LMFDB - Embedded newform 4031.2.a.c.1.18

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			1.18
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)

$f(q)$	$=$	$q-1.44025 q^{2} +1.49462 q^{3} +0.0743258 q^{4} +2.03878 q^{5} -2.15264 q^{6} -3.27699 q^{7} +2.77346 q^{8} -0.766098 q^{9} +O(q^{10})$	\(q-1.44025 q^{2} +1.49462 q^{3} +0.0743258 q^{4} +2.03878 q^{5} -2.15264 q^{6} -3.27699 q^{7} +2.77346 q^{8} -0.766098 q^{9} -2.93636 q^{10} +2.62027 q^{11} +0.111089 q^{12} -5.46435 q^{13} +4.71969 q^{14} +3.04721 q^{15} -4.14313 q^{16} +4.75105 q^{17} +1.10337 q^{18} +3.60952 q^{19} +0.151534 q^{20} -4.89787 q^{21} -3.77386 q^{22} -2.87506 q^{23} +4.14527 q^{24} -0.843373 q^{25} +7.87004 q^{26} -5.62890 q^{27} -0.243565 q^{28} -1.00000 q^{29} -4.38875 q^{30} +1.86600 q^{31} +0.420235 q^{32} +3.91633 q^{33} -6.84271 q^{34} -6.68106 q^{35} -0.0569409 q^{36} +9.24926 q^{37} -5.19862 q^{38} -8.16715 q^{39} +5.65447 q^{40} -7.59599 q^{41} +7.05416 q^{42} -2.51760 q^{43} +0.194754 q^{44} -1.56191 q^{45} +4.14082 q^{46} -0.622600 q^{47} -6.19242 q^{48} +3.73865 q^{49} +1.21467 q^{50} +7.10104 q^{51} -0.406142 q^{52} +3.25108 q^{53} +8.10704 q^{54} +5.34217 q^{55} -9.08858 q^{56} +5.39487 q^{57} +1.44025 q^{58} -1.92642 q^{59} +0.226486 q^{60} +0.556575 q^{61} -2.68751 q^{62} +2.51050 q^{63} +7.68101 q^{64} -11.1406 q^{65} -5.64050 q^{66} -8.15843 q^{67} +0.353126 q^{68} -4.29714 q^{69} +9.62241 q^{70} -0.307231 q^{71} -2.12474 q^{72} +4.83259 q^{73} -13.3213 q^{74} -1.26053 q^{75} +0.268280 q^{76} -8.58661 q^{77} +11.7628 q^{78} +0.0396770 q^{79} -8.44693 q^{80} -6.11480 q^{81} +10.9401 q^{82} -11.7301 q^{83} -0.364038 q^{84} +9.68636 q^{85} +3.62598 q^{86} -1.49462 q^{87} +7.26722 q^{88} -4.78814 q^{89} +2.24954 q^{90} +17.9066 q^{91} -0.213692 q^{92} +2.78897 q^{93} +0.896701 q^{94} +7.35902 q^{95} +0.628094 q^{96} -13.7411 q^{97} -5.38460 q^{98} -2.00739 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.44025	−1.01841	−0.509206	−	0.860645i	$-0.670061\pi$
$2$	−1.44025	−1.01841	−0.509206	+	0.860645i	$0.670061\pi$
$3$	1.49462	0.862922	0.431461	−	0.902132i	$-0.357998\pi$
$3$	1.49462	0.862922	0.431461	+	0.902132i	$0.357998\pi$
$4$	0.0743258	0.0371629
$5$	2.03878	0.911770	0.455885	−	0.890039i	$-0.349323\pi$
$5$	2.03878	0.911770	0.455885	+	0.890039i	$0.349323\pi$
$6$	−2.15264	−0.878810
$7$	−3.27699	−1.23859	−0.619293	−	0.785160i	$-0.712581\pi$
$7$	−3.27699	−1.23859	−0.619293	+	0.785160i	$0.712581\pi$
$8$	2.77346	0.980565
$9$	−0.766098	−0.255366
$10$	−2.93636	−0.928558
$11$	2.62027	0.790043	0.395021	−	0.918672i	$-0.370737\pi$
$11$	2.62027	0.790043	0.395021	+	0.918672i	$0.370737\pi$
$12$	0.111089	0.0320687
$13$	−5.46435	−1.51554	−0.757769	−	0.652523i	$-0.773711\pi$
$13$	−5.46435	−1.51554	−0.757769	+	0.652523i	$0.773711\pi$
$14$	4.71969	1.26139
$15$	3.04721	0.786787
$16$	−4.14313	−1.03578
$17$	4.75105	1.15230	0.576150	−	0.817344i	$-0.304554\pi$
$17$	4.75105	1.15230	0.576150	+	0.817344i	$0.304554\pi$
$18$	1.10337	0.260068
$19$	3.60952	0.828080	0.414040	−	0.910259i	$-0.364117\pi$
$19$	3.60952	0.828080	0.414040	+	0.910259i	$0.364117\pi$
$20$	0.151534	0.0338840
$21$	−4.89787	−1.06880
$22$	−3.77386	−0.804589
$23$	−2.87506	−0.599492	−0.299746	−	0.954019i	$-0.596902\pi$
$23$	−2.87506	−0.599492	−0.299746	+	0.954019i	$0.596902\pi$
$24$	4.14527	0.846151
$25$	−0.843373	−0.168675
$26$	7.87004	1.54344
$27$	−5.62890	−1.08328
$28$	−0.243565	−0.0460294
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−4.38875	−0.801273
$31$	1.86600	0.335143	0.167572	−	0.985860i	$-0.446407\pi$
$31$	1.86600	0.335143	0.167572	+	0.985860i	$0.446407\pi$
$32$	0.420235	0.0742878
$33$	3.91633	0.681745
$34$	−6.84271	−1.17352
$35$	−6.68106	−1.12931
$36$	−0.0569409	−0.00949015
$37$	9.24926	1.52057	0.760284	−	0.649591i	$-0.225060\pi$
$37$	9.24926	1.52057	0.760284	+	0.649591i	$0.225060\pi$
$38$	−5.19862	−0.843327
$39$	−8.16715	−1.30779
$40$	5.65447	0.894050
$41$	−7.59599	−1.18629	−0.593147	−	0.805094i	$-0.702115\pi$
$41$	−7.59599	−1.18629	−0.593147	+	0.805094i	$0.702115\pi$
$42$	7.05416	1.08848
$43$	−2.51760	−0.383931	−0.191965	−	0.981402i	$-0.561486\pi$
$43$	−2.51760	−0.383931	−0.191965	+	0.981402i	$0.561486\pi$
$44$	0.194754	0.0293603
$45$	−1.56191	−0.232835
$46$	4.14082	0.610530
$47$	−0.622600	−0.0908156	−0.0454078	−	0.998969i	$-0.514459\pi$
$47$	−0.622600	−0.0908156	−0.0454078	+	0.998969i	$0.514459\pi$
$48$	−6.19242	−0.893799
$49$	3.73865	0.534093
$50$	1.21467	0.171780
$51$	7.10104	0.994345
$52$	−0.406142	−0.0563218
$53$	3.25108	0.446570	0.223285	−	0.974753i	$-0.428322\pi$
$53$	3.25108	0.446570	0.223285	+	0.974753i	$0.428322\pi$
$54$	8.10704	1.10323
$55$	5.34217	0.720337
$56$	−9.08858	−1.21451
$57$	5.39487	0.714569
$58$	1.44025	0.189114
$59$	−1.92642	−0.250798	−0.125399	−	0.992106i	$-0.540021\pi$
$59$	−1.92642	−0.250798	−0.125399	+	0.992106i	$0.540021\pi$
$60$	0.226486	0.0292393
$61$	0.556575	0.0712621	0.0356310	−	0.999365i	$-0.488656\pi$
$61$	0.556575	0.0712621	0.0356310	+	0.999365i	$0.488656\pi$
$62$	−2.68751	−0.341314
$63$	2.51050	0.316293
$64$	7.68101	0.960126
$65$	−11.1406	−1.38182
$66$	−5.64050	−0.694297
$67$	−8.15843	−0.996711	−0.498355	−	0.866973i	$-0.666062\pi$
$67$	−8.15843	−0.996711	−0.498355	+	0.866973i	$0.666062\pi$
$68$	0.353126	0.0428228
$69$	−4.29714	−0.517315
$70$	9.62241	1.15010
$71$	−0.307231	−0.0364616	−0.0182308	−	0.999834i	$-0.505803\pi$
$71$	−0.307231	−0.0364616	−0.0182308	+	0.999834i	$0.505803\pi$
$72$	−2.12474	−0.250403
$73$	4.83259	0.565612	0.282806	−	0.959177i	$-0.408735\pi$
$73$	4.83259	0.565612	0.282806	+	0.959177i	$0.408735\pi$
$74$	−13.3213	−1.54856
$75$	−1.26053	−0.145553
$76$	0.268280	0.0307739
$77$	−8.58661	−0.978535
$78$	11.7628	1.33187
$79$	0.0396770	0.00446401	0.00223201	−	0.999998i	$-0.499290\pi$
$79$	0.0396770	0.00446401	0.00223201	+	0.999998i	$0.499290\pi$
$80$	−8.44693	−0.944395
$81$	−6.11480	−0.679422
$82$	10.9401	1.20814
$83$	−11.7301	−1.28755	−0.643774	−	0.765215i	$-0.722632\pi$
$83$	−11.7301	−1.28755	−0.643774	+	0.765215i	$0.722632\pi$
$84$	−0.364038	−0.0397198
$85$	9.68636	1.05063
$86$	3.62598	0.390999
$87$	−1.49462	−0.160241
$88$	7.26722	0.774688
$89$	−4.78814	−0.507542	−0.253771	−	0.967264i	$-0.581671\pi$
$89$	−4.78814	−0.507542	−0.253771	+	0.967264i	$0.581671\pi$
$90$	2.24954	0.237122
$91$	17.9066	1.87712
$92$	−0.213692	−0.0222789
$93$	2.78897	0.289202
$94$	0.896701	0.0924877
$95$	7.35902	0.755019
$96$	0.628094	0.0641046
$97$	−13.7411	−1.39520	−0.697598	−	0.716489i	$-0.745748\pi$
$97$	−13.7411	−1.39520	−0.697598	+	0.716489i	$0.745748\pi$
$98$	−5.38460	−0.543927
$99$	−2.00739	−0.201750
$100$	−0.0626844	−0.00626844
$101$	−8.43626	−0.839439	−0.419720	−	0.907654i	$-0.637872\pi$
$101$	−8.43626	−0.839439	−0.419720	+	0.907654i	$0.637872\pi$
$102$	−10.2273	−1.01265
$103$	6.94078	0.683895	0.341948	−	0.939719i	$-0.388913\pi$
$103$	6.94078	0.683895	0.341948	+	0.939719i	$0.388913\pi$
$104$	−15.1551	−1.48608
$105$	−9.98568	−0.974502
$106$	−4.68237	−0.454792
$107$	1.58175	0.152914	0.0764569	−	0.997073i	$-0.475639\pi$
$107$	1.58175	0.152914	0.0764569	+	0.997073i	$0.475639\pi$
$108$	−0.418373	−0.0402579
$109$	−3.50678	−0.335889	−0.167944	−	0.985796i	$-0.553713\pi$
$109$	−3.50678	−0.335889	−0.167944	+	0.985796i	$0.553713\pi$
$110$	−7.69406	−0.733600
$111$	13.8242	1.31213
$112$	13.5770	1.28290
$113$	−1.31139	−0.123365	−0.0616827	−	0.998096i	$-0.519647\pi$
$113$	−1.31139	−0.123365	−0.0616827	+	0.998096i	$0.519647\pi$
$114$	−7.76998	−0.727725
$115$	−5.86163	−0.546599
$116$	−0.0743258	−0.00690098
$117$	4.18623	0.387017
$118$	2.77453	0.255416
$119$	−15.5691	−1.42722
$120$	8.45131	0.771495
$121$	−4.13416	−0.375833
$122$	−0.801608	−0.0725742
$123$	−11.3531	−1.02368
$124$	0.138692	0.0124549
$125$	−11.9134	−1.06556
$126$	−3.61575	−0.322116
$127$	−8.66197	−0.768626	−0.384313	−	0.923203i	$-0.625562\pi$
$127$	−8.66197	−0.768626	−0.384313	+	0.923203i	$0.625562\pi$
$128$	−11.9031	−1.05209
$129$	−3.76287	−0.331302
$130$	16.0453	1.40727
$131$	−1.48069	−0.129368	−0.0646841	−	0.997906i	$-0.520604\pi$
$131$	−1.48069	−0.129368	−0.0646841	+	0.997906i	$0.520604\pi$
$132$	0.291084	0.0253356
$133$	−11.8284	−1.02565
$134$	11.7502	1.01506
$135$	−11.4761	−0.987705
$136$	13.1768	1.12990
$137$	−7.71079	−0.658777	−0.329388	−	0.944194i	$-0.606843\pi$
$137$	−7.71079	−0.658777	−0.329388	+	0.944194i	$0.606843\pi$
$138$	6.18897	0.526840
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	−0.496575	−0.0419683
$141$	−0.930553	−0.0783667
$142$	0.442490	0.0371329
$143$	−14.3181	−1.19734
$144$	3.17404	0.264504
$145$	−2.03878	−0.169312
$146$	−6.96015	−0.576026
$147$	5.58788	0.460881
$148$	0.687459	0.0565087
$149$	−3.90340	−0.319779	−0.159889	−	0.987135i	$-0.551114\pi$
$149$	−3.90340	−0.319779	−0.159889	+	0.987135i	$0.551114\pi$
$150$	1.81547	0.148233
$151$	−16.9343	−1.37810	−0.689048	−	0.724715i	$-0.741971\pi$
$151$	−16.9343	−1.37810	−0.689048	+	0.724715i	$0.741971\pi$
$152$	10.0108	0.811987
$153$	−3.63977	−0.294258
$154$	12.3669	0.996552
$155$	3.80436	0.305574
$156$	−0.607030	−0.0486013
$157$	−5.48315	−0.437603	−0.218802	−	0.975769i	$-0.570215\pi$
$157$	−5.48315	−0.437603	−0.218802	+	0.975769i	$0.570215\pi$
$158$	−0.0571449	−0.00454620
$159$	4.85914	0.385355
$160$	0.856768	0.0677334
$161$	9.42155	0.742522
$162$	8.80685	0.691932
$163$	17.8313	1.39665	0.698326	−	0.715780i	$-0.253929\pi$
$163$	17.8313	1.39665	0.698326	+	0.715780i	$0.253929\pi$
$164$	−0.564578	−0.0440861
$165$	7.98453	0.621595
$166$	16.8943	1.31125
$167$	17.5572	1.35862	0.679309	−	0.733852i	$-0.262280\pi$
$167$	17.5572	1.35862	0.679309	+	0.733852i	$0.262280\pi$
$168$	−13.5840	−1.04803
$169$	16.8591	1.29686
$170$	−13.9508	−1.06998
$171$	−2.76525	−0.211464
$172$	−0.187123	−0.0142680
$173$	−1.08352	−0.0823784	−0.0411892	−	0.999151i	$-0.513115\pi$
$173$	−1.08352	−0.0823784	−0.0411892	+	0.999151i	$0.513115\pi$
$174$	2.15264	0.163191
$175$	2.76372	0.208918
$176$	−10.8561	−0.818312
$177$	−2.87927	−0.216419
$178$	6.89613	0.516887
$179$	−9.97201	−0.745343	−0.372672	−	0.927963i	$-0.621558\pi$
$179$	−9.97201	−0.745343	−0.372672	+	0.927963i	$0.621558\pi$
$180$	−0.116090	−0.00865284
$181$	−0.387667	−0.0288151	−0.0144075	−	0.999896i	$-0.504586\pi$
$181$	−0.387667	−0.0288151	−0.0144075	+	0.999896i	$0.504586\pi$
$182$	−25.7900	−1.91169
$183$	0.831870	0.0614936
$184$	−7.97386	−0.587841
$185$	18.8572	1.38641
$186$	−4.01682	−0.294527
$187$	12.4491	0.910366
$188$	−0.0462753	−0.00337497
$189$	18.4458	1.34174
$190$	−10.5988	−0.768921
$191$	−15.9092	−1.15115	−0.575576	−	0.817749i	$-0.695222\pi$
$191$	−15.9092	−1.15115	−0.575576	+	0.817749i	$0.695222\pi$
$192$	11.4802	0.828514
$193$	1.38185	0.0994679	0.0497340	−	0.998763i	$-0.484163\pi$
$193$	1.38185	0.0994679	0.0497340	+	0.998763i	$0.484163\pi$
$194$	19.7906	1.42088
$195$	−16.6510	−1.19241
$196$	0.277878	0.0198485
$197$	−3.42754	−0.244202	−0.122101	−	0.992518i	$-0.538963\pi$
$197$	−3.42754	−0.244202	−0.122101	+	0.992518i	$0.538963\pi$
$198$	2.89114	0.205465
$199$	2.78137	0.197166	0.0985829	−	0.995129i	$-0.468569\pi$
$199$	2.78137	0.197166	0.0985829	+	0.995129i	$0.468569\pi$
$200$	−2.33906	−0.165396
$201$	−12.1938	−0.860083
$202$	12.1503	0.854895
$203$	3.27699	0.230000
$204$	0.527791	0.0369527
$205$	−15.4865	−1.08163
$206$	−9.99647	−0.696487
$207$	2.20258	0.153090
$208$	22.6395	1.56977
$209$	9.45793	0.654219
$210$	14.3819	0.992445
$211$	−3.24400	−0.223326	−0.111663	−	0.993746i	$-0.535618\pi$
$211$	−3.24400	−0.223326	−0.111663	+	0.993746i	$0.535618\pi$
$212$	0.241639	0.0165958
$213$	−0.459195	−0.0314635
$214$	−2.27812	−0.155729
$215$	−5.13284	−0.350056
$216$	−15.6115	−1.06223
$217$	−6.11486	−0.415104
$218$	5.05065	0.342073
$219$	7.22291	0.488079
$220$	0.397061	0.0267698
$221$	−25.9614	−1.74635
$222$	−19.9103	−1.33629
$223$	6.97597	0.467146	0.233573	−	0.972339i	$-0.424958\pi$
$223$	6.97597	0.467146	0.233573	+	0.972339i	$0.424958\pi$
$224$	−1.37711	−0.0920118
$225$	0.646107	0.0430738
$226$	1.88873	0.125637
$227$	2.69277	0.178725	0.0893627	−	0.995999i	$-0.471517\pi$
$227$	2.69277	0.178725	0.0893627	+	0.995999i	$0.471517\pi$
$228$	0.400979	0.0265555
$229$	1.37376	0.0907808	0.0453904	−	0.998969i	$-0.485547\pi$
$229$	1.37376	0.0907808	0.0453904	+	0.998969i	$0.485547\pi$
$230$	8.44222	0.556663
$231$	−12.8338	−0.844399
$232$	−2.77346	−0.182086
$233$	7.13599	0.467494	0.233747	−	0.972297i	$-0.424901\pi$
$233$	7.13599	0.467494	0.233747	+	0.972297i	$0.424901\pi$
$234$	−6.02923	−0.394143
$235$	−1.26935	−0.0828030
$236$	−0.143183	−0.00932039
$237$	0.0593022	0.00385209
$238$	22.4235	1.45350
$239$	12.3371	0.798023	0.399011	−	0.916946i	$-0.369353\pi$
$239$	12.3371	0.798023	0.399011	+	0.916946i	$0.369353\pi$
$240$	−12.6250	−0.814939
$241$	−27.0219	−1.74064	−0.870318	−	0.492490i	$-0.836087\pi$
$241$	−27.0219	−1.74064	−0.870318	+	0.492490i	$0.836087\pi$
$242$	5.95423	0.382753
$243$	7.74738	0.496995
$244$	0.0413679	0.00264831
$245$	7.62229	0.486971
$246$	16.3514	1.04253
$247$	−19.7237	−1.25499
$248$	5.17527	0.328630
$249$	−17.5321	−1.11105
$250$	17.1582	1.08518
$251$	−14.6599	−0.925327	−0.462664	−	0.886534i	$-0.653106\pi$
$251$	−14.6599	−0.925327	−0.462664	+	0.886534i	$0.653106\pi$
$252$	0.186595	0.0117544
$253$	−7.53346	−0.473624
$254$	12.4754	0.782778
$255$	14.4775	0.906614
$256$	1.78139	0.111337
$257$	−12.4837	−0.778711	−0.389355	−	0.921088i	$-0.627302\pi$
$257$	−12.4837	−0.778711	−0.389355	+	0.921088i	$0.627302\pi$
$258$	5.41948	0.337402
$259$	−30.3097	−1.88335
$260$	−0.828035	−0.0513526
$261$	0.766098	0.0474203
$262$	2.13256	0.131750
$263$	−6.88235	−0.424384	−0.212192	−	0.977228i	$-0.568060\pi$
$263$	−6.88235	−0.424384	−0.212192	+	0.977228i	$0.568060\pi$
$264$	10.8618	0.668495
$265$	6.62824	0.407169
$266$	17.0358	1.04453
$267$	−7.15648	−0.437969
$268$	−0.606382	−0.0370407
$269$	2.09390	0.127667	0.0638337	−	0.997961i	$-0.479667\pi$
$269$	2.09390	0.127667	0.0638337	+	0.997961i	$0.479667\pi$
$270$	16.5285	1.00589
$271$	−6.48705	−0.394060	−0.197030	−	0.980397i	$-0.563130\pi$
$271$	−6.48705	−0.394060	−0.197030	+	0.980397i	$0.563130\pi$
$272$	−19.6842	−1.19353
$273$	26.7637	1.61981
$274$	11.1055	0.670906
$275$	−2.20987	−0.133260
$276$	−0.319389	−0.0192249
$277$	0.612763	0.0368174	0.0184087	−	0.999831i	$-0.494140\pi$
$277$	0.612763	0.0368174	0.0184087	+	0.999831i	$0.494140\pi$
$278$	1.44025	0.0863806
$279$	−1.42954	−0.0855843
$280$	−18.5296	−1.10736
$281$	−8.04142	−0.479711	−0.239855	−	0.970809i	$-0.577100\pi$
$281$	−8.04142	−0.479711	−0.239855	+	0.970809i	$0.577100\pi$
$282$	1.34023	0.0798096
$283$	−4.81380	−0.286151	−0.143075	−	0.989712i	$-0.545699\pi$
$283$	−4.81380	−0.286151	−0.143075	+	0.989712i	$0.545699\pi$
$284$	−0.0228352	−0.00135502
$285$	10.9990	0.651523
$286$	20.6217	1.21939
$287$	24.8920	1.46933
$288$	−0.321942	−0.0189706
$289$	5.57251	0.327795
$290$	2.93636	0.172429
$291$	−20.5378	−1.20395
$292$	0.359187	0.0210198
$293$	31.3210	1.82979	0.914896	−	0.403689i	$-0.132272\pi$
$293$	31.3210	1.82979	0.914896	+	0.403689i	$0.132272\pi$
$294$	−8.04796	−0.469366
$295$	−3.92754	−0.228670
$296$	25.6524	1.49102
$297$	−14.7493	−0.855839
$298$	5.62188	0.325667
$299$	15.7104	0.908554
$300$	−0.0936896	−0.00540917
$301$	8.25015	0.475531
$302$	24.3897	1.40347
$303$	−12.6090	−0.724370
$304$	−14.9547	−0.857711
$305$	1.13473	0.0649747
$306$	5.24219	0.299676
$307$	−27.1027	−1.54683	−0.773415	−	0.633900i	$-0.781453\pi$
$307$	−27.1027	−1.54683	−0.773415	+	0.633900i	$0.781453\pi$
$308$	−0.638207	−0.0363652
$309$	10.3739	0.590148
$310$	−5.47924	−0.311200
$311$	2.04215	0.115800	0.0578998	−	0.998322i	$-0.481560\pi$
$311$	2.04215	0.115800	0.0578998	+	0.998322i	$0.481560\pi$
$312$	−22.6512	−1.28237
$313$	26.9743	1.52468	0.762340	−	0.647177i	$-0.224051\pi$
$313$	26.9743	1.52468	0.762340	+	0.647177i	$0.224051\pi$
$314$	7.89712	0.445660
$315$	5.11835	0.288386
$316$	0.00294903	0.000165896 0
$317$	−8.45186	−0.474704	−0.237352	−	0.971424i	$-0.576279\pi$
$317$	−8.45186	−0.474704	−0.237352	+	0.971424i	$0.576279\pi$
$318$	−6.99839	−0.392450
$319$	−2.62027	−0.146707
$320$	15.6599	0.875415
$321$	2.36413	0.131953
$322$	−13.5694	−0.756194
$323$	17.1490	0.954197
$324$	−0.454487	−0.0252493
$325$	4.60849	0.255633
$326$	−25.6815	−1.42237
$327$	−5.24132	−0.289846
$328$	−21.0671	−1.16324
$329$	2.04025	0.112483
$330$	−11.4997	−0.633040
$331$	24.3684	1.33941	0.669704	−	0.742628i	$-0.266421\pi$
$331$	24.3684	1.33941	0.669704	+	0.742628i	$0.266421\pi$
$332$	−0.871851	−0.0478491
$333$	−7.08584	−0.388302
$334$	−25.2868	−1.38363
$335$	−16.6332	−0.908771
$336$	20.2925	1.10705
$337$	−3.94436	−0.214863	−0.107431	−	0.994213i	$-0.534263\pi$
$337$	−3.94436	−0.214863	−0.107431	+	0.994213i	$0.534263\pi$
$338$	−24.2814	−1.32074
$339$	−1.96004	−0.106455
$340$	0.719946	0.0390446
$341$	4.88943	0.264777
$342$	3.98265	0.215357
$343$	10.6874	0.577065
$344$	−6.98245	−0.376469
$345$	−8.76093	−0.471672
$346$	1.56054	0.0838951
$347$	−13.7424	−0.737731	−0.368866	−	0.929483i	$-0.620254\pi$
$347$	−13.7424	−0.737731	−0.368866	+	0.929483i	$0.620254\pi$
$348$	−0.111089	−0.00595501
$349$	−17.4474	−0.933937	−0.466968	−	0.884274i	$-0.654654\pi$
$349$	−17.4474	−0.933937	−0.466968	+	0.884274i	$0.654654\pi$
$350$	−3.98046	−0.212764
$351$	30.7583	1.64176
$352$	1.10113	0.0586905
$353$	−18.6114	−0.990585	−0.495293	−	0.868726i	$-0.664939\pi$
$353$	−18.6114	−0.990585	−0.495293	+	0.868726i	$0.664939\pi$
$354$	4.14688	0.220404
$355$	−0.626376	−0.0332446
$356$	−0.355883	−0.0188618
$357$	−23.2700	−1.23158
$358$	14.3622	0.759066
$359$	−6.21832	−0.328190	−0.164095	−	0.986445i	$-0.552470\pi$
$359$	−6.21832	−0.328190	−0.164095	+	0.986445i	$0.552470\pi$
$360$	−4.33188	−0.228310
$361$	−5.97137	−0.314283
$362$	0.558339	0.0293456
$363$	−6.17902	−0.324314
$364$	1.33092	0.0697594
$365$	9.85260	0.515709
$366$	−1.19810	−0.0626258
$367$	−24.5931	−1.28375	−0.641875	−	0.766809i	$-0.721843\pi$
$367$	−24.5931	−1.28375	−0.641875	+	0.766809i	$0.721843\pi$
$368$	11.9118	0.620943
$369$	5.81927	0.302939
$370$	−27.1591	−1.41194
$371$	−10.6537	−0.553115
$372$	0.207292	0.0107476
$373$	32.5691	1.68636	0.843181	−	0.537630i	$-0.180680\pi$
$373$	32.5691	1.68636	0.843181	+	0.537630i	$0.180680\pi$
$374$	−17.9298	−0.927127
$375$	−17.8060	−0.919497
$376$	−1.72675	−0.0890506
$377$	5.46435	0.281428
$378$	−26.5667	−1.36644
$379$	−13.3120	−0.683793	−0.341897	−	0.939738i	$-0.611069\pi$
$379$	−13.3120	−0.683793	−0.341897	+	0.939738i	$0.611069\pi$
$380$	0.546965	0.0280587
$381$	−12.9464	−0.663264
$382$	22.9133	1.17235
$383$	−4.81382	−0.245975	−0.122987	−	0.992408i	$-0.539247\pi$
$383$	−4.81382	−0.245975	−0.122987	+	0.992408i	$0.539247\pi$
$384$	−17.7906	−0.907873
$385$	−17.5062	−0.892199
$386$	−1.99022	−0.101299
$387$	1.92873	0.0980428
$388$	−1.02132	−0.0518496
$389$	21.5087	1.09053	0.545266	−	0.838263i	$-0.316429\pi$
$389$	21.5087	1.09053	0.545266	+	0.838263i	$0.316429\pi$
$390$	23.9817	1.21436
$391$	−13.6596	−0.690795
$392$	10.3690	0.523713
$393$	−2.21307	−0.111635
$394$	4.93652	0.248698
$395$	0.0808927	0.00407015
$396$	−0.149201	−0.00749762
$397$	30.3454	1.52299	0.761495	−	0.648170i	$-0.224466\pi$
$397$	30.3454	1.52299	0.761495	+	0.648170i	$0.224466\pi$
$398$	−4.00587	−0.200796
$399$	−17.6789	−0.885054
$400$	3.49420	0.174710
$401$	16.9002	0.843958	0.421979	−	0.906606i	$-0.361336\pi$
$401$	16.9002	0.843958	0.421979	+	0.906606i	$0.361336\pi$
$402$	17.5621	0.875919
$403$	−10.1965	−0.507923
$404$	−0.627032	−0.0311960
$405$	−12.4667	−0.619477
$406$	−4.71969	−0.234234
$407$	24.2356	1.20131
$408$	19.6944	0.975019
$409$	5.52258	0.273074	0.136537	−	0.990635i	$-0.456403\pi$
$409$	5.52258	0.273074	0.136537	+	0.990635i	$0.456403\pi$
$410$	22.3045	1.10154
$411$	−11.5247	−0.568473
$412$	0.515879	0.0254155
$413$	6.31285	0.310635
$414$	−3.17227	−0.155909
$415$	−23.9152	−1.17395
$416$	−2.29631	−0.112586
$417$	−1.49462	−0.0731921
$418$	−13.6218	−0.666264
$419$	15.9483	0.779128	0.389564	−	0.920999i	$-0.372626\pi$
$419$	15.9483	0.779128	0.389564	+	0.920999i	$0.372626\pi$
$420$	−0.742194	−0.0362153
$421$	−18.3099	−0.892371	−0.446186	−	0.894940i	$-0.647218\pi$
$421$	−18.3099	−0.892371	−0.446186	+	0.894940i	$0.647218\pi$
$422$	4.67218	0.227438
$423$	0.476973	0.0231912
$424$	9.01673	0.437891
$425$	−4.00691	−0.194364
$426$	0.661356	0.0320428
$427$	−1.82389	−0.0882642
$428$	0.117565	0.00568272
$429$	−21.4002	−1.03321
$430$	7.39258	0.356502
$431$	−29.8400	−1.43734	−0.718670	−	0.695351i	$-0.755249\pi$
$431$	−29.8400	−1.43734	−0.718670	+	0.695351i	$0.755249\pi$
$432$	23.3213	1.12204
$433$	−4.86363	−0.233731	−0.116866	−	0.993148i	$-0.537285\pi$
$433$	−4.86363	−0.233731	−0.116866	+	0.993148i	$0.537285\pi$
$434$	8.80694	0.422746
$435$	−3.04721	−0.146103
$436$	−0.260644	−0.0124826
$437$	−10.3776	−0.496428
$438$	−10.4028	−0.497066
$439$	−23.0085	−1.09814	−0.549068	−	0.835778i	$-0.685017\pi$
$439$	−23.0085	−1.09814	−0.549068	+	0.835778i	$0.685017\pi$
$440$	14.8163	0.706338
$441$	−2.86418	−0.136389
$442$	37.3910	1.77851
$443$	−6.77027	−0.321665	−0.160833	−	0.986982i	$-0.551418\pi$
$443$	−6.77027	−0.321665	−0.160833	+	0.986982i	$0.551418\pi$
$444$	1.02749	0.0487626
$445$	−9.76198	−0.462762
$446$	−10.0472	−0.475747
$447$	−5.83411	−0.275944
$448$	−25.1706	−1.18920
$449$	−24.8000	−1.17038	−0.585192	−	0.810895i	$-0.698981\pi$
$449$	−24.8000	−1.17038	−0.585192	+	0.810895i	$0.698981\pi$
$450$	−0.930557	−0.0438669
$451$	−19.9036	−0.937222
$452$	−0.0974702	−0.00458461
$453$	−25.3105	−1.18919
$454$	−3.87827	−0.182016
$455$	36.5077	1.71151
$456$	14.9624	0.700681
$457$	10.0072	0.468117	0.234059	−	0.972222i	$-0.424799\pi$
$457$	10.0072	0.468117	0.234059	+	0.972222i	$0.424799\pi$
$458$	−1.97856	−0.0924523
$459$	−26.7432	−1.24827
$460$	−0.435670	−0.0203132
$461$	−13.3729	−0.622839	−0.311419	−	0.950273i	$-0.600804\pi$
$461$	−13.3729	−0.622839	−0.311419	+	0.950273i	$0.600804\pi$
$462$	18.4838	0.859946
$463$	10.4181	0.484170	0.242085	−	0.970255i	$-0.422169\pi$
$463$	10.4181	0.484170	0.242085	+	0.970255i	$0.422169\pi$
$464$	4.14313	0.192340
$465$	5.68609	0.263686
$466$	−10.2776	−0.476102
$467$	−9.04493	−0.418549	−0.209275	−	0.977857i	$-0.567110\pi$
$467$	−9.04493	−0.418549	−0.209275	+	0.977857i	$0.567110\pi$
$468$	0.311145	0.0143827
$469$	26.7351	1.23451
$470$	1.82818	0.0843275
$471$	−8.19525	−0.377617
$472$	−5.34283	−0.245924
$473$	−6.59681	−0.303321
$474$	−0.0854101	−0.00392302
$475$	−3.04417	−0.139676
$476$	−1.15719	−0.0530397
$477$	−2.49065	−0.114039
$478$	−17.7686	−0.812716
$479$	−1.94424	−0.0888345	−0.0444173	−	0.999013i	$-0.514143\pi$
$479$	−1.94424	−0.0888345	−0.0444173	+	0.999013i	$0.514143\pi$
$480$	1.28055	0.0584486
$481$	−50.5412	−2.30448
$482$	38.9184	1.77268
$483$	14.0817	0.640739
$484$	−0.307275	−0.0139670
$485$	−28.0151	−1.27210
$486$	−11.1582	−0.506145
$487$	34.1913	1.54936	0.774678	−	0.632356i	$-0.217912\pi$
$487$	34.1913	1.54936	0.774678	+	0.632356i	$0.217912\pi$
$488$	1.54364	0.0698771
$489$	26.6510	1.20520
$490$	−10.9780	−0.495937
$491$	−8.26060	−0.372796	−0.186398	−	0.982474i	$-0.559681\pi$
$491$	−8.26060	−0.372796	−0.186398	+	0.982474i	$0.559681\pi$
$492$	−0.843832	−0.0380429
$493$	−4.75105	−0.213977
$494$	28.4071	1.27809
$495$	−4.09262	−0.183950
$496$	−7.73107	−0.347135
$497$	1.00679	0.0451608
$498$	25.2507	1.13151
$499$	−26.1719	−1.17161	−0.585807	−	0.810451i	$-0.699222\pi$
$499$	−26.1719	−1.17161	−0.585807	+	0.810451i	$0.699222\pi$
$500$	−0.885470	−0.0395994
$501$	26.2414	1.17238
$502$	21.1140	0.942364
$503$	−29.1687	−1.30057	−0.650283	−	0.759692i	$-0.725350\pi$
$503$	−29.1687	−1.30057	−0.650283	+	0.759692i	$0.725350\pi$
$504$	6.96275	0.310145
$505$	−17.1997	−0.765376
$506$	10.8501	0.482345
$507$	25.1981	1.11909
$508$	−0.643808	−0.0285644
$509$	7.98212	0.353801	0.176901	−	0.984229i	$-0.443393\pi$
$509$	7.98212	0.353801	0.176901	+	0.984229i	$0.443393\pi$
$510$	−20.8512	−0.923306
$511$	−15.8364	−0.700559
$512$	21.2405	0.938705
$513$	−20.3176	−0.897045
$514$	17.9796	0.793048
$515$	14.1507	0.623556
$516$	−0.279678	−0.0123121
$517$	−1.63138	−0.0717482
$518$	43.6536	1.91803
$519$	−1.61945	−0.0710861
$520$	−30.8980	−1.35497
$521$	6.91447	0.302929	0.151464	−	0.988463i	$-0.451601\pi$
$521$	6.91447	0.302929	0.151464	+	0.988463i	$0.451601\pi$
$522$	−1.10337	−0.0482934
$523$	−42.3815	−1.85321	−0.926607	−	0.376032i	$-0.877288\pi$
$523$	−42.3815	−1.85321	−0.926607	+	0.376032i	$0.877288\pi$
$524$	−0.110053	−0.00480770
$525$	4.13073	0.180280
$526$	9.91232	0.432198
$527$	8.86546	0.386186
$528$	−16.2258	−0.706139
$529$	−14.7340	−0.640609
$530$	−9.54633	−0.414666
$531$	1.47583	0.0640454
$532$	−0.879152	−0.0381161
$533$	41.5071	1.79787
$534$	10.3071	0.446033
$535$	3.22485	0.139422
$536$	−22.6270	−0.977339
$537$	−14.9044	−0.643173
$538$	−3.01575	−0.130018
$539$	9.79630	0.421956
$540$	−0.852970	−0.0367060
$541$	−41.4801	−1.78337	−0.891684	−	0.452658i	$-0.850476\pi$
$541$	−41.4801	−1.78337	−0.891684	+	0.452658i	$0.850476\pi$
$542$	9.34298	0.401315
$543$	−0.579417	−0.0248652
$544$	1.99656	0.0856018
$545$	−7.14955	−0.306253
$546$	−38.5464	−1.64963
$547$	5.11870	0.218860	0.109430	−	0.993995i	$-0.465097\pi$
$547$	5.11870	0.218860	0.109430	+	0.993995i	$0.465097\pi$
$548$	−0.573111	−0.0244821
$549$	−0.426391	−0.0181979
$550$	3.18277	0.135714
$551$	−3.60952	−0.153771
$552$	−11.9179	−0.507261
$553$	−0.130021	−0.00552906
$554$	−0.882534	−0.0374953
$555$	28.1844	1.19636
$556$	−0.0743258	−0.00315212
$557$	−21.6503	−0.917354	−0.458677	−	0.888603i	$-0.651677\pi$
$557$	−21.6503	−0.917354	−0.458677	+	0.888603i	$0.651677\pi$
$558$	2.05890	0.0871600
$559$	13.7571	0.581862
$560$	27.6805	1.16971
$561$	18.6067	0.785574
$562$	11.5817	0.488543
$563$	15.6360	0.658980	0.329490	−	0.944159i	$-0.393123\pi$
$563$	15.6360	0.658980	0.329490	+	0.944159i	$0.393123\pi$
$564$	−0.0691641	−0.00291234
$565$	−2.67364	−0.112481
$566$	6.93308	0.291419
$567$	20.0381	0.841522
$568$	−0.852091	−0.0357529
$569$	11.0055	0.461374	0.230687	−	0.973028i	$-0.425903\pi$
$569$	11.0055	0.461374	0.230687	+	0.973028i	$0.425903\pi$
$570$	−15.8413	−0.663518
$571$	27.5562	1.15319	0.576595	−	0.817030i	$-0.304381\pi$
$571$	27.5562	1.15319	0.576595	+	0.817030i	$0.304381\pi$
$572$	−1.06420	−0.0444966
$573$	−23.7783	−0.993353
$574$	−35.8507	−1.49638
$575$	2.42475	0.101119
$576$	−5.88441	−0.245184
$577$	7.89082	0.328499	0.164250	−	0.986419i	$-0.447480\pi$
$577$	7.89082	0.328499	0.164250	+	0.986419i	$0.447480\pi$
$578$	−8.02582	−0.333830
$579$	2.06535	0.0858330
$580$	−0.151534	−0.00629211
$581$	38.4395	1.59474
$582$	29.5796	1.22611
$583$	8.51872	0.352809
$584$	13.4030	0.554620
$585$	8.53481	0.352871
$586$	−45.1101	−1.86348
$587$	0.971136	0.0400831	0.0200415	−	0.999799i	$-0.493620\pi$
$587$	0.971136	0.0400831	0.0200415	+	0.999799i	$0.493620\pi$
$588$	0.415324	0.0171277
$589$	6.73536	0.277526
$590$	5.65665	0.232881
$591$	−5.12288	−0.210727
$592$	−38.3208	−1.57498
$593$	−22.3549	−0.918007	−0.459004	−	0.888434i	$-0.651794\pi$
$593$	−22.3549	−0.918007	−0.459004	+	0.888434i	$0.651794\pi$
$594$	21.2427	0.871597
$595$	−31.7421	−1.30130
$596$	−0.290123	−0.0118839
$597$	4.15710	0.170139
$598$	−22.6269	−0.925282
$599$	0.156273	0.00638513	0.00319256	−	0.999995i	$-0.498984\pi$
$599$	0.156273	0.00638513	0.00319256	+	0.999995i	$0.498984\pi$
$600$	−3.49601	−0.142724
$601$	15.2662	0.622720	0.311360	−	0.950292i	$-0.399215\pi$
$601$	15.2662	0.622720	0.311360	+	0.950292i	$0.399215\pi$
$602$	−11.8823	−0.484286
$603$	6.25016	0.254526
$604$	−1.25866	−0.0512141
$605$	−8.42865	−0.342673
$606$	18.1602	0.737707
$607$	20.1916	0.819552	0.409776	−	0.912186i	$-0.365607\pi$
$607$	20.1916	0.819552	0.409776	+	0.912186i	$0.365607\pi$
$608$	1.51685	0.0615163
$609$	4.89787	0.198472
$610$	−1.63430	−0.0661710
$611$	3.40211	0.137635
$612$	−0.270529	−0.0109355
$613$	−13.6585	−0.551661	−0.275831	−	0.961206i	$-0.588953\pi$
$613$	−13.6585	−0.551661	−0.275831	+	0.961206i	$0.588953\pi$
$614$	39.0347	1.57531
$615$	−23.1466	−0.933360
$616$	−23.8146	−0.959517
$617$	−46.3670	−1.86667	−0.933333	−	0.359012i	$-0.883114\pi$
$617$	−46.3670	−1.86667	−0.933333	+	0.359012i	$0.883114\pi$
$618$	−14.9410	−0.601014
$619$	21.2940	0.855880	0.427940	−	0.903807i	$-0.359240\pi$
$619$	21.2940	0.855880	0.427940	+	0.903807i	$0.359240\pi$
$620$	0.282762	0.0113560
$621$	16.1835	0.649420
$622$	−2.94121	−0.117932
$623$	15.6907	0.628634
$624$	33.8376	1.35459
$625$	−20.0719	−0.802874
$626$	−38.8498	−1.55275
$627$	14.1361	0.564540
$628$	−0.407540	−0.0162626
$629$	43.9437	1.75215
$630$	−7.37171	−0.293696
$631$	31.1002	1.23808	0.619039	−	0.785360i	$-0.287522\pi$
$631$	31.1002	1.23808	0.619039	+	0.785360i	$0.287522\pi$
$632$	0.110042	0.00437725
$633$	−4.84857	−0.192713
$634$	12.1728	0.483444
$635$	−17.6599	−0.700810
$636$	0.361160	0.0143209
$637$	−20.4293	−0.809439
$638$	3.77386	0.149408
$639$	0.235369	0.00931105
$640$	−24.2677	−0.959266
$641$	14.4105	0.569181	0.284591	−	0.958649i	$-0.408142\pi$
$641$	14.4105	0.569181	0.284591	+	0.958649i	$0.408142\pi$
$642$	−3.40494	−0.134382
$643$	2.04280	0.0805600	0.0402800	−	0.999188i	$-0.487175\pi$
$643$	2.04280	0.0805600	0.0402800	+	0.999188i	$0.487175\pi$
$644$	0.700265	0.0275943
$645$	−7.67166	−0.302071
$646$	−24.6989	−0.971766
$647$	25.1709	0.989570	0.494785	−	0.869016i	$-0.335247\pi$
$647$	25.1709	0.989570	0.494785	+	0.869016i	$0.335247\pi$
$648$	−16.9591	−0.666217
$649$	−5.04774	−0.198141
$650$	−6.63738	−0.260340
$651$	−9.13941	−0.358202
$652$	1.32532	0.0519037
$653$	2.07204	0.0810851	0.0405425	−	0.999178i	$-0.487091\pi$
$653$	2.07204	0.0810851	0.0405425	+	0.999178i	$0.487091\pi$
$654$	7.54882	0.295182
$655$	−3.01880	−0.117954
$656$	31.4711	1.22874
$657$	−3.70224	−0.144438
$658$	−2.93848	−0.114554
$659$	29.1159	1.13419	0.567096	−	0.823651i	$-0.308067\pi$
$659$	29.1159	1.13419	0.567096	+	0.823651i	$0.308067\pi$
$660$	0.593457	0.0231003
$661$	−32.1182	−1.24925	−0.624627	−	0.780923i	$-0.714749\pi$
$661$	−32.1182	−1.24925	−0.624627	+	0.780923i	$0.714749\pi$
$662$	−35.0967	−1.36407
$663$	−38.8026	−1.50697
$664$	−32.5330	−1.26252
$665$	−24.1154	−0.935156
$666$	10.2054	0.395451
$667$	2.87506	0.111323
$668$	1.30495	0.0504902
$669$	10.4265	0.403110
$670$	23.9561	0.925504
$671$	1.45838	0.0563001
$672$	−2.05826	−0.0793990
$673$	−29.5198	−1.13791	−0.568953	−	0.822370i	$-0.692651\pi$
$673$	−29.5198	−1.13791	−0.568953	+	0.822370i	$0.692651\pi$
$674$	5.68087	0.218819
$675$	4.74726	0.182722
$676$	1.25307	0.0481950
$677$	−46.0987	−1.77172	−0.885859	−	0.463954i	$-0.846430\pi$
$677$	−46.0987	−1.77172	−0.885859	+	0.463954i	$0.846430\pi$
$678$	2.82295	0.108415
$679$	45.0294	1.72807
$680$	26.8647	1.03021
$681$	4.02468	0.154226
$682$	−7.04201	−0.269653
$683$	6.04029	0.231125	0.115563	−	0.993300i	$-0.463133\pi$
$683$	6.04029	0.231125	0.115563	+	0.993300i	$0.463133\pi$
$684$	−0.205529	−0.00785861
$685$	−15.7206	−0.600653
$686$	−15.3925	−0.587690
$687$	2.05326	0.0783367
$688$	10.4307	0.397668
$689$	−17.7650	−0.676794
$690$	12.6179	0.480357
$691$	−34.2188	−1.30175	−0.650873	−	0.759187i	$-0.725597\pi$
$691$	−34.2188	−1.30175	−0.650873	+	0.759187i	$0.725597\pi$
$692$	−0.0805334	−0.00306142
$693$	6.57819	0.249885
$694$	19.7925	0.751314
$695$	−2.03878	−0.0773354
$696$	−4.14527	−0.157126
$697$	−36.0889	−1.36697
$698$	25.1286	0.951132
$699$	10.6656	0.403411
$700$	0.205416	0.00776400
$701$	46.1790	1.74416	0.872079	−	0.489365i	$-0.162772\pi$
$701$	46.1790	1.74416	0.872079	+	0.489365i	$0.162772\pi$
$702$	−44.2997	−1.67198
$703$	33.3854	1.25915
$704$	20.1264	0.758541
$705$	−1.89719	−0.0714525
$706$	26.8051	1.00882
$707$	27.6455	1.03972
$708$	−0.214004	−0.00804277
$709$	23.2692	0.873893	0.436946	−	0.899488i	$-0.356060\pi$
$709$	23.2692	0.873893	0.436946	+	0.899488i	$0.356060\pi$
$710$	0.902139	0.0338567
$711$	−0.0303965	−0.00113996
$712$	−13.2797	−0.497678
$713$	−5.36487	−0.200916
$714$	33.5147	1.25426
$715$	−29.1915	−1.09170
$716$	−0.741178	−0.0276991
$717$	18.4394	0.688631
$718$	8.95595	0.334233
$719$	40.5027	1.51050	0.755248	−	0.655439i	$-0.227516\pi$
$719$	40.5027	1.51050	0.755248	+	0.655439i	$0.227516\pi$
$720$	6.47118	0.241167
$721$	−22.7449	−0.847063
$722$	8.60028	0.320069
$723$	−40.3876	−1.50203
$724$	−0.0288137	−0.00107085
$725$	0.843373	0.0313221
$726$	8.89934	0.330286
$727$	39.5308	1.46612	0.733059	−	0.680165i	$-0.238092\pi$
$727$	39.5308	1.46612	0.733059	+	0.680165i	$0.238092\pi$
$728$	49.6632	1.84064
$729$	29.9238	1.10829
$730$	−14.1902	−0.525204
$731$	−11.9613	−0.442403
$732$	0.0618294	0.00228528
$733$	−10.7227	−0.396051	−0.198026	−	0.980197i	$-0.563453\pi$
$733$	−10.7227	−0.396051	−0.198026	+	0.980197i	$0.563453\pi$
$734$	35.4203	1.30739
$735$	11.3925	0.420217
$736$	−1.20820	−0.0445350
$737$	−21.3773	−0.787444
$738$	−8.38122	−0.308517
$739$	−25.2786	−0.929888	−0.464944	−	0.885340i	$-0.653926\pi$
$739$	−25.2786	−0.929888	−0.464944	+	0.885340i	$0.653926\pi$
$740$	1.40158	0.0515230
$741$	−29.4795	−1.08296
$742$	15.3441	0.563299
$743$	−3.34340	−0.122657	−0.0613287	−	0.998118i	$-0.519534\pi$
$743$	−3.34340	−0.122657	−0.0613287	+	0.998118i	$0.519534\pi$
$744$	7.73508	0.283582
$745$	−7.95817	−0.291565
$746$	−46.9076	−1.71741
$747$	8.98643	0.328796
$748$	0.925287	0.0338318
$749$	−5.18339	−0.189397
$750$	25.6451	0.936427
$751$	−14.7805	−0.539349	−0.269675	−	0.962952i	$-0.586916\pi$
$751$	−14.7805	−0.539349	−0.269675	+	0.962952i	$0.586916\pi$
$752$	2.57951	0.0940651
$753$	−21.9111	−0.798485
$754$	−7.87004	−0.286610
$755$	−34.5254	−1.25651
$756$	1.37100	0.0498629
$757$	8.90434	0.323634	0.161817	−	0.986821i	$-0.448265\pi$
$757$	8.90434	0.323634	0.161817	+	0.986821i	$0.448265\pi$
$758$	19.1727	0.696383
$759$	−11.2597	−0.408701
$760$	20.4099	0.740345
$761$	4.07376	0.147674	0.0738369	−	0.997270i	$-0.476476\pi$
$761$	4.07376	0.147674	0.0738369	+	0.997270i	$0.476476\pi$
$762$	18.6461	0.675476
$763$	11.4917	0.416027
$764$	−1.18247	−0.0427801
$765$	−7.42070	−0.268296
$766$	6.93311	0.250504
$767$	10.5266	0.380094
$768$	2.66250	0.0960748
$769$	−14.3603	−0.517846	−0.258923	−	0.965898i	$-0.583368\pi$
$769$	−14.3603	−0.517846	−0.258923	+	0.965898i	$0.583368\pi$
$770$	25.2134	0.908626
$771$	−18.6584	−0.671966
$772$	0.102707	0.00369652
$773$	44.9513	1.61679	0.808393	−	0.588643i	$-0.200338\pi$
$773$	44.9513	1.61679	0.808393	+	0.588643i	$0.200338\pi$
$774$	−2.77786	−0.0998480
$775$	−1.57373	−0.0565302
$776$	−38.1103	−1.36808
$777$	−45.3016	−1.62519
$778$	−30.9779	−1.11061
$779$	−27.4179	−0.982347
$780$	−1.23760	−0.0443133
$781$	−0.805029	−0.0288062
$782$	19.6732	0.703514
$783$	5.62890	0.201161
$784$	−15.4897	−0.553204
$785$	−11.1789	−0.398994
$786$	3.18738	0.113690
$787$	11.9629	0.426431	0.213216	−	0.977005i	$-0.431606\pi$
$787$	11.9629	0.426431	0.213216	+	0.977005i	$0.431606\pi$
$788$	−0.254755	−0.00907525
$789$	−10.2865	−0.366210
$790$	−0.116506	−0.00414509
$791$	4.29741	0.152798
$792$	−5.56740	−0.197829
$793$	−3.04132	−0.108000
$794$	−43.7050	−1.55103
$795$	9.90673	0.351355
$796$	0.206727	0.00732726
$797$	7.01175	0.248369	0.124184	−	0.992259i	$-0.460369\pi$
$797$	7.01175	0.248369	0.124184	+	0.992259i	$0.460369\pi$
$798$	25.4621	0.901350
$799$	−2.95801	−0.104647
$800$	−0.354415	−0.0125305
$801$	3.66819	0.129609
$802$	−24.3406	−0.859497
$803$	12.6627	0.446858
$804$	−0.906313	−0.0319632
$805$	19.2085	0.677010
$806$	14.6855	0.517275
$807$	3.12960	0.110167
$808$	−23.3976	−0.823125
$809$	−25.0102	−0.879311	−0.439655	−	0.898167i	$-0.644899\pi$
$809$	−25.0102	−0.879311	−0.439655	+	0.898167i	$0.644899\pi$
$810$	17.9552	0.630883
$811$	−46.2665	−1.62464	−0.812318	−	0.583215i	$-0.801795\pi$
$811$	−46.2665	−1.62464	−0.812318	+	0.583215i	$0.801795\pi$
$812$	0.243565	0.00854745
$813$	−9.69570	−0.340043
$814$	−34.9054	−1.22343
$815$	36.3540	1.27343
$816$	−29.4205	−1.02992
$817$	−9.08733	−0.317925
$818$	−7.95391	−0.278102
$819$	−13.7182	−0.479354
$820$	−1.15105	−0.0401964
$821$	14.0006	0.488624	0.244312	−	0.969697i	$-0.421438\pi$
$821$	14.0006	0.488624	0.244312	+	0.969697i	$0.421438\pi$
$822$	16.5985	0.578940
$823$	42.5382	1.48279	0.741395	−	0.671069i	$-0.234165\pi$
$823$	42.5382	1.48279	0.741395	+	0.671069i	$0.234165\pi$
$824$	19.2499	0.670604
$825$	−3.30292	−0.114993
$826$	−9.09209	−0.316354
$827$	31.9840	1.11219	0.556097	−	0.831117i	$-0.312298\pi$
$827$	31.9840	1.11219	0.556097	+	0.831117i	$0.312298\pi$
$828$	0.163709	0.00568927
$829$	51.6858	1.79512	0.897560	−	0.440891i	$-0.145338\pi$
$829$	51.6858	1.79512	0.897560	+	0.440891i	$0.145338\pi$
$830$	34.4439	1.19556
$831$	0.915851	0.0317705
$832$	−41.9717	−1.45511
$833$	17.7625	0.615436
$834$	2.15264	0.0745397
$835$	35.7953	1.23875
$836$	0.702969	0.0243127
$837$	−10.5035	−0.363055
$838$	−22.9696	−0.793473
$839$	4.86296	0.167888	0.0839441	−	0.996470i	$-0.473248\pi$
$839$	4.86296	0.167888	0.0839441	+	0.996470i	$0.473248\pi$
$840$	−27.6948	−0.955563
$841$	1.00000	0.0344828
$842$	26.3709	0.908802
$843$	−12.0189	−0.413953
$844$	−0.241113	−0.00829946
$845$	34.3721	1.18244
$846$	−0.686961	−0.0236182
$847$	13.5476	0.465501
$848$	−13.4696	−0.462549
$849$	−7.19482	−0.246926
$850$	5.77096	0.197942
$851$	−26.5922	−0.911569
$852$	−0.0341300	−0.00116927
$853$	−18.0804	−0.619061	−0.309531	−	0.950889i	$-0.600172\pi$
$853$	−18.0804	−0.619061	−0.309531	+	0.950889i	$0.600172\pi$
$854$	2.62686	0.0898893
$855$	−5.63773	−0.192806
$856$	4.38692	0.149942
$857$	33.3957	1.14078	0.570388	−	0.821376i	$-0.306793\pi$
$857$	33.3957	1.14078	0.570388	+	0.821376i	$0.306793\pi$
$858$	30.8217	1.05223
$859$	40.4650	1.38065	0.690325	−	0.723500i	$-0.257468\pi$
$859$	40.4650	1.38065	0.690325	+	0.723500i	$0.257468\pi$
$860$	−0.381502	−0.0130091
$861$	37.2041	1.26791
$862$	42.9771	1.46380
$863$	−18.3769	−0.625558	−0.312779	−	0.949826i	$-0.601260\pi$
$863$	−18.3769	−0.625558	−0.312779	+	0.949826i	$0.601260\pi$
$864$	−2.36546	−0.0804747
$865$	−2.20906	−0.0751102
$866$	7.00486	0.238035
$867$	8.32881	0.282861
$868$	−0.454492	−0.0154265
$869$	0.103965	0.00352676
$870$	4.38875	0.148793
$871$	44.5805	1.51055
$872$	−9.72590	−0.329360
$873$	10.5270	0.356286
$874$	14.9464	0.505568
$875$	39.0399	1.31979
$876$	0.536849	0.0181384
$877$	−8.75942	−0.295784	−0.147892	−	0.989003i	$-0.547249\pi$
$877$	−8.75942	−0.295784	−0.147892	+	0.989003i	$0.547249\pi$
$878$	33.1380	1.11835
$879$	46.8131	1.57897
$880$	−22.1333	−0.746112
$881$	−11.6701	−0.393176	−0.196588	−	0.980486i	$-0.562986\pi$
$881$	−11.6701	−0.393176	−0.196588	+	0.980486i	$0.562986\pi$
$882$	4.12514	0.138901
$883$	7.97115	0.268251	0.134125	−	0.990964i	$-0.457178\pi$
$883$	7.97115	0.268251	0.134125	+	0.990964i	$0.457178\pi$
$884$	−1.92960	−0.0648996
$885$	−5.87020	−0.197325
$886$	9.75090	0.327588
$887$	10.6962	0.359144	0.179572	−	0.983745i	$-0.442529\pi$
$887$	10.6962	0.359144	0.179572	+	0.983745i	$0.442529\pi$
$888$	38.3407	1.28663
$889$	28.3852	0.952009
$890$	14.0597	0.471282
$891$	−16.0225	−0.536772
$892$	0.518495	0.0173605
$893$	−2.24729	−0.0752026
$894$	8.40259	0.281025
$895$	−20.3307	−0.679582
$896$	39.0062	1.30311
$897$	23.4811	0.784011
$898$	35.7182	1.19193
$899$	−1.86600	−0.0622346
$900$	0.0480224	0.00160075
$901$	15.4461	0.514583
$902$	28.6662	0.954479
$903$	12.3309	0.410346
$904$	−3.63709	−0.120968
$905$	−0.790369	−0.0262727
$906$	36.4535	1.21109
$907$	47.5710	1.57957	0.789784	−	0.613385i	$-0.210193\pi$
$907$	47.5710	1.57957	0.789784	+	0.613385i	$0.210193\pi$
$908$	0.200142	0.00664196
$909$	6.46301	0.214364
$910$	−52.5802	−1.74302
$911$	6.78371	0.224754	0.112377	−	0.993666i	$-0.464154\pi$
$911$	6.78371	0.224754	0.112377	+	0.993666i	$0.464154\pi$
$912$	−22.3517	−0.740137
$913$	−30.7362	−1.01722
$914$	−14.4129	−0.476736
$915$	1.69600	0.0560680
$916$	0.102106	0.00337368
$917$	4.85220	0.160234
$918$	38.5170	1.27125
$919$	46.1874	1.52358	0.761791	−	0.647823i	$-0.224320\pi$
$919$	46.1874	1.52358	0.761791	+	0.647823i	$0.224320\pi$
$920$	−16.2570	−0.535976
$921$	−40.5083	−1.33479
$922$	19.2604	0.634307
$923$	1.67882	0.0552589
$924$	−0.953879	−0.0313803
$925$	−7.80057	−0.256481
$926$	−15.0047	−0.493085
$927$	−5.31732	−0.174644
$928$	−0.420235	−0.0137949
$929$	35.1911	1.15458	0.577292	−	0.816538i	$-0.304110\pi$
$929$	35.1911	1.15458	0.577292	+	0.816538i	$0.304110\pi$
$930$	−8.18941	−0.268541
$931$	13.4947	0.442272
$932$	0.530388	0.0173734
$933$	3.05224	0.0999260
$934$	13.0270	0.426256
$935$	25.3809	0.830045
$936$	11.6103	0.379495
$937$	47.2913	1.54494	0.772469	−	0.635052i	$-0.219022\pi$
$937$	47.2913	1.54494	0.772469	+	0.635052i	$0.219022\pi$
$938$	−38.5052	−1.25724
$939$	40.3165	1.31568
$940$	−0.0943451	−0.00307720
$941$	5.46226	0.178065	0.0890323	−	0.996029i	$-0.471623\pi$
$941$	5.46226	0.178065	0.0890323	+	0.996029i	$0.471623\pi$
$942$	11.8032	0.384570
$943$	21.8389	0.711174
$944$	7.98139	0.259772
$945$	37.6070	1.22336
$946$	9.50106	0.308906
$947$	−30.8434	−1.00228	−0.501138	−	0.865368i	$-0.667085\pi$
$947$	−30.8434	−1.00228	−0.501138	+	0.865368i	$0.667085\pi$
$948$	0.00440768	0.000143155 0
$949$	−26.4070	−0.857208
$950$	4.38437	0.142248
$951$	−12.6324	−0.409632
$952$	−43.1803	−1.39948
$953$	−52.1811	−1.69031	−0.845155	−	0.534521i	$-0.820492\pi$
$953$	−52.1811	−1.69031	−0.845155	+	0.534521i	$0.820492\pi$
$954$	3.58716	0.116139
$955$	−32.4354	−1.04959
$956$	0.916968	0.0296569
$957$	−3.91633	−0.126597
$958$	2.80019	0.0904701
$959$	25.2682	0.815951
$960$	23.4057	0.755414
$961$	−27.5180	−0.887679
$962$	72.7921	2.34691
$963$	−1.21178	−0.0390490
$964$	−2.00843	−0.0646871
$965$	2.81729	0.0906919
$966$	−20.2812	−0.652536
$967$	−8.64602	−0.278037	−0.139019	−	0.990290i	$-0.544395\pi$
$967$	−8.64602	−0.278037	−0.139019	+	0.990290i	$0.544395\pi$
$968$	−11.4659	−0.368528
$969$	25.6313	0.823397
$970$	40.3488	1.29552
$971$	−3.69431	−0.118556	−0.0592781	−	0.998242i	$-0.518880\pi$
$971$	−3.69431	−0.118556	−0.0592781	+	0.998242i	$0.518880\pi$
$972$	0.575830	0.0184698
$973$	3.27699	0.105055
$974$	−49.2441	−1.57788
$975$	6.88796	0.220591
$976$	−2.30596	−0.0738120
$977$	−35.5479	−1.13728	−0.568639	−	0.822587i	$-0.692530\pi$
$977$	−35.5479	−1.13728	−0.568639	+	0.822587i	$0.692530\pi$
$978$	−38.3842	−1.22739
$979$	−12.5463	−0.400980
$980$	0.566533	0.0180972
$981$	2.68654	0.0857746
$982$	11.8973	0.379660
$983$	−41.1796	−1.31343	−0.656713	−	0.754141i	$-0.728054\pi$
$983$	−41.1796	−1.31343	−0.656713	+	0.754141i	$0.728054\pi$
$984$	−31.4874	−1.00378
$985$	−6.98800	−0.222656
$986$	6.84271	0.217916
$987$	3.04941	0.0970639
$988$	−1.46598	−0.0466390
$989$	7.23826	0.230163
$990$	5.89441	0.187337
$991$	42.6212	1.35391	0.676953	−	0.736026i	$-0.263300\pi$
$991$	42.6212	1.35391	0.676953	+	0.736026i	$0.263300\pi$
$992$	0.784159	0.0248971
$993$	36.4216	1.15581
$994$	−1.45003	−0.0459923
$995$	5.67059	0.179770
$996$	−1.30309	−0.0412900
$997$	22.5489	0.714132	0.357066	−	0.934079i	$-0.383777\pi$
$997$	22.5489	0.714132	0.357066	+	0.934079i	$0.383777\pi$
$998$	37.6941	1.19319
$999$	−52.0632	−1.64721

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.c.1.18	✓	61

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.c.1.18	✓	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)

\(f(q)\)	\(=\)	\(q-1.44025 q^{2} +1.49462 q^{3} +0.0743258 q^{4} +2.03878 q^{5} -2.15264 q^{6} -3.27699 q^{7} +2.77346 q^{8} -0.766098 q^{9} +O(q^{10})\)	\(q-1.44025 q^{2} +1.49462 q^{3} +0.0743258 q^{4} +2.03878 q^{5} -2.15264 q^{6} -3.27699 q^{7} +2.77346 q^{8} -0.766098 q^{9} -2.93636 q^{10} +2.62027 q^{11} +0.111089 q^{12} -5.46435 q^{13} +4.71969 q^{14} +3.04721 q^{15} -4.14313 q^{16} +4.75105 q^{17} +1.10337 q^{18} +3.60952 q^{19} +0.151534 q^{20} -4.89787 q^{21} -3.77386 q^{22} -2.87506 q^{23} +4.14527 q^{24} -0.843373 q^{25} +7.87004 q^{26} -5.62890 q^{27} -0.243565 q^{28} -1.00000 q^{29} -4.38875 q^{30} +1.86600 q^{31} +0.420235 q^{32} +3.91633 q^{33} -6.84271 q^{34} -6.68106 q^{35} -0.0569409 q^{36} +9.24926 q^{37} -5.19862 q^{38} -8.16715 q^{39} +5.65447 q^{40} -7.59599 q^{41} +7.05416 q^{42} -2.51760 q^{43} +0.194754 q^{44} -1.56191 q^{45} +4.14082 q^{46} -0.622600 q^{47} -6.19242 q^{48} +3.73865 q^{49} +1.21467 q^{50} +7.10104 q^{51} -0.406142 q^{52} +3.25108 q^{53} +8.10704 q^{54} +5.34217 q^{55} -9.08858 q^{56} +5.39487 q^{57} +1.44025 q^{58} -1.92642 q^{59} +0.226486 q^{60} +0.556575 q^{61} -2.68751 q^{62} +2.51050 q^{63} +7.68101 q^{64} -11.1406 q^{65} -5.64050 q^{66} -8.15843 q^{67} +0.353126 q^{68} -4.29714 q^{69} +9.62241 q^{70} -0.307231 q^{71} -2.12474 q^{72} +4.83259 q^{73} -13.3213 q^{74} -1.26053 q^{75} +0.268280 q^{76} -8.58661 q^{77} +11.7628 q^{78} +0.0396770 q^{79} -8.44693 q^{80} -6.11480 q^{81} +10.9401 q^{82} -11.7301 q^{83} -0.364038 q^{84} +9.68636 q^{85} +3.62598 q^{86} -1.49462 q^{87} +7.26722 q^{88} -4.78814 q^{89} +2.24954 q^{90} +17.9066 q^{91} -0.213692 q^{92} +2.78897 q^{93} +0.896701 q^{94} +7.35902 q^{95} +0.628094 q^{96} -13.7411 q^{97} -5.38460 q^{98} -2.00739 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) 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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