LMFDB - Embedded newform 111.2.a.b.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [111,2,Mod(1,111)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(111, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("111.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 111 = 3 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	111.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:	$0.886339462436$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.6224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 2x + 5 $ x^4 - 6x^2 - 2x + 5
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			1.4
Root			$-1.87228$ of defining polynomial
Character	$\chi$	$=$	111.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.87228 q^{2} +1.00000 q^{3} +1.50542 q^{4} -3.95712 q^{5} +1.87228 q^{6} +3.15885 q^{7} -0.925994 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.87228 q^{2} +1.00000 q^{3} +1.50542 q^{4} -3.95712 q^{5} +1.87228 q^{6} +3.15885 q^{7} -0.925994 q^{8} +1.00000 q^{9} -7.40882 q^{10} +1.01084 q^{11} +1.50542 q^{12} -5.91424 q^{13} +5.91424 q^{14} -3.95712 q^{15} -4.74455 q^{16} -2.94628 q^{17} +1.87228 q^{18} +6.75539 q^{19} -5.95712 q^{20} +3.15885 q^{21} +1.89257 q^{22} +2.10513 q^{23} -0.925994 q^{24} +10.6588 q^{25} -11.0731 q^{26} +1.00000 q^{27} +4.75539 q^{28} +4.54282 q^{29} -7.40882 q^{30} -1.15885 q^{31} -7.03113 q^{32} +1.01084 q^{33} -5.51625 q^{34} -12.4999 q^{35} +1.50542 q^{36} -1.00000 q^{37} +12.6480 q^{38} -5.91424 q^{39} +3.66427 q^{40} +1.41430 q^{41} +5.91424 q^{42} +6.33026 q^{43} +1.52173 q^{44} -3.95712 q^{45} +3.94139 q^{46} +2.90340 q^{47} -4.74455 q^{48} +2.97833 q^{49} +19.9562 q^{50} -2.94628 q^{51} -8.90340 q^{52} -6.49994 q^{53} +1.87228 q^{54} -4.00000 q^{55} -2.92507 q^{56} +6.75539 q^{57} +8.50542 q^{58} -7.54110 q^{59} -5.95712 q^{60} -9.48910 q^{61} -2.16969 q^{62} +3.15885 q^{63} -3.67510 q^{64} +23.4033 q^{65} +1.89257 q^{66} -3.15885 q^{67} -4.43539 q^{68} +2.10513 q^{69} -23.4033 q^{70} +4.92507 q^{71} -0.925994 q^{72} +4.14801 q^{73} -1.87228 q^{74} +10.6588 q^{75} +10.1697 q^{76} +3.19308 q^{77} -11.0731 q^{78} -4.73372 q^{79} +18.7748 q^{80} +1.00000 q^{81} +2.64795 q^{82} -11.9142 q^{83} +4.75539 q^{84} +11.6588 q^{85} +11.8520 q^{86} +4.54282 q^{87} -0.936028 q^{88} +14.2748 q^{89} -7.40882 q^{90} -18.6822 q^{91} +3.16910 q^{92} -1.15885 q^{93} +5.43597 q^{94} -26.7319 q^{95} -7.03113 q^{96} -10.2103 q^{97} +5.57625 q^{98} +1.01084 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{4} - 2 q^{5} + 4 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^4 - 2 * q^5 + 4 * q^7 - 6 * q^8 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{4} - 2 q^{5} + 4 q^{7} - 6 q^{8} + 4 q^{9} - 4 q^{10} + 4 q^{12} + 4 q^{13} - 4 q^{14} - 2 q^{15} - 4 q^{16} - 2 q^{17} + 8 q^{19} - 10 q^{20} + 4 q^{21} - 12 q^{22} - 10 q^{23} - 6 q^{24} - 8 q^{26} + 4 q^{27} - 2 q^{29} - 4 q^{30} + 4 q^{31} - 12 q^{32} - 16 q^{34} - 16 q^{35} + 4 q^{36} - 4 q^{37} + 12 q^{38} + 4 q^{39} + 4 q^{40} + 12 q^{41} - 4 q^{42} + 4 q^{43} + 32 q^{44} - 2 q^{45} + 12 q^{46} - 12 q^{47} - 4 q^{48} + 20 q^{49} + 32 q^{50} - 2 q^{51} - 12 q^{52} + 8 q^{53} - 16 q^{55} + 20 q^{56} + 8 q^{57} + 32 q^{58} - 10 q^{59} - 10 q^{60} - 8 q^{61} + 4 q^{62} + 4 q^{63} + 36 q^{65} - 12 q^{66} - 4 q^{67} + 22 q^{68} - 10 q^{69} - 36 q^{70} - 12 q^{71} - 6 q^{72} + 12 q^{73} + 28 q^{76} - 8 q^{77} - 8 q^{78} - 8 q^{79} + 10 q^{80} + 4 q^{81} - 28 q^{82} - 20 q^{83} + 4 q^{85} + 52 q^{86} - 2 q^{87} - 36 q^{88} + 26 q^{89} - 4 q^{90} - 24 q^{91} - 38 q^{92} + 4 q^{93} + 20 q^{94} - 28 q^{95} - 12 q^{96} - 4 q^{97} + 24 q^{98}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^4 - 2 * q^5 + 4 * q^7 - 6 * q^8 + 4 * q^9 - 4 * q^10 + 4 * q^12 + 4 * q^13 - 4 * q^14 - 2 * q^15 - 4 * q^16 - 2 * q^17 + 8 * q^19 - 10 * q^20 + 4 * q^21 - 12 * q^22 - 10 * q^23 - 6 * q^24 - 8 * q^26 + 4 * q^27 - 2 * q^29 - 4 * q^30 + 4 * q^31 - 12 * q^32 - 16 * q^34 - 16 * q^35 + 4 * q^36 - 4 * q^37 + 12 * q^38 + 4 * q^39 + 4 * q^40 + 12 * q^41 - 4 * q^42 + 4 * q^43 + 32 * q^44 - 2 * q^45 + 12 * q^46 - 12 * q^47 - 4 * q^48 + 20 * q^49 + 32 * q^50 - 2 * q^51 - 12 * q^52 + 8 * q^53 - 16 * q^55 + 20 * q^56 + 8 * q^57 + 32 * q^58 - 10 * q^59 - 10 * q^60 - 8 * q^61 + 4 * q^62 + 4 * q^63 + 36 * q^65 - 12 * q^66 - 4 * q^67 + 22 * q^68 - 10 * q^69 - 36 * q^70 - 12 * q^71 - 6 * q^72 + 12 * q^73 + 28 * q^76 - 8 * q^77 - 8 * q^78 - 8 * q^79 + 10 * q^80 + 4 * q^81 - 28 * q^82 - 20 * q^83 + 4 * q^85 + 52 * q^86 - 2 * q^87 - 36 * q^88 + 26 * q^89 - 4 * q^90 - 24 * q^91 - 38 * q^92 + 4 * q^93 + 20 * q^94 - 28 * q^95 - 12 * q^96 - 4 * q^97 + 24 * q^98

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 $

$2$	1.87228	1.32390	0.661950	−	0.749548i	$-0.269729\pi$
$2$	1.87228	1.32390	0.661950	+	0.749548i	$0.269729\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.50542	0.752709
$5$	−3.95712	−1.76968	−0.884839	−	0.465898i	$-0.845732\pi$
$5$	−3.95712	−1.76968	−0.884839	+	0.465898i	$0.845732\pi$
$6$	1.87228	0.764354
$7$	3.15885	1.19393	0.596966	−	0.802266i	$-0.296373\pi$
$7$	3.15885	1.19393	0.596966	+	0.802266i	$0.296373\pi$
$8$	−0.925994	−0.327388
$9$	1.00000	0.333333
$10$	−7.40882	−2.34287
$11$	1.01084	0.304779	0.152389	−	0.988321i	$-0.451303\pi$
$11$	1.01084	0.304779	0.152389	+	0.988321i	$0.451303\pi$
$12$	1.50542	0.434577
$13$	−5.91424	−1.64031	−0.820157	−	0.572138i	$-0.806114\pi$
$13$	−5.91424	−1.64031	−0.820157	+	0.572138i	$0.806114\pi$
$14$	5.91424	1.58065
$15$	−3.95712	−1.02172
$16$	−4.74455	−1.18614
$17$	−2.94628	−0.714578	−0.357289	−	0.933994i	$-0.616299\pi$
$17$	−2.94628	−0.714578	−0.357289	+	0.933994i	$0.616299\pi$
$18$	1.87228	0.441300
$19$	6.75539	1.54979	0.774896	−	0.632088i	$-0.217802\pi$
$19$	6.75539	1.54979	0.774896	+	0.632088i	$0.217802\pi$
$20$	−5.95712	−1.33205
$21$	3.15885	0.689317
$22$	1.89257	0.403496
$23$	2.10513	0.438950	0.219475	−	0.975618i	$-0.429566\pi$
$23$	2.10513	0.438950	0.219475	+	0.975618i	$0.429566\pi$
$24$	−0.925994	−0.189018
$25$	10.6588	2.13176
$26$	−11.0731	−2.17161
$27$	1.00000	0.192450
$28$	4.75539	0.898684
$29$	4.54282	0.843581	0.421790	−	0.906693i	$-0.361402\pi$
$29$	4.54282	0.843581	0.421790	+	0.906693i	$0.361402\pi$
$30$	−7.40882	−1.35266
$31$	−1.15885	−0.208135	−0.104068	−	0.994570i	$-0.533186\pi$
$31$	−1.15885	−0.208135	−0.104068	+	0.994570i	$0.533186\pi$
$32$	−7.03113	−1.24294
$33$	1.01084	0.175964
$34$	−5.51625	−0.946030
$35$	−12.4999	−2.11288
$36$	1.50542	0.250903
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	12.6480	2.05177
$39$	−5.91424	−0.947036
$40$	3.66427	0.579372
$41$	1.41430	0.220876	0.110438	−	0.993883i	$-0.464775\pi$
$41$	1.41430	0.220876	0.110438	+	0.993883i	$0.464775\pi$
$42$	5.91424	0.912587
$43$	6.33026	0.965355	0.482677	−	0.875798i	$-0.339664\pi$
$43$	6.33026	0.965355	0.482677	+	0.875798i	$0.339664\pi$
$44$	1.52173	0.229410
$45$	−3.95712	−0.589892
$46$	3.94139	0.581126
$47$	2.90340	0.423505	0.211752	−	0.977323i	$-0.432083\pi$
$47$	2.90340	0.423505	0.211752	+	0.977323i	$0.432083\pi$
$48$	−4.74455	−0.684817
$49$	2.97833	0.425475
$50$	19.9562	2.82223
$51$	−2.94628	−0.412562
$52$	−8.90340	−1.23468
$53$	−6.49994	−0.892836	−0.446418	−	0.894825i	$-0.647300\pi$
$53$	−6.49994	−0.892836	−0.446418	+	0.894825i	$0.647300\pi$
$54$	1.87228	0.254785
$55$	−4.00000	−0.539360
$56$	−2.92507	−0.390879
$57$	6.75539	0.894773
$58$	8.50542	1.11682
$59$	−7.54110	−0.981768	−0.490884	−	0.871225i	$-0.663326\pi$
$59$	−7.54110	−0.981768	−0.490884	+	0.871225i	$0.663326\pi$
$60$	−5.95712	−0.769061
$61$	−9.48910	−1.21496	−0.607478	−	0.794337i	$-0.707819\pi$
$61$	−9.48910	−1.21496	−0.607478	+	0.794337i	$0.707819\pi$
$62$	−2.16969	−0.275550
$63$	3.15885	0.397978
$64$	−3.67510	−0.459388
$65$	23.4033	2.90283
$66$	1.89257	0.232959
$67$	−3.15885	−0.385915	−0.192957	−	0.981207i	$-0.561808\pi$
$67$	−3.15885	−0.385915	−0.192957	+	0.981207i	$0.561808\pi$
$68$	−4.43539	−0.537870
$69$	2.10513	0.253428
$70$	−23.4033	−2.79723
$71$	4.92507	0.584499	0.292249	−	0.956342i	$-0.405596\pi$
$71$	4.92507	0.584499	0.292249	+	0.956342i	$0.405596\pi$
$72$	−0.925994	−0.109129
$73$	4.14801	0.485488	0.242744	−	0.970090i	$-0.421952\pi$
$73$	4.14801	0.485488	0.242744	+	0.970090i	$0.421952\pi$
$74$	−1.87228	−0.217648
$75$	10.6588	1.23077
$76$	10.1697	1.16654
$77$	3.19308	0.363885
$78$	−11.0731	−1.25378
$79$	−4.73372	−0.532585	−0.266292	−	0.963892i	$-0.585799\pi$
$79$	−4.73372	−0.532585	−0.266292	+	0.963892i	$0.585799\pi$
$80$	18.7748	2.09908
$81$	1.00000	0.111111
$82$	2.64795	0.292418
$83$	−11.9142	−1.30776	−0.653879	−	0.756599i	$-0.726859\pi$
$83$	−11.9142	−1.30776	−0.653879	+	0.756599i	$0.726859\pi$
$84$	4.75539	0.518855
$85$	11.6588	1.26457
$86$	11.8520	1.27803
$87$	4.54282	0.487042
$88$	−0.936028	−0.0997809
$89$	14.2748	1.51313	0.756564	−	0.653920i	$-0.226877\pi$
$89$	14.2748	1.51313	0.756564	+	0.653920i	$0.226877\pi$
$90$	−7.40882	−0.780958
$91$	−18.6822	−1.95843
$92$	3.16910	0.330402
$93$	−1.15885	−0.120167
$94$	5.43597	0.560677
$95$	−26.7319	−2.74263
$96$	−7.03113	−0.717611
$97$	−10.2103	−1.03670	−0.518348	−	0.855170i	$-0.673453\pi$
$97$	−10.2103	−1.03670	−0.518348	+	0.855170i	$0.673453\pi$
$98$	5.57625	0.563286
$99$	1.01084	0.101593
$100$	16.0459	1.60459
$101$	2.88173	0.286743	0.143371	−	0.989669i	$-0.454206\pi$
$101$	2.88173	0.286743	0.143371	+	0.989669i	$0.454206\pi$
$102$	−5.51625	−0.546191
$103$	5.07309	0.499866	0.249933	−	0.968263i	$-0.419591\pi$
$103$	5.07309	0.499866	0.249933	+	0.968263i	$0.419591\pi$
$104$	5.47655	0.537020
$105$	−12.4999	−1.21987
$106$	−12.1697	−1.18202
$107$	−8.58570	−0.830011	−0.415006	−	0.909819i	$-0.636220\pi$
$107$	−8.58570	−0.830011	−0.415006	+	0.909819i	$0.636220\pi$
$108$	1.50542	0.144859
$109$	−7.89257	−0.755971	−0.377985	−	0.925812i	$-0.623383\pi$
$109$	−7.89257	−0.755971	−0.377985	+	0.925812i	$0.623383\pi$
$110$	−7.48910	−0.714058
$111$	−1.00000	−0.0949158
$112$	−14.9873	−1.41617
$113$	−1.55366	−0.146156	−0.0730780	−	0.997326i	$-0.523282\pi$
$113$	−1.55366	−0.146156	−0.0730780	+	0.997326i	$0.523282\pi$
$114$	12.6480	1.18459
$115$	−8.33026	−0.776800
$116$	6.83885	0.634971
$117$	−5.91424	−0.546771
$118$	−14.1190	−1.29976
$119$	−9.30686	−0.853159
$120$	3.66427	0.334500
$121$	−9.97821	−0.907110
$122$	−17.7662	−1.60848
$123$	1.41430	0.127523
$124$	−1.74455	−0.156665
$125$	−22.3925	−2.00285
$126$	5.91424	0.526882
$127$	15.8285	1.40455	0.702275	−	0.711906i	$-0.252168\pi$
$127$	15.8285	1.40455	0.702275	+	0.711906i	$0.252168\pi$
$128$	7.18144	0.634756
$129$	6.33026	0.557348
$130$	43.8175	3.84305
$131$	−1.30917	−0.114382	−0.0571911	−	0.998363i	$-0.518214\pi$
$131$	−1.30917	−0.114382	−0.0571911	+	0.998363i	$0.518214\pi$
$132$	1.52173	0.132450
$133$	21.3393	1.85035
$134$	−5.91424	−0.510912
$135$	−3.95712	−0.340575
$136$	2.72824	0.233945
$137$	14.8393	1.26781	0.633904	−	0.773412i	$-0.281451\pi$
$137$	14.8393	1.26781	0.633904	+	0.773412i	$0.281451\pi$
$138$	3.94139	0.335513
$139$	10.3086	0.874363	0.437181	−	0.899373i	$-0.355977\pi$
$139$	10.3086	0.874363	0.437181	+	0.899373i	$0.355977\pi$
$140$	−18.8176	−1.59038
$141$	2.90340	0.244511
$142$	9.22110	0.773817
$143$	−5.97833	−0.499933
$144$	−4.74455	−0.395379
$145$	−17.9765	−1.49287
$146$	7.76623	0.642737
$147$	2.97833	0.245648
$148$	−1.50542	−0.123745
$149$	7.68230	0.629359	0.314679	−	0.949198i	$-0.398103\pi$
$149$	7.68230	0.629359	0.314679	+	0.949198i	$0.398103\pi$
$150$	19.9562	1.62942
$151$	−17.8501	−1.45262	−0.726312	−	0.687365i	$-0.758767\pi$
$151$	−17.8501	−1.45262	−0.726312	+	0.687365i	$0.758767\pi$
$152$	−6.25545	−0.507384
$153$	−2.94628	−0.238193
$154$	5.97833	0.481747
$155$	4.58570	0.368333
$156$	−8.90340	−0.712843
$157$	−23.1696	−1.84913	−0.924566	−	0.381021i	$-0.875573\pi$
$157$	−23.1696	−1.84913	−0.924566	+	0.381021i	$0.875573\pi$
$158$	−8.86282	−0.705088
$159$	−6.49994	−0.515479
$160$	27.8230	2.19960
$161$	6.64979	0.524077
$162$	1.87228	0.147100
$163$	0.819478	0.0641865	0.0320932	−	0.999485i	$-0.489783\pi$
$163$	0.819478	0.0641865	0.0320932	+	0.999485i	$0.489783\pi$
$164$	2.12911	0.166255
$165$	−4.00000	−0.311400
$166$	−22.3067	−1.73134
$167$	15.3262	1.18598	0.592990	−	0.805210i	$-0.297947\pi$
$167$	15.3262	1.18598	0.592990	+	0.805210i	$0.297947\pi$
$168$	−2.92507	−0.225674
$169$	21.9782	1.69063
$170$	21.8285	1.67417
$171$	6.75539	0.516597
$172$	9.52968	0.726631
$173$	−7.59654	−0.577554	−0.288777	−	0.957396i	$-0.593249\pi$
$173$	−7.59654	−0.577554	−0.288777	+	0.957396i	$0.593249\pi$
$174$	8.50542	0.644794
$175$	33.6695	2.54518
$176$	−4.79597	−0.361510
$177$	−7.54110	−0.566824
$178$	26.7264	2.00323
$179$	23.9553	1.79050	0.895251	−	0.445562i	$-0.146996\pi$
$179$	23.9553	1.79050	0.895251	+	0.445562i	$0.146996\pi$
$180$	−5.95712	−0.444017
$181$	9.65879	0.717932	0.358966	−	0.933351i	$-0.383129\pi$
$181$	9.65879	0.717932	0.358966	+	0.933351i	$0.383129\pi$
$182$	−34.9782	−2.59276
$183$	−9.48910	−0.701455
$184$	−1.94934	−0.143707
$185$	3.95712	0.290933
$186$	−2.16969	−0.159089
$187$	−2.97821	−0.217788
$188$	4.37083	0.318776
$189$	3.15885	0.229772
$190$	−50.0495	−3.63097
$191$	−3.11597	−0.225464	−0.112732	−	0.993625i	$-0.535960\pi$
$191$	−3.11597	−0.225464	−0.112732	+	0.993625i	$0.535960\pi$
$192$	−3.67510	−0.265228
$193$	15.5108	1.11649	0.558245	−	0.829676i	$-0.311475\pi$
$193$	15.5108	1.11649	0.558245	+	0.829676i	$0.311475\pi$
$194$	−19.1164	−1.37248
$195$	23.4033	1.67595
$196$	4.48363	0.320259
$197$	−3.51078	−0.250133	−0.125066	−	0.992148i	$-0.539914\pi$
$197$	−3.51078	−0.250133	−0.125066	+	0.992148i	$0.539914\pi$
$198$	1.89257	0.134499
$199$	0.647954	0.0459322	0.0229661	−	0.999736i	$-0.492689\pi$
$199$	0.647954	0.0459322	0.0229661	+	0.999736i	$0.492689\pi$
$200$	−9.86997	−0.697912
$201$	−3.15885	−0.222808
$202$	5.39539	0.379618
$203$	14.3501	1.00718
$204$	−4.43539	−0.310539
$205$	−5.59654	−0.390879
$206$	9.49822	0.661772
$207$	2.10513	0.146317
$208$	28.0604	1.94564
$209$	6.82859	0.472344
$210$	−23.4033	−1.61498
$211$	−1.80680	−0.124385	−0.0621927	−	0.998064i	$-0.519809\pi$
$211$	−1.80680	−0.124385	−0.0621927	+	0.998064i	$0.519809\pi$
$212$	−9.78513	−0.672045
$213$	4.92507	0.337461
$214$	−16.0748	−1.09885
$215$	−25.0496	−1.70837
$216$	−0.925994	−0.0630059
$217$	−3.66063	−0.248500
$218$	−14.7771	−1.00083
$219$	4.14801	0.280297
$220$	−6.02167	−0.405981
$221$	17.4250	1.17213
$222$	−1.87228	−0.125659
$223$	−4.66051	−0.312091	−0.156045	−	0.987750i	$-0.549875\pi$
$223$	−4.66051	−0.312091	−0.156045	+	0.987750i	$0.549875\pi$
$224$	−22.2103	−1.48399
$225$	10.6588	0.710586
$226$	−2.90888	−0.193496
$227$	7.70167	0.511178	0.255589	−	0.966786i	$-0.417731\pi$
$227$	7.70167	0.511178	0.255589	+	0.966786i	$0.417731\pi$
$228$	10.1697	0.673504
$229$	4.61372	0.304883	0.152442	−	0.988312i	$-0.451286\pi$
$229$	4.61372	0.304883	0.152442	+	0.988312i	$0.451286\pi$
$230$	−15.5965	−1.02841
$231$	3.19308	0.210089
$232$	−4.20662	−0.276178
$233$	−28.5604	−1.87105	−0.935525	−	0.353259i	$-0.885073\pi$
$233$	−28.5604	−1.87105	−0.935525	+	0.353259i	$0.885073\pi$
$234$	−11.0731	−0.723870
$235$	−11.4891	−0.749467
$236$	−11.3525	−0.738986
$237$	−4.73372	−0.307488
$238$	−17.4250	−1.12950
$239$	17.2549	1.11612	0.558062	−	0.829799i	$-0.311545\pi$
$239$	17.2549	1.11612	0.558062	+	0.829799i	$0.311545\pi$
$240$	18.7748	1.21191
$241$	−23.7644	−1.53080	−0.765399	−	0.643556i	$-0.777459\pi$
$241$	−23.7644	−1.53080	−0.765399	+	0.643556i	$0.777459\pi$
$242$	−18.6820	−1.20092
$243$	1.00000	0.0641500
$244$	−14.2851	−0.914508
$245$	−11.7856	−0.752954
$246$	2.64795	0.168827
$247$	−39.9530	−2.54215
$248$	1.07309	0.0681411
$249$	−11.9142	−0.755034
$250$	−41.9250	−2.65157
$251$	25.1267	1.58598	0.792991	−	0.609233i	$-0.208523\pi$
$251$	25.1267	1.58598	0.792991	+	0.609233i	$0.208523\pi$
$252$	4.75539	0.299561
$253$	2.12794	0.133783
$254$	29.6353	1.85948
$255$	11.6588	0.730102
$256$	20.7958	1.29974
$257$	8.29649	0.517521	0.258760	−	0.965942i	$-0.416686\pi$
$257$	8.29649	0.517521	0.258760	+	0.965942i	$0.416686\pi$
$258$	11.8520	0.737873
$259$	−3.15885	−0.196281
$260$	35.2318	2.18498
$261$	4.54282	0.281194
$262$	−2.45112	−0.151431
$263$	−22.6028	−1.39375	−0.696873	−	0.717194i	$-0.745426\pi$
$263$	−22.6028	−1.39375	−0.696873	+	0.717194i	$0.745426\pi$
$264$	−0.936028	−0.0576086
$265$	25.7210	1.58003
$266$	39.9530	2.44967
$267$	14.2748	0.873605
$268$	−4.75539	−0.290482
$269$	16.1713	0.985981	0.492990	−	0.870035i	$-0.335904\pi$
$269$	16.1713	0.985981	0.492990	+	0.870035i	$0.335904\pi$
$270$	−7.40882	−0.450886
$271$	10.0559	0.610853	0.305426	−	0.952216i	$-0.401201\pi$
$271$	10.0559	0.610853	0.305426	+	0.952216i	$0.401201\pi$
$272$	13.9788	0.847589
$273$	−18.6822	−1.13070
$274$	27.7833	1.67845
$275$	10.7743	0.649714
$276$	3.16910	0.190758
$277$	−6.74627	−0.405344	−0.202672	−	0.979247i	$-0.564963\pi$
$277$	−6.74627	−0.405344	−0.202672	+	0.979247i	$0.564963\pi$
$278$	19.3005	1.15757
$279$	−1.15885	−0.0693785
$280$	11.5749	0.691731
$281$	14.4463	0.861796	0.430898	−	0.902401i	$-0.358197\pi$
$281$	14.4463	0.861796	0.430898	+	0.902401i	$0.358197\pi$
$282$	5.43597	0.323707
$283$	−14.5405	−0.864344	−0.432172	−	0.901791i	$-0.642253\pi$
$283$	−14.5405	−0.864344	−0.432172	+	0.901791i	$0.642253\pi$
$284$	7.41430	0.439958
$285$	−26.7319	−1.58346
$286$	−11.1931	−0.661861
$287$	4.46755	0.263711
$288$	−7.03113	−0.414313
$289$	−8.31942	−0.489378
$290$	−33.6570	−1.97640
$291$	−10.2103	−0.598536
$292$	6.24449	0.365431
$293$	−11.3069	−0.660554	−0.330277	−	0.943884i	$-0.607142\pi$
$293$	−11.3069	−0.660554	−0.330277	+	0.943884i	$0.607142\pi$
$294$	5.57625	0.325214
$295$	29.8410	1.73741
$296$	0.925994	0.0538223
$297$	1.01084	0.0586547
$298$	14.3834	0.833207
$299$	−12.4502	−0.720016
$300$	16.0459	0.926413
$301$	19.9963	1.15257
$302$	−33.4204	−1.92313
$303$	2.88173	0.165551
$304$	−32.0513	−1.83827
$305$	37.5495	2.15008
$306$	−5.51625	−0.315343
$307$	21.4674	1.22521	0.612606	−	0.790389i	$-0.290121\pi$
$307$	21.4674	1.22521	0.612606	+	0.790389i	$0.290121\pi$
$308$	4.80692	0.273900
$309$	5.07309	0.288598
$310$	8.58570	0.487635
$311$	9.98063	0.565950	0.282975	−	0.959127i	$-0.408679\pi$
$311$	9.98063	0.565950	0.282975	+	0.959127i	$0.408679\pi$
$312$	5.47655	0.310048
$313$	−32.4422	−1.83374	−0.916871	−	0.399184i	$-0.869293\pi$
$313$	−32.4422	−1.83374	−0.916871	+	0.399184i	$0.869293\pi$
$314$	−43.3798	−2.44807
$315$	−12.4999	−0.704292
$316$	−7.12622	−0.400881
$317$	14.5496	0.817189	0.408594	−	0.912716i	$-0.366019\pi$
$317$	14.5496	0.817189	0.408594	+	0.912716i	$0.366019\pi$
$318$	−12.1697	−0.682442
$319$	4.59205	0.257105
$320$	14.5428	0.812968
$321$	−8.58570	−0.479207
$322$	12.4502	0.693825
$323$	−19.9033	−1.10745
$324$	1.50542	0.0836343
$325$	−63.0386	−3.49675
$326$	1.53429	0.0849764
$327$	−7.89257	−0.436460
$328$	−1.30963	−0.0723122
$329$	9.17141	0.505636
$330$	−7.48910	−0.412262
$331$	18.0296	0.990998	0.495499	−	0.868609i	$-0.334985\pi$
$331$	18.0296	0.990998	0.495499	+	0.868609i	$0.334985\pi$
$332$	−17.9359	−0.984361
$333$	−1.00000	−0.0547997
$334$	28.6949	1.57012
$335$	12.4999	0.682945
$336$	−14.9873	−0.817626
$337$	19.8068	1.07895	0.539473	−	0.842003i	$-0.318624\pi$
$337$	19.8068	1.07895	0.539473	+	0.842003i	$0.318624\pi$
$338$	41.1493	2.23823
$339$	−1.55366	−0.0843832
$340$	17.5514	0.951856
$341$	−1.17141	−0.0634352
$342$	12.6480	0.683923
$343$	−12.7039	−0.685944
$344$	−5.86178	−0.316046
$345$	−8.33026	−0.448486
$346$	−14.2228	−0.764624
$347$	−31.9867	−1.71714	−0.858569	−	0.512698i	$-0.828646\pi$
$347$	−31.9867	−1.71714	−0.858569	+	0.512698i	$0.828646\pi$
$348$	6.83885	0.366601
$349$	26.3193	1.40884	0.704420	−	0.709783i	$-0.251207\pi$
$349$	26.3193	1.40884	0.704420	+	0.709783i	$0.251207\pi$
$350$	63.0386	3.36956
$351$	−5.91424	−0.315679
$352$	−7.10732	−0.378821
$353$	5.01130	0.266725	0.133362	−	0.991067i	$-0.457423\pi$
$353$	5.01130	0.266725	0.133362	+	0.991067i	$0.457423\pi$
$354$	−14.1190	−0.750418
$355$	−19.4891	−1.03437
$356$	21.4896	1.13894
$357$	−9.30686	−0.492571
$358$	44.8509	2.37044
$359$	−18.5964	−0.981482	−0.490741	−	0.871306i	$-0.663274\pi$
$359$	−18.5964	−0.981482	−0.490741	+	0.871306i	$0.663274\pi$
$360$	3.66427	0.193124
$361$	26.6353	1.40186
$362$	18.0839	0.950470
$363$	−9.97821	−0.523720
$364$	−28.1245	−1.47412
$365$	−16.4142	−0.859157
$366$	−17.7662	−0.928655
$367$	22.6571	1.18269	0.591345	−	0.806419i	$-0.298597\pi$
$367$	22.6571	1.18269	0.591345	+	0.806419i	$0.298597\pi$
$368$	−9.98791	−0.520656
$369$	1.41430	0.0736253
$370$	7.40882	0.385166
$371$	−20.5323	−1.06599
$372$	−1.74455	−0.0904508
$373$	0.148013	0.00766380	0.00383190	−	0.999993i	$-0.498780\pi$
$373$	0.148013	0.00766380	0.00383190	+	0.999993i	$0.498780\pi$
$374$	−5.57603	−0.288330
$375$	−22.3925	−1.15634
$376$	−2.68853	−0.138650
$377$	−26.8673	−1.38374
$378$	5.91424	0.304196
$379$	18.1804	0.933865	0.466932	−	0.884293i	$-0.345359\pi$
$379$	18.1804	0.933865	0.466932	+	0.884293i	$0.345359\pi$
$380$	−40.2427	−2.06440
$381$	15.8285	0.810917
$382$	−5.83395	−0.298491
$383$	25.7982	1.31822	0.659112	−	0.752045i	$-0.270932\pi$
$383$	25.7982	1.31822	0.659112	+	0.752045i	$0.270932\pi$
$384$	7.18144	0.366476
$385$	−12.6354	−0.643959
$386$	29.0405	1.47812
$387$	6.33026	0.321785
$388$	−15.3707	−0.780330
$389$	−27.6565	−1.40224	−0.701120	−	0.713044i	$-0.747316\pi$
$389$	−27.6565	−1.40224	−0.701120	+	0.713044i	$0.747316\pi$
$390$	43.8175	2.21879
$391$	−6.20231	−0.313664
$392$	−2.75791	−0.139296
$393$	−1.30917	−0.0660386
$394$	−6.57315	−0.331150
$395$	18.7319	0.942503
$396$	1.52173	0.0764699
$397$	−17.6337	−0.885009	−0.442504	−	0.896766i	$-0.645910\pi$
$397$	−17.6337	−0.885009	−0.442504	+	0.896766i	$0.645910\pi$
$398$	1.21315	0.0608096
$399$	21.3393	1.06830
$400$	−50.5712	−2.52856
$401$	−11.6250	−0.580526	−0.290263	−	0.956947i	$-0.593743\pi$
$401$	−11.6250	−0.580526	−0.290263	+	0.956947i	$0.593743\pi$
$402$	−5.91424	−0.294975
$403$	6.85371	0.341408
$404$	4.33821	0.215834
$405$	−3.95712	−0.196631
$406$	26.8673	1.33340
$407$	−1.01084	−0.0501053
$408$	2.72824	0.135068
$409$	23.4250	1.15829	0.579146	−	0.815224i	$-0.303386\pi$
$409$	23.4250	1.15829	0.579146	+	0.815224i	$0.303386\pi$
$410$	−10.4783	−0.517485
$411$	14.8393	0.731969
$412$	7.63712	0.376254
$413$	−23.8212	−1.17216
$414$	3.94139	0.193709
$415$	47.1461	2.31431
$416$	41.5837	2.03881
$417$	10.3086	0.504813
$418$	12.7850	0.625335
$419$	−11.5749	−0.565469	−0.282735	−	0.959198i	$-0.591242\pi$
$419$	−11.5749	−0.565469	−0.282735	+	0.959198i	$0.591242\pi$
$420$	−18.8176	−0.918207
$421$	17.9576	0.875199	0.437600	−	0.899170i	$-0.355829\pi$
$421$	17.9576	0.875199	0.437600	+	0.899170i	$0.355829\pi$
$422$	−3.38283	−0.164674
$423$	2.90340	0.141168
$424$	6.01890	0.292304
$425$	−31.4038	−1.52331
$426$	9.22110	0.446764
$427$	−29.9746	−1.45057
$428$	−12.9251	−0.624757
$429$	−5.97833	−0.288636
$430$	−46.8997	−2.26171
$431$	20.4228	0.983733	0.491867	−	0.870671i	$-0.336315\pi$
$431$	20.4228	0.983733	0.491867	+	0.870671i	$0.336315\pi$
$432$	−4.74455	−0.228272
$433$	−21.0234	−1.01032	−0.505160	−	0.863026i	$-0.668566\pi$
$433$	−21.0234	−1.01032	−0.505160	+	0.863026i	$0.668566\pi$
$434$	−6.85371	−0.328989
$435$	−17.9765	−0.861907
$436$	−11.8816	−0.569026
$437$	14.2210	0.680282
$438$	7.76623	0.371085
$439$	−17.4982	−0.835144	−0.417572	−	0.908644i	$-0.637119\pi$
$439$	−17.4982	−0.835144	−0.417572	+	0.908644i	$0.637119\pi$
$440$	3.70397	0.176580
$441$	2.97833	0.141825
$442$	32.6244	1.55179
$443$	−11.3395	−0.538755	−0.269378	−	0.963035i	$-0.586818\pi$
$443$	−11.3395	−0.538755	−0.269378	+	0.963035i	$0.586818\pi$
$444$	−1.50542	−0.0714440
$445$	−56.4871	−2.67775
$446$	−8.72577	−0.413177
$447$	7.68230	0.363360
$448$	−11.6091	−0.548478
$449$	−33.7106	−1.59090	−0.795450	−	0.606020i	$-0.792765\pi$
$449$	−33.7106	−1.59090	−0.795450	+	0.606020i	$0.792765\pi$
$450$	19.9562	0.940744
$451$	1.42962	0.0673183
$452$	−2.33891	−0.110013
$453$	−17.8501	−0.838673
$454$	14.4197	0.676748
$455$	73.9276	3.46578
$456$	−6.25545	−0.292938
$457$	19.3566	0.905461	0.452731	−	0.891647i	$-0.350450\pi$
$457$	19.3566	0.905461	0.452731	+	0.891647i	$0.350450\pi$
$458$	8.63816	0.403635
$459$	−2.94628	−0.137521
$460$	−12.5405	−0.584705
$461$	6.93533	0.323010	0.161505	−	0.986872i	$-0.448365\pi$
$461$	6.93533	0.323010	0.161505	+	0.986872i	$0.448365\pi$
$462$	5.97833	0.278137
$463$	−7.96921	−0.370361	−0.185180	−	0.982705i	$-0.559287\pi$
$463$	−7.96921	−0.370361	−0.185180	+	0.982705i	$0.559287\pi$
$464$	−21.5537	−1.00060
$465$	4.58570	0.212657
$466$	−53.4729	−2.47708
$467$	21.5021	0.995000	0.497500	−	0.867464i	$-0.334251\pi$
$467$	21.5021	0.995000	0.497500	+	0.867464i	$0.334251\pi$
$468$	−8.90340	−0.411560
$469$	−9.97833	−0.460756
$470$	−21.5108	−0.992218
$471$	−23.1696	−1.06760
$472$	6.98301	0.321419
$473$	6.39885	0.294220
$474$	−8.86282	−0.407083
$475$	72.0043	3.30378
$476$	−14.0107	−0.642180
$477$	−6.49994	−0.297612
$478$	32.3059	1.47764
$479$	−32.0725	−1.46543	−0.732715	−	0.680536i	$-0.761747\pi$
$479$	−32.0725	−1.46543	−0.732715	+	0.680536i	$0.761747\pi$
$480$	27.8230	1.26994
$481$	5.91424	0.269666
$482$	−44.4935	−2.02662
$483$	6.64979	0.302576
$484$	−15.0214	−0.682790
$485$	40.4032	1.83462
$486$	1.87228	0.0849282
$487$	−9.43757	−0.427657	−0.213829	−	0.976871i	$-0.568593\pi$
$487$	−9.43757	−0.427657	−0.213829	+	0.976871i	$0.568593\pi$
$488$	8.78685	0.397762
$489$	0.819478	0.0370581
$490$	−22.0659	−0.996835
$491$	−3.33488	−0.150501	−0.0752505	−	0.997165i	$-0.523976\pi$
$491$	−3.33488	−0.150501	−0.0752505	+	0.997165i	$0.523976\pi$
$492$	2.12911	0.0959876
$493$	−13.3844	−0.602805
$494$	−74.8030	−3.36555
$495$	−4.00000	−0.179787
$496$	5.49822	0.246877
$497$	15.5576	0.697852
$498$	−22.3067	−0.999589
$499$	−29.6478	−1.32722	−0.663610	−	0.748079i	$-0.730976\pi$
$499$	−29.6478	−1.32722	−0.663610	+	0.748079i	$0.730976\pi$
$500$	−33.7101	−1.50756
$501$	15.3262	0.684726
$502$	47.0441	2.09968
$503$	−15.0302	−0.670164	−0.335082	−	0.942189i	$-0.608764\pi$
$503$	−15.0302	−0.670164	−0.335082	+	0.942189i	$0.608764\pi$
$504$	−2.92507	−0.130293
$505$	−11.4033	−0.507442
$506$	3.98410	0.177115
$507$	21.9782	0.976087
$508$	23.8285	1.05722
$509$	12.1074	0.536653	0.268326	−	0.963328i	$-0.413529\pi$
$509$	12.1074	0.536653	0.268326	+	0.963328i	$0.413529\pi$
$510$	21.8285	0.966581
$511$	13.1029	0.579640
$512$	24.5727	1.08597
$513$	6.75539	0.298258
$514$	15.5333	0.685145
$515$	−20.0748	−0.884602
$516$	9.52968	0.419521
$517$	2.93486	0.129075
$518$	−5.91424	−0.259857
$519$	−7.59654	−0.333451
$520$	−21.6713	−0.950351
$521$	−26.7102	−1.17020	−0.585098	−	0.810963i	$-0.698944\pi$
$521$	−26.7102	−1.17020	−0.585098	+	0.810963i	$0.698944\pi$
$522$	8.50542	0.372272
$523$	−6.88450	−0.301038	−0.150519	−	0.988607i	$-0.548094\pi$
$523$	−6.88450	−0.301038	−0.150519	+	0.988607i	$0.548094\pi$
$524$	−1.97084	−0.0860966
$525$	33.6695	1.46946
$526$	−42.3186	−1.84518
$527$	3.41430	0.148729
$528$	−4.79597	−0.208718
$529$	−18.5684	−0.807323
$530$	48.1569	2.09180
$531$	−7.54110	−0.327256
$532$	32.1245	1.39277
$533$	−8.36449	−0.362306
$534$	26.7264	1.15656
$535$	33.9746	1.46885
$536$	2.92507	0.126344
$537$	23.9553	1.03375
$538$	30.2771	1.30534
$539$	3.01060	0.129676
$540$	−5.95712	−0.256354
$541$	7.93591	0.341191	0.170596	−	0.985341i	$-0.445431\pi$
$541$	7.93591	0.341191	0.170596	+	0.985341i	$0.445431\pi$
$542$	18.8274	0.808707
$543$	9.65879	0.414498
$544$	20.7157	0.888178
$545$	31.2318	1.33782
$546$	−34.9782	−1.49693
$547$	4.15897	0.177825	0.0889123	−	0.996039i	$-0.471661\pi$
$547$	4.15897	0.177825	0.0889123	+	0.996039i	$0.471661\pi$
$548$	22.3394	0.954291
$549$	−9.48910	−0.404985
$550$	20.1725	0.860156
$551$	30.6885	1.30738
$552$	−1.94934	−0.0829694
$553$	−14.9531	−0.635870
$554$	−12.6309	−0.536635
$555$	3.95712	0.167970
$556$	15.5187	0.658141
$557$	4.94284	0.209435	0.104717	−	0.994502i	$-0.466606\pi$
$557$	4.94284	0.209435	0.104717	+	0.994502i	$0.466606\pi$
$558$	−2.16969	−0.0918501
$559$	−37.4386	−1.58349
$560$	59.3066	2.50616
$561$	−2.97821	−0.125740
$562$	27.0475	1.14093
$563$	−37.8912	−1.59692	−0.798462	−	0.602046i	$-0.794353\pi$
$563$	−37.8912	−1.59692	−0.798462	+	0.602046i	$0.794353\pi$
$564$	4.37083	0.184045
$565$	6.14801	0.258649
$566$	−27.2239	−1.14430
$567$	3.15885	0.132659
$568$	−4.56059	−0.191358
$569$	26.6393	1.11678	0.558389	−	0.829579i	$-0.311420\pi$
$569$	26.6393	1.11678	0.558389	+	0.829579i	$0.311420\pi$
$570$	−50.0495	−2.09634
$571$	−39.3267	−1.64577	−0.822885	−	0.568207i	$-0.807637\pi$
$571$	−39.3267	−1.64577	−0.822885	+	0.568207i	$0.807637\pi$
$572$	−8.99988	−0.376304
$573$	−3.11597	−0.130171
$574$	8.36449	0.349127
$575$	22.4382	0.935736
$576$	−3.67510	−0.153129
$577$	−26.5323	−1.10456	−0.552278	−	0.833660i	$-0.686241\pi$
$577$	−26.5323	−1.10456	−0.552278	+	0.833660i	$0.686241\pi$
$578$	−15.5763	−0.647887
$579$	15.5108	0.644606
$580$	−27.0621	−1.12369
$581$	−37.6353	−1.56137
$582$	−19.1164	−0.792402
$583$	−6.57038	−0.272117
$584$	−3.84103	−0.158943
$585$	23.4033	0.967609
$586$	−21.1696	−0.874507
$587$	−40.3804	−1.66668	−0.833339	−	0.552762i	$-0.813574\pi$
$587$	−40.3804	−1.66668	−0.833339	+	0.552762i	$0.813574\pi$
$588$	4.48363	0.184902
$589$	−7.82848	−0.322567
$590$	55.8707	2.30016
$591$	−3.51078	−0.144414
$592$	4.74455	0.195000
$593$	43.6786	1.79367	0.896833	−	0.442369i	$-0.145862\pi$
$593$	43.6786	1.79367	0.896833	+	0.442369i	$0.145862\pi$
$594$	1.89257	0.0776529
$595$	36.8284	1.50982
$596$	11.5651	0.473724
$597$	0.647954	0.0265190
$598$	−23.3103	−0.953229
$599$	−21.4360	−0.875850	−0.437925	−	0.899012i	$-0.644286\pi$
$599$	−21.4360	−0.875850	−0.437925	+	0.899012i	$0.644286\pi$
$600$	−9.86997	−0.402940
$601$	22.5091	0.918164	0.459082	−	0.888394i	$-0.348178\pi$
$601$	22.5091	0.918164	0.459082	+	0.888394i	$0.348178\pi$
$602$	37.4386	1.52589
$603$	−3.15885	−0.128638
$604$	−26.8719	−1.09340
$605$	39.4850	1.60529
$606$	5.39539	0.219173
$607$	14.2878	0.579926	0.289963	−	0.957038i	$-0.406357\pi$
$607$	14.2878	0.579926	0.289963	+	0.957038i	$0.406357\pi$
$608$	−47.4980	−1.92630
$609$	14.3501	0.581495
$610$	70.3031	2.84649
$611$	−17.1714	−0.694681
$612$	−4.43539	−0.179290
$613$	3.38628	0.136770	0.0683852	−	0.997659i	$-0.478215\pi$
$613$	3.38628	0.136770	0.0683852	+	0.997659i	$0.478215\pi$
$614$	40.1930	1.62206
$615$	−5.59654	−0.225674
$616$	−2.95677	−0.119132
$617$	−27.7644	−1.11775	−0.558876	−	0.829251i	$-0.688767\pi$
$617$	−27.7644	−1.11775	−0.558876	+	0.829251i	$0.688767\pi$
$618$	9.49822	0.382074
$619$	35.0776	1.40989	0.704944	−	0.709263i	$-0.250972\pi$
$619$	35.0776	1.40989	0.704944	+	0.709263i	$0.250972\pi$
$620$	6.90340	0.277247
$621$	2.10513	0.0844760
$622$	18.6865	0.749260
$623$	45.0920	1.80657
$624$	28.0604	1.12332
$625$	35.3159	1.41263
$626$	−60.7408	−2.42769
$627$	6.82859	0.272708
$628$	−34.8799	−1.39186
$629$	2.94628	0.117476
$630$	−23.4033	−0.932411
$631$	−37.9014	−1.50883	−0.754416	−	0.656396i	$-0.772080\pi$
$631$	−37.9014	−1.50883	−0.754416	+	0.656396i	$0.772080\pi$
$632$	4.38339	0.174362
$633$	−1.80680	−0.0718140
$634$	27.2409	1.08188
$635$	−62.6352	−2.48560
$636$	−9.78513	−0.388006
$637$	−17.6145	−0.697913
$638$	8.59759	0.340382
$639$	4.92507	0.194833
$640$	−28.4178	−1.12331
$641$	−31.0810	−1.22763	−0.613814	−	0.789451i	$-0.710365\pi$
$641$	−31.0810	−1.22763	−0.613814	+	0.789451i	$0.710365\pi$
$642$	−16.0748	−0.634422
$643$	−4.77706	−0.188389	−0.0941945	−	0.995554i	$-0.530028\pi$
$643$	−4.77706	−0.188389	−0.0941945	+	0.995554i	$0.530028\pi$
$644$	10.0107	0.394478
$645$	−25.0496	−0.986326
$646$	−37.2644	−1.46615
$647$	12.2874	0.483067	0.241533	−	0.970393i	$-0.422350\pi$
$647$	12.2874	0.483067	0.241533	+	0.970393i	$0.422350\pi$
$648$	−0.925994	−0.0363765
$649$	−7.62282	−0.299222
$650$	−118.026	−4.62935
$651$	−3.66063	−0.143471
$652$	1.23366	0.0483137
$653$	3.24231	0.126881	0.0634407	−	0.997986i	$-0.479793\pi$
$653$	3.24231	0.126881	0.0634407	+	0.997986i	$0.479793\pi$
$654$	−14.7771	−0.577829
$655$	5.18052	0.202420
$656$	−6.71020	−0.261989
$657$	4.14801	0.161829
$658$	17.1714	0.669411
$659$	21.6462	0.843218	0.421609	−	0.906778i	$-0.361466\pi$
$659$	21.6462	0.843218	0.421609	+	0.906778i	$0.361466\pi$
$660$	−6.02167	−0.234393
$661$	40.1895	1.56319	0.781596	−	0.623786i	$-0.214406\pi$
$661$	40.1895	1.56319	0.781596	+	0.623786i	$0.214406\pi$
$662$	33.7564	1.31198
$663$	17.4250	0.676732
$664$	11.0325	0.428144
$665$	−84.4420	−3.27452
$666$	−1.87228	−0.0725492
$667$	9.56324	0.370290
$668$	23.0724	0.892697
$669$	−4.66051	−0.180186
$670$	23.4033	0.904150
$671$	−9.59193	−0.370292
$672$	−22.2103	−0.856779
$673$	−4.51250	−0.173944	−0.0869720	−	0.996211i	$-0.527719\pi$
$673$	−4.51250	−0.173944	−0.0869720	+	0.996211i	$0.527719\pi$
$674$	37.0838	1.42841
$675$	10.6588	0.410257
$676$	33.0864	1.27255
$677$	16.4719	0.633067	0.316534	−	0.948581i	$-0.397481\pi$
$677$	16.4719	0.633067	0.316534	+	0.948581i	$0.397481\pi$
$678$	−2.90888	−0.111715
$679$	−32.2527	−1.23774
$680$	−10.7960	−0.414006
$681$	7.70167	0.295129
$682$	−2.19320	−0.0839819
$683$	−13.5194	−0.517307	−0.258653	−	0.965970i	$-0.583279\pi$
$683$	−13.5194	−0.517307	−0.258653	+	0.965970i	$0.583279\pi$
$684$	10.1697	0.388848
$685$	−58.7209	−2.24361
$686$	−23.7851	−0.908120
$687$	4.61372	0.176025
$688$	−30.0342	−1.14504
$689$	38.4422	1.46453
$690$	−15.5965	−0.593750
$691$	15.4549	0.587931	0.293966	−	0.955816i	$-0.405025\pi$
$691$	15.4549	0.587931	0.293966	+	0.955816i	$0.405025\pi$
$692$	−11.4360	−0.434730
$693$	3.19308	0.121295
$694$	−59.8880	−2.27332
$695$	−40.7923	−1.54734
$696$	−4.20662	−0.159452
$697$	−4.16692	−0.157833
$698$	49.2770	1.86516
$699$	−28.5604	−1.08025
$700$	50.6867	1.91578
$701$	−28.4146	−1.07321	−0.536603	−	0.843835i	$-0.680293\pi$
$701$	−28.4146	−1.07321	−0.536603	+	0.843835i	$0.680293\pi$
$702$	−11.0731	−0.417927
$703$	−6.75539	−0.254784
$704$	−3.71493	−0.140012
$705$	−11.4891	−0.432705
$706$	9.38254	0.353117
$707$	9.10295	0.342352
$708$	−11.3525	−0.426653
$709$	29.7678	1.11795	0.558977	−	0.829183i	$-0.311194\pi$
$709$	29.7678	1.11795	0.558977	+	0.829183i	$0.311194\pi$
$710$	−36.4890	−1.36941
$711$	−4.73372	−0.177528
$712$	−13.2184	−0.495380
$713$	−2.43953	−0.0913611
$714$	−17.4250	−0.652115
$715$	23.6570	0.884720
$716$	36.0627	1.34773
$717$	17.2549	0.644395
$718$	−34.8176	−1.29938
$719$	−16.2354	−0.605477	−0.302739	−	0.953074i	$-0.597901\pi$
$719$	−16.2354	−0.605477	−0.302739	+	0.953074i	$0.597901\pi$
$720$	18.7748	0.699694
$721$	16.0251	0.596806
$722$	49.8686	1.85592
$723$	−23.7644	−0.883807
$724$	14.5405	0.540394
$725$	48.4210	1.79831
$726$	−18.6820	−0.693353
$727$	36.4764	1.35284	0.676418	−	0.736518i	$-0.263531\pi$
$727$	36.4764	1.35284	0.676418	+	0.736518i	$0.263531\pi$
$728$	17.2996	0.641165
$729$	1.00000	0.0370370
$730$	−30.7319	−1.13744
$731$	−18.6507	−0.689822
$732$	−14.2851	−0.527991
$733$	−1.74178	−0.0643343	−0.0321671	−	0.999483i	$-0.510241\pi$
$733$	−1.74178	−0.0643343	−0.0321671	+	0.999483i	$0.510241\pi$
$734$	42.4203	1.56576
$735$	−11.7856	−0.434718
$736$	−14.8014	−0.545588
$737$	−3.19308	−0.117619
$738$	2.64795	0.0974725
$739$	19.8285	0.729402	0.364701	−	0.931125i	$-0.381171\pi$
$739$	19.8285	0.729402	0.364701	+	0.931125i	$0.381171\pi$
$740$	5.95712	0.218988
$741$	−39.9530	−1.46771
$742$	−38.4422	−1.41126
$743$	21.0072	0.770678	0.385339	−	0.922775i	$-0.374085\pi$
$743$	21.0072	0.770678	0.385339	+	0.922775i	$0.374085\pi$
$744$	1.07309	0.0393413
$745$	−30.3998	−1.11376
$746$	0.277120	0.0101461
$747$	−11.9142	−0.435919
$748$	−4.48345	−0.163931
$749$	−27.1209	−0.990978
$750$	−41.9250	−1.53088
$751$	−39.1369	−1.42813	−0.714064	−	0.700081i	$-0.753147\pi$
$751$	−39.1369	−1.42813	−0.714064	+	0.700081i	$0.753147\pi$
$752$	−13.7753	−0.502335
$753$	25.1267	0.915667
$754$	−50.3031	−1.83193
$755$	70.6352	2.57068
$756$	4.75539	0.172952
$757$	−31.5965	−1.14840	−0.574198	−	0.818717i	$-0.694686\pi$
$757$	−31.5965	−1.14840	−0.574198	+	0.818717i	$0.694686\pi$
$758$	34.0387	1.23634
$759$	2.12794	0.0772395
$760$	24.7536	0.897906
$761$	27.4684	0.995727	0.497864	−	0.867255i	$-0.334118\pi$
$761$	27.4684	0.995727	0.497864	+	0.867255i	$0.334118\pi$
$762$	29.6353	1.07357
$763$	−24.9314	−0.902578
$764$	−4.69083	−0.169708
$765$	11.6588	0.421524
$766$	48.3013	1.74520
$767$	44.5999	1.61041
$768$	20.7958	0.750406
$769$	46.3608	1.67181	0.835907	−	0.548871i	$-0.184942\pi$
$769$	46.3608	1.67181	0.835907	+	0.548871i	$0.184942\pi$
$770$	−23.6570	−0.852537
$771$	8.29649	0.298791
$772$	23.3502	0.840392
$773$	1.70397	0.0612877	0.0306439	−	0.999530i	$-0.490244\pi$
$773$	1.70397	0.0612877	0.0306439	+	0.999530i	$0.490244\pi$
$774$	11.8520	0.426011
$775$	−12.3519	−0.443694
$776$	9.45464	0.339402
$777$	−3.15885	−0.113323
$778$	−51.7806	−1.85642
$779$	9.55412	0.342312
$780$	35.2318	1.26150
$781$	4.97844	0.178143
$782$	−11.6124	−0.415260
$783$	4.54282	0.162347
$784$	−14.1308	−0.504672
$785$	91.6847	3.27237
$786$	−2.45112	−0.0874285
$787$	25.3863	0.904923	0.452462	−	0.891784i	$-0.350546\pi$
$787$	25.3863	0.904923	0.452462	+	0.891784i	$0.350546\pi$
$788$	−5.28519	−0.188277
$789$	−22.6028	−0.804680
$790$	35.0712	1.24778
$791$	−4.90777	−0.174500
$792$	−0.936028	−0.0332603
$793$	56.1208	1.99291
$794$	−33.0151	−1.17166
$795$	25.7210	0.912231
$796$	0.975442	0.0345736
$797$	−0.343513	−0.0121679	−0.00608393	−	0.999981i	$-0.501937\pi$
$797$	−0.343513	−0.0121679	−0.00608393	+	0.999981i	$0.501937\pi$
$798$	39.9530	1.41432
$799$	−8.55424	−0.302627
$800$	−74.9433	−2.64965
$801$	14.2748	0.504376
$802$	−21.7653	−0.768558
$803$	4.19296	0.147966
$804$	−4.75539	−0.167710
$805$	−26.3140	−0.927447
$806$	12.8320	0.451989
$807$	16.1713	0.569256
$808$	−2.66846	−0.0938762
$809$	−1.68277	−0.0591629	−0.0295815	−	0.999562i	$-0.509417\pi$
$809$	−1.68277	−0.0591629	−0.0295815	+	0.999562i	$0.509417\pi$
$810$	−7.40882	−0.260319
$811$	0.261795	0.00919285	0.00459642	−	0.999989i	$-0.498537\pi$
$811$	0.261795	0.00919285	0.00459642	+	0.999989i	$0.498537\pi$
$812$	21.6029	0.758113
$813$	10.0559	0.352676
$814$	−1.89257	−0.0663344
$815$	−3.24277	−0.113589
$816$	13.9788	0.489356
$817$	42.7633	1.49610
$818$	43.8581	1.53346
$819$	−18.6822	−0.652808
$820$	−8.42513	−0.294218
$821$	25.2382	0.880818	0.440409	−	0.897797i	$-0.354833\pi$
$821$	25.2382	0.880818	0.440409	+	0.897797i	$0.354833\pi$
$822$	27.7833	0.969054
$823$	−21.6823	−0.755798	−0.377899	−	0.925847i	$-0.623353\pi$
$823$	−21.6823	−0.755798	−0.377899	+	0.925847i	$0.623353\pi$
$824$	−4.69765	−0.163650
$825$	10.7743	0.375113
$826$	−44.5999	−1.55183
$827$	−13.4707	−0.468421	−0.234210	−	0.972186i	$-0.575250\pi$
$827$	−13.4707	−0.468421	−0.234210	+	0.972186i	$0.575250\pi$
$828$	3.16910	0.110134
$829$	−1.32219	−0.0459215	−0.0229607	−	0.999736i	$-0.507309\pi$
$829$	−1.32219	−0.0459215	−0.0229607	+	0.999736i	$0.507309\pi$
$830$	88.2704	3.06391
$831$	−6.74627	−0.234026
$832$	21.7354	0.753541
$833$	−8.77499	−0.304035
$834$	19.3005	0.668322
$835$	−60.6477	−2.09880
$836$	10.2799	0.355537
$837$	−1.15885	−0.0400557
$838$	−21.6713	−0.748624
$839$	18.6461	0.643735	0.321868	−	0.946785i	$-0.395689\pi$
$839$	18.6461	0.643735	0.321868	+	0.946785i	$0.395689\pi$
$840$	11.5749	0.399371
$841$	−8.36277	−0.288371
$842$	33.6216	1.15868
$843$	14.4463	0.497558
$844$	−2.71999	−0.0936261
$845$	−86.9704	−2.99187
$846$	5.43597	0.186892
$847$	−31.5197	−1.08303
$848$	30.8393	1.05903
$849$	−14.5405	−0.499029
$850$	−58.7966	−2.01671
$851$	−2.10513	−0.0721630
$852$	7.41430	0.254010
$853$	43.6353	1.49404	0.747022	−	0.664800i	$-0.231483\pi$
$853$	43.6353	1.49404	0.747022	+	0.664800i	$0.231483\pi$
$854$	−56.1208	−1.92042
$855$	−26.7319	−0.914211
$856$	7.95031	0.271736
$857$	−8.72530	−0.298051	−0.149025	−	0.988833i	$-0.547614\pi$
$857$	−8.72530	−0.298051	−0.149025	+	0.988833i	$0.547614\pi$
$858$	−11.1931	−0.382125
$859$	19.0902	0.651348	0.325674	−	0.945482i	$-0.394409\pi$
$859$	19.0902	0.651348	0.325674	+	0.945482i	$0.394409\pi$
$860$	−37.7101	−1.28590
$861$	4.46755	0.152254
$862$	38.2372	1.30236
$863$	−21.3999	−0.728461	−0.364231	−	0.931309i	$-0.618668\pi$
$863$	−21.3999	−0.728461	−0.364231	+	0.931309i	$0.618668\pi$
$864$	−7.03113	−0.239204
$865$	30.0604	1.02208
$866$	−39.3616	−1.33756
$867$	−8.31942	−0.282542
$868$	−5.51078	−0.187048
$869$	−4.78501	−0.162320
$870$	−33.6570	−1.14108
$871$	18.6822	0.633022
$872$	7.30847	0.247496
$873$	−10.2103	−0.345565
$874$	26.6256	0.900625
$875$	−70.7345	−2.39126
$876$	6.24449	0.210982
$877$	−8.10467	−0.273675	−0.136838	−	0.990593i	$-0.543694\pi$
$877$	−8.10467	−0.273675	−0.136838	+	0.990593i	$0.543694\pi$
$878$	−32.7615	−1.10565
$879$	−11.3069	−0.381371
$880$	18.9782	0.639755
$881$	20.0604	0.675852	0.337926	−	0.941173i	$-0.390275\pi$
$881$	20.0604	0.675852	0.337926	+	0.941173i	$0.390275\pi$
$882$	5.57625	0.187762
$883$	32.8582	1.10577	0.552884	−	0.833259i	$-0.313527\pi$
$883$	32.8582	1.10577	0.552884	+	0.833259i	$0.313527\pi$
$884$	26.2319	0.882275
$885$	29.8410	1.00310
$886$	−21.2307	−0.713258
$887$	−8.11722	−0.272550	−0.136275	−	0.990671i	$-0.543513\pi$
$887$	−8.11722	−0.272550	−0.136275	+	0.990671i	$0.543513\pi$
$888$	0.925994	0.0310743
$889$	49.9998	1.67694
$890$	−105.760	−3.54507
$891$	1.01084	0.0338643
$892$	−7.01602	−0.234914
$893$	19.6136	0.656344
$894$	14.3834	0.481053
$895$	−94.7939	−3.16861
$896$	22.6851	0.757856
$897$	−12.4502	−0.415702
$898$	−63.1155	−2.10619
$899$	−5.26445	−0.175579
$900$	16.0459	0.534865
$901$	19.1507	0.638001
$902$	2.67665	0.0891226
$903$	19.9963	0.665436
$904$	1.43868	0.0478497
$905$	−38.2210	−1.27051
$906$	−33.4204	−1.11032
$907$	−22.5829	−0.749854	−0.374927	−	0.927054i	$-0.622332\pi$
$907$	−22.5829	−0.749854	−0.374927	+	0.927054i	$0.622332\pi$
$908$	11.5942	0.384768
$909$	2.88173	0.0955809
$910$	138.413	4.58834
$911$	46.1124	1.52777	0.763886	−	0.645351i	$-0.223289\pi$
$911$	46.1124	1.52777	0.763886	+	0.645351i	$0.223289\pi$
$912$	−32.0513	−1.06132
$913$	−12.0433	−0.398577
$914$	36.2408	1.19874
$915$	37.5495	1.24135
$916$	6.94558	0.229489
$917$	−4.13545	−0.136565
$918$	−5.51625	−0.182064
$919$	−25.0480	−0.826256	−0.413128	−	0.910673i	$-0.635564\pi$
$919$	−25.0480	−0.826256	−0.413128	+	0.910673i	$0.635564\pi$
$920$	7.71376	0.254315
$921$	21.4674	0.707376
$922$	12.9849	0.427633
$923$	−29.1281	−0.958762
$924$	4.80692	0.158136
$925$	−10.6588	−0.350459
$926$	−14.9206	−0.490320
$927$	5.07309	0.166622
$928$	−31.9412	−1.04852
$929$	−23.9067	−0.784354	−0.392177	−	0.919890i	$-0.628278\pi$
$929$	−23.9067	−0.784354	−0.392177	+	0.919890i	$0.628278\pi$
$930$	8.58570	0.281536
$931$	20.1198	0.659398
$932$	−42.9953	−1.40836
$933$	9.98063	0.326751
$934$	40.2579	1.31728
$935$	11.7851	0.385415
$936$	5.47655	0.179007
$937$	20.6352	0.674121	0.337061	−	0.941483i	$-0.390567\pi$
$937$	20.6352	0.674121	0.337061	+	0.941483i	$0.390567\pi$
$938$	−18.6822	−0.609995
$939$	−32.4422	−1.05871
$940$	−17.2959	−0.564130
$941$	−28.5467	−0.930597	−0.465298	−	0.885154i	$-0.654053\pi$
$941$	−28.5467	−0.930597	−0.465298	+	0.885154i	$0.654053\pi$
$942$	−43.3798	−1.41339
$943$	2.97728	0.0969536
$944$	35.7791	1.16451
$945$	−12.4999	−0.406623
$946$	11.9804	0.389517
$947$	4.36679	0.141902	0.0709508	−	0.997480i	$-0.477397\pi$
$947$	4.36679	0.141902	0.0709508	+	0.997480i	$0.477397\pi$
$948$	−7.12622	−0.231449
$949$	−24.5323	−0.796353
$950$	134.812	4.37388
$951$	14.5496	0.471804
$952$	8.61810	0.279314
$953$	0.767947	0.0248762	0.0124381	−	0.999923i	$-0.496041\pi$
$953$	0.767947	0.0248762	0.0124381	+	0.999923i	$0.496041\pi$
$954$	−12.1697	−0.394008
$955$	12.3303	0.398998
$956$	25.9758	0.840117
$957$	4.59205	0.148440
$958$	−60.0486	−1.94008
$959$	46.8751	1.51368
$960$	14.5428	0.469368
$961$	−29.6571	−0.956680
$962$	11.0731	0.357011
$963$	−8.58570	−0.276670
$964$	−35.7753	−1.15225
$965$	−61.3780	−1.97583
$966$	12.4502	0.400580
$967$	14.4799	0.465641	0.232821	−	0.972520i	$-0.425204\pi$
$967$	14.4799	0.465641	0.232821	+	0.972520i	$0.425204\pi$
$968$	9.23976	0.296977
$969$	−19.9033	−0.639386
$970$	75.6460	2.42885
$971$	−24.2787	−0.779141	−0.389571	−	0.920997i	$-0.627377\pi$
$971$	−24.2787	−0.779141	−0.389571	+	0.920997i	$0.627377\pi$
$972$	1.50542	0.0482863
$973$	32.5633	1.04393
$974$	−17.6697	−0.566175
$975$	−63.0386	−2.01885
$976$	45.0216	1.44110
$977$	−52.4534	−1.67813	−0.839066	−	0.544029i	$-0.816898\pi$
$977$	−52.4534	−1.67813	−0.839066	+	0.544029i	$0.816898\pi$
$978$	1.53429	0.0490612
$979$	14.4295	0.461169
$980$	−17.7422	−0.566755
$981$	−7.89257	−0.251990
$982$	−6.24382	−0.199248
$983$	−50.3067	−1.60454	−0.802268	−	0.596964i	$-0.796373\pi$
$983$	−50.3067	−1.60454	−0.802268	+	0.596964i	$0.796373\pi$
$984$	−1.30963	−0.0417495
$985$	13.8926	0.442654
$986$	−25.0594	−0.798053
$987$	9.17141	0.291929
$988$	−60.1459	−1.91350
$989$	13.3260	0.423743
$990$	−7.48910	−0.238019
$991$	23.2434	0.738352	0.369176	−	0.929359i	$-0.379640\pi$
$991$	23.2434	0.738352	0.369176	+	0.929359i	$0.379640\pi$
$992$	8.14801	0.258700
$993$	18.0296	0.572153
$994$	29.1281	0.923886
$995$	−2.56403	−0.0812852
$996$	−17.9359	−0.568321
$997$	58.6568	1.85768	0.928840	−	0.370480i	$-0.120807\pi$
$997$	58.6568	1.85768	0.928840	+	0.370480i	$0.120807\pi$
$998$	−55.5089	−1.75710
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	111.2.a.b.1.4	✓	4
3.2	odd	2		333.2.a.g.1.1		4
4.3	odd	2		1776.2.a.u.1.1		4
5.4	even	2		2775.2.a.x.1.1		4
7.6	odd	2		5439.2.a.u.1.4		4
8.3	odd	2		7104.2.a.cf.1.4		4
8.5	even	2		7104.2.a.cc.1.4		4
12.11	even	2		5328.2.a.bs.1.4		4
15.14	odd	2		8325.2.a.bv.1.4		4
37.36	even	2		4107.2.a.i.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
111.2.a.b.1.4	✓	4	1.1	even	1	trivial
333.2.a.g.1.1		4	3.2	odd	2
1776.2.a.u.1.1		4	4.3	odd	2
2775.2.a.x.1.1		4	5.4	even	2
4107.2.a.i.1.1		4	37.36	even	2
5328.2.a.bs.1.4		4	12.11	even	2
5439.2.a.u.1.4		4	7.6	odd	2
7104.2.a.cc.1.4		4	8.5	even	2
7104.2.a.cf.1.4		4	8.3	odd	2
8325.2.a.bv.1.4		4	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 111 = 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	111.a (trivial)

\(f(q)\)	\(=\)	\(q+1.87228 q^{2} +1.00000 q^{3} +1.50542 q^{4} -3.95712 q^{5} +1.87228 q^{6} +3.15885 q^{7} -0.925994 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.87228 q^{2} +1.00000 q^{3} +1.50542 q^{4} -3.95712 q^{5} +1.87228 q^{6} +3.15885 q^{7} -0.925994 q^{8} +1.00000 q^{9} -7.40882 q^{10} +1.01084 q^{11} +1.50542 q^{12} -5.91424 q^{13} +5.91424 q^{14} -3.95712 q^{15} -4.74455 q^{16} -2.94628 q^{17} +1.87228 q^{18} +6.75539 q^{19} -5.95712 q^{20} +3.15885 q^{21} +1.89257 q^{22} +2.10513 q^{23} -0.925994 q^{24} +10.6588 q^{25} -11.0731 q^{26} +1.00000 q^{27} +4.75539 q^{28} +4.54282 q^{29} -7.40882 q^{30} -1.15885 q^{31} -7.03113 q^{32} +1.01084 q^{33} -5.51625 q^{34} -12.4999 q^{35} +1.50542 q^{36} -1.00000 q^{37} +12.6480 q^{38} -5.91424 q^{39} +3.66427 q^{40} +1.41430 q^{41} +5.91424 q^{42} +6.33026 q^{43} +1.52173 q^{44} -3.95712 q^{45} +3.94139 q^{46} +2.90340 q^{47} -4.74455 q^{48} +2.97833 q^{49} +19.9562 q^{50} -2.94628 q^{51} -8.90340 q^{52} -6.49994 q^{53} +1.87228 q^{54} -4.00000 q^{55} -2.92507 q^{56} +6.75539 q^{57} +8.50542 q^{58} -7.54110 q^{59} -5.95712 q^{60} -9.48910 q^{61} -2.16969 q^{62} +3.15885 q^{63} -3.67510 q^{64} +23.4033 q^{65} +1.89257 q^{66} -3.15885 q^{67} -4.43539 q^{68} +2.10513 q^{69} -23.4033 q^{70} +4.92507 q^{71} -0.925994 q^{72} +4.14801 q^{73} -1.87228 q^{74} +10.6588 q^{75} +10.1697 q^{76} +3.19308 q^{77} -11.0731 q^{78} -4.73372 q^{79} +18.7748 q^{80} +1.00000 q^{81} +2.64795 q^{82} -11.9142 q^{83} +4.75539 q^{84} +11.6588 q^{85} +11.8520 q^{86} +4.54282 q^{87} -0.936028 q^{88} +14.2748 q^{89} -7.40882 q^{90} -18.6822 q^{91} +3.16910 q^{92} -1.15885 q^{93} +5.43597 q^{94} -26.7319 q^{95} -7.03113 q^{96} -10.2103 q^{97} +5.57625 q^{98} +1.01084 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{4} - 2 q^{5} + 4 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^4 - 2 * q^5 + 4 * q^7 - 6 * q^8 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{4} - 2 q^{5} + 4 q^{7} - 6 q^{8} + 4 q^{9} - 4 q^{10} + 4 q^{12} + 4 q^{13} - 4 q^{14} - 2 q^{15} - 4 q^{16} - 2 q^{17} + 8 q^{19} - 10 q^{20} + 4 q^{21} - 12 q^{22} - 10 q^{23} - 6 q^{24} - 8 q^{26} + 4 q^{27} - 2 q^{29} - 4 q^{30} + 4 q^{31} - 12 q^{32} - 16 q^{34} - 16 q^{35} + 4 q^{36} - 4 q^{37} + 12 q^{38} + 4 q^{39} + 4 q^{40} + 12 q^{41} - 4 q^{42} + 4 q^{43} + 32 q^{44} - 2 q^{45} + 12 q^{46} - 12 q^{47} - 4 q^{48} + 20 q^{49} + 32 q^{50} - 2 q^{51} - 12 q^{52} + 8 q^{53} - 16 q^{55} + 20 q^{56} + 8 q^{57} + 32 q^{58} - 10 q^{59} - 10 q^{60} - 8 q^{61} + 4 q^{62} + 4 q^{63} + 36 q^{65} - 12 q^{66} - 4 q^{67} + 22 q^{68} - 10 q^{69} - 36 q^{70} - 12 q^{71} - 6 q^{72} + 12 q^{73} + 28 q^{76} - 8 q^{77} - 8 q^{78} - 8 q^{79} + 10 q^{80} + 4 q^{81} - 28 q^{82} - 20 q^{83} + 4 q^{85} + 52 q^{86} - 2 q^{87} - 36 q^{88} + 26 q^{89} - 4 q^{90} - 24 q^{91} - 38 q^{92} + 4 q^{93} + 20 q^{94} - 28 q^{95} - 12 q^{96} - 4 q^{97} + 24 q^{98}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^4 - 2 * q^5 + 4 * q^7 - 6 * q^8 + 4 * q^9 - 4 * q^10 + 4 * q^12 + 4 * q^13 - 4 * q^14 - 2 * q^15 - 4 * q^16 - 2 * q^17 + 8 * q^19 - 10 * q^20 + 4 * q^21 - 12 * q^22 - 10 * q^23 - 6 * q^24 - 8 * q^26 + 4 * q^27 - 2 * q^29 - 4 * q^30 + 4 * q^31 - 12 * q^32 - 16 * q^34 - 16 * q^35 + 4 * q^36 - 4 * q^37 + 12 * q^38 + 4 * q^39 + 4 * q^40 + 12 * q^41 - 4 * q^42 + 4 * q^43 + 32 * q^44 - 2 * q^45 + 12 * q^46 - 12 * q^47 - 4 * q^48 + 20 * q^49 + 32 * q^50 - 2 * q^51 - 12 * q^52 + 8 * q^53 - 16 * q^55 + 20 * q^56 + 8 * q^57 + 32 * q^58 - 10 * q^59 - 10 * q^60 - 8 * q^61 + 4 * q^62 + 4 * q^63 + 36 * q^65 - 12 * q^66 - 4 * q^67 + 22 * q^68 - 10 * q^69 - 36 * q^70 - 12 * q^71 - 6 * q^72 + 12 * q^73 + 28 * q^76 - 8 * q^77 - 8 * q^78 - 8 * q^79 + 10 * q^80 + 4 * q^81 - 28 * q^82 - 20 * q^83 + 4 * q^85 + 52 * q^86 - 2 * q^87 - 36 * q^88 + 26 * q^89 - 4 * q^90 - 24 * q^91 - 38 * q^92 + 4 * q^93 + 20 * q^94 - 28 * q^95 - 12 * q^96 - 4 * q^97 + 24 * q^98

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