LMFDB - Embedded newform 4031.2.a.c.1.45

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:	$61$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.45
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)

$f(q)$	$=$	$q+1.30614 q^{2} -2.79423 q^{3} -0.293985 q^{4} -1.11870 q^{5} -3.64967 q^{6} -3.78446 q^{7} -2.99628 q^{8} +4.80771 q^{9} +O(q^{10})$	\(q+1.30614 q^{2} -2.79423 q^{3} -0.293985 q^{4} -1.11870 q^{5} -3.64967 q^{6} -3.78446 q^{7} -2.99628 q^{8} +4.80771 q^{9} -1.46119 q^{10} +2.23379 q^{11} +0.821462 q^{12} +0.252424 q^{13} -4.94305 q^{14} +3.12592 q^{15} -3.32560 q^{16} +3.71154 q^{17} +6.27957 q^{18} +2.81070 q^{19} +0.328883 q^{20} +10.5746 q^{21} +2.91766 q^{22} +1.59419 q^{23} +8.37228 q^{24} -3.74850 q^{25} +0.329702 q^{26} -5.05115 q^{27} +1.11258 q^{28} -1.00000 q^{29} +4.08290 q^{30} +3.00352 q^{31} +1.64884 q^{32} -6.24172 q^{33} +4.84781 q^{34} +4.23369 q^{35} -1.41340 q^{36} -0.358919 q^{37} +3.67119 q^{38} -0.705329 q^{39} +3.35195 q^{40} -3.00508 q^{41} +13.8120 q^{42} +2.81576 q^{43} -0.656702 q^{44} -5.37841 q^{45} +2.08224 q^{46} +2.20963 q^{47} +9.29249 q^{48} +7.32214 q^{49} -4.89608 q^{50} -10.3709 q^{51} -0.0742089 q^{52} +12.5217 q^{53} -6.59754 q^{54} -2.49895 q^{55} +11.3393 q^{56} -7.85375 q^{57} -1.30614 q^{58} +1.32679 q^{59} -0.918974 q^{60} -12.0308 q^{61} +3.92303 q^{62} -18.1946 q^{63} +8.80482 q^{64} -0.282388 q^{65} -8.15260 q^{66} +6.52078 q^{67} -1.09114 q^{68} -4.45452 q^{69} +5.52982 q^{70} +2.32576 q^{71} -14.4052 q^{72} +7.70111 q^{73} -0.468800 q^{74} +10.4742 q^{75} -0.826306 q^{76} -8.45370 q^{77} -0.921262 q^{78} +5.44460 q^{79} +3.72037 q^{80} -0.309053 q^{81} -3.92507 q^{82} -7.46406 q^{83} -3.10879 q^{84} -4.15212 q^{85} +3.67779 q^{86} +2.79423 q^{87} -6.69306 q^{88} -12.6979 q^{89} -7.02498 q^{90} -0.955287 q^{91} -0.468668 q^{92} -8.39252 q^{93} +2.88609 q^{94} -3.14435 q^{95} -4.60723 q^{96} +10.2358 q^{97} +9.56378 q^{98} +10.7394 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) $ 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * 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 $

$2$	1.30614	0.923584	0.461792	−	0.886988i	$-0.347207\pi$
$2$	1.30614	0.923584	0.461792	+	0.886988i	$0.347207\pi$
$3$	−2.79423	−1.61325	−0.806624	−	0.591065i	$-0.798708\pi$
$3$	−2.79423	−1.61325	−0.806624	+	0.591065i	$0.798708\pi$
$4$	−0.293985	−0.146993
$5$	−1.11870	−0.500300	−0.250150	−	0.968207i	$-0.580480\pi$
$5$	−1.11870	−0.500300	−0.250150	+	0.968207i	$0.580480\pi$
$6$	−3.64967	−1.48997
$7$	−3.78446	−1.43039	−0.715196	−	0.698924i	$-0.753662\pi$
$7$	−3.78446	−1.43039	−0.715196	+	0.698924i	$0.753662\pi$
$8$	−2.99628	−1.05934
$9$	4.80771	1.60257
$10$	−1.46119	−0.462069
$11$	2.23379	0.673514	0.336757	−	0.941592i	$-0.390670\pi$
$11$	2.23379	0.673514	0.336757	+	0.941592i	$0.390670\pi$
$12$	0.821462	0.237136
$13$	0.252424	0.0700097	0.0350049	−	0.999387i	$-0.488855\pi$
$13$	0.252424	0.0700097	0.0350049	+	0.999387i	$0.488855\pi$
$14$	−4.94305	−1.32109
$15$	3.12592	0.807108
$16$	−3.32560	−0.831400
$17$	3.71154	0.900182	0.450091	−	0.892983i	$-0.351392\pi$
$17$	3.71154	0.900182	0.450091	+	0.892983i	$0.351392\pi$
$18$	6.27957	1.48011
$19$	2.81070	0.644820	0.322410	−	0.946600i	$-0.395507\pi$
$19$	2.81070	0.644820	0.322410	+	0.946600i	$0.395507\pi$
$20$	0.328883	0.0735405
$21$	10.5746	2.30758
$22$	2.91766	0.622046
$23$	1.59419	0.332411	0.166205	−	0.986091i	$-0.446848\pi$
$23$	1.59419	0.332411	0.166205	+	0.986091i	$0.446848\pi$
$24$	8.37228	1.70898
$25$	−3.74850	−0.749700
$26$	0.329702	0.0646598
$27$	−5.05115	−0.972095
$28$	1.11258	0.210257
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	4.08290	0.745432
$31$	3.00352	0.539448	0.269724	−	0.962938i	$-0.413068\pi$
$31$	3.00352	0.539448	0.269724	+	0.962938i	$0.413068\pi$
$32$	1.64884	0.291476
$33$	−6.24172	−1.08654
$34$	4.84781	0.831393
$35$	4.23369	0.715625
$36$	−1.41340	−0.235566
$37$	−0.358919	−0.0590060	−0.0295030	−	0.999565i	$-0.509392\pi$
$37$	−0.358919	−0.0590060	−0.0295030	+	0.999565i	$0.509392\pi$
$38$	3.67119	0.595545
$39$	−0.705329	−0.112943
$40$	3.35195	0.529990
$41$	−3.00508	−0.469315	−0.234657	−	0.972078i	$-0.575397\pi$
$41$	−3.00508	−0.469315	−0.234657	+	0.972078i	$0.575397\pi$
$42$	13.8120	2.13124
$43$	2.81576	0.429399	0.214700	−	0.976680i	$-0.431123\pi$
$43$	2.81576	0.429399	0.214700	+	0.976680i	$0.431123\pi$
$44$	−0.656702	−0.0990016
$45$	−5.37841	−0.801766
$46$	2.08224	0.307009
$47$	2.20963	0.322307	0.161153	−	0.986929i	$-0.448479\pi$
$47$	2.20963	0.322307	0.161153	+	0.986929i	$0.448479\pi$
$48$	9.29249	1.34126
$49$	7.32214	1.04602
$50$	−4.89608	−0.692411
$51$	−10.3709	−1.45222
$52$	−0.0742089	−0.0102909
$53$	12.5217	1.71999	0.859994	−	0.510304i	$-0.170467\pi$
$53$	12.5217	1.71999	0.859994	+	0.510304i	$0.170467\pi$
$54$	−6.59754	−0.897811
$55$	−2.49895	−0.336959
$56$	11.3393	1.51528
$57$	−7.85375	−1.04025
$58$	−1.30614	−0.171505
$59$	1.32679	0.172733	0.0863667	−	0.996263i	$-0.472474\pi$
$59$	1.32679	0.172733	0.0863667	+	0.996263i	$0.472474\pi$
$60$	−0.918974	−0.118639
$61$	−12.0308	−1.54039	−0.770195	−	0.637809i	$-0.779841\pi$
$61$	−12.0308	−1.54039	−0.770195	+	0.637809i	$0.779841\pi$
$62$	3.92303	0.498225
$63$	−18.1946	−2.29230
$64$	8.80482	1.10060
$65$	−0.282388	−0.0350259
$66$	−8.15260	−1.00352
$67$	6.52078	0.796640	0.398320	−	0.917247i	$-0.369593\pi$
$67$	6.52078	0.796640	0.398320	+	0.917247i	$0.369593\pi$
$68$	−1.09114	−0.132320
$69$	−4.45452	−0.536261
$70$	5.52982	0.660940
$71$	2.32576	0.276017	0.138008	−	0.990431i	$-0.455930\pi$
$71$	2.32576	0.276017	0.138008	+	0.990431i	$0.455930\pi$
$72$	−14.4052	−1.69767
$73$	7.70111	0.901346	0.450673	−	0.892689i	$-0.351184\pi$
$73$	7.70111	0.901346	0.450673	+	0.892689i	$0.351184\pi$
$74$	−0.468800	−0.0544969
$75$	10.4742	1.20945
$76$	−0.826306	−0.0947838
$77$	−8.45370	−0.963388
$78$	−0.921262	−0.104312
$79$	5.44460	0.612566	0.306283	−	0.951941i	$-0.400915\pi$
$79$	5.44460	0.612566	0.306283	+	0.951941i	$0.400915\pi$
$80$	3.72037	0.415950
$81$	−0.309053	−0.0343392
$82$	−3.92507	−0.433452
$83$	−7.46406	−0.819287	−0.409644	−	0.912246i	$-0.634347\pi$
$83$	−7.46406	−0.819287	−0.409644	+	0.912246i	$0.634347\pi$
$84$	−3.10879	−0.339197
$85$	−4.15212	−0.450361
$86$	3.67779	0.396586
$87$	2.79423	0.299573
$88$	−6.69306	−0.713483
$89$	−12.6979	−1.34597	−0.672986	−	0.739656i	$-0.734988\pi$
$89$	−12.6979	−1.34597	−0.672986	+	0.739656i	$0.734988\pi$
$90$	−7.02498	−0.740498
$91$	−0.955287	−0.100141
$92$	−0.468668	−0.0488620
$93$	−8.39252	−0.870263
$94$	2.88609	0.297678
$95$	−3.14435	−0.322603
$96$	−4.60723	−0.470223
$97$	10.2358	1.03928	0.519642	−	0.854384i	$-0.326065\pi$
$97$	10.2358	1.03928	0.519642	+	0.854384i	$0.326065\pi$
$98$	9.56378	0.966087
$99$	10.7394	1.07935
$100$	1.10200	0.110200
$101$	−13.2541	−1.31883	−0.659415	−	0.751779i	$-0.729196\pi$
$101$	−13.2541	−1.31883	−0.659415	+	0.751779i	$0.729196\pi$
$102$	−13.5459	−1.34124
$103$	−3.77382	−0.371846	−0.185923	−	0.982564i	$-0.559527\pi$
$103$	−3.77382	−0.371846	−0.185923	+	0.982564i	$0.559527\pi$
$104$	−0.756331	−0.0741644
$105$	−11.8299	−1.15448
$106$	16.3552	1.58855
$107$	−12.3238	−1.19138	−0.595692	−	0.803213i	$-0.703122\pi$
$107$	−12.3238	−1.19138	−0.595692	+	0.803213i	$0.703122\pi$
$108$	1.48497	0.142891
$109$	−17.1241	−1.64019	−0.820096	−	0.572226i	$-0.806080\pi$
$109$	−17.1241	−1.64019	−0.820096	+	0.572226i	$0.806080\pi$
$110$	−3.26400	−0.311210
$111$	1.00290	0.0951913
$112$	12.5856	1.18923
$113$	−8.21831	−0.773113	−0.386557	−	0.922266i	$-0.626336\pi$
$113$	−8.21831	−0.773113	−0.386557	+	0.922266i	$0.626336\pi$
$114$	−10.2581	−0.960762
$115$	−1.78342	−0.166305
$116$	0.293985	0.0272959
$117$	1.21358	0.112195
$118$	1.73298	0.159534
$119$	−14.0462	−1.28761
$120$	−9.36611	−0.855005
$121$	−6.01017	−0.546380
$122$	−15.7140	−1.42268
$123$	8.39688	0.757121
$124$	−0.882991	−0.0792949
$125$	9.78699	0.875375
$126$	−23.7648	−2.11713
$127$	−12.0815	−1.07206	−0.536028	−	0.844200i	$-0.680076\pi$
$127$	−12.0815	−1.07206	−0.536028	+	0.844200i	$0.680076\pi$
$128$	8.20270	0.725023
$129$	−7.86787	−0.692727
$130$	−0.368839	−0.0323493
$131$	10.4702	0.914785	0.457392	−	0.889265i	$-0.348784\pi$
$131$	10.4702	0.914785	0.457392	+	0.889265i	$0.348784\pi$
$132$	1.83498	0.159714
$133$	−10.6370	−0.922345
$134$	8.51708	0.735764
$135$	5.65075	0.486339
$136$	−11.1208	−0.953602
$137$	−18.8564	−1.61101	−0.805507	−	0.592587i	$-0.798107\pi$
$137$	−18.8564	−1.61101	−0.805507	+	0.592587i	$0.798107\pi$
$138$	−5.81825	−0.495282
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	−1.24464	−0.105192
$141$	−6.17420	−0.519961
$142$	3.03778	0.254925
$143$	0.563862	0.0471525
$144$	−15.9885	−1.33238
$145$	1.11870	0.0929034
$146$	10.0588	0.832469
$147$	−20.4597	−1.68749
$148$	0.105517	0.00867345
$149$	−8.56057	−0.701309	−0.350654	−	0.936505i	$-0.614041\pi$
$149$	−8.56057	−0.701309	−0.350654	+	0.936505i	$0.614041\pi$
$150$	13.6808	1.11703
$151$	10.0465	0.817574	0.408787	−	0.912630i	$-0.365952\pi$
$151$	10.0465	0.817574	0.408787	+	0.912630i	$0.365952\pi$
$152$	−8.42165	−0.683086
$153$	17.8440	1.44260
$154$	−11.0418	−0.889770
$155$	−3.36005	−0.269886
$156$	0.207356	0.0166018
$157$	17.2595	1.37746	0.688731	−	0.725017i	$-0.258168\pi$
$157$	17.2595	1.37746	0.688731	+	0.725017i	$0.258168\pi$
$158$	7.11144	0.565756
$159$	−34.9885	−2.77477
$160$	−1.84456	−0.145825
$161$	−6.03314	−0.475478
$162$	−0.403668	−0.0317152
$163$	−13.6721	−1.07088	−0.535440	−	0.844573i	$-0.679854\pi$
$163$	−13.6721	−1.07088	−0.535440	+	0.844573i	$0.679854\pi$
$164$	0.883450	0.0689859
$165$	6.98265	0.543598
$166$	−9.74915	−0.756681
$167$	16.0738	1.24383	0.621914	−	0.783086i	$-0.286355\pi$
$167$	16.0738	1.24383	0.621914	+	0.783086i	$0.286355\pi$
$168$	−31.6846	−2.44452
$169$	−12.9363	−0.995099
$170$	−5.42327	−0.415946
$171$	13.5131	1.03337
$172$	−0.827792	−0.0631186
$173$	20.0776	1.52647	0.763235	−	0.646122i	$-0.223610\pi$
$173$	20.0776	1.52647	0.763235	+	0.646122i	$0.223610\pi$
$174$	3.64967	0.276681
$175$	14.1860	1.07236
$176$	−7.42870	−0.559959
$177$	−3.70736	−0.278662
$178$	−16.5853	−1.24312
$179$	−2.64239	−0.197501	−0.0987506	−	0.995112i	$-0.531485\pi$
$179$	−2.64239	−0.197501	−0.0987506	+	0.995112i	$0.531485\pi$
$180$	1.58117	0.117854
$181$	−12.8485	−0.955023	−0.477512	−	0.878625i	$-0.658461\pi$
$181$	−12.8485	−0.955023	−0.477512	+	0.878625i	$0.658461\pi$
$182$	−1.24774	−0.0924889
$183$	33.6169	2.48503
$184$	−4.77663	−0.352138
$185$	0.401525	0.0295207
$186$	−10.9618	−0.803761
$187$	8.29081	0.606284
$188$	−0.649598	−0.0473768
$189$	19.1159	1.39048
$190$	−4.10697	−0.297951
$191$	9.60067	0.694680	0.347340	−	0.937739i	$-0.387085\pi$
$191$	9.60067	0.694680	0.347340	+	0.937739i	$0.387085\pi$
$192$	−24.6027	−1.77555
$193$	18.7349	1.34857	0.674285	−	0.738471i	$-0.264452\pi$
$193$	18.7349	1.34857	0.674285	+	0.738471i	$0.264452\pi$
$194$	13.3694	0.959866
$195$	0.789055	0.0565054
$196$	−2.15260	−0.153757
$197$	−5.89417	−0.419942	−0.209971	−	0.977708i	$-0.567337\pi$
$197$	−5.89417	−0.419942	−0.209971	+	0.977708i	$0.567337\pi$
$198$	14.0272	0.996873
$199$	27.7401	1.96645	0.983223	−	0.182410i	$-0.0583897\pi$
$199$	27.7401	1.96645	0.983223	+	0.182410i	$0.0583897\pi$
$200$	11.2315	0.794190
$201$	−18.2205	−1.28518
$202$	−17.3117	−1.21805
$203$	3.78446	0.265617
$204$	3.04889	0.213465
$205$	3.36180	0.234798
$206$	−4.92916	−0.343431
$207$	7.66439	0.532712
$208$	−0.839460	−0.0582061
$209$	6.27853	0.434295
$210$	−15.4516	−1.06626
$211$	3.79181	0.261039	0.130519	−	0.991446i	$-0.458335\pi$
$211$	3.79181	0.261039	0.130519	+	0.991446i	$0.458335\pi$
$212$	−3.68120	−0.252826
$213$	−6.49870	−0.445284
$214$	−16.0966	−1.10034
$215$	−3.15000	−0.214828
$216$	15.1347	1.02978
$217$	−11.3667	−0.771622
$218$	−22.3666	−1.51485
$219$	−21.5186	−1.45410
$220$	0.734656	0.0495305
$221$	0.936881	0.0630214
$222$	1.30994	0.0879171
$223$	5.37666	0.360048	0.180024	−	0.983662i	$-0.442382\pi$
$223$	5.37666	0.360048	0.180024	+	0.983662i	$0.442382\pi$
$224$	−6.23996	−0.416925
$225$	−18.0217	−1.20145
$226$	−10.7343	−0.714035
$227$	6.47520	0.429774	0.214887	−	0.976639i	$-0.431062\pi$
$227$	6.47520	0.429774	0.214887	+	0.976639i	$0.431062\pi$
$228$	2.30889	0.152910
$229$	12.6476	0.835778	0.417889	−	0.908498i	$-0.362770\pi$
$229$	12.6476	0.835778	0.417889	+	0.908498i	$0.362770\pi$
$230$	−2.32941	−0.153597
$231$	23.6216	1.55418
$232$	2.99628	0.196715
$233$	24.4679	1.60295	0.801474	−	0.598030i	$-0.204050\pi$
$233$	24.4679	1.60295	0.801474	+	0.598030i	$0.204050\pi$
$234$	1.58511	0.103622
$235$	−2.47192	−0.161250
$236$	−0.390057	−0.0253906
$237$	−15.2135	−0.988221
$238$	−18.3464	−1.18922
$239$	4.30697	0.278595	0.139298	−	0.990251i	$-0.455516\pi$
$239$	4.30697	0.278595	0.139298	+	0.990251i	$0.455516\pi$
$240$	−10.3956	−0.671030
$241$	−12.9234	−0.832467	−0.416234	−	0.909258i	$-0.636650\pi$
$241$	−12.9234	−0.832467	−0.416234	+	0.909258i	$0.636650\pi$
$242$	−7.85016	−0.504627
$243$	16.0170	1.02749
$244$	3.53689	0.226426
$245$	−8.19132	−0.523324
$246$	10.9675	0.699265
$247$	0.709488	0.0451436
$248$	−8.99938	−0.571461
$249$	20.8563	1.32171
$250$	12.7832	0.808482
$251$	−12.6480	−0.798334	−0.399167	−	0.916878i	$-0.630701\pi$
$251$	−12.6480	−0.798334	−0.399167	+	0.916878i	$0.630701\pi$
$252$	5.34894	0.336952
$253$	3.56108	0.223883
$254$	−15.7801	−0.990134
$255$	11.6020	0.726544
$256$	−6.89573	−0.430983
$257$	−16.3349	−1.01894	−0.509472	−	0.860487i	$-0.670159\pi$
$257$	−16.3349	−1.01894	−0.509472	+	0.860487i	$0.670159\pi$
$258$	−10.2766	−0.639792
$259$	1.35832	0.0844016
$260$	0.0830178	0.00514855
$261$	−4.80771	−0.297590
$262$	13.6756	0.844880
$263$	−17.6196	−1.08647	−0.543235	−	0.839581i	$-0.682801\pi$
$263$	−17.6196	−1.08647	−0.543235	+	0.839581i	$0.682801\pi$
$264$	18.7019	1.15102
$265$	−14.0081	−0.860510
$266$	−13.8935	−0.851863
$267$	35.4807	2.17139
$268$	−1.91701	−0.117100
$269$	−28.8184	−1.75709	−0.878543	−	0.477663i	$-0.841484\pi$
$269$	−28.8184	−1.75709	−0.878543	+	0.477663i	$0.841484\pi$
$270$	7.38070	0.449175
$271$	−12.7492	−0.774458	−0.387229	−	0.921983i	$-0.626568\pi$
$271$	−12.7492	−0.774458	−0.387229	+	0.921983i	$0.626568\pi$
$272$	−12.3431	−0.748411
$273$	2.66929	0.161553
$274$	−24.6292	−1.48791
$275$	−8.37337	−0.504933
$276$	1.30956	0.0788265
$277$	11.4070	0.685377	0.342689	−	0.939449i	$-0.388662\pi$
$277$	11.4070	0.685377	0.342689	+	0.939449i	$0.388662\pi$
$278$	−1.30614	−0.0783374
$279$	14.4400	0.864503
$280$	−12.6853	−0.758093
$281$	−13.4904	−0.804773	−0.402386	−	0.915470i	$-0.631819\pi$
$281$	−13.4904	−0.804773	−0.402386	+	0.915470i	$0.631819\pi$
$282$	−8.06439	−0.480228
$283$	−14.4968	−0.861743	−0.430871	−	0.902413i	$-0.641794\pi$
$283$	−14.4968	−0.861743	−0.430871	+	0.902413i	$0.641794\pi$
$284$	−0.683740	−0.0405725
$285$	8.78603	0.520439
$286$	0.736485	0.0435493
$287$	11.3726	0.671304
$288$	7.92713	0.467111
$289$	−3.22445	−0.189673
$290$	1.46119	0.0858041
$291$	−28.6011	−1.67662
$292$	−2.26401	−0.132491
$293$	−3.45960	−0.202112	−0.101056	−	0.994881i	$-0.532222\pi$
$293$	−3.45960	−0.202112	−0.101056	+	0.994881i	$0.532222\pi$
$294$	−26.7234	−1.55854
$295$	−1.48429	−0.0864185
$296$	1.07542	0.0625076
$297$	−11.2832	−0.654719
$298$	−11.1813	−0.647718
$299$	0.402410	0.0232720
$300$	−3.07925	−0.177781
$301$	−10.6561	−0.614209
$302$	13.1222	0.755098
$303$	37.0349	2.12760
$304$	−9.34728	−0.536103
$305$	13.4589	0.770657
$306$	23.3069	1.33237
$307$	−22.1992	−1.26697	−0.633487	−	0.773753i	$-0.718377\pi$
$307$	−22.1992	−1.26697	−0.633487	+	0.773753i	$0.718377\pi$
$308$	2.48526	0.141611
$309$	10.5449	0.599880
$310$	−4.38871	−0.249262
$311$	3.47265	0.196916	0.0984580	−	0.995141i	$-0.468609\pi$
$311$	3.47265	0.196916	0.0984580	+	0.995141i	$0.468609\pi$
$312$	2.11336	0.119646
$313$	−1.06154	−0.0600020	−0.0300010	−	0.999550i	$-0.509551\pi$
$313$	−1.06154	−0.0600020	−0.0300010	+	0.999550i	$0.509551\pi$
$314$	22.5435	1.27220
$315$	20.3544	1.14684
$316$	−1.60063	−0.0900427
$317$	−29.2539	−1.64307	−0.821533	−	0.570161i	$-0.806881\pi$
$317$	−29.2539	−1.64307	−0.821533	+	0.570161i	$0.806881\pi$
$318$	−45.7000	−2.56273
$319$	−2.23379	−0.125068
$320$	−9.85000	−0.550632
$321$	34.4354	1.92200
$322$	−7.88015	−0.439144
$323$	10.4321	0.580455
$324$	0.0908571	0.00504762
$325$	−0.946210	−0.0524863
$326$	−17.8577	−0.989048
$327$	47.8486	2.64604
$328$	9.00406	0.497166
$329$	−8.36224	−0.461025
$330$	9.12035	0.502059
$331$	22.8671	1.25689	0.628446	−	0.777853i	$-0.283691\pi$
$331$	22.8671	1.25689	0.628446	+	0.777853i	$0.283691\pi$
$332$	2.19433	0.120429
$333$	−1.72558	−0.0945612
$334$	20.9947	1.14878
$335$	−7.29483	−0.398559
$336$	−35.1671	−1.91852
$337$	17.7002	0.964192	0.482096	−	0.876118i	$-0.339876\pi$
$337$	17.7002	0.964192	0.482096	+	0.876118i	$0.339876\pi$
$338$	−16.8967	−0.919057
$339$	22.9638	1.24722
$340$	1.22066	0.0661998
$341$	6.70924	0.363325
$342$	17.6500	0.954403
$343$	−1.21913	−0.0658268
$344$	−8.43679	−0.454881
$345$	4.98330	0.268292
$346$	26.2242	1.40982
$347$	−7.43058	−0.398895	−0.199447	−	0.979909i	$-0.563915\pi$
$347$	−7.43058	−0.398895	−0.199447	+	0.979909i	$0.563915\pi$
$348$	−0.821462	−0.0440350
$349$	27.1571	1.45369	0.726843	−	0.686804i	$-0.240987\pi$
$349$	27.1571	1.45369	0.726843	+	0.686804i	$0.240987\pi$
$350$	18.5290	0.990418
$351$	−1.27503	−0.0680561
$352$	3.68316	0.196313
$353$	6.84781	0.364472	0.182236	−	0.983255i	$-0.441666\pi$
$353$	6.84781	0.364472	0.182236	+	0.983255i	$0.441666\pi$
$354$	−4.84234	−0.257368
$355$	−2.60184	−0.138091
$356$	3.73299	0.197848
$357$	39.2483	2.07724
$358$	−3.45134	−0.182409
$359$	−23.8949	−1.26112	−0.630562	−	0.776139i	$-0.717176\pi$
$359$	−23.8949	−1.26112	−0.630562	+	0.776139i	$0.717176\pi$
$360$	16.1152	0.849346
$361$	−11.0999	−0.584207
$362$	−16.7820	−0.882044
$363$	16.7938	0.881446
$364$	0.280841	0.0147200
$365$	−8.61526	−0.450944
$366$	43.9085	2.29513
$367$	−3.37960	−0.176414	−0.0882068	−	0.996102i	$-0.528114\pi$
$367$	−3.37960	−0.176414	−0.0882068	+	0.996102i	$0.528114\pi$
$368$	−5.30163	−0.276367
$369$	−14.4476	−0.752110
$370$	0.524449	0.0272648
$371$	−47.3879	−2.46026
$372$	2.46728	0.127922
$373$	4.29426	0.222348	0.111174	−	0.993801i	$-0.464539\pi$
$373$	4.29426	0.222348	0.111174	+	0.993801i	$0.464539\pi$
$374$	10.8290	0.559955
$375$	−27.3471	−1.41220
$376$	−6.62065	−0.341434
$377$	−0.252424	−0.0130005
$378$	24.9681	1.28422
$379$	−16.2310	−0.833728	−0.416864	−	0.908969i	$-0.636871\pi$
$379$	−16.2310	−0.833728	−0.416864	+	0.908969i	$0.636871\pi$
$380$	0.924393	0.0474203
$381$	33.7584	1.72949
$382$	12.5399	0.641595
$383$	−16.7115	−0.853920	−0.426960	−	0.904271i	$-0.640415\pi$
$383$	−16.7115	−0.853920	−0.426960	+	0.904271i	$0.640415\pi$
$384$	−22.9202	−1.16964
$385$	9.45719	0.481983
$386$	24.4705	1.24552
$387$	13.5374	0.688142
$388$	−3.00917	−0.152767
$389$	−3.17088	−0.160770	−0.0803849	−	0.996764i	$-0.525615\pi$
$389$	−3.17088	−0.160770	−0.0803849	+	0.996764i	$0.525615\pi$
$390$	1.03062	0.0521875
$391$	5.91689	0.299230
$392$	−21.9392	−1.10810
$393$	−29.2561	−1.47577
$394$	−7.69864	−0.387852
$395$	−6.09090	−0.306467
$396$	−3.15723	−0.158657
$397$	5.03835	0.252867	0.126434	−	0.991975i	$-0.459647\pi$
$397$	5.03835	0.252867	0.126434	+	0.991975i	$0.459647\pi$
$398$	36.2326	1.81618
$399$	29.7222	1.48797
$400$	12.4660	0.623301
$401$	15.9224	0.795129	0.397565	−	0.917574i	$-0.369855\pi$
$401$	15.9224	0.795129	0.397565	+	0.917574i	$0.369855\pi$
$402$	−23.7987	−1.18697
$403$	0.758159	0.0377666
$404$	3.89650	0.193858
$405$	0.345739	0.0171799
$406$	4.94305	0.245320
$407$	−0.801751	−0.0397413
$408$	31.0741	1.53840
$409$	−7.28600	−0.360269	−0.180135	−	0.983642i	$-0.557653\pi$
$409$	−7.28600	−0.360269	−0.180135	+	0.983642i	$0.557653\pi$
$410$	4.39100	0.216856
$411$	52.6891	2.59896
$412$	1.10945	0.0546586
$413$	−5.02119	−0.247076
$414$	10.0108	0.492004
$415$	8.35008	0.409890
$416$	0.416205	0.0204061
$417$	2.79423	0.136834
$418$	8.20067	0.401108
$419$	4.68634	0.228943	0.114471	−	0.993427i	$-0.463483\pi$
$419$	4.68634	0.228943	0.114471	+	0.993427i	$0.463483\pi$
$420$	3.47782	0.169700
$421$	−20.3540	−0.991993	−0.495997	−	0.868325i	$-0.665197\pi$
$421$	−20.3540	−0.991993	−0.495997	+	0.868325i	$0.665197\pi$
$422$	4.95265	0.241091
$423$	10.6232	0.516519
$424$	−37.5185	−1.82206
$425$	−13.9127	−0.674866
$426$	−8.48825	−0.411257
$427$	45.5302	2.20336
$428$	3.62301	0.175125
$429$	−1.57556	−0.0760687
$430$	−4.11436	−0.198412
$431$	17.7635	0.855639	0.427820	−	0.903864i	$-0.359282\pi$
$431$	17.7635	0.855639	0.427820	+	0.903864i	$0.359282\pi$
$432$	16.7981	0.808200
$433$	−10.1083	−0.485772	−0.242886	−	0.970055i	$-0.578094\pi$
$433$	−10.1083	−0.485772	−0.242886	+	0.970055i	$0.578094\pi$
$434$	−14.8466	−0.712657
$435$	−3.12592	−0.149876
$436$	5.03424	0.241096
$437$	4.48079	0.214345
$438$	−28.1065	−1.34298
$439$	−3.48595	−0.166375	−0.0831877	−	0.996534i	$-0.526510\pi$
$439$	−3.48595	−0.166375	−0.0831877	+	0.996534i	$0.526510\pi$
$440$	7.48756	0.356955
$441$	35.2027	1.67632
$442$	1.22370	0.0582056
$443$	15.8133	0.751314	0.375657	−	0.926759i	$-0.377417\pi$
$443$	15.8133	0.751314	0.375657	+	0.926759i	$0.377417\pi$
$444$	−0.294839	−0.0139924
$445$	14.2052	0.673389
$446$	7.02270	0.332535
$447$	23.9202	1.13139
$448$	−33.3215	−1.57429
$449$	10.6235	0.501353	0.250677	−	0.968071i	$-0.419347\pi$
$449$	10.6235	0.501353	0.250677	+	0.968071i	$0.419347\pi$
$450$	−23.5389	−1.10964
$451$	−6.71273	−0.316090
$452$	2.41606	0.113642
$453$	−28.0723	−1.31895
$454$	8.45755	0.396933
$455$	1.06868	0.0501007
$456$	23.5320	1.10199
$457$	−28.2105	−1.31963	−0.659817	−	0.751426i	$-0.729366\pi$
$457$	−28.2105	−1.31963	−0.659817	+	0.751426i	$0.729366\pi$
$458$	16.5196	0.771911
$459$	−18.7476	−0.875062
$460$	0.524301	0.0244457
$461$	−14.5897	−0.679512	−0.339756	−	0.940514i	$-0.610345\pi$
$461$	−14.5897	−0.679512	−0.339756	+	0.940514i	$0.610345\pi$
$462$	30.8532	1.43542
$463$	15.7438	0.731674	0.365837	−	0.930679i	$-0.380783\pi$
$463$	15.7438	0.731674	0.365837	+	0.930679i	$0.380783\pi$
$464$	3.32560	0.154387
$465$	9.38875	0.435393
$466$	31.9587	1.48046
$467$	37.1419	1.71872	0.859361	−	0.511370i	$-0.170862\pi$
$467$	37.1419	1.71872	0.859361	+	0.511370i	$0.170862\pi$
$468$	−0.356775	−0.0164919
$469$	−24.6776	−1.13951
$470$	−3.22868	−0.148928
$471$	−48.2271	−2.22219
$472$	−3.97543	−0.182984
$473$	6.28982	0.289206
$474$	−19.8710	−0.912705
$475$	−10.5359	−0.483421
$476$	4.12938	0.189270
$477$	60.2007	2.75640
$478$	5.62553	0.257306
$479$	−18.7912	−0.858593	−0.429297	−	0.903164i	$-0.641239\pi$
$479$	−18.7912	−0.858593	−0.429297	+	0.903164i	$0.641239\pi$
$480$	5.15413	0.235253
$481$	−0.0905997	−0.00413099
$482$	−16.8798	−0.768853
$483$	16.8580	0.767064
$484$	1.76690	0.0803138
$485$	−11.4508	−0.519954
$486$	20.9206	0.948976
$487$	−29.3834	−1.33149	−0.665744	−	0.746180i	$-0.731886\pi$
$487$	−29.3834	−1.33149	−0.665744	+	0.746180i	$0.731886\pi$
$488$	36.0477	1.63180
$489$	38.2029	1.72760
$490$	−10.6990	−0.483334
$491$	1.08069	0.0487708	0.0243854	−	0.999703i	$-0.492237\pi$
$491$	1.08069	0.0487708	0.0243854	+	0.999703i	$0.492237\pi$
$492$	−2.46856	−0.111291
$493$	−3.71154	−0.167160
$494$	0.926694	0.0416939
$495$	−12.0142	−0.540000
$496$	−9.98851	−0.448497
$497$	−8.80175	−0.394812
$498$	27.2413	1.22071
$499$	17.8236	0.797895	0.398948	−	0.916974i	$-0.369376\pi$
$499$	17.8236	0.797895	0.398948	+	0.916974i	$0.369376\pi$
$500$	−2.87723	−0.128674
$501$	−44.9139	−2.00660
$502$	−16.5201	−0.737328
$503$	10.0207	0.446803	0.223401	−	0.974727i	$-0.428284\pi$
$503$	10.0207	0.446803	0.223401	+	0.974727i	$0.428284\pi$
$504$	54.5160	2.42834
$505$	14.8274	0.659810
$506$	4.65129	0.206775
$507$	36.1469	1.60534
$508$	3.55177	0.157584
$509$	27.3731	1.21329	0.606647	−	0.794972i	$-0.292514\pi$
$509$	27.3731	1.21329	0.606647	+	0.794972i	$0.292514\pi$
$510$	15.1539	0.671024
$511$	−29.1445	−1.28928
$512$	−25.4122	−1.12307
$513$	−14.1973	−0.626826
$514$	−21.3358	−0.941080
$515$	4.22179	0.186034
$516$	2.31304	0.101826
$517$	4.93584	0.217078
$518$	1.77416	0.0779520
$519$	−56.1013	−2.46257
$520$	0.846111	0.0371044
$521$	−19.5621	−0.857029	−0.428515	−	0.903535i	$-0.640963\pi$
$521$	−19.5621	−0.857029	−0.428515	+	0.903535i	$0.640963\pi$
$522$	−6.27957	−0.274849
$523$	32.1253	1.40474	0.702370	−	0.711812i	$-0.252125\pi$
$523$	32.1253	1.40474	0.702370	+	0.711812i	$0.252125\pi$
$524$	−3.07808	−0.134467
$525$	−39.6391	−1.72999
$526$	−23.0137	−1.00345
$527$	11.1477	0.485601
$528$	20.7575	0.903354
$529$	−20.4586	−0.889503
$530$	−18.2966	−0.794753
$531$	6.37882	0.276817
$532$	3.12712	0.135578
$533$	−0.758553	−0.0328566
$534$	46.3430	2.00546
$535$	13.7867	0.596050
$536$	−19.5381	−0.843916
$537$	7.38343	0.318618
$538$	−37.6410	−1.62282
$539$	16.3561	0.704509
$540$	−1.66124	−0.0714883
$541$	−16.8417	−0.724081	−0.362041	−	0.932162i	$-0.617920\pi$
$541$	−16.8417	−0.724081	−0.362041	+	0.932162i	$0.617920\pi$
$542$	−16.6523	−0.715277
$543$	35.9017	1.54069
$544$	6.11973	0.262381
$545$	19.1568	0.820588
$546$	3.48648	0.149208
$547$	−4.43378	−0.189575	−0.0947874	−	0.995498i	$-0.530217\pi$
$547$	−4.43378	−0.189575	−0.0947874	+	0.995498i	$0.530217\pi$
$548$	5.54351	0.236807
$549$	−57.8407	−2.46858
$550$	−10.9368	−0.466348
$551$	−2.81070	−0.119740
$552$	13.3470	0.568085
$553$	−20.6049	−0.876209
$554$	14.8991	0.633003
$555$	−1.12195	−0.0476242
$556$	0.293985	0.0124678
$557$	−4.62202	−0.195841	−0.0979205	−	0.995194i	$-0.531219\pi$
$557$	−4.62202	−0.195841	−0.0979205	+	0.995194i	$0.531219\pi$
$558$	18.8608	0.798441
$559$	0.710764	0.0300621
$560$	−14.0796	−0.594971
$561$	−23.1664	−0.978087
$562$	−17.6205	−0.743275
$563$	−16.7682	−0.706697	−0.353349	−	0.935492i	$-0.614957\pi$
$563$	−16.7682	−0.706697	−0.353349	+	0.935492i	$0.614957\pi$
$564$	1.81512	0.0764305
$565$	9.19386	0.386789
$566$	−18.9349	−0.795892
$567$	1.16960	0.0491185
$568$	−6.96862	−0.292397
$569$	−27.1897	−1.13985	−0.569927	−	0.821696i	$-0.693028\pi$
$569$	−27.1897	−1.13985	−0.569927	+	0.821696i	$0.693028\pi$
$570$	11.4758	0.480669
$571$	−20.9007	−0.874666	−0.437333	−	0.899300i	$-0.644077\pi$
$571$	−20.9007	−0.874666	−0.437333	+	0.899300i	$0.644077\pi$
$572$	−0.165767	−0.00693107
$573$	−26.8265	−1.12069
$574$	14.8543	0.620006
$575$	−5.97581	−0.249208
$576$	42.3310	1.76379
$577$	−31.8470	−1.32581	−0.662903	−	0.748705i	$-0.730676\pi$
$577$	−31.8470	−1.32581	−0.662903	+	0.748705i	$0.730676\pi$
$578$	−4.21159	−0.175179
$579$	−52.3497	−2.17558
$580$	−0.328883	−0.0136561
$581$	28.2475	1.17190
$582$	−37.3571	−1.54850
$583$	27.9709	1.15844
$584$	−23.0746	−0.954836
$585$	−1.35764	−0.0561314
$586$	−4.51874	−0.186667
$587$	−2.74610	−0.113343	−0.0566717	−	0.998393i	$-0.518049\pi$
$587$	−2.74610	−0.113343	−0.0566717	+	0.998393i	$0.518049\pi$
$588$	6.01486	0.248049
$589$	8.44200	0.347847
$590$	−1.93869	−0.0798148
$591$	16.4696	0.677471
$592$	1.19362	0.0490576
$593$	−9.93931	−0.408158	−0.204079	−	0.978954i	$-0.565420\pi$
$593$	−9.93931	−0.408158	−0.204079	+	0.978954i	$0.565420\pi$
$594$	−14.7375	−0.604688
$595$	15.7135	0.644192
$596$	2.51668	0.103087
$597$	−77.5122	−3.17236
$598$	0.525606	0.0214936
$599$	14.1965	0.580052	0.290026	−	0.957019i	$-0.406336\pi$
$599$	14.1965	0.580052	0.290026	+	0.957019i	$0.406336\pi$
$600$	−31.3835	−1.28123
$601$	9.53919	0.389112	0.194556	−	0.980891i	$-0.437673\pi$
$601$	9.53919	0.389112	0.194556	+	0.980891i	$0.437673\pi$
$602$	−13.9184	−0.567274
$603$	31.3500	1.27667
$604$	−2.95353	−0.120177
$605$	6.72361	0.273354
$606$	48.3729	1.96502
$607$	−4.73680	−0.192261	−0.0961304	−	0.995369i	$-0.530647\pi$
$607$	−4.73680	−0.192261	−0.0961304	+	0.995369i	$0.530647\pi$
$608$	4.63439	0.187949
$609$	−10.5746	−0.428506
$610$	17.5793	0.711766
$611$	0.557761	0.0225646
$612$	−5.24588	−0.212052
$613$	0.0372771	0.00150561	0.000752804	−	1.00000i	$-0.499760\pi$
$613$	0.0372771	0.00150561	0.000752804		1.00000i	$0.499760\pi$
$614$	−28.9954	−1.17016
$615$	−9.39363	−0.378788
$616$	25.3296	1.02056
$617$	−6.17109	−0.248439	−0.124219	−	0.992255i	$-0.539643\pi$
$617$	−6.17109	−0.248439	−0.124219	+	0.992255i	$0.539643\pi$
$618$	13.7732	0.554039
$619$	8.46906	0.340400	0.170200	−	0.985409i	$-0.445559\pi$
$619$	8.46906	0.340400	0.170200	+	0.985409i	$0.445559\pi$
$620$	0.987806	0.0396713
$621$	−8.05249	−0.323135
$622$	4.53578	0.181868
$623$	48.0546	1.92527
$624$	2.34564	0.0939009
$625$	7.79374	0.311750
$626$	−1.38653	−0.0554169
$627$	−17.5436	−0.700625
$628$	−5.07405	−0.202477
$629$	−1.33214	−0.0531161
$630$	26.5858	1.05920
$631$	−48.1067	−1.91510	−0.957550	−	0.288269i	$-0.906920\pi$
$631$	−48.1067	−1.91510	−0.957550	+	0.288269i	$0.906920\pi$
$632$	−16.3135	−0.648918
$633$	−10.5952	−0.421121
$634$	−38.2099	−1.51751
$635$	13.5156	0.536350
$636$	10.2861	0.407871
$637$	1.84828	0.0732316
$638$	−2.91766	−0.115511
$639$	11.1816	0.442336
$640$	−9.17640	−0.362729
$641$	−21.6718	−0.855986	−0.427993	−	0.903782i	$-0.640779\pi$
$641$	−21.6718	−0.855986	−0.427993	+	0.903782i	$0.640779\pi$
$642$	44.9777	1.77513
$643$	19.4601	0.767430	0.383715	−	0.923452i	$-0.374645\pi$
$643$	19.4601	0.767430	0.383715	+	0.923452i	$0.374645\pi$
$644$	1.77365	0.0698918
$645$	8.80183	0.346572
$646$	13.6258	0.536099
$647$	−21.0427	−0.827273	−0.413636	−	0.910442i	$-0.635742\pi$
$647$	−21.0427	−0.827273	−0.413636	+	0.910442i	$0.635742\pi$
$648$	0.926009	0.0363771
$649$	2.96377	0.116338
$650$	−1.23589	−0.0484755
$651$	31.7611	1.24482
$652$	4.01940	0.157412
$653$	−27.9846	−1.09512	−0.547562	−	0.836765i	$-0.684444\pi$
$653$	−27.9846	−1.09512	−0.547562	+	0.836765i	$0.684444\pi$
$654$	62.4973	2.44384
$655$	−11.7131	−0.457667
$656$	9.99370	0.390189
$657$	37.0247	1.44447
$658$	−10.9223	−0.425795
$659$	−6.06475	−0.236249	−0.118125	−	0.992999i	$-0.537688\pi$
$659$	−6.06475	−0.236249	−0.118125	+	0.992999i	$0.537688\pi$
$660$	−2.05280	−0.0799050
$661$	−28.5472	−1.11036	−0.555179	−	0.831731i	$-0.687350\pi$
$661$	−28.5472	−1.11036	−0.555179	+	0.831731i	$0.687350\pi$
$662$	29.8678	1.16085
$663$	−2.61786	−0.101669
$664$	22.3644	0.867907
$665$	11.8997	0.461449
$666$	−2.25386	−0.0873352
$667$	−1.59419	−0.0617272
$668$	−4.72546	−0.182834
$669$	−15.0236	−0.580847
$670$	−9.52810	−0.368103
$671$	−26.8744	−1.03747
$672$	17.4359	0.672603
$673$	−10.6494	−0.410503	−0.205251	−	0.978709i	$-0.565801\pi$
$673$	−10.6494	−0.410503	−0.205251	+	0.978709i	$0.565801\pi$
$674$	23.1190	0.890512
$675$	18.9342	0.728780
$676$	3.80308	0.146272
$677$	1.74777	0.0671722	0.0335861	−	0.999436i	$-0.489307\pi$
$677$	1.74777	0.0671722	0.0335861	+	0.999436i	$0.489307\pi$
$678$	29.9941	1.15192
$679$	−38.7368	−1.48658
$680$	12.4409	0.477087
$681$	−18.0932	−0.693333
$682$	8.76323	0.335562
$683$	18.0506	0.690686	0.345343	−	0.938477i	$-0.387763\pi$
$683$	18.0506	0.690686	0.345343	+	0.938477i	$0.387763\pi$
$684$	−3.97264	−0.151898
$685$	21.0948	0.805990
$686$	−1.59236	−0.0607966
$687$	−35.3403	−1.34832
$688$	−9.36409	−0.357003
$689$	3.16077	0.120416
$690$	6.50891	0.247790
$691$	−19.0079	−0.723095	−0.361548	−	0.932354i	$-0.617751\pi$
$691$	−19.0079	−0.723095	−0.361548	+	0.932354i	$0.617751\pi$
$692$	−5.90251	−0.224380
$693$	−40.6429	−1.54390
$694$	−9.70542	−0.368413
$695$	1.11870	0.0424349
$696$	−8.37228	−0.317351
$697$	−11.1535	−0.422469
$698$	35.4711	1.34260
$699$	−68.3690	−2.58595
$700$	−4.17049	−0.157630
$701$	−25.9755	−0.981082	−0.490541	−	0.871418i	$-0.663201\pi$
$701$	−25.9755	−0.981082	−0.490541	+	0.871418i	$0.663201\pi$
$702$	−1.66537	−0.0628555
$703$	−1.00882	−0.0380482
$704$	19.6681	0.741271
$705$	6.90710	0.260137
$706$	8.94424	0.336621
$707$	50.1595	1.88644
$708$	1.08991	0.0409613
$709$	43.6784	1.64038	0.820189	−	0.572093i	$-0.193868\pi$
$709$	43.6784	1.64038	0.820189	+	0.572093i	$0.193868\pi$
$710$	−3.39838	−0.127539
$711$	26.1761	0.981679
$712$	38.0463	1.42585
$713$	4.78817	0.179318
$714$	51.2639	1.91850
$715$	−0.630795	−0.0235904
$716$	0.776823	0.0290312
$717$	−12.0347	−0.449443
$718$	−31.2102	−1.16475
$719$	−14.8673	−0.554457	−0.277229	−	0.960804i	$-0.589416\pi$
$719$	−14.8673	−0.554457	−0.277229	+	0.960804i	$0.589416\pi$
$720$	17.8864	0.666589
$721$	14.2819	0.531885
$722$	−14.4981	−0.539565
$723$	36.1108	1.34298
$724$	3.77728	0.140381
$725$	3.74850	0.139216
$726$	21.9351	0.814089
$727$	47.9511	1.77841	0.889203	−	0.457512i	$-0.151259\pi$
$727$	47.9511	1.77841	0.889203	+	0.457512i	$0.151259\pi$
$728$	2.86231	0.106084
$729$	−43.8281	−1.62326
$730$	−11.2528	−0.416484
$731$	10.4508	0.386537
$732$	−9.88287	−0.365281
$733$	29.4572	1.08803	0.544014	−	0.839076i	$-0.316904\pi$
$733$	29.4572	1.08803	0.544014	+	0.839076i	$0.316904\pi$
$734$	−4.41425	−0.162933
$735$	22.8884	0.844251
$736$	2.62855	0.0968898
$737$	14.5661	0.536548
$738$	−18.8706	−0.694637
$739$	−30.7803	−1.13227	−0.566137	−	0.824311i	$-0.691563\pi$
$739$	−30.7803	−1.13227	−0.566137	+	0.824311i	$0.691563\pi$
$740$	−0.118042	−0.00433933
$741$	−1.98247	−0.0728279
$742$	−61.8955	−2.27225
$743$	8.09360	0.296925	0.148463	−	0.988918i	$-0.452568\pi$
$743$	8.09360	0.296925	0.148463	+	0.988918i	$0.452568\pi$
$744$	25.1463	0.921908
$745$	9.57675	0.350865
$746$	5.60893	0.205357
$747$	−35.8851	−1.31297
$748$	−2.43738	−0.0891194
$749$	46.6388	1.70415
$750$	−35.7192	−1.30428
$751$	36.9839	1.34956	0.674781	−	0.738018i	$-0.264238\pi$
$751$	36.9839	1.34956	0.674781	+	0.738018i	$0.264238\pi$
$752$	−7.34833	−0.267966
$753$	35.3414	1.28791
$754$	−0.329702	−0.0120070
$755$	−11.2391	−0.409032
$756$	−5.61980	−0.204390
$757$	29.8650	1.08546	0.542731	−	0.839907i	$-0.317390\pi$
$757$	29.8650	1.08546	0.542731	+	0.839907i	$0.317390\pi$
$758$	−21.2000	−0.770018
$759$	−9.95047	−0.361179
$760$	9.42134	0.341748
$761$	22.4462	0.813674	0.406837	−	0.913501i	$-0.366632\pi$
$761$	22.4462	0.813674	0.406837	+	0.913501i	$0.366632\pi$
$762$	44.0933	1.59733
$763$	64.8055	2.34612
$764$	−2.82246	−0.102113
$765$	−19.9622	−0.721735
$766$	−21.8277	−0.788667
$767$	0.334913	0.0120930
$768$	19.2682	0.695283
$769$	−54.6555	−1.97093	−0.985463	−	0.169889i	$-0.945659\pi$
$769$	−54.6555	−1.97093	−0.985463	+	0.169889i	$0.945659\pi$
$770$	12.3525	0.445152
$771$	45.6435	1.64381
$772$	−5.50780	−0.198230
$773$	15.2531	0.548614	0.274307	−	0.961642i	$-0.411552\pi$
$773$	15.2531	0.548614	0.274307	+	0.961642i	$0.411552\pi$
$774$	17.6817	0.635557
$775$	−11.2587	−0.404424
$776$	−30.6692	−1.10096
$777$	−3.79544	−0.136161
$778$	−4.14162	−0.148484
$779$	−8.44640	−0.302624
$780$	−0.231971	−0.00830588
$781$	5.19526	0.185901
$782$	7.72832	0.276364
$783$	5.05115	0.180514
$784$	−24.3505	−0.869662
$785$	−19.3083	−0.689144
$786$	−38.2127	−1.36300
$787$	12.3386	0.439823	0.219911	−	0.975520i	$-0.429423\pi$
$787$	12.3386	0.439823	0.219911	+	0.975520i	$0.429423\pi$
$788$	1.73280	0.0617284
$789$	49.2331	1.75274
$790$	−7.95560	−0.283048
$791$	31.1019	1.10585
$792$	−32.1783	−1.14341
$793$	−3.03686	−0.107842
$794$	6.58081	0.233544
$795$	39.1418	1.38822
$796$	−8.15519	−0.289053
$797$	−31.5918	−1.11904	−0.559519	−	0.828817i	$-0.689014\pi$
$797$	−31.5918	−1.11904	−0.559519	+	0.828817i	$0.689014\pi$
$798$	38.8215	1.37427
$799$	8.20112	0.290135
$800$	−6.18067	−0.218520
$801$	−61.0477	−2.15701
$802$	20.7970	0.734368
$803$	17.2027	0.607069
$804$	5.35657	0.188912
$805$	6.74930	0.237882
$806$	0.990266	0.0348806
$807$	80.5251	2.83462
$808$	39.7129	1.39709
$809$	−23.6142	−0.830233	−0.415116	−	0.909768i	$-0.636259\pi$
$809$	−23.6142	−0.830233	−0.415116	+	0.909768i	$0.636259\pi$
$810$	0.451586	0.0158671
$811$	27.3970	0.962038	0.481019	−	0.876710i	$-0.340267\pi$
$811$	27.3970	0.962038	0.481019	+	0.876710i	$0.340267\pi$
$812$	−1.11258	−0.0390438
$813$	35.6242	1.24939
$814$	−1.04720	−0.0367044
$815$	15.2950	0.535762
$816$	34.4895	1.20737
$817$	7.91426	0.276885
$818$	−9.51657	−0.332739
$819$	−4.59274	−0.160483
$820$	−0.988320	−0.0345136
$821$	31.9506	1.11508	0.557542	−	0.830149i	$-0.311745\pi$
$821$	31.9506	1.11508	0.557542	+	0.830149i	$0.311745\pi$
$822$	68.8197	2.40036
$823$	−9.33365	−0.325351	−0.162675	−	0.986680i	$-0.552012\pi$
$823$	−9.33365	−0.325351	−0.162675	+	0.986680i	$0.552012\pi$
$824$	11.3074	0.393913
$825$	23.3971	0.814582
$826$	−6.55840	−0.228196
$827$	−33.1700	−1.15343	−0.576717	−	0.816944i	$-0.695667\pi$
$827$	−33.1700	−1.15343	−0.576717	+	0.816944i	$0.695667\pi$
$828$	−2.25322	−0.0783048
$829$	−7.31290	−0.253988	−0.126994	−	0.991904i	$-0.540533\pi$
$829$	−7.31290	−0.253988	−0.126994	+	0.991904i	$0.540533\pi$
$830$	10.9064	0.378567
$831$	−31.8736	−1.10568
$832$	2.22255	0.0770529
$833$	27.1764	0.941608
$834$	3.64967	0.126378
$835$	−17.9818	−0.622287
$836$	−1.84580	−0.0638382
$837$	−15.1712	−0.524395
$838$	6.12104	0.211448
$839$	−23.2601	−0.803026	−0.401513	−	0.915853i	$-0.631516\pi$
$839$	−23.2601	−0.803026	−0.401513	+	0.915853i	$0.631516\pi$
$840$	35.4457	1.22299
$841$	1.00000	0.0344828
$842$	−26.5853	−0.916189
$843$	37.6954	1.29830
$844$	−1.11474	−0.0383708
$845$	14.4719	0.497848
$846$	13.8755	0.477049
$847$	22.7453	0.781537
$848$	−41.6422	−1.43000
$849$	40.5072	1.39021
$850$	−18.1720	−0.623295
$851$	−0.572184	−0.0196142
$852$	1.91052	0.0654535
$853$	−8.54104	−0.292440	−0.146220	−	0.989252i	$-0.546711\pi$
$853$	−8.54104	−0.292440	−0.146220	+	0.989252i	$0.546711\pi$
$854$	59.4690	2.03499
$855$	−15.1171	−0.516994
$856$	36.9255	1.26209
$857$	−13.7904	−0.471069	−0.235535	−	0.971866i	$-0.575684\pi$
$857$	−13.7904	−0.471069	−0.235535	+	0.971866i	$0.575684\pi$
$858$	−2.05791	−0.0702558
$859$	−47.3549	−1.61573	−0.807864	−	0.589369i	$-0.799377\pi$
$859$	−47.3549	−1.61573	−0.807864	+	0.589369i	$0.799377\pi$
$860$	0.926055	0.0315782
$861$	−31.7777	−1.08298
$862$	23.2017	0.790255
$863$	−40.7141	−1.38592	−0.692961	−	0.720975i	$-0.743694\pi$
$863$	−40.7141	−1.38592	−0.692961	+	0.720975i	$0.743694\pi$
$864$	−8.32853	−0.283342
$865$	−22.4609	−0.763693
$866$	−13.2029	−0.448652
$867$	9.00984	0.305990
$868$	3.34164	0.113423
$869$	12.1621	0.412571
$870$	−4.08290	−0.138423
$871$	1.64600	0.0557725
$872$	51.3086	1.73753
$873$	49.2106	1.66553
$874$	5.85256	0.197966
$875$	−37.0385	−1.25213
$876$	6.32617	0.213741
$877$	−24.4174	−0.824518	−0.412259	−	0.911067i	$-0.635260\pi$
$877$	−24.4174	−0.824518	−0.412259	+	0.911067i	$0.635260\pi$
$878$	−4.55316	−0.153662
$879$	9.66691	0.326057
$880$	8.31052	0.280148
$881$	31.6730	1.06709	0.533545	−	0.845772i	$-0.320859\pi$
$881$	31.6730	1.06709	0.533545	+	0.845772i	$0.320859\pi$
$882$	45.9799	1.54822
$883$	−14.7751	−0.497222	−0.248611	−	0.968603i	$-0.579974\pi$
$883$	−14.7751	−0.497222	−0.248611	+	0.968603i	$0.579974\pi$
$884$	−0.275429	−0.00926369
$885$	4.14744	0.139415
$886$	20.6545	0.693902
$887$	−55.5482	−1.86513	−0.932563	−	0.361008i	$-0.882433\pi$
$887$	−55.5482	−1.86513	−0.932563	+	0.361008i	$0.882433\pi$
$888$	−3.00497	−0.100840
$889$	45.7218	1.53346
$890$	18.5540	0.621932
$891$	−0.690360	−0.0231279
$892$	−1.58066	−0.0529244
$893$	6.21060	0.207830
$894$	31.2432	1.04493
$895$	2.95605	0.0988099
$896$	−31.0428	−1.03707
$897$	−1.12443	−0.0375435
$898$	13.8758	0.463042
$899$	−3.00352	−0.100173
$900$	5.29812	0.176604
$901$	46.4748	1.54830
$902$	−8.76779	−0.291936
$903$	29.7756	0.990872
$904$	24.6243	0.818993
$905$	14.3737	0.477798
$906$	−36.6664	−1.21816
$907$	−9.85016	−0.327069	−0.163535	−	0.986538i	$-0.552290\pi$
$907$	−9.85016	−0.327069	−0.163535	+	0.986538i	$0.552290\pi$
$908$	−1.90362	−0.0631737
$909$	−63.7217	−2.11352
$910$	1.39586	0.0462722
$911$	−46.3724	−1.53639	−0.768193	−	0.640219i	$-0.778844\pi$
$911$	−46.3724	−1.53639	−0.768193	+	0.640219i	$0.778844\pi$
$912$	26.1184	0.864868
$913$	−16.6732	−0.551801
$914$	−36.8471	−1.21879
$915$	−37.6074	−1.24326
$916$	−3.71821	−0.122853
$917$	−39.6240	−1.30850
$918$	−24.4871	−0.808193
$919$	−45.4544	−1.49940	−0.749701	−	0.661777i	$-0.769803\pi$
$919$	−45.4544	−1.49940	−0.749701	+	0.661777i	$0.769803\pi$
$920$	5.34364	0.176174
$921$	62.0296	2.04394
$922$	−19.0563	−0.627587
$923$	0.587077	0.0193239
$924$	−6.94439	−0.228454
$925$	1.34541	0.0442368
$926$	20.5636	0.675763
$927$	−18.1434	−0.595909
$928$	−1.64884	−0.0541257
$929$	38.5507	1.26481	0.632404	−	0.774638i	$-0.282068\pi$
$929$	38.5507	1.26481	0.632404	+	0.774638i	$0.282068\pi$
$930$	12.2631	0.402122
$931$	20.5804	0.674494
$932$	−7.19322	−0.235622
$933$	−9.70338	−0.317674
$934$	48.5127	1.58738
$935$	−9.27498	−0.303324
$936$	−3.63622	−0.118854
$937$	−19.7287	−0.644510	−0.322255	−	0.946653i	$-0.604441\pi$
$937$	−19.7287	−0.644510	−0.322255	+	0.946653i	$0.604441\pi$
$938$	−32.2326	−1.05243
$939$	2.96619	0.0967981
$940$	0.726708	0.0237026
$941$	−26.0534	−0.849316	−0.424658	−	0.905354i	$-0.639606\pi$
$941$	−26.0534	−0.849316	−0.424658	+	0.905354i	$0.639606\pi$
$942$	−62.9916	−2.05238
$943$	−4.79066	−0.156005
$944$	−4.41238	−0.143611
$945$	−21.3850	−0.695656
$946$	8.21541	0.267106
$947$	16.8470	0.547455	0.273727	−	0.961807i	$-0.411743\pi$
$947$	16.8470	0.547455	0.273727	+	0.961807i	$0.411743\pi$
$948$	4.47254	0.145261
$949$	1.94394	0.0631030
$950$	−13.7614	−0.446480
$951$	81.7422	2.65067
$952$	42.0863	1.36402
$953$	29.5436	0.957011	0.478505	−	0.878085i	$-0.341179\pi$
$953$	29.5436	0.957011	0.478505	+	0.878085i	$0.341179\pi$
$954$	78.6309	2.54577
$955$	−10.7403	−0.347549
$956$	−1.26619	−0.0409514
$957$	6.24172	0.201766
$958$	−24.5441	−0.792983
$959$	71.3614	2.30438
$960$	27.5231	0.888306
$961$	−21.9789	−0.708996
$962$	−0.118336	−0.00381532
$963$	−59.2491	−1.90928
$964$	3.79928	0.122367
$965$	−20.9589	−0.674690
$966$	22.0189	0.708448
$967$	25.3657	0.815706	0.407853	−	0.913048i	$-0.366278\pi$
$967$	25.3657	0.815706	0.407853	+	0.913048i	$0.366278\pi$
$968$	18.0082	0.578804
$969$	−29.1495	−0.936418
$970$	−14.9564	−0.480221
$971$	24.6019	0.789512	0.394756	−	0.918786i	$-0.370829\pi$
$971$	24.6019	0.789512	0.394756	+	0.918786i	$0.370829\pi$
$972$	−4.70877	−0.151034
$973$	3.78446	0.121324
$974$	−38.3790	−1.22974
$975$	2.64393	0.0846734
$976$	40.0097	1.28068
$977$	−39.9083	−1.27678	−0.638389	−	0.769714i	$-0.720399\pi$
$977$	−39.9083	−1.27678	−0.638389	+	0.769714i	$0.720399\pi$
$978$	49.8986	1.59558
$979$	−28.3644	−0.906530
$980$	2.40813	0.0769248
$981$	−82.3277	−2.62852
$982$	1.41154	0.0450440
$983$	−11.8358	−0.377505	−0.188752	−	0.982025i	$-0.560444\pi$
$983$	−11.8358	−0.377505	−0.188752	+	0.982025i	$0.560444\pi$
$984$	−25.1594	−0.802052
$985$	6.59383	0.210097
$986$	−4.84781	−0.154386
$987$	23.3660	0.743748
$988$	−0.208579	−0.00663579
$989$	4.48885	0.142737
$990$	−15.6923	−0.498735
$991$	−6.91214	−0.219571	−0.109786	−	0.993955i	$-0.535016\pi$
$991$	−6.91214	−0.219571	−0.109786	+	0.993955i	$0.535016\pi$
$992$	4.95231	0.157236
$993$	−63.8960	−2.02768
$994$	−11.4964	−0.364642
$995$	−31.0330	−0.983813
$996$	−6.13145	−0.194282
$997$	4.97126	0.157441	0.0787206	−	0.996897i	$-0.474916\pi$
$997$	4.97126	0.157441	0.0787206	+	0.996897i	$0.474916\pi$
$998$	23.2802	0.736923
$999$	1.81296	0.0573594

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.c.1.45	✓	61

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.c.1.45	✓	61	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)

\(f(q)\)	\(=\)	\(q+1.30614 q^{2} -2.79423 q^{3} -0.293985 q^{4} -1.11870 q^{5} -3.64967 q^{6} -3.78446 q^{7} -2.99628 q^{8} +4.80771 q^{9} +O(q^{10})\)	\(q+1.30614 q^{2} -2.79423 q^{3} -0.293985 q^{4} -1.11870 q^{5} -3.64967 q^{6} -3.78446 q^{7} -2.99628 q^{8} +4.80771 q^{9} -1.46119 q^{10} +2.23379 q^{11} +0.821462 q^{12} +0.252424 q^{13} -4.94305 q^{14} +3.12592 q^{15} -3.32560 q^{16} +3.71154 q^{17} +6.27957 q^{18} +2.81070 q^{19} +0.328883 q^{20} +10.5746 q^{21} +2.91766 q^{22} +1.59419 q^{23} +8.37228 q^{24} -3.74850 q^{25} +0.329702 q^{26} -5.05115 q^{27} +1.11258 q^{28} -1.00000 q^{29} +4.08290 q^{30} +3.00352 q^{31} +1.64884 q^{32} -6.24172 q^{33} +4.84781 q^{34} +4.23369 q^{35} -1.41340 q^{36} -0.358919 q^{37} +3.67119 q^{38} -0.705329 q^{39} +3.35195 q^{40} -3.00508 q^{41} +13.8120 q^{42} +2.81576 q^{43} -0.656702 q^{44} -5.37841 q^{45} +2.08224 q^{46} +2.20963 q^{47} +9.29249 q^{48} +7.32214 q^{49} -4.89608 q^{50} -10.3709 q^{51} -0.0742089 q^{52} +12.5217 q^{53} -6.59754 q^{54} -2.49895 q^{55} +11.3393 q^{56} -7.85375 q^{57} -1.30614 q^{58} +1.32679 q^{59} -0.918974 q^{60} -12.0308 q^{61} +3.92303 q^{62} -18.1946 q^{63} +8.80482 q^{64} -0.282388 q^{65} -8.15260 q^{66} +6.52078 q^{67} -1.09114 q^{68} -4.45452 q^{69} +5.52982 q^{70} +2.32576 q^{71} -14.4052 q^{72} +7.70111 q^{73} -0.468800 q^{74} +10.4742 q^{75} -0.826306 q^{76} -8.45370 q^{77} -0.921262 q^{78} +5.44460 q^{79} +3.72037 q^{80} -0.309053 q^{81} -3.92507 q^{82} -7.46406 q^{83} -3.10879 q^{84} -4.15212 q^{85} +3.67779 q^{86} +2.79423 q^{87} -6.69306 q^{88} -12.6979 q^{89} -7.02498 q^{90} -0.955287 q^{91} -0.468668 q^{92} -8.39252 q^{93} +2.88609 q^{94} -3.14435 q^{95} -4.60723 q^{96} +10.2358 q^{97} +9.56378 q^{98} +10.7394 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * q^99

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