LMFDB - Embedded newform 4002.2.a.bi.1.5

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:	$1$
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} - 24x^{6} - 3x^{5} + 194x^{4} + 39x^{3} - 607x^{2} - 104x + 600 $ x^8 - 24x^6 - 3x^5 + 194x^4 + 39x^3 - 607x^2 - 104x + 600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$3.34107$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.464023 q^{5} -1.00000 q^{6} -2.35800 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.464023 q^{5} -1.00000 q^{6} -2.35800 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.464023 q^{10} +4.25869 q^{11} +1.00000 q^{12} -1.47447 q^{13} +2.35800 q^{14} -0.464023 q^{15} +1.00000 q^{16} -6.51602 q^{17} -1.00000 q^{18} +7.96818 q^{19} -0.464023 q^{20} -2.35800 q^{21} -4.25869 q^{22} -1.00000 q^{23} -1.00000 q^{24} -4.78468 q^{25} +1.47447 q^{26} +1.00000 q^{27} -2.35800 q^{28} +1.00000 q^{29} +0.464023 q^{30} -9.27083 q^{31} -1.00000 q^{32} +4.25869 q^{33} +6.51602 q^{34} +1.09416 q^{35} +1.00000 q^{36} -3.81883 q^{37} -7.96818 q^{38} -1.47447 q^{39} +0.464023 q^{40} +9.71069 q^{41} +2.35800 q^{42} +5.23918 q^{43} +4.25869 q^{44} -0.464023 q^{45} +1.00000 q^{46} -11.9675 q^{47} +1.00000 q^{48} -1.43986 q^{49} +4.78468 q^{50} -6.51602 q^{51} -1.47447 q^{52} -9.45843 q^{53} -1.00000 q^{54} -1.97613 q^{55} +2.35800 q^{56} +7.96818 q^{57} -1.00000 q^{58} -9.55233 q^{59} -0.464023 q^{60} +0.100329 q^{61} +9.27083 q^{62} -2.35800 q^{63} +1.00000 q^{64} +0.684187 q^{65} -4.25869 q^{66} +14.1044 q^{67} -6.51602 q^{68} -1.00000 q^{69} -1.09416 q^{70} +8.73384 q^{71} -1.00000 q^{72} +1.03806 q^{73} +3.81883 q^{74} -4.78468 q^{75} +7.96818 q^{76} -10.0420 q^{77} +1.47447 q^{78} -1.58911 q^{79} -0.464023 q^{80} +1.00000 q^{81} -9.71069 q^{82} +13.0499 q^{83} -2.35800 q^{84} +3.02359 q^{85} -5.23918 q^{86} +1.00000 q^{87} -4.25869 q^{88} -4.15176 q^{89} +0.464023 q^{90} +3.47678 q^{91} -1.00000 q^{92} -9.27083 q^{93} +11.9675 q^{94} -3.69742 q^{95} -1.00000 q^{96} +15.6619 q^{97} +1.43986 q^{98} +4.25869 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 3 q^{5} - 8 q^{6} - 6 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 3 * q^5 - 8 * q^6 - 6 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 3 q^{5} - 8 q^{6} - 6 q^{7} - 8 q^{8} + 8 q^{9} + 3 q^{10} - 7 q^{11} + 8 q^{12} - 3 q^{13} + 6 q^{14} - 3 q^{15} + 8 q^{16} - 12 q^{17} - 8 q^{18} - 4 q^{19} - 3 q^{20} - 6 q^{21} + 7 q^{22} - 8 q^{23} - 8 q^{24} + 13 q^{25} + 3 q^{26} + 8 q^{27} - 6 q^{28} + 8 q^{29} + 3 q^{30} - q^{31} - 8 q^{32} - 7 q^{33} + 12 q^{34} - 6 q^{35} + 8 q^{36} - 11 q^{37} + 4 q^{38} - 3 q^{39} + 3 q^{40} - 17 q^{41} + 6 q^{42} - 10 q^{43} - 7 q^{44} - 3 q^{45} + 8 q^{46} - 26 q^{47} + 8 q^{48} + 10 q^{49} - 13 q^{50} - 12 q^{51} - 3 q^{52} - 2 q^{53} - 8 q^{54} + q^{55} + 6 q^{56} - 4 q^{57} - 8 q^{58} - 25 q^{59} - 3 q^{60} + 3 q^{61} + q^{62} - 6 q^{63} + 8 q^{64} - 25 q^{65} + 7 q^{66} + q^{67} - 12 q^{68} - 8 q^{69} + 6 q^{70} - 27 q^{71} - 8 q^{72} + 16 q^{73} + 11 q^{74} + 13 q^{75} - 4 q^{76} - 16 q^{77} + 3 q^{78} - 6 q^{79} - 3 q^{80} + 8 q^{81} + 17 q^{82} - 44 q^{83} - 6 q^{84} - 20 q^{85} + 10 q^{86} + 8 q^{87} + 7 q^{88} - 52 q^{89} + 3 q^{90} - 18 q^{91} - 8 q^{92} - q^{93} + 26 q^{94} - 56 q^{95} - 8 q^{96} - 4 q^{97} - 10 q^{98} - 7 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 3 * q^5 - 8 * q^6 - 6 * q^7 - 8 * q^8 + 8 * q^9 + 3 * q^10 - 7 * q^11 + 8 * q^12 - 3 * q^13 + 6 * q^14 - 3 * q^15 + 8 * q^16 - 12 * q^17 - 8 * q^18 - 4 * q^19 - 3 * q^20 - 6 * q^21 + 7 * q^22 - 8 * q^23 - 8 * q^24 + 13 * q^25 + 3 * q^26 + 8 * q^27 - 6 * q^28 + 8 * q^29 + 3 * q^30 - q^31 - 8 * q^32 - 7 * q^33 + 12 * q^34 - 6 * q^35 + 8 * q^36 - 11 * q^37 + 4 * q^38 - 3 * q^39 + 3 * q^40 - 17 * q^41 + 6 * q^42 - 10 * q^43 - 7 * q^44 - 3 * q^45 + 8 * q^46 - 26 * q^47 + 8 * q^48 + 10 * q^49 - 13 * q^50 - 12 * q^51 - 3 * q^52 - 2 * q^53 - 8 * q^54 + q^55 + 6 * q^56 - 4 * q^57 - 8 * q^58 - 25 * q^59 - 3 * q^60 + 3 * q^61 + q^62 - 6 * q^63 + 8 * q^64 - 25 * q^65 + 7 * q^66 + q^67 - 12 * q^68 - 8 * q^69 + 6 * q^70 - 27 * q^71 - 8 * q^72 + 16 * q^73 + 11 * q^74 + 13 * q^75 - 4 * q^76 - 16 * q^77 + 3 * q^78 - 6 * q^79 - 3 * q^80 + 8 * q^81 + 17 * q^82 - 44 * q^83 - 6 * q^84 - 20 * q^85 + 10 * q^86 + 8 * q^87 + 7 * q^88 - 52 * q^89 + 3 * q^90 - 18 * q^91 - 8 * q^92 - q^93 + 26 * q^94 - 56 * q^95 - 8 * q^96 - 4 * q^97 - 10 * q^98 - 7 * 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
$5$	−0.464023	−0.207518	−0.103759	−	0.994602i	$-0.533087\pi$
$5$	−0.464023	−0.207518	−0.103759	+	0.994602i	$0.533087\pi$
$6$	−1.00000	−0.408248
$7$	−2.35800	−0.891238	−0.445619	−	0.895223i	$-0.647016\pi$
$7$	−2.35800	−0.891238	−0.445619	+	0.895223i	$0.647016\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0.464023	0.146737
$11$	4.25869	1.28404	0.642021	−	0.766687i	$-0.278096\pi$
$11$	4.25869	1.28404	0.642021	+	0.766687i	$0.278096\pi$
$12$	1.00000	0.288675
$13$	−1.47447	−0.408943	−0.204472	−	0.978872i	$-0.565548\pi$
$13$	−1.47447	−0.408943	−0.204472	+	0.978872i	$0.565548\pi$
$14$	2.35800	0.630201
$15$	−0.464023	−0.119810
$16$	1.00000	0.250000
$17$	−6.51602	−1.58037	−0.790184	−	0.612870i	$-0.790015\pi$
$17$	−6.51602	−1.58037	−0.790184	+	0.612870i	$0.790015\pi$
$18$	−1.00000	−0.235702
$19$	7.96818	1.82803	0.914013	−	0.405685i	$-0.132967\pi$
$19$	7.96818	1.82803	0.914013	+	0.405685i	$0.132967\pi$
$20$	−0.464023	−0.103759
$21$	−2.35800	−0.514557
$22$	−4.25869	−0.907955
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	−4.78468	−0.956936
$26$	1.47447	0.289167
$27$	1.00000	0.192450
$28$	−2.35800	−0.445619
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0.464023	0.0847187
$31$	−9.27083	−1.66509	−0.832545	−	0.553957i	$-0.813117\pi$
$31$	−9.27083	−1.66509	−0.832545	+	0.553957i	$0.813117\pi$
$32$	−1.00000	−0.176777
$33$	4.25869	0.741342
$34$	6.51602	1.11749
$35$	1.09416	0.184948
$36$	1.00000	0.166667
$37$	−3.81883	−0.627812	−0.313906	−	0.949454i	$-0.601638\pi$
$37$	−3.81883	−0.627812	−0.313906	+	0.949454i	$0.601638\pi$
$38$	−7.96818	−1.29261
$39$	−1.47447	−0.236104
$40$	0.464023	0.0733685
$41$	9.71069	1.51655	0.758277	−	0.651932i	$-0.226041\pi$
$41$	9.71069	1.51655	0.758277	+	0.651932i	$0.226041\pi$
$42$	2.35800	0.363847
$43$	5.23918	0.798968	0.399484	−	0.916740i	$-0.369189\pi$
$43$	5.23918	0.798968	0.399484	+	0.916740i	$0.369189\pi$
$44$	4.25869	0.642021
$45$	−0.464023	−0.0691725
$46$	1.00000	0.147442
$47$	−11.9675	−1.74564	−0.872820	−	0.488043i	$-0.837711\pi$
$47$	−11.9675	−1.74564	−0.872820	+	0.488043i	$0.837711\pi$
$48$	1.00000	0.144338
$49$	−1.43986	−0.205694
$50$	4.78468	0.676656
$51$	−6.51602	−0.912426
$52$	−1.47447	−0.204472
$53$	−9.45843	−1.29921	−0.649607	−	0.760270i	$-0.725067\pi$
$53$	−9.45843	−1.29921	−0.649607	+	0.760270i	$0.725067\pi$
$54$	−1.00000	−0.136083
$55$	−1.97613	−0.266461
$56$	2.35800	0.315100
$57$	7.96818	1.05541
$58$	−1.00000	−0.131306
$59$	−9.55233	−1.24361	−0.621804	−	0.783173i	$-0.713600\pi$
$59$	−9.55233	−1.24361	−0.621804	+	0.783173i	$0.713600\pi$
$60$	−0.464023	−0.0599052
$61$	0.100329	0.0128458	0.00642290	−	0.999979i	$-0.497956\pi$
$61$	0.100329	0.0128458	0.00642290	+	0.999979i	$0.497956\pi$
$62$	9.27083	1.17740
$63$	−2.35800	−0.297079
$64$	1.00000	0.125000
$65$	0.684187	0.0848629
$66$	−4.25869	−0.524208
$67$	14.1044	1.72313	0.861566	−	0.507646i	$-0.169484\pi$
$67$	14.1044	1.72313	0.861566	+	0.507646i	$0.169484\pi$
$68$	−6.51602	−0.790184
$69$	−1.00000	−0.120386
$70$	−1.09416	−0.130778
$71$	8.73384	1.03652	0.518258	−	0.855224i	$-0.326581\pi$
$71$	8.73384	1.03652	0.518258	+	0.855224i	$0.326581\pi$
$72$	−1.00000	−0.117851
$73$	1.03806	0.121496	0.0607479	−	0.998153i	$-0.480651\pi$
$73$	1.03806	0.121496	0.0607479	+	0.998153i	$0.480651\pi$
$74$	3.81883	0.443930
$75$	−4.78468	−0.552488
$76$	7.96818	0.914013
$77$	−10.0420	−1.14439
$78$	1.47447	0.166950
$79$	−1.58911	−0.178789	−0.0893944	−	0.995996i	$-0.528493\pi$
$79$	−1.58911	−0.178789	−0.0893944	+	0.995996i	$0.528493\pi$
$80$	−0.464023	−0.0518794
$81$	1.00000	0.111111
$82$	−9.71069	−1.07237
$83$	13.0499	1.43241	0.716207	−	0.697888i	$-0.245877\pi$
$83$	13.0499	1.43241	0.716207	+	0.697888i	$0.245877\pi$
$84$	−2.35800	−0.257278
$85$	3.02359	0.327954
$86$	−5.23918	−0.564956
$87$	1.00000	0.107211
$88$	−4.25869	−0.453978
$89$	−4.15176	−0.440086	−0.220043	−	0.975490i	$-0.570620\pi$
$89$	−4.15176	−0.440086	−0.220043	+	0.975490i	$0.570620\pi$
$90$	0.464023	0.0489124
$91$	3.47678	0.364466
$92$	−1.00000	−0.104257
$93$	−9.27083	−0.961340
$94$	11.9675	1.23435
$95$	−3.69742	−0.379348
$96$	−1.00000	−0.102062
$97$	15.6619	1.59023	0.795115	−	0.606459i	$-0.207411\pi$
$97$	15.6619	1.59023	0.795115	+	0.606459i	$0.207411\pi$
$98$	1.43986	0.145448
$99$	4.25869	0.428014
$100$	−4.78468	−0.478468
$101$	−13.1685	−1.31031	−0.655155	−	0.755494i	$-0.727397\pi$
$101$	−13.1685	−1.31031	−0.655155	+	0.755494i	$0.727397\pi$
$102$	6.51602	0.645182
$103$	−0.917126	−0.0903671	−0.0451835	−	0.998979i	$-0.514387\pi$
$103$	−0.917126	−0.0903671	−0.0451835	+	0.998979i	$0.514387\pi$
$104$	1.47447	0.144583
$105$	1.09416	0.106780
$106$	9.45843	0.918684
$107$	−13.8241	−1.33642	−0.668211	−	0.743972i	$-0.732940\pi$
$107$	−13.8241	−1.33642	−0.668211	+	0.743972i	$0.732940\pi$
$108$	1.00000	0.0962250
$109$	−19.1255	−1.83189	−0.915946	−	0.401303i	$-0.868558\pi$
$109$	−19.1255	−1.83189	−0.915946	+	0.401303i	$0.868558\pi$
$110$	1.97613	0.188417
$111$	−3.81883	−0.362467
$112$	−2.35800	−0.222810
$113$	−4.94300	−0.464998	−0.232499	−	0.972597i	$-0.574690\pi$
$113$	−4.94300	−0.464998	−0.232499	+	0.972597i	$0.574690\pi$
$114$	−7.96818	−0.746289
$115$	0.464023	0.0432704
$116$	1.00000	0.0928477
$117$	−1.47447	−0.136314
$118$	9.55233	0.879363
$119$	15.3647	1.40848
$120$	0.464023	0.0423593
$121$	7.13641	0.648765
$122$	−0.100329	−0.00908335
$123$	9.71069	0.875583
$124$	−9.27083	−0.832545
$125$	4.54032	0.406099
$126$	2.35800	0.210067
$127$	−3.32519	−0.295063	−0.147532	−	0.989057i	$-0.547133\pi$
$127$	−3.32519	−0.295063	−0.147532	+	0.989057i	$0.547133\pi$
$128$	−1.00000	−0.0883883
$129$	5.23918	0.461285
$130$	−0.684187	−0.0600072
$131$	8.81266	0.769966	0.384983	−	0.922924i	$-0.374207\pi$
$131$	8.81266	0.769966	0.384983	+	0.922924i	$0.374207\pi$
$132$	4.25869	0.370671
$133$	−18.7889	−1.62921
$134$	−14.1044	−1.21844
$135$	−0.464023	−0.0399368
$136$	6.51602	0.558744
$137$	−21.9768	−1.87761	−0.938804	−	0.344452i	$-0.888065\pi$
$137$	−21.9768	−1.87761	−0.938804	+	0.344452i	$0.888065\pi$
$138$	1.00000	0.0851257
$139$	−7.06678	−0.599396	−0.299698	−	0.954034i	$-0.596886\pi$
$139$	−7.06678	−0.599396	−0.299698	+	0.954034i	$0.596886\pi$
$140$	1.09416	0.0924738
$141$	−11.9675	−1.00785
$142$	−8.73384	−0.732928
$143$	−6.27929	−0.525101
$144$	1.00000	0.0833333
$145$	−0.464023	−0.0385350
$146$	−1.03806	−0.0859105
$147$	−1.43986	−0.118757
$148$	−3.81883	−0.313906
$149$	−6.52571	−0.534607	−0.267303	−	0.963612i	$-0.586133\pi$
$149$	−6.52571	−0.534607	−0.267303	+	0.963612i	$0.586133\pi$
$150$	4.78468	0.390668
$151$	4.49821	0.366059	0.183030	−	0.983107i	$-0.441410\pi$
$151$	4.49821	0.366059	0.183030	+	0.983107i	$0.441410\pi$
$152$	−7.96818	−0.646305
$153$	−6.51602	−0.526789
$154$	10.0420	0.809204
$155$	4.30188	0.345535
$156$	−1.47447	−0.118052
$157$	−18.0304	−1.43898	−0.719492	−	0.694501i	$-0.755625\pi$
$157$	−18.0304	−1.43898	−0.719492	+	0.694501i	$0.755625\pi$
$158$	1.58911	0.126423
$159$	−9.45843	−0.750102
$160$	0.464023	0.0366843
$161$	2.35800	0.185836
$162$	−1.00000	−0.0785674
$163$	11.1464	0.873056	0.436528	−	0.899691i	$-0.356208\pi$
$163$	11.1464	0.873056	0.436528	+	0.899691i	$0.356208\pi$
$164$	9.71069	0.758277
$165$	−1.97613	−0.153842
$166$	−13.0499	−1.01287
$167$	−13.9499	−1.07947	−0.539737	−	0.841834i	$-0.681476\pi$
$167$	−13.9499	−1.07947	−0.539737	+	0.841834i	$0.681476\pi$
$168$	2.35800	0.181923
$169$	−10.8259	−0.832765
$170$	−3.02359	−0.231898
$171$	7.96818	0.609342
$172$	5.23918	0.399484
$173$	−19.8698	−1.51067	−0.755336	−	0.655338i	$-0.772526\pi$
$173$	−19.8698	−1.51067	−0.755336	+	0.655338i	$0.772526\pi$
$174$	−1.00000	−0.0758098
$175$	11.2823	0.852859
$176$	4.25869	0.321011
$177$	−9.55233	−0.717997
$178$	4.15176	0.311188
$179$	−17.6806	−1.32151	−0.660757	−	0.750600i	$-0.729765\pi$
$179$	−17.6806	−1.32151	−0.660757	+	0.750600i	$0.729765\pi$
$180$	−0.464023	−0.0345863
$181$	17.2195	1.27991	0.639957	−	0.768411i	$-0.278952\pi$
$181$	17.2195	1.27991	0.639957	+	0.768411i	$0.278952\pi$
$182$	−3.47678	−0.257716
$183$	0.100329	0.00741652
$184$	1.00000	0.0737210
$185$	1.77203	0.130282
$186$	9.27083	0.679770
$187$	−27.7497	−2.02926
$188$	−11.9675	−0.872820
$189$	−2.35800	−0.171519
$190$	3.69742	0.268239
$191$	20.3059	1.46928	0.734641	−	0.678456i	$-0.237350\pi$
$191$	20.3059	1.46928	0.734641	+	0.678456i	$0.237350\pi$
$192$	1.00000	0.0721688
$193$	−3.08496	−0.222060	−0.111030	−	0.993817i	$-0.535415\pi$
$193$	−3.08496	−0.222060	−0.111030	+	0.993817i	$0.535415\pi$
$194$	−15.6619	−1.12446
$195$	0.684187	0.0489956
$196$	−1.43986	−0.102847
$197$	−19.5714	−1.39441	−0.697204	−	0.716873i	$-0.745573\pi$
$197$	−19.5714	−1.39441	−0.697204	+	0.716873i	$0.745573\pi$
$198$	−4.25869	−0.302652
$199$	−21.8115	−1.54618	−0.773088	−	0.634299i	$-0.781289\pi$
$199$	−21.8115	−1.54618	−0.773088	+	0.634299i	$0.781289\pi$
$200$	4.78468	0.338328
$201$	14.1044	0.994850
$202$	13.1685	0.926529
$203$	−2.35800	−0.165499
$204$	−6.51602	−0.456213
$205$	−4.50599	−0.314712
$206$	0.917126	0.0638992
$207$	−1.00000	−0.0695048
$208$	−1.47447	−0.102236
$209$	33.9340	2.34726
$210$	−1.09416	−0.0755046
$211$	14.0420	0.966691	0.483346	−	0.875430i	$-0.339421\pi$
$211$	14.0420	0.966691	0.483346	+	0.875430i	$0.339421\pi$
$212$	−9.45843	−0.649607
$213$	8.73384	0.598433
$214$	13.8241	0.944993
$215$	−2.43110	−0.165800
$216$	−1.00000	−0.0680414
$217$	21.8606	1.48399
$218$	19.1255	1.29534
$219$	1.03806	0.0701457
$220$	−1.97613	−0.133231
$221$	9.60765	0.646281
$222$	3.81883	0.256303
$223$	9.27010	0.620772	0.310386	−	0.950611i	$-0.399542\pi$
$223$	9.27010	0.620772	0.310386	+	0.950611i	$0.399542\pi$
$224$	2.35800	0.157550
$225$	−4.78468	−0.318979
$226$	4.94300	0.328803
$227$	−29.1466	−1.93453	−0.967264	−	0.253770i	$-0.918329\pi$
$227$	−29.1466	−1.93453	−0.967264	+	0.253770i	$0.918329\pi$
$228$	7.96818	0.527706
$229$	15.5618	1.02835	0.514175	−	0.857685i	$-0.328098\pi$
$229$	15.5618	1.02835	0.514175	+	0.857685i	$0.328098\pi$
$230$	−0.464023	−0.0305968
$231$	−10.0420	−0.660713
$232$	−1.00000	−0.0656532
$233$	−11.5210	−0.754767	−0.377383	−	0.926057i	$-0.623176\pi$
$233$	−11.5210	−0.754767	−0.377383	+	0.926057i	$0.623176\pi$
$234$	1.47447	0.0963889
$235$	5.55320	0.362251
$236$	−9.55233	−0.621804
$237$	−1.58911	−0.103224
$238$	−15.3647	−0.995949
$239$	15.3142	0.990590	0.495295	−	0.868725i	$-0.335060\pi$
$239$	15.3142	0.990590	0.495295	+	0.868725i	$0.335060\pi$
$240$	−0.464023	−0.0299526
$241$	−0.729295	−0.0469780	−0.0234890	−	0.999724i	$-0.507477\pi$
$241$	−0.729295	−0.0469780	−0.0234890	+	0.999724i	$0.507477\pi$
$242$	−7.13641	−0.458746
$243$	1.00000	0.0641500
$244$	0.100329	0.00642290
$245$	0.668128	0.0426851
$246$	−9.71069	−0.619131
$247$	−11.7488	−0.747559
$248$	9.27083	0.588698
$249$	13.0499	0.827004
$250$	−4.54032	−0.287155
$251$	−14.1551	−0.893464	−0.446732	−	0.894668i	$-0.647412\pi$
$251$	−14.1551	−0.893464	−0.446732	+	0.894668i	$0.647412\pi$
$252$	−2.35800	−0.148540
$253$	−4.25869	−0.267741
$254$	3.32519	0.208641
$255$	3.02359	0.189344
$256$	1.00000	0.0625000
$257$	9.96166	0.621392	0.310696	−	0.950509i	$-0.399438\pi$
$257$	9.96166	0.621392	0.310696	+	0.950509i	$0.399438\pi$
$258$	−5.23918	−0.326177
$259$	9.00478	0.559530
$260$	0.684187	0.0424315
$261$	1.00000	0.0618984
$262$	−8.81266	−0.544448
$263$	8.14617	0.502314	0.251157	−	0.967946i	$-0.419189\pi$
$263$	8.14617	0.502314	0.251157	+	0.967946i	$0.419189\pi$
$264$	−4.25869	−0.262104
$265$	4.38893	0.269610
$266$	18.7889	1.15202
$267$	−4.15176	−0.254084
$268$	14.1044	0.861566
$269$	−5.05591	−0.308264	−0.154132	−	0.988050i	$-0.549258\pi$
$269$	−5.05591	−0.308264	−0.154132	+	0.988050i	$0.549258\pi$
$270$	0.464023	0.0282396
$271$	13.8835	0.843364	0.421682	−	0.906744i	$-0.361440\pi$
$271$	13.8835	0.843364	0.421682	+	0.906744i	$0.361440\pi$
$272$	−6.51602	−0.395092
$273$	3.47678	0.210425
$274$	21.9768	1.32767
$275$	−20.3765	−1.22875
$276$	−1.00000	−0.0601929
$277$	−14.0987	−0.847109	−0.423554	−	0.905871i	$-0.639218\pi$
$277$	−14.0987	−0.847109	−0.423554	+	0.905871i	$0.639218\pi$
$278$	7.06678	0.423837
$279$	−9.27083	−0.555030
$280$	−1.09416	−0.0653889
$281$	−25.7608	−1.53676	−0.768380	−	0.639994i	$-0.778937\pi$
$281$	−25.7608	−1.53676	−0.768380	+	0.639994i	$0.778937\pi$
$282$	11.9675	0.712654
$283$	−15.6618	−0.930999	−0.465500	−	0.885048i	$-0.654125\pi$
$283$	−15.6618	−0.930999	−0.465500	+	0.885048i	$0.654125\pi$
$284$	8.73384	0.518258
$285$	−3.69742	−0.219016
$286$	6.27929	0.371302
$287$	−22.8978	−1.35161
$288$	−1.00000	−0.0589256
$289$	25.4585	1.49756
$290$	0.464023	0.0272484
$291$	15.6619	0.918119
$292$	1.03806	0.0607479
$293$	8.37521	0.489285	0.244643	−	0.969613i	$-0.421329\pi$
$293$	8.37521	0.489285	0.244643	+	0.969613i	$0.421329\pi$
$294$	1.43986	0.0839742
$295$	4.43250	0.258070
$296$	3.81883	0.221965
$297$	4.25869	0.247114
$298$	6.52571	0.378024
$299$	1.47447	0.0852706
$300$	−4.78468	−0.276244
$301$	−12.3540	−0.712071
$302$	−4.49821	−0.258843
$303$	−13.1685	−0.756508
$304$	7.96818	0.457007
$305$	−0.0465549	−0.00266573
$306$	6.51602	0.372496
$307$	25.6340	1.46301	0.731504	−	0.681837i	$-0.238819\pi$
$307$	25.6340	1.46301	0.731504	+	0.681837i	$0.238819\pi$
$308$	−10.0420	−0.572194
$309$	−0.917126	−0.0521735
$310$	−4.30188	−0.244330
$311$	10.8826	0.617093	0.308547	−	0.951209i	$-0.400157\pi$
$311$	10.8826	0.617093	0.308547	+	0.951209i	$0.400157\pi$
$312$	1.47447	0.0834752
$313$	−5.71391	−0.322970	−0.161485	−	0.986875i	$-0.551628\pi$
$313$	−5.71391	−0.322970	−0.161485	+	0.986875i	$0.551628\pi$
$314$	18.0304	1.01752
$315$	1.09416	0.0616492
$316$	−1.58911	−0.0893944
$317$	2.26469	0.127198	0.0635989	−	0.997976i	$-0.479742\pi$
$317$	2.26469	0.127198	0.0635989	+	0.997976i	$0.479742\pi$
$318$	9.45843	0.530402
$319$	4.25869	0.238441
$320$	−0.464023	−0.0259397
$321$	−13.8241	−0.771584
$322$	−2.35800	−0.131406
$323$	−51.9208	−2.88895
$324$	1.00000	0.0555556
$325$	7.05485	0.391333
$326$	−11.1464	−0.617344
$327$	−19.1255	−1.05764
$328$	−9.71069	−0.536183
$329$	28.2193	1.55578
$330$	1.97613	0.108782
$331$	−11.5459	−0.634620	−0.317310	−	0.948322i	$-0.602780\pi$
$331$	−11.5459	−0.634620	−0.317310	+	0.948322i	$0.602780\pi$
$332$	13.0499	0.716207
$333$	−3.81883	−0.209271
$334$	13.9499	0.763303
$335$	−6.54479	−0.357580
$336$	−2.35800	−0.128639
$337$	−7.63964	−0.416158	−0.208079	−	0.978112i	$-0.566721\pi$
$337$	−7.63964	−0.416158	−0.208079	+	0.978112i	$0.566721\pi$
$338$	10.8259	0.588854
$339$	−4.94300	−0.268467
$340$	3.02359	0.163977
$341$	−39.4816	−2.13805
$342$	−7.96818	−0.430870
$343$	19.9011	1.07456
$344$	−5.23918	−0.282478
$345$	0.464023	0.0249822
$346$	19.8698	1.06821
$347$	−29.2376	−1.56955	−0.784777	−	0.619778i	$-0.787223\pi$
$347$	−29.2376	−1.56955	−0.784777	+	0.619778i	$0.787223\pi$
$348$	1.00000	0.0536056
$349$	4.57499	0.244894	0.122447	−	0.992475i	$-0.460926\pi$
$349$	4.57499	0.244894	0.122447	+	0.992475i	$0.460926\pi$
$350$	−11.2823	−0.603062
$351$	−1.47447	−0.0787012
$352$	−4.25869	−0.226989
$353$	9.73503	0.518143	0.259072	−	0.965858i	$-0.416583\pi$
$353$	9.73503	0.518143	0.259072	+	0.965858i	$0.416583\pi$
$354$	9.55233	0.507701
$355$	−4.05271	−0.215095
$356$	−4.15176	−0.220043
$357$	15.3647	0.813189
$358$	17.6806	0.934451
$359$	−16.6817	−0.880427	−0.440214	−	0.897893i	$-0.645097\pi$
$359$	−16.6817	−0.880427	−0.440214	+	0.897893i	$0.645097\pi$
$360$	0.464023	0.0244562
$361$	44.4919	2.34168
$362$	−17.2195	−0.905036
$363$	7.13641	0.374564
$364$	3.47678	0.182233
$365$	−0.481685	−0.0252125
$366$	−0.100329	−0.00524427
$367$	32.6356	1.70356	0.851782	−	0.523897i	$-0.175522\pi$
$367$	32.6356	1.70356	0.851782	+	0.523897i	$0.175522\pi$
$368$	−1.00000	−0.0521286
$369$	9.71069	0.505518
$370$	−1.77203	−0.0921232
$371$	22.3029	1.15791
$372$	−9.27083	−0.480670
$373$	21.1617	1.09571	0.547855	−	0.836573i	$-0.315444\pi$
$373$	21.1617	1.09571	0.547855	+	0.836573i	$0.315444\pi$
$374$	27.7497	1.43490
$375$	4.54032	0.234461
$376$	11.9675	0.617177
$377$	−1.47447	−0.0759389
$378$	2.35800	0.121282
$379$	−30.9706	−1.59085	−0.795426	−	0.606051i	$-0.792753\pi$
$379$	−30.9706	−1.59085	−0.795426	+	0.606051i	$0.792753\pi$
$380$	−3.69742	−0.189674
$381$	−3.32519	−0.170355
$382$	−20.3059	−1.03894
$383$	−5.48854	−0.280451	−0.140226	−	0.990120i	$-0.544783\pi$
$383$	−5.48854	−0.280451	−0.140226	+	0.990120i	$0.544783\pi$
$384$	−1.00000	−0.0510310
$385$	4.65971	0.237481
$386$	3.08496	0.157020
$387$	5.23918	0.266323
$388$	15.6619	0.795115
$389$	−2.43548	−0.123484	−0.0617420	−	0.998092i	$-0.519666\pi$
$389$	−2.43548	−0.123484	−0.0617420	+	0.998092i	$0.519666\pi$
$390$	−0.684187	−0.0346451
$391$	6.51602	0.329529
$392$	1.43986	0.0727238
$393$	8.81266	0.444540
$394$	19.5714	0.985995
$395$	0.737384	0.0371018
$396$	4.25869	0.214007
$397$	21.1757	1.06278	0.531389	−	0.847128i	$-0.321670\pi$
$397$	21.1757	1.06278	0.531389	+	0.847128i	$0.321670\pi$
$398$	21.8115	1.09331
$399$	−18.7889	−0.940623
$400$	−4.78468	−0.239234
$401$	−2.14916	−0.107324	−0.0536620	−	0.998559i	$-0.517089\pi$
$401$	−2.14916	−0.107324	−0.0536620	+	0.998559i	$0.517089\pi$
$402$	−14.1044	−0.703465
$403$	13.6695	0.680928
$404$	−13.1685	−0.655155
$405$	−0.464023	−0.0230575
$406$	2.35800	0.117025
$407$	−16.2632	−0.806137
$408$	6.51602	0.322591
$409$	0.915866	0.0452867	0.0226433	−	0.999744i	$-0.492792\pi$
$409$	0.915866	0.0452867	0.0226433	+	0.999744i	$0.492792\pi$
$410$	4.50599	0.222535
$411$	−21.9768	−1.08404
$412$	−0.917126	−0.0451835
$413$	22.5244	1.10835
$414$	1.00000	0.0491473
$415$	−6.05546	−0.297251
$416$	1.47447	0.0722917
$417$	−7.06678	−0.346062
$418$	−33.9340	−1.65977
$419$	1.63718	0.0799814	0.0399907	−	0.999200i	$-0.487267\pi$
$419$	1.63718	0.0799814	0.0399907	+	0.999200i	$0.487267\pi$
$420$	1.09416	0.0533898
$421$	−5.35581	−0.261026	−0.130513	−	0.991447i	$-0.541662\pi$
$421$	−5.35581	−0.261026	−0.130513	+	0.991447i	$0.541662\pi$
$422$	−14.0420	−0.683554
$423$	−11.9675	−0.581880
$424$	9.45843	0.459342
$425$	31.1771	1.51231
$426$	−8.73384	−0.423156
$427$	−0.236575	−0.0114487
$428$	−13.8241	−0.668211
$429$	−6.27929	−0.303167
$430$	2.43110	0.117238
$431$	−30.8929	−1.48806	−0.744030	−	0.668146i	$-0.767088\pi$
$431$	−30.8929	−1.48806	−0.744030	+	0.668146i	$0.767088\pi$
$432$	1.00000	0.0481125
$433$	−25.0043	−1.20163	−0.600815	−	0.799388i	$-0.705157\pi$
$433$	−25.0043	−1.20163	−0.600815	+	0.799388i	$0.705157\pi$
$434$	−21.8606	−1.04934
$435$	−0.464023	−0.0222482
$436$	−19.1255	−0.915946
$437$	−7.96818	−0.381170
$438$	−1.03806	−0.0496005
$439$	−16.3561	−0.780632	−0.390316	−	0.920681i	$-0.627634\pi$
$439$	−16.3561	−0.780632	−0.390316	+	0.920681i	$0.627634\pi$
$440$	1.97613	0.0942083
$441$	−1.43986	−0.0685647
$442$	−9.60765	−0.456990
$443$	−9.64862	−0.458420	−0.229210	−	0.973377i	$-0.573614\pi$
$443$	−9.64862	−0.458420	−0.229210	+	0.973377i	$0.573614\pi$
$444$	−3.81883	−0.181234
$445$	1.92651	0.0913255
$446$	−9.27010	−0.438952
$447$	−6.52571	−0.308655
$448$	−2.35800	−0.111405
$449$	15.6790	0.739937	0.369969	−	0.929044i	$-0.379368\pi$
$449$	15.6790	0.739937	0.369969	+	0.929044i	$0.379368\pi$
$450$	4.78468	0.225552
$451$	41.3548	1.94732
$452$	−4.94300	−0.232499
$453$	4.49821	0.211344
$454$	29.1466	1.36792
$455$	−1.61331	−0.0756331
$456$	−7.96818	−0.373144
$457$	−2.51352	−0.117577	−0.0587887	−	0.998270i	$-0.518724\pi$
$457$	−2.51352	−0.117577	−0.0587887	+	0.998270i	$0.518724\pi$
$458$	−15.5618	−0.727153
$459$	−6.51602	−0.304142
$460$	0.464023	0.0216352
$461$	34.7409	1.61805	0.809023	−	0.587778i	$-0.199997\pi$
$461$	34.7409	1.61805	0.809023	+	0.587778i	$0.199997\pi$
$462$	10.0420	0.467194
$463$	−14.7451	−0.685265	−0.342632	−	0.939470i	$-0.611319\pi$
$463$	−14.7451	−0.685265	−0.342632	+	0.939470i	$0.611319\pi$
$464$	1.00000	0.0464238
$465$	4.30188	0.199495
$466$	11.5210	0.533701
$467$	−25.6704	−1.18788	−0.593942	−	0.804508i	$-0.702429\pi$
$467$	−25.6704	−1.18788	−0.593942	+	0.804508i	$0.702429\pi$
$468$	−1.47447	−0.0681572
$469$	−33.2582	−1.53572
$470$	−5.55320	−0.256150
$471$	−18.0304	−0.830798
$472$	9.55233	0.439682
$473$	22.3120	1.02591
$474$	1.58911	0.0729902
$475$	−38.1252	−1.74931
$476$	15.3647	0.704242
$477$	−9.45843	−0.433072
$478$	−15.3142	−0.700453
$479$	−26.1380	−1.19428	−0.597139	−	0.802138i	$-0.703696\pi$
$479$	−26.1380	−1.19428	−0.597139	+	0.802138i	$0.703696\pi$
$480$	0.464023	0.0211797
$481$	5.63073	0.256739
$482$	0.729295	0.0332185
$483$	2.35800	0.107293
$484$	7.13641	0.324382
$485$	−7.26751	−0.330001
$486$	−1.00000	−0.0453609
$487$	39.5607	1.79267	0.896334	−	0.443379i	$-0.146221\pi$
$487$	39.5607	1.79267	0.896334	+	0.443379i	$0.146221\pi$
$488$	−0.100329	−0.00454167
$489$	11.1464	0.504059
$490$	−0.668128	−0.0301829
$491$	−31.3153	−1.41324	−0.706621	−	0.707592i	$-0.749781\pi$
$491$	−31.3153	−1.41324	−0.706621	+	0.707592i	$0.749781\pi$
$492$	9.71069	0.437792
$493$	−6.51602	−0.293467
$494$	11.7488	0.528604
$495$	−1.97613	−0.0888204
$496$	−9.27083	−0.416273
$497$	−20.5944	−0.923783
$498$	−13.0499	−0.584780
$499$	−27.9966	−1.25330	−0.626650	−	0.779301i	$-0.715574\pi$
$499$	−27.9966	−1.25330	−0.626650	+	0.779301i	$0.715574\pi$
$500$	4.54032	0.203049
$501$	−13.9499	−0.623235
$502$	14.1551	0.631774
$503$	12.8674	0.573729	0.286865	−	0.957971i	$-0.407387\pi$
$503$	12.8674	0.573729	0.286865	+	0.957971i	$0.407387\pi$
$504$	2.35800	0.105033
$505$	6.11047	0.271912
$506$	4.25869	0.189322
$507$	−10.8259	−0.480797
$508$	−3.32519	−0.147532
$509$	35.2633	1.56302	0.781510	−	0.623893i	$-0.214450\pi$
$509$	35.2633	1.56302	0.781510	+	0.623893i	$0.214450\pi$
$510$	−3.02359	−0.133887
$511$	−2.44774	−0.108282
$512$	−1.00000	−0.0441942
$513$	7.96818	0.351804
$514$	−9.96166	−0.439390
$515$	0.425568	0.0187528
$516$	5.23918	0.230642
$517$	−50.9658	−2.24147
$518$	−9.00478	−0.395647
$519$	−19.8698	−0.872187
$520$	−0.684187	−0.0300036
$521$	32.9508	1.44360	0.721799	−	0.692102i	$-0.243315\pi$
$521$	32.9508	1.44360	0.721799	+	0.692102i	$0.243315\pi$
$522$	−1.00000	−0.0437688
$523$	−1.66264	−0.0727022	−0.0363511	−	0.999339i	$-0.511573\pi$
$523$	−1.66264	−0.0727022	−0.0363511	+	0.999339i	$0.511573\pi$
$524$	8.81266	0.384983
$525$	11.2823	0.492398
$526$	−8.14617	−0.355190
$527$	60.4089	2.63145
$528$	4.25869	0.185336
$529$	1.00000	0.0434783
$530$	−4.38893	−0.190643
$531$	−9.55233	−0.414536
$532$	−18.7889	−0.814604
$533$	−14.3181	−0.620185
$534$	4.15176	0.179664
$535$	6.41469	0.277331
$536$	−14.1044	−0.609219
$537$	−17.6806	−0.762976
$538$	5.05591	0.217976
$539$	−6.13190	−0.264120
$540$	−0.464023	−0.0199684
$541$	11.1755	0.480471	0.240236	−	0.970715i	$-0.422775\pi$
$541$	11.1755	0.480471	0.240236	+	0.970715i	$0.422775\pi$
$542$	−13.8835	−0.596348
$543$	17.2195	0.738958
$544$	6.51602	0.279372
$545$	8.87468	0.380150
$546$	−3.47678	−0.148793
$547$	0.0134387	0.000574596 0	0.000287298	−	1.00000i	$-0.499909\pi$
$547$	0.0134387	0.000574596 0	0.000287298		1.00000i	$0.499909\pi$
$548$	−21.9768	−0.938804
$549$	0.100329	0.00428193
$550$	20.3765	0.868855
$551$	7.96818	0.339456
$552$	1.00000	0.0425628
$553$	3.74711	0.159343
$554$	14.0987	0.598996
$555$	1.77203	0.0752183
$556$	−7.06678	−0.299698
$557$	−13.1184	−0.555846	−0.277923	−	0.960603i	$-0.589646\pi$
$557$	−13.1184	−0.555846	−0.277923	+	0.960603i	$0.589646\pi$
$558$	9.27083	0.392466
$559$	−7.72500	−0.326733
$560$	1.09416	0.0462369
$561$	−27.7497	−1.17159
$562$	25.7608	1.08665
$563$	5.43664	0.229127	0.114564	−	0.993416i	$-0.463453\pi$
$563$	5.43664	0.229127	0.114564	+	0.993416i	$0.463453\pi$
$564$	−11.9675	−0.503923
$565$	2.29367	0.0964952
$566$	15.6618	0.658316
$567$	−2.35800	−0.0990265
$568$	−8.73384	−0.366464
$569$	−14.4082	−0.604023	−0.302012	−	0.953304i	$-0.597658\pi$
$569$	−14.4082	−0.604023	−0.302012	+	0.953304i	$0.597658\pi$
$570$	3.69742	0.154868
$571$	43.1032	1.80381	0.901907	−	0.431929i	$-0.142167\pi$
$571$	43.1032	1.80381	0.901907	+	0.431929i	$0.142167\pi$
$572$	−6.27929	−0.262550
$573$	20.3059	0.848291
$574$	22.8978	0.955734
$575$	4.78468	0.199535
$576$	1.00000	0.0416667
$577$	−6.06518	−0.252497	−0.126248	−	0.991999i	$-0.540294\pi$
$577$	−6.06518	−0.252497	−0.126248	+	0.991999i	$0.540294\pi$
$578$	−25.4585	−1.05894
$579$	−3.08496	−0.128207
$580$	−0.464023	−0.0192675
$581$	−30.7716	−1.27662
$582$	−15.6619	−0.649208
$583$	−40.2805	−1.66825
$584$	−1.03806	−0.0429553
$585$	0.684187	0.0282876
$586$	−8.37521	−0.345977
$587$	−33.7101	−1.39136	−0.695682	−	0.718350i	$-0.744898\pi$
$587$	−33.7101	−1.39136	−0.695682	+	0.718350i	$0.744898\pi$
$588$	−1.43986	−0.0593787
$589$	−73.8717	−3.04383
$590$	−4.43250	−0.182483
$591$	−19.5714	−0.805061
$592$	−3.81883	−0.156953
$593$	28.2972	1.16203	0.581013	−	0.813894i	$-0.302657\pi$
$593$	28.2972	1.16203	0.581013	+	0.813894i	$0.302657\pi$
$594$	−4.25869	−0.174736
$595$	−7.12960	−0.292285
$596$	−6.52571	−0.267303
$597$	−21.8115	−0.892685
$598$	−1.47447	−0.0602954
$599$	−17.3190	−0.707635	−0.353817	−	0.935315i	$-0.615117\pi$
$599$	−17.3190	−0.707635	−0.353817	+	0.935315i	$0.615117\pi$
$600$	4.78468	0.195334
$601$	−30.1342	−1.22920	−0.614599	−	0.788839i	$-0.710682\pi$
$601$	−30.1342	−1.22920	−0.614599	+	0.788839i	$0.710682\pi$
$602$	12.3540	0.503510
$603$	14.1044	0.574377
$604$	4.49821	0.183030
$605$	−3.31146	−0.134630
$606$	13.1685	0.534932
$607$	12.0663	0.489758	0.244879	−	0.969554i	$-0.421252\pi$
$607$	12.0663	0.489758	0.244879	+	0.969554i	$0.421252\pi$
$608$	−7.96818	−0.323152
$609$	−2.35800	−0.0955508
$610$	0.0465549	0.00188495
$611$	17.6457	0.713868
$612$	−6.51602	−0.263395
$613$	5.36129	0.216540	0.108270	−	0.994122i	$-0.465469\pi$
$613$	5.36129	0.216540	0.108270	+	0.994122i	$0.465469\pi$
$614$	−25.6340	−1.03450
$615$	−4.50599	−0.181699
$616$	10.0420	0.404602
$617$	−0.112166	−0.00451564	−0.00225782	−	0.999997i	$-0.500719\pi$
$617$	−0.112166	−0.00451564	−0.00225782	+	0.999997i	$0.500719\pi$
$618$	0.917126	0.0368922
$619$	4.76718	0.191609	0.0958046	−	0.995400i	$-0.469458\pi$
$619$	4.76718	0.191609	0.0958046	+	0.995400i	$0.469458\pi$
$620$	4.30188	0.172768
$621$	−1.00000	−0.0401286
$622$	−10.8826	−0.436351
$623$	9.78983	0.392221
$624$	−1.47447	−0.0590259
$625$	21.8166	0.872664
$626$	5.71391	0.228374
$627$	33.9340	1.35519
$628$	−18.0304	−0.719492
$629$	24.8836	0.992173
$630$	−1.09416	−0.0435926
$631$	−4.81837	−0.191816	−0.0959082	−	0.995390i	$-0.530576\pi$
$631$	−4.81837	−0.191816	−0.0959082	+	0.995390i	$0.530576\pi$
$632$	1.58911	0.0632114
$633$	14.0420	0.558119
$634$	−2.26469	−0.0899424
$635$	1.54297	0.0612308
$636$	−9.45843	−0.375051
$637$	2.12302	0.0841172
$638$	−4.25869	−0.168603
$639$	8.73384	0.345505
$640$	0.464023	0.0183421
$641$	9.03591	0.356897	0.178448	−	0.983949i	$-0.442892\pi$
$641$	9.03591	0.356897	0.178448	+	0.983949i	$0.442892\pi$
$642$	13.8241	0.545592
$643$	−23.1711	−0.913779	−0.456889	−	0.889523i	$-0.651037\pi$
$643$	−23.1711	−0.913779	−0.456889	+	0.889523i	$0.651037\pi$
$644$	2.35800	0.0929180
$645$	−2.43110	−0.0957246
$646$	51.9208	2.04280
$647$	5.79213	0.227712	0.113856	−	0.993497i	$-0.463680\pi$
$647$	5.79213	0.227712	0.113856	+	0.993497i	$0.463680\pi$
$648$	−1.00000	−0.0392837
$649$	−40.6804	−1.59684
$650$	−7.05485	−0.276714
$651$	21.8606	0.856784
$652$	11.1464	0.436528
$653$	32.0401	1.25383	0.626913	−	0.779089i	$-0.284318\pi$
$653$	32.0401	1.25383	0.626913	+	0.779089i	$0.284318\pi$
$654$	19.1255	0.747866
$655$	−4.08928	−0.159781
$656$	9.71069	0.379139
$657$	1.03806	0.0404986
$658$	−28.2193	−1.10010
$659$	41.5073	1.61690	0.808448	−	0.588567i	$-0.200308\pi$
$659$	41.5073	1.61690	0.808448	+	0.588567i	$0.200308\pi$
$660$	−1.97613	−0.0769208
$661$	−22.5290	−0.876276	−0.438138	−	0.898908i	$-0.644362\pi$
$661$	−22.5290	−0.876276	−0.438138	+	0.898908i	$0.644362\pi$
$662$	11.5459	0.448744
$663$	9.60765	0.373130
$664$	−13.0499	−0.506435
$665$	8.71851	0.338089
$666$	3.81883	0.147977
$667$	−1.00000	−0.0387202
$668$	−13.9499	−0.539737
$669$	9.27010	0.358403
$670$	6.54479	0.252847
$671$	0.427269	0.0164945
$672$	2.35800	0.0909616
$673$	−17.1587	−0.661418	−0.330709	−	0.943733i	$-0.607288\pi$
$673$	−17.1587	−0.661418	−0.330709	+	0.943733i	$0.607288\pi$
$674$	7.63964	0.294268
$675$	−4.78468	−0.184163
$676$	−10.8259	−0.416383
$677$	−22.7270	−0.873468	−0.436734	−	0.899591i	$-0.643865\pi$
$677$	−22.7270	−0.873468	−0.436734	+	0.899591i	$0.643865\pi$
$678$	4.94300	0.189835
$679$	−36.9308	−1.41727
$680$	−3.02359	−0.115949
$681$	−29.1466	−1.11690
$682$	39.4816	1.51183
$683$	6.88772	0.263551	0.131776	−	0.991280i	$-0.457932\pi$
$683$	6.88772	0.263551	0.131776	+	0.991280i	$0.457932\pi$
$684$	7.96818	0.304671
$685$	10.1978	0.389637
$686$	−19.9011	−0.759829
$687$	15.5618	0.593718
$688$	5.23918	0.199742
$689$	13.9461	0.531305
$690$	−0.464023	−0.0176651
$691$	1.34817	0.0512868	0.0256434	−	0.999671i	$-0.491837\pi$
$691$	1.34817	0.0512868	0.0256434	+	0.999671i	$0.491837\pi$
$692$	−19.8698	−0.755336
$693$	−10.0420	−0.381463
$694$	29.2376	1.10984
$695$	3.27915	0.124385
$696$	−1.00000	−0.0379049
$697$	−63.2751	−2.39671
$698$	−4.57499	−0.173166
$699$	−11.5210	−0.435765
$700$	11.2823	0.426429
$701$	22.8514	0.863084	0.431542	−	0.902093i	$-0.357970\pi$
$701$	22.8514	0.863084	0.431542	+	0.902093i	$0.357970\pi$
$702$	1.47447	0.0556501
$703$	−30.4291	−1.14766
$704$	4.25869	0.160505
$705$	5.55320	0.209146
$706$	−9.73503	−0.366383
$707$	31.0512	1.16780
$708$	−9.55233	−0.358999
$709$	17.8223	0.669330	0.334665	−	0.942337i	$-0.391377\pi$
$709$	17.8223	0.669330	0.334665	+	0.942337i	$0.391377\pi$
$710$	4.05271	0.152095
$711$	−1.58911	−0.0595963
$712$	4.15176	0.155594
$713$	9.27083	0.347195
$714$	−15.3647	−0.575011
$715$	2.91374	0.108968
$716$	−17.6806	−0.660757
$717$	15.3142	0.571918
$718$	16.6817	0.622556
$719$	12.8812	0.480389	0.240194	−	0.970725i	$-0.422789\pi$
$719$	12.8812	0.480389	0.240194	+	0.970725i	$0.422789\pi$
$720$	−0.464023	−0.0172931
$721$	2.16258	0.0805386
$722$	−44.4919	−1.65582
$723$	−0.729295	−0.0271228
$724$	17.2195	0.639957
$725$	−4.78468	−0.177699
$726$	−7.13641	−0.264857
$727$	11.6980	0.433854	0.216927	−	0.976188i	$-0.430397\pi$
$727$	11.6980	0.433854	0.216927	+	0.976188i	$0.430397\pi$
$728$	−3.47678	−0.128858
$729$	1.00000	0.0370370
$730$	0.481685	0.0178279
$731$	−34.1386	−1.26266
$732$	0.100329	0.00370826
$733$	3.92528	0.144984	0.0724918	−	0.997369i	$-0.476905\pi$
$733$	3.92528	0.144984	0.0724918	+	0.997369i	$0.476905\pi$
$734$	−32.6356	−1.20460
$735$	0.668128	0.0246443
$736$	1.00000	0.0368605
$737$	60.0664	2.21257
$738$	−9.71069	−0.357455
$739$	−2.69549	−0.0991552	−0.0495776	−	0.998770i	$-0.515788\pi$
$739$	−2.69549	−0.0991552	−0.0495776	+	0.998770i	$0.515788\pi$
$740$	1.77203	0.0651410
$741$	−11.7488	−0.431604
$742$	−22.3029	−0.818766
$743$	3.66376	0.134410	0.0672051	−	0.997739i	$-0.478592\pi$
$743$	3.66376	0.134410	0.0672051	+	0.997739i	$0.478592\pi$
$744$	9.27083	0.339885
$745$	3.02808	0.110940
$746$	−21.1617	−0.774784
$747$	13.0499	0.477471
$748$	−27.7497	−1.01463
$749$	32.5971	1.19107
$750$	−4.54032	−0.165789
$751$	−13.1560	−0.480069	−0.240034	−	0.970764i	$-0.577159\pi$
$751$	−13.1560	−0.480069	−0.240034	+	0.970764i	$0.577159\pi$
$752$	−11.9675	−0.436410
$753$	−14.1551	−0.515842
$754$	1.47447	0.0536969
$755$	−2.08728	−0.0759637
$756$	−2.35800	−0.0857595
$757$	−0.762181	−0.0277019	−0.0138510	−	0.999904i	$-0.504409\pi$
$757$	−0.762181	−0.0277019	−0.0138510	+	0.999904i	$0.504409\pi$
$758$	30.9706	1.12490
$759$	−4.25869	−0.154581
$760$	3.69742	0.134120
$761$	19.5742	0.709565	0.354783	−	0.934949i	$-0.384555\pi$
$761$	19.5742	0.709565	0.354783	+	0.934949i	$0.384555\pi$
$762$	3.32519	0.120459
$763$	45.0978	1.63265
$764$	20.3059	0.734641
$765$	3.02359	0.109318
$766$	5.48854	0.198309
$767$	14.0846	0.508565
$768$	1.00000	0.0360844
$769$	−3.68597	−0.132919	−0.0664597	−	0.997789i	$-0.521170\pi$
$769$	−3.68597	−0.132919	−0.0664597	+	0.997789i	$0.521170\pi$
$770$	−4.65971	−0.167924
$771$	9.96166	0.358761
$772$	−3.08496	−0.111030
$773$	32.5087	1.16926	0.584628	−	0.811301i	$-0.301240\pi$
$773$	32.5087	1.16926	0.584628	+	0.811301i	$0.301240\pi$
$774$	−5.23918	−0.188319
$775$	44.3580	1.59339
$776$	−15.6619	−0.562231
$777$	9.00478	0.323045
$778$	2.43548	0.0873164
$779$	77.3765	2.77230
$780$	0.684187	0.0244978
$781$	37.1947	1.33093
$782$	−6.51602	−0.233012
$783$	1.00000	0.0357371
$784$	−1.43986	−0.0514235
$785$	8.36653	0.298614
$786$	−8.81266	−0.314337
$787$	44.3782	1.58191	0.790957	−	0.611872i	$-0.209583\pi$
$787$	44.3782	1.58191	0.790957	+	0.611872i	$0.209583\pi$
$788$	−19.5714	−0.697204
$789$	8.14617	0.290011
$790$	−0.737384	−0.0262349
$791$	11.6556	0.414424
$792$	−4.25869	−0.151326
$793$	−0.147932	−0.00525320
$794$	−21.1757	−0.751497
$795$	4.38893	0.155659
$796$	−21.8115	−0.773088
$797$	43.4038	1.53744	0.768720	−	0.639585i	$-0.220894\pi$
$797$	43.4038	1.53744	0.768720	+	0.639585i	$0.220894\pi$
$798$	18.7889	0.665121
$799$	77.9805	2.75875
$800$	4.78468	0.169164
$801$	−4.15176	−0.146695
$802$	2.14916	0.0758895
$803$	4.42078	0.156006
$804$	14.1044	0.497425
$805$	−1.09416	−0.0385642
$806$	−13.6695	−0.481489
$807$	−5.05591	−0.177977
$808$	13.1685	0.463265
$809$	28.2367	0.992749	0.496374	−	0.868109i	$-0.334664\pi$
$809$	28.2367	0.992749	0.496374	+	0.868109i	$0.334664\pi$
$810$	0.464023	0.0163041
$811$	−34.6295	−1.21601	−0.608003	−	0.793934i	$-0.708029\pi$
$811$	−34.6295	−1.21601	−0.608003	+	0.793934i	$0.708029\pi$
$812$	−2.35800	−0.0827494
$813$	13.8835	0.486916
$814$	16.2632	0.570025
$815$	−5.17221	−0.181174
$816$	−6.51602	−0.228106
$817$	41.7468	1.46053
$818$	−0.915866	−0.0320225
$819$	3.47678	0.121489
$820$	−4.50599	−0.157356
$821$	16.7600	0.584928	0.292464	−	0.956276i	$-0.405525\pi$
$821$	16.7600	0.584928	0.292464	+	0.956276i	$0.405525\pi$
$822$	21.9768	0.766530
$823$	24.8040	0.864612	0.432306	−	0.901727i	$-0.357700\pi$
$823$	24.8040	0.864612	0.432306	+	0.901727i	$0.357700\pi$
$824$	0.917126	0.0319496
$825$	−20.3765	−0.709417
$826$	−22.5244	−0.783722
$827$	−8.53265	−0.296709	−0.148355	−	0.988934i	$-0.547398\pi$
$827$	−8.53265	−0.296709	−0.148355	+	0.988934i	$0.547398\pi$
$828$	−1.00000	−0.0347524
$829$	26.2227	0.910751	0.455375	−	0.890299i	$-0.349505\pi$
$829$	26.2227	0.910751	0.455375	+	0.890299i	$0.349505\pi$
$830$	6.05546	0.210188
$831$	−14.0987	−0.489078
$832$	−1.47447	−0.0511179
$833$	9.38215	0.325072
$834$	7.06678	0.244702
$835$	6.47307	0.224010
$836$	33.9340	1.17363
$837$	−9.27083	−0.320447
$838$	−1.63718	−0.0565554
$839$	5.00605	0.172828	0.0864139	−	0.996259i	$-0.472459\pi$
$839$	5.00605	0.172828	0.0864139	+	0.996259i	$0.472459\pi$
$840$	−1.09416	−0.0377523
$841$	1.00000	0.0344828
$842$	5.35581	0.184573
$843$	−25.7608	−0.887249
$844$	14.0420	0.483346
$845$	5.02349	0.172813
$846$	11.9675	0.411451
$847$	−16.8276	−0.578204
$848$	−9.45843	−0.324804
$849$	−15.6618	−0.537513
$850$	−31.1771	−1.06937
$851$	3.81883	0.130908
$852$	8.73384	0.299217
$853$	−47.8064	−1.63686	−0.818430	−	0.574607i	$-0.805155\pi$
$853$	−47.8064	−1.63686	−0.818430	+	0.574607i	$0.805155\pi$
$854$	0.236575	0.00809543
$855$	−3.69742	−0.126449
$856$	13.8241	0.472497
$857$	−4.86054	−0.166033	−0.0830164	−	0.996548i	$-0.526455\pi$
$857$	−4.86054	−0.166033	−0.0830164	+	0.996548i	$0.526455\pi$
$858$	6.27929	0.214371
$859$	54.6105	1.86329	0.931643	−	0.363375i	$-0.118376\pi$
$859$	54.6105	1.86329	0.931643	+	0.363375i	$0.118376\pi$
$860$	−2.43110	−0.0829000
$861$	−22.8978	−0.780354
$862$	30.8929	1.05222
$863$	16.4570	0.560204	0.280102	−	0.959970i	$-0.409632\pi$
$863$	16.4570	0.560204	0.280102	+	0.959970i	$0.409632\pi$
$864$	−1.00000	−0.0340207
$865$	9.22004	0.313491
$866$	25.0043	0.849681
$867$	25.4585	0.864617
$868$	21.8606	0.741996
$869$	−6.76752	−0.229572
$870$	0.464023	0.0157319
$871$	−20.7965	−0.704663
$872$	19.1255	0.647671
$873$	15.6619	0.530076
$874$	7.96818	0.269528
$875$	−10.7061	−0.361931
$876$	1.03806	0.0350728
$877$	−13.3064	−0.449326	−0.224663	−	0.974436i	$-0.572128\pi$
$877$	−13.3064	−0.449326	−0.224663	+	0.974436i	$0.572128\pi$
$878$	16.3561	0.551990
$879$	8.37521	0.282489
$880$	−1.97613	−0.0666153
$881$	−30.0770	−1.01332	−0.506661	−	0.862146i	$-0.669120\pi$
$881$	−30.0770	−1.01332	−0.506661	+	0.862146i	$0.669120\pi$
$882$	1.43986	0.0484825
$883$	16.6503	0.560326	0.280163	−	0.959952i	$-0.409612\pi$
$883$	16.6503	0.560326	0.280163	+	0.959952i	$0.409612\pi$
$884$	9.60765	0.323140
$885$	4.43250	0.148997
$886$	9.64862	0.324152
$887$	18.1766	0.610311	0.305155	−	0.952303i	$-0.401292\pi$
$887$	18.1766	0.610311	0.305155	+	0.952303i	$0.401292\pi$
$888$	3.81883	0.128151
$889$	7.84079	0.262972
$890$	−1.92651	−0.0645769
$891$	4.25869	0.142671
$892$	9.27010	0.310386
$893$	−95.3592	−3.19107
$894$	6.52571	0.218252
$895$	8.20423	0.274237
$896$	2.35800	0.0787751
$897$	1.47447	0.0492310
$898$	−15.6790	−0.523215
$899$	−9.27083	−0.309200
$900$	−4.78468	−0.159489
$901$	61.6313	2.05324
$902$	−41.3548	−1.37696
$903$	−12.3540	−0.411115
$904$	4.94300	0.164402
$905$	−7.99024	−0.265605
$906$	−4.49821	−0.149443
$907$	17.5933	0.584177	0.292088	−	0.956391i	$-0.405650\pi$
$907$	17.5933	0.584177	0.292088	+	0.956391i	$0.405650\pi$
$908$	−29.1466	−0.967264
$909$	−13.1685	−0.436770
$910$	1.61331	0.0534807
$911$	−24.0851	−0.797974	−0.398987	−	0.916957i	$-0.630638\pi$
$911$	−24.0851	−0.797974	−0.398987	+	0.916957i	$0.630638\pi$
$912$	7.96818	0.263853
$913$	55.5755	1.83928
$914$	2.51352	0.0831398
$915$	−0.0465549	−0.00153906
$916$	15.5618	0.514175
$917$	−20.7802	−0.686223
$918$	6.51602	0.215061
$919$	−30.0913	−0.992619	−0.496310	−	0.868146i	$-0.665312\pi$
$919$	−30.0913	−0.992619	−0.496310	+	0.868146i	$0.665312\pi$
$920$	−0.464023	−0.0152984
$921$	25.6340	0.844668
$922$	−34.7409	−1.14413
$923$	−12.8778	−0.423877
$924$	−10.0420	−0.330356
$925$	18.2719	0.600776
$926$	14.7451	0.484555
$927$	−0.917126	−0.0301224
$928$	−1.00000	−0.0328266
$929$	−22.7827	−0.747475	−0.373737	−	0.927535i	$-0.621924\pi$
$929$	−22.7827	−0.747475	−0.373737	+	0.927535i	$0.621924\pi$
$930$	−4.30188	−0.141064
$931$	−11.4731	−0.376014
$932$	−11.5210	−0.377383
$933$	10.8826	0.356279
$934$	25.6704	0.839961
$935$	12.8765	0.421107
$936$	1.47447	0.0481944
$937$	46.6315	1.52338	0.761692	−	0.647939i	$-0.224369\pi$
$937$	46.6315	1.52338	0.761692	+	0.647939i	$0.224369\pi$
$938$	33.2582	1.08592
$939$	−5.71391	−0.186467
$940$	5.55320	0.181125
$941$	0.0577411	0.00188231	0.000941153	−	1.00000i	$-0.499700\pi$
$941$	0.0577411	0.00188231	0.000941153		1.00000i	$0.499700\pi$
$942$	18.0304	0.587463
$943$	−9.71069	−0.316224
$944$	−9.55233	−0.310902
$945$	1.09416	0.0355932
$946$	−22.3120	−0.725427
$947$	−33.2987	−1.08206	−0.541031	−	0.841002i	$-0.681966\pi$
$947$	−33.2987	−1.08206	−0.541031	+	0.841002i	$0.681966\pi$
$948$	−1.58911	−0.0516119
$949$	−1.53059	−0.0496849
$950$	38.1252	1.23695
$951$	2.26469	0.0734377
$952$	−15.3647	−0.497974
$953$	−29.7060	−0.962272	−0.481136	−	0.876646i	$-0.659776\pi$
$953$	−29.7060	−0.962272	−0.481136	+	0.876646i	$0.659776\pi$
$954$	9.45843	0.306228
$955$	−9.42241	−0.304902
$956$	15.3142	0.495295
$957$	4.25869	0.137664
$958$	26.1380	0.844481
$959$	51.8213	1.67340
$960$	−0.464023	−0.0149763
$961$	54.9483	1.77253
$962$	−5.63073	−0.181542
$963$	−13.8241	−0.445474
$964$	−0.729295	−0.0234890
$965$	1.43149	0.0460814
$966$	−2.35800	−0.0758673
$967$	31.0142	0.997349	0.498674	−	0.866789i	$-0.333820\pi$
$967$	31.0142	0.997349	0.498674	+	0.866789i	$0.333820\pi$
$968$	−7.13641	−0.229373
$969$	−51.9208	−1.66794
$970$	7.26751	0.233346
$971$	−48.6062	−1.55985	−0.779924	−	0.625875i	$-0.784742\pi$
$971$	−48.6062	−1.55985	−0.779924	+	0.625875i	$0.784742\pi$
$972$	1.00000	0.0320750
$973$	16.6634	0.534205
$974$	−39.5607	−1.26761
$975$	7.05485	0.225936
$976$	0.100329	0.00321145
$977$	−13.3352	−0.426631	−0.213315	−	0.976983i	$-0.568426\pi$
$977$	−13.3352	−0.426631	−0.213315	+	0.976983i	$0.568426\pi$
$978$	−11.1464	−0.356424
$979$	−17.6810	−0.565089
$980$	0.668128	0.0213426
$981$	−19.1255	−0.610630
$982$	31.3153	0.999313
$983$	30.8054	0.982540	0.491270	−	0.871007i	$-0.336533\pi$
$983$	30.8054	0.982540	0.491270	+	0.871007i	$0.336533\pi$
$984$	−9.71069	−0.309565
$985$	9.08160	0.289364
$986$	6.51602	0.207512
$987$	28.2193	0.898231
$988$	−11.7488	−0.373780
$989$	−5.23918	−0.166596
$990$	1.97613	0.0628055
$991$	−7.71487	−0.245071	−0.122535	−	0.992464i	$-0.539103\pi$
$991$	−7.71487	−0.245071	−0.122535	+	0.992464i	$0.539103\pi$
$992$	9.27083	0.294349
$993$	−11.5459	−0.366398
$994$	20.5944	0.653213
$995$	10.1210	0.320859
$996$	13.0499	0.413502
$997$	−1.69912	−0.0538115	−0.0269058	−	0.999638i	$-0.508565\pi$
$997$	−1.69912	−0.0538115	−0.0269058	+	0.999638i	$0.508565\pi$
$998$	27.9966	0.886216
$999$	−3.81883	−0.120822

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bi.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bi.1.5	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.464023 q^{5} -1.00000 q^{6} -2.35800 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.464023 q^{5} -1.00000 q^{6} -2.35800 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.464023 q^{10} +4.25869 q^{11} +1.00000 q^{12} -1.47447 q^{13} +2.35800 q^{14} -0.464023 q^{15} +1.00000 q^{16} -6.51602 q^{17} -1.00000 q^{18} +7.96818 q^{19} -0.464023 q^{20} -2.35800 q^{21} -4.25869 q^{22} -1.00000 q^{23} -1.00000 q^{24} -4.78468 q^{25} +1.47447 q^{26} +1.00000 q^{27} -2.35800 q^{28} +1.00000 q^{29} +0.464023 q^{30} -9.27083 q^{31} -1.00000 q^{32} +4.25869 q^{33} +6.51602 q^{34} +1.09416 q^{35} +1.00000 q^{36} -3.81883 q^{37} -7.96818 q^{38} -1.47447 q^{39} +0.464023 q^{40} +9.71069 q^{41} +2.35800 q^{42} +5.23918 q^{43} +4.25869 q^{44} -0.464023 q^{45} +1.00000 q^{46} -11.9675 q^{47} +1.00000 q^{48} -1.43986 q^{49} +4.78468 q^{50} -6.51602 q^{51} -1.47447 q^{52} -9.45843 q^{53} -1.00000 q^{54} -1.97613 q^{55} +2.35800 q^{56} +7.96818 q^{57} -1.00000 q^{58} -9.55233 q^{59} -0.464023 q^{60} +0.100329 q^{61} +9.27083 q^{62} -2.35800 q^{63} +1.00000 q^{64} +0.684187 q^{65} -4.25869 q^{66} +14.1044 q^{67} -6.51602 q^{68} -1.00000 q^{69} -1.09416 q^{70} +8.73384 q^{71} -1.00000 q^{72} +1.03806 q^{73} +3.81883 q^{74} -4.78468 q^{75} +7.96818 q^{76} -10.0420 q^{77} +1.47447 q^{78} -1.58911 q^{79} -0.464023 q^{80} +1.00000 q^{81} -9.71069 q^{82} +13.0499 q^{83} -2.35800 q^{84} +3.02359 q^{85} -5.23918 q^{86} +1.00000 q^{87} -4.25869 q^{88} -4.15176 q^{89} +0.464023 q^{90} +3.47678 q^{91} -1.00000 q^{92} -9.27083 q^{93} +11.9675 q^{94} -3.69742 q^{95} -1.00000 q^{96} +15.6619 q^{97} +1.43986 q^{98} +4.25869 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 3 q^{5} - 8 q^{6} - 6 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 3 * q^5 - 8 * q^6 - 6 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 3 q^{5} - 8 q^{6} - 6 q^{7} - 8 q^{8} + 8 q^{9} + 3 q^{10} - 7 q^{11} + 8 q^{12} - 3 q^{13} + 6 q^{14} - 3 q^{15} + 8 q^{16} - 12 q^{17} - 8 q^{18} - 4 q^{19} - 3 q^{20} - 6 q^{21} + 7 q^{22} - 8 q^{23} - 8 q^{24} + 13 q^{25} + 3 q^{26} + 8 q^{27} - 6 q^{28} + 8 q^{29} + 3 q^{30} - q^{31} - 8 q^{32} - 7 q^{33} + 12 q^{34} - 6 q^{35} + 8 q^{36} - 11 q^{37} + 4 q^{38} - 3 q^{39} + 3 q^{40} - 17 q^{41} + 6 q^{42} - 10 q^{43} - 7 q^{44} - 3 q^{45} + 8 q^{46} - 26 q^{47} + 8 q^{48} + 10 q^{49} - 13 q^{50} - 12 q^{51} - 3 q^{52} - 2 q^{53} - 8 q^{54} + q^{55} + 6 q^{56} - 4 q^{57} - 8 q^{58} - 25 q^{59} - 3 q^{60} + 3 q^{61} + q^{62} - 6 q^{63} + 8 q^{64} - 25 q^{65} + 7 q^{66} + q^{67} - 12 q^{68} - 8 q^{69} + 6 q^{70} - 27 q^{71} - 8 q^{72} + 16 q^{73} + 11 q^{74} + 13 q^{75} - 4 q^{76} - 16 q^{77} + 3 q^{78} - 6 q^{79} - 3 q^{80} + 8 q^{81} + 17 q^{82} - 44 q^{83} - 6 q^{84} - 20 q^{85} + 10 q^{86} + 8 q^{87} + 7 q^{88} - 52 q^{89} + 3 q^{90} - 18 q^{91} - 8 q^{92} - q^{93} + 26 q^{94} - 56 q^{95} - 8 q^{96} - 4 q^{97} - 10 q^{98} - 7 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 3 * q^5 - 8 * q^6 - 6 * q^7 - 8 * q^8 + 8 * q^9 + 3 * q^10 - 7 * q^11 + 8 * q^12 - 3 * q^13 + 6 * q^14 - 3 * q^15 + 8 * q^16 - 12 * q^17 - 8 * q^18 - 4 * q^19 - 3 * q^20 - 6 * q^21 + 7 * q^22 - 8 * q^23 - 8 * q^24 + 13 * q^25 + 3 * q^26 + 8 * q^27 - 6 * q^28 + 8 * q^29 + 3 * q^30 - q^31 - 8 * q^32 - 7 * q^33 + 12 * q^34 - 6 * q^35 + 8 * q^36 - 11 * q^37 + 4 * q^38 - 3 * q^39 + 3 * q^40 - 17 * q^41 + 6 * q^42 - 10 * q^43 - 7 * q^44 - 3 * q^45 + 8 * q^46 - 26 * q^47 + 8 * q^48 + 10 * q^49 - 13 * q^50 - 12 * q^51 - 3 * q^52 - 2 * q^53 - 8 * q^54 + q^55 + 6 * q^56 - 4 * q^57 - 8 * q^58 - 25 * q^59 - 3 * q^60 + 3 * q^61 + q^62 - 6 * q^63 + 8 * q^64 - 25 * q^65 + 7 * q^66 + q^67 - 12 * q^68 - 8 * q^69 + 6 * q^70 - 27 * q^71 - 8 * q^72 + 16 * q^73 + 11 * q^74 + 13 * q^75 - 4 * q^76 - 16 * q^77 + 3 * q^78 - 6 * q^79 - 3 * q^80 + 8 * q^81 + 17 * q^82 - 44 * q^83 - 6 * q^84 - 20 * q^85 + 10 * q^86 + 8 * q^87 + 7 * q^88 - 52 * q^89 + 3 * q^90 - 18 * q^91 - 8 * q^92 - q^93 + 26 * q^94 - 56 * q^95 - 8 * q^96 - 4 * q^97 - 10 * q^98 - 7 * q^99

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