LMFDB - Embedded newform 169.4.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	169.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:	$9.97132279097$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.438447 q^{2} -3.68466 q^{3} -7.80776 q^{4} -17.8078 q^{5} +1.61553 q^{6} +5.43845 q^{7} +6.93087 q^{8} -13.4233 q^{9} +O(q^{10})$	\(q-0.438447 q^{2} -3.68466 q^{3} -7.80776 q^{4} -17.8078 q^{5} +1.61553 q^{6} +5.43845 q^{7} +6.93087 q^{8} -13.4233 q^{9} +7.80776 q^{10} -22.4233 q^{11} +28.7689 q^{12} -2.38447 q^{14} +65.6155 q^{15} +59.4233 q^{16} +67.9848 q^{17} +5.88540 q^{18} -80.8078 q^{19} +139.039 q^{20} -20.0388 q^{21} +9.83143 q^{22} +140.531 q^{23} -25.5379 q^{24} +192.116 q^{25} +148.946 q^{27} -42.4621 q^{28} -106.693 q^{29} -28.7689 q^{30} -276.155 q^{31} -81.5009 q^{32} +82.6222 q^{33} -29.8078 q^{34} -96.8466 q^{35} +104.806 q^{36} -4.29168 q^{37} +35.4299 q^{38} -123.423 q^{40} +227.769 q^{41} +8.78596 q^{42} +27.5294 q^{43} +175.076 q^{44} +239.039 q^{45} -61.6155 q^{46} +318.617 q^{47} -218.955 q^{48} -313.423 q^{49} -84.2329 q^{50} -250.501 q^{51} -67.6562 q^{53} -65.3050 q^{54} +399.309 q^{55} +37.6932 q^{56} +297.749 q^{57} +46.7793 q^{58} -291.115 q^{59} -512.311 q^{60} +663.311 q^{61} +121.080 q^{62} -73.0019 q^{63} -439.652 q^{64} -36.2255 q^{66} -425.101 q^{67} -530.810 q^{68} -517.810 q^{69} +42.4621 q^{70} -152.963 q^{71} -93.0351 q^{72} +117.268 q^{73} +1.88167 q^{74} -707.884 q^{75} +630.928 q^{76} -121.948 q^{77} +202.462 q^{79} -1058.20 q^{80} -186.386 q^{81} -99.8647 q^{82} +336.155 q^{83} +156.458 q^{84} -1210.66 q^{85} -12.0702 q^{86} +393.128 q^{87} -155.413 q^{88} +718.194 q^{89} -104.806 q^{90} -1097.23 q^{92} +1017.54 q^{93} -139.697 q^{94} +1439.01 q^{95} +300.303 q^{96} +759.368 q^{97} +137.420 q^{98} +300.994 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5 q^{2} + 5 q^{3} + 5 q^{4} - 15 q^{5} - 38 q^{6} + 15 q^{7} - 15 q^{8} + 35 q^{9}+O(q^{10}) $ 2 * q - 5 * q^2 + 5 * q^3 + 5 * q^4 - 15 * q^5 - 38 * q^6 + 15 * q^7 - 15 * q^8 + 35 * q^9	\( 2 q - 5 q^{2} + 5 q^{3} + 5 q^{4} - 15 q^{5} - 38 q^{6} + 15 q^{7} - 15 q^{8} + 35 q^{9} - 5 q^{10} + 17 q^{11} + 140 q^{12} - 46 q^{14} + 90 q^{15} + 57 q^{16} + 70 q^{17} - 215 q^{18} - 141 q^{19} + 175 q^{20} + 63 q^{21} - 170 q^{22} + 145 q^{23} - 216 q^{24} + 75 q^{25} + 335 q^{27} + 80 q^{28} + 34 q^{29} - 140 q^{30} - 140 q^{31} + 105 q^{32} + 425 q^{33} - 39 q^{34} - 70 q^{35} + 725 q^{36} - 190 q^{37} + 310 q^{38} - 185 q^{40} + 538 q^{41} - 370 q^{42} + 455 q^{43} + 680 q^{44} + 375 q^{45} - 82 q^{46} + 60 q^{47} - 240 q^{48} - 565 q^{49} + 450 q^{50} - 233 q^{51} + 545 q^{53} - 914 q^{54} + 510 q^{55} - 172 q^{56} - 225 q^{57} - 595 q^{58} - 809 q^{59} - 200 q^{60} + 502 q^{61} - 500 q^{62} + 390 q^{63} - 1271 q^{64} - 1598 q^{66} - 475 q^{67} - 505 q^{68} - 479 q^{69} - 80 q^{70} + 127 q^{71} - 1155 q^{72} + 585 q^{73} + 849 q^{74} - 1725 q^{75} - 140 q^{76} + 255 q^{77} + 240 q^{79} - 1065 q^{80} + 122 q^{81} - 1515 q^{82} + 260 q^{83} + 1220 q^{84} - 1205 q^{85} - 1962 q^{86} + 1615 q^{87} - 1020 q^{88} + 921 q^{89} - 725 q^{90} - 1040 q^{92} + 2200 q^{93} + 1040 q^{94} + 1270 q^{95} + 1920 q^{96} - 415 q^{97} + 1285 q^{98} + 2210 q^{99}+O(q^{100}) \) 2 * q - 5 * q^2 + 5 * q^3 + 5 * q^4 - 15 * q^5 - 38 * q^6 + 15 * q^7 - 15 * q^8 + 35 * q^9 - 5 * q^10 + 17 * q^11 + 140 * q^12 - 46 * q^14 + 90 * q^15 + 57 * q^16 + 70 * q^17 - 215 * q^18 - 141 * q^19 + 175 * q^20 + 63 * q^21 - 170 * q^22 + 145 * q^23 - 216 * q^24 + 75 * q^25 + 335 * q^27 + 80 * q^28 + 34 * q^29 - 140 * q^30 - 140 * q^31 + 105 * q^32 + 425 * q^33 - 39 * q^34 - 70 * q^35 + 725 * q^36 - 190 * q^37 + 310 * q^38 - 185 * q^40 + 538 * q^41 - 370 * q^42 + 455 * q^43 + 680 * q^44 + 375 * q^45 - 82 * q^46 + 60 * q^47 - 240 * q^48 - 565 * q^49 + 450 * q^50 - 233 * q^51 + 545 * q^53 - 914 * q^54 + 510 * q^55 - 172 * q^56 - 225 * q^57 - 595 * q^58 - 809 * q^59 - 200 * q^60 + 502 * q^61 - 500 * q^62 + 390 * q^63 - 1271 * q^64 - 1598 * q^66 - 475 * q^67 - 505 * q^68 - 479 * q^69 - 80 * q^70 + 127 * q^71 - 1155 * q^72 + 585 * q^73 + 849 * q^74 - 1725 * q^75 - 140 * q^76 + 255 * q^77 + 240 * q^79 - 1065 * q^80 + 122 * q^81 - 1515 * q^82 + 260 * q^83 + 1220 * q^84 - 1205 * q^85 - 1962 * q^86 + 1615 * q^87 - 1020 * q^88 + 921 * q^89 - 725 * q^90 - 1040 * q^92 + 2200 * q^93 + 1040 * q^94 + 1270 * q^95 + 1920 * q^96 - 415 * q^97 + 1285 * q^98 + 2210 * 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 $

$2$	−0.438447	−0.155014	−0.0775072	−	0.996992i	$-0.524696\pi$
$2$	−0.438447	−0.155014	−0.0775072	+	0.996992i	$0.524696\pi$
$3$	−3.68466	−0.709113	−0.354556	−	0.935035i	$-0.615368\pi$
$3$	−3.68466	−0.709113	−0.354556	+	0.935035i	$0.615368\pi$
$4$	−7.80776	−0.975971
$5$	−17.8078	−1.59277	−0.796387	−	0.604787i	$-0.793258\pi$
$5$	−17.8078	−1.59277	−0.796387	+	0.604787i	$0.793258\pi$
$6$	1.61553	0.109923
$7$	5.43845	0.293649	0.146824	−	0.989163i	$-0.453095\pi$
$7$	5.43845	0.293649	0.146824	+	0.989163i	$0.453095\pi$
$8$	6.93087	0.306304
$9$	−13.4233	−0.497159
$10$	7.80776	0.246903
$11$	−22.4233	−0.614625	−0.307313	−	0.951609i	$-0.599430\pi$
$11$	−22.4233	−0.614625	−0.307313	+	0.951609i	$0.599430\pi$
$12$	28.7689	0.692073
$13$	0	0
$13$	0	0
$14$	−2.38447	−0.0455198
$15$	65.6155	1.12946
$16$	59.4233	0.928489
$17$	67.9848	0.969926	0.484963	−	0.874535i	$-0.338833\pi$
$17$	67.9848	0.969926	0.484963	+	0.874535i	$0.338833\pi$
$18$	5.88540	0.0770668
$19$	−80.8078	−0.975714	−0.487857	−	0.872923i	$-0.662221\pi$
$19$	−80.8078	−0.975714	−0.487857	+	0.872923i	$0.662221\pi$
$20$	139.039	1.55450
$21$	−20.0388	−0.208230
$22$	9.83143	0.0952758
$23$	140.531	1.27403	0.637017	−	0.770850i	$-0.280168\pi$
$23$	140.531	1.27403	0.637017	+	0.770850i	$0.280168\pi$
$24$	−25.5379	−0.217204
$25$	192.116	1.53693
$26$	0	0
$27$	148.946	1.06165
$28$	−42.4621	−0.286592
$29$	−106.693	−0.683187	−0.341594	−	0.939848i	$-0.610967\pi$
$29$	−106.693	−0.683187	−0.341594	+	0.939848i	$0.610967\pi$
$30$	−28.7689	−0.175082
$31$	−276.155	−1.59997	−0.799983	−	0.600023i	$-0.795158\pi$
$31$	−276.155	−1.59997	−0.799983	+	0.600023i	$0.795158\pi$
$32$	−81.5009	−0.450233
$33$	82.6222	0.435839
$34$	−29.8078	−0.150353
$35$	−96.8466	−0.467716
$36$	104.806	0.485212
$37$	−4.29168	−0.0190688	−0.00953442	−	0.999955i	$-0.503035\pi$
$37$	−4.29168	−0.0190688	−0.00953442	+	0.999955i	$0.503035\pi$
$38$	35.4299	0.151250
$39$	0	0
$40$	−123.423	−0.487873
$41$	227.769	0.867598	0.433799	−	0.901010i	$-0.357173\pi$
$41$	227.769	0.867598	0.433799	+	0.901010i	$0.357173\pi$
$42$	8.78596	0.0322787
$43$	27.5294	0.0976323	0.0488162	−	0.998808i	$-0.484455\pi$
$43$	27.5294	0.0976323	0.0488162	+	0.998808i	$0.484455\pi$
$44$	175.076	0.599856
$45$	239.039	0.791862
$46$	−61.6155	−0.197494
$47$	318.617	0.988832	0.494416	−	0.869225i	$-0.335382\pi$
$47$	318.617	0.988832	0.494416	+	0.869225i	$0.335382\pi$
$48$	−218.955	−0.658403
$49$	−313.423	−0.913771
$50$	−84.2329	−0.238247
$51$	−250.501	−0.687787
$52$	0	0
$53$	−67.6562	−0.175345	−0.0876726	−	0.996149i	$-0.527943\pi$
$53$	−67.6562	−0.175345	−0.0876726	+	0.996149i	$0.527943\pi$
$54$	−65.3050	−0.164572
$55$	399.309	0.978960
$56$	37.6932	0.0899457
$57$	297.749	0.691892
$58$	46.7793	0.105904
$59$	−291.115	−0.642371	−0.321186	−	0.947016i	$-0.604081\pi$
$59$	−291.115	−0.642371	−0.321186	+	0.947016i	$0.604081\pi$
$60$	−512.311	−1.10232
$61$	663.311	1.39227	0.696133	−	0.717913i	$-0.254902\pi$
$61$	663.311	1.39227	0.696133	+	0.717913i	$0.254902\pi$
$62$	121.080	0.248018
$63$	−73.0019	−0.145990
$64$	−439.652	−0.858696
$65$	0	0
$66$	−36.2255	−0.0675613
$67$	−425.101	−0.775140	−0.387570	−	0.921840i	$-0.626685\pi$
$67$	−425.101	−0.775140	−0.387570	+	0.921840i	$0.626685\pi$
$68$	−530.810	−0.946619
$69$	−517.810	−0.903434
$70$	42.4621	0.0725028
$71$	−152.963	−0.255681	−0.127841	−	0.991795i	$-0.540805\pi$
$71$	−152.963	−0.255681	−0.127841	+	0.991795i	$0.540805\pi$
$72$	−93.0351	−0.152282
$73$	117.268	0.188016	0.0940081	−	0.995571i	$-0.470032\pi$
$73$	117.268	0.188016	0.0940081	+	0.995571i	$0.470032\pi$
$74$	1.88167	0.00295595
$75$	−707.884	−1.08986
$76$	630.928	0.952268
$77$	−121.948	−0.180484
$78$	0	0
$79$	202.462	0.288339	0.144169	−	0.989553i	$-0.453949\pi$
$79$	202.462	0.288339	0.144169	+	0.989553i	$0.453949\pi$
$80$	−1058.20	−1.47887
$81$	−186.386	−0.255674
$82$	−99.8647	−0.134490
$83$	336.155	0.444552	0.222276	−	0.974984i	$-0.428651\pi$
$83$	336.155	0.444552	0.222276	+	0.974984i	$0.428651\pi$
$84$	156.458	0.203226
$85$	−1210.66	−1.54487
$86$	−12.0702	−0.0151344
$87$	393.128	0.484457
$88$	−155.413	−0.188262
$89$	718.194	0.855376	0.427688	−	0.903927i	$-0.359328\pi$
$89$	718.194	0.855376	0.427688	+	0.903927i	$0.359328\pi$
$90$	−104.806	−0.122750
$91$	0	0
$92$	−1097.23	−1.24342
$93$	1017.54	1.13456
$94$	−139.697	−0.153283
$95$	1439.01	1.55409
$96$	300.303	0.319266
$97$	759.368	0.794868	0.397434	−	0.917631i	$-0.369901\pi$
$97$	759.368	0.794868	0.397434	+	0.917631i	$0.369901\pi$
$98$	137.420	0.141648
$99$	300.994	0.305566
$100$	−1500.00	−1.50000
$101$	−348.697	−0.343531	−0.171766	−	0.985138i	$-0.554947\pi$
$101$	−348.697	−0.343531	−0.171766	+	0.985138i	$0.554947\pi$
$102$	109.831	0.106617
$103$	−580.303	−0.555136	−0.277568	−	0.960706i	$-0.589528\pi$
$103$	−580.303	−0.555136	−0.277568	+	0.960706i	$0.589528\pi$
$104$	0	0
$105$	356.847	0.331663
$106$	29.6637	0.0271810
$107$	571.493	0.516340	0.258170	−	0.966100i	$-0.416881\pi$
$107$	571.493	0.516340	0.258170	+	0.966100i	$0.416881\pi$
$108$	−1162.94	−1.03614
$109$	176.004	0.154661	0.0773307	−	0.997005i	$-0.475360\pi$
$109$	176.004	0.154661	0.0773307	+	0.997005i	$0.475360\pi$
$110$	−175.076	−0.151753
$111$	15.8134	0.0135220
$112$	323.170	0.272649
$113$	1264.88	1.05301	0.526505	−	0.850172i	$-0.323502\pi$
$113$	1264.88	1.05301	0.526505	+	0.850172i	$0.323502\pi$
$114$	−130.547	−0.107253
$115$	−2502.55	−2.02925
$116$	833.035	0.666770
$117$	0	0
$118$	127.638	0.0995768
$119$	369.732	0.284817
$120$	454.773	0.345957
$121$	−828.196	−0.622236
$122$	−290.827	−0.215821
$123$	−839.251	−0.615225
$124$	2156.16	1.56152
$125$	−1195.19	−0.855211
$126$	32.0075	0.0226306
$127$	2604.11	1.81950	0.909752	−	0.415151i	$-0.136271\pi$
$127$	2604.11	1.81950	0.909752	+	0.415151i	$0.136271\pi$
$128$	844.772	0.583344
$129$	−101.436	−0.0692323
$130$	0	0
$131$	2131.70	1.42174	0.710870	−	0.703324i	$-0.248302\pi$
$131$	2131.70	1.42174	0.710870	+	0.703324i	$0.248302\pi$
$132$	−645.094	−0.425366
$133$	−439.469	−0.286517
$134$	186.384	0.120158
$135$	−2652.40	−1.69098
$136$	471.194	0.297092
$137$	687.985	0.429040	0.214520	−	0.976720i	$-0.431181\pi$
$137$	687.985	0.429040	0.214520	+	0.976720i	$0.431181\pi$
$138$	227.032	0.140045
$139$	−679.580	−0.414685	−0.207343	−	0.978268i	$-0.566482\pi$
$139$	−679.580	−0.414685	−0.207343	+	0.978268i	$0.566482\pi$
$140$	756.155	0.456477
$141$	−1174.00	−0.701194
$142$	67.0662	0.0396343
$143$	0	0
$144$	−797.656	−0.461607
$145$	1899.97	1.08816
$146$	−51.4158	−0.0291452
$147$	1154.86	0.647966
$148$	33.5084	0.0186106
$149$	−1975.46	−1.08615	−0.543074	−	0.839685i	$-0.682740\pi$
$149$	−1975.46	−1.08615	−0.543074	+	0.839685i	$0.682740\pi$
$150$	310.370	0.168944
$151$	1803.24	0.971824	0.485912	−	0.874008i	$-0.338487\pi$
$151$	1803.24	0.971824	0.485912	+	0.874008i	$0.338487\pi$
$152$	−560.068	−0.298865
$153$	−912.580	−0.482208
$154$	53.4677	0.0279776
$155$	4917.71	2.54839
$156$	0	0
$157$	−397.168	−0.201894	−0.100947	−	0.994892i	$-0.532187\pi$
$157$	−397.168	−0.201894	−0.100947	+	0.994892i	$0.532187\pi$
$158$	−88.7689	−0.0446967
$159$	249.290	0.124340
$160$	1451.35	0.717120
$161$	764.272	0.374118
$162$	81.7206	0.0396332
$163$	941.393	0.452365	0.226183	−	0.974085i	$-0.427375\pi$
$163$	941.393	0.452365	0.226183	+	0.974085i	$0.427375\pi$
$164$	−1778.37	−0.846750
$165$	−1471.32	−0.694193
$166$	−147.386	−0.0689120
$167$	3680.43	1.70539	0.852696	−	0.522408i	$-0.174966\pi$
$167$	3680.43	1.70539	0.852696	+	0.522408i	$0.174966\pi$
$168$	−138.886	−0.0637817
$169$	0	0
$170$	530.810	0.239478
$171$	1084.71	0.485085
$172$	−214.943	−0.0952863
$173$	1422.77	0.625269	0.312634	−	0.949874i	$-0.398789\pi$
$173$	1422.77	0.625269	0.312634	+	0.949874i	$0.398789\pi$
$174$	−172.366	−0.0750978
$175$	1044.82	0.451318
$176$	−1332.47	−0.570673
$177$	1072.66	0.455514
$178$	−314.890	−0.132596
$179$	1167.89	0.487666	0.243833	−	0.969817i	$-0.421595\pi$
$179$	1167.89	0.487666	0.243833	+	0.969817i	$0.421595\pi$
$180$	−1866.36	−0.772834
$181$	−1133.96	−0.465673	−0.232836	−	0.972516i	$-0.574801\pi$
$181$	−1133.96	−0.465673	−0.232836	+	0.972516i	$0.574801\pi$
$182$	0	0
$183$	−2444.07	−0.987274
$184$	974.004	0.390242
$185$	76.4252	0.0303724
$186$	−446.137	−0.175873
$187$	−1524.44	−0.596141
$188$	−2487.69	−0.965071
$189$	810.035	0.311753
$190$	−630.928	−0.240907
$191$	2682.12	1.01608	0.508040	−	0.861333i	$-0.330370\pi$
$191$	2682.12	1.01608	0.508040	+	0.861333i	$0.330370\pi$
$192$	1619.97	0.608913
$193$	−1970.67	−0.734983	−0.367491	−	0.930027i	$-0.619783\pi$
$193$	−1970.67	−0.734983	−0.367491	+	0.930027i	$0.619783\pi$
$194$	−332.943	−0.123216
$195$	0	0
$196$	2447.14	0.891813
$197$	−4016.05	−1.45244	−0.726222	−	0.687460i	$-0.758726\pi$
$197$	−4016.05	−1.45244	−0.726222	+	0.687460i	$0.758726\pi$
$198$	−131.970	−0.0473672
$199$	−4226.06	−1.50541	−0.752707	−	0.658356i	$-0.771252\pi$
$199$	−4226.06	−1.50541	−0.752707	+	0.658356i	$0.771252\pi$
$200$	1331.53	0.470768
$201$	1566.35	0.549662
$202$	152.885	0.0532523
$203$	−580.245	−0.200617
$204$	1955.85	0.671260
$205$	−4056.06	−1.38189
$206$	254.432	0.0860541
$207$	−1886.39	−0.633398
$208$	0	0
$209$	1811.98	0.599699
$210$	−156.458	−0.0514126
$211$	1364.67	0.445249	0.222625	−	0.974904i	$-0.428538\pi$
$211$	1364.67	0.445249	0.222625	+	0.974904i	$0.428538\pi$
$212$	528.244	0.171132
$213$	563.617	0.181307
$214$	−250.570	−0.0800401
$215$	−490.237	−0.155506
$216$	1032.33	0.325189
$217$	−1501.86	−0.469828
$218$	−77.1683	−0.0239748
$219$	−432.093	−0.133325
$220$	−3117.71	−0.955436
$221$	0	0
$222$	−6.93332	−0.00209610
$223$	−1059.47	−0.318149	−0.159075	−	0.987267i	$-0.550851\pi$
$223$	−1059.47	−0.318149	−0.159075	+	0.987267i	$0.550851\pi$
$224$	−443.239	−0.132210
$225$	−2578.84	−0.764099
$226$	−554.584	−0.163232
$227$	3464.19	1.01289	0.506446	−	0.862272i	$-0.330959\pi$
$227$	3464.19	1.01289	0.506446	+	0.862272i	$0.330959\pi$
$228$	−2324.75	−0.675266
$229$	−2324.64	−0.670815	−0.335407	−	0.942073i	$-0.608874\pi$
$229$	−2324.64	−0.670815	−0.335407	+	0.942073i	$0.608874\pi$
$230$	1097.23	0.314563
$231$	449.336	0.127983
$232$	−739.476	−0.209263
$233$	−3731.01	−1.04904	−0.524521	−	0.851398i	$-0.675755\pi$
$233$	−3731.01	−1.04904	−0.524521	+	0.851398i	$0.675755\pi$
$234$	0	0
$235$	−5673.86	−1.57499
$236$	2272.95	0.626935
$237$	−746.004	−0.204465
$238$	−162.108	−0.0441508
$239$	6044.47	1.63592	0.817958	−	0.575278i	$-0.195106\pi$
$239$	6044.47	1.63592	0.817958	+	0.575278i	$0.195106\pi$
$240$	3899.09	1.04869
$241$	−5173.96	−1.38292	−0.691461	−	0.722414i	$-0.743033\pi$
$241$	−5173.96	−1.38292	−0.691461	+	0.722414i	$0.743033\pi$
$242$	363.120	0.0964556
$243$	−3334.77	−0.880353
$244$	−5178.97	−1.35881
$245$	5581.37	1.45543
$246$	367.967	0.0953688
$247$	0	0
$248$	−1914.00	−0.490076
$249$	−1238.62	−0.315238
$250$	524.029	0.132570
$251$	5620.73	1.41346	0.706728	−	0.707486i	$-0.250171\pi$
$251$	5620.73	1.41346	0.706728	+	0.707486i	$0.250171\pi$
$252$	569.981	0.142482
$253$	−3151.17	−0.783054
$254$	−1141.76	−0.282050
$255$	4460.86	1.09549
$256$	3146.83	0.768270
$257$	−1674.14	−0.406342	−0.203171	−	0.979143i	$-0.565125\pi$
$257$	−1674.14	−0.406342	−0.203171	+	0.979143i	$0.565125\pi$
$258$	44.4745	0.0107320
$259$	−23.3401	−0.00559954
$260$	0	0
$261$	1432.17	0.339653
$262$	−934.640	−0.220390
$263$	−6309.18	−1.47924	−0.739622	−	0.673023i	$-0.764996\pi$
$263$	−6309.18	−1.47924	−0.739622	+	0.673023i	$0.764996\pi$
$264$	572.644	0.133499
$265$	1204.81	0.279285
$266$	192.684	0.0444143
$267$	−2646.30	−0.606558
$268$	3319.09	0.756514
$269$	−2482.73	−0.562731	−0.281366	−	0.959601i	$-0.590787\pi$
$269$	−2482.73	−0.562731	−0.281366	+	0.959601i	$0.590787\pi$
$270$	1162.94	0.262126
$271$	2835.72	0.635638	0.317819	−	0.948151i	$-0.397050\pi$
$271$	2835.72	0.635638	0.317819	+	0.948151i	$0.397050\pi$
$272$	4039.88	0.900566
$273$	0	0
$274$	−301.645	−0.0665075
$275$	−4307.88	−0.944637
$276$	4042.94	0.881725
$277$	−3837.51	−0.832396	−0.416198	−	0.909274i	$-0.636638\pi$
$277$	−3837.51	−0.832396	−0.416198	+	0.909274i	$0.636638\pi$
$278$	297.960	0.0642822
$279$	3706.91	0.795438
$280$	−671.231	−0.143263
$281$	−9122.13	−1.93659	−0.968293	−	0.249819i	$-0.919629\pi$
$281$	−9122.13	−1.93659	−0.968293	+	0.249819i	$0.919629\pi$
$282$	514.735	0.108695
$283$	2127.85	0.446952	0.223476	−	0.974709i	$-0.428260\pi$
$283$	2127.85	0.446952	0.223476	+	0.974709i	$0.428260\pi$
$284$	1194.30	0.249537
$285$	−5302.24	−1.10203
$286$	0	0
$287$	1238.71	0.254769
$288$	1094.01	0.223838
$289$	−291.061	−0.0592430
$290$	−833.035	−0.168681
$291$	−2798.01	−0.563651
$292$	−915.601	−0.183498
$293$	8274.77	1.64989	0.824944	−	0.565215i	$-0.191207\pi$
$293$	8274.77	1.64989	0.824944	+	0.565215i	$0.191207\pi$
$294$	−506.344	−0.100444
$295$	5184.10	1.02315
$296$	−29.7450	−0.00584086
$297$	−3339.86	−0.652520
$298$	866.136	0.168369
$299$	0	0
$300$	5526.99	1.06367
$301$	149.717	0.0286696
$302$	−790.625	−0.150647
$303$	1284.83	0.243602
$304$	−4801.86	−0.905940
$305$	−11812.1	−2.21757
$306$	400.118	0.0747492
$307$	−3610.49	−0.671211	−0.335605	−	0.942003i	$-0.608941\pi$
$307$	−3610.49	−0.671211	−0.335605	+	0.942003i	$0.608941\pi$
$308$	952.140	0.176147
$309$	2138.22	0.393654
$310$	−2156.16	−0.395037
$311$	3331.06	0.607354	0.303677	−	0.952775i	$-0.401786\pi$
$311$	3331.06	0.607354	0.303677	+	0.952775i	$0.401786\pi$
$312$	0	0
$313$	−358.125	−0.0646724	−0.0323362	−	0.999477i	$-0.510295\pi$
$313$	−358.125	−0.0646724	−0.0323362	+	0.999477i	$0.510295\pi$
$314$	174.137	0.0312966
$315$	1300.00	0.232529
$316$	−1580.78	−0.281410
$317$	3047.46	0.539944	0.269972	−	0.962868i	$-0.412986\pi$
$317$	3047.46	0.539944	0.269972	+	0.962868i	$0.412986\pi$
$318$	−109.301	−0.0192744
$319$	2392.41	0.419904
$320$	7829.23	1.36771
$321$	−2105.76	−0.366143
$322$	−335.093	−0.0579938
$323$	−5493.70	−0.946371
$324$	1455.26	0.249530
$325$	0	0
$326$	−412.751	−0.0701232
$327$	−648.514	−0.109672
$328$	1578.64	0.265749
$329$	1732.78	0.290369
$330$	645.094	0.107610
$331$	7694.77	1.27777	0.638887	−	0.769301i	$-0.279395\pi$
$331$	7694.77	1.27777	0.638887	+	0.769301i	$0.279395\pi$
$332$	−2624.62	−0.433870
$333$	57.6084	0.00948025
$334$	−1613.68	−0.264360
$335$	7570.10	1.23462
$336$	−1190.77	−0.193339
$337$	4712.21	0.761693	0.380846	−	0.924638i	$-0.375633\pi$
$337$	4712.21	0.761693	0.380846	+	0.924638i	$0.375633\pi$
$338$	0	0
$339$	−4660.66	−0.746703
$340$	9452.53	1.50775
$341$	6192.31	0.983380
$342$	−475.586	−0.0751952
$343$	−3569.92	−0.561976
$344$	190.803	0.0299052
$345$	9221.03	1.43897
$346$	−623.811	−0.0969257
$347$	−5261.98	−0.814058	−0.407029	−	0.913415i	$-0.633435\pi$
$347$	−5261.98	−0.814058	−0.407029	+	0.913415i	$0.633435\pi$
$348$	−3069.45	−0.472815
$349$	50.3345	0.00772018	0.00386009	−	0.999993i	$-0.498771\pi$
$349$	50.3345	0.00772018	0.00386009	+	0.999993i	$0.498771\pi$
$350$	−458.096	−0.0699608
$351$	0	0
$352$	1827.52	0.276725
$353$	9057.64	1.36569	0.682846	−	0.730562i	$-0.260742\pi$
$353$	9057.64	1.36569	0.682846	+	0.730562i	$0.260742\pi$
$354$	−470.304	−0.0706112
$355$	2723.93	0.407243
$356$	−5607.49	−0.834821
$357$	−1362.34	−0.201968
$358$	−512.059	−0.0755953
$359$	7177.86	1.05525	0.527623	−	0.849479i	$-0.323083\pi$
$359$	7177.86	1.05525	0.527623	+	0.849479i	$0.323083\pi$
$360$	1656.75	0.242551
$361$	−329.105	−0.0479815
$362$	497.183	0.0721861
$363$	3051.62	0.441235
$364$	0	0
$365$	−2088.28	−0.299467
$366$	1071.60	0.153042
$367$	4004.14	0.569522	0.284761	−	0.958599i	$-0.408086\pi$
$367$	4004.14	0.569522	0.284761	+	0.958599i	$0.408086\pi$
$368$	8350.83	1.18293
$369$	−3057.41	−0.431334
$370$	−33.5084	−0.00470816
$371$	−367.945	−0.0514899
$372$	−7944.70	−1.10729
$373$	−10014.2	−1.39012	−0.695060	−	0.718952i	$-0.744622\pi$
$373$	−10014.2	−1.39012	−0.695060	+	0.718952i	$0.744622\pi$
$374$	668.388	0.0924105
$375$	4403.88	0.606441
$376$	2208.30	0.302883
$377$	0	0
$378$	−355.158	−0.0483263
$379$	−8169.12	−1.10717	−0.553587	−	0.832791i	$-0.686742\pi$
$379$	−8169.12	−1.10717	−0.553587	+	0.832791i	$0.686742\pi$
$380$	−11235.4	−1.51675
$381$	−9595.24	−1.29023
$382$	−1175.97	−0.157507
$383$	−7310.25	−0.975290	−0.487645	−	0.873042i	$-0.662144\pi$
$383$	−7310.25	−0.975290	−0.487645	+	0.873042i	$0.662144\pi$
$384$	−3112.70	−0.413656
$385$	2171.62	0.287470
$386$	864.033	0.113933
$387$	−369.535	−0.0485388
$388$	−5928.97	−0.775767
$389$	8785.47	1.14509	0.572546	−	0.819872i	$-0.305956\pi$
$389$	8785.47	1.14509	0.572546	+	0.819872i	$0.305956\pi$
$390$	0	0
$391$	9553.99	1.23572
$392$	−2172.30	−0.279892
$393$	−7854.60	−1.00817
$394$	1760.82	0.225150
$395$	−3605.40	−0.459259
$396$	−2350.09	−0.298224
$397$	11266.8	1.42434	0.712171	−	0.702006i	$-0.247712\pi$
$397$	11266.8	1.42434	0.712171	+	0.702006i	$0.247712\pi$
$398$	1852.90	0.233361
$399$	1619.29	0.203173
$400$	11416.2	1.42702
$401$	1576.23	0.196293	0.0981464	−	0.995172i	$-0.468709\pi$
$401$	1576.23	0.196293	0.0981464	+	0.995172i	$0.468709\pi$
$402$	−686.763	−0.0852055
$403$	0	0
$404$	2722.54	0.335276
$405$	3319.12	0.407231
$406$	254.407	0.0310985
$407$	96.2335	0.0117202
$408$	−1736.19	−0.210672
$409$	6755.78	0.816753	0.408377	−	0.912814i	$-0.366095\pi$
$409$	6755.78	0.816753	0.408377	+	0.912814i	$0.366095\pi$
$410$	1778.37	0.214213
$411$	−2534.99	−0.304238
$412$	4530.87	0.541796
$413$	−1583.21	−0.188631
$414$	827.083	0.0981858
$415$	−5986.17	−0.708072
$416$	0	0
$417$	2504.02	0.294059
$418$	−794.456	−0.0929620
$419$	10756.2	1.25411	0.627057	−	0.778973i	$-0.284259\pi$
$419$	10756.2	1.25411	0.627057	+	0.778973i	$0.284259\pi$
$420$	−2786.17	−0.323694
$421$	7886.03	0.912925	0.456463	−	0.889743i	$-0.349116\pi$
$421$	7886.03	0.912925	0.456463	+	0.889743i	$0.349116\pi$
$422$	−598.335	−0.0690201
$423$	−4276.89	−0.491607
$424$	−468.916	−0.0537089
$425$	13061.0	1.49071
$426$	−247.116	−0.0281052
$427$	3607.38	0.408837
$428$	−4462.08	−0.503932
$429$	0	0
$430$	214.943	0.0241057
$431$	−14084.6	−1.57409	−0.787044	−	0.616897i	$-0.788390\pi$
$431$	−14084.6	−1.57409	−0.787044	+	0.616897i	$0.788390\pi$
$432$	8850.86	0.985735
$433$	1864.14	0.206894	0.103447	−	0.994635i	$-0.467013\pi$
$433$	1864.14	0.206894	0.103447	+	0.994635i	$0.467013\pi$
$434$	658.485	0.0728301
$435$	−7000.73	−0.771630
$436$	−1374.20	−0.150945
$437$	−11356.0	−1.24309
$438$	189.450	0.0206673
$439$	6154.49	0.669106	0.334553	−	0.942377i	$-0.391415\pi$
$439$	6154.49	0.669106	0.334553	+	0.942377i	$0.391415\pi$
$440$	2767.56	0.299859
$441$	4207.17	0.454289
$442$	0	0
$443$	−14539.3	−1.55933	−0.779663	−	0.626200i	$-0.784609\pi$
$443$	−14539.3	−1.55933	−0.779663	+	0.626200i	$0.784609\pi$
$444$	−123.467	−0.0131970
$445$	−12789.4	−1.36242
$446$	464.521	0.0493177
$447$	7278.90	0.770202
$448$	−2391.03	−0.252155
$449$	−7043.87	−0.740358	−0.370179	−	0.928960i	$-0.620704\pi$
$449$	−7043.87	−0.740358	−0.370179	+	0.928960i	$0.620704\pi$
$450$	1130.68	0.118446
$451$	−5107.33	−0.533248
$452$	−9875.90	−1.02771
$453$	−6644.32	−0.689133
$454$	−1518.87	−0.157013
$455$	0	0
$456$	2063.66	0.211929
$457$	14098.9	1.44314	0.721572	−	0.692340i	$-0.243420\pi$
$457$	14098.9	1.44314	0.721572	+	0.692340i	$0.243420\pi$
$458$	1019.23	0.103986
$459$	10126.1	1.02973
$460$	19539.3	1.98049
$461$	14449.7	1.45985	0.729924	−	0.683529i	$-0.239556\pi$
$461$	14449.7	1.45985	0.729924	+	0.683529i	$0.239556\pi$
$462$	−197.010	−0.0198393
$463$	15806.5	1.58659	0.793293	−	0.608840i	$-0.208365\pi$
$463$	15806.5	1.58659	0.793293	+	0.608840i	$0.208365\pi$
$464$	−6340.06	−0.634332
$465$	−18120.1	−1.80709
$466$	1635.85	0.162617
$467$	−15071.3	−1.49340	−0.746699	−	0.665162i	$-0.768362\pi$
$467$	−15071.3	−1.49340	−0.746699	+	0.665162i	$0.768362\pi$
$468$	0	0
$469$	−2311.89	−0.227619
$470$	2487.69	0.244146
$471$	1463.43	0.143166
$472$	−2017.68	−0.196761
$473$	−617.299	−0.0600073
$474$	327.083	0.0316950
$475$	−15524.5	−1.49961
$476$	−2886.78	−0.277973
$477$	908.169	0.0871744
$478$	−2650.18	−0.253591
$479$	−392.545	−0.0374443	−0.0187222	−	0.999825i	$-0.505960\pi$
$479$	−392.545	−0.0374443	−0.0187222	+	0.999825i	$0.505960\pi$
$480$	−5347.73	−0.508519
$481$	0	0
$482$	2268.51	0.214373
$483$	−2816.08	−0.265292
$484$	6466.36	0.607284
$485$	−13522.7	−1.26605
$486$	1462.12	0.136467
$487$	9497.89	0.883758	0.441879	−	0.897075i	$-0.354312\pi$
$487$	9497.89	0.883758	0.441879	+	0.897075i	$0.354312\pi$
$488$	4597.32	0.426457
$489$	−3468.71	−0.320778
$490$	−2447.14	−0.225613
$491$	−1893.82	−0.174067	−0.0870337	−	0.996205i	$-0.527739\pi$
$491$	−1893.82	−0.174067	−0.0870337	+	0.996205i	$0.527739\pi$
$492$	6552.67	0.600442
$493$	−7253.52	−0.662641
$494$	0	0
$495$	−5360.04	−0.486699
$496$	−16410.1	−1.48555
$497$	−831.881	−0.0750804
$498$	543.068	0.0488664
$499$	−13370.1	−1.19945	−0.599727	−	0.800205i	$-0.704724\pi$
$499$	−13370.1	−1.19945	−0.599727	+	0.800205i	$0.704724\pi$
$500$	9331.79	0.834661
$501$	−13561.1	−1.20932
$502$	−2464.39	−0.219106
$503$	5554.71	0.492391	0.246195	−	0.969220i	$-0.420820\pi$
$503$	5554.71	0.492391	0.246195	+	0.969220i	$0.420820\pi$
$504$	−505.966	−0.0447173
$505$	6209.51	0.547168
$506$	1381.62	0.121385
$507$	0	0
$508$	−20332.3	−1.77578
$509$	2197.55	0.191365	0.0956824	−	0.995412i	$-0.469497\pi$
$509$	2197.55	0.191365	0.0956824	+	0.995412i	$0.469497\pi$
$510$	−1955.85	−0.169817
$511$	637.756	0.0552107
$512$	−8137.89	−0.702437
$513$	−12036.0	−1.03587
$514$	734.022	0.0629890
$515$	10333.9	0.884206
$516$	791.991	0.0675687
$517$	−7144.45	−0.607761
$518$	10.2334	0.000868010 0
$519$	−5242.44	−0.443386
$520$	0	0
$521$	17005.2	1.42997	0.714983	−	0.699142i	$-0.246435\pi$
$521$	17005.2	1.42997	0.714983	+	0.699142i	$0.246435\pi$
$522$	−627.932	−0.0526511
$523$	−14486.2	−1.21116	−0.605581	−	0.795783i	$-0.707059\pi$
$523$	−14486.2	−1.21116	−0.605581	+	0.795783i	$0.707059\pi$
$524$	−16643.8	−1.38758
$525$	−3849.79	−0.320035
$526$	2766.24	0.229304
$527$	−18774.4	−1.55185
$528$	4909.68	0.404671
$529$	7582.03	0.623163
$530$	−528.244	−0.0432933
$531$	3907.72	0.319361
$532$	3431.27	0.279632
$533$	0	0
$534$	1160.26	0.0940252
$535$	−10177.0	−0.822413
$536$	−2946.32	−0.237429
$537$	−4303.28	−0.345810
$538$	1088.55	0.0872315
$539$	7027.98	0.561626
$540$	20709.3	1.65034
$541$	−15266.7	−1.21325	−0.606623	−	0.794990i	$-0.707476\pi$
$541$	−15266.7	−1.21325	−0.606623	+	0.794990i	$0.707476\pi$
$542$	−1243.31	−0.0985330
$543$	4178.27	0.330215
$544$	−5540.83	−0.436693
$545$	−3134.23	−0.246341
$546$	0	0
$547$	15260.5	1.19286	0.596430	−	0.802665i	$-0.296586\pi$
$547$	15260.5	1.19286	0.596430	+	0.802665i	$0.296586\pi$
$548$	−5371.62	−0.418731
$549$	−8903.81	−0.692177
$550$	1888.78	0.146432
$551$	8621.64	0.666595
$552$	−3588.87	−0.276726
$553$	1101.08	0.0846703
$554$	1682.55	0.129033
$555$	−281.601	−0.0215374
$556$	5306.00	0.404721
$557$	−10442.1	−0.794337	−0.397169	−	0.917746i	$-0.630007\pi$
$557$	−10442.1	−0.794337	−0.397169	+	0.917746i	$0.630007\pi$
$558$	−1625.29	−0.123304
$559$	0	0
$560$	−5754.94	−0.434269
$561$	5617.06	0.422731
$562$	3999.57	0.300199
$563$	7145.26	0.534879	0.267440	−	0.963575i	$-0.413822\pi$
$563$	7145.26	0.534879	0.267440	+	0.963575i	$0.413822\pi$
$564$	9166.29	0.684344
$565$	−22524.7	−1.67721
$566$	−932.950	−0.0692841
$567$	−1013.65	−0.0750783
$568$	−1060.17	−0.0783162
$569$	4438.86	0.327042	0.163521	−	0.986540i	$-0.447715\pi$
$569$	4438.86	0.327042	0.163521	+	0.986540i	$0.447715\pi$
$570$	2324.75	0.170830
$571$	10117.3	0.741497	0.370748	−	0.928733i	$-0.379101\pi$
$571$	10117.3	0.741497	0.370748	+	0.928733i	$0.379101\pi$
$572$	0	0
$573$	−9882.70	−0.720516
$574$	−543.109	−0.0394929
$575$	26998.4	1.95810
$576$	5901.58	0.426909
$577$	3105.60	0.224069	0.112035	−	0.993704i	$-0.464263\pi$
$577$	3105.60	0.224069	0.112035	+	0.993704i	$0.464263\pi$
$578$	127.615	0.00918352
$579$	7261.23	0.521186
$580$	−14834.5	−1.06202
$581$	1828.16	0.130542
$582$	1226.78	0.0873741
$583$	1517.08	0.107772
$584$	812.769	0.0575901
$585$	0	0
$586$	−3628.05	−0.255757
$587$	19662.3	1.38254	0.691270	−	0.722597i	$-0.257052\pi$
$587$	19662.3	1.38254	0.691270	+	0.722597i	$0.257052\pi$
$588$	−9016.86	−0.632396
$589$	22315.5	1.56111
$590$	−2272.95	−0.158603
$591$	14797.8	1.02995
$592$	−255.026	−0.0177052
$593$	6395.51	0.442888	0.221444	−	0.975173i	$-0.428923\pi$
$593$	6395.51	0.442888	0.221444	+	0.975173i	$0.428923\pi$
$594$	1464.35	0.101150
$595$	−6584.10	−0.453650
$596$	15423.9	1.06005
$597$	15571.6	1.06751
$598$	0	0
$599$	8878.48	0.605618	0.302809	−	0.953051i	$-0.402076\pi$
$599$	8878.48	0.605618	0.302809	+	0.953051i	$0.402076\pi$
$600$	−4906.25	−0.333828
$601$	19100.6	1.29639	0.648194	−	0.761475i	$-0.275525\pi$
$601$	19100.6	1.29639	0.648194	+	0.761475i	$0.275525\pi$
$602$	−65.6430	−0.00444420
$603$	5706.26	0.385368
$604$	−14079.3	−0.948472
$605$	14748.3	0.991082
$606$	−563.330	−0.0377619
$607$	16595.8	1.10972	0.554861	−	0.831943i	$-0.312771\pi$
$607$	16595.8	1.10972	0.554861	+	0.831943i	$0.312771\pi$
$608$	6585.91	0.439299
$609$	2138.01	0.142260
$610$	5178.97	0.343755
$611$	0	0
$612$	7125.21	0.470620
$613$	16469.2	1.08513	0.542564	−	0.840015i	$-0.317454\pi$
$613$	16469.2	1.08513	0.542564	+	0.840015i	$0.317454\pi$
$614$	1583.01	0.104047
$615$	14945.2	0.979915
$616$	−845.205	−0.0552829
$617$	10116.0	0.660055	0.330027	−	0.943971i	$-0.392942\pi$
$617$	10116.0	0.660055	0.330027	+	0.943971i	$0.392942\pi$
$618$	−937.496	−0.0610220
$619$	18854.8	1.22430	0.612148	−	0.790743i	$-0.290306\pi$
$619$	18854.8	1.22430	0.612148	+	0.790743i	$0.290306\pi$
$620$	−38396.3	−2.48715
$621$	20931.6	1.35258
$622$	−1460.49	−0.0941487
$623$	3905.86	0.251180
$624$	0	0
$625$	−2730.82	−0.174773
$626$	157.019	0.0100252
$627$	−6676.51	−0.425254
$628$	3100.99	0.197043
$629$	−291.769	−0.0184954
$630$	−569.981	−0.0360454
$631$	18946.2	1.19531	0.597653	−	0.801755i	$-0.296100\pi$
$631$	18946.2	1.19531	0.597653	+	0.801755i	$0.296100\pi$
$632$	1403.24	0.0883194
$633$	−5028.33	−0.315732
$634$	−1336.15	−0.0836991
$635$	−46373.3	−2.89806
$636$	−1946.40	−0.121352
$637$	0	0
$638$	−1048.95	−0.0650912
$639$	2053.27	0.127114
$640$	−15043.5	−0.929135
$641$	−23586.9	−1.45340	−0.726698	−	0.686957i	$-0.758946\pi$
$641$	−23586.9	−1.45340	−0.726698	+	0.686957i	$0.758946\pi$
$642$	923.263	0.0567575
$643$	−27153.0	−1.66534	−0.832669	−	0.553772i	$-0.813188\pi$
$643$	−27153.0	−1.66534	−0.832669	+	0.553772i	$0.813188\pi$
$644$	−5967.25	−0.365128
$645$	1806.35	0.110272
$646$	2408.70	0.146701
$647$	6856.72	0.416639	0.208319	−	0.978061i	$-0.433201\pi$
$647$	6856.72	0.416639	0.208319	+	0.978061i	$0.433201\pi$
$648$	−1291.82	−0.0783140
$649$	6527.75	0.394817
$650$	0	0
$651$	5533.83	0.333161
$652$	−7350.17	−0.441495
$653$	−8073.89	−0.483853	−0.241926	−	0.970295i	$-0.577779\pi$
$653$	−8073.89	−0.483853	−0.241926	+	0.970295i	$0.577779\pi$
$654$	284.339	0.0170008
$655$	−37960.9	−2.26451
$656$	13534.8	0.805555
$657$	−1574.12	−0.0934739
$658$	−759.734	−0.0450114
$659$	5305.73	0.313629	0.156815	−	0.987628i	$-0.449877\pi$
$659$	5305.73	0.313629	0.156815	+	0.987628i	$0.449877\pi$
$660$	11487.7	0.677512
$661$	25848.3	1.52100	0.760502	−	0.649336i	$-0.224953\pi$
$661$	25848.3	1.52100	0.760502	+	0.649336i	$0.224953\pi$
$662$	−3373.75	−0.198073
$663$	0	0
$664$	2329.85	0.136168
$665$	7825.96	0.456357
$666$	−25.2583	−0.00146958
$667$	−14993.7	−0.870404
$668$	−28735.9	−1.66441
$669$	3903.78	0.225604
$670$	−3319.09	−0.191385
$671$	−14873.6	−0.855722
$672$	1633.18	0.0937521
$673$	−14529.1	−0.832177	−0.416089	−	0.909324i	$-0.636599\pi$
$673$	−14529.1	−0.832177	−0.416089	+	0.909324i	$0.636599\pi$
$674$	−2066.06	−0.118073
$675$	28615.0	1.63169
$676$	0	0
$677$	12058.1	0.684535	0.342267	−	0.939603i	$-0.388805\pi$
$677$	12058.1	0.684535	0.342267	+	0.939603i	$0.388805\pi$
$678$	2043.45	0.115750
$679$	4129.78	0.233412
$680$	−8390.91	−0.473201
$681$	−12764.4	−0.718255
$682$	−2715.00	−0.152438
$683$	30028.8	1.68231	0.841156	−	0.540792i	$-0.181875\pi$
$683$	30028.8	1.68231	0.841156	+	0.540792i	$0.181875\pi$
$684$	−8469.13	−0.473429
$685$	−12251.5	−0.683364
$686$	1565.22	0.0871144
$687$	8565.50	0.475683
$688$	1635.89	0.0906505
$689$	0	0
$690$	−4042.94	−0.223061
$691$	449.696	0.0247572	0.0123786	−	0.999923i	$-0.496060\pi$
$691$	449.696	0.0247572	0.0123786	+	0.999923i	$0.496060\pi$
$692$	−11108.7	−0.610244
$693$	1636.94	0.0897291
$694$	2307.10	0.126191
$695$	12101.8	0.660500
$696$	2724.72	0.148391
$697$	15484.8	0.841506
$698$	−22.0690	−0.00119674
$699$	13747.5	0.743889
$700$	−8157.67	−0.440473
$701$	−26986.0	−1.45399	−0.726994	−	0.686644i	$-0.759083\pi$
$701$	−26986.0	−1.45399	−0.726994	+	0.686644i	$0.759083\pi$
$702$	0	0
$703$	346.801	0.0186057
$704$	9858.46	0.527776
$705$	20906.2	1.11684
$706$	−3971.29	−0.211702
$707$	−1896.37	−0.100877
$708$	−8375.06	−0.444568
$709$	−9098.87	−0.481968	−0.240984	−	0.970529i	$-0.577470\pi$
$709$	−9098.87	−0.481968	−0.240984	+	0.970529i	$0.577470\pi$
$710$	−1194.30	−0.0631285
$711$	−2717.71	−0.143350
$712$	4977.71	0.262005
$713$	−38808.4	−2.03841
$714$	597.312	0.0313079
$715$	0	0
$716$	−9118.62	−0.475948
$717$	−22271.8	−1.16005
$718$	−3147.11	−0.163578
$719$	−6293.55	−0.326439	−0.163220	−	0.986590i	$-0.552188\pi$
$719$	−6293.55	−0.326439	−0.163220	+	0.986590i	$0.552188\pi$
$720$	14204.5	0.735235
$721$	−3155.95	−0.163015
$722$	144.295	0.00743783
$723$	19064.3	0.980648
$724$	8853.72	0.454483
$725$	−20497.5	−1.05001
$726$	−1337.97	−0.0683979
$727$	18070.7	0.921878	0.460939	−	0.887432i	$-0.347513\pi$
$727$	18070.7	0.921878	0.460939	+	0.887432i	$0.347513\pi$
$728$	0	0
$729$	17319.9	0.879944
$730$	915.601	0.0464218
$731$	1871.58	0.0946962
$732$	19082.7	0.963550
$733$	34771.5	1.75214	0.876068	−	0.482188i	$-0.160158\pi$
$733$	34771.5	1.75214	0.876068	+	0.482188i	$0.160158\pi$
$734$	−1755.61	−0.0882842
$735$	−20565.4	−1.03206
$736$	−11453.4	−0.573613
$737$	9532.17	0.476421
$738$	1340.51	0.0668631
$739$	23631.5	1.17632	0.588158	−	0.808746i	$-0.299853\pi$
$739$	23631.5	1.17632	0.588158	+	0.808746i	$0.299853\pi$
$740$	−596.710	−0.0296425
$741$	0	0
$742$	161.324	0.00798167
$743$	32502.8	1.60486	0.802431	−	0.596745i	$-0.203540\pi$
$743$	32502.8	1.60486	0.802431	+	0.596745i	$0.203540\pi$
$744$	7052.42	0.347519
$745$	35178.6	1.72999
$746$	4390.69	0.215489
$747$	−4512.31	−0.221013
$748$	11902.5	0.581816
$749$	3108.04	0.151622
$750$	−1930.87	−0.0940072
$751$	2020.86	0.0981920	0.0490960	−	0.998794i	$-0.484366\pi$
$751$	2020.86	0.0981920	0.0490960	+	0.998794i	$0.484366\pi$
$752$	18933.3	0.918120
$753$	−20710.5	−1.00230
$754$	0	0
$755$	−32111.6	−1.54790
$756$	−6324.56	−0.304262
$757$	12568.2	0.603434	0.301717	−	0.953398i	$-0.402440\pi$
$757$	12568.2	0.603434	0.301717	+	0.953398i	$0.402440\pi$
$758$	3581.73	0.171628
$759$	11611.0	0.555273
$760$	9973.56	0.476025
$761$	−8704.81	−0.414651	−0.207325	−	0.978272i	$-0.566476\pi$
$761$	−8704.81	−0.414651	−0.207325	+	0.978272i	$0.566476\pi$
$762$	4207.01	0.200005
$763$	957.187	0.0454161
$764$	−20941.4	−0.991665
$765$	16251.0	0.768048
$766$	3205.16	0.151184
$767$	0	0
$768$	−11595.0	−0.544790
$769$	−21915.9	−1.02771	−0.513853	−	0.857878i	$-0.671782\pi$
$769$	−21915.9	−1.02771	−0.513853	+	0.857878i	$0.671782\pi$
$770$	−952.140	−0.0445620
$771$	6168.63	0.288143
$772$	15386.5	0.717322
$773$	−23077.5	−1.07379	−0.536896	−	0.843649i	$-0.680403\pi$
$773$	−23077.5	−1.07379	−0.536896	+	0.843649i	$0.680403\pi$
$774$	162.022	0.00752422
$775$	−53054.0	−2.45904
$776$	5263.08	0.243471
$777$	86.0001	0.00397070
$778$	−3851.96	−0.177506
$779$	−18405.5	−0.846528
$780$	0	0
$781$	3429.94	0.157148
$782$	−4188.92	−0.191554
$783$	−15891.5	−0.725309
$784$	−18624.6	−0.848426
$785$	7072.67	0.321572
$786$	3443.83	0.156282
$787$	−16522.4	−0.748362	−0.374181	−	0.927356i	$-0.622076\pi$
$787$	−16522.4	−0.748362	−0.374181	+	0.927356i	$0.622076\pi$
$788$	31356.4	1.41754
$789$	23247.2	1.04895
$790$	1580.78	0.0711918
$791$	6879.00	0.309215
$792$	2086.15	0.0935962
$793$	0	0
$794$	−4939.89	−0.220794
$795$	−4439.30	−0.198045
$796$	32996.1	1.46924
$797$	−11719.4	−0.520855	−0.260427	−	0.965493i	$-0.583863\pi$
$797$	−11719.4	−0.520855	−0.260427	+	0.965493i	$0.583863\pi$
$798$	−709.974	−0.0314948
$799$	21661.2	0.959095
$800$	−15657.7	−0.691978
$801$	−9640.53	−0.425258
$802$	−691.096	−0.0304282
$803$	−2629.53	−0.115559
$804$	−12229.7	−0.536454
$805$	−13610.0	−0.595886
$806$	0	0
$807$	9148.01	0.399040
$808$	−2416.77	−0.105225
$809$	−24096.0	−1.04718	−0.523592	−	0.851969i	$-0.675408\pi$
$809$	−24096.0	−1.04718	−0.523592	+	0.851969i	$0.675408\pi$
$810$	−1455.26	−0.0631267
$811$	16622.6	0.719729	0.359864	−	0.933005i	$-0.382823\pi$
$811$	16622.6	0.719729	0.359864	+	0.933005i	$0.382823\pi$
$812$	4530.42	0.195796
$813$	−10448.7	−0.450739
$814$	−42.1933	−0.00181680
$815$	−16764.1	−0.720516
$816$	−14885.6	−0.638603
$817$	−2224.59	−0.0952613
$818$	−2962.05	−0.126609
$819$	0	0
$820$	31668.7	1.34868
$821$	−38005.5	−1.61559	−0.807797	−	0.589461i	$-0.799340\pi$
$821$	−38005.5	−1.61559	−0.807797	+	0.589461i	$0.799340\pi$
$822$	1111.46	0.0471613
$823$	−15859.5	−0.671722	−0.335861	−	0.941912i	$-0.609027\pi$
$823$	−15859.5	−0.671722	−0.335861	+	0.941912i	$0.609027\pi$
$824$	−4022.01	−0.170040
$825$	15873.1	0.669854
$826$	694.155	0.0292406
$827$	12201.0	0.513023	0.256512	−	0.966541i	$-0.417427\pi$
$827$	12201.0	0.513023	0.256512	+	0.966541i	$0.417427\pi$
$828$	14728.5	0.618177
$829$	−5431.41	−0.227552	−0.113776	−	0.993506i	$-0.536295\pi$
$829$	−5431.41	−0.227552	−0.113776	+	0.993506i	$0.536295\pi$
$830$	2624.62	0.109761
$831$	14139.9	0.590263
$832$	0	0
$833$	−21308.0	−0.886290
$834$	−1097.88	−0.0455834
$835$	−65540.3	−2.71630
$836$	−14147.5	−0.585288
$837$	−41132.2	−1.69861
$838$	−4716.02	−0.194406
$839$	−7960.90	−0.327582	−0.163791	−	0.986495i	$-0.552372\pi$
$839$	−7960.90	−0.327582	−0.163791	+	0.986495i	$0.552372\pi$
$840$	2473.26	0.101590
$841$	−13005.6	−0.533255
$842$	−3457.61	−0.141517
$843$	33611.9	1.37326
$844$	−10655.0	−0.434550
$845$	0	0
$846$	1875.19	0.0762062
$847$	−4504.10	−0.182719
$848$	−4020.35	−0.162806
$849$	−7840.40	−0.316940
$850$	−5726.56	−0.231082
$851$	−603.115	−0.0242944
$852$	−4400.59	−0.176950
$853$	13576.7	0.544969	0.272485	−	0.962160i	$-0.412155\pi$
$853$	13576.7	0.544969	0.272485	+	0.962160i	$0.412155\pi$
$854$	−1581.65	−0.0633756
$855$	−19316.2	−0.772631
$856$	3960.95	0.158157
$857$	−31223.9	−1.24456	−0.622281	−	0.782794i	$-0.713794\pi$
$857$	−31223.9	−1.24456	−0.622281	+	0.782794i	$0.713794\pi$
$858$	0	0
$859$	−11815.8	−0.469323	−0.234661	−	0.972077i	$-0.575398\pi$
$859$	−11815.8	−0.469323	−0.234661	+	0.972077i	$0.575398\pi$
$860$	3827.65	0.151770
$861$	−4564.22	−0.180660
$862$	6175.36	0.244006
$863$	−1790.84	−0.0706384	−0.0353192	−	0.999376i	$-0.511245\pi$
$863$	−1790.84	−0.0706384	−0.0353192	+	0.999376i	$0.511245\pi$
$864$	−12139.2	−0.477992
$865$	−25336.4	−0.995912
$866$	−817.328	−0.0320715
$867$	1072.46	0.0420100
$868$	11726.1	0.458538
$869$	−4539.87	−0.177220
$870$	3069.45	0.119614
$871$	0	0
$872$	1219.86	0.0473734
$873$	−10193.2	−0.395176
$874$	4979.01	0.192698
$875$	−6500.00	−0.251132
$876$	3373.68	0.130121
$877$	43542.5	1.67654	0.838270	−	0.545255i	$-0.183567\pi$
$877$	43542.5	1.67654	0.838270	+	0.545255i	$0.183567\pi$
$878$	−2698.42	−0.103721
$879$	−30489.7	−1.16996
$880$	23728.2	0.908953
$881$	−1020.04	−0.0390080	−0.0195040	−	0.999810i	$-0.506209\pi$
$881$	−1020.04	−0.0390080	−0.0195040	+	0.999810i	$0.506209\pi$
$882$	−1844.62	−0.0704214
$883$	−34781.9	−1.32560	−0.662800	−	0.748797i	$-0.730632\pi$
$883$	−34781.9	−1.32560	−0.662800	+	0.748797i	$0.730632\pi$
$884$	0	0
$885$	−19101.6	−0.725531
$886$	6374.70	0.241718
$887$	49785.1	1.88458	0.942288	−	0.334802i	$-0.108670\pi$
$887$	49785.1	1.88458	0.942288	+	0.334802i	$0.108670\pi$
$888$	109.600	0.00414183
$889$	14162.3	0.534295
$890$	5607.49	0.211195
$891$	4179.40	0.157144
$892$	8272.08	0.310504
$893$	−25746.8	−0.964818
$894$	−3191.41	−0.119392
$895$	−20797.5	−0.776743
$896$	4594.25	0.171298
$897$	0	0
$898$	3088.36	0.114766
$899$	29463.9	1.09308
$900$	20134.9	0.745738
$901$	−4599.60	−0.170072
$902$	2239.29	0.0826611
$903$	−551.656	−0.0203300
$904$	8766.74	0.322541
$905$	20193.3	0.741712
$906$	2913.18	0.106826
$907$	17389.9	0.636627	0.318314	−	0.947985i	$-0.396884\pi$
$907$	17389.9	0.636627	0.318314	+	0.947985i	$0.396884\pi$
$908$	−27047.6	−0.988553
$909$	4680.66	0.170790
$910$	0	0
$911$	20419.5	0.742621	0.371311	−	0.928509i	$-0.378909\pi$
$911$	20419.5	0.742621	0.371311	+	0.928509i	$0.378909\pi$
$912$	17693.2	0.642414
$913$	−7537.71	−0.273233
$914$	−6181.60	−0.223708
$915$	43523.5	1.57250
$916$	18150.2	0.654695
$917$	11593.2	0.417492
$918$	−4439.75	−0.159623
$919$	−33231.8	−1.19283	−0.596417	−	0.802674i	$-0.703410\pi$
$919$	−33231.8	−1.19283	−0.596417	+	0.802674i	$0.703410\pi$
$920$	−17344.8	−0.621567
$921$	13303.4	0.475964
$922$	−6335.43	−0.226297
$923$	0	0
$924$	−3508.31	−0.124908
$925$	−824.502	−0.0293075
$926$	−6930.31	−0.245944
$927$	7789.58	0.275991
$928$	8695.59	0.307594
$929$	−25222.8	−0.890780	−0.445390	−	0.895337i	$-0.646935\pi$
$929$	−25222.8	−0.890780	−0.445390	+	0.895337i	$0.646935\pi$
$930$	7944.70	0.280126
$931$	25327.0	0.891579
$932$	29130.9	1.02383
$933$	−12273.8	−0.430683
$934$	6607.97	0.231498
$935$	27146.9	0.949519
$936$	0	0
$937$	−26979.4	−0.940639	−0.470319	−	0.882496i	$-0.655861\pi$
$937$	−26979.4	−0.940639	−0.470319	+	0.882496i	$0.655861\pi$
$938$	1013.64	0.0352842
$939$	1319.57	0.0458600
$940$	44300.2	1.53714
$941$	7641.67	0.264730	0.132365	−	0.991201i	$-0.457743\pi$
$941$	7641.67	0.264730	0.132365	+	0.991201i	$0.457743\pi$
$942$	−641.635	−0.0221928
$943$	32008.7	1.10535
$944$	−17299.0	−0.596434
$945$	−14424.9	−0.496553
$946$	270.653	0.00930200
$947$	−2869.32	−0.0984587	−0.0492293	−	0.998788i	$-0.515677\pi$
$947$	−2869.32	−0.0984587	−0.0492293	+	0.998788i	$0.515677\pi$
$948$	5824.62	0.199552
$949$	0	0
$950$	6806.67	0.232461
$951$	−11228.8	−0.382881
$952$	2562.56	0.0872407
$953$	12313.6	0.418548	0.209274	−	0.977857i	$-0.432890\pi$
$953$	12313.6	0.418548	0.209274	+	0.977857i	$0.432890\pi$
$954$	−398.184	−0.0135133
$955$	−47762.6	−1.61839
$956$	−47193.8	−1.59661
$957$	−8815.22	−0.297759
$958$	172.110	0.00580442
$959$	3741.57	0.125987
$960$	−28848.0	−0.969861
$961$	46470.7	1.55989
$962$	0	0
$963$	−7671.32	−0.256703
$964$	40397.1	1.34969
$965$	35093.2	1.17066
$966$	1234.70	0.0411241
$967$	−17838.0	−0.593207	−0.296603	−	0.955001i	$-0.595854\pi$
$967$	−17838.0	−0.593207	−0.296603	+	0.955001i	$0.595854\pi$
$968$	−5740.12	−0.190593
$969$	20242.4	0.671084
$970$	5928.97	0.196255
$971$	41525.3	1.37241	0.686206	−	0.727408i	$-0.259275\pi$
$971$	41525.3	1.37241	0.686206	+	0.727408i	$0.259275\pi$
$972$	26037.1	0.859199
$973$	−3695.86	−0.121772
$974$	−4164.32	−0.136995
$975$	0	0
$976$	39416.1	1.29270
$977$	−31654.4	−1.03655	−0.518277	−	0.855213i	$-0.673426\pi$
$977$	−31654.4	−1.03655	−0.518277	+	0.855213i	$0.673426\pi$
$978$	1520.85	0.0497253
$979$	−16104.3	−0.525735
$980$	−43578.0	−1.42046
$981$	−2362.55	−0.0768913
$982$	830.342	0.0269830
$983$	−39913.2	−1.29505	−0.647525	−	0.762045i	$-0.724196\pi$
$983$	−39913.2	−1.29505	−0.647525	+	0.762045i	$0.724196\pi$
$984$	−5816.74	−0.188446
$985$	71516.8	2.31342
$986$	3180.28	0.102719
$987$	−6384.72	−0.205905
$988$	0	0
$989$	3868.74	0.124387
$990$	2350.09	0.0754453
$991$	−2700.94	−0.0865773	−0.0432887	−	0.999063i	$-0.513784\pi$
$991$	−2700.94	−0.0865773	−0.0432887	+	0.999063i	$0.513784\pi$
$992$	22506.9	0.720358
$993$	−28352.6	−0.906085
$994$	364.736	0.0116386
$995$	75256.6	2.39778
$996$	9670.83	0.307663
$997$	−9729.08	−0.309050	−0.154525	−	0.987989i	$-0.549385\pi$
$997$	−9729.08	−0.309050	−0.154525	+	0.987989i	$0.549385\pi$
$998$	5862.08	0.185933
$999$	−639.228	−0.0202445

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.a.f.1.2		2
3.2	odd	2		1521.4.a.t.1.1		2
13.2	odd	12		169.4.e.g.147.2		8
13.3	even	3		13.4.c.b.9.1	yes	4
13.4	even	6		169.4.c.f.146.2		4
13.5	odd	4		169.4.b.e.168.3		4
13.6	odd	12		169.4.e.g.23.3		8
13.7	odd	12		169.4.e.g.23.2		8
13.8	odd	4		169.4.b.e.168.2		4
13.9	even	3		13.4.c.b.3.1	✓	4
13.10	even	6		169.4.c.f.22.2		4
13.11	odd	12		169.4.e.g.147.3		8
13.12	even	2		169.4.a.j.1.1		2
39.29	odd	6		117.4.g.d.100.2		4
39.35	odd	6		117.4.g.d.55.2		4
39.38	odd	2		1521.4.a.l.1.2		2
52.3	odd	6		208.4.i.e.113.1		4
52.35	odd	6		208.4.i.e.81.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.c.b.3.1	✓	4	13.9	even	3
13.4.c.b.9.1	yes	4	13.3	even	3
117.4.g.d.55.2		4	39.35	odd	6
117.4.g.d.100.2		4	39.29	odd	6
169.4.a.f.1.2		2	1.1	even	1	trivial
169.4.a.j.1.1		2	13.12	even	2
169.4.b.e.168.2		4	13.8	odd	4
169.4.b.e.168.3		4	13.5	odd	4
169.4.c.f.22.2		4	13.10	even	6
169.4.c.f.146.2		4	13.4	even	6
169.4.e.g.23.2		8	13.7	odd	12
169.4.e.g.23.3		8	13.6	odd	12
169.4.e.g.147.2		8	13.2	odd	12
169.4.e.g.147.3		8	13.11	odd	12
208.4.i.e.81.1		4	52.35	odd	6
208.4.i.e.113.1		4	52.3	odd	6
1521.4.a.l.1.2		2	39.38	odd	2
1521.4.a.t.1.1		2	3.2	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:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

\(f(q)\)	\(=\)	\(q-0.438447 q^{2} -3.68466 q^{3} -7.80776 q^{4} -17.8078 q^{5} +1.61553 q^{6} +5.43845 q^{7} +6.93087 q^{8} -13.4233 q^{9} +O(q^{10})\)	\(q-0.438447 q^{2} -3.68466 q^{3} -7.80776 q^{4} -17.8078 q^{5} +1.61553 q^{6} +5.43845 q^{7} +6.93087 q^{8} -13.4233 q^{9} +7.80776 q^{10} -22.4233 q^{11} +28.7689 q^{12} -2.38447 q^{14} +65.6155 q^{15} +59.4233 q^{16} +67.9848 q^{17} +5.88540 q^{18} -80.8078 q^{19} +139.039 q^{20} -20.0388 q^{21} +9.83143 q^{22} +140.531 q^{23} -25.5379 q^{24} +192.116 q^{25} +148.946 q^{27} -42.4621 q^{28} -106.693 q^{29} -28.7689 q^{30} -276.155 q^{31} -81.5009 q^{32} +82.6222 q^{33} -29.8078 q^{34} -96.8466 q^{35} +104.806 q^{36} -4.29168 q^{37} +35.4299 q^{38} -123.423 q^{40} +227.769 q^{41} +8.78596 q^{42} +27.5294 q^{43} +175.076 q^{44} +239.039 q^{45} -61.6155 q^{46} +318.617 q^{47} -218.955 q^{48} -313.423 q^{49} -84.2329 q^{50} -250.501 q^{51} -67.6562 q^{53} -65.3050 q^{54} +399.309 q^{55} +37.6932 q^{56} +297.749 q^{57} +46.7793 q^{58} -291.115 q^{59} -512.311 q^{60} +663.311 q^{61} +121.080 q^{62} -73.0019 q^{63} -439.652 q^{64} -36.2255 q^{66} -425.101 q^{67} -530.810 q^{68} -517.810 q^{69} +42.4621 q^{70} -152.963 q^{71} -93.0351 q^{72} +117.268 q^{73} +1.88167 q^{74} -707.884 q^{75} +630.928 q^{76} -121.948 q^{77} +202.462 q^{79} -1058.20 q^{80} -186.386 q^{81} -99.8647 q^{82} +336.155 q^{83} +156.458 q^{84} -1210.66 q^{85} -12.0702 q^{86} +393.128 q^{87} -155.413 q^{88} +718.194 q^{89} -104.806 q^{90} -1097.23 q^{92} +1017.54 q^{93} -139.697 q^{94} +1439.01 q^{95} +300.303 q^{96} +759.368 q^{97} +137.420 q^{98} +300.994 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{2} + 5 q^{3} + 5 q^{4} - 15 q^{5} - 38 q^{6} + 15 q^{7} - 15 q^{8} + 35 q^{9}+O(q^{10}) \) 2 * q - 5 * q^2 + 5 * q^3 + 5 * q^4 - 15 * q^5 - 38 * q^6 + 15 * q^7 - 15 * q^8 + 35 * q^9	\( 2 q - 5 q^{2} + 5 q^{3} + 5 q^{4} - 15 q^{5} - 38 q^{6} + 15 q^{7} - 15 q^{8} + 35 q^{9} - 5 q^{10} + 17 q^{11} + 140 q^{12} - 46 q^{14} + 90 q^{15} + 57 q^{16} + 70 q^{17} - 215 q^{18} - 141 q^{19} + 175 q^{20} + 63 q^{21} - 170 q^{22} + 145 q^{23} - 216 q^{24} + 75 q^{25} + 335 q^{27} + 80 q^{28} + 34 q^{29} - 140 q^{30} - 140 q^{31} + 105 q^{32} + 425 q^{33} - 39 q^{34} - 70 q^{35} + 725 q^{36} - 190 q^{37} + 310 q^{38} - 185 q^{40} + 538 q^{41} - 370 q^{42} + 455 q^{43} + 680 q^{44} + 375 q^{45} - 82 q^{46} + 60 q^{47} - 240 q^{48} - 565 q^{49} + 450 q^{50} - 233 q^{51} + 545 q^{53} - 914 q^{54} + 510 q^{55} - 172 q^{56} - 225 q^{57} - 595 q^{58} - 809 q^{59} - 200 q^{60} + 502 q^{61} - 500 q^{62} + 390 q^{63} - 1271 q^{64} - 1598 q^{66} - 475 q^{67} - 505 q^{68} - 479 q^{69} - 80 q^{70} + 127 q^{71} - 1155 q^{72} + 585 q^{73} + 849 q^{74} - 1725 q^{75} - 140 q^{76} + 255 q^{77} + 240 q^{79} - 1065 q^{80} + 122 q^{81} - 1515 q^{82} + 260 q^{83} + 1220 q^{84} - 1205 q^{85} - 1962 q^{86} + 1615 q^{87} - 1020 q^{88} + 921 q^{89} - 725 q^{90} - 1040 q^{92} + 2200 q^{93} + 1040 q^{94} + 1270 q^{95} + 1920 q^{96} - 415 q^{97} + 1285 q^{98} + 2210 q^{99}+O(q^{100}) \) 2 * q - 5 * q^2 + 5 * q^3 + 5 * q^4 - 15 * q^5 - 38 * q^6 + 15 * q^7 - 15 * q^8 + 35 * q^9 - 5 * q^10 + 17 * q^11 + 140 * q^12 - 46 * q^14 + 90 * q^15 + 57 * q^16 + 70 * q^17 - 215 * q^18 - 141 * q^19 + 175 * q^20 + 63 * q^21 - 170 * q^22 + 145 * q^23 - 216 * q^24 + 75 * q^25 + 335 * q^27 + 80 * q^28 + 34 * q^29 - 140 * q^30 - 140 * q^31 + 105 * q^32 + 425 * q^33 - 39 * q^34 - 70 * q^35 + 725 * q^36 - 190 * q^37 + 310 * q^38 - 185 * q^40 + 538 * q^41 - 370 * q^42 + 455 * q^43 + 680 * q^44 + 375 * q^45 - 82 * q^46 + 60 * q^47 - 240 * q^48 - 565 * q^49 + 450 * q^50 - 233 * q^51 + 545 * q^53 - 914 * q^54 + 510 * q^55 - 172 * q^56 - 225 * q^57 - 595 * q^58 - 809 * q^59 - 200 * q^60 + 502 * q^61 - 500 * q^62 + 390 * q^63 - 1271 * q^64 - 1598 * q^66 - 475 * q^67 - 505 * q^68 - 479 * q^69 - 80 * q^70 + 127 * q^71 - 1155 * q^72 + 585 * q^73 + 849 * q^74 - 1725 * q^75 - 140 * q^76 + 255 * q^77 + 240 * q^79 - 1065 * q^80 + 122 * q^81 - 1515 * q^82 + 260 * q^83 + 1220 * q^84 - 1205 * q^85 - 1962 * q^86 + 1615 * q^87 - 1020 * q^88 + 921 * q^89 - 725 * q^90 - 1040 * q^92 + 2200 * q^93 + 1040 * q^94 + 1270 * q^95 + 1920 * q^96 - 415 * q^97 + 1285 * q^98 + 2210 * q^99

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