LMFDB - Embedded newform 5054.2.a.bf.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			$2.35877$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -0.717767 q^{3} +1.00000 q^{4} +2.91631 q^{5} +0.717767 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.48481 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.717767 q^{3} +1.00000 q^{4} +2.91631 q^{5} +0.717767 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.48481 q^{9} -2.91631 q^{10} +2.56382 q^{11} -0.717767 q^{12} -1.75896 q^{13} +1.00000 q^{14} -2.09323 q^{15} +1.00000 q^{16} +1.39477 q^{17} +2.48481 q^{18} +2.91631 q^{20} +0.717767 q^{21} -2.56382 q^{22} -3.44122 q^{23} +0.717767 q^{24} +3.50487 q^{25} +1.75896 q^{26} +3.93682 q^{27} -1.00000 q^{28} +0.789597 q^{29} +2.09323 q^{30} -4.99865 q^{31} -1.00000 q^{32} -1.84023 q^{33} -1.39477 q^{34} -2.91631 q^{35} -2.48481 q^{36} -7.90419 q^{37} +1.26252 q^{39} -2.91631 q^{40} -6.60934 q^{41} -0.717767 q^{42} +12.2294 q^{43} +2.56382 q^{44} -7.24648 q^{45} +3.44122 q^{46} -1.23536 q^{47} -0.717767 q^{48} +1.00000 q^{49} -3.50487 q^{50} -1.00112 q^{51} -1.75896 q^{52} +7.30992 q^{53} -3.93682 q^{54} +7.47689 q^{55} +1.00000 q^{56} -0.789597 q^{58} -1.68722 q^{59} -2.09323 q^{60} +4.80597 q^{61} +4.99865 q^{62} +2.48481 q^{63} +1.00000 q^{64} -5.12966 q^{65} +1.84023 q^{66} -2.63849 q^{67} +1.39477 q^{68} +2.46999 q^{69} +2.91631 q^{70} -8.70323 q^{71} +2.48481 q^{72} -4.32196 q^{73} +7.90419 q^{74} -2.51568 q^{75} -2.56382 q^{77} -1.26252 q^{78} +4.23183 q^{79} +2.91631 q^{80} +4.62871 q^{81} +6.60934 q^{82} -13.6474 q^{83} +0.717767 q^{84} +4.06760 q^{85} -12.2294 q^{86} -0.566747 q^{87} -2.56382 q^{88} -1.50287 q^{89} +7.24648 q^{90} +1.75896 q^{91} -3.44122 q^{92} +3.58787 q^{93} +1.23536 q^{94} +0.717767 q^{96} +11.0606 q^{97} -1.00000 q^{98} -6.37060 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 6 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 + 6 * q^5 + 4 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9} - 6 q^{10} + 4 q^{11} - 4 q^{12} - 6 q^{13} + 8 q^{14} - 8 q^{15} + 8 q^{16} + 2 q^{17} - 8 q^{18} + 6 q^{20} + 4 q^{21} - 4 q^{22} + 20 q^{23} + 4 q^{24} - 8 q^{25} + 6 q^{26} - 22 q^{27} - 8 q^{28} - 16 q^{29} + 8 q^{30} - 22 q^{31} - 8 q^{32} + 8 q^{33} - 2 q^{34} - 6 q^{35} + 8 q^{36} + 12 q^{37} + 8 q^{39} - 6 q^{40} - 36 q^{41} - 4 q^{42} + 4 q^{44} - 4 q^{45} - 20 q^{46} + 6 q^{47} - 4 q^{48} + 8 q^{49} + 8 q^{50} + 4 q^{51} - 6 q^{52} - 16 q^{53} + 22 q^{54} + 18 q^{55} + 8 q^{56} + 16 q^{58} - 14 q^{59} - 8 q^{60} + 10 q^{61} + 22 q^{62} - 8 q^{63} + 8 q^{64} - 32 q^{65} - 8 q^{66} + 20 q^{67} + 2 q^{68} - 40 q^{69} + 6 q^{70} - 8 q^{72} - 18 q^{73} - 12 q^{74} - 16 q^{75} - 4 q^{77} - 8 q^{78} - 4 q^{79} + 6 q^{80} + 12 q^{81} + 36 q^{82} + 36 q^{83} + 4 q^{84} - 16 q^{85} + 28 q^{87} - 4 q^{88} - 18 q^{89} + 4 q^{90} + 6 q^{91} + 20 q^{92} + 16 q^{93} - 6 q^{94} + 4 q^{96} - 26 q^{97} - 8 q^{98} + 4 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 6 * q^5 + 4 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9 - 6 * q^10 + 4 * q^11 - 4 * q^12 - 6 * q^13 + 8 * q^14 - 8 * q^15 + 8 * q^16 + 2 * q^17 - 8 * q^18 + 6 * q^20 + 4 * q^21 - 4 * q^22 + 20 * q^23 + 4 * q^24 - 8 * q^25 + 6 * q^26 - 22 * q^27 - 8 * q^28 - 16 * q^29 + 8 * q^30 - 22 * q^31 - 8 * q^32 + 8 * q^33 - 2 * q^34 - 6 * q^35 + 8 * q^36 + 12 * q^37 + 8 * q^39 - 6 * q^40 - 36 * q^41 - 4 * q^42 + 4 * q^44 - 4 * q^45 - 20 * q^46 + 6 * q^47 - 4 * q^48 + 8 * q^49 + 8 * q^50 + 4 * q^51 - 6 * q^52 - 16 * q^53 + 22 * q^54 + 18 * q^55 + 8 * q^56 + 16 * q^58 - 14 * q^59 - 8 * q^60 + 10 * q^61 + 22 * q^62 - 8 * q^63 + 8 * q^64 - 32 * q^65 - 8 * q^66 + 20 * q^67 + 2 * q^68 - 40 * q^69 + 6 * q^70 - 8 * q^72 - 18 * q^73 - 12 * q^74 - 16 * q^75 - 4 * q^77 - 8 * q^78 - 4 * q^79 + 6 * q^80 + 12 * q^81 + 36 * q^82 + 36 * q^83 + 4 * q^84 - 16 * q^85 + 28 * q^87 - 4 * q^88 - 18 * q^89 + 4 * q^90 + 6 * q^91 + 20 * q^92 + 16 * q^93 - 6 * q^94 + 4 * q^96 - 26 * q^97 - 8 * 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
$3$	−0.717767	−0.414403	−0.207202	−	0.978298i	$-0.566436\pi$
$3$	−0.717767	−0.414403	−0.207202	+	0.978298i	$0.566436\pi$
$4$	1.00000	0.500000
$5$	2.91631	1.30421	0.652107	−	0.758127i	$-0.273885\pi$
$5$	2.91631	1.30421	0.652107	+	0.758127i	$0.273885\pi$
$6$	0.717767	0.293027
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	−2.48481	−0.828270
$10$	−2.91631	−0.922219
$11$	2.56382	0.773020	0.386510	−	0.922285i	$-0.373680\pi$
$11$	2.56382	0.773020	0.386510	+	0.922285i	$0.373680\pi$
$12$	−0.717767	−0.207202
$13$	−1.75896	−0.487847	−0.243923	−	0.969795i	$-0.578435\pi$
$13$	−1.75896	−0.487847	−0.243923	+	0.969795i	$0.578435\pi$
$14$	1.00000	0.267261
$15$	−2.09323	−0.540471
$16$	1.00000	0.250000
$17$	1.39477	0.338283	0.169141	−	0.985592i	$-0.445901\pi$
$17$	1.39477	0.338283	0.169141	+	0.985592i	$0.445901\pi$
$18$	2.48481	0.585675
$19$	0	0
$19$	0	0
$20$	2.91631	0.652107
$21$	0.717767	0.156630
$22$	−2.56382	−0.546608
$23$	−3.44122	−0.717543	−0.358772	−	0.933425i	$-0.616804\pi$
$23$	−3.44122	−0.717543	−0.358772	+	0.933425i	$0.616804\pi$
$24$	0.717767	0.146514
$25$	3.50487	0.700974
$26$	1.75896	0.344960
$27$	3.93682	0.757641
$28$	−1.00000	−0.188982
$29$	0.789597	0.146625	0.0733123	−	0.997309i	$-0.476643\pi$
$29$	0.789597	0.146625	0.0733123	+	0.997309i	$0.476643\pi$
$30$	2.09323	0.382170
$31$	−4.99865	−0.897784	−0.448892	−	0.893586i	$-0.648181\pi$
$31$	−4.99865	−0.897784	−0.448892	+	0.893586i	$0.648181\pi$
$32$	−1.00000	−0.176777
$33$	−1.84023	−0.320342
$34$	−1.39477	−0.239202
$35$	−2.91631	−0.492947
$36$	−2.48481	−0.414135
$37$	−7.90419	−1.29944	−0.649721	−	0.760173i	$-0.725114\pi$
$37$	−7.90419	−1.29944	−0.649721	+	0.760173i	$0.725114\pi$
$38$	0	0
$39$	1.26252	0.202165
$40$	−2.91631	−0.461109
$41$	−6.60934	−1.03221	−0.516103	−	0.856527i	$-0.672618\pi$
$41$	−6.60934	−1.03221	−0.516103	+	0.856527i	$0.672618\pi$
$42$	−0.717767	−0.110754
$43$	12.2294	1.86497	0.932486	−	0.361205i	$-0.117635\pi$
$43$	12.2294	1.86497	0.932486	+	0.361205i	$0.117635\pi$
$44$	2.56382	0.386510
$45$	−7.24648	−1.08024
$46$	3.44122	0.507380
$47$	−1.23536	−0.180196	−0.0900981	−	0.995933i	$-0.528718\pi$
$47$	−1.23536	−0.180196	−0.0900981	+	0.995933i	$0.528718\pi$
$48$	−0.717767	−0.103601
$49$	1.00000	0.142857
$50$	−3.50487	−0.495664
$51$	−1.00112	−0.140185
$52$	−1.75896	−0.243923
$53$	7.30992	1.00409	0.502047	−	0.864840i	$-0.332580\pi$
$53$	7.30992	1.00409	0.502047	+	0.864840i	$0.332580\pi$
$54$	−3.93682	−0.535733
$55$	7.47689	1.00818
$56$	1.00000	0.133631
$57$	0	0
$58$	−0.789597	−0.103679
$59$	−1.68722	−0.219657	−0.109829	−	0.993951i	$-0.535030\pi$
$59$	−1.68722	−0.219657	−0.109829	+	0.993951i	$0.535030\pi$
$60$	−2.09323	−0.270235
$61$	4.80597	0.615342	0.307671	−	0.951493i	$-0.400450\pi$
$61$	4.80597	0.615342	0.307671	+	0.951493i	$0.400450\pi$
$62$	4.99865	0.634829
$63$	2.48481	0.313057
$64$	1.00000	0.125000
$65$	−5.12966	−0.636257
$66$	1.84023	0.226516
$67$	−2.63849	−0.322343	−0.161172	−	0.986926i	$-0.551527\pi$
$67$	−2.63849	−0.322343	−0.161172	+	0.986926i	$0.551527\pi$
$68$	1.39477	0.169141
$69$	2.46999	0.297352
$70$	2.91631	0.348566
$71$	−8.70323	−1.03288	−0.516442	−	0.856322i	$-0.672744\pi$
$71$	−8.70323	−1.03288	−0.516442	+	0.856322i	$0.672744\pi$
$72$	2.48481	0.292838
$73$	−4.32196	−0.505847	−0.252923	−	0.967486i	$-0.581392\pi$
$73$	−4.32196	−0.505847	−0.252923	+	0.967486i	$0.581392\pi$
$74$	7.90419	0.918844
$75$	−2.51568	−0.290486
$76$	0	0
$77$	−2.56382	−0.292174
$78$	−1.26252	−0.142952
$79$	4.23183	0.476118	0.238059	−	0.971251i	$-0.423489\pi$
$79$	4.23183	0.476118	0.238059	+	0.971251i	$0.423489\pi$
$80$	2.91631	0.326054
$81$	4.62871	0.514301
$82$	6.60934	0.729879
$83$	−13.6474	−1.49800	−0.748999	−	0.662571i	$-0.769465\pi$
$83$	−13.6474	−1.49800	−0.748999	+	0.662571i	$0.769465\pi$
$84$	0.717767	0.0783149
$85$	4.06760	0.441193
$86$	−12.2294	−1.31873
$87$	−0.566747	−0.0607617
$88$	−2.56382	−0.273304
$89$	−1.50287	−0.159304	−0.0796521	−	0.996823i	$-0.525381\pi$
$89$	−1.50287	−0.159304	−0.0796521	+	0.996823i	$0.525381\pi$
$90$	7.24648	0.763846
$91$	1.75896	0.184389
$92$	−3.44122	−0.358772
$93$	3.58787	0.372045
$94$	1.23536	0.127418
$95$	0	0
$96$	0.717767	0.0732568
$97$	11.0606	1.12303	0.561516	−	0.827466i	$-0.310218\pi$
$97$	11.0606	1.12303	0.561516	+	0.827466i	$0.310218\pi$
$98$	−1.00000	−0.101015
$99$	−6.37060	−0.640270
$100$	3.50487	0.350487
$101$	−15.0746	−1.49998	−0.749989	−	0.661450i	$-0.769941\pi$
$101$	−15.0746	−1.49998	−0.749989	+	0.661450i	$0.769941\pi$
$102$	1.00112	0.0991260
$103$	−13.0346	−1.28434	−0.642169	−	0.766563i	$-0.721965\pi$
$103$	−13.0346	−1.28434	−0.642169	+	0.766563i	$0.721965\pi$
$104$	1.75896	0.172480
$105$	2.09323	0.204279
$106$	−7.30992	−0.710002
$107$	−15.3752	−1.48638	−0.743188	−	0.669082i	$-0.766687\pi$
$107$	−15.3752	−1.48638	−0.743188	+	0.669082i	$0.766687\pi$
$108$	3.93682	0.378821
$109$	−19.9059	−1.90664	−0.953322	−	0.301957i	$-0.902360\pi$
$109$	−19.9059	−1.90664	−0.953322	+	0.301957i	$0.902360\pi$
$110$	−7.47689	−0.712894
$111$	5.67337	0.538493
$112$	−1.00000	−0.0944911
$113$	2.98600	0.280899	0.140450	−	0.990088i	$-0.455145\pi$
$113$	2.98600	0.280899	0.140450	+	0.990088i	$0.455145\pi$
$114$	0	0
$115$	−10.0357	−0.935830
$116$	0.789597	0.0733123
$117$	4.37067	0.404069
$118$	1.68722	0.155321
$119$	−1.39477	−0.127859
$120$	2.09323	0.191085
$121$	−4.42683	−0.402439
$122$	−4.80597	−0.435112
$123$	4.74397	0.427749
$124$	−4.99865	−0.448892
$125$	−4.36026	−0.389993
$126$	−2.48481	−0.221364
$127$	−4.15241	−0.368467	−0.184233	−	0.982883i	$-0.558980\pi$
$127$	−4.15241	−0.368467	−0.184233	+	0.982883i	$0.558980\pi$
$128$	−1.00000	−0.0883883
$129$	−8.77790	−0.772851
$130$	5.12966	0.449901
$131$	18.9379	1.65461	0.827305	−	0.561753i	$-0.189873\pi$
$131$	18.9379	1.65461	0.827305	+	0.561753i	$0.189873\pi$
$132$	−1.84023	−0.160171
$133$	0	0
$134$	2.63849	0.227931
$135$	11.4810	0.988126
$136$	−1.39477	−0.119601
$137$	−16.3088	−1.39336	−0.696678	−	0.717384i	$-0.745340\pi$
$137$	−16.3088	−1.39336	−0.696678	+	0.717384i	$0.745340\pi$
$138$	−2.46999	−0.210260
$139$	−3.62222	−0.307233	−0.153616	−	0.988131i	$-0.549092\pi$
$139$	−3.62222	−0.307233	−0.153616	+	0.988131i	$0.549092\pi$
$140$	−2.91631	−0.246473
$141$	0.886703	0.0746739
$142$	8.70323	0.730359
$143$	−4.50965	−0.377115
$144$	−2.48481	−0.207067
$145$	2.30271	0.191230
$146$	4.32196	0.357688
$147$	−0.717767	−0.0592005
$148$	−7.90419	−0.649721
$149$	−10.5028	−0.860420	−0.430210	−	0.902729i	$-0.641561\pi$
$149$	−10.5028	−0.860420	−0.430210	+	0.902729i	$0.641561\pi$
$150$	2.51568	0.205405
$151$	20.8828	1.69942	0.849708	−	0.527253i	$-0.176778\pi$
$151$	20.8828	1.69942	0.849708	+	0.527253i	$0.176778\pi$
$152$	0	0
$153$	−3.46575	−0.280189
$154$	2.56382	0.206598
$155$	−14.5776	−1.17090
$156$	1.26252	0.101083
$157$	14.4157	1.15049	0.575247	−	0.817980i	$-0.304906\pi$
$157$	14.4157	1.15049	0.575247	+	0.817980i	$0.304906\pi$
$158$	−4.23183	−0.336666
$159$	−5.24682	−0.416100
$160$	−2.91631	−0.230555
$161$	3.44122	0.271206
$162$	−4.62871	−0.363666
$163$	0.718973	0.0563143	0.0281572	−	0.999604i	$-0.491036\pi$
$163$	0.718973	0.0563143	0.0281572	+	0.999604i	$0.491036\pi$
$164$	−6.60934	−0.516103
$165$	−5.36667	−0.417795
$166$	13.6474	1.05924
$167$	14.5419	1.12528	0.562642	−	0.826701i	$-0.309785\pi$
$167$	14.5419	1.12528	0.562642	+	0.826701i	$0.309785\pi$
$168$	−0.717767	−0.0553770
$169$	−9.90607	−0.762006
$170$	−4.06760	−0.311970
$171$	0	0
$172$	12.2294	0.932486
$173$	24.2394	1.84289	0.921444	−	0.388510i	$-0.127010\pi$
$173$	24.2394	1.84289	0.921444	+	0.388510i	$0.127010\pi$
$174$	0.566747	0.0429650
$175$	−3.50487	−0.264943
$176$	2.56382	0.193255
$177$	1.21103	0.0910267
$178$	1.50287	0.112645
$179$	−9.38770	−0.701670	−0.350835	−	0.936437i	$-0.614102\pi$
$179$	−9.38770	−0.701670	−0.350835	+	0.936437i	$0.614102\pi$
$180$	−7.24648	−0.540121
$181$	−23.7807	−1.76760	−0.883801	−	0.467862i	$-0.845024\pi$
$181$	−23.7807	−1.76760	−0.883801	+	0.467862i	$0.845024\pi$
$182$	−1.75896	−0.130383
$183$	−3.44957	−0.255000
$184$	3.44122	0.253690
$185$	−23.0511	−1.69475
$186$	−3.58787	−0.263075
$187$	3.57595	0.261499
$188$	−1.23536	−0.0900981
$189$	−3.93682	−0.286361
$190$	0	0
$191$	−0.561930	−0.0406599	−0.0203299	−	0.999793i	$-0.506472\pi$
$191$	−0.561930	−0.0406599	−0.0203299	+	0.999793i	$0.506472\pi$
$192$	−0.717767	−0.0518004
$193$	17.3688	1.25024	0.625118	−	0.780530i	$-0.285051\pi$
$193$	17.3688	1.25024	0.625118	+	0.780530i	$0.285051\pi$
$194$	−11.0606	−0.794103
$195$	3.68191	0.263667
$196$	1.00000	0.0714286
$197$	20.4849	1.45949	0.729746	−	0.683719i	$-0.239638\pi$
$197$	20.4849	1.45949	0.729746	+	0.683719i	$0.239638\pi$
$198$	6.37060	0.452739
$199$	13.4529	0.953650	0.476825	−	0.878998i	$-0.341787\pi$
$199$	13.4529	0.953650	0.476825	+	0.878998i	$0.341787\pi$
$200$	−3.50487	−0.247832
$201$	1.89382	0.133580
$202$	15.0746	1.06064
$203$	−0.789597	−0.0554189
$204$	−1.00112	−0.0700927
$205$	−19.2749	−1.34622
$206$	13.0346	0.908164
$207$	8.55077	0.594319
$208$	−1.75896	−0.121962
$209$	0	0
$210$	−2.09323	−0.144447
$211$	5.21924	0.359308	0.179654	−	0.983730i	$-0.442502\pi$
$211$	5.21924	0.359308	0.179654	+	0.983730i	$0.442502\pi$
$212$	7.30992	0.502047
$213$	6.24689	0.428030
$214$	15.3752	1.05103
$215$	35.6649	2.43232
$216$	−3.93682	−0.267867
$217$	4.99865	0.339330
$218$	19.9059	1.34820
$219$	3.10216	0.209625
$220$	7.47689	0.504092
$221$	−2.45335	−0.165030
$222$	−5.67337	−0.380772
$223$	18.2290	1.22070	0.610352	−	0.792130i	$-0.291028\pi$
$223$	18.2290	1.22070	0.610352	+	0.792130i	$0.291028\pi$
$224$	1.00000	0.0668153
$225$	−8.70894	−0.580596
$226$	−2.98600	−0.198626
$227$	10.7390	0.712774	0.356387	−	0.934338i	$-0.384008\pi$
$227$	10.7390	0.712774	0.356387	+	0.934338i	$0.384008\pi$
$228$	0	0
$229$	−16.6977	−1.10341	−0.551707	−	0.834038i	$-0.686023\pi$
$229$	−16.6977	−1.10341	−0.551707	+	0.834038i	$0.686023\pi$
$230$	10.0357	0.661732
$231$	1.84023	0.121078
$232$	−0.789597	−0.0518396
$233$	−16.8460	−1.10362	−0.551810	−	0.833970i	$-0.686063\pi$
$233$	−16.8460	−1.10362	−0.551810	+	0.833970i	$0.686063\pi$
$234$	−4.37067	−0.285720
$235$	−3.60270	−0.235014
$236$	−1.68722	−0.109829
$237$	−3.03747	−0.197305
$238$	1.39477	0.0904098
$239$	−23.2019	−1.50081	−0.750403	−	0.660981i	$-0.770141\pi$
$239$	−23.2019	−1.50081	−0.750403	+	0.660981i	$0.770141\pi$
$240$	−2.09323	−0.135118
$241$	−1.76578	−0.113744	−0.0568720	−	0.998381i	$-0.518113\pi$
$241$	−1.76578	−0.113744	−0.0568720	+	0.998381i	$0.518113\pi$
$242$	4.42683	0.284568
$243$	−15.1328	−0.970769
$244$	4.80597	0.307671
$245$	2.91631	0.186316
$246$	−4.74397	−0.302464
$247$	0	0
$248$	4.99865	0.317414
$249$	9.79567	0.620775
$250$	4.36026	0.275767
$251$	4.47898	0.282711	0.141355	−	0.989959i	$-0.454854\pi$
$251$	4.47898	0.282711	0.141355	+	0.989959i	$0.454854\pi$
$252$	2.48481	0.156528
$253$	−8.82265	−0.554676
$254$	4.15241	0.260545
$255$	−2.91959	−0.182832
$256$	1.00000	0.0625000
$257$	−24.0895	−1.50266	−0.751330	−	0.659926i	$-0.770587\pi$
$257$	−24.0895	−1.50266	−0.751330	+	0.659926i	$0.770587\pi$
$258$	8.77790	0.546488
$259$	7.90419	0.491143
$260$	−5.12966	−0.318128
$261$	−1.96200	−0.121445
$262$	−18.9379	−1.16999
$263$	8.88831	0.548077	0.274038	−	0.961719i	$-0.411640\pi$
$263$	8.88831	0.548077	0.274038	+	0.961719i	$0.411640\pi$
$264$	1.84023	0.113258
$265$	21.3180	1.30955
$266$	0	0
$267$	1.07871	0.0660162
$268$	−2.63849	−0.161172
$269$	12.1748	0.742313	0.371156	−	0.928570i	$-0.378961\pi$
$269$	12.1748	0.742313	0.371156	+	0.928570i	$0.378961\pi$
$270$	−11.4810	−0.698711
$271$	10.8375	0.658332	0.329166	−	0.944272i	$-0.393232\pi$
$271$	10.8375	0.658332	0.329166	+	0.944272i	$0.393232\pi$
$272$	1.39477	0.0845706
$273$	−1.26252	−0.0764113
$274$	16.3088	0.985252
$275$	8.98586	0.541868
$276$	2.46999	0.148676
$277$	−11.3275	−0.680602	−0.340301	−	0.940317i	$-0.610529\pi$
$277$	−11.3275	−0.680602	−0.340301	+	0.940317i	$0.610529\pi$
$278$	3.62222	0.217247
$279$	12.4207	0.743607
$280$	2.91631	0.174283
$281$	−16.8113	−1.00288	−0.501438	−	0.865194i	$-0.667195\pi$
$281$	−16.8113	−1.00288	−0.501438	+	0.865194i	$0.667195\pi$
$282$	−0.886703	−0.0528024
$283$	−32.7713	−1.94805	−0.974025	−	0.226442i	$-0.927291\pi$
$283$	−32.7713	−1.94805	−0.974025	+	0.226442i	$0.927291\pi$
$284$	−8.70323	−0.516442
$285$	0	0
$286$	4.50965	0.266661
$287$	6.60934	0.390137
$288$	2.48481	0.146419
$289$	−15.0546	−0.885565
$290$	−2.30271	−0.135220
$291$	−7.93892	−0.465388
$292$	−4.32196	−0.252923
$293$	−16.5383	−0.966176	−0.483088	−	0.875572i	$-0.660485\pi$
$293$	−16.5383	−0.966176	−0.483088	+	0.875572i	$0.660485\pi$
$294$	0.717767	0.0418610
$295$	−4.92046	−0.286480
$296$	7.90419	0.459422
$297$	10.0933	0.585672
$298$	10.5028	0.608409
$299$	6.05295	0.350051
$300$	−2.51568	−0.145243
$301$	−12.2294	−0.704893
$302$	−20.8828	−1.20167
$303$	10.8201	0.621596
$304$	0	0
$305$	14.0157	0.802537
$306$	3.46575	0.198124
$307$	−10.7452	−0.613262	−0.306631	−	0.951828i	$-0.599202\pi$
$307$	−10.7452	−0.613262	−0.306631	+	0.951828i	$0.599202\pi$
$308$	−2.56382	−0.146087
$309$	9.35582	0.532234
$310$	14.5776	0.827953
$311$	−32.3009	−1.83161	−0.915807	−	0.401619i	$-0.868448\pi$
$311$	−32.3009	−1.83161	−0.915807	+	0.401619i	$0.868448\pi$
$312$	−1.26252	−0.0714762
$313$	−11.5120	−0.650697	−0.325349	−	0.945594i	$-0.605482\pi$
$313$	−11.5120	−0.650697	−0.325349	+	0.945594i	$0.605482\pi$
$314$	−14.4157	−0.813522
$315$	7.24648	0.408293
$316$	4.23183	0.238059
$317$	12.4053	0.696751	0.348376	−	0.937355i	$-0.386733\pi$
$317$	12.4053	0.696751	0.348376	+	0.937355i	$0.386733\pi$
$318$	5.24682	0.294227
$319$	2.02438	0.113344
$320$	2.91631	0.163027
$321$	11.0358	0.615959
$322$	−3.44122	−0.191771
$323$	0	0
$324$	4.62871	0.257151
$325$	−6.16492	−0.341968
$326$	−0.718973	−0.0398202
$327$	14.2878	0.790119
$328$	6.60934	0.364940
$329$	1.23536	0.0681077
$330$	5.36667	0.295426
$331$	−4.28173	−0.235345	−0.117673	−	0.993052i	$-0.537543\pi$
$331$	−4.28173	−0.235345	−0.117673	+	0.993052i	$0.537543\pi$
$332$	−13.6474	−0.748999
$333$	19.6404	1.07629
$334$	−14.5419	−0.795696
$335$	−7.69467	−0.420404
$336$	0.717767	0.0391574
$337$	−34.6005	−1.88481	−0.942404	−	0.334477i	$-0.891440\pi$
$337$	−34.6005	−1.88481	−0.942404	+	0.334477i	$0.891440\pi$
$338$	9.90607	0.538819
$339$	−2.14325	−0.116405
$340$	4.06760	0.220596
$341$	−12.8156	−0.694005
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−12.2294	−0.659367
$345$	7.20327	0.387811
$346$	−24.2394	−1.30312
$347$	−8.61568	−0.462514	−0.231257	−	0.972893i	$-0.574284\pi$
$347$	−8.61568	−0.462514	−0.231257	+	0.972893i	$0.574284\pi$
$348$	−0.566747	−0.0303808
$349$	−30.1120	−1.61186	−0.805930	−	0.592011i	$-0.798334\pi$
$349$	−30.1120	−1.61186	−0.805930	+	0.592011i	$0.798334\pi$
$350$	3.50487	0.187343
$351$	−6.92469	−0.369613
$352$	−2.56382	−0.136652
$353$	4.02781	0.214379	0.107189	−	0.994239i	$-0.465815\pi$
$353$	4.02781	0.214379	0.107189	+	0.994239i	$0.465815\pi$
$354$	−1.21103	−0.0643656
$355$	−25.3813	−1.34710
$356$	−1.50287	−0.0796521
$357$	1.00112	0.0529851
$358$	9.38770	0.496156
$359$	33.6389	1.77539	0.887696	−	0.460429i	$-0.152305\pi$
$359$	33.6389	1.77539	0.887696	+	0.460429i	$0.152305\pi$
$360$	7.24648	0.381923
$361$	0	0
$362$	23.7807	1.24988
$363$	3.17744	0.166772
$364$	1.75896	0.0921944
$365$	−12.6042	−0.659732
$366$	3.44957	0.180312
$367$	10.8822	0.568047	0.284024	−	0.958817i	$-0.408331\pi$
$367$	10.8822	0.568047	0.284024	+	0.958817i	$0.408331\pi$
$368$	−3.44122	−0.179386
$369$	16.4229	0.854944
$370$	23.0511	1.19837
$371$	−7.30992	−0.379512
$372$	3.58787	0.186022
$373$	10.9132	0.565062	0.282531	−	0.959258i	$-0.408826\pi$
$373$	10.9132	0.565062	0.282531	+	0.959258i	$0.408826\pi$
$374$	−3.57595	−0.184908
$375$	3.12965	0.161615
$376$	1.23536	0.0637090
$377$	−1.38887	−0.0715303
$378$	3.93682	0.202488
$379$	30.4270	1.56293	0.781465	−	0.623949i	$-0.214473\pi$
$379$	30.4270	1.56293	0.781465	+	0.623949i	$0.214473\pi$
$380$	0	0
$381$	2.98046	0.152694
$382$	0.561930	0.0287509
$383$	3.04363	0.155522	0.0777612	−	0.996972i	$-0.475223\pi$
$383$	3.04363	0.155522	0.0777612	+	0.996972i	$0.475223\pi$
$384$	0.717767	0.0366284
$385$	−7.47689	−0.381058
$386$	−17.3688	−0.884051
$387$	−30.3878	−1.54470
$388$	11.0606	0.561516
$389$	15.7728	0.799712	0.399856	−	0.916578i	$-0.369060\pi$
$389$	15.7728	0.799712	0.399856	+	0.916578i	$0.369060\pi$
$390$	−3.68191	−0.186441
$391$	−4.79972	−0.242732
$392$	−1.00000	−0.0505076
$393$	−13.5930	−0.685676
$394$	−20.4849	−1.03202
$395$	12.3413	0.620959
$396$	−6.37060	−0.320135
$397$	−29.8753	−1.49940	−0.749700	−	0.661778i	$-0.769802\pi$
$397$	−29.8753	−1.49940	−0.749700	+	0.661778i	$0.769802\pi$
$398$	−13.4529	−0.674333
$399$	0	0
$400$	3.50487	0.175244
$401$	10.0388	0.501316	0.250658	−	0.968076i	$-0.419353\pi$
$401$	10.0388	0.501316	0.250658	+	0.968076i	$0.419353\pi$
$402$	−1.89382	−0.0944554
$403$	8.79240	0.437981
$404$	−15.0746	−0.749989
$405$	13.4988	0.670759
$406$	0.789597	0.0391871
$407$	−20.2649	−1.00449
$408$	1.00112	0.0495630
$409$	10.0191	0.495412	0.247706	−	0.968835i	$-0.420323\pi$
$409$	10.0191	0.495412	0.247706	+	0.968835i	$0.420323\pi$
$410$	19.2749	0.951919
$411$	11.7059	0.577412
$412$	−13.0346	−0.642169
$413$	1.68722	0.0830227
$414$	−8.55077	−0.420247
$415$	−39.8001	−1.95371
$416$	1.75896	0.0862399
$417$	2.59991	0.127318
$418$	0	0
$419$	2.90178	0.141761	0.0708807	−	0.997485i	$-0.477419\pi$
$419$	2.90178	0.141761	0.0708807	+	0.997485i	$0.477419\pi$
$420$	2.09323	0.102139
$421$	−19.6306	−0.956738	−0.478369	−	0.878159i	$-0.658772\pi$
$421$	−19.6306	−0.956738	−0.478369	+	0.878159i	$0.658772\pi$
$422$	−5.21924	−0.254069
$423$	3.06964	0.149251
$424$	−7.30992	−0.355001
$425$	4.88851	0.237127
$426$	−6.24689	−0.302663
$427$	−4.80597	−0.232577
$428$	−15.3752	−0.743188
$429$	3.23688	0.156278
$430$	−35.6649	−1.71991
$431$	0.757246	0.0364752	0.0182376	−	0.999834i	$-0.494194\pi$
$431$	0.757246	0.0364752	0.0182376	+	0.999834i	$0.494194\pi$
$432$	3.93682	0.189410
$433$	2.51099	0.120670	0.0603351	−	0.998178i	$-0.480783\pi$
$433$	2.51099	0.120670	0.0603351	+	0.998178i	$0.480783\pi$
$434$	−4.99865	−0.239943
$435$	−1.65281	−0.0792462
$436$	−19.9059	−0.953322
$437$	0	0
$438$	−3.10216	−0.148227
$439$	−22.3934	−1.06878	−0.534389	−	0.845239i	$-0.679458\pi$
$439$	−22.3934	−1.06878	−0.534389	+	0.845239i	$0.679458\pi$
$440$	−7.47689	−0.356447
$441$	−2.48481	−0.118324
$442$	2.45335	0.116694
$443$	2.45606	0.116691	0.0583455	−	0.998296i	$-0.481418\pi$
$443$	2.45606	0.116691	0.0583455	+	0.998296i	$0.481418\pi$
$444$	5.67337	0.269246
$445$	−4.38285	−0.207767
$446$	−18.2290	−0.863169
$447$	7.53855	0.356561
$448$	−1.00000	−0.0472456
$449$	−4.08691	−0.192873	−0.0964367	−	0.995339i	$-0.530745\pi$
$449$	−4.08691	−0.192873	−0.0964367	+	0.995339i	$0.530745\pi$
$450$	8.70894	0.410543
$451$	−16.9451	−0.797916
$452$	2.98600	0.140450
$453$	−14.9890	−0.704244
$454$	−10.7390	−0.504008
$455$	5.12966	0.240482
$456$	0	0
$457$	10.5514	0.493576	0.246788	−	0.969069i	$-0.420625\pi$
$457$	10.5514	0.493576	0.246788	+	0.969069i	$0.420625\pi$
$458$	16.6977	0.780232
$459$	5.49097	0.256297
$460$	−10.0357	−0.467915
$461$	−17.6728	−0.823105	−0.411552	−	0.911386i	$-0.635013\pi$
$461$	−17.6728	−0.823105	−0.411552	+	0.911386i	$0.635013\pi$
$462$	−1.84023	−0.0856150
$463$	27.5515	1.28043	0.640214	−	0.768196i	$-0.278846\pi$
$463$	27.5515	1.28043	0.640214	+	0.768196i	$0.278846\pi$
$464$	0.789597	0.0366561
$465$	10.4633	0.485226
$466$	16.8460	0.780378
$467$	−34.2363	−1.58427	−0.792133	−	0.610349i	$-0.791029\pi$
$467$	−34.2363	−1.58427	−0.792133	+	0.610349i	$0.791029\pi$
$468$	4.37067	0.202034
$469$	2.63849	0.121834
$470$	3.60270	0.166180
$471$	−10.3471	−0.476769
$472$	1.68722	0.0776606
$473$	31.3541	1.44166
$474$	3.03747	0.139516
$475$	0	0
$476$	−1.39477	−0.0639294
$477$	−18.1638	−0.831661
$478$	23.2019	1.06123
$479$	24.7601	1.13132	0.565660	−	0.824639i	$-0.308622\pi$
$479$	24.7601	1.13132	0.565660	+	0.824639i	$0.308622\pi$
$480$	2.09323	0.0955426
$481$	13.9031	0.633928
$482$	1.76578	0.0804292
$483$	−2.46999	−0.112389
$484$	−4.42683	−0.201220
$485$	32.2561	1.46467
$486$	15.1328	0.686437
$487$	22.2019	1.00606	0.503032	−	0.864268i	$-0.332218\pi$
$487$	22.2019	1.00606	0.503032	+	0.864268i	$0.332218\pi$
$488$	−4.80597	−0.217556
$489$	−0.516056	−0.0233368
$490$	−2.91631	−0.131746
$491$	−31.0334	−1.40052	−0.700259	−	0.713888i	$-0.746932\pi$
$491$	−31.0334	−1.40052	−0.700259	+	0.713888i	$0.746932\pi$
$492$	4.74397	0.213875
$493$	1.10131	0.0496005
$494$	0	0
$495$	−18.5787	−0.835049
$496$	−4.99865	−0.224446
$497$	8.70323	0.390393
$498$	−9.79567	−0.438954
$499$	−24.0668	−1.07738	−0.538690	−	0.842504i	$-0.681080\pi$
$499$	−24.0668	−1.07738	−0.538690	+	0.842504i	$0.681080\pi$
$500$	−4.36026	−0.194997
$501$	−10.4377	−0.466321
$502$	−4.47898	−0.199907
$503$	−6.87199	−0.306407	−0.153203	−	0.988195i	$-0.548959\pi$
$503$	−6.87199	−0.306407	−0.153203	+	0.988195i	$0.548959\pi$
$504$	−2.48481	−0.110682
$505$	−43.9622	−1.95629
$506$	8.82265	0.392215
$507$	7.11026	0.315778
$508$	−4.15241	−0.184233
$509$	−4.26876	−0.189210	−0.0946048	−	0.995515i	$-0.530159\pi$
$509$	−4.26876	−0.189210	−0.0946048	+	0.995515i	$0.530159\pi$
$510$	2.91959	0.129282
$511$	4.32196	0.191192
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	24.0895	1.06254
$515$	−38.0130	−1.67505
$516$	−8.77790	−0.386425
$517$	−3.16725	−0.139295
$518$	−7.90419	−0.347290
$519$	−17.3983	−0.763699
$520$	5.12966	0.224951
$521$	−23.3507	−1.02301	−0.511506	−	0.859280i	$-0.670912\pi$
$521$	−23.3507	−1.02301	−0.511506	+	0.859280i	$0.670912\pi$
$522$	1.96200	0.0858744
$523$	−16.6780	−0.729278	−0.364639	−	0.931149i	$-0.618808\pi$
$523$	−16.6780	−0.729278	−0.364639	+	0.931149i	$0.618808\pi$
$524$	18.9379	0.827305
$525$	2.51568	0.109793
$526$	−8.88831	−0.387549
$527$	−6.97199	−0.303705
$528$	−1.84023	−0.0800855
$529$	−11.1580	−0.485132
$530$	−21.3180	−0.925995
$531$	4.19242	0.181936
$532$	0	0
$533$	11.6255	0.503558
$534$	−1.07871	−0.0466805
$535$	−44.8389	−1.93855
$536$	2.63849	0.113966
$537$	6.73819	0.290774
$538$	−12.1748	−0.524894
$539$	2.56382	0.110431
$540$	11.4810	0.494063
$541$	−30.8053	−1.32442	−0.662211	−	0.749318i	$-0.730382\pi$
$541$	−30.8053	−1.32442	−0.662211	+	0.749318i	$0.730382\pi$
$542$	−10.8375	−0.465511
$543$	17.0690	0.732500
$544$	−1.39477	−0.0598005
$545$	−58.0519	−2.48667
$546$	1.26252	0.0540309
$547$	−5.43027	−0.232182	−0.116091	−	0.993239i	$-0.537036\pi$
$547$	−5.43027	−0.232182	−0.116091	+	0.993239i	$0.537036\pi$
$548$	−16.3088	−0.696678
$549$	−11.9419	−0.509669
$550$	−8.98586	−0.383158
$551$	0	0
$552$	−2.46999	−0.105130
$553$	−4.23183	−0.179956
$554$	11.3275	0.481258
$555$	16.5453	0.702310
$556$	−3.62222	−0.153616
$557$	−10.0307	−0.425013	−0.212507	−	0.977160i	$-0.568163\pi$
$557$	−10.0307	−0.425013	−0.212507	+	0.977160i	$0.568163\pi$
$558$	−12.4207	−0.525810
$559$	−21.5111	−0.909821
$560$	−2.91631	−0.123237
$561$	−2.56670	−0.108366
$562$	16.8113	0.709140
$563$	1.13981	0.0480373	0.0240186	−	0.999712i	$-0.492354\pi$
$563$	1.13981	0.0480373	0.0240186	+	0.999712i	$0.492354\pi$
$564$	0.886703	0.0373369
$565$	8.70810	0.366353
$566$	32.7713	1.37748
$567$	−4.62871	−0.194388
$568$	8.70323	0.365179
$569$	−16.1500	−0.677041	−0.338521	−	0.940959i	$-0.609926\pi$
$569$	−16.1500	−0.677041	−0.338521	+	0.940959i	$0.609926\pi$
$570$	0	0
$571$	−13.3743	−0.559698	−0.279849	−	0.960044i	$-0.590284\pi$
$571$	−13.3743	−0.559698	−0.279849	+	0.960044i	$0.590284\pi$
$572$	−4.50965	−0.188558
$573$	0.403335	0.0168496
$574$	−6.60934	−0.275868
$575$	−12.0610	−0.502979
$576$	−2.48481	−0.103534
$577$	8.76918	0.365066	0.182533	−	0.983200i	$-0.441570\pi$
$577$	8.76918	0.365066	0.182533	+	0.983200i	$0.441570\pi$
$578$	15.0546	0.626189
$579$	−12.4668	−0.518102
$580$	2.30271	0.0956149
$581$	13.6474	0.566190
$582$	7.93892	0.329079
$583$	18.7413	0.776185
$584$	4.32196	0.178844
$585$	12.7462	0.526992
$586$	16.5383	0.683190
$587$	30.1409	1.24405	0.622024	−	0.782998i	$-0.286311\pi$
$587$	30.1409	1.24405	0.622024	+	0.782998i	$0.286311\pi$
$588$	−0.717767	−0.0296002
$589$	0	0
$590$	4.92046	0.202572
$591$	−14.7034	−0.604818
$592$	−7.90419	−0.324860
$593$	17.0917	0.701871	0.350935	−	0.936400i	$-0.385864\pi$
$593$	17.0917	0.701871	0.350935	+	0.936400i	$0.385864\pi$
$594$	−10.0933	−0.414133
$595$	−4.06760	−0.166755
$596$	−10.5028	−0.430210
$597$	−9.65605	−0.395196
$598$	−6.05295	−0.247524
$599$	3.36233	0.137381	0.0686906	−	0.997638i	$-0.478118\pi$
$599$	3.36233	0.137381	0.0686906	+	0.997638i	$0.478118\pi$
$600$	2.51568	0.102702
$601$	−32.7613	−1.33636	−0.668181	−	0.743999i	$-0.732927\pi$
$601$	−32.7613	−1.33636	−0.668181	+	0.743999i	$0.732927\pi$
$602$	12.2294	0.498435
$603$	6.55615	0.266987
$604$	20.8828	0.849708
$605$	−12.9100	−0.524867
$606$	−10.8201	−0.439534
$607$	−41.7569	−1.69486	−0.847431	−	0.530906i	$-0.821852\pi$
$607$	−41.7569	−1.69486	−0.847431	+	0.530906i	$0.821852\pi$
$608$	0	0
$609$	0.566747	0.0229658
$610$	−14.0157	−0.567480
$611$	2.17295	0.0879081
$612$	−3.46575	−0.140095
$613$	22.3997	0.904716	0.452358	−	0.891836i	$-0.350583\pi$
$613$	22.3997	0.904716	0.452358	+	0.891836i	$0.350583\pi$
$614$	10.7452	0.433642
$615$	13.8349	0.557876
$616$	2.56382	0.103299
$617$	−37.3977	−1.50558	−0.752788	−	0.658263i	$-0.771292\pi$
$617$	−37.3977	−1.50558	−0.752788	+	0.658263i	$0.771292\pi$
$618$	−9.35582	−0.376346
$619$	21.9167	0.880906	0.440453	−	0.897776i	$-0.354818\pi$
$619$	21.9167	0.880906	0.440453	+	0.897776i	$0.354818\pi$
$620$	−14.5776	−0.585451
$621$	−13.5474	−0.543640
$622$	32.3009	1.29515
$623$	1.50287	0.0602113
$624$	1.26252	0.0505413
$625$	−30.2402	−1.20961
$626$	11.5120	0.460113
$627$	0	0
$628$	14.4157	0.575247
$629$	−11.0246	−0.439578
$630$	−7.24648	−0.288707
$631$	17.8262	0.709650	0.354825	−	0.934933i	$-0.384541\pi$
$631$	17.8262	0.709650	0.354825	+	0.934933i	$0.384541\pi$
$632$	−4.23183	−0.168333
$633$	−3.74620	−0.148898
$634$	−12.4053	−0.492677
$635$	−12.1097	−0.480559
$636$	−5.24682	−0.208050
$637$	−1.75896	−0.0696924
$638$	−2.02438	−0.0801461
$639$	21.6259	0.855506
$640$	−2.91631	−0.115277
$641$	−9.79762	−0.386983	−0.193491	−	0.981102i	$-0.561981\pi$
$641$	−9.79762	−0.386983	−0.193491	+	0.981102i	$0.561981\pi$
$642$	−11.0358	−0.435549
$643$	27.1491	1.07066	0.535328	−	0.844645i	$-0.320188\pi$
$643$	27.1491	1.07066	0.535328	+	0.844645i	$0.320188\pi$
$644$	3.44122	0.135603
$645$	−25.5991	−1.00796
$646$	0	0
$647$	25.7144	1.01094	0.505469	−	0.862845i	$-0.331320\pi$
$647$	25.7144	1.01094	0.505469	+	0.862845i	$0.331320\pi$
$648$	−4.62871	−0.181833
$649$	−4.32573	−0.169800
$650$	6.16492	0.241808
$651$	−3.58787	−0.140620
$652$	0.718973	0.0281572
$653$	−5.60452	−0.219322	−0.109661	−	0.993969i	$-0.534976\pi$
$653$	−5.60452	−0.219322	−0.109661	+	0.993969i	$0.534976\pi$
$654$	−14.2878	−0.558699
$655$	55.2288	2.15797
$656$	−6.60934	−0.258051
$657$	10.7392	0.418978
$658$	−1.23536	−0.0481595
$659$	−24.3115	−0.947040	−0.473520	−	0.880783i	$-0.657017\pi$
$659$	−24.3115	−0.947040	−0.473520	+	0.880783i	$0.657017\pi$
$660$	−5.36667	−0.208897
$661$	10.6106	0.412705	0.206352	−	0.978478i	$-0.433841\pi$
$661$	10.6106	0.412705	0.206352	+	0.978478i	$0.433841\pi$
$662$	4.28173	0.166414
$663$	1.76093	0.0683890
$664$	13.6474	0.529622
$665$	0	0
$666$	−19.6404	−0.761051
$667$	−2.71717	−0.105209
$668$	14.5419	0.562642
$669$	−13.0842	−0.505864
$670$	7.69467	0.297271
$671$	12.3216	0.475672
$672$	−0.717767	−0.0276885
$673$	33.2603	1.28209	0.641045	−	0.767503i	$-0.278501\pi$
$673$	33.2603	1.28209	0.641045	+	0.767503i	$0.278501\pi$
$674$	34.6005	1.33276
$675$	13.7980	0.531087
$676$	−9.90607	−0.381003
$677$	51.6253	1.98412	0.992061	−	0.125758i	$-0.0401363\pi$
$677$	51.6253	1.98412	0.992061	+	0.125758i	$0.0401363\pi$
$678$	2.14325	0.0823111
$679$	−11.0606	−0.424466
$680$	−4.06760	−0.155985
$681$	−7.70813	−0.295376
$682$	12.8156	0.490736
$683$	10.1408	0.388026	0.194013	−	0.980999i	$-0.437850\pi$
$683$	10.1408	0.388026	0.194013	+	0.980999i	$0.437850\pi$
$684$	0	0
$685$	−47.5616	−1.81724
$686$	1.00000	0.0381802
$687$	11.9851	0.457259
$688$	12.2294	0.466243
$689$	−12.8578	−0.489844
$690$	−7.20327	−0.274224
$691$	7.25018	0.275810	0.137905	−	0.990445i	$-0.455963\pi$
$691$	7.25018	0.275810	0.137905	+	0.990445i	$0.455963\pi$
$692$	24.2394	0.921444
$693$	6.37060	0.241999
$694$	8.61568	0.327047
$695$	−10.5635	−0.400698
$696$	0.566747	0.0214825
$697$	−9.21853	−0.349177
$698$	30.1120	1.13976
$699$	12.0915	0.457344
$700$	−3.50487	−0.132472
$701$	−13.1064	−0.495022	−0.247511	−	0.968885i	$-0.579613\pi$
$701$	−13.1064	−0.495022	−0.247511	+	0.968885i	$0.579613\pi$
$702$	6.92469	0.261356
$703$	0	0
$704$	2.56382	0.0966276
$705$	2.58590	0.0973907
$706$	−4.02781	−0.151589
$707$	15.0746	0.566938
$708$	1.21103	0.0455134
$709$	−47.4592	−1.78237	−0.891184	−	0.453642i	$-0.850125\pi$
$709$	−47.4592	−1.78237	−0.891184	+	0.453642i	$0.850125\pi$
$710$	25.3813	0.952544
$711$	−10.5153	−0.394354
$712$	1.50287	0.0563225
$713$	17.2014	0.644199
$714$	−1.00112	−0.0374661
$715$	−13.1515	−0.491839
$716$	−9.38770	−0.350835
$717$	16.6536	0.621939
$718$	−33.6389	−1.25539
$719$	35.7207	1.33216	0.666080	−	0.745881i	$-0.267971\pi$
$719$	35.7207	1.33216	0.666080	+	0.745881i	$0.267971\pi$
$720$	−7.24648	−0.270060
$721$	13.0346	0.485434
$722$	0	0
$723$	1.26742	0.0471359
$724$	−23.7807	−0.883801
$725$	2.76744	0.102780
$726$	−3.17744	−0.117926
$727$	22.2094	0.823701	0.411851	−	0.911251i	$-0.364883\pi$
$727$	22.2094	0.823701	0.411851	+	0.911251i	$0.364883\pi$
$728$	−1.75896	−0.0651913
$729$	−3.02430	−0.112011
$730$	12.6042	0.466501
$731$	17.0573	0.630888
$732$	−3.44957	−0.127500
$733$	−35.2245	−1.30105	−0.650523	−	0.759486i	$-0.725450\pi$
$733$	−35.2245	−1.30105	−0.650523	+	0.759486i	$0.725450\pi$
$734$	−10.8822	−0.401670
$735$	−2.09323	−0.0772101
$736$	3.44122	0.126845
$737$	−6.76462	−0.249178
$738$	−16.4229	−0.604537
$739$	3.67115	0.135045	0.0675227	−	0.997718i	$-0.478490\pi$
$739$	3.67115	0.135045	0.0675227	+	0.997718i	$0.478490\pi$
$740$	−23.0511	−0.847375
$741$	0	0
$742$	7.30992	0.268356
$743$	36.1232	1.32523	0.662615	−	0.748960i	$-0.269446\pi$
$743$	36.1232	1.32523	0.662615	+	0.748960i	$0.269446\pi$
$744$	−3.58787	−0.131538
$745$	−30.6293	−1.12217
$746$	−10.9132	−0.399559
$747$	33.9112	1.24075
$748$	3.57595	0.130750
$749$	15.3752	0.561798
$750$	−3.12965	−0.114279
$751$	30.9862	1.13070	0.565351	−	0.824851i	$-0.308741\pi$
$751$	30.9862	1.13070	0.565351	+	0.824851i	$0.308741\pi$
$752$	−1.23536	−0.0450490
$753$	−3.21487	−0.117156
$754$	1.38887	0.0505796
$755$	60.9007	2.21640
$756$	−3.93682	−0.143181
$757$	53.0879	1.92951	0.964757	−	0.263143i	$-0.0847592\pi$
$757$	53.0879	1.92951	0.964757	+	0.263143i	$0.0847592\pi$
$758$	−30.4270	−1.10516
$759$	6.33261	0.229859
$760$	0	0
$761$	−4.65429	−0.168718	−0.0843589	−	0.996435i	$-0.526884\pi$
$761$	−4.65429	−0.168718	−0.0843589	+	0.996435i	$0.526884\pi$
$762$	−2.98046	−0.107971
$763$	19.9059	0.720643
$764$	−0.561930	−0.0203299
$765$	−10.1072	−0.365427
$766$	−3.04363	−0.109971
$767$	2.96775	0.107159
$768$	−0.717767	−0.0259002
$769$	−23.8596	−0.860399	−0.430200	−	0.902734i	$-0.641557\pi$
$769$	−23.8596	−0.860399	−0.430200	+	0.902734i	$0.641557\pi$
$770$	7.47689	0.269449
$771$	17.2906	0.622707
$772$	17.3688	0.625118
$773$	30.2120	1.08665	0.543324	−	0.839523i	$-0.317165\pi$
$773$	30.2120	1.08665	0.543324	+	0.839523i	$0.317165\pi$
$774$	30.3878	1.09227
$775$	−17.5196	−0.629323
$776$	−11.0606	−0.397051
$777$	−5.67337	−0.203531
$778$	−15.7728	−0.565482
$779$	0	0
$780$	3.68191	0.131833
$781$	−22.3135	−0.798440
$782$	4.79972	0.171638
$783$	3.10850	0.111089
$784$	1.00000	0.0357143
$785$	42.0405	1.50049
$786$	13.5930	0.484846
$787$	−2.65870	−0.0947723	−0.0473862	−	0.998877i	$-0.515089\pi$
$787$	−2.65870	−0.0947723	−0.0473862	+	0.998877i	$0.515089\pi$
$788$	20.4849	0.729746
$789$	−6.37974	−0.227125
$790$	−12.3413	−0.439085
$791$	−2.98600	−0.106170
$792$	6.37060	0.226369
$793$	−8.45350	−0.300192
$794$	29.8753	1.06024
$795$	−15.3014	−0.542683
$796$	13.4529	0.476825
$797$	12.1882	0.431729	0.215864	−	0.976423i	$-0.430743\pi$
$797$	12.1882	0.431729	0.215864	+	0.976423i	$0.430743\pi$
$798$	0	0
$799$	−1.72305	−0.0609572
$800$	−3.50487	−0.123916
$801$	3.73435	0.131947
$802$	−10.0388	−0.354484
$803$	−11.0807	−0.391030
$804$	1.89382	0.0667900
$805$	10.0357	0.353710
$806$	−8.79240	−0.309699
$807$	−8.73870	−0.307617
$808$	15.0746	0.530322
$809$	17.3289	0.609253	0.304626	−	0.952472i	$-0.401468\pi$
$809$	17.3289	0.609253	0.304626	+	0.952472i	$0.401468\pi$
$810$	−13.4988	−0.474298
$811$	−24.6678	−0.866203	−0.433102	−	0.901345i	$-0.642581\pi$
$811$	−24.6678	−0.866203	−0.433102	+	0.901345i	$0.642581\pi$
$812$	−0.789597	−0.0277094
$813$	−7.77881	−0.272815
$814$	20.2649	0.710285
$815$	2.09675	0.0734459
$816$	−1.00112	−0.0350463
$817$	0	0
$818$	−10.0191	−0.350309
$819$	−4.37067	−0.152724
$820$	−19.2749	−0.673108
$821$	−27.8333	−0.971389	−0.485694	−	0.874129i	$-0.661433\pi$
$821$	−27.8333	−0.971389	−0.485694	+	0.874129i	$0.661433\pi$
$822$	−11.7059	−0.408292
$823$	−19.0395	−0.663676	−0.331838	−	0.943336i	$-0.607669\pi$
$823$	−19.0395	−0.663676	−0.331838	+	0.943336i	$0.607669\pi$
$824$	13.0346	0.454082
$825$	−6.44976	−0.224552
$826$	−1.68722	−0.0587059
$827$	−21.3800	−0.743455	−0.371728	−	0.928342i	$-0.621234\pi$
$827$	−21.3800	−0.743455	−0.371728	+	0.928342i	$0.621234\pi$
$828$	8.55077	0.297160
$829$	−8.19845	−0.284744	−0.142372	−	0.989813i	$-0.545473\pi$
$829$	−8.19845	−0.284744	−0.142372	+	0.989813i	$0.545473\pi$
$830$	39.8001	1.38148
$831$	8.13050	0.282044
$832$	−1.75896	−0.0609808
$833$	1.39477	0.0483261
$834$	−2.59991	−0.0900277
$835$	42.4086	1.46761
$836$	0	0
$837$	−19.6788	−0.680198
$838$	−2.90178	−0.100240
$839$	−8.97637	−0.309899	−0.154949	−	0.987922i	$-0.549521\pi$
$839$	−8.97637	−0.309899	−0.154949	+	0.987922i	$0.549521\pi$
$840$	−2.09323	−0.0722234
$841$	−28.3765	−0.978501
$842$	19.6306	0.676516
$843$	12.0666	0.415595
$844$	5.21924	0.179654
$845$	−28.8892	−0.993818
$846$	−3.06964	−0.105536
$847$	4.42683	0.152108
$848$	7.30992	0.251024
$849$	23.5221	0.807278
$850$	−4.88851	−0.167674
$851$	27.2000	0.932405
$852$	6.24689	0.214015
$853$	−14.6720	−0.502360	−0.251180	−	0.967940i	$-0.580819\pi$
$853$	−14.6720	−0.502360	−0.251180	+	0.967940i	$0.580819\pi$
$854$	4.80597	0.164457
$855$	0	0
$856$	15.3752	0.525513
$857$	1.20492	0.0411594	0.0205797	−	0.999788i	$-0.493449\pi$
$857$	1.20492	0.0411594	0.0205797	+	0.999788i	$0.493449\pi$
$858$	−3.23688	−0.110505
$859$	6.65515	0.227071	0.113535	−	0.993534i	$-0.463782\pi$
$859$	6.65515	0.227071	0.113535	+	0.993534i	$0.463782\pi$
$860$	35.6649	1.21616
$861$	−4.74397	−0.161674
$862$	−0.757246	−0.0257919
$863$	12.4694	0.424462	0.212231	−	0.977220i	$-0.431927\pi$
$863$	12.4694	0.424462	0.212231	+	0.977220i	$0.431927\pi$
$864$	−3.93682	−0.133933
$865$	70.6897	2.40352
$866$	−2.51099	−0.0853267
$867$	10.8057	0.366981
$868$	4.99865	0.169665
$869$	10.8496	0.368049
$870$	1.65281	0.0560356
$871$	4.64099	0.157254
$872$	19.9059	0.674100
$873$	−27.4834	−0.930173
$874$	0	0
$875$	4.36026	0.147404
$876$	3.10216	0.104812
$877$	−32.2864	−1.09023	−0.545117	−	0.838360i	$-0.683515\pi$
$877$	−32.2864	−1.09023	−0.545117	+	0.838360i	$0.683515\pi$
$878$	22.3934	0.755740
$879$	11.8706	0.400387
$880$	7.47689	0.252046
$881$	44.5135	1.49970	0.749850	−	0.661608i	$-0.230126\pi$
$881$	44.5135	1.49970	0.749850	+	0.661608i	$0.230126\pi$
$882$	2.48481	0.0836679
$883$	−35.0278	−1.17878	−0.589389	−	0.807849i	$-0.700631\pi$
$883$	−35.0278	−1.17878	−0.589389	+	0.807849i	$0.700631\pi$
$884$	−2.45335	−0.0825150
$885$	3.53175	0.118718
$886$	−2.45606	−0.0825129
$887$	−12.1340	−0.407421	−0.203710	−	0.979031i	$-0.565300\pi$
$887$	−12.1340	−0.407421	−0.203710	+	0.979031i	$0.565300\pi$
$888$	−5.67337	−0.190386
$889$	4.15241	0.139267
$890$	4.38285	0.146913
$891$	11.8672	0.397565
$892$	18.2290	0.610352
$893$	0	0
$894$	−7.53855	−0.252127
$895$	−27.3775	−0.915128
$896$	1.00000	0.0334077
$897$	−4.34461	−0.145062
$898$	4.08691	0.136382
$899$	−3.94692	−0.131637
$900$	−8.70894	−0.290298
$901$	10.1957	0.339668
$902$	16.9451	0.564212
$903$	8.77790	0.292110
$904$	−2.98600	−0.0993128
$905$	−69.3518	−2.30533
$906$	14.9890	0.497976
$907$	8.50269	0.282327	0.141164	−	0.989986i	$-0.454916\pi$
$907$	8.50269	0.282327	0.141164	+	0.989986i	$0.454916\pi$
$908$	10.7390	0.356387
$909$	37.4575	1.24239
$910$	−5.12966	−0.170047
$911$	20.9608	0.694462	0.347231	−	0.937780i	$-0.387122\pi$
$911$	20.9608	0.694462	0.347231	+	0.937780i	$0.387122\pi$
$912$	0	0
$913$	−34.9895	−1.15798
$914$	−10.5514	−0.349011
$915$	−10.0600	−0.332574
$916$	−16.6977	−0.551707
$917$	−18.9379	−0.625384
$918$	−5.49097	−0.181229
$919$	4.33153	0.142884	0.0714420	−	0.997445i	$-0.477240\pi$
$919$	4.33153	0.142884	0.0714420	+	0.997445i	$0.477240\pi$
$920$	10.0357	0.330866
$921$	7.71257	0.254138
$922$	17.6728	0.582023
$923$	15.3086	0.503889
$924$	1.84023	0.0605390
$925$	−27.7032	−0.910875
$926$	−27.5515	−0.905400
$927$	32.3885	1.06378
$928$	−0.789597	−0.0259198
$929$	−33.7598	−1.10762	−0.553811	−	0.832642i	$-0.686827\pi$
$929$	−33.7598	−1.10762	−0.553811	+	0.832642i	$0.686827\pi$
$930$	−10.4633	−0.343106
$931$	0	0
$932$	−16.8460	−0.551810
$933$	23.1845	0.759027
$934$	34.2363	1.12024
$935$	10.4286	0.341051
$936$	−4.37067	−0.142860
$937$	−43.5661	−1.42324	−0.711621	−	0.702564i	$-0.752039\pi$
$937$	−43.5661	−1.42324	−0.711621	+	0.702564i	$0.752039\pi$
$938$	−2.63849	−0.0861498
$939$	8.26295	0.269651
$940$	−3.60270	−0.117507
$941$	30.6024	0.997609	0.498805	−	0.866714i	$-0.333773\pi$
$941$	30.6024	0.997609	0.498805	+	0.866714i	$0.333773\pi$
$942$	10.3471	0.337126
$943$	22.7442	0.740652
$944$	−1.68722	−0.0549143
$945$	−11.4810	−0.373477
$946$	−31.3541	−1.01941
$947$	7.73067	0.251213	0.125606	−	0.992080i	$-0.459912\pi$
$947$	7.73067	0.251213	0.125606	+	0.992080i	$0.459912\pi$
$948$	−3.03747	−0.0986524
$949$	7.60213	0.246776
$950$	0	0
$951$	−8.90412	−0.288736
$952$	1.39477	0.0452049
$953$	55.9729	1.81314	0.906571	−	0.422054i	$-0.138691\pi$
$953$	55.9729	1.81314	0.906571	+	0.422054i	$0.138691\pi$
$954$	18.1638	0.588073
$955$	−1.63876	−0.0530292
$956$	−23.2019	−0.750403
$957$	−1.45304	−0.0469700
$958$	−24.7601	−0.799963
$959$	16.3088	0.526639
$960$	−2.09323	−0.0675588
$961$	−6.01351	−0.193984
$962$	−13.9031	−0.448255
$963$	38.2044	1.23112
$964$	−1.76578	−0.0568720
$965$	50.6530	1.63058
$966$	2.46999	0.0794707
$967$	−35.4513	−1.14004	−0.570019	−	0.821631i	$-0.693064\pi$
$967$	−35.4513	−1.14004	−0.570019	+	0.821631i	$0.693064\pi$
$968$	4.42683	0.142284
$969$	0	0
$970$	−32.2561	−1.03568
$971$	28.3291	0.909123	0.454561	−	0.890715i	$-0.349796\pi$
$971$	28.3291	0.909123	0.454561	+	0.890715i	$0.349796\pi$
$972$	−15.1328	−0.485385
$973$	3.62222	0.116123
$974$	−22.2019	−0.711394
$975$	4.42498	0.141713
$976$	4.80597	0.153835
$977$	15.7603	0.504216	0.252108	−	0.967699i	$-0.418876\pi$
$977$	15.7603	0.504216	0.252108	+	0.967699i	$0.418876\pi$
$978$	0.516056	0.0165016
$979$	−3.85309	−0.123145
$980$	2.91631	0.0931581
$981$	49.4625	1.57922
$982$	31.0334	0.990316
$983$	20.4920	0.653594	0.326797	−	0.945094i	$-0.394031\pi$
$983$	20.4920	0.653594	0.326797	+	0.945094i	$0.394031\pi$
$984$	−4.74397	−0.151232
$985$	59.7405	1.90349
$986$	−1.10131	−0.0350729
$987$	−0.886703	−0.0282241
$988$	0	0
$989$	−42.0842	−1.33820
$990$	18.5787	0.590469
$991$	−6.85697	−0.217819	−0.108909	−	0.994052i	$-0.534736\pi$
$991$	−6.85697	−0.217819	−0.108909	+	0.994052i	$0.534736\pi$
$992$	4.99865	0.158707
$993$	3.07328	0.0975277
$994$	−8.70323	−0.276050
$995$	39.2328	1.24376
$996$	9.79567	0.310388
$997$	11.7171	0.371083	0.185541	−	0.982636i	$-0.440596\pi$
$997$	11.7171	0.371083	0.185541	+	0.982636i	$0.440596\pi$
$998$	24.0668	0.761822
$999$	−31.1174	−0.984510

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5054.2.a.bf.1.4	✓	8	1.1	even	1	trivial
5054.2.a.bi.1.5	yes	8	19.18	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:	\( N \)	\(=\)	\( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5054.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.717767 q^{3} +1.00000 q^{4} +2.91631 q^{5} +0.717767 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.48481 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.717767 q^{3} +1.00000 q^{4} +2.91631 q^{5} +0.717767 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.48481 q^{9} -2.91631 q^{10} +2.56382 q^{11} -0.717767 q^{12} -1.75896 q^{13} +1.00000 q^{14} -2.09323 q^{15} +1.00000 q^{16} +1.39477 q^{17} +2.48481 q^{18} +2.91631 q^{20} +0.717767 q^{21} -2.56382 q^{22} -3.44122 q^{23} +0.717767 q^{24} +3.50487 q^{25} +1.75896 q^{26} +3.93682 q^{27} -1.00000 q^{28} +0.789597 q^{29} +2.09323 q^{30} -4.99865 q^{31} -1.00000 q^{32} -1.84023 q^{33} -1.39477 q^{34} -2.91631 q^{35} -2.48481 q^{36} -7.90419 q^{37} +1.26252 q^{39} -2.91631 q^{40} -6.60934 q^{41} -0.717767 q^{42} +12.2294 q^{43} +2.56382 q^{44} -7.24648 q^{45} +3.44122 q^{46} -1.23536 q^{47} -0.717767 q^{48} +1.00000 q^{49} -3.50487 q^{50} -1.00112 q^{51} -1.75896 q^{52} +7.30992 q^{53} -3.93682 q^{54} +7.47689 q^{55} +1.00000 q^{56} -0.789597 q^{58} -1.68722 q^{59} -2.09323 q^{60} +4.80597 q^{61} +4.99865 q^{62} +2.48481 q^{63} +1.00000 q^{64} -5.12966 q^{65} +1.84023 q^{66} -2.63849 q^{67} +1.39477 q^{68} +2.46999 q^{69} +2.91631 q^{70} -8.70323 q^{71} +2.48481 q^{72} -4.32196 q^{73} +7.90419 q^{74} -2.51568 q^{75} -2.56382 q^{77} -1.26252 q^{78} +4.23183 q^{79} +2.91631 q^{80} +4.62871 q^{81} +6.60934 q^{82} -13.6474 q^{83} +0.717767 q^{84} +4.06760 q^{85} -12.2294 q^{86} -0.566747 q^{87} -2.56382 q^{88} -1.50287 q^{89} +7.24648 q^{90} +1.75896 q^{91} -3.44122 q^{92} +3.58787 q^{93} +1.23536 q^{94} +0.717767 q^{96} +11.0606 q^{97} -1.00000 q^{98} -6.37060 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 6 * q^5 + 4 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9} - 6 q^{10} + 4 q^{11} - 4 q^{12} - 6 q^{13} + 8 q^{14} - 8 q^{15} + 8 q^{16} + 2 q^{17} - 8 q^{18} + 6 q^{20} + 4 q^{21} - 4 q^{22} + 20 q^{23} + 4 q^{24} - 8 q^{25} + 6 q^{26} - 22 q^{27} - 8 q^{28} - 16 q^{29} + 8 q^{30} - 22 q^{31} - 8 q^{32} + 8 q^{33} - 2 q^{34} - 6 q^{35} + 8 q^{36} + 12 q^{37} + 8 q^{39} - 6 q^{40} - 36 q^{41} - 4 q^{42} + 4 q^{44} - 4 q^{45} - 20 q^{46} + 6 q^{47} - 4 q^{48} + 8 q^{49} + 8 q^{50} + 4 q^{51} - 6 q^{52} - 16 q^{53} + 22 q^{54} + 18 q^{55} + 8 q^{56} + 16 q^{58} - 14 q^{59} - 8 q^{60} + 10 q^{61} + 22 q^{62} - 8 q^{63} + 8 q^{64} - 32 q^{65} - 8 q^{66} + 20 q^{67} + 2 q^{68} - 40 q^{69} + 6 q^{70} - 8 q^{72} - 18 q^{73} - 12 q^{74} - 16 q^{75} - 4 q^{77} - 8 q^{78} - 4 q^{79} + 6 q^{80} + 12 q^{81} + 36 q^{82} + 36 q^{83} + 4 q^{84} - 16 q^{85} + 28 q^{87} - 4 q^{88} - 18 q^{89} + 4 q^{90} + 6 q^{91} + 20 q^{92} + 16 q^{93} - 6 q^{94} + 4 q^{96} - 26 q^{97} - 8 q^{98} + 4 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 6 * q^5 + 4 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9 - 6 * q^10 + 4 * q^11 - 4 * q^12 - 6 * q^13 + 8 * q^14 - 8 * q^15 + 8 * q^16 + 2 * q^17 - 8 * q^18 + 6 * q^20 + 4 * q^21 - 4 * q^22 + 20 * q^23 + 4 * q^24 - 8 * q^25 + 6 * q^26 - 22 * q^27 - 8 * q^28 - 16 * q^29 + 8 * q^30 - 22 * q^31 - 8 * q^32 + 8 * q^33 - 2 * q^34 - 6 * q^35 + 8 * q^36 + 12 * q^37 + 8 * q^39 - 6 * q^40 - 36 * q^41 - 4 * q^42 + 4 * q^44 - 4 * q^45 - 20 * q^46 + 6 * q^47 - 4 * q^48 + 8 * q^49 + 8 * q^50 + 4 * q^51 - 6 * q^52 - 16 * q^53 + 22 * q^54 + 18 * q^55 + 8 * q^56 + 16 * q^58 - 14 * q^59 - 8 * q^60 + 10 * q^61 + 22 * q^62 - 8 * q^63 + 8 * q^64 - 32 * q^65 - 8 * q^66 + 20 * q^67 + 2 * q^68 - 40 * q^69 + 6 * q^70 - 8 * q^72 - 18 * q^73 - 12 * q^74 - 16 * q^75 - 4 * q^77 - 8 * q^78 - 4 * q^79 + 6 * q^80 + 12 * q^81 + 36 * q^82 + 36 * q^83 + 4 * q^84 - 16 * q^85 + 28 * q^87 - 4 * q^88 - 18 * q^89 + 4 * q^90 + 6 * q^91 + 20 * q^92 + 16 * q^93 - 6 * q^94 + 4 * q^96 - 26 * q^97 - 8 * q^98 + 4 * q^99

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