LMFDB - Embedded newform 6027.2.a.bc.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6027,2,Mod(1,6027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6027, 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("6027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6027 = 3 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6027.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:	$48.1258372982$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.7457527933.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 6x^{6} + 23x^{5} - 4x^{4} - 27x^{3} + 8x^{2} + 8x + 1 $ x^8 - 3x^7 - 6x^6 + 23x^5 - 4x^4 - 27x^3 + 8x^2 + 8*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 861)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.978012$ of defining polynomial
Character	$\chi$	$=$	6027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.13860 q^{2} +1.00000 q^{3} -0.703600 q^{4} +2.27123 q^{5} -1.13860 q^{6} +3.07831 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.13860 q^{2} +1.00000 q^{3} -0.703600 q^{4} +2.27123 q^{5} -1.13860 q^{6} +3.07831 q^{8} +1.00000 q^{9} -2.58601 q^{10} +4.11356 q^{11} -0.703600 q^{12} -1.02235 q^{13} +2.27123 q^{15} -2.09775 q^{16} +1.01936 q^{17} -1.13860 q^{18} +0.361035 q^{19} -1.59804 q^{20} -4.68368 q^{22} -4.61988 q^{23} +3.07831 q^{24} +0.158481 q^{25} +1.16404 q^{26} +1.00000 q^{27} -1.50587 q^{29} -2.58601 q^{30} +3.87299 q^{31} -3.76813 q^{32} +4.11356 q^{33} -1.16064 q^{34} -0.703600 q^{36} +9.23719 q^{37} -0.411073 q^{38} -1.02235 q^{39} +6.99154 q^{40} +1.00000 q^{41} -2.19077 q^{43} -2.89430 q^{44} +2.27123 q^{45} +5.26018 q^{46} -1.32485 q^{47} -2.09775 q^{48} -0.180446 q^{50} +1.01936 q^{51} +0.719323 q^{52} +7.53796 q^{53} -1.13860 q^{54} +9.34284 q^{55} +0.361035 q^{57} +1.71458 q^{58} -1.63497 q^{59} -1.59804 q^{60} +14.8701 q^{61} -4.40977 q^{62} +8.48587 q^{64} -2.32198 q^{65} -4.68368 q^{66} +5.38619 q^{67} -0.717224 q^{68} -4.61988 q^{69} -1.61302 q^{71} +3.07831 q^{72} +2.22290 q^{73} -10.5174 q^{74} +0.158481 q^{75} -0.254024 q^{76} +1.16404 q^{78} -4.15827 q^{79} -4.76446 q^{80} +1.00000 q^{81} -1.13860 q^{82} +1.53726 q^{83} +2.31521 q^{85} +2.49440 q^{86} -1.50587 q^{87} +12.6628 q^{88} +1.06059 q^{89} -2.58601 q^{90} +3.25055 q^{92} +3.87299 q^{93} +1.50847 q^{94} +0.819993 q^{95} -3.76813 q^{96} +0.418003 q^{97} +4.11356 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + 8 q^{3} + 13 q^{4} + 7 q^{5} + q^{6} + 6 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + q^2 + 8 * q^3 + 13 * q^4 + 7 * q^5 + q^6 + 6 * q^8 + 8 * q^9	$ 8 q + q^{2} + 8 q^{3} + 13 q^{4} + 7 q^{5} + q^{6} + 6 q^{8} + 8 q^{9} + 8 q^{10} + 11 q^{11} + 13 q^{12} + 10 q^{13} + 7 q^{15} - 17 q^{16} + 3 q^{17} + q^{18} + 6 q^{19} + 11 q^{20} + 15 q^{22} + 14 q^{23} + 6 q^{24} + 25 q^{25} + 24 q^{26} + 8 q^{27} + 2 q^{29} + 8 q^{30} + 16 q^{31} + 3 q^{32} + 11 q^{33} - 4 q^{34} + 13 q^{36} - 20 q^{37} + 10 q^{38} + 10 q^{39} - 3 q^{40} + 8 q^{41} + 7 q^{43} + 7 q^{45} - 5 q^{46} + 14 q^{47} - 17 q^{48} - 5 q^{50} + 3 q^{51} + 23 q^{52} + 7 q^{53} + q^{54} + 48 q^{55} + 6 q^{57} - 20 q^{58} + 22 q^{59} + 11 q^{60} - 33 q^{62} - 10 q^{64} - 14 q^{65} + 15 q^{66} + 12 q^{67} - 27 q^{68} + 14 q^{69} - 5 q^{71} + 6 q^{72} + 2 q^{73} + 6 q^{74} + 25 q^{75} + 43 q^{76} + 24 q^{78} - 15 q^{79} - 7 q^{80} + 8 q^{81} + q^{82} + 15 q^{83} - 43 q^{85} + 31 q^{86} + 2 q^{87} + 17 q^{88} + 29 q^{89} + 8 q^{90} + 19 q^{92} + 16 q^{93} + 20 q^{94} + 14 q^{95} + 3 q^{96} + 19 q^{97} + 11 q^{99}+O(q^{100}) $ 8 * q + q^2 + 8 * q^3 + 13 * q^4 + 7 * q^5 + q^6 + 6 * q^8 + 8 * q^9 + 8 * q^10 + 11 * q^11 + 13 * q^12 + 10 * q^13 + 7 * q^15 - 17 * q^16 + 3 * q^17 + q^18 + 6 * q^19 + 11 * q^20 + 15 * q^22 + 14 * q^23 + 6 * q^24 + 25 * q^25 + 24 * q^26 + 8 * q^27 + 2 * q^29 + 8 * q^30 + 16 * q^31 + 3 * q^32 + 11 * q^33 - 4 * q^34 + 13 * q^36 - 20 * q^37 + 10 * q^38 + 10 * q^39 - 3 * q^40 + 8 * q^41 + 7 * q^43 + 7 * q^45 - 5 * q^46 + 14 * q^47 - 17 * q^48 - 5 * q^50 + 3 * q^51 + 23 * q^52 + 7 * q^53 + q^54 + 48 * q^55 + 6 * q^57 - 20 * q^58 + 22 * q^59 + 11 * q^60 - 33 * q^62 - 10 * q^64 - 14 * q^65 + 15 * q^66 + 12 * q^67 - 27 * q^68 + 14 * q^69 - 5 * q^71 + 6 * q^72 + 2 * q^73 + 6 * q^74 + 25 * q^75 + 43 * q^76 + 24 * q^78 - 15 * q^79 - 7 * q^80 + 8 * q^81 + q^82 + 15 * q^83 - 43 * q^85 + 31 * q^86 + 2 * q^87 + 17 * q^88 + 29 * q^89 + 8 * q^90 + 19 * q^92 + 16 * q^93 + 20 * q^94 + 14 * q^95 + 3 * q^96 + 19 * q^97 + 11 * q^99

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 $

$2$	−1.13860	−0.805109	−0.402554	−	0.915396i	$-0.631878\pi$
$2$	−1.13860	−0.805109	−0.402554	+	0.915396i	$0.631878\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.703600	−0.351800
$5$	2.27123	1.01572	0.507862	−	0.861438i	$-0.330436\pi$
$5$	2.27123	1.01572	0.507862	+	0.861438i	$0.330436\pi$
$6$	−1.13860	−0.464830
$7$	0	0
$7$	0	0
$8$	3.07831	1.08835
$9$	1.00000	0.333333
$10$	−2.58601	−0.817769
$11$	4.11356	1.24029	0.620143	−	0.784489i	$-0.287075\pi$
$11$	4.11356	1.24029	0.620143	+	0.784489i	$0.287075\pi$
$12$	−0.703600	−0.203112
$13$	−1.02235	−0.283548	−0.141774	−	0.989899i	$-0.545281\pi$
$13$	−1.02235	−0.283548	−0.141774	+	0.989899i	$0.545281\pi$
$14$	0	0
$15$	2.27123	0.586429
$16$	−2.09775	−0.524437
$17$	1.01936	0.247232	0.123616	−	0.992330i	$-0.460551\pi$
$17$	1.01936	0.247232	0.123616	+	0.992330i	$0.460551\pi$
$18$	−1.13860	−0.268370
$19$	0.361035	0.0828271	0.0414135	−	0.999142i	$-0.486814\pi$
$19$	0.361035	0.0828271	0.0414135	+	0.999142i	$0.486814\pi$
$20$	−1.59804	−0.357332
$21$	0	0
$22$	−4.68368	−0.998564
$23$	−4.61988	−0.963312	−0.481656	−	0.876360i	$-0.659965\pi$
$23$	−4.61988	−0.963312	−0.481656	+	0.876360i	$0.659965\pi$
$24$	3.07831	0.628357
$25$	0.158481	0.0316963
$26$	1.16404	0.228287
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.50587	−0.279633	−0.139816	−	0.990177i	$-0.544651\pi$
$29$	−1.50587	−0.279633	−0.139816	+	0.990177i	$0.544651\pi$
$30$	−2.58601	−0.472139
$31$	3.87299	0.695610	0.347805	−	0.937567i	$-0.386927\pi$
$31$	3.87299	0.695610	0.347805	+	0.937567i	$0.386927\pi$
$32$	−3.76813	−0.666117
$33$	4.11356	0.716079
$34$	−1.16064	−0.199049
$35$	0	0
$36$	−0.703600	−0.117267
$37$	9.23719	1.51858	0.759292	−	0.650750i	$-0.225545\pi$
$37$	9.23719	1.51858	0.759292	+	0.650750i	$0.225545\pi$
$38$	−0.411073	−0.0666848
$39$	−1.02235	−0.163706
$40$	6.99154	1.10546
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−2.19077	−0.334089	−0.167045	−	0.985949i	$-0.553422\pi$
$43$	−2.19077	−0.334089	−0.167045	+	0.985949i	$0.553422\pi$
$44$	−2.89430	−0.436332
$45$	2.27123	0.338575
$46$	5.26018	0.775571
$47$	−1.32485	−0.193249	−0.0966245	−	0.995321i	$-0.530805\pi$
$47$	−1.32485	−0.193249	−0.0966245	+	0.995321i	$0.530805\pi$
$48$	−2.09775	−0.302784
$49$	0	0
$50$	−0.180446	−0.0255190
$51$	1.01936	0.142739
$52$	0.719323	0.0997522
$53$	7.53796	1.03542	0.517709	−	0.855557i	$-0.326785\pi$
$53$	7.53796	1.03542	0.517709	+	0.855557i	$0.326785\pi$
$54$	−1.13860	−0.154943
$55$	9.34284	1.25979
$56$	0	0
$57$	0.361035	0.0478202
$58$	1.71458	0.225135
$59$	−1.63497	−0.212855	−0.106427	−	0.994320i	$-0.533941\pi$
$59$	−1.63497	−0.212855	−0.106427	+	0.994320i	$0.533941\pi$
$60$	−1.59804	−0.206306
$61$	14.8701	1.90392	0.951961	−	0.306218i	$-0.0990638\pi$
$61$	14.8701	1.90392	0.951961	+	0.306218i	$0.0990638\pi$
$62$	−4.40977	−0.560042
$63$	0	0
$64$	8.48587	1.06073
$65$	−2.32198	−0.288007
$66$	−4.68368	−0.576521
$67$	5.38619	0.658027	0.329014	−	0.944325i	$-0.393284\pi$
$67$	5.38619	0.658027	0.329014	+	0.944325i	$0.393284\pi$
$68$	−0.717224	−0.0869762
$69$	−4.61988	−0.556169
$70$	0	0
$71$	−1.61302	−0.191430	−0.0957149	−	0.995409i	$-0.530514\pi$
$71$	−1.61302	−0.191430	−0.0957149	+	0.995409i	$0.530514\pi$
$72$	3.07831	0.362782
$73$	2.22290	0.260171	0.130086	−	0.991503i	$-0.458475\pi$
$73$	2.22290	0.260171	0.130086	+	0.991503i	$0.458475\pi$
$74$	−10.5174	−1.22263
$75$	0.158481	0.0182999
$76$	−0.254024	−0.0291386
$77$	0	0
$78$	1.16404	0.131801
$79$	−4.15827	−0.467842	−0.233921	−	0.972256i	$-0.575156\pi$
$79$	−4.15827	−0.467842	−0.233921	+	0.972256i	$0.575156\pi$
$80$	−4.76446	−0.532683
$81$	1.00000	0.111111
$82$	−1.13860	−0.125737
$83$	1.53726	0.168736	0.0843679	−	0.996435i	$-0.473113\pi$
$83$	1.53726	0.168736	0.0843679	+	0.996435i	$0.473113\pi$
$84$	0	0
$85$	2.31521	0.251119
$86$	2.49440	0.268978
$87$	−1.50587	−0.161446
$88$	12.6628	1.34986
$89$	1.06059	0.112422	0.0562111	−	0.998419i	$-0.482098\pi$
$89$	1.06059	0.112422	0.0562111	+	0.998419i	$0.482098\pi$
$90$	−2.58601	−0.272590
$91$	0	0
$92$	3.25055	0.338893
$93$	3.87299	0.401611
$94$	1.50847	0.155586
$95$	0.819993	0.0841295
$96$	−3.76813	−0.384583
$97$	0.418003	0.0424418	0.0212209	−	0.999775i	$-0.493245\pi$
$97$	0.418003	0.0424418	0.0212209	+	0.999775i	$0.493245\pi$
$98$	0	0
$99$	4.11356	0.413428
$100$	−0.111508	−0.0111508
$101$	3.97091	0.395121	0.197560	−	0.980291i	$-0.436698\pi$
$101$	3.97091	0.395121	0.197560	+	0.980291i	$0.436698\pi$
$102$	−1.16064	−0.114921
$103$	−10.9705	−1.08096	−0.540479	−	0.841358i	$-0.681757\pi$
$103$	−10.9705	−1.08096	−0.540479	+	0.841358i	$0.681757\pi$
$104$	−3.14710	−0.308598
$105$	0	0
$106$	−8.58268	−0.833624
$107$	19.5498	1.88995	0.944974	−	0.327146i	$-0.106087\pi$
$107$	19.5498	1.88995	0.944974	+	0.327146i	$0.106087\pi$
$108$	−0.703600	−0.0677039
$109$	17.4207	1.66860	0.834299	−	0.551313i	$-0.185873\pi$
$109$	17.4207	1.66860	0.834299	+	0.551313i	$0.185873\pi$
$110$	−10.6377	−1.01427
$111$	9.23719	0.876755
$112$	0	0
$113$	−9.78704	−0.920687	−0.460344	−	0.887741i	$-0.652274\pi$
$113$	−9.78704	−0.920687	−0.460344	+	0.887741i	$0.652274\pi$
$114$	−0.411073	−0.0385005
$115$	−10.4928	−0.978460
$116$	1.05953	0.0983748
$117$	−1.02235	−0.0945160
$118$	1.86157	0.171371
$119$	0	0
$120$	6.99154	0.638237
$121$	5.92138	0.538307
$122$	−16.9310	−1.53286
$123$	1.00000	0.0901670
$124$	−2.72504	−0.244716
$125$	−10.9962	−0.983530
$126$	0	0
$127$	−16.9152	−1.50098	−0.750489	−	0.660883i	$-0.770182\pi$
$127$	−16.9152	−1.50098	−0.750489	+	0.660883i	$0.770182\pi$
$128$	−2.12572	−0.187888
$129$	−2.19077	−0.192887
$130$	2.64380	0.231877
$131$	−20.7104	−1.80948	−0.904738	−	0.425969i	$-0.859933\pi$
$131$	−20.7104	−1.80948	−0.904738	+	0.425969i	$0.859933\pi$
$132$	−2.89430	−0.251917
$133$	0	0
$134$	−6.13269	−0.529784
$135$	2.27123	0.195476
$136$	3.13791	0.269074
$137$	6.27873	0.536428	0.268214	−	0.963359i	$-0.413567\pi$
$137$	6.27873	0.536428	0.268214	+	0.963359i	$0.413567\pi$
$138$	5.26018	0.447776
$139$	−3.27698	−0.277950	−0.138975	−	0.990296i	$-0.544381\pi$
$139$	−3.27698	−0.277950	−0.138975	+	0.990296i	$0.544381\pi$
$140$	0	0
$141$	−1.32485	−0.111572
$142$	1.83657	0.154122
$143$	−4.20548	−0.351680
$144$	−2.09775	−0.174812
$145$	−3.42017	−0.284030
$146$	−2.53099	−0.209466
$147$	0	0
$148$	−6.49929	−0.534238
$149$	−12.5636	−1.02925	−0.514624	−	0.857416i	$-0.672068\pi$
$149$	−12.5636	−1.02925	−0.514624	+	0.857416i	$0.672068\pi$
$150$	−0.180446	−0.0147334
$151$	9.45156	0.769157	0.384578	−	0.923092i	$-0.374347\pi$
$151$	9.45156	0.769157	0.384578	+	0.923092i	$0.374347\pi$
$152$	1.11138	0.0901445
$153$	1.01936	0.0824106
$154$	0	0
$155$	8.79646	0.706548
$156$	0.719323	0.0575919
$157$	11.1900	0.893061	0.446530	−	0.894768i	$-0.352659\pi$
$157$	11.1900	0.893061	0.446530	+	0.894768i	$0.352659\pi$
$158$	4.73459	0.376664
$159$	7.53796	0.597799
$160$	−8.55828	−0.676592
$161$	0	0
$162$	−1.13860	−0.0894565
$163$	−5.24847	−0.411092	−0.205546	−	0.978647i	$-0.565897\pi$
$163$	−5.24847	−0.411092	−0.205546	+	0.978647i	$0.565897\pi$
$164$	−0.703600	−0.0549419
$165$	9.34284	0.727339
$166$	−1.75031	−0.135851
$167$	13.1078	1.01432	0.507158	−	0.861853i	$-0.330696\pi$
$167$	13.1078	1.01432	0.507158	+	0.861853i	$0.330696\pi$
$168$	0	0
$169$	−11.9548	−0.919601
$170$	−2.63608	−0.202178
$171$	0.361035	0.0276090
$172$	1.54143	0.117533
$173$	2.61929	0.199141	0.0995704	−	0.995031i	$-0.468253\pi$
$173$	2.61929	0.199141	0.0995704	+	0.995031i	$0.468253\pi$
$174$	1.71458	0.129982
$175$	0	0
$176$	−8.62921	−0.650451
$177$	−1.63497	−0.122892
$178$	−1.20758	−0.0905121
$179$	20.0966	1.50209	0.751044	−	0.660252i	$-0.229550\pi$
$179$	20.0966	1.50209	0.751044	+	0.660252i	$0.229550\pi$
$180$	−1.59804	−0.119111
$181$	−13.9771	−1.03891	−0.519456	−	0.854497i	$-0.673865\pi$
$181$	−13.9771	−1.03891	−0.519456	+	0.854497i	$0.673865\pi$
$182$	0	0
$183$	14.8701	1.09923
$184$	−14.2214	−1.04842
$185$	20.9798	1.54246
$186$	−4.40977	−0.323340
$187$	4.19321	0.306638
$188$	0.932164	0.0679850
$189$	0	0
$190$	−0.933640	−0.0677334
$191$	11.8333	0.856230	0.428115	−	0.903724i	$-0.359178\pi$
$191$	11.8333	0.856230	0.428115	+	0.903724i	$0.359178\pi$
$192$	8.48587	0.612415
$193$	9.09972	0.655012	0.327506	−	0.944849i	$-0.393792\pi$
$193$	9.09972	0.655012	0.327506	+	0.944849i	$0.393792\pi$
$194$	−0.475937	−0.0341703
$195$	−2.32198	−0.166281
$196$	0	0
$197$	0.846122	0.0602837	0.0301418	−	0.999546i	$-0.490404\pi$
$197$	0.846122	0.0602837	0.0301418	+	0.999546i	$0.490404\pi$
$198$	−4.68368	−0.332855
$199$	3.86730	0.274146	0.137073	−	0.990561i	$-0.456231\pi$
$199$	3.86730	0.274146	0.137073	+	0.990561i	$0.456231\pi$
$200$	0.487855	0.0344965
$201$	5.38619	0.379912
$202$	−4.52126	−0.318115
$203$	0	0
$204$	−0.717224	−0.0502157
$205$	2.27123	0.158630
$206$	12.4910	0.870288
$207$	−4.61988	−0.321104
$208$	2.14462	0.148703
$209$	1.48514	0.102729
$210$	0	0
$211$	−23.6238	−1.62633	−0.813166	−	0.582032i	$-0.802258\pi$
$211$	−23.6238	−1.62633	−0.813166	+	0.582032i	$0.802258\pi$
$212$	−5.30371	−0.364260
$213$	−1.61302	−0.110522
$214$	−22.2593	−1.52161
$215$	−4.97574	−0.339343
$216$	3.07831	0.209452
$217$	0	0
$218$	−19.8351	−1.34340
$219$	2.22290	0.150210
$220$	−6.57362	−0.443193
$221$	−1.04214	−0.0701021
$222$	−10.5174	−0.705883
$223$	14.5761	0.976089	0.488044	−	0.872819i	$-0.337710\pi$
$223$	14.5761	0.976089	0.488044	+	0.872819i	$0.337710\pi$
$224$	0	0
$225$	0.158481	0.0105654
$226$	11.1435	0.741253
$227$	5.42919	0.360348	0.180174	−	0.983635i	$-0.442334\pi$
$227$	5.42919	0.360348	0.180174	+	0.983635i	$0.442334\pi$
$228$	−0.254024	−0.0168232
$229$	−8.50896	−0.562288	−0.281144	−	0.959666i	$-0.590714\pi$
$229$	−8.50896	−0.562288	−0.281144	+	0.959666i	$0.590714\pi$
$230$	11.9471	0.787767
$231$	0	0
$232$	−4.63553	−0.304337
$233$	18.1099	1.18642	0.593210	−	0.805048i	$-0.297861\pi$
$233$	18.1099	1.18642	0.593210	+	0.805048i	$0.297861\pi$
$234$	1.16404	0.0760956
$235$	−3.00903	−0.196288
$236$	1.15036	0.0748823
$237$	−4.15827	−0.270109
$238$	0	0
$239$	2.22639	0.144013	0.0720066	−	0.997404i	$-0.477060\pi$
$239$	2.22639	0.144013	0.0720066	+	0.997404i	$0.477060\pi$
$240$	−4.76446	−0.307545
$241$	14.2272	0.916456	0.458228	−	0.888835i	$-0.348484\pi$
$241$	14.2272	0.916456	0.458228	+	0.888835i	$0.348484\pi$
$242$	−6.74206	−0.433396
$243$	1.00000	0.0641500
$244$	−10.4626	−0.669800
$245$	0	0
$246$	−1.13860	−0.0725942
$247$	−0.369103	−0.0234854
$248$	11.9223	0.757064
$249$	1.53726	0.0974197
$250$	12.5202	0.791848
$251$	−6.81117	−0.429918	−0.214959	−	0.976623i	$-0.568962\pi$
$251$	−6.81117	−0.429918	−0.214959	+	0.976623i	$0.568962\pi$
$252$	0	0
$253$	−19.0042	−1.19478
$254$	19.2595	1.20845
$255$	2.31521	0.144984
$256$	−14.5514	−0.909463
$257$	4.87985	0.304397	0.152198	−	0.988350i	$-0.451365\pi$
$257$	4.87985	0.304397	0.152198	+	0.988350i	$0.451365\pi$
$258$	2.49440	0.155295
$259$	0	0
$260$	1.63375	0.101321
$261$	−1.50587	−0.0932109
$262$	23.5808	1.45682
$263$	−16.5774	−1.02221	−0.511104	−	0.859519i	$-0.670763\pi$
$263$	−16.5774	−1.02221	−0.511104	+	0.859519i	$0.670763\pi$
$264$	12.6628	0.779342
$265$	17.1204	1.05170
$266$	0	0
$267$	1.06059	0.0649070
$268$	−3.78972	−0.231494
$269$	−7.34953	−0.448109	−0.224054	−	0.974577i	$-0.571929\pi$
$269$	−7.34953	−0.448109	−0.224054	+	0.974577i	$0.571929\pi$
$270$	−2.58601	−0.157380
$271$	14.2752	0.867158	0.433579	−	0.901116i	$-0.357250\pi$
$271$	14.2752	0.867158	0.433579	+	0.901116i	$0.357250\pi$
$272$	−2.13837	−0.129657
$273$	0	0
$274$	−7.14893	−0.431883
$275$	0.651923	0.0393124
$276$	3.25055	0.195660
$277$	−11.6467	−0.699781	−0.349891	−	0.936791i	$-0.613781\pi$
$277$	−11.6467	−0.699781	−0.349891	+	0.936791i	$0.613781\pi$
$278$	3.73116	0.223780
$279$	3.87299	0.231870
$280$	0	0
$281$	17.4871	1.04319	0.521595	−	0.853193i	$-0.325337\pi$
$281$	17.4871	1.04319	0.521595	+	0.853193i	$0.325337\pi$
$282$	1.50847	0.0898279
$283$	15.6673	0.931327	0.465663	−	0.884962i	$-0.345816\pi$
$283$	15.6673	0.931327	0.465663	+	0.884962i	$0.345816\pi$
$284$	1.13492	0.0673450
$285$	0.819993	0.0485722
$286$	4.78835	0.283141
$287$	0	0
$288$	−3.76813	−0.222039
$289$	−15.9609	−0.938876
$290$	3.89419	0.228675
$291$	0.418003	0.0245038
$292$	−1.56404	−0.0915282
$293$	−20.8334	−1.21710	−0.608550	−	0.793516i	$-0.708249\pi$
$293$	−20.8334	−1.21710	−0.608550	+	0.793516i	$0.708249\pi$
$294$	0	0
$295$	−3.71339	−0.216202
$296$	28.4349	1.65274
$297$	4.11356	0.238693
$298$	14.3048	0.828656
$299$	4.72312	0.273145
$300$	−0.111508	−0.00643789
$301$	0	0
$302$	−10.7615	−0.619255
$303$	3.97091	0.228123
$304$	−0.757360	−0.0434376
$305$	33.7734	1.93386
$306$	−1.16064	−0.0663495
$307$	4.35015	0.248276	0.124138	−	0.992265i	$-0.460383\pi$
$307$	4.35015	0.248276	0.124138	+	0.992265i	$0.460383\pi$
$308$	0	0
$309$	−10.9705	−0.624091
$310$	−10.0156	−0.568848
$311$	17.4860	0.991540	0.495770	−	0.868454i	$-0.334886\pi$
$311$	17.4860	0.991540	0.495770	+	0.868454i	$0.334886\pi$
$312$	−3.14710	−0.178169
$313$	27.3870	1.54800	0.774001	−	0.633184i	$-0.218252\pi$
$313$	27.3870	1.54800	0.774001	+	0.633184i	$0.218252\pi$
$314$	−12.7409	−0.719011
$315$	0	0
$316$	2.92576	0.164587
$317$	−31.8861	−1.79090	−0.895451	−	0.445161i	$-0.853147\pi$
$317$	−31.8861	−1.79090	−0.895451	+	0.445161i	$0.853147\pi$
$318$	−8.58268	−0.481293
$319$	−6.19448	−0.346824
$320$	19.2734	1.07741
$321$	19.5498	1.09116
$322$	0	0
$323$	0.368026	0.0204775
$324$	−0.703600	−0.0390889
$325$	−0.162023	−0.00898742
$326$	5.97589	0.330974
$327$	17.4207	0.963365
$328$	3.07831	0.169971
$329$	0	0
$330$	−10.6377	−0.585587
$331$	−16.1121	−0.885601	−0.442801	−	0.896620i	$-0.646015\pi$
$331$	−16.1121	−0.885601	−0.442801	+	0.896620i	$0.646015\pi$
$332$	−1.08161	−0.0593613
$333$	9.23719	0.506195
$334$	−14.9245	−0.816634
$335$	12.2333	0.668375
$336$	0	0
$337$	16.6380	0.906331	0.453166	−	0.891426i	$-0.350295\pi$
$337$	16.6380	0.906331	0.453166	+	0.891426i	$0.350295\pi$
$338$	13.6117	0.740378
$339$	−9.78704	−0.531559
$340$	−1.62898	−0.0883438
$341$	15.9318	0.862755
$342$	−0.411073	−0.0222283
$343$	0	0
$344$	−6.74386	−0.363605
$345$	−10.4928	−0.564914
$346$	−2.98231	−0.160330
$347$	25.4621	1.36688	0.683440	−	0.730007i	$-0.260483\pi$
$347$	25.4621	1.36688	0.683440	+	0.730007i	$0.260483\pi$
$348$	1.05953	0.0567967
$349$	−15.6721	−0.838911	−0.419455	−	0.907776i	$-0.637779\pi$
$349$	−15.6721	−0.838911	−0.419455	+	0.907776i	$0.637779\pi$
$350$	0	0
$351$	−1.02235	−0.0545688
$352$	−15.5004	−0.826175
$353$	10.9229	0.581366	0.290683	−	0.956819i	$-0.406117\pi$
$353$	10.9229	0.581366	0.290683	+	0.956819i	$0.406117\pi$
$354$	1.86157	0.0989413
$355$	−3.66353	−0.194440
$356$	−0.746231	−0.0395501
$357$	0	0
$358$	−22.8819	−1.20934
$359$	−1.37014	−0.0723134	−0.0361567	−	0.999346i	$-0.511512\pi$
$359$	−1.37014	−0.0723134	−0.0361567	+	0.999346i	$0.511512\pi$
$360$	6.99154	0.368487
$361$	−18.8697	−0.993140
$362$	15.9143	0.836437
$363$	5.92138	0.310792
$364$	0	0
$365$	5.04872	0.264262
$366$	−16.9310	−0.885000
$367$	33.2866	1.73755	0.868774	−	0.495209i	$-0.164908\pi$
$367$	33.2866	1.73755	0.868774	+	0.495209i	$0.164908\pi$
$368$	9.69135	0.505196
$369$	1.00000	0.0520579
$370$	−23.8875	−1.24185
$371$	0	0
$372$	−2.72504	−0.141287
$373$	−13.8615	−0.717719	−0.358859	−	0.933392i	$-0.616834\pi$
$373$	−13.8615	−0.717719	−0.358859	+	0.933392i	$0.616834\pi$
$374$	−4.77437	−0.246877
$375$	−10.9962	−0.567841
$376$	−4.07829	−0.210322
$377$	1.53952	0.0792893
$378$	0	0
$379$	1.47300	0.0756628	0.0378314	−	0.999284i	$-0.487955\pi$
$379$	1.47300	0.0756628	0.0378314	+	0.999284i	$0.487955\pi$
$380$	−0.576947	−0.0295968
$381$	−16.9152	−0.866590
$382$	−13.4734	−0.689359
$383$	−1.29949	−0.0664006	−0.0332003	−	0.999449i	$-0.510570\pi$
$383$	−1.29949	−0.0664006	−0.0332003	+	0.999449i	$0.510570\pi$
$384$	−2.12572	−0.108477
$385$	0	0
$386$	−10.3609	−0.527356
$387$	−2.19077	−0.111363
$388$	−0.294107	−0.0149310
$389$	16.8058	0.852089	0.426045	−	0.904702i	$-0.359907\pi$
$389$	16.8058	0.852089	0.426045	+	0.904702i	$0.359907\pi$
$390$	2.64380	0.133874
$391$	−4.70934	−0.238162
$392$	0	0
$393$	−20.7104	−1.04470
$394$	−0.963390	−0.0485349
$395$	−9.44438	−0.475198
$396$	−2.89430	−0.145444
$397$	17.7317	0.889930	0.444965	−	0.895548i	$-0.353216\pi$
$397$	17.7317	0.889930	0.444965	+	0.895548i	$0.353216\pi$
$398$	−4.40329	−0.220717
$399$	0	0
$400$	−0.332454	−0.0166227
$401$	26.9691	1.34677	0.673387	−	0.739290i	$-0.264839\pi$
$401$	26.9691	1.34677	0.673387	+	0.739290i	$0.264839\pi$
$402$	−6.13269	−0.305871
$403$	−3.95954	−0.197239
$404$	−2.79393	−0.139003
$405$	2.27123	0.112858
$406$	0	0
$407$	37.9977	1.88348
$408$	3.13791	0.155350
$409$	−22.8930	−1.13199	−0.565993	−	0.824410i	$-0.691507\pi$
$409$	−22.8930	−1.13199	−0.565993	+	0.824410i	$0.691507\pi$
$410$	−2.58601	−0.127714
$411$	6.27873	0.309707
$412$	7.71886	0.380281
$413$	0	0
$414$	5.26018	0.258524
$415$	3.49146	0.171389
$416$	3.85233	0.188876
$417$	−3.27698	−0.160474
$418$	−1.69097	−0.0827082
$419$	38.2790	1.87005	0.935026	−	0.354579i	$-0.115376\pi$
$419$	38.2790	1.87005	0.935026	+	0.354579i	$0.115376\pi$
$420$	0	0
$421$	−29.0730	−1.41693	−0.708465	−	0.705746i	$-0.750612\pi$
$421$	−29.0730	−1.41693	−0.708465	+	0.705746i	$0.750612\pi$
$422$	26.8980	1.30937
$423$	−1.32485	−0.0644164
$424$	23.2041	1.12689
$425$	0.161550	0.00783633
$426$	1.83657	0.0889823
$427$	0	0
$428$	−13.7552	−0.664884
$429$	−4.20548	−0.203043
$430$	5.66536	0.273208
$431$	−28.5861	−1.37694	−0.688472	−	0.725263i	$-0.741718\pi$
$431$	−28.5861	−1.37694	−0.688472	+	0.725263i	$0.741718\pi$
$432$	−2.09775	−0.100928
$433$	−25.3115	−1.21639	−0.608196	−	0.793787i	$-0.708106\pi$
$433$	−25.3115	−1.21639	−0.608196	+	0.793787i	$0.708106\pi$
$434$	0	0
$435$	−3.42017	−0.163985
$436$	−12.2572	−0.587012
$437$	−1.66794	−0.0797884
$438$	−2.53099	−0.120935
$439$	11.4408	0.546042	0.273021	−	0.962008i	$-0.411977\pi$
$439$	11.4408	0.546042	0.273021	+	0.962008i	$0.411977\pi$
$440$	28.7601	1.37109
$441$	0	0
$442$	1.18658	0.0564398
$443$	−28.2157	−1.34057	−0.670285	−	0.742104i	$-0.733828\pi$
$443$	−28.2157	−1.34057	−0.670285	+	0.742104i	$0.733828\pi$
$444$	−6.49929	−0.308442
$445$	2.40884	0.114190
$446$	−16.5963	−0.785858
$447$	−12.5636	−0.594237
$448$	0	0
$449$	37.0482	1.74841	0.874207	−	0.485553i	$-0.161382\pi$
$449$	37.0482	1.74841	0.874207	+	0.485553i	$0.161382\pi$
$450$	−0.180446	−0.00850632
$451$	4.11356	0.193700
$452$	6.88616	0.323898
$453$	9.45156	0.444073
$454$	−6.18165	−0.290119
$455$	0	0
$456$	1.11138	0.0520450
$457$	−3.13021	−0.146425	−0.0732126	−	0.997316i	$-0.523325\pi$
$457$	−3.13021	−0.146425	−0.0732126	+	0.997316i	$0.523325\pi$
$458$	9.68826	0.452703
$459$	1.01936	0.0475798
$460$	7.38274	0.344222
$461$	32.8502	1.52998	0.764992	−	0.644040i	$-0.222743\pi$
$461$	32.8502	1.52998	0.764992	+	0.644040i	$0.222743\pi$
$462$	0	0
$463$	15.8140	0.734937	0.367468	−	0.930036i	$-0.380225\pi$
$463$	15.8140	0.734937	0.367468	+	0.930036i	$0.380225\pi$
$464$	3.15893	0.146650
$465$	8.79646	0.407926
$466$	−20.6199	−0.955197
$467$	36.7438	1.70030	0.850151	−	0.526539i	$-0.176511\pi$
$467$	36.7438	1.70030	0.850151	+	0.526539i	$0.176511\pi$
$468$	0.719323	0.0332507
$469$	0	0
$470$	3.42607	0.158033
$471$	11.1900	0.515609
$472$	−5.03294	−0.231660
$473$	−9.01186	−0.414366
$474$	4.73459	0.217467
$475$	0.0572173	0.00262531
$476$	0	0
$477$	7.53796	0.345139
$478$	−2.53496	−0.115946
$479$	−34.6662	−1.58394	−0.791969	−	0.610561i	$-0.790944\pi$
$479$	−34.6662	−1.58394	−0.791969	+	0.610561i	$0.790944\pi$
$480$	−8.55828	−0.390630
$481$	−9.44361	−0.430591
$482$	−16.1991	−0.737847
$483$	0	0
$484$	−4.16628	−0.189376
$485$	0.949381	0.0431092
$486$	−1.13860	−0.0516477
$487$	−24.3473	−1.10328	−0.551640	−	0.834082i	$-0.685998\pi$
$487$	−24.3473	−1.10328	−0.551640	+	0.834082i	$0.685998\pi$
$488$	45.7748	2.07213
$489$	−5.24847	−0.237344
$490$	0	0
$491$	−23.6572	−1.06764	−0.533818	−	0.845600i	$-0.679243\pi$
$491$	−23.6572	−1.06764	−0.533818	+	0.845600i	$0.679243\pi$
$492$	−0.703600	−0.0317207
$493$	−1.53503	−0.0691341
$494$	0.420259	0.0189083
$495$	9.34284	0.419929
$496$	−8.12456	−0.364804
$497$	0	0
$498$	−1.75031	−0.0784335
$499$	31.4749	1.40901	0.704505	−	0.709699i	$-0.251169\pi$
$499$	31.4749	1.40901	0.704505	+	0.709699i	$0.251169\pi$
$500$	7.73692	0.346006
$501$	13.1078	0.585615
$502$	7.75517	0.346130
$503$	−17.3605	−0.774067	−0.387034	−	0.922066i	$-0.626500\pi$
$503$	−17.3605	−0.774067	−0.387034	+	0.922066i	$0.626500\pi$
$504$	0	0
$505$	9.01885	0.401334
$506$	21.6381	0.961929
$507$	−11.9548	−0.530932
$508$	11.9015	0.528044
$509$	−2.68874	−0.119176	−0.0595881	−	0.998223i	$-0.518979\pi$
$509$	−2.68874	−0.119176	−0.0595881	+	0.998223i	$0.518979\pi$
$510$	−2.63608	−0.116728
$511$	0	0
$512$	20.8196	0.920105
$513$	0.361035	0.0159401
$514$	−5.55618	−0.245072
$515$	−24.9166	−1.09795
$516$	1.54143	0.0678575
$517$	−5.44985	−0.239684
$518$	0	0
$519$	2.61929	0.114974
$520$	−7.14778	−0.313451
$521$	−39.2979	−1.72167	−0.860835	−	0.508884i	$-0.830058\pi$
$521$	−39.2979	−1.72167	−0.860835	+	0.508884i	$0.830058\pi$
$522$	1.71458	0.0750449
$523$	6.63116	0.289961	0.144980	−	0.989435i	$-0.453688\pi$
$523$	6.63116	0.289961	0.144980	+	0.989435i	$0.453688\pi$
$524$	14.5718	0.636573
$525$	0	0
$526$	18.8750	0.822988
$527$	3.94799	0.171977
$528$	−8.62921	−0.375538
$529$	−1.65667	−0.0720292
$530$	−19.4932	−0.846732
$531$	−1.63497	−0.0709516
$532$	0	0
$533$	−1.02235	−0.0442827
$534$	−1.20758	−0.0522572
$535$	44.4020	1.91967
$536$	16.5803	0.716162
$537$	20.0966	0.867231
$538$	8.36814	0.360776
$539$	0	0
$540$	−1.59804	−0.0687686
$541$	32.2640	1.38714	0.693568	−	0.720391i	$-0.256038\pi$
$541$	32.2640	1.38714	0.693568	+	0.720391i	$0.256038\pi$
$542$	−16.2537	−0.698157
$543$	−13.9771	−0.599816
$544$	−3.84109	−0.164685
$545$	39.5663	1.69484
$546$	0	0
$547$	−7.23994	−0.309558	−0.154779	−	0.987949i	$-0.549466\pi$
$547$	−7.23994	−0.309558	−0.154779	+	0.987949i	$0.549466\pi$
$548$	−4.41771	−0.188715
$549$	14.8701	0.634641
$550$	−0.742277	−0.0316508
$551$	−0.543671	−0.0231612
$552$	−14.2214	−0.605304
$553$	0	0
$554$	13.2609	0.563400
$555$	20.9798	0.890542
$556$	2.30568	0.0977828
$557$	−0.594345	−0.0251832	−0.0125916	−	0.999921i	$-0.504008\pi$
$557$	−0.594345	−0.0251832	−0.0125916	+	0.999921i	$0.504008\pi$
$558$	−4.40977	−0.186681
$559$	2.23973	0.0947303
$560$	0	0
$561$	4.19321	0.177038
$562$	−19.9107	−0.839882
$563$	5.25348	0.221408	0.110704	−	0.993853i	$-0.464690\pi$
$563$	5.25348	0.221408	0.110704	+	0.993853i	$0.464690\pi$
$564$	0.932164	0.0392512
$565$	−22.2286	−0.935165
$566$	−17.8388	−0.749819
$567$	0	0
$568$	−4.96536	−0.208342
$569$	−19.6362	−0.823194	−0.411597	−	0.911366i	$-0.635029\pi$
$569$	−19.6362	−0.823194	−0.411597	+	0.911366i	$0.635029\pi$
$570$	−0.933640	−0.0391059
$571$	−35.2958	−1.47708	−0.738542	−	0.674207i	$-0.764485\pi$
$571$	−35.2958	−1.47708	−0.738542	+	0.674207i	$0.764485\pi$
$572$	2.95898	0.123721
$573$	11.8333	0.494345
$574$	0	0
$575$	−0.732166	−0.0305334
$576$	8.48587	0.353578
$577$	34.3114	1.42840	0.714200	−	0.699941i	$-0.246790\pi$
$577$	34.3114	1.42840	0.714200	+	0.699941i	$0.246790\pi$
$578$	18.1730	0.755898
$579$	9.09972	0.378172
$580$	2.40643	0.0999217
$581$	0	0
$582$	−0.475937	−0.0197282
$583$	31.0078	1.28421
$584$	6.84278	0.283156
$585$	−2.32198	−0.0960022
$586$	23.7208	0.979898
$587$	−18.7338	−0.773228	−0.386614	−	0.922242i	$-0.626355\pi$
$587$	−18.7338	−0.773228	−0.386614	+	0.922242i	$0.626355\pi$
$588$	0	0
$589$	1.39829	0.0576154
$590$	4.22805	0.174066
$591$	0.846122	0.0348048
$592$	−19.3773	−0.796401
$593$	−5.35369	−0.219850	−0.109925	−	0.993940i	$-0.535061\pi$
$593$	−5.35369	−0.219850	−0.109925	+	0.993940i	$0.535061\pi$
$594$	−4.68368	−0.192174
$595$	0	0
$596$	8.83973	0.362089
$597$	3.86730	0.158278
$598$	−5.37773	−0.219912
$599$	38.4103	1.56940	0.784701	−	0.619874i	$-0.212816\pi$
$599$	38.4103	1.56940	0.784701	+	0.619874i	$0.212816\pi$
$600$	0.487855	0.0199166
$601$	−2.61417	−0.106634	−0.0533172	−	0.998578i	$-0.516979\pi$
$601$	−2.61417	−0.106634	−0.0533172	+	0.998578i	$0.516979\pi$
$602$	0	0
$603$	5.38619	0.219342
$604$	−6.65011	−0.270589
$605$	13.4488	0.546772
$606$	−4.52126	−0.183664
$607$	−34.7661	−1.41111	−0.705556	−	0.708654i	$-0.749303\pi$
$607$	−34.7661	−1.41111	−0.705556	+	0.708654i	$0.749303\pi$
$608$	−1.36043	−0.0551726
$609$	0	0
$610$	−38.4543	−1.55697
$611$	1.35445	0.0547954
$612$	−0.717224	−0.0289921
$613$	−7.23499	−0.292219	−0.146109	−	0.989268i	$-0.546675\pi$
$613$	−7.23499	−0.292219	−0.146109	+	0.989268i	$0.546675\pi$
$614$	−4.95306	−0.199889
$615$	2.27123	0.0915848
$616$	0	0
$617$	3.70944	0.149337	0.0746683	−	0.997208i	$-0.476210\pi$
$617$	3.70944	0.149337	0.0746683	+	0.997208i	$0.476210\pi$
$618$	12.4910	0.502461
$619$	−21.8562	−0.878473	−0.439237	−	0.898371i	$-0.644751\pi$
$619$	−21.8562	−0.878473	−0.439237	+	0.898371i	$0.644751\pi$
$620$	−6.18919	−0.248564
$621$	−4.61988	−0.185390
$622$	−19.9095	−0.798298
$623$	0	0
$624$	2.14462	0.0858537
$625$	−25.7673	−1.03069
$626$	−31.1827	−1.24631
$627$	1.48514	0.0593107
$628$	−7.87330	−0.314179
$629$	9.41605	0.375442
$630$	0	0
$631$	21.7931	0.867569	0.433784	−	0.901017i	$-0.357178\pi$
$631$	21.7931	0.867569	0.433784	+	0.901017i	$0.357178\pi$
$632$	−12.8004	−0.509174
$633$	−23.6238	−0.938963
$634$	36.3054	1.44187
$635$	−38.4182	−1.52458
$636$	−5.30371	−0.210306
$637$	0	0
$638$	7.05301	0.279231
$639$	−1.61302	−0.0638099
$640$	−4.82799	−0.190843
$641$	−27.4415	−1.08388	−0.541938	−	0.840419i	$-0.682309\pi$
$641$	−27.4415	−1.08388	−0.541938	+	0.840419i	$0.682309\pi$
$642$	−22.2593	−0.878504
$643$	0.581325	0.0229252	0.0114626	−	0.999934i	$-0.496351\pi$
$643$	0.581325	0.0229252	0.0114626	+	0.999934i	$0.496351\pi$
$644$	0	0
$645$	−4.97574	−0.195920
$646$	−0.419032	−0.0164866
$647$	29.9596	1.17783	0.588917	−	0.808194i	$-0.299555\pi$
$647$	29.9596	1.17783	0.588917	+	0.808194i	$0.299555\pi$
$648$	3.07831	0.120927
$649$	−6.72554	−0.264001
$650$	0.184479	0.00723585
$651$	0	0
$652$	3.69283	0.144622
$653$	8.27365	0.323773	0.161886	−	0.986809i	$-0.448242\pi$
$653$	8.27365	0.323773	0.161886	+	0.986809i	$0.448242\pi$
$654$	−19.8351	−0.775614
$655$	−47.0380	−1.83793
$656$	−2.09775	−0.0819033
$657$	2.22290	0.0867237
$658$	0	0
$659$	35.3248	1.37606	0.688030	−	0.725682i	$-0.258476\pi$
$659$	35.3248	1.37606	0.688030	+	0.725682i	$0.258476\pi$
$660$	−6.57362	−0.255878
$661$	32.2603	1.25478	0.627390	−	0.778705i	$-0.284123\pi$
$661$	32.2603	1.25478	0.627390	+	0.778705i	$0.284123\pi$
$662$	18.3452	0.713005
$663$	−1.04214	−0.0404735
$664$	4.73215	0.183643
$665$	0	0
$666$	−10.5174	−0.407542
$667$	6.95694	0.269374
$668$	−9.22268	−0.356836
$669$	14.5761	0.563545
$670$	−13.9287	−0.538114
$671$	61.1691	2.36141
$672$	0	0
$673$	−5.72480	−0.220675	−0.110337	−	0.993894i	$-0.535193\pi$
$673$	−5.72480	−0.220675	−0.110337	+	0.993894i	$0.535193\pi$
$674$	−18.9440	−0.729695
$675$	0.158481	0.00609995
$676$	8.41140	0.323515
$677$	51.3825	1.97479	0.987396	−	0.158270i	$-0.0505918\pi$
$677$	51.3825	1.97479	0.987396	+	0.158270i	$0.0505918\pi$
$678$	11.1435	0.427963
$679$	0	0
$680$	7.12692	0.273305
$681$	5.42919	0.208047
$682$	−18.1399	−0.694611
$683$	−12.9225	−0.494465	−0.247232	−	0.968956i	$-0.579521\pi$
$683$	−12.9225	−0.494465	−0.247232	+	0.968956i	$0.579521\pi$
$684$	−0.254024	−0.00971286
$685$	14.2604	0.544863
$686$	0	0
$687$	−8.50896	−0.324637
$688$	4.59568	0.175209
$689$	−7.70640	−0.293591
$690$	11.9471	0.454817
$691$	7.98422	0.303734	0.151867	−	0.988401i	$-0.451471\pi$
$691$	7.98422	0.303734	0.151867	+	0.988401i	$0.451471\pi$
$692$	−1.84293	−0.0700577
$693$	0	0
$694$	−28.9911	−1.10049
$695$	−7.44278	−0.282321
$696$	−4.63553	−0.175709
$697$	1.01936	0.0386111
$698$	17.8442	0.675414
$699$	18.1099	0.684980
$700$	0	0
$701$	−4.14530	−0.156566	−0.0782829	−	0.996931i	$-0.524944\pi$
$701$	−4.14530	−0.156566	−0.0782829	+	0.996931i	$0.524944\pi$
$702$	1.16404	0.0439338
$703$	3.33495	0.125780
$704$	34.9071	1.31561
$705$	−3.00903	−0.113327
$706$	−12.4367	−0.468063
$707$	0	0
$708$	1.15036	0.0432333
$709$	22.3009	0.837527	0.418764	−	0.908095i	$-0.362464\pi$
$709$	22.3009	0.837527	0.418764	+	0.908095i	$0.362464\pi$
$710$	4.17128	0.156545
$711$	−4.15827	−0.155947
$712$	3.26482	0.122354
$713$	−17.8928	−0.670090
$714$	0	0
$715$	−9.55162	−0.357210
$716$	−14.1400	−0.528435
$717$	2.22639	0.0831461
$718$	1.56004	0.0582201
$719$	−11.8846	−0.443221	−0.221610	−	0.975135i	$-0.571131\pi$
$719$	−11.8846	−0.443221	−0.221610	+	0.975135i	$0.571131\pi$
$720$	−4.76446	−0.177561
$721$	0	0
$722$	21.4849	0.799585
$723$	14.2272	0.529116
$724$	9.83431	0.365489
$725$	−0.238652	−0.00886332
$726$	−6.74206	−0.250221
$727$	52.7921	1.95795	0.978976	−	0.203977i	$-0.0653867\pi$
$727$	52.7921	1.95795	0.978976	+	0.203977i	$0.0653867\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−5.74846	−0.212760
$731$	−2.23319	−0.0825975
$732$	−10.4626	−0.386709
$733$	−8.77886	−0.324255	−0.162127	−	0.986770i	$-0.551836\pi$
$733$	−8.77886	−0.324255	−0.162127	+	0.986770i	$0.551836\pi$
$734$	−37.9000	−1.39892
$735$	0	0
$736$	17.4083	0.641679
$737$	22.1564	0.816142
$738$	−1.13860	−0.0419123
$739$	−49.4832	−1.82027	−0.910135	−	0.414311i	$-0.864023\pi$
$739$	−49.4832	−1.82027	−0.910135	+	0.414311i	$0.864023\pi$
$740$	−14.7614	−0.542639
$741$	−0.369103	−0.0135593
$742$	0	0
$743$	26.5917	0.975554	0.487777	−	0.872968i	$-0.337808\pi$
$743$	26.5917	0.975554	0.487777	+	0.872968i	$0.337808\pi$
$744$	11.9223	0.437091
$745$	−28.5348	−1.04543
$746$	15.7826	0.577842
$747$	1.53726	0.0562453
$748$	−2.95034	−0.107875
$749$	0	0
$750$	12.5202	0.457174
$751$	7.81219	0.285071	0.142535	−	0.989790i	$-0.454475\pi$
$751$	7.81219	0.285071	0.142535	+	0.989790i	$0.454475\pi$
$752$	2.77920	0.101347
$753$	−6.81117	−0.248213
$754$	−1.75289	−0.0638365
$755$	21.4666	0.781251
$756$	0	0
$757$	29.6870	1.07899	0.539497	−	0.841988i	$-0.318614\pi$
$757$	29.6870	1.07899	0.539497	+	0.841988i	$0.318614\pi$
$758$	−1.67715	−0.0609167
$759$	−19.0042	−0.689808
$760$	2.52419	0.0915620
$761$	12.7647	0.462720	0.231360	−	0.972868i	$-0.425683\pi$
$761$	12.7647	0.462720	0.231360	+	0.972868i	$0.425683\pi$
$762$	19.2595	0.697699
$763$	0	0
$764$	−8.32594	−0.301222
$765$	2.31521	0.0837065
$766$	1.47959	0.0534597
$767$	1.67151	0.0603546
$768$	−14.5514	−0.525079
$769$	−26.8186	−0.967104	−0.483552	−	0.875316i	$-0.660654\pi$
$769$	−26.8186	−0.967104	−0.483552	+	0.875316i	$0.660654\pi$
$770$	0	0
$771$	4.87985	0.175744
$772$	−6.40257	−0.230433
$773$	−19.7548	−0.710531	−0.355265	−	0.934766i	$-0.615609\pi$
$773$	−19.7548	−0.710531	−0.355265	+	0.934766i	$0.615609\pi$
$774$	2.49440	0.0896594
$775$	0.613798	0.0220483
$776$	1.28674	0.0461914
$777$	0	0
$778$	−19.1350	−0.686024
$779$	0.361035	0.0129354
$780$	1.63375	0.0584975
$781$	−6.63524	−0.237428
$782$	5.36203	0.191746
$783$	−1.50587	−0.0538153
$784$	0	0
$785$	25.4151	0.907104
$786$	23.5808	0.841098
$787$	38.0595	1.35667	0.678337	−	0.734751i	$-0.262701\pi$
$787$	38.0595	1.35667	0.678337	+	0.734751i	$0.262701\pi$
$788$	−0.595331	−0.0212078
$789$	−16.5774	−0.590172
$790$	10.7533	0.382586
$791$	0	0
$792$	12.6628	0.449953
$793$	−15.2024	−0.539853
$794$	−20.1893	−0.716490
$795$	17.1204	0.607199
$796$	−2.72103	−0.0964444
$797$	42.5173	1.50604	0.753019	−	0.657998i	$-0.228597\pi$
$797$	42.5173	1.50604	0.753019	+	0.657998i	$0.228597\pi$
$798$	0	0
$799$	−1.35050	−0.0477773
$800$	−0.597179	−0.0211135
$801$	1.06059	0.0374741
$802$	−30.7069	−1.08430
$803$	9.14405	0.322687
$804$	−3.78972	−0.133653
$805$	0	0
$806$	4.50832	0.158799
$807$	−7.34953	−0.258716
$808$	12.2237	0.430028
$809$	−38.2867	−1.34609	−0.673044	−	0.739603i	$-0.735013\pi$
$809$	−38.2867	−1.34609	−0.673044	+	0.739603i	$0.735013\pi$
$810$	−2.58601	−0.0908632
$811$	−35.9502	−1.26238	−0.631191	−	0.775628i	$-0.717434\pi$
$811$	−35.9502	−1.26238	−0.631191	+	0.775628i	$0.717434\pi$
$812$	0	0
$813$	14.2752	0.500654
$814$	−43.2640	−1.51640
$815$	−11.9205	−0.417556
$816$	−2.13837	−0.0748578
$817$	−0.790944	−0.0276716
$818$	26.0659	0.911372
$819$	0	0
$820$	−1.59804	−0.0558059
$821$	−18.3527	−0.640514	−0.320257	−	0.947331i	$-0.603769\pi$
$821$	−18.3527	−0.640514	−0.320257	+	0.947331i	$0.603769\pi$
$822$	−7.14893	−0.249348
$823$	−41.0862	−1.43218	−0.716088	−	0.698010i	$-0.754069\pi$
$823$	−41.0862	−1.43218	−0.716088	+	0.698010i	$0.754069\pi$
$824$	−33.7706	−1.17646
$825$	0.651923	0.0226970
$826$	0	0
$827$	23.0152	0.800316	0.400158	−	0.916446i	$-0.368955\pi$
$827$	23.0152	0.800316	0.400158	+	0.916446i	$0.368955\pi$
$828$	3.25055	0.112964
$829$	24.1424	0.838499	0.419249	−	0.907871i	$-0.362293\pi$
$829$	24.1424	0.838499	0.419249	+	0.907871i	$0.362293\pi$
$830$	−3.97536	−0.137987
$831$	−11.6467	−0.404019
$832$	−8.67550	−0.300769
$833$	0	0
$834$	3.73116	0.129199
$835$	29.7709	1.03027
$836$	−1.04494	−0.0361401
$837$	3.87299	0.133870
$838$	−43.5843	−1.50559
$839$	−15.6154	−0.539105	−0.269552	−	0.962986i	$-0.586876\pi$
$839$	−15.6154	−0.539105	−0.269552	+	0.962986i	$0.586876\pi$
$840$	0	0
$841$	−26.7324	−0.921806
$842$	33.1024	1.14078
$843$	17.4871	0.602286
$844$	16.6217	0.572144
$845$	−27.1521	−0.934061
$846$	1.50847	0.0518622
$847$	0	0
$848$	−15.8127	−0.543011
$849$	15.6673	0.537702
$850$	−0.183940	−0.00630910
$851$	−42.6747	−1.46287
$852$	1.13492	0.0388817
$853$	−16.3069	−0.558338	−0.279169	−	0.960242i	$-0.590059\pi$
$853$	−16.3069	−0.558338	−0.279169	+	0.960242i	$0.590059\pi$
$854$	0	0
$855$	0.819993	0.0280432
$856$	60.1802	2.05692
$857$	10.4701	0.357653	0.178827	−	0.983881i	$-0.442770\pi$
$857$	10.4701	0.357653	0.178827	+	0.983881i	$0.442770\pi$
$858$	4.78835	0.163471
$859$	29.4060	1.00332	0.501659	−	0.865065i	$-0.332723\pi$
$859$	29.4060	1.00332	0.501659	+	0.865065i	$0.332723\pi$
$860$	3.50093	0.119381
$861$	0	0
$862$	32.5480	1.10859
$863$	−11.8231	−0.402463	−0.201231	−	0.979544i	$-0.564494\pi$
$863$	−11.8231	−0.402463	−0.201231	+	0.979544i	$0.564494\pi$
$864$	−3.76813	−0.128194
$865$	5.94900	0.202272
$866$	28.8195	0.979327
$867$	−15.9609	−0.542061
$868$	0	0
$869$	−17.1053	−0.580257
$870$	3.89419	0.132026
$871$	−5.50655	−0.186582
$872$	53.6262	1.81601
$873$	0.418003	0.0141473
$874$	1.89911	0.0642383
$875$	0	0
$876$	−1.56404	−0.0528439
$877$	−35.9145	−1.21275	−0.606373	−	0.795181i	$-0.707376\pi$
$877$	−35.9145	−1.21275	−0.606373	+	0.795181i	$0.707376\pi$
$878$	−13.0265	−0.439623
$879$	−20.8334	−0.702693
$880$	−19.5989	−0.660679
$881$	41.3275	1.39236	0.696180	−	0.717868i	$-0.254882\pi$
$881$	41.3275	1.39236	0.696180	+	0.717868i	$0.254882\pi$
$882$	0	0
$883$	−29.3608	−0.988068	−0.494034	−	0.869443i	$-0.664478\pi$
$883$	−29.3608	−0.988068	−0.494034	+	0.869443i	$0.664478\pi$
$884$	0.733251	0.0246619
$885$	−3.71339	−0.124824
$886$	32.1263	1.07930
$887$	−25.4418	−0.854253	−0.427126	−	0.904192i	$-0.640474\pi$
$887$	−25.4418	−0.854253	−0.427126	+	0.904192i	$0.640474\pi$
$888$	28.4349	0.954213
$889$	0	0
$890$	−2.74270	−0.0919354
$891$	4.11356	0.137809
$892$	−10.2558	−0.343388
$893$	−0.478317	−0.0160063
$894$	14.3048	0.478425
$895$	45.6439	1.52571
$896$	0	0
$897$	4.72312	0.157700
$898$	−42.1830	−1.40766
$899$	−5.83222	−0.194515
$900$	−0.111508	−0.00371692
$901$	7.68391	0.255988
$902$	−4.68368	−0.155950
$903$	0	0
$904$	−30.1275	−1.00203
$905$	−31.7453	−1.05525
$906$	−10.7615	−0.357527
$907$	−21.3485	−0.708864	−0.354432	−	0.935082i	$-0.615326\pi$
$907$	−21.3485	−0.708864	−0.354432	+	0.935082i	$0.615326\pi$
$908$	−3.81998	−0.126770
$909$	3.97091	0.131707
$910$	0	0
$911$	−27.8400	−0.922381	−0.461190	−	0.887301i	$-0.652577\pi$
$911$	−27.8400	−0.922381	−0.461190	+	0.887301i	$0.652577\pi$
$912$	−0.757360	−0.0250787
$913$	6.32360	0.209281
$914$	3.56405	0.117888
$915$	33.7734	1.11652
$916$	5.98690	0.197813
$917$	0	0
$918$	−1.16064	−0.0383069
$919$	−0.843163	−0.0278134	−0.0139067	−	0.999903i	$-0.504427\pi$
$919$	−0.843163	−0.0278134	−0.0139067	+	0.999903i	$0.504427\pi$
$920$	−32.3001	−1.06490
$921$	4.35015	0.143342
$922$	−37.4030	−1.23180
$923$	1.64906	0.0542795
$924$	0	0
$925$	1.46392	0.0481335
$926$	−18.0057	−0.591704
$927$	−10.9705	−0.360319
$928$	5.67431	0.186268
$929$	−3.50232	−0.114907	−0.0574537	−	0.998348i	$-0.518298\pi$
$929$	−3.50232	−0.114907	−0.0574537	+	0.998348i	$0.518298\pi$
$930$	−10.0156	−0.328425
$931$	0	0
$932$	−12.7421	−0.417382
$933$	17.4860	0.572466
$934$	−41.8364	−1.36893
$935$	9.52375	0.311460
$936$	−3.14710	−0.102866
$937$	−31.5255	−1.02989	−0.514946	−	0.857222i	$-0.672188\pi$
$937$	−31.5255	−1.02989	−0.514946	+	0.857222i	$0.672188\pi$
$938$	0	0
$939$	27.3870	0.893740
$940$	2.11716	0.0690540
$941$	38.5977	1.25825	0.629124	−	0.777305i	$-0.283414\pi$
$941$	38.5977	1.25825	0.629124	+	0.777305i	$0.283414\pi$
$942$	−12.7409	−0.415121
$943$	−4.61988	−0.150444
$944$	3.42975	0.111629
$945$	0	0
$946$	10.2609	0.333610
$947$	−23.0815	−0.750047	−0.375023	−	0.927015i	$-0.622365\pi$
$947$	−23.0815	−0.750047	−0.375023	+	0.927015i	$0.622365\pi$
$948$	2.92576	0.0950242
$949$	−2.27258	−0.0737710
$950$	−0.0651474	−0.00211366
$951$	−31.8861	−1.03398
$952$	0	0
$953$	−0.506172	−0.0163965	−0.00819826	−	0.999966i	$-0.502610\pi$
$953$	−0.506172	−0.0163965	−0.00819826	+	0.999966i	$0.502610\pi$
$954$	−8.58268	−0.277875
$955$	26.8762	0.869694
$956$	−1.56649	−0.0506638
$957$	−6.19448	−0.200239
$958$	39.4707	1.27524
$959$	0	0
$960$	19.2734	0.622045
$961$	−15.9999	−0.516127
$962$	10.7524	0.346673
$963$	19.5498	0.629983
$964$	−10.0103	−0.322409
$965$	20.6676	0.665312
$966$	0	0
$967$	−48.6196	−1.56350	−0.781750	−	0.623592i	$-0.785673\pi$
$967$	−48.6196	−1.56350	−0.781750	+	0.623592i	$0.785673\pi$
$968$	18.2278	0.585864
$969$	0.368026	0.0118227
$970$	−1.08096	−0.0347076
$971$	−3.29543	−0.105755	−0.0528777	−	0.998601i	$-0.516839\pi$
$971$	−3.29543	−0.105755	−0.0528777	+	0.998601i	$0.516839\pi$
$972$	−0.703600	−0.0225680
$973$	0	0
$974$	27.7217	0.888260
$975$	−0.162023	−0.00518889
$976$	−31.1937	−0.998487
$977$	−3.50798	−0.112230	−0.0561151	−	0.998424i	$-0.517871\pi$
$977$	−3.50798	−0.112230	−0.0561151	+	0.998424i	$0.517871\pi$
$978$	5.97589	0.191088
$979$	4.36280	0.139436
$980$	0	0
$981$	17.4207	0.556199
$982$	26.9360	0.859562
$983$	−22.3364	−0.712422	−0.356211	−	0.934406i	$-0.615932\pi$
$983$	−22.3364	−0.712422	−0.356211	+	0.934406i	$0.615932\pi$
$984$	3.07831	0.0981328
$985$	1.92174	0.0612316
$986$	1.74777	0.0556605
$987$	0	0
$988$	0.259701	0.00826218
$989$	10.1211	0.321832
$990$	−10.6377	−0.338089
$991$	53.4713	1.69857	0.849286	−	0.527933i	$-0.177033\pi$
$991$	53.4713	1.69857	0.849286	+	0.527933i	$0.177033\pi$
$992$	−14.5939	−0.463358
$993$	−16.1121	−0.511302
$994$	0	0
$995$	8.78352	0.278456
$996$	−1.08161	−0.0342723
$997$	41.4921	1.31407	0.657033	−	0.753862i	$-0.271811\pi$
$997$	41.4921	1.31407	0.657033	+	0.753862i	$0.271811\pi$
$998$	−35.8372	−1.13441
$999$	9.23719	0.292252

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.bc.1.4		8
7.3	odd	6		861.2.i.d.247.5	✓	16
7.5	odd	6		861.2.i.d.739.5	yes	16
7.6	odd	2		6027.2.a.bb.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
861.2.i.d.247.5	✓	16	7.3	odd	6
861.2.i.d.739.5	yes	16	7.5	odd	6
6027.2.a.bb.1.4		8	7.6	odd	2
6027.2.a.bc.1.4		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:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

\(f(q)\)	\(=\)	\(q-1.13860 q^{2} +1.00000 q^{3} -0.703600 q^{4} +2.27123 q^{5} -1.13860 q^{6} +3.07831 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.13860 q^{2} +1.00000 q^{3} -0.703600 q^{4} +2.27123 q^{5} -1.13860 q^{6} +3.07831 q^{8} +1.00000 q^{9} -2.58601 q^{10} +4.11356 q^{11} -0.703600 q^{12} -1.02235 q^{13} +2.27123 q^{15} -2.09775 q^{16} +1.01936 q^{17} -1.13860 q^{18} +0.361035 q^{19} -1.59804 q^{20} -4.68368 q^{22} -4.61988 q^{23} +3.07831 q^{24} +0.158481 q^{25} +1.16404 q^{26} +1.00000 q^{27} -1.50587 q^{29} -2.58601 q^{30} +3.87299 q^{31} -3.76813 q^{32} +4.11356 q^{33} -1.16064 q^{34} -0.703600 q^{36} +9.23719 q^{37} -0.411073 q^{38} -1.02235 q^{39} +6.99154 q^{40} +1.00000 q^{41} -2.19077 q^{43} -2.89430 q^{44} +2.27123 q^{45} +5.26018 q^{46} -1.32485 q^{47} -2.09775 q^{48} -0.180446 q^{50} +1.01936 q^{51} +0.719323 q^{52} +7.53796 q^{53} -1.13860 q^{54} +9.34284 q^{55} +0.361035 q^{57} +1.71458 q^{58} -1.63497 q^{59} -1.59804 q^{60} +14.8701 q^{61} -4.40977 q^{62} +8.48587 q^{64} -2.32198 q^{65} -4.68368 q^{66} +5.38619 q^{67} -0.717224 q^{68} -4.61988 q^{69} -1.61302 q^{71} +3.07831 q^{72} +2.22290 q^{73} -10.5174 q^{74} +0.158481 q^{75} -0.254024 q^{76} +1.16404 q^{78} -4.15827 q^{79} -4.76446 q^{80} +1.00000 q^{81} -1.13860 q^{82} +1.53726 q^{83} +2.31521 q^{85} +2.49440 q^{86} -1.50587 q^{87} +12.6628 q^{88} +1.06059 q^{89} -2.58601 q^{90} +3.25055 q^{92} +3.87299 q^{93} +1.50847 q^{94} +0.819993 q^{95} -3.76813 q^{96} +0.418003 q^{97} +4.11356 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 8 q^{3} + 13 q^{4} + 7 q^{5} + q^{6} + 6 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + q^2 + 8 * q^3 + 13 * q^4 + 7 * q^5 + q^6 + 6 * q^8 + 8 * q^9	\( 8 q + q^{2} + 8 q^{3} + 13 q^{4} + 7 q^{5} + q^{6} + 6 q^{8} + 8 q^{9} + 8 q^{10} + 11 q^{11} + 13 q^{12} + 10 q^{13} + 7 q^{15} - 17 q^{16} + 3 q^{17} + q^{18} + 6 q^{19} + 11 q^{20} + 15 q^{22} + 14 q^{23} + 6 q^{24} + 25 q^{25} + 24 q^{26} + 8 q^{27} + 2 q^{29} + 8 q^{30} + 16 q^{31} + 3 q^{32} + 11 q^{33} - 4 q^{34} + 13 q^{36} - 20 q^{37} + 10 q^{38} + 10 q^{39} - 3 q^{40} + 8 q^{41} + 7 q^{43} + 7 q^{45} - 5 q^{46} + 14 q^{47} - 17 q^{48} - 5 q^{50} + 3 q^{51} + 23 q^{52} + 7 q^{53} + q^{54} + 48 q^{55} + 6 q^{57} - 20 q^{58} + 22 q^{59} + 11 q^{60} - 33 q^{62} - 10 q^{64} - 14 q^{65} + 15 q^{66} + 12 q^{67} - 27 q^{68} + 14 q^{69} - 5 q^{71} + 6 q^{72} + 2 q^{73} + 6 q^{74} + 25 q^{75} + 43 q^{76} + 24 q^{78} - 15 q^{79} - 7 q^{80} + 8 q^{81} + q^{82} + 15 q^{83} - 43 q^{85} + 31 q^{86} + 2 q^{87} + 17 q^{88} + 29 q^{89} + 8 q^{90} + 19 q^{92} + 16 q^{93} + 20 q^{94} + 14 q^{95} + 3 q^{96} + 19 q^{97} + 11 q^{99}+O(q^{100}) \) 8 * q + q^2 + 8 * q^3 + 13 * q^4 + 7 * q^5 + q^6 + 6 * q^8 + 8 * q^9 + 8 * q^10 + 11 * q^11 + 13 * q^12 + 10 * q^13 + 7 * q^15 - 17 * q^16 + 3 * q^17 + q^18 + 6 * q^19 + 11 * q^20 + 15 * q^22 + 14 * q^23 + 6 * q^24 + 25 * q^25 + 24 * q^26 + 8 * q^27 + 2 * q^29 + 8 * q^30 + 16 * q^31 + 3 * q^32 + 11 * q^33 - 4 * q^34 + 13 * q^36 - 20 * q^37 + 10 * q^38 + 10 * q^39 - 3 * q^40 + 8 * q^41 + 7 * q^43 + 7 * q^45 - 5 * q^46 + 14 * q^47 - 17 * q^48 - 5 * q^50 + 3 * q^51 + 23 * q^52 + 7 * q^53 + q^54 + 48 * q^55 + 6 * q^57 - 20 * q^58 + 22 * q^59 + 11 * q^60 - 33 * q^62 - 10 * q^64 - 14 * q^65 + 15 * q^66 + 12 * q^67 - 27 * q^68 + 14 * q^69 - 5 * q^71 + 6 * q^72 + 2 * q^73 + 6 * q^74 + 25 * q^75 + 43 * q^76 + 24 * q^78 - 15 * q^79 - 7 * q^80 + 8 * q^81 + q^82 + 15 * q^83 - 43 * q^85 + 31 * q^86 + 2 * q^87 + 17 * q^88 + 29 * q^89 + 8 * q^90 + 19 * q^92 + 16 * q^93 + 20 * q^94 + 14 * q^95 + 3 * q^96 + 19 * q^97 + 11 * q^99

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