LMFDB - Embedded newform 5054.2.a.bh.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5054,2,Mod(1,5054)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5054, 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("5054.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5054 = 2 \cdot 7 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5054.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:	$40.3563931816$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.19520000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 12x^{6} + 16x^{5} + 50x^{4} - 24x^{3} - 72x^{2} - 32x - 4 $ x^8 - 2x^7 - 12x^6 + 16x^5 + 50x^4 - 24x^3 - 72x^2 - 32*x - 4
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			1.4
Root			$3.02989$ of defining polynomial
Character	$\chi$	$=$	5054.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.02954 q^{3} +1.00000 q^{4} -1.38232 q^{5} -1.02954 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.94005 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.02954 q^{3} +1.00000 q^{4} -1.38232 q^{5} -1.02954 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.94005 q^{9} -1.38232 q^{10} +0.201562 q^{11} -1.02954 q^{12} +3.61000 q^{13} +1.00000 q^{14} +1.42315 q^{15} +1.00000 q^{16} -3.98785 q^{17} -1.94005 q^{18} -1.38232 q^{20} -1.02954 q^{21} +0.201562 q^{22} -1.95145 q^{23} -1.02954 q^{24} -3.08919 q^{25} +3.61000 q^{26} +5.08597 q^{27} +1.00000 q^{28} +4.85823 q^{29} +1.42315 q^{30} +3.87902 q^{31} +1.00000 q^{32} -0.207516 q^{33} -3.98785 q^{34} -1.38232 q^{35} -1.94005 q^{36} -8.95866 q^{37} -3.71663 q^{39} -1.38232 q^{40} -9.30590 q^{41} -1.02954 q^{42} +5.30315 q^{43} +0.201562 q^{44} +2.68177 q^{45} -1.95145 q^{46} +7.51513 q^{47} -1.02954 q^{48} +1.00000 q^{49} -3.08919 q^{50} +4.10564 q^{51} +3.61000 q^{52} -0.0971934 q^{53} +5.08597 q^{54} -0.278624 q^{55} +1.00000 q^{56} +4.85823 q^{58} -9.25161 q^{59} +1.42315 q^{60} -0.827230 q^{61} +3.87902 q^{62} -1.94005 q^{63} +1.00000 q^{64} -4.99017 q^{65} -0.207516 q^{66} -11.4197 q^{67} -3.98785 q^{68} +2.00909 q^{69} -1.38232 q^{70} +4.03775 q^{71} -1.94005 q^{72} -4.59223 q^{73} -8.95866 q^{74} +3.18044 q^{75} +0.201562 q^{77} -3.71663 q^{78} -13.2028 q^{79} -1.38232 q^{80} +0.583959 q^{81} -9.30590 q^{82} +10.1055 q^{83} -1.02954 q^{84} +5.51247 q^{85} +5.30315 q^{86} -5.00173 q^{87} +0.201562 q^{88} +6.42150 q^{89} +2.68177 q^{90} +3.61000 q^{91} -1.95145 q^{92} -3.99360 q^{93} +7.51513 q^{94} -1.02954 q^{96} -10.9140 q^{97} +1.00000 q^{98} -0.391042 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 - 4 * q^3 + 8 * q^4 - 2 * q^5 - 4 * q^6 + 8 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} - 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} + 8 q^{8} + 8 q^{9} - 2 q^{10} - 12 q^{11} - 4 q^{12} - 10 q^{13} + 8 q^{14} - 24 q^{15} + 8 q^{16} - 6 q^{17} + 8 q^{18} - 2 q^{20} - 4 q^{21} - 12 q^{22} - 20 q^{23} - 4 q^{24} + 8 q^{25} - 10 q^{26} - 22 q^{27} + 8 q^{28} - 8 q^{29} - 24 q^{30} - 18 q^{31} + 8 q^{32} + 16 q^{33} - 6 q^{34} - 2 q^{35} + 8 q^{36} - 36 q^{37} - 2 q^{40} - 4 q^{41} - 4 q^{42} - 16 q^{43} - 12 q^{44} + 8 q^{45} - 20 q^{46} - 10 q^{47} - 4 q^{48} + 8 q^{49} + 8 q^{50} - 12 q^{51} - 10 q^{52} - 32 q^{53} - 22 q^{54} - 22 q^{55} + 8 q^{56} - 8 q^{58} + 2 q^{59} - 24 q^{60} + 2 q^{61} - 18 q^{62} + 8 q^{63} + 8 q^{64} + 16 q^{66} - 44 q^{67} - 6 q^{68} - 2 q^{70} - 8 q^{71} + 8 q^{72} + 30 q^{73} - 36 q^{74} + 16 q^{75} - 12 q^{77} - 60 q^{79} - 2 q^{80} + 12 q^{81} - 4 q^{82} - 28 q^{83} - 4 q^{84} - 16 q^{85} - 16 q^{86} + 24 q^{87} - 12 q^{88} + 22 q^{89} + 8 q^{90} - 10 q^{91} - 20 q^{92} - 16 q^{93} - 10 q^{94} - 4 q^{96} + 6 q^{97} + 8 q^{98} - 52 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 4 * q^3 + 8 * q^4 - 2 * q^5 - 4 * q^6 + 8 * q^7 + 8 * q^8 + 8 * q^9 - 2 * q^10 - 12 * q^11 - 4 * q^12 - 10 * q^13 + 8 * q^14 - 24 * q^15 + 8 * q^16 - 6 * q^17 + 8 * q^18 - 2 * q^20 - 4 * q^21 - 12 * q^22 - 20 * q^23 - 4 * q^24 + 8 * q^25 - 10 * q^26 - 22 * q^27 + 8 * q^28 - 8 * q^29 - 24 * q^30 - 18 * q^31 + 8 * q^32 + 16 * q^33 - 6 * q^34 - 2 * q^35 + 8 * q^36 - 36 * q^37 - 2 * q^40 - 4 * q^41 - 4 * q^42 - 16 * q^43 - 12 * q^44 + 8 * q^45 - 20 * q^46 - 10 * q^47 - 4 * q^48 + 8 * q^49 + 8 * q^50 - 12 * q^51 - 10 * q^52 - 32 * q^53 - 22 * q^54 - 22 * q^55 + 8 * q^56 - 8 * q^58 + 2 * q^59 - 24 * q^60 + 2 * q^61 - 18 * q^62 + 8 * q^63 + 8 * q^64 + 16 * q^66 - 44 * q^67 - 6 * q^68 - 2 * q^70 - 8 * q^71 + 8 * q^72 + 30 * q^73 - 36 * q^74 + 16 * q^75 - 12 * q^77 - 60 * q^79 - 2 * q^80 + 12 * q^81 - 4 * q^82 - 28 * q^83 - 4 * q^84 - 16 * q^85 - 16 * q^86 + 24 * q^87 - 12 * q^88 + 22 * q^89 + 8 * q^90 - 10 * q^91 - 20 * q^92 - 16 * q^93 - 10 * q^94 - 4 * q^96 + 6 * q^97 + 8 * q^98 - 52 * 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
$3$	−1.02954	−0.594404	−0.297202	−	0.954815i	$-0.596053\pi$
$3$	−1.02954	−0.594404	−0.297202	+	0.954815i	$0.596053\pi$
$4$	1.00000	0.500000
$5$	−1.38232	−0.618192	−0.309096	−	0.951031i	$-0.600026\pi$
$5$	−1.38232	−0.618192	−0.309096	+	0.951031i	$0.600026\pi$
$6$	−1.02954	−0.420307
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	−1.94005	−0.646684
$10$	−1.38232	−0.437128
$11$	0.201562	0.0607734	0.0303867	−	0.999538i	$-0.490326\pi$
$11$	0.201562	0.0607734	0.0303867	+	0.999538i	$0.490326\pi$
$12$	−1.02954	−0.297202
$13$	3.61000	1.00123	0.500616	−	0.865669i	$-0.333107\pi$
$13$	3.61000	1.00123	0.500616	+	0.865669i	$0.333107\pi$
$14$	1.00000	0.267261
$15$	1.42315	0.367456
$16$	1.00000	0.250000
$17$	−3.98785	−0.967195	−0.483597	−	0.875291i	$-0.660670\pi$
$17$	−3.98785	−0.967195	−0.483597	+	0.875291i	$0.660670\pi$
$18$	−1.94005	−0.457275
$19$	0	0
$19$	0	0
$20$	−1.38232	−0.309096
$21$	−1.02954	−0.224664
$22$	0.201562	0.0429733
$23$	−1.95145	−0.406905	−0.203453	−	0.979085i	$-0.565216\pi$
$23$	−1.95145	−0.406905	−0.203453	+	0.979085i	$0.565216\pi$
$24$	−1.02954	−0.210153
$25$	−3.08919	−0.617839
$26$	3.61000	0.707978
$27$	5.08597	0.978795
$28$	1.00000	0.188982
$29$	4.85823	0.902150	0.451075	−	0.892486i	$-0.351041\pi$
$29$	4.85823	0.902150	0.451075	+	0.892486i	$0.351041\pi$
$30$	1.42315	0.259830
$31$	3.87902	0.696693	0.348347	−	0.937366i	$-0.386743\pi$
$31$	3.87902	0.696693	0.348347	+	0.937366i	$0.386743\pi$
$32$	1.00000	0.176777
$33$	−0.207516	−0.0361239
$34$	−3.98785	−0.683910
$35$	−1.38232	−0.233655
$36$	−1.94005	−0.323342
$37$	−8.95866	−1.47279	−0.736397	−	0.676550i	$-0.763474\pi$
$37$	−8.95866	−1.47279	−0.736397	+	0.676550i	$0.763474\pi$
$38$	0	0
$39$	−3.71663	−0.595137
$40$	−1.38232	−0.218564
$41$	−9.30590	−1.45334	−0.726669	−	0.686988i	$-0.758932\pi$
$41$	−9.30590	−1.45334	−0.726669	+	0.686988i	$0.758932\pi$
$42$	−1.02954	−0.158861
$43$	5.30315	0.808722	0.404361	−	0.914599i	$-0.367494\pi$
$43$	5.30315	0.808722	0.404361	+	0.914599i	$0.367494\pi$
$44$	0.201562	0.0303867
$45$	2.68177	0.399775
$46$	−1.95145	−0.287726
$47$	7.51513	1.09619	0.548097	−	0.836414i	$-0.315352\pi$
$47$	7.51513	1.09619	0.548097	+	0.836414i	$0.315352\pi$
$48$	−1.02954	−0.148601
$49$	1.00000	0.142857
$50$	−3.08919	−0.436878
$51$	4.10564	0.574904
$52$	3.61000	0.500616
$53$	−0.0971934	−0.0133505	−0.00667527	−	0.999978i	$-0.502125\pi$
$53$	−0.0971934	−0.0133505	−0.00667527	+	0.999978i	$0.502125\pi$
$54$	5.08597	0.692113
$55$	−0.278624	−0.0375696
$56$	1.00000	0.133631
$57$	0	0
$58$	4.85823	0.637917
$59$	−9.25161	−1.20446	−0.602229	−	0.798324i	$-0.705720\pi$
$59$	−9.25161	−1.20446	−0.602229	+	0.798324i	$0.705720\pi$
$60$	1.42315	0.183728
$61$	−0.827230	−0.105916	−0.0529580	−	0.998597i	$-0.516865\pi$
$61$	−0.827230	−0.105916	−0.0529580	+	0.998597i	$0.516865\pi$
$62$	3.87902	0.492637
$63$	−1.94005	−0.244424
$64$	1.00000	0.125000
$65$	−4.99017	−0.618954
$66$	−0.207516	−0.0255435
$67$	−11.4197	−1.39514	−0.697571	−	0.716516i	$-0.745736\pi$
$67$	−11.4197	−1.39514	−0.697571	+	0.716516i	$0.745736\pi$
$68$	−3.98785	−0.483597
$69$	2.00909	0.241866
$70$	−1.38232	−0.165219
$71$	4.03775	0.479193	0.239596	−	0.970873i	$-0.422985\pi$
$71$	4.03775	0.479193	0.239596	+	0.970873i	$0.422985\pi$
$72$	−1.94005	−0.228637
$73$	−4.59223	−0.537480	−0.268740	−	0.963213i	$-0.586607\pi$
$73$	−4.59223	−0.537480	−0.268740	+	0.963213i	$0.586607\pi$
$74$	−8.95866	−1.04142
$75$	3.18044	0.367246
$76$	0	0
$77$	0.201562	0.0229702
$78$	−3.71663	−0.420825
$79$	−13.2028	−1.48543	−0.742715	−	0.669607i	$-0.766462\pi$
$79$	−13.2028	−1.48543	−0.742715	+	0.669607i	$0.766462\pi$
$80$	−1.38232	−0.154548
$81$	0.583959	0.0648843
$82$	−9.30590	−1.02766
$83$	10.1055	1.10922	0.554610	−	0.832111i	$-0.312868\pi$
$83$	10.1055	1.10922	0.554610	+	0.832111i	$0.312868\pi$
$84$	−1.02954	−0.112332
$85$	5.51247	0.597912
$86$	5.30315	0.571853
$87$	−5.00173	−0.536242
$88$	0.201562	0.0214866
$89$	6.42150	0.680677	0.340339	−	0.940303i	$-0.389458\pi$
$89$	6.42150	0.680677	0.340339	+	0.940303i	$0.389458\pi$
$90$	2.68177	0.282683
$91$	3.61000	0.378430
$92$	−1.95145	−0.203453
$93$	−3.99360	−0.414117
$94$	7.51513	0.775127
$95$	0	0
$96$	−1.02954	−0.105077
$97$	−10.9140	−1.10815	−0.554074	−	0.832467i	$-0.686928\pi$
$97$	−10.9140	−1.10815	−0.554074	+	0.832467i	$0.686928\pi$
$98$	1.00000	0.101015
$99$	−0.391042	−0.0393012
$100$	−3.08919	−0.308919
$101$	−2.36077	−0.234905	−0.117453	−	0.993078i	$-0.537473\pi$
$101$	−2.36077	−0.234905	−0.117453	+	0.993078i	$0.537473\pi$
$102$	4.10564	0.406519
$103$	−7.52998	−0.741951	−0.370975	−	0.928643i	$-0.620977\pi$
$103$	−7.52998	−0.741951	−0.370975	+	0.928643i	$0.620977\pi$
$104$	3.61000	0.353989
$105$	1.42315	0.138885
$106$	−0.0971934	−0.00944026
$107$	3.07601	0.297369	0.148684	−	0.988885i	$-0.452496\pi$
$107$	3.07601	0.297369	0.148684	+	0.988885i	$0.452496\pi$
$108$	5.08597	0.489398
$109$	−2.31631	−0.221862	−0.110931	−	0.993828i	$-0.535383\pi$
$109$	−2.31631	−0.221862	−0.110931	+	0.993828i	$0.535383\pi$
$110$	−0.278624	−0.0265657
$111$	9.22327	0.875434
$112$	1.00000	0.0944911
$113$	−3.65946	−0.344253	−0.172127	−	0.985075i	$-0.555064\pi$
$113$	−3.65946	−0.344253	−0.172127	+	0.985075i	$0.555064\pi$
$114$	0	0
$115$	2.69753	0.251546
$116$	4.85823	0.451075
$117$	−7.00358	−0.647481
$118$	−9.25161	−0.851680
$119$	−3.98785	−0.365565
$120$	1.42315	0.129915
$121$	−10.9594	−0.996307
$122$	−0.827230	−0.0748939
$123$	9.58077	0.863869
$124$	3.87902	0.348347
$125$	11.1818	1.00013
$126$	−1.94005	−0.172834
$127$	−3.21620	−0.285392	−0.142696	−	0.989767i	$-0.545577\pi$
$127$	−3.21620	−0.285392	−0.142696	+	0.989767i	$0.545577\pi$
$128$	1.00000	0.0883883
$129$	−5.45979	−0.480708
$130$	−4.99017	−0.437666
$131$	−6.57612	−0.574559	−0.287279	−	0.957847i	$-0.592751\pi$
$131$	−6.57612	−0.574559	−0.287279	+	0.957847i	$0.592751\pi$
$132$	−0.207516	−0.0180620
$133$	0	0
$134$	−11.4197	−0.986514
$135$	−7.03043	−0.605083
$136$	−3.98785	−0.341955
$137$	−18.4291	−1.57450	−0.787250	−	0.616634i	$-0.788496\pi$
$137$	−18.4291	−1.57450	−0.787250	+	0.616634i	$0.788496\pi$
$138$	2.00909	0.171025
$139$	−3.46809	−0.294160	−0.147080	−	0.989125i	$-0.546987\pi$
$139$	−3.46809	−0.294160	−0.147080	+	0.989125i	$0.546987\pi$
$140$	−1.38232	−0.116827
$141$	−7.73711	−0.651583
$142$	4.03775	0.338840
$143$	0.727640	0.0608483
$144$	−1.94005	−0.161671
$145$	−6.71562	−0.557702
$146$	−4.59223	−0.380056
$147$	−1.02954	−0.0849148
$148$	−8.95866	−0.736397
$149$	13.8826	1.13731	0.568653	−	0.822578i	$-0.307465\pi$
$149$	13.8826	1.13731	0.568653	+	0.822578i	$0.307465\pi$
$150$	3.18044	0.259682
$151$	−19.5806	−1.59345	−0.796725	−	0.604342i	$-0.793436\pi$
$151$	−19.5806	−1.59345	−0.796725	+	0.604342i	$0.793436\pi$
$152$	0	0
$153$	7.73663	0.625469
$154$	0.201562	0.0162424
$155$	−5.36205	−0.430690
$156$	−3.71663	−0.297568
$157$	−4.95955	−0.395816	−0.197908	−	0.980221i	$-0.563415\pi$
$157$	−4.95955	−0.395816	−0.197908	+	0.980221i	$0.563415\pi$
$158$	−13.2028	−1.05036
$159$	0.100064	0.00793562
$160$	−1.38232	−0.109282
$161$	−1.95145	−0.153796
$162$	0.583959	0.0458801
$163$	10.2851	0.805594	0.402797	−	0.915289i	$-0.368038\pi$
$163$	10.2851	0.805594	0.402797	+	0.915289i	$0.368038\pi$
$164$	−9.30590	−0.726669
$165$	0.286853	0.0223315
$166$	10.1055	0.784336
$167$	0.132958	0.0102886	0.00514432	−	0.999987i	$-0.498363\pi$
$167$	0.132958	0.0102886	0.00514432	+	0.999987i	$0.498363\pi$
$168$	−1.02954	−0.0794306
$169$	0.0320700	0.00246692
$170$	5.51247	0.422787
$171$	0	0
$172$	5.30315	0.404361
$173$	19.5740	1.48818	0.744091	−	0.668078i	$-0.232883\pi$
$173$	19.5740	1.48818	0.744091	+	0.668078i	$0.232883\pi$
$174$	−5.00173	−0.379180
$175$	−3.08919	−0.233521
$176$	0.201562	0.0151933
$177$	9.52488	0.715934
$178$	6.42150	0.481312
$179$	−2.05705	−0.153751	−0.0768756	−	0.997041i	$-0.524494\pi$
$179$	−2.05705	−0.153751	−0.0768756	+	0.997041i	$0.524494\pi$
$180$	2.68177	0.199887
$181$	−4.47667	−0.332748	−0.166374	−	0.986063i	$-0.553206\pi$
$181$	−4.47667	−0.332748	−0.166374	+	0.986063i	$0.553206\pi$
$182$	3.61000	0.267591
$183$	0.851665	0.0629569
$184$	−1.95145	−0.143863
$185$	12.3837	0.910469
$186$	−3.99360	−0.292825
$187$	−0.803800	−0.0587797
$188$	7.51513	0.548097
$189$	5.08597	0.369950
$190$	0	0
$191$	−3.44067	−0.248958	−0.124479	−	0.992222i	$-0.539726\pi$
$191$	−3.44067	−0.248958	−0.124479	+	0.992222i	$0.539726\pi$
$192$	−1.02954	−0.0743005
$193$	−25.1998	−1.81392	−0.906959	−	0.421219i	$-0.861603\pi$
$193$	−25.1998	−1.81392	−0.906959	+	0.421219i	$0.861603\pi$
$194$	−10.9140	−0.783579
$195$	5.13756	0.367909
$196$	1.00000	0.0714286
$197$	−14.3027	−1.01902	−0.509512	−	0.860464i	$-0.670174\pi$
$197$	−14.3027	−1.01902	−0.509512	+	0.860464i	$0.670174\pi$
$198$	−0.391042	−0.0277901
$199$	12.3169	0.873121	0.436560	−	0.899675i	$-0.356197\pi$
$199$	12.3169	0.873121	0.436560	+	0.899675i	$0.356197\pi$
$200$	−3.08919	−0.218439
$201$	11.7570	0.829278
$202$	−2.36077	−0.166103
$203$	4.85823	0.340981
$204$	4.10564	0.287452
$205$	12.8637	0.898441
$206$	−7.52998	−0.524638
$207$	3.78591	0.263139
$208$	3.61000	0.250308
$209$	0	0
$210$	1.42315	0.0982066
$211$	−22.8505	−1.57309	−0.786547	−	0.617530i	$-0.788133\pi$
$211$	−22.8505	−1.57309	−0.786547	+	0.617530i	$0.788133\pi$
$212$	−0.0971934	−0.00667527
$213$	−4.15702	−0.284834
$214$	3.07601	0.210272
$215$	−7.33064	−0.499946
$216$	5.08597	0.346056
$217$	3.87902	0.263325
$218$	−2.31631	−0.156880
$219$	4.72788	0.319480
$220$	−0.278624	−0.0187848
$221$	−14.3961	−0.968387
$222$	9.22327	0.619026
$223$	7.72793	0.517501	0.258750	−	0.965944i	$-0.416689\pi$
$223$	7.72793	0.517501	0.258750	+	0.965944i	$0.416689\pi$
$224$	1.00000	0.0668153
$225$	5.99320	0.399547
$226$	−3.65946	−0.243424
$227$	−25.9981	−1.72556	−0.862779	−	0.505581i	$-0.831278\pi$
$227$	−25.9981	−1.72556	−0.862779	+	0.505581i	$0.831278\pi$
$228$	0	0
$229$	−24.1289	−1.59448	−0.797242	−	0.603660i	$-0.793708\pi$
$229$	−24.1289	−1.59448	−0.797242	+	0.603660i	$0.793708\pi$
$230$	2.69753	0.177870
$231$	−0.207516	−0.0136536
$232$	4.85823	0.318958
$233$	−27.2074	−1.78242	−0.891208	−	0.453595i	$-0.850141\pi$
$233$	−27.2074	−1.78242	−0.891208	+	0.453595i	$0.850141\pi$
$234$	−7.00358	−0.457838
$235$	−10.3883	−0.677659
$236$	−9.25161	−0.602229
$237$	13.5928	0.882946
$238$	−3.98785	−0.258494
$239$	26.5413	1.71681	0.858406	−	0.512971i	$-0.171455\pi$
$239$	26.5413	1.71681	0.858406	+	0.512971i	$0.171455\pi$
$240$	1.42315	0.0918639
$241$	11.8064	0.760517	0.380258	−	0.924880i	$-0.375835\pi$
$241$	11.8064	0.760517	0.380258	+	0.924880i	$0.375835\pi$
$242$	−10.9594	−0.704495
$243$	−15.8591	−1.01736
$244$	−0.827230	−0.0529580
$245$	−1.38232	−0.0883131
$246$	9.58077	0.610848
$247$	0	0
$248$	3.87902	0.246318
$249$	−10.4040	−0.659324
$250$	11.1818	0.707202
$251$	11.1809	0.705735	0.352867	−	0.935673i	$-0.385207\pi$
$251$	11.1809	0.705735	0.352867	+	0.935673i	$0.385207\pi$
$252$	−1.94005	−0.122212
$253$	−0.393339	−0.0247290
$254$	−3.21620	−0.201802
$255$	−5.67530	−0.355401
$256$	1.00000	0.0625000
$257$	29.5934	1.84599	0.922993	−	0.384817i	$-0.125736\pi$
$257$	29.5934	1.84599	0.922993	+	0.384817i	$0.125736\pi$
$258$	−5.45979	−0.339912
$259$	−8.95866	−0.556664
$260$	−4.99017	−0.309477
$261$	−9.42522	−0.583406
$262$	−6.57612	−0.406274
$263$	−24.6167	−1.51793	−0.758966	−	0.651131i	$-0.774295\pi$
$263$	−24.6167	−1.51793	−0.758966	+	0.651131i	$0.774295\pi$
$264$	−0.207516	−0.0127717
$265$	0.134352	0.00825320
$266$	0	0
$267$	−6.61117	−0.404597
$268$	−11.4197	−0.697571
$269$	7.74788	0.472397	0.236198	−	0.971705i	$-0.424098\pi$
$269$	7.74788	0.472397	0.236198	+	0.971705i	$0.424098\pi$
$270$	−7.03043	−0.427858
$271$	−11.5762	−0.703204	−0.351602	−	0.936150i	$-0.614363\pi$
$271$	−11.5762	−0.703204	−0.351602	+	0.936150i	$0.614363\pi$
$272$	−3.98785	−0.241799
$273$	−3.71663	−0.224940
$274$	−18.4291	−1.11334
$275$	−0.622666	−0.0375481
$276$	2.00909	0.120933
$277$	4.90088	0.294465	0.147233	−	0.989102i	$-0.452963\pi$
$277$	4.90088	0.294465	0.147233	+	0.989102i	$0.452963\pi$
$278$	−3.46809	−0.208002
$279$	−7.52551	−0.450541
$280$	−1.38232	−0.0826094
$281$	−16.3213	−0.973650	−0.486825	−	0.873500i	$-0.661845\pi$
$281$	−16.3213	−0.973650	−0.486825	+	0.873500i	$0.661845\pi$
$282$	−7.73711	−0.460738
$283$	−19.1772	−1.13997	−0.569983	−	0.821656i	$-0.693050\pi$
$283$	−19.1772	−1.13997	−0.569983	+	0.821656i	$0.693050\pi$
$284$	4.03775	0.239596
$285$	0	0
$286$	0.727640	0.0430262
$287$	−9.30590	−0.549310
$288$	−1.94005	−0.114319
$289$	−1.09709	−0.0645348
$290$	−6.71562	−0.394355
$291$	11.2364	0.658688
$292$	−4.59223	−0.268740
$293$	9.38811	0.548459	0.274230	−	0.961664i	$-0.411577\pi$
$293$	9.38811	0.548459	0.274230	+	0.961664i	$0.411577\pi$
$294$	−1.02954	−0.0600439
$295$	12.7887	0.744586
$296$	−8.95866	−0.520711
$297$	1.02514	0.0594847
$298$	13.8826	0.804196
$299$	−7.04473	−0.407407
$300$	3.18044	0.183623
$301$	5.30315	0.305668
$302$	−19.5806	−1.12674
$303$	2.43050	0.139629
$304$	0	0
$305$	1.14350	0.0654764
$306$	7.73663	0.442274
$307$	−24.5234	−1.39962	−0.699811	−	0.714328i	$-0.746733\pi$
$307$	−24.5234	−1.39962	−0.699811	+	0.714328i	$0.746733\pi$
$308$	0.201562	0.0114851
$309$	7.75240	0.441018
$310$	−5.36205	−0.304544
$311$	11.6440	0.660272	0.330136	−	0.943933i	$-0.392905\pi$
$311$	11.6440	0.660272	0.330136	+	0.943933i	$0.392905\pi$
$312$	−3.71663	−0.210413
$313$	−9.14724	−0.517033	−0.258516	−	0.966007i	$-0.583234\pi$
$313$	−9.14724	−0.517033	−0.258516	+	0.966007i	$0.583234\pi$
$314$	−4.95955	−0.279884
$315$	2.68177	0.151101
$316$	−13.2028	−0.742715
$317$	9.31007	0.522906	0.261453	−	0.965216i	$-0.415798\pi$
$317$	9.31007	0.522906	0.261453	+	0.965216i	$0.415798\pi$
$318$	0.100064	0.00561133
$319$	0.979237	0.0548267
$320$	−1.38232	−0.0772740
$321$	−3.16687	−0.176757
$322$	−1.95145	−0.108750
$323$	0	0
$324$	0.583959	0.0324422
$325$	−11.1520	−0.618601
$326$	10.2851	0.569641
$327$	2.38472	0.131876
$328$	−9.30590	−0.513832
$329$	7.51513	0.414323
$330$	0.286853	0.0157908
$331$	−26.6779	−1.46635	−0.733174	−	0.680041i	$-0.761962\pi$
$331$	−26.6779	−1.46635	−0.733174	+	0.680041i	$0.761962\pi$
$332$	10.1055	0.554610
$333$	17.3803	0.952432
$334$	0.132958	0.00727516
$335$	15.7857	0.862465
$336$	−1.02954	−0.0561659
$337$	−4.47140	−0.243573	−0.121786	−	0.992556i	$-0.538862\pi$
$337$	−4.47140	−0.243573	−0.121786	+	0.992556i	$0.538862\pi$
$338$	0.0320700	0.00174438
$339$	3.76755	0.204625
$340$	5.51247	0.298956
$341$	0.781866	0.0423404
$342$	0	0
$343$	1.00000	0.0539949
$344$	5.30315	0.285927
$345$	−2.77720	−0.149520
$346$	19.5740	1.05230
$347$	3.89738	0.209222	0.104611	−	0.994513i	$-0.466640\pi$
$347$	3.89738	0.209222	0.104611	+	0.994513i	$0.466640\pi$
$348$	−5.00173	−0.268121
$349$	13.5634	0.726031	0.363015	−	0.931783i	$-0.381747\pi$
$349$	13.5634	0.726031	0.363015	+	0.931783i	$0.381747\pi$
$350$	−3.08919	−0.165124
$351$	18.3603	0.980002
$352$	0.201562	0.0107433
$353$	27.5213	1.46481	0.732405	−	0.680869i	$-0.238398\pi$
$353$	27.5213	1.46481	0.732405	+	0.680869i	$0.238398\pi$
$354$	9.52488	0.506242
$355$	−5.58146	−0.296233
$356$	6.42150	0.340339
$357$	4.10564	0.217293
$358$	−2.05705	−0.108718
$359$	16.7795	0.885590	0.442795	−	0.896623i	$-0.353987\pi$
$359$	16.7795	0.885590	0.442795	+	0.896623i	$0.353987\pi$
$360$	2.68177	0.141342
$361$	0	0
$362$	−4.47667	−0.235288
$363$	11.2831	0.592208
$364$	3.61000	0.189215
$365$	6.34793	0.332266
$366$	0.851665	0.0445172
$367$	28.5583	1.49073	0.745365	−	0.666656i	$-0.232275\pi$
$367$	28.5583	1.49073	0.745365	+	0.666656i	$0.232275\pi$
$368$	−1.95145	−0.101726
$369$	18.0539	0.939850
$370$	12.3837	0.643799
$371$	−0.0971934	−0.00504603
$372$	−3.99360	−0.207059
$373$	−0.530273	−0.0274565	−0.0137282	−	0.999906i	$-0.504370\pi$
$373$	−0.530273	−0.0274565	−0.0137282	+	0.999906i	$0.504370\pi$
$374$	−0.803800	−0.0415635
$375$	−11.5121	−0.594484
$376$	7.51513	0.387563
$377$	17.5382	0.903263
$378$	5.08597	0.261594
$379$	0.442763	0.0227432	0.0113716	−	0.999935i	$-0.496380\pi$
$379$	0.442763	0.0227432	0.0113716	+	0.999935i	$0.496380\pi$
$380$	0	0
$381$	3.31120	0.169638
$382$	−3.44067	−0.176040
$383$	−3.06625	−0.156678	−0.0783390	−	0.996927i	$-0.524962\pi$
$383$	−3.06625	−0.156678	−0.0783390	+	0.996927i	$0.524962\pi$
$384$	−1.02954	−0.0525384
$385$	−0.278624	−0.0142000
$386$	−25.1998	−1.28263
$387$	−10.2884	−0.522988
$388$	−10.9140	−0.554074
$389$	−14.5166	−0.736021	−0.368010	−	0.929822i	$-0.619961\pi$
$389$	−14.5166	−0.736021	−0.368010	+	0.929822i	$0.619961\pi$
$390$	5.13756	0.260151
$391$	7.78208	0.393557
$392$	1.00000	0.0505076
$393$	6.77037	0.341520
$394$	−14.3027	−0.720559
$395$	18.2505	0.918281
$396$	−0.391042	−0.0196506
$397$	27.0449	1.35735	0.678674	−	0.734440i	$-0.262555\pi$
$397$	27.0449	1.35735	0.678674	+	0.734440i	$0.262555\pi$
$398$	12.3169	0.617390
$399$	0	0
$400$	−3.08919	−0.154460
$401$	−1.61913	−0.0808556	−0.0404278	−	0.999182i	$-0.512872\pi$
$401$	−1.61913	−0.0808556	−0.0404278	+	0.999182i	$0.512872\pi$
$402$	11.7570	0.586388
$403$	14.0033	0.697552
$404$	−2.36077	−0.117453
$405$	−0.807217	−0.0401110
$406$	4.85823	0.241110
$407$	−1.80573	−0.0895067
$408$	4.10564	0.203259
$409$	6.10992	0.302116	0.151058	−	0.988525i	$-0.451732\pi$
$409$	6.10992	0.302116	0.151058	+	0.988525i	$0.451732\pi$
$410$	12.8637	0.635294
$411$	18.9734	0.935889
$412$	−7.52998	−0.370975
$413$	−9.25161	−0.455242
$414$	3.78591	0.186068
$415$	−13.9690	−0.685710
$416$	3.61000	0.176995
$417$	3.57053	0.174850
$418$	0	0
$419$	1.92152	0.0938723	0.0469362	−	0.998898i	$-0.485054\pi$
$419$	1.92152	0.0938723	0.0469362	+	0.998898i	$0.485054\pi$
$420$	1.42315	0.0694426
$421$	31.5836	1.53929	0.769645	−	0.638472i	$-0.220433\pi$
$421$	31.5836	1.53929	0.769645	+	0.638472i	$0.220433\pi$
$422$	−22.8505	−1.11235
$423$	−14.5798	−0.708892
$424$	−0.0971934	−0.00472013
$425$	12.3192	0.597570
$426$	−4.15702	−0.201408
$427$	−0.827230	−0.0400325
$428$	3.07601	0.148684
$429$	−0.749132	−0.0361685
$430$	−7.33064	−0.353515
$431$	24.5430	1.18219	0.591097	−	0.806601i	$-0.298695\pi$
$431$	24.5430	1.18219	0.591097	+	0.806601i	$0.298695\pi$
$432$	5.08597	0.244699
$433$	−35.6753	−1.71444	−0.857222	−	0.514947i	$-0.827811\pi$
$433$	−35.6753	−1.71444	−0.857222	+	0.514947i	$0.827811\pi$
$434$	3.87902	0.186199
$435$	6.91399	0.331500
$436$	−2.31631	−0.110931
$437$	0	0
$438$	4.72788	0.225907
$439$	10.6745	0.509466	0.254733	−	0.967011i	$-0.418012\pi$
$439$	10.6745	0.509466	0.254733	+	0.967011i	$0.418012\pi$
$440$	−0.278624	−0.0132829
$441$	−1.94005	−0.0923834
$442$	−14.3961	−0.684753
$443$	−28.1790	−1.33882	−0.669412	−	0.742891i	$-0.733454\pi$
$443$	−28.1790	−1.33882	−0.669412	+	0.742891i	$0.733454\pi$
$444$	9.22327	0.437717
$445$	−8.87656	−0.420789
$446$	7.72793	0.365928
$447$	−14.2926	−0.676019
$448$	1.00000	0.0472456
$449$	−1.02365	−0.0483091	−0.0241545	−	0.999708i	$-0.507689\pi$
$449$	−1.02365	−0.0483091	−0.0241545	+	0.999708i	$0.507689\pi$
$450$	5.99320	0.282522
$451$	−1.87572	−0.0883242
$452$	−3.65946	−0.172127
$453$	20.1590	0.947153
$454$	−25.9981	−1.22015
$455$	−4.99017	−0.233943
$456$	0	0
$457$	30.0439	1.40539	0.702697	−	0.711489i	$-0.251979\pi$
$457$	30.0439	1.40539	0.702697	+	0.711489i	$0.251979\pi$
$458$	−24.1289	−1.12747
$459$	−20.2821	−0.946685
$460$	2.69753	0.125773
$461$	−11.9442	−0.556298	−0.278149	−	0.960538i	$-0.589721\pi$
$461$	−11.9442	−0.556298	−0.278149	+	0.960538i	$0.589721\pi$
$462$	−0.207516	−0.00965452
$463$	−27.3252	−1.26991	−0.634954	−	0.772550i	$-0.718981\pi$
$463$	−27.3252	−1.26991	−0.634954	+	0.772550i	$0.718981\pi$
$464$	4.85823	0.225538
$465$	5.52043	0.256004
$466$	−27.2074	−1.26036
$467$	−7.73250	−0.357818	−0.178909	−	0.983866i	$-0.557257\pi$
$467$	−7.73250	−0.357818	−0.178909	+	0.983866i	$0.557257\pi$
$468$	−7.00358	−0.323741
$469$	−11.4197	−0.527314
$470$	−10.3883	−0.479177
$471$	5.10605	0.235274
$472$	−9.25161	−0.425840
$473$	1.06892	0.0491488
$474$	13.5928	0.624337
$475$	0	0
$476$	−3.98785	−0.182783
$477$	0.188560	0.00863358
$478$	26.5413	1.21397
$479$	−1.52899	−0.0698614	−0.0349307	−	0.999390i	$-0.511121\pi$
$479$	−1.52899	−0.0698614	−0.0349307	+	0.999390i	$0.511121\pi$
$480$	1.42315	0.0649576
$481$	−32.3407	−1.47461
$482$	11.8064	0.537767
$483$	2.00909	0.0914168
$484$	−10.9594	−0.498153
$485$	15.0866	0.685048
$486$	−15.8591	−0.719384
$487$	14.6532	0.663999	0.332000	−	0.943280i	$-0.392277\pi$
$487$	14.6532	0.663999	0.332000	+	0.943280i	$0.392277\pi$
$488$	−0.827230	−0.0374469
$489$	−10.5889	−0.478848
$490$	−1.38232	−0.0624468
$491$	19.4632	0.878362	0.439181	−	0.898399i	$-0.355269\pi$
$491$	19.4632	0.878362	0.439181	+	0.898399i	$0.355269\pi$
$492$	9.58077	0.431935
$493$	−19.3739	−0.872555
$494$	0	0
$495$	0.540544	0.0242957
$496$	3.87902	0.174173
$497$	4.03775	0.181118
$498$	−10.4040	−0.466213
$499$	8.21283	0.367657	0.183828	−	0.982958i	$-0.441151\pi$
$499$	8.21283	0.367657	0.183828	+	0.982958i	$0.441151\pi$
$500$	11.1818	0.500067
$501$	−0.136886	−0.00611560
$502$	11.1809	0.499030
$503$	0.969968	0.0432488	0.0216244	−	0.999766i	$-0.493116\pi$
$503$	0.969968	0.0432488	0.0216244	+	0.999766i	$0.493116\pi$
$504$	−1.94005	−0.0864168
$505$	3.26333	0.145216
$506$	−0.393339	−0.0174861
$507$	−0.0330173	−0.00146635
$508$	−3.21620	−0.142696
$509$	−10.3220	−0.457516	−0.228758	−	0.973483i	$-0.573466\pi$
$509$	−10.3220	−0.457516	−0.228758	+	0.973483i	$0.573466\pi$
$510$	−5.67530	−0.251306
$511$	−4.59223	−0.203148
$512$	1.00000	0.0441942
$513$	0	0
$514$	29.5934	1.30531
$515$	10.4088	0.458668
$516$	−5.45979	−0.240354
$517$	1.51477	0.0666195
$518$	−8.95866	−0.393621
$519$	−20.1521	−0.884581
$520$	−4.99017	−0.218833
$521$	6.77463	0.296802	0.148401	−	0.988927i	$-0.452587\pi$
$521$	6.77463	0.296802	0.148401	+	0.988927i	$0.452587\pi$
$522$	−9.42522	−0.412531
$523$	41.6053	1.81927	0.909636	−	0.415405i	$-0.136360\pi$
$523$	41.6053	1.81927	0.909636	+	0.415405i	$0.136360\pi$
$524$	−6.57612	−0.287279
$525$	3.18044	0.138806
$526$	−24.6167	−1.07334
$527$	−15.4689	−0.673838
$528$	−0.207516	−0.00903098
$529$	−19.1918	−0.834428
$530$	0.134352	0.00583589
$531$	17.9486	0.778903
$532$	0	0
$533$	−33.5943	−1.45513
$534$	−6.61117	−0.286093
$535$	−4.25202	−0.183831
$536$	−11.4197	−0.493257
$537$	2.11781	0.0913903
$538$	7.74788	0.334035
$539$	0.201562	0.00868191
$540$	−7.03043	−0.302542
$541$	−40.8132	−1.75470	−0.877348	−	0.479855i	$-0.840689\pi$
$541$	−40.8132	−1.75470	−0.877348	+	0.479855i	$0.840689\pi$
$542$	−11.5762	−0.497241
$543$	4.60890	0.197787
$544$	−3.98785	−0.170977
$545$	3.20187	0.137153
$546$	−3.71663	−0.159057
$547$	−27.9469	−1.19492	−0.597461	−	0.801898i	$-0.703824\pi$
$547$	−27.9469	−1.19492	−0.597461	+	0.801898i	$0.703824\pi$
$548$	−18.4291	−0.787250
$549$	1.60487	0.0684942
$550$	−0.622666	−0.0265506
$551$	0	0
$552$	2.00909	0.0855126
$553$	−13.2028	−0.561440
$554$	4.90088	0.208218
$555$	−12.7495	−0.541186
$556$	−3.46809	−0.147080
$557$	22.0059	0.932421	0.466211	−	0.884674i	$-0.345619\pi$
$557$	22.0059	0.932421	0.466211	+	0.884674i	$0.345619\pi$
$558$	−7.52551	−0.318580
$559$	19.1443	0.809719
$560$	−1.38232	−0.0584136
$561$	0.827542	0.0349389
$562$	−16.3213	−0.688475
$563$	43.5246	1.83434	0.917171	−	0.398493i	$-0.130467\pi$
$563$	43.5246	1.83434	0.917171	+	0.398493i	$0.130467\pi$
$564$	−7.73711	−0.325791
$565$	5.05854	0.212814
$566$	−19.1772	−0.806078
$567$	0.583959	0.0245240
$568$	4.03775	0.169420
$569$	−27.9700	−1.17256	−0.586281	−	0.810108i	$-0.699409\pi$
$569$	−27.9700	−1.17256	−0.586281	+	0.810108i	$0.699409\pi$
$570$	0	0
$571$	29.3563	1.22852	0.614261	−	0.789103i	$-0.289454\pi$
$571$	29.3563	1.22852	0.614261	+	0.789103i	$0.289454\pi$
$572$	0.727640	0.0304241
$573$	3.54230	0.147982
$574$	−9.30590	−0.388421
$575$	6.02841	0.251402
$576$	−1.94005	−0.0808355
$577$	−5.18369	−0.215800	−0.107900	−	0.994162i	$-0.534413\pi$
$577$	−5.18369	−0.215800	−0.107900	+	0.994162i	$0.534413\pi$
$578$	−1.09709	−0.0456330
$579$	25.9441	1.07820
$580$	−6.71562	−0.278851
$581$	10.1055	0.419245
$582$	11.2364	0.465762
$583$	−0.0195905	−0.000811358 0
$584$	−4.59223	−0.190028
$585$	9.68118	0.400268
$586$	9.38811	0.387819
$587$	−15.2575	−0.629744	−0.314872	−	0.949134i	$-0.601962\pi$
$587$	−15.2575	−0.629744	−0.314872	+	0.949134i	$0.601962\pi$
$588$	−1.02954	−0.0424574
$589$	0	0
$590$	12.7887	0.526502
$591$	14.7252	0.605712
$592$	−8.95866	−0.368199
$593$	41.0813	1.68701	0.843504	−	0.537123i	$-0.180489\pi$
$593$	41.0813	1.68701	0.843504	+	0.537123i	$0.180489\pi$
$594$	1.02514	0.0420620
$595$	5.51247	0.225989
$596$	13.8826	0.568653
$597$	−12.6807	−0.518986
$598$	−7.04473	−0.288080
$599$	−25.9404	−1.05990	−0.529948	−	0.848030i	$-0.677789\pi$
$599$	−25.9404	−1.05990	−0.529948	+	0.848030i	$0.677789\pi$
$600$	3.18044	0.129841
$601$	−21.2064	−0.865026	−0.432513	−	0.901628i	$-0.642373\pi$
$601$	−21.2064	−0.865026	−0.432513	+	0.901628i	$0.642373\pi$
$602$	5.30315	0.216140
$603$	22.1549	0.902216
$604$	−19.5806	−0.796725
$605$	15.1493	0.615909
$606$	2.43050	0.0987323
$607$	−3.45922	−0.140405	−0.0702027	−	0.997533i	$-0.522365\pi$
$607$	−3.45922	−0.140405	−0.0702027	+	0.997533i	$0.522365\pi$
$608$	0	0
$609$	−5.00173	−0.202680
$610$	1.14350	0.0462988
$611$	27.1296	1.09755
$612$	7.73663	0.312735
$613$	40.4587	1.63411	0.817056	−	0.576558i	$-0.195604\pi$
$613$	40.4587	1.63411	0.817056	+	0.576558i	$0.195604\pi$
$614$	−24.5234	−0.989682
$615$	−13.2437	−0.534037
$616$	0.201562	0.00812118
$617$	−6.54854	−0.263634	−0.131817	−	0.991274i	$-0.542081\pi$
$617$	−6.54854	−0.263634	−0.131817	+	0.991274i	$0.542081\pi$
$618$	7.75240	0.311847
$619$	1.12076	0.0450470	0.0225235	−	0.999746i	$-0.492830\pi$
$619$	1.12076	0.0450470	0.0225235	+	0.999746i	$0.492830\pi$
$620$	−5.36205	−0.215345
$621$	−9.92501	−0.398277
$622$	11.6440	0.466883
$623$	6.42150	0.257272
$624$	−3.71663	−0.148784
$625$	−0.0109054	−0.000436216 0
$626$	−9.14724	−0.365597
$627$	0	0
$628$	−4.95955	−0.197908
$629$	35.7257	1.42448
$630$	2.68177	0.106844
$631$	46.4791	1.85030	0.925152	−	0.379598i	$-0.123938\pi$
$631$	46.4791	1.85030	0.925152	+	0.379598i	$0.123938\pi$
$632$	−13.2028	−0.525179
$633$	23.5255	0.935054
$634$	9.31007	0.369750
$635$	4.44581	0.176427
$636$	0.100064	0.00396781
$637$	3.61000	0.143033
$638$	0.979237	0.0387683
$639$	−7.83345	−0.309886
$640$	−1.38232	−0.0546410
$641$	−9.98955	−0.394563	−0.197282	−	0.980347i	$-0.563211\pi$
$641$	−9.98955	−0.394563	−0.197282	+	0.980347i	$0.563211\pi$
$642$	−3.16687	−0.124986
$643$	−18.7882	−0.740936	−0.370468	−	0.928845i	$-0.620803\pi$
$643$	−18.7882	−0.740936	−0.370468	+	0.928845i	$0.620803\pi$
$644$	−1.95145	−0.0768979
$645$	7.54717	0.297170
$646$	0	0
$647$	−6.40876	−0.251954	−0.125977	−	0.992033i	$-0.540207\pi$
$647$	−6.40876	−0.251954	−0.125977	+	0.992033i	$0.540207\pi$
$648$	0.583959	0.0229401
$649$	−1.86478	−0.0731989
$650$	−11.1520	−0.437417
$651$	−3.99360	−0.156522
$652$	10.2851	0.402797
$653$	38.4509	1.50470	0.752351	−	0.658763i	$-0.228920\pi$
$653$	38.4509	1.50470	0.752351	+	0.658763i	$0.228920\pi$
$654$	2.38472	0.0932501
$655$	9.09030	0.355187
$656$	−9.30590	−0.363334
$657$	8.90917	0.347580
$658$	7.51513	0.292970
$659$	−18.6345	−0.725897	−0.362948	−	0.931809i	$-0.618230\pi$
$659$	−18.6345	−0.725897	−0.362948	+	0.931809i	$0.618230\pi$
$660$	0.286853	0.0111658
$661$	−20.2296	−0.786840	−0.393420	−	0.919359i	$-0.628708\pi$
$661$	−20.2296	−0.786840	−0.393420	+	0.919359i	$0.628708\pi$
$662$	−26.6779	−1.03687
$663$	14.8213	0.575613
$664$	10.1055	0.392168
$665$	0	0
$666$	17.3803	0.673471
$667$	−9.48059	−0.367090
$668$	0.132958	0.00514432
$669$	−7.95620	−0.307604
$670$	15.7857	0.609855
$671$	−0.166739	−0.00643687
$672$	−1.02954	−0.0397153
$673$	27.1339	1.04593	0.522967	−	0.852353i	$-0.324825\pi$
$673$	27.1339	1.04593	0.522967	+	0.852353i	$0.324825\pi$
$674$	−4.47140	−0.172232
$675$	−15.7115	−0.604738
$676$	0.0320700	0.00123346
$677$	23.1322	0.889041	0.444521	−	0.895769i	$-0.353374\pi$
$677$	23.1322	0.889041	0.444521	+	0.895769i	$0.353374\pi$
$678$	3.76755	0.144692
$679$	−10.9140	−0.418841
$680$	5.51247	0.211394
$681$	26.7661	1.02568
$682$	0.781866	0.0299392
$683$	−24.3905	−0.933277	−0.466639	−	0.884448i	$-0.654535\pi$
$683$	−24.3905	−0.933277	−0.466639	+	0.884448i	$0.654535\pi$
$684$	0	0
$685$	25.4748	0.973343
$686$	1.00000	0.0381802
$687$	24.8416	0.947767
$688$	5.30315	0.202181
$689$	−0.350868	−0.0133670
$690$	−2.77720	−0.105726
$691$	−39.6683	−1.50905	−0.754527	−	0.656269i	$-0.772134\pi$
$691$	−39.6683	−1.50905	−0.754527	+	0.656269i	$0.772134\pi$
$692$	19.5740	0.744091
$693$	−0.391042	−0.0148544
$694$	3.89738	0.147942
$695$	4.79401	0.181847
$696$	−5.00173	−0.189590
$697$	37.1105	1.40566
$698$	13.5634	0.513381
$699$	28.0110	1.05947
$700$	−3.08919	−0.116761
$701$	46.0304	1.73854	0.869272	−	0.494334i	$-0.164588\pi$
$701$	46.0304	1.73854	0.869272	+	0.494334i	$0.164588\pi$
$702$	18.3603	0.692966
$703$	0	0
$704$	0.201562	0.00759667
$705$	10.6952	0.402803
$706$	27.5213	1.03578
$707$	−2.36077	−0.0887858
$708$	9.52488	0.357967
$709$	41.5265	1.55956	0.779780	−	0.626053i	$-0.215331\pi$
$709$	41.5265	1.55956	0.779780	+	0.626053i	$0.215331\pi$
$710$	−5.58146	−0.209468
$711$	25.6141	0.960604
$712$	6.42150	0.240656
$713$	−7.56972	−0.283488
$714$	4.10564	0.153650
$715$	−1.00583	−0.0376159
$716$	−2.05705	−0.0768756
$717$	−27.3252	−1.02048
$718$	16.7795	0.626207
$719$	−2.36816	−0.0883174	−0.0441587	−	0.999025i	$-0.514061\pi$
$719$	−2.36816	−0.0883174	−0.0441587	+	0.999025i	$0.514061\pi$
$720$	2.68177	0.0999437
$721$	−7.52998	−0.280431
$722$	0	0
$723$	−12.1551	−0.452054
$724$	−4.47667	−0.166374
$725$	−15.0080	−0.557384
$726$	11.2831	0.418755
$727$	50.4470	1.87098	0.935489	−	0.353357i	$-0.114960\pi$
$727$	50.4470	1.87098	0.935489	+	0.353357i	$0.114960\pi$
$728$	3.61000	0.133795
$729$	14.5757	0.539840
$730$	6.34793	0.234947
$731$	−21.1481	−0.782192
$732$	0.851665	0.0314784
$733$	−5.61817	−0.207512	−0.103756	−	0.994603i	$-0.533086\pi$
$733$	−5.61817	−0.207512	−0.103756	+	0.994603i	$0.533086\pi$
$734$	28.5583	1.05411
$735$	1.42315	0.0524937
$736$	−1.95145	−0.0719314
$737$	−2.30179	−0.0847875
$738$	18.0539	0.664574
$739$	21.1345	0.777443	0.388722	−	0.921355i	$-0.372917\pi$
$739$	21.1345	0.777443	0.388722	+	0.921355i	$0.372917\pi$
$740$	12.3837	0.455235
$741$	0	0
$742$	−0.0971934	−0.00356808
$743$	−5.74269	−0.210679	−0.105340	−	0.994436i	$-0.533593\pi$
$743$	−5.74269	−0.210679	−0.105340	+	0.994436i	$0.533593\pi$
$744$	−3.99360	−0.146413
$745$	−19.1902	−0.703073
$746$	−0.530273	−0.0194147
$747$	−19.6051	−0.717314
$748$	−0.803800	−0.0293898
$749$	3.07601	0.112395
$750$	−11.5121	−0.420364
$751$	−35.9013	−1.31006	−0.655029	−	0.755604i	$-0.727344\pi$
$751$	−35.9013	−1.31006	−0.655029	+	0.755604i	$0.727344\pi$
$752$	7.51513	0.274049
$753$	−11.5112	−0.419491
$754$	17.5382	0.638703
$755$	27.0667	0.985058
$756$	5.08597	0.184975
$757$	18.3088	0.665445	0.332722	−	0.943025i	$-0.392033\pi$
$757$	18.3088	0.665445	0.332722	+	0.943025i	$0.392033\pi$
$758$	0.442763	0.0160819
$759$	0.404957	0.0146990
$760$	0	0
$761$	−48.7667	−1.76779	−0.883896	−	0.467684i	$-0.845089\pi$
$761$	−48.7667	−1.76779	−0.883896	+	0.467684i	$0.845089\pi$
$762$	3.31120	0.119952
$763$	−2.31631	−0.0838559
$764$	−3.44067	−0.124479
$765$	−10.6945	−0.386660
$766$	−3.06625	−0.110788
$767$	−33.3983	−1.20594
$768$	−1.02954	−0.0371502
$769$	44.8502	1.61734	0.808671	−	0.588262i	$-0.200188\pi$
$769$	44.8502	1.61734	0.808671	+	0.588262i	$0.200188\pi$
$770$	−0.278624	−0.0100409
$771$	−30.4675	−1.09726
$772$	−25.1998	−0.906959
$773$	−40.8558	−1.46948	−0.734740	−	0.678349i	$-0.762696\pi$
$773$	−40.8558	−1.46948	−0.734740	+	0.678349i	$0.762696\pi$
$774$	−10.2884	−0.369808
$775$	−11.9831	−0.430444
$776$	−10.9140	−0.391790
$777$	9.22327	0.330883
$778$	−14.5166	−0.520445
$779$	0	0
$780$	5.13756	0.183954
$781$	0.813859	0.0291222
$782$	7.78208	0.278287
$783$	24.7088	0.883021
$784$	1.00000	0.0357143
$785$	6.85569	0.244690
$786$	6.77037	0.241491
$787$	3.83803	0.136811	0.0684056	−	0.997658i	$-0.478209\pi$
$787$	3.83803	0.136811	0.0684056	+	0.997658i	$0.478209\pi$
$788$	−14.3027	−0.509512
$789$	25.3438	0.902264
$790$	18.2505	0.649323
$791$	−3.65946	−0.130115
$792$	−0.391042	−0.0138951
$793$	−2.98630	−0.106047
$794$	27.0449	0.959789
$795$	−0.138321	−0.00490573
$796$	12.3169	0.436560
$797$	21.6035	0.765235	0.382617	−	0.923907i	$-0.375023\pi$
$797$	21.6035	0.765235	0.382617	+	0.923907i	$0.375023\pi$
$798$	0	0
$799$	−29.9692	−1.06023
$800$	−3.08919	−0.109220
$801$	−12.4580	−0.440183
$802$	−1.61913	−0.0571735
$803$	−0.925622	−0.0326645
$804$	11.7570	0.414639
$805$	2.69753	0.0950753
$806$	14.0033	0.493244
$807$	−7.97674	−0.280794
$808$	−2.36077	−0.0830515
$809$	−25.6743	−0.902660	−0.451330	−	0.892357i	$-0.649050\pi$
$809$	−25.6743	−0.902660	−0.451330	+	0.892357i	$0.649050\pi$
$810$	−0.807217	−0.0283627
$811$	−49.7923	−1.74844	−0.874222	−	0.485526i	$-0.838628\pi$
$811$	−49.7923	−1.74844	−0.874222	+	0.485526i	$0.838628\pi$
$812$	4.85823	0.170490
$813$	11.9181	0.417987
$814$	−1.80573	−0.0632908
$815$	−14.2173	−0.498011
$816$	4.10564	0.143726
$817$	0	0
$818$	6.10992	0.213628
$819$	−7.00358	−0.244725
$820$	12.8637	0.449221
$821$	−28.5028	−0.994756	−0.497378	−	0.867534i	$-0.665704\pi$
$821$	−28.5028	−0.994756	−0.497378	+	0.867534i	$0.665704\pi$
$822$	18.9734	0.661774
$823$	18.0495	0.629165	0.314582	−	0.949230i	$-0.398136\pi$
$823$	18.0495	0.629165	0.314582	+	0.949230i	$0.398136\pi$
$824$	−7.52998	−0.262319
$825$	0.641058	0.0223188
$826$	−9.25161	−0.321905
$827$	−26.1255	−0.908472	−0.454236	−	0.890881i	$-0.650088\pi$
$827$	−26.1255	−0.908472	−0.454236	+	0.890881i	$0.650088\pi$
$828$	3.78591	0.131570
$829$	27.1011	0.941261	0.470630	−	0.882330i	$-0.344027\pi$
$829$	27.1011	0.941261	0.470630	+	0.882330i	$0.344027\pi$
$830$	−13.9690	−0.484870
$831$	−5.04564	−0.175031
$832$	3.61000	0.125154
$833$	−3.98785	−0.138171
$834$	3.57053	0.123637
$835$	−0.183791	−0.00636035
$836$	0	0
$837$	19.7286	0.681920
$838$	1.92152	0.0663777
$839$	−37.2618	−1.28642	−0.643210	−	0.765690i	$-0.722398\pi$
$839$	−37.2618	−1.28642	−0.643210	+	0.765690i	$0.722398\pi$
$840$	1.42315	0.0491033
$841$	−5.39761	−0.186124
$842$	31.5836	1.08844
$843$	16.8034	0.578741
$844$	−22.8505	−0.786547
$845$	−0.0443310	−0.00152503
$846$	−14.5798	−0.501262
$847$	−10.9594	−0.376568
$848$	−0.0971934	−0.00333764
$849$	19.7437	0.677601
$850$	12.3192	0.422546
$851$	17.4824	0.599288
$852$	−4.15702	−0.142417
$853$	−3.05187	−0.104494	−0.0522471	−	0.998634i	$-0.516638\pi$
$853$	−3.05187	−0.104494	−0.0522471	+	0.998634i	$0.516638\pi$
$854$	−0.827230	−0.0283072
$855$	0	0
$856$	3.07601	0.105136
$857$	54.9863	1.87830	0.939148	−	0.343513i	$-0.111617\pi$
$857$	54.9863	1.87830	0.939148	+	0.343513i	$0.111617\pi$
$858$	−0.749132	−0.0255750
$859$	48.8210	1.66575	0.832876	−	0.553460i	$-0.186693\pi$
$859$	48.8210	1.66575	0.832876	+	0.553460i	$0.186693\pi$
$860$	−7.33064	−0.249973
$861$	9.58077	0.326512
$862$	24.5430	0.835937
$863$	14.1254	0.480833	0.240416	−	0.970670i	$-0.422716\pi$
$863$	14.1254	0.480833	0.240416	+	0.970670i	$0.422716\pi$
$864$	5.08597	0.173028
$865$	−27.0575	−0.919982
$866$	−35.6753	−1.21229
$867$	1.12950	0.0383597
$868$	3.87902	0.131663
$869$	−2.66119	−0.0902746
$870$	6.91399	0.234406
$871$	−41.2252	−1.39686
$872$	−2.31631	−0.0784400
$873$	21.1737	0.716622
$874$	0	0
$875$	11.1818	0.378015
$876$	4.72788	0.159740
$877$	13.0738	0.441472	0.220736	−	0.975334i	$-0.429154\pi$
$877$	13.0738	0.441472	0.220736	+	0.975334i	$0.429154\pi$
$878$	10.6745	0.360247
$879$	−9.66541	−0.326006
$880$	−0.278624	−0.00939240
$881$	55.8320	1.88103	0.940514	−	0.339754i	$-0.110344\pi$
$881$	55.8320	1.88103	0.940514	+	0.339754i	$0.110344\pi$
$882$	−1.94005	−0.0653250
$883$	−5.50018	−0.185096	−0.0925478	−	0.995708i	$-0.529501\pi$
$883$	−5.50018	−0.185096	−0.0925478	+	0.995708i	$0.529501\pi$
$884$	−14.3961	−0.484193
$885$	−13.1664	−0.442585
$886$	−28.1790	−0.946692
$887$	−42.0908	−1.41327	−0.706636	−	0.707578i	$-0.749788\pi$
$887$	−42.0908	−1.41327	−0.706636	+	0.707578i	$0.749788\pi$
$888$	9.22327	0.309513
$889$	−3.21620	−0.107868
$890$	−8.87656	−0.297543
$891$	0.117704	0.00394324
$892$	7.72793	0.258750
$893$	0	0
$894$	−14.2926	−0.478017
$895$	2.84350	0.0950477
$896$	1.00000	0.0334077
$897$	7.25281	0.242164
$898$	−1.02365	−0.0341597
$899$	18.8452	0.628522
$900$	5.99320	0.199773
$901$	0.387592	0.0129126
$902$	−1.87572	−0.0624546
$903$	−5.45979	−0.181690
$904$	−3.65946	−0.121712
$905$	6.18818	0.205702
$906$	20.1590	0.669738
$907$	−24.9876	−0.829700	−0.414850	−	0.909890i	$-0.636166\pi$
$907$	−24.9876	−0.829700	−0.414850	+	0.909890i	$0.636166\pi$
$908$	−25.9981	−0.862779
$909$	4.58001	0.151909
$910$	−4.99017	−0.165422
$911$	57.0687	1.89077	0.945385	−	0.325955i	$-0.105686\pi$
$911$	57.0687	1.89077	0.945385	+	0.325955i	$0.105686\pi$
$912$	0	0
$913$	2.03688	0.0674110
$914$	30.0439	0.993763
$915$	−1.17727	−0.0389194
$916$	−24.1289	−0.797242
$917$	−6.57612	−0.217163
$918$	−20.2821	−0.669408
$919$	−2.77343	−0.0914872	−0.0457436	−	0.998953i	$-0.514566\pi$
$919$	−2.77343	−0.0914872	−0.0457436	+	0.998953i	$0.514566\pi$
$920$	2.69753	0.0889348
$921$	25.2477	0.831941
$922$	−11.9442	−0.393362
$923$	14.5763	0.479783
$924$	−0.207516	−0.00682678
$925$	27.6750	0.909949
$926$	−27.3252	−0.897960
$927$	14.6085	0.479808
$928$	4.85823	0.159479
$929$	−12.7799	−0.419295	−0.209647	−	0.977777i	$-0.567232\pi$
$929$	−12.7799	−0.419295	−0.209647	+	0.977777i	$0.567232\pi$
$930$	5.52043	0.181022
$931$	0	0
$932$	−27.2074	−0.891208
$933$	−11.9880	−0.392468
$934$	−7.73250	−0.253015
$935$	1.11111	0.0363371
$936$	−7.00358	−0.228919
$937$	19.8246	0.647640	0.323820	−	0.946119i	$-0.395033\pi$
$937$	19.8246	0.647640	0.323820	+	0.946119i	$0.395033\pi$
$938$	−11.4197	−0.372867
$939$	9.41743	0.307326
$940$	−10.3883	−0.338829
$941$	−25.2379	−0.822732	−0.411366	−	0.911470i	$-0.634948\pi$
$941$	−25.2379	−0.822732	−0.411366	+	0.911470i	$0.634948\pi$
$942$	5.10605	0.166364
$943$	18.1600	0.591371
$944$	−9.25161	−0.301114
$945$	−7.03043	−0.228700
$946$	1.06892	0.0347534
$947$	−9.02597	−0.293305	−0.146652	−	0.989188i	$-0.546850\pi$
$947$	−9.02597	−0.293305	−0.146652	+	0.989188i	$0.546850\pi$
$948$	13.5928	0.441473
$949$	−16.5779	−0.538143
$950$	0	0
$951$	−9.58507	−0.310817
$952$	−3.98785	−0.129247
$953$	−23.2965	−0.754647	−0.377323	−	0.926082i	$-0.623155\pi$
$953$	−23.2965	−0.754647	−0.377323	+	0.926082i	$0.623155\pi$
$954$	0.188560	0.00610487
$955$	4.75610	0.153904
$956$	26.5413	0.858406
$957$	−1.00816	−0.0325892
$958$	−1.52899	−0.0493995
$959$	−18.4291	−0.595105
$960$	1.42315	0.0459319
$961$	−15.9532	−0.514618
$962$	−32.3407	−1.04271
$963$	−5.96762	−0.192304
$964$	11.8064	0.380258
$965$	34.8341	1.12135
$966$	2.00909	0.0646414
$967$	−23.9018	−0.768631	−0.384316	−	0.923202i	$-0.625563\pi$
$967$	−23.9018	−0.768631	−0.384316	+	0.923202i	$0.625563\pi$
$968$	−10.9594	−0.352248
$969$	0	0
$970$	15.0866	0.484402
$971$	58.0700	1.86356	0.931778	−	0.363029i	$-0.118258\pi$
$971$	58.0700	1.86356	0.931778	+	0.363029i	$0.118258\pi$
$972$	−15.8591	−0.508681
$973$	−3.46809	−0.111182
$974$	14.6532	0.469518
$975$	11.4814	0.367699
$976$	−0.827230	−0.0264790
$977$	45.3835	1.45195	0.725974	−	0.687722i	$-0.241389\pi$
$977$	45.3835	1.45195	0.725974	+	0.687722i	$0.241389\pi$
$978$	−10.5889	−0.338597
$979$	1.29433	0.0413671
$980$	−1.38232	−0.0441566
$981$	4.49376	0.143475
$982$	19.4632	0.621096
$983$	46.3756	1.47915	0.739576	−	0.673073i	$-0.235026\pi$
$983$	46.3756	1.47915	0.739576	+	0.673073i	$0.235026\pi$
$984$	9.58077	0.305424
$985$	19.7709	0.629952
$986$	−19.3739	−0.616990
$987$	−7.73711	−0.246275
$988$	0	0
$989$	−10.3488	−0.329073
$990$	0.540544	0.0171796
$991$	39.5347	1.25586	0.627931	−	0.778269i	$-0.283902\pi$
$991$	39.5347	1.25586	0.627931	+	0.778269i	$0.283902\pi$
$992$	3.87902	0.123159
$993$	27.4659	0.871603
$994$	4.03775	0.128070
$995$	−17.0259	−0.539756
$996$	−10.4040	−0.329662
$997$	−51.7281	−1.63825	−0.819123	−	0.573618i	$-0.805539\pi$
$997$	−51.7281	−1.63825	−0.819123	+	0.573618i	$0.805539\pi$
$998$	8.21283	0.259972
$999$	−45.5635	−1.44156

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5054.2.a.bh.1.4	yes	8
19.18	odd	2		5054.2.a.bg.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5054.2.a.bg.1.5	✓	8	19.18	odd	2
5054.2.a.bh.1.4	yes	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5054.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.02954 q^{3} +1.00000 q^{4} -1.38232 q^{5} -1.02954 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.94005 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.02954 q^{3} +1.00000 q^{4} -1.38232 q^{5} -1.02954 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.94005 q^{9} -1.38232 q^{10} +0.201562 q^{11} -1.02954 q^{12} +3.61000 q^{13} +1.00000 q^{14} +1.42315 q^{15} +1.00000 q^{16} -3.98785 q^{17} -1.94005 q^{18} -1.38232 q^{20} -1.02954 q^{21} +0.201562 q^{22} -1.95145 q^{23} -1.02954 q^{24} -3.08919 q^{25} +3.61000 q^{26} +5.08597 q^{27} +1.00000 q^{28} +4.85823 q^{29} +1.42315 q^{30} +3.87902 q^{31} +1.00000 q^{32} -0.207516 q^{33} -3.98785 q^{34} -1.38232 q^{35} -1.94005 q^{36} -8.95866 q^{37} -3.71663 q^{39} -1.38232 q^{40} -9.30590 q^{41} -1.02954 q^{42} +5.30315 q^{43} +0.201562 q^{44} +2.68177 q^{45} -1.95145 q^{46} +7.51513 q^{47} -1.02954 q^{48} +1.00000 q^{49} -3.08919 q^{50} +4.10564 q^{51} +3.61000 q^{52} -0.0971934 q^{53} +5.08597 q^{54} -0.278624 q^{55} +1.00000 q^{56} +4.85823 q^{58} -9.25161 q^{59} +1.42315 q^{60} -0.827230 q^{61} +3.87902 q^{62} -1.94005 q^{63} +1.00000 q^{64} -4.99017 q^{65} -0.207516 q^{66} -11.4197 q^{67} -3.98785 q^{68} +2.00909 q^{69} -1.38232 q^{70} +4.03775 q^{71} -1.94005 q^{72} -4.59223 q^{73} -8.95866 q^{74} +3.18044 q^{75} +0.201562 q^{77} -3.71663 q^{78} -13.2028 q^{79} -1.38232 q^{80} +0.583959 q^{81} -9.30590 q^{82} +10.1055 q^{83} -1.02954 q^{84} +5.51247 q^{85} +5.30315 q^{86} -5.00173 q^{87} +0.201562 q^{88} +6.42150 q^{89} +2.68177 q^{90} +3.61000 q^{91} -1.95145 q^{92} -3.99360 q^{93} +7.51513 q^{94} -1.02954 q^{96} -10.9140 q^{97} +1.00000 q^{98} -0.391042 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 - 4 * q^3 + 8 * q^4 - 2 * q^5 - 4 * q^6 + 8 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} - 4 q^{3} + 8 q^{4} - 2 q^{5} - 4 q^{6} + 8 q^{7} + 8 q^{8} + 8 q^{9} - 2 q^{10} - 12 q^{11} - 4 q^{12} - 10 q^{13} + 8 q^{14} - 24 q^{15} + 8 q^{16} - 6 q^{17} + 8 q^{18} - 2 q^{20} - 4 q^{21} - 12 q^{22} - 20 q^{23} - 4 q^{24} + 8 q^{25} - 10 q^{26} - 22 q^{27} + 8 q^{28} - 8 q^{29} - 24 q^{30} - 18 q^{31} + 8 q^{32} + 16 q^{33} - 6 q^{34} - 2 q^{35} + 8 q^{36} - 36 q^{37} - 2 q^{40} - 4 q^{41} - 4 q^{42} - 16 q^{43} - 12 q^{44} + 8 q^{45} - 20 q^{46} - 10 q^{47} - 4 q^{48} + 8 q^{49} + 8 q^{50} - 12 q^{51} - 10 q^{52} - 32 q^{53} - 22 q^{54} - 22 q^{55} + 8 q^{56} - 8 q^{58} + 2 q^{59} - 24 q^{60} + 2 q^{61} - 18 q^{62} + 8 q^{63} + 8 q^{64} + 16 q^{66} - 44 q^{67} - 6 q^{68} - 2 q^{70} - 8 q^{71} + 8 q^{72} + 30 q^{73} - 36 q^{74} + 16 q^{75} - 12 q^{77} - 60 q^{79} - 2 q^{80} + 12 q^{81} - 4 q^{82} - 28 q^{83} - 4 q^{84} - 16 q^{85} - 16 q^{86} + 24 q^{87} - 12 q^{88} + 22 q^{89} + 8 q^{90} - 10 q^{91} - 20 q^{92} - 16 q^{93} - 10 q^{94} - 4 q^{96} + 6 q^{97} + 8 q^{98} - 52 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 4 * q^3 + 8 * q^4 - 2 * q^5 - 4 * q^6 + 8 * q^7 + 8 * q^8 + 8 * q^9 - 2 * q^10 - 12 * q^11 - 4 * q^12 - 10 * q^13 + 8 * q^14 - 24 * q^15 + 8 * q^16 - 6 * q^17 + 8 * q^18 - 2 * q^20 - 4 * q^21 - 12 * q^22 - 20 * q^23 - 4 * q^24 + 8 * q^25 - 10 * q^26 - 22 * q^27 + 8 * q^28 - 8 * q^29 - 24 * q^30 - 18 * q^31 + 8 * q^32 + 16 * q^33 - 6 * q^34 - 2 * q^35 + 8 * q^36 - 36 * q^37 - 2 * q^40 - 4 * q^41 - 4 * q^42 - 16 * q^43 - 12 * q^44 + 8 * q^45 - 20 * q^46 - 10 * q^47 - 4 * q^48 + 8 * q^49 + 8 * q^50 - 12 * q^51 - 10 * q^52 - 32 * q^53 - 22 * q^54 - 22 * q^55 + 8 * q^56 - 8 * q^58 + 2 * q^59 - 24 * q^60 + 2 * q^61 - 18 * q^62 + 8 * q^63 + 8 * q^64 + 16 * q^66 - 44 * q^67 - 6 * q^68 - 2 * q^70 - 8 * q^71 + 8 * q^72 + 30 * q^73 - 36 * q^74 + 16 * q^75 - 12 * q^77 - 60 * q^79 - 2 * q^80 + 12 * q^81 - 4 * q^82 - 28 * q^83 - 4 * q^84 - 16 * q^85 - 16 * q^86 + 24 * q^87 - 12 * q^88 + 22 * q^89 + 8 * q^90 - 10 * q^91 - 20 * q^92 - 16 * q^93 - 10 * q^94 - 4 * q^96 + 6 * q^97 + 8 * q^98 - 52 * q^99

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