LMFDB - Embedded newform 538.2.a.c.1.4

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.4913.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + x + 1 $ x^4 - x^3 - 6*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.04948$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +3.04948 q^{3} +1.00000 q^{4} +1.48793 q^{5} -3.04948 q^{6} +0.344151 q^{7} -1.00000 q^{8} +6.29934 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +3.04948 q^{3} +1.00000 q^{4} +1.48793 q^{5} -3.04948 q^{6} +0.344151 q^{7} -1.00000 q^{8} +6.29934 q^{9} -1.48793 q^{10} -0.905704 q^{11} +3.04948 q^{12} -2.24985 q^{13} -0.344151 q^{14} +4.53741 q^{15} +1.00000 q^{16} +2.58570 q^{17} -6.29934 q^{18} -0.688301 q^{19} +1.48793 q^{20} +1.04948 q^{21} +0.905704 q^{22} +1.46259 q^{23} -3.04948 q^{24} -2.78607 q^{25} +2.24985 q^{26} +10.0613 q^{27} +0.344151 q^{28} -4.24274 q^{29} -4.53741 q^{30} +1.92985 q^{31} -1.00000 q^{32} -2.76193 q^{33} -2.58570 q^{34} +0.512072 q^{35} +6.29934 q^{36} -1.84911 q^{37} +0.688301 q^{38} -6.86089 q^{39} -1.48793 q^{40} +1.21740 q^{41} -1.04948 q^{42} -6.59036 q^{43} -0.905704 q^{44} +9.37296 q^{45} -1.46259 q^{46} -6.42244 q^{47} +3.04948 q^{48} -6.88156 q^{49} +2.78607 q^{50} +7.88503 q^{51} -2.24985 q^{52} +2.87689 q^{53} -10.0613 q^{54} -1.34762 q^{55} -0.344151 q^{56} -2.09896 q^{57} +4.24274 q^{58} +11.0118 q^{59} +4.53741 q^{60} -1.75015 q^{61} -1.92985 q^{62} +2.16792 q^{63} +1.00000 q^{64} -3.34762 q^{65} +2.76193 q^{66} +4.34415 q^{67} +2.58570 q^{68} +4.46014 q^{69} -0.512072 q^{70} +5.29586 q^{71} -6.29934 q^{72} +5.90206 q^{73} +1.84911 q^{74} -8.49607 q^{75} -0.688301 q^{76} -0.311699 q^{77} +6.86089 q^{78} -15.1107 q^{79} +1.48793 q^{80} +11.7836 q^{81} -1.21740 q^{82} +15.7701 q^{83} +1.04948 q^{84} +3.84733 q^{85} +6.59036 q^{86} -12.9382 q^{87} +0.905704 q^{88} -1.61934 q^{89} -9.37296 q^{90} -0.774289 q^{91} +1.46259 q^{92} +5.88503 q^{93} +6.42244 q^{94} -1.02414 q^{95} -3.04948 q^{96} +14.3436 q^{97} +6.88156 q^{98} -5.70533 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 3 q^{3} + 4 q^{4} + 5 q^{5} - 3 q^{6} - q^{7} - 4 q^{8} + 3 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 3 * q^3 + 4 * q^4 + 5 * q^5 - 3 * q^6 - q^7 - 4 * q^8 + 3 * q^9	\( 4 q - 4 q^{2} + 3 q^{3} + 4 q^{4} + 5 q^{5} - 3 q^{6} - q^{7} - 4 q^{8} + 3 q^{9} - 5 q^{10} + 7 q^{11} + 3 q^{12} + 4 q^{13} + q^{14} + 8 q^{15} + 4 q^{16} + 4 q^{17} - 3 q^{18} + 2 q^{19} + 5 q^{20} - 5 q^{21} - 7 q^{22} + 16 q^{23} - 3 q^{24} - q^{25} - 4 q^{26} + 6 q^{27} - q^{28} - 8 q^{30} - q^{31} - 4 q^{32} + q^{33} - 4 q^{34} + 3 q^{35} + 3 q^{36} - 2 q^{37} - 2 q^{38} + 3 q^{39} - 5 q^{40} - q^{41} + 5 q^{42} + 9 q^{43} + 7 q^{44} + 8 q^{45} - 16 q^{46} + 13 q^{47} + 3 q^{48} - 15 q^{49} + q^{50} + 3 q^{51} + 4 q^{52} + 28 q^{53} - 6 q^{54} + 13 q^{55} + q^{56} + 10 q^{57} + 19 q^{59} + 8 q^{60} - 20 q^{61} + q^{62} + 12 q^{63} + 4 q^{64} + 5 q^{65} - q^{66} + 15 q^{67} + 4 q^{68} - 5 q^{69} - 3 q^{70} + 15 q^{71} - 3 q^{72} - 7 q^{73} + 2 q^{74} + 12 q^{75} + 2 q^{76} - 6 q^{77} - 3 q^{78} - 17 q^{79} + 5 q^{80} + 4 q^{81} + q^{82} + 6 q^{83} - 5 q^{84} - 29 q^{85} - 9 q^{86} - 34 q^{87} - 7 q^{88} + 20 q^{89} - 8 q^{90} - 18 q^{91} + 16 q^{92} - 5 q^{93} - 13 q^{94} - 6 q^{95} - 3 q^{96} + 3 q^{97} + 15 q^{98} - 16 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 3 * q^3 + 4 * q^4 + 5 * q^5 - 3 * q^6 - q^7 - 4 * q^8 + 3 * q^9 - 5 * q^10 + 7 * q^11 + 3 * q^12 + 4 * q^13 + q^14 + 8 * q^15 + 4 * q^16 + 4 * q^17 - 3 * q^18 + 2 * q^19 + 5 * q^20 - 5 * q^21 - 7 * q^22 + 16 * q^23 - 3 * q^24 - q^25 - 4 * q^26 + 6 * q^27 - q^28 - 8 * q^30 - q^31 - 4 * q^32 + q^33 - 4 * q^34 + 3 * q^35 + 3 * q^36 - 2 * q^37 - 2 * q^38 + 3 * q^39 - 5 * q^40 - q^41 + 5 * q^42 + 9 * q^43 + 7 * q^44 + 8 * q^45 - 16 * q^46 + 13 * q^47 + 3 * q^48 - 15 * q^49 + q^50 + 3 * q^51 + 4 * q^52 + 28 * q^53 - 6 * q^54 + 13 * q^55 + q^56 + 10 * q^57 + 19 * q^59 + 8 * q^60 - 20 * q^61 + q^62 + 12 * q^63 + 4 * q^64 + 5 * q^65 - q^66 + 15 * q^67 + 4 * q^68 - 5 * q^69 - 3 * q^70 + 15 * q^71 - 3 * q^72 - 7 * q^73 + 2 * q^74 + 12 * q^75 + 2 * q^76 - 6 * q^77 - 3 * q^78 - 17 * q^79 + 5 * q^80 + 4 * q^81 + q^82 + 6 * q^83 - 5 * q^84 - 29 * q^85 - 9 * q^86 - 34 * q^87 - 7 * q^88 + 20 * q^89 - 8 * q^90 - 18 * q^91 + 16 * q^92 - 5 * q^93 - 13 * q^94 - 6 * q^95 - 3 * q^96 + 3 * q^97 + 15 * q^98 - 16 * 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
$3$	3.04948	1.76062	0.880309	−	0.474400i	$-0.157335\pi$
$3$	3.04948	1.76062	0.880309	+	0.474400i	$0.157335\pi$
$4$	1.00000	0.500000
$5$	1.48793	0.665422	0.332711	−	0.943029i	$-0.392037\pi$
$5$	1.48793	0.665422	0.332711	+	0.943029i	$0.392037\pi$
$6$	−3.04948	−1.24495
$7$	0.344151	0.130077	0.0650384	−	0.997883i	$-0.479283\pi$
$7$	0.344151	0.130077	0.0650384	+	0.997883i	$0.479283\pi$
$8$	−1.00000	−0.353553
$9$	6.29934	2.09978
$10$	−1.48793	−0.470524
$11$	−0.905704	−0.273080	−0.136540	−	0.990635i	$-0.543598\pi$
$11$	−0.905704	−0.273080	−0.136540	+	0.990635i	$0.543598\pi$
$12$	3.04948	0.880309
$13$	−2.24985	−0.623997	−0.311999	−	0.950083i	$-0.600998\pi$
$13$	−2.24985	−0.623997	−0.311999	+	0.950083i	$0.600998\pi$
$14$	−0.344151	−0.0919782
$15$	4.53741	1.17155
$16$	1.00000	0.250000
$17$	2.58570	0.627123	0.313562	−	0.949568i	$-0.398478\pi$
$17$	2.58570	0.627123	0.313562	+	0.949568i	$0.398478\pi$
$18$	−6.29934	−1.48477
$19$	−0.688301	−0.157907	−0.0789536	−	0.996878i	$-0.525158\pi$
$19$	−0.688301	−0.157907	−0.0789536	+	0.996878i	$0.525158\pi$
$20$	1.48793	0.332711
$21$	1.04948	0.229016
$22$	0.905704	0.193097
$23$	1.46259	0.304971	0.152486	−	0.988306i	$-0.451272\pi$
$23$	1.46259	0.304971	0.152486	+	0.988306i	$0.451272\pi$
$24$	−3.04948	−0.622473
$25$	−2.78607	−0.557214
$26$	2.24985	0.441233
$27$	10.0613	1.93629
$28$	0.344151	0.0650384
$29$	−4.24274	−0.787857	−0.393929	−	0.919141i	$-0.628884\pi$
$29$	−4.24274	−0.787857	−0.393929	+	0.919141i	$0.628884\pi$
$30$	−4.53741	−0.828414
$31$	1.92985	0.346611	0.173305	−	0.984868i	$-0.444555\pi$
$31$	1.92985	0.346611	0.173305	+	0.984868i	$0.444555\pi$
$32$	−1.00000	−0.176777
$33$	−2.76193	−0.480790
$34$	−2.58570	−0.443443
$35$	0.512072	0.0865559
$36$	6.29934	1.04989
$37$	−1.84911	−0.303991	−0.151996	−	0.988381i	$-0.548570\pi$
$37$	−1.84911	−0.303991	−0.151996	+	0.988381i	$0.548570\pi$
$38$	0.688301	0.111657
$39$	−6.86089	−1.09862
$40$	−1.48793	−0.235262
$41$	1.21740	0.190126	0.0950631	−	0.995471i	$-0.469695\pi$
$41$	1.21740	0.190126	0.0950631	+	0.995471i	$0.469695\pi$
$42$	−1.04948	−0.161938
$43$	−6.59036	−1.00502	−0.502510	−	0.864571i	$-0.667590\pi$
$43$	−6.59036	−1.00502	−0.502510	+	0.864571i	$0.667590\pi$
$44$	−0.905704	−0.136540
$45$	9.37296	1.39724
$46$	−1.46259	−0.215647
$47$	−6.42244	−0.936809	−0.468405	−	0.883514i	$-0.655171\pi$
$47$	−6.42244	−0.936809	−0.468405	+	0.883514i	$0.655171\pi$
$48$	3.04948	0.440155
$49$	−6.88156	−0.983080
$50$	2.78607	0.394010
$51$	7.88503	1.10413
$52$	−2.24985	−0.311999
$53$	2.87689	0.395172	0.197586	−	0.980286i	$-0.436690\pi$
$53$	2.87689	0.395172	0.197586	+	0.980286i	$0.436690\pi$
$54$	−10.0613	−1.36916
$55$	−1.34762	−0.181713
$56$	−0.344151	−0.0459891
$57$	−2.09896	−0.278014
$58$	4.24274	0.557099
$59$	11.0118	1.43361	0.716806	−	0.697273i	$-0.245603\pi$
$59$	11.0118	1.43361	0.716806	+	0.697273i	$0.245603\pi$
$60$	4.53741	0.585777
$61$	−1.75015	−0.224083	−0.112042	−	0.993704i	$-0.535739\pi$
$61$	−1.75015	−0.224083	−0.112042	+	0.993704i	$0.535739\pi$
$62$	−1.92985	−0.245091
$63$	2.16792	0.273132
$64$	1.00000	0.125000
$65$	−3.34762	−0.415221
$66$	2.76193	0.339970
$67$	4.34415	0.530722	0.265361	−	0.964149i	$-0.414509\pi$
$67$	4.34415	0.530722	0.265361	+	0.964149i	$0.414509\pi$
$68$	2.58570	0.313562
$69$	4.46014	0.536938
$70$	−0.512072	−0.0612043
$71$	5.29586	0.628503	0.314252	−	0.949340i	$-0.398246\pi$
$71$	5.29586	0.628503	0.314252	+	0.949340i	$0.398246\pi$
$72$	−6.29934	−0.742384
$73$	5.90206	0.690784	0.345392	−	0.938459i	$-0.387746\pi$
$73$	5.90206	0.690784	0.345392	+	0.938459i	$0.387746\pi$
$74$	1.84911	0.214954
$75$	−8.49607	−0.981041
$76$	−0.688301	−0.0789536
$77$	−0.311699	−0.0355213
$78$	6.86089	0.776843
$79$	−15.1107	−1.70009	−0.850046	−	0.526709i	$-0.823426\pi$
$79$	−15.1107	−1.70009	−0.850046	+	0.526709i	$0.823426\pi$
$80$	1.48793	0.166355
$81$	11.7836	1.30929
$82$	−1.21740	−0.134440
$83$	15.7701	1.73099	0.865495	−	0.500918i	$-0.167004\pi$
$83$	15.7701	1.73099	0.865495	+	0.500918i	$0.167004\pi$
$84$	1.04948	0.114508
$85$	3.84733	0.417302
$86$	6.59036	0.710657
$87$	−12.9382	−1.38712
$88$	0.905704	0.0965483
$89$	−1.61934	−0.171650	−0.0858250	−	0.996310i	$-0.527353\pi$
$89$	−1.61934	−0.171650	−0.0858250	+	0.996310i	$0.527353\pi$
$90$	−9.37296	−0.987997
$91$	−0.774289	−0.0811675
$92$	1.46259	0.152486
$93$	5.88503	0.610249
$94$	6.42244	0.662424
$95$	−1.02414	−0.105075
$96$	−3.04948	−0.311236
$97$	14.3436	1.45637	0.728184	−	0.685381i	$-0.240364\pi$
$97$	14.3436	1.45637	0.728184	+	0.685381i	$0.240364\pi$
$98$	6.88156	0.695143
$99$	−5.70533	−0.573407
$100$	−2.78607	−0.278607
$101$	2.90690	0.289247	0.144624	−	0.989487i	$-0.453803\pi$
$101$	2.90690	0.289247	0.144624	+	0.989487i	$0.453803\pi$
$102$	−7.88503	−0.780734
$103$	−12.0866	−1.19093	−0.595464	−	0.803382i	$-0.703032\pi$
$103$	−12.0866	−1.19093	−0.595464	+	0.803382i	$0.703032\pi$
$104$	2.24985	0.220616
$105$	1.56155	0.152392
$106$	−2.87689	−0.279429
$107$	−7.03712	−0.680304	−0.340152	−	0.940370i	$-0.610479\pi$
$107$	−7.03712	−0.680304	−0.340152	+	0.940370i	$0.610479\pi$
$108$	10.0613	0.968145
$109$	−4.91926	−0.471180	−0.235590	−	0.971853i	$-0.575702\pi$
$109$	−4.91926	−0.471180	−0.235590	+	0.971853i	$0.575702\pi$
$110$	1.34762	0.128491
$111$	−5.63882	−0.535213
$112$	0.344151	0.0325192
$113$	10.5108	0.988775	0.494387	−	0.869242i	$-0.335392\pi$
$113$	10.5108	0.988775	0.494387	+	0.869242i	$0.335392\pi$
$114$	2.09896	0.196586
$115$	2.17623	0.202934
$116$	−4.24274	−0.393929
$117$	−14.1726	−1.31026
$118$	−11.0118	−1.01372
$119$	0.889869	0.0815742
$120$	−4.53741	−0.414207
$121$	−10.1797	−0.925427
$122$	1.75015	0.158451
$123$	3.71244	0.334740
$124$	1.92985	0.173305
$125$	−11.5851	−1.03620
$126$	−2.16792	−0.193134
$127$	−20.0830	−1.78207	−0.891037	−	0.453930i	$-0.850021\pi$
$127$	−20.0830	−1.78207	−0.891037	+	0.453930i	$0.850021\pi$
$128$	−1.00000	−0.0883883
$129$	−20.0972	−1.76946
$130$	3.34762	0.293606
$131$	−8.54572	−0.746643	−0.373321	−	0.927702i	$-0.621781\pi$
$131$	−8.54572	−0.746643	−0.373321	+	0.927702i	$0.621781\pi$
$132$	−2.76193	−0.240395
$133$	−0.236879	−0.0205401
$134$	−4.34415	−0.375277
$135$	14.9704	1.28845
$136$	−2.58570	−0.221722
$137$	−15.0625	−1.28687	−0.643436	−	0.765500i	$-0.722492\pi$
$137$	−15.0625	−1.28687	−0.643436	+	0.765500i	$0.722492\pi$
$138$	−4.46014	−0.379673
$139$	−2.85030	−0.241760	−0.120880	−	0.992667i	$-0.538572\pi$
$139$	−2.85030	−0.241760	−0.120880	+	0.992667i	$0.538572\pi$
$140$	0.512072	0.0432780
$141$	−19.5851	−1.64936
$142$	−5.29586	−0.444419
$143$	2.03770	0.170401
$144$	6.29934	0.524945
$145$	−6.31289	−0.524257
$146$	−5.90206	−0.488458
$147$	−20.9852	−1.73083
$148$	−1.84911	−0.151996
$149$	−5.24502	−0.429689	−0.214844	−	0.976648i	$-0.568924\pi$
$149$	−5.24502	−0.429689	−0.214844	+	0.976648i	$0.568924\pi$
$150$	8.49607	0.693701
$151$	12.0613	0.981532	0.490766	−	0.871292i	$-0.336717\pi$
$151$	12.0613	0.981532	0.490766	+	0.871292i	$0.336717\pi$
$152$	0.688301	0.0558286
$153$	16.2882	1.31682
$154$	0.311699	0.0251174
$155$	2.87147	0.230642
$156$	−6.86089	−0.549311
$157$	−14.4190	−1.15076	−0.575380	−	0.817887i	$-0.695146\pi$
$157$	−14.4190	−1.15076	−0.575380	+	0.817887i	$0.695146\pi$
$158$	15.1107	1.20215
$159$	8.77304	0.695747
$160$	−1.48793	−0.117631
$161$	0.503352	0.0396697
$162$	−11.7836	−0.925809
$163$	20.9562	1.64142	0.820708	−	0.571347i	$-0.193579\pi$
$163$	20.9562	1.64142	0.820708	+	0.571347i	$0.193579\pi$
$164$	1.21740	0.0950631
$165$	−4.10955	−0.319928
$166$	−15.7701	−1.22399
$167$	−2.59520	−0.200823	−0.100411	−	0.994946i	$-0.532016\pi$
$167$	−2.59520	−0.200823	−0.100411	+	0.994946i	$0.532016\pi$
$168$	−1.04948	−0.0809692
$169$	−7.93816	−0.610627
$170$	−3.84733	−0.295077
$171$	−4.33584	−0.331570
$172$	−6.59036	−0.502510
$173$	−15.6629	−1.19083	−0.595414	−	0.803419i	$-0.703012\pi$
$173$	−15.6629	−1.19083	−0.595414	+	0.803419i	$0.703012\pi$
$174$	12.9382	0.980839
$175$	−0.958828	−0.0724806
$176$	−0.905704	−0.0682700
$177$	33.5802	2.52404
$178$	1.61934	0.121375
$179$	25.1868	1.88255	0.941273	−	0.337646i	$-0.109631\pi$
$179$	25.1868	1.88255	0.941273	+	0.337646i	$0.109631\pi$
$180$	9.37296	0.698619
$181$	6.06957	0.451148	0.225574	−	0.974226i	$-0.427574\pi$
$181$	6.06957	0.451148	0.225574	+	0.974226i	$0.427574\pi$
$182$	0.774289	0.0573941
$183$	−5.33704	−0.394525
$184$	−1.46259	−0.107824
$185$	−2.75134	−0.202283
$186$	−5.88503	−0.431511
$187$	−2.34187	−0.171255
$188$	−6.42244	−0.468405
$189$	3.46259	0.251866
$190$	1.02414	0.0742992
$191$	8.11319	0.587050	0.293525	−	0.955951i	$-0.405172\pi$
$191$	8.11319	0.587050	0.293525	+	0.955951i	$0.405172\pi$
$192$	3.04948	0.220077
$193$	−12.8968	−0.928333	−0.464166	−	0.885748i	$-0.653646\pi$
$193$	−12.8968	−0.928333	−0.464166	+	0.885748i	$0.653646\pi$
$194$	−14.3436	−1.02981
$195$	−10.2085	−0.731047
$196$	−6.88156	−0.491540
$197$	14.4920	1.03251	0.516257	−	0.856434i	$-0.327325\pi$
$197$	14.4920	1.03251	0.516257	+	0.856434i	$0.327325\pi$
$198$	5.70533	0.405460
$199$	20.5344	1.45564	0.727822	−	0.685766i	$-0.240533\pi$
$199$	20.5344	1.45564	0.727822	+	0.685766i	$0.240533\pi$
$200$	2.78607	0.197005
$201$	13.2474	0.934400
$202$	−2.90690	−0.204529
$203$	−1.46014	−0.102482
$204$	7.88503	0.552063
$205$	1.81141	0.126514
$206$	12.0866	0.842113
$207$	9.21335	0.640372
$208$	−2.24985	−0.155999
$209$	0.623397	0.0431213
$210$	−1.56155	−0.107757
$211$	−6.99356	−0.481456	−0.240728	−	0.970593i	$-0.577386\pi$
$211$	−6.99356	−0.481456	−0.240728	+	0.970593i	$0.577386\pi$
$212$	2.87689	0.197586
$213$	16.1496	1.10655
$214$	7.03712	0.481047
$215$	−9.80599	−0.668763
$216$	−10.0613	−0.684582
$217$	0.664158	0.0450860
$218$	4.91926	0.333174
$219$	17.9982	1.21621
$220$	−1.34762	−0.0908567
$221$	−5.81744	−0.391323
$222$	5.63882	0.378453
$223$	−8.26630	−0.553552	−0.276776	−	0.960934i	$-0.589266\pi$
$223$	−8.26630	−0.553552	−0.276776	+	0.960934i	$0.589266\pi$
$224$	−0.344151	−0.0229945
$225$	−17.5504	−1.17003
$226$	−10.5108	−0.699169
$227$	−3.27586	−0.217427	−0.108713	−	0.994073i	$-0.534673\pi$
$227$	−3.27586	−0.217427	−0.108713	+	0.994073i	$0.534673\pi$
$228$	−2.09896	−0.139007
$229$	−3.04601	−0.201286	−0.100643	−	0.994923i	$-0.532090\pi$
$229$	−3.04601	−0.201286	−0.100643	+	0.994923i	$0.532090\pi$
$230$	−2.17623	−0.143496
$231$	−0.950519	−0.0625395
$232$	4.24274	0.278550
$233$	12.9540	0.848644	0.424322	−	0.905511i	$-0.360512\pi$
$233$	12.9540	0.848644	0.424322	+	0.905511i	$0.360512\pi$
$234$	14.1726	0.926491
$235$	−9.55613	−0.623373
$236$	11.0118	0.716806
$237$	−46.0799	−2.99321
$238$	−0.889869	−0.0576817
$239$	8.68652	0.561885	0.280942	−	0.959725i	$-0.409353\pi$
$239$	8.68652	0.561885	0.280942	+	0.959725i	$0.409353\pi$
$240$	4.53741	0.292889
$241$	−15.8491	−1.02093	−0.510465	−	0.859898i	$-0.670527\pi$
$241$	−15.8491	−1.02093	−0.510465	+	0.859898i	$0.670527\pi$
$242$	10.1797	0.654376
$243$	5.75015	0.368872
$244$	−1.75015	−0.112042
$245$	−10.2393	−0.654163
$246$	−3.71244	−0.236697
$247$	1.54858	0.0985337
$248$	−1.92985	−0.122545
$249$	48.0905	3.04761
$250$	11.5851	0.732707
$251$	4.61103	0.291046	0.145523	−	0.989355i	$-0.453514\pi$
$251$	4.61103	0.291046	0.145523	+	0.989355i	$0.453514\pi$
$252$	2.16792	0.136566
$253$	−1.32467	−0.0832815
$254$	20.0830	1.26012
$255$	11.7324	0.734709
$256$	1.00000	0.0625000
$257$	20.2182	1.26118	0.630588	−	0.776118i	$-0.282814\pi$
$257$	20.2182	1.26118	0.630588	+	0.776118i	$0.282814\pi$
$258$	20.0972	1.25120
$259$	−0.636372	−0.0395422
$260$	−3.34762	−0.207611
$261$	−26.7264	−1.65433
$262$	8.54572	0.527956
$263$	15.7471	0.971009	0.485504	−	0.874234i	$-0.338636\pi$
$263$	15.7471	0.971009	0.485504	+	0.874234i	$0.338636\pi$
$264$	2.76193	0.169985
$265$	4.28061	0.262956
$266$	0.236879	0.0145240
$267$	−4.93816	−0.302210
$268$	4.34415	0.265361
$269$	1.00000	0.0609711
$269$	1.00000	0.0609711
$270$	−14.9704	−0.911072
$271$	17.3316	1.05282	0.526410	−	0.850231i	$-0.323538\pi$
$271$	17.3316	1.05282	0.526410	+	0.850231i	$0.323538\pi$
$272$	2.58570	0.156781
$273$	−2.36118	−0.142905
$274$	15.0625	0.909956
$275$	2.52335	0.152164
$276$	4.46014	0.268469
$277$	−23.1578	−1.39142	−0.695708	−	0.718325i	$-0.744909\pi$
$277$	−23.1578	−1.39142	−0.695708	+	0.718325i	$0.744909\pi$
$278$	2.85030	0.170950
$279$	12.1568	0.727806
$280$	−0.512072	−0.0306021
$281$	−14.0106	−0.835801	−0.417901	−	0.908493i	$-0.637234\pi$
$281$	−14.0106	−0.835801	−0.417901	+	0.908493i	$0.637234\pi$
$282$	19.5851	1.16628
$283$	26.1135	1.55229	0.776145	−	0.630555i	$-0.217173\pi$
$283$	26.1135	1.55229	0.776145	+	0.630555i	$0.217173\pi$
$284$	5.29586	0.314252
$285$	−3.12311	−0.184997
$286$	−2.03770	−0.120492
$287$	0.418970	0.0247310
$288$	−6.29934	−0.371192
$289$	−10.3142	−0.606716
$290$	6.31289	0.370706
$291$	43.7404	2.56411
$292$	5.90206	0.345392
$293$	−8.14964	−0.476107	−0.238053	−	0.971252i	$-0.576509\pi$
$293$	−8.14964	−0.476107	−0.238053	+	0.971252i	$0.576509\pi$
$294$	20.9852	1.22388
$295$	16.3847	0.953956
$296$	1.84911	0.107477
$297$	−9.11252	−0.528762
$298$	5.24502	0.303836
$299$	−3.29062	−0.190301
$300$	−8.49607	−0.490521
$301$	−2.26808	−0.130730
$302$	−12.0613	−0.694048
$303$	8.86453	0.509254
$304$	−0.688301	−0.0394768
$305$	−2.60409	−0.149110
$306$	−16.2882	−0.931133
$307$	0.809130	0.0461795	0.0230898	−	0.999733i	$-0.492650\pi$
$307$	0.809130	0.0461795	0.0230898	+	0.999733i	$0.492650\pi$
$308$	−0.311699	−0.0177607
$309$	−36.8579	−2.09677
$310$	−2.87147	−0.163089
$311$	−0.444367	−0.0251977	−0.0125989	−	0.999921i	$-0.504010\pi$
$311$	−0.444367	−0.0251977	−0.0125989	+	0.999921i	$0.504010\pi$
$312$	6.86089	0.388421
$313$	1.05085	0.0593974	0.0296987	−	0.999559i	$-0.490545\pi$
$313$	1.05085	0.0593974	0.0296987	+	0.999559i	$0.490545\pi$
$314$	14.4190	0.813710
$315$	3.22571	0.181748
$316$	−15.1107	−0.850046
$317$	6.46784	0.363270	0.181635	−	0.983366i	$-0.441861\pi$
$317$	6.46784	0.363270	0.181635	+	0.983366i	$0.441861\pi$
$318$	−8.77304	−0.491967
$319$	3.84266	0.215148
$320$	1.48793	0.0831777
$321$	−21.4596	−1.19776
$322$	−0.503352	−0.0280507
$323$	−1.77974	−0.0990273
$324$	11.7836	0.654646
$325$	6.26825	0.347700
$326$	−20.9562	−1.16066
$327$	−15.0012	−0.829568
$328$	−1.21740	−0.0672198
$329$	−2.21029	−0.121857
$330$	4.10955	0.226223
$331$	22.0153	1.21007	0.605034	−	0.796200i	$-0.293160\pi$
$331$	22.0153	1.21007	0.605034	+	0.796200i	$0.293160\pi$
$332$	15.7701	0.865495
$333$	−11.6482	−0.638315
$334$	2.59520	0.142003
$335$	6.46379	0.353154
$336$	1.04948	0.0572539
$337$	14.2647	0.777047	0.388524	−	0.921439i	$-0.372985\pi$
$337$	14.2647	0.777047	0.388524	+	0.921439i	$0.372985\pi$
$338$	7.93816	0.431779
$339$	32.0525	1.74086
$340$	3.84733	0.208651
$341$	−1.74787	−0.0946524
$342$	4.33584	0.234455
$343$	−4.77735	−0.257953
$344$	6.59036	0.355329
$345$	6.63637	0.357290
$346$	15.6629	0.842043
$347$	26.0825	1.40018	0.700092	−	0.714052i	$-0.253142\pi$
$347$	26.0825	1.40018	0.700092	+	0.714052i	$0.253142\pi$
$348$	−12.9382	−0.693558
$349$	24.5684	1.31512	0.657559	−	0.753403i	$-0.271589\pi$
$349$	24.5684	1.31512	0.657559	+	0.753403i	$0.271589\pi$
$350$	0.958828	0.0512515
$351$	−22.6364	−1.20824
$352$	0.905704	0.0482742
$353$	8.36823	0.445396	0.222698	−	0.974887i	$-0.428514\pi$
$353$	8.36823	0.445396	0.222698	+	0.974887i	$0.428514\pi$
$354$	−33.5802	−1.78477
$355$	7.87987	0.418220
$356$	−1.61934	−0.0858250
$357$	2.71364	0.143621
$358$	−25.1868	−1.33116
$359$	−2.12835	−0.112330	−0.0561651	−	0.998421i	$-0.517887\pi$
$359$	−2.12835	−0.112330	−0.0561651	+	0.998421i	$0.517887\pi$
$360$	−9.37296	−0.493998
$361$	−18.5262	−0.975065
$362$	−6.06957	−0.319010
$363$	−31.0428	−1.62932
$364$	−0.774289	−0.0405838
$365$	8.78184	0.459663
$366$	5.33704	0.278971
$367$	9.53496	0.497721	0.248860	−	0.968539i	$-0.419944\pi$
$367$	9.53496	0.497721	0.248860	+	0.968539i	$0.419944\pi$
$368$	1.46259	0.0762428
$369$	7.66882	0.399223
$370$	2.75134	0.143035
$371$	0.990085	0.0514027
$372$	5.88503	0.305125
$373$	16.7808	0.868875	0.434437	−	0.900702i	$-0.356947\pi$
$373$	16.7808	0.868875	0.434437	+	0.900702i	$0.356947\pi$
$374$	2.34187	0.121095
$375$	−35.3286	−1.82436
$376$	6.42244	0.331212
$377$	9.54555	0.491621
$378$	−3.46259	−0.178096
$379$	−29.0341	−1.49138	−0.745690	−	0.666293i	$-0.767880\pi$
$379$	−29.0341	−1.49138	−0.745690	+	0.666293i	$0.767880\pi$
$380$	−1.02414	−0.0525374
$381$	−61.2426	−3.13755
$382$	−8.11319	−0.415107
$383$	35.5798	1.81804	0.909022	−	0.416749i	$-0.136831\pi$
$383$	35.5798	1.81804	0.909022	+	0.416749i	$0.136831\pi$
$384$	−3.04948	−0.155618
$385$	−0.463785	−0.0236367
$386$	12.8968	0.656430
$387$	−41.5149	−2.11032
$388$	14.3436	0.728184
$389$	−37.5839	−1.90558	−0.952789	−	0.303634i	$-0.901800\pi$
$389$	−37.5839	−1.90558	−0.952789	+	0.303634i	$0.901800\pi$
$390$	10.2085	0.516928
$391$	3.78181	0.191255
$392$	6.88156	0.347571
$393$	−26.0600	−1.31455
$394$	−14.4920	−0.730097
$395$	−22.4837	−1.13128
$396$	−5.70533	−0.286704
$397$	−16.8913	−0.847750	−0.423875	−	0.905721i	$-0.639330\pi$
$397$	−16.8913	−0.847750	−0.423875	+	0.905721i	$0.639330\pi$
$398$	−20.5344	−1.02930
$399$	−0.722359	−0.0361632
$400$	−2.78607	−0.139303
$401$	3.68585	0.184063	0.0920314	−	0.995756i	$-0.470664\pi$
$401$	3.68585	0.184063	0.0920314	+	0.995756i	$0.470664\pi$
$402$	−13.2474	−0.660721
$403$	−4.34187	−0.216284
$404$	2.90690	0.144624
$405$	17.5332	0.871231
$406$	1.46014	0.0724656
$407$	1.67474	0.0830140
$408$	−7.88503	−0.390367
$409$	18.5784	0.918643	0.459322	−	0.888270i	$-0.348093\pi$
$409$	18.5784	0.918643	0.459322	+	0.888270i	$0.348093\pi$
$410$	−1.81141	−0.0894590
$411$	−45.9327	−2.26569
$412$	−12.0866	−0.595464
$413$	3.78971	0.186480
$414$	−9.21335	−0.452811
$415$	23.4647	1.15184
$416$	2.24985	0.110308
$417$	−8.69194	−0.425646
$418$	−0.623397	−0.0304913
$419$	3.30890	0.161650	0.0808251	−	0.996728i	$-0.474244\pi$
$419$	3.30890	0.161650	0.0808251	+	0.996728i	$0.474244\pi$
$420$	1.56155	0.0761960
$421$	7.23994	0.352853	0.176427	−	0.984314i	$-0.443546\pi$
$421$	7.23994	0.352853	0.176427	+	0.984314i	$0.443546\pi$
$422$	6.99356	0.340441
$423$	−40.4571	−1.96709
$424$	−2.87689	−0.139714
$425$	−7.20393	−0.349442
$426$	−16.1496	−0.782453
$427$	−0.602314	−0.0291480
$428$	−7.03712	−0.340152
$429$	6.21393	0.300011
$430$	9.80599	0.472887
$431$	−11.1645	−0.537777	−0.268888	−	0.963171i	$-0.586656\pi$
$431$	−11.1645	−0.537777	−0.268888	+	0.963171i	$0.586656\pi$
$432$	10.0613	0.484073
$433$	28.8143	1.38473	0.692363	−	0.721549i	$-0.256570\pi$
$433$	28.8143	1.38473	0.692363	+	0.721549i	$0.256570\pi$
$434$	−0.664158	−0.0318806
$435$	−19.2510	−0.923017
$436$	−4.91926	−0.235590
$437$	−1.00670	−0.0481571
$438$	−17.9982	−0.859988
$439$	20.4897	0.977922	0.488961	−	0.872306i	$-0.337376\pi$
$439$	20.4897	0.977922	0.488961	+	0.872306i	$0.337376\pi$
$440$	1.34762	0.0642454
$441$	−43.3493	−2.06425
$442$	5.81744	0.276707
$443$	9.58503	0.455398	0.227699	−	0.973732i	$-0.426880\pi$
$443$	9.58503	0.455398	0.227699	+	0.973732i	$0.426880\pi$
$444$	−5.63882	−0.267607
$445$	−2.40947	−0.114220
$446$	8.26630	0.391421
$447$	−15.9946	−0.756518
$448$	0.344151	0.0162596
$449$	−21.4476	−1.01218	−0.506088	−	0.862482i	$-0.668909\pi$
$449$	−21.4476	−1.01218	−0.506088	+	0.862482i	$0.668909\pi$
$450$	17.5504	0.827333
$451$	−1.10261	−0.0519197
$452$	10.5108	0.494387
$453$	36.7806	1.72810
$454$	3.27586	0.153744
$455$	−1.15209	−0.0540107
$456$	2.09896	0.0982929
$457$	10.7176	0.501350	0.250675	−	0.968071i	$-0.419347\pi$
$457$	10.7176	0.501350	0.250675	+	0.968071i	$0.419347\pi$
$458$	3.04601	0.142331
$459$	26.0154	1.21429
$460$	2.17623	0.101467
$461$	−19.6658	−0.915926	−0.457963	−	0.888971i	$-0.651421\pi$
$461$	−19.6658	−0.915926	−0.457963	+	0.888971i	$0.651421\pi$
$462$	0.950519	0.0442221
$463$	22.6226	1.05136	0.525682	−	0.850681i	$-0.323810\pi$
$463$	22.6226	1.05136	0.525682	+	0.850681i	$0.323810\pi$
$464$	−4.24274	−0.196964
$465$	8.75651	0.406073
$466$	−12.9540	−0.600082
$467$	30.0503	1.39056	0.695282	−	0.718737i	$-0.255280\pi$
$467$	30.0503	1.39056	0.695282	+	0.718737i	$0.255280\pi$
$468$	−14.1726	−0.655128
$469$	1.49504	0.0690347
$470$	9.55613	0.440792
$471$	−43.9704	−2.02605
$472$	−11.0118	−0.506858
$473$	5.96891	0.274451
$474$	46.0799	2.11652
$475$	1.91766	0.0879881
$476$	0.889869	0.0407871
$477$	18.1225	0.829773
$478$	−8.68652	−0.397312
$479$	−32.6251	−1.49068	−0.745339	−	0.666685i	$-0.767712\pi$
$479$	−32.6251	−1.49068	−0.745339	+	0.666685i	$0.767712\pi$
$480$	−4.53741	−0.207103
$481$	4.16022	0.189690
$482$	15.8491	0.721907
$483$	1.53496	0.0698432
$484$	−10.1797	−0.462714
$485$	21.3422	0.969099
$486$	−5.75015	−0.260832
$487$	−4.12981	−0.187139	−0.0935697	−	0.995613i	$-0.529828\pi$
$487$	−4.12981	−0.187139	−0.0935697	+	0.995613i	$0.529828\pi$
$488$	1.75015	0.0792254
$489$	63.9056	2.88991
$490$	10.2393	0.462563
$491$	−35.2225	−1.58957	−0.794785	−	0.606891i	$-0.792416\pi$
$491$	−35.2225	−1.58957	−0.794785	+	0.606891i	$0.792416\pi$
$492$	3.71244	0.167370
$493$	−10.9704	−0.494084
$494$	−1.54858	−0.0696738
$495$	−8.48912	−0.381558
$496$	1.92985	0.0866527
$497$	1.82258	0.0817537
$498$	−48.0905	−2.15499
$499$	16.3253	0.730819	0.365409	−	0.930847i	$-0.380929\pi$
$499$	16.3253	0.730819	0.365409	+	0.930847i	$0.380929\pi$
$500$	−11.5851	−0.518102
$501$	−7.91401	−0.353572
$502$	−4.61103	−0.205801
$503$	39.0571	1.74147	0.870736	−	0.491751i	$-0.163643\pi$
$503$	39.0571	1.74147	0.870736	+	0.491751i	$0.163643\pi$
$504$	−2.16792	−0.0965669
$505$	4.32526	0.192471
$506$	1.32467	0.0588889
$507$	−24.2073	−1.07508
$508$	−20.0830	−0.891037
$509$	8.19139	0.363077	0.181539	−	0.983384i	$-0.441892\pi$
$509$	8.19139	0.363077	0.181539	+	0.983384i	$0.441892\pi$
$510$	−11.7324	−0.519518
$511$	2.03120	0.0898549
$512$	−1.00000	−0.0441942
$513$	−6.92518	−0.305754
$514$	−20.2182	−0.891786
$515$	−17.9840	−0.792469
$516$	−20.0972	−0.884729
$517$	5.81683	0.255824
$518$	0.636372	0.0279606
$519$	−47.7637	−2.09660
$520$	3.34762	0.146803
$521$	−9.81960	−0.430205	−0.215102	−	0.976592i	$-0.569009\pi$
$521$	−9.81960	−0.430205	−0.215102	+	0.976592i	$0.569009\pi$
$522$	26.7264	1.16978
$523$	4.42973	0.193698	0.0968492	−	0.995299i	$-0.469124\pi$
$523$	4.42973	0.193698	0.0968492	+	0.995299i	$0.469124\pi$
$524$	−8.54572	−0.373321
$525$	−2.92393	−0.127611
$526$	−15.7471	−0.686607
$527$	4.99000	0.217368
$528$	−2.76193	−0.120197
$529$	−20.8608	−0.906993
$530$	−4.28061	−0.185938
$531$	69.3669	3.01027
$532$	−0.236879	−0.0102700
$533$	−2.73898	−0.118638
$534$	4.93816	0.213695
$535$	−10.4707	−0.452689
$536$	−4.34415	−0.187639
$537$	76.8065	3.31445
$538$	−1.00000	−0.0431131
$539$	6.23265	0.268459
$540$	14.9704	0.644225
$541$	3.01458	0.129607	0.0648035	−	0.997898i	$-0.479358\pi$
$541$	3.01458	0.129607	0.0648035	+	0.997898i	$0.479358\pi$
$542$	−17.3316	−0.744457
$543$	18.5090	0.794299
$544$	−2.58570	−0.110861
$545$	−7.31951	−0.313533
$546$	2.36118	0.101049
$547$	35.5411	1.51963	0.759813	−	0.650141i	$-0.225290\pi$
$547$	35.5411	1.51963	0.759813	+	0.650141i	$0.225290\pi$
$548$	−15.0625	−0.643436
$549$	−11.0248	−0.470525
$550$	−2.52335	−0.107596
$551$	2.92028	0.124408
$552$	−4.46014	−0.189836
$553$	−5.20037	−0.221142
$554$	23.1578	0.983880
$555$	−8.39016	−0.356142
$556$	−2.85030	−0.120880
$557$	21.5510	0.913144	0.456572	−	0.889687i	$-0.349077\pi$
$557$	21.5510	0.913144	0.456572	+	0.889687i	$0.349077\pi$
$558$	−12.1568	−0.514636
$559$	14.8274	0.627130
$560$	0.512072	0.0216390
$561$	−7.14150	−0.301514
$562$	14.0106	0.591001
$563$	30.3310	1.27830	0.639150	−	0.769082i	$-0.279286\pi$
$563$	30.3310	1.27830	0.639150	+	0.769082i	$0.279286\pi$
$564$	−19.5851	−0.824682
$565$	15.6393	0.657952
$566$	−26.1135	−1.09763
$567$	4.05534	0.170308
$568$	−5.29586	−0.222210
$569$	3.78038	0.158482	0.0792409	−	0.996855i	$-0.474750\pi$
$569$	3.78038	0.158482	0.0792409	+	0.996855i	$0.474750\pi$
$570$	3.12311	0.130812
$571$	18.4336	0.771424	0.385712	−	0.922619i	$-0.373956\pi$
$571$	18.4336	0.771424	0.385712	+	0.922619i	$0.373956\pi$
$572$	2.03770	0.0852006
$573$	24.7410	1.03357
$574$	−0.418970	−0.0174875
$575$	−4.07488	−0.169934
$576$	6.29934	0.262472
$577$	30.3922	1.26524	0.632622	−	0.774461i	$-0.281979\pi$
$577$	30.3922	1.26524	0.632622	+	0.774461i	$0.281979\pi$
$578$	10.3142	0.429013
$579$	−39.3286	−1.63444
$580$	−6.31289	−0.262129
$581$	5.42728	0.225161
$582$	−43.7404	−1.81310
$583$	−2.60561	−0.107913
$584$	−5.90206	−0.244229
$585$	−21.0878	−0.871873
$586$	8.14964	0.336658
$587$	−38.4248	−1.58596	−0.792981	−	0.609246i	$-0.791472\pi$
$587$	−38.4248	−1.58596	−0.792981	+	0.609246i	$0.791472\pi$
$588$	−20.9852	−0.865415
$589$	−1.32832	−0.0547323
$590$	−16.3847	−0.674549
$591$	44.1931	1.81786
$592$	−1.84911	−0.0759979
$593$	15.0010	0.616018	0.308009	−	0.951383i	$-0.400337\pi$
$593$	15.0010	0.616018	0.308009	+	0.951383i	$0.400337\pi$
$594$	9.11252	0.373891
$595$	1.32406	0.0542812
$596$	−5.24502	−0.214844
$597$	62.6192	2.56283
$598$	3.29062	0.134563
$599$	−8.67407	−0.354413	−0.177207	−	0.984174i	$-0.556706\pi$
$599$	−8.67407	−0.354413	−0.177207	+	0.984174i	$0.556706\pi$
$600$	8.49607	0.346850
$601$	−20.1244	−0.820891	−0.410445	−	0.911885i	$-0.634627\pi$
$601$	−20.1244	−0.820891	−0.410445	+	0.911885i	$0.634627\pi$
$602$	2.26808	0.0924400
$603$	27.3653	1.11440
$604$	12.0613	0.490766
$605$	−15.1467	−0.615800
$606$	−8.86453	−0.360097
$607$	5.35730	0.217446	0.108723	−	0.994072i	$-0.465324\pi$
$607$	5.35730	0.217446	0.108723	+	0.994072i	$0.465324\pi$
$608$	0.688301	0.0279143
$609$	−4.45268	−0.180432
$610$	2.60409	0.105437
$611$	14.4496	0.584566
$612$	16.2882	0.658410
$613$	21.6009	0.872454	0.436227	−	0.899837i	$-0.356314\pi$
$613$	21.6009	0.872454	0.436227	+	0.899837i	$0.356314\pi$
$614$	−0.809130	−0.0326538
$615$	5.52385	0.222743
$616$	0.311699	0.0125587
$617$	−36.2231	−1.45829	−0.729144	−	0.684360i	$-0.760082\pi$
$617$	−36.2231	−1.45829	−0.729144	+	0.684360i	$0.760082\pi$
$618$	36.8579	1.48264
$619$	31.9927	1.28590	0.642948	−	0.765910i	$-0.277711\pi$
$619$	31.9927	1.28590	0.642948	+	0.765910i	$0.277711\pi$
$620$	2.87147	0.115321
$621$	14.7155	0.590513
$622$	0.444367	0.0178175
$623$	−0.557298	−0.0223277
$624$	−6.86089	−0.274655
$625$	−3.30747	−0.132299
$626$	−1.05085	−0.0420003
$627$	1.90104	0.0759201
$628$	−14.4190	−0.575380
$629$	−4.78123	−0.190640
$630$	−3.22571	−0.128515
$631$	−22.3539	−0.889894	−0.444947	−	0.895557i	$-0.646778\pi$
$631$	−22.3539	−0.889894	−0.444947	+	0.895557i	$0.646778\pi$
$632$	15.1107	0.601073
$633$	−21.3267	−0.847661
$634$	−6.46784	−0.256871
$635$	−29.8820	−1.18583
$636$	8.77304	0.347873
$637$	15.4825	0.613439
$638$	−3.84266	−0.152133
$639$	33.3604	1.31972
$640$	−1.48793	−0.0588155
$641$	−8.18573	−0.323317	−0.161659	−	0.986847i	$-0.551684\pi$
$641$	−8.18573	−0.323317	−0.161659	+	0.986847i	$0.551684\pi$
$642$	21.4596	0.846941
$643$	17.4703	0.688961	0.344480	−	0.938793i	$-0.388055\pi$
$643$	17.4703	0.688961	0.344480	+	0.938793i	$0.388055\pi$
$644$	0.503352	0.0198348
$645$	−29.9032	−1.17744
$646$	1.77974	0.0700229
$647$	−44.5260	−1.75050	−0.875249	−	0.483673i	$-0.839303\pi$
$647$	−44.5260	−1.75050	−0.875249	+	0.483673i	$0.839303\pi$
$648$	−11.7836	−0.462904
$649$	−9.97341	−0.391491
$650$	−6.26825	−0.245861
$651$	2.02534	0.0793793
$652$	20.9562	0.820708
$653$	37.0875	1.45134	0.725672	−	0.688041i	$-0.241529\pi$
$653$	37.0875	1.45134	0.725672	+	0.688041i	$0.241529\pi$
$654$	15.0012	0.586593
$655$	−12.7154	−0.496832
$656$	1.21740	0.0475316
$657$	37.1791	1.45049
$658$	2.21029	0.0861660
$659$	15.6928	0.611305	0.305652	−	0.952143i	$-0.401126\pi$
$659$	15.6928	0.611305	0.305652	+	0.952143i	$0.401126\pi$
$660$	−4.10955	−0.159964
$661$	4.10083	0.159504	0.0797519	−	0.996815i	$-0.474587\pi$
$661$	4.10083	0.159504	0.0797519	+	0.996815i	$0.474587\pi$
$662$	−22.0153	−0.855647
$663$	−17.7402	−0.688971
$664$	−15.7701	−0.611997
$665$	−0.352460	−0.0136678
$666$	11.6482	0.451357
$667$	−6.20539	−0.240274
$668$	−2.59520	−0.100411
$669$	−25.2079	−0.974595
$670$	−6.46379	−0.249718
$671$	1.58511	0.0611926
$672$	−1.04948	−0.0404846
$673$	8.56587	0.330190	0.165095	−	0.986278i	$-0.447207\pi$
$673$	8.56587	0.330190	0.165095	+	0.986278i	$0.447207\pi$
$674$	−14.2647	−0.549456
$675$	−28.0314	−1.07893
$676$	−7.93816	−0.305314
$677$	15.2255	0.585165	0.292583	−	0.956240i	$-0.405485\pi$
$677$	15.2255	0.585165	0.292583	+	0.956240i	$0.405485\pi$
$678$	−32.0525	−1.23097
$679$	4.93635	0.189440
$680$	−3.84733	−0.147538
$681$	−9.98968	−0.382805
$682$	1.74787	0.0669294
$683$	−34.5265	−1.32112	−0.660560	−	0.750774i	$-0.729681\pi$
$683$	−34.5265	−1.32112	−0.660560	+	0.750774i	$0.729681\pi$
$684$	−4.33584	−0.165785
$685$	−22.4119	−0.856313
$686$	4.77735	0.182400
$687$	−9.28875	−0.354388
$688$	−6.59036	−0.251255
$689$	−6.47259	−0.246586
$690$	−6.63637	−0.252642
$691$	32.7649	1.24644	0.623218	−	0.782048i	$-0.285825\pi$
$691$	32.7649	1.24644	0.623218	+	0.782048i	$0.285825\pi$
$692$	−15.6629	−0.595414
$693$	−1.96349	−0.0745870
$694$	−26.0825	−0.990080
$695$	−4.24105	−0.160872
$696$	12.9382	0.490420
$697$	3.14783	0.119233
$698$	−24.5684	−0.929929
$699$	39.5029	1.49414
$700$	−0.958828	−0.0362403
$701$	−4.80166	−0.181356	−0.0906782	−	0.995880i	$-0.528903\pi$
$701$	−4.80166	−0.181356	−0.0906782	+	0.995880i	$0.528903\pi$
$702$	22.6364	0.854355
$703$	1.27274	0.0480024
$704$	−0.905704	−0.0341350
$705$	−29.1412	−1.09752
$706$	−8.36823	−0.314943
$707$	1.00041	0.0376243
$708$	33.5802	1.26202
$709$	−34.4618	−1.29424	−0.647120	−	0.762389i	$-0.724027\pi$
$709$	−34.4618	−1.29424	−0.647120	+	0.762389i	$0.724027\pi$
$710$	−7.87987	−0.295726
$711$	−95.1876	−3.56982
$712$	1.61934	0.0606874
$713$	2.82258	0.105706
$714$	−2.71364	−0.101555
$715$	3.03195	0.113389
$716$	25.1868	0.941273
$717$	26.4894	0.989264
$718$	2.12835	0.0794295
$719$	−41.1653	−1.53521	−0.767604	−	0.640925i	$-0.778551\pi$
$719$	−41.1653	−1.53521	−0.767604	+	0.640925i	$0.778551\pi$
$720$	9.37296	0.349310
$721$	−4.15961	−0.154912
$722$	18.5262	0.689475
$723$	−48.3316	−1.79747
$724$	6.06957	0.225574
$725$	11.8206	0.439005
$726$	31.0428	1.15211
$727$	−50.5549	−1.87498	−0.937489	−	0.348015i	$-0.886856\pi$
$727$	−50.5549	−1.87498	−0.937489	+	0.348015i	$0.886856\pi$
$728$	0.774289	0.0286971
$729$	−17.8159	−0.659848
$730$	−8.78184	−0.325031
$731$	−17.0407	−0.630272
$732$	−5.33704	−0.197263
$733$	−29.3797	−1.08516	−0.542581	−	0.840003i	$-0.682553\pi$
$733$	−29.3797	−1.08516	−0.542581	+	0.840003i	$0.682553\pi$
$734$	−9.53496	−0.351942
$735$	−31.2245	−1.15173
$736$	−1.46259	−0.0539118
$737$	−3.93451	−0.144930
$738$	−7.66882	−0.282293
$739$	−22.4975	−0.827584	−0.413792	−	0.910371i	$-0.635796\pi$
$739$	−22.4975	−0.827584	−0.413792	+	0.910371i	$0.635796\pi$
$740$	−2.75134	−0.101141
$741$	4.72236	0.173480
$742$	−0.990085	−0.0363472
$743$	−7.49123	−0.274826	−0.137413	−	0.990514i	$-0.543879\pi$
$743$	−7.49123	−0.274826	−0.137413	+	0.990514i	$0.543879\pi$
$744$	−5.88503	−0.215756
$745$	−7.80421	−0.285924
$746$	−16.7808	−0.614387
$747$	99.3409	3.63469
$748$	−2.34187	−0.0856274
$749$	−2.42183	−0.0884917
$750$	35.3286	1.29002
$751$	31.3441	1.14376	0.571881	−	0.820337i	$-0.306214\pi$
$751$	31.3441	1.14376	0.571881	+	0.820337i	$0.306214\pi$
$752$	−6.42244	−0.234202
$753$	14.0613	0.512421
$754$	−9.54555	−0.347628
$755$	17.9463	0.653132
$756$	3.46259	0.125933
$757$	−44.8626	−1.63056	−0.815280	−	0.579067i	$-0.803417\pi$
$757$	−44.8626	−1.63056	−0.815280	+	0.579067i	$0.803417\pi$
$758$	29.0341	1.05457
$759$	−4.03957	−0.146627
$760$	1.02414	0.0371496
$761$	−18.5983	−0.674189	−0.337094	−	0.941471i	$-0.609444\pi$
$761$	−18.5983	−0.674189	−0.337094	+	0.941471i	$0.609444\pi$
$762$	61.2426	2.21859
$763$	−1.69297	−0.0612895
$764$	8.11319	0.293525
$765$	24.2356	0.876241
$766$	−35.5798	−1.28555
$767$	−24.7749	−0.894570
$768$	3.04948	0.110039
$769$	52.1929	1.88213	0.941063	−	0.338232i	$-0.109829\pi$
$769$	52.1929	1.88213	0.941063	+	0.338232i	$0.109829\pi$
$770$	0.463785	0.0167137
$771$	61.6550	2.22045
$772$	−12.8968	−0.464166
$773$	41.7206	1.50059	0.750293	−	0.661105i	$-0.229912\pi$
$773$	41.7206	1.50059	0.750293	+	0.661105i	$0.229912\pi$
$774$	41.5149	1.49222
$775$	−5.37669	−0.193136
$776$	−14.3436	−0.514904
$777$	−1.94060	−0.0696188
$778$	37.5839	1.34745
$779$	−0.837940	−0.0300223
$780$	−10.2085	−0.365523
$781$	−4.79648	−0.171632
$782$	−3.78181	−0.135237
$783$	−42.6873	−1.52552
$784$	−6.88156	−0.245770
$785$	−21.4544	−0.765740
$786$	26.0600	0.929530
$787$	20.8879	0.744574	0.372287	−	0.928118i	$-0.378574\pi$
$787$	20.8879	0.744574	0.372287	+	0.928118i	$0.378574\pi$
$788$	14.4920	0.516257
$789$	48.0205	1.70958
$790$	22.4837	0.799934
$791$	3.61731	0.128617
$792$	5.70533	0.202730
$793$	3.93757	0.139827
$794$	16.8913	0.599450
$795$	13.0536	0.462965
$796$	20.5344	0.727822
$797$	20.6075	0.729955	0.364978	−	0.931016i	$-0.381077\pi$
$797$	20.6075	0.729955	0.364978	+	0.931016i	$0.381077\pi$
$798$	0.722359	0.0255712
$799$	−16.6065	−0.587495
$800$	2.78607	0.0985024
$801$	−10.2008	−0.360427
$802$	−3.68585	−0.130152
$803$	−5.34552	−0.188639
$804$	13.2474	0.467200
$805$	0.748951	0.0263971
$806$	4.34187	0.152936
$807$	3.04948	0.107347
$808$	−2.90690	−0.102264
$809$	8.27012	0.290762	0.145381	−	0.989376i	$-0.453559\pi$
$809$	8.27012	0.290762	0.145381	+	0.989376i	$0.453559\pi$
$810$	−17.5332	−0.616053
$811$	26.6200	0.934753	0.467377	−	0.884058i	$-0.345199\pi$
$811$	26.6200	0.934753	0.467377	+	0.884058i	$0.345199\pi$
$812$	−1.46014	−0.0512409
$813$	52.8524	1.85362
$814$	−1.67474	−0.0586997
$815$	31.1813	1.09223
$816$	7.88503	0.276031
$817$	4.53616	0.158700
$818$	−18.5784	−0.649579
$819$	−4.87751	−0.170434
$820$	1.81141	0.0632571
$821$	29.4026	1.02616	0.513078	−	0.858342i	$-0.328505\pi$
$821$	29.4026	1.02616	0.513078	+	0.858342i	$0.328505\pi$
$822$	45.9327	1.60209
$823$	15.4716	0.539305	0.269653	−	0.962958i	$-0.413091\pi$
$823$	15.4716	0.539305	0.269653	+	0.962958i	$0.413091\pi$
$824$	12.0866	0.421057
$825$	7.69492	0.267903
$826$	−3.78971	−0.131861
$827$	−24.5579	−0.853963	−0.426982	−	0.904260i	$-0.640423\pi$
$827$	−24.5579	−0.853963	−0.426982	+	0.904260i	$0.640423\pi$
$828$	9.21335	0.320186
$829$	20.1760	0.700741	0.350371	−	0.936611i	$-0.386056\pi$
$829$	20.1760	0.700741	0.350371	+	0.936611i	$0.386056\pi$
$830$	−23.4647	−0.814472
$831$	−70.6192	−2.44975
$832$	−2.24985	−0.0779997
$833$	−17.7936	−0.616513
$834$	8.69194	0.300977
$835$	−3.86147	−0.133632
$836$	0.623397	0.0215606
$837$	19.4167	0.671139
$838$	−3.30890	−0.114304
$839$	−39.9874	−1.38052	−0.690259	−	0.723562i	$-0.742504\pi$
$839$	−39.9874	−1.38052	−0.690259	+	0.723562i	$0.742504\pi$
$840$	−1.56155	−0.0538787
$841$	−10.9992	−0.379281
$842$	−7.23994	−0.249505
$843$	−42.7250	−1.47153
$844$	−6.99356	−0.240728
$845$	−11.8114	−0.406325
$846$	40.4571	1.39094
$847$	−3.50335	−0.120377
$848$	2.87689	0.0987930
$849$	79.6328	2.73299
$850$	7.20393	0.247093
$851$	−2.70449	−0.0927086
$852$	16.1496	0.553277
$853$	−38.2866	−1.31091	−0.655453	−	0.755236i	$-0.727522\pi$
$853$	−38.2866	−1.31091	−0.655453	+	0.755236i	$0.727522\pi$
$854$	0.602314	0.0206108
$855$	−6.45142	−0.220634
$856$	7.03712	0.240524
$857$	10.4287	0.356239	0.178120	−	0.984009i	$-0.442999\pi$
$857$	10.4287	0.356239	0.178120	+	0.984009i	$0.442999\pi$
$858$	−6.21393	−0.212140
$859$	−31.3752	−1.07051	−0.535254	−	0.844691i	$-0.679784\pi$
$859$	−31.3752	−1.07051	−0.535254	+	0.844691i	$0.679784\pi$
$860$	−9.80599	−0.334381
$861$	1.27764	0.0435419
$862$	11.1645	0.380266
$863$	22.9420	0.780954	0.390477	−	0.920613i	$-0.372310\pi$
$863$	22.9420	0.780954	0.390477	+	0.920613i	$0.372310\pi$
$864$	−10.0613	−0.342291
$865$	−23.3053	−0.792403
$866$	−28.8143	−0.979150
$867$	−31.4529	−1.06820
$868$	0.664158	0.0225430
$869$	13.6859	0.464261
$870$	19.2510	0.652672
$871$	−9.77371	−0.331169
$872$	4.91926	0.166587
$873$	90.3549	3.05805
$874$	1.00670	0.0340522
$875$	−3.98703	−0.134786
$876$	17.9982	0.608104
$877$	13.8521	0.467754	0.233877	−	0.972266i	$-0.424859\pi$
$877$	13.8521	0.467754	0.233877	+	0.972266i	$0.424859\pi$
$878$	−20.4897	−0.691495
$879$	−24.8522	−0.838243
$880$	−1.34762	−0.0454283
$881$	20.0782	0.676451	0.338225	−	0.941065i	$-0.390173\pi$
$881$	20.0782	0.676451	0.338225	+	0.941065i	$0.390173\pi$
$882$	43.3493	1.45965
$883$	55.3481	1.86261	0.931306	−	0.364237i	$-0.118670\pi$
$883$	55.3481	1.86261	0.931306	+	0.364237i	$0.118670\pi$
$884$	−5.81744	−0.195662
$885$	49.9650	1.67955
$886$	−9.58503	−0.322015
$887$	1.28823	0.0432544	0.0216272	−	0.999766i	$-0.493115\pi$
$887$	1.28823	0.0432544	0.0216272	+	0.999766i	$0.493115\pi$
$888$	5.63882	0.189226
$889$	−6.91156	−0.231806
$890$	2.40947	0.0807655
$891$	−10.6725	−0.357541
$892$	−8.26630	−0.276776
$893$	4.42058	0.147929
$894$	15.9946	0.534939
$895$	37.4761	1.25269
$896$	−0.344151	−0.0114973
$897$	−10.0347	−0.335048
$898$	21.4476	0.715716
$899$	−8.18784	−0.273080
$900$	−17.5504	−0.585013
$901$	7.43877	0.247822
$902$	1.10261	0.0367127
$903$	−6.91646	−0.230165
$904$	−10.5108	−0.349585
$905$	9.03109	0.300203
$906$	−36.7806	−1.22195
$907$	11.6760	0.387695	0.193848	−	0.981032i	$-0.437903\pi$
$907$	11.6760	0.387695	0.193848	+	0.981032i	$0.437903\pi$
$908$	−3.27586	−0.108713
$909$	18.3115	0.607355
$910$	1.15209	0.0381913
$911$	31.6479	1.04854	0.524271	−	0.851551i	$-0.324338\pi$
$911$	31.6479	1.04854	0.524271	+	0.851551i	$0.324338\pi$
$912$	−2.09896	−0.0695036
$913$	−14.2830	−0.472698
$914$	−10.7176	−0.354508
$915$	−7.94113	−0.262526
$916$	−3.04601	−0.100643
$917$	−2.94102	−0.0971209
$918$	−26.0154	−0.858635
$919$	−49.3145	−1.62674	−0.813368	−	0.581749i	$-0.802368\pi$
$919$	−49.3145	−1.62674	−0.813368	+	0.581749i	$0.802368\pi$
$920$	−2.17623	−0.0717482
$921$	2.46743	0.0813045
$922$	19.6658	0.647657
$923$	−11.9149	−0.392184
$924$	−0.950519	−0.0312698
$925$	5.15174	0.169388
$926$	−22.6226	−0.743426
$927$	−76.1375	−2.50069
$928$	4.24274	0.139275
$929$	−25.7689	−0.845450	−0.422725	−	0.906258i	$-0.638926\pi$
$929$	−25.7689	−0.845450	−0.422725	+	0.906258i	$0.638926\pi$
$930$	−8.75651	−0.287137
$931$	4.73659	0.155235
$932$	12.9540	0.424322
$933$	−1.35509	−0.0443636
$934$	−30.0503	−0.983277
$935$	−3.48454	−0.113957
$936$	14.1726	0.463245
$937$	−6.22816	−0.203465	−0.101733	−	0.994812i	$-0.532439\pi$
$937$	−6.22816	−0.203465	−0.101733	+	0.994812i	$0.532439\pi$
$938$	−1.49504	−0.0488149
$939$	3.20454	0.104576
$940$	−9.55613	−0.311687
$941$	15.3930	0.501799	0.250900	−	0.968013i	$-0.419274\pi$
$941$	15.3930	0.501799	0.250900	+	0.968013i	$0.419274\pi$
$942$	43.9704	1.43263
$943$	1.78056	0.0579830
$944$	11.0118	0.358403
$945$	5.15209	0.167597
$946$	−5.96891	−0.194066
$947$	−19.0683	−0.619635	−0.309818	−	0.950796i	$-0.600268\pi$
$947$	−19.0683	−0.619635	−0.309818	+	0.950796i	$0.600268\pi$
$948$	−46.0799	−1.49661
$949$	−13.2788	−0.431047
$950$	−1.91766	−0.0622170
$951$	19.7236	0.639580
$952$	−0.889869	−0.0288408
$953$	−39.7141	−1.28647	−0.643234	−	0.765670i	$-0.722408\pi$
$953$	−39.7141	−1.28647	−0.643234	+	0.765670i	$0.722408\pi$
$954$	−18.1225	−0.586738
$955$	12.0718	0.390636
$956$	8.68652	0.280942
$957$	11.7181	0.378793
$958$	32.6251	1.05407
$959$	−5.18376	−0.167392
$960$	4.53741	0.146444
$961$	−27.2757	−0.879861
$962$	−4.16022	−0.134131
$963$	−44.3292	−1.42849
$964$	−15.8491	−0.510465
$965$	−19.1895	−0.617733
$966$	−1.53496	−0.0493866
$967$	−54.5429	−1.75398	−0.876991	−	0.480507i	$-0.840453\pi$
$967$	−54.5429	−1.75398	−0.876991	+	0.480507i	$0.840453\pi$
$968$	10.1797	0.327188
$969$	−5.42728	−0.174349
$970$	−21.3422	−0.685257
$971$	−20.6894	−0.663955	−0.331977	−	0.943287i	$-0.607716\pi$
$971$	−20.6894	−0.663955	−0.331977	+	0.943287i	$0.607716\pi$
$972$	5.75015	0.184436
$973$	−0.980934	−0.0314473
$974$	4.12981	0.132328
$975$	19.1149	0.612167
$976$	−1.75015	−0.0560208
$977$	3.49749	0.111895	0.0559473	−	0.998434i	$-0.482182\pi$
$977$	3.49749	0.111895	0.0559473	+	0.998434i	$0.482182\pi$
$978$	−63.9056	−2.04347
$979$	1.46664	0.0468742
$980$	−10.2393	−0.327081
$981$	−30.9881	−0.989373
$982$	35.2225	1.12400
$983$	16.5884	0.529086	0.264543	−	0.964374i	$-0.414779\pi$
$983$	16.5884	0.529086	0.264543	+	0.964374i	$0.414779\pi$
$984$	−3.71244	−0.118348
$985$	21.5631	0.687057
$986$	10.9704	0.349370
$987$	−6.74023	−0.214544
$988$	1.54858	0.0492668
$989$	−9.63900	−0.306502
$990$	8.48912	0.269802
$991$	−27.1297	−0.861805	−0.430902	−	0.902399i	$-0.641805\pi$
$991$	−27.1297	−0.861805	−0.430902	+	0.902399i	$0.641805\pi$
$992$	−1.92985	−0.0612727
$993$	67.1351	2.13047
$994$	−1.82258	−0.0578086
$995$	30.5537	0.968617
$996$	48.0905	1.52381
$997$	56.5029	1.78946	0.894732	−	0.446603i	$-0.147366\pi$
$997$	56.5029	1.78946	0.894732	+	0.446603i	$0.147366\pi$
$998$	−16.3253	−0.516767
$999$	−18.6044	−0.588616

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.2.a.c.1.4	✓	4
3.2	odd	2		4842.2.a.j.1.2		4
4.3	odd	2		4304.2.a.e.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.c.1.4	✓	4	1.1	even	1	trivial
4304.2.a.e.1.1		4	4.3	odd	2
4842.2.a.j.1.2		4	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +3.04948 q^{3} +1.00000 q^{4} +1.48793 q^{5} -3.04948 q^{6} +0.344151 q^{7} -1.00000 q^{8} +6.29934 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +3.04948 q^{3} +1.00000 q^{4} +1.48793 q^{5} -3.04948 q^{6} +0.344151 q^{7} -1.00000 q^{8} +6.29934 q^{9} -1.48793 q^{10} -0.905704 q^{11} +3.04948 q^{12} -2.24985 q^{13} -0.344151 q^{14} +4.53741 q^{15} +1.00000 q^{16} +2.58570 q^{17} -6.29934 q^{18} -0.688301 q^{19} +1.48793 q^{20} +1.04948 q^{21} +0.905704 q^{22} +1.46259 q^{23} -3.04948 q^{24} -2.78607 q^{25} +2.24985 q^{26} +10.0613 q^{27} +0.344151 q^{28} -4.24274 q^{29} -4.53741 q^{30} +1.92985 q^{31} -1.00000 q^{32} -2.76193 q^{33} -2.58570 q^{34} +0.512072 q^{35} +6.29934 q^{36} -1.84911 q^{37} +0.688301 q^{38} -6.86089 q^{39} -1.48793 q^{40} +1.21740 q^{41} -1.04948 q^{42} -6.59036 q^{43} -0.905704 q^{44} +9.37296 q^{45} -1.46259 q^{46} -6.42244 q^{47} +3.04948 q^{48} -6.88156 q^{49} +2.78607 q^{50} +7.88503 q^{51} -2.24985 q^{52} +2.87689 q^{53} -10.0613 q^{54} -1.34762 q^{55} -0.344151 q^{56} -2.09896 q^{57} +4.24274 q^{58} +11.0118 q^{59} +4.53741 q^{60} -1.75015 q^{61} -1.92985 q^{62} +2.16792 q^{63} +1.00000 q^{64} -3.34762 q^{65} +2.76193 q^{66} +4.34415 q^{67} +2.58570 q^{68} +4.46014 q^{69} -0.512072 q^{70} +5.29586 q^{71} -6.29934 q^{72} +5.90206 q^{73} +1.84911 q^{74} -8.49607 q^{75} -0.688301 q^{76} -0.311699 q^{77} +6.86089 q^{78} -15.1107 q^{79} +1.48793 q^{80} +11.7836 q^{81} -1.21740 q^{82} +15.7701 q^{83} +1.04948 q^{84} +3.84733 q^{85} +6.59036 q^{86} -12.9382 q^{87} +0.905704 q^{88} -1.61934 q^{89} -9.37296 q^{90} -0.774289 q^{91} +1.46259 q^{92} +5.88503 q^{93} +6.42244 q^{94} -1.02414 q^{95} -3.04948 q^{96} +14.3436 q^{97} +6.88156 q^{98} -5.70533 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 3 q^{3} + 4 q^{4} + 5 q^{5} - 3 q^{6} - q^{7} - 4 q^{8} + 3 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 3 * q^3 + 4 * q^4 + 5 * q^5 - 3 * q^6 - q^7 - 4 * q^8 + 3 * q^9	\( 4 q - 4 q^{2} + 3 q^{3} + 4 q^{4} + 5 q^{5} - 3 q^{6} - q^{7} - 4 q^{8} + 3 q^{9} - 5 q^{10} + 7 q^{11} + 3 q^{12} + 4 q^{13} + q^{14} + 8 q^{15} + 4 q^{16} + 4 q^{17} - 3 q^{18} + 2 q^{19} + 5 q^{20} - 5 q^{21} - 7 q^{22} + 16 q^{23} - 3 q^{24} - q^{25} - 4 q^{26} + 6 q^{27} - q^{28} - 8 q^{30} - q^{31} - 4 q^{32} + q^{33} - 4 q^{34} + 3 q^{35} + 3 q^{36} - 2 q^{37} - 2 q^{38} + 3 q^{39} - 5 q^{40} - q^{41} + 5 q^{42} + 9 q^{43} + 7 q^{44} + 8 q^{45} - 16 q^{46} + 13 q^{47} + 3 q^{48} - 15 q^{49} + q^{50} + 3 q^{51} + 4 q^{52} + 28 q^{53} - 6 q^{54} + 13 q^{55} + q^{56} + 10 q^{57} + 19 q^{59} + 8 q^{60} - 20 q^{61} + q^{62} + 12 q^{63} + 4 q^{64} + 5 q^{65} - q^{66} + 15 q^{67} + 4 q^{68} - 5 q^{69} - 3 q^{70} + 15 q^{71} - 3 q^{72} - 7 q^{73} + 2 q^{74} + 12 q^{75} + 2 q^{76} - 6 q^{77} - 3 q^{78} - 17 q^{79} + 5 q^{80} + 4 q^{81} + q^{82} + 6 q^{83} - 5 q^{84} - 29 q^{85} - 9 q^{86} - 34 q^{87} - 7 q^{88} + 20 q^{89} - 8 q^{90} - 18 q^{91} + 16 q^{92} - 5 q^{93} - 13 q^{94} - 6 q^{95} - 3 q^{96} + 3 q^{97} + 15 q^{98} - 16 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 3 * q^3 + 4 * q^4 + 5 * q^5 - 3 * q^6 - q^7 - 4 * q^8 + 3 * q^9 - 5 * q^10 + 7 * q^11 + 3 * q^12 + 4 * q^13 + q^14 + 8 * q^15 + 4 * q^16 + 4 * q^17 - 3 * q^18 + 2 * q^19 + 5 * q^20 - 5 * q^21 - 7 * q^22 + 16 * q^23 - 3 * q^24 - q^25 - 4 * q^26 + 6 * q^27 - q^28 - 8 * q^30 - q^31 - 4 * q^32 + q^33 - 4 * q^34 + 3 * q^35 + 3 * q^36 - 2 * q^37 - 2 * q^38 + 3 * q^39 - 5 * q^40 - q^41 + 5 * q^42 + 9 * q^43 + 7 * q^44 + 8 * q^45 - 16 * q^46 + 13 * q^47 + 3 * q^48 - 15 * q^49 + q^50 + 3 * q^51 + 4 * q^52 + 28 * q^53 - 6 * q^54 + 13 * q^55 + q^56 + 10 * q^57 + 19 * q^59 + 8 * q^60 - 20 * q^61 + q^62 + 12 * q^63 + 4 * q^64 + 5 * q^65 - q^66 + 15 * q^67 + 4 * q^68 - 5 * q^69 - 3 * q^70 + 15 * q^71 - 3 * q^72 - 7 * q^73 + 2 * q^74 + 12 * q^75 + 2 * q^76 - 6 * q^77 - 3 * q^78 - 17 * q^79 + 5 * q^80 + 4 * q^81 + q^82 + 6 * q^83 - 5 * q^84 - 29 * q^85 - 9 * q^86 - 34 * q^87 - 7 * q^88 + 20 * q^89 - 8 * q^90 - 18 * q^91 + 16 * q^92 - 5 * q^93 - 13 * q^94 - 6 * q^95 - 3 * q^96 + 3 * q^97 + 15 * q^98 - 16 * q^99

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