LMFDB - Embedded newform 6038.2.a.d.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6038,2,Mod(1,6038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6038, 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("6038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6038 = 2 \cdot 3019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6038.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:	$48.2136727404$
Analytic rank:	$0$
Dimension:	$69$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	6038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.22571 q^{3} +1.00000 q^{4} +1.16705 q^{5} +2.22571 q^{6} +1.13637 q^{7} -1.00000 q^{8} +1.95376 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.22571 q^{3} +1.00000 q^{4} +1.16705 q^{5} +2.22571 q^{6} +1.13637 q^{7} -1.00000 q^{8} +1.95376 q^{9} -1.16705 q^{10} +2.98024 q^{11} -2.22571 q^{12} +3.21747 q^{13} -1.13637 q^{14} -2.59751 q^{15} +1.00000 q^{16} +0.222446 q^{17} -1.95376 q^{18} +6.49772 q^{19} +1.16705 q^{20} -2.52922 q^{21} -2.98024 q^{22} +2.94368 q^{23} +2.22571 q^{24} -3.63799 q^{25} -3.21747 q^{26} +2.32861 q^{27} +1.13637 q^{28} +5.19509 q^{29} +2.59751 q^{30} +5.20806 q^{31} -1.00000 q^{32} -6.63313 q^{33} -0.222446 q^{34} +1.32620 q^{35} +1.95376 q^{36} +8.74116 q^{37} -6.49772 q^{38} -7.16113 q^{39} -1.16705 q^{40} +7.75533 q^{41} +2.52922 q^{42} -11.3016 q^{43} +2.98024 q^{44} +2.28014 q^{45} -2.94368 q^{46} +9.92268 q^{47} -2.22571 q^{48} -5.70866 q^{49} +3.63799 q^{50} -0.495099 q^{51} +3.21747 q^{52} +6.11542 q^{53} -2.32861 q^{54} +3.47809 q^{55} -1.13637 q^{56} -14.4620 q^{57} -5.19509 q^{58} -13.6035 q^{59} -2.59751 q^{60} +2.09004 q^{61} -5.20806 q^{62} +2.22020 q^{63} +1.00000 q^{64} +3.75495 q^{65} +6.63313 q^{66} +11.5583 q^{67} +0.222446 q^{68} -6.55176 q^{69} -1.32620 q^{70} -5.36294 q^{71} -1.95376 q^{72} +5.41624 q^{73} -8.74116 q^{74} +8.09710 q^{75} +6.49772 q^{76} +3.38665 q^{77} +7.16113 q^{78} +11.2984 q^{79} +1.16705 q^{80} -11.0441 q^{81} -7.75533 q^{82} +0.729384 q^{83} -2.52922 q^{84} +0.259606 q^{85} +11.3016 q^{86} -11.5627 q^{87} -2.98024 q^{88} +8.57607 q^{89} -2.28014 q^{90} +3.65623 q^{91} +2.94368 q^{92} -11.5916 q^{93} -9.92268 q^{94} +7.58317 q^{95} +2.22571 q^{96} +2.83186 q^{97} +5.70866 q^{98} +5.82268 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) $ 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9} - 18 q^{10} - 4 q^{11} + 8 q^{12} + 40 q^{13} - 32 q^{14} + 6 q^{15} + 69 q^{16} + 6 q^{17} - 79 q^{18} + 13 q^{19} + 18 q^{20} + 23 q^{21} + 4 q^{22} + 11 q^{23} - 8 q^{24} + 111 q^{25} - 40 q^{26} + 32 q^{27} + 32 q^{28} + 23 q^{29} - 6 q^{30} + 30 q^{31} - 69 q^{32} + 37 q^{33} - 6 q^{34} - 12 q^{35} + 79 q^{36} + 81 q^{37} - 13 q^{38} - 8 q^{39} - 18 q^{40} - 13 q^{41} - 23 q^{42} + 42 q^{43} - 4 q^{44} + 89 q^{45} - 11 q^{46} + 35 q^{47} + 8 q^{48} + 115 q^{49} - 111 q^{50} - 21 q^{51} + 40 q^{52} + 41 q^{53} - 32 q^{54} + 28 q^{55} - 32 q^{56} + 39 q^{57} - 23 q^{58} - 24 q^{59} + 6 q^{60} + 69 q^{61} - 30 q^{62} + 79 q^{63} + 69 q^{64} + 15 q^{65} - 37 q^{66} + 87 q^{67} + 6 q^{68} + 51 q^{69} + 12 q^{70} - 22 q^{71} - 79 q^{72} + 104 q^{73} - 81 q^{74} + 39 q^{75} + 13 q^{76} + 47 q^{77} + 8 q^{78} + 16 q^{79} + 18 q^{80} + 105 q^{81} + 13 q^{82} + 30 q^{83} + 23 q^{84} + 74 q^{85} - 42 q^{86} + 47 q^{87} + 4 q^{88} - 4 q^{89} - 89 q^{90} + 70 q^{91} + 11 q^{92} + 104 q^{93} - 35 q^{94} - 36 q^{95} - 8 q^{96} + 158 q^{97} - 115 q^{98} + 34 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9 - 18 * q^10 - 4 * q^11 + 8 * q^12 + 40 * q^13 - 32 * q^14 + 6 * q^15 + 69 * q^16 + 6 * q^17 - 79 * q^18 + 13 * q^19 + 18 * q^20 + 23 * q^21 + 4 * q^22 + 11 * q^23 - 8 * q^24 + 111 * q^25 - 40 * q^26 + 32 * q^27 + 32 * q^28 + 23 * q^29 - 6 * q^30 + 30 * q^31 - 69 * q^32 + 37 * q^33 - 6 * q^34 - 12 * q^35 + 79 * q^36 + 81 * q^37 - 13 * q^38 - 8 * q^39 - 18 * q^40 - 13 * q^41 - 23 * q^42 + 42 * q^43 - 4 * q^44 + 89 * q^45 - 11 * q^46 + 35 * q^47 + 8 * q^48 + 115 * q^49 - 111 * q^50 - 21 * q^51 + 40 * q^52 + 41 * q^53 - 32 * q^54 + 28 * q^55 - 32 * q^56 + 39 * q^57 - 23 * q^58 - 24 * q^59 + 6 * q^60 + 69 * q^61 - 30 * q^62 + 79 * q^63 + 69 * q^64 + 15 * q^65 - 37 * q^66 + 87 * q^67 + 6 * q^68 + 51 * q^69 + 12 * q^70 - 22 * q^71 - 79 * q^72 + 104 * q^73 - 81 * q^74 + 39 * q^75 + 13 * q^76 + 47 * q^77 + 8 * q^78 + 16 * q^79 + 18 * q^80 + 105 * q^81 + 13 * q^82 + 30 * q^83 + 23 * q^84 + 74 * q^85 - 42 * q^86 + 47 * q^87 + 4 * q^88 - 4 * q^89 - 89 * q^90 + 70 * q^91 + 11 * q^92 + 104 * q^93 - 35 * q^94 - 36 * q^95 - 8 * q^96 + 158 * q^97 - 115 * q^98 + 34 * 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
$3$	−2.22571	−1.28501	−0.642506	−	0.766281i	$-0.722105\pi$
$3$	−2.22571	−1.28501	−0.642506	+	0.766281i	$0.722105\pi$
$4$	1.00000	0.500000
$5$	1.16705	0.521921	0.260961	−	0.965349i	$-0.415961\pi$
$5$	1.16705	0.521921	0.260961	+	0.965349i	$0.415961\pi$
$6$	2.22571	0.908640
$7$	1.13637	0.429507	0.214754	−	0.976668i	$-0.431105\pi$
$7$	1.13637	0.429507	0.214754	+	0.976668i	$0.431105\pi$
$8$	−1.00000	−0.353553
$9$	1.95376	0.651254
$10$	−1.16705	−0.369054
$11$	2.98024	0.898576	0.449288	−	0.893387i	$-0.351678\pi$
$11$	2.98024	0.898576	0.449288	+	0.893387i	$0.351678\pi$
$12$	−2.22571	−0.642506
$13$	3.21747	0.892365	0.446182	−	0.894942i	$-0.352783\pi$
$13$	3.21747	0.892365	0.446182	+	0.894942i	$0.352783\pi$
$14$	−1.13637	−0.303707
$15$	−2.59751	−0.670675
$16$	1.00000	0.250000
$17$	0.222446	0.0539510	0.0269755	−	0.999636i	$-0.491412\pi$
$17$	0.222446	0.0539510	0.0269755	+	0.999636i	$0.491412\pi$
$18$	−1.95376	−0.460506
$19$	6.49772	1.49068	0.745340	−	0.666685i	$-0.232287\pi$
$19$	6.49772	1.49068	0.745340	+	0.666685i	$0.232287\pi$
$20$	1.16705	0.260961
$21$	−2.52922	−0.551922
$22$	−2.98024	−0.635389
$23$	2.94368	0.613799	0.306900	−	0.951742i	$-0.400708\pi$
$23$	2.94368	0.613799	0.306900	+	0.951742i	$0.400708\pi$
$24$	2.22571	0.454320
$25$	−3.63799	−0.727598
$26$	−3.21747	−0.630997
$27$	2.32861	0.448142
$28$	1.13637	0.214754
$29$	5.19509	0.964704	0.482352	−	0.875977i	$-0.339783\pi$
$29$	5.19509	0.964704	0.482352	+	0.875977i	$0.339783\pi$
$30$	2.59751	0.474239
$31$	5.20806	0.935394	0.467697	−	0.883889i	$-0.345084\pi$
$31$	5.20806	0.935394	0.467697	+	0.883889i	$0.345084\pi$
$32$	−1.00000	−0.176777
$33$	−6.63313	−1.15468
$34$	−0.222446	−0.0381491
$35$	1.32620	0.224169
$36$	1.95376	0.325627
$37$	8.74116	1.43704	0.718519	−	0.695507i	$-0.244820\pi$
$37$	8.74116	1.43704	0.718519	+	0.695507i	$0.244820\pi$
$38$	−6.49772	−1.05407
$39$	−7.16113	−1.14670
$40$	−1.16705	−0.184527
$41$	7.75533	1.21118	0.605589	−	0.795777i	$-0.292937\pi$
$41$	7.75533	1.21118	0.605589	+	0.795777i	$0.292937\pi$
$42$	2.52922	0.390268
$43$	−11.3016	−1.72347	−0.861735	−	0.507358i	$-0.830622\pi$
$43$	−11.3016	−1.72347	−0.861735	+	0.507358i	$0.830622\pi$
$44$	2.98024	0.449288
$45$	2.28014	0.339904
$46$	−2.94368	−0.434022
$47$	9.92268	1.44737	0.723686	−	0.690130i	$-0.242446\pi$
$47$	9.92268	1.44737	0.723686	+	0.690130i	$0.242446\pi$
$48$	−2.22571	−0.321253
$49$	−5.70866	−0.815524
$50$	3.63799	0.514490
$51$	−0.495099	−0.0693277
$52$	3.21747	0.446182
$53$	6.11542	0.840017	0.420009	−	0.907520i	$-0.362027\pi$
$53$	6.11542	0.840017	0.420009	+	0.907520i	$0.362027\pi$
$54$	−2.32861	−0.316884
$55$	3.47809	0.468986
$56$	−1.13637	−0.151854
$57$	−14.4620	−1.91554
$58$	−5.19509	−0.682149
$59$	−13.6035	−1.77103	−0.885515	−	0.464610i	$-0.846195\pi$
$59$	−13.6035	−1.77103	−0.885515	+	0.464610i	$0.846195\pi$
$60$	−2.59751	−0.335337
$61$	2.09004	0.267603	0.133801	−	0.991008i	$-0.457282\pi$
$61$	2.09004	0.267603	0.133801	+	0.991008i	$0.457282\pi$
$62$	−5.20806	−0.661424
$63$	2.22020	0.279718
$64$	1.00000	0.125000
$65$	3.75495	0.465744
$66$	6.63313	0.816482
$67$	11.5583	1.41207	0.706035	−	0.708177i	$-0.250482\pi$
$67$	11.5583	1.41207	0.706035	+	0.708177i	$0.250482\pi$
$68$	0.222446	0.0269755
$69$	−6.55176	−0.788739
$70$	−1.32620	−0.158511
$71$	−5.36294	−0.636464	−0.318232	−	0.948013i	$-0.603089\pi$
$71$	−5.36294	−0.636464	−0.318232	+	0.948013i	$0.603089\pi$
$72$	−1.95376	−0.230253
$73$	5.41624	0.633923	0.316961	−	0.948438i	$-0.397337\pi$
$73$	5.41624	0.633923	0.316961	+	0.948438i	$0.397337\pi$
$74$	−8.74116	−1.01614
$75$	8.09710	0.934972
$76$	6.49772	0.745340
$77$	3.38665	0.385945
$78$	7.16113	0.810839
$79$	11.2984	1.27117	0.635584	−	0.772032i	$-0.280760\pi$
$79$	11.2984	1.27117	0.635584	+	0.772032i	$0.280760\pi$
$80$	1.16705	0.130480
$81$	−11.0441	−1.22712
$82$	−7.75533	−0.856433
$83$	0.729384	0.0800603	0.0400301	−	0.999198i	$-0.487255\pi$
$83$	0.729384	0.0800603	0.0400301	+	0.999198i	$0.487255\pi$
$84$	−2.52922	−0.275961
$85$	0.259606	0.0281582
$86$	11.3016	1.21868
$87$	−11.5627	−1.23966
$88$	−2.98024	−0.317695
$89$	8.57607	0.909062	0.454531	−	0.890731i	$-0.349807\pi$
$89$	8.57607	0.909062	0.454531	+	0.890731i	$0.349807\pi$
$90$	−2.28014	−0.240348
$91$	3.65623	0.383277
$92$	2.94368	0.306900
$93$	−11.5916	−1.20199
$94$	−9.92268	−1.02345
$95$	7.58317	0.778017
$96$	2.22571	0.227160
$97$	2.83186	0.287532	0.143766	−	0.989612i	$-0.454079\pi$
$97$	2.83186	0.287532	0.143766	+	0.989612i	$0.454079\pi$
$98$	5.70866	0.576662
$99$	5.82268	0.585202
$100$	−3.63799	−0.363799
$101$	−2.09296	−0.208257	−0.104129	−	0.994564i	$-0.533205\pi$
$101$	−2.09296	−0.208257	−0.104129	+	0.994564i	$0.533205\pi$
$102$	0.495099	0.0490221
$103$	−11.1722	−1.10083	−0.550415	−	0.834891i	$-0.685531\pi$
$103$	−11.1722	−1.10083	−0.550415	+	0.834891i	$0.685531\pi$
$104$	−3.21747	−0.315499
$105$	−2.95173	−0.288060
$106$	−6.11542	−0.593982
$107$	7.48394	0.723500	0.361750	−	0.932275i	$-0.382179\pi$
$107$	7.48394	0.723500	0.361750	+	0.932275i	$0.382179\pi$
$108$	2.32861	0.224071
$109$	−0.826407	−0.0791554	−0.0395777	−	0.999216i	$-0.512601\pi$
$109$	−0.826407	−0.0791554	−0.0395777	+	0.999216i	$0.512601\pi$
$110$	−3.47809	−0.331623
$111$	−19.4552	−1.84661
$112$	1.13637	0.107377
$113$	5.92412	0.557295	0.278647	−	0.960393i	$-0.410114\pi$
$113$	5.92412	0.557295	0.278647	+	0.960393i	$0.410114\pi$
$114$	14.4620	1.35449
$115$	3.43542	0.320355
$116$	5.19509	0.482352
$117$	6.28617	0.581157
$118$	13.6035	1.25231
$119$	0.252781	0.0231724
$120$	2.59751	0.237119
$121$	−2.11817	−0.192561
$122$	−2.09004	−0.189224
$123$	−17.2611	−1.55638
$124$	5.20806	0.467697
$125$	−10.0810	−0.901670
$126$	−2.22020	−0.197791
$127$	13.5232	1.19999	0.599996	−	0.800003i	$-0.295169\pi$
$127$	13.5232	1.19999	0.599996	+	0.800003i	$0.295169\pi$
$128$	−1.00000	−0.0883883
$129$	25.1539	2.21468
$130$	−3.75495	−0.329331
$131$	−21.0847	−1.84218	−0.921091	−	0.389346i	$-0.872701\pi$
$131$	−21.0847	−1.84218	−0.921091	+	0.389346i	$0.872701\pi$
$132$	−6.63313	−0.577340
$133$	7.38381	0.640257
$134$	−11.5583	−0.998485
$135$	2.71761	0.233895
$136$	−0.222446	−0.0190746
$137$	3.38930	0.289567	0.144784	−	0.989463i	$-0.453751\pi$
$137$	3.38930	0.289567	0.144784	+	0.989463i	$0.453751\pi$
$138$	6.55176	0.557723
$139$	−22.2135	−1.88412	−0.942061	−	0.335441i	$-0.891115\pi$
$139$	−22.2135	−1.88412	−0.942061	+	0.335441i	$0.891115\pi$
$140$	1.32620	0.112084
$141$	−22.0850	−1.85989
$142$	5.36294	0.450048
$143$	9.58882	0.801858
$144$	1.95376	0.162814
$145$	6.06294	0.503500
$146$	−5.41624	−0.448251
$147$	12.7058	1.04796
$148$	8.74116	0.718519
$149$	−18.2891	−1.49830	−0.749149	−	0.662401i	$-0.769537\pi$
$149$	−18.2891	−1.49830	−0.749149	+	0.662401i	$0.769537\pi$
$150$	−8.09710	−0.661125
$151$	7.02995	0.572089	0.286045	−	0.958216i	$-0.407659\pi$
$151$	7.02995	0.572089	0.286045	+	0.958216i	$0.407659\pi$
$152$	−6.49772	−0.527035
$153$	0.434606	0.0351359
$154$	−3.38665	−0.272904
$155$	6.07807	0.488202
$156$	−7.16113	−0.573349
$157$	12.4304	0.992057	0.496028	−	0.868306i	$-0.334791\pi$
$157$	12.4304	0.992057	0.496028	+	0.868306i	$0.334791\pi$
$158$	−11.2984	−0.898851
$159$	−13.6111	−1.07943
$160$	−1.16705	−0.0922635
$161$	3.34510	0.263631
$162$	11.0441	0.867706
$163$	0.336576	0.0263626	0.0131813	−	0.999913i	$-0.495804\pi$
$163$	0.336576	0.0263626	0.0131813	+	0.999913i	$0.495804\pi$
$164$	7.75533	0.605589
$165$	−7.74121	−0.602652
$166$	−0.729384	−0.0566112
$167$	−19.4521	−1.50525	−0.752624	−	0.658451i	$-0.771212\pi$
$167$	−19.4521	−1.50525	−0.752624	+	0.658451i	$0.771212\pi$
$168$	2.52922	0.195134
$169$	−2.64791	−0.203685
$170$	−0.259606	−0.0199108
$171$	12.6950	0.970811
$172$	−11.3016	−0.861735
$173$	−25.1320	−1.91075	−0.955376	−	0.295391i	$-0.904550\pi$
$173$	−25.1320	−1.91075	−0.955376	+	0.295391i	$0.904550\pi$
$174$	11.5627	0.876569
$175$	−4.13410	−0.312509
$176$	2.98024	0.224644
$177$	30.2775	2.27579
$178$	−8.57607	−0.642804
$179$	−15.2906	−1.14287	−0.571436	−	0.820647i	$-0.693613\pi$
$179$	−15.2906	−1.14287	−0.571436	+	0.820647i	$0.693613\pi$
$180$	2.28014	0.169952
$181$	−9.71814	−0.722343	−0.361172	−	0.932499i	$-0.617623\pi$
$181$	−9.71814	−0.722343	−0.361172	+	0.932499i	$0.617623\pi$
$182$	−3.65623	−0.271018
$183$	−4.65182	−0.343873
$184$	−2.94368	−0.217011
$185$	10.2014	0.750021
$186$	11.5916	0.849937
$187$	0.662942	0.0484791
$188$	9.92268	0.723686
$189$	2.64617	0.192480
$190$	−7.58317	−0.550141
$191$	−10.6931	−0.773729	−0.386865	−	0.922137i	$-0.626442\pi$
$191$	−10.6931	−0.773729	−0.386865	+	0.922137i	$0.626442\pi$
$192$	−2.22571	−0.160626
$193$	9.33818	0.672177	0.336088	−	0.941830i	$-0.390896\pi$
$193$	9.33818	0.672177	0.336088	+	0.941830i	$0.390896\pi$
$194$	−2.83186	−0.203316
$195$	−8.35741	−0.598487
$196$	−5.70866	−0.407762
$197$	−8.63562	−0.615262	−0.307631	−	0.951506i	$-0.599536\pi$
$197$	−8.63562	−0.615262	−0.307631	+	0.951506i	$0.599536\pi$
$198$	−5.82268	−0.413800
$199$	−2.30766	−0.163586	−0.0817929	−	0.996649i	$-0.526065\pi$
$199$	−2.30766	−0.163586	−0.0817929	+	0.996649i	$0.526065\pi$
$200$	3.63799	0.257245
$201$	−25.7254	−1.81453
$202$	2.09296	0.147260
$203$	5.90354	0.414347
$204$	−0.495099	−0.0346638
$205$	9.05087	0.632140
$206$	11.1722	0.778405
$207$	5.75125	0.399739
$208$	3.21747	0.223091
$209$	19.3648	1.33949
$210$	2.95173	0.203689
$211$	−15.2939	−1.05287	−0.526436	−	0.850215i	$-0.676472\pi$
$211$	−15.2939	−1.05287	−0.526436	+	0.850215i	$0.676472\pi$
$212$	6.11542	0.420009
$213$	11.9363	0.817863
$214$	−7.48394	−0.511591
$215$	−13.1895	−0.899516
$216$	−2.32861	−0.158442
$217$	5.91827	0.401759
$218$	0.826407	0.0559714
$219$	−12.0549	−0.814598
$220$	3.47809	0.234493
$221$	0.715712	0.0481440
$222$	19.4552	1.30575
$223$	−13.8814	−0.929569	−0.464784	−	0.885424i	$-0.653868\pi$
$223$	−13.8814	−0.929569	−0.464784	+	0.885424i	$0.653868\pi$
$224$	−1.13637	−0.0759269
$225$	−7.10777	−0.473852
$226$	−5.92412	−0.394067
$227$	24.8346	1.64833	0.824166	−	0.566349i	$-0.191645\pi$
$227$	24.8346	1.64833	0.824166	+	0.566349i	$0.191645\pi$
$228$	−14.4620	−0.957770
$229$	24.7816	1.63762	0.818808	−	0.574068i	$-0.194636\pi$
$229$	24.7816	1.63762	0.818808	+	0.574068i	$0.194636\pi$
$230$	−3.43542	−0.226525
$231$	−7.53769	−0.495944
$232$	−5.19509	−0.341074
$233$	5.85943	0.383864	0.191932	−	0.981408i	$-0.438525\pi$
$233$	5.85943	0.383864	0.191932	+	0.981408i	$0.438525\pi$
$234$	−6.28617	−0.410940
$235$	11.5803	0.755414
$236$	−13.6035	−0.885515
$237$	−25.1469	−1.63346
$238$	−0.252781	−0.0163853
$239$	−6.39294	−0.413525	−0.206763	−	0.978391i	$-0.566293\pi$
$239$	−6.39294	−0.413525	−0.206763	+	0.978391i	$0.566293\pi$
$240$	−2.59751	−0.167669
$241$	10.8331	0.697819	0.348909	−	0.937157i	$-0.386552\pi$
$241$	10.8331	0.697819	0.348909	+	0.937157i	$0.386552\pi$
$242$	2.11817	0.136161
$243$	17.5951	1.12872
$244$	2.09004	0.133801
$245$	−6.66231	−0.425639
$246$	17.2611	1.10053
$247$	20.9062	1.33023
$248$	−5.20806	−0.330712
$249$	−1.62339	−0.102878
$250$	10.0810	0.637577
$251$	30.3213	1.91386	0.956931	−	0.290316i	$-0.0937603\pi$
$251$	30.3213	1.91386	0.956931	+	0.290316i	$0.0937603\pi$
$252$	2.22020	0.139859
$253$	8.77286	0.551545
$254$	−13.5232	−0.848523
$255$	−0.577806	−0.0361836
$256$	1.00000	0.0625000
$257$	−9.89894	−0.617479	−0.308740	−	0.951147i	$-0.599907\pi$
$257$	−9.89894	−0.617479	−0.308740	+	0.951147i	$0.599907\pi$
$258$	−25.1539	−1.56602
$259$	9.93319	0.617218
$260$	3.75495	0.232872
$261$	10.1500	0.628268
$262$	21.0847	1.30262
$263$	13.9931	0.862849	0.431424	−	0.902149i	$-0.358011\pi$
$263$	13.9931	0.862849	0.431424	+	0.902149i	$0.358011\pi$
$264$	6.63313	0.408241
$265$	7.13701	0.438423
$266$	−7.38381	−0.452730
$267$	−19.0878	−1.16816
$268$	11.5583	0.706035
$269$	−23.3003	−1.42065	−0.710324	−	0.703875i	$-0.751451\pi$
$269$	−23.3003	−1.42065	−0.710324	+	0.703875i	$0.751451\pi$
$270$	−2.71761	−0.165389
$271$	−13.4422	−0.816555	−0.408278	−	0.912858i	$-0.633870\pi$
$271$	−13.4422	−0.816555	−0.408278	+	0.912858i	$0.633870\pi$
$272$	0.222446	0.0134878
$273$	−8.13769	−0.492515
$274$	−3.38930	−0.204755
$275$	−10.8421	−0.653802
$276$	−6.55176	−0.394369
$277$	11.0882	0.666223	0.333112	−	0.942887i	$-0.391901\pi$
$277$	11.0882	0.666223	0.333112	+	0.942887i	$0.391901\pi$
$278$	22.2135	1.33228
$279$	10.1753	0.609180
$280$	−1.32620	−0.0792557
$281$	8.77765	0.523630	0.261815	−	0.965118i	$-0.415679\pi$
$281$	8.77765	0.523630	0.261815	+	0.965118i	$0.415679\pi$
$282$	22.0850	1.31514
$283$	5.39594	0.320755	0.160378	−	0.987056i	$-0.448729\pi$
$283$	5.39594	0.320755	0.160378	+	0.987056i	$0.448729\pi$
$284$	−5.36294	−0.318232
$285$	−16.8779	−0.999761
$286$	−9.58882	−0.566999
$287$	8.81292	0.520210
$288$	−1.95376	−0.115127
$289$	−16.9505	−0.997089
$290$	−6.06294	−0.356028
$291$	−6.30289	−0.369482
$292$	5.41624	0.316961
$293$	−23.1293	−1.35123	−0.675614	−	0.737256i	$-0.736121\pi$
$293$	−23.1293	−1.35123	−0.675614	+	0.737256i	$0.736121\pi$
$294$	−12.7058	−0.741018
$295$	−15.8760	−0.924339
$296$	−8.74116	−0.508070
$297$	6.93983	0.402690
$298$	18.2891	1.05946
$299$	9.47118	0.547733
$300$	8.09710	0.467486
$301$	−12.8427	−0.740243
$302$	−7.02995	−0.404528
$303$	4.65831	0.267613
$304$	6.49772	0.372670
$305$	2.43919	0.139668
$306$	−0.434606	−0.0248448
$307$	−12.6045	−0.719379	−0.359689	−	0.933072i	$-0.617117\pi$
$307$	−12.6045	−0.719379	−0.359689	+	0.933072i	$0.617117\pi$
$308$	3.38665	0.192972
$309$	24.8660	1.41458
$310$	−6.07807	−0.345211
$311$	−2.94182	−0.166815	−0.0834076	−	0.996516i	$-0.526580\pi$
$311$	−2.94182	−0.166815	−0.0834076	+	0.996516i	$0.526580\pi$
$312$	7.16113	0.405419
$313$	18.4746	1.04425	0.522124	−	0.852869i	$-0.325140\pi$
$313$	18.4746	1.04425	0.522124	+	0.852869i	$0.325140\pi$
$314$	−12.4304	−0.701490
$315$	2.59108	0.145991
$316$	11.2984	0.635584
$317$	−16.0056	−0.898966	−0.449483	−	0.893289i	$-0.648392\pi$
$317$	−16.0056	−0.898966	−0.449483	+	0.893289i	$0.648392\pi$
$318$	13.6111	0.763273
$319$	15.4826	0.866860
$320$	1.16705	0.0652402
$321$	−16.6570	−0.929705
$322$	−3.34510	−0.186415
$323$	1.44539	0.0804237
$324$	−11.0441	−0.613561
$325$	−11.7051	−0.649283
$326$	−0.336576	−0.0186412
$327$	1.83934	0.101716
$328$	−7.75533	−0.428216
$329$	11.2758	0.621657
$330$	7.74121	0.426140
$331$	−5.71365	−0.314050	−0.157025	−	0.987595i	$-0.550190\pi$
$331$	−5.71365	−0.314050	−0.157025	+	0.987595i	$0.550190\pi$
$332$	0.729384	0.0400301
$333$	17.0782	0.935877
$334$	19.4521	1.06437
$335$	13.4891	0.736990
$336$	−2.52922	−0.137980
$337$	−9.34498	−0.509053	−0.254527	−	0.967066i	$-0.581920\pi$
$337$	−9.34498	−0.509053	−0.254527	+	0.967066i	$0.581920\pi$
$338$	2.64791	0.144027
$339$	−13.1854	−0.716130
$340$	0.259606	0.0140791
$341$	15.5213	0.840523
$342$	−12.6950	−0.686467
$343$	−14.4417	−0.779780
$344$	11.3016	0.609339
$345$	−7.64624	−0.411660
$346$	25.1320	1.35111
$347$	−6.59827	−0.354214	−0.177107	−	0.984192i	$-0.556674\pi$
$347$	−6.59827	−0.354214	−0.177107	+	0.984192i	$0.556674\pi$
$348$	−11.5627	−0.619828
$349$	21.4246	1.14683	0.573415	−	0.819265i	$-0.305618\pi$
$349$	21.4246	1.14683	0.573415	+	0.819265i	$0.305618\pi$
$350$	4.13410	0.220977
$351$	7.49224	0.399906
$352$	−2.98024	−0.158847
$353$	25.2064	1.34160	0.670800	−	0.741638i	$-0.265951\pi$
$353$	25.2064	1.34160	0.670800	+	0.741638i	$0.265951\pi$
$354$	−30.2775	−1.60923
$355$	−6.25883	−0.332184
$356$	8.57607	0.454531
$357$	−0.562615	−0.0297767
$358$	15.2906	0.808132
$359$	−23.5183	−1.24125	−0.620623	−	0.784109i	$-0.713120\pi$
$359$	−23.5183	−1.24125	−0.620623	+	0.784109i	$0.713120\pi$
$360$	−2.28014	−0.120174
$361$	23.2204	1.22212
$362$	9.71814	0.510774
$363$	4.71443	0.247443
$364$	3.65623	0.191639
$365$	6.32103	0.330858
$366$	4.65182	0.243155
$367$	3.88576	0.202835	0.101417	−	0.994844i	$-0.467662\pi$
$367$	3.88576	0.202835	0.101417	+	0.994844i	$0.467662\pi$
$368$	2.94368	0.153450
$369$	15.1521	0.788786
$370$	−10.2014	−0.530345
$371$	6.94937	0.360793
$372$	−11.5916	−0.600996
$373$	15.7942	0.817795	0.408897	−	0.912580i	$-0.365913\pi$
$373$	15.7942	0.817795	0.408897	+	0.912580i	$0.365913\pi$
$374$	−0.662942	−0.0342799
$375$	22.4373	1.15866
$376$	−9.92268	−0.511723
$377$	16.7150	0.860868
$378$	−2.64617	−0.136104
$379$	−12.5665	−0.645499	−0.322750	−	0.946484i	$-0.604607\pi$
$379$	−12.5665	−0.645499	−0.322750	+	0.946484i	$0.604607\pi$
$380$	7.58317	0.389009
$381$	−30.0987	−1.54200
$382$	10.6931	0.547109
$383$	−16.8231	−0.859623	−0.429811	−	0.902919i	$-0.641420\pi$
$383$	−16.8231	−0.859623	−0.429811	+	0.902919i	$0.641420\pi$
$384$	2.22571	0.113580
$385$	3.95240	0.201433
$386$	−9.33818	−0.475301
$387$	−22.0806	−1.12242
$388$	2.83186	0.143766
$389$	13.2507	0.671836	0.335918	−	0.941891i	$-0.390953\pi$
$389$	13.2507	0.671836	0.335918	+	0.941891i	$0.390953\pi$
$390$	8.35741	0.423194
$391$	0.654809	0.0331151
$392$	5.70866	0.288331
$393$	46.9284	2.36723
$394$	8.63562	0.435056
$395$	13.1858	0.663449
$396$	5.82268	0.292601
$397$	0.919129	0.0461298	0.0230649	−	0.999734i	$-0.492658\pi$
$397$	0.919129	0.0461298	0.0230649	+	0.999734i	$0.492658\pi$
$398$	2.30766	0.115673
$399$	−16.4342	−0.822738
$400$	−3.63799	−0.181900
$401$	24.7215	1.23453	0.617267	−	0.786754i	$-0.288240\pi$
$401$	24.7215	1.23453	0.617267	+	0.786754i	$0.288240\pi$
$402$	25.7254	1.28306
$403$	16.7567	0.834713
$404$	−2.09296	−0.104129
$405$	−12.8890	−0.640461
$406$	−5.90354	−0.292988
$407$	26.0508	1.29129
$408$	0.495099	0.0245110
$409$	31.1844	1.54197	0.770985	−	0.636853i	$-0.219764\pi$
$409$	31.1844	1.54197	0.770985	+	0.636853i	$0.219764\pi$
$410$	−9.05087	−0.446990
$411$	−7.54357	−0.372097
$412$	−11.1722	−0.550415
$413$	−15.4587	−0.760671
$414$	−5.75125	−0.282658
$415$	0.851228	0.0417852
$416$	−3.21747	−0.157749
$417$	49.4406	2.42112
$418$	−19.3648	−0.947161
$419$	−11.0710	−0.540853	−0.270427	−	0.962741i	$-0.587165\pi$
$419$	−11.0710	−0.540853	−0.270427	+	0.962741i	$0.587165\pi$
$420$	−2.95173	−0.144030
$421$	22.4026	1.09183	0.545917	−	0.837839i	$-0.316181\pi$
$421$	22.4026	1.09183	0.545917	+	0.837839i	$0.316181\pi$
$422$	15.2939	0.744493
$423$	19.3866	0.942607
$424$	−6.11542	−0.296991
$425$	−0.809256	−0.0392547
$426$	−11.9363	−0.578317
$427$	2.37506	0.114937
$428$	7.48394	0.361750
$429$	−21.3419	−1.03040
$430$	13.1895	0.636054
$431$	−13.1560	−0.633703	−0.316851	−	0.948475i	$-0.602626\pi$
$431$	−13.1560	−0.633703	−0.316851	+	0.948475i	$0.602626\pi$
$432$	2.32861	0.112036
$433$	−15.1121	−0.726241	−0.363120	−	0.931742i	$-0.618289\pi$
$433$	−15.1121	−0.726241	−0.363120	+	0.931742i	$0.618289\pi$
$434$	−5.91827	−0.284086
$435$	−13.4943	−0.647003
$436$	−0.826407	−0.0395777
$437$	19.1272	0.914978
$438$	12.0549	0.576008
$439$	−1.22872	−0.0586437	−0.0293218	−	0.999570i	$-0.509335\pi$
$439$	−1.22872	−0.0586437	−0.0293218	+	0.999570i	$0.509335\pi$
$440$	−3.47809	−0.165812
$441$	−11.1534	−0.531113
$442$	−0.715712	−0.0340430
$443$	−11.3893	−0.541120	−0.270560	−	0.962703i	$-0.587209\pi$
$443$	−11.3893	−0.541120	−0.270560	+	0.962703i	$0.587209\pi$
$444$	−19.4552	−0.923305
$445$	10.0087	0.474459
$446$	13.8814	0.657304
$447$	40.7061	1.92533
$448$	1.13637	0.0536884
$449$	4.21729	0.199026	0.0995132	−	0.995036i	$-0.468271\pi$
$449$	4.21729	0.199026	0.0995132	+	0.995036i	$0.468271\pi$
$450$	7.10777	0.335064
$451$	23.1127	1.08834
$452$	5.92412	0.278647
$453$	−15.6466	−0.735142
$454$	−24.8346	−1.16555
$455$	4.26701	0.200040
$456$	14.4620	0.677246
$457$	−0.220981	−0.0103371	−0.00516853	−	0.999987i	$-0.501645\pi$
$457$	−0.220981	−0.0103371	−0.00516853	+	0.999987i	$0.501645\pi$
$458$	−24.7816	−1.15797
$459$	0.517991	0.0241777
$460$	3.43542	0.160177
$461$	7.01233	0.326597	0.163298	−	0.986577i	$-0.447787\pi$
$461$	7.01233	0.326597	0.163298	+	0.986577i	$0.447787\pi$
$462$	7.53769	0.350685
$463$	−7.25612	−0.337220	−0.168610	−	0.985683i	$-0.553928\pi$
$463$	−7.25612	−0.337220	−0.168610	+	0.985683i	$0.553928\pi$
$464$	5.19509	0.241176
$465$	−13.5280	−0.627345
$466$	−5.85943	−0.271433
$467$	13.0754	0.605057	0.302528	−	0.953140i	$-0.402169\pi$
$467$	13.0754	0.605057	0.302528	+	0.953140i	$0.402169\pi$
$468$	6.28617	0.290578
$469$	13.1345	0.606494
$470$	−11.5803	−0.534158
$471$	−27.6665	−1.27480
$472$	13.6035	0.626154
$473$	−33.6813	−1.54867
$474$	25.1469	1.15503
$475$	−23.6386	−1.08462
$476$	0.252781	0.0115862
$477$	11.9481	0.547065
$478$	6.39294	0.292406
$479$	−5.79308	−0.264693	−0.132346	−	0.991204i	$-0.542251\pi$
$479$	−5.79308	−0.264693	−0.132346	+	0.991204i	$0.542251\pi$
$480$	2.59751	0.118560
$481$	28.1244	1.28236
$482$	−10.8331	−0.493432
$483$	−7.44522	−0.338769
$484$	−2.11817	−0.0962806
$485$	3.30493	0.150069
$486$	−17.5951	−0.798128
$487$	−33.0755	−1.49879	−0.749396	−	0.662122i	$-0.769656\pi$
$487$	−33.0755	−1.49879	−0.749396	+	0.662122i	$0.769656\pi$
$488$	−2.09004	−0.0946119
$489$	−0.749118	−0.0338763
$490$	6.66231	0.300972
$491$	30.9057	1.39476	0.697378	−	0.716703i	$-0.254350\pi$
$491$	30.9057	1.39476	0.697378	+	0.716703i	$0.254350\pi$
$492$	−17.2611	−0.778189
$493$	1.15563	0.0520468
$494$	−20.9062	−0.940614
$495$	6.79537	0.305429
$496$	5.20806	0.233849
$497$	−6.09428	−0.273366
$498$	1.62339	0.0727460
$499$	22.1208	0.990261	0.495131	−	0.868819i	$-0.335120\pi$
$499$	22.1208	0.990261	0.495131	+	0.868819i	$0.335120\pi$
$500$	−10.0810	−0.450835
$501$	43.2946	1.93426
$502$	−30.3213	−1.35330
$503$	6.94781	0.309787	0.154894	−	0.987931i	$-0.450497\pi$
$503$	6.94781	0.309787	0.154894	+	0.987931i	$0.450497\pi$
$504$	−2.22020	−0.0988954
$505$	−2.44259	−0.108694
$506$	−8.77286	−0.390001
$507$	5.89346	0.261738
$508$	13.5232	0.599996
$509$	0.388044	0.0171997	0.00859986	−	0.999963i	$-0.497263\pi$
$509$	0.388044	0.0171997	0.00859986	+	0.999963i	$0.497263\pi$
$510$	0.577806	0.0255857
$511$	6.15485	0.272274
$512$	−1.00000	−0.0441942
$513$	15.1307	0.668036
$514$	9.89894	0.436624
$515$	−13.0385	−0.574547
$516$	25.1539	1.10734
$517$	29.5720	1.30057
$518$	−9.93319	−0.436439
$519$	55.9365	2.45534
$520$	−3.75495	−0.164665
$521$	−12.1279	−0.531334	−0.265667	−	0.964065i	$-0.585592\pi$
$521$	−12.1279	−0.531334	−0.265667	+	0.964065i	$0.585592\pi$
$522$	−10.1500	−0.444252
$523$	−1.29163	−0.0564789	−0.0282395	−	0.999601i	$-0.508990\pi$
$523$	−1.29163	−0.0564789	−0.0282395	+	0.999601i	$0.508990\pi$
$524$	−21.0847	−0.921091
$525$	9.20129	0.401577
$526$	−13.9931	−0.610126
$527$	1.15851	0.0504655
$528$	−6.63313	−0.288670
$529$	−14.3348	−0.623251
$530$	−7.13701	−0.310012
$531$	−26.5781	−1.15339
$532$	7.38381	0.320129
$533$	24.9525	1.08081
$534$	19.0878	0.826010
$535$	8.73414	0.377610
$536$	−11.5583	−0.499242
$537$	34.0323	1.46860
$538$	23.3003	1.00455
$539$	−17.0132	−0.732810
$540$	2.71761	0.116947
$541$	10.2404	0.440268	0.220134	−	0.975470i	$-0.429350\pi$
$541$	10.2404	0.440268	0.220134	+	0.975470i	$0.429350\pi$
$542$	13.4422	0.577392
$543$	21.6297	0.928220
$544$	−0.222446	−0.00953729
$545$	−0.964460	−0.0413129
$546$	8.13769	0.348261
$547$	22.6653	0.969096	0.484548	−	0.874765i	$-0.338984\pi$
$547$	22.6653	0.969096	0.484548	+	0.874765i	$0.338984\pi$
$548$	3.38930	0.144784
$549$	4.08345	0.174278
$550$	10.8421	0.462308
$551$	33.7562	1.43806
$552$	6.55176	0.278861
$553$	12.8391	0.545975
$554$	−11.0882	−0.471091
$555$	−22.7053	−0.963785
$556$	−22.2135	−0.942061
$557$	−37.6973	−1.59728	−0.798642	−	0.601806i	$-0.794448\pi$
$557$	−37.6973	−1.59728	−0.798642	+	0.601806i	$0.794448\pi$
$558$	−10.1753	−0.430755
$559$	−36.3624	−1.53796
$560$	1.32620	0.0560422
$561$	−1.47551	−0.0622962
$562$	−8.77765	−0.370263
$563$	−28.4022	−1.19701	−0.598506	−	0.801118i	$-0.704239\pi$
$563$	−28.4022	−1.19701	−0.598506	+	0.801118i	$0.704239\pi$
$564$	−22.0850	−0.929945
$565$	6.91376	0.290864
$566$	−5.39594	−0.226808
$567$	−12.5502	−0.527058
$568$	5.36294	0.225024
$569$	33.0860	1.38704	0.693518	−	0.720439i	$-0.256060\pi$
$569$	33.0860	1.38704	0.693518	+	0.720439i	$0.256060\pi$
$570$	16.8779	0.706938
$571$	21.8864	0.915917	0.457959	−	0.888973i	$-0.348581\pi$
$571$	21.8864	0.915917	0.457959	+	0.888973i	$0.348581\pi$
$572$	9.58882	0.400929
$573$	23.7998	0.994251
$574$	−8.81292	−0.367844
$575$	−10.7091	−0.446599
$576$	1.95376	0.0814068
$577$	6.05203	0.251949	0.125975	−	0.992033i	$-0.459794\pi$
$577$	6.05203	0.251949	0.125975	+	0.992033i	$0.459794\pi$
$578$	16.9505	0.705049
$579$	−20.7840	−0.863755
$580$	6.06294	0.251750
$581$	0.828849	0.0343865
$582$	6.30289	0.261263
$583$	18.2254	0.754819
$584$	−5.41624	−0.224126
$585$	7.33628	0.303318
$586$	23.1293	0.955462
$587$	−11.9857	−0.494703	−0.247351	−	0.968926i	$-0.579560\pi$
$587$	−11.9857	−0.494703	−0.247351	+	0.968926i	$0.579560\pi$
$588$	12.7058	0.523979
$589$	33.8405	1.39437
$590$	15.8760	0.653606
$591$	19.2203	0.790619
$592$	8.74116	0.359259
$593$	1.61052	0.0661360	0.0330680	−	0.999453i	$-0.489472\pi$
$593$	1.61052	0.0661360	0.0330680	+	0.999453i	$0.489472\pi$
$594$	−6.93983	−0.284745
$595$	0.295008	0.0120941
$596$	−18.2891	−0.749149
$597$	5.13617	0.210210
$598$	−9.47118	−0.387306
$599$	−28.1192	−1.14892	−0.574460	−	0.818533i	$-0.694788\pi$
$599$	−28.1192	−1.14892	−0.574460	+	0.818533i	$0.694788\pi$
$600$	−8.09710	−0.330563
$601$	6.20875	0.253260	0.126630	−	0.991950i	$-0.459584\pi$
$601$	6.20875	0.253260	0.126630	+	0.991950i	$0.459584\pi$
$602$	12.8427	0.523431
$603$	22.5822	0.919617
$604$	7.02995	0.286045
$605$	−2.47202	−0.100502
$606$	−4.65831	−0.189231
$607$	−28.6574	−1.16317	−0.581583	−	0.813487i	$-0.697566\pi$
$607$	−28.6574	−1.16317	−0.581583	+	0.813487i	$0.697566\pi$
$608$	−6.49772	−0.263517
$609$	−13.1395	−0.532441
$610$	−2.43919	−0.0987599
$611$	31.9259	1.29158
$612$	0.434606	0.0175679
$613$	43.8458	1.77092	0.885458	−	0.464719i	$-0.153845\pi$
$613$	43.8458	1.77092	0.885458	+	0.464719i	$0.153845\pi$
$614$	12.6045	0.508678
$615$	−20.1446	−0.812307
$616$	−3.38665	−0.136452
$617$	1.90807	0.0768158	0.0384079	−	0.999262i	$-0.487771\pi$
$617$	1.90807	0.0768158	0.0384079	+	0.999262i	$0.487771\pi$
$618$	−24.8660	−1.00026
$619$	4.94137	0.198610	0.0993051	−	0.995057i	$-0.468338\pi$
$619$	4.94137	0.198610	0.0993051	+	0.995057i	$0.468338\pi$
$620$	6.07807	0.244101
$621$	6.85469	0.275069
$622$	2.94182	0.117956
$623$	9.74559	0.390449
$624$	−7.16113	−0.286675
$625$	6.42493	0.256997
$626$	−18.4746	−0.738395
$627$	−43.1002	−1.72126
$628$	12.4304	0.496028
$629$	1.94443	0.0775297
$630$	−2.59108	−0.103231
$631$	11.4017	0.453896	0.226948	−	0.973907i	$-0.427125\pi$
$631$	11.4017	0.453896	0.226948	+	0.973907i	$0.427125\pi$
$632$	−11.2984	−0.449425
$633$	34.0396	1.35295
$634$	16.0056	0.635665
$635$	15.7823	0.626301
$636$	−13.6111	−0.539716
$637$	−18.3674	−0.727744
$638$	−15.4826	−0.612963
$639$	−10.4779	−0.414500
$640$	−1.16705	−0.0461318
$641$	35.1480	1.38826	0.694131	−	0.719848i	$-0.255789\pi$
$641$	35.1480	1.38826	0.694131	+	0.719848i	$0.255789\pi$
$642$	16.6570	0.657401
$643$	24.9962	0.985752	0.492876	−	0.870100i	$-0.335946\pi$
$643$	24.9962	0.985752	0.492876	+	0.870100i	$0.335946\pi$
$644$	3.34510	0.131816
$645$	29.3559	1.15589
$646$	−1.44539	−0.0568681
$647$	3.53102	0.138819	0.0694093	−	0.997588i	$-0.477889\pi$
$647$	3.53102	0.138819	0.0694093	+	0.997588i	$0.477889\pi$
$648$	11.0441	0.433853
$649$	−40.5418	−1.59141
$650$	11.7051	0.459112
$651$	−13.1723	−0.516264
$652$	0.336576	0.0131813
$653$	50.1247	1.96153	0.980765	−	0.195194i	$-0.0625336\pi$
$653$	50.1247	1.96153	0.980765	+	0.195194i	$0.0625336\pi$
$654$	−1.83934	−0.0719238
$655$	−24.6070	−0.961474
$656$	7.75533	0.302795
$657$	10.5820	0.412845
$658$	−11.2758	−0.439578
$659$	−17.5207	−0.682509	−0.341255	−	0.939971i	$-0.610852\pi$
$659$	−17.5207	−0.682509	−0.341255	+	0.939971i	$0.610852\pi$
$660$	−7.74121	−0.301326
$661$	28.8917	1.12376	0.561879	−	0.827219i	$-0.310079\pi$
$661$	28.8917	1.12376	0.561879	+	0.827219i	$0.310079\pi$
$662$	5.71365	0.222067
$663$	−1.59296	−0.0618656
$664$	−0.729384	−0.0283056
$665$	8.61728	0.334164
$666$	−17.0782	−0.661765
$667$	15.2927	0.592135
$668$	−19.4521	−0.752624
$669$	30.8960	1.19451
$670$	−13.4891	−0.521130
$671$	6.22883	0.240461
$672$	2.52922	0.0975669
$673$	8.95127	0.345046	0.172523	−	0.985005i	$-0.444808\pi$
$673$	8.95127	0.345046	0.172523	+	0.985005i	$0.444808\pi$
$674$	9.34498	0.359955
$675$	−8.47148	−0.326067
$676$	−2.64791	−0.101843
$677$	12.6496	0.486162	0.243081	−	0.970006i	$-0.421842\pi$
$677$	12.6496	0.486162	0.243081	+	0.970006i	$0.421842\pi$
$678$	13.1854	0.506380
$679$	3.21804	0.123497
$680$	−0.259606	−0.00995542
$681$	−55.2745	−2.11812
$682$	−15.5213	−0.594339
$683$	6.40663	0.245143	0.122571	−	0.992460i	$-0.460886\pi$
$683$	6.40663	0.245143	0.122571	+	0.992460i	$0.460886\pi$
$684$	12.6950	0.485406
$685$	3.95548	0.151131
$686$	14.4417	0.551388
$687$	−55.1566	−2.10435
$688$	−11.3016	−0.430868
$689$	19.6762	0.749602
$690$	7.64624	0.291087
$691$	−22.3478	−0.850151	−0.425076	−	0.905158i	$-0.639753\pi$
$691$	−22.3478	−0.850151	−0.425076	+	0.905158i	$0.639753\pi$
$692$	−25.1320	−0.955376
$693$	6.61672	0.251348
$694$	6.59827	0.250467
$695$	−25.9243	−0.983363
$696$	11.5627	0.438285
$697$	1.72514	0.0653444
$698$	−21.4246	−0.810931
$699$	−13.0414	−0.493270
$700$	−4.13410	−0.156254
$701$	28.9322	1.09276	0.546378	−	0.837539i	$-0.316006\pi$
$701$	28.9322	1.09276	0.546378	+	0.837539i	$0.316006\pi$
$702$	−7.49224	−0.282776
$703$	56.7976	2.14216
$704$	2.98024	0.112322
$705$	−25.7743	−0.970716
$706$	−25.2064	−0.948654
$707$	−2.37838	−0.0894480
$708$	30.2775	1.13790
$709$	31.1836	1.17113	0.585563	−	0.810627i	$-0.300874\pi$
$709$	31.1836	1.17113	0.585563	+	0.810627i	$0.300874\pi$
$710$	6.25883	0.234890
$711$	22.0744	0.827853
$712$	−8.57607	−0.321402
$713$	15.3308	0.574144
$714$	0.562615	0.0210553
$715$	11.1906	0.418507
$716$	−15.2906	−0.571436
$717$	14.2288	0.531385
$718$	23.5183	0.877694
$719$	8.25066	0.307698	0.153849	−	0.988094i	$-0.450833\pi$
$719$	8.25066	0.307698	0.153849	+	0.988094i	$0.450833\pi$
$720$	2.28014	0.0849759
$721$	−12.6958	−0.472815
$722$	−23.2204	−0.864172
$723$	−24.1112	−0.896705
$724$	−9.71814	−0.361172
$725$	−18.8997	−0.701917
$726$	−4.71443	−0.174969
$727$	−46.1731	−1.71247	−0.856234	−	0.516589i	$-0.827202\pi$
$727$	−46.1731	−1.71247	−0.856234	+	0.516589i	$0.827202\pi$
$728$	−3.65623	−0.135509
$729$	−6.02913	−0.223301
$730$	−6.32103	−0.233952
$731$	−2.51398	−0.0929830
$732$	−4.65182	−0.171936
$733$	−1.63808	−0.0605037	−0.0302519	−	0.999542i	$-0.509631\pi$
$733$	−1.63808	−0.0605037	−0.0302519	+	0.999542i	$0.509631\pi$
$734$	−3.88576	−0.143426
$735$	14.8283	0.546951
$736$	−2.94368	−0.108505
$737$	34.4465	1.26885
$738$	−15.1521	−0.557756
$739$	−34.4746	−1.26817	−0.634085	−	0.773263i	$-0.718623\pi$
$739$	−34.4746	−1.26817	−0.634085	+	0.773263i	$0.718623\pi$
$740$	10.2014	0.375010
$741$	−46.5310	−1.70936
$742$	−6.94937	−0.255119
$743$	21.1165	0.774689	0.387344	−	0.921935i	$-0.373392\pi$
$743$	21.1165	0.774689	0.387344	+	0.921935i	$0.373392\pi$
$744$	11.5916	0.424969
$745$	−21.3443	−0.781994
$746$	−15.7942	−0.578268
$747$	1.42504	0.0521396
$748$	0.662942	0.0242396
$749$	8.50452	0.310748
$750$	−22.4373	−0.819294
$751$	−14.8703	−0.542623	−0.271312	−	0.962492i	$-0.587457\pi$
$751$	−14.8703	−0.542623	−0.271312	+	0.962492i	$0.587457\pi$
$752$	9.92268	0.361843
$753$	−67.4862	−2.45933
$754$	−16.7150	−0.608726
$755$	8.20432	0.298586
$756$	2.64617	0.0962401
$757$	35.0461	1.27377	0.636887	−	0.770957i	$-0.280222\pi$
$757$	35.0461	1.27377	0.636887	+	0.770957i	$0.280222\pi$
$758$	12.5665	0.456437
$759$	−19.5258	−0.708742
$760$	−7.58317	−0.275071
$761$	−35.1539	−1.27433	−0.637164	−	0.770728i	$-0.719893\pi$
$761$	−35.1539	−1.27433	−0.637164	+	0.770728i	$0.719893\pi$
$762$	30.0987	1.09036
$763$	−0.939104	−0.0339978
$764$	−10.6931	−0.386865
$765$	0.507208	0.0183381
$766$	16.8231	0.607845
$767$	−43.7690	−1.58041
$768$	−2.22571	−0.0803132
$769$	51.3069	1.85017	0.925087	−	0.379756i	$-0.123992\pi$
$769$	51.3069	1.85017	0.925087	+	0.379756i	$0.123992\pi$
$770$	−3.95240	−0.142435
$771$	22.0321	0.793468
$772$	9.33818	0.336088
$773$	43.3279	1.55840	0.779198	−	0.626777i	$-0.215626\pi$
$773$	43.3279	1.55840	0.779198	+	0.626777i	$0.215626\pi$
$774$	22.0806	0.793669
$775$	−18.9469	−0.680591
$776$	−2.83186	−0.101658
$777$	−22.1083	−0.793132
$778$	−13.2507	−0.475060
$779$	50.3919	1.80548
$780$	−8.35741	−0.299243
$781$	−15.9828	−0.571911
$782$	−0.654809	−0.0234159
$783$	12.0974	0.432325
$784$	−5.70866	−0.203881
$785$	14.5070	0.517775
$786$	−46.9284	−1.67388
$787$	−14.3873	−0.512852	−0.256426	−	0.966564i	$-0.582545\pi$
$787$	−14.3873	−0.512852	−0.256426	+	0.966564i	$0.582545\pi$
$788$	−8.63562	−0.307631
$789$	−31.1444	−1.10877
$790$	−13.1858	−0.469129
$791$	6.73199	0.239362
$792$	−5.82268	−0.206900
$793$	6.72465	0.238799
$794$	−0.919129	−0.0326187
$795$	−15.8849	−0.563378
$796$	−2.30766	−0.0817929
$797$	23.8249	0.843920	0.421960	−	0.906614i	$-0.361342\pi$
$797$	23.8249	0.843920	0.421960	+	0.906614i	$0.361342\pi$
$798$	16.4342	0.581764
$799$	2.20726	0.0780872
$800$	3.63799	0.128622
$801$	16.7556	0.592031
$802$	−24.7215	−0.872947
$803$	16.1417	0.569628
$804$	−25.7254	−0.907263
$805$	3.90391	0.137595
$806$	−16.7567	−0.590231
$807$	51.8597	1.82555
$808$	2.09296	0.0736301
$809$	−42.2060	−1.48388	−0.741942	−	0.670464i	$-0.766095\pi$
$809$	−42.2060	−1.48388	−0.741942	+	0.670464i	$0.766095\pi$
$810$	12.8890	0.452874
$811$	−23.0050	−0.807815	−0.403907	−	0.914800i	$-0.632348\pi$
$811$	−23.0050	−0.807815	−0.403907	+	0.914800i	$0.632348\pi$
$812$	5.90354	0.207174
$813$	29.9184	1.04928
$814$	−26.0508	−0.913078
$815$	0.392801	0.0137592
$816$	−0.495099	−0.0173319
$817$	−73.4343	−2.56914
$818$	−31.1844	−1.09034
$819$	7.14341	0.249611
$820$	9.05087	0.316070
$821$	−27.1066	−0.946028	−0.473014	−	0.881055i	$-0.656834\pi$
$821$	−27.1066	−0.946028	−0.473014	+	0.881055i	$0.656834\pi$
$822$	7.54357	0.263112
$823$	0.245941	0.00857298	0.00428649	−	0.999991i	$-0.498636\pi$
$823$	0.245941	0.00857298	0.00428649	+	0.999991i	$0.498636\pi$
$824$	11.1722	0.389202
$825$	24.1313	0.840143
$826$	15.4587	0.537875
$827$	18.7088	0.650570	0.325285	−	0.945616i	$-0.394540\pi$
$827$	18.7088	0.650570	0.325285	+	0.945616i	$0.394540\pi$
$828$	5.75125	0.199870
$829$	−34.5666	−1.20055	−0.600274	−	0.799794i	$-0.704942\pi$
$829$	−34.5666	−1.20055	−0.600274	+	0.799794i	$0.704942\pi$
$830$	−0.851228	−0.0295466
$831$	−24.6790	−0.856104
$832$	3.21747	0.111546
$833$	−1.26987	−0.0439983
$834$	−49.4406	−1.71199
$835$	−22.7016	−0.785621
$836$	19.3648	0.669744
$837$	12.1276	0.419190
$838$	11.0710	0.382441
$839$	−19.1664	−0.661697	−0.330849	−	0.943684i	$-0.607335\pi$
$839$	−19.1664	−0.661697	−0.330849	+	0.943684i	$0.607335\pi$
$840$	2.95173	0.101844
$841$	−2.01103	−0.0693457
$842$	−22.4026	−0.772044
$843$	−19.5365	−0.672871
$844$	−15.2939	−0.526436
$845$	−3.09024	−0.106308
$846$	−19.3866	−0.666524
$847$	−2.40703	−0.0827064
$848$	6.11542	0.210004
$849$	−12.0098	−0.412174
$850$	0.809256	0.0277572
$851$	25.7312	0.882053
$852$	11.9363	0.408932
$853$	−43.1888	−1.47876	−0.739378	−	0.673291i	$-0.764880\pi$
$853$	−43.1888	−1.47876	−0.739378	+	0.673291i	$0.764880\pi$
$854$	−2.37506	−0.0812730
$855$	14.8157	0.506687
$856$	−7.48394	−0.255796
$857$	5.54367	0.189368	0.0946841	−	0.995507i	$-0.469816\pi$
$857$	5.54367	0.189368	0.0946841	+	0.995507i	$0.469816\pi$
$858$	21.3419	0.728600
$859$	−0.882660	−0.0301160	−0.0150580	−	0.999887i	$-0.504793\pi$
$859$	−0.882660	−0.0301160	−0.0150580	+	0.999887i	$0.504793\pi$
$860$	−13.1895	−0.449758
$861$	−19.6150	−0.668476
$862$	13.1560	0.448096
$863$	−20.3566	−0.692947	−0.346474	−	0.938060i	$-0.612621\pi$
$863$	−20.3566	−0.692947	−0.346474	+	0.938060i	$0.612621\pi$
$864$	−2.32861	−0.0792211
$865$	−29.3304	−0.997262
$866$	15.1121	0.513530
$867$	37.7269	1.28127
$868$	5.91827	0.200879
$869$	33.6719	1.14224
$870$	13.4943	0.457500
$871$	37.1884	1.26008
$872$	0.826407	0.0279857
$873$	5.53279	0.187257
$874$	−19.1272	−0.646987
$875$	−11.4557	−0.387274
$876$	−12.0549	−0.407299
$877$	27.5417	0.930017	0.465009	−	0.885306i	$-0.346051\pi$
$877$	27.5417	0.930017	0.465009	+	0.885306i	$0.346051\pi$
$878$	1.22872	0.0414674
$879$	51.4790	1.73634
$880$	3.47809	0.117246
$881$	−24.8537	−0.837343	−0.418671	−	0.908138i	$-0.637504\pi$
$881$	−24.8537	−0.837343	−0.418671	+	0.908138i	$0.637504\pi$
$882$	11.1534	0.375554
$883$	23.9363	0.805520	0.402760	−	0.915306i	$-0.368051\pi$
$883$	23.9363	0.805520	0.402760	+	0.915306i	$0.368051\pi$
$884$	0.715712	0.0240720
$885$	35.3354	1.18779
$886$	11.3893	0.382630
$887$	34.8303	1.16949	0.584743	−	0.811219i	$-0.301195\pi$
$887$	34.8303	1.16949	0.584743	+	0.811219i	$0.301195\pi$
$888$	19.4552	0.652875
$889$	15.3674	0.515405
$890$	−10.0087	−0.335493
$891$	−32.9141	−1.10266
$892$	−13.8814	−0.464784
$893$	64.4748	2.15757
$894$	−40.7061	−1.36141
$895$	−17.8449	−0.596489
$896$	−1.13637	−0.0379634
$897$	−21.0801	−0.703843
$898$	−4.21729	−0.140733
$899$	27.0563	0.902379
$900$	−7.10777	−0.236926
$901$	1.36035	0.0453198
$902$	−23.1127	−0.769570
$903$	28.5842	0.951221
$904$	−5.92412	−0.197033
$905$	−11.3416	−0.377006
$906$	15.6466	0.519824
$907$	40.9638	1.36018	0.680091	−	0.733128i	$-0.261940\pi$
$907$	40.9638	1.36018	0.680091	+	0.733128i	$0.261940\pi$
$908$	24.8346	0.824166
$909$	−4.08915	−0.135628
$910$	−4.26701	−0.141450
$911$	−38.9775	−1.29138	−0.645692	−	0.763598i	$-0.723431\pi$
$911$	−38.9775	−1.29138	−0.645692	+	0.763598i	$0.723431\pi$
$912$	−14.4620	−0.478885
$913$	2.17374	0.0719402
$914$	0.220981	0.00730941
$915$	−5.42892	−0.179474
$916$	24.7816	0.818808
$917$	−23.9601	−0.791231
$918$	−0.517991	−0.0170962
$919$	40.5016	1.33602	0.668012	−	0.744151i	$-0.267146\pi$
$919$	40.5016	1.33602	0.668012	+	0.744151i	$0.267146\pi$
$920$	−3.43542	−0.113263
$921$	28.0540	0.924410
$922$	−7.01233	−0.230939
$923$	−17.2551	−0.567958
$924$	−7.53769	−0.247972
$925$	−31.8003	−1.04559
$926$	7.25612	0.238451
$927$	−21.8279	−0.716921
$928$	−5.19509	−0.170537
$929$	32.7481	1.07443	0.537216	−	0.843445i	$-0.319476\pi$
$929$	32.7481	1.07443	0.537216	+	0.843445i	$0.319476\pi$
$930$	13.5280	0.443600
$931$	−37.0933	−1.21568
$932$	5.85943	0.191932
$933$	6.54762	0.214359
$934$	−13.0754	−0.427840
$935$	0.773687	0.0253023
$936$	−6.28617	−0.205470
$937$	−10.1399	−0.331256	−0.165628	−	0.986188i	$-0.552965\pi$
$937$	−10.1399	−0.331256	−0.165628	+	0.986188i	$0.552965\pi$
$938$	−13.1345	−0.428856
$939$	−41.1191	−1.34187
$940$	11.5803	0.377707
$941$	−17.2830	−0.563410	−0.281705	−	0.959501i	$-0.590900\pi$
$941$	−17.2830	−0.563410	−0.281705	+	0.959501i	$0.590900\pi$
$942$	27.6665	0.901423
$943$	22.8292	0.743421
$944$	−13.6035	−0.442758
$945$	3.08821	0.100460
$946$	33.6813	1.09507
$947$	41.7891	1.35796	0.678981	−	0.734156i	$-0.262422\pi$
$947$	41.7891	1.35796	0.678981	+	0.734156i	$0.262422\pi$
$948$	−25.1469	−0.816732
$949$	17.4266	0.565690
$950$	23.6386	0.766939
$951$	35.6238	1.15518
$952$	−0.252781	−0.00819267
$953$	−39.3599	−1.27499	−0.637496	−	0.770454i	$-0.720030\pi$
$953$	−39.3599	−1.27499	−0.637496	+	0.770454i	$0.720030\pi$
$954$	−11.9481	−0.386833
$955$	−12.4795	−0.403826
$956$	−6.39294	−0.206763
$957$	−34.4597	−1.11393
$958$	5.79308	0.187166
$959$	3.85149	0.124371
$960$	−2.59751	−0.0838343
$961$	−3.87616	−0.125037
$962$	−28.1244	−0.906767
$963$	14.6218	0.471182
$964$	10.8331	0.348909
$965$	10.8981	0.350823
$966$	7.44522	0.239546
$967$	21.8800	0.703614	0.351807	−	0.936073i	$-0.385567\pi$
$967$	21.8800	0.703614	0.351807	+	0.936073i	$0.385567\pi$
$968$	2.11817	0.0680807
$969$	−3.21701	−0.103345
$970$	−3.30493	−0.106115
$971$	−48.6005	−1.55966	−0.779831	−	0.625990i	$-0.784695\pi$
$971$	−48.6005	−1.55966	−0.779831	+	0.625990i	$0.784695\pi$
$972$	17.5951	0.564362
$973$	−25.2427	−0.809244
$974$	33.0755	1.05981
$975$	26.0521	0.834336
$976$	2.09004	0.0669007
$977$	−36.2208	−1.15881	−0.579403	−	0.815041i	$-0.696714\pi$
$977$	−36.2208	−1.15881	−0.579403	+	0.815041i	$0.696714\pi$
$978$	0.749118	0.0239541
$979$	25.5588	0.816861
$980$	−6.66231	−0.212820
$981$	−1.61460	−0.0515503
$982$	−30.9057	−0.986242
$983$	−60.7606	−1.93796	−0.968981	−	0.247134i	$-0.920511\pi$
$983$	−60.7606	−1.93796	−0.968981	+	0.247134i	$0.920511\pi$
$984$	17.2611	0.550263
$985$	−10.0782	−0.321118
$986$	−1.15563	−0.0368026
$987$	−25.0967	−0.798836
$988$	20.9062	0.665115
$989$	−33.2681	−1.05786
$990$	−6.79537	−0.215971
$991$	−45.2909	−1.43871	−0.719356	−	0.694642i	$-0.755563\pi$
$991$	−45.2909	−1.43871	−0.719356	+	0.694642i	$0.755563\pi$
$992$	−5.20806	−0.165356
$993$	12.7169	0.403558
$994$	6.09428	0.193299
$995$	−2.69316	−0.0853789
$996$	−1.62339	−0.0514392
$997$	45.6107	1.44450	0.722252	−	0.691630i	$-0.243107\pi$
$997$	45.6107	1.44450	0.722252	+	0.691630i	$0.243107\pi$
$998$	−22.1208	−0.700220
$999$	20.3548	0.643997

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.d.1.12	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.d.1.12	✓	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:	\( N \)	\(=\)	\( 6038 = 2 \cdot 3019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.22571 q^{3} +1.00000 q^{4} +1.16705 q^{5} +2.22571 q^{6} +1.13637 q^{7} -1.00000 q^{8} +1.95376 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.22571 q^{3} +1.00000 q^{4} +1.16705 q^{5} +2.22571 q^{6} +1.13637 q^{7} -1.00000 q^{8} +1.95376 q^{9} -1.16705 q^{10} +2.98024 q^{11} -2.22571 q^{12} +3.21747 q^{13} -1.13637 q^{14} -2.59751 q^{15} +1.00000 q^{16} +0.222446 q^{17} -1.95376 q^{18} +6.49772 q^{19} +1.16705 q^{20} -2.52922 q^{21} -2.98024 q^{22} +2.94368 q^{23} +2.22571 q^{24} -3.63799 q^{25} -3.21747 q^{26} +2.32861 q^{27} +1.13637 q^{28} +5.19509 q^{29} +2.59751 q^{30} +5.20806 q^{31} -1.00000 q^{32} -6.63313 q^{33} -0.222446 q^{34} +1.32620 q^{35} +1.95376 q^{36} +8.74116 q^{37} -6.49772 q^{38} -7.16113 q^{39} -1.16705 q^{40} +7.75533 q^{41} +2.52922 q^{42} -11.3016 q^{43} +2.98024 q^{44} +2.28014 q^{45} -2.94368 q^{46} +9.92268 q^{47} -2.22571 q^{48} -5.70866 q^{49} +3.63799 q^{50} -0.495099 q^{51} +3.21747 q^{52} +6.11542 q^{53} -2.32861 q^{54} +3.47809 q^{55} -1.13637 q^{56} -14.4620 q^{57} -5.19509 q^{58} -13.6035 q^{59} -2.59751 q^{60} +2.09004 q^{61} -5.20806 q^{62} +2.22020 q^{63} +1.00000 q^{64} +3.75495 q^{65} +6.63313 q^{66} +11.5583 q^{67} +0.222446 q^{68} -6.55176 q^{69} -1.32620 q^{70} -5.36294 q^{71} -1.95376 q^{72} +5.41624 q^{73} -8.74116 q^{74} +8.09710 q^{75} +6.49772 q^{76} +3.38665 q^{77} +7.16113 q^{78} +11.2984 q^{79} +1.16705 q^{80} -11.0441 q^{81} -7.75533 q^{82} +0.729384 q^{83} -2.52922 q^{84} +0.259606 q^{85} +11.3016 q^{86} -11.5627 q^{87} -2.98024 q^{88} +8.57607 q^{89} -2.28014 q^{90} +3.65623 q^{91} +2.94368 q^{92} -11.5916 q^{93} -9.92268 q^{94} +7.58317 q^{95} +2.22571 q^{96} +2.83186 q^{97} +5.70866 q^{98} +5.82268 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9} - 18 q^{10} - 4 q^{11} + 8 q^{12} + 40 q^{13} - 32 q^{14} + 6 q^{15} + 69 q^{16} + 6 q^{17} - 79 q^{18} + 13 q^{19} + 18 q^{20} + 23 q^{21} + 4 q^{22} + 11 q^{23} - 8 q^{24} + 111 q^{25} - 40 q^{26} + 32 q^{27} + 32 q^{28} + 23 q^{29} - 6 q^{30} + 30 q^{31} - 69 q^{32} + 37 q^{33} - 6 q^{34} - 12 q^{35} + 79 q^{36} + 81 q^{37} - 13 q^{38} - 8 q^{39} - 18 q^{40} - 13 q^{41} - 23 q^{42} + 42 q^{43} - 4 q^{44} + 89 q^{45} - 11 q^{46} + 35 q^{47} + 8 q^{48} + 115 q^{49} - 111 q^{50} - 21 q^{51} + 40 q^{52} + 41 q^{53} - 32 q^{54} + 28 q^{55} - 32 q^{56} + 39 q^{57} - 23 q^{58} - 24 q^{59} + 6 q^{60} + 69 q^{61} - 30 q^{62} + 79 q^{63} + 69 q^{64} + 15 q^{65} - 37 q^{66} + 87 q^{67} + 6 q^{68} + 51 q^{69} + 12 q^{70} - 22 q^{71} - 79 q^{72} + 104 q^{73} - 81 q^{74} + 39 q^{75} + 13 q^{76} + 47 q^{77} + 8 q^{78} + 16 q^{79} + 18 q^{80} + 105 q^{81} + 13 q^{82} + 30 q^{83} + 23 q^{84} + 74 q^{85} - 42 q^{86} + 47 q^{87} + 4 q^{88} - 4 q^{89} - 89 q^{90} + 70 q^{91} + 11 q^{92} + 104 q^{93} - 35 q^{94} - 36 q^{95} - 8 q^{96} + 158 q^{97} - 115 q^{98} + 34 q^{99}+O(q^{100}) \) 69 * q - 69 * q^2 + 8 * q^3 + 69 * q^4 + 18 * q^5 - 8 * q^6 + 32 * q^7 - 69 * q^8 + 79 * q^9 - 18 * q^10 - 4 * q^11 + 8 * q^12 + 40 * q^13 - 32 * q^14 + 6 * q^15 + 69 * q^16 + 6 * q^17 - 79 * q^18 + 13 * q^19 + 18 * q^20 + 23 * q^21 + 4 * q^22 + 11 * q^23 - 8 * q^24 + 111 * q^25 - 40 * q^26 + 32 * q^27 + 32 * q^28 + 23 * q^29 - 6 * q^30 + 30 * q^31 - 69 * q^32 + 37 * q^33 - 6 * q^34 - 12 * q^35 + 79 * q^36 + 81 * q^37 - 13 * q^38 - 8 * q^39 - 18 * q^40 - 13 * q^41 - 23 * q^42 + 42 * q^43 - 4 * q^44 + 89 * q^45 - 11 * q^46 + 35 * q^47 + 8 * q^48 + 115 * q^49 - 111 * q^50 - 21 * q^51 + 40 * q^52 + 41 * q^53 - 32 * q^54 + 28 * q^55 - 32 * q^56 + 39 * q^57 - 23 * q^58 - 24 * q^59 + 6 * q^60 + 69 * q^61 - 30 * q^62 + 79 * q^63 + 69 * q^64 + 15 * q^65 - 37 * q^66 + 87 * q^67 + 6 * q^68 + 51 * q^69 + 12 * q^70 - 22 * q^71 - 79 * q^72 + 104 * q^73 - 81 * q^74 + 39 * q^75 + 13 * q^76 + 47 * q^77 + 8 * q^78 + 16 * q^79 + 18 * q^80 + 105 * q^81 + 13 * q^82 + 30 * q^83 + 23 * q^84 + 74 * q^85 - 42 * q^86 + 47 * q^87 + 4 * q^88 - 4 * q^89 - 89 * q^90 + 70 * q^91 + 11 * q^92 + 104 * q^93 - 35 * q^94 - 36 * q^95 - 8 * q^96 + 158 * q^97 - 115 * q^98 + 34 * q^99

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