LMFDB - Embedded newform 6038.2.a.d.1.11

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			1.11
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.23277 q^{3} +1.00000 q^{4} +1.71292 q^{5} +2.23277 q^{6} +0.0975688 q^{7} -1.00000 q^{8} +1.98527 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.23277 q^{3} +1.00000 q^{4} +1.71292 q^{5} +2.23277 q^{6} +0.0975688 q^{7} -1.00000 q^{8} +1.98527 q^{9} -1.71292 q^{10} -5.37342 q^{11} -2.23277 q^{12} +2.35898 q^{13} -0.0975688 q^{14} -3.82457 q^{15} +1.00000 q^{16} +7.42142 q^{17} -1.98527 q^{18} -2.33915 q^{19} +1.71292 q^{20} -0.217849 q^{21} +5.37342 q^{22} -8.18303 q^{23} +2.23277 q^{24} -2.06589 q^{25} -2.35898 q^{26} +2.26566 q^{27} +0.0975688 q^{28} +7.73258 q^{29} +3.82457 q^{30} +6.36520 q^{31} -1.00000 q^{32} +11.9976 q^{33} -7.42142 q^{34} +0.167128 q^{35} +1.98527 q^{36} -11.5883 q^{37} +2.33915 q^{38} -5.26707 q^{39} -1.71292 q^{40} -3.02575 q^{41} +0.217849 q^{42} +1.55387 q^{43} -5.37342 q^{44} +3.40062 q^{45} +8.18303 q^{46} -2.70920 q^{47} -2.23277 q^{48} -6.99048 q^{49} +2.06589 q^{50} -16.5703 q^{51} +2.35898 q^{52} +3.46516 q^{53} -2.26566 q^{54} -9.20425 q^{55} -0.0975688 q^{56} +5.22279 q^{57} -7.73258 q^{58} +3.53834 q^{59} -3.82457 q^{60} +4.36328 q^{61} -6.36520 q^{62} +0.193700 q^{63} +1.00000 q^{64} +4.04075 q^{65} -11.9976 q^{66} +11.2043 q^{67} +7.42142 q^{68} +18.2708 q^{69} -0.167128 q^{70} +4.35302 q^{71} -1.98527 q^{72} -1.62922 q^{73} +11.5883 q^{74} +4.61267 q^{75} -2.33915 q^{76} -0.524278 q^{77} +5.26707 q^{78} -11.2748 q^{79} +1.71292 q^{80} -11.0145 q^{81} +3.02575 q^{82} -12.0758 q^{83} -0.217849 q^{84} +12.7123 q^{85} -1.55387 q^{86} -17.2651 q^{87} +5.37342 q^{88} +6.23285 q^{89} -3.40062 q^{90} +0.230163 q^{91} -8.18303 q^{92} -14.2120 q^{93} +2.70920 q^{94} -4.00679 q^{95} +2.23277 q^{96} +14.5914 q^{97} +6.99048 q^{98} -10.6677 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.23277	−1.28909	−0.644546	−	0.764566i	$-0.722953\pi$
$3$	−2.23277	−1.28909	−0.644546	+	0.764566i	$0.722953\pi$
$4$	1.00000	0.500000
$5$	1.71292	0.766043	0.383021	−	0.923740i	$-0.374884\pi$
$5$	1.71292	0.766043	0.383021	+	0.923740i	$0.374884\pi$
$6$	2.23277	0.911525
$7$	0.0975688	0.0368775	0.0184388	−	0.999830i	$-0.494130\pi$
$7$	0.0975688	0.0368775	0.0184388	+	0.999830i	$0.494130\pi$
$8$	−1.00000	−0.353553
$9$	1.98527	0.661757
$10$	−1.71292	−0.541674
$11$	−5.37342	−1.62015	−0.810073	−	0.586329i	$-0.800573\pi$
$11$	−5.37342	−1.62015	−0.810073	+	0.586329i	$0.800573\pi$
$12$	−2.23277	−0.644546
$13$	2.35898	0.654264	0.327132	−	0.944979i	$-0.393918\pi$
$13$	2.35898	0.654264	0.327132	+	0.944979i	$0.393918\pi$
$14$	−0.0975688	−0.0260764
$15$	−3.82457	−0.987499
$16$	1.00000	0.250000
$17$	7.42142	1.79996	0.899979	−	0.435933i	$-0.143582\pi$
$17$	7.42142	1.79996	0.899979	+	0.435933i	$0.143582\pi$
$18$	−1.98527	−0.467933
$19$	−2.33915	−0.536638	−0.268319	−	0.963330i	$-0.586468\pi$
$19$	−2.33915	−0.536638	−0.268319	+	0.963330i	$0.586468\pi$
$20$	1.71292	0.383021
$21$	−0.217849	−0.0475385
$22$	5.37342	1.14562
$23$	−8.18303	−1.70628	−0.853139	−	0.521683i	$-0.825304\pi$
$23$	−8.18303	−1.70628	−0.853139	+	0.521683i	$0.825304\pi$
$24$	2.23277	0.455763
$25$	−2.06589	−0.413179
$26$	−2.35898	−0.462634
$27$	2.26566	0.436027
$28$	0.0975688	0.0184388
$29$	7.73258	1.43590	0.717952	−	0.696092i	$-0.245080\pi$
$29$	7.73258	1.43590	0.717952	+	0.696092i	$0.245080\pi$
$30$	3.82457	0.698267
$31$	6.36520	1.14322	0.571612	−	0.820524i	$-0.306318\pi$
$31$	6.36520	1.14322	0.571612	+	0.820524i	$0.306318\pi$
$32$	−1.00000	−0.176777
$33$	11.9976	2.08852
$34$	−7.42142	−1.27276
$35$	0.167128	0.0282498
$36$	1.98527	0.330878
$37$	−11.5883	−1.90511	−0.952556	−	0.304364i	$-0.901556\pi$
$37$	−11.5883	−1.90511	−0.952556	+	0.304364i	$0.901556\pi$
$38$	2.33915	0.379460
$39$	−5.26707	−0.843406
$40$	−1.71292	−0.270837
$41$	−3.02575	−0.472543	−0.236271	−	0.971687i	$-0.575925\pi$
$41$	−3.02575	−0.472543	−0.236271	+	0.971687i	$0.575925\pi$
$42$	0.217849	0.0336148
$43$	1.55387	0.236963	0.118481	−	0.992956i	$-0.462197\pi$
$43$	1.55387	0.236963	0.118481	+	0.992956i	$0.462197\pi$
$44$	−5.37342	−0.810073
$45$	3.40062	0.506934
$46$	8.18303	1.20652
$47$	−2.70920	−0.395178	−0.197589	−	0.980285i	$-0.563311\pi$
$47$	−2.70920	−0.395178	−0.197589	+	0.980285i	$0.563311\pi$
$48$	−2.23277	−0.322273
$49$	−6.99048	−0.998640
$50$	2.06589	0.292161
$51$	−16.5703	−2.32031
$52$	2.35898	0.327132
$53$	3.46516	0.475976	0.237988	−	0.971268i	$-0.423512\pi$
$53$	3.46516	0.475976	0.237988	+	0.971268i	$0.423512\pi$
$54$	−2.26566	−0.308317
$55$	−9.20425	−1.24110
$56$	−0.0975688	−0.0130382
$57$	5.22279	0.691775
$58$	−7.73258	−1.01534
$59$	3.53834	0.460653	0.230326	−	0.973113i	$-0.426021\pi$
$59$	3.53834	0.460653	0.230326	+	0.973113i	$0.426021\pi$
$60$	−3.82457	−0.493749
$61$	4.36328	0.558660	0.279330	−	0.960195i	$-0.409888\pi$
$61$	4.36328	0.558660	0.279330	+	0.960195i	$0.409888\pi$
$62$	−6.36520	−0.808381
$63$	0.193700	0.0244040
$64$	1.00000	0.125000
$65$	4.04075	0.501194
$66$	−11.9976	−1.47680
$67$	11.2043	1.36882	0.684410	−	0.729098i	$-0.260060\pi$
$67$	11.2043	1.36882	0.684410	+	0.729098i	$0.260060\pi$
$68$	7.42142	0.899979
$69$	18.2708	2.19955
$70$	−0.167128	−0.0199756
$71$	4.35302	0.516608	0.258304	−	0.966064i	$-0.416836\pi$
$71$	4.35302	0.516608	0.258304	+	0.966064i	$0.416836\pi$
$72$	−1.98527	−0.233966
$73$	−1.62922	−0.190686	−0.0953428	−	0.995444i	$-0.530395\pi$
$73$	−1.62922	−0.190686	−0.0953428	+	0.995444i	$0.530395\pi$
$74$	11.5883	1.34712
$75$	4.61267	0.532625
$76$	−2.33915	−0.268319
$77$	−0.524278	−0.0597470
$78$	5.26707	0.596378
$79$	−11.2748	−1.26851	−0.634257	−	0.773122i	$-0.718694\pi$
$79$	−11.2748	−1.26851	−0.634257	+	0.773122i	$0.718694\pi$
$80$	1.71292	0.191511
$81$	−11.0145	−1.22383
$82$	3.02575	0.334138
$83$	−12.0758	−1.32549	−0.662745	−	0.748845i	$-0.730609\pi$
$83$	−12.0758	−1.32549	−0.662745	+	0.748845i	$0.730609\pi$
$84$	−0.217849	−0.0237693
$85$	12.7123	1.37884
$86$	−1.55387	−0.167558
$87$	−17.2651	−1.85101
$88$	5.37342	0.572808
$89$	6.23285	0.660681	0.330341	−	0.943862i	$-0.392836\pi$
$89$	6.23285	0.660681	0.330341	+	0.943862i	$0.392836\pi$
$90$	−3.40062	−0.358456
$91$	0.230163	0.0241276
$92$	−8.18303	−0.853139
$93$	−14.2120	−1.47372
$94$	2.70920	0.279433
$95$	−4.00679	−0.411088
$96$	2.23277	0.227881
$97$	14.5914	1.48153	0.740766	−	0.671763i	$-0.234463\pi$
$97$	14.5914	1.48153	0.740766	+	0.671763i	$0.234463\pi$
$98$	6.99048	0.706145
$99$	−10.6677	−1.07214
$100$	−2.06589	−0.206589
$101$	2.90704	0.289261	0.144631	−	0.989486i	$-0.453801\pi$
$101$	2.90704	0.289261	0.144631	+	0.989486i	$0.453801\pi$
$102$	16.5703	1.64071
$103$	−6.32205	−0.622930	−0.311465	−	0.950258i	$-0.600820\pi$
$103$	−6.32205	−0.622930	−0.311465	+	0.950258i	$0.600820\pi$
$104$	−2.35898	−0.231317
$105$	−0.373158	−0.0364165
$106$	−3.46516	−0.336566
$107$	0.331824	0.0320787	0.0160393	−	0.999871i	$-0.494894\pi$
$107$	0.331824	0.0320787	0.0160393	+	0.999871i	$0.494894\pi$
$108$	2.26566	0.218013
$109$	17.4718	1.67349	0.836745	−	0.547592i	$-0.184456\pi$
$109$	17.4718	1.67349	0.836745	+	0.547592i	$0.184456\pi$
$110$	9.20425	0.877591
$111$	25.8741	2.45586
$112$	0.0975688	0.00921939
$113$	14.5545	1.36917	0.684584	−	0.728934i	$-0.259984\pi$
$113$	14.5545	1.36917	0.684584	+	0.728934i	$0.259984\pi$
$114$	−5.22279	−0.489159
$115$	−14.0169	−1.30708
$116$	7.73258	0.717952
$117$	4.68322	0.432963
$118$	−3.53834	−0.325731
$119$	0.724099	0.0663780
$120$	3.82457	0.349134
$121$	17.8736	1.62487
$122$	−4.36328	−0.395033
$123$	6.75581	0.609151
$124$	6.36520	0.571612
$125$	−12.1033	−1.08256
$126$	−0.193700	−0.0172562
$127$	9.99091	0.886550	0.443275	−	0.896386i	$-0.353816\pi$
$127$	9.99091	0.886550	0.443275	+	0.896386i	$0.353816\pi$
$128$	−1.00000	−0.0883883
$129$	−3.46943	−0.305466
$130$	−4.04075	−0.354398
$131$	−14.3490	−1.25368	−0.626840	−	0.779148i	$-0.715652\pi$
$131$	−14.3490	−1.25368	−0.626840	+	0.779148i	$0.715652\pi$
$132$	11.9976	1.04426
$133$	−0.228228	−0.0197899
$134$	−11.2043	−0.967901
$135$	3.88090	0.334015
$136$	−7.42142	−0.636381
$137$	−10.3387	−0.883294	−0.441647	−	0.897189i	$-0.645606\pi$
$137$	−10.3387	−0.883294	−0.441647	+	0.897189i	$0.645606\pi$
$138$	−18.2708	−1.55532
$139$	2.43867	0.206845	0.103422	−	0.994638i	$-0.467021\pi$
$139$	2.43867	0.206845	0.103422	+	0.994638i	$0.467021\pi$
$140$	0.167128	0.0141249
$141$	6.04903	0.509421
$142$	−4.35302	−0.365297
$143$	−12.6758	−1.06000
$144$	1.98527	0.165439
$145$	13.2453	1.09996
$146$	1.62922	0.134835
$147$	15.6081	1.28734
$148$	−11.5883	−0.952556
$149$	−15.7425	−1.28967	−0.644837	−	0.764320i	$-0.723075\pi$
$149$	−15.7425	−1.28967	−0.644837	+	0.764320i	$0.723075\pi$
$150$	−4.61267	−0.376623
$151$	20.0136	1.62869	0.814343	−	0.580385i	$-0.197098\pi$
$151$	20.0136	1.62869	0.814343	+	0.580385i	$0.197098\pi$
$152$	2.33915	0.189730
$153$	14.7335	1.19113
$154$	0.524278	0.0422475
$155$	10.9031	0.875758
$156$	−5.26707	−0.421703
$157$	−18.4201	−1.47008	−0.735040	−	0.678023i	$-0.762837\pi$
$157$	−18.4201	−1.47008	−0.735040	+	0.678023i	$0.762837\pi$
$158$	11.2748	0.896975
$159$	−7.73691	−0.613577
$160$	−1.71292	−0.135418
$161$	−0.798408	−0.0629234
$162$	11.0145	0.865382
$163$	−6.72147	−0.526466	−0.263233	−	0.964732i	$-0.584789\pi$
$163$	−6.72147	−0.526466	−0.263233	+	0.964732i	$0.584789\pi$
$164$	−3.02575	−0.236271
$165$	20.5510	1.59989
$166$	12.0758	0.937263
$167$	6.78204	0.524810	0.262405	−	0.964958i	$-0.415484\pi$
$167$	6.78204	0.524810	0.262405	+	0.964958i	$0.415484\pi$
$168$	0.217849	0.0168074
$169$	−7.43521	−0.571939
$170$	−12.7123	−0.974990
$171$	−4.64385	−0.355124
$172$	1.55387	0.118481
$173$	10.9929	0.835776	0.417888	−	0.908499i	$-0.362770\pi$
$173$	10.9929	0.835776	0.417888	+	0.908499i	$0.362770\pi$
$174$	17.2651	1.30886
$175$	−0.201567	−0.0152370
$176$	−5.37342	−0.405037
$177$	−7.90031	−0.593824
$178$	−6.23285	−0.467172
$179$	−13.5524	−1.01295	−0.506475	−	0.862254i	$-0.669052\pi$
$179$	−13.5524	−1.01295	−0.506475	+	0.862254i	$0.669052\pi$
$180$	3.40062	0.253467
$181$	0.988318	0.0734611	0.0367305	−	0.999325i	$-0.488306\pi$
$181$	0.988318	0.0734611	0.0367305	+	0.999325i	$0.488306\pi$
$182$	−0.230163	−0.0170608
$183$	−9.74220	−0.720164
$184$	8.18303	0.603261
$185$	−19.8499	−1.45940
$186$	14.2120	1.04208
$187$	−39.8784	−2.91620
$188$	−2.70920	−0.197589
$189$	0.221058	0.0160796
$190$	4.00679	0.290683
$191$	13.8078	0.999098	0.499549	−	0.866286i	$-0.333499\pi$
$191$	13.8078	0.999098	0.499549	+	0.866286i	$0.333499\pi$
$192$	−2.23277	−0.161136
$193$	−7.21519	−0.519361	−0.259680	−	0.965695i	$-0.583617\pi$
$193$	−7.21519	−0.519361	−0.259680	+	0.965695i	$0.583617\pi$
$194$	−14.5914	−1.04760
$195$	−9.02208	−0.646085
$196$	−6.99048	−0.499320
$197$	−4.20803	−0.299810	−0.149905	−	0.988700i	$-0.547897\pi$
$197$	−4.20803	−0.299810	−0.149905	+	0.988700i	$0.547897\pi$
$198$	10.6677	0.758119
$199$	−26.4490	−1.87492	−0.937461	−	0.348091i	$-0.886830\pi$
$199$	−26.4490	−1.87492	−0.937461	+	0.348091i	$0.886830\pi$
$200$	2.06589	0.146081
$201$	−25.0166	−1.76453
$202$	−2.90704	−0.204539
$203$	0.754459	0.0529526
$204$	−16.5703	−1.16016
$205$	−5.18288	−0.361988
$206$	6.32205	0.440478
$207$	−16.2455	−1.12914
$208$	2.35898	0.163566
$209$	12.5692	0.869432
$210$	0.373158	0.0257504
$211$	19.7483	1.35953	0.679766	−	0.733429i	$-0.262081\pi$
$211$	19.7483	1.35953	0.679766	+	0.733429i	$0.262081\pi$
$212$	3.46516	0.237988
$213$	−9.71930	−0.665955
$214$	−0.331824	−0.0226830
$215$	2.66166	0.181523
$216$	−2.26566	−0.154159
$217$	0.621045	0.0421593
$218$	−17.4718	−1.18334
$219$	3.63767	0.245811
$220$	−9.20425	−0.620551
$221$	17.5070	1.17765
$222$	−25.8741	−1.73656
$223$	−12.4069	−0.830828	−0.415414	−	0.909632i	$-0.636363\pi$
$223$	−12.4069	−0.830828	−0.415414	+	0.909632i	$0.636363\pi$
$224$	−0.0975688	−0.00651909
$225$	−4.10136	−0.273424
$226$	−14.5545	−0.968148
$227$	6.46671	0.429211	0.214605	−	0.976701i	$-0.431153\pi$
$227$	6.46671	0.429211	0.214605	+	0.976701i	$0.431153\pi$
$228$	5.22279	0.345888
$229$	20.7066	1.36833	0.684165	−	0.729328i	$-0.260167\pi$
$229$	20.7066	1.36833	0.684165	+	0.729328i	$0.260167\pi$
$230$	14.0169	0.924247
$231$	1.17059	0.0770194
$232$	−7.73258	−0.507669
$233$	−17.3364	−1.13575	−0.567874	−	0.823115i	$-0.692234\pi$
$233$	−17.3364	−1.13575	−0.567874	+	0.823115i	$0.692234\pi$
$234$	−4.68322	−0.306151
$235$	−4.64066	−0.302723
$236$	3.53834	0.230326
$237$	25.1740	1.63523
$238$	−0.724099	−0.0469364
$239$	4.92237	0.318402	0.159201	−	0.987246i	$-0.449108\pi$
$239$	4.92237	0.318402	0.159201	+	0.987246i	$0.449108\pi$
$240$	−3.82457	−0.246875
$241$	14.1910	0.914121	0.457061	−	0.889436i	$-0.348902\pi$
$241$	14.1910	0.914121	0.457061	+	0.889436i	$0.348902\pi$
$242$	−17.8736	−1.14896
$243$	17.7959	1.14161
$244$	4.36328	0.279330
$245$	−11.9742	−0.765001
$246$	−6.75581	−0.430735
$247$	−5.51801	−0.351103
$248$	−6.36520	−0.404190
$249$	26.9625	1.70868
$250$	12.1033	0.765482
$251$	−29.4402	−1.85825	−0.929124	−	0.369769i	$-0.879437\pi$
$251$	−29.4402	−1.85825	−0.929124	+	0.369769i	$0.879437\pi$
$252$	0.193700	0.0122020
$253$	43.9708	2.76442
$254$	−9.99091	−0.626886
$255$	−28.3837	−1.77746
$256$	1.00000	0.0625000
$257$	19.7734	1.23343	0.616717	−	0.787185i	$-0.288462\pi$
$257$	19.7734	1.23343	0.616717	+	0.787185i	$0.288462\pi$
$258$	3.46943	0.215997
$259$	−1.13066	−0.0702559
$260$	4.04075	0.250597
$261$	15.3513	0.950220
$262$	14.3490	0.886486
$263$	−18.4720	−1.13903	−0.569516	−	0.821980i	$-0.692869\pi$
$263$	−18.4720	−1.13903	−0.569516	+	0.821980i	$0.692869\pi$
$264$	−11.9976	−0.738402
$265$	5.93555	0.364618
$266$	0.228228	0.0139936
$267$	−13.9165	−0.851678
$268$	11.2043	0.684410
$269$	−5.54937	−0.338351	−0.169175	−	0.985586i	$-0.554110\pi$
$269$	−5.54937	−0.338351	−0.169175	+	0.985586i	$0.554110\pi$
$270$	−3.88090	−0.236184
$271$	30.7369	1.86713	0.933566	−	0.358404i	$-0.116679\pi$
$271$	30.7369	1.86713	0.933566	+	0.358404i	$0.116679\pi$
$272$	7.42142	0.449990
$273$	−0.513902	−0.0311027
$274$	10.3387	0.624583
$275$	11.1009	0.669410
$276$	18.2708	1.09977
$277$	32.2483	1.93761	0.968807	−	0.247818i	$-0.0797135\pi$
$277$	32.2483	1.93761	0.968807	+	0.247818i	$0.0797135\pi$
$278$	−2.43867	−0.146261
$279$	12.6366	0.756536
$280$	−0.167128	−0.00998780
$281$	−9.70871	−0.579173	−0.289587	−	0.957152i	$-0.593518\pi$
$281$	−9.70871	−0.579173	−0.289587	+	0.957152i	$0.593518\pi$
$282$	−6.04903	−0.360215
$283$	23.0144	1.36807	0.684033	−	0.729451i	$-0.260224\pi$
$283$	23.0144	1.36807	0.684033	+	0.729451i	$0.260224\pi$
$284$	4.35302	0.258304
$285$	8.94624	0.529929
$286$	12.6758	0.749535
$287$	−0.295219	−0.0174262
$288$	−1.98527	−0.116983
$289$	38.0774	2.23985
$290$	−13.2453	−0.777792
$291$	−32.5793	−1.90983
$292$	−1.62922	−0.0953428
$293$	10.3677	0.605690	0.302845	−	0.953040i	$-0.402064\pi$
$293$	10.3677	0.605690	0.302845	+	0.953040i	$0.402064\pi$
$294$	−15.6081	−0.910286
$295$	6.06091	0.352880
$296$	11.5883	0.673559
$297$	−12.1743	−0.706427
$298$	15.7425	0.911938
$299$	−19.3036	−1.11636
$300$	4.61267	0.266313
$301$	0.151609	0.00873860
$302$	−20.0136	−1.15165
$303$	−6.49076	−0.372884
$304$	−2.33915	−0.134160
$305$	7.47396	0.427958
$306$	−14.7335	−0.842259
$307$	30.4986	1.74065	0.870324	−	0.492480i	$-0.163909\pi$
$307$	30.4986	1.74065	0.870324	+	0.492480i	$0.163909\pi$
$308$	−0.524278	−0.0298735
$309$	14.1157	0.803014
$310$	−10.9031	−0.619254
$311$	−7.81146	−0.442948	−0.221474	−	0.975166i	$-0.571087\pi$
$311$	−7.81146	−0.442948	−0.221474	+	0.975166i	$0.571087\pi$
$312$	5.26707	0.298189
$313$	−2.05364	−0.116078	−0.0580392	−	0.998314i	$-0.518485\pi$
$313$	−2.05364	−0.116078	−0.0580392	+	0.998314i	$0.518485\pi$
$314$	18.4201	1.03950
$315$	0.331794	0.0186945
$316$	−11.2748	−0.634257
$317$	−22.5669	−1.26748	−0.633741	−	0.773545i	$-0.718482\pi$
$317$	−22.5669	−1.26748	−0.633741	+	0.773545i	$0.718482\pi$
$318$	7.73691	0.433864
$319$	−41.5504	−2.32638
$320$	1.71292	0.0957553
$321$	−0.740888	−0.0413523
$322$	0.798408	0.0444935
$323$	−17.3598	−0.965926
$324$	−11.0145	−0.611917
$325$	−4.87341	−0.270328
$326$	6.72147	0.372268
$327$	−39.0104	−2.15728
$328$	3.02575	0.167069
$329$	−0.264334	−0.0145732
$330$	−20.5510	−1.13130
$331$	2.94240	0.161729	0.0808645	−	0.996725i	$-0.474232\pi$
$331$	2.94240	0.161729	0.0808645	+	0.996725i	$0.474232\pi$
$332$	−12.0758	−0.662745
$333$	−23.0060	−1.26072
$334$	−6.78204	−0.371097
$335$	19.1921	1.04857
$336$	−0.217849	−0.0118846
$337$	−10.1220	−0.551380	−0.275690	−	0.961247i	$-0.588906\pi$
$337$	−10.1220	−0.551380	−0.275690	+	0.961247i	$0.588906\pi$
$338$	7.43521	0.404422
$339$	−32.4968	−1.76498
$340$	12.7123	0.689422
$341$	−34.2029	−1.85219
$342$	4.64385	0.251110
$343$	−1.36503	−0.0737049
$344$	−1.55387	−0.0837789
$345$	31.2965	1.68495
$346$	−10.9929	−0.590983
$347$	31.1775	1.67370	0.836849	−	0.547434i	$-0.184395\pi$
$347$	31.1775	1.67370	0.836849	+	0.547434i	$0.184395\pi$
$348$	−17.2651	−0.925506
$349$	−13.6397	−0.730116	−0.365058	−	0.930985i	$-0.618951\pi$
$349$	−13.6397	−0.730116	−0.365058	+	0.930985i	$0.618951\pi$
$350$	0.201567	0.0107742
$351$	5.34465	0.285276
$352$	5.37342	0.286404
$353$	−2.81644	−0.149904	−0.0749521	−	0.997187i	$-0.523880\pi$
$353$	−2.81644	−0.149904	−0.0749521	+	0.997187i	$0.523880\pi$
$354$	7.90031	0.419897
$355$	7.45639	0.395744
$356$	6.23285	0.330341
$357$	−1.61675	−0.0855674
$358$	13.5524	0.716264
$359$	19.6549	1.03734	0.518672	−	0.854973i	$-0.326427\pi$
$359$	19.6549	1.03734	0.518672	+	0.854973i	$0.326427\pi$
$360$	−3.40062	−0.179228
$361$	−13.5284	−0.712020
$362$	−0.988318	−0.0519448
$363$	−39.9077	−2.09461
$364$	0.230163	0.0120638
$365$	−2.79073	−0.146073
$366$	9.74220	0.509233
$367$	34.7192	1.81233	0.906164	−	0.422927i	$-0.138997\pi$
$367$	34.7192	1.81233	0.906164	+	0.422927i	$0.138997\pi$
$368$	−8.18303	−0.426570
$369$	−6.00693	−0.312708
$370$	19.8499	1.03195
$371$	0.338092	0.0175528
$372$	−14.2120	−0.736860
$373$	28.7126	1.48668	0.743341	−	0.668912i	$-0.233240\pi$
$373$	28.7126	1.48668	0.743341	+	0.668912i	$0.233240\pi$
$374$	39.8784	2.06206
$375$	27.0240	1.39551
$376$	2.70920	0.139717
$377$	18.2410	0.939460
$378$	−0.221058	−0.0113700
$379$	−19.0408	−0.978062	−0.489031	−	0.872266i	$-0.662649\pi$
$379$	−19.0408	−0.978062	−0.489031	+	0.872266i	$0.662649\pi$
$380$	−4.00679	−0.205544
$381$	−22.3074	−1.14284
$382$	−13.8078	−0.706469
$383$	15.0573	0.769390	0.384695	−	0.923044i	$-0.374307\pi$
$383$	15.0573	0.769390	0.384695	+	0.923044i	$0.374307\pi$
$384$	2.23277	0.113941
$385$	−0.898048	−0.0457688
$386$	7.21519	0.367244
$387$	3.08485	0.156812
$388$	14.5914	0.740766
$389$	15.1335	0.767299	0.383650	−	0.923479i	$-0.374667\pi$
$389$	15.1335	0.767299	0.383650	+	0.923479i	$0.374667\pi$
$390$	9.02208	0.456851
$391$	−60.7296	−3.07123
$392$	6.99048	0.353073
$393$	32.0381	1.61611
$394$	4.20803	0.211998
$395$	−19.3129	−0.971736
$396$	−10.6677	−0.536071
$397$	23.3975	1.17429	0.587143	−	0.809483i	$-0.300253\pi$
$397$	23.3975	1.17429	0.587143	+	0.809483i	$0.300253\pi$
$398$	26.4490	1.32577
$399$	0.509581	0.0255110
$400$	−2.06589	−0.103295
$401$	18.0761	0.902675	0.451338	−	0.892353i	$-0.350947\pi$
$401$	18.0761	0.902675	0.451338	+	0.892353i	$0.350947\pi$
$402$	25.0166	1.24771
$403$	15.0154	0.747970
$404$	2.90704	0.144631
$405$	−18.8670	−0.937510
$406$	−0.754459	−0.0374432
$407$	62.2690	3.08656
$408$	16.5703	0.820354
$409$	−5.12885	−0.253605	−0.126803	−	0.991928i	$-0.540471\pi$
$409$	−5.12885	−0.253605	−0.126803	+	0.991928i	$0.540471\pi$
$410$	5.18288	0.255964
$411$	23.0839	1.13865
$412$	−6.32205	−0.311465
$413$	0.345232	0.0169877
$414$	16.2455	0.798424
$415$	−20.6849	−1.01538
$416$	−2.35898	−0.115659
$417$	−5.44498	−0.266642
$418$	−12.5692	−0.614781
$419$	−0.200133	−0.00977713	−0.00488856	−	0.999988i	$-0.501556\pi$
$419$	−0.200133	−0.00977713	−0.00488856	+	0.999988i	$0.501556\pi$
$420$	−0.373158	−0.0182083
$421$	−24.3589	−1.18718	−0.593590	−	0.804768i	$-0.702290\pi$
$421$	−24.3589	−1.18718	−0.593590	+	0.804768i	$0.702290\pi$
$422$	−19.7483	−0.961334
$423$	−5.37850	−0.261512
$424$	−3.46516	−0.168283
$425$	−15.3319	−0.743704
$426$	9.71930	0.470901
$427$	0.425720	0.0206020
$428$	0.331824	0.0160393
$429$	28.3022	1.36644
$430$	−2.66166	−0.128356
$431$	16.4807	0.793846	0.396923	−	0.917852i	$-0.370078\pi$
$431$	16.4807	0.793846	0.396923	+	0.917852i	$0.370078\pi$
$432$	2.26566	0.109007
$433$	12.5160	0.601480	0.300740	−	0.953706i	$-0.402766\pi$
$433$	12.5160	0.601480	0.300740	+	0.953706i	$0.402766\pi$
$434$	−0.621045	−0.0298111
$435$	−29.5738	−1.41795
$436$	17.4718	0.836745
$437$	19.1413	0.915654
$438$	−3.63767	−0.173815
$439$	−25.3517	−1.20997	−0.604985	−	0.796237i	$-0.706821\pi$
$439$	−25.3517	−1.20997	−0.604985	+	0.796237i	$0.706821\pi$
$440$	9.20425	0.438796
$441$	−13.8780	−0.660857
$442$	−17.5070	−0.832722
$443$	10.9772	0.521541	0.260771	−	0.965401i	$-0.416023\pi$
$443$	10.9772	0.521541	0.260771	+	0.965401i	$0.416023\pi$
$444$	25.8741	1.22793
$445$	10.6764	0.506110
$446$	12.4069	0.587484
$447$	35.1494	1.66251
$448$	0.0975688	0.00460969
$449$	3.14705	0.148518	0.0742592	−	0.997239i	$-0.476341\pi$
$449$	3.14705	0.148518	0.0742592	+	0.997239i	$0.476341\pi$
$450$	4.10136	0.193340
$451$	16.2586	0.765588
$452$	14.5545	0.684584
$453$	−44.6858	−2.09952
$454$	−6.46671	−0.303498
$455$	0.394252	0.0184828
$456$	−5.22279	−0.244580
$457$	−3.15917	−0.147780	−0.0738899	−	0.997266i	$-0.523541\pi$
$457$	−3.15917	−0.147780	−0.0738899	+	0.997266i	$0.523541\pi$
$458$	−20.7066	−0.967555
$459$	16.8144	0.784830
$460$	−14.0169	−0.653541
$461$	17.3194	0.806647	0.403324	−	0.915057i	$-0.367855\pi$
$461$	17.3194	0.806647	0.403324	+	0.915057i	$0.367855\pi$
$462$	−1.17059	−0.0544609
$463$	−12.9978	−0.604059	−0.302030	−	0.953299i	$-0.597664\pi$
$463$	−12.9978	−0.604059	−0.302030	+	0.953299i	$0.597664\pi$
$464$	7.73258	0.358976
$465$	−24.3441	−1.12893
$466$	17.3364	0.803095
$467$	22.5355	1.04282	0.521409	−	0.853307i	$-0.325407\pi$
$467$	22.5355	1.04282	0.521409	+	0.853307i	$0.325407\pi$
$468$	4.68322	0.216482
$469$	1.09319	0.0504787
$470$	4.64066	0.214058
$471$	41.1278	1.89507
$472$	−3.53834	−0.162865
$473$	−8.34958	−0.383914
$474$	−25.1740	−1.15628
$475$	4.83244	0.221727
$476$	0.724099	0.0331890
$477$	6.87928	0.314981
$478$	−4.92237	−0.225144
$479$	−19.6484	−0.897760	−0.448880	−	0.893592i	$-0.648177\pi$
$479$	−19.6484	−0.897760	−0.448880	+	0.893592i	$0.648177\pi$
$480$	3.82457	0.174567
$481$	−27.3367	−1.24645
$482$	−14.1910	−0.646381
$483$	1.78266	0.0811140
$484$	17.8736	0.812437
$485$	24.9939	1.13492
$486$	−17.7959	−0.807239
$487$	9.30264	0.421543	0.210771	−	0.977535i	$-0.432402\pi$
$487$	9.30264	0.421543	0.210771	+	0.977535i	$0.432402\pi$
$488$	−4.36328	−0.197516
$489$	15.0075	0.678663
$490$	11.9742	0.540937
$491$	32.4114	1.46271	0.731354	−	0.681998i	$-0.238889\pi$
$491$	32.4114	1.46271	0.731354	+	0.681998i	$0.238889\pi$
$492$	6.75581	0.304575
$493$	57.3867	2.58457
$494$	5.51801	0.248267
$495$	−18.2729	−0.821307
$496$	6.36520	0.285806
$497$	0.424719	0.0190512
$498$	−26.9625	−1.20822
$499$	21.5774	0.965939	0.482969	−	0.875637i	$-0.339558\pi$
$499$	21.5774	0.965939	0.482969	+	0.875637i	$0.339558\pi$
$500$	−12.1033	−0.541278
$501$	−15.1427	−0.676528
$502$	29.4402	1.31398
$503$	−1.66802	−0.0743735	−0.0371868	−	0.999308i	$-0.511840\pi$
$503$	−1.66802	−0.0743735	−0.0371868	+	0.999308i	$0.511840\pi$
$504$	−0.193700	−0.00862810
$505$	4.97954	0.221587
$506$	−43.9708	−1.95474
$507$	16.6011	0.737281
$508$	9.99091	0.443275
$509$	−28.1824	−1.24916	−0.624582	−	0.780959i	$-0.714731\pi$
$509$	−28.1824	−1.24916	−0.624582	+	0.780959i	$0.714731\pi$
$510$	28.3837	1.25685
$511$	−0.158961	−0.00703202
$512$	−1.00000	−0.0441942
$513$	−5.29972	−0.233988
$514$	−19.7734	−0.872169
$515$	−10.8292	−0.477191
$516$	−3.46943	−0.152733
$517$	14.5577	0.640246
$518$	1.13066	0.0496784
$519$	−24.5447	−1.07739
$520$	−4.04075	−0.177199
$521$	−11.7429	−0.514465	−0.257232	−	0.966350i	$-0.582811\pi$
$521$	−11.7429	−0.514465	−0.257232	+	0.966350i	$0.582811\pi$
$522$	−15.3513	−0.671907
$523$	13.2453	0.579178	0.289589	−	0.957151i	$-0.406481\pi$
$523$	13.2453	0.579178	0.289589	+	0.957151i	$0.406481\pi$
$524$	−14.3490	−0.626840
$525$	0.450053	0.0196419
$526$	18.4720	0.805417
$527$	47.2388	2.05775
$528$	11.9976	0.522129
$529$	43.9619	1.91139
$530$	−5.93555	−0.257824
$531$	7.02456	0.304840
$532$	−0.228228	−0.00989495
$533$	−7.13769	−0.309168
$534$	13.9165	0.602228
$535$	0.568389	0.0245736
$536$	−11.2043	−0.483951
$537$	30.2593	1.30579
$538$	5.54937	0.239250
$539$	37.5628	1.61794
$540$	3.88090	0.167007
$541$	34.1334	1.46751	0.733754	−	0.679415i	$-0.237766\pi$
$541$	34.1334	1.46751	0.733754	+	0.679415i	$0.237766\pi$
$542$	−30.7369	−1.32026
$543$	−2.20669	−0.0946981
$544$	−7.42142	−0.318191
$545$	29.9278	1.28197
$546$	0.513902	0.0219930
$547$	−1.22828	−0.0525174	−0.0262587	−	0.999655i	$-0.508359\pi$
$547$	−1.22828	−0.0525174	−0.0262587	+	0.999655i	$0.508359\pi$
$548$	−10.3387	−0.441647
$549$	8.66228	0.369697
$550$	−11.1009	−0.473344
$551$	−18.0877	−0.770561
$552$	−18.2708	−0.777658
$553$	−1.10007	−0.0467797
$554$	−32.2483	−1.37010
$555$	44.3204	1.88130
$556$	2.43867	0.103422
$557$	33.2563	1.40911	0.704557	−	0.709648i	$-0.251146\pi$
$557$	33.2563	1.40911	0.704557	+	0.709648i	$0.251146\pi$
$558$	−12.6366	−0.534951
$559$	3.66554	0.155036
$560$	0.167128	0.00706244
$561$	89.0393	3.75924
$562$	9.70871	0.409537
$563$	−13.2120	−0.556820	−0.278410	−	0.960462i	$-0.589807\pi$
$563$	−13.2120	−0.556820	−0.278410	+	0.960462i	$0.589807\pi$
$564$	6.04903	0.254710
$565$	24.9307	1.04884
$566$	−23.0144	−0.967369
$567$	−1.07467	−0.0451320
$568$	−4.35302	−0.182649
$569$	−13.0996	−0.549162	−0.274581	−	0.961564i	$-0.588539\pi$
$569$	−13.0996	−0.549162	−0.274581	+	0.961564i	$0.588539\pi$
$570$	−8.94624	−0.374717
$571$	9.94312	0.416107	0.208053	−	0.978117i	$-0.433287\pi$
$571$	9.94312	0.416107	0.208053	+	0.978117i	$0.433287\pi$
$572$	−12.6758	−0.530002
$573$	−30.8297	−1.28793
$574$	0.295219	0.0123222
$575$	16.9053	0.704998
$576$	1.98527	0.0827196
$577$	11.6932	0.486793	0.243396	−	0.969927i	$-0.421738\pi$
$577$	11.6932	0.486793	0.243396	+	0.969927i	$0.421738\pi$
$578$	−38.0774	−1.58381
$579$	16.1099	0.669504
$580$	13.2453	0.549982
$581$	−1.17822	−0.0488808
$582$	32.5793	1.35045
$583$	−18.6198	−0.771151
$584$	1.62922	0.0674176
$585$	8.02199	0.331668
$586$	−10.3677	−0.428287
$587$	29.0588	1.19938	0.599692	−	0.800231i	$-0.295290\pi$
$587$	29.0588	1.19938	0.599692	+	0.800231i	$0.295290\pi$
$588$	15.6081	0.643669
$589$	−14.8892	−0.613497
$590$	−6.06091	−0.249524
$591$	9.39558	0.386482
$592$	−11.5883	−0.476278
$593$	10.8259	0.444565	0.222283	−	0.974982i	$-0.428649\pi$
$593$	10.8259	0.444565	0.222283	+	0.974982i	$0.428649\pi$
$594$	12.1743	0.499519
$595$	1.24033	0.0508484
$596$	−15.7425	−0.644837
$597$	59.0546	2.41695
$598$	19.3036	0.789383
$599$	39.2923	1.60544	0.802720	−	0.596356i	$-0.203385\pi$
$599$	39.2923	1.60544	0.802720	+	0.596356i	$0.203385\pi$
$600$	−4.61267	−0.188311
$601$	−9.18160	−0.374525	−0.187263	−	0.982310i	$-0.559962\pi$
$601$	−9.18160	−0.374525	−0.187263	+	0.982310i	$0.559962\pi$
$602$	−0.151609	−0.00617912
$603$	22.2435	0.905825
$604$	20.0136	0.814343
$605$	30.6161	1.24472
$606$	6.49076	0.263669
$607$	32.8302	1.33254	0.666268	−	0.745712i	$-0.267890\pi$
$607$	32.8302	1.33254	0.666268	+	0.745712i	$0.267890\pi$
$608$	2.33915	0.0948651
$609$	−1.68453	−0.0682608
$610$	−7.47396	−0.302612
$611$	−6.39096	−0.258551
$612$	14.7335	0.595567
$613$	39.4038	1.59151	0.795753	−	0.605622i	$-0.207075\pi$
$613$	39.4038	1.59151	0.795753	+	0.605622i	$0.207075\pi$
$614$	−30.4986	−1.23082
$615$	11.5722	0.466635
$616$	0.524278	0.0211238
$617$	−24.4997	−0.986322	−0.493161	−	0.869938i	$-0.664159\pi$
$617$	−24.4997	−0.986322	−0.493161	+	0.869938i	$0.664159\pi$
$618$	−14.1157	−0.567817
$619$	−1.35987	−0.0546580	−0.0273290	−	0.999626i	$-0.508700\pi$
$619$	−1.35987	−0.0546580	−0.0273290	+	0.999626i	$0.508700\pi$
$620$	10.9031	0.437879
$621$	−18.5400	−0.743983
$622$	7.81146	0.313211
$623$	0.608132	0.0243643
$624$	−5.26707	−0.210851
$625$	−10.4026	−0.416105
$626$	2.05364	0.0820798
$627$	−28.0642	−1.12078
$628$	−18.4201	−0.735040
$629$	−86.0019	−3.42912
$630$	−0.331794	−0.0132190
$631$	25.9144	1.03164	0.515819	−	0.856698i	$-0.327488\pi$
$631$	25.9144	1.03164	0.515819	+	0.856698i	$0.327488\pi$
$632$	11.2748	0.448487
$633$	−44.0935	−1.75256
$634$	22.5669	0.896246
$635$	17.1137	0.679135
$636$	−7.73691	−0.306789
$637$	−16.4904	−0.653374
$638$	41.5504	1.64500
$639$	8.64192	0.341869
$640$	−1.71292	−0.0677092
$641$	43.9871	1.73739	0.868693	−	0.495351i	$-0.164960\pi$
$641$	43.9871	1.73739	0.868693	+	0.495351i	$0.164960\pi$
$642$	0.740888	0.0292405
$643$	8.42262	0.332156	0.166078	−	0.986113i	$-0.446890\pi$
$643$	8.42262	0.332156	0.166078	+	0.986113i	$0.446890\pi$
$644$	−0.798408	−0.0314617
$645$	−5.94287	−0.234000
$646$	17.3598	0.683013
$647$	6.56737	0.258190	0.129095	−	0.991632i	$-0.458793\pi$
$647$	6.56737	0.258190	0.129095	+	0.991632i	$0.458793\pi$
$648$	11.0145	0.432691
$649$	−19.0130	−0.746325
$650$	4.87341	0.191151
$651$	−1.38665	−0.0543472
$652$	−6.72147	−0.263233
$653$	46.1303	1.80522	0.902609	−	0.430461i	$-0.141649\pi$
$653$	46.1303	1.80522	0.902609	+	0.430461i	$0.141649\pi$
$654$	39.0104	1.52543
$655$	−24.5788	−0.960373
$656$	−3.02575	−0.118136
$657$	−3.23444	−0.126187
$658$	0.264334	0.0103048
$659$	24.1380	0.940283	0.470141	−	0.882591i	$-0.344203\pi$
$659$	24.1380	0.940283	0.470141	+	0.882591i	$0.344203\pi$
$660$	20.5510	0.799946
$661$	35.4696	1.37961	0.689804	−	0.723996i	$-0.257697\pi$
$661$	35.4696	1.37961	0.689804	+	0.723996i	$0.257697\pi$
$662$	−2.94240	−0.114360
$663$	−39.0891	−1.51810
$664$	12.0758	0.468632
$665$	−0.390937	−0.0151599
$666$	23.0060	0.891464
$667$	−63.2759	−2.45005
$668$	6.78204	0.262405
$669$	27.7018	1.07101
$670$	−19.1921	−0.741454
$671$	−23.4457	−0.905112
$672$	0.217849	0.00840370
$673$	26.2595	1.01223	0.506114	−	0.862467i	$-0.331081\pi$
$673$	26.2595	1.01223	0.506114	+	0.862467i	$0.331081\pi$
$674$	10.1220	0.389884
$675$	−4.68061	−0.180157
$676$	−7.43521	−0.285969
$677$	−21.0540	−0.809170	−0.404585	−	0.914500i	$-0.632584\pi$
$677$	−21.0540	−0.809170	−0.404585	+	0.914500i	$0.632584\pi$
$678$	32.4968	1.24803
$679$	1.42367	0.0546353
$680$	−12.7123	−0.487495
$681$	−14.4387	−0.553292
$682$	34.2029	1.30970
$683$	10.1854	0.389735	0.194868	−	0.980830i	$-0.437572\pi$
$683$	10.1854	0.389735	0.194868	+	0.980830i	$0.437572\pi$
$684$	−4.64385	−0.177562
$685$	−17.7094	−0.676641
$686$	1.36503	0.0521173
$687$	−46.2331	−1.76390
$688$	1.55387	0.0592406
$689$	8.17425	0.311414
$690$	−31.2965	−1.19144
$691$	0.893066	0.0339738	0.0169869	−	0.999856i	$-0.494593\pi$
$691$	0.893066	0.0339738	0.0169869	+	0.999856i	$0.494593\pi$
$692$	10.9929	0.417888
$693$	−1.04083	−0.0395380
$694$	−31.1775	−1.18348
$695$	4.17725	0.158452
$696$	17.2651	0.654432
$697$	−22.4553	−0.850557
$698$	13.6397	0.516270
$699$	38.7083	1.46408
$700$	−0.201567	−0.00761851
$701$	36.2758	1.37012	0.685059	−	0.728487i	$-0.259776\pi$
$701$	36.2758	1.37012	0.685059	+	0.728487i	$0.259776\pi$
$702$	−5.34465	−0.201721
$703$	27.1069	1.02236
$704$	−5.37342	−0.202518
$705$	10.3615	0.390238
$706$	2.81644	0.105998
$707$	0.283637	0.0106673
$708$	−7.90031	−0.296912
$709$	−39.8268	−1.49573	−0.747863	−	0.663854i	$-0.768920\pi$
$709$	−39.8268	−1.49573	−0.747863	+	0.663854i	$0.768920\pi$
$710$	−7.45639	−0.279833
$711$	−22.3835	−0.839447
$712$	−6.23285	−0.233586
$713$	−52.0866	−1.95066
$714$	1.61675	0.0605053
$715$	−21.7127	−0.812008
$716$	−13.5524	−0.506475
$717$	−10.9905	−0.410449
$718$	−19.6549	−0.733513
$719$	−29.9582	−1.11725	−0.558626	−	0.829419i	$-0.688671\pi$
$719$	−29.9582	−1.11725	−0.558626	+	0.829419i	$0.688671\pi$
$720$	3.40062	0.126733
$721$	−0.616835	−0.0229721
$722$	13.5284	0.503474
$723$	−31.6852	−1.17839
$724$	0.988318	0.0367305
$725$	−15.9747	−0.593285
$726$	39.9077	1.48111
$727$	48.1014	1.78398	0.891991	−	0.452054i	$-0.149308\pi$
$727$	48.1014	1.78398	0.891991	+	0.452054i	$0.149308\pi$
$728$	−0.230163	−0.00853041
$729$	−6.69067	−0.247803
$730$	2.79073	0.103289
$731$	11.5319	0.426523
$732$	−9.74220	−0.360082
$733$	24.1199	0.890887	0.445444	−	0.895310i	$-0.353046\pi$
$733$	24.1199	0.890887	0.445444	+	0.895310i	$0.353046\pi$
$734$	−34.7192	−1.28151
$735$	26.7356	0.986156
$736$	8.18303	0.301630
$737$	−60.2052	−2.21769
$738$	6.00693	0.221118
$739$	11.0563	0.406711	0.203356	−	0.979105i	$-0.434815\pi$
$739$	11.0563	0.406711	0.203356	+	0.979105i	$0.434815\pi$
$740$	−19.8499	−0.729698
$741$	12.3205	0.452604
$742$	−0.338092	−0.0124117
$743$	−14.1119	−0.517714	−0.258857	−	0.965916i	$-0.583346\pi$
$743$	−14.1119	−0.517714	−0.258857	+	0.965916i	$0.583346\pi$
$744$	14.2120	0.521038
$745$	−26.9657	−0.987946
$746$	−28.7126	−1.05124
$747$	−23.9737	−0.877152
$748$	−39.8784	−1.45810
$749$	0.0323757	0.00118298
$750$	−27.0240	−0.986776
$751$	−27.6510	−1.00900	−0.504500	−	0.863412i	$-0.668323\pi$
$751$	−27.6510	−1.00900	−0.504500	+	0.863412i	$0.668323\pi$
$752$	−2.70920	−0.0987945
$753$	65.7332	2.39545
$754$	−18.2410	−0.664299
$755$	34.2818	1.24764
$756$	0.221058	0.00803980
$757$	40.8392	1.48432	0.742162	−	0.670220i	$-0.233800\pi$
$757$	40.8392	1.48432	0.742162	+	0.670220i	$0.233800\pi$
$758$	19.0408	0.691594
$759$	−98.1768	−3.56359
$760$	4.00679	0.145341
$761$	−52.7852	−1.91346	−0.956731	−	0.290975i	$-0.906020\pi$
$761$	−52.7852	−1.91346	−0.956731	+	0.290975i	$0.906020\pi$
$762$	22.3074	0.808113
$763$	1.70470	0.0617142
$764$	13.8078	0.499549
$765$	25.2374	0.912460
$766$	−15.0573	−0.544041
$767$	8.34688	0.301388
$768$	−2.23277	−0.0805682
$769$	−35.1173	−1.26636	−0.633181	−	0.774003i	$-0.718251\pi$
$769$	−35.1173	−1.26636	−0.633181	+	0.774003i	$0.718251\pi$
$770$	0.898048	0.0323634
$771$	−44.1496	−1.59001
$772$	−7.21519	−0.259680
$773$	−48.0415	−1.72793	−0.863965	−	0.503551i	$-0.832027\pi$
$773$	−48.0415	−1.72793	−0.863965	+	0.503551i	$0.832027\pi$
$774$	−3.08485	−0.110882
$775$	−13.1498	−0.472356
$776$	−14.5914	−0.523801
$777$	2.52451	0.0905662
$778$	−15.1335	−0.542562
$779$	7.07768	0.253584
$780$	−9.02208	−0.323042
$781$	−23.3906	−0.836981
$782$	60.7296	2.17169
$783$	17.5194	0.626093
$784$	−6.99048	−0.249660
$785$	−31.5522	−1.12614
$786$	−32.0381	−1.14276
$787$	37.4951	1.33656	0.668278	−	0.743912i	$-0.267032\pi$
$787$	37.4951	1.33656	0.668278	+	0.743912i	$0.267032\pi$
$788$	−4.20803	−0.149905
$789$	41.2438	1.46832
$790$	19.3129	0.687121
$791$	1.42006	0.0504916
$792$	10.6677	0.379060
$793$	10.2929	0.365511
$794$	−23.3975	−0.830346
$795$	−13.2527	−0.470026
$796$	−26.4490	−0.937461
$797$	18.6307	0.659933	0.329966	−	0.943993i	$-0.392963\pi$
$797$	18.6307	0.659933	0.329966	+	0.943993i	$0.392963\pi$
$798$	−0.509581	−0.0180390
$799$	−20.1061	−0.711304
$800$	2.06589	0.0730404
$801$	12.3739	0.437210
$802$	−18.0761	−0.638288
$803$	8.75447	0.308939
$804$	−25.0166	−0.882267
$805$	−1.36761	−0.0482020
$806$	−15.0154	−0.528894
$807$	12.3905	0.436165
$808$	−2.90704	−0.102269
$809$	−36.9013	−1.29738	−0.648690	−	0.761053i	$-0.724683\pi$
$809$	−36.9013	−1.29738	−0.648690	+	0.761053i	$0.724683\pi$
$810$	18.8670	0.662919
$811$	5.66167	0.198808	0.0994040	−	0.995047i	$-0.468306\pi$
$811$	5.66167	0.198808	0.0994040	+	0.995047i	$0.468306\pi$
$812$	0.754459	0.0264763
$813$	−68.6285	−2.40690
$814$	−62.2690	−2.18253
$815$	−11.5134	−0.403295
$816$	−16.5703	−0.580078
$817$	−3.63473	−0.127163
$818$	5.12885	0.179326
$819$	0.456936	0.0159666
$820$	−5.18288	−0.180994
$821$	−5.59199	−0.195162	−0.0975809	−	0.995228i	$-0.531110\pi$
$821$	−5.59199	−0.195162	−0.0975809	+	0.995228i	$0.531110\pi$
$822$	−23.0839	−0.805145
$823$	−13.9344	−0.485721	−0.242861	−	0.970061i	$-0.578086\pi$
$823$	−13.9344	−0.485721	−0.242861	+	0.970061i	$0.578086\pi$
$824$	6.32205	0.220239
$825$	−24.7858	−0.862931
$826$	−0.345232	−0.0120121
$827$	34.4514	1.19799	0.598997	−	0.800751i	$-0.295566\pi$
$827$	34.4514	1.19799	0.598997	+	0.800751i	$0.295566\pi$
$828$	−16.2455	−0.564571
$829$	−20.3852	−0.708009	−0.354004	−	0.935244i	$-0.615180\pi$
$829$	−20.3852	−0.708009	−0.354004	+	0.935244i	$0.615180\pi$
$830$	20.6849	0.717983
$831$	−72.0031	−2.49776
$832$	2.35898	0.0817830
$833$	−51.8793	−1.79751
$834$	5.44498	0.188544
$835$	11.6171	0.402027
$836$	12.5692	0.434716
$837$	14.4214	0.498476
$838$	0.200133	0.00691347
$839$	−35.5278	−1.22656	−0.613278	−	0.789867i	$-0.710149\pi$
$839$	−35.5278	−1.22656	−0.613278	+	0.789867i	$0.710149\pi$
$840$	0.373158	0.0128752
$841$	30.7928	1.06182
$842$	24.3589	0.839463
$843$	21.6773	0.746607
$844$	19.7483	0.679766
$845$	−12.7359	−0.438130
$846$	5.37850	0.184917
$847$	1.74391	0.0599214
$848$	3.46516	0.118994
$849$	−51.3860	−1.76356
$850$	15.3319	0.525878
$851$	94.8277	3.25065
$852$	−9.71930	−0.332978
$853$	24.1357	0.826390	0.413195	−	0.910643i	$-0.364413\pi$
$853$	24.1357	0.826390	0.413195	+	0.910643i	$0.364413\pi$
$854$	−0.425720	−0.0145678
$855$	−7.95455	−0.272040
$856$	−0.331824	−0.0113415
$857$	37.2349	1.27192	0.635961	−	0.771722i	$-0.280604\pi$
$857$	37.2349	1.27192	0.635961	+	0.771722i	$0.280604\pi$
$858$	−28.3022	−0.966220
$859$	−19.2666	−0.657367	−0.328683	−	0.944440i	$-0.606605\pi$
$859$	−19.2666	−0.657367	−0.328683	+	0.944440i	$0.606605\pi$
$860$	2.66166	0.0907617
$861$	0.659156	0.0224640
$862$	−16.4807	−0.561334
$863$	−27.1265	−0.923398	−0.461699	−	0.887037i	$-0.652760\pi$
$863$	−27.1265	−0.923398	−0.461699	+	0.887037i	$0.652760\pi$
$864$	−2.26566	−0.0770793
$865$	18.8300	0.640240
$866$	−12.5160	−0.425311
$867$	−85.0182	−2.88737
$868$	0.621045	0.0210796
$869$	60.5842	2.05518
$870$	29.5738	1.00265
$871$	26.4307	0.895569
$872$	−17.4718	−0.591668
$873$	28.9679	0.980414
$874$	−19.1413	−0.647465
$875$	−1.18091	−0.0399220
$876$	3.63767	0.122906
$877$	−8.54719	−0.288618	−0.144309	−	0.989533i	$-0.546096\pi$
$877$	−8.54719	−0.288618	−0.144309	+	0.989533i	$0.546096\pi$
$878$	25.3517	0.855577
$879$	−23.1488	−0.780790
$880$	−9.20425	−0.310275
$881$	−47.6192	−1.60433	−0.802166	−	0.597102i	$-0.796319\pi$
$881$	−47.6192	−1.60433	−0.802166	+	0.597102i	$0.796319\pi$
$882$	13.8780	0.467296
$883$	−5.18887	−0.174619	−0.0873096	−	0.996181i	$-0.527827\pi$
$883$	−5.18887	−0.174619	−0.0873096	+	0.996181i	$0.527827\pi$
$884$	17.5070	0.588824
$885$	−13.5326	−0.454894
$886$	−10.9772	−0.368785
$887$	−5.66440	−0.190192	−0.0950959	−	0.995468i	$-0.530316\pi$
$887$	−5.66440	−0.190192	−0.0950959	+	0.995468i	$0.530316\pi$
$888$	−25.8741	−0.868279
$889$	0.974802	0.0326938
$890$	−10.6764	−0.357874
$891$	59.1856	1.98279
$892$	−12.4069	−0.415414
$893$	6.33724	0.212068
$894$	−35.1494	−1.17557
$895$	−23.2141	−0.775963
$896$	−0.0975688	−0.00325955
$897$	43.1006	1.43909
$898$	−3.14705	−0.105018
$899$	49.2194	1.64156
$900$	−4.10136	−0.136712
$901$	25.7164	0.856738
$902$	−16.2586	−0.541353
$903$	−0.338508	−0.0112648
$904$	−14.5545	−0.484074
$905$	1.69291	0.0562743
$906$	44.6858	1.48459
$907$	−10.6393	−0.353272	−0.176636	−	0.984276i	$-0.556522\pi$
$907$	−10.6393	−0.353272	−0.176636	+	0.984276i	$0.556522\pi$
$908$	6.46671	0.214605
$909$	5.77126	0.191421
$910$	−0.394252	−0.0130693
$911$	−37.6874	−1.24864	−0.624319	−	0.781169i	$-0.714624\pi$
$911$	−37.6874	−1.24864	−0.624319	+	0.781169i	$0.714624\pi$
$912$	5.22279	0.172944
$913$	64.8883	2.14749
$914$	3.15917	0.104496
$915$	−16.6876	−0.551676
$916$	20.7066	0.684165
$917$	−1.40002	−0.0462327
$918$	−16.8144	−0.554958
$919$	−15.7601	−0.519876	−0.259938	−	0.965625i	$-0.583702\pi$
$919$	−15.7601	−0.519876	−0.259938	+	0.965625i	$0.583702\pi$
$920$	14.0169	0.462123
$921$	−68.0964	−2.24385
$922$	−17.3194	−0.570386
$923$	10.2687	0.337998
$924$	1.17059	0.0385097
$925$	23.9403	0.787152
$926$	12.9978	0.427134
$927$	−12.5510	−0.412228
$928$	−7.73258	−0.253834
$929$	−19.7287	−0.647278	−0.323639	−	0.946181i	$-0.604906\pi$
$929$	−19.7287	−0.647278	−0.323639	+	0.946181i	$0.604906\pi$
$930$	24.3441	0.798275
$931$	16.3518	0.535908
$932$	−17.3364	−0.567874
$933$	17.4412	0.571000
$934$	−22.5355	−0.737383
$935$	−68.3086	−2.23393
$936$	−4.68322	−0.153076
$937$	2.26797	0.0740912	0.0370456	−	0.999314i	$-0.488205\pi$
$937$	2.26797	0.0740912	0.0370456	+	0.999314i	$0.488205\pi$
$938$	−1.09319	−0.0356938
$939$	4.58530	0.149636
$940$	−4.64066	−0.151362
$941$	45.0306	1.46795	0.733977	−	0.679174i	$-0.237662\pi$
$941$	45.0306	1.46795	0.733977	+	0.679174i	$0.237662\pi$
$942$	−41.1278	−1.34002
$943$	24.7598	0.806290
$944$	3.53834	0.115163
$945$	0.378655	0.0123177
$946$	8.34958	0.271468
$947$	−32.7806	−1.06523	−0.532613	−	0.846359i	$-0.678790\pi$
$947$	−32.7806	−1.06523	−0.532613	+	0.846359i	$0.678790\pi$
$948$	25.1740	0.817615
$949$	−3.84330	−0.124759
$950$	−4.83244	−0.156785
$951$	50.3867	1.63390
$952$	−0.724099	−0.0234682
$953$	−38.8082	−1.25712	−0.628561	−	0.777761i	$-0.716356\pi$
$953$	−38.8082	−1.25712	−0.628561	+	0.777761i	$0.716356\pi$
$954$	−6.87928	−0.222725
$955$	23.6517	0.765352
$956$	4.92237	0.159201
$957$	92.7726	2.99891
$958$	19.6484	0.634812
$959$	−1.00873	−0.0325737
$960$	−3.82457	−0.123437
$961$	9.51574	0.306959
$962$	27.3367	0.881370
$963$	0.658761	0.0212283
$964$	14.1910	0.457061
$965$	−12.3591	−0.397852
$966$	−1.78266	−0.0573563
$967$	40.4229	1.29991	0.649956	−	0.759972i	$-0.274787\pi$
$967$	40.4229	1.29991	0.649956	+	0.759972i	$0.274787\pi$
$968$	−17.8736	−0.574480
$969$	38.7605	1.24517
$970$	−24.9939	−0.802507
$971$	−9.02478	−0.289619	−0.144809	−	0.989460i	$-0.546257\pi$
$971$	−9.02478	−0.289619	−0.144809	+	0.989460i	$0.546257\pi$
$972$	17.7959	0.570804
$973$	0.237938	0.00762793
$974$	−9.30264	−0.298076
$975$	10.8812	0.348477
$976$	4.36328	0.139665
$977$	41.4804	1.32708	0.663538	−	0.748143i	$-0.269054\pi$
$977$	41.4804	1.32708	0.663538	+	0.748143i	$0.269054\pi$
$978$	−15.0075	−0.479887
$979$	−33.4917	−1.07040
$980$	−11.9742	−0.382500
$981$	34.6861	1.10744
$982$	−32.4114	−1.03429
$983$	49.9297	1.59251	0.796256	−	0.604960i	$-0.206811\pi$
$983$	49.9297	1.59251	0.796256	+	0.604960i	$0.206811\pi$
$984$	−6.75581	−0.215367
$985$	−7.20804	−0.229667
$986$	−57.3867	−1.82757
$987$	0.590197	0.0187862
$988$	−5.51801	−0.175551
$989$	−12.7153	−0.404324
$990$	18.2729	0.580752
$991$	−58.0926	−1.84537	−0.922687	−	0.385550i	$-0.874012\pi$
$991$	−58.0926	−1.84537	−0.922687	+	0.385550i	$0.874012\pi$
$992$	−6.36520	−0.202095
$993$	−6.56971	−0.208483
$994$	−0.424719	−0.0134713
$995$	−45.3052	−1.43627
$996$	26.9625	0.854339
$997$	−42.9737	−1.36099	−0.680495	−	0.732753i	$-0.738235\pi$
$997$	−42.9737	−1.36099	−0.680495	+	0.732753i	$0.738235\pi$
$998$	−21.5774	−0.683022
$999$	−26.2553	−0.830679

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.d.1.11	✓	69

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.23277 q^{3} +1.00000 q^{4} +1.71292 q^{5} +2.23277 q^{6} +0.0975688 q^{7} -1.00000 q^{8} +1.98527 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.23277 q^{3} +1.00000 q^{4} +1.71292 q^{5} +2.23277 q^{6} +0.0975688 q^{7} -1.00000 q^{8} +1.98527 q^{9} -1.71292 q^{10} -5.37342 q^{11} -2.23277 q^{12} +2.35898 q^{13} -0.0975688 q^{14} -3.82457 q^{15} +1.00000 q^{16} +7.42142 q^{17} -1.98527 q^{18} -2.33915 q^{19} +1.71292 q^{20} -0.217849 q^{21} +5.37342 q^{22} -8.18303 q^{23} +2.23277 q^{24} -2.06589 q^{25} -2.35898 q^{26} +2.26566 q^{27} +0.0975688 q^{28} +7.73258 q^{29} +3.82457 q^{30} +6.36520 q^{31} -1.00000 q^{32} +11.9976 q^{33} -7.42142 q^{34} +0.167128 q^{35} +1.98527 q^{36} -11.5883 q^{37} +2.33915 q^{38} -5.26707 q^{39} -1.71292 q^{40} -3.02575 q^{41} +0.217849 q^{42} +1.55387 q^{43} -5.37342 q^{44} +3.40062 q^{45} +8.18303 q^{46} -2.70920 q^{47} -2.23277 q^{48} -6.99048 q^{49} +2.06589 q^{50} -16.5703 q^{51} +2.35898 q^{52} +3.46516 q^{53} -2.26566 q^{54} -9.20425 q^{55} -0.0975688 q^{56} +5.22279 q^{57} -7.73258 q^{58} +3.53834 q^{59} -3.82457 q^{60} +4.36328 q^{61} -6.36520 q^{62} +0.193700 q^{63} +1.00000 q^{64} +4.04075 q^{65} -11.9976 q^{66} +11.2043 q^{67} +7.42142 q^{68} +18.2708 q^{69} -0.167128 q^{70} +4.35302 q^{71} -1.98527 q^{72} -1.62922 q^{73} +11.5883 q^{74} +4.61267 q^{75} -2.33915 q^{76} -0.524278 q^{77} +5.26707 q^{78} -11.2748 q^{79} +1.71292 q^{80} -11.0145 q^{81} +3.02575 q^{82} -12.0758 q^{83} -0.217849 q^{84} +12.7123 q^{85} -1.55387 q^{86} -17.2651 q^{87} +5.37342 q^{88} +6.23285 q^{89} -3.40062 q^{90} +0.230163 q^{91} -8.18303 q^{92} -14.2120 q^{93} +2.70920 q^{94} -4.00679 q^{95} +2.23277 q^{96} +14.5914 q^{97} +6.99048 q^{98} -10.6677 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) \) 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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