LMFDB - Embedded newform 117.2.ba.a.71.1

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			71.1
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	117.71
Dual form			117.2.ba.a.89.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.93185 + 0.517638i) q^{2} +(1.73205 - 1.00000i) q^{4} +(0.517638 + 0.517638i) q^{5} +(-0.500000 + 1.86603i) q^{7} +(-1.26795 - 0.732051i) q^{10} +(1.60368 + 5.98502i) q^{11} +(-1.59808 + 3.23205i) q^{13} -3.86370i q^{14} +(-2.00000 + 3.46410i) q^{16} +(-0.896575 - 1.55291i) q^{17} +(3.36603 + 0.901924i) q^{19} +(1.41421 + 0.378937i) q^{20} +(-6.19615 - 10.7321i) q^{22} +(1.22474 - 2.12132i) q^{23} -4.46410i q^{25} +(1.41421 - 7.07107i) q^{26} +(1.00000 + 3.73205i) q^{28} +(1.22474 + 0.707107i) q^{29} +(6.36603 - 6.36603i) q^{31} +(2.07055 - 7.72741i) q^{32} +(2.53590 + 2.53590i) q^{34} +(-1.22474 + 0.707107i) q^{35} +(-7.83013 + 2.09808i) q^{37} -6.96953 q^{38} +(-5.27792 + 1.41421i) q^{41} +(3.69615 - 2.13397i) q^{43} +(8.76268 + 8.76268i) q^{44} +(-1.26795 + 4.73205i) q^{46} +(-4.76028 + 4.76028i) q^{47} +(2.83013 + 1.63397i) q^{49} +(2.31079 + 8.62398i) q^{50} +(0.464102 + 7.19615i) q^{52} -8.76268i q^{53} +(-2.26795 + 3.92820i) q^{55} +(-2.73205 - 0.732051i) q^{58} +(9.84873 + 2.63896i) q^{59} +(-1.40192 - 2.42820i) q^{61} +(-9.00292 + 15.5935i) q^{62} +8.00000i q^{64} +(-2.50026 + 0.845807i) q^{65} +(-2.40192 - 8.96410i) q^{67} +(-3.10583 - 1.79315i) q^{68} +(2.00000 - 2.00000i) q^{70} +(0.429705 - 1.60368i) q^{71} +(8.09808 + 8.09808i) q^{73} +(14.0406 - 8.10634i) q^{74} +(6.73205 - 1.80385i) q^{76} -11.9700 q^{77} -3.19615 q^{79} +(-2.82843 + 0.757875i) q^{80} +(9.46410 - 5.46410i) q^{82} +(-0.378937 - 0.378937i) q^{83} +(0.339746 - 1.26795i) q^{85} +(-6.03579 + 6.03579i) q^{86} +(-2.96713 - 11.0735i) q^{89} +(-5.23205 - 4.59808i) q^{91} -4.89898i q^{92} +(6.73205 - 11.6603i) q^{94} +(1.27551 + 2.20925i) q^{95} +(-11.6962 - 3.13397i) q^{97} +(-6.31319 - 1.69161i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{5}{12}\right)$	$-1$

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 $

$2$	−1.93185	+	0.517638i	−1.36603	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$2$	−1.93185	+	0.517638i	−1.36603	+	0.366025i	−0.500000	+	0.866025i	$0.666667\pi$
$3$		0			0
$3$		0			0
$4$	1.73205	−	1.00000i	0.866025	−	0.500000i
$5$	0.517638	+	0.517638i	0.231495	+	0.231495i	0.813316	−	0.581822i	$-0.197660\pi$
$5$	0.517638	+	0.517638i	0.231495	+	0.231495i	−0.581822	+	0.813316i	$0.697660\pi$
$6$		0			0
$7$	−0.500000	+	1.86603i	−0.188982	+	0.705291i	0.804761	+	0.593599i	$0.202294\pi$
$7$	−0.500000	+	1.86603i	−0.188982	+	0.705291i	−0.993743	+	0.111692i	$0.964373\pi$
$8$		0			0
$9$		0			0
$10$	−1.26795	−	0.732051i	−0.400961	−	0.231495i
$11$	1.60368	+	5.98502i	0.483528	+	1.80455i	0.586598	+	0.809878i	$0.300467\pi$
$11$	1.60368	+	5.98502i	0.483528	+	1.80455i	−0.103069	+	0.994674i	$0.532866\pi$
$12$		0			0
$13$	−1.59808	+	3.23205i	−0.443227	+	0.896410i
$13$	−1.59808	+	3.23205i	−0.443227	+	0.896410i
$14$		−	3.86370i		−	1.03262i
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$	−0.896575	−	1.55291i	−0.217451	−	0.376637i	0.736577	−	0.676354i	$-0.236441\pi$
$17$	−0.896575	−	1.55291i	−0.217451	−	0.376637i	−0.954028	+	0.299717i	$0.903108\pi$
$18$		0			0
$19$	3.36603	+	0.901924i	0.772219	+	0.206916i	0.623352	−	0.781942i	$-0.285771\pi$
$19$	3.36603	+	0.901924i	0.772219	+	0.206916i	0.148868	+	0.988857i	$0.452437\pi$
$20$	1.41421	+	0.378937i	0.316228	+	0.0847330i
$21$		0			0
$22$	−6.19615	−	10.7321i	−1.32102	−	2.28808i
$23$	1.22474	−	2.12132i	0.255377	−	0.442326i	−0.709621	−	0.704584i	$-0.751134\pi$
$23$	1.22474	−	2.12132i	0.255377	−	0.442326i	0.964998	+	0.262258i	$0.0844671\pi$
$24$		0			0
$25$		−	4.46410i		−	0.892820i
$26$	1.41421	−	7.07107i	0.277350	−	1.38675i
$27$		0			0
$28$	1.00000	+	3.73205i	0.188982	+	0.705291i
$29$	1.22474	+	0.707107i	0.227429	+	0.131306i	0.609386	−	0.792874i	$-0.291416\pi$
$29$	1.22474	+	0.707107i	0.227429	+	0.131306i	−0.381956	+	0.924180i	$0.624749\pi$
$30$		0			0
$31$	6.36603	−	6.36603i	1.14337	−	1.14337i	0.155543	−	0.987829i	$-0.450287\pi$
$31$	6.36603	−	6.36603i	1.14337	−	1.14337i	0.987829	−	0.155543i	$-0.0497126\pi$
$32$	2.07055	−	7.72741i	0.366025	−	1.36603i
$33$		0			0
$34$	2.53590	+	2.53590i	0.434903	+	0.434903i
$35$	−1.22474	+	0.707107i	−0.207020	+	0.119523i
$36$		0			0
$37$	−7.83013	+	2.09808i	−1.28726	+	0.344922i	−0.836621	−	0.547782i	$-0.815472\pi$
$37$	−7.83013	+	2.09808i	−1.28726	+	0.344922i	−0.450644	+	0.892704i	$0.648806\pi$
$38$	−6.96953			−1.13061
$39$		0			0
$40$		0			0
$41$	−5.27792	+	1.41421i	−0.824272	+	0.220863i	−0.646213	−	0.763157i	$-0.723648\pi$
$41$	−5.27792	+	1.41421i	−0.824272	+	0.220863i	−0.178059	+	0.984020i	$0.556982\pi$
$42$		0			0
$43$	3.69615	−	2.13397i	0.563658	−	0.325428i	−0.190954	−	0.981599i	$-0.561158\pi$
$43$	3.69615	−	2.13397i	0.563658	−	0.325428i	0.754612	+	0.656171i	$0.227825\pi$
$44$	8.76268	+	8.76268i	1.32102	+	1.32102i
$45$		0			0
$46$	−1.26795	+	4.73205i	−0.186949	+	0.697703i
$47$	−4.76028	+	4.76028i	−0.694358	+	0.694358i	−0.963188	−	0.268830i	$-0.913363\pi$
$47$	−4.76028	+	4.76028i	−0.694358	+	0.694358i	0.268830	+	0.963188i	$0.413363\pi$
$48$		0			0
$49$	2.83013	+	1.63397i	0.404304	+	0.233425i
$50$	2.31079	+	8.62398i	0.326795	+	1.21962i
$51$		0			0
$52$	0.464102	+	7.19615i	0.0643593	+	0.997927i
$53$		−	8.76268i		−	1.20365i	−0.798629	−	0.601824i	$-0.794441\pi$
$53$		−	8.76268i		−	1.20365i	0.798629	−	0.601824i	$-0.205559\pi$
$54$		0			0
$55$	−2.26795	+	3.92820i	−0.305810	+	0.529679i
$56$		0			0
$57$		0			0
$58$	−2.73205	−	0.732051i	−0.358736	−	0.0961230i
$59$	9.84873	+	2.63896i	1.28220	+	0.343563i	0.834691	−	0.550719i	$-0.185646\pi$
$59$	9.84873	+	2.63896i	1.28220	+	0.343563i	0.447504	+	0.894282i	$0.352313\pi$
$60$		0			0
$61$	−1.40192	−	2.42820i	−0.179498	−	0.310900i	0.762211	−	0.647329i	$-0.224114\pi$
$61$	−1.40192	−	2.42820i	−0.179498	−	0.310900i	−0.941709	+	0.336429i	$0.890781\pi$
$62$	−9.00292	+	15.5935i	−1.14337	+	1.98038i
$63$		0			0
$64$			8.00000i			1.00000i
$65$	−2.50026	+	0.845807i	−0.310119	+	0.104910i
$66$		0			0
$67$	−2.40192	−	8.96410i	−0.293442	−	1.09514i	−0.942447	−	0.334355i	$-0.891481\pi$
$67$	−2.40192	−	8.96410i	−0.293442	−	1.09514i	0.649005	−	0.760784i	$-0.275185\pi$
$68$	−3.10583	−	1.79315i	−0.376637	−	0.217451i
$69$		0			0
$70$	2.00000	−	2.00000i	0.239046	−	0.239046i
$71$	0.429705	−	1.60368i	0.0509966	−	0.190322i	−0.935729	−	0.352721i	$-0.885256\pi$
$71$	0.429705	−	1.60368i	0.0509966	−	0.190322i	0.986725	+	0.162399i	$0.0519231\pi$
$72$		0			0
$73$	8.09808	+	8.09808i	0.947808	+	0.947808i	0.998704	−	0.0508958i	$-0.0162077\pi$
$73$	8.09808	+	8.09808i	0.947808	+	0.947808i	−0.0508958	+	0.998704i	$0.516208\pi$
$74$	14.0406	−	8.10634i	1.63219	−	0.942343i
$75$		0			0
$76$	6.73205	−	1.80385i	0.772219	−	0.206916i
$77$	−11.9700			−1.36411
$78$		0			0
$79$	−3.19615			−0.359595			−0.179798	−	0.983704i	$-0.557544\pi$
$79$	−3.19615			−0.359595			−0.179798	+	0.983704i	$0.557544\pi$
$80$	−2.82843	+	0.757875i	−0.316228	+	0.0847330i
$81$		0			0
$82$	9.46410	−	5.46410i	1.04514	−	0.603409i
$83$	−0.378937	−	0.378937i	−0.0415938	−	0.0415938i	0.686004	−	0.727598i	$-0.259363\pi$
$83$	−0.378937	−	0.378937i	−0.0415938	−	0.0415938i	−0.727598	+	0.686004i	$0.759363\pi$
$84$		0			0
$85$	0.339746	−	1.26795i	0.0368506	−	0.137528i
$86$	−6.03579	+	6.03579i	−0.650856	+	0.650856i
$87$		0			0
$88$		0			0
$89$	−2.96713	−	11.0735i	−0.314515	−	1.17379i	−0.924440	−	0.381327i	$-0.875467\pi$
$89$	−2.96713	−	11.0735i	−0.314515	−	1.17379i	0.609925	−	0.792459i	$-0.291199\pi$
$90$		0			0
$91$	−5.23205	−	4.59808i	−0.548468	−	0.482009i
$92$		−	4.89898i		−	0.510754i
$93$		0			0
$94$	6.73205	−	11.6603i	0.694358	−	1.20266i
$95$	1.27551	+	2.20925i	0.130865	+	0.226665i
$96$		0			0
$97$	−11.6962	−	3.13397i	−1.18756	−	0.318207i	−0.389642	−	0.920967i	$-0.627401\pi$
$97$	−11.6962	−	3.13397i	−1.18756	−	0.318207i	−0.797923	+	0.602760i	$0.794068\pi$
$98$	−6.31319	−	1.69161i	−0.637729	−	0.170879i
$99$		0			0
$100$	−4.46410	−	7.73205i	−0.446410	−	0.773205i
$101$	7.91688	−	13.7124i	0.787759	−	1.36444i	−0.139579	−	0.990211i	$-0.544575\pi$
$101$	7.91688	−	13.7124i	0.787759	−	1.36444i	0.927337	−	0.374227i	$-0.122092\pi$
$102$		0			0
$103$			8.66025i			0.853320i	0.904412	+	0.426660i	$0.140310\pi$
$103$			8.66025i			0.853320i	−0.904412	+	0.426660i	$0.859690\pi$
$104$		0			0
$105$		0			0
$106$	4.53590	+	16.9282i	0.440565	+	1.64421i
$107$	11.2629	+	6.50266i	1.08883	+	0.628636i	0.933265	−	0.359190i	$-0.116947\pi$
$107$	11.2629	+	6.50266i	1.08883	+	0.628636i	0.155565	+	0.987826i	$0.450280\pi$
$108$		0			0
$109$	4.29423	−	4.29423i	0.411313	−	0.411313i	−0.470883	−	0.882196i	$-0.656065\pi$
$109$	4.29423	−	4.29423i	0.411313	−	0.411313i	0.882196	+	0.470883i	$0.156065\pi$
$110$	2.34795	−	8.76268i	0.223869	−	0.835489i
$111$		0			0
$112$	−5.46410	−	5.46410i	−0.516309	−	0.516309i
$113$	8.81345	−	5.08845i	0.829100	−	0.478681i	−0.0244446	−	0.999701i	$-0.507782\pi$
$113$	8.81345	−	5.08845i	0.829100	−	0.478681i	0.853544	+	0.521020i	$0.174448\pi$
$114$		0			0
$115$	1.73205	−	0.464102i	0.161515	−	0.0432777i
$116$	2.82843			0.262613
$117$		0			0
$118$	−20.3923			−1.87726
$119$	3.34607	−	0.896575i	0.306733	−	0.0821889i
$120$		0			0
$121$	−23.7224	+	13.6962i	−2.15658	+	1.24510i
$122$	3.96524	+	3.96524i	0.358996	+	0.358996i
$123$		0			0
$124$	4.66025	−	17.3923i	0.418503	−	1.56188i
$125$	4.89898	−	4.89898i	0.438178	−	0.438178i
$126$		0			0
$127$	3.40192	+	1.96410i	0.301872	+	0.174286i	0.643284	−	0.765628i	$-0.277572\pi$
$127$	3.40192	+	1.96410i	0.301872	+	0.174286i	−0.341412	+	0.939914i	$0.610905\pi$
$128$		0			0
$129$		0			0
$130$	4.39230	−	2.92820i	0.385231	−	0.256820i
$131$			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
$133$	−3.36603	+	5.83013i	−0.291871	+	0.505536i
$134$	9.28032	+	16.0740i	0.801698	+	1.38858i
$135$		0			0
$136$		0			0
$137$	1.03528	+	0.277401i	0.0884496	+	0.0237000i	0.302772	−	0.953063i	$-0.402088\pi$
$137$	1.03528	+	0.277401i	0.0884496	+	0.0237000i	−0.214323	+	0.976763i	$0.568754\pi$
$138$		0			0
$139$	8.69615	+	15.0622i	0.737598	+	1.27756i	0.953574	+	0.301159i	$0.0973734\pi$
$139$	8.69615	+	15.0622i	0.737598	+	1.27756i	−0.215976	+	0.976399i	$0.569293\pi$
$140$	−1.41421	+	2.44949i	−0.119523	+	0.207020i
$141$		0			0
$142$			3.32051i			0.278651i
$143$	−21.9067	−	4.38134i	−1.83193	−	0.366386i
$144$		0			0
$145$	0.267949	+	1.00000i	0.0222520	+	0.0830455i
$146$	−19.8362	−	11.4524i	−1.64165	−	0.947808i
$147$		0			0
$148$	−11.4641	+	11.4641i	−0.942343	+	0.942343i
$149$	−3.72500	+	13.9019i	−0.305164	+	1.13889i	0.627640	+	0.778504i	$0.284021\pi$
$149$	−3.72500	+	13.9019i	−0.305164	+	1.13889i	−0.932804	+	0.360384i	$0.882646\pi$
$150$		0			0
$151$	−1.53590	−	1.53590i	−0.124990	−	0.124990i	0.641845	−	0.766835i	$-0.278169\pi$
$151$	−1.53590	−	1.53590i	−0.124990	−	0.124990i	−0.766835	+	0.641845i	$0.778169\pi$
$152$		0			0
$153$		0			0
$154$	23.1244	−	6.19615i	1.86341	−	0.499300i
$155$	6.59059			0.529369
$156$		0			0
$157$	9.39230			0.749588			0.374794	−	0.927108i	$-0.377714\pi$
$157$	9.39230			0.749588			0.374794	+	0.927108i	$0.377714\pi$
$158$	6.17449	−	1.65445i	0.491216	−	0.131621i
$159$		0			0
$160$	5.07180	−	2.92820i	0.400961	−	0.231495i
$161$	3.34607	+	3.34607i	0.263707	+	0.263707i
$162$		0			0
$163$	3.93782	−	14.6962i	0.308434	−	1.15109i	−0.621515	−	0.783402i	$-0.713482\pi$
$163$	3.93782	−	14.6962i	0.308434	−	1.15109i	0.929949	−	0.367689i	$-0.119851\pi$
$164$	−7.72741	+	7.72741i	−0.603409	+	0.603409i
$165$		0			0
$166$	0.928203	+	0.535898i	0.0720425	+	0.0415938i
$167$	0.378937	+	1.41421i	0.0293231	+	0.109435i	0.979036	−	0.203686i	$-0.0652921\pi$
$167$	0.378937	+	1.41421i	0.0293231	+	0.109435i	−0.949713	+	0.313121i	$0.898625\pi$
$168$		0			0
$169$	−7.89230	−	10.3301i	−0.607100	−	0.794625i
$170$			2.62536i			0.201356i
$171$		0			0
$172$	4.26795	−	7.39230i	0.325428	−	0.563658i
$173$	−2.77766	−	4.81105i	−0.211182	−	0.365777i	0.740903	−	0.671612i	$-0.234398\pi$
$173$	−2.77766	−	4.81105i	−0.211182	−	0.365777i	−0.952085	+	0.305835i	$0.901064\pi$
$174$		0			0
$175$	8.33013	+	2.23205i	0.629698	+	0.168727i
$176$	−23.9401	−	6.41473i	−1.80455	−	0.483528i
$177$		0			0
$178$	11.4641	+	19.8564i	0.859271	+	1.48830i
$179$	4.00240	−	6.93237i	0.299154	−	0.518149i	−0.676789	−	0.736177i	$-0.736629\pi$
$179$	4.00240	−	6.93237i	0.299154	−	0.518149i	0.975943	+	0.218028i	$0.0699623\pi$
$180$		0			0
$181$		−	6.00000i		−	0.445976i	−0.974821	−	0.222988i	$-0.928419\pi$
$181$		−	6.00000i		−	0.445976i	0.974821	−	0.222988i	$-0.0715812\pi$
$182$	12.4877	+	6.17449i	0.925649	+	0.457684i
$183$		0			0
$184$		0			0
$185$	−5.13922	−	2.96713i	−0.377843	−	0.218148i
$186$		0			0
$187$	7.85641	−	7.85641i	0.574517	−	0.574517i
$188$	−3.48477	+	13.0053i	−0.254153	+	0.948511i
$189$		0			0
$190$	−3.60770	−	3.60770i	−0.261730	−	0.261730i
$191$	1.46498	−	0.845807i	0.106002	−	0.0612005i	−0.446062	−	0.895002i	$-0.647174\pi$
$191$	1.46498	−	0.845807i	0.106002	−	0.0612005i	0.552064	+	0.833802i	$0.313840\pi$
$192$		0			0
$193$	10.6962	−	2.86603i	0.769926	−	0.206301i	0.147587	−	0.989049i	$-0.452849\pi$
$193$	10.6962	−	2.86603i	0.769926	−	0.206301i	0.622339	+	0.782748i	$0.286183\pi$
$194$	24.2175			1.73871
$195$		0			0
$196$	6.53590			0.466850
$197$	−20.2151	+	5.41662i	−1.44027	+	0.385918i	−0.892627	−	0.450797i	$-0.851140\pi$
$197$	−20.2151	+	5.41662i	−1.44027	+	0.385918i	−0.547639	+	0.836715i	$0.684473\pi$
$198$		0			0
$199$	−5.59808	+	3.23205i	−0.396837	+	0.229114i	−0.685118	−	0.728432i	$-0.740250\pi$
$199$	−5.59808	+	3.23205i	−0.396837	+	0.229114i	0.288281	+	0.957546i	$0.406916\pi$
$200$		0			0
$201$		0			0
$202$	−8.19615	+	30.5885i	−0.576679	+	2.15220i
$203$	−1.93185	+	1.93185i	−0.135589	+	0.135589i
$204$		0			0
$205$	−3.46410	−	2.00000i	−0.241943	−	0.139686i
$206$	−4.48288	−	16.7303i	−0.312337	−	1.16566i
$207$		0			0
$208$	−8.00000	−	12.0000i	−0.554700	−	0.832050i
$209$			21.5921i			1.49356i
$210$		0			0
$211$	2.40192	−	4.16025i	0.165355	−	0.286404i	−0.771426	−	0.636319i	$-0.780456\pi$
$211$	2.40192	−	4.16025i	0.165355	−	0.286404i	0.936781	+	0.349915i	$0.113790\pi$
$212$	−8.76268	−	15.1774i	−0.601824	−	1.04239i
$213$		0			0
$214$	−25.1244	−	6.73205i	−1.71747	−	0.460194i
$215$	3.01790	+	0.808643i	0.205819	+	0.0551490i
$216$		0			0
$217$	8.69615	+	15.0622i	0.590333	+	1.02249i
$218$	−6.07296	+	10.5187i	−0.411313	+	0.712414i
$219$		0			0
$220$			9.07180i			0.611620i
$221$	6.45189	−	0.416102i	0.434001	−	0.0279901i
$222$		0			0
$223$	−1.24167	−	4.63397i	−0.0831484	−	0.310314i	0.911809	−	0.410615i	$-0.134686\pi$
$223$	−1.24167	−	4.63397i	−0.0831484	−	0.310314i	−0.994957	+	0.100301i	$0.968019\pi$
$224$	13.3843	+	7.72741i	0.894274	+	0.516309i
$225$		0			0
$226$	−14.3923	+	14.3923i	−0.957362	+	0.957362i
$227$	−3.48477	+	13.0053i	−0.231292	+	0.863194i	0.748493	+	0.663142i	$0.230778\pi$
$227$	−3.48477	+	13.0053i	−0.231292	+	0.863194i	−0.979785	+	0.200051i	$0.935889\pi$
$228$		0			0
$229$	−0.267949	−	0.267949i	−0.0177066	−	0.0177066i	0.698198	−	0.715905i	$-0.253985\pi$
$229$	−0.267949	−	0.267949i	−0.0177066	−	0.0177066i	−0.715905	+	0.698198i	$0.753985\pi$
$230$	−3.10583	+	1.79315i	−0.204792	+	0.118237i
$231$		0			0
$232$		0			0
$233$	10.4543			0.684884			0.342442	−	0.939539i	$-0.388746\pi$
$233$	10.4543			0.684884			0.342442	+	0.939539i	$0.388746\pi$
$234$		0			0
$235$	−4.92820			−0.321481
$236$	19.6975	−	5.27792i	1.28220	−	0.343563i
$237$		0			0
$238$	−6.00000	+	3.46410i	−0.388922	+	0.224544i
$239$	−9.52056	−	9.52056i	−0.615834	−	0.615834i	0.328626	−	0.944460i	$-0.393414\pi$
$239$	−9.52056	−	9.52056i	−0.615834	−	0.615834i	−0.944460	+	0.328626i	$0.893414\pi$
$240$		0			0
$241$	4.29423	−	16.0263i	0.276616	−	1.03234i	−0.678135	−	0.734937i	$-0.737212\pi$
$241$	4.29423	−	16.0263i	0.276616	−	1.03234i	0.954751	−	0.297406i	$-0.0961216\pi$
$242$	38.7386	−	38.7386i	2.49021	−	2.49021i
$243$		0			0
$244$	−4.85641	−	2.80385i	−0.310900	−	0.179498i
$245$	0.619174	+	2.31079i	0.0395576	+	0.147631i
$246$		0			0
$247$	−8.29423	+	9.43782i	−0.527749	+	0.600514i
$248$		0			0
$249$		0			0
$250$	−6.92820	+	12.0000i	−0.438178	+	0.758947i
$251$	6.45189	+	11.1750i	0.407240	+	0.705360i	0.994579	−	0.103980i	$-0.0331580\pi$
$251$	6.45189	+	11.1750i	0.407240	+	0.705360i	−0.587339	+	0.809341i	$0.699825\pi$
$252$		0			0
$253$	14.6603	+	3.92820i	0.921682	+	0.246964i
$254$	−7.58871	−	2.03339i	−0.476158	−	0.127586i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	9.14162	−	15.8338i	0.570239	−	0.987682i	−0.426302	−	0.904581i	$-0.640184\pi$
$257$	9.14162	−	15.8338i	0.570239	−	0.987682i	0.996541	−	0.0831016i	$-0.0264826\pi$
$258$		0			0
$259$		−	15.6603i		−	0.973081i
$260$	−3.48477	+	3.96524i	−0.216116	+	0.245914i
$261$		0			0
$262$	−3.66025	−	13.6603i	−0.226131	−	0.843933i
$263$	−17.8671	−	10.3156i	−1.10173	−	0.636087i	−0.165059	−	0.986284i	$-0.552781\pi$
$263$	−17.8671	−	10.3156i	−1.10173	−	0.636087i	−0.936676	+	0.350197i	$0.886115\pi$
$264$		0			0
$265$	4.53590	−	4.53590i	0.278638	−	0.278638i
$266$	3.48477	−	13.0053i	0.213665	−	0.797408i
$267$		0			0
$268$	−13.1244	−	13.1244i	−0.801698	−	0.801698i
$269$	−20.3166	+	11.7298i	−1.23873	+	0.715179i	−0.968834	−	0.247711i	$-0.920322\pi$
$269$	−20.3166	+	11.7298i	−1.23873	+	0.715179i	−0.269893	+	0.962890i	$0.586988\pi$
$270$		0			0
$271$	−21.6244	+	5.79423i	−1.31359	+	0.351974i	−0.846571	−	0.532275i	$-0.821337\pi$
$271$	−21.6244	+	5.79423i	−1.31359	+	0.351974i	−0.467015	+	0.884250i	$0.654670\pi$
$272$	7.17260			0.434903
$273$		0			0
$274$	−2.14359			−0.129499
$275$	26.7178	−	7.15900i	1.61114	−	0.431704i
$276$		0			0
$277$	6.80385	−	3.92820i	0.408804	−	0.236023i	−0.281472	−	0.959569i	$-0.590823\pi$
$277$	6.80385	−	3.92820i	0.408804	−	0.236023i	0.690276	+	0.723547i	$0.257489\pi$
$278$	−24.5964	−	24.5964i	−1.47520	−	1.47520i
$279$		0			0
$280$		0			0
$281$	−15.2146	+	15.2146i	−0.907626	+	0.907626i	−0.996080	−	0.0884547i	$-0.971807\pi$
$281$	−15.2146	+	15.2146i	−0.907626	+	0.907626i	0.0884547	+	0.996080i	$0.471807\pi$
$282$		0			0
$283$	4.20577	+	2.42820i	0.250007	+	0.144342i	0.619768	−	0.784785i	$-0.287227\pi$
$283$	4.20577	+	2.42820i	0.250007	+	0.144342i	−0.369760	+	0.929127i	$0.620560\pi$
$284$	−0.859411	−	3.20736i	−0.0509966	−	0.190322i
$285$		0			0
$286$	44.5885	−	2.87564i	2.63657	−	0.170040i
$287$		−	10.5558i		−	0.623091i
$288$		0			0
$289$	6.89230	−	11.9378i	0.405430	−	0.702225i
$290$	−1.03528	−	1.79315i	−0.0607935	−	0.105297i
$291$		0			0
$292$	22.1244	+	5.92820i	1.29473	+	0.346922i
$293$	21.4398	+	5.74479i	1.25253	+	0.335614i	0.823313	−	0.567588i	$-0.192123\pi$
$293$	21.4398	+	5.74479i	1.25253	+	0.335614i	0.429216	+	0.903202i	$0.358790\pi$
$294$		0			0
$295$	3.73205	+	6.46410i	0.217288	+	0.376355i
$296$		0			0
$297$		0			0
$298$		−	28.7846i		−	1.66745i
$299$	4.89898	+	7.34847i	0.283315	+	0.424973i
$300$		0			0
$301$	2.13397	+	7.96410i	0.123000	+	0.459043i
$302$	3.76217	+	2.17209i	0.216488	+	0.124990i
$303$		0			0
$304$	−9.85641	+	9.85641i	−0.565304	+	0.565304i
$305$	0.531241	−	1.98262i	0.0304188	−	0.113524i
$306$		0			0
$307$	3.70577	+	3.70577i	0.211500	+	0.211500i	0.804904	−	0.593405i	$-0.202217\pi$
$307$	3.70577	+	3.70577i	0.211500	+	0.211500i	−0.593405	+	0.804904i	$0.702217\pi$
$308$	−20.7327	+	11.9700i	−1.18136	+	0.682057i
$309$		0			0
$310$	−12.7321	+	3.41154i	−0.723132	+	0.193763i
$311$	−28.0812			−1.59234			−0.796169	−	0.605074i	$-0.793144\pi$
$311$	−28.0812			−1.59234			−0.796169	+	0.605074i	$0.793144\pi$
$312$		0			0
$313$	7.19615			0.406751			0.203375	−	0.979101i	$-0.434809\pi$
$313$	7.19615			0.406751			0.203375	+	0.979101i	$0.434809\pi$
$314$	−18.1445	+	4.86181i	−1.02396	+	0.274368i
$315$		0			0
$316$	−5.53590	+	3.19615i	−0.311419	+	0.179798i
$317$	0.757875	+	0.757875i	0.0425665	+	0.0425665i	0.728070	−	0.685503i	$-0.240418\pi$
$317$	0.757875	+	0.757875i	0.0425665	+	0.0425665i	−0.685503	+	0.728070i	$0.740418\pi$
$318$		0			0
$319$	−2.26795	+	8.46410i	−0.126981	+	0.473899i
$320$	−4.14110	+	4.14110i	−0.231495	+	0.231495i
$321$		0			0
$322$	−8.19615	−	4.73205i	−0.456754	−	0.263707i
$323$	−1.61729	−	6.03579i	−0.0899882	−	0.335840i
$324$		0			0
$325$	14.4282	+	7.13397i	0.800333	+	0.395722i
$326$			30.4292i			1.68531i
$327$		0			0
$328$		0			0
$329$	−6.50266	−	11.2629i	−0.358503	−	0.620946i
$330$		0			0
$331$	6.59808	+	1.76795i	0.362663	+	0.0971753i	0.435549	−	0.900165i	$-0.356554\pi$
$331$	6.59808	+	1.76795i	0.362663	+	0.0971753i	−0.0728860	+	0.997340i	$0.523221\pi$
$332$	−1.03528	−	0.277401i	−0.0568182	−	0.0152244i
$333$		0			0
$334$	−1.46410	−	2.53590i	−0.0801121	−	0.138758i
$335$	3.39683	−	5.88349i	0.185589	−	0.321449i
$336$		0			0
$337$			14.0718i			0.766540i	0.923636	+	0.383270i	$0.125202\pi$
$337$			14.0718i			0.766540i	−0.923636	+	0.383270i	$0.874798\pi$
$338$	20.5940	+	15.8709i	1.12017	+	0.863264i
$339$		0			0
$340$	−0.679492	−	2.53590i	−0.0368506	−	0.137528i
$341$	48.3099	+	27.8917i	2.61613	+	1.51042i
$342$		0			0
$343$	−14.0263	+	14.0263i	−0.757348	+	0.757348i
$344$		0			0
$345$		0			0
$346$	7.85641	+	7.85641i	0.422363	+	0.422363i
$347$	−12.9682	+	7.48717i	−0.696167	+	0.401932i	−0.805918	−	0.592027i	$-0.798328\pi$
$347$	−12.9682	+	7.48717i	−0.696167	+	0.401932i	0.109751	+	0.993959i	$0.464995\pi$
$348$		0			0
$349$	−33.4545	+	8.96410i	−1.79078	+	0.479837i	−0.992478	−	0.122423i	$-0.960933\pi$
$349$	−33.4545	+	8.96410i	−1.79078	+	0.479837i	−0.798299	+	0.602261i	$0.794267\pi$
$350$	−17.2480			−0.921942
$351$		0			0
$352$	49.5692			2.64205
$353$	11.7806	−	3.15660i	0.627017	−	0.168009i	0.0687013	−	0.997637i	$-0.478114\pi$
$353$	11.7806	−	3.15660i	0.627017	−	0.168009i	0.558316	+	0.829629i	$0.311448\pi$
$354$		0			0
$355$	1.05256	−	0.607695i	0.0558640	−	0.0322531i
$356$	−16.2127	−	16.2127i	−0.859271	−	0.859271i
$357$		0			0
$358$	−4.14359	+	15.4641i	−0.218996	+	0.817303i
$359$	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
$361$	−5.93782	−	3.42820i	−0.312517	−	0.180432i
$362$	3.10583	+	11.5911i	0.163239	+	0.609215i
$363$		0			0
$364$	−13.6603	−	2.73205i	−0.715992	−	0.143198i
$365$			8.38375i			0.438825i
$366$		0			0
$367$	−7.69615	+	13.3301i	−0.401736	+	0.695827i	−0.993936	−	0.109964i	$-0.964926\pi$
$367$	−7.69615	+	13.3301i	−0.401736	+	0.695827i	0.592200	+	0.805791i	$0.298260\pi$
$368$	4.89898	+	8.48528i	0.255377	+	0.442326i
$369$		0			0
$370$	11.4641	+	3.07180i	0.595990	+	0.159695i
$371$	16.3514	+	4.38134i	0.848922	+	0.227468i
$372$		0			0
$373$	−11.7942	−	20.4282i	−0.610682	−	1.05773i	−0.991126	−	0.132928i	$-0.957562\pi$
$373$	−11.7942	−	20.4282i	−0.610682	−	1.05773i	0.380444	−	0.924804i	$-0.375771\pi$
$374$	−11.1106	+	19.2442i	−0.574517	+	0.995093i
$375$		0			0
$376$		0			0
$377$	−4.24264	+	2.82843i	−0.218507	+	0.145671i
$378$		0			0
$379$	−5.35641	−	19.9904i	−0.275140	−	1.02684i	−0.955747	−	0.294188i	$-0.904951\pi$
$379$	−5.35641	−	19.9904i	−0.275140	−	1.02684i	0.680607	−	0.732648i	$-0.261716\pi$
$380$	4.41851	+	2.55103i	0.226665	+	0.130865i
$381$		0			0
$382$	−2.39230	+	2.39230i	−0.122401	+	0.122401i
$383$	−2.58819	+	9.65926i	−0.132250	+	0.493565i	−0.999994	−	0.00344689i	$-0.998903\pi$
$383$	−2.58819	+	9.65926i	−0.132250	+	0.493565i	0.867744	+	0.497012i	$0.165569\pi$
$384$		0			0
$385$	−6.19615	−	6.19615i	−0.315785	−	0.315785i
$386$	−19.1798	+	11.0735i	−0.976227	+	0.563625i
$387$		0			0
$388$	−23.3923	+	6.26795i	−1.18756	+	0.318207i
$389$	1.79315			0.0909164			0.0454582	−	0.998966i	$-0.485525\pi$
$389$	1.79315			0.0909164			0.0454582	+	0.998966i	$0.485525\pi$
$390$		0			0
$391$	−4.39230			−0.222128
$392$		0			0
$393$		0			0
$394$	36.2487	−	20.9282i	1.82618	−	1.05435i
$395$	−1.65445	−	1.65445i	−0.0832444	−	0.0832444i
$396$		0			0
$397$	−6.66987	+	24.8923i	−0.334751	+	1.24931i	0.569387	+	0.822069i	$0.307181\pi$
$397$	−6.66987	+	24.8923i	−0.334751	+	1.24931i	−0.904139	+	0.427239i	$0.859486\pi$
$398$	9.14162	−	9.14162i	0.458228	−	0.458228i
$399$		0			0
$400$	15.4641	+	8.92820i	0.773205	+	0.446410i
$401$	−3.14299	−	11.7298i	−0.156954	−	0.585759i	−0.998930	−	0.0462447i	$-0.985275\pi$
$401$	−3.14299	−	11.7298i	−0.156954	−	0.585759i	0.841977	−	0.539514i	$-0.181392\pi$
$402$		0			0
$403$	10.4019	+	30.7487i	0.518157	+	1.53170i
$404$		−	31.6675i		−	1.57552i
$405$		0			0
$406$	2.73205	−	4.73205i	0.135589	−	0.234848i
$407$	−25.1141	−	43.4988i	−1.24486	−	2.15616i
$408$		0			0
$409$	9.42820	+	2.52628i	0.466195	+	0.124916i	0.484267	−	0.874920i	$-0.339086\pi$
$409$	9.42820	+	2.52628i	0.466195	+	0.124916i	−0.0180725	+	0.999837i	$0.505753\pi$
$410$	7.72741	+	2.07055i	0.381629	+	0.102257i
$411$		0			0
$412$	8.66025	+	15.0000i	0.426660	+	0.738997i
$413$	−9.84873	+	17.0585i	−0.484624	+	0.839394i
$414$		0			0
$415$		−	0.392305i		−	0.0192575i
$416$	21.6665	+	19.0411i	1.06229	+	0.933567i
$417$		0			0
$418$	−11.1769	−	41.7128i	−0.546681	−	2.04024i
$419$	−22.5259	−	13.0053i	−1.10046	−	0.635352i	−0.164119	−	0.986441i	$-0.552478\pi$
$419$	−22.5259	−	13.0053i	−1.10046	−	0.635352i	−0.936342	+	0.351089i	$0.885811\pi$
$420$		0			0
$421$	13.6340	−	13.6340i	0.664479	−	0.664479i	−0.291953	−	0.956433i	$-0.594305\pi$
$421$	13.6340	−	13.6340i	0.664479	−	0.664479i	0.956433	+	0.291953i	$0.0943052\pi$
$422$	−2.48665	+	9.28032i	−0.121048	+	0.451759i
$423$		0			0
$424$		0			0
$425$	−6.93237	+	4.00240i	−0.336269	+	0.194145i
$426$		0			0
$427$	5.23205	−	1.40192i	0.253197	−	0.0678438i
$428$	26.0106			1.25727
$429$		0			0
$430$	−6.24871			−0.301340
$431$	11.4524	−	3.06866i	0.551643	−	0.147812i	0.0277789	−	0.999614i	$-0.491157\pi$
$431$	11.4524	−	3.06866i	0.551643	−	0.147812i	0.523864	+	0.851802i	$0.324490\pi$
$432$		0			0
$433$	9.99038	−	5.76795i	0.480107	−	0.277190i	−0.240354	−	0.970685i	$-0.577264\pi$
$433$	9.99038	−	5.76795i	0.480107	−	0.277190i	0.720461	+	0.693495i	$0.243930\pi$
$434$	−24.5964	−	24.5964i	−1.18067	−	1.18067i
$435$		0			0
$436$	3.14359	−	11.7321i	0.150551	−	0.561863i
$437$	6.03579	−	6.03579i	0.288731	−	0.288731i
$438$		0			0
$439$	−18.6962	−	10.7942i	−0.892319	−	0.515180i	−0.0176185	−	0.999845i	$-0.505608\pi$
$439$	−18.6962	−	10.7942i	−0.892319	−	0.515180i	−0.874700	+	0.484664i	$0.838942\pi$
$440$		0			0
$441$		0			0
$442$	−12.2487	+	4.14359i	−0.582612	+	0.197091i
$443$		−	35.0507i		−	1.66531i	−0.553792	−	0.832655i	$-0.686820\pi$
$443$		−	35.0507i		−	1.66531i	0.553792	−	0.832655i	$-0.313180\pi$
$444$		0			0
$445$	4.19615	−	7.26795i	0.198917	−	0.344534i
$446$	4.79744	+	8.30942i	0.227166	+	0.393462i
$447$		0			0
$448$	−14.9282	−	4.00000i	−0.705291	−	0.188982i
$449$	−5.08845	−	1.36345i	−0.240139	−	0.0643450i	0.136742	−	0.990607i	$-0.456337\pi$
$449$	−5.08845	−	1.36345i	−0.240139	−	0.0643450i	−0.376881	+	0.926262i	$0.623003\pi$
$450$		0			0
$451$	−16.9282	−	29.3205i	−0.797118	−	1.38065i
$452$	10.1769	−	17.6269i	0.478681	−	0.829100i
$453$		0			0
$454$		−	26.9282i		−	1.26380i
$455$	−0.328169	−	5.08845i	−0.0153848	−	0.238550i
$456$		0			0
$457$	1.06218	+	3.96410i	0.0496866	+	0.185433i	0.986309	−	0.164908i	$-0.0527326\pi$
$457$	1.06218	+	3.96410i	0.0496866	+	0.185433i	−0.936622	+	0.350340i	$0.886066\pi$
$458$	0.656339	+	0.378937i	0.0306687	+	0.0177066i
$459$		0			0
$460$	2.53590	−	2.53590i	0.118237	−	0.118237i
$461$	9.65926	−	36.0488i	0.449877	−	1.67896i	−0.252853	−	0.967505i	$-0.581369\pi$
$461$	9.65926	−	36.0488i	0.449877	−	1.67896i	0.702729	−	0.711457i	$-0.251964\pi$
$462$		0			0
$463$	3.83013	+	3.83013i	0.178001	+	0.178001i	0.790484	−	0.612483i	$-0.209829\pi$
$463$	3.83013	+	3.83013i	0.178001	+	0.178001i	−0.612483	+	0.790484i	$0.709829\pi$
$464$	−4.89898	+	2.82843i	−0.227429	+	0.131306i
$465$		0			0
$466$	−20.1962	+	5.41154i	−0.935569	+	0.250685i
$467$	22.0454			1.02014			0.510070	−	0.860133i	$-0.329620\pi$
$467$	22.0454			1.02014			0.510070	+	0.860133i	$0.329620\pi$
$468$		0			0
$469$	17.9282			0.827848
$470$	9.52056	−	2.55103i	0.439151	−	0.117670i
$471$		0			0
$472$		0			0
$473$	18.6993	+	18.6993i	0.859797	+	0.859797i
$474$		0			0
$475$	4.02628	−	15.0263i	0.184738	−	0.689453i
$476$	4.89898	−	4.89898i	0.224544	−	0.224544i
$477$		0			0
$478$	23.3205	+	13.4641i	1.06666	+	0.615834i
$479$	1.03528	+	3.86370i	0.0473030	+	0.176537i	0.985536	−	0.169468i	$-0.0542048\pi$
$479$	1.03528	+	3.86370i	0.0473030	+	0.176537i	−0.938233	+	0.346005i	$0.887538\pi$
$480$		0			0
$481$	5.73205	−	28.6603i	0.261359	−	1.30680i
$482$			33.1833i			1.51146i
$483$		0			0
$484$	−27.3923	+	47.4449i	−1.24510	+	2.15658i
$485$	−4.43211	−	7.67664i	−0.201252	−	0.348578i
$486$		0			0
$487$	−13.8301	−	3.70577i	−0.626703	−	0.167925i	−0.0685298	−	0.997649i	$-0.521831\pi$
$487$	−13.8301	−	3.70577i	−0.626703	−	0.167925i	−0.558173	+	0.829725i	$0.688497\pi$
$488$		0			0
$489$		0			0
$490$	−2.39230	−	4.14359i	−0.108073	−	0.187188i
$491$	8.15711	−	14.1285i	0.368125	−	0.637612i	−0.621147	−	0.783694i	$-0.713333\pi$
$491$	8.15711	−	14.1285i	0.368125	−	0.637612i	0.989272	+	0.146082i	$0.0466664\pi$
$492$		0			0
$493$		−	2.53590i		−	0.114211i
$494$	11.1378	−	22.5259i	0.501115	−	1.01349i
$495$		0			0
$496$	9.32051	+	34.7846i	0.418503	+	1.56188i
$497$	2.77766	+	1.60368i	0.124595	+	0.0719350i
$498$		0			0
$499$	−27.7321	+	27.7321i	−1.24146	+	1.24146i	−0.282060	+	0.959397i	$0.591018\pi$
$499$	−27.7321	+	27.7321i	−1.24146	+	1.24146i	−0.959397	+	0.282060i	$0.908982\pi$
$500$	3.58630	−	13.3843i	0.160384	−	0.598562i
$501$		0			0
$502$	−18.2487	−	18.2487i	−0.814480	−	0.814480i
$503$	−23.2702	+	13.4350i	−1.03756	+	0.599038i	−0.919143	−	0.393925i	$-0.871117\pi$
$503$	−23.2702	+	13.4350i	−1.03756	+	0.599038i	−0.118422	+	0.992963i	$0.537784\pi$
$504$		0			0
$505$	11.1962	−	3.00000i	0.498222	−	0.133498i
$506$	−30.3548			−1.34944
$507$		0			0
$508$	7.85641			0.348572
$509$	−24.7859	+	6.64136i	−1.09862	+	0.294373i	−0.762201	−	0.647341i	$-0.775881\pi$
$509$	−24.7859	+	6.64136i	−1.09862	+	0.294373i	−0.336415	+	0.941714i	$0.609214\pi$
$510$		0			0
$511$	−19.1603	+	11.0622i	−0.847600	+	0.489362i
$512$	22.6274	+	22.6274i	1.00000	+	1.00000i
$513$		0			0
$514$	−9.46410	+	35.3205i	−0.417444	+	1.55792i
$515$	−4.48288	+	4.48288i	−0.197539	+	0.197539i
$516$		0			0
$517$	−36.1244	−	20.8564i	−1.58875	−	0.917264i
$518$	8.10634	+	30.2533i	0.356172	+	1.32925i
$519$		0			0
$520$		0			0
$521$			32.2223i			1.41168i	0.708369	+	0.705842i	$0.249431\pi$
$521$			32.2223i			1.41168i	−0.708369	+	0.705842i	$0.750569\pi$
$522$		0			0
$523$	0.607695	−	1.05256i	0.0265727	−	0.0460252i	−0.852433	−	0.522836i	$-0.824874\pi$
$523$	0.607695	−	1.05256i	0.0265727	−	0.0460252i	0.879006	+	0.476811i	$0.158207\pi$
$524$	7.07107	+	12.2474i	0.308901	+	0.535032i
$525$		0			0
$526$	39.8564	+	10.6795i	1.73782	+	0.465648i
$527$	−15.5935	−	4.17827i	−0.679264	−	0.182008i
$528$		0			0
$529$	8.50000	+	14.7224i	0.369565	+	0.640106i
$530$	−6.41473	+	11.1106i	−0.278638	+	0.482615i
$531$		0			0
$532$			13.4641i			0.583743i
$533$	3.86370	−	19.3185i	0.167356	−	0.836778i
$534$		0			0
$535$	2.46410	+	9.19615i	0.106532	+	0.397584i
$536$		0			0
$537$		0			0
$538$	33.1769	−	33.1769i	1.43036	−	1.43036i
$539$	−5.24075	+	19.5588i	−0.225735	+	0.842455i
$540$		0			0
$541$	5.68653	+	5.68653i	0.244483	+	0.244483i	0.818702	−	0.574219i	$-0.194694\pi$
$541$	5.68653	+	5.68653i	0.244483	+	0.244483i	−0.574219	+	0.818702i	$0.694694\pi$
$542$	38.7757	−	22.3872i	1.66556	−	0.961612i
$543$		0			0
$544$	−13.8564	+	3.71281i	−0.594089	+	0.159186i
$545$	4.44571			0.190433
$546$		0			0
$547$	3.39230			0.145044			0.0725222	−	0.997367i	$-0.476895\pi$
$547$	3.39230			0.145044			0.0725222	+	0.997367i	$0.476895\pi$
$548$	2.07055	−	0.554803i	0.0884496	−	0.0237000i
$549$		0			0
$550$	−47.9090	+	27.6603i	−2.04285	+	1.17944i
$551$	3.48477	+	3.48477i	0.148456	+	0.148456i
$552$		0			0
$553$	1.59808	−	5.96410i	0.0679571	−	0.253619i
$554$	−11.1106	+	11.1106i	−0.472046	+	0.472046i
$555$		0			0
$556$	30.1244	+	17.3923i	1.27756	+	0.737598i
$557$	−2.96713	−	11.0735i	−0.125721	−	0.469198i	0.874143	−	0.485668i	$-0.161424\pi$
$557$	−2.96713	−	11.0735i	−0.125721	−	0.469198i	−0.999864	+	0.0164704i	$0.994757\pi$
$558$		0			0
$559$	0.990381	+	15.3564i	0.0418887	+	0.649507i
$560$		−	5.65685i		−	0.239046i
$561$		0			0
$562$	21.5167	−	37.2679i	0.907626	−	1.57205i
$563$	14.8492	+	25.7196i	0.625821	+	1.08395i	0.988381	+	0.151993i	$0.0485693\pi$
$563$	14.8492	+	25.7196i	0.625821	+	1.08395i	−0.362561	+	0.931960i	$0.618097\pi$
$564$		0			0
$565$	7.19615	+	1.92820i	0.302744	+	0.0811201i
$566$	−9.38186	−	2.51386i	−0.394349	−	0.105665i
$567$		0			0
$568$		0			0
$569$	3.67423	−	6.36396i	0.154032	−	0.266791i	−0.778674	−	0.627428i	$-0.784107\pi$
$569$	3.67423	−	6.36396i	0.154032	−	0.266791i	0.932706	+	0.360637i	$0.117441\pi$
$570$		0			0
$571$		−	33.7128i		−	1.41084i	−0.708791	−	0.705419i	$-0.750759\pi$
$571$		−	33.7128i		−	1.41084i	0.708791	−	0.705419i	$-0.249241\pi$
$572$	−42.3249	+	14.3180i	−1.76969	+	0.598666i
$573$		0			0
$574$	5.46410	+	20.3923i	0.228067	+	0.851158i
$575$	−9.46979	−	5.46739i	−0.394918	−	0.228006i
$576$		0			0
$577$	−1.19615	+	1.19615i	−0.0497965	+	0.0497965i	−0.731567	−	0.681770i	$-0.761210\pi$
$577$	−1.19615	+	1.19615i	−0.0497965	+	0.0497965i	0.681770	+	0.731567i	$0.261210\pi$
$578$	−7.13544	+	26.6298i	−0.296795	+	1.10765i
$579$		0			0
$580$	1.46410	+	1.46410i	0.0607935	+	0.0607935i
$581$	0.896575	−	0.517638i	0.0371962	−	0.0214752i
$582$		0			0
$583$	52.4449	−	14.0526i	2.17204	−	0.581998i
$584$		0			0
$585$		0			0
$586$	−44.3923			−1.83383
$587$	−7.15900	+	1.91825i	−0.295484	+	0.0791746i	−0.403515	−	0.914973i	$-0.632212\pi$
$587$	−7.15900	+	1.91825i	−0.295484	+	0.0791746i	0.108032	+	0.994147i	$0.465545\pi$
$588$		0			0
$589$	27.1699	−	15.6865i	1.11952	−	0.646352i
$590$	−10.5558	−	10.5558i	−0.434577	−	0.434577i
$591$		0			0
$592$	8.39230	−	31.3205i	0.344922	−	1.28726i
$593$	−11.4524	+	11.4524i	−0.470294	+	0.470294i	−0.902010	−	0.431716i	$-0.857908\pi$
$593$	−11.4524	+	11.4524i	−0.470294	+	0.470294i	0.431716	+	0.902010i	$0.357908\pi$
$594$		0			0
$595$	2.19615	+	1.26795i	0.0900335	+	0.0519808i
$596$	7.45001	+	27.8038i	0.305164	+	1.13889i
$597$		0			0
$598$	−13.2679	−	11.6603i	−0.542567	−	0.476823i
$599$		−	29.9759i		−	1.22478i	−0.790555	−	0.612391i	$-0.790208\pi$
$599$		−	29.9759i		−	1.22478i	0.790555	−	0.612391i	$-0.209792\pi$
$600$		0			0
$601$	−2.39230	+	4.14359i	−0.0975842	+	0.169021i	−0.910684	−	0.413103i	$-0.864445\pi$
$601$	−2.39230	+	4.14359i	−0.0975842	+	0.169021i	0.813100	+	0.582124i	$0.197778\pi$
$602$	−8.24504	−	14.2808i	−0.336043	−	0.582043i
$603$		0			0
$604$	−4.19615	−	1.12436i	−0.170739	−	0.0457494i
$605$	−19.3693	−	5.18998i	−0.787473	−	0.211003i
$606$		0			0
$607$	8.80385	+	15.2487i	0.357337	+	0.618926i	0.987515	−	0.157525i	$-0.0503514\pi$
$607$	8.80385	+	15.2487i	0.357337	+	0.618926i	−0.630178	+	0.776451i	$0.717018\pi$
$608$	13.9391	−	24.1432i	0.565304	−	0.979135i
$609$		0			0
$610$			4.10512i			0.166211i
$611$	−7.77817	−	22.9928i	−0.314671	−	0.930187i
$612$		0			0
$613$	2.33013	+	8.69615i	0.0941129	+	0.351234i	0.996883	−	0.0788882i	$-0.0251370\pi$
$613$	2.33013	+	8.69615i	0.0941129	+	0.351234i	−0.902771	+	0.430123i	$0.858470\pi$
$614$	−9.07725	−	5.24075i	−0.366328	−	0.211500i
$615$		0			0
$616$		0			0
$617$	−5.43022	+	20.2659i	−0.218612	+	0.815873i	0.766251	+	0.642541i	$0.222120\pi$
$617$	−5.43022	+	20.2659i	−0.218612	+	0.815873i	−0.984864	+	0.173332i	$0.944547\pi$
$618$		0			0
$619$	4.16987	+	4.16987i	0.167601	+	0.167601i	0.785924	−	0.618323i	$-0.212188\pi$
$619$	4.16987	+	4.16987i	0.167601	+	0.167601i	−0.618323	+	0.785924i	$0.712188\pi$
$620$	11.4152	−	6.59059i	0.458447	−	0.264685i
$621$		0			0
$622$	54.2487	−	14.5359i	2.17518	−	0.582836i
$623$	22.1469			0.887299
$624$		0			0
$625$	−17.2487			−0.689948
$626$	−13.9019	+	3.72500i	−0.555632	+	0.148881i
$627$		0			0
$628$	16.2679	−	9.39230i	0.649162	−	0.374794i
$629$	10.2784	+	10.2784i	0.409828	+	0.409828i
$630$		0			0
$631$	5.54552	−	20.6962i	0.220764	−	0.823901i	−0.763294	−	0.646051i	$-0.776419\pi$
$631$	5.54552	−	20.6962i	0.220764	−	0.823901i	0.984058	−	0.177850i	$-0.0569141\pi$
$632$		0			0
$633$		0			0
$634$	−1.85641	−	1.07180i	−0.0737273	−	0.0425665i
$635$	0.744272	+	2.77766i	0.0295355	+	0.110228i
$636$		0			0
$637$	−9.80385	+	6.53590i	−0.388443	+	0.258962i
$638$		−	17.5254i		−	0.693836i
$639$		0			0
$640$		0			0
$641$	13.4722	+	23.3345i	0.532120	+	0.921658i	0.999297	+	0.0374946i	$0.0119377\pi$
$641$	13.4722	+	23.3345i	0.532120	+	0.921658i	−0.467177	+	0.884164i	$0.654729\pi$
$642$		0			0
$643$	−0.964102	−	0.258330i	−0.0380205	−	0.0101876i	0.239759	−	0.970833i	$-0.422932\pi$
$643$	−0.964102	−	0.258330i	−0.0380205	−	0.0101876i	−0.277779	+	0.960645i	$0.589598\pi$
$644$	9.14162	+	2.44949i	0.360230	+	0.0965234i
$645$		0			0
$646$	6.24871	+	10.8231i	0.245852	+	0.425829i
$647$	−23.9909	+	41.5534i	−0.943178	+	1.63363i	−0.183820	+	0.982960i	$0.558846\pi$
$647$	−23.9909	+	41.5534i	−0.943178	+	1.63363i	−0.759359	+	0.650672i	$0.774487\pi$
$648$		0			0
$649$			63.1769i			2.47991i
$650$	−31.5660	−	6.31319i	−1.23812	−	0.247624i
$651$		0			0
$652$	−7.87564	−	29.3923i	−0.308434	−	1.15109i
$653$	−9.79796	−	5.65685i	−0.383424	−	0.221370i	0.295883	−	0.955224i	$-0.404386\pi$
$653$	−9.79796	−	5.65685i	−0.383424	−	0.221370i	−0.679307	+	0.733854i	$0.737719\pi$
$654$		0			0
$655$	−3.66025	+	3.66025i	−0.143018	+	0.143018i
$656$	5.65685	−	21.1117i	0.220863	−	0.824272i
$657$		0			0
$658$	18.3923	+	18.3923i	0.717007	+	0.717007i
$659$	23.2466	−	13.4214i	0.905559	−	0.522825i	0.0265591	−	0.999647i	$-0.491545\pi$
$659$	23.2466	−	13.4214i	0.905559	−	0.522825i	0.878999	+	0.476823i	$0.158212\pi$
$660$		0			0
$661$	4.40192	−	1.17949i	0.171215	−	0.0458769i	−0.172193	−	0.985063i	$-0.555085\pi$
$661$	4.40192	−	1.17949i	0.171215	−	0.0458769i	0.343408	+	0.939186i	$0.388419\pi$
$662$	−13.6617			−0.530976
$663$		0			0
$664$		0			0
$665$	−4.76028	+	1.27551i	−0.184596	+	0.0494623i
$666$		0			0
$667$	3.00000	−	1.73205i	0.116160	−	0.0670653i
$668$	2.07055	+	2.07055i	0.0801121	+	0.0801121i
$669$		0			0
$670$	−3.51666	+	13.1244i	−0.135860	+	0.507038i
$671$	12.2846	−	12.2846i	0.474242	−	0.474242i
$672$		0			0
$673$	−34.2846	−	19.7942i	−1.32157	−	0.763011i	−0.337595	−	0.941292i	$-0.609613\pi$
$673$	−34.2846	−	19.7942i	−1.32157	−	0.763011i	−0.983980	+	0.178280i	$0.942947\pi$
$674$	−7.28410	−	27.1846i	−0.280573	−	1.04711i
$675$		0			0
$676$	−24.0000	−	10.0000i	−0.923077	−	0.384615i
$677$			42.9812i			1.65190i	0.563742	+	0.825951i	$0.309361\pi$
$677$			42.9812i			1.65190i	−0.563742	+	0.825951i	$0.690639\pi$
$678$		0			0
$679$	11.6962	−	20.2583i	0.448857	−	0.777443i
$680$		0			0
$681$		0			0
$682$	−107.765	−	28.8756i	−4.12655	−	1.10571i
$683$	26.5791	+	7.12184i	1.01702	+	0.272509i	0.728559	−	0.684983i	$-0.240190\pi$
$683$	26.5791	+	7.12184i	1.01702	+	0.272509i	0.288460	+	0.957492i	$0.406857\pi$
$684$		0			0
$685$	0.392305	+	0.679492i	0.0149892	+	0.0259621i
$686$	19.8362	−	34.3572i	0.757348	−	1.31177i
$687$		0			0
$688$			17.0718i			0.650856i
$689$	28.3214	+	14.0034i	1.07896	+	0.533488i
$690$		0			0
$691$	10.5718	+	39.4545i	0.402170	+	1.50092i	0.809216	+	0.587511i	$0.199892\pi$
$691$	10.5718	+	39.4545i	0.402170	+	1.50092i	−0.407046	+	0.913408i	$0.633441\pi$
$692$	−9.62209	−	5.55532i	−0.365777	−	0.211182i
$693$		0			0
$694$	21.1769	−	21.1769i	0.803865	−	0.803865i
$695$	−3.29530	+	12.2982i	−0.124998	+	0.466498i
$696$		0			0
$697$	6.92820	+	6.92820i	0.262424	+	0.262424i
$698$	59.9889	−	34.6346i	2.27061	−	1.31094i
$699$		0			0
$700$	16.6603	−	4.46410i	0.629698	−	0.168727i
$701$	6.21166			0.234611			0.117306	−	0.993096i	$-0.462574\pi$
$701$	6.21166			0.234611			0.117306	+	0.993096i	$0.462574\pi$
$702$		0			0
$703$	−28.2487			−1.06542
$704$	−47.8802	+	12.8295i	−1.80455	+	0.483528i
$705$		0			0
$706$	−21.1244	+	12.1962i	−0.795026	+	0.459008i
$707$	21.6293	+	21.6293i	0.813454	+	0.813454i
$708$		0			0
$709$	−1.30385	+	4.86603i	−0.0489670	+	0.182747i	−0.986078	−	0.166285i	$-0.946823\pi$
$709$	−1.30385	+	4.86603i	−0.0489670	+	0.182747i	0.937111	+	0.349032i	$0.113490\pi$
$710$	−1.71882	+	1.71882i	−0.0645062	+	0.0645062i
$711$		0			0
$712$		0			0
$713$	−5.70762	−	21.3011i	−0.213752	−	0.797734i
$714$		0			0
$715$	−9.07180	−	13.6077i	−0.339266	−	0.508899i
$716$		−	16.0096i		−	0.598307i
$717$		0			0
$718$	−10.0000	+	17.3205i	−0.373197	+	0.646396i
$719$	−4.98691	−	8.63759i	−0.185980	−	0.322128i	0.757926	−	0.652341i	$-0.226213\pi$
$719$	−4.98691	−	8.63759i	−0.185980	−	0.322128i	−0.943906	+	0.330213i	$0.892879\pi$
$720$		0			0
$721$	−16.1603	−	4.33013i	−0.601839	−	0.161262i
$722$	13.2456	+	3.54914i	0.492949	+	0.132085i
$723$		0			0
$724$	−6.00000	−	10.3923i	−0.222988	−	0.386227i
$725$	3.15660	−	5.46739i	0.117233	−	0.203054i
$726$		0			0
$727$			41.1051i			1.52450i	0.647280	+	0.762252i	$0.275906\pi$
$727$			41.1051i			1.52450i	−0.647280	+	0.762252i	$0.724094\pi$
$728$		0			0
$729$		0			0
$730$	−4.33975	−	16.1962i	−0.160621	−	0.599446i
$731$	−6.62776	−	3.82654i	−0.245137	−	0.141530i
$732$		0			0
$733$	−10.2417	+	10.2417i	−0.378285	+	0.378285i	−0.870483	−	0.492198i	$-0.836193\pi$
$733$	−10.2417	+	10.2417i	−0.378285	+	0.378285i	0.492198	+	0.870483i	$0.336193\pi$
$734$	7.96764	−	29.7356i	0.294091	−	1.09756i
$735$		0			0
$736$	−13.8564	−	13.8564i	−0.510754	−	0.510754i
$737$	49.7984	−	28.7511i	1.83435	−	1.05906i
$738$		0			0
$739$	11.5622	−	3.09808i	0.425322	−	0.113965i	−0.0398077	−	0.999207i	$-0.512675\pi$
$739$	11.5622	−	3.09808i	0.425322	−	0.113965i	0.465129	+	0.885243i	$0.346008\pi$
$740$	−11.8685			−0.436295
$741$		0			0
$742$	−33.8564			−1.24291
$743$	30.0638	−	8.05558i	1.10293	−	0.295530i	0.338976	−	0.940795i	$-0.389920\pi$
$743$	30.0638	−	8.05558i	1.10293	−	0.295530i	0.763959	+	0.645265i	$0.223253\pi$
$744$		0			0
$745$	−9.12436	+	5.26795i	−0.334291	+	0.193003i
$746$	33.3591	+	33.3591i	1.22136	+	1.22136i
$747$		0			0
$748$	5.75129	−	21.4641i	0.210288	−	0.784805i
$749$	−17.7656	+	17.7656i	−0.649141	+	0.649141i
$750$		0			0
$751$	32.7846	+	18.9282i	1.19633	+	0.690700i	0.959734	−	0.280909i	$-0.0906359\pi$
$751$	32.7846	+	18.9282i	1.19633	+	0.690700i	0.236593	+	0.971609i	$0.423969\pi$
$752$	−6.96953	−	26.0106i	−0.254153	−	0.948511i
$753$		0			0
$754$	6.73205	−	7.66025i	0.245167	−	0.278970i
$755$		−	1.59008i		−	0.0578689i
$756$		0			0
$757$	25.1962	−	43.6410i	0.915770	−	1.58616i	0.109999	−	0.993932i	$-0.464915\pi$
$757$	25.1962	−	43.6410i	0.915770	−	1.58616i	0.805770	−	0.592228i	$-0.201752\pi$
$758$	20.6956	+	35.8458i	0.751697	+	1.30198i
$759$		0			0
$760$		0			0
$761$	−41.3268	−	11.0735i	−1.49809	−	0.401413i	−0.585633	−	0.810577i	$-0.699154\pi$
$761$	−41.3268	−	11.0735i	−1.49809	−	0.401413i	−0.912461	+	0.409163i	$0.865821\pi$
$762$		0			0
$763$	5.86603	+	10.1603i	0.212364	+	0.367826i
$764$	1.69161	−	2.92996i	0.0612005	−	0.106002i
$765$		0			0
$766$		−	20.0000i		−	0.722629i
$767$	−24.2683	+	27.6143i	−0.876276	+	0.997096i
$768$		0			0
$769$	7.50962	+	28.0263i	0.270804	+	1.01065i	0.958601	+	0.284751i	$0.0919109\pi$
$769$	7.50962	+	28.0263i	0.270804	+	1.01065i	−0.687798	+	0.725902i	$0.741422\pi$
$770$	15.1774	+	8.76268i	0.546956	+	0.315785i
$771$		0			0
$772$	15.6603	−	15.6603i	0.563625	−	0.563625i
$773$	10.8840	−	40.6197i	0.391470	−	1.46099i	−0.436239	−	0.899831i	$-0.643690\pi$
$773$	10.8840	−	40.6197i	0.391470	−	1.46099i	0.827710	−	0.561157i	$-0.189643\pi$
$774$		0			0
$775$	−28.4186	−	28.4186i	−1.02083	−	1.02083i
$776$		0			0
$777$		0			0
$778$	−3.46410	+	0.928203i	−0.124194	+	0.0332777i
$779$	−19.0411			−0.682219
$780$		0			0
$781$	10.2872			0.368104
$782$	8.48528	−	2.27362i	0.303433	−	0.0813046i
$783$		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)

\(f(q)\)	\(=\)	\(q+(-1.93185 + 0.517638i) q^{2} +(1.73205 - 1.00000i) q^{4} +(0.517638 + 0.517638i) q^{5} +(-0.500000 + 1.86603i) q^{7} +(-1.26795 - 0.732051i) q^{10} +(1.60368 + 5.98502i) q^{11} +(-1.59808 + 3.23205i) q^{13} -3.86370i q^{14} +(-2.00000 + 3.46410i) q^{16} +(-0.896575 - 1.55291i) q^{17} +(3.36603 + 0.901924i) q^{19} +(1.41421 + 0.378937i) q^{20} +(-6.19615 - 10.7321i) q^{22} +(1.22474 - 2.12132i) q^{23} -4.46410i q^{25} +(1.41421 - 7.07107i) q^{26} +(1.00000 + 3.73205i) q^{28} +(1.22474 + 0.707107i) q^{29} +(6.36603 - 6.36603i) q^{31} +(2.07055 - 7.72741i) q^{32} +(2.53590 + 2.53590i) q^{34} +(-1.22474 + 0.707107i) q^{35} +(-7.83013 + 2.09808i) q^{37} -6.96953 q^{38} +(-5.27792 + 1.41421i) q^{41} +(3.69615 - 2.13397i) q^{43} +(8.76268 + 8.76268i) q^{44} +(-1.26795 + 4.73205i) q^{46} +(-4.76028 + 4.76028i) q^{47} +(2.83013 + 1.63397i) q^{49} +(2.31079 + 8.62398i) q^{50} +(0.464102 + 7.19615i) q^{52} -8.76268i q^{53} +(-2.26795 + 3.92820i) q^{55} +(-2.73205 - 0.732051i) q^{58} +(9.84873 + 2.63896i) q^{59} +(-1.40192 - 2.42820i) q^{61} +(-9.00292 + 15.5935i) q^{62} +8.00000i q^{64} +(-2.50026 + 0.845807i) q^{65} +(-2.40192 - 8.96410i) q^{67} +(-3.10583 - 1.79315i) q^{68} +(2.00000 - 2.00000i) q^{70} +(0.429705 - 1.60368i) q^{71} +(8.09808 + 8.09808i) q^{73} +(14.0406 - 8.10634i) q^{74} +(6.73205 - 1.80385i) q^{76} -11.9700 q^{77} -3.19615 q^{79} +(-2.82843 + 0.757875i) q^{80} +(9.46410 - 5.46410i) q^{82} +(-0.378937 - 0.378937i) q^{83} +(0.339746 - 1.26795i) q^{85} +(-6.03579 + 6.03579i) q^{86} +(-2.96713 - 11.0735i) q^{89} +(-5.23205 - 4.59808i) q^{91} -4.89898i q^{92} +(6.73205 - 11.6603i) q^{94} +(1.27551 + 2.20925i) q^{95} +(-11.6962 - 3.13397i) q^{97} +(-6.31319 - 1.69161i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{7} - 24 q^{10} + 8 q^{13} - 16 q^{16} + 20 q^{19} - 8 q^{22} + 8 q^{28} + 44 q^{31} + 48 q^{34} - 28 q^{37} - 12 q^{43} - 24 q^{46} - 12 q^{49} - 24 q^{52} - 32 q^{55} - 8 q^{58} - 32 q^{61}+ \cdots - 52 q^{97}+O(q^{100}) \) 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

Introduction

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