LMFDB - Embedded newform 1338.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1338,2,Mod(1,1338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1338, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1338.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1338 = 2 \cdot 3 \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1338.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:	$10.6839837904$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.10273.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + x + 2 $ x^4 - 2x^3 - 5x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.35017$ of defining polynomial
Character	$\chi$	$=$	1338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.35017 q^{5} +1.00000 q^{6} -1.22988 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.35017 q^{5} +1.00000 q^{6} -1.22988 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.35017 q^{10} +5.87349 q^{11} -1.00000 q^{12} -3.29344 q^{13} +1.22988 q^{14} +2.35017 q^{15} +1.00000 q^{16} -1.69645 q^{17} -1.00000 q^{18} -0.613919 q^{19} -2.35017 q^{20} +1.22988 q^{21} -5.87349 q^{22} +5.52721 q^{23} +1.00000 q^{24} +0.523313 q^{25} +3.29344 q^{26} -1.00000 q^{27} -1.22988 q^{28} +6.27650 q^{29} -2.35017 q^{30} -1.38608 q^{31} -1.00000 q^{32} -5.87349 q^{33} +1.69645 q^{34} +2.89042 q^{35} +1.00000 q^{36} +2.93022 q^{37} +0.613919 q^{38} +3.29344 q^{39} +2.35017 q^{40} -2.48944 q^{41} -1.22988 q^{42} -3.47669 q^{43} +5.87349 q^{44} -2.35017 q^{45} -5.52721 q^{46} -0.843795 q^{47} -1.00000 q^{48} -5.48741 q^{49} -0.523313 q^{50} +1.69645 q^{51} -3.29344 q^{52} -3.77402 q^{53} +1.00000 q^{54} -13.8037 q^{55} +1.22988 q^{56} +0.613919 q^{57} -6.27650 q^{58} -4.43067 q^{59} +2.35017 q^{60} -7.93022 q^{61} +1.38608 q^{62} -1.22988 q^{63} +1.00000 q^{64} +7.74015 q^{65} +5.87349 q^{66} -8.35017 q^{67} -1.69645 q^{68} -5.52721 q^{69} -2.89042 q^{70} +7.83758 q^{71} -1.00000 q^{72} -6.71932 q^{73} -2.93022 q^{74} -0.523313 q^{75} -0.613919 q^{76} -7.22366 q^{77} -3.29344 q^{78} +0.124190 q^{79} -2.35017 q^{80} +1.00000 q^{81} +2.48944 q^{82} -0.936439 q^{83} +1.22988 q^{84} +3.98696 q^{85} +3.47669 q^{86} -6.27650 q^{87} -5.87349 q^{88} -9.09032 q^{89} +2.35017 q^{90} +4.05052 q^{91} +5.52721 q^{92} +1.38608 q^{93} +0.843795 q^{94} +1.44282 q^{95} +1.00000 q^{96} +1.28039 q^{97} +5.48741 q^{98} +5.87349 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} - 5 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 - 5 * q^7 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} - 5 q^{7} - 4 q^{8} + 4 q^{9} - 2 q^{10} + 4 q^{11} - 4 q^{12} - 5 q^{13} + 5 q^{14} - 2 q^{15} + 4 q^{16} - 2 q^{17} - 4 q^{18} - 7 q^{19} + 2 q^{20} + 5 q^{21} - 4 q^{22} - 4 q^{23} + 4 q^{24} - 6 q^{25} + 5 q^{26} - 4 q^{27} - 5 q^{28} + 9 q^{29} + 2 q^{30} - q^{31} - 4 q^{32} - 4 q^{33} + 2 q^{34} + 4 q^{36} - 11 q^{37} + 7 q^{38} + 5 q^{39} - 2 q^{40} + 14 q^{41} - 5 q^{42} - 22 q^{43} + 4 q^{44} + 2 q^{45} + 4 q^{46} - 8 q^{47} - 4 q^{48} - 7 q^{49} + 6 q^{50} + 2 q^{51} - 5 q^{52} + 3 q^{53} + 4 q^{54} - 13 q^{55} + 5 q^{56} + 7 q^{57} - 9 q^{58} - 6 q^{59} - 2 q^{60} - 9 q^{61} + q^{62} - 5 q^{63} + 4 q^{64} - 3 q^{65} + 4 q^{66} - 22 q^{67} - 2 q^{68} + 4 q^{69} + 5 q^{71} - 4 q^{72} - 3 q^{73} + 11 q^{74} + 6 q^{75} - 7 q^{76} + 2 q^{77} - 5 q^{78} - 29 q^{79} + 2 q^{80} + 4 q^{81} - 14 q^{82} - 12 q^{83} + 5 q^{84} - 10 q^{85} + 22 q^{86} - 9 q^{87} - 4 q^{88} + 9 q^{89} - 2 q^{90} - 18 q^{91} - 4 q^{92} + q^{93} + 8 q^{94} - 2 q^{95} + 4 q^{96} - 29 q^{97} + 7 q^{98} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 - 5 * q^7 - 4 * q^8 + 4 * q^9 - 2 * q^10 + 4 * q^11 - 4 * q^12 - 5 * q^13 + 5 * q^14 - 2 * q^15 + 4 * q^16 - 2 * q^17 - 4 * q^18 - 7 * q^19 + 2 * q^20 + 5 * q^21 - 4 * q^22 - 4 * q^23 + 4 * q^24 - 6 * q^25 + 5 * q^26 - 4 * q^27 - 5 * q^28 + 9 * q^29 + 2 * q^30 - q^31 - 4 * q^32 - 4 * q^33 + 2 * q^34 + 4 * q^36 - 11 * q^37 + 7 * q^38 + 5 * q^39 - 2 * q^40 + 14 * q^41 - 5 * q^42 - 22 * q^43 + 4 * q^44 + 2 * q^45 + 4 * q^46 - 8 * q^47 - 4 * q^48 - 7 * q^49 + 6 * q^50 + 2 * q^51 - 5 * q^52 + 3 * q^53 + 4 * q^54 - 13 * q^55 + 5 * q^56 + 7 * q^57 - 9 * q^58 - 6 * q^59 - 2 * q^60 - 9 * q^61 + q^62 - 5 * q^63 + 4 * q^64 - 3 * q^65 + 4 * q^66 - 22 * q^67 - 2 * q^68 + 4 * q^69 + 5 * q^71 - 4 * q^72 - 3 * q^73 + 11 * q^74 + 6 * q^75 - 7 * q^76 + 2 * q^77 - 5 * q^78 - 29 * q^79 + 2 * q^80 + 4 * q^81 - 14 * q^82 - 12 * q^83 + 5 * q^84 - 10 * q^85 + 22 * q^86 - 9 * q^87 - 4 * q^88 + 9 * q^89 - 2 * q^90 - 18 * q^91 - 4 * q^92 + q^93 + 8 * q^94 - 2 * q^95 + 4 * q^96 - 29 * q^97 + 7 * q^98 + 4 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−2.35017	−1.05103	−0.525515	−	0.850785i	$-0.676127\pi$
$5$	−2.35017	−1.05103	−0.525515	+	0.850785i	$0.676127\pi$
$6$	1.00000	0.408248
$7$	−1.22988	−0.464849	−0.232425	−	0.972614i	$-0.574666\pi$
$7$	−1.22988	−0.464849	−0.232425	+	0.972614i	$0.574666\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.35017	0.743190
$11$	5.87349	1.77092	0.885461	−	0.464713i	$-0.153843\pi$
$11$	5.87349	1.77092	0.885461	+	0.464713i	$0.153843\pi$
$12$	−1.00000	−0.288675
$13$	−3.29344	−0.913435	−0.456718	−	0.889612i	$-0.650975\pi$
$13$	−3.29344	−0.913435	−0.456718	+	0.889612i	$0.650975\pi$
$14$	1.22988	0.328698
$15$	2.35017	0.606812
$16$	1.00000	0.250000
$17$	−1.69645	−0.411450	−0.205725	−	0.978610i	$-0.565955\pi$
$17$	−1.69645	−0.411450	−0.205725	+	0.978610i	$0.565955\pi$
$18$	−1.00000	−0.235702
$19$	−0.613919	−0.140843	−0.0704214	−	0.997517i	$-0.522434\pi$
$19$	−0.613919	−0.140843	−0.0704214	+	0.997517i	$0.522434\pi$
$20$	−2.35017	−0.525515
$21$	1.22988	0.268381
$22$	−5.87349	−1.25223
$23$	5.52721	1.15250	0.576251	−	0.817273i	$-0.304515\pi$
$23$	5.52721	1.15250	0.576251	+	0.817273i	$0.304515\pi$
$24$	1.00000	0.204124
$25$	0.523313	0.104663
$26$	3.29344	0.645896
$27$	−1.00000	−0.192450
$28$	−1.22988	−0.232425
$29$	6.27650	1.16552	0.582759	−	0.812645i	$-0.301973\pi$
$29$	6.27650	1.16552	0.582759	+	0.812645i	$0.301973\pi$
$30$	−2.35017	−0.429081
$31$	−1.38608	−0.248947	−0.124474	−	0.992223i	$-0.539724\pi$
$31$	−1.38608	−0.248947	−0.124474	+	0.992223i	$0.539724\pi$
$32$	−1.00000	−0.176777
$33$	−5.87349	−1.02244
$34$	1.69645	0.290939
$35$	2.89042	0.488570
$36$	1.00000	0.166667
$37$	2.93022	0.481726	0.240863	−	0.970559i	$-0.422570\pi$
$37$	2.93022	0.481726	0.240863	+	0.970559i	$0.422570\pi$
$38$	0.613919	0.0995909
$39$	3.29344	0.527372
$40$	2.35017	0.371595
$41$	−2.48944	−0.388786	−0.194393	−	0.980924i	$-0.562274\pi$
$41$	−2.48944	−0.388786	−0.194393	+	0.980924i	$0.562274\pi$
$42$	−1.22988	−0.189774
$43$	−3.47669	−0.530190	−0.265095	−	0.964222i	$-0.585403\pi$
$43$	−3.47669	−0.530190	−0.265095	+	0.964222i	$0.585403\pi$
$44$	5.87349	0.885461
$45$	−2.35017	−0.350343
$46$	−5.52721	−0.814942
$47$	−0.843795	−0.123080	−0.0615401	−	0.998105i	$-0.519601\pi$
$47$	−0.843795	−0.123080	−0.0615401	+	0.998105i	$0.519601\pi$
$48$	−1.00000	−0.144338
$49$	−5.48741	−0.783915
$50$	−0.523313	−0.0740076
$51$	1.69645	0.237551
$52$	−3.29344	−0.456718
$53$	−3.77402	−0.518401	−0.259201	−	0.965824i	$-0.583459\pi$
$53$	−3.77402	−0.518401	−0.259201	+	0.965824i	$0.583459\pi$
$54$	1.00000	0.136083
$55$	−13.8037	−1.86129
$56$	1.22988	0.164349
$57$	0.613919	0.0813156
$58$	−6.27650	−0.824145
$59$	−4.43067	−0.576824	−0.288412	−	0.957506i	$-0.593127\pi$
$59$	−4.43067	−0.576824	−0.288412	+	0.957506i	$0.593127\pi$
$60$	2.35017	0.303406
$61$	−7.93022	−1.01536	−0.507680	−	0.861545i	$-0.669497\pi$
$61$	−7.93022	−1.01536	−0.507680	+	0.861545i	$0.669497\pi$
$62$	1.38608	0.176032
$63$	−1.22988	−0.154950
$64$	1.00000	0.125000
$65$	7.74015	0.960047
$66$	5.87349	0.722976
$67$	−8.35017	−1.02014	−0.510068	−	0.860134i	$-0.670380\pi$
$67$	−8.35017	−1.02014	−0.510068	+	0.860134i	$0.670380\pi$
$68$	−1.69645	−0.205725
$69$	−5.52721	−0.665397
$70$	−2.89042	−0.345471
$71$	7.83758	0.930149	0.465075	−	0.885271i	$-0.346027\pi$
$71$	7.83758	0.930149	0.465075	+	0.885271i	$0.346027\pi$
$72$	−1.00000	−0.117851
$73$	−6.71932	−0.786437	−0.393218	−	0.919445i	$-0.628638\pi$
$73$	−6.71932	−0.786437	−0.393218	+	0.919445i	$0.628638\pi$
$74$	−2.93022	−0.340631
$75$	−0.523313	−0.0604270
$76$	−0.613919	−0.0704214
$77$	−7.22366	−0.823212
$78$	−3.29344	−0.372908
$79$	0.124190	0.0139725	0.00698624	−	0.999976i	$-0.497776\pi$
$79$	0.124190	0.0139725	0.00698624	+	0.999976i	$0.497776\pi$
$80$	−2.35017	−0.262757
$81$	1.00000	0.111111
$82$	2.48944	0.274913
$83$	−0.936439	−0.102788	−0.0513938	−	0.998678i	$-0.516366\pi$
$83$	−0.936439	−0.102788	−0.0513938	+	0.998678i	$0.516366\pi$
$84$	1.22988	0.134190
$85$	3.98696	0.432446
$86$	3.47669	0.374901
$87$	−6.27650	−0.672912
$88$	−5.87349	−0.626116
$89$	−9.09032	−0.963572	−0.481786	−	0.876289i	$-0.660012\pi$
$89$	−9.09032	−0.963572	−0.481786	+	0.876289i	$0.660012\pi$
$90$	2.35017	0.247730
$91$	4.05052	0.424610
$92$	5.52721	0.576251
$93$	1.38608	0.143730
$94$	0.843795	0.0870308
$95$	1.44282	0.148030
$96$	1.00000	0.102062
$97$	1.28039	0.130004	0.0650022	−	0.997885i	$-0.479295\pi$
$97$	1.28039	0.130004	0.0650022	+	0.997885i	$0.479295\pi$
$98$	5.48741	0.554312
$99$	5.87349	0.590308
$100$	0.523313	0.0523313
$101$	−3.15806	−0.314239	−0.157119	−	0.987580i	$-0.550221\pi$
$101$	−3.15806	−0.314239	−0.157119	+	0.987580i	$0.550221\pi$
$102$	−1.69645	−0.167974
$103$	−16.9511	−1.67024	−0.835118	−	0.550070i	$-0.814601\pi$
$103$	−16.9511	−1.67024	−0.835118	+	0.550070i	$0.814601\pi$
$104$	3.29344	0.322948
$105$	−2.89042	−0.282076
$106$	3.77402	0.366565
$107$	−18.4405	−1.78271	−0.891355	−	0.453306i	$-0.850244\pi$
$107$	−18.4405	−1.78271	−0.891355	+	0.453306i	$0.850244\pi$
$108$	−1.00000	−0.0962250
$109$	6.56994	0.629286	0.314643	−	0.949210i	$-0.398115\pi$
$109$	6.56994	0.629286	0.314643	+	0.949210i	$0.398115\pi$
$110$	13.8037	1.31613
$111$	−2.93022	−0.278124
$112$	−1.22988	−0.116212
$113$	−5.19601	−0.488799	−0.244400	−	0.969675i	$-0.578591\pi$
$113$	−5.19601	−0.488799	−0.244400	+	0.969675i	$0.578591\pi$
$114$	−0.613919	−0.0574988
$115$	−12.9899	−1.21131
$116$	6.27650	0.582759
$117$	−3.29344	−0.304478
$118$	4.43067	0.407876
$119$	2.08643	0.191262
$120$	−2.35017	−0.214540
$121$	23.4978	2.13617
$122$	7.93022	0.717969
$123$	2.48944	0.224465
$124$	−1.38608	−0.124474
$125$	10.5210	0.941026
$126$	1.22988	0.109566
$127$	8.97063	0.796015	0.398007	−	0.917382i	$-0.369702\pi$
$127$	8.97063	0.796015	0.398007	+	0.917382i	$0.369702\pi$
$128$	−1.00000	−0.0883883
$129$	3.47669	0.306105
$130$	−7.74015	−0.678856
$131$	1.11347	0.0972845	0.0486422	−	0.998816i	$-0.484511\pi$
$131$	1.11347	0.0972845	0.0486422	+	0.998816i	$0.484511\pi$
$132$	−5.87349	−0.511221
$133$	0.755045	0.0654707
$134$	8.35017	0.721345
$135$	2.35017	0.202271
$136$	1.69645	0.145470
$137$	−1.61392	−0.137886	−0.0689432	−	0.997621i	$-0.521963\pi$
$137$	−1.61392	−0.137886	−0.0689432	+	0.997621i	$0.521963\pi$
$138$	5.52721	0.470507
$139$	−11.6626	−0.989207	−0.494604	−	0.869119i	$-0.664687\pi$
$139$	−11.6626	−0.989207	−0.494604	+	0.869119i	$0.664687\pi$
$140$	2.89042	0.244285
$141$	0.843795	0.0710604
$142$	−7.83758	−0.657715
$143$	−19.3440	−1.61762
$144$	1.00000	0.0833333
$145$	−14.7509	−1.22499
$146$	6.71932	0.556095
$147$	5.48741	0.452594
$148$	2.93022	0.240863
$149$	−10.0485	−0.823204	−0.411602	−	0.911364i	$-0.635031\pi$
$149$	−10.0485	−0.823204	−0.411602	+	0.911364i	$0.635031\pi$
$150$	0.523313	0.0427283
$151$	−12.3371	−1.00398	−0.501991	−	0.864873i	$-0.667399\pi$
$151$	−12.3371	−1.00398	−0.501991	+	0.864873i	$0.667399\pi$
$152$	0.613919	0.0497954
$153$	−1.69645	−0.137150
$154$	7.22366	0.582099
$155$	3.25753	0.261651
$156$	3.29344	0.263686
$157$	−15.2742	−1.21901	−0.609506	−	0.792781i	$-0.708632\pi$
$157$	−15.2742	−1.21901	−0.609506	+	0.792781i	$0.708632\pi$
$158$	−0.124190	−0.00988003
$159$	3.77402	0.299299
$160$	2.35017	0.185797
$161$	−6.79778	−0.535740
$162$	−1.00000	−0.0785674
$163$	−23.8503	−1.86810	−0.934051	−	0.357139i	$-0.883752\pi$
$163$	−23.8503	−1.86810	−0.934051	+	0.357139i	$0.883752\pi$
$164$	−2.48944	−0.194393
$165$	13.8037	1.07462
$166$	0.936439	0.0726818
$167$	−4.06356	−0.314448	−0.157224	−	0.987563i	$-0.550254\pi$
$167$	−4.06356	−0.314448	−0.157224	+	0.987563i	$0.550254\pi$
$168$	−1.22988	−0.0948870
$169$	−2.15327	−0.165636
$170$	−3.98696	−0.305786
$171$	−0.613919	−0.0469476
$172$	−3.47669	−0.265095
$173$	13.9269	1.05885	0.529423	−	0.848358i	$-0.322409\pi$
$173$	13.9269	1.05885	0.529423	+	0.848358i	$0.322409\pi$
$174$	6.27650	0.475820
$175$	−0.643610	−0.0486523
$176$	5.87349	0.442731
$177$	4.43067	0.333029
$178$	9.09032	0.681348
$179$	−20.8194	−1.55611	−0.778057	−	0.628193i	$-0.783795\pi$
$179$	−20.8194	−1.55611	−0.778057	+	0.628193i	$0.783795\pi$
$180$	−2.35017	−0.175172
$181$	24.7297	1.83815	0.919074	−	0.394085i	$-0.128939\pi$
$181$	24.7297	1.83815	0.919074	+	0.394085i	$0.128939\pi$
$182$	−4.05052	−0.300244
$183$	7.93022	0.586219
$184$	−5.52721	−0.407471
$185$	−6.88653	−0.506308
$186$	−1.38608	−0.101632
$187$	−9.96409	−0.728647
$188$	−0.843795	−0.0615401
$189$	1.22988	0.0894603
$190$	−1.44282	−0.104673
$191$	4.64779	0.336302	0.168151	−	0.985761i	$-0.446220\pi$
$191$	4.64779	0.336302	0.168151	+	0.985761i	$0.446220\pi$
$192$	−1.00000	−0.0721688
$193$	19.1419	1.37787	0.688933	−	0.724825i	$-0.258080\pi$
$193$	19.1419	1.37787	0.688933	+	0.724825i	$0.258080\pi$
$194$	−1.28039	−0.0919270
$195$	−7.74015	−0.554283
$196$	−5.48741	−0.391958
$197$	−20.5697	−1.46553	−0.732764	−	0.680483i	$-0.761770\pi$
$197$	−20.5697	−1.46553	−0.732764	+	0.680483i	$0.761770\pi$
$198$	−5.87349	−0.417410
$199$	6.35203	0.450283	0.225142	−	0.974326i	$-0.427715\pi$
$199$	6.35203	0.450283	0.225142	+	0.974326i	$0.427715\pi$
$200$	−0.523313	−0.0370038
$201$	8.35017	0.588976
$202$	3.15806	0.222200
$203$	−7.71932	−0.541790
$204$	1.69645	0.118775
$205$	5.85062	0.408625
$206$	16.9511	1.18104
$207$	5.52721	0.384167
$208$	−3.29344	−0.228359
$209$	−3.60585	−0.249422
$210$	2.89042	0.199458
$211$	5.17518	0.356274	0.178137	−	0.984006i	$-0.442993\pi$
$211$	5.17518	0.356274	0.178137	+	0.984006i	$0.442993\pi$
$212$	−3.77402	−0.259201
$213$	−7.83758	−0.537022
$214$	18.4405	1.26057
$215$	8.17082	0.557245
$216$	1.00000	0.0680414
$217$	1.70471	0.115723
$218$	−6.56994	−0.444972
$219$	6.71932	0.454049
$220$	−13.8037	−0.930646
$221$	5.58716	0.375833
$222$	2.93022	0.196664
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	1.22988	0.0821745
$225$	0.523313	0.0348875
$226$	5.19601	0.345633
$227$	7.53732	0.500269	0.250135	−	0.968211i	$-0.419525\pi$
$227$	7.53732	0.500269	0.250135	+	0.968211i	$0.419525\pi$
$228$	0.613919	0.0406578
$229$	−4.91389	−0.324719	−0.162360	−	0.986732i	$-0.551910\pi$
$229$	−4.91389	−0.324719	−0.162360	+	0.986732i	$0.551910\pi$
$230$	12.9899	0.856528
$231$	7.22366	0.475282
$232$	−6.27650	−0.412073
$233$	1.10815	0.0725973	0.0362987	−	0.999341i	$-0.488443\pi$
$233$	1.10815	0.0725973	0.0362987	+	0.999341i	$0.488443\pi$
$234$	3.29344	0.215299
$235$	1.98306	0.129361
$236$	−4.43067	−0.288412
$237$	−0.124190	−0.00806701
$238$	−2.08643	−0.135243
$239$	4.22509	0.273298	0.136649	−	0.990620i	$-0.456367\pi$
$239$	4.22509	0.273298	0.136649	+	0.990620i	$0.456367\pi$
$240$	2.35017	0.151703
$241$	15.9748	1.02903	0.514514	−	0.857482i	$-0.327972\pi$
$241$	15.9748	1.02903	0.514514	+	0.857482i	$0.327972\pi$
$242$	−23.4978	−1.51050
$243$	−1.00000	−0.0641500
$244$	−7.93022	−0.507680
$245$	12.8964	0.823918
$246$	−2.48944	−0.158721
$247$	2.02190	0.128651
$248$	1.38608	0.0880162
$249$	0.936439	0.0593444
$250$	−10.5210	−0.665406
$251$	−16.7746	−1.05880	−0.529402	−	0.848371i	$-0.677584\pi$
$251$	−16.7746	−1.05880	−0.529402	+	0.848371i	$0.677584\pi$
$252$	−1.22988	−0.0774749
$253$	32.4640	2.04099
$254$	−8.97063	−0.562867
$255$	−3.98696	−0.249673
$256$	1.00000	0.0625000
$257$	4.17907	0.260683	0.130342	−	0.991469i	$-0.458393\pi$
$257$	4.17907	0.260683	0.130342	+	0.991469i	$0.458393\pi$
$258$	−3.47669	−0.216449
$259$	−3.60381	−0.223930
$260$	7.74015	0.480024
$261$	6.27650	0.388506
$262$	−1.11347	−0.0687905
$263$	12.0157	0.740919	0.370460	−	0.928849i	$-0.379200\pi$
$263$	12.0157	0.740919	0.370460	+	0.928849i	$0.379200\pi$
$264$	5.87349	0.361488
$265$	8.86959	0.544855
$266$	−0.755045	−0.0462948
$267$	9.09032	0.556319
$268$	−8.35017	−0.510068
$269$	10.4075	0.634558	0.317279	−	0.948332i	$-0.397231\pi$
$269$	10.4075	0.634558	0.317279	+	0.948332i	$0.397231\pi$
$270$	−2.35017	−0.143027
$271$	2.26081	0.137335	0.0686673	−	0.997640i	$-0.478125\pi$
$271$	2.26081	0.137335	0.0686673	+	0.997640i	$0.478125\pi$
$272$	−1.69645	−0.102863
$273$	−4.05052	−0.245149
$274$	1.61392	0.0975004
$275$	3.07367	0.185349
$276$	−5.52721	−0.332699
$277$	23.4991	1.41192	0.705962	−	0.708250i	$-0.250515\pi$
$277$	23.4991	1.41192	0.705962	+	0.708250i	$0.250515\pi$
$278$	11.6626	0.699475
$279$	−1.38608	−0.0829825
$280$	−2.89042	−0.172736
$281$	6.98785	0.416860	0.208430	−	0.978037i	$-0.433165\pi$
$281$	6.98785	0.416860	0.208430	+	0.978037i	$0.433165\pi$
$282$	−0.843795	−0.0502473
$283$	−26.0886	−1.55081	−0.775403	−	0.631466i	$-0.782453\pi$
$283$	−26.0886	−1.55081	−0.775403	+	0.631466i	$0.782453\pi$
$284$	7.83758	0.465075
$285$	−1.44282	−0.0854651
$286$	19.3440	1.14383
$287$	3.06171	0.180727
$288$	−1.00000	−0.0589256
$289$	−14.1220	−0.830709
$290$	14.7509	0.866201
$291$	−1.28039	−0.0750581
$292$	−6.71932	−0.393218
$293$	22.0870	1.29034	0.645169	−	0.764040i	$-0.276787\pi$
$293$	22.0870	1.29034	0.645169	+	0.764040i	$0.276787\pi$
$294$	−5.48741	−0.320032
$295$	10.4128	0.606259
$296$	−2.93022	−0.170316
$297$	−5.87349	−0.340814
$298$	10.0485	0.582093
$299$	−18.2035	−1.05274
$300$	−0.523313	−0.0302135
$301$	4.27589	0.246458
$302$	12.3371	0.709922
$303$	3.15806	0.181426
$304$	−0.613919	−0.0352107
$305$	18.6374	1.06717
$306$	1.69645	0.0969798
$307$	−2.96613	−0.169286	−0.0846430	−	0.996411i	$-0.526975\pi$
$307$	−2.96613	−0.169286	−0.0846430	+	0.996411i	$0.526975\pi$
$308$	−7.22366	−0.411606
$309$	16.9511	0.964312
$310$	−3.25753	−0.185015
$311$	11.3392	0.642985	0.321493	−	0.946912i	$-0.395815\pi$
$311$	11.3392	0.642985	0.321493	+	0.946912i	$0.395815\pi$
$312$	−3.29344	−0.186454
$313$	17.2237	0.973539	0.486769	−	0.873531i	$-0.338175\pi$
$313$	17.2237	0.973539	0.486769	+	0.873531i	$0.338175\pi$
$314$	15.2742	0.861972
$315$	2.89042	0.162857
$316$	0.124190	0.00698624
$317$	2.82686	0.158772	0.0793861	−	0.996844i	$-0.474704\pi$
$317$	2.82686	0.158772	0.0793861	+	0.996844i	$0.474704\pi$
$318$	−3.77402	−0.211636
$319$	36.8649	2.06404
$320$	−2.35017	−0.131379
$321$	18.4405	1.02925
$322$	6.79778	0.378825
$323$	1.04149	0.0579498
$324$	1.00000	0.0555556
$325$	−1.72350	−0.0956025
$326$	23.8503	1.32095
$327$	−6.56994	−0.363318
$328$	2.48944	0.137456
$329$	1.03776	0.0572138
$330$	−13.8037	−0.759869
$331$	−21.7527	−1.19564	−0.597819	−	0.801631i	$-0.703966\pi$
$331$	−21.7527	−1.19564	−0.597819	+	0.801631i	$0.703966\pi$
$332$	−0.936439	−0.0513938
$333$	2.93022	0.160575
$334$	4.06356	0.222348
$335$	19.6244	1.07219
$336$	1.22988	0.0670952
$337$	−15.6351	−0.851696	−0.425848	−	0.904795i	$-0.640024\pi$
$337$	−15.6351	−0.851696	−0.425848	+	0.904795i	$0.640024\pi$
$338$	2.15327	0.117123
$339$	5.19601	0.282208
$340$	3.98696	0.216223
$341$	−8.14113	−0.440867
$342$	0.613919	0.0331970
$343$	15.3580	0.829252
$344$	3.47669	0.187450
$345$	12.9899	0.699352
$346$	−13.9269	−0.748717
$347$	29.0779	1.56098	0.780491	−	0.625167i	$-0.214969\pi$
$347$	29.0779	1.56098	0.780491	+	0.625167i	$0.214969\pi$
$348$	−6.27650	−0.336456
$349$	−28.9882	−1.55170	−0.775851	−	0.630917i	$-0.782679\pi$
$349$	−28.9882	−1.55170	−0.775851	+	0.630917i	$0.782679\pi$
$350$	0.643610	0.0344024
$351$	3.29344	0.175791
$352$	−5.87349	−0.313058
$353$	−1.07475	−0.0572030	−0.0286015	−	0.999591i	$-0.509105\pi$
$353$	−1.07475	−0.0572030	−0.0286015	+	0.999591i	$0.509105\pi$
$354$	−4.43067	−0.235487
$355$	−18.4197	−0.977614
$356$	−9.09032	−0.481786
$357$	−2.08643	−0.110425
$358$	20.8194	1.10034
$359$	18.6743	0.985590	0.492795	−	0.870145i	$-0.335975\pi$
$359$	18.6743	0.985590	0.492795	+	0.870145i	$0.335975\pi$
$360$	2.35017	0.123865
$361$	−18.6231	−0.980163
$362$	−24.7297	−1.29977
$363$	−23.4978	−1.23332
$364$	4.05052	0.212305
$365$	15.7916	0.826568
$366$	−7.93022	−0.414519
$367$	16.5536	0.864091	0.432046	−	0.901852i	$-0.357792\pi$
$367$	16.5536	0.864091	0.432046	+	0.901852i	$0.357792\pi$
$368$	5.52721	0.288126
$369$	−2.48944	−0.129595
$370$	6.88653	0.358014
$371$	4.64157	0.240978
$372$	1.38608	0.0718649
$373$	−26.2958	−1.36154	−0.680772	−	0.732495i	$-0.738356\pi$
$373$	−26.2958	−1.36154	−0.680772	+	0.732495i	$0.738356\pi$
$374$	9.96409	0.515231
$375$	−10.5210	−0.543302
$376$	0.843795	0.0435154
$377$	−20.6713	−1.06462
$378$	−1.22988	−0.0632580
$379$	−18.5670	−0.953723	−0.476862	−	0.878978i	$-0.658226\pi$
$379$	−18.5670	−0.953723	−0.476862	+	0.878978i	$0.658226\pi$
$380$	1.44282	0.0740149
$381$	−8.97063	−0.459579
$382$	−4.64779	−0.237802
$383$	13.3320	0.681233	0.340616	−	0.940202i	$-0.389364\pi$
$383$	13.3320	0.681233	0.340616	+	0.940202i	$0.389364\pi$
$384$	1.00000	0.0510310
$385$	16.9768	0.865220
$386$	−19.1419	−0.974298
$387$	−3.47669	−0.176730
$388$	1.28039	0.0650022
$389$	−21.8611	−1.10840	−0.554200	−	0.832384i	$-0.686976\pi$
$389$	−21.8611	−1.10840	−0.554200	+	0.832384i	$0.686976\pi$
$390$	7.74015	0.391938
$391$	−9.37664	−0.474197
$392$	5.48741	0.277156
$393$	−1.11347	−0.0561672
$394$	20.5697	1.03628
$395$	−0.291868	−0.0146855
$396$	5.87349	0.295154
$397$	−17.8783	−0.897285	−0.448642	−	0.893711i	$-0.648092\pi$
$397$	−17.8783	−0.897285	−0.448642	+	0.893711i	$0.648092\pi$
$398$	−6.35203	−0.318398
$399$	−0.755045	−0.0377995
$400$	0.523313	0.0261656
$401$	−20.0193	−0.999716	−0.499858	−	0.866107i	$-0.666614\pi$
$401$	−20.0193	−0.999716	−0.499858	+	0.866107i	$0.666614\pi$
$402$	−8.35017	−0.416469
$403$	4.56497	0.227397
$404$	−3.15806	−0.157119
$405$	−2.35017	−0.116781
$406$	7.71932	0.383103
$407$	17.2106	0.853099
$408$	−1.69645	−0.0839869
$409$	19.0633	0.942618	0.471309	−	0.881968i	$-0.343782\pi$
$409$	19.0633	0.942618	0.471309	+	0.881968i	$0.343782\pi$
$410$	−5.85062	−0.288942
$411$	1.61392	0.0796088
$412$	−16.9511	−0.835118
$413$	5.44917	0.268136
$414$	−5.52721	−0.271647
$415$	2.20079	0.108033
$416$	3.29344	0.161474
$417$	11.6626	0.571119
$418$	3.60585	0.176368
$419$	23.4539	1.14580	0.572898	−	0.819627i	$-0.305819\pi$
$419$	23.4539	1.14580	0.572898	+	0.819627i	$0.305819\pi$
$420$	−2.89042	−0.141038
$421$	19.0961	0.930685	0.465343	−	0.885131i	$-0.345931\pi$
$421$	19.0961	0.930685	0.465343	+	0.885131i	$0.345931\pi$
$422$	−5.17518	−0.251924
$423$	−0.843795	−0.0410267
$424$	3.77402	0.183282
$425$	−0.887776	−0.0430634
$426$	7.83758	0.379732
$427$	9.75319	0.471990
$428$	−18.4405	−0.891355
$429$	19.3440	0.933935
$430$	−8.17082	−0.394032
$431$	−31.9918	−1.54099	−0.770495	−	0.637446i	$-0.779991\pi$
$431$	−31.9918	−1.54099	−0.770495	+	0.637446i	$0.779991\pi$
$432$	−1.00000	−0.0481125
$433$	−2.02172	−0.0971578	−0.0485789	−	0.998819i	$-0.515469\pi$
$433$	−2.02172	−0.0971578	−0.0485789	+	0.998819i	$0.515469\pi$
$434$	−1.70471	−0.0818286
$435$	14.7509	0.707250
$436$	6.56994	0.314643
$437$	−3.39326	−0.162322
$438$	−6.71932	−0.321061
$439$	−13.7628	−0.656864	−0.328432	−	0.944528i	$-0.606520\pi$
$439$	−13.7628	−0.656864	−0.328432	+	0.944528i	$0.606520\pi$
$440$	13.8037	0.658066
$441$	−5.48741	−0.261305
$442$	−5.58716	−0.265754
$443$	−35.4448	−1.68403	−0.842017	−	0.539451i	$-0.818632\pi$
$443$	−35.4448	−1.68403	−0.842017	+	0.539451i	$0.818632\pi$
$444$	−2.93022	−0.139062
$445$	21.3638	1.01274
$446$	1.00000	0.0473514
$447$	10.0485	0.475277
$448$	−1.22988	−0.0581062
$449$	−17.2168	−0.812513	−0.406256	−	0.913759i	$-0.633166\pi$
$449$	−17.2168	−0.812513	−0.406256	+	0.913759i	$0.633166\pi$
$450$	−0.523313	−0.0246692
$451$	−14.6217	−0.688509
$452$	−5.19601	−0.244400
$453$	12.3371	0.579649
$454$	−7.53732	−0.353744
$455$	−9.51942	−0.446277
$456$	−0.613919	−0.0287494
$457$	29.3191	1.37149	0.685745	−	0.727842i	$-0.259477\pi$
$457$	29.3191	1.37149	0.685745	+	0.727842i	$0.259477\pi$
$458$	4.91389	0.229611
$459$	1.69645	0.0791836
$460$	−12.9899	−0.605657
$461$	19.2523	0.896668	0.448334	−	0.893866i	$-0.352018\pi$
$461$	19.2523	0.896668	0.448334	+	0.893866i	$0.352018\pi$
$462$	−7.22366	−0.336075
$463$	9.09450	0.422657	0.211329	−	0.977415i	$-0.432221\pi$
$463$	9.09450	0.422657	0.211329	+	0.977415i	$0.432221\pi$
$464$	6.27650	0.291379
$465$	−3.25753	−0.151064
$466$	−1.10815	−0.0513340
$467$	−19.6339	−0.908546	−0.454273	−	0.890862i	$-0.650101\pi$
$467$	−19.6339	−0.908546	−0.454273	+	0.890862i	$0.650101\pi$
$468$	−3.29344	−0.152239
$469$	10.2697	0.474210
$470$	−1.98306	−0.0914720
$471$	15.2742	0.703797
$472$	4.43067	0.203938
$473$	−20.4203	−0.938925
$474$	0.124190	0.00570424
$475$	−0.321272	−0.0147410
$476$	2.08643	0.0956312
$477$	−3.77402	−0.172800
$478$	−4.22509	−0.193251
$479$	5.33449	0.243739	0.121869	−	0.992546i	$-0.461111\pi$
$479$	5.33449	0.243739	0.121869	+	0.992546i	$0.461111\pi$
$480$	−2.35017	−0.107270
$481$	−9.65050	−0.440025
$482$	−15.9748	−0.727633
$483$	6.79778	0.309310
$484$	23.4978	1.06808
$485$	−3.00915	−0.136638
$486$	1.00000	0.0453609
$487$	1.69645	0.0768736	0.0384368	−	0.999261i	$-0.487762\pi$
$487$	1.69645	0.0768736	0.0384368	+	0.999261i	$0.487762\pi$
$488$	7.93022	0.358984
$489$	23.8503	1.07855
$490$	−12.8964	−0.582598
$491$	20.6650	0.932600	0.466300	−	0.884627i	$-0.345587\pi$
$491$	20.6650	0.932600	0.466300	+	0.884627i	$0.345587\pi$
$492$	2.48944	0.112233
$493$	−10.6478	−0.479552
$494$	−2.02190	−0.0909698
$495$	−13.8037	−0.620431
$496$	−1.38608	−0.0622369
$497$	−9.63925	−0.432379
$498$	−0.936439	−0.0419628
$499$	2.00275	0.0896554	0.0448277	−	0.998995i	$-0.485726\pi$
$499$	2.00275	0.0896554	0.0448277	+	0.998995i	$0.485726\pi$
$500$	10.5210	0.470513
$501$	4.06356	0.181547
$502$	16.7746	0.748688
$503$	−18.6915	−0.833412	−0.416706	−	0.909041i	$-0.636816\pi$
$503$	−18.6915	−0.833412	−0.416706	+	0.909041i	$0.636816\pi$
$504$	1.22988	0.0547830
$505$	7.42199	0.330274
$506$	−32.4640	−1.44320
$507$	2.15327	0.0956302
$508$	8.97063	0.398007
$509$	22.6404	1.00352	0.501759	−	0.865008i	$-0.332686\pi$
$509$	22.6404	1.00352	0.501759	+	0.865008i	$0.332686\pi$
$510$	3.98696	0.176545
$511$	8.26393	0.365575
$512$	−1.00000	−0.0441942
$513$	0.613919	0.0271052
$514$	−4.17907	−0.184331
$515$	39.8379	1.75547
$516$	3.47669	0.153053
$517$	−4.95602	−0.217965
$518$	3.60381	0.158342
$519$	−13.9269	−0.611325
$520$	−7.74015	−0.339428
$521$	−12.1290	−0.531380	−0.265690	−	0.964058i	$-0.585600\pi$
$521$	−12.1290	−0.531380	−0.265690	+	0.964058i	$0.585600\pi$
$522$	−6.27650	−0.274715
$523$	−23.3665	−1.02174	−0.510872	−	0.859657i	$-0.670677\pi$
$523$	−23.3665	−1.02174	−0.510872	+	0.859657i	$0.670677\pi$
$524$	1.11347	0.0486422
$525$	0.643610	0.0280894
$526$	−12.0157	−0.523909
$527$	2.35142	0.102429
$528$	−5.87349	−0.255611
$529$	7.55001	0.328261
$530$	−8.86959	−0.385270
$531$	−4.43067	−0.192275
$532$	0.755045	0.0327353
$533$	8.19882	0.355130
$534$	−9.09032	−0.393377
$535$	43.3383	1.87368
$536$	8.35017	0.360673
$537$	20.8194	0.898423
$538$	−10.4075	−0.448700
$539$	−32.2302	−1.38825
$540$	2.35017	0.101135
$541$	2.47997	0.106622	0.0533112	−	0.998578i	$-0.483022\pi$
$541$	2.47997	0.106622	0.0533112	+	0.998578i	$0.483022\pi$
$542$	−2.26081	−0.0971103
$543$	−24.7297	−1.06126
$544$	1.69645	0.0727348
$545$	−15.4405	−0.661398
$546$	4.05052	0.173346
$547$	−0.748219	−0.0319916	−0.0159958	−	0.999872i	$-0.505092\pi$
$547$	−0.748219	−0.0319916	−0.0159958	+	0.999872i	$0.505092\pi$
$548$	−1.61392	−0.0689432
$549$	−7.93022	−0.338454
$550$	−3.07367	−0.131062
$551$	−3.85327	−0.164155
$552$	5.52721	0.235254
$553$	−0.152738	−0.00649510
$554$	−23.4991	−0.998380
$555$	6.88653	0.292317
$556$	−11.6626	−0.494604
$557$	−5.21930	−0.221149	−0.110574	−	0.993868i	$-0.535269\pi$
$557$	−5.21930	−0.221149	−0.110574	+	0.993868i	$0.535269\pi$
$558$	1.38608	0.0586775
$559$	11.4502	0.484294
$560$	2.89042	0.122143
$561$	9.96409	0.420684
$562$	−6.98785	−0.294765
$563$	3.39805	0.143211	0.0716053	−	0.997433i	$-0.477188\pi$
$563$	3.39805	0.143211	0.0716053	+	0.997433i	$0.477188\pi$
$564$	0.843795	0.0355302
$565$	12.2115	0.513742
$566$	26.0886	1.09659
$567$	−1.22988	−0.0516499
$568$	−7.83758	−0.328857
$569$	−14.4845	−0.607221	−0.303610	−	0.952796i	$-0.598192\pi$
$569$	−14.4845	−0.607221	−0.303610	+	0.952796i	$0.598192\pi$
$570$	1.44282	0.0604329
$571$	−16.8460	−0.704982	−0.352491	−	0.935815i	$-0.614665\pi$
$571$	−16.8460	−0.704982	−0.352491	+	0.935815i	$0.614665\pi$
$572$	−19.3440	−0.808811
$573$	−4.64779	−0.194164
$574$	−3.06171	−0.127793
$575$	2.89246	0.120624
$576$	1.00000	0.0416667
$577$	−0.887489	−0.0369466	−0.0184733	−	0.999829i	$-0.505881\pi$
$577$	−0.887489	−0.0369466	−0.0184733	+	0.999829i	$0.505881\pi$
$578$	14.1220	0.587400
$579$	−19.1419	−0.795511
$580$	−14.7509	−0.612496
$581$	1.15170	0.0477807
$582$	1.28039	0.0530741
$583$	−22.1666	−0.918048
$584$	6.71932	0.278047
$585$	7.74015	0.320016
$586$	−22.0870	−0.912407
$587$	−33.0362	−1.36355	−0.681776	−	0.731561i	$-0.738792\pi$
$587$	−33.0362	−1.36355	−0.681776	+	0.731561i	$0.738792\pi$
$588$	5.48741	0.226297
$589$	0.850942	0.0350624
$590$	−10.4128	−0.428690
$591$	20.5697	0.846122
$592$	2.93022	0.120431
$593$	−12.6799	−0.520703	−0.260351	−	0.965514i	$-0.583838\pi$
$593$	−12.6799	−0.520703	−0.260351	+	0.965514i	$0.583838\pi$
$594$	5.87349	0.240992
$595$	−4.90346	−0.201022
$596$	−10.0485	−0.411602
$597$	−6.35203	−0.259971
$598$	18.2035	0.744397
$599$	−20.3036	−0.829582	−0.414791	−	0.909917i	$-0.636145\pi$
$599$	−20.3036	−0.829582	−0.414791	+	0.909917i	$0.636145\pi$
$600$	0.523313	0.0213642
$601$	29.5976	1.20731	0.603655	−	0.797245i	$-0.293710\pi$
$601$	29.5976	1.20731	0.603655	+	0.797245i	$0.293710\pi$
$602$	−4.27589	−0.174272
$603$	−8.35017	−0.340045
$604$	−12.3371	−0.501991
$605$	−55.2240	−2.24517
$606$	−3.15806	−0.128287
$607$	10.3987	0.422068	0.211034	−	0.977479i	$-0.432317\pi$
$607$	10.3987	0.422068	0.211034	+	0.977479i	$0.432317\pi$
$608$	0.613919	0.0248977
$609$	7.71932	0.312803
$610$	−18.6374	−0.754606
$611$	2.77899	0.112426
$612$	−1.69645	−0.0685750
$613$	2.74787	0.110985	0.0554926	−	0.998459i	$-0.482327\pi$
$613$	2.74787	0.110985	0.0554926	+	0.998459i	$0.482327\pi$
$614$	2.96613	0.119703
$615$	−5.85062	−0.235920
$616$	7.22366	0.291050
$617$	46.2786	1.86311	0.931553	−	0.363607i	$-0.118455\pi$
$617$	46.2786	1.86311	0.931553	+	0.363607i	$0.118455\pi$
$618$	−16.9511	−0.681871
$619$	−23.0901	−0.928071	−0.464036	−	0.885817i	$-0.653599\pi$
$619$	−23.0901	−0.928071	−0.464036	+	0.885817i	$0.653599\pi$
$620$	3.25753	0.130826
$621$	−5.52721	−0.221799
$622$	−11.3392	−0.454659
$623$	11.1800	0.447916
$624$	3.29344	0.131843
$625$	−27.3427	−1.09371
$626$	−17.2237	−0.688396
$627$	3.60585	0.144004
$628$	−15.2742	−0.609506
$629$	−4.97098	−0.198206
$630$	−2.89042	−0.115157
$631$	−48.9725	−1.94956	−0.974782	−	0.223159i	$-0.928363\pi$
$631$	−48.9725	−1.94956	−0.974782	+	0.223159i	$0.928363\pi$
$632$	−0.124190	−0.00494001
$633$	−5.17518	−0.205695
$634$	−2.82686	−0.112269
$635$	−21.0825	−0.836635
$636$	3.77402	0.149650
$637$	18.0724	0.716055
$638$	−36.8649	−1.45950
$639$	7.83758	0.310050
$640$	2.35017	0.0928987
$641$	35.1814	1.38958	0.694790	−	0.719213i	$-0.255497\pi$
$641$	35.1814	1.38958	0.694790	+	0.719213i	$0.255497\pi$
$642$	−18.4405	−0.727788
$643$	17.4912	0.689785	0.344893	−	0.938642i	$-0.387915\pi$
$643$	17.4912	0.689785	0.344893	+	0.938642i	$0.387915\pi$
$644$	−6.79778	−0.267870
$645$	−8.17082	−0.321726
$646$	−1.04149	−0.0409767
$647$	−41.5940	−1.63523	−0.817615	−	0.575766i	$-0.804704\pi$
$647$	−41.5940	−1.63523	−0.817615	+	0.575766i	$0.804704\pi$
$648$	−1.00000	−0.0392837
$649$	−26.0235	−1.02151
$650$	1.72350	0.0676012
$651$	−1.70471	−0.0668127
$652$	−23.8503	−0.934051
$653$	−14.0577	−0.550120	−0.275060	−	0.961427i	$-0.588698\pi$
$653$	−14.0577	−0.550120	−0.275060	+	0.961427i	$0.588698\pi$
$654$	6.56994	0.256905
$655$	−2.61685	−0.102249
$656$	−2.48944	−0.0971964
$657$	−6.71932	−0.262146
$658$	−1.03776	−0.0404562
$659$	−22.8293	−0.889304	−0.444652	−	0.895703i	$-0.646673\pi$
$659$	−22.8293	−0.889304	−0.444652	+	0.895703i	$0.646673\pi$
$660$	13.8037	0.537309
$661$	11.1420	0.433374	0.216687	−	0.976241i	$-0.430475\pi$
$661$	11.1420	0.433374	0.216687	+	0.976241i	$0.430475\pi$
$662$	21.7527	0.845443
$663$	−5.58716	−0.216987
$664$	0.936439	0.0363409
$665$	−1.77449	−0.0688116
$666$	−2.93022	−0.113544
$667$	34.6915	1.34326
$668$	−4.06356	−0.157224
$669$	1.00000	0.0386622
$670$	−19.6244	−0.758155
$671$	−46.5780	−1.79813
$672$	−1.22988	−0.0474435
$673$	3.67298	0.141583	0.0707915	−	0.997491i	$-0.477448\pi$
$673$	3.67298	0.141583	0.0707915	+	0.997491i	$0.477448\pi$
$674$	15.6351	0.602240
$675$	−0.523313	−0.0201423
$676$	−2.15327	−0.0828182
$677$	−11.9860	−0.460659	−0.230330	−	0.973113i	$-0.573980\pi$
$677$	−11.9860	−0.460659	−0.230330	+	0.973113i	$0.573980\pi$
$678$	−5.19601	−0.199551
$679$	−1.57473	−0.0604325
$680$	−3.98696	−0.152893
$681$	−7.53732	−0.288831
$682$	8.14113	0.311740
$683$	28.3794	1.08591	0.542954	−	0.839763i	$-0.317306\pi$
$683$	28.3794	1.08591	0.542954	+	0.839763i	$0.317306\pi$
$684$	−0.613919	−0.0234738
$685$	3.79299	0.144923
$686$	−15.3580	−0.586370
$687$	4.91389	0.187477
$688$	−3.47669	−0.132547
$689$	12.4295	0.473526
$690$	−12.9899	−0.494517
$691$	10.6173	0.403903	0.201951	−	0.979396i	$-0.435272\pi$
$691$	10.6173	0.403903	0.201951	+	0.979396i	$0.435272\pi$
$692$	13.9269	0.529423
$693$	−7.22366	−0.274404
$694$	−29.0779	−1.10378
$695$	27.4091	1.03969
$696$	6.27650	0.237910
$697$	4.22322	0.159966
$698$	28.9882	1.09722
$699$	−1.10815	−0.0419141
$700$	−0.643610	−0.0243262
$701$	−4.18060	−0.157899	−0.0789496	−	0.996879i	$-0.525157\pi$
$701$	−4.18060	−0.157899	−0.0789496	+	0.996879i	$0.525157\pi$
$702$	−3.29344	−0.124303
$703$	−1.79892	−0.0678475
$704$	5.87349	0.221365
$705$	−1.98306	−0.0746865
$706$	1.07475	0.0404487
$707$	3.88402	0.146074
$708$	4.43067	0.166515
$709$	−19.6965	−0.739716	−0.369858	−	0.929088i	$-0.620594\pi$
$709$	−19.6965	−0.739716	−0.369858	+	0.929088i	$0.620594\pi$
$710$	18.4197	0.691278
$711$	0.124190	0.00465749
$712$	9.09032	0.340674
$713$	−7.66115	−0.286912
$714$	2.08643	0.0780826
$715$	45.4616	1.70017
$716$	−20.8194	−0.778057
$717$	−4.22509	−0.157789
$718$	−18.6743	−0.696917
$719$	−9.17225	−0.342067	−0.171034	−	0.985265i	$-0.554711\pi$
$719$	−9.17225	−0.342067	−0.171034	+	0.985265i	$0.554711\pi$
$720$	−2.35017	−0.0875858
$721$	20.8477	0.776408
$722$	18.6231	0.693080
$723$	−15.9748	−0.594110
$724$	24.7297	0.919074
$725$	3.28457	0.121986
$726$	23.4978	0.872087
$727$	−22.6607	−0.840440	−0.420220	−	0.907422i	$-0.638047\pi$
$727$	−22.6607	−0.840440	−0.420220	+	0.907422i	$0.638047\pi$
$728$	−4.05052	−0.150122
$729$	1.00000	0.0370370
$730$	−15.7916	−0.584472
$731$	5.89804	0.218147
$732$	7.93022	0.293109
$733$	−22.6259	−0.835707	−0.417854	−	0.908514i	$-0.637218\pi$
$733$	−22.6259	−0.835707	−0.417854	+	0.908514i	$0.637218\pi$
$734$	−16.5536	−0.611005
$735$	−12.8964	−0.475689
$736$	−5.52721	−0.203736
$737$	−49.0446	−1.80658
$738$	2.48944	0.0916376
$739$	−9.64422	−0.354768	−0.177384	−	0.984142i	$-0.556764\pi$
$739$	−9.64422	−0.354768	−0.177384	+	0.984142i	$0.556764\pi$
$740$	−6.88653	−0.253154
$741$	−2.02190	−0.0742765
$742$	−4.64157	−0.170397
$743$	−24.7312	−0.907299	−0.453649	−	0.891180i	$-0.649878\pi$
$743$	−24.7312	−0.907299	−0.453649	+	0.891180i	$0.649878\pi$
$744$	−1.38608	−0.0508162
$745$	23.6157	0.865211
$746$	26.2958	0.962757
$747$	−0.936439	−0.0342625
$748$	−9.96409	−0.364323
$749$	22.6795	0.828692
$750$	10.5210	0.384172
$751$	44.0912	1.60891	0.804456	−	0.594012i	$-0.202457\pi$
$751$	44.0912	1.60891	0.804456	+	0.594012i	$0.202457\pi$
$752$	−0.843795	−0.0307700
$753$	16.7746	0.611301
$754$	20.6713	0.752803
$755$	28.9944	1.05521
$756$	1.22988	0.0447302
$757$	−24.2296	−0.880639	−0.440320	−	0.897841i	$-0.645135\pi$
$757$	−24.2296	−0.880639	−0.440320	+	0.897841i	$0.645135\pi$
$758$	18.5670	0.674384
$759$	−32.4640	−1.17837
$760$	−1.44282	−0.0523365
$761$	44.4206	1.61025	0.805123	−	0.593108i	$-0.202099\pi$
$761$	44.4206	1.61025	0.805123	+	0.593108i	$0.202099\pi$
$762$	8.97063	0.324972
$763$	−8.08021	−0.292523
$764$	4.64779	0.168151
$765$	3.98696	0.144149
$766$	−13.3320	−0.481704
$767$	14.5921	0.526891
$768$	−1.00000	−0.0360844
$769$	29.9804	1.08112	0.540560	−	0.841305i	$-0.318212\pi$
$769$	29.9804	1.08112	0.540560	+	0.841305i	$0.318212\pi$
$770$	−16.9768	−0.611803
$771$	−4.17907	−0.150506
$772$	19.1419	0.688933
$773$	13.7064	0.492986	0.246493	−	0.969145i	$-0.420722\pi$
$773$	13.7064	0.492986	0.246493	+	0.969145i	$0.420722\pi$
$774$	3.47669	0.124967
$775$	−0.725354	−0.0260555
$776$	−1.28039	−0.0459635
$777$	3.60381	0.129286
$778$	21.8611	0.783757
$779$	1.52832	0.0547576
$780$	−7.74015	−0.277142
$781$	46.0339	1.64722
$782$	9.37664	0.335308
$783$	−6.27650	−0.224304
$784$	−5.48741	−0.195979
$785$	35.8970	1.28122
$786$	1.11347	0.0397162
$787$	22.1364	0.789078	0.394539	−	0.918879i	$-0.370904\pi$
$787$	22.1364	0.789078	0.394539	+	0.918879i	$0.370904\pi$
$788$	−20.5697	−0.732764
$789$	−12.0157	−0.427770
$790$	0.291868	0.0103842
$791$	6.39044	0.227218
$792$	−5.87349	−0.208705
$793$	26.1177	0.927466
$794$	17.8783	0.634476
$795$	−8.86959	−0.314572
$796$	6.35203	0.225142
$797$	−50.5494	−1.79055	−0.895276	−	0.445511i	$-0.853022\pi$
$797$	−50.5494	−1.79055	−0.895276	+	0.445511i	$0.853022\pi$
$798$	0.755045	0.0267283
$799$	1.43146	0.0506414
$800$	−0.523313	−0.0185019
$801$	−9.09032	−0.321191
$802$	20.0193	0.706906
$803$	−39.4658	−1.39272
$804$	8.35017	0.294488
$805$	15.9760	0.563078
$806$	−4.56497	−0.160794
$807$	−10.4075	−0.366362
$808$	3.15806	0.111100
$809$	−35.1267	−1.23499	−0.617494	−	0.786576i	$-0.711852\pi$
$809$	−35.1267	−1.23499	−0.617494	+	0.786576i	$0.711852\pi$
$810$	2.35017	0.0825767
$811$	26.8154	0.941616	0.470808	−	0.882236i	$-0.343963\pi$
$811$	26.8154	0.941616	0.470808	+	0.882236i	$0.343963\pi$
$812$	−7.71932	−0.270895
$813$	−2.26081	−0.0792902
$814$	−17.2106	−0.603232
$815$	56.0524	1.96343
$816$	1.69645	0.0593877
$817$	2.13441	0.0746734
$818$	−19.0633	−0.666532
$819$	4.05052	0.141537
$820$	5.85062	0.204313
$821$	−0.589625	−0.0205780	−0.0102890	−	0.999947i	$-0.503275\pi$
$821$	−0.589625	−0.0205780	−0.0102890	+	0.999947i	$0.503275\pi$
$822$	−1.61392	−0.0562919
$823$	−7.50106	−0.261470	−0.130735	−	0.991417i	$-0.541734\pi$
$823$	−7.50106	−0.261470	−0.130735	+	0.991417i	$0.541734\pi$
$824$	16.9511	0.590518
$825$	−3.07367	−0.107011
$826$	−5.44917	−0.189601
$827$	−16.4319	−0.571393	−0.285696	−	0.958320i	$-0.592225\pi$
$827$	−16.4319	−0.571393	−0.285696	+	0.958320i	$0.592225\pi$
$828$	5.52721	0.192084
$829$	−28.0822	−0.975337	−0.487668	−	0.873029i	$-0.662152\pi$
$829$	−28.0822	−0.975337	−0.487668	+	0.873029i	$0.662152\pi$
$830$	−2.20079	−0.0763907
$831$	−23.4991	−0.815174
$832$	−3.29344	−0.114179
$833$	9.30912	0.322542
$834$	−11.6626	−0.403842
$835$	9.55007	0.330494
$836$	−3.60585	−0.124711
$837$	1.38608	0.0479100
$838$	−23.4539	−0.810200
$839$	−31.8314	−1.09894	−0.549470	−	0.835513i	$-0.685170\pi$
$839$	−31.8314	−1.09894	−0.549470	+	0.835513i	$0.685170\pi$
$840$	2.89042	0.0997290
$841$	10.3945	0.358430
$842$	−19.0961	−0.658094
$843$	−6.98785	−0.240674
$844$	5.17518	0.178137
$845$	5.06056	0.174089
$846$	0.843795	0.0290103
$847$	−28.8994	−0.992996
$848$	−3.77402	−0.129600
$849$	26.0886	0.895359
$850$	0.887776	0.0304505
$851$	16.1959	0.555190
$852$	−7.83758	−0.268511
$853$	27.4346	0.939342	0.469671	−	0.882842i	$-0.344373\pi$
$853$	27.4346	0.939342	0.469671	+	0.882842i	$0.344373\pi$
$854$	−9.75319	−0.333747
$855$	1.44282	0.0493433
$856$	18.4405	0.630283
$857$	7.08110	0.241886	0.120943	−	0.992659i	$-0.461408\pi$
$857$	7.08110	0.241886	0.120943	+	0.992659i	$0.461408\pi$
$858$	−19.3440	−0.660392
$859$	−11.6040	−0.395923	−0.197962	−	0.980210i	$-0.563432\pi$
$859$	−11.6040	−0.395923	−0.197962	+	0.980210i	$0.563432\pi$
$860$	8.17082	0.278623
$861$	−3.06171	−0.104343
$862$	31.9918	1.08964
$863$	39.9734	1.36071	0.680356	−	0.732882i	$-0.261825\pi$
$863$	39.9734	1.36071	0.680356	+	0.732882i	$0.261825\pi$
$864$	1.00000	0.0340207
$865$	−32.7307	−1.11288
$866$	2.02172	0.0687009
$867$	14.1220	0.479610
$868$	1.70471	0.0578615
$869$	0.729428	0.0247442
$870$	−14.7509	−0.500101
$871$	27.5008	0.931828
$872$	−6.56994	−0.222486
$873$	1.28039	0.0433348
$874$	3.39326	0.114779
$875$	−12.9395	−0.437435
$876$	6.71932	0.227025
$877$	31.1987	1.05350	0.526752	−	0.850019i	$-0.323410\pi$
$877$	31.1987	1.05350	0.526752	+	0.850019i	$0.323410\pi$
$878$	13.7628	0.464473
$879$	−22.0870	−0.744977
$880$	−13.8037	−0.465323
$881$	−14.5634	−0.490655	−0.245327	−	0.969440i	$-0.578895\pi$
$881$	−14.5634	−0.490655	−0.245327	+	0.969440i	$0.578895\pi$
$882$	5.48741	0.184771
$883$	−10.0488	−0.338168	−0.169084	−	0.985602i	$-0.554081\pi$
$883$	−10.0488	−0.338168	−0.169084	+	0.985602i	$0.554081\pi$
$884$	5.58716	0.187917
$885$	−10.4128	−0.350024
$886$	35.4448	1.19079
$887$	28.4399	0.954917	0.477458	−	0.878654i	$-0.341558\pi$
$887$	28.4399	0.954917	0.477458	+	0.878654i	$0.341558\pi$
$888$	2.93022	0.0983318
$889$	−11.0328	−0.370027
$890$	−21.3638	−0.716117
$891$	5.87349	0.196769
$892$	−1.00000	−0.0334825
$893$	0.518022	0.0173350
$894$	−10.0485	−0.336071
$895$	48.9292	1.63552
$896$	1.22988	0.0410873
$897$	18.2035	0.607797
$898$	17.2168	0.574533
$899$	−8.69974	−0.290153
$900$	0.523313	0.0174438
$901$	6.40244	0.213296
$902$	14.6217	0.486850
$903$	−4.27589	−0.142293
$904$	5.19601	0.172817
$905$	−58.1192	−1.93195
$906$	−12.3371	−0.409874
$907$	21.5378	0.715149	0.357575	−	0.933885i	$-0.383604\pi$
$907$	21.5378	0.715149	0.357575	+	0.933885i	$0.383604\pi$
$908$	7.53732	0.250135
$909$	−3.15806	−0.104746
$910$	9.51942	0.315566
$911$	−21.2338	−0.703507	−0.351754	−	0.936093i	$-0.614415\pi$
$911$	−21.2338	−0.703507	−0.351754	+	0.936093i	$0.614415\pi$
$912$	0.613919	0.0203289
$913$	−5.50016	−0.182029
$914$	−29.3191	−0.969789
$915$	−18.6374	−0.616133
$916$	−4.91389	−0.162360
$917$	−1.36943	−0.0452226
$918$	−1.69645	−0.0559913
$919$	−47.8688	−1.57905	−0.789524	−	0.613720i	$-0.789672\pi$
$919$	−47.8688	−1.57905	−0.789524	+	0.613720i	$0.789672\pi$
$920$	12.9899	0.428264
$921$	2.96613	0.0977373
$922$	−19.2523	−0.634040
$923$	−25.8126	−0.849631
$924$	7.22366	0.237641
$925$	1.53342	0.0504186
$926$	−9.09450	−0.298864
$927$	−16.9511	−0.556746
$928$	−6.27650	−0.206036
$929$	53.3852	1.75151	0.875755	−	0.482755i	$-0.160364\pi$
$929$	53.3852	1.75151	0.875755	+	0.482755i	$0.160364\pi$
$930$	3.25753	0.106819
$931$	3.36882	0.110409
$932$	1.10815	0.0362987
$933$	−11.3392	−0.371228
$934$	19.6339	0.642439
$935$	23.4173	0.765829
$936$	3.29344	0.107649
$937$	46.0746	1.50519	0.752596	−	0.658483i	$-0.228801\pi$
$937$	46.0746	1.50519	0.752596	+	0.658483i	$0.228801\pi$
$938$	−10.2697	−0.335317
$939$	−17.2237	−0.562073
$940$	1.98306	0.0646804
$941$	−57.8075	−1.88447	−0.942235	−	0.334954i	$-0.891279\pi$
$941$	−57.8075	−1.88447	−0.942235	+	0.334954i	$0.891279\pi$
$942$	−15.2742	−0.497660
$943$	−13.7597	−0.448076
$944$	−4.43067	−0.144206
$945$	−2.89042	−0.0940254
$946$	20.4203	0.663920
$947$	−49.0924	−1.59529	−0.797644	−	0.603128i	$-0.793921\pi$
$947$	−49.0924	−1.59529	−0.797644	+	0.603128i	$0.793921\pi$
$948$	−0.124190	−0.00403351
$949$	22.1297	0.718359
$950$	0.321272	0.0104234
$951$	−2.82686	−0.0916672
$952$	−2.08643	−0.0676215
$953$	−13.4296	−0.435028	−0.217514	−	0.976057i	$-0.569795\pi$
$953$	−13.4296	−0.435028	−0.217514	+	0.976057i	$0.569795\pi$
$954$	3.77402	0.122188
$955$	−10.9231	−0.353464
$956$	4.22509	0.136649
$957$	−36.8649	−1.19167
$958$	−5.33449	−0.172349
$959$	1.98492	0.0640964
$960$	2.35017	0.0758515
$961$	−29.0788	−0.938025
$962$	9.65050	0.311145
$963$	−18.4405	−0.594237
$964$	15.9748	0.514514
$965$	−44.9868	−1.44818
$966$	−6.79778	−0.218715
$967$	18.6164	0.598665	0.299332	−	0.954149i	$-0.403236\pi$
$967$	18.6164	0.598665	0.299332	+	0.954149i	$0.403236\pi$
$968$	−23.4978	−0.755249
$969$	−1.04149	−0.0334573
$970$	3.00915	0.0966180
$971$	−16.6628	−0.534734	−0.267367	−	0.963595i	$-0.586154\pi$
$971$	−16.6628	−0.534734	−0.267367	+	0.963595i	$0.586154\pi$
$972$	−1.00000	−0.0320750
$973$	14.3435	0.459832
$974$	−1.69645	−0.0543579
$975$	1.72350	0.0551961
$976$	−7.93022	−0.253840
$977$	45.0126	1.44008	0.720040	−	0.693932i	$-0.244123\pi$
$977$	45.0126	1.44008	0.720040	+	0.693932i	$0.244123\pi$
$978$	−23.8503	−0.762650
$979$	−53.3919	−1.70641
$980$	12.8964	0.411959
$981$	6.56994	0.209762
$982$	−20.6650	−0.659448
$983$	50.5203	1.61135	0.805674	−	0.592359i	$-0.201803\pi$
$983$	50.5203	1.61135	0.805674	+	0.592359i	$0.201803\pi$
$984$	−2.48944	−0.0793605
$985$	48.3422	1.54031
$986$	10.6478	0.339095
$987$	−1.03776	−0.0330324
$988$	2.02190	0.0643254
$989$	−19.2164	−0.611045
$990$	13.8037	0.438711
$991$	52.5979	1.67083	0.835414	−	0.549621i	$-0.185228\pi$
$991$	52.5979	1.67083	0.835414	+	0.549621i	$0.185228\pi$
$992$	1.38608	0.0440081
$993$	21.7527	0.690302
$994$	9.63925	0.305738
$995$	−14.9284	−0.473261
$996$	0.936439	0.0296722
$997$	−50.7575	−1.60751	−0.803754	−	0.594962i	$-0.797167\pi$
$997$	−50.7575	−1.60751	−0.803754	+	0.594962i	$0.797167\pi$
$998$	−2.00275	−0.0633960
$999$	−2.93022	−0.0927081

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1338.2.a.g.1.1	✓	4
3.2	odd	2		4014.2.a.q.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.g.1.1	✓	4	1.1	even	1	trivial
4014.2.a.q.1.4		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:	\( N \)	\(=\)	\( 1338 = 2 \cdot 3 \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1338.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.35017 q^{5} +1.00000 q^{6} -1.22988 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.35017 q^{5} +1.00000 q^{6} -1.22988 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.35017 q^{10} +5.87349 q^{11} -1.00000 q^{12} -3.29344 q^{13} +1.22988 q^{14} +2.35017 q^{15} +1.00000 q^{16} -1.69645 q^{17} -1.00000 q^{18} -0.613919 q^{19} -2.35017 q^{20} +1.22988 q^{21} -5.87349 q^{22} +5.52721 q^{23} +1.00000 q^{24} +0.523313 q^{25} +3.29344 q^{26} -1.00000 q^{27} -1.22988 q^{28} +6.27650 q^{29} -2.35017 q^{30} -1.38608 q^{31} -1.00000 q^{32} -5.87349 q^{33} +1.69645 q^{34} +2.89042 q^{35} +1.00000 q^{36} +2.93022 q^{37} +0.613919 q^{38} +3.29344 q^{39} +2.35017 q^{40} -2.48944 q^{41} -1.22988 q^{42} -3.47669 q^{43} +5.87349 q^{44} -2.35017 q^{45} -5.52721 q^{46} -0.843795 q^{47} -1.00000 q^{48} -5.48741 q^{49} -0.523313 q^{50} +1.69645 q^{51} -3.29344 q^{52} -3.77402 q^{53} +1.00000 q^{54} -13.8037 q^{55} +1.22988 q^{56} +0.613919 q^{57} -6.27650 q^{58} -4.43067 q^{59} +2.35017 q^{60} -7.93022 q^{61} +1.38608 q^{62} -1.22988 q^{63} +1.00000 q^{64} +7.74015 q^{65} +5.87349 q^{66} -8.35017 q^{67} -1.69645 q^{68} -5.52721 q^{69} -2.89042 q^{70} +7.83758 q^{71} -1.00000 q^{72} -6.71932 q^{73} -2.93022 q^{74} -0.523313 q^{75} -0.613919 q^{76} -7.22366 q^{77} -3.29344 q^{78} +0.124190 q^{79} -2.35017 q^{80} +1.00000 q^{81} +2.48944 q^{82} -0.936439 q^{83} +1.22988 q^{84} +3.98696 q^{85} +3.47669 q^{86} -6.27650 q^{87} -5.87349 q^{88} -9.09032 q^{89} +2.35017 q^{90} +4.05052 q^{91} +5.52721 q^{92} +1.38608 q^{93} +0.843795 q^{94} +1.44282 q^{95} +1.00000 q^{96} +1.28039 q^{97} +5.48741 q^{98} +5.87349 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} - 5 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 - 5 * q^7 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} - 5 q^{7} - 4 q^{8} + 4 q^{9} - 2 q^{10} + 4 q^{11} - 4 q^{12} - 5 q^{13} + 5 q^{14} - 2 q^{15} + 4 q^{16} - 2 q^{17} - 4 q^{18} - 7 q^{19} + 2 q^{20} + 5 q^{21} - 4 q^{22} - 4 q^{23} + 4 q^{24} - 6 q^{25} + 5 q^{26} - 4 q^{27} - 5 q^{28} + 9 q^{29} + 2 q^{30} - q^{31} - 4 q^{32} - 4 q^{33} + 2 q^{34} + 4 q^{36} - 11 q^{37} + 7 q^{38} + 5 q^{39} - 2 q^{40} + 14 q^{41} - 5 q^{42} - 22 q^{43} + 4 q^{44} + 2 q^{45} + 4 q^{46} - 8 q^{47} - 4 q^{48} - 7 q^{49} + 6 q^{50} + 2 q^{51} - 5 q^{52} + 3 q^{53} + 4 q^{54} - 13 q^{55} + 5 q^{56} + 7 q^{57} - 9 q^{58} - 6 q^{59} - 2 q^{60} - 9 q^{61} + q^{62} - 5 q^{63} + 4 q^{64} - 3 q^{65} + 4 q^{66} - 22 q^{67} - 2 q^{68} + 4 q^{69} + 5 q^{71} - 4 q^{72} - 3 q^{73} + 11 q^{74} + 6 q^{75} - 7 q^{76} + 2 q^{77} - 5 q^{78} - 29 q^{79} + 2 q^{80} + 4 q^{81} - 14 q^{82} - 12 q^{83} + 5 q^{84} - 10 q^{85} + 22 q^{86} - 9 q^{87} - 4 q^{88} + 9 q^{89} - 2 q^{90} - 18 q^{91} - 4 q^{92} + q^{93} + 8 q^{94} - 2 q^{95} + 4 q^{96} - 29 q^{97} + 7 q^{98} + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 - 5 * q^7 - 4 * q^8 + 4 * q^9 - 2 * q^10 + 4 * q^11 - 4 * q^12 - 5 * q^13 + 5 * q^14 - 2 * q^15 + 4 * q^16 - 2 * q^17 - 4 * q^18 - 7 * q^19 + 2 * q^20 + 5 * q^21 - 4 * q^22 - 4 * q^23 + 4 * q^24 - 6 * q^25 + 5 * q^26 - 4 * q^27 - 5 * q^28 + 9 * q^29 + 2 * q^30 - q^31 - 4 * q^32 - 4 * q^33 + 2 * q^34 + 4 * q^36 - 11 * q^37 + 7 * q^38 + 5 * q^39 - 2 * q^40 + 14 * q^41 - 5 * q^42 - 22 * q^43 + 4 * q^44 + 2 * q^45 + 4 * q^46 - 8 * q^47 - 4 * q^48 - 7 * q^49 + 6 * q^50 + 2 * q^51 - 5 * q^52 + 3 * q^53 + 4 * q^54 - 13 * q^55 + 5 * q^56 + 7 * q^57 - 9 * q^58 - 6 * q^59 - 2 * q^60 - 9 * q^61 + q^62 - 5 * q^63 + 4 * q^64 - 3 * q^65 + 4 * q^66 - 22 * q^67 - 2 * q^68 + 4 * q^69 + 5 * q^71 - 4 * q^72 - 3 * q^73 + 11 * q^74 + 6 * q^75 - 7 * q^76 + 2 * q^77 - 5 * q^78 - 29 * q^79 + 2 * q^80 + 4 * q^81 - 14 * q^82 - 12 * q^83 + 5 * q^84 - 10 * q^85 + 22 * q^86 - 9 * q^87 - 4 * q^88 + 9 * q^89 - 2 * q^90 - 18 * q^91 - 4 * q^92 + q^93 + 8 * q^94 - 2 * q^95 + 4 * q^96 - 29 * q^97 + 7 * q^98 + 4 * q^99

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