LMFDB - Embedded newform 6009.2.a.c.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6009,2,Mod(1,6009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6009, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6009 = 3 \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6009.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:	$47.9821065746$
Analytic rank:	$0$
Dimension:	$92$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	6009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.49751 q^{2} +1.00000 q^{3} +4.23755 q^{4} +1.03851 q^{5} -2.49751 q^{6} +3.59057 q^{7} -5.58831 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.49751 q^{2} +1.00000 q^{3} +4.23755 q^{4} +1.03851 q^{5} -2.49751 q^{6} +3.59057 q^{7} -5.58831 q^{8} +1.00000 q^{9} -2.59368 q^{10} +3.26925 q^{11} +4.23755 q^{12} -6.47501 q^{13} -8.96747 q^{14} +1.03851 q^{15} +5.48176 q^{16} -2.83011 q^{17} -2.49751 q^{18} +5.75723 q^{19} +4.40073 q^{20} +3.59057 q^{21} -8.16499 q^{22} +8.88748 q^{23} -5.58831 q^{24} -3.92151 q^{25} +16.1714 q^{26} +1.00000 q^{27} +15.2152 q^{28} +5.30574 q^{29} -2.59368 q^{30} +9.12036 q^{31} -2.51411 q^{32} +3.26925 q^{33} +7.06822 q^{34} +3.72882 q^{35} +4.23755 q^{36} -6.70483 q^{37} -14.3787 q^{38} -6.47501 q^{39} -5.80350 q^{40} +3.30609 q^{41} -8.96747 q^{42} -9.69454 q^{43} +13.8536 q^{44} +1.03851 q^{45} -22.1966 q^{46} -7.35956 q^{47} +5.48176 q^{48} +5.89217 q^{49} +9.79400 q^{50} -2.83011 q^{51} -27.4382 q^{52} +8.64314 q^{53} -2.49751 q^{54} +3.39514 q^{55} -20.0652 q^{56} +5.75723 q^{57} -13.2511 q^{58} -5.05582 q^{59} +4.40073 q^{60} -6.77839 q^{61} -22.7782 q^{62} +3.59057 q^{63} -4.68449 q^{64} -6.72434 q^{65} -8.16499 q^{66} -3.53678 q^{67} -11.9927 q^{68} +8.88748 q^{69} -9.31278 q^{70} +12.2921 q^{71} -5.58831 q^{72} +7.19047 q^{73} +16.7454 q^{74} -3.92151 q^{75} +24.3966 q^{76} +11.7385 q^{77} +16.1714 q^{78} +4.49239 q^{79} +5.69284 q^{80} +1.00000 q^{81} -8.25700 q^{82} +1.79287 q^{83} +15.2152 q^{84} -2.93908 q^{85} +24.2122 q^{86} +5.30574 q^{87} -18.2696 q^{88} -0.333276 q^{89} -2.59368 q^{90} -23.2490 q^{91} +37.6612 q^{92} +9.12036 q^{93} +18.3806 q^{94} +5.97892 q^{95} -2.51411 q^{96} +0.115527 q^{97} -14.7157 q^{98} +3.26925 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 92 q + 17 q^{2} + 92 q^{3} + 107 q^{4} + 34 q^{5} + 17 q^{6} + 22 q^{7} + 51 q^{8} + 92 q^{9}+O(q^{10}) $ 92 * q + 17 * q^2 + 92 * q^3 + 107 * q^4 + 34 * q^5 + 17 * q^6 + 22 * q^7 + 51 * q^8 + 92 * q^9	\( 92 q + 17 q^{2} + 92 q^{3} + 107 q^{4} + 34 q^{5} + 17 q^{6} + 22 q^{7} + 51 q^{8} + 92 q^{9} + 13 q^{10} + 40 q^{11} + 107 q^{12} + 6 q^{13} + 37 q^{14} + 34 q^{15} + 133 q^{16} + 77 q^{17} + 17 q^{18} + 34 q^{19} + 55 q^{20} + 22 q^{21} + 8 q^{22} + 83 q^{23} + 51 q^{24} + 110 q^{25} + 22 q^{26} + 92 q^{27} + 32 q^{28} + 97 q^{29} + 13 q^{30} + 44 q^{31} + 104 q^{32} + 40 q^{33} + 20 q^{34} + 80 q^{35} + 107 q^{36} + 12 q^{37} + 54 q^{38} + 6 q^{39} + 23 q^{40} + 67 q^{41} + 37 q^{42} + 30 q^{43} + 87 q^{44} + 34 q^{45} + 33 q^{46} + 69 q^{47} + 133 q^{48} + 112 q^{49} + 58 q^{50} + 77 q^{51} - 3 q^{52} + 113 q^{53} + 17 q^{54} + 42 q^{55} + 92 q^{56} + 34 q^{57} - 30 q^{58} + 72 q^{59} + 55 q^{60} + 19 q^{61} + 60 q^{62} + 22 q^{63} + 147 q^{64} + 74 q^{65} + 8 q^{66} + 26 q^{67} + 171 q^{68} + 83 q^{69} - 35 q^{70} + 134 q^{71} + 51 q^{72} - 17 q^{73} + 95 q^{74} + 110 q^{75} + 27 q^{76} + 108 q^{77} + 22 q^{78} + 159 q^{79} + 79 q^{80} + 92 q^{81} - 64 q^{82} + 73 q^{83} + 32 q^{84} - 4 q^{85} + 22 q^{86} + 97 q^{87} - 16 q^{88} + 50 q^{89} + 13 q^{90} + 17 q^{91} + 154 q^{92} + 44 q^{93} + 8 q^{94} + 155 q^{95} + 104 q^{96} - 20 q^{97} + 63 q^{98} + 40 q^{99}+O(q^{100}) \) 92 * q + 17 * q^2 + 92 * q^3 + 107 * q^4 + 34 * q^5 + 17 * q^6 + 22 * q^7 + 51 * q^8 + 92 * q^9 + 13 * q^10 + 40 * q^11 + 107 * q^12 + 6 * q^13 + 37 * q^14 + 34 * q^15 + 133 * q^16 + 77 * q^17 + 17 * q^18 + 34 * q^19 + 55 * q^20 + 22 * q^21 + 8 * q^22 + 83 * q^23 + 51 * q^24 + 110 * q^25 + 22 * q^26 + 92 * q^27 + 32 * q^28 + 97 * q^29 + 13 * q^30 + 44 * q^31 + 104 * q^32 + 40 * q^33 + 20 * q^34 + 80 * q^35 + 107 * q^36 + 12 * q^37 + 54 * q^38 + 6 * q^39 + 23 * q^40 + 67 * q^41 + 37 * q^42 + 30 * q^43 + 87 * q^44 + 34 * q^45 + 33 * q^46 + 69 * q^47 + 133 * q^48 + 112 * q^49 + 58 * q^50 + 77 * q^51 - 3 * q^52 + 113 * q^53 + 17 * q^54 + 42 * q^55 + 92 * q^56 + 34 * q^57 - 30 * q^58 + 72 * q^59 + 55 * q^60 + 19 * q^61 + 60 * q^62 + 22 * q^63 + 147 * q^64 + 74 * q^65 + 8 * q^66 + 26 * q^67 + 171 * q^68 + 83 * q^69 - 35 * q^70 + 134 * q^71 + 51 * q^72 - 17 * q^73 + 95 * q^74 + 110 * q^75 + 27 * q^76 + 108 * q^77 + 22 * q^78 + 159 * q^79 + 79 * q^80 + 92 * q^81 - 64 * q^82 + 73 * q^83 + 32 * q^84 - 4 * q^85 + 22 * q^86 + 97 * q^87 - 16 * q^88 + 50 * q^89 + 13 * q^90 + 17 * q^91 + 154 * q^92 + 44 * q^93 + 8 * q^94 + 155 * q^95 + 104 * q^96 - 20 * q^97 + 63 * q^98 + 40 * 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 $

$2$	−2.49751	−1.76601	−0.883003	−	0.469367i	$-0.844482\pi$
$2$	−2.49751	−1.76601	−0.883003	+	0.469367i	$0.844482\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.23755	2.11878
$5$	1.03851	0.464434	0.232217	−	0.972664i	$-0.425402\pi$
$5$	1.03851	0.464434	0.232217	+	0.972664i	$0.425402\pi$
$6$	−2.49751	−1.01960
$7$	3.59057	1.35711	0.678553	−	0.734551i	$-0.262607\pi$
$7$	3.59057	1.35711	0.678553	+	0.734551i	$0.262607\pi$
$8$	−5.58831	−1.97577
$9$	1.00000	0.333333
$10$	−2.59368	−0.820193
$11$	3.26925	0.985716	0.492858	−	0.870110i	$-0.335952\pi$
$11$	3.26925	0.985716	0.492858	+	0.870110i	$0.335952\pi$
$12$	4.23755	1.22328
$13$	−6.47501	−1.79585	−0.897923	−	0.440153i	$-0.854924\pi$
$13$	−6.47501	−1.79585	−0.897923	+	0.440153i	$0.854924\pi$
$14$	−8.96747	−2.39666
$15$	1.03851	0.268141
$16$	5.48176	1.37044
$17$	−2.83011	−0.686402	−0.343201	−	0.939262i	$-0.611511\pi$
$17$	−2.83011	−0.686402	−0.343201	+	0.939262i	$0.611511\pi$
$18$	−2.49751	−0.588669
$19$	5.75723	1.32080	0.660399	−	0.750914i	$-0.270387\pi$
$19$	5.75723	1.32080	0.660399	+	0.750914i	$0.270387\pi$
$20$	4.40073	0.984032
$21$	3.59057	0.783526
$22$	−8.16499	−1.74078
$23$	8.88748	1.85317	0.926584	−	0.376088i	$-0.122731\pi$
$23$	8.88748	1.85317	0.926584	+	0.376088i	$0.122731\pi$
$24$	−5.58831	−1.14071
$25$	−3.92151	−0.784301
$26$	16.1714	3.17147
$27$	1.00000	0.192450
$28$	15.2152	2.87541
$29$	5.30574	0.985251	0.492625	−	0.870242i	$-0.336037\pi$
$29$	5.30574	0.985251	0.492625	+	0.870242i	$0.336037\pi$
$30$	−2.59368	−0.473539
$31$	9.12036	1.63806	0.819032	−	0.573748i	$-0.194511\pi$
$31$	9.12036	1.63806	0.819032	+	0.573748i	$0.194511\pi$
$32$	−2.51411	−0.444436
$33$	3.26925	0.569104
$34$	7.06822	1.21219
$35$	3.72882	0.630286
$36$	4.23755	0.706259
$37$	−6.70483	−1.10227	−0.551134	−	0.834417i	$-0.685805\pi$
$37$	−6.70483	−1.10227	−0.551134	+	0.834417i	$0.685805\pi$
$38$	−14.3787	−2.33254
$39$	−6.47501	−1.03683
$40$	−5.80350	−0.917613
$41$	3.30609	0.516325	0.258163	−	0.966101i	$-0.416883\pi$
$41$	3.30609	0.516325	0.258163	+	0.966101i	$0.416883\pi$
$42$	−8.96747	−1.38371
$43$	−9.69454	−1.47840	−0.739202	−	0.673484i	$-0.764797\pi$
$43$	−9.69454	−1.47840	−0.739202	+	0.673484i	$0.764797\pi$
$44$	13.8536	2.08851
$45$	1.03851	0.154811
$46$	−22.1966	−3.27271
$47$	−7.35956	−1.07350	−0.536751	−	0.843741i	$-0.680349\pi$
$47$	−7.35956	−1.07350	−0.536751	+	0.843741i	$0.680349\pi$
$48$	5.48176	0.791223
$49$	5.89217	0.841738
$50$	9.79400	1.38508
$51$	−2.83011	−0.396294
$52$	−27.4382	−3.80500
$53$	8.64314	1.18723	0.593613	−	0.804750i	$-0.297701\pi$
$53$	8.64314	1.18723	0.593613	+	0.804750i	$0.297701\pi$
$54$	−2.49751	−0.339868
$55$	3.39514	0.457800
$56$	−20.0652	−2.68133
$57$	5.75723	0.762564
$58$	−13.2511	−1.73996
$59$	−5.05582	−0.658212	−0.329106	−	0.944293i	$-0.606747\pi$
$59$	−5.05582	−0.658212	−0.329106	+	0.944293i	$0.606747\pi$
$60$	4.40073	0.568131
$61$	−6.77839	−0.867883	−0.433942	−	0.900941i	$-0.642878\pi$
$61$	−6.77839	−0.867883	−0.433942	+	0.900941i	$0.642878\pi$
$62$	−22.7782	−2.89283
$63$	3.59057	0.452369
$64$	−4.68449	−0.585562
$65$	−6.72434	−0.834052
$66$	−8.16499	−1.00504
$67$	−3.53678	−0.432087	−0.216043	−	0.976384i	$-0.569315\pi$
$67$	−3.53678	−0.432087	−0.216043	+	0.976384i	$0.569315\pi$
$68$	−11.9927	−1.45433
$69$	8.88748	1.06993
$70$	−9.31278	−1.11309
$71$	12.2921	1.45880	0.729402	−	0.684085i	$-0.239798\pi$
$71$	12.2921	1.45880	0.729402	+	0.684085i	$0.239798\pi$
$72$	−5.58831	−0.658589
$73$	7.19047	0.841581	0.420790	−	0.907158i	$-0.361753\pi$
$73$	7.19047	0.841581	0.420790	+	0.907158i	$0.361753\pi$
$74$	16.7454	1.94661
$75$	−3.92151	−0.452816
$76$	24.3966	2.79848
$77$	11.7385	1.33772
$78$	16.1714	1.83105
$79$	4.49239	0.505433	0.252716	−	0.967540i	$-0.418676\pi$
$79$	4.49239	0.505433	0.252716	+	0.967540i	$0.418676\pi$
$80$	5.69284	0.636478
$81$	1.00000	0.111111
$82$	−8.25700	−0.911833
$83$	1.79287	0.196793	0.0983963	−	0.995147i	$-0.468629\pi$
$83$	1.79287	0.196793	0.0983963	+	0.995147i	$0.468629\pi$
$84$	15.2152	1.66012
$85$	−2.93908	−0.318788
$86$	24.2122	2.61087
$87$	5.30574	0.568835
$88$	−18.2696	−1.94755
$89$	−0.333276	−0.0353272	−0.0176636	−	0.999844i	$-0.505623\pi$
$89$	−0.333276	−0.0353272	−0.0176636	+	0.999844i	$0.505623\pi$
$90$	−2.59368	−0.273398
$91$	−23.2490	−2.43715
$92$	37.6612	3.92645
$93$	9.12036	0.945737
$94$	18.3806	1.89581
$95$	5.97892	0.613424
$96$	−2.51411	−0.256595
$97$	0.115527	0.0117300	0.00586498	−	0.999983i	$-0.498133\pi$
$97$	0.115527	0.0117300	0.00586498	+	0.999983i	$0.498133\pi$
$98$	−14.7157	−1.48651
$99$	3.26925	0.328572
$100$	−16.6176	−1.66176
$101$	10.2894	1.02383	0.511914	−	0.859036i	$-0.328936\pi$
$101$	10.2894	1.02383	0.511914	+	0.859036i	$0.328936\pi$
$102$	7.06822	0.699858
$103$	3.07723	0.303209	0.151604	−	0.988441i	$-0.451556\pi$
$103$	3.07723	0.303209	0.151604	+	0.988441i	$0.451556\pi$
$104$	36.1844	3.54817
$105$	3.72882	0.363896
$106$	−21.5863	−2.09665
$107$	8.32231	0.804548	0.402274	−	0.915519i	$-0.368220\pi$
$107$	8.32231	0.804548	0.402274	+	0.915519i	$0.368220\pi$
$108$	4.23755	0.407759
$109$	17.4130	1.66786	0.833930	−	0.551870i	$-0.186086\pi$
$109$	17.4130	1.66786	0.833930	+	0.551870i	$0.186086\pi$
$110$	−8.47939	−0.808478
$111$	−6.70483	−0.636394
$112$	19.6826	1.85983
$113$	−4.05787	−0.381732	−0.190866	−	0.981616i	$-0.561130\pi$
$113$	−4.05787	−0.381732	−0.190866	+	0.981616i	$0.561130\pi$
$114$	−14.3787	−1.34669
$115$	9.22970	0.860674
$116$	22.4833	2.08753
$117$	−6.47501	−0.598615
$118$	12.6270	1.16241
$119$	−10.1617	−0.931521
$120$	−5.80350	−0.529784
$121$	−0.311996	−0.0283633
$122$	16.9291	1.53269
$123$	3.30609	0.298100
$124$	38.6480	3.47069
$125$	−9.26504	−0.828690
$126$	−8.96747	−0.798886
$127$	−18.1318	−1.60894	−0.804470	−	0.593993i	$-0.797551\pi$
$127$	−18.1318	−1.60894	−0.804470	+	0.593993i	$0.797551\pi$
$128$	16.7278	1.47854
$129$	−9.69454	−0.853557
$130$	16.7941	1.47294
$131$	2.37992	0.207935	0.103967	−	0.994581i	$-0.466846\pi$
$131$	2.37992	0.207935	0.103967	+	0.994581i	$0.466846\pi$
$132$	13.8536	1.20580
$133$	20.6717	1.79246
$134$	8.83315	0.763068
$135$	1.03851	0.0893804
$136$	15.8155	1.35617
$137$	−3.82369	−0.326680	−0.163340	−	0.986570i	$-0.552227\pi$
$137$	−3.82369	−0.326680	−0.163340	+	0.986570i	$0.552227\pi$
$138$	−22.1966	−1.88950
$139$	22.9324	1.94510	0.972551	−	0.232688i	$-0.0747520\pi$
$139$	22.9324	1.94510	0.972551	+	0.232688i	$0.0747520\pi$
$140$	15.8011	1.33544
$141$	−7.35956	−0.619787
$142$	−30.6997	−2.57626
$143$	−21.1684	−1.77019
$144$	5.48176	0.456813
$145$	5.51004	0.457584
$146$	−17.9583	−1.48624
$147$	5.89217	0.485978
$148$	−28.4121	−2.33546
$149$	15.3927	1.26102	0.630510	−	0.776181i	$-0.282846\pi$
$149$	15.3927	1.26102	0.630510	+	0.776181i	$0.282846\pi$
$150$	9.79400	0.799676
$151$	8.42763	0.685831	0.342916	−	0.939366i	$-0.388586\pi$
$151$	8.42763	0.685831	0.342916	+	0.939366i	$0.388586\pi$
$152$	−32.1732	−2.60959
$153$	−2.83011	−0.228801
$154$	−29.3169	−2.36243
$155$	9.47154	0.760773
$156$	−27.4382	−2.19682
$157$	−11.0313	−0.880391	−0.440196	−	0.897902i	$-0.645091\pi$
$157$	−11.0313	−0.880391	−0.440196	+	0.897902i	$0.645091\pi$
$158$	−11.2198	−0.892597
$159$	8.64314	0.685446
$160$	−2.61092	−0.206411
$161$	31.9111	2.51495
$162$	−2.49751	−0.196223
$163$	6.33684	0.496340	0.248170	−	0.968717i	$-0.420171\pi$
$163$	6.33684	0.496340	0.248170	+	0.968717i	$0.420171\pi$
$164$	14.0098	1.09398
$165$	3.39514	0.264311
$166$	−4.47770	−0.347537
$167$	−16.2181	−1.25499	−0.627496	−	0.778620i	$-0.715920\pi$
$167$	−16.2181	−1.25499	−0.627496	+	0.778620i	$0.715920\pi$
$168$	−20.0652	−1.54806
$169$	28.9258	2.22506
$170$	7.34039	0.562982
$171$	5.75723	0.440266
$172$	−41.0811	−3.13241
$173$	−14.6141	−1.11109	−0.555545	−	0.831486i	$-0.687491\pi$
$173$	−14.6141	−1.11109	−0.555545	+	0.831486i	$0.687491\pi$
$174$	−13.2511	−1.00457
$175$	−14.0804	−1.06438
$176$	17.9212	1.35086
$177$	−5.05582	−0.380019
$178$	0.832360	0.0623880
$179$	1.43595	0.107328	0.0536638	−	0.998559i	$-0.482910\pi$
$179$	1.43595	0.107328	0.0536638	+	0.998559i	$0.482910\pi$
$180$	4.40073	0.328011
$181$	−16.4547	−1.22307	−0.611536	−	0.791217i	$-0.709448\pi$
$181$	−16.4547	−1.22307	−0.611536	+	0.791217i	$0.709448\pi$
$182$	58.0645	4.30403
$183$	−6.77839	−0.501073
$184$	−49.6660	−3.66143
$185$	−6.96301	−0.511930
$186$	−22.7782	−1.67018
$187$	−9.25233	−0.676598
$188$	−31.1865	−2.27451
$189$	3.59057	0.261175
$190$	−14.9324	−1.08331
$191$	−9.05642	−0.655300	−0.327650	−	0.944799i	$-0.606257\pi$
$191$	−9.05642	−0.655300	−0.327650	+	0.944799i	$0.606257\pi$
$192$	−4.68449	−0.338074
$193$	15.0117	1.08056	0.540281	−	0.841485i	$-0.318318\pi$
$193$	15.0117	1.08056	0.540281	+	0.841485i	$0.318318\pi$
$194$	−0.288529	−0.0207152
$195$	−6.72434	−0.481540
$196$	24.9684	1.78345
$197$	18.2379	1.29940	0.649698	−	0.760192i	$-0.274895\pi$
$197$	18.2379	1.29940	0.649698	+	0.760192i	$0.274895\pi$
$198$	−8.16499	−0.580260
$199$	−18.9414	−1.34272	−0.671360	−	0.741131i	$-0.734290\pi$
$199$	−18.9414	−1.34272	−0.671360	+	0.741131i	$0.734290\pi$
$200$	21.9146	1.54960
$201$	−3.53678	−0.249465
$202$	−25.6978	−1.80809
$203$	19.0506	1.33709
$204$	−11.9927	−0.839659
$205$	3.43340	0.239799
$206$	−7.68542	−0.535469
$207$	8.88748	0.617723
$208$	−35.4944	−2.46110
$209$	18.8218	1.30193
$210$	−9.31278	−0.642643
$211$	−11.3609	−0.782119	−0.391060	−	0.920365i	$-0.627891\pi$
$211$	−11.3609	−0.782119	−0.391060	+	0.920365i	$0.627891\pi$
$212$	36.6258	2.51547
$213$	12.2921	0.842241
$214$	−20.7851	−1.42084
$215$	−10.0678	−0.686621
$216$	−5.58831	−0.380236
$217$	32.7472	2.22303
$218$	−43.4891	−2.94545
$219$	7.19047	0.485887
$220$	14.3871	0.969977
$221$	18.3250	1.23267
$222$	16.7454	1.12388
$223$	9.30602	0.623177	0.311589	−	0.950217i	$-0.399139\pi$
$223$	9.30602	0.623177	0.311589	+	0.950217i	$0.399139\pi$
$224$	−9.02709	−0.603147
$225$	−3.92151	−0.261434
$226$	10.1346	0.674141
$227$	20.3456	1.35038	0.675191	−	0.737643i	$-0.264061\pi$
$227$	20.3456	1.35038	0.675191	+	0.737643i	$0.264061\pi$
$228$	24.3966	1.61570
$229$	−28.2211	−1.86490	−0.932452	−	0.361294i	$-0.882335\pi$
$229$	−28.2211	−1.86490	−0.932452	+	0.361294i	$0.882335\pi$
$230$	−23.0513	−1.51996
$231$	11.7385	0.772334
$232$	−29.6501	−1.94663
$233$	−5.95417	−0.390071	−0.195035	−	0.980796i	$-0.562482\pi$
$233$	−5.95417	−0.390071	−0.195035	+	0.980796i	$0.562482\pi$
$234$	16.1714	1.05716
$235$	−7.64295	−0.498571
$236$	−21.4243	−1.39460
$237$	4.49239	0.291812
$238$	25.3789	1.64507
$239$	−9.39985	−0.608026	−0.304013	−	0.952668i	$-0.598327\pi$
$239$	−9.39985	−0.608026	−0.304013	+	0.952668i	$0.598327\pi$
$240$	5.69284	0.367471
$241$	27.1187	1.74687	0.873434	−	0.486943i	$-0.161888\pi$
$241$	27.1187	1.74687	0.873434	+	0.486943i	$0.161888\pi$
$242$	0.779214	0.0500897
$243$	1.00000	0.0641500
$244$	−28.7238	−1.83885
$245$	6.11905	0.390932
$246$	−8.25700	−0.526447
$247$	−37.2781	−2.37195
$248$	−50.9674	−3.23643
$249$	1.79287	0.113618
$250$	23.1395	1.46347
$251$	16.1589	1.01994	0.509972	−	0.860191i	$-0.329656\pi$
$251$	16.1589	1.01994	0.509972	+	0.860191i	$0.329656\pi$
$252$	15.2152	0.958469
$253$	29.0554	1.82670
$254$	45.2844	2.84140
$255$	−2.93908	−0.184053
$256$	−32.4088	−2.02555
$257$	12.9525	0.807958	0.403979	−	0.914768i	$-0.367627\pi$
$257$	12.9525	0.807958	0.403979	+	0.914768i	$0.367627\pi$
$258$	24.2122	1.50739
$259$	−24.0741	−1.49589
$260$	−28.4948	−1.76717
$261$	5.30574	0.328417
$262$	−5.94388	−0.367214
$263$	24.1007	1.48611	0.743055	−	0.669230i	$-0.233376\pi$
$263$	24.1007	1.48611	0.743055	+	0.669230i	$0.233376\pi$
$264$	−18.2696	−1.12442
$265$	8.97595	0.551389
$266$	−51.6278	−3.16550
$267$	−0.333276	−0.0203962
$268$	−14.9873	−0.915496
$269$	−5.19253	−0.316594	−0.158297	−	0.987392i	$-0.550600\pi$
$269$	−5.19253	−0.316594	−0.158297	+	0.987392i	$0.550600\pi$
$270$	−2.59368	−0.157846
$271$	−1.44018	−0.0874848	−0.0437424	−	0.999043i	$-0.513928\pi$
$271$	−1.44018	−0.0874848	−0.0437424	+	0.999043i	$0.513928\pi$
$272$	−15.5140	−0.940672
$273$	−23.2490	−1.40709
$274$	9.54971	0.576919
$275$	−12.8204	−0.773098
$276$	37.6612	2.26694
$277$	23.2692	1.39811	0.699056	−	0.715067i	$-0.253604\pi$
$277$	23.2692	1.39811	0.699056	+	0.715067i	$0.253604\pi$
$278$	−57.2740	−3.43506
$279$	9.12036	0.546021
$280$	−20.8378	−1.24530
$281$	−18.8932	−1.12707	−0.563536	−	0.826091i	$-0.690560\pi$
$281$	−18.8932	−1.12707	−0.563536	+	0.826091i	$0.690560\pi$
$282$	18.3806	1.09455
$283$	−17.6763	−1.05075	−0.525375	−	0.850871i	$-0.676075\pi$
$283$	−17.6763	−1.05075	−0.525375	+	0.850871i	$0.676075\pi$
$284$	52.0885	3.09088
$285$	5.97892	0.354161
$286$	52.8684	3.12617
$287$	11.8707	0.700708
$288$	−2.51411	−0.148145
$289$	−8.99049	−0.528852
$290$	−13.7614	−0.808096
$291$	0.115527	0.00677229
$292$	30.4700	1.78312
$293$	−14.0558	−0.821151	−0.410576	−	0.911827i	$-0.634672\pi$
$293$	−14.0558	−0.821151	−0.410576	+	0.911827i	$0.634672\pi$
$294$	−14.7157	−0.858239
$295$	−5.25050	−0.305696
$296$	37.4687	2.17782
$297$	3.26925	0.189701
$298$	−38.4435	−2.22697
$299$	−57.5465	−3.32800
$300$	−16.6176	−0.959417
$301$	−34.8089	−2.00635
$302$	−21.0481	−1.21118
$303$	10.2894	0.591108
$304$	31.5597	1.81007
$305$	−7.03939	−0.403075
$306$	7.06822	0.404063
$307$	−1.87721	−0.107138	−0.0535689	−	0.998564i	$-0.517060\pi$
$307$	−1.87721	−0.107138	−0.0535689	+	0.998564i	$0.517060\pi$
$308$	49.7424	2.83433
$309$	3.07723	0.175058
$310$	−23.6553	−1.34353
$311$	−5.63137	−0.319326	−0.159663	−	0.987172i	$-0.551041\pi$
$311$	−5.63137	−0.319326	−0.159663	+	0.987172i	$0.551041\pi$
$312$	36.1844	2.04854
$313$	−10.6822	−0.603792	−0.301896	−	0.953341i	$-0.597620\pi$
$313$	−10.6822	−0.603792	−0.301896	+	0.953341i	$0.597620\pi$
$314$	27.5507	1.55478
$315$	3.72882	0.210095
$316$	19.0367	1.07090
$317$	−7.39685	−0.415449	−0.207724	−	0.978187i	$-0.566606\pi$
$317$	−7.39685	−0.415449	−0.207724	+	0.978187i	$0.566606\pi$
$318$	−21.5863	−1.21050
$319$	17.3458	0.971178
$320$	−4.86487	−0.271955
$321$	8.32231	0.464506
$322$	−79.6982	−4.44141
$323$	−16.2936	−0.906599
$324$	4.23755	0.235420
$325$	25.3918	1.40848
$326$	−15.8263	−0.876539
$327$	17.4130	0.962939
$328$	−18.4755	−1.02014
$329$	−26.4250	−1.45686
$330$	−8.47939	−0.466775
$331$	10.8155	0.594473	0.297236	−	0.954804i	$-0.403935\pi$
$331$	10.8155	0.594473	0.297236	+	0.954804i	$0.403935\pi$
$332$	7.59737	0.416960
$333$	−6.70483	−0.367422
$334$	40.5048	2.21632
$335$	−3.67297	−0.200676
$336$	19.6826	1.07377
$337$	21.5931	1.17625	0.588126	−	0.808770i	$-0.299866\pi$
$337$	21.5931	1.17625	0.588126	+	0.808770i	$0.299866\pi$
$338$	−72.2424	−3.92947
$339$	−4.05787	−0.220393
$340$	−12.4545	−0.675442
$341$	29.8167	1.61467
$342$	−14.3787	−0.777513
$343$	−3.97775	−0.214778
$344$	54.1761	2.92098
$345$	9.22970	0.496910
$346$	36.4989	1.96219
$347$	−0.770614	−0.0413687	−0.0206844	−	0.999786i	$-0.506585\pi$
$347$	−0.770614	−0.0413687	−0.0206844	+	0.999786i	$0.506585\pi$
$348$	22.4833	1.20523
$349$	−16.8669	−0.902862	−0.451431	−	0.892306i	$-0.649086\pi$
$349$	−16.8669	−0.902862	−0.451431	+	0.892306i	$0.649086\pi$
$350$	35.1660	1.87970
$351$	−6.47501	−0.345611
$352$	−8.21926	−0.438088
$353$	21.9207	1.16672	0.583361	−	0.812213i	$-0.301737\pi$
$353$	21.9207	1.16672	0.583361	+	0.812213i	$0.301737\pi$
$354$	12.6270	0.671116
$355$	12.7654	0.677519
$356$	−1.41227	−0.0748504
$357$	−10.1617	−0.537814
$358$	−3.58629	−0.189541
$359$	24.1579	1.27501	0.637503	−	0.770448i	$-0.279968\pi$
$359$	24.1579	1.27501	0.637503	+	0.770448i	$0.279968\pi$
$360$	−5.80350	−0.305871
$361$	14.1457	0.744510
$362$	41.0959	2.15995
$363$	−0.311996	−0.0163756
$364$	−98.5187	−5.16378
$365$	7.46735	0.390859
$366$	16.9291	0.884897
$367$	−21.8860	−1.14244	−0.571219	−	0.820798i	$-0.693529\pi$
$367$	−21.8860	−1.14244	−0.571219	+	0.820798i	$0.693529\pi$
$368$	48.7190	2.53965
$369$	3.30609	0.172108
$370$	17.3902	0.904072
$371$	31.0338	1.61119
$372$	38.6480	2.00381
$373$	9.20963	0.476857	0.238428	−	0.971160i	$-0.423368\pi$
$373$	9.20963	0.476857	0.238428	+	0.971160i	$0.423368\pi$
$374$	23.1078	1.19488
$375$	−9.26504	−0.478444
$376$	41.1275	2.12099
$377$	−34.3547	−1.76936
$378$	−8.96747	−0.461237
$379$	−8.81044	−0.452562	−0.226281	−	0.974062i	$-0.572657\pi$
$379$	−8.81044	−0.452562	−0.226281	+	0.974062i	$0.572657\pi$
$380$	25.3360	1.29971
$381$	−18.1318	−0.928922
$382$	22.6185	1.15726
$383$	−16.1291	−0.824160	−0.412080	−	0.911148i	$-0.635198\pi$
$383$	−16.1291	−0.824160	−0.412080	+	0.911148i	$0.635198\pi$
$384$	16.7278	0.853636
$385$	12.1905	0.621284
$386$	−37.4917	−1.90828
$387$	−9.69454	−0.492801
$388$	0.489550	0.0248532
$389$	0.447358	0.0226819	0.0113410	−	0.999936i	$-0.496390\pi$
$389$	0.447358	0.0226819	0.0113410	+	0.999936i	$0.496390\pi$
$390$	16.7941	0.850403
$391$	−25.1525	−1.27202
$392$	−32.9273	−1.66308
$393$	2.37992	0.120051
$394$	−45.5493	−2.29474
$395$	4.66537	0.234740
$396$	13.8536	0.696171
$397$	23.0482	1.15676	0.578378	−	0.815769i	$-0.303686\pi$
$397$	23.0482	1.15676	0.578378	+	0.815769i	$0.303686\pi$
$398$	47.3063	2.37125
$399$	20.6717	1.03488
$400$	−21.4967	−1.07484
$401$	10.8875	0.543697	0.271848	−	0.962340i	$-0.412365\pi$
$401$	10.8875	0.543697	0.271848	+	0.962340i	$0.412365\pi$
$402$	8.83315	0.440557
$403$	−59.0544	−2.94171
$404$	43.6017	2.16926
$405$	1.03851	0.0516038
$406$	−47.5791	−2.36131
$407$	−21.9198	−1.08652
$408$	15.8155	0.782985
$409$	34.4649	1.70418	0.852089	−	0.523398i	$-0.175336\pi$
$409$	34.4649	1.70418	0.852089	+	0.523398i	$0.175336\pi$
$410$	−8.57495	−0.423486
$411$	−3.82369	−0.188609
$412$	13.0399	0.642432
$413$	−18.1533	−0.893264
$414$	−22.1966	−1.09090
$415$	1.86190	0.0913972
$416$	16.2789	0.798139
$417$	22.9324	1.12301
$418$	−47.0077	−2.29922
$419$	−36.3512	−1.77587	−0.887937	−	0.459966i	$-0.847862\pi$
$419$	−36.3512	−1.77587	−0.887937	+	0.459966i	$0.847862\pi$
$420$	15.8011	0.771015
$421$	−18.8544	−0.918908	−0.459454	−	0.888202i	$-0.651955\pi$
$421$	−18.8544	−0.918908	−0.459454	+	0.888202i	$0.651955\pi$
$422$	28.3741	1.38123
$423$	−7.35956	−0.357834
$424$	−48.3006	−2.34568
$425$	11.0983	0.538346
$426$	−30.6997	−1.48740
$427$	−24.3382	−1.17781
$428$	35.2662	1.70466
$429$	−21.1684	−1.02202
$430$	25.1445	1.21258
$431$	15.0844	0.726591	0.363296	−	0.931674i	$-0.381652\pi$
$431$	15.0844	0.726591	0.363296	+	0.931674i	$0.381652\pi$
$432$	5.48176	0.263741
$433$	−33.1408	−1.59265	−0.796324	−	0.604871i	$-0.793225\pi$
$433$	−33.1408	−1.59265	−0.796324	+	0.604871i	$0.793225\pi$
$434$	−81.7865	−3.92588
$435$	5.51004	0.264186
$436$	73.7884	3.53382
$437$	51.1673	2.44766
$438$	−17.9583	−0.858079
$439$	−24.3230	−1.16087	−0.580437	−	0.814305i	$-0.697118\pi$
$439$	−24.3230	−1.16087	−0.580437	+	0.814305i	$0.697118\pi$
$440$	−18.9731	−0.904506
$441$	5.89217	0.280579
$442$	−45.7668	−2.17691
$443$	−0.410739	−0.0195148	−0.00975741	−	0.999952i	$-0.503106\pi$
$443$	−0.410739	−0.0195148	−0.00975741	+	0.999952i	$0.503106\pi$
$444$	−28.4121	−1.34838
$445$	−0.346109	−0.0164071
$446$	−23.2419	−1.10053
$447$	15.3927	0.728051
$448$	−16.8200	−0.794669
$449$	5.47078	0.258182	0.129091	−	0.991633i	$-0.458794\pi$
$449$	5.47078	0.258182	0.129091	+	0.991633i	$0.458794\pi$
$450$	9.79400	0.461693
$451$	10.8085	0.508950
$452$	−17.1954	−0.808805
$453$	8.42763	0.395965
$454$	−50.8132	−2.38478
$455$	−24.1442	−1.13190
$456$	−32.1732	−1.50665
$457$	33.9305	1.58720	0.793602	−	0.608437i	$-0.208203\pi$
$457$	33.9305	1.58720	0.793602	+	0.608437i	$0.208203\pi$
$458$	70.4825	3.29343
$459$	−2.83011	−0.132098
$460$	39.1114	1.82358
$461$	−6.80897	−0.317126	−0.158563	−	0.987349i	$-0.550686\pi$
$461$	−6.80897	−0.317126	−0.158563	+	0.987349i	$0.550686\pi$
$462$	−29.3169	−1.36395
$463$	−2.13700	−0.0993149	−0.0496574	−	0.998766i	$-0.515813\pi$
$463$	−2.13700	−0.0993149	−0.0496574	+	0.998766i	$0.515813\pi$
$464$	29.0847	1.35023
$465$	9.47154	0.439232
$466$	14.8706	0.688868
$467$	−31.3982	−1.45294	−0.726468	−	0.687200i	$-0.758839\pi$
$467$	−31.3982	−1.45294	−0.726468	+	0.687200i	$0.758839\pi$
$468$	−27.4382	−1.26833
$469$	−12.6991	−0.586388
$470$	19.0883	0.880479
$471$	−11.0313	−0.508294
$472$	28.2535	1.30047
$473$	−31.6939	−1.45729
$474$	−11.2198	−0.515341
$475$	−22.5770	−1.03590
$476$	−43.0607	−1.97368
$477$	8.64314	0.395742
$478$	23.4762	1.07378
$479$	−34.5545	−1.57883	−0.789417	−	0.613858i	$-0.789617\pi$
$479$	−34.5545	−1.57883	−0.789417	+	0.613858i	$0.789617\pi$
$480$	−2.61092	−0.119172
$481$	43.4139	1.97950
$482$	−67.7292	−3.08498
$483$	31.9111	1.45200
$484$	−1.32210	−0.0600955
$485$	0.119975	0.00544779
$486$	−2.49751	−0.113289
$487$	26.2103	1.18770	0.593851	−	0.804575i	$-0.297607\pi$
$487$	26.2103	1.18770	0.593851	+	0.804575i	$0.297607\pi$
$488$	37.8797	1.71474
$489$	6.33684	0.286562
$490$	−15.2824	−0.690388
$491$	−24.0218	−1.08409	−0.542045	−	0.840349i	$-0.682350\pi$
$491$	−24.0218	−1.08409	−0.542045	+	0.840349i	$0.682350\pi$
$492$	14.0098	0.631608
$493$	−15.0158	−0.676278
$494$	93.1025	4.18888
$495$	3.39514	0.152600
$496$	49.9956	2.24487
$497$	44.1356	1.97975
$498$	−4.47770	−0.200651
$499$	32.0280	1.43377	0.716886	−	0.697190i	$-0.245567\pi$
$499$	32.0280	1.43377	0.716886	+	0.697190i	$0.245567\pi$
$500$	−39.2611	−1.75581
$501$	−16.2181	−0.724570
$502$	−40.3571	−1.80123
$503$	−10.2060	−0.455062	−0.227531	−	0.973771i	$-0.573065\pi$
$503$	−10.2060	−0.455062	−0.227531	+	0.973771i	$0.573065\pi$
$504$	−20.0652	−0.893775
$505$	10.6856	0.475501
$506$	−72.5662	−3.22596
$507$	28.9258	1.28464
$508$	−76.8346	−3.40899
$509$	−29.6213	−1.31294	−0.656471	−	0.754351i	$-0.727952\pi$
$509$	−29.6213	−1.31294	−0.656471	+	0.754351i	$0.727952\pi$
$510$	7.34039	0.325038
$511$	25.8179	1.14212
$512$	47.4858	2.09859
$513$	5.75723	0.254188
$514$	−32.3491	−1.42686
$515$	3.19573	0.140821
$516$	−41.0811	−1.80850
$517$	−24.0602	−1.05817
$518$	60.1254	2.64176
$519$	−14.6141	−0.641489
$520$	37.5777	1.64789
$521$	23.5129	1.03012	0.515059	−	0.857155i	$-0.327770\pi$
$521$	23.5129	1.03012	0.515059	+	0.857155i	$0.327770\pi$
$522$	−13.2511	−0.579986
$523$	12.6891	0.554856	0.277428	−	0.960746i	$-0.410518\pi$
$523$	12.6891	0.554856	0.277428	+	0.960746i	$0.410518\pi$
$524$	10.0850	0.440567
$525$	−14.0804	−0.614520
$526$	−60.1916	−2.62448
$527$	−25.8116	−1.12437
$528$	17.9212	0.779922
$529$	55.9873	2.43423
$530$	−22.4175	−0.973755
$531$	−5.05582	−0.219404
$532$	87.5975	3.79783
$533$	−21.4070	−0.927240
$534$	0.832360	0.0360197
$535$	8.64277	0.373660
$536$	19.7646	0.853703
$537$	1.43595	0.0619657
$538$	12.9684	0.559107
$539$	19.2630	0.829715
$540$	4.40073	0.189377
$541$	−7.26637	−0.312406	−0.156203	−	0.987725i	$-0.549925\pi$
$541$	−7.26637	−0.312406	−0.156203	+	0.987725i	$0.549925\pi$
$542$	3.59687	0.154499
$543$	−16.4547	−0.706140
$544$	7.11521	0.305062
$545$	18.0835	0.774611
$546$	58.0645	2.48493
$547$	8.95477	0.382878	0.191439	−	0.981504i	$-0.438685\pi$
$547$	8.95477	0.382878	0.191439	+	0.981504i	$0.438685\pi$
$548$	−16.2031	−0.692162
$549$	−6.77839	−0.289294
$550$	32.0190	1.36530
$551$	30.5463	1.30132
$552$	−49.6660	−2.11393
$553$	16.1302	0.685926
$554$	−58.1151	−2.46907
$555$	−6.96301	−0.295563
$556$	97.1774	4.12124
$557$	0.751634	0.0318478	0.0159239	−	0.999873i	$-0.494931\pi$
$557$	0.751634	0.0318478	0.0159239	+	0.999873i	$0.494931\pi$
$558$	−22.7782	−0.964277
$559$	62.7723	2.65498
$560$	20.4405	0.863769
$561$	−9.25233	−0.390634
$562$	47.1859	1.99042
$563$	24.4478	1.03035	0.515177	−	0.857084i	$-0.327726\pi$
$563$	24.4478	1.03035	0.515177	+	0.857084i	$0.327726\pi$
$564$	−31.1865	−1.31319
$565$	−4.21412	−0.177289
$566$	44.1468	1.85563
$567$	3.59057	0.150790
$568$	−68.6921	−2.88226
$569$	20.3588	0.853484	0.426742	−	0.904373i	$-0.359661\pi$
$569$	20.3588	0.853484	0.426742	+	0.904373i	$0.359661\pi$
$570$	−14.9324	−0.625450
$571$	−19.8899	−0.832365	−0.416182	−	0.909281i	$-0.636632\pi$
$571$	−19.8899	−0.832365	−0.416182	+	0.909281i	$0.636632\pi$
$572$	−89.7024	−3.75065
$573$	−9.05642	−0.378337
$574$	−29.6473	−1.23745
$575$	−34.8523	−1.45344
$576$	−4.68449	−0.195187
$577$	10.9366	0.455297	0.227648	−	0.973743i	$-0.426896\pi$
$577$	10.9366	0.455297	0.227648	+	0.973743i	$0.426896\pi$
$578$	22.4538	0.933956
$579$	15.0117	0.623863
$580$	23.3491	0.969518
$581$	6.43741	0.267069
$582$	−0.288529	−0.0119599
$583$	28.2566	1.17027
$584$	−40.1826	−1.66277
$585$	−6.72434	−0.278017
$586$	35.1046	1.45016
$587$	−7.84429	−0.323769	−0.161884	−	0.986810i	$-0.551757\pi$
$587$	−7.84429	−0.323769	−0.161884	+	0.986810i	$0.551757\pi$
$588$	24.9684	1.02968
$589$	52.5080	2.16355
$590$	13.1132	0.539861
$591$	18.2379	0.750207
$592$	−36.7542	−1.51059
$593$	−11.4178	−0.468874	−0.234437	−	0.972131i	$-0.575325\pi$
$593$	−11.4178	−0.468874	−0.234437	+	0.972131i	$0.575325\pi$
$594$	−8.16499	−0.335013
$595$	−10.5530	−0.432630
$596$	65.2275	2.67182
$597$	−18.9414	−0.775220
$598$	143.723	5.87727
$599$	43.3412	1.77087	0.885437	−	0.464760i	$-0.153859\pi$
$599$	43.3412	1.77087	0.885437	+	0.464760i	$0.153859\pi$
$600$	21.9146	0.894660
$601$	−4.19078	−0.170946	−0.0854728	−	0.996341i	$-0.527240\pi$
$601$	−4.19078	−0.170946	−0.0854728	+	0.996341i	$0.527240\pi$
$602$	86.9355	3.54323
$603$	−3.53678	−0.144029
$604$	35.7126	1.45312
$605$	−0.324010	−0.0131729
$606$	−25.6978	−1.04390
$607$	−8.22345	−0.333780	−0.166890	−	0.985976i	$-0.553372\pi$
$607$	−8.22345	−0.333780	−0.166890	+	0.985976i	$0.553372\pi$
$608$	−14.4743	−0.587011
$609$	19.0506	0.771969
$610$	17.5810	0.711832
$611$	47.6532	1.92784
$612$	−11.9927	−0.484778
$613$	−9.39104	−0.379301	−0.189650	−	0.981852i	$-0.560735\pi$
$613$	−9.39104	−0.379301	−0.189650	+	0.981852i	$0.560735\pi$
$614$	4.68834	0.189206
$615$	3.43340	0.138448
$616$	−65.5982	−2.64303
$617$	−2.89668	−0.116616	−0.0583080	−	0.998299i	$-0.518571\pi$
$617$	−2.89668	−0.116616	−0.0583080	+	0.998299i	$0.518571\pi$
$618$	−7.68542	−0.309153
$619$	−25.5682	−1.02767	−0.513836	−	0.857889i	$-0.671776\pi$
$619$	−25.5682	−1.02767	−0.513836	+	0.857889i	$0.671776\pi$
$620$	40.1362	1.61191
$621$	8.88748	0.356642
$622$	14.0644	0.563931
$623$	−1.19665	−0.0479428
$624$	−35.4944	−1.42091
$625$	9.98573	0.399429
$626$	26.6788	1.06630
$627$	18.8218	0.751671
$628$	−46.7456	−1.86535
$629$	18.9754	0.756599
$630$	−9.31278	−0.371030
$631$	−32.5894	−1.29736	−0.648681	−	0.761060i	$-0.724679\pi$
$631$	−32.5894	−1.29736	−0.648681	+	0.761060i	$0.724679\pi$
$632$	−25.1049	−0.998617
$633$	−11.3609	−0.451557
$634$	18.4737	0.733685
$635$	−18.8300	−0.747247
$636$	36.6258	1.45231
$637$	−38.1518	−1.51163
$638$	−43.3213	−1.71511
$639$	12.2921	0.486268
$640$	17.3719	0.686685
$641$	−11.2301	−0.443563	−0.221781	−	0.975096i	$-0.571187\pi$
$641$	−11.2301	−0.443563	−0.221781	+	0.975096i	$0.571187\pi$
$642$	−20.7851	−0.820321
$643$	29.7029	1.17137	0.585685	−	0.810539i	$-0.300826\pi$
$643$	29.7029	1.17137	0.585685	+	0.810539i	$0.300826\pi$
$644$	135.225	5.32861
$645$	−10.0678	−0.396421
$646$	40.6934	1.60106
$647$	21.4133	0.841843	0.420922	−	0.907097i	$-0.361707\pi$
$647$	21.4133	0.841843	0.420922	+	0.907097i	$0.361707\pi$
$648$	−5.58831	−0.219530
$649$	−16.5288	−0.648810
$650$	−63.4163	−2.48739
$651$	32.7472	1.28347
$652$	26.8527	1.05163
$653$	8.96427	0.350799	0.175400	−	0.984497i	$-0.443878\pi$
$653$	8.96427	0.350799	0.175400	+	0.984497i	$0.443878\pi$
$654$	−43.4891	−1.70056
$655$	2.47156	0.0965720
$656$	18.1232	0.707592
$657$	7.19047	0.280527
$658$	65.9967	2.57282
$659$	31.6714	1.23374	0.616871	−	0.787064i	$-0.288400\pi$
$659$	31.6714	1.23374	0.616871	+	0.787064i	$0.288400\pi$
$660$	14.3871	0.560016
$661$	30.5802	1.18943	0.594716	−	0.803936i	$-0.297264\pi$
$661$	30.5802	1.18943	0.594716	+	0.803936i	$0.297264\pi$
$662$	−27.0118	−1.04984
$663$	18.3250	0.711683
$664$	−10.0191	−0.388816
$665$	21.4677	0.832482
$666$	16.7454	0.648870
$667$	47.1546	1.82583
$668$	−68.7250	−2.65905
$669$	9.30602	0.359792
$670$	9.17328	0.354395
$671$	−22.1602	−0.855487
$672$	−9.02709	−0.348227
$673$	−34.7537	−1.33966	−0.669828	−	0.742516i	$-0.733632\pi$
$673$	−34.7537	−1.33966	−0.669828	+	0.742516i	$0.733632\pi$
$674$	−53.9290	−2.07727
$675$	−3.92151	−0.150939
$676$	122.575	4.71441
$677$	−0.422304	−0.0162304	−0.00811522	−	0.999967i	$-0.502583\pi$
$677$	−0.422304	−0.0162304	−0.00811522	+	0.999967i	$0.502583\pi$
$678$	10.1346	0.389216
$679$	0.414806	0.0159188
$680$	16.4245	0.629852
$681$	20.3456	0.779643
$682$	−74.4676	−2.85151
$683$	25.4968	0.975607	0.487804	−	0.872953i	$-0.337798\pi$
$683$	25.4968	0.975607	0.487804	+	0.872953i	$0.337798\pi$
$684$	24.3966	0.932826
$685$	−3.97093	−0.151721
$686$	9.93448	0.379300
$687$	−28.2211	−1.07670
$688$	−53.1431	−2.02606
$689$	−55.9645	−2.13208
$690$	−23.0513	−0.877547
$691$	−11.7078	−0.445384	−0.222692	−	0.974889i	$-0.571484\pi$
$691$	−11.7078	−0.445384	−0.222692	+	0.974889i	$0.571484\pi$
$692$	−61.9281	−2.35415
$693$	11.7385	0.445907
$694$	1.92462	0.0730574
$695$	23.8155	0.903372
$696$	−29.6501	−1.12388
$697$	−9.35660	−0.354407
$698$	42.1251	1.59446
$699$	−5.95417	−0.225208
$700$	−59.6666	−2.25518
$701$	−19.7131	−0.744553	−0.372276	−	0.928122i	$-0.621423\pi$
$701$	−19.7131	−0.744553	−0.372276	+	0.928122i	$0.621423\pi$
$702$	16.1714	0.610350
$703$	−38.6012	−1.45587
$704$	−15.3148	−0.577198
$705$	−7.64295	−0.287850
$706$	−54.7472	−2.06044
$707$	36.9446	1.38944
$708$	−21.4243	−0.805175
$709$	−43.9021	−1.64878	−0.824389	−	0.566023i	$-0.808481\pi$
$709$	−43.9021	−1.64878	−0.824389	+	0.566023i	$0.808481\pi$
$710$	−31.8818	−1.19650
$711$	4.49239	0.168478
$712$	1.86245	0.0697983
$713$	81.0570	3.03561
$714$	25.3789	0.949782
$715$	−21.9836	−0.822138
$716$	6.08490	0.227403
$717$	−9.39985	−0.351044
$718$	−60.3346	−2.25167
$719$	−4.99056	−0.186117	−0.0930583	−	0.995661i	$-0.529664\pi$
$719$	−4.99056	−0.186117	−0.0930583	+	0.995661i	$0.529664\pi$
$720$	5.69284	0.212159
$721$	11.0490	0.411487
$722$	−35.3290	−1.31481
$723$	27.1187	1.00855
$724$	−69.7278	−2.59141
$725$	−20.8065	−0.772733
$726$	0.779214	0.0289193
$727$	6.59080	0.244439	0.122220	−	0.992503i	$-0.460999\pi$
$727$	6.59080	0.244439	0.122220	+	0.992503i	$0.460999\pi$
$728$	129.922	4.81525
$729$	1.00000	0.0370370
$730$	−18.6498	−0.690259
$731$	27.4366	1.01478
$732$	−28.7238	−1.06166
$733$	−3.59954	−0.132952	−0.0664760	−	0.997788i	$-0.521176\pi$
$733$	−3.59954	−0.132952	−0.0664760	+	0.997788i	$0.521176\pi$
$734$	54.6604	2.01755
$735$	6.11905	0.225705
$736$	−22.3441	−0.823615
$737$	−11.5626	−0.425915
$738$	−8.25700	−0.303944
$739$	−52.7946	−1.94208	−0.971040	−	0.238916i	$-0.923208\pi$
$739$	−52.7946	−1.94208	−0.971040	+	0.238916i	$0.923208\pi$
$740$	−29.5061	−1.08467
$741$	−37.2781	−1.36945
$742$	−77.5071	−2.84538
$743$	11.0013	0.403597	0.201799	−	0.979427i	$-0.435321\pi$
$743$	11.0013	0.403597	0.201799	+	0.979427i	$0.435321\pi$
$744$	−50.9674	−1.86856
$745$	15.9854	0.585661
$746$	−23.0011	−0.842132
$747$	1.79287	0.0655975
$748$	−39.2073	−1.43356
$749$	29.8818	1.09186
$750$	23.1395	0.844936
$751$	19.9161	0.726750	0.363375	−	0.931643i	$-0.381624\pi$
$751$	19.9161	0.726750	0.363375	+	0.931643i	$0.381624\pi$
$752$	−40.3433	−1.47117
$753$	16.1589	0.588864
$754$	85.8012	3.12470
$755$	8.75215	0.318523
$756$	15.2152	0.553372
$757$	−40.0155	−1.45439	−0.727194	−	0.686432i	$-0.759176\pi$
$757$	−40.0155	−1.45439	−0.727194	+	0.686432i	$0.759176\pi$
$758$	22.0042	0.799227
$759$	29.0554	1.05464
$760$	−33.4121	−1.21198
$761$	25.1410	0.911362	0.455681	−	0.890143i	$-0.349396\pi$
$761$	25.1410	0.911362	0.455681	+	0.890143i	$0.349396\pi$
$762$	45.2844	1.64048
$763$	62.5224	2.26346
$764$	−38.3771	−1.38843
$765$	−2.93908	−0.106263
$766$	40.2826	1.45547
$767$	32.7365	1.18205
$768$	−32.4088	−1.16945
$769$	22.5527	0.813271	0.406636	−	0.913590i	$-0.366702\pi$
$769$	22.5527	0.813271	0.406636	+	0.913590i	$0.366702\pi$
$770$	−30.4458	−1.09719
$771$	12.9525	0.466475
$772$	63.6127	2.28947
$773$	−24.9096	−0.895937	−0.447968	−	0.894049i	$-0.647852\pi$
$773$	−24.9096	−0.895937	−0.447968	+	0.894049i	$0.647852\pi$
$774$	24.2122	0.870290
$775$	−35.7655	−1.28474
$776$	−0.645599	−0.0231757
$777$	−24.0741	−0.863655
$778$	−1.11728	−0.0400564
$779$	19.0339	0.681962
$780$	−28.4948	−1.02028
$781$	40.1860	1.43797
$782$	62.8187	2.24639
$783$	5.30574	0.189612
$784$	32.2994	1.15355
$785$	−11.4560	−0.408884
$786$	−5.94388	−0.212011
$787$	−12.6206	−0.449877	−0.224938	−	0.974373i	$-0.572218\pi$
$787$	−12.6206	−0.449877	−0.224938	+	0.974373i	$0.572218\pi$
$788$	77.2841	2.75313
$789$	24.1007	0.858006
$790$	−11.6518	−0.414553
$791$	−14.5700	−0.518051
$792$	−18.2696	−0.649182
$793$	43.8901	1.55858
$794$	−57.5631	−2.04284
$795$	8.97595	0.318344
$796$	−80.2652	−2.84492
$797$	15.0234	0.532157	0.266079	−	0.963951i	$-0.414272\pi$
$797$	15.0234	0.532157	0.266079	+	0.963951i	$0.414272\pi$
$798$	−51.6278	−1.82760
$799$	20.8283	0.736854
$800$	9.85910	0.348572
$801$	−0.333276	−0.0117757
$802$	−27.1917	−0.960171
$803$	23.5075	0.829560
$804$	−14.9873	−0.528562
$805$	33.1399	1.16803
$806$	147.489	5.19508
$807$	−5.19253	−0.182786
$808$	−57.5001	−2.02285
$809$	−3.78123	−0.132941	−0.0664705	−	0.997788i	$-0.521174\pi$
$809$	−3.78123	−0.132941	−0.0664705	+	0.997788i	$0.521174\pi$
$810$	−2.59368	−0.0911326
$811$	5.09439	0.178888	0.0894441	−	0.995992i	$-0.471491\pi$
$811$	5.09439	0.178888	0.0894441	+	0.995992i	$0.471491\pi$
$812$	80.7279	2.83300
$813$	−1.44018	−0.0505094
$814$	54.7449	1.91881
$815$	6.58085	0.230517
$816$	−15.5140	−0.543097
$817$	−55.8137	−1.95267
$818$	−86.0763	−3.00959
$819$	−23.2490	−0.812385
$820$	14.5492	0.508081
$821$	−39.4819	−1.37793	−0.688964	−	0.724796i	$-0.741934\pi$
$821$	−39.4819	−1.37793	−0.688964	+	0.724796i	$0.741934\pi$
$822$	9.54971	0.333084
$823$	9.45532	0.329592	0.164796	−	0.986328i	$-0.447303\pi$
$823$	9.45532	0.329592	0.164796	+	0.986328i	$0.447303\pi$
$824$	−17.1965	−0.599070
$825$	−12.8204	−0.446349
$826$	45.3380	1.57751
$827$	32.0103	1.11311	0.556554	−	0.830811i	$-0.312123\pi$
$827$	32.0103	1.11311	0.556554	+	0.830811i	$0.312123\pi$
$828$	37.6612	1.30882
$829$	49.0904	1.70498	0.852489	−	0.522745i	$-0.175092\pi$
$829$	49.0904	1.70498	0.852489	+	0.522745i	$0.175092\pi$
$830$	−4.65012	−0.161408
$831$	23.2692	0.807200
$832$	30.3321	1.05158
$833$	−16.6755	−0.577771
$834$	−57.2740	−1.98323
$835$	−16.8426	−0.582861
$836$	79.7585	2.75851
$837$	9.12036	0.315246
$838$	90.7875	3.13620
$839$	16.2016	0.559340	0.279670	−	0.960096i	$-0.409775\pi$
$839$	16.2016	0.559340	0.279670	+	0.960096i	$0.409775\pi$
$840$	−20.8378	−0.718974
$841$	−0.849160	−0.0292814
$842$	47.0891	1.62280
$843$	−18.8932	−0.650716
$844$	−48.1426	−1.65714
$845$	30.0396	1.03339
$846$	18.3806	0.631937
$847$	−1.12024	−0.0384920
$848$	47.3796	1.62702
$849$	−17.6763	−0.606651
$850$	−27.7181	−0.950722
$851$	−59.5890	−2.04269
$852$	52.0885	1.78452
$853$	49.3031	1.68810	0.844052	−	0.536261i	$-0.180164\pi$
$853$	49.3031	1.68810	0.844052	+	0.536261i	$0.180164\pi$
$854$	60.7850	2.08002
$855$	5.97892	0.204475
$856$	−46.5077	−1.58960
$857$	−36.5390	−1.24815	−0.624074	−	0.781365i	$-0.714524\pi$
$857$	−36.5390	−1.24815	−0.624074	+	0.781365i	$0.714524\pi$
$858$	52.8684	1.80490
$859$	−51.1030	−1.74361	−0.871805	−	0.489852i	$-0.837051\pi$
$859$	−51.1030	−1.74361	−0.871805	+	0.489852i	$0.837051\pi$
$860$	−42.6630	−1.45480
$861$	11.8707	0.404554
$862$	−37.6735	−1.28316
$863$	37.4740	1.27563	0.637815	−	0.770189i	$-0.279838\pi$
$863$	37.4740	1.27563	0.637815	+	0.770189i	$0.279838\pi$
$864$	−2.51411	−0.0855318
$865$	−15.1769	−0.516028
$866$	82.7696	2.81262
$867$	−8.99049	−0.305333
$868$	138.768	4.71010
$869$	14.6867	0.498213
$870$	−13.7614	−0.466554
$871$	22.9007	0.775961
$872$	−97.3091	−3.29530
$873$	0.115527	0.00390999
$874$	−127.791	−4.32259
$875$	−33.2667	−1.12462
$876$	30.4700	1.02949
$877$	−15.4600	−0.522047	−0.261023	−	0.965332i	$-0.584060\pi$
$877$	−15.4600	−0.522047	−0.261023	+	0.965332i	$0.584060\pi$
$878$	60.7470	2.05011
$879$	−14.0558	−0.474092
$880$	18.6113	0.627387
$881$	−5.28948	−0.178207	−0.0891036	−	0.996022i	$-0.528400\pi$
$881$	−5.28948	−0.178207	−0.0891036	+	0.996022i	$0.528400\pi$
$882$	−14.7157	−0.495505
$883$	−21.5932	−0.726669	−0.363334	−	0.931659i	$-0.618362\pi$
$883$	−21.5932	−0.726669	−0.363334	+	0.931659i	$0.618362\pi$
$884$	77.6531	2.61176
$885$	−5.25050	−0.176494
$886$	1.02583	0.0344633
$887$	37.7214	1.26656	0.633280	−	0.773922i	$-0.281708\pi$
$887$	37.7214	1.26656	0.633280	+	0.773922i	$0.281708\pi$
$888$	37.4687	1.25737
$889$	−65.1036	−2.18350
$890$	0.864411	0.0289751
$891$	3.26925	0.109524
$892$	39.4348	1.32037
$893$	−42.3707	−1.41788
$894$	−38.4435	−1.28574
$895$	1.49124	0.0498466
$896$	60.0622	2.00654
$897$	−57.5465	−1.92142
$898$	−13.6633	−0.455951
$899$	48.3902	1.61390
$900$	−16.6176	−0.553920
$901$	−24.4610	−0.814915
$902$	−26.9942	−0.898809
$903$	−34.8089	−1.15837
$904$	22.6766	0.754213
$905$	−17.0883	−0.568036
$906$	−21.0481	−0.699276
$907$	−29.4498	−0.977864	−0.488932	−	0.872322i	$-0.662613\pi$
$907$	−29.4498	−0.977864	−0.488932	+	0.872322i	$0.662613\pi$
$908$	86.2154	2.86116
$909$	10.2894	0.341276
$910$	60.3003	1.99894
$911$	−59.5792	−1.97395	−0.986974	−	0.160879i	$-0.948567\pi$
$911$	−59.5792	−1.97395	−0.986974	+	0.160879i	$0.948567\pi$
$912$	31.5597	1.04505
$913$	5.86133	0.193982
$914$	−84.7419	−2.80301
$915$	−7.03939	−0.232715
$916$	−119.588	−3.95131
$917$	8.54527	0.282190
$918$	7.06822	0.233286
$919$	42.1975	1.39197	0.695984	−	0.718057i	$-0.254968\pi$
$919$	42.1975	1.39197	0.695984	+	0.718057i	$0.254968\pi$
$920$	−51.5785	−1.70049
$921$	−1.87721	−0.0618561
$922$	17.0055	0.560046
$923$	−79.5916	−2.61979
$924$	49.7424	1.63640
$925$	26.2930	0.864509
$926$	5.33718	0.175391
$927$	3.07723	0.101070
$928$	−13.3392	−0.437881
$929$	16.0800	0.527566	0.263783	−	0.964582i	$-0.415030\pi$
$929$	16.0800	0.527566	0.263783	+	0.964582i	$0.415030\pi$
$930$	−23.6553	−0.775687
$931$	33.9225	1.11177
$932$	−25.2311	−0.826473
$933$	−5.63137	−0.184363
$934$	78.4173	2.56589
$935$	−9.60860	−0.314235
$936$	36.1844	1.18272
$937$	31.3196	1.02317	0.511583	−	0.859234i	$-0.329059\pi$
$937$	31.3196	1.02317	0.511583	+	0.859234i	$0.329059\pi$
$938$	31.7160	1.03556
$939$	−10.6822	−0.348600
$940$	−32.3874	−1.05636
$941$	8.26857	0.269548	0.134774	−	0.990876i	$-0.456969\pi$
$941$	8.26857	0.269548	0.134774	+	0.990876i	$0.456969\pi$
$942$	27.5507	0.897651
$943$	29.3828	0.956837
$944$	−27.7148	−0.902039
$945$	3.72882	0.121299
$946$	79.1558	2.57358
$947$	10.6105	0.344796	0.172398	−	0.985027i	$-0.444848\pi$
$947$	10.6105	0.344796	0.172398	+	0.985027i	$0.444848\pi$
$948$	19.0367	0.618284
$949$	−46.5584	−1.51135
$950$	56.3863	1.82941
$951$	−7.39685	−0.239859
$952$	56.7867	1.84047
$953$	−33.6563	−1.09024	−0.545118	−	0.838359i	$-0.683515\pi$
$953$	−33.6563	−1.09024	−0.545118	+	0.838359i	$0.683515\pi$
$954$	−21.5863	−0.698883
$955$	−9.40515	−0.304344
$956$	−39.8324	−1.28827
$957$	17.3458	0.560710
$958$	86.3001	2.78823
$959$	−13.7292	−0.443340
$960$	−4.86487	−0.157013
$961$	52.1809	1.68325
$962$	−108.427	−3.49581
$963$	8.32231	0.268183
$964$	114.917	3.70122
$965$	15.5897	0.501850
$966$	−79.6982	−2.56425
$967$	6.99668	0.224998	0.112499	−	0.993652i	$-0.464114\pi$
$967$	6.99668	0.224998	0.112499	+	0.993652i	$0.464114\pi$
$968$	1.74353	0.0560393
$969$	−16.2936	−0.523425
$970$	−0.299639	−0.00962083
$971$	−1.21533	−0.0390018	−0.0195009	−	0.999810i	$-0.506208\pi$
$971$	−1.21533	−0.0390018	−0.0195009	+	0.999810i	$0.506208\pi$
$972$	4.23755	0.135920
$973$	82.3404	2.63971
$974$	−65.4605	−2.09749
$975$	25.3918	0.813188
$976$	−37.1575	−1.18938
$977$	34.8783	1.11586	0.557928	−	0.829889i	$-0.311596\pi$
$977$	34.8783	1.11586	0.557928	+	0.829889i	$0.311596\pi$
$978$	−15.8263	−0.506070
$979$	−1.08956	−0.0348226
$980$	25.9298	0.828297
$981$	17.4130	0.555953
$982$	59.9947	1.91451
$983$	42.7259	1.36275	0.681373	−	0.731936i	$-0.261383\pi$
$983$	42.7259	1.36275	0.681373	+	0.731936i	$0.261383\pi$
$984$	−18.4755	−0.588977
$985$	18.9402	0.603484
$986$	37.5021	1.19431
$987$	−26.4250	−0.841117
$988$	−157.968	−5.02563
$989$	−86.1600	−2.73973
$990$	−8.47939	−0.269493
$991$	−31.6956	−1.00685	−0.503423	−	0.864040i	$-0.667926\pi$
$991$	−31.6956	−1.00685	−0.503423	+	0.864040i	$0.667926\pi$
$992$	−22.9296	−0.728015
$993$	10.8155	0.343219
$994$	−110.229	−3.49626
$995$	−19.6708	−0.623605
$996$	7.59737	0.240732
$997$	−12.8500	−0.406962	−0.203481	−	0.979079i	$-0.565226\pi$
$997$	−12.8500	−0.406962	−0.203481	+	0.979079i	$0.565226\pi$
$998$	−79.9903	−2.53205
$999$	−6.70483	−0.212131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6009.2.a.c.1.5	✓	92

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6009.2.a.c.1.5	✓	92	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:	\( N \)	\(=\)	\( 6009 = 3 \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.49751 q^{2} +1.00000 q^{3} +4.23755 q^{4} +1.03851 q^{5} -2.49751 q^{6} +3.59057 q^{7} -5.58831 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.49751 q^{2} +1.00000 q^{3} +4.23755 q^{4} +1.03851 q^{5} -2.49751 q^{6} +3.59057 q^{7} -5.58831 q^{8} +1.00000 q^{9} -2.59368 q^{10} +3.26925 q^{11} +4.23755 q^{12} -6.47501 q^{13} -8.96747 q^{14} +1.03851 q^{15} +5.48176 q^{16} -2.83011 q^{17} -2.49751 q^{18} +5.75723 q^{19} +4.40073 q^{20} +3.59057 q^{21} -8.16499 q^{22} +8.88748 q^{23} -5.58831 q^{24} -3.92151 q^{25} +16.1714 q^{26} +1.00000 q^{27} +15.2152 q^{28} +5.30574 q^{29} -2.59368 q^{30} +9.12036 q^{31} -2.51411 q^{32} +3.26925 q^{33} +7.06822 q^{34} +3.72882 q^{35} +4.23755 q^{36} -6.70483 q^{37} -14.3787 q^{38} -6.47501 q^{39} -5.80350 q^{40} +3.30609 q^{41} -8.96747 q^{42} -9.69454 q^{43} +13.8536 q^{44} +1.03851 q^{45} -22.1966 q^{46} -7.35956 q^{47} +5.48176 q^{48} +5.89217 q^{49} +9.79400 q^{50} -2.83011 q^{51} -27.4382 q^{52} +8.64314 q^{53} -2.49751 q^{54} +3.39514 q^{55} -20.0652 q^{56} +5.75723 q^{57} -13.2511 q^{58} -5.05582 q^{59} +4.40073 q^{60} -6.77839 q^{61} -22.7782 q^{62} +3.59057 q^{63} -4.68449 q^{64} -6.72434 q^{65} -8.16499 q^{66} -3.53678 q^{67} -11.9927 q^{68} +8.88748 q^{69} -9.31278 q^{70} +12.2921 q^{71} -5.58831 q^{72} +7.19047 q^{73} +16.7454 q^{74} -3.92151 q^{75} +24.3966 q^{76} +11.7385 q^{77} +16.1714 q^{78} +4.49239 q^{79} +5.69284 q^{80} +1.00000 q^{81} -8.25700 q^{82} +1.79287 q^{83} +15.2152 q^{84} -2.93908 q^{85} +24.2122 q^{86} +5.30574 q^{87} -18.2696 q^{88} -0.333276 q^{89} -2.59368 q^{90} -23.2490 q^{91} +37.6612 q^{92} +9.12036 q^{93} +18.3806 q^{94} +5.97892 q^{95} -2.51411 q^{96} +0.115527 q^{97} -14.7157 q^{98} +3.26925 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q + 17 q^{2} + 92 q^{3} + 107 q^{4} + 34 q^{5} + 17 q^{6} + 22 q^{7} + 51 q^{8} + 92 q^{9}+O(q^{10}) \) 92 * q + 17 * q^2 + 92 * q^3 + 107 * q^4 + 34 * q^5 + 17 * q^6 + 22 * q^7 + 51 * q^8 + 92 * q^9	\( 92 q + 17 q^{2} + 92 q^{3} + 107 q^{4} + 34 q^{5} + 17 q^{6} + 22 q^{7} + 51 q^{8} + 92 q^{9} + 13 q^{10} + 40 q^{11} + 107 q^{12} + 6 q^{13} + 37 q^{14} + 34 q^{15} + 133 q^{16} + 77 q^{17} + 17 q^{18} + 34 q^{19} + 55 q^{20} + 22 q^{21} + 8 q^{22} + 83 q^{23} + 51 q^{24} + 110 q^{25} + 22 q^{26} + 92 q^{27} + 32 q^{28} + 97 q^{29} + 13 q^{30} + 44 q^{31} + 104 q^{32} + 40 q^{33} + 20 q^{34} + 80 q^{35} + 107 q^{36} + 12 q^{37} + 54 q^{38} + 6 q^{39} + 23 q^{40} + 67 q^{41} + 37 q^{42} + 30 q^{43} + 87 q^{44} + 34 q^{45} + 33 q^{46} + 69 q^{47} + 133 q^{48} + 112 q^{49} + 58 q^{50} + 77 q^{51} - 3 q^{52} + 113 q^{53} + 17 q^{54} + 42 q^{55} + 92 q^{56} + 34 q^{57} - 30 q^{58} + 72 q^{59} + 55 q^{60} + 19 q^{61} + 60 q^{62} + 22 q^{63} + 147 q^{64} + 74 q^{65} + 8 q^{66} + 26 q^{67} + 171 q^{68} + 83 q^{69} - 35 q^{70} + 134 q^{71} + 51 q^{72} - 17 q^{73} + 95 q^{74} + 110 q^{75} + 27 q^{76} + 108 q^{77} + 22 q^{78} + 159 q^{79} + 79 q^{80} + 92 q^{81} - 64 q^{82} + 73 q^{83} + 32 q^{84} - 4 q^{85} + 22 q^{86} + 97 q^{87} - 16 q^{88} + 50 q^{89} + 13 q^{90} + 17 q^{91} + 154 q^{92} + 44 q^{93} + 8 q^{94} + 155 q^{95} + 104 q^{96} - 20 q^{97} + 63 q^{98} + 40 q^{99}+O(q^{100}) \) 92 * q + 17 * q^2 + 92 * q^3 + 107 * q^4 + 34 * q^5 + 17 * q^6 + 22 * q^7 + 51 * q^8 + 92 * q^9 + 13 * q^10 + 40 * q^11 + 107 * q^12 + 6 * q^13 + 37 * q^14 + 34 * q^15 + 133 * q^16 + 77 * q^17 + 17 * q^18 + 34 * q^19 + 55 * q^20 + 22 * q^21 + 8 * q^22 + 83 * q^23 + 51 * q^24 + 110 * q^25 + 22 * q^26 + 92 * q^27 + 32 * q^28 + 97 * q^29 + 13 * q^30 + 44 * q^31 + 104 * q^32 + 40 * q^33 + 20 * q^34 + 80 * q^35 + 107 * q^36 + 12 * q^37 + 54 * q^38 + 6 * q^39 + 23 * q^40 + 67 * q^41 + 37 * q^42 + 30 * q^43 + 87 * q^44 + 34 * q^45 + 33 * q^46 + 69 * q^47 + 133 * q^48 + 112 * q^49 + 58 * q^50 + 77 * q^51 - 3 * q^52 + 113 * q^53 + 17 * q^54 + 42 * q^55 + 92 * q^56 + 34 * q^57 - 30 * q^58 + 72 * q^59 + 55 * q^60 + 19 * q^61 + 60 * q^62 + 22 * q^63 + 147 * q^64 + 74 * q^65 + 8 * q^66 + 26 * q^67 + 171 * q^68 + 83 * q^69 - 35 * q^70 + 134 * q^71 + 51 * q^72 - 17 * q^73 + 95 * q^74 + 110 * q^75 + 27 * q^76 + 108 * q^77 + 22 * q^78 + 159 * q^79 + 79 * q^80 + 92 * q^81 - 64 * q^82 + 73 * q^83 + 32 * q^84 - 4 * q^85 + 22 * q^86 + 97 * q^87 - 16 * q^88 + 50 * q^89 + 13 * q^90 + 17 * q^91 + 154 * q^92 + 44 * q^93 + 8 * q^94 + 155 * q^95 + 104 * q^96 - 20 * q^97 + 63 * q^98 + 40 * q^99

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