LMFDB - Embedded newform 1445.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1445,2,Mod(1,1445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1445, 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("1445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1445 = 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1445.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:	$11.5383830921$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 21 x^{8} - 232 x^{7} + 44 x^{6} + 424 x^{5} - 137 x^{4} + \cdots + 17 $ x^12 - 4x^11 - 10x^10 + 52x^9 + 21x^8 - 232x^7 + 44x^6 + 424x^5 - 137x^4 - 316x^3 + 46x^2 + 92*x + 17
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.73061$ of defining polynomial
Character	$\chi$	$=$	1445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73061 q^{2} +1.54691 q^{3} +5.45623 q^{4} +1.00000 q^{5} -4.22400 q^{6} -1.26445 q^{7} -9.43761 q^{8} -0.607075 q^{9} +O(q^{10})$	\(q-2.73061 q^{2} +1.54691 q^{3} +5.45623 q^{4} +1.00000 q^{5} -4.22400 q^{6} -1.26445 q^{7} -9.43761 q^{8} -0.607075 q^{9} -2.73061 q^{10} +0.417956 q^{11} +8.44028 q^{12} -5.66390 q^{13} +3.45273 q^{14} +1.54691 q^{15} +14.8580 q^{16} +1.65769 q^{18} -0.134144 q^{19} +5.45623 q^{20} -1.95600 q^{21} -1.14127 q^{22} +6.81922 q^{23} -14.5991 q^{24} +1.00000 q^{25} +15.4659 q^{26} -5.57981 q^{27} -6.89915 q^{28} +0.329510 q^{29} -4.22400 q^{30} -0.605659 q^{31} -21.6961 q^{32} +0.646540 q^{33} -1.26445 q^{35} -3.31234 q^{36} -8.83716 q^{37} +0.366294 q^{38} -8.76154 q^{39} -9.43761 q^{40} +1.63371 q^{41} +5.34106 q^{42} -1.99751 q^{43} +2.28046 q^{44} -0.607075 q^{45} -18.6206 q^{46} -5.06253 q^{47} +22.9839 q^{48} -5.40115 q^{49} -2.73061 q^{50} -30.9036 q^{52} -1.54296 q^{53} +15.2363 q^{54} +0.417956 q^{55} +11.9334 q^{56} -0.207508 q^{57} -0.899762 q^{58} -1.41104 q^{59} +8.44028 q^{60} -7.80498 q^{61} +1.65382 q^{62} +0.767619 q^{63} +29.5276 q^{64} -5.66390 q^{65} -1.76545 q^{66} -12.1014 q^{67} +10.5487 q^{69} +3.45273 q^{70} +4.82493 q^{71} +5.72934 q^{72} -2.90042 q^{73} +24.1308 q^{74} +1.54691 q^{75} -0.731919 q^{76} -0.528486 q^{77} +23.9243 q^{78} +4.33770 q^{79} +14.8580 q^{80} -6.81023 q^{81} -4.46102 q^{82} -4.39202 q^{83} -10.6724 q^{84} +5.45443 q^{86} +0.509721 q^{87} -3.94451 q^{88} +4.98704 q^{89} +1.65769 q^{90} +7.16175 q^{91} +37.2072 q^{92} -0.936899 q^{93} +13.8238 q^{94} -0.134144 q^{95} -33.5619 q^{96} -0.635381 q^{97} +14.7484 q^{98} -0.253731 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{2} - 8 q^{3} + 12 q^{4} + 12 q^{5} - 8 q^{6} - 16 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q - 4 * q^2 - 8 * q^3 + 12 * q^4 + 12 * q^5 - 8 * q^6 - 16 * q^7 - 12 * q^8 + 12 * q^9	$ 12 q - 4 q^{2} - 8 q^{3} + 12 q^{4} + 12 q^{5} - 8 q^{6} - 16 q^{7} - 12 q^{8} + 12 q^{9} - 4 q^{10} - 16 q^{11} - 16 q^{12} - 8 q^{13} + 16 q^{14} - 8 q^{15} + 12 q^{16} + 4 q^{18} + 12 q^{20} + 16 q^{21} - 16 q^{22} - 16 q^{23} + 12 q^{25} + 16 q^{26} - 32 q^{27} - 40 q^{28} - 16 q^{29} - 8 q^{30} - 24 q^{31} - 28 q^{32} - 16 q^{35} + 12 q^{36} - 24 q^{37} - 24 q^{38} - 8 q^{39} - 12 q^{40} - 8 q^{41} - 16 q^{43} - 8 q^{44} + 12 q^{45} - 40 q^{46} - 32 q^{47} + 24 q^{48} + 20 q^{49} - 4 q^{50} - 24 q^{52} - 8 q^{54} - 16 q^{55} + 24 q^{56} - 32 q^{57} + 16 q^{58} + 8 q^{59} - 16 q^{60} - 24 q^{61} - 8 q^{62} - 48 q^{63} + 36 q^{64} - 8 q^{65} + 40 q^{66} - 8 q^{67} + 48 q^{69} + 16 q^{70} + 16 q^{71} + 12 q^{72} - 16 q^{73} - 8 q^{75} + 16 q^{76} + 24 q^{77} + 24 q^{78} - 40 q^{79} + 12 q^{80} + 36 q^{81} + 16 q^{82} - 40 q^{83} + 32 q^{84} - 8 q^{86} - 32 q^{87} - 48 q^{88} + 8 q^{89} + 4 q^{90} - 72 q^{91} + 8 q^{92} + 24 q^{93} - 16 q^{94} - 8 q^{96} - 32 q^{97} - 60 q^{98} - 56 q^{99}+O(q^{100}) $ 12 * q - 4 * q^2 - 8 * q^3 + 12 * q^4 + 12 * q^5 - 8 * q^6 - 16 * q^7 - 12 * q^8 + 12 * q^9 - 4 * q^10 - 16 * q^11 - 16 * q^12 - 8 * q^13 + 16 * q^14 - 8 * q^15 + 12 * q^16 + 4 * q^18 + 12 * q^20 + 16 * q^21 - 16 * q^22 - 16 * q^23 + 12 * q^25 + 16 * q^26 - 32 * q^27 - 40 * q^28 - 16 * q^29 - 8 * q^30 - 24 * q^31 - 28 * q^32 - 16 * q^35 + 12 * q^36 - 24 * q^37 - 24 * q^38 - 8 * q^39 - 12 * q^40 - 8 * q^41 - 16 * q^43 - 8 * q^44 + 12 * q^45 - 40 * q^46 - 32 * q^47 + 24 * q^48 + 20 * q^49 - 4 * q^50 - 24 * q^52 - 8 * q^54 - 16 * q^55 + 24 * q^56 - 32 * q^57 + 16 * q^58 + 8 * q^59 - 16 * q^60 - 24 * q^61 - 8 * q^62 - 48 * q^63 + 36 * q^64 - 8 * q^65 + 40 * q^66 - 8 * q^67 + 48 * q^69 + 16 * q^70 + 16 * q^71 + 12 * q^72 - 16 * q^73 - 8 * q^75 + 16 * q^76 + 24 * q^77 + 24 * q^78 - 40 * q^79 + 12 * q^80 + 36 * q^81 + 16 * q^82 - 40 * q^83 + 32 * q^84 - 8 * q^86 - 32 * q^87 - 48 * q^88 + 8 * q^89 + 4 * q^90 - 72 * q^91 + 8 * q^92 + 24 * q^93 - 16 * q^94 - 8 * q^96 - 32 * q^97 - 60 * q^98 - 56 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.73061	−1.93083	−0.965416	−	0.260713i	$-0.916042\pi$
$2$	−2.73061	−1.93083	−0.965416	+	0.260713i	$0.916042\pi$
$3$	1.54691	0.893108	0.446554	−	0.894757i	$-0.352651\pi$
$3$	1.54691	0.893108	0.446554	+	0.894757i	$0.352651\pi$
$4$	5.45623	2.72811
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−4.22400	−1.72444
$7$	−1.26445	−0.477919	−0.238959	−	0.971030i	$-0.576806\pi$
$7$	−1.26445	−0.477919	−0.238959	+	0.971030i	$0.576806\pi$
$8$	−9.43761	−3.33670
$9$	−0.607075	−0.202358
$10$	−2.73061	−0.863495
$11$	0.417956	0.126018	0.0630092	−	0.998013i	$-0.479930\pi$
$11$	0.417956	0.126018	0.0630092	+	0.998013i	$0.479930\pi$
$12$	8.44028	2.43650
$13$	−5.66390	−1.57088	−0.785442	−	0.618935i	$-0.787564\pi$
$13$	−5.66390	−1.57088	−0.785442	+	0.618935i	$0.787564\pi$
$14$	3.45273	0.922781
$15$	1.54691	0.399410
$16$	14.8580	3.71449
$17$	0	0
$17$	0	0
$18$	1.65769	0.390720
$19$	−0.134144	−0.0307747	−0.0153873	−	0.999882i	$-0.504898\pi$
$19$	−0.134144	−0.0307747	−0.0153873	+	0.999882i	$0.504898\pi$
$20$	5.45623	1.22005
$21$	−1.95600	−0.426833
$22$	−1.14127	−0.243321
$23$	6.81922	1.42191	0.710953	−	0.703239i	$-0.248264\pi$
$23$	6.81922	1.42191	0.710953	+	0.703239i	$0.248264\pi$
$24$	−14.5991	−2.98003
$25$	1.00000	0.200000
$26$	15.4659	3.03311
$27$	−5.57981	−1.07384
$28$	−6.89915	−1.30382
$29$	0.329510	0.0611884	0.0305942	−	0.999532i	$-0.490260\pi$
$29$	0.329510	0.0611884	0.0305942	+	0.999532i	$0.490260\pi$
$30$	−4.22400	−0.771194
$31$	−0.605659	−0.108780	−0.0543898	−	0.998520i	$-0.517321\pi$
$31$	−0.605659	−0.108780	−0.0543898	+	0.998520i	$0.517321\pi$
$32$	−21.6961	−3.83537
$33$	0.646540	0.112548
$34$	0	0
$35$	−1.26445	−0.213732
$36$	−3.31234	−0.552057
$37$	−8.83716	−1.45282	−0.726410	−	0.687261i	$-0.758813\pi$
$37$	−8.83716	−1.45282	−0.726410	+	0.687261i	$0.758813\pi$
$38$	0.366294	0.0594207
$39$	−8.76154	−1.40297
$40$	−9.43761	−1.49222
$41$	1.63371	0.255143	0.127571	−	0.991829i	$-0.459282\pi$
$41$	1.63371	0.255143	0.127571	+	0.991829i	$0.459282\pi$
$42$	5.34106	0.824143
$43$	−1.99751	−0.304618	−0.152309	−	0.988333i	$-0.548671\pi$
$43$	−1.99751	−0.304618	−0.152309	+	0.988333i	$0.548671\pi$
$44$	2.28046	0.343793
$45$	−0.607075	−0.0904974
$46$	−18.6206	−2.74546
$47$	−5.06253	−0.738445	−0.369223	−	0.929341i	$-0.620376\pi$
$47$	−5.06253	−0.738445	−0.369223	+	0.929341i	$0.620376\pi$
$48$	22.9839	3.31744
$49$	−5.40115	−0.771594
$50$	−2.73061	−0.386167
$51$	0	0
$52$	−30.9036	−4.28555
$53$	−1.54296	−0.211942	−0.105971	−	0.994369i	$-0.533795\pi$
$53$	−1.54296	−0.211942	−0.105971	+	0.994369i	$0.533795\pi$
$54$	15.2363	2.07340
$55$	0.417956	0.0563572
$56$	11.9334	1.59467
$57$	−0.207508	−0.0274851
$58$	−0.899762	−0.118145
$59$	−1.41104	−0.183702	−0.0918510	−	0.995773i	$-0.529278\pi$
$59$	−1.41104	−0.183702	−0.0918510	+	0.995773i	$0.529278\pi$
$60$	8.44028	1.08964
$61$	−7.80498	−0.999325	−0.499663	−	0.866220i	$-0.666543\pi$
$61$	−7.80498	−0.999325	−0.499663	+	0.866220i	$0.666543\pi$
$62$	1.65382	0.210035
$63$	0.767619	0.0967109
$64$	29.5276	3.69095
$65$	−5.66390	−0.702521
$66$	−1.76545	−0.217312
$67$	−12.1014	−1.47842	−0.739209	−	0.673477i	$-0.764800\pi$
$67$	−12.1014	−1.47842	−0.739209	+	0.673477i	$0.764800\pi$
$68$	0	0
$69$	10.5487	1.26992
$70$	3.45273	0.412680
$71$	4.82493	0.572614	0.286307	−	0.958138i	$-0.407572\pi$
$71$	4.82493	0.572614	0.286307	+	0.958138i	$0.407572\pi$
$72$	5.72934	0.675209
$73$	−2.90042	−0.339468	−0.169734	−	0.985490i	$-0.554291\pi$
$73$	−2.90042	−0.339468	−0.169734	+	0.985490i	$0.554291\pi$
$74$	24.1308	2.80515
$75$	1.54691	0.178622
$76$	−0.731919	−0.0839568
$77$	−0.528486	−0.0602266
$78$	23.9243	2.70890
$79$	4.33770	0.488029	0.244015	−	0.969772i	$-0.421535\pi$
$79$	4.33770	0.488029	0.244015	+	0.969772i	$0.421535\pi$
$80$	14.8580	1.66117
$81$	−6.81023	−0.756693
$82$	−4.46102	−0.492638
$83$	−4.39202	−0.482087	−0.241043	−	0.970514i	$-0.577490\pi$
$83$	−4.39202	−0.482087	−0.241043	+	0.970514i	$0.577490\pi$
$84$	−10.6724	−1.16445
$85$	0	0
$86$	5.45443	0.588167
$87$	0.509721	0.0546478
$88$	−3.94451	−0.420486
$89$	4.98704	0.528625	0.264313	−	0.964437i	$-0.414855\pi$
$89$	4.98704	0.528625	0.264313	+	0.964437i	$0.414855\pi$
$90$	1.65769	0.174735
$91$	7.16175	0.750755
$92$	37.2072	3.87912
$93$	−0.936899	−0.0971519
$94$	13.8238	1.42581
$95$	−0.134144	−0.0137628
$96$	−33.5619	−3.42539
$97$	−0.635381	−0.0645131	−0.0322566	−	0.999480i	$-0.510269\pi$
$97$	−0.635381	−0.0645131	−0.0322566	+	0.999480i	$0.510269\pi$
$98$	14.7484	1.48982
$99$	−0.253731	−0.0255009
$100$	5.45623	0.545623
$101$	−7.45004	−0.741307	−0.370653	−	0.928771i	$-0.620866\pi$
$101$	−7.45004	−0.741307	−0.370653	+	0.928771i	$0.620866\pi$
$102$	0	0
$103$	13.5860	1.33866	0.669332	−	0.742963i	$-0.266580\pi$
$103$	13.5860	1.33866	0.669332	+	0.742963i	$0.266580\pi$
$104$	53.4537	5.24157
$105$	−1.95600	−0.190886
$106$	4.21322	0.409224
$107$	−9.27421	−0.896572	−0.448286	−	0.893890i	$-0.647965\pi$
$107$	−9.27421	−0.896572	−0.448286	+	0.893890i	$0.647965\pi$
$108$	−30.4447	−2.92955
$109$	−15.7066	−1.50442	−0.752209	−	0.658925i	$-0.771012\pi$
$109$	−15.7066	−1.50442	−0.752209	+	0.658925i	$0.771012\pi$
$110$	−1.14127	−0.108816
$111$	−13.6703	−1.29753
$112$	−18.7872	−1.77523
$113$	−1.24680	−0.117289	−0.0586447	−	0.998279i	$-0.518678\pi$
$113$	−1.24680	−0.117289	−0.0586447	+	0.998279i	$0.518678\pi$
$114$	0.566623	0.0530691
$115$	6.81922	0.635896
$116$	1.79788	0.166929
$117$	3.43842	0.317882
$118$	3.85301	0.354698
$119$	0	0
$120$	−14.5991	−1.33271
$121$	−10.8253	−0.984119
$122$	21.3124	1.92953
$123$	2.52720	0.227870
$124$	−3.30461	−0.296763
$125$	1.00000	0.0894427
$126$	−2.09607	−0.186733
$127$	17.8153	1.58085	0.790427	−	0.612556i	$-0.209859\pi$
$127$	17.8153	1.58085	0.790427	+	0.612556i	$0.209859\pi$
$128$	−37.2362	−3.29125
$129$	−3.08997	−0.272057
$130$	15.4659	1.35645
$131$	−4.47235	−0.390751	−0.195375	−	0.980729i	$-0.562593\pi$
$131$	−4.47235	−0.390751	−0.195375	+	0.980729i	$0.562593\pi$
$132$	3.52767	0.307044
$133$	0.169619	0.0147078
$134$	33.0441	2.85458
$135$	−5.57981	−0.480234
$136$	0	0
$137$	9.04916	0.773122	0.386561	−	0.922264i	$-0.373663\pi$
$137$	9.04916	0.773122	0.386561	+	0.922264i	$0.373663\pi$
$138$	−28.8044	−2.45199
$139$	−4.70076	−0.398713	−0.199357	−	0.979927i	$-0.563885\pi$
$139$	−4.70076	−0.398713	−0.199357	+	0.979927i	$0.563885\pi$
$140$	−6.89915	−0.583085
$141$	−7.83126	−0.659511
$142$	−13.1750	−1.10562
$143$	−2.36726	−0.197960
$144$	−9.01991	−0.751659
$145$	0.329510	0.0273643
$146$	7.91990	0.655456
$147$	−8.35509	−0.689116
$148$	−48.2176	−3.96346
$149$	−0.775949	−0.0635683	−0.0317841	−	0.999495i	$-0.510119\pi$
$149$	−0.775949	−0.0635683	−0.0317841	+	0.999495i	$0.510119\pi$
$150$	−4.22400	−0.344888
$151$	−2.28712	−0.186123	−0.0930617	−	0.995660i	$-0.529665\pi$
$151$	−2.28712	−0.186123	−0.0930617	+	0.995660i	$0.529665\pi$
$152$	1.26600	0.102686
$153$	0	0
$154$	1.44309	0.116288
$155$	−0.605659	−0.0486477
$156$	−47.8050	−3.82746
$157$	−19.1134	−1.52541	−0.762707	−	0.646745i	$-0.776130\pi$
$157$	−19.1134	−1.52541	−0.762707	+	0.646745i	$0.776130\pi$
$158$	−11.8446	−0.942303
$159$	−2.38682	−0.189287
$160$	−21.6961	−1.71523
$161$	−8.62260	−0.679556
$162$	18.5961	1.46105
$163$	−18.3295	−1.43568	−0.717839	−	0.696209i	$-0.754869\pi$
$163$	−18.3295	−1.43568	−0.717839	+	0.696209i	$0.754869\pi$
$164$	8.91389	0.696058
$165$	0.646540	0.0503330
$166$	11.9929	0.930829
$167$	−20.1484	−1.55913	−0.779564	−	0.626322i	$-0.784559\pi$
$167$	−20.1484	−1.55913	−0.779564	+	0.626322i	$0.784559\pi$
$168$	18.4599	1.42421
$169$	19.0798	1.46768
$170$	0	0
$171$	0.0814353	0.00622751
$172$	−10.8989	−0.831033
$173$	−19.1144	−1.45324	−0.726619	−	0.687041i	$-0.758909\pi$
$173$	−19.1144	−1.45324	−0.726619	+	0.687041i	$0.758909\pi$
$174$	−1.39185	−0.105516
$175$	−1.26445	−0.0955838
$176$	6.20998	0.468095
$177$	−2.18275	−0.164066
$178$	−13.6177	−1.02069
$179$	19.8154	1.48107	0.740535	−	0.672018i	$-0.234572\pi$
$179$	19.8154	1.48107	0.740535	+	0.672018i	$0.234572\pi$
$180$	−3.31234	−0.246887
$181$	8.86555	0.658971	0.329485	−	0.944161i	$-0.393125\pi$
$181$	8.86555	0.658971	0.329485	+	0.944161i	$0.393125\pi$
$182$	−19.5559	−1.44958
$183$	−12.0736	−0.892505
$184$	−64.3572	−4.74447
$185$	−8.83716	−0.649721
$186$	2.55831	0.187584
$187$	0	0
$188$	−27.6223	−2.01456
$189$	7.05542	0.513206
$190$	0.366294	0.0265738
$191$	10.6081	0.767576	0.383788	−	0.923421i	$-0.374619\pi$
$191$	10.6081	0.767576	0.383788	+	0.923421i	$0.374619\pi$
$192$	45.6765	3.29642
$193$	9.72535	0.700046	0.350023	−	0.936741i	$-0.386174\pi$
$193$	9.72535	0.700046	0.350023	+	0.936741i	$0.386174\pi$
$194$	1.73498	0.124564
$195$	−8.76154	−0.627427
$196$	−29.4699	−2.10500
$197$	23.7953	1.69534	0.847671	−	0.530521i	$-0.178004\pi$
$197$	23.7953	1.69534	0.847671	+	0.530521i	$0.178004\pi$
$198$	0.692839	0.0492380
$199$	13.2078	0.936276	0.468138	−	0.883655i	$-0.344925\pi$
$199$	13.2078	0.936276	0.468138	+	0.883655i	$0.344925\pi$
$200$	−9.43761	−0.667340
$201$	−18.7197	−1.32039
$202$	20.3431	1.43134
$203$	−0.416650	−0.0292431
$204$	0	0
$205$	1.63371	0.114103
$206$	−37.0979	−2.58474
$207$	−4.13978	−0.287735
$208$	−84.1541	−5.83504
$209$	−0.0560662	−0.00387818
$210$	5.34106	0.368568
$211$	−19.8864	−1.36904	−0.684518	−	0.728996i	$-0.739987\pi$
$211$	−19.8864	−1.36904	−0.684518	+	0.728996i	$0.739987\pi$
$212$	−8.41873	−0.578201
$213$	7.46373	0.511406
$214$	25.3243	1.73113
$215$	−1.99751	−0.136229
$216$	52.6601	3.58307
$217$	0.765828	0.0519878
$218$	42.8885	2.90478
$219$	−4.48668	−0.303182
$220$	2.28046	0.153749
$221$	0	0
$222$	37.3282	2.50530
$223$	6.36311	0.426106	0.213053	−	0.977041i	$-0.431659\pi$
$223$	6.36311	0.426106	0.213053	+	0.977041i	$0.431659\pi$
$224$	27.4337	1.83299
$225$	−0.607075	−0.0404717
$226$	3.40453	0.226466
$227$	−26.5359	−1.76125	−0.880625	−	0.473814i	$-0.842877\pi$
$227$	−26.5359	−1.76125	−0.880625	+	0.473814i	$0.842877\pi$
$228$	−1.13221	−0.0749825
$229$	15.5455	1.02728	0.513639	−	0.858006i	$-0.328297\pi$
$229$	15.5455	1.02728	0.513639	+	0.858006i	$0.328297\pi$
$230$	−18.6206	−1.22781
$231$	−0.817520	−0.0537889
$232$	−3.10978	−0.204167
$233$	3.80274	0.249126	0.124563	−	0.992212i	$-0.460247\pi$
$233$	3.80274	0.249126	0.124563	+	0.992212i	$0.460247\pi$
$234$	−9.38897	−0.613776
$235$	−5.06253	−0.330243
$236$	−7.69897	−0.501160
$237$	6.71002	0.435863
$238$	0	0
$239$	−5.80679	−0.375610	−0.187805	−	0.982206i	$-0.560137\pi$
$239$	−5.80679	−0.375610	−0.187805	+	0.982206i	$0.560137\pi$
$240$	22.9839	1.48361
$241$	−8.72796	−0.562218	−0.281109	−	0.959676i	$-0.590702\pi$
$241$	−8.72796	−0.562218	−0.281109	+	0.959676i	$0.590702\pi$
$242$	29.5597	1.90017
$243$	6.20463	0.398027
$244$	−42.5858	−2.72627
$245$	−5.40115	−0.345067
$246$	−6.90079	−0.439979
$247$	0.759777	0.0483434
$248$	5.71597	0.362965
$249$	−6.79405	−0.430556
$250$	−2.73061	−0.172699
$251$	15.2424	0.962093	0.481047	−	0.876695i	$-0.340257\pi$
$251$	15.2424	0.962093	0.481047	+	0.876695i	$0.340257\pi$
$252$	4.18830	0.263838
$253$	2.85014	0.179186
$254$	−48.6467	−3.05236
$255$	0	0
$256$	42.6224	2.66390
$257$	14.5342	0.906620	0.453310	−	0.891353i	$-0.350243\pi$
$257$	14.5342	0.906620	0.453310	+	0.891353i	$0.350243\pi$
$258$	8.43751	0.525296
$259$	11.1742	0.694331
$260$	−30.9036	−1.91656
$261$	−0.200037	−0.0123820
$262$	12.2122	0.754474
$263$	19.0283	1.17333	0.586666	−	0.809829i	$-0.300440\pi$
$263$	19.0283	1.17333	0.586666	+	0.809829i	$0.300440\pi$
$264$	−6.10179	−0.375539
$265$	−1.54296	−0.0947832
$266$	−0.463162	−0.0283983
$267$	7.71449	0.472119
$268$	−66.0278	−4.03329
$269$	22.6849	1.38312	0.691560	−	0.722319i	$-0.256924\pi$
$269$	22.6849	1.38312	0.691560	+	0.722319i	$0.256924\pi$
$270$	15.2363	0.927251
$271$	−9.96764	−0.605491	−0.302745	−	0.953071i	$-0.597903\pi$
$271$	−9.96764	−0.605491	−0.302745	+	0.953071i	$0.597903\pi$
$272$	0	0
$273$	11.0786	0.670506
$274$	−24.7097	−1.49277
$275$	0.417956	0.0252037
$276$	57.5562	3.46448
$277$	−11.7071	−0.703412	−0.351706	−	0.936110i	$-0.614398\pi$
$277$	−11.7071	−0.703412	−0.351706	+	0.936110i	$0.614398\pi$
$278$	12.8359	0.769848
$279$	0.367681	0.0220125
$280$	11.9334	0.713159
$281$	−1.18521	−0.0707038	−0.0353519	−	0.999375i	$-0.511255\pi$
$281$	−1.18521	−0.0707038	−0.0353519	+	0.999375i	$0.511255\pi$
$282$	21.3841	1.27341
$283$	4.61084	0.274086	0.137043	−	0.990565i	$-0.456240\pi$
$283$	4.61084	0.274086	0.137043	+	0.990565i	$0.456240\pi$
$284$	26.3259	1.56216
$285$	−0.207508	−0.0122917
$286$	6.46407	0.382229
$287$	−2.06575	−0.121937
$288$	13.1712	0.776118
$289$	0	0
$290$	−0.899762	−0.0528358
$291$	−0.982876	−0.0576172
$292$	−15.8253	−0.926108
$293$	−3.09527	−0.180828	−0.0904138	−	0.995904i	$-0.528819\pi$
$293$	−3.09527	−0.180828	−0.0904138	+	0.995904i	$0.528819\pi$
$294$	22.8145	1.33057
$295$	−1.41104	−0.0821541
$296$	83.4017	4.84763
$297$	−2.33212	−0.135323
$298$	2.11882	0.122740
$299$	−38.6234	−2.23365
$300$	8.44028	0.487300
$301$	2.52577	0.145583
$302$	6.24524	0.359373
$303$	−11.5245	−0.662067
$304$	−1.99310	−0.114312
$305$	−7.80498	−0.446912
$306$	0	0
$307$	11.8387	0.675668	0.337834	−	0.941206i	$-0.390306\pi$
$307$	11.8387	0.675668	0.337834	+	0.941206i	$0.390306\pi$
$308$	−2.88354	−0.164305
$309$	21.0162	1.19557
$310$	1.65382	0.0939306
$311$	23.4500	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$311$	23.4500	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$312$	82.6880	4.68129
$313$	27.6083	1.56051	0.780257	−	0.625459i	$-0.215088\pi$
$313$	27.6083	1.56051	0.780257	+	0.625459i	$0.215088\pi$
$314$	52.1912	2.94532
$315$	0.767619	0.0432504
$316$	23.6675	1.33140
$317$	−31.8609	−1.78949	−0.894744	−	0.446580i	$-0.852642\pi$
$317$	−31.8609	−1.78949	−0.894744	+	0.446580i	$0.852642\pi$
$318$	6.51746	0.365481
$319$	0.137720	0.00771087
$320$	29.5276	1.65065
$321$	−14.3464	−0.800736
$322$	23.5449	1.31211
$323$	0	0
$324$	−37.1582	−2.06434
$325$	−5.66390	−0.314177
$326$	50.0507	2.77205
$327$	−24.2966	−1.34361
$328$	−15.4183	−0.851334
$329$	6.40133	0.352917
$330$	−1.76545	−0.0971847
$331$	−26.0748	−1.43320	−0.716600	−	0.697484i	$-0.754303\pi$
$331$	−26.0748	−1.43320	−0.716600	+	0.697484i	$0.754303\pi$
$332$	−23.9639	−1.31519
$333$	5.36482	0.293990
$334$	55.0174	3.01042
$335$	−12.1014	−0.661168
$336$	−29.0621	−1.58547
$337$	16.5187	0.899833	0.449917	−	0.893071i	$-0.351454\pi$
$337$	16.5187	0.899833	0.449917	+	0.893071i	$0.351454\pi$
$338$	−52.0995	−2.83384
$339$	−1.92869	−0.104752
$340$	0	0
$341$	−0.253139	−0.0137082
$342$	−0.222368	−0.0120243
$343$	15.6807	0.846678
$344$	18.8518	1.01642
$345$	10.5487	0.567924
$346$	52.1938	2.80596
$347$	−20.3246	−1.09108	−0.545540	−	0.838085i	$-0.683675\pi$
$347$	−20.3246	−1.09108	−0.545540	+	0.838085i	$0.683675\pi$
$348$	2.78115	0.149086
$349$	4.88555	0.261517	0.130759	−	0.991414i	$-0.458259\pi$
$349$	4.88555	0.261517	0.130759	+	0.991414i	$0.458259\pi$
$350$	3.45273	0.184556
$351$	31.6035	1.68687
$352$	−9.06802	−0.483327
$353$	22.6733	1.20678	0.603389	−	0.797447i	$-0.293817\pi$
$353$	22.6733	1.20678	0.603389	+	0.797447i	$0.293817\pi$
$354$	5.96025	0.316783
$355$	4.82493	0.256081
$356$	27.2104	1.44215
$357$	0	0
$358$	−54.1080	−2.85970
$359$	13.4808	0.711490	0.355745	−	0.934583i	$-0.384227\pi$
$359$	13.4808	0.711490	0.355745	+	0.934583i	$0.384227\pi$
$360$	5.72934	0.301963
$361$	−18.9820	−0.999053
$362$	−24.2083	−1.27236
$363$	−16.7458	−0.878925
$364$	39.0761	2.04815
$365$	−2.90042	−0.151815
$366$	32.9683	1.72328
$367$	1.91088	0.0997470	0.0498735	−	0.998756i	$-0.484118\pi$
$367$	1.91088	0.0997470	0.0498735	+	0.998756i	$0.484118\pi$
$368$	101.320	5.28166
$369$	−0.991785	−0.0516302
$370$	24.1308	1.25450
$371$	1.95100	0.101291
$372$	−5.11193	−0.265041
$373$	24.9357	1.29112	0.645562	−	0.763708i	$-0.276623\pi$
$373$	24.9357	1.29112	0.645562	+	0.763708i	$0.276623\pi$
$374$	0	0
$375$	1.54691	0.0798820
$376$	47.7781	2.46397
$377$	−1.86631	−0.0961199
$378$	−19.2656	−0.990916
$379$	−16.6020	−0.852789	−0.426394	−	0.904537i	$-0.640216\pi$
$379$	−16.6020	−0.852789	−0.426394	+	0.904537i	$0.640216\pi$
$380$	−0.731919	−0.0375466
$381$	27.5587	1.41187
$382$	−28.9666	−1.48206
$383$	−20.0900	−1.02655	−0.513277	−	0.858223i	$-0.671568\pi$
$383$	−20.0900	−1.02655	−0.513277	+	0.858223i	$0.671568\pi$
$384$	−57.6010	−2.93944
$385$	−0.528486	−0.0269342
$386$	−26.5561	−1.35167
$387$	1.21264	0.0616420
$388$	−3.46678	−0.175999
$389$	−22.2783	−1.12955	−0.564777	−	0.825244i	$-0.691038\pi$
$389$	−22.2783	−1.12955	−0.564777	+	0.825244i	$0.691038\pi$
$390$	23.9243	1.21146
$391$	0	0
$392$	50.9740	2.57458
$393$	−6.91831	−0.348983
$394$	−64.9756	−3.27342
$395$	4.33770	0.218253
$396$	−1.38441	−0.0695694
$397$	26.2699	1.31845	0.659223	−	0.751947i	$-0.270885\pi$
$397$	26.2699	1.31845	0.659223	+	0.751947i	$0.270885\pi$
$398$	−36.0653	−1.80779
$399$	0.262384	0.0131356
$400$	14.8580	0.742899
$401$	−38.1569	−1.90547	−0.952733	−	0.303807i	$-0.901742\pi$
$401$	−38.1569	−1.90547	−0.952733	+	0.303807i	$0.901742\pi$
$402$	51.1162	2.54944
$403$	3.43040	0.170880
$404$	−40.6491	−2.02237
$405$	−6.81023	−0.338403
$406$	1.13771	0.0564635
$407$	−3.69355	−0.183082
$408$	0	0
$409$	−19.0995	−0.944411	−0.472206	−	0.881488i	$-0.656542\pi$
$409$	−19.0995	−0.944411	−0.472206	+	0.881488i	$0.656542\pi$
$410$	−4.46102	−0.220314
$411$	13.9982	0.690481
$412$	74.1281	3.65203
$413$	1.78420	0.0877947
$414$	11.3041	0.555567
$415$	−4.39202	−0.215596
$416$	122.885	6.02492
$417$	−7.27164	−0.356094
$418$	0.153095	0.00748811
$419$	−33.5509	−1.63907	−0.819533	−	0.573032i	$-0.805767\pi$
$419$	−33.5509	−1.63907	−0.819533	+	0.573032i	$0.805767\pi$
$420$	−10.6724	−0.520758
$421$	−18.9396	−0.923058	−0.461529	−	0.887125i	$-0.652699\pi$
$421$	−18.9396	−0.923058	−0.461529	+	0.887125i	$0.652699\pi$
$422$	54.3020	2.64338
$423$	3.07333	0.149431
$424$	14.5618	0.707186
$425$	0	0
$426$	−20.3805	−0.987440
$427$	9.86904	0.477596
$428$	−50.6022	−2.44595
$429$	−3.66194	−0.176800
$430$	5.45443	0.263036
$431$	−24.3871	−1.17469	−0.587343	−	0.809338i	$-0.699826\pi$
$431$	−24.3871	−1.17469	−0.587343	+	0.809338i	$0.699826\pi$
$432$	−82.9047	−3.98876
$433$	9.69125	0.465732	0.232866	−	0.972509i	$-0.425190\pi$
$433$	9.69125	0.465732	0.232866	+	0.972509i	$0.425190\pi$
$434$	−2.09118	−0.100380
$435$	0.509721	0.0244392
$436$	−85.6987	−4.10422
$437$	−0.914756	−0.0437587
$438$	12.2514	0.585393
$439$	3.22352	0.153850	0.0769251	−	0.997037i	$-0.475490\pi$
$439$	3.22352	0.153850	0.0769251	+	0.997037i	$0.475490\pi$
$440$	−3.94451	−0.188047
$441$	3.27891	0.156138
$442$	0	0
$443$	−17.5623	−0.834412	−0.417206	−	0.908812i	$-0.636991\pi$
$443$	−17.5623	−0.834412	−0.417206	+	0.908812i	$0.636991\pi$
$444$	−74.5882	−3.53980
$445$	4.98704	0.236408
$446$	−17.3752	−0.822739
$447$	−1.20032	−0.0567733
$448$	−37.3364	−1.76398
$449$	18.5233	0.874169	0.437084	−	0.899421i	$-0.356011\pi$
$449$	18.5233	0.874169	0.437084	+	0.899421i	$0.356011\pi$
$450$	1.65769	0.0781440
$451$	0.682819	0.0321527
$452$	−6.80284	−0.319979
$453$	−3.53797	−0.166228
$454$	72.4592	3.40068
$455$	7.16175	0.335748
$456$	1.95838	0.0917095
$457$	14.7077	0.687997	0.343998	−	0.938970i	$-0.388219\pi$
$457$	14.7077	0.687997	0.343998	+	0.938970i	$0.388219\pi$
$458$	−42.4488	−1.98350
$459$	0	0
$460$	37.2072	1.73480
$461$	17.9494	0.835985	0.417992	−	0.908451i	$-0.362734\pi$
$461$	17.9494	0.835985	0.417992	+	0.908451i	$0.362734\pi$
$462$	2.23233	0.103857
$463$	35.5792	1.65350	0.826752	−	0.562567i	$-0.190186\pi$
$463$	35.5792	1.65350	0.826752	+	0.562567i	$0.190186\pi$
$464$	4.89584	0.227284
$465$	−0.936899	−0.0434476
$466$	−10.3838	−0.481020
$467$	29.4411	1.36237	0.681185	−	0.732111i	$-0.261465\pi$
$467$	29.4411	1.36237	0.681185	+	0.732111i	$0.261465\pi$
$468$	18.7608	0.867217
$469$	15.3016	0.706564
$470$	13.8238	0.637643
$471$	−29.5666	−1.36236
$472$	13.3169	0.612959
$473$	−0.834873	−0.0383875
$474$	−18.3225	−0.841578
$475$	−0.134144	−0.00615493
$476$	0	0
$477$	0.936692	0.0428882
$478$	15.8561	0.725240
$479$	16.7360	0.764685	0.382343	−	0.924021i	$-0.375117\pi$
$479$	16.7360	0.764685	0.382343	+	0.924021i	$0.375117\pi$
$480$	−33.5619	−1.53188
$481$	50.0529	2.28221
$482$	23.8327	1.08555
$483$	−13.3384	−0.606917
$484$	−59.0654	−2.68479
$485$	−0.635381	−0.0288512
$486$	−16.9424	−0.768524
$487$	−23.4424	−1.06228	−0.531138	−	0.847285i	$-0.678235\pi$
$487$	−23.4424	−1.06228	−0.531138	+	0.847285i	$0.678235\pi$
$488$	73.6604	3.33445
$489$	−28.3541	−1.28222
$490$	14.7484	0.666267
$491$	25.4877	1.15025	0.575123	−	0.818067i	$-0.304954\pi$
$491$	25.4877	1.15025	0.575123	+	0.818067i	$0.304954\pi$
$492$	13.7890	0.621655
$493$	0	0
$494$	−2.07465	−0.0933431
$495$	−0.253731	−0.0114043
$496$	−8.99887	−0.404061
$497$	−6.10091	−0.273663
$498$	18.5519	0.831331
$499$	−28.7421	−1.28667	−0.643336	−	0.765584i	$-0.722450\pi$
$499$	−28.7421	−1.28667	−0.643336	+	0.765584i	$0.722450\pi$
$500$	5.45623	0.244010
$501$	−31.1677	−1.39247
$502$	−41.6211	−1.85764
$503$	−0.0102855	−0.000458606 0	−0.000229303	−	1.00000i	$-0.500073\pi$
$503$	−0.0102855	−0.000458606 0	−0.000229303		1.00000i	$0.500073\pi$
$504$	−7.24449	−0.322695
$505$	−7.45004	−0.331522
$506$	−7.78261	−0.345979
$507$	29.5147	1.31079
$508$	97.2044	4.31275
$509$	26.8066	1.18818	0.594091	−	0.804398i	$-0.297512\pi$
$509$	26.8066	1.18818	0.594091	+	0.804398i	$0.297512\pi$
$510$	0	0
$511$	3.66744	0.162238
$512$	−41.9126	−1.85229
$513$	0.748497	0.0330469
$514$	−39.6873	−1.75053
$515$	13.5860	0.598669
$516$	−16.8596	−0.742202
$517$	−2.11591	−0.0930578
$518$	−30.5124	−1.34064
$519$	−29.5681	−1.29790
$520$	53.4537	2.34410
$521$	10.1427	0.444359	0.222180	−	0.975006i	$-0.428683\pi$
$521$	10.1427	0.444359	0.222180	+	0.975006i	$0.428683\pi$
$522$	0.546223	0.0239075
$523$	−16.7568	−0.732724	−0.366362	−	0.930472i	$-0.619397\pi$
$523$	−16.7568	−0.732724	−0.366362	+	0.930472i	$0.619397\pi$
$524$	−24.4021	−1.06601
$525$	−1.95600	−0.0853666
$526$	−51.9587	−2.26551
$527$	0	0
$528$	9.60627	0.418059
$529$	23.5018	1.02182
$530$	4.21322	0.183010
$531$	0.856609	0.0371736
$532$	0.925478	0.0401245
$533$	−9.25318	−0.400800
$534$	−21.0653	−0.911583
$535$	−9.27421	−0.400959
$536$	114.208	4.93303
$537$	30.6525	1.32275
$538$	−61.9435	−2.67057
$539$	−2.25745	−0.0972351
$540$	−30.4447	−1.31013
$541$	−18.1364	−0.779745	−0.389873	−	0.920869i	$-0.627481\pi$
$541$	−18.1364	−0.779745	−0.389873	+	0.920869i	$0.627481\pi$
$542$	27.2177	1.16910
$543$	13.7142	0.588532
$544$	0	0
$545$	−15.7066	−0.672796
$546$	−30.2512	−1.29463
$547$	−22.5759	−0.965277	−0.482638	−	0.875820i	$-0.660321\pi$
$547$	−22.5759	−0.965277	−0.482638	+	0.875820i	$0.660321\pi$
$548$	49.3743	2.10916
$549$	4.73821	0.202222
$550$	−1.14127	−0.0486641
$551$	−0.0442016	−0.00188305
$552$	−99.5546	−4.23733
$553$	−5.48483	−0.233239
$554$	31.9676	1.35817
$555$	−13.6703	−0.580271
$556$	−25.6484	−1.08774
$557$	5.97545	0.253188	0.126594	−	0.991955i	$-0.459595\pi$
$557$	5.97545	0.253188	0.126594	+	0.991955i	$0.459595\pi$
$558$	−1.00399	−0.0425024
$559$	11.3137	0.478520
$560$	−18.7872	−0.793905
$561$	0	0
$562$	3.23635	0.136517
$563$	−3.56804	−0.150375	−0.0751874	−	0.997169i	$-0.523955\pi$
$563$	−3.56804	−0.150375	−0.0751874	+	0.997169i	$0.523955\pi$
$564$	−42.7292	−1.79922
$565$	−1.24680	−0.0524534
$566$	−12.5904	−0.529214
$567$	8.61123	0.361638
$568$	−45.5358	−1.91064
$569$	−44.1186	−1.84955	−0.924773	−	0.380518i	$-0.875746\pi$
$569$	−44.1186	−1.84955	−0.924773	+	0.380518i	$0.875746\pi$
$570$	0.566623	0.0237332
$571$	12.5732	0.526171	0.263085	−	0.964773i	$-0.415260\pi$
$571$	12.5732	0.526171	0.263085	+	0.964773i	$0.415260\pi$
$572$	−12.9163	−0.540059
$573$	16.4098	0.685528
$574$	5.64076	0.235441
$575$	6.81922	0.284381
$576$	−17.9255	−0.746896
$577$	−14.5344	−0.605075	−0.302538	−	0.953137i	$-0.597834\pi$
$577$	−14.5344	−0.605075	−0.302538	+	0.953137i	$0.597834\pi$
$578$	0	0
$579$	15.0442	0.625217
$580$	1.79788	0.0746529
$581$	5.55351	0.230398
$582$	2.68385	0.111249
$583$	−0.644889	−0.0267086
$584$	27.3730	1.13270
$585$	3.43842	0.142161
$586$	8.45197	0.349148
$587$	12.2048	0.503745	0.251872	−	0.967760i	$-0.418954\pi$
$587$	12.2048	0.503745	0.251872	+	0.967760i	$0.418954\pi$
$588$	−45.5873	−1.87999
$589$	0.0812453	0.00334766
$590$	3.85301	0.158626
$591$	36.8091	1.51412
$592$	−131.302	−5.39649
$593$	−11.4292	−0.469343	−0.234671	−	0.972075i	$-0.575401\pi$
$593$	−11.4292	−0.469343	−0.234671	+	0.972075i	$0.575401\pi$
$594$	6.36810	0.261286
$595$	0	0
$596$	−4.23376	−0.173421
$597$	20.4312	0.836195
$598$	105.465	4.31280
$599$	−27.2578	−1.11372	−0.556861	−	0.830606i	$-0.687994\pi$
$599$	−27.2578	−1.11372	−0.556861	+	0.830606i	$0.687994\pi$
$600$	−14.5991	−0.596006
$601$	11.2039	0.457016	0.228508	−	0.973542i	$-0.426615\pi$
$601$	11.2039	0.457016	0.228508	+	0.973542i	$0.426615\pi$
$602$	−6.89688	−0.281096
$603$	7.34644	0.299170
$604$	−12.4791	−0.507766
$605$	−10.8253	−0.440112
$606$	31.4690	1.27834
$607$	40.0476	1.62548	0.812742	−	0.582624i	$-0.197974\pi$
$607$	40.0476	1.62548	0.812742	+	0.582624i	$0.197974\pi$
$608$	2.91039	0.118032
$609$	−0.644519	−0.0261172
$610$	21.3124	0.862912
$611$	28.6737	1.16001
$612$	0	0
$613$	29.1043	1.17551	0.587755	−	0.809039i	$-0.300012\pi$
$613$	29.1043	1.17551	0.587755	+	0.809039i	$0.300012\pi$
$614$	−32.3268	−1.30460
$615$	2.52720	0.101907
$616$	4.98765	0.200958
$617$	16.4653	0.662869	0.331435	−	0.943478i	$-0.392467\pi$
$617$	16.4653	0.662869	0.331435	+	0.943478i	$0.392467\pi$
$618$	−57.3871	−2.30845
$619$	7.61375	0.306023	0.153011	−	0.988224i	$-0.451103\pi$
$619$	7.61375	0.306023	0.153011	+	0.988224i	$0.451103\pi$
$620$	−3.30461	−0.132717
$621$	−38.0500	−1.52689
$622$	−64.0327	−2.56748
$623$	−6.30589	−0.252640
$624$	−130.179	−5.21132
$625$	1.00000	0.0400000
$626$	−75.3875	−3.01309
$627$	−0.0867292	−0.00346363
$628$	−104.287	−4.16150
$629$	0	0
$630$	−2.09607	−0.0835093
$631$	−37.1952	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$631$	−37.1952	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$632$	−40.9375	−1.62841
$633$	−30.7624	−1.22270
$634$	86.9997	3.45520
$635$	17.8153	0.706979
$636$	−13.0230	−0.516396
$637$	30.5916	1.21208
$638$	−0.376061	−0.0148884
$639$	−2.92910	−0.115873
$640$	−37.2362	−1.47189
$641$	−8.61170	−0.340142	−0.170071	−	0.985432i	$-0.554400\pi$
$641$	−8.61170	−0.340142	−0.170071	+	0.985432i	$0.554400\pi$
$642$	39.1743	1.54609
$643$	19.1230	0.754136	0.377068	−	0.926186i	$-0.376932\pi$
$643$	19.1230	0.754136	0.377068	+	0.926186i	$0.376932\pi$
$644$	−47.0469	−1.85391
$645$	−3.08997	−0.121668
$646$	0	0
$647$	−25.5972	−1.00633	−0.503164	−	0.864191i	$-0.667831\pi$
$647$	−25.5972	−1.00633	−0.503164	+	0.864191i	$0.667831\pi$
$648$	64.2723	2.52486
$649$	−0.589754	−0.0231499
$650$	15.4659	0.606623
$651$	1.18467	0.0464307
$652$	−100.010	−3.91669
$653$	8.67406	0.339442	0.169721	−	0.985492i	$-0.445713\pi$
$653$	8.67406	0.339442	0.169721	+	0.985492i	$0.445713\pi$
$654$	66.3446	2.59428
$655$	−4.47235	−0.174749
$656$	24.2736	0.947726
$657$	1.76077	0.0686942
$658$	−17.4795	−0.681424
$659$	36.0328	1.40364	0.701819	−	0.712355i	$-0.252372\pi$
$659$	36.0328	1.40364	0.701819	+	0.712355i	$0.252372\pi$
$660$	3.52767	0.137314
$661$	−9.40443	−0.365790	−0.182895	−	0.983132i	$-0.558547\pi$
$661$	−9.40443	−0.365790	−0.182895	+	0.983132i	$0.558547\pi$
$662$	71.2001	2.76727
$663$	0	0
$664$	41.4502	1.60858
$665$	0.169619	0.00657753
$666$	−14.6492	−0.567646
$667$	2.24700	0.0870041
$668$	−109.934	−4.25348
$669$	9.84315	0.380558
$670$	33.0441	1.27661
$671$	−3.26214	−0.125933
$672$	42.4375	1.63706
$673$	4.55958	0.175759	0.0878794	−	0.996131i	$-0.471991\pi$
$673$	4.55958	0.175759	0.0878794	+	0.996131i	$0.471991\pi$
$674$	−45.1062	−1.73743
$675$	−5.57981	−0.214767
$676$	104.104	4.00399
$677$	35.6013	1.36827	0.684135	−	0.729355i	$-0.260180\pi$
$677$	35.6013	1.36827	0.684135	+	0.729355i	$0.260180\pi$
$678$	5.26650	0.202259
$679$	0.803410	0.0308321
$680$	0	0
$681$	−41.0486	−1.57299
$682$	0.691223	0.0264683
$683$	11.9719	0.458092	0.229046	−	0.973416i	$-0.426439\pi$
$683$	11.9719	0.458092	0.229046	+	0.973416i	$0.426439\pi$
$684$	0.444329	0.0169894
$685$	9.04916	0.345751
$686$	−42.8179	−1.63479
$687$	24.0475	0.917470
$688$	−29.6790	−1.13150
$689$	8.73917	0.332936
$690$	−28.8044	−1.09657
$691$	20.7951	0.791082	0.395541	−	0.918448i	$-0.370557\pi$
$691$	20.7951	0.791082	0.395541	+	0.918448i	$0.370557\pi$
$692$	−104.292	−3.96460
$693$	0.320831	0.0121874
$694$	55.4984	2.10669
$695$	−4.70076	−0.178310
$696$	−4.81055	−0.182343
$697$	0	0
$698$	−13.3405	−0.504946
$699$	5.88249	0.222496
$700$	−6.89915	−0.260763
$701$	−13.1645	−0.497215	−0.248607	−	0.968604i	$-0.579973\pi$
$701$	−13.1645	−0.497215	−0.248607	+	0.968604i	$0.579973\pi$
$702$	−86.2969	−3.25707
$703$	1.18545	0.0447101
$704$	12.3413	0.465129
$705$	−7.83126	−0.294942
$706$	−61.9119	−2.33009
$707$	9.42024	0.354284
$708$	−11.9096	−0.447590
$709$	−20.3126	−0.762854	−0.381427	−	0.924399i	$-0.624567\pi$
$709$	−20.3126	−0.762854	−0.381427	+	0.924399i	$0.624567\pi$
$710$	−13.1750	−0.494449
$711$	−2.63331	−0.0987568
$712$	−47.0657	−1.76386
$713$	−4.13012	−0.154674
$714$	0	0
$715$	−2.36726	−0.0885306
$716$	108.117	4.04053
$717$	−8.98257	−0.335460
$718$	−36.8108	−1.37377
$719$	−29.1456	−1.08695	−0.543474	−	0.839426i	$-0.682891\pi$
$719$	−29.1456	−1.08695	−0.543474	+	0.839426i	$0.682891\pi$
$720$	−9.01991	−0.336152
$721$	−17.1788	−0.639773
$722$	51.8324	1.92900
$723$	−13.5014	−0.502121
$724$	48.3724	1.79775
$725$	0.329510	0.0122377
$726$	45.7261	1.69706
$727$	−18.9928	−0.704402	−0.352201	−	0.935924i	$-0.614567\pi$
$727$	−18.9928	−0.704402	−0.352201	+	0.935924i	$0.614567\pi$
$728$	−67.5898	−2.50505
$729$	30.0287	1.11217
$730$	7.91990	0.293129
$731$	0	0
$732$	−65.8762	−2.43486
$733$	32.5640	1.20278	0.601390	−	0.798956i	$-0.294614\pi$
$733$	32.5640	1.20278	0.601390	+	0.798956i	$0.294614\pi$
$734$	−5.21786	−0.192595
$735$	−8.35509	−0.308182
$736$	−147.951	−5.45353
$737$	−5.05784	−0.186308
$738$	2.70818	0.0996894
$739$	30.7452	1.13098	0.565489	−	0.824756i	$-0.308687\pi$
$739$	30.7452	1.13098	0.565489	+	0.824756i	$0.308687\pi$
$740$	−48.2176	−1.77251
$741$	1.17531	0.0431759
$742$	−5.32742	−0.195576
$743$	9.77413	0.358578	0.179289	−	0.983796i	$-0.442620\pi$
$743$	9.77413	0.358578	0.179289	+	0.983796i	$0.442620\pi$
$744$	8.84209	0.324167
$745$	−0.775949	−0.0284286
$746$	−68.0898	−2.49294
$747$	2.66629	0.0975543
$748$	0	0
$749$	11.7268	0.428489
$750$	−4.22400	−0.154239
$751$	−12.9774	−0.473551	−0.236776	−	0.971564i	$-0.576091\pi$
$751$	−12.9774	−0.473551	−0.236776	+	0.971564i	$0.576091\pi$
$752$	−75.2189	−2.74295
$753$	23.5786	0.859253
$754$	5.09616	0.185591
$755$	−2.28712	−0.0832369
$756$	38.4960	1.40009
$757$	35.5081	1.29057	0.645283	−	0.763944i	$-0.276740\pi$
$757$	35.5081	1.29057	0.645283	+	0.763944i	$0.276740\pi$
$758$	45.3337	1.64659
$759$	4.40890	0.160033
$760$	1.26600	0.0459225
$761$	23.6745	0.858200	0.429100	−	0.903257i	$-0.358831\pi$
$761$	23.6745	0.858200	0.429100	+	0.903257i	$0.358831\pi$
$762$	−75.2519	−2.72609
$763$	19.8603	0.718990
$764$	57.8802	2.09403
$765$	0	0
$766$	54.8581	1.98210
$767$	7.99201	0.288575
$768$	65.9329	2.37915
$769$	5.83645	0.210468	0.105234	−	0.994447i	$-0.466441\pi$
$769$	5.83645	0.210468	0.105234	+	0.994447i	$0.466441\pi$
$770$	1.44309	0.0520054
$771$	22.4831	0.809709
$772$	53.0637	1.90981
$773$	−30.6187	−1.10128	−0.550639	−	0.834743i	$-0.685616\pi$
$773$	−30.6187	−1.10128	−0.550639	+	0.834743i	$0.685616\pi$
$774$	−3.31125	−0.119020
$775$	−0.605659	−0.0217559
$776$	5.99648	0.215261
$777$	17.2855	0.620112
$778$	60.8333	2.18098
$779$	−0.219152	−0.00785193
$780$	−47.8050	−1.71169
$781$	2.01661	0.0721600
$782$	0	0
$783$	−1.83860	−0.0657063
$784$	−80.2502	−2.86608
$785$	−19.1134	−0.682186
$786$	18.8912	0.673827
$787$	37.8438	1.34898	0.674492	−	0.738282i	$-0.264363\pi$
$787$	37.8438	1.34898	0.674492	+	0.738282i	$0.264363\pi$
$788$	129.832	4.62509
$789$	29.4350	1.04791
$790$	−11.8446	−0.421411
$791$	1.57653	0.0560548
$792$	2.39461	0.0850888
$793$	44.2067	1.56982
$794$	−71.7327	−2.54570
$795$	−2.38682	−0.0846516
$796$	72.0647	2.55427
$797$	−7.46090	−0.264279	−0.132139	−	0.991231i	$-0.542185\pi$
$797$	−7.46090	−0.264279	−0.132139	+	0.991231i	$0.542185\pi$
$798$	−0.716469	−0.0253627
$799$	0	0
$800$	−21.6961	−0.767073
$801$	−3.02751	−0.106972
$802$	104.192	3.67914
$803$	−1.21225	−0.0427793
$804$	−102.139	−3.60216
$805$	−8.62260	−0.303907
$806$	−9.36707	−0.329941
$807$	35.0914	1.23528
$808$	70.3106	2.47352
$809$	37.6824	1.32484	0.662421	−	0.749132i	$-0.269529\pi$
$809$	37.6824	1.32484	0.662421	+	0.749132i	$0.269529\pi$
$810$	18.5961	0.653400
$811$	6.87611	0.241453	0.120726	−	0.992686i	$-0.461478\pi$
$811$	6.87611	0.241453	0.120726	+	0.992686i	$0.461478\pi$
$812$	−2.27334	−0.0797785
$813$	−15.4190	−0.540769
$814$	10.0856	0.353501
$815$	−18.3295	−0.642055
$816$	0	0
$817$	0.267954	0.00937452
$818$	52.1534	1.82350
$819$	−4.34772	−0.151922
$820$	8.91389	0.311287
$821$	23.5090	0.820471	0.410236	−	0.911980i	$-0.365446\pi$
$821$	23.5090	0.820471	0.410236	+	0.911980i	$0.365446\pi$
$822$	−38.2237	−1.33320
$823$	54.8707	1.91267	0.956336	−	0.292268i	$-0.0944101\pi$
$823$	54.8707	1.91267	0.956336	+	0.292268i	$0.0944101\pi$
$824$	−128.219	−4.46672
$825$	0.646540	0.0225096
$826$	−4.87195	−0.169517
$827$	−17.6179	−0.612634	−0.306317	−	0.951930i	$-0.599097\pi$
$827$	−17.6179	−0.612634	−0.306317	+	0.951930i	$0.599097\pi$
$828$	−22.5876	−0.784973
$829$	11.9207	0.414022	0.207011	−	0.978339i	$-0.433626\pi$
$829$	11.9207	0.414022	0.207011	+	0.978339i	$0.433626\pi$
$830$	11.9929	0.416279
$831$	−18.1098	−0.628223
$832$	−167.242	−5.79806
$833$	0	0
$834$	19.8560	0.687558
$835$	−20.1484	−0.697263
$836$	−0.305910	−0.0105801
$837$	3.37946	0.116811
$838$	91.6143	3.16476
$839$	−42.3385	−1.46169	−0.730844	−	0.682545i	$-0.760873\pi$
$839$	−42.3385	−1.46169	−0.730844	+	0.682545i	$0.760873\pi$
$840$	18.4599	0.636928
$841$	−28.8914	−0.996256
$842$	51.7166	1.78227
$843$	−1.83341	−0.0631461
$844$	−108.505	−3.73489
$845$	19.0798	0.656366
$846$	−8.39207	−0.288525
$847$	13.6881	0.470329
$848$	−22.9252	−0.787256
$849$	7.13254	0.244788
$850$	0	0
$851$	−60.2626	−2.06578
$852$	40.7238	1.39517
$853$	−33.1729	−1.13582	−0.567909	−	0.823091i	$-0.692247\pi$
$853$	−33.1729	−1.13582	−0.567909	+	0.823091i	$0.692247\pi$
$854$	−26.9485	−0.922159
$855$	0.0814353	0.00278503
$856$	87.5264	2.99159
$857$	−15.3710	−0.525062	−0.262531	−	0.964924i	$-0.584557\pi$
$857$	−15.3710	−0.525062	−0.262531	+	0.964924i	$0.584557\pi$
$858$	9.99932	0.341371
$859$	−42.8054	−1.46050	−0.730250	−	0.683180i	$-0.760596\pi$
$859$	−42.8054	−1.46050	−0.730250	+	0.683180i	$0.760596\pi$
$860$	−10.8989	−0.371649
$861$	−3.19553	−0.108903
$862$	66.5917	2.26812
$863$	31.3841	1.06833	0.534163	−	0.845382i	$-0.320627\pi$
$863$	31.3841	1.06833	0.534163	+	0.845382i	$0.320627\pi$
$864$	121.060	4.11855
$865$	−19.1144	−0.649907
$866$	−26.4630	−0.899250
$867$	0	0
$868$	4.17853	0.141829
$869$	1.81297	0.0615007
$870$	−1.39185	−0.0471881
$871$	68.5410	2.32242
$872$	148.233	5.01979
$873$	0.385724	0.0130548
$874$	2.49784	0.0844907
$875$	−1.26445	−0.0427464
$876$	−24.4803	−0.827114
$877$	29.2384	0.987310	0.493655	−	0.869658i	$-0.335661\pi$
$877$	29.2384	0.987310	0.493655	+	0.869658i	$0.335661\pi$
$878$	−8.80217	−0.297059
$879$	−4.78810	−0.161499
$880$	6.20998	0.209338
$881$	−22.9441	−0.773006	−0.386503	−	0.922288i	$-0.626317\pi$
$881$	−22.9441	−0.773006	−0.386503	+	0.922288i	$0.626317\pi$
$882$	−8.95341	−0.301477
$883$	−21.1206	−0.710766	−0.355383	−	0.934721i	$-0.615649\pi$
$883$	−21.1206	−0.710766	−0.355383	+	0.934721i	$0.615649\pi$
$884$	0	0
$885$	−2.18275	−0.0733724
$886$	47.9559	1.61111
$887$	17.9732	0.603482	0.301741	−	0.953390i	$-0.402432\pi$
$887$	17.9732	0.603482	0.301741	+	0.953390i	$0.402432\pi$
$888$	129.015	4.32945
$889$	−22.5267	−0.755520
$890$	−13.6177	−0.456465
$891$	−2.84638	−0.0953573
$892$	34.7186	1.16246
$893$	0.679106	0.0227254
$894$	3.27761	0.109620
$895$	19.8154	0.662354
$896$	47.0835	1.57295
$897$	−59.7469	−1.99489
$898$	−50.5799	−1.68787
$899$	−0.199570	−0.00665605
$900$	−3.31234	−0.110411
$901$	0	0
$902$	−1.86451	−0.0620815
$903$	3.90713	0.130021
$904$	11.7668	0.391359
$905$	8.86555	0.294701
$906$	9.66081	0.320959
$907$	−55.8990	−1.85610	−0.928048	−	0.372461i	$-0.878514\pi$
$907$	−55.8990	−1.85610	−0.928048	+	0.372461i	$0.878514\pi$
$908$	−144.786	−4.80489
$909$	4.52273	0.150010
$910$	−19.5559	−0.648273
$911$	−0.561306	−0.0185969	−0.00929845	−	0.999957i	$-0.502960\pi$
$911$	−0.561306	−0.0185969	−0.00929845	+	0.999957i	$0.502960\pi$
$912$	−3.08315	−0.102093
$913$	−1.83567	−0.0607519
$914$	−40.1610	−1.32841
$915$	−12.0736	−0.399140
$916$	84.8200	2.80253
$917$	5.65508	0.186747
$918$	0	0
$919$	4.29583	0.141706	0.0708531	−	0.997487i	$-0.477428\pi$
$919$	4.29583	0.141706	0.0708531	+	0.997487i	$0.477428\pi$
$920$	−64.3572	−2.12179
$921$	18.3133	0.603445
$922$	−49.0127	−1.61415
$923$	−27.3280	−0.899511
$924$	−4.46058	−0.146742
$925$	−8.83716	−0.290564
$926$	−97.1528	−3.19264
$927$	−8.24769	−0.270890
$928$	−7.14907	−0.234680
$929$	9.55355	0.313442	0.156721	−	0.987643i	$-0.449908\pi$
$929$	9.55355	0.313442	0.156721	+	0.987643i	$0.449908\pi$
$930$	2.55831	0.0838901
$931$	0.724531	0.0237455
$932$	20.7486	0.679643
$933$	36.2750	1.18759
$934$	−80.3921	−2.63051
$935$	0	0
$936$	−32.4504	−1.06068
$937$	−55.7449	−1.82111	−0.910553	−	0.413392i	$-0.864344\pi$
$937$	−55.7449	−1.82111	−0.910553	+	0.413392i	$0.864344\pi$
$938$	−41.7828	−1.36426
$939$	42.7075	1.39371
$940$	−27.6223	−0.900940
$941$	29.4430	0.959813	0.479906	−	0.877320i	$-0.340671\pi$
$941$	29.4430	0.959813	0.479906	+	0.877320i	$0.340671\pi$
$942$	80.7349	2.63049
$943$	11.1406	0.362789
$944$	−20.9652	−0.682360
$945$	7.05542	0.229513
$946$	2.27971	0.0741199
$947$	31.5924	1.02662	0.513308	−	0.858205i	$-0.328420\pi$
$947$	31.5924	1.02662	0.513308	+	0.858205i	$0.328420\pi$
$948$	36.6114	1.18908
$949$	16.4277	0.533265
$950$	0.366294	0.0118841
$951$	−49.2859	−1.59821
$952$	0	0
$953$	−18.4486	−0.597610	−0.298805	−	0.954314i	$-0.596588\pi$
$953$	−18.4486	−0.597610	−0.298805	+	0.954314i	$0.596588\pi$
$954$	−2.55774	−0.0828099
$955$	10.6081	0.343270
$956$	−31.6832	−1.02471
$957$	0.213041	0.00688664
$958$	−45.6994	−1.47648
$959$	−11.4422	−0.369490
$960$	45.6765	1.47420
$961$	−30.6332	−0.988167
$962$	−136.675	−4.40657
$963$	5.63014	0.181429
$964$	−47.6218	−1.53379
$965$	9.72535	0.313070
$966$	36.4219	1.17185
$967$	−38.3638	−1.23370	−0.616848	−	0.787083i	$-0.711591\pi$
$967$	−38.3638	−1.23370	−0.616848	+	0.787083i	$0.711591\pi$
$968$	102.165	3.28371
$969$	0	0
$970$	1.73498	0.0557067
$971$	−52.5632	−1.68683	−0.843417	−	0.537260i	$-0.819459\pi$
$971$	−52.5632	−1.68683	−0.843417	+	0.537260i	$0.819459\pi$
$972$	33.8539	1.08586
$973$	5.94390	0.190553
$974$	64.0120	2.05108
$975$	−8.76154	−0.280594
$976$	−115.966	−3.71199
$977$	−34.3771	−1.09982	−0.549911	−	0.835224i	$-0.685338\pi$
$977$	−34.3771	−1.09982	−0.549911	+	0.835224i	$0.685338\pi$
$978$	77.4239	2.47574
$979$	2.08436	0.0666166
$980$	−29.4699	−0.941382
$981$	9.53507	0.304431
$982$	−69.5971	−2.22093
$983$	28.7282	0.916288	0.458144	−	0.888878i	$-0.348514\pi$
$983$	28.7282	0.916288	0.458144	+	0.888878i	$0.348514\pi$
$984$	−23.8507	−0.760333
$985$	23.7953	0.758180
$986$	0	0
$987$	9.90228	0.315193
$988$	4.14552	0.131886
$989$	−13.6215	−0.433138
$990$	0.692839	0.0220199
$991$	−47.8330	−1.51946	−0.759732	−	0.650236i	$-0.774670\pi$
$991$	−47.8330	−1.51946	−0.759732	+	0.650236i	$0.774670\pi$
$992$	13.1404	0.417209
$993$	−40.3353	−1.28000
$994$	16.6592	0.528398
$995$	13.2078	0.418715
$996$	−37.0699	−1.17460
$997$	7.81193	0.247406	0.123703	−	0.992319i	$-0.460523\pi$
$997$	7.81193	0.247406	0.123703	+	0.992319i	$0.460523\pi$
$998$	78.4833	2.48435
$999$	49.3097	1.56009

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1445.2.a.p.1.1		12
5.4	even	2		7225.2.a.bs.1.12		12
17.4	even	4		1445.2.d.j.866.24		24
17.11	odd	16		85.2.l.a.36.1	yes	24
17.13	even	4		1445.2.d.j.866.23		24
17.14	odd	16		85.2.l.a.26.1	✓	24
17.16	even	2		1445.2.a.q.1.1		12
51.11	even	16		765.2.be.b.631.6		24
51.14	even	16		765.2.be.b.451.6		24
85.14	odd	16		425.2.m.b.26.6		24
85.28	even	16		425.2.n.f.274.1		24
85.48	even	16		425.2.n.c.349.6		24
85.62	even	16		425.2.n.c.274.6		24
85.79	odd	16		425.2.m.b.376.6		24
85.82	even	16		425.2.n.f.349.1		24
85.84	even	2		7225.2.a.bq.1.12		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.l.a.26.1	✓	24	17.14	odd	16
85.2.l.a.36.1	yes	24	17.11	odd	16
425.2.m.b.26.6		24	85.14	odd	16
425.2.m.b.376.6		24	85.79	odd	16
425.2.n.c.274.6		24	85.62	even	16
425.2.n.c.349.6		24	85.48	even	16
425.2.n.f.274.1		24	85.28	even	16
425.2.n.f.349.1		24	85.82	even	16
765.2.be.b.451.6		24	51.14	even	16
765.2.be.b.631.6		24	51.11	even	16
1445.2.a.p.1.1		12	1.1	even	1	trivial
1445.2.a.q.1.1		12	17.16	even	2
1445.2.d.j.866.23		24	17.13	even	4
1445.2.d.j.866.24		24	17.4	even	4
7225.2.a.bq.1.12		12	85.84	even	2
7225.2.a.bs.1.12		12	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1445 = 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1445.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73061 q^{2} +1.54691 q^{3} +5.45623 q^{4} +1.00000 q^{5} -4.22400 q^{6} -1.26445 q^{7} -9.43761 q^{8} -0.607075 q^{9} +O(q^{10})\)	\(q-2.73061 q^{2} +1.54691 q^{3} +5.45623 q^{4} +1.00000 q^{5} -4.22400 q^{6} -1.26445 q^{7} -9.43761 q^{8} -0.607075 q^{9} -2.73061 q^{10} +0.417956 q^{11} +8.44028 q^{12} -5.66390 q^{13} +3.45273 q^{14} +1.54691 q^{15} +14.8580 q^{16} +1.65769 q^{18} -0.134144 q^{19} +5.45623 q^{20} -1.95600 q^{21} -1.14127 q^{22} +6.81922 q^{23} -14.5991 q^{24} +1.00000 q^{25} +15.4659 q^{26} -5.57981 q^{27} -6.89915 q^{28} +0.329510 q^{29} -4.22400 q^{30} -0.605659 q^{31} -21.6961 q^{32} +0.646540 q^{33} -1.26445 q^{35} -3.31234 q^{36} -8.83716 q^{37} +0.366294 q^{38} -8.76154 q^{39} -9.43761 q^{40} +1.63371 q^{41} +5.34106 q^{42} -1.99751 q^{43} +2.28046 q^{44} -0.607075 q^{45} -18.6206 q^{46} -5.06253 q^{47} +22.9839 q^{48} -5.40115 q^{49} -2.73061 q^{50} -30.9036 q^{52} -1.54296 q^{53} +15.2363 q^{54} +0.417956 q^{55} +11.9334 q^{56} -0.207508 q^{57} -0.899762 q^{58} -1.41104 q^{59} +8.44028 q^{60} -7.80498 q^{61} +1.65382 q^{62} +0.767619 q^{63} +29.5276 q^{64} -5.66390 q^{65} -1.76545 q^{66} -12.1014 q^{67} +10.5487 q^{69} +3.45273 q^{70} +4.82493 q^{71} +5.72934 q^{72} -2.90042 q^{73} +24.1308 q^{74} +1.54691 q^{75} -0.731919 q^{76} -0.528486 q^{77} +23.9243 q^{78} +4.33770 q^{79} +14.8580 q^{80} -6.81023 q^{81} -4.46102 q^{82} -4.39202 q^{83} -10.6724 q^{84} +5.45443 q^{86} +0.509721 q^{87} -3.94451 q^{88} +4.98704 q^{89} +1.65769 q^{90} +7.16175 q^{91} +37.2072 q^{92} -0.936899 q^{93} +13.8238 q^{94} -0.134144 q^{95} -33.5619 q^{96} -0.635381 q^{97} +14.7484 q^{98} -0.253731 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} - 8 q^{3} + 12 q^{4} + 12 q^{5} - 8 q^{6} - 16 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q - 4 * q^2 - 8 * q^3 + 12 * q^4 + 12 * q^5 - 8 * q^6 - 16 * q^7 - 12 * q^8 + 12 * q^9	\( 12 q - 4 q^{2} - 8 q^{3} + 12 q^{4} + 12 q^{5} - 8 q^{6} - 16 q^{7} - 12 q^{8} + 12 q^{9} - 4 q^{10} - 16 q^{11} - 16 q^{12} - 8 q^{13} + 16 q^{14} - 8 q^{15} + 12 q^{16} + 4 q^{18} + 12 q^{20} + 16 q^{21} - 16 q^{22} - 16 q^{23} + 12 q^{25} + 16 q^{26} - 32 q^{27} - 40 q^{28} - 16 q^{29} - 8 q^{30} - 24 q^{31} - 28 q^{32} - 16 q^{35} + 12 q^{36} - 24 q^{37} - 24 q^{38} - 8 q^{39} - 12 q^{40} - 8 q^{41} - 16 q^{43} - 8 q^{44} + 12 q^{45} - 40 q^{46} - 32 q^{47} + 24 q^{48} + 20 q^{49} - 4 q^{50} - 24 q^{52} - 8 q^{54} - 16 q^{55} + 24 q^{56} - 32 q^{57} + 16 q^{58} + 8 q^{59} - 16 q^{60} - 24 q^{61} - 8 q^{62} - 48 q^{63} + 36 q^{64} - 8 q^{65} + 40 q^{66} - 8 q^{67} + 48 q^{69} + 16 q^{70} + 16 q^{71} + 12 q^{72} - 16 q^{73} - 8 q^{75} + 16 q^{76} + 24 q^{77} + 24 q^{78} - 40 q^{79} + 12 q^{80} + 36 q^{81} + 16 q^{82} - 40 q^{83} + 32 q^{84} - 8 q^{86} - 32 q^{87} - 48 q^{88} + 8 q^{89} + 4 q^{90} - 72 q^{91} + 8 q^{92} + 24 q^{93} - 16 q^{94} - 8 q^{96} - 32 q^{97} - 60 q^{98} - 56 q^{99}+O(q^{100}) \) 12 * q - 4 * q^2 - 8 * q^3 + 12 * q^4 + 12 * q^5 - 8 * q^6 - 16 * q^7 - 12 * q^8 + 12 * q^9 - 4 * q^10 - 16 * q^11 - 16 * q^12 - 8 * q^13 + 16 * q^14 - 8 * q^15 + 12 * q^16 + 4 * q^18 + 12 * q^20 + 16 * q^21 - 16 * q^22 - 16 * q^23 + 12 * q^25 + 16 * q^26 - 32 * q^27 - 40 * q^28 - 16 * q^29 - 8 * q^30 - 24 * q^31 - 28 * q^32 - 16 * q^35 + 12 * q^36 - 24 * q^37 - 24 * q^38 - 8 * q^39 - 12 * q^40 - 8 * q^41 - 16 * q^43 - 8 * q^44 + 12 * q^45 - 40 * q^46 - 32 * q^47 + 24 * q^48 + 20 * q^49 - 4 * q^50 - 24 * q^52 - 8 * q^54 - 16 * q^55 + 24 * q^56 - 32 * q^57 + 16 * q^58 + 8 * q^59 - 16 * q^60 - 24 * q^61 - 8 * q^62 - 48 * q^63 + 36 * q^64 - 8 * q^65 + 40 * q^66 - 8 * q^67 + 48 * q^69 + 16 * q^70 + 16 * q^71 + 12 * q^72 - 16 * q^73 - 8 * q^75 + 16 * q^76 + 24 * q^77 + 24 * q^78 - 40 * q^79 + 12 * q^80 + 36 * q^81 + 16 * q^82 - 40 * q^83 + 32 * q^84 - 8 * q^86 - 32 * q^87 - 48 * q^88 + 8 * q^89 + 4 * q^90 - 72 * q^91 + 8 * q^92 + 24 * q^93 - 16 * q^94 - 8 * q^96 - 32 * q^97 - 60 * q^98 - 56 * q^99

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