LMFDB - Embedded newform 4009.2.a.c.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, 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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$71$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.08913 q^{2} +2.53881 q^{3} +2.36446 q^{4} +0.545075 q^{5} -5.30390 q^{6} +0.502260 q^{7} -0.761409 q^{8} +3.44555 q^{9} +O(q^{10})$	\(q-2.08913 q^{2} +2.53881 q^{3} +2.36446 q^{4} +0.545075 q^{5} -5.30390 q^{6} +0.502260 q^{7} -0.761409 q^{8} +3.44555 q^{9} -1.13873 q^{10} -0.886428 q^{11} +6.00292 q^{12} -1.49726 q^{13} -1.04929 q^{14} +1.38384 q^{15} -3.13824 q^{16} -7.27823 q^{17} -7.19820 q^{18} +1.00000 q^{19} +1.28881 q^{20} +1.27514 q^{21} +1.85186 q^{22} -3.29382 q^{23} -1.93307 q^{24} -4.70289 q^{25} +3.12796 q^{26} +1.13117 q^{27} +1.18758 q^{28} +1.34373 q^{29} -2.89103 q^{30} +10.9986 q^{31} +8.07901 q^{32} -2.25047 q^{33} +15.2052 q^{34} +0.273770 q^{35} +8.14688 q^{36} +7.35907 q^{37} -2.08913 q^{38} -3.80125 q^{39} -0.415026 q^{40} -12.1447 q^{41} -2.66394 q^{42} -6.10178 q^{43} -2.09593 q^{44} +1.87809 q^{45} +6.88122 q^{46} -7.55200 q^{47} -7.96740 q^{48} -6.74773 q^{49} +9.82495 q^{50} -18.4780 q^{51} -3.54021 q^{52} +2.77691 q^{53} -2.36316 q^{54} -0.483170 q^{55} -0.382426 q^{56} +2.53881 q^{57} -2.80722 q^{58} +10.6355 q^{59} +3.27204 q^{60} -13.0361 q^{61} -22.9775 q^{62} +1.73056 q^{63} -10.6016 q^{64} -0.816118 q^{65} +4.70153 q^{66} -4.04620 q^{67} -17.2091 q^{68} -8.36238 q^{69} -0.571940 q^{70} -4.91131 q^{71} -2.62348 q^{72} +1.17251 q^{73} -15.3740 q^{74} -11.9397 q^{75} +2.36446 q^{76} -0.445218 q^{77} +7.94130 q^{78} +6.89393 q^{79} -1.71058 q^{80} -7.46483 q^{81} +25.3718 q^{82} +4.36066 q^{83} +3.01503 q^{84} -3.96719 q^{85} +12.7474 q^{86} +3.41147 q^{87} +0.674935 q^{88} -6.11709 q^{89} -3.92356 q^{90} -0.752013 q^{91} -7.78811 q^{92} +27.9233 q^{93} +15.7771 q^{94} +0.545075 q^{95} +20.5111 q^{96} +2.25721 q^{97} +14.0969 q^{98} -3.05424 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) $ 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * 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 $

$2$	−2.08913	−1.47724	−0.738619	−	0.674123i	$-0.764522\pi$
$2$	−2.08913	−1.47724	−0.738619	+	0.674123i	$0.764522\pi$
$3$	2.53881	1.46578	0.732891	−	0.680346i	$-0.238170\pi$
$3$	2.53881	1.46578	0.732891	+	0.680346i	$0.238170\pi$
$4$	2.36446	1.18223
$5$	0.545075	0.243765	0.121883	−	0.992545i	$-0.461107\pi$
$5$	0.545075	0.243765	0.121883	+	0.992545i	$0.461107\pi$
$6$	−5.30390	−2.16531
$7$	0.502260	0.189837	0.0949183	−	0.995485i	$-0.469741\pi$
$7$	0.502260	0.189837	0.0949183	+	0.995485i	$0.469741\pi$
$8$	−0.761409	−0.269199
$9$	3.44555	1.14852
$10$	−1.13873	−0.360099
$11$	−0.886428	−0.267268	−0.133634	−	0.991031i	$-0.542665\pi$
$11$	−0.886428	−0.267268	−0.133634	+	0.991031i	$0.542665\pi$
$12$	6.00292	1.73289
$13$	−1.49726	−0.415264	−0.207632	−	0.978207i	$-0.566576\pi$
$13$	−1.49726	−0.415264	−0.207632	+	0.978207i	$0.566576\pi$
$14$	−1.04929	−0.280434
$15$	1.38384	0.357307
$16$	−3.13824	−0.784561
$17$	−7.27823	−1.76523	−0.882615	−	0.470096i	$-0.844219\pi$
$17$	−7.27823	−1.76523	−0.882615	+	0.470096i	$0.844219\pi$
$18$	−7.19820	−1.69663
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.28881	0.288187
$21$	1.27514	0.278259
$22$	1.85186	0.394819
$23$	−3.29382	−0.686809	−0.343404	−	0.939188i	$-0.611580\pi$
$23$	−3.29382	−0.686809	−0.343404	+	0.939188i	$0.611580\pi$
$24$	−1.93307	−0.394587
$25$	−4.70289	−0.940579
$26$	3.12796	0.613444
$27$	1.13117	0.217694
$28$	1.18758	0.224431
$29$	1.34373	0.249524	0.124762	−	0.992187i	$-0.460183\pi$
$29$	1.34373	0.249524	0.124762	+	0.992187i	$0.460183\pi$
$30$	−2.89103	−0.527827
$31$	10.9986	1.97540	0.987702	−	0.156348i	$-0.0499721\pi$
$31$	10.9986	1.97540	0.987702	+	0.156348i	$0.0499721\pi$
$32$	8.07901	1.42818
$33$	−2.25047	−0.391757
$34$	15.2052	2.60767
$35$	0.273770	0.0462755
$36$	8.14688	1.35781
$37$	7.35907	1.20982	0.604912	−	0.796293i	$-0.293208\pi$
$37$	7.35907	1.20982	0.604912	+	0.796293i	$0.293208\pi$
$38$	−2.08913	−0.338902
$39$	−3.80125	−0.608687
$40$	−0.415026	−0.0656213
$41$	−12.1447	−1.89668	−0.948341	−	0.317252i	$-0.897240\pi$
$41$	−12.1447	−1.89668	−0.948341	+	0.317252i	$0.897240\pi$
$42$	−2.66394	−0.411055
$43$	−6.10178	−0.930512	−0.465256	−	0.885176i	$-0.654038\pi$
$43$	−6.10178	−0.930512	−0.465256	+	0.885176i	$0.654038\pi$
$44$	−2.09593	−0.315973
$45$	1.87809	0.279968
$46$	6.88122	1.01458
$47$	−7.55200	−1.10157	−0.550786	−	0.834646i	$-0.685672\pi$
$47$	−7.55200	−1.10157	−0.550786	+	0.834646i	$0.685672\pi$
$48$	−7.96740	−1.14999
$49$	−6.74773	−0.963962
$50$	9.82495	1.38946
$51$	−18.4780	−2.58744
$52$	−3.54021	−0.490939
$53$	2.77691	0.381438	0.190719	−	0.981645i	$-0.438918\pi$
$53$	2.77691	0.381438	0.190719	+	0.981645i	$0.438918\pi$
$54$	−2.36316	−0.321586
$55$	−0.483170	−0.0651507
$56$	−0.382426	−0.0511038
$57$	2.53881	0.336273
$58$	−2.80722	−0.368607
$59$	10.6355	1.38463	0.692313	−	0.721598i	$-0.256592\pi$
$59$	10.6355	1.38463	0.692313	+	0.721598i	$0.256592\pi$
$60$	3.27204	0.422419
$61$	−13.0361	−1.66910	−0.834548	−	0.550935i	$-0.814271\pi$
$61$	−13.0361	−1.66910	−0.834548	+	0.550935i	$0.814271\pi$
$62$	−22.9775	−2.91814
$63$	1.73056	0.218031
$64$	−10.6016	−1.32520
$65$	−0.816118	−0.101227
$66$	4.70153	0.578718
$67$	−4.04620	−0.494322	−0.247161	−	0.968974i	$-0.579498\pi$
$67$	−4.04620	−0.494322	−0.247161	+	0.968974i	$0.579498\pi$
$68$	−17.2091	−2.08691
$69$	−8.36238	−1.00671
$70$	−0.571940	−0.0683600
$71$	−4.91131	−0.582866	−0.291433	−	0.956591i	$-0.594132\pi$
$71$	−4.91131	−0.582866	−0.291433	+	0.956591i	$0.594132\pi$
$72$	−2.62348	−0.309180
$73$	1.17251	0.137232	0.0686161	−	0.997643i	$-0.478142\pi$
$73$	1.17251	0.137232	0.0686161	+	0.997643i	$0.478142\pi$
$74$	−15.3740	−1.78720
$75$	−11.9397	−1.37868
$76$	2.36446	0.271222
$77$	−0.445218	−0.0507373
$78$	7.94130	0.899176
$79$	6.89393	0.775628	0.387814	−	0.921738i	$-0.373230\pi$
$79$	6.89393	0.775628	0.387814	+	0.921738i	$0.373230\pi$
$80$	−1.71058	−0.191249
$81$	−7.46483	−0.829425
$82$	25.3718	2.80185
$83$	4.36066	0.478645	0.239323	−	0.970940i	$-0.423075\pi$
$83$	4.36066	0.478645	0.239323	+	0.970940i	$0.423075\pi$
$84$	3.01503	0.328967
$85$	−3.96719	−0.430302
$86$	12.7474	1.37459
$87$	3.41147	0.365748
$88$	0.674935	0.0719483
$89$	−6.11709	−0.648410	−0.324205	−	0.945987i	$-0.605097\pi$
$89$	−6.11709	−0.648410	−0.324205	+	0.945987i	$0.605097\pi$
$90$	−3.92356	−0.413580
$91$	−0.752013	−0.0788324
$92$	−7.78811	−0.811967
$93$	27.9233	2.89551
$94$	15.7771	1.62728
$95$	0.545075	0.0559236
$96$	20.5111	2.09340
$97$	2.25721	0.229185	0.114592	−	0.993413i	$-0.463444\pi$
$97$	2.25721	0.229185	0.114592	+	0.993413i	$0.463444\pi$
$98$	14.0969	1.42400
$99$	−3.05424	−0.306962
$100$	−11.1198	−1.11198
$101$	−14.0586	−1.39888	−0.699441	−	0.714690i	$-0.746568\pi$
$101$	−14.0586	−1.39888	−0.699441	+	0.714690i	$0.746568\pi$
$102$	38.6030	3.82227
$103$	−2.33696	−0.230268	−0.115134	−	0.993350i	$-0.536730\pi$
$103$	−2.33696	−0.230268	−0.115134	+	0.993350i	$0.536730\pi$
$104$	1.14003	0.111789
$105$	0.695049	0.0678298
$106$	−5.80133	−0.563475
$107$	0.971779	0.0939454	0.0469727	−	0.998896i	$-0.485043\pi$
$107$	0.971779	0.0939454	0.0469727	+	0.998896i	$0.485043\pi$
$108$	2.67461	0.257364
$109$	−5.34939	−0.512378	−0.256189	−	0.966627i	$-0.582467\pi$
$109$	−5.34939	−0.512378	−0.256189	+	0.966627i	$0.582467\pi$
$110$	1.00941	0.0962430
$111$	18.6833	1.77334
$112$	−1.57621	−0.148938
$113$	9.49763	0.893462	0.446731	−	0.894668i	$-0.352588\pi$
$113$	9.49763	0.893462	0.446731	+	0.894668i	$0.352588\pi$
$114$	−5.30390	−0.496756
$115$	−1.79538	−0.167420
$116$	3.17720	0.294995
$117$	−5.15888	−0.476938
$118$	−22.2190	−2.04542
$119$	−3.65557	−0.335105
$120$	−1.05367	−0.0961865
$121$	−10.2142	−0.928568
$122$	27.2340	2.46565
$123$	−30.8331	−2.78012
$124$	26.0057	2.33538
$125$	−5.28881	−0.473045
$126$	−3.61537	−0.322083
$127$	−7.72323	−0.685326	−0.342663	−	0.939458i	$-0.611329\pi$
$127$	−7.72323	−0.685326	−0.342663	+	0.939458i	$0.611329\pi$
$128$	5.99013	0.529458
$129$	−15.4912	−1.36393
$130$	1.70498	0.149536
$131$	−14.6723	−1.28193	−0.640963	−	0.767572i	$-0.721465\pi$
$131$	−14.6723	−1.28193	−0.640963	+	0.767572i	$0.721465\pi$
$132$	−5.32116	−0.463147
$133$	0.502260	0.0435515
$134$	8.45304	0.730231
$135$	0.616573	0.0530662
$136$	5.54171	0.475198
$137$	−9.76110	−0.833947	−0.416974	−	0.908919i	$-0.636909\pi$
$137$	−9.76110	−0.833947	−0.416974	+	0.908919i	$0.636909\pi$
$138$	17.4701	1.48715
$139$	8.22954	0.698021	0.349010	−	0.937119i	$-0.386518\pi$
$139$	8.22954	0.698021	0.349010	+	0.937119i	$0.386518\pi$
$140$	0.647318	0.0547084
$141$	−19.1731	−1.61467
$142$	10.2604	0.861031
$143$	1.32721	0.110987
$144$	−10.8130	−0.901081
$145$	0.732434	0.0608253
$146$	−2.44953	−0.202724
$147$	−17.1312	−1.41296
$148$	17.4002	1.43029
$149$	7.50461	0.614802	0.307401	−	0.951580i	$-0.400541\pi$
$149$	7.50461	0.614802	0.307401	+	0.951580i	$0.400541\pi$
$150$	24.9437	2.03664
$151$	0.650776	0.0529594	0.0264797	−	0.999649i	$-0.491570\pi$
$151$	0.650776	0.0529594	0.0264797	+	0.999649i	$0.491570\pi$
$152$	−0.761409	−0.0617585
$153$	−25.0775	−2.02740
$154$	0.930118	0.0749510
$155$	5.99506	0.481535
$156$	−8.98791	−0.719609
$157$	2.24450	0.179131	0.0895653	−	0.995981i	$-0.471452\pi$
$157$	2.24450	0.179131	0.0895653	+	0.995981i	$0.471452\pi$
$158$	−14.4023	−1.14579
$159$	7.05005	0.559106
$160$	4.40367	0.348141
$161$	−1.65435	−0.130381
$162$	15.5950	1.22526
$163$	16.7436	1.31146	0.655730	−	0.754995i	$-0.272361\pi$
$163$	16.7436	1.31146	0.655730	+	0.754995i	$0.272361\pi$
$164$	−28.7157	−2.24232
$165$	−1.22668	−0.0954967
$166$	−9.10999	−0.707073
$167$	7.19056	0.556422	0.278211	−	0.960520i	$-0.410259\pi$
$167$	7.19056	0.556422	0.278211	+	0.960520i	$0.410259\pi$
$168$	−0.970906	−0.0749070
$169$	−10.7582	−0.827555
$170$	8.28797	0.635658
$171$	3.44555	0.263488
$172$	−14.4274	−1.10008
$173$	−7.21645	−0.548657	−0.274328	−	0.961636i	$-0.588455\pi$
$173$	−7.21645	−0.548657	−0.274328	+	0.961636i	$0.588455\pi$
$174$	−7.12701	−0.540297
$175$	−2.36208	−0.178556
$176$	2.78183	0.209688
$177$	27.0015	2.02956
$178$	12.7794	0.957856
$179$	1.92293	0.143727	0.0718633	−	0.997414i	$-0.477105\pi$
$179$	1.92293	0.143727	0.0718633	+	0.997414i	$0.477105\pi$
$180$	4.44066	0.330987
$181$	11.7533	0.873618	0.436809	−	0.899554i	$-0.356109\pi$
$181$	11.7533	0.873618	0.436809	+	0.899554i	$0.356109\pi$
$182$	1.57105	0.116454
$183$	−33.0961	−2.44653
$184$	2.50794	0.184888
$185$	4.01125	0.294913
$186$	−58.3354	−4.27736
$187$	6.45163	0.471790
$188$	−17.8564	−1.30231
$189$	0.568142	0.0413262
$190$	−1.13873	−0.0826124
$191$	3.58319	0.259270	0.129635	−	0.991562i	$-0.458619\pi$
$191$	3.58319	0.259270	0.129635	+	0.991562i	$0.458619\pi$
$192$	−26.9155	−1.94246
$193$	11.1120	0.799860	0.399930	−	0.916546i	$-0.369034\pi$
$193$	11.1120	0.799860	0.399930	+	0.916546i	$0.369034\pi$
$194$	−4.71560	−0.338561
$195$	−2.07197	−0.148377
$196$	−15.9548	−1.13963
$197$	0.415089	0.0295739	0.0147869	−	0.999891i	$-0.495293\pi$
$197$	0.415089	0.0295739	0.0147869	+	0.999891i	$0.495293\pi$
$198$	6.38069	0.453456
$199$	−7.11252	−0.504193	−0.252096	−	0.967702i	$-0.581120\pi$
$199$	−7.11252	−0.504193	−0.252096	+	0.967702i	$0.581120\pi$
$200$	3.58083	0.253203
$201$	−10.2725	−0.724568
$202$	29.3702	2.06648
$203$	0.674902	0.0473688
$204$	−43.6906	−3.05896
$205$	−6.61977	−0.462345
$206$	4.88222	0.340160
$207$	−11.3490	−0.788812
$208$	4.69876	0.325800
$209$	−0.886428	−0.0613155
$210$	−1.45205	−0.100201
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	6.56591	0.450948
$213$	−12.4689	−0.854354
$214$	−2.03017	−0.138780
$215$	−3.32593	−0.226826
$216$	−0.861284	−0.0586029
$217$	5.52415	0.375004
$218$	11.1756	0.756905
$219$	2.97678	0.201152
$220$	−1.14244	−0.0770232
$221$	10.8974	0.733038
$222$	−39.0318	−2.61964
$223$	15.6045	1.04495	0.522477	−	0.852653i	$-0.325008\pi$
$223$	15.6045	1.04495	0.522477	+	0.852653i	$0.325008\pi$
$224$	4.05777	0.271121
$225$	−16.2041	−1.08027
$226$	−19.8418	−1.31986
$227$	0.643938	0.0427396	0.0213698	−	0.999772i	$-0.493197\pi$
$227$	0.643938	0.0427396	0.0213698	+	0.999772i	$0.493197\pi$
$228$	6.00292	0.397553
$229$	−16.6315	−1.09904	−0.549519	−	0.835481i	$-0.685189\pi$
$229$	−16.6315	−1.09904	−0.549519	+	0.835481i	$0.685189\pi$
$230$	3.75078	0.247319
$231$	−1.13032	−0.0743698
$232$	−1.02313	−0.0671716
$233$	−16.6602	−1.09145	−0.545724	−	0.837965i	$-0.683745\pi$
$233$	−16.6602	−1.09145	−0.545724	+	0.837965i	$0.683745\pi$
$234$	10.7776	0.704551
$235$	−4.11641	−0.268525
$236$	25.1473	1.63695
$237$	17.5024	1.13690
$238$	7.63695	0.495030
$239$	16.5726	1.07199	0.535995	−	0.844221i	$-0.319937\pi$
$239$	16.5726	1.07199	0.535995	+	0.844221i	$0.319937\pi$
$240$	−4.34283	−0.280329
$241$	−9.26176	−0.596603	−0.298301	−	0.954472i	$-0.596420\pi$
$241$	−9.26176	−0.596603	−0.298301	+	0.954472i	$0.596420\pi$
$242$	21.3389	1.37172
$243$	−22.3453	−1.43345
$244$	−30.8233	−1.97326
$245$	−3.67802	−0.234980
$246$	64.4143	4.10690
$247$	−1.49726	−0.0952682
$248$	−8.37443	−0.531777
$249$	11.0709	0.701589
$250$	11.0490	0.698801
$251$	−9.06799	−0.572366	−0.286183	−	0.958175i	$-0.592387\pi$
$251$	−9.06799	−0.572366	−0.286183	+	0.958175i	$0.592387\pi$
$252$	4.09185	0.257763
$253$	2.91974	0.183562
$254$	16.1348	1.01239
$255$	−10.0719	−0.630729
$256$	8.68908	0.543067
$257$	3.44022	0.214595	0.107298	−	0.994227i	$-0.465780\pi$
$257$	3.44022	0.214595	0.107298	+	0.994227i	$0.465780\pi$
$258$	32.3632	2.01485
$259$	3.69617	0.229669
$260$	−1.92968	−0.119674
$261$	4.62989	0.286583
$262$	30.6524	1.89371
$263$	10.2535	0.632255	0.316128	−	0.948717i	$-0.397617\pi$
$263$	10.2535	0.632255	0.316128	+	0.948717i	$0.397617\pi$
$264$	1.71353	0.105461
$265$	1.51363	0.0929814
$266$	−1.04929	−0.0643359
$267$	−15.5301	−0.950428
$268$	−9.56709	−0.584403
$269$	−6.94192	−0.423257	−0.211628	−	0.977350i	$-0.567877\pi$
$269$	−6.94192	−0.423257	−0.211628	+	0.977350i	$0.567877\pi$
$270$	−1.28810	−0.0783913
$271$	20.7920	1.26302	0.631512	−	0.775366i	$-0.282435\pi$
$271$	20.7920	1.26302	0.631512	+	0.775366i	$0.282435\pi$
$272$	22.8409	1.38493
$273$	−1.90922	−0.115551
$274$	20.3922	1.23194
$275$	4.16878	0.251387
$276$	−19.7725	−1.19017
$277$	−2.59446	−0.155886	−0.0779431	−	0.996958i	$-0.524835\pi$
$277$	−2.59446	−0.155886	−0.0779431	+	0.996958i	$0.524835\pi$
$278$	−17.1926	−1.03114
$279$	37.8962	2.26879
$280$	−0.208451	−0.0124573
$281$	−30.7993	−1.83733	−0.918666	−	0.395036i	$-0.870732\pi$
$281$	−30.7993	−1.83733	−0.918666	+	0.395036i	$0.870732\pi$
$282$	40.0551	2.38525
$283$	0.0995144	0.00591552	0.00295776	−	0.999996i	$-0.499059\pi$
$283$	0.0995144	0.00591552	0.00295776	+	0.999996i	$0.499059\pi$
$284$	−11.6126	−0.689082
$285$	1.38384	0.0819718
$286$	−2.77272	−0.163954
$287$	−6.09980	−0.360060
$288$	27.8367	1.64029
$289$	35.9727	2.11604
$290$	−1.53015	−0.0898534
$291$	5.73062	0.335935
$292$	2.77236	0.162240
$293$	−11.8654	−0.693184	−0.346592	−	0.938016i	$-0.612661\pi$
$293$	−11.8654	−0.693184	−0.346592	+	0.938016i	$0.612661\pi$
$294$	35.7893	2.08728
$295$	5.79715	0.337523
$296$	−5.60326	−0.325683
$297$	−1.00270	−0.0581826
$298$	−15.6781	−0.908208
$299$	4.93170	0.285207
$300$	−28.2311	−1.62992
$301$	−3.06468	−0.176645
$302$	−1.35956	−0.0782336
$303$	−35.6921	−2.05046
$304$	−3.13824	−0.179991
$305$	−7.10564	−0.406867
$306$	52.3902	2.99495
$307$	27.8816	1.59129	0.795643	−	0.605766i	$-0.207133\pi$
$307$	27.8816	1.59129	0.795643	+	0.605766i	$0.207133\pi$
$308$	−1.05270	−0.0599832
$309$	−5.93310	−0.337522
$310$	−12.5245	−0.711341
$311$	16.9851	0.963139	0.481570	−	0.876408i	$-0.340067\pi$
$311$	16.9851	0.963139	0.481570	+	0.876408i	$0.340067\pi$
$312$	2.89431	0.163858
$313$	30.5752	1.72821	0.864107	−	0.503309i	$-0.167884\pi$
$313$	30.5752	1.72821	0.864107	+	0.503309i	$0.167884\pi$
$314$	−4.68905	−0.264618
$315$	0.943288	0.0531482
$316$	16.3004	0.916971
$317$	−5.64859	−0.317257	−0.158628	−	0.987338i	$-0.550707\pi$
$317$	−5.64859	−0.317257	−0.158628	+	0.987338i	$0.550707\pi$
$318$	−14.7285	−0.825932
$319$	−1.19112	−0.0666899
$320$	−5.77868	−0.323038
$321$	2.46716	0.137704
$322$	3.45616	0.192604
$323$	−7.27823	−0.404972
$324$	−17.6503	−0.980573
$325$	7.04144	0.390589
$326$	−34.9796	−1.93734
$327$	−13.5811	−0.751035
$328$	9.24708	0.510585
$329$	−3.79307	−0.209119
$330$	2.56269	0.141071
$331$	−18.9675	−1.04255	−0.521274	−	0.853389i	$-0.674543\pi$
$331$	−18.9675	−1.04255	−0.521274	+	0.853389i	$0.674543\pi$
$332$	10.3106	0.565869
$333$	25.3560	1.38950
$334$	−15.0220	−0.821967
$335$	−2.20548	−0.120498
$336$	−4.00171	−0.218311
$337$	−11.1788	−0.608945	−0.304473	−	0.952521i	$-0.598480\pi$
$337$	−11.1788	−0.608945	−0.304473	+	0.952521i	$0.598480\pi$
$338$	22.4753	1.22250
$339$	24.1127	1.30962
$340$	−9.38026	−0.508716
$341$	−9.74946	−0.527963
$342$	−7.19820	−0.389234
$343$	−6.90494	−0.372832
$344$	4.64595	0.250493
$345$	−4.55813	−0.245401
$346$	15.0761	0.810496
$347$	−20.2523	−1.08720	−0.543599	−	0.839345i	$-0.682939\pi$
$347$	−20.2523	−1.08720	−0.543599	+	0.839345i	$0.682939\pi$
$348$	8.06630	0.432399
$349$	31.4396	1.68292	0.841461	−	0.540317i	$-0.181696\pi$
$349$	31.4396	1.68292	0.841461	+	0.540317i	$0.181696\pi$
$350$	4.93468	0.263770
$351$	−1.69365	−0.0904005
$352$	−7.16147	−0.381707
$353$	−35.7646	−1.90356	−0.951779	−	0.306785i	$-0.900747\pi$
$353$	−35.7646	−1.90356	−0.951779	+	0.306785i	$0.900747\pi$
$354$	−56.4097	−2.99814
$355$	−2.67704	−0.142082
$356$	−14.4636	−0.766571
$357$	−9.28079	−0.491191
$358$	−4.01725	−0.212318
$359$	6.68138	0.352630	0.176315	−	0.984334i	$-0.443582\pi$
$359$	6.68138	0.352630	0.176315	+	0.984334i	$0.443582\pi$
$360$	−1.42999	−0.0753672
$361$	1.00000	0.0526316
$362$	−24.5542	−1.29054
$363$	−25.9320	−1.36108
$364$	−1.77811	−0.0931981
$365$	0.639107	0.0334524
$366$	69.1420	3.61411
$367$	37.6971	1.96777	0.983887	−	0.178792i	$-0.0572189\pi$
$367$	37.6971	1.96777	0.983887	+	0.178792i	$0.0572189\pi$
$368$	10.3368	0.538843
$369$	−41.8452	−2.17837
$370$	−8.38001	−0.435656
$371$	1.39473	0.0724110
$372$	66.0236	3.42316
$373$	−7.11293	−0.368293	−0.184147	−	0.982899i	$-0.558952\pi$
$373$	−7.11293	−0.368293	−0.184147	+	0.982899i	$0.558952\pi$
$374$	−13.4783	−0.696946
$375$	−13.4273	−0.693381
$376$	5.75017	0.296542
$377$	−2.01191	−0.103619
$378$	−1.18692	−0.0610487
$379$	−16.5868	−0.852007	−0.426003	−	0.904722i	$-0.640079\pi$
$379$	−16.5868	−0.852007	−0.426003	+	0.904722i	$0.640079\pi$
$380$	1.28881	0.0661146
$381$	−19.6078	−1.00454
$382$	−7.48574	−0.383004
$383$	−7.11316	−0.363466	−0.181733	−	0.983348i	$-0.558171\pi$
$383$	−7.11316	−0.363466	−0.181733	+	0.983348i	$0.558171\pi$
$384$	15.2078	0.776070
$385$	−0.242677	−0.0123680
$386$	−23.2144	−1.18158
$387$	−21.0240	−1.06871
$388$	5.33709	0.270950
$389$	23.9956	1.21663	0.608314	−	0.793697i	$-0.291846\pi$
$389$	23.9956	1.21663	0.608314	+	0.793697i	$0.291846\pi$
$390$	4.32861	0.219188
$391$	23.9732	1.21238
$392$	5.13779	0.259497
$393$	−37.2502	−1.87902
$394$	−0.867174	−0.0436876
$395$	3.75771	0.189071
$396$	−7.22162	−0.362900
$397$	−23.4761	−1.17823	−0.589115	−	0.808049i	$-0.700524\pi$
$397$	−23.4761	−1.17823	−0.589115	+	0.808049i	$0.700524\pi$
$398$	14.8590	0.744813
$399$	1.27514	0.0638370
$400$	14.7588	0.737941
$401$	22.4420	1.12070	0.560350	−	0.828256i	$-0.310666\pi$
$401$	22.4420	1.12070	0.560350	+	0.828256i	$0.310666\pi$
$402$	21.4606	1.07036
$403$	−16.4677	−0.820315
$404$	−33.2410	−1.65380
$405$	−4.06889	−0.202185
$406$	−1.40996	−0.0699750
$407$	−6.52329	−0.323347
$408$	14.0694	0.696537
$409$	8.20578	0.405750	0.202875	−	0.979205i	$-0.434972\pi$
$409$	8.20578	0.405750	0.202875	+	0.979205i	$0.434972\pi$
$410$	13.8296	0.682994
$411$	−24.7816	−1.22238
$412$	−5.52566	−0.272230
$413$	5.34179	0.262852
$414$	23.7096	1.16526
$415$	2.37689	0.116677
$416$	−12.0964	−0.593073
$417$	20.8932	1.02315
$418$	1.85186	0.0905776
$419$	−7.02038	−0.342968	−0.171484	−	0.985187i	$-0.554856\pi$
$419$	−7.02038	−0.342968	−0.171484	+	0.985187i	$0.554856\pi$
$420$	1.64342	0.0801906
$421$	6.72465	0.327739	0.163870	−	0.986482i	$-0.447602\pi$
$421$	6.72465	0.327739	0.163870	+	0.986482i	$0.447602\pi$
$422$	−2.08913	−0.101697
$423$	−26.0208	−1.26518
$424$	−2.11437	−0.102683
$425$	34.2288	1.66034
$426$	26.0491	1.26208
$427$	−6.54749	−0.316855
$428$	2.29774	0.111065
$429$	3.36954	0.162683
$430$	6.94830	0.335077
$431$	24.4434	1.17740	0.588699	−	0.808352i	$-0.299640\pi$
$431$	24.4434	1.17740	0.588699	+	0.808352i	$0.299640\pi$
$432$	−3.54989	−0.170794
$433$	−15.8218	−0.760345	−0.380172	−	0.924916i	$-0.624135\pi$
$433$	−15.8218	−0.760345	−0.380172	+	0.924916i	$0.624135\pi$
$434$	−11.5407	−0.553970
$435$	1.85951	0.0891566
$436$	−12.6484	−0.605750
$437$	−3.29382	−0.157565
$438$	−6.21889	−0.297150
$439$	−10.8332	−0.517041	−0.258520	−	0.966006i	$-0.583235\pi$
$439$	−10.8332	−0.517041	−0.258520	+	0.966006i	$0.583235\pi$
$440$	0.367890	0.0175385
$441$	−23.2497	−1.10713
$442$	−22.7661	−1.08287
$443$	−13.0347	−0.619299	−0.309650	−	0.950851i	$-0.600212\pi$
$443$	−13.0347	−0.619299	−0.309650	+	0.950851i	$0.600212\pi$
$444$	44.1759	2.09649
$445$	−3.33428	−0.158060
$446$	−32.5998	−1.54365
$447$	19.0528	0.901165
$448$	−5.32477	−0.251572
$449$	−1.11968	−0.0528409	−0.0264205	−	0.999651i	$-0.508411\pi$
$449$	−1.11968	−0.0528409	−0.0264205	+	0.999651i	$0.508411\pi$
$450$	33.8524	1.59582
$451$	10.7654	0.506923
$452$	22.4568	1.05628
$453$	1.65220	0.0776270
$454$	−1.34527	−0.0631366
$455$	−0.409904	−0.0192166
$456$	−1.93307	−0.0905244
$457$	8.39294	0.392605	0.196303	−	0.980543i	$-0.437107\pi$
$457$	8.39294	0.392605	0.196303	+	0.980543i	$0.437107\pi$
$458$	34.7453	1.62354
$459$	−8.23292	−0.384280
$460$	−4.24511	−0.197929
$461$	−14.7600	−0.687442	−0.343721	−	0.939072i	$-0.611688\pi$
$461$	−14.7600	−0.687442	−0.343721	+	0.939072i	$0.611688\pi$
$462$	2.36139	0.109862
$463$	−10.4446	−0.485403	−0.242701	−	0.970101i	$-0.578033\pi$
$463$	−10.4446	−0.485403	−0.242701	+	0.970101i	$0.578033\pi$
$464$	−4.21695	−0.195767
$465$	15.2203	0.705825
$466$	34.8053	1.61233
$467$	19.9157	0.921588	0.460794	−	0.887507i	$-0.347565\pi$
$467$	19.9157	0.921588	0.460794	+	0.887507i	$0.347565\pi$
$468$	−12.1980	−0.563851
$469$	−2.03225	−0.0938404
$470$	8.59972	0.396675
$471$	5.69835	0.262566
$472$	−8.09798	−0.372740
$473$	5.40879	0.248696
$474$	−36.5647	−1.67947
$475$	−4.70289	−0.215784
$476$	−8.64345	−0.396172
$477$	9.56800	0.438089
$478$	−34.6222	−1.58358
$479$	−39.4253	−1.80139	−0.900694	−	0.434453i	$-0.856942\pi$
$479$	−39.4253	−1.80139	−0.900694	+	0.434453i	$0.856942\pi$
$480$	11.1801	0.510299
$481$	−11.0184	−0.502397
$482$	19.3490	0.881324
$483$	−4.20009	−0.191111
$484$	−24.1512	−1.09778
$485$	1.23035	0.0558673
$486$	46.6822	2.11755
$487$	26.9105	1.21943	0.609716	−	0.792620i	$-0.291284\pi$
$487$	26.9105	1.21943	0.609716	+	0.792620i	$0.291284\pi$
$488$	9.92578	0.449319
$489$	42.5088	1.92232
$490$	7.68387	0.347122
$491$	−28.1432	−1.27009	−0.635043	−	0.772477i	$-0.719018\pi$
$491$	−28.1432	−1.27009	−0.635043	+	0.772477i	$0.719018\pi$
$492$	−72.9036	−3.28675
$493$	−9.77997	−0.440468
$494$	3.12796	0.140734
$495$	−1.66479	−0.0748267
$496$	−34.5162	−1.54982
$497$	−2.46676	−0.110649
$498$	−23.1285	−1.03641
$499$	5.60279	0.250815	0.125408	−	0.992105i	$-0.459976\pi$
$499$	5.60279	0.250815	0.125408	+	0.992105i	$0.459976\pi$
$500$	−12.5052	−0.559249
$501$	18.2554	0.815593
$502$	18.9442	0.845521
$503$	−4.67738	−0.208554	−0.104277	−	0.994548i	$-0.533253\pi$
$503$	−4.67738	−0.208554	−0.104277	+	0.994548i	$0.533253\pi$
$504$	−1.31767	−0.0586936
$505$	−7.66299	−0.340999
$506$	−6.09971	−0.271165
$507$	−27.3131	−1.21302
$508$	−18.2613	−0.810213
$509$	−32.3575	−1.43422	−0.717111	−	0.696959i	$-0.754536\pi$
$509$	−32.3575	−1.43422	−0.717111	+	0.696959i	$0.754536\pi$
$510$	21.0416	0.931736
$511$	0.588906	0.0260517
$512$	−30.1329	−1.33170
$513$	1.13117	0.0499424
$514$	−7.18707	−0.317008
$515$	−1.27382	−0.0561313
$516$	−36.6285	−1.61248
$517$	6.69431	0.294415
$518$	−7.72177	−0.339275
$519$	−18.3212	−0.804211
$520$	0.621400	0.0272502
$521$	−12.6607	−0.554676	−0.277338	−	0.960772i	$-0.589452\pi$
$521$	−12.6607	−0.554676	−0.277338	+	0.960772i	$0.589452\pi$
$522$	−9.67243	−0.423351
$523$	−6.30642	−0.275761	−0.137880	−	0.990449i	$-0.544029\pi$
$523$	−6.30642	−0.275761	−0.137880	+	0.990449i	$0.544029\pi$
$524$	−34.6921	−1.51553
$525$	−5.99686	−0.261724
$526$	−21.4208	−0.933992
$527$	−80.0503	−3.48704
$528$	7.06253	0.307357
$529$	−12.1508	−0.528294
$530$	−3.16216	−0.137356
$531$	36.6452	1.59027
$532$	1.18758	0.0514879
$533$	18.1837	0.787625
$534$	32.4444	1.40401
$535$	0.529693	0.0229006
$536$	3.08081	0.133071
$537$	4.88195	0.210672
$538$	14.5026	0.625251
$539$	5.98138	0.257636
$540$	1.45786	0.0627365
$541$	−2.25820	−0.0970876	−0.0485438	−	0.998821i	$-0.515458\pi$
$541$	−2.25820	−0.0970876	−0.0485438	+	0.998821i	$0.515458\pi$
$542$	−43.4372	−1.86579
$543$	29.8394	1.28053
$544$	−58.8009	−2.52107
$545$	−2.91582	−0.124900
$546$	3.98860	0.170696
$547$	5.45943	0.233428	0.116714	−	0.993166i	$-0.462764\pi$
$547$	5.45943	0.233428	0.116714	+	0.993166i	$0.462764\pi$
$548$	−23.0798	−0.985918
$549$	−44.9164	−1.91699
$550$	−8.70912	−0.371358
$551$	1.34373	0.0572448
$552$	6.36719	0.271006
$553$	3.46255	0.147242
$554$	5.42017	0.230281
$555$	10.1838	0.432278
$556$	19.4584	0.825222
$557$	−29.7576	−1.26087	−0.630436	−	0.776241i	$-0.717124\pi$
$557$	−29.7576	−1.26087	−0.630436	+	0.776241i	$0.717124\pi$
$558$	−79.1701	−3.35154
$559$	9.13593	0.386409
$560$	−0.859156	−0.0363060
$561$	16.3795	0.691542
$562$	64.3437	2.71417
$563$	0.0442812	0.00186623	0.000933115	−	1.00000i	$-0.499703\pi$
$563$	0.0442812	0.00186623	0.000933115		1.00000i	$0.499703\pi$
$564$	−45.3341	−1.90891
$565$	5.17693	0.217795
$566$	−0.207899	−0.00873863
$567$	−3.74929	−0.157455
$568$	3.73952	0.156907
$569$	−10.1196	−0.424234	−0.212117	−	0.977244i	$-0.568036\pi$
$569$	−10.1196	−0.424234	−0.212117	+	0.977244i	$0.568036\pi$
$570$	−2.89103	−0.121092
$571$	−36.7464	−1.53779	−0.768895	−	0.639375i	$-0.779193\pi$
$571$	−36.7464	−1.53779	−0.768895	+	0.639375i	$0.779193\pi$
$572$	3.13814	0.131212
$573$	9.09703	0.380034
$574$	12.7433	0.531894
$575$	15.4905	0.645998
$576$	−36.5284	−1.52202
$577$	30.3548	1.26369	0.631843	−	0.775097i	$-0.282299\pi$
$577$	30.3548	1.26369	0.631843	+	0.775097i	$0.282299\pi$
$578$	−75.1516	−3.12589
$579$	28.2113	1.17242
$580$	1.73181	0.0719096
$581$	2.19019	0.0908643
$582$	−11.9720	−0.496256
$583$	−2.46154	−0.101946
$584$	−0.892761	−0.0369427
$585$	−2.81198	−0.116261
$586$	24.7884	1.02400
$587$	−25.9241	−1.07000	−0.535002	−	0.844851i	$-0.679689\pi$
$587$	−25.9241	−1.07000	−0.535002	+	0.844851i	$0.679689\pi$
$588$	−40.5061	−1.67044
$589$	10.9986	0.453189
$590$	−12.1110	−0.498602
$591$	1.05383	0.0433488
$592$	−23.0945	−0.949179
$593$	25.0174	1.02734	0.513671	−	0.857987i	$-0.328285\pi$
$593$	25.0174	1.02734	0.513671	+	0.857987i	$0.328285\pi$
$594$	2.09477	0.0859496
$595$	−1.99256	−0.0816870
$596$	17.7444	0.726838
$597$	−18.0573	−0.739037
$598$	−10.3030	−0.421319
$599$	18.1978	0.743544	0.371772	−	0.928324i	$-0.378750\pi$
$599$	18.1978	0.743544	0.371772	+	0.928324i	$0.378750\pi$
$600$	9.09103	0.371140
$601$	−0.434167	−0.0177100	−0.00885501	−	0.999961i	$-0.502819\pi$
$601$	−0.434167	−0.0177100	−0.00885501	+	0.999961i	$0.502819\pi$
$602$	6.40251	0.260947
$603$	−13.9414	−0.567737
$604$	1.53874	0.0626103
$605$	−5.56753	−0.226352
$606$	74.5654	3.02901
$607$	31.7084	1.28700	0.643501	−	0.765445i	$-0.277481\pi$
$607$	31.7084	1.28700	0.643501	+	0.765445i	$0.277481\pi$
$608$	8.07901	0.327647
$609$	1.71345	0.0694324
$610$	14.8446	0.601040
$611$	11.3073	0.457444
$612$	−59.2949	−2.39685
$613$	1.83790	0.0742322	0.0371161	−	0.999311i	$-0.488183\pi$
$613$	1.83790	0.0742322	0.0371161	+	0.999311i	$0.488183\pi$
$614$	−58.2482	−2.35071
$615$	−16.8063	−0.677697
$616$	0.338993	0.0136584
$617$	−9.43200	−0.379718	−0.189859	−	0.981811i	$-0.560803\pi$
$617$	−9.43200	−0.379718	−0.189859	+	0.981811i	$0.560803\pi$
$618$	12.3950	0.498601
$619$	5.84887	0.235086	0.117543	−	0.993068i	$-0.462498\pi$
$619$	5.84887	0.235086	0.117543	+	0.993068i	$0.462498\pi$
$620$	14.1751	0.569285
$621$	−3.72587	−0.149514
$622$	−35.4842	−1.42279
$623$	−3.07237	−0.123092
$624$	11.9292	0.477552
$625$	20.6317	0.825267
$626$	−63.8756	−2.55298
$627$	−2.25047	−0.0898752
$628$	5.30703	0.211774
$629$	−53.5610	−2.13562
$630$	−1.97065	−0.0785126
$631$	−31.8250	−1.26693	−0.633467	−	0.773770i	$-0.718369\pi$
$631$	−31.8250	−1.26693	−0.633467	+	0.773770i	$0.718369\pi$
$632$	−5.24910	−0.208798
$633$	2.53881	0.100909
$634$	11.8006	0.468663
$635$	−4.20974	−0.167058
$636$	16.6696	0.660992
$637$	10.1031	0.400299
$638$	2.48840	0.0985168
$639$	−16.9222	−0.669431
$640$	3.26508	0.129063
$641$	10.8364	0.428011	0.214006	−	0.976832i	$-0.431349\pi$
$641$	10.8364	0.428011	0.214006	+	0.976832i	$0.431349\pi$
$642$	−5.15422	−0.203421
$643$	10.0693	0.397093	0.198546	−	0.980091i	$-0.436378\pi$
$643$	10.0693	0.397093	0.198546	+	0.980091i	$0.436378\pi$
$644$	−3.91166	−0.154141
$645$	−8.44390	−0.332478
$646$	15.2052	0.598240
$647$	−14.8226	−0.582736	−0.291368	−	0.956611i	$-0.594110\pi$
$647$	−14.8226	−0.582736	−0.291368	+	0.956611i	$0.594110\pi$
$648$	5.68379	0.223280
$649$	−9.42762	−0.370066
$650$	−14.7105	−0.576993
$651$	14.0248	0.549674
$652$	39.5896	1.55045
$653$	−31.7293	−1.24167	−0.620833	−	0.783943i	$-0.713205\pi$
$653$	−31.7293	−1.24167	−0.620833	+	0.783943i	$0.713205\pi$
$654$	28.3726	1.10946
$655$	−7.99752	−0.312489
$656$	38.1130	1.48806
$657$	4.03995	0.157613
$658$	7.92422	0.308918
$659$	−46.7350	−1.82054	−0.910269	−	0.414018i	$-0.864125\pi$
$659$	−46.7350	−1.82054	−0.910269	+	0.414018i	$0.864125\pi$
$660$	−2.90043	−0.112899
$661$	31.0533	1.20783	0.603916	−	0.797048i	$-0.293606\pi$
$661$	31.0533	1.20783	0.603916	+	0.797048i	$0.293606\pi$
$662$	39.6256	1.54009
$663$	27.6664	1.07447
$664$	−3.32025	−0.128851
$665$	0.273770	0.0106163
$666$	−52.9721	−2.05263
$667$	−4.42600	−0.171375
$668$	17.0018	0.657819
$669$	39.6168	1.53168
$670$	4.60754	0.178005
$671$	11.5555	0.446096
$672$	10.3019	0.397404
$673$	21.4003	0.824920	0.412460	−	0.910976i	$-0.364670\pi$
$673$	21.4003	0.824920	0.412460	+	0.910976i	$0.364670\pi$
$674$	23.3539	0.899557
$675$	−5.31977	−0.204758
$676$	−25.4374	−0.978362
$677$	−34.0647	−1.30921	−0.654606	−	0.755971i	$-0.727165\pi$
$677$	−34.0647	−1.30921	−0.654606	+	0.755971i	$0.727165\pi$
$678$	−50.3745	−1.93462
$679$	1.13371	0.0435077
$680$	3.02065	0.115837
$681$	1.63483	0.0626470
$682$	20.3679	0.779927
$683$	31.9822	1.22376	0.611882	−	0.790949i	$-0.290413\pi$
$683$	31.9822	1.22376	0.611882	+	0.790949i	$0.290413\pi$
$684$	8.14688	0.311504
$685$	−5.32054	−0.203287
$686$	14.4253	0.550761
$687$	−42.2241	−1.61095
$688$	19.1489	0.730043
$689$	−4.15775	−0.158398
$690$	9.52252	0.362516
$691$	3.43283	0.130591	0.0652955	−	0.997866i	$-0.479201\pi$
$691$	3.43283	0.130591	0.0652955	+	0.997866i	$0.479201\pi$
$692$	−17.0630	−0.648639
$693$	−1.53402	−0.0582726
$694$	42.3096	1.60605
$695$	4.48572	0.170153
$696$	−2.59753	−0.0984590
$697$	88.3919	3.34808
$698$	−65.6814	−2.48608
$699$	−42.2971	−1.59982
$700$	−5.58504	−0.211095
$701$	−8.09290	−0.305665	−0.152832	−	0.988252i	$-0.548839\pi$
$701$	−8.09290	−0.305665	−0.152832	+	0.988252i	$0.548839\pi$
$702$	3.53826	0.133543
$703$	7.35907	0.277552
$704$	9.39758	0.354185
$705$	−10.4508	−0.393599
$706$	74.7169	2.81201
$707$	−7.06107	−0.265559
$708$	63.8441	2.39941
$709$	22.1547	0.832037	0.416018	−	0.909356i	$-0.363425\pi$
$709$	22.1547	0.832037	0.416018	+	0.909356i	$0.363425\pi$
$710$	5.59268	0.209889
$711$	23.7534	0.890822
$712$	4.65761	0.174551
$713$	−36.2274	−1.35672
$714$	19.3888	0.725606
$715$	0.723430	0.0270548
$716$	4.54670	0.169918
$717$	42.0745	1.57130
$718$	−13.9583	−0.520918
$719$	−30.7062	−1.14515	−0.572574	−	0.819853i	$-0.694055\pi$
$719$	−30.7062	−1.14515	−0.572574	+	0.819853i	$0.694055\pi$
$720$	−5.89389	−0.219652
$721$	−1.17376	−0.0437132
$722$	−2.08913	−0.0777494
$723$	−23.5139	−0.874490
$724$	27.7903	1.03282
$725$	−6.31941	−0.234697
$726$	54.1753	2.01064
$727$	−47.6134	−1.76588	−0.882942	−	0.469482i	$-0.844441\pi$
$727$	−47.6134	−1.76588	−0.882942	+	0.469482i	$0.844441\pi$
$728$	0.572590	0.0212216
$729$	−34.3359	−1.27170
$730$	−1.33518	−0.0494172
$731$	44.4102	1.64257
$732$	−78.2544	−2.89237
$733$	−41.3484	−1.52724	−0.763619	−	0.645667i	$-0.776580\pi$
$733$	−41.3484	−1.52724	−0.763619	+	0.645667i	$0.776580\pi$
$734$	−78.7542	−2.90687
$735$	−9.33780	−0.344430
$736$	−26.6108	−0.980888
$737$	3.58667	0.132117
$738$	87.4200	3.21797
$739$	11.4151	0.419910	0.209955	−	0.977711i	$-0.432668\pi$
$739$	11.4151	0.419910	0.209955	+	0.977711i	$0.432668\pi$
$740$	9.48444	0.348655
$741$	−3.80125	−0.139642
$742$	−2.91378	−0.106968
$743$	43.8005	1.60689	0.803443	−	0.595382i	$-0.202999\pi$
$743$	43.8005	1.60689	0.803443	+	0.595382i	$0.202999\pi$
$744$	−21.2611	−0.779469
$745$	4.09058	0.149867
$746$	14.8598	0.544057
$747$	15.0249	0.549732
$748$	15.2546	0.557765
$749$	0.488086	0.0178343
$750$	28.0513	1.02429
$751$	44.0727	1.60823	0.804117	−	0.594471i	$-0.202638\pi$
$751$	44.0727	1.60823	0.804117	+	0.594471i	$0.202638\pi$
$752$	23.7000	0.864251
$753$	−23.0219	−0.838965
$754$	4.20314	0.153069
$755$	0.354722	0.0129097
$756$	1.34335	0.0488572
$757$	−25.9109	−0.941749	−0.470875	−	0.882200i	$-0.656062\pi$
$757$	−25.9109	−0.941749	−0.470875	+	0.882200i	$0.656062\pi$
$758$	34.6520	1.25862
$759$	7.41265	0.269062
$760$	−0.415026	−0.0150546
$761$	12.9657	0.470005	0.235002	−	0.971995i	$-0.424490\pi$
$761$	12.9657	0.470005	0.235002	+	0.971995i	$0.424490\pi$
$762$	40.9632	1.48394
$763$	−2.68679	−0.0972682
$764$	8.47231	0.306517
$765$	−13.6691	−0.494209
$766$	14.8603	0.536925
$767$	−15.9241	−0.574986
$768$	22.0599	0.796018
$769$	−16.6009	−0.598645	−0.299322	−	0.954152i	$-0.596761\pi$
$769$	−16.6009	−0.598645	−0.299322	+	0.954152i	$0.596761\pi$
$770$	0.506984	0.0182704
$771$	8.73406	0.314550
$772$	26.2739	0.945620
$773$	21.5445	0.774901	0.387450	−	0.921891i	$-0.373356\pi$
$773$	21.5445	0.774901	0.387450	+	0.921891i	$0.373356\pi$
$774$	43.9218	1.57874
$775$	−51.7252	−1.85802
$776$	−1.71866	−0.0616963
$777$	9.38386	0.336644
$778$	−50.1300	−1.79725
$779$	−12.1447	−0.435129
$780$	−4.89909	−0.175416
$781$	4.35353	0.155782
$782$	−50.0831	−1.79097
$783$	1.51999	0.0543199
$784$	21.1760	0.756287
$785$	1.22342	0.0436658
$786$	77.8205	2.77576
$787$	−4.22340	−0.150548	−0.0752740	−	0.997163i	$-0.523983\pi$
$787$	−4.22340	−0.150548	−0.0752740	+	0.997163i	$0.523983\pi$
$788$	0.981462	0.0349631
$789$	26.0316	0.926749
$790$	−7.85035	−0.279303
$791$	4.77028	0.169612
$792$	2.32552	0.0826339
$793$	19.5183	0.693116
$794$	49.0446	1.74053
$795$	3.84281	0.136290
$796$	−16.8173	−0.596073
$797$	5.28169	0.187087	0.0935436	−	0.995615i	$-0.470181\pi$
$797$	5.28169	0.187087	0.0935436	+	0.995615i	$0.470181\pi$
$798$	−2.66394	−0.0943024
$799$	54.9652	1.94453
$800$	−37.9947	−1.34332
$801$	−21.0767	−0.744710
$802$	−46.8843	−1.65554
$803$	−1.03935	−0.0366778
$804$	−24.2890	−0.856607
$805$	−0.901748	−0.0317824
$806$	34.4032	1.21180
$807$	−17.6242	−0.620402
$808$	10.7043	0.376578
$809$	2.96467	0.104232	0.0521161	−	0.998641i	$-0.483403\pi$
$809$	2.96467	0.104232	0.0521161	+	0.998641i	$0.483403\pi$
$810$	8.50045	0.298675
$811$	30.9700	1.08750	0.543751	−	0.839246i	$-0.317003\pi$
$811$	30.9700	1.08750	0.543751	+	0.839246i	$0.317003\pi$
$812$	1.59578	0.0560009
$813$	52.7869	1.85132
$814$	13.6280	0.477661
$815$	9.12653	0.319688
$816$	57.9886	2.03001
$817$	−6.10178	−0.213474
$818$	−17.1429	−0.599388
$819$	−2.59110	−0.0905403
$820$	−15.6522	−0.546599
$821$	13.4230	0.468468	0.234234	−	0.972180i	$-0.424742\pi$
$821$	13.4230	0.468468	0.234234	+	0.972180i	$0.424742\pi$
$822$	51.7719	1.80575
$823$	12.3681	0.431124	0.215562	−	0.976490i	$-0.430842\pi$
$823$	12.3681	0.431124	0.215562	+	0.976490i	$0.430842\pi$
$824$	1.77939	0.0619878
$825$	10.5837	0.368478
$826$	−11.1597	−0.388296
$827$	23.0041	0.799931	0.399966	−	0.916530i	$-0.369022\pi$
$827$	23.0041	0.799931	0.399966	+	0.916530i	$0.369022\pi$
$828$	−26.8343	−0.932558
$829$	20.9002	0.725893	0.362946	−	0.931810i	$-0.381771\pi$
$829$	20.9002	0.725893	0.362946	+	0.931810i	$0.381771\pi$
$830$	−4.96563	−0.172360
$831$	−6.58685	−0.228495
$832$	15.8734	0.550310
$833$	49.1116	1.70162
$834$	−43.6487	−1.51143
$835$	3.91940	0.135636
$836$	−2.09593	−0.0724891
$837$	12.4413	0.430033
$838$	14.6665	0.506645
$839$	−2.64740	−0.0913985	−0.0456993	−	0.998955i	$-0.514552\pi$
$839$	−2.64740	−0.0913985	−0.0456993	+	0.998955i	$0.514552\pi$
$840$	−0.529217	−0.0182597
$841$	−27.1944	−0.937738
$842$	−14.0487	−0.484149
$843$	−78.1935	−2.69313
$844$	2.36446	0.0813882
$845$	−5.86404	−0.201729
$846$	54.3609	1.86896
$847$	−5.13021	−0.176276
$848$	−8.71463	−0.299262
$849$	0.252648	0.00867086
$850$	−71.5083	−2.45271
$851$	−24.2394	−0.830917
$852$	−29.4822	−1.01004
$853$	−20.9211	−0.716324	−0.358162	−	0.933659i	$-0.616596\pi$
$853$	−20.9211	−0.716324	−0.358162	+	0.933659i	$0.616596\pi$
$854$	13.6786	0.468071
$855$	1.87809	0.0642292
$856$	−0.739922	−0.0252900
$857$	36.8618	1.25918	0.629588	−	0.776929i	$-0.283224\pi$
$857$	36.8618	1.25918	0.629588	+	0.776929i	$0.283224\pi$
$858$	−7.03940	−0.240321
$859$	25.4407	0.868024	0.434012	−	0.900907i	$-0.357098\pi$
$859$	25.4407	0.868024	0.434012	+	0.900907i	$0.357098\pi$
$860$	−7.86403	−0.268161
$861$	−15.4862	−0.527769
$862$	−51.0655	−1.73930
$863$	−26.2705	−0.894258	−0.447129	−	0.894470i	$-0.647553\pi$
$863$	−26.2705	−0.894258	−0.447129	+	0.894470i	$0.647553\pi$
$864$	9.13874	0.310906
$865$	−3.93351	−0.133743
$866$	33.0537	1.12321
$867$	91.3278	3.10165
$868$	13.0617	0.443341
$869$	−6.11097	−0.207301
$870$	−3.88476	−0.131706
$871$	6.05820	0.205274
$872$	4.07307	0.137932
$873$	7.77733	0.263223
$874$	6.88122	0.232761
$875$	−2.65636	−0.0898013
$876$	7.03849	0.237809
$877$	16.0157	0.540812	0.270406	−	0.962746i	$-0.412842\pi$
$877$	16.0157	0.540812	0.270406	+	0.962746i	$0.412842\pi$
$878$	22.6320	0.763792
$879$	−30.1240	−1.01606
$880$	1.51631	0.0511147
$881$	−19.8350	−0.668257	−0.334128	−	0.942528i	$-0.608442\pi$
$881$	−19.8350	−0.668257	−0.334128	+	0.942528i	$0.608442\pi$
$882$	48.5716	1.63549
$883$	−3.38233	−0.113825	−0.0569123	−	0.998379i	$-0.518126\pi$
$883$	−3.38233	−0.113825	−0.0569123	+	0.998379i	$0.518126\pi$
$884$	25.7665	0.866620
$885$	14.7179	0.494736
$886$	27.2313	0.914852
$887$	−8.29322	−0.278459	−0.139230	−	0.990260i	$-0.544463\pi$
$887$	−8.29322	−0.278459	−0.139230	+	0.990260i	$0.544463\pi$
$888$	−14.2256	−0.477380
$889$	−3.87907	−0.130100
$890$	6.96573	0.233492
$891$	6.61704	0.221679
$892$	36.8962	1.23538
$893$	−7.55200	−0.252718
$894$	−39.8037	−1.33124
$895$	1.04814	0.0350355
$896$	3.00861	0.100510
$897$	12.5206	0.418052
$898$	2.33915	0.0780586
$899$	14.7791	0.492911
$900$	−38.3139	−1.27713
$901$	−20.2110	−0.673327
$902$	−22.4903	−0.748846
$903$	−7.78064	−0.258923
$904$	−7.23159	−0.240519
$905$	6.40645	0.212958
$906$	−3.45165	−0.114673
$907$	−40.6705	−1.35044	−0.675221	−	0.737615i	$-0.735952\pi$
$907$	−40.6705	−1.35044	−0.675221	+	0.737615i	$0.735952\pi$
$908$	1.52257	0.0505281
$909$	−48.4396	−1.60664
$910$	0.856342	0.0283875
$911$	30.5709	1.01286	0.506430	−	0.862281i	$-0.330965\pi$
$911$	30.5709	1.01286	0.506430	+	0.862281i	$0.330965\pi$
$912$	−7.96740	−0.263827
$913$	−3.86542	−0.127927
$914$	−17.5339	−0.579971
$915$	−18.0399	−0.596379
$916$	−39.3245	−1.29932
$917$	−7.36932	−0.243356
$918$	17.1996	0.567673
$919$	−53.9539	−1.77978	−0.889888	−	0.456179i	$-0.849218\pi$
$919$	−53.9539	−1.77978	−0.889888	+	0.456179i	$0.849218\pi$
$920$	1.36702	0.0450693
$921$	70.7860	2.33248
$922$	30.8356	1.01552
$923$	7.35350	0.242043
$924$	−2.67261	−0.0879223
$925$	−34.6089	−1.13793
$926$	21.8202	0.717055
$927$	−8.05213	−0.264467
$928$	10.8560	0.356366
$929$	−55.6887	−1.82709	−0.913543	−	0.406742i	$-0.866665\pi$
$929$	−55.6887	−1.82709	−0.913543	+	0.406742i	$0.866665\pi$
$930$	−31.7972	−1.04267
$931$	−6.74773	−0.221148
$932$	−39.3925	−1.29034
$933$	43.1221	1.41175
$934$	−41.6064	−1.36140
$935$	3.51663	0.115006
$936$	3.92802	0.128391
$937$	60.9497	1.99114	0.995569	−	0.0940297i	$-0.0299749\pi$
$937$	60.9497	1.99114	0.995569	+	0.0940297i	$0.0299749\pi$
$938$	4.24562	0.138625
$939$	77.6246	2.53318
$940$	−9.73310	−0.317459
$941$	21.0671	0.686768	0.343384	−	0.939195i	$-0.388427\pi$
$941$	21.0671	0.686768	0.343384	+	0.939195i	$0.388427\pi$
$942$	−11.9046	−0.387873
$943$	40.0024	1.30266
$944$	−33.3768	−1.08632
$945$	0.309680	0.0100739
$946$	−11.2997	−0.367384
$947$	38.9283	1.26500	0.632500	−	0.774560i	$-0.282029\pi$
$947$	38.9283	1.26500	0.632500	+	0.774560i	$0.282029\pi$
$948$	41.3837	1.34408
$949$	−1.75555	−0.0569876
$950$	9.82495	0.318764
$951$	−14.3407	−0.465029
$952$	2.78338	0.0902100
$953$	10.8572	0.351701	0.175850	−	0.984417i	$-0.443733\pi$
$953$	10.8572	0.351701	0.175850	+	0.984417i	$0.443733\pi$
$954$	−19.9888	−0.647161
$955$	1.95311	0.0632010
$956$	39.1852	1.26734
$957$	−3.02403	−0.0977529
$958$	82.3646	2.66108
$959$	−4.90261	−0.158314
$960$	−14.6710	−0.473504
$961$	89.9689	2.90222
$962$	23.0189	0.742159
$963$	3.34832	0.107898
$964$	−21.8991	−0.705322
$965$	6.05689	0.194978
$966$	8.77453	0.282316
$967$	−14.1288	−0.454350	−0.227175	−	0.973854i	$-0.572949\pi$
$967$	−14.1288	−0.454350	−0.227175	+	0.973854i	$0.572949\pi$
$968$	7.77722	0.249969
$969$	−18.4780	−0.593600
$970$	−2.57036	−0.0825293
$971$	41.3006	1.32540	0.662700	−	0.748885i	$-0.269411\pi$
$971$	41.3006	1.32540	0.662700	+	0.748885i	$0.269411\pi$
$972$	−52.8346	−1.69467
$973$	4.13337	0.132510
$974$	−56.2195	−1.80139
$975$	17.8769	0.572518
$976$	40.9103	1.30951
$977$	32.7177	1.04673	0.523367	−	0.852108i	$-0.324676\pi$
$977$	32.7177	1.04673	0.523367	+	0.852108i	$0.324676\pi$
$978$	−88.8065	−2.83972
$979$	5.42236	0.173299
$980$	−8.69655	−0.277801
$981$	−18.4316	−0.588475
$982$	58.7948	1.87622
$983$	−6.59906	−0.210477	−0.105239	−	0.994447i	$-0.533561\pi$
$983$	−6.59906	−0.210477	−0.105239	+	0.994447i	$0.533561\pi$
$984$	23.4766	0.748406
$985$	0.226255	0.00720907
$986$	20.4316	0.650676
$987$	−9.62988	−0.306523
$988$	−3.54021	−0.112629
$989$	20.0982	0.639084
$990$	3.47796	0.110537
$991$	40.6842	1.29238	0.646188	−	0.763178i	$-0.276362\pi$
$991$	40.6842	1.29238	0.646188	+	0.763178i	$0.276362\pi$
$992$	88.8577	2.82124
$993$	−48.1549	−1.52815
$994$	5.15338	0.163455
$995$	−3.87686	−0.122905
$996$	26.1767	0.829441
$997$	15.7544	0.498946	0.249473	−	0.968382i	$-0.419743\pi$
$997$	15.7544	0.498946	0.249473	+	0.968382i	$0.419743\pi$
$998$	−11.7049	−0.370514
$999$	8.32436	0.263371

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.13	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.13	✓	71	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.08913 q^{2} +2.53881 q^{3} +2.36446 q^{4} +0.545075 q^{5} -5.30390 q^{6} +0.502260 q^{7} -0.761409 q^{8} +3.44555 q^{9} +O(q^{10})\)	\(q-2.08913 q^{2} +2.53881 q^{3} +2.36446 q^{4} +0.545075 q^{5} -5.30390 q^{6} +0.502260 q^{7} -0.761409 q^{8} +3.44555 q^{9} -1.13873 q^{10} -0.886428 q^{11} +6.00292 q^{12} -1.49726 q^{13} -1.04929 q^{14} +1.38384 q^{15} -3.13824 q^{16} -7.27823 q^{17} -7.19820 q^{18} +1.00000 q^{19} +1.28881 q^{20} +1.27514 q^{21} +1.85186 q^{22} -3.29382 q^{23} -1.93307 q^{24} -4.70289 q^{25} +3.12796 q^{26} +1.13117 q^{27} +1.18758 q^{28} +1.34373 q^{29} -2.89103 q^{30} +10.9986 q^{31} +8.07901 q^{32} -2.25047 q^{33} +15.2052 q^{34} +0.273770 q^{35} +8.14688 q^{36} +7.35907 q^{37} -2.08913 q^{38} -3.80125 q^{39} -0.415026 q^{40} -12.1447 q^{41} -2.66394 q^{42} -6.10178 q^{43} -2.09593 q^{44} +1.87809 q^{45} +6.88122 q^{46} -7.55200 q^{47} -7.96740 q^{48} -6.74773 q^{49} +9.82495 q^{50} -18.4780 q^{51} -3.54021 q^{52} +2.77691 q^{53} -2.36316 q^{54} -0.483170 q^{55} -0.382426 q^{56} +2.53881 q^{57} -2.80722 q^{58} +10.6355 q^{59} +3.27204 q^{60} -13.0361 q^{61} -22.9775 q^{62} +1.73056 q^{63} -10.6016 q^{64} -0.816118 q^{65} +4.70153 q^{66} -4.04620 q^{67} -17.2091 q^{68} -8.36238 q^{69} -0.571940 q^{70} -4.91131 q^{71} -2.62348 q^{72} +1.17251 q^{73} -15.3740 q^{74} -11.9397 q^{75} +2.36446 q^{76} -0.445218 q^{77} +7.94130 q^{78} +6.89393 q^{79} -1.71058 q^{80} -7.46483 q^{81} +25.3718 q^{82} +4.36066 q^{83} +3.01503 q^{84} -3.96719 q^{85} +12.7474 q^{86} +3.41147 q^{87} +0.674935 q^{88} -6.11709 q^{89} -3.92356 q^{90} -0.752013 q^{91} -7.78811 q^{92} +27.9233 q^{93} +15.7771 q^{94} +0.545075 q^{95} +20.5111 q^{96} +2.25721 q^{97} +14.0969 q^{98} -3.05424 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

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