LMFDB - Embedded newform 171.2.f.c.64.3

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			64.3
Root			$-1.27970 + 1.16721i$ of defining polynomial
Character	$\chi$	$=$	171.64
Dual form			171.2.f.c.163.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.370982 + 0.642559i) q^{2} +(0.724745 - 1.25529i) q^{4} +(-1.65068 - 2.85906i) q^{5} +1.44949 q^{7} +2.55940 q^{8} +O(q^{10})$	\(q+(0.370982 + 0.642559i) q^{2} +(0.724745 - 1.25529i) q^{4} +(-1.65068 - 2.85906i) q^{5} +1.44949 q^{7} +2.55940 q^{8} +(1.22474 - 2.12132i) q^{10} -1.81743 q^{11} +(0.500000 - 0.866025i) q^{13} +(0.537734 + 0.931383i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(3.30136 + 5.71812i) q^{17} +(1.00000 + 4.24264i) q^{19} -4.78529 q^{20} +(-0.674235 - 1.16781i) q^{22} +(-2.39264 + 4.14418i) q^{23} +(-2.94949 + 5.10867i) q^{25} +0.741964 q^{26} +(1.05051 - 1.81954i) q^{28} +(4.78529 - 8.28836i) q^{29} -4.55051 q^{31} +(2.93038 - 5.07556i) q^{32} +(-2.44949 + 4.24264i) q^{34} +(-2.39264 - 4.14418i) q^{35} -5.89898 q^{37} +(-2.35517 + 2.21650i) q^{38} +(-4.22474 - 7.31747i) q^{40} +(1.48393 + 2.57024i) q^{41} +(4.17423 + 7.22999i) q^{43} +(-1.31718 + 2.28141i) q^{44} -3.55051 q^{46} +(-1.48393 + 2.57024i) q^{47} -4.89898 q^{49} -4.37683 q^{50} +(-0.724745 - 1.25529i) q^{52} +(1.65068 - 2.85906i) q^{53} +(3.00000 + 5.19615i) q^{55} +3.70982 q^{56} +7.10102 q^{58} +(4.21008 + 7.29207i) q^{59} +(-2.50000 + 4.33013i) q^{61} +(-1.68816 - 2.92397i) q^{62} +2.34847 q^{64} -3.30136 q^{65} +(7.17423 - 12.4261i) q^{67} +9.57058 q^{68} +(1.77526 - 3.07483i) q^{70} +(-4.78529 - 8.28836i) q^{71} +(-2.50000 - 4.33013i) q^{73} +(-2.18841 - 3.79045i) q^{74} +(6.05051 + 1.81954i) q^{76} -2.63435 q^{77} +(7.17423 + 12.4261i) q^{79} +(-1.65068 + 2.85906i) q^{80} +(-1.10102 + 1.90702i) q^{82} +3.63487 q^{83} +(10.8990 - 18.8776i) q^{85} +(-3.09713 + 5.36439i) q^{86} -4.65153 q^{88} +(-8.25340 + 14.2953i) q^{89} +(0.724745 - 1.25529i) q^{91} +(3.46811 + 6.00695i) q^{92} -2.20204 q^{94} +(10.4793 - 9.86230i) q^{95} +(-6.44949 - 11.1708i) q^{97} +(-1.81743 - 3.14789i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.370982	+	0.642559i	0.262324	+	0.454358i	0.966859	−	0.255311i	$-0.0821778\pi$
$2$	0.370982	+	0.642559i	0.262324	+	0.454358i	−0.704535	+	0.709669i	$0.748844\pi$
$3$		0			0
$3$		0			0
$4$	0.724745	−	1.25529i	0.362372	−	0.627647i
$5$	−1.65068	−	2.85906i	−0.738207	−	1.27861i	−0.953302	−	0.302018i	$-0.902340\pi$
$5$	−1.65068	−	2.85906i	−0.738207	−	1.27861i	0.215096	−	0.976593i	$-0.430994\pi$
$6$		0			0
$7$	1.44949			0.547856			0.273928	−	0.961750i	$-0.411677\pi$
$7$	1.44949			0.547856			0.273928	+	0.961750i	$0.411677\pi$
$8$	2.55940			0.904883
$9$		0			0
$10$	1.22474	−	2.12132i	0.387298	−	0.670820i
$11$	−1.81743			−0.547977			−0.273988	−	0.961733i	$-0.588343\pi$
$11$	−1.81743			−0.547977			−0.273988	+	0.961733i	$0.588343\pi$
$12$		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	$0.122382\pi$
$14$	0.537734	+	0.931383i	0.143716	+	0.248923i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	3.30136	+	5.71812i	0.800697	+	1.38685i	0.919158	+	0.393890i	$0.128871\pi$
$17$	3.30136	+	5.71812i	0.800697	+	1.38685i	−0.118460	+	0.992959i	$0.537796\pi$
$18$		0			0
$19$	1.00000	+	4.24264i	0.229416	+	0.973329i
$19$	1.00000	+	4.24264i	0.229416	+	0.973329i
$20$	−4.78529			−1.07002
$21$		0			0
$22$	−0.674235	−	1.16781i	−0.143747	−	0.248978i
$23$	−2.39264	+	4.14418i	−0.498901	+	0.864121i	−0.999999	−	0.00126885i	$-0.999596\pi$
$23$	−2.39264	+	4.14418i	−0.498901	+	0.864121i	0.501098	+	0.865390i	$0.332929\pi$
$24$		0			0
$25$	−2.94949	+	5.10867i	−0.589898	+	1.02173i
$26$	0.741964			0.145511
$27$		0			0
$28$	1.05051	−	1.81954i	0.198528	−	0.343860i
$29$	4.78529	−	8.28836i	0.888606	−	1.53911i	0.0470812	−	0.998891i	$-0.485008\pi$
$29$	4.78529	−	8.28836i	0.888606	−	1.53911i	0.841524	−	0.540219i	$-0.181659\pi$
$30$		0			0
$31$	−4.55051			−0.817296			−0.408648	−	0.912692i	$-0.634000\pi$
$31$	−4.55051			−0.817296			−0.408648	+	0.912692i	$0.634000\pi$
$32$	2.93038	−	5.07556i	0.518023	−	0.897241i
$33$		0			0
$34$	−2.44949	+	4.24264i	−0.420084	+	0.727607i
$35$	−2.39264	−	4.14418i	−0.404431	−	0.700494i
$36$		0			0
$37$	−5.89898			−0.969786			−0.484893	−	0.874573i	$-0.661142\pi$
$37$	−5.89898			−0.969786			−0.484893	+	0.874573i	$0.661142\pi$
$38$	−2.35517	+	2.21650i	−0.382059	+	0.359564i
$39$		0			0
$40$	−4.22474	−	7.31747i	−0.667991	−	1.15699i
$41$	1.48393	+	2.57024i	0.231751	+	0.401404i	0.958323	−	0.285686i	$-0.0922214\pi$
$41$	1.48393	+	2.57024i	0.231751	+	0.401404i	−0.726573	+	0.687089i	$0.758888\pi$
$42$		0			0
$43$	4.17423	+	7.22999i	0.636565	+	1.10256i	0.986181	+	0.165670i	$0.0529786\pi$
$43$	4.17423	+	7.22999i	0.636565	+	1.10256i	−0.349616	+	0.936893i	$0.613688\pi$
$44$	−1.31718	+	2.28141i	−0.198572	+	0.343936i
$45$		0			0
$46$	−3.55051			−0.523494
$47$	−1.48393	+	2.57024i	−0.216453	+	0.374908i	−0.953721	−	0.300693i	$-0.902782\pi$
$47$	−1.48393	+	2.57024i	−0.216453	+	0.374908i	0.737268	+	0.675600i	$0.236116\pi$
$48$		0			0
$49$	−4.89898			−0.699854
$50$	−4.37683			−0.618977
$51$		0			0
$52$	−0.724745	−	1.25529i	−0.100504	−	0.174078i
$53$	1.65068	−	2.85906i	0.226738	−	0.392722i	−0.730101	−	0.683339i	$-0.760527\pi$
$53$	1.65068	−	2.85906i	0.226738	−	0.392722i	0.956840	+	0.290617i	$0.0938604\pi$
$54$		0			0
$55$	3.00000	+	5.19615i	0.404520	+	0.700649i
$56$	3.70982			0.495745
$57$		0			0
$58$	7.10102			0.932410
$59$	4.21008	+	7.29207i	0.548105	+	0.949346i	0.998404	+	0.0564684i	$0.0179840\pi$
$59$	4.21008	+	7.29207i	0.548105	+	0.949346i	−0.450299	+	0.892878i	$0.648683\pi$
$60$		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	$0.270381\pi$
$62$	−1.68816	−	2.92397i	−0.214396	−	0.371345i
$63$		0			0
$64$	2.34847			0.293559
$65$	−3.30136			−0.409483
$66$		0			0
$67$	7.17423	−	12.4261i	0.876472	−	1.51809i	0.0212861	−	0.999773i	$-0.493224\pi$
$67$	7.17423	−	12.4261i	0.876472	−	1.51809i	0.855186	−	0.518321i	$-0.173443\pi$
$68$	9.57058			1.16060
$69$		0			0
$70$	1.77526	−	3.07483i	0.212184	−	0.367513i
$71$	−4.78529	−	8.28836i	−0.567909	−	0.983648i	−0.996772	−	0.0802782i	$-0.974419\pi$
$71$	−4.78529	−	8.28836i	−0.567909	−	0.983648i	0.428863	−	0.903369i	$-0.358914\pi$
$72$		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	$0.927855\pi$
$74$	−2.18841	−	3.79045i	−0.254398	−	0.440630i
$75$		0			0
$76$	6.05051	+	1.81954i	0.694041	+	0.208715i
$77$	−2.63435			−0.300212
$78$		0			0
$79$	7.17423	+	12.4261i	0.807164	+	1.39805i	0.914820	+	0.403862i	$0.132332\pi$
$79$	7.17423	+	12.4261i	0.807164	+	1.39805i	−0.107656	+	0.994188i	$0.534334\pi$
$80$	−1.65068	+	2.85906i	−0.184552	+	0.319653i
$81$		0			0
$82$	−1.10102	+	1.90702i	−0.121587	+	0.210596i
$83$	3.63487			0.398978			0.199489	−	0.979900i	$-0.436072\pi$
$83$	3.63487			0.398978			0.199489	+	0.979900i	$0.436072\pi$
$84$		0			0
$85$	10.8990	−	18.8776i	1.18216	−	2.04756i
$86$	−3.09713	+	5.36439i	−0.333972	+	0.578457i
$87$		0			0
$88$	−4.65153			−0.495855
$89$	−8.25340	+	14.2953i	−0.874859	+	1.51530i	−0.0179455	+	0.999839i	$0.505713\pi$
$89$	−8.25340	+	14.2953i	−0.874859	+	1.51530i	−0.856913	+	0.515461i	$0.827621\pi$
$90$		0			0
$91$	0.724745	−	1.25529i	0.0759739	−	0.131591i
$92$	3.46811	+	6.00695i	0.361576	+	0.626268i
$93$		0			0
$94$	−2.20204			−0.227123
$95$	10.4793	−	9.86230i	1.07515	−	1.01185i
$96$		0			0
$97$	−6.44949	−	11.1708i	−0.654846	−	1.13423i	−0.981932	−	0.189233i	$-0.939400\pi$
$97$	−6.44949	−	11.1708i	−0.654846	−	1.13423i	0.327086	−	0.944995i	$-0.393933\pi$
$98$	−1.81743	−	3.14789i	−0.183588	−	0.317984i
$99$		0			0
$100$	4.27526	+	7.40496i	0.427526	+	0.740496i
$101$	−3.30136	+	5.71812i	−0.328498	+	0.568975i	−0.982214	−	0.187765i	$-0.939876\pi$
$101$	−3.30136	+	5.71812i	−0.328498	+	0.568975i	0.653716	+	0.756740i	$0.273209\pi$
$102$		0			0
$103$	1.44949			0.142822			0.0714112	−	0.997447i	$-0.477250\pi$
$103$	1.44949			0.142822			0.0714112	+	0.997447i	$0.477250\pi$
$104$	1.27970	−	2.21650i	0.125485	−	0.217346i
$105$		0			0
$106$	2.44949			0.237915
$107$	12.5384			1.21214			0.606068	−	0.795413i	$-0.292746\pi$
$107$	12.5384			1.21214			0.606068	+	0.795413i	$0.292746\pi$
$108$		0			0
$109$	−0.449490	−	0.778539i	−0.0430533	−	0.0745705i	0.843696	−	0.536822i	$-0.180375\pi$
$109$	−0.449490	−	0.778539i	−0.0430533	−	0.0745705i	−0.886749	+	0.462251i	$0.847042\pi$
$110$	−2.22589	+	3.85536i	−0.212230	+	0.367594i
$111$		0			0
$112$	−0.724745	−	1.25529i	−0.0684820	−	0.118614i
$113$	−6.26922			−0.589758			−0.294879	−	0.955535i	$-0.595279\pi$
$113$	−6.26922			−0.589758			−0.294879	+	0.955535i	$0.595279\pi$
$114$		0			0
$115$	15.7980			1.47317
$116$	−6.93623	−	12.0139i	−0.644012	−	1.11546i
$117$		0			0
$118$	−3.12372	+	5.41045i	−0.287562	+	0.498072i
$119$	4.78529	+	8.28836i	0.438667	+	0.759793i
$120$		0			0
$121$	−7.69694			−0.699722
$122$	−3.70982			−0.335871
$123$		0			0
$124$	−3.29796	+	5.71223i	−0.296165	+	0.512974i
$125$	2.96786			0.265453
$126$		0			0
$127$	6.89898	−	11.9494i	0.612185	−	1.06034i	−0.378686	−	0.925525i	$-0.623624\pi$
$127$	6.89898	−	11.9494i	0.612185	−	1.06034i	0.990871	−	0.134811i	$-0.0430427\pi$
$128$	−4.98952	−	8.64210i	−0.441015	−	0.763861i
$129$		0			0
$130$	−1.22474	−	2.12132i	−0.107417	−	0.186052i
$131$	−8.42015	−	14.5841i	−0.735672	−	1.27422i	−0.954428	−	0.298442i	$-0.903533\pi$
$131$	−8.42015	−	14.5841i	−0.735672	−	1.27422i	0.218756	−	0.975780i	$-0.429800\pi$
$132$		0			0
$133$	1.44949	+	6.14966i	0.125687	+	0.533244i
$134$	10.6460			0.919678
$135$		0			0
$136$	8.44949	+	14.6349i	0.724538	+	1.25494i
$137$	8.42015	−	14.5841i	0.719382	−	1.24601i	−0.241863	−	0.970311i	$-0.577758\pi$
$137$	8.42015	−	14.5841i	0.719382	−	1.24601i	0.961245	−	0.275696i	$-0.0889083\pi$
$138$		0			0
$139$	−6.17423	+	10.6941i	−0.523692	+	0.907061i	0.475928	+	0.879484i	$0.342112\pi$
$139$	−6.17423	+	10.6941i	−0.523692	+	0.907061i	−0.999620	+	0.0275764i	$0.991221\pi$
$140$	−6.93623			−0.586218
$141$		0			0
$142$	3.55051	−	6.14966i	0.297952	−	0.516068i
$143$	−0.908716	+	1.57394i	−0.0759907	+	0.131620i
$144$		0			0
$145$	−31.5959			−2.62390
$146$	1.85491	−	3.21280i	0.153513	−	0.265893i
$147$		0			0
$148$	−4.27526	+	7.40496i	−0.351424	+	0.608684i
$149$	−3.46811	−	6.00695i	−0.284119	−	0.492108i	0.688276	−	0.725449i	$-0.258368\pi$
$149$	−3.46811	−	6.00695i	−0.284119	−	0.492108i	−0.972395	+	0.233340i	$0.925034\pi$
$150$		0			0
$151$	−4.00000			−0.325515			−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000			−0.325515			−0.162758	+	0.986666i	$0.552039\pi$
$152$	2.55940	+	10.8586i	0.207594	+	0.880749i
$153$		0			0
$154$	−0.977296	−	1.69273i	−0.0787528	−	0.136404i
$155$	7.51144	+	13.0102i	0.603333	+	1.04500i
$156$		0			0
$157$	−6.84847	−	11.8619i	−0.546567	−	0.946682i	−0.998506	−	0.0546336i	$-0.982601\pi$
$157$	−6.84847	−	11.8619i	−0.546567	−	0.946682i	0.451939	−	0.892049i	$-0.350732\pi$
$158$	−5.32302	+	9.21975i	−0.423477	+	0.733484i
$159$		0			0
$160$	−19.3485			−1.52963
$161$	−3.46811	+	6.00695i	−0.273326	+	0.473414i
$162$		0			0
$163$	−4.55051			−0.356423			−0.178212	−	0.983992i	$-0.557031\pi$
$163$	−4.55051			−0.356423			−0.178212	+	0.983992i	$0.557031\pi$
$164$	4.30188			0.335920
$165$		0			0
$166$	1.34847	+	2.33562i	0.104662	+	0.181279i
$167$	−6.02751	+	10.4400i	−0.466423	+	0.807868i	−0.999264	−	0.0383471i	$-0.987791\pi$
$167$	−6.02751	+	10.4400i	−0.466423	+	0.807868i	0.532842	+	0.846215i	$0.321124\pi$
$168$		0			0
$169$	6.00000	+	10.3923i	0.461538	+	0.799408i
$170$	16.1733			1.24044
$171$		0			0
$172$	12.1010			0.922694
$173$	−4.78529	−	8.28836i	−0.363819	−	0.630152i	0.624767	−	0.780811i	$-0.285194\pi$
$173$	−4.78529	−	8.28836i	−0.363819	−	0.630152i	−0.988586	+	0.150659i	$0.951861\pi$
$174$		0			0
$175$	−4.27526	+	7.40496i	−0.323179	+	0.559762i
$176$	0.908716	+	1.57394i	0.0684971	+	0.118640i
$177$		0			0
$178$	−12.2474			−0.917985
$179$	−17.9907			−1.34469			−0.672345	−	0.740238i	$-0.734713\pi$
$179$	−17.9907			−1.34469			−0.672345	+	0.740238i	$0.734713\pi$
$180$		0			0
$181$	5.55051	−	9.61377i	0.412566	−	0.714586i	−0.582603	−	0.812757i	$-0.697966\pi$
$181$	5.55051	−	9.61377i	0.412566	−	0.714586i	0.995170	+	0.0981710i	$0.0312992\pi$
$182$	1.07547			0.0797191
$183$		0			0
$184$	−6.12372	+	10.6066i	−0.451447	+	0.781929i
$185$	9.73733	+	16.8655i	0.715903	+	1.23998i
$186$		0			0
$187$	−6.00000	−	10.3923i	−0.438763	−	0.759961i
$188$	2.15094	+	3.72553i	0.156873	+	0.271712i
$189$		0			0
$190$	10.2247	+	3.07483i	0.741781	+	0.223072i
$191$	−1.81743			−0.131505			−0.0657524	−	0.997836i	$-0.520945\pi$
$191$	−1.81743			−0.131505			−0.0657524	+	0.997836i	$0.520945\pi$
$192$		0			0
$193$	4.84847	+	8.39780i	0.349000	+	0.604487i	0.986072	−	0.166318i	$-0.0531878\pi$
$193$	4.84847	+	8.39780i	0.349000	+	0.604487i	−0.637072	+	0.770805i	$0.719854\pi$
$194$	4.78529	−	8.28836i	0.343564	−	0.595070i
$195$		0			0
$196$	−3.55051	+	6.14966i	−0.253608	+	0.439262i
$197$	−13.5389			−0.964610			−0.482305	−	0.876003i	$-0.660200\pi$
$197$	−13.5389			−0.964610			−0.482305	+	0.876003i	$0.660200\pi$
$198$		0			0
$199$	7.17423	−	12.4261i	0.508568	−	0.880866i	−0.491383	−	0.870944i	$-0.663508\pi$
$199$	7.17423	−	12.4261i	0.508568	−	0.880866i	0.999951	−	0.00992190i	$-0.00315829\pi$
$200$	−7.54891	+	13.0751i	−0.533789	+	0.924549i
$201$		0			0
$202$	−4.89898			−0.344691
$203$	6.93623	−	12.0139i	0.486828	−	0.843210i
$204$		0			0
$205$	4.89898	−	8.48528i	0.342160	−	0.592638i
$206$	0.537734	+	0.931383i	0.0374657	+	0.0648926i
$207$		0			0
$208$	−1.00000			−0.0693375
$209$	−1.81743	−	7.71071i	−0.125714	−	0.533361i
$210$		0			0
$211$	−0.174235	−	0.301783i	−0.0119948	−	0.0207756i	0.859966	−	0.510352i	$-0.170485\pi$
$211$	−0.174235	−	0.301783i	−0.0119948	−	0.0207756i	−0.871961	+	0.489576i	$0.837151\pi$
$212$	−2.39264	−	4.14418i	−0.164327	−	0.284624i
$213$		0			0
$214$	4.65153	+	8.05669i	0.317972	+	0.550744i
$215$	13.7807	−	23.8688i	0.939833	−	1.62784i
$216$		0			0
$217$	−6.59592			−0.447760
$218$	0.333505	−	0.577648i	0.0225878	−	0.0391232i
$219$		0			0
$220$	8.69694			0.586347
$221$	6.60272			0.444147
$222$		0			0
$223$	7.17423	+	12.4261i	0.480422	+	0.832116i	0.999748	−	0.0224607i	$-0.00715008\pi$
$223$	7.17423	+	12.4261i	0.480422	+	0.832116i	−0.519325	+	0.854577i	$0.673817\pi$
$224$	4.24755	−	7.35698i	0.283802	−	0.491559i
$225$		0			0
$226$	−2.32577	−	4.02834i	−0.154708	−	0.267961i
$227$	−14.3559			−0.952832			−0.476416	−	0.879220i	$-0.658064\pi$
$227$	−14.3559			−0.952832			−0.476416	+	0.879220i	$0.658064\pi$
$228$		0			0
$229$	8.79796			0.581385			0.290693	−	0.956816i	$-0.406114\pi$
$229$	8.79796			0.581385			0.290693	+	0.956816i	$0.406114\pi$
$230$	5.86076	+	10.1511i	0.386447	+	0.669346i
$231$		0			0
$232$	12.2474	−	21.2132i	0.804084	−	1.39272i
$233$	−0.333505	−	0.577648i	−0.0218486	−	0.0378430i	0.854894	−	0.518802i	$-0.173622\pi$
$233$	−0.333505	−	0.577648i	−0.0218486	−	0.0378430i	−0.876743	+	0.480959i	$0.840289\pi$
$234$		0			0
$235$	9.79796			0.639148
$236$	12.2049			0.794473
$237$		0			0
$238$	−3.55051	+	6.14966i	−0.230145	+	0.398624i
$239$	−14.3559			−0.928604			−0.464302	−	0.885677i	$-0.653695\pi$
$239$	−14.3559			−0.928604			−0.464302	+	0.885677i	$0.653695\pi$
$240$		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	$-0.0429475\pi$
$242$	−2.85542	−	4.94574i	−0.183554	−	0.317924i
$243$		0			0
$244$	3.62372	+	6.27647i	0.231985	+	0.401810i
$245$	8.08665	+	14.0065i	0.516637	+	0.894842i
$246$		0			0
$247$	4.17423	+	1.25529i	0.265600	+	0.0798725i
$248$	−11.6466			−0.739557
$249$		0			0
$250$	1.10102	+	1.90702i	0.0696347	+	0.120611i
$251$	−9.57058	+	16.5767i	−0.604089	+	1.04631i	0.388105	+	0.921615i	$0.373130\pi$
$251$	−9.57058	+	16.5767i	−0.604089	+	1.04631i	−0.992195	+	0.124698i	$0.960204\pi$
$252$		0			0
$253$	4.34847	−	7.53177i	0.273386	−	0.473518i
$254$	10.2376			0.642363
$255$		0			0
$256$	6.05051	−	10.4798i	0.378157	−	0.654987i
$257$	3.46811	−	6.00695i	0.216335	−	0.374703i	−0.737350	−	0.675511i	$-0.763923\pi$
$257$	3.46811	−	6.00695i	0.216335	−	0.374703i	0.953685	+	0.300808i	$0.0972564\pi$
$258$		0			0
$259$	−8.55051			−0.531303
$260$	−2.39264	+	4.14418i	−0.148385	+	0.257011i
$261$		0			0
$262$	6.24745	−	10.8209i	0.385969	−	0.668517i
$263$	7.75314	+	13.4288i	0.478079	+	0.828058i	0.999684	−	0.0251295i	$-0.00799983\pi$
$263$	7.75314	+	13.4288i	0.478079	+	0.828058i	−0.521605	+	0.853187i	$0.674666\pi$
$264$		0			0
$265$	−10.8990			−0.669519
$266$	−3.41379	+	3.21280i	−0.209313	+	0.196989i
$267$		0			0
$268$	−10.3990	−	18.0116i	−0.635219	−	1.10023i
$269$	−7.91990	−	13.7177i	−0.482885	−	0.836381i	0.516922	−	0.856032i	$-0.327078\pi$
$269$	−7.91990	−	13.7177i	−0.482885	−	0.836381i	−0.999807	+	0.0196517i	$0.993744\pi$
$270$		0			0
$271$	0.898979	+	1.55708i	0.0546091	+	0.0945858i	0.892038	−	0.451961i	$-0.149275\pi$
$271$	0.898979	+	1.55708i	0.0546091	+	0.0945858i	−0.837429	+	0.546547i	$0.815942\pi$
$272$	3.30136	−	5.71812i	0.200174	−	0.346712i
$273$		0			0
$274$	12.4949			0.754844
$275$	5.36050	−	9.28466i	0.323250	−	0.559886i
$276$		0			0
$277$	10.6969			0.642717			0.321358	−	0.946958i	$-0.395861\pi$
$277$	10.6969			0.642717			0.321358	+	0.946958i	$0.395861\pi$
$278$	−9.16212			−0.549507
$279$		0			0
$280$	−6.12372	−	10.6066i	−0.365963	−	0.633866i
$281$	3.46811	−	6.00695i	0.206890	−	0.358344i	−0.743843	−	0.668354i	$-0.766999\pi$
$281$	3.46811	−	6.00695i	0.206890	−	0.358344i	0.950733	+	0.310010i	$0.100332\pi$
$282$		0			0
$283$	12.8990	+	22.3417i	0.766765	+	1.32808i	0.939309	+	0.343074i	$0.111468\pi$
$283$	12.8990	+	22.3417i	0.766765	+	1.32808i	−0.172544	+	0.985002i	$0.555199\pi$
$284$	−13.8725			−0.823179
$285$		0			0
$286$	−1.34847			−0.0797367
$287$	2.15094	+	3.72553i	0.126966	+	0.219911i
$288$		0			0
$289$	−13.2980	+	23.0327i	−0.782233	+	1.35487i
$290$	−11.7215	−	20.3023i	−0.688311	−	1.19219i
$291$		0			0
$292$	−7.24745			−0.424125
$293$	32.3466			1.88971			0.944854	−	0.327492i	$-0.106203\pi$
$293$	32.3466			1.88971			0.944854	+	0.327492i	$0.106203\pi$
$294$		0			0
$295$	13.8990	−	24.0737i	0.809230	−	1.40163i
$296$	−15.0978			−0.877543
$297$		0			0
$298$	2.57321	−	4.45694i	0.149062	−	0.258183i
$299$	2.39264	+	4.14418i	0.138370	+	0.239664i
$300$		0			0
$301$	6.05051	+	10.4798i	0.348746	+	0.604045i
$302$	−1.48393	−	2.57024i	−0.0853904	−	0.147901i
$303$		0			0
$304$	3.17423	−	2.98735i	0.182055	−	0.171336i
$305$	16.5068			0.945177
$306$		0			0
$307$	−4.79796	−	8.31031i	−0.273834	−	0.474294i	0.696006	−	0.718036i	$-0.254959\pi$
$307$	−4.79796	−	8.31031i	−0.273834	−	0.474294i	−0.969840	+	0.243741i	$0.921625\pi$
$308$	−1.90923	+	3.30689i	−0.108789	+	0.188427i
$309$		0			0
$310$	−5.57321	+	9.65309i	−0.316537	+	0.548259i
$311$	25.2605			1.43239			0.716195	−	0.697901i	$-0.245882\pi$
$311$	25.2605			1.43239			0.716195	+	0.697901i	$0.245882\pi$
$312$		0			0
$313$	−6.44949	+	11.1708i	−0.364547	+	0.631413i	−0.988703	−	0.149886i	$-0.952109\pi$
$313$	−6.44949	+	11.1708i	−0.364547	+	0.631413i	0.624157	+	0.781299i	$0.285443\pi$
$314$	5.08132	−	8.80110i	0.286755	−	0.496675i
$315$		0			0
$316$	20.7980			1.16998
$317$	−0.166753	+	0.288824i	−0.00936576	+	0.0162220i	−0.870670	−	0.491867i	$-0.836315\pi$
$317$	−0.166753	+	0.288824i	−0.00936576	+	0.0162220i	0.861305	+	0.508089i	$0.169648\pi$
$318$		0			0
$319$	−8.69694	+	15.0635i	−0.486935	+	0.843396i
$320$	−3.87657	−	6.71442i	−0.216707	−	0.375347i
$321$		0			0
$322$	−5.14643			−0.286799
$323$	−20.9586	+	19.7246i	−1.16617	+	1.09751i
$324$		0			0
$325$	2.94949	+	5.10867i	0.163608	+	0.283378i
$326$	−1.68816	−	2.92397i	−0.0934984	−	0.161944i
$327$		0			0
$328$	3.79796	+	6.57826i	0.209707	+	0.363224i
$329$	−2.15094	+	3.72553i	−0.118585	+	0.205395i
$330$		0			0
$331$	1.44949			0.0796712			0.0398356	−	0.999206i	$-0.487317\pi$
$331$	1.44949			0.0796712			0.0398356	+	0.999206i	$0.487317\pi$
$332$	2.63435	−	4.56283i	0.144579	−	0.250418i
$333$		0			0
$334$	−8.94439			−0.489415
$335$	−47.3695			−2.58807
$336$		0			0
$337$	1.84847	+	3.20164i	0.100693	+	0.174405i	0.911970	−	0.410257i	$-0.134561\pi$
$337$	1.84847	+	3.20164i	0.100693	+	0.174405i	−0.811278	+	0.584661i	$0.801227\pi$
$338$	−4.45178	+	7.71071i	−0.242145	+	0.419408i
$339$		0			0
$340$	−15.7980	−	27.3629i	−0.856765	−	1.48396i
$341$	8.27025			0.447859
$342$		0			0
$343$	−17.2474			−0.931275
$344$	10.6835	+	18.5044i	0.576017	+	0.997690i
$345$		0			0
$346$	3.55051	−	6.14966i	0.190877	−	0.330608i
$347$	10.4793	+	18.1507i	0.562558	+	0.974379i	0.997272	+	0.0738104i	$0.0235160\pi$
$347$	10.4793	+	18.1507i	0.562558	+	0.974379i	−0.434714	+	0.900568i	$0.643151\pi$
$348$		0			0
$349$	2.79796			0.149771			0.0748857	−	0.997192i	$-0.476141\pi$
$349$	2.79796			0.149771			0.0748857	+	0.997192i	$0.476141\pi$
$350$	−6.34417			−0.339110
$351$		0			0
$352$	−5.32577	+	9.22450i	−0.283864	+	0.491667i
$353$	−2.63435			−0.140212			−0.0701062	−	0.997540i	$-0.522334\pi$
$353$	−2.63435			−0.140212			−0.0701062	+	0.997540i	$0.522334\pi$
$354$		0			0
$355$	−15.7980	+	27.3629i	−0.838469	+	1.45227i
$356$	11.9632	+	20.7209i	0.634049	+	1.09821i
$357$		0			0
$358$	−6.67423	−	11.5601i	−0.352744	−	0.610971i
$359$	9.57058	+	16.5767i	0.505116	+	0.874886i	0.999982	+	0.00591717i	$0.00188350\pi$
$359$	9.57058	+	16.5767i	0.505116	+	0.874886i	−0.494867	+	0.868969i	$0.664783\pi$
$360$		0			0
$361$	−17.0000	+	8.48528i	−0.894737	+	0.446594i
$362$	8.23656			0.432904
$363$		0			0
$364$	−1.05051	−	1.81954i	−0.0550617	−	0.0953697i
$365$	−8.25340	+	14.2953i	−0.432003	+	0.748251i
$366$		0			0
$367$	2.82577	−	4.89437i	0.147504	−	0.255484i	−0.782801	−	0.622273i	$-0.786209\pi$
$367$	2.82577	−	4.89437i	0.147504	−	0.255484i	0.930304	+	0.366789i	$0.119543\pi$
$368$	4.78529			0.249450
$369$		0			0
$370$	−7.22474	+	12.5136i	−0.375597	+	0.650552i
$371$	2.39264	−	4.14418i	0.124220	−	0.215155i
$372$		0			0
$373$	−1.30306			−0.0674700			−0.0337350	−	0.999431i	$-0.510740\pi$
$373$	−1.30306			−0.0674700			−0.0337350	+	0.999431i	$0.510740\pi$
$374$	4.45178	−	7.71071i	0.230196	−	0.398712i
$375$		0			0
$376$	−3.79796	+	6.57826i	−0.195865	+	0.339248i
$377$	−4.78529	−	8.28836i	−0.246455	−	0.426872i
$378$		0			0
$379$	1.44949			0.0744553			0.0372276	−	0.999307i	$-0.488147\pi$
$379$	1.44949			0.0744553			0.0372276	+	0.999307i	$0.488147\pi$
$380$	−4.78529	−	20.3023i	−0.245480	−	1.04148i
$381$		0			0
$382$	−0.674235	−	1.16781i	−0.0344968	−	0.0597503i
$383$	−7.51144	−	13.0102i	−0.383816	−	0.664790i	0.607788	−	0.794100i	$-0.292057\pi$
$383$	−7.51144	−	13.0102i	−0.383816	−	0.664790i	−0.991604	+	0.129310i	$0.958724\pi$
$384$		0			0
$385$	4.34847	+	7.53177i	0.221619	+	0.383855i
$386$	−3.59739	+	6.23086i	−0.183102	+	0.317142i
$387$		0			0
$388$	−18.6969			−0.949193
$389$	3.46811	−	6.00695i	0.175840	−	0.304564i	−0.764611	−	0.644492i	$-0.777069\pi$
$389$	3.46811	−	6.00695i	0.175840	−	0.304564i	0.940452	+	0.339927i	$0.110402\pi$
$390$		0			0
$391$	−31.5959			−1.59787
$392$	−12.5384			−0.633286
$393$		0			0
$394$	−5.02270	−	8.69958i	−0.253040	−	0.438278i
$395$	23.6847	−	41.0232i	1.19171	−	2.06410i
$396$		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	$0.777127\pi$
$398$	10.6460			0.533638
$399$		0			0
$400$	5.89898			0.294949
$401$	−11.5548	−	20.0134i	−0.577017	−	0.999423i	−0.995819	−	0.0913456i	$-0.970883\pi$
$401$	−11.5548	−	20.0134i	−0.577017	−	0.999423i	0.418802	−	0.908078i	$-0.362450\pi$
$402$		0			0
$403$	−2.27526	+	3.94086i	−0.113339	+	0.196308i
$404$	4.78529	+	8.28836i	0.238077	+	0.412361i
$405$		0			0
$406$	10.2929			0.510826
$407$	10.7210			0.531420
$408$		0			0
$409$	−6.44949	+	11.1708i	−0.318907	+	0.552363i	−0.980260	−	0.197712i	$-0.936649\pi$
$409$	−6.44949	+	11.1708i	−0.318907	+	0.552363i	0.661353	+	0.750074i	$0.269982\pi$
$410$	7.26973			0.359026
$411$		0			0
$412$	1.05051	−	1.81954i	0.0517549	−	0.0896422i
$413$	6.10246	+	10.5698i	0.300283	+	0.520105i
$414$		0			0
$415$	−6.00000	−	10.3923i	−0.294528	−	0.510138i
$416$	−2.93038	−	5.07556i	−0.143674	−	0.248850i
$417$		0			0
$418$	4.28036	−	4.02834i	0.209359	−	0.197033i
$419$	−17.9907			−0.878905			−0.439452	−	0.898266i	$-0.644827\pi$
$419$	−17.9907			−0.878905			−0.439452	+	0.898266i	$0.644827\pi$
$420$		0			0
$421$	−6.44949	−	11.1708i	−0.314329	−	0.544434i	0.664966	−	0.746874i	$-0.268446\pi$
$421$	−6.44949	−	11.1708i	−0.314329	−	0.544434i	−0.979295	+	0.202440i	$0.935113\pi$
$422$	0.129276	−	0.223912i	0.00629305	−	0.0108999i
$423$		0			0
$424$	4.22474	−	7.31747i	0.205172	−	0.355368i
$425$	−38.9493			−1.88932
$426$		0			0
$427$	−3.62372	+	6.27647i	−0.175364	+	0.303740i
$428$	9.08716	−	15.7394i	0.439245	−	0.760794i
$429$		0			0
$430$	20.4495			0.986162
$431$	−1.48393	+	2.57024i	−0.0714783	+	0.123804i	−0.899549	−	0.436819i	$-0.856105\pi$
$431$	−1.48393	+	2.57024i	−0.0714783	+	0.123804i	0.828071	+	0.560623i	$0.189438\pi$
$432$		0			0
$433$	−3.84847	+	6.66574i	−0.184946	+	0.320335i	−0.943558	−	0.331207i	$-0.892544\pi$
$433$	−3.84847	+	6.66574i	−0.184946	+	0.320335i	0.758613	+	0.651542i	$0.225878\pi$
$434$	−2.44697	−	4.23827i	−0.117458	−	0.203443i
$435$		0			0
$436$	−1.30306			−0.0624053
$437$	−19.9749	−	6.00695i	−0.955530	−	0.287351i
$438$		0			0
$439$	11.8258	+	20.4828i	0.564413	+	0.977592i	0.997104	+	0.0760497i	$0.0242308\pi$
$439$	11.8258	+	20.4828i	0.564413	+	0.977592i	−0.432691	+	0.901542i	$0.642436\pi$
$440$	7.67819	+	13.2990i	0.366043	+	0.634006i
$441$		0			0
$442$	2.44949	+	4.24264i	0.116510	+	0.201802i
$443$	9.23707	−	15.9991i	0.438866	−	0.760139i	−0.558736	−	0.829346i	$-0.688713\pi$
$443$	9.23707	−	15.9991i	0.438866	−	0.760139i	0.997602	+	0.0692066i	$0.0220468\pi$
$444$		0			0
$445$	54.4949			2.58331
$446$	−5.32302	+	9.21975i	−0.252052	+	0.436568i
$447$		0			0
$448$	3.40408			0.160828
$449$	28.7117			1.35499			0.677495	−	0.735527i	$-0.263066\pi$
$449$	28.7117			1.35499			0.677495	+	0.735527i	$0.263066\pi$
$450$		0			0
$451$	−2.69694	−	4.67123i	−0.126994	−	0.219960i
$452$	−4.54358	+	7.86971i	−0.213712	+	0.370160i
$453$		0			0
$454$	−5.32577	−	9.22450i	−0.249951	−	0.432927i
$455$	−4.78529			−0.224338
$456$		0			0
$457$	−29.8990			−1.39862			−0.699308	−	0.714821i	$-0.746508\pi$
$457$	−29.8990			−1.39862			−0.699308	+	0.714821i	$0.746508\pi$
$458$	3.26388	+	5.65321i	0.152511	+	0.264157i
$459$		0			0
$460$	11.4495	−	19.8311i	0.533835	−	0.924630i
$461$	4.61854	+	7.99954i	0.215107	+	0.372576i	0.953306	−	0.302007i	$-0.0976568\pi$
$461$	4.61854	+	7.99954i	0.215107	+	0.372576i	−0.738199	+	0.674583i	$0.764323\pi$
$462$		0			0
$463$	34.1464			1.58692			0.793460	−	0.608623i	$-0.208278\pi$
$463$	34.1464			1.58692			0.793460	+	0.608623i	$0.208278\pi$
$464$	−9.57058			−0.444303
$465$		0			0
$466$	0.247449	−	0.428594i	0.0114628	−	0.0198542i
$467$	16.1733			0.748411			0.374205	−	0.927346i	$-0.377915\pi$
$467$	16.1733			0.748411			0.374205	+	0.927346i	$0.377915\pi$
$468$		0			0
$469$	10.3990	−	18.0116i	0.480180	−	0.831697i
$470$	3.63487	+	6.29577i	0.167664	+	0.290402i
$471$		0			0
$472$	10.7753	+	18.6633i	0.495971	+	0.859048i
$473$	−7.58639	−	13.1400i	−0.348823	−	0.604178i
$474$		0			0
$475$	−24.6237	−	7.40496i	−1.12981	−	0.339763i
$476$	13.8725			0.635843
$477$		0			0
$478$	−5.32577	−	9.22450i	−0.243595	−	0.421919i
$479$	4.78529	−	8.28836i	0.218645	−	0.378705i	−0.735749	−	0.677255i	$-0.763170\pi$
$479$	4.78529	−	8.28836i	0.218645	−	0.378705i	0.954394	+	0.298550i	$0.0965029\pi$
$480$		0			0
$481$	−2.94949	+	5.10867i	−0.134485	+	0.232935i
$482$	14.0973			0.642115
$483$		0			0
$484$	−5.57832	+	9.66193i	−0.253560	+	0.439179i
$485$	−21.2921	+	36.8790i	−0.966824	+	1.67459i
$486$		0			0
$487$	8.00000			0.362515			0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000			0.362515			0.181257	+	0.983436i	$0.441983\pi$
$488$	−6.39849	+	11.0825i	−0.289646	+	0.501682i
$489$		0			0
$490$	−6.00000	+	10.3923i	−0.271052	+	0.469476i
$491$	15.0229	+	26.0204i	0.677973	+	1.17428i	0.975590	+	0.219598i	$0.0704746\pi$
$491$	15.0229	+	26.0204i	0.677973	+	1.17428i	−0.297618	+	0.954685i	$0.596192\pi$
$492$		0			0
$493$	63.1918			2.84602
$494$	0.741964	+	3.14789i	0.0333825	+	0.141630i
$495$		0			0
$496$	2.27526	+	3.94086i	0.102162	+	0.176950i
$497$	−6.93623	−	12.0139i	−0.311132	−	0.538897i
$498$		0			0
$499$	−21.1742	−	36.6749i	−0.947889	−	1.64179i	−0.749861	−	0.661595i	$-0.769880\pi$
$499$	−21.1742	−	36.6749i	−0.947889	−	1.64179i	−0.198028	−	0.980196i	$-0.563454\pi$
$500$	2.15094	−	3.72553i	0.0961929	−	0.166611i
$501$		0			0
$502$	−14.2020			−0.633868
$503$	0.333505	−	0.577648i	0.0148703	−	0.0257560i	−0.858495	−	0.512823i	$-0.828600\pi$
$503$	0.333505	−	0.577648i	0.0148703	−	0.0257560i	0.873365	+	0.487067i	$0.161933\pi$
$504$		0			0
$505$	21.7980			0.969996
$506$	6.45281			0.286863
$507$		0			0
$508$	−10.0000	−	17.3205i	−0.443678	−	0.768473i
$509$	−16.8403	+	29.1683i	−0.746433	+	1.29286i	0.203089	+	0.979160i	$0.434902\pi$
$509$	−16.8403	+	29.1683i	−0.746433	+	1.29286i	−0.949522	+	0.313700i	$0.898431\pi$
$510$		0			0
$511$	−3.62372	−	6.27647i	−0.160304	−	0.277655i
$512$	−10.9795			−0.485232
$513$		0			0
$514$	5.14643			0.226999
$515$	−2.39264	−	4.14418i	−0.105432	−	0.182614i
$516$		0			0
$517$	2.69694	−	4.67123i	0.118611	−	0.205441i
$518$	−3.17208	−	5.49421i	−0.139373	−	0.241402i
$519$		0			0
$520$	−8.44949			−0.370535
$521$	9.90408			0.433906			0.216953	−	0.976182i	$-0.430388\pi$
$521$	9.90408			0.433906			0.216953	+	0.976182i	$0.430388\pi$
$522$		0			0
$523$	2.82577	−	4.89437i	0.123562	−	0.214016i	−0.797608	−	0.603176i	$-0.793902\pi$
$523$	2.82577	−	4.89437i	0.123562	−	0.214016i	0.921170	+	0.389161i	$0.127235\pi$
$524$	−24.4099			−1.06635
$525$		0			0
$526$	−5.75255	+	9.96371i	−0.250823	+	0.434438i
$527$	−15.0229	−	26.0204i	−0.654407	−	1.13347i
$528$		0			0
$529$	0.0505103	+	0.0874863i	0.00219610	+	0.00380375i
$530$	−4.04332	−	7.00324i	−0.175631	−	0.304201i
$531$		0			0
$532$	8.77015	+	2.63740i	0.380234	+	0.114346i
$533$	2.96786			0.128552
$534$		0			0
$535$	−20.6969	−	35.8481i	−0.894807	−	1.54985i
$536$	18.3617	−	31.8034i	0.793105	−	1.37370i
$537$		0			0
$538$	5.87628	−	10.1780i	0.253344	−	0.438805i
$539$	8.90357			0.383504
$540$		0			0
$541$	−9.84847	+	17.0580i	−0.423419	+	0.733383i	−0.996271	−	0.0862756i	$-0.972503\pi$
$541$	−9.84847	+	17.0580i	−0.423419	+	0.733383i	0.572853	+	0.819658i	$0.305837\pi$
$542$	−0.667010	+	1.15530i	−0.0286505	+	0.0496242i
$543$		0			0
$544$	38.6969			1.65912
$545$	−1.48393	+	2.57024i	−0.0635645	+	0.110097i
$546$		0			0
$547$	16.1742	−	28.0146i	0.691560	−	1.19782i	−0.279766	−	0.960068i	$-0.590257\pi$
$547$	16.1742	−	28.0146i	0.691560	−	1.19782i	0.971327	−	0.237749i	$-0.0764097\pi$
$548$	−12.2049	−	21.1396i	−0.521369	−	0.903037i
$549$		0			0
$550$	7.95459			0.339185
$551$	39.9498	+	12.0139i	1.70192	+	0.511809i
$552$		0			0
$553$	10.3990	+	18.0116i	0.442210	+	0.765929i
$554$	3.96837	+	6.87342i	0.168600	+	0.292024i
$555$		0			0
$556$	8.94949	+	15.5010i	0.379543	+	0.657388i
$557$	−11.3880	+	19.7246i	−0.482525	+	0.835759i	−0.999799	−	0.0200617i	$-0.993614\pi$
$557$	−11.3880	+	19.7246i	−0.482525	+	0.835759i	0.517273	+	0.855820i	$0.326947\pi$
$558$		0			0
$559$	8.34847			0.353103
$560$	−2.39264	+	4.14418i	−0.101108	+	0.175124i
$561$		0			0
$562$	5.14643			0.217089
$563$	19.8082			0.834814			0.417407	−	0.908720i	$-0.362939\pi$
$563$	19.8082			0.834814			0.417407	+	0.908720i	$0.362939\pi$
$564$		0			0
$565$	10.3485	+	17.9241i	0.435363	+	0.754071i
$566$	−9.57058	+	16.5767i	−0.402281	+	0.696772i
$567$		0			0
$568$	−12.2474	−	21.2132i	−0.513892	−	0.890086i
$569$	8.90357			0.373257			0.186628	−	0.982431i	$-0.440244\pi$
$569$	8.90357			0.373257			0.186628	+	0.982431i	$0.440244\pi$
$570$		0			0
$571$	−13.2474			−0.554388			−0.277194	−	0.960814i	$-0.589405\pi$
$571$	−13.2474			−0.554388			−0.277194	+	0.960814i	$0.589405\pi$
$572$	1.31718	+	2.28141i	0.0550739	+	0.0953907i
$573$		0			0
$574$	−1.59592	+	2.76421i	−0.0666123	+	0.115376i
$575$	−14.1142	−	24.4464i	−0.588601	−	1.01949i
$576$		0			0
$577$	−6.69694			−0.278797			−0.139399	−	0.990236i	$-0.544517\pi$
$577$	−6.69694			−0.278797			−0.139399	+	0.990236i	$0.544517\pi$
$578$	−19.7332			−0.820793
$579$		0			0
$580$	−22.8990	+	39.6622i	−0.950828	+	1.64688i
$581$	5.26870			0.218583
$582$		0			0
$583$	−3.00000	+	5.19615i	−0.124247	+	0.215203i
$584$	−6.39849	−	11.0825i	−0.264771	−	0.458598i
$585$		0			0
$586$	12.0000	+	20.7846i	0.495715	+	0.858604i
$587$	−5.69400	−	9.86230i	−0.235017	−	0.407061i	0.724261	−	0.689526i	$-0.242181\pi$
$587$	−5.69400	−	9.86230i	−0.235017	−	0.407061i	−0.959278	+	0.282465i	$0.908848\pi$
$588$		0			0
$589$	−4.55051	−	19.3062i	−0.187501	−	0.795497i
$590$	20.6251			0.849121
$591$		0			0
$592$	2.94949	+	5.10867i	0.121223	+	0.209965i
$593$	5.28555	−	9.15483i	0.217051	−	0.375944i	−0.736854	−	0.676052i	$-0.763689\pi$
$593$	5.28555	−	9.15483i	0.217051	−	0.375944i	0.953905	+	0.300108i	$0.0970228\pi$
$594$		0			0
$595$	15.7980	−	27.3629i	0.647653	−	1.12177i
$596$	−10.0540			−0.411827
$597$		0			0
$598$	−1.77526	+	3.07483i	−0.0725956	+	0.125739i
$599$	−15.9316	+	27.5943i	−0.650947	+	1.12747i	0.331946	+	0.943298i	$0.392295\pi$
$599$	−15.9316	+	27.5943i	−0.650947	+	1.12747i	−0.982893	+	0.184175i	$0.941039\pi$
$600$		0			0
$601$	12.1010			0.493611			0.246805	−	0.969065i	$-0.420619\pi$
$601$	12.1010			0.493611			0.246805	+	0.969065i	$0.420619\pi$
$602$	−4.48926	+	7.77563i	−0.182969	+	0.316911i
$603$		0			0
$604$	−2.89898	+	5.02118i	−0.117958	+	0.204309i
$605$	12.7052	+	22.0060i	0.516539	+	0.894672i
$606$		0			0
$607$	−45.9444			−1.86483			−0.932413	−	0.361396i	$-0.882300\pi$
$607$	−45.9444			−1.86483			−0.932413	+	0.361396i	$0.882300\pi$
$608$	24.4642	+	7.35698i	0.992153	+	0.298365i
$609$		0			0
$610$	6.12372	+	10.6066i	0.247942	+	0.429449i
$611$	1.48393	+	2.57024i	0.0600333	+	0.103981i
$612$		0			0
$613$	−6.44949	−	11.1708i	−0.260492	−	0.451186i	0.705880	−	0.708331i	$-0.250552\pi$
$613$	−6.44949	−	11.1708i	−0.260492	−	0.451186i	−0.966373	+	0.257145i	$0.917218\pi$
$614$	3.55991	−	6.16595i	0.143666	−	0.248837i
$615$		0			0
$616$	−6.74235			−0.271657
$617$	11.5548	−	20.0134i	0.465177	−	0.805710i	−0.534032	−	0.845464i	$-0.679324\pi$
$617$	11.5548	−	20.0134i	0.465177	−	0.805710i	0.999210	+	0.0397536i	$0.0126573\pi$
$618$		0			0
$619$	−25.2474			−1.01478			−0.507390	−	0.861716i	$-0.669390\pi$
$619$	−25.2474			−1.01478			−0.507390	+	0.861716i	$0.669390\pi$
$620$	21.7755			0.874525
$621$		0			0
$622$	9.37117	+	16.2313i	0.375750	+	0.650818i
$623$	−11.9632	+	20.7209i	−0.479296	+	0.830165i
$624$		0			0
$625$	9.84847	+	17.0580i	0.393939	+	0.682322i
$626$	−9.57058			−0.382517
$627$		0			0
$628$	−19.8536			−0.792244
$629$	−19.4747	−	33.7311i	−0.776505	−	1.34495i
$630$		0			0
$631$	4.17423	−	7.22999i	0.166174	−	0.287821i	−0.770898	−	0.636959i	$-0.780192\pi$
$631$	4.17423	−	7.22999i	0.166174	−	0.287821i	0.937071	+	0.349138i	$0.113525\pi$
$632$	18.3617	+	31.8034i	0.730390	+	1.26507i
$633$		0			0
$634$	−0.247449			−0.00982744
$635$	−45.5520			−1.80768
$636$		0			0
$637$	−2.44949	+	4.24264i	−0.0970523	+	0.168100i
$638$	−12.9056			−0.510939
$639$		0			0
$640$	−16.4722	+	28.5307i	−0.651121	+	1.12777i
$641$	−7.41964	−	12.8512i	−0.293058	−	0.507591i	0.681473	−	0.731843i	$-0.261340\pi$
$641$	−7.41964	−	12.8512i	−0.293058	−	0.507591i	−0.974531	+	0.224252i	$0.928006\pi$
$642$		0			0
$643$	−9.17423	−	15.8902i	−0.361796	−	0.626650i	0.626460	−	0.779454i	$-0.284503\pi$
$643$	−9.17423	−	15.8902i	−0.361796	−	0.626650i	−0.988257	+	0.152804i	$0.951170\pi$
$644$	5.02699	+	8.70701i	0.198091	+	0.343104i
$645$		0			0
$646$	−20.4495	−	6.14966i	−0.804574	−	0.241955i
$647$	−12.5384			−0.492937			−0.246468	−	0.969151i	$-0.579270\pi$
$647$	−12.5384			−0.492937			−0.246468	+	0.969151i	$0.579270\pi$
$648$		0			0
$649$	−7.65153	−	13.2528i	−0.300349	−	0.520219i
$650$	−2.18841	+	3.79045i	−0.0858367	+	0.148673i
$651$		0			0
$652$	−3.29796	+	5.71223i	−0.129158	+	0.223708i
$653$	−5.26870			−0.206180			−0.103090	−	0.994672i	$-0.532873\pi$
$653$	−5.26870			−0.206180			−0.103090	+	0.994672i	$0.532873\pi$
$654$		0			0
$655$	−27.7980	+	48.1475i	−1.08616	+	1.88128i
$656$	1.48393	−	2.57024i	0.0579376	−	0.100351i
$657$		0			0
$658$	−3.19184			−0.124431
$659$	21.8673	−	37.8753i	0.851829	−	1.47541i	−0.0277267	−	0.999616i	$-0.508827\pi$
$659$	21.8673	−	37.8753i	0.851829	−	1.47541i	0.879556	−	0.475796i	$-0.157840\pi$
$660$		0			0
$661$	18.5959	−	32.2091i	0.723297	−	1.25279i	−0.236374	−	0.971662i	$-0.575959\pi$
$661$	18.5959	−	32.2091i	0.723297	−	1.25279i	0.959671	−	0.281126i	$-0.0907077\pi$
$662$	0.537734	+	0.931383i	0.0208996	+	0.0361992i
$663$		0			0
$664$	9.30306			0.361029
$665$	15.1896	−	14.2953i	0.589028	−	0.554348i
$666$		0			0
$667$	22.8990	+	39.6622i	0.886652	+	1.53573i
$668$	8.73681	+	15.1326i	0.338037	+	0.585498i
$669$		0			0
$670$	−17.5732	−	30.4377i	−0.678912	−	1.17591i
$671$	4.54358	−	7.86971i	0.175403	−	0.303807i
$672$		0			0
$673$	12.1010			0.466460			0.233230	−	0.972422i	$-0.425070\pi$
$673$	12.1010			0.466460			0.233230	+	0.972422i	$0.425070\pi$
$674$	−1.37150	+	2.37550i	−0.0528281	+	0.0915010i
$675$		0			0
$676$	17.3939			0.668995
$677$	7.26973			0.279398			0.139699	−	0.990194i	$-0.455386\pi$
$677$	7.26973			0.279398			0.139699	+	0.990194i	$0.455386\pi$
$678$		0			0
$679$	−9.34847	−	16.1920i	−0.358761	−	0.621393i
$680$	27.8948	−	48.3152i	1.06972	−	1.85280i
$681$		0			0
$682$	3.06811	+	5.31413i	0.117484	+	0.203488i
$683$	−17.9907			−0.688396			−0.344198	−	0.938897i	$-0.611849\pi$
$683$	−17.9907			−0.688396			−0.344198	+	0.938897i	$0.611849\pi$
$684$		0			0
$685$	−55.5959			−2.12421
$686$	−6.39849	−	11.0825i	−0.244296	−	0.423132i
$687$		0			0
$688$	4.17423	−	7.22999i	0.159141	−	0.275641i
$689$	−1.65068	−	2.85906i	−0.0628859	−	0.108922i
$690$		0			0
$691$	−33.3939			−1.27036			−0.635181	−	0.772363i	$-0.719075\pi$
$691$	−33.3939			−1.27036			−0.635181	+	0.772363i	$0.719075\pi$
$692$	−13.8725			−0.527351
$693$		0			0
$694$	−7.77526	+	13.4671i	−0.295145	+	0.511206i
$695$	40.7667			1.54637
$696$		0			0
$697$	−9.79796	+	16.9706i	−0.371124	+	0.642806i
$698$	1.03799	+	1.79786i	0.0392886	+	0.0680498i
$699$		0			0
$700$	6.19694	+	10.7334i	0.234222	+	0.405685i
$701$	−15.1896	−	26.3092i	−0.573704	−	0.993685i	−0.996181	−	0.0873112i	$-0.972173\pi$
$701$	−15.1896	−	26.3092i	−0.573704	−	0.993685i	0.422477	−	0.906374i	$-0.361161\pi$
$702$		0			0
$703$	−5.89898	−	25.0273i	−0.222484	−	0.943921i
$704$	−4.26818			−0.160863
$705$		0			0
$706$	−0.977296	−	1.69273i	−0.0367810	−	0.0637066i
$707$	−4.78529	+	8.28836i	−0.179969	+	0.311716i
$708$		0			0
$709$	15.1969	−	26.3219i	0.570733	−	0.988539i	−0.425758	−	0.904837i	$-0.639992\pi$
$709$	15.1969	−	26.3219i	0.570733	−	0.988539i	0.996491	−	0.0837014i	$-0.0266742\pi$
$710$	−23.4430			−0.879801
$711$		0			0
$712$	−21.1237	+	36.5874i	−0.791645	+	1.37117i
$713$	10.8878	−	18.8581i	0.407749	−	0.706243i
$714$		0			0
$715$	6.00000			0.224387
$716$	−13.0387	+	22.5837i	−0.487279	+	0.843991i
$717$		0			0
$718$	−7.10102	+	12.2993i	−0.265008	+	0.459007i
$719$	1.57573	+	2.72924i	0.0587647	+	0.101783i	0.893911	−	0.448244i	$-0.147951\pi$
$719$	1.57573	+	2.72924i	0.0587647	+	0.101783i	−0.835146	+	0.550028i	$0.814617\pi$
$720$		0			0
$721$	2.10102			0.0782461
$722$	−11.7590	−	7.77563i	−0.437624	−	0.289379i
$723$		0			0
$724$	−8.04541	−	13.9351i	−0.299005	−	0.517892i
$725$	28.2283	+	48.8929i	1.04837	+	1.81584i
$726$		0			0
$727$	−1.52270	−	2.63740i	−0.0564740	−	0.0978158i	0.836406	−	0.548110i	$-0.184652\pi$
$727$	−1.52270	−	2.63740i	−0.0564740	−	0.0978158i	−0.892880	+	0.450294i	$0.851319\pi$
$728$	1.85491	−	3.21280i	0.0687475	−	0.119074i
$729$		0			0
$730$	−12.2474			−0.453298
$731$	−27.5613	+	47.7376i	−1.01939	+	1.76564i
$732$		0			0
$733$	34.6969			1.28156			0.640780	−	0.767724i	$-0.278611\pi$
$733$	34.6969			1.28156			0.640780	+	0.767724i	$0.278611\pi$
$734$	4.19323			0.154775
$735$		0			0
$736$	14.0227	+	24.2880i	0.516884	+	0.895269i
$737$	−13.0387	+	22.5837i	−0.480286	+	0.831880i
$738$		0			0
$739$	−7.52270	−	13.0297i	−0.276727	−	0.479305i	0.693842	−	0.720127i	$-0.255916\pi$
$739$	−7.52270	−	13.0297i	−0.276727	−	0.479305i	−0.970569	+	0.240822i	$0.922583\pi$
$740$	28.2283			1.03769
$741$		0			0
$742$	3.55051			0.130343
$743$	6.84443	+	11.8549i	0.251098	+	0.434914i	0.963828	−	0.266524i	$-0.0858751\pi$
$743$	6.84443	+	11.8549i	0.251098	+	0.434914i	−0.712731	+	0.701438i	$0.752542\pi$
$744$		0			0
$745$	−11.4495	+	19.8311i	−0.419477	+	0.726555i
$746$	−0.483412	−	0.837295i	−0.0176990	−	0.0306555i
$747$		0			0
$748$	−17.3939			−0.635983
$749$	18.1743			0.664075
$750$		0			0
$751$	−4.52270	+	7.83355i	−0.165036	+	0.285850i	−0.936668	−	0.350219i	$-0.886107\pi$
$751$	−4.52270	+	7.83355i	−0.165036	+	0.285850i	0.771632	+	0.636069i	$0.219441\pi$
$752$	2.96786			0.108227
$753$		0			0
$754$	3.55051	−	6.14966i	0.129302	−	0.223958i
$755$	6.60272	+	11.4362i	0.240298	+	0.416208i
$756$		0			0
$757$	21.1969	+	36.7142i	0.770416	+	1.33440i	0.937335	+	0.348429i	$0.113285\pi$
$757$	21.1969	+	36.7142i	0.770416	+	1.33440i	−0.166919	+	0.985971i	$0.553382\pi$
$758$	0.537734	+	0.931383i	0.0195314	+	0.0338294i
$759$		0			0
$760$	26.8207	−	25.2415i	0.972888	−	0.915607i
$761$	−26.0774			−0.945304			−0.472652	−	0.881249i	$-0.656703\pi$
$761$	−26.0774			−0.945304			−0.472652	+	0.881249i	$0.656703\pi$
$762$		0			0
$763$	−0.651531	−	1.12848i	−0.0235870	−	0.0408539i
$764$	−1.31718	+	2.28141i	−0.0476537	+	0.0825387i
$765$		0			0
$766$	5.57321	−	9.65309i	0.201368	−	0.348780i
$767$	8.42015			0.304034
$768$		0			0
$769$	15.1969	−	26.3219i	0.548016	−	0.949191i	−0.450395	−	0.892829i	$-0.648717\pi$
$769$	15.1969	−	26.3219i	0.548016	−	0.949191i	0.998410	−	0.0563614i	$-0.0179499\pi$
$770$	−3.22641	+	5.58830i	−0.116272	+	0.201388i
$771$		0			0
$772$	14.0556			0.505873
$773$	−1.48393	+	2.57024i	−0.0533732	+	0.0924450i	−0.891478	−	0.453065i	$-0.850331\pi$
$773$	−1.48393	+	2.57024i	−0.0533732	+	0.0924450i	0.838104	+	0.545510i	$0.183664\pi$
$774$		0			0
$775$	13.4217	−	23.2470i	0.482121	−	0.835058i
$776$	−16.5068	−	28.5906i	−0.592560	−	1.02634i
$777$		0			0
$778$	5.14643			0.184508
$779$	−9.42067	+	8.86601i	−0.337530	+	0.317658i
$780$		0			0
$781$	8.69694	+	15.0635i	0.311201	+	0.539016i
$782$	−11.7215	−	20.3023i	−0.419160	−	0.726007i
$783$		0			0
$784$	2.44949	+	4.24264i	0.0874818	+	0.151523i
$785$	−22.6093

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)

\(f(q)\)	\(=\)	\(q+(0.370982 + 0.642559i) q^{2} +(0.724745 - 1.25529i) q^{4} +(-1.65068 - 2.85906i) q^{5} +1.44949 q^{7} +2.55940 q^{8} +O(q^{10})\)	\(q+(0.370982 + 0.642559i) q^{2} +(0.724745 - 1.25529i) q^{4} +(-1.65068 - 2.85906i) q^{5} +1.44949 q^{7} +2.55940 q^{8} +(1.22474 - 2.12132i) q^{10} -1.81743 q^{11} +(0.500000 - 0.866025i) q^{13} +(0.537734 + 0.931383i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(3.30136 + 5.71812i) q^{17} +(1.00000 + 4.24264i) q^{19} -4.78529 q^{20} +(-0.674235 - 1.16781i) q^{22} +(-2.39264 + 4.14418i) q^{23} +(-2.94949 + 5.10867i) q^{25} +0.741964 q^{26} +(1.05051 - 1.81954i) q^{28} +(4.78529 - 8.28836i) q^{29} -4.55051 q^{31} +(2.93038 - 5.07556i) q^{32} +(-2.44949 + 4.24264i) q^{34} +(-2.39264 - 4.14418i) q^{35} -5.89898 q^{37} +(-2.35517 + 2.21650i) q^{38} +(-4.22474 - 7.31747i) q^{40} +(1.48393 + 2.57024i) q^{41} +(4.17423 + 7.22999i) q^{43} +(-1.31718 + 2.28141i) q^{44} -3.55051 q^{46} +(-1.48393 + 2.57024i) q^{47} -4.89898 q^{49} -4.37683 q^{50} +(-0.724745 - 1.25529i) q^{52} +(1.65068 - 2.85906i) q^{53} +(3.00000 + 5.19615i) q^{55} +3.70982 q^{56} +7.10102 q^{58} +(4.21008 + 7.29207i) q^{59} +(-2.50000 + 4.33013i) q^{61} +(-1.68816 - 2.92397i) q^{62} +2.34847 q^{64} -3.30136 q^{65} +(7.17423 - 12.4261i) q^{67} +9.57058 q^{68} +(1.77526 - 3.07483i) q^{70} +(-4.78529 - 8.28836i) q^{71} +(-2.50000 - 4.33013i) q^{73} +(-2.18841 - 3.79045i) q^{74} +(6.05051 + 1.81954i) q^{76} -2.63435 q^{77} +(7.17423 + 12.4261i) q^{79} +(-1.65068 + 2.85906i) q^{80} +(-1.10102 + 1.90702i) q^{82} +3.63487 q^{83} +(10.8990 - 18.8776i) q^{85} +(-3.09713 + 5.36439i) q^{86} -4.65153 q^{88} +(-8.25340 + 14.2953i) q^{89} +(0.724745 - 1.25529i) q^{91} +(3.46811 + 6.00695i) q^{92} -2.20204 q^{94} +(10.4793 - 9.86230i) q^{95} +(-6.44949 - 11.1708i) q^{97} +(-1.81743 - 3.14789i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{4} - 8 q^{7}+O(q^{10}) \) 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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