LMFDB - Embedded newform 1849.4.a.l.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,4,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, 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("1849.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1849.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:	$109.094531601$
Analytic rank:	$0$
Dimension:	$60$
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.96101 q^{2} -0.861620 q^{3} +0.767595 q^{4} -10.7841 q^{5} +2.55127 q^{6} -8.60318 q^{7} +21.4152 q^{8} -26.2576 q^{9} +O(q^{10})$	\(q-2.96101 q^{2} -0.861620 q^{3} +0.767595 q^{4} -10.7841 q^{5} +2.55127 q^{6} -8.60318 q^{7} +21.4152 q^{8} -26.2576 q^{9} +31.9318 q^{10} +0.602394 q^{11} -0.661376 q^{12} +27.9803 q^{13} +25.4741 q^{14} +9.29177 q^{15} -69.5516 q^{16} +66.6048 q^{17} +77.7491 q^{18} +161.529 q^{19} -8.27780 q^{20} +7.41267 q^{21} -1.78370 q^{22} +45.1591 q^{23} -18.4518 q^{24} -8.70389 q^{25} -82.8501 q^{26} +45.8878 q^{27} -6.60376 q^{28} +47.4008 q^{29} -27.5131 q^{30} -12.0972 q^{31} +34.6211 q^{32} -0.519035 q^{33} -197.218 q^{34} +92.7773 q^{35} -20.1552 q^{36} -188.109 q^{37} -478.289 q^{38} -24.1084 q^{39} -230.943 q^{40} -64.2453 q^{41} -21.9490 q^{42} +0.462395 q^{44} +283.164 q^{45} -133.717 q^{46} -210.428 q^{47} +59.9270 q^{48} -268.985 q^{49} +25.7723 q^{50} -57.3881 q^{51} +21.4776 q^{52} -448.241 q^{53} -135.874 q^{54} -6.49626 q^{55} -184.239 q^{56} -139.176 q^{57} -140.354 q^{58} +167.665 q^{59} +7.13232 q^{60} +773.508 q^{61} +35.8200 q^{62} +225.899 q^{63} +453.899 q^{64} -301.742 q^{65} +1.53687 q^{66} -824.451 q^{67} +51.1255 q^{68} -38.9100 q^{69} -274.715 q^{70} -407.391 q^{71} -562.313 q^{72} +527.455 q^{73} +556.994 q^{74} +7.49945 q^{75} +123.989 q^{76} -5.18250 q^{77} +71.3854 q^{78} -726.737 q^{79} +750.049 q^{80} +669.418 q^{81} +190.231 q^{82} +646.859 q^{83} +5.68993 q^{84} -718.271 q^{85} -40.8415 q^{87} +12.9004 q^{88} -317.950 q^{89} -838.452 q^{90} -240.720 q^{91} +34.6639 q^{92} +10.4232 q^{93} +623.079 q^{94} -1741.94 q^{95} -29.8302 q^{96} +234.335 q^{97} +796.469 q^{98} -15.8174 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 60 q + 15 q^{2} + 37 q^{3} + 213 q^{4} + 51 q^{5} + 22 q^{6} + 54 q^{7} + 204 q^{8} + 387 q^{9}+O(q^{10}) $ 60 * q + 15 * q^2 + 37 * q^3 + 213 * q^4 + 51 * q^5 + 22 * q^6 + 54 * q^7 + 204 * q^8 + 387 * q^9	\( 60 q + 15 q^{2} + 37 q^{3} + 213 q^{4} + 51 q^{5} + 22 q^{6} + 54 q^{7} + 204 q^{8} + 387 q^{9} + 27 q^{10} + 43 q^{11} + 432 q^{12} + 13 q^{13} + 96 q^{14} + 65 q^{15} + 717 q^{16} - 138 q^{17} + 625 q^{18} + 610 q^{19} + 345 q^{20} + 611 q^{21} + 118 q^{22} - 243 q^{23} + 258 q^{24} + 899 q^{25} + 1071 q^{26} + 1609 q^{27} + 46 q^{28} + 773 q^{29} + 375 q^{30} - 97 q^{31} + 1967 q^{32} + 500 q^{33} + 217 q^{34} + 247 q^{35} + 175 q^{36} + 228 q^{37} + 1253 q^{38} + 1493 q^{39} + 2220 q^{40} - 951 q^{41} + 2643 q^{42} - 1378 q^{44} + 1086 q^{45} - 565 q^{46} - 2 q^{47} + 2303 q^{48} + 1264 q^{49} + 3273 q^{50} + 3076 q^{51} - 2825 q^{52} - 39 q^{53} + 5201 q^{54} + 1306 q^{55} + 3683 q^{56} + 1342 q^{57} + 2588 q^{58} - 1065 q^{59} + 2803 q^{60} + 2999 q^{61} + 5569 q^{62} + 2377 q^{63} + 2082 q^{64} + 5578 q^{65} + 3338 q^{66} + 961 q^{67} - 3754 q^{68} + 1817 q^{69} + 2738 q^{70} + 8003 q^{71} + 1412 q^{72} - 1011 q^{73} - 1413 q^{74} + 7457 q^{75} + 5516 q^{76} + 4052 q^{77} + 1091 q^{78} - 4422 q^{79} + 1610 q^{80} + 2108 q^{81} + 4676 q^{82} - 297 q^{83} - 54 q^{84} + 4333 q^{85} + 1377 q^{87} + 3652 q^{88} + 2480 q^{89} - 1414 q^{90} + 4551 q^{91} - 3286 q^{92} + 4 q^{93} + 4609 q^{94} - 835 q^{95} + 5864 q^{96} + 3785 q^{97} + 753 q^{98} + 5072 q^{99}+O(q^{100}) \) 60 * q + 15 * q^2 + 37 * q^3 + 213 * q^4 + 51 * q^5 + 22 * q^6 + 54 * q^7 + 204 * q^8 + 387 * q^9 + 27 * q^10 + 43 * q^11 + 432 * q^12 + 13 * q^13 + 96 * q^14 + 65 * q^15 + 717 * q^16 - 138 * q^17 + 625 * q^18 + 610 * q^19 + 345 * q^20 + 611 * q^21 + 118 * q^22 - 243 * q^23 + 258 * q^24 + 899 * q^25 + 1071 * q^26 + 1609 * q^27 + 46 * q^28 + 773 * q^29 + 375 * q^30 - 97 * q^31 + 1967 * q^32 + 500 * q^33 + 217 * q^34 + 247 * q^35 + 175 * q^36 + 228 * q^37 + 1253 * q^38 + 1493 * q^39 + 2220 * q^40 - 951 * q^41 + 2643 * q^42 - 1378 * q^44 + 1086 * q^45 - 565 * q^46 - 2 * q^47 + 2303 * q^48 + 1264 * q^49 + 3273 * q^50 + 3076 * q^51 - 2825 * q^52 - 39 * q^53 + 5201 * q^54 + 1306 * q^55 + 3683 * q^56 + 1342 * q^57 + 2588 * q^58 - 1065 * q^59 + 2803 * q^60 + 2999 * q^61 + 5569 * q^62 + 2377 * q^63 + 2082 * q^64 + 5578 * q^65 + 3338 * q^66 + 961 * q^67 - 3754 * q^68 + 1817 * q^69 + 2738 * q^70 + 8003 * q^71 + 1412 * q^72 - 1011 * q^73 - 1413 * q^74 + 7457 * q^75 + 5516 * q^76 + 4052 * q^77 + 1091 * q^78 - 4422 * q^79 + 1610 * q^80 + 2108 * q^81 + 4676 * q^82 - 297 * q^83 - 54 * q^84 + 4333 * q^85 + 1377 * q^87 + 3652 * q^88 + 2480 * q^89 - 1414 * q^90 + 4551 * q^91 - 3286 * q^92 + 4 * q^93 + 4609 * q^94 - 835 * q^95 + 5864 * q^96 + 3785 * q^97 + 753 * q^98 + 5072 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.96101	−1.04688	−0.523438	−	0.852064i	$-0.675351\pi$
$2$	−2.96101	−1.04688	−0.523438	+	0.852064i	$0.675351\pi$
$3$	−0.861620	−0.165819	−0.0829095	−	0.996557i	$-0.526421\pi$
$3$	−0.861620	−0.165819	−0.0829095	+	0.996557i	$0.526421\pi$
$4$	0.767595	0.0959494
$5$	−10.7841	−0.964556	−0.482278	−	0.876018i	$-0.660191\pi$
$5$	−10.7841	−0.964556	−0.482278	+	0.876018i	$0.660191\pi$
$6$	2.55127	0.173592
$7$	−8.60318	−0.464528	−0.232264	−	0.972653i	$-0.574613\pi$
$7$	−8.60318	−0.464528	−0.232264	+	0.972653i	$0.574613\pi$
$8$	21.4152	0.946429
$9$	−26.2576	−0.972504
$10$	31.9318	1.00977
$11$	0.602394	0.0165117	0.00825585	−	0.999966i	$-0.497372\pi$
$11$	0.602394	0.0165117	0.00825585	+	0.999966i	$0.497372\pi$
$12$	−0.661376	−0.0159102
$13$	27.9803	0.596950	0.298475	−	0.954417i	$-0.403522\pi$
$13$	27.9803	0.596950	0.298475	+	0.954417i	$0.403522\pi$
$14$	25.4741	0.486303
$15$	9.29177	0.159942
$16$	−69.5516	−1.08674
$17$	66.6048	0.950237	0.475119	−	0.879922i	$-0.342405\pi$
$17$	66.6048	0.950237	0.475119	+	0.879922i	$0.342405\pi$
$18$	77.7491	1.01809
$19$	161.529	1.95038	0.975191	−	0.221367i	$-0.0710518\pi$
$19$	161.529	1.95038	0.975191	+	0.221367i	$0.0710518\pi$
$20$	−8.27780	−0.0925486
$21$	7.41267	0.0770275
$22$	−1.78370	−0.0172857
$23$	45.1591	0.409406	0.204703	−	0.978824i	$-0.434377\pi$
$23$	45.1591	0.409406	0.204703	+	0.978824i	$0.434377\pi$
$24$	−18.4518	−0.156936
$25$	−8.70389	−0.0696311
$26$	−82.8501	−0.624933
$27$	45.8878	0.327079
$28$	−6.60376	−0.0445712
$29$	47.4008	0.303521	0.151761	−	0.988417i	$-0.451506\pi$
$29$	47.4008	0.303521	0.151761	+	0.988417i	$0.451506\pi$
$30$	−27.5131	−0.167439
$31$	−12.0972	−0.0700879	−0.0350439	−	0.999386i	$-0.511157\pi$
$31$	−12.0972	−0.0700879	−0.0350439	+	0.999386i	$0.511157\pi$
$32$	34.6211	0.191256
$33$	−0.519035	−0.00273795
$34$	−197.218	−0.994781
$35$	92.7773	0.448063
$36$	−20.1552	−0.0933112
$37$	−188.109	−0.835810	−0.417905	−	0.908491i	$-0.637236\pi$
$37$	−188.109	−0.835810	−0.417905	+	0.908491i	$0.637236\pi$
$38$	−478.289	−2.04181
$39$	−24.1084	−0.0989856
$40$	−230.943	−0.912884
$41$	−64.2453	−0.244718	−0.122359	−	0.992486i	$-0.539046\pi$
$41$	−64.2453	−0.244718	−0.122359	+	0.992486i	$0.539046\pi$
$42$	−21.9490	−0.0806383
$43$	0	0
$43$	0	0
$44$	0.462395	0.00158429
$45$	283.164	0.938035
$46$	−133.717	−0.428597
$47$	−210.428	−0.653065	−0.326532	−	0.945186i	$-0.605880\pi$
$47$	−210.428	−0.653065	−0.326532	+	0.945186i	$0.605880\pi$
$48$	59.9270	0.180203
$49$	−268.985	−0.784214
$50$	25.7723	0.0728951
$51$	−57.3881	−0.157567
$52$	21.4776	0.0572770
$53$	−448.241	−1.16171	−0.580855	−	0.814007i	$-0.697282\pi$
$53$	−448.241	−1.16171	−0.580855	+	0.814007i	$0.697282\pi$
$54$	−135.874	−0.342411
$55$	−6.49626	−0.0159265
$56$	−184.239	−0.439643
$57$	−139.176	−0.323410
$58$	−140.354	−0.317749
$59$	167.665	0.369968	0.184984	−	0.982742i	$-0.440777\pi$
$59$	167.665	0.369968	0.184984	+	0.982742i	$0.440777\pi$
$60$	7.13232	0.0153463
$61$	773.508	1.62357	0.811783	−	0.583959i	$-0.198497\pi$
$61$	773.508	1.62357	0.811783	+	0.583959i	$0.198497\pi$
$62$	35.8200	0.0733733
$63$	225.899	0.451755
$64$	453.899	0.886521
$65$	−301.742	−0.575792
$66$	1.53687	0.00286630
$67$	−824.451	−1.50332	−0.751662	−	0.659548i	$-0.770748\pi$
$67$	−824.451	−1.50332	−0.751662	+	0.659548i	$0.770748\pi$
$68$	51.1255	0.0911747
$69$	−38.9100	−0.0678872
$70$	−274.715	−0.469067
$71$	−407.391	−0.680964	−0.340482	−	0.940251i	$-0.610590\pi$
$71$	−407.391	−0.680964	−0.340482	+	0.940251i	$0.610590\pi$
$72$	−562.313	−0.920406
$73$	527.455	0.845670	0.422835	−	0.906207i	$-0.361035\pi$
$73$	527.455	0.845670	0.422835	+	0.906207i	$0.361035\pi$
$74$	556.994	0.874990
$75$	7.49945	0.0115462
$76$	123.989	0.187138
$77$	−5.18250	−0.00767014
$78$	71.3854	0.103626
$79$	−726.737	−1.03499	−0.517496	−	0.855686i	$-0.673136\pi$
$79$	−726.737	−1.03499	−0.517496	+	0.855686i	$0.673136\pi$
$80$	750.049	1.04822
$81$	669.418	0.918268
$82$	190.231	0.256189
$83$	646.859	0.855446	0.427723	−	0.903910i	$-0.359316\pi$
$83$	646.859	0.855446	0.427723	+	0.903910i	$0.359316\pi$
$84$	5.68993	0.00739074
$85$	−718.271	−0.916558
$86$	0	0
$87$	−40.8415	−0.0503296
$88$	12.9004	0.0156271
$89$	−317.950	−0.378681	−0.189341	−	0.981911i	$-0.560635\pi$
$89$	−317.950	−0.378681	−0.189341	+	0.981911i	$0.560635\pi$
$90$	−838.452	−0.982006
$91$	−240.720	−0.277300
$92$	34.6639	0.0392822
$93$	10.4232	0.0116219
$94$	623.079	0.683678
$95$	−1741.94	−1.88125
$96$	−29.8302	−0.0317139
$97$	234.335	0.245289	0.122645	−	0.992451i	$-0.460862\pi$
$97$	234.335	0.245289	0.122645	+	0.992451i	$0.460862\pi$
$98$	796.469	0.820975
$99$	−15.8174	−0.0160577
$100$	−6.68106	−0.00668106
$101$	−1580.72	−1.55730	−0.778652	−	0.627456i	$-0.784096\pi$
$101$	−1580.72	−1.55730	−0.778652	+	0.627456i	$0.784096\pi$
$102$	169.927	0.164953
$103$	−731.479	−0.699755	−0.349877	−	0.936795i	$-0.613777\pi$
$103$	−731.479	−0.699755	−0.349877	+	0.936795i	$0.613777\pi$
$104$	599.206	0.564971
$105$	−79.9388	−0.0742974
$106$	1327.25	1.21617
$107$	1578.91	1.42653	0.713265	−	0.700895i	$-0.247216\pi$
$107$	1578.91	1.42653	0.713265	+	0.700895i	$0.247216\pi$
$108$	35.2233	0.0313830
$109$	1429.34	1.25601	0.628007	−	0.778208i	$-0.283871\pi$
$109$	1429.34	1.25601	0.628007	+	0.778208i	$0.283871\pi$
$110$	19.2355	0.0166730
$111$	162.079	0.138593
$112$	598.364	0.504823
$113$	−862.538	−0.718060	−0.359030	−	0.933326i	$-0.616892\pi$
$113$	−862.538	−0.718060	−0.359030	+	0.933326i	$0.616892\pi$
$114$	412.103	0.338570
$115$	−486.999	−0.394895
$116$	36.3846	0.0291227
$117$	−734.697	−0.580536
$118$	−496.458	−0.387311
$119$	−573.013	−0.441412
$120$	198.986	0.151373
$121$	−1330.64	−0.999727
$122$	−2290.37	−1.69967
$123$	55.3551	0.0405788
$124$	−9.28576	−0.00672489
$125$	1441.87	1.03172
$126$	−668.890	−0.472932
$127$	209.146	0.146132	0.0730658	−	0.997327i	$-0.476722\pi$
$127$	209.146	0.146132	0.0730658	+	0.997327i	$0.476722\pi$
$128$	−1620.97	−1.11933
$129$	0	0
$130$	893.461	0.602783
$131$	2121.26	1.41477	0.707386	−	0.706828i	$-0.249874\pi$
$131$	2121.26	1.41477	0.707386	+	0.706828i	$0.249874\pi$
$132$	−0.398409	−0.000262705 0
$133$	−1389.66	−0.906007
$134$	2441.21	1.57379
$135$	−494.858	−0.315486
$136$	1426.36	0.899332
$137$	−2339.64	−1.45905	−0.729523	−	0.683956i	$-0.760258\pi$
$137$	−2339.64	−1.45905	−0.729523	+	0.683956i	$0.760258\pi$
$138$	115.213	0.0710695
$139$	−2251.97	−1.37417	−0.687085	−	0.726577i	$-0.741110\pi$
$139$	−2251.97	−1.37417	−0.687085	+	0.726577i	$0.741110\pi$
$140$	71.2154	0.0429914
$141$	181.309	0.108291
$142$	1206.29	0.712884
$143$	16.8552	0.00985666
$144$	1826.26	1.05686
$145$	−511.174	−0.292763
$146$	−1561.80	−0.885311
$147$	231.763	0.130037
$148$	−144.392	−0.0801955
$149$	1708.00	0.939091	0.469545	−	0.882908i	$-0.344418\pi$
$149$	1708.00	0.939091	0.469545	+	0.882908i	$0.344418\pi$
$150$	−22.2060	−0.0120874
$151$	562.070	0.302918	0.151459	−	0.988464i	$-0.451603\pi$
$151$	562.070	0.302918	0.151459	+	0.988464i	$0.451603\pi$
$152$	3459.18	1.84590
$153$	−1748.88	−0.924110
$154$	15.3455	0.00802969
$155$	130.457	0.0676037
$156$	−18.5055	−0.00949761
$157$	2593.63	1.31843	0.659216	−	0.751953i	$-0.270888\pi$
$157$	2593.63	1.31843	0.659216	+	0.751953i	$0.270888\pi$
$158$	2151.88	1.08351
$159$	386.214	0.192634
$160$	−373.356	−0.184478
$161$	−388.512	−0.190180
$162$	−1982.15	−0.961313
$163$	1292.48	0.621073	0.310536	−	0.950562i	$-0.399491\pi$
$163$	1292.48	0.621073	0.310536	+	0.950562i	$0.399491\pi$
$164$	−49.3144	−0.0234805
$165$	5.59731	0.00264091
$166$	−1915.36	−0.895546
$167$	−3979.37	−1.84391	−0.921956	−	0.387295i	$-0.873410\pi$
$167$	−3979.37	−1.84391	−0.921956	+	0.387295i	$0.873410\pi$
$168$	158.744	0.0729011
$169$	−1414.10	−0.643651
$170$	2126.81	0.959522
$171$	−4241.36	−1.89675
$172$	0	0
$173$	3543.99	1.55748	0.778742	−	0.627344i	$-0.215858\pi$
$173$	3543.99	1.55748	0.778742	+	0.627344i	$0.215858\pi$
$174$	120.932	0.0526888
$175$	74.8811	0.0323456
$176$	−41.8974	−0.0179440
$177$	−144.464	−0.0613477
$178$	941.454	0.396432
$179$	−3368.92	−1.40673	−0.703365	−	0.710828i	$-0.748320\pi$
$179$	−3368.92	−1.40673	−0.703365	+	0.710828i	$0.748320\pi$
$180$	217.355	0.0900039
$181$	−3009.59	−1.23592	−0.617960	−	0.786210i	$-0.712040\pi$
$181$	−3009.59	−1.23592	−0.617960	+	0.786210i	$0.712040\pi$
$182$	712.774	0.290299
$183$	−666.470	−0.269218
$184$	967.094	0.387473
$185$	2028.58	0.806186
$186$	−30.8632	−0.0121667
$187$	40.1223	0.0156900
$188$	−161.523	−0.0626612
$189$	−394.781	−0.151937
$190$	5157.90	1.96944
$191$	1771.05	0.670934	0.335467	−	0.942052i	$-0.391106\pi$
$191$	1771.05	0.670934	0.335467	+	0.942052i	$0.391106\pi$
$192$	−391.089	−0.147002
$193$	3570.23	1.33156	0.665779	−	0.746149i	$-0.268099\pi$
$193$	3570.23	1.33156	0.665779	+	0.746149i	$0.268099\pi$
$194$	−693.868	−0.256788
$195$	259.987	0.0954772
$196$	−206.472	−0.0752448
$197$	3939.57	1.42479	0.712393	−	0.701780i	$-0.247611\pi$
$197$	3939.57	1.42479	0.712393	+	0.701780i	$0.247611\pi$
$198$	46.8356	0.0168104
$199$	−2566.28	−0.914164	−0.457082	−	0.889425i	$-0.651105\pi$
$199$	−2566.28	−0.914164	−0.457082	+	0.889425i	$0.651105\pi$
$200$	−186.396	−0.0659009
$201$	710.364	0.249280
$202$	4680.53	1.63030
$203$	−407.798	−0.140994
$204$	−44.0508	−0.0151185
$205$	692.826	0.236044
$206$	2165.92	0.732557
$207$	−1185.77	−0.398149
$208$	−1946.08	−0.648731
$209$	97.3040	0.0322041
$210$	236.700	0.0777802
$211$	3792.42	1.23735	0.618675	−	0.785647i	$-0.287670\pi$
$211$	3792.42	1.23735	0.618675	+	0.785647i	$0.287670\pi$
$212$	−344.068	−0.111465
$213$	351.016	0.112917
$214$	−4675.16	−1.49340
$215$	0	0
$216$	982.699	0.309557
$217$	104.074	0.0325578
$218$	−4232.28	−1.31489
$219$	−454.466	−0.140228
$220$	−4.98650	−0.00152813
$221$	1863.62	0.567244
$222$	−479.917	−0.145090
$223$	1523.90	0.457613	0.228807	−	0.973472i	$-0.426518\pi$
$223$	1523.90	0.457613	0.228807	+	0.973472i	$0.426518\pi$
$224$	−297.852	−0.0888439
$225$	228.543	0.0677165
$226$	2553.99	0.751720
$227$	−3653.97	−1.06838	−0.534191	−	0.845364i	$-0.679384\pi$
$227$	−3653.97	−1.06838	−0.534191	+	0.845364i	$0.679384\pi$
$228$	−106.831	−0.0310310
$229$	406.283	0.117240	0.0586200	−	0.998280i	$-0.481330\pi$
$229$	406.283	0.117240	0.0586200	+	0.998280i	$0.481330\pi$
$230$	1442.01	0.413406
$231$	4.46535	0.00127186
$232$	1015.10	0.287261
$233$	2196.77	0.617662	0.308831	−	0.951117i	$-0.400062\pi$
$233$	2196.77	0.617662	0.308831	+	0.951117i	$0.400062\pi$
$234$	2175.45	0.607749
$235$	2269.27	0.629918
$236$	128.699	0.0354982
$237$	626.171	0.171621
$238$	1696.70	0.462104
$239$	−876.157	−0.237129	−0.118565	−	0.992946i	$-0.537829\pi$
$239$	−876.157	−0.237129	−0.118565	+	0.992946i	$0.537829\pi$
$240$	−646.257	−0.173816
$241$	−5816.03	−1.55454	−0.777268	−	0.629170i	$-0.783395\pi$
$241$	−5816.03	−1.55454	−0.777268	+	0.629170i	$0.783395\pi$
$242$	3940.03	1.04659
$243$	−1815.76	−0.479345
$244$	593.741	0.155780
$245$	2900.76	0.756418
$246$	−163.907	−0.0424810
$247$	4519.63	1.16428
$248$	−259.065	−0.0663332
$249$	−557.347	−0.141849
$250$	−4269.40	−1.08008
$251$	−4621.37	−1.16214	−0.581072	−	0.813852i	$-0.697367\pi$
$251$	−4621.37	−1.16214	−0.581072	+	0.813852i	$0.697367\pi$
$252$	173.399	0.0433456
$253$	27.2036	0.00675998
$254$	−619.284	−0.152982
$255$	618.877	0.151983
$256$	1168.52	0.285283
$257$	731.225	0.177481	0.0887404	−	0.996055i	$-0.471716\pi$
$257$	731.225	0.177481	0.0887404	+	0.996055i	$0.471716\pi$
$258$	0	0
$259$	1618.34	0.388257
$260$	−231.616	−0.0552469
$261$	−1244.63	−0.295176
$262$	−6281.07	−1.48109
$263$	4632.63	1.08616	0.543080	−	0.839681i	$-0.317258\pi$
$263$	4632.63	1.08616	0.543080	+	0.839681i	$0.317258\pi$
$264$	−11.1153	−0.00259128
$265$	4833.86	1.12054
$266$	4114.80	0.948477
$267$	273.952	0.0627925
$268$	−632.844	−0.144243
$269$	6447.04	1.46128	0.730638	−	0.682765i	$-0.239223\pi$
$269$	6447.04	1.46128	0.730638	+	0.682765i	$0.239223\pi$
$270$	1465.28	0.330274
$271$	−4561.23	−1.02242	−0.511209	−	0.859457i	$-0.670802\pi$
$271$	−4561.23	−1.02242	−0.511209	+	0.859457i	$0.670802\pi$
$272$	−4632.47	−1.03266
$273$	207.409	0.0459816
$274$	6927.72	1.52744
$275$	−5.24317	−0.00114973
$276$	−29.8672	−0.00651374
$277$	−7483.37	−1.62322	−0.811610	−	0.584199i	$-0.801409\pi$
$277$	−7483.37	−1.62322	−0.811610	+	0.584199i	$0.801409\pi$
$278$	6668.11	1.43859
$279$	317.644	0.0681607
$280$	1986.85	0.424060
$281$	−2026.60	−0.430237	−0.215119	−	0.976588i	$-0.569014\pi$
$281$	−2026.60	−0.430237	−0.215119	+	0.976588i	$0.569014\pi$
$282$	−536.858	−0.113367
$283$	3340.99	0.701771	0.350885	−	0.936418i	$-0.385881\pi$
$283$	3340.99	0.701771	0.350885	+	0.936418i	$0.385881\pi$
$284$	−312.711	−0.0653380
$285$	1500.89	0.311947
$286$	−49.9084	−0.0103187
$287$	552.714	0.113678
$288$	−909.067	−0.185998
$289$	−476.801	−0.0970488
$290$	1513.59	0.306487
$291$	−201.908	−0.0406736
$292$	404.872	0.0811415
$293$	3337.46	0.665448	0.332724	−	0.943024i	$-0.392032\pi$
$293$	3337.46	0.665448	0.332724	+	0.943024i	$0.392032\pi$
$294$	−686.254	−0.136133
$295$	−1808.11	−0.356855
$296$	−4028.41	−0.791035
$297$	27.6426	0.00540062
$298$	−5057.40	−0.983112
$299$	1263.57	0.244395
$300$	5.75654	0.00110785
$301$	0	0
$302$	−1664.30	−0.317117
$303$	1361.98	0.258230
$304$	−11234.6	−2.11956
$305$	−8341.56	−1.56602
$306$	5178.46	0.967428
$307$	−8083.45	−1.50276	−0.751379	−	0.659871i	$-0.770611\pi$
$307$	−8083.45	−1.50276	−0.751379	+	0.659871i	$0.770611\pi$
$308$	−3.97806	−0.000735946 0
$309$	630.257	0.116033
$310$	−386.285	−0.0707727
$311$	884.498	0.161271	0.0806355	−	0.996744i	$-0.474305\pi$
$311$	884.498	0.161271	0.0806355	+	0.996744i	$0.474305\pi$
$312$	−516.288	−0.0936828
$313$	−594.807	−0.107414	−0.0537069	−	0.998557i	$-0.517104\pi$
$313$	−594.807	−0.107414	−0.0537069	+	0.998557i	$0.517104\pi$
$314$	−7679.76	−1.38024
$315$	−2436.11	−0.435743
$316$	−557.840	−0.0993068
$317$	2072.18	0.367147	0.183573	−	0.983006i	$-0.441234\pi$
$317$	2072.18	0.367147	0.183573	+	0.983006i	$0.441234\pi$
$318$	−1143.58	−0.201664
$319$	28.5540	0.00501165
$320$	−4894.88	−0.855100
$321$	−1360.42	−0.236546
$322$	1150.39	0.199095
$323$	10758.6	1.85333
$324$	513.842	0.0881073
$325$	−243.538	−0.0415663
$326$	−3827.05	−0.650186
$327$	−1231.54	−0.208271
$328$	−1375.83	−0.231608
$329$	1810.35	0.303367
$330$	−16.5737	−0.00276470
$331$	3227.97	0.536027	0.268014	−	0.963415i	$-0.413633\pi$
$331$	3227.97	0.536027	0.268014	+	0.963415i	$0.413633\pi$
$332$	496.526	0.0820795
$333$	4939.30	0.812829
$334$	11783.0	1.93035
$335$	8890.93	1.45004
$336$	−515.563	−0.0837091
$337$	−4531.78	−0.732527	−0.366264	−	0.930511i	$-0.619363\pi$
$337$	−4531.78	−0.732527	−0.366264	+	0.930511i	$0.619363\pi$
$338$	4187.17	0.673823
$339$	743.181	0.119068
$340$	−551.341	−0.0879431
$341$	−7.28729	−0.00115727
$342$	12558.7	1.98567
$343$	5265.02	0.828817
$344$	0	0
$345$	419.609	0.0654811
$346$	−10493.8	−1.63049
$347$	7899.51	1.22210	0.611049	−	0.791593i	$-0.290748\pi$
$347$	7899.51	1.22210	0.611049	+	0.791593i	$0.290748\pi$
$348$	−31.3498	−0.00482909
$349$	−82.4130	−0.0126403	−0.00632015	−	0.999980i	$-0.502012\pi$
$349$	−82.4130	−0.0126403	−0.00632015	+	0.999980i	$0.502012\pi$
$350$	−221.724	−0.0338618
$351$	1283.96	0.195249
$352$	20.8555	0.00315797
$353$	−614.139	−0.0925986	−0.0462993	−	0.998928i	$-0.514743\pi$
$353$	−614.139	−0.0925986	−0.0462993	+	0.998928i	$0.514743\pi$
$354$	427.759	0.0642235
$355$	4393.33	0.656828
$356$	−244.057	−0.0363342
$357$	493.720	0.0731944
$358$	9975.41	1.47267
$359$	13278.2	1.95207	0.976037	−	0.217607i	$-0.0698251\pi$
$359$	13278.2	1.95207	0.976037	+	0.217607i	$0.0698251\pi$
$360$	6064.02	0.887783
$361$	19232.5	2.80399
$362$	8911.45	1.29385
$363$	1146.50	0.165774
$364$	−184.775	−0.0266068
$365$	−5688.11	−0.815696
$366$	1973.43	0.281838
$367$	−2031.01	−0.288877	−0.144438	−	0.989514i	$-0.546138\pi$
$367$	−2031.01	−0.288877	−0.144438	+	0.989514i	$0.546138\pi$
$368$	−3140.89	−0.444919
$369$	1686.93	0.237989
$370$	−6006.66	−0.843977
$371$	3856.30	0.539647
$372$	8.00080	0.00111511
$373$	9446.51	1.31132	0.655659	−	0.755057i	$-0.272391\pi$
$373$	9446.51	1.31132	0.655659	+	0.755057i	$0.272391\pi$
$374$	−118.803	−0.0164255
$375$	−1242.35	−0.171079
$376$	−4506.36	−0.618080
$377$	1326.29	0.181187
$378$	1168.95	0.159059
$379$	9460.95	1.28226	0.641130	−	0.767432i	$-0.278466\pi$
$379$	9460.95	1.28226	0.641130	+	0.767432i	$0.278466\pi$
$380$	−1337.10	−0.180505
$381$	−180.204	−0.0242314
$382$	−5244.09	−0.702385
$383$	−4923.81	−0.656906	−0.328453	−	0.944520i	$-0.606527\pi$
$383$	−4923.81	−0.656906	−0.328453	+	0.944520i	$0.606527\pi$
$384$	1396.66	0.185607
$385$	55.8885	0.00739829
$386$	−10571.5	−1.39398
$387$	0	0
$388$	179.874	0.0235354
$389$	753.932	0.0982671	0.0491335	−	0.998792i	$-0.484354\pi$
$389$	753.932	0.0982671	0.0491335	+	0.998792i	$0.484354\pi$
$390$	−769.825	−0.0999528
$391$	3007.82	0.389033
$392$	−5760.39	−0.742203
$393$	−1827.72	−0.234596
$394$	−11665.1	−1.49157
$395$	7837.18	0.998307
$396$	−12.1414	−0.00154073
$397$	12570.7	1.58919	0.794594	−	0.607141i	$-0.207684\pi$
$397$	12570.7	1.58919	0.794594	+	0.607141i	$0.207684\pi$
$398$	7598.78	0.957016
$399$	1197.36	0.150233
$400$	605.369	0.0756711
$401$	−4024.99	−0.501243	−0.250621	−	0.968085i	$-0.580635\pi$
$401$	−4024.99	−0.501243	−0.250621	+	0.968085i	$0.580635\pi$
$402$	−2103.40	−0.260965
$403$	−338.484	−0.0418389
$404$	−1213.35	−0.149422
$405$	−7219.04	−0.885721
$406$	1207.49	0.147603
$407$	−113.316	−0.0138006
$408$	−1228.98	−0.149126
$409$	11236.0	1.35840	0.679200	−	0.733953i	$-0.262327\pi$
$409$	11236.0	1.35840	0.679200	+	0.733953i	$0.262327\pi$
$410$	−2051.47	−0.247109
$411$	2015.89	0.241937
$412$	−561.479	−0.0671410
$413$	−1442.45	−0.171861
$414$	3511.08	0.416812
$415$	−6975.77	−0.825126
$416$	968.710	0.114170
$417$	1940.34	0.227863
$418$	−288.118	−0.0337137
$419$	−2749.76	−0.320607	−0.160304	−	0.987068i	$-0.551247\pi$
$419$	−2749.76	−0.320607	−0.160304	+	0.987068i	$0.551247\pi$
$420$	−61.3606	−0.00712879
$421$	3860.90	0.446957	0.223479	−	0.974709i	$-0.428259\pi$
$421$	3860.90	0.446957	0.223479	+	0.974709i	$0.428259\pi$
$422$	−11229.4	−1.29535
$423$	5525.33	0.635108
$424$	−9599.20	−1.09948
$425$	−579.721	−0.0661661
$426$	−1039.36	−0.118210
$427$	−6654.63	−0.754192
$428$	1211.96	0.136875
$429$	−14.5228	−0.00163442
$430$	0	0
$431$	11780.5	1.31659	0.658293	−	0.752762i	$-0.271279\pi$
$431$	11780.5	1.31659	0.658293	+	0.752762i	$0.271279\pi$
$432$	−3191.57	−0.355450
$433$	−9265.58	−1.02835	−0.514175	−	0.857686i	$-0.671902\pi$
$433$	−9265.58	−1.02835	−0.514175	+	0.857686i	$0.671902\pi$
$434$	−308.166	−0.0340840
$435$	440.438	0.0485457
$436$	1097.15	0.120514
$437$	7294.50	0.798497
$438$	1345.68	0.146801
$439$	1247.25	0.135600	0.0677998	−	0.997699i	$-0.478402\pi$
$439$	1247.25	0.135600	0.0677998	+	0.997699i	$0.478402\pi$
$440$	−139.119	−0.0150733
$441$	7062.91	0.762651
$442$	−5518.22	−0.593834
$443$	1302.97	0.139743	0.0698714	−	0.997556i	$-0.477741\pi$
$443$	1302.97	0.139743	0.0698714	+	0.997556i	$0.477741\pi$
$444$	124.411	0.0132979
$445$	3428.79	0.365259
$446$	−4512.28	−0.479064
$447$	−1471.64	−0.155719
$448$	−3904.97	−0.411814
$449$	814.381	0.0855970	0.0427985	−	0.999084i	$-0.486373\pi$
$449$	814.381	0.0855970	0.0427985	+	0.999084i	$0.486373\pi$
$450$	−676.720	−0.0708908
$451$	−38.7010	−0.00404071
$452$	−662.080	−0.0688974
$453$	−484.291	−0.0502295
$454$	10819.5	1.11846
$455$	2595.94	0.267471
$456$	−2980.50	−0.306085
$457$	10846.3	1.11022	0.555108	−	0.831778i	$-0.312677\pi$
$457$	10846.3	1.11022	0.555108	+	0.831778i	$0.312677\pi$
$458$	−1203.01	−0.122736
$459$	3056.35	0.310802
$460$	−373.818	−0.0378899
$461$	−17073.1	−1.72489	−0.862446	−	0.506149i	$-0.831069\pi$
$461$	−17073.1	−1.72489	−0.862446	+	0.506149i	$0.831069\pi$
$462$	−13.2220	−0.00133147
$463$	−777.106	−0.0780026	−0.0390013	−	0.999239i	$-0.512418\pi$
$463$	−777.106	−0.0780026	−0.0390013	+	0.999239i	$0.512418\pi$
$464$	−3296.80	−0.329850
$465$	−112.405	−0.0112100
$466$	−6504.66	−0.646615
$467$	−4755.48	−0.471215	−0.235608	−	0.971848i	$-0.575708\pi$
$467$	−4755.48	−0.471215	−0.235608	+	0.971848i	$0.575708\pi$
$468$	−563.950	−0.0557021
$469$	7092.90	0.698336
$470$	−6719.33	−0.659446
$471$	−2234.72	−0.218621
$472$	3590.59	0.350149
$473$	0	0
$474$	−1854.10	−0.179666
$475$	−1405.93	−0.135807
$476$	−439.842	−0.0423532
$477$	11769.7	1.12977
$478$	2594.31	0.248245
$479$	3202.71	0.305503	0.152751	−	0.988265i	$-0.451187\pi$
$479$	3202.71	0.305503	0.152751	+	0.988265i	$0.451187\pi$
$480$	321.691	0.0305899
$481$	−5263.36	−0.498937
$482$	17221.3	1.62741
$483$	334.750	0.0315355
$484$	−1021.39	−0.0959232
$485$	−2527.08	−0.236596
$486$	5376.48	0.501815
$487$	7485.92	0.696549	0.348274	−	0.937393i	$-0.386768\pi$
$487$	7485.92	0.696549	0.348274	+	0.937393i	$0.386768\pi$
$488$	16564.9	1.53659
$489$	−1113.63	−0.102986
$490$	−8589.17	−0.791876
$491$	−18220.4	−1.67469	−0.837346	−	0.546673i	$-0.815894\pi$
$491$	−18220.4	−1.67469	−0.837346	+	0.546673i	$0.815894\pi$
$492$	42.4903	0.00389352
$493$	3157.12	0.288417
$494$	−13382.7	−1.21886
$495$	170.576	0.0154885
$496$	841.380	0.0761675
$497$	3504.86	0.316327
$498$	1650.31	0.148498
$499$	−5674.55	−0.509074	−0.254537	−	0.967063i	$-0.581923\pi$
$499$	−5674.55	−0.509074	−0.254537	+	0.967063i	$0.581923\pi$
$500$	1106.77	0.0989928
$501$	3428.71	0.305755
$502$	13683.9	1.21662
$503$	15003.8	1.33000	0.664998	−	0.746846i	$-0.268433\pi$
$503$	15003.8	1.33000	0.664998	+	0.746846i	$0.268433\pi$
$504$	4837.68	0.427554
$505$	17046.6	1.50211
$506$	−80.5502	−0.00707687
$507$	1218.42	0.106729
$508$	160.539	0.0140212
$509$	14774.0	1.28653	0.643266	−	0.765643i	$-0.277579\pi$
$509$	14774.0	1.28653	0.643266	+	0.765643i	$0.277579\pi$
$510$	−1832.50	−0.159107
$511$	−4537.79	−0.392837
$512$	9507.76	0.820679
$513$	7412.21	0.637928
$514$	−2165.17	−0.185800
$515$	7888.32	0.674953
$516$	0	0
$517$	−126.760	−0.0107832
$518$	−4791.92	−0.406457
$519$	−3053.58	−0.258260
$520$	−6461.87	−0.544946
$521$	12098.9	1.01740	0.508698	−	0.860945i	$-0.330127\pi$
$521$	12098.9	1.01740	0.508698	+	0.860945i	$0.330127\pi$
$522$	3685.37	0.309012
$523$	111.880	0.00935408	0.00467704	−	0.999989i	$-0.498511\pi$
$523$	111.880	0.00935408	0.00467704	+	0.999989i	$0.498511\pi$
$524$	1628.27	0.135746
$525$	−64.5191	−0.00536351
$526$	−13717.3	−1.13707
$527$	−805.732	−0.0666001
$528$	36.0997	0.00297545
$529$	−10127.7	−0.832387
$530$	−14313.1	−1.17306
$531$	−4402.48	−0.359796
$532$	−1066.70	−0.0869308
$533$	−1797.61	−0.146084
$534$	−811.176	−0.0657360
$535$	−17027.0	−1.37597
$536$	−17655.8	−1.42279
$537$	2902.73	0.233263
$538$	−19089.8	−1.52977
$539$	−162.035	−0.0129487
$540$	−379.850	−0.0302707
$541$	−3173.52	−0.252200	−0.126100	−	0.992018i	$-0.540246\pi$
$541$	−3173.52	−0.252200	−0.126100	+	0.992018i	$0.540246\pi$
$542$	13505.9	1.07034
$543$	2593.13	0.204939
$544$	2305.93	0.181739
$545$	−15414.0	−1.21150
$546$	−614.141	−0.0481370
$547$	9970.28	0.779339	0.389669	−	0.920955i	$-0.372589\pi$
$547$	9970.28	0.779339	0.389669	+	0.920955i	$0.372589\pi$
$548$	−1795.90	−0.139995
$549$	−20310.5	−1.57892
$550$	15.5251	0.00120362
$551$	7656.60	0.591982
$552$	−833.268	−0.0642504
$553$	6252.25	0.480782
$554$	22158.4	1.69931
$555$	−1747.87	−0.133681
$556$	−1728.60	−0.131851
$557$	12143.8	0.923787	0.461894	−	0.886935i	$-0.347170\pi$
$557$	12143.8	0.923787	0.461894	+	0.886935i	$0.347170\pi$
$558$	−940.548	−0.0713558
$559$	0	0
$560$	−6452.80	−0.486930
$561$	−34.5702	−0.00260170
$562$	6000.78	0.450405
$563$	14652.7	1.09687	0.548433	−	0.836194i	$-0.315224\pi$
$563$	14652.7	1.09687	0.548433	+	0.836194i	$0.315224\pi$
$564$	139.172	0.0103904
$565$	9301.67	0.692609
$566$	−9892.71	−0.734667
$567$	−5759.12	−0.426561
$568$	−8724.38	−0.644484
$569$	17237.8	1.27003	0.635014	−	0.772501i	$-0.280994\pi$
$569$	17237.8	1.27003	0.635014	+	0.772501i	$0.280994\pi$
$570$	−4444.15	−0.326570
$571$	−4074.88	−0.298648	−0.149324	−	0.988788i	$-0.547710\pi$
$571$	−4074.88	−0.298648	−0.149324	+	0.988788i	$0.547710\pi$
$572$	12.9380	0.000945740 0
$573$	−1525.97	−0.111254
$574$	−1636.59	−0.119007
$575$	−393.060	−0.0285074
$576$	−11918.3	−0.862146
$577$	4716.44	0.340291	0.170146	−	0.985419i	$-0.445576\pi$
$577$	4716.44	0.340291	0.170146	+	0.985419i	$0.445576\pi$
$578$	1411.81	0.101598
$579$	−3076.18	−0.220798
$580$	−392.374	−0.0280905
$581$	−5565.05	−0.397379
$582$	597.851	0.0425803
$583$	−270.018	−0.0191818
$584$	11295.6	0.800366
$585$	7923.02	0.559960
$586$	−9882.25	−0.696642
$587$	7808.65	0.549059	0.274530	−	0.961579i	$-0.411478\pi$
$587$	7808.65	0.549059	0.274530	+	0.961579i	$0.411478\pi$
$588$	177.900	0.0124770
$589$	−1954.05	−0.136698
$590$	5353.84	0.373583
$591$	−3394.42	−0.236257
$592$	13083.3	0.908311
$593$	5384.85	0.372899	0.186450	−	0.982465i	$-0.440302\pi$
$593$	5384.85	0.372899	0.186450	+	0.982465i	$0.440302\pi$
$594$	−81.8500	−0.00565378
$595$	6179.41	0.425767
$596$	1311.05	0.0901052
$597$	2211.16	0.151586
$598$	−3741.44	−0.255851
$599$	1437.94	0.0980848	0.0490424	−	0.998797i	$-0.484383\pi$
$599$	1437.94	0.0980848	0.0490424	+	0.998797i	$0.484383\pi$
$600$	160.602	0.0109276
$601$	12358.1	0.838767	0.419383	−	0.907809i	$-0.362246\pi$
$601$	12358.1	0.838767	0.419383	+	0.907809i	$0.362246\pi$
$602$	0	0
$603$	21648.1	1.46199
$604$	431.442	0.0290648
$605$	14349.7	0.964293
$606$	−4032.84	−0.270335
$607$	−27345.5	−1.82853	−0.914265	−	0.405117i	$-0.867231\pi$
$607$	−27345.5	−1.82853	−0.914265	+	0.405117i	$0.867231\pi$
$608$	5592.30	0.373023
$609$	351.367	0.0233795
$610$	24699.5	1.63943
$611$	−5887.84	−0.389847
$612$	−1342.43	−0.0886678
$613$	−24663.4	−1.62503	−0.812515	−	0.582940i	$-0.801902\pi$
$613$	−24663.4	−1.62503	−0.812515	+	0.582940i	$0.801902\pi$
$614$	23935.2	1.57320
$615$	−596.953	−0.0391406
$616$	−110.985	−0.00725925
$617$	−13043.8	−0.851092	−0.425546	−	0.904937i	$-0.639918\pi$
$617$	−13043.8	−0.851092	−0.425546	+	0.904937i	$0.639918\pi$
$618$	−1866.20	−0.121472
$619$	14965.6	0.971756	0.485878	−	0.874027i	$-0.338500\pi$
$619$	14965.6	0.971756	0.485878	+	0.874027i	$0.338500\pi$
$620$	100.138	0.00648653
$621$	2072.26	0.133908
$622$	−2619.01	−0.168831
$623$	2735.38	0.175908
$624$	1676.78	0.107572
$625$	−14461.3	−0.925520
$626$	1761.23	0.112449
$627$	−83.8391	−0.00534005
$628$	1990.86	0.126503
$629$	−12529.0	−0.794218
$630$	7213.35	0.456169
$631$	−25299.7	−1.59614	−0.798069	−	0.602566i	$-0.794145\pi$
$631$	−25299.7	−1.59614	−0.798069	+	0.602566i	$0.794145\pi$
$632$	−15563.2	−0.979546
$633$	−3267.62	−0.205176
$634$	−6135.77	−0.384357
$635$	−2255.45	−0.140952
$636$	296.456	0.0184831
$637$	−7526.30	−0.468136
$638$	−84.5487	−0.00524658
$639$	10697.1	0.662240
$640$	17480.6	1.07966
$641$	−10950.1	−0.674734	−0.337367	−	0.941373i	$-0.609536\pi$
$641$	−10950.1	−0.674734	−0.337367	+	0.941373i	$0.609536\pi$
$642$	4028.21	0.247634
$643$	−23062.2	−1.41444	−0.707220	−	0.706993i	$-0.750051\pi$
$643$	−23062.2	−1.41444	−0.707220	+	0.706993i	$0.750051\pi$
$644$	−298.220	−0.0182477
$645$	0	0
$646$	−31856.3	−1.94020
$647$	9957.01	0.605024	0.302512	−	0.953146i	$-0.402175\pi$
$647$	9957.01	0.605024	0.302512	+	0.953146i	$0.402175\pi$
$648$	14335.7	0.869076
$649$	101.000	0.00610880
$650$	721.118	0.0435147
$651$	−89.6727	−0.00539870
$652$	992.101	0.0595915
$653$	−29817.1	−1.78688	−0.893442	−	0.449179i	$-0.851717\pi$
$653$	−29817.1	−1.78688	−0.893442	+	0.449179i	$0.851717\pi$
$654$	3646.62	0.218034
$655$	−22875.8	−1.36463
$656$	4468.36	0.265945
$657$	−13849.7	−0.822417
$658$	−5360.46	−0.317588
$659$	33462.5	1.97802	0.989009	−	0.147854i	$-0.0472367\pi$
$659$	33462.5	1.97802	0.989009	+	0.147854i	$0.0472367\pi$
$660$	4.29647	0.000253394 0
$661$	4390.53	0.258354	0.129177	−	0.991622i	$-0.458767\pi$
$661$	4390.53	0.258354	0.129177	+	0.991622i	$0.458767\pi$
$662$	−9558.05	−0.561154
$663$	−1605.74	−0.0940598
$664$	13852.6	0.809619
$665$	14986.2	0.873894
$666$	−14625.3	−0.850931
$667$	2140.58	0.124263
$668$	−3054.55	−0.176922
$669$	−1313.02	−0.0758809
$670$	−26326.2	−1.51801
$671$	465.956	0.0268078
$672$	256.635	0.0147320
$673$	−23321.7	−1.33579	−0.667893	−	0.744257i	$-0.732804\pi$
$673$	−23321.7	−1.33579	−0.667893	+	0.744257i	$0.732804\pi$
$674$	13418.6	0.766865
$675$	−399.403	−0.0227748
$676$	−1085.46	−0.0617579
$677$	−5401.07	−0.306617	−0.153309	−	0.988178i	$-0.548993\pi$
$677$	−5401.07	−0.306617	−0.153309	+	0.988178i	$0.548993\pi$
$678$	−2200.57	−0.124649
$679$	−2016.02	−0.113944
$680$	−15381.9	−0.867457
$681$	3148.34	0.177158
$682$	21.5778	0.00121152
$683$	−12861.1	−0.720520	−0.360260	−	0.932852i	$-0.617312\pi$
$683$	−12861.1	−0.720520	−0.360260	+	0.932852i	$0.617312\pi$
$684$	−3255.65	−0.181992
$685$	25230.9	1.40733
$686$	−15589.8	−0.867669
$687$	−350.062	−0.0194406
$688$	0	0
$689$	−12541.9	−0.693483
$690$	−1242.47	−0.0685505
$691$	5074.22	0.279352	0.139676	−	0.990197i	$-0.455394\pi$
$691$	5074.22	0.279352	0.139676	+	0.990197i	$0.455394\pi$
$692$	2720.35	0.149440
$693$	136.080	0.00745925
$694$	−23390.6	−1.27939
$695$	24285.4	1.32546
$696$	−874.631	−0.0476333
$697$	−4279.05	−0.232540
$698$	244.026	0.0132328
$699$	−1892.78	−0.102420
$700$	57.4784	0.00310354
$701$	31850.5	1.71608	0.858042	−	0.513580i	$-0.171681\pi$
$701$	31850.5	1.71608	0.858042	+	0.513580i	$0.171681\pi$
$702$	−3801.81	−0.204402
$703$	−30385.1	−1.63015
$704$	273.426	0.0146380
$705$	−1955.25	−0.104452
$706$	1818.47	0.0969393
$707$	13599.2	0.723411
$708$	−110.890	−0.00588628
$709$	12109.0	0.641416	0.320708	−	0.947178i	$-0.396079\pi$
$709$	12109.0	0.641416	0.320708	+	0.947178i	$0.396079\pi$
$710$	−13008.7	−0.687617
$711$	19082.4	1.00653
$712$	−6808.98	−0.358395
$713$	−546.300	−0.0286944
$714$	−1461.91	−0.0766255
$715$	−181.768	−0.00950730
$716$	−2585.97	−0.134975
$717$	754.915	0.0393205
$718$	−39316.8	−2.04358
$719$	5809.08	0.301310	0.150655	−	0.988586i	$-0.451862\pi$
$719$	5809.08	0.301310	0.150655	+	0.988586i	$0.451862\pi$
$720$	−19694.5	−1.01940
$721$	6293.04	0.325056
$722$	−56947.8	−2.93543
$723$	5011.21	0.257772
$724$	−2310.15	−0.118586
$725$	−412.572	−0.0211345
$726$	−3394.81	−0.173545
$727$	−26094.0	−1.33119	−0.665593	−	0.746315i	$-0.731821\pi$
$727$	−26094.0	−1.33119	−0.665593	+	0.746315i	$0.731821\pi$
$728$	−5155.07	−0.262445
$729$	−16509.8	−0.838784
$730$	16842.6	0.853933
$731$	0	0
$732$	−511.579	−0.0258313
$733$	25642.1	1.29211	0.646053	−	0.763292i	$-0.276418\pi$
$733$	25642.1	1.29211	0.646053	+	0.763292i	$0.276418\pi$
$734$	6013.84	0.302418
$735$	−2499.35	−0.125428
$736$	1563.46	0.0783015
$737$	−496.644	−0.0248224
$738$	−4995.02	−0.249145
$739$	−18682.6	−0.929972	−0.464986	−	0.885318i	$-0.653941\pi$
$739$	−18682.6	−0.929972	−0.464986	+	0.885318i	$0.653941\pi$
$740$	1557.13	0.0773531
$741$	−3894.21	−0.193060
$742$	−11418.6	−0.564944
$743$	18080.1	0.892724	0.446362	−	0.894853i	$-0.352719\pi$
$743$	18080.1	0.892724	0.446362	+	0.894853i	$0.352719\pi$
$744$	223.215	0.0109993
$745$	−18419.2	−0.905806
$746$	−27971.2	−1.37279
$747$	−16985.0	−0.831925
$748$	30.7977	0.00150545
$749$	−13583.6	−0.662663
$750$	3678.60	0.179098
$751$	7444.31	0.361713	0.180857	−	0.983509i	$-0.442113\pi$
$751$	7444.31	0.361713	0.180857	+	0.983509i	$0.442113\pi$
$752$	14635.6	0.709714
$753$	3981.87	0.192706
$754$	−3927.16	−0.189680
$755$	−6061.40	−0.292181
$756$	−303.032	−0.0145783
$757$	30069.0	1.44369	0.721846	−	0.692054i	$-0.243294\pi$
$757$	30069.0	1.44369	0.721846	+	0.692054i	$0.243294\pi$
$758$	−28014.0	−1.34237
$759$	−23.4392	−0.00112093
$760$	−37304.0	−1.78047
$761$	−21504.7	−1.02437	−0.512184	−	0.858875i	$-0.671164\pi$
$761$	−21504.7	−1.02437	−0.512184	+	0.858875i	$0.671164\pi$
$762$	533.588	0.0253673
$763$	−12296.8	−0.583453
$764$	1359.45	0.0643757
$765$	18860.1	0.891356
$766$	14579.5	0.687699
$767$	4691.32	0.220853
$768$	−1006.82	−0.0473053
$769$	−25948.5	−1.21681	−0.608405	−	0.793627i	$-0.708190\pi$
$769$	−25948.5	−1.21681	−0.608405	+	0.793627i	$0.708190\pi$
$770$	−165.486	−0.00774509
$771$	−630.039	−0.0294297
$772$	2740.49	0.127762
$773$	−14854.8	−0.691190	−0.345595	−	0.938384i	$-0.612323\pi$
$773$	−14854.8	−0.691190	−0.345595	+	0.938384i	$0.612323\pi$
$774$	0	0
$775$	105.293	0.00488029
$776$	5018.33	0.232149
$777$	−1394.39	−0.0643804
$778$	−2232.40	−0.102873
$779$	−10377.5	−0.477293
$780$	199.565	0.00916098
$781$	−245.410	−0.0112439
$782$	−8906.18	−0.407269
$783$	2175.12	0.0992752
$784$	18708.3	0.852239
$785$	−27969.8	−1.27170
$786$	5411.90	0.245593
$787$	−4869.66	−0.220565	−0.110282	−	0.993900i	$-0.535176\pi$
$787$	−4869.66	−0.220565	−0.110282	+	0.993900i	$0.535176\pi$
$788$	3024.00	0.136707
$789$	−3991.57	−0.180106
$790$	−23206.0	−1.04510
$791$	7420.57	0.333559
$792$	−338.734	−0.0151975
$793$	21643.0	0.969187
$794$	−37222.1	−1.66368
$795$	−4164.96	−0.185806
$796$	−1969.86	−0.0877135
$797$	4355.62	0.193581	0.0967904	−	0.995305i	$-0.469142\pi$
$797$	4355.62	0.193581	0.0967904	+	0.995305i	$0.469142\pi$
$798$	−3545.40	−0.157275
$799$	−14015.5	−0.620567
$800$	−301.338	−0.0133174
$801$	8348.61	0.368269
$802$	11918.0	0.524739
$803$	317.736	0.0139634
$804$	545.272	0.0239182
$805$	4189.74	0.183440
$806$	1002.26	0.0438002
$807$	−5554.90	−0.242307
$808$	−33851.5	−1.47388
$809$	35497.9	1.54269	0.771347	−	0.636415i	$-0.219583\pi$
$809$	35497.9	1.54269	0.771347	+	0.636415i	$0.219583\pi$
$810$	21375.7	0.927241
$811$	13077.5	0.566233	0.283116	−	0.959086i	$-0.408632\pi$
$811$	13077.5	0.566233	0.283116	+	0.959086i	$0.408632\pi$
$812$	−313.024	−0.0135283
$813$	3930.05	0.169536
$814$	335.530	0.0144476
$815$	−13938.2	−0.599060
$816$	3991.43	0.171235
$817$	0	0
$818$	−33270.0	−1.42208
$819$	6320.73	0.269675
$820$	531.810	0.0226483
$821$	32473.4	1.38043	0.690213	−	0.723606i	$-0.257517\pi$
$821$	32473.4	1.38043	0.690213	+	0.723606i	$0.257517\pi$
$822$	−5969.06	−0.253279
$823$	−31536.0	−1.33570	−0.667848	−	0.744298i	$-0.732784\pi$
$823$	−31536.0	−1.33570	−0.667848	+	0.744298i	$0.732784\pi$
$824$	−15664.8	−0.662268
$825$	4.51762	0.000190647 0
$826$	4271.12	0.179917
$827$	−19609.5	−0.824532	−0.412266	−	0.911064i	$-0.635262\pi$
$827$	−19609.5	−0.824532	−0.412266	+	0.911064i	$0.635262\pi$
$828$	−910.192	−0.0382021
$829$	−41250.9	−1.72823	−0.864116	−	0.503293i	$-0.832121\pi$
$829$	−41250.9	−1.72823	−0.864116	+	0.503293i	$0.832121\pi$
$830$	20655.4	0.863805
$831$	6447.82	0.269161
$832$	12700.2	0.529209
$833$	−17915.7	−0.745189
$834$	−5745.38	−0.238545
$835$	42913.8	1.77856
$836$	74.6901	0.00308996
$837$	−555.115	−0.0229242
$838$	8142.06	0.335636
$839$	−46406.6	−1.90958	−0.954788	−	0.297289i	$-0.903917\pi$
$839$	−46406.6	−1.90958	−0.954788	+	0.297289i	$0.903917\pi$
$840$	−1711.91	−0.0703172
$841$	−22142.2	−0.907875
$842$	−11432.2	−0.467909
$843$	1746.16	0.0713415
$844$	2911.04	0.118723
$845$	15249.8	0.620837
$846$	−16360.6	−0.664880
$847$	11447.7	0.464401
$848$	31175.9	1.26248
$849$	−2878.66	−0.116367
$850$	1716.56	0.0692677
$851$	−8494.86	−0.342186
$852$	269.438	0.0108343
$853$	−15973.5	−0.641174	−0.320587	−	0.947219i	$-0.603880\pi$
$853$	−15973.5	−0.641174	−0.320587	+	0.947219i	$0.603880\pi$
$854$	19704.4	0.789545
$855$	45739.1	1.82953
$856$	33812.7	1.35011
$857$	30203.3	1.20388	0.601939	−	0.798542i	$-0.294395\pi$
$857$	30203.3	1.20388	0.601939	+	0.798542i	$0.294395\pi$
$858$	43.0021	0.00171104
$859$	21285.6	0.845465	0.422732	−	0.906255i	$-0.361071\pi$
$859$	21285.6	0.845465	0.422732	+	0.906255i	$0.361071\pi$
$860$	0	0
$861$	−476.230	−0.0188500
$862$	−34882.3	−1.37830
$863$	17363.9	0.684904	0.342452	−	0.939535i	$-0.388743\pi$
$863$	17363.9	0.684904	0.342452	+	0.939535i	$0.388743\pi$
$864$	1588.69	0.0625558
$865$	−38218.7	−1.50228
$866$	27435.5	1.07655
$867$	410.821	0.0160925
$868$	79.8871	0.00312390
$869$	−437.782	−0.0170895
$870$	−1304.14	−0.0508213
$871$	−23068.4	−0.897409
$872$	30609.6	1.18873
$873$	−6153.07	−0.238545
$874$	−21599.1	−0.835928
$875$	−12404.7	−0.479263
$876$	−348.846	−0.0134548
$877$	7519.17	0.289515	0.144757	−	0.989467i	$-0.453760\pi$
$877$	7519.17	0.289515	0.144757	+	0.989467i	$0.453760\pi$
$878$	−3693.13	−0.141956
$879$	−2875.62	−0.110344
$880$	451.825	0.0173080
$881$	31964.7	1.22238	0.611190	−	0.791484i	$-0.290691\pi$
$881$	31964.7	1.22238	0.611190	+	0.791484i	$0.290691\pi$
$882$	−20913.4	−0.798401
$883$	−14116.8	−0.538014	−0.269007	−	0.963138i	$-0.586696\pi$
$883$	−14116.8	−0.538014	−0.269007	+	0.963138i	$0.586696\pi$
$884$	1430.51	0.0544267
$885$	1557.91	0.0591733
$886$	−3858.12	−0.146293
$887$	−37480.1	−1.41878	−0.709391	−	0.704815i	$-0.751030\pi$
$887$	−37480.1	−1.41878	−0.709391	+	0.704815i	$0.751030\pi$
$888$	3470.96	0.131169
$889$	−1799.32	−0.0678822
$890$	−10152.7	−0.382381
$891$	403.253	0.0151622
$892$	1169.74	0.0439077
$893$	−33990.1	−1.27373
$894$	4357.56	0.163019
$895$	36330.7	1.35687
$896$	13945.5	0.519962
$897$	−1088.72	−0.0405253
$898$	−2411.39	−0.0896094
$899$	−573.418	−0.0212731
$900$	175.429	0.00649736
$901$	−29855.0	−1.10390
$902$	114.594	0.00423012
$903$	0	0
$904$	−18471.5	−0.679593
$905$	32455.7	1.19211
$906$	1433.99	0.0525841
$907$	21529.5	0.788176	0.394088	−	0.919073i	$-0.371061\pi$
$907$	21529.5	0.788176	0.394088	+	0.919073i	$0.371061\pi$
$908$	−2804.77	−0.102511
$909$	41506.0	1.51448
$910$	−7686.61	−0.280009
$911$	11205.7	0.407531	0.203766	−	0.979020i	$-0.434682\pi$
$911$	11205.7	0.407531	0.203766	+	0.979020i	$0.434682\pi$
$912$	9679.94	0.351464
$913$	389.664	0.0141249
$914$	−32116.0	−1.16226
$915$	7187.26	0.259676
$916$	311.861	0.0112491
$917$	−18249.6	−0.657201
$918$	−9049.89	−0.325371
$919$	−16118.1	−0.578551	−0.289276	−	0.957246i	$-0.593414\pi$
$919$	−16118.1	−0.578551	−0.289276	+	0.957246i	$0.593414\pi$
$920$	−10429.2	−0.373740
$921$	6964.86	0.249186
$922$	50553.8	1.80575
$923$	−11398.9	−0.406501
$924$	3.42758	0.000122034 0
$925$	1637.28	0.0581984
$926$	2301.02	0.0816590
$927$	19206.9	0.680514
$928$	1641.07	0.0580504
$929$	9954.38	0.351553	0.175777	−	0.984430i	$-0.443756\pi$
$929$	9954.38	0.351553	0.175777	+	0.984430i	$0.443756\pi$
$930$	332.831	0.0117355
$931$	−43448.9	−1.52952
$932$	1686.23	0.0592643
$933$	−762.102	−0.0267418
$934$	14081.0	0.493304
$935$	−432.682	−0.0151339
$936$	−15733.7	−0.549436
$937$	−21886.8	−0.763084	−0.381542	−	0.924351i	$-0.624607\pi$
$937$	−21886.8	−0.763084	−0.381542	+	0.924351i	$0.624607\pi$
$938$	−21002.2	−0.731071
$939$	512.498	0.0178112
$940$	1741.88	0.0604402
$941$	19313.8	0.669089	0.334545	−	0.942380i	$-0.391418\pi$
$941$	19313.8	0.669089	0.334545	+	0.942380i	$0.391418\pi$
$942$	6617.04	0.228869
$943$	−2901.26	−0.100189
$944$	−11661.4	−0.402060
$945$	4257.35	0.146552
$946$	0	0
$947$	29898.4	1.02594	0.512971	−	0.858406i	$-0.328545\pi$
$947$	29898.4	1.02594	0.512971	+	0.858406i	$0.328545\pi$
$948$	480.646	0.0164669
$949$	14758.4	0.504823
$950$	4162.97	0.142173
$951$	−1785.44	−0.0608799
$952$	−12271.2	−0.417765
$953$	31248.7	1.06217	0.531084	−	0.847319i	$-0.321785\pi$
$953$	31248.7	1.06217	0.531084	+	0.847319i	$0.321785\pi$
$954$	−34850.4	−1.18273
$955$	−19099.1	−0.647154
$956$	−672.534	−0.0227524
$957$	−24.6027	−0.000831026 0
$958$	−9483.28	−0.319823
$959$	20128.4	0.677768
$960$	4217.53	0.141792
$961$	−29644.7	−0.995088
$962$	15584.9	0.522325
$963$	−41458.3	−1.38731
$964$	−4464.35	−0.149157
$965$	−38501.6	−1.28436
$966$	−991.199	−0.0330138
$967$	19566.6	0.650692	0.325346	−	0.945595i	$-0.394519\pi$
$967$	19566.6	0.650692	0.325346	+	0.945595i	$0.394519\pi$
$968$	−28495.9	−0.946171
$969$	−9269.82	−0.307316
$970$	7482.72	0.247686
$971$	17986.3	0.594448	0.297224	−	0.954808i	$-0.403939\pi$
$971$	17986.3	0.594448	0.297224	+	0.954808i	$0.403939\pi$
$972$	−1393.77	−0.0459928
$973$	19374.1	0.638341
$974$	−22165.9	−0.729200
$975$	209.837	0.00689248
$976$	−53798.7	−1.76440
$977$	−11777.6	−0.385669	−0.192834	−	0.981231i	$-0.561768\pi$
$977$	−11777.6	−0.385669	−0.192834	+	0.981231i	$0.561768\pi$
$978$	3297.46	0.107813
$979$	−191.531	−0.00625267
$980$	2226.61	0.0725779
$981$	−37530.9	−1.22148
$982$	53950.8	1.75320
$983$	−7196.21	−0.233493	−0.116746	−	0.993162i	$-0.537246\pi$
$983$	−7196.21	−0.233493	−0.116746	+	0.993162i	$0.537246\pi$
$984$	1185.44	0.0384050
$985$	−42484.6	−1.37429
$986$	−9348.28	−0.301937
$987$	−1559.83	−0.0503040
$988$	3469.25	0.111712
$989$	0	0
$990$	−505.078	−0.0162146
$991$	29829.6	0.956173	0.478087	−	0.878313i	$-0.341331\pi$
$991$	29829.6	0.956173	0.478087	+	0.878313i	$0.341331\pi$
$992$	−418.819	−0.0134048
$993$	−2781.28	−0.0888835
$994$	−10377.9	−0.331155
$995$	27674.9	0.881763
$996$	−427.817	−0.0136103
$997$	6363.30	0.202134	0.101067	−	0.994880i	$-0.467774\pi$
$997$	6363.30	0.202134	0.101067	+	0.994880i	$0.467774\pi$
$998$	16802.4	0.532937
$999$	−8631.93	−0.273376

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.l.1.15		60
43.14	even	21		43.4.g.a.24.3	yes	120
43.40	even	21		43.4.g.a.9.3	✓	120
43.42	odd	2		1849.4.a.k.1.46		60

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.g.a.9.3	✓	120	43.40	even	21
43.4.g.a.24.3	yes	120	43.14	even	21
1849.4.a.k.1.46		60	43.42	odd	2
1849.4.a.l.1.15		60	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q-2.96101 q^{2} -0.861620 q^{3} +0.767595 q^{4} -10.7841 q^{5} +2.55127 q^{6} -8.60318 q^{7} +21.4152 q^{8} -26.2576 q^{9} +O(q^{10})\)	\(q-2.96101 q^{2} -0.861620 q^{3} +0.767595 q^{4} -10.7841 q^{5} +2.55127 q^{6} -8.60318 q^{7} +21.4152 q^{8} -26.2576 q^{9} +31.9318 q^{10} +0.602394 q^{11} -0.661376 q^{12} +27.9803 q^{13} +25.4741 q^{14} +9.29177 q^{15} -69.5516 q^{16} +66.6048 q^{17} +77.7491 q^{18} +161.529 q^{19} -8.27780 q^{20} +7.41267 q^{21} -1.78370 q^{22} +45.1591 q^{23} -18.4518 q^{24} -8.70389 q^{25} -82.8501 q^{26} +45.8878 q^{27} -6.60376 q^{28} +47.4008 q^{29} -27.5131 q^{30} -12.0972 q^{31} +34.6211 q^{32} -0.519035 q^{33} -197.218 q^{34} +92.7773 q^{35} -20.1552 q^{36} -188.109 q^{37} -478.289 q^{38} -24.1084 q^{39} -230.943 q^{40} -64.2453 q^{41} -21.9490 q^{42} +0.462395 q^{44} +283.164 q^{45} -133.717 q^{46} -210.428 q^{47} +59.9270 q^{48} -268.985 q^{49} +25.7723 q^{50} -57.3881 q^{51} +21.4776 q^{52} -448.241 q^{53} -135.874 q^{54} -6.49626 q^{55} -184.239 q^{56} -139.176 q^{57} -140.354 q^{58} +167.665 q^{59} +7.13232 q^{60} +773.508 q^{61} +35.8200 q^{62} +225.899 q^{63} +453.899 q^{64} -301.742 q^{65} +1.53687 q^{66} -824.451 q^{67} +51.1255 q^{68} -38.9100 q^{69} -274.715 q^{70} -407.391 q^{71} -562.313 q^{72} +527.455 q^{73} +556.994 q^{74} +7.49945 q^{75} +123.989 q^{76} -5.18250 q^{77} +71.3854 q^{78} -726.737 q^{79} +750.049 q^{80} +669.418 q^{81} +190.231 q^{82} +646.859 q^{83} +5.68993 q^{84} -718.271 q^{85} -40.8415 q^{87} +12.9004 q^{88} -317.950 q^{89} -838.452 q^{90} -240.720 q^{91} +34.6639 q^{92} +10.4232 q^{93} +623.079 q^{94} -1741.94 q^{95} -29.8302 q^{96} +234.335 q^{97} +796.469 q^{98} -15.8174 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 15 q^{2} + 37 q^{3} + 213 q^{4} + 51 q^{5} + 22 q^{6} + 54 q^{7} + 204 q^{8} + 387 q^{9}+O(q^{10}) \) 60 * q + 15 * q^2 + 37 * q^3 + 213 * q^4 + 51 * q^5 + 22 * q^6 + 54 * q^7 + 204 * q^8 + 387 * q^9	\( 60 q + 15 q^{2} + 37 q^{3} + 213 q^{4} + 51 q^{5} + 22 q^{6} + 54 q^{7} + 204 q^{8} + 387 q^{9} + 27 q^{10} + 43 q^{11} + 432 q^{12} + 13 q^{13} + 96 q^{14} + 65 q^{15} + 717 q^{16} - 138 q^{17} + 625 q^{18} + 610 q^{19} + 345 q^{20} + 611 q^{21} + 118 q^{22} - 243 q^{23} + 258 q^{24} + 899 q^{25} + 1071 q^{26} + 1609 q^{27} + 46 q^{28} + 773 q^{29} + 375 q^{30} - 97 q^{31} + 1967 q^{32} + 500 q^{33} + 217 q^{34} + 247 q^{35} + 175 q^{36} + 228 q^{37} + 1253 q^{38} + 1493 q^{39} + 2220 q^{40} - 951 q^{41} + 2643 q^{42} - 1378 q^{44} + 1086 q^{45} - 565 q^{46} - 2 q^{47} + 2303 q^{48} + 1264 q^{49} + 3273 q^{50} + 3076 q^{51} - 2825 q^{52} - 39 q^{53} + 5201 q^{54} + 1306 q^{55} + 3683 q^{56} + 1342 q^{57} + 2588 q^{58} - 1065 q^{59} + 2803 q^{60} + 2999 q^{61} + 5569 q^{62} + 2377 q^{63} + 2082 q^{64} + 5578 q^{65} + 3338 q^{66} + 961 q^{67} - 3754 q^{68} + 1817 q^{69} + 2738 q^{70} + 8003 q^{71} + 1412 q^{72} - 1011 q^{73} - 1413 q^{74} + 7457 q^{75} + 5516 q^{76} + 4052 q^{77} + 1091 q^{78} - 4422 q^{79} + 1610 q^{80} + 2108 q^{81} + 4676 q^{82} - 297 q^{83} - 54 q^{84} + 4333 q^{85} + 1377 q^{87} + 3652 q^{88} + 2480 q^{89} - 1414 q^{90} + 4551 q^{91} - 3286 q^{92} + 4 q^{93} + 4609 q^{94} - 835 q^{95} + 5864 q^{96} + 3785 q^{97} + 753 q^{98} + 5072 q^{99}+O(q^{100}) \) 60 * q + 15 * q^2 + 37 * q^3 + 213 * q^4 + 51 * q^5 + 22 * q^6 + 54 * q^7 + 204 * q^8 + 387 * q^9 + 27 * q^10 + 43 * q^11 + 432 * q^12 + 13 * q^13 + 96 * q^14 + 65 * q^15 + 717 * q^16 - 138 * q^17 + 625 * q^18 + 610 * q^19 + 345 * q^20 + 611 * q^21 + 118 * q^22 - 243 * q^23 + 258 * q^24 + 899 * q^25 + 1071 * q^26 + 1609 * q^27 + 46 * q^28 + 773 * q^29 + 375 * q^30 - 97 * q^31 + 1967 * q^32 + 500 * q^33 + 217 * q^34 + 247 * q^35 + 175 * q^36 + 228 * q^37 + 1253 * q^38 + 1493 * q^39 + 2220 * q^40 - 951 * q^41 + 2643 * q^42 - 1378 * q^44 + 1086 * q^45 - 565 * q^46 - 2 * q^47 + 2303 * q^48 + 1264 * q^49 + 3273 * q^50 + 3076 * q^51 - 2825 * q^52 - 39 * q^53 + 5201 * q^54 + 1306 * q^55 + 3683 * q^56 + 1342 * q^57 + 2588 * q^58 - 1065 * q^59 + 2803 * q^60 + 2999 * q^61 + 5569 * q^62 + 2377 * q^63 + 2082 * q^64 + 5578 * q^65 + 3338 * q^66 + 961 * q^67 - 3754 * q^68 + 1817 * q^69 + 2738 * q^70 + 8003 * q^71 + 1412 * q^72 - 1011 * q^73 - 1413 * q^74 + 7457 * q^75 + 5516 * q^76 + 4052 * q^77 + 1091 * q^78 - 4422 * q^79 + 1610 * q^80 + 2108 * q^81 + 4676 * q^82 - 297 * q^83 - 54 * q^84 + 4333 * q^85 + 1377 * q^87 + 3652 * q^88 + 2480 * q^89 - 1414 * q^90 + 4551 * q^91 - 3286 * q^92 + 4 * q^93 + 4609 * q^94 - 835 * q^95 + 5864 * q^96 + 3785 * q^97 + 753 * q^98 + 5072 * q^99

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