LMFDB - Embedded newform 8043.2.a.u.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$0$
Dimension:	$53$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.31380 q^{2} +1.00000 q^{3} -0.273938 q^{4} -2.47121 q^{5} -1.31380 q^{6} +1.00000 q^{7} +2.98749 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.31380 q^{2} +1.00000 q^{3} -0.273938 q^{4} -2.47121 q^{5} -1.31380 q^{6} +1.00000 q^{7} +2.98749 q^{8} +1.00000 q^{9} +3.24666 q^{10} -3.23376 q^{11} -0.273938 q^{12} -0.997959 q^{13} -1.31380 q^{14} -2.47121 q^{15} -3.37708 q^{16} +0.610813 q^{17} -1.31380 q^{18} -0.730026 q^{19} +0.676959 q^{20} +1.00000 q^{21} +4.24851 q^{22} +2.85356 q^{23} +2.98749 q^{24} +1.10686 q^{25} +1.31112 q^{26} +1.00000 q^{27} -0.273938 q^{28} +4.76424 q^{29} +3.24666 q^{30} +5.76004 q^{31} -1.53819 q^{32} -3.23376 q^{33} -0.802484 q^{34} -2.47121 q^{35} -0.273938 q^{36} -9.78983 q^{37} +0.959106 q^{38} -0.997959 q^{39} -7.38271 q^{40} +4.12282 q^{41} -1.31380 q^{42} -0.218203 q^{43} +0.885852 q^{44} -2.47121 q^{45} -3.74900 q^{46} -5.48524 q^{47} -3.37708 q^{48} +1.00000 q^{49} -1.45420 q^{50} +0.610813 q^{51} +0.273379 q^{52} +13.1394 q^{53} -1.31380 q^{54} +7.99130 q^{55} +2.98749 q^{56} -0.730026 q^{57} -6.25924 q^{58} -10.0745 q^{59} +0.676959 q^{60} -0.871128 q^{61} -7.56752 q^{62} +1.00000 q^{63} +8.77503 q^{64} +2.46616 q^{65} +4.24851 q^{66} -7.72381 q^{67} -0.167325 q^{68} +2.85356 q^{69} +3.24666 q^{70} -7.46107 q^{71} +2.98749 q^{72} +3.40941 q^{73} +12.8618 q^{74} +1.10686 q^{75} +0.199982 q^{76} -3.23376 q^{77} +1.31112 q^{78} -12.0893 q^{79} +8.34547 q^{80} +1.00000 q^{81} -5.41655 q^{82} -6.33711 q^{83} -0.273938 q^{84} -1.50945 q^{85} +0.286674 q^{86} +4.76424 q^{87} -9.66084 q^{88} -10.3825 q^{89} +3.24666 q^{90} -0.997959 q^{91} -0.781701 q^{92} +5.76004 q^{93} +7.20649 q^{94} +1.80405 q^{95} -1.53819 q^{96} -0.165629 q^{97} -1.31380 q^{98} -3.23376 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 53 q + 11 q^{2} + 53 q^{3} + 63 q^{4} + 24 q^{5} + 11 q^{6} + 53 q^{7} + 30 q^{8} + 53 q^{9}+O(q^{10}) $ 53 * q + 11 * q^2 + 53 * q^3 + 63 * q^4 + 24 * q^5 + 11 * q^6 + 53 * q^7 + 30 * q^8 + 53 * q^9	\( 53 q + 11 q^{2} + 53 q^{3} + 63 q^{4} + 24 q^{5} + 11 q^{6} + 53 q^{7} + 30 q^{8} + 53 q^{9} + 2 q^{10} + 46 q^{11} + 63 q^{12} + 32 q^{13} + 11 q^{14} + 24 q^{15} + 67 q^{16} + 46 q^{17} + 11 q^{18} + 14 q^{19} + 53 q^{20} + 53 q^{21} + 13 q^{22} + 68 q^{23} + 30 q^{24} + 71 q^{25} + 11 q^{26} + 53 q^{27} + 63 q^{28} + 55 q^{29} + 2 q^{30} - 2 q^{31} + 51 q^{32} + 46 q^{33} - 7 q^{34} + 24 q^{35} + 63 q^{36} + 53 q^{37} + 16 q^{38} + 32 q^{39} - 20 q^{40} + 38 q^{41} + 11 q^{42} + 36 q^{43} + 70 q^{44} + 24 q^{45} + 4 q^{46} + 51 q^{47} + 67 q^{48} + 53 q^{49} + 32 q^{50} + 46 q^{51} + 10 q^{52} + 104 q^{53} + 11 q^{54} + 11 q^{55} + 30 q^{56} + 14 q^{57} + 4 q^{58} + 36 q^{59} + 53 q^{60} + 3 q^{61} + 25 q^{62} + 53 q^{63} + 82 q^{64} + 46 q^{65} + 13 q^{66} + 54 q^{67} + 88 q^{68} + 68 q^{69} + 2 q^{70} + 101 q^{71} + 30 q^{72} + q^{73} + 32 q^{74} + 71 q^{75} - 35 q^{76} + 46 q^{77} + 11 q^{78} + 14 q^{79} + 39 q^{80} + 53 q^{81} - 29 q^{82} + 38 q^{83} + 63 q^{84} + 16 q^{85} + 23 q^{86} + 55 q^{87} - 8 q^{88} + 52 q^{89} + 2 q^{90} + 32 q^{91} + 76 q^{92} - 2 q^{93} - 53 q^{94} + 46 q^{95} + 51 q^{96} - 3 q^{97} + 11 q^{98} + 46 q^{99}+O(q^{100}) \) 53 * q + 11 * q^2 + 53 * q^3 + 63 * q^4 + 24 * q^5 + 11 * q^6 + 53 * q^7 + 30 * q^8 + 53 * q^9 + 2 * q^10 + 46 * q^11 + 63 * q^12 + 32 * q^13 + 11 * q^14 + 24 * q^15 + 67 * q^16 + 46 * q^17 + 11 * q^18 + 14 * q^19 + 53 * q^20 + 53 * q^21 + 13 * q^22 + 68 * q^23 + 30 * q^24 + 71 * q^25 + 11 * q^26 + 53 * q^27 + 63 * q^28 + 55 * q^29 + 2 * q^30 - 2 * q^31 + 51 * q^32 + 46 * q^33 - 7 * q^34 + 24 * q^35 + 63 * q^36 + 53 * q^37 + 16 * q^38 + 32 * q^39 - 20 * q^40 + 38 * q^41 + 11 * q^42 + 36 * q^43 + 70 * q^44 + 24 * q^45 + 4 * q^46 + 51 * q^47 + 67 * q^48 + 53 * q^49 + 32 * q^50 + 46 * q^51 + 10 * q^52 + 104 * q^53 + 11 * q^54 + 11 * q^55 + 30 * q^56 + 14 * q^57 + 4 * q^58 + 36 * q^59 + 53 * q^60 + 3 * q^61 + 25 * q^62 + 53 * q^63 + 82 * q^64 + 46 * q^65 + 13 * q^66 + 54 * q^67 + 88 * q^68 + 68 * q^69 + 2 * q^70 + 101 * q^71 + 30 * q^72 + q^73 + 32 * q^74 + 71 * q^75 - 35 * q^76 + 46 * q^77 + 11 * q^78 + 14 * q^79 + 39 * q^80 + 53 * q^81 - 29 * q^82 + 38 * q^83 + 63 * q^84 + 16 * q^85 + 23 * q^86 + 55 * q^87 - 8 * q^88 + 52 * q^89 + 2 * q^90 + 32 * q^91 + 76 * q^92 - 2 * q^93 - 53 * q^94 + 46 * q^95 + 51 * q^96 - 3 * q^97 + 11 * q^98 + 46 * 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 $

$2$	−1.31380	−0.928995	−0.464497	−	0.885575i	$-0.653765\pi$
$2$	−1.31380	−0.928995	−0.464497	+	0.885575i	$0.653765\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.273938	−0.136969
$5$	−2.47121	−1.10516	−0.552579	−	0.833461i	$-0.686356\pi$
$5$	−2.47121	−1.10516	−0.552579	+	0.833461i	$0.686356\pi$
$6$	−1.31380	−0.536355
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.98749	1.05624
$9$	1.00000	0.333333
$10$	3.24666	1.02669
$11$	−3.23376	−0.975016	−0.487508	−	0.873118i	$-0.662094\pi$
$11$	−3.23376	−0.975016	−0.487508	+	0.873118i	$0.662094\pi$
$12$	−0.273938	−0.0790792
$13$	−0.997959	−0.276784	−0.138392	−	0.990378i	$-0.544193\pi$
$13$	−0.997959	−0.276784	−0.138392	+	0.990378i	$0.544193\pi$
$14$	−1.31380	−0.351127
$15$	−2.47121	−0.638063
$16$	−3.37708	−0.844270
$17$	0.610813	0.148144	0.0740720	−	0.997253i	$-0.476401\pi$
$17$	0.610813	0.148144	0.0740720	+	0.997253i	$0.476401\pi$
$18$	−1.31380	−0.309665
$19$	−0.730026	−0.167479	−0.0837397	−	0.996488i	$-0.526686\pi$
$19$	−0.730026	−0.167479	−0.0837397	+	0.996488i	$0.526686\pi$
$20$	0.676959	0.151373
$21$	1.00000	0.218218
$22$	4.24851	0.905785
$23$	2.85356	0.595009	0.297505	−	0.954720i	$-0.403846\pi$
$23$	2.85356	0.595009	0.297505	+	0.954720i	$0.403846\pi$
$24$	2.98749	0.609819
$25$	1.10686	0.221373
$26$	1.31112	0.257131
$27$	1.00000	0.192450
$28$	−0.273938	−0.0517695
$29$	4.76424	0.884697	0.442349	−	0.896843i	$-0.354145\pi$
$29$	4.76424	0.884697	0.442349	+	0.896843i	$0.354145\pi$
$30$	3.24666	0.592757
$31$	5.76004	1.03453	0.517267	−	0.855824i	$-0.326950\pi$
$31$	5.76004	1.03453	0.517267	+	0.855824i	$0.326950\pi$
$32$	−1.53819	−0.271916
$33$	−3.23376	−0.562926
$34$	−0.802484	−0.137625
$35$	−2.47121	−0.417710
$36$	−0.273938	−0.0456564
$37$	−9.78983	−1.60944	−0.804719	−	0.593656i	$-0.797684\pi$
$37$	−9.78983	−1.60944	−0.804719	+	0.593656i	$0.797684\pi$
$38$	0.959106	0.155588
$39$	−0.997959	−0.159801
$40$	−7.38271	−1.16731
$41$	4.12282	0.643876	0.321938	−	0.946761i	$-0.395666\pi$
$41$	4.12282	0.643876	0.321938	+	0.946761i	$0.395666\pi$
$42$	−1.31380	−0.202723
$43$	−0.218203	−0.0332756	−0.0166378	−	0.999862i	$-0.505296\pi$
$43$	−0.218203	−0.0332756	−0.0166378	+	0.999862i	$0.505296\pi$
$44$	0.885852	0.133547
$45$	−2.47121	−0.368386
$46$	−3.74900	−0.552760
$47$	−5.48524	−0.800105	−0.400052	−	0.916492i	$-0.631008\pi$
$47$	−5.48524	−0.800105	−0.400052	+	0.916492i	$0.631008\pi$
$48$	−3.37708	−0.487440
$49$	1.00000	0.142857
$50$	−1.45420	−0.205654
$51$	0.610813	0.0855310
$52$	0.273379	0.0379109
$53$	13.1394	1.80484	0.902418	−	0.430862i	$-0.141790\pi$
$53$	13.1394	1.80484	0.902418	+	0.430862i	$0.141790\pi$
$54$	−1.31380	−0.178785
$55$	7.99130	1.07755
$56$	2.98749	0.399220
$57$	−0.730026	−0.0966943
$58$	−6.25924	−0.821879
$59$	−10.0745	−1.31159	−0.655794	−	0.754940i	$-0.727666\pi$
$59$	−10.0745	−1.31159	−0.655794	+	0.754940i	$0.727666\pi$
$60$	0.676959	0.0873950
$61$	−0.871128	−0.111537	−0.0557683	−	0.998444i	$-0.517761\pi$
$61$	−0.871128	−0.111537	−0.0557683	+	0.998444i	$0.517761\pi$
$62$	−7.56752	−0.961076
$63$	1.00000	0.125988
$64$	8.77503	1.09688
$65$	2.46616	0.305890
$66$	4.24851	0.522955
$67$	−7.72381	−0.943613	−0.471807	−	0.881702i	$-0.656398\pi$
$67$	−7.72381	−0.943613	−0.471807	+	0.881702i	$0.656398\pi$
$68$	−0.167325	−0.0202912
$69$	2.85356	0.343529
$70$	3.24666	0.388051
$71$	−7.46107	−0.885466	−0.442733	−	0.896653i	$-0.645991\pi$
$71$	−7.46107	−0.885466	−0.442733	+	0.896653i	$0.645991\pi$
$72$	2.98749	0.352079
$73$	3.40941	0.399041	0.199521	−	0.979894i	$-0.436061\pi$
$73$	3.40941	0.399041	0.199521	+	0.979894i	$0.436061\pi$
$74$	12.8618	1.49516
$75$	1.10686	0.127810
$76$	0.199982	0.0229395
$77$	−3.23376	−0.368522
$78$	1.31112	0.148455
$79$	−12.0893	−1.36015	−0.680075	−	0.733143i	$-0.738053\pi$
$79$	−12.0893	−1.36015	−0.680075	+	0.733143i	$0.738053\pi$
$80$	8.34547	0.933052
$81$	1.00000	0.111111
$82$	−5.41655	−0.598158
$83$	−6.33711	−0.695589	−0.347794	−	0.937571i	$-0.613069\pi$
$83$	−6.33711	−0.695589	−0.347794	+	0.937571i	$0.613069\pi$
$84$	−0.273938	−0.0298891
$85$	−1.50945	−0.163722
$86$	0.286674	0.0309129
$87$	4.76424	0.510780
$88$	−9.66084	−1.02985
$89$	−10.3825	−1.10054	−0.550272	−	0.834985i	$-0.685476\pi$
$89$	−10.3825	−1.10054	−0.550272	+	0.834985i	$0.685476\pi$
$90$	3.24666	0.342228
$91$	−0.997959	−0.104615
$92$	−0.781701	−0.0814979
$93$	5.76004	0.597288
$94$	7.20649	0.743293
$95$	1.80405	0.185091
$96$	−1.53819	−0.156991
$97$	−0.165629	−0.0168171	−0.00840855	−	0.999965i	$-0.502677\pi$
$97$	−0.165629	−0.0168171	−0.00840855	+	0.999965i	$0.502677\pi$
$98$	−1.31380	−0.132714
$99$	−3.23376	−0.325005
$100$	−0.303213	−0.0303213
$101$	8.39578	0.835412	0.417706	−	0.908582i	$-0.362834\pi$
$101$	8.39578	0.835412	0.417706	+	0.908582i	$0.362834\pi$
$102$	−0.802484	−0.0794578
$103$	−8.13727	−0.801789	−0.400895	−	0.916124i	$-0.631301\pi$
$103$	−8.13727	−0.801789	−0.400895	+	0.916124i	$0.631301\pi$
$104$	−2.98139	−0.292350
$105$	−2.47121	−0.241165
$106$	−17.2625	−1.67668
$107$	19.6102	1.89579	0.947896	−	0.318579i	$-0.103206\pi$
$107$	19.6102	1.89579	0.947896	+	0.318579i	$0.103206\pi$
$108$	−0.273938	−0.0263597
$109$	−7.24314	−0.693767	−0.346883	−	0.937908i	$-0.612760\pi$
$109$	−7.24314	−0.693767	−0.346883	+	0.937908i	$0.612760\pi$
$110$	−10.4989	−1.00103
$111$	−9.78983	−0.929209
$112$	−3.37708	−0.319104
$113$	18.7732	1.76603	0.883015	−	0.469345i	$-0.155510\pi$
$113$	18.7732	1.76603	0.883015	+	0.469345i	$0.155510\pi$
$114$	0.959106	0.0898285
$115$	−7.05175	−0.657579
$116$	−1.30511	−0.121176
$117$	−0.997959	−0.0922613
$118$	13.2358	1.21846
$119$	0.610813	0.0559932
$120$	−7.38271	−0.673946
$121$	−0.542776	−0.0493433
$122$	1.14449	0.103617
$123$	4.12282	0.371742
$124$	−1.57790	−0.141699
$125$	9.62074	0.860505
$126$	−1.31380	−0.117042
$127$	2.39196	0.212252	0.106126	−	0.994353i	$-0.466155\pi$
$127$	2.39196	0.212252	0.106126	+	0.994353i	$0.466155\pi$
$128$	−8.45223	−0.747078
$129$	−0.218203	−0.0192117
$130$	−3.24004	−0.284170
$131$	6.27591	0.548329	0.274164	−	0.961683i	$-0.411599\pi$
$131$	6.27591	0.548329	0.274164	+	0.961683i	$0.411599\pi$
$132$	0.885852	0.0771035
$133$	−0.730026	−0.0633013
$134$	10.1475	0.876612
$135$	−2.47121	−0.212688
$136$	1.82480	0.156475
$137$	−11.6362	−0.994147	−0.497073	−	0.867708i	$-0.665592\pi$
$137$	−11.6362	−0.994147	−0.497073	+	0.867708i	$0.665592\pi$
$138$	−3.74900	−0.319136
$139$	18.4142	1.56187	0.780936	−	0.624611i	$-0.214742\pi$
$139$	18.4142	1.56187	0.780936	+	0.624611i	$0.214742\pi$
$140$	0.676959	0.0572134
$141$	−5.48524	−0.461941
$142$	9.80233	0.822593
$143$	3.22716	0.269869
$144$	−3.37708	−0.281423
$145$	−11.7734	−0.977730
$146$	−4.47927	−0.370707
$147$	1.00000	0.0824786
$148$	2.68181	0.220443
$149$	18.2987	1.49909	0.749545	−	0.661953i	$-0.230272\pi$
$149$	18.2987	1.49909	0.749545	+	0.661953i	$0.230272\pi$
$150$	−1.45420	−0.118735
$151$	7.42935	0.604592	0.302296	−	0.953214i	$-0.402247\pi$
$151$	7.42935	0.604592	0.302296	+	0.953214i	$0.402247\pi$
$152$	−2.18095	−0.176898
$153$	0.610813	0.0493813
$154$	4.24851	0.342354
$155$	−14.2342	−1.14332
$156$	0.273379	0.0218879
$157$	10.9748	0.875882	0.437941	−	0.899004i	$-0.355708\pi$
$157$	10.9748	0.875882	0.437941	+	0.899004i	$0.355708\pi$
$158$	15.8829	1.26357
$159$	13.1394	1.04202
$160$	3.80118	0.300510
$161$	2.85356	0.224892
$162$	−1.31380	−0.103222
$163$	−3.48187	−0.272721	−0.136361	−	0.990659i	$-0.543541\pi$
$163$	−3.48187	−0.272721	−0.136361	+	0.990659i	$0.543541\pi$
$164$	−1.12940	−0.0881912
$165$	7.99130	0.622122
$166$	8.32568	0.646198
$167$	5.57526	0.431427	0.215713	−	0.976457i	$-0.430792\pi$
$167$	5.57526	0.431427	0.215713	+	0.976457i	$0.430792\pi$
$168$	2.98749	0.230490
$169$	−12.0041	−0.923391
$170$	1.98310	0.152097
$171$	−0.730026	−0.0558265
$172$	0.0597742	0.00455774
$173$	5.09235	0.387165	0.193582	−	0.981084i	$-0.437989\pi$
$173$	5.09235	0.387165	0.193582	+	0.981084i	$0.437989\pi$
$174$	−6.25924	−0.474512
$175$	1.10686	0.0836711
$176$	10.9207	0.823177
$177$	−10.0745	−0.757246
$178$	13.6405	1.02240
$179$	−15.8795	−1.18689	−0.593446	−	0.804874i	$-0.702233\pi$
$179$	−15.8795	−1.18689	−0.593446	+	0.804874i	$0.702233\pi$
$180$	0.676959	0.0504575
$181$	−6.25237	−0.464735	−0.232367	−	0.972628i	$-0.574647\pi$
$181$	−6.25237	−0.464735	−0.232367	+	0.972628i	$0.574647\pi$
$182$	1.31112	0.0971863
$183$	−0.871128	−0.0643956
$184$	8.52500	0.628472
$185$	24.1927	1.77868
$186$	−7.56752	−0.554877
$187$	−1.97523	−0.144443
$188$	1.50262	0.109590
$189$	1.00000	0.0727393
$190$	−2.37015	−0.171949
$191$	12.1446	0.878753	0.439377	−	0.898303i	$-0.355199\pi$
$191$	12.1446	0.878753	0.439377	+	0.898303i	$0.355199\pi$
$192$	8.77503	0.633283
$193$	−5.07887	−0.365585	−0.182792	−	0.983152i	$-0.558514\pi$
$193$	−5.07887	−0.365585	−0.182792	+	0.983152i	$0.558514\pi$
$194$	0.217603	0.0156230
$195$	2.46616	0.176606
$196$	−0.273938	−0.0195670
$197$	3.70928	0.264275	0.132138	−	0.991231i	$-0.457816\pi$
$197$	3.70928	0.264275	0.132138	+	0.991231i	$0.457816\pi$
$198$	4.24851	0.301928
$199$	−11.5611	−0.819546	−0.409773	−	0.912188i	$-0.634392\pi$
$199$	−11.5611	−0.819546	−0.409773	+	0.912188i	$0.634392\pi$
$200$	3.30675	0.233823
$201$	−7.72381	−0.544795
$202$	−11.0303	−0.776093
$203$	4.76424	0.334384
$204$	−0.167325	−0.0117151
$205$	−10.1883	−0.711585
$206$	10.6907	0.744858
$207$	2.85356	0.198336
$208$	3.37019	0.233680
$209$	2.36073	0.163295
$210$	3.24666	0.224041
$211$	7.48393	0.515215	0.257607	−	0.966250i	$-0.417066\pi$
$211$	7.48393	0.515215	0.257607	+	0.966250i	$0.417066\pi$
$212$	−3.59939	−0.247207
$213$	−7.46107	−0.511224
$214$	−25.7639	−1.76118
$215$	0.539225	0.0367748
$216$	2.98749	0.203273
$217$	5.76004	0.391017
$218$	9.51601	0.644506
$219$	3.40941	0.230387
$220$	−2.18912	−0.147591
$221$	−0.609566	−0.0410039
$222$	12.8618	0.863230
$223$	−14.5539	−0.974598	−0.487299	−	0.873235i	$-0.662018\pi$
$223$	−14.5539	−0.974598	−0.487299	+	0.873235i	$0.662018\pi$
$224$	−1.53819	−0.102774
$225$	1.10686	0.0737910
$226$	−24.6641	−1.64063
$227$	−24.2842	−1.61180	−0.805900	−	0.592052i	$-0.798318\pi$
$227$	−24.2842	−1.61180	−0.805900	+	0.592052i	$0.798318\pi$
$228$	0.199982	0.0132441
$229$	−16.3939	−1.08334	−0.541670	−	0.840592i	$-0.682208\pi$
$229$	−16.3939	−1.08334	−0.541670	+	0.840592i	$0.682208\pi$
$230$	9.26456	0.610887
$231$	−3.23376	−0.212766
$232$	14.2331	0.934451
$233$	16.7045	1.09435	0.547173	−	0.837019i	$-0.315704\pi$
$233$	16.7045	1.09435	0.547173	+	0.837019i	$0.315704\pi$
$234$	1.31112	0.0857103
$235$	13.5552	0.884242
$236$	2.75979	0.179647
$237$	−12.0893	−0.785283
$238$	−0.802484	−0.0520173
$239$	−22.3195	−1.44373	−0.721865	−	0.692034i	$-0.756715\pi$
$239$	−22.3195	−1.44373	−0.721865	+	0.692034i	$0.756715\pi$
$240$	8.34547	0.538698
$241$	18.3355	1.18109	0.590545	−	0.807005i	$-0.298913\pi$
$241$	18.3355	1.18109	0.590545	+	0.807005i	$0.298913\pi$
$242$	0.713097	0.0458396
$243$	1.00000	0.0641500
$244$	0.238635	0.0152771
$245$	−2.47121	−0.157880
$246$	−5.41655	−0.345346
$247$	0.728536	0.0463556
$248$	17.2081	1.09271
$249$	−6.33711	−0.401598
$250$	−12.6397	−0.799405
$251$	29.4356	1.85796	0.928979	−	0.370133i	$-0.120688\pi$
$251$	29.4356	1.85796	0.928979	+	0.370133i	$0.120688\pi$
$252$	−0.273938	−0.0172565
$253$	−9.22775	−0.580144
$254$	−3.14255	−0.197181
$255$	−1.50945	−0.0945252
$256$	−6.44555	−0.402847
$257$	1.14094	0.0711696	0.0355848	−	0.999367i	$-0.488671\pi$
$257$	1.14094	0.0711696	0.0355848	+	0.999367i	$0.488671\pi$
$258$	0.286674	0.0178476
$259$	−9.78983	−0.608310
$260$	−0.675577	−0.0418975
$261$	4.76424	0.294899
$262$	−8.24527	−0.509395
$263$	5.96384	0.367746	0.183873	−	0.982950i	$-0.441136\pi$
$263$	5.96384	0.367746	0.183873	+	0.982950i	$0.441136\pi$
$264$	−9.66084	−0.594584
$265$	−32.4702	−1.99463
$266$	0.959106	0.0588066
$267$	−10.3825	−0.635400
$268$	2.11585	0.129246
$269$	9.56198	0.583004	0.291502	−	0.956570i	$-0.405845\pi$
$269$	9.56198	0.583004	0.291502	+	0.956570i	$0.405845\pi$
$270$	3.24666	0.197586
$271$	−20.5673	−1.24937	−0.624686	−	0.780876i	$-0.714773\pi$
$271$	−20.5673	−1.24937	−0.624686	+	0.780876i	$0.714773\pi$
$272$	−2.06277	−0.125074
$273$	−0.997959	−0.0603992
$274$	15.2876	0.923557
$275$	−3.57934	−0.215842
$276$	−0.781701	−0.0470529
$277$	−12.3009	−0.739091	−0.369545	−	0.929213i	$-0.620487\pi$
$277$	−12.3009	−0.739091	−0.369545	+	0.929213i	$0.620487\pi$
$278$	−24.1925	−1.45097
$279$	5.76004	0.344844
$280$	−7.38271	−0.441201
$281$	0.626461	0.0373715	0.0186858	−	0.999825i	$-0.494052\pi$
$281$	0.626461	0.0373715	0.0186858	+	0.999825i	$0.494052\pi$
$282$	7.20649	0.429140
$283$	24.8739	1.47860	0.739300	−	0.673376i	$-0.235157\pi$
$283$	24.8739	1.47860	0.739300	+	0.673376i	$0.235157\pi$
$284$	2.04387	0.121282
$285$	1.80405	0.106862
$286$	−4.23984	−0.250707
$287$	4.12282	0.243362
$288$	−1.53819	−0.0906386
$289$	−16.6269	−0.978053
$290$	15.4679	0.908306
$291$	−0.165629	−0.00970935
$292$	−0.933969	−0.0546564
$293$	−22.4399	−1.31096	−0.655478	−	0.755214i	$-0.727533\pi$
$293$	−22.4399	−1.31096	−0.655478	+	0.755214i	$0.727533\pi$
$294$	−1.31380	−0.0766222
$295$	24.8962	1.44951
$296$	−29.2470	−1.69995
$297$	−3.23376	−0.187642
$298$	−24.0408	−1.39265
$299$	−2.84774	−0.164689
$300$	−0.303213	−0.0175060
$301$	−0.218203	−0.0125770
$302$	−9.76065	−0.561662
$303$	8.39578	0.482325
$304$	2.46536	0.141398
$305$	2.15274	0.123265
$306$	−0.802484	−0.0458750
$307$	−4.04351	−0.230775	−0.115388	−	0.993321i	$-0.536811\pi$
$307$	−4.04351	−0.230775	−0.115388	+	0.993321i	$0.536811\pi$
$308$	0.885852	0.0504761
$309$	−8.13727	−0.462913
$310$	18.7009	1.06214
$311$	−28.9248	−1.64018	−0.820088	−	0.572238i	$-0.806076\pi$
$311$	−28.9248	−1.64018	−0.820088	+	0.572238i	$0.806076\pi$
$312$	−2.98139	−0.168788
$313$	28.5230	1.61222	0.806108	−	0.591769i	$-0.201570\pi$
$313$	28.5230	1.61222	0.806108	+	0.591769i	$0.201570\pi$
$314$	−14.4186	−0.813689
$315$	−2.47121	−0.139237
$316$	3.31172	0.186299
$317$	12.9302	0.726234	0.363117	−	0.931743i	$-0.381712\pi$
$317$	12.9302	0.726234	0.363117	+	0.931743i	$0.381712\pi$
$318$	−17.2625	−0.968033
$319$	−15.4064	−0.862594
$320$	−21.6849	−1.21222
$321$	19.6102	1.09454
$322$	−3.74900	−0.208924
$323$	−0.445910	−0.0248111
$324$	−0.273938	−0.0152188
$325$	−1.10461	−0.0612725
$326$	4.57447	0.253357
$327$	−7.24314	−0.400547
$328$	12.3169	0.680087
$329$	−5.48524	−0.302411
$330$	−10.4989	−0.577948
$331$	32.0232	1.76015	0.880077	−	0.474832i	$-0.157491\pi$
$331$	32.0232	1.76015	0.880077	+	0.474832i	$0.157491\pi$
$332$	1.73598	0.0952742
$333$	−9.78983	−0.536479
$334$	−7.32476	−0.400793
$335$	19.0871	1.04284
$336$	−3.37708	−0.184235
$337$	21.8196	1.18859	0.594295	−	0.804247i	$-0.297431\pi$
$337$	21.8196	1.18859	0.594295	+	0.804247i	$0.297431\pi$
$338$	15.7709	0.857825
$339$	18.7732	1.01962
$340$	0.413495	0.0224249
$341$	−18.6266	−1.00869
$342$	0.959106	0.0518625
$343$	1.00000	0.0539949
$344$	−0.651880	−0.0351470
$345$	−7.05175	−0.379653
$346$	−6.69032	−0.359674
$347$	7.08621	0.380408	0.190204	−	0.981745i	$-0.439085\pi$
$347$	7.08621	0.380408	0.190204	+	0.981745i	$0.439085\pi$
$348$	−1.30511	−0.0699612
$349$	36.4438	1.95079	0.975396	−	0.220462i	$-0.0707565\pi$
$349$	36.4438	1.95079	0.975396	+	0.220462i	$0.0707565\pi$
$350$	−1.45420	−0.0777300
$351$	−0.997959	−0.0532671
$352$	4.97413	0.265122
$353$	10.5439	0.561194	0.280597	−	0.959826i	$-0.409467\pi$
$353$	10.5439	0.561194	0.280597	+	0.959826i	$0.409467\pi$
$354$	13.2358	0.703477
$355$	18.4379	0.978580
$356$	2.84417	0.150741
$357$	0.610813	0.0323277
$358$	20.8625	1.10262
$359$	−5.38684	−0.284307	−0.142153	−	0.989845i	$-0.545403\pi$
$359$	−5.38684	−0.284307	−0.142153	+	0.989845i	$0.545403\pi$
$360$	−7.38271	−0.389103
$361$	−18.4671	−0.971951
$362$	8.21434	0.431736
$363$	−0.542776	−0.0284883
$364$	0.273379	0.0143290
$365$	−8.42536	−0.441004
$366$	1.14449	0.0598232
$367$	−14.9174	−0.778683	−0.389342	−	0.921093i	$-0.627297\pi$
$367$	−14.9174	−0.778683	−0.389342	+	0.921093i	$0.627297\pi$
$368$	−9.63672	−0.502349
$369$	4.12282	0.214625
$370$	−31.7843	−1.65239
$371$	13.1394	0.682164
$372$	−1.57790	−0.0818101
$373$	−22.0492	−1.14166	−0.570832	−	0.821067i	$-0.693379\pi$
$373$	−22.0492	−1.14166	−0.570832	+	0.821067i	$0.693379\pi$
$374$	2.59504	0.134187
$375$	9.62074	0.496813
$376$	−16.3871	−0.845101
$377$	−4.75452	−0.244870
$378$	−1.31380	−0.0675744
$379$	13.4866	0.692763	0.346381	−	0.938094i	$-0.387410\pi$
$379$	13.4866	0.692763	0.346381	+	0.938094i	$0.387410\pi$
$380$	−0.494197	−0.0253518
$381$	2.39196	0.122544
$382$	−15.9556	−0.816357
$383$	1.00000	0.0510976
$383$	1.00000	0.0510976
$384$	−8.45223	−0.431326
$385$	7.99130	0.407274
$386$	6.67260	0.339626
$387$	−0.218203	−0.0110919
$388$	0.0453722	0.00230342
$389$	32.6700	1.65644	0.828218	−	0.560406i	$-0.189355\pi$
$389$	32.6700	1.65644	0.828218	+	0.560406i	$0.189355\pi$
$390$	−3.24004	−0.164066
$391$	1.74299	0.0881470
$392$	2.98749	0.150891
$393$	6.27591	0.316578
$394$	−4.87324	−0.245510
$395$	29.8751	1.50318
$396$	0.885852	0.0445157
$397$	−5.45730	−0.273894	−0.136947	−	0.990578i	$-0.543729\pi$
$397$	−5.45730	−0.273894	−0.136947	+	0.990578i	$0.543729\pi$
$398$	15.1890	0.761354
$399$	−0.730026	−0.0365470
$400$	−3.73797	−0.186899
$401$	18.6905	0.933360	0.466680	−	0.884426i	$-0.345450\pi$
$401$	18.6905	0.933360	0.466680	+	0.884426i	$0.345450\pi$
$402$	10.1475	0.506112
$403$	−5.74828	−0.286342
$404$	−2.29993	−0.114426
$405$	−2.47121	−0.122795
$406$	−6.25924	−0.310641
$407$	31.6580	1.56923
$408$	1.82480	0.0903411
$409$	28.4100	1.40478	0.702391	−	0.711791i	$-0.252116\pi$
$409$	28.4100	1.40478	0.702391	+	0.711791i	$0.252116\pi$
$410$	13.3854	0.661058
$411$	−11.6362	−0.573971
$412$	2.22911	0.109820
$413$	−10.0745	−0.495734
$414$	−3.74900	−0.184253
$415$	15.6603	0.768735
$416$	1.53505	0.0752619
$417$	18.4142	0.901748
$418$	−3.10152	−0.151700
$419$	30.8331	1.50629	0.753147	−	0.657852i	$-0.228535\pi$
$419$	30.8331	1.50629	0.753147	+	0.657852i	$0.228535\pi$
$420$	0.676959	0.0330322
$421$	−25.6414	−1.24968	−0.624842	−	0.780752i	$-0.714836\pi$
$421$	−25.6414	−1.24968	−0.624842	+	0.780752i	$0.714836\pi$
$422$	−9.83236	−0.478632
$423$	−5.48524	−0.266702
$424$	39.2539	1.90634
$425$	0.676088	0.0327951
$426$	9.80233	0.474925
$427$	−0.871128	−0.0421568
$428$	−5.37200	−0.259665
$429$	3.22716	0.155809
$430$	−0.708432	−0.0341636
$431$	24.7250	1.19096	0.595480	−	0.803370i	$-0.296962\pi$
$431$	24.7250	1.19096	0.595480	+	0.803370i	$0.296962\pi$
$432$	−3.37708	−0.162480
$433$	27.7198	1.33213	0.666065	−	0.745894i	$-0.267978\pi$
$433$	27.7198	1.33213	0.666065	+	0.745894i	$0.267978\pi$
$434$	−7.56752	−0.363253
$435$	−11.7734	−0.564493
$436$	1.98417	0.0950247
$437$	−2.08318	−0.0996519
$438$	−4.47927	−0.214028
$439$	5.28951	0.252454	0.126227	−	0.992001i	$-0.459713\pi$
$439$	5.28951	0.252454	0.126227	+	0.992001i	$0.459713\pi$
$440$	23.8739	1.13815
$441$	1.00000	0.0476190
$442$	0.800846	0.0380924
$443$	−38.9352	−1.84987	−0.924934	−	0.380128i	$-0.875880\pi$
$443$	−38.9352	−1.84987	−0.924934	+	0.380128i	$0.875880\pi$
$444$	2.68181	0.127273
$445$	25.6573	1.21627
$446$	19.1208	0.905396
$447$	18.2987	0.865500
$448$	8.77503	0.414581
$449$	−10.6546	−0.502821	−0.251411	−	0.967881i	$-0.580894\pi$
$449$	−10.6546	−0.502821	−0.251411	+	0.967881i	$0.580894\pi$
$450$	−1.45420	−0.0685514
$451$	−13.3322	−0.627790
$452$	−5.14269	−0.241892
$453$	7.42935	0.349061
$454$	31.9045	1.49735
$455$	2.46616	0.115616
$456$	−2.18095	−0.102132
$457$	26.3755	1.23380	0.616898	−	0.787043i	$-0.288389\pi$
$457$	26.3755	1.23380	0.616898	+	0.787043i	$0.288389\pi$
$458$	21.5382	1.00642
$459$	0.610813	0.0285103
$460$	1.93174	0.0900681
$461$	30.8434	1.43652	0.718260	−	0.695775i	$-0.244939\pi$
$461$	30.8434	1.43652	0.718260	+	0.695775i	$0.244939\pi$
$462$	4.24851	0.197658
$463$	15.6240	0.726108	0.363054	−	0.931768i	$-0.381734\pi$
$463$	15.6240	0.726108	0.363054	+	0.931768i	$0.381734\pi$
$464$	−16.0892	−0.746924
$465$	−14.2342	−0.660097
$466$	−21.9463	−1.01664
$467$	−2.68590	−0.124289	−0.0621443	−	0.998067i	$-0.519794\pi$
$467$	−2.68590	−0.124289	−0.0621443	+	0.998067i	$0.519794\pi$
$468$	0.273379	0.0126370
$469$	−7.72381	−0.356652
$470$	−17.8087	−0.821456
$471$	10.9748	0.505690
$472$	−30.0975	−1.38535
$473$	0.705617	0.0324443
$474$	15.8829	0.729524
$475$	−0.808040	−0.0370754
$476$	−0.167325	−0.00766934
$477$	13.1394	0.601612
$478$	29.3233	1.34122
$479$	23.0973	1.05534	0.527671	−	0.849449i	$-0.323066\pi$
$479$	23.0973	1.05534	0.527671	+	0.849449i	$0.323066\pi$
$480$	3.80118	0.173499
$481$	9.76984	0.445466
$482$	−24.0891	−1.09723
$483$	2.85356	0.129842
$484$	0.148687	0.00675851
$485$	0.409304	0.0185855
$486$	−1.31380	−0.0595950
$487$	13.8026	0.625455	0.312727	−	0.949843i	$-0.398757\pi$
$487$	13.8026	0.625455	0.312727	+	0.949843i	$0.398757\pi$
$488$	−2.60249	−0.117809
$489$	−3.48187	−0.157456
$490$	3.24666	0.146669
$491$	43.0021	1.94066	0.970328	−	0.241791i	$-0.0777349\pi$
$491$	43.0021	1.94066	0.970328	+	0.241791i	$0.0777349\pi$
$492$	−1.12940	−0.0509172
$493$	2.91006	0.131063
$494$	−0.957148	−0.0430641
$495$	7.99130	0.359182
$496$	−19.4521	−0.873426
$497$	−7.46107	−0.334675
$498$	8.32568	0.373083
$499$	31.1866	1.39610	0.698052	−	0.716047i	$-0.254050\pi$
$499$	31.1866	1.39610	0.698052	+	0.716047i	$0.254050\pi$
$500$	−2.63549	−0.117863
$501$	5.57526	0.249084
$502$	−38.6724	−1.72603
$503$	−7.53917	−0.336155	−0.168078	−	0.985774i	$-0.553756\pi$
$503$	−7.53917	−0.336155	−0.168078	+	0.985774i	$0.553756\pi$
$504$	2.98749	0.133073
$505$	−20.7477	−0.923261
$506$	12.1234	0.538950
$507$	−12.0041	−0.533120
$508$	−0.655251	−0.0290720
$509$	18.7113	0.829362	0.414681	−	0.909967i	$-0.363893\pi$
$509$	18.7113	0.829362	0.414681	+	0.909967i	$0.363893\pi$
$510$	1.98310	0.0878134
$511$	3.40941	0.150823
$512$	25.3726	1.12132
$513$	−0.730026	−0.0322314
$514$	−1.49896	−0.0661162
$515$	20.1089	0.886103
$516$	0.0597742	0.00263141
$517$	17.7380	0.780115
$518$	12.8618	0.565117
$519$	5.09235	0.223530
$520$	7.36764	0.323093
$521$	43.8093	1.91932	0.959659	−	0.281166i	$-0.0907212\pi$
$521$	43.8093	1.91932	0.959659	+	0.281166i	$0.0907212\pi$
$522$	−6.25924	−0.273960
$523$	−18.4649	−0.807415	−0.403708	−	0.914888i	$-0.632279\pi$
$523$	−18.4649	−0.807415	−0.403708	+	0.914888i	$0.632279\pi$
$524$	−1.71921	−0.0751042
$525$	1.10686	0.0483075
$526$	−7.83527	−0.341634
$527$	3.51831	0.153260
$528$	10.9207	0.475262
$529$	−14.8572	−0.645964
$530$	42.6592	1.85300
$531$	−10.0745	−0.437196
$532$	0.199982	0.00867033
$533$	−4.11440	−0.178215
$534$	13.6405	0.590283
$535$	−48.4609	−2.09515
$536$	−23.0748	−0.996681
$537$	−15.8795	−0.685253
$538$	−12.5625	−0.541608
$539$	−3.23376	−0.139288
$540$	0.676959	0.0291317
$541$	20.2302	0.869765	0.434883	−	0.900487i	$-0.356790\pi$
$541$	20.2302	0.869765	0.434883	+	0.900487i	$0.356790\pi$
$542$	27.0212	1.16066
$543$	−6.25237	−0.268315
$544$	−0.939545	−0.0402827
$545$	17.8993	0.766722
$546$	1.31112	0.0561105
$547$	18.4294	0.787984	0.393992	−	0.919114i	$-0.371094\pi$
$547$	18.4294	0.787984	0.393992	+	0.919114i	$0.371094\pi$
$548$	3.18760	0.136168
$549$	−0.871128	−0.0371788
$550$	4.70252	0.200516
$551$	−3.47802	−0.148169
$552$	8.52500	0.362848
$553$	−12.0893	−0.514088
$554$	16.1609	0.686611
$555$	24.1927	1.02692
$556$	−5.04436	−0.213928
$557$	22.0997	0.936393	0.468197	−	0.883624i	$-0.344904\pi$
$557$	22.0997	0.936393	0.468197	+	0.883624i	$0.344904\pi$
$558$	−7.56752	−0.320359
$559$	0.217758	0.00921017
$560$	8.34547	0.352660
$561$	−1.97523	−0.0833941
$562$	−0.823042	−0.0347179
$563$	37.6938	1.58860	0.794302	−	0.607523i	$-0.207837\pi$
$563$	37.6938	1.58860	0.794302	+	0.607523i	$0.207837\pi$
$564$	1.50262	0.0632716
$565$	−46.3923	−1.95174
$566$	−32.6792	−1.37361
$567$	1.00000	0.0419961
$568$	−22.2899	−0.935263
$569$	−36.6198	−1.53518	−0.767590	−	0.640941i	$-0.778544\pi$
$569$	−36.6198	−1.53518	−0.767590	+	0.640941i	$0.778544\pi$
$570$	−2.37015	−0.0992746
$571$	35.5755	1.48879	0.744395	−	0.667739i	$-0.232738\pi$
$571$	35.5755	1.48879	0.744395	+	0.667739i	$0.232738\pi$
$572$	−0.884044	−0.0369637
$573$	12.1446	0.507348
$574$	−5.41655	−0.226082
$575$	3.15851	0.131719
$576$	8.77503	0.365626
$577$	−12.8378	−0.534444	−0.267222	−	0.963635i	$-0.586106\pi$
$577$	−12.8378	−0.534444	−0.267222	+	0.963635i	$0.586106\pi$
$578$	21.8444	0.908606
$579$	−5.07887	−0.211070
$580$	3.22519	0.133919
$581$	−6.33711	−0.262908
$582$	0.217603	0.00901993
$583$	−42.4897	−1.75974
$584$	10.1856	0.421483
$585$	2.46616	0.101963
$586$	29.4815	1.21787
$587$	−31.3270	−1.29300	−0.646501	−	0.762913i	$-0.723768\pi$
$587$	−31.3270	−1.29300	−0.646501	+	0.762913i	$0.723768\pi$
$588$	−0.273938	−0.0112970
$589$	−4.20498	−0.173263
$590$	−32.7085	−1.34659
$591$	3.70928	0.152579
$592$	33.0610	1.35880
$593$	−15.4391	−0.634008	−0.317004	−	0.948424i	$-0.602677\pi$
$593$	−15.4391	−0.634008	−0.317004	+	0.948424i	$0.602677\pi$
$594$	4.24851	0.174318
$595$	−1.50945	−0.0618812
$596$	−5.01272	−0.205329
$597$	−11.5611	−0.473165
$598$	3.74135	0.152995
$599$	30.7015	1.25443	0.627215	−	0.778846i	$-0.284195\pi$
$599$	30.7015	1.25443	0.627215	+	0.778846i	$0.284195\pi$
$600$	3.30675	0.134998
$601$	−24.4519	−0.997413	−0.498706	−	0.866771i	$-0.666191\pi$
$601$	−24.4519	−0.997413	−0.498706	+	0.866771i	$0.666191\pi$
$602$	0.286674	0.0116840
$603$	−7.72381	−0.314538
$604$	−2.03518	−0.0828104
$605$	1.34131	0.0545321
$606$	−11.0303	−0.448077
$607$	28.2934	1.14839	0.574196	−	0.818718i	$-0.305315\pi$
$607$	28.2934	1.14839	0.574196	+	0.818718i	$0.305315\pi$
$608$	1.12292	0.0455403
$609$	4.76424	0.193057
$610$	−2.82826	−0.114513
$611$	5.47405	0.221456
$612$	−0.167325	−0.00676372
$613$	14.6268	0.590772	0.295386	−	0.955378i	$-0.404552\pi$
$613$	14.6268	0.590772	0.295386	+	0.955378i	$0.404552\pi$
$614$	5.31235	0.214389
$615$	−10.1883	−0.410834
$616$	−9.66084	−0.389246
$617$	−29.9445	−1.20552	−0.602761	−	0.797922i	$-0.705933\pi$
$617$	−29.9445	−1.20552	−0.602761	+	0.797922i	$0.705933\pi$
$618$	10.6907	0.430044
$619$	−19.7558	−0.794053	−0.397027	−	0.917807i	$-0.629958\pi$
$619$	−19.7558	−0.794053	−0.397027	+	0.917807i	$0.629958\pi$
$620$	3.89931	0.156600
$621$	2.85356	0.114510
$622$	38.0013	1.52371
$623$	−10.3825	−0.415967
$624$	3.37019	0.134915
$625$	−29.3092	−1.17237
$626$	−37.4734	−1.49774
$627$	2.36073	0.0942785
$628$	−3.00641	−0.119969
$629$	−5.97975	−0.238428
$630$	3.24666	0.129350
$631$	−3.27075	−0.130206	−0.0651032	−	0.997879i	$-0.520738\pi$
$631$	−3.27075	−0.130206	−0.0651032	+	0.997879i	$0.520738\pi$
$632$	−36.1166	−1.43664
$633$	7.48393	0.297459
$634$	−16.9877	−0.674668
$635$	−5.91104	−0.234572
$636$	−3.59939	−0.142725
$637$	−0.997959	−0.0395406
$638$	20.2409	0.801345
$639$	−7.46107	−0.295155
$640$	20.8872	0.825639
$641$	44.1687	1.74456	0.872280	−	0.489007i	$-0.162641\pi$
$641$	44.1687	1.74456	0.872280	+	0.489007i	$0.162641\pi$
$642$	−25.7639	−1.01682
$643$	5.29362	0.208760	0.104380	−	0.994537i	$-0.466714\pi$
$643$	5.29362	0.208760	0.104380	+	0.994537i	$0.466714\pi$
$644$	−0.781701	−0.0308033
$645$	0.539225	0.0212320
$646$	0.585835	0.0230494
$647$	−18.5314	−0.728545	−0.364272	−	0.931292i	$-0.618682\pi$
$647$	−18.5314	−0.728545	−0.364272	+	0.931292i	$0.618682\pi$
$648$	2.98749	0.117360
$649$	32.5785	1.27882
$650$	1.45123	0.0569218
$651$	5.76004	0.225754
$652$	0.953819	0.0373544
$653$	26.0491	1.01938	0.509690	−	0.860358i	$-0.329760\pi$
$653$	26.0491	1.01938	0.509690	+	0.860358i	$0.329760\pi$
$654$	9.51601	0.372106
$655$	−15.5091	−0.605990
$656$	−13.9231	−0.543606
$657$	3.40941	0.133014
$658$	7.20649	0.280938
$659$	3.27079	0.127412	0.0637059	−	0.997969i	$-0.479708\pi$
$659$	3.27079	0.127412	0.0637059	+	0.997969i	$0.479708\pi$
$660$	−2.18912	−0.0852115
$661$	30.7933	1.19772	0.598860	−	0.800854i	$-0.295621\pi$
$661$	30.7933	1.19772	0.598860	+	0.800854i	$0.295621\pi$
$662$	−42.0719	−1.63517
$663$	−0.609566	−0.0236736
$664$	−18.9321	−0.734707
$665$	1.80405	0.0699579
$666$	12.8618	0.498386
$667$	13.5951	0.526403
$668$	−1.52728	−0.0590921
$669$	−14.5539	−0.562685
$670$	−25.0766	−0.968794
$671$	2.81702	0.108750
$672$	−1.53819	−0.0593369
$673$	−33.8052	−1.30309	−0.651547	−	0.758608i	$-0.725880\pi$
$673$	−33.8052	−1.30309	−0.651547	+	0.758608i	$0.725880\pi$
$674$	−28.6665	−1.10419
$675$	1.10686	0.0426032
$676$	3.28838	0.126476
$677$	−12.2138	−0.469413	−0.234706	−	0.972066i	$-0.575413\pi$
$677$	−12.2138	−0.469413	−0.234706	+	0.972066i	$0.575413\pi$
$678$	−24.6641	−0.947219
$679$	−0.165629	−0.00635626
$680$	−4.50946	−0.172930
$681$	−24.2842	−0.930573
$682$	24.4716	0.937065
$683$	26.1280	0.999762	0.499881	−	0.866094i	$-0.333377\pi$
$683$	26.1280	0.999762	0.499881	+	0.866094i	$0.333377\pi$
$684$	0.199982	0.00764651
$685$	28.7554	1.09869
$686$	−1.31380	−0.0501610
$687$	−16.3939	−0.625466
$688$	0.736889	0.0280936
$689$	−13.1126	−0.499550
$690$	9.26456	0.352696
$691$	2.07312	0.0788653	0.0394327	−	0.999222i	$-0.487445\pi$
$691$	2.07312	0.0788653	0.0394327	+	0.999222i	$0.487445\pi$
$692$	−1.39499	−0.0530296
$693$	−3.23376	−0.122841
$694$	−9.30984	−0.353397
$695$	−45.5053	−1.72612
$696$	14.2331	0.539506
$697$	2.51827	0.0953864
$698$	−47.8797	−1.81227
$699$	16.7045	0.631821
$700$	−0.303213	−0.0114604
$701$	50.9108	1.92288	0.961438	−	0.275023i	$-0.0886855\pi$
$701$	50.9108	1.92288	0.961438	+	0.275023i	$0.0886855\pi$
$702$	1.31112	0.0494848
$703$	7.14683	0.269548
$704$	−28.3764	−1.06947
$705$	13.5552	0.510517
$706$	−13.8525	−0.521347
$707$	8.39578	0.315756
$708$	2.75979	0.103719
$709$	−47.4710	−1.78281	−0.891406	−	0.453205i	$-0.850280\pi$
$709$	−47.4710	−1.78281	−0.891406	+	0.453205i	$0.850280\pi$
$710$	−24.2236	−0.909095
$711$	−12.0893	−0.453383
$712$	−31.0177	−1.16244
$713$	16.4366	0.615557
$714$	−0.802484	−0.0300322
$715$	−7.97499	−0.298248
$716$	4.35001	0.162568
$717$	−22.3195	−0.833538
$718$	7.07721	0.264119
$719$	−32.8066	−1.22348	−0.611739	−	0.791060i	$-0.709530\pi$
$719$	−32.8066	−1.22348	−0.611739	+	0.791060i	$0.709530\pi$
$720$	8.34547	0.311017
$721$	−8.13727	−0.303048
$722$	24.2620	0.902937
$723$	18.3355	0.681903
$724$	1.71276	0.0636544
$725$	5.27337	0.195848
$726$	0.713097	0.0264655
$727$	−22.8177	−0.846261	−0.423131	−	0.906069i	$-0.639069\pi$
$727$	−22.8177	−0.846261	−0.423131	+	0.906069i	$0.639069\pi$
$728$	−2.98139	−0.110498
$729$	1.00000	0.0370370
$730$	11.0692	0.409690
$731$	−0.133281	−0.00492959
$732$	0.238635	0.00882022
$733$	−0.940802	−0.0347493	−0.0173747	−	0.999849i	$-0.505531\pi$
$733$	−0.940802	−0.0347493	−0.0173747	+	0.999849i	$0.505531\pi$
$734$	19.5985	0.723392
$735$	−2.47121	−0.0911518
$736$	−4.38932	−0.161792
$737$	24.9770	0.920038
$738$	−5.41655	−0.199386
$739$	−20.6050	−0.757967	−0.378983	−	0.925404i	$-0.623726\pi$
$739$	−20.6050	−0.757967	−0.378983	+	0.925404i	$0.623726\pi$
$740$	−6.62731	−0.243625
$741$	0.728536	0.0267634
$742$	−17.2625	−0.633726
$743$	53.0050	1.94457	0.972283	−	0.233809i	$-0.0751190\pi$
$743$	53.0050	1.94457	0.972283	+	0.233809i	$0.0751190\pi$
$744$	17.2081	0.630879
$745$	−45.2199	−1.65673
$746$	28.9681	1.06060
$747$	−6.33711	−0.231863
$748$	0.541090	0.0197842
$749$	19.6102	0.716542
$750$	−12.6397	−0.461537
$751$	48.8355	1.78203	0.891016	−	0.453971i	$-0.149993\pi$
$751$	48.8355	1.78203	0.891016	+	0.453971i	$0.149993\pi$
$752$	18.5241	0.675505
$753$	29.4356	1.07269
$754$	6.24647	0.227483
$755$	−18.3595	−0.668169
$756$	−0.273938	−0.00996304
$757$	34.4446	1.25191	0.625954	−	0.779860i	$-0.284710\pi$
$757$	34.4446	1.25191	0.625954	+	0.779860i	$0.284710\pi$
$758$	−17.7187	−0.643573
$759$	−9.22775	−0.334946
$760$	5.38957	0.195500
$761$	−21.7939	−0.790028	−0.395014	−	0.918675i	$-0.629260\pi$
$761$	−21.7939	−0.790028	−0.395014	+	0.918675i	$0.629260\pi$
$762$	−3.14255	−0.113843
$763$	−7.24314	−0.262219
$764$	−3.32688	−0.120362
$765$	−1.50945	−0.0545741
$766$	−1.31380	−0.0474694
$767$	10.0539	0.363027
$768$	−6.44555	−0.232584
$769$	−50.9082	−1.83580	−0.917898	−	0.396817i	$-0.870115\pi$
$769$	−50.9082	−1.83580	−0.917898	+	0.396817i	$0.870115\pi$
$770$	−10.4989	−0.378356
$771$	1.14094	0.0410898
$772$	1.39130	0.0500739
$773$	30.8505	1.10962	0.554809	−	0.831978i	$-0.312792\pi$
$773$	30.8505	1.10962	0.554809	+	0.831978i	$0.312792\pi$
$774$	0.286674	0.0103043
$775$	6.37558	0.229018
$776$	−0.494816	−0.0177629
$777$	−9.78983	−0.351208
$778$	−42.9218	−1.53882
$779$	−3.00977	−0.107836
$780$	−0.675577	−0.0241895
$781$	24.1273	0.863344
$782$	−2.28994	−0.0818881
$783$	4.76424	0.170260
$784$	−3.37708	−0.120610
$785$	−27.1209	−0.967987
$786$	−8.24527	−0.294099
$787$	31.0686	1.10747	0.553737	−	0.832691i	$-0.313201\pi$
$787$	31.0686	1.10747	0.553737	+	0.832691i	$0.313201\pi$
$788$	−1.01611	−0.0361976
$789$	5.96384	0.212318
$790$	−39.2498	−1.39645
$791$	18.7732	0.667496
$792$	−9.66084	−0.343283
$793$	0.869350	0.0308715
$794$	7.16979	0.254446
$795$	−32.4702	−1.15160
$796$	3.16703	0.112253
$797$	−8.33212	−0.295139	−0.147569	−	0.989052i	$-0.547145\pi$
$797$	−8.33212	−0.295139	−0.147569	+	0.989052i	$0.547145\pi$
$798$	0.959106	0.0339520
$799$	−3.35046	−0.118531
$800$	−1.70257	−0.0601948
$801$	−10.3825	−0.366848
$802$	−24.5555	−0.867086
$803$	−11.0252	−0.389072
$804$	2.11585	0.0746202
$805$	−7.05175	−0.248541
$806$	7.55207	0.266010
$807$	9.56198	0.336598
$808$	25.0823	0.882393
$809$	40.9860	1.44099	0.720496	−	0.693459i	$-0.243914\pi$
$809$	40.9860	1.44099	0.720496	+	0.693459i	$0.243914\pi$
$810$	3.24666	0.114076
$811$	25.5796	0.898221	0.449111	−	0.893476i	$-0.351741\pi$
$811$	25.5796	0.898221	0.449111	+	0.893476i	$0.351741\pi$
$812$	−1.30511	−0.0458003
$813$	−20.5673	−0.721325
$814$	−41.5921	−1.45780
$815$	8.60443	0.301400
$816$	−2.06277	−0.0722112
$817$	0.159294	0.00557299
$818$	−37.3249	−1.30503
$819$	−0.997959	−0.0348715
$820$	2.79098	0.0974652
$821$	43.5601	1.52026	0.760129	−	0.649772i	$-0.225136\pi$
$821$	43.5601	1.52026	0.760129	+	0.649772i	$0.225136\pi$
$822$	15.2876	0.533216
$823$	−3.74584	−0.130572	−0.0652860	−	0.997867i	$-0.520796\pi$
$823$	−3.74584	−0.130572	−0.0652860	+	0.997867i	$0.520796\pi$
$824$	−24.3100	−0.846880
$825$	−3.57934	−0.124617
$826$	13.2358	0.460534
$827$	−15.7696	−0.548362	−0.274181	−	0.961678i	$-0.588407\pi$
$827$	−15.7696	−0.548362	−0.274181	+	0.961678i	$0.588407\pi$
$828$	−0.781701	−0.0271660
$829$	45.6807	1.58656	0.793279	−	0.608859i	$-0.208372\pi$
$829$	45.6807	1.58656	0.793279	+	0.608859i	$0.208372\pi$
$830$	−20.5745	−0.714150
$831$	−12.3009	−0.426714
$832$	−8.75712	−0.303598
$833$	0.610813	0.0211634
$834$	−24.1925	−0.837719
$835$	−13.7776	−0.476794
$836$	−0.646695	−0.0223664
$837$	5.76004	0.199096
$838$	−40.5084	−1.39934
$839$	−6.75109	−0.233074	−0.116537	−	0.993186i	$-0.537179\pi$
$839$	−6.75109	−0.233074	−0.116537	+	0.993186i	$0.537179\pi$
$840$	−7.38271	−0.254728
$841$	−6.30200	−0.217310
$842$	33.6875	1.16095
$843$	0.626461	0.0215765
$844$	−2.05014	−0.0705686
$845$	29.6646	1.02049
$846$	7.20649	0.247764
$847$	−0.542776	−0.0186500
$848$	−44.3728	−1.52377
$849$	24.8739	0.853670
$850$	−0.888242	−0.0304664
$851$	−27.9359	−0.957630
$852$	2.04387	0.0700220
$853$	9.62222	0.329459	0.164729	−	0.986339i	$-0.447325\pi$
$853$	9.62222	0.329459	0.164729	+	0.986339i	$0.447325\pi$
$854$	1.14449	0.0391635
$855$	1.80405	0.0616971
$856$	58.5854	2.00241
$857$	−17.6135	−0.601666	−0.300833	−	0.953677i	$-0.597265\pi$
$857$	−17.6135	−0.601666	−0.300833	+	0.953677i	$0.597265\pi$
$858$	−4.23984	−0.144746
$859$	−11.2296	−0.383149	−0.191575	−	0.981478i	$-0.561359\pi$
$859$	−11.2296	−0.383149	−0.191575	+	0.981478i	$0.561359\pi$
$860$	−0.147714	−0.00503702
$861$	4.12282	0.140505
$862$	−32.4836	−1.10639
$863$	−40.9518	−1.39402	−0.697008	−	0.717064i	$-0.745486\pi$
$863$	−40.9518	−1.39402	−0.697008	+	0.717064i	$0.745486\pi$
$864$	−1.53819	−0.0523302
$865$	−12.5843	−0.427878
$866$	−36.4182	−1.23754
$867$	−16.6269	−0.564679
$868$	−1.57790	−0.0535573
$869$	39.0939	1.32617
$870$	15.4679	0.524411
$871$	7.70804	0.261177
$872$	−21.6388	−0.732783
$873$	−0.165629	−0.00560570
$874$	2.73687	0.0925760
$875$	9.62074	0.325240
$876$	−0.933969	−0.0315559
$877$	−3.86405	−0.130480	−0.0652399	−	0.997870i	$-0.520781\pi$
$877$	−3.86405	−0.130480	−0.0652399	+	0.997870i	$0.520781\pi$
$878$	−6.94933	−0.234529
$879$	−22.4399	−0.756881
$880$	−26.9873	−0.909740
$881$	−5.61408	−0.189143	−0.0945716	−	0.995518i	$-0.530148\pi$
$881$	−5.61408	−0.189143	−0.0945716	+	0.995518i	$0.530148\pi$
$882$	−1.31380	−0.0442378
$883$	21.2376	0.714701	0.357350	−	0.933970i	$-0.383680\pi$
$883$	21.2376	0.714701	0.357350	+	0.933970i	$0.383680\pi$
$884$	0.166984	0.00561627
$885$	24.8962	0.836876
$886$	51.1530	1.71852
$887$	51.7143	1.73640	0.868199	−	0.496216i	$-0.165278\pi$
$887$	51.7143	1.73640	0.868199	+	0.496216i	$0.165278\pi$
$888$	−29.2470	−0.981466
$889$	2.39196	0.0802239
$890$	−33.7085	−1.12991
$891$	−3.23376	−0.108335
$892$	3.98686	0.133490
$893$	4.00437	0.134001
$894$	−24.0408	−0.804045
$895$	39.2416	1.31170
$896$	−8.45223	−0.282369
$897$	−2.84774	−0.0950833
$898$	13.9980	0.467118
$899$	27.4422	0.915249
$900$	−0.303213	−0.0101071
$901$	8.02572	0.267375
$902$	17.5158	0.583213
$903$	−0.218203	−0.00726134
$904$	56.0847	1.86535
$905$	15.4509	0.513605
$906$	−9.76065	−0.324276
$907$	−0.681749	−0.0226371	−0.0113186	−	0.999936i	$-0.503603\pi$
$907$	−0.681749	−0.0226371	−0.0113186	+	0.999936i	$0.503603\pi$
$908$	6.65238	0.220767
$909$	8.39578	0.278471
$910$	−3.24004	−0.107406
$911$	57.5003	1.90507	0.952535	−	0.304429i	$-0.0984656\pi$
$911$	57.5003	1.90507	0.952535	+	0.304429i	$0.0984656\pi$
$912$	2.46536	0.0816361
$913$	20.4927	0.678210
$914$	−34.6521	−1.14619
$915$	2.15274	0.0711673
$916$	4.49092	0.148384
$917$	6.27591	0.207249
$918$	−0.802484	−0.0264859
$919$	−30.5901	−1.00907	−0.504537	−	0.863390i	$-0.668337\pi$
$919$	−30.5901	−1.00907	−0.504537	+	0.863390i	$0.668337\pi$
$920$	−21.0670	−0.694560
$921$	−4.04351	−0.133238
$922$	−40.5220	−1.33452
$923$	7.44584	0.245083
$924$	0.885852	0.0291424
$925$	−10.8360	−0.356286
$926$	−20.5267	−0.674550
$927$	−8.13727	−0.267263
$928$	−7.32830	−0.240563
$929$	41.9842	1.37746	0.688729	−	0.725019i	$-0.258169\pi$
$929$	41.9842	1.37746	0.688729	+	0.725019i	$0.258169\pi$
$930$	18.7009	0.613227
$931$	−0.730026	−0.0239256
$932$	−4.57600	−0.149892
$933$	−28.9248	−0.946956
$934$	3.52873	0.115463
$935$	4.88119	0.159632
$936$	−2.98139	−0.0974499
$937$	−8.48621	−0.277233	−0.138616	−	0.990346i	$-0.544265\pi$
$937$	−8.48621	−0.277233	−0.138616	+	0.990346i	$0.544265\pi$
$938$	10.1475	0.331328
$939$	28.5230	0.930813
$940$	−3.71328	−0.121114
$941$	27.9268	0.910388	0.455194	−	0.890392i	$-0.349570\pi$
$941$	27.9268	0.910388	0.455194	+	0.890392i	$0.349570\pi$
$942$	−14.4186	−0.469784
$943$	11.7647	0.383112
$944$	34.0224	1.10733
$945$	−2.47121	−0.0803884
$946$	−0.927037	−0.0301406
$947$	−44.0541	−1.43157	−0.715783	−	0.698322i	$-0.753930\pi$
$947$	−44.0541	−1.43157	−0.715783	+	0.698322i	$0.753930\pi$
$948$	3.31172	0.107560
$949$	−3.40245	−0.110448
$950$	1.06160	0.0344429
$951$	12.9302	0.419292
$952$	1.82480	0.0591421
$953$	10.4728	0.339247	0.169624	−	0.985509i	$-0.445745\pi$
$953$	10.4728	0.339247	0.169624	+	0.985509i	$0.445745\pi$
$954$	−17.2625	−0.558894
$955$	−30.0119	−0.971161
$956$	6.11417	0.197747
$957$	−15.4064	−0.498019
$958$	−30.3451	−0.980406
$959$	−11.6362	−0.375752
$960$	−21.6849	−0.699877
$961$	2.17805	0.0702595
$962$	−12.8356	−0.413836
$963$	19.6102	0.631931
$964$	−5.02279	−0.161773
$965$	12.5509	0.404029
$966$	−3.74900	−0.120622
$967$	−8.03508	−0.258391	−0.129195	−	0.991619i	$-0.541239\pi$
$967$	−8.03508	−0.258391	−0.129195	+	0.991619i	$0.541239\pi$
$968$	−1.62154	−0.0521182
$969$	−0.445910	−0.0143247
$970$	−0.537742	−0.0172659
$971$	−59.1925	−1.89958	−0.949788	−	0.312893i	$-0.898702\pi$
$971$	−59.1925	−1.89958	−0.949788	+	0.312893i	$0.898702\pi$
$972$	−0.273938	−0.00878658
$973$	18.4142	0.590332
$974$	−18.1338	−0.581044
$975$	−1.10461	−0.0353757
$976$	2.94187	0.0941670
$977$	−54.9475	−1.75793	−0.878964	−	0.476889i	$-0.841764\pi$
$977$	−54.9475	−1.75793	−0.878964	+	0.476889i	$0.841764\pi$
$978$	4.57447	0.146276
$979$	33.5746	1.07305
$980$	0.676959	0.0216246
$981$	−7.24314	−0.231256
$982$	−56.4960	−1.80286
$983$	−45.8644	−1.46285	−0.731424	−	0.681923i	$-0.761144\pi$
$983$	−45.8644	−1.46285	−0.731424	+	0.681923i	$0.761144\pi$
$984$	12.3169	0.392648
$985$	−9.16640	−0.292066
$986$	−3.82323	−0.121756
$987$	−5.48524	−0.174597
$988$	−0.199574	−0.00634930
$989$	−0.622656	−0.0197993
$990$	−10.4989	−0.333678
$991$	−58.9300	−1.87197	−0.935986	−	0.352036i	$-0.885489\pi$
$991$	−58.9300	−1.87197	−0.935986	+	0.352036i	$0.885489\pi$
$992$	−8.86002	−0.281306
$993$	32.0232	1.01622
$994$	9.80233	0.310911
$995$	28.5699	0.905727
$996$	1.73598	0.0550066
$997$	−45.9353	−1.45478	−0.727392	−	0.686222i	$-0.759268\pi$
$997$	−45.9353	−1.45478	−0.727392	+	0.686222i	$0.759268\pi$
$998$	−40.9729	−1.29697
$999$	−9.78983	−0.309736

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.u.1.15	✓	53

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.u.1.15	✓	53	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q-1.31380 q^{2} +1.00000 q^{3} -0.273938 q^{4} -2.47121 q^{5} -1.31380 q^{6} +1.00000 q^{7} +2.98749 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.31380 q^{2} +1.00000 q^{3} -0.273938 q^{4} -2.47121 q^{5} -1.31380 q^{6} +1.00000 q^{7} +2.98749 q^{8} +1.00000 q^{9} +3.24666 q^{10} -3.23376 q^{11} -0.273938 q^{12} -0.997959 q^{13} -1.31380 q^{14} -2.47121 q^{15} -3.37708 q^{16} +0.610813 q^{17} -1.31380 q^{18} -0.730026 q^{19} +0.676959 q^{20} +1.00000 q^{21} +4.24851 q^{22} +2.85356 q^{23} +2.98749 q^{24} +1.10686 q^{25} +1.31112 q^{26} +1.00000 q^{27} -0.273938 q^{28} +4.76424 q^{29} +3.24666 q^{30} +5.76004 q^{31} -1.53819 q^{32} -3.23376 q^{33} -0.802484 q^{34} -2.47121 q^{35} -0.273938 q^{36} -9.78983 q^{37} +0.959106 q^{38} -0.997959 q^{39} -7.38271 q^{40} +4.12282 q^{41} -1.31380 q^{42} -0.218203 q^{43} +0.885852 q^{44} -2.47121 q^{45} -3.74900 q^{46} -5.48524 q^{47} -3.37708 q^{48} +1.00000 q^{49} -1.45420 q^{50} +0.610813 q^{51} +0.273379 q^{52} +13.1394 q^{53} -1.31380 q^{54} +7.99130 q^{55} +2.98749 q^{56} -0.730026 q^{57} -6.25924 q^{58} -10.0745 q^{59} +0.676959 q^{60} -0.871128 q^{61} -7.56752 q^{62} +1.00000 q^{63} +8.77503 q^{64} +2.46616 q^{65} +4.24851 q^{66} -7.72381 q^{67} -0.167325 q^{68} +2.85356 q^{69} +3.24666 q^{70} -7.46107 q^{71} +2.98749 q^{72} +3.40941 q^{73} +12.8618 q^{74} +1.10686 q^{75} +0.199982 q^{76} -3.23376 q^{77} +1.31112 q^{78} -12.0893 q^{79} +8.34547 q^{80} +1.00000 q^{81} -5.41655 q^{82} -6.33711 q^{83} -0.273938 q^{84} -1.50945 q^{85} +0.286674 q^{86} +4.76424 q^{87} -9.66084 q^{88} -10.3825 q^{89} +3.24666 q^{90} -0.997959 q^{91} -0.781701 q^{92} +5.76004 q^{93} +7.20649 q^{94} +1.80405 q^{95} -1.53819 q^{96} -0.165629 q^{97} -1.31380 q^{98} -3.23376 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 53 q + 11 q^{2} + 53 q^{3} + 63 q^{4} + 24 q^{5} + 11 q^{6} + 53 q^{7} + 30 q^{8} + 53 q^{9}+O(q^{10}) \) 53 * q + 11 * q^2 + 53 * q^3 + 63 * q^4 + 24 * q^5 + 11 * q^6 + 53 * q^7 + 30 * q^8 + 53 * q^9	\( 53 q + 11 q^{2} + 53 q^{3} + 63 q^{4} + 24 q^{5} + 11 q^{6} + 53 q^{7} + 30 q^{8} + 53 q^{9} + 2 q^{10} + 46 q^{11} + 63 q^{12} + 32 q^{13} + 11 q^{14} + 24 q^{15} + 67 q^{16} + 46 q^{17} + 11 q^{18} + 14 q^{19} + 53 q^{20} + 53 q^{21} + 13 q^{22} + 68 q^{23} + 30 q^{24} + 71 q^{25} + 11 q^{26} + 53 q^{27} + 63 q^{28} + 55 q^{29} + 2 q^{30} - 2 q^{31} + 51 q^{32} + 46 q^{33} - 7 q^{34} + 24 q^{35} + 63 q^{36} + 53 q^{37} + 16 q^{38} + 32 q^{39} - 20 q^{40} + 38 q^{41} + 11 q^{42} + 36 q^{43} + 70 q^{44} + 24 q^{45} + 4 q^{46} + 51 q^{47} + 67 q^{48} + 53 q^{49} + 32 q^{50} + 46 q^{51} + 10 q^{52} + 104 q^{53} + 11 q^{54} + 11 q^{55} + 30 q^{56} + 14 q^{57} + 4 q^{58} + 36 q^{59} + 53 q^{60} + 3 q^{61} + 25 q^{62} + 53 q^{63} + 82 q^{64} + 46 q^{65} + 13 q^{66} + 54 q^{67} + 88 q^{68} + 68 q^{69} + 2 q^{70} + 101 q^{71} + 30 q^{72} + q^{73} + 32 q^{74} + 71 q^{75} - 35 q^{76} + 46 q^{77} + 11 q^{78} + 14 q^{79} + 39 q^{80} + 53 q^{81} - 29 q^{82} + 38 q^{83} + 63 q^{84} + 16 q^{85} + 23 q^{86} + 55 q^{87} - 8 q^{88} + 52 q^{89} + 2 q^{90} + 32 q^{91} + 76 q^{92} - 2 q^{93} - 53 q^{94} + 46 q^{95} + 51 q^{96} - 3 q^{97} + 11 q^{98} + 46 q^{99}+O(q^{100}) \) 53 * q + 11 * q^2 + 53 * q^3 + 63 * q^4 + 24 * q^5 + 11 * q^6 + 53 * q^7 + 30 * q^8 + 53 * q^9 + 2 * q^10 + 46 * q^11 + 63 * q^12 + 32 * q^13 + 11 * q^14 + 24 * q^15 + 67 * q^16 + 46 * q^17 + 11 * q^18 + 14 * q^19 + 53 * q^20 + 53 * q^21 + 13 * q^22 + 68 * q^23 + 30 * q^24 + 71 * q^25 + 11 * q^26 + 53 * q^27 + 63 * q^28 + 55 * q^29 + 2 * q^30 - 2 * q^31 + 51 * q^32 + 46 * q^33 - 7 * q^34 + 24 * q^35 + 63 * q^36 + 53 * q^37 + 16 * q^38 + 32 * q^39 - 20 * q^40 + 38 * q^41 + 11 * q^42 + 36 * q^43 + 70 * q^44 + 24 * q^45 + 4 * q^46 + 51 * q^47 + 67 * q^48 + 53 * q^49 + 32 * q^50 + 46 * q^51 + 10 * q^52 + 104 * q^53 + 11 * q^54 + 11 * q^55 + 30 * q^56 + 14 * q^57 + 4 * q^58 + 36 * q^59 + 53 * q^60 + 3 * q^61 + 25 * q^62 + 53 * q^63 + 82 * q^64 + 46 * q^65 + 13 * q^66 + 54 * q^67 + 88 * q^68 + 68 * q^69 + 2 * q^70 + 101 * q^71 + 30 * q^72 + q^73 + 32 * q^74 + 71 * q^75 - 35 * q^76 + 46 * q^77 + 11 * q^78 + 14 * q^79 + 39 * q^80 + 53 * q^81 - 29 * q^82 + 38 * q^83 + 63 * q^84 + 16 * q^85 + 23 * q^86 + 55 * q^87 - 8 * q^88 + 52 * q^89 + 2 * q^90 + 32 * q^91 + 76 * q^92 - 2 * q^93 - 53 * q^94 + 46 * q^95 + 51 * q^96 - 3 * q^97 + 11 * q^98 + 46 * q^99

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