LMFDB - Embedded newform 117.2.ba.a.80.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,2,Mod(71,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 5]))

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

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

chi := DirichletCharacter("117.71");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	117.ba (of order $12$, degree $4$, 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:	$0.934249703649$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			80.2
Root			$-0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	117.80
Dual form			117.2.ba.a.98.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.517638 - 1.93185i) q^{2} +(-1.73205 - 1.00000i) q^{4} +(-1.93185 - 1.93185i) q^{5} +(-0.500000 + 0.133975i) q^{7} +(-4.73205 + 2.73205i) q^{10} +(4.05317 + 1.08604i) q^{11} +(3.59808 - 0.232051i) q^{13} +1.03528i q^{14} +(-2.00000 - 3.46410i) q^{16} +(-3.34607 + 5.79555i) q^{17} +(1.63397 + 6.09808i) q^{19} +(1.41421 + 5.27792i) q^{20} +(4.19615 - 7.26795i) q^{22} +(-1.22474 - 2.12132i) q^{23} +2.46410i q^{25} +(1.41421 - 7.07107i) q^{26} +(1.00000 + 0.267949i) q^{28} +(-1.22474 + 0.707107i) q^{29} +(4.63397 - 4.63397i) q^{31} +(-7.72741 + 2.07055i) q^{32} +(9.46410 + 9.46410i) q^{34} +(1.22474 + 0.707107i) q^{35} +(0.830127 - 3.09808i) q^{37} +12.6264 q^{38} +(-0.378937 + 1.41421i) q^{41} +(-6.69615 - 3.86603i) q^{43} +(-5.93426 - 5.93426i) q^{44} +(-4.73205 + 1.26795i) q^{46} +(-2.31079 + 2.31079i) q^{47} +(-5.83013 + 3.36603i) q^{49} +(4.76028 + 1.27551i) q^{50} +(-6.46410 - 3.19615i) q^{52} +5.93426i q^{53} +(-5.73205 - 9.92820i) q^{55} +(0.732051 + 2.73205i) q^{58} +(0.0507680 + 0.189469i) q^{59} +(-6.59808 + 11.4282i) q^{61} +(-6.55343 - 11.3509i) q^{62} +8.00000i q^{64} +(-7.39924 - 6.50266i) q^{65} +(-7.59808 - 2.03590i) q^{67} +(11.5911 - 6.69213i) q^{68} +(2.00000 - 2.00000i) q^{70} +(15.1266 - 4.05317i) q^{71} +(2.90192 + 2.90192i) q^{73} +(-5.55532 - 3.20736i) q^{74} +(3.26795 - 12.1962i) q^{76} -2.17209 q^{77} +7.19615 q^{79} +(-2.82843 + 10.5558i) q^{80} +(2.53590 + 1.46410i) q^{82} +(-5.27792 - 5.27792i) q^{83} +(17.6603 - 4.73205i) q^{85} +(-10.9348 + 10.9348i) q^{86} +(4.38134 + 1.17398i) q^{89} +(-1.76795 + 0.598076i) q^{91} +4.89898i q^{92} +(3.26795 + 5.66025i) q^{94} +(8.62398 - 14.9372i) q^{95} +(-1.30385 - 4.86603i) q^{97} +(3.48477 + 13.0053i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{7} - 24 q^{10} + 8 q^{13} - 16 q^{16} + 20 q^{19} - 8 q^{22} + 8 q^{28} + 44 q^{31} + 48 q^{34} - 28 q^{37} - 12 q^{43} - 24 q^{46} - 12 q^{49} - 24 q^{52} - 32 q^{55} - 8 q^{58} - 32 q^{61}+ \cdots - 52 q^{97}+O(q^{100}) $ 8 * q - 4 * q^7 - 24 * q^10 + 8 * q^13 - 16 * q^16 + 20 * q^19 - 8 * q^22 + 8 * q^28 + 44 * q^31 + 48 * q^34 - 28 * q^37 - 12 * q^43 - 24 * q^46 - 12 * q^49 - 24 * q^52 - 32 * q^55 - 8 * q^58 - 32 * q^61 - 40 * q^67 + 16 * q^70 + 44 * q^73 + 40 * q^76 + 16 * q^79 + 48 * q^82 + 72 * q^85 - 28 * q^91 + 40 * q^94 - 52 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{1}{12}\right)$	$-1$

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 $

$2$	0.517638	−	1.93185i	0.366025	−	1.36603i	−0.500000	−	0.866025i	$-0.666667\pi$
$2$	0.517638	−	1.93185i	0.366025	−	1.36603i	0.866025	−	0.500000i	$-0.166667\pi$
$3$		0			0
$3$		0			0
$4$	−1.73205	−	1.00000i	−0.866025	−	0.500000i
$5$	−1.93185	−	1.93185i	−0.863950	−	0.863950i	0.127844	−	0.991794i	$-0.459194\pi$
$5$	−1.93185	−	1.93185i	−0.863950	−	0.863950i	−0.991794	+	0.127844i	$0.959194\pi$
$6$		0			0
$7$	−0.500000	+	0.133975i	−0.188982	+	0.0506376i	−0.352069	−	0.935974i	$-0.614522\pi$
$7$	−0.500000	+	0.133975i	−0.188982	+	0.0506376i	0.163087	+	0.986612i	$0.447855\pi$
$8$		0			0
$9$		0			0
$10$	−4.73205	+	2.73205i	−1.49641	+	0.863950i
$11$	4.05317	+	1.08604i	1.22208	+	0.327455i	0.811490	−	0.584367i	$-0.198657\pi$
$11$	4.05317	+	1.08604i	1.22208	+	0.327455i	0.410588	+	0.911821i	$0.365324\pi$
$12$		0			0
$13$	3.59808	−	0.232051i	0.997927	−	0.0643593i
$13$	3.59808	−	0.232051i	0.997927	−	0.0643593i
$14$			1.03528i			0.276689i
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$	−3.34607	+	5.79555i	−0.811540	+	1.40563i	0.100246	+	0.994963i	$0.468037\pi$
$17$	−3.34607	+	5.79555i	−0.811540	+	1.40563i	−0.911786	+	0.410666i	$0.865296\pi$
$18$		0			0
$19$	1.63397	+	6.09808i	0.374859	+	1.39899i	0.853550	+	0.521011i	$0.174445\pi$
$19$	1.63397	+	6.09808i	0.374859	+	1.39899i	−0.478691	+	0.877984i	$0.658888\pi$
$20$	1.41421	+	5.27792i	0.316228	+	1.18018i
$21$		0			0
$22$	4.19615	−	7.26795i	0.894623	−	1.54953i
$23$	−1.22474	−	2.12132i	−0.255377	−	0.442326i	0.709621	−	0.704584i	$-0.248866\pi$
$23$	−1.22474	−	2.12132i	−0.255377	−	0.442326i	−0.964998	+	0.262258i	$0.915533\pi$
$24$		0			0
$25$			2.46410i			0.492820i
$26$	1.41421	−	7.07107i	0.277350	−	1.38675i
$27$		0			0
$28$	1.00000	+	0.267949i	0.188982	+	0.0506376i
$29$	−1.22474	+	0.707107i	−0.227429	+	0.131306i	−0.609386	−	0.792874i	$-0.708584\pi$
$29$	−1.22474	+	0.707107i	−0.227429	+	0.131306i	0.381956	+	0.924180i	$0.375251\pi$
$30$		0			0
$31$	4.63397	−	4.63397i	0.832286	−	0.832286i	−0.155543	−	0.987829i	$-0.549713\pi$
$31$	4.63397	−	4.63397i	0.832286	−	0.832286i	0.987829	+	0.155543i	$0.0497126\pi$
$32$	−7.72741	+	2.07055i	−1.36603	+	0.366025i
$33$		0			0
$34$	9.46410	+	9.46410i	1.62308	+	1.62308i
$35$	1.22474	+	0.707107i	0.207020	+	0.119523i
$36$		0			0
$37$	0.830127	−	3.09808i	0.136472	−	0.509321i	−0.863515	−	0.504322i	$-0.831742\pi$
$37$	0.830127	−	3.09808i	0.136472	−	0.509321i	0.999988	−	0.00499824i	$-0.00159099\pi$
$38$	12.6264			2.04827
$39$		0			0
$40$		0			0
$41$	−0.378937	+	1.41421i	−0.0591801	+	0.220863i	−0.989182	−	0.146690i	$-0.953138\pi$
$41$	−0.378937	+	1.41421i	−0.0591801	+	0.220863i	0.930002	+	0.367554i	$0.119805\pi$
$42$		0			0
$43$	−6.69615	−	3.86603i	−1.02115	−	0.589563i	−0.106716	−	0.994290i	$-0.534033\pi$
$43$	−6.69615	−	3.86603i	−1.02115	−	0.589563i	−0.914438	+	0.404726i	$0.867367\pi$
$44$	−5.93426	−	5.93426i	−0.894623	−	0.894623i
$45$		0			0
$46$	−4.73205	+	1.26795i	−0.697703	+	0.186949i
$47$	−2.31079	+	2.31079i	−0.337063	+	0.337063i	−0.855261	−	0.518198i	$-0.826603\pi$
$47$	−2.31079	+	2.31079i	−0.337063	+	0.337063i	0.518198	+	0.855261i	$0.326603\pi$
$48$		0			0
$49$	−5.83013	+	3.36603i	−0.832875	+	0.480861i
$50$	4.76028	+	1.27551i	0.673205	+	0.180385i
$51$		0			0
$52$	−6.46410	−	3.19615i	−0.896410	−	0.443227i
$53$			5.93426i			0.815133i	0.913176	+	0.407566i	$0.133622\pi$
$53$			5.93426i			0.815133i	−0.913176	+	0.407566i	$0.866378\pi$
$54$		0			0
$55$	−5.73205	−	9.92820i	−0.772910	−	1.33872i
$56$		0			0
$57$		0			0
$58$	0.732051	+	2.73205i	0.0961230	+	0.358736i
$59$	0.0507680	+	0.189469i	0.00660943	+	0.0246667i	0.969152	−	0.246465i	$-0.0792689\pi$
$59$	0.0507680	+	0.189469i	0.00660943	+	0.0246667i	−0.962542	+	0.271131i	$0.912602\pi$
$60$		0			0
$61$	−6.59808	+	11.4282i	−0.844797	+	1.46323i	0.0410002	+	0.999159i	$0.486946\pi$
$61$	−6.59808	+	11.4282i	−0.844797	+	1.46323i	−0.885797	+	0.464072i	$0.846388\pi$
$62$	−6.55343	−	11.3509i	−0.832286	−	1.44156i
$63$		0			0
$64$			8.00000i			1.00000i
$65$	−7.39924	−	6.50266i	−0.917762	−	0.806556i
$66$		0			0
$67$	−7.59808	−	2.03590i	−0.928253	−	0.248725i	−0.237143	−	0.971475i	$-0.576211\pi$
$67$	−7.59808	−	2.03590i	−0.928253	−	0.248725i	−0.691109	+	0.722750i	$0.742878\pi$
$68$	11.5911	−	6.69213i	1.40563	−	0.811540i
$69$		0			0
$70$	2.00000	−	2.00000i	0.239046	−	0.239046i
$71$	15.1266	−	4.05317i	1.79520	−	0.481023i	0.801990	−	0.597338i	$-0.203775\pi$
$71$	15.1266	−	4.05317i	1.79520	−	0.481023i	0.993212	+	0.116315i	$0.0371081\pi$
$72$		0			0
$73$	2.90192	+	2.90192i	0.339644	+	0.339644i	0.856233	−	0.516589i	$-0.172798\pi$
$73$	2.90192	+	2.90192i	0.339644	+	0.339644i	−0.516589	+	0.856233i	$0.672798\pi$
$74$	−5.55532	−	3.20736i	−0.645793	−	0.372849i
$75$		0			0
$76$	3.26795	−	12.1962i	0.374859	−	1.39899i
$77$	−2.17209			−0.247532
$78$		0			0
$79$	7.19615			0.809630			0.404815	−	0.914399i	$-0.367336\pi$
$79$	7.19615			0.809630			0.404815	+	0.914399i	$0.367336\pi$
$80$	−2.82843	+	10.5558i	−0.316228	+	1.18018i
$81$		0			0
$82$	2.53590	+	1.46410i	0.280043	+	0.161683i
$83$	−5.27792	−	5.27792i	−0.579327	−	0.579327i	0.355391	−	0.934718i	$-0.384348\pi$
$83$	−5.27792	−	5.27792i	−0.579327	−	0.579327i	−0.934718	+	0.355391i	$0.884348\pi$
$84$		0			0
$85$	17.6603	−	4.73205i	1.91552	−	0.513263i
$86$	−10.9348	+	10.9348i	−1.17913	+	1.17913i
$87$		0			0
$88$		0			0
$89$	4.38134	+	1.17398i	0.464421	+	0.124441i	0.483439	−	0.875378i	$-0.339387\pi$
$89$	4.38134	+	1.17398i	0.464421	+	0.124441i	−0.0190181	+	0.999819i	$0.506054\pi$
$90$		0			0
$91$	−1.76795	+	0.598076i	−0.185331	+	0.0626954i
$92$			4.89898i			0.510754i
$93$		0			0
$94$	3.26795	+	5.66025i	0.337063	+	0.583811i
$95$	8.62398	−	14.9372i	0.884802	−	1.53252i
$96$		0			0
$97$	−1.30385	−	4.86603i	−0.132386	−	0.494070i	0.867609	−	0.497247i	$-0.165656\pi$
$97$	−1.30385	−	4.86603i	−0.132386	−	0.494070i	−0.999995	+	0.00317651i	$0.998989\pi$
$98$	3.48477	+	13.0053i	0.352015	+	1.31374i
$99$		0			0
$100$	2.46410	−	4.26795i	0.246410	−	0.426795i
$101$	0.568406	+	0.984508i	0.0565585	+	0.0979622i	0.892918	−	0.450219i	$-0.148654\pi$
$101$	0.568406	+	0.984508i	0.0565585	+	0.0979622i	−0.836360	+	0.548181i	$0.815321\pi$
$102$		0			0
$103$		−	8.66025i		−	0.853320i	−0.904412	−	0.426660i	$-0.859690\pi$
$103$		−	8.66025i		−	0.853320i	0.904412	−	0.426660i	$-0.140310\pi$
$104$		0			0
$105$		0			0
$106$	11.4641	+	3.07180i	1.11349	+	0.298359i
$107$	1.46498	−	0.845807i	0.141625	−	0.0817673i	−0.427513	−	0.904009i	$-0.640610\pi$
$107$	1.46498	−	0.845807i	0.141625	−	0.0817673i	0.569138	+	0.822242i	$0.307277\pi$
$108$		0			0
$109$	−11.2942	+	11.2942i	−1.08179	+	1.08179i	−0.0854483	+	0.996343i	$0.527232\pi$
$109$	−11.2942	+	11.2942i	−1.08179	+	1.08179i	−0.996343	+	0.0854483i	$0.972768\pi$
$110$	−22.1469	+	5.93426i	−2.11163	+	0.565809i
$111$		0			0
$112$	1.46410	+	1.46410i	0.138345	+	0.138345i
$113$	3.91447	+	2.26002i	0.368242	+	0.212605i	0.672690	−	0.739924i	$-0.265139\pi$
$113$	3.91447	+	2.26002i	0.368242	+	0.212605i	−0.304448	+	0.952529i	$0.598472\pi$
$114$		0			0
$115$	−1.73205	+	6.46410i	−0.161515	+	0.602781i
$116$	2.82843			0.262613
$117$		0			0
$118$	0.392305			0.0361146
$119$	0.896575	−	3.34607i	0.0821889	−	0.306733i
$120$		0			0
$121$	5.72243	+	3.30385i	0.520221	+	0.300350i
$122$	18.6622	+	18.6622i	1.68959	+	1.68959i
$123$		0			0
$124$	−12.6603	+	3.39230i	−1.13692	+	0.304638i
$125$	−4.89898	+	4.89898i	−0.438178	+	0.438178i
$126$		0			0
$127$	8.59808	−	4.96410i	0.762956	−	0.440493i	−0.0674001	−	0.997726i	$-0.521470\pi$
$127$	8.59808	−	4.96410i	0.762956	−	0.440493i	0.830356	+	0.557233i	$0.188137\pi$
$128$		0			0
$129$		0			0
$130$	−16.3923	+	10.9282i	−1.43770	+	0.958467i
$131$			7.07107i			0.617802i	0.951094	+	0.308901i	$0.0999612\pi$
$131$			7.07107i			0.617802i	−0.951094	+	0.308901i	$0.900039\pi$
$132$		0			0
$133$	−1.63397	−	2.83013i	−0.141684	−	0.245403i
$134$	−7.86611	+	13.6245i	−0.679528	+	1.17698i
$135$		0			0
$136$		0			0
$137$	−3.86370	−	14.4195i	−0.330098	−	1.23194i	−0.909086	−	0.416608i	$-0.863219\pi$
$137$	−3.86370	−	14.4195i	−0.330098	−	1.23194i	0.578988	−	0.815336i	$-0.303448\pi$
$138$		0			0
$139$	−1.69615	+	2.93782i	−0.143866	+	0.249183i	−0.928949	−	0.370207i	$-0.879287\pi$
$139$	−1.69615	+	2.93782i	−0.143866	+	0.249183i	0.785083	+	0.619390i	$0.212620\pi$
$140$	−1.41421	−	2.44949i	−0.119523	−	0.207020i
$141$		0			0
$142$		−	31.3205i		−	2.62836i
$143$	14.8356	+	2.96713i	1.24062	+	0.248124i
$144$		0			0
$145$	3.73205	+	1.00000i	0.309930	+	0.0830455i
$146$	7.10823	−	4.10394i	0.588282	−	0.339644i
$147$		0			0
$148$	−4.53590	+	4.53590i	−0.372849	+	0.372849i
$149$	−6.17449	+	1.65445i	−0.505834	+	0.135538i	−0.502706	−	0.864457i	$-0.667662\pi$
$149$	−6.17449	+	1.65445i	−0.505834	+	0.135538i	−0.00312781	+	0.999995i	$0.500996\pi$
$150$		0			0
$151$	−8.46410	−	8.46410i	−0.688799	−	0.688799i	0.273168	−	0.961966i	$-0.411929\pi$
$151$	−8.46410	−	8.46410i	−0.688799	−	0.688799i	−0.961966	+	0.273168i	$0.911929\pi$
$152$		0			0
$153$		0			0
$154$	−1.12436	+	4.19615i	−0.0906032	+	0.338136i
$155$	−17.9043			−1.43811
$156$		0			0
$157$	−11.3923			−0.909205			−0.454602	−	0.890694i	$-0.650219\pi$
$157$	−11.3923			−0.909205			−0.454602	+	0.890694i	$0.650219\pi$
$158$	3.72500	−	13.9019i	0.296345	−	1.10598i
$159$		0			0
$160$	18.9282	+	10.9282i	1.49641	+	0.863950i
$161$	0.896575	+	0.896575i	0.0706600	+	0.0706600i
$162$		0			0
$163$	16.0622	−	4.30385i	1.25809	−	0.337103i	0.432632	−	0.901571i	$-0.357585\pi$
$163$	16.0622	−	4.30385i	1.25809	−	0.337103i	0.825455	+	0.564467i	$0.190918\pi$
$164$	2.07055	−	2.07055i	0.161683	−	0.161683i
$165$		0			0
$166$	−12.9282	+	7.46410i	−1.00342	+	0.579327i
$167$	5.27792	+	1.41421i	0.408417	+	0.109435i	0.457178	−	0.889375i	$-0.348860\pi$
$167$	5.27792	+	1.41421i	0.408417	+	0.109435i	−0.0487602	+	0.998811i	$0.515527\pi$
$168$		0			0
$169$	12.8923	−	1.66987i	0.991716	−	0.128452i
$170$		−	36.5665i		−	2.80452i
$171$		0			0
$172$	7.73205	+	13.3923i	0.589563	+	1.02115i
$173$	7.02030	−	12.1595i	0.533744	−	0.924471i	−0.465480	−	0.885059i	$-0.654118\pi$
$173$	7.02030	−	12.1595i	0.533744	−	0.924471i	0.999223	−	0.0394122i	$-0.0125486\pi$
$174$		0			0
$175$	−0.330127	−	1.23205i	−0.0249553	−	0.0931343i
$176$	−4.34418	−	16.2127i	−0.327455	−	1.22208i
$177$		0			0
$178$	4.53590	−	7.85641i	0.339980	−	0.588863i
$179$	−8.24504	−	14.2808i	−0.616264	−	1.06740i	−0.990161	−	0.139930i	$-0.955312\pi$
$179$	−8.24504	−	14.2808i	−0.616264	−	1.06740i	0.373898	−	0.927470i	$-0.378021\pi$
$180$		0			0
$181$		−	6.00000i		−	0.445976i	−0.974821	−	0.222988i	$-0.928419\pi$
$181$		−	6.00000i		−	0.445976i	0.974821	−	0.222988i	$-0.0715812\pi$
$182$	0.240237	+	3.72500i	0.0178075	+	0.276116i
$183$		0			0
$184$		0			0
$185$	−7.58871	+	4.38134i	−0.557933	+	0.322123i
$186$		0			0
$187$	−19.8564	+	19.8564i	−1.45204	+	1.45204i
$188$	6.31319	−	1.69161i	0.460437	−	0.123374i
$189$		0			0
$190$	−24.3923	−	24.3923i	−1.76960	−	1.76960i
$191$	11.2629	+	6.50266i	0.814958	+	0.470516i	0.848675	−	0.528915i	$-0.177401\pi$
$191$	11.2629	+	6.50266i	0.814958	+	0.470516i	−0.0337168	+	0.999431i	$0.510734\pi$
$192$		0			0
$193$	0.303848	−	1.13397i	0.0218714	−	0.0816253i	−0.954128	−	0.299401i	$-0.903213\pi$
$193$	0.303848	−	1.13397i	0.0218714	−	0.0816253i	0.975999	+	0.217775i	$0.0698800\pi$
$194$	−10.0754			−0.723369
$195$		0			0
$196$	13.4641			0.961722
$197$	1.83032	−	6.83083i	0.130405	−	0.486677i	−0.869570	−	0.493810i	$-0.835604\pi$
$197$	1.83032	−	6.83083i	0.130405	−	0.486677i	0.999975	+	0.00713319i	$0.00227058\pi$
$198$		0			0
$199$	−0.401924	−	0.232051i	−0.0284916	−	0.0164496i	0.485687	−	0.874133i	$-0.338570\pi$
$199$	−0.401924	−	0.232051i	−0.0284916	−	0.0164496i	−0.514178	+	0.857683i	$0.671903\pi$
$200$		0			0
$201$		0			0
$202$	2.19615	−	0.588457i	0.154521	−	0.0414037i
$203$	0.517638	−	0.517638i	0.0363311	−	0.0363311i
$204$		0			0
$205$	3.46410	−	2.00000i	0.241943	−	0.139686i
$206$	−16.7303	−	4.48288i	−1.16566	−	0.312337i
$207$		0			0
$208$	−8.00000	−	12.0000i	−0.554700	−	0.832050i
$209$			26.4911i			1.83243i
$210$		0			0
$211$	7.59808	+	13.1603i	0.523073	+	0.905989i	0.999639	+	0.0268507i	$0.00854788\pi$
$211$	7.59808	+	13.1603i	0.523073	+	0.905989i	−0.476566	+	0.879139i	$0.658119\pi$
$212$	5.93426	−	10.2784i	0.407566	−	0.705926i
$213$		0			0
$214$	−0.875644	−	3.26795i	−0.0598578	−	0.223392i
$215$	5.46739	+	20.4046i	0.372873	+	1.39158i
$216$		0			0
$217$	−1.69615	+	2.93782i	−0.115142	+	0.199432i
$218$	15.9725	+	27.6651i	1.08179	+	1.87372i
$219$		0			0
$220$			22.9282i			1.54582i
$221$	−10.6945	+	21.6293i	−0.719392	+	1.45494i
$222$		0			0
$223$	−23.7583	−	6.36603i	−1.59098	−	0.426301i	−0.648674	−	0.761066i	$-0.724676\pi$
$223$	−23.7583	−	6.36603i	−1.59098	−	0.426301i	−0.942301	+	0.334766i	$0.891343\pi$
$224$	3.58630	−	2.07055i	0.239620	−	0.138345i
$225$		0			0
$226$	6.39230	−	6.39230i	0.425210	−	0.425210i
$227$	6.31319	−	1.69161i	0.419021	−	0.112276i	−0.0431468	−	0.999069i	$-0.513738\pi$
$227$	6.31319	−	1.69161i	0.419021	−	0.112276i	0.462168	+	0.886792i	$0.347072\pi$
$228$		0			0
$229$	−3.73205	−	3.73205i	−0.246621	−	0.246621i	0.572961	−	0.819582i	$-0.305794\pi$
$229$	−3.73205	−	3.73205i	−0.246621	−	0.246621i	−0.819582	+	0.572961i	$0.805794\pi$
$230$	11.5911	+	6.69213i	0.764295	+	0.441266i
$231$		0			0
$232$		0			0
$233$	−18.9396			−1.24077			−0.620387	−	0.784296i	$-0.713024\pi$
$233$	−18.9396			−1.24077			−0.620387	+	0.784296i	$0.713024\pi$
$234$		0			0
$235$	8.92820			0.582412
$236$	0.101536	−	0.378937i	0.00660943	−	0.0246667i
$237$		0			0
$238$	−6.00000	−	3.46410i	−0.388922	−	0.224544i
$239$	−4.62158	−	4.62158i	−0.298945	−	0.298945i	0.541656	−	0.840601i	$-0.317798\pi$
$239$	−4.62158	−	4.62158i	−0.298945	−	0.298945i	−0.840601	+	0.541656i	$0.817798\pi$
$240$		0			0
$241$	−11.2942	+	3.02628i	−0.727525	+	0.194940i	−0.603527	−	0.797343i	$-0.706238\pi$
$241$	−11.2942	+	3.02628i	−0.727525	+	0.194940i	−0.123998	+	0.992282i	$0.539572\pi$
$242$	9.34469	−	9.34469i	0.600700	−	0.600700i
$243$		0			0
$244$	22.8564	−	13.1962i	1.46323	−	0.844797i
$245$	17.7656	+	4.76028i	1.13500	+	0.304123i
$246$		0			0
$247$	7.29423	+	21.5622i	0.464121	+	1.37197i
$248$		0			0
$249$		0			0
$250$	6.92820	+	12.0000i	0.438178	+	0.758947i
$251$	−10.6945	+	18.5235i	−0.675033	+	1.16919i	0.301426	+	0.953490i	$0.402537\pi$
$251$	−10.6945	+	18.5235i	−0.675033	+	1.16919i	−0.976459	+	0.215702i	$0.930796\pi$
$252$		0			0
$253$	−2.66025	−	9.92820i	−0.167249	−	0.624181i
$254$	−5.13922	−	19.1798i	−0.322463	−	1.20345i
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	−0.656339	−	1.13681i	−0.0409413	−	0.0709124i	0.844829	−	0.535037i	$-0.179702\pi$
$257$	−0.656339	−	1.13681i	−0.0409413	−	0.0709124i	−0.885770	+	0.464125i	$0.846369\pi$
$258$		0			0
$259$			1.66025i			0.103163i
$260$	6.31319	+	18.6622i	0.391528	+	1.15738i
$261$		0			0
$262$	13.6603	+	3.66025i	0.843933	+	0.226131i
$263$	−20.3166	+	11.7298i	−1.25278	+	0.723291i	−0.971660	−	0.236382i	$-0.924038\pi$
$263$	−20.3166	+	11.7298i	−1.25278	+	0.723291i	−0.281117	+	0.959674i	$0.590705\pi$
$264$		0			0
$265$	11.4641	−	11.4641i	0.704234	−	0.704234i
$266$	−6.31319	+	1.69161i	−0.387087	+	0.103720i
$267$		0			0
$268$	11.1244	+	11.1244i	0.679528	+	0.679528i
$269$	−17.8671	−	10.3156i	−1.08938	−	0.628953i	−0.155968	−	0.987762i	$-0.549850\pi$
$269$	−17.8671	−	10.3156i	−1.08938	−	0.628953i	−0.933411	+	0.358809i	$0.883183\pi$
$270$		0			0
$271$	2.62436	−	9.79423i	0.159418	−	0.594957i	−0.839268	−	0.543718i	$-0.817016\pi$
$271$	2.62436	−	9.79423i	0.159418	−	0.594957i	0.998686	−	0.0512393i	$-0.0163171\pi$
$272$	26.7685			1.62308
$273$		0			0
$274$	−29.8564			−1.80369
$275$	−2.67612	+	9.98743i	−0.161376	+	0.602265i
$276$		0			0
$277$	17.1962	+	9.92820i	1.03322	+	0.596528i	0.917904	−	0.396801i	$-0.129880\pi$
$277$	17.1962	+	9.92820i	1.03322	+	0.596528i	0.115312	+	0.993329i	$0.463213\pi$
$278$	4.79744	+	4.79744i	0.287732	+	0.287732i
$279$		0			0
$280$		0			0
$281$	16.6288	−	16.6288i	0.991990	−	0.991990i	−0.00797773	−	0.999968i	$-0.502539\pi$
$281$	16.6288	−	16.6288i	0.991990	−	0.991990i	0.999968	+	0.00797773i	$0.00253942\pi$
$282$		0			0
$283$	19.7942	−	11.4282i	1.17664	−	0.679336i	0.221409	−	0.975181i	$-0.428934\pi$
$283$	19.7942	−	11.4282i	1.17664	−	0.679336i	0.955236	+	0.295845i	$0.0956012\pi$
$284$	−30.2533	−	8.10634i	−1.79520	−	0.481023i
$285$		0			0
$286$	13.4115	−	27.1244i	0.793041	−	1.60390i
$287$		−	0.757875i		−	0.0447359i
$288$		0			0
$289$	−13.8923	−	24.0622i	−0.817194	−	1.41542i
$290$	3.86370	−	6.69213i	0.226884	−	0.392975i
$291$		0			0
$292$	−2.12436	−	7.92820i	−0.124319	−	0.463963i
$293$	−3.05506	−	11.4016i	−0.178479	−	0.666091i	−0.995933	−	0.0900977i	$-0.971282\pi$
$293$	−3.05506	−	11.4016i	−0.178479	−	0.666091i	0.817454	−	0.575993i	$-0.195385\pi$
$294$		0			0
$295$	0.267949	−	0.464102i	0.0156006	−	0.0270210i
$296$		0			0
$297$		0			0
$298$			12.7846i			0.740593i
$299$	−4.89898	−	7.34847i	−0.283315	−	0.424973i
$300$		0			0
$301$	3.86603	+	1.03590i	0.222834	+	0.0597082i
$302$	−20.7327	+	11.9700i	−1.19303	+	0.688799i
$303$		0			0
$304$	17.8564	−	17.8564i	1.02414	−	1.02414i
$305$	34.8241	−	9.33109i	1.99402	−	0.534297i
$306$		0			0
$307$	19.2942	+	19.2942i	1.10118	+	1.10118i	0.994269	+	0.106911i	$0.0340961\pi$
$307$	19.2942	+	19.2942i	1.10118	+	1.10118i	0.106911	+	0.994269i	$0.465904\pi$
$308$	3.76217	+	2.17209i	0.214369	+	0.123766i
$309$		0			0
$310$	−9.26795	+	34.5885i	−0.526384	+	1.96449i
$311$	11.1106			0.630026			0.315013	−	0.949087i	$-0.397991\pi$
$311$	11.1106			0.630026			0.315013	+	0.949087i	$0.397991\pi$
$312$		0			0
$313$	−3.19615			−0.180657			−0.0903286	−	0.995912i	$-0.528792\pi$
$313$	−3.19615			−0.180657			−0.0903286	+	0.995912i	$0.528792\pi$
$314$	−5.89709	+	22.0082i	−0.332792	+	1.24200i
$315$		0			0
$316$	−12.4641	−	7.19615i	−0.701160	−	0.404815i
$317$	10.5558	+	10.5558i	0.592875	+	0.592875i	0.938407	−	0.345532i	$-0.112302\pi$
$317$	10.5558	+	10.5558i	0.592875	+	0.592875i	−0.345532	+	0.938407i	$0.612302\pi$
$318$		0			0
$319$	−5.73205	+	1.53590i	−0.320933	+	0.0859938i
$320$	15.4548	−	15.4548i	0.863950	−	0.863950i
$321$		0			0
$322$	2.19615	−	1.26795i	0.122387	−	0.0706600i
$323$	−40.8091	−	10.9348i	−2.27068	−	0.608427i
$324$		0			0
$325$	0.571797	+	8.86603i	0.0317176	+	0.491799i
$326$		−	33.2576i		−	1.84197i
$327$		0			0
$328$		0			0
$329$	0.845807	−	1.46498i	0.0466309	−	0.0807670i
$330$		0			0
$331$	1.40192	+	5.23205i	0.0770567	+	0.287580i	0.993692	−	0.112146i	$-0.0357723\pi$
$331$	1.40192	+	5.23205i	0.0770567	+	0.287580i	−0.916635	+	0.399725i	$0.869106\pi$
$332$	3.86370	+	14.4195i	0.212048	+	0.791375i
$333$		0			0
$334$	5.46410	−	9.46410i	0.298982	−	0.517853i
$335$	10.7453	+	18.6114i	0.587079	+	1.01685i
$336$		0			0
$337$			27.9282i			1.52135i	0.649135	+	0.760673i	$0.275131\pi$
$337$			27.9282i			1.52135i	−0.649135	+	0.760673i	$0.724869\pi$
$338$	3.44760	−	25.7704i	0.187525	−	1.40173i
$339$		0			0
$340$	−35.3205	−	9.46410i	−1.91552	−	0.513263i
$341$	23.8150	−	13.7496i	1.28965	−	0.744582i
$342$		0			0
$343$	5.02628	−	5.02628i	0.271394	−	0.271394i
$344$		0			0
$345$		0			0
$346$	−19.8564	−	19.8564i	−1.06749	−	1.06749i
$347$	−25.2156	−	14.5582i	−1.35364	−	0.781527i	−0.364887	−	0.931052i	$-0.618892\pi$
$347$	−25.2156	−	14.5582i	−1.35364	−	0.781527i	−0.988758	+	0.149525i	$0.952226\pi$
$348$		0			0
$349$	−0.545517	+	2.03590i	−0.0292009	+	0.108979i	−0.978988	−	0.203917i	$-0.934633\pi$
$349$	−0.545517	+	2.03590i	−0.0292009	+	0.108979i	0.949787	+	0.312896i	$0.101299\pi$
$350$	−2.55103			−0.136358
$351$		0			0
$352$	−33.5692			−1.78925
$353$	−0.466870	+	1.74238i	−0.0248490	+	0.0927377i	−0.977237	−	0.212152i	$-0.931953\pi$
$353$	−0.466870	+	1.74238i	−0.0248490	+	0.0927377i	0.952388	+	0.304890i	$0.0986195\pi$
$354$		0			0
$355$	−37.0526	−	21.3923i	−1.96655	−	1.13539i
$356$	−6.41473	−	6.41473i	−0.339980	−	0.339980i
$357$		0			0
$358$	−31.8564	+	8.53590i	−1.68366	+	0.451136i
$359$	7.07107	−	7.07107i	0.373197	−	0.373197i	−0.495443	−	0.868640i	$-0.664994\pi$
$359$	7.07107	−	7.07107i	0.373197	−	0.373197i	0.868640	+	0.495443i	$0.164994\pi$
$360$		0			0
$361$	−18.0622	+	10.4282i	−0.950641	+	0.548853i
$362$	−11.5911	−	3.10583i	−0.609215	−	0.163239i
$363$		0			0
$364$	3.66025	+	0.732051i	0.191849	+	0.0383699i
$365$		−	11.2122i		−	0.586872i
$366$		0			0
$367$	2.69615	+	4.66987i	0.140738	+	0.243765i	0.927775	−	0.373141i	$-0.121719\pi$
$367$	2.69615	+	4.66987i	0.140738	+	0.243765i	−0.787037	+	0.616906i	$0.788386\pi$
$368$	−4.89898	+	8.48528i	−0.255377	+	0.442326i
$369$		0			0
$370$	4.53590	+	16.9282i	0.235810	+	0.880055i
$371$	−0.795040	−	2.96713i	−0.0412764	−	0.154046i
$372$		0			0
$373$	3.79423	−	6.57180i	0.196458	−	0.340275i	−0.750920	−	0.660394i	$-0.770389\pi$
$373$	3.79423	−	6.57180i	0.196458	−	0.340275i	0.947377	+	0.320119i	$0.103723\pi$
$374$	28.0812	+	48.6381i	1.45204	+	2.51501i
$375$		0			0
$376$		0			0
$377$	−4.24264	+	2.82843i	−0.218507	+	0.145671i
$378$		0			0
$379$	22.3564	+	5.99038i	1.14837	+	0.307705i	0.782312	−	0.622886i	$-0.214040\pi$
$379$	22.3564	+	5.99038i	1.14837	+	0.307705i	0.366059	+	0.930592i	$0.380707\pi$
$380$	−29.8744	+	17.2480i	−1.53252	+	0.884802i
$381$		0			0
$382$	18.3923	−	18.3923i	0.941032	−	0.941032i
$383$	9.65926	−	2.58819i	0.493565	−	0.132250i	−0.00344689	−	0.999994i	$-0.501097\pi$
$383$	9.65926	−	2.58819i	0.493565	−	0.132250i	0.497012	+	0.867744i	$0.334431\pi$
$384$		0			0
$385$	4.19615	+	4.19615i	0.213856	+	0.213856i
$386$	−2.03339	−	1.17398i	−0.103497	−	0.0597539i
$387$		0			0
$388$	−2.60770	+	9.73205i	−0.132386	+	0.494070i
$389$	6.69213			0.339304			0.169652	−	0.985504i	$-0.445736\pi$
$389$	6.69213			0.339304			0.169652	+	0.985504i	$0.445736\pi$
$390$		0			0
$391$	16.3923			0.828994
$392$		0			0
$393$		0			0
$394$	−12.2487	−	7.07180i	−0.617081	−	0.356272i
$395$	−13.9019	−	13.9019i	−0.699480	−	0.699480i
$396$		0			0
$397$	−15.3301	+	4.10770i	−0.769397	+	0.206159i	−0.622105	−	0.782934i	$-0.713722\pi$
$397$	−15.3301	+	4.10770i	−0.769397	+	0.206159i	−0.147292	+	0.989093i	$0.547056\pi$
$398$	−0.656339	+	0.656339i	−0.0328993	+	0.0328993i
$399$		0			0
$400$	8.53590	−	4.92820i	0.426795	−	0.246410i
$401$	38.4983	+	10.3156i	1.92251	+	0.515136i	0.986673	+	0.162717i	$0.0520258\pi$
$401$	38.4983	+	10.3156i	1.92251	+	0.515136i	0.935842	+	0.352419i	$0.114641\pi$
$402$		0			0
$403$	15.5981	−	17.7487i	0.776996	−	0.884126i
$404$		−	2.27362i		−	0.113117i
$405$		0			0
$406$	−0.732051	−	1.26795i	−0.0363311	−	0.0629273i
$407$	6.72930	−	11.6555i	0.333559	−	0.577741i
$408$		0			0
$409$	−4.42820	−	16.5263i	−0.218961	−	0.817172i	−0.984735	−	0.174062i	$-0.944311\pi$
$409$	−4.42820	−	16.5263i	−0.218961	−	0.817172i	0.765774	−	0.643110i	$-0.222356\pi$
$410$	−2.07055	−	7.72741i	−0.102257	−	0.381629i
$411$		0			0
$412$	−8.66025	+	15.0000i	−0.426660	+	0.738997i
$413$	−0.0507680	−	0.0879327i	−0.00249813	−	0.00432689i
$414$		0			0
$415$			20.3923i			1.00102i
$416$	−27.3233	+	9.24316i	−1.33964	+	0.453183i
$417$		0			0
$418$	51.1769	+	13.7128i	2.50314	+	0.670716i
$419$	−2.92996	+	1.69161i	−0.143138	+	0.0826408i	−0.569859	−	0.821743i	$-0.693002\pi$
$419$	−2.92996	+	1.69161i	−0.143138	+	0.0826408i	0.426721	+	0.904383i	$0.359669\pi$
$420$		0			0
$421$	15.3660	−	15.3660i	0.748894	−	0.748894i	−0.225377	−	0.974272i	$-0.572361\pi$
$421$	15.3660	−	15.3660i	0.748894	−	0.748894i	0.974272	+	0.225377i	$0.0723615\pi$
$422$	29.3567	−	7.86611i	1.42906	−	0.382916i
$423$		0			0
$424$		0			0
$425$	−14.2808	−	8.24504i	−0.692722	−	0.399943i
$426$		0			0
$427$	1.76795	−	6.59808i	0.0855571	−	0.319303i
$428$	−3.38323			−0.163535
$429$		0			0
$430$	42.2487			2.03741
$431$	4.10394	−	15.3161i	0.197680	−	0.737751i	−0.793877	−	0.608078i	$-0.791941\pi$
$431$	4.10394	−	15.3161i	0.197680	−	0.737751i	0.991557	−	0.129673i	$-0.0413927\pi$
$432$		0			0
$433$	−15.9904	−	9.23205i	−0.768449	−	0.443664i	0.0638723	−	0.997958i	$-0.479655\pi$
$433$	−15.9904	−	9.23205i	−0.768449	−	0.443664i	−0.832321	+	0.554294i	$0.812988\pi$
$434$	4.79744	+	4.79744i	0.230285	+	0.230285i
$435$		0			0
$436$	30.8564	−	8.26795i	1.47775	−	0.395963i
$437$	10.9348	−	10.9348i	0.523081	−	0.523081i
$438$		0			0
$439$	−8.30385	+	4.79423i	−0.396321	+	0.228816i	−0.684895	−	0.728641i	$-0.740152\pi$
$439$	−8.30385	+	4.79423i	−0.396321	+	0.228816i	0.288574	+	0.957457i	$0.406819\pi$
$440$		0			0
$441$		0			0
$442$	36.2487	+	31.8564i	1.72418	+	1.51525i
$443$			23.7370i			1.12778i	0.825850	+	0.563890i	$0.190696\pi$
$443$			23.7370i			1.12778i	−0.825850	+	0.563890i	$0.809304\pi$
$444$		0			0
$445$	−6.19615	−	10.7321i	−0.293726	−	0.508748i
$446$	−24.5964	+	42.6023i	−1.16467	+	2.01728i
$447$		0			0
$448$	−1.07180	−	4.00000i	−0.0506376	−	0.188982i
$449$	2.26002	+	8.43451i	0.106657	+	0.398049i	0.998528	−	0.0542408i	$-0.0172738\pi$
$449$	2.26002	+	8.43451i	0.106657	+	0.398049i	−0.891871	+	0.452290i	$0.850607\pi$
$450$		0			0
$451$	−3.07180	+	5.32051i	−0.144645	+	0.250533i
$452$	−4.52004	−	7.82894i	−0.212605	−	0.368242i
$453$		0			0
$454$		−	13.0718i		−	0.613490i
$455$	4.57081	+	2.26002i	0.214283	+	0.105951i
$456$		0			0
$457$	−11.0622	−	2.96410i	−0.517467	−	0.138655i	−0.00937223	−	0.999956i	$-0.502983\pi$
$457$	−11.0622	−	2.96410i	−0.517467	−	0.138655i	−0.508095	+	0.861301i	$0.669650\pi$
$458$	−9.14162	+	5.27792i	−0.427160	+	0.246621i
$459$		0			0
$460$	9.46410	−	9.46410i	0.441266	−	0.441266i
$461$	−2.58819	+	0.693504i	−0.120544	+	0.0322997i	−0.318587	−	0.947894i	$-0.603208\pi$
$461$	−2.58819	+	0.693504i	−0.120544	+	0.0322997i	0.198043	+	0.980193i	$0.436542\pi$
$462$		0			0
$463$	−4.83013	−	4.83013i	−0.224475	−	0.224475i	0.585905	−	0.810380i	$-0.300739\pi$
$463$	−4.83013	−	4.83013i	−0.224475	−	0.224475i	−0.810380	+	0.585905i	$0.800739\pi$
$464$	4.89898	+	2.82843i	0.227429	+	0.131306i
$465$		0			0
$466$	−9.80385	+	36.5885i	−0.454154	+	1.69493i
$467$	−22.0454			−1.02014			−0.510070	−	0.860133i	$-0.670380\pi$
$467$	−22.0454			−1.02014			−0.510070	+	0.860133i	$0.670380\pi$
$468$		0			0
$469$	4.07180			0.188018
$470$	4.62158	−	17.2480i	0.213177	−	0.795589i
$471$		0			0
$472$		0			0
$473$	−22.9420	−	22.9420i	−1.05487	−	1.05487i
$474$		0			0
$475$	−15.0263	+	4.02628i	−0.689453	+	0.184738i
$476$	−4.89898	+	4.89898i	−0.224544	+	0.224544i
$477$		0			0
$478$	−11.3205	+	6.53590i	−0.517788	+	0.298945i
$479$	−3.86370	−	1.03528i	−0.176537	−	0.0473030i	0.169468	−	0.985536i	$-0.445795\pi$
$479$	−3.86370	−	1.03528i	−0.176537	−	0.0473030i	−0.346005	+	0.938233i	$0.612462\pi$
$480$		0			0
$481$	2.26795	−	11.3397i	0.103410	−	0.517048i
$482$			23.3853i			1.06517i
$483$		0			0
$484$	−6.60770	−	11.4449i	−0.300350	−	0.520221i
$485$	−6.88160	+	11.9193i	−0.312477	+	0.541227i
$486$		0			0
$487$	−5.16987	−	19.2942i	−0.234269	−	0.874305i	−0.978477	−	0.206357i	$-0.933839\pi$
$487$	−5.16987	−	19.2942i	−0.234269	−	0.874305i	0.744208	−	0.667948i	$-0.232827\pi$
$488$		0			0
$489$		0			0
$490$	18.3923	−	31.8564i	0.830880	−	1.43913i
$491$	13.0561	+	22.6138i	0.589213	+	1.02055i	0.994336	+	0.106285i	$0.0338955\pi$
$491$	13.0561	+	22.6138i	0.589213	+	1.02055i	−0.405123	+	0.914262i	$0.632771\pi$
$492$		0			0
$493$		−	9.46410i		−	0.426242i
$494$	45.4307	−	2.92996i	2.04402	−	0.131825i
$495$		0			0
$496$	−25.3205	−	6.78461i	−1.13692	−	0.304638i
$497$	−7.02030	+	4.05317i	−0.314903	+	0.181810i
$498$		0			0
$499$	−24.2679	+	24.2679i	−1.08638	+	1.08638i	−0.0904848	+	0.995898i	$0.528842\pi$
$499$	−24.2679	+	24.2679i	−1.08638	+	1.08638i	−0.995898	+	0.0904848i	$0.971158\pi$
$500$	13.3843	−	3.58630i	0.598562	−	0.160384i
$501$		0			0
$502$	30.2487	+	30.2487i	1.35007	+	1.35007i
$503$	23.2702	+	13.4350i	1.03756	+	0.599038i	0.919143	−	0.393925i	$-0.128883\pi$
$503$	23.2702	+	13.4350i	1.03756	+	0.599038i	0.118422	+	0.992963i	$0.462216\pi$
$504$		0			0
$505$	0.803848	−	3.00000i	0.0357707	−	0.133498i
$506$	−20.5569			−0.913864
$507$		0			0
$508$	−19.8564			−0.880986
$509$	2.15849	−	8.05558i	0.0956732	−	0.357057i	−0.901447	−	0.432888i	$-0.857494\pi$
$509$	2.15849	−	8.05558i	0.0956732	−	0.357057i	0.997121	+	0.0758313i	$0.0241611\pi$
$510$		0			0
$511$	−1.83975	−	1.06218i	−0.0813856	−	0.0469880i
$512$	22.6274	+	22.6274i	1.00000	+	1.00000i
$513$		0			0
$514$	−2.53590	+	0.679492i	−0.111854	+	0.0299711i
$515$	−16.7303	+	16.7303i	−0.737226	+	0.737226i
$516$		0			0
$517$	−11.8756	+	6.85641i	−0.522290	+	0.301544i
$518$	3.20736	+	0.859411i	0.140924	+	0.0377603i
$519$		0			0
$520$		0			0
$521$		−	26.5654i		−	1.16385i	−0.813241	−	0.581927i	$-0.802299\pi$
$521$		−	26.5654i		−	1.16385i	0.813241	−	0.581927i	$-0.197701\pi$
$522$		0			0
$523$	21.3923	+	37.0526i	0.935420	+	1.62020i	0.773883	+	0.633329i	$0.218312\pi$
$523$	21.3923	+	37.0526i	0.935420	+	1.62020i	0.161537	+	0.986867i	$0.448355\pi$
$524$	7.07107	−	12.2474i	0.308901	−	0.535032i
$525$		0			0
$526$	12.1436	+	45.3205i	0.529486	+	1.97607i
$527$	11.3509	+	42.3620i	0.494452	+	1.84532i
$528$		0			0
$529$	8.50000	−	14.7224i	0.369565	−	0.640106i
$530$	−16.2127	−	28.0812i	−0.704234	−	1.21977i
$531$		0			0
$532$			6.53590i			0.283367i
$533$	−1.03528	+	5.17638i	−0.0448428	+	0.224214i
$534$		0			0
$535$	−4.46410	−	1.19615i	−0.193000	−	0.0517142i
$536$		0			0
$537$		0			0
$538$	−29.1769	+	29.1769i	−1.25791	+	1.25791i
$539$	−27.2862	+	7.31130i	−1.17530	+	0.314920i
$540$		0			0
$541$	−30.6865	−	30.6865i	−1.31932	−	1.31932i	−0.914316	−	0.405001i	$-0.867271\pi$
$541$	−30.6865	−	30.6865i	−1.31932	−	1.31932i	−0.405001	−	0.914316i	$-0.632729\pi$
$542$	−17.5625	−	10.1397i	−0.754375	−	0.435539i
$543$		0			0
$544$	13.8564	−	51.7128i	0.594089	−	2.21717i
$545$	43.6375			1.86923
$546$		0			0
$547$	−17.3923			−0.743641			−0.371821	−	0.928305i	$-0.621266\pi$
$547$	−17.3923			−0.743641			−0.371821	+	0.928305i	$0.621266\pi$
$548$	−7.72741	+	28.8391i	−0.330098	+	1.23194i
$549$		0			0
$550$	17.9090	+	10.3397i	0.763641	+	0.440888i
$551$	−6.31319	−	6.31319i	−0.268951	−	0.268951i
$552$		0			0
$553$	−3.59808	+	0.964102i	−0.153006	+	0.0409978i
$554$	28.0812	−	28.0812i	1.19306	−	1.19306i
$555$		0			0
$556$	5.87564	−	3.39230i	0.249183	−	0.143866i
$557$	4.38134	+	1.17398i	0.185643	+	0.0497430i	0.350443	−	0.936584i	$-0.386031\pi$
$557$	4.38134	+	1.17398i	0.185643	+	0.0497430i	−0.164800	+	0.986327i	$0.552698\pi$
$558$		0			0
$559$	−24.9904	−	12.3564i	−1.05698	−	0.522620i
$560$		−	5.65685i		−	0.239046i
$561$		0			0
$562$	−23.5167	−	40.7321i	−0.991990	−	1.71818i
$563$	14.8492	−	25.7196i	0.625821	−	1.08395i	−0.362561	−	0.931960i	$-0.618097\pi$
$563$	14.8492	−	25.7196i	0.625821	−	1.08395i	0.988381	−	0.151993i	$-0.0485693\pi$
$564$		0			0
$565$	−3.19615	−	11.9282i	−0.134463	−	0.501823i
$566$	−11.8313	−	44.1552i	−0.497309	−	1.85598i
$567$		0			0
$568$		0			0
$569$	−3.67423	−	6.36396i	−0.154032	−	0.266791i	0.778674	−	0.627428i	$-0.215893\pi$
$569$	−3.67423	−	6.36396i	−0.154032	−	0.266791i	−0.932706	+	0.360637i	$0.882559\pi$
$570$		0			0
$571$			21.7128i			0.908653i	0.890835	+	0.454326i	$0.150120\pi$
$571$			21.7128i			0.908653i	−0.890835	+	0.454326i	$0.849880\pi$
$572$	−22.7290	−	19.9749i	−0.950345	−	0.835191i
$573$		0			0
$574$	−1.46410	−	0.392305i	−0.0611104	−	0.0163745i
$575$	5.22715	−	3.01790i	0.217987	−	0.125855i
$576$		0			0
$577$	9.19615	−	9.19615i	0.382841	−	0.382841i	−0.489284	−	0.872125i	$-0.662742\pi$
$577$	9.19615	−	9.19615i	0.382841	−	0.382841i	0.872125	+	0.489284i	$0.162742\pi$
$578$	−53.6757	+	14.3824i	−2.23262	+	0.598228i
$579$		0			0
$580$	−5.46410	−	5.46410i	−0.226884	−	0.226884i
$581$	3.34607	+	1.93185i	0.138818	+	0.0801467i
$582$		0			0
$583$	−6.44486	+	24.0526i	−0.266919	+	0.996155i
$584$		0			0
$585$		0			0
$586$	−23.6077			−0.975225
$587$	9.98743	−	37.2736i	0.412225	−	1.53845i	−0.378104	−	0.925763i	$-0.623424\pi$
$587$	9.98743	−	37.2736i	0.412225	−	1.53845i	0.790329	−	0.612682i	$-0.209910\pi$
$588$		0			0
$589$	35.8301	+	20.6865i	1.47635	+	0.852374i
$590$	−0.757875	−	0.757875i	−0.0312012	−	0.0312012i
$591$		0			0
$592$	−12.3923	+	3.32051i	−0.509321	+	0.136472i
$593$	−4.10394	+	4.10394i	−0.168529	+	0.168529i	−0.786332	−	0.617804i	$-0.788023\pi$
$593$	−4.10394	+	4.10394i	−0.168529	+	0.168529i	0.617804	+	0.786332i	$0.288023\pi$
$594$		0			0
$595$	−8.19615	+	4.73205i	−0.336009	+	0.193995i
$596$	12.3490	+	3.30890i	0.505834	+	0.135538i
$597$		0			0
$598$	−16.7321	+	5.66025i	−0.684224	+	0.231465i
$599$		−	15.2789i		−	0.624281i	−0.950036	−	0.312140i	$-0.898954\pi$
$599$		−	15.2789i		−	0.624281i	0.950036	−	0.312140i	$-0.101046\pi$
$600$		0			0
$601$	18.3923	+	31.8564i	0.750238	+	1.29945i	0.947707	+	0.319141i	$0.103394\pi$
$601$	18.3923	+	31.8564i	0.750238	+	1.29945i	−0.197470	+	0.980309i	$0.563272\pi$
$602$	4.00240	−	6.93237i	0.163126	−	0.282542i
$603$		0			0
$604$	6.19615	+	23.1244i	0.252118	+	0.940917i
$605$	−4.67235	−	17.4374i	−0.189958	−	0.708932i
$606$		0			0
$607$	19.1962	−	33.2487i	0.779148	−	1.34952i	−0.153286	−	0.988182i	$-0.548985\pi$
$607$	19.1962	−	33.2487i	0.779148	−	1.34952i	0.932433	−	0.361342i	$-0.117681\pi$
$608$	−25.2528	−	43.7391i	−1.02414	−	1.77385i
$609$		0			0
$610$		−	72.1051i		−	2.91945i
$611$	−7.77817	+	8.85062i	−0.314671	+	0.358058i
$612$		0			0
$613$	−6.33013	−	1.69615i	−0.255671	−	0.0685070i	0.128707	−	0.991683i	$-0.458917\pi$
$613$	−6.33013	−	1.69615i	−0.255671	−	0.0685070i	−0.384378	+	0.923176i	$0.625584\pi$
$614$	47.2610	−	27.2862i	1.90730	−	1.10118i
$615$		0			0
$616$		0			0
$617$	−29.9251	+	8.01841i	−1.20474	+	0.322809i	−0.804696	−	0.593687i	$-0.797672\pi$
$617$	−29.9251	+	8.01841i	−1.20474	+	0.322809i	−0.400044	+	0.916496i	$0.631005\pi$
$618$		0			0
$619$	12.8301	+	12.8301i	0.515686	+	0.515686i	0.916263	−	0.400577i	$-0.131190\pi$
$619$	12.8301	+	12.8301i	0.515686	+	0.515686i	−0.400577	+	0.916263i	$0.631190\pi$
$620$	31.0112	+	17.9043i	1.24544	+	0.719054i
$621$		0			0
$622$	5.75129	−	21.4641i	0.230606	−	0.860632i
$623$	−2.34795			−0.0940688
$624$		0			0
$625$	31.2487			1.24995
$626$	−1.65445	+	6.17449i	−0.0661251	+	0.246782i
$627$		0			0
$628$	19.7321	+	11.3923i	0.787395	+	0.454602i
$629$	15.1774	+	15.1774i	0.605163	+	0.605163i
$630$		0			0
$631$	38.4545	−	10.3038i	1.53085	−	0.410190i	0.607553	−	0.794279i	$-0.292151\pi$
$631$	38.4545	−	10.3038i	1.53085	−	0.410190i	0.923296	+	0.384090i	$0.125485\pi$
$632$		0			0
$633$		0			0
$634$	25.8564	−	14.9282i	1.02689	−	0.592875i
$635$	−26.2001	−	7.02030i	−1.03972	−	0.278592i
$636$		0			0
$637$	−20.1962	+	13.4641i	−0.800201	+	0.533467i
$638$			11.8685i			0.469879i
$639$		0			0
$640$		0			0
$641$	−13.4722	+	23.3345i	−0.532120	+	0.921658i	0.467177	+	0.884164i	$0.345271\pi$
$641$	−13.4722	+	23.3345i	−0.532120	+	0.921658i	−0.999297	+	0.0374946i	$0.988062\pi$
$642$		0			0
$643$	5.96410	+	22.2583i	0.235201	+	0.877783i	0.978058	+	0.208333i	$0.0668036\pi$
$643$	5.96410	+	22.2583i	0.235201	+	0.877783i	−0.742857	+	0.669450i	$0.766530\pi$
$644$	−0.656339	−	2.44949i	−0.0258634	−	0.0965234i
$645$		0			0
$646$	−42.2487	+	73.1769i	−1.66225	+	2.87911i
$647$	−14.1929	−	24.5828i	−0.557981	−	0.966451i	−0.997665	−	0.0682984i	$-0.978243\pi$
$647$	−14.1929	−	24.5828i	−0.557981	−	0.966451i	0.439684	−	0.898152i	$-0.355090\pi$
$648$		0			0
$649$			0.823085i			0.0323089i
$650$	17.4238	+	3.48477i	0.683419	+	0.136684i
$651$		0			0
$652$	−32.1244	−	8.60770i	−1.25809	−	0.337103i
$653$	9.79796	−	5.65685i	0.383424	−	0.221370i	−0.295883	−	0.955224i	$-0.595614\pi$
$653$	9.79796	−	5.65685i	0.383424	−	0.221370i	0.679307	+	0.733854i	$0.262281\pi$
$654$		0			0
$655$	13.6603	−	13.6603i	0.533750	−	0.533750i
$656$	5.65685	−	1.51575i	0.220863	−	0.0591801i
$657$		0			0
$658$	−2.39230	−	2.39230i	−0.0932618	−	0.0932618i
$659$	40.3930	+	23.3209i	1.57349	+	0.908454i	0.995737	+	0.0922417i	$0.0294032\pi$
$659$	40.3930	+	23.3209i	1.57349	+	0.908454i	0.577752	+	0.816212i	$0.303930\pi$
$660$		0			0
$661$	9.59808	−	35.8205i	0.373322	−	1.39326i	−0.482459	−	0.875918i	$-0.660256\pi$
$661$	9.59808	−	35.8205i	0.373322	−	1.39326i	0.855781	−	0.517338i	$-0.173077\pi$
$662$	10.8332			0.421046
$663$		0			0
$664$		0			0
$665$	−2.31079	+	8.62398i	−0.0896086	+	0.334424i
$666$		0			0
$667$	3.00000	+	1.73205i	0.116160	+	0.0670653i
$668$	−7.72741	−	7.72741i	−0.298982	−	0.298982i
$669$		0			0
$670$	41.5167	−	11.1244i	1.60393	−	0.429771i
$671$	−39.1547	+	39.1547i	−1.51155	+	1.51155i
$672$		0			0
$673$	7.28461	−	4.20577i	0.280801	−	0.162121i	−0.352985	−	0.935629i	$-0.614833\pi$
$673$	7.28461	−	4.20577i	0.280801	−	0.162121i	0.633786	+	0.773508i	$0.281500\pi$
$674$	53.9531	+	14.4567i	2.07820	+	0.556851i
$675$		0			0
$676$	−24.0000	−	10.0000i	−0.923077	−	0.384615i
$677$			13.5873i			0.522204i	0.965311	+	0.261102i	$0.0840858\pi$
$677$			13.5873i			0.522204i	−0.965311	+	0.261102i	$0.915914\pi$
$678$		0			0
$679$	1.30385	+	2.25833i	0.0500371	+	0.0866668i
$680$		0			0
$681$		0			0
$682$	−14.2346	−	53.1244i	−0.545072	−	2.03424i
$683$	4.53365	+	16.9198i	0.173475	+	0.647418i	0.996806	+	0.0798568i	$0.0254463\pi$
$683$	4.53365	+	16.9198i	0.173475	+	0.647418i	−0.823331	+	0.567561i	$0.807887\pi$
$684$		0			0
$685$	−20.3923	+	35.3205i	−0.779150	+	1.34953i
$686$	−7.10823	−	12.3118i	−0.271394	−	0.470067i
$687$		0			0
$688$			30.9282i			1.17913i
$689$	1.37705	+	21.3519i	0.0524614	+	0.813443i
$690$		0			0
$691$	24.4282	+	6.54552i	0.929293	+	0.249003i	0.691553	−	0.722326i	$-0.256927\pi$
$691$	24.4282	+	6.54552i	0.929293	+	0.249003i	0.237740	+	0.971329i	$0.423594\pi$
$692$	−24.3190	+	14.0406i	−0.924471	+	0.533744i
$693$		0			0
$694$	−41.1769	+	41.1769i	−1.56305	+	1.56305i
$695$	8.95215	−	2.39872i	0.339574	−	0.0909887i
$696$		0			0
$697$	−6.92820	−	6.92820i	−0.262424	−	0.262424i
$698$	3.65067	+	2.10772i	0.138180	+	0.0797783i
$699$		0			0
$700$	−0.660254	+	2.46410i	−0.0249553	+	0.0931343i
$701$	−23.1822			−0.875580			−0.437790	−	0.899077i	$-0.644239\pi$
$701$	−23.1822			−0.875580			−0.437790	+	0.899077i	$0.644239\pi$
$702$		0			0
$703$	20.2487			0.763695
$704$	−8.68835	+	32.4254i	−0.327455	+	1.22208i
$705$		0			0
$706$	3.12436	+	1.80385i	0.117587	+	0.0678887i
$707$	−0.416102	−	0.416102i	−0.0156491	−	0.0156491i
$708$		0			0
$709$	−11.6962	+	3.13397i	−0.439258	+	0.117699i	−0.471669	−	0.881776i	$-0.656348\pi$
$709$	−11.6962	+	3.13397i	−0.439258	+	0.117699i	0.0324106	+	0.999475i	$0.489682\pi$
$710$	−60.5066	+	60.5066i	−2.27077	+	2.27077i
$711$		0			0
$712$		0			0
$713$	−15.5056	−	4.15471i	−0.580689	−	0.155595i
$714$		0			0
$715$	−22.9282	−	34.3923i	−0.857466	−	1.28620i
$716$			32.9802i			1.23253i
$717$		0			0
$718$	−10.0000	−	17.3205i	−0.373197	−	0.646396i
$719$	21.9575	−	38.0315i	0.818876	−	1.41833i	−0.0876356	−	0.996153i	$-0.527931\pi$
$719$	21.9575	−	38.0315i	0.818876	−	1.41833i	0.906511	−	0.422182i	$-0.138736\pi$
$720$		0			0
$721$	1.16025	+	4.33013i	0.0432101	+	0.161262i
$722$	10.7961	+	40.2915i	0.401788	+	1.49949i
$723$		0			0
$724$	−6.00000	+	10.3923i	−0.222988	+	0.386227i
$725$	−1.74238	−	3.01790i	−0.0647105	−	0.112082i
$726$		0			0
$727$		−	35.1051i		−	1.30198i	−0.759088	−	0.650988i	$-0.774355\pi$
$727$		−	35.1051i		−	1.30198i	0.759088	−	0.650988i	$-0.225645\pi$
$728$		0			0
$729$		0			0
$730$	−21.6603	−	5.80385i	−0.801682	−	0.214810i
$731$	44.8115	−	25.8719i	1.65741	−	0.956908i
$732$		0			0
$733$	−32.7583	+	32.7583i	−1.20996	+	1.20996i	−0.238916	+	0.971040i	$0.576792\pi$
$733$	−32.7583	+	32.7583i	−1.20996	+	1.20996i	−0.971040	+	0.238916i	$0.923208\pi$
$734$	10.4171	−	2.79126i	0.384503	−	0.103027i
$735$		0			0
$736$	13.8564	+	13.8564i	0.510754	+	0.510754i
$737$	−28.5852	−	16.5037i	−1.05295	−	0.607921i
$738$		0			0
$739$	−0.562178	+	2.09808i	−0.0206800	+	0.0771790i	−0.975495	−	0.220023i	$-0.929387\pi$
$739$	−0.562178	+	2.09808i	−0.0206800	+	0.0771790i	0.954815	+	0.297202i	$0.0960534\pi$
$740$	17.5254			0.644245
$741$		0			0
$742$	−6.14359			−0.225538
$743$	−1.77955	+	6.64136i	−0.0652853	+	0.243648i	−0.990855	−	0.134928i	$-0.956920\pi$
$743$	−1.77955	+	6.64136i	−0.0652853	+	0.243648i	0.925570	+	0.378576i	$0.123586\pi$
$744$		0			0
$745$	15.1244	+	8.73205i	0.554114	+	0.319918i
$746$	−10.7317	−	10.7317i	−0.392915	−	0.392915i
$747$		0			0
$748$	54.2487	−	14.5359i	1.98353	−	0.531485i
$749$	−0.619174	+	0.619174i	−0.0226241	+	0.0226241i
$750$		0			0
$751$	−8.78461	+	5.07180i	−0.320555	+	0.185072i	−0.651640	−	0.758528i	$-0.725919\pi$
$751$	−8.78461	+	5.07180i	−0.320555	+	0.185072i	0.331085	+	0.943601i	$0.392585\pi$
$752$	12.6264	+	3.38323i	0.460437	+	0.123374i
$753$		0			0
$754$	3.26795	+	9.66025i	0.119012	+	0.351806i
$755$			32.7028i			1.19018i
$756$		0			0
$757$	14.8038	+	25.6410i	0.538055	+	0.931939i	0.999009	+	0.0445144i	$0.0141741\pi$
$757$	14.8038	+	25.6410i	0.538055	+	0.931939i	−0.460954	+	0.887424i	$0.652493\pi$
$758$	23.1451	−	40.0884i	0.840666	−	1.45608i
$759$		0			0
$760$		0			0
$761$	0.314566	+	1.17398i	0.0114030	+	0.0425566i	0.971393	−	0.237478i	$-0.0763207\pi$
$761$	0.314566	+	1.17398i	0.0114030	+	0.0425566i	−0.959990	+	0.280035i	$0.909654\pi$
$762$		0			0
$763$	4.13397	−	7.16025i	0.149660	−	0.259219i
$764$	−13.0053	−	22.5259i	−0.470516	−	0.814958i
$765$		0			0
$766$		−	20.0000i		−	0.722629i
$767$	0.226633	+	0.669942i	0.00818326	+	0.0241902i
$768$		0			0
$769$	33.4904	+	8.97372i	1.20769	+	0.323601i	0.805857	−	0.592110i	$-0.201705\pi$
$769$	33.4904	+	8.97372i	1.20769	+	0.323601i	0.401837	+	0.915711i	$0.368372\pi$
$770$	10.2784	−	5.93426i	0.370409	−	0.213856i
$771$		0			0
$772$	−1.66025	+	1.66025i	−0.0597539	+	0.0597539i
$773$	−3.81294	+	1.02167i	−0.137142	+	0.0367470i	−0.326737	−	0.945115i	$-0.605949\pi$
$773$	−3.81294	+	1.02167i	−0.137142	+	0.0367470i	0.189595	+	0.981862i	$0.439282\pi$
$774$		0			0
$775$	11.4186	+	11.4186i	0.410168	+	0.410168i
$776$		0			0
$777$		0			0
$778$	3.46410	−	12.9282i	0.124194	−	0.463499i
$779$	−9.24316			−0.331170
$780$		0			0
$781$	65.7128			2.35139
$782$	8.48528	−	31.6675i	0.303433	−	1.13243i
$783$		0			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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	117.ba (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.517638 - 1.93185i) q^{2} +(-1.73205 - 1.00000i) q^{4} +(-1.93185 - 1.93185i) q^{5} +(-0.500000 + 0.133975i) q^{7} +(-4.73205 + 2.73205i) q^{10} +(4.05317 + 1.08604i) q^{11} +(3.59808 - 0.232051i) q^{13} +1.03528i q^{14} +(-2.00000 - 3.46410i) q^{16} +(-3.34607 + 5.79555i) q^{17} +(1.63397 + 6.09808i) q^{19} +(1.41421 + 5.27792i) q^{20} +(4.19615 - 7.26795i) q^{22} +(-1.22474 - 2.12132i) q^{23} +2.46410i q^{25} +(1.41421 - 7.07107i) q^{26} +(1.00000 + 0.267949i) q^{28} +(-1.22474 + 0.707107i) q^{29} +(4.63397 - 4.63397i) q^{31} +(-7.72741 + 2.07055i) q^{32} +(9.46410 + 9.46410i) q^{34} +(1.22474 + 0.707107i) q^{35} +(0.830127 - 3.09808i) q^{37} +12.6264 q^{38} +(-0.378937 + 1.41421i) q^{41} +(-6.69615 - 3.86603i) q^{43} +(-5.93426 - 5.93426i) q^{44} +(-4.73205 + 1.26795i) q^{46} +(-2.31079 + 2.31079i) q^{47} +(-5.83013 + 3.36603i) q^{49} +(4.76028 + 1.27551i) q^{50} +(-6.46410 - 3.19615i) q^{52} +5.93426i q^{53} +(-5.73205 - 9.92820i) q^{55} +(0.732051 + 2.73205i) q^{58} +(0.0507680 + 0.189469i) q^{59} +(-6.59808 + 11.4282i) q^{61} +(-6.55343 - 11.3509i) q^{62} +8.00000i q^{64} +(-7.39924 - 6.50266i) q^{65} +(-7.59808 - 2.03590i) q^{67} +(11.5911 - 6.69213i) q^{68} +(2.00000 - 2.00000i) q^{70} +(15.1266 - 4.05317i) q^{71} +(2.90192 + 2.90192i) q^{73} +(-5.55532 - 3.20736i) q^{74} +(3.26795 - 12.1962i) q^{76} -2.17209 q^{77} +7.19615 q^{79} +(-2.82843 + 10.5558i) q^{80} +(2.53590 + 1.46410i) q^{82} +(-5.27792 - 5.27792i) q^{83} +(17.6603 - 4.73205i) q^{85} +(-10.9348 + 10.9348i) q^{86} +(4.38134 + 1.17398i) q^{89} +(-1.76795 + 0.598076i) q^{91} +4.89898i q^{92} +(3.26795 + 5.66025i) q^{94} +(8.62398 - 14.9372i) q^{95} +(-1.30385 - 4.86603i) q^{97} +(3.48477 + 13.0053i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{7} - 24 q^{10} + 8 q^{13} - 16 q^{16} + 20 q^{19} - 8 q^{22} + 8 q^{28} + 44 q^{31} + 48 q^{34} - 28 q^{37} - 12 q^{43} - 24 q^{46} - 12 q^{49} - 24 q^{52} - 32 q^{55} - 8 q^{58} - 32 q^{61}+ \cdots - 52 q^{97}+O(q^{100}) \) 8 * q - 4 * q^7 - 24 * q^10 + 8 * q^13 - 16 * q^16 + 20 * q^19 - 8 * q^22 + 8 * q^28 + 44 * q^31 + 48 * q^34 - 28 * q^37 - 12 * q^43 - 24 * q^46 - 12 * q^49 - 24 * q^52 - 32 * q^55 - 8 * q^58 - 32 * q^61 - 40 * q^67 + 16 * q^70 + 44 * q^73 + 40 * q^76 + 16 * q^79 + 48 * q^82 + 72 * q^85 - 28 * q^91 + 40 * q^94 - 52 * q^97

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