LMFDB - Embedded newform 378.2.f.d.253.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(127,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("378.127");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.f (of order $3$, degree $2$, not 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:	$3.01834519640$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			253.1
Root			$1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	378.253
Dual form			378.2.f.d.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.724745 + 1.25529i) q^{5} +(0.500000 + 0.866025i) q^{7} -1.00000 q^{8} +O(q^{10})$	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.724745 + 1.25529i) q^{5} +(0.500000 + 0.866025i) q^{7} -1.00000 q^{8} -1.44949 q^{10} +(1.00000 + 1.73205i) q^{11} +(-2.44949 + 4.24264i) q^{13} +(-0.500000 + 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} -2.00000 q^{17} +2.55051 q^{19} +(-0.724745 - 1.25529i) q^{20} +(-1.00000 + 1.73205i) q^{22} +(-0.500000 + 0.866025i) q^{23} +(1.44949 + 2.51059i) q^{25} -4.89898 q^{26} -1.00000 q^{28} +(-3.44949 - 5.97469i) q^{29} +(-3.00000 + 5.19615i) q^{31} +(0.500000 - 0.866025i) q^{32} +(-1.00000 - 1.73205i) q^{34} -1.44949 q^{35} +11.7980 q^{37} +(1.27526 + 2.20881i) q^{38} +(0.724745 - 1.25529i) q^{40} +(4.89898 - 8.48528i) q^{41} +(-3.44949 - 5.97469i) q^{43} -2.00000 q^{44} -1.00000 q^{46} +(4.89898 + 8.48528i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(-1.44949 + 2.51059i) q^{50} +(-2.44949 - 4.24264i) q^{52} +10.8990 q^{53} -2.89898 q^{55} +(-0.500000 - 0.866025i) q^{56} +(3.44949 - 5.97469i) q^{58} +(-1.00000 + 1.73205i) q^{59} +(-3.27526 - 5.67291i) q^{61} -6.00000 q^{62} +1.00000 q^{64} +(-3.55051 - 6.14966i) q^{65} +(6.44949 - 11.1708i) q^{67} +(1.00000 - 1.73205i) q^{68} +(-0.724745 - 1.25529i) q^{70} -0.101021 q^{71} -6.89898 q^{73} +(5.89898 + 10.2173i) q^{74} +(-1.27526 + 2.20881i) q^{76} +(-1.00000 + 1.73205i) q^{77} +(0.949490 + 1.64456i) q^{79} +1.44949 q^{80} +9.79796 q^{82} +(1.00000 + 1.73205i) q^{83} +(1.44949 - 2.51059i) q^{85} +(3.44949 - 5.97469i) q^{86} +(-1.00000 - 1.73205i) q^{88} +16.8990 q^{89} -4.89898 q^{91} +(-0.500000 - 0.866025i) q^{92} +(-4.89898 + 8.48528i) q^{94} +(-1.84847 + 3.20164i) q^{95} +(-1.44949 - 2.51059i) q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 4 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^7 - 4 * q^8	$ 4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 4 q^{8} + 4 q^{10} + 4 q^{11} - 2 q^{14} - 2 q^{16} - 8 q^{17} + 20 q^{19} + 2 q^{20} - 4 q^{22} - 2 q^{23} - 4 q^{25} - 4 q^{28} - 4 q^{29} - 12 q^{31} + 2 q^{32} - 4 q^{34} + 4 q^{35} + 8 q^{37} + 10 q^{38} - 2 q^{40} - 4 q^{43} - 8 q^{44} - 4 q^{46} - 2 q^{49} + 4 q^{50} + 24 q^{53} + 8 q^{55} - 2 q^{56} + 4 q^{58} - 4 q^{59} - 18 q^{61} - 24 q^{62} + 4 q^{64} - 24 q^{65} + 16 q^{67} + 4 q^{68} + 2 q^{70} - 20 q^{71} - 8 q^{73} + 4 q^{74} - 10 q^{76} - 4 q^{77} - 6 q^{79} - 4 q^{80} + 4 q^{83} - 4 q^{85} + 4 q^{86} - 4 q^{88} + 48 q^{89} - 2 q^{92} + 22 q^{95} + 4 q^{97} - 4 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^7 - 4 * q^8 + 4 * q^10 + 4 * q^11 - 2 * q^14 - 2 * q^16 - 8 * q^17 + 20 * q^19 + 2 * q^20 - 4 * q^22 - 2 * q^23 - 4 * q^25 - 4 * q^28 - 4 * q^29 - 12 * q^31 + 2 * q^32 - 4 * q^34 + 4 * q^35 + 8 * q^37 + 10 * q^38 - 2 * q^40 - 4 * q^43 - 8 * q^44 - 4 * q^46 - 2 * q^49 + 4 * q^50 + 24 * q^53 + 8 * q^55 - 2 * q^56 + 4 * q^58 - 4 * q^59 - 18 * q^61 - 24 * q^62 + 4 * q^64 - 24 * q^65 + 16 * q^67 + 4 * q^68 + 2 * q^70 - 20 * q^71 - 8 * q^73 + 4 * q^74 - 10 * q^76 - 4 * q^77 - 6 * q^79 - 4 * q^80 + 4 * q^83 - 4 * q^85 + 4 * q^86 - 4 * q^88 + 48 * q^89 - 2 * q^92 + 22 * q^95 + 4 * q^97 - 4 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

Coefficient data

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

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

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

$2$	0.500000	+	0.866025i	0.353553	+	0.612372i
$2$	0.500000	+	0.866025i	0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$	−0.724745	+	1.25529i	−0.324116	+	0.561385i	−0.981333	−	0.192316i	$-0.938400\pi$
$5$	−0.724745	+	1.25529i	−0.324116	+	0.561385i	0.657217	+	0.753701i	$0.271733\pi$
$6$		0			0
$7$	0.500000	+	0.866025i	0.188982	+	0.327327i
$7$	0.500000	+	0.866025i	0.188982	+	0.327327i
$8$	−1.00000			−0.353553
$9$		0			0
$10$	−1.44949			−0.458369
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	0.976478	−	0.215615i	$-0.0691756\pi$
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	−0.674967	+	0.737848i	$0.735842\pi$
$12$		0			0
$13$	−2.44949	+	4.24264i	−0.679366	+	1.17670i	0.295806	+	0.955248i	$0.404412\pi$
$13$	−2.44949	+	4.24264i	−0.679366	+	1.17670i	−0.975172	+	0.221449i	$0.928921\pi$
$14$	−0.500000	+	0.866025i	−0.133631	+	0.231455i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$	2.55051			0.585127			0.292564	−	0.956246i	$-0.405492\pi$
$19$	2.55051			0.585127			0.292564	+	0.956246i	$0.405492\pi$
$20$	−0.724745	−	1.25529i	−0.162058	−	0.280692i
$21$		0			0
$22$	−1.00000	+	1.73205i	−0.213201	+	0.369274i
$23$	−0.500000	+	0.866025i	−0.104257	+	0.180579i	−0.913434	−	0.406986i	$-0.866580\pi$
$23$	−0.500000	+	0.866025i	−0.104257	+	0.180579i	0.809177	+	0.587565i	$0.199913\pi$
$24$		0			0
$25$	1.44949	+	2.51059i	0.289898	+	0.502118i
$26$	−4.89898			−0.960769
$27$		0			0
$28$	−1.00000			−0.188982
$29$	−3.44949	−	5.97469i	−0.640554	−	1.10947i	−0.985309	−	0.170780i	$-0.945371\pi$
$29$	−3.44949	−	5.97469i	−0.640554	−	1.10947i	0.344755	−	0.938693i	$-0.387962\pi$
$30$		0			0
$31$	−3.00000	+	5.19615i	−0.538816	+	0.933257i	0.460152	+	0.887840i	$0.347795\pi$
$31$	−3.00000	+	5.19615i	−0.538816	+	0.933257i	−0.998968	+	0.0454165i	$0.985539\pi$
$32$	0.500000	−	0.866025i	0.0883883	−	0.153093i
$33$		0			0
$34$	−1.00000	−	1.73205i	−0.171499	−	0.297044i
$35$	−1.44949			−0.245008
$36$		0			0
$37$	11.7980			1.93957			0.969786	−	0.243956i	$-0.0784453\pi$
$37$	11.7980			1.93957			0.969786	+	0.243956i	$0.0784453\pi$
$38$	1.27526	+	2.20881i	0.206874	+	0.358316i
$39$		0			0
$40$	0.724745	−	1.25529i	0.114592	−	0.198480i
$41$	4.89898	−	8.48528i	0.765092	−	1.32518i	−0.175106	−	0.984550i	$-0.556027\pi$
$41$	4.89898	−	8.48528i	0.765092	−	1.32518i	0.940198	−	0.340629i	$-0.110640\pi$
$42$		0			0
$43$	−3.44949	−	5.97469i	−0.526042	−	0.911132i	−0.999540	−	0.0303367i	$-0.990342\pi$
$43$	−3.44949	−	5.97469i	−0.526042	−	0.911132i	0.473497	−	0.880795i	$-0.342991\pi$
$44$	−2.00000			−0.301511
$45$		0			0
$46$	−1.00000			−0.147442
$47$	4.89898	+	8.48528i	0.714590	+	1.23771i	0.963118	+	0.269081i	$0.0867199\pi$
$47$	4.89898	+	8.48528i	0.714590	+	1.23771i	−0.248528	+	0.968625i	$0.579947\pi$
$48$		0			0
$49$	−0.500000	+	0.866025i	−0.0714286	+	0.123718i
$50$	−1.44949	+	2.51059i	−0.204989	+	0.355051i
$51$		0			0
$52$	−2.44949	−	4.24264i	−0.339683	−	0.588348i
$53$	10.8990			1.49709			0.748545	−	0.663084i	$-0.230753\pi$
$53$	10.8990			1.49709			0.748545	+	0.663084i	$0.230753\pi$
$54$		0			0
$55$	−2.89898			−0.390898
$56$	−0.500000	−	0.866025i	−0.0668153	−	0.115728i
$57$		0			0
$58$	3.44949	−	5.97469i	0.452940	−	0.784515i
$59$	−1.00000	+	1.73205i	−0.130189	+	0.225494i	−0.923749	−	0.382998i	$-0.874892\pi$
$59$	−1.00000	+	1.73205i	−0.130189	+	0.225494i	0.793560	+	0.608492i	$0.208225\pi$
$60$		0			0
$61$	−3.27526	−	5.67291i	−0.419353	−	0.726341i	0.576521	−	0.817082i	$-0.304410\pi$
$61$	−3.27526	−	5.67291i	−0.419353	−	0.726341i	−0.995875	+	0.0907408i	$0.971077\pi$
$62$	−6.00000			−0.762001
$63$		0			0
$64$	1.00000			0.125000
$65$	−3.55051	−	6.14966i	−0.440387	−	0.762772i
$66$		0			0
$67$	6.44949	−	11.1708i	0.787931	−	1.36474i	−0.139302	−	0.990250i	$-0.544486\pi$
$67$	6.44949	−	11.1708i	0.787931	−	1.36474i	0.927233	−	0.374486i	$-0.122181\pi$
$68$	1.00000	−	1.73205i	0.121268	−	0.210042i
$69$		0			0
$70$	−0.724745	−	1.25529i	−0.0866236	−	0.150036i
$71$	−0.101021			−0.0119889			−0.00599446	−	0.999982i	$-0.501908\pi$
$71$	−0.101021			−0.0119889			−0.00599446	+	0.999982i	$0.501908\pi$
$72$		0			0
$73$	−6.89898			−0.807464			−0.403732	−	0.914877i	$-0.632287\pi$
$73$	−6.89898			−0.807464			−0.403732	+	0.914877i	$0.632287\pi$
$74$	5.89898	+	10.2173i	0.685742	+	1.18774i
$75$		0			0
$76$	−1.27526	+	2.20881i	−0.146282	+	0.253368i
$77$	−1.00000	+	1.73205i	−0.113961	+	0.197386i
$78$		0			0
$79$	0.949490	+	1.64456i	0.106826	+	0.185028i	0.914483	−	0.404625i	$-0.132598\pi$
$79$	0.949490	+	1.64456i	0.106826	+	0.185028i	−0.807657	+	0.589653i	$0.799265\pi$
$80$	1.44949			0.162058
$81$		0			0
$82$	9.79796			1.08200
$83$	1.00000	+	1.73205i	0.109764	+	0.190117i	0.915675	−	0.401920i	$-0.131657\pi$
$83$	1.00000	+	1.73205i	0.109764	+	0.190117i	−0.805910	+	0.592037i	$0.798324\pi$
$84$		0			0
$85$	1.44949	−	2.51059i	0.157219	−	0.272312i
$86$	3.44949	−	5.97469i	0.371968	−	0.644268i
$87$		0			0
$88$	−1.00000	−	1.73205i	−0.106600	−	0.184637i
$89$	16.8990			1.79129			0.895644	−	0.444771i	$-0.146715\pi$
$89$	16.8990			1.79129			0.895644	+	0.444771i	$0.146715\pi$
$90$		0			0
$91$	−4.89898			−0.513553
$92$	−0.500000	−	0.866025i	−0.0521286	−	0.0902894i
$93$		0			0
$94$	−4.89898	+	8.48528i	−0.505291	+	0.875190i
$95$	−1.84847	+	3.20164i	−0.189649	+	0.328482i
$96$		0			0
$97$	−1.44949	−	2.51059i	−0.147173	−	0.254912i	0.783008	−	0.622011i	$-0.213684\pi$
$97$	−1.44949	−	2.51059i	−0.147173	−	0.254912i	−0.930182	+	0.367099i	$0.880351\pi$
$98$	−1.00000			−0.101015
$99$		0			0
$100$	−2.89898			−0.289898
$101$	−8.62372	−	14.9367i	−0.858093	−	1.48626i	−0.873746	−	0.486383i	$-0.838316\pi$
$101$	−8.62372	−	14.9367i	−0.858093	−	1.48626i	0.0156533	−	0.999877i	$-0.495017\pi$
$102$		0			0
$103$	−7.00000	+	12.1244i	−0.689730	+	1.19465i	0.282194	+	0.959357i	$0.408938\pi$
$103$	−7.00000	+	12.1244i	−0.689730	+	1.19465i	−0.971925	+	0.235291i	$0.924396\pi$
$104$	2.44949	−	4.24264i	0.240192	−	0.416025i
$105$		0			0
$106$	5.44949	+	9.43879i	0.529301	+	0.916777i
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$	12.6969			1.21615			0.608073	−	0.793881i	$-0.291943\pi$
$109$	12.6969			1.21615			0.608073	+	0.793881i	$0.291943\pi$
$110$	−1.44949	−	2.51059i	−0.138203	−	0.239375i
$111$		0			0
$112$	0.500000	−	0.866025i	0.0472456	−	0.0818317i
$113$	−3.05051	+	5.28364i	−0.286968	+	0.497043i	−0.973084	−	0.230449i	$-0.925981\pi$
$113$	−3.05051	+	5.28364i	−0.286968	+	0.497043i	0.686117	+	0.727492i	$0.259314\pi$
$114$		0			0
$115$	−0.724745	−	1.25529i	−0.0675828	−	0.117057i
$116$	6.89898			0.640554
$117$		0			0
$118$	−2.00000			−0.184115
$119$	−1.00000	−	1.73205i	−0.0916698	−	0.158777i
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$	3.27526	−	5.67291i	0.296528	−	0.513601i
$123$		0			0
$124$	−3.00000	−	5.19615i	−0.269408	−	0.466628i
$125$	−11.4495			−1.02407
$126$		0			0
$127$	−3.00000			−0.266207			−0.133103	−	0.991102i	$-0.542494\pi$
$127$	−3.00000			−0.266207			−0.133103	+	0.991102i	$0.542494\pi$
$128$	0.500000	+	0.866025i	0.0441942	+	0.0765466i
$129$		0			0
$130$	3.55051	−	6.14966i	0.311400	−	0.539361i
$131$	−4.27526	+	7.40496i	−0.373531	+	0.646974i	−0.990106	−	0.140322i	$-0.955186\pi$
$131$	−4.27526	+	7.40496i	−0.373531	+	0.646974i	0.616575	+	0.787296i	$0.288520\pi$
$132$		0			0
$133$	1.27526	+	2.20881i	0.110579	+	0.191528i
$134$	12.8990			1.11430
$135$		0			0
$136$	2.00000			0.171499
$137$	3.89898	+	6.75323i	0.333112	+	0.576967i	0.983120	−	0.182960i	$-0.0585678\pi$
$137$	3.89898	+	6.75323i	0.333112	+	0.576967i	−0.650008	+	0.759927i	$0.725235\pi$
$138$		0			0
$139$	−2.27526	+	3.94086i	−0.192985	+	0.334259i	−0.946238	−	0.323471i	$-0.895150\pi$
$139$	−2.27526	+	3.94086i	−0.192985	+	0.334259i	0.753253	+	0.657730i	$0.228483\pi$
$140$	0.724745	−	1.25529i	0.0612521	−	0.106092i
$141$		0			0
$142$	−0.0505103	−	0.0874863i	−0.00423873	−	0.00734169i
$143$	−9.79796			−0.819346
$144$		0			0
$145$	10.0000			0.830455
$146$	−3.44949	−	5.97469i	−0.285482	−	0.494469i
$147$		0			0
$148$	−5.89898	+	10.2173i	−0.484893	+	0.839860i
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	−0.962348	−	0.271821i	$-0.912374\pi$
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	0.716578	+	0.697507i	$0.245707\pi$
$150$		0			0
$151$	2.50000	+	4.33013i	0.203447	+	0.352381i	0.949637	−	0.313353i	$-0.101452\pi$
$151$	2.50000	+	4.33013i	0.203447	+	0.352381i	−0.746190	+	0.665733i	$0.768119\pi$
$152$	−2.55051			−0.206874
$153$		0			0
$154$	−2.00000			−0.161165
$155$	−4.34847	−	7.53177i	−0.349277	−	0.604966i
$156$		0			0
$157$	4.17423	−	7.22999i	0.333140	−	0.577016i	−0.649986	−	0.759947i	$-0.725225\pi$
$157$	4.17423	−	7.22999i	0.333140	−	0.577016i	0.983126	+	0.182931i	$0.0585584\pi$
$158$	−0.949490	+	1.64456i	−0.0755373	+	0.130835i
$159$		0			0
$160$	0.724745	+	1.25529i	0.0572961	+	0.0992398i
$161$	−1.00000			−0.0788110
$162$		0			0
$163$	−19.7980			−1.55070			−0.775348	−	0.631534i	$-0.782425\pi$
$163$	−19.7980			−1.55070			−0.775348	+	0.631534i	$0.782425\pi$
$164$	4.89898	+	8.48528i	0.382546	+	0.662589i
$165$		0			0
$166$	−1.00000	+	1.73205i	−0.0776151	+	0.134433i
$167$	−5.34847	+	9.26382i	−0.413877	+	0.716856i	−0.995310	−	0.0967384i	$-0.969159\pi$
$167$	−5.34847	+	9.26382i	−0.413877	+	0.716856i	0.581433	+	0.813594i	$0.302492\pi$
$168$		0			0
$169$	−5.50000	−	9.52628i	−0.423077	−	0.732791i
$170$	2.89898			0.222342
$171$		0			0
$172$	6.89898			0.526042
$173$	−1.55051	−	2.68556i	−0.117883	−	0.204180i	0.801045	−	0.598604i	$-0.204277\pi$
$173$	−1.55051	−	2.68556i	−0.117883	−	0.204180i	−0.918929	+	0.394424i	$0.870944\pi$
$174$		0			0
$175$	−1.44949	+	2.51059i	−0.109571	+	0.189783i
$176$	1.00000	−	1.73205i	0.0753778	−	0.130558i
$177$		0			0
$178$	8.44949	+	14.6349i	0.633316	+	1.09694i
$179$	−20.6969			−1.54696			−0.773481	−	0.633820i	$-0.781486\pi$
$179$	−20.6969			−1.54696			−0.773481	+	0.633820i	$0.781486\pi$
$180$		0			0
$181$	−10.3485			−0.769196			−0.384598	−	0.923084i	$-0.625660\pi$
$181$	−10.3485			−0.769196			−0.384598	+	0.923084i	$0.625660\pi$
$182$	−2.44949	−	4.24264i	−0.181568	−	0.314485i
$183$		0			0
$184$	0.500000	−	0.866025i	0.0368605	−	0.0638442i
$185$	−8.55051	+	14.8099i	−0.628646	+	1.08885i
$186$		0			0
$187$	−2.00000	−	3.46410i	−0.146254	−	0.253320i
$188$	−9.79796			−0.714590
$189$		0			0
$190$	−3.69694			−0.268204
$191$	2.05051	+	3.55159i	0.148370	+	0.256984i	0.930625	−	0.365974i	$-0.119264\pi$
$191$	2.05051	+	3.55159i	0.148370	+	0.256984i	−0.782255	+	0.622958i	$0.785931\pi$
$192$		0			0
$193$	8.94949	−	15.5010i	0.644198	−	1.11578i	−0.340288	−	0.940321i	$-0.610524\pi$
$193$	8.94949	−	15.5010i	0.644198	−	1.11578i	0.984486	−	0.175463i	$-0.0561422\pi$
$194$	1.44949	−	2.51059i	0.104067	−	0.180250i
$195$		0			0
$196$	−0.500000	−	0.866025i	−0.0357143	−	0.0618590i
$197$	−16.6969			−1.18961			−0.594804	−	0.803871i	$-0.702770\pi$
$197$	−16.6969			−1.18961			−0.594804	+	0.803871i	$0.702770\pi$
$198$		0			0
$199$	−2.89898			−0.205503			−0.102752	−	0.994707i	$-0.532765\pi$
$199$	−2.89898			−0.205503			−0.102752	+	0.994707i	$0.532765\pi$
$200$	−1.44949	−	2.51059i	−0.102494	−	0.177526i
$201$		0			0
$202$	8.62372	−	14.9367i	0.606763	−	1.05094i
$203$	3.44949	−	5.97469i	0.242107	−	0.419341i
$204$		0			0
$205$	7.10102	+	12.2993i	0.495957	+	0.859022i
$206$	−14.0000			−0.975426
$207$		0			0
$208$	4.89898			0.339683
$209$	2.55051	+	4.41761i	0.176422	+	0.305573i
$210$		0			0
$211$	−6.44949	+	11.1708i	−0.444001	+	0.769033i	−0.997982	−	0.0634968i	$-0.979775\pi$
$211$	−6.44949	+	11.1708i	−0.444001	+	0.769033i	0.553981	+	0.832529i	$0.313108\pi$
$212$	−5.44949	+	9.43879i	−0.374272	+	0.648259i
$213$		0			0
$214$	6.00000	+	10.3923i	0.410152	+	0.710403i
$215$	10.0000			0.681994
$216$		0			0
$217$	−6.00000			−0.407307
$218$	6.34847	+	10.9959i	0.429973	+	0.744734i
$219$		0			0
$220$	1.44949	−	2.51059i	0.0977246	−	0.169264i
$221$	4.89898	−	8.48528i	0.329541	−	0.570782i
$222$		0			0
$223$	5.55051	+	9.61377i	0.371690	+	0.643785i	0.989826	−	0.142286i	$-0.0454452\pi$
$223$	5.55051	+	9.61377i	0.371690	+	0.643785i	−0.618136	+	0.786071i	$0.712112\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$	−6.10102			−0.405834
$227$	−2.72474	−	4.71940i	−0.180848	−	0.313237i	0.761322	−	0.648374i	$-0.224551\pi$
$227$	−2.72474	−	4.71940i	−0.180848	−	0.313237i	−0.942169	+	0.335137i	$0.891217\pi$
$228$		0			0
$229$	−0.623724	+	1.08032i	−0.0412169	+	0.0713897i	−0.885898	−	0.463880i	$-0.846457\pi$
$229$	−0.623724	+	1.08032i	−0.0412169	+	0.0713897i	0.844681	+	0.535270i	$0.179790\pi$
$230$	0.724745	−	1.25529i	0.0477883	−	0.0827717i
$231$		0			0
$232$	3.44949	+	5.97469i	0.226470	+	0.392258i
$233$	7.00000			0.458585			0.229293	−	0.973358i	$-0.426359\pi$
$233$	7.00000			0.458585			0.229293	+	0.973358i	$0.426359\pi$
$234$		0			0
$235$	−14.2020			−0.926439
$236$	−1.00000	−	1.73205i	−0.0650945	−	0.112747i
$237$		0			0
$238$	1.00000	−	1.73205i	0.0648204	−	0.112272i
$239$	3.39898	−	5.88721i	0.219862	−	0.380812i	−0.734904	−	0.678171i	$-0.762773\pi$
$239$	3.39898	−	5.88721i	0.219862	−	0.380812i	0.954766	+	0.297360i	$0.0961061\pi$
$240$		0			0
$241$	−0.449490	−	0.778539i	−0.0289542	−	0.0501501i	0.851185	−	0.524865i	$-0.175884\pi$
$241$	−0.449490	−	0.778539i	−0.0289542	−	0.0501501i	−0.880139	+	0.474715i	$0.842551\pi$
$242$	7.00000			0.449977
$243$		0			0
$244$	6.55051			0.419353
$245$	−0.724745	−	1.25529i	−0.0463023	−	0.0801979i
$246$		0			0
$247$	−6.24745	+	10.8209i	−0.397516	+	0.688517i
$248$	3.00000	−	5.19615i	0.190500	−	0.329956i
$249$		0			0
$250$	−5.72474	−	9.91555i	−0.362065	−	0.627114i
$251$	−17.4495			−1.10140			−0.550701	−	0.834703i	$-0.685640\pi$
$251$	−17.4495			−1.10140			−0.550701	+	0.834703i	$0.685640\pi$
$252$		0			0
$253$	−2.00000			−0.125739
$254$	−1.50000	−	2.59808i	−0.0941184	−	0.163018i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	4.10102	−	7.10318i	0.255815	−	0.443084i	−0.709302	−	0.704905i	$-0.750990\pi$
$257$	4.10102	−	7.10318i	0.255815	−	0.443084i	0.965116	+	0.261821i	$0.0843230\pi$
$258$		0			0
$259$	5.89898	+	10.2173i	0.366545	+	0.634874i
$260$	7.10102			0.440387
$261$		0			0
$262$	−8.55051			−0.528252
$263$	12.9495	+	22.4292i	0.798500	+	1.38304i	0.920593	+	0.390523i	$0.127706\pi$
$263$	12.9495	+	22.4292i	0.798500	+	1.38304i	−0.122093	+	0.992519i	$0.538961\pi$
$264$		0			0
$265$	−7.89898	+	13.6814i	−0.485230	+	0.840444i
$266$	−1.27526	+	2.20881i	−0.0781909	+	0.135431i
$267$		0			0
$268$	6.44949	+	11.1708i	0.393965	+	0.682368i
$269$	18.3485			1.11873			0.559363	−	0.828923i	$-0.311046\pi$
$269$	18.3485			1.11873			0.559363	+	0.828923i	$0.311046\pi$
$270$		0			0
$271$	7.10102			0.431356			0.215678	−	0.976465i	$-0.430804\pi$
$271$	7.10102			0.431356			0.215678	+	0.976465i	$0.430804\pi$
$272$	1.00000	+	1.73205i	0.0606339	+	0.105021i
$273$		0			0
$274$	−3.89898	+	6.75323i	−0.235546	+	0.407978i
$275$	−2.89898	+	5.02118i	−0.174815	+	0.302789i
$276$		0			0
$277$	9.34847	+	16.1920i	0.561695	+	0.972884i	0.997349	+	0.0727700i	$0.0231839\pi$
$277$	9.34847	+	16.1920i	0.561695	+	0.972884i	−0.435654	+	0.900114i	$0.643483\pi$
$278$	−4.55051			−0.272921
$279$		0			0
$280$	1.44949			0.0866236
$281$	−9.50000	−	16.4545i	−0.566722	−	0.981592i	−0.996887	−	0.0788417i	$-0.974878\pi$
$281$	−9.50000	−	16.4545i	−0.566722	−	0.981592i	0.430165	−	0.902750i	$-0.358455\pi$
$282$		0			0
$283$	12.7247	−	22.0399i	0.756408	−	1.31014i	−0.188264	−	0.982118i	$-0.560286\pi$
$283$	12.7247	−	22.0399i	0.756408	−	1.31014i	0.944672	−	0.328018i	$-0.106381\pi$
$284$	0.0505103	−	0.0874863i	0.00299723	−	0.00519136i
$285$		0			0
$286$	−4.89898	−	8.48528i	−0.289683	−	0.501745i
$287$	9.79796			0.578355
$288$		0			0
$289$	−13.0000			−0.764706
$290$	5.00000	+	8.66025i	0.293610	+	0.508548i
$291$		0			0
$292$	3.44949	−	5.97469i	0.201866	−	0.349642i
$293$	1.37628	−	2.38378i	0.0804029	−	0.139262i	−0.823020	−	0.568012i	$-0.807713\pi$
$293$	1.37628	−	2.38378i	0.0804029	−	0.139262i	0.903423	+	0.428750i	$0.141046\pi$
$294$		0			0
$295$	−1.44949	−	2.51059i	−0.0843926	−	0.146172i
$296$	−11.7980			−0.685742
$297$		0			0
$298$	−6.00000			−0.347571
$299$	−2.44949	−	4.24264i	−0.141658	−	0.245358i
$300$		0			0
$301$	3.44949	−	5.97469i	0.198825	−	0.344375i
$302$	−2.50000	+	4.33013i	−0.143859	+	0.249171i
$303$		0			0
$304$	−1.27526	−	2.20881i	−0.0731409	−	0.126684i
$305$	9.49490			0.543676
$306$		0			0
$307$	25.2474			1.44095			0.720474	−	0.693482i	$-0.243924\pi$
$307$	25.2474			1.44095			0.720474	+	0.693482i	$0.243924\pi$
$308$	−1.00000	−	1.73205i	−0.0569803	−	0.0986928i
$309$		0			0
$310$	4.34847	−	7.53177i	0.246976	−	0.427776i
$311$	15.3485	−	26.5843i	0.870332	−	1.50746i	0.00867810	−	0.999962i	$-0.497238\pi$
$311$	15.3485	−	26.5843i	0.870332	−	1.50746i	0.861654	−	0.507497i	$-0.169429\pi$
$312$		0			0
$313$	2.34847	+	4.06767i	0.132743	+	0.229918i	0.924733	−	0.380616i	$-0.124288\pi$
$313$	2.34847	+	4.06767i	0.132743	+	0.229918i	−0.791990	+	0.610534i	$0.790955\pi$
$314$	8.34847			0.471131
$315$		0			0
$316$	−1.89898			−0.106826
$317$	10.3485	+	17.9241i	0.581228	+	1.00672i	0.995334	+	0.0964878i	$0.0307609\pi$
$317$	10.3485	+	17.9241i	0.581228	+	1.00672i	−0.414106	+	0.910229i	$0.635906\pi$
$318$		0			0
$319$	6.89898	−	11.9494i	0.386269	−	0.669037i
$320$	−0.724745	+	1.25529i	−0.0405145	+	0.0701731i
$321$		0			0
$322$	−0.500000	−	0.866025i	−0.0278639	−	0.0482617i
$323$	−5.10102			−0.283828
$324$		0			0
$325$	−14.2020			−0.787787
$326$	−9.89898	−	17.1455i	−0.548254	−	0.949603i
$327$		0			0
$328$	−4.89898	+	8.48528i	−0.270501	+	0.468521i
$329$	−4.89898	+	8.48528i	−0.270089	+	0.467809i
$330$		0			0
$331$	−2.34847	−	4.06767i	−0.129084	−	0.223579i	0.794238	−	0.607606i	$-0.207870\pi$
$331$	−2.34847	−	4.06767i	−0.129084	−	0.223579i	−0.923322	+	0.384027i	$0.874537\pi$
$332$	−2.00000			−0.109764
$333$		0			0
$334$	−10.6969			−0.585310
$335$	9.34847	+	16.1920i	0.510761	+	0.884665i
$336$		0			0
$337$	11.6969	−	20.2597i	0.637173	−	1.10362i	−0.348877	−	0.937168i	$-0.613437\pi$
$337$	11.6969	−	20.2597i	0.637173	−	1.10362i	0.986050	−	0.166447i	$-0.0532296\pi$
$338$	5.50000	−	9.52628i	0.299161	−	0.518161i
$339$		0			0
$340$	1.44949	+	2.51059i	0.0786096	+	0.136156i
$341$	−12.0000			−0.649836
$342$		0			0
$343$	−1.00000			−0.0539949
$344$	3.44949	+	5.97469i	0.185984	+	0.322134i
$345$		0			0
$346$	1.55051	−	2.68556i	0.0833559	−	0.144377i
$347$	9.79796	−	16.9706i	0.525982	−	0.911028i	−0.473560	−	0.880762i	$-0.657031\pi$
$347$	9.79796	−	16.9706i	0.525982	−	0.911028i	0.999542	−	0.0302659i	$-0.00963541\pi$
$348$		0			0
$349$	−5.55051	−	9.61377i	−0.297112	−	0.514613i	0.678362	−	0.734728i	$-0.262690\pi$
$349$	−5.55051	−	9.61377i	−0.297112	−	0.514613i	−0.975474	+	0.220115i	$0.929357\pi$
$350$	−2.89898			−0.154957
$351$		0			0
$352$	2.00000			0.106600
$353$	−3.00000	−	5.19615i	−0.159674	−	0.276563i	0.775077	−	0.631867i	$-0.217711\pi$
$353$	−3.00000	−	5.19615i	−0.159674	−	0.276563i	−0.934751	+	0.355303i	$0.884378\pi$
$354$		0			0
$355$	0.0732141	−	0.126811i	0.00388580	−	0.00673040i
$356$	−8.44949	+	14.6349i	−0.447822	+	0.775651i
$357$		0			0
$358$	−10.3485	−	17.9241i	−0.546934	−	0.947317i
$359$	8.79796			0.464339			0.232169	−	0.972675i	$-0.425418\pi$
$359$	8.79796			0.464339			0.232169	+	0.972675i	$0.425418\pi$
$360$		0			0
$361$	−12.4949			−0.657626
$362$	−5.17423	−	8.96204i	−0.271952	−	0.471034i
$363$		0			0
$364$	2.44949	−	4.24264i	0.128388	−	0.222375i
$365$	5.00000	−	8.66025i	0.261712	−	0.453298i
$366$		0			0
$367$	6.89898	+	11.9494i	0.360124	+	0.623753i	0.987981	−	0.154576i	$-0.0494011\pi$
$367$	6.89898	+	11.9494i	0.360124	+	0.623753i	−0.627857	+	0.778329i	$0.716068\pi$
$368$	1.00000			0.0521286
$369$		0			0
$370$	−17.1010			−0.889040
$371$	5.44949	+	9.43879i	0.282923	+	0.490038i
$372$		0			0
$373$	3.44949	−	5.97469i	0.178608	−	0.309358i	−0.762796	−	0.646639i	$-0.776174\pi$
$373$	3.44949	−	5.97469i	0.178608	−	0.309358i	0.941404	+	0.337281i	$0.109507\pi$
$374$	2.00000	−	3.46410i	0.103418	−	0.179124i
$375$		0			0
$376$	−4.89898	−	8.48528i	−0.252646	−	0.437595i
$377$	33.7980			1.74068
$378$		0			0
$379$	22.4949			1.15549			0.577743	−	0.816219i	$-0.303934\pi$
$379$	22.4949			1.15549			0.577743	+	0.816219i	$0.303934\pi$
$380$	−1.84847	−	3.20164i	−0.0948245	−	0.164241i
$381$		0			0
$382$	−2.05051	+	3.55159i	−0.104913	+	0.181715i
$383$	−1.44949	+	2.51059i	−0.0740655	+	0.128285i	−0.900679	−	0.434484i	$-0.856931\pi$
$383$	−1.44949	+	2.51059i	−0.0740655	+	0.128285i	0.826614	+	0.562769i	$0.190264\pi$
$384$		0			0
$385$	−1.44949	−	2.51059i	−0.0738728	−	0.127952i
$386$	17.8990			0.911034
$387$		0			0
$388$	2.89898			0.147173
$389$	−12.4495	−	21.5631i	−0.631214	−	1.09330i	−0.987304	−	0.158843i	$-0.949224\pi$
$389$	−12.4495	−	21.5631i	−0.631214	−	1.09330i	0.356090	−	0.934452i	$-0.384110\pi$
$390$		0			0
$391$	1.00000	−	1.73205i	0.0505722	−	0.0875936i
$392$	0.500000	−	0.866025i	0.0252538	−	0.0437409i
$393$		0			0
$394$	−8.34847	−	14.4600i	−0.420590	−	0.728483i
$395$	−2.75255			−0.138496
$396$		0			0
$397$	38.6969			1.94214			0.971072	−	0.238788i	$-0.0767500\pi$
$397$	38.6969			1.94214			0.971072	+	0.238788i	$0.0767500\pi$
$398$	−1.44949	−	2.51059i	−0.0726564	−	0.125844i
$399$		0			0
$400$	1.44949	−	2.51059i	0.0724745	−	0.125529i
$401$	−9.94949	+	17.2330i	−0.496854	+	0.860576i	−0.999993	−	0.00362911i	$-0.998845\pi$
$401$	−9.94949	+	17.2330i	−0.496854	+	0.860576i	0.503140	+	0.864205i	$0.332178\pi$
$402$		0			0
$403$	−14.6969	−	25.4558i	−0.732107	−	1.26805i
$404$	17.2474			0.858093
$405$		0			0
$406$	6.89898			0.342391
$407$	11.7980	+	20.4347i	0.584803	+	1.01291i
$408$		0			0
$409$	6.89898	−	11.9494i	0.341133	−	0.590859i	−0.643511	−	0.765437i	$-0.722523\pi$
$409$	6.89898	−	11.9494i	0.341133	−	0.590859i	0.984643	+	0.174578i	$0.0558562\pi$
$410$	−7.10102	+	12.2993i	−0.350694	+	0.607421i
$411$		0			0
$412$	−7.00000	−	12.1244i	−0.344865	−	0.597324i
$413$	−2.00000			−0.0984136
$414$		0			0
$415$	−2.89898			−0.142305
$416$	2.44949	+	4.24264i	0.120096	+	0.208013i
$417$		0			0
$418$	−2.55051	+	4.41761i	−0.124750	+	0.216073i
$419$	14.7247	−	25.5040i	0.719351	−	1.24595i	−0.241906	−	0.970300i	$-0.577773\pi$
$419$	14.7247	−	25.5040i	0.719351	−	1.24595i	0.961257	−	0.275653i	$-0.0888940\pi$
$420$		0			0
$421$	−11.4495	−	19.8311i	−0.558014	−	0.966509i	−0.997662	−	0.0683385i	$-0.978230\pi$
$421$	−11.4495	−	19.8311i	−0.558014	−	0.966509i	0.439648	−	0.898170i	$-0.355103\pi$
$422$	−12.8990			−0.627912
$423$		0			0
$424$	−10.8990			−0.529301
$425$	−2.89898	−	5.02118i	−0.140621	−	0.243563i
$426$		0			0
$427$	3.27526	−	5.67291i	0.158501	−	0.274531i
$428$	−6.00000	+	10.3923i	−0.290021	+	0.502331i
$429$		0			0
$430$	5.00000	+	8.66025i	0.241121	+	0.417635i
$431$	−31.5959			−1.52192			−0.760961	−	0.648798i	$-0.775272\pi$
$431$	−31.5959			−1.52192			−0.760961	+	0.648798i	$0.775272\pi$
$432$		0			0
$433$	−7.79796			−0.374746			−0.187373	−	0.982289i	$-0.559997\pi$
$433$	−7.79796			−0.374746			−0.187373	+	0.982289i	$0.559997\pi$
$434$	−3.00000	−	5.19615i	−0.144005	−	0.249423i
$435$		0			0
$436$	−6.34847	+	10.9959i	−0.304037	+	0.526607i
$437$	−1.27526	+	2.20881i	−0.0610037	+	0.105662i
$438$		0			0
$439$	−1.10102	−	1.90702i	−0.0525488	−	0.0910173i	0.838554	−	0.544818i	$-0.183401\pi$
$439$	−1.10102	−	1.90702i	−0.0525488	−	0.0910173i	−0.891103	+	0.453801i	$0.850068\pi$
$440$	2.89898			0.138203
$441$		0			0
$442$	9.79796			0.466041
$443$	−7.44949	−	12.9029i	−0.353936	−	0.613035i	0.632999	−	0.774152i	$-0.281824\pi$
$443$	−7.44949	−	12.9029i	−0.353936	−	0.613035i	−0.986935	+	0.161117i	$0.948490\pi$
$444$		0			0
$445$	−12.2474	+	21.2132i	−0.580585	+	1.00560i
$446$	−5.55051	+	9.61377i	−0.262824	+	0.455225i
$447$		0			0
$448$	0.500000	+	0.866025i	0.0236228	+	0.0409159i
$449$	−20.5959			−0.971981			−0.485991	−	0.873964i	$-0.661541\pi$
$449$	−20.5959			−0.971981			−0.485991	+	0.873964i	$0.661541\pi$
$450$		0			0
$451$	19.5959			0.922736
$452$	−3.05051	−	5.28364i	−0.143484	−	0.248521i
$453$		0			0
$454$	2.72474	−	4.71940i	0.127879	−	0.221492i
$455$	3.55051	−	6.14966i	0.166450	−	0.288301i
$456$		0			0
$457$	8.74745	+	15.1510i	0.409188	+	0.708735i	0.994799	−	0.101857i	$-0.0324785\pi$
$457$	8.74745	+	15.1510i	0.409188	+	0.708735i	−0.585611	+	0.810593i	$0.699145\pi$
$458$	−1.24745			−0.0582895
$459$		0			0
$460$	1.44949			0.0675828
$461$	2.82577	+	4.89437i	0.131609	+	0.227954i	0.924297	−	0.381674i	$-0.124652\pi$
$461$	2.82577	+	4.89437i	0.131609	+	0.227954i	−0.792688	+	0.609628i	$0.791319\pi$
$462$		0			0
$463$	−1.84847	+	3.20164i	−0.0859057	+	0.148793i	−0.905777	−	0.423755i	$-0.860712\pi$
$463$	−1.84847	+	3.20164i	−0.0859057	+	0.148793i	0.819871	+	0.572548i	$0.194045\pi$
$464$	−3.44949	+	5.97469i	−0.160139	+	0.277368i
$465$		0			0
$466$	3.50000	+	6.06218i	0.162134	+	0.280825i
$467$	−10.0000			−0.462745			−0.231372	−	0.972865i	$-0.574322\pi$
$467$	−10.0000			−0.462745			−0.231372	+	0.972865i	$0.574322\pi$
$468$		0			0
$469$	12.8990			0.595620
$470$	−7.10102	−	12.2993i	−0.327546	−	0.567326i
$471$		0			0
$472$	1.00000	−	1.73205i	0.0460287	−	0.0797241i
$473$	6.89898	−	11.9494i	0.317215	−	0.549433i
$474$		0			0
$475$	3.69694	+	6.40329i	0.169627	+	0.293803i
$476$	2.00000			0.0916698
$477$		0			0
$478$	6.79796			0.310931
$479$	4.79796	+	8.31031i	0.219224	+	0.379708i	0.954571	−	0.297983i	$-0.0963140\pi$
$479$	4.79796	+	8.31031i	0.219224	+	0.379708i	−0.735347	+	0.677691i	$0.762981\pi$
$480$		0			0
$481$	−28.8990	+	50.0545i	−1.31768	+	2.28229i
$482$	0.449490	−	0.778539i	0.0204737	−	0.0354615i
$483$		0			0
$484$	3.50000	+	6.06218i	0.159091	+	0.275554i
$485$	4.20204			0.190805
$486$		0			0
$487$	−36.3939			−1.64916			−0.824582	−	0.565742i	$-0.808590\pi$
$487$	−36.3939			−1.64916			−0.824582	+	0.565742i	$0.808590\pi$
$488$	3.27526	+	5.67291i	0.148264	+	0.256800i
$489$		0			0
$490$	0.724745	−	1.25529i	0.0327406	−	0.0567084i
$491$	7.89898	−	13.6814i	0.356476	−	0.617434i	−0.630893	−	0.775869i	$-0.717312\pi$
$491$	7.89898	−	13.6814i	0.356476	−	0.617434i	0.987369	+	0.158435i	$0.0506448\pi$
$492$		0			0
$493$	6.89898	+	11.9494i	0.310714	+	0.538173i
$494$	−12.4949			−0.562172
$495$		0			0
$496$	6.00000			0.269408
$497$	−0.0505103	−	0.0874863i	−0.00226569	−	0.00392430i
$498$		0			0
$499$	12.6969	−	21.9917i	0.568393	−	0.984486i	−0.428332	−	0.903621i	$-0.640899\pi$
$499$	12.6969	−	21.9917i	0.568393	−	0.984486i	0.996725	−	0.0808642i	$-0.0257680\pi$
$500$	5.72474	−	9.91555i	0.256018	−	0.443437i
$501$		0			0
$502$	−8.72474	−	15.1117i	−0.389404	−	0.674468i
$503$	24.4949			1.09217			0.546087	−	0.837729i	$-0.316117\pi$
$503$	24.4949			1.09217			0.546087	+	0.837729i	$0.316117\pi$
$504$		0			0
$505$	25.0000			1.11249
$506$	−1.00000	−	1.73205i	−0.0444554	−	0.0769991i
$507$		0			0
$508$	1.50000	−	2.59808i	0.0665517	−	0.115271i
$509$	−3.55051	+	6.14966i	−0.157374	+	0.272579i	−0.933921	−	0.357480i	$-0.883636\pi$
$509$	−3.55051	+	6.14966i	−0.157374	+	0.272579i	0.776547	+	0.630059i	$0.216969\pi$
$510$		0			0
$511$	−3.44949	−	5.97469i	−0.152596	−	0.264305i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	8.20204			0.361777
$515$	−10.1464	−	17.5741i	−0.447105	−	0.774409i
$516$		0			0
$517$	−9.79796	+	16.9706i	−0.430914	+	0.746364i
$518$	−5.89898	+	10.2173i	−0.259186	+	0.448924i
$519$		0			0
$520$	3.55051	+	6.14966i	0.155700	+	0.269681i
$521$	9.30306			0.407575			0.203787	−	0.979015i	$-0.434675\pi$
$521$	9.30306			0.407575			0.203787	+	0.979015i	$0.434675\pi$
$522$		0			0
$523$	−14.3485			−0.627415			−0.313707	−	0.949520i	$-0.601571\pi$
$523$	−14.3485			−0.627415			−0.313707	+	0.949520i	$0.601571\pi$
$524$	−4.27526	−	7.40496i	−0.186765	−	0.323487i
$525$		0			0
$526$	−12.9495	+	22.4292i	−0.564625	+	0.977958i
$527$	6.00000	−	10.3923i	0.261364	−	0.452696i
$528$		0			0
$529$	11.0000	+	19.0526i	0.478261	+	0.828372i
$530$	−15.7980			−0.686219
$531$		0			0
$532$	−2.55051			−0.110579
$533$	24.0000	+	41.5692i	1.03956	+	1.80056i
$534$		0			0
$535$	−8.69694	+	15.0635i	−0.376001	+	0.651254i
$536$	−6.44949	+	11.1708i	−0.278576	+	0.482507i
$537$		0			0
$538$	9.17423	+	15.8902i	0.395529	+	0.685077i
$539$	−2.00000			−0.0861461
$540$		0			0
$541$	−18.4949			−0.795158			−0.397579	−	0.917568i	$-0.630149\pi$
$541$	−18.4949			−0.795158			−0.397579	+	0.917568i	$0.630149\pi$
$542$	3.55051	+	6.14966i	0.152507	+	0.264151i
$543$		0			0
$544$	−1.00000	+	1.73205i	−0.0428746	+	0.0742611i
$545$	−9.20204	+	15.9384i	−0.394172	+	0.682726i
$546$		0			0
$547$	3.79796	+	6.57826i	0.162389	+	0.281266i	0.935725	−	0.352730i	$-0.114747\pi$
$547$	3.79796	+	6.57826i	0.162389	+	0.281266i	−0.773336	+	0.633996i	$0.781413\pi$
$548$	−7.79796			−0.333112
$549$		0			0
$550$	−5.79796			−0.247226
$551$	−8.79796	−	15.2385i	−0.374806	−	0.649182i
$552$		0			0
$553$	−0.949490	+	1.64456i	−0.0403764	+	0.0699340i
$554$	−9.34847	+	16.1920i	−0.397178	+	0.687933i
$555$		0			0
$556$	−2.27526	−	3.94086i	−0.0964923	−	0.167130i
$557$	12.8990			0.546547			0.273274	−	0.961936i	$-0.411894\pi$
$557$	12.8990			0.546547			0.273274	+	0.961936i	$0.411894\pi$
$558$		0			0
$559$	33.7980			1.42950
$560$	0.724745	+	1.25529i	0.0306261	+	0.0530459i
$561$		0			0
$562$	9.50000	−	16.4545i	0.400733	−	0.694090i
$563$	−19.9722	+	34.5929i	−0.841728	+	1.45791i	0.0467054	+	0.998909i	$0.485128\pi$
$563$	−19.9722	+	34.5929i	−0.841728	+	1.45791i	−0.888433	+	0.459006i	$0.848206\pi$
$564$		0			0
$565$	−4.42168	−	7.65858i	−0.186022	−	0.322199i
$566$	25.4495			1.06972
$567$		0			0
$568$	0.101021			0.00423873
$569$	−15.0000	−	25.9808i	−0.628833	−	1.08917i	−0.987786	−	0.155815i	$-0.950200\pi$
$569$	−15.0000	−	25.9808i	−0.628833	−	1.08917i	0.358954	−	0.933355i	$-0.383134\pi$
$570$		0			0
$571$	−16.8990	+	29.2699i	−0.707200	+	1.22491i	0.258691	+	0.965960i	$0.416709\pi$
$571$	−16.8990	+	29.2699i	−0.707200	+	1.22491i	−0.965892	+	0.258947i	$0.916625\pi$
$572$	4.89898	−	8.48528i	0.204837	−	0.354787i
$573$		0			0
$574$	4.89898	+	8.48528i	0.204479	+	0.354169i
$575$	−2.89898			−0.120896
$576$		0			0
$577$	−15.5959			−0.649267			−0.324633	−	0.945840i	$-0.605241\pi$
$577$	−15.5959			−0.649267			−0.324633	+	0.945840i	$0.605241\pi$
$578$	−6.50000	−	11.2583i	−0.270364	−	0.468285i
$579$		0			0
$580$	−5.00000	+	8.66025i	−0.207614	+	0.359597i
$581$	−1.00000	+	1.73205i	−0.0414870	+	0.0718576i
$582$		0			0
$583$	10.8990	+	18.8776i	0.451390	+	0.781830i
$584$	6.89898			0.285482
$585$		0			0
$586$	2.75255			0.113707
$587$	8.07321	+	13.9832i	0.333217	+	0.577149i	0.983141	−	0.182850i	$-0.0585324\pi$
$587$	8.07321	+	13.9832i	0.333217	+	0.577149i	−0.649924	+	0.760000i	$0.725199\pi$
$588$		0			0
$589$	−7.65153	+	13.2528i	−0.315276	+	0.546074i
$590$	1.44949	−	2.51059i	0.0596745	−	0.103359i
$591$		0			0
$592$	−5.89898	−	10.2173i	−0.242447	−	0.419930i
$593$	−14.6969			−0.603531			−0.301765	−	0.953382i	$-0.597576\pi$
$593$	−14.6969			−0.603531			−0.301765	+	0.953382i	$0.597576\pi$
$594$		0			0
$595$	2.89898			0.118847
$596$	−3.00000	−	5.19615i	−0.122885	−	0.212843i
$597$		0			0
$598$	2.44949	−	4.24264i	0.100167	−	0.173494i
$599$	−16.8990	+	29.2699i	−0.690474	+	1.19594i	0.281209	+	0.959646i	$0.409264\pi$
$599$	−16.8990	+	29.2699i	−0.690474	+	1.19594i	−0.971683	+	0.236289i	$0.924069\pi$
$600$		0			0
$601$	−8.34847	−	14.4600i	−0.340541	−	0.589835i	0.643992	−	0.765032i	$-0.277277\pi$
$601$	−8.34847	−	14.4600i	−0.340541	−	0.589835i	−0.984533	+	0.175198i	$0.943944\pi$
$602$	6.89898			0.281181
$603$		0			0
$604$	−5.00000			−0.203447
$605$	5.07321	+	8.78706i	0.206255	+	0.357245i
$606$		0			0
$607$	−10.3485	+	17.9241i	−0.420031	+	0.727516i	−0.995942	−	0.0899969i	$-0.971314\pi$
$607$	−10.3485	+	17.9241i	−0.420031	+	0.727516i	0.575911	+	0.817513i	$0.304648\pi$
$608$	1.27526	−	2.20881i	0.0517184	−	0.0895789i
$609$		0			0
$610$	4.74745	+	8.22282i	0.192219	+	0.332932i
$611$	−48.0000			−1.94187
$612$		0			0
$613$	−14.6969			−0.593604			−0.296802	−	0.954939i	$-0.595920\pi$
$613$	−14.6969			−0.593604			−0.296802	+	0.954939i	$0.595920\pi$
$614$	12.6237	+	21.8649i	0.509452	+	0.882397i
$615$		0			0
$616$	1.00000	−	1.73205i	0.0402911	−	0.0697863i
$617$	−7.69694	+	13.3315i	−0.309867	+	0.536706i	−0.978333	−	0.207037i	$-0.933618\pi$
$617$	−7.69694	+	13.3315i	−0.309867	+	0.536706i	0.668466	+	0.743743i	$0.266951\pi$
$618$		0			0
$619$	−15.0732	−	26.1076i	−0.605844	−	1.04935i	−0.991918	−	0.126884i	$-0.959502\pi$
$619$	−15.0732	−	26.1076i	−0.605844	−	1.04935i	0.386074	−	0.922468i	$-0.373831\pi$
$620$	8.69694			0.349277
$621$		0			0
$622$	30.6969			1.23084
$623$	8.44949	+	14.6349i	0.338522	+	0.586337i
$624$		0			0
$625$	1.05051	−	1.81954i	0.0420204	−	0.0727815i
$626$	−2.34847	+	4.06767i	−0.0938637	+	0.162577i
$627$		0			0
$628$	4.17423	+	7.22999i	0.166570	+	0.288508i
$629$	−23.5959			−0.940831
$630$		0			0
$631$	27.8990			1.11064			0.555320	−	0.831636i	$-0.312596\pi$
$631$	27.8990			1.11064			0.555320	+	0.831636i	$0.312596\pi$
$632$	−0.949490	−	1.64456i	−0.0377687	−	0.0654173i
$633$		0			0
$634$	−10.3485	+	17.9241i	−0.410990	+	0.711856i
$635$	2.17423	−	3.76588i	0.0862819	−	0.149445i
$636$		0			0
$637$	−2.44949	−	4.24264i	−0.0970523	−	0.168100i
$638$	13.7980			0.546266
$639$		0			0
$640$	−1.44949			−0.0572961
$641$	−3.74745	−	6.49077i	−0.148015	−	0.256370i	0.782479	−	0.622678i	$-0.213955\pi$
$641$	−3.74745	−	6.49077i	−0.148015	−	0.256370i	−0.930494	+	0.366308i	$0.880622\pi$
$642$		0			0
$643$	19.6969	−	34.1161i	0.776771	−	1.34541i	−0.157022	−	0.987595i	$-0.550189\pi$
$643$	19.6969	−	34.1161i	0.776771	−	1.34541i	0.933793	−	0.357812i	$-0.116477\pi$
$644$	0.500000	−	0.866025i	0.0197028	−	0.0341262i
$645$		0			0
$646$	−2.55051	−	4.41761i	−0.100348	−	0.173809i
$647$	50.6969			1.99310			0.996551	−	0.0829807i	$-0.0264440\pi$
$647$	50.6969			1.99310			0.996551	+	0.0829807i	$0.0264440\pi$
$648$		0			0
$649$	−4.00000			−0.157014
$650$	−7.10102	−	12.2993i	−0.278525	−	0.482419i
$651$		0			0
$652$	9.89898	−	17.1455i	0.387674	−	0.671471i
$653$	4.89898	−	8.48528i	0.191712	−	0.332055i	−0.754106	−	0.656753i	$-0.771929\pi$
$653$	4.89898	−	8.48528i	0.191712	−	0.332055i	0.945818	+	0.324698i	$0.105263\pi$
$654$		0			0
$655$	−6.19694	−	10.7334i	−0.242134	−	0.419389i
$656$	−9.79796			−0.382546
$657$		0			0
$658$	−9.79796			−0.381964
$659$	−12.3485	−	21.3882i	−0.481028	−	0.833165i	0.518735	−	0.854935i	$-0.326403\pi$
$659$	−12.3485	−	21.3882i	−0.481028	−	0.833165i	−0.999763	+	0.0217701i	$0.993070\pi$
$660$		0			0
$661$	−2.27526	+	3.94086i	−0.0884972	+	0.153282i	−0.906876	−	0.421397i	$-0.861540\pi$
$661$	−2.27526	+	3.94086i	−0.0884972	+	0.153282i	0.818379	+	0.574679i	$0.194873\pi$
$662$	2.34847	−	4.06767i	0.0912758	−	0.158094i
$663$		0			0
$664$	−1.00000	−	1.73205i	−0.0388075	−	0.0672166i
$665$	−3.69694			−0.143361
$666$		0			0
$667$	6.89898			0.267130
$668$	−5.34847	−	9.26382i	−0.206938	−	0.358428i
$669$		0			0
$670$	−9.34847	+	16.1920i	−0.361163	+	0.625552i
$671$	6.55051	−	11.3458i	0.252880	−	0.438000i
$672$		0			0
$673$	4.29796	+	7.44428i	0.165674	+	0.286956i	0.936894	−	0.349612i	$-0.113687\pi$
$673$	4.29796	+	7.44428i	0.165674	+	0.286956i	−0.771220	+	0.636568i	$0.780353\pi$
$674$	23.3939			0.901098
$675$		0			0
$676$	11.0000			0.423077
$677$	7.34847	+	12.7279i	0.282425	+	0.489174i	0.971981	−	0.235058i	$-0.0755280\pi$
$677$	7.34847	+	12.7279i	0.282425	+	0.489174i	−0.689557	+	0.724232i	$0.742195\pi$
$678$		0			0
$679$	1.44949	−	2.51059i	0.0556263	−	0.0963476i
$680$	−1.44949	+	2.51059i	−0.0555854	+	0.0962767i
$681$		0			0
$682$	−6.00000	−	10.3923i	−0.229752	−	0.397942i
$683$	−51.7980			−1.98199			−0.990997	−	0.133885i	$-0.957255\pi$
$683$	−51.7980			−1.98199			−0.990997	+	0.133885i	$0.957255\pi$
$684$		0			0
$685$	−11.3031			−0.431868
$686$	−0.500000	−	0.866025i	−0.0190901	−	0.0330650i
$687$		0			0
$688$	−3.44949	+	5.97469i	−0.131511	+	0.227783i
$689$	−26.6969	+	46.2405i	−1.01707	+	1.76162i
$690$		0			0
$691$	−25.5227	−	44.2066i	−0.970929	−	1.68170i	−0.692762	−	0.721167i	$-0.743606\pi$
$691$	−25.5227	−	44.2066i	−0.970929	−	1.68170i	−0.278168	−	0.960533i	$-0.589727\pi$
$692$	3.10102			0.117883
$693$		0			0
$694$	19.5959			0.743851
$695$	−3.29796	−	5.71223i	−0.125099	−	0.216677i
$696$		0			0
$697$	−9.79796	+	16.9706i	−0.371124	+	0.642806i
$698$	5.55051	−	9.61377i	0.210090	−	0.363886i
$699$		0			0
$700$	−1.44949	−	2.51059i	−0.0547856	−	0.0948914i
$701$	7.39388			0.279263			0.139631	−	0.990204i	$-0.455408\pi$
$701$	7.39388			0.279263			0.139631	+	0.990204i	$0.455408\pi$
$702$		0			0
$703$	30.0908			1.13490
$704$	1.00000	+	1.73205i	0.0376889	+	0.0652791i
$705$		0			0
$706$	3.00000	−	5.19615i	0.112906	−	0.195560i
$707$	8.62372	−	14.9367i	0.324329	−	0.561754i
$708$		0			0
$709$	−13.7980	−	23.8988i	−0.518193	−	0.897537i	−0.999777	−	0.0211367i	$-0.993271\pi$
$709$	−13.7980	−	23.8988i	−0.518193	−	0.897537i	0.481583	−	0.876400i	$-0.340062\pi$
$710$	0.146428			0.00549535
$711$		0			0
$712$	−16.8990			−0.633316
$713$	−3.00000	−	5.19615i	−0.112351	−	0.194597i
$714$		0			0
$715$	7.10102	−	12.2993i	0.265563	−	0.459969i
$716$	10.3485	−	17.9241i	0.386740	−	0.669854i
$717$		0			0
$718$	4.39898	+	7.61926i	0.164168	+	0.284348i
$719$	−9.79796			−0.365402			−0.182701	−	0.983169i	$-0.558484\pi$
$719$	−9.79796			−0.365402			−0.182701	+	0.983169i	$0.558484\pi$
$720$		0			0
$721$	−14.0000			−0.521387
$722$	−6.24745	−	10.8209i	−0.232506	−	0.402712i
$723$		0			0
$724$	5.17423	−	8.96204i	0.192299	−	0.333071i
$725$	10.0000	−	17.3205i	0.371391	−	0.643268i
$726$		0			0
$727$	4.24745	+	7.35680i	0.157529	+	0.272848i	0.933977	−	0.357333i	$-0.116314\pi$
$727$	4.24745	+	7.35680i	0.157529	+	0.272848i	−0.776448	+	0.630181i	$0.782981\pi$
$728$	4.89898			0.181568
$729$		0			0
$730$	10.0000			0.370117
$731$	6.89898	+	11.9494i	0.255168	+	0.441964i
$732$		0			0
$733$	8.72474	−	15.1117i	0.322256	−	0.558163i	−0.658697	−	0.752408i	$-0.728892\pi$
$733$	8.72474	−	15.1117i	0.322256	−	0.558163i	0.980953	+	0.194245i	$0.0622255\pi$
$734$	−6.89898	+	11.9494i	−0.254646	+	0.441060i
$735$		0			0
$736$	0.500000	+	0.866025i	0.0184302	+	0.0319221i
$737$	25.7980			0.950280
$738$		0			0
$739$	13.5959			0.500134			0.250067	−	0.968229i	$-0.419547\pi$
$739$	13.5959			0.500134			0.250067	+	0.968229i	$0.419547\pi$
$740$	−8.55051	−	14.8099i	−0.314323	−	0.544423i
$741$		0			0
$742$	−5.44949	+	9.43879i	−0.200057	+	0.346509i
$743$	18.0000	−	31.1769i	0.660356	−	1.14377i	−0.320166	−	0.947361i	$-0.603739\pi$
$743$	18.0000	−	31.1769i	0.660356	−	1.14377i	0.980522	−	0.196409i	$-0.0629279\pi$
$744$		0			0
$745$	−4.34847	−	7.53177i	−0.159316	−	0.275943i
$746$	6.89898			0.252590
$747$		0			0
$748$	4.00000			0.146254
$749$	6.00000	+	10.3923i	0.219235	+	0.379727i
$750$		0			0
$751$	−0.702041	+	1.21597i	−0.0256178	+	0.0443714i	−0.878550	−	0.477650i	$-0.841489\pi$
$751$	−0.702041	+	1.21597i	−0.0256178	+	0.0443714i	0.852932	+	0.522022i	$0.174822\pi$
$752$	4.89898	−	8.48528i	0.178647	−	0.309426i
$753$		0			0
$754$	16.8990	+	29.2699i	0.615425	+	1.06595i
$755$	−7.24745			−0.263762
$756$		0			0
$757$	−35.3939			−1.28641			−0.643206	−	0.765693i	$-0.722396\pi$
$757$	−35.3939			−1.28641			−0.643206	+	0.765693i	$0.722396\pi$
$758$	11.2474	+	19.4812i	0.408526	+	0.707587i
$759$		0			0
$760$	1.84847	−	3.20164i	0.0670510	−	0.116136i
$761$	1.00000	−	1.73205i	0.0362500	−	0.0627868i	−0.847331	−	0.531065i	$-0.821792\pi$
$761$	1.00000	−	1.73205i	0.0362500	−	0.0627868i	0.883581	+	0.468278i	$0.155125\pi$
$762$		0			0
$763$	6.34847	+	10.9959i	0.229830	+	0.398077i
$764$	−4.10102			−0.148370
$765$		0			0
$766$	−2.89898			−0.104744
$767$	−4.89898	−	8.48528i	−0.176892	−	0.306386i
$768$		0			0
$769$	17.0454	−	29.5235i	0.614673	−	1.06465i	−0.375769	−	0.926714i	$-0.622621\pi$
$769$	17.0454	−	29.5235i	0.614673	−	1.06465i	0.990442	−	0.137932i	$-0.0440454\pi$
$770$	1.44949	−	2.51059i	0.0522360	−	0.0904754i
$771$		0			0
$772$	8.94949	+	15.5010i	0.322099	+	0.557892i
$773$	−33.9444			−1.22089			−0.610447	−	0.792057i	$-0.709010\pi$
$773$	−33.9444			−1.22089			−0.610447	+	0.792057i	$0.709010\pi$
$774$		0			0
$775$	−17.3939			−0.624807
$776$	1.44949	+	2.51059i	0.0520336	+	0.0901249i
$777$		0			0
$778$	12.4495	−	21.5631i	0.446336	−	0.773076i
$779$	12.4949	−	21.6418i	0.447676	−	0.775398i
$780$		0			0
$781$	−0.101021	−	0.174973i	−0.00361480	−	0.00626101i
$782$	2.00000			0.0715199
$783$		0			0
$784$	1.00000			0.0357143
$785$	6.05051	+	10.4798i	0.215952	+	0.374040i

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.f (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.724745 + 1.25529i) q^{5} +(0.500000 + 0.866025i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.724745 + 1.25529i) q^{5} +(0.500000 + 0.866025i) q^{7} -1.00000 q^{8} -1.44949 q^{10} +(1.00000 + 1.73205i) q^{11} +(-2.44949 + 4.24264i) q^{13} +(-0.500000 + 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} -2.00000 q^{17} +2.55051 q^{19} +(-0.724745 - 1.25529i) q^{20} +(-1.00000 + 1.73205i) q^{22} +(-0.500000 + 0.866025i) q^{23} +(1.44949 + 2.51059i) q^{25} -4.89898 q^{26} -1.00000 q^{28} +(-3.44949 - 5.97469i) q^{29} +(-3.00000 + 5.19615i) q^{31} +(0.500000 - 0.866025i) q^{32} +(-1.00000 - 1.73205i) q^{34} -1.44949 q^{35} +11.7980 q^{37} +(1.27526 + 2.20881i) q^{38} +(0.724745 - 1.25529i) q^{40} +(4.89898 - 8.48528i) q^{41} +(-3.44949 - 5.97469i) q^{43} -2.00000 q^{44} -1.00000 q^{46} +(4.89898 + 8.48528i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(-1.44949 + 2.51059i) q^{50} +(-2.44949 - 4.24264i) q^{52} +10.8990 q^{53} -2.89898 q^{55} +(-0.500000 - 0.866025i) q^{56} +(3.44949 - 5.97469i) q^{58} +(-1.00000 + 1.73205i) q^{59} +(-3.27526 - 5.67291i) q^{61} -6.00000 q^{62} +1.00000 q^{64} +(-3.55051 - 6.14966i) q^{65} +(6.44949 - 11.1708i) q^{67} +(1.00000 - 1.73205i) q^{68} +(-0.724745 - 1.25529i) q^{70} -0.101021 q^{71} -6.89898 q^{73} +(5.89898 + 10.2173i) q^{74} +(-1.27526 + 2.20881i) q^{76} +(-1.00000 + 1.73205i) q^{77} +(0.949490 + 1.64456i) q^{79} +1.44949 q^{80} +9.79796 q^{82} +(1.00000 + 1.73205i) q^{83} +(1.44949 - 2.51059i) q^{85} +(3.44949 - 5.97469i) q^{86} +(-1.00000 - 1.73205i) q^{88} +16.8990 q^{89} -4.89898 q^{91} +(-0.500000 - 0.866025i) q^{92} +(-4.89898 + 8.48528i) q^{94} +(-1.84847 + 3.20164i) q^{95} +(-1.44949 - 2.51059i) q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 4 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^7 - 4 * q^8	\( 4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 4 q^{8} + 4 q^{10} + 4 q^{11} - 2 q^{14} - 2 q^{16} - 8 q^{17} + 20 q^{19} + 2 q^{20} - 4 q^{22} - 2 q^{23} - 4 q^{25} - 4 q^{28} - 4 q^{29} - 12 q^{31} + 2 q^{32} - 4 q^{34} + 4 q^{35} + 8 q^{37} + 10 q^{38} - 2 q^{40} - 4 q^{43} - 8 q^{44} - 4 q^{46} - 2 q^{49} + 4 q^{50} + 24 q^{53} + 8 q^{55} - 2 q^{56} + 4 q^{58} - 4 q^{59} - 18 q^{61} - 24 q^{62} + 4 q^{64} - 24 q^{65} + 16 q^{67} + 4 q^{68} + 2 q^{70} - 20 q^{71} - 8 q^{73} + 4 q^{74} - 10 q^{76} - 4 q^{77} - 6 q^{79} - 4 q^{80} + 4 q^{83} - 4 q^{85} + 4 q^{86} - 4 q^{88} + 48 q^{89} - 2 q^{92} + 22 q^{95} + 4 q^{97} - 4 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^7 - 4 * q^8 + 4 * q^10 + 4 * q^11 - 2 * q^14 - 2 * q^16 - 8 * q^17 + 20 * q^19 + 2 * q^20 - 4 * q^22 - 2 * q^23 - 4 * q^25 - 4 * q^28 - 4 * q^29 - 12 * q^31 + 2 * q^32 - 4 * q^34 + 4 * q^35 + 8 * q^37 + 10 * q^38 - 2 * q^40 - 4 * q^43 - 8 * q^44 - 4 * q^46 - 2 * q^49 + 4 * q^50 + 24 * q^53 + 8 * q^55 - 2 * q^56 + 4 * q^58 - 4 * q^59 - 18 * q^61 - 24 * q^62 + 4 * q^64 - 24 * q^65 + 16 * q^67 + 4 * q^68 + 2 * q^70 - 20 * q^71 - 8 * q^73 + 4 * q^74 - 10 * q^76 - 4 * q^77 - 6 * q^79 - 4 * q^80 + 4 * q^83 - 4 * q^85 + 4 * q^86 - 4 * q^88 + 48 * q^89 - 2 * q^92 + 22 * q^95 + 4 * q^97 - 4 * q^98

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