LMFDB - Embedded newform 8043.2.a.l.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:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
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)

$f(q)$	$=$	$q-2.56155 q^{2} +1.00000 q^{3} +4.56155 q^{4} -1.56155 q^{5} -2.56155 q^{6} +1.00000 q^{7} -6.56155 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.56155 q^{2} +1.00000 q^{3} +4.56155 q^{4} -1.56155 q^{5} -2.56155 q^{6} +1.00000 q^{7} -6.56155 q^{8} +1.00000 q^{9} +4.00000 q^{10} -2.43845 q^{11} +4.56155 q^{12} +2.00000 q^{13} -2.56155 q^{14} -1.56155 q^{15} +7.68466 q^{16} -2.00000 q^{17} -2.56155 q^{18} -2.43845 q^{19} -7.12311 q^{20} +1.00000 q^{21} +6.24621 q^{22} -3.12311 q^{23} -6.56155 q^{24} -2.56155 q^{25} -5.12311 q^{26} +1.00000 q^{27} +4.56155 q^{28} +1.12311 q^{29} +4.00000 q^{30} +1.56155 q^{31} -6.56155 q^{32} -2.43845 q^{33} +5.12311 q^{34} -1.56155 q^{35} +4.56155 q^{36} +8.24621 q^{37} +6.24621 q^{38} +2.00000 q^{39} +10.2462 q^{40} -8.68466 q^{41} -2.56155 q^{42} +4.68466 q^{43} -11.1231 q^{44} -1.56155 q^{45} +8.00000 q^{46} +9.56155 q^{47} +7.68466 q^{48} +1.00000 q^{49} +6.56155 q^{50} -2.00000 q^{51} +9.12311 q^{52} +7.12311 q^{53} -2.56155 q^{54} +3.80776 q^{55} -6.56155 q^{56} -2.43845 q^{57} -2.87689 q^{58} -8.68466 q^{59} -7.12311 q^{60} +4.24621 q^{61} -4.00000 q^{62} +1.00000 q^{63} +1.43845 q^{64} -3.12311 q^{65} +6.24621 q^{66} +4.68466 q^{67} -9.12311 q^{68} -3.12311 q^{69} +4.00000 q^{70} +8.00000 q^{71} -6.56155 q^{72} +0.438447 q^{73} -21.1231 q^{74} -2.56155 q^{75} -11.1231 q^{76} -2.43845 q^{77} -5.12311 q^{78} -5.12311 q^{79} -12.0000 q^{80} +1.00000 q^{81} +22.2462 q^{82} +4.00000 q^{83} +4.56155 q^{84} +3.12311 q^{85} -12.0000 q^{86} +1.12311 q^{87} +16.0000 q^{88} -10.4384 q^{89} +4.00000 q^{90} +2.00000 q^{91} -14.2462 q^{92} +1.56155 q^{93} -24.4924 q^{94} +3.80776 q^{95} -6.56155 q^{96} -2.87689 q^{97} -2.56155 q^{98} -2.43845 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 2 q^{3} + 5 q^{4} + q^{5} - q^{6} + 2 q^{7} - 9 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - q^2 + 2 * q^3 + 5 * q^4 + q^5 - q^6 + 2 * q^7 - 9 * q^8 + 2 * q^9	\( 2 q - q^{2} + 2 q^{3} + 5 q^{4} + q^{5} - q^{6} + 2 q^{7} - 9 q^{8} + 2 q^{9} + 8 q^{10} - 9 q^{11} + 5 q^{12} + 4 q^{13} - q^{14} + q^{15} + 3 q^{16} - 4 q^{17} - q^{18} - 9 q^{19} - 6 q^{20} + 2 q^{21} - 4 q^{22} + 2 q^{23} - 9 q^{24} - q^{25} - 2 q^{26} + 2 q^{27} + 5 q^{28} - 6 q^{29} + 8 q^{30} - q^{31} - 9 q^{32} - 9 q^{33} + 2 q^{34} + q^{35} + 5 q^{36} - 4 q^{38} + 4 q^{39} + 4 q^{40} - 5 q^{41} - q^{42} - 3 q^{43} - 14 q^{44} + q^{45} + 16 q^{46} + 15 q^{47} + 3 q^{48} + 2 q^{49} + 9 q^{50} - 4 q^{51} + 10 q^{52} + 6 q^{53} - q^{54} - 13 q^{55} - 9 q^{56} - 9 q^{57} - 14 q^{58} - 5 q^{59} - 6 q^{60} - 8 q^{61} - 8 q^{62} + 2 q^{63} + 7 q^{64} + 2 q^{65} - 4 q^{66} - 3 q^{67} - 10 q^{68} + 2 q^{69} + 8 q^{70} + 16 q^{71} - 9 q^{72} + 5 q^{73} - 34 q^{74} - q^{75} - 14 q^{76} - 9 q^{77} - 2 q^{78} - 2 q^{79} - 24 q^{80} + 2 q^{81} + 28 q^{82} + 8 q^{83} + 5 q^{84} - 2 q^{85} - 24 q^{86} - 6 q^{87} + 32 q^{88} - 25 q^{89} + 8 q^{90} + 4 q^{91} - 12 q^{92} - q^{93} - 16 q^{94} - 13 q^{95} - 9 q^{96} - 14 q^{97} - q^{98} - 9 q^{99}+O(q^{100}) \) 2 * q - q^2 + 2 * q^3 + 5 * q^4 + q^5 - q^6 + 2 * q^7 - 9 * q^8 + 2 * q^9 + 8 * q^10 - 9 * q^11 + 5 * q^12 + 4 * q^13 - q^14 + q^15 + 3 * q^16 - 4 * q^17 - q^18 - 9 * q^19 - 6 * q^20 + 2 * q^21 - 4 * q^22 + 2 * q^23 - 9 * q^24 - q^25 - 2 * q^26 + 2 * q^27 + 5 * q^28 - 6 * q^29 + 8 * q^30 - q^31 - 9 * q^32 - 9 * q^33 + 2 * q^34 + q^35 + 5 * q^36 - 4 * q^38 + 4 * q^39 + 4 * q^40 - 5 * q^41 - q^42 - 3 * q^43 - 14 * q^44 + q^45 + 16 * q^46 + 15 * q^47 + 3 * q^48 + 2 * q^49 + 9 * q^50 - 4 * q^51 + 10 * q^52 + 6 * q^53 - q^54 - 13 * q^55 - 9 * q^56 - 9 * q^57 - 14 * q^58 - 5 * q^59 - 6 * q^60 - 8 * q^61 - 8 * q^62 + 2 * q^63 + 7 * q^64 + 2 * q^65 - 4 * q^66 - 3 * q^67 - 10 * q^68 + 2 * q^69 + 8 * q^70 + 16 * q^71 - 9 * q^72 + 5 * q^73 - 34 * q^74 - q^75 - 14 * q^76 - 9 * q^77 - 2 * q^78 - 2 * q^79 - 24 * q^80 + 2 * q^81 + 28 * q^82 + 8 * q^83 + 5 * q^84 - 2 * q^85 - 24 * q^86 - 6 * q^87 + 32 * q^88 - 25 * q^89 + 8 * q^90 + 4 * q^91 - 12 * q^92 - q^93 - 16 * q^94 - 13 * q^95 - 9 * q^96 - 14 * q^97 - q^98 - 9 * 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 $

$2$	−2.56155	−1.81129	−0.905646	−	0.424035i	$-0.860613\pi$
$2$	−2.56155	−1.81129	−0.905646	+	0.424035i	$0.860613\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.56155	2.28078
$5$	−1.56155	−0.698348	−0.349174	−	0.937058i	$-0.613538\pi$
$5$	−1.56155	−0.698348	−0.349174	+	0.937058i	$0.613538\pi$
$6$	−2.56155	−1.04575
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−6.56155	−2.31986
$9$	1.00000	0.333333
$10$	4.00000	1.26491
$11$	−2.43845	−0.735219	−0.367610	−	0.929980i	$-0.619824\pi$
$11$	−2.43845	−0.735219	−0.367610	+	0.929980i	$0.619824\pi$
$12$	4.56155	1.31681
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−2.56155	−0.684604
$15$	−1.56155	−0.403191
$16$	7.68466	1.92116
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	−2.56155	−0.603764
$19$	−2.43845	−0.559418	−0.279709	−	0.960085i	$-0.590238\pi$
$19$	−2.43845	−0.559418	−0.279709	+	0.960085i	$0.590238\pi$
$20$	−7.12311	−1.59277
$21$	1.00000	0.218218
$22$	6.24621	1.33170
$23$	−3.12311	−0.651213	−0.325606	−	0.945505i	$-0.605568\pi$
$23$	−3.12311	−0.651213	−0.325606	+	0.945505i	$0.605568\pi$
$24$	−6.56155	−1.33937
$25$	−2.56155	−0.512311
$26$	−5.12311	−1.00472
$27$	1.00000	0.192450
$28$	4.56155	0.862052
$29$	1.12311	0.208555	0.104278	−	0.994548i	$-0.466747\pi$
$29$	1.12311	0.208555	0.104278	+	0.994548i	$0.466747\pi$
$30$	4.00000	0.730297
$31$	1.56155	0.280463	0.140232	−	0.990119i	$-0.455215\pi$
$31$	1.56155	0.280463	0.140232	+	0.990119i	$0.455215\pi$
$32$	−6.56155	−1.15993
$33$	−2.43845	−0.424479
$34$	5.12311	0.878605
$35$	−1.56155	−0.263951
$36$	4.56155	0.760259
$37$	8.24621	1.35567	0.677834	−	0.735215i	$-0.262919\pi$
$37$	8.24621	1.35567	0.677834	+	0.735215i	$0.262919\pi$
$38$	6.24621	1.01327
$39$	2.00000	0.320256
$40$	10.2462	1.62007
$41$	−8.68466	−1.35632	−0.678158	−	0.734916i	$-0.737221\pi$
$41$	−8.68466	−1.35632	−0.678158	+	0.734916i	$0.737221\pi$
$42$	−2.56155	−0.395256
$43$	4.68466	0.714404	0.357202	−	0.934027i	$-0.383731\pi$
$43$	4.68466	0.714404	0.357202	+	0.934027i	$0.383731\pi$
$44$	−11.1231	−1.67687
$45$	−1.56155	−0.232783
$46$	8.00000	1.17954
$47$	9.56155	1.39470	0.697348	−	0.716733i	$-0.254363\pi$
$47$	9.56155	1.39470	0.697348	+	0.716733i	$0.254363\pi$
$48$	7.68466	1.10918
$49$	1.00000	0.142857
$50$	6.56155	0.927944
$51$	−2.00000	−0.280056
$52$	9.12311	1.26515
$53$	7.12311	0.978434	0.489217	−	0.872162i	$-0.337283\pi$
$53$	7.12311	0.978434	0.489217	+	0.872162i	$0.337283\pi$
$54$	−2.56155	−0.348583
$55$	3.80776	0.513439
$56$	−6.56155	−0.876824
$57$	−2.43845	−0.322980
$58$	−2.87689	−0.377755
$59$	−8.68466	−1.13065	−0.565323	−	0.824870i	$-0.691249\pi$
$59$	−8.68466	−1.13065	−0.565323	+	0.824870i	$0.691249\pi$
$60$	−7.12311	−0.919589
$61$	4.24621	0.543672	0.271836	−	0.962344i	$-0.412369\pi$
$61$	4.24621	0.543672	0.271836	+	0.962344i	$0.412369\pi$
$62$	−4.00000	−0.508001
$63$	1.00000	0.125988
$64$	1.43845	0.179806
$65$	−3.12311	−0.387374
$66$	6.24621	0.768855
$67$	4.68466	0.572322	0.286161	−	0.958182i	$-0.407621\pi$
$67$	4.68466	0.572322	0.286161	+	0.958182i	$0.407621\pi$
$68$	−9.12311	−1.10634
$69$	−3.12311	−0.375978
$70$	4.00000	0.478091
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−6.56155	−0.773286
$73$	0.438447	0.0513164	0.0256582	−	0.999671i	$-0.491832\pi$
$73$	0.438447	0.0513164	0.0256582	+	0.999671i	$0.491832\pi$
$74$	−21.1231	−2.45551
$75$	−2.56155	−0.295783
$76$	−11.1231	−1.27591
$77$	−2.43845	−0.277887
$78$	−5.12311	−0.580077
$79$	−5.12311	−0.576394	−0.288197	−	0.957571i	$-0.593056\pi$
$79$	−5.12311	−0.576394	−0.288197	+	0.957571i	$0.593056\pi$
$80$	−12.0000	−1.34164
$81$	1.00000	0.111111
$82$	22.2462	2.45668
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	4.56155	0.497706
$85$	3.12311	0.338748
$86$	−12.0000	−1.29399
$87$	1.12311	0.120410
$88$	16.0000	1.70561
$89$	−10.4384	−1.10647	−0.553237	−	0.833024i	$-0.686607\pi$
$89$	−10.4384	−1.10647	−0.553237	+	0.833024i	$0.686607\pi$
$90$	4.00000	0.421637
$91$	2.00000	0.209657
$92$	−14.2462	−1.48527
$93$	1.56155	0.161925
$94$	−24.4924	−2.52620
$95$	3.80776	0.390668
$96$	−6.56155	−0.669686
$97$	−2.87689	−0.292104	−0.146052	−	0.989277i	$-0.546657\pi$
$97$	−2.87689	−0.292104	−0.146052	+	0.989277i	$0.546657\pi$
$98$	−2.56155	−0.258756
$99$	−2.43845	−0.245073
$100$	−11.6847	−1.16847
$101$	13.1231	1.30580	0.652899	−	0.757445i	$-0.273553\pi$
$101$	13.1231	1.30580	0.652899	+	0.757445i	$0.273553\pi$
$102$	5.12311	0.507263
$103$	−9.56155	−0.942128	−0.471064	−	0.882099i	$-0.656130\pi$
$103$	−9.56155	−0.942128	−0.471064	+	0.882099i	$0.656130\pi$
$104$	−13.1231	−1.28683
$105$	−1.56155	−0.152392
$106$	−18.2462	−1.77223
$107$	−2.24621	−0.217149	−0.108575	−	0.994088i	$-0.534629\pi$
$107$	−2.24621	−0.217149	−0.108575	+	0.994088i	$0.534629\pi$
$108$	4.56155	0.438936
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−9.75379	−0.929987
$111$	8.24621	0.782696
$112$	7.68466	0.726132
$113$	−19.3693	−1.82211	−0.911056	−	0.412283i	$-0.864732\pi$
$113$	−19.3693	−1.82211	−0.911056	+	0.412283i	$0.864732\pi$
$114$	6.24621	0.585011
$115$	4.87689	0.454773
$116$	5.12311	0.475668
$117$	2.00000	0.184900
$118$	22.2462	2.04793
$119$	−2.00000	−0.183340
$120$	10.2462	0.935347
$121$	−5.05398	−0.459452
$122$	−10.8769	−0.984748
$123$	−8.68466	−0.783069
$124$	7.12311	0.639674
$125$	11.8078	1.05612
$126$	−2.56155	−0.228201
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	9.43845	0.834249
$129$	4.68466	0.412461
$130$	8.00000	0.701646
$131$	3.80776	0.332686	0.166343	−	0.986068i	$-0.446804\pi$
$131$	3.80776	0.332686	0.166343	+	0.986068i	$0.446804\pi$
$132$	−11.1231	−0.968142
$133$	−2.43845	−0.211440
$134$	−12.0000	−1.03664
$135$	−1.56155	−0.134397
$136$	13.1231	1.12530
$137$	12.2462	1.04626	0.523132	−	0.852252i	$-0.324763\pi$
$137$	12.2462	1.04626	0.523132	+	0.852252i	$0.324763\pi$
$138$	8.00000	0.681005
$139$	9.56155	0.811000	0.405500	−	0.914095i	$-0.367097\pi$
$139$	9.56155	0.811000	0.405500	+	0.914095i	$0.367097\pi$
$140$	−7.12311	−0.602012
$141$	9.56155	0.805228
$142$	−20.4924	−1.71969
$143$	−4.87689	−0.407826
$144$	7.68466	0.640388
$145$	−1.75379	−0.145644
$146$	−1.12311	−0.0929489
$147$	1.00000	0.0824786
$148$	37.6155	3.09198
$149$	19.3693	1.58680	0.793398	−	0.608703i	$-0.208310\pi$
$149$	19.3693	1.58680	0.793398	+	0.608703i	$0.208310\pi$
$150$	6.56155	0.535749
$151$	−0.246211	−0.0200364	−0.0100182	−	0.999950i	$-0.503189\pi$
$151$	−0.246211	−0.0200364	−0.0100182	+	0.999950i	$0.503189\pi$
$152$	16.0000	1.29777
$153$	−2.00000	−0.161690
$154$	6.24621	0.503334
$155$	−2.43845	−0.195861
$156$	9.12311	0.730433
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	13.1231	1.04402
$159$	7.12311	0.564899
$160$	10.2462	0.810034
$161$	−3.12311	−0.246135
$162$	−2.56155	−0.201255
$163$	−21.1231	−1.65449	−0.827245	−	0.561842i	$-0.810093\pi$
$163$	−21.1231	−1.65449	−0.827245	+	0.561842i	$0.810093\pi$
$164$	−39.6155	−3.09345
$165$	3.80776	0.296434
$166$	−10.2462	−0.795260
$167$	−18.2462	−1.41193	−0.705967	−	0.708245i	$-0.749487\pi$
$167$	−18.2462	−1.41193	−0.705967	+	0.708245i	$0.749487\pi$
$168$	−6.56155	−0.506235
$169$	−9.00000	−0.692308
$170$	−8.00000	−0.613572
$171$	−2.43845	−0.186473
$172$	21.3693	1.62940
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	−2.87689	−0.218097
$175$	−2.56155	−0.193635
$176$	−18.7386	−1.41248
$177$	−8.68466	−0.652779
$178$	26.7386	2.00415
$179$	−19.8078	−1.48050	−0.740251	−	0.672331i	$-0.765293\pi$
$179$	−19.8078	−1.48050	−0.740251	+	0.672331i	$0.765293\pi$
$180$	−7.12311	−0.530925
$181$	−9.12311	−0.678115	−0.339058	−	0.940766i	$-0.610108\pi$
$181$	−9.12311	−0.678115	−0.339058	+	0.940766i	$0.610108\pi$
$182$	−5.12311	−0.379750
$183$	4.24621	0.313889
$184$	20.4924	1.51072
$185$	−12.8769	−0.946728
$186$	−4.00000	−0.293294
$187$	4.87689	0.356634
$188$	43.6155	3.18099
$189$	1.00000	0.0727393
$190$	−9.75379	−0.707614
$191$	−26.2462	−1.89911	−0.949555	−	0.313602i	$-0.898464\pi$
$191$	−26.2462	−1.89911	−0.949555	+	0.313602i	$0.898464\pi$
$192$	1.43845	0.103811
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	7.36932	0.529086
$195$	−3.12311	−0.223650
$196$	4.56155	0.325825
$197$	−14.0540	−1.00130	−0.500652	−	0.865649i	$-0.666906\pi$
$197$	−14.0540	−1.00130	−0.500652	+	0.865649i	$0.666906\pi$
$198$	6.24621	0.443899
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	16.8078	1.18849
$201$	4.68466	0.330430
$202$	−33.6155	−2.36518
$203$	1.12311	0.0788266
$204$	−9.12311	−0.638745
$205$	13.5616	0.947180
$206$	24.4924	1.70647
$207$	−3.12311	−0.217071
$208$	15.3693	1.06567
$209$	5.94602	0.411295
$210$	4.00000	0.276026
$211$	6.87689	0.473425	0.236712	−	0.971580i	$-0.423930\pi$
$211$	6.87689	0.473425	0.236712	+	0.971580i	$0.423930\pi$
$212$	32.4924	2.23159
$213$	8.00000	0.548151
$214$	5.75379	0.393321
$215$	−7.31534	−0.498902
$216$	−6.56155	−0.446457
$217$	1.56155	0.106005
$218$	5.12311	0.346980
$219$	0.438447	0.0296275
$220$	17.3693	1.17104
$221$	−4.00000	−0.269069
$222$	−21.1231	−1.41769
$223$	−4.68466	−0.313708	−0.156854	−	0.987622i	$-0.550135\pi$
$223$	−4.68466	−0.313708	−0.156854	+	0.987622i	$0.550135\pi$
$224$	−6.56155	−0.438412
$225$	−2.56155	−0.170770
$226$	49.6155	3.30038
$227$	10.2462	0.680065	0.340032	−	0.940414i	$-0.389562\pi$
$227$	10.2462	0.680065	0.340032	+	0.940414i	$0.389562\pi$
$228$	−11.1231	−0.736646
$229$	8.43845	0.557628	0.278814	−	0.960345i	$-0.410059\pi$
$229$	8.43845	0.557628	0.278814	+	0.960345i	$0.410059\pi$
$230$	−12.4924	−0.823726
$231$	−2.43845	−0.160438
$232$	−7.36932	−0.483819
$233$	−9.56155	−0.626398	−0.313199	−	0.949687i	$-0.601401\pi$
$233$	−9.56155	−0.626398	−0.313199	+	0.949687i	$0.601401\pi$
$234$	−5.12311	−0.334908
$235$	−14.9309	−0.973983
$236$	−39.6155	−2.57875
$237$	−5.12311	−0.332781
$238$	5.12311	0.332082
$239$	−11.3153	−0.731929	−0.365964	−	0.930629i	$-0.619261\pi$
$239$	−11.3153	−0.731929	−0.365964	+	0.930629i	$0.619261\pi$
$240$	−12.0000	−0.774597
$241$	0.246211	0.0158599	0.00792993	−	0.999969i	$-0.497476\pi$
$241$	0.246211	0.0158599	0.00792993	+	0.999969i	$0.497476\pi$
$242$	12.9460	0.832202
$243$	1.00000	0.0641500
$244$	19.3693	1.23999
$245$	−1.56155	−0.0997639
$246$	22.2462	1.41837
$247$	−4.87689	−0.310309
$248$	−10.2462	−0.650635
$249$	4.00000	0.253490
$250$	−30.2462	−1.91294
$251$	−7.12311	−0.449606	−0.224803	−	0.974404i	$-0.572174\pi$
$251$	−7.12311	−0.449606	−0.224803	+	0.974404i	$0.572174\pi$
$252$	4.56155	0.287351
$253$	7.61553	0.478784
$254$	15.3693	0.964357
$255$	3.12311	0.195576
$256$	−27.0540	−1.69087
$257$	−18.4384	−1.15016	−0.575079	−	0.818098i	$-0.695029\pi$
$257$	−18.4384	−1.15016	−0.575079	+	0.818098i	$0.695029\pi$
$258$	−12.0000	−0.747087
$259$	8.24621	0.512395
$260$	−14.2462	−0.883513
$261$	1.12311	0.0695185
$262$	−9.75379	−0.602591
$263$	6.63068	0.408865	0.204433	−	0.978881i	$-0.434465\pi$
$263$	6.63068	0.408865	0.204433	+	0.978881i	$0.434465\pi$
$264$	16.0000	0.984732
$265$	−11.1231	−0.683287
$266$	6.24621	0.382980
$267$	−10.4384	−0.638823
$268$	21.3693	1.30534
$269$	12.6847	0.773397	0.386699	−	0.922206i	$-0.373615\pi$
$269$	12.6847	0.773397	0.386699	+	0.922206i	$0.373615\pi$
$270$	4.00000	0.243432
$271$	28.2462	1.71584	0.857918	−	0.513787i	$-0.171758\pi$
$271$	28.2462	1.71584	0.857918	+	0.513787i	$0.171758\pi$
$272$	−15.3693	−0.931902
$273$	2.00000	0.121046
$274$	−31.3693	−1.89509
$275$	6.24621	0.376661
$276$	−14.2462	−0.857521
$277$	−28.9309	−1.73829	−0.869144	−	0.494560i	$-0.835329\pi$
$277$	−28.9309	−1.73829	−0.869144	+	0.494560i	$0.835329\pi$
$278$	−24.4924	−1.46896
$279$	1.56155	0.0934877
$280$	10.2462	0.612328
$281$	−22.0540	−1.31563	−0.657815	−	0.753180i	$-0.728519\pi$
$281$	−22.0540	−1.31563	−0.657815	+	0.753180i	$0.728519\pi$
$282$	−24.4924	−1.45850
$283$	21.1231	1.25564	0.627819	−	0.778359i	$-0.283948\pi$
$283$	21.1231	1.25564	0.627819	+	0.778359i	$0.283948\pi$
$284$	36.4924	2.16543
$285$	3.80776	0.225552
$286$	12.4924	0.738692
$287$	−8.68466	−0.512639
$288$	−6.56155	−0.386643
$289$	−13.0000	−0.764706
$290$	4.49242	0.263804
$291$	−2.87689	−0.168647
$292$	2.00000	0.117041
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	−2.56155	−0.149393
$295$	13.5616	0.789584
$296$	−54.1080	−3.14496
$297$	−2.43845	−0.141493
$298$	−49.6155	−2.87415
$299$	−6.24621	−0.361228
$300$	−11.6847	−0.674614
$301$	4.68466	0.270019
$302$	0.630683	0.0362917
$303$	13.1231	0.753903
$304$	−18.7386	−1.07473
$305$	−6.63068	−0.379672
$306$	5.12311	0.292868
$307$	−16.2462	−0.927220	−0.463610	−	0.886039i	$-0.653446\pi$
$307$	−16.2462	−0.927220	−0.463610	+	0.886039i	$0.653446\pi$
$308$	−11.1231	−0.633798
$309$	−9.56155	−0.543938
$310$	6.24621	0.354761
$311$	20.3002	1.15112	0.575559	−	0.817760i	$-0.304785\pi$
$311$	20.3002	1.15112	0.575559	+	0.817760i	$0.304785\pi$
$312$	−13.1231	−0.742950
$313$	−27.1771	−1.53614	−0.768070	−	0.640366i	$-0.778783\pi$
$313$	−27.1771	−1.53614	−0.768070	+	0.640366i	$0.778783\pi$
$314$	−35.8617	−2.02380
$315$	−1.56155	−0.0879835
$316$	−23.3693	−1.31463
$317$	31.3693	1.76188	0.880938	−	0.473231i	$-0.156913\pi$
$317$	31.3693	1.76188	0.880938	+	0.473231i	$0.156913\pi$
$318$	−18.2462	−1.02320
$319$	−2.73863	−0.153334
$320$	−2.24621	−0.125567
$321$	−2.24621	−0.125371
$322$	8.00000	0.445823
$323$	4.87689	0.271358
$324$	4.56155	0.253420
$325$	−5.12311	−0.284179
$326$	54.1080	2.99676
$327$	−2.00000	−0.110600
$328$	56.9848	3.14646
$329$	9.56155	0.527145
$330$	−9.75379	−0.536928
$331$	−6.24621	−0.343323	−0.171661	−	0.985156i	$-0.554914\pi$
$331$	−6.24621	−0.343323	−0.171661	+	0.985156i	$0.554914\pi$
$332$	18.2462	1.00139
$333$	8.24621	0.451890
$334$	46.7386	2.55742
$335$	−7.31534	−0.399680
$336$	7.68466	0.419232
$337$	−11.3693	−0.619326	−0.309663	−	0.950846i	$-0.600216\pi$
$337$	−11.3693	−0.619326	−0.309663	+	0.950846i	$0.600216\pi$
$338$	23.0540	1.25397
$339$	−19.3693	−1.05200
$340$	14.2462	0.772609
$341$	−3.80776	−0.206202
$342$	6.24621	0.337756
$343$	1.00000	0.0539949
$344$	−30.7386	−1.65732
$345$	4.87689	0.262563
$346$	25.6155	1.37710
$347$	7.80776	0.419143	0.209571	−	0.977793i	$-0.432793\pi$
$347$	7.80776	0.419143	0.209571	+	0.977793i	$0.432793\pi$
$348$	5.12311	0.274627
$349$	−31.8617	−1.70552	−0.852760	−	0.522303i	$-0.825073\pi$
$349$	−31.8617	−1.70552	−0.852760	+	0.522303i	$0.825073\pi$
$350$	6.56155	0.350730
$351$	2.00000	0.106752
$352$	16.0000	0.852803
$353$	−6.87689	−0.366020	−0.183010	−	0.983111i	$-0.558584\pi$
$353$	−6.87689	−0.366020	−0.183010	+	0.983111i	$0.558584\pi$
$354$	22.2462	1.18237
$355$	−12.4924	−0.663029
$356$	−47.6155	−2.52362
$357$	−2.00000	−0.105851
$358$	50.7386	2.68162
$359$	−28.6847	−1.51392	−0.756959	−	0.653462i	$-0.773316\pi$
$359$	−28.6847	−1.51392	−0.756959	+	0.653462i	$0.773316\pi$
$360$	10.2462	0.540023
$361$	−13.0540	−0.687051
$362$	23.3693	1.22826
$363$	−5.05398	−0.265265
$364$	9.12311	0.478181
$365$	−0.684658	−0.0358367
$366$	−10.8769	−0.568544
$367$	−17.6155	−0.919523	−0.459762	−	0.888042i	$-0.652065\pi$
$367$	−17.6155	−0.919523	−0.459762	+	0.888042i	$0.652065\pi$
$368$	−24.0000	−1.25109
$369$	−8.68466	−0.452105
$370$	32.9848	1.71480
$371$	7.12311	0.369813
$372$	7.12311	0.369316
$373$	−1.31534	−0.0681058	−0.0340529	−	0.999420i	$-0.510841\pi$
$373$	−1.31534	−0.0681058	−0.0340529	+	0.999420i	$0.510841\pi$
$374$	−12.4924	−0.645968
$375$	11.8078	0.609750
$376$	−62.7386	−3.23550
$377$	2.24621	0.115686
$378$	−2.56155	−0.131752
$379$	−22.4924	−1.15536	−0.577679	−	0.816264i	$-0.696041\pi$
$379$	−22.4924	−1.15536	−0.577679	+	0.816264i	$0.696041\pi$
$380$	17.3693	0.891027
$381$	−6.00000	−0.307389
$382$	67.2311	3.43984
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	9.43845	0.481654
$385$	3.80776	0.194062
$386$	−5.12311	−0.260759
$387$	4.68466	0.238135
$388$	−13.1231	−0.666225
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	8.00000	0.405096
$391$	6.24621	0.315884
$392$	−6.56155	−0.331408
$393$	3.80776	0.192076
$394$	36.0000	1.81365
$395$	8.00000	0.402524
$396$	−11.1231	−0.558957
$397$	−2.19224	−0.110025	−0.0550126	−	0.998486i	$-0.517520\pi$
$397$	−2.19224	−0.110025	−0.0550126	+	0.998486i	$0.517520\pi$
$398$	−25.6155	−1.28399
$399$	−2.43845	−0.122075
$400$	−19.6847	−0.984233
$401$	13.1231	0.655337	0.327668	−	0.944793i	$-0.393737\pi$
$401$	13.1231	0.655337	0.327668	+	0.944793i	$0.393737\pi$
$402$	−12.0000	−0.598506
$403$	3.12311	0.155573
$404$	59.8617	2.97823
$405$	−1.56155	−0.0775942
$406$	−2.87689	−0.142778
$407$	−20.1080	−0.996714
$408$	13.1231	0.649691
$409$	22.4924	1.11218	0.556089	−	0.831123i	$-0.312301\pi$
$409$	22.4924	1.11218	0.556089	+	0.831123i	$0.312301\pi$
$410$	−34.7386	−1.71562
$411$	12.2462	0.604061
$412$	−43.6155	−2.14878
$413$	−8.68466	−0.427344
$414$	8.00000	0.393179
$415$	−6.24621	−0.306614
$416$	−13.1231	−0.643413
$417$	9.56155	0.468231
$418$	−15.2311	−0.744975
$419$	−5.75379	−0.281091	−0.140545	−	0.990074i	$-0.544886\pi$
$419$	−5.75379	−0.281091	−0.140545	+	0.990074i	$0.544886\pi$
$420$	−7.12311	−0.347572
$421$	0.246211	0.0119996	0.00599980	−	0.999982i	$-0.498090\pi$
$421$	0.246211	0.0119996	0.00599980	+	0.999982i	$0.498090\pi$
$422$	−17.6155	−0.857510
$423$	9.56155	0.464899
$424$	−46.7386	−2.26983
$425$	5.12311	0.248507
$426$	−20.4924	−0.992861
$427$	4.24621	0.205489
$428$	−10.2462	−0.495269
$429$	−4.87689	−0.235459
$430$	18.7386	0.903657
$431$	27.1231	1.30647	0.653237	−	0.757153i	$-0.273411\pi$
$431$	27.1231	1.30647	0.653237	+	0.757153i	$0.273411\pi$
$432$	7.68466	0.369728
$433$	−18.0000	−0.865025	−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000	−0.865025	−0.432512	+	0.901628i	$0.642373\pi$
$434$	−4.00000	−0.192006
$435$	−1.75379	−0.0840877
$436$	−9.12311	−0.436918
$437$	7.61553	0.364300
$438$	−1.12311	−0.0536641
$439$	0.684658	0.0326770	0.0163385	−	0.999867i	$-0.494799\pi$
$439$	0.684658	0.0326770	0.0163385	+	0.999867i	$0.494799\pi$
$440$	−24.9848	−1.19111
$441$	1.00000	0.0476190
$442$	10.2462	0.487363
$443$	10.0540	0.477679	0.238839	−	0.971059i	$-0.423233\pi$
$443$	10.0540	0.477679	0.238839	+	0.971059i	$0.423233\pi$
$444$	37.6155	1.78515
$445$	16.3002	0.772703
$446$	12.0000	0.568216
$447$	19.3693	0.916137
$448$	1.43845	0.0679602
$449$	−13.3693	−0.630937	−0.315469	−	0.948936i	$-0.602162\pi$
$449$	−13.3693	−0.630937	−0.315469	+	0.948936i	$0.602162\pi$
$450$	6.56155	0.309315
$451$	21.1771	0.997190
$452$	−88.3542	−4.15583
$453$	−0.246211	−0.0115680
$454$	−26.2462	−1.23180
$455$	−3.12311	−0.146413
$456$	16.0000	0.749269
$457$	−16.2462	−0.759966	−0.379983	−	0.924994i	$-0.624070\pi$
$457$	−16.2462	−0.759966	−0.379983	+	0.924994i	$0.624070\pi$
$458$	−21.6155	−1.01003
$459$	−2.00000	−0.0933520
$460$	22.2462	1.03723
$461$	−11.1231	−0.518055	−0.259027	−	0.965870i	$-0.583402\pi$
$461$	−11.1231	−0.518055	−0.259027	+	0.965870i	$0.583402\pi$
$462$	6.24621	0.290600
$463$	30.0000	1.39422	0.697109	−	0.716965i	$-0.254469\pi$
$463$	30.0000	1.39422	0.697109	+	0.716965i	$0.254469\pi$
$464$	8.63068	0.400669
$465$	−2.43845	−0.113080
$466$	24.4924	1.13459
$467$	11.6155	0.537503	0.268751	−	0.963210i	$-0.413389\pi$
$467$	11.6155	0.537503	0.268751	+	0.963210i	$0.413389\pi$
$468$	9.12311	0.421716
$469$	4.68466	0.216317
$470$	38.2462	1.76417
$471$	14.0000	0.645086
$472$	56.9848	2.62294
$473$	−11.4233	−0.525244
$474$	13.1231	0.602764
$475$	6.24621	0.286596
$476$	−9.12311	−0.418157
$477$	7.12311	0.326145
$478$	28.9848	1.32574
$479$	25.3693	1.15915	0.579577	−	0.814918i	$-0.303218\pi$
$479$	25.3693	1.15915	0.579577	+	0.814918i	$0.303218\pi$
$480$	10.2462	0.467673
$481$	16.4924	0.751990
$482$	−0.630683	−0.0287268
$483$	−3.12311	−0.142106
$484$	−23.0540	−1.04791
$485$	4.49242	0.203990
$486$	−2.56155	−0.116194
$487$	19.7538	0.895130	0.447565	−	0.894251i	$-0.352291\pi$
$487$	19.7538	0.895130	0.447565	+	0.894251i	$0.352291\pi$
$488$	−27.8617	−1.26124
$489$	−21.1231	−0.955220
$490$	4.00000	0.180702
$491$	23.1231	1.04353	0.521766	−	0.853089i	$-0.325274\pi$
$491$	23.1231	1.04353	0.521766	+	0.853089i	$0.325274\pi$
$492$	−39.6155	−1.78601
$493$	−2.24621	−0.101164
$494$	12.4924	0.562061
$495$	3.80776	0.171146
$496$	12.0000	0.538816
$497$	8.00000	0.358849
$498$	−10.2462	−0.459144
$499$	26.7386	1.19699	0.598493	−	0.801128i	$-0.295767\pi$
$499$	26.7386	1.19699	0.598493	+	0.801128i	$0.295767\pi$
$500$	53.8617	2.40877
$501$	−18.2462	−0.815181
$502$	18.2462	0.814368
$503$	−11.3153	−0.504526	−0.252263	−	0.967659i	$-0.581175\pi$
$503$	−11.3153	−0.504526	−0.252263	+	0.967659i	$0.581175\pi$
$504$	−6.56155	−0.292275
$505$	−20.4924	−0.911901
$506$	−19.5076	−0.867218
$507$	−9.00000	−0.399704
$508$	−27.3693	−1.21432
$509$	3.36932	0.149342	0.0746712	−	0.997208i	$-0.476209\pi$
$509$	3.36932	0.149342	0.0746712	+	0.997208i	$0.476209\pi$
$510$	−8.00000	−0.354246
$511$	0.438447	0.0193958
$512$	50.4233	2.22842
$513$	−2.43845	−0.107660
$514$	47.2311	2.08327
$515$	14.9309	0.657933
$516$	21.3693	0.940732
$517$	−23.3153	−1.02541
$518$	−21.1231	−0.928096
$519$	−10.0000	−0.438951
$520$	20.4924	0.898652
$521$	−12.2462	−0.536516	−0.268258	−	0.963347i	$-0.586448\pi$
$521$	−12.2462	−0.536516	−0.268258	+	0.963347i	$0.586448\pi$
$522$	−2.87689	−0.125918
$523$	−29.6155	−1.29500	−0.647498	−	0.762067i	$-0.724185\pi$
$523$	−29.6155	−1.29500	−0.647498	+	0.762067i	$0.724185\pi$
$524$	17.3693	0.758782
$525$	−2.56155	−0.111795
$526$	−16.9848	−0.740574
$527$	−3.12311	−0.136045
$528$	−18.7386	−0.815494
$529$	−13.2462	−0.575922
$530$	28.4924	1.23763
$531$	−8.68466	−0.376882
$532$	−11.1231	−0.482248
$533$	−17.3693	−0.752349
$534$	26.7386	1.15709
$535$	3.50758	0.151646
$536$	−30.7386	−1.32771
$537$	−19.8078	−0.854768
$538$	−32.4924	−1.40085
$539$	−2.43845	−0.105031
$540$	−7.12311	−0.306530
$541$	16.7386	0.719650	0.359825	−	0.933020i	$-0.382836\pi$
$541$	16.7386	0.719650	0.359825	+	0.933020i	$0.382836\pi$
$542$	−72.3542	−3.10788
$543$	−9.12311	−0.391510
$544$	13.1231	0.562649
$545$	3.12311	0.133779
$546$	−5.12311	−0.219249
$547$	−20.2462	−0.865665	−0.432833	−	0.901474i	$-0.642486\pi$
$547$	−20.2462	−0.865665	−0.432833	+	0.901474i	$0.642486\pi$
$548$	55.8617	2.38630
$549$	4.24621	0.181224
$550$	−16.0000	−0.682242
$551$	−2.73863	−0.116670
$552$	20.4924	0.872215
$553$	−5.12311	−0.217857
$554$	74.1080	3.14855
$555$	−12.8769	−0.546594
$556$	43.6155	1.84971
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	−4.00000	−0.169334
$559$	9.36932	0.396280
$560$	−12.0000	−0.507093
$561$	4.87689	0.205903
$562$	56.4924	2.38299
$563$	−20.4924	−0.863653	−0.431826	−	0.901957i	$-0.642131\pi$
$563$	−20.4924	−0.863653	−0.431826	+	0.901957i	$0.642131\pi$
$564$	43.6155	1.83655
$565$	30.2462	1.27247
$566$	−54.1080	−2.27433
$567$	1.00000	0.0419961
$568$	−52.4924	−2.20253
$569$	29.6155	1.24155	0.620774	−	0.783990i	$-0.286819\pi$
$569$	29.6155	1.24155	0.620774	+	0.783990i	$0.286819\pi$
$570$	−9.75379	−0.408541
$571$	−20.6307	−0.863367	−0.431684	−	0.902025i	$-0.642080\pi$
$571$	−20.6307	−0.863367	−0.431684	+	0.902025i	$0.642080\pi$
$572$	−22.2462	−0.930161
$573$	−26.2462	−1.09645
$574$	22.2462	0.928539
$575$	8.00000	0.333623
$576$	1.43845	0.0599353
$577$	−37.1231	−1.54546	−0.772728	−	0.634738i	$-0.781108\pi$
$577$	−37.1231	−1.54546	−0.772728	+	0.634738i	$0.781108\pi$
$578$	33.3002	1.38511
$579$	2.00000	0.0831172
$580$	−8.00000	−0.332182
$581$	4.00000	0.165948
$582$	7.36932	0.305468
$583$	−17.3693	−0.719364
$584$	−2.87689	−0.119047
$585$	−3.12311	−0.129125
$586$	35.8617	1.48143
$587$	−35.2311	−1.45414	−0.727071	−	0.686563i	$-0.759119\pi$
$587$	−35.2311	−1.45414	−0.727071	+	0.686563i	$0.759119\pi$
$588$	4.56155	0.188115
$589$	−3.80776	−0.156896
$590$	−34.7386	−1.43017
$591$	−14.0540	−0.578103
$592$	63.3693	2.60446
$593$	−0.684658	−0.0281156	−0.0140578	−	0.999901i	$-0.504475\pi$
$593$	−0.684658	−0.0281156	−0.0140578	+	0.999901i	$0.504475\pi$
$594$	6.24621	0.256285
$595$	3.12311	0.128035
$596$	88.3542	3.61913
$597$	10.0000	0.409273
$598$	16.0000	0.654289
$599$	22.6307	0.924665	0.462332	−	0.886707i	$-0.347013\pi$
$599$	22.6307	0.924665	0.462332	+	0.886707i	$0.347013\pi$
$600$	16.8078	0.686174
$601$	−14.4924	−0.591158	−0.295579	−	0.955318i	$-0.595513\pi$
$601$	−14.4924	−0.591158	−0.295579	+	0.955318i	$0.595513\pi$
$602$	−12.0000	−0.489083
$603$	4.68466	0.190774
$604$	−1.12311	−0.0456985
$605$	7.89205	0.320857
$606$	−33.6155	−1.36554
$607$	21.5616	0.875156	0.437578	−	0.899180i	$-0.355836\pi$
$607$	21.5616	0.875156	0.437578	+	0.899180i	$0.355836\pi$
$608$	16.0000	0.648886
$609$	1.12311	0.0455105
$610$	16.9848	0.687696
$611$	19.1231	0.773638
$612$	−9.12311	−0.368780
$613$	21.1231	0.853154	0.426577	−	0.904451i	$-0.359719\pi$
$613$	21.1231	0.853154	0.426577	+	0.904451i	$0.359719\pi$
$614$	41.6155	1.67947
$615$	13.5616	0.546855
$616$	16.0000	0.644658
$617$	33.1771	1.33566	0.667829	−	0.744314i	$-0.267224\pi$
$617$	33.1771	1.33566	0.667829	+	0.744314i	$0.267224\pi$
$618$	24.4924	0.985230
$619$	22.4924	0.904047	0.452023	−	0.892006i	$-0.350702\pi$
$619$	22.4924	0.904047	0.452023	+	0.892006i	$0.350702\pi$
$620$	−11.1231	−0.446715
$621$	−3.12311	−0.125326
$622$	−52.0000	−2.08501
$623$	−10.4384	−0.418208
$624$	15.3693	0.615265
$625$	−5.63068	−0.225227
$626$	69.6155	2.78240
$627$	5.94602	0.237461
$628$	63.8617	2.54836
$629$	−16.4924	−0.657596
$630$	4.00000	0.159364
$631$	18.2462	0.726370	0.363185	−	0.931717i	$-0.381689\pi$
$631$	18.2462	0.726370	0.363185	+	0.931717i	$0.381689\pi$
$632$	33.6155	1.33715
$633$	6.87689	0.273332
$634$	−80.3542	−3.19127
$635$	9.36932	0.371810
$636$	32.4924	1.28841
$637$	2.00000	0.0792429
$638$	7.01515	0.277733
$639$	8.00000	0.316475
$640$	−14.7386	−0.582596
$641$	30.4924	1.20438	0.602189	−	0.798353i	$-0.294295\pi$
$641$	30.4924	1.20438	0.602189	+	0.798353i	$0.294295\pi$
$642$	5.75379	0.227084
$643$	−37.1771	−1.46612	−0.733060	−	0.680163i	$-0.761909\pi$
$643$	−37.1771	−1.46612	−0.733060	+	0.680163i	$0.761909\pi$
$644$	−14.2462	−0.561379
$645$	−7.31534	−0.288041
$646$	−12.4924	−0.491508
$647$	−13.5616	−0.533160	−0.266580	−	0.963813i	$-0.585894\pi$
$647$	−13.5616	−0.533160	−0.266580	+	0.963813i	$0.585894\pi$
$648$	−6.56155	−0.257762
$649$	21.1771	0.831273
$650$	13.1231	0.514731
$651$	1.56155	0.0612021
$652$	−96.3542	−3.77352
$653$	−35.6155	−1.39374	−0.696872	−	0.717196i	$-0.745425\pi$
$653$	−35.6155	−1.39374	−0.696872	+	0.717196i	$0.745425\pi$
$654$	5.12311	0.200329
$655$	−5.94602	−0.232330
$656$	−66.7386	−2.60571
$657$	0.438447	0.0171055
$658$	−24.4924	−0.954814
$659$	9.86174	0.384159	0.192079	−	0.981379i	$-0.438477\pi$
$659$	9.86174	0.384159	0.192079	+	0.981379i	$0.438477\pi$
$660$	17.3693	0.676100
$661$	42.0000	1.63361	0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000	1.63361	0.816805	+	0.576913i	$0.195743\pi$
$662$	16.0000	0.621858
$663$	−4.00000	−0.155347
$664$	−26.2462	−1.01855
$665$	3.80776	0.147659
$666$	−21.1231	−0.818504
$667$	−3.50758	−0.135814
$668$	−83.2311	−3.22031
$669$	−4.68466	−0.181119
$670$	18.7386	0.723937
$671$	−10.3542	−0.399718
$672$	−6.56155	−0.253117
$673$	20.7386	0.799416	0.399708	−	0.916643i	$-0.369112\pi$
$673$	20.7386	0.799416	0.399708	+	0.916643i	$0.369112\pi$
$674$	29.1231	1.12178
$675$	−2.56155	−0.0985942
$676$	−41.0540	−1.57900
$677$	−35.8617	−1.37828	−0.689139	−	0.724629i	$-0.742011\pi$
$677$	−35.8617	−1.37828	−0.689139	+	0.724629i	$0.742011\pi$
$678$	49.6155	1.90547
$679$	−2.87689	−0.110405
$680$	−20.4924	−0.785849
$681$	10.2462	0.392636
$682$	9.75379	0.373492
$683$	−51.2311	−1.96030	−0.980151	−	0.198253i	$-0.936473\pi$
$683$	−51.2311	−1.96030	−0.980151	+	0.198253i	$0.936473\pi$
$684$	−11.1231	−0.425303
$685$	−19.1231	−0.730656
$686$	−2.56155	−0.0978005
$687$	8.43845	0.321947
$688$	36.0000	1.37249
$689$	14.2462	0.542737
$690$	−12.4924	−0.475578
$691$	−12.6307	−0.480494	−0.240247	−	0.970712i	$-0.577228\pi$
$691$	−12.6307	−0.480494	−0.240247	+	0.970712i	$0.577228\pi$
$692$	−45.6155	−1.73404
$693$	−2.43845	−0.0926289
$694$	−20.0000	−0.759190
$695$	−14.9309	−0.566360
$696$	−7.36932	−0.279333
$697$	17.3693	0.657910
$698$	81.6155	3.08919
$699$	−9.56155	−0.361651
$700$	−11.6847	−0.441639
$701$	−20.1922	−0.762650	−0.381325	−	0.924441i	$-0.624532\pi$
$701$	−20.1922	−0.762650	−0.381325	+	0.924441i	$0.624532\pi$
$702$	−5.12311	−0.193359
$703$	−20.1080	−0.758386
$704$	−3.50758	−0.132197
$705$	−14.9309	−0.562329
$706$	17.6155	0.662969
$707$	13.1231	0.493545
$708$	−39.6155	−1.48884
$709$	18.8769	0.708937	0.354468	−	0.935068i	$-0.384662\pi$
$709$	18.8769	0.708937	0.354468	+	0.935068i	$0.384662\pi$
$710$	32.0000	1.20094
$711$	−5.12311	−0.192131
$712$	68.4924	2.56686
$713$	−4.87689	−0.182641
$714$	5.12311	0.191727
$715$	7.61553	0.284805
$716$	−90.3542	−3.37669
$717$	−11.3153	−0.422579
$718$	73.4773	2.74215
$719$	−19.1231	−0.713171	−0.356586	−	0.934263i	$-0.616059\pi$
$719$	−19.1231	−0.713171	−0.356586	+	0.934263i	$0.616059\pi$
$720$	−12.0000	−0.447214
$721$	−9.56155	−0.356091
$722$	33.4384	1.24445
$723$	0.246211	0.00915669
$724$	−41.6155	−1.54663
$725$	−2.87689	−0.106845
$726$	12.9460	0.480472
$727$	−1.56155	−0.0579148	−0.0289574	−	0.999581i	$-0.509219\pi$
$727$	−1.56155	−0.0579148	−0.0289574	+	0.999581i	$0.509219\pi$
$728$	−13.1231	−0.486375
$729$	1.00000	0.0370370
$730$	1.75379	0.0649106
$731$	−9.36932	−0.346537
$732$	19.3693	0.715911
$733$	0.738634	0.0272821	0.0136410	−	0.999907i	$-0.495658\pi$
$733$	0.738634	0.0272821	0.0136410	+	0.999907i	$0.495658\pi$
$734$	45.1231	1.66552
$735$	−1.56155	−0.0575987
$736$	20.4924	0.755361
$737$	−11.4233	−0.420782
$738$	22.2462	0.818894
$739$	−14.9848	−0.551226	−0.275613	−	0.961269i	$-0.588881\pi$
$739$	−14.9848	−0.551226	−0.275613	+	0.961269i	$0.588881\pi$
$740$	−58.7386	−2.15928
$741$	−4.87689	−0.179157
$742$	−18.2462	−0.669839
$743$	−5.56155	−0.204034	−0.102017	−	0.994783i	$-0.532530\pi$
$743$	−5.56155	−0.204034	−0.102017	+	0.994783i	$0.532530\pi$
$744$	−10.2462	−0.375644
$745$	−30.2462	−1.10814
$746$	3.36932	0.123359
$747$	4.00000	0.146352
$748$	22.2462	0.813402
$749$	−2.24621	−0.0820748
$750$	−30.2462	−1.10444
$751$	−31.3153	−1.14271	−0.571357	−	0.820702i	$-0.693583\pi$
$751$	−31.3153	−1.14271	−0.571357	+	0.820702i	$0.693583\pi$
$752$	73.4773	2.67944
$753$	−7.12311	−0.259580
$754$	−5.75379	−0.209541
$755$	0.384472	0.0139924
$756$	4.56155	0.165902
$757$	−13.5076	−0.490941	−0.245471	−	0.969404i	$-0.578942\pi$
$757$	−13.5076	−0.490941	−0.245471	+	0.969404i	$0.578942\pi$
$758$	57.6155	2.09269
$759$	7.61553	0.276426
$760$	−24.9848	−0.906296
$761$	−24.2462	−0.878924	−0.439462	−	0.898261i	$-0.644831\pi$
$761$	−24.2462	−0.878924	−0.439462	+	0.898261i	$0.644831\pi$
$762$	15.3693	0.556772
$763$	−2.00000	−0.0724049
$764$	−119.723	−4.33144
$765$	3.12311	0.112916
$766$	2.56155	0.0925527
$767$	−17.3693	−0.627170
$768$	−27.0540	−0.976226
$769$	−5.80776	−0.209433	−0.104717	−	0.994502i	$-0.533394\pi$
$769$	−5.80776	−0.209433	−0.104717	+	0.994502i	$0.533394\pi$
$770$	−9.75379	−0.351502
$771$	−18.4384	−0.664044
$772$	9.12311	0.328348
$773$	14.9848	0.538967	0.269484	−	0.963005i	$-0.413147\pi$
$773$	14.9848	0.538967	0.269484	+	0.963005i	$0.413147\pi$
$774$	−12.0000	−0.431331
$775$	−4.00000	−0.143684
$776$	18.8769	0.677641
$777$	8.24621	0.295831
$778$	−35.8617	−1.28571
$779$	21.1771	0.758748
$780$	−14.2462	−0.510096
$781$	−19.5076	−0.698036
$782$	−16.0000	−0.572159
$783$	1.12311	0.0401365
$784$	7.68466	0.274452
$785$	−21.8617	−0.780279
$786$	−9.75379	−0.347906
$787$	25.7538	0.918023	0.459012	−	0.888430i	$-0.348204\pi$
$787$	25.7538	0.918023	0.459012	+	0.888430i	$0.348204\pi$
$788$	−64.1080	−2.28375
$789$	6.63068	0.236059
$790$	−20.4924	−0.729088
$791$	−19.3693	−0.688694
$792$	16.0000	0.568535
$793$	8.49242	0.301575
$794$	5.61553	0.199288
$795$	−11.1231	−0.394496
$796$	45.6155	1.61680
$797$	−3.75379	−0.132966	−0.0664830	−	0.997788i	$-0.521178\pi$
$797$	−3.75379	−0.132966	−0.0664830	+	0.997788i	$0.521178\pi$
$798$	6.24621	0.221113
$799$	−19.1231	−0.676527
$800$	16.8078	0.594244
$801$	−10.4384	−0.368824
$802$	−33.6155	−1.18701
$803$	−1.06913	−0.0377288
$804$	21.3693	0.753638
$805$	4.87689	0.171888
$806$	−8.00000	−0.281788
$807$	12.6847	0.446521
$808$	−86.1080	−3.02927
$809$	22.1080	0.777274	0.388637	−	0.921391i	$-0.372946\pi$
$809$	22.1080	0.777274	0.388637	+	0.921391i	$0.372946\pi$
$810$	4.00000	0.140546
$811$	−46.9848	−1.64986	−0.824931	−	0.565234i	$-0.808786\pi$
$811$	−46.9848	−1.64986	−0.824931	+	0.565234i	$0.808786\pi$
$812$	5.12311	0.179786
$813$	28.2462	0.990638
$814$	51.5076	1.80534
$815$	32.9848	1.15541
$816$	−15.3693	−0.538034
$817$	−11.4233	−0.399650
$818$	−57.6155	−2.01448
$819$	2.00000	0.0698857
$820$	61.8617	2.16031
$821$	30.1080	1.05077	0.525387	−	0.850863i	$-0.323920\pi$
$821$	30.1080	1.05077	0.525387	+	0.850863i	$0.323920\pi$
$822$	−31.3693	−1.09413
$823$	−12.3002	−0.428758	−0.214379	−	0.976751i	$-0.568773\pi$
$823$	−12.3002	−0.428758	−0.214379	+	0.976751i	$0.568773\pi$
$824$	62.7386	2.18560
$825$	6.24621	0.217465
$826$	22.2462	0.774045
$827$	−13.7538	−0.478266	−0.239133	−	0.970987i	$-0.576863\pi$
$827$	−13.7538	−0.478266	−0.239133	+	0.970987i	$0.576863\pi$
$828$	−14.2462	−0.495090
$829$	16.9309	0.588033	0.294017	−	0.955800i	$-0.405008\pi$
$829$	16.9309	0.588033	0.294017	+	0.955800i	$0.405008\pi$
$830$	16.0000	0.555368
$831$	−28.9309	−1.00360
$832$	2.87689	0.0997384
$833$	−2.00000	−0.0692959
$834$	−24.4924	−0.848103
$835$	28.4924	0.986021
$836$	27.1231	0.938072
$837$	1.56155	0.0539752
$838$	14.7386	0.509138
$839$	−14.2462	−0.491834	−0.245917	−	0.969291i	$-0.579089\pi$
$839$	−14.2462	−0.491834	−0.245917	+	0.969291i	$0.579089\pi$
$840$	10.2462	0.353528
$841$	−27.7386	−0.956505
$842$	−0.630683	−0.0217348
$843$	−22.0540	−0.759579
$844$	31.3693	1.07978
$845$	14.0540	0.483471
$846$	−24.4924	−0.842067
$847$	−5.05398	−0.173657
$848$	54.7386	1.87973
$849$	21.1231	0.724943
$850$	−13.1231	−0.450119
$851$	−25.7538	−0.882829
$852$	36.4924	1.25021
$853$	0.822919	0.0281762	0.0140881	−	0.999901i	$-0.495515\pi$
$853$	0.822919	0.0281762	0.0140881	+	0.999901i	$0.495515\pi$
$854$	−10.8769	−0.372200
$855$	3.80776	0.130223
$856$	14.7386	0.503756
$857$	18.6307	0.636412	0.318206	−	0.948022i	$-0.396920\pi$
$857$	18.6307	0.636412	0.318206	+	0.948022i	$0.396920\pi$
$858$	12.4924	0.426484
$859$	−48.4924	−1.65454	−0.827270	−	0.561804i	$-0.810107\pi$
$859$	−48.4924	−1.65454	−0.827270	+	0.561804i	$0.810107\pi$
$860$	−33.3693	−1.13788
$861$	−8.68466	−0.295972
$862$	−69.4773	−2.36641
$863$	7.80776	0.265779	0.132890	−	0.991131i	$-0.457574\pi$
$863$	7.80776	0.265779	0.132890	+	0.991131i	$0.457574\pi$
$864$	−6.56155	−0.223229
$865$	15.6155	0.530944
$866$	46.1080	1.56681
$867$	−13.0000	−0.441503
$868$	7.12311	0.241774
$869$	12.4924	0.423776
$870$	4.49242	0.152307
$871$	9.36932	0.317467
$872$	13.1231	0.444404
$873$	−2.87689	−0.0973681
$874$	−19.5076	−0.659854
$875$	11.8078	0.399175
$876$	2.00000	0.0675737
$877$	−26.8769	−0.907568	−0.453784	−	0.891112i	$-0.649926\pi$
$877$	−26.8769	−0.907568	−0.453784	+	0.891112i	$0.649926\pi$
$878$	−1.75379	−0.0591875
$879$	−14.0000	−0.472208
$880$	29.2614	0.986400
$881$	−4.19224	−0.141240	−0.0706200	−	0.997503i	$-0.522498\pi$
$881$	−4.19224	−0.141240	−0.0706200	+	0.997503i	$0.522498\pi$
$882$	−2.56155	−0.0862520
$883$	20.7386	0.697911	0.348955	−	0.937139i	$-0.386537\pi$
$883$	20.7386	0.697911	0.348955	+	0.937139i	$0.386537\pi$
$884$	−18.2462	−0.613686
$885$	13.5616	0.455867
$886$	−25.7538	−0.865215
$887$	−28.4924	−0.956682	−0.478341	−	0.878174i	$-0.658762\pi$
$887$	−28.4924	−0.956682	−0.478341	+	0.878174i	$0.658762\pi$
$888$	−54.1080	−1.81574
$889$	−6.00000	−0.201234
$890$	−41.7538	−1.39959
$891$	−2.43845	−0.0816911
$892$	−21.3693	−0.715498
$893$	−23.3153	−0.780218
$894$	−49.6155	−1.65939
$895$	30.9309	1.03390
$896$	9.43845	0.315316
$897$	−6.24621	−0.208555
$898$	34.2462	1.14281
$899$	1.75379	0.0584921
$900$	−11.6847	−0.389489
$901$	−14.2462	−0.474610
$902$	−54.2462	−1.80620
$903$	4.68466	0.155896
$904$	127.093	4.22704
$905$	14.2462	0.473560
$906$	0.630683	0.0209530
$907$	−51.4773	−1.70927	−0.854637	−	0.519225i	$-0.826220\pi$
$907$	−51.4773	−1.70927	−0.854637	+	0.519225i	$0.826220\pi$
$908$	46.7386	1.55108
$909$	13.1231	0.435266
$910$	8.00000	0.265197
$911$	−1.06913	−0.0354219	−0.0177109	−	0.999843i	$-0.505638\pi$
$911$	−1.06913	−0.0354219	−0.0177109	+	0.999843i	$0.505638\pi$
$912$	−18.7386	−0.620498
$913$	−9.75379	−0.322803
$914$	41.6155	1.37652
$915$	−6.63068	−0.219204
$916$	38.4924	1.27183
$917$	3.80776	0.125743
$918$	5.12311	0.169088
$919$	−14.0540	−0.463598	−0.231799	−	0.972764i	$-0.574461\pi$
$919$	−14.0540	−0.463598	−0.231799	+	0.972764i	$0.574461\pi$
$920$	−32.0000	−1.05501
$921$	−16.2462	−0.535331
$922$	28.4924	0.938348
$923$	16.0000	0.526646
$924$	−11.1231	−0.365923
$925$	−21.1231	−0.694523
$926$	−76.8466	−2.52534
$927$	−9.56155	−0.314043
$928$	−7.36932	−0.241910
$929$	−40.8769	−1.34113	−0.670564	−	0.741852i	$-0.733948\pi$
$929$	−40.8769	−1.34113	−0.670564	+	0.741852i	$0.733948\pi$
$930$	6.24621	0.204821
$931$	−2.43845	−0.0799169
$932$	−43.6155	−1.42867
$933$	20.3002	0.664598
$934$	−29.7538	−0.973574
$935$	−7.61553	−0.249054
$936$	−13.1231	−0.428942
$937$	−22.4924	−0.734795	−0.367398	−	0.930064i	$-0.619751\pi$
$937$	−22.4924	−0.734795	−0.367398	+	0.930064i	$0.619751\pi$
$938$	−12.0000	−0.391814
$939$	−27.1771	−0.886891
$940$	−68.1080	−2.22144
$941$	−41.2311	−1.34409	−0.672047	−	0.740508i	$-0.734585\pi$
$941$	−41.2311	−1.34409	−0.672047	+	0.740508i	$0.734585\pi$
$942$	−35.8617	−1.16844
$943$	27.1231	0.883250
$944$	−66.7386	−2.17216
$945$	−1.56155	−0.0507973
$946$	29.2614	0.951369
$947$	−30.9309	−1.00512	−0.502559	−	0.864543i	$-0.667608\pi$
$947$	−30.9309	−1.00512	−0.502559	+	0.864543i	$0.667608\pi$
$948$	−23.3693	−0.759000
$949$	0.876894	0.0284652
$950$	−16.0000	−0.519109
$951$	31.3693	1.01722
$952$	13.1231	0.425322
$953$	−5.36932	−0.173929	−0.0869646	−	0.996211i	$-0.527717\pi$
$953$	−5.36932	−0.173929	−0.0869646	+	0.996211i	$0.527717\pi$
$954$	−18.2462	−0.590743
$955$	40.9848	1.32624
$956$	−51.6155	−1.66937
$957$	−2.73863	−0.0885275
$958$	−64.9848	−2.09957
$959$	12.2462	0.395451
$960$	−2.24621	−0.0724962
$961$	−28.5616	−0.921340
$962$	−42.2462	−1.36207
$963$	−2.24621	−0.0723831
$964$	1.12311	0.0361728
$965$	−3.12311	−0.100536
$966$	8.00000	0.257396
$967$	42.7386	1.37438	0.687191	−	0.726477i	$-0.258844\pi$
$967$	42.7386	1.37438	0.687191	+	0.726477i	$0.258844\pi$
$968$	33.1619	1.06586
$969$	4.87689	0.156668
$970$	−11.5076	−0.369486
$971$	6.73863	0.216253	0.108127	−	0.994137i	$-0.465515\pi$
$971$	6.73863	0.216253	0.108127	+	0.994137i	$0.465515\pi$
$972$	4.56155	0.146312
$973$	9.56155	0.306529
$974$	−50.6004	−1.62134
$975$	−5.12311	−0.164071
$976$	32.6307	1.04448
$977$	−43.3153	−1.38578	−0.692890	−	0.721043i	$-0.743663\pi$
$977$	−43.3153	−1.38578	−0.692890	+	0.721043i	$0.743663\pi$
$978$	54.1080	1.73018
$979$	25.4536	0.813501
$980$	−7.12311	−0.227539
$981$	−2.00000	−0.0638551
$982$	−59.2311	−1.89014
$983$	19.5076	0.622195	0.311098	−	0.950378i	$-0.399303\pi$
$983$	19.5076	0.622195	0.311098	+	0.950378i	$0.399303\pi$
$984$	56.9848	1.81661
$985$	21.9460	0.699258
$986$	5.75379	0.183238
$987$	9.56155	0.304348
$988$	−22.2462	−0.707746
$989$	−14.6307	−0.465229
$990$	−9.75379	−0.309996
$991$	−40.9848	−1.30193	−0.650963	−	0.759109i	$-0.725635\pi$
$991$	−40.9848	−1.30193	−0.650963	+	0.759109i	$0.725635\pi$
$992$	−10.2462	−0.325318
$993$	−6.24621	−0.198218
$994$	−20.4924	−0.649980
$995$	−15.6155	−0.495046
$996$	18.2462	0.578153
$997$	6.38447	0.202198	0.101099	−	0.994876i	$-0.467764\pi$
$997$	6.38447	0.202198	0.101099	+	0.994876i	$0.467764\pi$
$998$	−68.4924	−2.16809
$999$	8.24621	0.260899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.l.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.l.1.1	✓	2	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)

\(f(q)\)	\(=\)	\(q-2.56155 q^{2} +1.00000 q^{3} +4.56155 q^{4} -1.56155 q^{5} -2.56155 q^{6} +1.00000 q^{7} -6.56155 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.56155 q^{2} +1.00000 q^{3} +4.56155 q^{4} -1.56155 q^{5} -2.56155 q^{6} +1.00000 q^{7} -6.56155 q^{8} +1.00000 q^{9} +4.00000 q^{10} -2.43845 q^{11} +4.56155 q^{12} +2.00000 q^{13} -2.56155 q^{14} -1.56155 q^{15} +7.68466 q^{16} -2.00000 q^{17} -2.56155 q^{18} -2.43845 q^{19} -7.12311 q^{20} +1.00000 q^{21} +6.24621 q^{22} -3.12311 q^{23} -6.56155 q^{24} -2.56155 q^{25} -5.12311 q^{26} +1.00000 q^{27} +4.56155 q^{28} +1.12311 q^{29} +4.00000 q^{30} +1.56155 q^{31} -6.56155 q^{32} -2.43845 q^{33} +5.12311 q^{34} -1.56155 q^{35} +4.56155 q^{36} +8.24621 q^{37} +6.24621 q^{38} +2.00000 q^{39} +10.2462 q^{40} -8.68466 q^{41} -2.56155 q^{42} +4.68466 q^{43} -11.1231 q^{44} -1.56155 q^{45} +8.00000 q^{46} +9.56155 q^{47} +7.68466 q^{48} +1.00000 q^{49} +6.56155 q^{50} -2.00000 q^{51} +9.12311 q^{52} +7.12311 q^{53} -2.56155 q^{54} +3.80776 q^{55} -6.56155 q^{56} -2.43845 q^{57} -2.87689 q^{58} -8.68466 q^{59} -7.12311 q^{60} +4.24621 q^{61} -4.00000 q^{62} +1.00000 q^{63} +1.43845 q^{64} -3.12311 q^{65} +6.24621 q^{66} +4.68466 q^{67} -9.12311 q^{68} -3.12311 q^{69} +4.00000 q^{70} +8.00000 q^{71} -6.56155 q^{72} +0.438447 q^{73} -21.1231 q^{74} -2.56155 q^{75} -11.1231 q^{76} -2.43845 q^{77} -5.12311 q^{78} -5.12311 q^{79} -12.0000 q^{80} +1.00000 q^{81} +22.2462 q^{82} +4.00000 q^{83} +4.56155 q^{84} +3.12311 q^{85} -12.0000 q^{86} +1.12311 q^{87} +16.0000 q^{88} -10.4384 q^{89} +4.00000 q^{90} +2.00000 q^{91} -14.2462 q^{92} +1.56155 q^{93} -24.4924 q^{94} +3.80776 q^{95} -6.56155 q^{96} -2.87689 q^{97} -2.56155 q^{98} -2.43845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 2 q^{3} + 5 q^{4} + q^{5} - q^{6} + 2 q^{7} - 9 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - q^2 + 2 * q^3 + 5 * q^4 + q^5 - q^6 + 2 * q^7 - 9 * q^8 + 2 * q^9	\( 2 q - q^{2} + 2 q^{3} + 5 q^{4} + q^{5} - q^{6} + 2 q^{7} - 9 q^{8} + 2 q^{9} + 8 q^{10} - 9 q^{11} + 5 q^{12} + 4 q^{13} - q^{14} + q^{15} + 3 q^{16} - 4 q^{17} - q^{18} - 9 q^{19} - 6 q^{20} + 2 q^{21} - 4 q^{22} + 2 q^{23} - 9 q^{24} - q^{25} - 2 q^{26} + 2 q^{27} + 5 q^{28} - 6 q^{29} + 8 q^{30} - q^{31} - 9 q^{32} - 9 q^{33} + 2 q^{34} + q^{35} + 5 q^{36} - 4 q^{38} + 4 q^{39} + 4 q^{40} - 5 q^{41} - q^{42} - 3 q^{43} - 14 q^{44} + q^{45} + 16 q^{46} + 15 q^{47} + 3 q^{48} + 2 q^{49} + 9 q^{50} - 4 q^{51} + 10 q^{52} + 6 q^{53} - q^{54} - 13 q^{55} - 9 q^{56} - 9 q^{57} - 14 q^{58} - 5 q^{59} - 6 q^{60} - 8 q^{61} - 8 q^{62} + 2 q^{63} + 7 q^{64} + 2 q^{65} - 4 q^{66} - 3 q^{67} - 10 q^{68} + 2 q^{69} + 8 q^{70} + 16 q^{71} - 9 q^{72} + 5 q^{73} - 34 q^{74} - q^{75} - 14 q^{76} - 9 q^{77} - 2 q^{78} - 2 q^{79} - 24 q^{80} + 2 q^{81} + 28 q^{82} + 8 q^{83} + 5 q^{84} - 2 q^{85} - 24 q^{86} - 6 q^{87} + 32 q^{88} - 25 q^{89} + 8 q^{90} + 4 q^{91} - 12 q^{92} - q^{93} - 16 q^{94} - 13 q^{95} - 9 q^{96} - 14 q^{97} - q^{98} - 9 q^{99}+O(q^{100}) \) 2 * q - q^2 + 2 * q^3 + 5 * q^4 + q^5 - q^6 + 2 * q^7 - 9 * q^8 + 2 * q^9 + 8 * q^10 - 9 * q^11 + 5 * q^12 + 4 * q^13 - q^14 + q^15 + 3 * q^16 - 4 * q^17 - q^18 - 9 * q^19 - 6 * q^20 + 2 * q^21 - 4 * q^22 + 2 * q^23 - 9 * q^24 - q^25 - 2 * q^26 + 2 * q^27 + 5 * q^28 - 6 * q^29 + 8 * q^30 - q^31 - 9 * q^32 - 9 * q^33 + 2 * q^34 + q^35 + 5 * q^36 - 4 * q^38 + 4 * q^39 + 4 * q^40 - 5 * q^41 - q^42 - 3 * q^43 - 14 * q^44 + q^45 + 16 * q^46 + 15 * q^47 + 3 * q^48 + 2 * q^49 + 9 * q^50 - 4 * q^51 + 10 * q^52 + 6 * q^53 - q^54 - 13 * q^55 - 9 * q^56 - 9 * q^57 - 14 * q^58 - 5 * q^59 - 6 * q^60 - 8 * q^61 - 8 * q^62 + 2 * q^63 + 7 * q^64 + 2 * q^65 - 4 * q^66 - 3 * q^67 - 10 * q^68 + 2 * q^69 + 8 * q^70 + 16 * q^71 - 9 * q^72 + 5 * q^73 - 34 * q^74 - q^75 - 14 * q^76 - 9 * q^77 - 2 * q^78 - 2 * q^79 - 24 * q^80 + 2 * q^81 + 28 * q^82 + 8 * q^83 + 5 * q^84 - 2 * q^85 - 24 * q^86 - 6 * q^87 + 32 * q^88 - 25 * q^89 + 8 * q^90 + 4 * q^91 - 12 * q^92 - q^93 - 16 * q^94 - 13 * q^95 - 9 * q^96 - 14 * q^97 - q^98 - 9 * q^99

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