LMFDB - Embedded newform 39.4.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [39,4,Mod(1,39)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(39, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("39.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 39 = 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	39.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:	$2.30107449022$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3144.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 8 $ x^3 - x^2 - 16*x - 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.20905$ of defining polynomial
Character	$\chi$	$=$	39.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.20905 q^{2} +3.00000 q^{3} +9.71610 q^{4} -11.4322 q^{5} +12.6271 q^{6} -11.2543 q^{7} +7.22315 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+4.20905 q^{2} +3.00000 q^{3} +9.71610 q^{4} -11.4322 q^{5} +12.6271 q^{6} -11.2543 q^{7} +7.22315 q^{8} +9.00000 q^{9} -48.1187 q^{10} +25.8785 q^{11} +29.1483 q^{12} +13.0000 q^{13} -47.3699 q^{14} -34.2966 q^{15} -47.3262 q^{16} -20.3276 q^{17} +37.8814 q^{18} +154.712 q^{19} -111.076 q^{20} -33.7629 q^{21} +108.924 q^{22} -180.418 q^{23} +21.6695 q^{24} +5.69520 q^{25} +54.7176 q^{26} +27.0000 q^{27} -109.348 q^{28} -20.4522 q^{29} -144.356 q^{30} +266.424 q^{31} -256.984 q^{32} +77.6355 q^{33} -85.5599 q^{34} +128.661 q^{35} +87.4449 q^{36} +115.984 q^{37} +651.190 q^{38} +39.0000 q^{39} -82.5765 q^{40} +391.184 q^{41} -142.110 q^{42} +151.407 q^{43} +251.438 q^{44} -102.890 q^{45} -759.390 q^{46} -467.365 q^{47} -141.979 q^{48} -216.341 q^{49} +23.9714 q^{50} -60.9828 q^{51} +126.309 q^{52} +79.9842 q^{53} +113.644 q^{54} -295.848 q^{55} -81.2915 q^{56} +464.136 q^{57} -86.0843 q^{58} -873.710 q^{59} -333.229 q^{60} -187.068 q^{61} +1121.39 q^{62} -101.289 q^{63} -703.047 q^{64} -148.619 q^{65} +326.772 q^{66} -609.204 q^{67} -197.505 q^{68} -541.255 q^{69} +541.542 q^{70} +248.038 q^{71} +65.0084 q^{72} +852.765 q^{73} +488.181 q^{74} +17.0856 q^{75} +1503.20 q^{76} -291.244 q^{77} +164.153 q^{78} -331.221 q^{79} +541.043 q^{80} +81.0000 q^{81} +1646.51 q^{82} -435.432 q^{83} -328.044 q^{84} +232.389 q^{85} +637.281 q^{86} -61.3566 q^{87} +186.924 q^{88} +259.233 q^{89} -433.068 q^{90} -146.306 q^{91} -1752.96 q^{92} +799.273 q^{93} -1967.16 q^{94} -1768.70 q^{95} -770.951 q^{96} +1225.17 q^{97} -910.589 q^{98} +232.907 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 9 q^{3} + 10 q^{4} + 4 q^{5} + 6 q^{6} + 30 q^{7} - 6 q^{8} + 27 q^{9}+O(q^{10}) $ 3 * q + 2 * q^2 + 9 * q^3 + 10 * q^4 + 4 * q^5 + 6 * q^6 + 30 * q^7 - 6 * q^8 + 27 * q^9	\( 3 q + 2 q^{2} + 9 q^{3} + 10 q^{4} + 4 q^{5} + 6 q^{6} + 30 q^{7} - 6 q^{8} + 27 q^{9} - 4 q^{10} - 16 q^{11} + 30 q^{12} + 39 q^{13} - 176 q^{14} + 12 q^{15} - 110 q^{16} - 146 q^{17} + 18 q^{18} + 94 q^{19} - 244 q^{20} + 90 q^{21} - 56 q^{22} - 48 q^{23} - 18 q^{24} + 145 q^{25} + 26 q^{26} + 81 q^{27} + 80 q^{28} - 2 q^{29} - 12 q^{30} + 302 q^{31} + 154 q^{32} - 48 q^{33} + 164 q^{34} + 80 q^{35} + 90 q^{36} + 374 q^{37} + 312 q^{38} + 117 q^{39} - 516 q^{40} + 480 q^{41} - 528 q^{42} - 260 q^{43} + 712 q^{44} + 36 q^{45} - 1104 q^{46} - 24 q^{47} - 330 q^{48} + 447 q^{49} + 814 q^{50} - 438 q^{51} + 130 q^{52} - 678 q^{53} + 54 q^{54} - 1552 q^{55} + 96 q^{56} + 282 q^{57} - 628 q^{58} - 1788 q^{59} - 732 q^{60} + 230 q^{61} + 1952 q^{62} + 270 q^{63} - 750 q^{64} + 52 q^{65} - 168 q^{66} + 74 q^{67} - 460 q^{68} - 144 q^{69} + 1216 q^{70} - 948 q^{71} - 54 q^{72} - 222 q^{73} + 1724 q^{74} + 435 q^{75} + 2392 q^{76} + 112 q^{77} + 78 q^{78} - 24 q^{79} + 1100 q^{80} + 243 q^{81} + 564 q^{82} - 796 q^{83} + 240 q^{84} - 248 q^{85} + 1800 q^{86} - 6 q^{87} + 1608 q^{88} + 1436 q^{89} - 36 q^{90} + 390 q^{91} - 1296 q^{92} + 906 q^{93} - 1920 q^{94} - 4032 q^{95} + 462 q^{96} + 3242 q^{97} - 5070 q^{98} - 144 q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 + 9 * q^3 + 10 * q^4 + 4 * q^5 + 6 * q^6 + 30 * q^7 - 6 * q^8 + 27 * q^9 - 4 * q^10 - 16 * q^11 + 30 * q^12 + 39 * q^13 - 176 * q^14 + 12 * q^15 - 110 * q^16 - 146 * q^17 + 18 * q^18 + 94 * q^19 - 244 * q^20 + 90 * q^21 - 56 * q^22 - 48 * q^23 - 18 * q^24 + 145 * q^25 + 26 * q^26 + 81 * q^27 + 80 * q^28 - 2 * q^29 - 12 * q^30 + 302 * q^31 + 154 * q^32 - 48 * q^33 + 164 * q^34 + 80 * q^35 + 90 * q^36 + 374 * q^37 + 312 * q^38 + 117 * q^39 - 516 * q^40 + 480 * q^41 - 528 * q^42 - 260 * q^43 + 712 * q^44 + 36 * q^45 - 1104 * q^46 - 24 * q^47 - 330 * q^48 + 447 * q^49 + 814 * q^50 - 438 * q^51 + 130 * q^52 - 678 * q^53 + 54 * q^54 - 1552 * q^55 + 96 * q^56 + 282 * q^57 - 628 * q^58 - 1788 * q^59 - 732 * q^60 + 230 * q^61 + 1952 * q^62 + 270 * q^63 - 750 * q^64 + 52 * q^65 - 168 * q^66 + 74 * q^67 - 460 * q^68 - 144 * q^69 + 1216 * q^70 - 948 * q^71 - 54 * q^72 - 222 * q^73 + 1724 * q^74 + 435 * q^75 + 2392 * q^76 + 112 * q^77 + 78 * q^78 - 24 * q^79 + 1100 * q^80 + 243 * q^81 + 564 * q^82 - 796 * q^83 + 240 * q^84 - 248 * q^85 + 1800 * q^86 - 6 * q^87 + 1608 * q^88 + 1436 * q^89 - 36 * q^90 + 390 * q^91 - 1296 * q^92 + 906 * q^93 - 1920 * q^94 - 4032 * q^95 + 462 * q^96 + 3242 * q^97 - 5070 * q^98 - 144 * 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 $

$2$	4.20905	1.48812	0.744062	−	0.668111i	$-0.232897\pi$
$2$	4.20905	1.48812	0.744062	+	0.668111i	$0.232897\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	9.71610	1.21451
$5$	−11.4322	−1.02253	−0.511264	−	0.859424i	$-0.670822\pi$
$5$	−11.4322	−1.02253	−0.511264	+	0.859424i	$0.670822\pi$
$6$	12.6271	0.859169
$7$	−11.2543	−0.607675	−0.303838	−	0.952724i	$-0.598268\pi$
$7$	−11.2543	−0.607675	−0.303838	+	0.952724i	$0.598268\pi$
$8$	7.22315	0.319221
$9$	9.00000	0.333333
$10$	−48.1187	−1.52165
$11$	25.8785	0.709333	0.354666	−	0.934993i	$-0.384594\pi$
$11$	25.8785	0.709333	0.354666	+	0.934993i	$0.384594\pi$
$12$	29.1483	0.701199
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	−47.3699	−0.904296
$15$	−34.2966	−0.590356
$16$	−47.3262	−0.739472
$17$	−20.3276	−0.290010	−0.145005	−	0.989431i	$-0.546320\pi$
$17$	−20.3276	−0.290010	−0.145005	+	0.989431i	$0.546320\pi$
$18$	37.8814	0.496041
$19$	154.712	1.86807	0.934035	−	0.357181i	$-0.116262\pi$
$19$	154.712	1.86807	0.934035	+	0.357181i	$0.116262\pi$
$20$	−111.076	−1.24187
$21$	−33.7629	−0.350841
$22$	108.924	1.05558
$23$	−180.418	−1.63565	−0.817823	−	0.575471i	$-0.804819\pi$
$23$	−180.418	−1.63565	−0.817823	+	0.575471i	$0.804819\pi$
$24$	21.6695	0.184302
$25$	5.69520	0.0455616
$26$	54.7176	0.412731
$27$	27.0000	0.192450
$28$	−109.348	−0.738029
$29$	−20.4522	−0.130961	−0.0654806	−	0.997854i	$-0.520858\pi$
$29$	−20.4522	−0.130961	−0.0654806	+	0.997854i	$0.520858\pi$
$30$	−144.356	−0.878523
$31$	266.424	1.54359	0.771794	−	0.635873i	$-0.219360\pi$
$31$	266.424	1.54359	0.771794	+	0.635873i	$0.219360\pi$
$32$	−256.984	−1.41965
$33$	77.6355	0.409534
$34$	−85.5599	−0.431571
$35$	128.661	0.621364
$36$	87.4449	0.404837
$37$	115.984	0.515340	0.257670	−	0.966233i	$-0.417045\pi$
$37$	115.984	0.515340	0.257670	+	0.966233i	$0.417045\pi$
$38$	651.190	2.77992
$39$	39.0000	0.160128
$40$	−82.5765	−0.326412
$41$	391.184	1.49006	0.745032	−	0.667029i	$-0.232434\pi$
$41$	391.184	1.49006	0.745032	+	0.667029i	$0.232434\pi$
$42$	−142.110	−0.522095
$43$	151.407	0.536963	0.268482	−	0.963285i	$-0.413478\pi$
$43$	151.407	0.536963	0.268482	+	0.963285i	$0.413478\pi$
$44$	251.438	0.861494
$45$	−102.890	−0.340842
$46$	−759.390	−2.43404
$47$	−467.365	−1.45047	−0.725236	−	0.688500i	$-0.758269\pi$
$47$	−467.365	−1.45047	−0.725236	+	0.688500i	$0.758269\pi$
$48$	−141.979	−0.426934
$49$	−216.341	−0.630731
$50$	23.9714	0.0678012
$51$	−60.9828	−0.167437
$52$	126.309	0.336845
$53$	79.9842	0.207296	0.103648	−	0.994614i	$-0.466949\pi$
$53$	79.9842	0.207296	0.103648	+	0.994614i	$0.466949\pi$
$54$	113.644	0.286390
$55$	−295.848	−0.725312
$56$	−81.2915	−0.193983
$57$	464.136	1.07853
$58$	−86.0843	−0.194887
$59$	−873.710	−1.92792	−0.963960	−	0.266045i	$-0.914283\pi$
$59$	−873.710	−1.92792	−0.963960	+	0.266045i	$0.914283\pi$
$60$	−333.229	−0.716995
$61$	−187.068	−0.392649	−0.196325	−	0.980539i	$-0.562901\pi$
$61$	−187.068	−0.392649	−0.196325	+	0.980539i	$0.562901\pi$
$62$	1121.39	2.29705
$63$	−101.289	−0.202558
$64$	−703.047	−1.37314
$65$	−148.619	−0.283598
$66$	326.772	0.609437
$67$	−609.204	−1.11084	−0.555418	−	0.831571i	$-0.687442\pi$
$67$	−609.204	−1.11084	−0.555418	+	0.831571i	$0.687442\pi$
$68$	−197.505	−0.352221
$69$	−541.255	−0.944340
$70$	541.542	0.924667
$71$	248.038	0.414601	0.207301	−	0.978277i	$-0.433532\pi$
$71$	248.038	0.414601	0.207301	+	0.978277i	$0.433532\pi$
$72$	65.0084	0.106407
$73$	852.765	1.36724	0.683621	−	0.729838i	$-0.260404\pi$
$73$	852.765	1.36724	0.683621	+	0.729838i	$0.260404\pi$
$74$	488.181	0.766890
$75$	17.0856	0.0263050
$76$	1503.20	2.26880
$77$	−291.244	−0.431044
$78$	164.153	0.238291
$79$	−331.221	−0.471712	−0.235856	−	0.971788i	$-0.575789\pi$
$79$	−331.221	−0.471712	−0.235856	+	0.971788i	$0.575789\pi$
$80$	541.043	0.756130
$81$	81.0000	0.111111
$82$	1646.51	2.21740
$83$	−435.432	−0.575842	−0.287921	−	0.957654i	$-0.592964\pi$
$83$	−435.432	−0.575842	−0.287921	+	0.957654i	$0.592964\pi$
$84$	−328.044	−0.426101
$85$	232.389	0.296543
$86$	637.281	0.799067
$87$	−61.3566	−0.0756105
$88$	186.924	0.226434
$89$	259.233	0.308749	0.154375	−	0.988012i	$-0.450664\pi$
$89$	259.233	0.308749	0.154375	+	0.988012i	$0.450664\pi$
$90$	−433.068	−0.507216
$91$	−146.306	−0.168539
$92$	−1752.96	−1.98651
$93$	799.273	0.891191
$94$	−1967.16	−2.15848
$95$	−1768.70	−1.91015
$96$	−770.951	−0.819634
$97$	1225.17	1.28245	0.641223	−	0.767355i	$-0.278428\pi$
$97$	1225.17	1.28245	0.641223	+	0.767355i	$0.278428\pi$
$98$	−910.589	−0.938606
$99$	232.907	0.236444
$100$	55.3351	0.0553351
$101$	645.416	0.635855	0.317927	−	0.948115i	$-0.397013\pi$
$101$	645.416	0.635855	0.317927	+	0.948115i	$0.397013\pi$
$102$	−256.680	−0.249167
$103$	−511.137	−0.488969	−0.244484	−	0.969653i	$-0.578619\pi$
$103$	−511.137	−0.488969	−0.244484	+	0.969653i	$0.578619\pi$
$104$	93.9010	0.0885360
$105$	385.984	0.358745
$106$	336.657	0.308482
$107$	608.195	0.549499	0.274750	−	0.961516i	$-0.411405\pi$
$107$	608.195	0.549499	0.274750	+	0.961516i	$0.411405\pi$
$108$	262.335	0.233733
$109$	−1300.04	−1.14239	−0.571197	−	0.820813i	$-0.693521\pi$
$109$	−1300.04	−1.14239	−0.571197	+	0.820813i	$0.693521\pi$
$110$	−1245.24	−1.07935
$111$	347.951	0.297532
$112$	532.623	0.449359
$113$	42.1953	0.0351274	0.0175637	−	0.999846i	$-0.494409\pi$
$113$	42.1953	0.0351274	0.0175637	+	0.999846i	$0.494409\pi$
$114$	1953.57	1.60499
$115$	2062.58	1.67249
$116$	−198.716	−0.159054
$117$	117.000	0.0924500
$118$	−3677.49	−2.86899
$119$	228.773	0.176232
$120$	−247.729	−0.188454
$121$	−661.303	−0.496847
$122$	−787.378	−0.584311
$123$	1173.55	0.860289
$124$	2588.61	1.87471
$125$	1363.92	0.975939
$126$	−426.329	−0.301432
$127$	−311.018	−0.217310	−0.108655	−	0.994080i	$-0.534654\pi$
$127$	−311.018	−0.217310	−0.108655	+	0.994080i	$0.534654\pi$
$128$	−903.291	−0.623753
$129$	454.222	0.310016
$130$	−625.543	−0.422029
$131$	2000.98	1.33456	0.667278	−	0.744809i	$-0.267459\pi$
$131$	2000.98	1.33456	0.667278	+	0.744809i	$0.267459\pi$
$132$	754.314	0.497384
$133$	−1741.17	−1.13518
$134$	−2564.17	−1.65306
$135$	−308.669	−0.196785
$136$	−146.829	−0.0925773
$137$	1038.53	0.647644	0.323822	−	0.946118i	$-0.395032\pi$
$137$	1038.53	0.647644	0.323822	+	0.946118i	$0.395032\pi$
$138$	−2278.17	−1.40529
$139$	−2858.46	−1.74426	−0.872128	−	0.489277i	$-0.837261\pi$
$139$	−2858.46	−1.74426	−0.872128	+	0.489277i	$0.837261\pi$
$140$	1250.09	0.754655
$141$	−1402.09	−0.837430
$142$	1044.00	0.616978
$143$	336.421	0.196734
$144$	−425.936	−0.246491
$145$	233.814	0.133911
$146$	3589.33	2.03462
$147$	−649.022	−0.364153
$148$	1126.91	0.625887
$149$	743.479	0.408780	0.204390	−	0.978890i	$-0.434479\pi$
$149$	743.479	0.408780	0.204390	+	0.978890i	$0.434479\pi$
$150$	71.9141	0.0391451
$151$	2277.24	1.22728	0.613640	−	0.789586i	$-0.289705\pi$
$151$	2277.24	1.22728	0.613640	+	0.789586i	$0.289705\pi$
$152$	1117.51	0.596328
$153$	−182.948	−0.0966700
$154$	−1225.86	−0.641447
$155$	−3045.82	−1.57836
$156$	378.928	0.194478
$157$	3173.51	1.61321	0.806605	−	0.591091i	$-0.201303\pi$
$157$	3173.51	1.61321	0.806605	+	0.591091i	$0.201303\pi$
$158$	−1394.12	−0.701966
$159$	239.953	0.119682
$160$	2937.89	1.45163
$161$	2030.48	0.993941
$162$	340.933	0.165347
$163$	−2314.65	−1.11225	−0.556126	−	0.831098i	$-0.687713\pi$
$163$	−2314.65	−1.11225	−0.556126	+	0.831098i	$0.687713\pi$
$164$	3800.78	1.80970
$165$	−887.545	−0.418759
$166$	−1832.76	−0.856925
$167$	−2665.65	−1.23517	−0.617587	−	0.786502i	$-0.711890\pi$
$167$	−2665.65	−1.23517	−0.617587	+	0.786502i	$0.711890\pi$
$168$	−243.874	−0.111996
$169$	169.000	0.0769231
$170$	978.138	0.441293
$171$	1392.41	0.622690
$172$	1471.09	0.652148
$173$	−165.243	−0.0726198	−0.0363099	−	0.999341i	$-0.511560\pi$
$173$	−165.243	−0.0726198	−0.0363099	+	0.999341i	$0.511560\pi$
$174$	−258.253	−0.112518
$175$	−64.0954	−0.0276866
$176$	−1224.73	−0.524532
$177$	−2621.13	−1.11309
$178$	1091.13	0.459457
$179$	712.339	0.297446	0.148723	−	0.988879i	$-0.452484\pi$
$179$	712.339	0.297446	0.148723	+	0.988879i	$0.452484\pi$
$180$	−999.688	−0.413957
$181$	2206.53	0.906133	0.453066	−	0.891477i	$-0.350330\pi$
$181$	2206.53	0.906133	0.453066	+	0.891477i	$0.350330\pi$
$182$	−615.809	−0.250806
$183$	−561.204	−0.226696
$184$	−1303.19	−0.522132
$185$	−1325.95	−0.526949
$186$	3364.18	1.32620
$187$	−526.048	−0.205714
$188$	−4540.96	−1.76162
$189$	−303.866	−0.116947
$190$	−7444.54	−2.84254
$191$	1470.64	0.557129	0.278565	−	0.960417i	$-0.410141\pi$
$191$	1470.64	0.557129	0.278565	+	0.960417i	$0.410141\pi$
$192$	−2109.14	−0.792782
$193$	369.560	0.137832	0.0689158	−	0.997622i	$-0.478046\pi$
$193$	369.560	0.137832	0.0689158	+	0.997622i	$0.478046\pi$
$194$	5156.80	1.90844
$195$	−445.856	−0.163735
$196$	−2101.99	−0.766031
$197$	−4273.41	−1.54552	−0.772761	−	0.634697i	$-0.781125\pi$
$197$	−4273.41	−1.54552	−0.772761	+	0.634697i	$0.781125\pi$
$198$	980.315	0.351858
$199$	4154.31	1.47985	0.739927	−	0.672687i	$-0.234860\pi$
$199$	4154.31	1.47985	0.739927	+	0.672687i	$0.234860\pi$
$200$	41.1373	0.0145442
$201$	−1827.61	−0.641342
$202$	2716.59	0.946230
$203$	230.175	0.0795819
$204$	−592.515	−0.203355
$205$	−4472.09	−1.52363
$206$	−2151.40	−0.727646
$207$	−1623.77	−0.545215
$208$	−615.241	−0.205093
$209$	4003.71	1.32508
$210$	1624.63	0.533857
$211$	1231.59	0.401830	0.200915	−	0.979609i	$-0.435608\pi$
$211$	1231.59	0.401830	0.200915	+	0.979609i	$0.435608\pi$
$212$	777.134	0.251763
$213$	744.114	0.239370
$214$	2559.92	0.817723
$215$	−1730.92	−0.549059
$216$	195.025	0.0614341
$217$	−2998.42	−0.938000
$218$	−5471.92	−1.70002
$219$	2558.30	0.789377
$220$	−2874.49	−0.880901
$221$	−264.259	−0.0804343
$222$	1464.54	0.442764
$223$	−2187.24	−0.656809	−0.328404	−	0.944537i	$-0.606511\pi$
$223$	−2187.24	−0.656809	−0.328404	+	0.944537i	$0.606511\pi$
$224$	2892.17	0.862684
$225$	51.2568	0.0151872
$226$	177.602	0.0522739
$227$	4138.67	1.21010	0.605051	−	0.796187i	$-0.293153\pi$
$227$	4138.67	1.21010	0.605051	+	0.796187i	$0.293153\pi$
$228$	4509.59	1.30989
$229$	−835.354	−0.241056	−0.120528	−	0.992710i	$-0.538459\pi$
$229$	−835.354	−0.241056	−0.120528	+	0.992710i	$0.538459\pi$
$230$	8681.50	2.48887
$231$	−873.733	−0.248863
$232$	−147.729	−0.0418056
$233$	3685.51	1.03625	0.518124	−	0.855305i	$-0.326630\pi$
$233$	3685.51	1.03625	0.518124	+	0.855305i	$0.326630\pi$
$234$	492.459	0.137577
$235$	5343.01	1.48315
$236$	−8489.05	−2.34148
$237$	−993.662	−0.272343
$238$	962.917	0.262255
$239$	3026.21	0.819034	0.409517	−	0.912303i	$-0.365697\pi$
$239$	3026.21	0.819034	0.409517	+	0.912303i	$0.365697\pi$
$240$	1623.13	0.436552
$241$	3265.58	0.872839	0.436420	−	0.899743i	$-0.356246\pi$
$241$	3265.58	0.872839	0.436420	+	0.899743i	$0.356246\pi$
$242$	−2783.46	−0.739370
$243$	243.000	0.0641500
$244$	−1817.57	−0.476877
$245$	2473.25	0.644940
$246$	4939.53	1.28022
$247$	2011.25	0.518110
$248$	1924.42	0.492746
$249$	−1306.30	−0.332463
$250$	5740.79	1.45232
$251$	−6363.16	−1.60016	−0.800078	−	0.599897i	$-0.795208\pi$
$251$	−6363.16	−1.60016	−0.800078	+	0.599897i	$0.795208\pi$
$252$	−984.131	−0.246010
$253$	−4668.96	−1.16022
$254$	−1309.09	−0.323385
$255$	697.168	0.171209
$256$	1822.38	0.444917
$257$	−6085.36	−1.47702	−0.738511	−	0.674242i	$-0.764471\pi$
$257$	−6085.36	−1.47702	−0.738511	+	0.674242i	$0.764471\pi$
$258$	1911.84	0.461342
$259$	−1305.31	−0.313159
$260$	−1443.99	−0.344433
$261$	−184.070	−0.0436538
$262$	8422.24	1.98598
$263$	123.227	0.0288916	0.0144458	−	0.999896i	$-0.495402\pi$
$263$	123.227	0.0288916	0.0144458	+	0.999896i	$0.495402\pi$
$264$	560.773	0.130732
$265$	−914.395	−0.211965
$266$	−7328.69	−1.68929
$267$	777.700	0.178256
$268$	−5919.08	−1.34913
$269$	−1935.79	−0.438763	−0.219381	−	0.975639i	$-0.570404\pi$
$269$	−1935.79	−0.438763	−0.219381	+	0.975639i	$0.570404\pi$
$270$	−1299.20	−0.292841
$271$	−4612.69	−1.03395	−0.516976	−	0.856000i	$-0.672942\pi$
$271$	−4612.69	−1.03395	−0.516976	+	0.856000i	$0.672942\pi$
$272$	962.028	0.214454
$273$	−438.918	−0.0973059
$274$	4371.20	0.963774
$275$	147.383	0.0323183
$276$	−5258.89	−1.14691
$277$	−5834.30	−1.26552	−0.632761	−	0.774347i	$-0.718078\pi$
$277$	−5834.30	−1.26552	−0.632761	+	0.774347i	$0.718078\pi$
$278$	−12031.4	−2.59567
$279$	2397.82	0.514529
$280$	929.341	0.198353
$281$	4691.91	0.996071	0.498036	−	0.867157i	$-0.334055\pi$
$281$	4691.91	0.996071	0.498036	+	0.867157i	$0.334055\pi$
$282$	−5901.49	−1.24620
$283$	3465.60	0.727945	0.363973	−	0.931410i	$-0.381420\pi$
$283$	3465.60	0.727945	0.363973	+	0.931410i	$0.381420\pi$
$284$	2409.96	0.503539
$285$	−5306.09	−1.10283
$286$	1416.01	0.292764
$287$	−4402.50	−0.905475
$288$	−2312.85	−0.473216
$289$	−4499.79	−0.915894
$290$	984.133	0.199277
$291$	3675.51	0.740420
$292$	8285.55	1.66053
$293$	2677.31	0.533822	0.266911	−	0.963721i	$-0.413997\pi$
$293$	2677.31	0.533822	0.266911	+	0.963721i	$0.413997\pi$
$294$	−2731.77	−0.541904
$295$	9988.43	1.97135
$296$	837.767	0.164507
$297$	698.720	0.136511
$298$	3129.34	0.608315
$299$	−2345.44	−0.453646
$300$	166.005	0.0319477
$301$	−1703.98	−0.326299
$302$	9585.02	1.82634
$303$	1936.25	0.367111
$304$	−7321.93	−1.38139
$305$	2138.60	0.401494
$306$	−770.039	−0.143857
$307$	471.915	0.0877316	0.0438658	−	0.999037i	$-0.486033\pi$
$307$	471.915	0.0877316	0.0438658	+	0.999037i	$0.486033\pi$
$308$	−2829.76	−0.523508
$309$	−1533.41	−0.282306
$310$	−12820.0	−2.34880
$311$	−1518.52	−0.276872	−0.138436	−	0.990371i	$-0.544207\pi$
$311$	−1518.52	−0.276872	−0.138436	+	0.990371i	$0.544207\pi$
$312$	281.703	0.0511163
$313$	4049.86	0.731348	0.365674	−	0.930743i	$-0.380839\pi$
$313$	4049.86	0.731348	0.365674	+	0.930743i	$0.380839\pi$
$314$	13357.5	2.40066
$315$	1157.95	0.207121
$316$	−3218.17	−0.572900
$317$	3253.96	0.576532	0.288266	−	0.957550i	$-0.406921\pi$
$317$	3253.96	0.576532	0.288266	+	0.957550i	$0.406921\pi$
$318$	1009.97	0.178102
$319$	−529.272	−0.0928951
$320$	8037.37	1.40407
$321$	1824.59	0.317254
$322$	8546.40	1.47911
$323$	−3144.92	−0.541759
$324$	787.004	0.134946
$325$	74.0375	0.0126365
$326$	−9742.46	−1.65517
$327$	−3900.11	−0.659561
$328$	2825.58	0.475660
$329$	5259.86	0.881415
$330$	−3735.72	−0.623165
$331$	−3422.45	−0.568322	−0.284161	−	0.958777i	$-0.591715\pi$
$331$	−3422.45	−0.568322	−0.284161	+	0.958777i	$0.591715\pi$
$332$	−4230.71	−0.699368
$333$	1043.85	0.171780
$334$	−11219.8	−1.83809
$335$	6964.54	1.13586
$336$	1597.87	0.259437
$337$	−9301.67	−1.50354	−0.751772	−	0.659423i	$-0.770801\pi$
$337$	−9301.67	−1.50354	−0.751772	+	0.659423i	$0.770801\pi$
$338$	711.329	0.114471
$339$	126.586	0.0202808
$340$	2257.92	0.360155
$341$	6894.66	1.09492
$342$	5860.71	0.926640
$343$	6294.99	0.990955
$344$	1093.64	0.171410
$345$	6187.74	0.965613
$346$	−695.518	−0.108067
$347$	216.898	0.0335554	0.0167777	−	0.999859i	$-0.494659\pi$
$347$	216.898	0.0335554	0.0167777	+	0.999859i	$0.494659\pi$
$348$	−596.147	−0.0918299
$349$	−4809.84	−0.737721	−0.368861	−	0.929485i	$-0.620252\pi$
$349$	−4809.84	−0.737721	−0.368861	+	0.929485i	$0.620252\pi$
$350$	−269.781	−0.0412011
$351$	351.000	0.0533761
$352$	−6650.35	−1.00700
$353$	−2859.64	−0.431170	−0.215585	−	0.976485i	$-0.569166\pi$
$353$	−2859.64	−0.431170	−0.215585	+	0.976485i	$0.569166\pi$
$354$	−11032.5	−1.65641
$355$	−2835.62	−0.423941
$356$	2518.74	0.374980
$357$	686.319	0.101747
$358$	2998.27	0.442636
$359$	3686.04	0.541899	0.270949	−	0.962594i	$-0.412662\pi$
$359$	3686.04	0.541899	0.270949	+	0.962594i	$0.412662\pi$
$360$	−743.188	−0.108804
$361$	17076.8	2.48969
$362$	9287.39	1.34844
$363$	−1983.91	−0.286855
$364$	−1421.52	−0.204692
$365$	−9748.98	−1.39804
$366$	−2362.14	−0.337352
$367$	−3470.59	−0.493633	−0.246816	−	0.969062i	$-0.579384\pi$
$367$	−3470.59	−0.493633	−0.246816	+	0.969062i	$0.579384\pi$
$368$	8538.52	1.20951
$369$	3520.65	0.496688
$370$	−5580.98	−0.784166
$371$	−900.166	−0.125968
$372$	7765.82	1.08236
$373$	−11963.4	−1.66070	−0.830352	−	0.557240i	$-0.811860\pi$
$373$	−11963.4	−1.66070	−0.830352	+	0.557240i	$0.811860\pi$
$374$	−2214.16	−0.306127
$375$	4091.75	0.563459
$376$	−3375.85	−0.463021
$377$	−265.879	−0.0363221
$378$	−1278.99	−0.174032
$379$	345.604	0.0468403	0.0234202	−	0.999726i	$-0.492544\pi$
$379$	345.604	0.0468403	0.0234202	+	0.999726i	$0.492544\pi$
$380$	−17184.8	−2.31990
$381$	−933.055	−0.125464
$382$	6189.99	0.829078
$383$	−3386.40	−0.451793	−0.225897	−	0.974151i	$-0.572531\pi$
$383$	−3386.40	−0.451793	−0.225897	+	0.974151i	$0.572531\pi$
$384$	−2709.87	−0.360124
$385$	3329.56	0.440754
$386$	1555.49	0.205110
$387$	1362.67	0.178988
$388$	11903.9	1.55755
$389$	−1629.88	−0.212438	−0.106219	−	0.994343i	$-0.533874\pi$
$389$	−1629.88	−0.212438	−0.106219	+	0.994343i	$0.533874\pi$
$390$	−1876.63	−0.243659
$391$	3667.47	0.474353
$392$	−1562.66	−0.201343
$393$	6002.95	0.770506
$394$	−17987.0	−2.29993
$395$	3786.58	0.482338
$396$	2262.94	0.287165
$397$	7938.94	1.00364	0.501819	−	0.864973i	$-0.332664\pi$
$397$	7938.94	1.00364	0.501819	+	0.864973i	$0.332664\pi$
$398$	17485.7	2.20221
$399$	−5223.52	−0.655396
$400$	−269.532	−0.0336915
$401$	214.402	0.0267001	0.0133500	−	0.999911i	$-0.495750\pi$
$401$	214.402	0.0267001	0.0133500	+	0.999911i	$0.495750\pi$
$402$	−7692.51	−0.954396
$403$	3463.52	0.428114
$404$	6270.93	0.772253
$405$	−926.008	−0.113614
$406$	968.819	0.118428
$407$	3001.48	0.365548
$408$	−440.488	−0.0534495
$409$	−4783.73	−0.578338	−0.289169	−	0.957278i	$-0.593379\pi$
$409$	−4783.73	−0.578338	−0.289169	+	0.957278i	$0.593379\pi$
$410$	−18823.2	−2.26735
$411$	3115.58	0.373917
$412$	−4966.25	−0.593859
$413$	9832.99	1.17155
$414$	−6834.51	−0.811347
$415$	4977.95	0.588815
$416$	−3340.79	−0.393739
$417$	−8575.39	−1.00705
$418$	16851.8	1.97189
$419$	−9903.67	−1.15472	−0.577358	−	0.816491i	$-0.695916\pi$
$419$	−9903.67	−1.15472	−0.577358	+	0.816491i	$0.695916\pi$
$420$	3750.26	0.435700
$421$	−12120.6	−1.40314	−0.701572	−	0.712598i	$-0.747518\pi$
$421$	−12120.6	−1.40314	−0.701572	+	0.712598i	$0.747518\pi$
$422$	5183.82	0.597973
$423$	−4206.28	−0.483491
$424$	577.738	0.0661732
$425$	−115.770	−0.0132133
$426$	3132.01	0.356213
$427$	2105.32	0.238603
$428$	5909.28	0.667374
$429$	1009.26	0.113584
$430$	−7285.53	−0.817068
$431$	−13672.6	−1.52805	−0.764023	−	0.645189i	$-0.776779\pi$
$431$	−13672.6	−1.52805	−0.764023	+	0.645189i	$0.776779\pi$
$432$	−1277.81	−0.142311
$433$	7113.10	0.789455	0.394727	−	0.918798i	$-0.370839\pi$
$433$	7113.10	0.789455	0.394727	+	0.918798i	$0.370839\pi$
$434$	−12620.5	−1.39586
$435$	701.441	0.0773138
$436$	−12631.3	−1.38745
$437$	−27912.9	−3.05550
$438$	10768.0	1.17469
$439$	−6022.04	−0.654707	−0.327353	−	0.944902i	$-0.606157\pi$
$439$	−6022.04	−0.654707	−0.327353	+	0.944902i	$0.606157\pi$
$440$	−2136.96	−0.231535
$441$	−1947.07	−0.210244
$442$	−1112.28	−0.119696
$443$	−12994.4	−1.39364	−0.696821	−	0.717245i	$-0.745403\pi$
$443$	−12994.4	−1.39364	−0.696821	+	0.717245i	$0.745403\pi$
$444$	3380.72	0.361356
$445$	−2963.61	−0.315704
$446$	−9206.20	−0.977413
$447$	2230.44	0.236009
$448$	7912.30	0.834422
$449$	10984.3	1.15452	0.577260	−	0.816560i	$-0.304122\pi$
$449$	10984.3	1.15452	0.577260	+	0.816560i	$0.304122\pi$
$450$	215.742	0.0226004
$451$	10123.2	1.05695
$452$	409.973	0.0426627
$453$	6831.72	0.708570
$454$	17419.9	1.80078
$455$	1672.60	0.172335
$456$	3352.52	0.344290
$457$	9834.10	1.00661	0.503304	−	0.864109i	$-0.332118\pi$
$457$	9834.10	1.00661	0.503304	+	0.864109i	$0.332118\pi$
$458$	−3516.05	−0.358721
$459$	−548.845	−0.0558124
$460$	20040.2	2.03126
$461$	3401.42	0.343644	0.171822	−	0.985128i	$-0.445035\pi$
$461$	3401.42	0.343644	0.171822	+	0.985128i	$0.445035\pi$
$462$	−3677.59	−0.370339
$463$	1739.42	0.174596	0.0872979	−	0.996182i	$-0.472177\pi$
$463$	1739.42	0.174596	0.0872979	+	0.996182i	$0.472177\pi$
$464$	967.925	0.0968422
$465$	−9137.45	−0.911267
$466$	15512.5	1.54207
$467$	−7958.82	−0.788630	−0.394315	−	0.918975i	$-0.629018\pi$
$467$	−7958.82	−0.788630	−0.394315	+	0.918975i	$0.629018\pi$
$468$	1136.78	0.112282
$469$	6856.16	0.675028
$470$	22489.0	2.20711
$471$	9520.54	0.931387
$472$	−6310.94	−0.615433
$473$	3918.20	0.380886
$474$	−4182.37	−0.405280
$475$	881.114	0.0851122
$476$	2222.78	0.214036
$477$	719.858	0.0690986
$478$	12737.5	1.21882
$479$	8431.98	0.804315	0.402158	−	0.915570i	$-0.368260\pi$
$479$	8431.98	0.804315	0.402158	+	0.915570i	$0.368260\pi$
$480$	8813.66	0.838097
$481$	1507.79	0.142930
$482$	13745.0	1.29889
$483$	6091.45	0.573852
$484$	−6425.29	−0.603427
$485$	−14006.4	−1.31133
$486$	1022.80	0.0954632
$487$	−11684.7	−1.08723	−0.543617	−	0.839334i	$-0.682945\pi$
$487$	−11684.7	−1.08723	−0.543617	+	0.839334i	$0.682945\pi$
$488$	−1351.22	−0.125342
$489$	−6943.94	−0.642159
$490$	10410.0	0.959750
$491$	−3954.70	−0.363489	−0.181745	−	0.983346i	$-0.558174\pi$
$491$	−3954.70	−0.363489	−0.181745	+	0.983346i	$0.558174\pi$
$492$	11402.3	1.04483
$493$	415.744	0.0379801
$494$	8465.47	0.771011
$495$	−2662.63	−0.241771
$496$	−12608.8	−1.14144
$497$	−2791.49	−0.251943
$498$	−5498.27	−0.494746
$499$	−5690.37	−0.510493	−0.255246	−	0.966876i	$-0.582157\pi$
$499$	−5690.37	−0.510493	−0.255246	+	0.966876i	$0.582157\pi$
$500$	13251.9	1.18529
$501$	−7996.95	−0.713128
$502$	−26782.8	−2.38123
$503$	10859.1	0.962595	0.481298	−	0.876557i	$-0.340166\pi$
$503$	10859.1	0.962595	0.481298	+	0.876557i	$0.340166\pi$
$504$	−731.623	−0.0646609
$505$	−7378.53	−0.650178
$506$	−19651.9	−1.72655
$507$	507.000	0.0444116
$508$	−3021.88	−0.263926
$509$	−18558.6	−1.61610	−0.808049	−	0.589115i	$-0.799476\pi$
$509$	−18558.6	−1.61610	−0.808049	+	0.589115i	$0.799476\pi$
$510$	2934.41	0.254780
$511$	−9597.27	−0.830838
$512$	14896.8	1.28584
$513$	4177.22	0.359510
$514$	−25613.6	−2.19799
$515$	5843.42	0.499984
$516$	4413.27	0.376518
$517$	−12094.7	−1.02887
$518$	−5494.13	−0.466020
$519$	−495.730	−0.0419271
$520$	−1073.49	−0.0905305
$521$	17297.5	1.45454	0.727271	−	0.686350i	$-0.240788\pi$
$521$	17297.5	1.45454	0.727271	+	0.686350i	$0.240788\pi$
$522$	−774.759	−0.0649622
$523$	−5016.11	−0.419386	−0.209693	−	0.977767i	$-0.567247\pi$
$523$	−5016.11	−0.419386	−0.209693	+	0.977767i	$0.567247\pi$
$524$	19441.8	1.62084
$525$	−192.286	−0.0159849
$526$	518.667	0.0429942
$527$	−5415.77	−0.447656
$528$	−3674.19	−0.302839
$529$	20383.8	1.67533
$530$	−3848.73	−0.315431
$531$	−7863.39	−0.642640
$532$	−16917.4	−1.37869
$533$	5085.39	0.413269
$534$	3273.38	0.265268
$535$	−6953.01	−0.561878
$536$	−4400.37	−0.354603
$537$	2137.02	0.171730
$538$	−8147.83	−0.652933
$539$	−5598.57	−0.447398
$540$	−2999.06	−0.238998
$541$	17642.3	1.40204	0.701018	−	0.713144i	$-0.252729\pi$
$541$	17642.3	1.40204	0.701018	+	0.713144i	$0.252729\pi$
$542$	−19415.0	−1.53865
$543$	6619.59	0.523156
$544$	5223.86	0.411712
$545$	14862.3	1.16813
$546$	−1847.43	−0.144803
$547$	−18414.9	−1.43943	−0.719713	−	0.694271i	$-0.755727\pi$
$547$	−18414.9	−1.43943	−0.719713	+	0.694271i	$0.755727\pi$
$548$	10090.4	0.786571
$549$	−1683.61	−0.130883
$550$	620.343	0.0480937
$551$	−3164.20	−0.244645
$552$	−3909.57	−0.301453
$553$	3727.66	0.286648
$554$	−24556.9	−1.88325
$555$	−3977.84	−0.304234
$556$	−27773.1	−2.11842
$557$	8179.15	0.622193	0.311096	−	0.950378i	$-0.399304\pi$
$557$	8179.15	0.622193	0.311096	+	0.950378i	$0.399304\pi$
$558$	10092.5	0.765683
$559$	1968.30	0.148927
$560$	−6089.05	−0.459481
$561$	−1578.14	−0.118769
$562$	19748.5	1.48228
$563$	−1880.07	−0.140738	−0.0703690	−	0.997521i	$-0.522418\pi$
$563$	−1880.07	−0.140738	−0.0703690	+	0.997521i	$0.522418\pi$
$564$	−13622.9	−1.01707
$565$	−482.385	−0.0359187
$566$	14586.9	1.08327
$567$	−911.598	−0.0675194
$568$	1791.62	0.132350
$569$	10118.3	0.745485	0.372743	−	0.927935i	$-0.378417\pi$
$569$	10118.3	0.745485	0.372743	+	0.927935i	$0.378417\pi$
$570$	−22333.6	−1.64114
$571$	23428.9	1.71711	0.858555	−	0.512721i	$-0.171362\pi$
$571$	23428.9	1.71711	0.858555	+	0.512721i	$0.171362\pi$
$572$	3268.70	0.238935
$573$	4411.92	0.321659
$574$	−18530.3	−1.34746
$575$	−1027.52	−0.0745225
$576$	−6327.42	−0.457713
$577$	20508.1	1.47966	0.739831	−	0.672793i	$-0.234906\pi$
$577$	20508.1	1.47966	0.739831	+	0.672793i	$0.234906\pi$
$578$	−18939.8	−1.36296
$579$	1108.68	0.0795771
$580$	2271.76	0.162637
$581$	4900.49	0.349925
$582$	15470.4	1.10184
$583$	2069.87	0.147042
$584$	6159.65	0.436452
$585$	−1337.57	−0.0945327
$586$	11268.9	0.794394
$587$	−5968.43	−0.419665	−0.209833	−	0.977737i	$-0.567292\pi$
$587$	−5968.43	−0.419665	−0.209833	+	0.977737i	$0.567292\pi$
$588$	−6305.97	−0.442268
$589$	41219.0	2.88353
$590$	42041.8	2.93361
$591$	−12820.2	−0.892308
$592$	−5489.06	−0.381080
$593$	−14659.5	−1.01517	−0.507584	−	0.861602i	$-0.669461\pi$
$593$	−14659.5	−1.01517	−0.507584	+	0.861602i	$0.669461\pi$
$594$	2940.95	0.203146
$595$	−2615.38	−0.180202
$596$	7223.72	0.496468
$597$	12462.9	0.854394
$598$	−9872.07	−0.675082
$599$	23635.9	1.61225	0.806125	−	0.591746i	$-0.201561\pi$
$599$	23635.9	1.61225	0.806125	+	0.591746i	$0.201561\pi$
$600$	123.412	0.00839711
$601$	−11527.0	−0.782356	−0.391178	−	0.920315i	$-0.627932\pi$
$601$	−11527.0	−0.782356	−0.391178	+	0.920315i	$0.627932\pi$
$602$	−7172.15	−0.485573
$603$	−5482.83	−0.370279
$604$	22125.9	1.49055
$605$	7560.15	0.508039
$606$	8149.77	0.546306
$607$	−5098.56	−0.340930	−0.170465	−	0.985364i	$-0.554527\pi$
$607$	−5098.56	−0.340930	−0.170465	+	0.985364i	$0.554527\pi$
$608$	−39758.4	−2.65200
$609$	690.525	0.0459466
$610$	9001.47	0.597473
$611$	−6075.74	−0.402288
$612$	−1777.55	−0.117407
$613$	1516.39	0.0999128	0.0499564	−	0.998751i	$-0.484092\pi$
$613$	1516.39	0.0999128	0.0499564	+	0.998751i	$0.484092\pi$
$614$	1986.31	0.130556
$615$	−13416.3	−0.879668
$616$	−2103.70	−0.137598
$617$	18539.3	1.20966	0.604832	−	0.796353i	$-0.293240\pi$
$617$	18539.3	1.20966	0.604832	+	0.796353i	$0.293240\pi$
$618$	−6454.20	−0.420107
$619$	25684.9	1.66779	0.833897	−	0.551920i	$-0.186105\pi$
$619$	25684.9	1.66779	0.833897	+	0.551920i	$0.186105\pi$
$620$	−29593.5	−1.91694
$621$	−4871.30	−0.314780
$622$	−6391.51	−0.412020
$623$	−2917.49	−0.187619
$624$	−1845.72	−0.118410
$625$	−16304.5	−1.04349
$626$	17046.1	1.08834
$627$	12011.1	0.765038
$628$	30834.2	1.95926
$629$	−2357.67	−0.149454
$630$	4873.88	0.308222
$631$	−22410.9	−1.41389	−0.706945	−	0.707269i	$-0.749927\pi$
$631$	−22410.9	−1.41389	−0.706945	+	0.707269i	$0.749927\pi$
$632$	−2392.46	−0.150580
$633$	3694.77	0.231997
$634$	13696.1	0.857950
$635$	3555.62	0.222206
$636$	2331.40	0.145356
$637$	−2812.43	−0.174933
$638$	−2227.73	−0.138239
$639$	2232.34	0.138200
$640$	10326.6	0.637804
$641$	6827.81	0.420721	0.210361	−	0.977624i	$-0.432536\pi$
$641$	6827.81	0.420721	0.210361	+	0.977624i	$0.432536\pi$
$642$	7679.77	0.472113
$643$	−23264.3	−1.42684	−0.713418	−	0.700738i	$-0.752854\pi$
$643$	−23264.3	−1.42684	−0.713418	+	0.700738i	$0.752854\pi$
$644$	19728.4	1.20715
$645$	−5192.76	−0.316999
$646$	−13237.1	−0.806204
$647$	14745.9	0.896014	0.448007	−	0.894030i	$-0.352134\pi$
$647$	14745.9	0.896014	0.448007	+	0.894030i	$0.352134\pi$
$648$	585.075	0.0354690
$649$	−22610.3	−1.36754
$650$	311.628	0.0188047
$651$	−8995.26	−0.541554
$652$	−22489.3	−1.35084
$653$	10909.0	0.653755	0.326878	−	0.945067i	$-0.394003\pi$
$653$	10909.0	0.653755	0.326878	+	0.945067i	$0.394003\pi$
$654$	−16415.8	−0.981509
$655$	−22875.7	−1.36462
$656$	−18513.2	−1.10186
$657$	7674.89	0.455747
$658$	22139.0	1.31166
$659$	−4182.99	−0.247263	−0.123631	−	0.992328i	$-0.539454\pi$
$659$	−4182.99	−0.247263	−0.123631	+	0.992328i	$0.539454\pi$
$660$	−8623.47	−0.508588
$661$	2224.23	0.130881	0.0654406	−	0.997856i	$-0.479155\pi$
$661$	2224.23	0.130881	0.0654406	+	0.997856i	$0.479155\pi$
$662$	−14405.2	−0.845734
$663$	−792.776	−0.0464387
$664$	−3145.19	−0.183821
$665$	19905.4	1.16075
$666$	4393.63	0.255630
$667$	3689.95	0.214206
$668$	−25899.7	−1.50013
$669$	−6561.72	−0.379209
$670$	29314.1	1.69030
$671$	−4841.04	−0.278519
$672$	8676.51	0.498071
$673$	−24152.5	−1.38337	−0.691687	−	0.722197i	$-0.743132\pi$
$673$	−24152.5	−1.38337	−0.691687	+	0.722197i	$0.743132\pi$
$674$	−39151.2	−2.23746
$675$	153.770	0.00876833
$676$	1642.02	0.0934240
$677$	−15310.7	−0.869187	−0.434593	−	0.900627i	$-0.643108\pi$
$677$	−15310.7	−0.869187	−0.434593	+	0.900627i	$0.643108\pi$
$678$	532.806	0.0301804
$679$	−13788.4	−0.779310
$680$	1678.58	0.0946628
$681$	12416.0	0.698652
$682$	29020.0	1.62937
$683$	11399.6	0.638646	0.319323	−	0.947646i	$-0.396545\pi$
$683$	11399.6	0.638646	0.319323	+	0.947646i	$0.396545\pi$
$684$	13528.8	0.756265
$685$	−11872.6	−0.662233
$686$	26495.9	1.47466
$687$	−2506.06	−0.139174
$688$	−7165.54	−0.397069
$689$	1039.79	0.0574935
$690$	26044.5	1.43695
$691$	−3323.23	−0.182955	−0.0914773	−	0.995807i	$-0.529159\pi$
$691$	−3323.23	−0.182955	−0.0914773	+	0.995807i	$0.529159\pi$
$692$	−1605.52	−0.0881976
$693$	−2621.20	−0.143681
$694$	912.936	0.0499345
$695$	32678.5	1.78355
$696$	−443.188	−0.0241365
$697$	−7951.82	−0.432133
$698$	−20244.8	−1.09782
$699$	11056.5	0.598279
$700$	−622.758	−0.0336257
$701$	−12670.4	−0.682673	−0.341336	−	0.939941i	$-0.610880\pi$
$701$	−12670.4	−0.682673	−0.341336	+	0.939941i	$0.610880\pi$
$702$	1477.38	0.0794302
$703$	17944.0	0.962692
$704$	−18193.8	−0.974012
$705$	16029.0	0.856295
$706$	−12036.4	−0.641635
$707$	−7263.71	−0.386393
$708$	−25467.2	−1.35186
$709$	13075.2	0.692594	0.346297	−	0.938125i	$-0.387439\pi$
$709$	13075.2	0.692594	0.346297	+	0.938125i	$0.387439\pi$
$710$	−11935.3	−0.630877
$711$	−2980.99	−0.157237
$712$	1872.48	0.0985592
$713$	−48067.8	−2.52476
$714$	2888.75	0.151413
$715$	−3846.03	−0.201165
$716$	6921.16	0.361251
$717$	9078.62	0.472869
$718$	15514.7	0.806412
$719$	−2988.41	−0.155005	−0.0775026	−	0.996992i	$-0.524695\pi$
$719$	−2988.41	−0.155005	−0.0775026	+	0.996992i	$0.524695\pi$
$720$	4869.38	0.252043
$721$	5752.48	0.297134
$722$	71877.0	3.70496
$723$	9796.73	0.503934
$724$	21438.9	1.10051
$725$	−116.479	−0.00596680
$726$	−8350.37	−0.426875
$727$	−5507.46	−0.280963	−0.140482	−	0.990083i	$-0.544865\pi$
$727$	−5507.46	−0.280963	−0.140482	+	0.990083i	$0.544865\pi$
$728$	−1056.79	−0.0538011
$729$	729.000	0.0370370
$730$	−41033.9	−2.08046
$731$	−3077.75	−0.155725
$732$	−5452.71	−0.275325
$733$	−36585.2	−1.84353	−0.921764	−	0.387751i	$-0.873252\pi$
$733$	−36585.2	−1.84353	−0.921764	+	0.387751i	$0.873252\pi$
$734$	−14607.9	−0.734587
$735$	7419.75	0.372356
$736$	46364.6	2.32204
$737$	−15765.3	−0.787953
$738$	14818.6	0.739133
$739$	6425.89	0.319865	0.159933	−	0.987128i	$-0.448872\pi$
$739$	6425.89	0.319865	0.159933	+	0.987128i	$0.448872\pi$
$740$	−12883.0	−0.639986
$741$	6033.76	0.299131
$742$	−3788.84	−0.187457
$743$	20411.0	1.00782	0.503908	−	0.863757i	$-0.331895\pi$
$743$	20411.0	1.00782	0.503908	+	0.863757i	$0.331895\pi$
$744$	5773.27	0.284487
$745$	−8499.60	−0.417988
$746$	−50354.6	−2.47133
$747$	−3918.89	−0.191947
$748$	−5111.13	−0.249842
$749$	−6844.81	−0.333917
$750$	17222.4	0.838496
$751$	−24259.5	−1.17875	−0.589375	−	0.807860i	$-0.700626\pi$
$751$	−24259.5	−1.17875	−0.589375	+	0.807860i	$0.700626\pi$
$752$	22118.6	1.07258
$753$	−19089.5	−0.923850
$754$	−1119.10	−0.0540518
$755$	−26033.9	−1.25493
$756$	−2952.39	−0.142034
$757$	9295.39	0.446297	0.223148	−	0.974785i	$-0.428367\pi$
$757$	9295.39	0.446297	0.223148	+	0.974785i	$0.428367\pi$
$758$	1454.66	0.0697042
$759$	−14006.9	−0.669851
$760$	−12775.6	−0.609761
$761$	−21974.7	−1.04676	−0.523378	−	0.852101i	$-0.675328\pi$
$761$	−21974.7	−1.04676	−0.523378	+	0.852101i	$0.675328\pi$
$762$	−3927.27	−0.186706
$763$	14631.0	0.694204
$764$	14288.9	0.676641
$765$	2091.50	0.0988476
$766$	−14253.5	−0.672324
$767$	−11358.2	−0.534709
$768$	5467.13	0.256873
$769$	22987.4	1.07795	0.538977	−	0.842320i	$-0.318811\pi$
$769$	22987.4	1.07795	0.538977	+	0.842320i	$0.318811\pi$
$770$	14014.3	0.655897
$771$	−18256.1	−0.852759
$772$	3590.68	0.167398
$773$	31970.9	1.48760	0.743799	−	0.668404i	$-0.233022\pi$
$773$	31970.9	1.48760	0.743799	+	0.668404i	$0.233022\pi$
$774$	5735.53	0.266356
$775$	1517.34	0.0703283
$776$	8849.59	0.409384
$777$	−3915.94	−0.180803
$778$	−6860.25	−0.316133
$779$	60520.8	2.78354
$780$	−4331.98	−0.198859
$781$	6418.85	0.294090
$782$	15436.6	0.705896
$783$	−552.209	−0.0252035
$784$	10238.6	0.466408
$785$	−36280.2	−1.64955
$786$	25266.7	1.14661
$787$	6087.26	0.275715	0.137857	−	0.990452i	$-0.455978\pi$
$787$	6087.26	0.275715	0.137857	+	0.990452i	$0.455978\pi$
$788$	−41520.9	−1.87706
$789$	369.680	0.0166805
$790$	15937.9	0.717779
$791$	−474.878	−0.0213460
$792$	1682.32	0.0754780
$793$	−2431.88	−0.108901
$794$	33415.4	1.49354
$795$	−2743.19	−0.122378
$796$	40363.6	1.79730
$797$	23080.0	1.02577	0.512883	−	0.858458i	$-0.328577\pi$
$797$	23080.0	1.02577	0.512883	+	0.858458i	$0.328577\pi$
$798$	−21986.1	−0.975311
$799$	9500.41	0.420651
$800$	−1463.57	−0.0646813
$801$	2333.10	0.102916
$802$	902.429	0.0397330
$803$	22068.3	0.969829
$804$	−17757.3	−0.778918
$805$	−23212.9	−1.01633
$806$	14578.1	0.637087
$807$	−5807.37	−0.253320
$808$	4661.94	0.202978
$809$	−32377.8	−1.40710	−0.703550	−	0.710646i	$-0.748403\pi$
$809$	−32377.8	−1.40710	−0.703550	+	0.710646i	$0.748403\pi$
$810$	−3897.61	−0.169072
$811$	26352.8	1.14103	0.570513	−	0.821288i	$-0.306744\pi$
$811$	26352.8	1.14103	0.570513	+	0.821288i	$0.306744\pi$
$812$	2236.40	0.0966532
$813$	−13838.1	−0.596952
$814$	12633.4	0.543980
$815$	26461.5	1.13731
$816$	2886.08	0.123815
$817$	23424.5	1.00308
$818$	−20135.0	−0.860638
$819$	−1316.75	−0.0561796
$820$	−43451.3	−1.85047
$821$	35355.3	1.50294	0.751468	−	0.659770i	$-0.229346\pi$
$821$	35355.3	1.50294	0.751468	+	0.659770i	$0.229346\pi$
$822$	13113.6	0.556435
$823$	−12663.3	−0.536347	−0.268173	−	0.963371i	$-0.586420\pi$
$823$	−12663.3	−0.536347	−0.268173	+	0.963371i	$0.586420\pi$
$824$	−3692.02	−0.156089
$825$	442.149	0.0186590
$826$	41387.6	1.74341
$827$	−16295.2	−0.685176	−0.342588	−	0.939486i	$-0.611303\pi$
$827$	−16295.2	−0.685176	−0.342588	+	0.939486i	$0.611303\pi$
$828$	−15776.7	−0.662170
$829$	13638.9	0.571411	0.285705	−	0.958318i	$-0.407772\pi$
$829$	13638.9	0.571411	0.285705	+	0.958318i	$0.407772\pi$
$830$	20952.4	0.876229
$831$	−17502.9	−0.730649
$832$	−9139.61	−0.380840
$833$	4397.69	0.182918
$834$	−36094.2	−1.49861
$835$	30474.2	1.26300
$836$	38900.5	1.60933
$837$	7193.46	0.297064
$838$	−41685.0	−1.71836
$839$	−1890.31	−0.0777838	−0.0388919	−	0.999243i	$-0.512383\pi$
$839$	−1890.31	−0.0777838	−0.0388919	+	0.999243i	$0.512383\pi$
$840$	2788.02	0.114519
$841$	−23970.7	−0.982849
$842$	−51016.4	−2.08805
$843$	14075.7	0.575082
$844$	11966.3	0.488028
$845$	−1932.04	−0.0786559
$846$	−17704.5	−0.719494
$847$	7442.50	0.301921
$848$	−3785.35	−0.153289
$849$	10396.8	0.420279
$850$	−487.280	−0.0196630
$851$	−20925.6	−0.842914
$852$	7229.89	0.290718
$853$	1620.21	0.0650351	0.0325175	−	0.999471i	$-0.489648\pi$
$853$	1620.21	0.0650351	0.0325175	+	0.999471i	$0.489648\pi$
$854$	8861.39	0.355071
$855$	−15918.3	−0.636718
$856$	4393.08	0.175412
$857$	−14508.4	−0.578292	−0.289146	−	0.957285i	$-0.593371\pi$
$857$	−14508.4	−0.578292	−0.289146	+	0.957285i	$0.593371\pi$
$858$	4248.03	0.169027
$859$	29639.8	1.17730	0.588648	−	0.808389i	$-0.299660\pi$
$859$	29639.8	1.17730	0.588648	+	0.808389i	$0.299660\pi$
$860$	−16817.8	−0.666839
$861$	−13207.5	−0.522776
$862$	−57548.8	−2.27392
$863$	21528.8	0.849186	0.424593	−	0.905384i	$-0.360417\pi$
$863$	21528.8	0.849186	0.424593	+	0.905384i	$0.360417\pi$
$864$	−6938.56	−0.273211
$865$	1889.10	0.0742557
$866$	29939.4	1.17481
$867$	−13499.4	−0.528792
$868$	−29132.9	−1.13921
$869$	−8571.50	−0.334601
$870$	2952.40	0.115053
$871$	−7919.65	−0.308091
$872$	−9390.36	−0.364676
$873$	11026.5	0.427482
$874$	−117487.	−4.54696
$875$	−15349.9	−0.593054
$876$	24856.7	0.958708
$877$	14865.3	0.572366	0.286183	−	0.958175i	$-0.407613\pi$
$877$	14865.3	0.572366	0.286183	+	0.958175i	$0.407613\pi$
$878$	−25347.1	−0.974285
$879$	8031.92	0.308202
$880$	14001.4	0.536348
$881$	−21336.0	−0.815921	−0.407961	−	0.913000i	$-0.633760\pi$
$881$	−21336.0	−0.815921	−0.407961	+	0.913000i	$0.633760\pi$
$882$	−8195.30	−0.312869
$883$	37538.2	1.43065	0.715323	−	0.698794i	$-0.246280\pi$
$883$	37538.2	1.43065	0.715323	+	0.698794i	$0.246280\pi$
$884$	−2567.57	−0.0976884
$885$	29965.3	1.13816
$886$	−54694.1	−2.07391
$887$	34575.0	1.30881	0.654406	−	0.756144i	$-0.272919\pi$
$887$	34575.0	1.30881	0.654406	+	0.756144i	$0.272919\pi$
$888$	2513.30	0.0949784
$889$	3500.29	0.132054
$890$	−12474.0	−0.469807
$891$	2096.16	0.0788148
$892$	−21251.4	−0.797703
$893$	−72306.9	−2.70958
$894$	9388.02	0.351211
$895$	−8143.61	−0.304146
$896$	10165.9	0.379039
$897$	−7036.32	−0.261913
$898$	46233.3	1.71807
$899$	−5448.96	−0.202150
$900$	498.016	0.0184450
$901$	−1625.89	−0.0601178
$902$	42609.2	1.57287
$903$	−5111.95	−0.188389
$904$	304.783	0.0112134
$905$	−25225.5	−0.926545
$906$	28755.0	1.05444
$907$	−10424.8	−0.381641	−0.190820	−	0.981625i	$-0.561115\pi$
$907$	−10424.8	−0.381641	−0.190820	+	0.981625i	$0.561115\pi$
$908$	40211.7	1.46968
$909$	5808.75	0.211952
$910$	7040.05	0.256456
$911$	10961.8	0.398661	0.199331	−	0.979932i	$-0.436123\pi$
$911$	10961.8	0.398661	0.199331	+	0.979932i	$0.436123\pi$
$912$	−21965.8	−0.797543
$913$	−11268.3	−0.408464
$914$	41392.2	1.49796
$915$	6415.80	0.231803
$916$	−8116.38	−0.292765
$917$	−22519.7	−0.810976
$918$	−2310.12	−0.0830558
$919$	−10779.2	−0.386914	−0.193457	−	0.981109i	$-0.561970\pi$
$919$	−10779.2	−0.386914	−0.193457	+	0.981109i	$0.561970\pi$
$920$	14898.3	0.533895
$921$	1415.74	0.0506519
$922$	14316.7	0.511384
$923$	3224.49	0.114990
$924$	−8489.28	−0.302248
$925$	660.549	0.0234797
$926$	7321.32	0.259820
$927$	−4600.23	−0.162990
$928$	5255.88	0.185919
$929$	5429.07	0.191735	0.0958675	−	0.995394i	$-0.469437\pi$
$929$	5429.07	0.191735	0.0958675	+	0.995394i	$0.469437\pi$
$930$	−38460.0	−1.35608
$931$	−33470.5	−1.17825
$932$	35808.8	1.25854
$933$	−4555.55	−0.159852
$934$	−33499.1	−1.17358
$935$	6013.88	0.210348
$936$	845.109	0.0295120
$937$	−21300.1	−0.742631	−0.371315	−	0.928507i	$-0.621093\pi$
$937$	−21300.1	−0.742631	−0.371315	+	0.928507i	$0.621093\pi$
$938$	28857.9	1.00453
$939$	12149.6	0.422244
$940$	51913.2	1.80130
$941$	26851.2	0.930207	0.465103	−	0.885256i	$-0.346017\pi$
$941$	26851.2	0.930207	0.465103	+	0.885256i	$0.346017\pi$
$942$	40072.4	1.38602
$943$	−70576.7	−2.43722
$944$	41349.4	1.42564
$945$	3473.86	0.119582
$946$	16491.9	0.566805
$947$	−8021.68	−0.275258	−0.137629	−	0.990484i	$-0.543948\pi$
$947$	−8021.68	−0.275258	−0.137629	+	0.990484i	$0.543948\pi$
$948$	−9654.52	−0.330764
$949$	11085.9	0.379204
$950$	3708.65	0.126658
$951$	9761.88	0.332861
$952$	1652.46	0.0562569
$953$	35715.0	1.21398	0.606990	−	0.794709i	$-0.292377\pi$
$953$	35715.0	1.21398	0.606990	+	0.794709i	$0.292377\pi$
$954$	3029.92	0.102827
$955$	−16812.6	−0.569680
$956$	29402.9	0.994726
$957$	−1587.82	−0.0536330
$958$	35490.6	1.19692
$959$	−11687.9	−0.393557
$960$	24112.1	0.810641
$961$	41190.9	1.38266
$962$	6346.35	0.212697
$963$	5473.76	0.183166
$964$	31728.7	1.06007
$965$	−4224.88	−0.140936
$966$	25639.2	0.853963
$967$	53338.8	1.77380	0.886898	−	0.461965i	$-0.152855\pi$
$967$	53338.8	1.77380	0.886898	+	0.461965i	$0.152855\pi$
$968$	−4776.69	−0.158604
$969$	−9434.77	−0.312785
$970$	−58953.6	−1.95143
$971$	23112.9	0.763882	0.381941	−	0.924187i	$-0.375256\pi$
$971$	23112.9	0.763882	0.381941	+	0.924187i	$0.375256\pi$
$972$	2361.01	0.0779110
$973$	32170.0	1.05994
$974$	−49181.3	−1.61794
$975$	222.113	0.00729569
$976$	8853.22	0.290353
$977$	−52874.6	−1.73143	−0.865715	−	0.500538i	$-0.833136\pi$
$977$	−52874.6	−1.73143	−0.865715	+	0.500538i	$0.833136\pi$
$978$	−29227.4	−0.955612
$979$	6708.57	0.219006
$980$	24030.4	0.783287
$981$	−11700.3	−0.380798
$982$	−16645.5	−0.540917
$983$	45173.1	1.46572	0.732858	−	0.680381i	$-0.238186\pi$
$983$	45173.1	1.46572	0.732858	+	0.680381i	$0.238186\pi$
$984$	8476.73	0.274622
$985$	48854.5	1.58034
$986$	1749.89	0.0565190
$987$	15779.6	0.508885
$988$	19541.6	0.629251
$989$	−27316.7	−0.878281
$990$	−11207.2	−0.359785
$991$	60485.6	1.93884	0.969418	−	0.245414i	$-0.0789239\pi$
$991$	60485.6	1.93884	0.969418	+	0.245414i	$0.0789239\pi$
$992$	−68466.7	−2.19135
$993$	−10267.3	−0.328121
$994$	−11749.5	−0.374922
$995$	−47492.8	−1.51319
$996$	−12692.1	−0.403780
$997$	−18108.1	−0.575214	−0.287607	−	0.957749i	$-0.592860\pi$
$997$	−18108.1	−0.575214	−0.287607	+	0.957749i	$0.592860\pi$
$998$	−23951.1	−0.759677
$999$	3131.56	0.0991773

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.4.a.c.1.3	✓	3
3.2	odd	2		117.4.a.f.1.1		3
4.3	odd	2		624.4.a.t.1.1		3
5.4	even	2		975.4.a.l.1.1		3
7.6	odd	2		1911.4.a.k.1.3		3
8.3	odd	2		2496.4.a.bp.1.3		3
8.5	even	2		2496.4.a.bl.1.3		3
12.11	even	2		1872.4.a.bk.1.3		3
13.5	odd	4		507.4.b.g.337.1		6
13.8	odd	4		507.4.b.g.337.6		6
13.12	even	2		507.4.a.h.1.1		3
39.38	odd	2		1521.4.a.u.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.c.1.3	✓	3	1.1	even	1	trivial
117.4.a.f.1.1		3	3.2	odd	2
507.4.a.h.1.1		3	13.12	even	2
507.4.b.g.337.1		6	13.5	odd	4
507.4.b.g.337.6		6	13.8	odd	4
624.4.a.t.1.1		3	4.3	odd	2
975.4.a.l.1.1		3	5.4	even	2
1521.4.a.u.1.3		3	39.38	odd	2
1872.4.a.bk.1.3		3	12.11	even	2
1911.4.a.k.1.3		3	7.6	odd	2
2496.4.a.bl.1.3		3	8.5	even	2
2496.4.a.bp.1.3		3	8.3	odd	2

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:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	39.a (trivial)

\(f(q)\)	\(=\)	\(q+4.20905 q^{2} +3.00000 q^{3} +9.71610 q^{4} -11.4322 q^{5} +12.6271 q^{6} -11.2543 q^{7} +7.22315 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+4.20905 q^{2} +3.00000 q^{3} +9.71610 q^{4} -11.4322 q^{5} +12.6271 q^{6} -11.2543 q^{7} +7.22315 q^{8} +9.00000 q^{9} -48.1187 q^{10} +25.8785 q^{11} +29.1483 q^{12} +13.0000 q^{13} -47.3699 q^{14} -34.2966 q^{15} -47.3262 q^{16} -20.3276 q^{17} +37.8814 q^{18} +154.712 q^{19} -111.076 q^{20} -33.7629 q^{21} +108.924 q^{22} -180.418 q^{23} +21.6695 q^{24} +5.69520 q^{25} +54.7176 q^{26} +27.0000 q^{27} -109.348 q^{28} -20.4522 q^{29} -144.356 q^{30} +266.424 q^{31} -256.984 q^{32} +77.6355 q^{33} -85.5599 q^{34} +128.661 q^{35} +87.4449 q^{36} +115.984 q^{37} +651.190 q^{38} +39.0000 q^{39} -82.5765 q^{40} +391.184 q^{41} -142.110 q^{42} +151.407 q^{43} +251.438 q^{44} -102.890 q^{45} -759.390 q^{46} -467.365 q^{47} -141.979 q^{48} -216.341 q^{49} +23.9714 q^{50} -60.9828 q^{51} +126.309 q^{52} +79.9842 q^{53} +113.644 q^{54} -295.848 q^{55} -81.2915 q^{56} +464.136 q^{57} -86.0843 q^{58} -873.710 q^{59} -333.229 q^{60} -187.068 q^{61} +1121.39 q^{62} -101.289 q^{63} -703.047 q^{64} -148.619 q^{65} +326.772 q^{66} -609.204 q^{67} -197.505 q^{68} -541.255 q^{69} +541.542 q^{70} +248.038 q^{71} +65.0084 q^{72} +852.765 q^{73} +488.181 q^{74} +17.0856 q^{75} +1503.20 q^{76} -291.244 q^{77} +164.153 q^{78} -331.221 q^{79} +541.043 q^{80} +81.0000 q^{81} +1646.51 q^{82} -435.432 q^{83} -328.044 q^{84} +232.389 q^{85} +637.281 q^{86} -61.3566 q^{87} +186.924 q^{88} +259.233 q^{89} -433.068 q^{90} -146.306 q^{91} -1752.96 q^{92} +799.273 q^{93} -1967.16 q^{94} -1768.70 q^{95} -770.951 q^{96} +1225.17 q^{97} -910.589 q^{98} +232.907 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 9 q^{3} + 10 q^{4} + 4 q^{5} + 6 q^{6} + 30 q^{7} - 6 q^{8} + 27 q^{9}+O(q^{10}) \) 3 * q + 2 * q^2 + 9 * q^3 + 10 * q^4 + 4 * q^5 + 6 * q^6 + 30 * q^7 - 6 * q^8 + 27 * q^9	\( 3 q + 2 q^{2} + 9 q^{3} + 10 q^{4} + 4 q^{5} + 6 q^{6} + 30 q^{7} - 6 q^{8} + 27 q^{9} - 4 q^{10} - 16 q^{11} + 30 q^{12} + 39 q^{13} - 176 q^{14} + 12 q^{15} - 110 q^{16} - 146 q^{17} + 18 q^{18} + 94 q^{19} - 244 q^{20} + 90 q^{21} - 56 q^{22} - 48 q^{23} - 18 q^{24} + 145 q^{25} + 26 q^{26} + 81 q^{27} + 80 q^{28} - 2 q^{29} - 12 q^{30} + 302 q^{31} + 154 q^{32} - 48 q^{33} + 164 q^{34} + 80 q^{35} + 90 q^{36} + 374 q^{37} + 312 q^{38} + 117 q^{39} - 516 q^{40} + 480 q^{41} - 528 q^{42} - 260 q^{43} + 712 q^{44} + 36 q^{45} - 1104 q^{46} - 24 q^{47} - 330 q^{48} + 447 q^{49} + 814 q^{50} - 438 q^{51} + 130 q^{52} - 678 q^{53} + 54 q^{54} - 1552 q^{55} + 96 q^{56} + 282 q^{57} - 628 q^{58} - 1788 q^{59} - 732 q^{60} + 230 q^{61} + 1952 q^{62} + 270 q^{63} - 750 q^{64} + 52 q^{65} - 168 q^{66} + 74 q^{67} - 460 q^{68} - 144 q^{69} + 1216 q^{70} - 948 q^{71} - 54 q^{72} - 222 q^{73} + 1724 q^{74} + 435 q^{75} + 2392 q^{76} + 112 q^{77} + 78 q^{78} - 24 q^{79} + 1100 q^{80} + 243 q^{81} + 564 q^{82} - 796 q^{83} + 240 q^{84} - 248 q^{85} + 1800 q^{86} - 6 q^{87} + 1608 q^{88} + 1436 q^{89} - 36 q^{90} + 390 q^{91} - 1296 q^{92} + 906 q^{93} - 1920 q^{94} - 4032 q^{95} + 462 q^{96} + 3242 q^{97} - 5070 q^{98} - 144 q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 + 9 * q^3 + 10 * q^4 + 4 * q^5 + 6 * q^6 + 30 * q^7 - 6 * q^8 + 27 * q^9 - 4 * q^10 - 16 * q^11 + 30 * q^12 + 39 * q^13 - 176 * q^14 + 12 * q^15 - 110 * q^16 - 146 * q^17 + 18 * q^18 + 94 * q^19 - 244 * q^20 + 90 * q^21 - 56 * q^22 - 48 * q^23 - 18 * q^24 + 145 * q^25 + 26 * q^26 + 81 * q^27 + 80 * q^28 - 2 * q^29 - 12 * q^30 + 302 * q^31 + 154 * q^32 - 48 * q^33 + 164 * q^34 + 80 * q^35 + 90 * q^36 + 374 * q^37 + 312 * q^38 + 117 * q^39 - 516 * q^40 + 480 * q^41 - 528 * q^42 - 260 * q^43 + 712 * q^44 + 36 * q^45 - 1104 * q^46 - 24 * q^47 - 330 * q^48 + 447 * q^49 + 814 * q^50 - 438 * q^51 + 130 * q^52 - 678 * q^53 + 54 * q^54 - 1552 * q^55 + 96 * q^56 + 282 * q^57 - 628 * q^58 - 1788 * q^59 - 732 * q^60 + 230 * q^61 + 1952 * q^62 + 270 * q^63 - 750 * q^64 + 52 * q^65 - 168 * q^66 + 74 * q^67 - 460 * q^68 - 144 * q^69 + 1216 * q^70 - 948 * q^71 - 54 * q^72 - 222 * q^73 + 1724 * q^74 + 435 * q^75 + 2392 * q^76 + 112 * q^77 + 78 * q^78 - 24 * q^79 + 1100 * q^80 + 243 * q^81 + 564 * q^82 - 796 * q^83 + 240 * q^84 - 248 * q^85 + 1800 * q^86 - 6 * q^87 + 1608 * q^88 + 1436 * q^89 - 36 * q^90 + 390 * q^91 - 1296 * q^92 + 906 * q^93 - 1920 * q^94 - 4032 * q^95 + 462 * q^96 + 3242 * q^97 - 5070 * q^98 - 144 * q^99

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