LMFDB - Embedded newform 4002.2.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, 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("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.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:	$31.9561308889$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.16448.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 14 $ x^4 - 2x^3 - 7x^2 + 8*x + 14
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.18398$ of defining polynomial
Character	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.230234 q^{5} +1.00000 q^{6} +4.59819 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.230234 q^{5} +1.00000 q^{6} +4.59819 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.230234 q^{10} -4.50283 q^{11} -1.00000 q^{12} +1.15403 q^{13} -4.59819 q^{14} -0.230234 q^{15} +1.00000 q^{16} -7.42662 q^{17} -1.00000 q^{18} -5.94699 q^{19} +0.230234 q^{20} -4.59819 q^{21} +4.50283 q^{22} +1.00000 q^{23} +1.00000 q^{24} -4.94699 q^{25} -1.15403 q^{26} -1.00000 q^{27} +4.59819 q^{28} -1.00000 q^{29} +0.230234 q^{30} +6.50283 q^{31} -1.00000 q^{32} +4.50283 q^{33} +7.42662 q^{34} +1.05866 q^{35} +1.00000 q^{36} +7.29175 q^{37} +5.94699 q^{38} -1.15403 q^{39} -0.230234 q^{40} +0.0953661 q^{41} +4.59819 q^{42} +0.941339 q^{43} -4.50283 q^{44} +0.230234 q^{45} -1.00000 q^{46} -0.598193 q^{47} -1.00000 q^{48} +14.1434 q^{49} +4.94699 q^{50} +7.42662 q^{51} +1.15403 q^{52} +13.3736 q^{53} +1.00000 q^{54} -1.03670 q^{55} -4.59819 q^{56} +5.94699 q^{57} +1.00000 q^{58} -2.94134 q^{59} -0.230234 q^{60} +10.8883 q^{61} -6.50283 q^{62} +4.59819 q^{63} +1.00000 q^{64} +0.265697 q^{65} -4.50283 q^{66} +2.09251 q^{67} -7.42662 q^{68} -1.00000 q^{69} -1.05866 q^{70} +10.9633 q^{71} -1.00000 q^{72} -12.3928 q^{73} -7.29175 q^{74} +4.94699 q^{75} -5.94699 q^{76} -20.7049 q^{77} +1.15403 q^{78} +16.5276 q^{79} +0.230234 q^{80} +1.00000 q^{81} -0.0953661 q^{82} -1.71676 q^{83} -4.59819 q^{84} -1.70986 q^{85} -0.941339 q^{86} +1.00000 q^{87} +4.50283 q^{88} +12.5700 q^{89} -0.230234 q^{90} +5.30644 q^{91} +1.00000 q^{92} -6.50283 q^{93} +0.598193 q^{94} -1.36920 q^{95} +1.00000 q^{96} -1.82277 q^{97} -14.1434 q^{98} -4.50283 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 + 6 * q^7 - 4 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9} - 2 q^{10} - 4 q^{12} - 6 q^{14} - 2 q^{15} + 4 q^{16} - 6 q^{17} - 4 q^{18} + 2 q^{19} + 2 q^{20} - 6 q^{21} + 4 q^{23} + 4 q^{24} + 6 q^{25} - 4 q^{27} + 6 q^{28} - 4 q^{29} + 2 q^{30} + 8 q^{31} - 4 q^{32} + 6 q^{34} - 6 q^{35} + 4 q^{36} + 10 q^{37} - 2 q^{38} - 2 q^{40} + 6 q^{41} + 6 q^{42} + 14 q^{43} + 2 q^{45} - 4 q^{46} + 10 q^{47} - 4 q^{48} + 6 q^{49} - 6 q^{50} + 6 q^{51} + 4 q^{53} + 4 q^{54} - 20 q^{55} - 6 q^{56} - 2 q^{57} + 4 q^{58} - 22 q^{59} - 2 q^{60} + 28 q^{61} - 8 q^{62} + 6 q^{63} + 4 q^{64} + 12 q^{65} + 24 q^{67} - 6 q^{68} - 4 q^{69} + 6 q^{70} + 28 q^{71} - 4 q^{72} - 10 q^{74} - 6 q^{75} + 2 q^{76} - 4 q^{77} + 12 q^{79} + 2 q^{80} + 4 q^{81} - 6 q^{82} + 20 q^{83} - 6 q^{84} - 10 q^{85} - 14 q^{86} + 4 q^{87} - 24 q^{89} - 2 q^{90} + 28 q^{91} + 4 q^{92} - 8 q^{93} - 10 q^{94} + 2 q^{95} + 4 q^{96} - 32 q^{97} - 6 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 + 6 * q^7 - 4 * q^8 + 4 * q^9 - 2 * q^10 - 4 * q^12 - 6 * q^14 - 2 * q^15 + 4 * q^16 - 6 * q^17 - 4 * q^18 + 2 * q^19 + 2 * q^20 - 6 * q^21 + 4 * q^23 + 4 * q^24 + 6 * q^25 - 4 * q^27 + 6 * q^28 - 4 * q^29 + 2 * q^30 + 8 * q^31 - 4 * q^32 + 6 * q^34 - 6 * q^35 + 4 * q^36 + 10 * q^37 - 2 * q^38 - 2 * q^40 + 6 * q^41 + 6 * q^42 + 14 * q^43 + 2 * q^45 - 4 * q^46 + 10 * q^47 - 4 * q^48 + 6 * q^49 - 6 * q^50 + 6 * q^51 + 4 * q^53 + 4 * q^54 - 20 * q^55 - 6 * q^56 - 2 * q^57 + 4 * q^58 - 22 * q^59 - 2 * q^60 + 28 * q^61 - 8 * q^62 + 6 * q^63 + 4 * q^64 + 12 * q^65 + 24 * q^67 - 6 * q^68 - 4 * q^69 + 6 * q^70 + 28 * q^71 - 4 * q^72 - 10 * q^74 - 6 * q^75 + 2 * q^76 - 4 * q^77 + 12 * q^79 + 2 * q^80 + 4 * q^81 - 6 * q^82 + 20 * q^83 - 6 * q^84 - 10 * q^85 - 14 * q^86 + 4 * q^87 - 24 * q^89 - 2 * q^90 + 28 * q^91 + 4 * q^92 - 8 * q^93 - 10 * q^94 + 2 * q^95 + 4 * q^96 - 32 * q^97 - 6 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0.230234	0.102964	0.0514819	−	0.998674i	$-0.483606\pi$
$5$	0.230234	0.102964	0.0514819	+	0.998674i	$0.483606\pi$
$6$	1.00000	0.408248
$7$	4.59819	1.73795	0.868977	−	0.494853i	$-0.164778\pi$
$7$	4.59819	1.73795	0.868977	+	0.494853i	$0.164778\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−0.230234	−0.0728065
$11$	−4.50283	−1.35765	−0.678827	−	0.734299i	$-0.737511\pi$
$11$	−4.50283	−1.35765	−0.678827	+	0.734299i	$0.737511\pi$
$12$	−1.00000	−0.288675
$13$	1.15403	0.320070	0.160035	−	0.987111i	$-0.448839\pi$
$13$	1.15403	0.320070	0.160035	+	0.987111i	$0.448839\pi$
$14$	−4.59819	−1.22892
$15$	−0.230234	−0.0594462
$16$	1.00000	0.250000
$17$	−7.42662	−1.80122	−0.900610	−	0.434628i	$-0.856880\pi$
$17$	−7.42662	−1.80122	−0.900610	+	0.434628i	$0.856880\pi$
$18$	−1.00000	−0.235702
$19$	−5.94699	−1.36433	−0.682167	−	0.731197i	$-0.738962\pi$
$19$	−5.94699	−1.36433	−0.682167	+	0.731197i	$0.738962\pi$
$20$	0.230234	0.0514819
$21$	−4.59819	−1.00341
$22$	4.50283	0.960006
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	−4.94699	−0.989398
$26$	−1.15403	−0.226323
$27$	−1.00000	−0.192450
$28$	4.59819	0.868977
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0.230234	0.0420348
$31$	6.50283	1.16794	0.583971	−	0.811774i	$-0.301498\pi$
$31$	6.50283	1.16794	0.583971	+	0.811774i	$0.301498\pi$
$32$	−1.00000	−0.176777
$33$	4.50283	0.783842
$34$	7.42662	1.27365
$35$	1.05866	0.178946
$36$	1.00000	0.166667
$37$	7.29175	1.19876	0.599378	−	0.800466i	$-0.295415\pi$
$37$	7.29175	1.19876	0.599378	+	0.800466i	$0.295415\pi$
$38$	5.94699	0.964730
$39$	−1.15403	−0.184792
$40$	−0.230234	−0.0364032
$41$	0.0953661	0.0148937	0.00744684	−	0.999972i	$-0.497630\pi$
$41$	0.0953661	0.0148937	0.00744684	+	0.999972i	$0.497630\pi$
$42$	4.59819	0.709517
$43$	0.941339	0.143553	0.0717764	−	0.997421i	$-0.477133\pi$
$43$	0.941339	0.143553	0.0717764	+	0.997421i	$0.477133\pi$
$44$	−4.50283	−0.678827
$45$	0.230234	0.0343213
$46$	−1.00000	−0.147442
$47$	−0.598193	−0.0872554	−0.0436277	−	0.999048i	$-0.513892\pi$
$47$	−0.598193	−0.0872554	−0.0436277	+	0.999048i	$0.513892\pi$
$48$	−1.00000	−0.144338
$49$	14.1434	2.02048
$50$	4.94699	0.699610
$51$	7.42662	1.03993
$52$	1.15403	0.160035
$53$	13.3736	1.83701	0.918503	−	0.395413i	$-0.129399\pi$
$53$	13.3736	1.83701	0.918503	+	0.395413i	$0.129399\pi$
$54$	1.00000	0.136083
$55$	−1.03670	−0.139789
$56$	−4.59819	−0.614459
$57$	5.94699	0.787698
$58$	1.00000	0.131306
$59$	−2.94134	−0.382930	−0.191465	−	0.981499i	$-0.561324\pi$
$59$	−2.94134	−0.382930	−0.191465	+	0.981499i	$0.561324\pi$
$60$	−0.230234	−0.0297231
$61$	10.8883	1.39411	0.697054	−	0.717019i	$-0.254494\pi$
$61$	10.8883	1.39411	0.697054	+	0.717019i	$0.254494\pi$
$62$	−6.50283	−0.825860
$63$	4.59819	0.579318
$64$	1.00000	0.125000
$65$	0.265697	0.0329556
$66$	−4.50283	−0.554260
$67$	2.09251	0.255641	0.127820	−	0.991797i	$-0.459202\pi$
$67$	2.09251	0.255641	0.127820	+	0.991797i	$0.459202\pi$
$68$	−7.42662	−0.900610
$69$	−1.00000	−0.120386
$70$	−1.05866	−0.126534
$71$	10.9633	1.30110	0.650552	−	0.759462i	$-0.274538\pi$
$71$	10.9633	1.30110	0.650552	+	0.759462i	$0.274538\pi$
$72$	−1.00000	−0.117851
$73$	−12.3928	−1.45046	−0.725232	−	0.688504i	$-0.758268\pi$
$73$	−12.3928	−1.45046	−0.725232	+	0.688504i	$0.758268\pi$
$74$	−7.29175	−0.847649
$75$	4.94699	0.571229
$76$	−5.94699	−0.682167
$77$	−20.7049	−2.35954
$78$	1.15403	0.130668
$79$	16.5276	1.85950	0.929752	−	0.368185i	$-0.120021\pi$
$79$	16.5276	1.85950	0.929752	+	0.368185i	$0.120021\pi$
$80$	0.230234	0.0257410
$81$	1.00000	0.111111
$82$	−0.0953661	−0.0105314
$83$	−1.71676	−0.188439	−0.0942193	−	0.995551i	$-0.530035\pi$
$83$	−1.71676	−0.188439	−0.0942193	+	0.995551i	$0.530035\pi$
$84$	−4.59819	−0.501704
$85$	−1.70986	−0.185461
$86$	−0.941339	−0.101507
$87$	1.00000	0.107211
$88$	4.50283	0.480003
$89$	12.5700	1.33242	0.666209	−	0.745765i	$-0.267916\pi$
$89$	12.5700	1.33242	0.666209	+	0.745765i	$0.267916\pi$
$90$	−0.230234	−0.0242688
$91$	5.30644	0.556266
$92$	1.00000	0.104257
$93$	−6.50283	−0.674312
$94$	0.598193	0.0616989
$95$	−1.36920	−0.140477
$96$	1.00000	0.102062
$97$	−1.82277	−0.185075	−0.0925373	−	0.995709i	$-0.529498\pi$
$97$	−1.82277	−0.185075	−0.0925373	+	0.995709i	$0.529498\pi$
$98$	−14.1434	−1.42870
$99$	−4.50283	−0.452551
$100$	−4.94699	−0.494699
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	−7.42662	−0.735345
$103$	−11.9119	−1.17371	−0.586857	−	0.809690i	$-0.699635\pi$
$103$	−11.9119	−1.17371	−0.586857	+	0.809690i	$0.699635\pi$
$104$	−1.15403	−0.113162
$105$	−1.05866	−0.103315
$106$	−13.3736	−1.29896
$107$	12.0819	1.16800	0.583999	−	0.811755i	$-0.301487\pi$
$107$	12.0819	1.16800	0.583999	+	0.811755i	$0.301487\pi$
$108$	−1.00000	−0.0962250
$109$	7.71676	0.739131	0.369566	−	0.929205i	$-0.379506\pi$
$109$	7.71676	0.739131	0.369566	+	0.929205i	$0.379506\pi$
$110$	1.03670	0.0988459
$111$	−7.29175	−0.692102
$112$	4.59819	0.434488
$113$	8.65810	0.814485	0.407243	−	0.913320i	$-0.366490\pi$
$113$	8.65810	0.814485	0.407243	+	0.913320i	$0.366490\pi$
$114$	−5.94699	−0.556987
$115$	0.230234	0.0214695
$116$	−1.00000	−0.0928477
$117$	1.15403	0.106690
$118$	2.94134	0.270772
$119$	−34.1490	−3.13044
$120$	0.230234	0.0210174
$121$	9.27545	0.843223
$122$	−10.8883	−0.985783
$123$	−0.0953661	−0.00859887
$124$	6.50283	0.583971
$125$	−2.29014	−0.204836
$126$	−4.59819	−0.409640
$127$	5.11167	0.453587	0.226794	−	0.973943i	$-0.427176\pi$
$127$	5.11167	0.453587	0.226794	+	0.973943i	$0.427176\pi$
$128$	−1.00000	−0.0883883
$129$	−0.941339	−0.0828802
$130$	−0.265697	−0.0233031
$131$	10.3968	0.908374	0.454187	−	0.890906i	$-0.349930\pi$
$131$	10.3968	0.908374	0.454187	+	0.890906i	$0.349930\pi$
$132$	4.50283	0.391921
$133$	−27.3454	−2.37115
$134$	−2.09251	−0.180765
$135$	−0.230234	−0.0198154
$136$	7.42662	0.636827
$137$	12.7958	1.09322	0.546610	−	0.837387i	$-0.315918\pi$
$137$	12.7958	1.09322	0.546610	+	0.837387i	$0.315918\pi$
$138$	1.00000	0.0851257
$139$	−2.80361	−0.237799	−0.118900	−	0.992906i	$-0.537937\pi$
$139$	−2.80361	−0.237799	−0.118900	+	0.992906i	$0.537937\pi$
$140$	1.05866	0.0894732
$141$	0.598193	0.0503769
$142$	−10.9633	−0.920019
$143$	−5.19639	−0.434544
$144$	1.00000	0.0833333
$145$	−0.230234	−0.0191199
$146$	12.3928	1.02563
$147$	−14.1434	−1.16653
$148$	7.29175	0.599378
$149$	−6.92783	−0.567550	−0.283775	−	0.958891i	$-0.591587\pi$
$149$	−6.92783	−0.567550	−0.283775	+	0.958891i	$0.591587\pi$
$150$	−4.94699	−0.403920
$151$	17.5289	1.42648	0.713240	−	0.700920i	$-0.247227\pi$
$151$	17.5289	1.42648	0.713240	+	0.700920i	$0.247227\pi$
$152$	5.94699	0.482365
$153$	−7.42662	−0.600407
$154$	20.7049	1.66845
$155$	1.49717	0.120256
$156$	−1.15403	−0.0923961
$157$	6.09537	0.486463	0.243232	−	0.969968i	$-0.421792\pi$
$157$	6.09537	0.486463	0.243232	+	0.969968i	$0.421792\pi$
$158$	−16.5276	−1.31487
$159$	−13.3736	−1.06060
$160$	−0.230234	−0.0182016
$161$	4.59819	0.362388
$162$	−1.00000	−0.0785674
$163$	−1.25343	−0.0981765	−0.0490882	−	0.998794i	$-0.515632\pi$
$163$	−1.25343	−0.0981765	−0.0490882	+	0.998794i	$0.515632\pi$
$164$	0.0953661	0.00744684
$165$	1.03670	0.0807074
$166$	1.71676	0.133246
$167$	−2.49879	−0.193362	−0.0966810	−	0.995315i	$-0.530823\pi$
$167$	−2.49879	−0.193362	−0.0966810	+	0.995315i	$0.530823\pi$
$168$	4.59819	0.354758
$169$	−11.6682	−0.897555
$170$	1.70986	0.131140
$171$	−5.94699	−0.454778
$172$	0.941339	0.0717764
$173$	6.21269	0.472342	0.236171	−	0.971712i	$-0.424107\pi$
$173$	6.21269	0.472342	0.236171	+	0.971712i	$0.424107\pi$
$174$	−1.00000	−0.0758098
$175$	−22.7472	−1.71953
$176$	−4.50283	−0.339413
$177$	2.94134	0.221085
$178$	−12.5700	−0.942161
$179$	9.55084	0.713863	0.356932	−	0.934131i	$-0.383823\pi$
$179$	9.55084	0.713863	0.356932	+	0.934131i	$0.383823\pi$
$180$	0.230234	0.0171606
$181$	−14.2196	−1.05693	−0.528467	−	0.848954i	$-0.677233\pi$
$181$	−14.2196	−1.05693	−0.528467	+	0.848954i	$0.677233\pi$
$182$	−5.30644	−0.393340
$183$	−10.8883	−0.804889
$184$	−1.00000	−0.0737210
$185$	1.67881	0.123429
$186$	6.50283	0.476810
$187$	33.4408	2.44543
$188$	−0.598193	−0.0436277
$189$	−4.59819	−0.334469
$190$	1.36920	0.0993323
$191$	2.00286	0.144922	0.0724608	−	0.997371i	$-0.476915\pi$
$191$	2.00286	0.144922	0.0724608	+	0.997371i	$0.476915\pi$
$192$	−1.00000	−0.0721688
$193$	21.6242	1.55655	0.778274	−	0.627925i	$-0.216096\pi$
$193$	21.6242	1.55655	0.778274	+	0.627925i	$0.216096\pi$
$194$	1.82277	0.130868
$195$	−0.265697	−0.0190269
$196$	14.1434	1.01024
$197$	2.89898	0.206544	0.103272	−	0.994653i	$-0.467069\pi$
$197$	2.89898	0.206544	0.103272	+	0.994653i	$0.467069\pi$
$198$	4.50283	0.320002
$199$	12.2754	0.870184	0.435092	−	0.900386i	$-0.356716\pi$
$199$	12.2754	0.870184	0.435092	+	0.900386i	$0.356716\pi$
$200$	4.94699	0.349805
$201$	−2.09251	−0.147594
$202$	10.0000	0.703598
$203$	−4.59819	−0.322730
$204$	7.42662	0.519967
$205$	0.0219565	0.00153351
$206$	11.9119	0.829941
$207$	1.00000	0.0695048
$208$	1.15403	0.0800174
$209$	26.7783	1.85229
$210$	1.05866	0.0730546
$211$	−15.8369	−1.09026	−0.545130	−	0.838352i	$-0.683520\pi$
$211$	−15.8369	−1.09026	−0.545130	+	0.838352i	$0.683520\pi$
$212$	13.3736	0.918503
$213$	−10.9633	−0.751193
$214$	−12.0819	−0.825899
$215$	0.216728	0.0147808
$216$	1.00000	0.0680414
$217$	29.9013	2.02983
$218$	−7.71676	−0.522645
$219$	12.3928	0.837426
$220$	−1.03670	−0.0698946
$221$	−8.57052	−0.576516
$222$	7.29175	0.489390
$223$	−13.6569	−0.914531	−0.457265	−	0.889330i	$-0.651171\pi$
$223$	−13.6569	−0.914531	−0.457265	+	0.889330i	$0.651171\pi$
$224$	−4.59819	−0.307230
$225$	−4.94699	−0.329799
$226$	−8.65810	−0.575928
$227$	12.2726	0.814561	0.407280	−	0.913303i	$-0.366477\pi$
$227$	12.2726	0.814561	0.407280	+	0.913303i	$0.366477\pi$
$228$	5.94699	0.393849
$229$	−7.37647	−0.487451	−0.243725	−	0.969844i	$-0.578370\pi$
$229$	−7.37647	−0.487451	−0.243725	+	0.969844i	$0.578370\pi$
$230$	−0.230234	−0.0151812
$231$	20.7049	1.36228
$232$	1.00000	0.0656532
$233$	28.2020	1.84758	0.923789	−	0.382903i	$-0.125076\pi$
$233$	28.2020	1.84758	0.923789	+	0.382903i	$0.125076\pi$
$234$	−1.15403	−0.0754411
$235$	−0.137724	−0.00898415
$236$	−2.94134	−0.191465
$237$	−16.5276	−1.07359
$238$	34.1490	2.21355
$239$	1.14676	0.0741777	0.0370889	−	0.999312i	$-0.488192\pi$
$239$	1.14676	0.0741777	0.0370889	+	0.999312i	$0.488192\pi$
$240$	−0.230234	−0.0148616
$241$	12.6406	0.814249	0.407125	−	0.913373i	$-0.366531\pi$
$241$	12.6406	0.814249	0.407125	+	0.913373i	$0.366531\pi$
$242$	−9.27545	−0.596248
$243$	−1.00000	−0.0641500
$244$	10.8883	0.697054
$245$	3.25629	0.208037
$246$	0.0953661	0.00608032
$247$	−6.86299	−0.436682
$248$	−6.50283	−0.412930
$249$	1.71676	0.108795
$250$	2.29014	0.144841
$251$	−5.62425	−0.354999	−0.177500	−	0.984121i	$-0.556801\pi$
$251$	−5.62425	−0.354999	−0.177500	+	0.984121i	$0.556801\pi$
$252$	4.59819	0.289659
$253$	−4.50283	−0.283090
$254$	−5.11167	−0.320735
$255$	1.70986	0.107076
$256$	1.00000	0.0625000
$257$	−21.3177	−1.32976	−0.664882	−	0.746948i	$-0.731518\pi$
$257$	−21.3177	−1.32976	−0.664882	+	0.746948i	$0.731518\pi$
$258$	0.941339	0.0586052
$259$	33.5289	2.08338
$260$	0.265697	0.0164778
$261$	−1.00000	−0.0618984
$262$	−10.3968	−0.642317
$263$	−1.96205	−0.120985	−0.0604927	−	0.998169i	$-0.519267\pi$
$263$	−1.96205	−0.120985	−0.0604927	+	0.998169i	$0.519267\pi$
$264$	−4.50283	−0.277130
$265$	3.07906	0.189145
$266$	27.3454	1.67666
$267$	−12.5700	−0.769271
$268$	2.09251	0.127820
$269$	−3.85163	−0.234838	−0.117419	−	0.993082i	$-0.537462\pi$
$269$	−3.85163	−0.234838	−0.117419	+	0.993082i	$0.537462\pi$
$270$	0.230234	0.0140116
$271$	−1.47183	−0.0894076	−0.0447038	−	0.999000i	$-0.514234\pi$
$271$	−1.47183	−0.0894076	−0.0447038	+	0.999000i	$0.514234\pi$
$272$	−7.42662	−0.450305
$273$	−5.30644	−0.321160
$274$	−12.7958	−0.773024
$275$	22.2754	1.34326
$276$	−1.00000	−0.0601929
$277$	−24.6665	−1.48207	−0.741035	−	0.671467i	$-0.765665\pi$
$277$	−24.6665	−1.48207	−0.741035	+	0.671467i	$0.765665\pi$
$278$	2.80361	0.168150
$279$	6.50283	0.389314
$280$	−1.05866	−0.0632671
$281$	−12.3081	−0.734237	−0.367119	−	0.930174i	$-0.619656\pi$
$281$	−12.3081	−0.734237	−0.367119	+	0.930174i	$0.619656\pi$
$282$	−0.598193	−0.0356219
$283$	−23.8087	−1.41528	−0.707641	−	0.706572i	$-0.750241\pi$
$283$	−23.8087	−1.41528	−0.707641	+	0.706572i	$0.750241\pi$
$284$	10.9633	0.650552
$285$	1.36920	0.0811045
$286$	5.19639	0.307269
$287$	0.438512	0.0258845
$288$	−1.00000	−0.0589256
$289$	38.1547	2.24439
$290$	0.230234	0.0135198
$291$	1.82277	0.106853
$292$	−12.3928	−0.725232
$293$	17.3162	1.01162	0.505811	−	0.862644i	$-0.331193\pi$
$293$	17.3162	1.01162	0.505811	+	0.862644i	$0.331193\pi$
$294$	14.1434	0.824859
$295$	−0.677197	−0.0394279
$296$	−7.29175	−0.423824
$297$	4.50283	0.261281
$298$	6.92783	0.401319
$299$	1.15403	0.0667391
$300$	4.94699	0.285615
$301$	4.32846	0.249488
$302$	−17.5289	−1.00867
$303$	10.0000	0.574485
$304$	−5.94699	−0.341083
$305$	2.50687	0.143543
$306$	7.42662	0.424552
$307$	0.920937	0.0525606	0.0262803	−	0.999655i	$-0.491634\pi$
$307$	0.920937	0.0525606	0.0262803	+	0.999655i	$0.491634\pi$
$308$	−20.7049	−1.17977
$309$	11.9119	0.677644
$310$	−1.49717	−0.0850337
$311$	28.6829	1.62646	0.813229	−	0.581943i	$-0.197707\pi$
$311$	28.6829	1.62646	0.813229	+	0.581943i	$0.197707\pi$
$312$	1.15403	0.0653339
$313$	−21.1506	−1.19551	−0.597753	−	0.801681i	$-0.703939\pi$
$313$	−21.1506	−1.19551	−0.597753	+	0.801681i	$0.703939\pi$
$314$	−6.09537	−0.343981
$315$	1.05866	0.0596488
$316$	16.5276	0.929752
$317$	−27.2061	−1.52805	−0.764024	−	0.645187i	$-0.776779\pi$
$317$	−27.2061	−1.52805	−0.764024	+	0.645187i	$0.776779\pi$
$318$	13.3736	0.749955
$319$	4.50283	0.252110
$320$	0.230234	0.0128705
$321$	−12.0819	−0.674344
$322$	−4.59819	−0.256247
$323$	44.1661	2.45746
$324$	1.00000	0.0555556
$325$	−5.70896	−0.316676
$326$	1.25343	0.0694212
$327$	−7.71676	−0.426738
$328$	−0.0953661	−0.00526571
$329$	−2.75061	−0.151646
$330$	−1.03670	−0.0570687
$331$	21.1833	1.16434	0.582168	−	0.813068i	$-0.302204\pi$
$331$	21.1833	1.16434	0.582168	+	0.813068i	$0.302204\pi$
$332$	−1.71676	−0.0942193
$333$	7.29175	0.399586
$334$	2.49879	0.136728
$335$	0.481767	0.0263218
$336$	−4.59819	−0.250852
$337$	−21.5317	−1.17291	−0.586454	−	0.809982i	$-0.699477\pi$
$337$	−21.5317	−1.17291	−0.586454	+	0.809982i	$0.699477\pi$
$338$	11.6682	0.634668
$339$	−8.65810	−0.470243
$340$	−1.70986	−0.0927303
$341$	−29.2811	−1.58566
$342$	5.94699	0.321577
$343$	32.8466	1.77355
$344$	−0.941339	−0.0507536
$345$	−0.230234	−0.0123954
$346$	−6.21269	−0.333996
$347$	−13.2845	−0.713148	−0.356574	−	0.934267i	$-0.616055\pi$
$347$	−13.2845	−0.713148	−0.356574	+	0.934267i	$0.616055\pi$
$348$	1.00000	0.0536056
$349$	30.4392	1.62937	0.814686	−	0.579903i	$-0.196909\pi$
$349$	30.4392	1.62937	0.814686	+	0.579903i	$0.196909\pi$
$350$	22.7472	1.21589
$351$	−1.15403	−0.0615974
$352$	4.50283	0.240001
$353$	26.0847	1.38835	0.694175	−	0.719807i	$-0.255770\pi$
$353$	26.0847	1.38835	0.694175	+	0.719807i	$0.255770\pi$
$354$	−2.94134	−0.156330
$355$	2.52413	0.133967
$356$	12.5700	0.666209
$357$	34.1490	1.80736
$358$	−9.55084	−0.504778
$359$	29.2456	1.54353	0.771763	−	0.635910i	$-0.219375\pi$
$359$	29.2456	1.54353	0.771763	+	0.635910i	$0.219375\pi$
$360$	−0.230234	−0.0121344
$361$	16.3667	0.861406
$362$	14.2196	0.747365
$363$	−9.27545	−0.486835
$364$	5.30644	0.278133
$365$	−2.85324	−0.149345
$366$	10.8883	0.569142
$367$	25.0785	1.30909	0.654545	−	0.756023i	$-0.272860\pi$
$367$	25.0785	1.30909	0.654545	+	0.756023i	$0.272860\pi$
$368$	1.00000	0.0521286
$369$	0.0953661	0.00496456
$370$	−1.67881	−0.0872772
$371$	61.4944	3.19263
$372$	−6.50283	−0.337156
$373$	36.4257	1.88605	0.943025	−	0.332723i	$-0.107967\pi$
$373$	36.4257	1.88605	0.943025	+	0.332723i	$0.107967\pi$
$374$	−33.4408	−1.72918
$375$	2.29014	0.118262
$376$	0.598193	0.0308494
$377$	−1.15403	−0.0594354
$378$	4.59819	0.236506
$379$	−22.0847	−1.13442	−0.567208	−	0.823575i	$-0.691976\pi$
$379$	−22.0847	−1.13442	−0.567208	+	0.823575i	$0.691976\pi$
$380$	−1.36920	−0.0702385
$381$	−5.11167	−0.261879
$382$	−2.00286	−0.102475
$383$	30.2118	1.54375	0.771875	−	0.635774i	$-0.219319\pi$
$383$	30.2118	1.54375	0.771875	+	0.635774i	$0.219319\pi$
$384$	1.00000	0.0510310
$385$	−4.76697	−0.242947
$386$	−21.6242	−1.10065
$387$	0.941339	0.0478509
$388$	−1.82277	−0.0925373
$389$	−34.5622	−1.75237	−0.876187	−	0.481972i	$-0.839921\pi$
$389$	−34.5622	−1.75237	−0.876187	+	0.481972i	$0.839921\pi$
$390$	0.265697	0.0134541
$391$	−7.42662	−0.375580
$392$	−14.1434	−0.714348
$393$	−10.3968	−0.524450
$394$	−2.89898	−0.146049
$395$	3.80523	0.191462
$396$	−4.50283	−0.226276
$397$	−20.8981	−1.04884	−0.524422	−	0.851458i	$-0.675719\pi$
$397$	−20.8981	−1.04884	−0.524422	+	0.851458i	$0.675719\pi$
$398$	−12.2754	−0.615313
$399$	27.3454	1.36898
$400$	−4.94699	−0.247350
$401$	−6.97056	−0.348093	−0.174047	−	0.984737i	$-0.555684\pi$
$401$	−6.97056	−0.348093	−0.174047	+	0.984737i	$0.555684\pi$
$402$	2.09251	0.104365
$403$	7.50444	0.373823
$404$	−10.0000	−0.497519
$405$	0.230234	0.0114404
$406$	4.59819	0.228204
$407$	−32.8335	−1.62750
$408$	−7.42662	−0.367672
$409$	0.857280	0.0423897	0.0211949	−	0.999775i	$-0.493253\pi$
$409$	0.857280	0.0423897	0.0211949	+	0.999775i	$0.493253\pi$
$410$	−0.0219565	−0.00108436
$411$	−12.7958	−0.631171
$412$	−11.9119	−0.586857
$413$	−13.5248	−0.665514
$414$	−1.00000	−0.0491473
$415$	−0.395256	−0.0194024
$416$	−1.15403	−0.0565809
$417$	2.80361	0.137294
$418$	−26.7783	−1.30977
$419$	−3.35166	−0.163739	−0.0818695	−	0.996643i	$-0.526089\pi$
$419$	−3.35166	−0.163739	−0.0818695	+	0.996643i	$0.526089\pi$
$420$	−1.05866	−0.0516574
$421$	−28.7798	−1.40264	−0.701322	−	0.712845i	$-0.747406\pi$
$421$	−28.7798	−1.40264	−0.701322	+	0.712845i	$0.747406\pi$
$422$	15.8369	0.770930
$423$	−0.598193	−0.0290851
$424$	−13.3736	−0.649480
$425$	36.7394	1.78212
$426$	10.9633	0.531173
$427$	50.0666	2.42289
$428$	12.0819	0.583999
$429$	5.19639	0.250884
$430$	−0.216728	−0.0104516
$431$	8.84597	0.426096	0.213048	−	0.977042i	$-0.431661\pi$
$431$	8.84597	0.426096	0.213048	+	0.977042i	$0.431661\pi$
$432$	−1.00000	−0.0481125
$433$	2.74775	0.132048	0.0660242	−	0.997818i	$-0.478969\pi$
$433$	2.74775	0.132048	0.0660242	+	0.997818i	$0.478969\pi$
$434$	−29.9013	−1.43531
$435$	0.230234	0.0110389
$436$	7.71676	0.369566
$437$	−5.94699	−0.284483
$438$	−12.3928	−0.592150
$439$	16.4385	0.784567	0.392284	−	0.919844i	$-0.371685\pi$
$439$	16.4385	0.784567	0.392284	+	0.919844i	$0.371685\pi$
$440$	1.03670	0.0494230
$441$	14.1434	0.673494
$442$	8.57052	0.407658
$443$	−29.1751	−1.38615	−0.693075	−	0.720865i	$-0.743745\pi$
$443$	−29.1751	−1.38615	−0.693075	+	0.720865i	$0.743745\pi$
$444$	−7.29175	−0.346051
$445$	2.89404	0.137191
$446$	13.6569	0.646671
$447$	6.92783	0.327675
$448$	4.59819	0.217244
$449$	−30.8777	−1.45721	−0.728604	−	0.684935i	$-0.759830\pi$
$449$	−30.8777	−1.45721	−0.728604	+	0.684935i	$0.759830\pi$
$450$	4.94699	0.233203
$451$	−0.429417	−0.0202205
$452$	8.65810	0.407243
$453$	−17.5289	−0.823579
$454$	−12.2726	−0.575981
$455$	1.22172	0.0572753
$456$	−5.94699	−0.278493
$457$	−21.0797	−0.986068	−0.493034	−	0.870010i	$-0.664112\pi$
$457$	−21.0797	−0.986068	−0.493034	+	0.870010i	$0.664112\pi$
$458$	7.37647	0.344680
$459$	7.42662	0.346645
$460$	0.230234	0.0107347
$461$	−14.3504	−0.668365	−0.334183	−	0.942508i	$-0.608460\pi$
$461$	−14.3504	−0.668365	−0.334183	+	0.942508i	$0.608460\pi$
$462$	−20.7049	−0.963278
$463$	31.9436	1.48455	0.742273	−	0.670098i	$-0.233748\pi$
$463$	31.9436	1.48455	0.742273	+	0.670098i	$0.233748\pi$
$464$	−1.00000	−0.0464238
$465$	−1.49717	−0.0694297
$466$	−28.2020	−1.30643
$467$	−3.07180	−0.142146	−0.0710729	−	0.997471i	$-0.522642\pi$
$467$	−3.07180	−0.142146	−0.0710729	+	0.997471i	$0.522642\pi$
$468$	1.15403	0.0533449
$469$	9.62176	0.444292
$470$	0.137724	0.00635276
$471$	−6.09537	−0.280860
$472$	2.94134	0.135386
$473$	−4.23868	−0.194895
$474$	16.5276	0.759140
$475$	29.4197	1.34987
$476$	−34.1490	−1.56522
$477$	13.3736	0.612336
$478$	−1.14676	−0.0524516
$479$	−0.637695	−0.0291370	−0.0145685	−	0.999894i	$-0.504637\pi$
$479$	−0.637695	−0.0291370	−0.0145685	+	0.999894i	$0.504637\pi$
$480$	0.230234	0.0105087
$481$	8.41488	0.383686
$482$	−12.6406	−0.575761
$483$	−4.59819	−0.209225
$484$	9.27545	0.421611
$485$	−0.419665	−0.0190560
$486$	1.00000	0.0453609
$487$	20.6430	0.935425	0.467713	−	0.883881i	$-0.345078\pi$
$487$	20.6430	0.935425	0.467713	+	0.883881i	$0.345078\pi$
$488$	−10.8883	−0.492892
$489$	1.25343	0.0566822
$490$	−3.25629	−0.147104
$491$	17.5819	0.793460	0.396730	−	0.917935i	$-0.370145\pi$
$491$	17.5819	0.793460	0.396730	+	0.917935i	$0.370145\pi$
$492$	−0.0953661	−0.00429944
$493$	7.42662	0.334478
$494$	6.86299	0.308781
$495$	−1.03670	−0.0465964
$496$	6.50283	0.291986
$497$	50.4113	2.26126
$498$	−1.71676	−0.0769298
$499$	−9.35607	−0.418835	−0.209418	−	0.977826i	$-0.567157\pi$
$499$	−9.35607	−0.418835	−0.209418	+	0.977826i	$0.567157\pi$
$500$	−2.29014	−0.102418
$501$	2.49879	0.111638
$502$	5.62425	0.251022
$503$	−43.1722	−1.92495	−0.962477	−	0.271362i	$-0.912526\pi$
$503$	−43.1722	−1.92495	−0.962477	+	0.271362i	$0.912526\pi$
$504$	−4.59819	−0.204820
$505$	−2.30234	−0.102453
$506$	4.50283	0.200175
$507$	11.6682	0.518204
$508$	5.11167	0.226794
$509$	−31.7692	−1.40815	−0.704073	−	0.710127i	$-0.748637\pi$
$509$	−31.7692	−1.40815	−0.704073	+	0.710127i	$0.748637\pi$
$510$	−1.70986	−0.0757140
$511$	−56.9844	−2.52084
$512$	−1.00000	−0.0441942
$513$	5.94699	0.262566
$514$	21.3177	0.940286
$515$	−2.74253	−0.120850
$516$	−0.941339	−0.0414401
$517$	2.69356	0.118463
$518$	−33.5289	−1.47317
$519$	−6.21269	−0.272707
$520$	−0.265697	−0.0116516
$521$	−35.1440	−1.53969	−0.769844	−	0.638232i	$-0.779666\pi$
$521$	−35.1440	−1.53969	−0.769844	+	0.638232i	$0.779666\pi$
$522$	1.00000	0.0437688
$523$	30.6026	1.33816	0.669079	−	0.743191i	$-0.266689\pi$
$523$	30.6026	1.33816	0.669079	+	0.743191i	$0.266689\pi$
$524$	10.3968	0.454187
$525$	22.7472	0.992770
$526$	1.96205	0.0855496
$527$	−48.2940	−2.10372
$528$	4.50283	0.195960
$529$	1.00000	0.0434783
$530$	−3.07906	−0.133746
$531$	−2.94134	−0.127643
$532$	−27.3454	−1.18557
$533$	0.110055	0.00476702
$534$	12.5700	0.543957
$535$	2.78166	0.120262
$536$	−2.09251	−0.0903827
$537$	−9.55084	−0.412149
$538$	3.85163	0.166055
$539$	−63.6852	−2.74311
$540$	−0.230234	−0.00990770
$541$	−5.76197	−0.247727	−0.123863	−	0.992299i	$-0.539528\pi$
$541$	−5.76197	−0.247727	−0.123863	+	0.992299i	$0.539528\pi$
$542$	1.47183	0.0632207
$543$	14.2196	0.610221
$544$	7.42662	0.318414
$545$	1.77666	0.0761038
$546$	5.30644	0.227095
$547$	−19.5044	−0.833950	−0.416975	−	0.908918i	$-0.636910\pi$
$547$	−19.5044	−0.833950	−0.416975	+	0.908918i	$0.636910\pi$
$548$	12.7958	0.546610
$549$	10.8883	0.464703
$550$	−22.2754	−0.949828
$551$	5.94699	0.253350
$552$	1.00000	0.0425628
$553$	75.9973	3.23173
$554$	24.6665	1.04798
$555$	−1.67881	−0.0712615
$556$	−2.80361	−0.118900
$557$	32.8193	1.39060	0.695300	−	0.718720i	$-0.255272\pi$
$557$	32.8193	1.39060	0.695300	+	0.718720i	$0.255272\pi$
$558$	−6.50283	−0.275287
$559$	1.08633	0.0459469
$560$	1.05866	0.0447366
$561$	−33.4408	−1.41187
$562$	12.3081	0.519184
$563$	25.0440	1.05548	0.527739	−	0.849407i	$-0.323040\pi$
$563$	25.0440	1.05548	0.527739	+	0.849407i	$0.323040\pi$
$564$	0.598193	0.0251885
$565$	1.99339	0.0838626
$566$	23.8087	1.00076
$567$	4.59819	0.193106
$568$	−10.9633	−0.460010
$569$	3.80808	0.159643	0.0798216	−	0.996809i	$-0.474565\pi$
$569$	3.80808	0.159643	0.0798216	+	0.996809i	$0.474565\pi$
$570$	−1.36920	−0.0573495
$571$	−2.36392	−0.0989269	−0.0494635	−	0.998776i	$-0.515751\pi$
$571$	−2.36392	−0.0989269	−0.0494635	+	0.998776i	$0.515751\pi$
$572$	−5.19639	−0.217272
$573$	−2.00286	−0.0836705
$574$	−0.438512	−0.0183031
$575$	−4.94699	−0.206304
$576$	1.00000	0.0416667
$577$	22.8956	0.953156	0.476578	−	0.879132i	$-0.341877\pi$
$577$	22.8956	0.953156	0.476578	+	0.879132i	$0.341877\pi$
$578$	−38.1547	−1.58703
$579$	−21.6242	−0.898673
$580$	−0.230234	−0.00955996
$581$	−7.89398	−0.327498
$582$	−1.82277	−0.0755564
$583$	−60.2191	−2.49402
$584$	12.3928	0.512817
$585$	0.265697	0.0109852
$586$	−17.3162	−0.715325
$587$	14.2550	0.588369	0.294184	−	0.955749i	$-0.404952\pi$
$587$	14.2550	0.588369	0.294184	+	0.955749i	$0.404952\pi$
$588$	−14.1434	−0.583263
$589$	−38.6723	−1.59346
$590$	0.677197	0.0278798
$591$	−2.89898	−0.119248
$592$	7.29175	0.299689
$593$	13.5044	0.554561	0.277280	−	0.960789i	$-0.410567\pi$
$593$	13.5044	0.554561	0.277280	+	0.960789i	$0.410567\pi$
$594$	−4.50283	−0.184753
$595$	−7.86228	−0.322322
$596$	−6.92783	−0.283775
$597$	−12.2754	−0.502401
$598$	−1.15403	−0.0471917
$599$	−29.7855	−1.21700	−0.608501	−	0.793553i	$-0.708229\pi$
$599$	−29.7855	−1.21700	−0.608501	+	0.793553i	$0.708229\pi$
$600$	−4.94699	−0.201960
$601$	35.9819	1.46773	0.733867	−	0.679293i	$-0.237714\pi$
$601$	35.9819	1.46773	0.733867	+	0.679293i	$0.237714\pi$
$602$	−4.32846	−0.176415
$603$	2.09251	0.0852136
$604$	17.5289	0.713240
$605$	2.13553	0.0868215
$606$	−10.0000	−0.406222
$607$	30.7871	1.24961	0.624805	−	0.780781i	$-0.285179\pi$
$607$	30.7871	1.24961	0.624805	+	0.780781i	$0.285179\pi$
$608$	5.94699	0.241182
$609$	4.59819	0.186328
$610$	−2.50687	−0.101500
$611$	−0.690331	−0.0279278
$612$	−7.42662	−0.300203
$613$	18.2155	0.735719	0.367859	−	0.929881i	$-0.380091\pi$
$613$	18.2155	0.735719	0.367859	+	0.929881i	$0.380091\pi$
$614$	−0.920937	−0.0371660
$615$	−0.0219565	−0.000885373 0
$616$	20.7049	0.834223
$617$	−17.4592	−0.702882	−0.351441	−	0.936210i	$-0.614308\pi$
$617$	−17.4592	−0.702882	−0.351441	+	0.936210i	$0.614308\pi$
$618$	−11.9119	−0.479167
$619$	−14.3781	−0.577904	−0.288952	−	0.957344i	$-0.593307\pi$
$619$	−14.3781	−0.577904	−0.288952	+	0.957344i	$0.593307\pi$
$620$	1.49717	0.0601279
$621$	−1.00000	−0.0401286
$622$	−28.6829	−1.15008
$623$	57.7993	2.31568
$624$	−1.15403	−0.0461981
$625$	24.2077	0.968308
$626$	21.1506	0.845350
$627$	−26.7783	−1.06942
$628$	6.09537	0.243232
$629$	−54.1531	−2.15922
$630$	−1.05866	−0.0421781
$631$	−9.02357	−0.359223	−0.179611	−	0.983738i	$-0.557484\pi$
$631$	−9.02357	−0.359223	−0.179611	+	0.983738i	$0.557484\pi$
$632$	−16.5276	−0.657434
$633$	15.8369	0.629462
$634$	27.2061	1.08049
$635$	1.17688	0.0467031
$636$	−13.3736	−0.530298
$637$	16.3218	0.646695
$638$	−4.50283	−0.178269
$639$	10.9633	0.433701
$640$	−0.230234	−0.00910081
$641$	−16.3826	−0.647076	−0.323538	−	0.946215i	$-0.604872\pi$
$641$	−16.3826	−0.647076	−0.323538	+	0.946215i	$0.604872\pi$
$642$	12.0819	0.476833
$643$	−36.6733	−1.44625	−0.723127	−	0.690715i	$-0.757296\pi$
$643$	−36.6733	−1.44625	−0.723127	+	0.690715i	$0.757296\pi$
$644$	4.59819	0.181194
$645$	−0.216728	−0.00853367
$646$	−44.1661	−1.73769
$647$	30.5028	1.19919	0.599595	−	0.800304i	$-0.295328\pi$
$647$	30.5028	1.19919	0.599595	+	0.800304i	$0.295328\pi$
$648$	−1.00000	−0.0392837
$649$	13.2443	0.519886
$650$	5.70896	0.223924
$651$	−29.9013	−1.17192
$652$	−1.25343	−0.0490882
$653$	−17.6601	−0.691092	−0.345546	−	0.938402i	$-0.612306\pi$
$653$	−17.6601	−0.691092	−0.345546	+	0.938402i	$0.612306\pi$
$654$	7.71676	0.301749
$655$	2.39370	0.0935297
$656$	0.0953661	0.00372342
$657$	−12.3928	−0.483488
$658$	2.75061	0.107230
$659$	15.7473	0.613427	0.306714	−	0.951802i	$-0.400771\pi$
$659$	15.7473	0.613427	0.306714	+	0.951802i	$0.400771\pi$
$660$	1.03670	0.0403537
$661$	−34.4640	−1.34049	−0.670247	−	0.742138i	$-0.733812\pi$
$661$	−34.4640	−1.34049	−0.670247	+	0.742138i	$0.733812\pi$
$662$	−21.1833	−0.823310
$663$	8.57052	0.332852
$664$	1.71676	0.0666231
$665$	−6.29585	−0.244143
$666$	−7.29175	−0.282550
$667$	−1.00000	−0.0387202
$668$	−2.49879	−0.0966810
$669$	13.6569	0.528004
$670$	−0.481767	−0.0186123
$671$	−49.0283	−1.89272
$672$	4.59819	0.177379
$673$	47.2426	1.82107	0.910535	−	0.413432i	$-0.135670\pi$
$673$	47.2426	1.82107	0.910535	+	0.413432i	$0.135670\pi$
$674$	21.5317	0.829372
$675$	4.94699	0.190410
$676$	−11.6682	−0.448778
$677$	−5.38141	−0.206824	−0.103412	−	0.994639i	$-0.532976\pi$
$677$	−5.38141	−0.206824	−0.103412	+	0.994639i	$0.532976\pi$
$678$	8.65810	0.332512
$679$	−8.38146	−0.321651
$680$	1.70986	0.0655702
$681$	−12.2726	−0.470287
$682$	29.2811	1.12123
$683$	−14.7945	−0.566097	−0.283048	−	0.959106i	$-0.591346\pi$
$683$	−14.7945	−0.566097	−0.283048	+	0.959106i	$0.591346\pi$
$684$	−5.94699	−0.227389
$685$	2.94604	0.112562
$686$	−32.8466	−1.25409
$687$	7.37647	0.281430
$688$	0.941339	0.0358882
$689$	15.4335	0.587970
$690$	0.230234	0.00876487
$691$	−17.0310	−0.647889	−0.323945	−	0.946076i	$-0.605009\pi$
$691$	−17.0310	−0.647889	−0.323945	+	0.946076i	$0.605009\pi$
$692$	6.21269	0.236171
$693$	−20.7049	−0.786513
$694$	13.2845	0.504272
$695$	−0.645488	−0.0244848
$696$	−1.00000	−0.0379049
$697$	−0.708248	−0.0268268
$698$	−30.4392	−1.15214
$699$	−28.2020	−1.06670
$700$	−22.7472	−0.859764
$701$	38.9065	1.46948	0.734740	−	0.678349i	$-0.237304\pi$
$701$	38.9065	1.46948	0.734740	+	0.678349i	$0.237304\pi$
$702$	1.15403	0.0435560
$703$	−43.3640	−1.63550
$704$	−4.50283	−0.169707
$705$	0.137724	0.00518700
$706$	−26.0847	−0.981711
$707$	−45.9819	−1.72933
$708$	2.94134	0.110542
$709$	−13.2849	−0.498923	−0.249462	−	0.968385i	$-0.580254\pi$
$709$	−13.2849	−0.498923	−0.249462	+	0.968385i	$0.580254\pi$
$710$	−2.52413	−0.0947287
$711$	16.5276	0.619835
$712$	−12.5700	−0.471081
$713$	6.50283	0.243533
$714$	−34.1490	−1.27800
$715$	−1.19639	−0.0447423
$716$	9.55084	0.356932
$717$	−1.14676	−0.0428265
$718$	−29.2456	−1.09144
$719$	−3.73430	−0.139266	−0.0696330	−	0.997573i	$-0.522183\pi$
$719$	−3.73430	−0.139266	−0.0696330	+	0.997573i	$0.522183\pi$
$720$	0.230234	0.00858032
$721$	−54.7732	−2.03986
$722$	−16.3667	−0.609106
$723$	−12.6406	−0.470107
$724$	−14.2196	−0.528467
$725$	4.94699	0.183727
$726$	9.27545	0.344244
$727$	26.6547	0.988569	0.494284	−	0.869300i	$-0.335430\pi$
$727$	26.6547	0.988569	0.494284	+	0.869300i	$0.335430\pi$
$728$	−5.30644	−0.196670
$729$	1.00000	0.0370370
$730$	2.85324	0.105603
$731$	−6.99096	−0.258570
$732$	−10.8883	−0.402444
$733$	37.7912	1.39585	0.697925	−	0.716171i	$-0.254107\pi$
$733$	37.7912	1.39585	0.697925	+	0.716171i	$0.254107\pi$
$734$	−25.0785	−0.925666
$735$	−3.25629	−0.120110
$736$	−1.00000	−0.0368605
$737$	−9.42221	−0.347072
$738$	−0.0953661	−0.00351048
$739$	14.5477	0.535145	0.267572	−	0.963538i	$-0.413779\pi$
$739$	14.5477	0.535145	0.267572	+	0.963538i	$0.413779\pi$
$740$	1.67881	0.0617143
$741$	6.86299	0.252118
$742$	−61.4944	−2.25753
$743$	−27.2725	−1.00053	−0.500266	−	0.865872i	$-0.666764\pi$
$743$	−27.2725	−1.00053	−0.500266	+	0.865872i	$0.666764\pi$
$744$	6.50283	0.238405
$745$	−1.59502	−0.0584372
$746$	−36.4257	−1.33364
$747$	−1.71676	−0.0628129
$748$	33.4408	1.22272
$749$	55.5547	2.02993
$750$	−2.29014	−0.0836240
$751$	46.6547	1.70245	0.851227	−	0.524797i	$-0.175859\pi$
$751$	46.6547	1.70245	0.851227	+	0.524797i	$0.175859\pi$
$752$	−0.598193	−0.0218139
$753$	5.62425	0.204959
$754$	1.15403	0.0420272
$755$	4.03575	0.146876
$756$	−4.59819	−0.167235
$757$	−44.5955	−1.62085	−0.810425	−	0.585842i	$-0.800764\pi$
$757$	−44.5955	−1.62085	−0.810425	+	0.585842i	$0.800764\pi$
$758$	22.0847	0.802153
$759$	4.50283	0.163442
$760$	1.36920	0.0496661
$761$	12.0153	0.435556	0.217778	−	0.975998i	$-0.430119\pi$
$761$	12.0153	0.435556	0.217778	+	0.975998i	$0.430119\pi$
$762$	5.11167	0.185176
$763$	35.4831	1.28458
$764$	2.00286	0.0724608
$765$	−1.70986	−0.0618202
$766$	−30.2118	−1.09160
$767$	−3.39439	−0.122564
$768$	−1.00000	−0.0360844
$769$	−19.5603	−0.705363	−0.352681	−	0.935743i	$-0.614730\pi$
$769$	−19.5603	−0.705363	−0.352681	+	0.935743i	$0.614730\pi$
$770$	4.76697	0.171790
$771$	21.3177	0.767740
$772$	21.6242	0.778274
$773$	−14.7871	−0.531855	−0.265927	−	0.963993i	$-0.585678\pi$
$773$	−14.7871	−0.531855	−0.265927	+	0.963993i	$0.585678\pi$
$774$	−0.941339	−0.0338357
$775$	−32.1694	−1.15556
$776$	1.82277	0.0654338
$777$	−33.5289	−1.20284
$778$	34.5622	1.23912
$779$	−0.567142	−0.0203200
$780$	−0.265697	−0.00951346
$781$	−49.3658	−1.76645
$782$	7.42662	0.265575
$783$	1.00000	0.0357371
$784$	14.1434	0.505121
$785$	1.40336	0.0500881
$786$	10.3968	0.370842
$787$	4.83569	0.172374	0.0861869	−	0.996279i	$-0.472532\pi$
$787$	4.83569	0.172374	0.0861869	+	0.996279i	$0.472532\pi$
$788$	2.89898	0.103272
$789$	1.96205	0.0698509
$790$	−3.80523	−0.135384
$791$	39.8116	1.41554
$792$	4.50283	0.160001
$793$	12.5654	0.446212
$794$	20.8981	0.741645
$795$	−3.07906	−0.109203
$796$	12.2754	0.435092
$797$	−5.42128	−0.192032	−0.0960158	−	0.995380i	$-0.530610\pi$
$797$	−5.42128	−0.192032	−0.0960158	+	0.995380i	$0.530610\pi$
$798$	−27.3454	−0.968017
$799$	4.44255	0.157166
$800$	4.94699	0.174903
$801$	12.5700	0.444139
$802$	6.97056	0.246139
$803$	55.8025	1.96923
$804$	−2.09251	−0.0737971
$805$	1.05866	0.0373129
$806$	−7.50444	−0.264333
$807$	3.85163	0.135584
$808$	10.0000	0.351799
$809$	31.5215	1.10824	0.554118	−	0.832438i	$-0.313056\pi$
$809$	31.5215	1.10824	0.554118	+	0.832438i	$0.313056\pi$
$810$	−0.230234	−0.00808961
$811$	34.4856	1.21095	0.605476	−	0.795864i	$-0.292983\pi$
$811$	34.4856	1.21095	0.605476	+	0.795864i	$0.292983\pi$
$812$	−4.59819	−0.161365
$813$	1.47183	0.0516195
$814$	32.8335	1.15081
$815$	−0.288583	−0.0101086
$816$	7.42662	0.259984
$817$	−5.59813	−0.195854
$818$	−0.857280	−0.0299741
$819$	5.30644	0.185422
$820$	0.0219565	0.000766756 0
$821$	18.8206	0.656845	0.328422	−	0.944531i	$-0.393483\pi$
$821$	18.8206	0.656845	0.328422	+	0.944531i	$0.393483\pi$
$822$	12.7958	0.446306
$823$	29.0593	1.01294	0.506472	−	0.862256i	$-0.330949\pi$
$823$	29.0593	1.01294	0.506472	+	0.862256i	$0.330949\pi$
$824$	11.9119	0.414971
$825$	−22.2754	−0.775532
$826$	13.5248	0.470589
$827$	14.7447	0.512725	0.256362	−	0.966581i	$-0.417476\pi$
$827$	14.7447	0.512725	0.256362	+	0.966581i	$0.417476\pi$
$828$	1.00000	0.0347524
$829$	−45.6422	−1.58522	−0.792609	−	0.609730i	$-0.791278\pi$
$829$	−45.6422	−1.58522	−0.792609	+	0.609730i	$0.791278\pi$
$830$	0.395256	0.0137196
$831$	24.6665	0.855673
$832$	1.15403	0.0400087
$833$	−105.037	−3.63933
$834$	−2.80361	−0.0970812
$835$	−0.575306	−0.0199093
$836$	26.7783	0.926146
$837$	−6.50283	−0.224771
$838$	3.35166	0.115781
$839$	−5.37295	−0.185495	−0.0927475	−	0.995690i	$-0.529565\pi$
$839$	−5.37295	−0.185495	−0.0927475	+	0.995690i	$0.529565\pi$
$840$	1.05866	0.0365273
$841$	1.00000	0.0344828
$842$	28.7798	0.991818
$843$	12.3081	0.423912
$844$	−15.8369	−0.545130
$845$	−2.68642	−0.0924158
$846$	0.598193	0.0205663
$847$	42.6503	1.46548
$848$	13.3736	0.459252
$849$	23.8087	0.817114
$850$	−36.7394	−1.26015
$851$	7.29175	0.249958
$852$	−10.9633	−0.375596
$853$	42.6974	1.46193	0.730966	−	0.682414i	$-0.239070\pi$
$853$	42.6974	1.46193	0.730966	+	0.682414i	$0.239070\pi$
$854$	−50.0666	−1.71325
$855$	−1.36920	−0.0468257
$856$	−12.0819	−0.412949
$857$	48.5254	1.65760	0.828799	−	0.559547i	$-0.189025\pi$
$857$	48.5254	1.65760	0.828799	+	0.559547i	$0.189025\pi$
$858$	−5.19639	−0.177402
$859$	24.3112	0.829487	0.414744	−	0.909938i	$-0.363871\pi$
$859$	24.3112	0.829487	0.414744	+	0.909938i	$0.363871\pi$
$860$	0.216728	0.00739038
$861$	−0.438512	−0.0149444
$862$	−8.84597	−0.301295
$863$	41.2092	1.40278	0.701390	−	0.712778i	$-0.252563\pi$
$863$	41.2092	1.40278	0.701390	+	0.712778i	$0.252563\pi$
$864$	1.00000	0.0340207
$865$	1.43037	0.0486342
$866$	−2.74775	−0.0933723
$867$	−38.1547	−1.29580
$868$	29.9013	1.01491
$869$	−74.4211	−2.52456
$870$	−0.230234	−0.00780567
$871$	2.41481	0.0818228
$872$	−7.71676	−0.261322
$873$	−1.82277	−0.0616915
$874$	5.94699	0.201160
$875$	−10.5305	−0.355996
$876$	12.3928	0.418713
$877$	−24.3674	−0.822830	−0.411415	−	0.911448i	$-0.634965\pi$
$877$	−24.3674	−0.822830	−0.411415	+	0.911448i	$0.634965\pi$
$878$	−16.4385	−0.554773
$879$	−17.3162	−0.584061
$880$	−1.03670	−0.0349473
$881$	−11.9890	−0.403919	−0.201959	−	0.979394i	$-0.564731\pi$
$881$	−11.9890	−0.403919	−0.201959	+	0.979394i	$0.564731\pi$
$882$	−14.1434	−0.476232
$883$	−48.3787	−1.62807	−0.814037	−	0.580813i	$-0.802735\pi$
$883$	−48.3787	−1.62807	−0.814037	+	0.580813i	$0.802735\pi$
$884$	−8.57052	−0.288258
$885$	0.677197	0.0227637
$886$	29.1751	0.980156
$887$	41.8351	1.40469	0.702343	−	0.711839i	$-0.252137\pi$
$887$	41.8351	1.40469	0.702343	+	0.711839i	$0.252137\pi$
$888$	7.29175	0.244695
$889$	23.5044	0.788314
$890$	−2.89404	−0.0970086
$891$	−4.50283	−0.150850
$892$	−13.6569	−0.457265
$893$	3.55745	0.119045
$894$	−6.92783	−0.231701
$895$	2.19893	0.0735021
$896$	−4.59819	−0.153615
$897$	−1.15403	−0.0385319
$898$	30.8777	1.03040
$899$	−6.50283	−0.216881
$900$	−4.94699	−0.164900
$901$	−99.3207	−3.30885
$902$	0.429417	0.0142980
$903$	−4.32846	−0.144042
$904$	−8.65810	−0.287964
$905$	−3.27383	−0.108826
$906$	17.5289	0.582358
$907$	−0.443449	−0.0147245	−0.00736223	−	0.999973i	$-0.502343\pi$
$907$	−0.443449	−0.0147245	−0.00736223	+	0.999973i	$0.502343\pi$
$908$	12.2726	0.407280
$909$	−10.0000	−0.331679
$910$	−1.22172	−0.0404998
$911$	44.0010	1.45782	0.728910	−	0.684610i	$-0.240027\pi$
$911$	44.0010	1.45782	0.728910	+	0.684610i	$0.240027\pi$
$912$	5.94699	0.196925
$913$	7.73026	0.255834
$914$	21.0797	0.697255
$915$	−2.50687	−0.0828744
$916$	−7.37647	−0.243725
$917$	47.8065	1.57871
$918$	−7.42662	−0.245115
$919$	−52.5556	−1.73365	−0.866825	−	0.498613i	$-0.833843\pi$
$919$	−52.5556	−1.73365	−0.866825	+	0.498613i	$0.833843\pi$
$920$	−0.230234	−0.00759060
$921$	−0.920937	−0.0303459
$922$	14.3504	0.472606
$923$	12.6519	0.416444
$924$	20.7049	0.681140
$925$	−36.0722	−1.18605
$926$	−31.9436	−1.04973
$927$	−11.9119	−0.391238
$928$	1.00000	0.0328266
$929$	−51.6298	−1.69392	−0.846960	−	0.531656i	$-0.821570\pi$
$929$	−51.6298	−1.69392	−0.846960	+	0.531656i	$0.821570\pi$
$930$	1.49717	0.0490942
$931$	−84.1106	−2.75661
$932$	28.2020	0.923789
$933$	−28.6829	−0.939036
$934$	3.07180	0.100512
$935$	7.69921	0.251791
$936$	−1.15403	−0.0377206
$937$	12.2804	0.401183	0.200591	−	0.979675i	$-0.435714\pi$
$937$	12.2804	0.401183	0.200591	+	0.979675i	$0.435714\pi$
$938$	−9.62176	−0.314162
$939$	21.1506	0.690225
$940$	−0.137724	−0.00449208
$941$	−36.4257	−1.18744	−0.593721	−	0.804671i	$-0.702342\pi$
$941$	−36.4257	−1.18744	−0.593721	+	0.804671i	$0.702342\pi$
$942$	6.09537	0.198598
$943$	0.0953661	0.00310555
$944$	−2.94134	−0.0957324
$945$	−1.05866	−0.0344383
$946$	4.23868	0.137812
$947$	2.22738	0.0723801	0.0361900	−	0.999345i	$-0.488478\pi$
$947$	2.22738	0.0723801	0.0361900	+	0.999345i	$0.488478\pi$
$948$	−16.5276	−0.536793
$949$	−14.3016	−0.464250
$950$	−29.4197	−0.954502
$951$	27.2061	0.882219
$952$	34.1490	1.10678
$953$	16.1524	0.523228	0.261614	−	0.965173i	$-0.415745\pi$
$953$	16.1524	0.523228	0.261614	+	0.965173i	$0.415745\pi$
$954$	−13.3736	−0.432987
$955$	0.461126	0.0149217
$956$	1.14676	0.0370889
$957$	−4.50283	−0.145556
$958$	0.637695	0.0206030
$959$	58.8377	1.89997
$960$	−0.230234	−0.00743078
$961$	11.2868	0.364089
$962$	−8.41488	−0.271307
$963$	12.0819	0.389333
$964$	12.6406	0.407125
$965$	4.97864	0.160268
$966$	4.59819	0.147944
$967$	−1.46208	−0.0470174	−0.0235087	−	0.999724i	$-0.507484\pi$
$967$	−1.46208	−0.0470174	−0.0235087	+	0.999724i	$0.507484\pi$
$968$	−9.27545	−0.298124
$969$	−44.1661	−1.41882
$970$	0.419665	0.0134746
$971$	6.24440	0.200392	0.100196	−	0.994968i	$-0.468053\pi$
$971$	6.24440	0.200392	0.100196	+	0.994968i	$0.468053\pi$
$972$	−1.00000	−0.0320750
$973$	−12.8916	−0.413284
$974$	−20.6430	−0.661446
$975$	5.70896	0.182833
$976$	10.8883	0.348527
$977$	−13.8899	−0.444379	−0.222189	−	0.975004i	$-0.571320\pi$
$977$	−13.8899	−0.444379	−0.222189	+	0.975004i	$0.571320\pi$
$978$	−1.25343	−0.0400804
$979$	−56.6005	−1.80896
$980$	3.25629	0.104018
$981$	7.71676	0.246377
$982$	−17.5819	−0.561061
$983$	27.5406	0.878407	0.439204	−	0.898387i	$-0.355261\pi$
$983$	27.5406	0.878407	0.439204	+	0.898387i	$0.355261\pi$
$984$	0.0953661	0.00304016
$985$	0.667444	0.0212665
$986$	−7.42662	−0.236512
$987$	2.75061	0.0875528
$988$	−6.86299	−0.218341
$989$	0.941339	0.0299328
$990$	1.03670	0.0329486
$991$	−4.94289	−0.157016	−0.0785081	−	0.996913i	$-0.525016\pi$
$991$	−4.94289	−0.157016	−0.0785081	+	0.996913i	$0.525016\pi$
$992$	−6.50283	−0.206465
$993$	−21.1833	−0.672230
$994$	−50.4113	−1.59895
$995$	2.82623	0.0895975
$996$	1.71676	0.0543976
$997$	10.2437	0.324423	0.162211	−	0.986756i	$-0.448137\pi$
$997$	10.2437	0.324423	0.162211	+	0.986756i	$0.448137\pi$
$998$	9.35607	0.296161
$999$	−7.29175	−0.230701

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.ba.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.ba.1.2	✓	4	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:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.230234 q^{5} +1.00000 q^{6} +4.59819 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.230234 q^{5} +1.00000 q^{6} +4.59819 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.230234 q^{10} -4.50283 q^{11} -1.00000 q^{12} +1.15403 q^{13} -4.59819 q^{14} -0.230234 q^{15} +1.00000 q^{16} -7.42662 q^{17} -1.00000 q^{18} -5.94699 q^{19} +0.230234 q^{20} -4.59819 q^{21} +4.50283 q^{22} +1.00000 q^{23} +1.00000 q^{24} -4.94699 q^{25} -1.15403 q^{26} -1.00000 q^{27} +4.59819 q^{28} -1.00000 q^{29} +0.230234 q^{30} +6.50283 q^{31} -1.00000 q^{32} +4.50283 q^{33} +7.42662 q^{34} +1.05866 q^{35} +1.00000 q^{36} +7.29175 q^{37} +5.94699 q^{38} -1.15403 q^{39} -0.230234 q^{40} +0.0953661 q^{41} +4.59819 q^{42} +0.941339 q^{43} -4.50283 q^{44} +0.230234 q^{45} -1.00000 q^{46} -0.598193 q^{47} -1.00000 q^{48} +14.1434 q^{49} +4.94699 q^{50} +7.42662 q^{51} +1.15403 q^{52} +13.3736 q^{53} +1.00000 q^{54} -1.03670 q^{55} -4.59819 q^{56} +5.94699 q^{57} +1.00000 q^{58} -2.94134 q^{59} -0.230234 q^{60} +10.8883 q^{61} -6.50283 q^{62} +4.59819 q^{63} +1.00000 q^{64} +0.265697 q^{65} -4.50283 q^{66} +2.09251 q^{67} -7.42662 q^{68} -1.00000 q^{69} -1.05866 q^{70} +10.9633 q^{71} -1.00000 q^{72} -12.3928 q^{73} -7.29175 q^{74} +4.94699 q^{75} -5.94699 q^{76} -20.7049 q^{77} +1.15403 q^{78} +16.5276 q^{79} +0.230234 q^{80} +1.00000 q^{81} -0.0953661 q^{82} -1.71676 q^{83} -4.59819 q^{84} -1.70986 q^{85} -0.941339 q^{86} +1.00000 q^{87} +4.50283 q^{88} +12.5700 q^{89} -0.230234 q^{90} +5.30644 q^{91} +1.00000 q^{92} -6.50283 q^{93} +0.598193 q^{94} -1.36920 q^{95} +1.00000 q^{96} -1.82277 q^{97} -14.1434 q^{98} -4.50283 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 + 6 * q^7 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 6 q^{7} - 4 q^{8} + 4 q^{9} - 2 q^{10} - 4 q^{12} - 6 q^{14} - 2 q^{15} + 4 q^{16} - 6 q^{17} - 4 q^{18} + 2 q^{19} + 2 q^{20} - 6 q^{21} + 4 q^{23} + 4 q^{24} + 6 q^{25} - 4 q^{27} + 6 q^{28} - 4 q^{29} + 2 q^{30} + 8 q^{31} - 4 q^{32} + 6 q^{34} - 6 q^{35} + 4 q^{36} + 10 q^{37} - 2 q^{38} - 2 q^{40} + 6 q^{41} + 6 q^{42} + 14 q^{43} + 2 q^{45} - 4 q^{46} + 10 q^{47} - 4 q^{48} + 6 q^{49} - 6 q^{50} + 6 q^{51} + 4 q^{53} + 4 q^{54} - 20 q^{55} - 6 q^{56} - 2 q^{57} + 4 q^{58} - 22 q^{59} - 2 q^{60} + 28 q^{61} - 8 q^{62} + 6 q^{63} + 4 q^{64} + 12 q^{65} + 24 q^{67} - 6 q^{68} - 4 q^{69} + 6 q^{70} + 28 q^{71} - 4 q^{72} - 10 q^{74} - 6 q^{75} + 2 q^{76} - 4 q^{77} + 12 q^{79} + 2 q^{80} + 4 q^{81} - 6 q^{82} + 20 q^{83} - 6 q^{84} - 10 q^{85} - 14 q^{86} + 4 q^{87} - 24 q^{89} - 2 q^{90} + 28 q^{91} + 4 q^{92} - 8 q^{93} - 10 q^{94} + 2 q^{95} + 4 q^{96} - 32 q^{97} - 6 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 2 * q^5 + 4 * q^6 + 6 * q^7 - 4 * q^8 + 4 * q^9 - 2 * q^10 - 4 * q^12 - 6 * q^14 - 2 * q^15 + 4 * q^16 - 6 * q^17 - 4 * q^18 + 2 * q^19 + 2 * q^20 - 6 * q^21 + 4 * q^23 + 4 * q^24 + 6 * q^25 - 4 * q^27 + 6 * q^28 - 4 * q^29 + 2 * q^30 + 8 * q^31 - 4 * q^32 + 6 * q^34 - 6 * q^35 + 4 * q^36 + 10 * q^37 - 2 * q^38 - 2 * q^40 + 6 * q^41 + 6 * q^42 + 14 * q^43 + 2 * q^45 - 4 * q^46 + 10 * q^47 - 4 * q^48 + 6 * q^49 - 6 * q^50 + 6 * q^51 + 4 * q^53 + 4 * q^54 - 20 * q^55 - 6 * q^56 - 2 * q^57 + 4 * q^58 - 22 * q^59 - 2 * q^60 + 28 * q^61 - 8 * q^62 + 6 * q^63 + 4 * q^64 + 12 * q^65 + 24 * q^67 - 6 * q^68 - 4 * q^69 + 6 * q^70 + 28 * q^71 - 4 * q^72 - 10 * q^74 - 6 * q^75 + 2 * q^76 - 4 * q^77 + 12 * q^79 + 2 * q^80 + 4 * q^81 - 6 * q^82 + 20 * q^83 - 6 * q^84 - 10 * q^85 - 14 * q^86 + 4 * q^87 - 24 * q^89 - 2 * q^90 + 28 * q^91 + 4 * q^92 - 8 * q^93 - 10 * q^94 + 2 * q^95 + 4 * q^96 - 32 * q^97 - 6 * q^98

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