LMFDB - Embedded newform 4030.2.a.k.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4030,2,Mod(1,4030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4030, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4030 = 2 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4030.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:	$32.1797120146$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 18x^{6} + 12x^{5} + 98x^{4} - 18x^{3} - 173x^{2} - 48x + 32 $ x^8 - x^7 - 18x^6 + 12x^5 + 98x^4 - 18x^3 - 173x^2 - 48x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.21997$ of defining polynomial
Character	$\chi$	$=$	4030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.21997 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.21997 q^{6} -1.25636 q^{7} -1.00000 q^{8} -1.51167 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.21997 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.21997 q^{6} -1.25636 q^{7} -1.00000 q^{8} -1.51167 q^{9} +1.00000 q^{10} -6.19520 q^{11} +1.21997 q^{12} +1.00000 q^{13} +1.25636 q^{14} -1.21997 q^{15} +1.00000 q^{16} +3.94242 q^{17} +1.51167 q^{18} -4.85631 q^{19} -1.00000 q^{20} -1.53273 q^{21} +6.19520 q^{22} +8.13190 q^{23} -1.21997 q^{24} +1.00000 q^{25} -1.00000 q^{26} -5.50411 q^{27} -1.25636 q^{28} +0.0979954 q^{29} +1.21997 q^{30} -1.00000 q^{31} -1.00000 q^{32} -7.55798 q^{33} -3.94242 q^{34} +1.25636 q^{35} -1.51167 q^{36} +10.1251 q^{37} +4.85631 q^{38} +1.21997 q^{39} +1.00000 q^{40} -3.70981 q^{41} +1.53273 q^{42} -11.1300 q^{43} -6.19520 q^{44} +1.51167 q^{45} -8.13190 q^{46} +4.79248 q^{47} +1.21997 q^{48} -5.42156 q^{49} -1.00000 q^{50} +4.80965 q^{51} +1.00000 q^{52} +3.74006 q^{53} +5.50411 q^{54} +6.19520 q^{55} +1.25636 q^{56} -5.92457 q^{57} -0.0979954 q^{58} -5.37745 q^{59} -1.21997 q^{60} +11.3933 q^{61} +1.00000 q^{62} +1.89920 q^{63} +1.00000 q^{64} -1.00000 q^{65} +7.55798 q^{66} -7.70302 q^{67} +3.94242 q^{68} +9.92071 q^{69} -1.25636 q^{70} -2.68875 q^{71} +1.51167 q^{72} -0.389941 q^{73} -10.1251 q^{74} +1.21997 q^{75} -4.85631 q^{76} +7.78340 q^{77} -1.21997 q^{78} +1.52451 q^{79} -1.00000 q^{80} -2.17987 q^{81} +3.70981 q^{82} +6.14297 q^{83} -1.53273 q^{84} -3.94242 q^{85} +11.1300 q^{86} +0.119552 q^{87} +6.19520 q^{88} -1.25689 q^{89} -1.51167 q^{90} -1.25636 q^{91} +8.13190 q^{92} -1.21997 q^{93} -4.79248 q^{94} +4.85631 q^{95} -1.21997 q^{96} +7.44250 q^{97} +5.42156 q^{98} +9.36507 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} + q^{6} - 8 q^{7} - 8 q^{8} + 13 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 + q^6 - 8 * q^7 - 8 * q^8 + 13 * q^9	\( 8 q - 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} + q^{6} - 8 q^{7} - 8 q^{8} + 13 q^{9} + 8 q^{10} + 12 q^{11} - q^{12} + 8 q^{13} + 8 q^{14} + q^{15} + 8 q^{16} - 13 q^{18} - 3 q^{19} - 8 q^{20} + 27 q^{21} - 12 q^{22} + 17 q^{23} + q^{24} + 8 q^{25} - 8 q^{26} - 13 q^{27} - 8 q^{28} - 10 q^{29} - q^{30} - 8 q^{31} - 8 q^{32} + 4 q^{33} + 8 q^{35} + 13 q^{36} + 2 q^{37} + 3 q^{38} - q^{39} + 8 q^{40} + 14 q^{41} - 27 q^{42} - 19 q^{43} + 12 q^{44} - 13 q^{45} - 17 q^{46} + 8 q^{47} - q^{48} + 16 q^{49} - 8 q^{50} + 35 q^{51} + 8 q^{52} + 4 q^{53} + 13 q^{54} - 12 q^{55} + 8 q^{56} - 33 q^{57} + 10 q^{58} - 13 q^{59} + q^{60} + 11 q^{61} + 8 q^{62} - 27 q^{63} + 8 q^{64} - 8 q^{65} - 4 q^{66} - 34 q^{67} + 12 q^{69} - 8 q^{70} + 8 q^{71} - 13 q^{72} - 21 q^{73} - 2 q^{74} - q^{75} - 3 q^{76} + 19 q^{77} + q^{78} - 36 q^{79} - 8 q^{80} + 20 q^{81} - 14 q^{82} + 27 q^{84} + 19 q^{86} - 29 q^{87} - 12 q^{88} - 16 q^{89} + 13 q^{90} - 8 q^{91} + 17 q^{92} + q^{93} - 8 q^{94} + 3 q^{95} + q^{96} + q^{97} - 16 q^{98} + 65 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 + q^6 - 8 * q^7 - 8 * q^8 + 13 * q^9 + 8 * q^10 + 12 * q^11 - q^12 + 8 * q^13 + 8 * q^14 + q^15 + 8 * q^16 - 13 * q^18 - 3 * q^19 - 8 * q^20 + 27 * q^21 - 12 * q^22 + 17 * q^23 + q^24 + 8 * q^25 - 8 * q^26 - 13 * q^27 - 8 * q^28 - 10 * q^29 - q^30 - 8 * q^31 - 8 * q^32 + 4 * q^33 + 8 * q^35 + 13 * q^36 + 2 * q^37 + 3 * q^38 - q^39 + 8 * q^40 + 14 * q^41 - 27 * q^42 - 19 * q^43 + 12 * q^44 - 13 * q^45 - 17 * q^46 + 8 * q^47 - q^48 + 16 * q^49 - 8 * q^50 + 35 * q^51 + 8 * q^52 + 4 * q^53 + 13 * q^54 - 12 * q^55 + 8 * q^56 - 33 * q^57 + 10 * q^58 - 13 * q^59 + q^60 + 11 * q^61 + 8 * q^62 - 27 * q^63 + 8 * q^64 - 8 * q^65 - 4 * q^66 - 34 * q^67 + 12 * q^69 - 8 * q^70 + 8 * q^71 - 13 * q^72 - 21 * q^73 - 2 * q^74 - q^75 - 3 * q^76 + 19 * q^77 + q^78 - 36 * q^79 - 8 * q^80 + 20 * q^81 - 14 * q^82 + 27 * q^84 + 19 * q^86 - 29 * q^87 - 12 * q^88 - 16 * q^89 + 13 * q^90 - 8 * q^91 + 17 * q^92 + q^93 - 8 * q^94 + 3 * q^95 + q^96 + q^97 - 16 * q^98 + 65 * 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
$3$	1.21997	0.704352	0.352176	−	0.935934i	$-0.385442\pi$
$3$	1.21997	0.704352	0.352176	+	0.935934i	$0.385442\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.21997	−0.498052
$7$	−1.25636	−0.474859	−0.237430	−	0.971405i	$-0.576305\pi$
$7$	−1.25636	−0.474859	−0.237430	+	0.971405i	$0.576305\pi$
$8$	−1.00000	−0.353553
$9$	−1.51167	−0.503888
$10$	1.00000	0.316228
$11$	−6.19520	−1.86792	−0.933962	−	0.357373i	$-0.883672\pi$
$11$	−6.19520	−1.86792	−0.933962	+	0.357373i	$0.883672\pi$
$12$	1.21997	0.352176
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.25636	0.335776
$15$	−1.21997	−0.314996
$16$	1.00000	0.250000
$17$	3.94242	0.956177	0.478089	−	0.878312i	$-0.341330\pi$
$17$	3.94242	0.956177	0.478089	+	0.878312i	$0.341330\pi$
$18$	1.51167	0.356303
$19$	−4.85631	−1.11411	−0.557057	−	0.830474i	$-0.688069\pi$
$19$	−4.85631	−1.11411	−0.557057	+	0.830474i	$0.688069\pi$
$20$	−1.00000	−0.223607
$21$	−1.53273	−0.334468
$22$	6.19520	1.32082
$23$	8.13190	1.69562	0.847810	−	0.530301i	$-0.177921\pi$
$23$	8.13190	1.69562	0.847810	+	0.530301i	$0.177921\pi$
$24$	−1.21997	−0.249026
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	−5.50411	−1.05927
$28$	−1.25636	−0.237430
$29$	0.0979954	0.0181973	0.00909864	−	0.999959i	$-0.497104\pi$
$29$	0.0979954	0.0181973	0.00909864	+	0.999959i	$0.497104\pi$
$30$	1.21997	0.222736
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−1.00000	−0.176777
$33$	−7.55798	−1.31568
$34$	−3.94242	−0.676119
$35$	1.25636	0.212364
$36$	−1.51167	−0.251944
$37$	10.1251	1.66456	0.832280	−	0.554356i	$-0.187035\pi$
$37$	10.1251	1.66456	0.832280	+	0.554356i	$0.187035\pi$
$38$	4.85631	0.787797
$39$	1.21997	0.195352
$40$	1.00000	0.158114
$41$	−3.70981	−0.579375	−0.289688	−	0.957121i	$-0.593551\pi$
$41$	−3.70981	−0.579375	−0.289688	+	0.957121i	$0.593551\pi$
$42$	1.53273	0.236505
$43$	−11.1300	−1.69731	−0.848655	−	0.528946i	$-0.822587\pi$
$43$	−11.1300	−1.69731	−0.848655	+	0.528946i	$0.822587\pi$
$44$	−6.19520	−0.933962
$45$	1.51167	0.225346
$46$	−8.13190	−1.19898
$47$	4.79248	0.699055	0.349527	−	0.936926i	$-0.386342\pi$
$47$	4.79248	0.699055	0.349527	+	0.936926i	$0.386342\pi$
$48$	1.21997	0.176088
$49$	−5.42156	−0.774509
$50$	−1.00000	−0.141421
$51$	4.80965	0.673485
$52$	1.00000	0.138675
$53$	3.74006	0.513737	0.256868	−	0.966446i	$-0.417309\pi$
$53$	3.74006	0.513737	0.256868	+	0.966446i	$0.417309\pi$
$54$	5.50411	0.749015
$55$	6.19520	0.835361
$56$	1.25636	0.167888
$57$	−5.92457	−0.784728
$58$	−0.0979954	−0.0128674
$59$	−5.37745	−0.700084	−0.350042	−	0.936734i	$-0.613833\pi$
$59$	−5.37745	−0.700084	−0.350042	+	0.936734i	$0.613833\pi$
$60$	−1.21997	−0.157498
$61$	11.3933	1.45877	0.729384	−	0.684104i	$-0.239807\pi$
$61$	11.3933	1.45877	0.729384	+	0.684104i	$0.239807\pi$
$62$	1.00000	0.127000
$63$	1.89920	0.239276
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	7.55798	0.930323
$67$	−7.70302	−0.941074	−0.470537	−	0.882380i	$-0.655940\pi$
$67$	−7.70302	−0.941074	−0.470537	+	0.882380i	$0.655940\pi$
$68$	3.94242	0.478089
$69$	9.92071	1.19431
$70$	−1.25636	−0.150164
$71$	−2.68875	−0.319096	−0.159548	−	0.987190i	$-0.551004\pi$
$71$	−2.68875	−0.319096	−0.159548	+	0.987190i	$0.551004\pi$
$72$	1.51167	0.178151
$73$	−0.389941	−0.0456392	−0.0228196	−	0.999740i	$-0.507264\pi$
$73$	−0.389941	−0.0456392	−0.0228196	+	0.999740i	$0.507264\pi$
$74$	−10.1251	−1.17702
$75$	1.21997	0.140870
$76$	−4.85631	−0.557057
$77$	7.78340	0.887001
$78$	−1.21997	−0.138135
$79$	1.52451	0.171521	0.0857606	−	0.996316i	$-0.472668\pi$
$79$	1.52451	0.171521	0.0857606	+	0.996316i	$0.472668\pi$
$80$	−1.00000	−0.111803
$81$	−2.17987	−0.242208
$82$	3.70981	0.409680
$83$	6.14297	0.674279	0.337139	−	0.941455i	$-0.390541\pi$
$83$	6.14297	0.674279	0.337139	+	0.941455i	$0.390541\pi$
$84$	−1.53273	−0.167234
$85$	−3.94242	−0.427615
$86$	11.1300	1.20018
$87$	0.119552	0.0128173
$88$	6.19520	0.660411
$89$	−1.25689	−0.133230	−0.0666152	−	0.997779i	$-0.521220\pi$
$89$	−1.25689	−0.133230	−0.0666152	+	0.997779i	$0.521220\pi$
$90$	−1.51167	−0.159344
$91$	−1.25636	−0.131702
$92$	8.13190	0.847810
$93$	−1.21997	−0.126505
$94$	−4.79248	−0.494306
$95$	4.85631	0.498247
$96$	−1.21997	−0.124513
$97$	7.44250	0.755672	0.377836	−	0.925873i	$-0.376668\pi$
$97$	7.44250	0.755672	0.377836	+	0.925873i	$0.376668\pi$
$98$	5.42156	0.547660
$99$	9.36507	0.941225
$100$	1.00000	0.100000
$101$	8.38819	0.834656	0.417328	−	0.908756i	$-0.362967\pi$
$101$	8.38819	0.834656	0.417328	+	0.908756i	$0.362967\pi$
$102$	−4.80965	−0.476226
$103$	3.74657	0.369160	0.184580	−	0.982817i	$-0.440907\pi$
$103$	3.74657	0.369160	0.184580	+	0.982817i	$0.440907\pi$
$104$	−1.00000	−0.0980581
$105$	1.53273	0.149579
$106$	−3.74006	−0.363267
$107$	17.4669	1.68859	0.844294	−	0.535881i	$-0.180020\pi$
$107$	17.4669	1.68859	0.844294	+	0.535881i	$0.180020\pi$
$108$	−5.50411	−0.529633
$109$	14.8858	1.42580	0.712898	−	0.701267i	$-0.247382\pi$
$109$	14.8858	1.42580	0.712898	+	0.701267i	$0.247382\pi$
$110$	−6.19520	−0.590689
$111$	12.3524	1.17244
$112$	−1.25636	−0.118715
$113$	12.4761	1.17365	0.586827	−	0.809712i	$-0.300377\pi$
$113$	12.4761	1.17365	0.586827	+	0.809712i	$0.300377\pi$
$114$	5.92457	0.554886
$115$	−8.13190	−0.758304
$116$	0.0979954	0.00909864
$117$	−1.51167	−0.139754
$118$	5.37745	0.495034
$119$	−4.95310	−0.454050
$120$	1.21997	0.111368
$121$	27.3805	2.48914
$122$	−11.3933	−1.03151
$123$	−4.52587	−0.408084
$124$	−1.00000	−0.0898027
$125$	−1.00000	−0.0894427
$126$	−1.89920	−0.169194
$127$	−6.91591	−0.613687	−0.306844	−	0.951760i	$-0.599273\pi$
$127$	−6.91591	−0.613687	−0.306844	+	0.951760i	$0.599273\pi$
$128$	−1.00000	−0.0883883
$129$	−13.5783	−1.19550
$130$	1.00000	0.0877058
$131$	9.20313	0.804081	0.402040	−	0.915622i	$-0.368301\pi$
$131$	9.20313	0.804081	0.402040	+	0.915622i	$0.368301\pi$
$132$	−7.55798	−0.657838
$133$	6.10127	0.529047
$134$	7.70302	0.665440
$135$	5.50411	0.473718
$136$	−3.94242	−0.338060
$137$	12.6850	1.08376	0.541878	−	0.840457i	$-0.317713\pi$
$137$	12.6850	1.08376	0.541878	+	0.840457i	$0.317713\pi$
$138$	−9.92071	−0.844507
$139$	−7.23601	−0.613751	−0.306875	−	0.951750i	$-0.599283\pi$
$139$	−7.23601	−0.613751	−0.306875	+	0.951750i	$0.599283\pi$
$140$	1.25636	0.106182
$141$	5.84669	0.492381
$142$	2.68875	0.225635
$143$	−6.19520	−0.518069
$144$	−1.51167	−0.125972
$145$	−0.0979954	−0.00813807
$146$	0.389941	0.0322718
$147$	−6.61416	−0.545527
$148$	10.1251	0.832280
$149$	−1.64471	−0.134740	−0.0673699	−	0.997728i	$-0.521461\pi$
$149$	−1.64471	−0.134740	−0.0673699	+	0.997728i	$0.521461\pi$
$150$	−1.21997	−0.0996104
$151$	10.4223	0.848154	0.424077	−	0.905626i	$-0.360599\pi$
$151$	10.4223	0.848154	0.424077	+	0.905626i	$0.360599\pi$
$152$	4.85631	0.393899
$153$	−5.95962	−0.481807
$154$	−7.78340	−0.627204
$155$	1.00000	0.0803219
$156$	1.21997	0.0976760
$157$	−17.5740	−1.40256	−0.701280	−	0.712886i	$-0.747388\pi$
$157$	−17.5740	−1.40256	−0.701280	+	0.712886i	$0.747388\pi$
$158$	−1.52451	−0.121284
$159$	4.56278	0.361852
$160$	1.00000	0.0790569
$161$	−10.2166	−0.805181
$162$	2.17987	0.171267
$163$	−22.5160	−1.76359	−0.881796	−	0.471631i	$-0.843666\pi$
$163$	−22.5160	−1.76359	−0.881796	+	0.471631i	$0.843666\pi$
$164$	−3.70981	−0.289688
$165$	7.55798	0.588388
$166$	−6.14297	−0.476787
$167$	16.3877	1.26812	0.634058	−	0.773286i	$-0.281388\pi$
$167$	16.3877	1.26812	0.634058	+	0.773286i	$0.281388\pi$
$168$	1.53273	0.118252
$169$	1.00000	0.0769231
$170$	3.94242	0.302370
$171$	7.34111	0.561389
$172$	−11.1300	−0.848655
$173$	−2.32408	−0.176697	−0.0883483	−	0.996090i	$-0.528159\pi$
$173$	−2.32408	−0.176697	−0.0883483	+	0.996090i	$0.528159\pi$
$174$	−0.119552	−0.00906319
$175$	−1.25636	−0.0949719
$176$	−6.19520	−0.466981
$177$	−6.56035	−0.493106
$178$	1.25689	0.0942082
$179$	16.3828	1.22451	0.612253	−	0.790662i	$-0.290263\pi$
$179$	16.3828	1.22451	0.612253	+	0.790662i	$0.290263\pi$
$180$	1.51167	0.112673
$181$	22.3405	1.66056	0.830279	−	0.557349i	$-0.188181\pi$
$181$	22.3405	1.66056	0.830279	+	0.557349i	$0.188181\pi$
$182$	1.25636	0.0931276
$183$	13.8996	1.02749
$184$	−8.13190	−0.599492
$185$	−10.1251	−0.744414
$186$	1.21997	0.0894528
$187$	−24.4241	−1.78607
$188$	4.79248	0.349527
$189$	6.91514	0.503003
$190$	−4.85631	−0.352314
$191$	4.11474	0.297732	0.148866	−	0.988857i	$-0.452438\pi$
$191$	4.11474	0.297732	0.148866	+	0.988857i	$0.452438\pi$
$192$	1.21997	0.0880440
$193$	4.93210	0.355021	0.177510	−	0.984119i	$-0.443196\pi$
$193$	4.93210	0.355021	0.177510	+	0.984119i	$0.443196\pi$
$194$	−7.44250	−0.534341
$195$	−1.21997	−0.0873641
$196$	−5.42156	−0.387254
$197$	25.3535	1.80636	0.903180	−	0.429263i	$-0.141227\pi$
$197$	25.3535	1.80636	0.903180	+	0.429263i	$0.141227\pi$
$198$	−9.36507	−0.665547
$199$	−3.47528	−0.246356	−0.123178	−	0.992385i	$-0.539309\pi$
$199$	−3.47528	−0.246356	−0.123178	+	0.992385i	$0.539309\pi$
$200$	−1.00000	−0.0707107
$201$	−9.39748	−0.662847
$202$	−8.38819	−0.590191
$203$	−0.123117	−0.00864115
$204$	4.80965	0.336743
$205$	3.70981	0.259104
$206$	−3.74657	−0.261036
$207$	−12.2927	−0.854403
$208$	1.00000	0.0693375
$209$	30.0858	2.08108
$210$	−1.53273	−0.105768
$211$	0.277907	0.0191319	0.00956594	−	0.999954i	$-0.496955\pi$
$211$	0.277907	0.0191319	0.00956594	+	0.999954i	$0.496955\pi$
$212$	3.74006	0.256868
$213$	−3.28020	−0.224756
$214$	−17.4669	−1.19401
$215$	11.1300	0.759061
$216$	5.50411	0.374507
$217$	1.25636	0.0852872
$218$	−14.8858	−1.00819
$219$	−0.475718	−0.0321460
$220$	6.19520	0.417680
$221$	3.94242	0.265196
$222$	−12.3524	−0.829037
$223$	10.1358	0.678744	0.339372	−	0.940652i	$-0.389785\pi$
$223$	10.1358	0.678744	0.339372	+	0.940652i	$0.389785\pi$
$224$	1.25636	0.0839441
$225$	−1.51167	−0.100778
$226$	−12.4761	−0.829899
$227$	−21.8342	−1.44919	−0.724595	−	0.689175i	$-0.757973\pi$
$227$	−21.8342	−1.44919	−0.724595	+	0.689175i	$0.757973\pi$
$228$	−5.92457	−0.392364
$229$	−19.6008	−1.29526	−0.647629	−	0.761956i	$-0.724239\pi$
$229$	−19.6008	−1.29526	−0.647629	+	0.761956i	$0.724239\pi$
$230$	8.13190	0.536202
$231$	9.49554	0.624761
$232$	−0.0979954	−0.00643371
$233$	−12.8088	−0.839131	−0.419566	−	0.907725i	$-0.637818\pi$
$233$	−12.8088	−0.839131	−0.419566	+	0.907725i	$0.637818\pi$
$234$	1.51167	0.0988207
$235$	−4.79248	−0.312627
$236$	−5.37745	−0.350042
$237$	1.85987	0.120811
$238$	4.95310	0.321062
$239$	21.4524	1.38764	0.693821	−	0.720147i	$-0.255926\pi$
$239$	21.4524	1.38764	0.693821	+	0.720147i	$0.255926\pi$
$240$	−1.21997	−0.0787489
$241$	24.1039	1.55267	0.776335	−	0.630320i	$-0.217076\pi$
$241$	24.1039	1.55267	0.776335	+	0.630320i	$0.217076\pi$
$242$	−27.3805	−1.76009
$243$	13.8529	0.888667
$244$	11.3933	0.729384
$245$	5.42156	0.346371
$246$	4.52587	0.288559
$247$	−4.85631	−0.308999
$248$	1.00000	0.0635001
$249$	7.49426	0.474929
$250$	1.00000	0.0632456
$251$	7.02455	0.443386	0.221693	−	0.975117i	$-0.428842\pi$
$251$	7.02455	0.443386	0.221693	+	0.975117i	$0.428842\pi$
$252$	1.89920	0.119638
$253$	−50.3788	−3.16729
$254$	6.91591	0.433943
$255$	−4.80965	−0.301192
$256$	1.00000	0.0625000
$257$	−12.9723	−0.809188	−0.404594	−	0.914496i	$-0.632587\pi$
$257$	−12.9723	−0.809188	−0.404594	+	0.914496i	$0.632587\pi$
$258$	13.5783	0.845349
$259$	−12.7208	−0.790431
$260$	−1.00000	−0.0620174
$261$	−0.148136	−0.00916940
$262$	−9.20313	−0.568571
$263$	−7.74683	−0.477690	−0.238845	−	0.971058i	$-0.576769\pi$
$263$	−7.74683	−0.477690	−0.238845	+	0.971058i	$0.576769\pi$
$264$	7.55798	0.465161
$265$	−3.74006	−0.229750
$266$	−6.10127	−0.374093
$267$	−1.53338	−0.0938411
$268$	−7.70302	−0.470537
$269$	−10.4704	−0.638390	−0.319195	−	0.947689i	$-0.603412\pi$
$269$	−10.4704	−0.638390	−0.319195	+	0.947689i	$0.603412\pi$
$270$	−5.50411	−0.334970
$271$	1.39488	0.0847331	0.0423666	−	0.999102i	$-0.486510\pi$
$271$	1.39488	0.0847331	0.0423666	+	0.999102i	$0.486510\pi$
$272$	3.94242	0.239044
$273$	−1.53273	−0.0927647
$274$	−12.6850	−0.766331
$275$	−6.19520	−0.373585
$276$	9.92071	0.597156
$277$	20.8948	1.25544	0.627722	−	0.778437i	$-0.283987\pi$
$277$	20.8948	1.25544	0.627722	+	0.778437i	$0.283987\pi$
$278$	7.23601	0.433987
$279$	1.51167	0.0905010
$280$	−1.25636	−0.0750818
$281$	−25.4341	−1.51727	−0.758636	−	0.651515i	$-0.774134\pi$
$281$	−25.4341	−1.51727	−0.758636	+	0.651515i	$0.774134\pi$
$282$	−5.84669	−0.348166
$283$	15.7494	0.936206	0.468103	−	0.883674i	$-0.344938\pi$
$283$	15.7494	0.936206	0.468103	+	0.883674i	$0.344938\pi$
$284$	−2.68875	−0.159548
$285$	5.92457	0.350941
$286$	6.19520	0.366330
$287$	4.66086	0.275122
$288$	1.51167	0.0890757
$289$	−1.45733	−0.0857253
$290$	0.0979954	0.00575449
$291$	9.07965	0.532259
$292$	−0.389941	−0.0228196
$293$	25.1432	1.46888	0.734442	−	0.678671i	$-0.237444\pi$
$293$	25.1432	1.46888	0.734442	+	0.678671i	$0.237444\pi$
$294$	6.61416	0.385746
$295$	5.37745	0.313087
$296$	−10.1251	−0.588511
$297$	34.0991	1.97863
$298$	1.64471	0.0952755
$299$	8.13190	0.470280
$300$	1.21997	0.0704352
$301$	13.9833	0.805984
$302$	−10.4223	−0.599735
$303$	10.2334	0.587891
$304$	−4.85631	−0.278528
$305$	−11.3933	−0.652381
$306$	5.95962	0.340689
$307$	−28.0764	−1.60241	−0.801203	−	0.598392i	$-0.795806\pi$
$307$	−28.0764	−1.60241	−0.801203	+	0.598392i	$0.795806\pi$
$308$	7.78340	0.443500
$309$	4.57071	0.260019
$310$	−1.00000	−0.0567962
$311$	−11.7855	−0.668294	−0.334147	−	0.942521i	$-0.608448\pi$
$311$	−11.7855	−0.668294	−0.334147	+	0.942521i	$0.608448\pi$
$312$	−1.21997	−0.0690674
$313$	−20.3042	−1.14766	−0.573831	−	0.818974i	$-0.694543\pi$
$313$	−20.3042	−1.14766	−0.573831	+	0.818974i	$0.694543\pi$
$314$	17.5740	0.991760
$315$	−1.89920	−0.107008
$316$	1.52451	0.0857606
$317$	2.87817	0.161654	0.0808270	−	0.996728i	$-0.474244\pi$
$317$	2.87817	0.161654	0.0808270	+	0.996728i	$0.474244\pi$
$318$	−4.56278	−0.255868
$319$	−0.607101	−0.0339911
$320$	−1.00000	−0.0559017
$321$	21.3091	1.18936
$322$	10.2166	0.569349
$323$	−19.1456	−1.06529
$324$	−2.17987	−0.121104
$325$	1.00000	0.0554700
$326$	22.5160	1.24705
$327$	18.1602	1.00426
$328$	3.70981	0.204840
$329$	−6.02108	−0.331953
$330$	−7.55798	−0.416053
$331$	−19.4222	−1.06754	−0.533770	−	0.845630i	$-0.679225\pi$
$331$	−19.4222	−1.06754	−0.533770	+	0.845630i	$0.679225\pi$
$332$	6.14297	0.337139
$333$	−15.3058	−0.838752
$334$	−16.3877	−0.896693
$335$	7.70302	0.420861
$336$	−1.53273	−0.0836170
$337$	12.0215	0.654852	0.327426	−	0.944877i	$-0.393819\pi$
$337$	12.0215	0.654852	0.327426	+	0.944877i	$0.393819\pi$
$338$	−1.00000	−0.0543928
$339$	15.2205	0.826665
$340$	−3.94242	−0.213808
$341$	6.19520	0.335489
$342$	−7.34111	−0.396962
$343$	15.6059	0.842642
$344$	11.1300	0.600090
$345$	−9.92071	−0.534113
$346$	2.32408	0.124943
$347$	−0.399468	−0.0214446	−0.0107223	−	0.999943i	$-0.503413\pi$
$347$	−0.399468	−0.0214446	−0.0107223	+	0.999943i	$0.503413\pi$
$348$	0.119552	0.00640865
$349$	−18.1756	−0.972917	−0.486459	−	0.873704i	$-0.661712\pi$
$349$	−18.1756	−0.972917	−0.486459	+	0.873704i	$0.661712\pi$
$350$	1.25636	0.0671552
$351$	−5.50411	−0.293788
$352$	6.19520	0.330205
$353$	12.5752	0.669312	0.334656	−	0.942340i	$-0.391380\pi$
$353$	12.5752	0.669312	0.334656	+	0.942340i	$0.391380\pi$
$354$	6.56035	0.348678
$355$	2.68875	0.142704
$356$	−1.25689	−0.0666152
$357$	−6.04264	−0.319811
$358$	−16.3828	−0.865856
$359$	12.2589	0.646999	0.323499	−	0.946228i	$-0.395141\pi$
$359$	12.2589	0.646999	0.323499	+	0.946228i	$0.395141\pi$
$360$	−1.51167	−0.0796718
$361$	4.58373	0.241249
$362$	−22.3405	−1.17419
$363$	33.4035	1.75323
$364$	−1.25636	−0.0658511
$365$	0.389941	0.0204105
$366$	−13.8996	−0.726543
$367$	−21.0517	−1.09889	−0.549444	−	0.835530i	$-0.685160\pi$
$367$	−21.0517	−1.09889	−0.549444	+	0.835530i	$0.685160\pi$
$368$	8.13190	0.423905
$369$	5.60799	0.291940
$370$	10.1251	0.526380
$371$	−4.69886	−0.243953
$372$	−1.21997	−0.0632527
$373$	−2.27363	−0.117724	−0.0588620	−	0.998266i	$-0.518747\pi$
$373$	−2.27363	−0.117724	−0.0588620	+	0.998266i	$0.518747\pi$
$374$	24.4241	1.26294
$375$	−1.21997	−0.0629991
$376$	−4.79248	−0.247153
$377$	0.0979954	0.00504702
$378$	−6.91514	−0.355677
$379$	−20.1364	−1.03434	−0.517169	−	0.855883i	$-0.673014\pi$
$379$	−20.1364	−1.03434	−0.517169	+	0.855883i	$0.673014\pi$
$380$	4.85631	0.249123
$381$	−8.43722	−0.432252
$382$	−4.11474	−0.210529
$383$	30.7678	1.57216	0.786081	−	0.618123i	$-0.212107\pi$
$383$	30.7678	1.57216	0.786081	+	0.618123i	$0.212107\pi$
$384$	−1.21997	−0.0622565
$385$	−7.78340	−0.396679
$386$	−4.93210	−0.251037
$387$	16.8249	0.855255
$388$	7.44250	0.377836
$389$	−12.8122	−0.649606	−0.324803	−	0.945782i	$-0.605298\pi$
$389$	−12.8122	−0.649606	−0.324803	+	0.945782i	$0.605298\pi$
$390$	1.21997	0.0617757
$391$	32.0594	1.62131
$392$	5.42156	0.273830
$393$	11.2276	0.566356
$394$	−25.3535	−1.27729
$395$	−1.52451	−0.0767067
$396$	9.36507	0.470612
$397$	−17.8956	−0.898154	−0.449077	−	0.893493i	$-0.648247\pi$
$397$	−17.8956	−0.898154	−0.449077	+	0.893493i	$0.648247\pi$
$398$	3.47528	0.174200
$399$	7.44339	0.372635
$400$	1.00000	0.0500000
$401$	29.9697	1.49662	0.748309	−	0.663351i	$-0.230866\pi$
$401$	29.9697	1.49662	0.748309	+	0.663351i	$0.230866\pi$
$402$	9.39748	0.468704
$403$	−1.00000	−0.0498135
$404$	8.38819	0.417328
$405$	2.17987	0.108319
$406$	0.123117	0.00611022
$407$	−62.7271	−3.10927
$408$	−4.80965	−0.238113
$409$	−5.37068	−0.265563	−0.132782	−	0.991145i	$-0.542391\pi$
$409$	−5.37068	−0.265563	−0.132782	+	0.991145i	$0.542391\pi$
$410$	−3.70981	−0.183215
$411$	15.4754	0.763346
$412$	3.74657	0.184580
$413$	6.75601	0.332442
$414$	12.2927	0.604154
$415$	−6.14297	−0.301547
$416$	−1.00000	−0.0490290
$417$	−8.82774	−0.432296
$418$	−30.0858	−1.47154
$419$	4.75964	0.232524	0.116262	−	0.993219i	$-0.462909\pi$
$419$	4.75964	0.232524	0.116262	+	0.993219i	$0.462909\pi$
$420$	1.53273	0.0747893
$421$	−32.9054	−1.60371	−0.801855	−	0.597518i	$-0.796154\pi$
$421$	−32.9054	−1.60371	−0.801855	+	0.597518i	$0.796154\pi$
$422$	−0.277907	−0.0135283
$423$	−7.24462	−0.352246
$424$	−3.74006	−0.181633
$425$	3.94242	0.191235
$426$	3.28020	0.158926
$427$	−14.3141	−0.692710
$428$	17.4669	0.844294
$429$	−7.55798	−0.364903
$430$	−11.1300	−0.536737
$431$	22.1126	1.06513	0.532564	−	0.846390i	$-0.321229\pi$
$431$	22.1126	1.06513	0.532564	+	0.846390i	$0.321229\pi$
$432$	−5.50411	−0.264817
$433$	−24.6035	−1.18237	−0.591183	−	0.806537i	$-0.701339\pi$
$433$	−24.6035	−1.18237	−0.591183	+	0.806537i	$0.701339\pi$
$434$	−1.25636	−0.0603072
$435$	−0.119552	−0.00573207
$436$	14.8858	0.712898
$437$	−39.4910	−1.88911
$438$	0.475718	0.0227307
$439$	3.93282	0.187703	0.0938517	−	0.995586i	$-0.470082\pi$
$439$	3.93282	0.187703	0.0938517	+	0.995586i	$0.470082\pi$
$440$	−6.19520	−0.295345
$441$	8.19559	0.390266
$442$	−3.94242	−0.187522
$443$	−6.15845	−0.292597	−0.146298	−	0.989241i	$-0.546736\pi$
$443$	−6.15845	−0.292597	−0.146298	+	0.989241i	$0.546736\pi$
$444$	12.3524	0.586218
$445$	1.25689	0.0595825
$446$	−10.1358	−0.479945
$447$	−2.00650	−0.0949043
$448$	−1.25636	−0.0593574
$449$	8.74734	0.412812	0.206406	−	0.978466i	$-0.433823\pi$
$449$	8.74734	0.412812	0.206406	+	0.978466i	$0.433823\pi$
$450$	1.51167	0.0712606
$451$	22.9830	1.08223
$452$	12.4761	0.586827
$453$	12.7149	0.597399
$454$	21.8342	1.02473
$455$	1.25636	0.0588990
$456$	5.92457	0.277443
$457$	−5.32930	−0.249294	−0.124647	−	0.992201i	$-0.539780\pi$
$457$	−5.32930	−0.249294	−0.124647	+	0.992201i	$0.539780\pi$
$458$	19.6008	0.915885
$459$	−21.6995	−1.01285
$460$	−8.13190	−0.379152
$461$	−9.49165	−0.442070	−0.221035	−	0.975266i	$-0.570944\pi$
$461$	−9.49165	−0.442070	−0.221035	+	0.975266i	$0.570944\pi$
$462$	−9.49554	−0.441772
$463$	35.7240	1.66024	0.830118	−	0.557587i	$-0.188273\pi$
$463$	35.7240	1.66024	0.830118	+	0.557587i	$0.188273\pi$
$464$	0.0979954	0.00454932
$465$	1.21997	0.0565749
$466$	12.8088	0.593355
$467$	−8.31585	−0.384812	−0.192406	−	0.981315i	$-0.561629\pi$
$467$	−8.31585	−0.384812	−0.192406	+	0.981315i	$0.561629\pi$
$468$	−1.51167	−0.0698768
$469$	9.67777	0.446878
$470$	4.79248	0.221061
$471$	−21.4399	−0.987896
$472$	5.37745	0.247517
$473$	68.9527	3.17045
$474$	−1.85987	−0.0854265
$475$	−4.85631	−0.222823
$476$	−4.95310	−0.227025
$477$	−5.65372	−0.258866
$478$	−21.4524	−0.981211
$479$	−11.8759	−0.542625	−0.271312	−	0.962491i	$-0.587458\pi$
$479$	−11.8759	−0.542625	−0.271312	+	0.962491i	$0.587458\pi$
$480$	1.21997	0.0556839
$481$	10.1251	0.461666
$482$	−24.1039	−1.09790
$483$	−12.4640	−0.567130
$484$	27.3805	1.24457
$485$	−7.44250	−0.337947
$486$	−13.8529	−0.628382
$487$	28.5070	1.29177	0.645887	−	0.763433i	$-0.276488\pi$
$487$	28.5070	1.29177	0.645887	+	0.763433i	$0.276488\pi$
$488$	−11.3933	−0.515753
$489$	−27.4690	−1.24219
$490$	−5.42156	−0.244921
$491$	11.2618	0.508236	0.254118	−	0.967173i	$-0.418215\pi$
$491$	11.2618	0.508236	0.254118	+	0.967173i	$0.418215\pi$
$492$	−4.52587	−0.204042
$493$	0.386339	0.0173998
$494$	4.85631	0.218496
$495$	−9.36507	−0.420929
$496$	−1.00000	−0.0449013
$497$	3.37804	0.151526
$498$	−7.49426	−0.335826
$499$	1.70633	0.0763860	0.0381930	−	0.999270i	$-0.487840\pi$
$499$	1.70633	0.0763860	0.0381930	+	0.999270i	$0.487840\pi$
$500$	−1.00000	−0.0447214
$501$	19.9925	0.893199
$502$	−7.02455	−0.313521
$503$	17.4287	0.777105	0.388553	−	0.921427i	$-0.372975\pi$
$503$	17.4287	0.777105	0.388553	+	0.921427i	$0.372975\pi$
$504$	−1.89920	−0.0845969
$505$	−8.38819	−0.373269
$506$	50.3788	2.23961
$507$	1.21997	0.0541809
$508$	−6.91591	−0.306844
$509$	−6.28671	−0.278653	−0.139327	−	0.990246i	$-0.544494\pi$
$509$	−6.28671	−0.278653	−0.139327	+	0.990246i	$0.544494\pi$
$510$	4.80965	0.212975
$511$	0.489907	0.0216722
$512$	−1.00000	−0.0441942
$513$	26.7297	1.18014
$514$	12.9723	0.572182
$515$	−3.74657	−0.165093
$516$	−13.5783	−0.597752
$517$	−29.6904	−1.30578
$518$	12.7208	0.558919
$519$	−2.83532	−0.124457
$520$	1.00000	0.0438529
$521$	29.1450	1.27687	0.638433	−	0.769678i	$-0.279583\pi$
$521$	29.1450	1.27687	0.638433	+	0.769678i	$0.279583\pi$
$522$	0.148136	0.00648375
$523$	41.9384	1.83384	0.916920	−	0.399072i	$-0.130668\pi$
$523$	41.9384	1.83384	0.916920	+	0.399072i	$0.130668\pi$
$524$	9.20313	0.402040
$525$	−1.53273	−0.0668936
$526$	7.74683	0.337778
$527$	−3.94242	−0.171734
$528$	−7.55798	−0.328919
$529$	43.1279	1.87512
$530$	3.74006	0.162458
$531$	8.12891	0.352764
$532$	6.10127	0.264524
$533$	−3.70981	−0.160690
$534$	1.53338	0.0663557
$535$	−17.4669	−0.755159
$536$	7.70302	0.332720
$537$	19.9865	0.862483
$538$	10.4704	0.451410
$539$	33.5877	1.44672
$540$	5.50411	0.236859
$541$	29.9619	1.28816	0.644081	−	0.764957i	$-0.277240\pi$
$541$	29.9619	1.28816	0.644081	+	0.764957i	$0.277240\pi$
$542$	−1.39488	−0.0599154
$543$	27.2548	1.16962
$544$	−3.94242	−0.169030
$545$	−14.8858	−0.637636
$546$	1.53273	0.0655946
$547$	−32.2205	−1.37765	−0.688825	−	0.724928i	$-0.741873\pi$
$547$	−32.2205	−1.37765	−0.688825	+	0.724928i	$0.741873\pi$
$548$	12.6850	0.541878
$549$	−17.2229	−0.735057
$550$	6.19520	0.264164
$551$	−0.475896	−0.0202738
$552$	−9.92071	−0.422253
$553$	−1.91534	−0.0814485
$554$	−20.8948	−0.887733
$555$	−12.3524	−0.524329
$556$	−7.23601	−0.306875
$557$	−43.1338	−1.82764	−0.913819	−	0.406122i	$-0.866881\pi$
$557$	−43.1338	−1.82764	−0.913819	+	0.406122i	$0.866881\pi$
$558$	−1.51167	−0.0639939
$559$	−11.1300	−0.470749
$560$	1.25636	0.0530909
$561$	−29.7967	−1.25802
$562$	25.4341	1.07287
$563$	5.21783	0.219905	0.109953	−	0.993937i	$-0.464930\pi$
$563$	5.21783	0.219905	0.109953	+	0.993937i	$0.464930\pi$
$564$	5.84669	0.246190
$565$	−12.4761	−0.524874
$566$	−15.7494	−0.661997
$567$	2.73870	0.115015
$568$	2.68875	0.112818
$569$	−27.2589	−1.14275	−0.571376	−	0.820688i	$-0.693590\pi$
$569$	−27.2589	−1.14275	−0.571376	+	0.820688i	$0.693590\pi$
$570$	−5.92457	−0.248153
$571$	26.1074	1.09256	0.546280	−	0.837603i	$-0.316044\pi$
$571$	26.1074	1.09256	0.546280	+	0.837603i	$0.316044\pi$
$572$	−6.19520	−0.259034
$573$	5.01988	0.209708
$574$	−4.66086	−0.194540
$575$	8.13190	0.339124
$576$	−1.51167	−0.0629861
$577$	39.8441	1.65873	0.829366	−	0.558705i	$-0.188702\pi$
$577$	39.8441	1.65873	0.829366	+	0.558705i	$0.188702\pi$
$578$	1.45733	0.0606169
$579$	6.01703	0.250059
$580$	−0.0979954	−0.00406904
$581$	−7.71778	−0.320187
$582$	−9.07965	−0.376364
$583$	−23.1704	−0.959621
$584$	0.389941	0.0161359
$585$	1.51167	0.0624997
$586$	−25.1432	−1.03866
$587$	46.3225	1.91194	0.955968	−	0.293472i	$-0.0948108\pi$
$587$	46.3225	1.91194	0.955968	+	0.293472i	$0.0948108\pi$
$588$	−6.61416	−0.272763
$589$	4.85631	0.200101
$590$	−5.37745	−0.221386
$591$	30.9305	1.27231
$592$	10.1251	0.416140
$593$	−30.8985	−1.26885	−0.634426	−	0.772984i	$-0.718763\pi$
$593$	−30.8985	−1.26885	−0.634426	+	0.772984i	$0.718763\pi$
$594$	−34.0991	−1.39910
$595$	4.95310	0.203057
$596$	−1.64471	−0.0673699
$597$	−4.23975	−0.173522
$598$	−8.13190	−0.332538
$599$	5.46911	0.223462	0.111731	−	0.993739i	$-0.464361\pi$
$599$	5.46911	0.223462	0.111731	+	0.993739i	$0.464361\pi$
$600$	−1.21997	−0.0498052
$601$	−41.6901	−1.70057	−0.850287	−	0.526320i	$-0.823572\pi$
$601$	−41.6901	−1.70057	−0.850287	+	0.526320i	$0.823572\pi$
$602$	−13.9833	−0.569917
$603$	11.6444	0.474196
$604$	10.4223	0.424077
$605$	−27.3805	−1.11318
$606$	−10.2334	−0.415702
$607$	−22.5794	−0.916468	−0.458234	−	0.888832i	$-0.651518\pi$
$607$	−22.5794	−0.916468	−0.458234	+	0.888832i	$0.651518\pi$
$608$	4.85631	0.196949
$609$	−0.150200	−0.00608641
$610$	11.3933	0.461303
$611$	4.79248	0.193883
$612$	−5.95962	−0.240903
$613$	30.7692	1.24276	0.621378	−	0.783511i	$-0.286573\pi$
$613$	30.7692	1.24276	0.621378	+	0.783511i	$0.286573\pi$
$614$	28.0764	1.13307
$615$	4.52587	0.182501
$616$	−7.78340	−0.313602
$617$	7.03472	0.283207	0.141604	−	0.989923i	$-0.454774\pi$
$617$	7.03472	0.283207	0.141604	+	0.989923i	$0.454774\pi$
$618$	−4.57071	−0.183861
$619$	13.3944	0.538365	0.269182	−	0.963089i	$-0.413247\pi$
$619$	13.3944	0.538365	0.269182	+	0.963089i	$0.413247\pi$
$620$	1.00000	0.0401610
$621$	−44.7589	−1.79611
$622$	11.7855	0.472555
$623$	1.57911	0.0632657
$624$	1.21997	0.0488380
$625$	1.00000	0.0400000
$626$	20.3042	0.811520
$627$	36.7039	1.46581
$628$	−17.5740	−0.701280
$629$	39.9175	1.59161
$630$	1.89920	0.0756658
$631$	−6.38617	−0.254230	−0.127115	−	0.991888i	$-0.540572\pi$
$631$	−6.38617	−0.254230	−0.127115	+	0.991888i	$0.540572\pi$
$632$	−1.52451	−0.0606419
$633$	0.339039	0.0134756
$634$	−2.87817	−0.114307
$635$	6.91591	0.274449
$636$	4.56278	0.180926
$637$	−5.42156	−0.214810
$638$	0.607101	0.0240354
$639$	4.06449	0.160789
$640$	1.00000	0.0395285
$641$	−10.0299	−0.396156	−0.198078	−	0.980186i	$-0.563470\pi$
$641$	−10.0299	−0.396156	−0.198078	+	0.980186i	$0.563470\pi$
$642$	−21.3091	−0.841004
$643$	−28.9179	−1.14041	−0.570204	−	0.821503i	$-0.693136\pi$
$643$	−28.9179	−1.14041	−0.570204	+	0.821503i	$0.693136\pi$
$644$	−10.2166	−0.402590
$645$	13.5783	0.534646
$646$	19.1456	0.753274
$647$	−31.0042	−1.21890	−0.609451	−	0.792824i	$-0.708610\pi$
$647$	−31.0042	−1.21890	−0.609451	+	0.792824i	$0.708610\pi$
$648$	2.17987	0.0856334
$649$	33.3144	1.30770
$650$	−1.00000	−0.0392232
$651$	1.53273	0.0600722
$652$	−22.5160	−0.881796
$653$	−39.6812	−1.55285	−0.776424	−	0.630211i	$-0.782968\pi$
$653$	−39.6812	−1.55285	−0.776424	+	0.630211i	$0.782968\pi$
$654$	−18.1602	−0.710121
$655$	−9.20313	−0.359596
$656$	−3.70981	−0.144844
$657$	0.589461	0.0229971
$658$	6.02108	0.234726
$659$	−18.6436	−0.726251	−0.363126	−	0.931740i	$-0.618290\pi$
$659$	−18.6436	−0.726251	−0.363126	+	0.931740i	$0.618290\pi$
$660$	7.55798	0.294194
$661$	−26.8993	−1.04626	−0.523131	−	0.852252i	$-0.675236\pi$
$661$	−26.8993	−1.04626	−0.523131	+	0.852252i	$0.675236\pi$
$662$	19.4222	0.754864
$663$	4.80965	0.186791
$664$	−6.14297	−0.238394
$665$	−6.10127	−0.236597
$666$	15.3058	0.593087
$667$	0.796889	0.0308557
$668$	16.3877	0.634058
$669$	12.3654	0.478075
$670$	−7.70302	−0.297594
$671$	−70.5841	−2.72487
$672$	1.53273	0.0591262
$673$	37.0415	1.42784	0.713922	−	0.700225i	$-0.246917\pi$
$673$	37.0415	1.42784	0.713922	+	0.700225i	$0.246917\pi$
$674$	−12.0215	−0.463050
$675$	−5.50411	−0.211853
$676$	1.00000	0.0384615
$677$	6.87376	0.264180	0.132090	−	0.991238i	$-0.457831\pi$
$677$	6.87376	0.264180	0.132090	+	0.991238i	$0.457831\pi$
$678$	−15.2205	−0.584541
$679$	−9.35046	−0.358838
$680$	3.94242	0.151185
$681$	−26.6372	−1.02074
$682$	−6.19520	−0.237226
$683$	17.5904	0.673077	0.336539	−	0.941670i	$-0.390744\pi$
$683$	17.5904	0.673077	0.336539	+	0.941670i	$0.390744\pi$
$684$	7.34111	0.280694
$685$	−12.6850	−0.484670
$686$	−15.6059	−0.595838
$687$	−23.9124	−0.912317
$688$	−11.1300	−0.424328
$689$	3.74006	0.142485
$690$	9.92071	0.377675
$691$	−0.443152	−0.0168583	−0.00842914	−	0.999964i	$-0.502683\pi$
$691$	−0.443152	−0.0168583	−0.00842914	+	0.999964i	$0.502683\pi$
$692$	−2.32408	−0.0883483
$693$	−11.7659	−0.446949
$694$	0.399468	0.0151636
$695$	7.23601	0.274478
$696$	−0.119552	−0.00453160
$697$	−14.6256	−0.553985
$698$	18.1756	0.687956
$699$	−15.6264	−0.591044
$700$	−1.25636	−0.0474859
$701$	−8.81384	−0.332894	−0.166447	−	0.986050i	$-0.553229\pi$
$701$	−8.81384	−0.332894	−0.166447	+	0.986050i	$0.553229\pi$
$702$	5.50411	0.207739
$703$	−49.1707	−1.85451
$704$	−6.19520	−0.233490
$705$	−5.84669	−0.220199
$706$	−12.5752	−0.473275
$707$	−10.5386	−0.396344
$708$	−6.56035	−0.246553
$709$	−50.6941	−1.90386	−0.951928	−	0.306322i	$-0.900901\pi$
$709$	−50.6941	−1.90386	−0.951928	+	0.306322i	$0.900901\pi$
$710$	−2.68875	−0.100907
$711$	−2.30456	−0.0864276
$712$	1.25689	0.0471041
$713$	−8.13190	−0.304542
$714$	6.04264	0.226140
$715$	6.19520	0.231687
$716$	16.3828	0.612253
$717$	26.1714	0.977388
$718$	−12.2589	−0.457497
$719$	−0.258610	−0.00964454	−0.00482227	−	0.999988i	$-0.501535\pi$
$719$	−0.258610	−0.00964454	−0.00482227	+	0.999988i	$0.501535\pi$
$720$	1.51167	0.0563364
$721$	−4.70704	−0.175299
$722$	−4.58373	−0.170589
$723$	29.4061	1.09363
$724$	22.3405	0.830279
$725$	0.0979954	0.00363946
$726$	−33.4035	−1.23972
$727$	11.8164	0.438244	0.219122	−	0.975697i	$-0.429681\pi$
$727$	11.8164	0.438244	0.219122	+	0.975697i	$0.429681\pi$
$728$	1.25636	0.0465638
$729$	23.4398	0.868142
$730$	−0.389941	−0.0144324
$731$	−43.8792	−1.62293
$732$	13.8996	0.513743
$733$	24.1046	0.890324	0.445162	−	0.895450i	$-0.353146\pi$
$733$	24.1046	0.890324	0.445162	+	0.895450i	$0.353146\pi$
$734$	21.0517	0.777031
$735$	6.61416	0.243967
$736$	−8.13190	−0.299746
$737$	47.7218	1.75785
$738$	−5.60799	−0.206433
$739$	11.3058	0.415889	0.207945	−	0.978141i	$-0.433323\pi$
$739$	11.3058	0.415889	0.207945	+	0.978141i	$0.433323\pi$
$740$	−10.1251	−0.372207
$741$	−5.92457	−0.217644
$742$	4.69886	0.172501
$743$	41.3875	1.51836	0.759181	−	0.650880i	$-0.225600\pi$
$743$	41.3875	1.51836	0.759181	+	0.650880i	$0.225600\pi$
$744$	1.21997	0.0447264
$745$	1.64471	0.0602575
$746$	2.27363	0.0832434
$747$	−9.28612	−0.339761
$748$	−24.4241	−0.893033
$749$	−21.9447	−0.801841
$750$	1.21997	0.0445471
$751$	49.7494	1.81538	0.907691	−	0.419639i	$-0.137843\pi$
$751$	49.7494	1.81538	0.907691	+	0.419639i	$0.137843\pi$
$752$	4.79248	0.174764
$753$	8.56977	0.312300
$754$	−0.0979954	−0.00356878
$755$	−10.4223	−0.379306
$756$	6.91514	0.251501
$757$	43.5572	1.58311	0.791556	−	0.611097i	$-0.209271\pi$
$757$	43.5572	1.58311	0.791556	+	0.611097i	$0.209271\pi$
$758$	20.1364	0.731387
$759$	−61.4608	−2.23088
$760$	−4.85631	−0.176157
$761$	23.7787	0.861976	0.430988	−	0.902358i	$-0.358165\pi$
$761$	23.7787	0.861976	0.430988	+	0.902358i	$0.358165\pi$
$762$	8.43722	0.305648
$763$	−18.7019	−0.677053
$764$	4.11474	0.148866
$765$	5.95962	0.215470
$766$	−30.7678	−1.11169
$767$	−5.37745	−0.194168
$768$	1.21997	0.0440220
$769$	22.4435	0.809333	0.404666	−	0.914464i	$-0.367388\pi$
$769$	22.4435	0.809333	0.404666	+	0.914464i	$0.367388\pi$
$770$	7.78340	0.280494
$771$	−15.8258	−0.569953
$772$	4.93210	0.177510
$773$	15.9143	0.572399	0.286200	−	0.958170i	$-0.407608\pi$
$773$	15.9143	0.572399	0.286200	+	0.958170i	$0.407608\pi$
$774$	−16.8249	−0.604757
$775$	−1.00000	−0.0359211
$776$	−7.44250	−0.267170
$777$	−15.5190	−0.556742
$778$	12.8122	0.459341
$779$	18.0160	0.645490
$780$	−1.21997	−0.0436820
$781$	16.6574	0.596047
$782$	−32.0594	−1.14644
$783$	−0.539377	−0.0192758
$784$	−5.42156	−0.193627
$785$	17.5740	0.627244
$786$	−11.2276	−0.400474
$787$	28.3980	1.01228	0.506139	−	0.862452i	$-0.331072\pi$
$787$	28.3980	1.01228	0.506139	+	0.862452i	$0.331072\pi$
$788$	25.3535	0.903180
$789$	−9.45092	−0.336462
$790$	1.52451	0.0542398
$791$	−15.6745	−0.557320
$792$	−9.36507	−0.332773
$793$	11.3933	0.404590
$794$	17.8956	0.635091
$795$	−4.56278	−0.161825
$796$	−3.47528	−0.123178
$797$	−11.5766	−0.410065	−0.205033	−	0.978755i	$-0.565730\pi$
$797$	−11.5766	−0.410065	−0.205033	+	0.978755i	$0.565730\pi$
$798$	−7.44339	−0.263493
$799$	18.8940	0.668420
$800$	−1.00000	−0.0353553
$801$	1.90000	0.0671333
$802$	−29.9697	−1.05827
$803$	2.41577	0.0852505
$804$	−9.39748	−0.331424
$805$	10.2166	0.360088
$806$	1.00000	0.0352235
$807$	−12.7736	−0.449651
$808$	−8.38819	−0.295095
$809$	35.0074	1.23080	0.615398	−	0.788217i	$-0.288995\pi$
$809$	35.0074	1.23080	0.615398	+	0.788217i	$0.288995\pi$
$810$	−2.17987	−0.0765929
$811$	8.54016	0.299886	0.149943	−	0.988695i	$-0.452091\pi$
$811$	8.54016	0.299886	0.149943	+	0.988695i	$0.452091\pi$
$812$	−0.123117	−0.00432058
$813$	1.70172	0.0596819
$814$	62.7271	2.19859
$815$	22.5160	0.788702
$816$	4.80965	0.168371
$817$	54.0508	1.89100
$818$	5.37068	0.187782
$819$	1.89920	0.0663633
$820$	3.70981	0.129552
$821$	18.3125	0.639110	0.319555	−	0.947568i	$-0.396467\pi$
$821$	18.3125	0.639110	0.319555	+	0.947568i	$0.396467\pi$
$822$	−15.4754	−0.539767
$823$	−48.2494	−1.68187	−0.840934	−	0.541138i	$-0.817994\pi$
$823$	−48.2494	−1.68187	−0.840934	+	0.541138i	$0.817994\pi$
$824$	−3.74657	−0.130518
$825$	−7.55798	−0.263135
$826$	−6.75601	−0.235072
$827$	−26.9688	−0.937799	−0.468899	−	0.883252i	$-0.655349\pi$
$827$	−26.9688	−0.937799	−0.468899	+	0.883252i	$0.655349\pi$
$828$	−12.2927	−0.427201
$829$	−10.2040	−0.354399	−0.177200	−	0.984175i	$-0.556704\pi$
$829$	−10.2040	−0.354399	−0.177200	+	0.984175i	$0.556704\pi$
$830$	6.14297	0.213226
$831$	25.4911	0.884275
$832$	1.00000	0.0346688
$833$	−21.3741	−0.740567
$834$	8.82774	0.305680
$835$	−16.3877	−0.567118
$836$	30.0858	1.04054
$837$	5.50411	0.190250
$838$	−4.75964	−0.164419
$839$	−1.66489	−0.0574782	−0.0287391	−	0.999587i	$-0.509149\pi$
$839$	−1.66489	−0.0574782	−0.0287391	+	0.999587i	$0.509149\pi$
$840$	−1.53273	−0.0528840
$841$	−28.9904	−0.999669
$842$	32.9054	1.13399
$843$	−31.0289	−1.06869
$844$	0.277907	0.00956594
$845$	−1.00000	−0.0344010
$846$	7.24462	0.249075
$847$	−34.3998	−1.18199
$848$	3.74006	0.128434
$849$	19.2139	0.659418
$850$	−3.94242	−0.135224
$851$	82.3365	2.82246
$852$	−3.28020	−0.112378
$853$	−1.31607	−0.0450613	−0.0225306	−	0.999746i	$-0.507172\pi$
$853$	−1.31607	−0.0450613	−0.0225306	+	0.999746i	$0.507172\pi$
$854$	14.3141	0.489820
$855$	−7.34111	−0.251061
$856$	−17.4669	−0.597006
$857$	8.30555	0.283712	0.141856	−	0.989887i	$-0.454693\pi$
$857$	8.30555	0.283712	0.141856	+	0.989887i	$0.454693\pi$
$858$	7.55798	0.258025
$859$	−13.7885	−0.470458	−0.235229	−	0.971940i	$-0.575584\pi$
$859$	−13.7885	−0.470458	−0.235229	+	0.971940i	$0.575584\pi$
$860$	11.1300	0.379530
$861$	5.68612	0.193782
$862$	−22.1126	−0.753159
$863$	−18.9666	−0.645630	−0.322815	−	0.946462i	$-0.604629\pi$
$863$	−18.9666	−0.645630	−0.322815	+	0.946462i	$0.604629\pi$
$864$	5.50411	0.187254
$865$	2.32408	0.0790211
$866$	24.6035	0.836059
$867$	−1.77790	−0.0603808
$868$	1.25636	0.0426436
$869$	−9.44467	−0.320389
$870$	0.119552	0.00405318
$871$	−7.70302	−0.261007
$872$	−14.8858	−0.504095
$873$	−11.2506	−0.380774
$874$	39.4910	1.33580
$875$	1.25636	0.0424727
$876$	−0.475718	−0.0160730
$877$	−46.8828	−1.58312	−0.791561	−	0.611091i	$-0.790731\pi$
$877$	−46.8828	−1.58312	−0.791561	+	0.611091i	$0.790731\pi$
$878$	−3.93282	−0.132726
$879$	30.6741	1.03461
$880$	6.19520	0.208840
$881$	26.4927	0.892561	0.446280	−	0.894893i	$-0.352748\pi$
$881$	26.4927	0.892561	0.446280	+	0.894893i	$0.352748\pi$
$882$	−8.19559	−0.275960
$883$	26.8894	0.904900	0.452450	−	0.891790i	$-0.350550\pi$
$883$	26.8894	0.904900	0.452450	+	0.891790i	$0.350550\pi$
$884$	3.94242	0.132598
$885$	6.56035	0.220524
$886$	6.15845	0.206897
$887$	−39.2927	−1.31932	−0.659660	−	0.751564i	$-0.729300\pi$
$887$	−39.2927	−1.31932	−0.659660	+	0.751564i	$0.729300\pi$
$888$	−12.3524	−0.414519
$889$	8.68887	0.291415
$890$	−1.25689	−0.0421312
$891$	13.5047	0.452426
$892$	10.1358	0.339372
$893$	−23.2738	−0.778826
$894$	2.00650	0.0671075
$895$	−16.3828	−0.547616
$896$	1.25636	0.0419720
$897$	9.92071	0.331243
$898$	−8.74734	−0.291902
$899$	−0.0979954	−0.00326833
$900$	−1.51167	−0.0503888
$901$	14.7449	0.491224
$902$	−22.9830	−0.765251
$903$	17.0592	0.567696
$904$	−12.4761	−0.414949
$905$	−22.3405	−0.742624
$906$	−12.7149	−0.422425
$907$	−55.5786	−1.84546	−0.922728	−	0.385452i	$-0.874046\pi$
$907$	−55.5786	−1.84546	−0.922728	+	0.385452i	$0.874046\pi$
$908$	−21.8342	−0.724595
$909$	−12.6801	−0.420573
$910$	−1.25636	−0.0416479
$911$	−23.8537	−0.790309	−0.395154	−	0.918615i	$-0.629309\pi$
$911$	−23.8537	−0.790309	−0.395154	+	0.918615i	$0.629309\pi$
$912$	−5.92457	−0.196182
$913$	−38.0569	−1.25950
$914$	5.32930	0.176278
$915$	−13.8996	−0.459506
$916$	−19.6008	−0.647629
$917$	−11.5624	−0.381825
$918$	21.6995	0.716191
$919$	−29.3922	−0.969561	−0.484780	−	0.874636i	$-0.661100\pi$
$919$	−29.3922	−0.969561	−0.484780	+	0.874636i	$0.661100\pi$
$920$	8.13190	0.268101
$921$	−34.2525	−1.12866
$922$	9.49165	0.312591
$923$	−2.68875	−0.0885013
$924$	9.49554	0.312380
$925$	10.1251	0.332912
$926$	−35.7240	−1.17396
$927$	−5.66356	−0.186016
$928$	−0.0979954	−0.00321686
$929$	−12.9010	−0.423267	−0.211633	−	0.977349i	$-0.567878\pi$
$929$	−12.9010	−0.423267	−0.211633	+	0.977349i	$0.567878\pi$
$930$	−1.21997	−0.0400045
$931$	26.3288	0.862891
$932$	−12.8088	−0.419566
$933$	−14.3780	−0.470714
$934$	8.31585	0.272103
$935$	24.4241	0.798753
$936$	1.51167	0.0494103
$937$	−23.1548	−0.756435	−0.378218	−	0.925717i	$-0.623463\pi$
$937$	−23.1548	−0.756435	−0.378218	+	0.925717i	$0.623463\pi$
$938$	−9.67777	−0.315990
$939$	−24.7706	−0.808358
$940$	−4.79248	−0.156313
$941$	−23.9462	−0.780623	−0.390312	−	0.920683i	$-0.627633\pi$
$941$	−23.9462	−0.780623	−0.390312	+	0.920683i	$0.627633\pi$
$942$	21.4399	0.698548
$943$	−30.1678	−0.982400
$944$	−5.37745	−0.175021
$945$	−6.91514	−0.224950
$946$	−68.9527	−2.24184
$947$	27.5970	0.896783	0.448391	−	0.893837i	$-0.351997\pi$
$947$	27.5970	0.896783	0.448391	+	0.893837i	$0.351997\pi$
$948$	1.85987	0.0604057
$949$	−0.389941	−0.0126580
$950$	4.85631	0.157559
$951$	3.51129	0.113861
$952$	4.95310	0.160531
$953$	−32.5014	−1.05282	−0.526412	−	0.850229i	$-0.676463\pi$
$953$	−32.5014	−1.05282	−0.526412	+	0.850229i	$0.676463\pi$
$954$	5.65372	0.183046
$955$	−4.11474	−0.133150
$956$	21.4524	0.693821
$957$	−0.740647	−0.0239417
$958$	11.8759	0.383694
$959$	−15.9370	−0.514632
$960$	−1.21997	−0.0393745
$961$	1.00000	0.0322581
$962$	−10.1251	−0.326447
$963$	−26.4041	−0.850860
$964$	24.1039	0.776335
$965$	−4.93210	−0.158770
$966$	12.4640	0.401022
$967$	12.3305	0.396522	0.198261	−	0.980149i	$-0.436471\pi$
$967$	12.3305	0.396522	0.198261	+	0.980149i	$0.436471\pi$
$968$	−27.3805	−0.880043
$969$	−23.3571	−0.750339
$970$	7.44250	0.238964
$971$	−24.8685	−0.798068	−0.399034	−	0.916936i	$-0.630655\pi$
$971$	−24.8685	−0.798068	−0.399034	+	0.916936i	$0.630655\pi$
$972$	13.8529	0.444333
$973$	9.09103	0.291445
$974$	−28.5070	−0.913422
$975$	1.21997	0.0390704
$976$	11.3933	0.364692
$977$	−24.3680	−0.779603	−0.389801	−	0.920899i	$-0.627456\pi$
$977$	−24.3680	−0.779603	−0.389801	+	0.920899i	$0.627456\pi$
$978$	27.4690	0.878361
$979$	7.78671	0.248864
$980$	5.42156	0.173185
$981$	−22.5023	−0.718443
$982$	−11.2618	−0.359377
$983$	−6.58288	−0.209961	−0.104981	−	0.994474i	$-0.533478\pi$
$983$	−6.58288	−0.209961	−0.104981	+	0.994474i	$0.533478\pi$
$984$	4.52587	0.144279
$985$	−25.3535	−0.807828
$986$	−0.386339	−0.0123035
$987$	−7.34555	−0.233811
$988$	−4.85631	−0.154500
$989$	−90.5082	−2.87799
$990$	9.36507	0.297641
$991$	45.2085	1.43609	0.718047	−	0.695994i	$-0.245036\pi$
$991$	45.2085	1.43609	0.718047	+	0.695994i	$0.245036\pi$
$992$	1.00000	0.0317500
$993$	−23.6945	−0.751923
$994$	−3.37804	−0.107145
$995$	3.47528	0.110174
$996$	7.49426	0.237465
$997$	16.7302	0.529850	0.264925	−	0.964269i	$-0.414653\pi$
$997$	16.7302	0.529850	0.264925	+	0.964269i	$0.414653\pi$
$998$	−1.70633	−0.0540131
$999$	−55.7298	−1.76321

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.k.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.k.1.6	✓	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:	\( N \)	\(=\)	\( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4030.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.21997 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.21997 q^{6} -1.25636 q^{7} -1.00000 q^{8} -1.51167 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.21997 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.21997 q^{6} -1.25636 q^{7} -1.00000 q^{8} -1.51167 q^{9} +1.00000 q^{10} -6.19520 q^{11} +1.21997 q^{12} +1.00000 q^{13} +1.25636 q^{14} -1.21997 q^{15} +1.00000 q^{16} +3.94242 q^{17} +1.51167 q^{18} -4.85631 q^{19} -1.00000 q^{20} -1.53273 q^{21} +6.19520 q^{22} +8.13190 q^{23} -1.21997 q^{24} +1.00000 q^{25} -1.00000 q^{26} -5.50411 q^{27} -1.25636 q^{28} +0.0979954 q^{29} +1.21997 q^{30} -1.00000 q^{31} -1.00000 q^{32} -7.55798 q^{33} -3.94242 q^{34} +1.25636 q^{35} -1.51167 q^{36} +10.1251 q^{37} +4.85631 q^{38} +1.21997 q^{39} +1.00000 q^{40} -3.70981 q^{41} +1.53273 q^{42} -11.1300 q^{43} -6.19520 q^{44} +1.51167 q^{45} -8.13190 q^{46} +4.79248 q^{47} +1.21997 q^{48} -5.42156 q^{49} -1.00000 q^{50} +4.80965 q^{51} +1.00000 q^{52} +3.74006 q^{53} +5.50411 q^{54} +6.19520 q^{55} +1.25636 q^{56} -5.92457 q^{57} -0.0979954 q^{58} -5.37745 q^{59} -1.21997 q^{60} +11.3933 q^{61} +1.00000 q^{62} +1.89920 q^{63} +1.00000 q^{64} -1.00000 q^{65} +7.55798 q^{66} -7.70302 q^{67} +3.94242 q^{68} +9.92071 q^{69} -1.25636 q^{70} -2.68875 q^{71} +1.51167 q^{72} -0.389941 q^{73} -10.1251 q^{74} +1.21997 q^{75} -4.85631 q^{76} +7.78340 q^{77} -1.21997 q^{78} +1.52451 q^{79} -1.00000 q^{80} -2.17987 q^{81} +3.70981 q^{82} +6.14297 q^{83} -1.53273 q^{84} -3.94242 q^{85} +11.1300 q^{86} +0.119552 q^{87} +6.19520 q^{88} -1.25689 q^{89} -1.51167 q^{90} -1.25636 q^{91} +8.13190 q^{92} -1.21997 q^{93} -4.79248 q^{94} +4.85631 q^{95} -1.21997 q^{96} +7.44250 q^{97} +5.42156 q^{98} +9.36507 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} + q^{6} - 8 q^{7} - 8 q^{8} + 13 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 + q^6 - 8 * q^7 - 8 * q^8 + 13 * q^9	\( 8 q - 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} + q^{6} - 8 q^{7} - 8 q^{8} + 13 q^{9} + 8 q^{10} + 12 q^{11} - q^{12} + 8 q^{13} + 8 q^{14} + q^{15} + 8 q^{16} - 13 q^{18} - 3 q^{19} - 8 q^{20} + 27 q^{21} - 12 q^{22} + 17 q^{23} + q^{24} + 8 q^{25} - 8 q^{26} - 13 q^{27} - 8 q^{28} - 10 q^{29} - q^{30} - 8 q^{31} - 8 q^{32} + 4 q^{33} + 8 q^{35} + 13 q^{36} + 2 q^{37} + 3 q^{38} - q^{39} + 8 q^{40} + 14 q^{41} - 27 q^{42} - 19 q^{43} + 12 q^{44} - 13 q^{45} - 17 q^{46} + 8 q^{47} - q^{48} + 16 q^{49} - 8 q^{50} + 35 q^{51} + 8 q^{52} + 4 q^{53} + 13 q^{54} - 12 q^{55} + 8 q^{56} - 33 q^{57} + 10 q^{58} - 13 q^{59} + q^{60} + 11 q^{61} + 8 q^{62} - 27 q^{63} + 8 q^{64} - 8 q^{65} - 4 q^{66} - 34 q^{67} + 12 q^{69} - 8 q^{70} + 8 q^{71} - 13 q^{72} - 21 q^{73} - 2 q^{74} - q^{75} - 3 q^{76} + 19 q^{77} + q^{78} - 36 q^{79} - 8 q^{80} + 20 q^{81} - 14 q^{82} + 27 q^{84} + 19 q^{86} - 29 q^{87} - 12 q^{88} - 16 q^{89} + 13 q^{90} - 8 q^{91} + 17 q^{92} + q^{93} - 8 q^{94} + 3 q^{95} + q^{96} + q^{97} - 16 q^{98} + 65 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 + q^6 - 8 * q^7 - 8 * q^8 + 13 * q^9 + 8 * q^10 + 12 * q^11 - q^12 + 8 * q^13 + 8 * q^14 + q^15 + 8 * q^16 - 13 * q^18 - 3 * q^19 - 8 * q^20 + 27 * q^21 - 12 * q^22 + 17 * q^23 + q^24 + 8 * q^25 - 8 * q^26 - 13 * q^27 - 8 * q^28 - 10 * q^29 - q^30 - 8 * q^31 - 8 * q^32 + 4 * q^33 + 8 * q^35 + 13 * q^36 + 2 * q^37 + 3 * q^38 - q^39 + 8 * q^40 + 14 * q^41 - 27 * q^42 - 19 * q^43 + 12 * q^44 - 13 * q^45 - 17 * q^46 + 8 * q^47 - q^48 + 16 * q^49 - 8 * q^50 + 35 * q^51 + 8 * q^52 + 4 * q^53 + 13 * q^54 - 12 * q^55 + 8 * q^56 - 33 * q^57 + 10 * q^58 - 13 * q^59 + q^60 + 11 * q^61 + 8 * q^62 - 27 * q^63 + 8 * q^64 - 8 * q^65 - 4 * q^66 - 34 * q^67 + 12 * q^69 - 8 * q^70 + 8 * q^71 - 13 * q^72 - 21 * q^73 - 2 * q^74 - q^75 - 3 * q^76 + 19 * q^77 + q^78 - 36 * q^79 - 8 * q^80 + 20 * q^81 - 14 * q^82 + 27 * q^84 + 19 * q^86 - 29 * q^87 - 12 * q^88 - 16 * q^89 + 13 * q^90 - 8 * q^91 + 17 * q^92 + q^93 - 8 * q^94 + 3 * q^95 + q^96 + q^97 - 16 * q^98 + 65 * q^99

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