LMFDB - Embedded newform 171.2.f.c.64.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,2,Mod(64,171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(171, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("171.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 171 = 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	171.f (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.36544187456$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.764411904.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 6x^{6} + 21x^{4} - 54x^{2} + 81 $ x^8 - 6x^6 + 21x^4 - 54*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			64.4
Root			$1.69185 + 0.370982i$ of defining polynomial
Character	$\chi$	$=$	171.64
Dual form			171.2.f.c.163.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.16721 + 2.02166i) q^{2} +(-1.72474 + 2.98735i) q^{4} +(0.524648 + 0.908716i) q^{5} -3.44949 q^{7} -3.38371 q^{8} +O(q^{10})$	\(q+(1.16721 + 2.02166i) q^{2} +(-1.72474 + 2.98735i) q^{4} +(0.524648 + 0.908716i) q^{5} -3.44949 q^{7} -3.38371 q^{8} +(-1.22474 + 2.12132i) q^{10} +5.71812 q^{11} +(0.500000 - 0.866025i) q^{13} +(-4.02627 - 6.97370i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-1.04930 - 1.81743i) q^{17} +(1.00000 - 4.24264i) q^{19} -3.61953 q^{20} +(6.67423 + 11.5601i) q^{22} +(-1.80977 + 3.13461i) q^{23} +(1.94949 - 3.37662i) q^{25} +2.33441 q^{26} +(5.94949 - 10.3048i) q^{28} +(3.61953 - 6.26922i) q^{29} -9.44949 q^{31} +(-2.21650 + 3.83909i) q^{32} +(2.44949 - 4.24264i) q^{34} +(-1.80977 - 3.13461i) q^{35} +3.89898 q^{37} +(9.74439 - 2.93038i) q^{38} +(-1.77526 - 3.07483i) q^{40} +(4.66883 + 8.08665i) q^{41} +(-3.17423 - 5.49794i) q^{43} +(-9.86230 + 17.0820i) q^{44} -8.44949 q^{46} +(-4.66883 + 8.08665i) q^{47} +4.89898 q^{49} +9.10183 q^{50} +(1.72474 + 2.98735i) q^{52} +(-0.524648 + 0.908716i) q^{53} +(3.00000 + 5.19615i) q^{55} +11.6721 q^{56} +16.8990 q^{58} +(-3.90836 - 6.76947i) q^{59} +(-2.50000 + 4.33013i) q^{61} +(-11.0295 - 19.1037i) q^{62} -12.3485 q^{64} +1.04930 q^{65} +(-0.174235 + 0.301783i) q^{67} +7.23907 q^{68} +(4.22474 - 7.31747i) q^{70} +(-3.61953 - 6.26922i) q^{71} +(-2.50000 - 4.33013i) q^{73} +(4.55092 + 7.88242i) q^{74} +(10.9495 + 10.3048i) q^{76} -19.7246 q^{77} +(-0.174235 - 0.301783i) q^{79} +(0.524648 - 0.908716i) q^{80} +(-10.8990 + 18.8776i) q^{82} -11.4362 q^{83} +(1.10102 - 1.90702i) q^{85} +(7.40998 - 12.8345i) q^{86} -19.3485 q^{88} +(2.62324 - 4.54358i) q^{89} +(-1.72474 + 2.98735i) q^{91} +(-6.24277 - 10.8128i) q^{92} -21.7980 q^{94} +(4.38000 - 1.31718i) q^{95} +(-1.55051 - 2.68556i) q^{97} +(5.71812 + 9.90408i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{4} - 8 q^{7}+O(q^{10}) $ 8 * q - 4 * q^4 - 8 * q^7	$ 8 q - 4 q^{4} - 8 q^{7} + 4 q^{13} - 4 q^{16} + 8 q^{19} + 24 q^{22} - 4 q^{25} + 28 q^{28} - 56 q^{31} - 8 q^{37} - 24 q^{40} + 4 q^{43} - 48 q^{46} + 4 q^{52} + 24 q^{55} + 96 q^{58} - 20 q^{61} - 40 q^{64} + 28 q^{67} + 24 q^{70} - 20 q^{73} + 68 q^{76} + 28 q^{79} - 48 q^{82} + 48 q^{85} - 96 q^{88} - 4 q^{91} - 96 q^{94} - 32 q^{97}+O(q^{100}) $ 8 * q - 4 * q^4 - 8 * q^7 + 4 * q^13 - 4 * q^16 + 8 * q^19 + 24 * q^22 - 4 * q^25 + 28 * q^28 - 56 * q^31 - 8 * q^37 - 24 * q^40 + 4 * q^43 - 48 * q^46 + 4 * q^52 + 24 * q^55 + 96 * q^58 - 20 * q^61 - 40 * q^64 + 28 * q^67 + 24 * q^70 - 20 * q^73 + 68 * q^76 + 28 * q^79 - 48 * q^82 + 48 * q^85 - 96 * q^88 - 4 * q^91 - 96 * q^94 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/171\mathbb{Z}\right)^\times$.

$n$	$20$	$154$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$	1.16721	+	2.02166i	0.825340	+	1.42953i	0.901659	+	0.432447i	$0.142350\pi$
$2$	1.16721	+	2.02166i	0.825340	+	1.42953i	−0.0763191	+	0.997083i	$0.524317\pi$
$3$		0			0
$3$		0			0
$4$	−1.72474	+	2.98735i	−0.862372	+	1.49367i
$5$	0.524648	+	0.908716i	0.234630	+	0.406390i	0.959165	−	0.282847i	$-0.0912790\pi$
$5$	0.524648	+	0.908716i	0.234630	+	0.406390i	−0.724535	+	0.689238i	$0.757946\pi$
$6$		0			0
$7$	−3.44949			−1.30378			−0.651892	−	0.758312i	$-0.726025\pi$
$7$	−3.44949			−1.30378			−0.651892	+	0.758312i	$0.726025\pi$
$8$	−3.38371			−1.19632
$9$		0			0
$10$	−1.22474	+	2.12132i	−0.387298	+	0.670820i
$11$	5.71812			1.72408			0.862040	−	0.506841i	$-0.169187\pi$
$11$	5.71812			1.72408			0.862040	+	0.506841i	$0.169187\pi$
$12$		0			0
$13$	0.500000	−	0.866025i	0.138675	−	0.240192i	−0.788320	−	0.615265i	$-0.789049\pi$
$13$	0.500000	−	0.866025i	0.138675	−	0.240192i	0.926995	+	0.375073i	$0.122382\pi$
$14$	−4.02627	−	6.97370i	−1.07607	−	1.86380i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	−1.04930	−	1.81743i	−0.254491	−	0.440792i	0.710266	−	0.703934i	$-0.248575\pi$
$17$	−1.04930	−	1.81743i	−0.254491	−	0.440792i	−0.964757	+	0.263141i	$0.915241\pi$
$18$		0			0
$19$	1.00000	−	4.24264i	0.229416	−	0.973329i
$19$	1.00000	−	4.24264i	0.229416	−	0.973329i
$20$	−3.61953			−0.809352
$21$		0			0
$22$	6.67423	+	11.5601i	1.42295	+	2.46462i
$23$	−1.80977	+	3.13461i	−0.377362	+	0.653611i	−0.990678	−	0.136227i	$-0.956502\pi$
$23$	−1.80977	+	3.13461i	−0.377362	+	0.653611i	0.613315	+	0.789838i	$0.289836\pi$
$24$		0			0
$25$	1.94949	−	3.37662i	0.389898	−	0.675323i
$26$	2.33441			0.457816
$27$		0			0
$28$	5.94949	−	10.3048i	1.12435	−	1.94743i
$29$	3.61953	−	6.26922i	0.672130	−	1.16416i	−0.305168	−	0.952298i	$-0.598713\pi$
$29$	3.61953	−	6.26922i	0.672130	−	1.16416i	0.977299	−	0.211866i	$-0.0679539\pi$
$30$		0			0
$31$	−9.44949			−1.69718			−0.848589	−	0.529052i	$-0.822548\pi$
$31$	−9.44949			−1.69718			−0.848589	+	0.529052i	$0.822548\pi$
$32$	−2.21650	+	3.83909i	−0.391826	+	0.678662i
$33$		0			0
$34$	2.44949	−	4.24264i	0.420084	−	0.727607i
$35$	−1.80977	−	3.13461i	−0.305906	−	0.529845i
$36$		0			0
$37$	3.89898			0.640988			0.320494	−	0.947250i	$-0.396151\pi$
$37$	3.89898			0.640988			0.320494	+	0.947250i	$0.396151\pi$
$38$	9.74439	−	2.93038i	1.58075	−	0.475370i
$39$		0			0
$40$	−1.77526	−	3.07483i	−0.280692	−	0.486174i
$41$	4.66883	+	8.08665i	0.729149	+	1.26292i	0.957244	+	0.289283i	$0.0934170\pi$
$41$	4.66883	+	8.08665i	0.729149	+	1.26292i	−0.228095	+	0.973639i	$0.573250\pi$
$42$		0			0
$43$	−3.17423	−	5.49794i	−0.484066	−	0.838427i	0.515766	−	0.856729i	$-0.327507\pi$
$43$	−3.17423	−	5.49794i	−0.484066	−	0.838427i	−0.999833	+	0.0183020i	$0.994174\pi$
$44$	−9.86230	+	17.0820i	−1.48680	+	2.57521i
$45$		0			0
$46$	−8.44949			−1.24581
$47$	−4.66883	+	8.08665i	−0.681019	+	1.17956i	0.293652	+	0.955912i	$0.405129\pi$
$47$	−4.66883	+	8.08665i	−0.681019	+	1.17956i	−0.974670	+	0.223646i	$0.928204\pi$
$48$		0			0
$49$	4.89898			0.699854
$50$	9.10183			1.28719
$51$		0			0
$52$	1.72474	+	2.98735i	0.239179	+	0.414270i
$53$	−0.524648	+	0.908716i	−0.0720659	+	0.124822i	−0.899807	−	0.436289i	$-0.856293\pi$
$53$	−0.524648	+	0.908716i	−0.0720659	+	0.124822i	0.827741	+	0.561111i	$0.189626\pi$
$54$		0			0
$55$	3.00000	+	5.19615i	0.404520	+	0.700649i
$56$	11.6721			1.55975
$57$		0			0
$58$	16.8990			2.21894
$59$	−3.90836	−	6.76947i	−0.508825	−	0.881310i	−0.999948	−	0.0102201i	$-0.996747\pi$
$59$	−3.90836	−	6.76947i	−0.508825	−	0.881310i	0.491123	−	0.871090i	$-0.336587\pi$
$60$		0			0
$61$	−2.50000	+	4.33013i	−0.320092	+	0.554416i	−0.980507	−	0.196485i	$-0.937047\pi$
$61$	−2.50000	+	4.33013i	−0.320092	+	0.554416i	0.660415	+	0.750901i	$0.270381\pi$
$62$	−11.0295	−	19.1037i	−1.40075	−	2.42617i
$63$		0			0
$64$	−12.3485			−1.54356
$65$	1.04930			0.130149
$66$		0			0
$67$	−0.174235	+	0.301783i	−0.0212861	+	0.0368687i	−0.876472	−	0.481452i	$-0.840109\pi$
$67$	−0.174235	+	0.301783i	−0.0212861	+	0.0368687i	0.855186	+	0.518321i	$0.173443\pi$
$68$	7.23907			0.877866
$69$		0			0
$70$	4.22474	−	7.31747i	0.504954	−	0.874605i
$71$	−3.61953	−	6.26922i	−0.429560	−	0.744019i	0.567275	−	0.823529i	$-0.307998\pi$
$71$	−3.61953	−	6.26922i	−0.429560	−	0.744019i	−0.996834	+	0.0795098i	$0.974665\pi$
$72$		0			0
$73$	−2.50000	−	4.33013i	−0.292603	−	0.506803i	0.681822	−	0.731519i	$-0.261188\pi$
$73$	−2.50000	−	4.33013i	−0.292603	−	0.506803i	−0.974424	+	0.224716i	$0.927855\pi$
$74$	4.55092	+	7.88242i	0.529033	+	0.916313i
$75$		0			0
$76$	10.9495	+	10.3048i	1.25599	+	1.18204i
$77$	−19.7246			−2.24783
$78$		0			0
$79$	−0.174235	−	0.301783i	−0.0196029	−	0.0339533i	0.856058	−	0.516881i	$-0.172907\pi$
$79$	−0.174235	−	0.301783i	−0.0196029	−	0.0339533i	−0.875660	+	0.482927i	$0.839574\pi$
$80$	0.524648	−	0.908716i	0.0586574	−	0.101598i
$81$		0			0
$82$	−10.8990	+	18.8776i	−1.20359	+	2.08468i
$83$	−11.4362			−1.25529			−0.627646	−	0.778499i	$-0.715981\pi$
$83$	−11.4362			−1.25529			−0.627646	+	0.778499i	$0.715981\pi$
$84$		0			0
$85$	1.10102	−	1.90702i	0.119422	−	0.206846i
$86$	7.40998	−	12.8345i	0.799039	−	1.38398i
$87$		0			0
$88$	−19.3485			−2.06255
$89$	2.62324	−	4.54358i	0.278063	−	0.481619i	−0.692841	−	0.721091i	$-0.743641\pi$
$89$	2.62324	−	4.54358i	0.278063	−	0.481619i	0.970903	+	0.239472i	$0.0769744\pi$
$90$		0			0
$91$	−1.72474	+	2.98735i	−0.180802	+	0.313159i
$92$	−6.24277	−	10.8128i	−0.650854	−	1.12731i
$93$		0			0
$94$	−21.7980			−2.24829
$95$	4.38000	−	1.31718i	0.449379	−	0.135139i
$96$		0			0
$97$	−1.55051	−	2.68556i	−0.157430	−	0.272678i	0.776511	−	0.630104i	$-0.216988\pi$
$97$	−1.55051	−	2.68556i	−0.157430	−	0.272678i	−0.933941	+	0.357426i	$0.883654\pi$
$98$	5.71812	+	9.90408i	0.577618	+	1.00046i
$99$		0			0
$100$	6.72474	+	11.6476i	0.672474	+	1.16476i
$101$	1.04930	−	1.81743i	0.104409	−	0.180841i	−0.809088	−	0.587688i	$-0.800038\pi$
$101$	1.04930	−	1.81743i	0.104409	−	0.180841i	0.913497	+	0.406847i	$0.133372\pi$
$102$		0			0
$103$	−3.44949			−0.339888			−0.169944	−	0.985454i	$-0.554359\pi$
$103$	−3.44949			−0.339888			−0.169944	+	0.985454i	$0.554359\pi$
$104$	−1.69185	+	2.93038i	−0.165900	+	0.287347i
$105$		0			0
$106$	−2.44949			−0.237915
$107$	16.5767			1.60253			0.801266	−	0.598308i	$-0.204160\pi$
$107$	16.5767			1.60253			0.801266	+	0.598308i	$0.204160\pi$
$108$		0			0
$109$	4.44949	+	7.70674i	0.426184	+	0.738172i	0.996530	−	0.0832323i	$-0.0265243\pi$
$109$	4.44949	+	7.70674i	0.426184	+	0.738172i	−0.570346	+	0.821404i	$0.693191\pi$
$110$	−7.00324	+	12.1300i	−0.667733	+	1.15655i
$111$		0			0
$112$	1.72474	+	2.98735i	0.162973	+	0.282278i
$113$	−8.28836			−0.779703			−0.389852	−	0.920878i	$-0.627474\pi$
$113$	−8.28836			−0.779703			−0.389852	+	0.920878i	$0.627474\pi$
$114$		0			0
$115$	−3.79796			−0.354162
$116$	12.4855	+	21.6256i	1.15925	+	2.00789i
$117$		0			0
$118$	9.12372	−	15.8028i	0.839907	−	1.45476i
$119$	3.61953	+	6.26922i	0.331802	+	0.574698i
$120$		0			0
$121$	21.6969			1.97245
$122$	−11.6721			−1.05674
$123$		0			0
$124$	16.2980	−	28.2289i	1.46360	−	2.53503i
$125$	9.33766			0.835185
$126$		0			0
$127$	−2.89898	+	5.02118i	−0.257243	+	0.445558i	−0.965502	−	0.260395i	$-0.916147\pi$
$127$	−2.89898	+	5.02118i	−0.257243	+	0.445558i	0.708259	+	0.705952i	$0.249481\pi$
$128$	−9.98022	−	17.2862i	−0.882135	−	1.52790i
$129$		0			0
$130$	1.22474	+	2.12132i	0.107417	+	0.186052i
$131$	7.81671	+	13.5389i	0.682949	+	1.18290i	0.974077	+	0.226219i	$0.0726364\pi$
$131$	7.81671	+	13.5389i	0.682949	+	1.18290i	−0.291127	+	0.956684i	$0.594030\pi$
$132$		0			0
$133$	−3.44949	+	14.6349i	−0.299109	+	1.26901i
$134$	−0.813472			−0.0702732
$135$		0			0
$136$	3.55051	+	6.14966i	0.304454	+	0.527329i
$137$	−7.81671	+	13.5389i	−0.667827	+	1.15671i	0.310684	+	0.950513i	$0.399442\pi$
$137$	−7.81671	+	13.5389i	−0.667827	+	1.15671i	−0.978511	+	0.206197i	$0.933891\pi$
$138$		0			0
$139$	1.17423	−	2.03383i	0.0995973	−	0.172508i	−0.811921	−	0.583768i	$-0.801578\pi$
$139$	1.17423	−	2.03383i	0.0995973	−	0.172508i	0.911518	+	0.411260i	$0.134911\pi$
$140$	12.4855			1.05522
$141$		0			0
$142$	8.44949	−	14.6349i	0.709065	−	1.22814i
$143$	2.85906	−	4.95204i	0.239087	−	0.414110i
$144$		0			0
$145$	7.59592			0.630807
$146$	5.83604	−	10.1083i	0.482994	−	0.836570i
$147$		0			0
$148$	−6.72474	+	11.6476i	−0.552771	+	0.957427i
$149$	6.24277	+	10.8128i	0.511428	+	0.885819i	0.999912	+	0.0132463i	$0.00421655\pi$
$149$	6.24277	+	10.8128i	0.511428	+	0.885819i	−0.488485	+	0.872573i	$0.662450\pi$
$150$		0			0
$151$	−4.00000			−0.325515			−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000			−0.325515			−0.162758	+	0.986666i	$0.552039\pi$
$152$	−3.38371	+	14.3559i	−0.274455	+	1.16441i
$153$		0			0
$154$	−23.0227	−	39.8765i	−1.85522	−	3.21334i
$155$	−4.95765	−	8.58691i	−0.398208	−	0.689717i
$156$		0			0
$157$	7.84847	+	13.5939i	0.626376	+	1.08492i	0.988273	+	0.152697i	$0.0487959\pi$
$157$	7.84847	+	13.5939i	0.626376	+	1.08492i	−0.361897	+	0.932218i	$0.617871\pi$
$158$	0.406736	−	0.704487i	0.0323582	−	0.0560460i
$159$		0			0
$160$	−4.65153			−0.367736
$161$	6.24277	−	10.8128i	0.491999	−	0.852168i
$162$		0			0
$163$	−9.44949			−0.740141			−0.370071	−	0.929004i	$-0.620667\pi$
$163$	−9.44949			−0.740141			−0.370071	+	0.929004i	$0.620667\pi$
$164$	−32.2102			−2.51519
$165$		0			0
$166$	−13.3485	−	23.1202i	−1.03604	−	1.79448i
$167$	9.62648	−	16.6736i	0.744919	−	1.29024i	−0.205313	−	0.978696i	$-0.565821\pi$
$167$	9.62648	−	16.6736i	0.744919	−	1.29024i	0.950232	−	0.311542i	$-0.100845\pi$
$168$		0			0
$169$	6.00000	+	10.3923i	0.461538	+	0.799408i
$170$	5.14048			0.394257
$171$		0			0
$172$	21.8990			1.66978
$173$	−3.61953	−	6.26922i	−0.275188	−	0.476640i	0.694995	−	0.719015i	$-0.255407\pi$
$173$	−3.61953	−	6.26922i	−0.275188	−	0.476640i	−0.970183	+	0.242375i	$0.922073\pi$
$174$		0			0
$175$	−6.72474	+	11.6476i	−0.508343	+	0.880476i
$176$	−2.85906	−	4.95204i	−0.215510	−	0.373274i
$177$		0			0
$178$	12.2474			0.917985
$179$	0.577648			0.0431754			0.0215877	−	0.999767i	$-0.493128\pi$
$179$	0.577648			0.0431754			0.0215877	+	0.999767i	$0.493128\pi$
$180$		0			0
$181$	10.4495	−	18.0990i	0.776704	−	1.34529i	−0.157127	−	0.987578i	$-0.550223\pi$
$181$	10.4495	−	18.0990i	0.776704	−	1.34529i	0.933832	−	0.357713i	$-0.116443\pi$
$182$	−8.05254			−0.596894
$183$		0			0
$184$	6.12372	−	10.6066i	0.451447	−	0.781929i
$185$	2.04559	+	3.54307i	0.150395	+	0.260491i
$186$		0			0
$187$	−6.00000	−	10.3923i	−0.438763	−	0.759961i
$188$	−16.1051	−	27.8948i	−1.17458	−	2.03444i
$189$		0			0
$190$	7.77526	+	7.31747i	0.564076	+	0.530865i
$191$	5.71812			0.413749			0.206874	−	0.978367i	$-0.433671\pi$
$191$	5.71812			0.413749			0.206874	+	0.978367i	$0.433671\pi$
$192$		0			0
$193$	−9.84847	−	17.0580i	−0.708908	−	1.22787i	−0.965262	−	0.261282i	$-0.915855\pi$
$193$	−9.84847	−	17.0580i	−0.708908	−	1.22787i	0.256354	−	0.966583i	$-0.417479\pi$
$194$	3.61953	−	6.26922i	0.259867	−	0.450103i
$195$		0			0
$196$	−8.44949	+	14.6349i	−0.603535	+	1.04535i
$197$	14.5841			1.03908			0.519538	−	0.854447i	$-0.326104\pi$
$197$	14.5841			1.03908			0.519538	+	0.854447i	$0.326104\pi$
$198$		0			0
$199$	−0.174235	+	0.301783i	−0.0123512	+	0.0213928i	−0.872135	−	0.489265i	$-0.837265\pi$
$199$	−0.174235	+	0.301783i	−0.0123512	+	0.0213928i	0.859784	+	0.510658i	$0.170598\pi$
$200$	−6.59651	+	11.4255i	−0.466443	+	0.807904i
$201$		0			0
$202$	4.89898			0.344691
$203$	−12.4855	+	21.6256i	−0.876313	+	1.51782i
$204$		0			0
$205$	−4.89898	+	8.48528i	−0.342160	+	0.592638i
$206$	−4.02627	−	6.97370i	−0.280523	−	0.485881i
$207$		0			0
$208$	−1.00000			−0.0693375
$209$	5.71812	−	24.2599i	0.395531	−	1.67810i
$210$		0			0
$211$	7.17423	+	12.4261i	0.493895	+	0.855451i	0.999975	−	0.00703553i	$-0.00223950\pi$
$211$	7.17423	+	12.4261i	0.493895	+	0.855451i	−0.506081	+	0.862486i	$0.668906\pi$
$212$	−1.80977	−	3.13461i	−0.124295	−	0.215286i
$213$		0			0
$214$	19.3485	+	33.5125i	1.32263	+	2.29087i
$215$	3.33071	−	5.76896i	0.227152	−	0.393440i
$216$		0			0
$217$	32.5959			2.21276
$218$	−10.3870	+	17.9907i	−0.703493	+	1.21849i
$219$		0			0
$220$	−20.6969			−1.39539
$221$	−2.09859			−0.141166
$222$		0			0
$223$	−0.174235	−	0.301783i	−0.0116676	−	0.0202089i	0.860133	−	0.510070i	$-0.170381\pi$
$223$	−0.174235	−	0.301783i	−0.0116676	−	0.0202089i	−0.871800	+	0.489862i	$0.837047\pi$
$224$	7.64580	−	13.2429i	0.510857	−	0.884830i
$225$		0			0
$226$	−9.67423	−	16.7563i	−0.643521	−	1.11461i
$227$	−10.8586			−0.720711			−0.360355	−	0.932815i	$-0.617345\pi$
$227$	−10.8586			−0.720711			−0.360355	+	0.932815i	$0.617345\pi$
$228$		0			0
$229$	−10.7980			−0.713549			−0.356775	−	0.934190i	$-0.616124\pi$
$229$	−10.7980			−0.713549			−0.356775	+	0.934190i	$0.616124\pi$
$230$	−4.43300	−	7.67819i	−0.292304	−	0.506285i
$231$		0			0
$232$	−12.2474	+	21.2132i	−0.804084	+	1.39272i
$233$	10.3870	+	17.9907i	0.680472	+	1.17861i	0.974837	+	0.222919i	$0.0715585\pi$
$233$	10.3870	+	17.9907i	0.680472	+	1.17861i	−0.294365	+	0.955693i	$0.595108\pi$
$234$		0			0
$235$	−9.79796			−0.639148
$236$	26.9637			1.75519
$237$		0			0
$238$	−8.44949	+	14.6349i	−0.547699	+	0.948643i
$239$	−10.8586			−0.702384			−0.351192	−	0.936303i	$-0.614224\pi$
$239$	−10.8586			−0.702384			−0.351192	+	0.936303i	$0.614224\pi$
$240$		0			0
$241$	9.50000	−	16.4545i	0.611949	−	1.05993i	−0.378963	−	0.925412i	$-0.623719\pi$
$241$	9.50000	−	16.4545i	0.611949	−	1.05993i	0.990912	−	0.134515i	$-0.0429475\pi$
$242$	25.3248	+	43.8639i	1.62794	+	2.81968i
$243$		0			0
$244$	−8.62372	−	14.9367i	−0.552077	−	0.956226i
$245$	2.57024	+	4.45178i	0.164206	+	0.284414i
$246$		0			0
$247$	−3.17423	−	2.98735i	−0.201972	−	0.190080i
$248$	31.9743			2.03037
$249$		0			0
$250$	10.8990	+	18.8776i	0.689312	+	1.19392i
$251$	−7.23907	+	12.5384i	−0.456926	+	0.791419i	−0.998797	−	0.0490430i	$-0.984383\pi$
$251$	−7.23907	+	12.5384i	−0.456926	+	0.791419i	0.541871	+	0.840462i	$0.317716\pi$
$252$		0			0
$253$	−10.3485	+	17.9241i	−0.650603	+	1.12688i
$254$	−13.5348			−0.849251
$255$		0			0
$256$	10.9495	−	18.9651i	0.684343	−	1.18532i
$257$	−6.24277	+	10.8128i	−0.389413	+	0.674484i	−0.992371	−	0.123290i	$-0.960656\pi$
$257$	−6.24277	+	10.8128i	−0.389413	+	0.674484i	0.602957	+	0.797773i	$0.293989\pi$
$258$		0			0
$259$	−13.4495			−0.835711
$260$	−1.80977	+	3.13461i	−0.112237	+	0.194400i
$261$		0			0
$262$	−18.2474	+	31.6055i	−1.12733	+	1.95259i
$263$	12.9572	+	22.4425i	0.798975	+	1.38386i	0.920284	+	0.391250i	$0.127957\pi$
$263$	12.9572	+	22.4425i	0.798975	+	1.38386i	−0.121310	+	0.992615i	$0.538709\pi$
$264$		0			0
$265$	−1.10102			−0.0676352
$266$	−33.6132	+	10.1083i	−2.06096	+	0.619780i
$267$		0			0
$268$	−0.601021	−	1.04100i	−0.0367132	−	0.0635891i
$269$	−7.76371	−	13.4471i	−0.473362	−	0.819887i	0.526173	−	0.850378i	$-0.323626\pi$
$269$	−7.76371	−	13.4471i	−0.473362	−	0.819887i	−0.999535	+	0.0304905i	$0.990293\pi$
$270$		0			0
$271$	−8.89898	−	15.4135i	−0.540575	−	0.936303i	−0.998871	−	0.0475032i	$-0.984874\pi$
$271$	−8.89898	−	15.4135i	−0.540575	−	0.936303i	0.458297	−	0.888799i	$-0.348460\pi$
$272$	−1.04930	+	1.81743i	−0.0636229	+	0.110198i
$273$		0			0
$274$	−36.4949			−2.20474
$275$	11.1474	−	19.3079i	0.672215	−	1.16431i
$276$		0			0
$277$	−18.6969			−1.12339			−0.561695	−	0.827344i	$-0.689851\pi$
$277$	−18.6969			−1.12339			−0.561695	+	0.827344i	$0.689851\pi$
$278$	5.48230			0.328807
$279$		0			0
$280$	6.12372	+	10.6066i	0.365963	+	0.633866i
$281$	−6.24277	+	10.8128i	−0.372413	+	0.645037i	−0.989936	−	0.141515i	$-0.954803\pi$
$281$	−6.24277	+	10.8128i	−0.372413	+	0.645037i	0.617524	+	0.786552i	$0.288136\pi$
$282$		0			0
$283$	3.10102	+	5.37113i	0.184337	+	0.319280i	0.943353	−	0.331791i	$-0.107653\pi$
$283$	3.10102	+	5.37113i	0.184337	+	0.319280i	−0.759016	+	0.651072i	$0.774320\pi$
$284$	24.9711			1.48176
$285$		0			0
$286$	13.3485			0.789312
$287$	−16.1051	−	27.8948i	−0.950653	−	1.64658i
$288$		0			0
$289$	6.29796	−	10.9084i	0.370468	−	0.641670i
$290$	8.86601	+	15.3564i	0.520630	+	0.901758i
$291$		0			0
$292$	17.2474			1.00933
$293$	10.2810			0.600620			0.300310	−	0.953842i	$-0.402910\pi$
$293$	10.2810			0.600620			0.300310	+	0.953842i	$0.402910\pi$
$294$		0			0
$295$	4.10102	−	7.10318i	0.238771	−	0.413563i
$296$	−13.1930			−0.766828
$297$		0			0
$298$	−14.5732	+	25.2415i	−0.844204	+	1.46220i
$299$	1.80977	+	3.13461i	0.104662	+	0.181279i
$300$		0			0
$301$	10.9495	+	18.9651i	0.631118	+	1.09313i
$302$	−4.66883	−	8.08665i	−0.268661	−	0.465334i
$303$		0			0
$304$	−4.17423	+	1.25529i	−0.239409	+	0.0719961i
$305$	−5.24648			−0.300412
$306$		0			0
$307$	14.7980	+	25.6308i	0.844564	+	1.46283i	0.885999	+	0.463687i	$0.153474\pi$
$307$	14.7980	+	25.6308i	0.844564	+	1.46283i	−0.0414351	+	0.999141i	$0.513193\pi$
$308$	34.0199	−	58.9242i	1.93846	−	3.35752i
$309$		0			0
$310$	11.5732	−	20.0454i	0.657314	−	1.13850i
$311$	−23.4501			−1.32974			−0.664868	−	0.746961i	$-0.731512\pi$
$311$	−23.4501			−1.32974			−0.664868	+	0.746961i	$0.731512\pi$
$312$		0			0
$313$	−1.55051	+	2.68556i	−0.0876400	+	0.151797i	−0.906513	−	0.422178i	$-0.861266\pi$
$313$	−1.55051	+	2.68556i	−0.0876400	+	0.151797i	0.818873	+	0.573975i	$0.194599\pi$
$314$	−18.3216	+	31.7339i	−1.03395	+	1.79085i
$315$		0			0
$316$	1.20204			0.0676201
$317$	5.19348	−	8.99536i	0.291695	−	0.505230i	−0.682516	−	0.730871i	$-0.739114\pi$
$317$	5.19348	−	8.99536i	0.291695	−	0.505230i	0.974211	+	0.225641i	$0.0724475\pi$
$318$		0			0
$319$	20.6969	−	35.8481i	1.15881	−	2.00711i
$320$	−6.47860	−	11.2213i	−0.362164	−	0.627287i
$321$		0			0
$322$	29.1464			1.62427
$323$	−8.76001	+	2.63435i	−0.487420	+	0.146579i
$324$		0			0
$325$	−1.94949	−	3.37662i	−0.108138	−	0.187301i
$326$	−11.0295	−	19.1037i	−0.610868	−	1.05805i
$327$		0			0
$328$	−15.7980	−	27.3629i	−0.872296	−	1.51086i
$329$	16.1051	−	27.8948i	0.887902	−	1.53789i
$330$		0			0
$331$	−3.44949			−0.189601			−0.0948006	−	0.995496i	$-0.530221\pi$
$331$	−3.44949			−0.189601			−0.0948006	+	0.995496i	$0.530221\pi$
$332$	19.7246	−	34.1640i	1.08253	−	1.87499i
$333$		0			0
$334$	44.9444			2.45925
$335$	−0.365647			−0.0199774
$336$		0			0
$337$	−12.8485	−	22.2542i	−0.699901	−	1.21226i	−0.968500	−	0.249012i	$-0.919894\pi$
$337$	−12.8485	−	22.2542i	−0.699901	−	1.21226i	0.268600	−	0.963252i	$-0.413439\pi$
$338$	−14.0065	+	24.2599i	−0.761852	+	1.31957i
$339$		0			0
$340$	3.79796	+	6.57826i	0.205973	+	0.356756i
$341$	−54.0334			−2.92607
$342$		0			0
$343$	7.24745			0.391325
$344$	10.7407	+	18.6034i	0.579099	+	1.00303i
$345$		0			0
$346$	8.44949	−	14.6349i	0.454247	−	0.786780i
$347$	4.38000	+	7.58639i	0.235131	+	0.407259i	0.959311	−	0.282352i	$-0.0911148\pi$
$347$	4.38000	+	7.58639i	0.235131	+	0.407259i	−0.724180	+	0.689611i	$0.757781\pi$
$348$		0			0
$349$	−16.7980			−0.899174			−0.449587	−	0.893237i	$-0.648429\pi$
$349$	−16.7980			−0.899174			−0.449587	+	0.893237i	$0.648429\pi$
$350$	−31.3967			−1.67822
$351$		0			0
$352$	−12.6742	+	21.9524i	−0.675539	+	1.17007i
$353$	−19.7246			−1.04984			−0.524918	−	0.851153i	$-0.675904\pi$
$353$	−19.7246			−1.04984			−0.524918	+	0.851153i	$0.675904\pi$
$354$		0			0
$355$	3.79796	−	6.57826i	0.201575	−	0.349138i
$356$	9.04883	+	15.6730i	0.479587	+	0.830669i
$357$		0			0
$358$	0.674235	+	1.16781i	0.0356344	+	0.0617206i
$359$	7.23907	+	12.5384i	0.382063	+	0.661753i	0.991357	−	0.131192i	$-0.0418803\pi$
$359$	7.23907	+	12.5384i	0.382063	+	0.661753i	−0.609294	+	0.792945i	$0.708547\pi$
$360$		0			0
$361$	−17.0000	−	8.48528i	−0.894737	−	0.446594i
$362$	48.7869			2.56418
$363$		0			0
$364$	−5.94949	−	10.3048i	−0.311838	−	0.540119i
$365$	2.62324	−	4.54358i	0.137307	−	0.237822i
$366$		0			0
$367$	10.1742	−	17.6223i	0.531091	−	0.919876i	−0.468251	−	0.883596i	$-0.655116\pi$
$367$	10.1742	−	17.6223i	0.531091	−	0.919876i	0.999342	−	0.0362806i	$-0.0115510\pi$
$368$	3.61953			0.188681
$369$		0			0
$370$	−4.77526	+	8.27098i	−0.248254	+	0.429988i
$371$	1.80977	−	3.13461i	0.0939584	−	0.162741i
$372$		0			0
$373$	−30.6969			−1.58943			−0.794714	−	0.606985i	$-0.792379\pi$
$373$	−30.6969			−1.58943			−0.794714	+	0.606985i	$0.792379\pi$
$374$	14.0065	−	24.2599i	0.724258	−	1.25445i
$375$		0			0
$376$	15.7980	−	27.3629i	0.814718	−	1.41113i
$377$	−3.61953	−	6.26922i	−0.186415	−	0.322881i
$378$		0			0
$379$	−3.44949			−0.177188			−0.0885942	−	0.996068i	$-0.528237\pi$
$379$	−3.44949			−0.177188			−0.0885942	+	0.996068i	$0.528237\pi$
$380$	−3.61953	+	15.3564i	−0.185678	+	0.787766i
$381$		0			0
$382$	6.67423	+	11.5601i	0.341484	+	0.591467i
$383$	4.95765	+	8.58691i	0.253324	+	0.438770i	0.964439	−	0.264306i	$-0.0851427\pi$
$383$	4.95765	+	8.58691i	0.253324	+	0.438770i	−0.711115	+	0.703076i	$0.751809\pi$
$384$		0			0
$385$	−10.3485	−	17.9241i	−0.527407	−	0.913495i
$386$	22.9904	−	39.8206i	1.17018	−	2.02681i
$387$		0			0
$388$	10.6969			0.543055
$389$	−6.24277	+	10.8128i	−0.316521	+	0.548231i	−0.979760	−	0.200177i	$-0.935848\pi$
$389$	−6.24277	+	10.8128i	−0.316521	+	0.548231i	0.663239	+	0.748408i	$0.269181\pi$
$390$		0			0
$391$	7.59592			0.384142
$392$	−16.5767			−0.837251
$393$		0			0
$394$	17.0227	+	29.4842i	0.857591	+	1.48539i
$395$	0.182824	−	0.316660i	0.00919885	−	0.0159329i
$396$		0			0
$397$	3.50000	+	6.06218i	0.175660	+	0.304252i	0.940389	−	0.340099i	$-0.110461\pi$
$397$	3.50000	+	6.06218i	0.175660	+	0.304252i	−0.764730	+	0.644351i	$0.777127\pi$
$398$	−0.813472			−0.0407756
$399$		0			0
$400$	−3.89898			−0.194949
$401$	3.67253	+	6.36101i	0.183398	+	0.317654i	0.943035	−	0.332692i	$-0.107957\pi$
$401$	3.67253	+	6.36101i	0.183398	+	0.317654i	−0.759638	+	0.650346i	$0.774624\pi$
$402$		0			0
$403$	−4.72474	+	8.18350i	−0.235356	+	0.407649i
$404$	3.61953	+	6.26922i	0.180079	+	0.311905i
$405$		0			0
$406$	−58.2929			−2.89303
$407$	22.2948			1.10511
$408$		0			0
$409$	−1.55051	+	2.68556i	−0.0766678	+	0.132793i	−0.901810	−	0.432132i	$-0.857761\pi$
$409$	−1.55051	+	2.68556i	−0.0766678	+	0.132793i	0.825143	+	0.564925i	$0.191095\pi$
$410$	−22.8725			−1.12959
$411$		0			0
$412$	5.94949	−	10.3048i	0.293110	−	0.507682i
$413$	13.4818	+	23.3512i	0.663398	+	1.14904i
$414$		0			0
$415$	−6.00000	−	10.3923i	−0.294528	−	0.510138i
$416$	2.21650	+	3.83909i	0.108673	+	0.188227i
$417$		0			0
$418$	55.7196	−	16.7563i	2.72534	−	0.819576i
$419$	0.577648			0.0282199			0.0141100	−	0.999900i	$-0.495509\pi$
$419$	0.577648			0.0282199			0.0141100	+	0.999900i	$0.495509\pi$
$420$		0			0
$421$	−1.55051	−	2.68556i	−0.0755672	−	0.130886i	0.825766	−	0.564014i	$-0.190743\pi$
$421$	−1.55051	−	2.68556i	−0.0755672	−	0.130886i	−0.901333	+	0.433127i	$0.857410\pi$
$422$	−16.7476	+	29.0078i	−0.815262	+	1.41208i
$423$		0			0
$424$	1.77526	−	3.07483i	0.0862140	−	0.149327i
$425$	−8.18236			−0.396903
$426$		0			0
$427$	8.62372	−	14.9367i	0.417331	−	0.722839i
$428$	−28.5906	+	49.5204i	−1.38198	+	2.39366i
$429$		0			0
$430$	15.5505			0.749912
$431$	−4.66883	+	8.08665i	−0.224890	+	0.389520i	−0.956286	−	0.292432i	$-0.905535\pi$
$431$	−4.66883	+	8.08665i	−0.224890	+	0.389520i	0.731397	+	0.681952i	$0.238869\pi$
$432$		0			0
$433$	10.8485	−	18.7901i	0.521344	−	0.902995i	−0.478348	−	0.878171i	$-0.658764\pi$
$433$	10.8485	−	18.7901i	0.521344	−	0.902995i	0.999692	−	0.0248240i	$-0.00790255\pi$
$434$	38.0462	+	65.8979i	1.82628	+	3.16320i
$435$		0			0
$436$	−30.6969			−1.47012
$437$	11.4892	+	10.8128i	0.549605	+	0.517246i
$438$		0			0
$439$	19.1742	+	33.2107i	0.915136	+	1.58506i	0.806702	+	0.590959i	$0.201250\pi$
$439$	19.1742	+	33.2107i	0.915136	+	1.58506i	0.108435	+	0.994104i	$0.465416\pi$
$440$	−10.1511	−	17.5823i	−0.483936	−	0.838202i
$441$		0			0
$442$	−2.44949	−	4.24264i	−0.116510	−	0.201802i
$443$	17.6260	−	30.5292i	0.837437	−	1.45048i	−0.0545930	−	0.998509i	$-0.517386\pi$
$443$	17.6260	−	30.5292i	0.837437	−	1.45048i	0.892030	−	0.451975i	$-0.149281\pi$
$444$		0			0
$445$	5.50510			0.260967
$446$	0.406736	−	0.704487i	0.0192595	−	0.0333584i
$447$		0			0
$448$	42.5959			2.01247
$449$	21.7172			1.02490			0.512449	−	0.858718i	$-0.328738\pi$
$449$	21.7172			1.02490			0.512449	+	0.858718i	$0.328738\pi$
$450$		0			0
$451$	26.6969	+	46.2405i	1.25711	+	2.17738i
$452$	14.2953	−	24.7602i	0.672395	−	1.16462i
$453$		0			0
$454$	−12.6742	−	21.9524i	−0.594831	−	1.03028i
$455$	−3.61953			−0.169686
$456$		0			0
$457$	−20.1010			−0.940286			−0.470143	−	0.882590i	$-0.655798\pi$
$457$	−20.1010			−0.940286			−0.470143	+	0.882590i	$0.655798\pi$
$458$	−12.6035	−	21.8298i	−0.588921	−	1.02004i
$459$		0			0
$460$	6.55051	−	11.3458i	0.305419	−	0.529001i
$461$	8.81301	+	15.2646i	0.410463	+	0.710942i	0.994940	−	0.100468i	$-0.0320338\pi$
$461$	8.81301	+	15.2646i	0.410463	+	0.710942i	−0.584478	+	0.811410i	$0.698701\pi$
$462$		0			0
$463$	−0.146428			−0.00680510			−0.00340255	−	0.999994i	$-0.501083\pi$
$463$	−0.146428			−0.00680510			−0.00340255	+	0.999994i	$0.501083\pi$
$464$	−7.23907			−0.336065
$465$		0			0
$466$	−24.2474	+	41.9978i	−1.12324	+	1.94551i
$467$	5.14048			0.237873			0.118936	−	0.992902i	$-0.462052\pi$
$467$	5.14048			0.237873			0.118936	+	0.992902i	$0.462052\pi$
$468$		0			0
$469$	0.601021	−	1.04100i	0.0277525	−	0.0480688i
$470$	−11.4362	−	19.8082i	−0.527515	−	0.913682i
$471$		0			0
$472$	13.2247	+	22.9059i	0.608718	+	1.05433i
$473$	−18.1507	−	31.4379i	−0.834569	−	1.44552i
$474$		0			0
$475$	−12.3763	−	11.6476i	−0.567862	−	0.534429i
$476$	−24.9711			−1.14455
$477$		0			0
$478$	−12.6742	−	21.9524i	−0.579706	−	1.00408i
$479$	3.61953	−	6.26922i	0.165381	−	0.286448i	−0.771410	−	0.636339i	$-0.780448\pi$
$479$	3.61953	−	6.26922i	0.165381	−	0.286448i	0.936790	+	0.349891i	$0.113781\pi$
$480$		0			0
$481$	1.94949	−	3.37662i	0.0888891	−	0.153960i
$482$	44.3539			2.02026
$483$		0			0
$484$	−37.4217	+	64.8163i	−1.70099	+	2.94619i
$485$	1.62694	−	2.81795i	0.0738757	−	0.127956i
$486$		0			0
$487$	8.00000			0.362515			0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000			0.362515			0.181257	+	0.983436i	$0.441983\pi$
$488$	8.45927	−	14.6519i	0.382933	−	0.663260i
$489$		0			0
$490$	−6.00000	+	10.3923i	−0.271052	+	0.469476i
$491$	−9.91530	−	17.1738i	−0.447471	−	0.775043i	0.550749	−	0.834671i	$-0.314342\pi$
$491$	−9.91530	−	17.1738i	−0.447471	−	0.775043i	−0.998221	+	0.0596275i	$0.981009\pi$
$492$		0			0
$493$	−15.1918			−0.684206
$494$	2.33441	−	9.90408i	0.105030	−	0.445606i
$495$		0			0
$496$	4.72474	+	8.18350i	0.212147	+	0.367450i
$497$	12.4855	+	21.6256i	0.560053	+	0.970040i
$498$		0			0
$499$	−13.8258	−	23.9469i	−0.618926	−	1.07201i	−0.989682	−	0.143281i	$-0.954235\pi$
$499$	−13.8258	−	23.9469i	−0.618926	−	1.07201i	0.370756	−	0.928730i	$-0.379099\pi$
$500$	−16.1051	+	27.8948i	−0.720241	+	1.24749i
$501$		0			0
$502$	−33.7980			−1.50848
$503$	−10.3870	+	17.9907i	−0.463131	+	0.802167i	−0.999115	−	0.0420614i	$-0.986608\pi$
$503$	−10.3870	+	17.9907i	−0.463131	+	0.802167i	0.535984	+	0.844228i	$0.319941\pi$
$504$		0			0
$505$	2.20204			0.0979895
$506$	−48.3152			−2.14787
$507$		0			0
$508$	−10.0000	−	17.3205i	−0.443678	−	0.768473i
$509$	15.6334	−	27.0779i	0.692940	−	1.20021i	−0.277931	−	0.960601i	$-0.589649\pi$
$509$	15.6334	−	27.0779i	0.692940	−	1.20021i	0.970870	−	0.239605i	$-0.0770181\pi$
$510$		0			0
$511$	8.62372	+	14.9367i	0.381491	+	0.660762i
$512$	11.2004			0.494993
$513$		0			0
$514$	−29.1464			−1.28559
$515$	−1.80977	−	3.13461i	−0.0797478	−	0.138127i
$516$		0			0
$517$	−26.6969	+	46.2405i	−1.17413	+	2.03365i
$518$	−15.6983	−	27.1903i	−0.689745	−	1.19467i
$519$		0			0
$520$	−3.55051			−0.155700
$521$	−3.14789			−0.137911			−0.0689557	−	0.997620i	$-0.521967\pi$
$521$	−3.14789			−0.137911			−0.0689557	+	0.997620i	$0.521967\pi$
$522$		0			0
$523$	10.1742	−	17.6223i	0.444888	−	0.770569i	−0.553156	−	0.833078i	$-0.686577\pi$
$523$	10.1742	−	17.6223i	0.444888	−	0.770569i	0.998044	+	0.0625086i	$0.0199101\pi$
$524$	−53.9274			−2.35583
$525$		0			0
$526$	−30.2474	+	52.3901i	−1.31885	+	2.28432i
$527$	9.91530	+	17.1738i	0.431917	+	0.748103i
$528$		0			0
$529$	4.94949	+	8.57277i	0.215195	+	0.372729i
$530$	−1.28512	−	2.22589i	−0.0558220	−	0.0966865i
$531$		0			0
$532$	−37.7702	−	35.5464i	−1.63754	−	1.54113i
$533$	9.33766			0.404459
$534$		0			0
$535$	8.69694	+	15.0635i	0.376001	+	0.651254i
$536$	0.589559	−	1.02115i	0.0254651	−	0.0441068i
$537$		0			0
$538$	18.1237	−	31.3912i	0.781369	−	1.35337i
$539$	28.0130			1.20660
$540$		0			0
$541$	4.84847	−	8.39780i	0.208452	−	0.361049i	−0.742775	−	0.669541i	$-0.766491\pi$
$541$	4.84847	−	8.39780i	0.208452	−	0.361049i	0.951227	+	0.308492i	$0.0998242\pi$
$542$	20.7739	−	35.9815i	0.892316	−	1.54554i
$543$		0			0
$544$	9.30306			0.398865
$545$	−4.66883	+	8.08665i	−0.199991	+	0.346394i
$546$		0			0
$547$	8.82577	−	15.2867i	0.377362	−	0.653611i	−0.613315	−	0.789838i	$-0.710164\pi$
$547$	8.82577	−	15.2867i	0.377362	−	0.653611i	0.990678	+	0.136227i	$0.0434978\pi$
$548$	−26.9637	−	46.7025i	−1.15183	−	1.99503i
$549$		0			0
$550$	52.0454			2.21922
$551$	−22.9785	−	21.6256i	−0.978917	−	0.921281i
$552$		0			0
$553$	0.601021	+	1.04100i	0.0255580	+	0.0442677i
$554$	−21.8232	−	37.7989i	−0.927179	−	1.60592i
$555$		0			0
$556$	4.05051	+	7.01569i	0.171780	+	0.297532i
$557$	−1.52094	+	2.63435i	−0.0644444	+	0.111621i	−0.896447	−	0.443150i	$-0.853861\pi$
$557$	−1.52094	+	2.63435i	−0.0644444	+	0.111621i	0.832003	+	0.554771i	$0.187194\pi$
$558$		0			0
$559$	−6.34847			−0.268512
$560$	−1.80977	+	3.13461i	−0.0764766	+	0.132461i
$561$		0			0
$562$	−29.1464			−1.22947
$563$	−6.29577			−0.265335			−0.132668	−	0.991161i	$-0.542354\pi$
$563$	−6.29577			−0.265335			−0.132668	+	0.991161i	$0.542354\pi$
$564$		0			0
$565$	−4.34847	−	7.53177i	−0.182941	−	0.316864i
$566$	−7.23907	+	12.5384i	−0.304281	+	0.527030i
$567$		0			0
$568$	12.2474	+	21.2132i	0.513892	+	0.890086i
$569$	28.0130			1.17436			0.587182	−	0.809455i	$-0.300237\pi$
$569$	28.0130			1.17436			0.587182	+	0.809455i	$0.300237\pi$
$570$		0			0
$571$	11.2474			0.470691			0.235346	−	0.971912i	$-0.424378\pi$
$571$	11.2474			0.470691			0.235346	+	0.971912i	$0.424378\pi$
$572$	9.86230	+	17.0820i	0.412364	+	0.714235i
$573$		0			0
$574$	37.5959	−	65.1180i	1.56922	−	2.71797i
$575$	7.05624	+	12.2218i	0.294266	+	0.509683i
$576$		0			0
$577$	22.6969			0.944886			0.472443	−	0.881361i	$-0.343372\pi$
$577$	22.6969			0.944886			0.472443	+	0.881361i	$0.343372\pi$
$578$	29.4041			1.22305
$579$		0			0
$580$	−13.1010	+	22.6916i	−0.543990	+	0.942219i
$581$	39.4492			1.63663
$582$		0			0
$583$	−3.00000	+	5.19615i	−0.124247	+	0.215203i
$584$	8.45927	+	14.6519i	0.350047	+	0.606300i
$585$		0			0
$586$	12.0000	+	20.7846i	0.495715	+	0.858604i
$587$	−0.760471	−	1.31718i	−0.0313880	−	0.0543656i	0.849905	−	0.526936i	$-0.176659\pi$
$587$	−0.760471	−	1.31718i	−0.0313880	−	0.0543656i	−0.881293	+	0.472571i	$0.843326\pi$
$588$		0			0
$589$	−9.44949	+	40.0908i	−0.389359	+	1.65191i
$590$	19.1470			0.788268
$591$		0			0
$592$	−1.94949	−	3.37662i	−0.0801235	−	0.138778i
$593$	−11.9609	+	20.7169i	−0.491175	+	0.850740i	−0.999948	−	0.0101603i	$-0.996766\pi$
$593$	−11.9609	+	20.7169i	−0.491175	+	0.850740i	0.508773	+	0.860901i	$0.330099\pi$
$594$		0			0
$595$	−3.79796	+	6.57826i	−0.155701	+	0.269682i
$596$	−43.0688			−1.76416
$597$		0			0
$598$	−4.22474	+	7.31747i	−0.172763	+	0.299234i
$599$	12.7744	−	22.1259i	0.521946	−	0.904038i	−0.477728	−	0.878508i	$-0.658539\pi$
$599$	12.7744	−	22.1259i	0.521946	−	0.904038i	0.999674	−	0.0255298i	$-0.00812726\pi$
$600$		0			0
$601$	21.8990			0.893278			0.446639	−	0.894714i	$-0.352621\pi$
$601$	21.8990			0.893278			0.446639	+	0.894714i	$0.352621\pi$
$602$	−25.5606	+	44.2723i	−1.04177	+	1.80441i
$603$		0			0
$604$	6.89898	−	11.9494i	0.280715	−	0.486213i
$605$	11.3832	+	19.7164i	0.462795	+	0.801584i
$606$		0			0
$607$	7.94439			0.322453			0.161226	−	0.986917i	$-0.448455\pi$
$607$	7.94439			0.322453			0.161226	+	0.986917i	$0.448455\pi$
$608$	14.0714	+	13.2429i	0.570671	+	0.537071i
$609$		0			0
$610$	−6.12372	−	10.6066i	−0.247942	−	0.429449i
$611$	4.66883	+	8.08665i	0.188881	+	0.327151i
$612$		0			0
$613$	−1.55051	−	2.68556i	−0.0626245	−	0.108469i	0.833013	−	0.553253i	$-0.186614\pi$
$613$	−1.55051	−	2.68556i	−0.0626245	−	0.108469i	−0.895638	+	0.444784i	$0.853280\pi$
$614$	−34.5446	+	59.8329i	−1.39411	+	2.41466i
$615$		0			0
$616$	66.7423			2.68913
$617$	−3.67253	+	6.36101i	−0.147851	+	0.256085i	−0.930433	−	0.366462i	$-0.880569\pi$
$617$	−3.67253	+	6.36101i	−0.147851	+	0.256085i	0.782582	+	0.622547i	$0.213902\pi$
$618$		0			0
$619$	−0.752551			−0.0302476			−0.0151238	−	0.999886i	$-0.504814\pi$
$619$	−0.752551			−0.0302476			−0.0151238	+	0.999886i	$0.504814\pi$
$620$	34.2027			1.37362
$621$		0			0
$622$	−27.3712	−	47.4083i	−1.09748	−	1.90090i
$623$	−9.04883	+	15.6730i	−0.362534	+	0.627927i
$624$		0			0
$625$	−4.84847	−	8.39780i	−0.193939	−	0.335912i
$626$	−7.23907			−0.289331
$627$		0			0
$628$	−54.1464			−2.16068
$629$	−4.09118	−	7.08613i	−0.163126	−	0.282543i
$630$		0			0
$631$	−3.17423	+	5.49794i	−0.126364	+	0.218869i	−0.922265	−	0.386557i	$-0.873664\pi$
$631$	−3.17423	+	5.49794i	−0.126364	+	0.218869i	0.795901	+	0.605427i	$0.206997\pi$
$632$	0.589559	+	1.02115i	0.0234514	+	0.0406190i
$633$		0			0
$634$	24.2474			0.962989
$635$	−6.08377			−0.241427
$636$		0			0
$637$	2.44949	−	4.24264i	0.0970523	−	0.168100i
$638$	96.6305			3.82564
$639$		0			0
$640$	10.4722	−	18.1384i	0.413950	−	0.716982i
$641$	−23.3441	−	40.4332i	−0.922038	−	1.59702i	−0.796257	−	0.604959i	$-0.793189\pi$
$641$	−23.3441	−	40.4332i	−0.922038	−	1.59702i	−0.125782	−	0.992058i	$-0.540144\pi$
$642$		0			0
$643$	−1.82577	−	3.16232i	−0.0720012	−	0.124710i	0.827777	−	0.561057i	$-0.189605\pi$
$643$	−1.82577	−	3.16232i	−0.0720012	−	0.124710i	−0.899778	+	0.436347i	$0.856272\pi$
$644$	21.5344	+	37.2986i	0.848573	+	1.46977i
$645$		0			0
$646$	−15.5505	−	14.6349i	−0.611827	−	0.575804i
$647$	−16.5767			−0.651698			−0.325849	−	0.945422i	$-0.605650\pi$
$647$	−16.5767			−0.651698			−0.325849	+	0.945422i	$0.605650\pi$
$648$		0			0
$649$	−22.3485	−	38.7087i	−0.877254	−	1.51945i
$650$	4.55092	−	7.88242i	0.178502	−	0.309174i
$651$		0			0
$652$	16.2980	−	28.2289i	0.638277	−	1.10553i
$653$	−39.4492			−1.54377			−0.771884	−	0.635764i	$-0.780685\pi$
$653$	−39.4492			−1.54377			−0.771884	+	0.635764i	$0.780685\pi$
$654$		0			0
$655$	−8.20204	+	14.2064i	−0.320480	+	0.555088i
$656$	4.66883	−	8.08665i	0.182287	−	0.315731i
$657$		0			0
$658$	75.1918			2.93128
$659$	5.90095	−	10.2207i	0.229868	−	0.398144i	−0.727901	−	0.685683i	$-0.759504\pi$
$659$	5.90095	−	10.2207i	0.229868	−	0.398144i	0.957769	+	0.287539i	$0.0928371\pi$
$660$		0			0
$661$	−20.5959	+	35.6732i	−0.801088	+	1.38753i	0.117812	+	0.993036i	$0.462412\pi$
$661$	−20.5959	+	35.6732i	−0.801088	+	1.38753i	−0.918900	+	0.394490i	$0.870921\pi$
$662$	−4.02627	−	6.97370i	−0.156485	−	0.271041i
$663$		0			0
$664$	38.6969			1.50173
$665$	−15.1088	+	4.54358i	−0.585893	+	0.176193i
$666$		0			0
$667$	13.1010	+	22.6916i	0.507274	+	0.878624i
$668$	33.2064	+	57.5153i	1.28480	+	2.22533i
$669$		0			0
$670$	−0.426786	−	0.739215i	−0.0164882	−	0.0285584i
$671$	−14.2953	+	24.7602i	−0.551864	+	0.955857i
$672$		0			0
$673$	21.8990			0.844144			0.422072	−	0.906562i	$-0.361303\pi$
$673$	21.8990			0.844144			0.422072	+	0.906562i	$0.361303\pi$
$674$	29.9936	−	51.9505i	1.15531	−	2.00106i
$675$		0			0
$676$	−41.3939			−1.59207
$677$	−22.8725			−0.879061			−0.439531	−	0.898228i	$-0.644855\pi$
$677$	−22.8725			−0.879061			−0.439531	+	0.898228i	$0.644855\pi$
$678$		0			0
$679$	5.34847	+	9.26382i	0.205255	+	0.355513i
$680$	−3.72553	+	6.45281i	−0.142868	+	0.247454i
$681$		0			0
$682$	−63.0681	−	109.237i	−2.41500	−	4.18291i
$683$	0.577648			0.0221031			0.0110515	−	0.999939i	$-0.496482\pi$
$683$	0.577648			0.0221031			0.0110515	+	0.999939i	$0.496482\pi$
$684$		0			0
$685$	−16.4041			−0.626768
$686$	8.45927	+	14.6519i	0.322977	+	0.559412i
$687$		0			0
$688$	−3.17423	+	5.49794i	−0.121017	+	0.209607i
$689$	0.524648	+	0.908716i	0.0199875	+	0.0346193i
$690$		0			0
$691$	25.3939			0.966029			0.483014	−	0.875612i	$-0.339542\pi$
$691$	25.3939			0.966029			0.483014	+	0.875612i	$0.339542\pi$
$692$	24.9711			0.949258
$693$		0			0
$694$	−10.2247	+	17.7098i	−0.388126	+	0.672254i
$695$	2.46424			0.0934739
$696$		0			0
$697$	9.79796	−	16.9706i	0.371124	−	0.642806i
$698$	−19.6067	−	33.9598i	−0.742124	−	1.28540i
$699$		0			0
$700$	−23.1969	−	40.1783i	−0.876762	−	1.51860i
$701$	15.1088	+	26.1692i	0.570651	+	0.988396i	0.996499	+	0.0836016i	$0.0266423\pi$
$701$	15.1088	+	26.1692i	0.570651	+	0.988396i	−0.425848	+	0.904794i	$0.640024\pi$
$702$		0			0
$703$	3.89898	−	16.5420i	0.147053	−	0.623892i
$704$	−70.6101			−2.66122
$705$		0			0
$706$	−23.0227	−	39.8765i	−0.866471	−	1.50077i
$707$	−3.61953	+	6.26922i	−0.136127	+	0.235778i
$708$		0			0
$709$	−14.1969	+	24.5898i	−0.533177	+	0.923490i	0.466072	+	0.884747i	$0.345669\pi$
$709$	−14.1969	+	24.5898i	−0.533177	+	0.923490i	−0.999249	+	0.0387432i	$0.987665\pi$
$710$	17.7320			0.665471
$711$		0			0
$712$	−8.87628	+	15.3742i	−0.332652	+	0.576171i
$713$	17.1014	−	29.6204i	0.640451	−	1.10929i
$714$		0			0
$715$	6.00000			0.224387
$716$	−0.996295	+	1.72563i	−0.0372333	+	0.0644900i
$717$		0			0
$718$	−16.8990	+	29.2699i	−0.630664	+	1.09234i
$719$	−23.6330	−	40.9335i	−0.881361	−	1.52656i	−0.849829	−	0.527059i	$-0.823295\pi$
$719$	−23.6330	−	40.9335i	−0.881361	−	1.52656i	−0.0315323	−	0.999503i	$-0.510039\pi$
$720$		0			0
$721$	11.8990			0.443141
$722$	−2.68815	−	44.2723i	−0.100043	−	1.64765i
$723$		0			0
$724$	36.0454	+	62.4325i	1.33962	+	2.32028i
$725$	−14.1125	−	24.4435i	−0.524125	−	0.907810i
$726$		0			0
$727$	20.5227	+	35.5464i	0.761145	+	1.31834i	0.942261	+	0.334880i	$0.108696\pi$
$727$	20.5227	+	35.5464i	0.761145	+	1.31834i	−0.181116	+	0.983462i	$0.557971\pi$
$728$	5.83604	−	10.1083i	0.216298	−	0.374639i
$729$		0			0
$730$	12.2474			0.453298
$731$	−6.66142	+	11.5379i	−0.246381	+	0.426745i
$732$		0			0
$733$	5.30306			0.195873			0.0979365	−	0.995193i	$-0.468776\pi$
$733$	5.30306			0.195873			0.0979365	+	0.995193i	$0.468776\pi$
$734$	47.5018			1.75332
$735$		0			0
$736$	−8.02270	−	13.8957i	−0.295721	−	0.512203i
$737$	−0.996295	+	1.72563i	−0.0366990	+	0.0635645i
$738$		0			0
$739$	14.5227	+	25.1541i	0.534226	+	0.925307i	0.999200	+	0.0399828i	$0.0127303\pi$
$739$	14.5227	+	25.1541i	0.534226	+	0.925307i	−0.464974	+	0.885324i	$0.653936\pi$
$740$	−14.1125			−0.518785
$741$		0			0
$742$	8.44949			0.310191
$743$	15.8163	+	27.3946i	0.580242	+	1.00501i	0.995450	+	0.0952823i	$0.0303754\pi$
$743$	15.8163	+	27.3946i	0.580242	+	1.00501i	−0.415208	+	0.909726i	$0.636291\pi$
$744$		0			0
$745$	−6.55051	+	11.3458i	−0.239992	+	0.415679i
$746$	−35.8297	−	62.0588i	−1.31182	−	2.27214i
$747$		0			0
$748$	41.3939			1.51351
$749$	−57.1812			−2.08936
$750$		0			0
$751$	17.5227	−	30.3502i	0.639413	−	1.10750i	−0.346149	−	0.938179i	$-0.612511\pi$
$751$	17.5227	−	30.3502i	0.639413	−	1.10750i	0.985562	−	0.169316i	$-0.0541557\pi$
$752$	9.33766			0.340509
$753$		0			0
$754$	8.44949	−	14.6349i	0.307712	−	0.532973i
$755$	−2.09859	−	3.63487i	−0.0763755	−	0.132286i
$756$		0			0
$757$	−8.19694	−	14.1975i	−0.297923	−	0.516017i	0.677738	−	0.735304i	$-0.262960\pi$
$757$	−8.19694	−	14.1975i	−0.297923	−	0.516017i	−0.975661	+	0.219286i	$0.929627\pi$
$758$	−4.02627	−	6.97370i	−0.146241	−	0.253296i
$759$		0			0
$760$	−14.8207	+	4.45694i	−0.537602	+	0.161670i
$761$	−1.99259			−0.0722313			−0.0361157	−	0.999348i	$-0.511498\pi$
$761$	−1.99259			−0.0722313			−0.0361157	+	0.999348i	$0.511498\pi$
$762$		0			0
$763$	−15.3485	−	26.5843i	−0.555652	−	0.962417i
$764$	−9.86230	+	17.0820i	−0.356806	+	0.618006i
$765$		0			0
$766$	−11.5732	+	20.0454i	−0.418157	+	0.724270i
$767$	−7.81671			−0.282245
$768$		0			0
$769$	−14.1969	+	24.5898i	−0.511955	+	0.886732i	0.487949	+	0.872872i	$0.337745\pi$
$769$	−14.1969	+	24.5898i	−0.511955	+	0.886732i	−0.999904	+	0.0138595i	$0.995588\pi$
$770$	24.1576	−	41.8422i	0.870580	−	1.50789i
$771$		0			0
$772$	67.9444			2.44537
$773$	−4.66883	+	8.08665i	−0.167926	+	0.290857i	−0.937691	−	0.347472i	$-0.887040\pi$
$773$	−4.66883	+	8.08665i	−0.167926	+	0.290857i	0.769764	+	0.638328i	$0.220374\pi$
$774$		0			0
$775$	−18.4217	+	31.9073i	−0.661726	+	1.14614i
$776$	5.24648	+	9.08716i	0.188338	+	0.326210i
$777$		0			0
$778$	−29.1464			−1.04495
$779$	38.9776	−	11.7215i	1.39652	−	0.419967i
$780$		0			0
$781$	−20.6969	−	35.8481i	−0.740595	−	1.28275i
$782$	8.86601	+	15.3564i	0.317048	+	0.549143i
$783$		0			0
$784$	−2.44949	−	4.24264i	−0.0874818	−	0.151523i
$785$	−8.23536

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:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	171.f (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.16721 + 2.02166i) q^{2} +(-1.72474 + 2.98735i) q^{4} +(0.524648 + 0.908716i) q^{5} -3.44949 q^{7} -3.38371 q^{8} +O(q^{10})\)	\(q+(1.16721 + 2.02166i) q^{2} +(-1.72474 + 2.98735i) q^{4} +(0.524648 + 0.908716i) q^{5} -3.44949 q^{7} -3.38371 q^{8} +(-1.22474 + 2.12132i) q^{10} +5.71812 q^{11} +(0.500000 - 0.866025i) q^{13} +(-4.02627 - 6.97370i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-1.04930 - 1.81743i) q^{17} +(1.00000 - 4.24264i) q^{19} -3.61953 q^{20} +(6.67423 + 11.5601i) q^{22} +(-1.80977 + 3.13461i) q^{23} +(1.94949 - 3.37662i) q^{25} +2.33441 q^{26} +(5.94949 - 10.3048i) q^{28} +(3.61953 - 6.26922i) q^{29} -9.44949 q^{31} +(-2.21650 + 3.83909i) q^{32} +(2.44949 - 4.24264i) q^{34} +(-1.80977 - 3.13461i) q^{35} +3.89898 q^{37} +(9.74439 - 2.93038i) q^{38} +(-1.77526 - 3.07483i) q^{40} +(4.66883 + 8.08665i) q^{41} +(-3.17423 - 5.49794i) q^{43} +(-9.86230 + 17.0820i) q^{44} -8.44949 q^{46} +(-4.66883 + 8.08665i) q^{47} +4.89898 q^{49} +9.10183 q^{50} +(1.72474 + 2.98735i) q^{52} +(-0.524648 + 0.908716i) q^{53} +(3.00000 + 5.19615i) q^{55} +11.6721 q^{56} +16.8990 q^{58} +(-3.90836 - 6.76947i) q^{59} +(-2.50000 + 4.33013i) q^{61} +(-11.0295 - 19.1037i) q^{62} -12.3485 q^{64} +1.04930 q^{65} +(-0.174235 + 0.301783i) q^{67} +7.23907 q^{68} +(4.22474 - 7.31747i) q^{70} +(-3.61953 - 6.26922i) q^{71} +(-2.50000 - 4.33013i) q^{73} +(4.55092 + 7.88242i) q^{74} +(10.9495 + 10.3048i) q^{76} -19.7246 q^{77} +(-0.174235 - 0.301783i) q^{79} +(0.524648 - 0.908716i) q^{80} +(-10.8990 + 18.8776i) q^{82} -11.4362 q^{83} +(1.10102 - 1.90702i) q^{85} +(7.40998 - 12.8345i) q^{86} -19.3485 q^{88} +(2.62324 - 4.54358i) q^{89} +(-1.72474 + 2.98735i) q^{91} +(-6.24277 - 10.8128i) q^{92} -21.7980 q^{94} +(4.38000 - 1.31718i) q^{95} +(-1.55051 - 2.68556i) q^{97} +(5.71812 + 9.90408i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{4} - 8 q^{7}+O(q^{10}) \) 8 * q - 4 * q^4 - 8 * q^7	\( 8 q - 4 q^{4} - 8 q^{7} + 4 q^{13} - 4 q^{16} + 8 q^{19} + 24 q^{22} - 4 q^{25} + 28 q^{28} - 56 q^{31} - 8 q^{37} - 24 q^{40} + 4 q^{43} - 48 q^{46} + 4 q^{52} + 24 q^{55} + 96 q^{58} - 20 q^{61} - 40 q^{64} + 28 q^{67} + 24 q^{70} - 20 q^{73} + 68 q^{76} + 28 q^{79} - 48 q^{82} + 48 q^{85} - 96 q^{88} - 4 q^{91} - 96 q^{94} - 32 q^{97}+O(q^{100}) \) 8 * q - 4 * q^4 - 8 * q^7 + 4 * q^13 - 4 * q^16 + 8 * q^19 + 24 * q^22 - 4 * q^25 + 28 * q^28 - 56 * q^31 - 8 * q^37 - 24 * q^40 + 4 * q^43 - 48 * q^46 + 4 * q^52 + 24 * q^55 + 96 * q^58 - 20 * q^61 - 40 * q^64 + 28 * q^67 + 24 * q^70 - 20 * q^73 + 68 * q^76 + 28 * q^79 - 48 * q^82 + 48 * q^85 - 96 * q^88 - 4 * q^91 - 96 * q^94 - 32 * q^97

Introduction

L-functions

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