LMFDB - Embedded newform 5082.2.a.ca.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5082,2,Mod(1,5082)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5082, 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("5082.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5082.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:	$40.5799743072$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 11 $ x^4 - 7*x^2 + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.54336$ of defining polynomial
Character	$\chi$	$=$	5082.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.11525 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.11525 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.11525 q^{10} +1.00000 q^{12} +2.16140 q^{13} +1.00000 q^{14} -3.11525 q^{15} +1.00000 q^{16} -0.710334 q^{17} -1.00000 q^{18} -0.307294 q^{19} -3.11525 q^{20} -1.00000 q^{21} +0.879178 q^{23} -1.00000 q^{24} +4.70476 q^{25} -2.16140 q^{26} +1.00000 q^{27} -1.00000 q^{28} -6.57188 q^{29} +3.11525 q^{30} +7.68713 q^{31} -1.00000 q^{32} +0.710334 q^{34} +3.11525 q^{35} +1.00000 q^{36} -3.87918 q^{37} +0.307294 q^{38} +2.16140 q^{39} +3.11525 q^{40} -4.28967 q^{41} +1.00000 q^{42} +8.24254 q^{43} -3.11525 q^{45} -0.879178 q^{46} +10.0384 q^{47} +1.00000 q^{48} +1.00000 q^{49} -4.70476 q^{50} -0.710334 q^{51} +2.16140 q^{52} -6.58394 q^{53} -1.00000 q^{54} +1.00000 q^{56} -0.307294 q^{57} +6.57188 q^{58} -5.75091 q^{59} -3.11525 q^{60} -3.12427 q^{61} -7.68713 q^{62} -1.00000 q^{63} +1.00000 q^{64} -6.73328 q^{65} -6.67624 q^{67} -0.710334 q^{68} +0.879178 q^{69} -3.11525 q^{70} +10.8451 q^{71} -1.00000 q^{72} -10.7453 q^{73} +3.87918 q^{74} +4.70476 q^{75} -0.307294 q^{76} -2.16140 q^{78} +9.71565 q^{79} -3.11525 q^{80} +1.00000 q^{81} +4.28967 q^{82} -15.7847 q^{83} -1.00000 q^{84} +2.21286 q^{85} -8.24254 q^{86} -6.57188 q^{87} +14.8726 q^{89} +3.11525 q^{90} -2.16140 q^{91} +0.879178 q^{92} +7.68713 q^{93} -10.0384 q^{94} +0.957296 q^{95} -1.00000 q^{96} -0.261144 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 2 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 2 * q^5 - 4 * q^6 - 4 * q^7 - 4 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 2 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} + 4 q^{9} - 2 q^{10} + 4 q^{12} - 2 q^{13} + 4 q^{14} + 2 q^{15} + 4 q^{16} - 6 q^{17} - 4 q^{18} - 4 q^{19} + 2 q^{20} - 4 q^{21} - 2 q^{23} - 4 q^{24} + 2 q^{25} + 2 q^{26} + 4 q^{27} - 4 q^{28} - 18 q^{29} - 2 q^{30} + 8 q^{31} - 4 q^{32} + 6 q^{34} - 2 q^{35} + 4 q^{36} - 10 q^{37} + 4 q^{38} - 2 q^{39} - 2 q^{40} - 14 q^{41} + 4 q^{42} - 10 q^{43} + 2 q^{45} + 2 q^{46} - 6 q^{47} + 4 q^{48} + 4 q^{49} - 2 q^{50} - 6 q^{51} - 2 q^{52} - 4 q^{53} - 4 q^{54} + 4 q^{56} - 4 q^{57} + 18 q^{58} - 10 q^{59} + 2 q^{60} - 8 q^{61} - 8 q^{62} - 4 q^{63} + 4 q^{64} - 8 q^{65} - 12 q^{67} - 6 q^{68} - 2 q^{69} + 2 q^{70} + 20 q^{71} - 4 q^{72} - 10 q^{73} + 10 q^{74} + 2 q^{75} - 4 q^{76} + 2 q^{78} + 14 q^{79} + 2 q^{80} + 4 q^{81} + 14 q^{82} - 20 q^{83} - 4 q^{84} - 26 q^{85} + 10 q^{86} - 18 q^{87} - 2 q^{89} - 2 q^{90} + 2 q^{91} - 2 q^{92} + 8 q^{93} + 6 q^{94} - 10 q^{95} - 4 q^{96} - 4 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 2 * q^5 - 4 * q^6 - 4 * q^7 - 4 * q^8 + 4 * q^9 - 2 * q^10 + 4 * q^12 - 2 * q^13 + 4 * q^14 + 2 * q^15 + 4 * q^16 - 6 * q^17 - 4 * q^18 - 4 * q^19 + 2 * q^20 - 4 * q^21 - 2 * q^23 - 4 * q^24 + 2 * q^25 + 2 * q^26 + 4 * q^27 - 4 * q^28 - 18 * q^29 - 2 * q^30 + 8 * q^31 - 4 * q^32 + 6 * q^34 - 2 * q^35 + 4 * q^36 - 10 * q^37 + 4 * q^38 - 2 * q^39 - 2 * q^40 - 14 * q^41 + 4 * q^42 - 10 * q^43 + 2 * q^45 + 2 * q^46 - 6 * q^47 + 4 * q^48 + 4 * q^49 - 2 * q^50 - 6 * q^51 - 2 * q^52 - 4 * q^53 - 4 * q^54 + 4 * q^56 - 4 * q^57 + 18 * q^58 - 10 * q^59 + 2 * q^60 - 8 * q^61 - 8 * q^62 - 4 * q^63 + 4 * q^64 - 8 * q^65 - 12 * q^67 - 6 * q^68 - 2 * q^69 + 2 * q^70 + 20 * q^71 - 4 * q^72 - 10 * q^73 + 10 * q^74 + 2 * q^75 - 4 * q^76 + 2 * q^78 + 14 * q^79 + 2 * q^80 + 4 * q^81 + 14 * q^82 - 20 * q^83 - 4 * q^84 - 26 * q^85 + 10 * q^86 - 18 * q^87 - 2 * q^89 - 2 * q^90 + 2 * q^91 - 2 * q^92 + 8 * q^93 + 6 * q^94 - 10 * q^95 - 4 * q^96 - 4 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−3.11525	−1.39318	−0.696590	−	0.717469i	$-0.745300\pi$
$5$	−3.11525	−1.39318	−0.696590	+	0.717469i	$0.745300\pi$
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	3.11525	0.985127
$11$	0	0
$11$	0	0
$12$	1.00000	0.288675
$13$	2.16140	0.599463	0.299732	−	0.954024i	$-0.403103\pi$
$13$	2.16140	0.599463	0.299732	+	0.954024i	$0.403103\pi$
$14$	1.00000	0.267261
$15$	−3.11525	−0.804353
$16$	1.00000	0.250000
$17$	−0.710334	−0.172281	−0.0861406	−	0.996283i	$-0.527453\pi$
$17$	−0.710334	−0.172281	−0.0861406	+	0.996283i	$0.527453\pi$
$18$	−1.00000	−0.235702
$19$	−0.307294	−0.0704981	−0.0352490	−	0.999379i	$-0.511222\pi$
$19$	−0.307294	−0.0704981	−0.0352490	+	0.999379i	$0.511222\pi$
$20$	−3.11525	−0.696590
$21$	−1.00000	−0.218218
$22$	0	0
$23$	0.879178	0.183321	0.0916606	−	0.995790i	$-0.470782\pi$
$23$	0.879178	0.183321	0.0916606	+	0.995790i	$0.470782\pi$
$24$	−1.00000	−0.204124
$25$	4.70476	0.940952
$26$	−2.16140	−0.423885
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	−6.57188	−1.22037	−0.610184	−	0.792260i	$-0.708905\pi$
$29$	−6.57188	−1.22037	−0.610184	+	0.792260i	$0.708905\pi$
$30$	3.11525	0.568764
$31$	7.68713	1.38065	0.690325	−	0.723500i	$-0.257468\pi$
$31$	7.68713	1.38065	0.690325	+	0.723500i	$0.257468\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0.710334	0.121821
$35$	3.11525	0.526573
$36$	1.00000	0.166667
$37$	−3.87918	−0.637733	−0.318866	−	0.947800i	$-0.603302\pi$
$37$	−3.87918	−0.637733	−0.318866	+	0.947800i	$0.603302\pi$
$38$	0.307294	0.0498497
$39$	2.16140	0.346100
$40$	3.11525	0.492564
$41$	−4.28967	−0.669933	−0.334967	−	0.942230i	$-0.608725\pi$
$41$	−4.28967	−0.669933	−0.334967	+	0.942230i	$0.608725\pi$
$42$	1.00000	0.154303
$43$	8.24254	1.25698	0.628488	−	0.777819i	$-0.283674\pi$
$43$	8.24254	1.25698	0.628488	+	0.777819i	$0.283674\pi$
$44$	0	0
$45$	−3.11525	−0.464393
$46$	−0.879178	−0.129628
$47$	10.0384	1.46426	0.732129	−	0.681166i	$-0.238527\pi$
$47$	10.0384	1.46426	0.732129	+	0.681166i	$0.238527\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	−4.70476	−0.665353
$51$	−0.710334	−0.0994666
$52$	2.16140	0.299732
$53$	−6.58394	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$53$	−6.58394	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	−0.307294	−0.0407021
$58$	6.57188	0.862931
$59$	−5.75091	−0.748704	−0.374352	−	0.927287i	$-0.622135\pi$
$59$	−5.75091	−0.748704	−0.374352	+	0.927287i	$0.622135\pi$
$60$	−3.11525	−0.402177
$61$	−3.12427	−0.400022	−0.200011	−	0.979794i	$-0.564098\pi$
$61$	−3.12427	−0.400022	−0.200011	+	0.979794i	$0.564098\pi$
$62$	−7.68713	−0.976267
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−6.73328	−0.835161
$66$	0	0
$67$	−6.67624	−0.815632	−0.407816	−	0.913064i	$-0.633710\pi$
$67$	−6.67624	−0.815632	−0.407816	+	0.913064i	$0.633710\pi$
$68$	−0.710334	−0.0861406
$69$	0.879178	0.105841
$70$	−3.11525	−0.372343
$71$	10.8451	1.28707	0.643537	−	0.765415i	$-0.277466\pi$
$71$	10.8451	1.28707	0.643537	+	0.765415i	$0.277466\pi$
$72$	−1.00000	−0.117851
$73$	−10.7453	−1.25765	−0.628823	−	0.777548i	$-0.716463\pi$
$73$	−10.7453	−1.25765	−0.628823	+	0.777548i	$0.716463\pi$
$74$	3.87918	0.450945
$75$	4.70476	0.543259
$76$	−0.307294	−0.0352490
$77$	0	0
$78$	−2.16140	−0.244730
$79$	9.71565	1.09310	0.546548	−	0.837428i	$-0.315942\pi$
$79$	9.71565	1.09310	0.546548	+	0.837428i	$0.315942\pi$
$80$	−3.11525	−0.348295
$81$	1.00000	0.111111
$82$	4.28967	0.473714
$83$	−15.7847	−1.73260	−0.866301	−	0.499523i	$-0.833509\pi$
$83$	−15.7847	−1.73260	−0.866301	+	0.499523i	$0.833509\pi$
$84$	−1.00000	−0.109109
$85$	2.21286	0.240019
$86$	−8.24254	−0.888816
$87$	−6.57188	−0.704580
$88$	0	0
$89$	14.8726	1.57650	0.788248	−	0.615358i	$-0.210989\pi$
$89$	14.8726	1.57650	0.788248	+	0.615358i	$0.210989\pi$
$90$	3.11525	0.328376
$91$	−2.16140	−0.226576
$92$	0.879178	0.0916606
$93$	7.68713	0.797118
$94$	−10.0384	−1.03539
$95$	0.957296	0.0982165
$96$	−1.00000	−0.102062
$97$	−0.261144	−0.0265152	−0.0132576	−	0.999912i	$-0.504220\pi$
$97$	−0.261144	−0.0265152	−0.0132576	+	0.999912i	$0.504220\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	4.70476	0.470476
$101$	−2.55081	−0.253815	−0.126908	−	0.991915i	$-0.540505\pi$
$101$	−2.55081	−0.253815	−0.126908	+	0.991915i	$0.540505\pi$
$102$	0.710334	0.0703335
$103$	18.8420	1.85656	0.928278	−	0.371888i	$-0.121289\pi$
$103$	18.8420	1.85656	0.928278	+	0.371888i	$0.121289\pi$
$104$	−2.16140	−0.211942
$105$	3.11525	0.304017
$106$	6.58394	0.639488
$107$	3.40091	0.328778	0.164389	−	0.986396i	$-0.447435\pi$
$107$	3.40091	0.328778	0.164389	+	0.986396i	$0.447435\pi$
$108$	1.00000	0.0962250
$109$	−6.96590	−0.667212	−0.333606	−	0.942713i	$-0.608266\pi$
$109$	−6.96590	−0.667212	−0.333606	+	0.942713i	$0.608266\pi$
$110$	0	0
$111$	−3.87918	−0.368195
$112$	−1.00000	−0.0944911
$113$	−1.39802	−0.131515	−0.0657573	−	0.997836i	$-0.520946\pi$
$113$	−1.39802	−0.131515	−0.0657573	+	0.997836i	$0.520946\pi$
$114$	0.307294	0.0287807
$115$	−2.73886	−0.255400
$116$	−6.57188	−0.610184
$117$	2.16140	0.199821
$118$	5.75091	0.529414
$119$	0.710334	0.0651162
$120$	3.11525	0.284382
$121$	0	0
$122$	3.12427	0.282858
$123$	−4.28967	−0.386786
$124$	7.68713	0.690325
$125$	0.919752	0.0822651
$126$	1.00000	0.0890871
$127$	−16.0430	−1.42359	−0.711795	−	0.702387i	$-0.752118\pi$
$127$	−16.0430	−1.42359	−0.711795	+	0.702387i	$0.752118\pi$
$128$	−1.00000	−0.0883883
$129$	8.24254	0.725716
$130$	6.73328	0.590548
$131$	−19.8696	−1.73601	−0.868007	−	0.496551i	$-0.834600\pi$
$131$	−19.8696	−1.73601	−0.868007	+	0.496551i	$0.834600\pi$
$132$	0	0
$133$	0.307294	0.0266458
$134$	6.67624	0.576739
$135$	−3.11525	−0.268118
$136$	0.710334	0.0609106
$137$	2.32335	0.198497	0.0992485	−	0.995063i	$-0.468356\pi$
$137$	2.32335	0.198497	0.0992485	+	0.995063i	$0.468356\pi$
$138$	−0.879178	−0.0748406
$139$	−7.48206	−0.634620	−0.317310	−	0.948322i	$-0.602780\pi$
$139$	−7.48206	−0.634620	−0.317310	+	0.948322i	$0.602780\pi$
$140$	3.11525	0.263286
$141$	10.0384	0.845390
$142$	−10.8451	−0.910099
$143$	0	0
$144$	1.00000	0.0833333
$145$	20.4730	1.70019
$146$	10.7453	0.889290
$147$	1.00000	0.0824786
$148$	−3.87918	−0.318866
$149$	−1.42756	−0.116950	−0.0584751	−	0.998289i	$-0.518624\pi$
$149$	−1.42756	−0.116950	−0.0584751	+	0.998289i	$0.518624\pi$
$150$	−4.70476	−0.384142
$151$	9.96190	0.810688	0.405344	−	0.914164i	$-0.367152\pi$
$151$	9.96190	0.810688	0.405344	+	0.914164i	$0.367152\pi$
$152$	0.307294	0.0249248
$153$	−0.710334	−0.0574271
$154$	0	0
$155$	−23.9473	−1.92349
$156$	2.16140	0.173050
$157$	−24.6413	−1.96659	−0.983296	−	0.182014i	$-0.941738\pi$
$157$	−24.6413	−1.96659	−0.983296	+	0.182014i	$0.941738\pi$
$158$	−9.71565	−0.772936
$159$	−6.58394	−0.522140
$160$	3.11525	0.246282
$161$	−0.879178	−0.0692889
$162$	−1.00000	−0.0785674
$163$	14.3469	1.12374	0.561868	−	0.827227i	$-0.310083\pi$
$163$	14.3469	1.12374	0.561868	+	0.827227i	$0.310083\pi$
$164$	−4.28967	−0.334967
$165$	0	0
$166$	15.7847	1.22513
$167$	−14.0316	−1.08579	−0.542897	−	0.839799i	$-0.682673\pi$
$167$	−14.0316	−1.08579	−0.542897	+	0.839799i	$0.682673\pi$
$168$	1.00000	0.0771517
$169$	−8.32837	−0.640644
$170$	−2.21286	−0.169719
$171$	−0.307294	−0.0234994
$172$	8.24254	0.628488
$173$	−14.1774	−1.07789	−0.538946	−	0.842340i	$-0.681177\pi$
$173$	−14.1774	−1.07789	−0.538946	+	0.842340i	$0.681177\pi$
$174$	6.57188	0.498213
$175$	−4.70476	−0.355646
$176$	0	0
$177$	−5.75091	−0.432265
$178$	−14.8726	−1.11475
$179$	−15.8912	−1.18777	−0.593883	−	0.804551i	$-0.702406\pi$
$179$	−15.8912	−1.18777	−0.593883	+	0.804551i	$0.702406\pi$
$180$	−3.11525	−0.232197
$181$	−17.7593	−1.32004	−0.660018	−	0.751250i	$-0.729451\pi$
$181$	−17.7593	−1.32004	−0.660018	+	0.751250i	$0.729451\pi$
$182$	2.16140	0.160213
$183$	−3.12427	−0.230953
$184$	−0.879178	−0.0648139
$185$	12.0846	0.888477
$186$	−7.68713	−0.563648
$187$	0	0
$188$	10.0384	0.732129
$189$	−1.00000	−0.0727393
$190$	−0.957296	−0.0694496
$191$	−26.2906	−1.90232	−0.951159	−	0.308700i	$-0.900106\pi$
$191$	−26.2906	−1.90232	−0.951159	+	0.308700i	$0.900106\pi$
$192$	1.00000	0.0721688
$193$	−5.56788	−0.400785	−0.200392	−	0.979716i	$-0.564222\pi$
$193$	−5.56788	−0.400785	−0.200392	+	0.979716i	$0.564222\pi$
$194$	0.261144	0.0187491
$195$	−6.73328	−0.482180
$196$	1.00000	0.0714286
$197$	11.8089	0.841346	0.420673	−	0.907212i	$-0.361794\pi$
$197$	11.8089	0.841346	0.420673	+	0.907212i	$0.361794\pi$
$198$	0	0
$199$	9.53591	0.675983	0.337991	−	0.941149i	$-0.390252\pi$
$199$	9.53591	0.675983	0.337991	+	0.941149i	$0.390252\pi$
$200$	−4.70476	−0.332677
$201$	−6.67624	−0.470905
$202$	2.55081	0.179474
$203$	6.57188	0.461256
$204$	−0.710334	−0.0497333
$205$	13.3634	0.933338
$206$	−18.8420	−1.31278
$207$	0.879178	0.0611071
$208$	2.16140	0.149866
$209$	0	0
$210$	−3.11525	−0.214972
$211$	11.9545	0.822978	0.411489	−	0.911415i	$-0.365009\pi$
$211$	11.9545	0.822978	0.411489	+	0.911415i	$0.365009\pi$
$212$	−6.58394	−0.452187
$213$	10.8451	0.743093
$214$	−3.40091	−0.232481
$215$	−25.6776	−1.75119
$216$	−1.00000	−0.0680414
$217$	−7.68713	−0.521836
$218$	6.96590	0.471790
$219$	−10.7453	−0.726102
$220$	0	0
$221$	−1.53531	−0.103276
$222$	3.87918	0.260353
$223$	−8.77887	−0.587877	−0.293938	−	0.955824i	$-0.594966\pi$
$223$	−8.77887	−0.587877	−0.293938	+	0.955824i	$0.594966\pi$
$224$	1.00000	0.0668153
$225$	4.70476	0.313651
$226$	1.39802	0.0929949
$227$	19.5027	1.29444	0.647220	−	0.762303i	$-0.275931\pi$
$227$	19.5027	1.29444	0.647220	+	0.762303i	$0.275931\pi$
$228$	−0.307294	−0.0203510
$229$	−7.27967	−0.481054	−0.240527	−	0.970642i	$-0.577320\pi$
$229$	−7.27967	−0.481054	−0.240527	+	0.970642i	$0.577320\pi$
$230$	2.73886	0.180595
$231$	0	0
$232$	6.57188	0.431465
$233$	2.73238	0.179004	0.0895021	−	0.995987i	$-0.471472\pi$
$233$	2.73238	0.179004	0.0895021	+	0.995987i	$0.471472\pi$
$234$	−2.16140	−0.141295
$235$	−31.2722	−2.03997
$236$	−5.75091	−0.374352
$237$	9.71565	0.631099
$238$	−0.710334	−0.0460441
$239$	−25.8977	−1.67518	−0.837592	−	0.546297i	$-0.816037\pi$
$239$	−25.8977	−1.67518	−0.837592	+	0.546297i	$0.816037\pi$
$240$	−3.11525	−0.201088
$241$	6.36034	0.409705	0.204853	−	0.978793i	$-0.434328\pi$
$241$	6.36034	0.409705	0.204853	+	0.978793i	$0.434328\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−3.12427	−0.200011
$245$	−3.11525	−0.199026
$246$	4.28967	0.273499
$247$	−0.664184	−0.0422610
$248$	−7.68713	−0.488133
$249$	−15.7847	−1.00032
$250$	−0.919752	−0.0581702
$251$	−12.8826	−0.813144	−0.406572	−	0.913619i	$-0.633276\pi$
$251$	−12.8826	−0.813144	−0.406572	+	0.913619i	$0.633276\pi$
$252$	−1.00000	−0.0629941
$253$	0	0
$254$	16.0430	1.00663
$255$	2.21286	0.138575
$256$	1.00000	0.0625000
$257$	5.22836	0.326136	0.163068	−	0.986615i	$-0.447861\pi$
$257$	5.22836	0.326136	0.163068	+	0.986615i	$0.447861\pi$
$258$	−8.24254	−0.513158
$259$	3.87918	0.241040
$260$	−6.73328	−0.417580
$261$	−6.57188	−0.406789
$262$	19.8696	1.22755
$263$	−26.8248	−1.65409	−0.827043	−	0.562139i	$-0.809979\pi$
$263$	−26.8248	−1.65409	−0.827043	+	0.562139i	$0.809979\pi$
$264$	0	0
$265$	20.5106	1.25995
$266$	−0.307294	−0.0188414
$267$	14.8726	0.910190
$268$	−6.67624	−0.407816
$269$	−11.0403	−0.673140	−0.336570	−	0.941658i	$-0.609267\pi$
$269$	−11.0403	−0.673140	−0.336570	+	0.941658i	$0.609267\pi$
$270$	3.11525	0.189588
$271$	18.1134	1.10031	0.550155	−	0.835063i	$-0.314569\pi$
$271$	18.1134	1.10031	0.550155	+	0.835063i	$0.314569\pi$
$272$	−0.710334	−0.0430703
$273$	−2.16140	−0.130814
$274$	−2.32335	−0.140359
$275$	0	0
$276$	0.879178	0.0529203
$277$	−11.3212	−0.680226	−0.340113	−	0.940385i	$-0.610465\pi$
$277$	−11.3212	−0.680226	−0.340113	+	0.940385i	$0.610465\pi$
$278$	7.48206	0.448744
$279$	7.68713	0.460216
$280$	−3.11525	−0.186172
$281$	−25.1590	−1.50086	−0.750430	−	0.660950i	$-0.770154\pi$
$281$	−25.1590	−1.50086	−0.750430	+	0.660950i	$0.770154\pi$
$282$	−10.0384	−0.597781
$283$	−5.61748	−0.333924	−0.166962	−	0.985963i	$-0.553396\pi$
$283$	−5.61748	−0.333924	−0.166962	+	0.985963i	$0.553396\pi$
$284$	10.8451	0.643537
$285$	0.957296	0.0567053
$286$	0	0
$287$	4.28967	0.253211
$288$	−1.00000	−0.0589256
$289$	−16.4954	−0.970319
$290$	−20.4730	−1.20222
$291$	−0.261144	−0.0153085
$292$	−10.7453	−0.628823
$293$	−8.30116	−0.484959	−0.242480	−	0.970157i	$-0.577961\pi$
$293$	−8.30116	−0.484959	−0.242480	+	0.970157i	$0.577961\pi$
$294$	−1.00000	−0.0583212
$295$	17.9155	1.04308
$296$	3.87918	0.225473
$297$	0	0
$298$	1.42756	0.0826963
$299$	1.90025	0.109894
$300$	4.70476	0.271629
$301$	−8.24254	−0.475092
$302$	−9.96190	−0.573243
$303$	−2.55081	−0.146540
$304$	−0.307294	−0.0176245
$305$	9.73286	0.557302
$306$	0.710334	0.0406071
$307$	−1.44088	−0.0822355	−0.0411178	−	0.999154i	$-0.513092\pi$
$307$	−1.44088	−0.0822355	−0.0411178	+	0.999154i	$0.513092\pi$
$308$	0	0
$309$	18.8420	1.07188
$310$	23.9473	1.36012
$311$	9.89426	0.561052	0.280526	−	0.959846i	$-0.409491\pi$
$311$	9.89426	0.561052	0.280526	+	0.959846i	$0.409491\pi$
$312$	−2.16140	−0.122365
$313$	−5.55886	−0.314205	−0.157103	−	0.987582i	$-0.550215\pi$
$313$	−5.55886	−0.314205	−0.157103	+	0.987582i	$0.550215\pi$
$314$	24.6413	1.39059
$315$	3.11525	0.175524
$316$	9.71565	0.546548
$317$	−0.694280	−0.0389946	−0.0194973	−	0.999810i	$-0.506207\pi$
$317$	−0.694280	−0.0389946	−0.0194973	+	0.999810i	$0.506207\pi$
$318$	6.58394	0.369209
$319$	0	0
$320$	−3.11525	−0.174148
$321$	3.40091	0.189820
$322$	0.879178	0.0489947
$323$	0.218281	0.0121455
$324$	1.00000	0.0555556
$325$	10.1688	0.564066
$326$	−14.3469	−0.794601
$327$	−6.96590	−0.385215
$328$	4.28967	0.236857
$329$	−10.0384	−0.553437
$330$	0	0
$331$	−31.0269	−1.70539	−0.852696	−	0.522408i	$-0.825034\pi$
$331$	−31.0269	−1.70539	−0.852696	+	0.522408i	$0.825034\pi$
$332$	−15.7847	−0.866301
$333$	−3.87918	−0.212578
$334$	14.0316	0.767772
$335$	20.7981	1.13632
$336$	−1.00000	−0.0545545
$337$	−16.4371	−0.895384	−0.447692	−	0.894188i	$-0.647754\pi$
$337$	−16.4371	−0.895384	−0.447692	+	0.894188i	$0.647754\pi$
$338$	8.32837	0.453003
$339$	−1.39802	−0.0759300
$340$	2.21286	0.120009
$341$	0	0
$342$	0.307294	0.0166166
$343$	−1.00000	−0.0539949
$344$	−8.24254	−0.444408
$345$	−2.73886	−0.147455
$346$	14.1774	0.762184
$347$	−35.0591	−1.88207	−0.941036	−	0.338308i	$-0.890146\pi$
$347$	−35.0591	−1.88207	−0.941036	+	0.338308i	$0.890146\pi$
$348$	−6.57188	−0.352290
$349$	−8.62698	−0.461792	−0.230896	−	0.972978i	$-0.574166\pi$
$349$	−8.62698	−0.461792	−0.230896	+	0.972978i	$0.574166\pi$
$350$	4.70476	0.251480
$351$	2.16140	0.115367
$352$	0	0
$353$	−28.0551	−1.49322	−0.746611	−	0.665261i	$-0.768320\pi$
$353$	−28.0551	−1.49322	−0.746611	+	0.665261i	$0.768320\pi$
$354$	5.75091	0.305657
$355$	−33.7851	−1.79313
$356$	14.8726	0.788248
$357$	0.710334	0.0375948
$358$	15.8912	0.839878
$359$	18.6575	0.984703	0.492351	−	0.870397i	$-0.336138\pi$
$359$	18.6575	0.984703	0.492351	+	0.870397i	$0.336138\pi$
$360$	3.11525	0.164188
$361$	−18.9056	−0.995030
$362$	17.7593	0.933406
$363$	0	0
$364$	−2.16140	−0.113288
$365$	33.4744	1.75213
$366$	3.12427	0.163308
$367$	−1.50679	−0.0786538	−0.0393269	−	0.999226i	$-0.512521\pi$
$367$	−1.50679	−0.0786538	−0.0393269	+	0.999226i	$0.512521\pi$
$368$	0.879178	0.0458303
$369$	−4.28967	−0.223311
$370$	−12.0846	−0.628248
$371$	6.58394	0.341821
$372$	7.68713	0.398559
$373$	34.9467	1.80947	0.904737	−	0.425971i	$-0.140067\pi$
$373$	34.9467	1.80947	0.904737	+	0.425971i	$0.140067\pi$
$374$	0	0
$375$	0.919752	0.0474958
$376$	−10.0384	−0.517693
$377$	−14.2044	−0.731566
$378$	1.00000	0.0514344
$379$	11.9312	0.612867	0.306434	−	0.951892i	$-0.400864\pi$
$379$	11.9312	0.612867	0.306434	+	0.951892i	$0.400864\pi$
$380$	0.957296	0.0491083
$381$	−16.0430	−0.821910
$382$	26.2906	1.34514
$383$	18.0769	0.923686	0.461843	−	0.886962i	$-0.347188\pi$
$383$	18.0769	0.923686	0.461843	+	0.886962i	$0.347188\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	5.56788	0.283398
$387$	8.24254	0.418992
$388$	−0.261144	−0.0132576
$389$	−1.51581	−0.0768547	−0.0384274	−	0.999261i	$-0.512235\pi$
$389$	−1.51581	−0.0768547	−0.0384274	+	0.999261i	$0.512235\pi$
$390$	6.73328	0.340953
$391$	−0.624510	−0.0315828
$392$	−1.00000	−0.0505076
$393$	−19.8696	−1.00229
$394$	−11.8089	−0.594921
$395$	−30.2666	−1.52288
$396$	0	0
$397$	32.2351	1.61783	0.808917	−	0.587924i	$-0.200054\pi$
$397$	32.2351	1.61783	0.808917	+	0.587924i	$0.200054\pi$
$398$	−9.53591	−0.477992
$399$	0.307294	0.0153839
$400$	4.70476	0.235238
$401$	8.93278	0.446082	0.223041	−	0.974809i	$-0.428402\pi$
$401$	8.93278	0.446082	0.223041	+	0.974809i	$0.428402\pi$
$402$	6.67624	0.332980
$403$	16.6149	0.827649
$404$	−2.55081	−0.126908
$405$	−3.11525	−0.154798
$406$	−6.57188	−0.326157
$407$	0	0
$408$	0.710334	0.0351668
$409$	21.4864	1.06243	0.531217	−	0.847236i	$-0.321735\pi$
$409$	21.4864	1.06243	0.531217	+	0.847236i	$0.321735\pi$
$410$	−13.3634	−0.659970
$411$	2.32335	0.114602
$412$	18.8420	0.928278
$413$	5.75091	0.282984
$414$	−0.879178	−0.0432092
$415$	49.1734	2.41383
$416$	−2.16140	−0.105971
$417$	−7.48206	−0.366398
$418$	0	0
$419$	22.3380	1.09128	0.545642	−	0.838018i	$-0.316286\pi$
$419$	22.3380	1.09128	0.545642	+	0.838018i	$0.316286\pi$
$420$	3.11525	0.152008
$421$	9.90055	0.482523	0.241262	−	0.970460i	$-0.422439\pi$
$421$	9.90055	0.482523	0.241262	+	0.970460i	$0.422439\pi$
$422$	−11.9545	−0.581934
$423$	10.0384	0.488086
$424$	6.58394	0.319744
$425$	−3.34195	−0.162108
$426$	−10.8451	−0.525446
$427$	3.12427	0.151194
$428$	3.40091	0.164389
$429$	0	0
$430$	25.6776	1.23828
$431$	−1.62634	−0.0783381	−0.0391690	−	0.999233i	$-0.512471\pi$
$431$	−1.62634	−0.0783381	−0.0391690	+	0.999233i	$0.512471\pi$
$432$	1.00000	0.0481125
$433$	−16.4527	−0.790665	−0.395333	−	0.918538i	$-0.629371\pi$
$433$	−16.4527	−0.790665	−0.395333	+	0.918538i	$0.629371\pi$
$434$	7.68713	0.368994
$435$	20.4730	0.981607
$436$	−6.96590	−0.333606
$437$	−0.270166	−0.0129238
$438$	10.7453	0.513432
$439$	6.45481	0.308071	0.154036	−	0.988065i	$-0.450773\pi$
$439$	6.45481	0.308071	0.154036	+	0.988065i	$0.450773\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	1.53531	0.0730274
$443$	34.6921	1.64827	0.824135	−	0.566393i	$-0.191662\pi$
$443$	34.6921	1.64827	0.824135	+	0.566393i	$0.191662\pi$
$444$	−3.87918	−0.184098
$445$	−46.3319	−2.19634
$446$	8.77887	0.415692
$447$	−1.42756	−0.0675213
$448$	−1.00000	−0.0472456
$449$	−36.2968	−1.71295	−0.856476	−	0.516188i	$-0.827351\pi$
$449$	−36.2968	−1.71295	−0.856476	+	0.516188i	$0.827351\pi$
$450$	−4.70476	−0.221784
$451$	0	0
$452$	−1.39802	−0.0657573
$453$	9.96190	0.468051
$454$	−19.5027	−0.915308
$455$	6.73328	0.315661
$456$	0.307294	0.0143904
$457$	−21.2889	−0.995851	−0.497925	−	0.867220i	$-0.665905\pi$
$457$	−21.2889	−0.995851	−0.497925	+	0.867220i	$0.665905\pi$
$458$	7.27967	0.340157
$459$	−0.710334	−0.0331555
$460$	−2.73886	−0.127700
$461$	34.3997	1.60215	0.801076	−	0.598562i	$-0.204261\pi$
$461$	34.3997	1.60215	0.801076	+	0.598562i	$0.204261\pi$
$462$	0	0
$463$	33.4916	1.55648	0.778242	−	0.627964i	$-0.216111\pi$
$463$	33.4916	1.55648	0.778242	+	0.627964i	$0.216111\pi$
$464$	−6.57188	−0.305092
$465$	−23.9473	−1.11053
$466$	−2.73238	−0.126575
$467$	−24.9067	−1.15255	−0.576273	−	0.817257i	$-0.695493\pi$
$467$	−24.9067	−1.15255	−0.576273	+	0.817257i	$0.695493\pi$
$468$	2.16140	0.0999106
$469$	6.67624	0.308280
$470$	31.2722	1.44248
$471$	−24.6413	−1.13541
$472$	5.75091	0.264707
$473$	0	0
$474$	−9.71565	−0.446255
$475$	−1.44574	−0.0663353
$476$	0.710334	0.0325581
$477$	−6.58394	−0.301458
$478$	25.8977	1.18453
$479$	26.0657	1.19097	0.595487	−	0.803365i	$-0.296959\pi$
$479$	26.0657	1.19097	0.595487	+	0.803365i	$0.296959\pi$
$480$	3.11525	0.142191
$481$	−8.38444	−0.382298
$482$	−6.36034	−0.289705
$483$	−0.879178	−0.0400040
$484$	0	0
$485$	0.813528	0.0369404
$486$	−1.00000	−0.0453609
$487$	−11.0362	−0.500099	−0.250050	−	0.968233i	$-0.580447\pi$
$487$	−11.0362	−0.500099	−0.250050	+	0.968233i	$0.580447\pi$
$488$	3.12427	0.141429
$489$	14.3469	0.648789
$490$	3.11525	0.140732
$491$	−0.182886	−0.00825353	−0.00412677	−	0.999991i	$-0.501314\pi$
$491$	−0.182886	−0.00825353	−0.00412677	+	0.999991i	$0.501314\pi$
$492$	−4.28967	−0.193393
$493$	4.66823	0.210247
$494$	0.664184	0.0298830
$495$	0	0
$496$	7.68713	0.345162
$497$	−10.8451	−0.486468
$498$	15.7847	0.707331
$499$	16.8881	0.756016	0.378008	−	0.925802i	$-0.376609\pi$
$499$	16.8881	0.756016	0.378008	+	0.925802i	$0.376609\pi$
$500$	0.919752	0.0411326
$501$	−14.0316	−0.626884
$502$	12.8826	0.574980
$503$	−4.08395	−0.182094	−0.0910472	−	0.995847i	$-0.529021\pi$
$503$	−4.08395	−0.182094	−0.0910472	+	0.995847i	$0.529021\pi$
$504$	1.00000	0.0445435
$505$	7.94640	0.353610
$506$	0	0
$507$	−8.32837	−0.369876
$508$	−16.0430	−0.711795
$509$	37.2974	1.65318	0.826589	−	0.562806i	$-0.190278\pi$
$509$	37.2974	1.65318	0.826589	+	0.562806i	$0.190278\pi$
$510$	−2.21286	−0.0979873
$511$	10.7453	0.475345
$512$	−1.00000	−0.0441942
$513$	−0.307294	−0.0135674
$514$	−5.22836	−0.230613
$515$	−58.6974	−2.58652
$516$	8.24254	0.362858
$517$	0	0
$518$	−3.87918	−0.170441
$519$	−14.1774	−0.622321
$520$	6.73328	0.295274
$521$	−0.942142	−0.0412760	−0.0206380	−	0.999787i	$-0.506570\pi$
$521$	−0.942142	−0.0412760	−0.0206380	+	0.999787i	$0.506570\pi$
$522$	6.57188	0.287644
$523$	−22.9919	−1.00536	−0.502682	−	0.864471i	$-0.667653\pi$
$523$	−22.9919	−1.00536	−0.502682	+	0.864471i	$0.667653\pi$
$524$	−19.8696	−0.868007
$525$	−4.70476	−0.205332
$526$	26.8248	1.16962
$527$	−5.46043	−0.237860
$528$	0	0
$529$	−22.2270	−0.966393
$530$	−20.5106	−0.890923
$531$	−5.75091	−0.249568
$532$	0.307294	0.0133229
$533$	−9.27167	−0.401600
$534$	−14.8726	−0.643602
$535$	−10.5947	−0.458048
$536$	6.67624	0.288369
$537$	−15.8912	−0.685757
$538$	11.0403	0.475982
$539$	0	0
$540$	−3.11525	−0.134059
$541$	−18.7441	−0.805872	−0.402936	−	0.915228i	$-0.632010\pi$
$541$	−18.7441	−0.805872	−0.402936	+	0.915228i	$0.632010\pi$
$542$	−18.1134	−0.778036
$543$	−17.7593	−0.762123
$544$	0.710334	0.0304553
$545$	21.7005	0.929547
$546$	2.16140	0.0924992
$547$	−26.3033	−1.12465	−0.562325	−	0.826917i	$-0.690093\pi$
$547$	−26.3033	−1.12465	−0.562325	+	0.826917i	$0.690093\pi$
$548$	2.32335	0.0992485
$549$	−3.12427	−0.133341
$550$	0	0
$551$	2.01950	0.0860336
$552$	−0.879178	−0.0374203
$553$	−9.71565	−0.413152
$554$	11.3212	0.480992
$555$	12.0846	0.512962
$556$	−7.48206	−0.317310
$557$	7.66494	0.324774	0.162387	−	0.986727i	$-0.448081\pi$
$557$	7.66494	0.324774	0.162387	+	0.986727i	$0.448081\pi$
$558$	−7.68713	−0.325422
$559$	17.8154	0.753511
$560$	3.11525	0.131643
$561$	0	0
$562$	25.1590	1.06127
$563$	40.2363	1.69576	0.847880	−	0.530189i	$-0.177879\pi$
$563$	40.2363	1.69576	0.847880	+	0.530189i	$0.177879\pi$
$564$	10.0384	0.422695
$565$	4.35518	0.183224
$566$	5.61748	0.236120
$567$	−1.00000	−0.0419961
$568$	−10.8451	−0.455049
$569$	30.0648	1.26038	0.630192	−	0.776440i	$-0.282976\pi$
$569$	30.0648	1.26038	0.630192	+	0.776440i	$0.282976\pi$
$570$	−0.957296	−0.0400967
$571$	21.9318	0.917817	0.458909	−	0.888483i	$-0.348240\pi$
$571$	21.9318	0.917817	0.458909	+	0.888483i	$0.348240\pi$
$572$	0	0
$573$	−26.2906	−1.09830
$574$	−4.28967	−0.179047
$575$	4.13632	0.172496
$576$	1.00000	0.0416667
$577$	−36.3013	−1.51124	−0.755622	−	0.655007i	$-0.772665\pi$
$577$	−36.3013	−1.51124	−0.755622	+	0.655007i	$0.772665\pi$
$578$	16.4954	0.686119
$579$	−5.56788	−0.231393
$580$	20.4730	0.850097
$581$	15.7847	0.654862
$582$	0.261144	0.0108248
$583$	0	0
$584$	10.7453	0.444645
$585$	−6.73328	−0.278387
$586$	8.30116	0.342918
$587$	−5.88591	−0.242938	−0.121469	−	0.992595i	$-0.538760\pi$
$587$	−5.88591	−0.242938	−0.121469	+	0.992595i	$0.538760\pi$
$588$	1.00000	0.0412393
$589$	−2.36221	−0.0973331
$590$	−17.9155	−0.737569
$591$	11.8089	0.485751
$592$	−3.87918	−0.159433
$593$	10.5795	0.434449	0.217224	−	0.976122i	$-0.430300\pi$
$593$	10.5795	0.434449	0.217224	+	0.976122i	$0.430300\pi$
$594$	0	0
$595$	−2.21286	−0.0907186
$596$	−1.42756	−0.0584751
$597$	9.53591	0.390279
$598$	−1.90025	−0.0777071
$599$	−12.0731	−0.493293	−0.246647	−	0.969105i	$-0.579329\pi$
$599$	−12.0731	−0.493293	−0.246647	+	0.969105i	$0.579329\pi$
$600$	−4.70476	−0.192071
$601$	43.0464	1.75590	0.877949	−	0.478754i	$-0.158911\pi$
$601$	43.0464	1.75590	0.877949	+	0.478754i	$0.158911\pi$
$602$	8.24254	0.335941
$603$	−6.67624	−0.271877
$604$	9.96190	0.405344
$605$	0	0
$606$	2.55081	0.103620
$607$	−11.8346	−0.480352	−0.240176	−	0.970729i	$-0.577205\pi$
$607$	−11.8346	−0.480352	−0.240176	+	0.970729i	$0.577205\pi$
$608$	0.307294	0.0124624
$609$	6.57188	0.266306
$610$	−9.73286	−0.394072
$611$	21.6971	0.877769
$612$	−0.710334	−0.0287135
$613$	−37.6731	−1.52160	−0.760800	−	0.648986i	$-0.775193\pi$
$613$	−37.6731	−1.52160	−0.760800	+	0.648986i	$0.775193\pi$
$614$	1.44088	0.0581493
$615$	13.3634	0.538863
$616$	0	0
$617$	−40.5797	−1.63368	−0.816838	−	0.576867i	$-0.804275\pi$
$617$	−40.5797	−1.63368	−0.816838	+	0.576867i	$0.804275\pi$
$618$	−18.8420	−0.757935
$619$	−23.1193	−0.929243	−0.464621	−	0.885509i	$-0.653810\pi$
$619$	−23.1193	−0.929243	−0.464621	+	0.885509i	$0.653810\pi$
$620$	−23.9473	−0.961747
$621$	0.879178	0.0352802
$622$	−9.89426	−0.396724
$623$	−14.8726	−0.595859
$624$	2.16140	0.0865251
$625$	−26.3890	−1.05556
$626$	5.55886	0.222177
$627$	0	0
$628$	−24.6413	−0.983296
$629$	2.75551	0.109869
$630$	−3.11525	−0.124114
$631$	−18.4438	−0.734236	−0.367118	−	0.930174i	$-0.619655\pi$
$631$	−18.4438	−0.734236	−0.367118	+	0.930174i	$0.619655\pi$
$632$	−9.71565	−0.386468
$633$	11.9545	0.475147
$634$	0.694280	0.0275734
$635$	49.9780	1.98332
$636$	−6.58394	−0.261070
$637$	2.16140	0.0856376
$638$	0	0
$639$	10.8451	0.429025
$640$	3.11525	0.123141
$641$	−7.69072	−0.303765	−0.151883	−	0.988399i	$-0.548534\pi$
$641$	−7.69072	−0.303765	−0.151883	+	0.988399i	$0.548534\pi$
$642$	−3.40091	−0.134223
$643$	16.2348	0.640240	0.320120	−	0.947377i	$-0.396277\pi$
$643$	16.2348	0.640240	0.320120	+	0.947377i	$0.396277\pi$
$644$	−0.879178	−0.0346445
$645$	−25.6776	−1.01105
$646$	−0.218281	−0.00858816
$647$	−14.6071	−0.574266	−0.287133	−	0.957891i	$-0.592702\pi$
$647$	−14.6071	−0.574266	−0.287133	+	0.957891i	$0.592702\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	−10.1688	−0.398855
$651$	−7.68713	−0.301282
$652$	14.3469	0.561868
$653$	−14.1173	−0.552453	−0.276226	−	0.961093i	$-0.589084\pi$
$653$	−14.1173	−0.552453	−0.276226	+	0.961093i	$0.589084\pi$
$654$	6.96590	0.272388
$655$	61.8987	2.41858
$656$	−4.28967	−0.167483
$657$	−10.7453	−0.419215
$658$	10.0384	0.391339
$659$	−33.9283	−1.32166	−0.660829	−	0.750536i	$-0.729795\pi$
$659$	−33.9283	−1.32166	−0.660829	+	0.750536i	$0.729795\pi$
$660$	0	0
$661$	18.7723	0.730157	0.365079	−	0.930977i	$-0.381042\pi$
$661$	18.7723	0.730157	0.365079	+	0.930977i	$0.381042\pi$
$662$	31.0269	1.20589
$663$	−1.53531	−0.0596266
$664$	15.7847	0.612567
$665$	−0.957296	−0.0371224
$666$	3.87918	0.150315
$667$	−5.77786	−0.223719
$668$	−14.0316	−0.542897
$669$	−8.77887	−0.339411
$670$	−20.7981	−0.803501
$671$	0	0
$672$	1.00000	0.0385758
$673$	−26.4633	−1.02009	−0.510043	−	0.860149i	$-0.670370\pi$
$673$	−26.4633	−1.02009	−0.510043	+	0.860149i	$0.670370\pi$
$674$	16.4371	0.633132
$675$	4.70476	0.181086
$676$	−8.32837	−0.320322
$677$	−11.8779	−0.456505	−0.228253	−	0.973602i	$-0.573301\pi$
$677$	−11.8779	−0.456505	−0.228253	+	0.973602i	$0.573301\pi$
$678$	1.39802	0.0536906
$679$	0.261144	0.0100218
$680$	−2.21286	−0.0848595
$681$	19.5027	0.747346
$682$	0	0
$683$	−35.6132	−1.36270	−0.681351	−	0.731957i	$-0.738607\pi$
$683$	−35.6132	−1.36270	−0.681351	+	0.731957i	$0.738607\pi$
$684$	−0.307294	−0.0117497
$685$	−7.23780	−0.276542
$686$	1.00000	0.0381802
$687$	−7.27967	−0.277737
$688$	8.24254	0.314244
$689$	−14.2305	−0.542139
$690$	2.73886	0.104266
$691$	3.88337	0.147730	0.0738651	−	0.997268i	$-0.476467\pi$
$691$	3.88337	0.147730	0.0738651	+	0.997268i	$0.476467\pi$
$692$	−14.1774	−0.538946
$693$	0	0
$694$	35.0591	1.33083
$695$	23.3085	0.884140
$696$	6.57188	0.249107
$697$	3.04709	0.115417
$698$	8.62698	0.326536
$699$	2.73238	0.103348
$700$	−4.70476	−0.177823
$701$	−34.1646	−1.29038	−0.645189	−	0.764023i	$-0.723221\pi$
$701$	−34.1646	−1.29038	−0.645189	+	0.764023i	$0.723221\pi$
$702$	−2.16140	−0.0815766
$703$	1.19205	0.0449589
$704$	0	0
$705$	−31.2722	−1.17778
$706$	28.0551	1.05587
$707$	2.55081	0.0959331
$708$	−5.75091	−0.216132
$709$	−47.7715	−1.79410	−0.897048	−	0.441934i	$-0.854293\pi$
$709$	−47.7715	−1.79410	−0.897048	+	0.441934i	$0.854293\pi$
$710$	33.7851	1.26793
$711$	9.71565	0.364365
$712$	−14.8726	−0.557375
$713$	6.75836	0.253102
$714$	−0.710334	−0.0265836
$715$	0	0
$716$	−15.8912	−0.593883
$717$	−25.8977	−0.967168
$718$	−18.6575	−0.696290
$719$	3.28656	0.122568	0.0612841	−	0.998120i	$-0.480480\pi$
$719$	3.28656	0.122568	0.0612841	+	0.998120i	$0.480480\pi$
$720$	−3.11525	−0.116098
$721$	−18.8420	−0.701712
$722$	18.9056	0.703592
$723$	6.36034	0.236543
$724$	−17.7593	−0.660018
$725$	−30.9191	−1.14831
$726$	0	0
$727$	−6.49508	−0.240889	−0.120445	−	0.992720i	$-0.538432\pi$
$727$	−6.49508	−0.240889	−0.120445	+	0.992720i	$0.538432\pi$
$728$	2.16140	0.0801067
$729$	1.00000	0.0370370
$730$	−33.4744	−1.23894
$731$	−5.85496	−0.216553
$732$	−3.12427	−0.115476
$733$	50.1314	1.85165	0.925823	−	0.377957i	$-0.123373\pi$
$733$	50.1314	1.85165	0.925823	+	0.377957i	$0.123373\pi$
$734$	1.50679	0.0556166
$735$	−3.11525	−0.114908
$736$	−0.879178	−0.0324069
$737$	0	0
$738$	4.28967	0.157905
$739$	−15.2238	−0.560017	−0.280008	−	0.959998i	$-0.590337\pi$
$739$	−15.2238	−0.560017	−0.280008	+	0.959998i	$0.590337\pi$
$740$	12.0846	0.444239
$741$	−0.664184	−0.0243994
$742$	−6.58394	−0.241704
$743$	23.5025	0.862224	0.431112	−	0.902298i	$-0.358121\pi$
$743$	23.5025	0.862224	0.431112	+	0.902298i	$0.358121\pi$
$744$	−7.68713	−0.281824
$745$	4.44720	0.162933
$746$	−34.9467	−1.27949
$747$	−15.7847	−0.577534
$748$	0	0
$749$	−3.40091	−0.124267
$750$	−0.919752	−0.0335846
$751$	16.5983	0.605680	0.302840	−	0.953041i	$-0.402065\pi$
$751$	16.5983	0.605680	0.302840	+	0.953041i	$0.402065\pi$
$752$	10.0384	0.366064
$753$	−12.8826	−0.469469
$754$	14.2044	0.517295
$755$	−31.0338	−1.12943
$756$	−1.00000	−0.0363696
$757$	−1.59580	−0.0580004	−0.0290002	−	0.999579i	$-0.509232\pi$
$757$	−1.59580	−0.0580004	−0.0290002	+	0.999579i	$0.509232\pi$
$758$	−11.9312	−0.433362
$759$	0	0
$760$	−0.957296	−0.0347248
$761$	−3.60901	−0.130827	−0.0654133	−	0.997858i	$-0.520837\pi$
$761$	−3.60901	−0.130827	−0.0654133	+	0.997858i	$0.520837\pi$
$762$	16.0430	0.581178
$763$	6.96590	0.252183
$764$	−26.2906	−0.951159
$765$	2.21286	0.0800063
$766$	−18.0769	−0.653145
$767$	−12.4300	−0.448821
$768$	1.00000	0.0360844
$769$	38.6906	1.39522	0.697609	−	0.716479i	$-0.254247\pi$
$769$	38.6906	1.39522	0.697609	+	0.716479i	$0.254247\pi$
$770$	0	0
$771$	5.22836	0.188295
$772$	−5.56788	−0.200392
$773$	1.73051	0.0622420	0.0311210	−	0.999516i	$-0.490092\pi$
$773$	1.73051	0.0622420	0.0311210	+	0.999516i	$0.490092\pi$
$774$	−8.24254	−0.296272
$775$	36.1661	1.29912
$776$	0.261144	0.00937453
$777$	3.87918	0.139165
$778$	1.51581	0.0543445
$779$	1.31819	0.0472290
$780$	−6.73328	−0.241090
$781$	0	0
$782$	0.624510	0.0223324
$783$	−6.57188	−0.234860
$784$	1.00000	0.0357143
$785$	76.7638	2.73982
$786$	19.8696	0.708725
$787$	−2.59580	−0.0925304	−0.0462652	−	0.998929i	$-0.514732\pi$
$787$	−2.59580	−0.0925304	−0.0462652	+	0.998929i	$0.514732\pi$
$788$	11.8089	0.420673
$789$	−26.8248	−0.954987
$790$	30.2666	1.07684
$791$	1.39802	0.0497079
$792$	0	0
$793$	−6.75278	−0.239798
$794$	−32.2351	−1.14398
$795$	20.5106	0.727435
$796$	9.53591	0.337991
$797$	30.0576	1.06469	0.532347	−	0.846526i	$-0.321310\pi$
$797$	30.0576	1.06469	0.532347	+	0.846526i	$0.321310\pi$
$798$	−0.307294	−0.0108781
$799$	−7.13065	−0.252264
$800$	−4.70476	−0.166338
$801$	14.8726	0.525499
$802$	−8.93278	−0.315427
$803$	0	0
$804$	−6.67624	−0.235453
$805$	2.73886	0.0965320
$806$	−16.6149	−0.585236
$807$	−11.0403	−0.388638
$808$	2.55081	0.0897372
$809$	−30.9471	−1.08804	−0.544021	−	0.839071i	$-0.683099\pi$
$809$	−30.9471	−1.08804	−0.544021	+	0.839071i	$0.683099\pi$
$810$	3.11525	0.109459
$811$	−12.6734	−0.445024	−0.222512	−	0.974930i	$-0.571426\pi$
$811$	−12.6734	−0.445024	−0.222512	+	0.974930i	$0.571426\pi$
$812$	6.57188	0.230628
$813$	18.1134	0.635264
$814$	0	0
$815$	−44.6941	−1.56557
$816$	−0.710334	−0.0248667
$817$	−2.53288	−0.0886144
$818$	−21.4864	−0.751254
$819$	−2.16140	−0.0755253
$820$	13.3634	0.466669
$821$	3.82516	0.133499	0.0667496	−	0.997770i	$-0.478737\pi$
$821$	3.82516	0.133499	0.0667496	+	0.997770i	$0.478737\pi$
$822$	−2.32335	−0.0810361
$823$	−48.0740	−1.67576	−0.837878	−	0.545858i	$-0.816204\pi$
$823$	−48.0740	−1.67576	−0.837878	+	0.545858i	$0.816204\pi$
$824$	−18.8420	−0.656391
$825$	0	0
$826$	−5.75091	−0.200100
$827$	43.7937	1.52286	0.761428	−	0.648249i	$-0.224499\pi$
$827$	43.7937	1.52286	0.761428	+	0.648249i	$0.224499\pi$
$828$	0.879178	0.0305535
$829$	49.7535	1.72801	0.864006	−	0.503482i	$-0.167948\pi$
$829$	49.7535	1.72801	0.864006	+	0.503482i	$0.167948\pi$
$830$	−49.1734	−1.70683
$831$	−11.3212	−0.392729
$832$	2.16140	0.0749329
$833$	−0.710334	−0.0246116
$834$	7.48206	0.259082
$835$	43.7117	1.51271
$836$	0	0
$837$	7.68713	0.265706
$838$	−22.3380	−0.771655
$839$	−19.1826	−0.662255	−0.331128	−	0.943586i	$-0.607429\pi$
$839$	−19.1826	−0.662255	−0.331128	+	0.943586i	$0.607429\pi$
$840$	−3.11525	−0.107486
$841$	14.1897	0.489299
$842$	−9.90055	−0.341196
$843$	−25.1590	−0.866522
$844$	11.9545	0.411489
$845$	25.9449	0.892532
$846$	−10.0384	−0.345129
$847$	0	0
$848$	−6.58394	−0.226093
$849$	−5.61748	−0.192791
$850$	3.34195	0.114628
$851$	−3.41049	−0.116910
$852$	10.8451	0.371546
$853$	34.5644	1.18346	0.591732	−	0.806135i	$-0.298444\pi$
$853$	34.5644	1.18346	0.591732	+	0.806135i	$0.298444\pi$
$854$	−3.12427	−0.106910
$855$	0.957296	0.0327388
$856$	−3.40091	−0.116241
$857$	1.15867	0.0395792	0.0197896	−	0.999804i	$-0.493700\pi$
$857$	1.15867	0.0395792	0.0197896	+	0.999804i	$0.493700\pi$
$858$	0	0
$859$	−13.5036	−0.460737	−0.230369	−	0.973103i	$-0.573993\pi$
$859$	−13.5036	−0.460737	−0.230369	+	0.973103i	$0.573993\pi$
$860$	−25.6776	−0.875597
$861$	4.28967	0.146191
$862$	1.62634	0.0553934
$863$	31.1526	1.06045	0.530223	−	0.847858i	$-0.322108\pi$
$863$	31.1526	1.06045	0.530223	+	0.847858i	$0.322108\pi$
$864$	−1.00000	−0.0340207
$865$	44.1662	1.50170
$866$	16.4527	0.559085
$867$	−16.4954	−0.560214
$868$	−7.68713	−0.260918
$869$	0	0
$870$	−20.4730	−0.694101
$871$	−14.4300	−0.488942
$872$	6.96590	0.235895
$873$	−0.261144	−0.00883839
$874$	0.270166	0.00913850
$875$	−0.919752	−0.0310933
$876$	−10.7453	−0.363051
$877$	−30.9140	−1.04389	−0.521945	−	0.852979i	$-0.674793\pi$
$877$	−30.9140	−1.04389	−0.521945	+	0.852979i	$0.674793\pi$
$878$	−6.45481	−0.217839
$879$	−8.30116	−0.279991
$880$	0	0
$881$	23.1676	0.780535	0.390268	−	0.920701i	$-0.372382\pi$
$881$	23.1676	0.780535	0.390268	+	0.920701i	$0.372382\pi$
$882$	−1.00000	−0.0336718
$883$	−15.1154	−0.508675	−0.254337	−	0.967116i	$-0.581857\pi$
$883$	−15.1154	−0.508675	−0.254337	+	0.967116i	$0.581857\pi$
$884$	−1.53531	−0.0516381
$885$	17.9155	0.602223
$886$	−34.6921	−1.16550
$887$	9.61961	0.322995	0.161497	−	0.986873i	$-0.448368\pi$
$887$	9.61961	0.322995	0.161497	+	0.986873i	$0.448368\pi$
$888$	3.87918	0.130177
$889$	16.0430	0.538067
$890$	46.3319	1.55305
$891$	0	0
$892$	−8.77887	−0.293938
$893$	−3.08475	−0.103227
$894$	1.42756	0.0477447
$895$	49.5051	1.65477
$896$	1.00000	0.0334077
$897$	1.90025	0.0634476
$898$	36.2968	1.21124
$899$	−50.5189	−1.68490
$900$	4.70476	0.156825
$901$	4.67679	0.155807
$902$	0	0
$903$	−8.24254	−0.274295
$904$	1.39802	0.0464975
$905$	55.3245	1.83905
$906$	−9.96190	−0.330962
$907$	−22.8743	−0.759528	−0.379764	−	0.925083i	$-0.623995\pi$
$907$	−22.8743	−0.759528	−0.379764	+	0.925083i	$0.623995\pi$
$908$	19.5027	0.647220
$909$	−2.55081	−0.0846050
$910$	−6.73328	−0.223206
$911$	23.5631	0.780681	0.390340	−	0.920671i	$-0.372357\pi$
$911$	23.5631	0.780681	0.390340	+	0.920671i	$0.372357\pi$
$912$	−0.307294	−0.0101755
$913$	0	0
$914$	21.2889	0.704173
$915$	9.73286	0.321759
$916$	−7.27967	−0.240527
$917$	19.8696	0.656152
$918$	0.710334	0.0234445
$919$	−30.8825	−1.01872	−0.509359	−	0.860554i	$-0.670118\pi$
$919$	−30.8825	−1.01872	−0.509359	+	0.860554i	$0.670118\pi$
$920$	2.73886	0.0902974
$921$	−1.44088	−0.0474787
$922$	−34.3997	−1.13289
$923$	23.4405	0.771554
$924$	0	0
$925$	−18.2506	−0.600076
$926$	−33.4916	−1.10060
$927$	18.8420	0.618852
$928$	6.57188	0.215733
$929$	−41.0166	−1.34571	−0.672856	−	0.739774i	$-0.734932\pi$
$929$	−41.0166	−1.34571	−0.672856	+	0.739774i	$0.734932\pi$
$930$	23.9473	0.785263
$931$	−0.307294	−0.0100712
$932$	2.73238	0.0895021
$933$	9.89426	0.323924
$934$	24.9067	0.814973
$935$	0	0
$936$	−2.16140	−0.0706474
$937$	3.61272	0.118022	0.0590111	−	0.998257i	$-0.481205\pi$
$937$	3.61272	0.118022	0.0590111	+	0.998257i	$0.481205\pi$
$938$	−6.67624	−0.217987
$939$	−5.55886	−0.181407
$940$	−31.2722	−1.01999
$941$	40.1503	1.30886	0.654431	−	0.756122i	$-0.272908\pi$
$941$	40.1503	1.30886	0.654431	+	0.756122i	$0.272908\pi$
$942$	24.6413	0.802858
$943$	−3.77138	−0.122813
$944$	−5.75091	−0.187176
$945$	3.11525	0.101339
$946$	0	0
$947$	−33.8239	−1.09913	−0.549564	−	0.835452i	$-0.685206\pi$
$947$	−33.8239	−1.09913	−0.549564	+	0.835452i	$0.685206\pi$
$948$	9.71565	0.315550
$949$	−23.2249	−0.753913
$950$	1.44574	0.0469061
$951$	−0.694280	−0.0225136
$952$	−0.710334	−0.0230220
$953$	−33.3691	−1.08093	−0.540466	−	0.841366i	$-0.681752\pi$
$953$	−33.3691	−1.08093	−0.540466	+	0.841366i	$0.681752\pi$
$954$	6.58394	0.213163
$955$	81.9016	2.65027
$956$	−25.8977	−0.837592
$957$	0	0
$958$	−26.0657	−0.842146
$959$	−2.32335	−0.0750248
$960$	−3.11525	−0.100544
$961$	28.0920	0.906193
$962$	8.38444	0.270325
$963$	3.40091	0.109593
$964$	6.36034	0.204853
$965$	17.3453	0.558366
$966$	0.879178	0.0282871
$967$	−48.0391	−1.54483	−0.772417	−	0.635116i	$-0.780952\pi$
$967$	−48.0391	−1.54483	−0.772417	+	0.635116i	$0.780952\pi$
$968$	0	0
$969$	0.218281	0.00701220
$970$	−0.813528	−0.0261208
$971$	−40.7119	−1.30651	−0.653253	−	0.757140i	$-0.726596\pi$
$971$	−40.7119	−1.30651	−0.653253	+	0.757140i	$0.726596\pi$
$972$	1.00000	0.0320750
$973$	7.48206	0.239864
$974$	11.0362	0.353624
$975$	10.1688	0.325664
$976$	−3.12427	−0.100005
$977$	2.79780	0.0895095	0.0447548	−	0.998998i	$-0.485749\pi$
$977$	2.79780	0.0895095	0.0447548	+	0.998998i	$0.485749\pi$
$978$	−14.3469	−0.458763
$979$	0	0
$980$	−3.11525	−0.0995129
$981$	−6.96590	−0.222404
$982$	0.182886	0.00583613
$983$	29.1939	0.931141	0.465571	−	0.885011i	$-0.345849\pi$
$983$	29.1939	0.931141	0.465571	+	0.885011i	$0.345849\pi$
$984$	4.28967	0.136750
$985$	−36.7875	−1.17215
$986$	−4.66823	−0.148667
$987$	−10.0384	−0.319527
$988$	−0.664184	−0.0211305
$989$	7.24666	0.230430
$990$	0	0
$991$	49.6877	1.57838	0.789191	−	0.614148i	$-0.210500\pi$
$991$	49.6877	1.57838	0.789191	+	0.614148i	$0.210500\pi$
$992$	−7.68713	−0.244067
$993$	−31.0269	−0.984608
$994$	10.8451	0.343985
$995$	−29.7067	−0.941766
$996$	−15.7847	−0.500159
$997$	32.7150	1.03610	0.518048	−	0.855352i	$-0.326659\pi$
$997$	32.7150	1.03610	0.518048	+	0.855352i	$0.326659\pi$
$998$	−16.8881	−0.534584
$999$	−3.87918	−0.122732

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5082.2.a.ca.1.1		4
11.2	odd	10		462.2.j.e.169.2	✓	8
11.6	odd	10		462.2.j.e.421.2	yes	8
11.10	odd	2		5082.2.a.cf.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.j.e.169.2	✓	8	11.2	odd	10
462.2.j.e.421.2	yes	8	11.6	odd	10
5082.2.a.ca.1.1		4	1.1	even	1	trivial
5082.2.a.cf.1.1		4	11.10	odd	2

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:	\( N \)	\(=\)	\( 5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5082.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.11525 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.11525 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.11525 q^{10} +1.00000 q^{12} +2.16140 q^{13} +1.00000 q^{14} -3.11525 q^{15} +1.00000 q^{16} -0.710334 q^{17} -1.00000 q^{18} -0.307294 q^{19} -3.11525 q^{20} -1.00000 q^{21} +0.879178 q^{23} -1.00000 q^{24} +4.70476 q^{25} -2.16140 q^{26} +1.00000 q^{27} -1.00000 q^{28} -6.57188 q^{29} +3.11525 q^{30} +7.68713 q^{31} -1.00000 q^{32} +0.710334 q^{34} +3.11525 q^{35} +1.00000 q^{36} -3.87918 q^{37} +0.307294 q^{38} +2.16140 q^{39} +3.11525 q^{40} -4.28967 q^{41} +1.00000 q^{42} +8.24254 q^{43} -3.11525 q^{45} -0.879178 q^{46} +10.0384 q^{47} +1.00000 q^{48} +1.00000 q^{49} -4.70476 q^{50} -0.710334 q^{51} +2.16140 q^{52} -6.58394 q^{53} -1.00000 q^{54} +1.00000 q^{56} -0.307294 q^{57} +6.57188 q^{58} -5.75091 q^{59} -3.11525 q^{60} -3.12427 q^{61} -7.68713 q^{62} -1.00000 q^{63} +1.00000 q^{64} -6.73328 q^{65} -6.67624 q^{67} -0.710334 q^{68} +0.879178 q^{69} -3.11525 q^{70} +10.8451 q^{71} -1.00000 q^{72} -10.7453 q^{73} +3.87918 q^{74} +4.70476 q^{75} -0.307294 q^{76} -2.16140 q^{78} +9.71565 q^{79} -3.11525 q^{80} +1.00000 q^{81} +4.28967 q^{82} -15.7847 q^{83} -1.00000 q^{84} +2.21286 q^{85} -8.24254 q^{86} -6.57188 q^{87} +14.8726 q^{89} +3.11525 q^{90} -2.16140 q^{91} +0.879178 q^{92} +7.68713 q^{93} -10.0384 q^{94} +0.957296 q^{95} -1.00000 q^{96} -0.261144 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 2 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 2 * q^5 - 4 * q^6 - 4 * q^7 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 2 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} + 4 q^{9} - 2 q^{10} + 4 q^{12} - 2 q^{13} + 4 q^{14} + 2 q^{15} + 4 q^{16} - 6 q^{17} - 4 q^{18} - 4 q^{19} + 2 q^{20} - 4 q^{21} - 2 q^{23} - 4 q^{24} + 2 q^{25} + 2 q^{26} + 4 q^{27} - 4 q^{28} - 18 q^{29} - 2 q^{30} + 8 q^{31} - 4 q^{32} + 6 q^{34} - 2 q^{35} + 4 q^{36} - 10 q^{37} + 4 q^{38} - 2 q^{39} - 2 q^{40} - 14 q^{41} + 4 q^{42} - 10 q^{43} + 2 q^{45} + 2 q^{46} - 6 q^{47} + 4 q^{48} + 4 q^{49} - 2 q^{50} - 6 q^{51} - 2 q^{52} - 4 q^{53} - 4 q^{54} + 4 q^{56} - 4 q^{57} + 18 q^{58} - 10 q^{59} + 2 q^{60} - 8 q^{61} - 8 q^{62} - 4 q^{63} + 4 q^{64} - 8 q^{65} - 12 q^{67} - 6 q^{68} - 2 q^{69} + 2 q^{70} + 20 q^{71} - 4 q^{72} - 10 q^{73} + 10 q^{74} + 2 q^{75} - 4 q^{76} + 2 q^{78} + 14 q^{79} + 2 q^{80} + 4 q^{81} + 14 q^{82} - 20 q^{83} - 4 q^{84} - 26 q^{85} + 10 q^{86} - 18 q^{87} - 2 q^{89} - 2 q^{90} + 2 q^{91} - 2 q^{92} + 8 q^{93} + 6 q^{94} - 10 q^{95} - 4 q^{96} - 4 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 2 * q^5 - 4 * q^6 - 4 * q^7 - 4 * q^8 + 4 * q^9 - 2 * q^10 + 4 * q^12 - 2 * q^13 + 4 * q^14 + 2 * q^15 + 4 * q^16 - 6 * q^17 - 4 * q^18 - 4 * q^19 + 2 * q^20 - 4 * q^21 - 2 * q^23 - 4 * q^24 + 2 * q^25 + 2 * q^26 + 4 * q^27 - 4 * q^28 - 18 * q^29 - 2 * q^30 + 8 * q^31 - 4 * q^32 + 6 * q^34 - 2 * q^35 + 4 * q^36 - 10 * q^37 + 4 * q^38 - 2 * q^39 - 2 * q^40 - 14 * q^41 + 4 * q^42 - 10 * q^43 + 2 * q^45 + 2 * q^46 - 6 * q^47 + 4 * q^48 + 4 * q^49 - 2 * q^50 - 6 * q^51 - 2 * q^52 - 4 * q^53 - 4 * q^54 + 4 * q^56 - 4 * q^57 + 18 * q^58 - 10 * q^59 + 2 * q^60 - 8 * q^61 - 8 * q^62 - 4 * q^63 + 4 * q^64 - 8 * q^65 - 12 * q^67 - 6 * q^68 - 2 * q^69 + 2 * q^70 + 20 * q^71 - 4 * q^72 - 10 * q^73 + 10 * q^74 + 2 * q^75 - 4 * q^76 + 2 * q^78 + 14 * q^79 + 2 * q^80 + 4 * q^81 + 14 * q^82 - 20 * q^83 - 4 * q^84 - 26 * q^85 + 10 * q^86 - 18 * q^87 - 2 * q^89 - 2 * q^90 + 2 * q^91 - 2 * q^92 + 8 * q^93 + 6 * q^94 - 10 * q^95 - 4 * q^96 - 4 * q^98

Introduction

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