LMFDB - Embedded newform 4031.2.a.b.1.15

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:	$1$
Dimension:	$59$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	$q-1.54716 q^{2} -2.20924 q^{3} +0.393702 q^{4} +3.10213 q^{5} +3.41805 q^{6} -0.403081 q^{7} +2.48520 q^{8} +1.88074 q^{9} +O(q^{10})$	\(q-1.54716 q^{2} -2.20924 q^{3} +0.393702 q^{4} +3.10213 q^{5} +3.41805 q^{6} -0.403081 q^{7} +2.48520 q^{8} +1.88074 q^{9} -4.79950 q^{10} +4.11840 q^{11} -0.869782 q^{12} +2.34927 q^{13} +0.623630 q^{14} -6.85336 q^{15} -4.63240 q^{16} -4.47742 q^{17} -2.90980 q^{18} +3.90241 q^{19} +1.22132 q^{20} +0.890501 q^{21} -6.37182 q^{22} +1.73861 q^{23} -5.49040 q^{24} +4.62323 q^{25} -3.63470 q^{26} +2.47272 q^{27} -0.158694 q^{28} +1.00000 q^{29} +10.6032 q^{30} -7.67771 q^{31} +2.19667 q^{32} -9.09853 q^{33} +6.92729 q^{34} -1.25041 q^{35} +0.740450 q^{36} -2.11197 q^{37} -6.03765 q^{38} -5.19011 q^{39} +7.70942 q^{40} -5.91537 q^{41} -1.37775 q^{42} -9.74382 q^{43} +1.62142 q^{44} +5.83430 q^{45} -2.68991 q^{46} -5.49237 q^{47} +10.2341 q^{48} -6.83753 q^{49} -7.15288 q^{50} +9.89170 q^{51} +0.924914 q^{52} -3.43964 q^{53} -3.82569 q^{54} +12.7758 q^{55} -1.00174 q^{56} -8.62136 q^{57} -1.54716 q^{58} -4.94333 q^{59} -2.69818 q^{60} -6.07693 q^{61} +11.8786 q^{62} -0.758089 q^{63} +5.86621 q^{64} +7.28776 q^{65} +14.0769 q^{66} +3.76245 q^{67} -1.76277 q^{68} -3.84101 q^{69} +1.93458 q^{70} -13.6384 q^{71} +4.67401 q^{72} +7.18853 q^{73} +3.26756 q^{74} -10.2138 q^{75} +1.53639 q^{76} -1.66005 q^{77} +8.02993 q^{78} -5.44675 q^{79} -14.3703 q^{80} -11.1050 q^{81} +9.15203 q^{82} -1.60444 q^{83} +0.350592 q^{84} -13.8896 q^{85} +15.0752 q^{86} -2.20924 q^{87} +10.2350 q^{88} -3.76861 q^{89} -9.02659 q^{90} -0.946947 q^{91} +0.684495 q^{92} +16.9619 q^{93} +8.49757 q^{94} +12.1058 q^{95} -4.85296 q^{96} -0.150331 q^{97} +10.5787 q^{98} +7.74563 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) $ 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9} - 18 q^{10} - 27 q^{11} - 8 q^{12} - 22 q^{13} - 24 q^{14} - 18 q^{15} + 5 q^{16} - 23 q^{17} + q^{18} - 32 q^{19} - 14 q^{20} - 36 q^{21} - 6 q^{22} - 3 q^{23} - 18 q^{24} - 8 q^{25} - q^{26} - 12 q^{27} - 9 q^{28} + 59 q^{29} - 18 q^{30} - 32 q^{31} - 39 q^{32} - 12 q^{33} - 18 q^{34} - 9 q^{35} + 10 q^{36} - 44 q^{37} + 5 q^{38} - 27 q^{39} - 68 q^{40} - 44 q^{41} - 25 q^{42} - 40 q^{43} - 56 q^{44} - 39 q^{45} - 40 q^{46} - 20 q^{47} - 9 q^{48} - 39 q^{49} - 21 q^{50} - 28 q^{51} - 49 q^{52} - 31 q^{53} - 32 q^{54} - 32 q^{55} - 48 q^{56} - 58 q^{57} - 5 q^{58} + 6 q^{59} - 44 q^{60} - 88 q^{61} + 35 q^{62} - 22 q^{63} - 10 q^{64} - 43 q^{65} - 31 q^{66} - 45 q^{67} - 29 q^{68} - 60 q^{69} - 14 q^{70} - 20 q^{71} - 4 q^{72} - 90 q^{73} - 25 q^{74} + 15 q^{75} - 64 q^{76} - 39 q^{77} - 28 q^{78} - 120 q^{79} + 24 q^{80} - 77 q^{81} - 71 q^{82} - 33 q^{83} - 14 q^{84} - 71 q^{85} - 61 q^{86} - 6 q^{87} - 34 q^{88} - 78 q^{89} - 88 q^{90} - 28 q^{91} - 31 q^{92} - 36 q^{93} - 4 q^{94} - 12 q^{95} - 29 q^{96} - 48 q^{97} - 4 q^{98} - 43 q^{99}+O(q^{100}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9 - 18 * q^10 - 27 * q^11 - 8 * q^12 - 22 * q^13 - 24 * q^14 - 18 * q^15 + 5 * q^16 - 23 * q^17 + q^18 - 32 * q^19 - 14 * q^20 - 36 * q^21 - 6 * q^22 - 3 * q^23 - 18 * q^24 - 8 * q^25 - q^26 - 12 * q^27 - 9 * q^28 + 59 * q^29 - 18 * q^30 - 32 * q^31 - 39 * q^32 - 12 * q^33 - 18 * q^34 - 9 * q^35 + 10 * q^36 - 44 * q^37 + 5 * q^38 - 27 * q^39 - 68 * q^40 - 44 * q^41 - 25 * q^42 - 40 * q^43 - 56 * q^44 - 39 * q^45 - 40 * q^46 - 20 * q^47 - 9 * q^48 - 39 * q^49 - 21 * q^50 - 28 * q^51 - 49 * q^52 - 31 * q^53 - 32 * q^54 - 32 * q^55 - 48 * q^56 - 58 * q^57 - 5 * q^58 + 6 * q^59 - 44 * q^60 - 88 * q^61 + 35 * q^62 - 22 * q^63 - 10 * q^64 - 43 * q^65 - 31 * q^66 - 45 * q^67 - 29 * q^68 - 60 * q^69 - 14 * q^70 - 20 * q^71 - 4 * q^72 - 90 * q^73 - 25 * q^74 + 15 * q^75 - 64 * q^76 - 39 * q^77 - 28 * q^78 - 120 * q^79 + 24 * q^80 - 77 * q^81 - 71 * q^82 - 33 * q^83 - 14 * q^84 - 71 * q^85 - 61 * q^86 - 6 * q^87 - 34 * q^88 - 78 * q^89 - 88 * q^90 - 28 * q^91 - 31 * q^92 - 36 * q^93 - 4 * q^94 - 12 * q^95 - 29 * q^96 - 48 * q^97 - 4 * q^98 - 43 * 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 $

$2$	−1.54716	−1.09401	−0.547003	−	0.837130i	$-0.684232\pi$
$2$	−1.54716	−1.09401	−0.547003	+	0.837130i	$0.684232\pi$
$3$	−2.20924	−1.27550	−0.637752	−	0.770241i	$-0.720136\pi$
$3$	−2.20924	−1.27550	−0.637752	+	0.770241i	$0.720136\pi$
$4$	0.393702	0.196851
$5$	3.10213	1.38732	0.693658	−	0.720304i	$-0.255998\pi$
$5$	3.10213	1.38732	0.693658	+	0.720304i	$0.255998\pi$
$6$	3.41805	1.39541
$7$	−0.403081	−0.152350	−0.0761751	−	0.997094i	$-0.524271\pi$
$7$	−0.403081	−0.152350	−0.0761751	+	0.997094i	$0.524271\pi$
$8$	2.48520	0.878651
$9$	1.88074	0.626913
$10$	−4.79950	−1.51773
$11$	4.11840	1.24174	0.620872	−	0.783912i	$-0.286779\pi$
$11$	4.11840	1.24174	0.620872	+	0.783912i	$0.286779\pi$
$12$	−0.869782	−0.251084
$13$	2.34927	0.651571	0.325786	−	0.945444i	$-0.394371\pi$
$13$	2.34927	0.651571	0.325786	+	0.945444i	$0.394371\pi$
$14$	0.623630	0.166672
$15$	−6.85336	−1.76953
$16$	−4.63240	−1.15810
$17$	−4.47742	−1.08593	−0.542967	−	0.839754i	$-0.682699\pi$
$17$	−4.47742	−1.08593	−0.542967	+	0.839754i	$0.682699\pi$
$18$	−2.90980	−0.685847
$19$	3.90241	0.895275	0.447637	−	0.894215i	$-0.352266\pi$
$19$	3.90241	0.895275	0.447637	+	0.894215i	$0.352266\pi$
$20$	1.22132	0.273095
$21$	0.890501	0.194323
$22$	−6.37182	−1.35848
$23$	1.73861	0.362526	0.181263	−	0.983435i	$-0.441981\pi$
$23$	1.73861	0.362526	0.181263	+	0.983435i	$0.441981\pi$
$24$	−5.49040	−1.12072
$25$	4.62323	0.924647
$26$	−3.63470	−0.712824
$27$	2.47272	0.475875
$28$	−0.158694	−0.0299903
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	10.6032	1.93588
$31$	−7.67771	−1.37896	−0.689478	−	0.724306i	$-0.742160\pi$
$31$	−7.67771	−1.37896	−0.689478	+	0.724306i	$0.742160\pi$
$32$	2.19667	0.388320
$33$	−9.09853	−1.58385
$34$	6.92729	1.18802
$35$	−1.25041	−0.211358
$36$	0.740450	0.123408
$37$	−2.11197	−0.347206	−0.173603	−	0.984816i	$-0.555541\pi$
$37$	−2.11197	−0.347206	−0.173603	+	0.984816i	$0.555541\pi$
$38$	−6.03765	−0.979437
$39$	−5.19011	−0.831083
$40$	7.70942	1.21897
$41$	−5.91537	−0.923826	−0.461913	−	0.886925i	$-0.652837\pi$
$41$	−5.91537	−0.923826	−0.461913	+	0.886925i	$0.652837\pi$
$42$	−1.37775	−0.212591
$43$	−9.74382	−1.48592	−0.742959	−	0.669337i	$-0.766578\pi$
$43$	−9.74382	−1.48592	−0.742959	+	0.669337i	$0.766578\pi$
$44$	1.62142	0.244439
$45$	5.83430	0.869726
$46$	−2.68991	−0.396606
$47$	−5.49237	−0.801144	−0.400572	−	0.916265i	$-0.631189\pi$
$47$	−5.49237	−0.801144	−0.400572	+	0.916265i	$0.631189\pi$
$48$	10.2341	1.47716
$49$	−6.83753	−0.976789
$50$	−7.15288	−1.01157
$51$	9.89170	1.38512
$52$	0.924914	0.128262
$53$	−3.43964	−0.472471	−0.236236	−	0.971696i	$-0.575914\pi$
$53$	−3.43964	−0.472471	−0.236236	+	0.971696i	$0.575914\pi$
$54$	−3.82569	−0.520610
$55$	12.7758	1.72269
$56$	−1.00174	−0.133863
$57$	−8.62136	−1.14193
$58$	−1.54716	−0.203152
$59$	−4.94333	−0.643566	−0.321783	−	0.946813i	$-0.604282\pi$
$59$	−4.94333	−0.643566	−0.321783	+	0.946813i	$0.604282\pi$
$60$	−2.69818	−0.348334
$61$	−6.07693	−0.778072	−0.389036	−	0.921223i	$-0.627192\pi$
$61$	−6.07693	−0.778072	−0.389036	+	0.921223i	$0.627192\pi$
$62$	11.8786	1.50859
$63$	−0.758089	−0.0955102
$64$	5.86621	0.733276
$65$	7.28776	0.903936
$66$	14.0769	1.73274
$67$	3.76245	0.459656	0.229828	−	0.973231i	$-0.426184\pi$
$67$	3.76245	0.459656	0.229828	+	0.973231i	$0.426184\pi$
$68$	−1.76277	−0.213767
$69$	−3.84101	−0.462404
$70$	1.93458	0.231227
$71$	−13.6384	−1.61859	−0.809293	−	0.587406i	$-0.800149\pi$
$71$	−13.6384	−1.61859	−0.809293	+	0.587406i	$0.800149\pi$
$72$	4.67401	0.550837
$73$	7.18853	0.841354	0.420677	−	0.907211i	$-0.361793\pi$
$73$	7.18853	0.841354	0.420677	+	0.907211i	$0.361793\pi$
$74$	3.26756	0.379846
$75$	−10.2138	−1.17939
$76$	1.53639	0.176236
$77$	−1.66005	−0.189180
$78$	8.02993	0.909210
$79$	−5.44675	−0.612807	−0.306404	−	0.951902i	$-0.599126\pi$
$79$	−5.44675	−0.612807	−0.306404	+	0.951902i	$0.599126\pi$
$80$	−14.3703	−1.60665
$81$	−11.1050	−1.23389
$82$	9.15203	1.01067
$83$	−1.60444	−0.176110	−0.0880551	−	0.996116i	$-0.528065\pi$
$83$	−1.60444	−0.176110	−0.0880551	+	0.996116i	$0.528065\pi$
$84$	0.350592	0.0382527
$85$	−13.8896	−1.50653
$86$	15.0752	1.62560
$87$	−2.20924	−0.236855
$88$	10.2350	1.09106
$89$	−3.76861	−0.399472	−0.199736	−	0.979850i	$-0.564008\pi$
$89$	−3.76861	−0.399472	−0.199736	+	0.979850i	$0.564008\pi$
$90$	−9.02659	−0.951487
$91$	−0.946947	−0.0992670
$92$	0.684495	0.0713636
$93$	16.9619	1.75887
$94$	8.49757	0.876458
$95$	12.1058	1.24203
$96$	−4.85296	−0.495304
$97$	−0.150331	−0.0152638	−0.00763188	−	0.999971i	$-0.502429\pi$
$97$	−0.150331	−0.0152638	−0.00763188	+	0.999971i	$0.502429\pi$
$98$	10.5787	1.06861
$99$	7.74563	0.778466
$100$	1.82018	0.182018
$101$	−3.74186	−0.372329	−0.186165	−	0.982519i	$-0.559606\pi$
$101$	−3.74186	−0.372329	−0.186165	+	0.982519i	$0.559606\pi$
$102$	−15.3040	−1.51533
$103$	12.3483	1.21672	0.608359	−	0.793662i	$-0.291828\pi$
$103$	12.3483	1.21672	0.608359	+	0.793662i	$0.291828\pi$
$104$	5.83841	0.572504
$105$	2.76245	0.269588
$106$	5.32168	0.516887
$107$	9.37089	0.905918	0.452959	−	0.891531i	$-0.350368\pi$
$107$	9.37089	0.905918	0.452959	+	0.891531i	$0.350368\pi$
$108$	0.973513	0.0936764
$109$	0.610685	0.0584931	0.0292465	−	0.999572i	$-0.490689\pi$
$109$	0.610685	0.0584931	0.0292465	+	0.999572i	$0.490689\pi$
$110$	−19.7662	−1.88464
$111$	4.66586	0.442863
$112$	1.86723	0.176437
$113$	−13.0639	−1.22894	−0.614472	−	0.788939i	$-0.710631\pi$
$113$	−13.0639	−1.22894	−0.614472	+	0.788939i	$0.710631\pi$
$114$	13.3386	1.24928
$115$	5.39341	0.502938
$116$	0.393702	0.0365543
$117$	4.41837	0.408478
$118$	7.64811	0.704066
$119$	1.80476	0.165442
$120$	−17.0320	−1.55480
$121$	5.96122	0.541929
$122$	9.40198	0.851216
$123$	13.0685	1.17834
$124$	−3.02273	−0.271449
$125$	−1.16878	−0.104539
$126$	1.17288	0.104489
$127$	3.56323	0.316185	0.158093	−	0.987424i	$-0.449466\pi$
$127$	3.56323	0.316185	0.158093	+	0.987424i	$0.449466\pi$
$128$	−13.4693	−1.19053
$129$	21.5264	1.89530
$130$	−11.2753	−0.988912
$131$	−1.19895	−0.104753	−0.0523764	−	0.998627i	$-0.516680\pi$
$131$	−1.19895	−0.104753	−0.0523764	+	0.998627i	$0.516680\pi$
$132$	−3.58211	−0.311783
$133$	−1.57299	−0.136395
$134$	−5.82111	−0.502867
$135$	7.67070	0.660189
$136$	−11.1273	−0.954157
$137$	15.8164	1.35129	0.675644	−	0.737228i	$-0.263866\pi$
$137$	15.8164	1.35129	0.675644	+	0.737228i	$0.263866\pi$
$138$	5.94266	0.505873
$139$	1.00000	0.0848189
$139$	1.00000	0.0848189
$140$	−0.492289	−0.0416060
$141$	12.1340	1.02186
$142$	21.1008	1.77074
$143$	9.67525	0.809085
$144$	−8.71234	−0.726028
$145$	3.10213	0.257618
$146$	−11.1218	−0.920447
$147$	15.1057	1.24590
$148$	−0.831488	−0.0683479
$149$	−5.70689	−0.467527	−0.233763	−	0.972294i	$-0.575104\pi$
$149$	−5.70689	−0.467527	−0.233763	+	0.972294i	$0.575104\pi$
$150$	15.8024	1.29026
$151$	23.5478	1.91629	0.958145	−	0.286283i	$-0.0924197\pi$
$151$	23.5478	1.91629	0.958145	+	0.286283i	$0.0924197\pi$
$152$	9.69827	0.786634
$153$	−8.42086	−0.680786
$154$	2.56836	0.206964
$155$	−23.8173	−1.91305
$156$	−2.04336	−0.163599
$157$	−9.30160	−0.742349	−0.371174	−	0.928563i	$-0.621045\pi$
$157$	−9.30160	−0.742349	−0.371174	+	0.928563i	$0.621045\pi$
$158$	8.42699	0.670415
$159$	7.59900	0.602640
$160$	6.81436	0.538722
$161$	−0.700801	−0.0552309
$162$	17.1813	1.34989
$163$	1.44455	0.113146	0.0565730	−	0.998398i	$-0.481983\pi$
$163$	1.44455	0.113146	0.0565730	+	0.998398i	$0.481983\pi$
$164$	−2.32889	−0.181856
$165$	−28.2249	−2.19730
$166$	2.48233	0.192666
$167$	19.6540	1.52087	0.760437	−	0.649412i	$-0.224985\pi$
$167$	19.6540	1.52087	0.760437	+	0.649412i	$0.224985\pi$
$168$	2.21307	0.170742
$169$	−7.48091	−0.575455
$170$	21.4894	1.64816
$171$	7.33942	0.561259
$172$	−3.83616	−0.292504
$173$	−14.0744	−1.07006	−0.535028	−	0.844834i	$-0.679699\pi$
$173$	−14.0744	−1.07006	−0.535028	+	0.844834i	$0.679699\pi$
$174$	3.41805	0.259121
$175$	−1.86354	−0.140870
$176$	−19.0781	−1.43807
$177$	10.9210	0.820872
$178$	5.83064	0.437025
$179$	−16.4202	−1.22730	−0.613651	−	0.789577i	$-0.710300\pi$
$179$	−16.4202	−1.22730	−0.613651	+	0.789577i	$0.710300\pi$
$180$	2.29698	0.171207
$181$	−6.68406	−0.496822	−0.248411	−	0.968655i	$-0.579908\pi$
$181$	−6.68406	−0.496822	−0.248411	+	0.968655i	$0.579908\pi$
$182$	1.46508	0.108599
$183$	13.4254	0.992434
$184$	4.32080	0.318534
$185$	−6.55163	−0.481685
$186$	−26.2427	−1.92421
$187$	−18.4398	−1.34845
$188$	−2.16236	−0.157706
$189$	−0.996704	−0.0724995
$190$	−18.7296	−1.35879
$191$	−2.88528	−0.208772	−0.104386	−	0.994537i	$-0.533288\pi$
$191$	−2.88528	−0.208772	−0.104386	+	0.994537i	$0.533288\pi$
$192$	−12.9599	−0.935298
$193$	4.80047	0.345545	0.172773	−	0.984962i	$-0.444727\pi$
$193$	4.80047	0.345545	0.172773	+	0.984962i	$0.444727\pi$
$194$	0.232585	0.0166987
$195$	−16.1004	−1.15297
$196$	−2.69195	−0.192282
$197$	−16.6777	−1.18824	−0.594118	−	0.804378i	$-0.702499\pi$
$197$	−16.6777	−1.18824	−0.594118	+	0.804378i	$0.702499\pi$
$198$	−11.9837	−0.851647
$199$	13.2710	0.940754	0.470377	−	0.882466i	$-0.344118\pi$
$199$	13.2710	0.940754	0.470377	+	0.882466i	$0.344118\pi$
$200$	11.4897	0.812441
$201$	−8.31215	−0.586294
$202$	5.78926	0.407331
$203$	−0.403081	−0.0282907
$204$	3.89438	0.272661
$205$	−18.3503	−1.28164
$206$	−19.1048	−1.33110
$207$	3.26988	0.227272
$208$	−10.8828	−0.754585
$209$	16.0717	1.11170
$210$	−4.27396	−0.294931
$211$	3.31853	0.228457	0.114228	−	0.993455i	$-0.463560\pi$
$211$	3.31853	0.228457	0.114228	+	0.993455i	$0.463560\pi$
$212$	−1.35419	−0.0930065
$213$	30.1306	2.06451
$214$	−14.4983	−0.991080
$215$	−30.2266	−2.06144
$216$	6.14519	0.418127
$217$	3.09473	0.210084
$218$	−0.944828	−0.0639918
$219$	−15.8812	−1.07315
$220$	5.02987	0.339114
$221$	−10.5187	−0.707564
$222$	−7.21882	−0.484496
$223$	−19.8004	−1.32593	−0.662966	−	0.748650i	$-0.730703\pi$
$223$	−19.8004	−1.32593	−0.662966	+	0.748650i	$0.730703\pi$
$224$	−0.885434	−0.0591605
$225$	8.69509	0.579673
$226$	20.2119	1.34447
$227$	−0.470883	−0.0312536	−0.0156268	−	0.999878i	$-0.504974\pi$
$227$	−0.470883	−0.0312536	−0.0156268	+	0.999878i	$0.504974\pi$
$228$	−3.39425	−0.224790
$229$	−2.37341	−0.156839	−0.0784197	−	0.996920i	$-0.524987\pi$
$229$	−2.37341	−0.156839	−0.0784197	+	0.996920i	$0.524987\pi$
$230$	−8.34447	−0.550218
$231$	3.66744	0.241300
$232$	2.48520	0.163161
$233$	−1.84049	−0.120574	−0.0602872	−	0.998181i	$-0.519202\pi$
$233$	−1.84049	−0.120574	−0.0602872	+	0.998181i	$0.519202\pi$
$234$	−6.83592	−0.446878
$235$	−17.0381	−1.11144
$236$	−1.94620	−0.126687
$237$	12.0332	0.781639
$238$	−2.79225	−0.180995
$239$	−4.85234	−0.313872	−0.156936	−	0.987609i	$-0.550162\pi$
$239$	−4.85234	−0.313872	−0.156936	+	0.987609i	$0.550162\pi$
$240$	31.7475	2.04929
$241$	−13.8724	−0.893598	−0.446799	−	0.894634i	$-0.647436\pi$
$241$	−13.8724	−0.893598	−0.446799	+	0.894634i	$0.647436\pi$
$242$	−9.22296	−0.592875
$243$	17.1155	1.09796
$244$	−2.39250	−0.153164
$245$	−21.2109	−1.35512
$246$	−20.2190	−1.28912
$247$	9.16783	0.583335
$248$	−19.0806	−1.21162
$249$	3.54459	0.224629
$250$	1.80829	0.114366
$251$	−4.17161	−0.263310	−0.131655	−	0.991296i	$-0.542029\pi$
$251$	−4.17161	−0.263310	−0.131655	+	0.991296i	$0.542029\pi$
$252$	−0.298461	−0.0188013
$253$	7.16031	0.450165
$254$	−5.51288	−0.345909
$255$	30.6854	1.92159
$256$	9.10673	0.569170
$257$	−1.91327	−0.119347	−0.0596733	−	0.998218i	$-0.519006\pi$
$257$	−1.91327	−0.119347	−0.0596733	+	0.998218i	$0.519006\pi$
$258$	−33.3048	−2.07347
$259$	0.851296	0.0528969
$260$	2.86921	0.177941
$261$	1.88074	0.116415
$262$	1.85497	0.114600
$263$	10.8885	0.671414	0.335707	−	0.941967i	$-0.391025\pi$
$263$	10.8885	0.671414	0.335707	+	0.941967i	$0.391025\pi$
$264$	−22.6117	−1.39165
$265$	−10.6702	−0.655467
$266$	2.43366	0.149217
$267$	8.32576	0.509528
$268$	1.48128	0.0904838
$269$	6.46063	0.393911	0.196956	−	0.980412i	$-0.436894\pi$
$269$	6.46063	0.393911	0.196956	+	0.980412i	$0.436894\pi$
$270$	−11.8678	−0.722251
$271$	−20.6869	−1.25664	−0.628320	−	0.777955i	$-0.716257\pi$
$271$	−20.6869	−1.25664	−0.628320	+	0.777955i	$0.716257\pi$
$272$	20.7412	1.25762
$273$	2.09203	0.126616
$274$	−24.4705	−1.47832
$275$	19.0403	1.14817
$276$	−1.51221	−0.0910246
$277$	1.03274	0.0620515	0.0310257	−	0.999519i	$-0.490123\pi$
$277$	1.03274	0.0620515	0.0310257	+	0.999519i	$0.490123\pi$
$278$	−1.54716	−0.0927925
$279$	−14.4398	−0.864486
$280$	−3.10752	−0.185710
$281$	19.3386	1.15364	0.576822	−	0.816870i	$-0.304292\pi$
$281$	19.3386	1.15364	0.576822	+	0.816870i	$0.304292\pi$
$282$	−18.7732	−1.11793
$283$	−1.43524	−0.0853162	−0.0426581	−	0.999090i	$-0.513583\pi$
$283$	−1.43524	−0.0853162	−0.0426581	+	0.999090i	$0.513583\pi$
$284$	−5.36948	−0.318620
$285$	−26.7446	−1.58421
$286$	−14.9692	−0.885145
$287$	2.38437	0.140745
$288$	4.13136	0.243443
$289$	3.04732	0.179254
$290$	−4.79950	−0.281836
$291$	0.332116	0.0194690
$292$	2.83014	0.165621
$293$	−3.97101	−0.231989	−0.115994	−	0.993250i	$-0.537005\pi$
$293$	−3.97101	−0.231989	−0.115994	+	0.993250i	$0.537005\pi$
$294$	−23.3710	−1.36302
$295$	−15.3349	−0.892830
$296$	−5.24867	−0.305073
$297$	10.1836	0.590915
$298$	8.82947	0.511478
$299$	4.08448	0.236212
$300$	−4.02120	−0.232164
$301$	3.92754	0.226380
$302$	−36.4321	−2.09643
$303$	8.26667	0.474908
$304$	−18.0775	−1.03682
$305$	−18.8515	−1.07943
$306$	13.0284	0.744785
$307$	9.27642	0.529433	0.264717	−	0.964326i	$-0.414722\pi$
$307$	9.27642	0.529433	0.264717	+	0.964326i	$0.414722\pi$
$308$	−0.653564	−0.0372403
$309$	−27.2804	−1.55193
$310$	36.8491	2.09289
$311$	7.27782	0.412687	0.206344	−	0.978480i	$-0.433844\pi$
$311$	7.27782	0.412687	0.206344	+	0.978480i	$0.433844\pi$
$312$	−12.8985	−0.730231
$313$	26.7197	1.51029	0.755144	−	0.655558i	$-0.227567\pi$
$313$	26.7197	1.51029	0.755144	+	0.655558i	$0.227567\pi$
$314$	14.3911	0.812135
$315$	−2.35169	−0.132503
$316$	−2.14440	−0.120632
$317$	−27.9212	−1.56821	−0.784104	−	0.620629i	$-0.786877\pi$
$317$	−27.9212	−1.56821	−0.784104	+	0.620629i	$0.786877\pi$
$318$	−11.7569	−0.659292
$319$	4.11840	0.230586
$320$	18.1978	1.01729
$321$	−20.7025	−1.15550
$322$	1.08425	0.0604229
$323$	−17.4727	−0.972210
$324$	−4.37208	−0.242893
$325$	10.8612	0.602473
$326$	−2.23495	−0.123783
$327$	−1.34915	−0.0746082
$328$	−14.7009	−0.811720
$329$	2.21387	0.122054
$330$	43.6684	2.40386
$331$	−5.17250	−0.284306	−0.142153	−	0.989845i	$-0.545403\pi$
$331$	−5.17250	−0.284306	−0.142153	+	0.989845i	$0.545403\pi$
$332$	−0.631672	−0.0346675
$333$	−3.97207	−0.217668
$334$	−30.4079	−1.66385
$335$	11.6716	0.637689
$336$	−4.12516	−0.225046
$337$	−34.3273	−1.86993	−0.934964	−	0.354742i	$-0.884569\pi$
$337$	−34.3273	−1.86993	−0.934964	+	0.354742i	$0.884569\pi$
$338$	11.5742	0.629551
$339$	28.8612	1.56752
$340$	−5.46835	−0.296563
$341$	−31.6199	−1.71231
$342$	−11.3552	−0.614021
$343$	5.57764	0.301164
$344$	−24.2153	−1.30560
$345$	−11.9153	−0.641500
$346$	21.7753	1.17065
$347$	1.00298	0.0538430	0.0269215	−	0.999638i	$-0.491430\pi$
$347$	1.00298	0.0538430	0.0269215	+	0.999638i	$0.491430\pi$
$348$	−0.869782	−0.0466252
$349$	−4.13613	−0.221402	−0.110701	−	0.993854i	$-0.535310\pi$
$349$	−4.13613	−0.221402	−0.110701	+	0.993854i	$0.535310\pi$
$350$	2.88319	0.154113
$351$	5.80909	0.310066
$352$	9.04676	0.482194
$353$	18.4956	0.984419	0.492210	−	0.870477i	$-0.336189\pi$
$353$	18.4956	0.984419	0.492210	+	0.870477i	$0.336189\pi$
$354$	−16.8965	−0.898039
$355$	−42.3083	−2.24549
$356$	−1.48371	−0.0786365
$357$	−3.98715	−0.211022
$358$	25.4047	1.34268
$359$	6.78653	0.358179	0.179090	−	0.983833i	$-0.442685\pi$
$359$	6.78653	0.358179	0.179090	+	0.983833i	$0.442685\pi$
$360$	14.4994	0.764186
$361$	−3.77118	−0.198483
$362$	10.3413	0.543527
$363$	−13.1698	−0.691234
$364$	−0.372815	−0.0195408
$365$	22.2998	1.16722
$366$	−20.7712	−1.08573
$367$	−8.68007	−0.453096	−0.226548	−	0.974000i	$-0.572744\pi$
$367$	−8.68007	−0.453096	−0.226548	+	0.974000i	$0.572744\pi$
$368$	−8.05396	−0.419841
$369$	−11.1253	−0.579158
$370$	10.1364	0.526967
$371$	1.38645	0.0719811
$372$	6.67793	0.346235
$373$	1.64341	0.0850927	0.0425464	−	0.999094i	$-0.486453\pi$
$373$	1.64341	0.0850927	0.0425464	+	0.999094i	$0.486453\pi$
$374$	28.5293	1.47522
$375$	2.58211	0.133340
$376$	−13.6496	−0.703926
$377$	2.34927	0.120994
$378$	1.54206	0.0793150
$379$	−9.41847	−0.483794	−0.241897	−	0.970302i	$-0.577770\pi$
$379$	−9.41847	−0.483794	−0.241897	+	0.970302i	$0.577770\pi$
$380$	4.76608	0.244495
$381$	−7.87202	−0.403296
$382$	4.46399	0.228398
$383$	33.3062	1.70186	0.850932	−	0.525275i	$-0.176038\pi$
$383$	33.3062	1.70186	0.850932	+	0.525275i	$0.176038\pi$
$384$	29.7569	1.51853
$385$	−5.14969	−0.262452
$386$	−7.42709	−0.378029
$387$	−18.3256	−0.931541
$388$	−0.0591855	−0.00300469
$389$	−7.39361	−0.374871	−0.187435	−	0.982277i	$-0.560018\pi$
$389$	−7.39361	−0.374871	−0.187435	+	0.982277i	$0.560018\pi$
$390$	24.9099	1.26136
$391$	−7.78451	−0.393679
$392$	−16.9926	−0.858257
$393$	2.64877	0.133613
$394$	25.8030	1.29994
$395$	−16.8965	−0.850157
$396$	3.04947	0.153242
$397$	13.1732	0.661146	0.330573	−	0.943780i	$-0.392758\pi$
$397$	13.1732	0.661146	0.330573	+	0.943780i	$0.392758\pi$
$398$	−20.5323	−1.02919
$399$	3.47510	0.173973
$400$	−21.4167	−1.07083
$401$	−18.5267	−0.925177	−0.462589	−	0.886573i	$-0.653079\pi$
$401$	−18.5267	−0.925177	−0.462589	+	0.886573i	$0.653079\pi$
$402$	12.8602	0.641410
$403$	−18.0370	−0.898489
$404$	−1.47318	−0.0732934
$405$	−34.4493	−1.71180
$406$	0.623630	0.0309502
$407$	−8.69795	−0.431142
$408$	24.5828	1.21703
$409$	−14.6894	−0.726343	−0.363172	−	0.931722i	$-0.618306\pi$
$409$	−14.6894	−0.726343	−0.363172	+	0.931722i	$0.618306\pi$
$410$	28.3908	1.40212
$411$	−34.9422	−1.72357
$412$	4.86157	0.239512
$413$	1.99256	0.0980474
$414$	−5.05902	−0.248637
$415$	−4.97719	−0.244321
$416$	5.16057	0.253018
$417$	−2.20924	−0.108187
$418$	−24.8655	−1.21621
$419$	4.31142	0.210627	0.105313	−	0.994439i	$-0.466415\pi$
$419$	4.31142	0.210627	0.105313	+	0.994439i	$0.466415\pi$
$420$	1.08758	0.0530687
$421$	−22.2808	−1.08590	−0.542949	−	0.839766i	$-0.682692\pi$
$421$	−22.2808	−1.08590	−0.542949	+	0.839766i	$0.682692\pi$
$422$	−5.13429	−0.249933
$423$	−10.3297	−0.502248
$424$	−8.54820	−0.415137
$425$	−20.7002	−1.00411
$426$	−46.6168	−2.25859
$427$	2.44949	0.118539
$428$	3.68934	0.178331
$429$	−21.3749	−1.03199
$430$	46.7654	2.25523
$431$	19.6451	0.946269	0.473135	−	0.880990i	$-0.343122\pi$
$431$	19.6451	0.946269	0.473135	+	0.880990i	$0.343122\pi$
$432$	−11.4546	−0.551111
$433$	−38.2464	−1.83800	−0.919002	−	0.394254i	$-0.871003\pi$
$433$	−38.2464	−1.83800	−0.919002	+	0.394254i	$0.871003\pi$
$434$	−4.78805	−0.229834
$435$	−6.85336	−0.328593
$436$	0.240428	0.0115144
$437$	6.78478	0.324560
$438$	24.5707	1.17403
$439$	4.73599	0.226036	0.113018	−	0.993593i	$-0.463948\pi$
$439$	4.73599	0.226036	0.113018	+	0.993593i	$0.463948\pi$
$440$	31.7505	1.51364
$441$	−12.8596	−0.612362
$442$	16.2741	0.774080
$443$	−3.57040	−0.169635	−0.0848174	−	0.996397i	$-0.527031\pi$
$443$	−3.57040	−0.169635	−0.0848174	+	0.996397i	$0.527031\pi$
$444$	1.83696	0.0871781
$445$	−11.6907	−0.554194
$446$	30.6343	1.45058
$447$	12.6079	0.596333
$448$	−2.36456	−0.111715
$449$	14.6933	0.693418	0.346709	−	0.937973i	$-0.387299\pi$
$449$	14.6933	0.693418	0.346709	+	0.937973i	$0.387299\pi$
$450$	−13.4527	−0.634166
$451$	−24.3619	−1.14716
$452$	−5.14326	−0.241919
$453$	−52.0227	−2.44424
$454$	0.728531	0.0341916
$455$	−2.93756	−0.137715
$456$	−21.4258	−1.00335
$457$	26.3511	1.23265	0.616326	−	0.787491i	$-0.288621\pi$
$457$	26.3511	1.23265	0.616326	+	0.787491i	$0.288621\pi$
$458$	3.67205	0.171583
$459$	−11.0714	−0.516769
$460$	2.12340	0.0990039
$461$	−33.8138	−1.57486	−0.787432	−	0.616401i	$-0.788590\pi$
$461$	−33.8138	−1.57486	−0.787432	+	0.616401i	$0.788590\pi$
$462$	−5.67412	−0.263984
$463$	−30.8743	−1.43485	−0.717424	−	0.696636i	$-0.754679\pi$
$463$	−30.8743	−1.43485	−0.717424	+	0.696636i	$0.754679\pi$
$464$	−4.63240	−0.215054
$465$	52.6180	2.44010
$466$	2.84753	0.131909
$467$	15.8915	0.735371	0.367686	−	0.929950i	$-0.380150\pi$
$467$	15.8915	0.735371	0.367686	+	0.929950i	$0.380150\pi$
$468$	1.73952	0.0804094
$469$	−1.51657	−0.0700287
$470$	26.3606	1.21592
$471$	20.5495	0.946870
$472$	−12.2851	−0.565470
$473$	−40.1289	−1.84513
$474$	−18.6172	−0.855118
$475$	18.0418	0.827813
$476$	0.710538	0.0325675
$477$	−6.46907	−0.296198
$478$	7.50734	0.343378
$479$	15.2191	0.695379	0.347689	−	0.937610i	$-0.386966\pi$
$479$	15.2191	0.695379	0.347689	+	0.937610i	$0.386966\pi$
$480$	−15.0545	−0.687143
$481$	−4.96161	−0.226230
$482$	21.4628	0.977603
$483$	1.54824	0.0704472
$484$	2.34695	0.106679
$485$	−0.466346	−0.0211757
$486$	−26.4805	−1.20118
$487$	20.1320	0.912268	0.456134	−	0.889911i	$-0.349234\pi$
$487$	20.1320	0.912268	0.456134	+	0.889911i	$0.349234\pi$
$488$	−15.1024	−0.683653
$489$	−3.19136	−0.144318
$490$	32.8167	1.48251
$491$	1.43675	0.0648398	0.0324199	−	0.999474i	$-0.489679\pi$
$491$	1.43675	0.0648398	0.0324199	+	0.999474i	$0.489679\pi$
$492$	5.14508	0.231958
$493$	−4.47742	−0.201653
$494$	−14.1841	−0.638173
$495$	24.0280	1.07998
$496$	35.5662	1.59697
$497$	5.49739	0.246592
$498$	−5.48405	−0.245746
$499$	4.26581	0.190964	0.0954820	−	0.995431i	$-0.469561\pi$
$499$	4.26581	0.190964	0.0954820	+	0.995431i	$0.469561\pi$
$500$	−0.460151	−0.0205786
$501$	−43.4204	−1.93988
$502$	6.45415	0.288063
$503$	−33.5016	−1.49376	−0.746881	−	0.664957i	$-0.768450\pi$
$503$	−33.5016	−1.49376	−0.746881	+	0.664957i	$0.768450\pi$
$504$	−1.88400	−0.0839201
$505$	−11.6078	−0.516539
$506$	−11.0781	−0.492483
$507$	16.5271	0.733995
$508$	1.40285	0.0622414
$509$	41.7970	1.85262	0.926310	−	0.376763i	$-0.122963\pi$
$509$	41.7970	1.85262	0.926310	+	0.376763i	$0.122963\pi$
$510$	−47.4752	−2.10224
$511$	−2.89756	−0.128180
$512$	12.8490	0.567853
$513$	9.64956	0.426038
$514$	2.96013	0.130566
$515$	38.3062	1.68797
$516$	8.47500	0.373091
$517$	−22.6198	−0.994817
$518$	−1.31709	−0.0578696
$519$	31.0937	1.36486
$520$	18.1115	0.794244
$521$	−27.9775	−1.22572	−0.612858	−	0.790193i	$-0.709980\pi$
$521$	−27.9775	−1.22572	−0.612858	+	0.790193i	$0.709980\pi$
$522$	−2.90980	−0.127359
$523$	0.222632	0.00973501	0.00486751	−	0.999988i	$-0.498451\pi$
$523$	0.222632	0.00973501	0.00486751	+	0.999988i	$0.498451\pi$
$524$	−0.472029	−0.0206207
$525$	4.11700	0.179680
$526$	−16.8462	−0.734531
$527$	34.3763	1.49746
$528$	42.1481	1.83426
$529$	−19.9772	−0.868575
$530$	16.5086	0.717086
$531$	−9.29710	−0.403460
$532$	−0.619288	−0.0268495
$533$	−13.8968	−0.601939
$534$	−12.8813	−0.557428
$535$	29.0697	1.25679
$536$	9.35044	0.403877
$537$	36.2761	1.56543
$538$	−9.99562	−0.430942
$539$	−28.1597	−1.21292
$540$	3.01997	0.129959
$541$	6.47376	0.278329	0.139164	−	0.990269i	$-0.455558\pi$
$541$	6.47376	0.278329	0.139164	+	0.990269i	$0.455558\pi$
$542$	32.0059	1.37477
$543$	14.7667	0.633699
$544$	−9.83541	−0.421690
$545$	1.89443	0.0811484
$546$	−3.23671	−0.138518
$547$	4.97587	0.212753	0.106376	−	0.994326i	$-0.466075\pi$
$547$	4.97587	0.212753	0.106376	+	0.994326i	$0.466075\pi$
$548$	6.22695	0.266002
$549$	−11.4291	−0.487783
$550$	−29.4584	−1.25611
$551$	3.90241	0.166248
$552$	−9.54568	−0.406291
$553$	2.19548	0.0933613
$554$	−1.59782	−0.0678847
$555$	14.4741	0.614392
$556$	0.393702	0.0166967
$557$	42.6838	1.80857	0.904284	−	0.426930i	$-0.140405\pi$
$557$	42.6838	1.80857	0.904284	+	0.426930i	$0.140405\pi$
$558$	22.3406	0.945753
$559$	−22.8909	−0.968182
$560$	5.79240	0.244774
$561$	40.7380	1.71996
$562$	−29.9199	−1.26210
$563$	−40.5210	−1.70776	−0.853879	−	0.520472i	$-0.825756\pi$
$563$	−40.5210	−1.70776	−0.853879	+	0.520472i	$0.825756\pi$
$564$	4.77716	0.201155
$565$	−40.5258	−1.70493
$566$	2.22055	0.0933365
$567$	4.47622	0.187984
$568$	−33.8942	−1.42217
$569$	23.5326	0.986539	0.493270	−	0.869877i	$-0.335802\pi$
$569$	23.5326	0.986539	0.493270	+	0.869877i	$0.335802\pi$
$570$	41.3782	1.73314
$571$	−29.7964	−1.24694	−0.623470	−	0.781847i	$-0.714278\pi$
$571$	−29.7964	−1.24694	−0.623470	+	0.781847i	$0.714278\pi$
$572$	3.80917	0.159269
$573$	6.37428	0.266289
$574$	−3.68900	−0.153976
$575$	8.03801	0.335208
$576$	11.0328	0.459700
$577$	14.3064	0.595581	0.297791	−	0.954631i	$-0.403750\pi$
$577$	14.3064	0.595581	0.297791	+	0.954631i	$0.403750\pi$
$578$	−4.71469	−0.196105
$579$	−10.6054	−0.440745
$580$	1.22132	0.0507124
$581$	0.646719	0.0268304
$582$	−0.513837	−0.0212992
$583$	−14.1658	−0.586689
$584$	17.8649	0.739256
$585$	13.7064	0.566689
$586$	6.14378	0.253797
$587$	4.32371	0.178459	0.0892293	−	0.996011i	$-0.471560\pi$
$587$	4.32371	0.178459	0.0892293	+	0.996011i	$0.471560\pi$
$588$	5.94716	0.245257
$589$	−29.9616	−1.23454
$590$	23.7255	0.976762
$591$	36.8450	1.51560
$592$	9.78351	0.402100
$593$	3.61741	0.148549	0.0742746	−	0.997238i	$-0.476336\pi$
$593$	3.61741	0.148549	0.0742746	+	0.997238i	$0.476336\pi$
$594$	−15.7557	−0.646465
$595$	5.59861	0.229521
$596$	−2.24681	−0.0920331
$597$	−29.3187	−1.19994
$598$	−6.31934	−0.258417
$599$	−42.0742	−1.71911	−0.859553	−	0.511047i	$-0.829258\pi$
$599$	−42.0742	−1.71911	−0.859553	+	0.511047i	$0.829258\pi$
$600$	−25.3834	−1.03627
$601$	−1.91586	−0.0781496	−0.0390748	−	0.999236i	$-0.512441\pi$
$601$	−1.91586	−0.0781496	−0.0390748	+	0.999236i	$0.512441\pi$
$602$	−6.07653	−0.247661
$603$	7.07618	0.288164
$604$	9.27080	0.377224
$605$	18.4925	0.751828
$606$	−12.7899	−0.519552
$607$	9.97088	0.404705	0.202353	−	0.979313i	$-0.435141\pi$
$607$	9.97088	0.404705	0.202353	+	0.979313i	$0.435141\pi$
$608$	8.57230	0.347653
$609$	0.890501	0.0360849
$610$	29.1662	1.18091
$611$	−12.9031	−0.522003
$612$	−3.31531	−0.134013
$613$	−14.6388	−0.591255	−0.295627	−	0.955303i	$-0.595529\pi$
$613$	−14.6388	−0.591255	−0.295627	+	0.955303i	$0.595529\pi$
$614$	−14.3521	−0.579204
$615$	40.5402	1.63474
$616$	−4.12555	−0.166223
$617$	−24.1302	−0.971444	−0.485722	−	0.874113i	$-0.661443\pi$
$617$	−24.1302	−0.971444	−0.485722	+	0.874113i	$0.661443\pi$
$618$	42.2072	1.69782
$619$	15.7860	0.634493	0.317246	−	0.948343i	$-0.397242\pi$
$619$	15.7860	0.634493	0.317246	+	0.948343i	$0.397242\pi$
$620$	−9.37691	−0.376586
$621$	4.29910	0.172517
$622$	−11.2599	−0.451483
$623$	1.51905	0.0608596
$624$	24.0427	0.962477
$625$	−26.7419	−1.06968
$626$	−41.3397	−1.65227
$627$	−35.5062	−1.41798
$628$	−3.66206	−0.146132
$629$	9.45620	0.377043
$630$	3.63844	0.144959
$631$	5.50337	0.219086	0.109543	−	0.993982i	$-0.465061\pi$
$631$	5.50337	0.219086	0.109543	+	0.993982i	$0.465061\pi$
$632$	−13.5363	−0.538443
$633$	−7.33142	−0.291398
$634$	43.1985	1.71563
$635$	11.0536	0.438649
$636$	2.99174	0.118630
$637$	−16.0632	−0.636448
$638$	−6.37182	−0.252263
$639$	−25.6503	−1.01471
$640$	−41.7836	−1.65164
$641$	−2.62615	−0.103727	−0.0518634	−	0.998654i	$-0.516516\pi$
$641$	−2.62615	−0.103727	−0.0518634	+	0.998654i	$0.516516\pi$
$642$	32.0301	1.26413
$643$	−9.78787	−0.385996	−0.192998	−	0.981199i	$-0.561821\pi$
$643$	−9.78787	−0.385996	−0.192998	+	0.981199i	$0.561821\pi$
$644$	−0.275907	−0.0108723
$645$	66.7778	2.62937
$646$	27.0331	1.06360
$647$	1.63922	0.0644444	0.0322222	−	0.999481i	$-0.489742\pi$
$647$	1.63922	0.0644444	0.0322222	+	0.999481i	$0.489742\pi$
$648$	−27.5982	−1.08416
$649$	−20.3586	−0.799145
$650$	−16.8041	−0.659110
$651$	−6.83701	−0.267963
$652$	0.568723	0.0222729
$653$	32.0366	1.25369	0.626844	−	0.779145i	$-0.284346\pi$
$653$	32.0366	1.25369	0.626844	+	0.779145i	$0.284346\pi$
$654$	2.08735	0.0816219
$655$	−3.71930	−0.145325
$656$	27.4024	1.06988
$657$	13.5197	0.527455
$658$	−3.42521	−0.133528
$659$	−0.274852	−0.0107067	−0.00535336	−	0.999986i	$-0.501704\pi$
$659$	−0.274852	−0.0107067	−0.00535336	+	0.999986i	$0.501704\pi$
$660$	−11.1122	−0.432541
$661$	42.5396	1.65460	0.827300	−	0.561761i	$-0.189876\pi$
$661$	42.5396	1.65460	0.827300	+	0.561761i	$0.189876\pi$
$662$	8.00268	0.311033
$663$	23.2383	0.902501
$664$	−3.98735	−0.154739
$665$	−4.87961	−0.189223
$666$	6.14543	0.238130
$667$	1.73861	0.0673194
$668$	7.73783	0.299386
$669$	43.7438	1.69123
$670$	−18.0579	−0.697636
$671$	−25.0272	−0.966166
$672$	1.95614	0.0754596
$673$	21.3929	0.824634	0.412317	−	0.911040i	$-0.364720\pi$
$673$	21.3929	0.824634	0.412317	+	0.911040i	$0.364720\pi$
$674$	53.1098	2.04571
$675$	11.4319	0.440016
$676$	−2.94525	−0.113279
$677$	−18.8618	−0.724919	−0.362460	−	0.931999i	$-0.618063\pi$
$677$	−18.8618	−0.724919	−0.362460	+	0.931999i	$0.618063\pi$
$678$	−44.6528	−1.71488
$679$	0.0605954	0.00232544
$680$	−34.5183	−1.32372
$681$	1.04029	0.0398641
$682$	48.9210	1.87328
$683$	22.9501	0.878162	0.439081	−	0.898447i	$-0.355304\pi$
$683$	22.9501	0.878162	0.439081	+	0.898447i	$0.355304\pi$
$684$	2.88954	0.110484
$685$	49.0646	1.87466
$686$	−8.62949	−0.329476
$687$	5.24343	0.200049
$688$	45.1373	1.72084
$689$	−8.08067	−0.307849
$690$	18.4349	0.701805
$691$	22.2657	0.847028	0.423514	−	0.905890i	$-0.360796\pi$
$691$	22.2657	0.847028	0.423514	+	0.905890i	$0.360796\pi$
$692$	−5.54112	−0.210642
$693$	−3.12211	−0.118599
$694$	−1.55178	−0.0589046
$695$	3.10213	0.117671
$696$	−5.49040	−0.208113
$697$	26.4856	1.00321
$698$	6.39925	0.242215
$699$	4.06608	0.153793
$700$	−0.733678	−0.0277304
$701$	40.8653	1.54346	0.771731	−	0.635950i	$-0.219391\pi$
$701$	40.8653	1.54346	0.771731	+	0.635950i	$0.219391\pi$
$702$	−8.98759	−0.339215
$703$	−8.24179	−0.310845
$704$	24.1594	0.910542
$705$	37.6412	1.41765
$706$	−28.6156	−1.07696
$707$	1.50827	0.0567244
$708$	4.29962	0.161589
$709$	−19.8066	−0.743852	−0.371926	−	0.928262i	$-0.621302\pi$
$709$	−19.8066	−0.743852	−0.371926	+	0.928262i	$0.621302\pi$
$710$	65.4576	2.45658
$711$	−10.2439	−0.384177
$712$	−9.36575	−0.350996
$713$	−13.3486	−0.499907
$714$	6.16876	0.230860
$715$	30.0139	1.12246
$716$	−6.46466	−0.241596
$717$	10.7200	0.400345
$718$	−10.4998	−0.391851
$719$	24.0240	0.895944	0.447972	−	0.894047i	$-0.352146\pi$
$719$	24.0240	0.895944	0.447972	+	0.894047i	$0.352146\pi$
$720$	−27.0268	−1.00723
$721$	−4.97738	−0.185367
$722$	5.83462	0.217142
$723$	30.6474	1.13979
$724$	−2.63153	−0.0978000
$725$	4.62323	0.171703
$726$	20.3757	0.756214
$727$	37.2752	1.38246	0.691230	−	0.722635i	$-0.257069\pi$
$727$	37.2752	1.38246	0.691230	+	0.722635i	$0.257069\pi$
$728$	−2.35335	−0.0872210
$729$	−4.49720	−0.166563
$730$	−34.5013	−1.27695
$731$	43.6272	1.61361
$732$	5.28561	0.195362
$733$	−43.0034	−1.58837	−0.794183	−	0.607678i	$-0.792101\pi$
$733$	−43.0034	−1.58837	−0.794183	+	0.607678i	$0.792101\pi$
$734$	13.4294	0.495690
$735$	46.8600	1.72846
$736$	3.81915	0.140776
$737$	15.4953	0.570776
$738$	17.2126	0.633603
$739$	−14.6438	−0.538681	−0.269340	−	0.963045i	$-0.586806\pi$
$739$	−14.6438	−0.538681	−0.269340	+	0.963045i	$0.586806\pi$
$740$	−2.57939	−0.0948202
$741$	−20.2539	−0.744047
$742$	−2.14506	−0.0787478
$743$	25.2145	0.925029	0.462514	−	0.886612i	$-0.346947\pi$
$743$	25.2145	0.925029	0.462514	+	0.886612i	$0.346947\pi$
$744$	42.1537	1.54543
$745$	−17.7035	−0.648608
$746$	−2.54262	−0.0930920
$747$	−3.01753	−0.110406
$748$	−7.25980	−0.265444
$749$	−3.77722	−0.138017
$750$	−3.99494	−0.145875
$751$	−16.7254	−0.610320	−0.305160	−	0.952301i	$-0.598710\pi$
$751$	−16.7254	−0.610320	−0.305160	+	0.952301i	$0.598710\pi$
$752$	25.4429	0.927806
$753$	9.21609	0.335853
$754$	−3.63470	−0.132368
$755$	73.0483	2.65850
$756$	−0.392404	−0.0142716
$757$	31.1750	1.13308	0.566538	−	0.824036i	$-0.308282\pi$
$757$	31.1750	1.13308	0.566538	+	0.824036i	$0.308282\pi$
$758$	14.5719	0.529274
$759$	−15.8188	−0.574187
$760$	30.0853	1.09131
$761$	0.0778152	0.00282080	0.00141040	−	0.999999i	$-0.499551\pi$
$761$	0.0778152	0.00282080	0.00141040	+	0.999999i	$0.499551\pi$
$762$	12.1793	0.441208
$763$	−0.246155	−0.00891142
$764$	−1.13594	−0.0410969
$765$	−26.1226	−0.944466
$766$	−51.5299	−1.86185
$767$	−11.6132	−0.419329
$768$	−20.1189	−0.725980
$769$	−29.3989	−1.06015	−0.530075	−	0.847951i	$-0.677836\pi$
$769$	−29.3989	−1.06015	−0.530075	+	0.847951i	$0.677836\pi$
$770$	7.96739	0.287125
$771$	4.22687	0.152227
$772$	1.88995	0.0680209
$773$	28.5161	1.02565	0.512826	−	0.858492i	$-0.328598\pi$
$773$	28.5161	1.02565	0.512826	+	0.858492i	$0.328598\pi$
$774$	28.3526	1.01911
$775$	−35.4958	−1.27505
$776$	−0.373602	−0.0134115
$777$	−1.88072	−0.0674703
$778$	11.4391	0.410111
$779$	−23.0842	−0.827078
$780$	−6.33876	−0.226964
$781$	−56.1686	−2.00987
$782$	12.0439	0.430688
$783$	2.47272	0.0883677
$784$	31.6742	1.13122
$785$	−28.8548	−1.02987
$786$	−4.09807	−0.146173
$787$	51.0637	1.82022	0.910112	−	0.414362i	$-0.135995\pi$
$787$	51.0637	1.82022	0.910112	+	0.414362i	$0.135995\pi$
$788$	−6.56604	−0.233905
$789$	−24.0553	−0.856391
$790$	26.1416	0.930078
$791$	5.26578	0.187230
$792$	19.2494	0.683999
$793$	−14.2764	−0.506969
$794$	−20.3811	−0.723299
$795$	23.5731	0.836052
$796$	5.22480	0.185188
$797$	24.7712	0.877439	0.438720	−	0.898624i	$-0.355432\pi$
$797$	24.7712	0.877439	0.438720	+	0.898624i	$0.355432\pi$
$798$	−5.37654	−0.190327
$799$	24.5917	0.869991
$800$	10.1557	0.359058
$801$	−7.08777	−0.250434
$802$	28.6637	1.01215
$803$	29.6052	1.04475
$804$	−3.27251	−0.115413
$805$	−2.17398	−0.0766227
$806$	27.9062	0.982953
$807$	−14.2731	−0.502436
$808$	−9.29927	−0.327147
$809$	−15.0958	−0.530740	−0.265370	−	0.964147i	$-0.585494\pi$
$809$	−15.0958	−0.530740	−0.265370	+	0.964147i	$0.585494\pi$
$810$	53.2986	1.87272
$811$	4.63122	0.162624	0.0813120	−	0.996689i	$-0.474089\pi$
$811$	4.63122	0.162624	0.0813120	+	0.996689i	$0.474089\pi$
$812$	−0.158694	−0.00556905
$813$	45.7023	1.60285
$814$	13.4571	0.471672
$815$	4.48119	0.156969
$816$	−45.8223	−1.60410
$817$	−38.0244	−1.33030
$818$	22.7268	0.794625
$819$	−1.78096	−0.0622317
$820$	−7.22454	−0.252292
$821$	14.6909	0.512717	0.256359	−	0.966582i	$-0.417477\pi$
$821$	14.6909	0.512717	0.256359	+	0.966582i	$0.417477\pi$
$822$	54.0612	1.88560
$823$	−42.6789	−1.48769	−0.743847	−	0.668350i	$-0.767001\pi$
$823$	−42.6789	−1.48769	−0.743847	+	0.668350i	$0.767001\pi$
$824$	30.6881	1.06907
$825$	−42.0646	−1.46450
$826$	−3.08281	−0.107265
$827$	−1.46780	−0.0510404	−0.0255202	−	0.999674i	$-0.508124\pi$
$827$	−1.46780	−0.0510404	−0.0255202	+	0.999674i	$0.508124\pi$
$828$	1.28736	0.0447387
$829$	−21.0784	−0.732082	−0.366041	−	0.930599i	$-0.619287\pi$
$829$	−21.0784	−0.732082	−0.366041	+	0.930599i	$0.619287\pi$
$830$	7.70051	0.267288
$831$	−2.28158	−0.0791470
$832$	13.7813	0.477782
$833$	30.6145	1.06073
$834$	3.41805	0.118357
$835$	60.9694	2.10993
$836$	6.32746	0.218840
$837$	−18.9848	−0.656210
$838$	−6.67046	−0.230427
$839$	33.9321	1.17147	0.585734	−	0.810503i	$-0.300806\pi$
$839$	33.9321	1.17147	0.585734	+	0.810503i	$0.300806\pi$
$840$	6.86525	0.236874
$841$	1.00000	0.0344828
$842$	34.4719	1.18798
$843$	−42.7236	−1.47148
$844$	1.30651	0.0449719
$845$	−23.2068	−0.798338
$846$	15.9817	0.549462
$847$	−2.40285	−0.0825630
$848$	15.9338	0.547169
$849$	3.17079	0.108821
$850$	32.0265	1.09850
$851$	−3.67191	−0.125871
$852$	11.8625	0.406401
$853$	−8.55782	−0.293014	−0.146507	−	0.989210i	$-0.546803\pi$
$853$	−8.55782	−0.293014	−0.146507	+	0.989210i	$0.546803\pi$
$854$	−3.78976	−0.129683
$855$	22.7678	0.778644
$856$	23.2885	0.795985
$857$	35.6393	1.21742	0.608708	−	0.793394i	$-0.291688\pi$
$857$	35.6393	1.21742	0.608708	+	0.793394i	$0.291688\pi$
$858$	33.0704	1.12901
$859$	1.40841	0.0480542	0.0240271	−	0.999711i	$-0.492351\pi$
$859$	1.40841	0.0480542	0.0240271	+	0.999711i	$0.492351\pi$
$860$	−11.9003	−0.405796
$861$	−5.26765	−0.179521
$862$	−30.3940	−1.03522
$863$	−53.2886	−1.81397	−0.906983	−	0.421167i	$-0.861621\pi$
$863$	−53.2886	−1.81397	−0.906983	+	0.421167i	$0.861621\pi$
$864$	5.43174	0.184791
$865$	−43.6606	−1.48451
$866$	59.1732	2.01079
$867$	−6.73226	−0.228640
$868$	1.21840	0.0413553
$869$	−22.4319	−0.760950
$870$	10.6032	0.359483
$871$	8.83902	0.299499
$872$	1.51767	0.0513950
$873$	−0.282733	−0.00956905
$874$	−10.4971	−0.355071
$875$	0.471112	0.0159265
$876$	−6.25245	−0.211251
$877$	−38.9463	−1.31512	−0.657562	−	0.753400i	$-0.728412\pi$
$877$	−38.9463	−1.31512	−0.657562	+	0.753400i	$0.728412\pi$
$878$	−7.32733	−0.247285
$879$	8.77291	0.295903
$880$	−59.1828	−1.99505
$881$	−32.0193	−1.07876	−0.539379	−	0.842063i	$-0.681341\pi$
$881$	−32.0193	−1.07876	−0.539379	+	0.842063i	$0.681341\pi$
$882$	19.8958	0.669928
$883$	1.41115	0.0474889	0.0237445	−	0.999718i	$-0.492441\pi$
$883$	1.41115	0.0474889	0.0237445	+	0.999718i	$0.492441\pi$
$884$	−4.14123	−0.139285
$885$	33.8784	1.13881
$886$	5.52398	0.185582
$887$	−0.186636	−0.00626661	−0.00313330	−	0.999995i	$-0.500997\pi$
$887$	−0.186636	−0.00626661	−0.00313330	+	0.999995i	$0.500997\pi$
$888$	11.5956	0.389122
$889$	−1.43627	−0.0481708
$890$	18.0874	0.606292
$891$	−45.7350	−1.53218
$892$	−7.79545	−0.261011
$893$	−21.4335	−0.717244
$894$	−19.5064	−0.652392
$895$	−50.9376	−1.70266
$896$	5.42921	0.181377
$897$	−9.02359	−0.301289
$898$	−22.7328	−0.758604
$899$	−7.67771	−0.256066
$900$	3.42328	0.114109
$901$	15.4007	0.513073
$902$	37.6917	1.25500
$903$	−8.67688	−0.288749
$904$	−32.4663	−1.07981
$905$	−20.7349	−0.689250
$906$	80.4873	2.67401
$907$	28.8116	0.956672	0.478336	−	0.878177i	$-0.341240\pi$
$907$	28.8116	0.956672	0.478336	+	0.878177i	$0.341240\pi$
$908$	−0.185387	−0.00615230
$909$	−7.03747	−0.233418
$910$	4.54487	0.150661
$911$	−13.9811	−0.463215	−0.231607	−	0.972809i	$-0.574398\pi$
$911$	−13.9811	−0.463215	−0.231607	+	0.972809i	$0.574398\pi$
$912$	39.9376	1.32247
$913$	−6.60773	−0.218684
$914$	−40.7693	−1.34853
$915$	41.6474	1.37682
$916$	−0.934417	−0.0308740
$917$	0.483273	0.0159591
$918$	17.1292	0.565348
$919$	35.9836	1.18699	0.593495	−	0.804838i	$-0.297748\pi$
$919$	35.9836	1.18699	0.593495	+	0.804838i	$0.297748\pi$
$920$	13.4037	0.441907
$921$	−20.4938	−0.675295
$922$	52.3153	1.72291
$923$	−32.0404	−1.05462
$924$	1.44388	0.0475001
$925$	−9.76415	−0.321043
$926$	47.7674	1.56973
$927$	23.2240	0.762776
$928$	2.19667	0.0721091
$929$	52.1625	1.71140	0.855698	−	0.517476i	$-0.173128\pi$
$929$	52.1625	1.71140	0.855698	+	0.517476i	$0.173128\pi$
$930$	−81.4085	−2.66949
$931$	−26.6828	−0.874495
$932$	−0.724604	−0.0237352
$933$	−16.0784	−0.526385
$934$	−24.5867	−0.804501
$935$	−57.2028	−1.87073
$936$	10.9805	0.358910
$937$	−9.45435	−0.308860	−0.154430	−	0.988004i	$-0.549354\pi$
$937$	−9.45435	−0.308860	−0.154430	+	0.988004i	$0.549354\pi$
$938$	2.34638	0.0766119
$939$	−59.0303	−1.92638
$940$	−6.70792	−0.218788
$941$	4.03918	0.131673	0.0658367	−	0.997830i	$-0.479028\pi$
$941$	4.03918	0.131673	0.0658367	+	0.997830i	$0.479028\pi$
$942$	−31.7933	−1.03588
$943$	−10.2845	−0.334911
$944$	22.8995	0.745314
$945$	−3.09191	−0.100580
$946$	62.0859	2.01859
$947$	−53.8353	−1.74941	−0.874706	−	0.484654i	$-0.838946\pi$
$947$	−53.8353	−1.74941	−0.874706	+	0.484654i	$0.838946\pi$
$948$	4.73748	0.153866
$949$	16.8878	0.548202
$950$	−27.9135	−0.905633
$951$	61.6845	2.00026
$952$	4.48519	0.145366
$953$	−16.6735	−0.540107	−0.270053	−	0.962845i	$-0.587041\pi$
$953$	−16.6735	−0.540107	−0.270053	+	0.962845i	$0.587041\pi$
$954$	10.0087	0.324043
$955$	−8.95053	−0.289632
$956$	−1.91037	−0.0617859
$957$	−9.09853	−0.294114
$958$	−23.5464	−0.760749
$959$	−6.37529	−0.205869
$960$	−40.2032	−1.29755
$961$	27.9472	0.901521
$962$	7.67639	0.247497
$963$	17.6242	0.567932
$964$	−5.46158	−0.175906
$965$	14.8917	0.479380
$966$	−2.39537	−0.0770698
$967$	6.57681	0.211496	0.105748	−	0.994393i	$-0.466276\pi$
$967$	6.57681	0.211496	0.105748	+	0.994393i	$0.466276\pi$
$968$	14.8148	0.476167
$969$	38.6015	1.24006
$970$	0.721511	0.0231663
$971$	−11.9564	−0.383699	−0.191849	−	0.981424i	$-0.561449\pi$
$971$	−11.9564	−0.383699	−0.191849	+	0.981424i	$0.561449\pi$
$972$	6.73842	0.216135
$973$	−0.403081	−0.0129222
$974$	−31.1474	−0.998027
$975$	−23.9951	−0.768458
$976$	28.1508	0.901085
$977$	25.5013	0.815858	0.407929	−	0.913014i	$-0.366251\pi$
$977$	25.5013	0.815858	0.407929	+	0.913014i	$0.366251\pi$
$978$	4.93754	0.157885
$979$	−15.5207	−0.496042
$980$	−8.35078	−0.266756
$981$	1.14854	0.0366700
$982$	−2.22289	−0.0709352
$983$	41.3117	1.31764	0.658820	−	0.752301i	$-0.271056\pi$
$983$	41.3117	1.31764	0.658820	+	0.752301i	$0.271056\pi$
$984$	32.4778	1.03535
$985$	−51.7364	−1.64846
$986$	6.92729	0.220610
$987$	−4.89096	−0.155681
$988$	3.60939	0.114830
$989$	−16.9407	−0.538684
$990$	−37.1751	−1.18150
$991$	9.78606	0.310864	0.155432	−	0.987847i	$-0.450323\pi$
$991$	9.78606	0.310864	0.155432	+	0.987847i	$0.450323\pi$
$992$	−16.8654	−0.535476
$993$	11.4273	0.362634
$994$	−8.50534	−0.269773
$995$	41.1683	1.30512
$996$	1.39551	0.0442185
$997$	−59.3760	−1.88046	−0.940229	−	0.340543i	$-0.889389\pi$
$997$	−59.3760	−1.88046	−0.940229	+	0.340543i	$0.889389\pi$
$998$	−6.59989	−0.208916
$999$	−5.22231	−0.165227

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.b.1.15	✓	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.b.1.15	✓	59	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)

\(f(q)\)	\(=\)	\(q-1.54716 q^{2} -2.20924 q^{3} +0.393702 q^{4} +3.10213 q^{5} +3.41805 q^{6} -0.403081 q^{7} +2.48520 q^{8} +1.88074 q^{9} +O(q^{10})\)	\(q-1.54716 q^{2} -2.20924 q^{3} +0.393702 q^{4} +3.10213 q^{5} +3.41805 q^{6} -0.403081 q^{7} +2.48520 q^{8} +1.88074 q^{9} -4.79950 q^{10} +4.11840 q^{11} -0.869782 q^{12} +2.34927 q^{13} +0.623630 q^{14} -6.85336 q^{15} -4.63240 q^{16} -4.47742 q^{17} -2.90980 q^{18} +3.90241 q^{19} +1.22132 q^{20} +0.890501 q^{21} -6.37182 q^{22} +1.73861 q^{23} -5.49040 q^{24} +4.62323 q^{25} -3.63470 q^{26} +2.47272 q^{27} -0.158694 q^{28} +1.00000 q^{29} +10.6032 q^{30} -7.67771 q^{31} +2.19667 q^{32} -9.09853 q^{33} +6.92729 q^{34} -1.25041 q^{35} +0.740450 q^{36} -2.11197 q^{37} -6.03765 q^{38} -5.19011 q^{39} +7.70942 q^{40} -5.91537 q^{41} -1.37775 q^{42} -9.74382 q^{43} +1.62142 q^{44} +5.83430 q^{45} -2.68991 q^{46} -5.49237 q^{47} +10.2341 q^{48} -6.83753 q^{49} -7.15288 q^{50} +9.89170 q^{51} +0.924914 q^{52} -3.43964 q^{53} -3.82569 q^{54} +12.7758 q^{55} -1.00174 q^{56} -8.62136 q^{57} -1.54716 q^{58} -4.94333 q^{59} -2.69818 q^{60} -6.07693 q^{61} +11.8786 q^{62} -0.758089 q^{63} +5.86621 q^{64} +7.28776 q^{65} +14.0769 q^{66} +3.76245 q^{67} -1.76277 q^{68} -3.84101 q^{69} +1.93458 q^{70} -13.6384 q^{71} +4.67401 q^{72} +7.18853 q^{73} +3.26756 q^{74} -10.2138 q^{75} +1.53639 q^{76} -1.66005 q^{77} +8.02993 q^{78} -5.44675 q^{79} -14.3703 q^{80} -11.1050 q^{81} +9.15203 q^{82} -1.60444 q^{83} +0.350592 q^{84} -13.8896 q^{85} +15.0752 q^{86} -2.20924 q^{87} +10.2350 q^{88} -3.76861 q^{89} -9.02659 q^{90} -0.946947 q^{91} +0.684495 q^{92} +16.9619 q^{93} +8.49757 q^{94} +12.1058 q^{95} -4.85296 q^{96} -0.150331 q^{97} +10.5787 q^{98} +7.74563 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9	\( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9} - 18 q^{10} - 27 q^{11} - 8 q^{12} - 22 q^{13} - 24 q^{14} - 18 q^{15} + 5 q^{16} - 23 q^{17} + q^{18} - 32 q^{19} - 14 q^{20} - 36 q^{21} - 6 q^{22} - 3 q^{23} - 18 q^{24} - 8 q^{25} - q^{26} - 12 q^{27} - 9 q^{28} + 59 q^{29} - 18 q^{30} - 32 q^{31} - 39 q^{32} - 12 q^{33} - 18 q^{34} - 9 q^{35} + 10 q^{36} - 44 q^{37} + 5 q^{38} - 27 q^{39} - 68 q^{40} - 44 q^{41} - 25 q^{42} - 40 q^{43} - 56 q^{44} - 39 q^{45} - 40 q^{46} - 20 q^{47} - 9 q^{48} - 39 q^{49} - 21 q^{50} - 28 q^{51} - 49 q^{52} - 31 q^{53} - 32 q^{54} - 32 q^{55} - 48 q^{56} - 58 q^{57} - 5 q^{58} + 6 q^{59} - 44 q^{60} - 88 q^{61} + 35 q^{62} - 22 q^{63} - 10 q^{64} - 43 q^{65} - 31 q^{66} - 45 q^{67} - 29 q^{68} - 60 q^{69} - 14 q^{70} - 20 q^{71} - 4 q^{72} - 90 q^{73} - 25 q^{74} + 15 q^{75} - 64 q^{76} - 39 q^{77} - 28 q^{78} - 120 q^{79} + 24 q^{80} - 77 q^{81} - 71 q^{82} - 33 q^{83} - 14 q^{84} - 71 q^{85} - 61 q^{86} - 6 q^{87} - 34 q^{88} - 78 q^{89} - 88 q^{90} - 28 q^{91} - 31 q^{92} - 36 q^{93} - 4 q^{94} - 12 q^{95} - 29 q^{96} - 48 q^{97} - 4 q^{98} - 43 q^{99}+O(q^{100}) \) 59 * q - 5 * q^2 - 6 * q^3 + 41 * q^4 - 5 * q^5 - 7 * q^6 - 10 * q^7 - 12 * q^8 + 23 * q^9 - 18 * q^10 - 27 * q^11 - 8 * q^12 - 22 * q^13 - 24 * q^14 - 18 * q^15 + 5 * q^16 - 23 * q^17 + q^18 - 32 * q^19 - 14 * q^20 - 36 * q^21 - 6 * q^22 - 3 * q^23 - 18 * q^24 - 8 * q^25 - q^26 - 12 * q^27 - 9 * q^28 + 59 * q^29 - 18 * q^30 - 32 * q^31 - 39 * q^32 - 12 * q^33 - 18 * q^34 - 9 * q^35 + 10 * q^36 - 44 * q^37 + 5 * q^38 - 27 * q^39 - 68 * q^40 - 44 * q^41 - 25 * q^42 - 40 * q^43 - 56 * q^44 - 39 * q^45 - 40 * q^46 - 20 * q^47 - 9 * q^48 - 39 * q^49 - 21 * q^50 - 28 * q^51 - 49 * q^52 - 31 * q^53 - 32 * q^54 - 32 * q^55 - 48 * q^56 - 58 * q^57 - 5 * q^58 + 6 * q^59 - 44 * q^60 - 88 * q^61 + 35 * q^62 - 22 * q^63 - 10 * q^64 - 43 * q^65 - 31 * q^66 - 45 * q^67 - 29 * q^68 - 60 * q^69 - 14 * q^70 - 20 * q^71 - 4 * q^72 - 90 * q^73 - 25 * q^74 + 15 * q^75 - 64 * q^76 - 39 * q^77 - 28 * q^78 - 120 * q^79 + 24 * q^80 - 77 * q^81 - 71 * q^82 - 33 * q^83 - 14 * q^84 - 71 * q^85 - 61 * q^86 - 6 * q^87 - 34 * q^88 - 78 * q^89 - 88 * q^90 - 28 * q^91 - 31 * q^92 - 36 * q^93 - 4 * q^94 - 12 * q^95 - 29 * q^96 - 48 * q^97 - 4 * q^98 - 43 * q^99

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