LMFDB - Embedded newform 39.4.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("39.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$2.30107449022$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{14}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 14 $ x^2 - 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.74166$ of defining polynomial
Character	$\chi$	$=$	39.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.74166 q^{2} -3.00000 q^{3} +14.4833 q^{4} +4.51669 q^{5} -14.2250 q^{6} -7.48331 q^{7} +30.7417 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+4.74166 q^{2} -3.00000 q^{3} +14.4833 q^{4} +4.51669 q^{5} -14.2250 q^{6} -7.48331 q^{7} +30.7417 q^{8} +9.00000 q^{9} +21.4166 q^{10} -66.8999 q^{11} -43.4499 q^{12} -13.0000 q^{13} -35.4833 q^{14} -13.5501 q^{15} +29.8999 q^{16} +96.9666 q^{17} +42.6749 q^{18} +31.4833 q^{19} +65.4166 q^{20} +22.4499 q^{21} -317.216 q^{22} +183.600 q^{23} -92.2250 q^{24} -104.600 q^{25} -61.6415 q^{26} -27.0000 q^{27} -108.383 q^{28} +112.200 q^{29} -64.2497 q^{30} -77.2831 q^{31} -104.158 q^{32} +200.700 q^{33} +459.783 q^{34} -33.7998 q^{35} +130.350 q^{36} +54.7664 q^{37} +149.283 q^{38} +39.0000 q^{39} +138.850 q^{40} +451.716 q^{41} +106.450 q^{42} -113.434 q^{43} -968.932 q^{44} +40.6502 q^{45} +870.566 q^{46} -42.2670 q^{47} -89.6997 q^{48} -287.000 q^{49} -495.975 q^{50} -290.900 q^{51} -188.283 q^{52} -530.999 q^{53} -128.025 q^{54} -302.166 q^{55} -230.049 q^{56} -94.4499 q^{57} +532.015 q^{58} +219.666 q^{59} -196.250 q^{60} +822.865 q^{61} -366.450 q^{62} -67.3498 q^{63} -733.082 q^{64} -58.7169 q^{65} +951.649 q^{66} -872.082 q^{67} +1404.40 q^{68} -550.799 q^{69} -160.267 q^{70} -100.299 q^{71} +276.675 q^{72} -165.634 q^{73} +259.684 q^{74} +313.799 q^{75} +455.983 q^{76} +500.633 q^{77} +184.925 q^{78} -545.266 q^{79} +135.048 q^{80} +81.0000 q^{81} +2141.88 q^{82} -454.534 q^{83} +325.150 q^{84} +437.968 q^{85} -537.864 q^{86} -336.601 q^{87} -2056.61 q^{88} -230.915 q^{89} +192.749 q^{90} +97.2831 q^{91} +2659.13 q^{92} +231.849 q^{93} -200.415 q^{94} +142.200 q^{95} +312.475 q^{96} -1089.16 q^{97} -1360.86 q^{98} -602.099 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 6 q^{3} + 14 q^{4} + 24 q^{5} - 6 q^{6} + 54 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 6 * q^3 + 14 * q^4 + 24 * q^5 - 6 * q^6 + 54 * q^8 + 18 * q^9	\( 2 q + 2 q^{2} - 6 q^{3} + 14 q^{4} + 24 q^{5} - 6 q^{6} + 54 q^{8} + 18 q^{9} - 32 q^{10} - 44 q^{11} - 42 q^{12} - 26 q^{13} - 56 q^{14} - 72 q^{15} - 30 q^{16} + 164 q^{17} + 18 q^{18} + 48 q^{19} + 56 q^{20} - 380 q^{22} + 8 q^{23} - 162 q^{24} + 150 q^{25} - 26 q^{26} - 54 q^{27} - 112 q^{28} + 404 q^{29} + 96 q^{30} + 40 q^{31} - 126 q^{32} + 132 q^{33} + 276 q^{34} + 112 q^{35} + 126 q^{36} - 100 q^{37} + 104 q^{38} + 78 q^{39} + 592 q^{40} + 200 q^{41} + 168 q^{42} - 616 q^{43} - 980 q^{44} + 216 q^{45} + 1352 q^{46} - 324 q^{47} + 90 q^{48} - 574 q^{49} - 1194 q^{50} - 492 q^{51} - 182 q^{52} - 164 q^{53} - 54 q^{54} + 144 q^{55} - 56 q^{56} - 144 q^{57} - 268 q^{58} + 140 q^{59} - 168 q^{60} + 628 q^{61} - 688 q^{62} - 194 q^{64} - 312 q^{65} + 1140 q^{66} - 472 q^{67} + 1372 q^{68} - 24 q^{69} - 560 q^{70} + 428 q^{71} + 486 q^{72} - 900 q^{73} + 684 q^{74} - 450 q^{75} + 448 q^{76} + 672 q^{77} + 78 q^{78} - 432 q^{79} - 1032 q^{80} + 162 q^{81} + 2832 q^{82} - 1388 q^{83} + 336 q^{84} + 1744 q^{85} + 840 q^{86} - 1212 q^{87} - 1524 q^{88} + 960 q^{89} - 288 q^{90} + 2744 q^{92} - 120 q^{93} + 572 q^{94} + 464 q^{95} + 378 q^{96} - 532 q^{97} - 574 q^{98} - 396 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 6 * q^3 + 14 * q^4 + 24 * q^5 - 6 * q^6 + 54 * q^8 + 18 * q^9 - 32 * q^10 - 44 * q^11 - 42 * q^12 - 26 * q^13 - 56 * q^14 - 72 * q^15 - 30 * q^16 + 164 * q^17 + 18 * q^18 + 48 * q^19 + 56 * q^20 - 380 * q^22 + 8 * q^23 - 162 * q^24 + 150 * q^25 - 26 * q^26 - 54 * q^27 - 112 * q^28 + 404 * q^29 + 96 * q^30 + 40 * q^31 - 126 * q^32 + 132 * q^33 + 276 * q^34 + 112 * q^35 + 126 * q^36 - 100 * q^37 + 104 * q^38 + 78 * q^39 + 592 * q^40 + 200 * q^41 + 168 * q^42 - 616 * q^43 - 980 * q^44 + 216 * q^45 + 1352 * q^46 - 324 * q^47 + 90 * q^48 - 574 * q^49 - 1194 * q^50 - 492 * q^51 - 182 * q^52 - 164 * q^53 - 54 * q^54 + 144 * q^55 - 56 * q^56 - 144 * q^57 - 268 * q^58 + 140 * q^59 - 168 * q^60 + 628 * q^61 - 688 * q^62 - 194 * q^64 - 312 * q^65 + 1140 * q^66 - 472 * q^67 + 1372 * q^68 - 24 * q^69 - 560 * q^70 + 428 * q^71 + 486 * q^72 - 900 * q^73 + 684 * q^74 - 450 * q^75 + 448 * q^76 + 672 * q^77 + 78 * q^78 - 432 * q^79 - 1032 * q^80 + 162 * q^81 + 2832 * q^82 - 1388 * q^83 + 336 * q^84 + 1744 * q^85 + 840 * q^86 - 1212 * q^87 - 1524 * q^88 + 960 * q^89 - 288 * q^90 + 2744 * q^92 - 120 * q^93 + 572 * q^94 + 464 * q^95 + 378 * q^96 - 532 * q^97 - 574 * q^98 - 396 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	4.74166	1.67643	0.838215	−	0.545341i	$-0.183600\pi$
$2$	4.74166	1.67643	0.838215	+	0.545341i	$0.183600\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	14.4833	1.81041
$5$	4.51669	0.403985	0.201992	−	0.979387i	$-0.435258\pi$
$5$	4.51669	0.403985	0.201992	+	0.979387i	$0.435258\pi$
$6$	−14.2250	−0.967887
$7$	−7.48331	−0.404061	−0.202031	−	0.979379i	$-0.564754\pi$
$7$	−7.48331	−0.404061	−0.202031	+	0.979379i	$0.564754\pi$
$8$	30.7417	1.35860
$9$	9.00000	0.333333
$10$	21.4166	0.677252
$11$	−66.8999	−1.83373	−0.916867	−	0.399193i	$-0.869290\pi$
$11$	−66.8999	−1.83373	−0.916867	+	0.399193i	$0.869290\pi$
$12$	−43.4499	−1.04524
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−35.4833	−0.677380
$15$	−13.5501	−0.233241
$16$	29.8999	0.467186
$17$	96.9666	1.38340	0.691702	−	0.722183i	$-0.256861\pi$
$17$	96.9666	1.38340	0.691702	+	0.722183i	$0.256861\pi$
$18$	42.6749	0.558810
$19$	31.4833	0.380146	0.190073	−	0.981770i	$-0.439128\pi$
$19$	31.4833	0.380146	0.190073	+	0.981770i	$0.439128\pi$
$20$	65.4166	0.731380
$21$	22.4499	0.233285
$22$	−317.216	−3.07413
$23$	183.600	1.66448	0.832242	−	0.554412i	$-0.187057\pi$
$23$	183.600	1.66448	0.832242	+	0.554412i	$0.187057\pi$
$24$	−92.2250	−0.784389
$25$	−104.600	−0.836796
$26$	−61.6415	−0.464958
$27$	−27.0000	−0.192450
$28$	−108.383	−0.731518
$29$	112.200	0.718450	0.359225	−	0.933251i	$-0.383041\pi$
$29$	112.200	0.718450	0.359225	+	0.933251i	$0.383041\pi$
$30$	−64.2497	−0.391011
$31$	−77.2831	−0.447757	−0.223878	−	0.974617i	$-0.571872\pi$
$31$	−77.2831	−0.447757	−0.223878	+	0.974617i	$0.571872\pi$
$32$	−104.158	−0.575398
$33$	200.700	1.05871
$34$	459.783	2.31918
$35$	−33.7998	−0.163234
$36$	130.350	0.603471
$37$	54.7664	0.243339	0.121669	−	0.992571i	$-0.461175\pi$
$37$	54.7664	0.243339	0.121669	+	0.992571i	$0.461175\pi$
$38$	149.283	0.637287
$39$	39.0000	0.160128
$40$	138.850	0.548854
$41$	451.716	1.72064	0.860319	−	0.509756i	$-0.170264\pi$
$41$	451.716	1.72064	0.860319	+	0.509756i	$0.170264\pi$
$42$	106.450	0.391085
$43$	−113.434	−0.402291	−0.201145	−	0.979561i	$-0.564466\pi$
$43$	−113.434	−0.402291	−0.201145	+	0.979561i	$0.564466\pi$
$44$	−968.932	−3.31982
$45$	40.6502	0.134662
$46$	870.566	2.79039
$47$	−42.2670	−0.131176	−0.0655880	−	0.997847i	$-0.520892\pi$
$47$	−42.2670	−0.131176	−0.0655880	+	0.997847i	$0.520892\pi$
$48$	−89.6997	−0.269730
$49$	−287.000	−0.836735
$50$	−495.975	−1.40283
$51$	−290.900	−0.798708
$52$	−188.283	−0.502119
$53$	−530.999	−1.37619	−0.688097	−	0.725618i	$-0.741554\pi$
$53$	−530.999	−1.37619	−0.688097	+	0.725618i	$0.741554\pi$
$54$	−128.025	−0.322629
$55$	−302.166	−0.740800
$56$	−230.049	−0.548958
$57$	−94.4499	−0.219477
$58$	532.015	1.20443
$59$	219.666	0.484714	0.242357	−	0.970187i	$-0.422080\pi$
$59$	219.666	0.484714	0.242357	+	0.970187i	$0.422080\pi$
$60$	−196.250	−0.422262
$61$	822.865	1.72717	0.863583	−	0.504207i	$-0.168215\pi$
$61$	822.865	1.72717	0.863583	+	0.504207i	$0.168215\pi$
$62$	−366.450	−0.750632
$63$	−67.3498	−0.134687
$64$	−733.082	−1.43180
$65$	−58.7169	−0.112045
$66$	951.649	1.77485
$67$	−872.082	−1.59018	−0.795088	−	0.606495i	$-0.792575\pi$
$67$	−872.082	−1.59018	−0.795088	+	0.606495i	$0.792575\pi$
$68$	1404.40	2.50453
$69$	−550.799	−0.960991
$70$	−160.267	−0.273651
$71$	−100.299	−0.167653	−0.0838263	−	0.996480i	$-0.526714\pi$
$71$	−100.299	−0.167653	−0.0838263	+	0.996480i	$0.526714\pi$
$72$	276.675	0.452867
$73$	−165.634	−0.265562	−0.132781	−	0.991145i	$-0.542391\pi$
$73$	−165.634	−0.265562	−0.132781	+	0.991145i	$0.542391\pi$
$74$	259.684	0.407941
$75$	313.799	0.483125
$76$	455.983	0.688221
$77$	500.633	0.740940
$78$	184.925	0.268443
$79$	−545.266	−0.776547	−0.388273	−	0.921544i	$-0.626928\pi$
$79$	−545.266	−0.776547	−0.388273	+	0.921544i	$0.626928\pi$
$80$	135.048	0.188736
$81$	81.0000	0.111111
$82$	2141.88	2.88453
$83$	−454.534	−0.601103	−0.300552	−	0.953766i	$-0.597171\pi$
$83$	−454.534	−0.601103	−0.300552	+	0.953766i	$0.597171\pi$
$84$	325.150	0.422342
$85$	437.968	0.558874
$86$	−537.864	−0.674412
$87$	−336.601	−0.414797
$88$	−2056.61	−2.49132
$89$	−230.915	−0.275022	−0.137511	−	0.990500i	$-0.543910\pi$
$89$	−230.915	−0.275022	−0.137511	+	0.990500i	$0.543910\pi$
$90$	192.749	0.225751
$91$	97.2831	0.112066
$92$	2659.13	3.01341
$93$	231.849	0.258512
$94$	−200.415	−0.219907
$95$	142.200	0.153573
$96$	312.475	0.332206
$97$	−1089.16	−1.14008	−0.570041	−	0.821616i	$-0.693073\pi$
$97$	−1089.16	−1.14008	−0.570041	+	0.821616i	$0.693073\pi$
$98$	−1360.86	−1.40273
$99$	−602.099	−0.611245
$100$	−1514.95	−1.51495
$101$	−77.2336	−0.0760894	−0.0380447	−	0.999276i	$-0.512113\pi$
$101$	−77.2336	−0.0760894	−0.0380447	+	0.999276i	$0.512113\pi$
$102$	−1379.35	−1.33898
$103$	1351.36	1.29275	0.646377	−	0.763018i	$-0.276283\pi$
$103$	1351.36	1.29275	0.646377	+	0.763018i	$0.276283\pi$
$104$	−399.642	−0.376808
$105$	101.399	0.0942434
$106$	−2517.81	−2.30709
$107$	1133.67	1.02426	0.512129	−	0.858908i	$-0.328857\pi$
$107$	1133.67	1.02426	0.512129	+	0.858908i	$0.328857\pi$
$108$	−391.049	−0.348414
$109$	1017.66	0.894262	0.447131	−	0.894469i	$-0.352446\pi$
$109$	1017.66	0.894262	0.447131	+	0.894469i	$0.352446\pi$
$110$	−1432.77	−1.24190
$111$	−164.299	−0.140492
$112$	−223.750	−0.188772
$113$	1570.06	1.30707	0.653536	−	0.756895i	$-0.273285\pi$
$113$	1570.06	1.30707	0.653536	+	0.756895i	$0.273285\pi$
$114$	−447.849	−0.367938
$115$	829.261	0.672426
$116$	1625.03	1.30069
$117$	−117.000	−0.0924500
$118$	1041.58	0.812588
$119$	−725.632	−0.558979
$120$	−416.551	−0.316881
$121$	3144.60	2.36258
$122$	3901.75	2.89547
$123$	−1355.15	−0.993411
$124$	−1119.32	−0.810625
$125$	−1037.03	−0.742037
$126$	−319.350	−0.225793
$127$	−1248.16	−0.872099	−0.436050	−	0.899923i	$-0.643623\pi$
$127$	−1248.16	−0.872099	−0.436050	+	0.899923i	$0.643623\pi$
$128$	−2642.76	−1.82491
$129$	340.301	0.232263
$130$	−278.415	−0.187836
$131$	−1274.80	−0.850227	−0.425113	−	0.905140i	$-0.639766\pi$
$131$	−1274.80	−0.850227	−0.425113	+	0.905140i	$0.639766\pi$
$132$	2906.80	1.91670
$133$	−235.600	−0.153602
$134$	−4135.11	−2.66582
$135$	−121.951	−0.0777469
$136$	2980.91	1.87950
$137$	874.915	0.545613	0.272807	−	0.962069i	$-0.412048\pi$
$137$	874.915	0.545613	0.272807	+	0.962069i	$0.412048\pi$
$138$	−2611.70	−1.61103
$139$	310.334	0.189368	0.0946840	−	0.995507i	$-0.469816\pi$
$139$	310.334	0.189368	0.0946840	+	0.995507i	$0.469816\pi$
$140$	−489.533	−0.295522
$141$	126.801	0.0757345
$142$	−475.585	−0.281058
$143$	869.699	0.508586
$144$	269.099	0.155729
$145$	506.773	0.290243
$146$	−785.380	−0.445195
$147$	861.000	0.483089
$148$	793.199	0.440544
$149$	5.08064	0.00279344	0.00139672	−	0.999999i	$-0.499555\pi$
$149$	5.08064	0.00279344	0.00139672	+	0.999999i	$0.499555\pi$
$150$	1487.93	0.809924
$151$	6.54894	0.00352944	0.00176472	−	0.999998i	$-0.499438\pi$
$151$	6.54894	0.00352944	0.00176472	+	0.999998i	$0.499438\pi$
$152$	967.849	0.516467
$153$	872.700	0.461135
$154$	2373.83	1.24213
$155$	−349.063	−0.180887
$156$	564.849	0.289898
$157$	−2297.60	−1.16795	−0.583975	−	0.811772i	$-0.698503\pi$
$157$	−2297.60	−1.16795	−0.583975	+	0.811772i	$0.698503\pi$
$158$	−2585.46	−1.30183
$159$	1593.00	0.794546
$160$	−470.450	−0.232452
$161$	−1373.93	−0.672553
$162$	384.074	0.186270
$163$	−1085.49	−0.521606	−0.260803	−	0.965392i	$-0.583987\pi$
$163$	−1085.49	−0.521606	−0.260803	+	0.965392i	$0.583987\pi$
$164$	6542.34	3.11507
$165$	906.497	0.427701
$166$	−2155.24	−1.00771
$167$	−109.066	−0.0505374	−0.0252687	−	0.999681i	$-0.508044\pi$
$167$	−109.066	−0.0505374	−0.0252687	+	0.999681i	$0.508044\pi$
$168$	690.148	0.316941
$169$	169.000	0.0769231
$170$	2076.69	0.936912
$171$	283.350	0.126715
$172$	−1642.90	−0.728313
$173$	−1889.17	−0.830236	−0.415118	−	0.909768i	$-0.636260\pi$
$173$	−1889.17	−0.830236	−0.415118	+	0.909768i	$0.636260\pi$
$174$	−1596.05	−0.695379
$175$	782.751	0.338117
$176$	−2000.30	−0.856694
$177$	−658.999	−0.279850
$178$	−1094.92	−0.461054
$179$	3427.86	1.43134	0.715672	−	0.698437i	$-0.246121\pi$
$179$	3427.86	1.43134	0.715672	+	0.698437i	$0.246121\pi$
$180$	588.749	0.243793
$181$	208.403	0.0855826	0.0427913	−	0.999084i	$-0.486375\pi$
$181$	208.403	0.0855826	0.0427913	+	0.999084i	$0.486375\pi$
$182$	461.283	0.187871
$183$	−2468.60	−0.997180
$184$	5644.15	2.26137
$185$	247.363	0.0983052
$186$	1099.35	0.433378
$187$	−6487.06	−2.53679
$188$	−612.166	−0.237483
$189$	202.049	0.0777616
$190$	674.265	0.257454
$191$	957.735	0.362824	0.181412	−	0.983407i	$-0.441933\pi$
$191$	957.735	0.362824	0.181412	+	0.983407i	$0.441933\pi$
$192$	2199.25	0.826650
$193$	−512.730	−0.191228	−0.0956142	−	0.995418i	$-0.530482\pi$
$193$	−512.730	−0.191228	−0.0956142	+	0.995418i	$0.530482\pi$
$194$	−5164.45	−1.91127
$195$	176.151	0.0646893
$196$	−4156.71	−1.51484
$197$	−3870.35	−1.39975	−0.699876	−	0.714265i	$-0.746761\pi$
$197$	−3870.35	−1.39975	−0.699876	+	0.714265i	$0.746761\pi$
$198$	−2854.95	−1.02471
$199$	2305.83	0.821388	0.410694	−	0.911773i	$-0.365286\pi$
$199$	2305.83	0.821388	0.410694	+	0.911773i	$0.365286\pi$
$200$	−3215.56	−1.13687
$201$	2616.25	0.918088
$202$	−366.215	−0.127558
$203$	−839.630	−0.290298
$204$	−4213.19	−1.44599
$205$	2040.26	0.695111
$206$	6407.70	2.16721
$207$	1652.40	0.554828
$208$	−388.699	−0.129574
$209$	−2106.23	−0.697086
$210$	480.801	0.157992
$211$	−3672.40	−1.19819	−0.599096	−	0.800677i	$-0.704473\pi$
$211$	−3672.40	−1.19819	−0.599096	+	0.800677i	$0.704473\pi$
$212$	−7690.62	−2.49148
$213$	300.898	0.0967942
$214$	5375.46	1.71710
$215$	−512.345	−0.162519
$216$	−830.025	−0.261463
$217$	578.334	0.180921
$218$	4825.41	1.49917
$219$	496.902	0.153322
$220$	−4376.36	−1.34116
$221$	−1260.57	−0.383687
$222$	−779.051	−0.235525
$223$	5087.05	1.52760	0.763798	−	0.645455i	$-0.223332\pi$
$223$	5087.05	1.52760	0.763798	+	0.645455i	$0.223332\pi$
$224$	779.449	0.232496
$225$	−941.396	−0.278932
$226$	7444.71	2.19122
$227$	−2625.83	−0.767763	−0.383882	−	0.923382i	$-0.625413\pi$
$227$	−2625.83	−0.767763	−0.383882	+	0.923382i	$0.625413\pi$
$228$	−1367.95	−0.397345
$229$	1678.73	0.484425	0.242213	−	0.970223i	$-0.422127\pi$
$229$	1678.73	0.484425	0.242213	+	0.970223i	$0.422127\pi$
$230$	3932.07	1.12727
$231$	−1501.90	−0.427782
$232$	3449.22	0.976088
$233$	648.506	0.182339	0.0911696	−	0.995835i	$-0.470939\pi$
$233$	648.506	0.182339	0.0911696	+	0.995835i	$0.470939\pi$
$234$	−554.774	−0.154986
$235$	−190.907	−0.0529931
$236$	3181.50	0.877533
$237$	1635.80	0.448340
$238$	−3440.70	−0.937089
$239$	5219.69	1.41269	0.706347	−	0.707866i	$-0.250342\pi$
$239$	5219.69	1.41269	0.706347	+	0.707866i	$0.250342\pi$
$240$	−405.145	−0.108967
$241$	6103.56	1.63139	0.815695	−	0.578483i	$-0.196355\pi$
$241$	6103.56	1.63139	0.815695	+	0.578483i	$0.196355\pi$
$242$	14910.6	3.96070
$243$	−243.000	−0.0641500
$244$	11917.8	3.12689
$245$	−1296.29	−0.338028
$246$	−6425.64	−1.66538
$247$	−409.283	−0.105433
$248$	−2375.81	−0.608323
$249$	1363.60	0.347047
$250$	−4917.24	−1.24397
$251$	6423.40	1.61530	0.807652	−	0.589660i	$-0.200738\pi$
$251$	6423.40	1.61530	0.807652	+	0.589660i	$0.200738\pi$
$252$	−975.449	−0.243839
$253$	−12282.8	−3.05222
$254$	−5918.36	−1.46201
$255$	−1313.90	−0.322666
$256$	−6666.39	−1.62754
$257$	−1230.23	−0.298597	−0.149299	−	0.988792i	$-0.547702\pi$
$257$	−1230.23	−0.298597	−0.149299	+	0.988792i	$0.547702\pi$
$258$	1613.59	0.389372
$259$	−409.834	−0.0983238
$260$	−850.415	−0.202848
$261$	1009.80	0.239483
$262$	−6044.66	−1.42534
$263$	−514.992	−0.120744	−0.0603722	−	0.998176i	$-0.519229\pi$
$263$	−514.992	−0.120744	−0.0603722	+	0.998176i	$0.519229\pi$
$264$	6169.84	1.43836
$265$	−2398.35	−0.555961
$266$	−1117.13	−0.257503
$267$	692.745	0.158784
$268$	−12630.6	−2.87888
$269$	−5132.60	−1.16335	−0.581673	−	0.813423i	$-0.697602\pi$
$269$	−5132.60	−1.16335	−0.581673	+	0.813423i	$0.697602\pi$
$270$	−578.247	−0.130337
$271$	−4300.00	−0.963862	−0.481931	−	0.876209i	$-0.660064\pi$
$271$	−4300.00	−0.963862	−0.481931	+	0.876209i	$0.660064\pi$
$272$	2899.29	0.646306
$273$	−291.849	−0.0647015
$274$	4148.55	0.914682
$275$	6997.70	1.53446
$276$	−7977.39	−1.73979
$277$	−1812.80	−0.393215	−0.196607	−	0.980482i	$-0.562992\pi$
$277$	−1812.80	−0.393215	−0.196607	+	0.980482i	$0.562992\pi$
$278$	1471.50	0.317462
$279$	−695.548	−0.149252
$280$	−1039.06	−0.221771
$281$	−4073.08	−0.864696	−0.432348	−	0.901707i	$-0.642315\pi$
$281$	−4073.08	−0.864696	−0.432348	+	0.901707i	$0.642315\pi$
$282$	601.246	0.126963
$283$	6346.29	1.33303	0.666516	−	0.745491i	$-0.267785\pi$
$283$	6346.29	1.33303	0.666516	+	0.745491i	$0.267785\pi$
$284$	−1452.67	−0.303520
$285$	−426.601	−0.0886654
$286$	4123.81	0.852609
$287$	−3380.33	−0.695243
$288$	−937.424	−0.191799
$289$	4489.53	0.913806
$290$	2402.94	0.486572
$291$	3267.49	0.658226
$292$	−2398.93	−0.480777
$293$	−8390.97	−1.67306	−0.836529	−	0.547923i	$-0.815419\pi$
$293$	−8390.97	−1.67306	−0.836529	+	0.547923i	$0.815419\pi$
$294$	4082.57	0.809864
$295$	992.164	0.195817
$296$	1683.61	0.330601
$297$	1806.30	0.352902
$298$	24.0907	0.00468300
$299$	−2386.79	−0.461645
$300$	4544.84	0.874656
$301$	848.861	0.162550
$302$	31.0528	0.00591685
$303$	231.701	0.0439302
$304$	941.348	0.177599
$305$	3716.62	0.697748
$306$	4138.04	0.773059
$307$	4005.27	0.744603	0.372301	−	0.928112i	$-0.378569\pi$
$307$	4005.27	0.744603	0.372301	+	0.928112i	$0.378569\pi$
$308$	7250.82	1.34141
$309$	−4054.09	−0.746372
$310$	−1655.14	−0.303244
$311$	5836.53	1.06418	0.532088	−	0.846689i	$-0.321407\pi$
$311$	5836.53	1.06418	0.532088	+	0.846689i	$0.321407\pi$
$312$	1198.92	0.217550
$313$	1763.19	0.318407	0.159204	−	0.987246i	$-0.449107\pi$
$313$	1763.19	0.318407	0.159204	+	0.987246i	$0.449107\pi$
$314$	−10894.4	−1.95798
$315$	−304.198	−0.0544115
$316$	−7897.26	−1.40587
$317$	−6106.35	−1.08191	−0.540957	−	0.841050i	$-0.681938\pi$
$317$	−6106.35	−1.08191	−0.540957	+	0.841050i	$0.681938\pi$
$318$	7553.44	1.33200
$319$	−7506.18	−1.31745
$320$	−3311.10	−0.578425
$321$	−3401.00	−0.591356
$322$	−6514.72	−1.12749
$323$	3052.83	0.525895
$324$	1173.15	0.201157
$325$	1359.79	0.232086
$326$	−5147.00	−0.874436
$327$	−3052.99	−0.516302
$328$	13886.5	2.33766
$329$	316.297	0.0530031
$330$	4298.30	0.717011
$331$	7490.38	1.24383	0.621916	−	0.783084i	$-0.286354\pi$
$331$	7490.38	1.24383	0.621916	+	0.783084i	$0.286354\pi$
$332$	−6583.16	−1.08825
$333$	492.898	0.0811130
$334$	−517.152	−0.0847224
$335$	−3938.92	−0.642406
$336$	671.251	0.108987
$337$	9462.46	1.52953	0.764767	−	0.644307i	$-0.222854\pi$
$337$	9462.46	1.52953	0.764767	+	0.644307i	$0.222854\pi$
$338$	801.340	0.128956
$339$	−4710.19	−0.754639
$340$	6343.22	1.01179
$341$	5170.23	0.821066
$342$	1343.55	0.212429
$343$	4714.49	0.742153
$344$	−3487.14	−0.546553
$345$	−2487.78	−0.388226
$346$	−8957.79	−1.39183
$347$	11460.3	1.77297	0.886487	−	0.462753i	$-0.153138\pi$
$347$	11460.3	1.77297	0.886487	+	0.462753i	$0.153138\pi$
$348$	−4875.09	−0.750955
$349$	−3673.96	−0.563503	−0.281751	−	0.959487i	$-0.590915\pi$
$349$	−3673.96	−0.563503	−0.281751	+	0.959487i	$0.590915\pi$
$350$	3711.54	0.566829
$351$	351.000	0.0533761
$352$	6968.17	1.05513
$353$	3388.65	0.510935	0.255467	−	0.966818i	$-0.417771\pi$
$353$	3388.65	0.510935	0.255467	+	0.966818i	$0.417771\pi$
$354$	−3124.75	−0.469148
$355$	−453.020	−0.0677290
$356$	−3344.41	−0.497903
$357$	2176.90	0.322727
$358$	16253.7	2.39955
$359$	−9673.98	−1.42221	−0.711105	−	0.703086i	$-0.751805\pi$
$359$	−9673.98	−1.42221	−0.711105	+	0.703086i	$0.751805\pi$
$360$	1249.65	0.182951
$361$	−5867.80	−0.855489
$362$	988.174	0.143473
$363$	−9433.79	−1.36404
$364$	1408.98	0.202887
$365$	−748.117	−0.107283
$366$	−11705.2	−1.67170
$367$	−8715.98	−1.23970	−0.619851	−	0.784720i	$-0.712807\pi$
$367$	−8715.98	−1.23970	−0.619851	+	0.784720i	$0.712807\pi$
$368$	5489.61	0.777624
$369$	4065.44	0.573546
$370$	1172.91	0.164802
$371$	3973.63	0.556067
$372$	3357.95	0.468014
$373$	4667.99	0.647987	0.323994	−	0.946059i	$-0.394974\pi$
$373$	4667.99	0.647987	0.323994	+	0.946059i	$0.394974\pi$
$374$	−30759.4	−4.25276
$375$	3111.09	0.428416
$376$	−1299.36	−0.178216
$377$	−1458.60	−0.199262
$378$	958.049	0.130362
$379$	10862.7	1.47225	0.736123	−	0.676848i	$-0.236655\pi$
$379$	10862.7	1.47225	0.736123	+	0.676848i	$0.236655\pi$
$380$	2059.53	0.278031
$381$	3744.49	0.503507
$382$	4541.25	0.608248
$383$	10054.2	1.34137	0.670686	−	0.741742i	$-0.266000\pi$
$383$	10054.2	1.34137	0.670686	+	0.741742i	$0.266000\pi$
$384$	7928.27	1.05361
$385$	2261.20	0.299329
$386$	−2431.19	−0.320581
$387$	−1020.90	−0.134097
$388$	−15774.7	−2.06402
$389$	6418.50	0.836584	0.418292	−	0.908313i	$-0.362629\pi$
$389$	6418.50	0.836584	0.418292	+	0.908313i	$0.362629\pi$
$390$	835.246	0.108447
$391$	17803.0	2.30265
$392$	−8822.86	−1.13679
$393$	3824.40	0.490879
$394$	−18351.9	−2.34658
$395$	−2462.79	−0.313713
$396$	−8720.39	−1.10661
$397$	−12019.9	−1.51955	−0.759775	−	0.650186i	$-0.774691\pi$
$397$	−12019.9	−1.51955	−0.759775	+	0.650186i	$0.774691\pi$
$398$	10933.5	1.37700
$399$	706.799	0.0886822
$400$	−3127.52	−0.390939
$401$	3599.80	0.448293	0.224147	−	0.974555i	$-0.428041\pi$
$401$	3599.80	0.448293	0.224147	+	0.974555i	$0.428041\pi$
$402$	12405.3	1.53911
$403$	1004.68	0.124185
$404$	−1118.60	−0.137753
$405$	365.852	0.0448872
$406$	−3981.24	−0.486664
$407$	−3663.87	−0.446219
$408$	−8942.74	−1.08513
$409$	−48.0968	−0.00581475	−0.00290737	−	0.999996i	$-0.500925\pi$
$409$	−48.0968	−0.00581475	−0.00290737	+	0.999996i	$0.500925\pi$
$410$	9674.20	1.16530
$411$	−2624.74	−0.315010
$412$	19572.2	2.34042
$413$	−1643.83	−0.195854
$414$	7835.10	0.930130
$415$	−2052.99	−0.242837
$416$	1354.06	0.159587
$417$	−931.001	−0.109332
$418$	−9987.02	−1.16862
$419$	723.462	0.0843518	0.0421759	−	0.999110i	$-0.486571\pi$
$419$	723.462	0.0843518	0.0421759	+	0.999110i	$0.486571\pi$
$420$	1468.60	0.170620
$421$	−14845.5	−1.71859	−0.859295	−	0.511481i	$-0.829097\pi$
$421$	−14845.5	−1.71859	−0.859295	+	0.511481i	$0.829097\pi$
$422$	−17413.3	−2.00868
$423$	−380.403	−0.0437253
$424$	−16323.8	−1.86970
$425$	−10142.7	−1.15763
$426$	1426.75	0.162269
$427$	−6157.76	−0.697880
$428$	16419.2	1.85433
$429$	−2609.10	−0.293632
$430$	−2429.36	−0.272452
$431$	1103.11	0.123283	0.0616417	−	0.998098i	$-0.480366\pi$
$431$	1103.11	0.123283	0.0616417	+	0.998098i	$0.480366\pi$
$432$	−807.297	−0.0899099
$433$	8893.53	0.987057	0.493528	−	0.869730i	$-0.335707\pi$
$433$	8893.53	0.987057	0.493528	+	0.869730i	$0.335707\pi$
$434$	2742.26	0.303301
$435$	−1520.32	−0.167572
$436$	14739.1	1.61898
$437$	5780.32	0.632747
$438$	2356.14	0.257034
$439$	−10901.7	−1.18521	−0.592607	−	0.805492i	$-0.701901\pi$
$439$	−10901.7	−1.18521	−0.592607	+	0.805492i	$0.701901\pi$
$440$	−9289.08	−1.00645
$441$	−2583.00	−0.278912
$442$	−5977.17	−0.643224
$443$	−3781.37	−0.405550	−0.202775	−	0.979225i	$-0.564996\pi$
$443$	−3781.37	−0.405550	−0.202775	+	0.979225i	$0.564996\pi$
$444$	−2379.60	−0.254348
$445$	−1042.97	−0.111105
$446$	24121.0	2.56091
$447$	−15.2419	−0.00161279
$448$	5485.88	0.578535
$449$	106.834	0.0112289	0.00561447	−	0.999984i	$-0.498213\pi$
$449$	106.834	0.0112289	0.00561447	+	0.999984i	$0.498213\pi$
$450$	−4463.78	−0.467610
$451$	−30219.7	−3.15519
$452$	22739.7	2.36634
$453$	−19.6468	−0.00203772
$454$	−12450.8	−1.28710
$455$	439.397	0.0452731
$456$	−2903.55	−0.298182
$457$	−1237.64	−0.126684	−0.0633419	−	0.997992i	$-0.520176\pi$
$457$	−1237.64	−0.126684	−0.0633419	+	0.997992i	$0.520176\pi$
$458$	7959.95	0.812105
$459$	−2618.10	−0.266236
$460$	12010.5	1.21737
$461$	8790.90	0.888141	0.444071	−	0.895992i	$-0.353534\pi$
$461$	8790.90	0.888141	0.444071	+	0.895992i	$0.353534\pi$
$462$	−7121.49	−0.717146
$463$	−3861.55	−0.387606	−0.193803	−	0.981040i	$-0.562082\pi$
$463$	−3861.55	−0.387606	−0.193803	+	0.981040i	$0.562082\pi$
$464$	3354.77	0.335650
$465$	1047.19	0.104435
$466$	3074.99	0.305679
$467$	−8991.38	−0.890945	−0.445473	−	0.895296i	$-0.646964\pi$
$467$	−8991.38	−0.890945	−0.445473	+	0.895296i	$0.646964\pi$
$468$	−1694.55	−0.167373
$469$	6526.06	0.642528
$470$	−905.214	−0.0888391
$471$	6892.79	0.674316
$472$	6752.91	0.658533
$473$	7588.71	0.737694
$474$	7756.39	0.751609
$475$	−3293.14	−0.318105
$476$	−10509.6	−1.01198
$477$	−4778.99	−0.458731
$478$	24750.0	2.36828
$479$	4179.82	0.398707	0.199354	−	0.979928i	$-0.436116\pi$
$479$	4179.82	0.398707	0.199354	+	0.979928i	$0.436116\pi$
$480$	1411.35	0.134206
$481$	−711.963	−0.0674901
$482$	28941.0	2.73491
$483$	4121.80	0.388299
$484$	45544.2	4.27725
$485$	−4919.41	−0.460575
$486$	−1152.22	−0.107543
$487$	−18443.8	−1.71616	−0.858078	−	0.513519i	$-0.828342\pi$
$487$	−18443.8	−1.71616	−0.858078	+	0.513519i	$0.828342\pi$
$488$	25296.2	2.34653
$489$	3256.46	0.301149
$490$	−6146.56	−0.566680
$491$	8093.26	0.743877	0.371939	−	0.928257i	$-0.378693\pi$
$491$	8093.26	0.743877	0.371939	+	0.928257i	$0.378693\pi$
$492$	−19627.0	−1.79849
$493$	10879.7	0.993907
$494$	−1940.68	−0.176752
$495$	−2719.49	−0.246933
$496$	−2310.76	−0.209185
$497$	750.571	0.0677418
$498$	6465.73	0.581800
$499$	−10941.6	−0.981591	−0.490796	−	0.871275i	$-0.663294\pi$
$499$	−10941.6	−0.981591	−0.490796	+	0.871275i	$0.663294\pi$
$500$	−15019.6	−1.34340
$501$	327.197	0.0291778
$502$	30457.5	2.70794
$503$	−9260.11	−0.820851	−0.410425	−	0.911894i	$-0.634620\pi$
$503$	−9260.11	−0.820851	−0.410425	+	0.911894i	$0.634620\pi$
$504$	−2070.45	−0.182986
$505$	−348.840	−0.0307389
$506$	−58240.8	−5.11684
$507$	−507.000	−0.0444116
$508$	−18077.5	−1.57886
$509$	−9996.40	−0.870497	−0.435248	−	0.900310i	$-0.643339\pi$
$509$	−9996.40	−0.870497	−0.435248	+	0.900310i	$0.643339\pi$
$510$	−6230.08	−0.540927
$511$	1239.49	0.107303
$512$	−10467.7	−0.903538
$513$	−850.049	−0.0731591
$514$	−5833.32	−0.500577
$515$	6103.68	0.522253
$516$	4928.69	0.420491
$517$	2827.66	0.240542
$518$	−1943.29	−0.164833
$519$	5667.51	0.479337
$520$	−1805.06	−0.152225
$521$	11427.9	0.960973	0.480486	−	0.877002i	$-0.340460\pi$
$521$	11427.9	0.960973	0.480486	+	0.877002i	$0.340460\pi$
$522$	4788.14	0.401477
$523$	−4810.47	−0.402193	−0.201097	−	0.979571i	$-0.564451\pi$
$523$	−4810.47	−0.402193	−0.201097	+	0.979571i	$0.564451\pi$
$524$	−18463.3	−1.53926
$525$	−2348.25	−0.195212
$526$	−2441.92	−0.202419
$527$	−7493.88	−0.619428
$528$	6000.90	0.494613
$529$	21541.8	1.77051
$530$	−11372.2	−0.932030
$531$	1977.00	0.161571
$532$	−3412.26	−0.278083
$533$	−5872.31	−0.477219
$534$	3284.76	0.266190
$535$	5120.41	0.413785
$536$	−26809.2	−2.16042
$537$	−10283.6	−0.826386
$538$	−24337.0	−1.95027
$539$	19200.3	1.53435
$540$	−1766.25	−0.140754
$541$	2411.99	0.191681	0.0958406	−	0.995397i	$-0.469446\pi$
$541$	2411.99	0.191681	0.0958406	+	0.995397i	$0.469446\pi$
$542$	−20389.1	−1.61585
$543$	−625.208	−0.0494111
$544$	−10099.9	−0.796008
$545$	4596.47	0.361268
$546$	−1383.85	−0.108468
$547$	−4396.34	−0.343646	−0.171823	−	0.985128i	$-0.554966\pi$
$547$	−4396.34	−0.343646	−0.171823	+	0.985128i	$0.554966\pi$
$548$	12671.7	0.987786
$549$	7405.79	0.575722
$550$	33180.7	2.57242
$551$	3532.43	0.273116
$552$	−16932.5	−1.30560
$553$	4080.40	0.313772
$554$	−8595.67	−0.659197
$555$	−742.088	−0.0567565
$556$	4494.66	0.342835
$557$	−17488.0	−1.33032	−0.665160	−	0.746701i	$-0.731637\pi$
$557$	−17488.0	−1.33032	−0.665160	+	0.746701i	$0.731637\pi$
$558$	−3298.05	−0.250211
$559$	1474.64	0.111575
$560$	−1010.61	−0.0762608
$561$	19461.2	1.46462
$562$	−19313.2	−1.44960
$563$	−6881.77	−0.515154	−0.257577	−	0.966258i	$-0.582924\pi$
$563$	−6881.77	−0.515154	−0.257577	+	0.966258i	$0.582924\pi$
$564$	1836.50	0.137111
$565$	7091.49	0.528037
$566$	30092.0	2.23473
$567$	−606.148	−0.0448957
$568$	−3083.36	−0.227773
$569$	14733.5	1.08552	0.542758	−	0.839889i	$-0.317380\pi$
$569$	14733.5	1.08552	0.542758	+	0.839889i	$0.317380\pi$
$570$	−2022.79	−0.148641
$571$	4488.51	0.328964	0.164482	−	0.986380i	$-0.447405\pi$
$571$	4488.51	0.328964	0.164482	+	0.986380i	$0.447405\pi$
$572$	12596.1	0.920752
$573$	−2873.21	−0.209476
$574$	−16028.4	−1.16553
$575$	−19204.4	−1.39284
$576$	−6597.74	−0.477267
$577$	−10552.2	−0.761338	−0.380669	−	0.924711i	$-0.624306\pi$
$577$	−10552.2	−0.761338	−0.380669	+	0.924711i	$0.624306\pi$
$578$	21287.8	1.53193
$579$	1538.19	0.110406
$580$	7339.75	0.525460
$581$	3401.42	0.242882
$582$	15493.3	1.10347
$583$	35523.8	2.52357
$584$	−5091.86	−0.360793
$585$	−528.452	−0.0373484
$586$	−39787.1	−2.80476
$587$	−1637.20	−0.115118	−0.0575591	−	0.998342i	$-0.518332\pi$
$587$	−1637.20	−0.115118	−0.0575591	+	0.998342i	$0.518332\pi$
$588$	12470.1	0.874591
$589$	−2433.13	−0.170213
$590$	4704.50	0.328273
$591$	11611.1	0.808147
$592$	1637.51	0.113685
$593$	14024.0	0.971155	0.485577	−	0.874194i	$-0.338609\pi$
$593$	14024.0	0.971155	0.485577	+	0.874194i	$0.338609\pi$
$594$	8564.84	0.591616
$595$	−3277.45	−0.225819
$596$	73.5845	0.00505728
$597$	−6917.50	−0.474229
$598$	−11317.4	−0.773915
$599$	365.860	0.0249560	0.0124780	−	0.999922i	$-0.496028\pi$
$599$	365.860	0.0249560	0.0124780	+	0.999922i	$0.496028\pi$
$600$	9646.69	0.656374
$601$	−10128.9	−0.687465	−0.343733	−	0.939068i	$-0.611691\pi$
$601$	−10128.9	−0.687465	−0.343733	+	0.939068i	$0.611691\pi$
$602$	4025.01	0.272503
$603$	−7848.74	−0.530058
$604$	94.8504	0.00638975
$605$	14203.1	0.954446
$606$	1098.65	0.0736459
$607$	2324.97	0.155465	0.0777327	−	0.996974i	$-0.475232\pi$
$607$	2324.97	0.155465	0.0777327	+	0.996974i	$0.475232\pi$
$608$	−3279.25	−0.218735
$609$	2518.89	0.167603
$610$	17623.0	1.16973
$611$	549.471	0.0363817
$612$	12639.6	0.834845
$613$	633.133	0.0417162	0.0208581	−	0.999782i	$-0.493360\pi$
$613$	633.133	0.0417162	0.0208581	+	0.999782i	$0.493360\pi$
$614$	18991.6	1.24827
$615$	−6120.77	−0.401323
$616$	15390.3	1.00664
$617$	−2981.85	−0.194562	−0.0972810	−	0.995257i	$-0.531015\pi$
$617$	−2981.85	−0.194562	−0.0972810	+	0.995257i	$0.531015\pi$
$618$	−19223.1	−1.25124
$619$	15158.6	0.984292	0.492146	−	0.870513i	$-0.336213\pi$
$619$	15158.6	0.984292	0.492146	+	0.870513i	$0.336213\pi$
$620$	−5055.60	−0.327480
$621$	−4957.19	−0.320330
$622$	27674.8	1.78402
$623$	1728.01	0.111126
$624$	1166.10	0.0748096
$625$	8391.01	0.537025
$626$	8360.45	0.533787
$627$	6318.69	0.402463
$628$	−33276.8	−2.11447
$629$	5310.51	0.336636
$630$	−1442.40	−0.0912170
$631$	5562.30	0.350922	0.175461	−	0.984486i	$-0.443858\pi$
$631$	5562.30	0.350922	0.175461	+	0.984486i	$0.443858\pi$
$632$	−16762.4	−1.05502
$633$	11017.2	0.691776
$634$	−28954.2	−1.81375
$635$	−5637.56	−0.352315
$636$	23071.9	1.43846
$637$	3731.00	0.232068
$638$	−35591.7	−2.20861
$639$	−902.693	−0.0558842
$640$	−11936.5	−0.737237
$641$	−24140.7	−1.48752	−0.743761	−	0.668446i	$-0.766960\pi$
$641$	−24140.7	−1.48752	−0.743761	+	0.668446i	$0.766960\pi$
$642$	−16126.4	−0.991366
$643$	−1749.69	−0.107311	−0.0536555	−	0.998560i	$-0.517087\pi$
$643$	−1749.69	−0.107311	−0.0536555	+	0.998560i	$0.517087\pi$
$644$	−19899.1	−1.21760
$645$	1537.03	0.0938305
$646$	14475.5	0.881626
$647$	1489.51	0.0905083	0.0452542	−	0.998976i	$-0.485590\pi$
$647$	1489.51	0.0905083	0.0452542	+	0.998976i	$0.485590\pi$
$648$	2490.07	0.150956
$649$	−14695.7	−0.888836
$650$	6447.68	0.389075
$651$	−1735.00	−0.104455
$652$	−15721.4	−0.944323
$653$	10668.2	0.639327	0.319663	−	0.947531i	$-0.396430\pi$
$653$	10668.2	0.639327	0.319663	+	0.947531i	$0.396430\pi$
$654$	−14476.2	−0.865544
$655$	−5757.86	−0.343478
$656$	13506.3	0.803858
$657$	−1490.71	−0.0885205
$658$	1499.77	0.0888559
$659$	−13933.1	−0.823608	−0.411804	−	0.911272i	$-0.635101\pi$
$659$	−13933.1	−0.823608	−0.411804	+	0.911272i	$0.635101\pi$
$660$	13129.1	0.774317
$661$	−2349.69	−0.138264	−0.0691320	−	0.997608i	$-0.522023\pi$
$661$	−2349.69	−0.138264	−0.0691320	+	0.997608i	$0.522023\pi$
$662$	35516.8	2.08520
$663$	3781.70	0.221522
$664$	−13973.1	−0.816660
$665$	−1064.13	−0.0620529
$666$	2337.15	0.135980
$667$	20599.9	1.19585
$668$	−1579.63	−0.0914937
$669$	−15261.1	−0.881958
$670$	−18677.0	−1.07695
$671$	−55049.6	−3.16716
$672$	−2338.35	−0.134232
$673$	−32421.0	−1.85697	−0.928483	−	0.371374i	$-0.878887\pi$
$673$	−32421.0	−1.85697	−0.928483	+	0.371374i	$0.878887\pi$
$674$	44867.7	2.56415
$675$	2824.19	0.161042
$676$	2447.68	0.139263
$677$	−1071.74	−0.0608421	−0.0304211	−	0.999537i	$-0.509685\pi$
$677$	−1071.74	−0.0608421	−0.0304211	+	0.999537i	$0.509685\pi$
$678$	−22334.1	−1.26510
$679$	8150.56	0.460663
$680$	13463.9	0.759287
$681$	7877.48	0.443268
$682$	24515.5	1.37646
$683$	−305.487	−0.0171144	−0.00855721	−	0.999963i	$-0.502724\pi$
$683$	−305.487	−0.0171144	−0.00855721	+	0.999963i	$0.502724\pi$
$684$	4103.84	0.229407
$685$	3951.72	0.220419
$686$	22354.5	1.24417
$687$	−5036.18	−0.279683
$688$	−3391.66	−0.187944
$689$	6902.99	0.381688
$690$	−11796.2	−0.650833
$691$	2180.81	0.120061	0.0600303	−	0.998197i	$-0.480880\pi$
$691$	2180.81	0.120061	0.0600303	+	0.998197i	$0.480880\pi$
$692$	−27361.4	−1.50307
$693$	4505.70	0.246980
$694$	54340.9	2.97227
$695$	1401.68	0.0765018
$696$	−10347.7	−0.563545
$697$	43801.4	2.38034
$698$	−17420.6	−0.944672
$699$	−1945.52	−0.105274
$700$	11336.8	0.612132
$701$	−15168.3	−0.817259	−0.408629	−	0.912700i	$-0.633993\pi$
$701$	−15168.3	−0.817259	−0.408629	+	0.912700i	$0.633993\pi$
$702$	1664.32	0.0894812
$703$	1724.23	0.0925043
$704$	49043.1	2.62554
$705$	572.720	0.0305956
$706$	16067.8	0.856545
$707$	577.963	0.0307448
$708$	−9544.49	−0.506644
$709$	−8988.17	−0.476104	−0.238052	−	0.971252i	$-0.576509\pi$
$709$	−8988.17	−0.476104	−0.238052	+	0.971252i	$0.576509\pi$
$710$	−2148.07	−0.113543
$711$	−4907.39	−0.258849
$712$	−7098.71	−0.373645
$713$	−14189.1	−0.745284
$714$	10322.1	0.541029
$715$	3928.15	0.205461
$716$	49646.8	2.59132
$717$	−15659.1	−0.815619
$718$	−45870.7	−2.38423
$719$	8448.18	0.438198	0.219099	−	0.975703i	$-0.429688\pi$
$719$	8448.18	0.438198	0.219099	+	0.975703i	$0.429688\pi$
$720$	1215.44	0.0629120
$721$	−10112.7	−0.522352
$722$	−27823.1	−1.43417
$723$	−18310.7	−0.941883
$724$	3018.36	0.154940
$725$	−11736.1	−0.601197
$726$	−44731.8	−2.28671
$727$	−8624.18	−0.439963	−0.219982	−	0.975504i	$-0.570600\pi$
$727$	−8624.18	−0.439963	−0.219982	+	0.975504i	$0.570600\pi$
$728$	2990.64	0.152254
$729$	729.000	0.0370370
$730$	−3547.31	−0.179852
$731$	−10999.3	−0.556530
$732$	−35753.5	−1.80531
$733$	31124.2	1.56835	0.784174	−	0.620541i	$-0.213087\pi$
$733$	31124.2	1.56835	0.784174	+	0.620541i	$0.213087\pi$
$734$	−41328.2	−2.07827
$735$	3888.87	0.195161
$736$	−19123.4	−0.957742
$737$	58342.2	2.91596
$738$	19276.9	0.961509
$739$	−17671.1	−0.879626	−0.439813	−	0.898089i	$-0.644955\pi$
$739$	−17671.1	−0.879626	−0.439813	+	0.898089i	$0.644955\pi$
$740$	3582.63	0.177973
$741$	1227.85	0.0608720
$742$	18841.6	0.932206
$743$	−21331.1	−1.05325	−0.526623	−	0.850099i	$-0.676542\pi$
$743$	−21331.1	−1.05325	−0.526623	+	0.850099i	$0.676542\pi$
$744$	7127.43	0.351215
$745$	22.9477	0.00112851
$746$	22134.0	1.08630
$747$	−4090.81	−0.200368
$748$	−93954.1	−4.59265
$749$	−8483.58	−0.413863
$750$	14751.7	0.718208
$751$	11712.9	0.569122	0.284561	−	0.958658i	$-0.408152\pi$
$751$	11712.9	0.569122	0.284561	+	0.958658i	$0.408152\pi$
$752$	−1263.78	−0.0612835
$753$	−19270.2	−0.932596
$754$	−6916.20	−0.334049
$755$	29.5795	0.00142584
$756$	2926.35	0.140781
$757$	−16610.9	−0.797537	−0.398768	−	0.917052i	$-0.630562\pi$
$757$	−16610.9	−0.797537	−0.398768	+	0.917052i	$0.630562\pi$
$758$	51507.4	2.46812
$759$	36848.4	1.76220
$760$	4371.47	0.208645
$761$	−29365.5	−1.39882	−0.699408	−	0.714723i	$-0.746553\pi$
$761$	−29365.5	−1.39882	−0.699408	+	0.714723i	$0.746553\pi$
$762$	17755.1	0.844093
$763$	−7615.50	−0.361336
$764$	13871.2	0.656861
$765$	3941.71	0.186291
$766$	47673.5	2.24871
$767$	−2855.66	−0.134435
$768$	19999.2	0.939659
$769$	−28599.9	−1.34114	−0.670572	−	0.741844i	$-0.733951\pi$
$769$	−28599.9	−1.34114	−0.670572	+	0.741844i	$0.733951\pi$
$770$	10721.8	0.501803
$771$	3690.68	0.172395
$772$	−7426.03	−0.346203
$773$	13491.8	0.627772	0.313886	−	0.949461i	$-0.398369\pi$
$773$	13491.8	0.627772	0.313886	+	0.949461i	$0.398369\pi$
$774$	−4840.78	−0.224804
$775$	8083.78	0.374681
$776$	−33482.7	−1.54892
$777$	1229.50	0.0567673
$778$	30434.3	1.40247
$779$	14221.5	0.654093
$780$	2551.25	0.117114
$781$	6710.01	0.307430
$782$	84415.9	3.86024
$783$	−3029.41	−0.138266
$784$	−8581.27	−0.390911
$785$	−10377.5	−0.471834
$786$	18134.0	0.822923
$787$	25876.7	1.17205	0.586025	−	0.810293i	$-0.300692\pi$
$787$	25876.7	1.17205	0.586025	+	0.810293i	$0.300692\pi$
$788$	−56055.5	−2.53413
$789$	1544.98	0.0697118
$790$	−11677.7	−0.525918
$791$	−11749.3	−0.528137
$792$	−18509.5	−0.830438
$793$	−10697.3	−0.479030
$794$	−56994.2	−2.54742
$795$	7195.06	0.320984
$796$	33396.1	1.48705
$797$	21936.4	0.974938	0.487469	−	0.873140i	$-0.337920\pi$
$797$	21936.4	0.974938	0.487469	+	0.873140i	$0.337920\pi$
$798$	3351.40	0.148669
$799$	−4098.49	−0.181469
$800$	10894.9	0.481491
$801$	−2078.23	−0.0916739
$802$	17069.0	0.751531
$803$	11080.9	0.486969
$804$	37891.9	1.66212
$805$	−6205.62	−0.271701
$806$	4763.85	0.208188
$807$	15397.8	0.671658
$808$	−2374.29	−0.103375
$809$	5583.23	0.242640	0.121320	−	0.992613i	$-0.461287\pi$
$809$	5583.23	0.242640	0.121320	+	0.992613i	$0.461287\pi$
$810$	1734.74	0.0752502
$811$	12925.4	0.559647	0.279823	−	0.960052i	$-0.409724\pi$
$811$	12925.4	0.559647	0.279823	+	0.960052i	$0.409724\pi$
$812$	−12160.6	−0.525559
$813$	12900.0	0.556486
$814$	−17372.8	−0.748054
$815$	−4902.80	−0.210721
$816$	−8697.87	−0.373145
$817$	−3571.27	−0.152929
$818$	−228.058	−0.00974801
$819$	875.548	0.0373555
$820$	29549.7	1.25844
$821$	−10153.2	−0.431608	−0.215804	−	0.976437i	$-0.569237\pi$
$821$	−10153.2	−0.431608	−0.215804	+	0.976437i	$0.569237\pi$
$822$	−12445.6	−0.528092
$823$	3282.93	0.139047	0.0695235	−	0.997580i	$-0.477852\pi$
$823$	3282.93	0.139047	0.0695235	+	0.997580i	$0.477852\pi$
$824$	41543.1	1.75634
$825$	−20993.1	−0.885922
$826$	−7794.49	−0.328335
$827$	−17689.3	−0.743795	−0.371897	−	0.928274i	$-0.621293\pi$
$827$	−17689.3	−0.743795	−0.371897	+	0.928274i	$0.621293\pi$
$828$	23932.2	1.00447
$829$	38181.5	1.59964	0.799818	−	0.600243i	$-0.204930\pi$
$829$	38181.5	1.59964	0.799818	+	0.600243i	$0.204930\pi$
$830$	−9734.56	−0.407098
$831$	5438.40	0.227023
$832$	9530.06	0.397110
$833$	−27829.4	−1.15754
$834$	−4414.49	−0.183287
$835$	−492.615	−0.0204163
$836$	−30505.2	−1.26201
$837$	2086.64	0.0861708
$838$	3430.41	0.141410
$839$	43895.2	1.80623	0.903117	−	0.429395i	$-0.141273\pi$
$839$	43895.2	1.80623	0.903117	+	0.429395i	$0.141273\pi$
$840$	3117.18	0.128039
$841$	−11800.1	−0.483829
$842$	−70392.4	−2.88109
$843$	12219.2	0.499233
$844$	−53188.5	−2.16922
$845$	763.320	0.0310757
$846$	−1803.74	−0.0733024
$847$	−23532.0	−0.954627
$848$	−15876.8	−0.642938
$849$	−19038.9	−0.769626
$850$	−48093.0	−1.94068
$851$	10055.1	0.405034
$852$	4358.00	0.175238
$853$	−19955.2	−0.800998	−0.400499	−	0.916297i	$-0.631163\pi$
$853$	−19955.2	−0.800998	−0.400499	+	0.916297i	$0.631163\pi$
$854$	−29198.0	−1.16995
$855$	1279.80	0.0511910
$856$	34850.8	1.39156
$857$	26030.4	1.03755	0.518776	−	0.854910i	$-0.326388\pi$
$857$	26030.4	1.03755	0.518776	+	0.854910i	$0.326388\pi$
$858$	−12371.4	−0.492254
$859$	−45617.4	−1.81193	−0.905964	−	0.423354i	$-0.860853\pi$
$859$	−45617.4	−1.81193	−0.905964	+	0.423354i	$0.860853\pi$
$860$	−7420.45	−0.294227
$861$	10141.0	0.401399
$862$	5230.59	0.206676
$863$	−2010.93	−0.0793195	−0.0396597	−	0.999213i	$-0.512627\pi$
$863$	−2010.93	−0.0793195	−0.0396597	+	0.999213i	$0.512627\pi$
$864$	2812.27	0.110735
$865$	−8532.78	−0.335403
$866$	42170.1	1.65473
$867$	−13468.6	−0.527586
$868$	8376.19	0.327542
$869$	36478.2	1.42398
$870$	−7208.83	−0.280922
$871$	11337.1	0.441035
$872$	31284.7	1.21495
$873$	−9802.48	−0.380027
$874$	27408.3	1.06076
$875$	7760.41	0.299828
$876$	7196.79	0.277576
$877$	−36767.3	−1.41567	−0.707836	−	0.706376i	$-0.750329\pi$
$877$	−36767.3	−1.41567	−0.707836	+	0.706376i	$0.750329\pi$
$878$	−51692.0	−1.98693
$879$	25172.9	0.965940
$880$	−9034.72	−0.346091
$881$	35401.2	1.35380	0.676899	−	0.736076i	$-0.263323\pi$
$881$	35401.2	1.35380	0.676899	+	0.736076i	$0.263323\pi$
$882$	−12247.7	−0.467575
$883$	−11928.0	−0.454596	−0.227298	−	0.973825i	$-0.572989\pi$
$883$	−11928.0	−0.454596	−0.227298	+	0.973825i	$0.572989\pi$
$884$	−18257.2	−0.694633
$885$	−2976.49	−0.113055
$886$	−17930.0	−0.679875
$887$	−32939.3	−1.24689	−0.623447	−	0.781866i	$-0.714268\pi$
$887$	−32939.3	−1.24689	−0.623447	+	0.781866i	$0.714268\pi$
$888$	−5050.83	−0.190872
$889$	9340.40	0.352381
$890$	−4945.41	−0.186259
$891$	−5418.89	−0.203748
$892$	73677.3	2.76558
$893$	−1330.70	−0.0498660
$894$	−72.2720	−0.00270373
$895$	15482.6	0.578241
$896$	19776.6	0.737376
$897$	7160.38	0.266531
$898$	506.569	0.0188245
$899$	−8671.18	−0.321691
$900$	−13634.5	−0.504983
$901$	−51489.2	−1.90383
$902$	−143292.	−5.28946
$903$	−2546.58	−0.0938483
$904$	48266.4	1.77579
$905$	941.289	0.0345740
$906$	−93.1585	−0.00341610
$907$	1500.89	0.0549461	0.0274731	−	0.999623i	$-0.491254\pi$
$907$	1500.89	0.0549461	0.0274731	+	0.999623i	$0.491254\pi$
$908$	−38030.7	−1.38997
$909$	−695.102	−0.0253631
$910$	2083.47	0.0758971
$911$	19728.2	0.717481	0.358740	−	0.933437i	$-0.383206\pi$
$911$	19728.2	0.717481	0.358740	+	0.933437i	$0.383206\pi$
$912$	−2824.04	−0.102537
$913$	30408.3	1.10226
$914$	−5868.48	−0.212376
$915$	−11149.9	−0.402845
$916$	24313.5	0.877011
$917$	9539.72	0.343543
$918$	−12414.1	−0.446326
$919$	19992.0	0.717602	0.358801	−	0.933414i	$-0.383186\pi$
$919$	19992.0	0.717602	0.358801	+	0.933414i	$0.383186\pi$
$920$	25492.9	0.913560
$921$	−12015.8	−0.429897
$922$	41683.4	1.48891
$923$	1303.89	0.0464984
$924$	−21752.5	−0.774463
$925$	−5728.54	−0.203625
$926$	−18310.2	−0.649794
$927$	12162.3	0.430918
$928$	−11686.6	−0.413395
$929$	−1923.62	−0.0679354	−0.0339677	−	0.999423i	$-0.510814\pi$
$929$	−1923.62	−0.0679354	−0.0339677	+	0.999423i	$0.510814\pi$
$930$	4965.42	0.175078
$931$	−9035.71	−0.318081
$932$	9392.52	0.330110
$933$	−17509.6	−0.614403
$934$	−42634.0	−1.49361
$935$	−29300.0	−1.02483
$936$	−3596.77	−0.125603
$937$	−10252.9	−0.357468	−0.178734	−	0.983897i	$-0.557200\pi$
$937$	−10252.9	−0.357468	−0.178734	+	0.983897i	$0.557200\pi$
$938$	30944.4	1.07715
$939$	−5289.58	−0.183833
$940$	−2764.96	−0.0959394
$941$	11043.0	0.382563	0.191282	−	0.981535i	$-0.438736\pi$
$941$	11043.0	0.382563	0.191282	+	0.981535i	$0.438736\pi$
$942$	32683.2	1.13044
$943$	82934.8	2.86398
$944$	6568.00	0.226451
$945$	912.594	0.0314145
$946$	35983.1	1.23669
$947$	32105.2	1.10167	0.550833	−	0.834615i	$-0.314310\pi$
$947$	32105.2	1.10167	0.550833	+	0.834615i	$0.314310\pi$
$948$	23691.8	0.811680
$949$	2153.24	0.0736535
$950$	−15614.9	−0.533280
$951$	18319.0	0.624643
$952$	−22307.1	−0.759431
$953$	−9473.37	−0.322007	−0.161003	−	0.986954i	$-0.551473\pi$
$953$	−9473.37	−0.322007	−0.161003	+	0.986954i	$0.551473\pi$
$954$	−22660.3	−0.769031
$955$	4325.79	0.146575
$956$	75598.4	2.55756
$957$	22518.5	0.760628
$958$	19819.3	0.668404
$959$	−6547.26	−0.220461
$960$	9933.30	0.333954
$961$	−23818.3	−0.799514
$962$	−3375.89	−0.113142
$963$	10203.0	0.341419
$964$	88399.8	2.95349
$965$	−2315.84	−0.0772534
$966$	19544.2	0.650956
$967$	−5310.75	−0.176610	−0.0883052	−	0.996093i	$-0.528145\pi$
$967$	−5310.75	−0.176610	−0.0883052	+	0.996093i	$0.528145\pi$
$968$	96670.1	3.20981
$969$	−9158.49	−0.303626
$970$	−23326.2	−0.772122
$971$	24271.2	0.802164	0.401082	−	0.916042i	$-0.368634\pi$
$971$	24271.2	0.802164	0.401082	+	0.916042i	$0.368634\pi$
$972$	−3519.45	−0.116138
$973$	−2322.32	−0.0765163
$974$	−87454.2	−2.87702
$975$	−4079.38	−0.133995
$976$	24603.6	0.806907
$977$	−49602.5	−1.62428	−0.812142	−	0.583460i	$-0.801699\pi$
$977$	−49602.5	−1.62428	−0.812142	+	0.583460i	$0.801699\pi$
$978$	15441.0	0.504856
$979$	15448.2	0.504317
$980$	−18774.6	−0.611971
$981$	9158.98	0.298087
$982$	38375.5	1.24706
$983$	47385.7	1.53751	0.768753	−	0.639545i	$-0.220877\pi$
$983$	47385.7	1.53751	0.768753	+	0.639545i	$0.220877\pi$
$984$	−41659.5	−1.34965
$985$	−17481.2	−0.565478
$986$	51587.7	1.66621
$987$	−948.891	−0.0306014
$988$	−5927.78	−0.190878
$989$	−20826.4	−0.669607
$990$	−12894.9	−0.413966
$991$	−8947.33	−0.286802	−0.143401	−	0.989665i	$-0.545804\pi$
$991$	−8947.33	−0.286802	−0.143401	+	0.989665i	$0.545804\pi$
$992$	8049.67	0.257638
$993$	−22471.1	−0.718127
$994$	3558.95	0.113564
$995$	10414.7	0.331828
$996$	19749.5	0.628299
$997$	−14908.6	−0.473582	−0.236791	−	0.971561i	$-0.576096\pi$
$997$	−14908.6	−0.473582	−0.236791	+	0.971561i	$0.576096\pi$
$998$	−51881.4	−1.64557
$999$	−1478.69	−0.0468306

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.4.a.b.1.2	✓	2
3.2	odd	2		117.4.a.c.1.1		2
4.3	odd	2		624.4.a.r.1.1		2
5.4	even	2		975.4.a.j.1.1		2
7.6	odd	2		1911.4.a.h.1.2		2
8.3	odd	2		2496.4.a.s.1.2		2
8.5	even	2		2496.4.a.bc.1.2		2
12.11	even	2		1872.4.a.t.1.2		2
13.5	odd	4		507.4.b.f.337.1		4
13.8	odd	4		507.4.b.f.337.4		4
13.12	even	2		507.4.a.f.1.1		2
39.38	odd	2		1521.4.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.b.1.2	✓	2	1.1	even	1	trivial
117.4.a.c.1.1		2	3.2	odd	2
507.4.a.f.1.1		2	13.12	even	2
507.4.b.f.337.1		4	13.5	odd	4
507.4.b.f.337.4		4	13.8	odd	4
624.4.a.r.1.1		2	4.3	odd	2
975.4.a.j.1.1		2	5.4	even	2
1521.4.a.s.1.2		2	39.38	odd	2
1872.4.a.t.1.2		2	12.11	even	2
1911.4.a.h.1.2		2	7.6	odd	2
2496.4.a.s.1.2		2	8.3	odd	2
2496.4.a.bc.1.2		2	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+4.74166 q^{2} -3.00000 q^{3} +14.4833 q^{4} +4.51669 q^{5} -14.2250 q^{6} -7.48331 q^{7} +30.7417 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+4.74166 q^{2} -3.00000 q^{3} +14.4833 q^{4} +4.51669 q^{5} -14.2250 q^{6} -7.48331 q^{7} +30.7417 q^{8} +9.00000 q^{9} +21.4166 q^{10} -66.8999 q^{11} -43.4499 q^{12} -13.0000 q^{13} -35.4833 q^{14} -13.5501 q^{15} +29.8999 q^{16} +96.9666 q^{17} +42.6749 q^{18} +31.4833 q^{19} +65.4166 q^{20} +22.4499 q^{21} -317.216 q^{22} +183.600 q^{23} -92.2250 q^{24} -104.600 q^{25} -61.6415 q^{26} -27.0000 q^{27} -108.383 q^{28} +112.200 q^{29} -64.2497 q^{30} -77.2831 q^{31} -104.158 q^{32} +200.700 q^{33} +459.783 q^{34} -33.7998 q^{35} +130.350 q^{36} +54.7664 q^{37} +149.283 q^{38} +39.0000 q^{39} +138.850 q^{40} +451.716 q^{41} +106.450 q^{42} -113.434 q^{43} -968.932 q^{44} +40.6502 q^{45} +870.566 q^{46} -42.2670 q^{47} -89.6997 q^{48} -287.000 q^{49} -495.975 q^{50} -290.900 q^{51} -188.283 q^{52} -530.999 q^{53} -128.025 q^{54} -302.166 q^{55} -230.049 q^{56} -94.4499 q^{57} +532.015 q^{58} +219.666 q^{59} -196.250 q^{60} +822.865 q^{61} -366.450 q^{62} -67.3498 q^{63} -733.082 q^{64} -58.7169 q^{65} +951.649 q^{66} -872.082 q^{67} +1404.40 q^{68} -550.799 q^{69} -160.267 q^{70} -100.299 q^{71} +276.675 q^{72} -165.634 q^{73} +259.684 q^{74} +313.799 q^{75} +455.983 q^{76} +500.633 q^{77} +184.925 q^{78} -545.266 q^{79} +135.048 q^{80} +81.0000 q^{81} +2141.88 q^{82} -454.534 q^{83} +325.150 q^{84} +437.968 q^{85} -537.864 q^{86} -336.601 q^{87} -2056.61 q^{88} -230.915 q^{89} +192.749 q^{90} +97.2831 q^{91} +2659.13 q^{92} +231.849 q^{93} -200.415 q^{94} +142.200 q^{95} +312.475 q^{96} -1089.16 q^{97} -1360.86 q^{98} -602.099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 6 q^{3} + 14 q^{4} + 24 q^{5} - 6 q^{6} + 54 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 6 * q^3 + 14 * q^4 + 24 * q^5 - 6 * q^6 + 54 * q^8 + 18 * q^9	\( 2 q + 2 q^{2} - 6 q^{3} + 14 q^{4} + 24 q^{5} - 6 q^{6} + 54 q^{8} + 18 q^{9} - 32 q^{10} - 44 q^{11} - 42 q^{12} - 26 q^{13} - 56 q^{14} - 72 q^{15} - 30 q^{16} + 164 q^{17} + 18 q^{18} + 48 q^{19} + 56 q^{20} - 380 q^{22} + 8 q^{23} - 162 q^{24} + 150 q^{25} - 26 q^{26} - 54 q^{27} - 112 q^{28} + 404 q^{29} + 96 q^{30} + 40 q^{31} - 126 q^{32} + 132 q^{33} + 276 q^{34} + 112 q^{35} + 126 q^{36} - 100 q^{37} + 104 q^{38} + 78 q^{39} + 592 q^{40} + 200 q^{41} + 168 q^{42} - 616 q^{43} - 980 q^{44} + 216 q^{45} + 1352 q^{46} - 324 q^{47} + 90 q^{48} - 574 q^{49} - 1194 q^{50} - 492 q^{51} - 182 q^{52} - 164 q^{53} - 54 q^{54} + 144 q^{55} - 56 q^{56} - 144 q^{57} - 268 q^{58} + 140 q^{59} - 168 q^{60} + 628 q^{61} - 688 q^{62} - 194 q^{64} - 312 q^{65} + 1140 q^{66} - 472 q^{67} + 1372 q^{68} - 24 q^{69} - 560 q^{70} + 428 q^{71} + 486 q^{72} - 900 q^{73} + 684 q^{74} - 450 q^{75} + 448 q^{76} + 672 q^{77} + 78 q^{78} - 432 q^{79} - 1032 q^{80} + 162 q^{81} + 2832 q^{82} - 1388 q^{83} + 336 q^{84} + 1744 q^{85} + 840 q^{86} - 1212 q^{87} - 1524 q^{88} + 960 q^{89} - 288 q^{90} + 2744 q^{92} - 120 q^{93} + 572 q^{94} + 464 q^{95} + 378 q^{96} - 532 q^{97} - 574 q^{98} - 396 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 6 * q^3 + 14 * q^4 + 24 * q^5 - 6 * q^6 + 54 * q^8 + 18 * q^9 - 32 * q^10 - 44 * q^11 - 42 * q^12 - 26 * q^13 - 56 * q^14 - 72 * q^15 - 30 * q^16 + 164 * q^17 + 18 * q^18 + 48 * q^19 + 56 * q^20 - 380 * q^22 + 8 * q^23 - 162 * q^24 + 150 * q^25 - 26 * q^26 - 54 * q^27 - 112 * q^28 + 404 * q^29 + 96 * q^30 + 40 * q^31 - 126 * q^32 + 132 * q^33 + 276 * q^34 + 112 * q^35 + 126 * q^36 - 100 * q^37 + 104 * q^38 + 78 * q^39 + 592 * q^40 + 200 * q^41 + 168 * q^42 - 616 * q^43 - 980 * q^44 + 216 * q^45 + 1352 * q^46 - 324 * q^47 + 90 * q^48 - 574 * q^49 - 1194 * q^50 - 492 * q^51 - 182 * q^52 - 164 * q^53 - 54 * q^54 + 144 * q^55 - 56 * q^56 - 144 * q^57 - 268 * q^58 + 140 * q^59 - 168 * q^60 + 628 * q^61 - 688 * q^62 - 194 * q^64 - 312 * q^65 + 1140 * q^66 - 472 * q^67 + 1372 * q^68 - 24 * q^69 - 560 * q^70 + 428 * q^71 + 486 * q^72 - 900 * q^73 + 684 * q^74 - 450 * q^75 + 448 * q^76 + 672 * q^77 + 78 * q^78 - 432 * q^79 - 1032 * q^80 + 162 * q^81 + 2832 * q^82 - 1388 * q^83 + 336 * q^84 + 1744 * q^85 + 840 * q^86 - 1212 * q^87 - 1524 * q^88 + 960 * q^89 - 288 * q^90 + 2744 * q^92 - 120 * q^93 + 572 * q^94 + 464 * q^95 + 378 * q^96 - 532 * q^97 - 574 * q^98 - 396 * q^99

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