LMFDB - Embedded newform 8002.2.a.e.1.35

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.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:	$63.8962916974$
Analytic rank:	$0$
Dimension:	$77$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.35
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.200784 q^{3} +1.00000 q^{4} -2.29596 q^{5} +0.200784 q^{6} -0.443206 q^{7} -1.00000 q^{8} -2.95969 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.200784 q^{3} +1.00000 q^{4} -2.29596 q^{5} +0.200784 q^{6} -0.443206 q^{7} -1.00000 q^{8} -2.95969 q^{9} +2.29596 q^{10} +0.0657130 q^{11} -0.200784 q^{12} -5.22204 q^{13} +0.443206 q^{14} +0.460992 q^{15} +1.00000 q^{16} -6.05974 q^{17} +2.95969 q^{18} +0.987451 q^{19} -2.29596 q^{20} +0.0889888 q^{21} -0.0657130 q^{22} +5.95236 q^{23} +0.200784 q^{24} +0.271430 q^{25} +5.22204 q^{26} +1.19661 q^{27} -0.443206 q^{28} -3.90713 q^{29} -0.460992 q^{30} -6.48171 q^{31} -1.00000 q^{32} -0.0131941 q^{33} +6.05974 q^{34} +1.01758 q^{35} -2.95969 q^{36} -1.23889 q^{37} -0.987451 q^{38} +1.04850 q^{39} +2.29596 q^{40} +6.05668 q^{41} -0.0889888 q^{42} -4.24894 q^{43} +0.0657130 q^{44} +6.79532 q^{45} -5.95236 q^{46} -2.96424 q^{47} -0.200784 q^{48} -6.80357 q^{49} -0.271430 q^{50} +1.21670 q^{51} -5.22204 q^{52} -11.7849 q^{53} -1.19661 q^{54} -0.150874 q^{55} +0.443206 q^{56} -0.198264 q^{57} +3.90713 q^{58} -10.7757 q^{59} +0.460992 q^{60} -2.55019 q^{61} +6.48171 q^{62} +1.31175 q^{63} +1.00000 q^{64} +11.9896 q^{65} +0.0131941 q^{66} -3.49450 q^{67} -6.05974 q^{68} -1.19514 q^{69} -1.01758 q^{70} -7.48186 q^{71} +2.95969 q^{72} +4.47735 q^{73} +1.23889 q^{74} -0.0544988 q^{75} +0.987451 q^{76} -0.0291244 q^{77} -1.04850 q^{78} -13.1542 q^{79} -2.29596 q^{80} +8.63880 q^{81} -6.05668 q^{82} -10.9279 q^{83} +0.0889888 q^{84} +13.9129 q^{85} +4.24894 q^{86} +0.784490 q^{87} -0.0657130 q^{88} -1.90357 q^{89} -6.79532 q^{90} +2.31444 q^{91} +5.95236 q^{92} +1.30143 q^{93} +2.96424 q^{94} -2.26715 q^{95} +0.200784 q^{96} +3.05381 q^{97} +6.80357 q^{98} -0.194490 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) $ 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * 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
$3$	−0.200784	−0.115923	−0.0579614	−	0.998319i	$-0.518460\pi$
$3$	−0.200784	−0.115923	−0.0579614	+	0.998319i	$0.518460\pi$
$4$	1.00000	0.500000
$5$	−2.29596	−1.02678	−0.513392	−	0.858154i	$-0.671611\pi$
$5$	−2.29596	−1.02678	−0.513392	+	0.858154i	$0.671611\pi$
$6$	0.200784	0.0819698
$7$	−0.443206	−0.167516	−0.0837581	−	0.996486i	$-0.526692\pi$
$7$	−0.443206	−0.167516	−0.0837581	+	0.996486i	$0.526692\pi$
$8$	−1.00000	−0.353553
$9$	−2.95969	−0.986562
$10$	2.29596	0.726046
$11$	0.0657130	0.0198132	0.00990661	−	0.999951i	$-0.496847\pi$
$11$	0.0657130	0.0198132	0.00990661	+	0.999951i	$0.496847\pi$
$12$	−0.200784	−0.0579614
$13$	−5.22204	−1.44833	−0.724167	−	0.689625i	$-0.757775\pi$
$13$	−5.22204	−1.44833	−0.724167	+	0.689625i	$0.757775\pi$
$14$	0.443206	0.118452
$15$	0.460992	0.119028
$16$	1.00000	0.250000
$17$	−6.05974	−1.46970	−0.734851	−	0.678229i	$-0.762748\pi$
$17$	−6.05974	−1.46970	−0.734851	+	0.678229i	$0.762748\pi$
$18$	2.95969	0.697605
$19$	0.987451	0.226537	0.113268	−	0.993564i	$-0.463868\pi$
$19$	0.987451	0.226537	0.113268	+	0.993564i	$0.463868\pi$
$20$	−2.29596	−0.513392
$21$	0.0889888	0.0194190
$22$	−0.0657130	−0.0140101
$23$	5.95236	1.24115	0.620577	−	0.784146i	$-0.286898\pi$
$23$	5.95236	1.24115	0.620577	+	0.784146i	$0.286898\pi$
$24$	0.200784	0.0409849
$25$	0.271430	0.0542859
$26$	5.22204	1.02413
$27$	1.19661	0.230288
$28$	−0.443206	−0.0837581
$29$	−3.90713	−0.725536	−0.362768	−	0.931879i	$-0.618168\pi$
$29$	−3.90713	−0.725536	−0.362768	+	0.931879i	$0.618168\pi$
$30$	−0.460992	−0.0841653
$31$	−6.48171	−1.16415	−0.582075	−	0.813135i	$-0.697759\pi$
$31$	−6.48171	−1.16415	−0.582075	+	0.813135i	$0.697759\pi$
$32$	−1.00000	−0.176777
$33$	−0.0131941	−0.00229680
$34$	6.05974	1.03924
$35$	1.01758	0.172003
$36$	−2.95969	−0.493281
$37$	−1.23889	−0.203672	−0.101836	−	0.994801i	$-0.532472\pi$
$37$	−1.23889	−0.203672	−0.101836	+	0.994801i	$0.532472\pi$
$38$	−0.987451	−0.160186
$39$	1.04850	0.167895
$40$	2.29596	0.363023
$41$	6.05668	0.945894	0.472947	−	0.881091i	$-0.343190\pi$
$41$	6.05668	0.945894	0.472947	+	0.881091i	$0.343190\pi$
$42$	−0.0889888	−0.0137313
$43$	−4.24894	−0.647957	−0.323979	−	0.946064i	$-0.605021\pi$
$43$	−4.24894	−0.647957	−0.323979	+	0.946064i	$0.605021\pi$
$44$	0.0657130	0.00990661
$45$	6.79532	1.01299
$46$	−5.95236	−0.877628
$47$	−2.96424	−0.432380	−0.216190	−	0.976351i	$-0.569363\pi$
$47$	−2.96424	−0.432380	−0.216190	+	0.976351i	$0.569363\pi$
$48$	−0.200784	−0.0289807
$49$	−6.80357	−0.971938
$50$	−0.271430	−0.0383860
$51$	1.21670	0.170372
$52$	−5.22204	−0.724167
$53$	−11.7849	−1.61878	−0.809391	−	0.587270i	$-0.800203\pi$
$53$	−11.7849	−1.61878	−0.809391	+	0.587270i	$0.800203\pi$
$54$	−1.19661	−0.162838
$55$	−0.150874	−0.0203439
$56$	0.443206	0.0592260
$57$	−0.198264	−0.0262608
$58$	3.90713	0.513031
$59$	−10.7757	−1.40287	−0.701436	−	0.712732i	$-0.747457\pi$
$59$	−10.7757	−1.40287	−0.701436	+	0.712732i	$0.747457\pi$
$60$	0.460992	0.0595138
$61$	−2.55019	−0.326519	−0.163259	−	0.986583i	$-0.552201\pi$
$61$	−2.55019	−0.326519	−0.163259	+	0.986583i	$0.552201\pi$
$62$	6.48171	0.823179
$63$	1.31175	0.165265
$64$	1.00000	0.125000
$65$	11.9896	1.48713
$66$	0.0131941	0.00162408
$67$	−3.49450	−0.426922	−0.213461	−	0.976952i	$-0.568474\pi$
$67$	−3.49450	−0.426922	−0.213461	+	0.976952i	$0.568474\pi$
$68$	−6.05974	−0.734851
$69$	−1.19514	−0.143878
$70$	−1.01758	−0.121625
$71$	−7.48186	−0.887933	−0.443967	−	0.896043i	$-0.646429\pi$
$71$	−7.48186	−0.887933	−0.443967	+	0.896043i	$0.646429\pi$
$72$	2.95969	0.348802
$73$	4.47735	0.524034	0.262017	−	0.965063i	$-0.415612\pi$
$73$	4.47735	0.524034	0.262017	+	0.965063i	$0.415612\pi$
$74$	1.23889	0.144018
$75$	−0.0544988	−0.00629298
$76$	0.987451	0.113268
$77$	−0.0291244	−0.00331904
$78$	−1.04850	−0.118720
$79$	−13.1542	−1.47997	−0.739984	−	0.672624i	$-0.765167\pi$
$79$	−13.1542	−1.47997	−0.739984	+	0.672624i	$0.765167\pi$
$80$	−2.29596	−0.256696
$81$	8.63880	0.959866
$82$	−6.05668	−0.668848
$83$	−10.9279	−1.19949	−0.599744	−	0.800192i	$-0.704731\pi$
$83$	−10.9279	−1.19949	−0.599744	+	0.800192i	$0.704731\pi$
$84$	0.0889888	0.00970948
$85$	13.9129	1.50907
$86$	4.24894	0.458175
$87$	0.784490	0.0841062
$88$	−0.0657130	−0.00700503
$89$	−1.90357	−0.201778	−0.100889	−	0.994898i	$-0.532169\pi$
$89$	−1.90357	−0.201778	−0.100889	+	0.994898i	$0.532169\pi$
$90$	−6.79532	−0.716289
$91$	2.31444	0.242619
$92$	5.95236	0.620577
$93$	1.30143	0.134952
$94$	2.96424	0.305738
$95$	−2.26715	−0.232604
$96$	0.200784	0.0204924
$97$	3.05381	0.310068	0.155034	−	0.987909i	$-0.450451\pi$
$97$	3.05381	0.310068	0.155034	+	0.987909i	$0.450451\pi$
$98$	6.80357	0.687264
$99$	−0.194490	−0.0195470
$100$	0.271430	0.0271430
$101$	−1.62540	−0.161733	−0.0808666	−	0.996725i	$-0.525769\pi$
$101$	−1.62540	−0.161733	−0.0808666	+	0.996725i	$0.525769\pi$
$102$	−1.21670	−0.120471
$103$	−0.800128	−0.0788390	−0.0394195	−	0.999223i	$-0.512551\pi$
$103$	−0.800128	−0.0788390	−0.0394195	+	0.999223i	$0.512551\pi$
$104$	5.22204	0.512063
$105$	−0.204315	−0.0199391
$106$	11.7849	1.14465
$107$	−8.61465	−0.832809	−0.416405	−	0.909179i	$-0.636710\pi$
$107$	−8.61465	−0.832809	−0.416405	+	0.909179i	$0.636710\pi$
$108$	1.19661	0.115144
$109$	−19.3946	−1.85766	−0.928831	−	0.370505i	$-0.879185\pi$
$109$	−19.3946	−1.85766	−0.928831	+	0.370505i	$0.879185\pi$
$110$	0.150874	0.0143853
$111$	0.248749	0.0236102
$112$	−0.443206	−0.0418791
$113$	6.30110	0.592758	0.296379	−	0.955070i	$-0.404221\pi$
$113$	6.30110	0.592758	0.296379	+	0.955070i	$0.404221\pi$
$114$	0.198264	0.0185692
$115$	−13.6664	−1.27440
$116$	−3.90713	−0.362768
$117$	15.4556	1.42887
$118$	10.7757	0.991981
$119$	2.68571	0.246199
$120$	−0.460992	−0.0420826
$121$	−10.9957	−0.999607
$122$	2.55019	0.230884
$123$	−1.21608	−0.109651
$124$	−6.48171	−0.582075
$125$	10.8566	0.971044
$126$	−1.31175	−0.116860
$127$	5.98262	0.530872	0.265436	−	0.964129i	$-0.414484\pi$
$127$	5.98262	0.530872	0.265436	+	0.964129i	$0.414484\pi$
$128$	−1.00000	−0.0883883
$129$	0.853120	0.0751130
$130$	−11.9896	−1.05156
$131$	10.0759	0.880334	0.440167	−	0.897916i	$-0.354919\pi$
$131$	10.0759	0.880334	0.440167	+	0.897916i	$0.354919\pi$
$132$	−0.0131941	−0.00114840
$133$	−0.437644	−0.0379486
$134$	3.49450	0.301879
$135$	−2.74737	−0.236456
$136$	6.05974	0.519618
$137$	−9.32577	−0.796755	−0.398377	−	0.917222i	$-0.630427\pi$
$137$	−9.32577	−0.796755	−0.398377	+	0.917222i	$0.630427\pi$
$138$	1.19514	0.101737
$139$	6.79434	0.576288	0.288144	−	0.957587i	$-0.406962\pi$
$139$	6.79434	0.576288	0.288144	+	0.957587i	$0.406962\pi$
$140$	1.01758	0.0860015
$141$	0.595173	0.0501226
$142$	7.48186	0.627864
$143$	−0.343156	−0.0286961
$144$	−2.95969	−0.246640
$145$	8.97061	0.744969
$146$	−4.47735	−0.370548
$147$	1.36605	0.112670
$148$	−1.23889	−0.101836
$149$	6.30000	0.516116	0.258058	−	0.966129i	$-0.416917\pi$
$149$	6.30000	0.516116	0.258058	+	0.966129i	$0.416917\pi$
$150$	0.0544988	0.00444981
$151$	6.92286	0.563374	0.281687	−	0.959506i	$-0.409106\pi$
$151$	6.92286	0.563374	0.281687	+	0.959506i	$0.409106\pi$
$152$	−0.987451	−0.0800928
$153$	17.9349	1.44995
$154$	0.0291244	0.00234691
$155$	14.8818	1.19533
$156$	1.04850	0.0839474
$157$	14.3252	1.14327	0.571637	−	0.820507i	$-0.306309\pi$
$157$	14.3252	1.14327	0.571637	+	0.820507i	$0.306309\pi$
$158$	13.1542	1.04650
$159$	2.36622	0.187654
$160$	2.29596	0.181512
$161$	−2.63813	−0.207913
$162$	−8.63880	−0.678728
$163$	−9.94788	−0.779178	−0.389589	−	0.920989i	$-0.627383\pi$
$163$	−9.94788	−0.779178	−0.389589	+	0.920989i	$0.627383\pi$
$164$	6.05668	0.472947
$165$	0.0302932	0.00235832
$166$	10.9279	0.848167
$167$	12.4132	0.960561	0.480280	−	0.877115i	$-0.340535\pi$
$167$	12.4132	0.960561	0.480280	+	0.877115i	$0.340535\pi$
$168$	−0.0889888	−0.00686564
$169$	14.2697	1.09767
$170$	−13.9129	−1.06707
$171$	−2.92254	−0.223492
$172$	−4.24894	−0.323979
$173$	−1.69219	−0.128655	−0.0643273	−	0.997929i	$-0.520490\pi$
$173$	−1.69219	−0.128655	−0.0643273	+	0.997929i	$0.520490\pi$
$174$	−0.784490	−0.0594720
$175$	−0.120299	−0.00909378
$176$	0.0657130	0.00495330
$177$	2.16358	0.162625
$178$	1.90357	0.142679
$179$	−18.2511	−1.36415	−0.682074	−	0.731283i	$-0.738922\pi$
$179$	−18.2511	−1.36415	−0.682074	+	0.731283i	$0.738922\pi$
$180$	6.79532	0.506493
$181$	2.58244	0.191951	0.0959755	−	0.995384i	$-0.469403\pi$
$181$	2.58244	0.191951	0.0959755	+	0.995384i	$0.469403\pi$
$182$	−2.31444	−0.171558
$183$	0.512038	0.0378509
$184$	−5.95236	−0.438814
$185$	2.84443	0.209127
$186$	−1.30143	−0.0954251
$187$	−0.398203	−0.0291195
$188$	−2.96424	−0.216190
$189$	−0.530345	−0.0385770
$190$	2.26715	0.164476
$191$	4.32691	0.313084	0.156542	−	0.987671i	$-0.449965\pi$
$191$	4.32691	0.313084	0.156542	+	0.987671i	$0.449965\pi$
$192$	−0.200784	−0.0144903
$193$	0.877843	0.0631885	0.0315942	−	0.999501i	$-0.489942\pi$
$193$	0.877843	0.0631885	0.0315942	+	0.999501i	$0.489942\pi$
$194$	−3.05381	−0.219251
$195$	−2.40732	−0.172392
$196$	−6.80357	−0.485969
$197$	3.66518	0.261133	0.130567	−	0.991440i	$-0.458320\pi$
$197$	3.66518	0.261133	0.130567	+	0.991440i	$0.458320\pi$
$198$	0.194490	0.0138218
$199$	−5.62262	−0.398577	−0.199288	−	0.979941i	$-0.563863\pi$
$199$	−5.62262	−0.398577	−0.199288	+	0.979941i	$0.563863\pi$
$200$	−0.271430	−0.0191930
$201$	0.701641	0.0494899
$202$	1.62540	0.114363
$203$	1.73167	0.121539
$204$	1.21670	0.0851860
$205$	−13.9059	−0.971229
$206$	0.800128	0.0557476
$207$	−17.6171	−1.22448
$208$	−5.22204	−0.362083
$209$	0.0648883	0.00448842
$210$	0.204315	0.0140991
$211$	−7.70814	−0.530650	−0.265325	−	0.964159i	$-0.585479\pi$
$211$	−7.70814	−0.530650	−0.265325	+	0.964159i	$0.585479\pi$
$212$	−11.7849	−0.809391
$213$	1.50224	0.102932
$214$	8.61465	0.588885
$215$	9.75539	0.665312
$216$	−1.19661	−0.0814190
$217$	2.87274	0.195014
$218$	19.3946	1.31356
$219$	−0.898980	−0.0607474
$220$	−0.150874	−0.0101719
$221$	31.6442	2.12862
$222$	−0.248749	−0.0166949
$223$	−15.3558	−1.02830	−0.514151	−	0.857700i	$-0.671893\pi$
$223$	−15.3558	−1.02830	−0.514151	+	0.857700i	$0.671893\pi$
$224$	0.443206	0.0296130
$225$	−0.803347	−0.0535564
$226$	−6.30110	−0.419143
$227$	−21.8972	−1.45337	−0.726683	−	0.686972i	$-0.758939\pi$
$227$	−21.8972	−1.45337	−0.726683	+	0.686972i	$0.758939\pi$
$228$	−0.198264	−0.0131304
$229$	23.9817	1.58475	0.792376	−	0.610033i	$-0.208844\pi$
$229$	23.9817	1.58475	0.792376	+	0.610033i	$0.208844\pi$
$230$	13.6664	0.901135
$231$	0.00584772	0.000384752 0
$232$	3.90713	0.256516
$233$	26.2413	1.71913	0.859564	−	0.511029i	$-0.170735\pi$
$233$	26.2413	1.71913	0.859564	+	0.511029i	$0.170735\pi$
$234$	−15.4556	−1.01036
$235$	6.80579	0.443960
$236$	−10.7757	−0.701436
$237$	2.64116	0.171562
$238$	−2.68571	−0.174089
$239$	0.0739620	0.00478420	0.00239210	−	0.999997i	$-0.499239\pi$
$239$	0.0739620	0.00478420	0.00239210	+	0.999997i	$0.499239\pi$
$240$	0.460992	0.0297569
$241$	−15.1583	−0.976430	−0.488215	−	0.872724i	$-0.662352\pi$
$241$	−15.1583	−0.976430	−0.488215	+	0.872724i	$0.662352\pi$
$242$	10.9957	0.706829
$243$	−5.32436	−0.341558
$244$	−2.55019	−0.163259
$245$	15.6207	0.997971
$246$	1.21608	0.0775347
$247$	−5.15651	−0.328101
$248$	6.48171	0.411589
$249$	2.19414	0.139048
$250$	−10.8566	−0.686632
$251$	−10.4349	−0.658643	−0.329322	−	0.944218i	$-0.606820\pi$
$251$	−10.4349	−0.658643	−0.329322	+	0.944218i	$0.606820\pi$
$252$	1.31175	0.0826326
$253$	0.391148	0.0245912
$254$	−5.98262	−0.375383
$255$	−2.79349	−0.174935
$256$	1.00000	0.0625000
$257$	−12.6077	−0.786444	−0.393222	−	0.919443i	$-0.628640\pi$
$257$	−12.6077	−0.786444	−0.393222	+	0.919443i	$0.628640\pi$
$258$	−0.853120	−0.0531129
$259$	0.549082	0.0341183
$260$	11.9896	0.743563
$261$	11.5639	0.715786
$262$	−10.0759	−0.622490
$263$	9.51627	0.586799	0.293399	−	0.955990i	$-0.405213\pi$
$263$	9.51627	0.586799	0.293399	+	0.955990i	$0.405213\pi$
$264$	0.0131941	0.000812042 0
$265$	27.0577	1.66214
$266$	0.437644	0.0268337
$267$	0.382207	0.0233907
$268$	−3.49450	−0.213461
$269$	−1.30081	−0.0793117	−0.0396558	−	0.999213i	$-0.512626\pi$
$269$	−1.30081	−0.0793117	−0.0396558	+	0.999213i	$0.512626\pi$
$270$	2.74737	0.167200
$271$	−25.5515	−1.55214	−0.776072	−	0.630644i	$-0.782791\pi$
$271$	−25.5515	−1.55214	−0.776072	+	0.630644i	$0.782791\pi$
$272$	−6.05974	−0.367425
$273$	−0.464703	−0.0281251
$274$	9.32577	0.563391
$275$	0.0178365	0.00107558
$276$	−1.19514	−0.0719390
$277$	12.2527	0.736194	0.368097	−	0.929787i	$-0.380009\pi$
$277$	12.2527	0.736194	0.368097	+	0.929787i	$0.380009\pi$
$278$	−6.79434	−0.407497
$279$	19.1838	1.14851
$280$	−1.01758	−0.0608123
$281$	28.6694	1.71027	0.855136	−	0.518404i	$-0.173474\pi$
$281$	28.6694	1.71027	0.855136	+	0.518404i	$0.173474\pi$
$282$	−0.595173	−0.0354421
$283$	−7.43240	−0.441810	−0.220905	−	0.975295i	$-0.570901\pi$
$283$	−7.43240	−0.441810	−0.220905	+	0.975295i	$0.570901\pi$
$284$	−7.48186	−0.443967
$285$	0.455207	0.0269641
$286$	0.343156	0.0202912
$287$	−2.68436	−0.158453
$288$	2.95969	0.174401
$289$	19.7204	1.16002
$290$	−8.97061	−0.526773
$291$	−0.613158	−0.0359439
$292$	4.47735	0.262017
$293$	−3.32565	−0.194286	−0.0971432	−	0.995270i	$-0.530970\pi$
$293$	−3.32565	−0.194286	−0.0971432	+	0.995270i	$0.530970\pi$
$294$	−1.36605	−0.0796696
$295$	24.7405	1.44045
$296$	1.23889	0.0720088
$297$	0.0786328	0.00456274
$298$	−6.30000	−0.364949
$299$	−31.0835	−1.79760
$300$	−0.0544988	−0.00314649
$301$	1.88316	0.108543
$302$	−6.92286	−0.398366
$303$	0.326354	0.0187486
$304$	0.987451	0.0566342
$305$	5.85514	0.335264
$306$	−17.9349	−1.02527
$307$	−1.30079	−0.0742399	−0.0371199	−	0.999311i	$-0.511818\pi$
$307$	−1.30079	−0.0742399	−0.0371199	+	0.999311i	$0.511818\pi$
$308$	−0.0291244	−0.00165952
$309$	0.160653	0.00913923
$310$	−14.8818	−0.845227
$311$	20.0764	1.13843	0.569214	−	0.822189i	$-0.307248\pi$
$311$	20.0764	1.13843	0.569214	+	0.822189i	$0.307248\pi$
$312$	−1.04850	−0.0593598
$313$	−24.8721	−1.40586	−0.702928	−	0.711261i	$-0.748125\pi$
$313$	−24.8721	−1.40586	−0.702928	+	0.711261i	$0.748125\pi$
$314$	−14.3252	−0.808417
$315$	−3.01173	−0.169692
$316$	−13.1542	−0.739984
$317$	20.8234	1.16956	0.584779	−	0.811193i	$-0.301181\pi$
$317$	20.8234	1.16956	0.584779	+	0.811193i	$0.301181\pi$
$318$	−2.36622	−0.132691
$319$	−0.256749	−0.0143752
$320$	−2.29596	−0.128348
$321$	1.72968	0.0965416
$322$	2.63813	0.147017
$323$	−5.98369	−0.332941
$324$	8.63880	0.479933
$325$	−1.41742	−0.0786242
$326$	9.94788	0.550962
$327$	3.89412	0.215345
$328$	−6.05668	−0.334424
$329$	1.31377	0.0724306
$330$	−0.0302932	−0.00166758
$331$	9.65365	0.530613	0.265306	−	0.964164i	$-0.414527\pi$
$331$	9.65365	0.530613	0.265306	+	0.964164i	$0.414527\pi$
$332$	−10.9279	−0.599744
$333$	3.66671	0.200935
$334$	−12.4132	−0.679219
$335$	8.02324	0.438356
$336$	0.0889888	0.00485474
$337$	−26.7884	−1.45926	−0.729629	−	0.683844i	$-0.760307\pi$
$337$	−26.7884	−1.45926	−0.729629	+	0.683844i	$0.760307\pi$
$338$	−14.2697	−0.776170
$339$	−1.26516	−0.0687142
$340$	13.9129	0.754533
$341$	−0.425933	−0.0230656
$342$	2.92254	0.158033
$343$	6.11783	0.330332
$344$	4.24894	0.229087
$345$	2.74399	0.147732
$346$	1.69219	0.0909725
$347$	28.4888	1.52936	0.764679	−	0.644411i	$-0.222898\pi$
$347$	28.4888	1.52936	0.764679	+	0.644411i	$0.222898\pi$
$348$	0.784490	0.0420531
$349$	−11.6593	−0.624108	−0.312054	−	0.950064i	$-0.601017\pi$
$349$	−11.6593	−0.624108	−0.312054	+	0.950064i	$0.601017\pi$
$350$	0.120299	0.00643027
$351$	−6.24875	−0.333534
$352$	−0.0657130	−0.00350251
$353$	18.6552	0.992916	0.496458	−	0.868061i	$-0.334634\pi$
$353$	18.6552	0.992916	0.496458	+	0.868061i	$0.334634\pi$
$354$	−2.16358	−0.114993
$355$	17.1780	0.911716
$356$	−1.90357	−0.100889
$357$	−0.539249	−0.0285401
$358$	18.2511	0.964598
$359$	31.2009	1.64672	0.823360	−	0.567519i	$-0.192097\pi$
$359$	31.2009	1.64672	0.823360	+	0.567519i	$0.192097\pi$
$360$	−6.79532	−0.358145
$361$	−18.0249	−0.948681
$362$	−2.58244	−0.135730
$363$	2.20776	0.115877
$364$	2.31444	0.121310
$365$	−10.2798	−0.538070
$366$	−0.512038	−0.0267647
$367$	34.7404	1.81343	0.906717	−	0.421740i	$-0.138581\pi$
$367$	34.7404	1.81343	0.906717	+	0.421740i	$0.138581\pi$
$368$	5.95236	0.310288
$369$	−17.9259	−0.933183
$370$	−2.84443	−0.147875
$371$	5.22315	0.271172
$372$	1.30143	0.0674758
$373$	−21.2826	−1.10197	−0.550987	−	0.834514i	$-0.685748\pi$
$373$	−21.2826	−1.10197	−0.550987	+	0.834514i	$0.685748\pi$
$374$	0.398203	0.0205906
$375$	−2.17983	−0.112566
$376$	2.96424	0.152869
$377$	20.4032	1.05082
$378$	0.530345	0.0272780
$379$	−18.6158	−0.956230	−0.478115	−	0.878297i	$-0.658680\pi$
$379$	−18.6158	−0.956230	−0.478115	+	0.878297i	$0.658680\pi$
$380$	−2.26715	−0.116302
$381$	−1.20122	−0.0615401
$382$	−4.32691	−0.221384
$383$	−13.4246	−0.685965	−0.342983	−	0.939342i	$-0.611437\pi$
$383$	−13.4246	−0.685965	−0.342983	+	0.939342i	$0.611437\pi$
$384$	0.200784	0.0102462
$385$	0.0668685	0.00340793
$386$	−0.877843	−0.0446810
$387$	12.5755	0.639250
$388$	3.05381	0.155034
$389$	−1.44707	−0.0733695	−0.0366847	−	0.999327i	$-0.511680\pi$
$389$	−1.44707	−0.0733695	−0.0366847	+	0.999327i	$0.511680\pi$
$390$	2.40732	0.121899
$391$	−36.0698	−1.82413
$392$	6.80357	0.343632
$393$	−2.02308	−0.102051
$394$	−3.66518	−0.184649
$395$	30.2016	1.51961
$396$	−0.194490	−0.00977348
$397$	35.7067	1.79207	0.896034	−	0.443985i	$-0.146436\pi$
$397$	35.7067	1.79207	0.896034	+	0.443985i	$0.146436\pi$
$398$	5.62262	0.281836
$399$	0.0878721	0.00439911
$400$	0.271430	0.0135715
$401$	5.48471	0.273893	0.136947	−	0.990578i	$-0.456271\pi$
$401$	5.48471	0.273893	0.136947	+	0.990578i	$0.456271\pi$
$402$	−0.701641	−0.0349947
$403$	33.8478	1.68608
$404$	−1.62540	−0.0808666
$405$	−19.8343	−0.985576
$406$	−1.73167	−0.0859411
$407$	−0.0814109	−0.00403539
$408$	−1.21670	−0.0602356
$409$	−34.6578	−1.71372	−0.856858	−	0.515552i	$-0.827587\pi$
$409$	−34.6578	−1.71372	−0.856858	+	0.515552i	$0.827587\pi$
$410$	13.9059	0.686763
$411$	1.87247	0.0923620
$412$	−0.800128	−0.0394195
$413$	4.77585	0.235004
$414$	17.6171	0.865835
$415$	25.0899	1.23162
$416$	5.22204	0.256032
$417$	−1.36420	−0.0668049
$418$	−0.0648883	−0.00317379
$419$	−0.221011	−0.0107971	−0.00539855	−	0.999985i	$-0.501718\pi$
$419$	−0.221011	−0.0107971	−0.00539855	+	0.999985i	$0.501718\pi$
$420$	−0.204315	−0.00996954
$421$	−18.1784	−0.885962	−0.442981	−	0.896531i	$-0.646079\pi$
$421$	−18.1784	−0.885962	−0.442981	+	0.896531i	$0.646079\pi$
$422$	7.70814	0.375226
$423$	8.77323	0.426569
$424$	11.7849	0.572326
$425$	−1.64479	−0.0797842
$426$	−1.50224	−0.0727837
$427$	1.13026	0.0546972
$428$	−8.61465	−0.416405
$429$	0.0689003	0.00332654
$430$	−9.75539	−0.470447
$431$	−12.2671	−0.590887	−0.295443	−	0.955360i	$-0.595467\pi$
$431$	−12.2671	−0.590887	−0.295443	+	0.955360i	$0.595467\pi$
$432$	1.19661	0.0575719
$433$	−7.33472	−0.352484	−0.176242	−	0.984347i	$-0.556394\pi$
$433$	−7.33472	−0.352484	−0.176242	+	0.984347i	$0.556394\pi$
$434$	−2.87274	−0.137896
$435$	−1.80116	−0.0863589
$436$	−19.3946	−0.928831
$437$	5.87767	0.281167
$438$	0.898980	0.0429549
$439$	−13.6906	−0.653418	−0.326709	−	0.945125i	$-0.605940\pi$
$439$	−13.6906	−0.653418	−0.326709	+	0.945125i	$0.605940\pi$
$440$	0.150874	0.00719265
$441$	20.1364	0.958877
$442$	−31.6442	−1.50516
$443$	24.6137	1.16943	0.584715	−	0.811239i	$-0.301206\pi$
$443$	24.6137	1.16943	0.584715	+	0.811239i	$0.301206\pi$
$444$	0.248749	0.0118051
$445$	4.37053	0.207183
$446$	15.3558	0.727119
$447$	−1.26494	−0.0598296
$448$	−0.443206	−0.0209395
$449$	−39.4678	−1.86260	−0.931299	−	0.364255i	$-0.881324\pi$
$449$	−39.4678	−1.86260	−0.931299	+	0.364255i	$0.881324\pi$
$450$	0.803347	0.0378701
$451$	0.398002	0.0187412
$452$	6.30110	0.296379
$453$	−1.39000	−0.0653079
$454$	21.8972	1.02769
$455$	−5.31387	−0.249118
$456$	0.198264	0.00928458
$457$	13.0389	0.609933	0.304966	−	0.952363i	$-0.401355\pi$
$457$	13.0389	0.609933	0.304966	+	0.952363i	$0.401355\pi$
$458$	−23.9817	−1.12059
$459$	−7.25114	−0.338454
$460$	−13.6664	−0.637199
$461$	−29.2682	−1.36316	−0.681578	−	0.731746i	$-0.738706\pi$
$461$	−29.2682	−1.36316	−0.681578	+	0.731746i	$0.738706\pi$
$462$	−0.00584772	−0.000272061 0
$463$	−7.13135	−0.331422	−0.165711	−	0.986174i	$-0.552992\pi$
$463$	−7.13135	−0.331422	−0.165711	+	0.986174i	$0.552992\pi$
$464$	−3.90713	−0.181384
$465$	−2.98802	−0.138566
$466$	−26.2413	−1.21561
$467$	8.86217	0.410092	0.205046	−	0.978752i	$-0.434266\pi$
$467$	8.86217	0.410092	0.205046	+	0.978752i	$0.434266\pi$
$468$	15.4556	0.714435
$469$	1.54879	0.0715163
$470$	−6.80579	−0.313927
$471$	−2.87627	−0.132532
$472$	10.7757	0.495990
$473$	−0.279211	−0.0128381
$474$	−2.64116	−0.121313
$475$	0.268023	0.0122978
$476$	2.68571	0.123099
$477$	34.8796	1.59703
$478$	−0.0739620	−0.00338294
$479$	−11.5049	−0.525672	−0.262836	−	0.964840i	$-0.584658\pi$
$479$	−11.5049	−0.525672	−0.262836	+	0.964840i	$0.584658\pi$
$480$	−0.460992	−0.0210413
$481$	6.46952	0.294985
$482$	15.1583	0.690440
$483$	0.529694	0.0241019
$484$	−10.9957	−0.499804
$485$	−7.01143	−0.318373
$486$	5.32436	0.241518
$487$	23.2218	1.05228	0.526141	−	0.850397i	$-0.323638\pi$
$487$	23.2218	1.05228	0.526141	+	0.850397i	$0.323638\pi$
$488$	2.55019	0.115442
$489$	1.99738	0.0903244
$490$	−15.6207	−0.705672
$491$	4.66706	0.210622	0.105311	−	0.994439i	$-0.466416\pi$
$491$	4.66706	0.210622	0.105311	+	0.994439i	$0.466416\pi$
$492$	−1.21608	−0.0548253
$493$	23.6762	1.06632
$494$	5.15651	0.232002
$495$	0.446541	0.0200705
$496$	−6.48171	−0.291038
$497$	3.31601	0.148743
$498$	−2.19414	−0.0983218
$499$	−31.4924	−1.40979	−0.704896	−	0.709311i	$-0.749006\pi$
$499$	−31.4924	−1.40979	−0.704896	+	0.709311i	$0.749006\pi$
$500$	10.8566	0.485522
$501$	−2.49237	−0.111351
$502$	10.4349	0.465731
$503$	5.24643	0.233927	0.116963	−	0.993136i	$-0.462684\pi$
$503$	5.24643	0.233927	0.116963	+	0.993136i	$0.462684\pi$
$504$	−1.31175	−0.0584301
$505$	3.73185	0.166065
$506$	−0.391148	−0.0173886
$507$	−2.86513	−0.127245
$508$	5.98262	0.265436
$509$	14.0221	0.621517	0.310759	−	0.950489i	$-0.399417\pi$
$509$	14.0221	0.621517	0.310759	+	0.950489i	$0.399417\pi$
$510$	2.79349	0.123698
$511$	−1.98439	−0.0877842
$512$	−1.00000	−0.0441942
$513$	1.18159	0.0521686
$514$	12.6077	0.556100
$515$	1.83706	0.0809506
$516$	0.853120	0.0375565
$517$	−0.194789	−0.00856683
$518$	−0.549082	−0.0241253
$519$	0.339764	0.0149140
$520$	−11.9896	−0.525778
$521$	43.7531	1.91686	0.958429	−	0.285331i	$-0.0921035\pi$
$521$	43.7531	1.91686	0.958429	+	0.285331i	$0.0921035\pi$
$522$	−11.5639	−0.506137
$523$	13.6581	0.597226	0.298613	−	0.954374i	$-0.403476\pi$
$523$	13.6581	0.597226	0.298613	+	0.954374i	$0.403476\pi$
$524$	10.0759	0.440167
$525$	0.0241542	0.00105418
$526$	−9.51627	−0.414929
$527$	39.2775	1.71095
$528$	−0.0131941	−0.000574201 0
$529$	12.4306	0.540463
$530$	−27.0577	−1.17531
$531$	31.8926	1.38402
$532$	−0.437644	−0.0189743
$533$	−31.6282	−1.36997
$534$	−0.382207	−0.0165397
$535$	19.7789	0.855115
$536$	3.49450	0.150940
$537$	3.66452	0.158136
$538$	1.30081	0.0560818
$539$	−0.447083	−0.0192572
$540$	−2.74737	−0.118228
$541$	−33.3820	−1.43520	−0.717602	−	0.696454i	$-0.754760\pi$
$541$	−33.3820	−1.43520	−0.717602	+	0.696454i	$0.754760\pi$
$542$	25.5515	1.09753
$543$	−0.518512	−0.0222515
$544$	6.05974	0.259809
$545$	44.5291	1.90742
$546$	0.464703	0.0198875
$547$	−34.4635	−1.47355	−0.736775	−	0.676138i	$-0.763652\pi$
$547$	−34.4635	−1.47355	−0.736775	+	0.676138i	$0.763652\pi$
$548$	−9.32577	−0.398377
$549$	7.54777	0.322131
$550$	−0.0178365	−0.000760549 0
$551$	−3.85810	−0.164361
$552$	1.19514	0.0508686
$553$	5.83005	0.247919
$554$	−12.2527	−0.520568
$555$	−0.571117	−0.0242426
$556$	6.79434	0.288144
$557$	12.5151	0.530280	0.265140	−	0.964210i	$-0.414582\pi$
$557$	12.5151	0.530280	0.265140	+	0.964210i	$0.414582\pi$
$558$	−19.1838	−0.812117
$559$	22.1881	0.938458
$560$	1.01758	0.0430008
$561$	0.0799529	0.00337561
$562$	−28.6694	−1.20934
$563$	1.29185	0.0544451	0.0272225	−	0.999629i	$-0.491334\pi$
$563$	1.29185	0.0544451	0.0272225	+	0.999629i	$0.491334\pi$
$564$	0.595173	0.0250613
$565$	−14.4671	−0.608635
$566$	7.43240	0.312407
$567$	−3.82877	−0.160793
$568$	7.48186	0.313932
$569$	9.05930	0.379786	0.189893	−	0.981805i	$-0.439186\pi$
$569$	9.05930	0.379786	0.189893	+	0.981805i	$0.439186\pi$
$570$	−0.455207	−0.0190665
$571$	2.84133	0.118906	0.0594529	−	0.998231i	$-0.481064\pi$
$571$	2.84133	0.118906	0.0594529	+	0.998231i	$0.481064\pi$
$572$	−0.343156	−0.0143481
$573$	−0.868775	−0.0362936
$574$	2.68436	0.112043
$575$	1.61565	0.0673772
$576$	−2.95969	−0.123320
$577$	10.5198	0.437945	0.218972	−	0.975731i	$-0.429730\pi$
$577$	10.5198	0.437945	0.218972	+	0.975731i	$0.429730\pi$
$578$	−19.7204	−0.820260
$579$	−0.176257	−0.00732499
$580$	8.97061	0.372485
$581$	4.84330	0.200934
$582$	0.613158	0.0254162
$583$	−0.774422	−0.0320733
$584$	−4.47735	−0.185274
$585$	−35.4854	−1.46714
$586$	3.32565	0.137381
$587$	30.6011	1.26304	0.631521	−	0.775358i	$-0.282431\pi$
$587$	30.6011	1.26304	0.631521	+	0.775358i	$0.282431\pi$
$588$	1.36605	0.0563349
$589$	−6.40037	−0.263723
$590$	−24.7405	−1.01855
$591$	−0.735910	−0.0302713
$592$	−1.23889	−0.0509179
$593$	11.6793	0.479611	0.239806	−	0.970821i	$-0.422916\pi$
$593$	11.6793	0.479611	0.239806	+	0.970821i	$0.422916\pi$
$594$	−0.0786328	−0.00322634
$595$	−6.16629	−0.252793
$596$	6.30000	0.258058
$597$	1.12893	0.0462041
$598$	31.0835	1.27110
$599$	6.82646	0.278922	0.139461	−	0.990228i	$-0.455463\pi$
$599$	6.82646	0.278922	0.139461	+	0.990228i	$0.455463\pi$
$600$	0.0544988	0.00222490
$601$	9.91559	0.404465	0.202233	−	0.979338i	$-0.435180\pi$
$601$	9.91559	0.404465	0.202233	+	0.979338i	$0.435180\pi$
$602$	−1.88316	−0.0767518
$603$	10.3426	0.421185
$604$	6.92286	0.281687
$605$	25.2456	1.02638
$606$	−0.326354	−0.0132572
$607$	2.66158	0.108030	0.0540151	−	0.998540i	$-0.482798\pi$
$607$	2.66158	0.108030	0.0540151	+	0.998540i	$0.482798\pi$
$608$	−0.987451	−0.0400464
$609$	−0.347691	−0.0140892
$610$	−5.85514	−0.237068
$611$	15.4794	0.626230
$612$	17.9349	0.724976
$613$	−11.4946	−0.464263	−0.232131	−	0.972684i	$-0.574570\pi$
$613$	−11.4946	−0.464263	−0.232131	+	0.972684i	$0.574570\pi$
$614$	1.30079	0.0524955
$615$	2.79208	0.112588
$616$	0.0291244	0.00117346
$617$	12.9990	0.523319	0.261660	−	0.965160i	$-0.415730\pi$
$617$	12.9990	0.523319	0.261660	+	0.965160i	$0.415730\pi$
$618$	−0.160653	−0.00646241
$619$	−9.87474	−0.396899	−0.198450	−	0.980111i	$-0.563591\pi$
$619$	−9.87474	−0.396899	−0.198450	+	0.980111i	$0.563591\pi$
$620$	14.8818	0.597666
$621$	7.12266	0.285823
$622$	−20.0764	−0.804990
$623$	0.843676	0.0338012
$624$	1.04850	0.0419737
$625$	−26.2835	−1.05134
$626$	24.8721	0.994091
$627$	−0.0130285	−0.000520310 0
$628$	14.3252	0.571637
$629$	7.50732	0.299337
$630$	3.01173	0.119990
$631$	23.7102	0.943886	0.471943	−	0.881629i	$-0.343553\pi$
$631$	23.7102	0.943886	0.471943	+	0.881629i	$0.343553\pi$
$632$	13.1542	0.523248
$633$	1.54767	0.0615145
$634$	−20.8234	−0.827002
$635$	−13.7359	−0.545091
$636$	2.36622	0.0938269
$637$	35.5285	1.40769
$638$	0.256749	0.0101648
$639$	22.1440	0.876001
$640$	2.29596	0.0907558
$641$	−2.82716	−0.111666	−0.0558331	−	0.998440i	$-0.517781\pi$
$641$	−2.82716	−0.111666	−0.0558331	+	0.998440i	$0.517781\pi$
$642$	−1.72968	−0.0682652
$643$	27.3536	1.07872	0.539360	−	0.842075i	$-0.318666\pi$
$643$	27.3536	1.07872	0.539360	+	0.842075i	$0.318666\pi$
$644$	−2.63813	−0.103957
$645$	−1.95873	−0.0771249
$646$	5.98369	0.235425
$647$	−7.87764	−0.309702	−0.154851	−	0.987938i	$-0.549490\pi$
$647$	−7.87764	−0.309702	−0.154851	+	0.987938i	$0.549490\pi$
$648$	−8.63880	−0.339364
$649$	−0.708101	−0.0277954
$650$	1.41742	0.0555957
$651$	−0.576800	−0.0226066
$652$	−9.94788	−0.389589
$653$	10.4871	0.410392	0.205196	−	0.978721i	$-0.434217\pi$
$653$	10.4871	0.410392	0.205196	+	0.978721i	$0.434217\pi$
$654$	−3.89412	−0.152272
$655$	−23.1338	−0.903913
$656$	6.05668	0.236474
$657$	−13.2515	−0.516992
$658$	−1.31377	−0.0512162
$659$	21.7892	0.848787	0.424393	−	0.905478i	$-0.360487\pi$
$659$	21.7892	0.848787	0.424393	+	0.905478i	$0.360487\pi$
$660$	0.0302932	0.00117916
$661$	−16.9777	−0.660355	−0.330178	−	0.943919i	$-0.607109\pi$
$661$	−16.9777	−0.660355	−0.330178	+	0.943919i	$0.607109\pi$
$662$	−9.65365	−0.375200
$663$	−6.35365	−0.246755
$664$	10.9279	0.424083
$665$	1.00481	0.0389650
$666$	−3.66671	−0.142082
$667$	−23.2567	−0.900502
$668$	12.4132	0.480280
$669$	3.08320	0.119204
$670$	−8.02324	−0.309965
$671$	−0.167581	−0.00646938
$672$	−0.0889888	−0.00343282
$673$	−10.7092	−0.412808	−0.206404	−	0.978467i	$-0.566176\pi$
$673$	−10.7092	−0.412808	−0.206404	+	0.978467i	$0.566176\pi$
$674$	26.7884	1.03185
$675$	0.324796	0.0125014
$676$	14.2697	0.548835
$677$	−17.1672	−0.659790	−0.329895	−	0.944018i	$-0.607013\pi$
$677$	−17.1672	−0.659790	−0.329895	+	0.944018i	$0.607013\pi$
$678$	1.26516	0.0485882
$679$	−1.35347	−0.0519414
$680$	−13.9129	−0.533536
$681$	4.39661	0.168478
$682$	0.425933	0.0163098
$683$	−27.0195	−1.03387	−0.516935	−	0.856024i	$-0.672927\pi$
$683$	−27.0195	−1.03387	−0.516935	+	0.856024i	$0.672927\pi$
$684$	−2.92254	−0.111746
$685$	21.4116	0.818095
$686$	−6.11783	−0.233580
$687$	−4.81514	−0.183709
$688$	−4.24894	−0.161989
$689$	61.5413	2.34454
$690$	−2.74399	−0.104462
$691$	3.46776	0.131920	0.0659599	−	0.997822i	$-0.478989\pi$
$691$	3.46776	0.131920	0.0659599	+	0.997822i	$0.478989\pi$
$692$	−1.69219	−0.0643273
$693$	0.0861991	0.00327443
$694$	−28.4888	−1.08142
$695$	−15.5995	−0.591724
$696$	−0.784490	−0.0297360
$697$	−36.7019	−1.39018
$698$	11.6593	0.441311
$699$	−5.26885	−0.199286
$700$	−0.120299	−0.00454689
$701$	−17.1971	−0.649525	−0.324763	−	0.945796i	$-0.605284\pi$
$701$	−17.1971	−0.649525	−0.324763	+	0.945796i	$0.605284\pi$
$702$	6.24875	0.235844
$703$	−1.22334	−0.0461391
$704$	0.0657130	0.00247665
$705$	−1.36649	−0.0514651
$706$	−18.6552	−0.702098
$707$	0.720387	0.0270929
$708$	2.16358	0.0813124
$709$	−25.2893	−0.949760	−0.474880	−	0.880051i	$-0.657509\pi$
$709$	−25.2893	−0.949760	−0.474880	+	0.880051i	$0.657509\pi$
$710$	−17.1780	−0.644681
$711$	38.9324	1.46008
$712$	1.90357	0.0713394
$713$	−38.5815	−1.44489
$714$	0.539249	0.0201809
$715$	0.787872	0.0294647
$716$	−18.2511	−0.682074
$717$	−0.0148504	−0.000554598 0
$718$	−31.2009	−1.16441
$719$	−4.37890	−0.163306	−0.0816528	−	0.996661i	$-0.526020\pi$
$719$	−4.37890	−0.163306	−0.0816528	+	0.996661i	$0.526020\pi$
$720$	6.79532	0.253247
$721$	0.354622	0.0132068
$722$	18.0249	0.670819
$723$	3.04354	0.113190
$724$	2.58244	0.0959755
$725$	−1.06051	−0.0393864
$726$	−2.20776	−0.0819376
$727$	−14.5756	−0.540578	−0.270289	−	0.962779i	$-0.587119\pi$
$727$	−14.5756	−0.540578	−0.270289	+	0.962779i	$0.587119\pi$
$728$	−2.31444	−0.0857789
$729$	−24.8473	−0.920272
$730$	10.2798	0.380473
$731$	25.7475	0.952304
$732$	0.512038	0.0189255
$733$	−21.7938	−0.804971	−0.402486	−	0.915426i	$-0.631854\pi$
$733$	−21.7938	−0.804971	−0.402486	+	0.915426i	$0.631854\pi$
$734$	−34.7404	−1.28229
$735$	−3.13639	−0.115688
$736$	−5.95236	−0.219407
$737$	−0.229634	−0.00845869
$738$	17.9259	0.659860
$739$	21.6251	0.795493	0.397746	−	0.917495i	$-0.369792\pi$
$739$	21.6251	0.795493	0.397746	+	0.917495i	$0.369792\pi$
$740$	2.84443	0.104563
$741$	1.03535	0.0380343
$742$	−5.22315	−0.191748
$743$	−40.8632	−1.49913	−0.749563	−	0.661933i	$-0.769737\pi$
$743$	−40.8632	−1.49913	−0.749563	+	0.661933i	$0.769737\pi$
$744$	−1.30143	−0.0477126
$745$	−14.4645	−0.529940
$746$	21.2826	0.779213
$747$	32.3430	1.18337
$748$	−0.398203	−0.0145598
$749$	3.81807	0.139509
$750$	2.17983	0.0795963
$751$	−19.2893	−0.703875	−0.351937	−	0.936024i	$-0.614477\pi$
$751$	−19.2893	−0.703875	−0.351937	+	0.936024i	$0.614477\pi$
$752$	−2.96424	−0.108095
$753$	2.09516	0.0763518
$754$	−20.4032	−0.743041
$755$	−15.8946	−0.578464
$756$	−0.530345	−0.0192885
$757$	−15.2005	−0.552472	−0.276236	−	0.961090i	$-0.589087\pi$
$757$	−15.2005	−0.552472	−0.276236	+	0.961090i	$0.589087\pi$
$758$	18.6158	0.676156
$759$	−0.0785362	−0.00285069
$760$	2.26715	0.0822381
$761$	−21.0512	−0.763104	−0.381552	−	0.924347i	$-0.624610\pi$
$761$	−21.0512	−0.763104	−0.381552	+	0.924347i	$0.624610\pi$
$762$	1.20122	0.0435154
$763$	8.59579	0.311188
$764$	4.32691	0.156542
$765$	−41.1778	−1.48879
$766$	13.4246	0.485051
$767$	56.2710	2.03183
$768$	−0.200784	−0.00724517
$769$	11.2519	0.405753	0.202876	−	0.979204i	$-0.434971\pi$
$769$	11.2519	0.405753	0.202876	+	0.979204i	$0.434971\pi$
$770$	−0.0668685	−0.00240977
$771$	2.53142	0.0911668
$772$	0.877843	0.0315942
$773$	23.2601	0.836608	0.418304	−	0.908307i	$-0.362625\pi$
$773$	23.2601	0.836608	0.418304	+	0.908307i	$0.362625\pi$
$774$	−12.5755	−0.452018
$775$	−1.75933	−0.0631970
$776$	−3.05381	−0.109626
$777$	−0.110247	−0.00395509
$778$	1.44707	0.0518801
$779$	5.98067	0.214280
$780$	−2.40732	−0.0861959
$781$	−0.491655	−0.0175928
$782$	36.0698	1.28985
$783$	−4.67531	−0.167082
$784$	−6.80357	−0.242985
$785$	−32.8900	−1.17390
$786$	2.02308	0.0721608
$787$	−43.5652	−1.55293	−0.776465	−	0.630160i	$-0.782989\pi$
$787$	−43.5652	−1.55293	−0.776465	+	0.630160i	$0.782989\pi$
$788$	3.66518	0.130567
$789$	−1.91072	−0.0680233
$790$	−30.2016	−1.07453
$791$	−2.79269	−0.0992966
$792$	0.194490	0.00691089
$793$	13.3172	0.472908
$794$	−35.7067	−1.26718
$795$	−5.43275	−0.192680
$796$	−5.62262	−0.199288
$797$	−24.0864	−0.853183	−0.426592	−	0.904444i	$-0.640286\pi$
$797$	−24.0864	−0.853183	−0.426592	+	0.904444i	$0.640286\pi$
$798$	−0.0878721	−0.00311064
$799$	17.9625	0.635469
$800$	−0.271430	−0.00959649
$801$	5.63398	0.199067
$802$	−5.48471	−0.193672
$803$	0.294220	0.0103828
$804$	0.701641	0.0247450
$805$	6.05703	0.213482
$806$	−33.8478	−1.19224
$807$	0.261182	0.00919403
$808$	1.62540	0.0571813
$809$	−24.6476	−0.866562	−0.433281	−	0.901259i	$-0.642644\pi$
$809$	−24.6476	−0.866562	−0.433281	+	0.901259i	$0.642644\pi$
$810$	19.8343	0.696907
$811$	−5.43408	−0.190816	−0.0954082	−	0.995438i	$-0.530416\pi$
$811$	−5.43408	−0.190816	−0.0954082	+	0.995438i	$0.530416\pi$
$812$	1.73167	0.0607696
$813$	5.13034	0.179929
$814$	0.0814109	0.00285345
$815$	22.8399	0.800047
$816$	1.21670	0.0425930
$817$	−4.19562	−0.146786
$818$	34.6578	1.21178
$819$	−6.85002	−0.239359
$820$	−13.9059	−0.485615
$821$	34.6219	1.20831	0.604156	−	0.796866i	$-0.293510\pi$
$821$	34.6219	1.20831	0.604156	+	0.796866i	$0.293510\pi$
$822$	−1.87247	−0.0653098
$823$	−52.4113	−1.82694	−0.913472	−	0.406902i	$-0.866609\pi$
$823$	−52.4113	−1.82694	−0.913472	+	0.406902i	$0.866609\pi$
$824$	0.800128	0.0278738
$825$	−0.00358128	−0.000124684 0
$826$	−4.77585	−0.166173
$827$	−24.3231	−0.845797	−0.422898	−	0.906177i	$-0.638987\pi$
$827$	−24.3231	−0.845797	−0.422898	+	0.906177i	$0.638987\pi$
$828$	−17.6171	−0.612238
$829$	30.5861	1.06230	0.531149	−	0.847278i	$-0.321760\pi$
$829$	30.5861	1.06230	0.531149	+	0.847278i	$0.321760\pi$
$830$	−25.0899	−0.870884
$831$	−2.46015	−0.0853417
$832$	−5.22204	−0.181042
$833$	41.2278	1.42846
$834$	1.36420	0.0472382
$835$	−28.5002	−0.986289
$836$	0.0648883	0.00224421
$837$	−7.75609	−0.268090
$838$	0.221011	0.00763470
$839$	0.379687	0.0131082	0.00655412	−	0.999979i	$-0.497914\pi$
$839$	0.379687	0.0131082	0.00655412	+	0.999979i	$0.497914\pi$
$840$	0.204315	0.00704953
$841$	−13.7343	−0.473597
$842$	18.1784	0.626470
$843$	−5.75635	−0.198259
$844$	−7.70814	−0.265325
$845$	−32.7627	−1.12707
$846$	−8.77323	−0.301630
$847$	4.87336	0.167451
$848$	−11.7849	−0.404696
$849$	1.49231	0.0512158
$850$	1.64479	0.0564159
$851$	−7.37430	−0.252788
$852$	1.50224	0.0514659
$853$	7.19130	0.246225	0.123113	−	0.992393i	$-0.460712\pi$
$853$	7.19130	0.246225	0.123113	+	0.992393i	$0.460712\pi$
$854$	−1.13026	−0.0386768
$855$	6.71004	0.229479
$856$	8.61465	0.294443
$857$	−4.31879	−0.147527	−0.0737635	−	0.997276i	$-0.523501\pi$
$857$	−4.31879	−0.147527	−0.0737635	+	0.997276i	$0.523501\pi$
$858$	−0.0689003	−0.00235222
$859$	37.7536	1.28814	0.644069	−	0.764968i	$-0.277245\pi$
$859$	37.7536	1.28814	0.644069	+	0.764968i	$0.277245\pi$
$860$	9.75539	0.332656
$861$	0.538977	0.0183683
$862$	12.2671	0.417820
$863$	−6.52775	−0.222207	−0.111104	−	0.993809i	$-0.535439\pi$
$863$	−6.52775	−0.222207	−0.111104	+	0.993809i	$0.535439\pi$
$864$	−1.19661	−0.0407095
$865$	3.88519	0.132101
$866$	7.33472	0.249244
$867$	−3.95954	−0.134473
$868$	2.87274	0.0975071
$869$	−0.864405	−0.0293229
$870$	1.80116	0.0610650
$871$	18.2484	0.618325
$872$	19.3946	0.656782
$873$	−9.03833	−0.305901
$874$	−5.87767	−0.198815
$875$	−4.81172	−0.162666
$876$	−0.898980	−0.0303737
$877$	16.5833	0.559977	0.279988	−	0.960003i	$-0.409669\pi$
$877$	16.5833	0.559977	0.279988	+	0.960003i	$0.409669\pi$
$878$	13.6906	0.462036
$879$	0.667737	0.0225222
$880$	−0.150874	−0.00508597
$881$	−6.22524	−0.209734	−0.104867	−	0.994486i	$-0.533442\pi$
$881$	−6.22524	−0.209734	−0.104867	+	0.994486i	$0.533442\pi$
$882$	−20.1364	−0.678029
$883$	40.3638	1.35835	0.679175	−	0.733976i	$-0.262338\pi$
$883$	40.3638	1.35835	0.679175	+	0.733976i	$0.262338\pi$
$884$	31.6442	1.06431
$885$	−4.96750	−0.166981
$886$	−24.6137	−0.826912
$887$	−25.4303	−0.853865	−0.426932	−	0.904284i	$-0.640406\pi$
$887$	−25.4303	−0.853865	−0.426932	+	0.904284i	$0.640406\pi$
$888$	−0.248749	−0.00834746
$889$	−2.65154	−0.0889296
$890$	−4.37053	−0.146500
$891$	0.567681	0.0190180
$892$	−15.3558	−0.514151
$893$	−2.92705	−0.0979498
$894$	1.26494	0.0423059
$895$	41.9037	1.40069
$896$	0.443206	0.0148065
$897$	6.24107	0.208383
$898$	39.4678	1.31706
$899$	25.3249	0.844633
$900$	−0.803347	−0.0267782
$901$	71.4135	2.37913
$902$	−0.398002	−0.0132520
$903$	−0.378108	−0.0125827
$904$	−6.30110	−0.209572
$905$	−5.92917	−0.197092
$906$	1.39000	0.0461797
$907$	−3.34039	−0.110916	−0.0554579	−	0.998461i	$-0.517662\pi$
$907$	−3.34039	−0.110916	−0.0554579	+	0.998461i	$0.517662\pi$
$908$	−21.8972	−0.726683
$909$	4.81067	0.159560
$910$	5.31387	0.176153
$911$	−0.828856	−0.0274612	−0.0137306	−	0.999906i	$-0.504371\pi$
$911$	−0.828856	−0.0274612	−0.0137306	+	0.999906i	$0.504371\pi$
$912$	−0.198264	−0.00656519
$913$	−0.718102	−0.0237657
$914$	−13.0389	−0.431287
$915$	−1.17562	−0.0388648
$916$	23.9817	0.792376
$917$	−4.46570	−0.147470
$918$	7.25114	0.239323
$919$	20.9157	0.689945	0.344972	−	0.938613i	$-0.387888\pi$
$919$	20.9157	0.689945	0.344972	+	0.938613i	$0.387888\pi$
$920$	13.6664	0.450567
$921$	0.261178	0.00860610
$922$	29.2682	0.963897
$923$	39.0706	1.28602
$924$	0.00584772	0.000192376 0
$925$	−0.336271	−0.0110565
$926$	7.13135	0.234351
$927$	2.36813	0.0777795
$928$	3.90713	0.128258
$929$	19.1309	0.627665	0.313833	−	0.949478i	$-0.398387\pi$
$929$	19.1309	0.627665	0.313833	+	0.949478i	$0.398387\pi$
$930$	2.98802	0.0979810
$931$	−6.71819	−0.220180
$932$	26.2413	0.859564
$933$	−4.03102	−0.131970
$934$	−8.86217	−0.289979
$935$	0.914259	0.0298995
$936$	−15.4556	−0.505182
$937$	−2.98421	−0.0974900	−0.0487450	−	0.998811i	$-0.515522\pi$
$937$	−2.98421	−0.0974900	−0.0487450	+	0.998811i	$0.515522\pi$
$938$	−1.54879	−0.0505697
$939$	4.99393	0.162971
$940$	6.80579	0.221980
$941$	−19.7437	−0.643626	−0.321813	−	0.946803i	$-0.604292\pi$
$941$	−19.7437	−0.643626	−0.321813	+	0.946803i	$0.604292\pi$
$942$	2.87627	0.0937139
$943$	36.0516	1.17400
$944$	−10.7757	−0.350718
$945$	1.21765	0.0396102
$946$	0.279211	0.00907792
$947$	−23.5295	−0.764605	−0.382303	−	0.924037i	$-0.624869\pi$
$947$	−23.5295	−0.764605	−0.382303	+	0.924037i	$0.624869\pi$
$948$	2.64116	0.0857810
$949$	−23.3809	−0.758976
$950$	−0.268023	−0.00869583
$951$	−4.18100	−0.135578
$952$	−2.68571	−0.0870445
$953$	36.0159	1.16667	0.583336	−	0.812231i	$-0.301747\pi$
$953$	36.0159	1.16667	0.583336	+	0.812231i	$0.301747\pi$
$954$	−34.8796	−1.12927
$955$	−9.93441	−0.321470
$956$	0.0739620	0.00239210
$957$	0.0515512	0.00166641
$958$	11.5049	0.371707
$959$	4.13324	0.133469
$960$	0.460992	0.0148785
$961$	11.0126	0.355246
$962$	−6.46952	−0.208586
$963$	25.4966	0.821618
$964$	−15.1583	−0.488215
$965$	−2.01549	−0.0648810
$966$	−0.529694	−0.0170426
$967$	53.7919	1.72983	0.864915	−	0.501919i	$-0.167372\pi$
$967$	53.7919	1.72983	0.864915	+	0.501919i	$0.167372\pi$
$968$	10.9957	0.353415
$969$	1.20143	0.0385955
$970$	7.01143	0.225124
$971$	2.52562	0.0810512	0.0405256	−	0.999179i	$-0.487097\pi$
$971$	2.52562	0.0810512	0.0405256	+	0.999179i	$0.487097\pi$
$972$	−5.32436	−0.170779
$973$	−3.01129	−0.0965377
$974$	−23.2218	−0.744076
$975$	0.284595	0.00911433
$976$	−2.55019	−0.0816297
$977$	0.833970	0.0266811	0.0133405	−	0.999911i	$-0.495753\pi$
$977$	0.833970	0.0266811	0.0133405	+	0.999911i	$0.495753\pi$
$978$	−1.99738	−0.0638690
$979$	−0.125089	−0.00399788
$980$	15.6207	0.498985
$981$	57.4018	1.83270
$982$	−4.66706	−0.148932
$983$	36.9676	1.17908	0.589541	−	0.807739i	$-0.299309\pi$
$983$	36.9676	1.17908	0.589541	+	0.807739i	$0.299309\pi$
$984$	1.21608	0.0387674
$985$	−8.41511	−0.268128
$986$	−23.6762	−0.754003
$987$	−0.263785	−0.00839636
$988$	−5.15651	−0.164050
$989$	−25.2912	−0.804215
$990$	−0.446541	−0.0141920
$991$	−13.2317	−0.420319	−0.210159	−	0.977667i	$-0.567398\pi$
$991$	−13.2317	−0.420319	−0.210159	+	0.977667i	$0.567398\pi$
$992$	6.48171	0.205795
$993$	−1.93830	−0.0615101
$994$	−3.31601	−0.105177
$995$	12.9093	0.409252
$996$	2.19414	0.0695240
$997$	13.2405	0.419331	0.209666	−	0.977773i	$-0.432762\pi$
$997$	13.2405	0.419331	0.209666	+	0.977773i	$0.432762\pi$
$998$	31.4924	0.996873
$999$	−1.48246	−0.0469031

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.e.1.35	✓	77

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.e.1.35	✓	77	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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.200784 q^{3} +1.00000 q^{4} -2.29596 q^{5} +0.200784 q^{6} -0.443206 q^{7} -1.00000 q^{8} -2.95969 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.200784 q^{3} +1.00000 q^{4} -2.29596 q^{5} +0.200784 q^{6} -0.443206 q^{7} -1.00000 q^{8} -2.95969 q^{9} +2.29596 q^{10} +0.0657130 q^{11} -0.200784 q^{12} -5.22204 q^{13} +0.443206 q^{14} +0.460992 q^{15} +1.00000 q^{16} -6.05974 q^{17} +2.95969 q^{18} +0.987451 q^{19} -2.29596 q^{20} +0.0889888 q^{21} -0.0657130 q^{22} +5.95236 q^{23} +0.200784 q^{24} +0.271430 q^{25} +5.22204 q^{26} +1.19661 q^{27} -0.443206 q^{28} -3.90713 q^{29} -0.460992 q^{30} -6.48171 q^{31} -1.00000 q^{32} -0.0131941 q^{33} +6.05974 q^{34} +1.01758 q^{35} -2.95969 q^{36} -1.23889 q^{37} -0.987451 q^{38} +1.04850 q^{39} +2.29596 q^{40} +6.05668 q^{41} -0.0889888 q^{42} -4.24894 q^{43} +0.0657130 q^{44} +6.79532 q^{45} -5.95236 q^{46} -2.96424 q^{47} -0.200784 q^{48} -6.80357 q^{49} -0.271430 q^{50} +1.21670 q^{51} -5.22204 q^{52} -11.7849 q^{53} -1.19661 q^{54} -0.150874 q^{55} +0.443206 q^{56} -0.198264 q^{57} +3.90713 q^{58} -10.7757 q^{59} +0.460992 q^{60} -2.55019 q^{61} +6.48171 q^{62} +1.31175 q^{63} +1.00000 q^{64} +11.9896 q^{65} +0.0131941 q^{66} -3.49450 q^{67} -6.05974 q^{68} -1.19514 q^{69} -1.01758 q^{70} -7.48186 q^{71} +2.95969 q^{72} +4.47735 q^{73} +1.23889 q^{74} -0.0544988 q^{75} +0.987451 q^{76} -0.0291244 q^{77} -1.04850 q^{78} -13.1542 q^{79} -2.29596 q^{80} +8.63880 q^{81} -6.05668 q^{82} -10.9279 q^{83} +0.0889888 q^{84} +13.9129 q^{85} +4.24894 q^{86} +0.784490 q^{87} -0.0657130 q^{88} -1.90357 q^{89} -6.79532 q^{90} +2.31444 q^{91} +5.95236 q^{92} +1.30143 q^{93} +2.96424 q^{94} -2.26715 q^{95} +0.200784 q^{96} +3.05381 q^{97} +6.80357 q^{98} -0.194490 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * q^99

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