LMFDB - Embedded newform 8006.2.a.a.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8006,2,Mod(1,8006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8006, 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("8006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8006 = 2 \cdot 4003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8006.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:	$63.9282318582$
Analytic rank:	$1$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	8006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.51689 q^{3} +1.00000 q^{4} -2.29473 q^{5} -2.51689 q^{6} +0.132144 q^{7} +1.00000 q^{8} +3.33476 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.51689 q^{3} +1.00000 q^{4} -2.29473 q^{5} -2.51689 q^{6} +0.132144 q^{7} +1.00000 q^{8} +3.33476 q^{9} -2.29473 q^{10} -2.47079 q^{11} -2.51689 q^{12} +0.119573 q^{13} +0.132144 q^{14} +5.77560 q^{15} +1.00000 q^{16} +2.82300 q^{17} +3.33476 q^{18} +0.420153 q^{19} -2.29473 q^{20} -0.332594 q^{21} -2.47079 q^{22} -2.16027 q^{23} -2.51689 q^{24} +0.265799 q^{25} +0.119573 q^{26} -0.842555 q^{27} +0.132144 q^{28} -0.106054 q^{29} +5.77560 q^{30} -1.98449 q^{31} +1.00000 q^{32} +6.21871 q^{33} +2.82300 q^{34} -0.303236 q^{35} +3.33476 q^{36} +0.402570 q^{37} +0.420153 q^{38} -0.300953 q^{39} -2.29473 q^{40} +1.12505 q^{41} -0.332594 q^{42} +5.52572 q^{43} -2.47079 q^{44} -7.65238 q^{45} -2.16027 q^{46} -2.26857 q^{47} -2.51689 q^{48} -6.98254 q^{49} +0.265799 q^{50} -7.10519 q^{51} +0.119573 q^{52} +7.02431 q^{53} -0.842555 q^{54} +5.66980 q^{55} +0.132144 q^{56} -1.05748 q^{57} -0.106054 q^{58} +8.73108 q^{59} +5.77560 q^{60} +10.5602 q^{61} -1.98449 q^{62} +0.440670 q^{63} +1.00000 q^{64} -0.274389 q^{65} +6.21871 q^{66} -9.00325 q^{67} +2.82300 q^{68} +5.43717 q^{69} -0.303236 q^{70} +2.60909 q^{71} +3.33476 q^{72} -0.648832 q^{73} +0.402570 q^{74} -0.668988 q^{75} +0.420153 q^{76} -0.326501 q^{77} -0.300953 q^{78} +4.72525 q^{79} -2.29473 q^{80} -7.88366 q^{81} +1.12505 q^{82} -1.90966 q^{83} -0.332594 q^{84} -6.47803 q^{85} +5.52572 q^{86} +0.266928 q^{87} -2.47079 q^{88} +9.72874 q^{89} -7.65238 q^{90} +0.0158009 q^{91} -2.16027 q^{92} +4.99475 q^{93} -2.26857 q^{94} -0.964139 q^{95} -2.51689 q^{96} +4.94732 q^{97} -6.98254 q^{98} -8.23948 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9}+O(q^{10}) $ 69 * q + 69 * q^2 - 15 * q^3 + 69 * q^4 - 9 * q^5 - 15 * q^6 - 29 * q^7 + 69 * q^8 + 40 * q^9	\( 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9} - 9 q^{10} - 48 q^{11} - 15 q^{12} - 30 q^{13} - 29 q^{14} - 51 q^{15} + 69 q^{16} - 37 q^{17} + 40 q^{18} - 72 q^{19} - 9 q^{20} - 38 q^{21} - 48 q^{22} - 75 q^{23} - 15 q^{24} + 18 q^{25} - 30 q^{26} - 48 q^{27} - 29 q^{28} - 27 q^{29} - 51 q^{30} - 61 q^{31} + 69 q^{32} - 29 q^{33} - 37 q^{34} - 64 q^{35} + 40 q^{36} - 42 q^{37} - 72 q^{38} - 68 q^{39} - 9 q^{40} - 49 q^{41} - 38 q^{42} - 95 q^{43} - 48 q^{44} - 20 q^{45} - 75 q^{46} - 62 q^{47} - 15 q^{48} - 4 q^{49} + 18 q^{50} - 76 q^{51} - 30 q^{52} - 28 q^{53} - 48 q^{54} - 76 q^{55} - 29 q^{56} - 44 q^{57} - 27 q^{58} - 68 q^{59} - 51 q^{60} - 62 q^{61} - 61 q^{62} - 91 q^{63} + 69 q^{64} - 79 q^{65} - 29 q^{66} - 116 q^{67} - 37 q^{68} - 23 q^{69} - 64 q^{70} - 89 q^{71} + 40 q^{72} - 60 q^{73} - 42 q^{74} - 47 q^{75} - 72 q^{76} + 5 q^{77} - 68 q^{78} - 170 q^{79} - 9 q^{80} - 3 q^{81} - 49 q^{82} - 82 q^{83} - 38 q^{84} - 81 q^{85} - 95 q^{86} - 51 q^{87} - 48 q^{88} - 78 q^{89} - 20 q^{90} - 85 q^{91} - 75 q^{92} - 21 q^{93} - 62 q^{94} - 70 q^{95} - 15 q^{96} - 60 q^{97} - 4 q^{98} - 148 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 15 * q^3 + 69 * q^4 - 9 * q^5 - 15 * q^6 - 29 * q^7 + 69 * q^8 + 40 * q^9 - 9 * q^10 - 48 * q^11 - 15 * q^12 - 30 * q^13 - 29 * q^14 - 51 * q^15 + 69 * q^16 - 37 * q^17 + 40 * q^18 - 72 * q^19 - 9 * q^20 - 38 * q^21 - 48 * q^22 - 75 * q^23 - 15 * q^24 + 18 * q^25 - 30 * q^26 - 48 * q^27 - 29 * q^28 - 27 * q^29 - 51 * q^30 - 61 * q^31 + 69 * q^32 - 29 * q^33 - 37 * q^34 - 64 * q^35 + 40 * q^36 - 42 * q^37 - 72 * q^38 - 68 * q^39 - 9 * q^40 - 49 * q^41 - 38 * q^42 - 95 * q^43 - 48 * q^44 - 20 * q^45 - 75 * q^46 - 62 * q^47 - 15 * q^48 - 4 * q^49 + 18 * q^50 - 76 * q^51 - 30 * q^52 - 28 * q^53 - 48 * q^54 - 76 * q^55 - 29 * q^56 - 44 * q^57 - 27 * q^58 - 68 * q^59 - 51 * q^60 - 62 * q^61 - 61 * q^62 - 91 * q^63 + 69 * q^64 - 79 * q^65 - 29 * q^66 - 116 * q^67 - 37 * q^68 - 23 * q^69 - 64 * q^70 - 89 * q^71 + 40 * q^72 - 60 * q^73 - 42 * q^74 - 47 * q^75 - 72 * q^76 + 5 * q^77 - 68 * q^78 - 170 * q^79 - 9 * q^80 - 3 * q^81 - 49 * q^82 - 82 * q^83 - 38 * q^84 - 81 * q^85 - 95 * q^86 - 51 * q^87 - 48 * q^88 - 78 * q^89 - 20 * q^90 - 85 * q^91 - 75 * q^92 - 21 * q^93 - 62 * q^94 - 70 * q^95 - 15 * q^96 - 60 * q^97 - 4 * q^98 - 148 * 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
$3$	−2.51689	−1.45313	−0.726565	−	0.687098i	$-0.758884\pi$
$3$	−2.51689	−1.45313	−0.726565	+	0.687098i	$0.758884\pi$
$4$	1.00000	0.500000
$5$	−2.29473	−1.02624	−0.513118	−	0.858318i	$-0.671510\pi$
$5$	−2.29473	−1.02624	−0.513118	+	0.858318i	$0.671510\pi$
$6$	−2.51689	−1.02752
$7$	0.132144	0.0499459	0.0249729	−	0.999688i	$-0.492050\pi$
$7$	0.132144	0.0499459	0.0249729	+	0.999688i	$0.492050\pi$
$8$	1.00000	0.353553
$9$	3.33476	1.11159
$10$	−2.29473	−0.725658
$11$	−2.47079	−0.744971	−0.372485	−	0.928038i	$-0.621494\pi$
$11$	−2.47079	−0.744971	−0.372485	+	0.928038i	$0.621494\pi$
$12$	−2.51689	−0.726565
$13$	0.119573	0.0331636	0.0165818	−	0.999863i	$-0.494722\pi$
$13$	0.119573	0.0331636	0.0165818	+	0.999863i	$0.494722\pi$
$14$	0.132144	0.0353171
$15$	5.77560	1.49125
$16$	1.00000	0.250000
$17$	2.82300	0.684678	0.342339	−	0.939577i	$-0.388781\pi$
$17$	2.82300	0.684678	0.342339	+	0.939577i	$0.388781\pi$
$18$	3.33476	0.786010
$19$	0.420153	0.0963897	0.0481948	−	0.998838i	$-0.484653\pi$
$19$	0.420153	0.0963897	0.0481948	+	0.998838i	$0.484653\pi$
$20$	−2.29473	−0.513118
$21$	−0.332594	−0.0725779
$22$	−2.47079	−0.526774
$23$	−2.16027	−0.450447	−0.225224	−	0.974307i	$-0.572311\pi$
$23$	−2.16027	−0.450447	−0.225224	+	0.974307i	$0.572311\pi$
$24$	−2.51689	−0.513759
$25$	0.265799	0.0531598
$26$	0.119573	0.0234502
$27$	−0.842555	−0.162150
$28$	0.132144	0.0249729
$29$	−0.106054	−0.0196938	−0.00984691	−	0.999952i	$-0.503134\pi$
$29$	−0.106054	−0.0196938	−0.00984691	+	0.999952i	$0.503134\pi$
$30$	5.77560	1.05448
$31$	−1.98449	−0.356424	−0.178212	−	0.983992i	$-0.557031\pi$
$31$	−1.98449	−0.356424	−0.178212	+	0.983992i	$0.557031\pi$
$32$	1.00000	0.176777
$33$	6.21871	1.08254
$34$	2.82300	0.484140
$35$	−0.303236	−0.0512562
$36$	3.33476	0.555793
$37$	0.402570	0.0661822	0.0330911	−	0.999452i	$-0.489465\pi$
$37$	0.402570	0.0661822	0.0330911	+	0.999452i	$0.489465\pi$
$38$	0.420153	0.0681578
$39$	−0.300953	−0.0481911
$40$	−2.29473	−0.362829
$41$	1.12505	0.175704	0.0878520	−	0.996134i	$-0.472000\pi$
$41$	1.12505	0.175704	0.0878520	+	0.996134i	$0.472000\pi$
$42$	−0.332594	−0.0513203
$43$	5.52572	0.842664	0.421332	−	0.906906i	$-0.361563\pi$
$43$	5.52572	0.842664	0.421332	+	0.906906i	$0.361563\pi$
$44$	−2.47079	−0.372485
$45$	−7.65238	−1.14075
$46$	−2.16027	−0.318514
$47$	−2.26857	−0.330904	−0.165452	−	0.986218i	$-0.552908\pi$
$47$	−2.26857	−0.330904	−0.165452	+	0.986218i	$0.552908\pi$
$48$	−2.51689	−0.363282
$49$	−6.98254	−0.997505
$50$	0.265799	0.0375897
$51$	−7.10519	−0.994926
$52$	0.119573	0.0165818
$53$	7.02431	0.964863	0.482431	−	0.875934i	$-0.339754\pi$
$53$	7.02431	0.964863	0.482431	+	0.875934i	$0.339754\pi$
$54$	−0.842555	−0.114657
$55$	5.66980	0.764516
$56$	0.132144	0.0176585
$57$	−1.05748	−0.140067
$58$	−0.106054	−0.0139256
$59$	8.73108	1.13669	0.568345	−	0.822790i	$-0.307584\pi$
$59$	8.73108	1.13669	0.568345	+	0.822790i	$0.307584\pi$
$60$	5.77560	0.745627
$61$	10.5602	1.35209	0.676046	−	0.736859i	$-0.263692\pi$
$61$	10.5602	1.35209	0.676046	+	0.736859i	$0.263692\pi$
$62$	−1.98449	−0.252030
$63$	0.440670	0.0555192
$64$	1.00000	0.125000
$65$	−0.274389	−0.0340337
$66$	6.21871	0.765471
$67$	−9.00325	−1.09992	−0.549961	−	0.835190i	$-0.685357\pi$
$67$	−9.00325	−1.09992	−0.549961	+	0.835190i	$0.685357\pi$
$68$	2.82300	0.342339
$69$	5.43717	0.654558
$70$	−0.303236	−0.0362436
$71$	2.60909	0.309642	0.154821	−	0.987943i	$-0.450520\pi$
$71$	2.60909	0.309642	0.154821	+	0.987943i	$0.450520\pi$
$72$	3.33476	0.393005
$73$	−0.648832	−0.0759400	−0.0379700	−	0.999279i	$-0.512089\pi$
$73$	−0.648832	−0.0759400	−0.0379700	+	0.999279i	$0.512089\pi$
$74$	0.402570	0.0467979
$75$	−0.668988	−0.0772481
$76$	0.420153	0.0481948
$77$	−0.326501	−0.0372082
$78$	−0.300953	−0.0340762
$79$	4.72525	0.531632	0.265816	−	0.964024i	$-0.414359\pi$
$79$	4.72525	0.531632	0.265816	+	0.964024i	$0.414359\pi$
$80$	−2.29473	−0.256559
$81$	−7.88366	−0.875962
$82$	1.12505	0.124241
$83$	−1.90966	−0.209613	−0.104806	−	0.994493i	$-0.533422\pi$
$83$	−1.90966	−0.209613	−0.104806	+	0.994493i	$0.533422\pi$
$84$	−0.332594	−0.0362889
$85$	−6.47803	−0.702641
$86$	5.52572	0.595854
$87$	0.266928	0.0286177
$88$	−2.47079	−0.263387
$89$	9.72874	1.03124	0.515622	−	0.856816i	$-0.327561\pi$
$89$	9.72874	1.03124	0.515622	+	0.856816i	$0.327561\pi$
$90$	−7.65238	−0.806632
$91$	0.0158009	0.00165639
$92$	−2.16027	−0.225224
$93$	4.99475	0.517931
$94$	−2.26857	−0.233985
$95$	−0.964139	−0.0989185
$96$	−2.51689	−0.256880
$97$	4.94732	0.502324	0.251162	−	0.967945i	$-0.419187\pi$
$97$	4.94732	0.502324	0.251162	+	0.967945i	$0.419187\pi$
$98$	−6.98254	−0.705343
$99$	−8.23948	−0.828099
$100$	0.265799	0.0265799
$101$	−1.87397	−0.186467	−0.0932335	−	0.995644i	$-0.529720\pi$
$101$	−1.87397	−0.186467	−0.0932335	+	0.995644i	$0.529720\pi$
$102$	−7.10519	−0.703519
$103$	−13.6593	−1.34589	−0.672947	−	0.739691i	$-0.734972\pi$
$103$	−13.6593	−1.34589	−0.672947	+	0.739691i	$0.734972\pi$
$104$	0.119573	0.0117251
$105$	0.763213	0.0744820
$106$	7.02431	0.682261
$107$	6.02672	0.582625	0.291312	−	0.956628i	$-0.405908\pi$
$107$	6.02672	0.582625	0.291312	+	0.956628i	$0.405908\pi$
$108$	−0.842555	−0.0810748
$109$	11.4504	1.09675	0.548376	−	0.836232i	$-0.315246\pi$
$109$	11.4504	1.09675	0.548376	+	0.836232i	$0.315246\pi$
$110$	5.66980	0.540594
$111$	−1.01323	−0.0961713
$112$	0.132144	0.0124865
$113$	−9.88513	−0.929915	−0.464958	−	0.885333i	$-0.653930\pi$
$113$	−9.88513	−0.929915	−0.464958	+	0.885333i	$0.653930\pi$
$114$	−1.05748	−0.0990421
$115$	4.95724	0.462265
$116$	−0.106054	−0.00984691
$117$	0.398748	0.0368643
$118$	8.73108	0.803761
$119$	0.373044	0.0341968
$120$	5.77560	0.527238
$121$	−4.89521	−0.445019
$122$	10.5602	0.956074
$123$	−2.83164	−0.255321
$124$	−1.98449	−0.178212
$125$	10.8637	0.971681
$126$	0.440670	0.0392580
$127$	14.9763	1.32893	0.664465	−	0.747320i	$-0.268660\pi$
$127$	14.9763	1.32893	0.664465	+	0.747320i	$0.268660\pi$
$128$	1.00000	0.0883883
$129$	−13.9077	−1.22450
$130$	−0.274389	−0.0240655
$131$	−21.0756	−1.84138	−0.920692	−	0.390289i	$-0.872375\pi$
$131$	−21.0756	−1.84138	−0.920692	+	0.390289i	$0.872375\pi$
$132$	6.21871	0.541270
$133$	0.0555208	0.00481427
$134$	−9.00325	−0.777762
$135$	1.93344	0.166404
$136$	2.82300	0.242070
$137$	19.7621	1.68839	0.844194	−	0.536037i	$-0.180079\pi$
$137$	19.7621	1.68839	0.844194	+	0.536037i	$0.180079\pi$
$138$	5.43717	0.462842
$139$	13.1190	1.11274	0.556368	−	0.830936i	$-0.312194\pi$
$139$	13.1190	1.11274	0.556368	+	0.830936i	$0.312194\pi$
$140$	−0.303236	−0.0256281
$141$	5.70974	0.480847
$142$	2.60909	0.218950
$143$	−0.295440	−0.0247059
$144$	3.33476	0.277897
$145$	0.243367	0.0202105
$146$	−0.648832	−0.0536977
$147$	17.5743	1.44950
$148$	0.402570	0.0330911
$149$	−15.2489	−1.24924	−0.624619	−	0.780930i	$-0.714746\pi$
$149$	−15.2489	−1.24924	−0.624619	+	0.780930i	$0.714746\pi$
$150$	−0.668988	−0.0546227
$151$	−3.93686	−0.320377	−0.160189	−	0.987086i	$-0.551210\pi$
$151$	−3.93686	−0.320377	−0.160189	+	0.987086i	$0.551210\pi$
$152$	0.420153	0.0340789
$153$	9.41402	0.761079
$154$	−0.326501	−0.0263102
$155$	4.55387	0.365775
$156$	−0.300953	−0.0240955
$157$	−17.7520	−1.41676	−0.708380	−	0.705831i	$-0.750574\pi$
$157$	−17.7520	−1.41676	−0.708380	+	0.705831i	$0.750574\pi$
$158$	4.72525	0.375920
$159$	−17.6794	−1.40207
$160$	−2.29473	−0.181415
$161$	−0.285467	−0.0224980
$162$	−7.88366	−0.619399
$163$	−22.9423	−1.79698	−0.898490	−	0.438995i	$-0.855335\pi$
$163$	−22.9423	−1.79698	−0.898490	+	0.438995i	$0.855335\pi$
$164$	1.12505	0.0878520
$165$	−14.2703	−1.11094
$166$	−1.90966	−0.148219
$167$	−14.0502	−1.08724	−0.543618	−	0.839333i	$-0.682946\pi$
$167$	−14.0502	−1.08724	−0.543618	+	0.839333i	$0.682946\pi$
$168$	−0.332594	−0.0256601
$169$	−12.9857	−0.998900
$170$	−6.47803	−0.496842
$171$	1.40111	0.107145
$172$	5.52572	0.421332
$173$	−2.17684	−0.165502	−0.0827512	−	0.996570i	$-0.526371\pi$
$173$	−2.17684	−0.165502	−0.0827512	+	0.996570i	$0.526371\pi$
$174$	0.266928	0.0202358
$175$	0.0351239	0.00265511
$176$	−2.47079	−0.186243
$177$	−21.9752	−1.65176
$178$	9.72874	0.729200
$179$	0.350654	0.0262091	0.0131046	−	0.999914i	$-0.495829\pi$
$179$	0.350654	0.0262091	0.0131046	+	0.999914i	$0.495829\pi$
$180$	−7.65238	−0.570375
$181$	3.11337	0.231415	0.115707	−	0.993283i	$-0.463087\pi$
$181$	3.11337	0.231415	0.115707	+	0.993283i	$0.463087\pi$
$182$	0.0158009	0.00117124
$183$	−26.5789	−1.96477
$184$	−2.16027	−0.159257
$185$	−0.923791	−0.0679185
$186$	4.99475	0.366232
$187$	−6.97503	−0.510065
$188$	−2.26857	−0.165452
$189$	−0.111339	−0.00809871
$190$	−0.964139	−0.0699460
$191$	13.4312	0.971850	0.485925	−	0.874001i	$-0.338483\pi$
$191$	13.4312	0.971850	0.485925	+	0.874001i	$0.338483\pi$
$192$	−2.51689	−0.181641
$193$	19.3725	1.39447	0.697233	−	0.716845i	$-0.254414\pi$
$193$	19.3725	1.39447	0.697233	+	0.716845i	$0.254414\pi$
$194$	4.94732	0.355197
$195$	0.690607	0.0494554
$196$	−6.98254	−0.498753
$197$	−7.54152	−0.537311	−0.268655	−	0.963236i	$-0.586579\pi$
$197$	−7.54152	−0.537311	−0.268655	+	0.963236i	$0.586579\pi$
$198$	−8.23948	−0.585555
$199$	−6.24408	−0.442631	−0.221315	−	0.975202i	$-0.571035\pi$
$199$	−6.24408	−0.442631	−0.221315	+	0.975202i	$0.571035\pi$
$200$	0.265799	0.0187948
$201$	22.6602	1.59833
$202$	−1.87397	−0.131852
$203$	−0.0140145	−0.000983625 0
$204$	−7.10519	−0.497463
$205$	−2.58170	−0.180314
$206$	−13.6593	−0.951690
$207$	−7.20397	−0.500711
$208$	0.119573	0.00829091
$209$	−1.03811	−0.0718075
$210$	0.763213	0.0526667
$211$	−7.76655	−0.534671	−0.267336	−	0.963604i	$-0.586143\pi$
$211$	−7.76655	−0.534671	−0.267336	+	0.963604i	$0.586143\pi$
$212$	7.02431	0.482431
$213$	−6.56681	−0.449950
$214$	6.02672	0.411978
$215$	−12.6801	−0.864772
$216$	−0.842555	−0.0573286
$217$	−0.262239	−0.0178019
$218$	11.4504	0.775520
$219$	1.63304	0.110351
$220$	5.66980	0.382258
$221$	0.337555	0.0227064
$222$	−1.01323	−0.0680034
$223$	−17.2263	−1.15356	−0.576780	−	0.816900i	$-0.695691\pi$
$223$	−17.2263	−1.15356	−0.576780	+	0.816900i	$0.695691\pi$
$224$	0.132144	0.00882927
$225$	0.886376	0.0590917
$226$	−9.88513	−0.657549
$227$	16.2768	1.08033	0.540166	−	0.841559i	$-0.318361\pi$
$227$	16.2768	1.08033	0.540166	+	0.841559i	$0.318361\pi$
$228$	−1.05748	−0.0700334
$229$	12.8541	0.849422	0.424711	−	0.905329i	$-0.360376\pi$
$229$	12.8541	0.849422	0.424711	+	0.905329i	$0.360376\pi$
$230$	4.95724	0.326871
$231$	0.821768	0.0540684
$232$	−0.106054	−0.00696282
$233$	4.91281	0.321849	0.160924	−	0.986967i	$-0.448552\pi$
$233$	4.91281	0.321849	0.160924	+	0.986967i	$0.448552\pi$
$234$	0.398748	0.0260670
$235$	5.20575	0.339586
$236$	8.73108	0.568345
$237$	−11.8929	−0.772530
$238$	0.373044	0.0241808
$239$	−19.4051	−1.25521	−0.627605	−	0.778532i	$-0.715965\pi$
$239$	−19.4051	−1.25521	−0.627605	+	0.778532i	$0.715965\pi$
$240$	5.77560	0.372813
$241$	−25.4999	−1.64260	−0.821298	−	0.570500i	$-0.806749\pi$
$241$	−25.4999	−1.64260	−0.821298	+	0.570500i	$0.806749\pi$
$242$	−4.89521	−0.314676
$243$	22.3700	1.43504
$244$	10.5602	0.676046
$245$	16.0231	1.02368
$246$	−2.83164	−0.180539
$247$	0.0502390	0.00319663
$248$	−1.98449	−0.126015
$249$	4.80642	0.304595
$250$	10.8637	0.687082
$251$	21.1186	1.33299	0.666497	−	0.745508i	$-0.267793\pi$
$251$	21.1186	1.33299	0.666497	+	0.745508i	$0.267793\pi$
$252$	0.440670	0.0277596
$253$	5.33756	0.335570
$254$	14.9763	0.939695
$255$	16.3045	1.02103
$256$	1.00000	0.0625000
$257$	−18.4309	−1.14969	−0.574845	−	0.818262i	$-0.694938\pi$
$257$	−18.4309	−1.14969	−0.574845	+	0.818262i	$0.694938\pi$
$258$	−13.9077	−0.865853
$259$	0.0531974	0.00330553
$260$	−0.274389	−0.0170169
$261$	−0.353666	−0.0218914
$262$	−21.0756	−1.30206
$263$	−20.1771	−1.24417	−0.622087	−	0.782948i	$-0.713715\pi$
$263$	−20.1771	−1.24417	−0.622087	+	0.782948i	$0.713715\pi$
$264$	6.21871	0.382735
$265$	−16.1189	−0.990177
$266$	0.0555208	0.00340420
$267$	−24.4862	−1.49853
$268$	−9.00325	−0.549961
$269$	32.1655	1.96117	0.980583	−	0.196107i	$-0.0628298\pi$
$269$	32.1655	1.96117	0.980583	+	0.196107i	$0.0628298\pi$
$270$	1.93344	0.117665
$271$	−31.9495	−1.94079	−0.970397	−	0.241513i	$-0.922356\pi$
$271$	−31.9495	−1.94079	−0.970397	+	0.241513i	$0.922356\pi$
$272$	2.82300	0.171169
$273$	−0.0397693	−0.00240695
$274$	19.7621	1.19387
$275$	−0.656733	−0.0396025
$276$	5.43717	0.327279
$277$	−8.21612	−0.493659	−0.246829	−	0.969059i	$-0.579389\pi$
$277$	−8.21612	−0.493659	−0.246829	+	0.969059i	$0.579389\pi$
$278$	13.1190	0.786823
$279$	−6.61779	−0.396197
$280$	−0.303236	−0.0181218
$281$	−29.8191	−1.77886	−0.889431	−	0.457070i	$-0.848899\pi$
$281$	−29.8191	−1.77886	−0.889431	+	0.457070i	$0.848899\pi$
$282$	5.70974	0.340010
$283$	−4.05174	−0.240851	−0.120426	−	0.992722i	$-0.538426\pi$
$283$	−4.05174	−0.240851	−0.120426	+	0.992722i	$0.538426\pi$
$284$	2.60909	0.154821
$285$	2.42664	0.143741
$286$	−0.295440	−0.0174697
$287$	0.148670	0.00877569
$288$	3.33476	0.196503
$289$	−9.03067	−0.531216
$290$	0.243367	0.0142910
$291$	−12.4519	−0.729942
$292$	−0.648832	−0.0379700
$293$	−16.2615	−0.950008	−0.475004	−	0.879984i	$-0.657553\pi$
$293$	−16.2615	−0.950008	−0.475004	+	0.879984i	$0.657553\pi$
$294$	17.5743	1.02495
$295$	−20.0355	−1.16651
$296$	0.402570	0.0233989
$297$	2.08177	0.120797
$298$	−15.2489	−0.883345
$299$	−0.258310	−0.0149385
$300$	−0.668988	−0.0386241
$301$	0.730193	0.0420876
$302$	−3.93686	−0.226541
$303$	4.71658	0.270961
$304$	0.420153	0.0240974
$305$	−24.2328	−1.38757
$306$	9.41402	0.538164
$307$	−17.3531	−0.990395	−0.495197	−	0.868781i	$-0.664904\pi$
$307$	−17.3531	−0.990395	−0.495197	+	0.868781i	$0.664904\pi$
$308$	−0.326501	−0.0186041
$309$	34.3791	1.95576
$310$	4.55387	0.258642
$311$	−9.67616	−0.548685	−0.274342	−	0.961632i	$-0.588460\pi$
$311$	−9.67616	−0.548685	−0.274342	+	0.961632i	$0.588460\pi$
$312$	−0.300953	−0.0170381
$313$	23.6451	1.33650	0.668250	−	0.743937i	$-0.267044\pi$
$313$	23.6451	1.33650	0.668250	+	0.743937i	$0.267044\pi$
$314$	−17.7520	−1.00180
$315$	−1.01122	−0.0569758
$316$	4.72525	0.265816
$317$	−31.0828	−1.74578	−0.872891	−	0.487915i	$-0.837758\pi$
$317$	−31.0828	−1.74578	−0.872891	+	0.487915i	$0.837758\pi$
$318$	−17.6794	−0.991414
$319$	0.262038	0.0146713
$320$	−2.29473	−0.128279
$321$	−15.1686	−0.846629
$322$	−0.285467	−0.0159085
$323$	1.18609	0.0659959
$324$	−7.88366	−0.437981
$325$	0.0317825	0.00176297
$326$	−22.9423	−1.27066
$327$	−28.8195	−1.59372
$328$	1.12505	0.0621207
$329$	−0.299778	−0.0165273
$330$	−14.2703	−0.785553
$331$	−10.7741	−0.592201	−0.296100	−	0.955157i	$-0.595686\pi$
$331$	−10.7741	−0.592201	−0.296100	+	0.955157i	$0.595686\pi$
$332$	−1.90966	−0.104806
$333$	1.34248	0.0735672
$334$	−14.0502	−0.768792
$335$	20.6601	1.12878
$336$	−0.332594	−0.0181445
$337$	0.468074	0.0254976	0.0127488	−	0.999919i	$-0.495942\pi$
$337$	0.468074	0.0254976	0.0127488	+	0.999919i	$0.495942\pi$
$338$	−12.9857	−0.706329
$339$	24.8798	1.35129
$340$	−6.47803	−0.351320
$341$	4.90325	0.265526
$342$	1.40111	0.0757633
$343$	−1.84771	−0.0997672
$344$	5.52572	0.297927
$345$	−12.4768	−0.671731
$346$	−2.17684	−0.117028
$347$	30.4799	1.63625	0.818123	−	0.575044i	$-0.195015\pi$
$347$	30.4799	1.63625	0.818123	+	0.575044i	$0.195015\pi$
$348$	0.266928	0.0143088
$349$	−22.1633	−1.18638	−0.593188	−	0.805064i	$-0.702131\pi$
$349$	−22.1633	−1.18638	−0.593188	+	0.805064i	$0.702131\pi$
$350$	0.0351239	0.00187745
$351$	−0.100747	−0.00537748
$352$	−2.47079	−0.131693
$353$	19.5175	1.03881	0.519407	−	0.854527i	$-0.326153\pi$
$353$	19.5175	1.03881	0.519407	+	0.854527i	$0.326153\pi$
$354$	−21.9752	−1.16797
$355$	−5.98717	−0.317766
$356$	9.72874	0.515622
$357$	−0.938911	−0.0496925
$358$	0.350654	0.0185327
$359$	−10.1657	−0.536528	−0.268264	−	0.963345i	$-0.586450\pi$
$359$	−10.1657	−0.536528	−0.268264	+	0.963345i	$0.586450\pi$
$360$	−7.65238	−0.403316
$361$	−18.8235	−0.990709
$362$	3.11337	0.163635
$363$	12.3207	0.646670
$364$	0.0158009	0.000828194 0
$365$	1.48890	0.0779324
$366$	−26.5789	−1.38930
$367$	7.14271	0.372846	0.186423	−	0.982470i	$-0.440310\pi$
$367$	7.14271	0.372846	0.186423	+	0.982470i	$0.440310\pi$
$368$	−2.16027	−0.112612
$369$	3.75178	0.195310
$370$	−0.923791	−0.0480256
$371$	0.928223	0.0481909
$372$	4.99475	0.258965
$373$	−17.4276	−0.902366	−0.451183	−	0.892431i	$-0.648998\pi$
$373$	−17.4276	−0.902366	−0.451183	+	0.892431i	$0.648998\pi$
$374$	−6.97503	−0.360670
$375$	−27.3429	−1.41198
$376$	−2.26857	−0.116992
$377$	−0.0126813	−0.000653119 0
$378$	−0.111339	−0.00572665
$379$	−29.5499	−1.51788	−0.758938	−	0.651163i	$-0.774281\pi$
$379$	−29.5499	−1.51788	−0.758938	+	0.651163i	$0.774281\pi$
$380$	−0.964139	−0.0494593
$381$	−37.6937	−1.93111
$382$	13.4312	0.687201
$383$	−21.6527	−1.10640	−0.553202	−	0.833047i	$-0.686594\pi$
$383$	−21.6527	−1.10640	−0.553202	+	0.833047i	$0.686594\pi$
$384$	−2.51689	−0.128440
$385$	0.749232	0.0381844
$386$	19.3725	0.986036
$387$	18.4269	0.936694
$388$	4.94732	0.251162
$389$	17.8334	0.904192	0.452096	−	0.891969i	$-0.350676\pi$
$389$	17.8334	0.904192	0.452096	+	0.891969i	$0.350676\pi$
$390$	0.690607	0.0349703
$391$	−6.09844	−0.308411
$392$	−6.98254	−0.352671
$393$	53.0451	2.67577
$394$	−7.54152	−0.379936
$395$	−10.8432	−0.545579
$396$	−8.23948	−0.414050
$397$	36.0595	1.80977	0.904887	−	0.425653i	$-0.139955\pi$
$397$	36.0595	1.80977	0.904887	+	0.425653i	$0.139955\pi$
$398$	−6.24408	−0.312987
$399$	−0.139740	−0.00699576
$400$	0.265799	0.0132900
$401$	−26.5674	−1.32671	−0.663356	−	0.748304i	$-0.730868\pi$
$401$	−26.5674	−1.32671	−0.663356	+	0.748304i	$0.730868\pi$
$402$	22.6602	1.13019
$403$	−0.237292	−0.0118203
$404$	−1.87397	−0.0932335
$405$	18.0909	0.898943
$406$	−0.0140145	−0.000695528 0
$407$	−0.994666	−0.0493038
$408$	−7.10519	−0.351759
$409$	1.82782	0.0903801	0.0451900	−	0.998978i	$-0.485611\pi$
$409$	1.82782	0.0903801	0.0451900	+	0.998978i	$0.485611\pi$
$410$	−2.58170	−0.127501
$411$	−49.7391	−2.45345
$412$	−13.6593	−0.672947
$413$	1.15376	0.0567730
$414$	−7.20397	−0.354056
$415$	4.38217	0.215112
$416$	0.119573	0.00586256
$417$	−33.0191	−1.61695
$418$	−1.03811	−0.0507756
$419$	−20.8830	−1.02020	−0.510101	−	0.860114i	$-0.670392\pi$
$419$	−20.8830	−1.02020	−0.510101	+	0.860114i	$0.670392\pi$
$420$	0.763213	0.0372410
$421$	17.1889	0.837738	0.418869	−	0.908047i	$-0.362427\pi$
$421$	17.1889	0.837738	0.418869	+	0.908047i	$0.362427\pi$
$422$	−7.76655	−0.378069
$423$	−7.56512	−0.367829
$424$	7.02431	0.341131
$425$	0.750351	0.0363974
$426$	−6.56681	−0.318163
$427$	1.39547	0.0675315
$428$	6.02672	0.291312
$429$	0.743592	0.0359009
$430$	−12.6801	−0.611486
$431$	−2.55160	−0.122906	−0.0614531	−	0.998110i	$-0.519573\pi$
$431$	−2.55160	−0.122906	−0.0614531	+	0.998110i	$0.519573\pi$
$432$	−0.842555	−0.0405374
$433$	30.9676	1.48821	0.744103	−	0.668064i	$-0.232877\pi$
$433$	30.9676	1.48821	0.744103	+	0.668064i	$0.232877\pi$
$434$	−0.262239	−0.0125879
$435$	−0.612528	−0.0293685
$436$	11.4504	0.548376
$437$	−0.907643	−0.0434184
$438$	1.63304	0.0780298
$439$	7.88547	0.376353	0.188176	−	0.982135i	$-0.439742\pi$
$439$	7.88547	0.376353	0.188176	+	0.982135i	$0.439742\pi$
$440$	5.66980	0.270297
$441$	−23.2851	−1.10881
$442$	0.337555	0.0160559
$443$	−2.42001	−0.114978	−0.0574890	−	0.998346i	$-0.518309\pi$
$443$	−2.42001	−0.114978	−0.0574890	+	0.998346i	$0.518309\pi$
$444$	−1.01323	−0.0480856
$445$	−22.3249	−1.05830
$446$	−17.2263	−0.815689
$447$	38.3799	1.81531
$448$	0.132144	0.00624324
$449$	24.2065	1.14237	0.571187	−	0.820820i	$-0.306483\pi$
$449$	24.2065	1.14237	0.571187	+	0.820820i	$0.306483\pi$
$450$	0.886376	0.0417842
$451$	−2.77977	−0.130894
$452$	−9.88513	−0.464958
$453$	9.90867	0.465550
$454$	16.2768	0.763910
$455$	−0.0362589	−0.00169984
$456$	−1.05748	−0.0495211
$457$	−5.98923	−0.280164	−0.140082	−	0.990140i	$-0.544737\pi$
$457$	−5.98923	−0.280164	−0.140082	+	0.990140i	$0.544737\pi$
$458$	12.8541	0.600632
$459$	−2.37853	−0.111020
$460$	4.95724	0.231132
$461$	1.20065	0.0559199	0.0279599	−	0.999609i	$-0.491099\pi$
$461$	1.20065	0.0559199	0.0279599	+	0.999609i	$0.491099\pi$
$462$	0.821768	0.0382321
$463$	−27.9661	−1.29969	−0.649846	−	0.760066i	$-0.725167\pi$
$463$	−27.9661	−1.29969	−0.649846	+	0.760066i	$0.725167\pi$
$464$	−0.106054	−0.00492346
$465$	−11.4616	−0.531519
$466$	4.91281	0.227581
$467$	−12.9847	−0.600860	−0.300430	−	0.953804i	$-0.597130\pi$
$467$	−12.9847	−0.600860	−0.300430	+	0.953804i	$0.597130\pi$
$468$	0.398748	0.0184321
$469$	−1.18973	−0.0549366
$470$	5.20575	0.240124
$471$	44.6798	2.05874
$472$	8.73108	0.401881
$473$	−13.6529	−0.627760
$474$	−11.8929	−0.546261
$475$	0.111676	0.00512406
$476$	0.373044	0.0170984
$477$	23.4244	1.07253
$478$	−19.4051	−0.887568
$479$	13.0541	0.596457	0.298228	−	0.954495i	$-0.403604\pi$
$479$	13.0541	0.596457	0.298228	+	0.954495i	$0.403604\pi$
$480$	5.77560	0.263619
$481$	0.0481366	0.00219484
$482$	−25.4999	−1.16149
$483$	0.718491	0.0326925
$484$	−4.89521	−0.222509
$485$	−11.3528	−0.515503
$486$	22.3700	1.01472
$487$	−25.6856	−1.16393	−0.581963	−	0.813216i	$-0.697715\pi$
$487$	−25.6856	−1.16393	−0.581963	+	0.813216i	$0.697715\pi$
$488$	10.5602	0.478037
$489$	57.7433	2.61124
$490$	16.0231	0.723848
$491$	−16.3359	−0.737231	−0.368615	−	0.929582i	$-0.620168\pi$
$491$	−16.3359	−0.737231	−0.368615	+	0.929582i	$0.620168\pi$
$492$	−2.83164	−0.127660
$493$	−0.299392	−0.0134839
$494$	0.0502390	0.00226036
$495$	18.9074	0.849825
$496$	−1.98449	−0.0891061
$497$	0.344777	0.0154654
$498$	4.80642	0.215381
$499$	−24.0348	−1.07595	−0.537974	−	0.842962i	$-0.680810\pi$
$499$	−24.0348	−1.07595	−0.537974	+	0.842962i	$0.680810\pi$
$500$	10.8637	0.485841
$501$	35.3629	1.57990
$502$	21.1186	0.942569
$503$	40.3015	1.79696	0.898478	−	0.439019i	$-0.144674\pi$
$503$	40.3015	1.79696	0.898478	+	0.439019i	$0.144674\pi$
$504$	0.440670	0.0196290
$505$	4.30026	0.191359
$506$	5.33756	0.237284
$507$	32.6836	1.45153
$508$	14.9763	0.664465
$509$	−3.60821	−0.159931	−0.0799654	−	0.996798i	$-0.525481\pi$
$509$	−3.60821	−0.159931	−0.0799654	+	0.996798i	$0.525481\pi$
$510$	16.3045	0.721976
$511$	−0.0857395	−0.00379289
$512$	1.00000	0.0441942
$513$	−0.354002	−0.0156296
$514$	−18.4309	−0.812954
$515$	31.3445	1.38120
$516$	−13.9077	−0.612250
$517$	5.60515	0.246514
$518$	0.0531974	0.00233736
$519$	5.47889	0.240497
$520$	−0.274389	−0.0120327
$521$	16.8387	0.737716	0.368858	−	0.929486i	$-0.379749\pi$
$521$	16.8387	0.737716	0.368858	+	0.929486i	$0.379749\pi$
$522$	−0.353666	−0.0154796
$523$	−23.0506	−1.00793	−0.503965	−	0.863724i	$-0.668126\pi$
$523$	−23.0506	−1.00793	−0.503965	+	0.863724i	$0.668126\pi$
$524$	−21.0756	−0.920692
$525$	−0.0884030	−0.00385823
$526$	−20.1771	−0.879764
$527$	−5.60221	−0.244036
$528$	6.21871	0.270635
$529$	−18.3332	−0.797097
$530$	−16.1189	−0.700161
$531$	29.1161	1.26353
$532$	0.0555208	0.00240713
$533$	0.134526	0.00582698
$534$	−24.4862	−1.05962
$535$	−13.8297	−0.597910
$536$	−9.00325	−0.388881
$537$	−0.882560	−0.0380853
$538$	32.1655	1.38675
$539$	17.2524	0.743112
$540$	1.93344	0.0832019
$541$	9.98933	0.429475	0.214737	−	0.976672i	$-0.431110\pi$
$541$	9.98933	0.429475	0.214737	+	0.976672i	$0.431110\pi$
$542$	−31.9495	−1.37235
$543$	−7.83602	−0.336276
$544$	2.82300	0.121035
$545$	−26.2757	−1.12553
$546$	−0.0397693	−0.00170197
$547$	19.1208	0.817548	0.408774	−	0.912636i	$-0.365956\pi$
$547$	19.1208	0.817548	0.408774	+	0.912636i	$0.365956\pi$
$548$	19.7621	0.844194
$549$	35.2157	1.50297
$550$	−0.656733	−0.0280032
$551$	−0.0445591	−0.00189828
$552$	5.43717	0.231421
$553$	0.624415	0.0265528
$554$	−8.21612	−0.349069
$555$	2.32509	0.0986944
$556$	13.1190	0.556368
$557$	−24.8091	−1.05119	−0.525597	−	0.850734i	$-0.676158\pi$
$557$	−24.8091	−1.05119	−0.525597	+	0.850734i	$0.676158\pi$
$558$	−6.61779	−0.280153
$559$	0.660728	0.0279458
$560$	−0.303236	−0.0128141
$561$	17.5554	0.741191
$562$	−29.8191	−1.25784
$563$	−10.3672	−0.436927	−0.218464	−	0.975845i	$-0.570105\pi$
$563$	−10.3672	−0.436927	−0.218464	+	0.975845i	$0.570105\pi$
$564$	5.70974	0.240424
$565$	22.6837	0.954312
$566$	−4.05174	−0.170307
$567$	−1.04178	−0.0437507
$568$	2.60909	0.109475
$569$	30.1707	1.26482	0.632411	−	0.774633i	$-0.282065\pi$
$569$	30.1707	1.26482	0.632411	+	0.774633i	$0.282065\pi$
$570$	2.42664	0.101641
$571$	15.8999	0.665389	0.332694	−	0.943035i	$-0.392042\pi$
$571$	15.8999	0.665389	0.332694	+	0.943035i	$0.392042\pi$
$572$	−0.295440	−0.0123530
$573$	−33.8050	−1.41222
$574$	0.148670	0.00620535
$575$	−0.574197	−0.0239457
$576$	3.33476	0.138948
$577$	3.19689	0.133088	0.0665441	−	0.997783i	$-0.478803\pi$
$577$	3.19689	0.133088	0.0665441	+	0.997783i	$0.478803\pi$
$578$	−9.03067	−0.375627
$579$	−48.7586	−2.02634
$580$	0.243367	0.0101053
$581$	−0.252351	−0.0104693
$582$	−12.4519	−0.516147
$583$	−17.3556	−0.718795
$584$	−0.648832	−0.0268489
$585$	−0.915020	−0.0378314
$586$	−16.2615	−0.671757
$587$	11.2277	0.463416	0.231708	−	0.972785i	$-0.425569\pi$
$587$	11.2277	0.463416	0.231708	+	0.972785i	$0.425569\pi$
$588$	17.5743	0.724752
$589$	−0.833788	−0.0343556
$590$	−20.0355	−0.824848
$591$	18.9812	0.780783
$592$	0.402570	0.0165455
$593$	−31.6234	−1.29862	−0.649308	−	0.760525i	$-0.724941\pi$
$593$	−31.6234	−1.29862	−0.649308	+	0.760525i	$0.724941\pi$
$594$	2.08177	0.0854162
$595$	−0.856035	−0.0350940
$596$	−15.2489	−0.624619
$597$	15.7157	0.643200
$598$	−0.258310	−0.0105631
$599$	−3.91648	−0.160023	−0.0800114	−	0.996794i	$-0.525496\pi$
$599$	−3.91648	−0.160023	−0.0800114	+	0.996794i	$0.525496\pi$
$600$	−0.668988	−0.0273113
$601$	−18.2464	−0.744287	−0.372144	−	0.928175i	$-0.621377\pi$
$601$	−18.2464	−0.744287	−0.372144	+	0.928175i	$0.621377\pi$
$602$	0.730193	0.0297604
$603$	−30.0237	−1.22266
$604$	−3.93686	−0.160189
$605$	11.2332	0.456694
$606$	4.71658	0.191598
$607$	4.58685	0.186175	0.0930873	−	0.995658i	$-0.470326\pi$
$607$	4.58685	0.186175	0.0930873	+	0.995658i	$0.470326\pi$
$608$	0.420153	0.0170394
$609$	0.0352730	0.00142934
$610$	−24.2328	−0.981157
$611$	−0.271260	−0.0109740
$612$	9.41402	0.380539
$613$	−38.9896	−1.57478	−0.787388	−	0.616458i	$-0.788567\pi$
$613$	−38.9896	−1.57478	−0.787388	+	0.616458i	$0.788567\pi$
$614$	−17.3531	−0.700315
$615$	6.49786	0.262019
$616$	−0.326501	−0.0131551
$617$	−30.1527	−1.21390	−0.606952	−	0.794739i	$-0.707608\pi$
$617$	−30.1527	−1.21390	−0.606952	+	0.794739i	$0.707608\pi$
$618$	34.3791	1.38293
$619$	−6.17002	−0.247994	−0.123997	−	0.992283i	$-0.539571\pi$
$619$	−6.17002	−0.247994	−0.123997	+	0.992283i	$0.539571\pi$
$620$	4.55387	0.182888
$621$	1.82014	0.0730398
$622$	−9.67616	−0.387979
$623$	1.28560	0.0515064
$624$	−0.300953	−0.0120478
$625$	−26.2583	−1.05033
$626$	23.6451	0.945048
$627$	2.61281	0.104346
$628$	−17.7520	−0.708380
$629$	1.13646	0.0453135
$630$	−1.01122	−0.0402879
$631$	30.4809	1.21343	0.606713	−	0.794921i	$-0.292488\pi$
$631$	30.4809	1.21343	0.606713	+	0.794921i	$0.292488\pi$
$632$	4.72525	0.187960
$633$	19.5476	0.776946
$634$	−31.0828	−1.23445
$635$	−34.3665	−1.36379
$636$	−17.6794	−0.701036
$637$	−0.834925	−0.0330809
$638$	0.262038	0.0103742
$639$	8.70069	0.344194
$640$	−2.29473	−0.0907073
$641$	17.6779	0.698235	0.349118	−	0.937079i	$-0.386481\pi$
$641$	17.6779	0.698235	0.349118	+	0.937079i	$0.386481\pi$
$642$	−15.1686	−0.598657
$643$	−5.24103	−0.206686	−0.103343	−	0.994646i	$-0.532954\pi$
$643$	−5.24103	−0.206686	−0.103343	+	0.994646i	$0.532954\pi$
$644$	−0.285467	−0.0112490
$645$	31.9144	1.25663
$646$	1.18609	0.0466661
$647$	21.6030	0.849303	0.424651	−	0.905357i	$-0.360397\pi$
$647$	21.6030	0.849303	0.424651	+	0.905357i	$0.360397\pi$
$648$	−7.88366	−0.309699
$649$	−21.5726	−0.846801
$650$	0.0317825	0.00124661
$651$	0.660028	0.0258685
$652$	−22.9423	−0.898490
$653$	−4.20970	−0.164738	−0.0823692	−	0.996602i	$-0.526249\pi$
$653$	−4.20970	−0.164738	−0.0823692	+	0.996602i	$0.526249\pi$
$654$	−28.8195	−1.12693
$655$	48.3629	1.88969
$656$	1.12505	0.0439260
$657$	−2.16370	−0.0844139
$658$	−0.299778	−0.0116866
$659$	14.6472	0.570573	0.285287	−	0.958442i	$-0.407911\pi$
$659$	14.6472	0.570573	0.285287	+	0.958442i	$0.407911\pi$
$660$	−14.2703	−0.555470
$661$	19.9308	0.775219	0.387610	−	0.921824i	$-0.373301\pi$
$661$	19.9308	0.775219	0.387610	+	0.921824i	$0.373301\pi$
$662$	−10.7741	−0.418749
$663$	−0.849591	−0.0329954
$664$	−1.90966	−0.0741093
$665$	−0.127405	−0.00494057
$666$	1.34248	0.0520199
$667$	0.229106	0.00887103
$668$	−14.0502	−0.543618
$669$	43.3568	1.67627
$670$	20.6601	0.798168
$671$	−26.0920	−1.00727
$672$	−0.332594	−0.0128301
$673$	−34.6229	−1.33461	−0.667307	−	0.744782i	$-0.732553\pi$
$673$	−34.6229	−1.33461	−0.667307	+	0.744782i	$0.732553\pi$
$674$	0.468074	0.0180295
$675$	−0.223950	−0.00861985
$676$	−12.9857	−0.499450
$677$	41.6361	1.60020	0.800102	−	0.599864i	$-0.204779\pi$
$677$	41.6361	1.60020	0.800102	+	0.599864i	$0.204779\pi$
$678$	24.8798	0.955505
$679$	0.653760	0.0250890
$680$	−6.47803	−0.248421
$681$	−40.9671	−1.56986
$682$	4.90325	0.187755
$683$	1.04484	0.0399798	0.0199899	−	0.999800i	$-0.493637\pi$
$683$	1.04484	0.0399798	0.0199899	+	0.999800i	$0.493637\pi$
$684$	1.40111	0.0535727
$685$	−45.3487	−1.73268
$686$	−1.84771	−0.0705460
$687$	−32.3524	−1.23432
$688$	5.52572	0.210666
$689$	0.839919	0.0319984
$690$	−12.4768	−0.474985
$691$	1.55482	0.0591480	0.0295740	−	0.999563i	$-0.490585\pi$
$691$	1.55482	0.0591480	0.0295740	+	0.999563i	$0.490585\pi$
$692$	−2.17684	−0.0827512
$693$	−1.08880	−0.0413602
$694$	30.4799	1.15700
$695$	−30.1045	−1.14193
$696$	0.266928	0.0101179
$697$	3.17603	0.120301
$698$	−22.1633	−0.838894
$699$	−12.3650	−0.467688
$700$	0.0351239	0.00132756
$701$	−9.19917	−0.347448	−0.173724	−	0.984794i	$-0.555580\pi$
$701$	−9.19917	−0.347448	−0.173724	+	0.984794i	$0.555580\pi$
$702$	−0.100747	−0.00380245
$703$	0.169141	0.00637928
$704$	−2.47079	−0.0931213
$705$	−13.1023	−0.493462
$706$	19.5175	0.734553
$707$	−0.247635	−0.00931325
$708$	−21.9752	−0.825879
$709$	−2.94609	−0.110643	−0.0553213	−	0.998469i	$-0.517618\pi$
$709$	−2.94609	−0.110643	−0.0553213	+	0.998469i	$0.517618\pi$
$710$	−5.98717	−0.224694
$711$	15.7576	0.590955
$712$	9.72874	0.364600
$713$	4.28702	0.160550
$714$	−0.938911	−0.0351379
$715$	0.677956	0.0253541
$716$	0.350654	0.0131046
$717$	48.8405	1.82398
$718$	−10.1657	−0.379382
$719$	−15.5457	−0.579756	−0.289878	−	0.957064i	$-0.593615\pi$
$719$	−15.5457	−0.579756	−0.289878	+	0.957064i	$0.593615\pi$
$720$	−7.65238	−0.285187
$721$	−1.80500	−0.0672218
$722$	−18.8235	−0.700537
$723$	64.1807	2.38690
$724$	3.11337	0.115707
$725$	−0.0281892	−0.00104692
$726$	12.3207	0.457265
$727$	−45.9418	−1.70389	−0.851944	−	0.523633i	$-0.824576\pi$
$727$	−45.9418	−1.70389	−0.851944	+	0.523633i	$0.824576\pi$
$728$	0.0158009	0.000585621 0
$729$	−32.6520	−1.20933
$730$	1.48890	0.0551065
$731$	15.5991	0.576954
$732$	−26.5789	−0.982383
$733$	27.7189	1.02382	0.511911	−	0.859039i	$-0.328938\pi$
$733$	27.7189	1.02382	0.511911	+	0.859039i	$0.328938\pi$
$734$	7.14271	0.263642
$735$	−40.3284	−1.48753
$736$	−2.16027	−0.0796285
$737$	22.2451	0.819410
$738$	3.75178	0.138105
$739$	−38.4466	−1.41428	−0.707140	−	0.707074i	$-0.750015\pi$
$739$	−38.4466	−1.41428	−0.707140	+	0.707074i	$0.750015\pi$
$740$	−0.923791	−0.0339592
$741$	−0.126446	−0.00464512
$742$	0.928223	0.0340761
$743$	−30.7310	−1.12741	−0.563705	−	0.825976i	$-0.690624\pi$
$743$	−30.7310	−1.12741	−0.563705	+	0.825976i	$0.690624\pi$
$744$	4.99475	0.183116
$745$	34.9921	1.28201
$746$	−17.4276	−0.638069
$747$	−6.36827	−0.233003
$748$	−6.97503	−0.255032
$749$	0.796397	0.0290997
$750$	−27.3429	−0.998420
$751$	−24.2851	−0.886174	−0.443087	−	0.896479i	$-0.646117\pi$
$751$	−24.2851	−0.886174	−0.443087	+	0.896479i	$0.646117\pi$
$752$	−2.26857	−0.0827261
$753$	−53.1533	−1.93701
$754$	−0.0126813	−0.000461825 0
$755$	9.03405	0.328783
$756$	−0.111339	−0.00404935
$757$	−23.5455	−0.855775	−0.427887	−	0.903832i	$-0.640742\pi$
$757$	−23.5455	−0.855775	−0.427887	+	0.903832i	$0.640742\pi$
$758$	−29.5499	−1.07330
$759$	−13.4341	−0.487627
$760$	−0.964139	−0.0349730
$761$	1.10985	0.0402321	0.0201161	−	0.999798i	$-0.493596\pi$
$761$	1.10985	0.0402321	0.0201161	+	0.999798i	$0.493596\pi$
$762$	−37.6937	−1.36550
$763$	1.51311	0.0547782
$764$	13.4312	0.485925
$765$	−21.6027	−0.781046
$766$	−21.6527	−0.782345
$767$	1.04400	0.0376968
$768$	−2.51689	−0.0908206
$769$	−46.6440	−1.68203	−0.841013	−	0.541016i	$-0.818040\pi$
$769$	−46.6440	−1.68203	−0.841013	+	0.541016i	$0.818040\pi$
$770$	0.749232	0.0270005
$771$	46.3887	1.67065
$772$	19.3725	0.697233
$773$	11.0155	0.396201	0.198100	−	0.980182i	$-0.436523\pi$
$773$	11.0155	0.396201	0.198100	+	0.980182i	$0.436523\pi$
$774$	18.4269	0.662343
$775$	−0.527475	−0.0189475
$776$	4.94732	0.177598
$777$	−0.133892	−0.00480336
$778$	17.8334	0.639360
$779$	0.472695	0.0169360
$780$	0.690607	0.0247277
$781$	−6.44651	−0.230674
$782$	−6.09844	−0.218080
$783$	0.0893567	0.00319335
$784$	−6.98254	−0.249376
$785$	40.7360	1.45393
$786$	53.0451	1.89206
$787$	−5.39879	−0.192446	−0.0962230	−	0.995360i	$-0.530676\pi$
$787$	−5.39879	−0.192446	−0.0962230	+	0.995360i	$0.530676\pi$
$788$	−7.54152	−0.268655
$789$	50.7837	1.80795
$790$	−10.8432	−0.385783
$791$	−1.30626	−0.0464454
$792$	−8.23948	−0.292777
$793$	1.26271	0.0448403
$794$	36.0595	1.27970
$795$	40.5696	1.43886
$796$	−6.24408	−0.221315
$797$	3.85461	0.136537	0.0682687	−	0.997667i	$-0.478252\pi$
$797$	3.85461	0.136537	0.0682687	+	0.997667i	$0.478252\pi$
$798$	−0.139740	−0.00494675
$799$	−6.40416	−0.226563
$800$	0.265799	0.00939742
$801$	32.4430	1.14632
$802$	−26.5674	−0.938127
$803$	1.60313	0.0565731
$804$	22.6602	0.799165
$805$	0.655071	0.0230882
$806$	−0.237292	−0.00835824
$807$	−80.9572	−2.84983
$808$	−1.87397	−0.0659260
$809$	−0.935094	−0.0328761	−0.0164381	−	0.999865i	$-0.505233\pi$
$809$	−0.935094	−0.0328761	−0.0164381	+	0.999865i	$0.505233\pi$
$810$	18.0909	0.635649
$811$	24.1180	0.846896	0.423448	−	0.905920i	$-0.360820\pi$
$811$	24.1180	0.846896	0.423448	+	0.905920i	$0.360820\pi$
$812$	−0.0140145	−0.000491813 0
$813$	80.4136	2.82023
$814$	−0.994666	−0.0348630
$815$	52.6464	1.84412
$816$	−7.10519	−0.248731
$817$	2.32165	0.0812241
$818$	1.82782	0.0639084
$819$	0.0526923	0.00184122
$820$	−2.58170	−0.0901568
$821$	27.9510	0.975495	0.487747	−	0.872985i	$-0.337819\pi$
$821$	27.9510	0.975495	0.487747	+	0.872985i	$0.337819\pi$
$822$	−49.7391	−1.73485
$823$	33.6913	1.17440	0.587202	−	0.809440i	$-0.300229\pi$
$823$	33.6913	1.17440	0.587202	+	0.809440i	$0.300229\pi$
$824$	−13.6593	−0.475845
$825$	1.65293	0.0575476
$826$	1.15376	0.0401446
$827$	−6.23618	−0.216853	−0.108427	−	0.994104i	$-0.534581\pi$
$827$	−6.23618	−0.216853	−0.108427	+	0.994104i	$0.534581\pi$
$828$	−7.20397	−0.250355
$829$	22.6694	0.787341	0.393671	−	0.919252i	$-0.371205\pi$
$829$	22.6694	0.787341	0.393671	+	0.919252i	$0.371205\pi$
$830$	4.38217	0.152107
$831$	20.6791	0.717350
$832$	0.119573	0.00414546
$833$	−19.7117	−0.682970
$834$	−33.0191	−1.14336
$835$	32.2414	1.11576
$836$	−1.03811	−0.0359037
$837$	1.67204	0.0577941
$838$	−20.8830	−0.721392
$839$	6.97094	0.240664	0.120332	−	0.992734i	$-0.461604\pi$
$839$	6.97094	0.240664	0.120332	+	0.992734i	$0.461604\pi$
$840$	0.763213	0.0263334
$841$	−28.9888	−0.999612
$842$	17.1889	0.592370
$843$	75.0517	2.58492
$844$	−7.76655	−0.267336
$845$	29.7987	1.02511
$846$	−7.56512	−0.260094
$847$	−0.646874	−0.0222269
$848$	7.02431	0.241216
$849$	10.1978	0.349988
$850$	0.750351	0.0257368
$851$	−0.869660	−0.0298116
$852$	−6.56681	−0.224975
$853$	3.55609	0.121758	0.0608790	−	0.998145i	$-0.480610\pi$
$853$	3.55609	0.121758	0.0608790	+	0.998145i	$0.480610\pi$
$854$	1.39547	0.0477520
$855$	−3.21517	−0.109957
$856$	6.02672	0.205989
$857$	−39.1155	−1.33616	−0.668080	−	0.744089i	$-0.732884\pi$
$857$	−39.1155	−1.33616	−0.668080	+	0.744089i	$0.732884\pi$
$858$	0.743592	0.0253858
$859$	19.1589	0.653694	0.326847	−	0.945077i	$-0.394014\pi$
$859$	19.1589	0.653694	0.326847	+	0.945077i	$0.394014\pi$
$860$	−12.6801	−0.432386
$861$	−0.374186	−0.0127522
$862$	−2.55160	−0.0869078
$863$	15.9659	0.543484	0.271742	−	0.962370i	$-0.412400\pi$
$863$	15.9659	0.543484	0.271742	+	0.962370i	$0.412400\pi$
$864$	−0.842555	−0.0286643
$865$	4.99528	0.169845
$866$	30.9676	1.05232
$867$	22.7293	0.771926
$868$	−0.262239	−0.00890097
$869$	−11.6751	−0.396050
$870$	−0.612528	−0.0207667
$871$	−1.07655	−0.0364774
$872$	11.4504	0.387760
$873$	16.4981	0.558377
$874$	−0.907643	−0.0307015
$875$	1.43558	0.0485315
$876$	1.63304	0.0551754
$877$	48.7616	1.64656	0.823281	−	0.567634i	$-0.192141\pi$
$877$	48.7616	1.64656	0.823281	+	0.567634i	$0.192141\pi$
$878$	7.88547	0.266122
$879$	40.9285	1.38048
$880$	5.66980	0.191129
$881$	51.0174	1.71882	0.859410	−	0.511287i	$-0.170831\pi$
$881$	51.0174	1.71882	0.859410	+	0.511287i	$0.170831\pi$
$882$	−23.2851	−0.784050
$883$	−17.7942	−0.598823	−0.299411	−	0.954124i	$-0.596790\pi$
$883$	−17.7942	−0.598823	−0.299411	+	0.954124i	$0.596790\pi$
$884$	0.337555	0.0113532
$885$	50.4272	1.69509
$886$	−2.42001	−0.0813017
$887$	−7.42627	−0.249350	−0.124675	−	0.992198i	$-0.539789\pi$
$887$	−7.42627	−0.249350	−0.124675	+	0.992198i	$0.539789\pi$
$888$	−1.01323	−0.0340017
$889$	1.97903	0.0663745
$890$	−22.3249	−0.748331
$891$	19.4788	0.652566
$892$	−17.2263	−0.576780
$893$	−0.953145	−0.0318958
$894$	38.3799	1.28361
$895$	−0.804658	−0.0268967
$896$	0.132144	0.00441463
$897$	0.650140	0.0217075
$898$	24.2065	0.807780
$899$	0.210464	0.00701936
$900$	0.886376	0.0295459
$901$	19.8296	0.660620
$902$	−2.77977	−0.0925562
$903$	−1.83782	−0.0611588
$904$	−9.88513	−0.328775
$905$	−7.14435	−0.237486
$906$	9.90867	0.329194
$907$	−38.4345	−1.27620	−0.638098	−	0.769955i	$-0.720279\pi$
$907$	−38.4345	−1.27620	−0.638098	+	0.769955i	$0.720279\pi$
$908$	16.2768	0.540166
$909$	−6.24924	−0.207274
$910$	−0.0362589	−0.00120197
$911$	−43.8038	−1.45128	−0.725642	−	0.688072i	$-0.758457\pi$
$911$	−43.8038	−1.45128	−0.725642	+	0.688072i	$0.758457\pi$
$912$	−1.05748	−0.0350167
$913$	4.71837	0.156155
$914$	−5.98923	−0.198106
$915$	60.9914	2.01631
$916$	12.8541	0.424711
$917$	−2.78502	−0.0919696
$918$	−2.37853	−0.0785032
$919$	8.58264	0.283115	0.141558	−	0.989930i	$-0.454789\pi$
$919$	8.58264	0.283115	0.141558	+	0.989930i	$0.454789\pi$
$920$	4.95724	0.163435
$921$	43.6760	1.43917
$922$	1.20065	0.0395413
$923$	0.311978	0.0102689
$924$	0.821768	0.0270342
$925$	0.107003	0.00351823
$926$	−27.9661	−0.919022
$927$	−45.5506	−1.49608
$928$	−0.106054	−0.00348141
$929$	−40.5922	−1.33179	−0.665894	−	0.746046i	$-0.731950\pi$
$929$	−40.5922	−1.33179	−0.665894	+	0.746046i	$0.731950\pi$
$930$	−11.4616	−0.375841
$931$	−2.93373	−0.0961492
$932$	4.91281	0.160924
$933$	24.3539	0.797310
$934$	−12.9847	−0.424872
$935$	16.0058	0.523447
$936$	0.398748	0.0130335
$937$	−21.2289	−0.693519	−0.346760	−	0.937954i	$-0.612718\pi$
$937$	−21.2289	−0.693519	−0.346760	+	0.937954i	$0.612718\pi$
$938$	−1.18973	−0.0388460
$939$	−59.5122	−1.94211
$940$	5.20575	0.169793
$941$	53.9507	1.75874	0.879371	−	0.476137i	$-0.157964\pi$
$941$	53.9507	1.75874	0.879371	+	0.476137i	$0.157964\pi$
$942$	44.6798	1.45575
$943$	−2.43042	−0.0791453
$944$	8.73108	0.284172
$945$	0.255493	0.00831119
$946$	−13.6529	−0.443894
$947$	−61.1137	−1.98593	−0.992963	−	0.118422i	$-0.962217\pi$
$947$	−61.1137	−1.98593	−0.992963	+	0.118422i	$0.962217\pi$
$948$	−11.8929	−0.386265
$949$	−0.0775829	−0.00251845
$950$	0.111676	0.00362326
$951$	78.2321	2.53685
$952$	0.373044	0.0120904
$953$	29.3707	0.951412	0.475706	−	0.879604i	$-0.342193\pi$
$953$	29.3707	0.951412	0.475706	+	0.879604i	$0.342193\pi$
$954$	23.4244	0.758392
$955$	−30.8211	−0.997347
$956$	−19.4051	−0.627605
$957$	−0.659523	−0.0213193
$958$	13.0541	0.421759
$959$	2.61145	0.0843281
$960$	5.77560	0.186407
$961$	−27.0618	−0.872962
$962$	0.0481366	0.00155199
$963$	20.0977	0.647638
$964$	−25.4999	−0.821298
$965$	−44.4548	−1.43105
$966$	0.718491	0.0231171
$967$	−27.5539	−0.886074	−0.443037	−	0.896503i	$-0.646099\pi$
$967$	−27.5539	−0.886074	−0.443037	+	0.896503i	$0.646099\pi$
$968$	−4.89521	−0.157338
$969$	−2.98527	−0.0959006
$970$	−11.3528	−0.364516
$971$	−8.94409	−0.287030	−0.143515	−	0.989648i	$-0.545840\pi$
$971$	−8.94409	−0.287030	−0.143515	+	0.989648i	$0.545840\pi$
$972$	22.3700	0.717518
$973$	1.73360	0.0555766
$974$	−25.6856	−0.823019
$975$	−0.0799931	−0.00256183
$976$	10.5602	0.338023
$977$	49.4501	1.58205	0.791025	−	0.611784i	$-0.209548\pi$
$977$	49.4501	1.58205	0.791025	+	0.611784i	$0.209548\pi$
$978$	57.7433	1.84643
$979$	−24.0376	−0.768247
$980$	16.0231	0.511838
$981$	38.1844	1.21913
$982$	−16.3359	−0.521301
$983$	19.9105	0.635046	0.317523	−	0.948251i	$-0.397149\pi$
$983$	19.9105	0.635046	0.317523	+	0.948251i	$0.397149\pi$
$984$	−2.83164	−0.0902695
$985$	17.3058	0.551408
$986$	−0.299392	−0.00953458
$987$	0.754510	0.0240163
$988$	0.0502390	0.00159832
$989$	−11.9370	−0.379576
$990$	18.9074	0.600917
$991$	−29.6016	−0.940326	−0.470163	−	0.882580i	$-0.655805\pi$
$991$	−29.6016	−0.940326	−0.470163	+	0.882580i	$0.655805\pi$
$992$	−1.98449	−0.0630075
$993$	27.1174	0.860544
$994$	0.344777	0.0109357
$995$	14.3285	0.454244
$996$	4.80642	0.152297
$997$	22.9016	0.725302	0.362651	−	0.931925i	$-0.381872\pi$
$997$	22.9016	0.725302	0.362651	+	0.931925i	$0.381872\pi$
$998$	−24.0348	−0.760810
$999$	−0.339187	−0.0107314

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8006.2.a.a.1.11	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8006.2.a.a.1.11	✓	69	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:	\( N \)	\(=\)	\( 8006 = 2 \cdot 4003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8006.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.51689 q^{3} +1.00000 q^{4} -2.29473 q^{5} -2.51689 q^{6} +0.132144 q^{7} +1.00000 q^{8} +3.33476 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.51689 q^{3} +1.00000 q^{4} -2.29473 q^{5} -2.51689 q^{6} +0.132144 q^{7} +1.00000 q^{8} +3.33476 q^{9} -2.29473 q^{10} -2.47079 q^{11} -2.51689 q^{12} +0.119573 q^{13} +0.132144 q^{14} +5.77560 q^{15} +1.00000 q^{16} +2.82300 q^{17} +3.33476 q^{18} +0.420153 q^{19} -2.29473 q^{20} -0.332594 q^{21} -2.47079 q^{22} -2.16027 q^{23} -2.51689 q^{24} +0.265799 q^{25} +0.119573 q^{26} -0.842555 q^{27} +0.132144 q^{28} -0.106054 q^{29} +5.77560 q^{30} -1.98449 q^{31} +1.00000 q^{32} +6.21871 q^{33} +2.82300 q^{34} -0.303236 q^{35} +3.33476 q^{36} +0.402570 q^{37} +0.420153 q^{38} -0.300953 q^{39} -2.29473 q^{40} +1.12505 q^{41} -0.332594 q^{42} +5.52572 q^{43} -2.47079 q^{44} -7.65238 q^{45} -2.16027 q^{46} -2.26857 q^{47} -2.51689 q^{48} -6.98254 q^{49} +0.265799 q^{50} -7.10519 q^{51} +0.119573 q^{52} +7.02431 q^{53} -0.842555 q^{54} +5.66980 q^{55} +0.132144 q^{56} -1.05748 q^{57} -0.106054 q^{58} +8.73108 q^{59} +5.77560 q^{60} +10.5602 q^{61} -1.98449 q^{62} +0.440670 q^{63} +1.00000 q^{64} -0.274389 q^{65} +6.21871 q^{66} -9.00325 q^{67} +2.82300 q^{68} +5.43717 q^{69} -0.303236 q^{70} +2.60909 q^{71} +3.33476 q^{72} -0.648832 q^{73} +0.402570 q^{74} -0.668988 q^{75} +0.420153 q^{76} -0.326501 q^{77} -0.300953 q^{78} +4.72525 q^{79} -2.29473 q^{80} -7.88366 q^{81} +1.12505 q^{82} -1.90966 q^{83} -0.332594 q^{84} -6.47803 q^{85} +5.52572 q^{86} +0.266928 q^{87} -2.47079 q^{88} +9.72874 q^{89} -7.65238 q^{90} +0.0158009 q^{91} -2.16027 q^{92} +4.99475 q^{93} -2.26857 q^{94} -0.964139 q^{95} -2.51689 q^{96} +4.94732 q^{97} -6.98254 q^{98} -8.23948 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9}+O(q^{10}) \) 69 * q + 69 * q^2 - 15 * q^3 + 69 * q^4 - 9 * q^5 - 15 * q^6 - 29 * q^7 + 69 * q^8 + 40 * q^9	\( 69 q + 69 q^{2} - 15 q^{3} + 69 q^{4} - 9 q^{5} - 15 q^{6} - 29 q^{7} + 69 q^{8} + 40 q^{9} - 9 q^{10} - 48 q^{11} - 15 q^{12} - 30 q^{13} - 29 q^{14} - 51 q^{15} + 69 q^{16} - 37 q^{17} + 40 q^{18} - 72 q^{19} - 9 q^{20} - 38 q^{21} - 48 q^{22} - 75 q^{23} - 15 q^{24} + 18 q^{25} - 30 q^{26} - 48 q^{27} - 29 q^{28} - 27 q^{29} - 51 q^{30} - 61 q^{31} + 69 q^{32} - 29 q^{33} - 37 q^{34} - 64 q^{35} + 40 q^{36} - 42 q^{37} - 72 q^{38} - 68 q^{39} - 9 q^{40} - 49 q^{41} - 38 q^{42} - 95 q^{43} - 48 q^{44} - 20 q^{45} - 75 q^{46} - 62 q^{47} - 15 q^{48} - 4 q^{49} + 18 q^{50} - 76 q^{51} - 30 q^{52} - 28 q^{53} - 48 q^{54} - 76 q^{55} - 29 q^{56} - 44 q^{57} - 27 q^{58} - 68 q^{59} - 51 q^{60} - 62 q^{61} - 61 q^{62} - 91 q^{63} + 69 q^{64} - 79 q^{65} - 29 q^{66} - 116 q^{67} - 37 q^{68} - 23 q^{69} - 64 q^{70} - 89 q^{71} + 40 q^{72} - 60 q^{73} - 42 q^{74} - 47 q^{75} - 72 q^{76} + 5 q^{77} - 68 q^{78} - 170 q^{79} - 9 q^{80} - 3 q^{81} - 49 q^{82} - 82 q^{83} - 38 q^{84} - 81 q^{85} - 95 q^{86} - 51 q^{87} - 48 q^{88} - 78 q^{89} - 20 q^{90} - 85 q^{91} - 75 q^{92} - 21 q^{93} - 62 q^{94} - 70 q^{95} - 15 q^{96} - 60 q^{97} - 4 q^{98} - 148 q^{99}+O(q^{100}) \) 69 * q + 69 * q^2 - 15 * q^3 + 69 * q^4 - 9 * q^5 - 15 * q^6 - 29 * q^7 + 69 * q^8 + 40 * q^9 - 9 * q^10 - 48 * q^11 - 15 * q^12 - 30 * q^13 - 29 * q^14 - 51 * q^15 + 69 * q^16 - 37 * q^17 + 40 * q^18 - 72 * q^19 - 9 * q^20 - 38 * q^21 - 48 * q^22 - 75 * q^23 - 15 * q^24 + 18 * q^25 - 30 * q^26 - 48 * q^27 - 29 * q^28 - 27 * q^29 - 51 * q^30 - 61 * q^31 + 69 * q^32 - 29 * q^33 - 37 * q^34 - 64 * q^35 + 40 * q^36 - 42 * q^37 - 72 * q^38 - 68 * q^39 - 9 * q^40 - 49 * q^41 - 38 * q^42 - 95 * q^43 - 48 * q^44 - 20 * q^45 - 75 * q^46 - 62 * q^47 - 15 * q^48 - 4 * q^49 + 18 * q^50 - 76 * q^51 - 30 * q^52 - 28 * q^53 - 48 * q^54 - 76 * q^55 - 29 * q^56 - 44 * q^57 - 27 * q^58 - 68 * q^59 - 51 * q^60 - 62 * q^61 - 61 * q^62 - 91 * q^63 + 69 * q^64 - 79 * q^65 - 29 * q^66 - 116 * q^67 - 37 * q^68 - 23 * q^69 - 64 * q^70 - 89 * q^71 + 40 * q^72 - 60 * q^73 - 42 * q^74 - 47 * q^75 - 72 * q^76 + 5 * q^77 - 68 * q^78 - 170 * q^79 - 9 * q^80 - 3 * q^81 - 49 * q^82 - 82 * q^83 - 38 * q^84 - 81 * q^85 - 95 * q^86 - 51 * q^87 - 48 * q^88 - 78 * q^89 - 20 * q^90 - 85 * q^91 - 75 * q^92 - 21 * q^93 - 62 * q^94 - 70 * q^95 - 15 * q^96 - 60 * q^97 - 4 * q^98 - 148 * q^99

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