LMFDB - Embedded newform 3024.2.t.l.289.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3024,2,Mod(289,3024)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3024, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 2])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3024.289"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$3024 = 2^{4} \cdot 3^{3} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3024.t (of order $3$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [22,0,0,0,6,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$24.1467615712$
Analytic rank:	$0$
Dimension:	$22$
Relative dimension:	$11$ over $\Q(\zeta_{3})$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.3
Character	$\chi$	$=$	3024.289
Dual form			3024.2.t.l.1873.3

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.85591 q^{5} +(2.60465 - 0.464545i) q^{7} -2.57601 q^{11} +(2.82227 + 4.88832i) q^{13} +(-3.57951 - 6.19989i) q^{17} +(-0.636599 + 1.10262i) q^{19} +0.241277 q^{23} -1.55558 q^{25} +(-0.923571 + 1.59967i) q^{29} +(-1.49552 + 2.59031i) q^{31} +(-4.83401 + 0.862156i) q^{35} +(0.338260 - 0.585884i) q^{37} +(0.733933 + 1.27121i) q^{41} +(-4.14269 + 7.17535i) q^{43} +(6.15723 + 10.6646i) q^{47} +(6.56840 - 2.41995i) q^{49} +(-3.35508 - 5.81117i) q^{53} +4.78085 q^{55} +(-1.04139 + 1.80375i) q^{59} +(-6.47973 - 11.2232i) q^{61} +(-5.23789 - 9.07230i) q^{65} +(-2.41551 + 4.18379i) q^{67} -1.53621 q^{71} +(-6.55954 - 11.3615i) q^{73} +(-6.70960 + 1.19667i) q^{77} +(-1.86009 - 3.22177i) q^{79} +(-3.00173 + 5.19915i) q^{83} +(6.64326 + 11.5065i) q^{85} +(-6.60349 + 11.4376i) q^{89} +(9.62187 + 11.4213i) q^{91} +(1.18147 - 2.04637i) q^{95} +(6.40860 - 11.1000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$22 q + 6 q^{5} - 7 q^{7} + 6 q^{11} - 3 q^{13} - 7 q^{17} + q^{19} - 4 q^{23} + 20 q^{25} - 9 q^{29} + 4 q^{31} + 14 q^{35} + 2 q^{37} - 16 q^{41} + 5 q^{47} - 15 q^{49} - 11 q^{53} - 22 q^{55} - 19 q^{59}+ \cdots - 29 q^{97}+O(q^{100})$ 22 * q + 6 * q^5 - 7 * q^7 + 6 * q^11 - 3 * q^13 - 7 * q^17 + q^19 - 4 * q^23 + 20 * q^25 - 9 * q^29 + 4 * q^31 + 14 * q^35 + 2 * q^37 - 16 * q^41 + 5 * q^47 - 15 * q^49 - 11 * q^53 - 22 * q^55 - 19 * q^59 - 13 * q^61 - 13 * q^65 - 26 * q^67 - 48 * q^71 - 35 * q^73 + 4 * q^77 - 10 * q^79 - 28 * q^83 - 20 * q^85 - 6 * q^89 + 37 * q^91 + 12 * q^95 - 29 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$ .

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−1.85591			−0.829990			−0.414995	−	0.909824i	$-0.636217\pi$
$5$	−1.85591			−0.829990			−0.414995	+	0.909824i	$0.636217\pi$
$6$		0			0
$7$	2.60465	−	0.464545i	0.984465	−	0.175582i
$7$	2.60465	−	0.464545i	0.984465	−	0.175582i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.57601			−0.776696			−0.388348	−	0.921513i	$-0.626954\pi$
$11$	−2.57601			−0.776696			−0.388348	+	0.921513i	$0.626954\pi$
$12$		0			0
$13$	2.82227	+	4.88832i	0.782757	+	1.35578i	0.930330	+	0.366724i	$0.119521\pi$
$13$	2.82227	+	4.88832i	0.782757	+	1.35578i	−0.147573	+	0.989051i	$0.547146\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.57951	−	6.19989i	−0.868158	−	1.50369i	−0.863876	−	0.503704i	$-0.831970\pi$
$17$	−3.57951	−	6.19989i	−0.868158	−	1.50369i	−0.00428199	−	0.999991i	$-0.501363\pi$
$18$		0			0
$19$	−0.636599	+	1.10262i	−0.146046	+	0.252959i	−0.929763	−	0.368160i	$-0.879988\pi$
$19$	−0.636599	+	1.10262i	−0.146046	+	0.252959i	0.783717	+	0.621118i	$0.213321\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.241277			0.0503098			0.0251549	−	0.999684i	$-0.491992\pi$
$23$	0.241277			0.0503098			0.0251549	+	0.999684i	$0.491992\pi$
$24$		0			0
$25$	−1.55558			−0.311116
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.923571	+	1.59967i	−0.171503	+	0.297051i	−0.938945	−	0.344066i	$-0.888196\pi$
$29$	−0.923571	+	1.59967i	−0.171503	+	0.297051i	0.767443	+	0.641118i	$0.221529\pi$
$30$		0			0
$31$	−1.49552	+	2.59031i	−0.268602	+	0.465233i	−0.968501	−	0.249009i	$-0.919895\pi$
$31$	−1.49552	+	2.59031i	−0.268602	+	0.465233i	0.699899	+	0.714242i	$0.253228\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.83401	+	0.862156i	−0.817096	+	0.145731i
$36$		0			0
$37$	0.338260	−	0.585884i	0.0556097	−	0.0963188i	−0.836880	−	0.547386i	$-0.815623\pi$
$37$	0.338260	−	0.585884i	0.0556097	−	0.0963188i	0.892490	+	0.451067i	$0.148956\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.733933	+	1.27121i	0.114621	+	0.198529i	0.917628	−	0.397440i	$-0.130101\pi$
$41$	0.733933	+	1.27121i	0.114621	+	0.198529i	−0.803007	+	0.595969i	$0.796768\pi$
$42$		0			0
$43$	−4.14269	+	7.17535i	−0.631754	+	1.09423i	0.355439	+	0.934700i	$0.384331\pi$
$43$	−4.14269	+	7.17535i	−0.631754	+	1.09423i	−0.987193	+	0.159531i	$0.949002\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.15723	+	10.6646i	0.898124	+	1.55560i	0.829890	+	0.557928i	$0.188403\pi$
$47$	6.15723	+	10.6646i	0.898124	+	1.55560i	0.0682346	+	0.997669i	$0.478263\pi$
$48$		0			0
$49$	6.56840	−	2.41995i	0.938342	−	0.345708i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.35508	−	5.81117i	−0.460856	−	0.798226i	0.538148	−	0.842851i	$-0.319124\pi$
$53$	−3.35508	−	5.81117i	−0.460856	−	0.798226i	−0.999004	+	0.0446243i	$0.985791\pi$
$54$		0			0
$55$	4.78085			0.644650
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.04139	+	1.80375i	−0.135578	+	0.234828i	−0.925818	−	0.377969i	$-0.876622\pi$
$59$	−1.04139	+	1.80375i	−0.135578	+	0.234828i	0.790240	+	0.612797i	$0.209956\pi$
$60$		0			0
$61$	−6.47973	−	11.2232i	−0.829644	−	1.43699i	−0.898317	−	0.439347i	$-0.855210\pi$
$61$	−6.47973	−	11.2232i	−0.829644	−	1.43699i	0.0686730	−	0.997639i	$-0.478123\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−5.23789	−	9.07230i	−0.649681	−	1.12528i
$66$		0			0
$67$	−2.41551	+	4.18379i	−0.295102	+	0.511131i	−0.975009	−	0.222167i	$-0.928687\pi$
$67$	−2.41551	+	4.18379i	−0.295102	+	0.511131i	0.679907	+	0.733298i	$0.262020\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.53621			−0.182314			−0.0911572	−	0.995837i	$-0.529057\pi$
$71$	−1.53621			−0.182314			−0.0911572	+	0.995837i	$0.529057\pi$
$72$		0			0
$73$	−6.55954	−	11.3615i	−0.767736	−	1.32976i	−0.938788	−	0.344496i	$-0.888050\pi$
$73$	−6.55954	−	11.3615i	−0.767736	−	1.32976i	0.171052	−	0.985262i	$-0.445283\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.70960	+	1.19667i	−0.764630	+	0.136373i
$78$		0			0
$79$	−1.86009	−	3.22177i	−0.209277	−	0.362478i	0.742210	−	0.670167i	$-0.233778\pi$
$79$	−1.86009	−	3.22177i	−0.209277	−	0.362478i	−0.951487	+	0.307689i	$0.900444\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.00173	+	5.19915i	−0.329483	+	0.570681i	−0.982409	−	0.186740i	$-0.940208\pi$
$83$	−3.00173	+	5.19915i	−0.329483	+	0.570681i	0.652926	+	0.757421i	$0.273541\pi$
$84$		0			0
$85$	6.64326	+	11.5065i	0.720563	+	1.24805i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.60349	+	11.4376i	−0.699968	+	1.21238i	0.268509	+	0.963277i	$0.413469\pi$
$89$	−6.60349	+	11.4376i	−0.699968	+	1.21238i	−0.968477	+	0.249103i	$0.919864\pi$
$90$		0			0
$91$	9.62187	+	11.4213i	1.00865	+	1.19728i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.18147	−	2.04637i	0.121217	−	0.209953i
$96$		0			0
$97$	6.40860	−	11.1000i	0.650695	−	1.12704i	−0.332260	−	0.943188i	$-0.607811\pi$
$97$	6.40860	−	11.1000i	0.650695	−	1.12704i	0.982955	−	0.183848i	$-0.0588556\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12.2013			−1.21408			−0.607039	−	0.794672i	$-0.707643\pi$
$101$	−12.2013			−1.21408			−0.607039	+	0.794672i	$0.707643\pi$
$102$		0			0
$103$	−13.6433			−1.34431			−0.672155	−	0.740411i	$-0.734631\pi$
$103$	−13.6433			−1.34431			−0.672155	+	0.740411i	$0.734631\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.48002	+	11.2237i	−0.626448	+	1.08504i	0.361811	+	0.932251i	$0.382158\pi$
$107$	−6.48002	+	11.2237i	−0.626448	+	1.08504i	−0.988259	+	0.152788i	$0.951175\pi$
$108$		0			0
$109$	7.70089	+	13.3383i	0.737612	+	1.27758i	0.953568	+	0.301178i	$0.0973799\pi$
$109$	7.70089	+	13.3383i	0.737612	+	1.27758i	−0.215956	+	0.976403i	$0.569287\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.73446	+	13.3965i	0.727597	+	1.26023i	0.957896	+	0.287115i	$0.0926961\pi$
$113$	7.73446	+	13.3965i	0.727597	+	1.26023i	−0.230299	+	0.973120i	$0.573971\pi$
$114$		0			0
$115$	−0.447790			−0.0417566
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−12.2035	−	14.4857i	−1.11869	−	1.32790i
$120$		0			0
$121$	−4.36418			−0.396744
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.1666			1.08821
$126$		0			0
$127$	3.19404			0.283425			0.141713	−	0.989908i	$-0.454739\pi$
$127$	3.19404			0.283425			0.141713	+	0.989908i	$0.454739\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	14.0868			1.23077			0.615383	−	0.788229i	$-0.289002\pi$
$131$	14.0868			1.23077			0.615383	+	0.788229i	$0.289002\pi$
$132$		0			0
$133$	−1.14590	+	3.16767i	−0.0993620	+	0.274672i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−13.6964			−1.17016			−0.585079	−	0.810976i	$-0.698937\pi$
$137$	−13.6964			−1.17016			−0.585079	+	0.810976i	$0.698937\pi$
$138$		0			0
$139$	4.94131	+	8.55859i	0.419116	+	0.725931i	0.995851	−	0.0910010i	$-0.0290067\pi$
$139$	4.94131	+	8.55859i	0.419116	+	0.725931i	−0.576735	+	0.816932i	$0.695673\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−7.27019	−	12.5923i	−0.607964	−	1.05302i
$144$		0			0
$145$	1.71407	−	2.96885i	0.142346	−	0.246550i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.92029			0.321163			0.160581	−	0.987023i	$-0.448663\pi$
$149$	3.92029			0.321163			0.160581	+	0.987023i	$0.448663\pi$
$150$		0			0
$151$	−19.5784			−1.59327			−0.796634	−	0.604462i	$-0.793388\pi$
$151$	−19.5784			−1.59327			−0.796634	+	0.604462i	$0.793388\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.77555	−	4.80739i	0.222937	−	0.386139i
$156$		0			0
$157$	−7.39637	+	12.8109i	−0.590295	+	1.02242i	0.403898	+	0.914804i	$0.367655\pi$
$157$	−7.39637	+	12.8109i	−0.590295	+	1.02242i	−0.994193	+	0.107616i	$0.965678\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.628443	−	0.112084i	0.0495282	−	0.00883347i
$162$		0			0
$163$	−7.54686	+	13.0715i	−0.591116	+	1.02384i	0.402967	+	0.915215i	$0.367979\pi$
$163$	−7.54686	+	13.0715i	−0.591116	+	1.02384i	−0.994082	+	0.108628i	$0.965354\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.92946	+	3.34192i	0.149306	+	0.258605i	0.930971	−	0.365093i	$-0.118963\pi$
$167$	1.92946	+	3.34192i	0.149306	+	0.258605i	−0.781665	+	0.623698i	$0.785629\pi$
$168$		0			0
$169$	−9.43043	+	16.3340i	−0.725418	+	1.25646i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.325786	+	0.564277i	0.0247690	+	0.0429012i	0.878144	−	0.478396i	$-0.158782\pi$
$173$	0.325786	+	0.564277i	0.0247690	+	0.0429012i	−0.853375	+	0.521297i	$0.825448\pi$
$174$		0			0
$175$	−4.05174	+	0.722638i	−0.306283	+	0.0546263i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	10.9059	+	18.8896i	0.815145	+	1.41187i	0.909223	+	0.416308i	$0.136676\pi$
$179$	10.9059	+	18.8896i	0.815145	+	1.41187i	−0.0940781	+	0.995565i	$0.529990\pi$
$180$		0			0
$181$	−25.0338			−1.86075			−0.930374	−	0.366613i	$-0.880517\pi$
$181$	−25.0338			−1.86075			−0.930374	+	0.366613i	$0.880517\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.627782	+	1.08735i	−0.0461555	+	0.0799436i
$186$		0			0
$187$	9.22085	+	15.9710i	0.674295	+	1.16791i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.33036	−	7.50041i	−0.313334	−	0.542711i	0.665748	−	0.746177i	$-0.268113\pi$
$191$	−4.33036	−	7.50041i	−0.313334	−	0.542711i	−0.979082	+	0.203466i	$0.934779\pi$
$192$		0			0
$193$	−0.808322	+	1.40006i	−0.0581843	+	0.100778i	−0.893650	−	0.448764i	$-0.851864\pi$
$193$	−0.808322	+	1.40006i	−0.0581843	+	0.100778i	0.835466	+	0.549542i	$0.185198\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−10.7746			−0.767659			−0.383829	−	0.923404i	$-0.625395\pi$
$197$	−10.7746			−0.767659			−0.383829	+	0.923404i	$0.625395\pi$
$198$		0			0
$199$	−2.38768	−	4.13558i	−0.169258	−	0.293163i	0.768901	−	0.639368i	$-0.220804\pi$
$199$	−2.38768	−	4.13558i	−0.169258	−	0.293163i	−0.938159	+	0.346204i	$0.887470\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.66246	+	4.59562i	−0.116682	+	0.322549i
$204$		0			0
$205$	−1.36212	−	2.35925i	−0.0951343	−	0.164777i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.63988	−	2.84036i	0.113433	−	0.196472i
$210$		0			0
$211$	−2.42787	−	4.20520i	−0.167142	−	0.289498i	0.770272	−	0.637715i	$-0.220120\pi$
$211$	−2.42787	−	4.20520i	−0.167142	−	0.289498i	−0.937414	+	0.348218i	$0.886787\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	7.68848	−	13.3168i	0.524350	−	0.908200i
$216$		0			0
$217$	−2.69198	+	7.44158i	−0.182743	+	0.505167i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	20.2047	−	34.9955i	1.35911	−	2.35406i
$222$		0			0
$223$	−3.86187	+	6.68896i	−0.258610	+	0.447926i	−0.965870	−	0.259028i	$-0.916598\pi$
$223$	−3.86187	+	6.68896i	−0.258610	+	0.447926i	0.707260	+	0.706954i	$0.249931\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−13.9491			−0.925837			−0.462919	−	0.886401i	$-0.653198\pi$
$227$	−13.9491			−0.925837			−0.462919	+	0.886401i	$0.653198\pi$
$228$		0			0
$229$	1.60027			0.105749			0.0528745	−	0.998601i	$-0.483162\pi$
$229$	1.60027			0.105749			0.0528745	+	0.998601i	$0.483162\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3.69939	+	6.40753i	−0.242355	+	0.419771i	−0.961385	−	0.275208i	$-0.911253\pi$
$233$	−3.69939	+	6.40753i	−0.242355	+	0.419771i	0.719030	+	0.694979i	$0.244587\pi$
$234$		0			0
$235$	−11.4273	−	19.7926i	−0.745434	−	1.29113i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.25117	−	2.16709i	−0.0809316	−	0.140178i	0.822719	−	0.568449i	$-0.192456\pi$
$239$	−1.25117	−	2.16709i	−0.0809316	−	0.140178i	−0.903650	+	0.428271i	$0.859123\pi$
$240$		0			0
$241$	4.24297			0.273314			0.136657	−	0.990618i	$-0.456364\pi$
$241$	4.24297			0.273314			0.136657	+	0.990618i	$0.456364\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−12.1904	+	4.49123i	−0.778815	+	0.286934i
$246$		0			0
$247$	−7.18661			−0.457273
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−13.5381			−0.854516			−0.427258	−	0.904130i	$-0.640520\pi$
$251$	−13.5381			−0.854516			−0.427258	+	0.904130i	$0.640520\pi$
$252$		0			0
$253$	−0.621532			−0.0390754
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.15495			0.383935			0.191968	−	0.981401i	$-0.438513\pi$
$257$	6.15495			0.383935			0.191968	+	0.981401i	$0.438513\pi$
$258$		0			0
$259$	0.608880	−	1.68316i	0.0378340	−	0.104586i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−25.3411			−1.56260			−0.781300	−	0.624156i	$-0.785443\pi$
$263$	−25.3411			−1.56260			−0.781300	+	0.624156i	$0.785443\pi$
$264$		0			0
$265$	6.22675	+	10.7850i	0.382506	+	0.662520i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5.42092	+	9.38931i	0.330519	+	0.572476i	0.982614	−	0.185662i	$-0.0594428\pi$
$269$	5.42092	+	9.38931i	0.330519	+	0.572476i	−0.652095	+	0.758138i	$0.726109\pi$
$270$		0			0
$271$	15.0184	−	26.0127i	0.912306	−	1.58016i	0.101507	−	0.994835i	$-0.467634\pi$
$271$	15.0184	−	26.0127i	0.912306	−	1.58016i	0.810799	−	0.585325i	$-0.199033\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.00719			0.241643
$276$		0			0
$277$	19.7629			1.18744			0.593720	−	0.804672i	$-0.297659\pi$
$277$	19.7629			1.18744			0.593720	+	0.804672i	$0.297659\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3.98596	−	6.90388i	0.237782	−	0.411851i	−0.722295	−	0.691585i	$-0.756913\pi$
$281$	3.98596	−	6.90388i	0.237782	−	0.411851i	0.960078	+	0.279734i	$0.0902462\pi$
$282$		0			0
$283$	11.6063	−	20.1028i	0.689926	−	1.19499i	−0.281936	−	0.959433i	$-0.590977\pi$
$283$	11.6063	−	20.1028i	0.689926	−	1.19499i	0.971861	−	0.235553i	$-0.0756901\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.50217	+	2.97011i	0.147698	+	0.175320i
$288$		0			0
$289$	−17.1258	+	29.6627i	−1.00740	+	1.74486i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−11.8556	−	20.5345i	−0.692612	−	1.19964i	−0.970979	−	0.239164i	$-0.923126\pi$
$293$	−11.8556	−	20.5345i	−0.692612	−	1.19964i	0.278367	−	0.960475i	$-0.410207\pi$
$294$		0			0
$295$	1.93274	−	3.34760i	0.112528	−	0.194905i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.680950	+	1.17944i	0.0393803	+	0.0682088i
$300$		0			0
$301$	−7.45698	+	20.6137i	−0.429813	+	1.18816i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	12.0258	+	20.8293i	0.688597	+	1.19268i
$306$		0			0
$307$	−3.87810			−0.221335			−0.110668	−	0.993857i	$-0.535299\pi$
$307$	−3.87810			−0.221335			−0.110668	+	0.993857i	$0.535299\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.46220	−	5.99670i	0.196323	−	0.340042i	−0.751010	−	0.660290i	$-0.770433\pi$
$311$	3.46220	−	5.99670i	0.196323	−	0.340042i	0.947333	+	0.320249i	$0.103767\pi$
$312$		0			0
$313$	−15.1157	−	26.1811i	−0.854388	−	1.47984i	−0.877212	−	0.480104i	$-0.840599\pi$
$313$	−15.1157	−	26.1811i	−0.854388	−	1.47984i	0.0228236	−	0.999740i	$-0.492734\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4.68699	+	8.11811i	0.263248	+	0.455959i	0.967103	−	0.254385i	$-0.0818730\pi$
$317$	4.68699	+	8.11811i	0.263248	+	0.455959i	−0.703855	+	0.710343i	$0.748540\pi$
$318$		0			0
$319$	2.37913	−	4.12077i	0.133205	−	0.230719i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	9.11484			0.507163
$324$		0			0
$325$	−4.39027	−	7.60418i	−0.243529	−	0.421804i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	20.9916	+	24.9173i	1.15731	+	1.37374i
$330$		0			0
$331$	13.7720	+	23.8539i	0.756979	+	1.31113i	0.944384	+	0.328844i	$0.106659\pi$
$331$	13.7720	+	23.8539i	0.756979	+	1.31113i	−0.187405	+	0.982283i	$0.560008\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.48298	−	7.76475i	0.244931	−	0.424234i
$336$		0			0
$337$	−3.41673	−	5.91796i	−0.186121	−	0.322372i	0.757832	−	0.652449i	$-0.226258\pi$
$337$	−3.41673	−	5.91796i	−0.186121	−	0.322372i	−0.943954	+	0.330078i	$0.892925\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.85246	−	6.67266i	0.208622	−	0.361345i
$342$		0			0
$343$	15.9842	−	9.35445i	0.863065	−	0.505093i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.0959	+	17.4867i	−0.541979	+	0.938735i	0.456812	+	0.889564i	$0.348991\pi$
$347$	−10.0959	+	17.4867i	−0.541979	+	0.938735i	−0.998790	+	0.0491714i	$0.984342\pi$
$348$		0			0
$349$	−4.25154	+	7.36388i	−0.227580	+	0.394180i	−0.957090	−	0.289790i	$-0.906415\pi$
$349$	−4.25154	+	7.36388i	−0.227580	+	0.394180i	0.729511	+	0.683970i	$0.239748\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4.70904			−0.250637			−0.125318	−	0.992117i	$-0.539995\pi$
$353$	−4.70904			−0.250637			−0.125318	+	0.992117i	$0.539995\pi$
$354$		0			0
$355$	2.85107			0.151319
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6.03357	−	10.4504i	0.318440	−	0.551554i	−0.661723	−	0.749748i	$-0.730175\pi$
$359$	6.03357	−	10.4504i	0.318440	−	0.551554i	0.980163	+	0.198195i	$0.0635079\pi$
$360$		0			0
$361$	8.68948	+	15.0506i	0.457341	+	0.792138i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	12.1739	+	21.0859i	0.637213	+	1.10369i
$366$		0			0
$367$	−0.960711			−0.0501487			−0.0250744	−	0.999686i	$-0.507982\pi$
$367$	−0.960711			−0.0501487			−0.0250744	+	0.999686i	$0.507982\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−11.4384	−	13.5775i	−0.593850	−	0.704908i
$372$		0			0
$373$	−7.04998			−0.365034			−0.182517	−	0.983203i	$-0.558425\pi$
$373$	−7.04998			−0.365034			−0.182517	+	0.983203i	$0.558425\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−10.4263			−0.536980
$378$		0			0
$379$	37.1330			1.90739			0.953697	−	0.300769i	$-0.0972434\pi$
$379$	37.1330			1.90739			0.953697	+	0.300769i	$0.0972434\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	32.1975			1.64522			0.822608	−	0.568609i	$-0.192518\pi$
$383$	32.1975			1.64522			0.822608	+	0.568609i	$0.192518\pi$
$384$		0			0
$385$	12.4524	−	2.22092i	0.634635	−	0.113189i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	25.7426			1.30520			0.652600	−	0.757702i	$-0.273678\pi$
$389$	25.7426			1.30520			0.652600	+	0.757702i	$0.273678\pi$
$390$		0			0
$391$	−0.863654	−	1.49589i	−0.0436769	−	0.0756506i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.45217	+	5.97933i	0.173697	+	0.300853i
$396$		0			0
$397$	9.44903	−	16.3662i	0.474233	−	0.821396i	−0.525332	−	0.850898i	$-0.676059\pi$
$397$	9.44903	−	16.3662i	0.474233	−	0.821396i	0.999565	+	0.0295016i	$0.00939202\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−15.2039			−0.759245			−0.379622	−	0.925142i	$-0.623946\pi$
$401$	−15.2039			−0.759245			−0.379622	+	0.925142i	$0.623946\pi$
$402$		0			0
$403$	−16.8830			−0.841002
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.871362	+	1.50924i	−0.0431918	+	0.0748104i
$408$		0			0
$409$	14.9729	−	25.9339i	0.740363	−	1.28235i	−0.211967	−	0.977277i	$-0.567987\pi$
$409$	14.9729	−	25.9339i	0.740363	−	1.28235i	0.952330	−	0.305070i	$-0.0986798\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.87454	+	5.18191i	−0.0922403	+	0.254985i
$414$		0			0
$415$	5.57096	−	9.64918i	0.273468	−	0.473660i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.2660	−	21.2453i	−0.599231	−	1.03790i	−0.992935	−	0.118661i	$-0.962140\pi$
$419$	−12.2660	−	21.2453i	−0.599231	−	1.03790i	0.393704	−	0.919237i	$-0.371194\pi$
$420$		0			0
$421$	−2.37791	+	4.11866i	−0.115892	+	0.200731i	−0.918136	−	0.396265i	$-0.870306\pi$
$421$	−2.37791	+	4.11866i	−0.115892	+	0.200731i	0.802244	+	0.596996i	$0.203639\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5.56822	+	9.64444i	0.270098	+	0.467824i
$426$		0			0
$427$	−22.0911	−	26.2224i	−1.06906	−	1.26899i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.36446	+	2.36331i	0.0657237	+	0.113837i	0.897015	−	0.442000i	$-0.145731\pi$
$431$	1.36446	+	2.36331i	0.0657237	+	0.113837i	−0.831291	+	0.555837i	$0.812398\pi$
$432$		0			0
$433$	14.5592			0.699672			0.349836	−	0.936811i	$-0.386237\pi$
$433$	14.5592			0.699672			0.349836	+	0.936811i	$0.386237\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.153597	+	0.266037i	−0.00734753	+	0.0127263i
$438$		0			0
$439$	−1.44066	−	2.49529i	−0.0687587	−	0.119094i	0.829596	−	0.558363i	$-0.188571\pi$
$439$	−1.44066	−	2.49529i	−0.0687587	−	0.119094i	−0.898355	+	0.439270i	$0.855237\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.4865	−	21.6273i	−0.593254	−	1.02755i	−0.993791	−	0.111265i	$-0.964510\pi$
$443$	−12.4865	−	21.6273i	−0.593254	−	1.02755i	0.400537	−	0.916281i	$-0.368824\pi$
$444$		0			0
$445$	12.2555	−	21.2272i	0.580967	−	1.00626i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.99154			0.141180			0.0705898	−	0.997505i	$-0.477512\pi$
$449$	2.99154			0.141180			0.0705898	+	0.997505i	$0.477512\pi$
$450$		0			0
$451$	−1.89062	−	3.27464i	−0.0890257	−	0.154197i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−17.8574	−	21.1969i	−0.837166	−	0.993727i
$456$		0			0
$457$	12.8085	+	22.1850i	0.599158	+	1.03777i	0.992946	+	0.118571i	$0.0378312\pi$
$457$	12.8085	+	22.1850i	0.599158	+	1.03777i	−0.393788	+	0.919201i	$0.628835\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.45759	−	11.1849i	0.300760	−	0.520931i	−0.675548	−	0.737316i	$-0.736093\pi$
$461$	6.45759	−	11.1849i	0.300760	−	0.520931i	0.976308	+	0.216384i	$0.0694264\pi$
$462$		0			0
$463$	12.2457	+	21.2102i	0.569108	+	0.985724i	0.996654	+	0.0817305i	$0.0260447\pi$
$463$	12.2457	+	21.2102i	0.569108	+	0.985724i	−0.427547	+	0.903993i	$0.640622\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.4087	+	18.0283i	−0.481655	+	0.834251i	−0.999778	−	0.0210550i	$-0.993297\pi$
$467$	−10.4087	+	18.0283i	−0.481655	+	0.834251i	0.518123	+	0.855306i	$0.326631\pi$
$468$		0			0
$469$	−4.34800	+	12.0194i	−0.200772	+	0.555005i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10.6716	−	18.4838i	0.490681	−	0.849884i
$474$		0			0
$475$	0.990281	−	1.71522i	0.0454372	−	0.0786996i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	27.4873			1.25592			0.627962	−	0.778244i	$-0.283889\pi$
$479$	27.4873			1.25592			0.627962	+	0.778244i	$0.283889\pi$
$480$		0			0
$481$	3.81865			0.174115
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−11.8938	+	20.6007i	−0.540070	+	0.935429i
$486$		0			0
$487$	6.32927	+	10.9626i	0.286807	+	0.496763i	0.973046	−	0.230613i	$-0.0740730\pi$
$487$	6.32927	+	10.9626i	0.286807	+	0.496763i	−0.686239	+	0.727376i	$0.740740\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.40618	−	2.43557i	−0.0634598	−	0.109916i	0.832550	−	0.553950i	$-0.186880\pi$
$491$	−1.40618	−	2.43557i	−0.0634598	−	0.109916i	−0.896010	+	0.444034i	$0.853547\pi$
$492$		0			0
$493$	13.2237			0.595566
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.00128	+	0.713638i	−0.179482	+	0.0320110i
$498$		0			0
$499$	4.24205			0.189900			0.0949502	−	0.995482i	$-0.469731\pi$
$499$	4.24205			0.189900			0.0949502	+	0.995482i	$0.469731\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	22.2162			0.990570			0.495285	−	0.868730i	$-0.335064\pi$
$503$	22.2162			0.990570			0.495285	+	0.868730i	$0.335064\pi$
$504$		0			0
$505$	22.6446			1.00767
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4.85469			0.215180			0.107590	−	0.994195i	$-0.465687\pi$
$509$	4.85469			0.215180			0.107590	+	0.994195i	$0.465687\pi$
$510$		0			0
$511$	−22.3632	−	26.5454i	−0.989290	−	1.17430i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	25.3207			1.11576
$516$		0			0
$517$	−15.8611	−	27.4722i	−0.697569	−	1.20823i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−7.92316	−	13.7233i	−0.347120	−	0.601229i	0.638617	−	0.769525i	$-0.279507\pi$
$521$	−7.92316	−	13.7233i	−0.347120	−	0.601229i	−0.985737	+	0.168296i	$0.946174\pi$
$522$		0			0
$523$	−10.7605	+	18.6377i	−0.470524	+	0.814972i	−0.999432	−	0.0337078i	$-0.989268\pi$
$523$	−10.7605	+	18.6377i	−0.470524	+	0.814972i	0.528908	+	0.848679i	$0.322602\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	21.4128			0.932758
$528$		0			0
$529$	−22.9418			−0.997469
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.14271	+	7.17539i	−0.179441	+	0.310801i
$534$		0			0
$535$	12.0264	−	20.8303i	0.519945	−	0.900572i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−16.9202	+	6.23382i	−0.728806	+	0.268510i
$540$		0			0
$541$	7.55977	−	13.0939i	0.325020	−	0.562951i	−0.656497	−	0.754329i	$-0.727962\pi$
$541$	7.55977	−	13.0939i	0.325020	−	0.562951i	0.981516	+	0.191378i	$0.0612957\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−14.2922	−	24.7548i	−0.612211	−	1.06038i
$546$		0			0
$547$	19.4532	−	33.6939i	0.831757	−	1.44065i	−0.0648863	−	0.997893i	$-0.520668\pi$
$547$	19.4532	−	33.6939i	0.831757	−	1.44065i	0.896644	−	0.442753i	$-0.145998\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.17589	−	2.03670i	−0.0500945	−	0.0867662i
$552$		0			0
$553$	−6.34154	−	7.52749i	−0.269670	−	0.320101i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5.37036	+	9.30173i	0.227549	+	0.394127i	0.957081	−	0.289820i	$-0.0935954\pi$
$557$	5.37036	+	9.30173i	0.227549	+	0.394127i	−0.729532	+	0.683947i	$0.760262\pi$
$558$		0			0
$559$	−46.7672			−1.97804
$560$		0			0
$561$		0			0
$562$		0			0
$563$	11.7380	−	20.3308i	0.494697	−	0.856840i	−0.505285	−	0.862953i	$-0.668612\pi$
$563$	11.7380	−	20.3308i	0.494697	−	0.856840i	0.999981	+	0.00611281i	$0.00194578\pi$
$564$		0			0
$565$	−14.3545	−	24.8627i	−0.603898	−	1.04598i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−18.9681	−	32.8537i	−0.795183	−	1.37730i	−0.922723	−	0.385463i	$-0.874042\pi$
$569$	−18.9681	−	32.8537i	−0.795183	−	1.37730i	0.127540	−	0.991833i	$-0.459292\pi$
$570$		0			0
$571$	−2.15815	+	3.73803i	−0.0903158	+	0.156432i	−0.907644	−	0.419741i	$-0.862121\pi$
$571$	−2.15815	+	3.73803i	−0.0903158	+	0.156432i	0.817328	+	0.576172i	$0.195454\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.375326			−0.0156522
$576$		0			0
$577$	−5.05923	−	8.76284i	−0.210618	−	0.364802i	0.741290	−	0.671185i	$-0.234214\pi$
$577$	−5.05923	−	8.76284i	−0.210618	−	0.364802i	−0.951908	+	0.306383i	$0.900881\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−5.40322	+	14.9364i	−0.224163	+	0.619667i
$582$		0			0
$583$	8.64272	+	14.9696i	0.357945	+	0.619979i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.10992	+	7.11859i	−0.169635	+	0.293816i	−0.938291	−	0.345846i	$-0.887592\pi$
$587$	−4.10992	+	7.11859i	−0.169635	+	0.293816i	0.768657	+	0.639661i	$0.220925\pi$
$588$		0			0
$589$	−1.90409	−	3.29797i	−0.0784565	−	0.135891i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−21.8434	+	37.8339i	−0.897002	+	1.55365i	−0.0656957	+	0.997840i	$0.520927\pi$
$593$	−21.8434	+	37.8339i	−0.897002	+	1.55365i	−0.831307	+	0.555814i	$0.812407\pi$
$594$		0			0
$595$	22.6486	+	26.8842i	0.928504	+	1.10215i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7.63946	−	13.2319i	0.312140	−	0.540642i	−0.666686	−	0.745339i	$-0.732288\pi$
$599$	7.63946	−	13.2319i	0.312140	−	0.540642i	0.978825	+	0.204697i	$0.0656209\pi$
$600$		0			0
$601$	7.65696	−	13.2622i	0.312334	−	0.540978i	−0.666533	−	0.745475i	$-0.732223\pi$
$601$	7.65696	−	13.2622i	0.312334	−	0.540978i	0.978867	+	0.204497i	$0.0655559\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	8.09955			0.329293
$606$		0			0
$607$	2.66981			0.108364			0.0541821	−	0.998531i	$-0.482745\pi$
$607$	2.66981			0.108364			0.0541821	+	0.998531i	$0.482745\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−34.7547	+	60.1970i	−1.40603	+	2.43531i
$612$		0			0
$613$	−13.5875	−	23.5343i	−0.548796	−	0.950542i	−0.998357	−	0.0572929i	$-0.981753\pi$
$613$	−13.5875	−	23.5343i	−0.548796	−	0.950542i	0.449562	−	0.893249i	$-0.351580\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	17.6058	+	30.4942i	0.708785	+	1.22765i	0.965308	+	0.261113i	$0.0840895\pi$
$617$	17.6058	+	30.4942i	0.708785	+	1.22765i	−0.256524	+	0.966538i	$0.582577\pi$
$618$		0			0
$619$	31.2681			1.25677			0.628385	−	0.777902i	$-0.283716\pi$
$619$	31.2681			1.25677			0.628385	+	0.777902i	$0.283716\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−11.8865	+	32.8585i	−0.476223	+	1.31645i
$624$		0			0
$625$	−14.8023			−0.592090
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4.84322			−0.193112
$630$		0			0
$631$	−15.5090			−0.617403			−0.308702	−	0.951159i	$-0.599894\pi$
$631$	−15.5090			−0.617403			−0.308702	+	0.951159i	$0.599894\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.92787			−0.235240
$636$		0			0
$637$	30.3673	+	25.2786i	1.20320	+	1.00158i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	33.5310			1.32440			0.662198	−	0.749329i	$-0.269624\pi$
$641$	33.5310			1.32440			0.662198	+	0.749329i	$0.269624\pi$
$642$		0			0
$643$	10.2721	+	17.7918i	0.405093	+	0.701641i	0.994332	−	0.106317i	$-0.0339059\pi$
$643$	10.2721	+	17.7918i	0.405093	+	0.701641i	−0.589239	+	0.807958i	$0.700573\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−16.8855	−	29.2465i	−0.663836	−	1.14980i	−0.979599	−	0.200960i	$-0.935594\pi$
$647$	−16.8855	−	29.2465i	−0.663836	−	1.14980i	0.315763	−	0.948838i	$-0.397739\pi$
$648$		0			0
$649$	2.68264	−	4.64647i	0.105303	−	0.182390i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−18.0115			−0.704845			−0.352423	−	0.935841i	$-0.614642\pi$
$653$	−18.0115			−0.704845			−0.352423	+	0.935841i	$0.614642\pi$
$654$		0			0
$655$	−26.1438			−1.02152
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.42710	−	2.47180i	0.0555918	−	0.0962878i	−0.836890	−	0.547371i	$-0.815629\pi$
$659$	1.42710	−	2.47180i	0.0555918	−	0.0962878i	0.892482	+	0.451083i	$0.148962\pi$
$660$		0			0
$661$	−7.02746	+	12.1719i	−0.273337	+	0.473433i	−0.969714	−	0.244243i	$-0.921461\pi$
$661$	−7.02746	+	12.1719i	−0.273337	+	0.473433i	0.696378	+	0.717676i	$0.254794\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2.12669	−	5.87892i	0.0824695	−	0.227975i
$666$		0			0
$667$	−0.222837	+	0.385964i	−0.00862827	+	0.0149446i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	16.6918	+	28.9111i	0.644381	+	1.11610i
$672$		0			0
$673$	−7.54157	+	13.0624i	−0.290706	+	0.503518i	−0.973977	−	0.226647i	$-0.927223\pi$
$673$	−7.54157	+	13.0624i	−0.290706	+	0.503518i	0.683271	+	0.730165i	$0.260557\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.1093	−	31.3663i	−0.695998	−	1.20550i	−0.969843	−	0.243731i	$-0.921629\pi$
$677$	−18.1093	−	31.3663i	−0.695998	−	1.20550i	0.273845	−	0.961774i	$-0.411705\pi$
$678$		0			0
$679$	11.5357	−	31.8887i	0.442699	−	1.22378i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	8.84350	+	15.3174i	0.338387	+	0.586104i	0.984130	−	0.177452i	$-0.0567853\pi$
$683$	8.84350	+	15.3174i	0.338387	+	0.586104i	−0.645742	+	0.763555i	$0.723452\pi$
$684$		0			0
$685$	25.4193			0.971220
$686$		0			0
$687$		0			0
$688$		0			0
$689$	18.9379	−	32.8014i	0.721477	−	1.24963i
$690$		0			0
$691$	11.2049	+	19.4074i	0.426253	+	0.738292i	0.996537	−	0.0831559i	$-0.0265000\pi$
$691$	11.2049	+	19.4074i	0.426253	+	0.738292i	−0.570283	+	0.821448i	$0.693167\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−9.17064	−	15.8840i	−0.347862	−	0.602515i
$696$		0			0
$697$	5.25424	−	9.10061i	0.199018	−	0.344710i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	31.1776			1.17756			0.588781	−	0.808293i	$-0.299608\pi$
$701$	31.1776			1.17756			0.588781	+	0.808293i	$0.299608\pi$
$702$		0			0
$703$	0.430672	+	0.745946i	0.0162431	+	0.0281339i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−31.7802	+	5.66806i	−1.19522	+	0.213170i
$708$		0			0
$709$	4.02492	+	6.97137i	0.151159	+	0.261815i	0.931654	−	0.363347i	$-0.118366\pi$
$709$	4.02492	+	6.97137i	0.151159	+	0.261815i	−0.780495	+	0.625162i	$0.785033\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.360834	+	0.624982i	−0.0135133	+	0.0234058i
$714$		0			0
$715$	13.4929	+	23.3703i	0.504604	+	0.874000i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−20.9980	+	36.3696i	−0.783093	+	1.35636i	0.147039	+	0.989131i	$0.453026\pi$
$719$	−20.9980	+	36.3696i	−0.783093	+	1.35636i	−0.930132	+	0.367226i	$0.880308\pi$
$720$		0			0
$721$	−35.5359	+	6.33791i	−1.32343	+	0.236036i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.43669	−	2.48842i	0.0533573	−	0.0924176i
$726$		0			0
$727$	−0.668774	+	1.15835i	−0.0248035	+	0.0429609i	−0.878161	−	0.478366i	$-0.841229\pi$
$727$	−0.668774	+	1.15835i	−0.0248035	+	0.0429609i	0.853357	+	0.521327i	$0.174563\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	59.3152			2.19385
$732$		0			0
$733$	29.4749			1.08868			0.544340	−	0.838865i	$-0.316780\pi$
$733$	29.4749			1.08868			0.544340	+	0.838865i	$0.316780\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	6.22238	−	10.7775i	0.229204	−	0.396993i
$738$		0			0
$739$	−9.52146	−	16.4916i	−0.350252	−	0.606655i	0.636041	−	0.771655i	$-0.280571\pi$
$739$	−9.52146	−	16.4916i	−0.350252	−	0.606655i	−0.986294	+	0.165000i	$0.947238\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	21.6613	+	37.5185i	0.794676	+	1.37642i	0.923045	+	0.384693i	$0.125693\pi$
$743$	21.6613	+	37.5185i	0.794676	+	1.37642i	−0.128369	+	0.991726i	$0.540974\pi$
$744$		0			0
$745$	−7.27573			−0.266562
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−11.6643	+	32.2441i	−0.426203	+	1.17818i
$750$		0			0
$751$	−34.8763			−1.27265			−0.636327	−	0.771420i	$-0.719547\pi$
$751$	−34.8763			−1.27265			−0.636327	+	0.771420i	$0.719547\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	36.3358			1.32240
$756$		0			0
$757$	8.67255			0.315209			0.157605	−	0.987502i	$-0.449623\pi$
$757$	8.67255			0.315209			0.157605	+	0.987502i	$0.449623\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	5.48977			0.199004			0.0995021	−	0.995037i	$-0.468275\pi$
$761$	5.48977			0.199004			0.0995021	+	0.995037i	$0.468275\pi$
$762$		0			0
$763$	26.2544	+	31.1643i	0.950473	+	1.12822i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−11.7564			−0.424499
$768$		0			0
$769$	1.81365	+	3.14134i	0.0654021	+	0.113280i	0.896872	−	0.442290i	$-0.145834\pi$
$769$	1.81365	+	3.14134i	0.0654021	+	0.113280i	−0.831470	+	0.555569i	$0.812500\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6.96717	+	12.0675i	0.250592	+	0.434037i	0.963689	−	0.267028i	$-0.0860415\pi$
$773$	6.96717	+	12.0675i	0.250592	+	0.434037i	−0.713097	+	0.701065i	$0.752708\pi$
$774$		0			0
$775$	2.32640	−	4.02944i	0.0835666	−	0.144742i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.86888			−0.0669597
$780$		0			0
$781$	3.95729			0.141603
$782$		0			0
$783$		0			0
$784$		0			0
$785$	13.7270	−	23.7759i	0.489939	−	0.848599i
$786$		0			0
$787$	8.78923	−	15.2234i	0.313302	−	0.542655i	−0.665773	−	0.746154i	$-0.731898\pi$
$787$	8.78923	−	15.2234i	0.313302	−	0.542655i	0.979075	+	0.203499i	$0.0652314\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	26.3688	+	31.3001i	0.937567	+	1.11290i
$792$		0			0
$793$	36.5751	−	63.3499i	1.29882	−	2.24962i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−5.57971	−	9.66434i	−0.197644	−	0.342329i	0.750120	−	0.661301i	$-0.229996\pi$
$797$	−5.57971	−	9.66434i	−0.197644	−	0.342329i	−0.947764	+	0.318973i	$0.896662\pi$
$798$		0			0
$799$	44.0797	−	76.3483i	1.55943	−	2.70101i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	16.8974	+	29.2672i	0.596297	+	1.03282i
$804$		0			0
$805$	−1.16634	+	0.208019i	−0.0411079	+	0.00733169i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−14.3481	−	24.8517i	−0.504453	−	0.873738i	−0.999987	−	0.00514935i	$-0.998361\pi$
$809$	−14.3481	−	24.8517i	−0.504453	−	0.873738i	0.495534	−	0.868589i	$-0.334972\pi$
$810$		0			0
$811$	−47.1695			−1.65635			−0.828173	−	0.560473i	$-0.810620\pi$
$811$	−47.1695			−1.65635			−0.828173	+	0.560473i	$0.810620\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	14.0063	−	24.2597i	0.490620	−	0.849779i
$816$		0			0
$817$	−5.27446	−	9.13563i	−0.184530	−	0.319615i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−19.3854	−	33.5765i	−0.676554	−	1.17183i	−0.976012	−	0.217717i	$-0.930139\pi$
$821$	−19.3854	−	33.5765i	−0.676554	−	1.17183i	0.299458	−	0.954110i	$-0.403194\pi$
$822$		0			0
$823$	−11.3920	+	19.7316i	−0.397101	+	0.687799i	−0.993367	−	0.114988i	$-0.963317\pi$
$823$	−11.3920	+	19.7316i	−0.397101	+	0.687799i	0.596266	+	0.802787i	$0.296650\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.55152			−0.0539515			−0.0269758	−	0.999636i	$-0.508588\pi$
$827$	−1.55152			−0.0539515			−0.0269758	+	0.999636i	$0.508588\pi$
$828$		0			0
$829$	−23.8972	−	41.3911i	−0.829983	−	1.43757i	−0.898051	−	0.439892i	$-0.855017\pi$
$829$	−23.8972	−	41.3911i	−0.829983	−	1.43757i	0.0680673	−	0.997681i	$-0.478317\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−38.5151	−	32.0611i	−1.33447	−	1.11085i
$834$		0			0
$835$	−3.58090	−	6.20231i	−0.123922	−	0.214640i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	19.5804	−	33.9142i	0.675990	−	1.17085i	−0.300188	−	0.953880i	$-0.597049\pi$
$839$	19.5804	−	33.9142i	0.675990	−	1.17085i	0.976178	−	0.216970i	$-0.0696173\pi$
$840$		0			0
$841$	12.7940	+	22.1599i	0.441174	+	0.764135i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	17.5021	−	30.3145i	0.602089	−	1.04285i
$846$		0			0
$847$	−11.3672	+	2.02736i	−0.390580	+	0.0696609i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	0.0816145	−	0.141361i	0.00279771	−	0.00484578i
$852$		0			0
$853$	14.5234	−	25.1552i	0.497270	−	0.861298i	−0.502725	−	0.864447i	$-0.667669\pi$
$853$	14.5234	−	25.1552i	0.497270	−	0.861298i	0.999995	+	0.00314895i	$0.00100234\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	48.2584			1.64848			0.824239	−	0.566243i	$-0.191603\pi$
$857$	48.2584			1.64848			0.824239	+	0.566243i	$0.191603\pi$
$858$		0			0
$859$	10.1506			0.346332			0.173166	−	0.984893i	$-0.444600\pi$
$859$	10.1506			0.346332			0.173166	+	0.984893i	$0.444600\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	25.7981	−	44.6837i	0.878179	−	1.52105i	0.0248411	−	0.999691i	$-0.492092\pi$
$863$	25.7981	−	44.6837i	0.878179	−	1.52105i	0.853338	−	0.521359i	$-0.174575\pi$
$864$		0			0
$865$	−0.604630	−	1.04725i	−0.0205580	−	0.0356076i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4.79161	+	8.29931i	0.162544	+	0.281535i
$870$		0			0
$871$	−27.2689			−0.923972
$872$		0			0
$873$		0			0
$874$		0			0
$875$	31.6897	−	5.65193i	1.07131	−	0.191070i
$876$		0			0
$877$	−10.3978			−0.351109			−0.175555	−	0.984470i	$-0.556172\pi$
$877$	−10.3978			−0.351109			−0.175555	+	0.984470i	$0.556172\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−23.6562			−0.796996			−0.398498	−	0.917169i	$-0.630468\pi$
$881$	−23.6562			−0.796996			−0.398498	+	0.917169i	$0.630468\pi$
$882$		0			0
$883$	−41.8601			−1.40871			−0.704353	−	0.709850i	$-0.748763\pi$
$883$	−41.8601			−1.40871			−0.704353	+	0.709850i	$0.748763\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	42.2505			1.41863			0.709316	−	0.704891i	$-0.249004\pi$
$887$	42.2505			1.41863			0.709316	+	0.704891i	$0.249004\pi$
$888$		0			0
$889$	8.31936	−	1.48378i	0.279022	−	0.0497643i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−15.6787			−0.524669
$894$		0			0
$895$	−20.2404	−	35.0574i	−0.676563	−	1.17184i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.76243	−	4.78467i	−0.0921321	−	0.159578i
$900$		0			0
$901$	−24.0191	+	41.6023i	−0.800192	+	1.38597i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	46.4606			1.54440
$906$		0			0
$907$	40.8807			1.35742			0.678711	−	0.734405i	$-0.262539\pi$
$907$	40.8807			1.35742			0.678711	+	0.734405i	$0.262539\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	14.4235	−	24.9823i	0.477873	−	0.827701i	−0.521805	−	0.853065i	$-0.674741\pi$
$911$	14.4235	−	24.9823i	0.477873	−	0.827701i	0.999678	+	0.0253641i	$0.00807452\pi$
$912$		0			0
$913$	7.73249	−	13.3931i	0.255908	−	0.443246i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	36.6910	−	6.54393i	1.21165	−	0.216100i
$918$		0			0
$919$	−21.8195	+	37.7925i	−0.719760	+	1.24666i	0.241335	+	0.970442i	$0.422415\pi$
$919$	−21.8195	+	37.7925i	−0.719760	+	1.24666i	−0.961095	+	0.276219i	$0.910919\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−4.33560	−	7.50947i	−0.142708	−	0.247177i
$924$		0			0
$925$	−0.526192	+	0.911391i	−0.0173011	+	0.0299663i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−13.7377	−	23.7944i	−0.450719	−	0.780668i	0.547712	−	0.836667i	$-0.315499\pi$
$929$	−13.7377	−	23.7944i	−0.450719	−	0.780668i	−0.998431	+	0.0559990i	$0.982166\pi$
$930$		0			0
$931$	−1.51314	+	8.78299i	−0.0495911	+	0.287851i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−17.1131	−	29.6408i	−0.559658	−	0.969356i
$936$		0			0
$937$	−3.11920			−0.101900			−0.0509500	−	0.998701i	$-0.516225\pi$
$937$	−3.11920			−0.101900			−0.0509500	+	0.998701i	$0.516225\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−13.0608	+	22.6219i	−0.425769	+	0.737454i	−0.996492	−	0.0836892i	$-0.973330\pi$
$941$	−13.0608	+	22.6219i	−0.425769	+	0.737454i	0.570723	+	0.821143i	$0.306663\pi$
$942$		0			0
$943$	0.177081	+	0.306714i	0.00576656	+	0.00998797i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	3.59015	+	6.21833i	0.116664	+	0.202068i	0.918444	−	0.395551i	$-0.129447\pi$
$947$	3.59015	+	6.21833i	0.116664	+	0.202068i	−0.801780	+	0.597620i	$0.796113\pi$
$948$		0			0
$949$	37.0256	−	64.1302i	1.20190	−	2.08176i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	32.1187			1.04043			0.520214	−	0.854036i	$-0.325852\pi$
$953$	32.1187			1.04043			0.520214	+	0.854036i	$0.325852\pi$
$954$		0			0
$955$	8.03678	+	13.9201i	0.260064	+	0.450444i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−35.6742	+	6.36257i	−1.15198	+	0.205458i
$960$		0			0
$961$	11.0269	+	19.0991i	0.355705	+	0.616100i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.50018	−	2.59838i	0.0482924	−	0.0836449i
$966$		0			0
$967$	27.4860	+	47.6071i	0.883890	+	1.53094i	0.846981	+	0.531623i	$0.178418\pi$
$967$	27.4860	+	47.6071i	0.883890	+	1.53094i	0.0369085	+	0.999319i	$0.488249\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	20.8518	−	36.1163i	0.669165	−	1.15903i	−0.308973	−	0.951071i	$-0.599985\pi$
$971$	20.8518	−	36.1163i	0.669165	−	1.15903i	0.978138	−	0.207957i	$-0.0666813\pi$
$972$		0			0
$973$	16.8462	+	19.9967i	0.540065	+	0.641064i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	2.61437	−	4.52823i	0.0836412	−	0.144871i	−0.821170	−	0.570683i	$-0.806678\pi$
$977$	2.61437	−	4.52823i	0.0836412	−	0.144871i	0.904811	+	0.425813i	$0.140012\pi$
$978$		0			0
$979$	17.0106	−	29.4633i	0.543662	−	0.941651i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	6.90697			0.220298			0.110149	−	0.993915i	$-0.464867\pi$
$983$	6.90697			0.220298			0.110149	+	0.993915i	$0.464867\pi$
$984$		0			0
$985$	19.9967			0.637149
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.999537	+	1.73125i	−0.0317834	+	0.0550505i
$990$		0			0
$991$	−2.19861	−	3.80811i	−0.0698412	−	0.120968i	0.828990	−	0.559263i	$-0.188916\pi$
$991$	−2.19861	−	3.80811i	−0.0698412	−	0.120968i	−0.898831	+	0.438295i	$0.855583\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4.43133	+	7.67528i	0.140482	+	0.243323i
$996$		0			0
$997$	−28.0359			−0.887907			−0.443954	−	0.896050i	$-0.646425\pi$
$997$	−28.0359			−0.887907			−0.443954	+	0.896050i	$0.646425\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.t.l.289.3		22
3.2	odd	2		1008.2.t.k.961.4		22
4.3	odd	2		1512.2.t.d.289.3		22
7.4	even	3		3024.2.q.k.2881.9		22
9.4	even	3		3024.2.q.k.2305.9		22
9.5	odd	6		1008.2.q.k.625.5		22
12.11	even	2		504.2.t.d.457.8	yes	22
21.11	odd	6		1008.2.q.k.529.5		22
28.11	odd	6		1512.2.q.c.1369.9		22
36.23	even	6		504.2.q.d.121.7	yes	22
36.31	odd	6		1512.2.q.c.793.9		22
63.4	even	3	inner	3024.2.t.l.1873.3		22
63.32	odd	6		1008.2.t.k.193.4		22
84.11	even	6		504.2.q.d.25.7	✓	22
252.67	odd	6		1512.2.t.d.361.3		22
252.95	even	6		504.2.t.d.193.8	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.q.d.25.7	✓	22	84.11	even	6
504.2.q.d.121.7	yes	22	36.23	even	6
504.2.t.d.193.8	yes	22	252.95	even	6
504.2.t.d.457.8	yes	22	12.11	even	2
1008.2.q.k.529.5		22	21.11	odd	6
1008.2.q.k.625.5		22	9.5	odd	6
1008.2.t.k.193.4		22	63.32	odd	6
1008.2.t.k.961.4		22	3.2	odd	2
1512.2.q.c.793.9		22	36.31	odd	6
1512.2.q.c.1369.9		22	28.11	odd	6
1512.2.t.d.289.3		22	4.3	odd	2
1512.2.t.d.361.3		22	252.67	odd	6
3024.2.q.k.2305.9		22	9.4	even	3
3024.2.q.k.2881.9		22	7.4	even	3
3024.2.t.l.289.3		22	1.1	even	1	trivial
3024.2.t.l.1873.3		22	63.4	even	3	inner

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 $\F_{q}$
Belyi maps

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Modular forms

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3024.2.t.l.289.3

Level	$3024$
Weight	$2$
Character	3024.289
Analytic conductor	$24.147$
Analytic rank	$0$
Dimension	$22$
Inner twists	$2$