LMFDB - Embedded newform 8038.2.a.c.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.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:	$64.1837531447$
Analytic rank:	$1$
Dimension:	$84$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.71043 q^{3} +1.00000 q^{4} +1.06338 q^{5} +2.71043 q^{6} +1.86356 q^{7} -1.00000 q^{8} +4.34646 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.71043 q^{3} +1.00000 q^{4} +1.06338 q^{5} +2.71043 q^{6} +1.86356 q^{7} -1.00000 q^{8} +4.34646 q^{9} -1.06338 q^{10} +3.09787 q^{11} -2.71043 q^{12} -6.26833 q^{13} -1.86356 q^{14} -2.88221 q^{15} +1.00000 q^{16} +6.57064 q^{17} -4.34646 q^{18} -0.849992 q^{19} +1.06338 q^{20} -5.05105 q^{21} -3.09787 q^{22} -0.270544 q^{23} +2.71043 q^{24} -3.86923 q^{25} +6.26833 q^{26} -3.64948 q^{27} +1.86356 q^{28} +9.10439 q^{29} +2.88221 q^{30} -9.85940 q^{31} -1.00000 q^{32} -8.39659 q^{33} -6.57064 q^{34} +1.98166 q^{35} +4.34646 q^{36} -2.63446 q^{37} +0.849992 q^{38} +16.9899 q^{39} -1.06338 q^{40} +7.17827 q^{41} +5.05105 q^{42} +9.97111 q^{43} +3.09787 q^{44} +4.62192 q^{45} +0.270544 q^{46} -5.13007 q^{47} -2.71043 q^{48} -3.52715 q^{49} +3.86923 q^{50} -17.8093 q^{51} -6.26833 q^{52} -8.64281 q^{53} +3.64948 q^{54} +3.29421 q^{55} -1.86356 q^{56} +2.30385 q^{57} -9.10439 q^{58} +5.03485 q^{59} -2.88221 q^{60} -10.9833 q^{61} +9.85940 q^{62} +8.09987 q^{63} +1.00000 q^{64} -6.66560 q^{65} +8.39659 q^{66} -9.50889 q^{67} +6.57064 q^{68} +0.733291 q^{69} -1.98166 q^{70} +8.31024 q^{71} -4.34646 q^{72} -8.35978 q^{73} +2.63446 q^{74} +10.4873 q^{75} -0.849992 q^{76} +5.77307 q^{77} -16.9899 q^{78} -7.14604 q^{79} +1.06338 q^{80} -3.14769 q^{81} -7.17827 q^{82} +2.45042 q^{83} -5.05105 q^{84} +6.98707 q^{85} -9.97111 q^{86} -24.6769 q^{87} -3.09787 q^{88} -3.07992 q^{89} -4.62192 q^{90} -11.6814 q^{91} -0.270544 q^{92} +26.7233 q^{93} +5.13007 q^{94} -0.903862 q^{95} +2.71043 q^{96} +3.31786 q^{97} +3.52715 q^{98} +13.4648 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9}+O(q^{10}) $ 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9} + 32 q^{10} - 6 q^{11} - 19 q^{12} - 29 q^{13} - q^{14} - 4 q^{15} + 84 q^{16} - 36 q^{17} - 77 q^{18} + 33 q^{19} - 32 q^{20} - 21 q^{21} + 6 q^{22} - 62 q^{23} + 19 q^{24} + 82 q^{25} + 29 q^{26} - 82 q^{27} + q^{28} - 51 q^{29} + 4 q^{30} + 39 q^{31} - 84 q^{32} - 32 q^{33} + 36 q^{34} - 34 q^{35} + 77 q^{36} - 32 q^{37} - 33 q^{38} + 29 q^{39} + 32 q^{40} - 38 q^{41} + 21 q^{42} - 6 q^{44} - 91 q^{45} + 62 q^{46} - 58 q^{47} - 19 q^{48} + 83 q^{49} - 82 q^{50} - q^{51} - 29 q^{52} - 106 q^{53} + 82 q^{54} + 32 q^{55} - q^{56} - 44 q^{57} + 51 q^{58} - 42 q^{59} - 4 q^{60} - 41 q^{61} - 39 q^{62} - 9 q^{63} + 84 q^{64} - 49 q^{65} + 32 q^{66} - 16 q^{67} - 36 q^{68} - 45 q^{69} + 34 q^{70} - 62 q^{71} - 77 q^{72} - 16 q^{73} + 32 q^{74} - 80 q^{75} + 33 q^{76} - 134 q^{77} - 29 q^{78} + 53 q^{79} - 32 q^{80} + 56 q^{81} + 38 q^{82} - 90 q^{83} - 21 q^{84} - 60 q^{85} - 3 q^{87} + 6 q^{88} - 54 q^{89} + 91 q^{90} + 33 q^{91} - 62 q^{92} - 69 q^{93} + 58 q^{94} - 47 q^{95} + 19 q^{96} - 31 q^{97} - 83 q^{98} + 23 q^{99}+O(q^{100}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9 + 32 * q^10 - 6 * q^11 - 19 * q^12 - 29 * q^13 - q^14 - 4 * q^15 + 84 * q^16 - 36 * q^17 - 77 * q^18 + 33 * q^19 - 32 * q^20 - 21 * q^21 + 6 * q^22 - 62 * q^23 + 19 * q^24 + 82 * q^25 + 29 * q^26 - 82 * q^27 + q^28 - 51 * q^29 + 4 * q^30 + 39 * q^31 - 84 * q^32 - 32 * q^33 + 36 * q^34 - 34 * q^35 + 77 * q^36 - 32 * q^37 - 33 * q^38 + 29 * q^39 + 32 * q^40 - 38 * q^41 + 21 * q^42 - 6 * q^44 - 91 * q^45 + 62 * q^46 - 58 * q^47 - 19 * q^48 + 83 * q^49 - 82 * q^50 - q^51 - 29 * q^52 - 106 * q^53 + 82 * q^54 + 32 * q^55 - q^56 - 44 * q^57 + 51 * q^58 - 42 * q^59 - 4 * q^60 - 41 * q^61 - 39 * q^62 - 9 * q^63 + 84 * q^64 - 49 * q^65 + 32 * q^66 - 16 * q^67 - 36 * q^68 - 45 * q^69 + 34 * q^70 - 62 * q^71 - 77 * q^72 - 16 * q^73 + 32 * q^74 - 80 * q^75 + 33 * q^76 - 134 * q^77 - 29 * q^78 + 53 * q^79 - 32 * q^80 + 56 * q^81 + 38 * q^82 - 90 * q^83 - 21 * q^84 - 60 * q^85 - 3 * q^87 + 6 * q^88 - 54 * q^89 + 91 * q^90 + 33 * q^91 - 62 * q^92 - 69 * q^93 + 58 * q^94 - 47 * q^95 + 19 * q^96 - 31 * q^97 - 83 * q^98 + 23 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−2.71043	−1.56487	−0.782435	−	0.622732i	$-0.786023\pi$
$3$	−2.71043	−1.56487	−0.782435	+	0.622732i	$0.786023\pi$
$4$	1.00000	0.500000
$5$	1.06338	0.475557	0.237778	−	0.971319i	$-0.423581\pi$
$5$	1.06338	0.475557	0.237778	+	0.971319i	$0.423581\pi$
$6$	2.71043	1.10653
$7$	1.86356	0.704358	0.352179	−	0.935933i	$-0.385441\pi$
$7$	1.86356	0.704358	0.352179	+	0.935933i	$0.385441\pi$
$8$	−1.00000	−0.353553
$9$	4.34646	1.44882
$10$	−1.06338	−0.336269
$11$	3.09787	0.934044	0.467022	−	0.884246i	$-0.345327\pi$
$11$	3.09787	0.934044	0.467022	+	0.884246i	$0.345327\pi$
$12$	−2.71043	−0.782435
$13$	−6.26833	−1.73852	−0.869261	−	0.494354i	$-0.835405\pi$
$13$	−6.26833	−1.73852	−0.869261	+	0.494354i	$0.835405\pi$
$14$	−1.86356	−0.498057
$15$	−2.88221	−0.744185
$16$	1.00000	0.250000
$17$	6.57064	1.59361	0.796807	−	0.604233i	$-0.206521\pi$
$17$	6.57064	1.59361	0.796807	+	0.604233i	$0.206521\pi$
$18$	−4.34646	−1.02447
$19$	−0.849992	−0.195002	−0.0975008	−	0.995235i	$-0.531085\pi$
$19$	−0.849992	−0.195002	−0.0975008	+	0.995235i	$0.531085\pi$
$20$	1.06338	0.237778
$21$	−5.05105	−1.10223
$22$	−3.09787	−0.660469
$23$	−0.270544	−0.0564123	−0.0282061	−	0.999602i	$-0.508979\pi$
$23$	−0.270544	−0.0564123	−0.0282061	+	0.999602i	$0.508979\pi$
$24$	2.71043	0.553265
$25$	−3.86923	−0.773846
$26$	6.26833	1.22932
$27$	−3.64948	−0.702343
$28$	1.86356	0.352179
$29$	9.10439	1.69064	0.845321	−	0.534258i	$-0.179409\pi$
$29$	9.10439	1.69064	0.845321	+	0.534258i	$0.179409\pi$
$30$	2.88221	0.526218
$31$	−9.85940	−1.77080	−0.885400	−	0.464830i	$-0.846116\pi$
$31$	−9.85940	−1.77080	−0.885400	+	0.464830i	$0.846116\pi$
$32$	−1.00000	−0.176777
$33$	−8.39659	−1.46166
$34$	−6.57064	−1.12686
$35$	1.98166	0.334962
$36$	4.34646	0.724409
$37$	−2.63446	−0.433103	−0.216552	−	0.976271i	$-0.569481\pi$
$37$	−2.63446	−0.433103	−0.216552	+	0.976271i	$0.569481\pi$
$38$	0.849992	0.137887
$39$	16.9899	2.72056
$40$	−1.06338	−0.168135
$41$	7.17827	1.12106	0.560528	−	0.828135i	$-0.310598\pi$
$41$	7.17827	1.12106	0.560528	+	0.828135i	$0.310598\pi$
$42$	5.05105	0.779394
$43$	9.97111	1.52058	0.760290	−	0.649584i	$-0.225057\pi$
$43$	9.97111	1.52058	0.760290	+	0.649584i	$0.225057\pi$
$44$	3.09787	0.467022
$45$	4.62192	0.688996
$46$	0.270544	0.0398895
$47$	−5.13007	−0.748298	−0.374149	−	0.927369i	$-0.622065\pi$
$47$	−5.13007	−0.748298	−0.374149	+	0.927369i	$0.622065\pi$
$48$	−2.71043	−0.391218
$49$	−3.52715	−0.503879
$50$	3.86923	0.547192
$51$	−17.8093	−2.49380
$52$	−6.26833	−0.869261
$53$	−8.64281	−1.18718	−0.593591	−	0.804767i	$-0.702290\pi$
$53$	−8.64281	−1.18718	−0.593591	+	0.804767i	$0.702290\pi$
$54$	3.64948	0.496631
$55$	3.29421	0.444191
$56$	−1.86356	−0.249028
$57$	2.30385	0.305152
$58$	−9.10439	−1.19546
$59$	5.03485	0.655481	0.327741	−	0.944768i	$-0.393713\pi$
$59$	5.03485	0.655481	0.327741	+	0.944768i	$0.393713\pi$
$60$	−2.88221	−0.372092
$61$	−10.9833	−1.40627	−0.703136	−	0.711055i	$-0.748218\pi$
$61$	−10.9833	−1.40627	−0.703136	+	0.711055i	$0.748218\pi$
$62$	9.85940	1.25214
$63$	8.09987	1.02049
$64$	1.00000	0.125000
$65$	−6.66560	−0.826766
$66$	8.39659	1.03355
$67$	−9.50889	−1.16170	−0.580848	−	0.814012i	$-0.697279\pi$
$67$	−9.50889	−1.16170	−0.580848	+	0.814012i	$0.697279\pi$
$68$	6.57064	0.796807
$69$	0.733291	0.0882779
$70$	−1.98166	−0.236854
$71$	8.31024	0.986244	0.493122	−	0.869960i	$-0.335856\pi$
$71$	8.31024	0.986244	0.493122	+	0.869960i	$0.335856\pi$
$72$	−4.34646	−0.512235
$73$	−8.35978	−0.978439	−0.489219	−	0.872161i	$-0.662718\pi$
$73$	−8.35978	−0.978439	−0.489219	+	0.872161i	$0.662718\pi$
$74$	2.63446	0.306250
$75$	10.4873	1.21097
$76$	−0.849992	−0.0975008
$77$	5.77307	0.657902
$78$	−16.9899	−1.92373
$79$	−7.14604	−0.803992	−0.401996	−	0.915641i	$-0.631683\pi$
$79$	−7.14604	−0.803992	−0.401996	+	0.915641i	$0.631683\pi$
$80$	1.06338	0.118889
$81$	−3.14769	−0.349743
$82$	−7.17827	−0.792707
$83$	2.45042	0.268968	0.134484	−	0.990916i	$-0.457062\pi$
$83$	2.45042	0.268968	0.134484	+	0.990916i	$0.457062\pi$
$84$	−5.05105	−0.551115
$85$	6.98707	0.757854
$86$	−9.97111	−1.07521
$87$	−24.6769	−2.64564
$88$	−3.09787	−0.330235
$89$	−3.07992	−0.326471	−0.163236	−	0.986587i	$-0.552193\pi$
$89$	−3.07992	−0.326471	−0.163236	+	0.986587i	$0.552193\pi$
$90$	−4.62192	−0.487193
$91$	−11.6814	−1.22454
$92$	−0.270544	−0.0282061
$93$	26.7233	2.77107
$94$	5.13007	0.529127
$95$	−0.903862	−0.0927343
$96$	2.71043	0.276633
$97$	3.31786	0.336877	0.168439	−	0.985712i	$-0.446127\pi$
$97$	3.31786	0.336877	0.168439	+	0.985712i	$0.446127\pi$
$98$	3.52715	0.356296
$99$	13.4648	1.35326
$100$	−3.86923	−0.386923
$101$	−6.58211	−0.654944	−0.327472	−	0.944861i	$-0.606197\pi$
$101$	−6.58211	−0.654944	−0.327472	+	0.944861i	$0.606197\pi$
$102$	17.8093	1.76338
$103$	−3.75855	−0.370341	−0.185171	−	0.982706i	$-0.559284\pi$
$103$	−3.75855	−0.370341	−0.185171	+	0.982706i	$0.559284\pi$
$104$	6.26833	0.614660
$105$	−5.37117	−0.524173
$106$	8.64281	0.839464
$107$	−6.12568	−0.592192	−0.296096	−	0.955158i	$-0.595685\pi$
$107$	−6.12568	−0.592192	−0.296096	+	0.955158i	$0.595685\pi$
$108$	−3.64948	−0.351171
$109$	−8.20635	−0.786026	−0.393013	−	0.919533i	$-0.628567\pi$
$109$	−8.20635	−0.786026	−0.393013	+	0.919533i	$0.628567\pi$
$110$	−3.29421	−0.314091
$111$	7.14054	0.677750
$112$	1.86356	0.176090
$113$	7.60779	0.715680	0.357840	−	0.933783i	$-0.383513\pi$
$113$	7.60779	0.715680	0.357840	+	0.933783i	$0.383513\pi$
$114$	−2.30385	−0.215775
$115$	−0.287690	−0.0268273
$116$	9.10439	0.845321
$117$	−27.2450	−2.51880
$118$	−5.03485	−0.463495
$119$	12.2448	1.12248
$120$	2.88221	0.263109
$121$	−1.40317	−0.127561
$122$	10.9833	0.994385
$123$	−19.4562	−1.75431
$124$	−9.85940	−0.885400
$125$	−9.43134	−0.843564
$126$	−8.09987	−0.721594
$127$	3.45787	0.306836	0.153418	−	0.988161i	$-0.450972\pi$
$127$	3.45787	0.306836	0.153418	+	0.988161i	$0.450972\pi$
$128$	−1.00000	−0.0883883
$129$	−27.0260	−2.37951
$130$	6.66560	0.584612
$131$	−17.6890	−1.54550	−0.772749	−	0.634712i	$-0.781119\pi$
$131$	−17.6890	−1.54550	−0.772749	+	0.634712i	$0.781119\pi$
$132$	−8.39659	−0.730829
$133$	−1.58401	−0.137351
$134$	9.50889	0.821443
$135$	−3.88077	−0.334004
$136$	−6.57064	−0.563428
$137$	6.66978	0.569838	0.284919	−	0.958552i	$-0.408033\pi$
$137$	6.66978	0.569838	0.284919	+	0.958552i	$0.408033\pi$
$138$	−0.733291	−0.0624219
$139$	2.59234	0.219879	0.109939	−	0.993938i	$-0.464934\pi$
$139$	2.59234	0.219879	0.109939	+	0.993938i	$0.464934\pi$
$140$	1.98166	0.167481
$141$	13.9047	1.17099
$142$	−8.31024	−0.697380
$143$	−19.4185	−1.62386
$144$	4.34646	0.362205
$145$	9.68140	0.803997
$146$	8.35978	0.691861
$147$	9.56012	0.788506
$148$	−2.63446	−0.216552
$149$	7.63303	0.625322	0.312661	−	0.949865i	$-0.398780\pi$
$149$	7.63303	0.625322	0.312661	+	0.949865i	$0.398780\pi$
$150$	−10.4873	−0.856284
$151$	11.5850	0.942774	0.471387	−	0.881926i	$-0.343753\pi$
$151$	11.5850	0.942774	0.471387	+	0.881926i	$0.343753\pi$
$152$	0.849992	0.0689434
$153$	28.5590	2.30886
$154$	−5.77307	−0.465207
$155$	−10.4843	−0.842116
$156$	16.9899	1.36028
$157$	−9.52836	−0.760446	−0.380223	−	0.924895i	$-0.624153\pi$
$157$	−9.52836	−0.760446	−0.380223	+	0.924895i	$0.624153\pi$
$158$	7.14604	0.568508
$159$	23.4258	1.85779
$160$	−1.06338	−0.0840674
$161$	−0.504174	−0.0397345
$162$	3.14769	0.247306
$163$	−18.4703	−1.44671	−0.723354	−	0.690477i	$-0.757401\pi$
$163$	−18.4703	−1.44671	−0.723354	+	0.690477i	$0.757401\pi$
$164$	7.17827	0.560528
$165$	−8.92874	−0.695102
$166$	−2.45042	−0.190189
$167$	−7.56596	−0.585472	−0.292736	−	0.956193i	$-0.594566\pi$
$167$	−7.56596	−0.585472	−0.292736	+	0.956193i	$0.594566\pi$
$168$	5.05105	0.389697
$169$	26.2919	2.02246
$170$	−6.98707	−0.535884
$171$	−3.69445	−0.282522
$172$	9.97111	0.760290
$173$	−20.7948	−1.58100	−0.790500	−	0.612461i	$-0.790179\pi$
$173$	−20.7948	−1.58100	−0.790500	+	0.612461i	$0.790179\pi$
$174$	24.6769	1.87075
$175$	−7.21053	−0.545065
$176$	3.09787	0.233511
$177$	−13.6466	−1.02574
$178$	3.07992	0.230850
$179$	14.1701	1.05912	0.529562	−	0.848271i	$-0.322356\pi$
$179$	14.1701	1.05912	0.529562	+	0.848271i	$0.322356\pi$
$180$	4.62192	0.344498
$181$	18.6843	1.38880	0.694398	−	0.719591i	$-0.255671\pi$
$181$	18.6843	1.38880	0.694398	+	0.719591i	$0.255671\pi$
$182$	11.6814	0.865882
$183$	29.7696	2.20063
$184$	0.270544	0.0199448
$185$	−2.80143	−0.205965
$186$	−26.7233	−1.95944
$187$	20.3550	1.48851
$188$	−5.13007	−0.374149
$189$	−6.80102	−0.494701
$190$	0.903862	0.0655730
$191$	8.58204	0.620975	0.310487	−	0.950578i	$-0.399508\pi$
$191$	8.58204	0.620975	0.310487	+	0.950578i	$0.399508\pi$
$192$	−2.71043	−0.195609
$193$	6.60065	0.475126	0.237563	−	0.971372i	$-0.423651\pi$
$193$	6.60065	0.475126	0.237563	+	0.971372i	$0.423651\pi$
$194$	−3.31786	−0.238208
$195$	18.0667	1.29378
$196$	−3.52715	−0.251940
$197$	23.3022	1.66022	0.830108	−	0.557602i	$-0.188279\pi$
$197$	23.3022	1.66022	0.830108	+	0.557602i	$0.188279\pi$
$198$	−13.4648	−0.956900
$199$	7.62372	0.540431	0.270216	−	0.962800i	$-0.412905\pi$
$199$	7.62372	0.540431	0.270216	+	0.962800i	$0.412905\pi$
$200$	3.86923	0.273596
$201$	25.7732	1.81790
$202$	6.58211	0.463115
$203$	16.9666	1.19082
$204$	−17.8093	−1.24690
$205$	7.63321	0.533126
$206$	3.75855	0.261871
$207$	−1.17591	−0.0817312
$208$	−6.26833	−0.434630
$209$	−2.63317	−0.182140
$210$	5.37117	0.370646
$211$	18.4791	1.27215	0.636077	−	0.771626i	$-0.280556\pi$
$211$	18.4791	1.27215	0.636077	+	0.771626i	$0.280556\pi$
$212$	−8.64281	−0.593591
$213$	−22.5244	−1.54334
$214$	6.12568	0.418743
$215$	10.6031	0.723122
$216$	3.64948	0.248316
$217$	−18.3736	−1.24728
$218$	8.20635	0.555804
$219$	22.6586	1.53113
$220$	3.29421	0.222096
$221$	−41.1869	−2.77053
$222$	−7.14054	−0.479242
$223$	−22.4092	−1.50063	−0.750314	−	0.661081i	$-0.770098\pi$
$223$	−22.4092	−1.50063	−0.750314	+	0.661081i	$0.770098\pi$
$224$	−1.86356	−0.124514
$225$	−16.8174	−1.12116
$226$	−7.60779	−0.506062
$227$	−2.27015	−0.150675	−0.0753375	−	0.997158i	$-0.524003\pi$
$227$	−2.27015	−0.150675	−0.0753375	+	0.997158i	$0.524003\pi$
$228$	2.30385	0.152576
$229$	1.99611	0.131906	0.0659532	−	0.997823i	$-0.478991\pi$
$229$	1.99611	0.131906	0.0659532	+	0.997823i	$0.478991\pi$
$230$	0.287690	0.0189697
$231$	−15.6475	−1.02953
$232$	−9.10439	−0.597732
$233$	−0.163947	−0.0107406	−0.00537028	−	0.999986i	$-0.501709\pi$
$233$	−0.163947	−0.0107406	−0.00537028	+	0.999986i	$0.501709\pi$
$234$	27.2450	1.78106
$235$	−5.45520	−0.355858
$236$	5.03485	0.327741
$237$	19.3689	1.25814
$238$	−12.2448	−0.793710
$239$	13.3738	0.865076	0.432538	−	0.901616i	$-0.357618\pi$
$239$	13.3738	0.865076	0.432538	+	0.901616i	$0.357618\pi$
$240$	−2.88221	−0.186046
$241$	−25.5540	−1.64608	−0.823039	−	0.567985i	$-0.807723\pi$
$241$	−25.5540	−1.64608	−0.823039	+	0.567985i	$0.807723\pi$
$242$	1.40317	0.0901993
$243$	19.4800	1.24965
$244$	−10.9833	−0.703136
$245$	−3.75070	−0.239623
$246$	19.4562	1.24048
$247$	5.32803	0.339014
$248$	9.85940	0.626072
$249$	−6.64169	−0.420900
$250$	9.43134	0.596490
$251$	−21.1415	−1.33444	−0.667220	−	0.744861i	$-0.732516\pi$
$251$	−21.1415	−1.33444	−0.667220	+	0.744861i	$0.732516\pi$
$252$	8.09987	0.510244
$253$	−0.838111	−0.0526916
$254$	−3.45787	−0.216966
$255$	−18.9380	−1.18594
$256$	1.00000	0.0625000
$257$	1.56557	0.0976574	0.0488287	−	0.998807i	$-0.484451\pi$
$257$	1.56557	0.0976574	0.0488287	+	0.998807i	$0.484451\pi$
$258$	27.0260	1.68257
$259$	−4.90947	−0.305060
$260$	−6.66560	−0.413383
$261$	39.5718	2.44943
$262$	17.6890	1.09283
$263$	−8.77079	−0.540830	−0.270415	−	0.962744i	$-0.587161\pi$
$263$	−8.77079	−0.540830	−0.270415	+	0.962744i	$0.587161\pi$
$264$	8.39659	0.516774
$265$	−9.19057	−0.564572
$266$	1.58401	0.0971218
$267$	8.34793	0.510885
$268$	−9.50889	−0.580848
$269$	10.1698	0.620065	0.310032	−	0.950726i	$-0.399660\pi$
$269$	10.1698	0.620065	0.310032	+	0.950726i	$0.399660\pi$
$270$	3.88077	0.236176
$271$	28.8219	1.75080	0.875402	−	0.483395i	$-0.160597\pi$
$271$	28.8219	1.75080	0.875402	+	0.483395i	$0.160597\pi$
$272$	6.57064	0.398404
$273$	31.6616	1.91625
$274$	−6.66978	−0.402936
$275$	−11.9864	−0.722806
$276$	0.733291	0.0441390
$277$	−10.0398	−0.603232	−0.301616	−	0.953429i	$-0.597526\pi$
$277$	−10.0398	−0.603232	−0.301616	+	0.953429i	$0.597526\pi$
$278$	−2.59234	−0.155478
$279$	−42.8534	−2.56557
$280$	−1.98166	−0.118427
$281$	−16.0888	−0.959780	−0.479890	−	0.877329i	$-0.659323\pi$
$281$	−16.0888	−0.959780	−0.479890	+	0.877329i	$0.659323\pi$
$282$	−13.9047	−0.828014
$283$	28.2850	1.68137	0.840685	−	0.541525i	$-0.182153\pi$
$283$	28.2850	1.68137	0.840685	+	0.541525i	$0.182153\pi$
$284$	8.31024	0.493122
$285$	2.44986	0.145117
$286$	19.4185	1.14824
$287$	13.3771	0.789626
$288$	−4.34646	−0.256117
$289$	26.1733	1.53961
$290$	−9.68140	−0.568512
$291$	−8.99284	−0.527169
$292$	−8.35978	−0.489219
$293$	−14.6485	−0.855777	−0.427889	−	0.903832i	$-0.640742\pi$
$293$	−14.6485	−0.855777	−0.427889	+	0.903832i	$0.640742\pi$
$294$	−9.56012	−0.557558
$295$	5.35394	0.311719
$296$	2.63446	0.153125
$297$	−11.3056	−0.656019
$298$	−7.63303	−0.442169
$299$	1.69586	0.0980740
$300$	10.4873	0.605484
$301$	18.5817	1.07103
$302$	−11.5850	−0.666642
$303$	17.8404	1.02490
$304$	−0.849992	−0.0487504
$305$	−11.6794	−0.668763
$306$	−28.5590	−1.63261
$307$	−32.7968	−1.87181	−0.935907	−	0.352247i	$-0.885418\pi$
$307$	−32.7968	−1.87181	−0.935907	+	0.352247i	$0.885418\pi$
$308$	5.77307	0.328951
$309$	10.1873	0.579536
$310$	10.4843	0.595466
$311$	31.0234	1.75917	0.879587	−	0.475739i	$-0.157819\pi$
$311$	31.0234	1.75917	0.879587	+	0.475739i	$0.157819\pi$
$312$	−16.9899	−0.961863
$313$	−8.48590	−0.479651	−0.239826	−	0.970816i	$-0.577090\pi$
$313$	−8.48590	−0.479651	−0.239826	+	0.970816i	$0.577090\pi$
$314$	9.52836	0.537717
$315$	8.61322	0.485300
$316$	−7.14604	−0.401996
$317$	−25.0837	−1.40884	−0.704419	−	0.709784i	$-0.748793\pi$
$317$	−25.0837	−1.40884	−0.704419	+	0.709784i	$0.748793\pi$
$318$	−23.4258	−1.31365
$319$	28.2043	1.57914
$320$	1.06338	0.0594446
$321$	16.6033	0.926704
$322$	0.504174	0.0280965
$323$	−5.58499	−0.310757
$324$	−3.14769	−0.174872
$325$	24.2536	1.34535
$326$	18.4703	1.02298
$327$	22.2428	1.23003
$328$	−7.17827	−0.396353
$329$	−9.56018	−0.527070
$330$	8.92874	0.491511
$331$	3.50865	0.192853	0.0964265	−	0.995340i	$-0.469259\pi$
$331$	3.50865	0.192853	0.0964265	+	0.995340i	$0.469259\pi$
$332$	2.45042	0.134484
$333$	−11.4506	−0.627488
$334$	7.56596	0.413991
$335$	−10.1115	−0.552453
$336$	−5.05105	−0.275557
$337$	−18.2161	−0.992295	−0.496148	−	0.868238i	$-0.665253\pi$
$337$	−18.2161	−0.992295	−0.496148	+	0.868238i	$0.665253\pi$
$338$	−26.2919	−1.43009
$339$	−20.6204	−1.11995
$340$	6.98707	0.378927
$341$	−30.5432	−1.65401
$342$	3.69445	0.199773
$343$	−19.6180	−1.05927
$344$	−9.97111	−0.537606
$345$	0.779766	0.0419812
$346$	20.7948	1.11794
$347$	29.7233	1.59563	0.797815	−	0.602903i	$-0.205989\pi$
$347$	29.7233	1.59563	0.797815	+	0.602903i	$0.205989\pi$
$348$	−24.6769	−1.32282
$349$	29.4024	1.57387	0.786937	−	0.617033i	$-0.211666\pi$
$349$	29.4024	1.57387	0.786937	+	0.617033i	$0.211666\pi$
$350$	7.21053	0.385419
$351$	22.8761	1.22104
$352$	−3.09787	−0.165117
$353$	19.8979	1.05906	0.529530	−	0.848291i	$-0.322368\pi$
$353$	19.8979	1.05906	0.529530	+	0.848291i	$0.322368\pi$
$354$	13.6466	0.725310
$355$	8.83692	0.469015
$356$	−3.07992	−0.163236
$357$	−33.1886	−1.75653
$358$	−14.1701	−0.748914
$359$	21.7881	1.14993	0.574965	−	0.818178i	$-0.305016\pi$
$359$	21.7881	1.14993	0.574965	+	0.818178i	$0.305016\pi$
$360$	−4.62192	−0.243597
$361$	−18.2775	−0.961974
$362$	−18.6843	−0.982027
$363$	3.80320	0.199616
$364$	−11.6814	−0.612271
$365$	−8.88960	−0.465303
$366$	−29.7696	−1.55608
$367$	2.02654	0.105785	0.0528923	−	0.998600i	$-0.483156\pi$
$367$	2.02654	0.105785	0.0528923	+	0.998600i	$0.483156\pi$
$368$	−0.270544	−0.0141031
$369$	31.2000	1.62421
$370$	2.80143	0.145639
$371$	−16.1064	−0.836202
$372$	26.7233	1.38554
$373$	−23.0535	−1.19366	−0.596832	−	0.802366i	$-0.703574\pi$
$373$	−23.0535	−1.19366	−0.596832	+	0.802366i	$0.703574\pi$
$374$	−20.3550	−1.05253
$375$	25.5630	1.32007
$376$	5.13007	0.264563
$377$	−57.0693	−2.93922
$378$	6.80102	0.349806
$379$	−19.4627	−0.999730	−0.499865	−	0.866103i	$-0.666617\pi$
$379$	−19.4627	−0.999730	−0.499865	+	0.866103i	$0.666617\pi$
$380$	−0.903862	−0.0463671
$381$	−9.37233	−0.480159
$382$	−8.58204	−0.439095
$383$	−5.11988	−0.261614	−0.130807	−	0.991408i	$-0.541757\pi$
$383$	−5.11988	−0.261614	−0.130807	+	0.991408i	$0.541757\pi$
$384$	2.71043	0.138316
$385$	6.13895	0.312870
$386$	−6.60065	−0.335964
$387$	43.3390	2.20305
$388$	3.31786	0.168439
$389$	−1.36383	−0.0691487	−0.0345744	−	0.999402i	$-0.511008\pi$
$389$	−1.36383	−0.0691487	−0.0345744	+	0.999402i	$0.511008\pi$
$390$	−18.0667	−0.914841
$391$	−1.77765	−0.0898995
$392$	3.52715	0.178148
$393$	47.9449	2.41850
$394$	−23.3022	−1.17395
$395$	−7.59893	−0.382344
$396$	13.4648	0.676630
$397$	−10.8409	−0.544090	−0.272045	−	0.962285i	$-0.587700\pi$
$397$	−10.8409	−0.544090	−0.272045	+	0.962285i	$0.587700\pi$
$398$	−7.62372	−0.382142
$399$	4.29335	0.214936
$400$	−3.86923	−0.193461
$401$	7.94722	0.396865	0.198432	−	0.980115i	$-0.436415\pi$
$401$	7.94722	0.396865	0.198432	+	0.980115i	$0.436415\pi$
$402$	−25.7732	−1.28545
$403$	61.8020	3.07857
$404$	−6.58211	−0.327472
$405$	−3.34718	−0.166323
$406$	−16.9666	−0.842036
$407$	−8.16124	−0.404538
$408$	17.8093	0.881691
$409$	−27.0399	−1.33704	−0.668519	−	0.743695i	$-0.733072\pi$
$409$	−27.0399	−1.33704	−0.668519	+	0.743695i	$0.733072\pi$
$410$	−7.63321	−0.376977
$411$	−18.0780	−0.891722
$412$	−3.75855	−0.185171
$413$	9.38273	0.461694
$414$	1.17591	0.0577927
$415$	2.60572	0.127910
$416$	6.26833	0.307330
$417$	−7.02635	−0.344082
$418$	2.63317	0.128792
$419$	18.1270	0.885563	0.442781	−	0.896630i	$-0.353992\pi$
$419$	18.1270	0.885563	0.442781	+	0.896630i	$0.353992\pi$
$420$	−5.37117	−0.262086
$421$	−33.8752	−1.65098	−0.825488	−	0.564419i	$-0.809100\pi$
$421$	−33.8752	−1.65098	−0.825488	+	0.564419i	$0.809100\pi$
$422$	−18.4791	−0.899549
$423$	−22.2976	−1.08415
$424$	8.64281	0.419732
$425$	−25.4233	−1.23321
$426$	22.5244	1.09131
$427$	−20.4681	−0.990520
$428$	−6.12568	−0.296096
$429$	52.6326	2.54112
$430$	−10.6031	−0.511325
$431$	−1.06105	−0.0511090	−0.0255545	−	0.999673i	$-0.508135\pi$
$431$	−1.06105	−0.0511090	−0.0255545	+	0.999673i	$0.508135\pi$
$432$	−3.64948	−0.175586
$433$	23.1207	1.11111	0.555554	−	0.831480i	$-0.312506\pi$
$433$	23.1207	1.11111	0.555554	+	0.831480i	$0.312506\pi$
$434$	18.3736	0.881959
$435$	−26.2408	−1.25815
$436$	−8.20635	−0.393013
$437$	0.229960	0.0110005
$438$	−22.6586	−1.08267
$439$	−17.5575	−0.837976	−0.418988	−	0.907992i	$-0.637615\pi$
$439$	−17.5575	−0.837976	−0.418988	+	0.907992i	$0.637615\pi$
$440$	−3.29421	−0.157045
$441$	−15.3306	−0.730030
$442$	41.1869	1.95906
$443$	−9.85140	−0.468054	−0.234027	−	0.972230i	$-0.575190\pi$
$443$	−9.85140	−0.468054	−0.234027	+	0.972230i	$0.575190\pi$
$444$	7.14054	0.338875
$445$	−3.27512	−0.155256
$446$	22.4092	1.06110
$447$	−20.6888	−0.978548
$448$	1.86356	0.0880448
$449$	22.1453	1.04510	0.522550	−	0.852609i	$-0.324981\pi$
$449$	22.1453	1.04510	0.522550	+	0.852609i	$0.324981\pi$
$450$	16.8174	0.792781
$451$	22.2374	1.04712
$452$	7.60779	0.357840
$453$	−31.4004	−1.47532
$454$	2.27015	0.106543
$455$	−12.4217	−0.582339
$456$	−2.30385	−0.107888
$457$	−1.28020	−0.0598851	−0.0299425	−	0.999552i	$-0.509532\pi$
$457$	−1.28020	−0.0598851	−0.0299425	+	0.999552i	$0.509532\pi$
$458$	−1.99611	−0.0932719
$459$	−23.9794	−1.11926
$460$	−0.287690	−0.0134136
$461$	−1.36841	−0.0637330	−0.0318665	−	0.999492i	$-0.510145\pi$
$461$	−1.36841	−0.0637330	−0.0318665	+	0.999492i	$0.510145\pi$
$462$	15.6475	0.727989
$463$	22.5493	1.04795	0.523976	−	0.851733i	$-0.324448\pi$
$463$	22.5493	1.04795	0.523976	+	0.851733i	$0.324448\pi$
$464$	9.10439	0.422661
$465$	28.4169	1.31780
$466$	0.163947	0.00759472
$467$	18.5799	0.859776	0.429888	−	0.902882i	$-0.358553\pi$
$467$	18.5799	0.859776	0.429888	+	0.902882i	$0.358553\pi$
$468$	−27.2450	−1.25940
$469$	−17.7204	−0.818250
$470$	5.45520	0.251630
$471$	25.8260	1.19000
$472$	−5.03485	−0.231748
$473$	30.8893	1.42029
$474$	−19.3689	−0.889642
$475$	3.28881	0.150901
$476$	12.2448	0.561238
$477$	−37.5656	−1.72001
$478$	−13.3738	−0.611701
$479$	−22.4388	−1.02525	−0.512627	−	0.858612i	$-0.671327\pi$
$479$	−22.4388	−1.02525	−0.512627	+	0.858612i	$0.671327\pi$
$480$	2.88221	0.131555
$481$	16.5137	0.752959
$482$	25.5540	1.16395
$483$	1.36653	0.0621793
$484$	−1.40317	−0.0637805
$485$	3.52813	0.160204
$486$	−19.4800	−0.883633
$487$	1.62414	0.0735966	0.0367983	−	0.999323i	$-0.488284\pi$
$487$	1.62414	0.0735966	0.0367983	+	0.999323i	$0.488284\pi$
$488$	10.9833	0.497193
$489$	50.0627	2.26391
$490$	3.75070	0.169439
$491$	−11.7970	−0.532391	−0.266196	−	0.963919i	$-0.585767\pi$
$491$	−11.7970	−0.532391	−0.266196	+	0.963919i	$0.585767\pi$
$492$	−19.4562	−0.877154
$493$	59.8217	2.69423
$494$	−5.32803	−0.239719
$495$	14.3181	0.643552
$496$	−9.85940	−0.442700
$497$	15.4866	0.694669
$498$	6.64169	0.297621
$499$	31.9954	1.43231	0.716156	−	0.697940i	$-0.245900\pi$
$499$	31.9954	1.43231	0.716156	+	0.697940i	$0.245900\pi$
$500$	−9.43134	−0.421782
$501$	20.5070	0.916187
$502$	21.1415	0.943591
$503$	4.15491	0.185258	0.0926292	−	0.995701i	$-0.470473\pi$
$503$	4.15491	0.185258	0.0926292	+	0.995701i	$0.470473\pi$
$504$	−8.09987	−0.360797
$505$	−6.99926	−0.311463
$506$	0.838111	0.0372586
$507$	−71.2626	−3.16488
$508$	3.45787	0.153418
$509$	1.55464	0.0689083	0.0344541	−	0.999406i	$-0.489031\pi$
$509$	1.55464	0.0689083	0.0344541	+	0.999406i	$0.489031\pi$
$510$	18.9380	0.838589
$511$	−15.5789	−0.689171
$512$	−1.00000	−0.0441942
$513$	3.10203	0.136958
$514$	−1.56557	−0.0690542
$515$	−3.99676	−0.176118
$516$	−27.0260	−1.18976
$517$	−15.8923	−0.698944
$518$	4.90947	0.215710
$519$	56.3630	2.47406
$520$	6.66560	0.292306
$521$	−20.6056	−0.902750	−0.451375	−	0.892334i	$-0.649066\pi$
$521$	−20.6056	−0.902750	−0.451375	+	0.892334i	$0.649066\pi$
$522$	−39.5718	−1.73201
$523$	−34.2183	−1.49626	−0.748130	−	0.663552i	$-0.769048\pi$
$523$	−34.2183	−1.49626	−0.748130	+	0.663552i	$0.769048\pi$
$524$	−17.6890	−0.772749
$525$	19.5437	0.852956
$526$	8.77079	0.382425
$527$	−64.7826	−2.82197
$528$	−8.39659	−0.365415
$529$	−22.9268	−0.996818
$530$	9.19057	0.399213
$531$	21.8837	0.949673
$532$	−1.58401	−0.0686755
$533$	−44.9957	−1.94898
$534$	−8.34793	−0.361250
$535$	−6.51391	−0.281621
$536$	9.50889	0.410722
$537$	−38.4072	−1.65739
$538$	−10.1698	−0.438452
$539$	−10.9267	−0.470646
$540$	−3.88077	−0.167002
$541$	−15.8482	−0.681366	−0.340683	−	0.940178i	$-0.610658\pi$
$541$	−15.8482	−0.681366	−0.340683	+	0.940178i	$0.610658\pi$
$542$	−28.8219	−1.23801
$543$	−50.6427	−2.17328
$544$	−6.57064	−0.281714
$545$	−8.72645	−0.373800
$546$	−31.6616	−1.35499
$547$	−43.2967	−1.85123	−0.925616	−	0.378464i	$-0.876452\pi$
$547$	−43.2967	−1.85123	−0.925616	+	0.378464i	$0.876452\pi$
$548$	6.66978	0.284919
$549$	−47.7386	−2.03743
$550$	11.9864	0.511101
$551$	−7.73866	−0.329678
$552$	−0.733291	−0.0312110
$553$	−13.3170	−0.566299
$554$	10.0398	0.426549
$555$	7.59309	0.322309
$556$	2.59234	0.109939
$557$	−44.5019	−1.88561	−0.942803	−	0.333350i	$-0.891821\pi$
$557$	−44.5019	−1.88561	−0.942803	+	0.333350i	$0.891821\pi$
$558$	42.8534	1.81413
$559$	−62.5022	−2.64356
$560$	1.98166	0.0837406
$561$	−55.1710	−2.32932
$562$	16.0888	0.678667
$563$	−18.3422	−0.773031	−0.386515	−	0.922283i	$-0.626321\pi$
$563$	−18.3422	−0.773031	−0.386515	+	0.922283i	$0.626321\pi$
$564$	13.9047	0.585495
$565$	8.08995	0.340347
$566$	−28.2850	−1.18891
$567$	−5.86590	−0.246345
$568$	−8.31024	−0.348690
$569$	13.4149	0.562383	0.281192	−	0.959652i	$-0.409270\pi$
$569$	13.4149	0.562383	0.281192	+	0.959652i	$0.409270\pi$
$570$	−2.44986	−0.102613
$571$	−10.2451	−0.428744	−0.214372	−	0.976752i	$-0.568771\pi$
$571$	−10.2451	−0.428744	−0.214372	+	0.976752i	$0.568771\pi$
$572$	−19.4185	−0.811928
$573$	−23.2611	−0.971745
$574$	−13.3771	−0.558350
$575$	1.04680	0.0436544
$576$	4.34646	0.181102
$577$	−4.58647	−0.190937	−0.0954686	−	0.995432i	$-0.530435\pi$
$577$	−4.58647	−0.190937	−0.0954686	+	0.995432i	$0.530435\pi$
$578$	−26.1733	−1.08867
$579$	−17.8906	−0.743510
$580$	9.68140	0.401998
$581$	4.56649	0.189450
$582$	8.99284	0.372765
$583$	−26.7744	−1.10888
$584$	8.35978	0.345930
$585$	−28.9717	−1.19783
$586$	14.6485	0.605126
$587$	0.793952	0.0327699	0.0163850	−	0.999866i	$-0.494784\pi$
$587$	0.793952	0.0327699	0.0163850	+	0.999866i	$0.494784\pi$
$588$	9.56012	0.394253
$589$	8.38041	0.345309
$590$	−5.35394	−0.220418
$591$	−63.1592	−2.59802
$592$	−2.63446	−0.108276
$593$	−28.0159	−1.15047	−0.575237	−	0.817987i	$-0.695090\pi$
$593$	−28.0159	−1.15047	−0.575237	+	0.817987i	$0.695090\pi$
$594$	11.3056	0.463876
$595$	13.0208	0.533801
$596$	7.63303	0.312661
$597$	−20.6636	−0.845704
$598$	−1.69586	−0.0693488
$599$	−33.5512	−1.37086	−0.685432	−	0.728137i	$-0.740387\pi$
$599$	−33.5512	−1.37086	−0.685432	+	0.728137i	$0.740387\pi$
$600$	−10.4873	−0.428142
$601$	30.8653	1.25902	0.629511	−	0.776992i	$-0.283255\pi$
$601$	30.8653	1.25902	0.629511	+	0.776992i	$0.283255\pi$
$602$	−18.5817	−0.757335
$603$	−41.3300	−1.68309
$604$	11.5850	0.471387
$605$	−1.49210	−0.0606625
$606$	−17.8404	−0.724716
$607$	29.3295	1.19045	0.595224	−	0.803560i	$-0.297063\pi$
$607$	29.3295	1.19045	0.595224	+	0.803560i	$0.297063\pi$
$608$	0.849992	0.0344717
$609$	−45.9867	−1.86348
$610$	11.6794	0.472887
$611$	32.1570	1.30093
$612$	28.5590	1.15443
$613$	−13.2330	−0.534476	−0.267238	−	0.963631i	$-0.586111\pi$
$613$	−13.2330	−0.534476	−0.267238	+	0.963631i	$0.586111\pi$
$614$	32.7968	1.32357
$615$	−20.6893	−0.834273
$616$	−5.77307	−0.232603
$617$	−35.4681	−1.42789	−0.713945	−	0.700201i	$-0.753094\pi$
$617$	−35.4681	−1.42789	−0.713945	+	0.700201i	$0.753094\pi$
$618$	−10.1873	−0.409794
$619$	−2.35729	−0.0947474	−0.0473737	−	0.998877i	$-0.515085\pi$
$619$	−2.35729	−0.0947474	−0.0473737	+	0.998877i	$0.515085\pi$
$620$	−10.4843	−0.421058
$621$	0.987344	0.0396208
$622$	−31.0234	−1.24392
$623$	−5.73961	−0.229953
$624$	16.9899	0.680140
$625$	9.31707	0.372683
$626$	8.48590	0.339165
$627$	7.13703	0.285026
$628$	−9.52836	−0.380223
$629$	−17.3101	−0.690200
$630$	−8.61322	−0.343159
$631$	18.8449	0.750203	0.375101	−	0.926984i	$-0.377608\pi$
$631$	18.8449	0.750203	0.375101	+	0.926984i	$0.377608\pi$
$632$	7.14604	0.284254
$633$	−50.0864	−1.99076
$634$	25.0837	0.996200
$635$	3.67702	0.145918
$636$	23.4258	0.928893
$637$	22.1094	0.876005
$638$	−28.2043	−1.11662
$639$	36.1201	1.42889
$640$	−1.06338	−0.0420337
$641$	−40.2214	−1.58865	−0.794325	−	0.607493i	$-0.792175\pi$
$641$	−40.2214	−1.58865	−0.794325	+	0.607493i	$0.792175\pi$
$642$	−16.6033	−0.655279
$643$	1.92738	0.0760085	0.0380043	−	0.999278i	$-0.487900\pi$
$643$	1.92738	0.0760085	0.0380043	+	0.999278i	$0.487900\pi$
$644$	−0.504174	−0.0198672
$645$	−28.7389	−1.13159
$646$	5.58499	0.219739
$647$	37.5222	1.47515	0.737575	−	0.675266i	$-0.235971\pi$
$647$	37.5222	1.47515	0.737575	+	0.675266i	$0.235971\pi$
$648$	3.14769	0.123653
$649$	15.5973	0.612249
$650$	−24.2536	−0.951304
$651$	49.8003	1.95183
$652$	−18.4703	−0.723354
$653$	−27.9507	−1.09380	−0.546899	−	0.837199i	$-0.684192\pi$
$653$	−27.9507	−1.09380	−0.546899	+	0.837199i	$0.684192\pi$
$654$	−22.2428	−0.869761
$655$	−18.8101	−0.734972
$656$	7.17827	0.280264
$657$	−36.3354	−1.41758
$658$	9.56018	0.372695
$659$	−18.5999	−0.724549	−0.362275	−	0.932071i	$-0.618000\pi$
$659$	−18.5999	−0.724549	−0.362275	+	0.932071i	$0.618000\pi$
$660$	−8.92874	−0.347551
$661$	−10.8869	−0.423451	−0.211725	−	0.977329i	$-0.567908\pi$
$661$	−10.8869	−0.423451	−0.211725	+	0.977329i	$0.567908\pi$
$662$	−3.50865	−0.136368
$663$	111.635	4.33553
$664$	−2.45042	−0.0950946
$665$	−1.68440	−0.0653182
$666$	11.4506	0.443701
$667$	−2.46314	−0.0953730
$668$	−7.56596	−0.292736
$669$	60.7386	2.34829
$670$	10.1115	0.390643
$671$	−34.0250	−1.31352
$672$	5.05105	0.194848
$673$	20.8374	0.803222	0.401611	−	0.915810i	$-0.368450\pi$
$673$	20.8374	0.803222	0.401611	+	0.915810i	$0.368450\pi$
$674$	18.2161	0.701659
$675$	14.1207	0.543505
$676$	26.2919	1.01123
$677$	−46.5902	−1.79061	−0.895304	−	0.445456i	$-0.853042\pi$
$677$	−46.5902	−1.79061	−0.895304	+	0.445456i	$0.853042\pi$
$678$	20.6204	0.791922
$679$	6.18302	0.237282
$680$	−6.98707	−0.267942
$681$	6.15309	0.235787
$682$	30.5432	1.16956
$683$	12.5568	0.480474	0.240237	−	0.970714i	$-0.422775\pi$
$683$	12.5568	0.480474	0.240237	+	0.970714i	$0.422775\pi$
$684$	−3.69445	−0.141261
$685$	7.09249	0.270990
$686$	19.6180	0.749017
$687$	−5.41031	−0.206416
$688$	9.97111	0.380145
$689$	54.1760	2.06394
$690$	−0.779766	−0.0296852
$691$	0.137163	0.00521792	0.00260896	−	0.999997i	$-0.499170\pi$
$691$	0.137163	0.00521792	0.00260896	+	0.999997i	$0.499170\pi$
$692$	−20.7948	−0.790500
$693$	25.0924	0.953181
$694$	−29.7233	−1.12828
$695$	2.75663	0.104565
$696$	24.6769	0.935374
$697$	47.1658	1.78653
$698$	−29.4024	−1.11290
$699$	0.444369	0.0168076
$700$	−7.21053	−0.272532
$701$	−13.7786	−0.520410	−0.260205	−	0.965553i	$-0.583790\pi$
$701$	−13.7786	−0.520410	−0.260205	+	0.965553i	$0.583790\pi$
$702$	−22.8761	−0.863404
$703$	2.23927	0.0844558
$704$	3.09787	0.116756
$705$	14.7860	0.556872
$706$	−19.8979	−0.748868
$707$	−12.2661	−0.461315
$708$	−13.6466	−0.512872
$709$	−24.6974	−0.927529	−0.463764	−	0.885959i	$-0.653502\pi$
$709$	−24.6974	−0.927529	−0.463764	+	0.885959i	$0.653502\pi$
$710$	−8.83692	−0.331644
$711$	−31.0599	−1.16484
$712$	3.07992	0.115425
$713$	2.66740	0.0998949
$714$	33.1886	1.24205
$715$	−20.6492	−0.772236
$716$	14.1701	0.529562
$717$	−36.2487	−1.35373
$718$	−21.7881	−0.813123
$719$	−47.5177	−1.77211	−0.886055	−	0.463580i	$-0.846565\pi$
$719$	−47.5177	−1.77211	−0.886055	+	0.463580i	$0.846565\pi$
$720$	4.62192	0.172249
$721$	−7.00428	−0.260853
$722$	18.2775	0.680219
$723$	69.2624	2.57590
$724$	18.6843	0.694398
$725$	−35.2270	−1.30830
$726$	−3.80320	−0.141150
$727$	−38.9408	−1.44423	−0.722117	−	0.691771i	$-0.756831\pi$
$727$	−38.9408	−1.44423	−0.722117	+	0.691771i	$0.756831\pi$
$728$	11.6814	0.432941
$729$	−43.3563	−1.60579
$730$	8.88960	0.329019
$731$	65.5166	2.42322
$732$	29.7696	1.10032
$733$	20.1008	0.742439	0.371220	−	0.928545i	$-0.378940\pi$
$733$	20.1008	0.742439	0.371220	+	0.928545i	$0.378940\pi$
$734$	−2.02654	−0.0748011
$735$	10.1660	0.374979
$736$	0.270544	0.00997238
$737$	−29.4574	−1.08508
$738$	−31.2000	−1.14849
$739$	8.30382	0.305461	0.152731	−	0.988268i	$-0.451193\pi$
$739$	8.30382	0.305461	0.152731	+	0.988268i	$0.451193\pi$
$740$	−2.80143	−0.102983
$741$	−14.4413	−0.530513
$742$	16.1064	0.591284
$743$	31.4043	1.15211	0.576055	−	0.817411i	$-0.304591\pi$
$743$	31.4043	1.15211	0.576055	+	0.817411i	$0.304591\pi$
$744$	−26.7233	−0.979722
$745$	8.11679	0.297376
$746$	23.0535	0.844048
$747$	10.6506	0.389686
$748$	20.3550	0.744253
$749$	−11.4156	−0.417116
$750$	−25.5630	−0.933430
$751$	−8.58126	−0.313135	−0.156567	−	0.987667i	$-0.550043\pi$
$751$	−8.58126	−0.313135	−0.156567	+	0.987667i	$0.550043\pi$
$752$	−5.13007	−0.187074
$753$	57.3026	2.08822
$754$	57.0693	2.07834
$755$	12.3192	0.448343
$756$	−6.80102	−0.247351
$757$	−41.7466	−1.51731	−0.758654	−	0.651494i	$-0.774142\pi$
$757$	−41.7466	−1.51731	−0.758654	+	0.651494i	$0.774142\pi$
$758$	19.4627	0.706916
$759$	2.27165	0.0824555
$760$	0.903862	0.0327865
$761$	47.3620	1.71687	0.858435	−	0.512922i	$-0.171437\pi$
$761$	47.3620	1.71687	0.858435	+	0.512922i	$0.171437\pi$
$762$	9.37233	0.339524
$763$	−15.2930	−0.553644
$764$	8.58204	0.310487
$765$	30.3690	1.09799
$766$	5.11988	0.184989
$767$	−31.5601	−1.13957
$768$	−2.71043	−0.0978044
$769$	14.6212	0.527254	0.263627	−	0.964625i	$-0.415081\pi$
$769$	14.6212	0.527254	0.263627	+	0.964625i	$0.415081\pi$
$770$	−6.13895	−0.221232
$771$	−4.24337	−0.152821
$772$	6.60065	0.237563
$773$	−26.9923	−0.970845	−0.485422	−	0.874280i	$-0.661334\pi$
$773$	−26.9923	−0.970845	−0.485422	+	0.874280i	$0.661334\pi$
$774$	−43.3390	−1.55779
$775$	38.1483	1.37033
$776$	−3.31786	−0.119104
$777$	13.3068	0.477379
$778$	1.36383	0.0488955
$779$	−6.10147	−0.218608
$780$	18.0667	0.646891
$781$	25.7441	0.921196
$782$	1.77765	0.0635685
$783$	−33.2263	−1.18741
$784$	−3.52715	−0.125970
$785$	−10.1322	−0.361635
$786$	−47.9449	−1.71014
$787$	16.8913	0.602110	0.301055	−	0.953607i	$-0.402661\pi$
$787$	16.8913	0.602110	0.301055	+	0.953607i	$0.402661\pi$
$788$	23.3022	0.830108
$789$	23.7727	0.846329
$790$	7.59893	0.270358
$791$	14.1775	0.504095
$792$	−13.4648	−0.478450
$793$	68.8472	2.44484
$794$	10.8409	0.384730
$795$	24.9104	0.883483
$796$	7.62372	0.270216
$797$	−5.30211	−0.187810	−0.0939052	−	0.995581i	$-0.529935\pi$
$797$	−5.30211	−0.187810	−0.0939052	+	0.995581i	$0.529935\pi$
$798$	−4.29335	−0.151983
$799$	−33.7079	−1.19250
$800$	3.86923	0.136798
$801$	−13.3867	−0.472997
$802$	−7.94722	−0.280626
$803$	−25.8976	−0.913905
$804$	25.7732	0.908952
$805$	−0.536127	−0.0188960
$806$	−61.8020	−2.17688
$807$	−27.5646	−0.970321
$808$	6.58211	0.231558
$809$	12.6414	0.444448	0.222224	−	0.974996i	$-0.428668\pi$
$809$	12.6414	0.444448	0.222224	+	0.974996i	$0.428668\pi$
$810$	3.34718	0.117608
$811$	−44.1273	−1.54952	−0.774760	−	0.632255i	$-0.782130\pi$
$811$	−44.1273	−1.54952	−0.774760	+	0.632255i	$0.782130\pi$
$812$	16.9666	0.595409
$813$	−78.1198	−2.73978
$814$	8.16124	0.286051
$815$	−19.6409	−0.687992
$816$	−17.8093	−0.623450
$817$	−8.47536	−0.296515
$818$	27.0399	0.945429
$819$	−50.7726	−1.77414
$820$	7.63321	0.266563
$821$	−52.0460	−1.81642	−0.908208	−	0.418519i	$-0.862549\pi$
$821$	−52.0460	−1.81642	−0.908208	+	0.418519i	$0.862549\pi$
$822$	18.0780	0.630543
$823$	48.3345	1.68484	0.842418	−	0.538824i	$-0.181131\pi$
$823$	48.3345	1.68484	0.842418	+	0.538824i	$0.181131\pi$
$824$	3.75855	0.130935
$825$	32.4883	1.13110
$826$	−9.38273	−0.326467
$827$	−35.5806	−1.23726	−0.618629	−	0.785683i	$-0.712312\pi$
$827$	−35.5806	−1.23726	−0.618629	+	0.785683i	$0.712312\pi$
$828$	−1.17591	−0.0408656
$829$	−0.555387	−0.0192894	−0.00964469	−	0.999953i	$-0.503070\pi$
$829$	−0.555387	−0.0192894	−0.00964469	+	0.999953i	$0.503070\pi$
$830$	−2.60572	−0.0904458
$831$	27.2122	0.943980
$832$	−6.26833	−0.217315
$833$	−23.1757	−0.802989
$834$	7.02635	0.243303
$835$	−8.04547	−0.278425
$836$	−2.63317	−0.0910700
$837$	35.9817	1.24371
$838$	−18.1270	−0.626187
$839$	35.1656	1.21405	0.607026	−	0.794682i	$-0.292362\pi$
$839$	35.1656	1.21405	0.607026	+	0.794682i	$0.292362\pi$
$840$	5.37117	0.185323
$841$	53.8899	1.85827
$842$	33.8752	1.16742
$843$	43.6078	1.50193
$844$	18.4791	0.636077
$845$	27.9583	0.961793
$846$	22.2976	0.766608
$847$	−2.61489	−0.0898487
$848$	−8.64281	−0.296795
$849$	−76.6647	−2.63112
$850$	25.4233	0.872012
$851$	0.712738	0.0244323
$852$	−22.5244	−0.771672
$853$	−28.2781	−0.968223	−0.484112	−	0.875006i	$-0.660857\pi$
$853$	−28.2781	−0.968223	−0.484112	+	0.875006i	$0.660857\pi$
$854$	20.4681	0.700403
$855$	−3.92860	−0.134355
$856$	6.12568	0.209372
$857$	4.42623	0.151197	0.0755985	−	0.997138i	$-0.475913\pi$
$857$	4.42623	0.151197	0.0755985	+	0.997138i	$0.475913\pi$
$858$	−52.6326	−1.79685
$859$	−5.90919	−0.201619	−0.100809	−	0.994906i	$-0.532143\pi$
$859$	−5.90919	−0.201619	−0.100809	+	0.994906i	$0.532143\pi$
$860$	10.6031	0.361561
$861$	−36.2578	−1.23566
$862$	1.06105	0.0361395
$863$	11.3835	0.387499	0.193749	−	0.981051i	$-0.437935\pi$
$863$	11.3835	0.387499	0.193749	+	0.981051i	$0.437935\pi$
$864$	3.64948	0.124158
$865$	−22.1127	−0.751856
$866$	−23.1207	−0.785672
$867$	−70.9411	−2.40929
$868$	−18.3736	−0.623639
$869$	−22.1375	−0.750964
$870$	26.2408	0.889647
$871$	59.6049	2.01963
$872$	8.20635	0.277902
$873$	14.4209	0.488074
$874$	−0.229960	−0.00777852
$875$	−17.5758	−0.594172
$876$	22.6586	0.765565
$877$	−8.40553	−0.283835	−0.141917	−	0.989879i	$-0.545327\pi$
$877$	−8.40553	−0.283835	−0.141917	+	0.989879i	$0.545327\pi$
$878$	17.5575	0.592538
$879$	39.7039	1.33918
$880$	3.29421	0.111048
$881$	34.5557	1.16421	0.582106	−	0.813113i	$-0.302229\pi$
$881$	34.5557	1.16421	0.582106	+	0.813113i	$0.302229\pi$
$882$	15.3306	0.516209
$883$	−46.8671	−1.57720	−0.788601	−	0.614905i	$-0.789194\pi$
$883$	−46.8671	−1.57720	−0.788601	+	0.614905i	$0.789194\pi$
$884$	−41.1869	−1.38527
$885$	−14.5115	−0.487799
$886$	9.85140	0.330964
$887$	10.9002	0.365994	0.182997	−	0.983113i	$-0.441420\pi$
$887$	10.9002	0.365994	0.182997	+	0.983113i	$0.441420\pi$
$888$	−7.14054	−0.239621
$889$	6.44394	0.216123
$890$	3.27512	0.109782
$891$	−9.75115	−0.326676
$892$	−22.4092	−0.750314
$893$	4.36052	0.145919
$894$	20.6888	0.691938
$895$	15.0682	0.503674
$896$	−1.86356	−0.0622571
$897$	−4.59651	−0.153473
$898$	−22.1453	−0.738997
$899$	−89.7638	−2.99379
$900$	−16.8174	−0.560581
$901$	−56.7888	−1.89191
$902$	−22.2374	−0.740423
$903$	−50.3646	−1.67603
$904$	−7.60779	−0.253031
$905$	19.8685	0.660451
$906$	31.4004	1.04321
$907$	49.0149	1.62751	0.813756	−	0.581207i	$-0.197419\pi$
$907$	49.0149	1.62751	0.813756	+	0.581207i	$0.197419\pi$
$908$	−2.27015	−0.0753375
$909$	−28.6088	−0.948895
$910$	12.4217	0.411776
$911$	1.41067	0.0467377	0.0233689	−	0.999727i	$-0.492561\pi$
$911$	1.41067	0.0467377	0.0233689	+	0.999727i	$0.492561\pi$
$912$	2.30385	0.0762880
$913$	7.59108	0.251228
$914$	1.28020	0.0423452
$915$	31.6564	1.04653
$916$	1.99611	0.0659532
$917$	−32.9645	−1.08858
$918$	23.9794	0.791439
$919$	−46.4217	−1.53131	−0.765655	−	0.643251i	$-0.777585\pi$
$919$	−46.4217	−1.53131	−0.765655	+	0.643251i	$0.777585\pi$
$920$	0.287690	0.00948487
$921$	88.8937	2.92915
$922$	1.36841	0.0450660
$923$	−52.0913	−1.71461
$924$	−15.6475	−0.514766
$925$	10.1933	0.335155
$926$	−22.5493	−0.741015
$927$	−16.3364	−0.536557
$928$	−9.10439	−0.298866
$929$	−49.1776	−1.61346	−0.806732	−	0.590918i	$-0.798766\pi$
$929$	−49.1776	−1.61346	−0.806732	+	0.590918i	$0.798766\pi$
$930$	−28.4169	−0.931827
$931$	2.99805	0.0982572
$932$	−0.163947	−0.00537028
$933$	−84.0868	−2.75288
$934$	−18.5799	−0.607954
$935$	21.6451	0.707870
$936$	27.2450	0.890531
$937$	−14.9022	−0.486832	−0.243416	−	0.969922i	$-0.578268\pi$
$937$	−14.9022	−0.486832	−0.243416	+	0.969922i	$0.578268\pi$
$938$	17.7204	0.578590
$939$	23.0005	0.750592
$940$	−5.45520	−0.177929
$941$	46.1480	1.50438	0.752191	−	0.658946i	$-0.228997\pi$
$941$	46.1480	1.50438	0.752191	+	0.658946i	$0.228997\pi$
$942$	−25.8260	−0.841457
$943$	−1.94204	−0.0632414
$944$	5.03485	0.163870
$945$	−7.23205	−0.235258
$946$	−30.8893	−1.00430
$947$	11.8880	0.386309	0.193154	−	0.981168i	$-0.438128\pi$
$947$	11.8880	0.386309	0.193154	+	0.981168i	$0.438128\pi$
$948$	19.3689	0.629072
$949$	52.4019	1.70104
$950$	−3.28881	−0.106703
$951$	67.9876	2.20465
$952$	−12.2448	−0.396855
$953$	55.8640	1.80961	0.904806	−	0.425824i	$-0.140016\pi$
$953$	55.8640	1.80961	0.904806	+	0.425824i	$0.140016\pi$
$954$	37.5656	1.21623
$955$	9.12595	0.295309
$956$	13.3738	0.432538
$957$	−76.4458	−2.47114
$958$	22.4388	0.724963
$959$	12.4295	0.401370
$960$	−2.88221	−0.0930231
$961$	66.2078	2.13573
$962$	−16.5137	−0.532423
$963$	−26.6250	−0.857979
$964$	−25.5540	−0.823039
$965$	7.01899	0.225949
$966$	−1.36653	−0.0439674
$967$	−44.1860	−1.42093	−0.710463	−	0.703735i	$-0.751514\pi$
$967$	−44.1860	−1.42093	−0.710463	+	0.703735i	$0.751514\pi$
$968$	1.40317	0.0450996
$969$	15.1378	0.486295
$970$	−3.52813	−0.113282
$971$	24.9263	0.799924	0.399962	−	0.916532i	$-0.369023\pi$
$971$	24.9263	0.799924	0.399962	+	0.916532i	$0.369023\pi$
$972$	19.4800	0.624823
$973$	4.83096	0.154874
$974$	−1.62414	−0.0520407
$975$	−65.7378	−2.10529
$976$	−10.9833	−0.351568
$977$	−6.35678	−0.203371	−0.101686	−	0.994817i	$-0.532424\pi$
$977$	−6.35678	−0.203371	−0.101686	+	0.994817i	$0.532424\pi$
$978$	−50.0627	−1.60083
$979$	−9.54121	−0.304938
$980$	−3.75070	−0.119812
$981$	−35.6685	−1.13881
$982$	11.7970	0.376457
$983$	−15.9598	−0.509039	−0.254520	−	0.967068i	$-0.581917\pi$
$983$	−15.9598	−0.509039	−0.254520	+	0.967068i	$0.581917\pi$
$984$	19.4562	0.620242
$985$	24.7791	0.789527
$986$	−59.8217	−1.90511
$987$	25.9123	0.824796
$988$	5.32803	0.169507
$989$	−2.69762	−0.0857794
$990$	−14.3181	−0.455060
$991$	5.60794	0.178142	0.0890711	−	0.996025i	$-0.471610\pi$
$991$	5.60794	0.178142	0.0890711	+	0.996025i	$0.471610\pi$
$992$	9.85940	0.313036
$993$	−9.50998	−0.301790
$994$	−15.4866	−0.491205
$995$	8.10689	0.257006
$996$	−6.64169	−0.210450
$997$	−57.5099	−1.82136	−0.910678	−	0.413117i	$-0.864440\pi$
$997$	−57.5099	−1.82136	−0.910678	+	0.413117i	$0.864440\pi$
$998$	−31.9954	−1.01280
$999$	9.61442	0.304187

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.c.1.11	✓	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.c.1.11	✓	84	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.71043 q^{3} +1.00000 q^{4} +1.06338 q^{5} +2.71043 q^{6} +1.86356 q^{7} -1.00000 q^{8} +4.34646 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.71043 q^{3} +1.00000 q^{4} +1.06338 q^{5} +2.71043 q^{6} +1.86356 q^{7} -1.00000 q^{8} +4.34646 q^{9} -1.06338 q^{10} +3.09787 q^{11} -2.71043 q^{12} -6.26833 q^{13} -1.86356 q^{14} -2.88221 q^{15} +1.00000 q^{16} +6.57064 q^{17} -4.34646 q^{18} -0.849992 q^{19} +1.06338 q^{20} -5.05105 q^{21} -3.09787 q^{22} -0.270544 q^{23} +2.71043 q^{24} -3.86923 q^{25} +6.26833 q^{26} -3.64948 q^{27} +1.86356 q^{28} +9.10439 q^{29} +2.88221 q^{30} -9.85940 q^{31} -1.00000 q^{32} -8.39659 q^{33} -6.57064 q^{34} +1.98166 q^{35} +4.34646 q^{36} -2.63446 q^{37} +0.849992 q^{38} +16.9899 q^{39} -1.06338 q^{40} +7.17827 q^{41} +5.05105 q^{42} +9.97111 q^{43} +3.09787 q^{44} +4.62192 q^{45} +0.270544 q^{46} -5.13007 q^{47} -2.71043 q^{48} -3.52715 q^{49} +3.86923 q^{50} -17.8093 q^{51} -6.26833 q^{52} -8.64281 q^{53} +3.64948 q^{54} +3.29421 q^{55} -1.86356 q^{56} +2.30385 q^{57} -9.10439 q^{58} +5.03485 q^{59} -2.88221 q^{60} -10.9833 q^{61} +9.85940 q^{62} +8.09987 q^{63} +1.00000 q^{64} -6.66560 q^{65} +8.39659 q^{66} -9.50889 q^{67} +6.57064 q^{68} +0.733291 q^{69} -1.98166 q^{70} +8.31024 q^{71} -4.34646 q^{72} -8.35978 q^{73} +2.63446 q^{74} +10.4873 q^{75} -0.849992 q^{76} +5.77307 q^{77} -16.9899 q^{78} -7.14604 q^{79} +1.06338 q^{80} -3.14769 q^{81} -7.17827 q^{82} +2.45042 q^{83} -5.05105 q^{84} +6.98707 q^{85} -9.97111 q^{86} -24.6769 q^{87} -3.09787 q^{88} -3.07992 q^{89} -4.62192 q^{90} -11.6814 q^{91} -0.270544 q^{92} +26.7233 q^{93} +5.13007 q^{94} -0.903862 q^{95} +2.71043 q^{96} +3.31786 q^{97} +3.52715 q^{98} +13.4648 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9}+O(q^{10}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9} + 32 q^{10} - 6 q^{11} - 19 q^{12} - 29 q^{13} - q^{14} - 4 q^{15} + 84 q^{16} - 36 q^{17} - 77 q^{18} + 33 q^{19} - 32 q^{20} - 21 q^{21} + 6 q^{22} - 62 q^{23} + 19 q^{24} + 82 q^{25} + 29 q^{26} - 82 q^{27} + q^{28} - 51 q^{29} + 4 q^{30} + 39 q^{31} - 84 q^{32} - 32 q^{33} + 36 q^{34} - 34 q^{35} + 77 q^{36} - 32 q^{37} - 33 q^{38} + 29 q^{39} + 32 q^{40} - 38 q^{41} + 21 q^{42} - 6 q^{44} - 91 q^{45} + 62 q^{46} - 58 q^{47} - 19 q^{48} + 83 q^{49} - 82 q^{50} - q^{51} - 29 q^{52} - 106 q^{53} + 82 q^{54} + 32 q^{55} - q^{56} - 44 q^{57} + 51 q^{58} - 42 q^{59} - 4 q^{60} - 41 q^{61} - 39 q^{62} - 9 q^{63} + 84 q^{64} - 49 q^{65} + 32 q^{66} - 16 q^{67} - 36 q^{68} - 45 q^{69} + 34 q^{70} - 62 q^{71} - 77 q^{72} - 16 q^{73} + 32 q^{74} - 80 q^{75} + 33 q^{76} - 134 q^{77} - 29 q^{78} + 53 q^{79} - 32 q^{80} + 56 q^{81} + 38 q^{82} - 90 q^{83} - 21 q^{84} - 60 q^{85} - 3 q^{87} + 6 q^{88} - 54 q^{89} + 91 q^{90} + 33 q^{91} - 62 q^{92} - 69 q^{93} + 58 q^{94} - 47 q^{95} + 19 q^{96} - 31 q^{97} - 83 q^{98} + 23 q^{99}+O(q^{100}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9 + 32 * q^10 - 6 * q^11 - 19 * q^12 - 29 * q^13 - q^14 - 4 * q^15 + 84 * q^16 - 36 * q^17 - 77 * q^18 + 33 * q^19 - 32 * q^20 - 21 * q^21 + 6 * q^22 - 62 * q^23 + 19 * q^24 + 82 * q^25 + 29 * q^26 - 82 * q^27 + q^28 - 51 * q^29 + 4 * q^30 + 39 * q^31 - 84 * q^32 - 32 * q^33 + 36 * q^34 - 34 * q^35 + 77 * q^36 - 32 * q^37 - 33 * q^38 + 29 * q^39 + 32 * q^40 - 38 * q^41 + 21 * q^42 - 6 * q^44 - 91 * q^45 + 62 * q^46 - 58 * q^47 - 19 * q^48 + 83 * q^49 - 82 * q^50 - q^51 - 29 * q^52 - 106 * q^53 + 82 * q^54 + 32 * q^55 - q^56 - 44 * q^57 + 51 * q^58 - 42 * q^59 - 4 * q^60 - 41 * q^61 - 39 * q^62 - 9 * q^63 + 84 * q^64 - 49 * q^65 + 32 * q^66 - 16 * q^67 - 36 * q^68 - 45 * q^69 + 34 * q^70 - 62 * q^71 - 77 * q^72 - 16 * q^73 + 32 * q^74 - 80 * q^75 + 33 * q^76 - 134 * q^77 - 29 * q^78 + 53 * q^79 - 32 * q^80 + 56 * q^81 + 38 * q^82 - 90 * q^83 - 21 * q^84 - 60 * q^85 - 3 * q^87 + 6 * q^88 - 54 * q^89 + 91 * q^90 + 33 * q^91 - 62 * q^92 - 69 * q^93 + 58 * q^94 - 47 * q^95 + 19 * q^96 - 31 * q^97 - 83 * q^98 + 23 * q^99

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