LMFDB - Embedded newform 538.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [538,2,Mod(1,538)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(538, 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("538.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 538 = 2 \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	538.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:	$4.29595162874$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	538.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} -3.61803 q^{5} +0.618034 q^{6} -4.23607 q^{7} +1.00000 q^{8} -2.61803 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} -3.61803 q^{5} +0.618034 q^{6} -4.23607 q^{7} +1.00000 q^{8} -2.61803 q^{9} -3.61803 q^{10} +0.236068 q^{11} +0.618034 q^{12} -1.00000 q^{13} -4.23607 q^{14} -2.23607 q^{15} +1.00000 q^{16} +0.236068 q^{17} -2.61803 q^{18} +2.00000 q^{19} -3.61803 q^{20} -2.61803 q^{21} +0.236068 q^{22} -8.23607 q^{23} +0.618034 q^{24} +8.09017 q^{25} -1.00000 q^{26} -3.47214 q^{27} -4.23607 q^{28} +0.854102 q^{29} -2.23607 q^{30} +7.47214 q^{31} +1.00000 q^{32} +0.145898 q^{33} +0.236068 q^{34} +15.3262 q^{35} -2.61803 q^{36} -8.70820 q^{37} +2.00000 q^{38} -0.618034 q^{39} -3.61803 q^{40} +7.70820 q^{41} -2.61803 q^{42} +2.23607 q^{43} +0.236068 q^{44} +9.47214 q^{45} -8.23607 q^{46} -6.85410 q^{47} +0.618034 q^{48} +10.9443 q^{49} +8.09017 q^{50} +0.145898 q^{51} -1.00000 q^{52} -3.47214 q^{54} -0.854102 q^{55} -4.23607 q^{56} +1.23607 q^{57} +0.854102 q^{58} -6.56231 q^{59} -2.23607 q^{60} -5.47214 q^{61} +7.47214 q^{62} +11.0902 q^{63} +1.00000 q^{64} +3.61803 q^{65} +0.145898 q^{66} +1.76393 q^{67} +0.236068 q^{68} -5.09017 q^{69} +15.3262 q^{70} +9.70820 q^{71} -2.61803 q^{72} -10.7082 q^{73} -8.70820 q^{74} +5.00000 q^{75} +2.00000 q^{76} -1.00000 q^{77} -0.618034 q^{78} -13.7984 q^{79} -3.61803 q^{80} +5.70820 q^{81} +7.70820 q^{82} -5.23607 q^{83} -2.61803 q^{84} -0.854102 q^{85} +2.23607 q^{86} +0.527864 q^{87} +0.236068 q^{88} -9.79837 q^{89} +9.47214 q^{90} +4.23607 q^{91} -8.23607 q^{92} +4.61803 q^{93} -6.85410 q^{94} -7.23607 q^{95} +0.618034 q^{96} +12.6525 q^{97} +10.9443 q^{98} -0.618034 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 5 q^{5} - q^{6} - 4 q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - q^3 + 2 * q^4 - 5 * q^5 - q^6 - 4 * q^7 + 2 * q^8 - 3 * q^9	$ 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 5 q^{5} - q^{6} - 4 q^{7} + 2 q^{8} - 3 q^{9} - 5 q^{10} - 4 q^{11} - q^{12} - 2 q^{13} - 4 q^{14} + 2 q^{16} - 4 q^{17} - 3 q^{18} + 4 q^{19} - 5 q^{20} - 3 q^{21} - 4 q^{22} - 12 q^{23} - q^{24} + 5 q^{25} - 2 q^{26} + 2 q^{27} - 4 q^{28} - 5 q^{29} + 6 q^{31} + 2 q^{32} + 7 q^{33} - 4 q^{34} + 15 q^{35} - 3 q^{36} - 4 q^{37} + 4 q^{38} + q^{39} - 5 q^{40} + 2 q^{41} - 3 q^{42} - 4 q^{44} + 10 q^{45} - 12 q^{46} - 7 q^{47} - q^{48} + 4 q^{49} + 5 q^{50} + 7 q^{51} - 2 q^{52} + 2 q^{54} + 5 q^{55} - 4 q^{56} - 2 q^{57} - 5 q^{58} + 7 q^{59} - 2 q^{61} + 6 q^{62} + 11 q^{63} + 2 q^{64} + 5 q^{65} + 7 q^{66} + 8 q^{67} - 4 q^{68} + q^{69} + 15 q^{70} + 6 q^{71} - 3 q^{72} - 8 q^{73} - 4 q^{74} + 10 q^{75} + 4 q^{76} - 2 q^{77} + q^{78} - 3 q^{79} - 5 q^{80} - 2 q^{81} + 2 q^{82} - 6 q^{83} - 3 q^{84} + 5 q^{85} + 10 q^{87} - 4 q^{88} + 5 q^{89} + 10 q^{90} + 4 q^{91} - 12 q^{92} + 7 q^{93} - 7 q^{94} - 10 q^{95} - q^{96} - 6 q^{97} + 4 q^{98} + q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - q^3 + 2 * q^4 - 5 * q^5 - q^6 - 4 * q^7 + 2 * q^8 - 3 * q^9 - 5 * q^10 - 4 * q^11 - q^12 - 2 * q^13 - 4 * q^14 + 2 * q^16 - 4 * q^17 - 3 * q^18 + 4 * q^19 - 5 * q^20 - 3 * q^21 - 4 * q^22 - 12 * q^23 - q^24 + 5 * q^25 - 2 * q^26 + 2 * q^27 - 4 * q^28 - 5 * q^29 + 6 * q^31 + 2 * q^32 + 7 * q^33 - 4 * q^34 + 15 * q^35 - 3 * q^36 - 4 * q^37 + 4 * q^38 + q^39 - 5 * q^40 + 2 * q^41 - 3 * q^42 - 4 * q^44 + 10 * q^45 - 12 * q^46 - 7 * q^47 - q^48 + 4 * q^49 + 5 * q^50 + 7 * q^51 - 2 * q^52 + 2 * q^54 + 5 * q^55 - 4 * q^56 - 2 * q^57 - 5 * q^58 + 7 * q^59 - 2 * q^61 + 6 * q^62 + 11 * q^63 + 2 * q^64 + 5 * q^65 + 7 * q^66 + 8 * q^67 - 4 * q^68 + q^69 + 15 * q^70 + 6 * q^71 - 3 * q^72 - 8 * q^73 - 4 * q^74 + 10 * q^75 + 4 * q^76 - 2 * q^77 + q^78 - 3 * q^79 - 5 * q^80 - 2 * q^81 + 2 * q^82 - 6 * q^83 - 3 * q^84 + 5 * q^85 + 10 * q^87 - 4 * q^88 + 5 * q^89 + 10 * q^90 + 4 * q^91 - 12 * q^92 + 7 * q^93 - 7 * q^94 - 10 * q^95 - q^96 - 6 * q^97 + 4 * q^98 + 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.618034	0.356822	0.178411	−	0.983956i	$-0.442904\pi$
$3$	0.618034	0.356822	0.178411	+	0.983956i	$0.442904\pi$
$4$	1.00000	0.500000
$5$	−3.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$5$	−3.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$6$	0.618034	0.252311
$7$	−4.23607	−1.60108	−0.800542	−	0.599277i	$-0.795455\pi$
$7$	−4.23607	−1.60108	−0.800542	+	0.599277i	$0.795455\pi$
$8$	1.00000	0.353553
$9$	−2.61803	−0.872678
$10$	−3.61803	−1.14412
$11$	0.236068	0.0711772	0.0355886	−	0.999367i	$-0.488669\pi$
$11$	0.236068	0.0711772	0.0355886	+	0.999367i	$0.488669\pi$
$12$	0.618034	0.178411
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	−4.23607	−1.13214
$15$	−2.23607	−0.577350
$16$	1.00000	0.250000
$17$	0.236068	0.0572549	0.0286274	−	0.999590i	$-0.490886\pi$
$17$	0.236068	0.0572549	0.0286274	+	0.999590i	$0.490886\pi$
$18$	−2.61803	−0.617077
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−3.61803	−0.809017
$21$	−2.61803	−0.571302
$22$	0.236068	0.0503299
$23$	−8.23607	−1.71734	−0.858669	−	0.512530i	$-0.828708\pi$
$23$	−8.23607	−1.71734	−0.858669	+	0.512530i	$0.828708\pi$
$24$	0.618034	0.126156
$25$	8.09017	1.61803
$26$	−1.00000	−0.196116
$27$	−3.47214	−0.668213
$28$	−4.23607	−0.800542
$29$	0.854102	0.158603	0.0793014	−	0.996851i	$-0.474731\pi$
$29$	0.854102	0.158603	0.0793014	+	0.996851i	$0.474731\pi$
$30$	−2.23607	−0.408248
$31$	7.47214	1.34204	0.671018	−	0.741441i	$-0.265857\pi$
$31$	7.47214	1.34204	0.671018	+	0.741441i	$0.265857\pi$
$32$	1.00000	0.176777
$33$	0.145898	0.0253976
$34$	0.236068	0.0404853
$35$	15.3262	2.59061
$36$	−2.61803	−0.436339
$37$	−8.70820	−1.43162	−0.715810	−	0.698295i	$-0.753942\pi$
$37$	−8.70820	−1.43162	−0.715810	+	0.698295i	$0.753942\pi$
$38$	2.00000	0.324443
$39$	−0.618034	−0.0989646
$40$	−3.61803	−0.572061
$41$	7.70820	1.20382	0.601910	−	0.798564i	$-0.294407\pi$
$41$	7.70820	1.20382	0.601910	+	0.798564i	$0.294407\pi$
$42$	−2.61803	−0.403971
$43$	2.23607	0.340997	0.170499	−	0.985358i	$-0.445462\pi$
$43$	2.23607	0.340997	0.170499	+	0.985358i	$0.445462\pi$
$44$	0.236068	0.0355886
$45$	9.47214	1.41202
$46$	−8.23607	−1.21434
$47$	−6.85410	−0.999774	−0.499887	−	0.866091i	$-0.666625\pi$
$47$	−6.85410	−0.999774	−0.499887	+	0.866091i	$0.666625\pi$
$48$	0.618034	0.0892055
$49$	10.9443	1.56347
$50$	8.09017	1.14412
$51$	0.145898	0.0204298
$52$	−1.00000	−0.138675
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	−3.47214	−0.472498
$55$	−0.854102	−0.115167
$56$	−4.23607	−0.566068
$57$	1.23607	0.163721
$58$	0.854102	0.112149
$59$	−6.56231	−0.854339	−0.427170	−	0.904171i	$-0.640489\pi$
$59$	−6.56231	−0.854339	−0.427170	+	0.904171i	$0.640489\pi$
$60$	−2.23607	−0.288675
$61$	−5.47214	−0.700635	−0.350318	−	0.936631i	$-0.613926\pi$
$61$	−5.47214	−0.700635	−0.350318	+	0.936631i	$0.613926\pi$
$62$	7.47214	0.948962
$63$	11.0902	1.39723
$64$	1.00000	0.125000
$65$	3.61803	0.448762
$66$	0.145898	0.0179588
$67$	1.76393	0.215499	0.107749	−	0.994178i	$-0.465636\pi$
$67$	1.76393	0.215499	0.107749	+	0.994178i	$0.465636\pi$
$68$	0.236068	0.0286274
$69$	−5.09017	−0.612784
$70$	15.3262	1.83184
$71$	9.70820	1.15215	0.576076	−	0.817396i	$-0.304583\pi$
$71$	9.70820	1.15215	0.576076	+	0.817396i	$0.304583\pi$
$72$	−2.61803	−0.308538
$73$	−10.7082	−1.25330	−0.626650	−	0.779301i	$-0.715575\pi$
$73$	−10.7082	−1.25330	−0.626650	+	0.779301i	$0.715575\pi$
$74$	−8.70820	−1.01231
$75$	5.00000	0.577350
$76$	2.00000	0.229416
$77$	−1.00000	−0.113961
$78$	−0.618034	−0.0699786
$79$	−13.7984	−1.55244	−0.776219	−	0.630463i	$-0.782865\pi$
$79$	−13.7984	−1.55244	−0.776219	+	0.630463i	$0.782865\pi$
$80$	−3.61803	−0.404508
$81$	5.70820	0.634245
$82$	7.70820	0.851229
$83$	−5.23607	−0.574733	−0.287367	−	0.957821i	$-0.592780\pi$
$83$	−5.23607	−0.574733	−0.287367	+	0.957821i	$0.592780\pi$
$84$	−2.61803	−0.285651
$85$	−0.854102	−0.0926404
$86$	2.23607	0.241121
$87$	0.527864	0.0565930
$88$	0.236068	0.0251649
$89$	−9.79837	−1.03863	−0.519313	−	0.854584i	$-0.673812\pi$
$89$	−9.79837	−1.03863	−0.519313	+	0.854584i	$0.673812\pi$
$90$	9.47214	0.998451
$91$	4.23607	0.444061
$92$	−8.23607	−0.858669
$93$	4.61803	0.478868
$94$	−6.85410	−0.706947
$95$	−7.23607	−0.742405
$96$	0.618034	0.0630778
$97$	12.6525	1.28466	0.642332	−	0.766426i	$-0.277967\pi$
$97$	12.6525	1.28466	0.642332	+	0.766426i	$0.277967\pi$
$98$	10.9443	1.10554
$99$	−0.618034	−0.0621148
$100$	8.09017	0.809017
$101$	−5.09017	−0.506491	−0.253245	−	0.967402i	$-0.581498\pi$
$101$	−5.09017	−0.506491	−0.253245	+	0.967402i	$0.581498\pi$
$102$	0.145898	0.0144461
$103$	10.8541	1.06949	0.534743	−	0.845015i	$-0.320408\pi$
$103$	10.8541	1.06949	0.534743	+	0.845015i	$0.320408\pi$
$104$	−1.00000	−0.0980581
$105$	9.47214	0.924386
$106$	0	0
$107$	−12.2361	−1.18291	−0.591453	−	0.806340i	$-0.701445\pi$
$107$	−12.2361	−1.18291	−0.591453	+	0.806340i	$0.701445\pi$
$108$	−3.47214	−0.334106
$109$	−4.70820	−0.450964	−0.225482	−	0.974247i	$-0.572396\pi$
$109$	−4.70820	−0.450964	−0.225482	+	0.974247i	$0.572396\pi$
$110$	−0.854102	−0.0814354
$111$	−5.38197	−0.510834
$112$	−4.23607	−0.400271
$113$	−15.3820	−1.44701	−0.723507	−	0.690317i	$-0.757471\pi$
$113$	−15.3820	−1.44701	−0.723507	+	0.690317i	$0.757471\pi$
$114$	1.23607	0.115768
$115$	29.7984	2.77871
$116$	0.854102	0.0793014
$117$	2.61803	0.242037
$118$	−6.56231	−0.604109
$119$	−1.00000	−0.0916698
$120$	−2.23607	−0.204124
$121$	−10.9443	−0.994934
$122$	−5.47214	−0.495424
$123$	4.76393	0.429549
$124$	7.47214	0.671018
$125$	−11.1803	−1.00000
$126$	11.0902	0.987991
$127$	20.7984	1.84556	0.922779	−	0.385331i	$-0.125913\pi$
$127$	20.7984	1.84556	0.922779	+	0.385331i	$0.125913\pi$
$128$	1.00000	0.0883883
$129$	1.38197	0.121675
$130$	3.61803	0.317323
$131$	2.76393	0.241486	0.120743	−	0.992684i	$-0.461472\pi$
$131$	2.76393	0.241486	0.120743	+	0.992684i	$0.461472\pi$
$132$	0.145898	0.0126988
$133$	−8.47214	−0.734627
$134$	1.76393	0.152381
$135$	12.5623	1.08119
$136$	0.236068	0.0202427
$137$	−15.6180	−1.33434	−0.667169	−	0.744906i	$-0.732494\pi$
$137$	−15.6180	−1.33434	−0.667169	+	0.744906i	$0.732494\pi$
$138$	−5.09017	−0.433304
$139$	10.5623	0.895883	0.447942	−	0.894063i	$-0.352157\pi$
$139$	10.5623	0.895883	0.447942	+	0.894063i	$0.352157\pi$
$140$	15.3262	1.29530
$141$	−4.23607	−0.356741
$142$	9.70820	0.814694
$143$	−0.236068	−0.0197410
$144$	−2.61803	−0.218169
$145$	−3.09017	−0.256625
$146$	−10.7082	−0.886217
$147$	6.76393	0.557880
$148$	−8.70820	−0.715810
$149$	0.326238	0.0267265	0.0133632	−	0.999911i	$-0.495746\pi$
$149$	0.326238	0.0267265	0.0133632	+	0.999911i	$0.495746\pi$
$150$	5.00000	0.408248
$151$	−9.94427	−0.809253	−0.404627	−	0.914482i	$-0.632599\pi$
$151$	−9.94427	−0.809253	−0.404627	+	0.914482i	$0.632599\pi$
$152$	2.00000	0.162221
$153$	−0.618034	−0.0499651
$154$	−1.00000	−0.0805823
$155$	−27.0344	−2.17146
$156$	−0.618034	−0.0494823
$157$	22.6525	1.80786	0.903932	−	0.427676i	$-0.140668\pi$
$157$	22.6525	1.80786	0.903932	+	0.427676i	$0.140668\pi$
$158$	−13.7984	−1.09774
$159$	0	0
$160$	−3.61803	−0.286031
$161$	34.8885	2.74960
$162$	5.70820	0.448479
$163$	17.3262	1.35710	0.678548	−	0.734556i	$-0.262610\pi$
$163$	17.3262	1.35710	0.678548	+	0.734556i	$0.262610\pi$
$164$	7.70820	0.601910
$165$	−0.527864	−0.0410942
$166$	−5.23607	−0.406398
$167$	3.61803	0.279972	0.139986	−	0.990153i	$-0.455294\pi$
$167$	3.61803	0.279972	0.139986	+	0.990153i	$0.455294\pi$
$168$	−2.61803	−0.201986
$169$	−12.0000	−0.923077
$170$	−0.854102	−0.0655066
$171$	−5.23607	−0.400412
$172$	2.23607	0.170499
$173$	−7.47214	−0.568096	−0.284048	−	0.958810i	$-0.591678\pi$
$173$	−7.47214	−0.568096	−0.284048	+	0.958810i	$0.591678\pi$
$174$	0.527864	0.0400173
$175$	−34.2705	−2.59061
$176$	0.236068	0.0177943
$177$	−4.05573	−0.304847
$178$	−9.79837	−0.734419
$179$	18.7082	1.39832	0.699158	−	0.714967i	$-0.253558\pi$
$179$	18.7082	1.39832	0.699158	+	0.714967i	$0.253558\pi$
$180$	9.47214	0.706011
$181$	19.4164	1.44321	0.721605	−	0.692305i	$-0.243405\pi$
$181$	19.4164	1.44321	0.721605	+	0.692305i	$0.243405\pi$
$182$	4.23607	0.313998
$183$	−3.38197	−0.250002
$184$	−8.23607	−0.607171
$185$	31.5066	2.31641
$186$	4.61803	0.338611
$187$	0.0557281	0.00407524
$188$	−6.85410	−0.499887
$189$	14.7082	1.06986
$190$	−7.23607	−0.524960
$191$	−5.05573	−0.365820	−0.182910	−	0.983130i	$-0.558552\pi$
$191$	−5.05573	−0.365820	−0.182910	+	0.983130i	$0.558552\pi$
$192$	0.618034	0.0446028
$193$	−8.23607	−0.592845	−0.296423	−	0.955057i	$-0.595794\pi$
$193$	−8.23607	−0.592845	−0.296423	+	0.955057i	$0.595794\pi$
$194$	12.6525	0.908395
$195$	2.23607	0.160128
$196$	10.9443	0.781734
$197$	−22.3820	−1.59465	−0.797325	−	0.603551i	$-0.793752\pi$
$197$	−22.3820	−1.59465	−0.797325	+	0.603551i	$0.793752\pi$
$198$	−0.618034	−0.0439218
$199$	−5.38197	−0.381517	−0.190759	−	0.981637i	$-0.561095\pi$
$199$	−5.38197	−0.381517	−0.190759	+	0.981637i	$0.561095\pi$
$200$	8.09017	0.572061
$201$	1.09017	0.0768947
$202$	−5.09017	−0.358143
$203$	−3.61803	−0.253936
$204$	0.145898	0.0102149
$205$	−27.8885	−1.94782
$206$	10.8541	0.756241
$207$	21.5623	1.49868
$208$	−1.00000	−0.0693375
$209$	0.472136	0.0326583
$210$	9.47214	0.653639
$211$	0.854102	0.0587988	0.0293994	−	0.999568i	$-0.490641\pi$
$211$	0.854102	0.0587988	0.0293994	+	0.999568i	$0.490641\pi$
$212$	0	0
$213$	6.00000	0.411113
$214$	−12.2361	−0.836440
$215$	−8.09017	−0.551745
$216$	−3.47214	−0.236249
$217$	−31.6525	−2.14871
$218$	−4.70820	−0.318880
$219$	−6.61803	−0.447205
$220$	−0.854102	−0.0575835
$221$	−0.236068	−0.0158797
$222$	−5.38197	−0.361214
$223$	−5.32624	−0.356671	−0.178336	−	0.983970i	$-0.557071\pi$
$223$	−5.32624	−0.356671	−0.178336	+	0.983970i	$0.557071\pi$
$224$	−4.23607	−0.283034
$225$	−21.1803	−1.41202
$226$	−15.3820	−1.02319
$227$	−24.0344	−1.59522	−0.797611	−	0.603172i	$-0.793903\pi$
$227$	−24.0344	−1.59522	−0.797611	+	0.603172i	$0.793903\pi$
$228$	1.23607	0.0818606
$229$	−10.7082	−0.707618	−0.353809	−	0.935318i	$-0.615114\pi$
$229$	−10.7082	−0.707618	−0.353809	+	0.935318i	$0.615114\pi$
$230$	29.7984	1.96485
$231$	−0.618034	−0.0406637
$232$	0.854102	0.0560745
$233$	4.23607	0.277514	0.138757	−	0.990326i	$-0.455689\pi$
$233$	4.23607	0.277514	0.138757	+	0.990326i	$0.455689\pi$
$234$	2.61803	0.171146
$235$	24.7984	1.61767
$236$	−6.56231	−0.427170
$237$	−8.52786	−0.553944
$238$	−1.00000	−0.0648204
$239$	−19.2705	−1.24651	−0.623253	−	0.782020i	$-0.714189\pi$
$239$	−19.2705	−1.24651	−0.623253	+	0.782020i	$0.714189\pi$
$240$	−2.23607	−0.144338
$241$	−4.70820	−0.303282	−0.151641	−	0.988436i	$-0.548456\pi$
$241$	−4.70820	−0.303282	−0.151641	+	0.988436i	$0.548456\pi$
$242$	−10.9443	−0.703524
$243$	13.9443	0.894525
$244$	−5.47214	−0.350318
$245$	−39.5967	−2.52974
$246$	4.76393	0.303737
$247$	−2.00000	−0.127257
$248$	7.47214	0.474481
$249$	−3.23607	−0.205077
$250$	−11.1803	−0.707107
$251$	24.0902	1.52056	0.760279	−	0.649597i	$-0.225062\pi$
$251$	24.0902	1.52056	0.760279	+	0.649597i	$0.225062\pi$
$252$	11.0902	0.698615
$253$	−1.94427	−0.122235
$254$	20.7984	1.30501
$255$	−0.527864	−0.0330561
$256$	1.00000	0.0625000
$257$	−5.47214	−0.341342	−0.170671	−	0.985328i	$-0.554594\pi$
$257$	−5.47214	−0.341342	−0.170671	+	0.985328i	$0.554594\pi$
$258$	1.38197	0.0860374
$259$	36.8885	2.29214
$260$	3.61803	0.224381
$261$	−2.23607	−0.138409
$262$	2.76393	0.170756
$263$	23.2705	1.43492	0.717461	−	0.696599i	$-0.245304\pi$
$263$	23.2705	1.43492	0.717461	+	0.696599i	$0.245304\pi$
$264$	0.145898	0.00897940
$265$	0	0
$266$	−8.47214	−0.519460
$267$	−6.05573	−0.370605
$268$	1.76393	0.107749
$269$	1.00000	0.0609711
$269$	1.00000	0.0609711
$270$	12.5623	0.764518
$271$	24.1246	1.46547	0.732733	−	0.680516i	$-0.238244\pi$
$271$	24.1246	1.46547	0.732733	+	0.680516i	$0.238244\pi$
$272$	0.236068	0.0143137
$273$	2.61803	0.158451
$274$	−15.6180	−0.943520
$275$	1.90983	0.115167
$276$	−5.09017	−0.306392
$277$	−21.3820	−1.28472	−0.642359	−	0.766404i	$-0.722044\pi$
$277$	−21.3820	−1.28472	−0.642359	+	0.766404i	$0.722044\pi$
$278$	10.5623	0.633485
$279$	−19.5623	−1.17116
$280$	15.3262	0.915918
$281$	−29.1803	−1.74075	−0.870377	−	0.492387i	$-0.836125\pi$
$281$	−29.1803	−1.74075	−0.870377	+	0.492387i	$0.836125\pi$
$282$	−4.23607	−0.252254
$283$	5.76393	0.342630	0.171315	−	0.985216i	$-0.445198\pi$
$283$	5.76393	0.342630	0.171315	+	0.985216i	$0.445198\pi$
$284$	9.70820	0.576076
$285$	−4.47214	−0.264906
$286$	−0.236068	−0.0139590
$287$	−32.6525	−1.92741
$288$	−2.61803	−0.154269
$289$	−16.9443	−0.996722
$290$	−3.09017	−0.181461
$291$	7.81966	0.458397
$292$	−10.7082	−0.626650
$293$	7.52786	0.439783	0.219891	−	0.975524i	$-0.429430\pi$
$293$	7.52786	0.439783	0.219891	+	0.975524i	$0.429430\pi$
$294$	6.76393	0.394481
$295$	23.7426	1.38235
$296$	−8.70820	−0.506154
$297$	−0.819660	−0.0475615
$298$	0.326238	0.0188985
$299$	8.23607	0.476304
$300$	5.00000	0.288675
$301$	−9.47214	−0.545965
$302$	−9.94427	−0.572229
$303$	−3.14590	−0.180727
$304$	2.00000	0.114708
$305$	19.7984	1.13365
$306$	−0.618034	−0.0353307
$307$	32.1246	1.83345	0.916724	−	0.399521i	$-0.130823\pi$
$307$	32.1246	1.83345	0.916724	+	0.399521i	$0.130823\pi$
$308$	−1.00000	−0.0569803
$309$	6.70820	0.381616
$310$	−27.0344	−1.53545
$311$	−17.2361	−0.977368	−0.488684	−	0.872461i	$-0.662523\pi$
$311$	−17.2361	−0.977368	−0.488684	+	0.872461i	$0.662523\pi$
$312$	−0.618034	−0.0349893
$313$	13.1459	0.743050	0.371525	−	0.928423i	$-0.378835\pi$
$313$	13.1459	0.743050	0.371525	+	0.928423i	$0.378835\pi$
$314$	22.6525	1.27835
$315$	−40.1246	−2.26077
$316$	−13.7984	−0.776219
$317$	−30.2361	−1.69823	−0.849113	−	0.528211i	$-0.822863\pi$
$317$	−30.2361	−1.69823	−0.849113	+	0.528211i	$0.822863\pi$
$318$	0	0
$319$	0.201626	0.0112889
$320$	−3.61803	−0.202254
$321$	−7.56231	−0.422087
$322$	34.8885	1.94426
$323$	0.472136	0.0262703
$324$	5.70820	0.317122
$325$	−8.09017	−0.448762
$326$	17.3262	0.959612
$327$	−2.90983	−0.160914
$328$	7.70820	0.425614
$329$	29.0344	1.60072
$330$	−0.527864	−0.0290580
$331$	0.819660	0.0450526	0.0225263	−	0.999746i	$-0.492829\pi$
$331$	0.819660	0.0450526	0.0225263	+	0.999746i	$0.492829\pi$
$332$	−5.23607	−0.287367
$333$	22.7984	1.24934
$334$	3.61803	0.197970
$335$	−6.38197	−0.348684
$336$	−2.61803	−0.142825
$337$	−27.9787	−1.52410	−0.762049	−	0.647520i	$-0.775806\pi$
$337$	−27.9787	−1.52410	−0.762049	+	0.647520i	$0.775806\pi$
$338$	−12.0000	−0.652714
$339$	−9.50658	−0.516326
$340$	−0.854102	−0.0463202
$341$	1.76393	0.0955223
$342$	−5.23607	−0.283134
$343$	−16.7082	−0.902158
$344$	2.23607	0.120561
$345$	18.4164	0.991506
$346$	−7.47214	−0.401705
$347$	13.3607	0.717239	0.358619	−	0.933484i	$-0.383248\pi$
$347$	13.3607	0.717239	0.358619	+	0.933484i	$0.383248\pi$
$348$	0.527864	0.0282965
$349$	21.2705	1.13858	0.569292	−	0.822135i	$-0.307217\pi$
$349$	21.2705	1.13858	0.569292	+	0.822135i	$0.307217\pi$
$350$	−34.2705	−1.83184
$351$	3.47214	0.185329
$352$	0.236068	0.0125825
$353$	−9.85410	−0.524481	−0.262240	−	0.965003i	$-0.584461\pi$
$353$	−9.85410	−0.524481	−0.262240	+	0.965003i	$0.584461\pi$
$354$	−4.05573	−0.215560
$355$	−35.1246	−1.86422
$356$	−9.79837	−0.519313
$357$	−0.618034	−0.0327098
$358$	18.7082	0.988759
$359$	−16.9443	−0.894284	−0.447142	−	0.894463i	$-0.647558\pi$
$359$	−16.9443	−0.894284	−0.447142	+	0.894463i	$0.647558\pi$
$360$	9.47214	0.499225
$361$	−15.0000	−0.789474
$362$	19.4164	1.02050
$363$	−6.76393	−0.355014
$364$	4.23607	0.222030
$365$	38.7426	2.02788
$366$	−3.38197	−0.176778
$367$	−9.20163	−0.480321	−0.240160	−	0.970733i	$-0.577200\pi$
$367$	−9.20163	−0.480321	−0.240160	+	0.970733i	$0.577200\pi$
$368$	−8.23607	−0.429335
$369$	−20.1803	−1.05055
$370$	31.5066	1.63795
$371$	0	0
$372$	4.61803	0.239434
$373$	6.94427	0.359561	0.179780	−	0.983707i	$-0.442461\pi$
$373$	6.94427	0.359561	0.179780	+	0.983707i	$0.442461\pi$
$374$	0.0557281	0.00288163
$375$	−6.90983	−0.356822
$376$	−6.85410	−0.353473
$377$	−0.854102	−0.0439885
$378$	14.7082	0.756508
$379$	6.14590	0.315694	0.157847	−	0.987464i	$-0.449545\pi$
$379$	6.14590	0.315694	0.157847	+	0.987464i	$0.449545\pi$
$380$	−7.23607	−0.371202
$381$	12.8541	0.658536
$382$	−5.05573	−0.258674
$383$	−17.3820	−0.888177	−0.444088	−	0.895983i	$-0.646472\pi$
$383$	−17.3820	−0.888177	−0.444088	+	0.895983i	$0.646472\pi$
$384$	0.618034	0.0315389
$385$	3.61803	0.184392
$386$	−8.23607	−0.419205
$387$	−5.85410	−0.297581
$388$	12.6525	0.642332
$389$	28.9443	1.46753	0.733766	−	0.679402i	$-0.237761\pi$
$389$	28.9443	1.46753	0.733766	+	0.679402i	$0.237761\pi$
$390$	2.23607	0.113228
$391$	−1.94427	−0.0983261
$392$	10.9443	0.552769
$393$	1.70820	0.0861675
$394$	−22.3820	−1.12759
$395$	49.9230	2.51190
$396$	−0.618034	−0.0310574
$397$	−34.2705	−1.71999	−0.859994	−	0.510304i	$-0.829533\pi$
$397$	−34.2705	−1.71999	−0.859994	+	0.510304i	$0.829533\pi$
$398$	−5.38197	−0.269774
$399$	−5.23607	−0.262131
$400$	8.09017	0.404508
$401$	−4.27051	−0.213259	−0.106630	−	0.994299i	$-0.534006\pi$
$401$	−4.27051	−0.213259	−0.106630	+	0.994299i	$0.534006\pi$
$402$	1.09017	0.0543727
$403$	−7.47214	−0.372214
$404$	−5.09017	−0.253245
$405$	−20.6525	−1.02623
$406$	−3.61803	−0.179560
$407$	−2.05573	−0.101899
$408$	0.145898	0.00722303
$409$	−11.6525	−0.576178	−0.288089	−	0.957604i	$-0.593020\pi$
$409$	−11.6525	−0.576178	−0.288089	+	0.957604i	$0.593020\pi$
$410$	−27.8885	−1.37732
$411$	−9.65248	−0.476122
$412$	10.8541	0.534743
$413$	27.7984	1.36787
$414$	21.5623	1.05973
$415$	18.9443	0.929938
$416$	−1.00000	−0.0490290
$417$	6.52786	0.319671
$418$	0.472136	0.0230929
$419$	−17.4377	−0.851887	−0.425944	−	0.904750i	$-0.640058\pi$
$419$	−17.4377	−0.851887	−0.425944	+	0.904750i	$0.640058\pi$
$420$	9.47214	0.462193
$421$	11.4721	0.559118	0.279559	−	0.960129i	$-0.409812\pi$
$421$	11.4721	0.559118	0.279559	+	0.960129i	$0.409812\pi$
$422$	0.854102	0.0415770
$423$	17.9443	0.872480
$424$	0	0
$425$	1.90983	0.0926404
$426$	6.00000	0.290701
$427$	23.1803	1.12178
$428$	−12.2361	−0.591453
$429$	−0.145898	−0.00704402
$430$	−8.09017	−0.390143
$431$	−15.4721	−0.745267	−0.372633	−	0.927979i	$-0.621545\pi$
$431$	−15.4721	−0.745267	−0.372633	+	0.927979i	$0.621545\pi$
$432$	−3.47214	−0.167053
$433$	12.2016	0.586373	0.293186	−	0.956055i	$-0.405284\pi$
$433$	12.2016	0.586373	0.293186	+	0.956055i	$0.405284\pi$
$434$	−31.6525	−1.51937
$435$	−1.90983	−0.0915693
$436$	−4.70820	−0.225482
$437$	−16.4721	−0.787969
$438$	−6.61803	−0.316222
$439$	19.1459	0.913784	0.456892	−	0.889522i	$-0.348963\pi$
$439$	19.1459	0.913784	0.456892	+	0.889522i	$0.348963\pi$
$440$	−0.854102	−0.0407177
$441$	−28.6525	−1.36440
$442$	−0.236068	−0.0112286
$443$	33.4164	1.58766	0.793831	−	0.608139i	$-0.208084\pi$
$443$	33.4164	1.58766	0.793831	+	0.608139i	$0.208084\pi$
$444$	−5.38197	−0.255417
$445$	35.4508	1.68053
$446$	−5.32624	−0.252205
$447$	0.201626	0.00953659
$448$	−4.23607	−0.200135
$449$	2.90983	0.137323	0.0686617	−	0.997640i	$-0.478127\pi$
$449$	2.90983	0.137323	0.0686617	+	0.997640i	$0.478127\pi$
$450$	−21.1803	−0.998451
$451$	1.81966	0.0856844
$452$	−15.3820	−0.723507
$453$	−6.14590	−0.288759
$454$	−24.0344	−1.12799
$455$	−15.3262	−0.718505
$456$	1.23607	0.0578842
$457$	26.7984	1.25358	0.626788	−	0.779190i	$-0.284369\pi$
$457$	26.7984	1.25358	0.626788	+	0.779190i	$0.284369\pi$
$458$	−10.7082	−0.500362
$459$	−0.819660	−0.0382585
$460$	29.7984	1.38936
$461$	21.1246	0.983871	0.491936	−	0.870632i	$-0.336290\pi$
$461$	21.1246	0.983871	0.491936	+	0.870632i	$0.336290\pi$
$462$	−0.618034	−0.0287535
$463$	−20.0902	−0.933669	−0.466835	−	0.884345i	$-0.654606\pi$
$463$	−20.0902	−0.933669	−0.466835	+	0.884345i	$0.654606\pi$
$464$	0.854102	0.0396507
$465$	−16.7082	−0.774824
$466$	4.23607	0.196232
$467$	−6.58359	−0.304652	−0.152326	−	0.988330i	$-0.548676\pi$
$467$	−6.58359	−0.304652	−0.152326	+	0.988330i	$0.548676\pi$
$468$	2.61803	0.121019
$469$	−7.47214	−0.345031
$470$	24.7984	1.14386
$471$	14.0000	0.645086
$472$	−6.56231	−0.302055
$473$	0.527864	0.0242712
$474$	−8.52786	−0.391698
$475$	16.1803	0.742405
$476$	−1.00000	−0.0458349
$477$	0	0
$478$	−19.2705	−0.881413
$479$	−36.5967	−1.67215	−0.836074	−	0.548617i	$-0.815155\pi$
$479$	−36.5967	−1.67215	−0.836074	+	0.548617i	$0.815155\pi$
$480$	−2.23607	−0.102062
$481$	8.70820	0.397060
$482$	−4.70820	−0.214453
$483$	21.5623	0.981119
$484$	−10.9443	−0.497467
$485$	−45.7771	−2.07863
$486$	13.9443	0.632525
$487$	−14.8328	−0.672139	−0.336070	−	0.941837i	$-0.609098\pi$
$487$	−14.8328	−0.672139	−0.336070	+	0.941837i	$0.609098\pi$
$488$	−5.47214	−0.247712
$489$	10.7082	0.484242
$490$	−39.5967	−1.78880
$491$	6.76393	0.305252	0.152626	−	0.988284i	$-0.451227\pi$
$491$	6.76393	0.305252	0.152626	+	0.988284i	$0.451227\pi$
$492$	4.76393	0.214775
$493$	0.201626	0.00908078
$494$	−2.00000	−0.0899843
$495$	2.23607	0.100504
$496$	7.47214	0.335509
$497$	−41.1246	−1.84469
$498$	−3.23607	−0.145012
$499$	11.9443	0.534699	0.267350	−	0.963600i	$-0.413852\pi$
$499$	11.9443	0.534699	0.267350	+	0.963600i	$0.413852\pi$
$500$	−11.1803	−0.500000
$501$	2.23607	0.0999001
$502$	24.0902	1.07520
$503$	34.3607	1.53207	0.766033	−	0.642801i	$-0.222228\pi$
$503$	34.3607	1.53207	0.766033	+	0.642801i	$0.222228\pi$
$504$	11.0902	0.493995
$505$	18.4164	0.819519
$506$	−1.94427	−0.0864334
$507$	−7.41641	−0.329374
$508$	20.7984	0.922779
$509$	26.6738	1.18229	0.591147	−	0.806564i	$-0.298675\pi$
$509$	26.6738	1.18229	0.591147	+	0.806564i	$0.298675\pi$
$510$	−0.527864	−0.0233742
$511$	45.3607	2.00664
$512$	1.00000	0.0441942
$513$	−6.94427	−0.306597
$514$	−5.47214	−0.241366
$515$	−39.2705	−1.73047
$516$	1.38197	0.0608377
$517$	−1.61803	−0.0711611
$518$	36.8885	1.62079
$519$	−4.61803	−0.202709
$520$	3.61803	0.158661
$521$	30.4164	1.33257	0.666284	−	0.745699i	$-0.267884\pi$
$521$	30.4164	1.33257	0.666284	+	0.745699i	$0.267884\pi$
$522$	−2.23607	−0.0978700
$523$	27.6180	1.20765	0.603826	−	0.797116i	$-0.293642\pi$
$523$	27.6180	1.20765	0.603826	+	0.797116i	$0.293642\pi$
$524$	2.76393	0.120743
$525$	−21.1803	−0.924386
$526$	23.2705	1.01464
$527$	1.76393	0.0768381
$528$	0.145898	0.00634940
$529$	44.8328	1.94925
$530$	0	0
$531$	17.1803	0.745563
$532$	−8.47214	−0.367314
$533$	−7.70820	−0.333879
$534$	−6.05573	−0.262057
$535$	44.2705	1.91398
$536$	1.76393	0.0761903
$537$	11.5623	0.498950
$538$	1.00000	0.0431131
$539$	2.58359	0.111283
$540$	12.5623	0.540596
$541$	20.4721	0.880166	0.440083	−	0.897957i	$-0.354949\pi$
$541$	20.4721	0.880166	0.440083	+	0.897957i	$0.354949\pi$
$542$	24.1246	1.03624
$543$	12.0000	0.514969
$544$	0.236068	0.0101213
$545$	17.0344	0.729675
$546$	2.61803	0.112042
$547$	3.20163	0.136892	0.0684458	−	0.997655i	$-0.478196\pi$
$547$	3.20163	0.136892	0.0684458	+	0.997655i	$0.478196\pi$
$548$	−15.6180	−0.667169
$549$	14.3262	0.611429
$550$	1.90983	0.0814354
$551$	1.70820	0.0727719
$552$	−5.09017	−0.216652
$553$	58.4508	2.48558
$554$	−21.3820	−0.908433
$555$	19.4721	0.826546
$556$	10.5623	0.447942
$557$	28.2918	1.19876	0.599381	−	0.800464i	$-0.295413\pi$
$557$	28.2918	1.19876	0.599381	+	0.800464i	$0.295413\pi$
$558$	−19.5623	−0.828138
$559$	−2.23607	−0.0945756
$560$	15.3262	0.647652
$561$	0.0344419	0.00145414
$562$	−29.1803	−1.23090
$563$	7.29180	0.307313	0.153656	−	0.988124i	$-0.450895\pi$
$563$	7.29180	0.307313	0.153656	+	0.988124i	$0.450895\pi$
$564$	−4.23607	−0.178371
$565$	55.6525	2.34132
$566$	5.76393	0.242276
$567$	−24.1803	−1.01548
$568$	9.70820	0.407347
$569$	−5.90983	−0.247753	−0.123876	−	0.992298i	$-0.539533\pi$
$569$	−5.90983	−0.247753	−0.123876	+	0.992298i	$0.539533\pi$
$570$	−4.47214	−0.187317
$571$	10.0000	0.418487	0.209243	−	0.977864i	$-0.432900\pi$
$571$	10.0000	0.418487	0.209243	+	0.977864i	$0.432900\pi$
$572$	−0.236068	−0.00987050
$573$	−3.12461	−0.130533
$574$	−32.6525	−1.36289
$575$	−66.6312	−2.77871
$576$	−2.61803	−0.109085
$577$	3.00000	0.124892	0.0624458	−	0.998048i	$-0.480110\pi$
$577$	3.00000	0.124892	0.0624458	+	0.998048i	$0.480110\pi$
$578$	−16.9443	−0.704789
$579$	−5.09017	−0.211540
$580$	−3.09017	−0.128312
$581$	22.1803	0.920196
$582$	7.81966	0.324135
$583$	0	0
$584$	−10.7082	−0.443109
$585$	−9.47214	−0.391625
$586$	7.52786	0.310973
$587$	−10.6738	−0.440553	−0.220277	−	0.975437i	$-0.570696\pi$
$587$	−10.6738	−0.440553	−0.220277	+	0.975437i	$0.570696\pi$
$588$	6.76393	0.278940
$589$	14.9443	0.615768
$590$	23.7426	0.977469
$591$	−13.8328	−0.569006
$592$	−8.70820	−0.357905
$593$	−0.583592	−0.0239653	−0.0119826	−	0.999928i	$-0.503814\pi$
$593$	−0.583592	−0.0239653	−0.0119826	+	0.999928i	$0.503814\pi$
$594$	−0.819660	−0.0336311
$595$	3.61803	0.148325
$596$	0.326238	0.0133632
$597$	−3.32624	−0.136134
$598$	8.23607	0.336798
$599$	−15.5967	−0.637266	−0.318633	−	0.947878i	$-0.603224\pi$
$599$	−15.5967	−0.637266	−0.318633	+	0.947878i	$0.603224\pi$
$600$	5.00000	0.204124
$601$	−25.6869	−1.04779	−0.523896	−	0.851782i	$-0.675522\pi$
$601$	−25.6869	−1.04779	−0.523896	+	0.851782i	$0.675522\pi$
$602$	−9.47214	−0.386055
$603$	−4.61803	−0.188061
$604$	−9.94427	−0.404627
$605$	39.5967	1.60984
$606$	−3.14590	−0.127793
$607$	47.5755	1.93103	0.965514	−	0.260350i	$-0.0838381\pi$
$607$	47.5755	1.93103	0.965514	+	0.260350i	$0.0838381\pi$
$608$	2.00000	0.0811107
$609$	−2.23607	−0.0906100
$610$	19.7984	0.801613
$611$	6.85410	0.277287
$612$	−0.618034	−0.0249825
$613$	−26.5279	−1.07145	−0.535725	−	0.844392i	$-0.679962\pi$
$613$	−26.5279	−1.07145	−0.535725	+	0.844392i	$0.679962\pi$
$614$	32.1246	1.29644
$615$	−17.2361	−0.695025
$616$	−1.00000	−0.0402911
$617$	36.1591	1.45571	0.727854	−	0.685732i	$-0.240518\pi$
$617$	36.1591	1.45571	0.727854	+	0.685732i	$0.240518\pi$
$618$	6.70820	0.269844
$619$	−40.8328	−1.64121	−0.820605	−	0.571496i	$-0.806363\pi$
$619$	−40.8328	−1.64121	−0.820605	+	0.571496i	$0.806363\pi$
$620$	−27.0344	−1.08573
$621$	28.5967	1.14755
$622$	−17.2361	−0.691103
$623$	41.5066	1.66293
$624$	−0.618034	−0.0247412
$625$	0	0
$626$	13.1459	0.525416
$627$	0.291796	0.0116532
$628$	22.6525	0.903932
$629$	−2.05573	−0.0819672
$630$	−40.1246	−1.59860
$631$	5.20163	0.207073	0.103537	−	0.994626i	$-0.466984\pi$
$631$	5.20163	0.207073	0.103537	+	0.994626i	$0.466984\pi$
$632$	−13.7984	−0.548870
$633$	0.527864	0.0209807
$634$	−30.2361	−1.20083
$635$	−75.2492	−2.98617
$636$	0	0
$637$	−10.9443	−0.433628
$638$	0.201626	0.00798245
$639$	−25.4164	−1.00546
$640$	−3.61803	−0.143015
$641$	3.18034	0.125616	0.0628079	−	0.998026i	$-0.479994\pi$
$641$	3.18034	0.125616	0.0628079	+	0.998026i	$0.479994\pi$
$642$	−7.56231	−0.298460
$643$	−4.14590	−0.163498	−0.0817491	−	0.996653i	$-0.526051\pi$
$643$	−4.14590	−0.163498	−0.0817491	+	0.996653i	$0.526051\pi$
$644$	34.8885	1.37480
$645$	−5.00000	−0.196875
$646$	0.472136	0.0185759
$647$	−2.81966	−0.110852	−0.0554261	−	0.998463i	$-0.517652\pi$
$647$	−2.81966	−0.110852	−0.0554261	+	0.998463i	$0.517652\pi$
$648$	5.70820	0.224239
$649$	−1.54915	−0.0608095
$650$	−8.09017	−0.317323
$651$	−19.5623	−0.766707
$652$	17.3262	0.678548
$653$	−27.5410	−1.07776	−0.538882	−	0.842381i	$-0.681153\pi$
$653$	−27.5410	−1.07776	−0.538882	+	0.842381i	$0.681153\pi$
$654$	−2.90983	−0.113783
$655$	−10.0000	−0.390732
$656$	7.70820	0.300955
$657$	28.0344	1.09373
$658$	29.0344	1.13188
$659$	−31.3262	−1.22030	−0.610148	−	0.792287i	$-0.708890\pi$
$659$	−31.3262	−1.22030	−0.610148	+	0.792287i	$0.708890\pi$
$660$	−0.527864	−0.0205471
$661$	−20.8541	−0.811131	−0.405565	−	0.914066i	$-0.632925\pi$
$661$	−20.8541	−0.811131	−0.405565	+	0.914066i	$0.632925\pi$
$662$	0.819660	0.0318570
$663$	−0.145898	−0.00566621
$664$	−5.23607	−0.203199
$665$	30.6525	1.18865
$666$	22.7984	0.883419
$667$	−7.03444	−0.272375
$668$	3.61803	0.139986
$669$	−3.29180	−0.127268
$670$	−6.38197	−0.246557
$671$	−1.29180	−0.0498692
$672$	−2.61803	−0.100993
$673$	10.7082	0.412771	0.206385	−	0.978471i	$-0.433830\pi$
$673$	10.7082	0.412771	0.206385	+	0.978471i	$0.433830\pi$
$674$	−27.9787	−1.07770
$675$	−28.0902	−1.08119
$676$	−12.0000	−0.461538
$677$	−17.4377	−0.670185	−0.335093	−	0.942185i	$-0.608768\pi$
$677$	−17.4377	−0.670185	−0.335093	+	0.942185i	$0.608768\pi$
$678$	−9.50658	−0.365098
$679$	−53.5967	−2.05685
$680$	−0.854102	−0.0327533
$681$	−14.8541	−0.569210
$682$	1.76393	0.0675444
$683$	−21.2361	−0.812576	−0.406288	−	0.913745i	$-0.633177\pi$
$683$	−21.2361	−0.812576	−0.406288	+	0.913745i	$0.633177\pi$
$684$	−5.23607	−0.200206
$685$	56.5066	2.15901
$686$	−16.7082	−0.637922
$687$	−6.61803	−0.252494
$688$	2.23607	0.0852493
$689$	0	0
$690$	18.4164	0.701101
$691$	−15.1246	−0.575367	−0.287684	−	0.957725i	$-0.592885\pi$
$691$	−15.1246	−0.575367	−0.287684	+	0.957725i	$0.592885\pi$
$692$	−7.47214	−0.284048
$693$	2.61803	0.0994509
$694$	13.3607	0.507164
$695$	−38.2148	−1.44957
$696$	0.527864	0.0200086
$697$	1.81966	0.0689245
$698$	21.2705	0.805101
$699$	2.61803	0.0990231
$700$	−34.2705	−1.29530
$701$	−10.9787	−0.414660	−0.207330	−	0.978271i	$-0.566477\pi$
$701$	−10.9787	−0.414660	−0.207330	+	0.978271i	$0.566477\pi$
$702$	3.47214	0.131047
$703$	−17.4164	−0.656872
$704$	0.236068	0.00889715
$705$	15.3262	0.577220
$706$	−9.85410	−0.370864
$707$	21.5623	0.810934
$708$	−4.05573	−0.152424
$709$	−37.9443	−1.42503	−0.712514	−	0.701658i	$-0.752443\pi$
$709$	−37.9443	−1.42503	−0.712514	+	0.701658i	$0.752443\pi$
$710$	−35.1246	−1.31820
$711$	36.1246	1.35478
$712$	−9.79837	−0.367210
$713$	−61.5410	−2.30473
$714$	−0.618034	−0.0231293
$715$	0.854102	0.0319416
$716$	18.7082	0.699158
$717$	−11.9098	−0.444781
$718$	−16.9443	−0.632355
$719$	42.1803	1.57306	0.786531	−	0.617551i	$-0.211875\pi$
$719$	42.1803	1.57306	0.786531	+	0.617551i	$0.211875\pi$
$720$	9.47214	0.353006
$721$	−45.9787	−1.71234
$722$	−15.0000	−0.558242
$723$	−2.90983	−0.108218
$724$	19.4164	0.721605
$725$	6.90983	0.256625
$726$	−6.76393	−0.251033
$727$	−44.7082	−1.65814	−0.829068	−	0.559148i	$-0.811128\pi$
$727$	−44.7082	−1.65814	−0.829068	+	0.559148i	$0.811128\pi$
$728$	4.23607	0.156999
$729$	−8.50658	−0.315058
$730$	38.7426	1.43393
$731$	0.527864	0.0195238
$732$	−3.38197	−0.125001
$733$	−33.3607	−1.23220	−0.616102	−	0.787666i	$-0.711289\pi$
$733$	−33.3607	−1.23220	−0.616102	+	0.787666i	$0.711289\pi$
$734$	−9.20163	−0.339638
$735$	−24.4721	−0.902668
$736$	−8.23607	−0.303585
$737$	0.416408	0.0153386
$738$	−20.1803	−0.742849
$739$	35.0344	1.28876	0.644381	−	0.764704i	$-0.277115\pi$
$739$	35.0344	1.28876	0.644381	+	0.764704i	$0.277115\pi$
$740$	31.5066	1.15820
$741$	−1.23607	−0.0454081
$742$	0	0
$743$	−9.20163	−0.337575	−0.168787	−	0.985652i	$-0.553985\pi$
$743$	−9.20163	−0.337575	−0.168787	+	0.985652i	$0.553985\pi$
$744$	4.61803	0.169305
$745$	−1.18034	−0.0432443
$746$	6.94427	0.254248
$747$	13.7082	0.501557
$748$	0.0557281	0.00203762
$749$	51.8328	1.89393
$750$	−6.90983	−0.252311
$751$	39.8673	1.45478	0.727388	−	0.686226i	$-0.240734\pi$
$751$	39.8673	1.45478	0.727388	+	0.686226i	$0.240734\pi$
$752$	−6.85410	−0.249943
$753$	14.8885	0.542569
$754$	−0.854102	−0.0311046
$755$	35.9787	1.30940
$756$	14.7082	0.534932
$757$	−38.9443	−1.41545	−0.707727	−	0.706486i	$-0.750279\pi$
$757$	−38.9443	−1.41545	−0.707727	+	0.706486i	$0.750279\pi$
$758$	6.14590	0.223229
$759$	−1.20163	−0.0436163
$760$	−7.23607	−0.262480
$761$	−1.41641	−0.0513447	−0.0256724	−	0.999670i	$-0.508173\pi$
$761$	−1.41641	−0.0513447	−0.0256724	+	0.999670i	$0.508173\pi$
$762$	12.8541	0.465655
$763$	19.9443	0.722031
$764$	−5.05573	−0.182910
$765$	2.23607	0.0808452
$766$	−17.3820	−0.628036
$767$	6.56231	0.236951
$768$	0.618034	0.0223014
$769$	−33.3262	−1.20177	−0.600887	−	0.799334i	$-0.705186\pi$
$769$	−33.3262	−1.20177	−0.600887	+	0.799334i	$0.705186\pi$
$770$	3.61803	0.130385
$771$	−3.38197	−0.121799
$772$	−8.23607	−0.296423
$773$	47.5967	1.71194	0.855968	−	0.517029i	$-0.172962\pi$
$773$	47.5967	1.71194	0.855968	+	0.517029i	$0.172962\pi$
$774$	−5.85410	−0.210421
$775$	60.4508	2.17146
$776$	12.6525	0.454197
$777$	22.7984	0.817887
$778$	28.9443	1.03770
$779$	15.4164	0.552350
$780$	2.23607	0.0800641
$781$	2.29180	0.0820069
$782$	−1.94427	−0.0695270
$783$	−2.96556	−0.105980
$784$	10.9443	0.390867
$785$	−81.9574	−2.92519
$786$	1.70820	0.0609296
$787$	−2.34752	−0.0836802	−0.0418401	−	0.999124i	$-0.513322\pi$
$787$	−2.34752	−0.0836802	−0.0418401	+	0.999124i	$0.513322\pi$
$788$	−22.3820	−0.797325
$789$	14.3820	0.512012
$790$	49.9230	1.77618
$791$	65.1591	2.31679
$792$	−0.618034	−0.0219609
$793$	5.47214	0.194321
$794$	−34.2705	−1.21621
$795$	0	0
$796$	−5.38197	−0.190759
$797$	13.2016	0.467626	0.233813	−	0.972282i	$-0.424880\pi$
$797$	13.2016	0.467626	0.233813	+	0.972282i	$0.424880\pi$
$798$	−5.23607	−0.185355
$799$	−1.61803	−0.0572419
$800$	8.09017	0.286031
$801$	25.6525	0.906386
$802$	−4.27051	−0.150797
$803$	−2.52786	−0.0892064
$804$	1.09017	0.0384473
$805$	−126.228	−4.44895
$806$	−7.47214	−0.263195
$807$	0.618034	0.0217558
$808$	−5.09017	−0.179072
$809$	15.5279	0.545931	0.272965	−	0.962024i	$-0.411996\pi$
$809$	15.5279	0.545931	0.272965	+	0.962024i	$0.411996\pi$
$810$	−20.6525	−0.725654
$811$	−47.5967	−1.67135	−0.835674	−	0.549226i	$-0.814923\pi$
$811$	−47.5967	−1.67135	−0.835674	+	0.549226i	$0.814923\pi$
$812$	−3.61803	−0.126968
$813$	14.9098	0.522911
$814$	−2.05573	−0.0720532
$815$	−62.6869	−2.19583
$816$	0.145898	0.00510745
$817$	4.47214	0.156460
$818$	−11.6525	−0.407419
$819$	−11.0902	−0.387522
$820$	−27.8885	−0.973910
$821$	38.3607	1.33880	0.669398	−	0.742904i	$-0.266552\pi$
$821$	38.3607	1.33880	0.669398	+	0.742904i	$0.266552\pi$
$822$	−9.65248	−0.336669
$823$	−50.8885	−1.77386	−0.886932	−	0.461901i	$-0.847168\pi$
$823$	−50.8885	−1.77386	−0.886932	+	0.461901i	$0.847168\pi$
$824$	10.8541	0.378121
$825$	1.18034	0.0410942
$826$	27.7984	0.967229
$827$	−37.7426	−1.31244	−0.656220	−	0.754569i	$-0.727846\pi$
$827$	−37.7426	−1.31244	−0.656220	+	0.754569i	$0.727846\pi$
$828$	21.5623	0.749342
$829$	43.7984	1.52118	0.760590	−	0.649232i	$-0.224910\pi$
$829$	43.7984	1.52118	0.760590	+	0.649232i	$0.224910\pi$
$830$	18.9443	0.657565
$831$	−13.2148	−0.458416
$832$	−1.00000	−0.0346688
$833$	2.58359	0.0895162
$834$	6.52786	0.226041
$835$	−13.0902	−0.453004
$836$	0.472136	0.0163292
$837$	−25.9443	−0.896765
$838$	−17.4377	−0.602375
$839$	14.4508	0.498899	0.249449	−	0.968388i	$-0.419750\pi$
$839$	14.4508	0.498899	0.249449	+	0.968388i	$0.419750\pi$
$840$	9.47214	0.326820
$841$	−28.2705	−0.974845
$842$	11.4721	0.395356
$843$	−18.0344	−0.621139
$844$	0.854102	0.0293994
$845$	43.4164	1.49357
$846$	17.9443	0.616937
$847$	46.3607	1.59297
$848$	0	0
$849$	3.56231	0.122258
$850$	1.90983	0.0655066
$851$	71.7214	2.45858
$852$	6.00000	0.205557
$853$	51.8115	1.77399	0.886996	−	0.461776i	$-0.152788\pi$
$853$	51.8115	1.77399	0.886996	+	0.461776i	$0.152788\pi$
$854$	23.1803	0.793215
$855$	18.9443	0.647880
$856$	−12.2361	−0.418220
$857$	−42.0689	−1.43705	−0.718523	−	0.695503i	$-0.755181\pi$
$857$	−42.0689	−1.43705	−0.718523	+	0.695503i	$0.755181\pi$
$858$	−0.145898	−0.00498088
$859$	−12.7426	−0.434773	−0.217387	−	0.976086i	$-0.569753\pi$
$859$	−12.7426	−0.434773	−0.217387	+	0.976086i	$0.569753\pi$
$860$	−8.09017	−0.275873
$861$	−20.1803	−0.687744
$862$	−15.4721	−0.526983
$863$	17.3262	0.589792	0.294896	−	0.955529i	$-0.404715\pi$
$863$	17.3262	0.589792	0.294896	+	0.955529i	$0.404715\pi$
$864$	−3.47214	−0.118124
$865$	27.0344	0.919199
$866$	12.2016	0.414628
$867$	−10.4721	−0.355652
$868$	−31.6525	−1.07436
$869$	−3.25735	−0.110498
$870$	−1.90983	−0.0647493
$871$	−1.76393	−0.0597686
$872$	−4.70820	−0.159440
$873$	−33.1246	−1.12110
$874$	−16.4721	−0.557178
$875$	47.3607	1.60108
$876$	−6.61803	−0.223603
$877$	42.5755	1.43767	0.718836	−	0.695180i	$-0.244675\pi$
$877$	42.5755	1.43767	0.718836	+	0.695180i	$0.244675\pi$
$878$	19.1459	0.646143
$879$	4.65248	0.156924
$880$	−0.854102	−0.0287918
$881$	−44.5066	−1.49946	−0.749732	−	0.661741i	$-0.769818\pi$
$881$	−44.5066	−1.49946	−0.749732	+	0.661741i	$0.769818\pi$
$882$	−28.6525	−0.964779
$883$	−50.1935	−1.68915	−0.844573	−	0.535441i	$-0.820146\pi$
$883$	−50.1935	−1.68915	−0.844573	+	0.535441i	$0.820146\pi$
$884$	−0.236068	−0.00793983
$885$	14.6738	0.493253
$886$	33.4164	1.12265
$887$	−13.8328	−0.464460	−0.232230	−	0.972661i	$-0.574602\pi$
$887$	−13.8328	−0.464460	−0.232230	+	0.972661i	$0.574602\pi$
$888$	−5.38197	−0.180607
$889$	−88.1033	−2.95489
$890$	35.4508	1.18832
$891$	1.34752	0.0451438
$892$	−5.32624	−0.178336
$893$	−13.7082	−0.458728
$894$	0.201626	0.00674339
$895$	−67.6869	−2.26252
$896$	−4.23607	−0.141517
$897$	5.09017	0.169956
$898$	2.90983	0.0971023
$899$	6.38197	0.212850
$900$	−21.1803	−0.706011
$901$	0	0
$902$	1.81966	0.0605881
$903$	−5.85410	−0.194812
$904$	−15.3820	−0.511597
$905$	−70.2492	−2.33516
$906$	−6.14590	−0.204184
$907$	−14.1246	−0.469000	−0.234500	−	0.972116i	$-0.575345\pi$
$907$	−14.1246	−0.469000	−0.234500	+	0.972116i	$0.575345\pi$
$908$	−24.0344	−0.797611
$909$	13.3262	0.442003
$910$	−15.3262	−0.508060
$911$	47.3394	1.56842	0.784212	−	0.620493i	$-0.213067\pi$
$911$	47.3394	1.56842	0.784212	+	0.620493i	$0.213067\pi$
$912$	1.23607	0.0409303
$913$	−1.23607	−0.0409079
$914$	26.7984	0.886411
$915$	12.2361	0.404512
$916$	−10.7082	−0.353809
$917$	−11.7082	−0.386639
$918$	−0.819660	−0.0270528
$919$	−5.34752	−0.176399	−0.0881993	−	0.996103i	$-0.528111\pi$
$919$	−5.34752	−0.176399	−0.0881993	+	0.996103i	$0.528111\pi$
$920$	29.7984	0.982423
$921$	19.8541	0.654215
$922$	21.1246	0.695702
$923$	−9.70820	−0.319549
$924$	−0.618034	−0.0203318
$925$	−70.4508	−2.31641
$926$	−20.0902	−0.660204
$927$	−28.4164	−0.933317
$928$	0.854102	0.0280373
$929$	17.8197	0.584644	0.292322	−	0.956320i	$-0.405572\pi$
$929$	17.8197	0.584644	0.292322	+	0.956320i	$0.405572\pi$
$930$	−16.7082	−0.547884
$931$	21.8885	0.717368
$932$	4.23607	0.138757
$933$	−10.6525	−0.348746
$934$	−6.58359	−0.215422
$935$	−0.201626	−0.00659388
$936$	2.61803	0.0855731
$937$	−33.0902	−1.08101	−0.540504	−	0.841341i	$-0.681767\pi$
$937$	−33.0902	−1.08101	−0.540504	+	0.841341i	$0.681767\pi$
$938$	−7.47214	−0.243974
$939$	8.12461	0.265137
$940$	24.7984	0.808834
$941$	−46.9787	−1.53146	−0.765731	−	0.643161i	$-0.777623\pi$
$941$	−46.9787	−1.53146	−0.765731	+	0.643161i	$0.777623\pi$
$942$	14.0000	0.456145
$943$	−63.4853	−2.06737
$944$	−6.56231	−0.213585
$945$	−53.2148	−1.73108
$946$	0.527864	0.0171623
$947$	−50.2148	−1.63176	−0.815881	−	0.578220i	$-0.803747\pi$
$947$	−50.2148	−1.63176	−0.815881	+	0.578220i	$0.803747\pi$
$948$	−8.52786	−0.276972
$949$	10.7082	0.347603
$950$	16.1803	0.524960
$951$	−18.6869	−0.605965
$952$	−1.00000	−0.0324102
$953$	25.3050	0.819708	0.409854	−	0.912151i	$-0.365580\pi$
$953$	25.3050	0.819708	0.409854	+	0.912151i	$0.365580\pi$
$954$	0	0
$955$	18.2918	0.591909
$956$	−19.2705	−0.623253
$957$	0.124612	0.00402813
$958$	−36.5967	−1.18239
$959$	66.1591	2.13639
$960$	−2.23607	−0.0721688
$961$	24.8328	0.801059
$962$	8.70820	0.280764
$963$	32.0344	1.03230
$964$	−4.70820	−0.151641
$965$	29.7984	0.959244
$966$	21.5623	0.693756
$967$	1.85410	0.0596239	0.0298119	−	0.999556i	$-0.490509\pi$
$967$	1.85410	0.0596239	0.0298119	+	0.999556i	$0.490509\pi$
$968$	−10.9443	−0.351762
$969$	0.291796	0.00937384
$970$	−45.7771	−1.46981
$971$	−0.652476	−0.0209389	−0.0104695	−	0.999945i	$-0.503333\pi$
$971$	−0.652476	−0.0209389	−0.0104695	+	0.999945i	$0.503333\pi$
$972$	13.9443	0.447263
$973$	−44.7426	−1.43438
$974$	−14.8328	−0.475274
$975$	−5.00000	−0.160128
$976$	−5.47214	−0.175159
$977$	31.1459	0.996446	0.498223	−	0.867049i	$-0.333986\pi$
$977$	31.1459	0.996446	0.498223	+	0.867049i	$0.333986\pi$
$978$	10.7082	0.342411
$979$	−2.31308	−0.0739264
$980$	−39.5967	−1.26487
$981$	12.3262	0.393546
$982$	6.76393	0.215846
$983$	−31.4377	−1.00271	−0.501353	−	0.865243i	$-0.667164\pi$
$983$	−31.4377	−1.00271	−0.501353	+	0.865243i	$0.667164\pi$
$984$	4.76393	0.151869
$985$	80.9787	2.58020
$986$	0.201626	0.00642108
$987$	17.9443	0.571172
$988$	−2.00000	−0.0636285
$989$	−18.4164	−0.585608
$990$	2.23607	0.0710669
$991$	−16.3820	−0.520390	−0.260195	−	0.965556i	$-0.583787\pi$
$991$	−16.3820	−0.520390	−0.260195	+	0.965556i	$0.583787\pi$
$992$	7.47214	0.237241
$993$	0.506578	0.0160758
$994$	−41.1246	−1.30439
$995$	19.4721	0.617308
$996$	−3.23607	−0.102539
$997$	−17.5836	−0.556878	−0.278439	−	0.960454i	$-0.589817\pi$
$997$	−17.5836	−0.556878	−0.278439	+	0.960454i	$0.589817\pi$
$998$	11.9443	0.378089
$999$	30.2361	0.956627

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.2.a.a.1.2	✓	2
3.2	odd	2		4842.2.a.f.1.2		2
4.3	odd	2		4304.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.a.1.2	✓	2	1.1	even	1	trivial
4304.2.a.d.1.1		2	4.3	odd	2
4842.2.a.f.1.2		2	3.2	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:	\( N \)	\(=\)	\( 538 = 2 \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	538.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} -3.61803 q^{5} +0.618034 q^{6} -4.23607 q^{7} +1.00000 q^{8} -2.61803 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} -3.61803 q^{5} +0.618034 q^{6} -4.23607 q^{7} +1.00000 q^{8} -2.61803 q^{9} -3.61803 q^{10} +0.236068 q^{11} +0.618034 q^{12} -1.00000 q^{13} -4.23607 q^{14} -2.23607 q^{15} +1.00000 q^{16} +0.236068 q^{17} -2.61803 q^{18} +2.00000 q^{19} -3.61803 q^{20} -2.61803 q^{21} +0.236068 q^{22} -8.23607 q^{23} +0.618034 q^{24} +8.09017 q^{25} -1.00000 q^{26} -3.47214 q^{27} -4.23607 q^{28} +0.854102 q^{29} -2.23607 q^{30} +7.47214 q^{31} +1.00000 q^{32} +0.145898 q^{33} +0.236068 q^{34} +15.3262 q^{35} -2.61803 q^{36} -8.70820 q^{37} +2.00000 q^{38} -0.618034 q^{39} -3.61803 q^{40} +7.70820 q^{41} -2.61803 q^{42} +2.23607 q^{43} +0.236068 q^{44} +9.47214 q^{45} -8.23607 q^{46} -6.85410 q^{47} +0.618034 q^{48} +10.9443 q^{49} +8.09017 q^{50} +0.145898 q^{51} -1.00000 q^{52} -3.47214 q^{54} -0.854102 q^{55} -4.23607 q^{56} +1.23607 q^{57} +0.854102 q^{58} -6.56231 q^{59} -2.23607 q^{60} -5.47214 q^{61} +7.47214 q^{62} +11.0902 q^{63} +1.00000 q^{64} +3.61803 q^{65} +0.145898 q^{66} +1.76393 q^{67} +0.236068 q^{68} -5.09017 q^{69} +15.3262 q^{70} +9.70820 q^{71} -2.61803 q^{72} -10.7082 q^{73} -8.70820 q^{74} +5.00000 q^{75} +2.00000 q^{76} -1.00000 q^{77} -0.618034 q^{78} -13.7984 q^{79} -3.61803 q^{80} +5.70820 q^{81} +7.70820 q^{82} -5.23607 q^{83} -2.61803 q^{84} -0.854102 q^{85} +2.23607 q^{86} +0.527864 q^{87} +0.236068 q^{88} -9.79837 q^{89} +9.47214 q^{90} +4.23607 q^{91} -8.23607 q^{92} +4.61803 q^{93} -6.85410 q^{94} -7.23607 q^{95} +0.618034 q^{96} +12.6525 q^{97} +10.9443 q^{98} -0.618034 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 5 q^{5} - q^{6} - 4 q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - q^3 + 2 * q^4 - 5 * q^5 - q^6 - 4 * q^7 + 2 * q^8 - 3 * q^9	\( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 5 q^{5} - q^{6} - 4 q^{7} + 2 q^{8} - 3 q^{9} - 5 q^{10} - 4 q^{11} - q^{12} - 2 q^{13} - 4 q^{14} + 2 q^{16} - 4 q^{17} - 3 q^{18} + 4 q^{19} - 5 q^{20} - 3 q^{21} - 4 q^{22} - 12 q^{23} - q^{24} + 5 q^{25} - 2 q^{26} + 2 q^{27} - 4 q^{28} - 5 q^{29} + 6 q^{31} + 2 q^{32} + 7 q^{33} - 4 q^{34} + 15 q^{35} - 3 q^{36} - 4 q^{37} + 4 q^{38} + q^{39} - 5 q^{40} + 2 q^{41} - 3 q^{42} - 4 q^{44} + 10 q^{45} - 12 q^{46} - 7 q^{47} - q^{48} + 4 q^{49} + 5 q^{50} + 7 q^{51} - 2 q^{52} + 2 q^{54} + 5 q^{55} - 4 q^{56} - 2 q^{57} - 5 q^{58} + 7 q^{59} - 2 q^{61} + 6 q^{62} + 11 q^{63} + 2 q^{64} + 5 q^{65} + 7 q^{66} + 8 q^{67} - 4 q^{68} + q^{69} + 15 q^{70} + 6 q^{71} - 3 q^{72} - 8 q^{73} - 4 q^{74} + 10 q^{75} + 4 q^{76} - 2 q^{77} + q^{78} - 3 q^{79} - 5 q^{80} - 2 q^{81} + 2 q^{82} - 6 q^{83} - 3 q^{84} + 5 q^{85} + 10 q^{87} - 4 q^{88} + 5 q^{89} + 10 q^{90} + 4 q^{91} - 12 q^{92} + 7 q^{93} - 7 q^{94} - 10 q^{95} - q^{96} - 6 q^{97} + 4 q^{98} + q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - q^3 + 2 * q^4 - 5 * q^5 - q^6 - 4 * q^7 + 2 * q^8 - 3 * q^9 - 5 * q^10 - 4 * q^11 - q^12 - 2 * q^13 - 4 * q^14 + 2 * q^16 - 4 * q^17 - 3 * q^18 + 4 * q^19 - 5 * q^20 - 3 * q^21 - 4 * q^22 - 12 * q^23 - q^24 + 5 * q^25 - 2 * q^26 + 2 * q^27 - 4 * q^28 - 5 * q^29 + 6 * q^31 + 2 * q^32 + 7 * q^33 - 4 * q^34 + 15 * q^35 - 3 * q^36 - 4 * q^37 + 4 * q^38 + q^39 - 5 * q^40 + 2 * q^41 - 3 * q^42 - 4 * q^44 + 10 * q^45 - 12 * q^46 - 7 * q^47 - q^48 + 4 * q^49 + 5 * q^50 + 7 * q^51 - 2 * q^52 + 2 * q^54 + 5 * q^55 - 4 * q^56 - 2 * q^57 - 5 * q^58 + 7 * q^59 - 2 * q^61 + 6 * q^62 + 11 * q^63 + 2 * q^64 + 5 * q^65 + 7 * q^66 + 8 * q^67 - 4 * q^68 + q^69 + 15 * q^70 + 6 * q^71 - 3 * q^72 - 8 * q^73 - 4 * q^74 + 10 * q^75 + 4 * q^76 - 2 * q^77 + q^78 - 3 * q^79 - 5 * q^80 - 2 * q^81 + 2 * q^82 - 6 * q^83 - 3 * q^84 + 5 * q^85 + 10 * q^87 - 4 * q^88 + 5 * q^89 + 10 * q^90 + 4 * q^91 - 12 * q^92 + 7 * q^93 - 7 * q^94 - 10 * q^95 - q^96 - 6 * q^97 + 4 * q^98 + q^99

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