LMFDB - Embedded newform 8042.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, 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("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.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:	$64.2156933055$
Analytic rank:	$0$
Dimension:	$86$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.40447 q^{3} +1.00000 q^{4} -1.12046 q^{5} +3.40447 q^{6} +3.83806 q^{7} -1.00000 q^{8} +8.59041 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.40447 q^{3} +1.00000 q^{4} -1.12046 q^{5} +3.40447 q^{6} +3.83806 q^{7} -1.00000 q^{8} +8.59041 q^{9} +1.12046 q^{10} +0.103454 q^{11} -3.40447 q^{12} +3.82445 q^{13} -3.83806 q^{14} +3.81458 q^{15} +1.00000 q^{16} -3.12423 q^{17} -8.59041 q^{18} -1.69585 q^{19} -1.12046 q^{20} -13.0666 q^{21} -0.103454 q^{22} -1.75653 q^{23} +3.40447 q^{24} -3.74456 q^{25} -3.82445 q^{26} -19.0324 q^{27} +3.83806 q^{28} +1.73082 q^{29} -3.81458 q^{30} +1.88344 q^{31} -1.00000 q^{32} -0.352205 q^{33} +3.12423 q^{34} -4.30040 q^{35} +8.59041 q^{36} +5.06635 q^{37} +1.69585 q^{38} -13.0202 q^{39} +1.12046 q^{40} -1.44472 q^{41} +13.0666 q^{42} +11.9263 q^{43} +0.103454 q^{44} -9.62524 q^{45} +1.75653 q^{46} -1.12399 q^{47} -3.40447 q^{48} +7.73070 q^{49} +3.74456 q^{50} +10.6363 q^{51} +3.82445 q^{52} -0.556433 q^{53} +19.0324 q^{54} -0.115916 q^{55} -3.83806 q^{56} +5.77347 q^{57} -1.73082 q^{58} -6.44203 q^{59} +3.81458 q^{60} +11.3879 q^{61} -1.88344 q^{62} +32.9705 q^{63} +1.00000 q^{64} -4.28516 q^{65} +0.352205 q^{66} +14.5779 q^{67} -3.12423 q^{68} +5.98007 q^{69} +4.30040 q^{70} +11.9703 q^{71} -8.59041 q^{72} -9.36331 q^{73} -5.06635 q^{74} +12.7482 q^{75} -1.69585 q^{76} +0.397061 q^{77} +13.0202 q^{78} -4.17319 q^{79} -1.12046 q^{80} +39.0240 q^{81} +1.44472 q^{82} +9.41280 q^{83} -13.0666 q^{84} +3.50058 q^{85} -11.9263 q^{86} -5.89254 q^{87} -0.103454 q^{88} +0.195299 q^{89} +9.62524 q^{90} +14.6785 q^{91} -1.75653 q^{92} -6.41211 q^{93} +1.12399 q^{94} +1.90014 q^{95} +3.40447 q^{96} -11.2817 q^{97} -7.73070 q^{98} +0.888709 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) $ 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9 + 4 * q^10 + 13 * q^11 + 12 * q^12 + 45 * q^13 - 35 * q^14 + 17 * q^15 + 86 * q^16 + 5 * q^17 - 72 * q^18 + 47 * q^19 - 4 * q^20 + 15 * q^21 - 13 * q^22 + 6 * q^23 - 12 * q^24 + 112 * q^25 - 45 * q^26 + 51 * q^27 + 35 * q^28 - 14 * q^29 - 17 * q^30 + 24 * q^31 - 86 * q^32 + 43 * q^33 - 5 * q^34 + 42 * q^35 + 72 * q^36 + 61 * q^37 - 47 * q^38 + 20 * q^39 + 4 * q^40 - 16 * q^41 - 15 * q^42 + 72 * q^43 + 13 * q^44 + 6 * q^45 - 6 * q^46 + 11 * q^47 + 12 * q^48 + 89 * q^49 - 112 * q^50 + 56 * q^51 + 45 * q^52 - 7 * q^53 - 51 * q^54 + 48 * q^55 - 35 * q^56 + 65 * q^57 + 14 * q^58 + 24 * q^59 + 17 * q^60 + 31 * q^61 - 24 * q^62 + 98 * q^63 + 86 * q^64 - 9 * q^65 - 43 * q^66 + 157 * q^67 + 5 * q^68 + q^69 - 42 * q^70 - 11 * q^71 - 72 * q^72 + 74 * q^73 - 61 * q^74 + 76 * q^75 + 47 * q^76 - 13 * q^77 - 20 * q^78 + 57 * q^79 - 4 * q^80 + 34 * q^81 + 16 * q^82 + 65 * q^83 + 15 * q^84 + 102 * q^85 - 72 * q^86 + 49 * q^87 - 13 * q^88 - 34 * q^89 - 6 * q^90 + 91 * q^91 + 6 * q^92 + 57 * q^93 - 11 * q^94 - 13 * q^95 - 12 * q^96 + 64 * q^97 - 89 * q^98 + 56 * 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
$3$	−3.40447	−1.96557	−0.982786	−	0.184749i	$-0.940853\pi$
$3$	−3.40447	−1.96557	−0.982786	+	0.184749i	$0.940853\pi$
$4$	1.00000	0.500000
$5$	−1.12046	−0.501086	−0.250543	−	0.968105i	$-0.580609\pi$
$5$	−1.12046	−0.501086	−0.250543	+	0.968105i	$0.580609\pi$
$6$	3.40447	1.38987
$7$	3.83806	1.45065	0.725325	−	0.688407i	$-0.241689\pi$
$7$	3.83806	1.45065	0.725325	+	0.688407i	$0.241689\pi$
$8$	−1.00000	−0.353553
$9$	8.59041	2.86347
$10$	1.12046	0.354322
$11$	0.103454	0.0311924	0.0155962	−	0.999878i	$-0.495035\pi$
$11$	0.103454	0.0311924	0.0155962	+	0.999878i	$0.495035\pi$
$12$	−3.40447	−0.982786
$13$	3.82445	1.06071	0.530356	−	0.847775i	$-0.322058\pi$
$13$	3.82445	1.06071	0.530356	+	0.847775i	$0.322058\pi$
$14$	−3.83806	−1.02576
$15$	3.81458	0.984921
$16$	1.00000	0.250000
$17$	−3.12423	−0.757736	−0.378868	−	0.925451i	$-0.623687\pi$
$17$	−3.12423	−0.757736	−0.378868	+	0.925451i	$0.623687\pi$
$18$	−8.59041	−2.02478
$19$	−1.69585	−0.389055	−0.194527	−	0.980897i	$-0.562317\pi$
$19$	−1.69585	−0.389055	−0.194527	+	0.980897i	$0.562317\pi$
$20$	−1.12046	−0.250543
$21$	−13.0666	−2.85136
$22$	−0.103454	−0.0220564
$23$	−1.75653	−0.366263	−0.183131	−	0.983088i	$-0.558623\pi$
$23$	−1.75653	−0.366263	−0.183131	+	0.983088i	$0.558623\pi$
$24$	3.40447	0.694934
$25$	−3.74456	−0.748912
$26$	−3.82445	−0.750037
$27$	−19.0324	−3.66279
$28$	3.83806	0.725325
$29$	1.73082	0.321406	0.160703	−	0.987003i	$-0.448624\pi$
$29$	1.73082	0.321406	0.160703	+	0.987003i	$0.448624\pi$
$30$	−3.81458	−0.696445
$31$	1.88344	0.338276	0.169138	−	0.985592i	$-0.445902\pi$
$31$	1.88344	0.338276	0.169138	+	0.985592i	$0.445902\pi$
$32$	−1.00000	−0.176777
$33$	−0.352205	−0.0613110
$34$	3.12423	0.535800
$35$	−4.30040	−0.726901
$36$	8.59041	1.43174
$37$	5.06635	0.832903	0.416451	−	0.909158i	$-0.363274\pi$
$37$	5.06635	0.832903	0.416451	+	0.909158i	$0.363274\pi$
$38$	1.69585	0.275103
$39$	−13.0202	−2.08491
$40$	1.12046	0.177161
$41$	−1.44472	−0.225627	−0.112814	−	0.993616i	$-0.535986\pi$
$41$	−1.44472	−0.225627	−0.112814	+	0.993616i	$0.535986\pi$
$42$	13.0666	2.01621
$43$	11.9263	1.81875	0.909373	−	0.415981i	$-0.136562\pi$
$43$	11.9263	1.81875	0.909373	+	0.415981i	$0.136562\pi$
$44$	0.103454	0.0155962
$45$	−9.62524	−1.43485
$46$	1.75653	0.258987
$47$	−1.12399	−0.163950	−0.0819751	−	0.996634i	$-0.526123\pi$
$47$	−1.12399	−0.163950	−0.0819751	+	0.996634i	$0.526123\pi$
$48$	−3.40447	−0.491393
$49$	7.73070	1.10439
$50$	3.74456	0.529561
$51$	10.6363	1.48938
$52$	3.82445	0.530356
$53$	−0.556433	−0.0764319	−0.0382160	−	0.999270i	$-0.512167\pi$
$53$	−0.556433	−0.0764319	−0.0382160	+	0.999270i	$0.512167\pi$
$54$	19.0324	2.58998
$55$	−0.115916	−0.0156301
$56$	−3.83806	−0.512882
$57$	5.77347	0.764715
$58$	−1.73082	−0.227268
$59$	−6.44203	−0.838681	−0.419340	−	0.907829i	$-0.637739\pi$
$59$	−6.44203	−0.838681	−0.419340	+	0.907829i	$0.637739\pi$
$60$	3.81458	0.492461
$61$	11.3879	1.45807	0.729034	−	0.684477i	$-0.239969\pi$
$61$	11.3879	1.45807	0.729034	+	0.684477i	$0.239969\pi$
$62$	−1.88344	−0.239197
$63$	32.9705	4.15389
$64$	1.00000	0.125000
$65$	−4.28516	−0.531509
$66$	0.352205	0.0433534
$67$	14.5779	1.78097	0.890485	−	0.455012i	$-0.150365\pi$
$67$	14.5779	1.78097	0.890485	+	0.455012i	$0.150365\pi$
$68$	−3.12423	−0.378868
$69$	5.98007	0.719915
$70$	4.30040	0.513997
$71$	11.9703	1.42062	0.710309	−	0.703890i	$-0.248555\pi$
$71$	11.9703	1.42062	0.710309	+	0.703890i	$0.248555\pi$
$72$	−8.59041	−1.01239
$73$	−9.36331	−1.09589	−0.547946	−	0.836514i	$-0.684590\pi$
$73$	−9.36331	−1.09589	−0.547946	+	0.836514i	$0.684590\pi$
$74$	−5.06635	−0.588951
$75$	12.7482	1.47204
$76$	−1.69585	−0.194527
$77$	0.397061	0.0452493
$78$	13.0202	1.47425
$79$	−4.17319	−0.469520	−0.234760	−	0.972053i	$-0.575430\pi$
$79$	−4.17319	−0.469520	−0.234760	+	0.972053i	$0.575430\pi$
$80$	−1.12046	−0.125272
$81$	39.0240	4.33600
$82$	1.44472	0.159543
$83$	9.41280	1.03319	0.516594	−	0.856230i	$-0.327200\pi$
$83$	9.41280	1.03319	0.516594	+	0.856230i	$0.327200\pi$
$84$	−13.0666	−1.42568
$85$	3.50058	0.379691
$86$	−11.9263	−1.28605
$87$	−5.89254	−0.631746
$88$	−0.103454	−0.0110282
$89$	0.195299	0.0207016	0.0103508	−	0.999946i	$-0.496705\pi$
$89$	0.195299	0.0207016	0.0103508	+	0.999946i	$0.496705\pi$
$90$	9.62524	1.01459
$91$	14.6785	1.53872
$92$	−1.75653	−0.183131
$93$	−6.41211	−0.664905
$94$	1.12399	0.115930
$95$	1.90014	0.194950
$96$	3.40447	0.347467
$97$	−11.2817	−1.14548	−0.572741	−	0.819736i	$-0.694120\pi$
$97$	−11.2817	−1.14548	−0.572741	+	0.819736i	$0.694120\pi$
$98$	−7.73070	−0.780918
$99$	0.888709	0.0893186
$100$	−3.74456	−0.374456
$101$	−16.3651	−1.62838	−0.814192	−	0.580595i	$-0.802820\pi$
$101$	−16.3651	−1.62838	−0.814192	+	0.580595i	$0.802820\pi$
$102$	−10.6363	−1.05315
$103$	−2.72133	−0.268140	−0.134070	−	0.990972i	$-0.542805\pi$
$103$	−2.72133	−0.268140	−0.134070	+	0.990972i	$0.542805\pi$
$104$	−3.82445	−0.375018
$105$	14.6406	1.42878
$106$	0.556433	0.0540455
$107$	−0.571223	−0.0552222	−0.0276111	−	0.999619i	$-0.508790\pi$
$107$	−0.571223	−0.0552222	−0.0276111	+	0.999619i	$0.508790\pi$
$108$	−19.0324	−1.83139
$109$	1.32179	0.126605	0.0633024	−	0.997994i	$-0.479837\pi$
$109$	1.32179	0.126605	0.0633024	+	0.997994i	$0.479837\pi$
$110$	0.115916	0.0110522
$111$	−17.2482	−1.63713
$112$	3.83806	0.362662
$113$	−4.40607	−0.414488	−0.207244	−	0.978289i	$-0.566449\pi$
$113$	−4.40607	−0.414488	−0.207244	+	0.978289i	$0.566449\pi$
$114$	−5.77347	−0.540735
$115$	1.96813	0.183529
$116$	1.73082	0.160703
$117$	32.8536	3.03732
$118$	6.44203	0.593037
$119$	−11.9910	−1.09921
$120$	−3.81458	−0.348222
$121$	−10.9893	−0.999027
$122$	−11.3879	−1.03101
$123$	4.91851	0.443487
$124$	1.88344	0.169138
$125$	9.79796	0.876356
$126$	−32.9705	−2.93725
$127$	12.5153	1.11055	0.555276	−	0.831666i	$-0.312612\pi$
$127$	12.5153	1.11055	0.555276	+	0.831666i	$0.312612\pi$
$128$	−1.00000	−0.0883883
$129$	−40.6028	−3.57488
$130$	4.28516	0.375833
$131$	−16.9230	−1.47857	−0.739283	−	0.673395i	$-0.764836\pi$
$131$	−16.9230	−1.47857	−0.739283	+	0.673395i	$0.764836\pi$
$132$	−0.352205	−0.0306555
$133$	−6.50877	−0.564382
$134$	−14.5779	−1.25934
$135$	21.3251	1.83537
$136$	3.12423	0.267900
$137$	14.6461	1.25130	0.625651	−	0.780103i	$-0.284833\pi$
$137$	14.6461	1.25130	0.625651	+	0.780103i	$0.284833\pi$
$138$	−5.98007	−0.509057
$139$	18.4316	1.56335	0.781675	−	0.623686i	$-0.214365\pi$
$139$	18.4316	1.56335	0.781675	+	0.623686i	$0.214365\pi$
$140$	−4.30040	−0.363451
$141$	3.82658	0.322256
$142$	−11.9703	−1.00453
$143$	0.395653	0.0330862
$144$	8.59041	0.715868
$145$	−1.93932	−0.161052
$146$	9.36331	0.774913
$147$	−26.3189	−2.17075
$148$	5.06635	0.416451
$149$	−8.36090	−0.684951	−0.342476	−	0.939527i	$-0.611265\pi$
$149$	−8.36090	−0.684951	−0.342476	+	0.939527i	$0.611265\pi$
$150$	−12.7482	−1.04089
$151$	−11.7406	−0.955440	−0.477720	−	0.878512i	$-0.658537\pi$
$151$	−11.7406	−0.955440	−0.477720	+	0.878512i	$0.658537\pi$
$152$	1.69585	0.137552
$153$	−26.8384	−2.16976
$154$	−0.397061	−0.0319961
$155$	−2.11033	−0.169505
$156$	−13.0202	−1.04245
$157$	5.57613	0.445024	0.222512	−	0.974930i	$-0.428574\pi$
$157$	5.57613	0.445024	0.222512	+	0.974930i	$0.428574\pi$
$158$	4.17319	0.332001
$159$	1.89436	0.150232
$160$	1.12046	0.0885804
$161$	−6.74168	−0.531319
$162$	−39.0240	−3.06601
$163$	0.689524	0.0540077	0.0270038	−	0.999635i	$-0.491403\pi$
$163$	0.689524	0.0540077	0.0270038	+	0.999635i	$0.491403\pi$
$164$	−1.44472	−0.112814
$165$	0.394632	0.0307221
$166$	−9.41280	−0.730575
$167$	6.56375	0.507918	0.253959	−	0.967215i	$-0.418267\pi$
$167$	6.56375	0.507918	0.253959	+	0.967215i	$0.418267\pi$
$168$	13.0666	1.00811
$169$	1.62644	0.125110
$170$	−3.50058	−0.268482
$171$	−14.5681	−1.11405
$172$	11.9263	0.909373
$173$	6.91163	0.525482	0.262741	−	0.964866i	$-0.415374\pi$
$173$	6.91163	0.525482	0.262741	+	0.964866i	$0.415374\pi$
$174$	5.89254	0.446712
$175$	−14.3718	−1.08641
$176$	0.103454	0.00779811
$177$	21.9317	1.64849
$178$	−0.195299	−0.0146383
$179$	14.6092	1.09194	0.545972	−	0.837803i	$-0.316160\pi$
$179$	14.6092	1.09194	0.545972	+	0.837803i	$0.316160\pi$
$180$	−9.62524	−0.717423
$181$	0.464401	0.0345187	0.0172593	−	0.999851i	$-0.494506\pi$
$181$	0.464401	0.0345187	0.0172593	+	0.999851i	$0.494506\pi$
$182$	−14.6785	−1.08804
$183$	−38.7697	−2.86594
$184$	1.75653	0.129493
$185$	−5.67666	−0.417356
$186$	6.41211	0.470159
$187$	−0.323213	−0.0236356
$188$	−1.12399	−0.0819751
$189$	−73.0475	−5.31342
$190$	−1.90014	−0.137851
$191$	0.0538546	0.00389678	0.00194839	−	0.999998i	$-0.499380\pi$
$191$	0.0538546	0.00389678	0.00194839	+	0.999998i	$0.499380\pi$
$192$	−3.40447	−0.245696
$193$	7.36228	0.529949	0.264974	−	0.964255i	$-0.414637\pi$
$193$	7.36228	0.529949	0.264974	+	0.964255i	$0.414637\pi$
$194$	11.2817	0.809978
$195$	14.5887	1.04472
$196$	7.73070	0.552193
$197$	19.1876	1.36706	0.683528	−	0.729924i	$-0.260445\pi$
$197$	19.1876	1.36706	0.683528	+	0.729924i	$0.260445\pi$
$198$	−0.888709	−0.0631578
$199$	−10.0373	−0.711523	−0.355761	−	0.934577i	$-0.615778\pi$
$199$	−10.0373	−0.711523	−0.355761	+	0.934577i	$0.615778\pi$
$200$	3.74456	0.264780
$201$	−49.6299	−3.50063
$202$	16.3651	1.15144
$203$	6.64300	0.466247
$204$	10.6363	0.744692
$205$	1.61876	0.113059
$206$	2.72133	0.189604
$207$	−15.0894	−1.04878
$208$	3.82445	0.265178
$209$	−0.175442	−0.0121356
$210$	−14.6406	−1.01030
$211$	0.217298	0.0149594	0.00747972	−	0.999972i	$-0.497619\pi$
$211$	0.217298	0.0149594	0.00747972	+	0.999972i	$0.497619\pi$
$212$	−0.556433	−0.0382160
$213$	−40.7527	−2.79233
$214$	0.571223	0.0390480
$215$	−13.3630	−0.911349
$216$	19.0324	1.29499
$217$	7.22875	0.490720
$218$	−1.32179	−0.0895232
$219$	31.8771	2.15405
$220$	−0.115916	−0.00781505
$221$	−11.9485	−0.803740
$222$	17.2482	1.15763
$223$	8.46482	0.566847	0.283423	−	0.958995i	$-0.408530\pi$
$223$	8.46482	0.566847	0.283423	+	0.958995i	$0.408530\pi$
$224$	−3.83806	−0.256441
$225$	−32.1673	−2.14449
$226$	4.40607	0.293088
$227$	−13.6199	−0.903986	−0.451993	−	0.892022i	$-0.649287\pi$
$227$	−13.6199	−0.903986	−0.451993	+	0.892022i	$0.649287\pi$
$228$	5.77347	0.382357
$229$	−16.8844	−1.11575	−0.557876	−	0.829924i	$-0.688383\pi$
$229$	−16.8844	−1.11575	−0.557876	+	0.829924i	$0.688383\pi$
$230$	−1.96813	−0.129775
$231$	−1.35178	−0.0889408
$232$	−1.73082	−0.113634
$233$	−10.1780	−0.666783	−0.333392	−	0.942788i	$-0.608193\pi$
$233$	−10.1780	−0.666783	−0.333392	+	0.942788i	$0.608193\pi$
$234$	−32.8536	−2.14771
$235$	1.25939	0.0821532
$236$	−6.44203	−0.419340
$237$	14.2075	0.922875
$238$	11.9910	0.777259
$239$	5.56786	0.360155	0.180077	−	0.983652i	$-0.442365\pi$
$239$	5.56786	0.360155	0.180077	+	0.983652i	$0.442365\pi$
$240$	3.81458	0.246230
$241$	17.7321	1.14223	0.571114	−	0.820871i	$-0.306512\pi$
$241$	17.7321	1.14223	0.571114	+	0.820871i	$0.306512\pi$
$242$	10.9893	0.706419
$243$	−75.7587	−4.85992
$244$	11.3879	0.729034
$245$	−8.66196	−0.553393
$246$	−4.91851	−0.313593
$247$	−6.48570	−0.412675
$248$	−1.88344	−0.119599
$249$	−32.0456	−2.03081
$250$	−9.79796	−0.619678
$251$	7.39420	0.466717	0.233359	−	0.972391i	$-0.425028\pi$
$251$	7.39420	0.466717	0.233359	+	0.972391i	$0.425028\pi$
$252$	32.9705	2.07695
$253$	−0.181720	−0.0114246
$254$	−12.5153	−0.785279
$255$	−11.9176	−0.746311
$256$	1.00000	0.0625000
$257$	−16.0060	−0.998427	−0.499214	−	0.866479i	$-0.666378\pi$
$257$	−16.0060	−0.998427	−0.499214	+	0.866479i	$0.666378\pi$
$258$	40.6028	2.52782
$259$	19.4450	1.20825
$260$	−4.28516	−0.265754
$261$	14.8685	0.920336
$262$	16.9230	1.04550
$263$	15.5834	0.960916	0.480458	−	0.877018i	$-0.340470\pi$
$263$	15.5834	0.960916	0.480458	+	0.877018i	$0.340470\pi$
$264$	0.352205	0.0216767
$265$	0.623463	0.0382990
$266$	6.50877	0.399078
$267$	−0.664889	−0.0406905
$268$	14.5779	0.890485
$269$	15.9836	0.974535	0.487267	−	0.873253i	$-0.337994\pi$
$269$	15.9836	0.974535	0.487267	+	0.873253i	$0.337994\pi$
$270$	−21.3251	−1.29780
$271$	−12.1165	−0.736023	−0.368011	−	0.929821i	$-0.619961\pi$
$271$	−12.1165	−0.736023	−0.368011	+	0.929821i	$0.619961\pi$
$272$	−3.12423	−0.189434
$273$	−49.9724	−3.02447
$274$	−14.6461	−0.884804
$275$	−0.387388	−0.0233604
$276$	5.98007	0.359958
$277$	−17.9231	−1.07689	−0.538447	−	0.842659i	$-0.680989\pi$
$277$	−17.9231	−1.07689	−0.538447	+	0.842659i	$0.680989\pi$
$278$	−18.4316	−1.10546
$279$	16.1795	0.968643
$280$	4.30040	0.256998
$281$	20.3334	1.21299	0.606495	−	0.795087i	$-0.292575\pi$
$281$	20.3334	1.21299	0.606495	+	0.795087i	$0.292575\pi$
$282$	−3.82658	−0.227869
$283$	1.61413	0.0959504	0.0479752	−	0.998849i	$-0.484723\pi$
$283$	1.61413	0.0959504	0.0479752	+	0.998849i	$0.484723\pi$
$284$	11.9703	0.710309
$285$	−6.46896	−0.383188
$286$	−0.395653	−0.0233955
$287$	−5.54492	−0.327307
$288$	−8.59041	−0.506195
$289$	−7.23921	−0.425836
$290$	1.93932	0.113881
$291$	38.4082	2.25153
$292$	−9.36331	−0.547946
$293$	0.643565	0.0375975	0.0187987	−	0.999823i	$-0.494016\pi$
$293$	0.643565	0.0375975	0.0187987	+	0.999823i	$0.494016\pi$
$294$	26.3189	1.53495
$295$	7.21806	0.420252
$296$	−5.06635	−0.294476
$297$	−1.96897	−0.114251
$298$	8.36090	0.484334
$299$	−6.71778	−0.388499
$300$	12.7482	0.736020
$301$	45.7739	2.63836
$302$	11.7406	0.675598
$303$	55.7144	3.20071
$304$	−1.69585	−0.0972637
$305$	−12.7597	−0.730618
$306$	26.8384	1.53425
$307$	5.29202	0.302031	0.151016	−	0.988531i	$-0.451746\pi$
$307$	5.29202	0.302031	0.151016	+	0.988531i	$0.451746\pi$
$308$	0.397061	0.0226247
$309$	9.26468	0.527049
$310$	2.11033	0.119858
$311$	13.1005	0.742863	0.371431	−	0.928460i	$-0.378867\pi$
$311$	13.1005	0.742863	0.371431	+	0.928460i	$0.378867\pi$
$312$	13.0202	0.737125
$313$	−10.9173	−0.617080	−0.308540	−	0.951211i	$-0.599840\pi$
$313$	−10.9173	−0.617080	−0.308540	+	0.951211i	$0.599840\pi$
$314$	−5.57613	−0.314679
$315$	−36.9423	−2.08146
$316$	−4.17319	−0.234760
$317$	−26.7093	−1.50015	−0.750073	−	0.661355i	$-0.769982\pi$
$317$	−26.7093	−1.50015	−0.750073	+	0.661355i	$0.769982\pi$
$318$	−1.89436	−0.106230
$319$	0.179060	0.0100254
$320$	−1.12046	−0.0626358
$321$	1.94471	0.108543
$322$	6.74168	0.375699
$323$	5.29822	0.294801
$324$	39.0240	2.16800
$325$	−14.3209	−0.794380
$326$	−0.689524	−0.0381892
$327$	−4.50001	−0.248851
$328$	1.44472	0.0797714
$329$	−4.31392	−0.237834
$330$	−0.394632	−0.0217238
$331$	−17.9560	−0.986949	−0.493474	−	0.869760i	$-0.664273\pi$
$331$	−17.9560	−0.986949	−0.493474	+	0.869760i	$0.664273\pi$
$332$	9.41280	0.516594
$333$	43.5220	2.38499
$334$	−6.56375	−0.359152
$335$	−16.3340	−0.892420
$336$	−13.0666	−0.712839
$337$	17.0461	0.928563	0.464281	−	0.885688i	$-0.346313\pi$
$337$	17.0461	0.928563	0.464281	+	0.885688i	$0.346313\pi$
$338$	−1.62644	−0.0884665
$339$	15.0003	0.814707
$340$	3.50058	0.189846
$341$	0.194849	0.0105516
$342$	14.5681	0.787750
$343$	2.80446	0.151427
$344$	−11.9263	−0.643024
$345$	−6.70044	−0.360740
$346$	−6.91163	−0.371572
$347$	10.0841	0.541341	0.270670	−	0.962672i	$-0.412755\pi$
$347$	10.0841	0.541341	0.270670	+	0.962672i	$0.412755\pi$
$348$	−5.89254	−0.315873
$349$	−1.68331	−0.0901056	−0.0450528	−	0.998985i	$-0.514346\pi$
$349$	−1.68331	−0.0901056	−0.0450528	+	0.998985i	$0.514346\pi$
$350$	14.3718	0.768208
$351$	−72.7885	−3.88516
$352$	−0.103454	−0.00551410
$353$	29.5254	1.57148	0.785738	−	0.618559i	$-0.212283\pi$
$353$	29.5254	1.57148	0.785738	+	0.618559i	$0.212283\pi$
$354$	−21.9317	−1.16566
$355$	−13.4123	−0.711853
$356$	0.195299	0.0103508
$357$	40.8229	2.16058
$358$	−14.6092	−0.772121
$359$	−10.9526	−0.578058	−0.289029	−	0.957320i	$-0.593332\pi$
$359$	−10.9526	−0.578058	−0.289029	+	0.957320i	$0.593332\pi$
$360$	9.62524	0.507295
$361$	−16.1241	−0.848636
$362$	−0.464401	−0.0244084
$363$	37.4127	1.96366
$364$	14.6785	0.769361
$365$	10.4912	0.549137
$366$	38.7697	2.02652
$367$	16.3364	0.852752	0.426376	−	0.904546i	$-0.359790\pi$
$367$	16.3364	0.852752	0.426376	+	0.904546i	$0.359790\pi$
$368$	−1.75653	−0.0915656
$369$	−12.4108	−0.646078
$370$	5.67666	0.295115
$371$	−2.13562	−0.110876
$372$	−6.41211	−0.332453
$373$	38.1215	1.97386	0.986929	−	0.161157i	$-0.0515226\pi$
$373$	38.1215	1.97386	0.986929	+	0.161157i	$0.0515226\pi$
$374$	0.323213	0.0167129
$375$	−33.3569	−1.72254
$376$	1.12399	0.0579652
$377$	6.61945	0.340919
$378$	73.0475	3.75716
$379$	7.40020	0.380123	0.190061	−	0.981772i	$-0.439131\pi$
$379$	7.40020	0.380123	0.190061	+	0.981772i	$0.439131\pi$
$380$	1.90014	0.0974750
$381$	−42.6079	−2.18287
$382$	−0.0538546	−0.00275544
$383$	2.61367	0.133552	0.0667760	−	0.997768i	$-0.478729\pi$
$383$	2.61367	0.133552	0.0667760	+	0.997768i	$0.478729\pi$
$384$	3.40447	0.173734
$385$	−0.444892	−0.0226738
$386$	−7.36228	−0.374730
$387$	102.452	5.20793
$388$	−11.2817	−0.572741
$389$	2.92045	0.148073	0.0740363	−	0.997256i	$-0.476412\pi$
$389$	2.92045	0.148073	0.0740363	+	0.997256i	$0.476412\pi$
$390$	−14.5887	−0.738727
$391$	5.48781	0.277530
$392$	−7.73070	−0.390459
$393$	57.6137	2.90623
$394$	−19.1876	−0.966655
$395$	4.67590	0.235270
$396$	0.888709	0.0446593
$397$	17.4895	0.877774	0.438887	−	0.898542i	$-0.355373\pi$
$397$	17.4895	0.877774	0.438887	+	0.898542i	$0.355373\pi$
$398$	10.0373	0.503123
$399$	22.1589	1.10933
$400$	−3.74456	−0.187228
$401$	0.619684	0.0309455	0.0154728	−	0.999880i	$-0.495075\pi$
$401$	0.619684	0.0309455	0.0154728	+	0.999880i	$0.495075\pi$
$402$	49.6299	2.47532
$403$	7.20313	0.358813
$404$	−16.3651	−0.814192
$405$	−43.7249	−2.17271
$406$	−6.64300	−0.329687
$407$	0.524132	0.0259803
$408$	−10.6363	−0.526577
$409$	−17.8248	−0.881379	−0.440689	−	0.897660i	$-0.645266\pi$
$409$	−17.8248	−0.881379	−0.440689	+	0.897660i	$0.645266\pi$
$410$	−1.61876	−0.0799447
$411$	−49.8623	−2.45952
$412$	−2.72133	−0.134070
$413$	−24.7249	−1.21663
$414$	15.0894	0.741601
$415$	−10.5467	−0.517717
$416$	−3.82445	−0.187509
$417$	−62.7499	−3.07288
$418$	0.175442	0.00858114
$419$	19.7361	0.964172	0.482086	−	0.876124i	$-0.339879\pi$
$419$	19.7361	0.964172	0.482086	+	0.876124i	$0.339879\pi$
$420$	14.6406	0.714388
$421$	7.14855	0.348399	0.174200	−	0.984710i	$-0.444266\pi$
$421$	7.14855	0.348399	0.174200	+	0.984710i	$0.444266\pi$
$422$	−0.217298	−0.0105779
$423$	−9.65550	−0.469467
$424$	0.556433	0.0270228
$425$	11.6989	0.567478
$426$	40.7527	1.97447
$427$	43.7074	2.11515
$428$	−0.571223	−0.0276111
$429$	−1.34699	−0.0650333
$430$	13.3630	0.644421
$431$	−28.3249	−1.36436	−0.682181	−	0.731183i	$-0.738968\pi$
$431$	−28.3249	−1.36436	−0.682181	+	0.731183i	$0.738968\pi$
$432$	−19.0324	−0.915696
$433$	−1.41383	−0.0679443	−0.0339721	−	0.999423i	$-0.510816\pi$
$433$	−1.41383	−0.0679443	−0.0339721	+	0.999423i	$0.510816\pi$
$434$	−7.22875	−0.346991
$435$	6.60237	0.316559
$436$	1.32179	0.0633024
$437$	2.97882	0.142496
$438$	−31.8771	−1.52315
$439$	−0.221097	−0.0105524	−0.00527618	−	0.999986i	$-0.501679\pi$
$439$	−0.221097	−0.0105524	−0.00527618	+	0.999986i	$0.501679\pi$
$440$	0.115916	0.00552608
$441$	66.4099	3.16238
$442$	11.9485	0.568330
$443$	−29.8839	−1.41983	−0.709914	−	0.704289i	$-0.751266\pi$
$443$	−29.8839	−1.41983	−0.709914	+	0.704289i	$0.751266\pi$
$444$	−17.2482	−0.818565
$445$	−0.218825	−0.0103733
$446$	−8.46482	−0.400821
$447$	28.4644	1.34632
$448$	3.83806	0.181331
$449$	−19.5468	−0.922469	−0.461234	−	0.887278i	$-0.652593\pi$
$449$	−19.5468	−0.922469	−0.461234	+	0.887278i	$0.652593\pi$
$450$	32.1673	1.51638
$451$	−0.149462	−0.00703787
$452$	−4.40607	−0.207244
$453$	39.9706	1.87798
$454$	13.6199	0.639214
$455$	−16.4467	−0.771033
$456$	−5.77347	−0.270368
$457$	−16.4176	−0.767985	−0.383992	−	0.923336i	$-0.625451\pi$
$457$	−16.4176	−0.767985	−0.383992	+	0.923336i	$0.625451\pi$
$458$	16.8844	0.788956
$459$	59.4615	2.77543
$460$	1.96813	0.0917646
$461$	−25.8489	−1.20390	−0.601951	−	0.798533i	$-0.705610\pi$
$461$	−25.8489	−1.20390	−0.601951	+	0.798533i	$0.705610\pi$
$462$	1.35178	0.0628906
$463$	18.2560	0.848428	0.424214	−	0.905562i	$-0.360550\pi$
$463$	18.2560	0.848428	0.424214	+	0.905562i	$0.360550\pi$
$464$	1.73082	0.0803515
$465$	7.18454	0.333175
$466$	10.1780	0.471487
$467$	−13.4379	−0.621834	−0.310917	−	0.950437i	$-0.600636\pi$
$467$	−13.4379	−0.621834	−0.310917	+	0.950437i	$0.600636\pi$
$468$	32.8536	1.51866
$469$	55.9507	2.58357
$470$	−1.25939	−0.0580911
$471$	−18.9838	−0.874726
$472$	6.44203	0.296518
$473$	1.23382	0.0567311
$474$	−14.2075	−0.652571
$475$	6.35022	0.291368
$476$	−11.9910	−0.549605
$477$	−4.77999	−0.218861
$478$	−5.56786	−0.254668
$479$	31.0748	1.41984	0.709921	−	0.704281i	$-0.248731\pi$
$479$	31.0748	1.41984	0.709921	+	0.704281i	$0.248731\pi$
$480$	−3.81458	−0.174111
$481$	19.3760	0.883470
$482$	−17.7321	−0.807677
$483$	22.9518	1.04435
$484$	−10.9893	−0.499514
$485$	12.6407	0.573986
$486$	75.7587	3.43649
$487$	−6.65137	−0.301402	−0.150701	−	0.988579i	$-0.548153\pi$
$487$	−6.65137	−0.301402	−0.150701	+	0.988579i	$0.548153\pi$
$488$	−11.3879	−0.515505
$489$	−2.34746	−0.106156
$490$	8.66196	0.391308
$491$	−1.68704	−0.0761349	−0.0380675	−	0.999275i	$-0.512120\pi$
$491$	−1.68704	−0.0761349	−0.0380675	+	0.999275i	$0.512120\pi$
$492$	4.91851	0.221743
$493$	−5.40748	−0.243541
$494$	6.48570	0.291805
$495$	−0.995766	−0.0447564
$496$	1.88344	0.0845689
$497$	45.9429	2.06082
$498$	32.0456	1.43600
$499$	−20.7511	−0.928947	−0.464474	−	0.885587i	$-0.653756\pi$
$499$	−20.7511	−0.928947	−0.464474	+	0.885587i	$0.653756\pi$
$500$	9.79796	0.438178
$501$	−22.3461	−0.998350
$502$	−7.39420	−0.330019
$503$	−21.2123	−0.945809	−0.472904	−	0.881114i	$-0.656794\pi$
$503$	−21.2123	−0.945809	−0.472904	+	0.881114i	$0.656794\pi$
$504$	−32.9705	−1.46862
$505$	18.3365	0.815961
$506$	0.181720	0.00807843
$507$	−5.53715	−0.245914
$508$	12.5153	0.555276
$509$	17.2262	0.763539	0.381770	−	0.924258i	$-0.375315\pi$
$509$	17.2262	0.763539	0.381770	+	0.924258i	$0.375315\pi$
$510$	11.9176	0.527721
$511$	−35.9369	−1.58976
$512$	−1.00000	−0.0441942
$513$	32.2761	1.42502
$514$	16.0060	0.705995
$515$	3.04915	0.134362
$516$	−40.6028	−1.78744
$517$	−0.116280	−0.00511401
$518$	−19.4450	−0.854362
$519$	−23.5304	−1.03287
$520$	4.28516	0.187917
$521$	−19.8601	−0.870088	−0.435044	−	0.900409i	$-0.643267\pi$
$521$	−19.8601	−0.870088	−0.435044	+	0.900409i	$0.643267\pi$
$522$	−14.8685	−0.650776
$523$	−0.834827	−0.0365044	−0.0182522	−	0.999833i	$-0.505810\pi$
$523$	−0.834827	−0.0365044	−0.0182522	+	0.999833i	$0.505810\pi$
$524$	−16.9230	−0.739283
$525$	48.9285	2.13542
$526$	−15.5834	−0.679470
$527$	−5.88429	−0.256324
$528$	−0.352205	−0.0153277
$529$	−19.9146	−0.865852
$530$	−0.623463	−0.0270815
$531$	−55.3397	−2.40154
$532$	−6.50877	−0.282191
$533$	−5.52527	−0.239326
$534$	0.664889	0.0287726
$535$	0.640034	0.0276711
$536$	−14.5779	−0.629668
$537$	−49.7367	−2.14630
$538$	−15.9836	−0.689100
$539$	0.799769	0.0344485
$540$	21.3251	0.917686
$541$	−12.0615	−0.518564	−0.259282	−	0.965802i	$-0.583486\pi$
$541$	−12.0615	−0.518564	−0.259282	+	0.965802i	$0.583486\pi$
$542$	12.1165	0.520447
$543$	−1.58104	−0.0678489
$544$	3.12423	0.133950
$545$	−1.48102	−0.0634400
$546$	49.9724	2.13862
$547$	20.8571	0.891786	0.445893	−	0.895086i	$-0.352886\pi$
$547$	20.8571	0.891786	0.445893	+	0.895086i	$0.352886\pi$
$548$	14.6461	0.625651
$549$	97.8266	4.17514
$550$	0.387388	0.0165183
$551$	−2.93522	−0.125044
$552$	−5.98007	−0.254528
$553$	−16.0169	−0.681109
$554$	17.9231	0.761479
$555$	19.3260	0.820344
$556$	18.4316	0.781675
$557$	12.8057	0.542596	0.271298	−	0.962495i	$-0.412547\pi$
$557$	12.8057	0.542596	0.271298	+	0.962495i	$0.412547\pi$
$558$	−16.1795	−0.684934
$559$	45.6116	1.92917
$560$	−4.30040	−0.181725
$561$	1.10037	0.0464575
$562$	−20.3334	−0.857714
$563$	−29.0983	−1.22635	−0.613173	−	0.789949i	$-0.710107\pi$
$563$	−29.0983	−1.22635	−0.613173	+	0.789949i	$0.710107\pi$
$564$	3.82658	0.161128
$565$	4.93684	0.207695
$566$	−1.61413	−0.0678472
$567$	149.776	6.29001
$568$	−11.9703	−0.502264
$569$	0.149402	0.00626327	0.00313164	−	0.999995i	$-0.499003\pi$
$569$	0.149402	0.00626327	0.00313164	+	0.999995i	$0.499003\pi$
$570$	6.46896	0.270955
$571$	20.9906	0.878428	0.439214	−	0.898382i	$-0.355257\pi$
$571$	20.9906	0.878428	0.439214	+	0.898382i	$0.355257\pi$
$572$	0.395653	0.0165431
$573$	−0.183346	−0.00765940
$574$	5.54492	0.231441
$575$	6.57745	0.274299
$576$	8.59041	0.357934
$577$	−35.1608	−1.46377	−0.731883	−	0.681431i	$-0.761358\pi$
$577$	−35.1608	−1.46377	−0.731883	+	0.681431i	$0.761358\pi$
$578$	7.23921	0.301111
$579$	−25.0647	−1.04165
$580$	−1.93932	−0.0805261
$581$	36.1269	1.49879
$582$	−38.4082	−1.59207
$583$	−0.0575650	−0.00238410
$584$	9.36331	0.387456
$585$	−36.8113	−1.52196
$586$	−0.643565	−0.0265854
$587$	36.5805	1.50984	0.754919	−	0.655818i	$-0.227676\pi$
$587$	36.5805	1.50984	0.754919	+	0.655818i	$0.227676\pi$
$588$	−26.3189	−1.08537
$589$	−3.19403	−0.131608
$590$	−7.21806	−0.297163
$591$	−65.3234	−2.68705
$592$	5.06635	0.208226
$593$	22.3199	0.916570	0.458285	−	0.888805i	$-0.348464\pi$
$593$	22.3199	0.916570	0.458285	+	0.888805i	$0.348464\pi$
$594$	1.96897	0.0807878
$595$	13.4354	0.550799
$596$	−8.36090	−0.342476
$597$	34.1716	1.39855
$598$	6.71778	0.274710
$599$	−13.1602	−0.537710	−0.268855	−	0.963181i	$-0.586645\pi$
$599$	−13.1602	−0.537710	−0.268855	+	0.963181i	$0.586645\pi$
$600$	−12.7482	−0.520445
$601$	−42.2678	−1.72414	−0.862070	−	0.506789i	$-0.830832\pi$
$601$	−42.2678	−1.72414	−0.862070	+	0.506789i	$0.830832\pi$
$602$	−45.7739	−1.86561
$603$	125.230	5.09976
$604$	−11.7406	−0.477720
$605$	12.3131	0.500599
$606$	−55.7144	−2.26324
$607$	43.6465	1.77156	0.885778	−	0.464109i	$-0.153625\pi$
$607$	43.6465	1.77156	0.885778	+	0.464109i	$0.153625\pi$
$608$	1.69585	0.0687758
$609$	−22.6159	−0.916442
$610$	12.7597	0.516625
$611$	−4.29863	−0.173904
$612$	−26.8384	−1.08488
$613$	−36.0549	−1.45625	−0.728123	−	0.685447i	$-0.759607\pi$
$613$	−36.0549	−1.45625	−0.728123	+	0.685447i	$0.759607\pi$
$614$	−5.29202	−0.213568
$615$	−5.51101	−0.222225
$616$	−0.397061	−0.0159980
$617$	−43.8146	−1.76391	−0.881955	−	0.471334i	$-0.843773\pi$
$617$	−43.8146	−1.76391	−0.881955	+	0.471334i	$0.843773\pi$
$618$	−9.26468	−0.372680
$619$	−1.64726	−0.0662091	−0.0331045	−	0.999452i	$-0.510539\pi$
$619$	−1.64726	−0.0662091	−0.0331045	+	0.999452i	$0.510539\pi$
$620$	−2.11033	−0.0847527
$621$	33.4310	1.34154
$622$	−13.1005	−0.525283
$623$	0.749568	0.0300308
$624$	−13.0202	−0.521226
$625$	7.74455	0.309782
$626$	10.9173	0.436341
$627$	0.597286	0.0238533
$628$	5.57613	0.222512
$629$	−15.8284	−0.631121
$630$	36.9423	1.47181
$631$	12.9588	0.515881	0.257941	−	0.966161i	$-0.416956\pi$
$631$	12.9588	0.515881	0.257941	+	0.966161i	$0.416956\pi$
$632$	4.17319	0.166000
$633$	−0.739786	−0.0294039
$634$	26.7093	1.06076
$635$	−14.0229	−0.556483
$636$	1.89436	0.0751162
$637$	29.5657	1.17144
$638$	−0.179060	−0.00708905
$639$	102.830	4.06790
$640$	1.12046	0.0442902
$641$	17.5748	0.694162	0.347081	−	0.937835i	$-0.387173\pi$
$641$	17.5748	0.694162	0.347081	+	0.937835i	$0.387173\pi$
$642$	−1.94471	−0.0767516
$643$	−25.3767	−1.00076	−0.500380	−	0.865806i	$-0.666806\pi$
$643$	−25.3767	−1.00076	−0.500380	+	0.865806i	$0.666806\pi$
$644$	−6.74168	−0.265659
$645$	45.4939	1.79132
$646$	−5.29822	−0.208456
$647$	6.83324	0.268642	0.134321	−	0.990938i	$-0.457115\pi$
$647$	6.83324	0.268642	0.134321	+	0.990938i	$0.457115\pi$
$648$	−39.0240	−1.53301
$649$	−0.666451	−0.0261605
$650$	14.3209	0.561712
$651$	−24.6101	−0.964545
$652$	0.689524	0.0270038
$653$	20.4040	0.798469	0.399235	−	0.916849i	$-0.369276\pi$
$653$	20.4040	0.798469	0.399235	+	0.916849i	$0.369276\pi$
$654$	4.50001	0.175964
$655$	18.9616	0.740890
$656$	−1.44472	−0.0564069
$657$	−80.4347	−3.13806
$658$	4.31392	0.168174
$659$	36.2105	1.41056	0.705280	−	0.708929i	$-0.250821\pi$
$659$	36.2105	1.41056	0.705280	+	0.708929i	$0.250821\pi$
$660$	0.394632	0.0153610
$661$	36.3919	1.41548	0.707740	−	0.706473i	$-0.249715\pi$
$661$	36.3919	1.41548	0.707740	+	0.706473i	$0.249715\pi$
$662$	17.9560	0.697878
$663$	40.6782	1.57981
$664$	−9.41280	−0.365287
$665$	7.29284	0.282804
$666$	−43.5220	−1.68644
$667$	−3.04025	−0.117719
$668$	6.56375	0.253959
$669$	−28.8182	−1.11418
$670$	16.3340	0.631037
$671$	1.17812	0.0454807
$672$	13.0666	0.504053
$673$	−10.8815	−0.419450	−0.209725	−	0.977760i	$-0.567257\pi$
$673$	−10.8815	−0.419450	−0.209725	+	0.977760i	$0.567257\pi$
$674$	−17.0461	−0.656593
$675$	71.2680	2.74311
$676$	1.62644	0.0625552
$677$	−2.38205	−0.0915496	−0.0457748	−	0.998952i	$-0.514576\pi$
$677$	−2.38205	−0.0915496	−0.0457748	+	0.998952i	$0.514576\pi$
$678$	−15.0003	−0.576085
$679$	−43.2998	−1.66169
$680$	−3.50058	−0.134241
$681$	46.3686	1.77685
$682$	−0.194849	−0.00746114
$683$	15.6160	0.597528	0.298764	−	0.954327i	$-0.403426\pi$
$683$	15.6160	0.597528	0.298764	+	0.954327i	$0.403426\pi$
$684$	−14.5681	−0.557023
$685$	−16.4104	−0.627011
$686$	−2.80446	−0.107075
$687$	57.4824	2.19309
$688$	11.9263	0.454687
$689$	−2.12805	−0.0810723
$690$	6.70044	0.255082
$691$	11.2577	0.428262	0.214131	−	0.976805i	$-0.431308\pi$
$691$	11.2577	0.428262	0.214131	+	0.976805i	$0.431308\pi$
$692$	6.91163	0.262741
$693$	3.41092	0.129570
$694$	−10.0841	−0.382786
$695$	−20.6520	−0.783374
$696$	5.89254	0.223356
$697$	4.51364	0.170966
$698$	1.68331	0.0637143
$699$	34.6507	1.31061
$700$	−14.3718	−0.543205
$701$	22.3718	0.844971	0.422486	−	0.906370i	$-0.361158\pi$
$701$	22.3718	0.844971	0.422486	+	0.906370i	$0.361158\pi$
$702$	72.7885	2.74722
$703$	−8.59177	−0.324045
$704$	0.103454	0.00389905
$705$	−4.28754	−0.161478
$706$	−29.5254	−1.11120
$707$	−62.8101	−2.36222
$708$	21.9317	0.824244
$709$	52.1103	1.95704	0.978522	−	0.206141i	$-0.0660905\pi$
$709$	52.1103	1.95704	0.978522	+	0.206141i	$0.0660905\pi$
$710$	13.4123	0.503356
$711$	−35.8494	−1.34446
$712$	−0.195299	−0.00731913
$713$	−3.30833	−0.123898
$714$	−40.8229	−1.52776
$715$	−0.443315	−0.0165790
$716$	14.6092	0.545972
$717$	−18.9556	−0.707910
$718$	10.9526	0.408749
$719$	33.7289	1.25788	0.628938	−	0.777456i	$-0.283490\pi$
$719$	33.7289	1.25788	0.628938	+	0.777456i	$0.283490\pi$
$720$	−9.62524	−0.358712
$721$	−10.4446	−0.388978
$722$	16.1241	0.600077
$723$	−60.3685	−2.24513
$724$	0.464401	0.0172593
$725$	−6.48117	−0.240705
$726$	−37.4127	−1.38852
$727$	48.2376	1.78903	0.894517	−	0.447033i	$-0.147519\pi$
$727$	48.2376	1.78903	0.894517	+	0.447033i	$0.147519\pi$
$728$	−14.6785	−0.544020
$729$	140.846	5.21653
$730$	−10.4912	−0.388298
$731$	−37.2605	−1.37813
$732$	−38.7697	−1.43297
$733$	30.7900	1.13725	0.568626	−	0.822596i	$-0.307475\pi$
$733$	30.7900	1.13725	0.568626	+	0.822596i	$0.307475\pi$
$734$	−16.3364	−0.602987
$735$	29.4894	1.08773
$736$	1.75653	0.0647467
$737$	1.50813	0.0555528
$738$	12.4108	0.456846
$739$	26.1911	0.963454	0.481727	−	0.876321i	$-0.340010\pi$
$739$	26.1911	0.963454	0.481727	+	0.876321i	$0.340010\pi$
$740$	−5.67666	−0.208678
$741$	22.0804	0.811142
$742$	2.13562	0.0784011
$743$	−8.62158	−0.316295	−0.158148	−	0.987415i	$-0.550552\pi$
$743$	−8.62158	−0.316295	−0.158148	+	0.987415i	$0.550552\pi$
$744$	6.41211	0.235079
$745$	9.36808	0.343220
$746$	−38.1215	−1.39573
$747$	80.8598	2.95851
$748$	−0.323213	−0.0118178
$749$	−2.19239	−0.0801081
$750$	33.3569	1.21802
$751$	−7.22212	−0.263539	−0.131769	−	0.991280i	$-0.542066\pi$
$751$	−7.22212	−0.263539	−0.131769	+	0.991280i	$0.542066\pi$
$752$	−1.12399	−0.0409876
$753$	−25.1733	−0.917367
$754$	−6.61945	−0.241066
$755$	13.1550	0.478758
$756$	−73.0475	−2.65671
$757$	−14.4122	−0.523819	−0.261910	−	0.965092i	$-0.584352\pi$
$757$	−14.4122	−0.523819	−0.261910	+	0.965092i	$0.584352\pi$
$758$	−7.40020	−0.268787
$759$	0.618659	0.0224559
$760$	−1.90014	−0.0689253
$761$	−24.9852	−0.905713	−0.452857	−	0.891583i	$-0.649595\pi$
$761$	−24.9852	−0.905713	−0.452857	+	0.891583i	$0.649595\pi$
$762$	42.6079	1.54352
$763$	5.07312	0.183659
$764$	0.0538546	0.00194839
$765$	30.0714	1.08724
$766$	−2.61367	−0.0944356
$767$	−24.6372	−0.889599
$768$	−3.40447	−0.122848
$769$	50.8227	1.83272	0.916358	−	0.400361i	$-0.131115\pi$
$769$	50.8227	1.83272	0.916358	+	0.400361i	$0.131115\pi$
$770$	0.444892	0.0160328
$771$	54.4920	1.96248
$772$	7.36228	0.264974
$773$	35.3377	1.27101	0.635505	−	0.772097i	$-0.280792\pi$
$773$	35.3377	1.27101	0.635505	+	0.772097i	$0.280792\pi$
$774$	−102.452	−3.68256
$775$	−7.05266	−0.253339
$776$	11.2817	0.404989
$777$	−66.1997	−2.37490
$778$	−2.92045	−0.104703
$779$	2.45003	0.0877814
$780$	14.5887	0.522359
$781$	1.23838	0.0443126
$782$	−5.48781	−0.196244
$783$	−32.9417	−1.17724
$784$	7.73070	0.276096
$785$	−6.24785	−0.222995
$786$	−57.6137	−2.05501
$787$	−19.8830	−0.708752	−0.354376	−	0.935103i	$-0.615307\pi$
$787$	−19.8830	−0.708752	−0.354376	+	0.935103i	$0.615307\pi$
$788$	19.1876	0.683528
$789$	−53.0534	−1.88875
$790$	−4.67590	−0.166361
$791$	−16.9108	−0.601278
$792$	−0.888709	−0.0315789
$793$	43.5524	1.54659
$794$	−17.4895	−0.620680
$795$	−2.12256	−0.0752794
$796$	−10.0373	−0.355761
$797$	−25.7452	−0.911943	−0.455971	−	0.889994i	$-0.650708\pi$
$797$	−25.7452	−0.911943	−0.455971	+	0.889994i	$0.650708\pi$
$798$	−22.1589	−0.784417
$799$	3.51159	0.124231
$800$	3.74456	0.132390
$801$	1.67770	0.0592785
$802$	−0.619684	−0.0218818
$803$	−0.968668	−0.0341835
$804$	−49.6299	−1.75031
$805$	7.55381	0.266237
$806$	−7.20313	−0.253719
$807$	−54.4155	−1.91552
$808$	16.3651	0.575721
$809$	10.2074	0.358873	0.179437	−	0.983770i	$-0.442572\pi$
$809$	10.2074	0.358873	0.179437	+	0.983770i	$0.442572\pi$
$810$	43.7249	1.53634
$811$	−6.72203	−0.236042	−0.118021	−	0.993011i	$-0.537655\pi$
$811$	−6.72203	−0.236042	−0.118021	+	0.993011i	$0.537655\pi$
$812$	6.64300	0.233124
$813$	41.2501	1.44671
$814$	−0.524132	−0.0183708
$815$	−0.772586	−0.0270625
$816$	10.6363	0.372346
$817$	−20.2252	−0.707592
$818$	17.8248	0.623229
$819$	126.094	4.40609
$820$	1.61876	0.0565294
$821$	−38.4075	−1.34043	−0.670216	−	0.742167i	$-0.733798\pi$
$821$	−38.4075	−1.34043	−0.670216	+	0.742167i	$0.733798\pi$
$822$	49.8623	1.73915
$823$	−5.44865	−0.189928	−0.0949640	−	0.995481i	$-0.530274\pi$
$823$	−5.44865	−0.189928	−0.0949640	+	0.995481i	$0.530274\pi$
$824$	2.72133	0.0948020
$825$	1.31885	0.0459165
$826$	24.7249	0.860289
$827$	27.7143	0.963720	0.481860	−	0.876248i	$-0.339961\pi$
$827$	27.7143	0.963720	0.481860	+	0.876248i	$0.339961\pi$
$828$	−15.0894	−0.524391
$829$	4.61777	0.160382	0.0801909	−	0.996780i	$-0.474447\pi$
$829$	4.61777	0.160382	0.0801909	+	0.996780i	$0.474447\pi$
$830$	10.5467	0.366081
$831$	61.0186	2.11671
$832$	3.82445	0.132589
$833$	−24.1525	−0.836833
$834$	62.7499	2.17285
$835$	−7.35444	−0.254511
$836$	−0.175442	−0.00606778
$837$	−35.8464	−1.23903
$838$	−19.7361	−0.681772
$839$	−14.2444	−0.491770	−0.245885	−	0.969299i	$-0.579079\pi$
$839$	−14.2444	−0.491770	−0.245885	+	0.969299i	$0.579079\pi$
$840$	−14.6406	−0.505149
$841$	−26.0043	−0.896698
$842$	−7.14855	−0.246355
$843$	−69.2245	−2.38422
$844$	0.217298	0.00747972
$845$	−1.82236	−0.0626912
$846$	9.65550	0.331963
$847$	−42.1776	−1.44924
$848$	−0.556433	−0.0191080
$849$	−5.49527	−0.188597
$850$	−11.6989	−0.401268
$851$	−8.89921	−0.305061
$852$	−40.7527	−1.39616
$853$	34.2844	1.17388	0.586938	−	0.809632i	$-0.300333\pi$
$853$	34.2844	1.17388	0.586938	+	0.809632i	$0.300333\pi$
$854$	−43.7074	−1.49563
$855$	16.3230	0.558234
$856$	0.571223	0.0195240
$857$	4.90205	0.167451	0.0837253	−	0.996489i	$-0.473318\pi$
$857$	4.90205	0.167451	0.0837253	+	0.996489i	$0.473318\pi$
$858$	1.34699	0.0459855
$859$	48.6695	1.66058	0.830292	−	0.557329i	$-0.188174\pi$
$859$	48.6695	1.66058	0.830292	+	0.557329i	$0.188174\pi$
$860$	−13.3630	−0.455675
$861$	18.8775	0.643344
$862$	28.3249	0.964750
$863$	−15.3208	−0.521526	−0.260763	−	0.965403i	$-0.583974\pi$
$863$	−15.3208	−0.521526	−0.260763	+	0.965403i	$0.583974\pi$
$864$	19.0324	0.647495
$865$	−7.74423	−0.263312
$866$	1.41383	0.0480438
$867$	24.6457	0.837011
$868$	7.22875	0.245360
$869$	−0.431731	−0.0146455
$870$	−6.60237	−0.223841
$871$	55.7524	1.88910
$872$	−1.32179	−0.0447616
$873$	−96.9144	−3.28006
$874$	−2.97882	−0.100760
$875$	37.6052	1.27129
$876$	31.8771	1.07703
$877$	37.8015	1.27646	0.638232	−	0.769844i	$-0.279666\pi$
$877$	37.8015	1.27646	0.638232	+	0.769844i	$0.279666\pi$
$878$	0.221097	0.00746165
$879$	−2.19100	−0.0739005
$880$	−0.115916	−0.00390753
$881$	−44.4516	−1.49761	−0.748806	−	0.662789i	$-0.769373\pi$
$881$	−44.4516	−1.49761	−0.748806	+	0.662789i	$0.769373\pi$
$882$	−66.4099	−2.23614
$883$	30.9777	1.04248	0.521241	−	0.853409i	$-0.325469\pi$
$883$	30.9777	1.04248	0.521241	+	0.853409i	$0.325469\pi$
$884$	−11.9485	−0.401870
$885$	−24.5737	−0.826035
$886$	29.8839	1.00397
$887$	33.2633	1.11687	0.558436	−	0.829547i	$-0.311401\pi$
$887$	33.2633	1.11687	0.558436	+	0.829547i	$0.311401\pi$
$888$	17.2482	0.578813
$889$	48.0344	1.61102
$890$	0.218825	0.00733504
$891$	4.03717	0.135250
$892$	8.46482	0.283423
$893$	1.90611	0.0637856
$894$	−28.4644	−0.951992
$895$	−16.3691	−0.547159
$896$	−3.83806	−0.128221
$897$	22.8705	0.763623
$898$	19.5468	0.652284
$899$	3.25990	0.108724
$900$	−32.1673	−1.07224
$901$	1.73842	0.0579152
$902$	0.149462	0.00497653
$903$	−155.836	−5.18589
$904$	4.40607	0.146544
$905$	−0.520345	−0.0172968
$906$	−39.9706	−1.32794
$907$	10.9486	0.363542	0.181771	−	0.983341i	$-0.441817\pi$
$907$	10.9486	0.363542	0.181771	+	0.983341i	$0.441817\pi$
$908$	−13.6199	−0.451993
$909$	−140.583	−4.66283
$910$	16.4467	0.545203
$911$	33.2020	1.10003	0.550015	−	0.835155i	$-0.314622\pi$
$911$	33.2020	1.10003	0.550015	+	0.835155i	$0.314622\pi$
$912$	5.77347	0.191179
$913$	0.973788	0.0322277
$914$	16.4176	0.543047
$915$	43.4400	1.43608
$916$	−16.8844	−0.557876
$917$	−64.9514	−2.14488
$918$	−59.4615	−1.96252
$919$	−18.8565	−0.622017	−0.311009	−	0.950407i	$-0.600667\pi$
$919$	−18.8565	−0.622017	−0.311009	+	0.950407i	$0.600667\pi$
$920$	−1.96813	−0.0648874
$921$	−18.0165	−0.593664
$922$	25.8489	0.851288
$923$	45.7800	1.50687
$924$	−1.35178	−0.0444704
$925$	−18.9713	−0.623771
$926$	−18.2560	−0.599929
$927$	−23.3773	−0.767812
$928$	−1.73082	−0.0568171
$929$	−18.9793	−0.622690	−0.311345	−	0.950297i	$-0.600780\pi$
$929$	−18.9793	−0.622690	−0.311345	+	0.950297i	$0.600780\pi$
$930$	−7.18454	−0.235590
$931$	−13.1101	−0.429666
$932$	−10.1780	−0.333392
$933$	−44.6003	−1.46015
$934$	13.4379	0.439703
$935$	0.362148	0.0118435
$936$	−32.8536	−1.07385
$937$	−6.18280	−0.201983	−0.100992	−	0.994887i	$-0.532202\pi$
$937$	−6.18280	−0.201983	−0.100992	+	0.994887i	$0.532202\pi$
$938$	−55.9507	−1.82686
$939$	37.1675	1.21291
$940$	1.25939	0.0410766
$941$	9.39988	0.306427	0.153214	−	0.988193i	$-0.451038\pi$
$941$	9.39988	0.306427	0.153214	+	0.988193i	$0.451038\pi$
$942$	18.9838	0.618525
$943$	2.53770	0.0826389
$944$	−6.44203	−0.209670
$945$	81.8470	2.66248
$946$	−1.23382	−0.0401150
$947$	50.3046	1.63468	0.817341	−	0.576155i	$-0.195447\pi$
$947$	50.3046	1.63468	0.817341	+	0.576155i	$0.195447\pi$
$948$	14.2075	0.461438
$949$	−35.8095	−1.16243
$950$	−6.35022	−0.206028
$951$	90.9311	2.94864
$952$	11.9910	0.388629
$953$	21.2584	0.688628	0.344314	−	0.938855i	$-0.388112\pi$
$953$	21.2584	0.688628	0.344314	+	0.938855i	$0.388112\pi$
$954$	4.77999	0.154758
$955$	−0.0603421	−0.00195262
$956$	5.56786	0.180077
$957$	−0.609604	−0.0197057
$958$	−31.0748	−1.00398
$959$	56.2127	1.81520
$960$	3.81458	0.123115
$961$	−27.4527	−0.885570
$962$	−19.3760	−0.624708
$963$	−4.90704	−0.158127
$964$	17.7321	0.571114
$965$	−8.24917	−0.265550
$966$	−22.9518	−0.738463
$967$	−27.1321	−0.872509	−0.436254	−	0.899823i	$-0.643695\pi$
$967$	−27.1321	−0.872509	−0.436254	+	0.899823i	$0.643695\pi$
$968$	10.9893	0.353209
$969$	−18.0376	−0.579452
$970$	−12.6407	−0.405869
$971$	12.3740	0.397100	0.198550	−	0.980091i	$-0.436377\pi$
$971$	12.3740	0.397100	0.198550	+	0.980091i	$0.436377\pi$
$972$	−75.7587	−2.42996
$973$	70.7417	2.26787
$974$	6.65137	0.213124
$975$	48.7551	1.56141
$976$	11.3879	0.364517
$977$	−29.3625	−0.939388	−0.469694	−	0.882829i	$-0.655636\pi$
$977$	−29.3625	−0.939388	−0.469694	+	0.882829i	$0.655636\pi$
$978$	2.34746	0.0750636
$979$	0.0202044	0.000645734 0
$980$	−8.66196	−0.276696
$981$	11.3548	0.362529
$982$	1.68704	0.0538355
$983$	58.2021	1.85636	0.928180	−	0.372132i	$-0.121373\pi$
$983$	58.2021	1.85636	0.928180	+	0.372132i	$0.121373\pi$
$984$	−4.91851	−0.156796
$985$	−21.4990	−0.685014
$986$	5.40748	0.172209
$987$	14.6866	0.467480
$988$	−6.48570	−0.206338
$989$	−20.9490	−0.666139
$990$	0.995766	0.0316475
$991$	36.4659	1.15838	0.579189	−	0.815193i	$-0.303369\pi$
$991$	36.4659	1.15838	0.579189	+	0.815193i	$0.303369\pi$
$992$	−1.88344	−0.0597993
$993$	61.1305	1.93992
$994$	−45.9429	−1.45722
$995$	11.2464	0.356535
$996$	−32.0456	−1.01540
$997$	27.1476	0.859772	0.429886	−	0.902883i	$-0.358554\pi$
$997$	27.1476	0.859772	0.429886	+	0.902883i	$0.358554\pi$
$998$	20.7511	0.656865
$999$	−96.4248	−3.05074

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.c.1.1	✓	86

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.c.1.1	✓	86	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.40447 q^{3} +1.00000 q^{4} -1.12046 q^{5} +3.40447 q^{6} +3.83806 q^{7} -1.00000 q^{8} +8.59041 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.40447 q^{3} +1.00000 q^{4} -1.12046 q^{5} +3.40447 q^{6} +3.83806 q^{7} -1.00000 q^{8} +8.59041 q^{9} +1.12046 q^{10} +0.103454 q^{11} -3.40447 q^{12} +3.82445 q^{13} -3.83806 q^{14} +3.81458 q^{15} +1.00000 q^{16} -3.12423 q^{17} -8.59041 q^{18} -1.69585 q^{19} -1.12046 q^{20} -13.0666 q^{21} -0.103454 q^{22} -1.75653 q^{23} +3.40447 q^{24} -3.74456 q^{25} -3.82445 q^{26} -19.0324 q^{27} +3.83806 q^{28} +1.73082 q^{29} -3.81458 q^{30} +1.88344 q^{31} -1.00000 q^{32} -0.352205 q^{33} +3.12423 q^{34} -4.30040 q^{35} +8.59041 q^{36} +5.06635 q^{37} +1.69585 q^{38} -13.0202 q^{39} +1.12046 q^{40} -1.44472 q^{41} +13.0666 q^{42} +11.9263 q^{43} +0.103454 q^{44} -9.62524 q^{45} +1.75653 q^{46} -1.12399 q^{47} -3.40447 q^{48} +7.73070 q^{49} +3.74456 q^{50} +10.6363 q^{51} +3.82445 q^{52} -0.556433 q^{53} +19.0324 q^{54} -0.115916 q^{55} -3.83806 q^{56} +5.77347 q^{57} -1.73082 q^{58} -6.44203 q^{59} +3.81458 q^{60} +11.3879 q^{61} -1.88344 q^{62} +32.9705 q^{63} +1.00000 q^{64} -4.28516 q^{65} +0.352205 q^{66} +14.5779 q^{67} -3.12423 q^{68} +5.98007 q^{69} +4.30040 q^{70} +11.9703 q^{71} -8.59041 q^{72} -9.36331 q^{73} -5.06635 q^{74} +12.7482 q^{75} -1.69585 q^{76} +0.397061 q^{77} +13.0202 q^{78} -4.17319 q^{79} -1.12046 q^{80} +39.0240 q^{81} +1.44472 q^{82} +9.41280 q^{83} -13.0666 q^{84} +3.50058 q^{85} -11.9263 q^{86} -5.89254 q^{87} -0.103454 q^{88} +0.195299 q^{89} +9.62524 q^{90} +14.6785 q^{91} -1.75653 q^{92} -6.41211 q^{93} +1.12399 q^{94} +1.90014 q^{95} +3.40447 q^{96} -11.2817 q^{97} -7.73070 q^{98} +0.888709 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9 + 4 * q^10 + 13 * q^11 + 12 * q^12 + 45 * q^13 - 35 * q^14 + 17 * q^15 + 86 * q^16 + 5 * q^17 - 72 * q^18 + 47 * q^19 - 4 * q^20 + 15 * q^21 - 13 * q^22 + 6 * q^23 - 12 * q^24 + 112 * q^25 - 45 * q^26 + 51 * q^27 + 35 * q^28 - 14 * q^29 - 17 * q^30 + 24 * q^31 - 86 * q^32 + 43 * q^33 - 5 * q^34 + 42 * q^35 + 72 * q^36 + 61 * q^37 - 47 * q^38 + 20 * q^39 + 4 * q^40 - 16 * q^41 - 15 * q^42 + 72 * q^43 + 13 * q^44 + 6 * q^45 - 6 * q^46 + 11 * q^47 + 12 * q^48 + 89 * q^49 - 112 * q^50 + 56 * q^51 + 45 * q^52 - 7 * q^53 - 51 * q^54 + 48 * q^55 - 35 * q^56 + 65 * q^57 + 14 * q^58 + 24 * q^59 + 17 * q^60 + 31 * q^61 - 24 * q^62 + 98 * q^63 + 86 * q^64 - 9 * q^65 - 43 * q^66 + 157 * q^67 + 5 * q^68 + q^69 - 42 * q^70 - 11 * q^71 - 72 * q^72 + 74 * q^73 - 61 * q^74 + 76 * q^75 + 47 * q^76 - 13 * q^77 - 20 * q^78 + 57 * q^79 - 4 * q^80 + 34 * q^81 + 16 * q^82 + 65 * q^83 + 15 * q^84 + 102 * q^85 - 72 * q^86 + 49 * q^87 - 13 * q^88 - 34 * q^89 - 6 * q^90 + 91 * q^91 + 6 * q^92 + 57 * q^93 - 11 * q^94 - 13 * q^95 - 12 * q^96 + 64 * q^97 - 89 * q^98 + 56 * q^99

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