LMFDB - Embedded newform 8049.2.a.d.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8049,2,Mod(1,8049)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8049, 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("8049.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8049 = 3 \cdot 2683 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8049.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:	$64.2715885869$
Analytic rank:	$0$
Dimension:	$129$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.62578 q^{2} +1.00000 q^{3} +4.89474 q^{4} -0.512843 q^{5} -2.62578 q^{6} +1.48852 q^{7} -7.60096 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.62578 q^{2} +1.00000 q^{3} +4.89474 q^{4} -0.512843 q^{5} -2.62578 q^{6} +1.48852 q^{7} -7.60096 q^{8} +1.00000 q^{9} +1.34662 q^{10} -3.52059 q^{11} +4.89474 q^{12} +5.56534 q^{13} -3.90853 q^{14} -0.512843 q^{15} +10.1690 q^{16} +0.724523 q^{17} -2.62578 q^{18} -2.38418 q^{19} -2.51023 q^{20} +1.48852 q^{21} +9.24432 q^{22} -3.45147 q^{23} -7.60096 q^{24} -4.73699 q^{25} -14.6134 q^{26} +1.00000 q^{27} +7.28591 q^{28} -7.25957 q^{29} +1.34662 q^{30} +8.34151 q^{31} -11.4997 q^{32} -3.52059 q^{33} -1.90244 q^{34} -0.763376 q^{35} +4.89474 q^{36} -11.8958 q^{37} +6.26033 q^{38} +5.56534 q^{39} +3.89810 q^{40} +0.448168 q^{41} -3.90853 q^{42} +6.03831 q^{43} -17.2324 q^{44} -0.512843 q^{45} +9.06283 q^{46} -9.64567 q^{47} +10.1690 q^{48} -4.78431 q^{49} +12.4383 q^{50} +0.724523 q^{51} +27.2409 q^{52} +11.5963 q^{53} -2.62578 q^{54} +1.80551 q^{55} -11.3142 q^{56} -2.38418 q^{57} +19.0621 q^{58} +1.65924 q^{59} -2.51023 q^{60} +1.82823 q^{61} -21.9030 q^{62} +1.48852 q^{63} +9.85763 q^{64} -2.85414 q^{65} +9.24432 q^{66} +11.7540 q^{67} +3.54635 q^{68} -3.45147 q^{69} +2.00446 q^{70} +1.36179 q^{71} -7.60096 q^{72} -3.49325 q^{73} +31.2358 q^{74} -4.73699 q^{75} -11.6699 q^{76} -5.24047 q^{77} -14.6134 q^{78} +0.789718 q^{79} -5.21510 q^{80} +1.00000 q^{81} -1.17679 q^{82} +15.3946 q^{83} +7.28591 q^{84} -0.371567 q^{85} -15.8553 q^{86} -7.25957 q^{87} +26.7599 q^{88} +5.68030 q^{89} +1.34662 q^{90} +8.28411 q^{91} -16.8941 q^{92} +8.34151 q^{93} +25.3275 q^{94} +1.22271 q^{95} -11.4997 q^{96} +1.12851 q^{97} +12.5626 q^{98} -3.52059 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 129 q + 8 q^{2} + 129 q^{3} + 158 q^{4} + 11 q^{5} + 8 q^{6} + 40 q^{7} + 18 q^{8} + 129 q^{9}+O(q^{10}) $ 129 * q + 8 * q^2 + 129 * q^3 + 158 * q^4 + 11 * q^5 + 8 * q^6 + 40 * q^7 + 18 * q^8 + 129 * q^9	\( 129 q + 8 q^{2} + 129 q^{3} + 158 q^{4} + 11 q^{5} + 8 q^{6} + 40 q^{7} + 18 q^{8} + 129 q^{9} + 20 q^{10} + 48 q^{11} + 158 q^{12} + 77 q^{13} + 13 q^{14} + 11 q^{15} + 212 q^{16} + 9 q^{17} + 8 q^{18} + 68 q^{19} + 19 q^{20} + 40 q^{21} + 45 q^{22} + 64 q^{23} + 18 q^{24} + 188 q^{25} + 19 q^{26} + 129 q^{27} + 69 q^{28} + 23 q^{29} + 20 q^{30} + 133 q^{31} + 24 q^{32} + 48 q^{33} + 63 q^{34} + 26 q^{35} + 158 q^{36} + 147 q^{37} + 9 q^{38} + 77 q^{39} + 58 q^{40} + 21 q^{41} + 13 q^{42} + 76 q^{43} + 110 q^{44} + 11 q^{45} + 48 q^{46} + 85 q^{47} + 212 q^{48} + 213 q^{49} + 17 q^{50} + 9 q^{51} + 139 q^{52} + 30 q^{53} + 8 q^{54} + 103 q^{55} + 19 q^{56} + 68 q^{57} + 94 q^{58} + 64 q^{59} + 19 q^{60} + 110 q^{61} - 10 q^{62} + 40 q^{63} + 288 q^{64} - 8 q^{65} + 45 q^{66} + 118 q^{67} - 15 q^{68} + 64 q^{69} + 75 q^{70} + 154 q^{71} + 18 q^{72} + 137 q^{73} + 28 q^{74} + 188 q^{75} + 156 q^{76} + 17 q^{77} + 19 q^{78} + 157 q^{79} + 2 q^{80} + 129 q^{81} + 72 q^{82} + 39 q^{83} + 69 q^{84} + 127 q^{85} + 54 q^{86} + 23 q^{87} + 97 q^{88} + 31 q^{89} + 20 q^{90} + 137 q^{91} + 82 q^{92} + 133 q^{93} + 40 q^{94} + 68 q^{95} + 24 q^{96} + 170 q^{97} - 21 q^{98} + 48 q^{99}+O(q^{100}) \) 129 * q + 8 * q^2 + 129 * q^3 + 158 * q^4 + 11 * q^5 + 8 * q^6 + 40 * q^7 + 18 * q^8 + 129 * q^9 + 20 * q^10 + 48 * q^11 + 158 * q^12 + 77 * q^13 + 13 * q^14 + 11 * q^15 + 212 * q^16 + 9 * q^17 + 8 * q^18 + 68 * q^19 + 19 * q^20 + 40 * q^21 + 45 * q^22 + 64 * q^23 + 18 * q^24 + 188 * q^25 + 19 * q^26 + 129 * q^27 + 69 * q^28 + 23 * q^29 + 20 * q^30 + 133 * q^31 + 24 * q^32 + 48 * q^33 + 63 * q^34 + 26 * q^35 + 158 * q^36 + 147 * q^37 + 9 * q^38 + 77 * q^39 + 58 * q^40 + 21 * q^41 + 13 * q^42 + 76 * q^43 + 110 * q^44 + 11 * q^45 + 48 * q^46 + 85 * q^47 + 212 * q^48 + 213 * q^49 + 17 * q^50 + 9 * q^51 + 139 * q^52 + 30 * q^53 + 8 * q^54 + 103 * q^55 + 19 * q^56 + 68 * q^57 + 94 * q^58 + 64 * q^59 + 19 * q^60 + 110 * q^61 - 10 * q^62 + 40 * q^63 + 288 * q^64 - 8 * q^65 + 45 * q^66 + 118 * q^67 - 15 * q^68 + 64 * q^69 + 75 * q^70 + 154 * q^71 + 18 * q^72 + 137 * q^73 + 28 * q^74 + 188 * q^75 + 156 * q^76 + 17 * q^77 + 19 * q^78 + 157 * q^79 + 2 * q^80 + 129 * q^81 + 72 * q^82 + 39 * q^83 + 69 * q^84 + 127 * q^85 + 54 * q^86 + 23 * q^87 + 97 * q^88 + 31 * q^89 + 20 * q^90 + 137 * q^91 + 82 * q^92 + 133 * q^93 + 40 * q^94 + 68 * q^95 + 24 * q^96 + 170 * q^97 - 21 * q^98 + 48 * 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 $

$2$	−2.62578	−1.85671	−0.928355	−	0.371696i	$-0.878777\pi$
$2$	−2.62578	−1.85671	−0.928355	+	0.371696i	$0.878777\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.89474	2.44737
$5$	−0.512843	−0.229350	−0.114675	−	0.993403i	$-0.536583\pi$
$5$	−0.512843	−0.229350	−0.114675	+	0.993403i	$0.536583\pi$
$6$	−2.62578	−1.07197
$7$	1.48852	0.562607	0.281304	−	0.959619i	$-0.409233\pi$
$7$	1.48852	0.562607	0.281304	+	0.959619i	$0.409233\pi$
$8$	−7.60096	−2.68734
$9$	1.00000	0.333333
$10$	1.34662	0.425837
$11$	−3.52059	−1.06150	−0.530749	−	0.847529i	$-0.678089\pi$
$11$	−3.52059	−1.06150	−0.530749	+	0.847529i	$0.678089\pi$
$12$	4.89474	1.41299
$13$	5.56534	1.54355	0.771773	−	0.635898i	$-0.219370\pi$
$13$	5.56534	1.54355	0.771773	+	0.635898i	$0.219370\pi$
$14$	−3.90853	−1.04460
$15$	−0.512843	−0.132416
$16$	10.1690	2.54225
$17$	0.724523	0.175723	0.0878614	−	0.996133i	$-0.471997\pi$
$17$	0.724523	0.175723	0.0878614	+	0.996133i	$0.471997\pi$
$18$	−2.62578	−0.618903
$19$	−2.38418	−0.546967	−0.273484	−	0.961877i	$-0.588176\pi$
$19$	−2.38418	−0.546967	−0.273484	+	0.961877i	$0.588176\pi$
$20$	−2.51023	−0.561305
$21$	1.48852	0.324821
$22$	9.24432	1.97089
$23$	−3.45147	−0.719682	−0.359841	−	0.933014i	$-0.617169\pi$
$23$	−3.45147	−0.719682	−0.359841	+	0.933014i	$0.617169\pi$
$24$	−7.60096	−1.55154
$25$	−4.73699	−0.947398
$26$	−14.6134	−2.86592
$27$	1.00000	0.192450
$28$	7.28591	1.37691
$29$	−7.25957	−1.34807	−0.674034	−	0.738700i	$-0.735440\pi$
$29$	−7.25957	−1.34807	−0.674034	+	0.738700i	$0.735440\pi$
$30$	1.34662	0.245857
$31$	8.34151	1.49818	0.749089	−	0.662469i	$-0.230491\pi$
$31$	8.34151	1.49818	0.749089	+	0.662469i	$0.230491\pi$
$32$	−11.4997	−2.03287
$33$	−3.52059	−0.612857
$34$	−1.90244	−0.326266
$35$	−0.763376	−0.129034
$36$	4.89474	0.815790
$37$	−11.8958	−1.95566	−0.977829	−	0.209404i	$-0.932848\pi$
$37$	−11.8958	−1.95566	−0.977829	+	0.209404i	$0.932848\pi$
$38$	6.26033	1.01556
$39$	5.56534	0.891167
$40$	3.89810	0.616344
$41$	0.448168	0.0699921	0.0349960	−	0.999387i	$-0.488858\pi$
$41$	0.448168	0.0699921	0.0349960	+	0.999387i	$0.488858\pi$
$42$	−3.90853	−0.603099
$43$	6.03831	0.920834	0.460417	−	0.887703i	$-0.347700\pi$
$43$	6.03831	0.920834	0.460417	+	0.887703i	$0.347700\pi$
$44$	−17.2324	−2.59788
$45$	−0.512843	−0.0764501
$46$	9.06283	1.33624
$47$	−9.64567	−1.40697	−0.703483	−	0.710712i	$-0.748373\pi$
$47$	−9.64567	−1.40697	−0.703483	+	0.710712i	$0.748373\pi$
$48$	10.1690	1.46777
$49$	−4.78431	−0.683473
$50$	12.4383	1.75904
$51$	0.724523	0.101454
$52$	27.2409	3.77763
$53$	11.5963	1.59288	0.796440	−	0.604718i	$-0.206714\pi$
$53$	11.5963	1.59288	0.796440	+	0.604718i	$0.206714\pi$
$54$	−2.62578	−0.357324
$55$	1.80551	0.243455
$56$	−11.3142	−1.51192
$57$	−2.38418	−0.315792
$58$	19.0621	2.50297
$59$	1.65924	0.216014	0.108007	−	0.994150i	$-0.465553\pi$
$59$	1.65924	0.216014	0.108007	+	0.994150i	$0.465553\pi$
$60$	−2.51023	−0.324070
$61$	1.82823	0.234081	0.117041	−	0.993127i	$-0.462659\pi$
$61$	1.82823	0.234081	0.117041	+	0.993127i	$0.462659\pi$
$62$	−21.9030	−2.78168
$63$	1.48852	0.187536
$64$	9.85763	1.23220
$65$	−2.85414	−0.354013
$66$	9.24432	1.13790
$67$	11.7540	1.43597	0.717987	−	0.696057i	$-0.245064\pi$
$67$	11.7540	1.43597	0.717987	+	0.696057i	$0.245064\pi$
$68$	3.54635	0.430058
$69$	−3.45147	−0.415509
$70$	2.00446	0.239579
$71$	1.36179	0.161614	0.0808072	−	0.996730i	$-0.474250\pi$
$71$	1.36179	0.161614	0.0808072	+	0.996730i	$0.474250\pi$
$72$	−7.60096	−0.895782
$73$	−3.49325	−0.408854	−0.204427	−	0.978882i	$-0.565533\pi$
$73$	−3.49325	−0.408854	−0.204427	+	0.978882i	$0.565533\pi$
$74$	31.2358	3.63109
$75$	−4.73699	−0.546981
$76$	−11.6699	−1.33863
$77$	−5.24047	−0.597207
$78$	−14.6134	−1.65464
$79$	0.789718	0.0888502	0.0444251	−	0.999013i	$-0.485854\pi$
$79$	0.789718	0.0888502	0.0444251	+	0.999013i	$0.485854\pi$
$80$	−5.21510	−0.583066
$81$	1.00000	0.111111
$82$	−1.17679	−0.129955
$83$	15.3946	1.68978	0.844891	−	0.534939i	$-0.179665\pi$
$83$	15.3946	1.68978	0.844891	+	0.534939i	$0.179665\pi$
$84$	7.28591	0.794958
$85$	−0.371567	−0.0403021
$86$	−15.8553	−1.70972
$87$	−7.25957	−0.778308
$88$	26.7599	2.85261
$89$	5.68030	0.602110	0.301055	−	0.953607i	$-0.402661\pi$
$89$	5.68030	0.602110	0.301055	+	0.953607i	$0.402661\pi$
$90$	1.34662	0.141946
$91$	8.28411	0.868410
$92$	−16.8941	−1.76133
$93$	8.34151	0.864974
$94$	25.3275	2.61233
$95$	1.22271	0.125447
$96$	−11.4997	−1.17368
$97$	1.12851	0.114583	0.0572915	−	0.998357i	$-0.481754\pi$
$97$	1.12851	0.114583	0.0572915	+	0.998357i	$0.481754\pi$
$98$	12.5626	1.26901
$99$	−3.52059	−0.353833
$100$	−23.1863	−2.31863
$101$	14.3838	1.43124	0.715622	−	0.698487i	$-0.246143\pi$
$101$	14.3838	1.43124	0.715622	+	0.698487i	$0.246143\pi$
$102$	−1.90244	−0.188370
$103$	−16.4285	−1.61874	−0.809372	−	0.587297i	$-0.800192\pi$
$103$	−16.4285	−1.61874	−0.809372	+	0.587297i	$0.800192\pi$
$104$	−42.3019	−4.14804
$105$	−0.763376	−0.0744979
$106$	−30.4495	−2.95751
$107$	5.09072	0.492139	0.246069	−	0.969252i	$-0.420861\pi$
$107$	5.09072	0.492139	0.246069	+	0.969252i	$0.420861\pi$
$108$	4.89474	0.470997
$109$	17.9102	1.71549	0.857745	−	0.514076i	$-0.171865\pi$
$109$	17.9102	1.71549	0.857745	+	0.514076i	$0.171865\pi$
$110$	−4.74088	−0.452026
$111$	−11.8958	−1.12910
$112$	15.1367	1.43029
$113$	3.54735	0.333707	0.166853	−	0.985982i	$-0.446639\pi$
$113$	3.54735	0.333707	0.166853	+	0.985982i	$0.446639\pi$
$114$	6.26033	0.586334
$115$	1.77007	0.165059
$116$	−35.5337	−3.29922
$117$	5.56534	0.514516
$118$	−4.35680	−0.401076
$119$	1.07847	0.0988628
$120$	3.89810	0.355846
$121$	1.39458	0.126780
$122$	−4.80055	−0.434621
$123$	0.448168	0.0404099
$124$	40.8295	3.66660
$125$	4.99355	0.446637
$126$	−3.90853	−0.348199
$127$	17.4517	1.54859	0.774293	−	0.632828i	$-0.218106\pi$
$127$	17.4517	1.54859	0.774293	+	0.632828i	$0.218106\pi$
$128$	−2.88469	−0.254973
$129$	6.03831	0.531644
$130$	7.49437	0.657299
$131$	−22.1381	−1.93421	−0.967106	−	0.254375i	$-0.918130\pi$
$131$	−22.1381	−1.93421	−0.967106	+	0.254375i	$0.918130\pi$
$132$	−17.2324	−1.49989
$133$	−3.54889	−0.307728
$134$	−30.8633	−2.66619
$135$	−0.512843	−0.0441385
$136$	−5.50707	−0.472228
$137$	3.33383	0.284828	0.142414	−	0.989807i	$-0.454514\pi$
$137$	3.33383	0.284828	0.142414	+	0.989807i	$0.454514\pi$
$138$	9.06283	0.771479
$139$	−11.9080	−1.01002	−0.505011	−	0.863113i	$-0.668512\pi$
$139$	−11.9080	−1.01002	−0.505011	+	0.863113i	$0.668512\pi$
$140$	−3.73653	−0.315794
$141$	−9.64567	−0.812312
$142$	−3.57576	−0.300071
$143$	−19.5933	−1.63847
$144$	10.1690	0.847416
$145$	3.72302	0.309180
$146$	9.17252	0.759123
$147$	−4.78431	−0.394603
$148$	−58.2269	−4.78622
$149$	1.00308	0.0821752	0.0410876	−	0.999156i	$-0.486918\pi$
$149$	1.00308	0.0821752	0.0410876	+	0.999156i	$0.486918\pi$
$150$	12.4383	1.01558
$151$	−0.437071	−0.0355683	−0.0177842	−	0.999842i	$-0.505661\pi$
$151$	−0.437071	−0.0355683	−0.0177842	+	0.999842i	$0.505661\pi$
$152$	18.1220	1.46989
$153$	0.724523	0.0585742
$154$	13.7603	1.10884
$155$	−4.27788	−0.343608
$156$	27.2409	2.18102
$157$	7.52817	0.600813	0.300407	−	0.953811i	$-0.402878\pi$
$157$	7.52817	0.600813	0.300407	+	0.953811i	$0.402878\pi$
$158$	−2.07363	−0.164969
$159$	11.5963	0.919649
$160$	5.89752	0.466240
$161$	−5.13758	−0.404898
$162$	−2.62578	−0.206301
$163$	1.23427	0.0966751	0.0483376	−	0.998831i	$-0.484608\pi$
$163$	1.23427	0.0966751	0.0483376	+	0.998831i	$0.484608\pi$
$164$	2.19367	0.171296
$165$	1.80551	0.140559
$166$	−40.4230	−3.13743
$167$	−3.90265	−0.301996	−0.150998	−	0.988534i	$-0.548249\pi$
$167$	−3.90265	−0.301996	−0.150998	+	0.988534i	$0.548249\pi$
$168$	−11.3142	−0.872907
$169$	17.9730	1.38254
$170$	0.975654	0.0748292
$171$	−2.38418	−0.182322
$172$	29.5560	2.25362
$173$	23.1694	1.76154	0.880770	−	0.473545i	$-0.157026\pi$
$173$	23.1694	1.76154	0.880770	+	0.473545i	$0.157026\pi$
$174$	19.0621	1.44509
$175$	−7.05110	−0.533013
$176$	−35.8009	−2.69859
$177$	1.65924	0.124716
$178$	−14.9152	−1.11794
$179$	9.88048	0.738501	0.369251	−	0.929330i	$-0.379614\pi$
$179$	9.88048	0.738501	0.369251	+	0.929330i	$0.379614\pi$
$180$	−2.51023	−0.187102
$181$	−8.83167	−0.656453	−0.328226	−	0.944599i	$-0.606451\pi$
$181$	−8.83167	−0.656453	−0.328226	+	0.944599i	$0.606451\pi$
$182$	−21.7523	−1.61239
$183$	1.82823	0.135147
$184$	26.2345	1.93403
$185$	6.10068	0.448531
$186$	−21.9030	−1.60600
$187$	−2.55075	−0.186529
$188$	−47.2131	−3.44337
$189$	1.48852	0.108274
$190$	−3.21057	−0.232919
$191$	10.5628	0.764301	0.382150	−	0.924100i	$-0.375184\pi$
$191$	10.5628	0.764301	0.382150	+	0.924100i	$0.375184\pi$
$192$	9.85763	0.711413
$193$	−0.0684308	−0.00492576	−0.00246288	−	0.999997i	$-0.500784\pi$
$193$	−0.0684308	−0.00492576	−0.00246288	+	0.999997i	$0.500784\pi$
$194$	−2.96323	−0.212747
$195$	−2.85414	−0.204390
$196$	−23.4180	−1.67271
$197$	24.2333	1.72655	0.863277	−	0.504731i	$-0.168408\pi$
$197$	24.2333	1.72655	0.863277	+	0.504731i	$0.168408\pi$
$198$	9.24432	0.656965
$199$	19.0038	1.34715	0.673573	−	0.739121i	$-0.264759\pi$
$199$	19.0038	1.34715	0.673573	+	0.739121i	$0.264759\pi$
$200$	36.0057	2.54599
$201$	11.7540	0.829060
$202$	−37.7688	−2.65741
$203$	−10.8060	−0.758433
$204$	3.54635	0.248294
$205$	−0.229840	−0.0160527
$206$	43.1376	3.00554
$207$	−3.45147	−0.239894
$208$	56.5939	3.92408
$209$	8.39371	0.580605
$210$	2.00446	0.138321
$211$	−12.5023	−0.860696	−0.430348	−	0.902663i	$-0.641609\pi$
$211$	−12.5023	−0.860696	−0.430348	+	0.902663i	$0.641609\pi$
$212$	56.7610	3.89836
$213$	1.36179	0.0933081
$214$	−13.3671	−0.913758
$215$	−3.09671	−0.211194
$216$	−7.60096	−0.517180
$217$	12.4165	0.842886
$218$	−47.0284	−3.18516
$219$	−3.49325	−0.236052
$220$	8.83751	0.595825
$221$	4.03222	0.271236
$222$	31.2358	2.09641
$223$	13.9293	0.932775	0.466387	−	0.884581i	$-0.345555\pi$
$223$	13.9293	0.932775	0.466387	+	0.884581i	$0.345555\pi$
$224$	−17.1175	−1.14371
$225$	−4.73699	−0.315799
$226$	−9.31458	−0.619596
$227$	9.17776	0.609149	0.304575	−	0.952488i	$-0.401486\pi$
$227$	9.17776	0.609149	0.304575	+	0.952488i	$0.401486\pi$
$228$	−11.6699	−0.772859
$229$	2.95535	0.195295	0.0976476	−	0.995221i	$-0.468868\pi$
$229$	2.95535	0.195295	0.0976476	+	0.995221i	$0.468868\pi$
$230$	−4.64781	−0.306467
$231$	−5.24047	−0.344797
$232$	55.1797	3.62273
$233$	11.3384	0.742802	0.371401	−	0.928473i	$-0.378878\pi$
$233$	11.3384	0.742802	0.371401	+	0.928473i	$0.378878\pi$
$234$	−14.6134	−0.955306
$235$	4.94672	0.322688
$236$	8.12153	0.528667
$237$	0.789718	0.0512977
$238$	−2.83182	−0.183560
$239$	−3.31280	−0.214287	−0.107144	−	0.994244i	$-0.534170\pi$
$239$	−3.31280	−0.214287	−0.107144	+	0.994244i	$0.534170\pi$
$240$	−5.21510	−0.336633
$241$	14.4472	0.930628	0.465314	−	0.885146i	$-0.345941\pi$
$241$	14.4472	0.930628	0.465314	+	0.885146i	$0.345941\pi$
$242$	−3.66186	−0.235393
$243$	1.00000	0.0641500
$244$	8.94873	0.572884
$245$	2.45360	0.156755
$246$	−1.17679	−0.0750295
$247$	−13.2687	−0.844270
$248$	−63.4034	−4.02612
$249$	15.3946	0.975596
$250$	−13.1120	−0.829274
$251$	10.8076	0.682171	0.341085	−	0.940032i	$-0.389206\pi$
$251$	10.8076	0.682171	0.341085	+	0.940032i	$0.389206\pi$
$252$	7.28591	0.458969
$253$	12.1512	0.763942
$254$	−45.8243	−2.87527
$255$	−0.371567	−0.0232684
$256$	−12.1407	−0.758794
$257$	8.13340	0.507348	0.253674	−	0.967290i	$-0.418361\pi$
$257$	8.13340	0.507348	0.253674	+	0.967290i	$0.418361\pi$
$258$	−15.8553	−0.987108
$259$	−17.7071	−1.10027
$260$	−13.9703	−0.866401
$261$	−7.25957	−0.449356
$262$	58.1298	3.59127
$263$	−8.25552	−0.509057	−0.254529	−	0.967065i	$-0.581920\pi$
$263$	−8.25552	−0.509057	−0.254529	+	0.967065i	$0.581920\pi$
$264$	26.7599	1.64696
$265$	−5.94710	−0.365328
$266$	9.31862	0.571361
$267$	5.68030	0.347628
$268$	57.5325	3.51436
$269$	−17.6807	−1.07801	−0.539005	−	0.842302i	$-0.681200\pi$
$269$	−17.6807	−1.07801	−0.539005	+	0.842302i	$0.681200\pi$
$270$	1.34662	0.0819524
$271$	17.7186	1.07633	0.538164	−	0.842840i	$-0.319118\pi$
$271$	17.7186	1.07633	0.538164	+	0.842840i	$0.319118\pi$
$272$	7.36767	0.446731
$273$	8.28411	0.501377
$274$	−8.75391	−0.528843
$275$	16.6770	1.00566
$276$	−16.8941	−1.01690
$277$	−8.63283	−0.518697	−0.259348	−	0.965784i	$-0.583508\pi$
$277$	−8.63283	−0.518697	−0.259348	+	0.965784i	$0.583508\pi$
$278$	31.2678	1.87532
$279$	8.34151	0.499393
$280$	5.80239	0.346759
$281$	14.0154	0.836087	0.418044	−	0.908427i	$-0.362716\pi$
$281$	14.0154	0.836087	0.418044	+	0.908427i	$0.362716\pi$
$282$	25.3275	1.50823
$283$	20.3114	1.20739	0.603695	−	0.797215i	$-0.293694\pi$
$283$	20.3114	1.20739	0.603695	+	0.797215i	$0.293694\pi$
$284$	6.66559	0.395530
$285$	1.22271	0.0724270
$286$	51.4477	3.04217
$287$	0.667106	0.0393780
$288$	−11.4997	−0.677624
$289$	−16.4751	−0.969122
$290$	−9.77585	−0.574058
$291$	1.12851	0.0661546
$292$	−17.0986	−1.00062
$293$	−16.9270	−0.988884	−0.494442	−	0.869211i	$-0.664628\pi$
$293$	−16.9270	−0.988884	−0.494442	+	0.869211i	$0.664628\pi$
$294$	12.5626	0.732664
$295$	−0.850928	−0.0495430
$296$	90.4195	5.25553
$297$	−3.52059	−0.204286
$298$	−2.63386	−0.152575
$299$	−19.2086	−1.11086
$300$	−23.1863	−1.33866
$301$	8.98814	0.518068
$302$	1.14765	0.0660401
$303$	14.3838	0.826329
$304$	−24.2447	−1.39053
$305$	−0.937597	−0.0536867
$306$	−1.90244	−0.108755
$307$	14.7635	0.842595	0.421298	−	0.906922i	$-0.361575\pi$
$307$	14.7635	0.842595	0.421298	+	0.906922i	$0.361575\pi$
$308$	−25.6507	−1.46159
$309$	−16.4285	−0.934582
$310$	11.2328	0.637980
$311$	−0.591578	−0.0335453	−0.0167727	−	0.999859i	$-0.505339\pi$
$311$	−0.591578	−0.0335453	−0.0167727	+	0.999859i	$0.505339\pi$
$312$	−42.3019	−2.39487
$313$	4.83572	0.273331	0.136666	−	0.990617i	$-0.456361\pi$
$313$	4.83572	0.273331	0.136666	+	0.990617i	$0.456361\pi$
$314$	−19.7673	−1.11554
$315$	−0.763376	−0.0430114
$316$	3.86546	0.217449
$317$	6.27398	0.352382	0.176191	−	0.984356i	$-0.443622\pi$
$317$	6.27398	0.352382	0.176191	+	0.984356i	$0.443622\pi$
$318$	−30.4495	−1.70752
$319$	25.5580	1.43097
$320$	−5.05542	−0.282607
$321$	5.09072	0.284136
$322$	13.4902	0.751779
$323$	−1.72739	−0.0961146
$324$	4.89474	0.271930
$325$	−26.3630	−1.46235
$326$	−3.24091	−0.179498
$327$	17.9102	0.990438
$328$	−3.40651	−0.188093
$329$	−14.3578	−0.791569
$330$	−4.74088	−0.260977
$331$	15.6961	0.862734	0.431367	−	0.902177i	$-0.358031\pi$
$331$	15.6961	0.862734	0.431367	+	0.902177i	$0.358031\pi$
$332$	75.3528	4.13552
$333$	−11.8958	−0.651886
$334$	10.2475	0.560719
$335$	−6.02793	−0.329341
$336$	15.1367	0.825777
$337$	−24.7578	−1.34864	−0.674321	−	0.738438i	$-0.735564\pi$
$337$	−24.7578	−1.34864	−0.674321	+	0.738438i	$0.735564\pi$
$338$	−47.1931	−2.56697
$339$	3.54735	0.192666
$340$	−1.81872	−0.0986341
$341$	−29.3670	−1.59031
$342$	6.26033	0.338520
$343$	−17.5412	−0.947134
$344$	−45.8970	−2.47460
$345$	1.77007	0.0952971
$346$	−60.8379	−3.27067
$347$	−31.5231	−1.69225	−0.846125	−	0.532984i	$-0.821071\pi$
$347$	−31.5231	−1.69225	−0.846125	+	0.532984i	$0.821071\pi$
$348$	−35.5337	−1.90481
$349$	−5.76463	−0.308573	−0.154287	−	0.988026i	$-0.549308\pi$
$349$	−5.76463	−0.308573	−0.154287	+	0.988026i	$0.549308\pi$
$350$	18.5147	0.989650
$351$	5.56534	0.297056
$352$	40.4856	2.15789
$353$	−4.29745	−0.228730	−0.114365	−	0.993439i	$-0.536483\pi$
$353$	−4.29745	−0.228730	−0.114365	+	0.993439i	$0.536483\pi$
$354$	−4.35680	−0.231561
$355$	−0.698383	−0.0370663
$356$	27.8036	1.47359
$357$	1.07847	0.0570785
$358$	−25.9440	−1.37118
$359$	−35.7012	−1.88424	−0.942118	−	0.335282i	$-0.891168\pi$
$359$	−35.7012	−1.88424	−0.942118	+	0.335282i	$0.891168\pi$
$360$	3.89810	0.205448
$361$	−13.3157	−0.700827
$362$	23.1901	1.21884
$363$	1.39458	0.0731963
$364$	40.5485	2.12532
$365$	1.79149	0.0937709
$366$	−4.80055	−0.250929
$367$	−5.69948	−0.297510	−0.148755	−	0.988874i	$-0.547527\pi$
$367$	−5.69948	−0.297510	−0.148755	+	0.988874i	$0.547527\pi$
$368$	−35.0980	−1.82961
$369$	0.448168	0.0233307
$370$	−16.0191	−0.832792
$371$	17.2614	0.896165
$372$	40.8295	2.11691
$373$	23.5329	1.21849	0.609244	−	0.792983i	$-0.291473\pi$
$373$	23.5329	1.21849	0.609244	+	0.792983i	$0.291473\pi$
$374$	6.69772	0.346331
$375$	4.99355	0.257866
$376$	73.3164	3.78100
$377$	−40.4020	−2.08081
$378$	−3.90853	−0.201033
$379$	21.2003	1.08899	0.544494	−	0.838765i	$-0.316722\pi$
$379$	21.2003	1.08899	0.544494	+	0.838765i	$0.316722\pi$
$380$	5.98484	0.307016
$381$	17.4517	0.894076
$382$	−27.7358	−1.41908
$383$	2.96748	0.151631	0.0758156	−	0.997122i	$-0.475844\pi$
$383$	2.96748	0.151631	0.0758156	+	0.997122i	$0.475844\pi$
$384$	−2.88469	−0.147209
$385$	2.68754	0.136970
$386$	0.179685	0.00914570
$387$	6.03831	0.306945
$388$	5.52377	0.280427
$389$	29.2435	1.48270	0.741351	−	0.671117i	$-0.234185\pi$
$389$	29.2435	1.48270	0.741351	+	0.671117i	$0.234185\pi$
$390$	7.49437	0.379492
$391$	−2.50067	−0.126465
$392$	36.3654	1.83673
$393$	−22.1381	−1.11672
$394$	−63.6315	−3.20571
$395$	−0.405001	−0.0203778
$396$	−17.2324	−0.865960
$397$	−31.3696	−1.57439	−0.787197	−	0.616702i	$-0.788468\pi$
$397$	−31.3696	−1.57439	−0.787197	+	0.616702i	$0.788468\pi$
$398$	−49.8999	−2.50126
$399$	−3.54889	−0.177667
$400$	−48.1704	−2.40852
$401$	17.1485	0.856357	0.428179	−	0.903694i	$-0.359155\pi$
$401$	17.1485	0.856357	0.428179	+	0.903694i	$0.359155\pi$
$402$	−30.8633	−1.53932
$403$	46.4233	2.31251
$404$	70.4051	3.50278
$405$	−0.512843	−0.0254834
$406$	28.3742	1.40819
$407$	41.8803	2.07593
$408$	−5.50707	−0.272641
$409$	−8.67410	−0.428907	−0.214453	−	0.976734i	$-0.568797\pi$
$409$	−8.67410	−0.428907	−0.214453	+	0.976734i	$0.568797\pi$
$410$	0.603510	0.0298052
$411$	3.33383	0.164446
$412$	−80.4130	−3.96166
$413$	2.46980	0.121531
$414$	9.06283	0.445414
$415$	−7.89504	−0.387552
$416$	−63.9995	−3.13783
$417$	−11.9080	−0.583137
$418$	−22.0401	−1.07802
$419$	33.6300	1.64293	0.821466	−	0.570258i	$-0.193157\pi$
$419$	33.6300	1.64293	0.821466	+	0.570258i	$0.193157\pi$
$420$	−3.73653	−0.182324
$421$	31.2738	1.52419	0.762096	−	0.647464i	$-0.224170\pi$
$421$	31.2738	1.52419	0.762096	+	0.647464i	$0.224170\pi$
$422$	32.8284	1.59806
$423$	−9.64567	−0.468989
$424$	−88.1433	−4.28062
$425$	−3.43206	−0.166479
$426$	−3.57576	−0.173246
$427$	2.72136	0.131696
$428$	24.9177	1.20444
$429$	−19.5933	−0.945973
$430$	8.13128	0.392125
$431$	−4.62618	−0.222835	−0.111418	−	0.993774i	$-0.535539\pi$
$431$	−4.62618	−0.222835	−0.111418	+	0.993774i	$0.535539\pi$
$432$	10.1690	0.489256
$433$	7.47343	0.359150	0.179575	−	0.983744i	$-0.442528\pi$
$433$	7.47343	0.359150	0.179575	+	0.983744i	$0.442528\pi$
$434$	−32.6030	−1.56499
$435$	3.72302	0.178505
$436$	87.6659	4.19844
$437$	8.22892	0.393643
$438$	9.17252	0.438280
$439$	−15.6216	−0.745577	−0.372788	−	0.927916i	$-0.621598\pi$
$439$	−15.6216	−0.745577	−0.372788	+	0.927916i	$0.621598\pi$
$440$	−13.7236	−0.654248
$441$	−4.78431	−0.227824
$442$	−10.5877	−0.503607
$443$	14.7430	0.700463	0.350232	−	0.936663i	$-0.386103\pi$
$443$	14.7430	0.700463	0.350232	+	0.936663i	$0.386103\pi$
$444$	−58.2269	−2.76332
$445$	−2.91310	−0.138094
$446$	−36.5753	−1.73189
$447$	1.00308	0.0474439
$448$	14.6733	0.693247
$449$	−20.1808	−0.952392	−0.476196	−	0.879339i	$-0.657985\pi$
$449$	−20.1808	−0.952392	−0.476196	+	0.879339i	$0.657985\pi$
$450$	12.4383	0.586348
$451$	−1.57782	−0.0742965
$452$	17.3634	0.816704
$453$	−0.437071	−0.0205354
$454$	−24.0988	−1.13101
$455$	−4.24845	−0.199170
$456$	18.1220	0.848641
$457$	10.6887	0.499994	0.249997	−	0.968247i	$-0.419570\pi$
$457$	10.6887	0.499994	0.249997	+	0.968247i	$0.419570\pi$
$458$	−7.76012	−0.362606
$459$	0.724523	0.0338179
$460$	8.66401	0.403961
$461$	23.4106	1.09034	0.545171	−	0.838325i	$-0.316465\pi$
$461$	23.4106	1.09034	0.545171	+	0.838325i	$0.316465\pi$
$462$	13.7603	0.640189
$463$	−28.6425	−1.33113	−0.665564	−	0.746340i	$-0.731809\pi$
$463$	−28.6425	−1.33113	−0.665564	+	0.746340i	$0.731809\pi$
$464$	−73.8225	−3.42713
$465$	−4.27788	−0.198382
$466$	−29.7721	−1.37917
$467$	−28.0090	−1.29610	−0.648051	−	0.761597i	$-0.724416\pi$
$467$	−28.0090	−1.29610	−0.648051	+	0.761597i	$0.724416\pi$
$468$	27.2409	1.25921
$469$	17.4960	0.807889
$470$	−12.9890	−0.599138
$471$	7.52817	0.346880
$472$	−12.6118	−0.580505
$473$	−21.2584	−0.977464
$474$	−2.07363	−0.0952449
$475$	11.2938	0.518196
$476$	5.27881	0.241954
$477$	11.5963	0.530960
$478$	8.69870	0.397869
$479$	23.0747	1.05431	0.527156	−	0.849769i	$-0.323258\pi$
$479$	23.0747	1.05431	0.527156	+	0.849769i	$0.323258\pi$
$480$	5.89752	0.269184
$481$	−66.2042	−3.01865
$482$	−37.9353	−1.72791
$483$	−5.13758	−0.233768
$484$	6.82609	0.310277
$485$	−0.578750	−0.0262797
$486$	−2.62578	−0.119108
$487$	−15.4914	−0.701982	−0.350991	−	0.936379i	$-0.614155\pi$
$487$	−15.4914	−0.701982	−0.350991	+	0.936379i	$0.614155\pi$
$488$	−13.8963	−0.629057
$489$	1.23427	0.0558154
$490$	−6.44263	−0.291048
$491$	−2.10376	−0.0949415	−0.0474707	−	0.998873i	$-0.515116\pi$
$491$	−2.10376	−0.0949415	−0.0474707	+	0.998873i	$0.515116\pi$
$492$	2.19367	0.0988981
$493$	−5.25973	−0.236886
$494$	34.8408	1.56756
$495$	1.80551	0.0811517
$496$	84.8247	3.80874
$497$	2.02705	0.0909254
$498$	−40.4230	−1.81140
$499$	34.7065	1.55367	0.776837	−	0.629701i	$-0.216823\pi$
$499$	34.7065	1.55367	0.776837	+	0.629701i	$0.216823\pi$
$500$	24.4421	1.09309
$501$	−3.90265	−0.174357
$502$	−28.3785	−1.26659
$503$	−6.60682	−0.294583	−0.147292	−	0.989093i	$-0.547056\pi$
$503$	−6.60682	−0.294583	−0.147292	+	0.989093i	$0.547056\pi$
$504$	−11.3142	−0.503973
$505$	−7.37665	−0.328257
$506$	−31.9065	−1.41842
$507$	17.9730	0.798207
$508$	85.4214	3.78996
$509$	−14.3949	−0.638042	−0.319021	−	0.947748i	$-0.603354\pi$
$509$	−14.3949	−0.638042	−0.319021	+	0.947748i	$0.603354\pi$
$510$	0.975654	0.0432027
$511$	−5.19977	−0.230024
$512$	37.6482	1.66383
$513$	−2.38418	−0.105264
$514$	−21.3566	−0.941997
$515$	8.42522	0.371259
$516$	29.5560	1.30113
$517$	33.9585	1.49349
$518$	46.4951	2.04288
$519$	23.1694	1.01703
$520$	21.6942	0.951355
$521$	22.7094	0.994915	0.497458	−	0.867488i	$-0.334267\pi$
$521$	22.7094	0.994915	0.497458	+	0.867488i	$0.334267\pi$
$522$	19.0621	0.834324
$523$	35.4153	1.54860	0.774301	−	0.632818i	$-0.218102\pi$
$523$	35.4153	1.54860	0.774301	+	0.632818i	$0.218102\pi$
$524$	−108.360	−4.73373
$525$	−7.05110	−0.307735
$526$	21.6772	0.945171
$527$	6.04361	0.263264
$528$	−35.8009	−1.55803
$529$	−11.0873	−0.482057
$530$	15.6158	0.678307
$531$	1.65924	0.0720047
$532$	−17.3709	−0.753124
$533$	2.49421	0.108036
$534$	−14.9152	−0.645445
$535$	−2.61074	−0.112872
$536$	−89.3413	−3.85896
$537$	9.88048	0.426374
$538$	46.4257	2.00155
$539$	16.8436	0.725506
$540$	−2.51023	−0.108023
$541$	−34.5273	−1.48445	−0.742223	−	0.670153i	$-0.766229\pi$
$541$	−34.5273	−1.48445	−0.742223	+	0.670153i	$0.766229\pi$
$542$	−46.5252	−1.99843
$543$	−8.83167	−0.379003
$544$	−8.33177	−0.357222
$545$	−9.18514	−0.393448
$546$	−21.7523	−0.930911
$547$	2.88866	0.123510	0.0617550	−	0.998091i	$-0.480330\pi$
$547$	2.88866	0.123510	0.0617550	+	0.998091i	$0.480330\pi$
$548$	16.3182	0.697080
$549$	1.82823	0.0780271
$550$	−43.7903	−1.86722
$551$	17.3081	0.737350
$552$	26.2345	1.11662
$553$	1.17551	0.0499878
$554$	22.6679	0.963069
$555$	6.10068	0.258960
$556$	−58.2865	−2.47190
$557$	−43.6156	−1.84805	−0.924026	−	0.382331i	$-0.875122\pi$
$557$	−43.6156	−1.84805	−0.924026	+	0.382331i	$0.875122\pi$
$558$	−21.9030	−0.927227
$559$	33.6052	1.42135
$560$	−7.76277	−0.328037
$561$	−2.55075	−0.107693
$562$	−36.8014	−1.55237
$563$	−6.61836	−0.278931	−0.139465	−	0.990227i	$-0.544538\pi$
$563$	−6.61836	−0.278931	−0.139465	+	0.990227i	$0.544538\pi$
$564$	−47.2131	−1.98803
$565$	−1.81923	−0.0765358
$566$	−53.3335	−2.24177
$567$	1.48852	0.0625119
$568$	−10.3509	−0.434314
$569$	7.41292	0.310766	0.155383	−	0.987854i	$-0.450339\pi$
$569$	7.41292	0.310766	0.155383	+	0.987854i	$0.450339\pi$
$570$	−3.21057	−0.134476
$571$	−30.8065	−1.28921	−0.644607	−	0.764514i	$-0.722979\pi$
$571$	−30.8065	−1.28921	−0.644607	+	0.764514i	$0.722979\pi$
$572$	−95.9040	−4.00995
$573$	10.5628	0.441269
$574$	−1.75168	−0.0731136
$575$	16.3496	0.681826
$576$	9.85763	0.410735
$577$	−5.62535	−0.234186	−0.117093	−	0.993121i	$-0.537358\pi$
$577$	−5.62535	−0.234186	−0.117093	+	0.993121i	$0.537358\pi$
$578$	43.2600	1.79938
$579$	−0.0684308	−0.00284389
$580$	18.2232	0.756678
$581$	22.9152	0.950683
$582$	−2.96323	−0.122830
$583$	−40.8260	−1.69084
$584$	26.5521	1.09873
$585$	−2.85414	−0.118004
$586$	44.4466	1.83607
$587$	−20.2568	−0.836089	−0.418044	−	0.908427i	$-0.637284\pi$
$587$	−20.2568	−0.836089	−0.418044	+	0.908427i	$0.637284\pi$
$588$	−23.4180	−0.965741
$589$	−19.8876	−0.819455
$590$	2.23435	0.0919869
$591$	24.2333	0.996826
$592$	−120.968	−4.97177
$593$	−15.7130	−0.645254	−0.322627	−	0.946526i	$-0.604566\pi$
$593$	−15.7130	−0.645254	−0.322627	+	0.946526i	$0.604566\pi$
$594$	9.24432	0.379299
$595$	−0.553084	−0.0226742
$596$	4.90980	0.201113
$597$	19.0038	0.777775
$598$	50.4377	2.06255
$599$	5.31680	0.217239	0.108619	−	0.994083i	$-0.465357\pi$
$599$	5.31680	0.217239	0.108619	+	0.994083i	$0.465357\pi$
$600$	36.0057	1.46993
$601$	−35.9353	−1.46583	−0.732915	−	0.680320i	$-0.761841\pi$
$601$	−35.9353	−1.46583	−0.732915	+	0.680320i	$0.761841\pi$
$602$	−23.6009	−0.961901
$603$	11.7540	0.478658
$604$	−2.13935	−0.0870489
$605$	−0.715199	−0.0290770
$606$	−37.7688	−1.53425
$607$	−11.3260	−0.459708	−0.229854	−	0.973225i	$-0.573825\pi$
$607$	−11.3260	−0.459708	−0.229854	+	0.973225i	$0.573825\pi$
$608$	27.4172	1.11191
$609$	−10.8060	−0.437881
$610$	2.46193	0.0996805
$611$	−53.6814	−2.17172
$612$	3.54635	0.143353
$613$	32.0428	1.29420	0.647099	−	0.762406i	$-0.275982\pi$
$613$	32.0428	1.29420	0.647099	+	0.762406i	$0.275982\pi$
$614$	−38.7656	−1.56445
$615$	−0.229840	−0.00926804
$616$	39.8326	1.60490
$617$	14.1753	0.570675	0.285337	−	0.958427i	$-0.407894\pi$
$617$	14.1753	0.570675	0.285337	+	0.958427i	$0.407894\pi$
$618$	43.1376	1.73525
$619$	−10.4528	−0.420135	−0.210067	−	0.977687i	$-0.567368\pi$
$619$	−10.4528	−0.420135	−0.210067	+	0.977687i	$0.567368\pi$
$620$	−20.9391	−0.840936
$621$	−3.45147	−0.138503
$622$	1.55336	0.0622839
$623$	8.45522	0.338751
$624$	56.5939	2.26557
$625$	21.1241	0.844962
$626$	−12.6976	−0.507496
$627$	8.39371	0.335213
$628$	36.8484	1.47041
$629$	−8.61879	−0.343654
$630$	2.00446	0.0798597
$631$	−3.59792	−0.143231	−0.0716155	−	0.997432i	$-0.522815\pi$
$631$	−3.59792	−0.143231	−0.0716155	+	0.997432i	$0.522815\pi$
$632$	−6.00261	−0.238771
$633$	−12.5023	−0.496923
$634$	−16.4741	−0.654271
$635$	−8.94997	−0.355169
$636$	56.7610	2.25072
$637$	−26.6263	−1.05497
$638$	−67.1098	−2.65690
$639$	1.36179	0.0538715
$640$	1.47939	0.0584781
$641$	39.6629	1.56659	0.783294	−	0.621651i	$-0.213538\pi$
$641$	39.6629	1.56659	0.783294	+	0.621651i	$0.213538\pi$
$642$	−13.3671	−0.527559
$643$	8.37893	0.330433	0.165216	−	0.986257i	$-0.447168\pi$
$643$	8.37893	0.330433	0.165216	+	0.986257i	$0.447168\pi$
$644$	−25.1471	−0.990936
$645$	−3.09671	−0.121933
$646$	4.53575	0.178457
$647$	−5.16456	−0.203040	−0.101520	−	0.994834i	$-0.532371\pi$
$647$	−5.16456	−0.203040	−0.101520	+	0.994834i	$0.532371\pi$
$648$	−7.60096	−0.298594
$649$	−5.84150	−0.229299
$650$	69.2234	2.71517
$651$	12.4165	0.486640
$652$	6.04141	0.236600
$653$	−16.3389	−0.639392	−0.319696	−	0.947520i	$-0.603581\pi$
$653$	−16.3389	−0.639392	−0.319696	+	0.947520i	$0.603581\pi$
$654$	−47.0284	−1.83896
$655$	11.3534	0.443612
$656$	4.55742	0.177937
$657$	−3.49325	−0.136285
$658$	37.7004	1.46971
$659$	−1.48616	−0.0578924	−0.0289462	−	0.999581i	$-0.509215\pi$
$659$	−1.48616	−0.0578924	−0.0289462	+	0.999581i	$0.509215\pi$
$660$	8.83751	0.344000
$661$	20.4045	0.793641	0.396821	−	0.917896i	$-0.370114\pi$
$661$	20.4045	0.793641	0.396821	+	0.917896i	$0.370114\pi$
$662$	−41.2145	−1.60185
$663$	4.03222	0.156598
$664$	−117.014	−4.54103
$665$	1.82002	0.0705775
$666$	31.2358	1.21036
$667$	25.0562	0.970181
$668$	−19.1024	−0.739096
$669$	13.9293	0.538538
$670$	15.8281	0.611491
$671$	−6.43647	−0.248477
$672$	−17.1175	−0.660320
$673$	28.6175	1.10312	0.551561	−	0.834134i	$-0.314032\pi$
$673$	28.6175	1.10312	0.551561	+	0.834134i	$0.314032\pi$
$674$	65.0085	2.50404
$675$	−4.73699	−0.182327
$676$	87.9730	3.38358
$677$	−26.4014	−1.01469	−0.507343	−	0.861744i	$-0.669372\pi$
$677$	−26.4014	−1.01469	−0.507343	+	0.861744i	$0.669372\pi$
$678$	−9.31458	−0.357724
$679$	1.67981	0.0644652
$680$	2.82426	0.108306
$681$	9.17776	0.351692
$682$	77.1115	2.95275
$683$	−24.2404	−0.927531	−0.463766	−	0.885958i	$-0.653502\pi$
$683$	−24.2404	−0.927531	−0.463766	+	0.885958i	$0.653502\pi$
$684$	−11.6699	−0.446211
$685$	−1.70973	−0.0653255
$686$	46.0593	1.75855
$687$	2.95535	0.112754
$688$	61.4036	2.34099
$689$	64.5375	2.45868
$690$	−4.64781	−0.176939
$691$	30.3612	1.15500	0.577498	−	0.816392i	$-0.304029\pi$
$691$	30.3612	1.15500	0.577498	+	0.816392i	$0.304029\pi$
$692$	113.408	4.31114
$693$	−5.24047	−0.199069
$694$	82.7729	3.14202
$695$	6.10693	0.231649
$696$	55.1797	2.09158
$697$	0.324708	0.0122992
$698$	15.1367	0.572931
$699$	11.3384	0.428857
$700$	−34.5133	−1.30448
$701$	−45.3021	−1.71104	−0.855519	−	0.517771i	$-0.826762\pi$
$701$	−45.3021	−1.71104	−0.855519	+	0.517771i	$0.826762\pi$
$702$	−14.6134	−0.551546
$703$	28.3617	1.06968
$704$	−34.7047	−1.30798
$705$	4.94672	0.186304
$706$	11.2842	0.424685
$707$	21.4106	0.805228
$708$	8.12153	0.305226
$709$	−11.7523	−0.441367	−0.220684	−	0.975345i	$-0.570829\pi$
$709$	−11.7523	−0.441367	−0.220684	+	0.975345i	$0.570829\pi$
$710$	1.83380	0.0688214
$711$	0.789718	0.0296167
$712$	−43.1757	−1.61808
$713$	−28.7905	−1.07821
$714$	−2.83182	−0.105978
$715$	10.0483	0.375784
$716$	48.3624	1.80739
$717$	−3.31280	−0.123719
$718$	93.7435	3.49848
$719$	−2.02369	−0.0754710	−0.0377355	−	0.999288i	$-0.512014\pi$
$719$	−2.02369	−0.0754710	−0.0377355	+	0.999288i	$0.512014\pi$
$720$	−5.21510	−0.194355
$721$	−24.4541	−0.910717
$722$	34.9642	1.30123
$723$	14.4472	0.537298
$724$	−43.2287	−1.60658
$725$	34.3885	1.27716
$726$	−3.66186	−0.135904
$727$	32.2155	1.19481	0.597403	−	0.801941i	$-0.296199\pi$
$727$	32.2155	1.19481	0.597403	+	0.801941i	$0.296199\pi$
$728$	−62.9671	−2.33372
$729$	1.00000	0.0370370
$730$	−4.70407	−0.174105
$731$	4.37490	0.161811
$732$	8.94873	0.330755
$733$	−22.3878	−0.826913	−0.413457	−	0.910524i	$-0.635679\pi$
$733$	−22.3878	−0.826913	−0.413457	+	0.910524i	$0.635679\pi$
$734$	14.9656	0.552390
$735$	2.45360	0.0905025
$736$	39.6908	1.46302
$737$	−41.3809	−1.52428
$738$	−1.17679	−0.0433183
$739$	14.6496	0.538895	0.269447	−	0.963015i	$-0.413159\pi$
$739$	14.6496	0.538895	0.269447	+	0.963015i	$0.413159\pi$
$740$	29.8613	1.09772
$741$	−13.2687	−0.487439
$742$	−45.3246	−1.66392
$743$	−14.6172	−0.536252	−0.268126	−	0.963384i	$-0.586404\pi$
$743$	−14.6172	−0.536252	−0.268126	+	0.963384i	$0.586404\pi$
$744$	−63.4034	−2.32448
$745$	−0.514421	−0.0188469
$746$	−61.7924	−2.26238
$747$	15.3946	0.563261
$748$	−12.4853	−0.456507
$749$	7.57763	0.276881
$750$	−13.1120	−0.478782
$751$	−19.0736	−0.696004	−0.348002	−	0.937494i	$-0.613140\pi$
$751$	−19.0736	−0.696004	−0.348002	+	0.937494i	$0.613140\pi$
$752$	−98.0868	−3.57686
$753$	10.8076	0.393851
$754$	106.087	3.86345
$755$	0.224149	0.00815762
$756$	7.28591	0.264986
$757$	−27.1331	−0.986168	−0.493084	−	0.869982i	$-0.664130\pi$
$757$	−27.1331	−0.986168	−0.493084	+	0.869982i	$0.664130\pi$
$758$	−55.6675	−2.02193
$759$	12.1512	0.441062
$760$	−9.29376	−0.337120
$761$	0.419839	0.0152191	0.00760957	−	0.999971i	$-0.497578\pi$
$761$	0.419839	0.0152191	0.00760957	+	0.999971i	$0.497578\pi$
$762$	−45.8243	−1.66004
$763$	26.6597	0.965146
$764$	51.7024	1.87053
$765$	−0.371567	−0.0134340
$766$	−7.79196	−0.281535
$767$	9.23421	0.333428
$768$	−12.1407	−0.438090
$769$	23.0788	0.832242	0.416121	−	0.909309i	$-0.363389\pi$
$769$	23.0788	0.832242	0.416121	+	0.909309i	$0.363389\pi$
$770$	−7.05689	−0.254313
$771$	8.13340	0.292917
$772$	−0.334951	−0.0120552
$773$	52.5502	1.89010	0.945050	−	0.326925i	$-0.106013\pi$
$773$	52.5502	1.89010	0.945050	+	0.326925i	$0.106013\pi$
$774$	−15.8553	−0.569907
$775$	−39.5136	−1.41937
$776$	−8.57777	−0.307924
$777$	−17.7071	−0.635240
$778$	−76.7870	−2.75295
$779$	−1.06851	−0.0382834
$780$	−13.9703	−0.500217
$781$	−4.79430	−0.171554
$782$	6.56623	0.234808
$783$	−7.25957	−0.259436
$784$	−48.6517	−1.73756
$785$	−3.86077	−0.137797
$786$	58.1298	2.07342
$787$	19.4221	0.692322	0.346161	−	0.938175i	$-0.387485\pi$
$787$	19.4221	0.692322	0.346161	+	0.938175i	$0.387485\pi$
$788$	118.616	4.22551
$789$	−8.25552	−0.293904
$790$	1.06345	0.0378357
$791$	5.28030	0.187746
$792$	26.7599	0.950871
$793$	10.1747	0.361316
$794$	82.3697	2.92319
$795$	−5.94710	−0.210922
$796$	93.0188	3.29696
$797$	−23.0029	−0.814803	−0.407401	−	0.913249i	$-0.633565\pi$
$797$	−23.0029	−0.814803	−0.407401	+	0.913249i	$0.633565\pi$
$798$	9.31862	0.329875
$799$	−6.98852	−0.247236
$800$	54.4738	1.92594
$801$	5.68030	0.200703
$802$	−45.0284	−1.59001
$803$	12.2983	0.433998
$804$	57.5325	2.02902
$805$	2.63477	0.0928636
$806$	−121.897	−4.29366
$807$	−17.6807	−0.622390
$808$	−109.331	−3.84625
$809$	−0.835205	−0.0293643	−0.0146821	−	0.999892i	$-0.504674\pi$
$809$	−0.835205	−0.0293643	−0.0146821	+	0.999892i	$0.504674\pi$
$810$	1.34662	0.0473152
$811$	22.2473	0.781209	0.390604	−	0.920559i	$-0.372266\pi$
$811$	22.2473	0.781209	0.390604	+	0.920559i	$0.372266\pi$
$812$	−52.8926	−1.85617
$813$	17.7186	0.621418
$814$	−109.969	−3.85440
$815$	−0.632985	−0.0221725
$816$	7.36767	0.257920
$817$	−14.3964	−0.503666
$818$	22.7763	0.796355
$819$	8.28411	0.289470
$820$	−1.12501	−0.0392869
$821$	14.7409	0.514460	0.257230	−	0.966350i	$-0.417190\pi$
$821$	14.7409	0.514460	0.257230	+	0.966350i	$0.417190\pi$
$822$	−8.75391	−0.305328
$823$	8.52954	0.297321	0.148661	−	0.988888i	$-0.452504\pi$
$823$	8.52954	0.297321	0.148661	+	0.988888i	$0.452504\pi$
$824$	124.872	4.35012
$825$	16.6770	0.580619
$826$	−6.48517	−0.225648
$827$	45.1685	1.57066	0.785332	−	0.619075i	$-0.212492\pi$
$827$	45.1685	1.57066	0.785332	+	0.619075i	$0.212492\pi$
$828$	−16.8941	−0.587110
$829$	20.5514	0.713779	0.356889	−	0.934147i	$-0.383837\pi$
$829$	20.5514	0.713779	0.356889	+	0.934147i	$0.383837\pi$
$830$	20.7307	0.719572
$831$	−8.63283	−0.299470
$832$	54.8610	1.90196
$833$	−3.46635	−0.120102
$834$	31.2678	1.08272
$835$	2.00145	0.0692629
$836$	41.0850	1.42096
$837$	8.34151	0.288325
$838$	−88.3050	−3.05045
$839$	−12.0818	−0.417109	−0.208554	−	0.978011i	$-0.566876\pi$
$839$	−12.0818	−0.417109	−0.208554	+	0.978011i	$0.566876\pi$
$840$	5.80239	0.200202
$841$	23.7014	0.817289
$842$	−82.1183	−2.82998
$843$	14.0154	0.482715
$844$	−61.1957	−2.10644
$845$	−9.21731	−0.317085
$846$	25.3275	0.870776
$847$	2.07585	0.0713272
$848$	117.923	4.04950
$849$	20.3114	0.697087
$850$	9.01185	0.309104
$851$	41.0581	1.40745
$852$	6.66559	0.228360
$853$	22.0064	0.753485	0.376743	−	0.926318i	$-0.377044\pi$
$853$	22.0064	0.753485	0.376743	+	0.926318i	$0.377044\pi$
$854$	−7.14570	−0.244521
$855$	1.22271	0.0418157
$856$	−38.6944	−1.32255
$857$	−18.0295	−0.615875	−0.307937	−	0.951407i	$-0.599639\pi$
$857$	−18.0295	−0.615875	−0.307937	+	0.951407i	$0.599639\pi$
$858$	51.4477	1.75640
$859$	−23.6107	−0.805586	−0.402793	−	0.915291i	$-0.631961\pi$
$859$	−23.6107	−0.805586	−0.402793	+	0.915291i	$0.631961\pi$
$860$	−15.1576	−0.516869
$861$	0.667106	0.0227349
$862$	12.1473	0.413740
$863$	23.5983	0.803296	0.401648	−	0.915794i	$-0.368438\pi$
$863$	23.5983	0.803296	0.401648	+	0.915794i	$0.368438\pi$
$864$	−11.4997	−0.391226
$865$	−11.8823	−0.404010
$866$	−19.6236	−0.666838
$867$	−16.4751	−0.559523
$868$	60.7755	2.06285
$869$	−2.78028	−0.0943144
$870$	−9.77585	−0.331432
$871$	65.4147	2.21649
$872$	−136.135	−4.61011
$873$	1.12851	0.0381943
$874$	−21.6074	−0.730880
$875$	7.43299	0.251281
$876$	−17.0986	−0.577707
$877$	−25.4944	−0.860883	−0.430442	−	0.902618i	$-0.641642\pi$
$877$	−25.4944	−0.860883	−0.430442	+	0.902618i	$0.641642\pi$
$878$	41.0189	1.38432
$879$	−16.9270	−0.570932
$880$	18.3602	0.618924
$881$	38.9668	1.31283	0.656413	−	0.754402i	$-0.272073\pi$
$881$	38.9668	1.31283	0.656413	+	0.754402i	$0.272073\pi$
$882$	12.5626	0.423004
$883$	−0.477590	−0.0160722	−0.00803610	−	0.999968i	$-0.502558\pi$
$883$	−0.477590	−0.0160722	−0.00803610	+	0.999968i	$0.502558\pi$
$884$	19.7366	0.663815
$885$	−0.850928	−0.0286036
$886$	−38.7120	−1.30056
$887$	40.7089	1.36687	0.683436	−	0.730010i	$-0.260485\pi$
$887$	40.7089	1.36687	0.683436	+	0.730010i	$0.260485\pi$
$888$	90.4195	3.03428
$889$	25.9771	0.871245
$890$	7.64917	0.256401
$891$	−3.52059	−0.117944
$892$	68.1803	2.28284
$893$	22.9970	0.769565
$894$	−2.63386	−0.0880895
$895$	−5.06713	−0.169376
$896$	−4.29391	−0.143449
$897$	−19.2086	−0.641357
$898$	52.9905	1.76831
$899$	−60.5558	−2.01965
$900$	−23.1863	−0.772878
$901$	8.40182	0.279905
$902$	4.14301	0.137947
$903$	8.98814	0.299107
$904$	−26.9633	−0.896785
$905$	4.52926	0.150558
$906$	1.14765	0.0381283
$907$	−16.5792	−0.550502	−0.275251	−	0.961372i	$-0.588761\pi$
$907$	−16.5792	−0.550502	−0.275251	+	0.961372i	$0.588761\pi$
$908$	44.9227	1.49081
$909$	14.3838	0.477082
$910$	11.1555	0.369801
$911$	53.8871	1.78536	0.892680	−	0.450692i	$-0.148823\pi$
$911$	53.8871	1.78536	0.892680	+	0.450692i	$0.148823\pi$
$912$	−24.2447	−0.802821
$913$	−54.1983	−1.79370
$914$	−28.0661	−0.928344
$915$	−0.937597	−0.0309960
$916$	14.4657	0.477960
$917$	−32.9529	−1.08820
$918$	−1.90244	−0.0627899
$919$	47.8047	1.57693	0.788466	−	0.615079i	$-0.210876\pi$
$919$	47.8047	1.57693	0.788466	+	0.615079i	$0.210876\pi$
$920$	−13.4542	−0.443572
$921$	14.7635	0.486473
$922$	−61.4712	−2.02445
$923$	7.57881	0.249459
$924$	−25.6507	−0.843847
$925$	56.3503	1.85279
$926$	75.2089	2.47152
$927$	−16.4285	−0.539581
$928$	83.4826	2.74045
$929$	53.1529	1.74389	0.871944	−	0.489605i	$-0.162859\pi$
$929$	53.1529	1.74389	0.871944	+	0.489605i	$0.162859\pi$
$930$	11.2328	0.368338
$931$	11.4066	0.373838
$932$	55.4984	1.81791
$933$	−0.591578	−0.0193674
$934$	73.5456	2.40649
$935$	1.30814	0.0427806
$936$	−42.3019	−1.38268
$937$	14.2116	0.464274	0.232137	−	0.972683i	$-0.425428\pi$
$937$	14.2116	0.464274	0.232137	+	0.972683i	$0.425428\pi$
$938$	−45.9406	−1.50002
$939$	4.83572	0.157808
$940$	24.2129	0.789738
$941$	24.1522	0.787340	0.393670	−	0.919252i	$-0.371205\pi$
$941$	24.1522	0.787340	0.393670	+	0.919252i	$0.371205\pi$
$942$	−19.7673	−0.644055
$943$	−1.54684	−0.0503721
$944$	16.8728	0.549162
$945$	−0.763376	−0.0248326
$946$	55.8201	1.81487
$947$	−33.7074	−1.09534	−0.547671	−	0.836694i	$-0.684486\pi$
$947$	−33.7074	−1.09534	−0.547671	+	0.836694i	$0.684486\pi$
$948$	3.86546	0.125544
$949$	−19.4411	−0.631086
$950$	−29.6551	−0.962139
$951$	6.27398	0.203448
$952$	−8.19738	−0.265679
$953$	−34.8626	−1.12931	−0.564656	−	0.825327i	$-0.690991\pi$
$953$	−34.8626	−1.12931	−0.564656	+	0.825327i	$0.690991\pi$
$954$	−30.4495	−0.985838
$955$	−5.41708	−0.175293
$956$	−16.2153	−0.524440
$957$	25.5580	0.826173
$958$	−60.5893	−1.95755
$959$	4.96247	0.160246
$960$	−5.05542	−0.163163
$961$	38.5807	1.24454
$962$	173.838	5.60476
$963$	5.09072	0.164046
$964$	70.7154	2.27759
$965$	0.0350943	0.00112972
$966$	13.4902	0.434040
$967$	−11.9065	−0.382886	−0.191443	−	0.981504i	$-0.561317\pi$
$967$	−11.9065	−0.382886	−0.191443	+	0.981504i	$0.561317\pi$
$968$	−10.6001	−0.340701
$969$	−1.72739	−0.0554918
$970$	1.51967	0.0487937
$971$	13.1482	0.421944	0.210972	−	0.977492i	$-0.432337\pi$
$971$	13.1482	0.421944	0.210972	+	0.977492i	$0.432337\pi$
$972$	4.89474	0.156999
$973$	−17.7253	−0.568246
$974$	40.6770	1.30338
$975$	−26.3630	−0.844290
$976$	18.5913	0.595093
$977$	20.3065	0.649663	0.324831	−	0.945772i	$-0.394692\pi$
$977$	20.3065	0.649663	0.324831	+	0.945772i	$0.394692\pi$
$978$	−3.24091	−0.103633
$979$	−19.9980	−0.639139
$980$	12.0097	0.383637
$981$	17.9102	0.571830
$982$	5.52402	0.176279
$983$	32.3400	1.03149	0.515743	−	0.856743i	$-0.327516\pi$
$983$	32.3400	1.03149	0.515743	+	0.856743i	$0.327516\pi$
$984$	−3.40651	−0.108595
$985$	−12.4279	−0.395986
$986$	13.8109	0.439829
$987$	−14.3578	−0.457013
$988$	−64.9470	−2.06624
$989$	−20.8411	−0.662708
$990$	−4.74088	−0.150675
$991$	33.7887	1.07333	0.536667	−	0.843794i	$-0.319683\pi$
$991$	33.7887	1.07333	0.536667	+	0.843794i	$0.319683\pi$
$992$	−95.9245	−3.04561
$993$	15.6961	0.498100
$994$	−5.32258	−0.168822
$995$	−9.74598	−0.308968
$996$	75.3528	2.38764
$997$	−23.0393	−0.729663	−0.364832	−	0.931073i	$-0.618873\pi$
$997$	−23.0393	−0.729663	−0.364832	+	0.931073i	$0.618873\pi$
$998$	−91.1317	−2.88472
$999$	−11.8958	−0.376367

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8049.2.a.d.1.9	✓	129

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8049.2.a.d.1.9	✓	129	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:	\( N \)	\(=\)	\( 8049 = 3 \cdot 2683 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8049.a (trivial)

\(f(q)\)	\(=\)	\(q-2.62578 q^{2} +1.00000 q^{3} +4.89474 q^{4} -0.512843 q^{5} -2.62578 q^{6} +1.48852 q^{7} -7.60096 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.62578 q^{2} +1.00000 q^{3} +4.89474 q^{4} -0.512843 q^{5} -2.62578 q^{6} +1.48852 q^{7} -7.60096 q^{8} +1.00000 q^{9} +1.34662 q^{10} -3.52059 q^{11} +4.89474 q^{12} +5.56534 q^{13} -3.90853 q^{14} -0.512843 q^{15} +10.1690 q^{16} +0.724523 q^{17} -2.62578 q^{18} -2.38418 q^{19} -2.51023 q^{20} +1.48852 q^{21} +9.24432 q^{22} -3.45147 q^{23} -7.60096 q^{24} -4.73699 q^{25} -14.6134 q^{26} +1.00000 q^{27} +7.28591 q^{28} -7.25957 q^{29} +1.34662 q^{30} +8.34151 q^{31} -11.4997 q^{32} -3.52059 q^{33} -1.90244 q^{34} -0.763376 q^{35} +4.89474 q^{36} -11.8958 q^{37} +6.26033 q^{38} +5.56534 q^{39} +3.89810 q^{40} +0.448168 q^{41} -3.90853 q^{42} +6.03831 q^{43} -17.2324 q^{44} -0.512843 q^{45} +9.06283 q^{46} -9.64567 q^{47} +10.1690 q^{48} -4.78431 q^{49} +12.4383 q^{50} +0.724523 q^{51} +27.2409 q^{52} +11.5963 q^{53} -2.62578 q^{54} +1.80551 q^{55} -11.3142 q^{56} -2.38418 q^{57} +19.0621 q^{58} +1.65924 q^{59} -2.51023 q^{60} +1.82823 q^{61} -21.9030 q^{62} +1.48852 q^{63} +9.85763 q^{64} -2.85414 q^{65} +9.24432 q^{66} +11.7540 q^{67} +3.54635 q^{68} -3.45147 q^{69} +2.00446 q^{70} +1.36179 q^{71} -7.60096 q^{72} -3.49325 q^{73} +31.2358 q^{74} -4.73699 q^{75} -11.6699 q^{76} -5.24047 q^{77} -14.6134 q^{78} +0.789718 q^{79} -5.21510 q^{80} +1.00000 q^{81} -1.17679 q^{82} +15.3946 q^{83} +7.28591 q^{84} -0.371567 q^{85} -15.8553 q^{86} -7.25957 q^{87} +26.7599 q^{88} +5.68030 q^{89} +1.34662 q^{90} +8.28411 q^{91} -16.8941 q^{92} +8.34151 q^{93} +25.3275 q^{94} +1.22271 q^{95} -11.4997 q^{96} +1.12851 q^{97} +12.5626 q^{98} -3.52059 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 129 q + 8 q^{2} + 129 q^{3} + 158 q^{4} + 11 q^{5} + 8 q^{6} + 40 q^{7} + 18 q^{8} + 129 q^{9}+O(q^{10}) \) 129 * q + 8 * q^2 + 129 * q^3 + 158 * q^4 + 11 * q^5 + 8 * q^6 + 40 * q^7 + 18 * q^8 + 129 * q^9	\( 129 q + 8 q^{2} + 129 q^{3} + 158 q^{4} + 11 q^{5} + 8 q^{6} + 40 q^{7} + 18 q^{8} + 129 q^{9} + 20 q^{10} + 48 q^{11} + 158 q^{12} + 77 q^{13} + 13 q^{14} + 11 q^{15} + 212 q^{16} + 9 q^{17} + 8 q^{18} + 68 q^{19} + 19 q^{20} + 40 q^{21} + 45 q^{22} + 64 q^{23} + 18 q^{24} + 188 q^{25} + 19 q^{26} + 129 q^{27} + 69 q^{28} + 23 q^{29} + 20 q^{30} + 133 q^{31} + 24 q^{32} + 48 q^{33} + 63 q^{34} + 26 q^{35} + 158 q^{36} + 147 q^{37} + 9 q^{38} + 77 q^{39} + 58 q^{40} + 21 q^{41} + 13 q^{42} + 76 q^{43} + 110 q^{44} + 11 q^{45} + 48 q^{46} + 85 q^{47} + 212 q^{48} + 213 q^{49} + 17 q^{50} + 9 q^{51} + 139 q^{52} + 30 q^{53} + 8 q^{54} + 103 q^{55} + 19 q^{56} + 68 q^{57} + 94 q^{58} + 64 q^{59} + 19 q^{60} + 110 q^{61} - 10 q^{62} + 40 q^{63} + 288 q^{64} - 8 q^{65} + 45 q^{66} + 118 q^{67} - 15 q^{68} + 64 q^{69} + 75 q^{70} + 154 q^{71} + 18 q^{72} + 137 q^{73} + 28 q^{74} + 188 q^{75} + 156 q^{76} + 17 q^{77} + 19 q^{78} + 157 q^{79} + 2 q^{80} + 129 q^{81} + 72 q^{82} + 39 q^{83} + 69 q^{84} + 127 q^{85} + 54 q^{86} + 23 q^{87} + 97 q^{88} + 31 q^{89} + 20 q^{90} + 137 q^{91} + 82 q^{92} + 133 q^{93} + 40 q^{94} + 68 q^{95} + 24 q^{96} + 170 q^{97} - 21 q^{98} + 48 q^{99}+O(q^{100}) \) 129 * q + 8 * q^2 + 129 * q^3 + 158 * q^4 + 11 * q^5 + 8 * q^6 + 40 * q^7 + 18 * q^8 + 129 * q^9 + 20 * q^10 + 48 * q^11 + 158 * q^12 + 77 * q^13 + 13 * q^14 + 11 * q^15 + 212 * q^16 + 9 * q^17 + 8 * q^18 + 68 * q^19 + 19 * q^20 + 40 * q^21 + 45 * q^22 + 64 * q^23 + 18 * q^24 + 188 * q^25 + 19 * q^26 + 129 * q^27 + 69 * q^28 + 23 * q^29 + 20 * q^30 + 133 * q^31 + 24 * q^32 + 48 * q^33 + 63 * q^34 + 26 * q^35 + 158 * q^36 + 147 * q^37 + 9 * q^38 + 77 * q^39 + 58 * q^40 + 21 * q^41 + 13 * q^42 + 76 * q^43 + 110 * q^44 + 11 * q^45 + 48 * q^46 + 85 * q^47 + 212 * q^48 + 213 * q^49 + 17 * q^50 + 9 * q^51 + 139 * q^52 + 30 * q^53 + 8 * q^54 + 103 * q^55 + 19 * q^56 + 68 * q^57 + 94 * q^58 + 64 * q^59 + 19 * q^60 + 110 * q^61 - 10 * q^62 + 40 * q^63 + 288 * q^64 - 8 * q^65 + 45 * q^66 + 118 * q^67 - 15 * q^68 + 64 * q^69 + 75 * q^70 + 154 * q^71 + 18 * q^72 + 137 * q^73 + 28 * q^74 + 188 * q^75 + 156 * q^76 + 17 * q^77 + 19 * q^78 + 157 * q^79 + 2 * q^80 + 129 * q^81 + 72 * q^82 + 39 * q^83 + 69 * q^84 + 127 * q^85 + 54 * q^86 + 23 * q^87 + 97 * q^88 + 31 * q^89 + 20 * q^90 + 137 * q^91 + 82 * q^92 + 133 * q^93 + 40 * q^94 + 68 * q^95 + 24 * q^96 + 170 * q^97 - 21 * q^98 + 48 * q^99

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