LMFDB - Embedded newform 6038.2.a.d.1.8

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			1.8
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)

$f(q)$	$=$	$q-1.00000 q^{2} -2.63337 q^{3} +1.00000 q^{4} -0.783693 q^{5} +2.63337 q^{6} +4.82371 q^{7} -1.00000 q^{8} +3.93463 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.63337 q^{3} +1.00000 q^{4} -0.783693 q^{5} +2.63337 q^{6} +4.82371 q^{7} -1.00000 q^{8} +3.93463 q^{9} +0.783693 q^{10} +0.639247 q^{11} -2.63337 q^{12} -0.779883 q^{13} -4.82371 q^{14} +2.06375 q^{15} +1.00000 q^{16} -4.03702 q^{17} -3.93463 q^{18} +5.34960 q^{19} -0.783693 q^{20} -12.7026 q^{21} -0.639247 q^{22} +8.73612 q^{23} +2.63337 q^{24} -4.38583 q^{25} +0.779883 q^{26} -2.46121 q^{27} +4.82371 q^{28} -0.728405 q^{29} -2.06375 q^{30} +6.47433 q^{31} -1.00000 q^{32} -1.68337 q^{33} +4.03702 q^{34} -3.78031 q^{35} +3.93463 q^{36} -2.47595 q^{37} -5.34960 q^{38} +2.05372 q^{39} +0.783693 q^{40} -8.20492 q^{41} +12.7026 q^{42} +9.19075 q^{43} +0.639247 q^{44} -3.08354 q^{45} -8.73612 q^{46} -5.64658 q^{47} -2.63337 q^{48} +16.2682 q^{49} +4.38583 q^{50} +10.6310 q^{51} -0.779883 q^{52} +4.42478 q^{53} +2.46121 q^{54} -0.500973 q^{55} -4.82371 q^{56} -14.0875 q^{57} +0.728405 q^{58} +0.747616 q^{59} +2.06375 q^{60} +11.2755 q^{61} -6.47433 q^{62} +18.9795 q^{63} +1.00000 q^{64} +0.611188 q^{65} +1.68337 q^{66} +14.8982 q^{67} -4.03702 q^{68} -23.0054 q^{69} +3.78031 q^{70} +3.45258 q^{71} -3.93463 q^{72} -1.13343 q^{73} +2.47595 q^{74} +11.5495 q^{75} +5.34960 q^{76} +3.08354 q^{77} -2.05372 q^{78} -6.05867 q^{79} -0.783693 q^{80} -5.32260 q^{81} +8.20492 q^{82} +7.77691 q^{83} -12.7026 q^{84} +3.16378 q^{85} -9.19075 q^{86} +1.91816 q^{87} -0.639247 q^{88} -2.69832 q^{89} +3.08354 q^{90} -3.76193 q^{91} +8.73612 q^{92} -17.0493 q^{93} +5.64658 q^{94} -4.19244 q^{95} +2.63337 q^{96} -8.01215 q^{97} -16.2682 q^{98} +2.51520 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.63337	−1.52038	−0.760188	−	0.649703i	$-0.774893\pi$
$3$	−2.63337	−1.52038	−0.760188	+	0.649703i	$0.774893\pi$
$4$	1.00000	0.500000
$5$	−0.783693	−0.350478	−0.175239	−	0.984526i	$-0.556070\pi$
$5$	−0.783693	−0.350478	−0.175239	+	0.984526i	$0.556070\pi$
$6$	2.63337	1.07507
$7$	4.82371	1.82319	0.911595	−	0.411089i	$-0.134851\pi$
$7$	4.82371	1.82319	0.911595	+	0.411089i	$0.134851\pi$
$8$	−1.00000	−0.353553
$9$	3.93463	1.31154
$10$	0.783693	0.247825
$11$	0.639247	0.192740	0.0963701	−	0.995346i	$-0.469277\pi$
$11$	0.639247	0.192740	0.0963701	+	0.995346i	$0.469277\pi$
$12$	−2.63337	−0.760188
$13$	−0.779883	−0.216301	−0.108150	−	0.994135i	$-0.534493\pi$
$13$	−0.779883	−0.216301	−0.108150	+	0.994135i	$0.534493\pi$
$14$	−4.82371	−1.28919
$15$	2.06375	0.532858
$16$	1.00000	0.250000
$17$	−4.03702	−0.979121	−0.489561	−	0.871969i	$-0.662843\pi$
$17$	−4.03702	−0.979121	−0.489561	+	0.871969i	$0.662843\pi$
$18$	−3.93463	−0.927400
$19$	5.34960	1.22728	0.613641	−	0.789585i	$-0.289704\pi$
$19$	5.34960	1.22728	0.613641	+	0.789585i	$0.289704\pi$
$20$	−0.783693	−0.175239
$21$	−12.7026	−2.77193
$22$	−0.639247	−0.136288
$23$	8.73612	1.82161	0.910804	−	0.412840i	$-0.135463\pi$
$23$	8.73612	1.82161	0.910804	+	0.412840i	$0.135463\pi$
$24$	2.63337	0.537534
$25$	−4.38583	−0.877165
$26$	0.779883	0.152948
$27$	−2.46121	−0.473661
$28$	4.82371	0.911595
$29$	−0.728405	−0.135261	−0.0676307	−	0.997710i	$-0.521544\pi$
$29$	−0.728405	−0.135261	−0.0676307	+	0.997710i	$0.521544\pi$
$30$	−2.06375	−0.376788
$31$	6.47433	1.16282	0.581412	−	0.813610i	$-0.302501\pi$
$31$	6.47433	1.16282	0.581412	+	0.813610i	$0.302501\pi$
$32$	−1.00000	−0.176777
$33$	−1.68337	−0.293037
$34$	4.03702	0.692343
$35$	−3.78031	−0.638988
$36$	3.93463	0.655771
$37$	−2.47595	−0.407044	−0.203522	−	0.979070i	$-0.565239\pi$
$37$	−2.47595	−0.407044	−0.203522	+	0.979070i	$0.565239\pi$
$38$	−5.34960	−0.867819
$39$	2.05372	0.328858
$40$	0.783693	0.123913
$41$	−8.20492	−1.28139	−0.640697	−	0.767794i	$-0.721354\pi$
$41$	−8.20492	−1.28139	−0.640697	+	0.767794i	$0.721354\pi$
$42$	12.7026	1.96005
$43$	9.19075	1.40158	0.700788	−	0.713369i	$-0.252832\pi$
$43$	9.19075	1.40158	0.700788	+	0.713369i	$0.252832\pi$
$44$	0.639247	0.0963701
$45$	−3.08354	−0.459667
$46$	−8.73612	−1.28807
$47$	−5.64658	−0.823638	−0.411819	−	0.911266i	$-0.635106\pi$
$47$	−5.64658	−0.823638	−0.411819	+	0.911266i	$0.635106\pi$
$48$	−2.63337	−0.380094
$49$	16.2682	2.32402
$50$	4.38583	0.620249
$51$	10.6310	1.48863
$52$	−0.779883	−0.108150
$53$	4.42478	0.607790	0.303895	−	0.952705i	$-0.401713\pi$
$53$	4.42478	0.607790	0.303895	+	0.952705i	$0.401713\pi$
$54$	2.46121	0.334929
$55$	−0.500973	−0.0675512
$56$	−4.82371	−0.644595
$57$	−14.0875	−1.86593
$58$	0.728405	0.0956443
$59$	0.747616	0.0973313	0.0486657	−	0.998815i	$-0.484503\pi$
$59$	0.747616	0.0973313	0.0486657	+	0.998815i	$0.484503\pi$
$60$	2.06375	0.266429
$61$	11.2755	1.44367	0.721837	−	0.692063i	$-0.243298\pi$
$61$	11.2755	1.44367	0.721837	+	0.692063i	$0.243298\pi$
$62$	−6.47433	−0.822240
$63$	18.9795	2.39119
$64$	1.00000	0.125000
$65$	0.611188	0.0758086
$66$	1.68337	0.207209
$67$	14.8982	1.82011	0.910053	−	0.414492i	$-0.136041\pi$
$67$	14.8982	1.82011	0.910053	+	0.414492i	$0.136041\pi$
$68$	−4.03702	−0.489561
$69$	−23.0054	−2.76953
$70$	3.78031	0.451833
$71$	3.45258	0.409746	0.204873	−	0.978789i	$-0.434322\pi$
$71$	3.45258	0.409746	0.204873	+	0.978789i	$0.434322\pi$
$72$	−3.93463	−0.463700
$73$	−1.13343	−0.132658	−0.0663288	−	0.997798i	$-0.521129\pi$
$73$	−1.13343	−0.132658	−0.0663288	+	0.997798i	$0.521129\pi$
$74$	2.47595	0.287824
$75$	11.5495	1.33362
$76$	5.34960	0.613641
$77$	3.08354	0.351402
$78$	−2.05372	−0.232538
$79$	−6.05867	−0.681653	−0.340827	−	0.940126i	$-0.610707\pi$
$79$	−6.05867	−0.681653	−0.340827	+	0.940126i	$0.610707\pi$
$80$	−0.783693	−0.0876195
$81$	−5.32260	−0.591399
$82$	8.20492	0.906082
$83$	7.77691	0.853626	0.426813	−	0.904340i	$-0.359636\pi$
$83$	7.77691	0.853626	0.426813	+	0.904340i	$0.359636\pi$
$84$	−12.7026	−1.38597
$85$	3.16378	0.343160
$86$	−9.19075	−0.991064
$87$	1.91816	0.205648
$88$	−0.639247	−0.0681439
$89$	−2.69832	−0.286021	−0.143010	−	0.989721i	$-0.545678\pi$
$89$	−2.69832	−0.286021	−0.143010	+	0.989721i	$0.545678\pi$
$90$	3.08354	0.325033
$91$	−3.76193	−0.394357
$92$	8.73612	0.910804
$93$	−17.0493	−1.76793
$94$	5.64658	0.582400
$95$	−4.19244	−0.430135
$96$	2.63337	0.268767
$97$	−8.01215	−0.813510	−0.406755	−	0.913537i	$-0.633340\pi$
$97$	−8.01215	−0.813510	−0.406755	+	0.913537i	$0.633340\pi$
$98$	−16.2682	−1.64333
$99$	2.51520	0.252787
$100$	−4.38583	−0.438583
$101$	−3.10956	−0.309413	−0.154706	−	0.987960i	$-0.549443\pi$
$101$	−3.10956	−0.309413	−0.154706	+	0.987960i	$0.549443\pi$
$102$	−10.6310	−1.05262
$103$	16.5914	1.63480	0.817399	−	0.576072i	$-0.195415\pi$
$103$	16.5914	1.63480	0.817399	+	0.576072i	$0.195415\pi$
$104$	0.779883	0.0764738
$105$	9.95494	0.971502
$106$	−4.42478	−0.429773
$107$	−6.84601	−0.661829	−0.330914	−	0.943661i	$-0.607357\pi$
$107$	−6.84601	−0.661829	−0.330914	+	0.943661i	$0.607357\pi$
$108$	−2.46121	−0.236830
$109$	5.86786	0.562039	0.281020	−	0.959702i	$-0.409327\pi$
$109$	5.86786	0.562039	0.281020	+	0.959702i	$0.409327\pi$
$110$	0.500973	0.0477659
$111$	6.52009	0.618860
$112$	4.82371	0.455798
$113$	4.32497	0.406859	0.203429	−	0.979090i	$-0.434791\pi$
$113$	4.32497	0.406859	0.203429	+	0.979090i	$0.434791\pi$
$114$	14.0875	1.31941
$115$	−6.84643	−0.638433
$116$	−0.728405	−0.0676307
$117$	−3.06855	−0.283687
$118$	−0.747616	−0.0688236
$119$	−19.4734	−1.78513
$120$	−2.06375	−0.188394
$121$	−10.5914	−0.962851
$122$	−11.2755	−1.02083
$123$	21.6066	1.94820
$124$	6.47433	0.581412
$125$	7.35560	0.657905
$126$	−18.9795	−1.69083
$127$	−13.9043	−1.23380	−0.616902	−	0.787040i	$-0.711613\pi$
$127$	−13.9043	−1.23380	−0.616902	+	0.787040i	$0.711613\pi$
$128$	−1.00000	−0.0883883
$129$	−24.2026	−2.13092
$130$	−0.611188	−0.0536048
$131$	−0.102525	−0.00895763	−0.00447882	−	0.999990i	$-0.501426\pi$
$131$	−0.102525	−0.00895763	−0.00447882	+	0.999990i	$0.501426\pi$
$132$	−1.68337	−0.146519
$133$	25.8049	2.23757
$134$	−14.8982	−1.28701
$135$	1.92884	0.166008
$136$	4.03702	0.346172
$137$	3.82932	0.327161	0.163581	−	0.986530i	$-0.447696\pi$
$137$	3.82932	0.327161	0.163581	+	0.986530i	$0.447696\pi$
$138$	23.0054	1.95835
$139$	9.44936	0.801484	0.400742	−	0.916191i	$-0.368752\pi$
$139$	9.44936	0.801484	0.400742	+	0.916191i	$0.368752\pi$
$140$	−3.78031	−0.319494
$141$	14.8695	1.25224
$142$	−3.45258	−0.289734
$143$	−0.498538	−0.0416898
$144$	3.93463	0.327886
$145$	0.570846	0.0474062
$146$	1.13343	0.0938030
$147$	−42.8401	−3.53339
$148$	−2.47595	−0.203522
$149$	−3.26550	−0.267520	−0.133760	−	0.991014i	$-0.542705\pi$
$149$	−3.26550	−0.267520	−0.133760	+	0.991014i	$0.542705\pi$
$150$	−11.5495	−0.943012
$151$	−14.5555	−1.18451	−0.592256	−	0.805750i	$-0.701763\pi$
$151$	−14.5555	−1.18451	−0.592256	+	0.805750i	$0.701763\pi$
$152$	−5.34960	−0.433910
$153$	−15.8842	−1.28416
$154$	−3.08354	−0.248479
$155$	−5.07388	−0.407544
$156$	2.05372	0.164429
$157$	3.61173	0.288248	0.144124	−	0.989560i	$-0.453964\pi$
$157$	3.61173	0.288248	0.144124	+	0.989560i	$0.453964\pi$
$158$	6.05867	0.482002
$159$	−11.6521	−0.924070
$160$	0.783693	0.0619563
$161$	42.1405	3.32114
$162$	5.32260	0.418183
$163$	−24.8362	−1.94532	−0.972659	−	0.232238i	$-0.925395\pi$
$163$	−24.8362	−1.94532	−0.972659	+	0.232238i	$0.925395\pi$
$164$	−8.20492	−0.640697
$165$	1.31925	0.102703
$166$	−7.77691	−0.603605
$167$	18.0272	1.39499	0.697493	−	0.716592i	$-0.254299\pi$
$167$	18.0272	1.39499	0.697493	+	0.716592i	$0.254299\pi$
$168$	12.7026	0.980027
$169$	−12.3918	−0.953214
$170$	−3.16378	−0.242651
$171$	21.0487	1.60963
$172$	9.19075	0.700788
$173$	2.73645	0.208049	0.104024	−	0.994575i	$-0.466828\pi$
$173$	2.73645	0.208049	0.104024	+	0.994575i	$0.466828\pi$
$174$	−1.91816	−0.145415
$175$	−21.1560	−1.59924
$176$	0.639247	0.0481850
$177$	−1.96875	−0.147980
$178$	2.69832	0.202247
$179$	−5.82639	−0.435485	−0.217742	−	0.976006i	$-0.569869\pi$
$179$	−5.82639	−0.435485	−0.217742	+	0.976006i	$0.569869\pi$
$180$	−3.08354	−0.229833
$181$	−8.05317	−0.598588	−0.299294	−	0.954161i	$-0.596751\pi$
$181$	−8.05317	−0.598588	−0.299294	+	0.954161i	$0.596751\pi$
$182$	3.76193	0.278853
$183$	−29.6924	−2.19493
$184$	−8.73612	−0.644035
$185$	1.94039	0.142660
$186$	17.0493	1.25011
$187$	−2.58065	−0.188716
$188$	−5.64658	−0.411819
$189$	−11.8722	−0.863574
$190$	4.19244	0.304151
$191$	−15.6790	−1.13449	−0.567245	−	0.823549i	$-0.691991\pi$
$191$	−15.6790	−1.13449	−0.567245	+	0.823549i	$0.691991\pi$
$192$	−2.63337	−0.190047
$193$	−5.30050	−0.381538	−0.190769	−	0.981635i	$-0.561098\pi$
$193$	−5.30050	−0.381538	−0.190769	+	0.981635i	$0.561098\pi$
$194$	8.01215	0.575239
$195$	−1.60948	−0.115258
$196$	16.2682	1.16201
$197$	−23.2132	−1.65388	−0.826938	−	0.562294i	$-0.809919\pi$
$197$	−23.2132	−1.65388	−0.826938	+	0.562294i	$0.809919\pi$
$198$	−2.51520	−0.178747
$199$	10.1906	0.722395	0.361197	−	0.932489i	$-0.382368\pi$
$199$	10.1906	0.722395	0.361197	+	0.932489i	$0.382368\pi$
$200$	4.38583	0.310125
$201$	−39.2325	−2.76724
$202$	3.10956	0.218788
$203$	−3.51362	−0.246607
$204$	10.6310	0.744316
$205$	6.43014	0.449100
$206$	−16.5914	−1.15598
$207$	34.3734	2.38911
$208$	−0.779883	−0.0540751
$209$	3.41971	0.236546
$210$	−9.95494	−0.686956
$211$	−25.8622	−1.78043	−0.890213	−	0.455545i	$-0.849444\pi$
$211$	−25.8622	−1.78043	−0.890213	+	0.455545i	$0.849444\pi$
$212$	4.42478	0.303895
$213$	−9.09192	−0.622968
$214$	6.84601	0.467984
$215$	−7.20273	−0.491222
$216$	2.46121	0.167464
$217$	31.2303	2.12005
$218$	−5.86786	−0.397422
$219$	2.98473	0.201689
$220$	−0.500973	−0.0337756
$221$	3.14840	0.211785
$222$	−6.52009	−0.437600
$223$	−26.1177	−1.74897	−0.874485	−	0.485053i	$-0.838800\pi$
$223$	−26.1177	−1.74897	−0.874485	+	0.485053i	$0.838800\pi$
$224$	−4.82371	−0.322298
$225$	−17.2566	−1.15044
$226$	−4.32497	−0.287692
$227$	−1.03381	−0.0686161	−0.0343080	−	0.999411i	$-0.510923\pi$
$227$	−1.03381	−0.0686161	−0.0343080	+	0.999411i	$0.510923\pi$
$228$	−14.0875	−0.932964
$229$	−6.85550	−0.453024	−0.226512	−	0.974008i	$-0.572732\pi$
$229$	−6.85550	−0.453024	−0.226512	+	0.974008i	$0.572732\pi$
$230$	6.84643	0.451440
$231$	−8.12010	−0.534263
$232$	0.728405	0.0478222
$233$	3.24011	0.212267	0.106133	−	0.994352i	$-0.466153\pi$
$233$	3.24011	0.212267	0.106133	+	0.994352i	$0.466153\pi$
$234$	3.06855	0.200597
$235$	4.42518	0.288667
$236$	0.747616	0.0486657
$237$	15.9547	1.03637
$238$	19.4734	1.26227
$239$	30.5529	1.97630	0.988150	−	0.153492i	$-0.0490518\pi$
$239$	30.5529	1.97630	0.988150	+	0.153492i	$0.0490518\pi$
$240$	2.06375	0.133215
$241$	6.84419	0.440873	0.220436	−	0.975401i	$-0.429252\pi$
$241$	6.84419	0.440873	0.220436	+	0.975401i	$0.429252\pi$
$242$	10.5914	0.680839
$243$	21.4000	1.37281
$244$	11.2755	0.721837
$245$	−12.7492	−0.814520
$246$	−21.6066	−1.37759
$247$	−4.17206	−0.265462
$248$	−6.47433	−0.411120
$249$	−20.4795	−1.29783
$250$	−7.35560	−0.465209
$251$	−2.30778	−0.145666	−0.0728328	−	0.997344i	$-0.523204\pi$
$251$	−2.30778	−0.145666	−0.0728328	+	0.997344i	$0.523204\pi$
$252$	18.9795	1.19560
$253$	5.58454	0.351097
$254$	13.9043	0.872431
$255$	−8.33141	−0.521733
$256$	1.00000	0.0625000
$257$	−3.90956	−0.243871	−0.121936	−	0.992538i	$-0.538910\pi$
$257$	−3.90956	−0.243871	−0.121936	+	0.992538i	$0.538910\pi$
$258$	24.2026	1.50679
$259$	−11.9433	−0.742119
$260$	0.611188	0.0379043
$261$	−2.86600	−0.177401
$262$	0.102525	0.00633400
$263$	−22.9626	−1.41593	−0.707966	−	0.706247i	$-0.750387\pi$
$263$	−22.9626	−1.41593	−0.707966	+	0.706247i	$0.750387\pi$
$264$	1.68337	0.103604
$265$	−3.46767	−0.213017
$266$	−25.8049	−1.58220
$267$	7.10566	0.434859
$268$	14.8982	0.910053
$269$	0.566684	0.0345513	0.0172757	−	0.999851i	$-0.494501\pi$
$269$	0.566684	0.0345513	0.0172757	+	0.999851i	$0.494501\pi$
$270$	−1.92884	−0.117385
$271$	26.7335	1.62395	0.811974	−	0.583694i	$-0.198393\pi$
$271$	26.7335	1.62395	0.811974	+	0.583694i	$0.198393\pi$
$272$	−4.03702	−0.244780
$273$	9.90654	0.599571
$274$	−3.82932	−0.231338
$275$	−2.80362	−0.169065
$276$	−23.0054	−1.38476
$277$	20.4709	1.22997	0.614987	−	0.788537i	$-0.289161\pi$
$277$	20.4709	1.22997	0.614987	+	0.788537i	$0.289161\pi$
$278$	−9.44936	−0.566735
$279$	25.4741	1.52509
$280$	3.78031	0.225916
$281$	2.16983	0.129441	0.0647206	−	0.997903i	$-0.479384\pi$
$281$	2.16983	0.129441	0.0647206	+	0.997903i	$0.479384\pi$
$282$	−14.8695	−0.885467
$283$	31.4420	1.86903	0.934516	−	0.355921i	$-0.115833\pi$
$283$	31.4420	1.86903	0.934516	+	0.355921i	$0.115833\pi$
$284$	3.45258	0.204873
$285$	11.0402	0.653967
$286$	0.498538	0.0294791
$287$	−39.5782	−2.33622
$288$	−3.93463	−0.231850
$289$	−0.702463	−0.0413214
$290$	−0.570846	−0.0335212
$291$	21.0989	1.23684
$292$	−1.13343	−0.0663288
$293$	13.8708	0.810342	0.405171	−	0.914241i	$-0.367212\pi$
$293$	13.8708	0.810342	0.405171	+	0.914241i	$0.367212\pi$
$294$	42.8401	2.49848
$295$	−0.585901	−0.0341125
$296$	2.47595	0.143912
$297$	−1.57332	−0.0912935
$298$	3.26550	0.189165
$299$	−6.81315	−0.394015
$300$	11.5495	0.666810
$301$	44.3335	2.55534
$302$	14.5555	0.837576
$303$	8.18862	0.470424
$304$	5.34960	0.306820
$305$	−8.83649	−0.505976
$306$	15.8842	0.908037
$307$	−6.63086	−0.378443	−0.189222	−	0.981934i	$-0.560596\pi$
$307$	−6.63086	−0.378443	−0.189222	+	0.981934i	$0.560596\pi$
$308$	3.08354	0.175701
$309$	−43.6912	−2.48551
$310$	5.07388	0.288177
$311$	21.0103	1.19139	0.595694	−	0.803212i	$-0.296877\pi$
$311$	21.0103	1.19139	0.595694	+	0.803212i	$0.296877\pi$
$312$	−2.05372	−0.116269
$313$	19.0949	1.07931	0.539653	−	0.841888i	$-0.318555\pi$
$313$	19.0949	1.07931	0.539653	+	0.841888i	$0.318555\pi$
$314$	−3.61173	−0.203822
$315$	−14.8741	−0.838060
$316$	−6.05867	−0.340827
$317$	1.32754	0.0745622	0.0372811	−	0.999305i	$-0.488130\pi$
$317$	1.32754	0.0745622	0.0372811	+	0.999305i	$0.488130\pi$
$318$	11.6521	0.653416
$319$	−0.465631	−0.0260703
$320$	−0.783693	−0.0438097
$321$	18.0281	1.00623
$322$	−42.1405	−2.34840
$323$	−21.5964	−1.20166
$324$	−5.32260	−0.295700
$325$	3.42043	0.189731
$326$	24.8362	1.37555
$327$	−15.4522	−0.854511
$328$	8.20492	0.453041
$329$	−27.2374	−1.50165
$330$	−1.31925	−0.0726221
$331$	33.7759	1.85649	0.928245	−	0.371970i	$-0.121318\pi$
$331$	33.7759	1.85649	0.928245	+	0.371970i	$0.121318\pi$
$332$	7.77691	0.426813
$333$	−9.74195	−0.533855
$334$	−18.0272	−0.986404
$335$	−11.6756	−0.637907
$336$	−12.7026	−0.692984
$337$	−2.78896	−0.151925	−0.0759623	−	0.997111i	$-0.524203\pi$
$337$	−2.78896	−0.151925	−0.0759623	+	0.997111i	$0.524203\pi$
$338$	12.3918	0.674024
$339$	−11.3892	−0.618578
$340$	3.16378	0.171580
$341$	4.13869	0.224123
$342$	−21.0487	−1.13818
$343$	44.7070	2.41395
$344$	−9.19075	−0.495532
$345$	18.0292	0.970658
$346$	−2.73645	−0.147113
$347$	3.18475	0.170966	0.0854832	−	0.996340i	$-0.472757\pi$
$347$	3.18475	0.170966	0.0854832	+	0.996340i	$0.472757\pi$
$348$	1.91816	0.102824
$349$	−6.42023	−0.343667	−0.171833	−	0.985126i	$-0.554969\pi$
$349$	−6.42023	−0.343667	−0.171833	+	0.985126i	$0.554969\pi$
$350$	21.1560	1.13083
$351$	1.91946	0.102453
$352$	−0.639247	−0.0340720
$353$	28.6133	1.52293	0.761466	−	0.648205i	$-0.224480\pi$
$353$	28.6133	1.52293	0.761466	+	0.648205i	$0.224480\pi$
$354$	1.96875	0.104638
$355$	−2.70576	−0.143607
$356$	−2.69832	−0.143010
$357$	51.2807	2.71406
$358$	5.82639	0.307934
$359$	2.80054	0.147807	0.0739035	−	0.997265i	$-0.476454\pi$
$359$	2.80054	0.147807	0.0739035	+	0.997265i	$0.476454\pi$
$360$	3.08354	0.162517
$361$	9.61818	0.506220
$362$	8.05317	0.423265
$363$	27.8910	1.46390
$364$	−3.76193	−0.197179
$365$	0.888258	0.0464935
$366$	29.6924	1.55205
$367$	27.4481	1.43278	0.716390	−	0.697700i	$-0.245793\pi$
$367$	27.4481	1.43278	0.716390	+	0.697700i	$0.245793\pi$
$368$	8.73612	0.455402
$369$	−32.2833	−1.68060
$370$	−1.94039	−0.100876
$371$	21.3439	1.10812
$372$	−17.0493	−0.883964
$373$	28.7310	1.48763	0.743817	−	0.668383i	$-0.233013\pi$
$373$	28.7310	1.48763	0.743817	+	0.668383i	$0.233013\pi$
$374$	2.58065	0.133442
$375$	−19.3700	−1.00026
$376$	5.64658	0.291200
$377$	0.568071	0.0292571
$378$	11.8722	0.610639
$379$	23.5247	1.20838	0.604191	−	0.796839i	$-0.293496\pi$
$379$	23.5247	1.20838	0.604191	+	0.796839i	$0.293496\pi$
$380$	−4.19244	−0.215068
$381$	36.6150	1.87585
$382$	15.6790	0.802205
$383$	28.1474	1.43827	0.719133	−	0.694872i	$-0.244539\pi$
$383$	28.1474	1.43827	0.719133	+	0.694872i	$0.244539\pi$
$384$	2.63337	0.134383
$385$	−2.41655	−0.123159
$386$	5.30050	0.269788
$387$	36.1622	1.83823
$388$	−8.01215	−0.406755
$389$	−14.9920	−0.760127	−0.380063	−	0.924960i	$-0.624098\pi$
$389$	−14.9920	−0.760127	−0.380063	+	0.924960i	$0.624098\pi$
$390$	1.60948	0.0814994
$391$	−35.2679	−1.78357
$392$	−16.2682	−0.821667
$393$	0.269986	0.0136190
$394$	23.2132	1.16947
$395$	4.74813	0.238904
$396$	2.51520	0.126393
$397$	6.81154	0.341862	0.170931	−	0.985283i	$-0.445323\pi$
$397$	6.81154	0.341862	0.170931	+	0.985283i	$0.445323\pi$
$398$	−10.1906	−0.510810
$399$	−67.9538	−3.40194
$400$	−4.38583	−0.219291
$401$	−6.62350	−0.330762	−0.165381	−	0.986230i	$-0.552885\pi$
$401$	−6.62350	−0.330762	−0.165381	+	0.986230i	$0.552885\pi$
$402$	39.2325	1.95674
$403$	−5.04922	−0.251519
$404$	−3.10956	−0.154706
$405$	4.17128	0.207272
$406$	3.51362	0.174378
$407$	−1.58274	−0.0784537
$408$	−10.6310	−0.526311
$409$	−6.64483	−0.328566	−0.164283	−	0.986413i	$-0.552531\pi$
$409$	−6.64483	−0.328566	−0.164283	+	0.986413i	$0.552531\pi$
$410$	−6.43014	−0.317562
$411$	−10.0840	−0.497408
$412$	16.5914	0.817399
$413$	3.60628	0.177454
$414$	−34.3734	−1.68936
$415$	−6.09470	−0.299177
$416$	0.779883	0.0382369
$417$	−24.8836	−1.21856
$418$	−3.41971	−0.167264
$419$	14.7786	0.721980	0.360990	−	0.932570i	$-0.382439\pi$
$419$	14.7786	0.721980	0.360990	+	0.932570i	$0.382439\pi$
$420$	9.95494	0.485751
$421$	2.43998	0.118917	0.0594586	−	0.998231i	$-0.481063\pi$
$421$	2.43998	0.118917	0.0594586	+	0.998231i	$0.481063\pi$
$422$	25.8622	1.25895
$423$	−22.2172	−1.08024
$424$	−4.42478	−0.214886
$425$	17.7057	0.858851
$426$	9.09192	0.440505
$427$	54.3895	2.63209
$428$	−6.84601	−0.330914
$429$	1.31283	0.0633842
$430$	7.20273	0.347346
$431$	−13.8144	−0.665417	−0.332709	−	0.943030i	$-0.607963\pi$
$431$	−13.8144	−0.665417	−0.332709	+	0.943030i	$0.607963\pi$
$432$	−2.46121	−0.118415
$433$	−10.2175	−0.491022	−0.245511	−	0.969394i	$-0.578956\pi$
$433$	−10.2175	−0.491022	−0.245511	+	0.969394i	$0.578956\pi$
$434$	−31.2303	−1.49910
$435$	−1.50325	−0.0720752
$436$	5.86786	0.281020
$437$	46.7347	2.23562
$438$	−2.98473	−0.142616
$439$	−11.1717	−0.533197	−0.266599	−	0.963808i	$-0.585900\pi$
$439$	−11.1717	−0.533197	−0.266599	+	0.963808i	$0.585900\pi$
$440$	0.500973	0.0238829
$441$	64.0092	3.04806
$442$	−3.14840	−0.149754
$443$	36.2605	1.72279	0.861394	−	0.507937i	$-0.169592\pi$
$443$	36.2605	1.72279	0.861394	+	0.507937i	$0.169592\pi$
$444$	6.52009	0.309430
$445$	2.11465	0.100244
$446$	26.1177	1.23671
$447$	8.59925	0.406731
$448$	4.82371	0.227899
$449$	−18.8920	−0.891568	−0.445784	−	0.895141i	$-0.647075\pi$
$449$	−18.8920	−0.891568	−0.445784	+	0.895141i	$0.647075\pi$
$450$	17.2566	0.813483
$451$	−5.24497	−0.246976
$452$	4.32497	0.203429
$453$	38.3301	1.80090
$454$	1.03381	0.0485189
$455$	2.94820	0.138214
$456$	14.0875	0.659705
$457$	5.20825	0.243631	0.121816	−	0.992553i	$-0.461128\pi$
$457$	5.20825	0.243631	0.121816	+	0.992553i	$0.461128\pi$
$458$	6.85550	0.320336
$459$	9.93597	0.463772
$460$	−6.84643	−0.319217
$461$	−18.8480	−0.877839	−0.438919	−	0.898526i	$-0.644639\pi$
$461$	−18.8480	−0.877839	−0.438919	+	0.898526i	$0.644639\pi$
$462$	8.12010	0.377781
$463$	−30.3051	−1.40840	−0.704198	−	0.710003i	$-0.748693\pi$
$463$	−30.3051	−1.40840	−0.704198	+	0.710003i	$0.748693\pi$
$464$	−0.728405	−0.0338154
$465$	13.3614	0.619620
$466$	−3.24011	−0.150095
$467$	21.5407	0.996783	0.498391	−	0.866952i	$-0.333924\pi$
$467$	21.5407	0.996783	0.498391	+	0.866952i	$0.333924\pi$
$468$	−3.06855	−0.141844
$469$	71.8646	3.31840
$470$	−4.42518	−0.204118
$471$	−9.51102	−0.438245
$472$	−0.747616	−0.0344118
$473$	5.87516	0.270140
$474$	−15.9547	−0.732823
$475$	−23.4624	−1.07653
$476$	−19.4734	−0.892563
$477$	17.4099	0.797143
$478$	−30.5529	−1.39746
$479$	−30.0350	−1.37233	−0.686167	−	0.727444i	$-0.740708\pi$
$479$	−30.0350	−1.37233	−0.686167	+	0.727444i	$0.740708\pi$
$480$	−2.06375	−0.0941969
$481$	1.93095	0.0880439
$482$	−6.84419	−0.311744
$483$	−110.971	−5.04938
$484$	−10.5914	−0.481426
$485$	6.27906	0.285118
$486$	−21.4000	−0.970723
$487$	6.72085	0.304551	0.152275	−	0.988338i	$-0.451340\pi$
$487$	6.72085	0.304551	0.152275	+	0.988338i	$0.451340\pi$
$488$	−11.2755	−0.510416
$489$	65.4027	2.95761
$490$	12.7492	0.575952
$491$	−8.03542	−0.362634	−0.181317	−	0.983425i	$-0.558036\pi$
$491$	−8.03542	−0.362634	−0.181317	+	0.983425i	$0.558036\pi$
$492$	21.6066	0.974100
$493$	2.94059	0.132437
$494$	4.17206	0.187710
$495$	−1.97114	−0.0885962
$496$	6.47433	0.290706
$497$	16.6543	0.747046
$498$	20.4795	0.917706
$499$	5.37367	0.240559	0.120279	−	0.992740i	$-0.461621\pi$
$499$	5.37367	0.240559	0.120279	+	0.992740i	$0.461621\pi$
$500$	7.35560	0.328953
$501$	−47.4722	−2.12090
$502$	2.30778	0.103001
$503$	−18.7421	−0.835670	−0.417835	−	0.908523i	$-0.637211\pi$
$503$	−18.7421	−0.835670	−0.417835	+	0.908523i	$0.637211\pi$
$504$	−18.9795	−0.845414
$505$	2.43694	0.108442
$506$	−5.58454	−0.248263
$507$	32.6321	1.44924
$508$	−13.9043	−0.616902
$509$	1.38857	0.0615472	0.0307736	−	0.999526i	$-0.490203\pi$
$509$	1.38857	0.0615472	0.0307736	+	0.999526i	$0.490203\pi$
$510$	8.33141	0.368921
$511$	−5.46732	−0.241860
$512$	−1.00000	−0.0441942
$513$	−13.1665	−0.581315
$514$	3.90956	0.172443
$515$	−13.0025	−0.572960
$516$	−24.2026	−1.06546
$517$	−3.60956	−0.158748
$518$	11.9433	0.524757
$519$	−7.20608	−0.316312
$520$	−0.611188	−0.0268024
$521$	23.4475	1.02725	0.513627	−	0.858014i	$-0.328302\pi$
$521$	23.4475	1.02725	0.513627	+	0.858014i	$0.328302\pi$
$522$	2.86600	0.125442
$523$	−31.6351	−1.38330	−0.691652	−	0.722231i	$-0.743117\pi$
$523$	−31.6351	−1.38330	−0.691652	+	0.722231i	$0.743117\pi$
$524$	−0.102525	−0.00447882
$525$	55.7114	2.43144
$526$	22.9626	1.00121
$527$	−26.1370	−1.13855
$528$	−1.68337	−0.0732594
$529$	53.3198	2.31825
$530$	3.46767	0.150626
$531$	2.94159	0.127654
$532$	25.8049	1.11878
$533$	6.39888	0.277166
$534$	−7.10566	−0.307492
$535$	5.36517	0.231956
$536$	−14.8982	−0.643505
$537$	15.3430	0.662101
$538$	−0.566684	−0.0244315
$539$	10.3994	0.447933
$540$	1.92884	0.0830039
$541$	4.17516	0.179504	0.0897520	−	0.995964i	$-0.471393\pi$
$541$	4.17516	0.179504	0.0897520	+	0.995964i	$0.471393\pi$
$542$	−26.7335	−1.14830
$543$	21.2070	0.910078
$544$	4.03702	0.173086
$545$	−4.59860	−0.196982
$546$	−9.90654	−0.423961
$547$	13.6180	0.582265	0.291132	−	0.956683i	$-0.405968\pi$
$547$	13.6180	0.582265	0.291132	+	0.956683i	$0.405968\pi$
$548$	3.82932	0.163581
$549$	44.3647	1.89344
$550$	2.80362	0.119547
$551$	−3.89667	−0.166004
$552$	23.0054	0.979176
$553$	−29.2252	−1.24278
$554$	−20.4709	−0.869723
$555$	−5.10975	−0.216897
$556$	9.44936	0.400742
$557$	−4.32394	−0.183211	−0.0916057	−	0.995795i	$-0.529200\pi$
$557$	−4.32394	−0.183211	−0.0916057	+	0.995795i	$0.529200\pi$
$558$	−25.4741	−1.07840
$559$	−7.16771	−0.303162
$560$	−3.78031	−0.159747
$561$	6.79581	0.286919
$562$	−2.16983	−0.0915288
$563$	26.7827	1.12876	0.564378	−	0.825516i	$-0.309116\pi$
$563$	26.7827	1.12876	0.564378	+	0.825516i	$0.309116\pi$
$564$	14.8695	0.626119
$565$	−3.38944	−0.142595
$566$	−31.4420	−1.32161
$567$	−25.6747	−1.07823
$568$	−3.45258	−0.144867
$569$	−35.2686	−1.47854	−0.739268	−	0.673411i	$-0.764828\pi$
$569$	−35.2686	−1.47854	−0.739268	+	0.673411i	$0.764828\pi$
$570$	−11.0402	−0.462425
$571$	−0.965925	−0.0404227	−0.0202113	−	0.999796i	$-0.506434\pi$
$571$	−0.965925	−0.0404227	−0.0202113	+	0.999796i	$0.506434\pi$
$572$	−0.498538	−0.0208449
$573$	41.2885	1.72485
$574$	39.5782	1.65196
$575$	−38.3151	−1.59785
$576$	3.93463	0.163943
$577$	6.43414	0.267857	0.133928	−	0.990991i	$-0.457241\pi$
$577$	6.43414	0.267857	0.133928	+	0.990991i	$0.457241\pi$
$578$	0.702463	0.0292186
$579$	13.9582	0.580082
$580$	0.570846	0.0237031
$581$	37.5135	1.55632
$582$	−21.0989	−0.874579
$583$	2.82853	0.117146
$584$	1.13343	0.0469015
$585$	2.40480	0.0994262
$586$	−13.8708	−0.572998
$587$	13.9792	0.576983	0.288491	−	0.957483i	$-0.406846\pi$
$587$	13.9792	0.576983	0.288491	+	0.957483i	$0.406846\pi$
$588$	−42.8401	−1.76670
$589$	34.6350	1.42711
$590$	0.585901	0.0241212
$591$	61.1290	2.51451
$592$	−2.47595	−0.101761
$593$	−7.96466	−0.327069	−0.163535	−	0.986538i	$-0.552290\pi$
$593$	−7.96466	−0.327069	−0.163535	+	0.986538i	$0.552290\pi$
$594$	1.57332	0.0645542
$595$	15.2612	0.625647
$596$	−3.26550	−0.133760
$597$	−26.8357	−1.09831
$598$	6.81315	0.278610
$599$	−7.40918	−0.302731	−0.151366	−	0.988478i	$-0.548367\pi$
$599$	−7.40918	−0.302731	−0.151366	+	0.988478i	$0.548367\pi$
$600$	−11.5495	−0.471506
$601$	−37.8661	−1.54459	−0.772296	−	0.635263i	$-0.780892\pi$
$601$	−37.8661	−1.54459	−0.772296	+	0.635263i	$0.780892\pi$
$602$	−44.3335	−1.80690
$603$	58.6189	2.38715
$604$	−14.5555	−0.592256
$605$	8.30037	0.337458
$606$	−8.18862	−0.332640
$607$	17.3876	0.705741	0.352871	−	0.935672i	$-0.385206\pi$
$607$	17.3876	0.705741	0.352871	+	0.935672i	$0.385206\pi$
$608$	−5.34960	−0.216955
$609$	9.25264	0.374936
$610$	8.83649	0.357779
$611$	4.40367	0.178153
$612$	−15.8842	−0.642079
$613$	−3.60915	−0.145772	−0.0728860	−	0.997340i	$-0.523221\pi$
$613$	−3.60915	−0.145772	−0.0728860	+	0.997340i	$0.523221\pi$
$614$	6.63086	0.267600
$615$	−16.9329	−0.682801
$616$	−3.08354	−0.124239
$617$	26.6891	1.07446	0.537232	−	0.843435i	$-0.319470\pi$
$617$	26.6891	1.07446	0.537232	+	0.843435i	$0.319470\pi$
$618$	43.6912	1.75752
$619$	4.74724	0.190808	0.0954038	−	0.995439i	$-0.469586\pi$
$619$	4.74724	0.190808	0.0954038	+	0.995439i	$0.469586\pi$
$620$	−5.07388	−0.203772
$621$	−21.5015	−0.862824
$622$	−21.0103	−0.842438
$623$	−13.0159	−0.521471
$624$	2.05372	0.0822145
$625$	16.1646	0.646584
$626$	−19.0949	−0.763185
$627$	−9.00536	−0.359639
$628$	3.61173	0.144124
$629$	9.99547	0.398545
$630$	14.8741	0.592598
$631$	23.5161	0.936161	0.468080	−	0.883686i	$-0.344946\pi$
$631$	23.5161	0.936161	0.468080	+	0.883686i	$0.344946\pi$
$632$	6.05867	0.241001
$633$	68.1046	2.70692
$634$	−1.32754	−0.0527234
$635$	10.8967	0.432421
$636$	−11.6521	−0.462035
$637$	−12.6873	−0.502688
$638$	0.465631	0.0184345
$639$	13.5846	0.537400
$640$	0.783693	0.0309782
$641$	−23.0519	−0.910494	−0.455247	−	0.890365i	$-0.650449\pi$
$641$	−23.0519	−0.910494	−0.455247	+	0.890365i	$0.650449\pi$
$642$	−18.0281	−0.711511
$643$	−2.47201	−0.0974864	−0.0487432	−	0.998811i	$-0.515522\pi$
$643$	−2.47201	−0.0974864	−0.0487432	+	0.998811i	$0.515522\pi$
$644$	42.1405	1.66057
$645$	18.9674	0.746842
$646$	21.5964	0.849700
$647$	−5.27717	−0.207467	−0.103733	−	0.994605i	$-0.533079\pi$
$647$	−5.27717	−0.207467	−0.103733	+	0.994605i	$0.533079\pi$
$648$	5.32260	0.209091
$649$	0.477911	0.0187597
$650$	−3.42043	−0.134160
$651$	−82.2408	−3.22327
$652$	−24.8362	−0.972659
$653$	−45.4090	−1.77699	−0.888495	−	0.458886i	$-0.848249\pi$
$653$	−45.4090	−1.77699	−0.888495	+	0.458886i	$0.848249\pi$
$654$	15.4522	0.604230
$655$	0.0803479	0.00313945
$656$	−8.20492	−0.320348
$657$	−4.45961	−0.173986
$658$	27.2374	1.06183
$659$	21.6983	0.845245	0.422623	−	0.906306i	$-0.361110\pi$
$659$	21.6983	0.845245	0.422623	+	0.906306i	$0.361110\pi$
$660$	1.31925	0.0513516
$661$	11.8215	0.459804	0.229902	−	0.973214i	$-0.426159\pi$
$661$	11.8215	0.459804	0.229902	+	0.973214i	$0.426159\pi$
$662$	−33.7759	−1.31274
$663$	−8.29090	−0.321992
$664$	−7.77691	−0.301802
$665$	−20.2231	−0.784218
$666$	9.74195	0.377493
$667$	−6.36344	−0.246393
$668$	18.0272	0.697493
$669$	68.7775	2.65909
$670$	11.6756	0.451068
$671$	7.20780	0.278254
$672$	12.7026	0.490013
$673$	−47.2502	−1.82136	−0.910681	−	0.413111i	$-0.864442\pi$
$673$	−47.2502	−1.82136	−0.910681	+	0.413111i	$0.864442\pi$
$674$	2.78896	0.107427
$675$	10.7945	0.415479
$676$	−12.3918	−0.476607
$677$	−3.30027	−0.126840	−0.0634199	−	0.997987i	$-0.520201\pi$
$677$	−3.30027	−0.126840	−0.0634199	+	0.997987i	$0.520201\pi$
$678$	11.3892	0.437401
$679$	−38.6483	−1.48318
$680$	−3.16378	−0.121326
$681$	2.72239	0.104322
$682$	−4.13869	−0.158479
$683$	−19.5370	−0.747563	−0.373781	−	0.927517i	$-0.621939\pi$
$683$	−19.5370	−0.747563	−0.373781	+	0.927517i	$0.621939\pi$
$684$	21.0487	0.804816
$685$	−3.00101	−0.114663
$686$	−44.7070	−1.70692
$687$	18.0530	0.688767
$688$	9.19075	0.350394
$689$	−3.45081	−0.131465
$690$	−18.0292	−0.686359
$691$	5.29860	0.201568	0.100784	−	0.994908i	$-0.467865\pi$
$691$	5.29860	0.201568	0.100784	+	0.994908i	$0.467865\pi$
$692$	2.73645	0.104024
$693$	12.1326	0.460879
$694$	−3.18475	−0.120892
$695$	−7.40539	−0.280903
$696$	−1.91816	−0.0727076
$697$	33.1234	1.25464
$698$	6.42023	0.243009
$699$	−8.53241	−0.322725
$700$	−21.1560	−0.799620
$701$	−35.7889	−1.35173	−0.675863	−	0.737027i	$-0.736229\pi$
$701$	−35.7889	−1.35173	−0.675863	+	0.737027i	$0.736229\pi$
$702$	−1.91946	−0.0724453
$703$	−13.2453	−0.499558
$704$	0.639247	0.0240925
$705$	−11.6531	−0.438882
$706$	−28.6133	−1.07688
$707$	−14.9996	−0.564119
$708$	−1.96875	−0.0739901
$709$	23.8283	0.894891	0.447445	−	0.894311i	$-0.352334\pi$
$709$	23.8283	0.894891	0.447445	+	0.894311i	$0.352334\pi$
$710$	2.70576	0.101546
$711$	−23.8386	−0.894017
$712$	2.69832	0.101124
$713$	56.5605	2.11821
$714$	−51.2807	−1.91913
$715$	0.390700	0.0146114
$716$	−5.82639	−0.217742
$717$	−80.4569	−3.00472
$718$	−2.80054	−0.104515
$719$	−10.0896	−0.376277	−0.188139	−	0.982142i	$-0.560245\pi$
$719$	−10.0896	−0.376277	−0.188139	+	0.982142i	$0.560245\pi$
$720$	−3.08354	−0.114917
$721$	80.0320	2.98055
$722$	−9.61818	−0.357952
$723$	−18.0233	−0.670292
$724$	−8.05317	−0.299294
$725$	3.19466	0.118647
$726$	−27.8910	−1.03513
$727$	24.0841	0.893229	0.446615	−	0.894726i	$-0.352629\pi$
$727$	24.0841	0.893229	0.446615	+	0.894726i	$0.352629\pi$
$728$	3.76193	0.139426
$729$	−40.3863	−1.49579
$730$	−0.888258	−0.0328759
$731$	−37.1033	−1.37231
$732$	−29.6924	−1.09746
$733$	9.47048	0.349800	0.174900	−	0.984586i	$-0.444040\pi$
$733$	9.47048	0.349800	0.174900	+	0.984586i	$0.444040\pi$
$734$	−27.4481	−1.01313
$735$	33.5735	1.23838
$736$	−8.73612	−0.322018
$737$	9.52363	0.350807
$738$	32.2833	1.18836
$739$	20.0815	0.738709	0.369355	−	0.929289i	$-0.379579\pi$
$739$	20.0815	0.738709	0.369355	+	0.929289i	$0.379579\pi$
$740$	1.94039	0.0713300
$741$	10.9866	0.403601
$742$	−21.3439	−0.783558
$743$	25.1345	0.922096	0.461048	−	0.887375i	$-0.347474\pi$
$743$	25.1345	0.922096	0.461048	+	0.887375i	$0.347474\pi$
$744$	17.0493	0.625057
$745$	2.55914	0.0937598
$746$	−28.7310	−1.05192
$747$	30.5992	1.11957
$748$	−2.58065	−0.0943580
$749$	−33.0232	−1.20664
$750$	19.3700	0.707293
$751$	−41.3103	−1.50743	−0.753716	−	0.657200i	$-0.771741\pi$
$751$	−41.3103	−1.50743	−0.753716	+	0.657200i	$0.771741\pi$
$752$	−5.64658	−0.205909
$753$	6.07723	0.221467
$754$	−0.568071	−0.0206879
$755$	11.4071	0.415145
$756$	−11.8722	−0.431787
$757$	34.1769	1.24218	0.621090	−	0.783739i	$-0.286690\pi$
$757$	34.1769	1.24218	0.621090	+	0.783739i	$0.286690\pi$
$758$	−23.5247	−0.854456
$759$	−14.7061	−0.533799
$760$	4.19244	0.152076
$761$	−49.8327	−1.80643	−0.903217	−	0.429184i	$-0.858801\pi$
$761$	−49.8327	−1.80643	−0.903217	+	0.429184i	$0.858801\pi$
$762$	−36.6150	−1.32642
$763$	28.3049	1.02470
$764$	−15.6790	−0.567245
$765$	12.4483	0.450069
$766$	−28.1474	−1.01701
$767$	−0.583053	−0.0210528
$768$	−2.63337	−0.0950235
$769$	18.9569	0.683604	0.341802	−	0.939772i	$-0.388963\pi$
$769$	18.9569	0.683604	0.341802	+	0.939772i	$0.388963\pi$
$770$	2.41655	0.0870863
$771$	10.2953	0.370776
$772$	−5.30050	−0.190769
$773$	32.4142	1.16586	0.582929	−	0.812523i	$-0.301907\pi$
$773$	32.4142	1.16586	0.582929	+	0.812523i	$0.301907\pi$
$774$	−36.1622	−1.29982
$775$	−28.3953	−1.01999
$776$	8.01215	0.287619
$777$	31.4510	1.12830
$778$	14.9920	0.537491
$779$	−43.8930	−1.57263
$780$	−1.60948	−0.0576288
$781$	2.20705	0.0789746
$782$	35.2679	1.26118
$783$	1.79276	0.0640681
$784$	16.2682	0.581006
$785$	−2.83049	−0.101024
$786$	−0.269986	−0.00963007
$787$	35.7745	1.27522	0.637612	−	0.770358i	$-0.279923\pi$
$787$	35.7745	1.27522	0.637612	+	0.770358i	$0.279923\pi$
$788$	−23.2132	−0.826938
$789$	60.4688	2.15275
$790$	−4.74813	−0.168931
$791$	20.8624	0.741781
$792$	−2.51520	−0.0893736
$793$	−8.79353	−0.312268
$794$	−6.81154	−0.241733
$795$	9.13164	0.323866
$796$	10.1906	0.361197
$797$	34.3508	1.21677	0.608383	−	0.793643i	$-0.291818\pi$
$797$	34.3508	1.21677	0.608383	+	0.793643i	$0.291818\pi$
$798$	67.9538	2.40554
$799$	22.7953	0.806441
$800$	4.38583	0.155062
$801$	−10.6169	−0.375128
$802$	6.62350	0.233884
$803$	−0.724539	−0.0255684
$804$	−39.2325	−1.38362
$805$	−33.0252	−1.16399
$806$	5.04922	0.177851
$807$	−1.49229	−0.0525310
$808$	3.10956	0.109394
$809$	6.33939	0.222881	0.111441	−	0.993771i	$-0.464454\pi$
$809$	6.33939	0.222881	0.111441	+	0.993771i	$0.464454\pi$
$810$	−4.17128	−0.146564
$811$	−15.1352	−0.531470	−0.265735	−	0.964046i	$-0.585615\pi$
$811$	−15.1352	−0.531470	−0.265735	+	0.964046i	$0.585615\pi$
$812$	−3.51362	−0.123304
$813$	−70.3993	−2.46901
$814$	1.58274	0.0554752
$815$	19.4639	0.681791
$816$	10.6310	0.372158
$817$	49.1668	1.72013
$818$	6.64483	0.232331
$819$	−14.8018	−0.517216
$820$	6.43014	0.224550
$821$	−13.0471	−0.455347	−0.227674	−	0.973738i	$-0.573112\pi$
$821$	−13.0471	−0.455347	−0.227674	+	0.973738i	$0.573112\pi$
$822$	10.0840	0.351721
$823$	−3.20352	−0.111668	−0.0558338	−	0.998440i	$-0.517782\pi$
$823$	−3.20352	−0.111668	−0.0558338	+	0.998440i	$0.517782\pi$
$824$	−16.5914	−0.577988
$825$	7.38298	0.257042
$826$	−3.60628	−0.125479
$827$	21.2920	0.740396	0.370198	−	0.928953i	$-0.379290\pi$
$827$	21.2920	0.740396	0.370198	+	0.928953i	$0.379290\pi$
$828$	34.3734	1.19456
$829$	−5.86501	−0.203700	−0.101850	−	0.994800i	$-0.532476\pi$
$829$	−5.86501	−0.203700	−0.101850	+	0.994800i	$0.532476\pi$
$830$	6.09470	0.211550
$831$	−53.9073	−1.87002
$832$	−0.779883	−0.0270376
$833$	−65.6750	−2.27550
$834$	24.8836	0.861650
$835$	−14.1278	−0.488912
$836$	3.41971	0.118273
$837$	−15.9347	−0.550784
$838$	−14.7786	−0.510517
$839$	52.9641	1.82852	0.914262	−	0.405124i	$-0.132772\pi$
$839$	52.9641	1.82852	0.914262	+	0.405124i	$0.132772\pi$
$840$	−9.95494	−0.343478
$841$	−28.4694	−0.981704
$842$	−2.43998	−0.0840871
$843$	−5.71396	−0.196799
$844$	−25.8622	−0.890213
$845$	9.71135	0.334081
$846$	22.2172	0.763842
$847$	−51.0897	−1.75546
$848$	4.42478	0.151948
$849$	−82.7983	−2.84163
$850$	−17.7057	−0.607299
$851$	−21.6302	−0.741474
$852$	−9.09192	−0.311484
$853$	−46.9232	−1.60662	−0.803309	−	0.595562i	$-0.796929\pi$
$853$	−46.9232	−1.60662	−0.803309	+	0.595562i	$0.796929\pi$
$854$	−54.3895	−1.86117
$855$	−16.4957	−0.564140
$856$	6.84601	0.233992
$857$	21.3906	0.730691	0.365345	−	0.930872i	$-0.380951\pi$
$857$	21.3906	0.730691	0.365345	+	0.930872i	$0.380951\pi$
$858$	−1.31283	−0.0448194
$859$	−7.18504	−0.245150	−0.122575	−	0.992459i	$-0.539115\pi$
$859$	−7.18504	−0.245150	−0.122575	+	0.992459i	$0.539115\pi$
$860$	−7.20273	−0.245611
$861$	104.224	3.55194
$862$	13.8144	0.470521
$863$	−46.3938	−1.57926	−0.789632	−	0.613581i	$-0.789728\pi$
$863$	−46.3938	−1.57926	−0.789632	+	0.613581i	$0.789728\pi$
$864$	2.46121	0.0837322
$865$	−2.14454	−0.0729164
$866$	10.2175	0.347205
$867$	1.84984	0.0628240
$868$	31.2303	1.06002
$869$	−3.87298	−0.131382
$870$	1.50325	0.0509649
$871$	−11.6189	−0.393690
$872$	−5.86786	−0.198711
$873$	−31.5248	−1.06695
$874$	−46.7347	−1.58083
$875$	35.4813	1.19949
$876$	2.98473	0.100845
$877$	−31.6125	−1.06748	−0.533738	−	0.845650i	$-0.679213\pi$
$877$	−31.6125	−1.06748	−0.533738	+	0.845650i	$0.679213\pi$
$878$	11.1717	0.377027
$879$	−36.5270	−1.23202
$880$	−0.500973	−0.0168878
$881$	42.3386	1.42642	0.713212	−	0.700949i	$-0.247240\pi$
$881$	42.3386	1.42642	0.713212	+	0.700949i	$0.247240\pi$
$882$	−64.0092	−2.15530
$883$	46.1277	1.55232	0.776161	−	0.630535i	$-0.217164\pi$
$883$	46.1277	1.55232	0.776161	+	0.630535i	$0.217164\pi$
$884$	3.14840	0.105892
$885$	1.54289	0.0518638
$886$	−36.2605	−1.21820
$887$	8.60550	0.288944	0.144472	−	0.989509i	$-0.453852\pi$
$887$	8.60550	0.288944	0.144472	+	0.989509i	$0.453852\pi$
$888$	−6.52009	−0.218800
$889$	−67.0701	−2.24946
$890$	−2.11465	−0.0708832
$891$	−3.40245	−0.113986
$892$	−26.1177	−0.874485
$893$	−30.2069	−1.01084
$894$	−8.59925	−0.287602
$895$	4.56610	0.152628
$896$	−4.82371	−0.161149
$897$	17.9415	0.599050
$898$	18.8920	0.630434
$899$	−4.71593	−0.157285
$900$	−17.2566	−0.575220
$901$	−17.8629	−0.595100
$902$	5.24497	0.174638
$903$	−116.746	−3.88508
$904$	−4.32497	−0.143846
$905$	6.31121	0.209792
$906$	−38.3301	−1.27343
$907$	46.9695	1.55960	0.779798	−	0.626032i	$-0.215322\pi$
$907$	46.9695	1.55960	0.779798	+	0.626032i	$0.215322\pi$
$908$	−1.03381	−0.0343080
$909$	−12.2350	−0.405808
$910$	−2.94820	−0.0977317
$911$	5.02122	0.166361	0.0831803	−	0.996535i	$-0.473492\pi$
$911$	5.02122	0.166361	0.0831803	+	0.996535i	$0.473492\pi$
$912$	−14.0875	−0.466482
$913$	4.97136	0.164528
$914$	−5.20825	−0.172273
$915$	23.2697	0.769274
$916$	−6.85550	−0.226512
$917$	−0.494550	−0.0163315
$918$	−9.93597	−0.327936
$919$	−4.75419	−0.156826	−0.0784132	−	0.996921i	$-0.524985\pi$
$919$	−4.75419	−0.156826	−0.0784132	+	0.996921i	$0.524985\pi$
$920$	6.84643	0.225720
$921$	17.4615	0.575376
$922$	18.8480	0.620726
$923$	−2.69261	−0.0886284
$924$	−8.12010	−0.267132
$925$	10.8591	0.357045
$926$	30.3051	0.995887
$927$	65.2809	2.14411
$928$	0.728405	0.0239111
$929$	−1.91013	−0.0626694	−0.0313347	−	0.999509i	$-0.509976\pi$
$929$	−1.91013	−0.0626694	−0.0313347	+	0.999509i	$0.509976\pi$
$930$	−13.3614	−0.438137
$931$	87.0282	2.85223
$932$	3.24011	0.106133
$933$	−55.3280	−1.81136
$934$	−21.5407	−0.704832
$935$	2.02244	0.0661408
$936$	3.06855	0.100299
$937$	8.19224	0.267629	0.133814	−	0.991006i	$-0.457277\pi$
$937$	8.19224	0.267629	0.133814	+	0.991006i	$0.457277\pi$
$938$	−71.8646	−2.34646
$939$	−50.2838	−1.64095
$940$	4.42518	0.144333
$941$	53.5838	1.74678	0.873390	−	0.487021i	$-0.161916\pi$
$941$	53.5838	1.74678	0.873390	+	0.487021i	$0.161916\pi$
$942$	9.51102	0.309886
$943$	−71.6792	−2.33420
$944$	0.747616	0.0243328
$945$	9.30414	0.302664
$946$	−5.87516	−0.191018
$947$	15.2052	0.494102	0.247051	−	0.969002i	$-0.420538\pi$
$947$	15.2052	0.494102	0.247051	+	0.969002i	$0.420538\pi$
$948$	15.9547	0.518184
$949$	0.883940	0.0286939
$950$	23.4624	0.761221
$951$	−3.49591	−0.113363
$952$	19.4734	0.631137
$953$	25.1076	0.813313	0.406657	−	0.913581i	$-0.366695\pi$
$953$	25.1076	0.813313	0.406657	+	0.913581i	$0.366695\pi$
$954$	−17.4099	−0.563665
$955$	12.2875	0.397614
$956$	30.5529	0.988150
$957$	1.22618	0.0396367
$958$	30.0350	0.970387
$959$	18.4715	0.596477
$960$	2.06375	0.0666073
$961$	10.9169	0.352158
$962$	−1.93095	−0.0622564
$963$	−26.9365	−0.868017
$964$	6.84419	0.220436
$965$	4.15396	0.133721
$966$	110.971	3.57045
$967$	38.1368	1.22640	0.613198	−	0.789929i	$-0.289883\pi$
$967$	38.1368	1.22640	0.613198	+	0.789929i	$0.289883\pi$
$968$	10.5914	0.340419
$969$	56.8713	1.82697
$970$	−6.27906	−0.201609
$971$	−32.7639	−1.05144	−0.525722	−	0.850657i	$-0.676205\pi$
$971$	−32.7639	−1.05144	−0.525722	+	0.850657i	$0.676205\pi$
$972$	21.4000	0.686405
$973$	45.5810	1.46126
$974$	−6.72085	−0.215350
$975$	−9.00725	−0.288463
$976$	11.2755	0.360919
$977$	−59.6344	−1.90787	−0.953937	−	0.300006i	$-0.903011\pi$
$977$	−59.6344	−1.90787	−0.953937	+	0.300006i	$0.903011\pi$
$978$	−65.4027	−2.09135
$979$	−1.72489	−0.0551277
$980$	−12.7492	−0.407260
$981$	23.0878	0.737138
$982$	8.03542	0.256421
$983$	−38.6932	−1.23412	−0.617061	−	0.786915i	$-0.711677\pi$
$983$	−38.6932	−1.23412	−0.617061	+	0.786915i	$0.711677\pi$
$984$	−21.6066	−0.688793
$985$	18.1921	0.579647
$986$	−2.94059	−0.0936474
$987$	71.7262	2.28307
$988$	−4.17206	−0.132731
$989$	80.2915	2.55312
$990$	1.97114	0.0626470
$991$	−26.8087	−0.851608	−0.425804	−	0.904815i	$-0.640009\pi$
$991$	−26.8087	−0.851608	−0.425804	+	0.904815i	$0.640009\pi$
$992$	−6.47433	−0.205560
$993$	−88.9443	−2.82256
$994$	−16.6543	−0.528241
$995$	−7.98632	−0.253183
$996$	−20.4795	−0.648916
$997$	−19.9780	−0.632709	−0.316354	−	0.948641i	$-0.602459\pi$
$997$	−19.9780	−0.632709	−0.316354	+	0.948641i	$0.602459\pi$
$998$	−5.37367	−0.170101
$999$	6.09385	0.192801

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6038.2.a.d.1.8	✓	69

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6038.2.a.d.1.8	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.63337 q^{3} +1.00000 q^{4} -0.783693 q^{5} +2.63337 q^{6} +4.82371 q^{7} -1.00000 q^{8} +3.93463 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.63337 q^{3} +1.00000 q^{4} -0.783693 q^{5} +2.63337 q^{6} +4.82371 q^{7} -1.00000 q^{8} +3.93463 q^{9} +0.783693 q^{10} +0.639247 q^{11} -2.63337 q^{12} -0.779883 q^{13} -4.82371 q^{14} +2.06375 q^{15} +1.00000 q^{16} -4.03702 q^{17} -3.93463 q^{18} +5.34960 q^{19} -0.783693 q^{20} -12.7026 q^{21} -0.639247 q^{22} +8.73612 q^{23} +2.63337 q^{24} -4.38583 q^{25} +0.779883 q^{26} -2.46121 q^{27} +4.82371 q^{28} -0.728405 q^{29} -2.06375 q^{30} +6.47433 q^{31} -1.00000 q^{32} -1.68337 q^{33} +4.03702 q^{34} -3.78031 q^{35} +3.93463 q^{36} -2.47595 q^{37} -5.34960 q^{38} +2.05372 q^{39} +0.783693 q^{40} -8.20492 q^{41} +12.7026 q^{42} +9.19075 q^{43} +0.639247 q^{44} -3.08354 q^{45} -8.73612 q^{46} -5.64658 q^{47} -2.63337 q^{48} +16.2682 q^{49} +4.38583 q^{50} +10.6310 q^{51} -0.779883 q^{52} +4.42478 q^{53} +2.46121 q^{54} -0.500973 q^{55} -4.82371 q^{56} -14.0875 q^{57} +0.728405 q^{58} +0.747616 q^{59} +2.06375 q^{60} +11.2755 q^{61} -6.47433 q^{62} +18.9795 q^{63} +1.00000 q^{64} +0.611188 q^{65} +1.68337 q^{66} +14.8982 q^{67} -4.03702 q^{68} -23.0054 q^{69} +3.78031 q^{70} +3.45258 q^{71} -3.93463 q^{72} -1.13343 q^{73} +2.47595 q^{74} +11.5495 q^{75} +5.34960 q^{76} +3.08354 q^{77} -2.05372 q^{78} -6.05867 q^{79} -0.783693 q^{80} -5.32260 q^{81} +8.20492 q^{82} +7.77691 q^{83} -12.7026 q^{84} +3.16378 q^{85} -9.19075 q^{86} +1.91816 q^{87} -0.639247 q^{88} -2.69832 q^{89} +3.08354 q^{90} -3.76193 q^{91} +8.73612 q^{92} -17.0493 q^{93} +5.64658 q^{94} -4.19244 q^{95} +2.63337 q^{96} -8.01215 q^{97} -16.2682 q^{98} +2.51520 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) \) 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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