pi$
$31$	−1.17157	−0.210421	−0.105210	+	0.994450i	$0.533552\pi$
$32$	1.00000	0.176777
$33$	6.00000	1.04447
$34$	7.65685	1.31314
$35$	9.65685	1.63231
$36$	−1.00000	−0.166667
$37$	8.48528	1.39497	0.697486	−	0.716599i	$-0.254302\pi$
$37$	8.48528	1.39497	0.697486	+	0.716599i	$0.254302\pi$
$38$	−5.41421	−0.878301
$39$	0	0
$40$	−2.82843	−0.447214
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	−4.82843	−0.745042
$43$	−1.65685	−0.252668	−0.126334	−	0.991988i	$-0.540321\pi$
$43$	−1.65685	−0.252668	−0.126334	+	0.991988i	$0.540321\pi$
$44$	4.24264	0.639602
$45$	2.82843	0.421637
$46$	1.17157	0.172739
$47$	−9.07107	−1.32315	−0.661576	−	0.749878i	$-0.730112\pi$
$47$	−9.07107	−1.32315	−0.661576	+	0.749878i	$0.730112\pi$
$48$	1.41421	0.204124
$49$	4.65685	0.665265
$50$	3.00000	0.424264
$51$	10.8284	1.51628
$52$	0	0
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	−5.65685	−0.769800
$55$	−12.0000	−1.61808
$56$	−3.41421	−0.456243
$57$	−7.65685	−1.01418
$58$	−1.65685	−0.217556
$59$	−1.17157	−0.152526	−0.0762629	−	0.997088i	$-0.524299\pi$
$59$	−1.17157	−0.152526	−0.0762629	+	0.997088i	$0.524299\pi$
$60$	−4.00000	−0.516398
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−1.17157	−0.148790
$63$	3.41421	0.430150
$64$	1.00000	0.125000
$65$	0	0
$66$	6.00000	0.738549
$67$	−8.24264	−1.00700	−0.503499	−	0.863996i	$-0.667954\pi$
$67$	−8.24264	−1.00700	−0.503499	+	0.863996i	$0.667954\pi$
$68$	7.65685	0.928530
$69$	1.65685	0.199462
$70$	9.65685	1.15421
$71$	−0.585786	−0.0695201	−0.0347600	−	0.999396i	$-0.511067\pi$
$71$	−0.585786	−0.0695201	−0.0347600	+	0.999396i	$0.511067\pi$
$72$	−1.00000	−0.117851
$73$	−13.6569	−1.59841	−0.799207	−	0.601056i	$-0.794747\pi$
$73$	−13.6569	−1.59841	−0.799207	+	0.601056i	$0.794747\pi$
$74$	8.48528	0.986394
$75$	4.24264	0.489898
$76$	−5.41421	−0.621053
$77$	−14.4853	−1.65075
$78$	0	0
$79$	−10.2426	−1.15239	−0.576194	−	0.817313i	$-0.695463\pi$
$79$	−10.2426	−1.15239	−0.576194	+	0.817313i	$0.695463\pi$
$80$	−2.82843	−0.316228
$81$	−5.00000	−0.555556
$82$	−1.00000	−0.110432
$83$	17.6569	1.93809	0.969046	−	0.246881i	$-0.0794057\pi$
$83$	17.6569	1.93809	0.969046	+	0.246881i	$0.0794057\pi$
$84$	−4.82843	−0.526825
$85$	−21.6569	−2.34902
$86$	−1.65685	−0.178663
$87$	−2.34315	−0.251212
$88$	4.24264	0.452267
$89$	−11.6569	−1.23562	−0.617812	−	0.786326i	$-0.711981\pi$
$89$	−11.6569	−1.23562	−0.617812	+	0.786326i	$0.711981\pi$
$90$	2.82843	0.298142
$91$	0	0
$92$	1.17157	0.122145
$93$	−1.65685	−0.171808
$94$	−9.07107	−0.935609
$95$	15.3137	1.57115
$96$	1.41421	0.144338
$97$	3.65685	0.371297	0.185649	−	0.982616i	$-0.440561\pi$
$97$	3.65685	0.371297	0.185649	+	0.982616i	$0.440561\pi$
$98$	4.65685	0.470413
$99$	−4.24264	−0.426401
$100$	3.00000	0.300000
$101$	−6.34315	−0.631167	−0.315583	−	0.948898i	$-0.602200\pi$
$101$	−6.34315	−0.631167	−0.315583	+	0.948898i	$0.602200\pi$
$102$	10.8284	1.07217
$103$	4.48528	0.441948	0.220974	−	0.975280i	$-0.429076\pi$
$103$	4.48528	0.441948	0.220974	+	0.975280i	$0.429076\pi$
$104$	0	0
$105$	13.6569	1.33277
$106$	12.0000	1.16554
$107$	6.34315	0.613215	0.306608	−	0.951836i	$-0.400806\pi$
$107$	6.34315	0.613215	0.306608	+	0.951836i	$0.400806\pi$
$108$	−5.65685	−0.544331
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	−12.0000	−1.14416
$111$	12.0000	1.13899
$112$	−3.41421	−0.322613
$113$	−1.65685	−0.155864	−0.0779319	−	0.996959i	$-0.524832\pi$
$113$	−1.65685	−0.155864	−0.0779319	+	0.996959i	$0.524832\pi$
$114$	−7.65685	−0.717130
$115$	−3.31371	−0.309005
$116$	−1.65685	−0.153835
$117$	0	0
$118$	−1.17157	−0.107852
$119$	−26.1421	−2.39645
$120$	−4.00000	−0.365148
$121$	7.00000	0.636364
$122$	6.00000	0.543214
$123$	−1.41421	−0.127515
$124$	−1.17157	−0.105210
$125$	5.65685	0.505964
$126$	3.41421	0.304162
$127$	18.1421	1.60985	0.804927	−	0.593374i	$-0.202204\pi$
$127$	18.1421	1.60985	0.804927	+	0.593374i	$0.202204\pi$
$128$	1.00000	0.0883883
$129$	−2.34315	−0.206302
$130$	0	0
$131$	−10.1421	−0.886123	−0.443061	−	0.896491i	$-0.646108\pi$
$131$	−10.1421	−0.886123	−0.443061	+	0.896491i	$0.646108\pi$
$132$	6.00000	0.522233
$133$	18.4853	1.60288
$134$	−8.24264	−0.712056
$135$	16.0000	1.37706
$136$	7.65685	0.656570
$137$	5.31371	0.453981	0.226990	−	0.973897i	$-0.427111\pi$
$137$	5.31371	0.453981	0.226990	+	0.973897i	$0.427111\pi$
$138$	1.65685	0.141041
$139$	12.4853	1.05899	0.529494	−	0.848314i	$-0.322382\pi$
$139$	12.4853	1.05899	0.529494	+	0.848314i	$0.322382\pi$
$140$	9.65685	0.816153
$141$	−12.8284	−1.08035
$142$	−0.585786	−0.0491581
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	4.68629	0.389175
$146$	−13.6569	−1.13025
$147$	6.58579	0.543187
$148$	8.48528	0.697486
$149$	3.31371	0.271470	0.135735	−	0.990745i	$-0.456660\pi$
$149$	3.31371	0.271470	0.135735	+	0.990745i	$0.456660\pi$
$150$	4.24264	0.346410
$151$	−11.8995	−0.968367	−0.484184	−	0.874966i	$-0.660883\pi$
$151$	−11.8995	−0.968367	−0.484184	+	0.874966i	$0.660883\pi$
$152$	−5.41421	−0.439151
$153$	−7.65685	−0.619020
$154$	−14.4853	−1.16726
$155$	3.31371	0.266163
$156$	0	0
$157$	−20.9706	−1.67363	−0.836817	−	0.547483i	$-0.815586\pi$
$157$	−20.9706	−1.67363	−0.836817	+	0.547483i	$0.815586\pi$
$158$	−10.2426	−0.814861
$159$	16.9706	1.34585
$160$	−2.82843	−0.223607
$161$	−4.00000	−0.315244
$162$	−5.00000	−0.392837
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−1.00000	−0.0780869
$165$	−16.9706	−1.32116
$166$	17.6569	1.37044
$167$	−18.7279	−1.44921	−0.724605	−	0.689164i	$-0.757978\pi$
$167$	−18.7279	−1.44921	−0.724605	+	0.689164i	$0.757978\pi$
$168$	−4.82843	−0.372521
$169$	−13.0000	−1.00000
$170$	−21.6569	−1.66100
$171$	5.41421	0.414035
$172$	−1.65685	−0.126334
$173$	13.3137	1.01222	0.506111	−	0.862468i	$-0.331083\pi$
$173$	13.3137	1.01222	0.506111	+	0.862468i	$0.331083\pi$
$174$	−2.34315	−0.177633
$175$	−10.2426	−0.774271
$176$	4.24264	0.319801
$177$	−1.65685	−0.124537
$178$	−11.6569	−0.873718
$179$	−15.5563	−1.16274	−0.581368	−	0.813641i	$-0.697482\pi$
$179$	−15.5563	−1.16274	−0.581368	+	0.813641i	$0.697482\pi$
$180$	2.82843	0.210819
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	8.48528	0.627250
$184$	1.17157	0.0863695
$185$	−24.0000	−1.76452
$186$	−1.65685	−0.121486
$187$	32.4853	2.37556
$188$	−9.07107	−0.661576
$189$	19.3137	1.40487
$190$	15.3137	1.11097
$191$	−2.92893	−0.211930	−0.105965	−	0.994370i	$-0.533793\pi$
$191$	−2.92893	−0.211930	−0.105965	+	0.994370i	$0.533793\pi$
$192$	1.41421	0.102062
$193$	15.6569	1.12701	0.563503	−	0.826114i	$-0.309454\pi$
$193$	15.6569	1.12701	0.563503	+	0.826114i	$0.309454\pi$
$194$	3.65685	0.262547
$195$	0	0
$196$	4.65685	0.332632
$197$	9.31371	0.663574	0.331787	−	0.943354i	$-0.392348\pi$
$197$	9.31371	0.663574	0.331787	+	0.943354i	$0.392348\pi$
$198$	−4.24264	−0.301511
$199$	−2.24264	−0.158977	−0.0794883	−	0.996836i	$-0.525329\pi$
$199$	−2.24264	−0.158977	−0.0794883	+	0.996836i	$0.525329\pi$
$200$	3.00000	0.212132
$201$	−11.6569	−0.822211
$202$	−6.34315	−0.446302
$203$	5.65685	0.397033
$204$	10.8284	0.758142
$205$	2.82843	0.197546
$206$	4.48528	0.312504
$207$	−1.17157	−0.0814299
$208$	0	0
$209$	−22.9706	−1.58891
$210$	13.6569	0.942412
$211$	3.07107	0.211421	0.105711	−	0.994397i	$-0.466288\pi$
$211$	3.07107	0.211421	0.105711	+	0.994397i	$0.466288\pi$
$212$	12.0000	0.824163
$213$	−0.828427	−0.0567629
$214$	6.34315	0.433609
$215$	4.68629	0.319602
$216$	−5.65685	−0.384900
$217$	4.00000	0.271538
$218$	12.0000	0.812743
$219$	−19.3137	−1.30510
$220$	−12.0000	−0.809040
$221$	0	0
$222$	12.0000	0.805387
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	−3.41421	−0.228122
$225$	−3.00000	−0.200000
$226$	−1.65685	−0.110212
$227$	13.8995	0.922542	0.461271	−	0.887259i	$-0.347394\pi$
$227$	13.8995	0.922542	0.461271	+	0.887259i	$0.347394\pi$
$228$	−7.65685	−0.507088
$229$	−8.00000	−0.528655	−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000	−0.528655	−0.264327	+	0.964433i	$0.585150\pi$
$230$	−3.31371	−0.218499
$231$	−20.4853	−1.34783
$232$	−1.65685	−0.108778
$233$	27.6569	1.81186	0.905930	−	0.423427i	$-0.139173\pi$
$233$	27.6569	1.81186	0.905930	+	0.423427i	$0.139173\pi$
$234$	0	0
$235$	25.6569	1.67367
$236$	−1.17157	−0.0762629
$237$	−14.4853	−0.940920
$238$	−26.1421	−1.69454
$239$	−0.585786	−0.0378914	−0.0189457	−	0.999821i	$-0.506031\pi$
$239$	−0.585786	−0.0378914	−0.0189457	+	0.999821i	$0.506031\pi$
$240$	−4.00000	−0.258199
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	7.00000	0.449977
$243$	9.89949	0.635053
$244$	6.00000	0.384111
$245$	−13.1716	−0.841501
$246$	−1.41421	−0.0901670
$247$	0	0
$248$	−1.17157	−0.0743950
$249$	24.9706	1.58245
$250$	5.65685	0.357771
$251$	2.14214	0.135210	0.0676052	−	0.997712i	$-0.478464\pi$
$251$	2.14214	0.135210	0.0676052	+	0.997712i	$0.478464\pi$
$252$	3.41421	0.215075
$253$	4.97056	0.312497
$254$	18.1421	1.13834
$255$	−30.6274	−1.91796
$256$	1.00000	0.0625000
$257$	13.3137	0.830486	0.415243	−	0.909710i	$-0.363696\pi$
$257$	13.3137	0.830486	0.415243	+	0.909710i	$0.363696\pi$
$258$	−2.34315	−0.145878
$259$	−28.9706	−1.80014
$260$	0	0
$261$	1.65685	0.102557
$262$	−10.1421	−0.626583
$263$	−6.72792	−0.414861	−0.207431	−	0.978250i	$-0.566510\pi$
$263$	−6.72792	−0.414861	−0.207431	+	0.978250i	$0.566510\pi$
$264$	6.00000	0.369274
$265$	−33.9411	−2.08499
$266$	18.4853	1.13341
$267$	−16.4853	−1.00888
$268$	−8.24264	−0.503499
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	16.0000	0.973729
$271$	−20.4853	−1.24439	−0.622196	−	0.782861i	$-0.713759\pi$
$271$	−20.4853	−1.24439	−0.622196	+	0.782861i	$0.713759\pi$
$272$	7.65685	0.464265
$273$	0	0
$274$	5.31371	0.321013
$275$	12.7279	0.767523
$276$	1.65685	0.0997309
$277$	24.4853	1.47118	0.735589	−	0.677428i	$-0.236906\pi$
$277$	24.4853	1.47118	0.735589	+	0.677428i	$0.236906\pi$
$278$	12.4853	0.748817
$279$	1.17157	0.0701402
$280$	9.65685	0.577107
$281$	−22.9706	−1.37031	−0.685154	−	0.728398i	$-0.740265\pi$
$281$	−22.9706	−1.37031	−0.685154	+	0.728398i	$0.740265\pi$
$282$	−12.8284	−0.763922
$283$	−11.5147	−0.684479	−0.342239	−	0.939613i	$-0.611185\pi$
$283$	−11.5147	−0.684479	−0.342239	+	0.939613i	$0.611185\pi$
$284$	−0.585786	−0.0347600
$285$	21.6569	1.28284
$286$	0	0
$287$	3.41421	0.201535
$288$	−1.00000	−0.0589256
$289$	41.6274	2.44867
$290$	4.68629	0.275189
$291$	5.17157	0.303163
$292$	−13.6569	−0.799207
$293$	−17.6569	−1.03152	−0.515762	−	0.856732i	$-0.672491\pi$
$293$	−17.6569	−1.03152	−0.515762	+	0.856732i	$0.672491\pi$
$294$	6.58579	0.384091
$295$	3.31371	0.192932
$296$	8.48528	0.493197
$297$	−24.0000	−1.39262
$298$	3.31371	0.191958
$299$	0	0
$300$	4.24264	0.244949
$301$	5.65685	0.326056
$302$	−11.8995	−0.684739
$303$	−8.97056	−0.515345
$304$	−5.41421	−0.310526
$305$	−16.9706	−0.971732
$306$	−7.65685	−0.437713
$307$	12.4853	0.712573	0.356286	−	0.934377i	$-0.384043\pi$
$307$	12.4853	0.712573	0.356286	+	0.934377i	$0.384043\pi$
$308$	−14.4853	−0.825376
$309$	6.34315	0.360849
$310$	3.31371	0.188206
$311$	5.07107	0.287554	0.143777	−	0.989610i	$-0.454075\pi$
$311$	5.07107	0.287554	0.143777	+	0.989610i	$0.454075\pi$
$312$	0	0
$313$	−1.31371	−0.0742552	−0.0371276	−	0.999311i	$-0.511821\pi$
$313$	−1.31371	−0.0742552	−0.0371276	+	0.999311i	$0.511821\pi$
$314$	−20.9706	−1.18344
$315$	−9.65685	−0.544102
$316$	−10.2426	−0.576194
$317$	−11.3137	−0.635441	−0.317721	−	0.948184i	$-0.602917\pi$
$317$	−11.3137	−0.635441	−0.317721	+	0.948184i	$0.602917\pi$
$318$	16.9706	0.951662
$319$	−7.02944	−0.393573
$320$	−2.82843	−0.158114
$321$	8.97056	0.500688
$322$	−4.00000	−0.222911
$323$	−41.4558	−2.30666
$324$	−5.00000	−0.277778
$325$	0	0
$326$	−4.00000	−0.221540
$327$	16.9706	0.938474
$328$	−1.00000	−0.0552158
$329$	30.9706	1.70746
$330$	−16.9706	−0.934199
$331$	5.89949	0.324266	0.162133	−	0.986769i	$-0.448163\pi$
$331$	5.89949	0.324266	0.162133	+	0.986769i	$0.448163\pi$
$332$	17.6569	0.969046
$333$	−8.48528	−0.464991
$334$	−18.7279	−1.02475
$335$	23.3137	1.27376
$336$	−4.82843	−0.263412
$337$	−12.0000	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$338$	−13.0000	−0.707107
$339$	−2.34315	−0.127262
$340$	−21.6569	−1.17451
$341$	−4.97056	−0.269171
$342$	5.41421	0.292767
$343$	8.00000	0.431959
$344$	−1.65685	−0.0893316
$345$	−4.68629	−0.252301
$346$	13.3137	0.715749
$347$	21.8995	1.17563	0.587813	−	0.808997i	$-0.299989\pi$
$347$	21.8995	1.17563	0.587813	+	0.808997i	$0.299989\pi$
$348$	−2.34315	−0.125606
$349$	15.5147	0.830484	0.415242	−	0.909711i	$-0.363697\pi$
$349$	15.5147	0.830484	0.415242	+	0.909711i	$0.363697\pi$
$350$	−10.2426	−0.547492
$351$	0	0
$352$	4.24264	0.226134
$353$	−3.31371	−0.176371	−0.0881855	−	0.996104i	$-0.528107\pi$
$353$	−3.31371	−0.176371	−0.0881855	+	0.996104i	$0.528107\pi$
$354$	−1.65685	−0.0880608
$355$	1.65685	0.0879367
$356$	−11.6569	−0.617812
$357$	−36.9706	−1.95669
$358$	−15.5563	−0.822179
$359$	13.6569	0.720781	0.360391	−	0.932801i	$-0.382643\pi$
$359$	13.6569	0.720781	0.360391	+	0.932801i	$0.382643\pi$
$360$	2.82843	0.149071
$361$	10.3137	0.542827
$362$	0	0
$363$	9.89949	0.519589
$364$	0	0
$365$	38.6274	2.02185
$366$	8.48528	0.443533
$367$	−13.6569	−0.712882	−0.356441	−	0.934318i	$-0.616010\pi$
$367$	−13.6569	−0.712882	−0.356441	+	0.934318i	$0.616010\pi$
$368$	1.17157	0.0610725
$369$	1.00000	0.0520579
$370$	−24.0000	−1.24770
$371$	−40.9706	−2.12709
$372$	−1.65685	−0.0859039
$373$	−32.6274	−1.68938	−0.844692	−	0.535253i	$-0.820216\pi$
$373$	−32.6274	−1.68938	−0.844692	+	0.535253i	$0.820216\pi$
$374$	32.4853	1.67977
$375$	8.00000	0.413118
$376$	−9.07107	−0.467805
$377$	0	0
$378$	19.3137	0.993390
$379$	−21.4558	−1.10211	−0.551056	−	0.834468i	$-0.685775\pi$
$379$	−21.4558	−1.10211	−0.551056	+	0.834468i	$0.685775\pi$
$380$	15.3137	0.785577
$381$	25.6569	1.31444
$382$	−2.92893	−0.149857
$383$	−3.89949	−0.199255	−0.0996274	−	0.995025i	$-0.531765\pi$
$383$	−3.89949	−0.199255	−0.0996274	+	0.995025i	$0.531765\pi$
$384$	1.41421	0.0721688
$385$	40.9706	2.08805
$386$	15.6569	0.796913
$387$	1.65685	0.0842226
$388$	3.65685	0.185649
$389$	−9.31371	−0.472224	−0.236112	−	0.971726i	$-0.575873\pi$
$389$	−9.31371	−0.472224	−0.236112	+	0.971726i	$0.575873\pi$
$390$	0	0
$391$	8.97056	0.453661
$392$	4.65685	0.235207
$393$	−14.3431	−0.723516
$394$	9.31371	0.469218
$395$	28.9706	1.45767
$396$	−4.24264	−0.213201
$397$	−4.97056	−0.249465	−0.124733	−	0.992190i	$-0.539807\pi$
$397$	−4.97056	−0.249465	−0.124733	+	0.992190i	$0.539807\pi$
$398$	−2.24264	−0.112413
$399$	26.1421	1.30874
$400$	3.00000	0.150000
$401$	−11.3137	−0.564980	−0.282490	−	0.959270i	$-0.591160\pi$
$401$	−11.3137	−0.564980	−0.282490	+	0.959270i	$0.591160\pi$
$402$	−11.6569	−0.581391
$403$	0	0
$404$	−6.34315	−0.315583
$405$	14.1421	0.702728
$406$	5.65685	0.280745
$407$	36.0000	1.78445
$408$	10.8284	0.536087
$409$	−13.6569	−0.675288	−0.337644	−	0.941274i	$-0.609630\pi$
$409$	−13.6569	−0.675288	−0.337644	+	0.941274i	$0.609630\pi$
$410$	2.82843	0.139686
$411$	7.51472	0.370674
$412$	4.48528	0.220974
$413$	4.00000	0.196827
$414$	−1.17157	−0.0575797
$415$	−49.9411	−2.45151
$416$	0	0
$417$	17.6569	0.864660
$418$	−22.9706	−1.12353
$419$	17.6569	0.862594	0.431297	−	0.902210i	$-0.358056\pi$
$419$	17.6569	0.862594	0.431297	+	0.902210i	$0.358056\pi$
$420$	13.6569	0.666386
$421$	−8.97056	−0.437198	−0.218599	−	0.975815i	$-0.570149\pi$
$421$	−8.97056	−0.437198	−0.218599	+	0.975815i	$0.570149\pi$
$422$	3.07107	0.149497
$423$	9.07107	0.441050
$424$	12.0000	0.582772
$425$	22.9706	1.11424
$426$	−0.828427	−0.0401374
$427$	−20.4853	−0.991352
$428$	6.34315	0.306608
$429$	0	0
$430$	4.68629	0.225993
$431$	−5.65685	−0.272481	−0.136241	−	0.990676i	$-0.543502\pi$
$431$	−5.65685	−0.272481	−0.136241	+	0.990676i	$0.543502\pi$
$432$	−5.65685	−0.272166
$433$	−35.9411	−1.72722	−0.863610	−	0.504160i	$-0.831802\pi$
$433$	−35.9411	−1.72722	−0.863610	+	0.504160i	$0.831802\pi$
$434$	4.00000	0.192006
$435$	6.62742	0.317760
$436$	12.0000	0.574696
$437$	−6.34315	−0.303434
$438$	−19.3137	−0.922845
$439$	8.58579	0.409777	0.204889	−	0.978785i	$-0.434317\pi$
$439$	8.58579	0.409777	0.204889	+	0.978785i	$0.434317\pi$
$440$	−12.0000	−0.572078
$441$	−4.65685	−0.221755
$442$	0	0
$443$	−9.17157	−0.435755	−0.217877	−	0.975976i	$-0.569913\pi$
$443$	−9.17157	−0.435755	−0.217877	+	0.975976i	$0.569913\pi$
$444$	12.0000	0.569495
$445$	32.9706	1.56295
$446$	0	0
$447$	4.68629	0.221654
$448$	−3.41421	−0.161306
$449$	−5.31371	−0.250769	−0.125385	−	0.992108i	$-0.540017\pi$
$449$	−5.31371	−0.250769	−0.125385	+	0.992108i	$0.540017\pi$
$450$	−3.00000	−0.141421
$451$	−4.24264	−0.199778
$452$	−1.65685	−0.0779319
$453$	−16.8284	−0.790668
$454$	13.8995	0.652336
$455$	0	0
$456$	−7.65685	−0.358565
$457$	24.6274	1.15202	0.576011	−	0.817442i	$-0.304609\pi$
$457$	24.6274	1.15202	0.576011	+	0.817442i	$0.304609\pi$
$458$	−8.00000	−0.373815
$459$	−43.3137	−2.02171
$460$	−3.31371	−0.154502
$461$	19.7990	0.922131	0.461065	−	0.887366i	$-0.347467\pi$
$461$	19.7990	0.922131	0.461065	+	0.887366i	$0.347467\pi$
$462$	−20.4853	−0.953062
$463$	−3.61522	−0.168014	−0.0840068	−	0.996465i	$-0.526772\pi$
$463$	−3.61522	−0.168014	−0.0840068	+	0.996465i	$0.526772\pi$
$464$	−1.65685	−0.0769175
$465$	4.68629	0.217322
$466$	27.6569	1.28118
$467$	−5.85786	−0.271070	−0.135535	−	0.990773i	$-0.543275\pi$
$467$	−5.85786	−0.271070	−0.135535	+	0.990773i	$0.543275\pi$
$468$	0	0
$469$	28.1421	1.29948
$470$	25.6569	1.18346
$471$	−29.6569	−1.36652
$472$	−1.17157	−0.0539260
$473$	−7.02944	−0.323214
$474$	−14.4853	−0.665331
$475$	−16.2426	−0.745263
$476$	−26.1421	−1.19822
$477$	−12.0000	−0.549442
$478$	−0.585786	−0.0267932
$479$	20.5858	0.940589	0.470294	−	0.882510i	$-0.344148\pi$
$479$	20.5858	0.940589	0.470294	+	0.882510i	$0.344148\pi$
$480$	−4.00000	−0.182574
$481$	0	0
$482$	18.0000	0.819878
$483$	−5.65685	−0.257396
$484$	7.00000	0.318182
$485$	−10.3431	−0.469658
$486$	9.89949	0.449050
$487$	−3.51472	−0.159267	−0.0796336	−	0.996824i	$-0.525375\pi$
$487$	−3.51472	−0.159267	−0.0796336	+	0.996824i	$0.525375\pi$
$488$	6.00000	0.271607
$489$	−5.65685	−0.255812
$490$	−13.1716	−0.595031
$491$	−31.7990	−1.43507	−0.717534	−	0.696523i	$-0.754729\pi$
$491$	−31.7990	−1.43507	−0.717534	+	0.696523i	$0.754729\pi$
$492$	−1.41421	−0.0637577
$493$	−12.6863	−0.571362
$494$	0	0
$495$	12.0000	0.539360
$496$	−1.17157	−0.0526052
$497$	2.00000	0.0897123
$498$	24.9706	1.11896
$499$	−28.5269	−1.27704	−0.638520	−	0.769605i	$-0.720453\pi$
$499$	−28.5269	−1.27704	−0.638520	+	0.769605i	$0.720453\pi$
$500$	5.65685	0.252982
$501$	−26.4853	−1.18328
$502$	2.14214	0.0956082
$503$	1.07107	0.0477566	0.0238783	−	0.999715i	$-0.492399\pi$
$503$	1.07107	0.0477566	0.0238783	+	0.999715i	$0.492399\pi$
$504$	3.41421	0.152081
$505$	17.9411	0.798370
$506$	4.97056	0.220968
$507$	−18.3848	−0.816497
$508$	18.1421	0.804927
$509$	−0.686292	−0.0304193	−0.0152097	−	0.999884i	$-0.504842\pi$
$509$	−0.686292	−0.0304193	−0.0152097	+	0.999884i	$0.504842\pi$
$510$	−30.6274	−1.35620
$511$	46.6274	2.06268
$512$	1.00000	0.0441942
$513$	30.6274	1.35223
$514$	13.3137	0.587243
$515$	−12.6863	−0.559025
$516$	−2.34315	−0.103151
$517$	−38.4853	−1.69258
$518$	−28.9706	−1.27289
$519$	18.8284	0.826476
$520$	0	0
$521$	−2.68629	−0.117689	−0.0588443	−	0.998267i	$-0.518742\pi$
$521$	−2.68629	−0.117689	−0.0588443	+	0.998267i	$0.518742\pi$
$522$	1.65685	0.0725185
$523$	−15.3137	−0.669622	−0.334811	−	0.942285i	$-0.608672\pi$
$523$	−15.3137	−0.669622	−0.334811	+	0.942285i	$0.608672\pi$
$524$	−10.1421	−0.443061
$525$	−14.4853	−0.632190
$526$	−6.72792	−0.293351
$527$	−8.97056	−0.390764
$528$	6.00000	0.261116
$529$	−21.6274	−0.940322
$530$	−33.9411	−1.47431
$531$	1.17157	0.0508419
$532$	18.4853	0.801439
$533$	0	0
$534$	−16.4853	−0.713388
$535$	−17.9411	−0.775662
$536$	−8.24264	−0.356028
$537$	−22.0000	−0.949370
$538$	18.0000	0.776035
$539$	19.7574	0.851010
$540$	16.0000	0.688530
$541$	8.48528	0.364811	0.182405	−	0.983223i	$-0.441612\pi$
$541$	8.48528	0.364811	0.182405	+	0.983223i	$0.441612\pi$
$542$	−20.4853	−0.879918
$543$	0	0
$544$	7.65685	0.328285
$545$	−33.9411	−1.45388
$546$	0	0
$547$	−9.41421	−0.402523	−0.201261	−	0.979538i	$-0.564504\pi$
$547$	−9.41421	−0.402523	−0.201261	+	0.979538i	$0.564504\pi$
$548$	5.31371	0.226990
$549$	−6.00000	−0.256074
$550$	12.7279	0.542720
$551$	8.97056	0.382159
$552$	1.65685	0.0705204
$553$	34.9706	1.48710
$554$	24.4853	1.04028
$555$	−33.9411	−1.44072
$556$	12.4853	0.529494
$557$	−5.65685	−0.239689	−0.119844	−	0.992793i	$-0.538240\pi$
$557$	−5.65685	−0.239689	−0.119844	+	0.992793i	$0.538240\pi$
$558$	1.17157	0.0495966
$559$	0	0
$560$	9.65685	0.408077
$561$	45.9411	1.93964
$562$	−22.9706	−0.968955
$563$	−0.727922	−0.0306783	−0.0153391	−	0.999882i	$-0.504883\pi$
$563$	−0.727922	−0.0306783	−0.0153391	+	0.999882i	$0.504883\pi$
$564$	−12.8284	−0.540174
$565$	4.68629	0.197154
$566$	−11.5147	−0.484000
$567$	17.0711	0.716917
$568$	−0.585786	−0.0245791
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	21.6569	0.907106
$571$	32.2426	1.34931	0.674656	−	0.738132i	$-0.264292\pi$
$571$	32.2426	1.34931	0.674656	+	0.738132i	$0.264292\pi$
$572$	0	0
$573$	−4.14214	−0.173040
$574$	3.41421	0.142507
$575$	3.51472	0.146574
$576$	−1.00000	−0.0416667
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	41.6274	1.73147
$579$	22.1421	0.920196
$580$	4.68629	0.194588
$581$	−60.2843	−2.50101
$582$	5.17157	0.214369
$583$	50.9117	2.10855
$584$	−13.6569	−0.565125
$585$	0	0
$586$	−17.6569	−0.729398
$587$	9.89949	0.408596	0.204298	−	0.978909i	$-0.434509\pi$
$587$	9.89949	0.408596	0.204298	+	0.978909i	$0.434509\pi$
$588$	6.58579	0.271593
$589$	6.34315	0.261365
$590$	3.31371	0.136423
$591$	13.1716	0.541806
$592$	8.48528	0.348743
$593$	13.3137	0.546728	0.273364	−	0.961911i	$-0.411864\pi$
$593$	13.3137	0.546728	0.273364	+	0.961911i	$0.411864\pi$
$594$	−24.0000	−0.984732
$595$	73.9411	3.03129
$596$	3.31371	0.135735
$597$	−3.17157	−0.129804
$598$	0	0
$599$	−3.51472	−0.143608	−0.0718038	−	0.997419i	$-0.522876\pi$
$599$	−3.51472	−0.143608	−0.0718038	+	0.997419i	$0.522876\pi$
$600$	4.24264	0.173205
$601$	−10.9706	−0.447499	−0.223749	−	0.974647i	$-0.571830\pi$
$601$	−10.9706	−0.447499	−0.223749	+	0.974647i	$0.571830\pi$
$602$	5.65685	0.230556
$603$	8.24264	0.335666
$604$	−11.8995	−0.484184
$605$	−19.7990	−0.804943
$606$	−8.97056	−0.364404
$607$	27.3137	1.10863	0.554315	−	0.832307i	$-0.312980\pi$
$607$	27.3137	1.10863	0.554315	+	0.832307i	$0.312980\pi$
$608$	−5.41421	−0.219575
$609$	8.00000	0.324176
$610$	−16.9706	−0.687118
$611$	0	0
$612$	−7.65685	−0.309510
$613$	−10.8284	−0.437356	−0.218678	−	0.975797i	$-0.570174\pi$
$613$	−10.8284	−0.437356	−0.218678	+	0.975797i	$0.570174\pi$
$614$	12.4853	0.503865
$615$	4.00000	0.161296
$616$	−14.4853	−0.583629
$617$	−33.3137	−1.34116	−0.670580	−	0.741837i	$-0.733955\pi$
$617$	−33.3137	−1.34116	−0.670580	+	0.741837i	$0.733955\pi$
$618$	6.34315	0.255159
$619$	38.4264	1.54449	0.772244	−	0.635326i	$-0.219134\pi$
$619$	38.4264	1.54449	0.772244	+	0.635326i	$0.219134\pi$
$620$	3.31371	0.133082
$621$	−6.62742	−0.265949
$622$	5.07107	0.203331
$623$	39.7990	1.59451
$624$	0	0
$625$	−31.0000	−1.24000
$626$	−1.31371	−0.0525064
$627$	−32.4853	−1.29734
$628$	−20.9706	−0.836817
$629$	64.9706	2.59055
$630$	−9.65685	−0.384738
$631$	−33.9411	−1.35117	−0.675587	−	0.737280i	$-0.736110\pi$
$631$	−33.9411	−1.35117	−0.675587	+	0.737280i	$0.736110\pi$
$632$	−10.2426	−0.407430
$633$	4.34315	0.172625
$634$	−11.3137	−0.449325
$635$	−51.3137	−2.03632
$636$	16.9706	0.672927
$637$	0	0
$638$	−7.02944	−0.278298
$639$	0.585786	0.0231734
$640$	−2.82843	−0.111803
$641$	1.02944	0.0406603	0.0203302	−	0.999793i	$-0.493528\pi$
$641$	1.02944	0.0406603	0.0203302	+	0.999793i	$0.493528\pi$
$642$	8.97056	0.354040
$643$	6.87006	0.270929	0.135464	−	0.990782i	$-0.456747\pi$
$643$	6.87006	0.270929	0.135464	+	0.990782i	$0.456747\pi$
$644$	−4.00000	−0.157622
$645$	6.62742	0.260954
$646$	−41.4558	−1.63106
$647$	44.4853	1.74890	0.874448	−	0.485118i	$-0.161224\pi$
$647$	44.4853	1.74890	0.874448	+	0.485118i	$0.161224\pi$
$648$	−5.00000	−0.196419
$649$	−4.97056	−0.195112
$650$	0	0
$651$	5.65685	0.221710
$652$	−4.00000	−0.156652
$653$	−11.3137	−0.442740	−0.221370	−	0.975190i	$-0.571053\pi$
$653$	−11.3137	−0.442740	−0.221370	+	0.975190i	$0.571053\pi$
$654$	16.9706	0.663602
$655$	28.6863	1.12087
$656$	−1.00000	−0.0390434
$657$	13.6569	0.532805
$658$	30.9706	1.20736
$659$	13.4142	0.522544	0.261272	−	0.965265i	$-0.415858\pi$
$659$	13.4142	0.522544	0.261272	+	0.965265i	$0.415858\pi$
$660$	−16.9706	−0.660578
$661$	41.4558	1.61245	0.806223	−	0.591612i	$-0.201508\pi$
$661$	41.4558	1.61245	0.806223	+	0.591612i	$0.201508\pi$
$662$	5.89949	0.229290
$663$	0	0
$664$	17.6569	0.685219
$665$	−52.2843	−2.02750
$666$	−8.48528	−0.328798
$667$	−1.94113	−0.0751607
$668$	−18.7279	−0.724605
$669$	0	0
$670$	23.3137	0.900687
$671$	25.4558	0.982712
$672$	−4.82843	−0.186261
$673$	−50.9706	−1.96477	−0.982385	−	0.186866i	$-0.940167\pi$
$673$	−50.9706	−1.96477	−0.982385	+	0.186866i	$0.940167\pi$
$674$	−12.0000	−0.462223
$675$	−16.9706	−0.653197
$676$	−13.0000	−0.500000
$677$	−38.1421	−1.46592	−0.732961	−	0.680271i	$-0.761862\pi$
$677$	−38.1421	−1.46592	−0.732961	+	0.680271i	$0.761862\pi$
$678$	−2.34315	−0.0899880
$679$	−12.4853	−0.479141
$680$	−21.6569	−0.830502
$681$	19.6569	0.753252
$682$	−4.97056	−0.190333
$683$	−41.4142	−1.58467	−0.792335	−	0.610086i	$-0.791135\pi$
$683$	−41.4142	−1.58467	−0.792335	+	0.610086i	$0.791135\pi$
$684$	5.41421	0.207018
$685$	−15.0294	−0.574245
$686$	8.00000	0.305441
$687$	−11.3137	−0.431645
$688$	−1.65685	−0.0631670
$689$	0	0
$690$	−4.68629	−0.178404
$691$	12.2426	0.465732	0.232866	−	0.972509i	$-0.425190\pi$
$691$	12.2426	0.465732	0.232866	+	0.972509i	$0.425190\pi$
$692$	13.3137	0.506111
$693$	14.4853	0.550250
$694$	21.8995	0.831293
$695$	−35.3137	−1.33953
$696$	−2.34315	−0.0888167
$697$	−7.65685	−0.290024
$698$	15.5147	0.587241
$699$	39.1127	1.47938
$700$	−10.2426	−0.387135
$701$	25.4558	0.961454	0.480727	−	0.876870i	$-0.340373\pi$
$701$	25.4558	0.961454	0.480727	+	0.876870i	$0.340373\pi$
$702$	0	0
$703$	−45.9411	−1.73270
$704$	4.24264	0.159901
$705$	36.2843	1.36654
$706$	−3.31371	−0.124713
$707$	21.6569	0.814490
$708$	−1.65685	−0.0622684
$709$	−21.6569	−0.813340	−0.406670	−	0.913575i	$-0.633310\pi$
$709$	−21.6569	−0.813340	−0.406670	+	0.913575i	$0.633310\pi$
$710$	1.65685	0.0621806
$711$	10.2426	0.384129
$712$	−11.6569	−0.436859
$713$	−1.37258	−0.0514036
$714$	−36.9706	−1.38359
$715$	0	0
$716$	−15.5563	−0.581368
$717$	−0.828427	−0.0309382
$718$	13.6569	0.509669
$719$	−12.1005	−0.451273	−0.225636	−	0.974212i	$-0.572446\pi$
$719$	−12.1005	−0.451273	−0.225636	+	0.974212i	$0.572446\pi$
$720$	2.82843	0.105409
$721$	−15.3137	−0.570312
$722$	10.3137	0.383836
$723$	25.4558	0.946713
$724$	0	0
$725$	−4.97056	−0.184602
$726$	9.89949	0.367405
$727$	29.0711	1.07819	0.539093	−	0.842246i	$-0.318767\pi$
$727$	29.0711	1.07819	0.539093	+	0.842246i	$0.318767\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	38.6274	1.42966
$731$	−12.6863	−0.469219
$732$	8.48528	0.313625
$733$	26.6863	0.985681	0.492840	−	0.870120i	$-0.335959\pi$
$733$	26.6863	0.985681	0.492840	+	0.870120i	$0.335959\pi$
$734$	−13.6569	−0.504084
$735$	−18.6274	−0.687083
$736$	1.17157	0.0431847
$737$	−34.9706	−1.28816
$738$	1.00000	0.0368105
$739$	−19.0294	−0.700009	−0.350005	−	0.936748i	$-0.613820\pi$
$739$	−19.0294	−0.700009	−0.350005	+	0.936748i	$0.613820\pi$
$740$	−24.0000	−0.882258
$741$	0	0
$742$	−40.9706	−1.50408
$743$	3.51472	0.128943	0.0644713	−	0.997920i	$-0.479464\pi$
$743$	3.51472	0.128943	0.0644713	+	0.997920i	$0.479464\pi$
$744$	−1.65685	−0.0607432
$745$	−9.37258	−0.343385
$746$	−32.6274	−1.19457
$747$	−17.6569	−0.646031
$748$	32.4853	1.18778
$749$	−21.6569	−0.791324
$750$	8.00000	0.292119
$751$	−50.7279	−1.85109	−0.925544	−	0.378640i	$-0.876392\pi$
$751$	−50.7279	−1.85109	−0.925544	+	0.378640i	$0.876392\pi$
$752$	−9.07107	−0.330788
$753$	3.02944	0.110399
$754$	0	0
$755$	33.6569	1.22490
$756$	19.3137	0.702433
$757$	30.6274	1.11317	0.556586	−	0.830790i	$-0.312111\pi$
$757$	30.6274	1.11317	0.556586	+	0.830790i	$0.312111\pi$
$758$	−21.4558	−0.779311
$759$	7.02944	0.255152
$760$	15.3137	0.555487
$761$	−36.0000	−1.30500	−0.652499	−	0.757789i	$-0.726280\pi$
$761$	−36.0000	−1.30500	−0.652499	+	0.757789i	$0.726280\pi$
$762$	25.6569	0.929450
$763$	−40.9706	−1.48323
$764$	−2.92893	−0.105965
$765$	21.6569	0.783005
$766$	−3.89949	−0.140894
$767$	0	0
$768$	1.41421	0.0510310
$769$	0.627417	0.0226252	0.0113126	−	0.999936i	$-0.496399\pi$
$769$	0.627417	0.0226252	0.0113126	+	0.999936i	$0.496399\pi$
$770$	40.9706	1.47648
$771$	18.8284	0.678089
$772$	15.6569	0.563503
$773$	11.3137	0.406926	0.203463	−	0.979083i	$-0.434780\pi$
$773$	11.3137	0.406926	0.203463	+	0.979083i	$0.434780\pi$
$774$	1.65685	0.0595544
$775$	−3.51472	−0.126252
$776$	3.65685	0.131273
$777$	−40.9706	−1.46981
$778$	−9.31371	−0.333913
$779$	5.41421	0.193984
$780$	0	0
$781$	−2.48528	−0.0889304
$782$	8.97056	0.320787
$783$	9.37258	0.334949
$784$	4.65685	0.166316
$785$	59.3137	2.11700
$786$	−14.3431	−0.511603
$787$	19.5147	0.695625	0.347812	−	0.937564i	$-0.386925\pi$
$787$	19.5147	0.695625	0.347812	+	0.937564i	$0.386925\pi$
$788$	9.31371	0.331787
$789$	−9.51472	−0.338733
$790$	28.9706	1.03073
$791$	5.65685	0.201135
$792$	−4.24264	−0.150756
$793$	0	0
$794$	−4.97056	−0.176399
$795$	−48.0000	−1.70238
$796$	−2.24264	−0.0794883
$797$	21.3137	0.754970	0.377485	−	0.926016i	$-0.376789\pi$
$797$	21.3137	0.754970	0.377485	+	0.926016i	$0.376789\pi$
$798$	26.1421	0.925422
$799$	−69.4558	−2.45717
$800$	3.00000	0.106066
$801$	11.6569	0.411875
$802$	−11.3137	−0.399501
$803$	−57.9411	−2.04470
$804$	−11.6569	−0.411106
$805$	11.3137	0.398756
$806$	0	0
$807$	25.4558	0.896088
$808$	−6.34315	−0.223151
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	14.1421	0.496904
$811$	8.20101	0.287976	0.143988	−	0.989579i	$-0.454007\pi$
$811$	8.20101	0.287976	0.143988	+	0.989579i	$0.454007\pi$
$812$	5.65685	0.198517
$813$	−28.9706	−1.01604
$814$	36.0000	1.26180
$815$	11.3137	0.396302
$816$	10.8284	0.379071
$817$	8.97056	0.313840
$818$	−13.6569	−0.477501
$819$	0	0
$820$	2.82843	0.0987730
$821$	−6.14214	−0.214362	−0.107181	−	0.994240i	$-0.534182\pi$
$821$	−6.14214	−0.214362	−0.107181	+	0.994240i	$0.534182\pi$
$822$	7.51472	0.262106
$823$	−47.2132	−1.64575	−0.822874	−	0.568223i	$-0.807631\pi$
$823$	−47.2132	−1.64575	−0.822874	+	0.568223i	$0.807631\pi$
$824$	4.48528	0.156252
$825$	18.0000	0.626680
$826$	4.00000	0.139178
$827$	28.9289	1.00596	0.502979	−	0.864299i	$-0.332237\pi$
$827$	28.9289	1.00596	0.502979	+	0.864299i	$0.332237\pi$
$828$	−1.17157	−0.0407150
$829$	−18.8284	−0.653938	−0.326969	−	0.945035i	$-0.606027\pi$
$829$	−18.8284	−0.653938	−0.326969	+	0.945035i	$0.606027\pi$
$830$	−49.9411	−1.73348
$831$	34.6274	1.20121
$832$	0	0
$833$	35.6569	1.23544
$834$	17.6569	0.611407
$835$	52.9706	1.83312
$836$	−22.9706	−0.794454
$837$	6.62742	0.229077
$838$	17.6569	0.609946
$839$	−38.5269	−1.33010	−0.665048	−	0.746800i	$-0.731589\pi$
$839$	−38.5269	−1.33010	−0.665048	+	0.746800i	$0.731589\pi$
$840$	13.6569	0.471206
$841$	−26.2548	−0.905339
$842$	−8.97056	−0.309146
$843$	−32.4853	−1.11885
$844$	3.07107	0.105711
$845$	36.7696	1.26491
$846$	9.07107	0.311870
$847$	−23.8995	−0.821196
$848$	12.0000	0.412082
$849$	−16.2843	−0.558875
$850$	22.9706	0.787884
$851$	9.94113	0.340777
$852$	−0.828427	−0.0283814
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	−20.4853	−0.700992
$855$	−15.3137	−0.523718
$856$	6.34315	0.216804
$857$	42.3431	1.44641	0.723207	−	0.690631i	$-0.242667\pi$
$857$	42.3431	1.44641	0.723207	+	0.690631i	$0.242667\pi$
$858$	0	0
$859$	29.4558	1.00502	0.502510	−	0.864571i	$-0.332410\pi$
$859$	29.4558	1.00502	0.502510	+	0.864571i	$0.332410\pi$
$860$	4.68629	0.159801
$861$	4.82843	0.164552
$862$	−5.65685	−0.192673
$863$	53.2548	1.81282	0.906408	−	0.422404i	$-0.138814\pi$
$863$	53.2548	1.81282	0.906408	+	0.422404i	$0.138814\pi$
$864$	−5.65685	−0.192450
$865$	−37.6569	−1.28037
$866$	−35.9411	−1.22133
$867$	58.8701	1.99933
$868$	4.00000	0.135769
$869$	−43.4558	−1.47414
$870$	6.62742	0.224690
$871$	0	0
$872$	12.0000	0.406371
$873$	−3.65685	−0.123766
$874$	−6.34315	−0.214560
$875$	−19.3137	−0.652923
$876$	−19.3137	−0.652550
$877$	10.0000	0.337676	0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000	0.337676	0.168838	+	0.985644i	$0.445999\pi$
$878$	8.58579	0.289756
$879$	−24.9706	−0.842236
$880$	−12.0000	−0.404520
$881$	28.6274	0.964482	0.482241	−	0.876039i	$-0.339823\pi$
$881$	28.6274	0.964482	0.482241	+	0.876039i	$0.339823\pi$
$882$	−4.65685	−0.156804
$883$	−34.3848	−1.15714	−0.578570	−	0.815633i	$-0.696389\pi$
$883$	−34.3848	−1.15714	−0.578570	+	0.815633i	$0.696389\pi$
$884$	0	0
$885$	4.68629	0.157528
$886$	−9.17157	−0.308125
$887$	−21.5563	−0.723791	−0.361896	−	0.932219i	$-0.617870\pi$
$887$	−21.5563	−0.723791	−0.361896	+	0.932219i	$0.617870\pi$
$888$	12.0000	0.402694
$889$	−61.9411	−2.07744
$890$	32.9706	1.10518
$891$	−21.2132	−0.710669
$892$	0	0
$893$	49.1127	1.64349
$894$	4.68629	0.156733
$895$	44.0000	1.47076
$896$	−3.41421	−0.114061
$897$	0	0
$898$	−5.31371	−0.177321
$899$	1.94113	0.0647402
$900$	−3.00000	−0.100000
$901$	91.8823	3.06104
$902$	−4.24264	−0.141264
$903$	8.00000	0.266223
$904$	−1.65685	−0.0551062
$905$	0	0
$906$	−16.8284	−0.559087
$907$	6.82843	0.226734	0.113367	−	0.993553i	$-0.463836\pi$
$907$	6.82843	0.226734	0.113367	+	0.993553i	$0.463836\pi$
$908$	13.8995	0.461271
$909$	6.34315	0.210389
$910$	0	0
$911$	51.3137	1.70010	0.850050	−	0.526703i	$-0.176572\pi$
$911$	51.3137	1.70010	0.850050	+	0.526703i	$0.176572\pi$
$912$	−7.65685	−0.253544
$913$	74.9117	2.47922
$914$	24.6274	0.814603
$915$	−24.0000	−0.793416
$916$	−8.00000	−0.264327
$917$	34.6274	1.14350
$918$	−43.3137	−1.42957
$919$	11.6985	0.385897	0.192949	−	0.981209i	$-0.438195\pi$
$919$	11.6985	0.385897	0.192949	+	0.981209i	$0.438195\pi$
$920$	−3.31371	−0.109250
$921$	17.6569	0.581813
$922$	19.7990	0.652045
$923$	0	0
$924$	−20.4853	−0.673916
$925$	25.4558	0.836983
$926$	−3.61522	−0.118804
$927$	−4.48528	−0.147316
$928$	−1.65685	−0.0543889
$929$	51.6569	1.69481	0.847403	−	0.530950i	$-0.178165\pi$
$929$	51.6569	1.69481	0.847403	+	0.530950i	$0.178165\pi$
$930$	4.68629	0.153670
$931$	−25.2132	−0.826329
$932$	27.6569	0.905930
$933$	7.17157	0.234787
$934$	−5.85786	−0.191675
$935$	−91.8823	−3.00487
$936$	0	0
$937$	−10.9706	−0.358393	−0.179196	−	0.983813i	$-0.557350\pi$
$937$	−10.9706	−0.358393	−0.179196	+	0.983813i	$0.557350\pi$
$938$	28.1421	0.918873
$939$	−1.85786	−0.0606291
$940$	25.6569	0.836834
$941$	32.6274	1.06362	0.531812	−	0.846863i	$-0.321511\pi$
$941$	32.6274	1.06362	0.531812	+	0.846863i	$0.321511\pi$
$942$	−29.6569	−0.966273
$943$	−1.17157	−0.0381517
$944$	−1.17157	−0.0381314
$945$	−54.6274	−1.77703
$946$	−7.02944	−0.228547
$947$	−13.4558	−0.437256	−0.218628	−	0.975808i	$-0.570158\pi$
$947$	−13.4558	−0.437256	−0.218628	+	0.975808i	$0.570158\pi$
$948$	−14.4853	−0.470460
$949$	0	0
$950$	−16.2426	−0.526981
$951$	−16.0000	−0.518836
$952$	−26.1421	−0.847271
$953$	26.3431	0.853338	0.426669	−	0.904408i	$-0.359687\pi$
$953$	26.3431	0.853338	0.426669	+	0.904408i	$0.359687\pi$
$954$	−12.0000	−0.388514
$955$	8.28427	0.268073
$956$	−0.585786	−0.0189457
$957$	−9.94113	−0.321351
$958$	20.5858	0.665097
$959$	−18.1421	−0.585840
$960$	−4.00000	−0.129099
$961$	−29.6274	−0.955723
$962$	0	0
$963$	−6.34315	−0.204405
$964$	18.0000	0.579741
$965$	−44.2843	−1.42556
$966$	−5.65685	−0.182006
$967$	56.6690	1.82235	0.911177	−	0.412015i	$-0.135175\pi$
$967$	56.6690	1.82235	0.911177	+	0.412015i	$0.135175\pi$
$968$	7.00000	0.224989
$969$	−58.6274	−1.88338
$970$	−10.3431	−0.332098
$971$	−32.5269	−1.04384	−0.521919	−	0.852995i	$-0.674784\pi$
$971$	−32.5269	−1.04384	−0.521919	+	0.852995i	$0.674784\pi$
$972$	9.89949	0.317526
$973$	−42.6274	−1.36657
$974$	−3.51472	−0.112619
$975$	0	0
$976$	6.00000	0.192055
$977$	8.34315	0.266921	0.133460	−	0.991054i	$-0.457391\pi$
$977$	8.34315	0.266921	0.133460	+	0.991054i	$0.457391\pi$
$978$	−5.65685	−0.180886
$979$	−49.4558	−1.58062
$980$	−13.1716	−0.420750
$981$	−12.0000	−0.383131
$982$	−31.7990	−1.01475
$983$	23.7990	0.759070	0.379535	−	0.925177i	$-0.376084\pi$
$983$	23.7990	0.759070	0.379535	+	0.925177i	$0.376084\pi$
$984$	−1.41421	−0.0450835
$985$	−26.3431	−0.839362
$986$	−12.6863	−0.404014
$987$	43.7990	1.39414
$988$	0	0
$989$	−1.94113	−0.0617242
$990$	12.0000	0.381385
$991$	2.72792	0.0866553	0.0433277	−	0.999061i	$-0.486204\pi$
$991$	2.72792	0.0866553	0.0433277	+	0.999061i	$0.486204\pi$
$992$	−1.17157	−0.0371975
$993$	8.34315	0.264762
$994$	2.00000	0.0634361
$995$	6.34315	0.201091
$996$	24.9706	0.791223
$997$	−45.9411	−1.45497	−0.727485	−	0.686124i	$-0.759311\pi$
$997$	−45.9411	−1.45497	−0.727485	+	0.686124i	$0.759311\pi$
$998$	−28.5269	−0.903004
$999$	−48.0000	−1.51865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	82.2.a.b.1.2	✓	2
3.2	odd	2		738.2.a.k.1.2		2
4.3	odd	2		656.2.a.e.1.1		2
5.2	odd	4		2050.2.c.l.1149.3		4
5.3	odd	4		2050.2.c.l.1149.2		4
5.4	even	2		2050.2.a.h.1.1		2
7.6	odd	2		4018.2.a.ba.1.1		2
8.3	odd	2		2624.2.a.l.1.2		2
8.5	even	2		2624.2.a.j.1.1		2
11.10	odd	2		9922.2.a.i.1.2		2
12.11	even	2		5904.2.a.z.1.2		2
41.40	even	2		3362.2.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
82.2.a.b.1.2	✓	2	1.1	even	1	trivial
656.2.a.e.1.1		2	4.3	odd	2
738.2.a.k.1.2		2	3.2	odd	2
2050.2.a.h.1.1		2	5.4	even	2
2050.2.c.l.1149.2		4	5.3	odd	4
2050.2.c.l.1149.3		4	5.2	odd	4
2624.2.a.j.1.1		2	8.5	even	2
2624.2.a.l.1.2		2	8.3	odd	2
3362.2.a.m.1.1		2	41.40	even	2
4018.2.a.ba.1.1		2	7.6	odd	2
5904.2.a.z.1.2		2	12.11	even	2
9922.2.a.i.1.2		2	11.10	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 82 = 2 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	82.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -2.82843 q^{5} +1.41421 q^{6} -3.41421 q^{7} +1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -2.82843 q^{5} +1.41421 q^{6} -3.41421 q^{7} +1.00000 q^{8} -1.00000 q^{9} -2.82843 q^{10} +4.24264 q^{11} +1.41421 q^{12} -3.41421 q^{14} -4.00000 q^{15} +1.00000 q^{16} +7.65685 q^{17} -1.00000 q^{18} -5.41421 q^{19} -2.82843 q^{20} -4.82843 q^{21} +4.24264 q^{22} +1.17157 q^{23} +1.41421 q^{24} +3.00000 q^{25} -5.65685 q^{27} -3.41421 q^{28} -1.65685 q^{29} -4.00000 q^{30} -1.17157 q^{31} +1.00000 q^{32} +6.00000 q^{33} +7.65685 q^{34} +9.65685 q^{35} -1.00000 q^{36} +8.48528 q^{37} -5.41421 q^{38} -2.82843 q^{40} -1.00000 q^{41} -4.82843 q^{42} -1.65685 q^{43} +4.24264 q^{44} +2.82843 q^{45} +1.17157 q^{46} -9.07107 q^{47} +1.41421 q^{48} +4.65685 q^{49} +3.00000 q^{50} +10.8284 q^{51} +12.0000 q^{53} -5.65685 q^{54} -12.0000 q^{55} -3.41421 q^{56} -7.65685 q^{57} -1.65685 q^{58} -1.17157 q^{59} -4.00000 q^{60} +6.00000 q^{61} -1.17157 q^{62} +3.41421 q^{63} +1.00000 q^{64} +6.00000 q^{66} -8.24264 q^{67} +7.65685 q^{68} +1.65685 q^{69} +9.65685 q^{70} -0.585786 q^{71} -1.00000 q^{72} -13.6569 q^{73} +8.48528 q^{74} +4.24264 q^{75} -5.41421 q^{76} -14.4853 q^{77} -10.2426 q^{79} -2.82843 q^{80} -5.00000 q^{81} -1.00000 q^{82} +17.6569 q^{83} -4.82843 q^{84} -21.6569 q^{85} -1.65685 q^{86} -2.34315 q^{87} +4.24264 q^{88} -11.6569 q^{89} +2.82843 q^{90} +1.17157 q^{92} -1.65685 q^{93} -9.07107 q^{94} +15.3137 q^{95} +1.41421 q^{96} +3.65685 q^{97} +4.65685 q^{98} -4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{7} + 2 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^7 + 2 * q^8 - 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{7} + 2 q^{8} - 2 q^{9} - 4 q^{14} - 8 q^{15} + 2 q^{16} + 4 q^{17} - 2 q^{18} - 8 q^{19} - 4 q^{21} + 8 q^{23} + 6 q^{25} - 4 q^{28} + 8 q^{29} - 8 q^{30} - 8 q^{31} + 2 q^{32} + 12 q^{33} + 4 q^{34} + 8 q^{35} - 2 q^{36} - 8 q^{38} - 2 q^{41} - 4 q^{42} + 8 q^{43} + 8 q^{46} - 4 q^{47} - 2 q^{49} + 6 q^{50} + 16 q^{51} + 24 q^{53} - 24 q^{55} - 4 q^{56} - 4 q^{57} + 8 q^{58} - 8 q^{59} - 8 q^{60} + 12 q^{61} - 8 q^{62} + 4 q^{63} + 2 q^{64} + 12 q^{66} - 8 q^{67} + 4 q^{68} - 8 q^{69} + 8 q^{70} - 4 q^{71} - 2 q^{72} - 16 q^{73} - 8 q^{76} - 12 q^{77} - 12 q^{79} - 10 q^{81} - 2 q^{82} + 24 q^{83} - 4 q^{84} - 32 q^{85} + 8 q^{86} - 16 q^{87} - 12 q^{89} + 8 q^{92} + 8 q^{93} - 4 q^{94} + 8 q^{95} - 4 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^7 + 2 * q^8 - 2 * q^9 - 4 * q^14 - 8 * q^15 + 2 * q^16 + 4 * q^17 - 2 * q^18 - 8 * q^19 - 4 * q^21 + 8 * q^23 + 6 * q^25 - 4 * q^28 + 8 * q^29 - 8 * q^30 - 8 * q^31 + 2 * q^32 + 12 * q^33 + 4 * q^34 + 8 * q^35 - 2 * q^36 - 8 * q^38 - 2 * q^41 - 4 * q^42 + 8 * q^43 + 8 * q^46 - 4 * q^47 - 2 * q^49 + 6 * q^50 + 16 * q^51 + 24 * q^53 - 24 * q^55 - 4 * q^56 - 4 * q^57 + 8 * q^58 - 8 * q^59 - 8 * q^60 + 12 * q^61 - 8 * q^62 + 4 * q^63 + 2 * q^64 + 12 * q^66 - 8 * q^67 + 4 * q^68 - 8 * q^69 + 8 * q^70 - 4 * q^71 - 2 * q^72 - 16 * q^73 - 8 * q^76 - 12 * q^77 - 12 * q^79 - 10 * q^81 - 2 * q^82 + 24 * q^83 - 4 * q^84 - 32 * q^85 + 8 * q^86 - 16 * q^87 - 12 * q^89 + 8 * q^92 + 8 * q^93 - 4 * q^94 + 8 * q^95 - 4 * q^97 - 2 * q^98

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