LMFDB - Embedded newform 4031.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4031,2,Mod(1,4031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4031, 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("4031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4031 = 29 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4031.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:	$32.1876970548$
Analytic rank:	$0$
Dimension:	$103$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	4031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.80086 q^{2} -1.28750 q^{3} +5.84483 q^{4} -1.26767 q^{5} +3.60610 q^{6} -2.24450 q^{7} -10.7688 q^{8} -1.34235 q^{9} +O(q^{10})$	\(q-2.80086 q^{2} -1.28750 q^{3} +5.84483 q^{4} -1.26767 q^{5} +3.60610 q^{6} -2.24450 q^{7} -10.7688 q^{8} -1.34235 q^{9} +3.55056 q^{10} -3.60673 q^{11} -7.52519 q^{12} -0.255387 q^{13} +6.28652 q^{14} +1.63211 q^{15} +18.4723 q^{16} -6.73564 q^{17} +3.75975 q^{18} +7.52746 q^{19} -7.40929 q^{20} +2.88978 q^{21} +10.1019 q^{22} +7.98821 q^{23} +13.8648 q^{24} -3.39302 q^{25} +0.715305 q^{26} +5.59076 q^{27} -13.1187 q^{28} -1.00000 q^{29} -4.57133 q^{30} -0.488796 q^{31} -30.2008 q^{32} +4.64365 q^{33} +18.8656 q^{34} +2.84527 q^{35} -7.84583 q^{36} -5.31384 q^{37} -21.0834 q^{38} +0.328810 q^{39} +13.6513 q^{40} -9.66689 q^{41} -8.09387 q^{42} -6.73298 q^{43} -21.0807 q^{44} +1.70166 q^{45} -22.3739 q^{46} -9.53386 q^{47} -23.7831 q^{48} -1.96224 q^{49} +9.50339 q^{50} +8.67211 q^{51} -1.49269 q^{52} +1.47632 q^{53} -15.6590 q^{54} +4.57213 q^{55} +24.1706 q^{56} -9.69157 q^{57} +2.80086 q^{58} -10.4343 q^{59} +9.53943 q^{60} +5.52699 q^{61} +1.36905 q^{62} +3.01291 q^{63} +47.6437 q^{64} +0.323746 q^{65} -13.0062 q^{66} -15.6729 q^{67} -39.3687 q^{68} -10.2848 q^{69} -7.96921 q^{70} +0.700236 q^{71} +14.4556 q^{72} -6.68135 q^{73} +14.8833 q^{74} +4.36850 q^{75} +43.9967 q^{76} +8.09529 q^{77} -0.920952 q^{78} +3.27505 q^{79} -23.4168 q^{80} -3.17102 q^{81} +27.0756 q^{82} -11.6678 q^{83} +16.8903 q^{84} +8.53855 q^{85} +18.8582 q^{86} +1.28750 q^{87} +38.8402 q^{88} -5.88234 q^{89} -4.76611 q^{90} +0.573216 q^{91} +46.6897 q^{92} +0.629323 q^{93} +26.7030 q^{94} -9.54230 q^{95} +38.8834 q^{96} -3.42284 q^{97} +5.49596 q^{98} +4.84151 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 103 q + q^{2} + 2 q^{3} + 127 q^{4} + 9 q^{5} + 19 q^{6} + 18 q^{7} + 149 q^{9}+O(q^{10}) $ 103 * q + q^2 + 2 * q^3 + 127 * q^4 + 9 * q^5 + 19 * q^6 + 18 * q^7 + 149 * q^9	\( 103 q + q^{2} + 2 q^{3} + 127 q^{4} + 9 q^{5} + 19 q^{6} + 18 q^{7} + 149 q^{9} + 20 q^{10} + 9 q^{11} + 36 q^{13} - 10 q^{14} + 16 q^{15} + 179 q^{16} + 21 q^{17} + 7 q^{18} + 42 q^{19} + 24 q^{20} + 28 q^{21} + 32 q^{22} + 25 q^{23} + 68 q^{24} + 194 q^{25} - 5 q^{26} + 14 q^{27} + 59 q^{28} - 103 q^{29} + 84 q^{30} + 34 q^{31} + 11 q^{32} + 42 q^{33} + 54 q^{34} + 35 q^{35} + 214 q^{36} + 34 q^{37} + 9 q^{38} + 23 q^{39} + 46 q^{40} + 16 q^{41} + 13 q^{42} + 68 q^{43} - 6 q^{44} + 25 q^{45} + 60 q^{46} + 6 q^{47} + 5 q^{48} + 257 q^{49} - 51 q^{50} + 68 q^{51} + 37 q^{52} + 35 q^{53} + 30 q^{54} + 66 q^{55} - 54 q^{56} + 78 q^{57} - q^{58} + 10 q^{59} - 24 q^{60} + 70 q^{61} + 29 q^{62} + 26 q^{63} + 276 q^{64} + 95 q^{65} + 77 q^{66} + 71 q^{67} - 21 q^{68} - 20 q^{69} + 48 q^{70} + 32 q^{71} + 32 q^{72} + 94 q^{73} + 35 q^{74} + 7 q^{75} + 134 q^{76} + 17 q^{77} + 58 q^{78} + 110 q^{79} + 78 q^{80} + 267 q^{81} - 71 q^{82} + 35 q^{83} + 96 q^{84} + 71 q^{85} + 33 q^{86} - 2 q^{87} + 100 q^{88} + 22 q^{89} - 134 q^{90} + 108 q^{91} - 11 q^{92} + 78 q^{93} + 90 q^{94} + 12 q^{95} + 177 q^{96} + 44 q^{97} - 18 q^{98} + 83 q^{99}+O(q^{100}) \) 103 * q + q^2 + 2 * q^3 + 127 * q^4 + 9 * q^5 + 19 * q^6 + 18 * q^7 + 149 * q^9 + 20 * q^10 + 9 * q^11 + 36 * q^13 - 10 * q^14 + 16 * q^15 + 179 * q^16 + 21 * q^17 + 7 * q^18 + 42 * q^19 + 24 * q^20 + 28 * q^21 + 32 * q^22 + 25 * q^23 + 68 * q^24 + 194 * q^25 - 5 * q^26 + 14 * q^27 + 59 * q^28 - 103 * q^29 + 84 * q^30 + 34 * q^31 + 11 * q^32 + 42 * q^33 + 54 * q^34 + 35 * q^35 + 214 * q^36 + 34 * q^37 + 9 * q^38 + 23 * q^39 + 46 * q^40 + 16 * q^41 + 13 * q^42 + 68 * q^43 - 6 * q^44 + 25 * q^45 + 60 * q^46 + 6 * q^47 + 5 * q^48 + 257 * q^49 - 51 * q^50 + 68 * q^51 + 37 * q^52 + 35 * q^53 + 30 * q^54 + 66 * q^55 - 54 * q^56 + 78 * q^57 - q^58 + 10 * q^59 - 24 * q^60 + 70 * q^61 + 29 * q^62 + 26 * q^63 + 276 * q^64 + 95 * q^65 + 77 * q^66 + 71 * q^67 - 21 * q^68 - 20 * q^69 + 48 * q^70 + 32 * q^71 + 32 * q^72 + 94 * q^73 + 35 * q^74 + 7 * q^75 + 134 * q^76 + 17 * q^77 + 58 * q^78 + 110 * q^79 + 78 * q^80 + 267 * q^81 - 71 * q^82 + 35 * q^83 + 96 * q^84 + 71 * q^85 + 33 * q^86 - 2 * q^87 + 100 * q^88 + 22 * q^89 - 134 * q^90 + 108 * q^91 - 11 * q^92 + 78 * q^93 + 90 * q^94 + 12 * q^95 + 177 * q^96 + 44 * q^97 - 18 * q^98 + 83 * 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 $

$2$	−2.80086	−1.98051	−0.990254	−	0.139272i	$-0.955524\pi$
$2$	−2.80086	−1.98051	−0.990254	+	0.139272i	$0.955524\pi$
$3$	−1.28750	−0.743336	−0.371668	−	0.928366i	$-0.621214\pi$
$3$	−1.28750	−0.743336	−0.371668	+	0.928366i	$0.621214\pi$
$4$	5.84483	2.92241
$5$	−1.26767	−0.566918	−0.283459	−	0.958984i	$-0.591482\pi$
$5$	−1.26767	−0.566918	−0.283459	+	0.958984i	$0.591482\pi$
$6$	3.60610	1.47218
$7$	−2.24450	−0.848340	−0.424170	−	0.905583i	$-0.639434\pi$
$7$	−2.24450	−0.848340	−0.424170	+	0.905583i	$0.639434\pi$
$8$	−10.7688	−3.80736
$9$	−1.34235	−0.447452
$10$	3.55056	1.12278
$11$	−3.60673	−1.08747	−0.543735	−	0.839257i	$-0.682990\pi$
$11$	−3.60673	−1.08747	−0.543735	+	0.839257i	$0.682990\pi$
$12$	−7.52519	−2.17234
$13$	−0.255387	−0.0708317	−0.0354158	−	0.999373i	$-0.511276\pi$
$13$	−0.255387	−0.0708317	−0.0354158	+	0.999373i	$0.511276\pi$
$14$	6.28652	1.68014
$15$	1.63211	0.421410
$16$	18.4723	4.61809
$17$	−6.73564	−1.63363	−0.816817	−	0.576897i	$-0.804263\pi$
$17$	−6.73564	−1.63363	−0.816817	+	0.576897i	$0.804263\pi$
$18$	3.75975	0.886181
$19$	7.52746	1.72692	0.863459	−	0.504419i	$-0.168293\pi$
$19$	7.52746	1.72692	0.863459	+	0.504419i	$0.168293\pi$
$20$	−7.40929	−1.65677
$21$	2.88978	0.630602
$22$	10.1019	2.15374
$23$	7.98821	1.66566	0.832829	−	0.553530i	$-0.186720\pi$
$23$	7.98821	1.66566	0.832829	+	0.553530i	$0.186720\pi$
$24$	13.8648	2.83014
$25$	−3.39302	−0.678605
$26$	0.715305	0.140283
$27$	5.59076	1.07594
$28$	−13.1187	−2.47920
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−4.57133	−0.834606
$31$	−0.488796	−0.0877904	−0.0438952	−	0.999036i	$-0.513977\pi$
$31$	−0.488796	−0.0877904	−0.0438952	+	0.999036i	$0.513977\pi$
$32$	−30.2008	−5.33880
$33$	4.64365	0.808355
$34$	18.8656	3.23543
$35$	2.84527	0.480939
$36$	−7.84583	−1.30764
$37$	−5.31384	−0.873589	−0.436795	−	0.899561i	$-0.643886\pi$
$37$	−5.31384	−0.873589	−0.436795	+	0.899561i	$0.643886\pi$
$38$	−21.0834	−3.42017
$39$	0.328810	0.0526518
$40$	13.6513	2.15846
$41$	−9.66689	−1.50971	−0.754857	−	0.655889i	$-0.772294\pi$
$41$	−9.66689	−1.50971	−0.754857	+	0.655889i	$0.772294\pi$
$42$	−8.09387	−1.24891
$43$	−6.73298	−1.02677	−0.513385	−	0.858158i	$-0.671609\pi$
$43$	−6.73298	−1.02677	−0.513385	+	0.858158i	$0.671609\pi$
$44$	−21.0807	−3.17804
$45$	1.70166	0.253668
$46$	−22.3739	−3.29885
$47$	−9.53386	−1.39066	−0.695329	−	0.718692i	$-0.744741\pi$
$47$	−9.53386	−1.39066	−0.695329	+	0.718692i	$0.744741\pi$
$48$	−23.7831	−3.43279
$49$	−1.96224	−0.280320
$50$	9.50339	1.34398
$51$	8.67211	1.21434
$52$	−1.49269	−0.206999
$53$	1.47632	0.202788	0.101394	−	0.994846i	$-0.467670\pi$
$53$	1.47632	0.202788	0.101394	+	0.994846i	$0.467670\pi$
$54$	−15.6590	−2.13091
$55$	4.57213	0.616506
$56$	24.1706	3.22993
$57$	−9.69157	−1.28368
$58$	2.80086	0.367771
$59$	−10.4343	−1.35843	−0.679213	−	0.733941i	$-0.737679\pi$
$59$	−10.4343	−1.35843	−0.679213	+	0.733941i	$0.737679\pi$
$60$	9.53943	1.23153
$61$	5.52699	0.707658	0.353829	−	0.935310i	$-0.384879\pi$
$61$	5.52699	0.707658	0.353829	+	0.935310i	$0.384879\pi$
$62$	1.36905	0.173870
$63$	3.01291	0.379591
$64$	47.6437	5.95546
$65$	0.323746	0.0401557
$66$	−13.0062	−1.60095
$67$	−15.6729	−1.91475	−0.957375	−	0.288847i	$-0.906728\pi$
$67$	−15.6729	−1.91475	−0.957375	+	0.288847i	$0.906728\pi$
$68$	−39.3687	−4.77415
$69$	−10.2848	−1.23814
$70$	−7.96921	−0.952503
$71$	0.700236	0.0831027	0.0415514	−	0.999136i	$-0.486770\pi$
$71$	0.700236	0.0831027	0.0415514	+	0.999136i	$0.486770\pi$
$72$	14.4556	1.70361
$73$	−6.68135	−0.781993	−0.390996	−	0.920392i	$-0.627869\pi$
$73$	−6.68135	−0.781993	−0.390996	+	0.920392i	$0.627869\pi$
$74$	14.8833	1.73015
$75$	4.36850	0.504431
$76$	43.9967	5.04677
$77$	8.09529	0.922544
$78$	−0.920952	−0.104277
$79$	3.27505	0.368472	0.184236	−	0.982882i	$-0.441019\pi$
$79$	3.27505	0.368472	0.184236	+	0.982882i	$0.441019\pi$
$80$	−23.4168	−2.61807
$81$	−3.17102	−0.352336
$82$	27.0756	2.99000
$83$	−11.6678	−1.28070	−0.640351	−	0.768082i	$-0.721211\pi$
$83$	−11.6678	−1.28070	−0.640351	+	0.768082i	$0.721211\pi$
$84$	16.8903	1.84288
$85$	8.53855	0.926136
$86$	18.8582	2.03353
$87$	1.28750	0.138034
$88$	38.8402	4.14038
$89$	−5.88234	−0.623527	−0.311763	−	0.950160i	$-0.600920\pi$
$89$	−5.88234	−0.623527	−0.311763	+	0.950160i	$0.600920\pi$
$90$	−4.76611	−0.502392
$91$	0.573216	0.0600893
$92$	46.6897	4.86774
$93$	0.629323	0.0652577
$94$	26.7030	2.75421
$95$	−9.54230	−0.979020
$96$	38.8834	3.96852
$97$	−3.42284	−0.347537	−0.173768	−	0.984787i	$-0.555594\pi$
$97$	−3.42284	−0.347537	−0.173768	+	0.984787i	$0.555594\pi$
$98$	5.49596	0.555175
$99$	4.84151	0.486590
$100$	−19.8316	−1.98316
$101$	−17.1193	−1.70343	−0.851717	−	0.524003i	$-0.824438\pi$
$101$	−17.1193	−1.70343	−0.851717	+	0.524003i	$0.824438\pi$
$102$	−24.2894	−2.40501
$103$	−17.6615	−1.74024	−0.870120	−	0.492841i	$-0.835959\pi$
$103$	−17.6615	−1.74024	−0.870120	+	0.492841i	$0.835959\pi$
$104$	2.75022	0.269681
$105$	−3.66328	−0.357499
$106$	−4.13497	−0.401624
$107$	17.8330	1.72398	0.861992	−	0.506921i	$-0.169217\pi$
$107$	17.8330	1.72398	0.861992	+	0.506921i	$0.169217\pi$
$108$	32.6770	3.14435
$109$	−10.2191	−0.978813	−0.489406	−	0.872056i	$-0.662786\pi$
$109$	−10.2191	−0.978813	−0.489406	+	0.872056i	$0.662786\pi$
$110$	−12.8059	−1.22099
$111$	6.84154	0.649370
$112$	−41.4611	−3.91771
$113$	−9.60030	−0.903121	−0.451560	−	0.892241i	$-0.649132\pi$
$113$	−9.60030	−0.903121	−0.451560	+	0.892241i	$0.649132\pi$
$114$	27.1448	2.54234
$115$	−10.1264	−0.944291
$116$	−5.84483	−0.542679
$117$	0.342820	0.0316938
$118$	29.2249	2.69037
$119$	15.1181	1.38588
$120$	−17.5760	−1.60446
$121$	2.00849	0.182590
$122$	−15.4803	−1.40152
$123$	12.4461	1.12222
$124$	−2.85693	−0.256560
$125$	10.6396	0.951630
$126$	−8.43874	−0.751783
$127$	−2.69346	−0.239006	−0.119503	−	0.992834i	$-0.538130\pi$
$127$	−2.69346	−0.239006	−0.119503	+	0.992834i	$0.538130\pi$
$128$	−73.0417	−6.45603
$129$	8.66869	0.763235
$130$	−0.906767	−0.0795288
$131$	6.09889	0.532863	0.266431	−	0.963854i	$-0.414155\pi$
$131$	6.09889	0.532863	0.266431	+	0.963854i	$0.414155\pi$
$132$	27.1413	2.36235
$133$	−16.8954	−1.46501
$134$	43.8977	3.79218
$135$	−7.08722	−0.609971
$136$	72.5350	6.21982
$137$	−13.3454	−1.14017	−0.570086	−	0.821585i	$-0.693090\pi$
$137$	−13.3454	−1.14017	−0.570086	+	0.821585i	$0.693090\pi$
$138$	28.8063	2.45215
$139$	1.00000	0.0848189
$139$	1.00000	0.0848189
$140$	16.6301	1.40550
$141$	12.2748	1.03373
$142$	−1.96126	−0.164586
$143$	0.921113	0.0770273
$144$	−24.7964	−2.06637
$145$	1.26767	0.105274
$146$	18.7135	1.54874
$147$	2.52637	0.208372
$148$	−31.0584	−2.55299
$149$	6.47916	0.530794	0.265397	−	0.964139i	$-0.414497\pi$
$149$	6.47916	0.530794	0.265397	+	0.964139i	$0.414497\pi$
$150$	−12.2356	−0.999030
$151$	13.7697	1.12057	0.560283	−	0.828301i	$-0.310692\pi$
$151$	13.7697	1.12057	0.560283	+	0.828301i	$0.310692\pi$
$152$	−81.0619	−6.57499
$153$	9.04162	0.730972
$154$	−22.6738	−1.82711
$155$	0.619630	0.0497699
$156$	1.92184	0.153870
$157$	20.5778	1.64229	0.821145	−	0.570719i	$-0.193335\pi$
$157$	20.5778	1.64229	0.821145	+	0.570719i	$0.193335\pi$
$158$	−9.17297	−0.729762
$159$	−1.90076	−0.150740
$160$	38.2846	3.02666
$161$	−17.9295	−1.41304
$162$	8.88159	0.697804
$163$	−9.93852	−0.778445	−0.389223	−	0.921144i	$-0.627256\pi$
$163$	−9.93852	−0.778445	−0.389223	+	0.921144i	$0.627256\pi$
$164$	−56.5013	−4.41201
$165$	−5.88659	−0.458271
$166$	32.6798	2.53644
$167$	3.94140	0.304995	0.152497	−	0.988304i	$-0.451268\pi$
$167$	3.94140	0.304995	0.152497	+	0.988304i	$0.451268\pi$
$168$	−31.1195	−2.40092
$169$	−12.9348	−0.994983
$170$	−23.9153	−1.83422
$171$	−10.1045	−0.772712
$172$	−39.3531	−3.00065
$173$	1.36702	0.103933	0.0519664	−	0.998649i	$-0.483451\pi$
$173$	1.36702	0.103933	0.0519664	+	0.998649i	$0.483451\pi$
$174$	−3.60610	−0.273378
$175$	7.61563	0.575687
$176$	−66.6247	−5.02203
$177$	13.4341	1.00977
$178$	16.4756	1.23490
$179$	7.41410	0.554156	0.277078	−	0.960847i	$-0.410634\pi$
$179$	7.41410	0.554156	0.277078	+	0.960847i	$0.410634\pi$
$180$	9.94589	0.741323
$181$	5.05386	0.375650	0.187825	−	0.982202i	$-0.439856\pi$
$181$	5.05386	0.375650	0.187825	+	0.982202i	$0.439856\pi$
$182$	−1.60550	−0.119007
$183$	−7.11597	−0.526028
$184$	−86.0237	−6.34175
$185$	6.73617	0.495253
$186$	−1.76265	−0.129244
$187$	24.2936	1.77653
$188$	−55.7238	−4.06407
$189$	−12.5484	−0.912765
$190$	26.7267	1.93896
$191$	8.03078	0.581087	0.290543	−	0.956862i	$-0.406164\pi$
$191$	8.03078	0.581087	0.290543	+	0.956862i	$0.406164\pi$
$192$	−61.3410	−4.42691
$193$	5.95013	0.428300	0.214150	−	0.976801i	$-0.431302\pi$
$193$	5.95013	0.428300	0.214150	+	0.976801i	$0.431302\pi$
$194$	9.58690	0.688299
$195$	−0.416821	−0.0298492
$196$	−11.4689	−0.819210
$197$	8.49542	0.605274	0.302637	−	0.953106i	$-0.402133\pi$
$197$	8.49542	0.605274	0.302637	+	0.953106i	$0.402133\pi$
$198$	−13.5604	−0.963695
$199$	−6.65851	−0.472009	−0.236005	−	0.971752i	$-0.575838\pi$
$199$	−6.65851	−0.472009	−0.236005	+	0.971752i	$0.575838\pi$
$200$	36.5389	2.58369
$201$	20.1788	1.42330
$202$	47.9488	3.37366
$203$	2.24450	0.157533
$204$	50.6870	3.54880
$205$	12.2544	0.855883
$206$	49.4674	3.44656
$207$	−10.7230	−0.745301
$208$	−4.71760	−0.327107
$209$	−27.1495	−1.87797
$210$	10.2603	0.708030
$211$	−15.7026	−1.08101	−0.540507	−	0.841339i	$-0.681768\pi$
$211$	−15.7026	−1.08101	−0.540507	+	0.841339i	$0.681768\pi$
$212$	8.62884	0.592631
$213$	−0.901551	−0.0617732
$214$	−49.9479	−3.41437
$215$	8.53518	0.582094
$216$	−60.2060	−4.09650
$217$	1.09710	0.0744761
$218$	28.6223	1.93855
$219$	8.60221	0.581283
$220$	26.7233	1.80168
$221$	1.72020	0.115713
$222$	−19.1622	−1.28608
$223$	−15.5129	−1.03882	−0.519412	−	0.854524i	$-0.673849\pi$
$223$	−15.5129	−1.03882	−0.519412	+	0.854524i	$0.673849\pi$
$224$	67.7857	4.52912
$225$	4.55464	0.303643
$226$	26.8891	1.78864
$227$	11.7560	0.780272	0.390136	−	0.920757i	$-0.372428\pi$
$227$	11.7560	0.780272	0.390136	+	0.920757i	$0.372428\pi$
$228$	−56.6456	−3.75144
$229$	−9.68789	−0.640194	−0.320097	−	0.947385i	$-0.603715\pi$
$229$	−9.68789	−0.640194	−0.320097	+	0.947385i	$0.603715\pi$
$230$	28.3626	1.87018
$231$	−10.4226	−0.685760
$232$	10.7688	0.707008
$233$	−5.87176	−0.384672	−0.192336	−	0.981329i	$-0.561606\pi$
$233$	−5.87176	−0.384672	−0.192336	+	0.981329i	$0.561606\pi$
$234$	−0.960192	−0.0627697
$235$	12.0858	0.788388
$236$	−60.9865	−3.96988
$237$	−4.21662	−0.273899
$238$	−42.3438	−2.74474
$239$	5.74430	0.371568	0.185784	−	0.982591i	$-0.440518\pi$
$239$	5.74430	0.371568	0.185784	+	0.982591i	$0.440518\pi$
$240$	30.1490	1.94611
$241$	−5.27523	−0.339807	−0.169904	−	0.985461i	$-0.554346\pi$
$241$	−5.27523	−0.339807	−0.169904	+	0.985461i	$0.554346\pi$
$242$	−5.62550	−0.361621
$243$	−12.6896	−0.814039
$244$	32.3043	2.06807
$245$	2.48746	0.158918
$246$	−34.8597	−2.22258
$247$	−1.92242	−0.122320
$248$	5.26376	0.334249
$249$	15.0222	0.951993
$250$	−29.7999	−1.88471
$251$	−12.2630	−0.774031	−0.387016	−	0.922073i	$-0.626494\pi$
$251$	−12.2630	−0.774031	−0.387016	+	0.922073i	$0.626494\pi$
$252$	17.6099	1.10932
$253$	−28.8113	−1.81135
$254$	7.54400	0.473353
$255$	−10.9933	−0.688430
$256$	109.292	6.83077
$257$	−14.4191	−0.899441	−0.449720	−	0.893169i	$-0.648476\pi$
$257$	−14.4191	−0.899441	−0.449720	+	0.893169i	$0.648476\pi$
$258$	−24.2798	−1.51159
$259$	11.9269	0.741100
$260$	1.89224	0.117352
$261$	1.34235	0.0830897
$262$	−17.0822	−1.05534
$263$	−22.2968	−1.37488	−0.687440	−	0.726241i	$-0.741266\pi$
$263$	−22.2968	−1.37488	−0.687440	+	0.726241i	$0.741266\pi$
$264$	−50.0066	−3.07770
$265$	−1.87148	−0.114964
$266$	47.3215	2.90147
$267$	7.57349	0.463490
$268$	−91.6054	−5.59569
$269$	−7.58173	−0.462266	−0.231133	−	0.972922i	$-0.574243\pi$
$269$	−7.58173	−0.462266	−0.231133	+	0.972922i	$0.574243\pi$
$270$	19.8503	1.20805
$271$	24.9207	1.51382	0.756912	−	0.653517i	$-0.226707\pi$
$271$	24.9207	1.51382	0.756912	+	0.653517i	$0.226707\pi$
$272$	−124.423	−7.54426
$273$	−0.738013	−0.0446666
$274$	37.3785	2.25812
$275$	12.2377	0.737962
$276$	−60.1128	−3.61837
$277$	18.9483	1.13849	0.569247	−	0.822167i	$-0.307235\pi$
$277$	18.9483	1.13849	0.569247	+	0.822167i	$0.307235\pi$
$278$	−2.80086	−0.167985
$279$	0.656138	0.0392819
$280$	−30.6402	−1.83110
$281$	0.976913	0.0582777	0.0291389	−	0.999575i	$-0.490723\pi$
$281$	0.976913	0.0582777	0.0291389	+	0.999575i	$0.490723\pi$
$282$	−34.3800	−2.04730
$283$	8.68700	0.516389	0.258194	−	0.966093i	$-0.416872\pi$
$283$	8.68700	0.516389	0.258194	+	0.966093i	$0.416872\pi$
$284$	4.09276	0.242861
$285$	12.2857	0.727741
$286$	−2.57991	−0.152553
$287$	21.6973	1.28075
$288$	40.5402	2.38886
$289$	28.3689	1.66876
$290$	−3.55056	−0.208496
$291$	4.40689	0.258336
$292$	−39.0513	−2.28531
$293$	17.4631	1.02021	0.510104	−	0.860113i	$-0.329607\pi$
$293$	17.4631	1.02021	0.510104	+	0.860113i	$0.329607\pi$
$294$	−7.07602	−0.412682
$295$	13.2272	0.770115
$296$	57.2238	3.32606
$297$	−20.1644	−1.17006
$298$	−18.1472	−1.05124
$299$	−2.04009	−0.117981
$300$	25.5331	1.47416
$301$	15.1122	0.871050
$302$	−38.5672	−2.21929
$303$	22.0410	1.26622
$304$	139.050	7.97505
$305$	−7.00637	−0.401184
$306$	−25.3243	−1.44770
$307$	−3.87458	−0.221134	−0.110567	−	0.993869i	$-0.535267\pi$
$307$	−3.87458	−0.221134	−0.110567	+	0.993869i	$0.535267\pi$
$308$	47.3156	2.69605
$309$	22.7391	1.29358
$310$	−1.73550	−0.0985697
$311$	3.42075	0.193973	0.0969864	−	0.995286i	$-0.469080\pi$
$311$	3.42075	0.193973	0.0969864	+	0.995286i	$0.469080\pi$
$312$	−3.54090	−0.200464
$313$	−22.6985	−1.28299	−0.641497	−	0.767126i	$-0.721686\pi$
$313$	−22.6985	−1.28299	−0.641497	+	0.767126i	$0.721686\pi$
$314$	−57.6357	−3.25257
$315$	−3.81936	−0.215197
$316$	19.1421	1.07683
$317$	24.8020	1.39302	0.696510	−	0.717547i	$-0.254735\pi$
$317$	24.8020	1.39302	0.696510	+	0.717547i	$0.254735\pi$
$318$	5.32376	0.298541
$319$	3.60673	0.201938
$320$	−60.3963	−3.37625
$321$	−22.9600	−1.28150
$322$	50.2181	2.79855
$323$	−50.7023	−2.82115
$324$	−18.5341	−1.02967
$325$	0.866535	0.0480667
$326$	27.8364	1.54172
$327$	13.1571	0.727587
$328$	104.101	5.74802
$329$	21.3987	1.17975
$330$	16.4875	0.907609
$331$	−30.8677	−1.69664	−0.848322	−	0.529481i	$-0.822387\pi$
$331$	−30.8677	−1.69664	−0.848322	+	0.529481i	$0.822387\pi$
$332$	−68.1960	−3.74274
$333$	7.13305	0.390889
$334$	−11.0393	−0.604045
$335$	19.8680	1.08551
$336$	53.3810	2.91217
$337$	−2.76308	−0.150514	−0.0752572	−	0.997164i	$-0.523978\pi$
$337$	−2.76308	−0.150514	−0.0752572	+	0.997164i	$0.523978\pi$
$338$	36.2285	1.97057
$339$	12.3604	0.671322
$340$	49.9063	2.70655
$341$	1.76295	0.0954694
$342$	28.3014	1.53036
$343$	20.1157	1.08615
$344$	72.5063	3.90928
$345$	13.0377	0.701925
$346$	−3.82884	−0.205840
$347$	−2.38415	−0.127988	−0.0639940	−	0.997950i	$-0.520384\pi$
$347$	−2.38415	−0.127988	−0.0639940	+	0.997950i	$0.520384\pi$
$348$	7.52519	0.403393
$349$	−27.7695	−1.48646	−0.743232	−	0.669033i	$-0.766708\pi$
$349$	−27.7695	−1.48646	−0.743232	+	0.669033i	$0.766708\pi$
$350$	−21.3303	−1.14015
$351$	−1.42781	−0.0762109
$352$	108.926	5.80579
$353$	−1.71135	−0.0910861	−0.0455431	−	0.998962i	$-0.514502\pi$
$353$	−1.71135	−0.0910861	−0.0455431	+	0.998962i	$0.514502\pi$
$354$	−37.6270	−1.99985
$355$	−0.887665	−0.0471124
$356$	−34.3813	−1.82220
$357$	−19.4645	−1.03017
$358$	−20.7659	−1.09751
$359$	−19.6102	−1.03498	−0.517492	−	0.855688i	$-0.673134\pi$
$359$	−19.6102	−1.03498	−0.517492	+	0.855688i	$0.673134\pi$
$360$	−18.3249	−0.965805
$361$	37.6626	1.98224
$362$	−14.1552	−0.743979
$363$	−2.58592	−0.135726
$364$	3.35035	0.175606
$365$	8.46972	0.443325
$366$	19.9309	1.04180
$367$	11.8857	0.620430	0.310215	−	0.950666i	$-0.399599\pi$
$367$	11.8857	0.620430	0.310215	+	0.950666i	$0.399599\pi$
$368$	147.561	7.69215
$369$	12.9764	0.675524
$370$	−18.8671	−0.980853
$371$	−3.31360	−0.172033
$372$	3.67828	0.190710
$373$	29.7118	1.53842	0.769208	−	0.638998i	$-0.220651\pi$
$373$	29.7118	1.53842	0.769208	+	0.638998i	$0.220651\pi$
$374$	−68.0431	−3.51843
$375$	−13.6984	−0.707381
$376$	102.669	5.29473
$377$	0.255387	0.0131531
$378$	35.1465	1.80774
$379$	6.41364	0.329446	0.164723	−	0.986340i	$-0.447327\pi$
$379$	6.41364	0.329446	0.164723	+	0.986340i	$0.447327\pi$
$380$	−55.7731	−2.86110
$381$	3.46782	0.177662
$382$	−22.4931	−1.15085
$383$	11.5668	0.591037	0.295519	−	0.955337i	$-0.404508\pi$
$383$	11.5668	0.591037	0.295519	+	0.955337i	$0.404508\pi$
$384$	94.0408	4.79900
$385$	−10.2621	−0.523006
$386$	−16.6655	−0.848251
$387$	9.03805	0.459430
$388$	−20.0059	−1.01565
$389$	−27.1289	−1.37549	−0.687744	−	0.725953i	$-0.741399\pi$
$389$	−27.1289	−1.37549	−0.687744	+	0.725953i	$0.741399\pi$
$390$	1.16746	0.0591166
$391$	−53.8058	−2.72107
$392$	21.1310	1.06728
$393$	−7.85230	−0.396096
$394$	−23.7945	−1.19875
$395$	−4.15167	−0.208893
$396$	28.2978	1.42202
$397$	2.27588	0.114223	0.0571116	−	0.998368i	$-0.481811\pi$
$397$	2.27588	0.114223	0.0571116	+	0.998368i	$0.481811\pi$
$398$	18.6496	0.934819
$399$	21.7527	1.08900
$400$	−62.6771	−3.13385
$401$	−10.3736	−0.518032	−0.259016	−	0.965873i	$-0.583398\pi$
$401$	−10.3736	−0.518032	−0.259016	+	0.965873i	$0.583398\pi$
$402$	−56.5180	−2.81886
$403$	0.124832	0.00621834
$404$	−100.059	−4.97814
$405$	4.01980	0.199745
$406$	−6.28652	−0.311995
$407$	19.1656	0.950002
$408$	−93.3885	−4.62342
$409$	35.9993	1.78005	0.890026	−	0.455909i	$-0.150686\pi$
$409$	35.9993	1.78005	0.890026	+	0.455909i	$0.150686\pi$
$410$	−34.3228	−1.69508
$411$	17.1821	0.847530
$412$	−103.228	−5.08570
$413$	23.4197	1.15241
$414$	30.0337	1.47608
$415$	14.7908	0.726053
$416$	7.71291	0.378156
$417$	−1.28750	−0.0630489
$418$	76.0420	3.71934
$419$	25.4173	1.24172	0.620858	−	0.783923i	$-0.286784\pi$
$419$	25.4173	1.24172	0.620858	+	0.783923i	$0.286784\pi$
$420$	−21.4112	−1.04476
$421$	17.7106	0.863164	0.431582	−	0.902074i	$-0.357956\pi$
$421$	17.7106	0.863164	0.431582	+	0.902074i	$0.357956\pi$
$422$	43.9810	2.14096
$423$	12.7978	0.622252
$424$	−15.8982	−0.772087
$425$	22.8542	1.10859
$426$	2.52512	0.122342
$427$	−12.4053	−0.600335
$428$	104.231	5.03820
$429$	−1.18593	−0.0572572
$430$	−23.9058	−1.15284
$431$	29.3696	1.41469	0.707343	−	0.706871i	$-0.249894\pi$
$431$	29.3696	1.41469	0.707343	+	0.706871i	$0.249894\pi$
$432$	103.275	4.96880
$433$	−0.111866	−0.00537594	−0.00268797	−	0.999996i	$-0.500856\pi$
$433$	−0.111866	−0.00537594	−0.00268797	+	0.999996i	$0.500856\pi$
$434$	−3.07283	−0.147500
$435$	−1.63211	−0.0782539
$436$	−59.7289	−2.86050
$437$	60.1310	2.87645
$438$	−24.0936	−1.15124
$439$	2.18508	0.104288	0.0521441	−	0.998640i	$-0.483394\pi$
$439$	2.18508	0.104288	0.0521441	+	0.998640i	$0.483394\pi$
$440$	−49.2365	−2.34726
$441$	2.63402	0.125429
$442$	−4.81804	−0.229171
$443$	−30.2186	−1.43573	−0.717865	−	0.696183i	$-0.754880\pi$
$443$	−30.2186	−1.43573	−0.717865	+	0.696183i	$0.754880\pi$
$444$	39.9876	1.89773
$445$	7.45684	0.353488
$446$	43.4496	2.05740
$447$	−8.34190	−0.394558
$448$	−106.936	−5.05225
$449$	38.9446	1.83791	0.918954	−	0.394365i	$-0.129036\pi$
$449$	38.9446	1.83791	0.918954	+	0.394365i	$0.129036\pi$
$450$	−12.7569	−0.601367
$451$	34.8658	1.64177
$452$	−56.1121	−2.63929
$453$	−17.7285	−0.832957
$454$	−32.9269	−1.54533
$455$	−0.726646	−0.0340657
$456$	104.367	4.88743
$457$	1.07493	0.0502833	0.0251416	−	0.999684i	$-0.491996\pi$
$457$	1.07493	0.0502833	0.0251416	+	0.999684i	$0.491996\pi$
$458$	27.1344	1.26791
$459$	−37.6574	−1.75770
$460$	−59.1870	−2.75961
$461$	−14.7098	−0.685102	−0.342551	−	0.939499i	$-0.611291\pi$
$461$	−14.7098	−0.685102	−0.342551	+	0.939499i	$0.611291\pi$
$462$	29.1924	1.35815
$463$	−1.43433	−0.0666590	−0.0333295	−	0.999444i	$-0.510611\pi$
$463$	−1.43433	−0.0666590	−0.0333295	+	0.999444i	$0.510611\pi$
$464$	−18.4723	−0.857557
$465$	−0.797771	−0.0369958
$466$	16.4460	0.761845
$467$	−29.0040	−1.34214	−0.671071	−	0.741393i	$-0.734166\pi$
$467$	−29.0040	−1.34214	−0.671071	+	0.741393i	$0.734166\pi$
$468$	2.00373	0.0926222
$469$	35.1778	1.62436
$470$	−33.8505	−1.56141
$471$	−26.4939	−1.22077
$472$	112.365	5.17201
$473$	24.2840	1.11658
$474$	11.8102	0.542459
$475$	−25.5408	−1.17189
$476$	88.3628	4.05010
$477$	−1.98175	−0.0907379
$478$	−16.0890	−0.735894
$479$	17.3724	0.793766	0.396883	−	0.917869i	$-0.370092\pi$
$479$	17.3724	0.793766	0.396883	+	0.917869i	$0.370092\pi$
$480$	−49.2912	−2.24983
$481$	1.35709	0.0618778
$482$	14.7752	0.672991
$483$	23.0842	1.05037
$484$	11.7393	0.533603
$485$	4.33902	0.197025
$486$	35.5419	1.61221
$487$	−15.8762	−0.719418	−0.359709	−	0.933065i	$-0.617124\pi$
$487$	−15.8762	−0.719418	−0.359709	+	0.933065i	$0.617124\pi$
$488$	−59.5192	−2.69431
$489$	12.7958	0.578646
$490$	−6.96704	−0.314739
$491$	7.11634	0.321156	0.160578	−	0.987023i	$-0.448664\pi$
$491$	7.11634	0.321156	0.160578	+	0.987023i	$0.448664\pi$
$492$	72.7452	3.27961
$493$	6.73564	0.303358
$494$	5.38443	0.242257
$495$	−6.13742	−0.275856
$496$	−9.02921	−0.405424
$497$	−1.57168	−0.0704993
$498$	−42.0751	−1.88543
$499$	−24.5273	−1.09799	−0.548997	−	0.835824i	$-0.684990\pi$
$499$	−24.5273	−1.09799	−0.548997	+	0.835824i	$0.684990\pi$
$500$	62.1863	2.78106
$501$	−5.07454	−0.226714
$502$	34.3469	1.53298
$503$	−18.9369	−0.844356	−0.422178	−	0.906513i	$-0.638734\pi$
$503$	−18.9369	−0.844356	−0.422178	+	0.906513i	$0.638734\pi$
$504$	−32.4455	−1.44524
$505$	21.7015	0.965706
$506$	80.6965	3.58740
$507$	16.6535	0.739607
$508$	−15.7428	−0.698473
$509$	27.8773	1.23564	0.617820	−	0.786320i	$-0.288016\pi$
$509$	27.8773	1.23564	0.617820	+	0.786320i	$0.288016\pi$
$510$	30.7908	1.36344
$511$	14.9963	0.663395
$512$	−160.029	−7.07236
$513$	42.0842	1.85806
$514$	40.3860	1.78135
$515$	22.3889	0.986572
$516$	50.6670	2.23049
$517$	34.3861	1.51230
$518$	−33.4055	−1.46776
$519$	−1.76004	−0.0772570
$520$	−3.48636	−0.152887
$521$	−1.21574	−0.0532625	−0.0266312	−	0.999645i	$-0.508478\pi$
$521$	−1.21574	−0.0532625	−0.0266312	+	0.999645i	$0.508478\pi$
$522$	−3.75975	−0.164560
$523$	5.34586	0.233758	0.116879	−	0.993146i	$-0.462711\pi$
$523$	5.34586	0.233758	0.116879	+	0.993146i	$0.462711\pi$
$524$	35.6470	1.55725
$525$	−9.80509	−0.427929
$526$	62.4503	2.72296
$527$	3.29236	0.143417
$528$	85.7791	3.73305
$529$	40.8116	1.77442
$530$	5.24176	0.227688
$531$	14.0065	0.607830
$532$	−98.7504	−4.28137
$533$	2.46880	0.106936
$534$	−21.2123	−0.917946
$535$	−22.6063	−0.977357
$536$	168.779	7.29014
$537$	−9.54562	−0.411924
$538$	21.2354	0.915522
$539$	7.07726	0.304839
$540$	−41.4236	−1.78259
$541$	−28.6163	−1.23031	−0.615156	−	0.788406i	$-0.710907\pi$
$541$	−28.6163	−1.23031	−0.615156	+	0.788406i	$0.710907\pi$
$542$	−69.7994	−2.99814
$543$	−6.50682	−0.279234
$544$	203.422	8.72165
$545$	12.9544	0.554906
$546$	2.06707	0.0884625
$547$	−36.1887	−1.54732	−0.773658	−	0.633603i	$-0.781575\pi$
$547$	−36.1887	−1.54732	−0.773658	+	0.633603i	$0.781575\pi$
$548$	−78.0013	−3.33205
$549$	−7.41918	−0.316643
$550$	−34.2761	−1.46154
$551$	−7.52746	−0.320681
$552$	110.755	4.71405
$553$	−7.35084	−0.312590
$554$	−53.0716	−2.25480
$555$	−8.67279	−0.368139
$556$	5.84483	0.247876
$557$	14.3102	0.606343	0.303172	−	0.952936i	$-0.401954\pi$
$557$	14.3102	0.606343	0.303172	+	0.952936i	$0.401954\pi$
$558$	−1.83775	−0.0777982
$559$	1.71952	0.0727279
$560$	52.5588	2.22102
$561$	−31.2780	−1.32056
$562$	−2.73620	−0.115420
$563$	−17.8771	−0.753428	−0.376714	−	0.926330i	$-0.622946\pi$
$563$	−17.8771	−0.753428	−0.376714	+	0.926330i	$0.622946\pi$
$564$	71.7441	3.02097
$565$	12.1700	0.511995
$566$	−24.3311	−1.02271
$567$	7.11734	0.298900
$568$	−7.54072	−0.316402
$569$	−8.97747	−0.376355	−0.188178	−	0.982135i	$-0.560258\pi$
$569$	−8.97747	−0.376355	−0.188178	+	0.982135i	$0.560258\pi$
$570$	−34.4105	−1.44130
$571$	10.2213	0.427750	0.213875	−	0.976861i	$-0.431392\pi$
$571$	10.2213	0.427750	0.213875	+	0.976861i	$0.431392\pi$
$572$	5.38374	0.225106
$573$	−10.3396	−0.431943
$574$	−60.7711	−2.53654
$575$	−27.1042	−1.13032
$576$	−63.9547	−2.66478
$577$	−29.4318	−1.22526	−0.612630	−	0.790370i	$-0.709888\pi$
$577$	−29.4318	−1.22526	−0.612630	+	0.790370i	$0.709888\pi$
$578$	−79.4574	−3.30499
$579$	−7.66077	−0.318371
$580$	7.40929	0.307654
$581$	26.1882	1.08647
$582$	−12.3431	−0.511638
$583$	−5.32469	−0.220526
$584$	71.9503	2.97732
$585$	−0.434582	−0.0179677
$586$	−48.9119	−2.02053
$587$	33.8478	1.39705	0.698524	−	0.715587i	$-0.253841\pi$
$587$	33.8478	1.39705	0.698524	+	0.715587i	$0.253841\pi$
$588$	14.7662	0.608948
$589$	−3.67939	−0.151607
$590$	−37.0475	−1.52522
$591$	−10.9378	−0.449922
$592$	−98.1590	−4.03431
$593$	−30.9315	−1.27021	−0.635103	−	0.772428i	$-0.719042\pi$
$593$	−30.9315	−1.27021	−0.635103	+	0.772428i	$0.719042\pi$
$594$	56.4776	2.31730
$595$	−19.1647	−0.785678
$596$	37.8696	1.55120
$597$	8.57281	0.350862
$598$	5.71401	0.233663
$599$	7.68210	0.313882	0.156941	−	0.987608i	$-0.449837\pi$
$599$	7.68210	0.313882	0.156941	+	0.987608i	$0.449837\pi$
$600$	−47.0437	−1.92055
$601$	−3.30010	−0.134614	−0.0673069	−	0.997732i	$-0.521441\pi$
$601$	−3.30010	−0.134614	−0.0673069	+	0.997732i	$0.521441\pi$
$602$	−42.3271	−1.72512
$603$	21.0386	0.856758
$604$	80.4818	3.27476
$605$	−2.54609	−0.103513
$606$	−61.7338	−2.50777
$607$	−6.37458	−0.258736	−0.129368	−	0.991597i	$-0.541295\pi$
$607$	−6.37458	−0.258736	−0.129368	+	0.991597i	$0.541295\pi$
$608$	−227.336	−9.21967
$609$	−2.88978	−0.117100
$610$	19.6239	0.794548
$611$	2.43483	0.0985026
$612$	52.8467	2.13620
$613$	5.06113	0.204417	0.102209	−	0.994763i	$-0.467409\pi$
$613$	5.06113	0.204417	0.102209	+	0.994763i	$0.467409\pi$
$614$	10.8522	0.437958
$615$	−15.7775	−0.636209
$616$	−87.1768	−3.51245
$617$	−7.79944	−0.313994	−0.156997	−	0.987599i	$-0.550181\pi$
$617$	−7.79944	−0.313994	−0.156997	+	0.987599i	$0.550181\pi$
$618$	−63.6891	−2.56195
$619$	20.7632	0.834544	0.417272	−	0.908782i	$-0.362986\pi$
$619$	20.7632	0.834544	0.417272	+	0.908782i	$0.362986\pi$
$620$	3.62163	0.145448
$621$	44.6602	1.79215
$622$	−9.58104	−0.384165
$623$	13.2029	0.528962
$624$	6.07389	0.243150
$625$	3.47772	0.139109
$626$	63.5753	2.54098
$627$	34.9549	1.39596
$628$	120.274	4.79945
$629$	35.7921	1.42712
$630$	10.6975	0.426199
$631$	−49.7203	−1.97933	−0.989667	−	0.143387i	$-0.954201\pi$
$631$	−49.7203	−1.97933	−0.989667	+	0.143387i	$0.954201\pi$
$632$	−35.2685	−1.40291
$633$	20.2171	0.803557
$634$	−69.4670	−2.75889
$635$	3.41440	0.135497
$636$	−11.1096	−0.440524
$637$	0.501130	0.0198555
$638$	−10.1019	−0.399940
$639$	−0.939965	−0.0371844
$640$	92.5924	3.66004
$641$	−45.9298	−1.81412	−0.907060	−	0.421002i	$-0.861679\pi$
$641$	−45.9298	−1.81412	−0.907060	+	0.421002i	$0.861679\pi$
$642$	64.3077	2.53802
$643$	−21.8193	−0.860470	−0.430235	−	0.902717i	$-0.641569\pi$
$643$	−21.8193	−0.860470	−0.430235	+	0.902717i	$0.641569\pi$
$644$	−104.795	−4.12950
$645$	−10.9890	−0.432692
$646$	142.010	5.58731
$647$	−38.1597	−1.50021	−0.750107	−	0.661316i	$-0.769998\pi$
$647$	−38.1597	−1.50021	−0.750107	+	0.661316i	$0.769998\pi$
$648$	34.1482	1.34147
$649$	37.6336	1.47725
$650$	−2.42704	−0.0951965
$651$	−1.41251	−0.0553607
$652$	−58.0890	−2.27494
$653$	7.47260	0.292425	0.146213	−	0.989253i	$-0.453292\pi$
$653$	7.47260	0.292425	0.146213	+	0.989253i	$0.453292\pi$
$654$	−36.8511	−1.44099
$655$	−7.73136	−0.302089
$656$	−178.570	−6.97199
$657$	8.96874	0.349904
$658$	−59.9349	−2.33650
$659$	−4.17397	−0.162595	−0.0812973	−	0.996690i	$-0.525906\pi$
$659$	−4.17397	−0.162595	−0.0812973	+	0.996690i	$0.525906\pi$
$660$	−34.4061	−1.33926
$661$	−33.4038	−1.29926	−0.649629	−	0.760251i	$-0.725076\pi$
$661$	−33.4038	−1.29926	−0.649629	+	0.760251i	$0.725076\pi$
$662$	86.4562	3.36022
$663$	−2.21475	−0.0860137
$664$	125.648	4.87609
$665$	21.4177	0.830541
$666$	−19.9787	−0.774158
$667$	−7.98821	−0.309305
$668$	23.0368	0.891321
$669$	19.9729	0.772195
$670$	−55.6476	−2.14985
$671$	−19.9343	−0.769557
$672$	−87.2737	−3.36666
$673$	28.3551	1.09301	0.546504	−	0.837457i	$-0.315958\pi$
$673$	28.3551	1.09301	0.546504	+	0.837457i	$0.315958\pi$
$674$	7.73900	0.298095
$675$	−18.9696	−0.730140
$676$	−75.6015	−2.90775
$677$	15.8234	0.608142	0.304071	−	0.952649i	$-0.401654\pi$
$677$	15.8234	0.608142	0.304071	+	0.952649i	$0.401654\pi$
$678$	−34.6196	−1.32956
$679$	7.68255	0.294829
$680$	−91.9502	−3.52613
$681$	−15.1358	−0.580004
$682$	−4.93779	−0.189078
$683$	−44.9604	−1.72036	−0.860181	−	0.509989i	$-0.829649\pi$
$683$	−44.9604	−1.72036	−0.860181	+	0.509989i	$0.829649\pi$
$684$	−59.0592	−2.25818
$685$	16.9175	0.646383
$686$	−56.3413	−2.15112
$687$	12.4731	0.475879
$688$	−124.374	−4.74171
$689$	−0.377034	−0.0143638
$690$	−36.5167	−1.39017
$691$	−0.0803508	−0.00305669	−0.00152834	−	0.999999i	$-0.500486\pi$
$691$	−0.0803508	−0.00305669	−0.00152834	+	0.999999i	$0.500486\pi$
$692$	7.99001	0.303734
$693$	−10.8667	−0.412794
$694$	6.67768	0.253481
$695$	−1.26767	−0.0480853
$696$	−13.8648	−0.525545
$697$	65.1127	2.46632
$698$	77.7784	2.94396
$699$	7.55986	0.285940
$700$	44.5120	1.68240
$701$	43.5813	1.64604	0.823022	−	0.568010i	$-0.192286\pi$
$701$	43.5813	1.64604	0.823022	+	0.568010i	$0.192286\pi$
$702$	3.99910	0.150936
$703$	−39.9997	−1.50862
$704$	−171.838	−6.47638
$705$	−15.5604	−0.586037
$706$	4.79326	0.180397
$707$	38.4242	1.44509
$708$	78.5198	2.95096
$709$	−50.1204	−1.88231	−0.941156	−	0.337971i	$-0.890259\pi$
$709$	−50.1204	−1.88231	−0.941156	+	0.337971i	$0.890259\pi$
$710$	2.48623	0.0933065
$711$	−4.39628	−0.164873
$712$	63.3459	2.37399
$713$	−3.90461	−0.146229
$714$	54.5174	2.04026
$715$	−1.16766	−0.0436681
$716$	43.3341	1.61947
$717$	−7.39577	−0.276200
$718$	54.9254	2.04980
$719$	5.31962	0.198388	0.0991941	−	0.995068i	$-0.468374\pi$
$719$	5.31962	0.198388	0.0991941	+	0.995068i	$0.468374\pi$
$720$	31.4336	1.17146
$721$	39.6412	1.47631
$722$	−105.488	−3.92585
$723$	6.79183	0.252591
$724$	29.5389	1.09781
$725$	3.39302	0.126014
$726$	7.24281	0.268806
$727$	14.1495	0.524775	0.262387	−	0.964963i	$-0.415490\pi$
$727$	14.1495	0.524775	0.262387	+	0.964963i	$0.415490\pi$
$728$	−6.17286	−0.228782
$729$	25.8509	0.957440
$730$	−23.7225	−0.878009
$731$	45.3510	1.67737
$732$	−41.5916	−1.53727
$733$	20.0976	0.742323	0.371161	−	0.928568i	$-0.378960\pi$
$733$	20.0976	0.742323	0.371161	+	0.928568i	$0.378960\pi$
$734$	−33.2903	−1.22877
$735$	−3.20260	−0.118130
$736$	−241.251	−8.89262
$737$	56.5279	2.08223
$738$	−36.3451	−1.33788
$739$	18.5066	0.680776	0.340388	−	0.940285i	$-0.389442\pi$
$739$	18.5066	0.680776	0.340388	+	0.940285i	$0.389442\pi$
$740$	39.3717	1.44733
$741$	2.47510	0.0909252
$742$	9.28092	0.340713
$743$	−45.3608	−1.66413	−0.832064	−	0.554680i	$-0.812841\pi$
$743$	−45.3608	−1.66413	−0.832064	+	0.554680i	$0.812841\pi$
$744$	−6.77707	−0.248459
$745$	−8.21342	−0.300916
$746$	−83.2185	−3.04685
$747$	15.6623	0.573052
$748$	141.992	5.19175
$749$	−40.0262	−1.46252
$750$	38.3673	1.40097
$751$	28.9917	1.05792	0.528961	−	0.848646i	$-0.322582\pi$
$751$	28.9917	1.05792	0.528961	+	0.848646i	$0.322582\pi$
$752$	−176.113	−6.42217
$753$	15.7885	0.575365
$754$	−0.715305	−0.0260499
$755$	−17.4554	−0.635269
$756$	−73.3435	−2.66748
$757$	−8.43917	−0.306727	−0.153363	−	0.988170i	$-0.549011\pi$
$757$	−8.43917	−0.306727	−0.153363	+	0.988170i	$0.549011\pi$
$758$	−17.9637	−0.652471
$759$	37.0945	1.34644
$760$	102.759	3.72748
$761$	−6.24183	−0.226266	−0.113133	−	0.993580i	$-0.536089\pi$
$761$	−6.24183	−0.226266	−0.113133	+	0.993580i	$0.536089\pi$
$762$	−9.71287	−0.351860
$763$	22.9367	0.830366
$764$	46.9385	1.69818
$765$	−11.4618	−0.414401
$766$	−32.3971	−1.17055
$767$	2.66478	0.0962196
$768$	−140.713	−5.07755
$769$	−31.3531	−1.13062	−0.565311	−	0.824878i	$-0.691244\pi$
$769$	−31.3531	−1.13062	−0.565311	+	0.824878i	$0.691244\pi$
$770$	28.7428	1.03582
$771$	18.5646	0.668587
$772$	34.7775	1.25167
$773$	44.9568	1.61698	0.808491	−	0.588508i	$-0.200285\pi$
$773$	44.9568	1.61698	0.808491	+	0.588508i	$0.200285\pi$
$774$	−25.3143	−0.909905
$775$	1.65850	0.0595749
$776$	36.8600	1.32320
$777$	−15.3558	−0.550887
$778$	75.9842	2.72416
$779$	−72.7671	−2.60715
$780$	−2.43625	−0.0872317
$781$	−2.52556	−0.0903717
$782$	150.703	5.38911
$783$	−5.59076	−0.199798
$784$	−36.2471	−1.29454
$785$	−26.0858	−0.931043
$786$	21.9932	0.784472
$787$	6.22557	0.221918	0.110959	−	0.993825i	$-0.464608\pi$
$787$	6.22557	0.221918	0.110959	+	0.993825i	$0.464608\pi$
$788$	49.6543	1.76886
$789$	28.7071	1.02200
$790$	11.6283	0.413715
$791$	21.5478	0.766153
$792$	−52.1374	−1.85262
$793$	−1.41152	−0.0501246
$794$	−6.37443	−0.226220
$795$	2.40952	0.0854570
$796$	−38.9178	−1.37941
$797$	3.39805	0.120365	0.0601825	−	0.998187i	$-0.480832\pi$
$797$	3.39805	0.120365	0.0601825	+	0.998187i	$0.480832\pi$
$798$	−60.9263	−2.15677
$799$	64.2167	2.27182
$800$	102.472	3.62294
$801$	7.89618	0.278998
$802$	29.0550	1.02597
$803$	24.0978	0.850393
$804$	117.942	4.15948
$805$	22.7286	0.801079
$806$	−0.349638	−0.0123155
$807$	9.76145	0.343619
$808$	184.355	6.48558
$809$	3.74489	0.131663	0.0658316	−	0.997831i	$-0.479030\pi$
$809$	3.74489	0.131663	0.0658316	+	0.997831i	$0.479030\pi$
$810$	−11.2589	−0.395597
$811$	−0.925035	−0.0324824	−0.0162412	−	0.999868i	$-0.505170\pi$
$811$	−0.925035	−0.0324824	−0.0162412	+	0.999868i	$0.505170\pi$
$812$	13.1187	0.460376
$813$	−32.0853	−1.12528
$814$	−53.6801	−1.88149
$815$	12.5987	0.441314
$816$	160.194	5.60792
$817$	−50.6823	−1.77315
$818$	−100.829	−3.52541
$819$	−0.769459	−0.0268871
$820$	71.6248	2.50125
$821$	0.0161586	0.000563938 0	0.000281969	−	1.00000i	$-0.499910\pi$
$821$	0.0161586	0.000563938 0	0.000281969		1.00000i	$0.499910\pi$
$822$	−48.1247	−1.67854
$823$	9.14920	0.318921	0.159460	−	0.987204i	$-0.449025\pi$
$823$	9.14920	0.318921	0.159460	+	0.987204i	$0.449025\pi$
$824$	190.194	6.62571
$825$	−15.7560	−0.548554
$826$	−65.5953	−2.28235
$827$	22.5525	0.784229	0.392114	−	0.919916i	$-0.371744\pi$
$827$	22.5525	0.784229	0.392114	+	0.919916i	$0.371744\pi$
$828$	−62.6742	−2.17808
$829$	−31.3526	−1.08892	−0.544461	−	0.838786i	$-0.683266\pi$
$829$	−31.3526	−1.08892	−0.544461	+	0.838786i	$0.683266\pi$
$830$	−41.4270	−1.43795
$831$	−24.3959	−0.846283
$832$	−12.1676	−0.421835
$833$	13.2169	0.457940
$834$	3.60610	0.124869
$835$	−4.99638	−0.172907
$836$	−158.684	−5.48821
$837$	−2.73274	−0.0944574
$838$	−71.1903	−2.45923
$839$	−51.6164	−1.78200	−0.890999	−	0.454006i	$-0.849995\pi$
$839$	−51.6164	−1.78200	−0.890999	+	0.454006i	$0.849995\pi$
$840$	39.4492	1.36113
$841$	1.00000	0.0344828
$842$	−49.6051	−1.70950
$843$	−1.25777	−0.0433199
$844$	−91.7793	−3.15917
$845$	16.3970	0.564073
$846$	−35.8449	−1.23237
$847$	−4.50805	−0.154898
$848$	27.2711	0.936494
$849$	−11.1845	−0.383850
$850$	−64.0114	−2.19557
$851$	−42.4481	−1.45510
$852$	−5.26941	−0.180527
$853$	11.3500	0.388615	0.194308	−	0.980941i	$-0.437754\pi$
$853$	11.3500	0.388615	0.194308	+	0.980941i	$0.437754\pi$
$854$	34.7455	1.18897
$855$	12.8092	0.438064
$856$	−192.041	−6.56382
$857$	17.6282	0.602168	0.301084	−	0.953598i	$-0.402652\pi$
$857$	17.6282	0.602168	0.301084	+	0.953598i	$0.402652\pi$
$858$	3.32162	0.113398
$859$	37.8704	1.29212	0.646061	−	0.763285i	$-0.276415\pi$
$859$	37.8704	1.29212	0.646061	+	0.763285i	$0.276415\pi$
$860$	49.8866	1.70112
$861$	−27.9352	−0.952028
$862$	−82.2603	−2.80180
$863$	−2.54389	−0.0865950	−0.0432975	−	0.999062i	$-0.513786\pi$
$863$	−2.54389	−0.0865950	−0.0432975	+	0.999062i	$0.513786\pi$
$864$	−168.846	−5.74425
$865$	−1.73293	−0.0589213
$866$	0.313321	0.0106471
$867$	−36.5248	−1.24045
$868$	6.41237	0.217650
$869$	−11.8122	−0.400702
$870$	4.57133	0.154983
$871$	4.00266	0.135625
$872$	110.048	3.72669
$873$	4.59466	0.155506
$874$	−168.418	−5.69684
$875$	−23.8804	−0.807306
$876$	50.2784	1.69875
$877$	5.38599	0.181872	0.0909360	−	0.995857i	$-0.471014\pi$
$877$	5.38599	0.181872	0.0909360	+	0.995857i	$0.471014\pi$
$878$	−6.12010	−0.206544
$879$	−22.4837	−0.758357
$880$	84.4579	2.84708
$881$	−41.1013	−1.38474	−0.692370	−	0.721543i	$-0.743433\pi$
$881$	−41.1013	−1.38474	−0.692370	+	0.721543i	$0.743433\pi$
$882$	−7.37752	−0.248414
$883$	−2.41457	−0.0812568	−0.0406284	−	0.999174i	$-0.512936\pi$
$883$	−2.41457	−0.0812568	−0.0406284	+	0.999174i	$0.512936\pi$
$884$	10.0543	0.338161
$885$	−17.0299	−0.572454
$886$	84.6381	2.84347
$887$	7.35040	0.246802	0.123401	−	0.992357i	$-0.460620\pi$
$887$	7.35040	0.246802	0.123401	+	0.992357i	$0.460620\pi$
$888$	−73.6754	−2.47238
$889$	6.04545	0.202758
$890$	−20.8856	−0.700086
$891$	11.4370	0.383154
$892$	−90.6705	−3.03587
$893$	−71.7658	−2.40155
$894$	23.3645	0.781426
$895$	−9.39860	−0.314161
$896$	163.942	5.47691
$897$	2.62661	0.0876998
$898$	−109.078	−3.63999
$899$	0.488796	0.0163023
$900$	26.6211	0.887369
$901$	−9.94397	−0.331282
$902$	−97.6544	−3.25154
$903$	−19.4568	−0.647483
$904$	103.384	3.43850
$905$	−6.40660	−0.212963
$906$	49.6551	1.64968
$907$	−55.5964	−1.84605	−0.923024	−	0.384742i	$-0.874291\pi$
$907$	−55.5964	−1.84605	−0.923024	+	0.384742i	$0.874291\pi$
$908$	68.7117	2.28028
$909$	22.9802	0.762204
$910$	2.03524	0.0674674
$911$	25.0319	0.829344	0.414672	−	0.909971i	$-0.363896\pi$
$911$	25.0319	0.829344	0.414672	+	0.909971i	$0.363896\pi$
$912$	−179.026	−5.92815
$913$	42.0824	1.39273
$914$	−3.01074	−0.0995864
$915$	9.02068	0.298214
$916$	−56.6240	−1.87091
$917$	−13.6889	−0.452049
$918$	105.473	3.48113
$919$	43.4472	1.43319	0.716595	−	0.697490i	$-0.245700\pi$
$919$	43.4472	1.43319	0.716595	+	0.697490i	$0.245700\pi$
$920$	109.049	3.59525
$921$	4.98851	0.164377
$922$	41.2000	1.35685
$923$	−0.178831	−0.00588631
$924$	−60.9186	−2.00407
$925$	18.0300	0.592822
$926$	4.01736	0.132019
$927$	23.7080	0.778673
$928$	30.2008	0.991391
$929$	48.0355	1.57599	0.787997	−	0.615679i	$-0.211118\pi$
$929$	48.0355	1.57599	0.787997	+	0.615679i	$0.211118\pi$
$930$	2.23445	0.0732704
$931$	−14.7707	−0.484089
$932$	−34.3194	−1.12417
$933$	−4.40420	−0.144187
$934$	81.2361	2.65813
$935$	−30.7962	−1.00714
$936$	−3.69177	−0.120669
$937$	7.90586	0.258273	0.129137	−	0.991627i	$-0.458779\pi$
$937$	7.90586	0.258273	0.129137	+	0.991627i	$0.458779\pi$
$938$	−98.5281	−3.21706
$939$	29.2242	0.953695
$940$	70.6392	2.30400
$941$	0.808864	0.0263682	0.0131841	−	0.999913i	$-0.495803\pi$
$941$	0.808864	0.0263682	0.0131841	+	0.999913i	$0.495803\pi$
$942$	74.2057	2.41775
$943$	−77.2212	−2.51467
$944$	−192.745	−6.27333
$945$	15.9072	0.517463
$946$	−68.0162	−2.21140
$947$	20.6038	0.669532	0.334766	−	0.942301i	$-0.391343\pi$
$947$	20.6038	0.669532	0.334766	+	0.942301i	$0.391343\pi$
$948$	−24.6454	−0.800445
$949$	1.70633	0.0553899
$950$	71.5364	2.32095
$951$	−31.9325	−1.03548
$952$	−162.805	−5.27652
$953$	43.5705	1.41139	0.705693	−	0.708518i	$-0.250636\pi$
$953$	43.5705	1.41139	0.705693	+	0.708518i	$0.250636\pi$
$954$	5.55060	0.179707
$955$	−10.1803	−0.329428
$956$	33.5745	1.08588
$957$	−4.64365	−0.150108
$958$	−48.6577	−1.57206
$959$	29.9536	0.967253
$960$	77.7599	2.50969
$961$	−30.7611	−0.992293
$962$	−3.80101	−0.122549
$963$	−23.9383	−0.771400
$964$	−30.8328	−0.993057
$965$	−7.54278	−0.242811
$966$	−64.6556	−2.08026
$967$	−2.68832	−0.0864504	−0.0432252	−	0.999065i	$-0.513763\pi$
$967$	−2.68832	−0.0864504	−0.0432252	+	0.999065i	$0.513763\pi$
$968$	−21.6291	−0.695185
$969$	65.2790	2.09706
$970$	−12.1530	−0.390209
$971$	14.8856	0.477701	0.238851	−	0.971056i	$-0.423229\pi$
$971$	14.8856	0.477701	0.238851	+	0.971056i	$0.423229\pi$
$972$	−74.1686	−2.37896
$973$	−2.24450	−0.0719552
$974$	44.4669	1.42481
$975$	−1.11566	−0.0357297
$976$	102.096	3.26803
$977$	51.0705	1.63389	0.816944	−	0.576716i	$-0.195666\pi$
$977$	51.0705	1.63389	0.816944	+	0.576716i	$0.195666\pi$
$978$	−35.8393	−1.14601
$979$	21.2160	0.678066
$980$	14.5388	0.464424
$981$	13.7177	0.437971
$982$	−19.9319	−0.636052
$983$	−10.9323	−0.348686	−0.174343	−	0.984685i	$-0.555780\pi$
$983$	−10.9323	−0.348686	−0.174343	+	0.984685i	$0.555780\pi$
$984$	−134.030	−4.27271
$985$	−10.7694	−0.343140
$986$	−18.8656	−0.600803
$987$	−27.5508	−0.876950
$988$	−11.2362	−0.357471
$989$	−53.7845	−1.71025
$990$	17.1901	0.546336
$991$	−13.7923	−0.438127	−0.219064	−	0.975711i	$-0.570300\pi$
$991$	−13.7923	−0.438127	−0.219064	+	0.975711i	$0.570300\pi$
$992$	14.7620	0.468696
$993$	39.7421	1.26118
$994$	4.40205	0.139625
$995$	8.44077	0.267590
$996$	87.8021	2.78212
$997$	7.02525	0.222492	0.111246	−	0.993793i	$-0.464516\pi$
$997$	7.02525	0.222492	0.111246	+	0.993793i	$0.464516\pi$
$998$	68.6977	2.17459
$999$	−29.7084	−0.939932

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.e.1.2	✓	103

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.e.1.2	✓	103	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:	\( N \)	\(=\)	\( 4031 = 29 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.80086 q^{2} -1.28750 q^{3} +5.84483 q^{4} -1.26767 q^{5} +3.60610 q^{6} -2.24450 q^{7} -10.7688 q^{8} -1.34235 q^{9} +O(q^{10})\)	\(q-2.80086 q^{2} -1.28750 q^{3} +5.84483 q^{4} -1.26767 q^{5} +3.60610 q^{6} -2.24450 q^{7} -10.7688 q^{8} -1.34235 q^{9} +3.55056 q^{10} -3.60673 q^{11} -7.52519 q^{12} -0.255387 q^{13} +6.28652 q^{14} +1.63211 q^{15} +18.4723 q^{16} -6.73564 q^{17} +3.75975 q^{18} +7.52746 q^{19} -7.40929 q^{20} +2.88978 q^{21} +10.1019 q^{22} +7.98821 q^{23} +13.8648 q^{24} -3.39302 q^{25} +0.715305 q^{26} +5.59076 q^{27} -13.1187 q^{28} -1.00000 q^{29} -4.57133 q^{30} -0.488796 q^{31} -30.2008 q^{32} +4.64365 q^{33} +18.8656 q^{34} +2.84527 q^{35} -7.84583 q^{36} -5.31384 q^{37} -21.0834 q^{38} +0.328810 q^{39} +13.6513 q^{40} -9.66689 q^{41} -8.09387 q^{42} -6.73298 q^{43} -21.0807 q^{44} +1.70166 q^{45} -22.3739 q^{46} -9.53386 q^{47} -23.7831 q^{48} -1.96224 q^{49} +9.50339 q^{50} +8.67211 q^{51} -1.49269 q^{52} +1.47632 q^{53} -15.6590 q^{54} +4.57213 q^{55} +24.1706 q^{56} -9.69157 q^{57} +2.80086 q^{58} -10.4343 q^{59} +9.53943 q^{60} +5.52699 q^{61} +1.36905 q^{62} +3.01291 q^{63} +47.6437 q^{64} +0.323746 q^{65} -13.0062 q^{66} -15.6729 q^{67} -39.3687 q^{68} -10.2848 q^{69} -7.96921 q^{70} +0.700236 q^{71} +14.4556 q^{72} -6.68135 q^{73} +14.8833 q^{74} +4.36850 q^{75} +43.9967 q^{76} +8.09529 q^{77} -0.920952 q^{78} +3.27505 q^{79} -23.4168 q^{80} -3.17102 q^{81} +27.0756 q^{82} -11.6678 q^{83} +16.8903 q^{84} +8.53855 q^{85} +18.8582 q^{86} +1.28750 q^{87} +38.8402 q^{88} -5.88234 q^{89} -4.76611 q^{90} +0.573216 q^{91} +46.6897 q^{92} +0.629323 q^{93} +26.7030 q^{94} -9.54230 q^{95} +38.8834 q^{96} -3.42284 q^{97} +5.49596 q^{98} +4.84151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 103 q + q^{2} + 2 q^{3} + 127 q^{4} + 9 q^{5} + 19 q^{6} + 18 q^{7} + 149 q^{9}+O(q^{10}) \) 103 * q + q^2 + 2 * q^3 + 127 * q^4 + 9 * q^5 + 19 * q^6 + 18 * q^7 + 149 * q^9	\( 103 q + q^{2} + 2 q^{3} + 127 q^{4} + 9 q^{5} + 19 q^{6} + 18 q^{7} + 149 q^{9} + 20 q^{10} + 9 q^{11} + 36 q^{13} - 10 q^{14} + 16 q^{15} + 179 q^{16} + 21 q^{17} + 7 q^{18} + 42 q^{19} + 24 q^{20} + 28 q^{21} + 32 q^{22} + 25 q^{23} + 68 q^{24} + 194 q^{25} - 5 q^{26} + 14 q^{27} + 59 q^{28} - 103 q^{29} + 84 q^{30} + 34 q^{31} + 11 q^{32} + 42 q^{33} + 54 q^{34} + 35 q^{35} + 214 q^{36} + 34 q^{37} + 9 q^{38} + 23 q^{39} + 46 q^{40} + 16 q^{41} + 13 q^{42} + 68 q^{43} - 6 q^{44} + 25 q^{45} + 60 q^{46} + 6 q^{47} + 5 q^{48} + 257 q^{49} - 51 q^{50} + 68 q^{51} + 37 q^{52} + 35 q^{53} + 30 q^{54} + 66 q^{55} - 54 q^{56} + 78 q^{57} - q^{58} + 10 q^{59} - 24 q^{60} + 70 q^{61} + 29 q^{62} + 26 q^{63} + 276 q^{64} + 95 q^{65} + 77 q^{66} + 71 q^{67} - 21 q^{68} - 20 q^{69} + 48 q^{70} + 32 q^{71} + 32 q^{72} + 94 q^{73} + 35 q^{74} + 7 q^{75} + 134 q^{76} + 17 q^{77} + 58 q^{78} + 110 q^{79} + 78 q^{80} + 267 q^{81} - 71 q^{82} + 35 q^{83} + 96 q^{84} + 71 q^{85} + 33 q^{86} - 2 q^{87} + 100 q^{88} + 22 q^{89} - 134 q^{90} + 108 q^{91} - 11 q^{92} + 78 q^{93} + 90 q^{94} + 12 q^{95} + 177 q^{96} + 44 q^{97} - 18 q^{98} + 83 q^{99}+O(q^{100}) \) 103 * q + q^2 + 2 * q^3 + 127 * q^4 + 9 * q^5 + 19 * q^6 + 18 * q^7 + 149 * q^9 + 20 * q^10 + 9 * q^11 + 36 * q^13 - 10 * q^14 + 16 * q^15 + 179 * q^16 + 21 * q^17 + 7 * q^18 + 42 * q^19 + 24 * q^20 + 28 * q^21 + 32 * q^22 + 25 * q^23 + 68 * q^24 + 194 * q^25 - 5 * q^26 + 14 * q^27 + 59 * q^28 - 103 * q^29 + 84 * q^30 + 34 * q^31 + 11 * q^32 + 42 * q^33 + 54 * q^34 + 35 * q^35 + 214 * q^36 + 34 * q^37 + 9 * q^38 + 23 * q^39 + 46 * q^40 + 16 * q^41 + 13 * q^42 + 68 * q^43 - 6 * q^44 + 25 * q^45 + 60 * q^46 + 6 * q^47 + 5 * q^48 + 257 * q^49 - 51 * q^50 + 68 * q^51 + 37 * q^52 + 35 * q^53 + 30 * q^54 + 66 * q^55 - 54 * q^56 + 78 * q^57 - q^58 + 10 * q^59 - 24 * q^60 + 70 * q^61 + 29 * q^62 + 26 * q^63 + 276 * q^64 + 95 * q^65 + 77 * q^66 + 71 * q^67 - 21 * q^68 - 20 * q^69 + 48 * q^70 + 32 * q^71 + 32 * q^72 + 94 * q^73 + 35 * q^74 + 7 * q^75 + 134 * q^76 + 17 * q^77 + 58 * q^78 + 110 * q^79 + 78 * q^80 + 267 * q^81 - 71 * q^82 + 35 * q^83 + 96 * q^84 + 71 * q^85 + 33 * q^86 - 2 * q^87 + 100 * q^88 + 22 * q^89 - 134 * q^90 + 108 * q^91 - 11 * q^92 + 78 * q^93 + 90 * q^94 + 12 * q^95 + 177 * q^96 + 44 * q^97 - 18 * q^98 + 83 * q^99

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