LMFDB - Embedded newform 4002.2.a.bj.1.8

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 26x^{6} + 4x^{5} + 209x^{4} + 113x^{3} - 436x^{2} - 360x - 16 $ x^8 - x^7 - 26x^6 + 4x^5 + 209x^4 + 113x^3 - 436x^2 - 360x - 16
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			1.8
Root			$4.15688$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.15688 q^{5} -1.00000 q^{6} +0.461913 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.15688 q^{5} -1.00000 q^{6} +0.461913 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.15688 q^{10} +2.29353 q^{11} -1.00000 q^{12} -2.16373 q^{13} +0.461913 q^{14} -4.15688 q^{15} +1.00000 q^{16} +3.68899 q^{17} +1.00000 q^{18} +0.461913 q^{19} +4.15688 q^{20} -0.461913 q^{21} +2.29353 q^{22} +1.00000 q^{23} -1.00000 q^{24} +12.2797 q^{25} -2.16373 q^{26} -1.00000 q^{27} +0.461913 q^{28} +1.00000 q^{29} -4.15688 q^{30} -2.61314 q^{31} +1.00000 q^{32} -2.29353 q^{33} +3.68899 q^{34} +1.92012 q^{35} +1.00000 q^{36} -7.35359 q^{37} +0.461913 q^{38} +2.16373 q^{39} +4.15688 q^{40} +1.98532 q^{41} -0.461913 q^{42} -4.73656 q^{43} +2.29353 q^{44} +4.15688 q^{45} +1.00000 q^{46} +12.4440 q^{47} -1.00000 q^{48} -6.78664 q^{49} +12.2797 q^{50} -3.68899 q^{51} -2.16373 q^{52} -7.91682 q^{53} -1.00000 q^{54} +9.53392 q^{55} +0.461913 q^{56} -0.461913 q^{57} +1.00000 q^{58} +13.9439 q^{59} -4.15688 q^{60} +7.56872 q^{61} -2.61314 q^{62} +0.461913 q^{63} +1.00000 q^{64} -8.99437 q^{65} -2.29353 q^{66} -1.07490 q^{67} +3.68899 q^{68} -1.00000 q^{69} +1.92012 q^{70} -7.96922 q^{71} +1.00000 q^{72} +14.6021 q^{73} -7.35359 q^{74} -12.2797 q^{75} +0.461913 q^{76} +1.05941 q^{77} +2.16373 q^{78} -16.1867 q^{79} +4.15688 q^{80} +1.00000 q^{81} +1.98532 q^{82} +1.22708 q^{83} -0.461913 q^{84} +15.3347 q^{85} -4.73656 q^{86} -1.00000 q^{87} +2.29353 q^{88} +1.14905 q^{89} +4.15688 q^{90} -0.999454 q^{91} +1.00000 q^{92} +2.61314 q^{93} +12.4440 q^{94} +1.92012 q^{95} -1.00000 q^{96} -13.6303 q^{97} -6.78664 q^{98} +2.29353 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + q^5 - 8 * q^6 + 8 * q^8 + 8 * q^9	$ 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9} + q^{10} + 3 q^{11} - 8 q^{12} + 9 q^{13} - q^{15} + 8 q^{16} + 8 q^{17} + 8 q^{18} + q^{20} + 3 q^{22} + 8 q^{23} - 8 q^{24} + 13 q^{25} + 9 q^{26} - 8 q^{27} + 8 q^{29} - q^{30} - 9 q^{31} + 8 q^{32} - 3 q^{33} + 8 q^{34} - 2 q^{35} + 8 q^{36} + 7 q^{37} - 9 q^{39} + q^{40} + 3 q^{41} + 16 q^{43} + 3 q^{44} + q^{45} + 8 q^{46} + 24 q^{47} - 8 q^{48} + 6 q^{49} + 13 q^{50} - 8 q^{51} + 9 q^{52} + 8 q^{53} - 8 q^{54} + 13 q^{55} + 8 q^{58} - 3 q^{59} - q^{60} + 31 q^{61} - 9 q^{62} + 8 q^{64} + 13 q^{65} - 3 q^{66} - 11 q^{67} + 8 q^{68} - 8 q^{69} - 2 q^{70} + 7 q^{71} + 8 q^{72} + 14 q^{73} + 7 q^{74} - 13 q^{75} + 10 q^{77} - 9 q^{78} + 12 q^{79} + q^{80} + 8 q^{81} + 3 q^{82} - 8 q^{83} + 22 q^{85} + 16 q^{86} - 8 q^{87} + 3 q^{88} - 12 q^{89} + q^{90} + 28 q^{91} + 8 q^{92} + 9 q^{93} + 24 q^{94} - 2 q^{95} - 8 q^{96} + 16 q^{97} + 6 q^{98} + 3 q^{99}+O(q^{100}) $ 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + q^5 - 8 * q^6 + 8 * q^8 + 8 * q^9 + q^10 + 3 * q^11 - 8 * q^12 + 9 * q^13 - q^15 + 8 * q^16 + 8 * q^17 + 8 * q^18 + q^20 + 3 * q^22 + 8 * q^23 - 8 * q^24 + 13 * q^25 + 9 * q^26 - 8 * q^27 + 8 * q^29 - q^30 - 9 * q^31 + 8 * q^32 - 3 * q^33 + 8 * q^34 - 2 * q^35 + 8 * q^36 + 7 * q^37 - 9 * q^39 + q^40 + 3 * q^41 + 16 * q^43 + 3 * q^44 + q^45 + 8 * q^46 + 24 * q^47 - 8 * q^48 + 6 * q^49 + 13 * q^50 - 8 * q^51 + 9 * q^52 + 8 * q^53 - 8 * q^54 + 13 * q^55 + 8 * q^58 - 3 * q^59 - q^60 + 31 * q^61 - 9 * q^62 + 8 * q^64 + 13 * q^65 - 3 * q^66 - 11 * q^67 + 8 * q^68 - 8 * q^69 - 2 * q^70 + 7 * q^71 + 8 * q^72 + 14 * q^73 + 7 * q^74 - 13 * q^75 + 10 * q^77 - 9 * q^78 + 12 * q^79 + q^80 + 8 * q^81 + 3 * q^82 - 8 * q^83 + 22 * q^85 + 16 * q^86 - 8 * q^87 + 3 * q^88 - 12 * q^89 + q^90 + 28 * q^91 + 8 * q^92 + 9 * q^93 + 24 * q^94 - 2 * q^95 - 8 * q^96 + 16 * q^97 + 6 * q^98 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	4.15688	1.85901	0.929507	−	0.368804i	$-0.120233\pi$
$5$	4.15688	1.85901	0.929507	+	0.368804i	$0.120233\pi$
$6$	−1.00000	−0.408248
$7$	0.461913	0.174587	0.0872933	−	0.996183i	$-0.472178\pi$
$7$	0.461913	0.174587	0.0872933	+	0.996183i	$0.472178\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	4.15688	1.31452
$11$	2.29353	0.691524	0.345762	−	0.938322i	$-0.387620\pi$
$11$	2.29353	0.691524	0.345762	+	0.938322i	$0.387620\pi$
$12$	−1.00000	−0.288675
$13$	−2.16373	−0.600110	−0.300055	−	0.953922i	$-0.597005\pi$
$13$	−2.16373	−0.600110	−0.300055	+	0.953922i	$0.597005\pi$
$14$	0.461913	0.123451
$15$	−4.15688	−1.07330
$16$	1.00000	0.250000
$17$	3.68899	0.894713	0.447356	−	0.894356i	$-0.352366\pi$
$17$	3.68899	0.894713	0.447356	+	0.894356i	$0.352366\pi$
$18$	1.00000	0.235702
$19$	0.461913	0.105970	0.0529850	−	0.998595i	$-0.483126\pi$
$19$	0.461913	0.105970	0.0529850	+	0.998595i	$0.483126\pi$
$20$	4.15688	0.929507
$21$	−0.461913	−0.100798
$22$	2.29353	0.488982
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	12.2797	2.45594
$26$	−2.16373	−0.424342
$27$	−1.00000	−0.192450
$28$	0.461913	0.0872933
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−4.15688	−0.758940
$31$	−2.61314	−0.469334	−0.234667	−	0.972076i	$-0.575400\pi$
$31$	−2.61314	−0.469334	−0.234667	+	0.972076i	$0.575400\pi$
$32$	1.00000	0.176777
$33$	−2.29353	−0.399252
$34$	3.68899	0.632657
$35$	1.92012	0.324559
$36$	1.00000	0.166667
$37$	−7.35359	−1.20892	−0.604462	−	0.796634i	$-0.706612\pi$
$37$	−7.35359	−1.20892	−0.604462	+	0.796634i	$0.706612\pi$
$38$	0.461913	0.0749322
$39$	2.16373	0.346474
$40$	4.15688	0.657261
$41$	1.98532	0.310056	0.155028	−	0.987910i	$-0.450453\pi$
$41$	1.98532	0.310056	0.155028	+	0.987910i	$0.450453\pi$
$42$	−0.461913	−0.0712747
$43$	−4.73656	−0.722319	−0.361160	−	0.932504i	$-0.617619\pi$
$43$	−4.73656	−0.722319	−0.361160	+	0.932504i	$0.617619\pi$
$44$	2.29353	0.345762
$45$	4.15688	0.619672
$46$	1.00000	0.147442
$47$	12.4440	1.81514	0.907568	−	0.419904i	$-0.137936\pi$
$47$	12.4440	1.81514	0.907568	+	0.419904i	$0.137936\pi$
$48$	−1.00000	−0.144338
$49$	−6.78664	−0.969520
$50$	12.2797	1.73661
$51$	−3.68899	−0.516563
$52$	−2.16373	−0.300055
$53$	−7.91682	−1.08746	−0.543730	−	0.839260i	$-0.682988\pi$
$53$	−7.91682	−1.08746	−0.543730	+	0.839260i	$0.682988\pi$
$54$	−1.00000	−0.136083
$55$	9.53392	1.28555
$56$	0.461913	0.0617257
$57$	−0.461913	−0.0611819
$58$	1.00000	0.131306
$59$	13.9439	1.81534	0.907671	−	0.419682i	$-0.137858\pi$
$59$	13.9439	1.81534	0.907671	+	0.419682i	$0.137858\pi$
$60$	−4.15688	−0.536651
$61$	7.56872	0.969076	0.484538	−	0.874770i	$-0.338988\pi$
$61$	7.56872	0.969076	0.484538	+	0.874770i	$0.338988\pi$
$62$	−2.61314	−0.331870
$63$	0.461913	0.0581955
$64$	1.00000	0.125000
$65$	−8.99437	−1.11561
$66$	−2.29353	−0.282314
$67$	−1.07490	−0.131320	−0.0656600	−	0.997842i	$-0.520915\pi$
$67$	−1.07490	−0.131320	−0.0656600	+	0.997842i	$0.520915\pi$
$68$	3.68899	0.447356
$69$	−1.00000	−0.120386
$70$	1.92012	0.229498
$71$	−7.96922	−0.945772	−0.472886	−	0.881124i	$-0.656788\pi$
$71$	−7.96922	−0.945772	−0.472886	+	0.881124i	$0.656788\pi$
$72$	1.00000	0.117851
$73$	14.6021	1.70905	0.854524	−	0.519412i	$-0.173849\pi$
$73$	14.6021	1.70905	0.854524	+	0.519412i	$0.173849\pi$
$74$	−7.35359	−0.854838
$75$	−12.2797	−1.41793
$76$	0.461913	0.0529850
$77$	1.05941	0.120731
$78$	2.16373	0.244994
$79$	−16.1867	−1.82114	−0.910571	−	0.413353i	$-0.864357\pi$
$79$	−16.1867	−1.82114	−0.910571	+	0.413353i	$0.864357\pi$
$80$	4.15688	0.464754
$81$	1.00000	0.111111
$82$	1.98532	0.219242
$83$	1.22708	0.134690	0.0673448	−	0.997730i	$-0.478547\pi$
$83$	1.22708	0.134690	0.0673448	+	0.997730i	$0.478547\pi$
$84$	−0.461913	−0.0503988
$85$	15.3347	1.66328
$86$	−4.73656	−0.510757
$87$	−1.00000	−0.107211
$88$	2.29353	0.244491
$89$	1.14905	0.121799	0.0608997	−	0.998144i	$-0.480603\pi$
$89$	1.14905	0.121799	0.0608997	+	0.998144i	$0.480603\pi$
$90$	4.15688	0.438174
$91$	−0.999454	−0.104771
$92$	1.00000	0.104257
$93$	2.61314	0.270970
$94$	12.4440	1.28350
$95$	1.92012	0.197000
$96$	−1.00000	−0.102062
$97$	−13.6303	−1.38394	−0.691972	−	0.721924i	$-0.743258\pi$
$97$	−13.6303	−1.38394	−0.691972	+	0.721924i	$0.743258\pi$
$98$	−6.78664	−0.685554
$99$	2.29353	0.230508
$100$	12.2797	1.22797
$101$	7.27014	0.723406	0.361703	−	0.932293i	$-0.382195\pi$
$101$	7.27014	0.723406	0.361703	+	0.932293i	$0.382195\pi$
$102$	−3.68899	−0.365265
$103$	−9.69167	−0.954949	−0.477475	−	0.878646i	$-0.658448\pi$
$103$	−9.69167	−0.954949	−0.477475	+	0.878646i	$0.658448\pi$
$104$	−2.16373	−0.212171
$105$	−1.92012	−0.187384
$106$	−7.91682	−0.768950
$107$	0.728742	0.0704502	0.0352251	−	0.999379i	$-0.488785\pi$
$107$	0.728742	0.0704502	0.0352251	+	0.999379i	$0.488785\pi$
$108$	−1.00000	−0.0962250
$109$	4.33632	0.415344	0.207672	−	0.978199i	$-0.433411\pi$
$109$	4.33632	0.415344	0.207672	+	0.978199i	$0.433411\pi$
$110$	9.53392	0.909024
$111$	7.35359	0.697972
$112$	0.461913	0.0436467
$113$	10.0087	0.941535	0.470768	−	0.882257i	$-0.343977\pi$
$113$	10.0087	0.941535	0.470768	+	0.882257i	$0.343977\pi$
$114$	−0.461913	−0.0432621
$115$	4.15688	0.387631
$116$	1.00000	0.0928477
$117$	−2.16373	−0.200037
$118$	13.9439	1.28364
$119$	1.70399	0.156205
$120$	−4.15688	−0.379470
$121$	−5.73974	−0.521794
$122$	7.56872	0.685240
$123$	−1.98532	−0.179011
$124$	−2.61314	−0.234667
$125$	30.2608	2.70661
$126$	0.461913	0.0411505
$127$	−13.1060	−1.16297	−0.581486	−	0.813557i	$-0.697528\pi$
$127$	−13.1060	−1.16297	−0.581486	+	0.813557i	$0.697528\pi$
$128$	1.00000	0.0883883
$129$	4.73656	0.417031
$130$	−8.99437	−0.788858
$131$	−19.0985	−1.66865	−0.834323	−	0.551275i	$-0.814141\pi$
$131$	−19.0985	−1.66865	−0.834323	+	0.551275i	$0.814141\pi$
$132$	−2.29353	−0.199626
$133$	0.213363	0.0185010
$134$	−1.07490	−0.0928573
$135$	−4.15688	−0.357768
$136$	3.68899	0.316329
$137$	20.0123	1.70977	0.854883	−	0.518821i	$-0.173629\pi$
$137$	20.0123	1.70977	0.854883	+	0.518821i	$0.173629\pi$
$138$	−1.00000	−0.0851257
$139$	−21.7534	−1.84510	−0.922550	−	0.385876i	$-0.873899\pi$
$139$	−21.7534	−1.84510	−0.922550	+	0.385876i	$0.873899\pi$
$140$	1.92012	0.162280
$141$	−12.4440	−1.04797
$142$	−7.96922	−0.668762
$143$	−4.96257	−0.414991
$144$	1.00000	0.0833333
$145$	4.15688	0.345210
$146$	14.6021	1.20848
$147$	6.78664	0.559752
$148$	−7.35359	−0.604462
$149$	−1.08290	−0.0887144	−0.0443572	−	0.999016i	$-0.514124\pi$
$149$	−1.08290	−0.0887144	−0.0443572	+	0.999016i	$0.514124\pi$
$150$	−12.2797	−1.00263
$151$	−9.66170	−0.786258	−0.393129	−	0.919483i	$-0.628607\pi$
$151$	−9.66170	−0.786258	−0.393129	+	0.919483i	$0.628607\pi$
$152$	0.461913	0.0374661
$153$	3.68899	0.298238
$154$	1.05941	0.0853696
$155$	−10.8625	−0.872500
$156$	2.16373	0.173237
$157$	2.14945	0.171545	0.0857726	−	0.996315i	$-0.472664\pi$
$157$	2.14945	0.171545	0.0857726	+	0.996315i	$0.472664\pi$
$158$	−16.1867	−1.28774
$159$	7.91682	0.627845
$160$	4.15688	0.328630
$161$	0.461913	0.0364038
$162$	1.00000	0.0785674
$163$	4.33739	0.339730	0.169865	−	0.985467i	$-0.445667\pi$
$163$	4.33739	0.339730	0.169865	+	0.985467i	$0.445667\pi$
$164$	1.98532	0.155028
$165$	−9.53392	−0.742215
$166$	1.22708	0.0952400
$167$	−17.8394	−1.38045	−0.690226	−	0.723594i	$-0.742489\pi$
$167$	−17.8394	−1.38045	−0.690226	+	0.723594i	$0.742489\pi$
$168$	−0.461913	−0.0356374
$169$	−8.31828	−0.639867
$170$	15.3347	1.17612
$171$	0.461913	0.0353234
$172$	−4.73656	−0.361160
$173$	19.1513	1.45604	0.728021	−	0.685554i	$-0.240440\pi$
$173$	19.1513	1.45604	0.728021	+	0.685554i	$0.240440\pi$
$174$	−1.00000	−0.0758098
$175$	5.67214	0.428774
$176$	2.29353	0.172881
$177$	−13.9439	−1.04809
$178$	1.14905	0.0861252
$179$	18.1871	1.35936	0.679682	−	0.733507i	$-0.262118\pi$
$179$	18.1871	1.35936	0.679682	+	0.733507i	$0.262118\pi$
$180$	4.15688	0.309836
$181$	6.09503	0.453040	0.226520	−	0.974007i	$-0.427265\pi$
$181$	6.09503	0.453040	0.226520	+	0.974007i	$0.427265\pi$
$182$	−0.999454	−0.0740845
$183$	−7.56872	−0.559496
$184$	1.00000	0.0737210
$185$	−30.5680	−2.24741
$186$	2.61314	0.191605
$187$	8.46081	0.618715
$188$	12.4440	0.907568
$189$	−0.461913	−0.0335992
$190$	1.92012	0.139300
$191$	−20.6701	−1.49564	−0.747818	−	0.663904i	$-0.768898\pi$
$191$	−20.6701	−1.49564	−0.747818	+	0.663904i	$0.768898\pi$
$192$	−1.00000	−0.0721688
$193$	23.3525	1.68095	0.840474	−	0.541852i	$-0.182277\pi$
$193$	23.3525	1.68095	0.840474	+	0.541852i	$0.182277\pi$
$194$	−13.6303	−0.978596
$195$	8.99437	0.644100
$196$	−6.78664	−0.484760
$197$	−15.5939	−1.11102	−0.555508	−	0.831511i	$-0.687476\pi$
$197$	−15.5939	−1.11102	−0.555508	+	0.831511i	$0.687476\pi$
$198$	2.29353	0.162994
$199$	2.44620	0.173406	0.0867032	−	0.996234i	$-0.472367\pi$
$199$	2.44620	0.173406	0.0867032	+	0.996234i	$0.472367\pi$
$200$	12.2797	0.868304
$201$	1.07490	0.0758177
$202$	7.27014	0.511525
$203$	0.461913	0.0324199
$204$	−3.68899	−0.258281
$205$	8.25276	0.576398
$206$	−9.69167	−0.675251
$207$	1.00000	0.0695048
$208$	−2.16373	−0.150028
$209$	1.05941	0.0732809
$210$	−1.92012	−0.132501
$211$	−17.3972	−1.19767	−0.598835	−	0.800872i	$-0.704370\pi$
$211$	−17.3972	−1.19767	−0.598835	+	0.800872i	$0.704370\pi$
$212$	−7.91682	−0.543730
$213$	7.96922	0.546042
$214$	0.728742	0.0498158
$215$	−19.6893	−1.34280
$216$	−1.00000	−0.0680414
$217$	−1.20704	−0.0819395
$218$	4.33632	0.293692
$219$	−14.6021	−0.986719
$220$	9.53392	0.642777
$221$	−7.98198	−0.536926
$222$	7.35359	0.493541
$223$	−26.5315	−1.77668	−0.888342	−	0.459183i	$-0.848142\pi$
$223$	−26.5315	−1.77668	−0.888342	+	0.459183i	$0.848142\pi$
$224$	0.461913	0.0308629
$225$	12.2797	0.818645
$226$	10.0087	0.665766
$227$	15.5943	1.03503	0.517515	−	0.855674i	$-0.326857\pi$
$227$	15.5943	1.03503	0.517515	+	0.855674i	$0.326857\pi$
$228$	−0.461913	−0.0305909
$229$	24.3201	1.60712	0.803558	−	0.595226i	$-0.202937\pi$
$229$	24.3201	1.60712	0.803558	+	0.595226i	$0.202937\pi$
$230$	4.15688	0.274097
$231$	−1.05941	−0.0697040
$232$	1.00000	0.0656532
$233$	24.3411	1.59464	0.797318	−	0.603559i	$-0.206251\pi$
$233$	24.3411	1.59464	0.797318	+	0.603559i	$0.206251\pi$
$234$	−2.16373	−0.141447
$235$	51.7281	3.37437
$236$	13.9439	0.907671
$237$	16.1867	1.05144
$238$	1.70399	0.110454
$239$	22.5843	1.46086	0.730430	−	0.682988i	$-0.239320\pi$
$239$	22.5843	1.46086	0.730430	+	0.682988i	$0.239320\pi$
$240$	−4.15688	−0.268326
$241$	29.1961	1.88068	0.940342	−	0.340232i	$-0.110506\pi$
$241$	29.1961	1.88068	0.940342	+	0.340232i	$0.110506\pi$
$242$	−5.73974	−0.368964
$243$	−1.00000	−0.0641500
$244$	7.56872	0.484538
$245$	−28.2113	−1.80235
$246$	−1.98532	−0.126580
$247$	−0.999454	−0.0635938
$248$	−2.61314	−0.165935
$249$	−1.22708	−0.0777631
$250$	30.2608	1.91386
$251$	−15.7167	−0.992027	−0.496014	−	0.868315i	$-0.665203\pi$
$251$	−15.7167	−0.992027	−0.496014	+	0.868315i	$0.665203\pi$
$252$	0.461913	0.0290978
$253$	2.29353	0.144193
$254$	−13.1060	−0.822345
$255$	−15.3347	−0.960297
$256$	1.00000	0.0625000
$257$	16.1079	1.00478	0.502391	−	0.864640i	$-0.332454\pi$
$257$	16.1079	1.00478	0.502391	+	0.864640i	$0.332454\pi$
$258$	4.73656	0.294886
$259$	−3.39672	−0.211062
$260$	−8.99437	−0.557807
$261$	1.00000	0.0618984
$262$	−19.0985	−1.17991
$263$	20.2205	1.24685	0.623424	−	0.781884i	$-0.285741\pi$
$263$	20.2205	1.24685	0.623424	+	0.781884i	$0.285741\pi$
$264$	−2.29353	−0.141157
$265$	−32.9093	−2.02160
$266$	0.213363	0.0130822
$267$	−1.14905	−0.0703209
$268$	−1.07490	−0.0656600
$269$	7.47540	0.455783	0.227891	−	0.973687i	$-0.426817\pi$
$269$	7.47540	0.455783	0.227891	+	0.973687i	$0.426817\pi$
$270$	−4.15688	−0.252980
$271$	−12.5168	−0.760339	−0.380169	−	0.924917i	$-0.624134\pi$
$271$	−12.5168	−0.760339	−0.380169	+	0.924917i	$0.624134\pi$
$272$	3.68899	0.223678
$273$	0.999454	0.0604897
$274$	20.0123	1.20899
$275$	28.1638	1.69834
$276$	−1.00000	−0.0601929
$277$	−7.21591	−0.433562	−0.216781	−	0.976220i	$-0.569556\pi$
$277$	−7.21591	−0.433562	−0.216781	+	0.976220i	$0.569556\pi$
$278$	−21.7534	−1.30468
$279$	−2.61314	−0.156445
$280$	1.92012	0.114749
$281$	12.3483	0.736636	0.368318	−	0.929700i	$-0.379934\pi$
$281$	12.3483	0.736636	0.368318	+	0.929700i	$0.379934\pi$
$282$	−12.4440	−0.741027
$283$	12.3915	0.736599	0.368300	−	0.929707i	$-0.379940\pi$
$283$	12.3915	0.736599	0.368300	+	0.929707i	$0.379940\pi$
$284$	−7.96922	−0.472886
$285$	−1.92012	−0.113738
$286$	−4.96257	−0.293443
$287$	0.917047	0.0541316
$288$	1.00000	0.0589256
$289$	−3.39132	−0.199490
$290$	4.15688	0.244101
$291$	13.6303	0.799020
$292$	14.6021	0.854524
$293$	−26.6792	−1.55862	−0.779309	−	0.626640i	$-0.784430\pi$
$293$	−26.6792	−1.55862	−0.779309	+	0.626640i	$0.784430\pi$
$294$	6.78664	0.395805
$295$	57.9632	3.37475
$296$	−7.35359	−0.427419
$297$	−2.29353	−0.133084
$298$	−1.08290	−0.0627305
$299$	−2.16373	−0.125132
$300$	−12.2797	−0.708967
$301$	−2.18788	−0.126107
$302$	−9.66170	−0.555968
$303$	−7.27014	−0.417659
$304$	0.461913	0.0264925
$305$	31.4623	1.80153
$306$	3.68899	0.210886
$307$	18.2765	1.04310	0.521548	−	0.853222i	$-0.325355\pi$
$307$	18.2765	1.04310	0.521548	+	0.853222i	$0.325355\pi$
$308$	1.05941	0.0603655
$309$	9.69167	0.551340
$310$	−10.8625	−0.616950
$311$	−5.47998	−0.310741	−0.155371	−	0.987856i	$-0.549657\pi$
$311$	−5.47998	−0.310741	−0.155371	+	0.987856i	$0.549657\pi$
$312$	2.16373	0.122497
$313$	9.94247	0.561982	0.280991	−	0.959710i	$-0.409337\pi$
$313$	9.94247	0.561982	0.280991	+	0.959710i	$0.409337\pi$
$314$	2.14945	0.121301
$315$	1.92012	0.108186
$316$	−16.1867	−0.910571
$317$	−0.528602	−0.0296893	−0.0148446	−	0.999890i	$-0.504725\pi$
$317$	−0.528602	−0.0296893	−0.0148446	+	0.999890i	$0.504725\pi$
$318$	7.91682	0.443953
$319$	2.29353	0.128413
$320$	4.15688	0.232377
$321$	−0.728742	−0.0406744
$322$	0.461913	0.0257414
$323$	1.70399	0.0948128
$324$	1.00000	0.0555556
$325$	−26.5699	−1.47383
$326$	4.33739	0.240226
$327$	−4.33632	−0.239799
$328$	1.98532	0.109621
$329$	5.74802	0.316899
$330$	−9.53392	−0.524825
$331$	18.7648	1.03140	0.515702	−	0.856768i	$-0.327531\pi$
$331$	18.7648	1.03140	0.515702	+	0.856768i	$0.327531\pi$
$332$	1.22708	0.0673448
$333$	−7.35359	−0.402974
$334$	−17.8394	−0.976127
$335$	−4.46824	−0.244126
$336$	−0.461913	−0.0251994
$337$	−14.8937	−0.811313	−0.405657	−	0.914025i	$-0.632957\pi$
$337$	−14.8937	−0.811313	−0.405657	+	0.914025i	$0.632957\pi$
$338$	−8.31828	−0.452455
$339$	−10.0087	−0.543596
$340$	15.3347	0.831642
$341$	−5.99331	−0.324556
$342$	0.461913	0.0249774
$343$	−6.36822	−0.343852
$344$	−4.73656	−0.255378
$345$	−4.15688	−0.223799
$346$	19.1513	1.02958
$347$	26.8489	1.44132	0.720662	−	0.693287i	$-0.243838\pi$
$347$	26.8489	1.44132	0.720662	+	0.693287i	$0.243838\pi$
$348$	−1.00000	−0.0536056
$349$	−1.95595	−0.104699	−0.0523497	−	0.998629i	$-0.516671\pi$
$349$	−1.95595	−0.104699	−0.0523497	+	0.998629i	$0.516671\pi$
$350$	5.67214	0.303189
$351$	2.16373	0.115491
$352$	2.29353	0.122245
$353$	−16.3310	−0.869210	−0.434605	−	0.900621i	$-0.643112\pi$
$353$	−16.3310	−0.869210	−0.434605	+	0.900621i	$0.643112\pi$
$354$	−13.9439	−0.741110
$355$	−33.1271	−1.75820
$356$	1.14905	0.0608997
$357$	−1.70399	−0.0901849
$358$	18.1871	0.961216
$359$	−6.59012	−0.347813	−0.173907	−	0.984762i	$-0.555639\pi$
$359$	−6.59012	−0.347813	−0.173907	+	0.984762i	$0.555639\pi$
$360$	4.15688	0.219087
$361$	−18.7866	−0.988770
$362$	6.09503	0.320347
$363$	5.73974	0.301258
$364$	−0.999454	−0.0523856
$365$	60.6993	3.17714
$366$	−7.56872	−0.395623
$367$	18.8635	0.984668	0.492334	−	0.870406i	$-0.336144\pi$
$367$	18.8635	0.984668	0.492334	+	0.870406i	$0.336144\pi$
$368$	1.00000	0.0521286
$369$	1.98532	0.103352
$370$	−30.5680	−1.58916
$371$	−3.65688	−0.189856
$372$	2.61314	0.135485
$373$	2.94634	0.152555	0.0762777	−	0.997087i	$-0.475696\pi$
$373$	2.94634	0.152555	0.0762777	+	0.997087i	$0.475696\pi$
$374$	8.46081	0.437498
$375$	−30.2608	−1.56266
$376$	12.4440	0.641748
$377$	−2.16373	−0.111438
$378$	−0.461913	−0.0237582
$379$	29.8372	1.53264	0.766318	−	0.642461i	$-0.222087\pi$
$379$	29.8372	1.53264	0.766318	+	0.642461i	$0.222087\pi$
$380$	1.92012	0.0985000
$381$	13.1060	0.671442
$382$	−20.6701	−1.05757
$383$	−31.8983	−1.62993	−0.814964	−	0.579511i	$-0.803243\pi$
$383$	−31.8983	−1.62993	−0.814964	+	0.579511i	$0.803243\pi$
$384$	−1.00000	−0.0510310
$385$	4.40384	0.224441
$386$	23.3525	1.18861
$387$	−4.73656	−0.240773
$388$	−13.6303	−0.691972
$389$	1.87244	0.0949367	0.0474684	−	0.998873i	$-0.484885\pi$
$389$	1.87244	0.0949367	0.0474684	+	0.998873i	$0.484885\pi$
$390$	8.99437	0.455448
$391$	3.68899	0.186560
$392$	−6.78664	−0.342777
$393$	19.0985	0.963394
$394$	−15.5939	−0.785607
$395$	−67.2861	−3.38553
$396$	2.29353	0.115254
$397$	−31.7326	−1.59262	−0.796308	−	0.604892i	$-0.793216\pi$
$397$	−31.7326	−1.59262	−0.796308	+	0.604892i	$0.793216\pi$
$398$	2.44620	0.122617
$399$	−0.213363	−0.0106815
$400$	12.2797	0.613984
$401$	−25.1940	−1.25813	−0.629063	−	0.777354i	$-0.716561\pi$
$401$	−25.1940	−1.25813	−0.629063	+	0.777354i	$0.716561\pi$
$402$	1.07490	0.0536112
$403$	5.65413	0.281653
$404$	7.27014	0.361703
$405$	4.15688	0.206557
$406$	0.461913	0.0229243
$407$	−16.8657	−0.836000
$408$	−3.68899	−0.182632
$409$	−36.3432	−1.79705	−0.898527	−	0.438918i	$-0.855362\pi$
$409$	−36.3432	−1.79705	−0.898527	+	0.438918i	$0.855362\pi$
$410$	8.25276	0.407575
$411$	−20.0123	−0.987134
$412$	−9.69167	−0.477475
$413$	6.44087	0.316935
$414$	1.00000	0.0491473
$415$	5.10083	0.250390
$416$	−2.16373	−0.106086
$417$	21.7534	1.06527
$418$	1.05941	0.0518174
$419$	−23.5521	−1.15060	−0.575298	−	0.817944i	$-0.695114\pi$
$419$	−23.5521	−1.15060	−0.575298	+	0.817944i	$0.695114\pi$
$420$	−1.92012	−0.0936922
$421$	−27.8590	−1.35777	−0.678883	−	0.734246i	$-0.737536\pi$
$421$	−27.8590	−1.35777	−0.678883	+	0.734246i	$0.737536\pi$
$422$	−17.3972	−0.846881
$423$	12.4440	0.605046
$424$	−7.91682	−0.384475
$425$	45.2997	2.19736
$426$	7.96922	0.386110
$427$	3.49609	0.169188
$428$	0.728742	0.0352251
$429$	4.96257	0.239595
$430$	−19.6893	−0.949504
$431$	−18.7957	−0.905355	−0.452678	−	0.891674i	$-0.649531\pi$
$431$	−18.7957	−0.905355	−0.452678	+	0.891674i	$0.649531\pi$
$432$	−1.00000	−0.0481125
$433$	5.89718	0.283400	0.141700	−	0.989910i	$-0.454743\pi$
$433$	5.89718	0.283400	0.141700	+	0.989910i	$0.454743\pi$
$434$	−1.20704	−0.0579400
$435$	−4.15688	−0.199307
$436$	4.33632	0.207672
$437$	0.461913	0.0220963
$438$	−14.6021	−0.697716
$439$	−1.21719	−0.0580931	−0.0290466	−	0.999578i	$-0.509247\pi$
$439$	−1.21719	−0.0580931	−0.0290466	+	0.999578i	$0.509247\pi$
$440$	9.53392	0.454512
$441$	−6.78664	−0.323173
$442$	−7.98198	−0.379664
$443$	−0.190584	−0.00905492	−0.00452746	−	0.999990i	$-0.501441\pi$
$443$	−0.190584	−0.00905492	−0.00452746	+	0.999990i	$0.501441\pi$
$444$	7.35359	0.348986
$445$	4.77648	0.226427
$446$	−26.5315	−1.25630
$447$	1.08290	0.0512193
$448$	0.461913	0.0218233
$449$	20.2172	0.954110	0.477055	−	0.878874i	$-0.341704\pi$
$449$	20.2172	0.954110	0.477055	+	0.878874i	$0.341704\pi$
$450$	12.2797	0.578870
$451$	4.55339	0.214411
$452$	10.0087	0.470768
$453$	9.66170	0.453946
$454$	15.5943	0.731877
$455$	−4.15461	−0.194771
$456$	−0.461913	−0.0216311
$457$	−15.9356	−0.745436	−0.372718	−	0.927945i	$-0.621574\pi$
$457$	−15.9356	−0.745436	−0.372718	+	0.927945i	$0.621574\pi$
$458$	24.3201	1.13640
$459$	−3.68899	−0.172188
$460$	4.15688	0.193816
$461$	−25.9025	−1.20640	−0.603200	−	0.797590i	$-0.706108\pi$
$461$	−25.9025	−1.20640	−0.603200	+	0.797590i	$0.706108\pi$
$462$	−1.05941	−0.0492882
$463$	9.65635	0.448768	0.224384	−	0.974501i	$-0.427963\pi$
$463$	9.65635	0.448768	0.224384	+	0.974501i	$0.427963\pi$
$464$	1.00000	0.0464238
$465$	10.8625	0.503738
$466$	24.3411	1.12758
$467$	6.92345	0.320379	0.160189	−	0.987086i	$-0.448789\pi$
$467$	6.92345	0.320379	0.160189	+	0.987086i	$0.448789\pi$
$468$	−2.16373	−0.100018
$469$	−0.496511	−0.0229267
$470$	51.7281	2.38604
$471$	−2.14945	−0.0990417
$472$	13.9439	0.641820
$473$	−10.8634	−0.499501
$474$	16.1867	0.743478
$475$	5.67214	0.260256
$476$	1.70399	0.0781024
$477$	−7.91682	−0.362486
$478$	22.5843	1.03298
$479$	−1.79876	−0.0821872	−0.0410936	−	0.999155i	$-0.513084\pi$
$479$	−1.79876	−0.0821872	−0.0410936	+	0.999155i	$0.513084\pi$
$480$	−4.15688	−0.189735
$481$	15.9112	0.725488
$482$	29.1961	1.32984
$483$	−0.461913	−0.0210178
$484$	−5.73974	−0.260897
$485$	−56.6594	−2.57277
$486$	−1.00000	−0.0453609
$487$	−34.3279	−1.55554	−0.777772	−	0.628546i	$-0.783650\pi$
$487$	−34.3279	−1.55554	−0.777772	+	0.628546i	$0.783650\pi$
$488$	7.56872	0.342620
$489$	−4.33739	−0.196143
$490$	−28.2113	−1.27445
$491$	15.8749	0.716422	0.358211	−	0.933641i	$-0.383387\pi$
$491$	15.8749	0.716422	0.358211	+	0.933641i	$0.383387\pi$
$492$	−1.98532	−0.0895053
$493$	3.68899	0.166144
$494$	−0.999454	−0.0449676
$495$	9.53392	0.428518
$496$	−2.61314	−0.117334
$497$	−3.68108	−0.165119
$498$	−1.22708	−0.0549868
$499$	8.06965	0.361247	0.180624	−	0.983552i	$-0.442188\pi$
$499$	8.06965	0.361247	0.180624	+	0.983552i	$0.442188\pi$
$500$	30.2608	1.35330
$501$	17.8394	0.797004
$502$	−15.7167	−0.701469
$503$	8.91440	0.397473	0.198737	−	0.980053i	$-0.436316\pi$
$503$	8.91440	0.397473	0.198737	+	0.980053i	$0.436316\pi$
$504$	0.461913	0.0205752
$505$	30.2211	1.34482
$506$	2.29353	0.101960
$507$	8.31828	0.369428
$508$	−13.1060	−0.581486
$509$	−12.0368	−0.533523	−0.266762	−	0.963763i	$-0.585954\pi$
$509$	−12.0368	−0.533523	−0.266762	+	0.963763i	$0.585954\pi$
$510$	−15.3347	−0.679033
$511$	6.74490	0.298377
$512$	1.00000	0.0441942
$513$	−0.461913	−0.0203940
$514$	16.1079	0.710488
$515$	−40.2872	−1.77526
$516$	4.73656	0.208516
$517$	28.5405	1.25521
$518$	−3.39672	−0.149243
$519$	−19.1513	−0.840647
$520$	−8.99437	−0.394429
$521$	−33.7157	−1.47711	−0.738556	−	0.674193i	$-0.764492\pi$
$521$	−33.7157	−1.47711	−0.738556	+	0.674193i	$0.764492\pi$
$522$	1.00000	0.0437688
$523$	−19.1624	−0.837915	−0.418958	−	0.908006i	$-0.637604\pi$
$523$	−19.1624	−0.837915	−0.418958	+	0.908006i	$0.637604\pi$
$524$	−19.0985	−0.834323
$525$	−5.67214	−0.247553
$526$	20.2205	0.881655
$527$	−9.63987	−0.419919
$528$	−2.29353	−0.0998129
$529$	1.00000	0.0434783
$530$	−32.9093	−1.42949
$531$	13.9439	0.605114
$532$	0.213363	0.00925048
$533$	−4.29570	−0.186068
$534$	−1.14905	−0.0497244
$535$	3.02930	0.130968
$536$	−1.07490	−0.0464286
$537$	−18.1871	−0.784830
$538$	7.47540	0.322287
$539$	−15.5653	−0.670446
$540$	−4.15688	−0.178884
$541$	39.0286	1.67797	0.838984	−	0.544156i	$-0.183150\pi$
$541$	39.0286	1.67797	0.838984	+	0.544156i	$0.183150\pi$
$542$	−12.5168	−0.537641
$543$	−6.09503	−0.261563
$544$	3.68899	0.158164
$545$	18.0256	0.772130
$546$	0.999454	0.0427727
$547$	3.89417	0.166503	0.0832514	−	0.996529i	$-0.473470\pi$
$547$	3.89417	0.166503	0.0832514	+	0.996529i	$0.473470\pi$
$548$	20.0123	0.854883
$549$	7.56872	0.323025
$550$	28.1638	1.20091
$551$	0.461913	0.0196781
$552$	−1.00000	−0.0425628
$553$	−7.47683	−0.317947
$554$	−7.21591	−0.306575
$555$	30.5680	1.29754
$556$	−21.7534	−0.922550
$557$	−28.5076	−1.20791	−0.603953	−	0.797020i	$-0.706408\pi$
$557$	−28.5076	−1.20791	−0.603953	+	0.797020i	$0.706408\pi$
$558$	−2.61314	−0.110623
$559$	10.2486	0.433471
$560$	1.92012	0.0811398
$561$	−8.46081	−0.357216
$562$	12.3483	0.520881
$563$	−14.2653	−0.601213	−0.300606	−	0.953748i	$-0.597189\pi$
$563$	−14.2653	−0.601213	−0.300606	+	0.953748i	$0.597189\pi$
$564$	−12.4440	−0.523985
$565$	41.6048	1.75033
$566$	12.3915	0.520854
$567$	0.461913	0.0193985
$568$	−7.96922	−0.334381
$569$	−1.33127	−0.0558097	−0.0279049	−	0.999611i	$-0.508884\pi$
$569$	−1.33127	−0.0558097	−0.0279049	+	0.999611i	$0.508884\pi$
$570$	−1.92012	−0.0804249
$571$	−41.8433	−1.75109	−0.875544	−	0.483138i	$-0.839497\pi$
$571$	−41.8433	−1.75109	−0.875544	+	0.483138i	$0.839497\pi$
$572$	−4.96257	−0.207495
$573$	20.6701	0.863506
$574$	0.917047	0.0382768
$575$	12.2797	0.512098
$576$	1.00000	0.0416667
$577$	32.1496	1.33840	0.669202	−	0.743080i	$-0.266636\pi$
$577$	32.1496	1.33840	0.669202	+	0.743080i	$0.266636\pi$
$578$	−3.39132	−0.141060
$579$	−23.3525	−0.970496
$580$	4.15688	0.172605
$581$	0.566805	0.0235150
$582$	13.6303	0.564993
$583$	−18.1574	−0.752004
$584$	14.6021	0.604240
$585$	−8.99437	−0.371871
$586$	−26.6792	−1.10211
$587$	−41.3320	−1.70595	−0.852976	−	0.521949i	$-0.825205\pi$
$587$	−41.3320	−1.70595	−0.852976	+	0.521949i	$0.825205\pi$
$588$	6.78664	0.279876
$589$	−1.20704	−0.0497354
$590$	57.9632	2.38631
$591$	15.5939	0.641446
$592$	−7.35359	−0.302231
$593$	−16.6122	−0.682181	−0.341090	−	0.940031i	$-0.610796\pi$
$593$	−16.6122	−0.682181	−0.341090	+	0.940031i	$0.610796\pi$
$594$	−2.29353	−0.0941045
$595$	7.08330	0.290387
$596$	−1.08290	−0.0443572
$597$	−2.44620	−0.100116
$598$	−2.16373	−0.0884815
$599$	−30.4205	−1.24295	−0.621474	−	0.783435i	$-0.713466\pi$
$599$	−30.4205	−1.24295	−0.621474	+	0.783435i	$0.713466\pi$
$600$	−12.2797	−0.501316
$601$	−22.4176	−0.914433	−0.457216	−	0.889355i	$-0.651154\pi$
$601$	−22.4176	−0.914433	−0.457216	+	0.889355i	$0.651154\pi$
$602$	−2.18788	−0.0891713
$603$	−1.07490	−0.0437733
$604$	−9.66170	−0.393129
$605$	−23.8594	−0.970023
$606$	−7.27014	−0.295329
$607$	27.9891	1.13604	0.568021	−	0.823014i	$-0.307709\pi$
$607$	27.9891	1.13604	0.568021	+	0.823014i	$0.307709\pi$
$608$	0.461913	0.0187330
$609$	−0.461913	−0.0187177
$610$	31.4623	1.27387
$611$	−26.9253	−1.08928
$612$	3.68899	0.149119
$613$	−9.67901	−0.390931	−0.195466	−	0.980711i	$-0.562622\pi$
$613$	−9.67901	−0.390931	−0.195466	+	0.980711i	$0.562622\pi$
$614$	18.2765	0.737581
$615$	−8.25276	−0.332783
$616$	1.05941	0.0426848
$617$	−3.72668	−0.150031	−0.0750153	−	0.997182i	$-0.523901\pi$
$617$	−3.72668	−0.150031	−0.0750153	+	0.997182i	$0.523901\pi$
$618$	9.69167	0.389856
$619$	13.7364	0.552112	0.276056	−	0.961142i	$-0.410972\pi$
$619$	13.7364	0.552112	0.276056	+	0.961142i	$0.410972\pi$
$620$	−10.8625	−0.436250
$621$	−1.00000	−0.0401286
$622$	−5.47998	−0.219727
$623$	0.530762	0.0212645
$624$	2.16373	0.0866185
$625$	64.3921	2.57568
$626$	9.94247	0.397381
$627$	−1.05941	−0.0423087
$628$	2.14945	0.0857726
$629$	−27.1274	−1.08164
$630$	1.92012	0.0764993
$631$	7.66355	0.305081	0.152541	−	0.988297i	$-0.451255\pi$
$631$	7.66355	0.305081	0.152541	+	0.988297i	$0.451255\pi$
$632$	−16.1867	−0.643871
$633$	17.3972	0.691475
$634$	−0.528602	−0.0209935
$635$	−54.4802	−2.16198
$636$	7.91682	0.313922
$637$	14.6844	0.581819
$638$	2.29353	0.0908016
$639$	−7.96922	−0.315257
$640$	4.15688	0.164315
$641$	43.4274	1.71528	0.857639	−	0.514252i	$-0.171930\pi$
$641$	43.4274	1.71528	0.857639	+	0.514252i	$0.171930\pi$
$642$	−0.728742	−0.0287612
$643$	−42.1097	−1.66064	−0.830322	−	0.557284i	$-0.811843\pi$
$643$	−42.1097	−1.66064	−0.830322	+	0.557284i	$0.811843\pi$
$644$	0.461913	0.0182019
$645$	19.6893	0.775267
$646$	1.70399	0.0670427
$647$	10.4651	0.411426	0.205713	−	0.978612i	$-0.434049\pi$
$647$	10.4651	0.411426	0.205713	+	0.978612i	$0.434049\pi$
$648$	1.00000	0.0392837
$649$	31.9807	1.25535
$650$	−26.5699	−1.04216
$651$	1.20704	0.0473078
$652$	4.33739	0.169865
$653$	24.0670	0.941814	0.470907	−	0.882183i	$-0.343927\pi$
$653$	24.0670	0.941814	0.470907	+	0.882183i	$0.343927\pi$
$654$	−4.33632	−0.169563
$655$	−79.3904	−3.10204
$656$	1.98532	0.0775139
$657$	14.6021	0.569683
$658$	5.74802	0.224081
$659$	3.90566	0.152143	0.0760715	−	0.997102i	$-0.475762\pi$
$659$	3.90566	0.152143	0.0760715	+	0.997102i	$0.475762\pi$
$660$	−9.53392	−0.371107
$661$	26.2025	1.01916	0.509580	−	0.860423i	$-0.329801\pi$
$661$	26.2025	1.01916	0.509580	+	0.860423i	$0.329801\pi$
$662$	18.7648	0.729313
$663$	7.98198	0.309995
$664$	1.22708	0.0476200
$665$	0.886927	0.0343936
$666$	−7.35359	−0.284946
$667$	1.00000	0.0387202
$668$	−17.8394	−0.690226
$669$	26.5315	1.02577
$670$	−4.46824	−0.172623
$671$	17.3591	0.670139
$672$	−0.461913	−0.0178187
$673$	−22.0573	−0.850245	−0.425122	−	0.905136i	$-0.639769\pi$
$673$	−22.0573	−0.850245	−0.425122	+	0.905136i	$0.639769\pi$
$674$	−14.8937	−0.573685
$675$	−12.2797	−0.472645
$676$	−8.31828	−0.319934
$677$	−10.2724	−0.394800	−0.197400	−	0.980323i	$-0.563250\pi$
$677$	−10.2724	−0.394800	−0.197400	+	0.980323i	$0.563250\pi$
$678$	−10.0087	−0.384380
$679$	−6.29600	−0.241618
$680$	15.3347	0.588060
$681$	−15.5943	−0.597575
$682$	−5.99331	−0.229496
$683$	−5.59566	−0.214112	−0.107056	−	0.994253i	$-0.534142\pi$
$683$	−5.59566	−0.214112	−0.107056	+	0.994253i	$0.534142\pi$
$684$	0.461913	0.0176617
$685$	83.1888	3.17848
$686$	−6.36822	−0.243140
$687$	−24.3201	−0.927869
$688$	−4.73656	−0.180580
$689$	17.1299	0.652596
$690$	−4.15688	−0.158250
$691$	4.46860	0.169994	0.0849968	−	0.996381i	$-0.472912\pi$
$691$	4.46860	0.169994	0.0849968	+	0.996381i	$0.472912\pi$
$692$	19.1513	0.728021
$693$	1.05941	0.0402436
$694$	26.8489	1.01917
$695$	−90.4264	−3.43007
$696$	−1.00000	−0.0379049
$697$	7.32385	0.277411
$698$	−1.95595	−0.0740337
$699$	−24.3411	−0.920664
$700$	5.67214	0.214387
$701$	−14.5795	−0.550661	−0.275331	−	0.961350i	$-0.588787\pi$
$701$	−14.5795	−0.550661	−0.275331	+	0.961350i	$0.588787\pi$
$702$	2.16373	0.0816647
$703$	−3.39672	−0.128110
$704$	2.29353	0.0864405
$705$	−51.7281	−1.94819
$706$	−16.3310	−0.614624
$707$	3.35817	0.126297
$708$	−13.9439	−0.524044
$709$	−44.2346	−1.66127	−0.830633	−	0.556820i	$-0.812021\pi$
$709$	−44.2346	−1.66127	−0.830633	+	0.556820i	$0.812021\pi$
$710$	−33.1271	−1.24324
$711$	−16.1867	−0.607047
$712$	1.14905	0.0430626
$713$	−2.61314	−0.0978630
$714$	−1.70399	−0.0637704
$715$	−20.6288	−0.771474
$716$	18.1871	0.679682
$717$	−22.5843	−0.843428
$718$	−6.59012	−0.245941
$719$	−19.0329	−0.709808	−0.354904	−	0.934903i	$-0.615486\pi$
$719$	−19.0329	−0.709808	−0.354904	+	0.934903i	$0.615486\pi$
$720$	4.15688	0.154918
$721$	−4.47671	−0.166721
$722$	−18.7866	−0.699166
$723$	−29.1961	−1.08581
$724$	6.09503	0.226520
$725$	12.2797	0.456056
$726$	5.73974	0.213022
$727$	−0.437151	−0.0162130	−0.00810652	−	0.999967i	$-0.502580\pi$
$727$	−0.437151	−0.0162130	−0.00810652	+	0.999967i	$0.502580\pi$
$728$	−0.999454	−0.0370422
$729$	1.00000	0.0370370
$730$	60.6993	2.24658
$731$	−17.4732	−0.646268
$732$	−7.56872	−0.279748
$733$	−23.9362	−0.884104	−0.442052	−	0.896989i	$-0.645749\pi$
$733$	−23.9362	−0.884104	−0.442052	+	0.896989i	$0.645749\pi$
$734$	18.8635	0.696265
$735$	28.2113	1.04059
$736$	1.00000	0.0368605
$737$	−2.46531	−0.0908110
$738$	1.98532	0.0730808
$739$	32.8844	1.20967	0.604836	−	0.796350i	$-0.293239\pi$
$739$	32.8844	1.20967	0.604836	+	0.796350i	$0.293239\pi$
$740$	−30.5680	−1.12370
$741$	0.999454	0.0367159
$742$	−3.65688	−0.134248
$743$	51.0078	1.87129	0.935647	−	0.352937i	$-0.114817\pi$
$743$	51.0078	1.87129	0.935647	+	0.352937i	$0.114817\pi$
$744$	2.61314	0.0958025
$745$	−4.50148	−0.164921
$746$	2.94634	0.107873
$747$	1.22708	0.0448966
$748$	8.46081	0.309358
$749$	0.336615	0.0122997
$750$	−30.2608	−1.10497
$751$	−0.271820	−0.00991884	−0.00495942	−	0.999988i	$-0.501579\pi$
$751$	−0.271820	−0.00991884	−0.00495942	+	0.999988i	$0.501579\pi$
$752$	12.4440	0.453784
$753$	15.7167	0.572747
$754$	−2.16373	−0.0787984
$755$	−40.1625	−1.46166
$756$	−0.461913	−0.0167996
$757$	−13.8906	−0.504863	−0.252431	−	0.967615i	$-0.581230\pi$
$757$	−13.8906	−0.504863	−0.252431	+	0.967615i	$0.581230\pi$
$758$	29.8372	1.08374
$759$	−2.29353	−0.0832497
$760$	1.92012	0.0696500
$761$	49.1271	1.78086	0.890428	−	0.455123i	$-0.150405\pi$
$761$	49.1271	1.78086	0.890428	+	0.455123i	$0.150405\pi$
$762$	13.1060	0.474781
$763$	2.00300	0.0725135
$764$	−20.6701	−0.747818
$765$	15.3347	0.554428
$766$	−31.8983	−1.15253
$767$	−30.1708	−1.08941
$768$	−1.00000	−0.0360844
$769$	−5.52476	−0.199228	−0.0996140	−	0.995026i	$-0.531761\pi$
$769$	−5.52476	−0.199228	−0.0996140	+	0.995026i	$0.531761\pi$
$770$	4.40384	0.158703
$771$	−16.1079	−0.580111
$772$	23.3525	0.840474
$773$	−26.0924	−0.938479	−0.469240	−	0.883071i	$-0.655472\pi$
$773$	−26.0924	−0.938479	−0.469240	+	0.883071i	$0.655472\pi$
$774$	−4.73656	−0.170252
$775$	−32.0886	−1.15266
$776$	−13.6303	−0.489298
$777$	3.39672	0.121857
$778$	1.87244	0.0671304
$779$	0.917047	0.0328566
$780$	8.99437	0.322050
$781$	−18.2776	−0.654024
$782$	3.68899	0.131918
$783$	−1.00000	−0.0357371
$784$	−6.78664	−0.242380
$785$	8.93503	0.318905
$786$	19.0985	0.681222
$787$	23.4089	0.834438	0.417219	−	0.908806i	$-0.363005\pi$
$787$	23.4089	0.834438	0.417219	+	0.908806i	$0.363005\pi$
$788$	−15.5939	−0.555508
$789$	−20.2205	−0.719868
$790$	−67.2861	−2.39393
$791$	4.62313	0.164379
$792$	2.29353	0.0814969
$793$	−16.3767	−0.581552
$794$	−31.7326	−1.12615
$795$	32.9093	1.16717
$796$	2.44620	0.0867032
$797$	−10.2260	−0.362223	−0.181111	−	0.983463i	$-0.557969\pi$
$797$	−10.2260	−0.362223	−0.181111	+	0.983463i	$0.557969\pi$
$798$	−0.213363	−0.00755299
$799$	45.9057	1.62403
$800$	12.2797	0.434152
$801$	1.14905	0.0405998
$802$	−25.1940	−0.889629
$803$	33.4903	1.18185
$804$	1.07490	0.0379088
$805$	1.92012	0.0676753
$806$	5.65413	0.199158
$807$	−7.47540	−0.263146
$808$	7.27014	0.255763
$809$	2.71618	0.0954957	0.0477478	−	0.998859i	$-0.484796\pi$
$809$	2.71618	0.0954957	0.0477478	+	0.998859i	$0.484796\pi$
$810$	4.15688	0.146058
$811$	−5.21689	−0.183190	−0.0915950	−	0.995796i	$-0.529197\pi$
$811$	−5.21689	−0.183190	−0.0915950	+	0.995796i	$0.529197\pi$
$812$	0.461913	0.0162100
$813$	12.5168	0.438982
$814$	−16.8657	−0.591141
$815$	18.0300	0.631564
$816$	−3.68899	−0.129141
$817$	−2.18788	−0.0765442
$818$	−36.3432	−1.27071
$819$	−0.999454	−0.0349238
$820$	8.25276	0.288199
$821$	−29.5785	−1.03230	−0.516148	−	0.856500i	$-0.672634\pi$
$821$	−29.5785	−1.03230	−0.516148	+	0.856500i	$0.672634\pi$
$822$	−20.0123	−0.698009
$823$	−48.1796	−1.67943	−0.839717	−	0.543024i	$-0.817279\pi$
$823$	−48.1796	−1.67943	−0.839717	+	0.543024i	$0.817279\pi$
$824$	−9.69167	−0.337625
$825$	−28.1638	−0.980536
$826$	6.44087	0.224107
$827$	−10.1065	−0.351439	−0.175719	−	0.984440i	$-0.556225\pi$
$827$	−10.1065	−0.351439	−0.175719	+	0.984440i	$0.556225\pi$
$828$	1.00000	0.0347524
$829$	−2.23728	−0.0777040	−0.0388520	−	0.999245i	$-0.512370\pi$
$829$	−2.23728	−0.0777040	−0.0388520	+	0.999245i	$0.512370\pi$
$830$	5.10083	0.177053
$831$	7.21591	0.250317
$832$	−2.16373	−0.0750138
$833$	−25.0359	−0.867441
$834$	21.7534	0.753259
$835$	−74.1562	−2.56628
$836$	1.05941	0.0366404
$837$	2.61314	0.0903235
$838$	−23.5521	−0.813594
$839$	−35.3953	−1.22198	−0.610991	−	0.791638i	$-0.709229\pi$
$839$	−35.3953	−1.22198	−0.610991	+	0.791638i	$0.709229\pi$
$840$	−1.92012	−0.0662504
$841$	1.00000	0.0344828
$842$	−27.8590	−0.960086
$843$	−12.3483	−0.425297
$844$	−17.3972	−0.598835
$845$	−34.5781	−1.18952
$846$	12.4440	0.427832
$847$	−2.65126	−0.0910983
$848$	−7.91682	−0.271865
$849$	−12.3915	−0.425276
$850$	45.2997	1.55377
$851$	−7.35359	−0.252078
$852$	7.96922	0.273021
$853$	22.4439	0.768463	0.384232	−	0.923237i	$-0.374466\pi$
$853$	22.4439	0.768463	0.384232	+	0.923237i	$0.374466\pi$
$854$	3.49609	0.119634
$855$	1.92012	0.0656666
$856$	0.728742	0.0249079
$857$	−22.5813	−0.771362	−0.385681	−	0.922632i	$-0.626034\pi$
$857$	−22.5813	−0.771362	−0.385681	+	0.922632i	$0.626034\pi$
$858$	4.96257	0.169419
$859$	−32.5018	−1.10895	−0.554473	−	0.832201i	$-0.687080\pi$
$859$	−32.5018	−1.10895	−0.554473	+	0.832201i	$0.687080\pi$
$860$	−19.6893	−0.671401
$861$	−0.917047	−0.0312529
$862$	−18.7957	−0.640183
$863$	24.3498	0.828878	0.414439	−	0.910077i	$-0.363978\pi$
$863$	24.3498	0.828878	0.414439	+	0.910077i	$0.363978\pi$
$864$	−1.00000	−0.0340207
$865$	79.6095	2.70680
$866$	5.89718	0.200394
$867$	3.39132	0.115175
$868$	−1.20704	−0.0409698
$869$	−37.1245	−1.25936
$870$	−4.15688	−0.140932
$871$	2.32579	0.0788065
$872$	4.33632	0.146846
$873$	−13.6303	−0.461315
$874$	0.461913	0.0156244
$875$	13.9778	0.472537
$876$	−14.6021	−0.493360
$877$	−5.06049	−0.170880	−0.0854402	−	0.996343i	$-0.527230\pi$
$877$	−5.06049	−0.170880	−0.0854402	+	0.996343i	$0.527230\pi$
$878$	−1.21719	−0.0410780
$879$	26.6792	0.899869
$880$	9.53392	0.321388
$881$	−34.5655	−1.16454	−0.582270	−	0.812995i	$-0.697835\pi$
$881$	−34.5655	−1.16454	−0.582270	+	0.812995i	$0.697835\pi$
$882$	−6.78664	−0.228518
$883$	27.4681	0.924374	0.462187	−	0.886782i	$-0.347065\pi$
$883$	27.4681	0.924374	0.462187	+	0.886782i	$0.347065\pi$
$884$	−7.98198	−0.268463
$885$	−57.9632	−1.94841
$886$	−0.190584	−0.00640280
$887$	4.10201	0.137732	0.0688661	−	0.997626i	$-0.478062\pi$
$887$	4.10201	0.137732	0.0688661	+	0.997626i	$0.478062\pi$
$888$	7.35359	0.246770
$889$	−6.05384	−0.203039
$890$	4.77648	0.160108
$891$	2.29353	0.0768360
$892$	−26.5315	−0.888342
$893$	5.74802	0.192350
$894$	1.08290	0.0362175
$895$	75.6015	2.52708
$896$	0.461913	0.0154314
$897$	2.16373	0.0722448
$898$	20.2172	0.674657
$899$	−2.61314	−0.0871532
$900$	12.2797	0.409323
$901$	−29.2051	−0.972963
$902$	4.55339	0.151611
$903$	2.18788	0.0728081
$904$	10.0087	0.332883
$905$	25.3363	0.842208
$906$	9.66170	0.320988
$907$	33.4330	1.11012	0.555062	−	0.831809i	$-0.312695\pi$
$907$	33.4330	1.11012	0.555062	+	0.831809i	$0.312695\pi$
$908$	15.5943	0.517515
$909$	7.27014	0.241135
$910$	−4.15461	−0.137724
$911$	−10.2241	−0.338741	−0.169371	−	0.985552i	$-0.554173\pi$
$911$	−10.2241	−0.338741	−0.169371	+	0.985552i	$0.554173\pi$
$912$	−0.461913	−0.0152955
$913$	2.81434	0.0931412
$914$	−15.9356	−0.527103
$915$	−31.4623	−1.04011
$916$	24.3201	0.803558
$917$	−8.82186	−0.291323
$918$	−3.68899	−0.121755
$919$	17.2072	0.567615	0.283807	−	0.958881i	$-0.408402\pi$
$919$	17.2072	0.567615	0.283807	+	0.958881i	$0.408402\pi$
$920$	4.15688	0.137048
$921$	−18.2765	−0.602232
$922$	−25.9025	−0.853053
$923$	17.2432	0.567568
$924$	−1.05941	−0.0348520
$925$	−90.2998	−2.96904
$926$	9.65635	0.317327
$927$	−9.69167	−0.318316
$928$	1.00000	0.0328266
$929$	−44.7854	−1.46936	−0.734681	−	0.678412i	$-0.762668\pi$
$929$	−44.7854	−1.46936	−0.734681	+	0.678412i	$0.762668\pi$
$930$	10.8625	0.356196
$931$	−3.13483	−0.102740
$932$	24.3411	0.797318
$933$	5.47998	0.179407
$934$	6.92345	0.226542
$935$	35.1706	1.15020
$936$	−2.16373	−0.0707237
$937$	−55.1344	−1.80116	−0.900581	−	0.434688i	$-0.856859\pi$
$937$	−55.1344	−1.80116	−0.900581	+	0.434688i	$0.856859\pi$
$938$	−0.496511	−0.0162116
$939$	−9.94247	−0.324460
$940$	51.7281	1.68718
$941$	−40.0574	−1.30583	−0.652917	−	0.757429i	$-0.726455\pi$
$941$	−40.0574	−1.30583	−0.652917	+	0.757429i	$0.726455\pi$
$942$	−2.14945	−0.0700330
$943$	1.98532	0.0646510
$944$	13.9439	0.453836
$945$	−1.92012	−0.0624614
$946$	−10.8634	−0.353201
$947$	−25.7928	−0.838154	−0.419077	−	0.907951i	$-0.637646\pi$
$947$	−25.7928	−0.838154	−0.419077	+	0.907951i	$0.637646\pi$
$948$	16.1867	0.525718
$949$	−31.5950	−1.02562
$950$	5.67214	0.184029
$951$	0.528602	0.0171411
$952$	1.70399	0.0552268
$953$	17.1018	0.553983	0.276992	−	0.960872i	$-0.410663\pi$
$953$	17.1018	0.553983	0.276992	+	0.960872i	$0.410663\pi$
$954$	−7.91682	−0.256317
$955$	−85.9232	−2.78041
$956$	22.5843	0.730430
$957$	−2.29353	−0.0741392
$958$	−1.79876	−0.0581151
$959$	9.24394	0.298502
$960$	−4.15688	−0.134163
$961$	−24.1715	−0.779725
$962$	15.9112	0.512997
$963$	0.728742	0.0234834
$964$	29.1961	0.940342
$965$	97.0735	3.12491
$966$	−0.461913	−0.0148618
$967$	−24.8857	−0.800271	−0.400136	−	0.916456i	$-0.631037\pi$
$967$	−24.8857	−0.800271	−0.400136	+	0.916456i	$0.631037\pi$
$968$	−5.73974	−0.184482
$969$	−1.70399	−0.0547402
$970$	−56.6594	−1.81922
$971$	−21.7672	−0.698542	−0.349271	−	0.937022i	$-0.613571\pi$
$971$	−21.7672	−0.698542	−0.349271	+	0.937022i	$0.613571\pi$
$972$	−1.00000	−0.0320750
$973$	−10.0482	−0.322130
$974$	−34.3279	−1.09994
$975$	26.5699	0.850918
$976$	7.56872	0.242269
$977$	−11.9994	−0.383896	−0.191948	−	0.981405i	$-0.561481\pi$
$977$	−11.9994	−0.383896	−0.191948	+	0.981405i	$0.561481\pi$
$978$	−4.33739	−0.138694
$979$	2.63538	0.0842272
$980$	−28.2113	−0.901175
$981$	4.33632	0.138448
$982$	15.8749	0.506587
$983$	26.3179	0.839412	0.419706	−	0.907660i	$-0.362133\pi$
$983$	26.3179	0.839412	0.419706	+	0.907660i	$0.362133\pi$
$984$	−1.98532	−0.0632898
$985$	−64.8218	−2.06540
$986$	3.68899	0.117482
$987$	−5.74802	−0.182962
$988$	−0.999454	−0.0317969
$989$	−4.73656	−0.150614
$990$	9.53392	0.303008
$991$	41.3491	1.31350	0.656749	−	0.754110i	$-0.271931\pi$
$991$	41.3491	1.31350	0.656749	+	0.754110i	$0.271931\pi$
$992$	−2.61314	−0.0829674
$993$	−18.7648	−0.595482
$994$	−3.68108	−0.116757
$995$	10.1686	0.322365
$996$	−1.22708	−0.0388816
$997$	−9.78327	−0.309839	−0.154920	−	0.987927i	$-0.549512\pi$
$997$	−9.78327	−0.309839	−0.154920	+	0.987927i	$0.549512\pi$
$998$	8.06965	0.255440
$999$	7.35359	0.232657

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bj.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bj.1.8	✓	8	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.15688 q^{5} -1.00000 q^{6} +0.461913 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +4.15688 q^{5} -1.00000 q^{6} +0.461913 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.15688 q^{10} +2.29353 q^{11} -1.00000 q^{12} -2.16373 q^{13} +0.461913 q^{14} -4.15688 q^{15} +1.00000 q^{16} +3.68899 q^{17} +1.00000 q^{18} +0.461913 q^{19} +4.15688 q^{20} -0.461913 q^{21} +2.29353 q^{22} +1.00000 q^{23} -1.00000 q^{24} +12.2797 q^{25} -2.16373 q^{26} -1.00000 q^{27} +0.461913 q^{28} +1.00000 q^{29} -4.15688 q^{30} -2.61314 q^{31} +1.00000 q^{32} -2.29353 q^{33} +3.68899 q^{34} +1.92012 q^{35} +1.00000 q^{36} -7.35359 q^{37} +0.461913 q^{38} +2.16373 q^{39} +4.15688 q^{40} +1.98532 q^{41} -0.461913 q^{42} -4.73656 q^{43} +2.29353 q^{44} +4.15688 q^{45} +1.00000 q^{46} +12.4440 q^{47} -1.00000 q^{48} -6.78664 q^{49} +12.2797 q^{50} -3.68899 q^{51} -2.16373 q^{52} -7.91682 q^{53} -1.00000 q^{54} +9.53392 q^{55} +0.461913 q^{56} -0.461913 q^{57} +1.00000 q^{58} +13.9439 q^{59} -4.15688 q^{60} +7.56872 q^{61} -2.61314 q^{62} +0.461913 q^{63} +1.00000 q^{64} -8.99437 q^{65} -2.29353 q^{66} -1.07490 q^{67} +3.68899 q^{68} -1.00000 q^{69} +1.92012 q^{70} -7.96922 q^{71} +1.00000 q^{72} +14.6021 q^{73} -7.35359 q^{74} -12.2797 q^{75} +0.461913 q^{76} +1.05941 q^{77} +2.16373 q^{78} -16.1867 q^{79} +4.15688 q^{80} +1.00000 q^{81} +1.98532 q^{82} +1.22708 q^{83} -0.461913 q^{84} +15.3347 q^{85} -4.73656 q^{86} -1.00000 q^{87} +2.29353 q^{88} +1.14905 q^{89} +4.15688 q^{90} -0.999454 q^{91} +1.00000 q^{92} +2.61314 q^{93} +12.4440 q^{94} +1.92012 q^{95} -1.00000 q^{96} -13.6303 q^{97} -6.78664 q^{98} +2.29353 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + q^5 - 8 * q^6 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9} + q^{10} + 3 q^{11} - 8 q^{12} + 9 q^{13} - q^{15} + 8 q^{16} + 8 q^{17} + 8 q^{18} + q^{20} + 3 q^{22} + 8 q^{23} - 8 q^{24} + 13 q^{25} + 9 q^{26} - 8 q^{27} + 8 q^{29} - q^{30} - 9 q^{31} + 8 q^{32} - 3 q^{33} + 8 q^{34} - 2 q^{35} + 8 q^{36} + 7 q^{37} - 9 q^{39} + q^{40} + 3 q^{41} + 16 q^{43} + 3 q^{44} + q^{45} + 8 q^{46} + 24 q^{47} - 8 q^{48} + 6 q^{49} + 13 q^{50} - 8 q^{51} + 9 q^{52} + 8 q^{53} - 8 q^{54} + 13 q^{55} + 8 q^{58} - 3 q^{59} - q^{60} + 31 q^{61} - 9 q^{62} + 8 q^{64} + 13 q^{65} - 3 q^{66} - 11 q^{67} + 8 q^{68} - 8 q^{69} - 2 q^{70} + 7 q^{71} + 8 q^{72} + 14 q^{73} + 7 q^{74} - 13 q^{75} + 10 q^{77} - 9 q^{78} + 12 q^{79} + q^{80} + 8 q^{81} + 3 q^{82} - 8 q^{83} + 22 q^{85} + 16 q^{86} - 8 q^{87} + 3 q^{88} - 12 q^{89} + q^{90} + 28 q^{91} + 8 q^{92} + 9 q^{93} + 24 q^{94} - 2 q^{95} - 8 q^{96} + 16 q^{97} + 6 q^{98} + 3 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + q^5 - 8 * q^6 + 8 * q^8 + 8 * q^9 + q^10 + 3 * q^11 - 8 * q^12 + 9 * q^13 - q^15 + 8 * q^16 + 8 * q^17 + 8 * q^18 + q^20 + 3 * q^22 + 8 * q^23 - 8 * q^24 + 13 * q^25 + 9 * q^26 - 8 * q^27 + 8 * q^29 - q^30 - 9 * q^31 + 8 * q^32 - 3 * q^33 + 8 * q^34 - 2 * q^35 + 8 * q^36 + 7 * q^37 - 9 * q^39 + q^40 + 3 * q^41 + 16 * q^43 + 3 * q^44 + q^45 + 8 * q^46 + 24 * q^47 - 8 * q^48 + 6 * q^49 + 13 * q^50 - 8 * q^51 + 9 * q^52 + 8 * q^53 - 8 * q^54 + 13 * q^55 + 8 * q^58 - 3 * q^59 - q^60 + 31 * q^61 - 9 * q^62 + 8 * q^64 + 13 * q^65 - 3 * q^66 - 11 * q^67 + 8 * q^68 - 8 * q^69 - 2 * q^70 + 7 * q^71 + 8 * q^72 + 14 * q^73 + 7 * q^74 - 13 * q^75 + 10 * q^77 - 9 * q^78 + 12 * q^79 + q^80 + 8 * q^81 + 3 * q^82 - 8 * q^83 + 22 * q^85 + 16 * q^86 - 8 * q^87 + 3 * q^88 - 12 * q^89 + q^90 + 28 * q^91 + 8 * q^92 + 9 * q^93 + 24 * q^94 - 2 * q^95 - 8 * q^96 + 16 * q^97 + 6 * q^98 + 3 * q^99

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