LMFDB - Embedded newform 1849.2.a.p.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,2,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, 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("1849.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1849.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:	$14.7643393337$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 5 x^{19} - 19 x^{18} + 127 x^{17} + 95 x^{16} - 1293 x^{15} + 329 x^{14} + 6765 x^{13} + \cdots - 43 $ x^20 - 5x^19 - 19x^18 + 127x^17 + 95x^16 - 1293x^15 + 329x^14 + 6765x^13 - 5068x^12 - 19360x^11 + 20357x^10 + 29618x^9 - 37889x^8 - 21700x^7 + 32885x^6 + 6093x^5 - 11330x^4 - 1004x^3 + 1386x^2 + 87*x - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Root			$-2.64020$ of defining polynomial
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64020 q^{2} +0.354022 q^{3} +4.97064 q^{4} +1.23228 q^{5} +0.934688 q^{6} +3.87263 q^{7} +7.84307 q^{8} -2.87467 q^{9} +O(q^{10})$	\(q+2.64020 q^{2} +0.354022 q^{3} +4.97064 q^{4} +1.23228 q^{5} +0.934688 q^{6} +3.87263 q^{7} +7.84307 q^{8} -2.87467 q^{9} +3.25345 q^{10} +2.01275 q^{11} +1.75972 q^{12} -3.33076 q^{13} +10.2245 q^{14} +0.436253 q^{15} +10.7660 q^{16} -1.76500 q^{17} -7.58969 q^{18} +2.88182 q^{19} +6.12520 q^{20} +1.37100 q^{21} +5.31406 q^{22} -7.07373 q^{23} +2.77662 q^{24} -3.48149 q^{25} -8.79387 q^{26} -2.07976 q^{27} +19.2494 q^{28} -1.16122 q^{29} +1.15179 q^{30} -7.21474 q^{31} +12.7381 q^{32} +0.712558 q^{33} -4.65996 q^{34} +4.77215 q^{35} -14.2889 q^{36} -0.396953 q^{37} +7.60858 q^{38} -1.17916 q^{39} +9.66483 q^{40} -1.94531 q^{41} +3.61970 q^{42} +10.0047 q^{44} -3.54239 q^{45} -18.6760 q^{46} -2.91101 q^{47} +3.81139 q^{48} +7.99728 q^{49} -9.19183 q^{50} -0.624850 q^{51} -16.5560 q^{52} +5.58304 q^{53} -5.49098 q^{54} +2.48027 q^{55} +30.3733 q^{56} +1.02023 q^{57} -3.06585 q^{58} +6.47805 q^{59} +2.16846 q^{60} +12.6104 q^{61} -19.0483 q^{62} -11.1325 q^{63} +12.0992 q^{64} -4.10442 q^{65} +1.88129 q^{66} -5.78439 q^{67} -8.77320 q^{68} -2.50426 q^{69} +12.5994 q^{70} -3.37541 q^{71} -22.5462 q^{72} +2.47282 q^{73} -1.04803 q^{74} -1.23253 q^{75} +14.3245 q^{76} +7.79464 q^{77} -3.11322 q^{78} +8.62549 q^{79} +13.2666 q^{80} +7.88772 q^{81} -5.13600 q^{82} +3.06803 q^{83} +6.81473 q^{84} -2.17497 q^{85} -0.411098 q^{87} +15.7861 q^{88} +14.6882 q^{89} -9.35260 q^{90} -12.8988 q^{91} -35.1609 q^{92} -2.55418 q^{93} -7.68564 q^{94} +3.55120 q^{95} +4.50958 q^{96} -18.2946 q^{97} +21.1144 q^{98} -5.78599 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 5 q^{2} + q^{3} + 23 q^{4} + 3 q^{5} + 5 q^{6} + 2 q^{7} - 9 q^{8} + 27 q^{9}+O(q^{10}) $ 20 * q - 5 * q^2 + q^3 + 23 * q^4 + 3 * q^5 + 5 * q^6 + 2 * q^7 - 9 * q^8 + 27 * q^9	\( 20 q - 5 q^{2} + q^{3} + 23 q^{4} + 3 q^{5} + 5 q^{6} + 2 q^{7} - 9 q^{8} + 27 q^{9} + 21 q^{11} + 12 q^{12} + 11 q^{13} + 8 q^{14} + 4 q^{15} + 29 q^{16} + 15 q^{17} - 9 q^{18} + 7 q^{19} + 17 q^{20} + 8 q^{21} - 47 q^{22} + 26 q^{23} - q^{24} + 11 q^{25} - 58 q^{26} + 22 q^{27} + 28 q^{28} + 12 q^{29} + 24 q^{30} + 26 q^{31} - 3 q^{32} + 17 q^{33} - 54 q^{34} + 26 q^{35} + 14 q^{36} + 19 q^{37} + 5 q^{38} + 7 q^{39} - 16 q^{40} + 27 q^{41} + 52 q^{42} + 32 q^{44} - 75 q^{45} - 59 q^{46} + 45 q^{47} + 66 q^{48} + 20 q^{49} - 75 q^{51} + 11 q^{52} + 3 q^{53} + 57 q^{54} + 2 q^{55} + 87 q^{56} - 24 q^{57} - 46 q^{58} + 66 q^{59} + 29 q^{60} - 30 q^{61} - 72 q^{62} + 21 q^{63} - 7 q^{64} + 6 q^{65} + 41 q^{66} - 6 q^{67} + 28 q^{68} + 35 q^{69} + 80 q^{70} + 31 q^{71} + 15 q^{72} + 26 q^{73} + 87 q^{74} - 73 q^{75} + 68 q^{76} + 21 q^{77} - 50 q^{78} + 39 q^{79} + 60 q^{80} + 16 q^{81} - 45 q^{82} + 13 q^{83} + 31 q^{84} + 22 q^{85} + 61 q^{87} - 50 q^{88} + 4 q^{89} - 13 q^{90} + 25 q^{91} + 5 q^{92} - 67 q^{93} - 42 q^{94} + 79 q^{95} + 36 q^{96} - 2 q^{97} + 35 q^{98} + 13 q^{99}+O(q^{100}) \) 20 * q - 5 * q^2 + q^3 + 23 * q^4 + 3 * q^5 + 5 * q^6 + 2 * q^7 - 9 * q^8 + 27 * q^9 + 21 * q^11 + 12 * q^12 + 11 * q^13 + 8 * q^14 + 4 * q^15 + 29 * q^16 + 15 * q^17 - 9 * q^18 + 7 * q^19 + 17 * q^20 + 8 * q^21 - 47 * q^22 + 26 * q^23 - q^24 + 11 * q^25 - 58 * q^26 + 22 * q^27 + 28 * q^28 + 12 * q^29 + 24 * q^30 + 26 * q^31 - 3 * q^32 + 17 * q^33 - 54 * q^34 + 26 * q^35 + 14 * q^36 + 19 * q^37 + 5 * q^38 + 7 * q^39 - 16 * q^40 + 27 * q^41 + 52 * q^42 + 32 * q^44 - 75 * q^45 - 59 * q^46 + 45 * q^47 + 66 * q^48 + 20 * q^49 - 75 * q^51 + 11 * q^52 + 3 * q^53 + 57 * q^54 + 2 * q^55 + 87 * q^56 - 24 * q^57 - 46 * q^58 + 66 * q^59 + 29 * q^60 - 30 * q^61 - 72 * q^62 + 21 * q^63 - 7 * q^64 + 6 * q^65 + 41 * q^66 - 6 * q^67 + 28 * q^68 + 35 * q^69 + 80 * q^70 + 31 * q^71 + 15 * q^72 + 26 * q^73 + 87 * q^74 - 73 * q^75 + 68 * q^76 + 21 * q^77 - 50 * q^78 + 39 * q^79 + 60 * q^80 + 16 * q^81 - 45 * q^82 + 13 * q^83 + 31 * q^84 + 22 * q^85 + 61 * q^87 - 50 * q^88 + 4 * q^89 - 13 * q^90 + 25 * q^91 + 5 * q^92 - 67 * q^93 - 42 * q^94 + 79 * q^95 + 36 * q^96 - 2 * q^97 + 35 * q^98 + 13 * 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 $

$2$	2.64020	1.86690	0.933450	−	0.358706i	$-0.116782\pi$
$2$	2.64020	1.86690	0.933450	+	0.358706i	$0.116782\pi$
$3$	0.354022	0.204395	0.102197	−	0.994764i	$-0.467413\pi$
$3$	0.354022	0.204395	0.102197	+	0.994764i	$0.467413\pi$
$4$	4.97064	2.48532
$5$	1.23228	0.551091	0.275545	−	0.961288i	$-0.411142\pi$
$5$	1.23228	0.551091	0.275545	+	0.961288i	$0.411142\pi$
$6$	0.934688	0.381585
$7$	3.87263	1.46372	0.731859	−	0.681457i	$-0.238653\pi$
$7$	3.87263	1.46372	0.731859	+	0.681457i	$0.238653\pi$
$8$	7.84307	2.77294
$9$	−2.87467	−0.958223
$10$	3.25345	1.02883
$11$	2.01275	0.606867	0.303434	−	0.952853i	$-0.401867\pi$
$11$	2.01275	0.606867	0.303434	+	0.952853i	$0.401867\pi$
$12$	1.75972	0.507986
$13$	−3.33076	−0.923787	−0.461894	−	0.886935i	$-0.652830\pi$
$13$	−3.33076	−0.923787	−0.461894	+	0.886935i	$0.652830\pi$
$14$	10.2245	2.73261
$15$	0.436253	0.112640
$16$	10.7660	2.69149
$17$	−1.76500	−0.428076	−0.214038	−	0.976825i	$-0.568662\pi$
$17$	−1.76500	−0.428076	−0.214038	+	0.976825i	$0.568662\pi$
$18$	−7.58969	−1.78891
$19$	2.88182	0.661135	0.330568	−	0.943782i	$-0.392760\pi$
$19$	2.88182	0.661135	0.330568	+	0.943782i	$0.392760\pi$
$20$	6.12520	1.36964
$21$	1.37100	0.299176
$22$	5.31406	1.13296
$23$	−7.07373	−1.47497	−0.737487	−	0.675361i	$-0.763988\pi$
$23$	−7.07373	−1.47497	−0.737487	+	0.675361i	$0.763988\pi$
$24$	2.77662	0.566775
$25$	−3.48149	−0.696299
$26$	−8.79387	−1.72462
$27$	−2.07976	−0.400251
$28$	19.2494	3.63780
$29$	−1.16122	−0.215633	−0.107817	−	0.994171i	$-0.534386\pi$
$29$	−1.16122	−0.215633	−0.107817	+	0.994171i	$0.534386\pi$
$30$	1.15179	0.210288
$31$	−7.21474	−1.29581	−0.647903	−	0.761723i	$-0.724354\pi$
$31$	−7.21474	−1.29581	−0.647903	+	0.761723i	$0.724354\pi$
$32$	12.7381	2.25180
$33$	0.712558	0.124040
$34$	−4.65996	−0.799176
$35$	4.77215	0.806641
$36$	−14.2889	−2.38149
$37$	−0.396953	−0.0652587	−0.0326293	−	0.999468i	$-0.510388\pi$
$37$	−0.396953	−0.0652587	−0.0326293	+	0.999468i	$0.510388\pi$
$38$	7.60858	1.23427
$39$	−1.17916	−0.188817
$40$	9.66483	1.52814
$41$	−1.94531	−0.303807	−0.151903	−	0.988395i	$-0.548540\pi$
$41$	−1.94531	−0.303807	−0.151903	+	0.988395i	$0.548540\pi$
$42$	3.61970	0.558532
$43$	0	0
$43$	0	0
$44$	10.0047	1.50826
$45$	−3.54239	−0.528068
$46$	−18.6760	−2.75363
$47$	−2.91101	−0.424614	−0.212307	−	0.977203i	$-0.568098\pi$
$47$	−2.91101	−0.424614	−0.212307	+	0.977203i	$0.568098\pi$
$48$	3.81139	0.550127
$49$	7.99728	1.14247
$50$	−9.19183	−1.29992
$51$	−0.624850	−0.0874966
$52$	−16.5560	−2.29591
$53$	5.58304	0.766889	0.383445	−	0.923564i	$-0.374738\pi$
$53$	5.58304	0.766889	0.383445	+	0.923564i	$0.374738\pi$
$54$	−5.49098	−0.747228
$55$	2.48027	0.334439
$56$	30.3733	4.05880
$57$	1.02023	0.135133
$58$	−3.06585	−0.402566
$59$	6.47805	0.843371	0.421685	−	0.906742i	$-0.361439\pi$
$59$	6.47805	0.843371	0.421685	+	0.906742i	$0.361439\pi$
$60$	2.16846	0.279947
$61$	12.6104	1.61459	0.807295	−	0.590147i	$-0.200930\pi$
$61$	12.6104	1.61459	0.807295	+	0.590147i	$0.200930\pi$
$62$	−19.0483	−2.41914
$63$	−11.1325	−1.40257
$64$	12.0992	1.51240
$65$	−4.10442	−0.509091
$66$	1.88129	0.231571
$67$	−5.78439	−0.706675	−0.353338	−	0.935496i	$-0.614953\pi$
$67$	−5.78439	−0.706675	−0.353338	+	0.935496i	$0.614953\pi$
$68$	−8.77320	−1.06391
$69$	−2.50426	−0.301477
$70$	12.5994	1.50592
$71$	−3.37541	−0.400588	−0.200294	−	0.979736i	$-0.564190\pi$
$71$	−3.37541	−0.400588	−0.200294	+	0.979736i	$0.564190\pi$
$72$	−22.5462	−2.65710
$73$	2.47282	0.289422	0.144711	−	0.989474i	$-0.453775\pi$
$73$	2.47282	0.289422	0.144711	+	0.989474i	$0.453775\pi$
$74$	−1.04803	−0.121832
$75$	−1.23253	−0.142320
$76$	14.3245	1.64313
$77$	7.79464	0.888282
$78$	−3.11322	−0.352503
$79$	8.62549	0.970443	0.485222	−	0.874391i	$-0.338739\pi$
$79$	8.62549	0.970443	0.485222	+	0.874391i	$0.338739\pi$
$80$	13.2666	1.48326
$81$	7.88772	0.876414
$82$	−5.13600	−0.567177
$83$	3.06803	0.336760	0.168380	−	0.985722i	$-0.446146\pi$
$83$	3.06803	0.336760	0.168380	+	0.985722i	$0.446146\pi$
$84$	6.81473	0.743548
$85$	−2.17497	−0.235909
$86$	0	0
$87$	−0.411098	−0.0440743
$88$	15.7861	1.68281
$89$	14.6882	1.55695	0.778473	−	0.627678i	$-0.215994\pi$
$89$	14.6882	1.55695	0.778473	+	0.627678i	$0.215994\pi$
$90$	−9.35260	−0.985850
$91$	−12.8988	−1.35216
$92$	−35.1609	−3.66578
$93$	−2.55418	−0.264856
$94$	−7.68564	−0.792713
$95$	3.55120	0.364346
$96$	4.50958	0.460257
$97$	−18.2946	−1.85753	−0.928767	−	0.370665i	$-0.879130\pi$
$97$	−18.2946	−1.85753	−0.928767	+	0.370665i	$0.879130\pi$
$98$	21.1144	2.13287
$99$	−5.78599	−0.581514
$100$	−17.3052	−1.73052
$101$	17.5215	1.74345	0.871727	−	0.489991i	$-0.163000\pi$
$101$	17.5215	1.74345	0.871727	+	0.489991i	$0.163000\pi$
$102$	−1.64973	−0.163347
$103$	−10.2855	−1.01346	−0.506732	−	0.862103i	$-0.669147\pi$
$103$	−10.2855	−1.01346	−0.506732	+	0.862103i	$0.669147\pi$
$104$	−26.1234	−2.56161
$105$	1.68945	0.164873
$106$	14.7403	1.43171
$107$	−11.9961	−1.15971	−0.579853	−	0.814721i	$-0.696890\pi$
$107$	−11.9961	−1.15971	−0.579853	+	0.814721i	$0.696890\pi$
$108$	−10.3377	−0.994750
$109$	0.989870	0.0948124	0.0474062	−	0.998876i	$-0.484904\pi$
$109$	0.989870	0.0948124	0.0474062	+	0.998876i	$0.484904\pi$
$110$	6.54839	0.624364
$111$	−0.140530	−0.0133385
$112$	41.6926	3.93958
$113$	10.5953	0.996721	0.498360	−	0.866970i	$-0.333936\pi$
$113$	10.5953	0.996721	0.498360	+	0.866970i	$0.333936\pi$
$114$	2.69360	0.252279
$115$	−8.71679	−0.812845
$116$	−5.77201	−0.535918
$117$	9.57484	0.885194
$118$	17.1033	1.57449
$119$	−6.83521	−0.626583
$120$	3.42156	0.312345
$121$	−6.94883	−0.631712
$122$	33.2938	3.01428
$123$	−0.688683	−0.0620965
$124$	−35.8619	−3.22049
$125$	−10.4515	−0.934815
$126$	−29.3921	−2.61845
$127$	7.69558	0.682872	0.341436	−	0.939905i	$-0.389087\pi$
$127$	7.69558	0.682872	0.341436	+	0.939905i	$0.389087\pi$
$128$	6.46810	0.571705
$129$	0	0
$130$	−10.8365	−0.950422
$131$	−10.2319	−0.893967	−0.446984	−	0.894542i	$-0.647502\pi$
$131$	−10.2319	−0.893967	−0.446984	+	0.894542i	$0.647502\pi$
$132$	3.54187	0.308280
$133$	11.1602	0.967715
$134$	−15.2719	−1.31929
$135$	−2.56284	−0.220574
$136$	−13.8430	−1.18703
$137$	−4.67121	−0.399089	−0.199544	−	0.979889i	$-0.563946\pi$
$137$	−4.67121	−0.399089	−0.199544	+	0.979889i	$0.563946\pi$
$138$	−6.61173	−0.562828
$139$	11.6953	0.991984	0.495992	−	0.868327i	$-0.334805\pi$
$139$	11.6953	0.991984	0.495992	+	0.868327i	$0.334805\pi$
$140$	23.7206	2.00476
$141$	−1.03056	−0.0867889
$142$	−8.91176	−0.747858
$143$	−6.70400	−0.560616
$144$	−30.9486	−2.57905
$145$	−1.43095	−0.118834
$146$	6.52874	0.540322
$147$	2.83121	0.233514
$148$	−1.97311	−0.162189
$149$	−9.86933	−0.808527	−0.404263	−	0.914643i	$-0.632472\pi$
$149$	−9.86933	−0.808527	−0.404263	+	0.914643i	$0.632472\pi$
$150$	−3.25411	−0.265697
$151$	11.4912	0.935143	0.467572	−	0.883955i	$-0.345129\pi$
$151$	11.4912	0.935143	0.467572	+	0.883955i	$0.345129\pi$
$152$	22.6023	1.83329
$153$	5.07380	0.410193
$154$	20.5794	1.65833
$155$	−8.89056	−0.714107
$156$	−5.86120	−0.469271
$157$	−6.87153	−0.548407	−0.274204	−	0.961672i	$-0.588414\pi$
$157$	−6.87153	−0.548407	−0.274204	+	0.961672i	$0.588414\pi$
$158$	22.7730	1.81172
$159$	1.97652	0.156748
$160$	15.6969	1.24095
$161$	−27.3939	−2.15895
$162$	20.8251	1.63618
$163$	20.1403	1.57751	0.788756	−	0.614706i	$-0.210725\pi$
$163$	20.1403	1.57751	0.788756	+	0.614706i	$0.210725\pi$
$164$	−9.66944	−0.755056
$165$	0.878069	0.0683576
$166$	8.10021	0.628698
$167$	9.27297	0.717564	0.358782	−	0.933421i	$-0.383192\pi$
$167$	9.27297	0.717564	0.358782	+	0.933421i	$0.383192\pi$
$168$	10.7528	0.829598
$169$	−1.90602	−0.146617
$170$	−5.74236	−0.440419
$171$	−8.28428	−0.633515
$172$	0	0
$173$	23.3849	1.77792	0.888962	−	0.457980i	$-0.151427\pi$
$173$	23.3849	1.77792	0.888962	+	0.457980i	$0.151427\pi$
$174$	−1.08538	−0.0822824
$175$	−13.4825	−1.01918
$176$	21.6692	1.63338
$177$	2.29337	0.172381
$178$	38.7797	2.90666
$179$	14.1785	1.05975	0.529874	−	0.848076i	$-0.322239\pi$
$179$	14.1785	1.05975	0.529874	+	0.848076i	$0.322239\pi$
$180$	−17.6079	−1.31242
$181$	17.0329	1.26604	0.633021	−	0.774134i	$-0.281815\pi$
$181$	17.0329	1.26604	0.633021	+	0.774134i	$0.281815\pi$
$182$	−34.0554	−2.52436
$183$	4.46435	0.330014
$184$	−55.4797	−4.09002
$185$	−0.489156	−0.0359635
$186$	−6.74353	−0.494460
$187$	−3.55251	−0.259785
$188$	−14.4696	−1.05530
$189$	−8.05415	−0.585854
$190$	9.37587	0.680197
$191$	7.88382	0.570453	0.285227	−	0.958460i	$-0.407931\pi$
$191$	7.88382	0.570453	0.285227	+	0.958460i	$0.407931\pi$
$192$	4.28340	0.309128
$193$	−0.566215	−0.0407570	−0.0203785	−	0.999792i	$-0.506487\pi$
$193$	−0.566215	−0.0407570	−0.0203785	+	0.999792i	$0.506487\pi$
$194$	−48.3013	−3.46783
$195$	−1.45306	−0.104056
$196$	39.7516	2.83940
$197$	−14.8974	−1.06140	−0.530699	−	0.847560i	$-0.678071\pi$
$197$	−14.8974	−1.06140	−0.530699	+	0.847560i	$0.678071\pi$
$198$	−15.2762	−1.08563
$199$	−16.1063	−1.14175	−0.570873	−	0.821038i	$-0.693395\pi$
$199$	−16.1063	−1.14175	−0.570873	+	0.821038i	$0.693395\pi$
$200$	−27.3056	−1.93080
$201$	−2.04780	−0.144441
$202$	46.2602	3.25486
$203$	−4.49698	−0.315626
$204$	−3.10591	−0.217457
$205$	−2.39716	−0.167425
$206$	−27.1559	−1.89204
$207$	20.3346	1.41335
$208$	−35.8589	−2.48637
$209$	5.80039	0.401221
$210$	4.46047	0.307802
$211$	9.60083	0.660948	0.330474	−	0.943815i	$-0.392791\pi$
$211$	9.60083	0.660948	0.330474	+	0.943815i	$0.392791\pi$
$212$	27.7513	1.90596
$213$	−1.19497	−0.0818781
$214$	−31.6720	−2.16506
$215$	0	0
$216$	−16.3117	−1.10987
$217$	−27.9400	−1.89669
$218$	2.61345	0.177005
$219$	0.875434	0.0591564
$220$	12.3285	0.831188
$221$	5.87881	0.395452
$222$	−0.371027	−0.0249017
$223$	−17.0660	−1.14283	−0.571413	−	0.820663i	$-0.693605\pi$
$223$	−17.0660	−1.14283	−0.571413	+	0.820663i	$0.693605\pi$
$224$	49.3301	3.29601
$225$	10.0081	0.667209
$226$	27.9736	1.86078
$227$	−17.5322	−1.16365	−0.581825	−	0.813314i	$-0.697661\pi$
$227$	−17.5322	−1.16365	−0.581825	+	0.813314i	$0.697661\pi$
$228$	5.07119	0.335848
$229$	28.8052	1.90350	0.951752	−	0.306869i	$-0.0992815\pi$
$229$	28.8052	1.90350	0.951752	+	0.306869i	$0.0992815\pi$
$230$	−23.0140	−1.51750
$231$	2.75948	0.181560
$232$	−9.10754	−0.597939
$233$	−9.06399	−0.593802	−0.296901	−	0.954908i	$-0.595953\pi$
$233$	−9.06399	−0.593802	−0.296901	+	0.954908i	$0.595953\pi$
$234$	25.2795	1.65257
$235$	−3.58717	−0.234001
$236$	32.2001	2.09605
$237$	3.05361	0.198353
$238$	−18.0463	−1.16977
$239$	−13.7684	−0.890601	−0.445300	−	0.895381i	$-0.646903\pi$
$239$	−13.7684	−0.890601	−0.445300	+	0.895381i	$0.646903\pi$
$240$	4.69669	0.303170
$241$	−1.82258	−0.117403	−0.0587014	−	0.998276i	$-0.518696\pi$
$241$	−1.82258	−0.117403	−0.0587014	+	0.998276i	$0.518696\pi$
$242$	−18.3463	−1.17934
$243$	9.03172	0.579385
$244$	62.6815	4.01277
$245$	9.85486	0.629604
$246$	−1.81826	−0.115928
$247$	−9.59867	−0.610749
$248$	−56.5857	−3.59320
$249$	1.08615	0.0688321
$250$	−27.5941	−1.74521
$251$	11.4746	0.724272	0.362136	−	0.932125i	$-0.382048\pi$
$251$	11.4746	0.724272	0.362136	+	0.932125i	$0.382048\pi$
$252$	−55.3358	−3.48583
$253$	−14.2377	−0.895114
$254$	20.3178	1.27485
$255$	−0.769989	−0.0482186
$256$	−7.12141	−0.445088
$257$	26.9482	1.68098	0.840491	−	0.541826i	$-0.182267\pi$
$257$	26.9482	1.68098	0.840491	+	0.541826i	$0.182267\pi$
$258$	0	0
$259$	−1.53725	−0.0955203
$260$	−20.4016	−1.26525
$261$	3.33813	0.206625
$262$	−27.0143	−1.66895
$263$	−19.4419	−1.19884	−0.599420	−	0.800435i	$-0.704602\pi$
$263$	−19.4419	−1.19884	−0.599420	+	0.800435i	$0.704602\pi$
$264$	5.58864	0.343957
$265$	6.87985	0.422626
$266$	29.4652	1.80663
$267$	5.19995	0.318232
$268$	−28.7521	−1.75631
$269$	5.33971	0.325568	0.162784	−	0.986662i	$-0.447953\pi$
$269$	5.33971	0.325568	0.162784	+	0.986662i	$0.447953\pi$
$270$	−6.76641	−0.411791
$271$	−25.8968	−1.57312	−0.786561	−	0.617513i	$-0.788140\pi$
$271$	−25.8968	−1.57312	−0.786561	+	0.617513i	$0.788140\pi$
$272$	−19.0020	−1.15216
$273$	−4.56647	−0.276375
$274$	−12.3329	−0.745059
$275$	−7.00738	−0.422561
$276$	−12.4478	−0.749267
$277$	−10.9528	−0.658091	−0.329046	−	0.944314i	$-0.606727\pi$
$277$	−10.9528	−0.658091	−0.329046	+	0.944314i	$0.606727\pi$
$278$	30.8779	1.85194
$279$	20.7400	1.24167
$280$	37.4283	2.23677
$281$	−18.8817	−1.12639	−0.563193	−	0.826325i	$-0.690427\pi$
$281$	−18.8817	−1.12639	−0.563193	+	0.826325i	$0.690427\pi$
$282$	−2.72088	−0.162026
$283$	−6.41809	−0.381516	−0.190758	−	0.981637i	$-0.561095\pi$
$283$	−6.41809	−0.381516	−0.190758	+	0.981637i	$0.561095\pi$
$284$	−16.7780	−0.995589
$285$	1.25720	0.0744703
$286$	−17.6999	−1.04662
$287$	−7.53347	−0.444687
$288$	−36.6179	−2.15773
$289$	−13.8848	−0.816751
$290$	−3.77798	−0.221850
$291$	−6.47669	−0.379670
$292$	12.2915	0.719306
$293$	5.29449	0.309307	0.154654	−	0.987969i	$-0.450574\pi$
$293$	5.29449	0.309307	0.154654	+	0.987969i	$0.450574\pi$
$294$	7.47496	0.435948
$295$	7.98276	0.464774
$296$	−3.11333	−0.180959
$297$	−4.18604	−0.242899
$298$	−26.0570	−1.50944
$299$	23.5609	1.36256
$300$	−6.12644	−0.353710
$301$	0	0
$302$	30.3391	1.74582
$303$	6.20300	0.356353
$304$	31.0256	1.77944
$305$	15.5394	0.889786
$306$	13.3958	0.765789
$307$	1.32505	0.0756246	0.0378123	−	0.999285i	$-0.487961\pi$
$307$	1.32505	0.0756246	0.0378123	+	0.999285i	$0.487961\pi$
$308$	38.7443	2.20766
$309$	−3.64131	−0.207147
$310$	−23.4728	−1.33317
$311$	22.8600	1.29627	0.648136	−	0.761525i	$-0.275549\pi$
$311$	22.8600	1.29627	0.648136	+	0.761525i	$0.275549\pi$
$312$	−9.24826	−0.523580
$313$	8.35596	0.472307	0.236153	−	0.971716i	$-0.424113\pi$
$313$	8.35596	0.472307	0.236153	+	0.971716i	$0.424113\pi$
$314$	−18.1422	−1.02382
$315$	−13.7184	−0.772942
$316$	42.8742	2.41186
$317$	−2.72735	−0.153183	−0.0765916	−	0.997063i	$-0.524404\pi$
$317$	−2.72735	−0.153183	−0.0765916	+	0.997063i	$0.524404\pi$
$318$	5.21840	0.292633
$319$	−2.33725	−0.130861
$320$	14.9096	0.833472
$321$	−4.24688	−0.237038
$322$	−72.3254	−4.03054
$323$	−5.08643	−0.283016
$324$	39.2070	2.17817
$325$	11.5960	0.643232
$326$	53.1744	2.94506
$327$	0.350436	0.0193792
$328$	−15.2572	−0.842439
$329$	−11.2733	−0.621515
$330$	2.31827	0.127617
$331$	−29.4946	−1.62117	−0.810585	−	0.585620i	$-0.800851\pi$
$331$	−29.4946	−1.62117	−0.810585	+	0.585620i	$0.800851\pi$
$332$	15.2501	0.836957
$333$	1.14111	0.0625324
$334$	24.4825	1.33962
$335$	−7.12796	−0.389442
$336$	14.7601	0.805230
$337$	−18.9379	−1.03162	−0.515808	−	0.856704i	$-0.672508\pi$
$337$	−18.9379	−1.03162	−0.515808	+	0.856704i	$0.672508\pi$
$338$	−5.03226	−0.273719
$339$	3.75097	0.203725
$340$	−10.8110	−0.586309
$341$	−14.5215	−0.786382
$342$	−21.8721	−1.18271
$343$	3.86208	0.208533
$344$	0	0
$345$	−3.08594	−0.166141
$346$	61.7409	3.31921
$347$	2.17271	0.116637	0.0583187	−	0.998298i	$-0.481426\pi$
$347$	2.17271	0.116637	0.0583187	+	0.998298i	$0.481426\pi$
$348$	−2.04342	−0.109539
$349$	−0.653626	−0.0349878	−0.0174939	−	0.999847i	$-0.505569\pi$
$349$	−0.653626	−0.0349878	−0.0174939	+	0.999847i	$0.505569\pi$
$350$	−35.5966	−1.90272
$351$	6.92720	0.369746
$352$	25.6387	1.36655
$353$	−7.33995	−0.390666	−0.195333	−	0.980737i	$-0.562579\pi$
$353$	−7.33995	−0.390666	−0.195333	+	0.980737i	$0.562579\pi$
$354$	6.05496	0.321818
$355$	−4.15944	−0.220760
$356$	73.0097	3.86951
$357$	−2.41982	−0.128070
$358$	37.4340	1.97845
$359$	9.30210	0.490946	0.245473	−	0.969403i	$-0.421057\pi$
$359$	9.30210	0.490946	0.245473	+	0.969403i	$0.421057\pi$
$360$	−27.7832	−1.46430
$361$	−10.6951	−0.562900
$362$	44.9701	2.36358
$363$	−2.46004	−0.129119
$364$	−64.1154	−3.36056
$365$	3.04720	0.159498
$366$	11.7868	0.616103
$367$	8.35151	0.435945	0.217973	−	0.975955i	$-0.430056\pi$
$367$	8.35151	0.435945	0.217973	+	0.975955i	$0.430056\pi$
$368$	−76.1555	−3.96988
$369$	5.59213	0.291114
$370$	−1.29147	−0.0671402
$371$	21.6210	1.12251
$372$	−12.6959	−0.658252
$373$	−36.4568	−1.88766	−0.943831	−	0.330430i	$-0.892806\pi$
$373$	−36.4568	−1.88766	−0.943831	+	0.330430i	$0.892806\pi$
$374$	−9.37933	−0.484994
$375$	−3.70008	−0.191071
$376$	−22.8312	−1.17743
$377$	3.86775	0.199199
$378$	−21.2646	−1.09373
$379$	−7.42192	−0.381239	−0.190619	−	0.981664i	$-0.561050\pi$
$379$	−7.42192	−0.381239	−0.190619	+	0.981664i	$0.561050\pi$
$380$	17.6517	0.905515
$381$	2.72440	0.139575
$382$	20.8148	1.06498
$383$	1.29783	0.0663159	0.0331580	−	0.999450i	$-0.489444\pi$
$383$	1.29783	0.0663159	0.0331580	+	0.999450i	$0.489444\pi$
$384$	2.28985	0.116853
$385$	9.60516	0.489524
$386$	−1.49492	−0.0760894
$387$	0	0
$388$	−90.9357	−4.61656
$389$	9.92974	0.503458	0.251729	−	0.967798i	$-0.419001\pi$
$389$	9.92974	0.503458	0.251729	+	0.967798i	$0.419001\pi$
$390$	−3.83635	−0.194261
$391$	12.4852	0.631402
$392$	62.7232	3.16800
$393$	−3.62233	−0.182722
$394$	−39.3322	−1.98153
$395$	10.6290	0.534802
$396$	−28.7601	−1.44525
$397$	−15.2455	−0.765151	−0.382576	−	0.923924i	$-0.624963\pi$
$397$	−15.2455	−0.765151	−0.382576	+	0.923924i	$0.624963\pi$
$398$	−42.5238	−2.13153
$399$	3.95097	0.197796
$400$	−37.4817	−1.87408
$401$	−16.9609	−0.846989	−0.423495	−	0.905899i	$-0.639197\pi$
$401$	−16.9609	−0.846989	−0.423495	+	0.905899i	$0.639197\pi$
$402$	−5.40660	−0.269657
$403$	24.0306	1.19705
$404$	87.0930	4.33304
$405$	9.71986	0.482984
$406$	−11.8729	−0.589243
$407$	−0.798968	−0.0396034
$408$	−4.90075	−0.242623
$409$	4.36360	0.215766	0.107883	−	0.994164i	$-0.465593\pi$
$409$	4.36360	0.215766	0.107883	+	0.994164i	$0.465593\pi$
$410$	−6.32898	−0.312566
$411$	−1.65371	−0.0815717
$412$	−51.1257	−2.51878
$413$	25.0871	1.23446
$414$	53.6874	2.63859
$415$	3.78067	0.185586
$416$	−42.4277	−2.08019
$417$	4.14040	0.202756
$418$	15.3142	0.749041
$419$	18.9773	0.927102	0.463551	−	0.886070i	$-0.346575\pi$
$419$	18.9773	0.927102	0.463551	+	0.886070i	$0.346575\pi$
$420$	8.39763	0.409763
$421$	9.92401	0.483667	0.241833	−	0.970318i	$-0.422251\pi$
$421$	9.92401	0.483667	0.241833	+	0.970318i	$0.422251\pi$
$422$	25.3481	1.23393
$423$	8.36818	0.406875
$424$	43.7881	2.12654
$425$	6.14485	0.298069
$426$	−3.15496	−0.152858
$427$	48.8353	2.36330
$428$	−59.6282	−2.88224
$429$	−2.37336	−0.114587
$430$	0	0
$431$	−16.2572	−0.783080	−0.391540	−	0.920161i	$-0.628058\pi$
$431$	−16.2572	−0.783080	−0.391540	+	0.920161i	$0.628058\pi$
$432$	−22.3907	−1.07727
$433$	18.1960	0.874443	0.437221	−	0.899354i	$-0.355963\pi$
$433$	18.1960	0.874443	0.437221	+	0.899354i	$0.355963\pi$
$434$	−73.7672	−3.54094
$435$	−0.506586	−0.0242890
$436$	4.92029	0.235639
$437$	−20.3852	−0.975158
$438$	2.31132	0.110439
$439$	11.8004	0.563203	0.281602	−	0.959531i	$-0.409134\pi$
$439$	11.8004	0.563203	0.281602	+	0.959531i	$0.409134\pi$
$440$	19.4529	0.927380
$441$	−22.9895	−1.09474
$442$	15.5212	0.738269
$443$	40.1372	1.90697	0.953487	−	0.301435i	$-0.0974656\pi$
$443$	40.1372	1.90697	0.953487	+	0.301435i	$0.0974656\pi$
$444$	−0.698525	−0.0331505
$445$	18.0999	0.858019
$446$	−45.0577	−2.13354
$447$	−3.49396	−0.165259
$448$	46.8559	2.21373
$449$	14.1271	0.666697	0.333349	−	0.942804i	$-0.391821\pi$
$449$	14.1271	0.666697	0.333349	+	0.942804i	$0.391821\pi$
$450$	26.4235	1.24561
$451$	−3.91543	−0.184370
$452$	52.6653	2.47717
$453$	4.06815	0.191138
$454$	−46.2883	−2.17242
$455$	−15.8949	−0.745165
$456$	8.00172	0.374715
$457$	−20.9251	−0.978834	−0.489417	−	0.872050i	$-0.662790\pi$
$457$	−20.9251	−0.978834	−0.489417	+	0.872050i	$0.662790\pi$
$458$	76.0515	3.55365
$459$	3.67079	0.171338
$460$	−43.3280	−2.02018
$461$	−13.9343	−0.648984	−0.324492	−	0.945889i	$-0.605193\pi$
$461$	−13.9343	−0.648984	−0.324492	+	0.945889i	$0.605193\pi$
$462$	7.28556	0.338955
$463$	30.3421	1.41012	0.705060	−	0.709148i	$-0.250920\pi$
$463$	30.3421	1.41012	0.705060	+	0.709148i	$0.250920\pi$
$464$	−12.5017	−0.580375
$465$	−3.14745	−0.145960
$466$	−23.9307	−1.10857
$467$	−27.8085	−1.28682	−0.643412	−	0.765520i	$-0.722482\pi$
$467$	−27.8085	−1.28682	−0.643412	+	0.765520i	$0.722482\pi$
$468$	47.5931	2.19999
$469$	−22.4008	−1.03437
$470$	−9.47083	−0.436857
$471$	−2.43267	−0.112092
$472$	50.8078	2.33862
$473$	0	0
$474$	8.06214	0.370306
$475$	−10.0330	−0.460348
$476$	−33.9754	−1.55726
$477$	−16.0494	−0.734851
$478$	−36.3512	−1.66266
$479$	−2.32175	−0.106084	−0.0530418	−	0.998592i	$-0.516892\pi$
$479$	−2.32175	−0.106084	−0.0530418	+	0.998592i	$0.516892\pi$
$480$	5.55705	0.253643
$481$	1.32216	0.0602852
$482$	−4.81198	−0.219179
$483$	−9.69806	−0.441277
$484$	−34.5401	−1.57001
$485$	−22.5440	−1.02367
$486$	23.8455	1.08165
$487$	−22.8313	−1.03458	−0.517292	−	0.855809i	$-0.673060\pi$
$487$	−22.8313	−1.03458	−0.517292	+	0.855809i	$0.673060\pi$
$488$	98.9039	4.47717
$489$	7.13012	0.322435
$490$	26.0188	1.17541
$491$	31.3422	1.41446	0.707228	−	0.706986i	$-0.249945\pi$
$491$	31.3422	1.41446	0.707228	+	0.706986i	$0.249945\pi$
$492$	−3.42320	−0.154330
$493$	2.04956	0.0923075
$494$	−25.3424	−1.14021
$495$	−7.12994	−0.320467
$496$	−77.6737	−3.48765
$497$	−13.0717	−0.586348
$498$	2.86765	0.128503
$499$	5.24166	0.234649	0.117324	−	0.993094i	$-0.462568\pi$
$499$	5.24166	0.234649	0.117324	+	0.993094i	$0.462568\pi$
$500$	−51.9509	−2.32331
$501$	3.28284	0.146666
$502$	30.2953	1.35214
$503$	−12.7431	−0.568185	−0.284092	−	0.958797i	$-0.591692\pi$
$503$	−12.7431	−0.568185	−0.284092	+	0.958797i	$0.591692\pi$
$504$	−87.3132	−3.88924
$505$	21.5913	0.960802
$506$	−37.5902	−1.67109
$507$	−0.674772	−0.0299677
$508$	38.2519	1.69715
$509$	−27.0581	−1.19933	−0.599664	−	0.800252i	$-0.704699\pi$
$509$	−27.0581	−1.19933	−0.599664	+	0.800252i	$0.704699\pi$
$510$	−2.03292	−0.0900193
$511$	9.57634	0.423632
$512$	−31.7381	−1.40264
$513$	−5.99351	−0.264620
$514$	71.1485	3.13823
$515$	−12.6746	−0.558511
$516$	0	0
$517$	−5.85914	−0.257684
$518$	−4.05865	−0.178327
$519$	8.27879	0.363399
$520$	−32.1913	−1.41168
$521$	33.9186	1.48600	0.743001	−	0.669290i	$-0.233402\pi$
$521$	33.9186	1.48600	0.743001	+	0.669290i	$0.233402\pi$
$522$	8.81331	0.385748
$523$	−7.29596	−0.319030	−0.159515	−	0.987196i	$-0.550993\pi$
$523$	−7.29596	−0.319030	−0.159515	+	0.987196i	$0.550993\pi$
$524$	−50.8592	−2.22179
$525$	−4.77312	−0.208316
$526$	−51.3305	−2.23811
$527$	12.7340	0.554704
$528$	7.67138	0.333854
$529$	27.0376	1.17555
$530$	18.1641	0.789000
$531$	−18.6223	−0.808137
$532$	55.4735	2.40508
$533$	6.47937	0.280653
$534$	13.7289	0.594107
$535$	−14.7825	−0.639104
$536$	−45.3673	−1.95957
$537$	5.01949	0.216607
$538$	14.0979	0.607803
$539$	16.0965	0.693326
$540$	−12.7390	−0.548198
$541$	−4.37476	−0.188086	−0.0940429	−	0.995568i	$-0.529979\pi$
$541$	−4.37476	−0.188086	−0.0940429	+	0.995568i	$0.529979\pi$
$542$	−68.3728	−2.93686
$543$	6.03001	0.258772
$544$	−22.4829	−0.963944
$545$	1.21979	0.0522502
$546$	−12.0564	−0.515965
$547$	30.0494	1.28482	0.642409	−	0.766362i	$-0.277935\pi$
$547$	30.0494	1.28482	0.642409	+	0.766362i	$0.277935\pi$
$548$	−23.2189	−0.991863
$549$	−36.2506	−1.54714
$550$	−18.5009	−0.788879
$551$	−3.34643	−0.142563
$552$	−19.6411	−0.835979
$553$	33.4033	1.42045
$554$	−28.9176	−1.22859
$555$	−0.173172	−0.00735074
$556$	58.1332	2.46540
$557$	−41.4828	−1.75768	−0.878841	−	0.477115i	$-0.841683\pi$
$557$	−41.4828	−1.75768	−0.878841	+	0.477115i	$0.841683\pi$
$558$	54.7576	2.31808
$559$	0	0
$560$	51.3768	2.17107
$561$	−1.25767	−0.0530988
$562$	−49.8513	−2.10285
$563$	10.9508	0.461520	0.230760	−	0.973011i	$-0.425879\pi$
$563$	10.9508	0.461520	0.230760	+	0.973011i	$0.425879\pi$
$564$	−5.12255	−0.215698
$565$	13.0563	0.549284
$566$	−16.9450	−0.712252
$567$	30.5462	1.28282
$568$	−26.4736	−1.11081
$569$	23.0491	0.966267	0.483133	−	0.875547i	$-0.339499\pi$
$569$	23.0491	0.966267	0.483133	+	0.875547i	$0.339499\pi$
$570$	3.31927	0.139029
$571$	24.6585	1.03193	0.515963	−	0.856611i	$-0.327434\pi$
$571$	24.6585	1.03193	0.515963	+	0.856611i	$0.327434\pi$
$572$	−33.3231	−1.39331
$573$	2.79105	0.116598
$574$	−19.8899	−0.830186
$575$	24.6271	1.02702
$576$	−34.7813	−1.44922
$577$	−34.5271	−1.43738	−0.718692	−	0.695329i	$-0.755259\pi$
$577$	−34.5271	−1.43738	−0.718692	+	0.695329i	$0.755259\pi$
$578$	−36.6585	−1.52479
$579$	−0.200453	−0.00833053
$580$	−7.11271	−0.295339
$581$	11.8814	0.492922
$582$	−17.0997	−0.708806
$583$	11.2373	0.465400
$584$	19.3945	0.802551
$585$	11.7989	0.487822
$586$	13.9785	0.577446
$587$	−15.8934	−0.655991	−0.327995	−	0.944679i	$-0.606373\pi$
$587$	−15.8934	−0.655991	−0.327995	+	0.944679i	$0.606373\pi$
$588$	14.0729	0.580358
$589$	−20.7916	−0.856703
$590$	21.0760	0.867687
$591$	−5.27402	−0.216944
$592$	−4.27358	−0.175643
$593$	30.4491	1.25039	0.625197	−	0.780467i	$-0.285019\pi$
$593$	30.4491	1.25039	0.625197	+	0.780467i	$0.285019\pi$
$594$	−11.0520	−0.453468
$595$	−8.42287	−0.345304
$596$	−49.0569	−2.00945
$597$	−5.70199	−0.233367
$598$	62.2055	2.54377
$599$	3.07992	0.125842	0.0629211	−	0.998019i	$-0.479958\pi$
$599$	3.07992	0.125842	0.0629211	+	0.998019i	$0.479958\pi$
$600$	−9.66679	−0.394645
$601$	0.873893	0.0356468	0.0178234	−	0.999841i	$-0.494326\pi$
$601$	0.873893	0.0356468	0.0178234	+	0.999841i	$0.494326\pi$
$602$	0	0
$603$	16.6282	0.677152
$604$	57.1187	2.32413
$605$	−8.56289	−0.348131
$606$	16.3771	0.665276
$607$	38.7768	1.57390	0.786952	−	0.617014i	$-0.211658\pi$
$607$	38.7768	1.57390	0.786952	+	0.617014i	$0.211658\pi$
$608$	36.7090	1.48875
$609$	−1.59203	−0.0645124
$610$	41.0272	1.66114
$611$	9.69588	0.392253
$612$	25.2200	1.01946
$613$	41.1253	1.66103	0.830517	−	0.556993i	$-0.188045\pi$
$613$	41.1253	1.66103	0.830517	+	0.556993i	$0.188045\pi$
$614$	3.49839	0.141184
$615$	−0.848648	−0.0342208
$616$	61.1339	2.46316
$617$	−16.3836	−0.659578	−0.329789	−	0.944055i	$-0.606978\pi$
$617$	−16.3836	−0.659578	−0.329789	+	0.944055i	$0.606978\pi$
$618$	−9.61377	−0.386723
$619$	12.3790	0.497553	0.248776	−	0.968561i	$-0.419972\pi$
$619$	12.3790	0.497553	0.248776	+	0.968561i	$0.419972\pi$
$620$	−44.1917	−1.77478
$621$	14.7117	0.590359
$622$	60.3549	2.42001
$623$	56.8820	2.27893
$624$	−12.6948	−0.508200
$625$	4.52827	0.181131
$626$	22.0614	0.881750
$627$	2.05347	0.0820076
$628$	−34.1559	−1.36297
$629$	0.700624	0.0279357
$630$	−36.2192	−1.44301
$631$	25.8426	1.02878	0.514389	−	0.857557i	$-0.328019\pi$
$631$	25.8426	1.02878	0.514389	+	0.857557i	$0.328019\pi$
$632$	67.6503	2.69098
$633$	3.39891	0.135094
$634$	−7.20074	−0.285978
$635$	9.48308	0.376324
$636$	9.82456	0.389569
$637$	−26.6370	−1.05540
$638$	−6.17080	−0.244304
$639$	9.70320	0.383853
$640$	7.97049	0.315061
$641$	33.3990	1.31918	0.659590	−	0.751625i	$-0.270730\pi$
$641$	33.3990	1.31918	0.659590	+	0.751625i	$0.270730\pi$
$642$	−11.2126	−0.442526
$643$	17.2011	0.678346	0.339173	−	0.940724i	$-0.389853\pi$
$643$	17.2011	0.678346	0.339173	+	0.940724i	$0.389853\pi$
$644$	−136.165	−5.36567
$645$	0	0
$646$	−13.4292	−0.528364
$647$	19.6984	0.774424	0.387212	−	0.921991i	$-0.373438\pi$
$647$	19.6984	0.774424	0.387212	+	0.921991i	$0.373438\pi$
$648$	61.8640	2.43025
$649$	13.0387	0.511814
$650$	30.6158	1.20085
$651$	−9.89139	−0.387674
$652$	100.110	3.92062
$653$	29.4327	1.15179	0.575895	−	0.817524i	$-0.304654\pi$
$653$	29.4327	1.15179	0.575895	+	0.817524i	$0.304654\pi$
$654$	0.925220	0.0361790
$655$	−12.6086	−0.492657
$656$	−20.9432	−0.817693
$657$	−7.10855	−0.277331
$658$	−29.7636	−1.16031
$659$	25.2342	0.982983	0.491492	−	0.870882i	$-0.336452\pi$
$659$	25.2342	0.982983	0.491492	+	0.870882i	$0.336452\pi$
$660$	4.36456	0.169890
$661$	−33.5839	−1.30626	−0.653131	−	0.757245i	$-0.726545\pi$
$661$	−33.5839	−1.30626	−0.653131	+	0.757245i	$0.726545\pi$
$662$	−77.8716	−3.02657
$663$	2.08123	0.0808282
$664$	24.0628	0.933818
$665$	13.7525	0.533299
$666$	3.01275	0.116742
$667$	8.21416	0.318054
$668$	46.0926	1.78338
$669$	−6.04175	−0.233588
$670$	−18.8192	−0.727050
$671$	25.3815	0.979842
$672$	17.4639	0.673686
$673$	10.8275	0.417371	0.208685	−	0.977983i	$-0.433082\pi$
$673$	10.8275	0.417371	0.208685	+	0.977983i	$0.433082\pi$
$674$	−49.9999	−1.92592
$675$	7.24068	0.278694
$676$	−9.47412	−0.364389
$677$	33.6851	1.29462	0.647311	−	0.762226i	$-0.275894\pi$
$677$	33.6851	1.29462	0.647311	+	0.762226i	$0.275894\pi$
$678$	9.90329	0.380333
$679$	−70.8482	−2.71890
$680$	−17.0585	−0.654162
$681$	−6.20677	−0.237844
$682$	−38.3396	−1.46810
$683$	−16.3061	−0.623935	−0.311967	−	0.950093i	$-0.600988\pi$
$683$	−16.3061	−0.623935	−0.311967	+	0.950093i	$0.600988\pi$
$684$	−41.1782	−1.57449
$685$	−5.75623	−0.219934
$686$	10.1966	0.389310
$687$	10.1977	0.389066
$688$	0	0
$689$	−18.5958	−0.708443
$690$	−8.14748	−0.310169
$691$	12.6997	0.483120	0.241560	−	0.970386i	$-0.422341\pi$
$691$	12.6997	0.483120	0.241560	+	0.970386i	$0.422341\pi$
$692$	116.238	4.41871
$693$	−22.4070	−0.851172
$694$	5.73639	0.217750
$695$	14.4119	0.546673
$696$	−3.22427	−0.122216
$697$	3.43348	0.130052
$698$	−1.72570	−0.0653188
$699$	−3.20885	−0.121370
$700$	−67.0168	−2.53300
$701$	−15.5441	−0.587094	−0.293547	−	0.955945i	$-0.594836\pi$
$701$	−15.5441	−0.587094	−0.293547	+	0.955945i	$0.594836\pi$
$702$	18.2892	0.690280
$703$	−1.14395	−0.0431448
$704$	24.3527	0.917829
$705$	−1.26994	−0.0478286
$706$	−19.3789	−0.729335
$707$	67.8543	2.55192
$708$	11.3995	0.428421
$709$	6.94298	0.260749	0.130375	−	0.991465i	$-0.458382\pi$
$709$	6.94298	0.260749	0.130375	+	0.991465i	$0.458382\pi$
$710$	−10.9818	−0.412138
$711$	−24.7954	−0.929901
$712$	115.201	4.31732
$713$	51.0351	1.91128
$714$	−6.38879	−0.239094
$715$	−8.26118	−0.308951
$716$	70.4761	2.63381
$717$	−4.87430	−0.182034
$718$	24.5594	0.916547
$719$	41.9754	1.56542	0.782710	−	0.622387i	$-0.213837\pi$
$719$	41.9754	1.56542	0.782710	+	0.622387i	$0.213837\pi$
$720$	−38.1372	−1.42129
$721$	−39.8321	−1.48343
$722$	−28.2372	−1.05088
$723$	−0.645235	−0.0239965
$724$	84.6642	3.14652
$725$	4.04278	0.150145
$726$	−6.49499	−0.241052
$727$	25.4912	0.945417	0.472708	−	0.881219i	$-0.343276\pi$
$727$	25.4912	0.945417	0.472708	+	0.881219i	$0.343276\pi$
$728$	−101.166	−3.74947
$729$	−20.4657	−0.757990
$730$	8.04521	0.297767
$731$	0	0
$732$	22.1906	0.820190
$733$	5.82076	0.214994	0.107497	−	0.994205i	$-0.465716\pi$
$733$	5.82076	0.214994	0.107497	+	0.994205i	$0.465716\pi$
$734$	22.0496	0.813867
$735$	3.48884	0.128688
$736$	−90.1061	−3.32135
$737$	−11.6425	−0.428858
$738$	14.7643	0.543482
$739$	−18.6097	−0.684568	−0.342284	−	0.939597i	$-0.611200\pi$
$739$	−18.6097	−0.684568	−0.342284	+	0.939597i	$0.611200\pi$
$740$	−2.43142	−0.0893807
$741$	−3.39814	−0.124834
$742$	57.0838	2.09561
$743$	3.49660	0.128278	0.0641390	−	0.997941i	$-0.479570\pi$
$743$	3.49660	0.128278	0.0641390	+	0.997941i	$0.479570\pi$
$744$	−20.0326	−0.734431
$745$	−12.1617	−0.445572
$746$	−96.2531	−3.52408
$747$	−8.81958	−0.322692
$748$	−17.6583	−0.645650
$749$	−46.4565	−1.69748
$750$	−9.76894	−0.356711
$751$	−3.46346	−0.126383	−0.0631917	−	0.998001i	$-0.520128\pi$
$751$	−3.46346	−0.126383	−0.0631917	+	0.998001i	$0.520128\pi$
$752$	−31.3398	−1.14285
$753$	4.06227	0.148037
$754$	10.2116	0.371886
$755$	14.1604	0.515349
$756$	−40.0343	−1.45603
$757$	28.8554	1.04877	0.524384	−	0.851482i	$-0.324296\pi$
$757$	28.8554	1.04877	0.524384	+	0.851482i	$0.324296\pi$
$758$	−19.5953	−0.711735
$759$	−5.04045	−0.182957
$760$	27.8523	1.01031
$761$	25.3496	0.918922	0.459461	−	0.888198i	$-0.348043\pi$
$761$	25.3496	0.918922	0.459461	+	0.888198i	$0.348043\pi$
$762$	7.19296	0.260574
$763$	3.83340	0.138779
$764$	39.1876	1.41776
$765$	6.25233	0.226053
$766$	3.42652	0.123805
$767$	−21.5769	−0.779095
$768$	−2.52114	−0.0909737
$769$	−38.4737	−1.38740	−0.693698	−	0.720266i	$-0.744020\pi$
$769$	−38.4737	−1.38740	−0.693698	+	0.720266i	$0.744020\pi$
$770$	25.3595	0.913893
$771$	9.54025	0.343584
$772$	−2.81445	−0.101294
$773$	34.4168	1.23789	0.618943	−	0.785436i	$-0.287561\pi$
$773$	34.4168	1.23789	0.618943	+	0.785436i	$0.287561\pi$
$774$	0	0
$775$	25.1181	0.902268
$776$	−143.486	−5.15083
$777$	−0.544222	−0.0195238
$778$	26.2165	0.939906
$779$	−5.60604	−0.200857
$780$	−7.22262	−0.258611
$781$	−6.79387	−0.243104
$782$	32.9633	1.17876
$783$	2.41506	0.0863074
$784$	86.0984	3.07494
$785$	−8.46762	−0.302222
$786$	−9.56365	−0.341124
$787$	−14.5634	−0.519130	−0.259565	−	0.965726i	$-0.583579\pi$
$787$	−14.5634	−0.519130	−0.259565	+	0.965726i	$0.583579\pi$
$788$	−74.0498	−2.63791
$789$	−6.88287	−0.245037
$790$	28.0626	0.998423
$791$	41.0316	1.45892
$792$	−45.3799	−1.61251
$793$	−42.0021	−1.49154
$794$	−40.2512	−1.42846
$795$	2.43562	0.0863825
$796$	−80.0586	−2.83760
$797$	−1.81667	−0.0643496	−0.0321748	−	0.999482i	$-0.510243\pi$
$797$	−1.81667	−0.0643496	−0.0321748	+	0.999482i	$0.510243\pi$
$798$	10.4313	0.369265
$799$	5.13794	0.181767
$800$	−44.3477	−1.56793
$801$	−42.2237	−1.49190
$802$	−44.7802	−1.58125
$803$	4.97718	0.175641
$804$	−10.1789	−0.358981
$805$	−33.7569	−1.18978
$806$	63.4455	2.23477
$807$	1.89037	0.0665443
$808$	137.422	4.83450
$809$	32.7007	1.14969	0.574847	−	0.818261i	$-0.305061\pi$
$809$	32.7007	1.14969	0.574847	+	0.818261i	$0.305061\pi$
$810$	25.6623	0.901682
$811$	−19.6087	−0.688556	−0.344278	−	0.938868i	$-0.611876\pi$
$811$	−19.6087	−0.688556	−0.344278	+	0.938868i	$0.611876\pi$
$812$	−22.3529	−0.784432
$813$	−9.16806	−0.321538
$814$	−2.10943	−0.0739355
$815$	24.8185	0.869353
$816$	−6.72712	−0.235496
$817$	0	0
$818$	11.5208	0.402814
$819$	37.0798	1.29567
$820$	−11.9154	−0.416105
$821$	−44.4856	−1.55256	−0.776280	−	0.630389i	$-0.782895\pi$
$821$	−44.4856	−1.55256	−0.776280	+	0.630389i	$0.782895\pi$
$822$	−4.36613	−0.152286
$823$	−13.2025	−0.460211	−0.230105	−	0.973166i	$-0.573907\pi$
$823$	−13.2025	−0.460211	−0.230105	+	0.973166i	$0.573907\pi$
$824$	−80.6702	−2.81028
$825$	−2.48077	−0.0863693
$826$	66.2349	2.30461
$827$	−44.1702	−1.53595	−0.767974	−	0.640481i	$-0.778735\pi$
$827$	−44.1702	−1.53595	−0.767974	+	0.640481i	$0.778735\pi$
$828$	101.076	3.51264
$829$	−41.1538	−1.42933	−0.714665	−	0.699467i	$-0.753421\pi$
$829$	−41.1538	−1.42933	−0.714665	+	0.699467i	$0.753421\pi$
$830$	9.98170	0.346470
$831$	−3.87754	−0.134510
$832$	−40.2997	−1.39714
$833$	−14.1152	−0.489063
$834$	10.9315	0.378526
$835$	11.4269	0.395443
$836$	28.8316	0.997163
$837$	15.0049	0.518647
$838$	50.1038	1.73081
$839$	−22.4874	−0.776350	−0.388175	−	0.921586i	$-0.626894\pi$
$839$	−22.4874	−0.776350	−0.388175	+	0.921586i	$0.626894\pi$
$840$	13.2505	0.457184
$841$	−27.6516	−0.953502
$842$	26.2013	0.902958
$843$	−6.68453	−0.230227
$844$	47.7222	1.64267
$845$	−2.34874	−0.0807991
$846$	22.0937	0.759595
$847$	−26.9103	−0.924648
$848$	60.1068	2.06408
$849$	−2.27214	−0.0779798
$850$	16.2236	0.556465
$851$	2.80794	0.0962549
$852$	−5.93977	−0.203493
$853$	−27.3398	−0.936097	−0.468048	−	0.883703i	$-0.655043\pi$
$853$	−27.3398	−0.936097	−0.468048	+	0.883703i	$0.655043\pi$
$854$	128.935	4.41206
$855$	−10.2085	−0.349124
$856$	−94.0862	−3.21580
$857$	−11.3186	−0.386636	−0.193318	−	0.981136i	$-0.561925\pi$
$857$	−11.3186	−0.386636	−0.193318	+	0.981136i	$0.561925\pi$
$858$	−6.26615	−0.213923
$859$	28.3173	0.966174	0.483087	−	0.875572i	$-0.339515\pi$
$859$	28.3173	0.966174	0.483087	+	0.875572i	$0.339515\pi$
$860$	0	0
$861$	−2.66702	−0.0908917
$862$	−42.9221	−1.46193
$863$	−46.3789	−1.57875	−0.789377	−	0.613908i	$-0.789596\pi$
$863$	−46.3789	−1.57875	−0.789377	+	0.613908i	$0.789596\pi$
$864$	−26.4923	−0.901286
$865$	28.8167	0.979798
$866$	48.0409	1.63250
$867$	−4.91551	−0.166940
$868$	−138.880	−4.71389
$869$	17.3610	0.588930
$870$	−1.33749	−0.0453451
$871$	19.2664	0.652818
$872$	7.76362	0.262909
$873$	52.5909	1.77993
$874$	−53.8210	−1.82052
$875$	−40.4750	−1.36830
$876$	4.35147	0.147022
$877$	−15.1650	−0.512085	−0.256043	−	0.966665i	$-0.582419\pi$
$877$	−15.1650	−0.512085	−0.256043	+	0.966665i	$0.582419\pi$
$878$	31.1554	1.05144
$879$	1.87437	0.0632208
$880$	26.7025	0.900140
$881$	−44.0304	−1.48342	−0.741711	−	0.670720i	$-0.765985\pi$
$881$	−44.0304	−1.48342	−0.741711	+	0.670720i	$0.765985\pi$
$882$	−60.6968	−2.04377
$883$	38.2864	1.28844	0.644220	−	0.764840i	$-0.277182\pi$
$883$	38.2864	1.28844	0.644220	+	0.764840i	$0.277182\pi$
$884$	29.2214	0.982823
$885$	2.82607	0.0949974
$886$	105.970	3.56013
$887$	−15.1269	−0.507911	−0.253956	−	0.967216i	$-0.581732\pi$
$887$	−15.1269	−0.507911	−0.253956	+	0.967216i	$0.581732\pi$
$888$	−1.10219	−0.0369870
$889$	29.8021	0.999531
$890$	47.7874	1.60184
$891$	15.8760	0.531867
$892$	−84.8291	−2.84029
$893$	−8.38901	−0.280727
$894$	−9.22474	−0.308522
$895$	17.4718	0.584018
$896$	25.0486	0.836814
$897$	8.34109	0.278501
$898$	37.2982	1.24466
$899$	8.37791	0.279419
$900$	49.7468	1.65823
$901$	−9.85408	−0.328287
$902$	−10.3375	−0.344201
$903$	0	0
$904$	83.0996	2.76385
$905$	20.9892	0.697705
$906$	10.7407	0.356836
$907$	21.2339	0.705061	0.352531	−	0.935800i	$-0.385321\pi$
$907$	21.2339	0.705061	0.352531	+	0.935800i	$0.385321\pi$
$908$	−87.1460	−2.89204
$909$	−50.3685	−1.67062
$910$	−41.9657	−1.39115
$911$	−55.4368	−1.83670	−0.918351	−	0.395766i	$-0.870479\pi$
$911$	−55.4368	−1.83670	−0.918351	+	0.395766i	$0.870479\pi$
$912$	10.9837	0.363708
$913$	6.17519	0.204369
$914$	−55.2463	−1.82739
$915$	5.50131	0.181868
$916$	143.180	4.73081
$917$	−39.6245	−1.30852
$918$	9.69161	0.319871
$919$	−37.5058	−1.23720	−0.618600	−	0.785706i	$-0.712300\pi$
$919$	−37.5058	−1.23720	−0.618600	+	0.785706i	$0.712300\pi$
$920$	−68.3664	−2.25397
$921$	0.469097	0.0154573
$922$	−36.7892	−1.21159
$923$	11.2427	0.370058
$924$	13.7164	0.451235
$925$	1.38199	0.0454395
$926$	80.1092	2.63255
$927$	29.5675	0.971125
$928$	−14.7918	−0.485564
$929$	7.11367	0.233392	0.116696	−	0.993168i	$-0.462770\pi$
$929$	7.11367	0.233392	0.116696	+	0.993168i	$0.462770\pi$
$930$	−8.30990	−0.272492
$931$	23.0467	0.755326
$932$	−45.0538	−1.47579
$933$	8.09294	0.264951
$934$	−73.4199	−2.40237
$935$	−4.37768	−0.143165
$936$	75.0961	2.45459
$937$	−23.9242	−0.781568	−0.390784	−	0.920482i	$-0.627796\pi$
$937$	−23.9242	−0.781568	−0.390784	+	0.920482i	$0.627796\pi$
$938$	−59.1425	−1.93107
$939$	2.95819	0.0965370
$940$	−17.8305	−0.581567
$941$	31.8161	1.03718	0.518588	−	0.855024i	$-0.326458\pi$
$941$	31.8161	1.03718	0.518588	+	0.855024i	$0.326458\pi$
$942$	−6.42273	−0.209264
$943$	13.7606	0.448107
$944$	69.7425	2.26993
$945$	−9.92495	−0.322859
$946$	0	0
$947$	−58.2996	−1.89448	−0.947241	−	0.320523i	$-0.896141\pi$
$947$	−58.2996	−1.89448	−0.947241	+	0.320523i	$0.896141\pi$
$948$	15.1784	0.492972
$949$	−8.23639	−0.267365
$950$	−26.4892	−0.859424
$951$	−0.965542	−0.0313098
$952$	−53.6090	−1.73748
$953$	−26.6689	−0.863889	−0.431944	−	0.901900i	$-0.642172\pi$
$953$	−26.6689	−0.863889	−0.431944	+	0.901900i	$0.642172\pi$
$954$	−42.3735	−1.37189
$955$	9.71505	0.314372
$956$	−68.4375	−2.21343
$957$	−0.827438	−0.0267473
$958$	−6.12989	−0.198048
$959$	−18.0899	−0.584153
$960$	5.27833	0.170357
$961$	21.0525	0.679113
$962$	3.49075	0.112546
$963$	34.4848	1.11126
$964$	−9.05940	−0.291784
$965$	−0.697733	−0.0224608
$966$	−25.6048	−0.823821
$967$	36.5173	1.17432	0.587158	−	0.809472i	$-0.300247\pi$
$967$	36.5173	1.17432	0.587158	+	0.809472i	$0.300247\pi$
$968$	−54.5002	−1.75170
$969$	−1.80071	−0.0578471
$970$	−59.5205	−1.91109
$971$	−43.2825	−1.38900	−0.694501	−	0.719492i	$-0.744375\pi$
$971$	−43.2825	−1.38900	−0.694501	+	0.719492i	$0.744375\pi$
$972$	44.8934	1.43996
$973$	45.2917	1.45198
$974$	−60.2790	−1.93146
$975$	4.10525	0.131473
$976$	135.763	4.34566
$977$	21.4720	0.686949	0.343474	−	0.939162i	$-0.388396\pi$
$977$	21.4720	0.686949	0.343474	+	0.939162i	$0.388396\pi$
$978$	18.8249	0.601955
$979$	29.5637	0.944860
$980$	48.9849	1.56477
$981$	−2.84555	−0.0908514
$982$	82.7497	2.64065
$983$	−27.5273	−0.877984	−0.438992	−	0.898491i	$-0.644664\pi$
$983$	−27.5273	−0.877984	−0.438992	+	0.898491i	$0.644664\pi$
$984$	−5.40139	−0.172190
$985$	−18.3578	−0.584927
$986$	5.41124	0.172329
$987$	−3.99098	−0.127034
$988$	−47.7115	−1.51791
$989$	0	0
$990$	−18.8244	−0.598280
$991$	−7.84451	−0.249189	−0.124595	−	0.992208i	$-0.539763\pi$
$991$	−7.84451	−0.249189	−0.124595	+	0.992208i	$0.539763\pi$
$992$	−91.9023	−2.91790
$993$	−10.4417	−0.331359
$994$	−34.5120	−1.09465
$995$	−19.8474	−0.629206
$996$	5.39887	0.171070
$997$	49.2281	1.55907	0.779535	−	0.626359i	$-0.215456\pi$
$997$	49.2281	1.55907	0.779535	+	0.626359i	$0.215456\pi$
$998$	13.8390	0.438066
$999$	0.825568	0.0261198

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.2.a.p.1.20	✓	20
43.42	odd	2		1849.2.a.r.1.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.2.a.p.1.20	✓	20	1.1	even	1	trivial
1849.2.a.r.1.1	yes	20	43.42	odd	2

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:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q+2.64020 q^{2} +0.354022 q^{3} +4.97064 q^{4} +1.23228 q^{5} +0.934688 q^{6} +3.87263 q^{7} +7.84307 q^{8} -2.87467 q^{9} +O(q^{10})\)	\(q+2.64020 q^{2} +0.354022 q^{3} +4.97064 q^{4} +1.23228 q^{5} +0.934688 q^{6} +3.87263 q^{7} +7.84307 q^{8} -2.87467 q^{9} +3.25345 q^{10} +2.01275 q^{11} +1.75972 q^{12} -3.33076 q^{13} +10.2245 q^{14} +0.436253 q^{15} +10.7660 q^{16} -1.76500 q^{17} -7.58969 q^{18} +2.88182 q^{19} +6.12520 q^{20} +1.37100 q^{21} +5.31406 q^{22} -7.07373 q^{23} +2.77662 q^{24} -3.48149 q^{25} -8.79387 q^{26} -2.07976 q^{27} +19.2494 q^{28} -1.16122 q^{29} +1.15179 q^{30} -7.21474 q^{31} +12.7381 q^{32} +0.712558 q^{33} -4.65996 q^{34} +4.77215 q^{35} -14.2889 q^{36} -0.396953 q^{37} +7.60858 q^{38} -1.17916 q^{39} +9.66483 q^{40} -1.94531 q^{41} +3.61970 q^{42} +10.0047 q^{44} -3.54239 q^{45} -18.6760 q^{46} -2.91101 q^{47} +3.81139 q^{48} +7.99728 q^{49} -9.19183 q^{50} -0.624850 q^{51} -16.5560 q^{52} +5.58304 q^{53} -5.49098 q^{54} +2.48027 q^{55} +30.3733 q^{56} +1.02023 q^{57} -3.06585 q^{58} +6.47805 q^{59} +2.16846 q^{60} +12.6104 q^{61} -19.0483 q^{62} -11.1325 q^{63} +12.0992 q^{64} -4.10442 q^{65} +1.88129 q^{66} -5.78439 q^{67} -8.77320 q^{68} -2.50426 q^{69} +12.5994 q^{70} -3.37541 q^{71} -22.5462 q^{72} +2.47282 q^{73} -1.04803 q^{74} -1.23253 q^{75} +14.3245 q^{76} +7.79464 q^{77} -3.11322 q^{78} +8.62549 q^{79} +13.2666 q^{80} +7.88772 q^{81} -5.13600 q^{82} +3.06803 q^{83} +6.81473 q^{84} -2.17497 q^{85} -0.411098 q^{87} +15.7861 q^{88} +14.6882 q^{89} -9.35260 q^{90} -12.8988 q^{91} -35.1609 q^{92} -2.55418 q^{93} -7.68564 q^{94} +3.55120 q^{95} +4.50958 q^{96} -18.2946 q^{97} +21.1144 q^{98} -5.78599 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{2} + q^{3} + 23 q^{4} + 3 q^{5} + 5 q^{6} + 2 q^{7} - 9 q^{8} + 27 q^{9}+O(q^{10}) \) 20 * q - 5 * q^2 + q^3 + 23 * q^4 + 3 * q^5 + 5 * q^6 + 2 * q^7 - 9 * q^8 + 27 * q^9	\( 20 q - 5 q^{2} + q^{3} + 23 q^{4} + 3 q^{5} + 5 q^{6} + 2 q^{7} - 9 q^{8} + 27 q^{9} + 21 q^{11} + 12 q^{12} + 11 q^{13} + 8 q^{14} + 4 q^{15} + 29 q^{16} + 15 q^{17} - 9 q^{18} + 7 q^{19} + 17 q^{20} + 8 q^{21} - 47 q^{22} + 26 q^{23} - q^{24} + 11 q^{25} - 58 q^{26} + 22 q^{27} + 28 q^{28} + 12 q^{29} + 24 q^{30} + 26 q^{31} - 3 q^{32} + 17 q^{33} - 54 q^{34} + 26 q^{35} + 14 q^{36} + 19 q^{37} + 5 q^{38} + 7 q^{39} - 16 q^{40} + 27 q^{41} + 52 q^{42} + 32 q^{44} - 75 q^{45} - 59 q^{46} + 45 q^{47} + 66 q^{48} + 20 q^{49} - 75 q^{51} + 11 q^{52} + 3 q^{53} + 57 q^{54} + 2 q^{55} + 87 q^{56} - 24 q^{57} - 46 q^{58} + 66 q^{59} + 29 q^{60} - 30 q^{61} - 72 q^{62} + 21 q^{63} - 7 q^{64} + 6 q^{65} + 41 q^{66} - 6 q^{67} + 28 q^{68} + 35 q^{69} + 80 q^{70} + 31 q^{71} + 15 q^{72} + 26 q^{73} + 87 q^{74} - 73 q^{75} + 68 q^{76} + 21 q^{77} - 50 q^{78} + 39 q^{79} + 60 q^{80} + 16 q^{81} - 45 q^{82} + 13 q^{83} + 31 q^{84} + 22 q^{85} + 61 q^{87} - 50 q^{88} + 4 q^{89} - 13 q^{90} + 25 q^{91} + 5 q^{92} - 67 q^{93} - 42 q^{94} + 79 q^{95} + 36 q^{96} - 2 q^{97} + 35 q^{98} + 13 q^{99}+O(q^{100}) \) 20 * q - 5 * q^2 + q^3 + 23 * q^4 + 3 * q^5 + 5 * q^6 + 2 * q^7 - 9 * q^8 + 27 * q^9 + 21 * q^11 + 12 * q^12 + 11 * q^13 + 8 * q^14 + 4 * q^15 + 29 * q^16 + 15 * q^17 - 9 * q^18 + 7 * q^19 + 17 * q^20 + 8 * q^21 - 47 * q^22 + 26 * q^23 - q^24 + 11 * q^25 - 58 * q^26 + 22 * q^27 + 28 * q^28 + 12 * q^29 + 24 * q^30 + 26 * q^31 - 3 * q^32 + 17 * q^33 - 54 * q^34 + 26 * q^35 + 14 * q^36 + 19 * q^37 + 5 * q^38 + 7 * q^39 - 16 * q^40 + 27 * q^41 + 52 * q^42 + 32 * q^44 - 75 * q^45 - 59 * q^46 + 45 * q^47 + 66 * q^48 + 20 * q^49 - 75 * q^51 + 11 * q^52 + 3 * q^53 + 57 * q^54 + 2 * q^55 + 87 * q^56 - 24 * q^57 - 46 * q^58 + 66 * q^59 + 29 * q^60 - 30 * q^61 - 72 * q^62 + 21 * q^63 - 7 * q^64 + 6 * q^65 + 41 * q^66 - 6 * q^67 + 28 * q^68 + 35 * q^69 + 80 * q^70 + 31 * q^71 + 15 * q^72 + 26 * q^73 + 87 * q^74 - 73 * q^75 + 68 * q^76 + 21 * q^77 - 50 * q^78 + 39 * q^79 + 60 * q^80 + 16 * q^81 - 45 * q^82 + 13 * q^83 + 31 * q^84 + 22 * q^85 + 61 * q^87 - 50 * q^88 + 4 * q^89 - 13 * q^90 + 25 * q^91 + 5 * q^92 - 67 * q^93 - 42 * q^94 + 79 * q^95 + 36 * q^96 - 2 * q^97 + 35 * q^98 + 13 * q^99

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