LMFDB - Embedded newform 8043.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.2236783457$
Analytic rank:	$1$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.75032 q^{2} +1.00000 q^{3} +5.56428 q^{4} +0.854023 q^{5} -2.75032 q^{6} +1.00000 q^{7} -9.80291 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.75032 q^{2} +1.00000 q^{3} +5.56428 q^{4} +0.854023 q^{5} -2.75032 q^{6} +1.00000 q^{7} -9.80291 q^{8} +1.00000 q^{9} -2.34884 q^{10} +4.74593 q^{11} +5.56428 q^{12} -1.71642 q^{13} -2.75032 q^{14} +0.854023 q^{15} +15.8326 q^{16} -7.45667 q^{17} -2.75032 q^{18} +0.607673 q^{19} +4.75202 q^{20} +1.00000 q^{21} -13.0529 q^{22} -1.05212 q^{23} -9.80291 q^{24} -4.27065 q^{25} +4.72071 q^{26} +1.00000 q^{27} +5.56428 q^{28} -5.60590 q^{29} -2.34884 q^{30} +1.97279 q^{31} -23.9390 q^{32} +4.74593 q^{33} +20.5082 q^{34} +0.854023 q^{35} +5.56428 q^{36} -8.12508 q^{37} -1.67130 q^{38} -1.71642 q^{39} -8.37191 q^{40} +6.62182 q^{41} -2.75032 q^{42} +12.4343 q^{43} +26.4077 q^{44} +0.854023 q^{45} +2.89367 q^{46} -3.21971 q^{47} +15.8326 q^{48} +1.00000 q^{49} +11.7457 q^{50} -7.45667 q^{51} -9.55063 q^{52} -0.926059 q^{53} -2.75032 q^{54} +4.05313 q^{55} -9.80291 q^{56} +0.607673 q^{57} +15.4180 q^{58} -5.22147 q^{59} +4.75202 q^{60} -6.34053 q^{61} -5.42580 q^{62} +1.00000 q^{63} +34.1748 q^{64} -1.46586 q^{65} -13.0529 q^{66} -10.1668 q^{67} -41.4910 q^{68} -1.05212 q^{69} -2.34884 q^{70} -14.9675 q^{71} -9.80291 q^{72} +9.16404 q^{73} +22.3466 q^{74} -4.27065 q^{75} +3.38126 q^{76} +4.74593 q^{77} +4.72071 q^{78} +2.09803 q^{79} +13.5214 q^{80} +1.00000 q^{81} -18.2121 q^{82} -5.48491 q^{83} +5.56428 q^{84} -6.36816 q^{85} -34.1982 q^{86} -5.60590 q^{87} -46.5240 q^{88} -4.86707 q^{89} -2.34884 q^{90} -1.71642 q^{91} -5.85429 q^{92} +1.97279 q^{93} +8.85525 q^{94} +0.518966 q^{95} -23.9390 q^{96} +13.7470 q^{97} -2.75032 q^{98} +4.74593 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9}+O(q^{10}) $ 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9	\( 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9} - 14 q^{10} - 29 q^{11} + 33 q^{12} - 40 q^{13} - 9 q^{14} - 27 q^{15} + 23 q^{16} - 46 q^{17} - 9 q^{18} + 17 q^{19} - 53 q^{20} + 40 q^{21} - 15 q^{22} - 62 q^{23} - 18 q^{24} + 35 q^{25} - 29 q^{26} + 40 q^{27} + 33 q^{28} - 47 q^{29} - 14 q^{30} + q^{31} - 45 q^{32} - 29 q^{33} + 9 q^{34} - 27 q^{35} + 33 q^{36} - 49 q^{37} - 52 q^{38} - 40 q^{39} - 12 q^{40} - 49 q^{41} - 9 q^{42} - 45 q^{43} - 68 q^{44} - 27 q^{45} - 36 q^{46} - 88 q^{47} + 23 q^{48} + 40 q^{49} - 32 q^{50} - 46 q^{51} - 34 q^{52} - 96 q^{53} - 9 q^{54} + 26 q^{55} - 18 q^{56} + 17 q^{57} + 12 q^{58} - 45 q^{59} - 53 q^{60} - 23 q^{61} - 15 q^{62} + 40 q^{63} + 50 q^{64} - 32 q^{65} - 15 q^{66} - 31 q^{67} - 76 q^{68} - 62 q^{69} - 14 q^{70} - 89 q^{71} - 18 q^{72} + 4 q^{73} + 10 q^{74} + 35 q^{75} + 51 q^{76} - 29 q^{77} - 29 q^{78} - 44 q^{79} - 53 q^{80} + 40 q^{81} - 15 q^{82} - 60 q^{83} + 33 q^{84} - 24 q^{85} - 9 q^{86} - 47 q^{87} - 64 q^{88} - 39 q^{89} - 14 q^{90} - 40 q^{91} - 80 q^{92} + q^{93} + 51 q^{94} - 71 q^{95} - 45 q^{96} - 21 q^{97} - 9 q^{98} - 29 q^{99}+O(q^{100}) \) 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9 - 14 * q^10 - 29 * q^11 + 33 * q^12 - 40 * q^13 - 9 * q^14 - 27 * q^15 + 23 * q^16 - 46 * q^17 - 9 * q^18 + 17 * q^19 - 53 * q^20 + 40 * q^21 - 15 * q^22 - 62 * q^23 - 18 * q^24 + 35 * q^25 - 29 * q^26 + 40 * q^27 + 33 * q^28 - 47 * q^29 - 14 * q^30 + q^31 - 45 * q^32 - 29 * q^33 + 9 * q^34 - 27 * q^35 + 33 * q^36 - 49 * q^37 - 52 * q^38 - 40 * q^39 - 12 * q^40 - 49 * q^41 - 9 * q^42 - 45 * q^43 - 68 * q^44 - 27 * q^45 - 36 * q^46 - 88 * q^47 + 23 * q^48 + 40 * q^49 - 32 * q^50 - 46 * q^51 - 34 * q^52 - 96 * q^53 - 9 * q^54 + 26 * q^55 - 18 * q^56 + 17 * q^57 + 12 * q^58 - 45 * q^59 - 53 * q^60 - 23 * q^61 - 15 * q^62 + 40 * q^63 + 50 * q^64 - 32 * q^65 - 15 * q^66 - 31 * q^67 - 76 * q^68 - 62 * q^69 - 14 * q^70 - 89 * q^71 - 18 * q^72 + 4 * q^73 + 10 * q^74 + 35 * q^75 + 51 * q^76 - 29 * q^77 - 29 * q^78 - 44 * q^79 - 53 * q^80 + 40 * q^81 - 15 * q^82 - 60 * q^83 + 33 * q^84 - 24 * q^85 - 9 * q^86 - 47 * q^87 - 64 * q^88 - 39 * q^89 - 14 * q^90 - 40 * q^91 - 80 * q^92 + q^93 + 51 * q^94 - 71 * q^95 - 45 * q^96 - 21 * q^97 - 9 * q^98 - 29 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.75032	−1.94477	−0.972386	−	0.233378i	$-0.925022\pi$
$2$	−2.75032	−1.94477	−0.972386	+	0.233378i	$0.925022\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.56428	2.78214
$5$	0.854023	0.381930	0.190965	−	0.981597i	$-0.438838\pi$
$5$	0.854023	0.381930	0.190965	+	0.981597i	$0.438838\pi$
$6$	−2.75032	−1.12281
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−9.80291	−3.46585
$9$	1.00000	0.333333
$10$	−2.34884	−0.742768
$11$	4.74593	1.43095	0.715477	−	0.698637i	$-0.246210\pi$
$11$	4.74593	1.43095	0.715477	+	0.698637i	$0.246210\pi$
$12$	5.56428	1.60627
$13$	−1.71642	−0.476049	−0.238024	−	0.971259i	$-0.576500\pi$
$13$	−1.71642	−0.476049	−0.238024	+	0.971259i	$0.576500\pi$
$14$	−2.75032	−0.735055
$15$	0.854023	0.220508
$16$	15.8326	3.95816
$17$	−7.45667	−1.80851	−0.904254	−	0.426996i	$-0.859572\pi$
$17$	−7.45667	−1.80851	−0.904254	+	0.426996i	$0.859572\pi$
$18$	−2.75032	−0.648257
$19$	0.607673	0.139410	0.0697049	−	0.997568i	$-0.477794\pi$
$19$	0.607673	0.139410	0.0697049	+	0.997568i	$0.477794\pi$
$20$	4.75202	1.06258
$21$	1.00000	0.218218
$22$	−13.0529	−2.78288
$23$	−1.05212	−0.219382	−0.109691	−	0.993966i	$-0.534986\pi$
$23$	−1.05212	−0.219382	−0.109691	+	0.993966i	$0.534986\pi$
$24$	−9.80291	−2.00101
$25$	−4.27065	−0.854129
$26$	4.72071	0.925807
$27$	1.00000	0.192450
$28$	5.56428	1.05155
$29$	−5.60590	−1.04099	−0.520494	−	0.853865i	$-0.674252\pi$
$29$	−5.60590	−1.04099	−0.520494	+	0.853865i	$0.674252\pi$
$30$	−2.34884	−0.428837
$31$	1.97279	0.354323	0.177161	−	0.984182i	$-0.443309\pi$
$31$	1.97279	0.354323	0.177161	+	0.984182i	$0.443309\pi$
$32$	−23.9390	−4.23186
$33$	4.74593	0.826161
$34$	20.5082	3.51713
$35$	0.854023	0.144356
$36$	5.56428	0.927380
$37$	−8.12508	−1.33576	−0.667878	−	0.744271i	$-0.732797\pi$
$37$	−8.12508	−1.33576	−0.667878	+	0.744271i	$0.732797\pi$
$38$	−1.67130	−0.271120
$39$	−1.71642	−0.274847
$40$	−8.37191	−1.32372
$41$	6.62182	1.03415	0.517077	−	0.855939i	$-0.327020\pi$
$41$	6.62182	1.03415	0.517077	+	0.855939i	$0.327020\pi$
$42$	−2.75032	−0.424384
$43$	12.4343	1.89621	0.948103	−	0.317964i	$-0.102999\pi$
$43$	12.4343	1.89621	0.948103	+	0.317964i	$0.102999\pi$
$44$	26.4077	3.98111
$45$	0.854023	0.127310
$46$	2.89367	0.426649
$47$	−3.21971	−0.469643	−0.234822	−	0.972038i	$-0.575451\pi$
$47$	−3.21971	−0.469643	−0.234822	+	0.972038i	$0.575451\pi$
$48$	15.8326	2.28524
$49$	1.00000	0.142857
$50$	11.7457	1.66109
$51$	−7.45667	−1.04414
$52$	−9.55063	−1.32443
$53$	−0.926059	−0.127204	−0.0636020	−	0.997975i	$-0.520259\pi$
$53$	−0.926059	−0.127204	−0.0636020	+	0.997975i	$0.520259\pi$
$54$	−2.75032	−0.374272
$55$	4.05313	0.546525
$56$	−9.80291	−1.30997
$57$	0.607673	0.0804882
$58$	15.4180	2.02449
$59$	−5.22147	−0.679777	−0.339888	−	0.940466i	$-0.610389\pi$
$59$	−5.22147	−0.679777	−0.339888	+	0.940466i	$0.610389\pi$
$60$	4.75202	0.613483
$61$	−6.34053	−0.811822	−0.405911	−	0.913913i	$-0.633046\pi$
$61$	−6.34053	−0.811822	−0.405911	+	0.913913i	$0.633046\pi$
$62$	−5.42580	−0.689077
$63$	1.00000	0.125988
$64$	34.1748	4.27185
$65$	−1.46586	−0.181818
$66$	−13.0529	−1.60670
$67$	−10.1668	−1.24207	−0.621036	−	0.783782i	$-0.713288\pi$
$67$	−10.1668	−1.24207	−0.621036	+	0.783782i	$0.713288\pi$
$68$	−41.4910	−5.03152
$69$	−1.05212	−0.126660
$70$	−2.34884	−0.280740
$71$	−14.9675	−1.77631	−0.888156	−	0.459541i	$-0.848014\pi$
$71$	−14.9675	−1.77631	−0.888156	+	0.459541i	$0.848014\pi$
$72$	−9.80291	−1.15528
$73$	9.16404	1.07257	0.536285	−	0.844037i	$-0.319827\pi$
$73$	9.16404	1.07257	0.536285	+	0.844037i	$0.319827\pi$
$74$	22.3466	2.59774
$75$	−4.27065	−0.493132
$76$	3.38126	0.387857
$77$	4.74593	0.540849
$78$	4.72071	0.534515
$79$	2.09803	0.236047	0.118024	−	0.993011i	$-0.462344\pi$
$79$	2.09803	0.236047	0.118024	+	0.993011i	$0.462344\pi$
$80$	13.5214	1.51174
$81$	1.00000	0.111111
$82$	−18.2121	−2.01119
$83$	−5.48491	−0.602047	−0.301024	−	0.953617i	$-0.597328\pi$
$83$	−5.48491	−0.602047	−0.301024	+	0.953617i	$0.597328\pi$
$84$	5.56428	0.607112
$85$	−6.36816	−0.690724
$86$	−34.1982	−3.68769
$87$	−5.60590	−0.601015
$88$	−46.5240	−4.95947
$89$	−4.86707	−0.515908	−0.257954	−	0.966157i	$-0.583048\pi$
$89$	−4.86707	−0.515908	−0.257954	+	0.966157i	$0.583048\pi$
$90$	−2.34884	−0.247589
$91$	−1.71642	−0.179930
$92$	−5.85429	−0.610352
$93$	1.97279	0.204568
$94$	8.85525	0.913349
$95$	0.518966	0.0532448
$96$	−23.9390	−2.44327
$97$	13.7470	1.39580	0.697901	−	0.716195i	$-0.254118\pi$
$97$	13.7470	1.39580	0.697901	+	0.716195i	$0.254118\pi$
$98$	−2.75032	−0.277825
$99$	4.74593	0.476984
$100$	−23.7631	−2.37631
$101$	9.53998	0.949263	0.474632	−	0.880185i	$-0.342581\pi$
$101$	9.53998	0.949263	0.474632	+	0.880185i	$0.342581\pi$
$102$	20.5082	2.03062
$103$	−2.73317	−0.269307	−0.134654	−	0.990893i	$-0.542992\pi$
$103$	−2.73317	−0.269307	−0.134654	+	0.990893i	$0.542992\pi$
$104$	16.8259	1.64992
$105$	0.854023	0.0833441
$106$	2.54696	0.247383
$107$	4.85054	0.468920	0.234460	−	0.972126i	$-0.424668\pi$
$107$	4.85054	0.468920	0.234460	+	0.972126i	$0.424668\pi$
$108$	5.56428	0.535423
$109$	7.38359	0.707219	0.353610	−	0.935393i	$-0.384954\pi$
$109$	7.38359	0.707219	0.353610	+	0.935393i	$0.384954\pi$
$110$	−11.1474	−1.06287
$111$	−8.12508	−0.771199
$112$	15.8326	1.49604
$113$	−20.3300	−1.91249	−0.956243	−	0.292574i	$-0.905488\pi$
$113$	−20.3300	−1.91249	−0.956243	+	0.292574i	$0.905488\pi$
$114$	−1.67130	−0.156531
$115$	−0.898535	−0.0837888
$116$	−31.1928	−2.89618
$117$	−1.71642	−0.158683
$118$	14.3607	1.32201
$119$	−7.45667	−0.683551
$120$	−8.37191	−0.764247
$121$	11.5239	1.04763
$122$	17.4385	1.57881
$123$	6.62182	0.597069
$124$	10.9771	0.985776
$125$	−7.91734	−0.708148
$126$	−2.75032	−0.245018
$127$	15.5686	1.38149	0.690743	−	0.723100i	$-0.257284\pi$
$127$	15.5686	1.38149	0.690743	+	0.723100i	$0.257284\pi$
$128$	−46.1136	−4.07591
$129$	12.4343	1.09477
$130$	4.03159	0.353594
$131$	−14.3482	−1.25361	−0.626803	−	0.779178i	$-0.715637\pi$
$131$	−14.3482	−1.25361	−0.626803	+	0.779178i	$0.715637\pi$
$132$	26.4077	2.29849
$133$	0.607673	0.0526919
$134$	27.9620	2.41555
$135$	0.854023	0.0735026
$136$	73.0971	6.26802
$137$	7.46065	0.637406	0.318703	−	0.947855i	$-0.396753\pi$
$137$	7.46065	0.637406	0.318703	+	0.947855i	$0.396753\pi$
$138$	2.89367	0.246326
$139$	11.7357	0.995410	0.497705	−	0.867346i	$-0.334176\pi$
$139$	11.7357	0.995410	0.497705	+	0.867346i	$0.334176\pi$
$140$	4.75202	0.401619
$141$	−3.21971	−0.271149
$142$	41.1654	3.45452
$143$	−8.14601	−0.681204
$144$	15.8326	1.31939
$145$	−4.78756	−0.397585
$146$	−25.2041	−2.08590
$147$	1.00000	0.0824786
$148$	−45.2102	−3.71626
$149$	2.67691	0.219301	0.109651	−	0.993970i	$-0.465027\pi$
$149$	2.67691	0.219301	0.109651	+	0.993970i	$0.465027\pi$
$150$	11.7457	0.959029
$151$	−4.88270	−0.397349	−0.198674	−	0.980066i	$-0.563664\pi$
$151$	−4.88270	−0.397349	−0.198674	+	0.980066i	$0.563664\pi$
$152$	−5.95696	−0.483174
$153$	−7.45667	−0.602836
$154$	−13.0529	−1.05183
$155$	1.68480	0.135327
$156$	−9.55063	−0.764662
$157$	−7.78436	−0.621260	−0.310630	−	0.950531i	$-0.600540\pi$
$157$	−7.78436	−0.621260	−0.310630	+	0.950531i	$0.600540\pi$
$158$	−5.77027	−0.459058
$159$	−0.926059	−0.0734413
$160$	−20.4445	−1.61628
$161$	−1.05212	−0.0829187
$162$	−2.75032	−0.216086
$163$	−12.3715	−0.969009	−0.484505	−	0.874789i	$-0.661000\pi$
$163$	−12.3715	−0.969009	−0.484505	+	0.874789i	$0.661000\pi$
$164$	36.8456	2.87716
$165$	4.05313	0.315536
$166$	15.0853	1.17084
$167$	−23.3068	−1.80353	−0.901766	−	0.432224i	$-0.857729\pi$
$167$	−23.3068	−1.80353	−0.901766	+	0.432224i	$0.857729\pi$
$168$	−9.80291	−0.756311
$169$	−10.0539	−0.773377
$170$	17.5145	1.34330
$171$	0.607673	0.0464699
$172$	69.1876	5.27551
$173$	−12.8639	−0.978022	−0.489011	−	0.872278i	$-0.662642\pi$
$173$	−12.8639	−0.978022	−0.489011	+	0.872278i	$0.662642\pi$
$174$	15.4180	1.16884
$175$	−4.27065	−0.322830
$176$	75.1406	5.66394
$177$	−5.22147	−0.392469
$178$	13.3860	1.00332
$179$	12.9600	0.968676	0.484338	−	0.874881i	$-0.339060\pi$
$179$	12.9600	0.968676	0.484338	+	0.874881i	$0.339060\pi$
$180$	4.75202	0.354195
$181$	−7.99964	−0.594609	−0.297304	−	0.954783i	$-0.596088\pi$
$181$	−7.99964	−0.594609	−0.297304	+	0.954783i	$0.596088\pi$
$182$	4.72071	0.349922
$183$	−6.34053	−0.468706
$184$	10.3139	0.760347
$185$	−6.93900	−0.510166
$186$	−5.42580	−0.397839
$187$	−35.3888	−2.58789
$188$	−17.9154	−1.30661
$189$	1.00000	0.0727393
$190$	−1.42733	−0.103549
$191$	−4.52477	−0.327401	−0.163700	−	0.986510i	$-0.552343\pi$
$191$	−4.52477	−0.327401	−0.163700	+	0.986510i	$0.552343\pi$
$192$	34.1748	2.46635
$193$	−27.3893	−1.97152	−0.985761	−	0.168150i	$-0.946221\pi$
$193$	−27.3893	−1.97152	−0.985761	+	0.168150i	$0.946221\pi$
$194$	−37.8088	−2.71451
$195$	−1.46586	−0.104972
$196$	5.56428	0.397448
$197$	−13.5507	−0.965447	−0.482724	−	0.875773i	$-0.660352\pi$
$197$	−13.5507	−0.965447	−0.482724	+	0.875773i	$0.660352\pi$
$198$	−13.0529	−0.927626
$199$	−18.7070	−1.32611	−0.663054	−	0.748572i	$-0.730740\pi$
$199$	−18.7070	−1.32611	−0.663054	+	0.748572i	$0.730740\pi$
$200$	41.8648	2.96029
$201$	−10.1668	−0.717111
$202$	−26.2380	−1.84610
$203$	−5.60590	−0.393457
$204$	−41.4910	−2.90495
$205$	5.65518	0.394975
$206$	7.51710	0.523741
$207$	−1.05212	−0.0731275
$208$	−27.1754	−1.88428
$209$	2.88398	0.199489
$210$	−2.34884	−0.162085
$211$	−28.2125	−1.94223	−0.971114	−	0.238618i	$-0.923306\pi$
$211$	−28.2125	−1.94223	−0.971114	+	0.238618i	$0.923306\pi$
$212$	−5.15285	−0.353899
$213$	−14.9675	−1.02555
$214$	−13.3406	−0.911942
$215$	10.6191	0.724219
$216$	−9.80291	−0.667004
$217$	1.97279	0.133921
$218$	−20.3072	−1.37538
$219$	9.16404	0.619248
$220$	22.5528	1.52051
$221$	12.7988	0.860938
$222$	22.3466	1.49981
$223$	−18.1527	−1.21560	−0.607799	−	0.794091i	$-0.707947\pi$
$223$	−18.1527	−1.21560	−0.607799	+	0.794091i	$0.707947\pi$
$224$	−23.9390	−1.59949
$225$	−4.27065	−0.284710
$226$	55.9141	3.71935
$227$	−9.33228	−0.619405	−0.309703	−	0.950833i	$-0.600230\pi$
$227$	−9.33228	−0.619405	−0.309703	+	0.950833i	$0.600230\pi$
$228$	3.38126	0.223929
$229$	27.5946	1.82350	0.911750	−	0.410745i	$-0.134731\pi$
$229$	27.5946	1.82350	0.911750	+	0.410745i	$0.134731\pi$
$230$	2.47126	0.162950
$231$	4.74593	0.312260
$232$	54.9541	3.60792
$233$	25.7212	1.68505	0.842525	−	0.538657i	$-0.181068\pi$
$233$	25.7212	1.68505	0.842525	+	0.538657i	$0.181068\pi$
$234$	4.72071	0.308602
$235$	−2.74971	−0.179371
$236$	−29.0537	−1.89123
$237$	2.09803	0.136282
$238$	20.5082	1.32935
$239$	15.1424	0.979482	0.489741	−	0.871868i	$-0.337091\pi$
$239$	15.1424	0.979482	0.489741	+	0.871868i	$0.337091\pi$
$240$	13.5214	0.872804
$241$	12.2755	0.790732	0.395366	−	0.918524i	$-0.370618\pi$
$241$	12.2755	0.790732	0.395366	+	0.918524i	$0.370618\pi$
$242$	−31.6944	−2.03740
$243$	1.00000	0.0641500
$244$	−35.2805	−2.25860
$245$	0.854023	0.0545615
$246$	−18.2121	−1.16116
$247$	−1.04302	−0.0663658
$248$	−19.3391	−1.22803
$249$	−5.48491	−0.347592
$250$	21.7752	1.37719
$251$	−1.98377	−0.125215	−0.0626073	−	0.998038i	$-0.519942\pi$
$251$	−1.98377	−0.125215	−0.0626073	+	0.998038i	$0.519942\pi$
$252$	5.56428	0.350517
$253$	−4.99330	−0.313926
$254$	−42.8186	−2.68668
$255$	−6.36816	−0.398790
$256$	58.4778	3.65486
$257$	−31.6093	−1.97173	−0.985866	−	0.167537i	$-0.946419\pi$
$257$	−31.6093	−1.97173	−0.985866	+	0.167537i	$0.946419\pi$
$258$	−34.1982	−2.12909
$259$	−8.12508	−0.504868
$260$	−8.15645	−0.505842
$261$	−5.60590	−0.346996
$262$	39.4621	2.43798
$263$	5.86871	0.361880	0.180940	−	0.983494i	$-0.442086\pi$
$263$	5.86871	0.361880	0.180940	+	0.983494i	$0.442086\pi$
$264$	−46.5240	−2.86335
$265$	−0.790876	−0.0485831
$266$	−1.67130	−0.102474
$267$	−4.86707	−0.297860
$268$	−56.5709	−3.45562
$269$	17.2404	1.05117	0.525584	−	0.850742i	$-0.323847\pi$
$269$	17.2404	1.05117	0.525584	+	0.850742i	$0.323847\pi$
$270$	−2.34884	−0.142946
$271$	−26.1144	−1.58634	−0.793169	−	0.609002i	$-0.791570\pi$
$271$	−26.1144	−1.58634	−0.793169	+	0.609002i	$0.791570\pi$
$272$	−118.059	−7.15836
$273$	−1.71642	−0.103882
$274$	−20.5192	−1.23961
$275$	−20.2682	−1.22222
$276$	−5.85429	−0.352387
$277$	21.7232	1.30522	0.652611	−	0.757693i	$-0.273673\pi$
$277$	21.7232	1.30522	0.652611	+	0.757693i	$0.273673\pi$
$278$	−32.2770	−1.93585
$279$	1.97279	0.118108
$280$	−8.37191	−0.500317
$281$	14.2447	0.849767	0.424883	−	0.905248i	$-0.360315\pi$
$281$	14.2447	0.849767	0.424883	+	0.905248i	$0.360315\pi$
$282$	8.85525	0.527322
$283$	−8.55267	−0.508403	−0.254202	−	0.967151i	$-0.581813\pi$
$283$	−8.55267	−0.508403	−0.254202	+	0.967151i	$0.581813\pi$
$284$	−83.2832	−4.94195
$285$	0.518966	0.0307409
$286$	22.4042	1.32479
$287$	6.62182	0.390874
$288$	−23.9390	−1.41062
$289$	38.6019	2.27070
$290$	13.1673	0.773213
$291$	13.7470	0.805866
$292$	50.9912	2.98404
$293$	6.68289	0.390419	0.195209	−	0.980762i	$-0.437461\pi$
$293$	6.68289	0.390419	0.195209	+	0.980762i	$0.437461\pi$
$294$	−2.75032	−0.160402
$295$	−4.45925	−0.259628
$296$	79.6495	4.62953
$297$	4.74593	0.275387
$298$	−7.36237	−0.426491
$299$	1.80588	0.104437
$300$	−23.7631	−1.37196
$301$	12.4343	0.716698
$302$	13.4290	0.772753
$303$	9.53998	0.548057
$304$	9.62106	0.551806
$305$	−5.41496	−0.310060
$306$	20.5082	1.17238
$307$	10.3288	0.589493	0.294747	−	0.955575i	$-0.404765\pi$
$307$	10.3288	0.589493	0.294747	+	0.955575i	$0.404765\pi$
$308$	26.4077	1.50472
$309$	−2.73317	−0.155485
$310$	−4.63376	−0.263180
$311$	8.06168	0.457136	0.228568	−	0.973528i	$-0.426596\pi$
$311$	8.06168	0.457136	0.228568	+	0.973528i	$0.426596\pi$
$312$	16.8259	0.952579
$313$	4.92617	0.278444	0.139222	−	0.990261i	$-0.455540\pi$
$313$	4.92617	0.278444	0.139222	+	0.990261i	$0.455540\pi$
$314$	21.4095	1.20821
$315$	0.854023	0.0481187
$316$	11.6740	0.656716
$317$	−33.8929	−1.90362	−0.951808	−	0.306693i	$-0.900777\pi$
$317$	−33.8929	−1.90362	−0.951808	+	0.306693i	$0.900777\pi$
$318$	2.54696	0.142827
$319$	−26.6052	−1.48961
$320$	29.1860	1.63155
$321$	4.85054	0.270731
$322$	2.89367	0.161258
$323$	−4.53121	−0.252123
$324$	5.56428	0.309127
$325$	7.33022	0.406607
$326$	34.0256	1.88450
$327$	7.38359	0.408313
$328$	−64.9131	−3.58423
$329$	−3.21971	−0.177508
$330$	−11.1474	−0.613646
$331$	−6.19964	−0.340763	−0.170382	−	0.985378i	$-0.554500\pi$
$331$	−6.19964	−0.340763	−0.170382	+	0.985378i	$0.554500\pi$
$332$	−30.5196	−1.67498
$333$	−8.12508	−0.445252
$334$	64.1012	3.50746
$335$	−8.68267	−0.474385
$336$	15.8326	0.863741
$337$	14.7281	0.802288	0.401144	−	0.916015i	$-0.368613\pi$
$337$	14.7281	0.802288	0.401144	+	0.916015i	$0.368613\pi$
$338$	27.6515	1.50404
$339$	−20.3300	−1.10417
$340$	−35.4342	−1.92169
$341$	9.36272	0.507019
$342$	−1.67130	−0.0903734
$343$	1.00000	0.0539949
$344$	−121.892	−6.57197
$345$	−0.898535	−0.0483755
$346$	35.3798	1.90203
$347$	−3.26606	−0.175331	−0.0876657	−	0.996150i	$-0.527941\pi$
$347$	−3.26606	−0.175331	−0.0876657	+	0.996150i	$0.527941\pi$
$348$	−31.1928	−1.67211
$349$	5.55268	0.297228	0.148614	−	0.988895i	$-0.452519\pi$
$349$	5.55268	0.297228	0.148614	+	0.988895i	$0.452519\pi$
$350$	11.7457	0.627832
$351$	−1.71642	−0.0916157
$352$	−113.613	−6.05559
$353$	−24.9265	−1.32670	−0.663351	−	0.748309i	$-0.730866\pi$
$353$	−24.9265	−1.32670	−0.663351	+	0.748309i	$0.730866\pi$
$354$	14.3607	0.763264
$355$	−12.7826	−0.678428
$356$	−27.0817	−1.43533
$357$	−7.45667	−0.394649
$358$	−35.6442	−1.88385
$359$	−19.1494	−1.01067	−0.505333	−	0.862924i	$-0.668630\pi$
$359$	−19.1494	−1.01067	−0.505333	+	0.862924i	$0.668630\pi$
$360$	−8.37191	−0.441238
$361$	−18.6307	−0.980565
$362$	22.0016	1.15638
$363$	11.5239	0.604848
$364$	−9.55063	−0.500589
$365$	7.82629	0.409647
$366$	17.4385	0.911526
$367$	4.39543	0.229439	0.114720	−	0.993398i	$-0.463403\pi$
$367$	4.39543	0.229439	0.114720	+	0.993398i	$0.463403\pi$
$368$	−16.6578	−0.868350
$369$	6.62182	0.344718
$370$	19.0845	0.992156
$371$	−0.926059	−0.0480786
$372$	10.9771	0.569138
$373$	−25.9920	−1.34582	−0.672908	−	0.739727i	$-0.734955\pi$
$373$	−25.9920	−1.34582	−0.672908	+	0.739727i	$0.734955\pi$
$374$	97.3308	5.03285
$375$	−7.91734	−0.408850
$376$	31.5626	1.62771
$377$	9.62207	0.495562
$378$	−2.75032	−0.141461
$379$	−13.5259	−0.694779	−0.347390	−	0.937721i	$-0.612932\pi$
$379$	−13.5259	−0.694779	−0.347390	+	0.937721i	$0.612932\pi$
$380$	2.88767	0.148134
$381$	15.5686	0.797601
$382$	12.4446	0.636720
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	−46.1136	−2.35323
$385$	4.05313	0.206567
$386$	75.3294	3.83416
$387$	12.4343	0.632069
$388$	76.4924	3.88331
$389$	−5.30664	−0.269057	−0.134529	−	0.990910i	$-0.542952\pi$
$389$	−5.30664	−0.269057	−0.134529	+	0.990910i	$0.542952\pi$
$390$	4.03159	0.204147
$391$	7.84531	0.396755
$392$	−9.80291	−0.495122
$393$	−14.3482	−0.723770
$394$	37.2688	1.87758
$395$	1.79177	0.0901536
$396$	26.4077	1.32704
$397$	−7.22679	−0.362702	−0.181351	−	0.983418i	$-0.558047\pi$
$397$	−7.22679	−0.362702	−0.181351	+	0.983418i	$0.558047\pi$
$398$	51.4504	2.57898
$399$	0.607673	0.0304217
$400$	−67.6155	−3.38078
$401$	−3.45753	−0.172661	−0.0863304	−	0.996267i	$-0.527514\pi$
$401$	−3.45753	−0.172661	−0.0863304	+	0.996267i	$0.527514\pi$
$402$	27.9620	1.39462
$403$	−3.38613	−0.168675
$404$	53.0831	2.64098
$405$	0.854023	0.0424367
$406$	15.4180	0.765184
$407$	−38.5611	−1.91140
$408$	73.0971	3.61884
$409$	35.6269	1.76164	0.880819	−	0.473453i	$-0.156993\pi$
$409$	35.6269	1.76164	0.880819	+	0.473453i	$0.156993\pi$
$410$	−15.5536	−0.768137
$411$	7.46065	0.368007
$412$	−15.2081	−0.749250
$413$	−5.22147	−0.256932
$414$	2.89367	0.142216
$415$	−4.68424	−0.229940
$416$	41.0894	2.01457
$417$	11.7357	0.574700
$418$	−7.93186	−0.387960
$419$	33.6178	1.64234	0.821170	−	0.570684i	$-0.193322\pi$
$419$	33.6178	1.64234	0.821170	+	0.570684i	$0.193322\pi$
$420$	4.75202	0.231875
$421$	25.6872	1.25192	0.625958	−	0.779857i	$-0.284708\pi$
$421$	25.6872	1.25192	0.625958	+	0.779857i	$0.284708\pi$
$422$	77.5934	3.77719
$423$	−3.21971	−0.156548
$424$	9.07808	0.440871
$425$	31.8448	1.54470
$426$	41.1654	1.99447
$427$	−6.34053	−0.306840
$428$	26.9898	1.30460
$429$	−8.14601	−0.393293
$430$	−29.2060	−1.40844
$431$	28.5308	1.37428	0.687139	−	0.726526i	$-0.258866\pi$
$431$	28.5308	1.37428	0.687139	+	0.726526i	$0.258866\pi$
$432$	15.8326	0.761748
$433$	17.4870	0.840370	0.420185	−	0.907438i	$-0.361965\pi$
$433$	17.4870	0.840370	0.420185	+	0.907438i	$0.361965\pi$
$434$	−5.42580	−0.260447
$435$	−4.78756	−0.229546
$436$	41.0843	1.96758
$437$	−0.639345	−0.0305840
$438$	−25.2041	−1.20430
$439$	−4.91022	−0.234352	−0.117176	−	0.993111i	$-0.537384\pi$
$439$	−4.91022	−0.234352	−0.117176	+	0.993111i	$0.537384\pi$
$440$	−39.7325	−1.89417
$441$	1.00000	0.0476190
$442$	−35.2007	−1.67433
$443$	−4.63812	−0.220364	−0.110182	−	0.993911i	$-0.535143\pi$
$443$	−4.63812	−0.220364	−0.110182	+	0.993911i	$0.535143\pi$
$444$	−45.2102	−2.14558
$445$	−4.15658	−0.197041
$446$	49.9259	2.36406
$447$	2.67691	0.126614
$448$	34.1748	1.61461
$449$	−16.4430	−0.775992	−0.387996	−	0.921661i	$-0.626833\pi$
$449$	−16.4430	−0.775992	−0.387996	+	0.921661i	$0.626833\pi$
$450$	11.7457	0.553695
$451$	31.4267	1.47983
$452$	−113.122	−5.32080
$453$	−4.88270	−0.229409
$454$	25.6668	1.20460
$455$	−1.46586	−0.0687206
$456$	−5.95696	−0.278960
$457$	37.1758	1.73901	0.869504	−	0.493925i	$-0.164438\pi$
$457$	37.1758	1.73901	0.869504	+	0.493925i	$0.164438\pi$
$458$	−75.8940	−3.54629
$459$	−7.45667	−0.348047
$460$	−4.99970	−0.233112
$461$	−22.6819	−1.05640	−0.528202	−	0.849119i	$-0.677134\pi$
$461$	−22.6819	−1.05640	−0.528202	+	0.849119i	$0.677134\pi$
$462$	−13.0529	−0.607274
$463$	−13.5071	−0.627730	−0.313865	−	0.949468i	$-0.601624\pi$
$463$	−13.5071	−0.627730	−0.313865	+	0.949468i	$0.601624\pi$
$464$	−88.7561	−4.12040
$465$	1.68480	0.0781309
$466$	−70.7416	−3.27704
$467$	17.4776	0.808767	0.404384	−	0.914589i	$-0.367486\pi$
$467$	17.4776	0.808767	0.404384	+	0.914589i	$0.367486\pi$
$468$	−9.55063	−0.441478
$469$	−10.1668	−0.469459
$470$	7.56258	0.348836
$471$	−7.78436	−0.358684
$472$	51.1856	2.35601
$473$	59.0122	2.71338
$474$	−5.77027	−0.265037
$475$	−2.59516	−0.119074
$476$	−41.4910	−1.90173
$477$	−0.926059	−0.0424013
$478$	−41.6466	−1.90487
$479$	−21.2318	−0.970104	−0.485052	−	0.874485i	$-0.661199\pi$
$479$	−21.2318	−0.970104	−0.485052	+	0.874485i	$0.661199\pi$
$480$	−20.4445	−0.933158
$481$	13.9460	0.635885
$482$	−33.7615	−1.53779
$483$	−1.05212	−0.0478732
$484$	64.1221	2.91464
$485$	11.7403	0.533099
$486$	−2.75032	−0.124757
$487$	−9.69893	−0.439500	−0.219750	−	0.975556i	$-0.570524\pi$
$487$	−9.69893	−0.439500	−0.219750	+	0.975556i	$0.570524\pi$
$488$	62.1557	2.81366
$489$	−12.3715	−0.559458
$490$	−2.34884	−0.106110
$491$	15.8927	0.717228	0.358614	−	0.933486i	$-0.383249\pi$
$491$	15.8927	0.717228	0.358614	+	0.933486i	$0.383249\pi$
$492$	36.8456	1.66113
$493$	41.8013	1.88264
$494$	2.86865	0.129066
$495$	4.05313	0.182175
$496$	31.2344	1.40247
$497$	−14.9675	−0.671383
$498$	15.0853	0.675987
$499$	−19.4092	−0.868875	−0.434438	−	0.900702i	$-0.643053\pi$
$499$	−19.4092	−0.868875	−0.434438	+	0.900702i	$0.643053\pi$
$500$	−44.0543	−1.97017
$501$	−23.3068	−1.04127
$502$	5.45602	0.243514
$503$	−32.6461	−1.45562	−0.727809	−	0.685780i	$-0.759461\pi$
$503$	−32.6461	−1.45562	−0.727809	+	0.685780i	$0.759461\pi$
$504$	−9.80291	−0.436657
$505$	8.14736	0.362553
$506$	13.7332	0.610514
$507$	−10.0539	−0.446510
$508$	86.6278	3.84349
$509$	−6.92931	−0.307136	−0.153568	−	0.988138i	$-0.549076\pi$
$509$	−6.92931	−0.307136	−0.153568	+	0.988138i	$0.549076\pi$
$510$	17.5145	0.775555
$511$	9.16404	0.405393
$512$	−68.6057	−3.03197
$513$	0.607673	0.0268294
$514$	86.9357	3.83457
$515$	−2.33419	−0.102857
$516$	69.1876	3.04582
$517$	−15.2805	−0.672037
$518$	22.3466	0.981853
$519$	−12.8639	−0.564661
$520$	14.3697	0.630153
$521$	34.6614	1.51854	0.759271	−	0.650774i	$-0.225556\pi$
$521$	34.6614	1.51854	0.759271	+	0.650774i	$0.225556\pi$
$522$	15.4180	0.674829
$523$	−11.6427	−0.509101	−0.254551	−	0.967059i	$-0.581928\pi$
$523$	−11.6427	−0.509101	−0.254551	+	0.967059i	$0.581928\pi$
$524$	−79.8373	−3.48771
$525$	−4.27065	−0.186386
$526$	−16.1408	−0.703774
$527$	−14.7104	−0.640796
$528$	75.1406	3.27008
$529$	−21.8930	−0.951871
$530$	2.17516	0.0944831
$531$	−5.22147	−0.226592
$532$	3.38126	0.146596
$533$	−11.3658	−0.492308
$534$	13.3860	0.579269
$535$	4.14247	0.179095
$536$	99.6642	4.30484
$537$	12.9600	0.559266
$538$	−47.4168	−2.04428
$539$	4.74593	0.204422
$540$	4.75202	0.204494
$541$	−37.0296	−1.59203	−0.796014	−	0.605279i	$-0.793062\pi$
$541$	−37.0296	−1.59203	−0.796014	+	0.605279i	$0.793062\pi$
$542$	71.8231	3.08506
$543$	−7.99964	−0.343298
$544$	178.505	7.65335
$545$	6.30575	0.270109
$546$	4.72071	0.202028
$547$	17.0681	0.729780	0.364890	−	0.931051i	$-0.381107\pi$
$547$	17.0681	0.729780	0.364890	+	0.931051i	$0.381107\pi$
$548$	41.5131	1.77335
$549$	−6.34053	−0.270607
$550$	55.7441	2.37694
$551$	−3.40655	−0.145124
$552$	10.3139	0.438987
$553$	2.09803	0.0892175
$554$	−59.7459	−2.53836
$555$	−6.93900	−0.294544
$556$	65.3008	2.76937
$557$	28.9219	1.22546	0.612730	−	0.790293i	$-0.290071\pi$
$557$	28.9219	1.22546	0.612730	+	0.790293i	$0.290071\pi$
$558$	−5.42580	−0.229692
$559$	−21.3424	−0.902687
$560$	13.5214	0.571384
$561$	−35.3888	−1.49412
$562$	−39.1775	−1.65260
$563$	−40.0468	−1.68777	−0.843886	−	0.536523i	$-0.819738\pi$
$563$	−40.0468	−1.68777	−0.843886	+	0.536523i	$0.819738\pi$
$564$	−17.9154	−0.754373
$565$	−17.3623	−0.730437
$566$	23.5226	0.988729
$567$	1.00000	0.0419961
$568$	146.725	6.15644
$569$	10.2917	0.431451	0.215725	−	0.976454i	$-0.430788\pi$
$569$	10.2917	0.431451	0.215725	+	0.976454i	$0.430788\pi$
$570$	−1.42733	−0.0597841
$571$	7.24028	0.302996	0.151498	−	0.988458i	$-0.451590\pi$
$571$	7.24028	0.302996	0.151498	+	0.988458i	$0.451590\pi$
$572$	−45.3267	−1.89520
$573$	−4.52477	−0.189025
$574$	−18.2121	−0.760160
$575$	4.49323	0.187381
$576$	34.1748	1.42395
$577$	22.2378	0.925772	0.462886	−	0.886418i	$-0.346814\pi$
$577$	22.2378	0.925772	0.462886	+	0.886418i	$0.346814\pi$
$578$	−106.168	−4.41599
$579$	−27.3893	−1.13826
$580$	−26.6393	−1.10614
$581$	−5.48491	−0.227552
$582$	−37.8088	−1.56723
$583$	−4.39502	−0.182023
$584$	−89.8343	−3.71737
$585$	−1.46586	−0.0606059
$586$	−18.3801	−0.759275
$587$	30.2145	1.24709	0.623544	−	0.781788i	$-0.285692\pi$
$587$	30.2145	1.24709	0.623544	+	0.781788i	$0.285692\pi$
$588$	5.56428	0.229467
$589$	1.19881	0.0493961
$590$	12.2644	0.504916
$591$	−13.5507	−0.557401
$592$	−128.641	−5.28713
$593$	43.5550	1.78859	0.894296	−	0.447477i	$-0.147677\pi$
$593$	43.5550	1.78859	0.894296	+	0.447477i	$0.147677\pi$
$594$	−13.0529	−0.535565
$595$	−6.36816	−0.261069
$596$	14.8951	0.610126
$597$	−18.7070	−0.765628
$598$	−4.96675	−0.203106
$599$	−23.9195	−0.977324	−0.488662	−	0.872473i	$-0.662515\pi$
$599$	−23.9195	−0.977324	−0.488662	+	0.872473i	$0.662515\pi$
$600$	41.8648	1.70912
$601$	5.94157	0.242362	0.121181	−	0.992630i	$-0.461332\pi$
$601$	5.94157	0.242362	0.121181	+	0.992630i	$0.461332\pi$
$602$	−34.1982	−1.39382
$603$	−10.1668	−0.414024
$604$	−27.1687	−1.10548
$605$	9.84166	0.400121
$606$	−26.2380	−1.06585
$607$	46.9704	1.90647	0.953235	−	0.302231i	$-0.0977314\pi$
$607$	46.9704	1.90647	0.953235	+	0.302231i	$0.0977314\pi$
$608$	−14.5471	−0.589962
$609$	−5.60590	−0.227162
$610$	14.8929	0.602995
$611$	5.52637	0.223573
$612$	−41.4910	−1.67717
$613$	−21.8023	−0.880586	−0.440293	−	0.897854i	$-0.645125\pi$
$613$	−21.8023	−0.880586	−0.440293	+	0.897854i	$0.645125\pi$
$614$	−28.4074	−1.14643
$615$	5.65518	0.228039
$616$	−46.5240	−1.87450
$617$	−2.08492	−0.0839356	−0.0419678	−	0.999119i	$-0.513363\pi$
$617$	−2.08492	−0.0839356	−0.0419678	+	0.999119i	$0.513363\pi$
$618$	7.51710	0.302382
$619$	32.6841	1.31368	0.656842	−	0.754028i	$-0.271892\pi$
$619$	32.6841	1.31368	0.656842	+	0.754028i	$0.271892\pi$
$620$	9.37472	0.376498
$621$	−1.05212	−0.0422202
$622$	−22.1722	−0.889025
$623$	−4.86707	−0.194995
$624$	−27.1754	−1.08789
$625$	14.5916	0.583666
$626$	−13.5486	−0.541510
$627$	2.88398	0.115175
$628$	−43.3143	−1.72843
$629$	60.5860	2.41572
$630$	−2.34884	−0.0935799
$631$	8.68197	0.345624	0.172812	−	0.984955i	$-0.444715\pi$
$631$	8.68197	0.345624	0.172812	+	0.984955i	$0.444715\pi$
$632$	−20.5668	−0.818105
$633$	−28.2125	−1.12135
$634$	93.2165	3.70210
$635$	13.2959	0.527632
$636$	−5.15285	−0.204324
$637$	−1.71642	−0.0680070
$638$	73.1729	2.89694
$639$	−14.9675	−0.592104
$640$	−39.3821	−1.55671
$641$	−36.0498	−1.42388	−0.711940	−	0.702240i	$-0.752183\pi$
$641$	−36.0498	−1.42388	−0.711940	+	0.702240i	$0.752183\pi$
$642$	−13.3406	−0.526510
$643$	17.1189	0.675102	0.337551	−	0.941307i	$-0.390401\pi$
$643$	17.1189	0.675102	0.337551	+	0.941307i	$0.390401\pi$
$644$	−5.85429	−0.230691
$645$	10.6191	0.418128
$646$	12.4623	0.490323
$647$	41.7554	1.64157	0.820787	−	0.571234i	$-0.193535\pi$
$647$	41.7554	1.64157	0.820787	+	0.571234i	$0.193535\pi$
$648$	−9.80291	−0.385095
$649$	−24.7807	−0.972729
$650$	−20.1605	−0.790758
$651$	1.97279	0.0773196
$652$	−68.8384	−2.69592
$653$	20.6614	0.808544	0.404272	−	0.914639i	$-0.367525\pi$
$653$	20.6614	0.808544	0.404272	+	0.914639i	$0.367525\pi$
$654$	−20.3072	−0.794076
$655$	−12.2537	−0.478790
$656$	104.841	4.09335
$657$	9.16404	0.357523
$658$	8.85525	0.345213
$659$	37.6705	1.46743	0.733717	−	0.679455i	$-0.237784\pi$
$659$	37.6705	1.46743	0.733717	+	0.679455i	$0.237784\pi$
$660$	22.5528	0.877865
$661$	−16.0568	−0.624538	−0.312269	−	0.949994i	$-0.601089\pi$
$661$	−16.0568	−0.624538	−0.312269	+	0.949994i	$0.601089\pi$
$662$	17.0510	0.662707
$663$	12.7988	0.497063
$664$	53.7681	2.08661
$665$	0.518966	0.0201246
$666$	22.3466	0.865913
$667$	5.89808	0.228375
$668$	−129.685	−5.01768
$669$	−18.1527	−0.701825
$670$	23.8802	0.922571
$671$	−30.0918	−1.16168
$672$	−23.9390	−0.923467
$673$	−12.3717	−0.476895	−0.238448	−	0.971155i	$-0.576639\pi$
$673$	−12.3717	−0.476895	−0.238448	+	0.971155i	$0.576639\pi$
$674$	−40.5069	−1.56027
$675$	−4.27065	−0.164377
$676$	−55.9427	−2.15164
$677$	1.04784	0.0402718	0.0201359	−	0.999797i	$-0.493590\pi$
$677$	1.04784	0.0402718	0.0201359	+	0.999797i	$0.493590\pi$
$678$	55.9141	2.14737
$679$	13.7470	0.527563
$680$	62.4265	2.39395
$681$	−9.33228	−0.357614
$682$	−25.7505	−0.986037
$683$	21.9053	0.838182	0.419091	−	0.907944i	$-0.362349\pi$
$683$	21.9053	0.838182	0.419091	+	0.907944i	$0.362349\pi$
$684$	3.38126	0.129286
$685$	6.37156	0.243445
$686$	−2.75032	−0.105008
$687$	27.5946	1.05280
$688$	196.867	7.50548
$689$	1.58951	0.0605553
$690$	2.47126	0.0940793
$691$	−37.3627	−1.42134	−0.710672	−	0.703524i	$-0.751609\pi$
$691$	−37.3627	−1.42134	−0.710672	+	0.703524i	$0.751609\pi$
$692$	−71.5782	−2.72099
$693$	4.74593	0.180283
$694$	8.98272	0.340980
$695$	10.0226	0.380177
$696$	54.9541	2.08303
$697$	−49.3767	−1.87028
$698$	−15.2717	−0.578041
$699$	25.7212	0.972865
$700$	−23.7631	−0.898159
$701$	−49.1462	−1.85622	−0.928112	−	0.372300i	$-0.878569\pi$
$701$	−49.1462	−1.85622	−0.928112	+	0.372300i	$0.878569\pi$
$702$	4.72071	0.178172
$703$	−4.93739	−0.186217
$704$	162.191	6.11281
$705$	−2.74971	−0.103560
$706$	68.5558	2.58013
$707$	9.53998	0.358788
$708$	−29.0537	−1.09190
$709$	−23.5759	−0.885413	−0.442707	−	0.896667i	$-0.645982\pi$
$709$	−23.5759	−0.885413	−0.442707	+	0.896667i	$0.645982\pi$
$710$	35.1562	1.31939
$711$	2.09803	0.0786824
$712$	47.7114	1.78806
$713$	−2.07561	−0.0777322
$714$	20.5082	0.767502
$715$	−6.95688	−0.260172
$716$	72.1131	2.69499
$717$	15.1424	0.565504
$718$	52.6670	1.96552
$719$	−1.29523	−0.0483039	−0.0241520	−	0.999708i	$-0.507689\pi$
$719$	−1.29523	−0.0483039	−0.0241520	+	0.999708i	$0.507689\pi$
$720$	13.5214	0.503914
$721$	−2.73317	−0.101789
$722$	51.2405	1.90698
$723$	12.2755	0.456529
$724$	−44.5122	−1.65428
$725$	23.9408	0.889139
$726$	−31.6944	−1.17629
$727$	−21.9682	−0.814756	−0.407378	−	0.913260i	$-0.633557\pi$
$727$	−21.9682	−0.814756	−0.407378	+	0.913260i	$0.633557\pi$
$728$	16.8259	0.623610
$729$	1.00000	0.0370370
$730$	−21.5248	−0.796670
$731$	−92.7181	−3.42930
$732$	−35.2805	−1.30400
$733$	−24.2577	−0.895977	−0.447989	−	0.894039i	$-0.647859\pi$
$733$	−24.2577	−0.895977	−0.447989	+	0.894039i	$0.647859\pi$
$734$	−12.0888	−0.446208
$735$	0.854023	0.0315011
$736$	25.1867	0.928395
$737$	−48.2509	−1.77735
$738$	−18.2121	−0.670398
$739$	16.9737	0.624389	0.312194	−	0.950018i	$-0.398936\pi$
$739$	16.9737	0.624389	0.312194	+	0.950018i	$0.398936\pi$
$740$	−38.6105	−1.41935
$741$	−1.04302	−0.0383163
$742$	2.54696	0.0935019
$743$	27.8513	1.02177	0.510883	−	0.859650i	$-0.329318\pi$
$743$	27.8513	1.02177	0.510883	+	0.859650i	$0.329318\pi$
$744$	−19.3391	−0.709004
$745$	2.28614	0.0837578
$746$	71.4864	2.61730
$747$	−5.48491	−0.200682
$748$	−196.913	−7.19987
$749$	4.85054	0.177235
$750$	21.7752	0.795119
$751$	−17.3536	−0.633243	−0.316622	−	0.948552i	$-0.602549\pi$
$751$	−17.3536	−0.633243	−0.316622	+	0.948552i	$0.602549\pi$
$752$	−50.9765	−1.85892
$753$	−1.98377	−0.0722927
$754$	−26.4638	−0.963755
$755$	−4.16994	−0.151760
$756$	5.56428	0.202371
$757$	−15.1338	−0.550047	−0.275024	−	0.961437i	$-0.588686\pi$
$757$	−15.1338	−0.550047	−0.275024	+	0.961437i	$0.588686\pi$
$758$	37.2006	1.35119
$759$	−4.99330	−0.181245
$760$	−5.08738	−0.184539
$761$	44.5964	1.61662	0.808310	−	0.588757i	$-0.200383\pi$
$761$	44.5964	1.61662	0.808310	+	0.588757i	$0.200383\pi$
$762$	−42.8186	−1.55115
$763$	7.38359	0.267304
$764$	−25.1771	−0.910874
$765$	−6.36816	−0.230241
$766$	2.75032	0.0993732
$767$	8.96222	0.323607
$768$	58.4778	2.11014
$769$	24.3796	0.879152	0.439576	−	0.898205i	$-0.355129\pi$
$769$	24.3796	0.879152	0.439576	+	0.898205i	$0.355129\pi$
$770$	−11.1474	−0.401726
$771$	−31.6093	−1.13838
$772$	−152.402	−5.48505
$773$	43.3720	1.55998	0.779991	−	0.625790i	$-0.215223\pi$
$773$	43.3720	1.55998	0.779991	+	0.625790i	$0.215223\pi$
$774$	−34.1982	−1.22923
$775$	−8.42507	−0.302638
$776$	−134.761	−4.83764
$777$	−8.12508	−0.291486
$778$	14.5950	0.523255
$779$	4.02390	0.144171
$780$	−8.15645	−0.292048
$781$	−71.0347	−2.54182
$782$	−21.5771	−0.771597
$783$	−5.60590	−0.200338
$784$	15.8326	0.565451
$785$	−6.64802	−0.237278
$786$	39.4621	1.40757
$787$	−7.67030	−0.273417	−0.136708	−	0.990611i	$-0.543652\pi$
$787$	−7.67030	−0.273417	−0.136708	+	0.990611i	$0.543652\pi$
$788$	−75.3998	−2.68601
$789$	5.86871	0.208931
$790$	−4.92794	−0.175328
$791$	−20.3300	−0.722852
$792$	−46.5240	−1.65316
$793$	10.8830	0.386467
$794$	19.8760	0.705373
$795$	−0.790876	−0.0280495
$796$	−104.091	−3.68941
$797$	−39.6351	−1.40395	−0.701973	−	0.712203i	$-0.747697\pi$
$797$	−39.6351	−1.40395	−0.701973	+	0.712203i	$0.747697\pi$
$798$	−1.67130	−0.0591633
$799$	24.0083	0.849353
$800$	102.235	3.61455
$801$	−4.86707	−0.171969
$802$	9.50932	0.335786
$803$	43.4919	1.53480
$804$	−56.5709	−1.99510
$805$	−0.898535	−0.0316692
$806$	9.31295	0.328035
$807$	17.2404	0.606892
$808$	−93.5196	−3.29001
$809$	19.1034	0.671638	0.335819	−	0.941927i	$-0.390987\pi$
$809$	19.1034	0.671638	0.335819	+	0.941927i	$0.390987\pi$
$810$	−2.34884	−0.0825298
$811$	−25.3020	−0.888472	−0.444236	−	0.895910i	$-0.646525\pi$
$811$	−25.3020	−0.888472	−0.444236	+	0.895910i	$0.646525\pi$
$812$	−31.1928	−1.09465
$813$	−26.1144	−0.915872
$814$	106.055	3.71724
$815$	−10.5655	−0.370094
$816$	−118.059	−4.13288
$817$	7.55596	0.264350
$818$	−97.9856	−3.42598
$819$	−1.71642	−0.0599765
$820$	31.4670	1.09888
$821$	0.154306	0.00538531	0.00269266	−	0.999996i	$-0.499143\pi$
$821$	0.154306	0.00538531	0.00269266	+	0.999996i	$0.499143\pi$
$822$	−20.5192	−0.715689
$823$	−13.7971	−0.480937	−0.240468	−	0.970657i	$-0.577301\pi$
$823$	−13.7971	−0.480937	−0.240468	+	0.970657i	$0.577301\pi$
$824$	26.7930	0.933379
$825$	−20.2682	−0.705648
$826$	14.3607	0.499673
$827$	31.3959	1.09174	0.545872	−	0.837869i	$-0.316199\pi$
$827$	31.3959	1.09174	0.545872	+	0.837869i	$0.316199\pi$
$828$	−5.85429	−0.203451
$829$	−30.6396	−1.06416	−0.532078	−	0.846695i	$-0.678589\pi$
$829$	−30.6396	−1.06416	−0.532078	+	0.846695i	$0.678589\pi$
$830$	12.8832	0.447181
$831$	21.7232	0.753571
$832$	−58.6582	−2.03361
$833$	−7.45667	−0.258358
$834$	−32.2770	−1.11766
$835$	−19.9045	−0.688824
$836$	16.0472	0.555005
$837$	1.97279	0.0681895
$838$	−92.4599	−3.19398
$839$	9.79443	0.338141	0.169071	−	0.985604i	$-0.445923\pi$
$839$	9.79443	0.338141	0.169071	+	0.985604i	$0.445923\pi$
$840$	−8.37191	−0.288858
$841$	2.42608	0.0836579
$842$	−70.6480	−2.43469
$843$	14.2447	0.490613
$844$	−156.982	−5.40355
$845$	−8.58626	−0.295376
$846$	8.85525	0.304450
$847$	11.5239	0.395966
$848$	−14.6620	−0.503494
$849$	−8.55267	−0.293527
$850$	−87.5834	−3.00409
$851$	8.54857	0.293041
$852$	−83.2832	−2.85324
$853$	−36.3728	−1.24538	−0.622690	−	0.782469i	$-0.713960\pi$
$853$	−36.3728	−1.24538	−0.622690	+	0.782469i	$0.713960\pi$
$854$	17.4385	0.596734
$855$	0.518966	0.0177483
$856$	−47.5495	−1.62521
$857$	−18.7362	−0.640015	−0.320007	−	0.947415i	$-0.603685\pi$
$857$	−18.7362	−0.640015	−0.320007	+	0.947415i	$0.603685\pi$
$858$	22.4042	0.764866
$859$	16.0827	0.548736	0.274368	−	0.961625i	$-0.411531\pi$
$859$	16.0827	0.548736	0.274368	+	0.961625i	$0.411531\pi$
$860$	59.0878	2.01488
$861$	6.62182	0.225671
$862$	−78.4688	−2.67266
$863$	29.6249	1.00844	0.504221	−	0.863574i	$-0.331780\pi$
$863$	29.6249	1.00844	0.504221	+	0.863574i	$0.331780\pi$
$864$	−23.9390	−0.814422
$865$	−10.9860	−0.373537
$866$	−48.0948	−1.63433
$867$	38.6019	1.31099
$868$	10.9771	0.372588
$869$	9.95713	0.337772
$870$	13.1673	0.446415
$871$	17.4505	0.591287
$872$	−72.3807	−2.45112
$873$	13.7470	0.465267
$874$	1.75841	0.0594790
$875$	−7.91734	−0.267655
$876$	50.9912	1.72283
$877$	9.31457	0.314531	0.157265	−	0.987556i	$-0.449732\pi$
$877$	9.31457	0.314531	0.157265	+	0.987556i	$0.449732\pi$
$878$	13.5047	0.455761
$879$	6.68289	0.225408
$880$	64.1718	2.16323
$881$	−24.2543	−0.817149	−0.408575	−	0.912725i	$-0.633974\pi$
$881$	−24.2543	−0.817149	−0.408575	+	0.912725i	$0.633974\pi$
$882$	−2.75032	−0.0926082
$883$	−44.4053	−1.49436	−0.747178	−	0.664624i	$-0.768592\pi$
$883$	−44.4053	−1.49436	−0.747178	+	0.664624i	$0.768592\pi$
$884$	71.2159	2.39525
$885$	−4.45925	−0.149896
$886$	12.7563	0.428557
$887$	−15.0567	−0.505556	−0.252778	−	0.967524i	$-0.581344\pi$
$887$	−15.0567	−0.505556	−0.252778	+	0.967524i	$0.581344\pi$
$888$	79.6495	2.67286
$889$	15.5686	0.522153
$890$	11.4319	0.383200
$891$	4.74593	0.158995
$892$	−101.007	−3.38196
$893$	−1.95653	−0.0654728
$894$	−7.36237	−0.246235
$895$	11.0681	0.369967
$896$	−46.1136	−1.54055
$897$	1.80588	0.0602966
$898$	45.2235	1.50913
$899$	−11.0592	−0.368846
$900$	−23.7631	−0.792102
$901$	6.90532	0.230049
$902$	−86.4336	−2.87793
$903$	12.4343	0.413786
$904$	199.293	6.62840
$905$	−6.83188	−0.227099
$906$	13.4290	0.446149
$907$	43.4793	1.44371	0.721854	−	0.692045i	$-0.243290\pi$
$907$	43.4793	1.44371	0.721854	+	0.692045i	$0.243290\pi$
$908$	−51.9274	−1.72327
$909$	9.53998	0.316421
$910$	4.03159	0.133646
$911$	1.83637	0.0608418	0.0304209	−	0.999537i	$-0.490315\pi$
$911$	1.83637	0.0608418	0.0304209	+	0.999537i	$0.490315\pi$
$912$	9.62106	0.318585
$913$	−26.0310	−0.861501
$914$	−102.245	−3.38198
$915$	−5.41496	−0.179013
$916$	153.544	5.07323
$917$	−14.3482	−0.473819
$918$	20.5082	0.676873
$919$	10.3464	0.341296	0.170648	−	0.985332i	$-0.445414\pi$
$919$	10.3464	0.341296	0.170648	+	0.985332i	$0.445414\pi$
$920$	8.80826	0.290400
$921$	10.3288	0.340344
$922$	62.3827	2.05446
$923$	25.6905	0.845612
$924$	26.4077	0.868749
$925$	34.6993	1.14091
$926$	37.1490	1.22079
$927$	−2.73317	−0.0897690
$928$	134.200	4.40532
$929$	−0.601258	−0.0197266	−0.00986332	−	0.999951i	$-0.503140\pi$
$929$	−0.601258	−0.0197266	−0.00986332	+	0.999951i	$0.503140\pi$
$930$	−4.63376	−0.151947
$931$	0.607673	0.0199157
$932$	143.120	4.68805
$933$	8.06168	0.263928
$934$	−48.0691	−1.57287
$935$	−30.2229	−0.988394
$936$	16.8259	0.549972
$937$	−23.0109	−0.751733	−0.375866	−	0.926674i	$-0.622655\pi$
$937$	−23.0109	−0.751733	−0.375866	+	0.926674i	$0.622655\pi$
$938$	27.9620	0.912991
$939$	4.92617	0.160760
$940$	−15.3001	−0.499035
$941$	5.06021	0.164958	0.0824791	−	0.996593i	$-0.473716\pi$
$941$	5.06021	0.164958	0.0824791	+	0.996593i	$0.473716\pi$
$942$	21.4095	0.697560
$943$	−6.96695	−0.226875
$944$	−82.6695	−2.69066
$945$	0.854023	0.0277814
$946$	−162.302	−5.27691
$947$	−26.2875	−0.854229	−0.427115	−	0.904197i	$-0.640470\pi$
$947$	−26.2875	−0.854229	−0.427115	+	0.904197i	$0.640470\pi$
$948$	11.6740	0.379155
$949$	−15.7293	−0.510596
$950$	7.13752	0.231572
$951$	−33.8929	−1.09905
$952$	73.0971	2.36909
$953$	−18.6697	−0.604771	−0.302386	−	0.953186i	$-0.597783\pi$
$953$	−18.6697	−0.604771	−0.302386	+	0.953186i	$0.597783\pi$
$954$	2.54696	0.0824610
$955$	−3.86425	−0.125044
$956$	84.2567	2.72506
$957$	−26.6052	−0.860025
$958$	58.3942	1.88663
$959$	7.46065	0.240917
$960$	29.1860	0.941975
$961$	−27.1081	−0.874455
$962$	−38.3561	−1.23665
$963$	4.85054	0.156307
$964$	68.3041	2.19993
$965$	−23.3911	−0.752985
$966$	2.89367	0.0931024
$967$	−60.9597	−1.96033	−0.980167	−	0.198176i	$-0.936498\pi$
$967$	−60.9597	−1.96033	−0.980167	+	0.198176i	$0.936498\pi$
$968$	−112.968	−3.63092
$969$	−4.53121	−0.145564
$970$	−32.2896	−1.03676
$971$	−42.4475	−1.36221	−0.681103	−	0.732188i	$-0.738499\pi$
$971$	−42.4475	−1.36221	−0.681103	+	0.732188i	$0.738499\pi$
$972$	5.56428	0.178474
$973$	11.7357	0.376230
$974$	26.6752	0.854728
$975$	7.33022	0.234755
$976$	−100.387	−3.21332
$977$	2.80075	0.0896040	0.0448020	−	0.998996i	$-0.485734\pi$
$977$	2.80075	0.0896040	0.0448020	+	0.998996i	$0.485734\pi$
$978$	34.0256	1.08802
$979$	−23.0988	−0.738240
$980$	4.75202	0.151798
$981$	7.38359	0.235740
$982$	−43.7101	−1.39485
$983$	7.47115	0.238293	0.119146	−	0.992877i	$-0.461984\pi$
$983$	7.47115	0.238293	0.119146	+	0.992877i	$0.461984\pi$
$984$	−64.9131	−2.06935
$985$	−11.5726	−0.368734
$986$	−114.967	−3.66130
$987$	−3.21971	−0.102485
$988$	−5.80366	−0.184639
$989$	−13.0823	−0.415994
$990$	−11.1474	−0.354289
$991$	−24.0138	−0.762824	−0.381412	−	0.924405i	$-0.624562\pi$
$991$	−24.0138	−0.762824	−0.381412	+	0.924405i	$0.624562\pi$
$992$	−47.2266	−1.49944
$993$	−6.19964	−0.196740
$994$	41.1654	1.30569
$995$	−15.9762	−0.506481
$996$	−30.5196	−0.967049
$997$	44.5662	1.41143	0.705713	−	0.708498i	$-0.250627\pi$
$997$	44.5662	1.41143	0.705713	+	0.708498i	$0.250627\pi$
$998$	53.3816	1.68976
$999$	−8.12508	−0.257066

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.n.1.1	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.n.1.1	✓	40	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q-2.75032 q^{2} +1.00000 q^{3} +5.56428 q^{4} +0.854023 q^{5} -2.75032 q^{6} +1.00000 q^{7} -9.80291 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.75032 q^{2} +1.00000 q^{3} +5.56428 q^{4} +0.854023 q^{5} -2.75032 q^{6} +1.00000 q^{7} -9.80291 q^{8} +1.00000 q^{9} -2.34884 q^{10} +4.74593 q^{11} +5.56428 q^{12} -1.71642 q^{13} -2.75032 q^{14} +0.854023 q^{15} +15.8326 q^{16} -7.45667 q^{17} -2.75032 q^{18} +0.607673 q^{19} +4.75202 q^{20} +1.00000 q^{21} -13.0529 q^{22} -1.05212 q^{23} -9.80291 q^{24} -4.27065 q^{25} +4.72071 q^{26} +1.00000 q^{27} +5.56428 q^{28} -5.60590 q^{29} -2.34884 q^{30} +1.97279 q^{31} -23.9390 q^{32} +4.74593 q^{33} +20.5082 q^{34} +0.854023 q^{35} +5.56428 q^{36} -8.12508 q^{37} -1.67130 q^{38} -1.71642 q^{39} -8.37191 q^{40} +6.62182 q^{41} -2.75032 q^{42} +12.4343 q^{43} +26.4077 q^{44} +0.854023 q^{45} +2.89367 q^{46} -3.21971 q^{47} +15.8326 q^{48} +1.00000 q^{49} +11.7457 q^{50} -7.45667 q^{51} -9.55063 q^{52} -0.926059 q^{53} -2.75032 q^{54} +4.05313 q^{55} -9.80291 q^{56} +0.607673 q^{57} +15.4180 q^{58} -5.22147 q^{59} +4.75202 q^{60} -6.34053 q^{61} -5.42580 q^{62} +1.00000 q^{63} +34.1748 q^{64} -1.46586 q^{65} -13.0529 q^{66} -10.1668 q^{67} -41.4910 q^{68} -1.05212 q^{69} -2.34884 q^{70} -14.9675 q^{71} -9.80291 q^{72} +9.16404 q^{73} +22.3466 q^{74} -4.27065 q^{75} +3.38126 q^{76} +4.74593 q^{77} +4.72071 q^{78} +2.09803 q^{79} +13.5214 q^{80} +1.00000 q^{81} -18.2121 q^{82} -5.48491 q^{83} +5.56428 q^{84} -6.36816 q^{85} -34.1982 q^{86} -5.60590 q^{87} -46.5240 q^{88} -4.86707 q^{89} -2.34884 q^{90} -1.71642 q^{91} -5.85429 q^{92} +1.97279 q^{93} +8.85525 q^{94} +0.518966 q^{95} -23.9390 q^{96} +13.7470 q^{97} -2.75032 q^{98} +4.74593 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9}+O(q^{10}) \) 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9	\( 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9} - 14 q^{10} - 29 q^{11} + 33 q^{12} - 40 q^{13} - 9 q^{14} - 27 q^{15} + 23 q^{16} - 46 q^{17} - 9 q^{18} + 17 q^{19} - 53 q^{20} + 40 q^{21} - 15 q^{22} - 62 q^{23} - 18 q^{24} + 35 q^{25} - 29 q^{26} + 40 q^{27} + 33 q^{28} - 47 q^{29} - 14 q^{30} + q^{31} - 45 q^{32} - 29 q^{33} + 9 q^{34} - 27 q^{35} + 33 q^{36} - 49 q^{37} - 52 q^{38} - 40 q^{39} - 12 q^{40} - 49 q^{41} - 9 q^{42} - 45 q^{43} - 68 q^{44} - 27 q^{45} - 36 q^{46} - 88 q^{47} + 23 q^{48} + 40 q^{49} - 32 q^{50} - 46 q^{51} - 34 q^{52} - 96 q^{53} - 9 q^{54} + 26 q^{55} - 18 q^{56} + 17 q^{57} + 12 q^{58} - 45 q^{59} - 53 q^{60} - 23 q^{61} - 15 q^{62} + 40 q^{63} + 50 q^{64} - 32 q^{65} - 15 q^{66} - 31 q^{67} - 76 q^{68} - 62 q^{69} - 14 q^{70} - 89 q^{71} - 18 q^{72} + 4 q^{73} + 10 q^{74} + 35 q^{75} + 51 q^{76} - 29 q^{77} - 29 q^{78} - 44 q^{79} - 53 q^{80} + 40 q^{81} - 15 q^{82} - 60 q^{83} + 33 q^{84} - 24 q^{85} - 9 q^{86} - 47 q^{87} - 64 q^{88} - 39 q^{89} - 14 q^{90} - 40 q^{91} - 80 q^{92} + q^{93} + 51 q^{94} - 71 q^{95} - 45 q^{96} - 21 q^{97} - 9 q^{98} - 29 q^{99}+O(q^{100}) \) 40 * q - 9 * q^2 + 40 * q^3 + 33 * q^4 - 27 * q^5 - 9 * q^6 + 40 * q^7 - 18 * q^8 + 40 * q^9 - 14 * q^10 - 29 * q^11 + 33 * q^12 - 40 * q^13 - 9 * q^14 - 27 * q^15 + 23 * q^16 - 46 * q^17 - 9 * q^18 + 17 * q^19 - 53 * q^20 + 40 * q^21 - 15 * q^22 - 62 * q^23 - 18 * q^24 + 35 * q^25 - 29 * q^26 + 40 * q^27 + 33 * q^28 - 47 * q^29 - 14 * q^30 + q^31 - 45 * q^32 - 29 * q^33 + 9 * q^34 - 27 * q^35 + 33 * q^36 - 49 * q^37 - 52 * q^38 - 40 * q^39 - 12 * q^40 - 49 * q^41 - 9 * q^42 - 45 * q^43 - 68 * q^44 - 27 * q^45 - 36 * q^46 - 88 * q^47 + 23 * q^48 + 40 * q^49 - 32 * q^50 - 46 * q^51 - 34 * q^52 - 96 * q^53 - 9 * q^54 + 26 * q^55 - 18 * q^56 + 17 * q^57 + 12 * q^58 - 45 * q^59 - 53 * q^60 - 23 * q^61 - 15 * q^62 + 40 * q^63 + 50 * q^64 - 32 * q^65 - 15 * q^66 - 31 * q^67 - 76 * q^68 - 62 * q^69 - 14 * q^70 - 89 * q^71 - 18 * q^72 + 4 * q^73 + 10 * q^74 + 35 * q^75 + 51 * q^76 - 29 * q^77 - 29 * q^78 - 44 * q^79 - 53 * q^80 + 40 * q^81 - 15 * q^82 - 60 * q^83 + 33 * q^84 - 24 * q^85 - 9 * q^86 - 47 * q^87 - 64 * q^88 - 39 * q^89 - 14 * q^90 - 40 * q^91 - 80 * q^92 + q^93 + 51 * q^94 - 71 * q^95 - 45 * q^96 - 21 * q^97 - 9 * q^98 - 29 * q^99

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