LMFDB - Embedded newform 855.2.k.j.676.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,2,Mod(406,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("855.406");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.k (of order $3$, degree $2$, minimal)

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:	no
Analytic conductor:	$6.82720937282$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
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} - 3 x^{11} + 13 x^{10} - 10 x^{9} + 44 x^{8} - 20 x^{7} + 119 x^{6} + 13 x^{5} + 83 x^{4} + 14 x^{3} + 46 x^{2} + 12 x + 4 $ x^12 - 3x^11 + 13x^10 - 10x^9 + 44x^8 - 20x^7 + 119x^6 + 13x^5 + 83x^4 + 14x^3 + 46x^2 + 12*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			676.6
Root			$1.50733 - 2.61078i$ of defining polynomial
Character	$\chi$	$=$	855.676
Dual form			855.2.k.j.406.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00733 - 1.74475i) q^{2} +(-1.02944 - 1.78305i) q^{4} +(-0.500000 + 0.866025i) q^{5} +1.66469 q^{7} -0.118641 q^{8} +O(q^{10})$	\(q+(1.00733 - 1.74475i) q^{2} +(-1.02944 - 1.78305i) q^{4} +(-0.500000 + 0.866025i) q^{5} +1.66469 q^{7} -0.118641 q^{8} +(1.00733 + 1.74475i) q^{10} +0.745446 q^{11} +(0.269815 + 0.467333i) q^{13} +(1.67690 - 2.90447i) q^{14} +(1.93938 - 3.35910i) q^{16} +(0.705067 - 1.22121i) q^{17} +(4.17330 - 1.25841i) q^{19} +2.05889 q^{20} +(0.750913 - 1.30062i) q^{22} +(0.437471 + 0.757722i) q^{23} +(-0.500000 - 0.866025i) q^{25} +1.08718 q^{26} +(-1.71370 - 2.96822i) q^{28} +(-3.70083 - 6.41003i) q^{29} +7.02934 q^{31} +(-4.02584 - 6.97297i) q^{32} +(-1.42048 - 2.46034i) q^{34} +(-0.832344 + 1.44166i) q^{35} +7.84476 q^{37} +(2.00828 - 8.54902i) q^{38} +(0.0593204 - 0.102746i) q^{40} +(-4.36591 + 7.56198i) q^{41} +(-2.93324 + 5.08052i) q^{43} +(-0.767395 - 1.32917i) q^{44} +1.76272 q^{46} +(-3.29875 - 5.71360i) q^{47} -4.22881 q^{49} -2.01467 q^{50} +(0.555519 - 0.962187i) q^{52} +(-3.85018 - 6.66871i) q^{53} +(-0.372723 + 0.645575i) q^{55} -0.197500 q^{56} -14.9119 q^{58} +(-2.45356 + 4.24969i) q^{59} +(2.60473 + 4.51153i) q^{61} +(7.08089 - 12.2645i) q^{62} -8.46397 q^{64} -0.539630 q^{65} +(-0.443531 - 0.768219i) q^{67} -2.90331 q^{68} +(1.67690 + 2.90447i) q^{70} +(-1.41061 + 2.44324i) q^{71} +(-0.524369 + 0.908233i) q^{73} +(7.90230 - 13.6872i) q^{74} +(-6.53999 - 6.14573i) q^{76} +1.24094 q^{77} +(-2.88624 + 4.99911i) q^{79} +(1.93938 + 3.35910i) q^{80} +(8.79587 + 15.2349i) q^{82} +3.61555 q^{83} +(0.705067 + 1.22121i) q^{85} +(5.90950 + 10.2356i) q^{86} -0.0884403 q^{88} +(0.464416 + 0.804392i) q^{89} +(0.449158 + 0.777964i) q^{91} +(0.900704 - 1.56007i) q^{92} -13.2918 q^{94} +(-0.996831 + 4.24339i) q^{95} +(-0.00920110 + 0.0159368i) q^{97} +(-4.25983 + 7.37824i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} - 5 q^{4} - 6 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 - 5 * q^4 - 6 * q^5 + 4 * q^7 + 12 * q^8	$ 12 q - 3 q^{2} - 5 q^{4} - 6 q^{5} + 4 q^{7} + 12 q^{8} - 3 q^{10} - 8 q^{13} - 10 q^{14} - 3 q^{16} - 4 q^{17} - 6 q^{19} + 10 q^{20} + 2 q^{23} - 6 q^{25} + 40 q^{26} - 26 q^{28} - 4 q^{29} + 24 q^{31} - 15 q^{32} + 7 q^{34} - 2 q^{35} + 29 q^{38} - 6 q^{40} - 12 q^{41} - 4 q^{43} - 6 q^{44} + 48 q^{46} - 6 q^{47} + 32 q^{49} + 6 q^{50} - 20 q^{52} - 26 q^{53} + 44 q^{56} - 20 q^{58} - 16 q^{59} + 20 q^{61} + 25 q^{62} + 28 q^{64} + 16 q^{65} - 12 q^{67} - 54 q^{68} - 10 q^{70} + 8 q^{71} - 4 q^{73} + 16 q^{74} - 66 q^{76} - 48 q^{77} - 12 q^{79} - 3 q^{80} + 26 q^{82} + 44 q^{83} - 4 q^{85} + 44 q^{86} - 32 q^{88} + 8 q^{89} + 2 q^{91} - 36 q^{92} - 14 q^{94} + 6 q^{95} + 30 q^{97} - 49 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 - 5 * q^4 - 6 * q^5 + 4 * q^7 + 12 * q^8 - 3 * q^10 - 8 * q^13 - 10 * q^14 - 3 * q^16 - 4 * q^17 - 6 * q^19 + 10 * q^20 + 2 * q^23 - 6 * q^25 + 40 * q^26 - 26 * q^28 - 4 * q^29 + 24 * q^31 - 15 * q^32 + 7 * q^34 - 2 * q^35 + 29 * q^38 - 6 * q^40 - 12 * q^41 - 4 * q^43 - 6 * q^44 + 48 * q^46 - 6 * q^47 + 32 * q^49 + 6 * q^50 - 20 * q^52 - 26 * q^53 + 44 * q^56 - 20 * q^58 - 16 * q^59 + 20 * q^61 + 25 * q^62 + 28 * q^64 + 16 * q^65 - 12 * q^67 - 54 * q^68 - 10 * q^70 + 8 * q^71 - 4 * q^73 + 16 * q^74 - 66 * q^76 - 48 * q^77 - 12 * q^79 - 3 * q^80 + 26 * q^82 + 44 * q^83 - 4 * q^85 + 44 * q^86 - 32 * q^88 + 8 * q^89 + 2 * q^91 - 36 * q^92 - 14 * q^94 + 6 * q^95 + 30 * q^97 - 49 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/855\mathbb{Z}\right)^\times$.

$n$	$172$	$191$	$496$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	1.00733	−	1.74475i	0.712293	−	1.23373i	−0.251702	−	0.967805i	$-0.580990\pi$
$2$	1.00733	−	1.74475i	0.712293	−	1.23373i	0.963994	−	0.265922i	$-0.0856765\pi$
$3$		0			0
$3$		0			0
$4$	−1.02944	−	1.78305i	−0.514722	−	0.891525i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	1.66469			0.629193			0.314596	−	0.949226i	$-0.398131\pi$
$7$	1.66469			0.629193			0.314596	+	0.949226i	$0.398131\pi$
$8$	−0.118641			−0.0419458
$9$		0			0
$10$	1.00733	+	1.74475i	0.318547	+	0.551740i
$11$	0.745446			0.224760			0.112380	−	0.993665i	$-0.464153\pi$
$11$	0.745446			0.224760			0.112380	+	0.993665i	$0.464153\pi$
$12$		0			0
$13$	0.269815	+	0.467333i	0.0748332	+	0.129615i	0.901014	−	0.433791i	$-0.142824\pi$
$13$	0.269815	+	0.467333i	0.0748332	+	0.129615i	−0.826181	+	0.563405i	$0.809491\pi$
$14$	1.67690	−	2.90447i	0.448170	−	0.776252i
$15$		0			0
$16$	1.93938	−	3.35910i	0.484844	−	0.839775i
$17$	0.705067	−	1.22121i	0.171004	−	0.296187i	−0.767767	−	0.640729i	$-0.778632\pi$
$17$	0.705067	−	1.22121i	0.171004	−	0.296187i	0.938771	+	0.344542i	$0.111966\pi$
$18$		0			0
$19$	4.17330	−	1.25841i	0.957420	−	0.288700i
$19$	4.17330	−	1.25841i	0.957420	−	0.288700i
$20$	2.05889			0.460381
$21$		0			0
$22$	0.750913	−	1.30062i	0.160095	−	0.277293i
$23$	0.437471	+	0.757722i	0.0912190	+	0.157996i	0.908024	−	0.418917i	$-0.137590\pi$
$23$	0.437471	+	0.757722i	0.0912190	+	0.157996i	−0.816805	+	0.576913i	$0.804257\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$	1.08718			0.213213
$27$		0			0
$28$	−1.71370	−	2.96822i	−0.323859	−	0.560941i
$29$	−3.70083	−	6.41003i	−0.687228	−	1.19031i	−0.972731	−	0.231935i	$-0.925494\pi$
$29$	−3.70083	−	6.41003i	−0.687228	−	1.19031i	0.285504	−	0.958378i	$-0.407839\pi$
$30$		0			0
$31$	7.02934			1.26251			0.631253	−	0.775577i	$-0.282541\pi$
$31$	7.02934			1.26251			0.631253	+	0.775577i	$0.282541\pi$
$32$	−4.02584	−	6.97297i	−0.711675	−	1.23266i
$33$		0			0
$34$	−1.42048	−	2.46034i	−0.243610	−	0.421944i
$35$	−0.832344	+	1.44166i	−0.140692	+	0.243685i
$36$		0			0
$37$	7.84476			1.28967			0.644836	−	0.764321i	$-0.276926\pi$
$37$	7.84476			1.28967			0.644836	+	0.764321i	$0.276926\pi$
$38$	2.00828	−	8.54902i	0.325787	−	1.38683i
$39$		0			0
$40$	0.0593204	−	0.102746i	0.00937937	−	0.0162455i
$41$	−4.36591	+	7.56198i	−0.681841	+	1.18098i	0.292577	+	0.956242i	$0.405487\pi$
$41$	−4.36591	+	7.56198i	−0.681841	+	1.18098i	−0.974418	+	0.224742i	$0.927846\pi$
$42$		0			0
$43$	−2.93324	+	5.08052i	−0.447315	+	0.774772i	−0.998210	−	0.0598025i	$-0.980953\pi$
$43$	−2.93324	+	5.08052i	−0.447315	+	0.774772i	0.550896	+	0.834574i	$0.314286\pi$
$44$	−0.767395	−	1.32917i	−0.115689	−	0.200380i
$45$		0			0
$46$	1.76272			0.259899
$47$	−3.29875	−	5.71360i	−0.481172	−	0.833414i	0.518595	−	0.855020i	$-0.326455\pi$
$47$	−3.29875	−	5.71360i	−0.481172	−	0.833414i	−0.999767	+	0.0216064i	$0.993122\pi$
$48$		0			0
$49$	−4.22881			−0.604116
$50$	−2.01467			−0.284917
$51$		0			0
$52$	0.555519	−	0.962187i	0.0770366	−	0.133431i
$53$	−3.85018	−	6.66871i	−0.528863	−	0.916018i	−0.999434	−	0.0336552i	$-0.989285\pi$
$53$	−3.85018	−	6.66871i	−0.528863	−	0.916018i	0.470570	−	0.882362i	$-0.344048\pi$
$54$		0			0
$55$	−0.372723	+	0.645575i	−0.0502580	+	0.0870494i
$56$	−0.197500			−0.0263920
$57$		0			0
$58$	−14.9119			−1.95803
$59$	−2.45356	+	4.24969i	−0.319426	+	0.553263i	−0.980368	−	0.197174i	$-0.936824\pi$
$59$	−2.45356	+	4.24969i	−0.319426	+	0.553263i	0.660942	+	0.750437i	$0.270157\pi$
$60$		0			0
$61$	2.60473	+	4.51153i	0.333502	+	0.577643i	0.983196	−	0.182553i	$-0.0584362\pi$
$61$	2.60473	+	4.51153i	0.333502	+	0.577643i	−0.649694	+	0.760196i	$0.725103\pi$
$62$	7.08089	−	12.2645i	0.899274	−	1.55759i
$63$		0			0
$64$	−8.46397			−1.05800
$65$	−0.539630			−0.0669329
$66$		0			0
$67$	−0.443531	−	0.768219i	−0.0541860	−	0.0938528i	0.837660	−	0.546192i	$-0.183923\pi$
$67$	−0.443531	−	0.768219i	−0.0541860	−	0.0938528i	−0.891846	+	0.452339i	$0.850590\pi$
$68$	−2.90331			−0.352078
$69$		0			0
$70$	1.67690	+	2.90447i	0.200428	+	0.347151i
$71$	−1.41061	+	2.44324i	−0.167408	+	0.289959i	−0.937508	−	0.347964i	$-0.886873\pi$
$71$	−1.41061	+	2.44324i	−0.167408	+	0.289959i	0.770100	+	0.637923i	$0.220206\pi$
$72$		0			0
$73$	−0.524369	+	0.908233i	−0.0613727	+	0.106301i	−0.895079	−	0.445907i	$-0.852881\pi$
$73$	−0.524369	+	0.908233i	−0.0613727	+	0.106301i	0.833707	+	0.552208i	$0.186214\pi$
$74$	7.90230	−	13.6872i	0.918624	−	1.59110i
$75$		0			0
$76$	−6.53999	−	6.14573i	−0.750188	−	0.704963i
$77$	1.24094			0.141418
$78$		0			0
$79$	−2.88624	+	4.99911i	−0.324727	+	0.562444i	−0.981457	−	0.191682i	$-0.938606\pi$
$79$	−2.88624	+	4.99911i	−0.324727	+	0.562444i	0.656730	+	0.754126i	$0.271939\pi$
$80$	1.93938	+	3.35910i	0.216829	+	0.375559i
$81$		0			0
$82$	8.79587	+	15.2349i	0.971341	+	1.68241i
$83$	3.61555			0.396858			0.198429	−	0.980115i	$-0.436416\pi$
$83$	3.61555			0.396858			0.198429	+	0.980115i	$0.436416\pi$
$84$		0			0
$85$	0.705067	+	1.22121i	0.0764752	+	0.132459i
$86$	5.90950	+	10.2356i	0.637238	+	1.10373i
$87$		0			0
$88$	−0.0884403			−0.00942776
$89$	0.464416	+	0.804392i	0.0492280	+	0.0852654i	0.889589	−	0.456761i	$-0.150991\pi$
$89$	0.464416	+	0.804392i	0.0492280	+	0.0852654i	−0.840361	+	0.542026i	$0.817657\pi$
$90$		0			0
$91$	0.449158	+	0.777964i	0.0470845	+	0.0815528i
$92$	0.900704	−	1.56007i	0.0939049	−	0.162648i
$93$		0			0
$94$	−13.2918			−1.37094
$95$	−0.996831	+	4.24339i	−0.102273	+	0.435362i
$96$		0			0
$97$	−0.00920110	+	0.0159368i	−0.000934230	+	0.00161813i	−0.866492	−	0.499191i	$-0.833631\pi$
$97$	−0.00920110	+	0.0159368i	−0.000934230	+	0.00161813i	0.865558	+	0.500809i	$0.166964\pi$
$98$	−4.25983	+	7.37824i	−0.430308	+	0.745315i
$99$		0			0
$100$	−1.02944	+	1.78305i	−0.102944	+	0.178305i
$101$	7.93725	+	13.7477i	0.789786	+	1.36795i	0.926098	+	0.377284i	$0.123142\pi$
$101$	7.93725	+	13.7477i	0.789786	+	1.36795i	−0.136312	+	0.990666i	$0.543525\pi$
$102$		0			0
$103$	−8.55819			−0.843264			−0.421632	−	0.906767i	$-0.638543\pi$
$103$	−8.55819			−0.843264			−0.421632	+	0.906767i	$0.638543\pi$
$104$	−0.0320110	−	0.0554448i	−0.00313894	−	0.00543681i
$105$		0			0
$106$	−15.5137			−1.50682
$107$	−11.6508			−1.12632			−0.563162	−	0.826347i	$-0.690415\pi$
$107$	−11.6508			−1.12632			−0.563162	+	0.826347i	$0.690415\pi$
$108$		0			0
$109$	1.12075	−	1.94119i	0.107348	−	0.185932i	−0.807347	−	0.590077i	$-0.799097\pi$
$109$	1.12075	−	1.94119i	0.107348	−	0.185932i	0.914695	+	0.404145i	$0.132431\pi$
$110$	0.750913	+	1.30062i	0.0715968	+	0.124009i
$111$		0			0
$112$	3.22846	−	5.59185i	0.305061	−	0.528380i
$113$	8.94383			0.841364			0.420682	−	0.907208i	$-0.361791\pi$
$113$	8.94383			0.841364			0.420682	+	0.907208i	$0.361791\pi$
$114$		0			0
$115$	−0.874942			−0.0815888
$116$	−7.61960	+	13.1975i	−0.707462	+	1.22536i
$117$		0			0
$118$	4.94311	+	8.56172i	0.455050	+	0.788170i
$119$	1.17372	−	2.03294i	0.107594	−	0.186359i
$120$		0			0
$121$	−10.4443			−0.949483
$122$	10.4954			0.950205
$123$		0			0
$124$	−7.23631	−	12.5337i	−0.649840	−	1.12556i
$125$	1.00000			0.0894427
$126$		0			0
$127$	9.91270	+	17.1693i	0.879609	+	1.52353i	0.851770	+	0.523916i	$0.175529\pi$
$127$	9.91270	+	17.1693i	0.879609	+	1.52353i	0.0278395	+	0.999612i	$0.491137\pi$
$128$	−0.474357	+	0.821611i	−0.0419276	+	0.0726208i
$129$		0			0
$130$	−0.543588	+	0.941522i	−0.0476758	+	0.0825769i
$131$	−0.917977	+	1.58998i	−0.0802040	+	0.138917i	−0.903337	−	0.428931i	$-0.858891\pi$
$131$	−0.917977	+	1.58998i	−0.0802040	+	0.138917i	0.823133	+	0.567848i	$0.192224\pi$
$132$		0			0
$133$	6.94723	−	2.09486i	0.602402	−	0.181648i
$134$	−1.78714			−0.154385
$135$		0			0
$136$	−0.0836496	+	0.144885i	−0.00717290	+	0.0124238i
$137$	−1.98094	−	3.43109i	−0.169243	−	0.293138i	0.768911	−	0.639356i	$-0.220799\pi$
$137$	−1.98094	−	3.43109i	−0.169243	−	0.293138i	−0.938154	+	0.346218i	$0.887466\pi$
$138$		0			0
$139$	0.889046	+	1.53987i	0.0754079	+	0.130610i	0.901264	−	0.433271i	$-0.142641\pi$
$139$	0.889046	+	1.53987i	0.0754079	+	0.130610i	−0.825856	+	0.563882i	$0.809307\pi$
$140$	3.42741			0.289669
$141$		0			0
$142$	2.84190	+	4.92232i	0.238487	+	0.413072i
$143$	0.201133	+	0.348372i	0.0168196	+	0.0291323i
$144$		0			0
$145$	7.40167			0.614675
$146$	1.05643	+	1.82979i	0.0874307	+	0.151434i
$147$		0			0
$148$	−8.07575	−	13.9876i	−0.663822	−	1.14977i
$149$	−9.55320	+	16.5466i	−0.782629	+	1.35555i	0.147777	+	0.989021i	$0.452788\pi$
$149$	−9.55320	+	16.5466i	−0.782629	+	1.35555i	−0.930405	+	0.366532i	$0.880545\pi$
$150$		0			0
$151$	2.44644			0.199088			0.0995442	−	0.995033i	$-0.468262\pi$
$151$	2.44644			0.199088			0.0995442	+	0.995033i	$0.468262\pi$
$152$	−0.495123	+	0.149299i	−0.0401598	+	0.0121097i
$153$		0			0
$154$	1.25004	−	2.16513i	0.100731	−	0.174471i
$155$	−3.51467	+	6.08758i	−0.282305	+	0.488967i
$156$		0			0
$157$	−4.60586	+	7.97759i	−0.367588	+	0.636681i	−0.989188	−	0.146654i	$-0.953150\pi$
$157$	−4.60586	+	7.97759i	−0.367588	+	0.636681i	0.621600	+	0.783335i	$0.286483\pi$
$158$	5.81481	+	10.0715i	0.462601	+	0.801249i
$159$		0			0
$160$	8.05169			0.636542
$161$	0.728253	+	1.26137i	0.0573944	+	0.0994100i
$162$		0			0
$163$	1.09691			0.0859164			0.0429582	−	0.999077i	$-0.486322\pi$
$163$	1.09691			0.0859164			0.0429582	+	0.999077i	$0.486322\pi$
$164$	17.9779			1.40383
$165$		0			0
$166$	3.64207	−	6.30824i	0.282679	−	0.489615i
$167$	−6.24438	−	10.8156i	−0.483205	−	0.836935i	0.516609	−	0.856221i	$-0.327194\pi$
$167$	−6.24438	−	10.8156i	−0.483205	−	0.836935i	−0.999814	+	0.0192861i	$0.993861\pi$
$168$		0			0
$169$	6.35440	−	11.0061i	0.488800	−	0.846626i
$170$	2.84095			0.217891
$171$		0			0
$172$	12.0784			0.920971
$173$	−2.15558	+	3.73357i	−0.163886	+	0.283858i	−0.936259	−	0.351310i	$-0.885736\pi$
$173$	−2.15558	+	3.73357i	−0.163886	+	0.283858i	0.772373	+	0.635169i	$0.219069\pi$
$174$		0			0
$175$	−0.832344	−	1.44166i	−0.0629193	−	0.108979i
$176$	1.44570	−	2.50403i	0.108974	−	0.188748i
$177$		0			0
$178$	1.87129			0.140259
$179$	−12.6038			−0.942052			−0.471026	−	0.882119i	$-0.656116\pi$
$179$	−12.6038			−0.942052			−0.471026	+	0.882119i	$0.656116\pi$
$180$		0			0
$181$	−1.84534	−	3.19623i	−0.137163	−	0.237574i	0.789259	−	0.614061i	$-0.210465\pi$
$181$	−1.84534	−	3.19623i	−0.137163	−	0.237574i	−0.926422	+	0.376487i	$0.877132\pi$
$182$	1.80981			0.134152
$183$		0			0
$184$	−0.0519019	−	0.0898967i	−0.00382626	−	0.00662727i
$185$	−3.92238	+	6.79377i	−0.288379	+	0.499488i
$186$		0			0
$187$	0.525589	−	0.910348i	0.0384349	−	0.0665712i
$188$	−6.79175	+	11.7637i	−0.495339	+	0.857953i
$189$		0			0
$190$	6.39952	+	6.01373i	0.464270	+	0.436282i
$191$	−6.23634			−0.451246			−0.225623	−	0.974215i	$-0.572442\pi$
$191$	−6.23634			−0.451246			−0.225623	+	0.974215i	$0.572442\pi$
$192$		0			0
$193$	9.46275	−	16.3900i	0.681143	−	1.17978i	−0.293489	−	0.955963i	$-0.594816\pi$
$193$	9.46275	−	16.3900i	0.681143	−	1.17978i	0.974632	−	0.223813i	$-0.0718503\pi$
$194$	0.0185372	+	0.0321073i	0.00133089	+	0.00230517i
$195$		0			0
$196$	4.35333	+	7.54019i	0.310952	+	0.538585i
$197$	6.85139			0.488141			0.244071	−	0.969757i	$-0.421517\pi$
$197$	6.85139			0.488141			0.244071	+	0.969757i	$0.421517\pi$
$198$		0			0
$199$	−9.99420	−	17.3105i	−0.708470	−	1.22711i	−0.965424	−	0.260683i	$-0.916052\pi$
$199$	−9.99420	−	17.3105i	−0.708470	−	1.22711i	0.256954	−	0.966424i	$-0.417281\pi$
$200$	0.0593204	+	0.102746i	0.00419458	+	0.00726523i
$201$		0			0
$202$	31.9819			2.25024
$203$	−6.16073	−	10.6707i	−0.432399	−	0.748936i
$204$		0			0
$205$	−4.36591	−	7.56198i	−0.304929	−	0.528152i
$206$	−8.62096	+	14.9319i	−0.600651	+	1.04036i
$207$		0			0
$208$	2.09309			0.145130
$209$	3.11097	−	0.938079i	0.215190	−	0.0648883i
$210$		0			0
$211$	−7.41451	+	12.8423i	−0.510436	+	0.884101i	0.489491	+	0.872008i	$0.337183\pi$
$211$	−7.41451	+	12.8423i	−0.510436	+	0.884101i	−0.999927	+	0.0120925i	$0.996151\pi$
$212$	−7.92709	+	13.7301i	−0.544435	+	0.942989i
$213$		0			0
$214$	−11.7362	+	20.3278i	−0.802272	+	1.38958i
$215$	−2.93324	−	5.08052i	−0.200045	−	0.346488i
$216$		0			0
$217$	11.7017			0.794360
$218$	−2.25793	−	3.91085i	−0.152926	−	0.264876i
$219$		0			0
$220$	1.53479			0.103476
$221$	0.760951			0.0511871
$222$		0			0
$223$	9.44388	−	16.3573i	0.632409	−	1.09536i	−0.354649	−	0.934999i	$-0.615400\pi$
$223$	9.44388	−	16.3573i	0.632409	−	1.09536i	0.987058	−	0.160365i	$-0.0512670\pi$
$224$	−6.70177	−	11.6078i	−0.447781	−	0.775579i
$225$		0			0
$226$	9.00942	−	15.6048i	0.599298	−	1.03801i
$227$	15.7322			1.04418			0.522090	−	0.852890i	$-0.325152\pi$
$227$	15.7322			1.04418			0.522090	+	0.852890i	$0.325152\pi$
$228$		0			0
$229$	−19.0417			−1.25831			−0.629157	−	0.777278i	$-0.716600\pi$
$229$	−19.0417			−1.25831			−0.629157	+	0.777278i	$0.716600\pi$
$230$	−0.881359	+	1.52656i	−0.0581151	+	0.100658i
$231$		0			0
$232$	0.439070	+	0.760491i	0.0288263	+	0.0499287i
$233$	−0.594442	+	1.02960i	−0.0389432	+	0.0674515i	−0.884840	−	0.465895i	$-0.845732\pi$
$233$	−0.594442	+	1.02960i	−0.0389432	+	0.0674515i	0.845897	+	0.533346i	$0.179066\pi$
$234$		0			0
$235$	6.59749			0.430373
$236$	10.1032			0.657663
$237$		0			0
$238$	−2.36465	−	4.09569i	−0.153277	−	0.265484i
$239$	−25.9446			−1.67822			−0.839108	−	0.543965i	$-0.816922\pi$
$239$	−25.9446			−1.67822			−0.839108	+	0.543965i	$0.816922\pi$
$240$		0			0
$241$	4.52815	+	7.84299i	0.291684	+	0.505211i	0.974208	−	0.225652i	$-0.0724512\pi$
$241$	4.52815	+	7.84299i	0.291684	+	0.505211i	−0.682524	+	0.730863i	$0.739118\pi$
$242$	−10.5209	+	18.2228i	−0.676310	+	1.17140i
$243$		0			0
$244$	5.36286	−	9.28874i	0.343322	−	0.594651i
$245$	2.11441	−	3.66226i	0.135085	−	0.233973i
$246$		0			0
$247$	1.71412	+	1.61078i	0.109067	+	0.102492i
$248$	−0.833966			−0.0529569
$249$		0			0
$250$	1.00733	−	1.74475i	0.0637094	−	0.110348i
$251$	1.46528	+	2.53794i	0.0924876	+	0.160193i	0.908557	−	0.417760i	$-0.137185\pi$
$251$	1.46528	+	2.53794i	0.0924876	+	0.160193i	−0.816070	+	0.577954i	$0.803851\pi$
$252$		0			0
$253$	0.326111	+	0.564841i	0.0205024	+	0.0355113i
$254$	39.9416			2.50616
$255$		0			0
$256$	−7.50830	−	13.0047i	−0.469268	−	0.812797i
$257$	5.42535	+	9.39698i	0.338424	+	0.586168i	0.984136	−	0.177413i	$-0.0567728\pi$
$257$	5.42535	+	9.39698i	0.338424	+	0.586168i	−0.645712	+	0.763581i	$0.723439\pi$
$258$		0			0
$259$	13.0591			0.811452
$260$	0.555519	+	0.962187i	0.0344518	+	0.0596723i
$261$		0			0
$262$	1.84942	+	3.20329i	0.114257	+	0.197900i
$263$	−8.42898	+	14.5994i	−0.519753	+	0.900239i	0.479983	+	0.877278i	$0.340643\pi$
$263$	−8.42898	+	14.5994i	−0.519753	+	0.900239i	−0.999736	+	0.0229610i	$0.992691\pi$
$264$		0			0
$265$	7.70036			0.473029
$266$	3.34317	−	14.2314i	0.204983	−	0.872586i
$267$		0			0
$268$	−0.913181	+	1.58168i	−0.0557814	+	0.0966163i
$269$	−14.7731	+	25.5878i	−0.900733	+	1.56012i	−0.0741891	+	0.997244i	$0.523637\pi$
$269$	−14.7731	+	25.5878i	−0.900733	+	1.56012i	−0.826544	+	0.562872i	$0.809696\pi$
$270$		0			0
$271$	−10.6224	+	18.3985i	−0.645262	+	1.11763i	0.338979	+	0.940794i	$0.389919\pi$
$271$	−10.6224	+	18.3985i	−0.645262	+	1.11763i	−0.984241	+	0.176833i	$0.943415\pi$
$272$	−2.73478	−	4.73678i	−0.165821	−	0.287210i
$273$		0			0
$274$	−7.98188			−0.482203
$275$	−0.372723	−	0.645575i	−0.0224760	−	0.0389297i
$276$		0			0
$277$	−28.3656			−1.70432			−0.852161	−	0.523279i	$-0.824708\pi$
$277$	−28.3656			−1.70432			−0.852161	+	0.523279i	$0.824708\pi$
$278$	3.58226			0.214850
$279$		0			0
$280$	0.0987499	−	0.171040i	0.00590143	−	0.0102216i
$281$	−8.80909	−	15.2578i	−0.525506	−	0.910204i	−0.999559	−	0.0297069i	$-0.990543\pi$
$281$	−8.80909	−	15.2578i	−0.525506	−	0.910204i	0.474052	−	0.880497i	$-0.342791\pi$
$282$		0			0
$283$	8.54552	−	14.8013i	0.507978	−	0.879844i	−0.491979	−	0.870607i	$-0.663726\pi$
$283$	8.54552	−	14.8013i	0.507978	−	0.879844i	0.999957	−	0.00923712i	$-0.00294031\pi$
$284$	5.80856			0.344675
$285$		0			0
$286$	0.810431			0.0479218
$287$	−7.26788	+	12.5883i	−0.429010	+	0.743066i
$288$		0			0
$289$	7.50576	+	13.0004i	0.441515	+	0.764727i
$290$	7.45595	−	12.9141i	0.437829	−	0.758341i
$291$		0			0
$292$	2.15923			0.126360
$293$	22.2555			1.30018			0.650089	−	0.759858i	$-0.274732\pi$
$293$	22.2555			1.30018			0.650089	+	0.759858i	$0.274732\pi$
$294$		0			0
$295$	−2.45356	−	4.24969i	−0.142852	−	0.247427i
$296$	−0.930708			−0.0540963
$297$		0			0
$298$	19.2465	+	33.3360i	1.11492	+	1.93110i
$299$	−0.236073	+	0.408890i	−0.0136524	+	0.0236467i
$300$		0			0
$301$	−4.88293	+	8.45747i	−0.281447	+	0.487481i
$302$	2.46438	−	4.26843i	0.141809	−	0.245621i
$303$		0			0
$304$	3.86646	−	16.4591i	0.221757	−	0.943992i
$305$	−5.20947			−0.298293
$306$		0			0
$307$	13.2964	−	23.0300i	0.758863	−	1.31439i	−0.184568	−	0.982820i	$-0.559088\pi$
$307$	13.2964	−	23.0300i	0.758863	−	1.31439i	0.943431	−	0.331570i	$-0.107578\pi$
$308$	−1.27747	−	2.21265i	−0.0727908	−	0.126077i
$309$		0			0
$310$	7.08089	+	12.2645i	0.402168	+	0.696575i
$311$	11.1894			0.634493			0.317247	−	0.948343i	$-0.397242\pi$
$311$	11.1894			0.634493			0.317247	+	0.948343i	$0.397242\pi$
$312$		0			0
$313$	−10.2321	−	17.7225i	−0.578351	−	1.00173i	−0.995669	−	0.0929730i	$-0.970363\pi$
$313$	−10.2321	−	17.7225i	−0.578351	−	1.00173i	0.417317	−	0.908761i	$-0.362970\pi$
$314$	9.27928	+	16.0722i	0.523660	+	0.907006i
$315$		0			0
$316$	11.8849			0.668577
$317$	7.76650	+	13.4520i	0.436210	+	0.755538i	0.997394	−	0.0721532i	$-0.0229870\pi$
$317$	7.76650	+	13.4520i	0.436210	+	0.755538i	−0.561183	+	0.827692i	$0.689654\pi$
$318$		0			0
$319$	−2.75877	−	4.77833i	−0.154462	−	0.267535i
$320$	4.23198	−	7.33001i	0.236575	−	0.409760i
$321$		0			0
$322$	2.93438			0.163526
$323$	1.40567	−	5.98374i	0.0782133	−	0.332944i
$324$		0			0
$325$	0.269815	−	0.467333i	0.0149666	−	0.0259230i
$326$	1.10495	−	1.91383i	0.0611976	−	0.105997i
$327$		0			0
$328$	0.517975	−	0.897159i	0.0286004	−	0.0495373i
$329$	−5.49138	−	9.51135i	−0.302750	−	0.524378i
$330$		0			0
$331$	−16.3063			−0.896278			−0.448139	−	0.893964i	$-0.647913\pi$
$331$	−16.3063			−0.896278			−0.448139	+	0.893964i	$0.647913\pi$
$332$	−3.72201	−	6.44670i	−0.204272	−	0.353809i
$333$		0			0
$334$	−25.1607			−1.37673
$335$	0.887063			0.0484654
$336$		0			0
$337$	−14.5604	+	25.2193i	−0.793155	+	1.37378i	0.130849	+	0.991402i	$0.458230\pi$
$337$	−14.5604	+	25.2193i	−0.793155	+	1.37378i	−0.924004	+	0.382382i	$0.875104\pi$
$338$	−12.8020	−	22.1737i	−0.696337	−	1.20609i
$339$		0			0
$340$	1.45165	−	2.51434i	0.0787270	−	0.136359i
$341$	5.23999			0.283761
$342$		0			0
$343$	−18.6925			−1.00930
$344$	0.348001	−	0.602756i	0.0187630	−	0.0324984i
$345$		0			0
$346$	4.34278	+	7.52191i	0.233469	+	0.404381i
$347$	−1.36152	+	2.35823i	−0.0730904	+	0.126596i	−0.900254	−	0.435364i	$-0.856620\pi$
$347$	−1.36152	+	2.35823i	−0.0730904	+	0.126596i	0.827164	+	0.561961i	$0.189953\pi$
$348$		0			0
$349$	−5.70995			−0.305647			−0.152823	−	0.988254i	$-0.548837\pi$
$349$	−5.70995			−0.305647			−0.152823	+	0.988254i	$0.548837\pi$
$350$	−3.35379			−0.179268
$351$		0			0
$352$	−3.00105	−	5.19797i	−0.159956	−	0.277053i
$353$	27.6435			1.47131			0.735657	−	0.677354i	$-0.236874\pi$
$353$	27.6435			1.47131			0.735657	+	0.677354i	$0.236874\pi$
$354$		0			0
$355$	−1.41061	−	2.44324i	−0.0748672	−	0.129674i
$356$	0.956181	−	1.65615i	0.0506775	−	0.0877760i
$357$		0			0
$358$	−12.6962	+	21.9905i	−0.671017	+	1.16224i
$359$	−2.11633	+	3.66559i	−0.111696	+	0.193463i	−0.916454	−	0.400140i	$-0.868962\pi$
$359$	−2.11633	+	3.66559i	−0.111696	+	0.193463i	0.804758	+	0.593603i	$0.202295\pi$
$360$		0			0
$361$	15.8328	−	10.5035i	0.833305	−	0.552813i
$362$	−7.43550			−0.390801
$363$		0			0
$364$	0.924766	−	1.60174i	0.0484709	−	0.0839540i
$365$	−0.524369	−	0.908233i	−0.0274467	−	0.0475391i
$366$		0			0
$367$	−0.769642	−	1.33306i	−0.0401750	−	0.0695852i	0.845239	−	0.534389i	$-0.179458\pi$
$367$	−0.769642	−	1.33306i	−0.0401750	−	0.0695852i	−0.885414	+	0.464804i	$0.846125\pi$
$368$	3.39369			0.176908
$369$		0			0
$370$	7.90230	+	13.6872i	0.410821	+	0.711563i
$371$	−6.40935	−	11.1013i	−0.332757	−	0.576352i
$372$		0			0
$373$	0.490289			0.0253862			0.0126931	−	0.999919i	$-0.495960\pi$
$373$	0.490289			0.0253862			0.0126931	+	0.999919i	$0.495960\pi$
$374$	−1.05889	−	1.83405i	−0.0547538	−	0.0948364i
$375$		0			0
$376$	0.391366	+	0.677865i	0.0201831	+	0.0349582i
$377$	1.99708	−	3.45905i	0.102855	−	0.178150i
$378$		0			0
$379$	−34.6250			−1.77857			−0.889283	−	0.457358i	$-0.848796\pi$
$379$	−34.6250			−1.77857			−0.889283	+	0.457358i	$0.848796\pi$
$380$	8.59235	−	2.59093i	0.440778	−	0.132912i
$381$		0			0
$382$	−6.28208	+	10.8809i	−0.321419	+	0.556714i
$383$	11.9762	−	20.7433i	0.611953	−	1.05993i	−0.378958	−	0.925414i	$-0.623717\pi$
$383$	11.9762	−	20.7433i	0.611953	−	1.05993i	0.990911	−	0.134520i	$-0.0429492\pi$
$384$		0			0
$385$	−0.620468	+	1.07468i	−0.0316220	+	0.0547708i
$386$	−19.0643	−	33.0203i	−0.970347	−	1.68069i
$387$		0			0
$388$	0.0378881			0.00192348
$389$	8.75864	+	15.1704i	0.444081	+	0.769171i	0.997988	−	0.0634081i	$-0.0201970\pi$
$389$	8.75864	+	15.1704i	0.444081	+	0.769171i	−0.553907	+	0.832579i	$0.686864\pi$
$390$		0			0
$391$	1.23379			0.0623952
$392$	0.501710			0.0253402
$393$		0			0
$394$	6.90164	−	11.9540i	0.347699	−	0.602233i
$395$	−2.88624	−	4.99911i	−0.145222	−	0.251532i
$396$		0			0
$397$	−4.44472	+	7.69848i	−0.223074	+	0.386376i	−0.955740	−	0.294213i	$-0.904943\pi$
$397$	−4.44472	+	7.69848i	−0.223074	+	0.386376i	0.732666	+	0.680589i	$0.238276\pi$
$398$	−40.2700			−2.01855
$399$		0			0
$400$	−3.87876			−0.193938
$401$	18.1428	−	31.4242i	0.906007	−	1.56925i	0.0864475	−	0.996256i	$-0.472449\pi$
$401$	18.1428	−	31.4242i	0.906007	−	1.56925i	0.819560	−	0.572994i	$-0.194218\pi$
$402$		0			0
$403$	1.89662	+	3.28504i	0.0944774	+	0.163640i
$404$	16.3419	−	28.3050i	0.813041	−	1.40823i
$405$		0			0
$406$	−24.8237			−1.23198
$407$	5.84785			0.289867
$408$		0			0
$409$	8.69791	+	15.0652i	0.430084	+	0.744928i	0.996880	−	0.0789306i	$-0.0251505\pi$
$409$	8.69791	+	15.0652i	0.430084	+	0.744928i	−0.566796	+	0.823858i	$0.691817\pi$
$410$	−17.5917			−0.868794
$411$		0			0
$412$	8.81018	+	15.2597i	0.434046	+	0.751790i
$413$	−4.08441	+	7.07441i	−0.200981	+	0.348109i
$414$		0			0
$415$	−1.80777	+	3.13116i	−0.0887402	+	0.153702i
$416$	2.17247	−	3.76282i	0.106514	−	0.184487i
$417$		0			0
$418$	1.49707	−	6.37283i	0.0732240	−	0.311705i
$419$	−30.9259			−1.51083			−0.755414	−	0.655248i	$-0.772564\pi$
$419$	−30.9259			−1.51083			−0.755414	+	0.655248i	$0.772564\pi$
$420$		0			0
$421$	5.81345	−	10.0692i	0.283330	−	0.490742i	−0.688873	−	0.724882i	$-0.741894\pi$
$421$	5.81345	−	10.0692i	0.283330	−	0.490742i	0.972203	+	0.234140i	$0.0752275\pi$
$422$	14.9378	+	25.8730i	0.727160	+	1.25948i
$423$		0			0
$424$	0.456788	+	0.791180i	0.0221836	+	0.0384231i
$425$	−1.41013			−0.0684015
$426$		0			0
$427$	4.33607	+	7.51029i	0.209837	+	0.363449i
$428$	11.9938	+	20.7739i	0.579744	+	1.00415i
$429$		0			0
$430$	−11.8190			−0.569963
$431$	−12.2150	−	21.1569i	−0.588374	−	1.01909i	−0.994446	−	0.105252i	$-0.966435\pi$
$431$	−12.2150	−	21.1569i	−0.588374	−	1.01909i	0.406072	−	0.913841i	$-0.366898\pi$
$432$		0			0
$433$	−10.4503	−	18.1004i	−0.502208	−	0.869851i	−0.999997	−	0.00255203i	$-0.999188\pi$
$433$	−10.4503	−	18.1004i	−0.502208	−	0.869851i	0.497788	−	0.867299i	$-0.334146\pi$
$434$	11.7875	−	20.4165i	0.565817	−	0.980023i
$435$		0			0
$436$	−4.61498			−0.221017
$437$	2.77922	+	2.61168i	0.132948	+	0.124934i
$438$		0			0
$439$	−8.21019	+	14.2205i	−0.391851	+	0.678706i	−0.992694	−	0.120661i	$-0.961498\pi$
$439$	−8.21019	+	14.2205i	−0.391851	+	0.678706i	0.600843	+	0.799367i	$0.294832\pi$
$440$	0.0442201	−	0.0765915i	0.00210811	−	0.00365136i
$441$		0			0
$442$	0.766531	−	1.32767i	0.0364602	−	0.0631509i
$443$	−1.32291	−	2.29135i	−0.0628533	−	0.108865i	0.832886	−	0.553444i	$-0.186687\pi$
$443$	−1.32291	−	2.29135i	−0.0628533	−	0.108865i	−0.895740	+	0.444579i	$0.853353\pi$
$444$		0			0
$445$	−0.928832			−0.0440309
$446$	−19.0263	−	32.9545i	−0.900920	−	1.56044i
$447$		0			0
$448$	−14.0899			−0.665683
$449$	8.17261			0.385689			0.192845	−	0.981229i	$-0.438229\pi$
$449$	8.17261			0.385689			0.192845	+	0.981229i	$0.438229\pi$
$450$		0			0
$451$	−3.25455	+	5.63705i	−0.153251	+	0.265438i
$452$	−9.20717	−	15.9473i	−0.433069	−	0.750097i
$453$		0			0
$454$	15.8476	−	27.4488i	0.743763	−	1.28823i
$455$	−0.898315			−0.0421137
$456$		0			0
$457$	−8.94164			−0.418272			−0.209136	−	0.977887i	$-0.567065\pi$
$457$	−8.94164			−0.418272			−0.209136	+	0.977887i	$0.567065\pi$
$458$	−19.1814	+	33.2232i	−0.896288	+	1.55242i
$459$		0			0
$460$	0.900704	+	1.56007i	0.0419956	+	0.0727384i
$461$	8.82526	−	15.2858i	0.411033	−	0.711930i	−0.583970	−	0.811775i	$-0.698501\pi$
$461$	8.82526	−	15.2858i	0.411033	−	0.711930i	0.995003	+	0.0998450i	$0.0318347\pi$
$462$		0			0
$463$	−20.8670			−0.969771			−0.484885	−	0.874578i	$-0.661139\pi$
$463$	−20.8670			−0.969771			−0.484885	+	0.874578i	$0.661139\pi$
$464$	−28.7093			−1.33279
$465$		0			0
$466$	1.19760	+	2.07431i	0.0554779	+	0.0960905i
$467$	−24.4820			−1.13289			−0.566445	−	0.824099i	$-0.691682\pi$
$467$	−24.4820			−1.13289			−0.566445	+	0.824099i	$0.691682\pi$
$468$		0			0
$469$	−0.738341	−	1.27884i	−0.0340934	−	0.0590515i
$470$	6.64588	−	11.5110i	0.306552	−	0.530963i
$471$		0			0
$472$	0.291092	−	0.504186i	0.0133986	−	0.0232071i
$473$	−2.18657	+	3.78725i	−0.100539	+	0.174138i
$474$		0			0
$475$	−3.17646	−	2.98497i	−0.145746	−	0.136960i
$476$	−4.83310			−0.221525
$477$		0			0
$478$	−26.1349	+	45.2669i	−1.19538	+	2.07046i
$479$	−1.65218	−	2.86165i	−0.0754898	−	0.130752i	0.825809	−	0.563949i	$-0.190719\pi$
$479$	−1.65218	−	2.86165i	−0.0754898	−	0.130752i	−0.901299	+	0.433197i	$0.857385\pi$
$480$		0			0
$481$	2.11664	+	3.66612i	0.0965103	+	0.167161i
$482$	18.2454			0.831057
$483$		0			0
$484$	10.7518	+	18.6227i	0.488720	+	0.846487i
$485$	−0.00920110	−	0.0159368i	−0.000417800	−	0.000723652i
$486$		0			0
$487$	32.6648			1.48019			0.740093	−	0.672505i	$-0.234782\pi$
$487$	32.6648			1.48019			0.740093	+	0.672505i	$0.234782\pi$
$488$	−0.309028	−	0.535252i	−0.0139890	−	0.0242297i
$489$		0			0
$490$	−4.25983	−	7.37824i	−0.192439	−	0.333315i
$491$	6.51626	−	11.2865i	0.294075	−	0.509352i	−0.680694	−	0.732567i	$-0.738322\pi$
$491$	6.51626	−	11.2865i	0.294075	−	0.509352i	0.974769	+	0.223215i	$0.0716552\pi$
$492$		0			0
$493$	−10.4373			−0.470074
$494$	4.53710	−	1.36811i	0.204134	−	0.0615544i
$495$		0			0
$496$	13.6325	−	23.6122i	0.612119	−	1.06022i
$497$	−2.34822	+	4.06723i	−0.105332	+	0.182440i
$498$		0			0
$499$	−15.1552	+	26.2496i	−0.678441	+	1.17509i	0.297009	+	0.954875i	$0.404011\pi$
$499$	−15.1552	+	26.2496i	−0.678441	+	1.17509i	−0.975450	+	0.220220i	$0.929323\pi$
$500$	−1.02944	−	1.78305i	−0.0460381	−	0.0797404i
$501$		0			0
$502$	5.90411			0.263513
$503$	−19.0259	−	32.9539i	−0.848324	−	1.46934i	−0.882703	−	0.469931i	$-0.844279\pi$
$503$	−19.0259	−	32.9539i	−0.848324	−	1.46934i	0.0343796	−	0.999409i	$-0.489054\pi$
$504$		0			0
$505$	−15.8745			−0.706406
$506$	1.31401			0.0584150
$507$		0			0
$508$	20.4091	−	35.3497i	0.905509	−	1.56839i
$509$	11.8719	+	20.5627i	0.526212	+	0.911427i	0.999534	+	0.0305368i	$0.00972167\pi$
$509$	11.8719	+	20.5627i	0.526212	+	0.911427i	−0.473321	+	0.880890i	$0.656945\pi$
$510$		0			0
$511$	−0.872910	+	1.51192i	−0.0386153	+	0.0668836i
$512$	−32.1509			−1.42088
$513$		0			0
$514$	21.8606			0.964228
$515$	4.27910	−	7.41161i	0.188559	−	0.326595i
$516$		0			0
$517$	−2.45904	−	4.25918i	−0.108148	−	0.187318i
$518$	13.1549	−	22.7849i	0.577991	−	1.00111i
$519$		0			0
$520$	0.0640221			0.00280755
$521$	29.8052			1.30579			0.652896	−	0.757448i	$-0.273554\pi$
$521$	29.8052			1.30579			0.652896	+	0.757448i	$0.273554\pi$
$522$		0			0
$523$	−2.97546	−	5.15365i	−0.130108	−	0.225353i	0.793610	−	0.608427i	$-0.208199\pi$
$523$	−2.97546	−	5.15365i	−0.130108	−	0.225353i	−0.923718	+	0.383073i	$0.874866\pi$
$524$	3.78002			0.165131
$525$		0			0
$526$	16.9816	+	29.4130i	0.740433	+	1.28247i
$527$	4.95615	−	8.58431i	0.215893	−	0.373938i
$528$		0			0
$529$	11.1172	−	19.2556i	0.483358	−	0.837201i
$530$	7.75684	−	13.4352i	0.336936	−	0.583589i
$531$		0			0
$532$	−10.8870	−	10.2307i	−0.472013	−	0.443558i
$533$	−4.71196			−0.204097
$534$		0			0
$535$	5.82539	−	10.0899i	0.251854	−	0.436223i
$536$	0.0526209	+	0.0911420i	0.00227288	+	0.00393674i
$537$		0			0
$538$	29.7629	+	51.5509i	1.28317	+	2.22252i
$539$	−3.15235			−0.135781
$540$		0			0
$541$	−2.20614	−	3.82115i	−0.0948496	−	0.164284i	0.814696	−	0.579888i	$-0.196904\pi$
$541$	−2.20614	−	3.82115i	−0.0948496	−	0.164284i	−0.909546	+	0.415604i	$0.863570\pi$
$542$	21.4005	+	37.0668i	0.919231	+	1.59216i
$543$		0			0
$544$	−11.3540			−0.486797
$545$	1.12075	+	1.94119i	0.0480075	+	0.0831514i
$546$		0			0
$547$	−22.2986	−	38.6224i	−0.953421	−	1.65137i	−0.737942	−	0.674865i	$-0.764202\pi$
$547$	−22.2986	−	38.6224i	−0.953421	−	1.65137i	−0.215479	−	0.976508i	$-0.569131\pi$
$548$	−4.07854	+	7.06424i	−0.174227	+	0.301769i
$549$		0			0
$550$	−1.50183			−0.0640381
$551$	−23.5111	−	22.0938i	−1.00161	−	0.941227i
$552$		0			0
$553$	−4.80468	+	8.32195i	−0.204316	+	0.353885i
$554$	−28.5736	+	49.4910i	−1.21398	+	2.10267i
$555$		0			0
$556$	1.83045	−	3.17043i	0.0776282	−	0.134456i
$557$	6.46476	+	11.1973i	0.273921	+	0.474444i	0.969862	−	0.243654i	$-0.0783461\pi$
$557$	6.46476	+	11.1973i	0.273921	+	0.474444i	−0.695942	+	0.718098i	$0.745013\pi$
$558$		0			0
$559$	−3.16573			−0.133896
$560$	3.22846	+	5.59185i	0.136427	+	0.236299i
$561$		0			0
$562$	−35.4948			−1.49726
$563$	31.7105			1.33644			0.668219	−	0.743964i	$-0.267057\pi$
$563$	31.7105			1.33644			0.668219	+	0.743964i	$0.267057\pi$
$564$		0			0
$565$	−4.47191	+	7.74558i	−0.188135	+	0.325859i
$566$	−17.2164	−	29.8196i	−0.723659	−	1.25341i
$567$		0			0
$568$	0.167355	−	0.289868i	0.00702207	−	0.0121626i
$569$	28.4382			1.19219			0.596095	−	0.802914i	$-0.296718\pi$
$569$	28.4382			1.19219			0.596095	+	0.802914i	$0.296718\pi$
$570$		0			0
$571$	−7.32088			−0.306369			−0.153185	−	0.988198i	$-0.548953\pi$
$571$	−7.32088			−0.306369			−0.153185	+	0.988198i	$0.548953\pi$
$572$	0.414110	−	0.717259i	0.0173148	−	0.0299901i
$573$		0			0
$574$	14.6424	+	25.3613i	0.611161	+	1.05856i
$575$	0.437471	−	0.757722i	0.0182438	−	0.0315992i
$576$		0			0
$577$	−13.8157			−0.575156			−0.287578	−	0.957757i	$-0.592850\pi$
$577$	−13.8157			−0.575156			−0.287578	+	0.957757i	$0.592850\pi$
$578$	30.2432			1.25795
$579$		0			0
$580$	−7.61960	−	13.1975i	−0.316387	−	0.547998i
$581$	6.01876			0.249700
$582$		0			0
$583$	−2.87010	−	4.97116i	−0.118868	−	0.205885i
$584$	0.0622115	−	0.107753i	0.00257433	−	0.00445887i
$585$		0			0
$586$	22.4187	−	38.8303i	0.926108	−	1.60407i
$587$	−9.94338	+	17.2224i	−0.410407	+	0.710846i	−0.994934	−	0.100528i	$-0.967947\pi$
$587$	−9.94338	+	17.2224i	−0.410407	+	0.710846i	0.584527	+	0.811374i	$0.301280\pi$
$588$		0			0
$589$	29.3355	−	8.84580i	1.20875	−	0.364485i
$590$	−9.88622			−0.407009
$591$		0			0
$592$	15.2140	−	26.3514i	0.625290	−	1.08303i
$593$	4.92452	+	8.52952i	0.202226	+	0.350266i	0.949245	−	0.314537i	$-0.101849\pi$
$593$	4.92452	+	8.52952i	0.202226	+	0.350266i	−0.747019	+	0.664802i	$0.768516\pi$
$594$		0			0
$595$	1.17372	+	2.03294i	0.0481177	+	0.0833423i
$596$	39.3380			1.61135
$597$		0			0
$598$	0.475608	+	0.823777i	0.0194491	+	0.0336867i
$599$	3.58661	+	6.21220i	0.146545	+	0.253823i	0.929948	−	0.367690i	$-0.119851\pi$
$599$	3.58661	+	6.21220i	0.146545	+	0.253823i	−0.783403	+	0.621514i	$0.786518\pi$
$600$		0			0
$601$	39.2040			1.59916			0.799582	−	0.600557i	$-0.205054\pi$
$601$	39.2040			1.59916			0.799582	+	0.600557i	$0.205054\pi$
$602$	9.83747	+	17.0390i	0.400946	+	0.694458i
$603$		0			0
$604$	−2.51847	−	4.36212i	−0.102475	−	0.177492i
$605$	5.22215	−	9.04504i	0.212311	−	0.367733i
$606$		0			0
$607$	44.1741			1.79297			0.896485	−	0.443074i	$-0.146112\pi$
$607$	44.1741			1.79297			0.896485	+	0.443074i	$0.146112\pi$
$608$	−25.5759	−	24.0341i	−1.03724	−	0.974711i
$609$		0			0
$610$	−5.24768	+	9.08924i	−0.212472	+	0.368013i
$611$	1.78010	−	3.08323i	0.0720152	−	0.124734i
$612$		0			0
$613$	3.75407	−	6.50224i	0.151625	−	0.262623i	−0.780200	−	0.625531i	$-0.784883\pi$
$613$	3.75407	−	6.50224i	0.151625	−	0.262623i	0.931825	+	0.362907i	$0.118216\pi$
$614$	−26.7877	−	46.3977i	−1.08107	−	1.87246i
$615$		0			0
$616$	−0.147225			−0.00593188
$617$	−7.53673	−	13.0540i	−0.303417	−	0.525534i	0.673490	−	0.739196i	$-0.264794\pi$
$617$	−7.53673	−	13.0540i	−0.303417	−	0.525534i	−0.976908	+	0.213662i	$0.931461\pi$
$618$		0			0
$619$	43.5536			1.75057			0.875283	−	0.483610i	$-0.160675\pi$
$619$	43.5536			1.75057			0.875283	+	0.483610i	$0.160675\pi$
$620$	14.4726			0.581234
$621$		0			0
$622$	11.2715	−	19.5228i	0.451945	−	0.782792i
$623$	0.773108	+	1.33906i	0.0309739	+	0.0536484i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$	−41.2285			−1.64782
$627$		0			0
$628$	18.9659			0.756822
$629$	5.53108	−	9.58012i	0.220539	−	0.381984i
$630$		0			0
$631$	−13.9200	−	24.1101i	−0.554146	−	0.959809i	−0.997969	−	0.0636948i	$-0.979712\pi$
$631$	−13.9200	−	24.1101i	−0.554146	−	0.959809i	0.443823	−	0.896114i	$-0.353622\pi$
$632$	0.342425	−	0.593098i	0.0136209	−	0.0235922i
$633$		0			0
$634$	31.2939			1.24284
$635$	−19.8254			−0.786747
$636$		0			0
$637$	−1.14100	−	1.97627i	−0.0452080	−	0.0783025i
$638$	−11.1160			−0.440088
$639$		0			0
$640$	−0.474357	−	0.821611i	−0.0187506	−	0.0324770i
$641$	−2.21833	+	3.84226i	−0.0876187	+	0.151760i	−0.906504	−	0.422197i	$-0.861259\pi$
$641$	−2.21833	+	3.84226i	−0.0876187	+	0.151760i	0.818885	+	0.573957i	$0.194592\pi$
$642$		0			0
$643$	3.32952	−	5.76690i	0.131304	−	0.227425i	−0.792876	−	0.609383i	$-0.791417\pi$
$643$	3.32952	−	5.76690i	0.131304	−	0.227425i	0.924179	+	0.381959i	$0.124750\pi$
$644$	1.49939	−	2.59702i	0.0590843	−	0.102337i
$645$		0			0
$646$	−9.02418	−	8.48017i	−0.355052	−	0.333648i
$647$	1.27795			0.0502415			0.0251208	−	0.999684i	$-0.492003\pi$
$647$	1.27795			0.0502415			0.0251208	+	0.999684i	$0.492003\pi$
$648$		0			0
$649$	−1.82900	+	3.16792i	−0.0717944	+	0.124352i
$650$	−0.543588	−	0.941522i	−0.0213213	−	0.0369295i
$651$		0			0
$652$	−1.12920	−	1.95584i	−0.0442231	−	0.0765966i
$653$	45.2709			1.77159			0.885795	−	0.464078i	$-0.153614\pi$
$653$	45.2709			1.77159			0.885795	+	0.464078i	$0.153614\pi$
$654$		0			0
$655$	−0.917977	−	1.58998i	−0.0358683	−	0.0621257i
$656$	16.9343	+	29.3311i	0.661174	+	1.14519i
$657$		0			0
$658$	−22.1266			−0.862586
$659$	6.80538	+	11.7873i	0.265100	+	0.459167i	0.967590	−	0.252527i	$-0.0812617\pi$
$659$	6.80538	+	11.7873i	0.265100	+	0.459167i	−0.702490	+	0.711694i	$0.747928\pi$
$660$		0			0
$661$	14.5259	+	25.1596i	0.564991	+	0.978594i	0.997050	+	0.0767486i	$0.0244539\pi$
$661$	14.5259	+	25.1596i	0.564991	+	0.978594i	−0.432059	+	0.901845i	$0.642213\pi$
$662$	−16.4259	+	28.4506i	−0.638412	+	1.10576i
$663$		0			0
$664$	−0.428951			−0.0166465
$665$	−1.65941	+	7.06391i	−0.0643493	+	0.273927i
$666$		0			0
$667$	3.23802	−	5.60841i	0.125376	−	0.217158i
$668$	−12.8565	+	22.2681i	−0.497432	+	0.861578i
$669$		0			0
$670$	0.893568	−	1.54771i	0.0345216	−	0.0597931i
$671$	1.94169	+	3.36311i	0.0749581	+	0.129831i
$672$		0			0
$673$	−13.4534			−0.518591			−0.259296	−	0.965798i	$-0.583490\pi$
$673$	−13.4534			−0.518591			−0.259296	+	0.965798i	$0.583490\pi$
$674$	29.3344	+	50.8086i	1.12992	+	1.95707i
$675$		0			0
$676$	−26.1660			−1.00638
$677$	48.1048			1.84882			0.924409	−	0.381404i	$-0.124559\pi$
$677$	48.1048			1.84882			0.924409	+	0.381404i	$0.124559\pi$
$678$		0			0
$679$	−0.0153170	+	0.0265298i	−0.000587811	+	0.00101812i
$680$	−0.0836496	−	0.144885i	−0.00320782	−	0.00555610i
$681$		0			0
$682$	5.27842	−	9.14250i	0.202121	−	0.350084i
$683$	−31.7434			−1.21463			−0.607314	−	0.794462i	$-0.707753\pi$
$683$	−31.7434			−1.21463			−0.607314	+	0.794462i	$0.707753\pi$
$684$		0			0
$685$	3.96188			0.151376
$686$	−18.8296	+	32.6138i	−0.718916	+	1.24520i
$687$		0			0
$688$	11.3773	+	19.7061i	0.433756	+	0.751287i
$689$	2.07767	−	3.59864i	0.0791530	−	0.137097i
$690$		0			0
$691$	5.63348			0.214308			0.107154	−	0.994242i	$-0.465826\pi$
$691$	5.63348			0.214308			0.107154	+	0.994242i	$0.465826\pi$
$692$	8.87620			0.337422
$693$		0			0
$694$	2.74302	+	4.75105i	0.104124	+	0.180347i
$695$	−1.77809			−0.0674469
$696$		0			0
$697$	6.15652	+	10.6634i	0.233195	+	0.403905i
$698$	−5.75183	+	9.96246i	−0.217710	+	0.377085i
$699$		0			0
$700$	−1.71370	+	2.96822i	−0.0647719	+	0.112188i
$701$	−14.9878	+	25.9596i	−0.566080	+	0.980480i	0.430868	+	0.902415i	$0.358207\pi$
$701$	−14.9878	+	25.9596i	−0.566080	+	0.980480i	−0.996948	+	0.0780649i	$0.975126\pi$
$702$		0			0
$703$	32.7385	−	9.87195i	1.23476	−	0.372328i
$704$	−6.30943			−0.237796
$705$		0			0
$706$	27.8462	−	48.2311i	1.04801	−	1.81520i
$707$	13.2130	+	22.8857i	0.496928	+	0.860704i
$708$		0			0
$709$	−1.14344	−	1.98050i	−0.0429429	−	0.0743793i	0.843755	−	0.536728i	$-0.180340\pi$
$709$	−1.14344	−	1.98050i	−0.0429429	−	0.0743793i	−0.886698	+	0.462349i	$0.847007\pi$
$710$	−5.68380			−0.213309
$711$		0			0
$712$	−0.0550987	−	0.0954337i	−0.00206491	−	0.00357653i
$713$	3.07513	+	5.32628i	0.115165	+	0.199471i
$714$		0			0
$715$	−0.402265			−0.0150439
$716$	12.9749	+	22.4732i	0.484895	+	0.839863i
$717$		0			0
$718$	4.26370	+	7.38495i	0.159120	+	0.275604i
$719$	22.1981	−	38.4483i	0.827850	−	1.43388i	−0.0718716	−	0.997414i	$-0.522897\pi$
$719$	22.1981	−	38.4483i	0.827850	−	1.43388i	0.899722	−	0.436464i	$-0.143769\pi$
$720$		0			0
$721$	−14.2467			−0.530575
$722$	−2.37702	−	38.2048i	−0.0884636	−	1.42184i
$723$		0			0
$724$	−3.79935	+	6.58067i	−0.141202	+	0.244569i
$725$	−3.70083	+	6.41003i	−0.137446	+	0.238063i
$726$		0			0
$727$	4.39956	−	7.62026i	0.163171	−	0.282620i	−0.772834	−	0.634609i	$-0.781161\pi$
$727$	4.39956	−	7.62026i	0.163171	−	0.282620i	0.936004	+	0.351989i	$0.114495\pi$
$728$	−0.0532884	−	0.0922982i	−0.00197500	−	0.00342080i
$729$		0			0
$730$	−2.11286			−0.0782004
$731$	4.13626	+	7.16421i	0.152985	+	0.264978i
$732$		0			0
$733$	28.5385			1.05409			0.527047	−	0.849836i	$-0.323299\pi$
$733$	28.5385			1.05409			0.527047	+	0.849836i	$0.323299\pi$
$734$	−3.10115			−0.114465
$735$		0			0
$736$	3.52238	−	6.10094i	0.129837	−	0.224884i
$737$	−0.330629	−	0.572666i	−0.0121789	−	0.0210944i
$738$		0			0
$739$	5.16208	−	8.94099i	0.189890	−	0.328900i	−0.755323	−	0.655352i	$-0.772520\pi$
$739$	5.16208	−	8.94099i	0.189890	−	0.328900i	0.945213	+	0.326453i	$0.105853\pi$
$740$	16.1515			0.593741
$741$		0			0
$742$	−25.8254			−0.948081
$743$	13.5295	−	23.4339i	0.496351	−	0.859705i	−0.503640	−	0.863914i	$-0.668006\pi$
$743$	13.5295	−	23.4339i	0.496351	−	0.859705i	0.999991	+	0.00420839i	$0.00133958\pi$
$744$		0			0
$745$	−9.55320	−	16.5466i	−0.350002	−	0.606222i
$746$	0.493885	−	0.855434i	0.0180824	−	0.0313196i
$747$		0			0
$748$	−2.16426			−0.0791332
$749$	−19.3949			−0.708675
$750$		0			0
$751$	25.9995	+	45.0325i	0.948735	+	1.64326i	0.748093	+	0.663593i	$0.230969\pi$
$751$	25.9995	+	45.0325i	0.948735	+	1.64326i	0.200642	+	0.979665i	$0.435697\pi$
$752$	−25.5901			−0.933173
$753$		0			0
$754$	−4.02346	−	6.96883i	−0.146526	−	0.253790i
$755$	−1.22322	+	2.11868i	−0.0445175	+	0.0771066i
$756$		0			0
$757$	−26.7487	+	46.3301i	−0.972199	+	1.68390i	−0.283313	+	0.959028i	$0.591433\pi$
$757$	−26.7487	+	46.3301i	−0.972199	+	1.68390i	−0.688886	+	0.724870i	$0.741900\pi$
$758$	−34.8789	+	60.4121i	−1.26686	+	2.19427i
$759$		0			0
$760$	0.118265	−	0.503438i	0.00428991	−	0.0182616i
$761$	33.8123			1.22569			0.612847	−	0.790201i	$-0.290024\pi$
$761$	33.8123			1.22569			0.612847	+	0.790201i	$0.290024\pi$
$762$		0			0
$763$	1.86569	−	3.23147i	0.0675426	−	0.116987i
$764$	6.41996	+	11.1197i	0.232266	+	0.402297i
$765$		0			0
$766$	−24.1280	−	41.7909i	−0.871780	−	1.50997i
$767$	−2.64803			−0.0956148
$768$		0			0
$769$	3.09718	+	5.36447i	0.111687	+	0.193448i	0.916451	−	0.400148i	$-0.131041\pi$
$769$	3.09718	+	5.36447i	0.111687	+	0.193448i	−0.804763	+	0.593596i	$0.797708\pi$
$770$	1.25004	+	2.16513i	0.0450482	+	0.0780257i
$771$		0			0
$772$	−38.9655			−1.40240
$773$	4.36013	+	7.55197i	0.156823	+	0.271626i	0.933721	−	0.358001i	$-0.116541\pi$
$773$	4.36013	+	7.55197i	0.156823	+	0.271626i	−0.776898	+	0.629626i	$0.783208\pi$
$774$		0			0
$775$	−3.51467	−	6.08758i	−0.126251	−	0.218672i
$776$	0.00109163	−	0.00189075i	3.91871e−5	−	6.78740e-5i
$777$		0			0
$778$	35.2915			1.26526
$779$	−8.70416	+	37.0525i	−0.311859	+	1.32754i
$780$		0			0
$781$	−1.05153	+	1.82130i	−0.0376267	+	0.0651714i
$782$	1.24283	−	2.15265i	0.0444437	−	0.0769787i
$783$		0

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:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00733 - 1.74475i) q^{2} +(-1.02944 - 1.78305i) q^{4} +(-0.500000 + 0.866025i) q^{5} +1.66469 q^{7} -0.118641 q^{8} +O(q^{10})\)	\(q+(1.00733 - 1.74475i) q^{2} +(-1.02944 - 1.78305i) q^{4} +(-0.500000 + 0.866025i) q^{5} +1.66469 q^{7} -0.118641 q^{8} +(1.00733 + 1.74475i) q^{10} +0.745446 q^{11} +(0.269815 + 0.467333i) q^{13} +(1.67690 - 2.90447i) q^{14} +(1.93938 - 3.35910i) q^{16} +(0.705067 - 1.22121i) q^{17} +(4.17330 - 1.25841i) q^{19} +2.05889 q^{20} +(0.750913 - 1.30062i) q^{22} +(0.437471 + 0.757722i) q^{23} +(-0.500000 - 0.866025i) q^{25} +1.08718 q^{26} +(-1.71370 - 2.96822i) q^{28} +(-3.70083 - 6.41003i) q^{29} +7.02934 q^{31} +(-4.02584 - 6.97297i) q^{32} +(-1.42048 - 2.46034i) q^{34} +(-0.832344 + 1.44166i) q^{35} +7.84476 q^{37} +(2.00828 - 8.54902i) q^{38} +(0.0593204 - 0.102746i) q^{40} +(-4.36591 + 7.56198i) q^{41} +(-2.93324 + 5.08052i) q^{43} +(-0.767395 - 1.32917i) q^{44} +1.76272 q^{46} +(-3.29875 - 5.71360i) q^{47} -4.22881 q^{49} -2.01467 q^{50} +(0.555519 - 0.962187i) q^{52} +(-3.85018 - 6.66871i) q^{53} +(-0.372723 + 0.645575i) q^{55} -0.197500 q^{56} -14.9119 q^{58} +(-2.45356 + 4.24969i) q^{59} +(2.60473 + 4.51153i) q^{61} +(7.08089 - 12.2645i) q^{62} -8.46397 q^{64} -0.539630 q^{65} +(-0.443531 - 0.768219i) q^{67} -2.90331 q^{68} +(1.67690 + 2.90447i) q^{70} +(-1.41061 + 2.44324i) q^{71} +(-0.524369 + 0.908233i) q^{73} +(7.90230 - 13.6872i) q^{74} +(-6.53999 - 6.14573i) q^{76} +1.24094 q^{77} +(-2.88624 + 4.99911i) q^{79} +(1.93938 + 3.35910i) q^{80} +(8.79587 + 15.2349i) q^{82} +3.61555 q^{83} +(0.705067 + 1.22121i) q^{85} +(5.90950 + 10.2356i) q^{86} -0.0884403 q^{88} +(0.464416 + 0.804392i) q^{89} +(0.449158 + 0.777964i) q^{91} +(0.900704 - 1.56007i) q^{92} -13.2918 q^{94} +(-0.996831 + 4.24339i) q^{95} +(-0.00920110 + 0.0159368i) q^{97} +(-4.25983 + 7.37824i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} - 5 q^{4} - 6 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 - 5 * q^4 - 6 * q^5 + 4 * q^7 + 12 * q^8	\( 12 q - 3 q^{2} - 5 q^{4} - 6 q^{5} + 4 q^{7} + 12 q^{8} - 3 q^{10} - 8 q^{13} - 10 q^{14} - 3 q^{16} - 4 q^{17} - 6 q^{19} + 10 q^{20} + 2 q^{23} - 6 q^{25} + 40 q^{26} - 26 q^{28} - 4 q^{29} + 24 q^{31} - 15 q^{32} + 7 q^{34} - 2 q^{35} + 29 q^{38} - 6 q^{40} - 12 q^{41} - 4 q^{43} - 6 q^{44} + 48 q^{46} - 6 q^{47} + 32 q^{49} + 6 q^{50} - 20 q^{52} - 26 q^{53} + 44 q^{56} - 20 q^{58} - 16 q^{59} + 20 q^{61} + 25 q^{62} + 28 q^{64} + 16 q^{65} - 12 q^{67} - 54 q^{68} - 10 q^{70} + 8 q^{71} - 4 q^{73} + 16 q^{74} - 66 q^{76} - 48 q^{77} - 12 q^{79} - 3 q^{80} + 26 q^{82} + 44 q^{83} - 4 q^{85} + 44 q^{86} - 32 q^{88} + 8 q^{89} + 2 q^{91} - 36 q^{92} - 14 q^{94} + 6 q^{95} + 30 q^{97} - 49 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 - 5 * q^4 - 6 * q^5 + 4 * q^7 + 12 * q^8 - 3 * q^10 - 8 * q^13 - 10 * q^14 - 3 * q^16 - 4 * q^17 - 6 * q^19 + 10 * q^20 + 2 * q^23 - 6 * q^25 + 40 * q^26 - 26 * q^28 - 4 * q^29 + 24 * q^31 - 15 * q^32 + 7 * q^34 - 2 * q^35 + 29 * q^38 - 6 * q^40 - 12 * q^41 - 4 * q^43 - 6 * q^44 + 48 * q^46 - 6 * q^47 + 32 * q^49 + 6 * q^50 - 20 * q^52 - 26 * q^53 + 44 * q^56 - 20 * q^58 - 16 * q^59 + 20 * q^61 + 25 * q^62 + 28 * q^64 + 16 * q^65 - 12 * q^67 - 54 * q^68 - 10 * q^70 + 8 * q^71 - 4 * q^73 + 16 * q^74 - 66 * q^76 - 48 * q^77 - 12 * q^79 - 3 * q^80 + 26 * q^82 + 44 * q^83 - 4 * q^85 + 44 * q^86 - 32 * q^88 + 8 * q^89 + 2 * q^91 - 36 * q^92 - 14 * q^94 + 6 * q^95 + 30 * q^97 - 49 * q^98

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