LMFDB - Embedded newform 6045.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
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			$2.17009$ of defining polynomial
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.539189 q^{2} +1.00000 q^{3} -1.70928 q^{4} -1.00000 q^{5} -0.539189 q^{6} +4.17009 q^{7} +2.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.539189 q^{2} +1.00000 q^{3} -1.70928 q^{4} -1.00000 q^{5} -0.539189 q^{6} +4.17009 q^{7} +2.00000 q^{8} +1.00000 q^{9} +0.539189 q^{10} +0.921622 q^{11} -1.70928 q^{12} +1.00000 q^{13} -2.24846 q^{14} -1.00000 q^{15} +2.34017 q^{16} +5.07838 q^{17} -0.539189 q^{18} +7.80098 q^{19} +1.70928 q^{20} +4.17009 q^{21} -0.496928 q^{22} +3.07838 q^{23} +2.00000 q^{24} +1.00000 q^{25} -0.539189 q^{26} +1.00000 q^{27} -7.12783 q^{28} +0.800984 q^{29} +0.539189 q^{30} +1.00000 q^{31} -5.26180 q^{32} +0.921622 q^{33} -2.73820 q^{34} -4.17009 q^{35} -1.70928 q^{36} -7.60197 q^{37} -4.20620 q^{38} +1.00000 q^{39} -2.00000 q^{40} +10.3763 q^{41} -2.24846 q^{42} -0.800984 q^{43} -1.57531 q^{44} -1.00000 q^{45} -1.65983 q^{46} +3.46081 q^{47} +2.34017 q^{48} +10.3896 q^{49} -0.539189 q^{50} +5.07838 q^{51} -1.70928 q^{52} +4.72261 q^{53} -0.539189 q^{54} -0.921622 q^{55} +8.34017 q^{56} +7.80098 q^{57} -0.431882 q^{58} +2.32684 q^{59} +1.70928 q^{60} -4.38243 q^{61} -0.539189 q^{62} +4.17009 q^{63} -1.84324 q^{64} -1.00000 q^{65} -0.496928 q^{66} +0.460811 q^{67} -8.68035 q^{68} +3.07838 q^{69} +2.24846 q^{70} +1.26180 q^{71} +2.00000 q^{72} -12.0072 q^{73} +4.09890 q^{74} +1.00000 q^{75} -13.3340 q^{76} +3.84324 q^{77} -0.539189 q^{78} -4.19902 q^{79} -2.34017 q^{80} +1.00000 q^{81} -5.59478 q^{82} -1.68035 q^{83} -7.12783 q^{84} -5.07838 q^{85} +0.431882 q^{86} +0.800984 q^{87} +1.84324 q^{88} -10.4391 q^{89} +0.539189 q^{90} +4.17009 q^{91} -5.26180 q^{92} +1.00000 q^{93} -1.86603 q^{94} -7.80098 q^{95} -5.26180 q^{96} +10.7165 q^{97} -5.60197 q^{98} +0.921622 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 2 q^{4} - 3 q^{5} + 7 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 2 * q^4 - 3 * q^5 + 7 * q^7 + 6 * q^8 + 3 * q^9	$ 3 q + 3 q^{3} + 2 q^{4} - 3 q^{5} + 7 q^{7} + 6 q^{8} + 3 q^{9} + 6 q^{11} + 2 q^{12} + 3 q^{13} + 2 q^{14} - 3 q^{15} - 4 q^{16} + 12 q^{17} + 14 q^{19} - 2 q^{20} + 7 q^{21} + 16 q^{22} + 6 q^{23} + 6 q^{24} + 3 q^{25} + 3 q^{27} - 7 q^{29} + 3 q^{31} - 8 q^{32} + 6 q^{33} - 16 q^{34} - 7 q^{35} + 2 q^{36} - 4 q^{37} + 12 q^{38} + 3 q^{39} - 6 q^{40} + q^{41} + 2 q^{42} + 7 q^{43} + 16 q^{44} - 3 q^{45} - 16 q^{46} + 12 q^{47} - 4 q^{48} + 2 q^{49} + 12 q^{51} + 2 q^{52} + 8 q^{53} - 6 q^{55} + 14 q^{56} + 14 q^{57} + 12 q^{58} - 5 q^{59} - 2 q^{60} - 18 q^{61} + 7 q^{63} - 12 q^{64} - 3 q^{65} + 16 q^{66} + 3 q^{67} - 4 q^{68} + 6 q^{69} - 2 q^{70} - 4 q^{71} + 6 q^{72} - 2 q^{73} - 24 q^{74} + 3 q^{75} + 6 q^{76} + 18 q^{77} - 22 q^{79} + 4 q^{80} + 3 q^{81} - 32 q^{82} + 17 q^{83} - 12 q^{85} - 12 q^{86} - 7 q^{87} + 12 q^{88} + 16 q^{89} + 7 q^{91} - 8 q^{92} + 3 q^{93} + 8 q^{94} - 14 q^{95} - 8 q^{96} - 9 q^{97} + 2 q^{98} + 6 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 2 * q^4 - 3 * q^5 + 7 * q^7 + 6 * q^8 + 3 * q^9 + 6 * q^11 + 2 * q^12 + 3 * q^13 + 2 * q^14 - 3 * q^15 - 4 * q^16 + 12 * q^17 + 14 * q^19 - 2 * q^20 + 7 * q^21 + 16 * q^22 + 6 * q^23 + 6 * q^24 + 3 * q^25 + 3 * q^27 - 7 * q^29 + 3 * q^31 - 8 * q^32 + 6 * q^33 - 16 * q^34 - 7 * q^35 + 2 * q^36 - 4 * q^37 + 12 * q^38 + 3 * q^39 - 6 * q^40 + q^41 + 2 * q^42 + 7 * q^43 + 16 * q^44 - 3 * q^45 - 16 * q^46 + 12 * q^47 - 4 * q^48 + 2 * q^49 + 12 * q^51 + 2 * q^52 + 8 * q^53 - 6 * q^55 + 14 * q^56 + 14 * q^57 + 12 * q^58 - 5 * q^59 - 2 * q^60 - 18 * q^61 + 7 * q^63 - 12 * q^64 - 3 * q^65 + 16 * q^66 + 3 * q^67 - 4 * q^68 + 6 * q^69 - 2 * q^70 - 4 * q^71 + 6 * q^72 - 2 * q^73 - 24 * q^74 + 3 * q^75 + 6 * q^76 + 18 * q^77 - 22 * q^79 + 4 * q^80 + 3 * q^81 - 32 * q^82 + 17 * q^83 - 12 * q^85 - 12 * q^86 - 7 * q^87 + 12 * q^88 + 16 * q^89 + 7 * q^91 - 8 * q^92 + 3 * q^93 + 8 * q^94 - 14 * q^95 - 8 * q^96 - 9 * q^97 + 2 * q^98 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.539189	−0.381264	−0.190632	−	0.981662i	$-0.561054\pi$
$2$	−0.539189	−0.381264	−0.190632	+	0.981662i	$0.561054\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.70928	−0.854638
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.539189	−0.220123
$7$	4.17009	1.57614	0.788072	−	0.615583i	$-0.211079\pi$
$7$	4.17009	1.57614	0.788072	+	0.615583i	$0.211079\pi$
$8$	2.00000	0.707107
$9$	1.00000	0.333333
$10$	0.539189	0.170506
$11$	0.921622	0.277880	0.138940	−	0.990301i	$-0.455631\pi$
$11$	0.921622	0.277880	0.138940	+	0.990301i	$0.455631\pi$
$12$	−1.70928	−0.493425
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−2.24846	−0.600927
$15$	−1.00000	−0.258199
$16$	2.34017	0.585043
$17$	5.07838	1.23169	0.615844	−	0.787868i	$-0.288815\pi$
$17$	5.07838	1.23169	0.615844	+	0.787868i	$0.288815\pi$
$18$	−0.539189	−0.127088
$19$	7.80098	1.78967	0.894834	−	0.446399i	$-0.147294\pi$
$19$	7.80098	1.78967	0.894834	+	0.446399i	$0.147294\pi$
$20$	1.70928	0.382206
$21$	4.17009	0.909987
$22$	−0.496928	−0.105946
$23$	3.07838	0.641886	0.320943	−	0.947098i	$-0.396000\pi$
$23$	3.07838	0.641886	0.320943	+	0.947098i	$0.396000\pi$
$24$	2.00000	0.408248
$25$	1.00000	0.200000
$26$	−0.539189	−0.105744
$27$	1.00000	0.192450
$28$	−7.12783	−1.34703
$29$	0.800984	0.148739	0.0743695	−	0.997231i	$-0.476306\pi$
$29$	0.800984	0.148739	0.0743695	+	0.997231i	$0.476306\pi$
$30$	0.539189	0.0984420
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−5.26180	−0.930163
$33$	0.921622	0.160434
$34$	−2.73820	−0.469598
$35$	−4.17009	−0.704873
$36$	−1.70928	−0.284879
$37$	−7.60197	−1.24976	−0.624878	−	0.780722i	$-0.714851\pi$
$37$	−7.60197	−1.24976	−0.624878	+	0.780722i	$0.714851\pi$
$38$	−4.20620	−0.682336
$39$	1.00000	0.160128
$40$	−2.00000	−0.316228
$41$	10.3763	1.62050	0.810252	−	0.586081i	$-0.199330\pi$
$41$	10.3763	1.62050	0.810252	+	0.586081i	$0.199330\pi$
$42$	−2.24846	−0.346946
$43$	−0.800984	−0.122149	−0.0610745	−	0.998133i	$-0.519453\pi$
$43$	−0.800984	−0.122149	−0.0610745	+	0.998133i	$0.519453\pi$
$44$	−1.57531	−0.237486
$45$	−1.00000	−0.149071
$46$	−1.65983	−0.244728
$47$	3.46081	0.504811	0.252406	−	0.967622i	$-0.418778\pi$
$47$	3.46081	0.504811	0.252406	+	0.967622i	$0.418778\pi$
$48$	2.34017	0.337775
$49$	10.3896	1.48423
$50$	−0.539189	−0.0762528
$51$	5.07838	0.711115
$52$	−1.70928	−0.237034
$53$	4.72261	0.648700	0.324350	−	0.945937i	$-0.394854\pi$
$53$	4.72261	0.648700	0.324350	+	0.945937i	$0.394854\pi$
$54$	−0.539189	−0.0733743
$55$	−0.921622	−0.124272
$56$	8.34017	1.11450
$57$	7.80098	1.03327
$58$	−0.431882	−0.0567088
$59$	2.32684	0.302929	0.151465	−	0.988463i	$-0.451601\pi$
$59$	2.32684	0.302929	0.151465	+	0.988463i	$0.451601\pi$
$60$	1.70928	0.220667
$61$	−4.38243	−0.561113	−0.280557	−	0.959837i	$-0.590519\pi$
$61$	−4.38243	−0.561113	−0.280557	+	0.959837i	$0.590519\pi$
$62$	−0.539189	−0.0684771
$63$	4.17009	0.525382
$64$	−1.84324	−0.230406
$65$	−1.00000	−0.124035
$66$	−0.496928	−0.0611677
$67$	0.460811	0.0562970	0.0281485	−	0.999604i	$-0.491039\pi$
$67$	0.460811	0.0562970	0.0281485	+	0.999604i	$0.491039\pi$
$68$	−8.68035	−1.05265
$69$	3.07838	0.370593
$70$	2.24846	0.268743
$71$	1.26180	0.149748	0.0748738	−	0.997193i	$-0.476145\pi$
$71$	1.26180	0.149748	0.0748738	+	0.997193i	$0.476145\pi$
$72$	2.00000	0.235702
$73$	−12.0072	−1.40534	−0.702668	−	0.711518i	$-0.748008\pi$
$73$	−12.0072	−1.40534	−0.702668	+	0.711518i	$0.748008\pi$
$74$	4.09890	0.476487
$75$	1.00000	0.115470
$76$	−13.3340	−1.52952
$77$	3.84324	0.437978
$78$	−0.539189	−0.0610511
$79$	−4.19902	−0.472426	−0.236213	−	0.971701i	$-0.575906\pi$
$79$	−4.19902	−0.472426	−0.236213	+	0.971701i	$0.575906\pi$
$80$	−2.34017	−0.261639
$81$	1.00000	0.111111
$82$	−5.59478	−0.617840
$83$	−1.68035	−0.184442	−0.0922210	−	0.995739i	$-0.529397\pi$
$83$	−1.68035	−0.184442	−0.0922210	+	0.995739i	$0.529397\pi$
$84$	−7.12783	−0.777710
$85$	−5.07838	−0.550827
$86$	0.431882	0.0465710
$87$	0.800984	0.0858745
$88$	1.84324	0.196491
$89$	−10.4391	−1.10654	−0.553270	−	0.833002i	$-0.686620\pi$
$89$	−10.4391	−1.10654	−0.553270	+	0.833002i	$0.686620\pi$
$90$	0.539189	0.0568355
$91$	4.17009	0.437144
$92$	−5.26180	−0.548580
$93$	1.00000	0.103695
$94$	−1.86603	−0.192466
$95$	−7.80098	−0.800364
$96$	−5.26180	−0.537030
$97$	10.7165	1.08809	0.544046	−	0.839055i	$-0.316892\pi$
$97$	10.7165	1.08809	0.544046	+	0.839055i	$0.316892\pi$
$98$	−5.60197	−0.565884
$99$	0.921622	0.0926265
$100$	−1.70928	−0.170928
$101$	−6.74539	−0.671192	−0.335596	−	0.942006i	$-0.608938\pi$
$101$	−6.74539	−0.671192	−0.335596	+	0.942006i	$0.608938\pi$
$102$	−2.73820	−0.271123
$103$	−11.6670	−1.14959	−0.574793	−	0.818299i	$-0.694917\pi$
$103$	−11.6670	−1.14959	−0.574793	+	0.818299i	$0.694917\pi$
$104$	2.00000	0.196116
$105$	−4.17009	−0.406959
$106$	−2.54638	−0.247326
$107$	4.55252	0.440109	0.220054	−	0.975488i	$-0.429377\pi$
$107$	4.55252	0.440109	0.220054	+	0.975488i	$0.429377\pi$
$108$	−1.70928	−0.164475
$109$	−11.5597	−1.10722	−0.553610	−	0.832776i	$-0.686750\pi$
$109$	−11.5597	−1.10722	−0.553610	+	0.832776i	$0.686750\pi$
$110$	0.496928	0.0473803
$111$	−7.60197	−0.721547
$112$	9.75872	0.922113
$113$	−3.04945	−0.286868	−0.143434	−	0.989660i	$-0.545814\pi$
$113$	−3.04945	−0.286868	−0.143434	+	0.989660i	$0.545814\pi$
$114$	−4.20620	−0.393947
$115$	−3.07838	−0.287060
$116$	−1.36910	−0.127118
$117$	1.00000	0.0924500
$118$	−1.25461	−0.115496
$119$	21.1773	1.94132
$120$	−2.00000	−0.182574
$121$	−10.1506	−0.922783
$122$	2.36296	0.213932
$123$	10.3763	0.935599
$124$	−1.70928	−0.153497
$125$	−1.00000	−0.0894427
$126$	−2.24846	−0.200309
$127$	−2.70209	−0.239771	−0.119886	−	0.992788i	$-0.538253\pi$
$127$	−2.70209	−0.239771	−0.119886	+	0.992788i	$0.538253\pi$
$128$	11.5174	1.01801
$129$	−0.800984	−0.0705227
$130$	0.539189	0.0472900
$131$	−0.248464	−0.0217084	−0.0108542	−	0.999941i	$-0.503455\pi$
$131$	−0.248464	−0.0217084	−0.0108542	+	0.999941i	$0.503455\pi$
$132$	−1.57531	−0.137113
$133$	32.5308	2.82078
$134$	−0.248464	−0.0214640
$135$	−1.00000	−0.0860663
$136$	10.1568	0.870935
$137$	14.7877	1.26339	0.631697	−	0.775215i	$-0.282359\pi$
$137$	14.7877	1.26339	0.631697	+	0.775215i	$0.282359\pi$
$138$	−1.65983	−0.141294
$139$	−18.3968	−1.56040	−0.780198	−	0.625532i	$-0.784882\pi$
$139$	−18.3968	−1.56040	−0.780198	+	0.625532i	$0.784882\pi$
$140$	7.12783	0.602411
$141$	3.46081	0.291453
$142$	−0.680346	−0.0570934
$143$	0.921622	0.0770699
$144$	2.34017	0.195014
$145$	−0.800984	−0.0665181
$146$	6.47414	0.535804
$147$	10.3896	0.856922
$148$	12.9939	1.06809
$149$	−1.27513	−0.104462	−0.0522312	−	0.998635i	$-0.516633\pi$
$149$	−1.27513	−0.104462	−0.0522312	+	0.998635i	$0.516633\pi$
$150$	−0.539189	−0.0440246
$151$	−23.3402	−1.89940	−0.949698	−	0.313167i	$-0.898610\pi$
$151$	−23.3402	−1.89940	−0.949698	+	0.313167i	$0.898610\pi$
$152$	15.6020	1.26549
$153$	5.07838	0.410563
$154$	−2.07223	−0.166985
$155$	−1.00000	−0.0803219
$156$	−1.70928	−0.136852
$157$	15.3607	1.22592	0.612958	−	0.790115i	$-0.289979\pi$
$157$	15.3607	1.22592	0.612958	+	0.790115i	$0.289979\pi$
$158$	2.26406	0.180119
$159$	4.72261	0.374527
$160$	5.26180	0.415981
$161$	12.8371	1.01171
$162$	−0.539189	−0.0423627
$163$	12.6803	0.993201	0.496601	−	0.867979i	$-0.334581\pi$
$163$	12.6803	0.993201	0.496601	+	0.867979i	$0.334581\pi$
$164$	−17.7359	−1.38494
$165$	−0.921622	−0.0717482
$166$	0.906024	0.0703211
$167$	−10.5464	−0.816103	−0.408052	−	0.912959i	$-0.633792\pi$
$167$	−10.5464	−0.816103	−0.408052	+	0.912959i	$0.633792\pi$
$168$	8.34017	0.643458
$169$	1.00000	0.0769231
$170$	2.73820	0.210011
$171$	7.80098	0.596556
$172$	1.36910	0.104393
$173$	4.62475	0.351614	0.175807	−	0.984425i	$-0.443747\pi$
$173$	4.62475	0.351614	0.175807	+	0.984425i	$0.443747\pi$
$174$	−0.431882	−0.0327409
$175$	4.17009	0.315229
$176$	2.15676	0.162572
$177$	2.32684	0.174896
$178$	5.62863	0.421884
$179$	−4.64423	−0.347126	−0.173563	−	0.984823i	$-0.555528\pi$
$179$	−4.64423	−0.347126	−0.173563	+	0.984823i	$0.555528\pi$
$180$	1.70928	0.127402
$181$	−1.05172	−0.0781734	−0.0390867	−	0.999236i	$-0.512445\pi$
$181$	−1.05172	−0.0781734	−0.0390867	+	0.999236i	$0.512445\pi$
$182$	−2.24846	−0.166667
$183$	−4.38243	−0.323959
$184$	6.15676	0.453882
$185$	7.60197	0.558908
$186$	−0.539189	−0.0395352
$187$	4.68035	0.342261
$188$	−5.91548	−0.431431
$189$	4.17009	0.303329
$190$	4.20620	0.305150
$191$	23.6020	1.70778	0.853889	−	0.520455i	$-0.174238\pi$
$191$	23.6020	1.70778	0.853889	+	0.520455i	$0.174238\pi$
$192$	−1.84324	−0.133025
$193$	−6.17009	−0.444133	−0.222066	−	0.975032i	$-0.571280\pi$
$193$	−6.17009	−0.444133	−0.222066	+	0.975032i	$0.571280\pi$
$194$	−5.77820	−0.414850
$195$	−1.00000	−0.0716115
$196$	−17.7587	−1.26848
$197$	−23.1978	−1.65277	−0.826387	−	0.563102i	$-0.809608\pi$
$197$	−23.1978	−1.65277	−0.826387	+	0.563102i	$0.809608\pi$
$198$	−0.496928	−0.0353152
$199$	16.1568	1.14532	0.572661	−	0.819792i	$-0.305911\pi$
$199$	16.1568	1.14532	0.572661	+	0.819792i	$0.305911\pi$
$200$	2.00000	0.141421
$201$	0.460811	0.0325031
$202$	3.63704	0.255901
$203$	3.34017	0.234434
$204$	−8.68035	−0.607746
$205$	−10.3763	−0.724712
$206$	6.29072	0.438296
$207$	3.07838	0.213962
$208$	2.34017	0.162262
$209$	7.18956	0.497312
$210$	2.24846	0.155159
$211$	25.4040	1.74888	0.874442	−	0.485131i	$-0.161228\pi$
$211$	25.4040	1.74888	0.874442	+	0.485131i	$0.161228\pi$
$212$	−8.07223	−0.554403
$213$	1.26180	0.0864568
$214$	−2.45467	−0.167798
$215$	0.800984	0.0546267
$216$	2.00000	0.136083
$217$	4.17009	0.283084
$218$	6.23287	0.422143
$219$	−12.0072	−0.811371
$220$	1.57531	0.106207
$221$	5.07838	0.341609
$222$	4.09890	0.275100
$223$	−14.3474	−0.960770	−0.480385	−	0.877058i	$-0.659503\pi$
$223$	−14.3474	−0.960770	−0.480385	+	0.877058i	$0.659503\pi$
$224$	−21.9421	−1.46607
$225$	1.00000	0.0666667
$226$	1.64423	0.109372
$227$	21.8310	1.44897	0.724486	−	0.689290i	$-0.242077\pi$
$227$	21.8310	1.44897	0.724486	+	0.689290i	$0.242077\pi$
$228$	−13.3340	−0.883068
$229$	6.20620	0.410117	0.205059	−	0.978750i	$-0.434261\pi$
$229$	6.20620	0.410117	0.205059	+	0.978750i	$0.434261\pi$
$230$	1.65983	0.109446
$231$	3.84324	0.252867
$232$	1.60197	0.105174
$233$	−25.8059	−1.69060	−0.845301	−	0.534291i	$-0.820579\pi$
$233$	−25.8059	−1.69060	−0.845301	+	0.534291i	$0.820579\pi$
$234$	−0.539189	−0.0352479
$235$	−3.46081	−0.225758
$236$	−3.97721	−0.258895
$237$	−4.19902	−0.272755
$238$	−11.4186	−0.740155
$239$	1.26898	0.0820837	0.0410418	−	0.999157i	$-0.486932\pi$
$239$	1.26898	0.0820837	0.0410418	+	0.999157i	$0.486932\pi$
$240$	−2.34017	−0.151058
$241$	−4.34244	−0.279721	−0.139861	−	0.990171i	$-0.544665\pi$
$241$	−4.34244	−0.279721	−0.139861	+	0.990171i	$0.544665\pi$
$242$	5.47310	0.351824
$243$	1.00000	0.0641500
$244$	7.49079	0.479548
$245$	−10.3896	−0.663769
$246$	−5.59478	−0.356710
$247$	7.80098	0.496365
$248$	2.00000	0.127000
$249$	−1.68035	−0.106488
$250$	0.539189	0.0341013
$251$	−4.35350	−0.274791	−0.137395	−	0.990516i	$-0.543873\pi$
$251$	−4.35350	−0.274791	−0.137395	+	0.990516i	$0.543873\pi$
$252$	−7.12783	−0.449011
$253$	2.83710	0.178367
$254$	1.45694	0.0914163
$255$	−5.07838	−0.318020
$256$	−2.52359	−0.157724
$257$	8.14834	0.508280	0.254140	−	0.967167i	$-0.418208\pi$
$257$	8.14834	0.508280	0.254140	+	0.967167i	$0.418208\pi$
$258$	0.431882	0.0268878
$259$	−31.7009	−1.96980
$260$	1.70928	0.106005
$261$	0.800984	0.0495797
$262$	0.133969	0.00827664
$263$	−26.2667	−1.61968	−0.809838	−	0.586654i	$-0.800445\pi$
$263$	−26.2667	−1.61968	−0.809838	+	0.586654i	$0.800445\pi$
$264$	1.84324	0.113444
$265$	−4.72261	−0.290107
$266$	−17.5402	−1.07546
$267$	−10.4391	−0.638861
$268$	−0.787653	−0.0481136
$269$	28.3545	1.72881	0.864404	−	0.502798i	$-0.167696\pi$
$269$	28.3545	1.72881	0.864404	+	0.502798i	$0.167696\pi$
$270$	0.539189	0.0328140
$271$	−31.6719	−1.92393	−0.961967	−	0.273167i	$-0.911929\pi$
$271$	−31.6719	−1.92393	−0.961967	+	0.273167i	$0.911929\pi$
$272$	11.8843	0.720590
$273$	4.17009	0.252385
$274$	−7.97334	−0.481687
$275$	0.921622	0.0555759
$276$	−5.26180	−0.316723
$277$	−13.9555	−0.838503	−0.419252	−	0.907870i	$-0.637707\pi$
$277$	−13.9555	−0.838503	−0.419252	+	0.907870i	$0.637707\pi$
$278$	9.91935	0.594923
$279$	1.00000	0.0598684
$280$	−8.34017	−0.498421
$281$	6.93495	0.413705	0.206852	−	0.978372i	$-0.433678\pi$
$281$	6.93495	0.413705	0.206852	+	0.978372i	$0.433678\pi$
$282$	−1.86603	−0.111121
$283$	−0.118371	−0.00703641	−0.00351820	−	0.999994i	$-0.501120\pi$
$283$	−0.118371	−0.00703641	−0.00351820	+	0.999994i	$0.501120\pi$
$284$	−2.15676	−0.127980
$285$	−7.80098	−0.462090
$286$	−0.496928	−0.0293840
$287$	43.2700	2.55415
$288$	−5.26180	−0.310054
$289$	8.78992	0.517054
$290$	0.431882	0.0253610
$291$	10.7165	0.628210
$292$	20.5236	1.20105
$293$	19.2195	1.12282	0.561409	−	0.827539i	$-0.310260\pi$
$293$	19.2195	1.12282	0.561409	+	0.827539i	$0.310260\pi$
$294$	−5.60197	−0.326713
$295$	−2.32684	−0.135474
$296$	−15.2039	−0.883711
$297$	0.921622	0.0534779
$298$	0.687534	0.0398278
$299$	3.07838	0.178027
$300$	−1.70928	−0.0986851
$301$	−3.34017	−0.192524
$302$	12.5848	0.724172
$303$	−6.74539	−0.387513
$304$	18.2557	1.04703
$305$	4.38243	0.250937
$306$	−2.73820	−0.156533
$307$	3.10116	0.176993	0.0884964	−	0.996076i	$-0.471794\pi$
$307$	3.10116	0.176993	0.0884964	+	0.996076i	$0.471794\pi$
$308$	−6.56916	−0.374313
$309$	−11.6670	−0.663713
$310$	0.539189	0.0306239
$311$	−31.1917	−1.76872	−0.884358	−	0.466809i	$-0.845404\pi$
$311$	−31.1917	−1.76872	−0.884358	+	0.466809i	$0.845404\pi$
$312$	2.00000	0.113228
$313$	13.2257	0.747560	0.373780	−	0.927517i	$-0.378062\pi$
$313$	13.2257	0.747560	0.373780	+	0.927517i	$0.378062\pi$
$314$	−8.28231	−0.467398
$315$	−4.17009	−0.234958
$316$	7.17727	0.403753
$317$	−5.06278	−0.284354	−0.142177	−	0.989841i	$-0.545410\pi$
$317$	−5.06278	−0.284354	−0.142177	+	0.989841i	$0.545410\pi$
$318$	−2.54638	−0.142794
$319$	0.738205	0.0413315
$320$	1.84324	0.103041
$321$	4.55252	0.254097
$322$	−6.92162	−0.385727
$323$	39.6163	2.20431
$324$	−1.70928	−0.0949597
$325$	1.00000	0.0554700
$326$	−6.83710	−0.378672
$327$	−11.5597	−0.639253
$328$	20.7526	1.14587
$329$	14.4319	0.795655
$330$	0.496928	0.0273550
$331$	−10.0556	−0.552705	−0.276353	−	0.961056i	$-0.589126\pi$
$331$	−10.0556	−0.552705	−0.276353	+	0.961056i	$0.589126\pi$
$332$	2.87217	0.157631
$333$	−7.60197	−0.416585
$334$	5.68649	0.311151
$335$	−0.460811	−0.0251768
$336$	9.75872	0.532382
$337$	21.5174	1.17213	0.586065	−	0.810264i	$-0.300676\pi$
$337$	21.5174	1.17213	0.586065	+	0.810264i	$0.300676\pi$
$338$	−0.539189	−0.0293280
$339$	−3.04945	−0.165623
$340$	8.68035	0.470758
$341$	0.921622	0.0499086
$342$	−4.20620	−0.227445
$343$	14.1350	0.763219
$344$	−1.60197	−0.0863723
$345$	−3.07838	−0.165734
$346$	−2.49362	−0.134058
$347$	34.5802	1.85636	0.928182	−	0.372127i	$-0.121371\pi$
$347$	34.5802	1.85636	0.928182	+	0.372127i	$0.121371\pi$
$348$	−1.36910	−0.0733916
$349$	11.8576	0.634724	0.317362	−	0.948304i	$-0.397203\pi$
$349$	11.8576	0.634724	0.317362	+	0.948304i	$0.397203\pi$
$350$	−2.24846	−0.120185
$351$	1.00000	0.0533761
$352$	−4.84939	−0.258473
$353$	−26.7321	−1.42280	−0.711402	−	0.702785i	$-0.751939\pi$
$353$	−26.7321	−1.42280	−0.711402	+	0.702785i	$0.751939\pi$
$354$	−1.25461	−0.0666816
$355$	−1.26180	−0.0669691
$356$	17.8432	0.945690
$357$	21.1773	1.12082
$358$	2.50412	0.132347
$359$	5.30632	0.280057	0.140029	−	0.990147i	$-0.455281\pi$
$359$	5.30632	0.280057	0.140029	+	0.990147i	$0.455281\pi$
$360$	−2.00000	−0.105409
$361$	41.8554	2.20291
$362$	0.567073	0.0298047
$363$	−10.1506	−0.532769
$364$	−7.12783	−0.373600
$365$	12.0072	0.628485
$366$	2.36296	0.123514
$367$	25.0433	1.30725	0.653625	−	0.756819i	$-0.273247\pi$
$367$	25.0433	1.30725	0.653625	+	0.756819i	$0.273247\pi$
$368$	7.20394	0.375531
$369$	10.3763	0.540168
$370$	−4.09890	−0.213091
$371$	19.6937	1.02244
$372$	−1.70928	−0.0886218
$373$	19.5441	1.01196	0.505978	−	0.862546i	$-0.331132\pi$
$373$	19.5441	1.01196	0.505978	+	0.862546i	$0.331132\pi$
$374$	−2.52359	−0.130492
$375$	−1.00000	−0.0516398
$376$	6.92162	0.356955
$377$	0.800984	0.0412528
$378$	−2.24846	−0.115649
$379$	−15.2918	−0.785485	−0.392743	−	0.919648i	$-0.628474\pi$
$379$	−15.2918	−0.785485	−0.392743	+	0.919648i	$0.628474\pi$
$380$	13.3340	0.684021
$381$	−2.70209	−0.138432
$382$	−12.7259	−0.651115
$383$	−20.1278	−1.02848	−0.514242	−	0.857645i	$-0.671927\pi$
$383$	−20.1278	−1.02848	−0.514242	+	0.857645i	$0.671927\pi$
$384$	11.5174	0.587747
$385$	−3.84324	−0.195870
$386$	3.32684	0.169332
$387$	−0.800984	−0.0407163
$388$	−18.3174	−0.929924
$389$	15.8794	0.805116	0.402558	−	0.915395i	$-0.368121\pi$
$389$	15.8794	0.805116	0.402558	+	0.915395i	$0.368121\pi$
$390$	0.539189	0.0273029
$391$	15.6332	0.790603
$392$	20.7792	1.04951
$393$	−0.248464	−0.0125334
$394$	12.5080	0.630144
$395$	4.19902	0.211275
$396$	−1.57531	−0.0791621
$397$	−19.7587	−0.991662	−0.495831	−	0.868419i	$-0.665136\pi$
$397$	−19.7587	−0.991662	−0.495831	+	0.868419i	$0.665136\pi$
$398$	−8.71154	−0.436670
$399$	32.5308	1.62858
$400$	2.34017	0.117009
$401$	−20.8710	−1.04225	−0.521123	−	0.853482i	$-0.674487\pi$
$401$	−20.8710	−1.04225	−0.521123	+	0.853482i	$0.674487\pi$
$402$	−0.248464	−0.0123923
$403$	1.00000	0.0498135
$404$	11.5297	0.573626
$405$	−1.00000	−0.0496904
$406$	−1.80098	−0.0893813
$407$	−7.00614	−0.347282
$408$	10.1568	0.502834
$409$	4.20847	0.208096	0.104048	−	0.994572i	$-0.466821\pi$
$409$	4.20847	0.208096	0.104048	+	0.994572i	$0.466821\pi$
$410$	5.59478	0.276307
$411$	14.7877	0.729421
$412$	19.9421	0.982479
$413$	9.70313	0.477460
$414$	−1.65983	−0.0815760
$415$	1.68035	0.0824849
$416$	−5.26180	−0.257981
$417$	−18.3968	−0.900896
$418$	−3.87653	−0.189607
$419$	11.5609	0.564788	0.282394	−	0.959298i	$-0.408871\pi$
$419$	11.5609	0.564788	0.282394	+	0.959298i	$0.408871\pi$
$420$	7.12783	0.347802
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	−13.6976	−0.666786
$423$	3.46081	0.168270
$424$	9.44521	0.458700
$425$	5.07838	0.246338
$426$	−0.680346	−0.0329629
$427$	−18.2751	−0.884395
$428$	−7.78151	−0.376133
$429$	0.921622	0.0444963
$430$	−0.431882	−0.0208272
$431$	6.38470	0.307540	0.153770	−	0.988107i	$-0.450858\pi$
$431$	6.38470	0.307540	0.153770	+	0.988107i	$0.450858\pi$
$432$	2.34017	0.112592
$433$	−28.4173	−1.36565	−0.682825	−	0.730582i	$-0.739249\pi$
$433$	−28.4173	−1.36565	−0.682825	+	0.730582i	$0.739249\pi$
$434$	−2.24846	−0.107930
$435$	−0.800984	−0.0384042
$436$	19.7587	0.946271
$437$	24.0144	1.14876
$438$	6.47414	0.309346
$439$	−30.1506	−1.43901	−0.719505	−	0.694487i	$-0.755631\pi$
$439$	−30.1506	−1.43901	−0.719505	+	0.694487i	$0.755631\pi$
$440$	−1.84324	−0.0878732
$441$	10.3896	0.494744
$442$	−2.73820	−0.130243
$443$	−14.1194	−0.670834	−0.335417	−	0.942070i	$-0.608877\pi$
$443$	−14.1194	−0.670834	−0.335417	+	0.942070i	$0.608877\pi$
$444$	12.9939	0.616661
$445$	10.4391	0.494859
$446$	7.73594	0.366307
$447$	−1.27513	−0.0603114
$448$	−7.68649	−0.363152
$449$	−0.106085	−0.00500644	−0.00250322	−	0.999997i	$-0.500797\pi$
$449$	−0.106085	−0.00500644	−0.00250322	+	0.999997i	$0.500797\pi$
$450$	−0.539189	−0.0254176
$451$	9.56302	0.450305
$452$	5.21235	0.245168
$453$	−23.3402	−1.09662
$454$	−11.7710	−0.552441
$455$	−4.17009	−0.195497
$456$	15.6020	0.730629
$457$	37.0010	1.73084	0.865418	−	0.501051i	$-0.167053\pi$
$457$	37.0010	1.73084	0.865418	+	0.501051i	$0.167053\pi$
$458$	−3.34632	−0.156363
$459$	5.07838	0.237038
$460$	5.26180	0.245332
$461$	−15.9421	−0.742500	−0.371250	−	0.928533i	$-0.621071\pi$
$461$	−15.9421	−0.742500	−0.371250	+	0.928533i	$0.621071\pi$
$462$	−2.07223	−0.0964091
$463$	18.9672	0.881480	0.440740	−	0.897635i	$-0.354716\pi$
$463$	18.9672	0.881480	0.440740	+	0.897635i	$0.354716\pi$
$464$	1.87444	0.0870188
$465$	−1.00000	−0.0463739
$466$	13.9143	0.644565
$467$	31.0905	1.43870	0.719348	−	0.694650i	$-0.244441\pi$
$467$	31.0905	1.43870	0.719348	+	0.694650i	$0.244441\pi$
$468$	−1.70928	−0.0790113
$469$	1.92162	0.0887323
$470$	1.86603	0.0860736
$471$	15.3607	0.707783
$472$	4.65368	0.214203
$473$	−0.738205	−0.0339427
$474$	2.26406	0.103992
$475$	7.80098	0.357934
$476$	−36.1978	−1.65912
$477$	4.72261	0.216233
$478$	−0.684222	−0.0312956
$479$	7.42347	0.339187	0.169594	−	0.985514i	$-0.445755\pi$
$479$	7.42347	0.339187	0.169594	+	0.985514i	$0.445755\pi$
$480$	5.26180	0.240167
$481$	−7.60197	−0.346620
$482$	2.34140	0.106648
$483$	12.8371	0.584108
$484$	17.3502	0.788645
$485$	−10.7165	−0.486610
$486$	−0.539189	−0.0244581
$487$	13.1701	0.596794	0.298397	−	0.954442i	$-0.403548\pi$
$487$	13.1701	0.596794	0.298397	+	0.954442i	$0.403548\pi$
$488$	−8.76487	−0.396767
$489$	12.6803	0.573425
$490$	5.60197	0.253071
$491$	23.7682	1.07264	0.536321	−	0.844014i	$-0.319813\pi$
$491$	23.7682	1.07264	0.536321	+	0.844014i	$0.319813\pi$
$492$	−17.7359	−0.799598
$493$	4.06770	0.183200
$494$	−4.20620	−0.189246
$495$	−0.921622	−0.0414238
$496$	2.34017	0.105077
$497$	5.26180	0.236024
$498$	0.906024	0.0405999
$499$	−23.4391	−1.04928	−0.524638	−	0.851325i	$-0.675799\pi$
$499$	−23.4391	−1.04928	−0.524638	+	0.851325i	$0.675799\pi$
$500$	1.70928	0.0764411
$501$	−10.5464	−0.471177
$502$	2.34736	0.104768
$503$	−20.4764	−0.912998	−0.456499	−	0.889724i	$-0.650897\pi$
$503$	−20.4764	−0.912998	−0.456499	+	0.889724i	$0.650897\pi$
$504$	8.34017	0.371501
$505$	6.74539	0.300166
$506$	−1.52973	−0.0680050
$507$	1.00000	0.0444116
$508$	4.61861	0.204918
$509$	31.3412	1.38918	0.694588	−	0.719408i	$-0.255587\pi$
$509$	31.3412	1.38918	0.694588	+	0.719408i	$0.255587\pi$
$510$	2.73820	0.121250
$511$	−50.0710	−2.21501
$512$	−21.6742	−0.957873
$513$	7.80098	0.344422
$514$	−4.39350	−0.193789
$515$	11.6670	0.514110
$516$	1.36910	0.0602714
$517$	3.18956	0.140277
$518$	17.0928	0.751012
$519$	4.62475	0.203004
$520$	−2.00000	−0.0877058
$521$	−0.547599	−0.0239908	−0.0119954	−	0.999928i	$-0.503818\pi$
$521$	−0.547599	−0.0239908	−0.0119954	+	0.999928i	$0.503818\pi$
$522$	−0.431882	−0.0189029
$523$	−11.5802	−0.506368	−0.253184	−	0.967418i	$-0.581478\pi$
$523$	−11.5802	−0.506368	−0.253184	+	0.967418i	$0.581478\pi$
$524$	0.424694	0.0185528
$525$	4.17009	0.181997
$526$	14.1627	0.617524
$527$	5.07838	0.221218
$528$	2.15676	0.0938607
$529$	−13.5236	−0.587982
$530$	2.54638	0.110608
$531$	2.32684	0.100976
$532$	−55.6041	−2.41074
$533$	10.3763	0.449447
$534$	5.62863	0.243575
$535$	−4.55252	−0.196823
$536$	0.921622	0.0398080
$537$	−4.64423	−0.200413
$538$	−15.2885	−0.659132
$539$	9.57531	0.412438
$540$	1.70928	0.0735555
$541$	11.8432	0.509181	0.254590	−	0.967049i	$-0.418059\pi$
$541$	11.8432	0.509181	0.254590	+	0.967049i	$0.418059\pi$
$542$	17.0772	0.733527
$543$	−1.05172	−0.0451334
$544$	−26.7214	−1.14567
$545$	11.5597	0.495163
$546$	−2.24846	−0.0962254
$547$	−41.0687	−1.75597	−0.877986	−	0.478686i	$-0.841113\pi$
$547$	−41.0687	−1.75597	−0.877986	+	0.478686i	$0.841113\pi$
$548$	−25.2762	−1.07974
$549$	−4.38243	−0.187038
$550$	−0.496928	−0.0211891
$551$	6.24846	0.266194
$552$	6.15676	0.262049
$553$	−17.5103	−0.744612
$554$	7.52464	0.319691
$555$	7.60197	0.322686
$556$	31.4452	1.33357
$557$	16.1795	0.685549	0.342775	−	0.939418i	$-0.388633\pi$
$557$	16.1795	0.685549	0.342775	+	0.939418i	$0.388633\pi$
$558$	−0.539189	−0.0228257
$559$	−0.800984	−0.0338780
$560$	−9.75872	−0.412381
$561$	4.68035	0.197604
$562$	−3.73925	−0.157731
$563$	1.02439	0.0431731	0.0215865	−	0.999767i	$-0.493128\pi$
$563$	1.02439	0.0431731	0.0215865	+	0.999767i	$0.493128\pi$
$564$	−5.91548	−0.249087
$565$	3.04945	0.128291
$566$	0.0638242	0.00268273
$567$	4.17009	0.175127
$568$	2.52359	0.105888
$569$	−0.505339	−0.0211849	−0.0105925	−	0.999944i	$-0.503372\pi$
$569$	−0.505339	−0.0211849	−0.0105925	+	0.999944i	$0.503372\pi$
$570$	4.20620	0.176178
$571$	4.83710	0.202426	0.101213	−	0.994865i	$-0.467728\pi$
$571$	4.83710	0.202426	0.101213	+	0.994865i	$0.467728\pi$
$572$	−1.57531	−0.0658669
$573$	23.6020	0.985986
$574$	−23.3307	−0.973805
$575$	3.07838	0.128377
$576$	−1.84324	−0.0768019
$577$	−11.6286	−0.484106	−0.242053	−	0.970263i	$-0.577821\pi$
$577$	−11.6286	−0.484106	−0.242053	+	0.970263i	$0.577821\pi$
$578$	−4.73943	−0.197134
$579$	−6.17009	−0.256420
$580$	1.36910	0.0568489
$581$	−7.00719	−0.290707
$582$	−5.77820	−0.239514
$583$	4.35246	0.180260
$584$	−24.0144	−0.993722
$585$	−1.00000	−0.0413449
$586$	−10.3630	−0.428090
$587$	42.2762	1.74492	0.872462	−	0.488682i	$-0.162522\pi$
$587$	42.2762	1.74492	0.872462	+	0.488682i	$0.162522\pi$
$588$	−17.7587	−0.732357
$589$	7.80098	0.321434
$590$	1.25461	0.0516514
$591$	−23.1978	−0.954230
$592$	−17.7899	−0.731161
$593$	−20.7948	−0.853942	−0.426971	−	0.904265i	$-0.640419\pi$
$593$	−20.7948	−0.853942	−0.426971	+	0.904265i	$0.640419\pi$
$594$	−0.496928	−0.0203892
$595$	−21.1773	−0.868184
$596$	2.17954	0.0892775
$597$	16.1568	0.661252
$598$	−1.65983	−0.0678754
$599$	−2.66597	−0.108929	−0.0544643	−	0.998516i	$-0.517345\pi$
$599$	−2.66597	−0.108929	−0.0544643	+	0.998516i	$0.517345\pi$
$600$	2.00000	0.0816497
$601$	−21.8732	−0.892227	−0.446114	−	0.894976i	$-0.647192\pi$
$601$	−21.8732	−0.892227	−0.446114	+	0.894976i	$0.647192\pi$
$602$	1.80098	0.0734026
$603$	0.460811	0.0187657
$604$	39.8948	1.62330
$605$	10.1506	0.412681
$606$	3.63704	0.147745
$607$	−34.1978	−1.38805	−0.694023	−	0.719953i	$-0.744163\pi$
$607$	−34.1978	−1.38805	−0.694023	+	0.719953i	$0.744163\pi$
$608$	−41.0472	−1.66468
$609$	3.34017	0.135351
$610$	−2.36296	−0.0956734
$611$	3.46081	0.140009
$612$	−8.68035	−0.350882
$613$	6.09890	0.246332	0.123166	−	0.992386i	$-0.460695\pi$
$613$	6.09890	0.246332	0.123166	+	0.992386i	$0.460695\pi$
$614$	−1.67211	−0.0674810
$615$	−10.3763	−0.418412
$616$	7.68649	0.309697
$617$	41.8141	1.68337	0.841687	−	0.539966i	$-0.181563\pi$
$617$	41.8141	1.68337	0.841687	+	0.539966i	$0.181563\pi$
$618$	6.29072	0.253050
$619$	−7.29072	−0.293039	−0.146519	−	0.989208i	$-0.546807\pi$
$619$	−7.29072	−0.293039	−0.146519	+	0.989208i	$0.546807\pi$
$620$	1.70928	0.0686462
$621$	3.07838	0.123531
$622$	16.8182	0.674348
$623$	−43.5318	−1.74407
$624$	2.34017	0.0936819
$625$	1.00000	0.0400000
$626$	−7.13114	−0.285018
$627$	7.18956	0.287123
$628$	−26.2557	−1.04771
$629$	−38.6057	−1.53931
$630$	2.24846	0.0895810
$631$	−34.6453	−1.37921	−0.689603	−	0.724187i	$-0.742215\pi$
$631$	−34.6453	−1.37921	−0.689603	+	0.724187i	$0.742215\pi$
$632$	−8.39803	−0.334056
$633$	25.4040	1.00972
$634$	2.72979	0.108414
$635$	2.70209	0.107229
$636$	−8.07223	−0.320085
$637$	10.3896	0.411652
$638$	−0.398032	−0.0157582
$639$	1.26180	0.0499158
$640$	−11.5174	−0.455267
$641$	26.9532	1.06459	0.532294	−	0.846560i	$-0.321330\pi$
$641$	26.9532	1.06459	0.532294	+	0.846560i	$0.321330\pi$
$642$	−2.45467	−0.0968780
$643$	3.80939	0.150228	0.0751139	−	0.997175i	$-0.476068\pi$
$643$	3.80939	0.150228	0.0751139	+	0.997175i	$0.476068\pi$
$644$	−21.9421	−0.864641
$645$	0.800984	0.0315387
$646$	−21.3607	−0.840425
$647$	−47.2027	−1.85573	−0.927865	−	0.372916i	$-0.878358\pi$
$647$	−47.2027	−1.85573	−0.927865	+	0.372916i	$0.878358\pi$
$648$	2.00000	0.0785674
$649$	2.14447	0.0841778
$650$	−0.539189	−0.0211487
$651$	4.17009	0.163439
$652$	−21.6742	−0.848827
$653$	27.7610	1.08637	0.543186	−	0.839613i	$-0.317218\pi$
$653$	27.7610	1.08637	0.543186	+	0.839613i	$0.317218\pi$
$654$	6.23287	0.243724
$655$	0.248464	0.00970830
$656$	24.2823	0.948065
$657$	−12.0072	−0.468445
$658$	−7.78151	−0.303355
$659$	9.14342	0.356177	0.178089	−	0.984014i	$-0.443009\pi$
$659$	9.14342	0.356177	0.178089	+	0.984014i	$0.443009\pi$
$660$	1.57531	0.0613187
$661$	5.84324	0.227276	0.113638	−	0.993522i	$-0.463750\pi$
$661$	5.84324	0.227276	0.113638	+	0.993522i	$0.463750\pi$
$662$	5.42186	0.210727
$663$	5.07838	0.197228
$664$	−3.36069	−0.130420
$665$	−32.5308	−1.26149
$666$	4.09890	0.158829
$667$	2.46573	0.0954735
$668$	18.0267	0.697472
$669$	−14.3474	−0.554701
$670$	0.248464	0.00959901
$671$	−4.03895	−0.155922
$672$	−21.9421	−0.846436
$673$	8.59478	0.331304	0.165652	−	0.986184i	$-0.447027\pi$
$673$	8.59478	0.331304	0.165652	+	0.986184i	$0.447027\pi$
$674$	−11.6020	−0.446891
$675$	1.00000	0.0384900
$676$	−1.70928	−0.0657414
$677$	12.9828	0.498969	0.249485	−	0.968379i	$-0.419739\pi$
$677$	12.9828	0.498969	0.249485	+	0.968379i	$0.419739\pi$
$678$	1.64423	0.0631462
$679$	44.6886	1.71499
$680$	−10.1568	−0.389494
$681$	21.8310	0.836564
$682$	−0.496928	−0.0190284
$683$	−18.1568	−0.694749	−0.347374	−	0.937726i	$-0.612927\pi$
$683$	−18.1568	−0.694749	−0.347374	+	0.937726i	$0.612927\pi$
$684$	−13.3340	−0.509839
$685$	−14.7877	−0.565007
$686$	−7.62144	−0.290988
$687$	6.20620	0.236781
$688$	−1.87444	−0.0714624
$689$	4.72261	0.179917
$690$	1.65983	0.0631885
$691$	9.26180	0.352335	0.176168	−	0.984360i	$-0.443630\pi$
$691$	9.26180	0.352335	0.176168	+	0.984360i	$0.443630\pi$
$692$	−7.90498	−0.300502
$693$	3.84324	0.145993
$694$	−18.6453	−0.707765
$695$	18.3968	0.697831
$696$	1.60197	0.0607224
$697$	52.6947	1.99595
$698$	−6.39350	−0.241997
$699$	−25.8059	−0.976069
$700$	−7.12783	−0.269407
$701$	−29.4063	−1.11066	−0.555330	−	0.831630i	$-0.687408\pi$
$701$	−29.4063	−1.11066	−0.555330	+	0.831630i	$0.687408\pi$
$702$	−0.539189	−0.0203504
$703$	−59.3028	−2.23665
$704$	−1.69878	−0.0640250
$705$	−3.46081	−0.130342
$706$	14.4136	0.542464
$707$	−28.1289	−1.05790
$708$	−3.97721	−0.149473
$709$	9.79380	0.367814	0.183907	−	0.982944i	$-0.441126\pi$
$709$	9.79380	0.367814	0.183907	+	0.982944i	$0.441126\pi$
$710$	0.680346	0.0255329
$711$	−4.19902	−0.157475
$712$	−20.8781	−0.782441
$713$	3.07838	0.115286
$714$	−11.4186	−0.427329
$715$	−0.921622	−0.0344667
$716$	7.93827	0.296667
$717$	1.26898	0.0473910
$718$	−2.86111	−0.106776
$719$	−18.0772	−0.674164	−0.337082	−	0.941475i	$-0.609440\pi$
$719$	−18.0772	−0.674164	−0.337082	+	0.941475i	$0.609440\pi$
$720$	−2.34017	−0.0872131
$721$	−48.6525	−1.81191
$722$	−22.5679	−0.839892
$723$	−4.34244	−0.161497
$724$	1.79767	0.0668099
$725$	0.800984	0.0297478
$726$	5.47310	0.203126
$727$	−31.9539	−1.18510	−0.592552	−	0.805532i	$-0.701879\pi$
$727$	−31.9539	−1.18510	−0.592552	+	0.805532i	$0.701879\pi$
$728$	8.34017	0.309107
$729$	1.00000	0.0370370
$730$	−6.47414	−0.239619
$731$	−4.06770	−0.150449
$732$	7.49079	0.276867
$733$	37.2095	1.37436	0.687182	−	0.726485i	$-0.258847\pi$
$733$	37.2095	1.37436	0.687182	+	0.726485i	$0.258847\pi$
$734$	−13.5031	−0.498407
$735$	−10.3896	−0.383227
$736$	−16.1978	−0.597059
$737$	0.424694	0.0156438
$738$	−5.59478	−0.205947
$739$	19.7237	0.725546	0.362773	−	0.931877i	$-0.381830\pi$
$739$	19.7237	0.725546	0.362773	+	0.931877i	$0.381830\pi$
$740$	−12.9939	−0.477664
$741$	7.80098	0.286576
$742$	−10.6186	−0.389821
$743$	−33.0183	−1.21132	−0.605661	−	0.795723i	$-0.707091\pi$
$743$	−33.0183	−1.21132	−0.605661	+	0.795723i	$0.707091\pi$
$744$	2.00000	0.0733236
$745$	1.27513	0.0467170
$746$	−10.5380	−0.385822
$747$	−1.68035	−0.0614806
$748$	−8.00000	−0.292509
$749$	18.9844	0.693675
$750$	0.539189	0.0196884
$751$	27.1445	0.990516	0.495258	−	0.868746i	$-0.335074\pi$
$751$	27.1445	0.990516	0.495258	+	0.868746i	$0.335074\pi$
$752$	8.09890	0.295336
$753$	−4.35350	−0.158651
$754$	−0.431882	−0.0157282
$755$	23.3402	0.849436
$756$	−7.12783	−0.259237
$757$	−18.1845	−0.660926	−0.330463	−	0.943819i	$-0.607205\pi$
$757$	−18.1845	−0.660926	−0.330463	+	0.943819i	$0.607205\pi$
$758$	8.24515	0.299477
$759$	2.83710	0.102980
$760$	−15.6020	−0.565943
$761$	−26.8587	−0.973626	−0.486813	−	0.873506i	$-0.661841\pi$
$761$	−26.8587	−0.973626	−0.486813	+	0.873506i	$0.661841\pi$
$762$	1.45694	0.0527792
$763$	−48.2050	−1.74514
$764$	−40.3423	−1.45953
$765$	−5.07838	−0.183609
$766$	10.8527	0.392124
$767$	2.32684	0.0840174
$768$	−2.52359	−0.0910622
$769$	53.7165	1.93707	0.968533	−	0.248887i	$-0.0800647\pi$
$769$	53.7165	1.93707	0.968533	+	0.248887i	$0.0800647\pi$
$770$	2.07223	0.0746782
$771$	8.14834	0.293455
$772$	10.5464	0.379572
$773$	−24.1194	−0.867515	−0.433758	−	0.901030i	$-0.642813\pi$
$773$	−24.1194	−0.867515	−0.433758	+	0.901030i	$0.642813\pi$
$774$	0.431882	0.0155237
$775$	1.00000	0.0359211
$776$	21.4329	0.769397
$777$	−31.7009	−1.13726
$778$	−8.56198	−0.306962
$779$	80.9453	2.90017
$780$	1.70928	0.0612019
$781$	1.16290	0.0416118
$782$	−8.42923	−0.301429
$783$	0.800984	0.0286248
$784$	24.3135	0.868340
$785$	−15.3607	−0.548247
$786$	0.133969	0.00477852
$787$	−52.7552	−1.88052	−0.940260	−	0.340456i	$-0.889419\pi$
$787$	−52.7552	−1.88052	−0.940260	+	0.340456i	$0.889419\pi$
$788$	39.6514	1.41252
$789$	−26.2667	−0.935120
$790$	−2.26406	−0.0805517
$791$	−12.7165	−0.452145
$792$	1.84324	0.0654968
$793$	−4.38243	−0.155625
$794$	10.6537	0.378085
$795$	−4.72261	−0.167494
$796$	−27.6163	−0.978835
$797$	8.61142	0.305032	0.152516	−	0.988301i	$-0.451262\pi$
$797$	8.61142	0.305032	0.152516	+	0.988301i	$0.451262\pi$
$798$	−17.5402	−0.620918
$799$	17.5753	0.621770
$800$	−5.26180	−0.186033
$801$	−10.4391	−0.368846
$802$	11.2534	0.397371
$803$	−11.0661	−0.390514
$804$	−0.787653	−0.0277784
$805$	−12.8371	−0.452448
$806$	−0.539189	−0.0189921
$807$	28.3545	0.998127
$808$	−13.4908	−0.474604
$809$	−12.9227	−0.454337	−0.227168	−	0.973855i	$-0.572947\pi$
$809$	−12.9227	−0.454337	−0.227168	+	0.973855i	$0.572947\pi$
$810$	0.539189	0.0189452
$811$	31.0894	1.09170	0.545849	−	0.837884i	$-0.316207\pi$
$811$	31.0894	1.09170	0.545849	+	0.837884i	$0.316207\pi$
$812$	−5.70928	−0.200356
$813$	−31.6719	−1.11078
$814$	3.77763	0.132406
$815$	−12.6803	−0.444173
$816$	11.8843	0.416033
$817$	−6.24846	−0.218606
$818$	−2.26916	−0.0793393
$819$	4.17009	0.145715
$820$	17.7359	0.619366
$821$	27.9733	0.976276	0.488138	−	0.872766i	$-0.337676\pi$
$821$	27.9733	0.976276	0.488138	+	0.872766i	$0.337676\pi$
$822$	−7.97334	−0.278102
$823$	−0.908291	−0.0316610	−0.0158305	−	0.999875i	$-0.505039\pi$
$823$	−0.908291	−0.0316610	−0.0158305	+	0.999875i	$0.505039\pi$
$824$	−23.3340	−0.812879
$825$	0.921622	0.0320868
$826$	−5.23182	−0.182038
$827$	41.0638	1.42793	0.713965	−	0.700182i	$-0.246898\pi$
$827$	41.0638	1.42793	0.713965	+	0.700182i	$0.246898\pi$
$828$	−5.26180	−0.182860
$829$	15.4042	0.535009	0.267505	−	0.963557i	$-0.413801\pi$
$829$	15.4042	0.535009	0.267505	+	0.963557i	$0.413801\pi$
$830$	−0.906024	−0.0314485
$831$	−13.9555	−0.484110
$832$	−1.84324	−0.0639030
$833$	52.7624	1.82811
$834$	9.91935	0.343479
$835$	10.5464	0.364972
$836$	−12.2889	−0.425022
$837$	1.00000	0.0345651
$838$	−6.23353	−0.215334
$839$	32.6442	1.12700	0.563502	−	0.826115i	$-0.309454\pi$
$839$	32.6442	1.12700	0.563502	+	0.826115i	$0.309454\pi$
$840$	−8.34017	−0.287763
$841$	−28.3584	−0.977877
$842$	0	0
$843$	6.93495	0.238852
$844$	−43.4224	−1.49466
$845$	−1.00000	−0.0344010
$846$	−1.86603	−0.0641555
$847$	−42.3289	−1.45444
$848$	11.0517	0.379517
$849$	−0.118371	−0.00406247
$850$	−2.73820	−0.0939196
$851$	−23.4017	−0.802201
$852$	−2.15676	−0.0738892
$853$	12.5909	0.431104	0.215552	−	0.976492i	$-0.430845\pi$
$853$	12.5909	0.431104	0.215552	+	0.976492i	$0.430845\pi$
$854$	9.85374	0.337188
$855$	−7.80098	−0.266788
$856$	9.10504	0.311204
$857$	−43.7959	−1.49604	−0.748019	−	0.663677i	$-0.768995\pi$
$857$	−43.7959	−1.49604	−0.748019	+	0.663677i	$0.768995\pi$
$858$	−0.496928	−0.0169649
$859$	−4.24128	−0.144710	−0.0723552	−	0.997379i	$-0.523052\pi$
$859$	−4.24128	−0.144710	−0.0723552	+	0.997379i	$0.523052\pi$
$860$	−1.36910	−0.0466860
$861$	43.2700	1.47464
$862$	−3.44256	−0.117254
$863$	−6.14011	−0.209012	−0.104506	−	0.994524i	$-0.533326\pi$
$863$	−6.14011	−0.209012	−0.104506	+	0.994524i	$0.533326\pi$
$864$	−5.26180	−0.179010
$865$	−4.62475	−0.157246
$866$	15.3223	0.520673
$867$	8.78992	0.298521
$868$	−7.12783	−0.241934
$869$	−3.86991	−0.131278
$870$	0.431882	0.0146422
$871$	0.460811	0.0156140
$872$	−23.1194	−0.782922
$873$	10.7165	0.362697
$874$	−12.9483	−0.437982
$875$	−4.17009	−0.140975
$876$	20.5236	0.693428
$877$	−1.98562	−0.0670498	−0.0335249	−	0.999438i	$-0.510673\pi$
$877$	−1.98562	−0.0670498	−0.0335249	+	0.999438i	$0.510673\pi$
$878$	16.2569	0.548643
$879$	19.2195	0.648259
$880$	−2.15676	−0.0727042
$881$	−3.50412	−0.118057	−0.0590283	−	0.998256i	$-0.518800\pi$
$881$	−3.50412	−0.118057	−0.0590283	+	0.998256i	$0.518800\pi$
$882$	−5.60197	−0.188628
$883$	−27.8033	−0.935654	−0.467827	−	0.883820i	$-0.654963\pi$
$883$	−27.8033	−0.935654	−0.467827	+	0.883820i	$0.654963\pi$
$884$	−8.68035	−0.291952
$885$	−2.32684	−0.0782159
$886$	7.61303	0.255765
$887$	21.4931	0.721666	0.360833	−	0.932630i	$-0.382492\pi$
$887$	21.4931	0.721666	0.360833	+	0.932630i	$0.382492\pi$
$888$	−15.2039	−0.510211
$889$	−11.2679	−0.377915
$890$	−5.62863	−0.188672
$891$	0.921622	0.0308755
$892$	24.5236	0.821111
$893$	26.9977	0.903445
$894$	0.687534	0.0229946
$895$	4.64423	0.155239
$896$	48.0288	1.60453
$897$	3.07838	0.102784
$898$	0.0571996	0.00190878
$899$	0.800984	0.0267143
$900$	−1.70928	−0.0569758
$901$	23.9832	0.798996
$902$	−5.15627	−0.171685
$903$	−3.34017	−0.111154
$904$	−6.09890	−0.202846
$905$	1.05172	0.0349602
$906$	12.5848	0.418101
$907$	−16.5548	−0.549693	−0.274846	−	0.961488i	$-0.588627\pi$
$907$	−16.5548	−0.549693	−0.274846	+	0.961488i	$0.588627\pi$
$908$	−37.3151	−1.23835
$909$	−6.74539	−0.223731
$910$	2.24846	0.0745359
$911$	8.33072	0.276009	0.138004	−	0.990432i	$-0.455931\pi$
$911$	8.33072	0.276009	0.138004	+	0.990432i	$0.455931\pi$
$912$	18.2557	0.604505
$913$	−1.54864	−0.0512526
$914$	−19.9506	−0.659906
$915$	4.38243	0.144879
$916$	−10.6081	−0.350502
$917$	−1.03612	−0.0342156
$918$	−2.73820	−0.0903742
$919$	3.10731	0.102501	0.0512503	−	0.998686i	$-0.483679\pi$
$919$	3.10731	0.102501	0.0512503	+	0.998686i	$0.483679\pi$
$920$	−6.15676	−0.202982
$921$	3.10116	0.102187
$922$	8.59583	0.283088
$923$	1.26180	0.0415325
$924$	−6.56916	−0.216110
$925$	−7.60197	−0.249951
$926$	−10.2269	−0.336077
$927$	−11.6670	−0.383195
$928$	−4.21461	−0.138351
$929$	−23.3802	−0.767078	−0.383539	−	0.923525i	$-0.625295\pi$
$929$	−23.3802	−0.767078	−0.383539	+	0.923525i	$0.625295\pi$
$930$	0.539189	0.0176807
$931$	81.0493	2.65628
$932$	44.1094	1.44485
$933$	−31.1917	−1.02117
$934$	−16.7636	−0.548523
$935$	−4.68035	−0.153064
$936$	2.00000	0.0653720
$937$	44.3090	1.44751	0.723756	−	0.690056i	$-0.242414\pi$
$937$	44.3090	1.44751	0.723756	+	0.690056i	$0.242414\pi$
$938$	−1.03612	−0.0338304
$939$	13.2257	0.431604
$940$	5.91548	0.192942
$941$	−3.45694	−0.112693	−0.0563464	−	0.998411i	$-0.517945\pi$
$941$	−3.45694	−0.112693	−0.0563464	+	0.998411i	$0.517945\pi$
$942$	−8.28231	−0.269852
$943$	31.9421	1.04018
$944$	5.44521	0.177227
$945$	−4.17009	−0.135653
$946$	0.398032	0.0129411
$947$	39.4557	1.28214	0.641069	−	0.767483i	$-0.278491\pi$
$947$	39.4557	1.28214	0.641069	+	0.767483i	$0.278491\pi$
$948$	7.17727	0.233107
$949$	−12.0072	−0.389770
$950$	−4.20620	−0.136467
$951$	−5.06278	−0.164172
$952$	42.3545	1.37272
$953$	−13.6007	−0.440571	−0.220286	−	0.975435i	$-0.570699\pi$
$953$	−13.6007	−0.440571	−0.220286	+	0.975435i	$0.570699\pi$
$954$	−2.54638	−0.0824420
$955$	−23.6020	−0.763742
$956$	−2.16904	−0.0701518
$957$	0.738205	0.0238628
$958$	−4.00265	−0.129320
$959$	61.6658	1.99129
$960$	1.84324	0.0594905
$961$	1.00000	0.0322581
$962$	4.09890	0.132154
$963$	4.55252	0.146703
$964$	7.42243	0.239060
$965$	6.17009	0.198622
$966$	−6.92162	−0.222700
$967$	−11.3295	−0.364332	−0.182166	−	0.983268i	$-0.558311\pi$
$967$	−11.3295	−0.364332	−0.182166	+	0.983268i	$0.558311\pi$
$968$	−20.3012	−0.652506
$969$	39.6163	1.27266
$970$	5.77820	0.185527
$971$	−45.7854	−1.46932	−0.734661	−	0.678434i	$-0.762659\pi$
$971$	−45.7854	−1.46932	−0.734661	+	0.678434i	$0.762659\pi$
$972$	−1.70928	−0.0548250
$973$	−76.7163	−2.45941
$974$	−7.10116	−0.227536
$975$	1.00000	0.0320256
$976$	−10.2557	−0.328275
$977$	45.7587	1.46395	0.731976	−	0.681331i	$-0.238598\pi$
$977$	45.7587	1.46395	0.731976	+	0.681331i	$0.238598\pi$
$978$	−6.83710	−0.218626
$979$	−9.62088	−0.307485
$980$	17.7587	0.567282
$981$	−11.5597	−0.369073
$982$	−12.8155	−0.408960
$983$	−9.49693	−0.302905	−0.151452	−	0.988465i	$-0.548395\pi$
$983$	−9.49693	−0.302905	−0.151452	+	0.988465i	$0.548395\pi$
$984$	20.7526	0.661568
$985$	23.1978	0.739143
$986$	−2.19326	−0.0698476
$987$	14.4319	0.459372
$988$	−13.3340	−0.424212
$989$	−2.46573	−0.0784057
$990$	0.496928	0.0157934
$991$	6.31229	0.200516	0.100258	−	0.994961i	$-0.468033\pi$
$991$	6.31229	0.200516	0.100258	+	0.994961i	$0.468033\pi$
$992$	−5.26180	−0.167062
$993$	−10.0556	−0.319105
$994$	−2.83710	−0.0899874
$995$	−16.1568	−0.512204
$996$	2.87217	0.0910083
$997$	−28.3330	−0.897315	−0.448657	−	0.893704i	$-0.648098\pi$
$997$	−28.3330	−0.897315	−0.448657	+	0.893704i	$0.648098\pi$
$998$	12.6381	0.400052
$999$	−7.60197	−0.240516

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.q.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.q.1.2	✓	3	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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

\(f(q)\)	\(=\)	\(q-0.539189 q^{2} +1.00000 q^{3} -1.70928 q^{4} -1.00000 q^{5} -0.539189 q^{6} +4.17009 q^{7} +2.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.539189 q^{2} +1.00000 q^{3} -1.70928 q^{4} -1.00000 q^{5} -0.539189 q^{6} +4.17009 q^{7} +2.00000 q^{8} +1.00000 q^{9} +0.539189 q^{10} +0.921622 q^{11} -1.70928 q^{12} +1.00000 q^{13} -2.24846 q^{14} -1.00000 q^{15} +2.34017 q^{16} +5.07838 q^{17} -0.539189 q^{18} +7.80098 q^{19} +1.70928 q^{20} +4.17009 q^{21} -0.496928 q^{22} +3.07838 q^{23} +2.00000 q^{24} +1.00000 q^{25} -0.539189 q^{26} +1.00000 q^{27} -7.12783 q^{28} +0.800984 q^{29} +0.539189 q^{30} +1.00000 q^{31} -5.26180 q^{32} +0.921622 q^{33} -2.73820 q^{34} -4.17009 q^{35} -1.70928 q^{36} -7.60197 q^{37} -4.20620 q^{38} +1.00000 q^{39} -2.00000 q^{40} +10.3763 q^{41} -2.24846 q^{42} -0.800984 q^{43} -1.57531 q^{44} -1.00000 q^{45} -1.65983 q^{46} +3.46081 q^{47} +2.34017 q^{48} +10.3896 q^{49} -0.539189 q^{50} +5.07838 q^{51} -1.70928 q^{52} +4.72261 q^{53} -0.539189 q^{54} -0.921622 q^{55} +8.34017 q^{56} +7.80098 q^{57} -0.431882 q^{58} +2.32684 q^{59} +1.70928 q^{60} -4.38243 q^{61} -0.539189 q^{62} +4.17009 q^{63} -1.84324 q^{64} -1.00000 q^{65} -0.496928 q^{66} +0.460811 q^{67} -8.68035 q^{68} +3.07838 q^{69} +2.24846 q^{70} +1.26180 q^{71} +2.00000 q^{72} -12.0072 q^{73} +4.09890 q^{74} +1.00000 q^{75} -13.3340 q^{76} +3.84324 q^{77} -0.539189 q^{78} -4.19902 q^{79} -2.34017 q^{80} +1.00000 q^{81} -5.59478 q^{82} -1.68035 q^{83} -7.12783 q^{84} -5.07838 q^{85} +0.431882 q^{86} +0.800984 q^{87} +1.84324 q^{88} -10.4391 q^{89} +0.539189 q^{90} +4.17009 q^{91} -5.26180 q^{92} +1.00000 q^{93} -1.86603 q^{94} -7.80098 q^{95} -5.26180 q^{96} +10.7165 q^{97} -5.60197 q^{98} +0.921622 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 2 q^{4} - 3 q^{5} + 7 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 2 * q^4 - 3 * q^5 + 7 * q^7 + 6 * q^8 + 3 * q^9	\( 3 q + 3 q^{3} + 2 q^{4} - 3 q^{5} + 7 q^{7} + 6 q^{8} + 3 q^{9} + 6 q^{11} + 2 q^{12} + 3 q^{13} + 2 q^{14} - 3 q^{15} - 4 q^{16} + 12 q^{17} + 14 q^{19} - 2 q^{20} + 7 q^{21} + 16 q^{22} + 6 q^{23} + 6 q^{24} + 3 q^{25} + 3 q^{27} - 7 q^{29} + 3 q^{31} - 8 q^{32} + 6 q^{33} - 16 q^{34} - 7 q^{35} + 2 q^{36} - 4 q^{37} + 12 q^{38} + 3 q^{39} - 6 q^{40} + q^{41} + 2 q^{42} + 7 q^{43} + 16 q^{44} - 3 q^{45} - 16 q^{46} + 12 q^{47} - 4 q^{48} + 2 q^{49} + 12 q^{51} + 2 q^{52} + 8 q^{53} - 6 q^{55} + 14 q^{56} + 14 q^{57} + 12 q^{58} - 5 q^{59} - 2 q^{60} - 18 q^{61} + 7 q^{63} - 12 q^{64} - 3 q^{65} + 16 q^{66} + 3 q^{67} - 4 q^{68} + 6 q^{69} - 2 q^{70} - 4 q^{71} + 6 q^{72} - 2 q^{73} - 24 q^{74} + 3 q^{75} + 6 q^{76} + 18 q^{77} - 22 q^{79} + 4 q^{80} + 3 q^{81} - 32 q^{82} + 17 q^{83} - 12 q^{85} - 12 q^{86} - 7 q^{87} + 12 q^{88} + 16 q^{89} + 7 q^{91} - 8 q^{92} + 3 q^{93} + 8 q^{94} - 14 q^{95} - 8 q^{96} - 9 q^{97} + 2 q^{98} + 6 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 2 * q^4 - 3 * q^5 + 7 * q^7 + 6 * q^8 + 3 * q^9 + 6 * q^11 + 2 * q^12 + 3 * q^13 + 2 * q^14 - 3 * q^15 - 4 * q^16 + 12 * q^17 + 14 * q^19 - 2 * q^20 + 7 * q^21 + 16 * q^22 + 6 * q^23 + 6 * q^24 + 3 * q^25 + 3 * q^27 - 7 * q^29 + 3 * q^31 - 8 * q^32 + 6 * q^33 - 16 * q^34 - 7 * q^35 + 2 * q^36 - 4 * q^37 + 12 * q^38 + 3 * q^39 - 6 * q^40 + q^41 + 2 * q^42 + 7 * q^43 + 16 * q^44 - 3 * q^45 - 16 * q^46 + 12 * q^47 - 4 * q^48 + 2 * q^49 + 12 * q^51 + 2 * q^52 + 8 * q^53 - 6 * q^55 + 14 * q^56 + 14 * q^57 + 12 * q^58 - 5 * q^59 - 2 * q^60 - 18 * q^61 + 7 * q^63 - 12 * q^64 - 3 * q^65 + 16 * q^66 + 3 * q^67 - 4 * q^68 + 6 * q^69 - 2 * q^70 - 4 * q^71 + 6 * q^72 - 2 * q^73 - 24 * q^74 + 3 * q^75 + 6 * q^76 + 18 * q^77 - 22 * q^79 + 4 * q^80 + 3 * q^81 - 32 * q^82 + 17 * q^83 - 12 * q^85 - 12 * q^86 - 7 * q^87 + 12 * q^88 + 16 * q^89 + 7 * q^91 - 8 * q^92 + 3 * q^93 + 8 * q^94 - 14 * q^95 - 8 * q^96 - 9 * q^97 + 2 * q^98 + 6 * q^99

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