LMFDB - Embedded newform 2166.2.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2166,2,Mod(1,2166)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2166, 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("2166.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2166 = 2 \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2166.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:	$17.2955970778$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.17557$ of defining polynomial
Character	$\chi$	$=$	2166.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.17557 q^{5} -1.00000 q^{6} +0.381966 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.17557 q^{5} -1.00000 q^{6} +0.381966 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.17557 q^{10} +2.66605 q^{11} -1.00000 q^{12} -7.04029 q^{13} +0.381966 q^{14} -3.17557 q^{15} +1.00000 q^{16} +4.68915 q^{17} +1.00000 q^{18} +3.17557 q^{20} -0.381966 q^{21} +2.66605 q^{22} +1.43184 q^{23} -1.00000 q^{24} +5.08425 q^{25} -7.04029 q^{26} -1.00000 q^{27} +0.381966 q^{28} +8.20524 q^{29} -3.17557 q^{30} +8.62750 q^{31} +1.00000 q^{32} -2.66605 q^{33} +4.68915 q^{34} +1.21296 q^{35} +1.00000 q^{36} -2.70228 q^{37} +7.04029 q^{39} +3.17557 q^{40} +3.25731 q^{41} -0.381966 q^{42} -8.72353 q^{43} +2.66605 q^{44} +3.17557 q^{45} +1.43184 q^{46} +5.35706 q^{47} -1.00000 q^{48} -6.85410 q^{49} +5.08425 q^{50} -4.68915 q^{51} -7.04029 q^{52} -8.20524 q^{53} -1.00000 q^{54} +8.46621 q^{55} +0.381966 q^{56} +8.20524 q^{58} +11.8137 q^{59} -3.17557 q^{60} -5.27044 q^{61} +8.62750 q^{62} +0.381966 q^{63} +1.00000 q^{64} -22.3570 q^{65} -2.66605 q^{66} -12.2147 q^{67} +4.68915 q^{68} -1.43184 q^{69} +1.21296 q^{70} +15.0996 q^{71} +1.00000 q^{72} -3.84818 q^{73} -2.70228 q^{74} -5.08425 q^{75} +1.01834 q^{77} +7.04029 q^{78} +1.39560 q^{79} +3.17557 q^{80} +1.00000 q^{81} +3.25731 q^{82} -4.31562 q^{83} -0.381966 q^{84} +14.8907 q^{85} -8.72353 q^{86} -8.20524 q^{87} +2.66605 q^{88} -7.80423 q^{89} +3.17557 q^{90} -2.68915 q^{91} +1.43184 q^{92} -8.62750 q^{93} +5.35706 q^{94} -1.00000 q^{96} +12.7448 q^{97} -6.85410 q^{98} +2.66605 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 8 q^{5} - 4 q^{6} + 6 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 8 * q^5 - 4 * q^6 + 6 * q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 8 q^{5} - 4 q^{6} + 6 q^{7} + 4 q^{8} + 4 q^{9} + 8 q^{10} + 12 q^{11} - 4 q^{12} - 4 q^{13} + 6 q^{14} - 8 q^{15} + 4 q^{16} + 4 q^{17} + 4 q^{18} + 8 q^{20} - 6 q^{21} + 12 q^{22} + 12 q^{23} - 4 q^{24} + 6 q^{25} - 4 q^{26} - 4 q^{27} + 6 q^{28} + 10 q^{29} - 8 q^{30} - 8 q^{31} + 4 q^{32} - 12 q^{33} + 4 q^{34} + 12 q^{35} + 4 q^{36} + 8 q^{37} + 4 q^{39} + 8 q^{40} - 8 q^{41} - 6 q^{42} - 4 q^{43} + 12 q^{44} + 8 q^{45} + 12 q^{46} + 4 q^{47} - 4 q^{48} - 14 q^{49} + 6 q^{50} - 4 q^{51} - 4 q^{52} - 10 q^{53} - 4 q^{54} + 24 q^{55} + 6 q^{56} + 10 q^{58} - 6 q^{59} - 8 q^{60} + 4 q^{61} - 8 q^{62} + 6 q^{63} + 4 q^{64} - 8 q^{65} - 12 q^{66} + 12 q^{67} + 4 q^{68} - 12 q^{69} + 12 q^{70} + 4 q^{72} - 10 q^{73} + 8 q^{74} - 6 q^{75} + 28 q^{77} + 4 q^{78} + 32 q^{79} + 8 q^{80} + 4 q^{81} - 8 q^{82} + 8 q^{83} - 6 q^{84} - 12 q^{85} - 4 q^{86} - 10 q^{87} + 12 q^{88} - 16 q^{89} + 8 q^{90} + 4 q^{91} + 12 q^{92} + 8 q^{93} + 4 q^{94} - 4 q^{96} + 8 q^{97} - 14 q^{98} + 12 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 8 * q^5 - 4 * q^6 + 6 * q^7 + 4 * q^8 + 4 * q^9 + 8 * q^10 + 12 * q^11 - 4 * q^12 - 4 * q^13 + 6 * q^14 - 8 * q^15 + 4 * q^16 + 4 * q^17 + 4 * q^18 + 8 * q^20 - 6 * q^21 + 12 * q^22 + 12 * q^23 - 4 * q^24 + 6 * q^25 - 4 * q^26 - 4 * q^27 + 6 * q^28 + 10 * q^29 - 8 * q^30 - 8 * q^31 + 4 * q^32 - 12 * q^33 + 4 * q^34 + 12 * q^35 + 4 * q^36 + 8 * q^37 + 4 * q^39 + 8 * q^40 - 8 * q^41 - 6 * q^42 - 4 * q^43 + 12 * q^44 + 8 * q^45 + 12 * q^46 + 4 * q^47 - 4 * q^48 - 14 * q^49 + 6 * q^50 - 4 * q^51 - 4 * q^52 - 10 * q^53 - 4 * q^54 + 24 * q^55 + 6 * q^56 + 10 * q^58 - 6 * q^59 - 8 * q^60 + 4 * q^61 - 8 * q^62 + 6 * q^63 + 4 * q^64 - 8 * q^65 - 12 * q^66 + 12 * q^67 + 4 * q^68 - 12 * q^69 + 12 * q^70 + 4 * q^72 - 10 * q^73 + 8 * q^74 - 6 * q^75 + 28 * q^77 + 4 * q^78 + 32 * q^79 + 8 * q^80 + 4 * q^81 - 8 * q^82 + 8 * q^83 - 6 * q^84 - 12 * q^85 - 4 * q^86 - 10 * q^87 + 12 * q^88 - 16 * q^89 + 8 * q^90 + 4 * q^91 + 12 * q^92 + 8 * q^93 + 4 * q^94 - 4 * q^96 + 8 * q^97 - 14 * q^98 + 12 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	3.17557	1.42016	0.710079	−	0.704122i	$-0.248659\pi$
$5$	3.17557	1.42016	0.710079	+	0.704122i	$0.248659\pi$
$6$	−1.00000	−0.408248
$7$	0.381966	0.144370	0.0721848	−	0.997391i	$-0.477003\pi$
$7$	0.381966	0.144370	0.0721848	+	0.997391i	$0.477003\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	3.17557	1.00420
$11$	2.66605	0.803843	0.401921	−	0.915674i	$-0.368342\pi$
$11$	2.66605	0.803843	0.401921	+	0.915674i	$0.368342\pi$
$12$	−1.00000	−0.288675
$13$	−7.04029	−1.95263	−0.976313	−	0.216362i	$-0.930581\pi$
$13$	−7.04029	−1.95263	−0.976313	+	0.216362i	$0.930581\pi$
$14$	0.381966	0.102085
$15$	−3.17557	−0.819929
$16$	1.00000	0.250000
$17$	4.68915	1.13729	0.568643	−	0.822584i	$-0.307468\pi$
$17$	4.68915	1.13729	0.568643	+	0.822584i	$0.307468\pi$
$18$	1.00000	0.235702
$19$	0	0
$19$	0	0
$20$	3.17557	0.710079
$21$	−0.381966	−0.0833518
$22$	2.66605	0.568403
$23$	1.43184	0.298560	0.149280	−	0.988795i	$-0.452304\pi$
$23$	1.43184	0.298560	0.149280	+	0.988795i	$0.452304\pi$
$24$	−1.00000	−0.204124
$25$	5.08425	1.01685
$26$	−7.04029	−1.38072
$27$	−1.00000	−0.192450
$28$	0.381966	0.0721848
$29$	8.20524	1.52368	0.761838	−	0.647768i	$-0.224297\pi$
$29$	8.20524	1.52368	0.761838	+	0.647768i	$0.224297\pi$
$30$	−3.17557	−0.579777
$31$	8.62750	1.54955	0.774773	−	0.632240i	$-0.217864\pi$
$31$	8.62750	1.54955	0.774773	+	0.632240i	$0.217864\pi$
$32$	1.00000	0.176777
$33$	−2.66605	−0.464099
$34$	4.68915	0.804183
$35$	1.21296	0.205028
$36$	1.00000	0.166667
$37$	−2.70228	−0.444252	−0.222126	−	0.975018i	$-0.571300\pi$
$37$	−2.70228	−0.444252	−0.222126	+	0.975018i	$0.571300\pi$
$38$	0	0
$39$	7.04029	1.12735
$40$	3.17557	0.502102
$41$	3.25731	0.508707	0.254353	−	0.967111i	$-0.418137\pi$
$41$	3.25731	0.508707	0.254353	+	0.967111i	$0.418137\pi$
$42$	−0.381966	−0.0589386
$43$	−8.72353	−1.33033	−0.665163	−	0.746699i	$-0.731638\pi$
$43$	−8.72353	−1.33033	−0.665163	+	0.746699i	$0.731638\pi$
$44$	2.66605	0.401921
$45$	3.17557	0.473386
$46$	1.43184	0.211114
$47$	5.35706	0.781408	0.390704	−	0.920516i	$-0.372232\pi$
$47$	5.35706	0.781408	0.390704	+	0.920516i	$0.372232\pi$
$48$	−1.00000	−0.144338
$49$	−6.85410	−0.979157
$50$	5.08425	0.719021
$51$	−4.68915	−0.656613
$52$	−7.04029	−0.976313
$53$	−8.20524	−1.12708	−0.563538	−	0.826090i	$-0.690560\pi$
$53$	−8.20524	−1.12708	−0.563538	+	0.826090i	$0.690560\pi$
$54$	−1.00000	−0.136083
$55$	8.46621	1.14158
$56$	0.381966	0.0510424
$57$	0	0
$58$	8.20524	1.07740
$59$	11.8137	1.53801	0.769006	−	0.639241i	$-0.220752\pi$
$59$	11.8137	1.53801	0.769006	+	0.639241i	$0.220752\pi$
$60$	−3.17557	−0.409964
$61$	−5.27044	−0.674811	−0.337405	−	0.941359i	$-0.609549\pi$
$61$	−5.27044	−0.674811	−0.337405	+	0.941359i	$0.609549\pi$
$62$	8.62750	1.09569
$63$	0.381966	0.0481232
$64$	1.00000	0.125000
$65$	−22.3570	−2.77304
$66$	−2.66605	−0.328167
$67$	−12.2147	−1.49226	−0.746132	−	0.665798i	$-0.768091\pi$
$67$	−12.2147	−1.49226	−0.746132	+	0.665798i	$0.768091\pi$
$68$	4.68915	0.568643
$69$	−1.43184	−0.172374
$70$	1.21296	0.144976
$71$	15.0996	1.79200	0.895999	−	0.444057i	$-0.146461\pi$
$71$	15.0996	1.79200	0.895999	+	0.444057i	$0.146461\pi$
$72$	1.00000	0.117851
$73$	−3.84818	−0.450395	−0.225198	−	0.974313i	$-0.572303\pi$
$73$	−3.84818	−0.450395	−0.225198	+	0.974313i	$0.572303\pi$
$74$	−2.70228	−0.314134
$75$	−5.08425	−0.587078
$76$	0	0
$77$	1.01834	0.116050
$78$	7.04029	0.797156
$79$	1.39560	0.157018	0.0785089	−	0.996913i	$-0.474984\pi$
$79$	1.39560	0.157018	0.0785089	+	0.996913i	$0.474984\pi$
$80$	3.17557	0.355040
$81$	1.00000	0.111111
$82$	3.25731	0.359710
$83$	−4.31562	−0.473700	−0.236850	−	0.971546i	$-0.576115\pi$
$83$	−4.31562	−0.473700	−0.236850	+	0.971546i	$0.576115\pi$
$84$	−0.381966	−0.0416759
$85$	14.8907	1.61513
$86$	−8.72353	−0.940682
$87$	−8.20524	−0.879694
$88$	2.66605	0.284201
$89$	−7.80423	−0.827246	−0.413623	−	0.910448i	$-0.635737\pi$
$89$	−7.80423	−0.827246	−0.413623	+	0.910448i	$0.635737\pi$
$90$	3.17557	0.334735
$91$	−2.68915	−0.281900
$92$	1.43184	0.149280
$93$	−8.62750	−0.894630
$94$	5.35706	0.552539
$95$	0	0
$96$	−1.00000	−0.102062
$97$	12.7448	1.29404	0.647018	−	0.762475i	$-0.276016\pi$
$97$	12.7448	1.29404	0.647018	+	0.762475i	$0.276016\pi$
$98$	−6.85410	−0.692369
$99$	2.66605	0.267948
$100$	5.08425	0.508425
$101$	3.56597	0.354827	0.177413	−	0.984136i	$-0.443227\pi$
$101$	3.56597	0.354827	0.177413	+	0.984136i	$0.443227\pi$
$102$	−4.68915	−0.464295
$103$	7.04216	0.693885	0.346942	−	0.937886i	$-0.387220\pi$
$103$	7.04216	0.693885	0.346942	+	0.937886i	$0.387220\pi$
$104$	−7.04029	−0.690358
$105$	−1.21296	−0.118373
$106$	−8.20524	−0.796964
$107$	3.63116	0.351038	0.175519	−	0.984476i	$-0.443840\pi$
$107$	3.63116	0.351038	0.175519	+	0.984476i	$0.443840\pi$
$108$	−1.00000	−0.0962250
$109$	3.46621	0.332003	0.166002	−	0.986125i	$-0.446914\pi$
$109$	3.46621	0.332003	0.166002	+	0.986125i	$0.446914\pi$
$110$	8.46621	0.807222
$111$	2.70228	0.256489
$112$	0.381966	0.0360924
$113$	−10.2764	−0.966719	−0.483359	−	0.875422i	$-0.660584\pi$
$113$	−10.2764	−0.966719	−0.483359	+	0.875422i	$0.660584\pi$
$114$	0	0
$115$	4.54691	0.424002
$116$	8.20524	0.761838
$117$	−7.04029	−0.650875
$118$	11.8137	1.08754
$119$	1.79110	0.164190
$120$	−3.17557	−0.289889
$121$	−3.89220	−0.353837
$122$	−5.27044	−0.477163
$123$	−3.25731	−0.293702
$124$	8.62750	0.774773
$125$	0.267536	0.0239291
$126$	0.381966	0.0340282
$127$	21.2723	1.88761	0.943806	−	0.330500i	$-0.107217\pi$
$127$	21.2723	1.88761	0.943806	+	0.330500i	$0.107217\pi$
$128$	1.00000	0.0883883
$129$	8.72353	0.768064
$130$	−22.3570	−1.96083
$131$	2.93731	0.256634	0.128317	−	0.991733i	$-0.459043\pi$
$131$	2.93731	0.256634	0.128317	+	0.991733i	$0.459043\pi$
$132$	−2.66605	−0.232049
$133$	0	0
$134$	−12.2147	−1.05519
$135$	−3.17557	−0.273310
$136$	4.68915	0.402092
$137$	7.61218	0.650353	0.325176	−	0.945653i	$-0.394576\pi$
$137$	7.61218	0.650353	0.325176	+	0.945653i	$0.394576\pi$
$138$	−1.43184	−0.121886
$139$	0.434034	0.0368143	0.0184071	−	0.999831i	$-0.494140\pi$
$139$	0.434034	0.0368143	0.0184071	+	0.999831i	$0.494140\pi$
$140$	1.21296	0.102514
$141$	−5.35706	−0.451146
$142$	15.0996	1.26713
$143$	−18.7697	−1.56960
$144$	1.00000	0.0833333
$145$	26.0563	2.16386
$146$	−3.84818	−0.318478
$147$	6.85410	0.565317
$148$	−2.70228	−0.222126
$149$	17.1352	1.40377	0.701884	−	0.712292i	$-0.252343\pi$
$149$	17.1352	1.40377	0.701884	+	0.712292i	$0.252343\pi$
$150$	−5.08425	−0.415127
$151$	−9.31852	−0.758330	−0.379165	−	0.925329i	$-0.623789\pi$
$151$	−9.31852	−0.758330	−0.379165	+	0.925329i	$0.623789\pi$
$152$	0	0
$153$	4.68915	0.379096
$154$	1.01834	0.0820601
$155$	27.3972	2.20060
$156$	7.04029	0.563675
$157$	−17.6428	−1.40805	−0.704025	−	0.710175i	$-0.748616\pi$
$157$	−17.6428	−1.40805	−0.704025	+	0.710175i	$0.748616\pi$
$158$	1.39560	0.111028
$159$	8.20524	0.650718
$160$	3.17557	0.251051
$161$	0.546915	0.0431029
$162$	1.00000	0.0785674
$163$	−7.25731	−0.568436	−0.284218	−	0.958760i	$-0.591734\pi$
$163$	−7.25731	−0.568436	−0.284218	+	0.958760i	$0.591734\pi$
$164$	3.25731	0.254353
$165$	−8.46621	−0.659094
$166$	−4.31562	−0.334957
$167$	−11.8386	−0.916098	−0.458049	−	0.888927i	$-0.651452\pi$
$167$	−11.8386	−0.916098	−0.458049	+	0.888927i	$0.651452\pi$
$168$	−0.381966	−0.0294693
$169$	36.5657	2.81275
$170$	14.8907	1.14207
$171$	0	0
$172$	−8.72353	−0.665163
$173$	−15.0711	−1.14584	−0.572918	−	0.819613i	$-0.694189\pi$
$173$	−15.0711	−1.14584	−0.572918	+	0.819613i	$0.694189\pi$
$174$	−8.20524	−0.622038
$175$	1.94201	0.146802
$176$	2.66605	0.200961
$177$	−11.8137	−0.887972
$178$	−7.80423	−0.584951
$179$	−0.0630049	−0.00470921	−0.00235460	−	0.999997i	$-0.500749\pi$
$179$	−0.0630049	−0.00470921	−0.00235460	+	0.999997i	$0.500749\pi$
$180$	3.17557	0.236693
$181$	−3.79110	−0.281790	−0.140895	−	0.990025i	$-0.544998\pi$
$181$	−3.79110	−0.281790	−0.140895	+	0.990025i	$0.544998\pi$
$182$	−2.68915	−0.199333
$183$	5.27044	0.389602
$184$	1.43184	0.105557
$185$	−8.58129	−0.630909
$186$	−8.62750	−0.632599
$187$	12.5015	0.914200
$188$	5.35706	0.390704
$189$	−0.381966	−0.0277839
$190$	0	0
$191$	26.2572	1.89990	0.949952	−	0.312396i	$-0.101131\pi$
$191$	26.2572	1.89990	0.949952	+	0.312396i	$0.101131\pi$
$192$	−1.00000	−0.0721688
$193$	11.8897	0.855839	0.427920	−	0.903817i	$-0.359247\pi$
$193$	11.8897	0.855839	0.427920	+	0.903817i	$0.359247\pi$
$194$	12.7448	0.915021
$195$	22.3570	1.60101
$196$	−6.85410	−0.489579
$197$	−8.00186	−0.570109	−0.285055	−	0.958511i	$-0.592012\pi$
$197$	−8.00186	−0.570109	−0.285055	+	0.958511i	$0.592012\pi$
$198$	2.66605	0.189468
$199$	−24.3452	−1.72578	−0.862892	−	0.505389i	$-0.831349\pi$
$199$	−24.3452	−1.72578	−0.862892	+	0.505389i	$0.831349\pi$
$200$	5.08425	0.359511
$201$	12.2147	0.861559
$202$	3.56597	0.250900
$203$	3.13412	0.219972
$204$	−4.68915	−0.328306
$205$	10.3438	0.722444
$206$	7.04216	0.490650
$207$	1.43184	0.0995199
$208$	−7.04029	−0.488157
$209$	0	0
$210$	−1.21296	−0.0837022
$211$	−20.2016	−1.39073	−0.695367	−	0.718655i	$-0.744758\pi$
$211$	−20.2016	−1.39073	−0.695367	+	0.718655i	$0.744758\pi$
$212$	−8.20524	−0.563538
$213$	−15.0996	−1.03461
$214$	3.63116	0.248221
$215$	−27.7022	−1.88927
$216$	−1.00000	−0.0680414
$217$	3.29541	0.223707
$218$	3.46621	0.234762
$219$	3.84818	0.260036
$220$	8.46621	0.570792
$221$	−33.0130	−2.22070
$222$	2.70228	0.181365
$223$	−18.3581	−1.22935	−0.614675	−	0.788781i	$-0.710713\pi$
$223$	−18.3581	−1.22935	−0.614675	+	0.788781i	$0.710713\pi$
$224$	0.381966	0.0255212
$225$	5.08425	0.338950
$226$	−10.2764	−0.683573
$227$	1.53141	0.101643	0.0508217	−	0.998708i	$-0.483816\pi$
$227$	1.53141	0.101643	0.0508217	+	0.998708i	$0.483816\pi$
$228$	0	0
$229$	−21.2381	−1.40346	−0.701728	−	0.712445i	$-0.747588\pi$
$229$	−21.2381	−1.40346	−0.701728	+	0.712445i	$0.747588\pi$
$230$	4.54691	0.299815
$231$	−1.01834	−0.0670018
$232$	8.20524	0.538701
$233$	−10.9655	−0.718375	−0.359187	−	0.933265i	$-0.616946\pi$
$233$	−10.9655	−0.718375	−0.359187	+	0.933265i	$0.616946\pi$
$234$	−7.04029	−0.460238
$235$	17.0117	1.10972
$236$	11.8137	0.769006
$237$	−1.39560	−0.0906543
$238$	1.79110	0.116100
$239$	−6.85765	−0.443584	−0.221792	−	0.975094i	$-0.571191\pi$
$239$	−6.85765	−0.443584	−0.221792	+	0.975094i	$0.571191\pi$
$240$	−3.17557	−0.204982
$241$	−19.1109	−1.23104	−0.615521	−	0.788121i	$-0.711054\pi$
$241$	−19.1109	−1.23104	−0.615521	+	0.788121i	$0.711054\pi$
$242$	−3.89220	−0.250200
$243$	−1.00000	−0.0641500
$244$	−5.27044	−0.337405
$245$	−21.7657	−1.39056
$246$	−3.25731	−0.207679
$247$	0	0
$248$	8.62750	0.547847
$249$	4.31562	0.273491
$250$	0.267536	0.0169204
$251$	10.5165	0.663795	0.331897	−	0.943315i	$-0.392311\pi$
$251$	10.5165	0.663795	0.331897	+	0.943315i	$0.392311\pi$
$252$	0.381966	0.0240616
$253$	3.81736	0.239995
$254$	21.2723	1.33474
$255$	−14.8907	−0.932494
$256$	1.00000	0.0625000
$257$	−1.60637	−0.100203	−0.0501014	−	0.998744i	$-0.515954\pi$
$257$	−1.60637	−0.100203	−0.0501014	+	0.998744i	$0.515954\pi$
$258$	8.72353	0.543103
$259$	−1.03218	−0.0641365
$260$	−22.3570	−1.38652
$261$	8.20524	0.507892
$262$	2.93731	0.181468
$263$	13.4471	0.829181	0.414590	−	0.910008i	$-0.363925\pi$
$263$	13.4471	0.829181	0.414590	+	0.910008i	$0.363925\pi$
$264$	−2.66605	−0.164084
$265$	−26.0563	−1.60063
$266$	0	0
$267$	7.80423	0.477611
$268$	−12.2147	−0.746132
$269$	−22.5492	−1.37485	−0.687424	−	0.726257i	$-0.741258\pi$
$269$	−22.5492	−1.37485	−0.687424	+	0.726257i	$0.741258\pi$
$270$	−3.17557	−0.193259
$271$	−5.02260	−0.305101	−0.152551	−	0.988296i	$-0.548749\pi$
$271$	−5.02260	−0.305101	−0.152551	+	0.988296i	$0.548749\pi$
$272$	4.68915	0.284322
$273$	2.68915	0.162755
$274$	7.61218	0.459869
$275$	13.5548	0.817387
$276$	−1.43184	−0.0861868
$277$	2.11288	0.126951	0.0634754	−	0.997983i	$-0.479782\pi$
$277$	2.11288	0.126951	0.0634754	+	0.997983i	$0.479782\pi$
$278$	0.434034	0.0260316
$279$	8.62750	0.516515
$280$	1.21296	0.0724882
$281$	−18.7616	−1.11922	−0.559612	−	0.828754i	$-0.689050\pi$
$281$	−18.7616	−1.11922	−0.559612	+	0.828754i	$0.689050\pi$
$282$	−5.35706	−0.319008
$283$	−1.38434	−0.0822905	−0.0411453	−	0.999153i	$-0.513101\pi$
$283$	−1.38434	−0.0822905	−0.0411453	+	0.999153i	$0.513101\pi$
$284$	15.0996	0.895999
$285$	0	0
$286$	−18.7697	−1.10988
$287$	1.24418	0.0734417
$288$	1.00000	0.0589256
$289$	4.98816	0.293421
$290$	26.0563	1.53008
$291$	−12.7448	−0.747112
$292$	−3.84818	−0.225198
$293$	−0.226599	−0.0132381	−0.00661903	−	0.999978i	$-0.502107\pi$
$293$	−0.226599	−0.0132381	−0.00661903	+	0.999978i	$0.502107\pi$
$294$	6.85410	0.399739
$295$	37.5152	2.18422
$296$	−2.70228	−0.157067
$297$	−2.66605	−0.154700
$298$	17.1352	0.992613
$299$	−10.0806	−0.582975
$300$	−5.08425	−0.293539
$301$	−3.33209	−0.192058
$302$	−9.31852	−0.536221
$303$	−3.56597	−0.204859
$304$	0	0
$305$	−16.7367	−0.958338
$306$	4.68915	0.268061
$307$	22.4853	1.28330	0.641651	−	0.766997i	$-0.278250\pi$
$307$	22.4853	1.28330	0.641651	+	0.766997i	$0.278250\pi$
$308$	1.01834	0.0580252
$309$	−7.04216	−0.400614
$310$	27.3972	1.55606
$311$	−4.22513	−0.239585	−0.119793	−	0.992799i	$-0.538223\pi$
$311$	−4.22513	−0.239585	−0.119793	+	0.992799i	$0.538223\pi$
$312$	7.04029	0.398578
$313$	2.09455	0.118391	0.0591956	−	0.998246i	$-0.481146\pi$
$313$	2.09455	0.118391	0.0591956	+	0.998246i	$0.481146\pi$
$314$	−17.6428	−0.995642
$315$	1.21296	0.0683426
$316$	1.39560	0.0785089
$317$	−1.95616	−0.109869	−0.0549344	−	0.998490i	$-0.517495\pi$
$317$	−1.95616	−0.109869	−0.0549344	+	0.998490i	$0.517495\pi$
$318$	8.20524	0.460127
$319$	21.8755	1.22480
$320$	3.17557	0.177520
$321$	−3.63116	−0.202672
$322$	0.546915	0.0304784
$323$	0	0
$324$	1.00000	0.0555556
$325$	−35.7946	−1.98553
$326$	−7.25731	−0.401945
$327$	−3.46621	−0.191682
$328$	3.25731	0.179855
$329$	2.04622	0.112812
$330$	−8.46621	−0.466050
$331$	8.23015	0.452370	0.226185	−	0.974084i	$-0.427375\pi$
$331$	8.23015	0.452370	0.226185	+	0.974084i	$0.427375\pi$
$332$	−4.31562	−0.236850
$333$	−2.70228	−0.148084
$334$	−11.8386	−0.647779
$335$	−38.7887	−2.11925
$336$	−0.381966	−0.0208380
$337$	−7.56597	−0.412144	−0.206072	−	0.978537i	$-0.566068\pi$
$337$	−7.56597	−0.412144	−0.206072	+	0.978537i	$0.566068\pi$
$338$	36.5657	1.98891
$339$	10.2764	0.558135
$340$	14.8907	0.807564
$341$	23.0013	1.24559
$342$	0	0
$343$	−5.29180	−0.285730
$344$	−8.72353	−0.470341
$345$	−4.54691	−0.244798
$346$	−15.0711	−0.810228
$347$	−24.5084	−1.31568	−0.657839	−	0.753158i	$-0.728529\pi$
$347$	−24.5084	−1.31568	−0.657839	+	0.753158i	$0.728529\pi$
$348$	−8.20524	−0.439847
$349$	−10.5278	−0.563538	−0.281769	−	0.959482i	$-0.590921\pi$
$349$	−10.5278	−0.563538	−0.281769	+	0.959482i	$0.590921\pi$
$350$	1.94201	0.103805
$351$	7.04029	0.375783
$352$	2.66605	0.142101
$353$	11.0249	0.586794	0.293397	−	0.955991i	$-0.405214\pi$
$353$	11.0249	0.586794	0.293397	+	0.955991i	$0.405214\pi$
$354$	−11.8137	−0.627891
$355$	47.9500	2.54492
$356$	−7.80423	−0.413623
$357$	−1.79110	−0.0947949
$358$	−0.0630049	−0.00332991
$359$	−5.60034	−0.295575	−0.147787	−	0.989019i	$-0.547215\pi$
$359$	−5.60034	−0.295575	−0.147787	+	0.989019i	$0.547215\pi$
$360$	3.17557	0.167367
$361$	0	0
$362$	−3.79110	−0.199256
$363$	3.89220	0.204288
$364$	−2.68915	−0.140950
$365$	−12.2202	−0.639633
$366$	5.27044	0.275490
$367$	37.3238	1.94828	0.974142	−	0.225937i	$-0.0725443\pi$
$367$	37.3238	1.94828	0.974142	+	0.225937i	$0.0725443\pi$
$368$	1.43184	0.0746399
$369$	3.25731	0.169569
$370$	−8.58129	−0.446120
$371$	−3.13412	−0.162716
$372$	−8.62750	−0.447315
$373$	29.4141	1.52300	0.761502	−	0.648162i	$-0.224462\pi$
$373$	29.4141	1.52300	0.761502	+	0.648162i	$0.224462\pi$
$374$	12.5015	0.646437
$375$	−0.267536	−0.0138155
$376$	5.35706	0.276269
$377$	−57.7673	−2.97517
$378$	−0.381966	−0.0196462
$379$	28.2454	1.45087	0.725433	−	0.688293i	$-0.241639\pi$
$379$	28.2454	1.45087	0.725433	+	0.688293i	$0.241639\pi$
$380$	0	0
$381$	−21.2723	−1.08981
$382$	26.2572	1.34344
$383$	−16.8254	−0.859736	−0.429868	−	0.902892i	$-0.641440\pi$
$383$	−16.8254	−0.859736	−0.429868	+	0.902892i	$0.641440\pi$
$384$	−1.00000	−0.0510310
$385$	3.23381	0.164810
$386$	11.8897	0.605170
$387$	−8.72353	−0.443442
$388$	12.7448	0.647018
$389$	1.57035	0.0796199	0.0398100	−	0.999207i	$-0.487325\pi$
$389$	1.57035	0.0796199	0.0398100	+	0.999207i	$0.487325\pi$
$390$	22.3570	1.13209
$391$	6.71413	0.339548
$392$	−6.85410	−0.346184
$393$	−2.93731	−0.148168
$394$	−8.00186	−0.403128
$395$	4.43184	0.222990
$396$	2.66605	0.133974
$397$	3.08780	0.154972	0.0774860	−	0.996993i	$-0.475311\pi$
$397$	3.08780	0.154972	0.0774860	+	0.996993i	$0.475311\pi$
$398$	−24.3452	−1.22031
$399$	0	0
$400$	5.08425	0.254212
$401$	3.92895	0.196202	0.0981012	−	0.995176i	$-0.468723\pi$
$401$	3.92895	0.196202	0.0981012	+	0.995176i	$0.468723\pi$
$402$	12.2147	0.609214
$403$	−60.7402	−3.02568
$404$	3.56597	0.177413
$405$	3.17557	0.157795
$406$	3.13412	0.155544
$407$	−7.20441	−0.357109
$408$	−4.68915	−0.232148
$409$	−28.7032	−1.41928	−0.709641	−	0.704563i	$-0.751143\pi$
$409$	−28.7032	−1.41928	−0.709641	+	0.704563i	$0.751143\pi$
$410$	10.3438	0.510845
$411$	−7.61218	−0.375481
$412$	7.04216	0.346942
$413$	4.51243	0.222042
$414$	1.43184	0.0703712
$415$	−13.7045	−0.672730
$416$	−7.04029	−0.345179
$417$	−0.434034	−0.0212547
$418$	0	0
$419$	−3.73663	−0.182547	−0.0912733	−	0.995826i	$-0.529094\pi$
$419$	−3.73663	−0.182547	−0.0912733	+	0.995826i	$0.529094\pi$
$420$	−1.21296	−0.0591864
$421$	−21.6928	−1.05724	−0.528620	−	0.848858i	$-0.677291\pi$
$421$	−21.6928	−1.05724	−0.528620	+	0.848858i	$0.677291\pi$
$422$	−20.2016	−0.983398
$423$	5.35706	0.260469
$424$	−8.20524	−0.398482
$425$	23.8408	1.15645
$426$	−15.0996	−0.731580
$427$	−2.01313	−0.0974221
$428$	3.63116	0.175519
$429$	18.7697	0.906212
$430$	−27.7022	−1.33592
$431$	−5.09756	−0.245541	−0.122770	−	0.992435i	$-0.539178\pi$
$431$	−5.09756	−0.245541	−0.122770	+	0.992435i	$0.539178\pi$
$432$	−1.00000	−0.0481125
$433$	−18.1903	−0.874171	−0.437085	−	0.899420i	$-0.643989\pi$
$433$	−18.1903	−0.874171	−0.437085	+	0.899420i	$0.643989\pi$
$434$	3.29541	0.158185
$435$	−26.0563	−1.24931
$436$	3.46621	0.166002
$437$	0	0
$438$	3.84818	0.183873
$439$	−29.4760	−1.40681	−0.703406	−	0.710789i	$-0.748338\pi$
$439$	−29.4760	−1.40681	−0.703406	+	0.710789i	$0.748338\pi$
$440$	8.46621	0.403611
$441$	−6.85410	−0.326386
$442$	−33.0130	−1.57027
$443$	−20.6001	−0.978740	−0.489370	−	0.872076i	$-0.662773\pi$
$443$	−20.6001	−0.978740	−0.489370	+	0.872076i	$0.662773\pi$
$444$	2.70228	0.128245
$445$	−24.7829	−1.17482
$446$	−18.3581	−0.869281
$447$	−17.1352	−0.810465
$448$	0.381966	0.0180462
$449$	−5.32606	−0.251352	−0.125676	−	0.992071i	$-0.540110\pi$
$449$	−5.32606	−0.251352	−0.125676	+	0.992071i	$0.540110\pi$
$450$	5.08425	0.239674
$451$	8.68414	0.408920
$452$	−10.2764	−0.483359
$453$	9.31852	0.437822
$454$	1.53141	0.0718727
$455$	−8.53960	−0.400342
$456$	0	0
$457$	−14.5527	−0.680748	−0.340374	−	0.940290i	$-0.610554\pi$
$457$	−14.5527	−0.680748	−0.340374	+	0.940290i	$0.610554\pi$
$458$	−21.2381	−0.992393
$459$	−4.68915	−0.218871
$460$	4.54691	0.212001
$461$	−27.7283	−1.29144	−0.645718	−	0.763576i	$-0.723442\pi$
$461$	−27.7283	−1.29144	−0.645718	+	0.763576i	$0.723442\pi$
$462$	−1.01834	−0.0473774
$463$	3.33216	0.154859	0.0774293	−	0.996998i	$-0.475329\pi$
$463$	3.33216	0.154859	0.0774293	+	0.996998i	$0.475329\pi$
$464$	8.20524	0.380919
$465$	−27.3972	−1.27052
$466$	−10.9655	−0.507968
$467$	1.33094	0.0615884	0.0307942	−	0.999526i	$-0.490196\pi$
$467$	1.33094	0.0615884	0.0307942	+	0.999526i	$0.490196\pi$
$468$	−7.04029	−0.325438
$469$	−4.66560	−0.215438
$470$	17.0117	0.784693
$471$	17.6428	0.812938
$472$	11.8137	0.543769
$473$	−23.2573	−1.06937
$474$	−1.39560	−0.0641023
$475$	0	0
$476$	1.79110	0.0820948
$477$	−8.20524	−0.375692
$478$	−6.85765	−0.313662
$479$	−37.1883	−1.69918	−0.849589	−	0.527445i	$-0.823150\pi$
$479$	−37.1883	−1.69918	−0.849589	+	0.527445i	$0.823150\pi$
$480$	−3.17557	−0.144944
$481$	19.0249	0.867459
$482$	−19.1109	−0.870478
$483$	−0.546915	−0.0248855
$484$	−3.89220	−0.176918
$485$	40.4719	1.83773
$486$	−1.00000	−0.0453609
$487$	−25.6013	−1.16010	−0.580052	−	0.814579i	$-0.696968\pi$
$487$	−25.6013	−1.16010	−0.580052	+	0.814579i	$0.696968\pi$
$488$	−5.27044	−0.238582
$489$	7.25731	0.328187
$490$	−21.7657	−0.983273
$491$	25.3711	1.14498	0.572491	−	0.819911i	$-0.305977\pi$
$491$	25.3711	1.14498	0.572491	+	0.819911i	$0.305977\pi$
$492$	−3.25731	−0.146851
$493$	38.4756	1.73286
$494$	0	0
$495$	8.46621	0.380528
$496$	8.62750	0.387386
$497$	5.76755	0.258710
$498$	4.31562	0.193387
$499$	21.0996	0.944550	0.472275	−	0.881451i	$-0.343433\pi$
$499$	21.0996	0.944550	0.472275	+	0.881451i	$0.343433\pi$
$500$	0.267536	0.0119646
$501$	11.8386	0.528910
$502$	10.5165	0.469374
$503$	−26.7154	−1.19118	−0.595591	−	0.803288i	$-0.703082\pi$
$503$	−26.7154	−1.19118	−0.595591	+	0.803288i	$0.703082\pi$
$504$	0.381966	0.0170141
$505$	11.3240	0.503910
$506$	3.81736	0.169702
$507$	−36.5657	−1.62394
$508$	21.2723	0.943806
$509$	−4.44943	−0.197217	−0.0986087	−	0.995126i	$-0.531439\pi$
$509$	−4.44943	−0.197217	−0.0986087	+	0.995126i	$0.531439\pi$
$510$	−14.8907	−0.659373
$511$	−1.46987	−0.0650234
$512$	1.00000	0.0441942
$513$	0	0
$514$	−1.60637	−0.0708541
$515$	22.3629	0.985426
$516$	8.72353	0.384032
$517$	14.2822	0.628129
$518$	−1.03218	−0.0453514
$519$	15.0711	0.661549
$520$	−22.3570	−0.980417
$521$	−4.52798	−0.198374	−0.0991871	−	0.995069i	$-0.531624\pi$
$521$	−4.52798	−0.198374	−0.0991871	+	0.995069i	$0.531624\pi$
$522$	8.20524	0.359134
$523$	−3.84233	−0.168013	−0.0840066	−	0.996465i	$-0.526772\pi$
$523$	−3.84233	−0.168013	−0.0840066	+	0.996465i	$0.526772\pi$
$524$	2.93731	0.128317
$525$	−1.94201	−0.0847563
$526$	13.4471	0.586319
$527$	40.4557	1.76228
$528$	−2.66605	−0.116025
$529$	−20.9498	−0.910862
$530$	−26.0563	−1.13181
$531$	11.8137	0.512671
$532$	0	0
$533$	−22.9324	−0.993314
$534$	7.80423	0.337722
$535$	11.5310	0.498529
$536$	−12.2147	−0.527595
$537$	0.0630049	0.00271886
$538$	−22.5492	−0.972164
$539$	−18.2733	−0.787089
$540$	−3.17557	−0.136655
$541$	−2.76766	−0.118991	−0.0594955	−	0.998229i	$-0.518949\pi$
$541$	−2.76766	−0.118991	−0.0594955	+	0.998229i	$0.518949\pi$
$542$	−5.02260	−0.215739
$543$	3.79110	0.162692
$544$	4.68915	0.201046
$545$	11.0072	0.471497
$546$	2.68915	0.115085
$547$	−4.23955	−0.181270	−0.0906349	−	0.995884i	$-0.528890\pi$
$547$	−4.23955	−0.181270	−0.0906349	+	0.995884i	$0.528890\pi$
$548$	7.61218	0.325176
$549$	−5.27044	−0.224937
$550$	13.5548	0.577980
$551$	0	0
$552$	−1.43184	−0.0609432
$553$	0.533074	0.0226686
$554$	2.11288	0.0897677
$555$	8.58129	0.364255
$556$	0.434034	0.0184071
$557$	−17.5035	−0.741646	−0.370823	−	0.928704i	$-0.620924\pi$
$557$	−17.5035	−0.741646	−0.370823	+	0.928704i	$0.620924\pi$
$558$	8.62750	0.365231
$559$	61.4162	2.59763
$560$	1.21296	0.0512569
$561$	−12.5015	−0.527813
$562$	−18.7616	−0.791411
$563$	6.45162	0.271903	0.135952	−	0.990715i	$-0.456591\pi$
$563$	6.45162	0.271903	0.135952	+	0.990715i	$0.456591\pi$
$564$	−5.35706	−0.225573
$565$	−32.6333	−1.37289
$566$	−1.38434	−0.0581882
$567$	0.381966	0.0160411
$568$	15.0996	0.633567
$569$	−0.506508	−0.0212339	−0.0106170	−	0.999944i	$-0.503380\pi$
$569$	−0.506508	−0.0212339	−0.0106170	+	0.999944i	$0.503380\pi$
$570$	0	0
$571$	2.08999	0.0874633	0.0437316	−	0.999043i	$-0.486075\pi$
$571$	2.08999	0.0874633	0.0437316	+	0.999043i	$0.486075\pi$
$572$	−18.7697	−0.784802
$573$	−26.2572	−1.09691
$574$	1.24418	0.0519312
$575$	7.27984	0.303590
$576$	1.00000	0.0416667
$577$	43.6345	1.81653	0.908264	−	0.418398i	$-0.137408\pi$
$577$	43.6345	1.81653	0.908264	+	0.418398i	$0.137408\pi$
$578$	4.98816	0.207480
$579$	−11.8897	−0.494119
$580$	26.0563	1.08193
$581$	−1.64842	−0.0683879
$582$	−12.7448	−0.528288
$583$	−21.8755	−0.905993
$584$	−3.84818	−0.159239
$585$	−22.3570	−0.924346
$586$	−0.226599	−0.00936072
$587$	44.2051	1.82454	0.912270	−	0.409590i	$-0.134328\pi$
$587$	44.2051	1.82454	0.912270	+	0.409590i	$0.134328\pi$
$588$	6.85410	0.282658
$589$	0	0
$590$	37.5152	1.54448
$591$	8.00186	0.329153
$592$	−2.70228	−0.111063
$593$	5.92278	0.243219	0.121610	−	0.992578i	$-0.461194\pi$
$593$	5.92278	0.243219	0.121610	+	0.992578i	$0.461194\pi$
$594$	−2.66605	−0.109389
$595$	5.68776	0.233175
$596$	17.1352	0.701884
$597$	24.3452	0.996382
$598$	−10.0806	−0.412226
$599$	26.3593	1.07701	0.538505	−	0.842622i	$-0.318989\pi$
$599$	26.3593	1.07701	0.538505	+	0.842622i	$0.318989\pi$
$600$	−5.08425	−0.207564
$601$	−2.85069	−0.116282	−0.0581410	−	0.998308i	$-0.518517\pi$
$601$	−2.85069	−0.116282	−0.0581410	+	0.998308i	$0.518517\pi$
$602$	−3.33209	−0.135806
$603$	−12.2147	−0.497422
$604$	−9.31852	−0.379165
$605$	−12.3600	−0.502504
$606$	−3.56597	−0.144857
$607$	6.77821	0.275119	0.137560	−	0.990493i	$-0.456074\pi$
$607$	6.77821	0.275119	0.137560	+	0.990493i	$0.456074\pi$
$608$	0	0
$609$	−3.13412	−0.127001
$610$	−16.7367	−0.677647
$611$	−37.7153	−1.52580
$612$	4.68915	0.189548
$613$	−20.2191	−0.816642	−0.408321	−	0.912838i	$-0.633886\pi$
$613$	−20.2191	−0.816642	−0.408321	+	0.912838i	$0.633886\pi$
$614$	22.4853	0.907431
$615$	−10.3438	−0.417103
$616$	1.01834	0.0410300
$617$	−10.9133	−0.439351	−0.219676	−	0.975573i	$-0.570500\pi$
$617$	−10.9133	−0.439351	−0.219676	+	0.975573i	$0.570500\pi$
$618$	−7.04216	−0.283277
$619$	13.8029	0.554787	0.277394	−	0.960756i	$-0.410529\pi$
$619$	13.8029	0.554787	0.277394	+	0.960756i	$0.410529\pi$
$620$	27.3972	1.10030
$621$	−1.43184	−0.0574578
$622$	−4.22513	−0.169412
$623$	−2.98095	−0.119429
$624$	7.04029	0.281837
$625$	−24.5717	−0.982866
$626$	2.09455	0.0837152
$627$	0	0
$628$	−17.6428	−0.704025
$629$	−12.6714	−0.505242
$630$	1.21296	0.0483255
$631$	28.2587	1.12496	0.562480	−	0.826811i	$-0.309847\pi$
$631$	28.2587	1.12496	0.562480	+	0.826811i	$0.309847\pi$
$632$	1.39560	0.0555142
$633$	20.2016	0.802941
$634$	−1.95616	−0.0776890
$635$	67.5517	2.68071
$636$	8.20524	0.325359
$637$	48.2549	1.91193
$638$	21.8755	0.866061
$639$	15.0996	0.597332
$640$	3.17557	0.125525
$641$	10.1379	0.400421	0.200211	−	0.979753i	$-0.435837\pi$
$641$	10.1379	0.400421	0.200211	+	0.979753i	$0.435837\pi$
$642$	−3.63116	−0.143311
$643$	−8.53379	−0.336540	−0.168270	−	0.985741i	$-0.553818\pi$
$643$	−8.53379	−0.336540	−0.168270	+	0.985741i	$0.553818\pi$
$644$	0.546915	0.0215515
$645$	27.7022	1.09077
$646$	0	0
$647$	−14.2383	−0.559764	−0.279882	−	0.960034i	$-0.590295\pi$
$647$	−14.2383	−0.559764	−0.279882	+	0.960034i	$0.590295\pi$
$648$	1.00000	0.0392837
$649$	31.4958	1.23632
$650$	−35.7946	−1.40398
$651$	−3.29541	−0.129157
$652$	−7.25731	−0.284218
$653$	28.5091	1.11565	0.557823	−	0.829960i	$-0.311637\pi$
$653$	28.5091	1.11565	0.557823	+	0.829960i	$0.311637\pi$
$654$	−3.46621	−0.135540
$655$	9.32764	0.364461
$656$	3.25731	0.127177
$657$	−3.84818	−0.150132
$658$	2.04622	0.0797698
$659$	40.5659	1.58022	0.790112	−	0.612963i	$-0.210022\pi$
$659$	40.5659	1.58022	0.790112	+	0.612963i	$0.210022\pi$
$660$	−8.46621	−0.329547
$661$	4.47342	0.173996	0.0869980	−	0.996208i	$-0.472273\pi$
$661$	4.47342	0.173996	0.0869980	+	0.996208i	$0.472273\pi$
$662$	8.23015	0.319874
$663$	33.0130	1.28212
$664$	−4.31562	−0.167478
$665$	0	0
$666$	−2.70228	−0.104711
$667$	11.7486	0.454908
$668$	−11.8386	−0.458049
$669$	18.3581	0.709765
$670$	−38.7887	−1.49854
$671$	−14.0512	−0.542442
$672$	−0.381966	−0.0147347
$673$	4.55272	0.175495	0.0877473	−	0.996143i	$-0.472033\pi$
$673$	4.55272	0.175495	0.0877473	+	0.996143i	$0.472033\pi$
$674$	−7.56597	−0.291430
$675$	−5.08425	−0.195693
$676$	36.5657	1.40637
$677$	24.4472	0.939583	0.469792	−	0.882777i	$-0.344329\pi$
$677$	24.4472	0.939583	0.469792	+	0.882777i	$0.344329\pi$
$678$	10.2764	0.394661
$679$	4.86807	0.186819
$680$	14.8907	0.571034
$681$	−1.53141	−0.0586838
$682$	23.0013	0.880766
$683$	−25.3627	−0.970476	−0.485238	−	0.874382i	$-0.661267\pi$
$683$	−25.3627	−0.970476	−0.485238	+	0.874382i	$0.661267\pi$
$684$	0	0
$685$	24.1730	0.923604
$686$	−5.29180	−0.202042
$687$	21.2381	0.810286
$688$	−8.72353	−0.332581
$689$	57.7673	2.20076
$690$	−4.54691	−0.173098
$691$	−19.0686	−0.725405	−0.362702	−	0.931905i	$-0.618146\pi$
$691$	−19.0686	−0.725405	−0.362702	+	0.931905i	$0.618146\pi$
$692$	−15.0711	−0.572918
$693$	1.01834	0.0386835
$694$	−24.5084	−0.930325
$695$	1.37831	0.0522821
$696$	−8.20524	−0.311019
$697$	15.2740	0.578545
$698$	−10.5278	−0.398481
$699$	10.9655	0.414754
$700$	1.94201	0.0734011
$701$	−16.5346	−0.624504	−0.312252	−	0.949999i	$-0.601083\pi$
$701$	−16.5346	−0.624504	−0.312252	+	0.949999i	$0.601083\pi$
$702$	7.04029	0.265719
$703$	0	0
$704$	2.66605	0.100480
$705$	−17.0117	−0.640699
$706$	11.0249	0.414926
$707$	1.36208	0.0512262
$708$	−11.8137	−0.443986
$709$	28.3262	1.06381	0.531906	−	0.846804i	$-0.321476\pi$
$709$	28.3262	1.06381	0.531906	+	0.846804i	$0.321476\pi$
$710$	47.9500	1.79953
$711$	1.39560	0.0523393
$712$	−7.80423	−0.292476
$713$	12.3532	0.462632
$714$	−1.79110	−0.0670301
$715$	−59.6046	−2.22909
$716$	−0.0630049	−0.00235460
$717$	6.85765	0.256104
$718$	−5.60034	−0.209003
$719$	−20.6450	−0.769929	−0.384965	−	0.922931i	$-0.625786\pi$
$719$	−20.6450	−0.769929	−0.384965	+	0.922931i	$0.625786\pi$
$720$	3.17557	0.118347
$721$	2.68987	0.100176
$722$	0	0
$723$	19.1109	0.710742
$724$	−3.79110	−0.140895
$725$	41.7175	1.54935
$726$	3.89220	0.144453
$727$	−18.2148	−0.675549	−0.337774	−	0.941227i	$-0.609674\pi$
$727$	−18.2148	−0.675549	−0.337774	+	0.941227i	$0.609674\pi$
$728$	−2.68915	−0.0996666
$729$	1.00000	0.0370370
$730$	−12.2202	−0.452289
$731$	−40.9059	−1.51296
$732$	5.27044	0.194801
$733$	−28.0081	−1.03450	−0.517252	−	0.855833i	$-0.673045\pi$
$733$	−28.0081	−1.03450	−0.517252	+	0.855833i	$0.673045\pi$
$734$	37.3238	1.37764
$735$	21.7657	0.802839
$736$	1.43184	0.0527784
$737$	−32.5650	−1.19955
$738$	3.25731	0.119903
$739$	−46.2549	−1.70151	−0.850757	−	0.525559i	$-0.823856\pi$
$739$	−46.2549	−1.70151	−0.850757	+	0.525559i	$0.823856\pi$
$740$	−8.58129	−0.315454
$741$	0	0
$742$	−3.13412	−0.115057
$743$	30.8503	1.13179	0.565894	−	0.824478i	$-0.308531\pi$
$743$	30.8503	1.13179	0.565894	+	0.824478i	$0.308531\pi$
$744$	−8.62750	−0.316300
$745$	54.4139	1.99357
$746$	29.4141	1.07693
$747$	−4.31562	−0.157900
$748$	12.5015	0.457100
$749$	1.38698	0.0506792
$750$	−0.267536	−0.00976902
$751$	−12.9512	−0.472597	−0.236299	−	0.971680i	$-0.575934\pi$
$751$	−12.9512	−0.472597	−0.236299	+	0.971680i	$0.575934\pi$
$752$	5.35706	0.195352
$753$	−10.5165	−0.383242
$754$	−57.7673	−2.10376
$755$	−29.5916	−1.07695
$756$	−0.381966	−0.0138920
$757$	0.294101	0.0106893	0.00534464	−	0.999986i	$-0.498299\pi$
$757$	0.294101	0.0106893	0.00534464	+	0.999986i	$0.498299\pi$
$758$	28.2454	1.02592
$759$	−3.81736	−0.138561
$760$	0	0
$761$	15.4993	0.561849	0.280925	−	0.959730i	$-0.409359\pi$
$761$	15.4993	0.561849	0.280925	+	0.959730i	$0.409359\pi$
$762$	−21.2723	−0.770614
$763$	1.32398	0.0479311
$764$	26.2572	0.949952
$765$	14.8907	0.538376
$766$	−16.8254	−0.607925
$767$	−83.1719	−3.00316
$768$	−1.00000	−0.0360844
$769$	−21.7968	−0.786014	−0.393007	−	0.919535i	$-0.628565\pi$
$769$	−21.7968	−0.786014	−0.393007	+	0.919535i	$0.628565\pi$
$770$	3.23381	0.116538
$771$	1.60637	0.0578521
$772$	11.8897	0.427920
$773$	−7.46267	−0.268413	−0.134207	−	0.990953i	$-0.542849\pi$
$773$	−7.46267	−0.268413	−0.134207	+	0.990953i	$0.542849\pi$
$774$	−8.72353	−0.313561
$775$	43.8644	1.57565
$776$	12.7448	0.457511
$777$	1.03218	0.0370292
$778$	1.57035	0.0562998
$779$	0	0
$780$	22.3570	0.800507
$781$	40.2563	1.44048
$782$	6.71413	0.240097
$783$	−8.20524	−0.293231
$784$	−6.85410	−0.244789
$785$	−56.0260	−1.99965
$786$	−2.93731	−0.104770
$787$	−15.8788	−0.566017	−0.283009	−	0.959117i	$-0.591333\pi$
$787$	−15.8788	−0.566017	−0.283009	+	0.959117i	$0.591333\pi$
$788$	−8.00186	−0.285055
$789$	−13.4471	−0.478728
$790$	4.43184	0.157678
$791$	−3.92522	−0.139565
$792$	2.66605	0.0947338
$793$	37.1054	1.31765
$794$	3.08780	0.109582
$795$	26.0563	0.924123
$796$	−24.3452	−0.862892
$797$	−17.0673	−0.604556	−0.302278	−	0.953220i	$-0.597747\pi$
$797$	−17.0673	−0.604556	−0.302278	+	0.953220i	$0.597747\pi$
$798$	0	0
$799$	25.1201	0.888685
$800$	5.08425	0.179755
$801$	−7.80423	−0.275749
$802$	3.92895	0.138736
$803$	−10.2594	−0.362047
$804$	12.2147	0.430780
$805$	1.73677	0.0612130
$806$	−60.7402	−2.13948
$807$	22.5492	0.793769
$808$	3.56597	0.125450
$809$	33.6132	1.18178	0.590889	−	0.806753i	$-0.298777\pi$
$809$	33.6132	1.18178	0.590889	+	0.806753i	$0.298777\pi$
$810$	3.17557	0.111578
$811$	−19.6047	−0.688415	−0.344207	−	0.938894i	$-0.611852\pi$
$811$	−19.6047	−0.688415	−0.344207	+	0.938894i	$0.611852\pi$
$812$	3.13412	0.109986
$813$	5.02260	0.176150
$814$	−7.20441	−0.252514
$815$	−23.0461	−0.807270
$816$	−4.68915	−0.164153
$817$	0	0
$818$	−28.7032	−1.00358
$819$	−2.68915	−0.0939666
$820$	10.3438	0.361222
$821$	2.63979	0.0921292	0.0460646	−	0.998938i	$-0.485332\pi$
$821$	2.63979	0.0921292	0.0460646	+	0.998938i	$0.485332\pi$
$822$	−7.61218	−0.265505
$823$	−1.49912	−0.0522560	−0.0261280	−	0.999659i	$-0.508318\pi$
$823$	−1.49912	−0.0522560	−0.0261280	+	0.999659i	$0.508318\pi$
$824$	7.04216	0.245325
$825$	−13.5548	−0.471919
$826$	4.51243	0.157008
$827$	31.6406	1.10025	0.550126	−	0.835082i	$-0.314580\pi$
$827$	31.6406	1.10025	0.550126	+	0.835082i	$0.314580\pi$
$828$	1.43184	0.0497599
$829$	48.5942	1.68775	0.843873	−	0.536543i	$-0.180270\pi$
$829$	48.5942	1.68775	0.843873	+	0.536543i	$0.180270\pi$
$830$	−13.7045	−0.475692
$831$	−2.11288	−0.0732950
$832$	−7.04029	−0.244078
$833$	−32.1399	−1.11358
$834$	−0.434034	−0.0150294
$835$	−37.5943	−1.30100
$836$	0	0
$837$	−8.62750	−0.298210
$838$	−3.73663	−0.129080
$839$	−49.0971	−1.69502	−0.847510	−	0.530779i	$-0.821899\pi$
$839$	−49.0971	−1.69502	−0.847510	+	0.530779i	$0.821899\pi$
$840$	−1.21296	−0.0418511
$841$	38.3260	1.32159
$842$	−21.6928	−0.747582
$843$	18.7616	0.646185
$844$	−20.2016	−0.695367
$845$	116.117	3.99455
$846$	5.35706	0.184180
$847$	−1.48669	−0.0510833
$848$	−8.20524	−0.281769
$849$	1.38434	0.0475105
$850$	23.8408	0.817733
$851$	−3.86924	−0.132636
$852$	−15.0996	−0.517305
$853$	9.55514	0.327162	0.163581	−	0.986530i	$-0.447696\pi$
$853$	9.55514	0.327162	0.163581	+	0.986530i	$0.447696\pi$
$854$	−2.01313	−0.0688879
$855$	0	0
$856$	3.63116	0.124111
$857$	−2.77720	−0.0948674	−0.0474337	−	0.998874i	$-0.515104\pi$
$857$	−2.77720	−0.0948674	−0.0474337	+	0.998874i	$0.515104\pi$
$858$	18.7697	0.640788
$859$	−38.1187	−1.30059	−0.650296	−	0.759681i	$-0.725355\pi$
$859$	−38.1187	−1.30059	−0.650296	+	0.759681i	$0.725355\pi$
$860$	−27.7022	−0.944636
$861$	−1.24418	−0.0424016
$862$	−5.09756	−0.173624
$863$	−28.2441	−0.961439	−0.480720	−	0.876874i	$-0.659625\pi$
$863$	−28.2441	−0.961439	−0.480720	+	0.876874i	$0.659625\pi$
$864$	−1.00000	−0.0340207
$865$	−47.8594	−1.62727
$866$	−18.1903	−0.618132
$867$	−4.98816	−0.169407
$868$	3.29541	0.111854
$869$	3.72075	0.126218
$870$	−26.0563	−0.883392
$871$	85.9952	2.91383
$872$	3.46621	0.117381
$873$	12.7448	0.431345
$874$	0	0
$875$	0.102189	0.00345464
$876$	3.84818	0.130018
$877$	−22.1722	−0.748703	−0.374351	−	0.927287i	$-0.622135\pi$
$877$	−22.1722	−0.748703	−0.374351	+	0.927287i	$0.622135\pi$
$878$	−29.4760	−0.994766
$879$	0.226599	0.00764299
$880$	8.46621	0.285396
$881$	55.1257	1.85723	0.928616	−	0.371043i	$-0.121000\pi$
$881$	55.1257	1.85723	0.928616	+	0.371043i	$0.121000\pi$
$882$	−6.85410	−0.230790
$883$	20.5446	0.691381	0.345691	−	0.938349i	$-0.387645\pi$
$883$	20.5446	0.691381	0.345691	+	0.938349i	$0.387645\pi$
$884$	−33.0130	−1.11035
$885$	−37.5152	−1.26106
$886$	−20.6001	−0.692074
$887$	50.7843	1.70517	0.852585	−	0.522589i	$-0.175034\pi$
$887$	50.7843	1.70517	0.852585	+	0.522589i	$0.175034\pi$
$888$	2.70228	0.0906826
$889$	8.12530	0.272514
$890$	−24.7829	−0.830724
$891$	2.66605	0.0893159
$892$	−18.3581	−0.614675
$893$	0	0
$894$	−17.1352	−0.573086
$895$	−0.200076	−0.00668782
$896$	0.381966	0.0127606
$897$	10.0806	0.336581
$898$	−5.32606	−0.177733
$899$	70.7908	2.36100
$900$	5.08425	0.169475
$901$	−38.4756	−1.28181
$902$	8.68414	0.289150
$903$	3.33209	0.110885
$904$	−10.2764	−0.341787
$905$	−12.0389	−0.400186
$906$	9.31852	0.309587
$907$	−29.1553	−0.968084	−0.484042	−	0.875045i	$-0.660832\pi$
$907$	−29.1553	−0.968084	−0.484042	+	0.875045i	$0.660832\pi$
$908$	1.53141	0.0508217
$909$	3.56597	0.118276
$910$	−8.53960	−0.283085
$911$	−4.87180	−0.161410	−0.0807049	−	0.996738i	$-0.525717\pi$
$911$	−4.87180	−0.161410	−0.0807049	+	0.996738i	$0.525717\pi$
$912$	0	0
$913$	−11.5056	−0.380781
$914$	−14.5527	−0.481361
$915$	16.7367	0.553297
$916$	−21.2381	−0.701728
$917$	1.12195	0.0370501
$918$	−4.68915	−0.154765
$919$	−13.1897	−0.435088	−0.217544	−	0.976051i	$-0.569805\pi$
$919$	−13.1897	−0.435088	−0.217544	+	0.976051i	$0.569805\pi$
$920$	4.54691	0.149907
$921$	−22.4853	−0.740915
$922$	−27.7283	−0.913183
$923$	−106.306	−3.49910
$924$	−1.01834	−0.0335009
$925$	−13.7391	−0.451738
$926$	3.33216	0.109502
$927$	7.04216	0.231295
$928$	8.20524	0.269350
$929$	−16.0439	−0.526384	−0.263192	−	0.964744i	$-0.584775\pi$
$929$	−16.0439	−0.526384	−0.263192	+	0.964744i	$0.584775\pi$
$930$	−27.3972	−0.898391
$931$	0	0
$932$	−10.9655	−0.359187
$933$	4.22513	0.138325
$934$	1.33094	0.0435496
$935$	39.6994	1.29831
$936$	−7.04029	−0.230119
$937$	−48.2717	−1.57697	−0.788484	−	0.615056i	$-0.789133\pi$
$937$	−48.2717	−1.57697	−0.788484	+	0.615056i	$0.789133\pi$
$938$	−4.66560	−0.152337
$939$	−2.09455	−0.0683532
$940$	17.0117	0.554861
$941$	−11.1612	−0.363845	−0.181923	−	0.983313i	$-0.558232\pi$
$941$	−11.1612	−0.363845	−0.181923	+	0.983313i	$0.558232\pi$
$942$	17.6428	0.574834
$943$	4.66395	0.151879
$944$	11.8137	0.384503
$945$	−1.21296	−0.0394576
$946$	−23.2573	−0.756160
$947$	0.116113	0.00377316	0.00188658	−	0.999998i	$-0.499399\pi$
$947$	0.116113	0.00377316	0.00188658	+	0.999998i	$0.499399\pi$
$948$	−1.39560	−0.0453271
$949$	27.0923	0.879454
$950$	0	0
$951$	1.95616	0.0634328
$952$	1.79110	0.0580498
$953$	16.9191	0.548063	0.274031	−	0.961721i	$-0.411643\pi$
$953$	16.9191	0.548063	0.274031	+	0.961721i	$0.411643\pi$
$954$	−8.20524	−0.265655
$955$	83.3816	2.69816
$956$	−6.85765	−0.221792
$957$	−21.8755	−0.707136
$958$	−37.1883	−1.20150
$959$	2.90759	0.0938911
$960$	−3.17557	−0.102491
$961$	43.4338	1.40109
$962$	19.0249	0.613386
$963$	3.63116	0.117013
$964$	−19.1109	−0.615521
$965$	37.7566	1.21543
$966$	−0.546915	−0.0175967
$967$	31.2513	1.00498	0.502488	−	0.864584i	$-0.332418\pi$
$967$	31.2513	1.00498	0.502488	+	0.864584i	$0.332418\pi$
$968$	−3.89220	−0.125100
$969$	0	0
$970$	40.4719	1.29947
$971$	21.1207	0.677796	0.338898	−	0.940823i	$-0.389946\pi$
$971$	21.1207	0.677796	0.338898	+	0.940823i	$0.389946\pi$
$972$	−1.00000	−0.0320750
$973$	0.165786	0.00531486
$974$	−25.6013	−0.820317
$975$	35.7946	1.14634
$976$	−5.27044	−0.168703
$977$	46.4685	1.48666	0.743330	−	0.668925i	$-0.233245\pi$
$977$	46.4685	1.48666	0.743330	+	0.668925i	$0.233245\pi$
$978$	7.25731	0.232063
$979$	−20.8064	−0.664976
$980$	−21.7657	−0.695279
$981$	3.46621	0.110668
$982$	25.3711	0.809625
$983$	−8.96344	−0.285889	−0.142945	−	0.989731i	$-0.545657\pi$
$983$	−8.96344	−0.285889	−0.142945	+	0.989731i	$0.545657\pi$
$984$	−3.25731	−0.103839
$985$	−25.4105	−0.809645
$986$	38.4756	1.22531
$987$	−2.04622	−0.0651318
$988$	0	0
$989$	−12.4907	−0.397181
$990$	8.46621	0.269074
$991$	23.2216	0.737657	0.368829	−	0.929497i	$-0.379759\pi$
$991$	23.2216	0.737657	0.368829	+	0.929497i	$0.379759\pi$
$992$	8.62750	0.273923
$993$	−8.23015	−0.261176
$994$	5.76755	0.182936
$995$	−77.3098	−2.45089
$996$	4.31562	0.136746
$997$	37.6024	1.19088	0.595440	−	0.803400i	$-0.296978\pi$
$997$	37.6024	1.19088	0.595440	+	0.803400i	$0.296978\pi$
$998$	21.0996	0.667897
$999$	2.70228	0.0854964

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.2.a.x.1.3	yes	4
3.2	odd	2		6498.2.a.bv.1.2		4
19.18	odd	2		2166.2.a.w.1.3	✓	4
57.56	even	2		6498.2.a.by.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2166.2.a.w.1.3	✓	4	19.18	odd	2
2166.2.a.x.1.3	yes	4	1.1	even	1	trivial
6498.2.a.bv.1.2		4	3.2	odd	2
6498.2.a.by.1.2		4	57.56	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:	\( N \)	\(=\)	\( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2166.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.17557 q^{5} -1.00000 q^{6} +0.381966 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.17557 q^{5} -1.00000 q^{6} +0.381966 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.17557 q^{10} +2.66605 q^{11} -1.00000 q^{12} -7.04029 q^{13} +0.381966 q^{14} -3.17557 q^{15} +1.00000 q^{16} +4.68915 q^{17} +1.00000 q^{18} +3.17557 q^{20} -0.381966 q^{21} +2.66605 q^{22} +1.43184 q^{23} -1.00000 q^{24} +5.08425 q^{25} -7.04029 q^{26} -1.00000 q^{27} +0.381966 q^{28} +8.20524 q^{29} -3.17557 q^{30} +8.62750 q^{31} +1.00000 q^{32} -2.66605 q^{33} +4.68915 q^{34} +1.21296 q^{35} +1.00000 q^{36} -2.70228 q^{37} +7.04029 q^{39} +3.17557 q^{40} +3.25731 q^{41} -0.381966 q^{42} -8.72353 q^{43} +2.66605 q^{44} +3.17557 q^{45} +1.43184 q^{46} +5.35706 q^{47} -1.00000 q^{48} -6.85410 q^{49} +5.08425 q^{50} -4.68915 q^{51} -7.04029 q^{52} -8.20524 q^{53} -1.00000 q^{54} +8.46621 q^{55} +0.381966 q^{56} +8.20524 q^{58} +11.8137 q^{59} -3.17557 q^{60} -5.27044 q^{61} +8.62750 q^{62} +0.381966 q^{63} +1.00000 q^{64} -22.3570 q^{65} -2.66605 q^{66} -12.2147 q^{67} +4.68915 q^{68} -1.43184 q^{69} +1.21296 q^{70} +15.0996 q^{71} +1.00000 q^{72} -3.84818 q^{73} -2.70228 q^{74} -5.08425 q^{75} +1.01834 q^{77} +7.04029 q^{78} +1.39560 q^{79} +3.17557 q^{80} +1.00000 q^{81} +3.25731 q^{82} -4.31562 q^{83} -0.381966 q^{84} +14.8907 q^{85} -8.72353 q^{86} -8.20524 q^{87} +2.66605 q^{88} -7.80423 q^{89} +3.17557 q^{90} -2.68915 q^{91} +1.43184 q^{92} -8.62750 q^{93} +5.35706 q^{94} -1.00000 q^{96} +12.7448 q^{97} -6.85410 q^{98} +2.66605 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 8 q^{5} - 4 q^{6} + 6 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 8 * q^5 - 4 * q^6 + 6 * q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} + 8 q^{5} - 4 q^{6} + 6 q^{7} + 4 q^{8} + 4 q^{9} + 8 q^{10} + 12 q^{11} - 4 q^{12} - 4 q^{13} + 6 q^{14} - 8 q^{15} + 4 q^{16} + 4 q^{17} + 4 q^{18} + 8 q^{20} - 6 q^{21} + 12 q^{22} + 12 q^{23} - 4 q^{24} + 6 q^{25} - 4 q^{26} - 4 q^{27} + 6 q^{28} + 10 q^{29} - 8 q^{30} - 8 q^{31} + 4 q^{32} - 12 q^{33} + 4 q^{34} + 12 q^{35} + 4 q^{36} + 8 q^{37} + 4 q^{39} + 8 q^{40} - 8 q^{41} - 6 q^{42} - 4 q^{43} + 12 q^{44} + 8 q^{45} + 12 q^{46} + 4 q^{47} - 4 q^{48} - 14 q^{49} + 6 q^{50} - 4 q^{51} - 4 q^{52} - 10 q^{53} - 4 q^{54} + 24 q^{55} + 6 q^{56} + 10 q^{58} - 6 q^{59} - 8 q^{60} + 4 q^{61} - 8 q^{62} + 6 q^{63} + 4 q^{64} - 8 q^{65} - 12 q^{66} + 12 q^{67} + 4 q^{68} - 12 q^{69} + 12 q^{70} + 4 q^{72} - 10 q^{73} + 8 q^{74} - 6 q^{75} + 28 q^{77} + 4 q^{78} + 32 q^{79} + 8 q^{80} + 4 q^{81} - 8 q^{82} + 8 q^{83} - 6 q^{84} - 12 q^{85} - 4 q^{86} - 10 q^{87} + 12 q^{88} - 16 q^{89} + 8 q^{90} + 4 q^{91} + 12 q^{92} + 8 q^{93} + 4 q^{94} - 4 q^{96} + 8 q^{97} - 14 q^{98} + 12 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 - 4 * q^3 + 4 * q^4 + 8 * q^5 - 4 * q^6 + 6 * q^7 + 4 * q^8 + 4 * q^9 + 8 * q^10 + 12 * q^11 - 4 * q^12 - 4 * q^13 + 6 * q^14 - 8 * q^15 + 4 * q^16 + 4 * q^17 + 4 * q^18 + 8 * q^20 - 6 * q^21 + 12 * q^22 + 12 * q^23 - 4 * q^24 + 6 * q^25 - 4 * q^26 - 4 * q^27 + 6 * q^28 + 10 * q^29 - 8 * q^30 - 8 * q^31 + 4 * q^32 - 12 * q^33 + 4 * q^34 + 12 * q^35 + 4 * q^36 + 8 * q^37 + 4 * q^39 + 8 * q^40 - 8 * q^41 - 6 * q^42 - 4 * q^43 + 12 * q^44 + 8 * q^45 + 12 * q^46 + 4 * q^47 - 4 * q^48 - 14 * q^49 + 6 * q^50 - 4 * q^51 - 4 * q^52 - 10 * q^53 - 4 * q^54 + 24 * q^55 + 6 * q^56 + 10 * q^58 - 6 * q^59 - 8 * q^60 + 4 * q^61 - 8 * q^62 + 6 * q^63 + 4 * q^64 - 8 * q^65 - 12 * q^66 + 12 * q^67 + 4 * q^68 - 12 * q^69 + 12 * q^70 + 4 * q^72 - 10 * q^73 + 8 * q^74 - 6 * q^75 + 28 * q^77 + 4 * q^78 + 32 * q^79 + 8 * q^80 + 4 * q^81 - 8 * q^82 + 8 * q^83 - 6 * q^84 - 12 * q^85 - 4 * q^86 - 10 * q^87 + 12 * q^88 - 16 * q^89 + 8 * q^90 + 4 * q^91 + 12 * q^92 + 8 * q^93 + 4 * q^94 - 4 * q^96 + 8 * q^97 - 14 * q^98 + 12 * q^99

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