LMFDB - Embedded newform 507.4.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, 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("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.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:	$29.9139683729$
Analytic rank:	$1$
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:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.20905$ of defining polynomial
Character	$\chi$	$=$	507.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.20905 q^{2} +3.00000 q^{3} +9.71610 q^{4} +11.4322 q^{5} -12.6271 q^{6} +11.2543 q^{7} -7.22315 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-4.20905 q^{2} +3.00000 q^{3} +9.71610 q^{4} +11.4322 q^{5} -12.6271 q^{6} +11.2543 q^{7} -7.22315 q^{8} +9.00000 q^{9} -48.1187 q^{10} -25.8785 q^{11} +29.1483 q^{12} -47.3699 q^{14} +34.2966 q^{15} -47.3262 q^{16} -20.3276 q^{17} -37.8814 q^{18} -154.712 q^{19} +111.076 q^{20} +33.7629 q^{21} +108.924 q^{22} -180.418 q^{23} -21.6695 q^{24} +5.69520 q^{25} +27.0000 q^{27} +109.348 q^{28} -20.4522 q^{29} -144.356 q^{30} -266.424 q^{31} +256.984 q^{32} -77.6355 q^{33} +85.5599 q^{34} +128.661 q^{35} +87.4449 q^{36} -115.984 q^{37} +651.190 q^{38} -82.5765 q^{40} -391.184 q^{41} -142.110 q^{42} +151.407 q^{43} -251.438 q^{44} +102.890 q^{45} +759.390 q^{46} +467.365 q^{47} -141.979 q^{48} -216.341 q^{49} -23.9714 q^{50} -60.9828 q^{51} +79.9842 q^{53} -113.644 q^{54} -295.848 q^{55} -81.2915 q^{56} -464.136 q^{57} +86.0843 q^{58} +873.710 q^{59} +333.229 q^{60} -187.068 q^{61} +1121.39 q^{62} +101.289 q^{63} -703.047 q^{64} +326.772 q^{66} +609.204 q^{67} -197.505 q^{68} -541.255 q^{69} -541.542 q^{70} -248.038 q^{71} -65.0084 q^{72} -852.765 q^{73} +488.181 q^{74} +17.0856 q^{75} -1503.20 q^{76} -291.244 q^{77} -331.221 q^{79} -541.043 q^{80} +81.0000 q^{81} +1646.51 q^{82} +435.432 q^{83} +328.044 q^{84} -232.389 q^{85} -637.281 q^{86} -61.3566 q^{87} +186.924 q^{88} -259.233 q^{89} -433.068 q^{90} -1752.96 q^{92} -799.273 q^{93} -1967.16 q^{94} -1768.70 q^{95} +770.951 q^{96} -1225.17 q^{97} +910.589 q^{98} -232.907 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	\( 3 q - 2 q^{2} + 9 q^{3} + 10 q^{4} - 4 q^{5} - 6 q^{6} - 30 q^{7} + 6 q^{8} + 27 q^{9} - 4 q^{10} + 16 q^{11} + 30 q^{12} - 176 q^{14} - 12 q^{15} - 110 q^{16} - 146 q^{17} - 18 q^{18} - 94 q^{19} + 244 q^{20} - 90 q^{21} - 56 q^{22} - 48 q^{23} + 18 q^{24} + 145 q^{25} + 81 q^{27} - 80 q^{28} - 2 q^{29} - 12 q^{30} - 302 q^{31} - 154 q^{32} + 48 q^{33} - 164 q^{34} + 80 q^{35} + 90 q^{36} - 374 q^{37} + 312 q^{38} - 516 q^{40} - 480 q^{41} - 528 q^{42} - 260 q^{43} - 712 q^{44} - 36 q^{45} + 1104 q^{46} + 24 q^{47} - 330 q^{48} + 447 q^{49} - 814 q^{50} - 438 q^{51} - 678 q^{53} - 54 q^{54} - 1552 q^{55} + 96 q^{56} - 282 q^{57} + 628 q^{58} + 1788 q^{59} + 732 q^{60} + 230 q^{61} + 1952 q^{62} - 270 q^{63} - 750 q^{64} - 168 q^{66} - 74 q^{67} - 460 q^{68} - 144 q^{69} - 1216 q^{70} + 948 q^{71} + 54 q^{72} + 222 q^{73} + 1724 q^{74} + 435 q^{75} - 2392 q^{76} + 112 q^{77} - 24 q^{79} - 1100 q^{80} + 243 q^{81} + 564 q^{82} + 796 q^{83} - 240 q^{84} + 248 q^{85} - 1800 q^{86} - 6 q^{87} + 1608 q^{88} - 1436 q^{89} - 36 q^{90} - 1296 q^{92} - 906 q^{93} - 1920 q^{94} - 4032 q^{95} - 462 q^{96} - 3242 q^{97} + 5070 q^{98} + 144 q^{99}+O(q^{100}) \) 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 - 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 + 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 - 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 - 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 - 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 - 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 - 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	$-0.767103\pi$
$2$	−4.20905	−1.48812	−0.744062	+	0.668111i	$0.767103\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	9.71610	1.21451
$5$	11.4322	1.02253	0.511264	−	0.859424i	$-0.329178\pi$
$5$	11.4322	1.02253	0.511264	+	0.859424i	$0.329178\pi$
$6$	−12.6271	−0.859169
$7$	11.2543	0.607675	0.303838	−	0.952724i	$-0.401732\pi$
$7$	11.2543	0.607675	0.303838	+	0.952724i	$0.401732\pi$
$8$	−7.22315	−0.319221
$9$	9.00000	0.333333
$10$	−48.1187	−1.52165
$11$	−25.8785	−0.709333	−0.354666	−	0.934993i	$-0.615406\pi$
$11$	−25.8785	−0.709333	−0.354666	+	0.934993i	$0.615406\pi$
$12$	29.1483	0.701199
$13$	0	0
$13$	0	0
$14$	−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	$-0.883738\pi$
$19$	−154.712	−1.86807	−0.934035	+	0.357181i	$0.883738\pi$
$20$	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
$26$	0	0
$27$	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	$-0.780640\pi$
$31$	−266.424	−1.54359	−0.771794	+	0.635873i	$0.780640\pi$
$32$	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	$-0.582955\pi$
$37$	−115.984	−0.515340	−0.257670	+	0.966233i	$0.582955\pi$
$38$	651.190	2.77992
$39$	0	0
$40$	−82.5765	−0.326412
$41$	−391.184	−1.49006	−0.745032	−	0.667029i	$-0.767566\pi$
$41$	−391.184	−1.49006	−0.745032	+	0.667029i	$0.767566\pi$
$42$	−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	$-0.241731\pi$
$47$	467.365	1.45047	0.725236	+	0.688500i	$0.241731\pi$
$48$	−141.979	−0.426934
$49$	−216.341	−0.630731
$50$	−23.9714	−0.0678012
$51$	−60.9828	−0.167437
$52$	0	0
$53$	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	$-0.0857171\pi$
$59$	873.710	1.92792	0.963960	+	0.266045i	$0.0857171\pi$
$60$	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
$65$	0	0
$66$	326.772	0.609437
$67$	609.204	1.11084	0.555418	−	0.831571i	$-0.312558\pi$
$67$	609.204	1.11084	0.555418	+	0.831571i	$0.312558\pi$
$68$	−197.505	−0.352221
$69$	−541.255	−0.944340
$70$	−541.542	−0.924667
$71$	−248.038	−0.414601	−0.207301	−	0.978277i	$-0.566468\pi$
$71$	−248.038	−0.414601	−0.207301	+	0.978277i	$0.566468\pi$
$72$	−65.0084	−0.106407
$73$	−852.765	−1.36724	−0.683621	−	0.729838i	$-0.739596\pi$
$73$	−852.765	−1.36724	−0.683621	+	0.729838i	$0.739596\pi$
$74$	488.181	0.766890
$75$	17.0856	0.0263050
$76$	−1503.20	−2.26880
$77$	−291.244	−0.431044
$78$	0	0
$79$	−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	$-0.407036\pi$
$83$	435.432	0.575842	0.287921	+	0.957654i	$0.407036\pi$
$84$	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	$-0.549336\pi$
$89$	−259.233	−0.308749	−0.154375	+	0.988012i	$0.549336\pi$
$90$	−433.068	−0.507216
$91$	0	0
$92$	−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	$-0.721572\pi$
$97$	−1225.17	−1.28245	−0.641223	+	0.767355i	$0.721572\pi$
$98$	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	$0.578619\pi$
$104$	0	0
$105$	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	$-0.306479\pi$
$109$	1300.04	1.14239	0.571197	+	0.820813i	$0.306479\pi$
$110$	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
$117$	0	0
$118$	−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
$130$	0	0
$131$	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	$-0.604968\pi$
$137$	−1038.53	−0.647644	−0.323822	+	0.946118i	$0.604968\pi$
$138$	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
$143$	0	0
$144$	−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	$-0.565521\pi$
$149$	−743.479	−0.408780	−0.204390	+	0.978890i	$0.565521\pi$
$150$	−71.9141	−0.0391451
$151$	−2277.24	−1.22728	−0.613640	−	0.789586i	$-0.710295\pi$
$151$	−2277.24	−1.22728	−0.613640	+	0.789586i	$0.710295\pi$
$152$	1117.51	0.596328
$153$	−182.948	−0.0966700
$154$	1225.86	0.641447
$155$	−3045.82	−1.57836
$156$	0	0
$157$	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	$-0.312287\pi$
$163$	2314.65	1.11225	0.556126	+	0.831098i	$0.312287\pi$
$164$	−3800.78	−1.80970
$165$	−887.545	−0.418759
$166$	−1832.76	−0.856925
$167$	2665.65	1.23517	0.617587	−	0.786502i	$-0.288110\pi$
$167$	2665.65	1.23517	0.617587	+	0.786502i	$0.288110\pi$
$168$	−243.874	−0.111996
$169$	0	0
$170$	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	$0.350330\pi$
$182$	0	0
$183$	−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	$-0.521954\pi$
$193$	−369.560	−0.137832	−0.0689158	+	0.997622i	$0.521954\pi$
$194$	5156.80	1.90844
$195$	0	0
$196$	−2101.99	−0.766031
$197$	4273.41	1.54552	0.772761	−	0.634697i	$-0.218875\pi$
$197$	4273.41	1.54552	0.772761	+	0.634697i	$0.218875\pi$
$198$	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
$208$	0	0
$209$	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
$221$	0	0
$222$	1464.54	0.442764
$223$	2187.24	0.656809	0.328404	−	0.944537i	$-0.393489\pi$
$223$	2187.24	0.656809	0.328404	+	0.944537i	$0.393489\pi$
$224$	2892.17	0.862684
$225$	51.2568	0.0151872
$226$	−177.602	−0.0522739
$227$	−4138.67	−1.21010	−0.605051	−	0.796187i	$-0.706847\pi$
$227$	−4138.67	−1.21010	−0.605051	+	0.796187i	$0.706847\pi$
$228$	−4509.59	−1.30989
$229$	835.354	0.241056	0.120528	−	0.992710i	$-0.461541\pi$
$229$	835.354	0.241056	0.120528	+	0.992710i	$0.461541\pi$
$230$	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	$0.326630\pi$
$234$	0	0
$235$	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	$-0.634303\pi$
$239$	−3026.21	−0.819034	−0.409517	+	0.912303i	$0.634303\pi$
$240$	−1623.13	−0.436552
$241$	−3265.58	−0.872839	−0.436420	−	0.899743i	$-0.643754\pi$
$241$	−3265.58	−0.872839	−0.436420	+	0.899743i	$0.643754\pi$
$242$	2783.46	0.739370
$243$	243.000	0.0641500
$244$	−1817.57	−0.476877
$245$	−2473.25	−0.644940
$246$	4939.53	1.28022
$247$	0	0
$248$	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
$260$	0	0
$261$	−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	$-0.327058\pi$
$271$	4612.69	1.03395	0.516976	+	0.856000i	$0.327058\pi$
$272$	962.028	0.214454
$273$	0	0
$274$	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	$-0.665945\pi$
$281$	−4691.91	−0.996071	−0.498036	+	0.867157i	$0.665945\pi$
$282$	−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
$286$	0	0
$287$	−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	$-0.586003\pi$
$293$	−2677.31	−0.533822	−0.266911	+	0.963721i	$0.586003\pi$
$294$	2731.77	0.541904
$295$	9988.43	1.97135
$296$	837.767	0.164507
$297$	−698.720	−0.136511
$298$	3129.34	0.608315
$299$	0	0
$300$	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	$-0.513967\pi$
$307$	−471.915	−0.0877316	−0.0438658	+	0.999037i	$0.513967\pi$
$308$	−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	$0.544207\pi$
$312$	0	0
$313$	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	$-0.593079\pi$
$317$	−3253.96	−0.576532	−0.288266	+	0.957550i	$0.593079\pi$
$318$	−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
$325$	0	0
$326$	−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	$-0.408285\pi$
$331$	3422.45	0.568322	0.284161	+	0.958777i	$0.408285\pi$
$332$	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	$0.770801\pi$
$338$	0	0
$339$	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	$-0.379748\pi$
$349$	4809.84	0.737721	0.368861	+	0.929485i	$0.379748\pi$
$350$	−269.781	−0.0412011
$351$	0	0
$352$	−6650.35	−1.00700
$353$	2859.64	0.431170	0.215585	−	0.976485i	$-0.430834\pi$
$353$	2859.64	0.431170	0.215585	+	0.976485i	$0.430834\pi$
$354$	−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	$-0.587338\pi$
$359$	−3686.04	−0.541899	−0.270949	+	0.962594i	$0.587338\pi$
$360$	−743.188	−0.108804
$361$	17076.8	2.48969
$362$	−9287.39	−1.34844
$363$	−1983.91	−0.286855
$364$	0	0
$365$	−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
$377$	0	0
$378$	−1278.99	−0.174032
$379$	−345.604	−0.0468403	−0.0234202	−	0.999726i	$-0.507456\pi$
$379$	−345.604	−0.0468403	−0.0234202	+	0.999726i	$0.507456\pi$
$380$	−17184.8	−2.31990
$381$	−933.055	−0.125464
$382$	−6189.99	−0.829078
$383$	3386.40	0.451793	0.225897	−	0.974151i	$-0.427469\pi$
$383$	3386.40	0.451793	0.225897	+	0.974151i	$0.427469\pi$
$384$	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	$0.533874\pi$
$390$	0	0
$391$	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	$-0.667336\pi$
$397$	−7938.94	−1.00364	−0.501819	+	0.864973i	$0.667336\pi$
$398$	−17485.7	−2.20221
$399$	−5223.52	−0.655396
$400$	−269.532	−0.0336915
$401$	−214.402	−0.0267001	−0.0133500	−	0.999911i	$-0.504250\pi$
$401$	−214.402	−0.0267001	−0.0133500	+	0.999911i	$0.504250\pi$
$402$	−7692.51	−0.954396
$403$	0	0
$404$	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	$-0.406621\pi$
$409$	4783.73	0.578338	0.289169	+	0.957278i	$0.406621\pi$
$410$	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
$416$	0	0
$417$	−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	$-0.252482\pi$
$421$	12120.6	1.40314	0.701572	+	0.712598i	$0.252482\pi$
$422$	−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
$429$	0	0
$430$	−7285.53	−0.817068
$431$	13672.6	1.52805	0.764023	−	0.645189i	$-0.223221\pi$
$431$	13672.6	1.52805	0.764023	+	0.645189i	$0.223221\pi$
$432$	−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
$442$	0	0
$443$	−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	$-0.695878\pi$
$449$	−10984.3	−1.15452	−0.577260	+	0.816560i	$0.695878\pi$
$450$	−215.742	−0.0226004
$451$	10123.2	1.05695
$452$	409.973	0.0426627
$453$	−6831.72	−0.708570
$454$	17419.9	1.80078
$455$	0	0
$456$	3352.52	0.344290
$457$	−9834.10	−1.00661	−0.503304	−	0.864109i	$-0.667882\pi$
$457$	−9834.10	−1.00661	−0.503304	+	0.864109i	$0.667882\pi$
$458$	−3516.05	−0.358721
$459$	−548.845	−0.0558124
$460$	−20040.2	−2.03126
$461$	−3401.42	−0.343644	−0.171822	−	0.985128i	$-0.554965\pi$
$461$	−3401.42	−0.343644	−0.171822	+	0.985128i	$0.554965\pi$
$462$	3677.59	0.370339
$463$	−1739.42	−0.174596	−0.0872979	−	0.996182i	$-0.527823\pi$
$463$	−1739.42	−0.174596	−0.0872979	+	0.996182i	$0.527823\pi$
$464$	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	$0.629018\pi$
$468$	0	0
$469$	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	$-0.631740\pi$
$479$	−8431.98	−0.804315	−0.402158	+	0.915570i	$0.631740\pi$
$480$	8813.66	0.838097
$481$	0	0
$482$	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	$-0.317055\pi$
$487$	11684.7	1.08723	0.543617	+	0.839334i	$0.317055\pi$
$488$	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
$494$	0	0
$495$	−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	$-0.417843\pi$
$499$	5690.37	0.510493	0.255246	+	0.966876i	$0.417843\pi$
$500$	−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
$507$	0	0
$508$	−3021.88	−0.263926
$509$	18558.6	1.61610	0.808049	−	0.589115i	$-0.200524\pi$
$509$	18558.6	1.61610	0.808049	+	0.589115i	$0.200524\pi$
$510$	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
$520$	0	0
$521$	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
$533$	0	0
$534$	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	$-0.747271\pi$
$541$	−17642.3	−1.40204	−0.701018	+	0.713144i	$0.747271\pi$
$542$	−19415.0	−1.53865
$543$	6619.59	0.523156
$544$	−5223.86	−0.411712
$545$	14862.3	1.16813
$546$	0	0
$547$	−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	$-0.600696\pi$
$557$	−8179.15	−0.622193	−0.311096	+	0.950378i	$0.600696\pi$
$558$	10092.5	0.765683
$559$	0	0
$560$	−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	$0.171362\pi$
$572$	0	0
$573$	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	$-0.765094\pi$
$577$	−20508.1	−1.47966	−0.739831	+	0.672793i	$0.765094\pi$
$578$	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
$585$	0	0
$586$	11268.9	0.794394
$587$	5968.43	0.419665	0.209833	−	0.977737i	$-0.432708\pi$
$587$	5968.43	0.419665	0.209833	+	0.977737i	$0.432708\pi$
$588$	−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	$-0.330539\pi$
$593$	14659.5	1.01517	0.507584	+	0.861602i	$0.330539\pi$
$594$	2940.95	0.203146
$595$	−2615.38	−0.180202
$596$	−7223.72	−0.496468
$597$	12462.9	0.854394
$598$	0	0
$599$	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
$611$	0	0
$612$	−1777.55	−0.117407
$613$	−1516.39	−0.0999128	−0.0499564	−	0.998751i	$-0.515908\pi$
$613$	−1516.39	−0.0999128	−0.0499564	+	0.998751i	$0.515908\pi$
$614$	1986.31	0.130556
$615$	−13416.3	−0.879668
$616$	2103.70	0.137598
$617$	−18539.3	−1.20966	−0.604832	−	0.796353i	$-0.706760\pi$
$617$	−18539.3	−1.20966	−0.604832	+	0.796353i	$0.706760\pi$
$618$	6454.20	0.420107
$619$	−25684.9	−1.66779	−0.833897	−	0.551920i	$-0.813895\pi$
$619$	−25684.9	−1.66779	−0.833897	+	0.551920i	$0.813895\pi$
$620$	−29593.5	−1.91694
$621$	−4871.30	−0.314780
$622$	6391.51	0.412020
$623$	−2917.49	−0.187619
$624$	0	0
$625$	−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	$-0.250073\pi$
$631$	22410.9	1.41389	0.706945	+	0.707269i	$0.250073\pi$
$632$	2392.46	0.150580
$633$	3694.77	0.231997
$634$	13696.1	0.857950
$635$	−3555.62	−0.222206
$636$	2331.40	0.145356
$637$	0	0
$638$	−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	$-0.247146\pi$
$643$	23264.3	1.42684	0.713418	+	0.700738i	$0.247146\pi$
$644$	−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
$650$	0	0
$651$	−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	$-0.520845\pi$
$661$	−2224.23	−0.130881	−0.0654406	+	0.997856i	$0.520845\pi$
$662$	−14405.2	−0.845734
$663$	0	0
$664$	−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
$676$	0	0
$677$	−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	$-0.603455\pi$
$683$	−11399.6	−0.638646	−0.319323	+	0.947646i	$0.603455\pi$
$684$	−13528.8	−0.756265
$685$	−11872.6	−0.662233
$686$	26495.9	1.47466
$687$	2506.06	0.139174
$688$	−7165.54	−0.397069
$689$	0	0
$690$	26044.5	1.43695
$691$	3323.23	0.182955	0.0914773	−	0.995807i	$-0.470841\pi$
$691$	3323.23	0.182955	0.0914773	+	0.995807i	$0.470841\pi$
$692$	−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	$0.610880\pi$
$702$	0	0
$703$	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	$-0.612561\pi$
$709$	−13075.2	−0.692594	−0.346297	+	0.938125i	$0.612561\pi$
$710$	11935.3	0.630877
$711$	−2980.99	−0.157237
$712$	1872.48	0.0985592
$713$	48067.8	2.52476
$714$	2888.75	0.151413
$715$	0	0
$716$	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	$0.544865\pi$
$728$	0	0
$729$	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	$-0.126748\pi$
$733$	36585.2	1.84353	0.921764	+	0.387751i	$0.126748\pi$
$734$	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	$-0.551128\pi$
$739$	−6425.89	−0.319865	−0.159933	+	0.987128i	$0.551128\pi$
$740$	−12883.0	−0.639986
$741$	0	0
$742$	−3788.84	−0.187457
$743$	−20411.0	−1.00782	−0.503908	−	0.863757i	$-0.668105\pi$
$743$	−20411.0	−1.00782	−0.503908	+	0.863757i	$0.668105\pi$
$744$	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
$754$	0	0
$755$	−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	$-0.324672\pi$
$761$	21974.7	1.04676	0.523378	+	0.852101i	$0.324672\pi$
$762$	3927.27	0.186706
$763$	14631.0	0.694204
$764$	14288.9	0.676641
$765$	−2091.50	−0.0988476
$766$	−14253.5	−0.672324
$767$	0	0
$768$	5467.13	0.256873
$769$	−22987.4	−1.07795	−0.538977	−	0.842320i	$-0.681189\pi$
$769$	−22987.4	−1.07795	−0.538977	+	0.842320i	$0.681189\pi$
$770$	14014.3	0.655897
$771$	−18256.1	−0.852759
$772$	−3590.68	−0.167398
$773$	−31970.9	−1.48760	−0.743799	−	0.668404i	$-0.766978\pi$
$773$	−31970.9	−1.48760	−0.743799	+	0.668404i	$0.766978\pi$
$774$	−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
$780$	0	0
$781$	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	$-0.544022\pi$
$787$	−6087.26	−0.275715	−0.137857	+	0.990452i	$0.544022\pi$
$788$	41520.9	1.87706
$789$	369.680	0.0166805
$790$	15937.9	0.717779
$791$	474.878	0.0213460
$792$	1682.32	0.0754780
$793$	0	0
$794$	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
$806$	0	0
$807$	−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	$-0.693256\pi$
$811$	−26352.8	−1.14103	−0.570513	+	0.821288i	$0.693256\pi$
$812$	−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
$819$	0	0
$820$	−43451.3	−1.85047
$821$	−35355.3	−1.50294	−0.751468	−	0.659770i	$-0.770654\pi$
$821$	−35355.3	−1.50294	−0.751468	+	0.659770i	$0.770654\pi$
$822$	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	$-0.388697\pi$
$827$	16295.2	0.685176	0.342588	+	0.939486i	$0.388697\pi$
$828$	−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
$832$	0	0
$833$	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	$-0.487617\pi$
$839$	1890.31	0.0777838	0.0388919	+	0.999243i	$0.487617\pi$
$840$	−2788.02	−0.114519
$841$	−23970.7	−0.982849
$842$	−51016.4	−2.08805
$843$	−14075.7	−0.575082
$844$	11966.3	0.488028
$845$	0	0
$846$	−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	$-0.510352\pi$
$853$	−1620.21	−0.0650351	−0.0325175	+	0.999471i	$0.510352\pi$
$854$	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	$0.593371\pi$
$858$	0	0
$859$	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	$-0.639583\pi$
$863$	−21528.8	−0.849186	−0.424593	+	0.905384i	$0.639583\pi$
$864$	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
$871$	0	0
$872$	−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	$-0.592387\pi$
$877$	−14865.3	−0.572366	−0.286183	+	0.958175i	$0.592387\pi$
$878$	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	$0.246280\pi$
$884$	0	0
$885$	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
$897$	0	0
$898$	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
$910$	0	0
$911$	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
$923$	0	0
$924$	−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	$-0.530563\pi$
$929$	−5429.07	−0.191735	−0.0958675	+	0.995394i	$0.530563\pi$
$930$	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
$936$	0	0
$937$	−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	$-0.653983\pi$
$941$	−26851.2	−0.930207	−0.465103	+	0.885256i	$0.653983\pi$
$942$	−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	$-0.456052\pi$
$947$	8021.68	0.275258	0.137629	+	0.990484i	$0.456052\pi$
$948$	−9654.52	−0.330764
$949$	0	0
$950$	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
$962$	0	0
$963$	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	$-0.847145\pi$
$967$	−53338.8	−1.77380	−0.886898	+	0.461965i	$0.847145\pi$
$968$	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
$975$	0	0
$976$	8853.22	0.290353
$977$	52874.6	1.73143	0.865715	−	0.500538i	$-0.166864\pi$
$977$	52874.6	1.73143	0.865715	+	0.500538i	$0.166864\pi$
$978$	−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	$-0.761814\pi$
$983$	−45173.1	−1.46572	−0.732858	+	0.680381i	$0.761814\pi$
$984$	8476.73	0.274622
$985$	48854.5	1.58034
$986$	−1749.89	−0.0565190
$987$	15779.6	0.508885
$988$	0	0
$989$	−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	507.4.a.h.1.1		3
3.2	odd	2		1521.4.a.u.1.3		3
13.5	odd	4		507.4.b.g.337.6		6
13.8	odd	4		507.4.b.g.337.1		6
13.12	even	2		39.4.a.c.1.3	✓	3
39.38	odd	2		117.4.a.f.1.1		3
52.51	odd	2		624.4.a.t.1.1		3
65.64	even	2		975.4.a.l.1.1		3
91.90	odd	2		1911.4.a.k.1.3		3
104.51	odd	2		2496.4.a.bp.1.3		3
104.77	even	2		2496.4.a.bl.1.3		3
156.155	even	2		1872.4.a.bk.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.c.1.3	✓	3	13.12	even	2
117.4.a.f.1.1		3	39.38	odd	2
507.4.a.h.1.1		3	1.1	even	1	trivial
507.4.b.g.337.1		6	13.8	odd	4
507.4.b.g.337.6		6	13.5	odd	4
624.4.a.t.1.1		3	52.51	odd	2
975.4.a.l.1.1		3	65.64	even	2
1521.4.a.u.1.3		3	3.2	odd	2
1872.4.a.bk.1.3		3	156.155	even	2
1911.4.a.k.1.3		3	91.90	odd	2
2496.4.a.bl.1.3		3	104.77	even	2
2496.4.a.bp.1.3		3	104.51	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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

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