LMFDB - Embedded newform 1620.2.i.f.1081.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,2,Mod(541,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

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

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

chi := DirichletCharacter("1620.541");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1620.i (of order $3$, degree $2$, not 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:	$12.9357651274$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1081.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1620.1081
Dual form			1620.2.i.f.541.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{5} +(2.00000 + 3.46410i) q^{7} +O(q^{10})$	$q+(-0.500000 + 0.866025i) q^{5} +(2.00000 + 3.46410i) q^{7} +(3.00000 + 5.19615i) q^{11} +(2.00000 - 3.46410i) q^{13} +3.00000 q^{17} -7.00000 q^{19} +(-4.50000 + 7.79423i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(3.50000 - 6.06218i) q^{31} -4.00000 q^{35} +2.00000 q^{37} +(3.00000 - 5.19615i) q^{41} +(-1.00000 - 1.73205i) q^{43} +(-4.50000 + 7.79423i) q^{49} -9.00000 q^{53} -6.00000 q^{55} +(-6.00000 + 10.3923i) q^{59} +(3.50000 + 6.06218i) q^{61} +(2.00000 + 3.46410i) q^{65} +(-1.00000 + 1.73205i) q^{67} -6.00000 q^{71} +2.00000 q^{73} +(-12.0000 + 20.7846i) q^{77} +(0.500000 + 0.866025i) q^{79} +(4.50000 + 7.79423i) q^{83} +(-1.50000 + 2.59808i) q^{85} +6.00000 q^{89} +16.0000 q^{91} +(3.50000 - 6.06218i) q^{95} +(-4.00000 - 6.92820i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q - q^5 + 4 * q^7	$ 2 q - q^{5} + 4 q^{7} + 6 q^{11} + 4 q^{13} + 6 q^{17} - 14 q^{19} - 9 q^{23} - q^{25} + 7 q^{31} - 8 q^{35} + 4 q^{37} + 6 q^{41} - 2 q^{43} - 9 q^{49} - 18 q^{53} - 12 q^{55} - 12 q^{59} + 7 q^{61} + 4 q^{65} - 2 q^{67} - 12 q^{71} + 4 q^{73} - 24 q^{77} + q^{79} + 9 q^{83} - 3 q^{85} + 12 q^{89} + 32 q^{91} + 7 q^{95} - 8 q^{97}+O(q^{100}) $ 2 * q - q^5 + 4 * q^7 + 6 * q^11 + 4 * q^13 + 6 * q^17 - 14 * q^19 - 9 * q^23 - q^25 + 7 * q^31 - 8 * q^35 + 4 * q^37 + 6 * q^41 - 2 * q^43 - 9 * q^49 - 18 * q^53 - 12 * q^55 - 12 * q^59 + 7 * q^61 + 4 * q^65 - 2 * q^67 - 12 * q^71 + 4 * q^73 - 24 * q^77 + q^79 + 9 * q^83 - 3 * q^85 + 12 * q^89 + 32 * q^91 + 7 * q^95 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	2.00000	+	3.46410i	0.755929	+	1.30931i	0.944911	+	0.327327i	$0.106148\pi$
$7$	2.00000	+	3.46410i	0.755929	+	1.30931i	−0.188982	+	0.981981i	$0.560519\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.00000	+	5.19615i	0.904534	+	1.56670i	0.821541	+	0.570149i	$0.193114\pi$
$11$	3.00000	+	5.19615i	0.904534	+	1.56670i	0.0829925	+	0.996550i	$0.473552\pi$
$12$		0			0
$13$	2.00000	−	3.46410i	0.554700	−	0.960769i	−0.443227	−	0.896410i	$-0.646166\pi$
$13$	2.00000	−	3.46410i	0.554700	−	0.960769i	0.997927	−	0.0643593i	$-0.0205004\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.00000			0.727607			0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000			0.727607			0.363803	+	0.931476i	$0.381478\pi$
$18$		0			0
$19$	−7.00000			−1.60591			−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000			−1.60591			−0.802955	+	0.596040i	$0.796740\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.50000	+	7.79423i	−0.938315	+	1.62521i	−0.169701	+	0.985496i	$0.554280\pi$
$23$	−4.50000	+	7.79423i	−0.938315	+	1.62521i	−0.768613	+	0.639713i	$0.779053\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$		0			0
$31$	3.50000	−	6.06218i	0.628619	−	1.08880i	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	3.50000	−	6.06218i	0.628619	−	1.08880i	0.987829	−	0.155543i	$-0.0497126\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.00000			−0.676123
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.00000	−	5.19615i	0.468521	−	0.811503i	−0.530831	−	0.847477i	$-0.678120\pi$
$41$	3.00000	−	5.19615i	0.468521	−	0.811503i	0.999353	+	0.0359748i	$0.0114536\pi$
$42$		0			0
$43$	−1.00000	−	1.73205i	−0.152499	−	0.264135i	0.779647	−	0.626219i	$-0.215399\pi$
$43$	−1.00000	−	1.73205i	−0.152499	−	0.264135i	−0.932145	+	0.362084i	$0.882065\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	−4.50000	+	7.79423i	−0.642857	+	1.11346i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.00000			−1.23625			−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000			−1.23625			−0.618123	+	0.786082i	$0.712106\pi$
$54$		0			0
$55$	−6.00000			−0.809040
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	0.150148	+	0.988663i	$0.452025\pi$
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	−0.931282	+	0.364299i	$0.881308\pi$
$60$		0			0
$61$	3.50000	+	6.06218i	0.448129	+	0.776182i	0.998264	−	0.0588933i	$-0.0187572\pi$
$61$	3.50000	+	6.06218i	0.448129	+	0.776182i	−0.550135	+	0.835076i	$0.685424\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.00000	+	3.46410i	0.248069	+	0.429669i
$66$		0			0
$67$	−1.00000	+	1.73205i	−0.122169	+	0.211604i	−0.920623	−	0.390453i	$-0.872318\pi$
$67$	−1.00000	+	1.73205i	−0.122169	+	0.211604i	0.798454	+	0.602056i	$0.205652\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$	2.00000			0.234082			0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000			0.234082			0.117041	+	0.993127i	$0.462659\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.0000	+	20.7846i	−1.36753	+	2.36863i
$78$		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	$0.815418\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.50000	+	7.79423i	0.493939	+	0.855528i	0.999976	−	0.00698436i	$-0.00222321\pi$
$83$	4.50000	+	7.79423i	0.493939	+	0.855528i	−0.506036	+	0.862512i	$0.668890\pi$
$84$		0			0
$85$	−1.50000	+	2.59808i	−0.162698	+	0.281801i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		0			0
$91$	16.0000			1.67726
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.50000	−	6.06218i	0.359092	−	0.621966i
$96$		0			0
$97$	−4.00000	−	6.92820i	−0.406138	−	0.703452i	0.588315	−	0.808632i	$-0.299792\pi$
$97$	−4.00000	−	6.92820i	−0.406138	−	0.703452i	−0.994453	+	0.105180i	$0.966458\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.00000	+	10.3923i	0.597022	+	1.03407i	0.993258	+	0.115924i	$0.0369830\pi$
$101$	6.00000	+	10.3923i	0.597022	+	1.03407i	−0.396236	+	0.918149i	$0.629684\pi$
$102$		0			0
$103$	2.00000	−	3.46410i	0.197066	−	0.341328i	−0.750510	−	0.660859i	$-0.770192\pi$
$103$	2.00000	−	3.46410i	0.197066	−	0.341328i	0.947576	+	0.319531i	$0.103525\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$	5.00000			0.478913			0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000			0.478913			0.239457	+	0.970907i	$0.423031\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.00000	−	5.19615i	0.282216	−	0.488813i	−0.689714	−	0.724082i	$-0.742264\pi$
$113$	3.00000	−	5.19615i	0.282216	−	0.488813i	0.971930	+	0.235269i	$0.0755971\pi$
$114$		0			0
$115$	−4.50000	−	7.79423i	−0.419627	−	0.726816i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.00000	+	10.3923i	0.550019	+	0.952661i
$120$		0			0
$121$	−12.5000	+	21.6506i	−1.13636	+	1.96824i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	2.00000			0.177471			0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000			0.177471			0.0887357	+	0.996055i	$0.471717\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.00000	+	5.19615i	−0.262111	+	0.453990i	−0.966803	−	0.255524i	$-0.917752\pi$
$131$	−3.00000	+	5.19615i	−0.262111	+	0.453990i	0.704692	+	0.709514i	$0.251085\pi$
$132$		0			0
$133$	−14.0000	−	24.2487i	−1.21395	−	2.10263i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.50000	−	7.79423i	−0.384461	−	0.665906i	0.607233	−	0.794524i	$-0.292279\pi$
$137$	−4.50000	−	7.79423i	−0.384461	−	0.665906i	−0.991694	+	0.128618i	$0.958946\pi$
$138$		0			0
$139$	2.00000	−	3.46410i	0.169638	−	0.293821i	−0.768655	−	0.639664i	$-0.779074\pi$
$139$	2.00000	−	3.46410i	0.169638	−	0.293821i	0.938293	+	0.345843i	$0.112407\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	24.0000			2.00698
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	−0.962348	−	0.271821i	$-0.912374\pi$
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	0.716578	+	0.697507i	$0.245707\pi$
$150$		0			0
$151$	8.00000	+	13.8564i	0.651031	+	1.12762i	0.982873	+	0.184284i	$0.0589965\pi$
$151$	8.00000	+	13.8564i	0.651031	+	1.12762i	−0.331842	+	0.943335i	$0.607670\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.50000	+	6.06218i	0.281127	+	0.486926i
$156$		0			0
$157$	5.00000	−	8.66025i	0.399043	−	0.691164i	−0.594565	−	0.804048i	$-0.702676\pi$
$157$	5.00000	−	8.66025i	0.399043	−	0.691164i	0.993608	+	0.112884i	$0.0360089\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−36.0000			−2.83720
$162$		0			0
$163$	−22.0000			−1.72317			−0.861586	−	0.507611i	$-0.830529\pi$
$163$	−22.0000			−1.72317			−0.861586	+	0.507611i	$0.830529\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.50000	−	2.59808i	0.116073	−	0.201045i	−0.802135	−	0.597143i	$-0.796303\pi$
$167$	1.50000	−	2.59808i	0.116073	−	0.201045i	0.918208	+	0.396098i	$0.129636\pi$
$168$		0			0
$169$	−1.50000	−	2.59808i	−0.115385	−	0.199852i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.50000	−	7.79423i	−0.342129	−	0.592584i	0.642699	−	0.766119i	$-0.277815\pi$
$173$	−4.50000	−	7.79423i	−0.342129	−	0.592584i	−0.984828	+	0.173534i	$0.944481\pi$
$174$		0			0
$175$	2.00000	−	3.46410i	0.151186	−	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	5.00000			0.371647			0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000			0.371647			0.185824	+	0.982583i	$0.440505\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.00000	+	1.73205i	−0.0735215	+	0.127343i
$186$		0			0
$187$	9.00000	+	15.5885i	0.658145	+	1.13994i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	9.00000	+	15.5885i	0.651217	+	1.12794i	0.982828	+	0.184525i	$0.0590746\pi$
$191$	9.00000	+	15.5885i	0.651217	+	1.12794i	−0.331611	+	0.943416i	$0.607592\pi$
$192$		0			0
$193$	−4.00000	+	6.92820i	−0.287926	+	0.498703i	−0.973315	−	0.229475i	$-0.926299\pi$
$193$	−4.00000	+	6.92820i	−0.287926	+	0.498703i	0.685388	+	0.728178i	$0.259632\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.0000			−1.06871			−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000			−1.06871			−0.534353	+	0.845262i	$0.679445\pi$
$198$		0			0
$199$	20.0000			1.41776			0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000			1.41776			0.708881	+	0.705328i	$0.249200\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	3.00000	+	5.19615i	0.209529	+	0.362915i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−21.0000	−	36.3731i	−1.45260	−	2.51598i
$210$		0			0
$211$	6.50000	−	11.2583i	0.447478	−	0.775055i	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	6.50000	−	11.2583i	0.447478	−	0.775055i	0.998221	+	0.0596196i	$0.0189888\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.00000			0.136399
$216$		0			0
$217$	28.0000			1.90076
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.00000	−	10.3923i	0.403604	−	0.699062i
$222$		0			0
$223$	−7.00000	−	12.1244i	−0.468755	−	0.811907i	0.530607	−	0.847618i	$-0.321964\pi$
$223$	−7.00000	−	12.1244i	−0.468755	−	0.811907i	−0.999362	+	0.0357107i	$0.988630\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.50000	−	12.9904i	−0.497792	−	0.862202i	0.502204	−	0.864749i	$-0.332523\pi$
$227$	−7.50000	−	12.9904i	−0.497792	−	0.862202i	−0.999997	+	0.00254715i	$0.999189\pi$
$228$		0			0
$229$	12.5000	−	21.6506i	0.826023	−	1.43071i	−0.0751115	−	0.997175i	$-0.523931\pi$
$229$	12.5000	−	21.6506i	0.826023	−	1.43071i	0.901135	−	0.433539i	$-0.142735\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	9.00000	−	15.5885i	0.582162	−	1.00833i	−0.413061	−	0.910703i	$-0.635540\pi$
$239$	9.00000	−	15.5885i	0.582162	−	1.00833i	0.995223	−	0.0976302i	$-0.0311262\pi$
$240$		0			0
$241$	12.5000	+	21.6506i	0.805196	+	1.39464i	0.916159	+	0.400815i	$0.131273\pi$
$241$	12.5000	+	21.6506i	0.805196	+	1.39464i	−0.110963	+	0.993825i	$0.535394\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.50000	−	7.79423i	−0.287494	−	0.497955i
$246$		0			0
$247$	−14.0000	+	24.2487i	−0.890799	+	1.54291i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$	−54.0000			−3.39495
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.50000	−	12.9904i	0.467837	−	0.810318i	−0.531487	−	0.847066i	$-0.678367\pi$
$257$	7.50000	−	12.9904i	0.467837	−	0.810318i	0.999325	+	0.0367485i	$0.0117000\pi$
$258$		0			0
$259$	4.00000	+	6.92820i	0.248548	+	0.430498i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.00000	−	10.3923i	−0.369976	−	0.640817i	0.619586	−	0.784929i	$-0.287301\pi$
$263$	−6.00000	−	10.3923i	−0.369976	−	0.640817i	−0.989561	+	0.144112i	$0.953967\pi$
$264$		0			0
$265$	4.50000	−	7.79423i	0.276433	−	0.478796i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.00000			−0.365826			−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000			−0.365826			−0.182913	+	0.983129i	$0.558553\pi$
$270$		0			0
$271$	−7.00000			−0.425220			−0.212610	−	0.977137i	$-0.568196\pi$
$271$	−7.00000			−0.425220			−0.212610	+	0.977137i	$0.568196\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.00000	−	5.19615i	0.180907	−	0.313340i
$276$		0			0
$277$	11.0000	+	19.0526i	0.660926	+	1.14476i	0.980373	+	0.197153i	$0.0631696\pi$
$277$	11.0000	+	19.0526i	0.660926	+	1.14476i	−0.319447	+	0.947604i	$0.603497\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.0000	−	20.7846i	−0.715860	−	1.23991i	−0.962627	−	0.270831i	$-0.912702\pi$
$281$	−12.0000	−	20.7846i	−0.715860	−	1.23991i	0.246767	−	0.969075i	$-0.420632\pi$
$282$		0			0
$283$	−1.00000	+	1.73205i	−0.0594438	+	0.102960i	−0.894216	−	0.447636i	$-0.852266\pi$
$283$	−1.00000	+	1.73205i	−0.0594438	+	0.102960i	0.834772	+	0.550596i	$0.185599\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	24.0000			1.41668
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$	7.50000	−	12.9904i	0.438155	−	0.758906i	−0.559393	−	0.828903i	$-0.688966\pi$
$293$	7.50000	−	12.9904i	0.438155	−	0.758906i	0.997547	+	0.0699967i	$0.0222989\pi$
$294$		0			0
$295$	−6.00000	−	10.3923i	−0.349334	−	0.605063i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	18.0000	+	31.1769i	1.04097	+	1.80301i
$300$		0			0
$301$	4.00000	−	6.92820i	0.230556	−	0.399335i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−7.00000			−0.400819
$306$		0			0
$307$	−34.0000			−1.94048			−0.970241	−	0.242140i	$-0.922151\pi$
$307$	−34.0000			−1.94048			−0.970241	+	0.242140i	$0.922151\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	−0.294384	−	0.955687i	$-0.595114\pi$
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	0.974841	−	0.222900i	$-0.0715523\pi$
$312$		0			0
$313$	−10.0000	−	17.3205i	−0.565233	−	0.979013i	−0.997028	−	0.0770410i	$-0.975453\pi$
$313$	−10.0000	−	17.3205i	−0.565233	−	0.979013i	0.431795	−	0.901972i	$-0.357881\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4.50000	+	7.79423i	0.252745	+	0.437767i	0.964281	−	0.264883i	$-0.0853332\pi$
$317$	4.50000	+	7.79423i	0.252745	+	0.437767i	−0.711535	+	0.702650i	$0.752000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−21.0000			−1.16847
$324$		0			0
$325$	−4.00000			−0.221880
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4.00000	−	6.92820i	−0.219860	−	0.380808i	0.734905	−	0.678170i	$-0.237227\pi$
$331$	−4.00000	−	6.92820i	−0.219860	−	0.380808i	−0.954765	+	0.297361i	$0.903893\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.00000	−	1.73205i	−0.0546358	−	0.0946320i
$336$		0			0
$337$	14.0000	−	24.2487i	0.762629	−	1.32091i	−0.178863	−	0.983874i	$-0.557242\pi$
$337$	14.0000	−	24.2487i	0.762629	−	1.32091i	0.941491	−	0.337037i	$-0.109425\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	42.0000			2.27443
$342$		0			0
$343$	−8.00000			−0.431959
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	−0.980921	−	0.194409i	$-0.937721\pi$
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	0.658824	+	0.752297i	$0.271054\pi$
$348$		0			0
$349$	9.50000	+	16.4545i	0.508523	+	0.880788i	0.999951	+	0.00987003i	$0.00314178\pi$
$349$	9.50000	+	16.4545i	0.508523	+	0.880788i	−0.491428	+	0.870918i	$0.663525\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.00000	+	15.5885i	0.479022	+	0.829690i	0.999711	−	0.0240566i	$-0.00765819\pi$
$353$	9.00000	+	15.5885i	0.479022	+	0.829690i	−0.520689	+	0.853746i	$0.674325\pi$
$354$		0			0
$355$	3.00000	−	5.19615i	0.159223	−	0.275783i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6.00000			0.316668			0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000			0.316668			0.158334	+	0.987386i	$0.449388\pi$
$360$		0			0
$361$	30.0000			1.57895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.00000	+	1.73205i	−0.0523424	+	0.0906597i
$366$		0			0
$367$	2.00000	+	3.46410i	0.104399	+	0.180825i	0.913493	−	0.406855i	$-0.133375\pi$
$367$	2.00000	+	3.46410i	0.104399	+	0.180825i	−0.809093	+	0.587680i	$0.800041\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−18.0000	−	31.1769i	−0.934513	−	1.61862i
$372$		0			0
$373$	2.00000	−	3.46410i	0.103556	−	0.179364i	−0.809591	−	0.586994i	$-0.800311\pi$
$373$	2.00000	−	3.46410i	0.103556	−	0.179364i	0.913147	+	0.407630i	$0.133645\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	23.0000			1.18143			0.590715	−	0.806880i	$-0.298846\pi$
$379$	23.0000			1.18143			0.590715	+	0.806880i	$0.298846\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.50000	+	7.79423i	−0.229939	+	0.398266i	−0.957790	−	0.287469i	$-0.907186\pi$
$383$	−4.50000	+	7.79423i	−0.229939	+	0.398266i	0.727851	+	0.685736i	$0.240519\pi$
$384$		0			0
$385$	−12.0000	−	20.7846i	−0.611577	−	1.05928i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−15.0000	−	25.9808i	−0.760530	−	1.31728i	−0.942578	−	0.333987i	$-0.891606\pi$
$389$	−15.0000	−	25.9808i	−0.760530	−	1.31728i	0.182047	−	0.983290i	$-0.441728\pi$
$390$		0			0
$391$	−13.5000	+	23.3827i	−0.682724	+	1.18251i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.00000			−0.0503155
$396$		0			0
$397$	8.00000			0.401508			0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000			0.401508			0.200754	+	0.979642i	$0.435661\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.00000	−	5.19615i	0.149813	−	0.259483i	−0.781345	−	0.624099i	$-0.785466\pi$
$401$	3.00000	−	5.19615i	0.149813	−	0.259483i	0.931158	+	0.364615i	$0.118800\pi$
$402$		0			0
$403$	−14.0000	−	24.2487i	−0.697390	−	1.20791i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.00000	+	10.3923i	0.297409	+	0.515127i
$408$		0			0
$409$	−8.50000	+	14.7224i	−0.420298	+	0.727977i	−0.995968	−	0.0897044i	$-0.971408\pi$
$409$	−8.50000	+	14.7224i	−0.420298	+	0.727977i	0.575670	+	0.817682i	$0.304741\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−48.0000			−2.36193
$414$		0			0
$415$	−9.00000			−0.441793
$416$		0			0
$417$		0			0
$418$		0			0
$419$	15.0000	−	25.9808i	0.732798	−	1.26924i	−0.222885	−	0.974845i	$-0.571547\pi$
$419$	15.0000	−	25.9808i	0.732798	−	1.26924i	0.955683	−	0.294398i	$-0.0951193\pi$
$420$		0			0
$421$	18.5000	+	32.0429i	0.901635	+	1.56168i	0.825372	+	0.564590i	$0.190966\pi$
$421$	18.5000	+	32.0429i	0.901635	+	1.56168i	0.0762630	+	0.997088i	$0.475701\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.50000	−	2.59808i	−0.0727607	−	0.126025i
$426$		0			0
$427$	−14.0000	+	24.2487i	−0.677507	+	1.17348i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	36.0000			1.73406			0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000			1.73406			0.867029	+	0.498257i	$0.166026\pi$
$432$		0			0
$433$	−4.00000			−0.192228			−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000			−0.192228			−0.0961139	+	0.995370i	$0.530641\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	31.5000	−	54.5596i	1.50685	−	2.60994i
$438$		0			0
$439$	9.50000	+	16.4545i	0.453410	+	0.785330i	0.998595	−	0.0529862i	$-0.0168739\pi$
$439$	9.50000	+	16.4545i	0.453410	+	0.785330i	−0.545185	+	0.838316i	$0.683541\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−13.5000	−	23.3827i	−0.641404	−	1.11094i	−0.985119	−	0.171871i	$-0.945019\pi$
$443$	−13.5000	−	23.3827i	−0.641404	−	1.11094i	0.343715	−	0.939074i	$-0.388315\pi$
$444$		0			0
$445$	−3.00000	+	5.19615i	−0.142214	+	0.246321i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	24.0000			1.13263			0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000			1.13263			0.566315	+	0.824189i	$0.308369\pi$
$450$		0			0
$451$	36.0000			1.69517
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.00000	+	13.8564i	−0.375046	+	0.649598i
$456$		0			0
$457$	−4.00000	−	6.92820i	−0.187112	−	0.324088i	0.757174	−	0.653213i	$-0.226579\pi$
$457$	−4.00000	−	6.92820i	−0.187112	−	0.324088i	−0.944286	+	0.329125i	$0.893246\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.00000	−	5.19615i	−0.139724	−	0.242009i	0.787668	−	0.616100i	$-0.211288\pi$
$461$	−3.00000	−	5.19615i	−0.139724	−	0.242009i	−0.927392	+	0.374091i	$0.877955\pi$
$462$		0			0
$463$	11.0000	−	19.0526i	0.511213	−	0.885448i	−0.488702	−	0.872451i	$-0.662530\pi$
$463$	11.0000	−	19.0526i	0.511213	−	0.885448i	0.999916	−	0.0129968i	$-0.00413714\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−33.0000			−1.52706			−0.763529	−	0.645774i	$-0.776535\pi$
$467$	−33.0000			−1.52706			−0.763529	+	0.645774i	$0.776535\pi$
$468$		0			0
$469$	−8.00000			−0.369406
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.00000	−	10.3923i	0.275880	−	0.477839i
$474$		0			0
$475$	3.50000	+	6.06218i	0.160591	+	0.278152i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3.00000	+	5.19615i	0.137073	+	0.237418i	0.926388	−	0.376571i	$-0.122897\pi$
$479$	3.00000	+	5.19615i	0.137073	+	0.237418i	−0.789314	+	0.613990i	$0.789564\pi$
$480$		0			0
$481$	4.00000	−	6.92820i	0.182384	−	0.315899i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.00000			0.363261
$486$		0			0
$487$	38.0000			1.72194			0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000			1.72194			0.860972	+	0.508652i	$0.169856\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−15.0000	+	25.9808i	−0.676941	+	1.17250i	0.298957	+	0.954267i	$0.403361\pi$
$491$	−15.0000	+	25.9808i	−0.676941	+	1.17250i	−0.975898	+	0.218229i	$0.929972\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12.0000	−	20.7846i	−0.538274	−	0.932317i
$498$		0			0
$499$	−2.50000	+	4.33013i	−0.111915	+	0.193843i	−0.916542	−	0.399937i	$-0.869032\pi$
$499$	−2.50000	+	4.33013i	−0.111915	+	0.193843i	0.804627	+	0.593780i	$0.202365\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	33.0000			1.47140			0.735699	−	0.677309i	$-0.236854\pi$
$503$	33.0000			1.47140			0.735699	+	0.677309i	$0.236854\pi$
$504$		0			0
$505$	−12.0000			−0.533993
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6.00000	+	10.3923i	−0.265945	+	0.460631i	−0.967811	−	0.251679i	$-0.919017\pi$
$509$	−6.00000	+	10.3923i	−0.265945	+	0.460631i	0.701866	+	0.712309i	$0.252351\pi$
$510$		0			0
$511$	4.00000	+	6.92820i	0.176950	+	0.306486i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.00000	+	3.46410i	0.0881305	+	0.152647i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−30.0000			−1.31432			−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000			−1.31432			−0.657162	+	0.753749i	$0.728243\pi$
$522$		0			0
$523$	32.0000			1.39926			0.699631	−	0.714504i	$-0.253348\pi$
$523$	32.0000			1.39926			0.699631	+	0.714504i	$0.253348\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	10.5000	−	18.1865i	0.457387	−	0.792218i
$528$		0			0
$529$	−29.0000	−	50.2295i	−1.26087	−	2.18389i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−12.0000	−	20.7846i	−0.519778	−	0.900281i
$534$		0			0
$535$	−6.00000	+	10.3923i	−0.259403	+	0.449299i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−54.0000			−2.32594
$540$		0			0
$541$	−10.0000			−0.429934			−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000			−0.429934			−0.214967	+	0.976621i	$0.568964\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.50000	+	4.33013i	−0.107088	+	0.185482i
$546$		0			0
$547$	−13.0000	−	22.5167i	−0.555840	−	0.962743i	−0.997838	−	0.0657267i	$-0.979063\pi$
$547$	−13.0000	−	22.5167i	−0.555840	−	0.962743i	0.441998	−	0.897016i	$-0.354270\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−2.00000	+	3.46410i	−0.0850487	+	0.147309i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6.00000			−0.254228			−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000			−0.254228			−0.127114	+	0.991888i	$0.540571\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.00000	+	10.3923i	−0.252870	+	0.437983i	−0.964315	−	0.264758i	$-0.914708\pi$
$563$	−6.00000	+	10.3923i	−0.252870	+	0.437983i	0.711445	+	0.702742i	$0.248041\pi$
$564$		0			0
$565$	3.00000	+	5.19615i	0.126211	+	0.218604i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18.0000	+	31.1769i	0.754599	+	1.30700i	0.945573	+	0.325409i	$0.105502\pi$
$569$	18.0000	+	31.1769i	0.754599	+	1.30700i	−0.190974	+	0.981595i	$0.561165\pi$
$570$		0			0
$571$	3.50000	−	6.06218i	0.146470	−	0.253694i	−0.783450	−	0.621455i	$-0.786542\pi$
$571$	3.50000	−	6.06218i	0.146470	−	0.253694i	0.929921	+	0.367760i	$0.119875\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	9.00000			0.375326
$576$		0			0
$577$	14.0000			0.582828			0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000			0.582828			0.291414	+	0.956597i	$0.405874\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−18.0000	+	31.1769i	−0.746766	+	1.29344i
$582$		0			0
$583$	−27.0000	−	46.7654i	−1.11823	−	1.93682i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	19.5000	+	33.7750i	0.804851	+	1.39404i	0.916392	+	0.400283i	$0.131088\pi$
$587$	19.5000	+	33.7750i	0.804851	+	1.39404i	−0.111540	+	0.993760i	$0.535578\pi$
$588$		0			0
$589$	−24.5000	+	42.4352i	−1.00950	+	1.74851i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	21.0000			0.862367			0.431183	−	0.902264i	$-0.358096\pi$
$593$	21.0000			0.862367			0.431183	+	0.902264i	$0.358096\pi$
$594$		0			0
$595$	−12.0000			−0.491952
$596$		0			0
$597$		0			0
$598$		0			0
$599$	18.0000	−	31.1769i	0.735460	−	1.27385i	−0.219061	−	0.975711i	$-0.570299\pi$
$599$	18.0000	−	31.1769i	0.735460	−	1.27385i	0.954521	−	0.298143i	$-0.0963673\pi$
$600$		0			0
$601$	3.50000	+	6.06218i	0.142768	+	0.247281i	0.928538	−	0.371237i	$-0.121066\pi$
$601$	3.50000	+	6.06218i	0.142768	+	0.247281i	−0.785770	+	0.618519i	$0.787733\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−12.5000	−	21.6506i	−0.508197	−	0.880223i
$606$		0			0
$607$	11.0000	−	19.0526i	0.446476	−	0.773320i	−0.551678	−	0.834058i	$-0.686012\pi$
$607$	11.0000	−	19.0526i	0.446476	−	0.773320i	0.998154	+	0.0607380i	$0.0193454\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−16.0000			−0.646234			−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000			−0.646234			−0.323117	+	0.946359i	$0.604731\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7.50000	+	12.9904i	−0.301939	+	0.522973i	−0.976575	−	0.215177i	$-0.930967\pi$
$617$	−7.50000	+	12.9904i	−0.301939	+	0.522973i	0.674636	+	0.738150i	$0.264300\pi$
$618$		0			0
$619$	−10.0000	−	17.3205i	−0.401934	−	0.696170i	0.592025	−	0.805919i	$-0.298329\pi$
$619$	−10.0000	−	17.3205i	−0.401934	−	0.696170i	−0.993959	+	0.109749i	$0.964995\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	12.0000	+	20.7846i	0.480770	+	0.832718i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	6.00000			0.239236
$630$		0			0
$631$	23.0000			0.915616			0.457808	−	0.889051i	$-0.348635\pi$
$631$	23.0000			0.915616			0.457808	+	0.889051i	$0.348635\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.00000	+	1.73205i	−0.0396838	+	0.0687343i
$636$		0			0
$637$	18.0000	+	31.1769i	0.713186	+	1.23527i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$641$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$642$		0			0
$643$	2.00000	−	3.46410i	0.0788723	−	0.136611i	−0.823891	−	0.566748i	$-0.808201\pi$
$643$	2.00000	−	3.46410i	0.0788723	−	0.136611i	0.902764	+	0.430137i	$0.141535\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−27.0000			−1.06148			−0.530740	−	0.847535i	$-0.678086\pi$
$647$	−27.0000			−1.06148			−0.530740	+	0.847535i	$0.678086\pi$
$648$		0			0
$649$	−72.0000			−2.82625
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4.50000	−	7.79423i	0.176099	−	0.305012i	−0.764442	−	0.644692i	$-0.776986\pi$
$653$	4.50000	−	7.79423i	0.176099	−	0.305012i	0.940541	+	0.339680i	$0.110319\pi$
$654$		0			0
$655$	−3.00000	−	5.19615i	−0.117220	−	0.203030i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−12.0000	−	20.7846i	−0.467454	−	0.809653i	0.531855	−	0.846836i	$-0.321495\pi$
$659$	−12.0000	−	20.7846i	−0.467454	−	0.809653i	−0.999309	+	0.0371821i	$0.988162\pi$
$660$		0			0
$661$	−7.00000	+	12.1244i	−0.272268	+	0.471583i	−0.969442	−	0.245319i	$-0.921107\pi$
$661$	−7.00000	+	12.1244i	−0.272268	+	0.471583i	0.697174	+	0.716902i	$0.254441\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	28.0000			1.08579
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−21.0000	+	36.3731i	−0.810696	+	1.40417i
$672$		0			0
$673$	14.0000	+	24.2487i	0.539660	+	0.934719i	0.998922	+	0.0464181i	$0.0147807\pi$
$673$	14.0000	+	24.2487i	0.539660	+	0.934719i	−0.459262	+	0.888301i	$0.651886\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15.0000	+	25.9808i	0.576497	+	0.998522i	0.995877	+	0.0907112i	$0.0289140\pi$
$677$	15.0000	+	25.9808i	0.576497	+	0.998522i	−0.419380	+	0.907811i	$0.637753\pi$
$678$		0			0
$679$	16.0000	−	27.7128i	0.614024	−	1.06352i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	39.0000			1.49229			0.746147	−	0.665782i	$-0.231902\pi$
$683$	39.0000			1.49229			0.746147	+	0.665782i	$0.231902\pi$
$684$		0			0
$685$	9.00000			0.343872
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−18.0000	+	31.1769i	−0.685745	+	1.18775i
$690$		0			0
$691$	18.5000	+	32.0429i	0.703773	+	1.21897i	0.967132	+	0.254273i	$0.0818362\pi$
$691$	18.5000	+	32.0429i	0.703773	+	1.21897i	−0.263359	+	0.964698i	$0.584830\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.00000	+	3.46410i	0.0758643	+	0.131401i
$696$		0			0
$697$	9.00000	−	15.5885i	0.340899	−	0.590455i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	6.00000			0.226617			0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000			0.226617			0.113308	+	0.993560i	$0.463855\pi$
$702$		0			0
$703$	−14.0000			−0.528020
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−24.0000	+	41.5692i	−0.902613	+	1.56337i
$708$		0			0
$709$	23.0000	+	39.8372i	0.863783	+	1.49612i	0.868250	+	0.496126i	$0.165245\pi$
$709$	23.0000	+	39.8372i	0.863783	+	1.49612i	−0.00446726	+	0.999990i	$0.501422\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	31.5000	+	54.5596i	1.17968	+	2.04327i
$714$		0			0
$715$	−12.0000	+	20.7846i	−0.448775	+	0.777300i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	6.00000			0.223762			0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000			0.223762			0.111881	+	0.993722i	$0.464312\pi$
$720$		0			0
$721$	16.0000			0.595871
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	14.0000	+	24.2487i	0.519231	+	0.899335i	0.999750	+	0.0223506i	$0.00711500\pi$
$727$	14.0000	+	24.2487i	0.519231	+	0.899335i	−0.480519	+	0.876984i	$0.659552\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3.00000	−	5.19615i	−0.110959	−	0.192187i
$732$		0			0
$733$	20.0000	−	34.6410i	0.738717	−	1.27950i	−0.214356	−	0.976756i	$-0.568765\pi$
$733$	20.0000	−	34.6410i	0.738717	−	1.27950i	0.953073	−	0.302740i	$-0.0979013\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−12.0000			−0.442026
$738$		0			0
$739$	11.0000			0.404642			0.202321	−	0.979319i	$-0.435152\pi$
$739$	11.0000			0.404642			0.202321	+	0.979319i	$0.435152\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	12.0000	−	20.7846i	0.440237	−	0.762513i	−0.557470	−	0.830197i	$-0.688228\pi$
$743$	12.0000	−	20.7846i	0.440237	−	0.762513i	0.997707	+	0.0676840i	$0.0215610\pi$
$744$		0			0
$745$	−3.00000	−	5.19615i	−0.109911	−	0.190372i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	24.0000	+	41.5692i	0.876941	+	1.51891i
$750$		0			0
$751$	−11.5000	+	19.9186i	−0.419641	+	0.726839i	−0.995903	−	0.0904254i	$-0.971177\pi$
$751$	−11.5000	+	19.9186i	−0.419641	+	0.726839i	0.576262	+	0.817265i	$0.304511\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−16.0000			−0.582300
$756$		0			0
$757$	14.0000			0.508839			0.254419	−	0.967094i	$-0.418116\pi$
$757$	14.0000			0.508839			0.254419	+	0.967094i	$0.418116\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$	10.0000	+	17.3205i	0.362024	+	0.627044i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	24.0000	+	41.5692i	0.866590	+	1.50098i
$768$		0			0
$769$	−2.50000	+	4.33013i	−0.0901523	+	0.156148i	−0.907575	−	0.419890i	$-0.862069\pi$
$769$	−2.50000	+	4.33013i	−0.0901523	+	0.156148i	0.817423	+	0.576038i	$0.195402\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−9.00000			−0.323708			−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000			−0.323708			−0.161854	+	0.986815i	$0.551747\pi$
$774$		0			0
$775$	−7.00000			−0.251447
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−21.0000	+	36.3731i	−0.752403	+	1.30320i
$780$		0			0
$781$	−18.0000	−	31.1769i	−0.644091	−	1.11560i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	5.00000	+	8.66025i	0.178458	+	0.309098i
$786$		0			0
$787$	−13.0000	+	22.5167i	−0.463400	+	0.802632i	−0.999128	−	0.0417585i	$-0.986704\pi$
$787$	−13.0000	+	22.5167i	−0.463400	+	0.802632i	0.535728	+	0.844391i	$0.320037\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	24.0000			0.853342
$792$		0			0
$793$	28.0000			0.994309
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.50000	−	2.59808i	0.0531327	−	0.0920286i	−0.838236	−	0.545308i	$-0.816413\pi$
$797$	1.50000	−	2.59808i	0.0531327	−	0.0920286i	0.891368	+	0.453279i	$0.149746\pi$
$798$		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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1620.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{5} +(2.00000 + 3.46410i) q^{7} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{5} +(2.00000 + 3.46410i) q^{7} +(3.00000 + 5.19615i) q^{11} +(2.00000 - 3.46410i) q^{13} +3.00000 q^{17} -7.00000 q^{19} +(-4.50000 + 7.79423i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(3.50000 - 6.06218i) q^{31} -4.00000 q^{35} +2.00000 q^{37} +(3.00000 - 5.19615i) q^{41} +(-1.00000 - 1.73205i) q^{43} +(-4.50000 + 7.79423i) q^{49} -9.00000 q^{53} -6.00000 q^{55} +(-6.00000 + 10.3923i) q^{59} +(3.50000 + 6.06218i) q^{61} +(2.00000 + 3.46410i) q^{65} +(-1.00000 + 1.73205i) q^{67} -6.00000 q^{71} +2.00000 q^{73} +(-12.0000 + 20.7846i) q^{77} +(0.500000 + 0.866025i) q^{79} +(4.50000 + 7.79423i) q^{83} +(-1.50000 + 2.59808i) q^{85} +6.00000 q^{89} +16.0000 q^{91} +(3.50000 - 6.06218i) q^{95} +(-4.00000 - 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q - q^5 + 4 * q^7	\( 2 q - q^{5} + 4 q^{7} + 6 q^{11} + 4 q^{13} + 6 q^{17} - 14 q^{19} - 9 q^{23} - q^{25} + 7 q^{31} - 8 q^{35} + 4 q^{37} + 6 q^{41} - 2 q^{43} - 9 q^{49} - 18 q^{53} - 12 q^{55} - 12 q^{59} + 7 q^{61} + 4 q^{65} - 2 q^{67} - 12 q^{71} + 4 q^{73} - 24 q^{77} + q^{79} + 9 q^{83} - 3 q^{85} + 12 q^{89} + 32 q^{91} + 7 q^{95} - 8 q^{97}+O(q^{100}) \) 2 * q - q^5 + 4 * q^7 + 6 * q^11 + 4 * q^13 + 6 * q^17 - 14 * q^19 - 9 * q^23 - q^25 + 7 * q^31 - 8 * q^35 + 4 * q^37 + 6 * q^41 - 2 * q^43 - 9 * q^49 - 18 * q^53 - 12 * q^55 - 12 * q^59 + 7 * q^61 + 4 * q^65 - 2 * q^67 - 12 * q^71 + 4 * q^73 - 24 * q^77 + q^79 + 9 * q^83 - 3 * q^85 + 12 * q^89 + 32 * q^91 + 7 * q^95 - 8 * q^97

Introduction

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