LMFDB - Embedded newform 855.2.be.a.64.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,2,Mod(64,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("855.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.be (of order $6$, 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:	$6.82720937282$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			64.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	855.64
Dual form			855.2.be.a.334.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.73205 - 1.00000i) q^{2} +(1.00000 - 1.73205i) q^{4} +(-2.23205 + 0.133975i) q^{5} +4.00000i q^{7} +O(q^{10})$	\(q+(1.73205 - 1.00000i) q^{2} +(1.00000 - 1.73205i) q^{4} +(-2.23205 + 0.133975i) q^{5} +4.00000i q^{7} +(-3.73205 + 2.46410i) q^{10} +1.00000 q^{11} +(1.73205 + 1.00000i) q^{13} +(4.00000 + 6.92820i) q^{14} +(2.00000 + 3.46410i) q^{16} +(-1.73205 + 1.00000i) q^{17} +(3.50000 - 2.59808i) q^{19} +(-2.00000 + 4.00000i) q^{20} +(1.73205 - 1.00000i) q^{22} +(5.19615 + 3.00000i) q^{23} +(4.96410 - 0.598076i) q^{25} +4.00000 q^{26} +(6.92820 + 4.00000i) q^{28} +(-4.50000 + 7.79423i) q^{29} -7.00000 q^{31} +(6.92820 + 4.00000i) q^{32} +(-2.00000 + 3.46410i) q^{34} +(-0.535898 - 8.92820i) q^{35} -2.00000i q^{37} +(3.46410 - 8.00000i) q^{38} +(1.00000 + 1.73205i) q^{41} +(1.73205 - 1.00000i) q^{43} +(1.00000 - 1.73205i) q^{44} +12.0000 q^{46} +(-5.19615 - 3.00000i) q^{47} -9.00000 q^{49} +(8.00000 - 6.00000i) q^{50} +(3.46410 - 2.00000i) q^{52} +(3.46410 + 2.00000i) q^{53} +(-2.23205 + 0.133975i) q^{55} +18.0000i q^{58} +(-4.50000 - 7.79423i) q^{59} +(3.50000 - 6.06218i) q^{61} +(-12.1244 + 7.00000i) q^{62} +8.00000 q^{64} +(-4.00000 - 2.00000i) q^{65} +(-8.66025 - 5.00000i) q^{67} +4.00000i q^{68} +(-9.85641 - 14.9282i) q^{70} +(0.500000 + 0.866025i) q^{71} +(8.66025 - 5.00000i) q^{73} +(-2.00000 - 3.46410i) q^{74} +(-1.00000 - 8.66025i) q^{76} +4.00000i q^{77} +(0.500000 + 0.866025i) q^{79} +(-4.92820 - 7.46410i) q^{80} +(3.46410 + 2.00000i) q^{82} +6.00000i q^{83} +(3.73205 - 2.46410i) q^{85} +(2.00000 - 3.46410i) q^{86} +(5.50000 - 9.52628i) q^{89} +(-4.00000 + 6.92820i) q^{91} +(10.3923 - 6.00000i) q^{92} -12.0000 q^{94} +(-7.46410 + 6.26795i) q^{95} +(5.19615 - 3.00000i) q^{97} +(-15.5885 + 9.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 2 q^{5}+O(q^{10}) $ 4 * q + 4 * q^4 - 2 * q^5	$ 4 q + 4 q^{4} - 2 q^{5} - 8 q^{10} + 4 q^{11} + 16 q^{14} + 8 q^{16} + 14 q^{19} - 8 q^{20} + 6 q^{25} + 16 q^{26} - 18 q^{29} - 28 q^{31} - 8 q^{34} - 16 q^{35} + 4 q^{41} + 4 q^{44} + 48 q^{46} - 36 q^{49} + 32 q^{50} - 2 q^{55} - 18 q^{59} + 14 q^{61} + 32 q^{64} - 16 q^{65} + 16 q^{70} + 2 q^{71} - 8 q^{74} - 4 q^{76} + 2 q^{79} + 8 q^{80} + 8 q^{85} + 8 q^{86} + 22 q^{89} - 16 q^{91} - 48 q^{94} - 16 q^{95}+O(q^{100}) $ 4 * q + 4 * q^4 - 2 * q^5 - 8 * q^10 + 4 * q^11 + 16 * q^14 + 8 * q^16 + 14 * q^19 - 8 * q^20 + 6 * q^25 + 16 * q^26 - 18 * q^29 - 28 * q^31 - 8 * q^34 - 16 * q^35 + 4 * q^41 + 4 * q^44 + 48 * q^46 - 36 * q^49 + 32 * q^50 - 2 * q^55 - 18 * q^59 + 14 * q^61 + 32 * q^64 - 16 * q^65 + 16 * q^70 + 2 * q^71 - 8 * q^74 - 4 * q^76 + 2 * q^79 + 8 * q^80 + 8 * q^85 + 8 * q^86 + 22 * q^89 - 16 * q^91 - 48 * q^94 - 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/855\mathbb{Z}\right)^\times$.

$n$	$172$	$191$	$496$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	1.73205	−	1.00000i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$2$	1.73205	−	1.00000i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$3$		0			0
$3$		0			0
$4$	1.00000	−	1.73205i	0.500000	−	0.866025i
$5$	−2.23205	+	0.133975i	−0.998203	+	0.0599153i
$5$	−2.23205	+	0.133975i	−0.998203	+	0.0599153i
$6$		0			0
$7$			4.00000i			1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$			4.00000i			1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$		0			0
$9$		0			0
$10$	−3.73205	+	2.46410i	−1.18018	+	0.779217i
$11$	1.00000			0.301511			0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000			0.301511			0.150756	+	0.988571i	$0.451829\pi$
$12$		0			0
$13$	1.73205	+	1.00000i	0.480384	+	0.277350i	0.720577	−	0.693375i	$-0.243877\pi$
$13$	1.73205	+	1.00000i	0.480384	+	0.277350i	−0.240192	+	0.970725i	$0.577210\pi$
$14$	4.00000	+	6.92820i	1.06904	+	1.85164i
$15$		0			0
$16$	2.00000	+	3.46410i	0.500000	+	0.866025i
$17$	−1.73205	+	1.00000i	−0.420084	+	0.242536i	−0.695113	−	0.718900i	$-0.744646\pi$
$17$	−1.73205	+	1.00000i	−0.420084	+	0.242536i	0.275029	+	0.961436i	$0.411312\pi$
$18$		0			0
$19$	3.50000	−	2.59808i	0.802955	−	0.596040i
$19$	3.50000	−	2.59808i	0.802955	−	0.596040i
$20$	−2.00000	+	4.00000i	−0.447214	+	0.894427i
$21$		0			0
$22$	1.73205	−	1.00000i	0.369274	−	0.213201i
$23$	5.19615	+	3.00000i	1.08347	+	0.625543i	0.931831	−	0.362892i	$-0.118211\pi$
$23$	5.19615	+	3.00000i	1.08347	+	0.625543i	0.151642	+	0.988436i	$0.451544\pi$
$24$		0			0
$25$	4.96410	−	0.598076i	0.992820	−	0.119615i
$26$	4.00000			0.784465
$27$		0			0
$28$	6.92820	+	4.00000i	1.30931	+	0.755929i
$29$	−4.50000	+	7.79423i	−0.835629	+	1.44735i	0.0578882	+	0.998323i	$0.481563\pi$
$29$	−4.50000	+	7.79423i	−0.835629	+	1.44735i	−0.893517	+	0.449029i	$0.851770\pi$
$30$		0			0
$31$	−7.00000			−1.25724			−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000			−1.25724			−0.628619	+	0.777714i	$0.716379\pi$
$32$	6.92820	+	4.00000i	1.22474	+	0.707107i
$33$		0			0
$34$	−2.00000	+	3.46410i	−0.342997	+	0.594089i
$35$	−0.535898	−	8.92820i	−0.0905834	−	1.50914i
$36$		0			0
$37$		−	2.00000i		−	0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$		−	2.00000i		−	0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	3.46410	−	8.00000i	0.561951	−	1.29777i
$39$		0			0
$40$		0			0
$41$	1.00000	+	1.73205i	0.156174	+	0.270501i	0.933486	−	0.358614i	$-0.116751\pi$
$41$	1.00000	+	1.73205i	0.156174	+	0.270501i	−0.777312	+	0.629115i	$0.783417\pi$
$42$		0			0
$43$	1.73205	−	1.00000i	0.264135	−	0.152499i	−0.362084	−	0.932145i	$-0.617935\pi$
$43$	1.73205	−	1.00000i	0.264135	−	0.152499i	0.626219	+	0.779647i	$0.284601\pi$
$44$	1.00000	−	1.73205i	0.150756	−	0.261116i
$45$		0			0
$46$	12.0000			1.76930
$47$	−5.19615	−	3.00000i	−0.757937	−	0.437595i	0.0706177	−	0.997503i	$-0.477503\pi$
$47$	−5.19615	−	3.00000i	−0.757937	−	0.437595i	−0.828554	+	0.559908i	$0.810836\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$	8.00000	−	6.00000i	1.13137	−	0.848528i
$51$		0			0
$52$	3.46410	−	2.00000i	0.480384	−	0.277350i
$53$	3.46410	+	2.00000i	0.475831	+	0.274721i	0.718677	−	0.695344i	$-0.244748\pi$
$53$	3.46410	+	2.00000i	0.475831	+	0.274721i	−0.242846	+	0.970065i	$0.578081\pi$
$54$		0			0
$55$	−2.23205	+	0.133975i	−0.300970	+	0.0180651i
$56$		0			0
$57$		0			0
$58$			18.0000i			2.36352i
$59$	−4.50000	−	7.79423i	−0.585850	−	1.01472i	−0.994769	−	0.102151i	$-0.967427\pi$
$59$	−4.50000	−	7.79423i	−0.585850	−	1.01472i	0.408919	−	0.912571i	$-0.365906\pi$
$60$		0			0
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	−0.550135	−	0.835076i	$-0.685424\pi$
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	0.998264	+	0.0588933i	$0.0187572\pi$
$62$	−12.1244	+	7.00000i	−1.53979	+	0.889001i
$63$		0			0
$64$	8.00000			1.00000
$65$	−4.00000	−	2.00000i	−0.496139	−	0.248069i
$66$		0			0
$67$	−8.66025	−	5.00000i	−1.05802	−	0.610847i	−0.133135	−	0.991098i	$-0.542504\pi$
$67$	−8.66025	−	5.00000i	−1.05802	−	0.610847i	−0.924883	+	0.380251i	$0.875838\pi$
$68$			4.00000i			0.485071i
$69$		0			0
$70$	−9.85641	−	14.9282i	−1.17807	−	1.78426i
$71$	0.500000	+	0.866025i	0.0593391	+	0.102778i	0.894169	−	0.447730i	$-0.147767\pi$
$71$	0.500000	+	0.866025i	0.0593391	+	0.102778i	−0.834830	+	0.550508i	$0.814434\pi$
$72$		0			0
$73$	8.66025	−	5.00000i	1.01361	−	0.585206i	0.101361	−	0.994850i	$-0.467680\pi$
$73$	8.66025	−	5.00000i	1.01361	−	0.585206i	0.912245	+	0.409644i	$0.134347\pi$
$74$	−2.00000	−	3.46410i	−0.232495	−	0.402694i
$75$		0			0
$76$	−1.00000	−	8.66025i	−0.114708	−	0.993399i
$77$			4.00000i			0.455842i
$78$		0			0
$79$	0.500000	+	0.866025i	0.0562544	+	0.0974355i	0.892781	−	0.450490i	$-0.148751\pi$
$79$	0.500000	+	0.866025i	0.0562544	+	0.0974355i	−0.836527	+	0.547926i	$0.815418\pi$
$80$	−4.92820	−	7.46410i	−0.550990	−	0.834512i
$81$		0			0
$82$	3.46410	+	2.00000i	0.382546	+	0.220863i
$83$			6.00000i			0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$			6.00000i			0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$		0			0
$85$	3.73205	−	2.46410i	0.404798	−	0.267269i
$86$	2.00000	−	3.46410i	0.215666	−	0.373544i
$87$		0			0
$88$		0			0
$89$	5.50000	−	9.52628i	0.582999	−	1.00978i	−0.412123	−	0.911128i	$-0.635213\pi$
$89$	5.50000	−	9.52628i	0.582999	−	1.00978i	0.995122	−	0.0986553i	$-0.0314541\pi$
$90$		0			0
$91$	−4.00000	+	6.92820i	−0.419314	+	0.726273i
$92$	10.3923	−	6.00000i	1.08347	−	0.625543i
$93$		0			0
$94$	−12.0000			−1.23771
$95$	−7.46410	+	6.26795i	−0.765801	+	0.643078i
$96$		0			0
$97$	5.19615	−	3.00000i	0.527589	−	0.304604i	−0.212445	−	0.977173i	$-0.568143\pi$
$97$	5.19615	−	3.00000i	0.527589	−	0.304604i	0.740034	+	0.672569i	$0.234809\pi$
$98$	−15.5885	+	9.00000i	−1.57467	+	0.909137i
$99$		0			0
$100$	3.92820	−	9.19615i	0.392820	−	0.919615i
$101$	7.50000	−	12.9904i	0.746278	−	1.29259i	−0.203317	−	0.979113i	$-0.565172\pi$
$101$	7.50000	−	12.9904i	0.746278	−	1.29259i	0.949595	−	0.313478i	$-0.101494\pi$
$102$		0			0
$103$			16.0000i			1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$			16.0000i			1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$		0			0
$105$		0			0
$106$	8.00000			0.777029
$107$		−	10.0000i		−	0.966736i	−0.875417	−	0.483368i	$-0.839413\pi$
$107$		−	10.0000i		−	0.966736i	0.875417	−	0.483368i	$-0.160587\pi$
$108$		0			0
$109$	7.50000	+	12.9904i	0.718370	+	1.24425i	0.961645	+	0.274296i	$0.0884447\pi$
$109$	7.50000	+	12.9904i	0.718370	+	1.24425i	−0.243276	+	0.969957i	$0.578222\pi$
$110$	−3.73205	+	2.46410i	−0.355837	+	0.234943i
$111$		0			0
$112$	−13.8564	+	8.00000i	−1.30931	+	0.755929i
$113$			12.0000i			1.12887i	0.825479	+	0.564433i	$0.190905\pi$
$113$			12.0000i			1.12887i	−0.825479	+	0.564433i	$0.809095\pi$
$114$		0			0
$115$	−12.0000	−	6.00000i	−1.11901	−	0.559503i
$116$	9.00000	+	15.5885i	0.835629	+	1.44735i
$117$		0			0
$118$	−15.5885	−	9.00000i	−1.43503	−	0.828517i
$119$	−4.00000	−	6.92820i	−0.366679	−	0.635107i
$120$		0			0
$121$	−10.0000			−0.909091
$122$		−	14.0000i		−	1.26750i
$123$		0			0
$124$	−7.00000	+	12.1244i	−0.628619	+	1.08880i
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$	−5.19615	−	3.00000i	−0.461084	−	0.266207i	0.251416	−	0.967879i	$-0.419104\pi$
$127$	−5.19615	−	3.00000i	−0.461084	−	0.266207i	−0.712500	+	0.701672i	$0.752437\pi$
$128$		0			0
$129$		0			0
$130$	−8.92820	+	0.535898i	−0.783055	+	0.0470014i
$131$	6.00000	+	10.3923i	0.524222	+	0.907980i	0.999602	+	0.0281993i	$0.00897729\pi$
$131$	6.00000	+	10.3923i	0.524222	+	0.907980i	−0.475380	+	0.879781i	$0.657689\pi$
$132$		0			0
$133$	10.3923	+	14.0000i	0.901127	+	1.21395i
$134$	−20.0000			−1.72774
$135$		0			0
$136$		0			0
$137$	−10.3923	−	6.00000i	−0.887875	−	0.512615i	−0.0146279	−	0.999893i	$-0.504656\pi$
$137$	−10.3923	−	6.00000i	−0.887875	−	0.512615i	−0.873247	+	0.487278i	$0.837990\pi$
$138$		0			0
$139$	10.0000	−	17.3205i	0.848189	−	1.46911i	−0.0346338	−	0.999400i	$-0.511026\pi$
$139$	10.0000	−	17.3205i	0.848189	−	1.46911i	0.882823	−	0.469706i	$-0.155640\pi$
$140$	−16.0000	−	8.00000i	−1.35225	−	0.676123i
$141$		0			0
$142$	1.73205	+	1.00000i	0.145350	+	0.0839181i
$143$	1.73205	+	1.00000i	0.144841	+	0.0836242i
$144$		0			0
$145$	9.00000	−	18.0000i	0.747409	−	1.49482i
$146$	10.0000	−	17.3205i	0.827606	−	1.43346i
$147$		0			0
$148$	−3.46410	−	2.00000i	−0.284747	−	0.164399i
$149$	0.500000	+	0.866025i	0.0409616	+	0.0709476i	0.885779	−	0.464107i	$-0.153625\pi$
$149$	0.500000	+	0.866025i	0.0409616	+	0.0709476i	−0.844818	+	0.535054i	$0.820291\pi$
$150$		0			0
$151$	9.00000			0.732410			0.366205	−	0.930534i	$-0.380657\pi$
$151$	9.00000			0.732410			0.366205	+	0.930534i	$0.380657\pi$
$152$		0			0
$153$		0			0
$154$	4.00000	+	6.92820i	0.322329	+	0.558291i
$155$	15.6244	−	0.937822i	1.25498	−	0.0753277i
$156$		0			0
$157$	−3.46410	+	2.00000i	−0.276465	+	0.159617i	−0.631822	−	0.775113i	$-0.717693\pi$
$157$	−3.46410	+	2.00000i	−0.276465	+	0.159617i	0.355357	+	0.934731i	$0.384359\pi$
$158$	1.73205	+	1.00000i	0.137795	+	0.0795557i
$159$		0			0
$160$	−16.0000	−	8.00000i	−1.26491	−	0.632456i
$161$	−12.0000	+	20.7846i	−0.945732	+	1.63806i
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	4.00000			0.312348
$165$		0			0
$166$	6.00000	+	10.3923i	0.465690	+	0.806599i
$167$	10.3923	+	6.00000i	0.804181	+	0.464294i	0.844931	−	0.534875i	$-0.179641\pi$
$167$	10.3923	+	6.00000i	0.804181	+	0.464294i	−0.0407502	+	0.999169i	$0.512975\pi$
$168$		0			0
$169$	−4.50000	−	7.79423i	−0.346154	−	0.599556i
$170$	4.00000	−	8.00000i	0.306786	−	0.613572i
$171$		0			0
$172$		−	4.00000i		−	0.304997i
$173$	20.7846	−	12.0000i	1.58022	−	0.912343i	0.585399	−	0.810745i	$-0.300938\pi$
$173$	20.7846	−	12.0000i	1.58022	−	0.912343i	0.994826	−	0.101598i	$-0.0323955\pi$
$174$		0			0
$175$	2.39230	+	19.8564i	0.180841	+	1.50100i
$176$	2.00000	+	3.46410i	0.150756	+	0.261116i
$177$		0			0
$178$		−	22.0000i		−	1.64897i
$179$	15.0000			1.12115			0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000			1.12115			0.560576	+	0.828103i	$0.310580\pi$
$180$		0			0
$181$	3.00000	−	5.19615i	0.222988	−	0.386227i	−0.732726	−	0.680524i	$-0.761752\pi$
$181$	3.00000	−	5.19615i	0.222988	−	0.386227i	0.955714	+	0.294297i	$0.0950855\pi$
$182$			16.0000i			1.18600i
$183$		0			0
$184$		0			0
$185$	0.267949	+	4.46410i	0.0197000	+	0.328207i
$186$		0			0
$187$	−1.73205	+	1.00000i	−0.126660	+	0.0731272i
$188$	−10.3923	+	6.00000i	−0.757937	+	0.437595i
$189$		0			0
$190$	−6.66025	+	18.3205i	−0.483186	+	1.32911i
$191$	3.00000			0.217072			0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000			0.217072			0.108536	+	0.994092i	$0.465384\pi$
$192$		0			0
$193$	13.8564	−	8.00000i	0.997406	−	0.575853i	0.0899262	−	0.995948i	$-0.471337\pi$
$193$	13.8564	−	8.00000i	0.997406	−	0.575853i	0.907480	+	0.420096i	$0.138004\pi$
$194$	6.00000	−	10.3923i	0.430775	−	0.746124i
$195$		0			0
$196$	−9.00000	+	15.5885i	−0.642857	+	1.11346i
$197$		−	18.0000i		−	1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$		−	18.0000i		−	1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$		0			0
$199$	6.50000	−	11.2583i	0.460773	−	0.798082i	−0.538227	−	0.842800i	$-0.680906\pi$
$199$	6.50000	−	11.2583i	0.460773	−	0.798082i	0.999000	+	0.0447181i	$0.0142390\pi$
$200$		0			0
$201$		0			0
$202$		−	30.0000i		−	2.11079i
$203$	−31.1769	−	18.0000i	−2.18819	−	1.26335i
$204$		0			0
$205$	−2.46410	−	3.73205i	−0.172100	−	0.260658i
$206$	16.0000	+	27.7128i	1.11477	+	1.93084i
$207$		0			0
$208$			8.00000i			0.554700i
$209$	3.50000	−	2.59808i	0.242100	−	0.179713i
$210$		0			0
$211$	−2.50000	−	4.33013i	−0.172107	−	0.298098i	0.767049	−	0.641588i	$-0.221724\pi$
$211$	−2.50000	−	4.33013i	−0.172107	−	0.298098i	−0.939156	+	0.343490i	$0.888391\pi$
$212$	6.92820	−	4.00000i	0.475831	−	0.274721i
$213$		0			0
$214$	−10.0000	−	17.3205i	−0.683586	−	1.18401i
$215$	−3.73205	+	2.46410i	−0.254524	+	0.168050i
$216$		0			0
$217$		−	28.0000i		−	1.90076i
$218$	25.9808	+	15.0000i	1.75964	+	1.01593i
$219$		0			0
$220$	−2.00000	+	4.00000i	−0.134840	+	0.269680i
$221$	−4.00000			−0.269069
$222$		0			0
$223$	−1.73205	+	1.00000i	−0.115987	+	0.0669650i	−0.556871	−	0.830599i	$-0.687998\pi$
$223$	−1.73205	+	1.00000i	−0.115987	+	0.0669650i	0.440884	+	0.897564i	$0.354665\pi$
$224$	−16.0000	+	27.7128i	−1.06904	+	1.85164i
$225$		0			0
$226$	12.0000	+	20.7846i	0.798228	+	1.38257i
$227$			14.0000i			0.929213i	0.885517	+	0.464606i	$0.153804\pi$
$227$			14.0000i			0.929213i	−0.885517	+	0.464606i	$0.846196\pi$
$228$		0			0
$229$	−17.0000			−1.12339			−0.561696	−	0.827344i	$-0.689851\pi$
$229$	−17.0000			−1.12339			−0.561696	+	0.827344i	$0.689851\pi$
$230$	−26.7846	+	1.60770i	−1.76612	+	0.106008i
$231$		0			0
$232$		0			0
$233$	6.92820	−	4.00000i	0.453882	−	0.262049i	−0.255586	−	0.966786i	$-0.582269\pi$
$233$	6.92820	−	4.00000i	0.453882	−	0.262049i	0.709468	+	0.704737i	$0.248935\pi$
$234$		0			0
$235$	12.0000	+	6.00000i	0.782794	+	0.391397i
$236$	−18.0000			−1.17170
$237$		0			0
$238$	−13.8564	−	8.00000i	−0.898177	−	0.518563i
$239$	−19.0000			−1.22901			−0.614504	−	0.788914i	$-0.710644\pi$
$239$	−19.0000			−1.22901			−0.614504	+	0.788914i	$0.710644\pi$
$240$		0			0
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	−0.881680	−	0.471848i	$-0.843587\pi$
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	0.849472	+	0.527633i	$0.176921\pi$
$242$	−17.3205	+	10.0000i	−1.11340	+	0.642824i
$243$		0			0
$244$	−7.00000	−	12.1244i	−0.448129	−	0.776182i
$245$	20.0885	−	1.20577i	1.28340	−	0.0770339i
$246$		0			0
$247$	8.66025	−	1.00000i	0.551039	−	0.0636285i
$248$		0			0
$249$		0			0
$250$	−17.0526	+	14.4641i	−1.07850	+	0.914790i
$251$	6.50000	−	11.2583i	0.410276	−	0.710620i	−0.584643	−	0.811290i	$-0.698766\pi$
$251$	6.50000	−	11.2583i	0.410276	−	0.710620i	0.994920	+	0.100671i	$0.0320989\pi$
$252$		0			0
$253$	5.19615	+	3.00000i	0.326679	+	0.188608i
$254$	−12.0000			−0.752947
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	5.19615	+	3.00000i	0.324127	+	0.187135i	0.653231	−	0.757159i	$-0.273413\pi$
$257$	5.19615	+	3.00000i	0.324127	+	0.187135i	−0.329104	+	0.944294i	$0.606747\pi$
$258$		0			0
$259$	8.00000			0.497096
$260$	−7.46410	+	4.92820i	−0.462904	+	0.305634i
$261$		0			0
$262$	20.7846	+	12.0000i	1.28408	+	0.741362i
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$	−8.00000	−	4.00000i	−0.491436	−	0.245718i
$266$	32.0000	+	13.8564i	1.96205	+	0.849591i
$267$		0			0
$268$	−17.3205	+	10.0000i	−1.05802	+	0.610847i
$269$	−1.50000	−	2.59808i	−0.0914566	−	0.158408i	0.816668	−	0.577108i	$-0.195819\pi$
$269$	−1.50000	−	2.59808i	−0.0914566	−	0.158408i	−0.908124	+	0.418701i	$0.862486\pi$
$270$		0			0
$271$	−1.50000	−	2.59808i	−0.0911185	−	0.157822i	0.816864	−	0.576831i	$-0.195711\pi$
$271$	−1.50000	−	2.59808i	−0.0911185	−	0.157822i	−0.907982	+	0.419009i	$0.862378\pi$
$272$	−6.92820	−	4.00000i	−0.420084	−	0.242536i
$273$		0			0
$274$	−24.0000			−1.44989
$275$	4.96410	−	0.598076i	0.299347	−	0.0360654i
$276$		0			0
$277$			28.0000i			1.68236i	0.540758	+	0.841178i	$0.318138\pi$
$277$			28.0000i			1.68236i	−0.540758	+	0.841178i	$0.681862\pi$
$278$		−	40.0000i		−	2.39904i
$279$		0			0
$280$		0			0
$281$	5.00000	−	8.66025i	0.298275	−	0.516627i	−0.677466	−	0.735554i	$-0.736922\pi$
$281$	5.00000	−	8.66025i	0.298275	−	0.516627i	0.975741	+	0.218926i	$0.0702554\pi$
$282$		0			0
$283$	12.1244	−	7.00000i	0.720718	−	0.416107i	−0.0942988	−	0.995544i	$-0.530061\pi$
$283$	12.1244	−	7.00000i	0.720718	−	0.416107i	0.815017	+	0.579437i	$0.196728\pi$
$284$	2.00000			0.118678
$285$		0			0
$286$	4.00000			0.236525
$287$	−6.92820	+	4.00000i	−0.408959	+	0.236113i
$288$		0			0
$289$	−6.50000	+	11.2583i	−0.382353	+	0.662255i
$290$	−2.41154	−	40.1769i	−0.141611	−	2.35927i
$291$		0			0
$292$		−	20.0000i		−	1.17041i
$293$		−	4.00000i		−	0.233682i	−0.993151	−	0.116841i	$-0.962723\pi$
$293$		−	4.00000i		−	0.233682i	0.993151	−	0.116841i	$-0.0372769\pi$
$294$		0			0
$295$	11.0885	+	16.7942i	0.645595	+	0.977798i
$296$		0			0
$297$		0			0
$298$	1.73205	+	1.00000i	0.100335	+	0.0579284i
$299$	6.00000	+	10.3923i	0.346989	+	0.601003i
$300$		0			0
$301$	4.00000	+	6.92820i	0.230556	+	0.399335i
$302$	15.5885	−	9.00000i	0.897015	−	0.517892i
$303$		0			0
$304$	16.0000	+	6.92820i	0.917663	+	0.397360i
$305$	−7.00000	+	14.0000i	−0.400819	+	0.801638i
$306$		0			0
$307$	−13.8564	+	8.00000i	−0.790827	+	0.456584i	−0.840254	−	0.542194i	$-0.817594\pi$
$307$	−13.8564	+	8.00000i	−0.790827	+	0.456584i	0.0494267	+	0.998778i	$0.484261\pi$
$308$	6.92820	+	4.00000i	0.394771	+	0.227921i
$309$		0			0
$310$	26.1244	−	17.2487i	1.48376	−	0.979661i
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−25.9808	−	15.0000i	−1.46852	−	0.847850i	−0.469142	−	0.883123i	$-0.655437\pi$
$313$	−25.9808	−	15.0000i	−1.46852	−	0.847850i	−0.999378	+	0.0352727i	$0.988770\pi$
$314$	−4.00000	+	6.92820i	−0.225733	+	0.390981i
$315$		0			0
$316$	2.00000			0.112509
$317$	1.73205	+	1.00000i	0.0972817	+	0.0561656i	0.547852	−	0.836576i	$-0.315446\pi$
$317$	1.73205	+	1.00000i	0.0972817	+	0.0561656i	−0.450570	+	0.892741i	$0.648779\pi$
$318$		0			0
$319$	−4.50000	+	7.79423i	−0.251952	+	0.436393i
$320$	−17.8564	+	1.07180i	−0.998203	+	0.0599153i
$321$		0			0
$322$			48.0000i			2.67494i
$323$	−3.46410	+	8.00000i	−0.192748	+	0.445132i
$324$		0			0
$325$	9.19615	+	3.92820i	0.510111	+	0.217898i
$326$	4.00000	+	6.92820i	0.221540	+	0.383718i
$327$		0			0
$328$		0			0
$329$	12.0000	−	20.7846i	0.661581	−	1.14589i
$330$		0			0
$331$	20.0000			1.09930			0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000			1.09930			0.549650	+	0.835395i	$0.314761\pi$
$332$	10.3923	+	6.00000i	0.570352	+	0.329293i
$333$		0			0
$334$	24.0000			1.31322
$335$	20.0000	+	10.0000i	1.09272	+	0.546358i
$336$		0			0
$337$	−29.4449	+	17.0000i	−1.60396	+	0.926049i	−0.613280	+	0.789865i	$0.710150\pi$
$337$	−29.4449	+	17.0000i	−1.60396	+	0.926049i	−0.990684	+	0.136184i	$0.956516\pi$
$338$	−15.5885	−	9.00000i	−0.847900	−	0.489535i
$339$		0			0
$340$	−0.535898	−	8.92820i	−0.0290632	−	0.484200i
$341$	−7.00000			−0.379071
$342$		0			0
$343$		−	8.00000i		−	0.431959i
$344$		0			0
$345$		0			0
$346$	24.0000	−	41.5692i	1.29025	−	2.23478i
$347$	−10.3923	+	6.00000i	−0.557888	+	0.322097i	−0.752297	−	0.658824i	$-0.771054\pi$
$347$	−10.3923	+	6.00000i	−0.557888	+	0.322097i	0.194409	+	0.980921i	$0.437721\pi$
$348$		0			0
$349$	−14.0000			−0.749403			−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000			−0.749403			−0.374701	+	0.927146i	$0.622255\pi$
$350$	24.0000	+	32.0000i	1.28285	+	1.71047i
$351$		0			0
$352$	6.92820	+	4.00000i	0.369274	+	0.213201i
$353$			8.00000i			0.425797i	0.977074	+	0.212899i	$0.0682904\pi$
$353$			8.00000i			0.425797i	−0.977074	+	0.212899i	$0.931710\pi$
$354$		0			0
$355$	−1.23205	−	1.86603i	−0.0653905	−	0.0990383i
$356$	−11.0000	−	19.0526i	−0.582999	−	1.00978i
$357$		0			0
$358$	25.9808	−	15.0000i	1.37313	−	0.792775i
$359$	10.0000	+	17.3205i	0.527780	+	0.914141i	0.999476	+	0.0323801i	$0.0103087\pi$
$359$	10.0000	+	17.3205i	0.527780	+	0.914141i	−0.471696	+	0.881761i	$0.656358\pi$
$360$		0			0
$361$	5.50000	−	18.1865i	0.289474	−	0.957186i
$362$		−	12.0000i		−	0.630706i
$363$		0			0
$364$	8.00000	+	13.8564i	0.419314	+	0.726273i
$365$	−18.6603	+	12.3205i	−0.976722	+	0.644885i
$366$		0			0
$367$	−13.8564	−	8.00000i	−0.723299	−	0.417597i	0.0926670	−	0.995697i	$-0.470461\pi$
$367$	−13.8564	−	8.00000i	−0.723299	−	0.417597i	−0.815966	+	0.578101i	$0.803794\pi$
$368$			24.0000i			1.25109i
$369$		0			0
$370$	4.92820	+	7.46410i	0.256205	+	0.388040i
$371$	−8.00000	+	13.8564i	−0.415339	+	0.719389i
$372$		0			0
$373$		−	12.0000i		−	0.621336i	−0.950518	−	0.310668i	$-0.899447\pi$
$373$		−	12.0000i		−	0.621336i	0.950518	−	0.310668i	$-0.100553\pi$
$374$	−2.00000	+	3.46410i	−0.103418	+	0.179124i
$375$		0			0
$376$		0			0
$377$	−15.5885	+	9.00000i	−0.802846	+	0.463524i
$378$		0			0
$379$	29.0000			1.48963			0.744815	−	0.667271i	$-0.232538\pi$
$379$	29.0000			1.48963			0.744815	+	0.667271i	$0.232538\pi$
$380$	3.39230	+	19.1962i	0.174022	+	0.984742i
$381$		0			0
$382$	5.19615	−	3.00000i	0.265858	−	0.153493i
$383$	−5.19615	+	3.00000i	−0.265511	+	0.153293i	−0.626846	−	0.779143i	$-0.715654\pi$
$383$	−5.19615	+	3.00000i	−0.265511	+	0.153293i	0.361335	+	0.932436i	$0.382321\pi$
$384$		0			0
$385$	−0.535898	−	8.92820i	−0.0273119	−	0.455023i
$386$	16.0000	−	27.7128i	0.814379	−	1.41055i
$387$		0			0
$388$		−	12.0000i		−	0.609208i
$389$	16.5000	−	28.5788i	0.836583	−	1.44900i	−0.0561516	−	0.998422i	$-0.517883\pi$
$389$	16.5000	−	28.5788i	0.836583	−	1.44900i	0.892735	−	0.450582i	$-0.148784\pi$
$390$		0			0
$391$	−12.0000			−0.606866
$392$		0			0
$393$		0			0
$394$	−18.0000	−	31.1769i	−0.906827	−	1.57067i
$395$	−1.23205	−	1.86603i	−0.0619912	−	0.0938899i
$396$		0			0
$397$	6.92820	−	4.00000i	0.347717	−	0.200754i	−0.315963	−	0.948772i	$-0.602327\pi$
$397$	6.92820	−	4.00000i	0.347717	−	0.200754i	0.663679	+	0.748017i	$0.268994\pi$
$398$		−	26.0000i		−	1.30326i
$399$		0			0
$400$	12.0000	+	16.0000i	0.600000	+	0.800000i
$401$	1.50000	+	2.59808i	0.0749064	+	0.129742i	0.901046	−	0.433724i	$-0.142801\pi$
$401$	1.50000	+	2.59808i	0.0749064	+	0.129742i	−0.826139	+	0.563466i	$0.809468\pi$
$402$		0			0
$403$	−12.1244	−	7.00000i	−0.603957	−	0.348695i
$404$	−15.0000	−	25.9808i	−0.746278	−	1.29259i
$405$		0			0
$406$	−72.0000			−3.57330
$407$		−	2.00000i		−	0.0991363i
$408$		0			0
$409$	−2.50000	+	4.33013i	−0.123617	+	0.214111i	−0.921192	−	0.389109i	$-0.872783\pi$
$409$	−2.50000	+	4.33013i	−0.123617	+	0.214111i	0.797574	+	0.603220i	$0.206116\pi$
$410$	−8.00000	−	4.00000i	−0.395092	−	0.197546i
$411$		0			0
$412$	27.7128	+	16.0000i	1.36531	+	0.788263i
$413$	31.1769	−	18.0000i	1.53412	−	0.885722i
$414$		0			0
$415$	−0.803848	−	13.3923i	−0.0394593	−	0.657402i
$416$	8.00000	+	13.8564i	0.392232	+	0.679366i
$417$		0			0
$418$	3.46410	−	8.00000i	0.169435	−	0.391293i
$419$	9.00000			0.439679			0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000			0.439679			0.219839	+	0.975536i	$0.429447\pi$
$420$		0			0
$421$	7.50000	+	12.9904i	0.365528	+	0.633112i	0.988861	−	0.148844i	$-0.0475552\pi$
$421$	7.50000	+	12.9904i	0.365528	+	0.633112i	−0.623333	+	0.781956i	$0.714222\pi$
$422$	−8.66025	−	5.00000i	−0.421575	−	0.243396i
$423$		0			0
$424$		0			0
$425$	−8.00000	+	6.00000i	−0.388057	+	0.291043i
$426$		0			0
$427$	24.2487	+	14.0000i	1.17348	+	0.677507i
$428$	−17.3205	−	10.0000i	−0.837218	−	0.483368i
$429$		0			0
$430$	−4.00000	+	8.00000i	−0.192897	+	0.385794i
$431$	10.5000	−	18.1865i	0.505767	−	0.876014i	−0.494211	−	0.869342i	$-0.664543\pi$
$431$	10.5000	−	18.1865i	0.505767	−	0.876014i	0.999978	−	0.00667224i	$-0.00212386\pi$
$432$		0			0
$433$	3.46410	+	2.00000i	0.166474	+	0.0961139i	0.580922	−	0.813959i	$-0.302692\pi$
$433$	3.46410	+	2.00000i	0.166474	+	0.0961139i	−0.414448	+	0.910073i	$0.636025\pi$
$434$	−28.0000	−	48.4974i	−1.34404	−	2.32795i
$435$		0			0
$436$	30.0000			1.43674
$437$	25.9808	−	3.00000i	1.24283	−	0.143509i
$438$		0			0
$439$	−6.50000	−	11.2583i	−0.310228	−	0.537331i	0.668184	−	0.743996i	$-0.267072\pi$
$439$	−6.50000	−	11.2583i	−0.310228	−	0.537331i	−0.978412	+	0.206666i	$0.933739\pi$
$440$		0			0
$441$		0			0
$442$	−6.92820	+	4.00000i	−0.329541	+	0.190261i
$443$	3.46410	+	2.00000i	0.164584	+	0.0950229i	0.580030	−	0.814595i	$-0.303041\pi$
$443$	3.46410	+	2.00000i	0.164584	+	0.0950229i	−0.415445	+	0.909618i	$0.636374\pi$
$444$		0			0
$445$	−11.0000	+	22.0000i	−0.521450	+	1.04290i
$446$	−2.00000	+	3.46410i	−0.0947027	+	0.164030i
$447$		0			0
$448$			32.0000i			1.51186i
$449$	31.0000			1.46298			0.731490	−	0.681852i	$-0.238825\pi$
$449$	31.0000			1.46298			0.731490	+	0.681852i	$0.238825\pi$
$450$		0			0
$451$	1.00000	+	1.73205i	0.0470882	+	0.0815591i
$452$	20.7846	+	12.0000i	0.977626	+	0.564433i
$453$		0			0
$454$	14.0000	+	24.2487i	0.657053	+	1.13805i
$455$	8.00000	−	16.0000i	0.375046	−	0.750092i
$456$		0			0
$457$		−	34.0000i		−	1.59045i	−0.606313	−	0.795226i	$-0.707352\pi$
$457$		−	34.0000i		−	1.59045i	0.606313	−	0.795226i	$-0.292648\pi$
$458$	−29.4449	+	17.0000i	−1.37587	+	0.794358i
$459$		0			0
$460$	−22.3923	+	14.7846i	−1.04405	+	0.689336i
$461$	−12.5000	−	21.6506i	−0.582183	−	1.00837i	−0.995220	−	0.0976564i	$-0.968865\pi$
$461$	−12.5000	−	21.6506i	−0.582183	−	1.00837i	0.413037	−	0.910714i	$-0.364468\pi$
$462$		0			0
$463$		−	4.00000i		−	0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$		−	4.00000i		−	0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	−36.0000			−1.67126
$465$		0			0
$466$	8.00000	−	13.8564i	0.370593	−	0.641886i
$467$			10.0000i			0.462745i	0.972865	+	0.231372i	$0.0743216\pi$
$467$			10.0000i			0.462745i	−0.972865	+	0.231372i	$0.925678\pi$
$468$		0			0
$469$	20.0000	−	34.6410i	0.923514	−	1.59957i
$470$	26.7846	−	1.60770i	1.23548	−	0.0741574i
$471$		0			0
$472$		0			0
$473$	1.73205	−	1.00000i	0.0796398	−	0.0459800i
$474$		0			0
$475$	15.8205	−	14.9904i	0.725895	−	0.687806i
$476$	−16.0000			−0.733359
$477$		0			0
$478$	−32.9090	+	19.0000i	−1.50522	+	0.869040i
$479$	−7.50000	+	12.9904i	−0.342684	+	0.593546i	−0.984930	−	0.172953i	$-0.944669\pi$
$479$	−7.50000	+	12.9904i	−0.342684	+	0.593546i	0.642246	+	0.766498i	$0.278003\pi$
$480$		0			0
$481$	2.00000	−	3.46410i	0.0911922	−	0.157949i
$482$			2.00000i			0.0910975i
$483$		0			0
$484$	−10.0000	+	17.3205i	−0.454545	+	0.787296i
$485$	−11.1962	+	7.39230i	−0.508391	+	0.335667i
$486$		0			0
$487$			38.0000i			1.72194i	0.508652	+	0.860972i	$0.330144\pi$
$487$			38.0000i			1.72194i	−0.508652	+	0.860972i	$0.669856\pi$
$488$		0			0
$489$		0			0
$490$	33.5885	−	22.1769i	1.51737	−	1.00185i
$491$	6.50000	+	11.2583i	0.293341	+	0.508081i	0.974598	−	0.223963i	$-0.0718996\pi$
$491$	6.50000	+	11.2583i	0.293341	+	0.508081i	−0.681257	+	0.732045i	$0.738566\pi$
$492$		0			0
$493$		−	18.0000i		−	0.810679i
$494$	14.0000	−	10.3923i	0.629890	−	0.467572i
$495$		0			0
$496$	−14.0000	−	24.2487i	−0.628619	−	1.08880i
$497$	−3.46410	+	2.00000i	−0.155386	+	0.0897123i
$498$		0			0
$499$	14.0000	+	24.2487i	0.626726	+	1.08552i	0.988204	+	0.153141i	$0.0489388\pi$
$499$	14.0000	+	24.2487i	0.626726	+	1.08552i	−0.361478	+	0.932381i	$0.617728\pi$
$500$	−7.53590	+	21.0526i	−0.337016	+	0.941499i
$501$		0			0
$502$		−	26.0000i		−	1.16044i
$503$	−22.5167	−	13.0000i	−1.00397	−	0.579641i	−0.0945483	−	0.995520i	$-0.530141\pi$
$503$	−22.5167	−	13.0000i	−1.00397	−	0.579641i	−0.909420	+	0.415879i	$0.863474\pi$
$504$		0			0
$505$	−15.0000	+	30.0000i	−0.667491	+	1.33498i
$506$	12.0000			0.533465
$507$		0			0
$508$	−10.3923	+	6.00000i	−0.461084	+	0.266207i
$509$	−9.00000	+	15.5885i	−0.398918	+	0.690946i	−0.993593	−	0.113020i	$-0.963948\pi$
$509$	−9.00000	+	15.5885i	−0.398918	+	0.690946i	0.594675	+	0.803966i	$0.297281\pi$
$510$		0			0
$511$	20.0000	+	34.6410i	0.884748	+	1.53243i
$512$			32.0000i			1.41421i
$513$		0			0
$514$	12.0000			0.529297
$515$	−2.14359	−	35.7128i	−0.0944580	−	1.57369i
$516$		0			0
$517$	−5.19615	−	3.00000i	−0.228527	−	0.131940i
$518$	13.8564	−	8.00000i	0.608816	−	0.351500i
$519$		0			0
$520$		0			0
$521$	−15.0000			−0.657162			−0.328581	−	0.944476i	$-0.606570\pi$
$521$	−15.0000			−0.657162			−0.328581	+	0.944476i	$0.606570\pi$
$522$		0			0
$523$	−13.8564	−	8.00000i	−0.605898	−	0.349816i	0.165460	−	0.986216i	$-0.447089\pi$
$523$	−13.8564	−	8.00000i	−0.605898	−	0.349816i	−0.771358	+	0.636401i	$0.780422\pi$
$524$	24.0000			1.04844
$525$		0			0
$526$		0			0
$527$	12.1244	−	7.00000i	0.528145	−	0.304925i
$528$		0			0
$529$	6.50000	+	11.2583i	0.282609	+	0.489493i
$530$	−17.8564	+	1.07180i	−0.775633	+	0.0465559i
$531$		0			0
$532$	34.6410	−	4.00000i	1.50188	−	0.173422i
$533$			4.00000i			0.173259i
$534$		0			0
$535$	1.33975	+	22.3205i	0.0579223	+	0.965000i
$536$		0			0
$537$		0			0
$538$	−5.19615	−	3.00000i	−0.224022	−	0.129339i
$539$	−9.00000			−0.387657
$540$		0			0
$541$	−11.5000	+	19.9186i	−0.494424	+	0.856367i	−0.999979	−	0.00642713i	$-0.997954\pi$
$541$	−11.5000	+	19.9186i	−0.494424	+	0.856367i	0.505556	+	0.862794i	$0.331288\pi$
$542$	−5.19615	−	3.00000i	−0.223194	−	0.128861i
$543$		0			0
$544$	−16.0000			−0.685994
$545$	−18.4808	−	27.9904i	−0.791629	−	1.19898i
$546$		0			0
$547$	19.0526	+	11.0000i	0.814629	+	0.470326i	0.848561	−	0.529098i	$-0.177470\pi$
$547$	19.0526	+	11.0000i	0.814629	+	0.470326i	−0.0339321	+	0.999424i	$0.510803\pi$
$548$	−20.7846	+	12.0000i	−0.887875	+	0.512615i
$549$		0			0
$550$	8.00000	−	6.00000i	0.341121	−	0.255841i
$551$	4.50000	+	38.9711i	0.191706	+	1.66023i
$552$		0			0
$553$	−3.46410	+	2.00000i	−0.147309	+	0.0850487i
$554$	28.0000	+	48.4974i	1.18961	+	2.06046i
$555$		0			0
$556$	−20.0000	−	34.6410i	−0.848189	−	1.46911i
$557$	−12.1244	−	7.00000i	−0.513725	−	0.296600i	0.220638	−	0.975356i	$-0.429186\pi$
$557$	−12.1244	−	7.00000i	−0.513725	−	0.296600i	−0.734364	+	0.678756i	$0.762519\pi$
$558$		0			0
$559$	4.00000			0.169182
$560$	29.8564	−	19.7128i	1.26166	−	0.833018i
$561$		0			0
$562$		−	20.0000i		−	0.843649i
$563$			24.0000i			1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$			24.0000i			1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$		0			0
$565$	−1.60770	−	26.7846i	−0.0676362	−	1.12684i
$566$	14.0000	−	24.2487i	0.588464	−	1.01925i
$567$		0			0
$568$		0			0
$569$	15.0000			0.628833			0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000			0.628833			0.314416	+	0.949285i	$0.398191\pi$
$570$		0			0
$571$	13.0000			0.544033			0.272017	−	0.962293i	$-0.412309\pi$
$571$	13.0000			0.544033			0.272017	+	0.962293i	$0.412309\pi$
$572$	3.46410	−	2.00000i	0.144841	−	0.0836242i
$573$		0			0
$574$	−8.00000	+	13.8564i	−0.333914	+	0.578355i
$575$	27.5885	+	11.7846i	1.15052	+	0.491452i
$576$		0			0
$577$		−	6.00000i		−	0.249783i	−0.992170	−	0.124892i	$-0.960142\pi$
$577$		−	6.00000i		−	0.249783i	0.992170	−	0.124892i	$-0.0398583\pi$
$578$			26.0000i			1.08146i
$579$		0			0
$580$	−22.1769	−	33.5885i	−0.920846	−	1.39468i
$581$	−24.0000			−0.995688
$582$		0			0
$583$	3.46410	+	2.00000i	0.143468	+	0.0828315i
$584$		0			0
$585$		0			0
$586$	−4.00000	−	6.92820i	−0.165238	−	0.286201i
$587$	−20.7846	+	12.0000i	−0.857873	+	0.495293i	−0.863299	−	0.504692i	$-0.831606\pi$
$587$	−20.7846	+	12.0000i	−0.857873	+	0.495293i	0.00542667	+	0.999985i	$0.498273\pi$
$588$		0			0
$589$	−24.5000	+	18.1865i	−1.00950	+	0.749363i
$590$	36.0000	+	18.0000i	1.48210	+	0.741048i
$591$		0			0
$592$	6.92820	−	4.00000i	0.284747	−	0.164399i
$593$	−27.7128	−	16.0000i	−1.13803	−	0.657041i	−0.192087	−	0.981378i	$-0.561526\pi$
$593$	−27.7128	−	16.0000i	−1.13803	−	0.657041i	−0.945942	+	0.324337i	$0.894859\pi$
$594$		0			0
$595$	9.85641	+	14.9282i	0.404073	+	0.611997i
$596$	2.00000			0.0819232
$597$		0			0
$598$	20.7846	+	12.0000i	0.849946	+	0.490716i
$599$	6.00000	−	10.3923i	0.245153	−	0.424618i	−0.717021	−	0.697051i	$-0.754495\pi$
$599$	6.00000	−	10.3923i	0.245153	−	0.424618i	0.962175	+	0.272433i	$0.0878284\pi$
$600$		0			0
$601$	17.0000			0.693444			0.346722	−	0.937968i	$-0.387295\pi$
$601$	17.0000			0.693444			0.346722	+	0.937968i	$0.387295\pi$
$602$	13.8564	+	8.00000i	0.564745	+	0.326056i
$603$		0			0
$604$	9.00000	−	15.5885i	0.366205	−	0.634285i
$605$	22.3205	−	1.33975i	0.907458	−	0.0544684i
$606$		0			0
$607$			14.0000i			0.568242i	0.958788	+	0.284121i	$0.0917018\pi$
$607$			14.0000i			0.568242i	−0.958788	+	0.284121i	$0.908298\pi$
$608$	34.6410	−	4.00000i	1.40488	−	0.162221i
$609$		0			0
$610$	1.87564	+	31.2487i	0.0759426	+	1.26522i
$611$	−6.00000	−	10.3923i	−0.242734	−	0.420428i
$612$		0			0
$613$	20.7846	−	12.0000i	0.839482	−	0.484675i	−0.0176058	−	0.999845i	$-0.505604\pi$
$613$	20.7846	−	12.0000i	0.839482	−	0.484675i	0.857088	+	0.515170i	$0.172271\pi$
$614$	−16.0000	+	27.7128i	−0.645707	+	1.11840i
$615$		0			0
$616$		0			0
$617$	−15.5885	−	9.00000i	−0.627568	−	0.362326i	0.152242	−	0.988343i	$-0.451351\pi$
$617$	−15.5885	−	9.00000i	−0.627568	−	0.362326i	−0.779809	+	0.626017i	$0.784684\pi$
$618$		0			0
$619$	−4.00000			−0.160774			−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000			−0.160774			−0.0803868	+	0.996764i	$0.525616\pi$
$620$	14.0000	−	28.0000i	0.562254	−	1.12451i
$621$		0			0
$622$		0			0
$623$	38.1051	+	22.0000i	1.52665	+	0.881411i
$624$		0			0
$625$	24.2846	−	5.93782i	0.971384	−	0.237513i
$626$	−60.0000			−2.39808
$627$		0			0
$628$			8.00000i			0.319235i
$629$	2.00000	+	3.46410i	0.0797452	+	0.138123i
$630$		0			0
$631$	−0.500000	+	0.866025i	−0.0199047	+	0.0344759i	−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−0.500000	+	0.866025i	−0.0199047	+	0.0344759i	0.855901	+	0.517139i	$0.173003\pi$
$632$		0			0
$633$		0			0
$634$	4.00000			0.158860
$635$	12.0000	+	6.00000i	0.476205	+	0.238103i
$636$		0			0
$637$	−15.5885	−	9.00000i	−0.617637	−	0.356593i
$638$			18.0000i			0.712627i
$639$		0			0
$640$		0			0
$641$	10.5000	+	18.1865i	0.414725	+	0.718325i	0.995400	−	0.0958109i	$-0.0305444\pi$
$641$	10.5000	+	18.1865i	0.414725	+	0.718325i	−0.580674	+	0.814136i	$0.697211\pi$
$642$		0			0
$643$	−39.8372	+	23.0000i	−1.57102	+	0.907031i	−0.574981	+	0.818167i	$0.694991\pi$
$643$	−39.8372	+	23.0000i	−1.57102	+	0.907031i	−0.996044	+	0.0888646i	$0.971676\pi$
$644$	24.0000	+	41.5692i	0.945732	+	1.63806i
$645$		0			0
$646$	2.00000	+	17.3205i	0.0786889	+	0.681466i
$647$		−	6.00000i		−	0.235884i	−0.993020	−	0.117942i	$-0.962370\pi$
$647$		−	6.00000i		−	0.235884i	0.993020	−	0.117942i	$-0.0376297\pi$
$648$		0			0
$649$	−4.50000	−	7.79423i	−0.176640	−	0.305950i
$650$	19.8564	−	2.39230i	0.778832	−	0.0938339i
$651$		0			0
$652$	6.92820	+	4.00000i	0.271329	+	0.156652i
$653$		−	10.0000i		−	0.391330i	−0.980671	−	0.195665i	$-0.937313\pi$
$653$		−	10.0000i		−	0.391330i	0.980671	−	0.195665i	$-0.0626866\pi$
$654$		0			0
$655$	−14.7846	−	22.3923i	−0.577683	−	0.874940i
$656$	−4.00000	+	6.92820i	−0.156174	+	0.270501i
$657$		0			0
$658$		−	48.0000i		−	1.87123i
$659$	−10.0000	+	17.3205i	−0.389545	+	0.674711i	−0.992388	−	0.123148i	$-0.960701\pi$
$659$	−10.0000	+	17.3205i	−0.389545	+	0.674711i	0.602844	+	0.797859i	$0.294034\pi$
$660$		0			0
$661$	7.50000	−	12.9904i	0.291716	−	0.505267i	−0.682499	−	0.730886i	$-0.739107\pi$
$661$	7.50000	−	12.9904i	0.291716	−	0.505267i	0.974216	+	0.225619i	$0.0724404\pi$
$662$	34.6410	−	20.0000i	1.34636	−	0.777322i
$663$		0			0
$664$		0			0
$665$	−25.0718	−	29.8564i	−0.972243	−	1.15778i
$666$		0			0
$667$	−46.7654	+	27.0000i	−1.81076	+	1.04544i
$668$	20.7846	−	12.0000i	0.804181	−	0.464294i
$669$		0			0
$670$	44.6410	−	2.67949i	1.72463	−	0.103518i
$671$	3.50000	−	6.06218i	0.135116	−	0.234028i
$672$		0			0
$673$		−	20.0000i		−	0.770943i	−0.922720	−	0.385472i	$-0.874039\pi$
$673$		−	20.0000i		−	0.770943i	0.922720	−	0.385472i	$-0.125961\pi$
$674$	−34.0000	+	58.8897i	−1.30963	+	2.26835i
$675$		0			0
$676$	−18.0000			−0.692308
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	12.0000	+	20.7846i	0.460518	+	0.797640i
$680$		0			0
$681$		0			0
$682$	−12.1244	+	7.00000i	−0.464266	+	0.268044i
$683$		−	30.0000i		−	1.14792i	−0.818884	−	0.573959i	$-0.805407\pi$
$683$		−	30.0000i		−	1.14792i	0.818884	−	0.573959i	$-0.194593\pi$
$684$		0			0
$685$	24.0000	+	12.0000i	0.916993	+	0.458496i
$686$	−8.00000	−	13.8564i	−0.305441	−	0.529040i
$687$		0			0
$688$	6.92820	+	4.00000i	0.264135	+	0.152499i
$689$	4.00000	+	6.92820i	0.152388	+	0.263944i
$690$		0			0
$691$	3.00000			0.114125			0.0570627	−	0.998371i	$-0.481827\pi$
$691$	3.00000			0.114125			0.0570627	+	0.998371i	$0.481827\pi$
$692$		−	48.0000i		−	1.82469i
$693$		0			0
$694$	−12.0000	+	20.7846i	−0.455514	+	0.788973i
$695$	−20.0000	+	40.0000i	−0.758643	+	1.51729i
$696$		0			0
$697$	−3.46410	−	2.00000i	−0.131212	−	0.0757554i
$698$	−24.2487	+	14.0000i	−0.917827	+	0.529908i
$699$		0			0
$700$	36.7846	+	15.7128i	1.39033	+	0.593889i
$701$	3.00000	+	5.19615i	0.113308	+	0.196256i	0.917102	−	0.398652i	$-0.130522\pi$
$701$	3.00000	+	5.19615i	0.113308	+	0.196256i	−0.803794	+	0.594908i	$0.797189\pi$
$702$		0			0
$703$	−5.19615	−	7.00000i	−0.195977	−	0.264010i
$704$	8.00000			0.301511
$705$		0			0
$706$	8.00000	+	13.8564i	0.301084	+	0.521493i
$707$	51.9615	+	30.0000i	1.95421	+	1.12827i
$708$		0			0
$709$	12.5000	−	21.6506i	0.469447	−	0.813107i	−0.529943	−	0.848034i	$-0.677787\pi$
$709$	12.5000	−	21.6506i	0.469447	−	0.813107i	0.999390	+	0.0349269i	$0.0111198\pi$
$710$	−4.00000	−	2.00000i	−0.150117	−	0.0750587i
$711$		0			0
$712$		0			0
$713$	−36.3731	−	21.0000i	−1.36218	−	0.786456i
$714$		0			0
$715$	−4.00000	−	2.00000i	−0.149592	−	0.0747958i
$716$	15.0000	−	25.9808i	0.560576	−	0.970947i
$717$		0			0
$718$	34.6410	+	20.0000i	1.29279	+	0.746393i
$719$	10.5000	+	18.1865i	0.391584	+	0.678243i	0.992659	−	0.120950i	$-0.0385939\pi$
$719$	10.5000	+	18.1865i	0.391584	+	0.678243i	−0.601075	+	0.799193i	$0.705261\pi$
$720$		0			0
$721$	−64.0000			−2.38348
$722$	−8.66025	−	37.0000i	−0.322301	−	1.37700i
$723$		0			0
$724$	−6.00000	−	10.3923i	−0.222988	−	0.386227i
$725$	−17.6769	+	41.3827i	−0.656504	+	1.53691i
$726$		0			0
$727$	−6.92820	+	4.00000i	−0.256953	+	0.148352i	−0.622944	−	0.782267i	$-0.714063\pi$
$727$	−6.92820	+	4.00000i	−0.256953	+	0.148352i	0.365991	+	0.930618i	$0.380730\pi$
$728$		0			0
$729$		0			0
$730$	−20.0000	+	40.0000i	−0.740233	+	1.48047i
$731$	−2.00000	+	3.46410i	−0.0739727	+	0.128124i
$732$		0			0
$733$		−	6.00000i		−	0.221615i	−0.993842	−	0.110808i	$-0.964656\pi$
$733$		−	6.00000i		−	0.221615i	0.993842	−	0.110808i	$-0.0353437\pi$
$734$	−32.0000			−1.18114
$735$		0			0
$736$	24.0000	+	41.5692i	0.884652	+	1.53226i
$737$	−8.66025	−	5.00000i	−0.319005	−	0.184177i
$738$		0			0
$739$	−14.5000	−	25.1147i	−0.533391	−	0.923861i	−0.999239	−	0.0389959i	$-0.987584\pi$
$739$	−14.5000	−	25.1147i	−0.533391	−	0.923861i	0.465848	−	0.884865i	$-0.345749\pi$
$740$	8.00000	+	4.00000i	0.294086	+	0.147043i
$741$		0			0
$742$			32.0000i			1.17476i
$743$	−43.3013	+	25.0000i	−1.58857	+	0.917161i	−0.595026	+	0.803706i	$0.702858\pi$
$743$	−43.3013	+	25.0000i	−1.58857	+	0.917161i	−0.993543	+	0.113455i	$0.963808\pi$
$744$		0			0
$745$	−1.23205	−	1.86603i	−0.0451388	−	0.0683659i
$746$	−12.0000	−	20.7846i	−0.439351	−	0.760979i
$747$		0			0
$748$			4.00000i			0.146254i
$749$	40.0000			1.46157
$750$		0			0
$751$	−20.5000	+	35.5070i	−0.748056	+	1.29567i	0.200698	+	0.979653i	$0.435679\pi$
$751$	−20.5000	+	35.5070i	−0.748056	+	1.29567i	−0.948753	+	0.316017i	$0.897654\pi$
$752$		−	24.0000i		−	0.875190i
$753$		0			0
$754$	−18.0000	+	31.1769i	−0.655521	+	1.13540i
$755$	−20.0885	+	1.20577i	−0.731094	+	0.0438825i
$756$		0			0
$757$	−22.5167	+	13.0000i	−0.818382	+	0.472493i	−0.849858	−	0.527011i	$-0.823312\pi$
$757$	−22.5167	+	13.0000i	−0.818382	+	0.472493i	0.0314762	+	0.999505i	$0.489979\pi$
$758$	50.2295	−	29.0000i	1.82442	−	1.05333i
$759$		0			0
$760$		0			0
$761$	18.0000			0.652499			0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000			0.652499			0.326250	+	0.945284i	$0.394215\pi$
$762$		0			0
$763$	−51.9615	+	30.0000i	−1.88113	+	1.08607i
$764$	3.00000	−	5.19615i	0.108536	−	0.187990i
$765$		0			0
$766$	−6.00000	+	10.3923i	−0.216789	+	0.375489i
$767$		−	18.0000i		−	0.649942i
$768$		0			0
$769$	2.50000	−	4.33013i	0.0901523	−	0.156148i	−0.817423	−	0.576038i	$-0.804598\pi$
$769$	2.50000	−	4.33013i	0.0901523	−	0.156148i	0.907575	+	0.419890i	$0.137931\pi$
$770$	−9.85641	−	14.9282i	−0.355200	−	0.537975i
$771$		0			0
$772$		−	32.0000i		−	1.15171i
$773$	29.4449	+	17.0000i	1.05906	+	0.611448i	0.925172	−	0.379549i	$-0.123921\pi$
$773$	29.4449	+	17.0000i	1.05906	+	0.611448i	0.133887	+	0.990997i	$0.457254\pi$
$774$		0			0
$775$	−34.7487	+	4.18653i	−1.24821	+	0.150385i
$776$		0			0
$777$		0			0
$778$		−	66.0000i		−	2.36621i
$779$	8.00000	+	3.46410i	0.286630	+	0.124114i
$780$		0			0
$781$	0.500000	+	0.866025i	0.0178914	+	0.0309888i
$782$	−20.7846	+	12.0000i	−0.743256	+	0.429119i
$783$		0			0

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.be (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.73205 - 1.00000i) q^{2} +(1.00000 - 1.73205i) q^{4} +(-2.23205 + 0.133975i) q^{5} +4.00000i q^{7} +O(q^{10})\)	\(q+(1.73205 - 1.00000i) q^{2} +(1.00000 - 1.73205i) q^{4} +(-2.23205 + 0.133975i) q^{5} +4.00000i q^{7} +(-3.73205 + 2.46410i) q^{10} +1.00000 q^{11} +(1.73205 + 1.00000i) q^{13} +(4.00000 + 6.92820i) q^{14} +(2.00000 + 3.46410i) q^{16} +(-1.73205 + 1.00000i) q^{17} +(3.50000 - 2.59808i) q^{19} +(-2.00000 + 4.00000i) q^{20} +(1.73205 - 1.00000i) q^{22} +(5.19615 + 3.00000i) q^{23} +(4.96410 - 0.598076i) q^{25} +4.00000 q^{26} +(6.92820 + 4.00000i) q^{28} +(-4.50000 + 7.79423i) q^{29} -7.00000 q^{31} +(6.92820 + 4.00000i) q^{32} +(-2.00000 + 3.46410i) q^{34} +(-0.535898 - 8.92820i) q^{35} -2.00000i q^{37} +(3.46410 - 8.00000i) q^{38} +(1.00000 + 1.73205i) q^{41} +(1.73205 - 1.00000i) q^{43} +(1.00000 - 1.73205i) q^{44} +12.0000 q^{46} +(-5.19615 - 3.00000i) q^{47} -9.00000 q^{49} +(8.00000 - 6.00000i) q^{50} +(3.46410 - 2.00000i) q^{52} +(3.46410 + 2.00000i) q^{53} +(-2.23205 + 0.133975i) q^{55} +18.0000i q^{58} +(-4.50000 - 7.79423i) q^{59} +(3.50000 - 6.06218i) q^{61} +(-12.1244 + 7.00000i) q^{62} +8.00000 q^{64} +(-4.00000 - 2.00000i) q^{65} +(-8.66025 - 5.00000i) q^{67} +4.00000i q^{68} +(-9.85641 - 14.9282i) q^{70} +(0.500000 + 0.866025i) q^{71} +(8.66025 - 5.00000i) q^{73} +(-2.00000 - 3.46410i) q^{74} +(-1.00000 - 8.66025i) q^{76} +4.00000i q^{77} +(0.500000 + 0.866025i) q^{79} +(-4.92820 - 7.46410i) q^{80} +(3.46410 + 2.00000i) q^{82} +6.00000i q^{83} +(3.73205 - 2.46410i) q^{85} +(2.00000 - 3.46410i) q^{86} +(5.50000 - 9.52628i) q^{89} +(-4.00000 + 6.92820i) q^{91} +(10.3923 - 6.00000i) q^{92} -12.0000 q^{94} +(-7.46410 + 6.26795i) q^{95} +(5.19615 - 3.00000i) q^{97} +(-15.5885 + 9.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 2 q^{5}+O(q^{10}) \) 4 * q + 4 * q^4 - 2 * q^5	\( 4 q + 4 q^{4} - 2 q^{5} - 8 q^{10} + 4 q^{11} + 16 q^{14} + 8 q^{16} + 14 q^{19} - 8 q^{20} + 6 q^{25} + 16 q^{26} - 18 q^{29} - 28 q^{31} - 8 q^{34} - 16 q^{35} + 4 q^{41} + 4 q^{44} + 48 q^{46} - 36 q^{49} + 32 q^{50} - 2 q^{55} - 18 q^{59} + 14 q^{61} + 32 q^{64} - 16 q^{65} + 16 q^{70} + 2 q^{71} - 8 q^{74} - 4 q^{76} + 2 q^{79} + 8 q^{80} + 8 q^{85} + 8 q^{86} + 22 q^{89} - 16 q^{91} - 48 q^{94} - 16 q^{95}+O(q^{100}) \) 4 * q + 4 * q^4 - 2 * q^5 - 8 * q^10 + 4 * q^11 + 16 * q^14 + 8 * q^16 + 14 * q^19 - 8 * q^20 + 6 * q^25 + 16 * q^26 - 18 * q^29 - 28 * q^31 - 8 * q^34 - 16 * q^35 + 4 * q^41 + 4 * q^44 + 48 * q^46 - 36 * q^49 + 32 * q^50 - 2 * q^55 - 18 * q^59 + 14 * q^61 + 32 * q^64 - 16 * q^65 + 16 * q^70 + 2 * q^71 - 8 * q^74 - 4 * q^76 + 2 * q^79 + 8 * q^80 + 8 * q^85 + 8 * q^86 + 22 * q^89 - 16 * q^91 - 48 * q^94 - 16 * q^95

Introduction

L-functions

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