LMFDB - Embedded newform 585.2.b.g.181.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [585,2,Mod(181,585)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(585, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("585.181"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$585 = 3^{2} \cdot 5 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	585.b (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,-10,0,0,0,0,0,2,0,0,-8,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$4.67124851824$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5089536.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18$ x^6 - 2x^5 + 2x^4 + 2x^3 + 16x^2 - 24*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.4
Root			$-1.33641 + 1.33641i$ of defining polynomial
Character	$\chi$	$=$	585.181
Dual form			585.2.b.g.181.3

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.571993i q^{2} +1.67282 q^{4} -1.00000i q^{5} -2.67282i q^{7} +2.10083i q^{8} +0.571993 q^{10} -5.10083i q^{11} +(-1.57199 + 3.24482i) q^{13} +1.52884 q^{14} +2.14399 q^{16} +5.34565 q^{17} -6.24482i q^{19} -1.67282i q^{20} +2.91764 q^{22} -2.42801 q^{23} -1.00000 q^{25} +(-1.85601 - 0.899170i) q^{26} -4.47116i q^{28} -2.67282 q^{29} +0.244817i q^{31} +5.42801i q^{32} +3.05767i q^{34} -2.67282 q^{35} +3.32718i q^{37} +3.57199 q^{38} +2.10083 q^{40} +6.48963i q^{41} +10.9176 q^{43} -8.53279i q^{44} -1.38880i q^{46} -2.67282i q^{47} -0.143987 q^{49} -0.571993i q^{50} +(-2.62967 + 5.42801i) q^{52} +4.20166 q^{53} -5.10083 q^{55} +5.61515 q^{56} -1.52884i q^{58} +0.899170i q^{59} -5.81681 q^{61} -0.140034 q^{62} +1.18319 q^{64} +(3.24482 + 1.57199i) q^{65} -2.18319i q^{67} +8.94233 q^{68} -1.52884i q^{70} +6.24482i q^{71} +10.9608i q^{73} -1.90312 q^{74} -10.4465i q^{76} -13.6336 q^{77} -3.63362 q^{79} -2.14399i q^{80} -3.71203 q^{82} +9.81681i q^{83} -5.34565i q^{85} +6.24482i q^{86} +10.7160 q^{88} -7.63362i q^{89} +(8.67282 + 4.20166i) q^{91} -4.06163 q^{92} +1.52884 q^{94} -6.24482 q^{95} +11.3456i q^{97} -0.0823593i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 10 q^{4} + 2 q^{10} - 8 q^{13} - 8 q^{14} + 10 q^{16} - 8 q^{17} - 24 q^{22} - 16 q^{23} - 6 q^{25} - 14 q^{26} + 4 q^{29} + 4 q^{35} + 20 q^{38} - 6 q^{40} + 24 q^{43} + 2 q^{49} + 20 q^{52} - 12 q^{53}+ \cdots - 16 q^{95}+O(q^{100})$ 6 * q - 10 * q^4 + 2 * q^10 - 8 * q^13 - 8 * q^14 + 10 * q^16 - 8 * q^17 - 24 * q^22 - 16 * q^23 - 6 * q^25 - 14 * q^26 + 4 * q^29 + 4 * q^35 + 20 * q^38 - 6 * q^40 + 24 * q^43 + 2 * q^49 + 20 * q^52 - 12 * q^53 - 12 * q^55 + 48 * q^56 - 12 * q^61 - 8 * q^62 + 30 * q^64 - 2 * q^65 + 88 * q^68 - 20 * q^74 - 36 * q^77 + 24 * q^79 - 28 * q^82 + 60 * q^88 + 32 * q^91 + 20 * q^92 - 8 * q^94 - 16 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/585\mathbb{Z}\right)^\times$ .

$n$	$326$	$352$	$496$
$\chi(n)$	$1$	$1$	$-1$

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

)

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

$2$			0.571993i			0.404460i	0.979338	+	0.202230i	$0.0648189\pi$
$2$			0.571993i			0.404460i	−0.979338	+	0.202230i	$0.935181\pi$
$3$		0			0
$3$		0			0
$4$	1.67282			0.836412
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$		−	2.67282i		−	1.01023i	−0.863051	−	0.505116i	$-0.831450\pi$
$7$		−	2.67282i		−	1.01023i	0.863051	−	0.505116i	$-0.168550\pi$
$8$			2.10083i			0.742756i
$9$		0			0
$10$	0.571993			0.180880
$11$		−	5.10083i		−	1.53796i	−0.639274	−	0.768979i	$-0.720765\pi$
$11$		−	5.10083i		−	1.53796i	0.639274	−	0.768979i	$-0.279235\pi$
$12$		0			0
$13$	−1.57199	+	3.24482i	−0.435992	+	0.899950i
$13$	−1.57199	+	3.24482i	−0.435992	+	0.899950i
$14$	1.52884			0.408599
$15$		0			0
$16$	2.14399			0.535997
$17$	5.34565			1.29651			0.648255	−	0.761423i	$-0.275499\pi$
$17$	5.34565			1.29651			0.648255	+	0.761423i	$0.275499\pi$
$18$		0			0
$19$		−	6.24482i		−	1.43266i	−0.697762	−	0.716330i	$-0.745821\pi$
$19$		−	6.24482i		−	1.43266i	0.697762	−	0.716330i	$-0.254179\pi$
$20$		−	1.67282i		−	0.374055i
$21$		0			0
$22$	2.91764			0.622043
$23$	−2.42801			−0.506274			−0.253137	−	0.967430i	$-0.581462\pi$
$23$	−2.42801			−0.506274			−0.253137	+	0.967430i	$0.581462\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$	−1.85601	−	0.899170i	−0.363994	−	0.176342i
$27$		0			0
$28$		−	4.47116i		−	0.844970i
$29$	−2.67282			−0.496331			−0.248165	−	0.968718i	$-0.579828\pi$
$29$	−2.67282			−0.496331			−0.248165	+	0.968718i	$0.579828\pi$
$30$		0			0
$31$			0.244817i			0.0439704i	0.999758	+	0.0219852i	$0.00699867\pi$
$31$			0.244817i			0.0439704i	−0.999758	+	0.0219852i	$0.993001\pi$
$32$			5.42801i			0.959545i
$33$		0			0
$34$			3.05767i			0.524387i
$35$	−2.67282			−0.451790
$36$		0			0
$37$			3.32718i			0.546984i	0.961874	+	0.273492i	$0.0881788\pi$
$37$			3.32718i			0.546984i	−0.961874	+	0.273492i	$0.911821\pi$
$38$	3.57199			0.579454
$39$		0			0
$40$	2.10083			0.332170
$41$			6.48963i			1.01351i	0.862090	+	0.506755i	$0.169155\pi$
$41$			6.48963i			1.01351i	−0.862090	+	0.506755i	$0.830845\pi$
$42$		0			0
$43$	10.9176			1.66492			0.832462	−	0.554082i	$-0.186930\pi$
$43$	10.9176			1.66492			0.832462	+	0.554082i	$0.186930\pi$
$44$		−	8.53279i		−	1.28637i
$45$		0			0
$46$		−	1.38880i		−	0.204768i
$47$		−	2.67282i		−	0.389871i	−0.980816	−	0.194936i	$-0.937550\pi$
$47$		−	2.67282i		−	0.389871i	0.980816	−	0.194936i	$-0.0624498\pi$
$48$		0			0
$49$	−0.143987			−0.0205695
$50$		−	0.571993i		−	0.0808921i
$51$		0			0
$52$	−2.62967	+	5.42801i	−0.364669	+	0.752729i
$53$	4.20166			0.577143			0.288571	−	0.957458i	$-0.406820\pi$
$53$	4.20166			0.577143			0.288571	+	0.957458i	$0.406820\pi$
$54$		0			0
$55$	−5.10083			−0.687796
$56$	5.61515			0.750356
$57$		0			0
$58$		−	1.52884i		−	0.200746i
$59$			0.899170i			0.117062i	0.998286	+	0.0585310i	$0.0186416\pi$
$59$			0.899170i			0.117062i	−0.998286	+	0.0585310i	$0.981358\pi$
$60$		0			0
$61$	−5.81681			−0.744766			−0.372383	−	0.928079i	$-0.621459\pi$
$61$	−5.81681			−0.744766			−0.372383	+	0.928079i	$0.621459\pi$
$62$	−0.140034			−0.0177843
$63$		0			0
$64$	1.18319			0.147899
$65$	3.24482	+	1.57199i	0.402470	+	0.194982i
$66$		0			0
$67$		−	2.18319i		−	0.266719i	−0.991068	−	0.133360i	$-0.957423\pi$
$67$		−	2.18319i		−	0.266719i	0.991068	−	0.133360i	$-0.0425765\pi$
$68$	8.94233			1.08442
$69$		0			0
$70$		−	1.52884i		−	0.182731i
$71$			6.24482i			0.741123i	0.928808	+	0.370562i	$0.120835\pi$
$71$			6.24482i			0.741123i	−0.928808	+	0.370562i	$0.879165\pi$
$72$		0			0
$73$			10.9608i			1.28286i	0.767180	+	0.641432i	$0.221659\pi$
$73$			10.9608i			1.28286i	−0.767180	+	0.641432i	$0.778341\pi$
$74$	−1.90312			−0.221233
$75$		0			0
$76$		−	10.4465i		−	1.19829i
$77$	−13.6336			−1.55370
$78$		0			0
$79$	−3.63362			−0.408814			−0.204407	−	0.978886i	$-0.565527\pi$
$79$	−3.63362			−0.408814			−0.204407	+	0.978886i	$0.565527\pi$
$80$		−	2.14399i		−	0.239705i
$81$		0			0
$82$	−3.71203			−0.409925
$83$			9.81681i			1.07753i	0.842455	+	0.538767i	$0.181110\pi$
$83$			9.81681i			1.07753i	−0.842455	+	0.538767i	$0.818890\pi$
$84$		0			0
$85$		−	5.34565i		−	0.579817i
$86$			6.24482i			0.673396i
$87$		0			0
$88$	10.7160			1.14233
$89$		−	7.63362i		−	0.809162i	−0.914502	−	0.404581i	$-0.867417\pi$
$89$		−	7.63362i		−	0.809162i	0.914502	−	0.404581i	$-0.132583\pi$
$90$		0			0
$91$	8.67282	+	4.20166i	0.909159	+	0.440454i
$92$	−4.06163			−0.423454
$93$		0			0
$94$	1.52884			0.157688
$95$	−6.24482			−0.640705
$96$		0			0
$97$			11.3456i			1.15198i	0.817458	+	0.575988i	$0.195383\pi$
$97$			11.3456i			1.15198i	−0.817458	+	0.575988i	$0.804617\pi$
$98$		−	0.0823593i		−	0.00831955i
$99$		0			0
$100$	−1.67282			−0.167282
$101$	10.8560			1.08021			0.540107	−	0.841596i	$-0.318384\pi$
$101$	10.8560			1.08021			0.540107	+	0.841596i	$0.318384\pi$
$102$		0			0
$103$	−10.6297			−1.04737			−0.523686	−	0.851911i	$-0.675444\pi$
$103$	−10.6297			−1.04737			−0.523686	+	0.851911i	$0.675444\pi$
$104$	−6.81681	−	3.30249i	−0.668443	−	0.323836i
$105$		0			0
$106$			2.40332i			0.233431i
$107$	−0.140034			−0.0135376			−0.00676878	−	0.999977i	$-0.502155\pi$
$107$	−0.140034			−0.0135376			−0.00676878	+	0.999977i	$0.502155\pi$
$108$		0			0
$109$		−	9.05767i		−	0.867568i	−0.901017	−	0.433784i	$-0.857178\pi$
$109$		−	9.05767i		−	0.867568i	0.901017	−	0.433784i	$-0.142822\pi$
$110$		−	2.91764i		−	0.278186i
$111$		0			0
$112$		−	5.73050i		−	0.541481i
$113$	−12.9793			−1.22099			−0.610493	−	0.792021i	$-0.709029\pi$
$113$	−12.9793			−1.22099			−0.610493	+	0.792021i	$0.709029\pi$
$114$		0			0
$115$			2.42801i			0.226413i
$116$	−4.47116			−0.415137
$117$		0			0
$118$	−0.514319			−0.0473469
$119$		−	14.2880i		−	1.30978i
$120$		0			0
$121$	−15.0185			−1.36532
$122$		−	3.32718i		−	0.301228i
$123$		0			0
$124$			0.409536i			0.0367774i
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	−12.0616			−1.07030			−0.535148	−	0.844758i	$-0.679744\pi$
$127$	−12.0616			−1.07030			−0.535148	+	0.844758i	$0.679744\pi$
$128$			11.5328i			1.01936i
$129$		0			0
$130$	−0.899170	+	1.85601i	−0.0788624	+	0.162783i
$131$	−7.63362			−0.666953			−0.333476	−	0.942758i	$-0.608222\pi$
$131$	−7.63362			−0.666953			−0.333476	+	0.942758i	$0.608222\pi$
$132$		0			0
$133$	−16.6913			−1.44732
$134$	1.24877			0.107877
$135$		0			0
$136$			11.2303i			0.962990i
$137$			3.43196i			0.293212i	0.989195	+	0.146606i	$0.0468350\pi$
$137$			3.43196i			0.293212i	−0.989195	+	0.146606i	$0.953165\pi$
$138$		0			0
$139$	−0.942326			−0.0799270			−0.0399635	−	0.999201i	$-0.512724\pi$
$139$	−0.942326			−0.0799270			−0.0399635	+	0.999201i	$0.512724\pi$
$140$	−4.47116			−0.377882
$141$		0			0
$142$	−3.57199			−0.299755
$143$	16.5513	+	8.01847i	1.38409	+	0.670538i
$144$		0			0
$145$			2.67282i			0.221966i
$146$	−6.26950			−0.518868
$147$		0			0
$148$			5.56578i			0.457504i
$149$		−	11.3456i		−	0.929472i	−0.885449	−	0.464736i	$-0.846149\pi$
$149$		−	11.3456i		−	0.929472i	0.885449	−	0.464736i	$-0.153851\pi$
$150$		0			0
$151$			21.0224i			1.71078i	0.517984	+	0.855390i	$0.326683\pi$
$151$			21.0224i			1.71078i	−0.517984	+	0.855390i	$0.673317\pi$
$152$	13.1193			1.06412
$153$		0			0
$154$		−	7.79834i		−	0.628408i
$155$	0.244817			0.0196642
$156$		0			0
$157$	−9.83528			−0.784941			−0.392470	−	0.919765i	$-0.628379\pi$
$157$	−9.83528			−0.784941			−0.392470	+	0.919765i	$0.628379\pi$
$158$		−	2.07841i		−	0.165349i
$159$		0			0
$160$	5.42801			0.429122
$161$			6.48963i			0.511455i
$162$		0			0
$163$			12.3849i			0.970056i	0.874499	+	0.485028i	$0.161191\pi$
$163$			12.3849i			0.970056i	−0.874499	+	0.485028i	$0.838809\pi$
$164$			10.8560i			0.847712i
$165$		0			0
$166$	−5.61515			−0.435820
$167$			8.50811i			0.658377i	0.944264	+	0.329188i	$0.106775\pi$
$167$			8.50811i			0.658377i	−0.944264	+	0.329188i	$0.893225\pi$
$168$		0			0
$169$	−8.05767	−	10.2017i	−0.619821	−	0.784743i
$170$	3.05767			0.234513
$171$		0			0
$172$	18.2633			1.39256
$173$	−21.5473			−1.63821			−0.819106	−	0.573643i	$-0.805530\pi$
$173$	−21.5473			−1.63821			−0.819106	+	0.573643i	$0.805530\pi$
$174$		0			0
$175$			2.67282i			0.202046i
$176$		−	10.9361i		−	0.824340i
$177$		0			0
$178$	4.36638			0.327274
$179$	2.56804			0.191944			0.0959722	−	0.995384i	$-0.469404\pi$
$179$	2.56804			0.191944			0.0959722	+	0.995384i	$0.469404\pi$
$180$		0			0
$181$	13.7305			1.02058			0.510290	−	0.860002i	$-0.329538\pi$
$181$	13.7305			1.02058			0.510290	+	0.860002i	$0.329538\pi$
$182$	−2.40332	+	4.96080i	−0.178146	+	0.367719i
$183$		0			0
$184$		−	5.10083i		−	0.376038i
$185$	3.32718			0.244619
$186$		0			0
$187$		−	27.2672i		−	1.99398i
$188$		−	4.47116i		−	0.326093i
$189$		0			0
$190$		−	3.57199i		−	0.259140i
$191$	10.6913			0.773595			0.386797	−	0.922165i	$-0.373581\pi$
$191$	10.6913			0.773595			0.386797	+	0.922165i	$0.373581\pi$
$192$		0			0
$193$		−	19.2593i		−	1.38632i	−0.720785	−	0.693159i	$-0.756219\pi$
$193$		−	19.2593i		−	1.38632i	0.720785	−	0.693159i	$-0.243781\pi$
$194$	−6.48963			−0.465929
$195$		0			0
$196$	−0.240864			−0.0172046
$197$		−	17.1809i		−	1.22409i	−0.790823	−	0.612045i	$-0.790347\pi$
$197$		−	17.1809i		−	1.22409i	0.790823	−	0.612045i	$-0.209653\pi$
$198$		0			0
$199$	−11.5473			−0.818567			−0.409283	−	0.912407i	$-0.634221\pi$
$199$	−11.5473			−0.818567			−0.409283	+	0.912407i	$0.634221\pi$
$200$		−	2.10083i		−	0.148551i
$201$		0			0
$202$			6.20957i			0.436904i
$203$			7.14399i			0.501410i
$204$		0			0
$205$	6.48963			0.453256
$206$		−	6.08010i		−	0.423621i
$207$		0			0
$208$	−3.37033	+	6.95684i	−0.233691	+	0.482370i
$209$	−31.8538			−2.20337
$210$		0			0
$211$	24.1233			1.66071			0.830357	−	0.557232i	$-0.188137\pi$
$211$	24.1233			1.66071			0.830357	+	0.557232i	$0.188137\pi$
$212$	7.02864			0.482729
$213$		0			0
$214$		−	0.0800983i		−	0.00547541i
$215$		−	10.9176i		−	0.744577i
$216$		0			0
$217$	0.654353			0.0444203
$218$	5.18093			0.350897
$219$		0			0
$220$	−8.53279			−0.575281
$221$	−8.40332	+	17.3456i	−0.565269	+	1.16679i
$222$		0			0
$223$		−	5.73050i		−	0.383743i	−0.981420	−	0.191871i	$-0.938544\pi$
$223$		−	5.73050i		−	0.383743i	0.981420	−	0.191871i	$-0.0614556\pi$
$224$	14.5081			0.969364
$225$		0			0
$226$		−	7.42405i		−	0.493841i
$227$			2.95289i			0.195990i	0.995187	+	0.0979951i	$0.0312430\pi$
$227$			2.95289i			0.195990i	−0.995187	+	0.0979951i	$0.968757\pi$
$228$		0			0
$229$			25.2593i			1.66918i	0.550869	+	0.834592i	$0.314297\pi$
$229$			25.2593i			1.66918i	−0.550869	+	0.834592i	$0.685703\pi$
$230$	−1.38880			−0.0915750
$231$		0			0
$232$		−	5.61515i		−	0.368653i
$233$	23.3456			1.52942			0.764712	−	0.644372i	$-0.222881\pi$
$233$	23.3456			1.52942			0.764712	+	0.644372i	$0.222881\pi$
$234$		0			0
$235$	−2.67282			−0.174356
$236$			1.50415i			0.0979120i
$237$		0			0
$238$	8.17262			0.529753
$239$		−	3.79213i		−	0.245292i	−0.992450	−	0.122646i	$-0.960862\pi$
$239$		−	3.79213i		−	0.245292i	0.992450	−	0.122646i	$-0.0391380\pi$
$240$		0			0
$241$		−	3.43196i		−	0.221072i	−0.993872	−	0.110536i	$-0.964743\pi$
$241$		−	3.43196i		−	0.221072i	0.993872	−	0.110536i	$-0.0352567\pi$
$242$		−	8.59046i		−	0.552216i
$243$		0			0
$244$	−9.73050			−0.622931
$245$			0.143987i			0.00919896i
$246$		0			0
$247$	20.2633	+	9.81681i	1.28932	+	0.624629i
$248$	−0.514319			−0.0326593
$249$		0			0
$250$	−0.571993			−0.0361760
$251$	25.4689			1.60758			0.803791	−	0.594911i	$-0.202813\pi$
$251$	25.4689			1.60758			0.803791	+	0.594911i	$0.202813\pi$
$252$		0			0
$253$			12.3849i			0.778629i
$254$		−	6.89917i		−	0.432892i
$255$		0			0
$256$	−4.23030			−0.264394
$257$	−9.26724			−0.578075			−0.289037	−	0.957318i	$-0.593335\pi$
$257$	−9.26724			−0.578075			−0.289037	+	0.957318i	$0.593335\pi$
$258$		0			0
$259$	8.89296			0.552581
$260$	5.42801	+	2.62967i	0.336631	+	0.163085i
$261$		0			0
$262$		−	4.36638i		−	0.269756i
$263$	8.14794			0.502423			0.251212	−	0.967932i	$-0.419171\pi$
$263$	8.14794			0.502423			0.251212	+	0.967932i	$0.419171\pi$
$264$		0			0
$265$		−	4.20166i		−	0.258106i
$266$		−	9.54731i		−	0.585383i
$267$		0			0
$268$		−	3.65209i		−	0.223087i
$269$	−1.14399			−0.0697501			−0.0348750	−	0.999392i	$-0.511103\pi$
$269$	−1.14399			−0.0697501			−0.0348750	+	0.999392i	$0.511103\pi$
$270$		0			0
$271$		−	3.18714i		−	0.193605i	−0.995304	−	0.0968026i	$-0.969138\pi$
$271$		−	3.18714i		−	0.193605i	0.995304	−	0.0968026i	$-0.0308615\pi$
$272$	11.4610			0.694925
$273$		0			0
$274$	−1.96306			−0.118593
$275$			5.10083i			0.307592i
$276$		0			0
$277$	7.62571			0.458185			0.229092	−	0.973405i	$-0.426424\pi$
$277$	7.62571			0.458185			0.229092	+	0.973405i	$0.426424\pi$
$278$		−	0.539004i		−	0.0323273i
$279$		0			0
$280$		−	5.61515i		−	0.335569i
$281$			20.2386i			1.20733i	0.797237	+	0.603667i	$0.206294\pi$
$281$			20.2386i			1.20733i	−0.797237	+	0.603667i	$0.793706\pi$
$282$		0			0
$283$	19.8969			1.18275			0.591374	−	0.806397i	$-0.298586\pi$
$283$	19.8969			1.18275			0.591374	+	0.806397i	$0.298586\pi$
$284$			10.4465i			0.619884i
$285$		0			0
$286$	−4.58651	+	9.46721i	−0.271206	+	0.559808i
$287$	17.3456			1.02388
$288$		0			0
$289$	11.5759			0.680938
$290$	−1.52884			−0.0897764
$291$		0			0
$292$			18.3355i			1.07300i
$293$			26.0185i			1.52002i	0.649914	+	0.760008i	$0.274805\pi$
$293$			26.0185i			1.52002i	−0.649914	+	0.760008i	$0.725195\pi$
$294$		0			0
$295$	0.899170			0.0523517
$296$	−6.98983			−0.406276
$297$		0			0
$298$	6.48963			0.375934
$299$	3.81681	−	7.87844i	0.220732	−	0.455622i
$300$		0			0
$301$		−	29.1809i		−	1.68196i
$302$	−12.0247			−0.691943
$303$		0			0
$304$		−	13.3888i		−	0.767901i
$305$			5.81681i			0.333070i
$306$		0			0
$307$		−	2.39276i		−	0.136562i	−0.997666	−	0.0682809i	$-0.978249\pi$
$307$		−	2.39276i		−	0.136562i	0.997666	−	0.0682809i	$-0.0217514\pi$
$308$	−22.8066			−1.29953
$309$		0			0
$310$			0.140034i			0.00795338i
$311$	−3.54731			−0.201149			−0.100575	−	0.994930i	$-0.532068\pi$
$311$	−3.54731			−0.201149			−0.100575	+	0.994930i	$0.532068\pi$
$312$		0			0
$313$	4.97927			0.281445			0.140722	−	0.990049i	$-0.455057\pi$
$313$	4.97927			0.281445			0.140722	+	0.990049i	$0.455057\pi$
$314$		−	5.62571i		−	0.317477i
$315$		0			0
$316$	−6.07841			−0.341937
$317$		−	18.5944i		−	1.04437i	−0.852833	−	0.522183i	$-0.825118\pi$
$317$		−	18.5944i		−	1.04437i	0.852833	−	0.522183i	$-0.174882\pi$
$318$		0			0
$319$			13.6336i			0.763336i
$320$		−	1.18319i		−	0.0661423i
$321$		0			0
$322$	−3.71203			−0.206863
$323$		−	33.3826i		−	1.85746i
$324$		0			0
$325$	1.57199	−	3.24482i	0.0871985	−	0.179990i
$326$	−7.08405			−0.392349
$327$		0			0
$328$	−13.6336			−0.752791
$329$	−7.14399			−0.393861
$330$		0			0
$331$		−	5.91990i		−	0.325387i	−0.986677	−	0.162694i	$-0.947982\pi$
$331$		−	5.91990i		−	0.325387i	0.986677	−	0.162694i	$-0.0520182\pi$
$332$			16.4218i			0.901263i
$333$		0			0
$334$	−4.86658			−0.266287
$335$	−2.18319			−0.119280
$336$		0			0
$337$	13.9216			0.758358			0.379179	−	0.925323i	$-0.376207\pi$
$337$	13.9216			0.758358			0.379179	+	0.925323i	$0.376207\pi$
$338$	5.83528	−	4.60894i	0.317397	−	0.250693i
$339$		0			0
$340$		−	8.94233i		−	0.484966i
$341$	1.24877			0.0676247
$342$		0			0
$343$		−	18.3249i		−	0.989452i
$344$			22.9361i			1.23663i
$345$		0			0
$346$		−	12.3249i		−	0.662592i
$347$	−7.65831			−0.411119			−0.205560	−	0.978645i	$-0.565901\pi$
$347$	−7.65831			−0.411119			−0.205560	+	0.978645i	$0.565901\pi$
$348$		0			0
$349$			11.1809i			0.598501i	0.954175	+	0.299251i	$0.0967367\pi$
$349$			11.1809i			0.598501i	−0.954175	+	0.299251i	$0.903263\pi$
$350$	−1.52884			−0.0817198
$351$		0			0
$352$	27.6873			1.47574
$353$		−	9.65209i		−	0.513729i	−0.966447	−	0.256864i	$-0.917311\pi$
$353$		−	9.65209i		−	0.513729i	0.966447	−	0.256864i	$-0.0826894\pi$
$354$		0			0
$355$	6.24482			0.331440
$356$		−	12.7697i		−	0.676793i
$357$		0			0
$358$			1.46890i			0.0776339i
$359$		−	9.51206i		−	0.502027i	−0.967984	−	0.251014i	$-0.919236\pi$
$359$		−	9.51206i		−	0.502027i	0.967984	−	0.251014i	$-0.0807639\pi$
$360$		0			0
$361$	−19.9977			−1.05251
$362$			7.85375i			0.412784i
$363$		0			0
$364$	14.5081	+	7.02864i	0.760431	+	0.368401i
$365$	10.9608			0.573714
$366$		0			0
$367$	−17.0040			−0.887599			−0.443800	−	0.896126i	$-0.646370\pi$
$367$	−17.0040			−0.887599			−0.443800	+	0.896126i	$0.646370\pi$
$368$	−5.20561			−0.271361
$369$		0			0
$370$			1.90312i			0.0989386i
$371$		−	11.2303i		−	0.583048i
$372$		0			0
$373$	−1.83528			−0.0950273			−0.0475136	−	0.998871i	$-0.515130\pi$
$373$	−1.83528			−0.0950273			−0.0475136	+	0.998871i	$0.515130\pi$
$374$	15.5967			0.806485
$375$		0			0
$376$	5.61515			0.289579
$377$	4.20166	−	8.67282i	0.216397	−	0.446673i
$378$		0			0
$379$		−	32.3681i		−	1.66264i	−0.555797	−	0.831318i	$-0.687587\pi$
$379$		−	32.3681i		−	1.66264i	0.555797	−	0.831318i	$-0.312413\pi$
$380$	−10.4465			−0.535893
$381$		0			0
$382$			6.11535i			0.312888i
$383$			28.1417i			1.43798i	0.695023	+	0.718988i	$0.255394\pi$
$383$			28.1417i			1.43798i	−0.695023	+	0.718988i	$0.744606\pi$
$384$		0			0
$385$			13.6336i			0.694834i
$386$	11.0162			0.560710
$387$		0			0
$388$			18.9793i			0.963526i
$389$	31.9585			1.62036			0.810181	−	0.586180i	$-0.199369\pi$
$389$	31.9585			1.62036			0.810181	+	0.586180i	$0.199369\pi$
$390$		0			0
$391$	−12.9793			−0.656390
$392$		−	0.302491i		−	0.0152781i
$393$		0			0
$394$	9.82738			0.495096
$395$			3.63362i			0.182827i
$396$		0			0
$397$			12.7098i			0.637885i	0.947774	+	0.318942i	$0.103328\pi$
$397$			12.7098i			0.637885i	−0.947774	+	0.318942i	$0.896672\pi$
$398$		−	6.60498i		−	0.331078i
$399$		0			0
$400$	−2.14399			−0.107199
$401$			0.979268i			0.0489023i	0.999701	+	0.0244512i	$0.00778382\pi$
$401$			0.979268i			0.0489023i	−0.999701	+	0.0244512i	$0.992216\pi$
$402$		0			0
$403$	−0.794386	−	0.384851i	−0.0395712	−	0.0191708i
$404$	18.1602			0.903504
$405$		0			0
$406$	−4.08631			−0.202800
$407$	16.9714			0.841239
$408$		0			0
$409$			7.79834i			0.385603i	0.981238	+	0.192802i	$0.0617574\pi$
$409$			7.79834i			0.385603i	−0.981238	+	0.192802i	$0.938243\pi$
$410$			3.71203i			0.183324i
$411$		0			0
$412$	−17.7816			−0.876035
$413$	2.40332			0.118260
$414$		0			0
$415$	9.81681			0.481888
$416$	−17.6129	−	8.53279i	−0.863543	−	0.418354i
$417$		0			0
$418$		−	18.2201i		−	0.891176i
$419$	−10.2017			−0.498384			−0.249192	−	0.968454i	$-0.580165\pi$
$419$	−10.2017			−0.498384			−0.249192	+	0.968454i	$0.580165\pi$
$420$		0			0
$421$			31.1888i			1.52005i	0.649893	+	0.760025i	$0.274814\pi$
$421$			31.1888i			1.52005i	−0.649893	+	0.760025i	$0.725186\pi$
$422$			13.7983i			0.671693i
$423$		0			0
$424$			8.82698i			0.428676i
$425$	−5.34565			−0.259302
$426$		0			0
$427$			15.5473i			0.752387i
$428$	−0.234252			−0.0113230
$429$		0			0
$430$	6.24482			0.301152
$431$		−	22.9361i		−	1.10479i	−0.833581	−	0.552397i	$-0.813713\pi$
$431$		−	22.9361i		−	1.10479i	0.833581	−	0.552397i	$-0.186287\pi$
$432$		0			0
$433$	−16.2880			−0.782750			−0.391375	−	0.920231i	$-0.628000\pi$
$433$	−16.2880			−0.782750			−0.391375	+	0.920231i	$0.628000\pi$
$434$			0.374285i			0.0179663i
$435$		0			0
$436$		−	15.1519i		−	0.725644i
$437$			15.1625i			0.725319i
$438$		0			0
$439$	23.7569			1.13385			0.566927	−	0.823768i	$-0.308132\pi$
$439$	23.7569			1.13385			0.566927	+	0.823768i	$0.308132\pi$
$440$		−	10.7160i		−	0.510864i
$441$		0			0
$442$	−9.92159	−	4.80664i	−0.471922	−	0.228629i
$443$	2.75292			0.130795			0.0653976	−	0.997859i	$-0.479168\pi$
$443$	2.75292			0.130795			0.0653976	+	0.997859i	$0.479168\pi$
$444$		0			0
$445$	−7.63362			−0.361868
$446$	3.27781			0.155209
$447$		0			0
$448$		−	3.16246i		−	0.149412i
$449$			27.1025i			1.27905i	0.768772	+	0.639524i	$0.220868\pi$
$449$			27.1025i			1.27905i	−0.768772	+	0.639524i	$0.779132\pi$
$450$		0			0
$451$	33.1025			1.55874
$452$	−21.7120			−1.02125
$453$		0			0
$454$	−1.68903			−0.0792703
$455$	4.20166	−	8.67282i	0.196977	−	0.406588i
$456$		0			0
$457$		−	40.1523i		−	1.87824i	−0.343583	−	0.939122i	$-0.611641\pi$
$457$		−	40.1523i		−	1.87824i	0.343583	−	0.939122i	$-0.388359\pi$
$458$	−14.4482			−0.675119
$459$		0			0
$460$			4.06163i			0.189374i
$461$			2.00791i			0.0935175i	0.998906	+	0.0467587i	$0.0148892\pi$
$461$			2.00791i			0.0935175i	−0.998906	+	0.0467587i	$0.985111\pi$
$462$		0			0
$463$			2.67282i			0.124217i	0.998069	+	0.0621083i	$0.0197824\pi$
$463$			2.67282i			0.124217i	−0.998069	+	0.0621083i	$0.980218\pi$
$464$	−5.73050			−0.266032
$465$		0			0
$466$			13.3536i			0.618591i
$467$	−31.8890			−1.47565			−0.737824	−	0.674994i	$-0.764146\pi$
$467$	−31.8890			−1.47565			−0.737824	+	0.674994i	$0.764146\pi$
$468$		0			0
$469$	−5.83528			−0.269448
$470$		−	1.52884i		−	0.0705200i
$471$		0			0
$472$	−1.88900			−0.0869484
$473$		−	55.6890i		−	2.56058i
$474$		0			0
$475$			6.24482i			0.286532i
$476$		−	23.9013i		−	1.09551i
$477$		0			0
$478$	2.16907			0.0992110
$479$			6.73445i			0.307705i	0.988094	+	0.153852i	$0.0491680\pi$
$479$			6.73445i			0.307705i	−0.988094	+	0.153852i	$0.950832\pi$
$480$		0			0
$481$	−10.7961	−	5.23030i	−0.492259	−	0.238481i
$482$	1.96306			0.0894148
$483$		0			0
$484$	−25.1233			−1.14197
$485$	11.3456			0.515179
$486$		0			0
$487$		−	39.6521i		−	1.79681i	−0.439170	−	0.898404i	$-0.644727\pi$
$487$		−	39.6521i		−	1.79681i	0.439170	−	0.898404i	$-0.355273\pi$
$488$		−	12.2201i		−	0.553179i
$489$		0			0
$490$	−0.0823593			−0.00372062
$491$	−31.0946			−1.40328			−0.701640	−	0.712531i	$-0.747549\pi$
$491$	−31.0946			−1.40328			−0.701640	+	0.712531i	$0.747549\pi$
$492$		0			0
$493$	−14.2880			−0.643498
$494$	−5.61515	+	11.5905i	−0.252638	+	0.521480i
$495$		0			0
$496$			0.524884i			0.0235680i
$497$	16.6913			0.748707
$498$		0			0
$499$		−	13.1087i		−	0.586828i	−0.955986	−	0.293414i	$-0.905209\pi$
$499$		−	13.1087i		−	0.586828i	0.955986	−	0.293414i	$-0.0947914\pi$
$500$			1.67282i			0.0748110i
$501$		0			0
$502$			14.5680i			0.650203i
$503$	8.59273			0.383131			0.191565	−	0.981480i	$-0.438644\pi$
$503$	8.59273			0.383131			0.191565	+	0.981480i	$0.438644\pi$
$504$		0			0
$505$		−	10.8560i		−	0.483086i
$506$	−7.08405			−0.314924
$507$		0			0
$508$	−20.1770			−0.895209
$509$			29.8353i			1.32243i	0.750198	+	0.661213i	$0.229958\pi$
$509$			29.8353i			1.32243i	−0.750198	+	0.661213i	$0.770042\pi$
$510$		0			0
$511$	29.2963			1.29599
$512$			20.6459i			0.912428i
$513$		0			0
$514$		−	5.30080i		−	0.233808i
$515$			10.6297i			0.468399i
$516$		0			0
$517$	−13.6336			−0.599606
$518$			5.08671i			0.223497i
$519$		0			0
$520$	−3.30249	+	6.81681i	−0.144824	+	0.298937i
$521$	−10.0969			−0.442352			−0.221176	−	0.975234i	$-0.570990\pi$
$521$	−10.0969			−0.442352			−0.221176	+	0.975234i	$0.570990\pi$
$522$		0			0
$523$	16.1479			0.706100			0.353050	−	0.935604i	$-0.385145\pi$
$523$	16.1479			0.706100			0.353050	+	0.935604i	$0.385145\pi$
$524$	−12.7697			−0.557847
$525$		0			0
$526$			4.66057i			0.203210i
$527$			1.30871i			0.0570081i
$528$		0			0
$529$	−17.1048			−0.743686
$530$	2.40332			0.104394
$531$		0			0
$532$	−27.9216			−1.21055
$533$	−21.0577	−	10.2017i	−0.912109	−	0.441883i
$534$		0			0
$535$			0.140034i			0.00605418i
$536$	4.58651			0.198107
$537$		0			0
$538$		−	0.654353i		−	0.0282111i
$539$			0.734451i			0.0316350i
$540$		0			0
$541$			26.7776i			1.15126i	0.817711	+	0.575630i	$0.195243\pi$
$541$			26.7776i			1.15126i	−0.817711	+	0.575630i	$0.804757\pi$
$542$	1.82302			0.0783056
$543$		0			0
$544$			29.0162i			1.24406i
$545$	−9.05767			−0.387988
$546$		0			0
$547$	12.3865			0.529610			0.264805	−	0.964302i	$-0.414692\pi$
$547$	12.3865			0.529610			0.264805	+	0.964302i	$0.414692\pi$
$548$			5.74106i			0.245246i
$549$		0			0
$550$	−2.91764			−0.124409
$551$			16.6913i			0.711073i
$552$		0			0
$553$			9.71203i			0.412997i
$554$			4.36186i			0.185318i
$555$		0			0
$556$	−1.57634			−0.0668519
$557$		−	10.4218i		−	0.441586i	−0.975321	−	0.220793i	$-0.929136\pi$
$557$		−	10.4218i		−	0.441586i	0.975321	−	0.220793i	$-0.0708644\pi$
$558$		0			0
$559$	−17.1625	+	35.4257i	−0.725895	+	1.49835i
$560$	−5.73050			−0.242158
$561$		0			0
$562$	−11.5763			−0.488319
$563$	−9.78156			−0.412244			−0.206122	−	0.978526i	$-0.566084\pi$
$563$	−9.78156			−0.412244			−0.206122	+	0.978526i	$0.566084\pi$
$564$		0			0
$565$			12.9793i			0.546042i
$566$			11.3809i			0.478375i
$567$		0			0
$568$	−13.1193			−0.550474
$569$	30.2201			1.26689			0.633447	−	0.773786i	$-0.281639\pi$
$569$	30.2201			1.26689			0.633447	+	0.773786i	$0.281639\pi$
$570$		0			0
$571$	−39.8643			−1.66827			−0.834135	−	0.551561i	$-0.814033\pi$
$571$	−39.8643			−1.66827			−0.834135	+	0.551561i	$0.814033\pi$
$572$	27.6873	+	13.4135i	1.15767	+	0.560846i
$573$		0			0
$574$			9.92159i			0.414119i
$575$	2.42801			0.101255
$576$		0			0
$577$			8.46326i			0.352330i	0.984361	+	0.176165i	$0.0563692\pi$
$577$			8.46326i			0.352330i	−0.984361	+	0.176165i	$0.943631\pi$
$578$			6.62136i			0.275412i
$579$		0			0
$580$			4.47116i			0.185655i
$581$	26.2386			1.08856
$582$		0			0
$583$		−	21.4320i		−	0.887621i
$584$	−23.0268			−0.952855
$585$		0			0
$586$	−14.8824			−0.614786
$587$		−	9.32718i		−	0.384974i	−0.981300	−	0.192487i	$-0.938345\pi$
$587$		−	9.32718i		−	0.384974i	0.981300	−	0.192487i	$-0.0616553\pi$
$588$		0			0
$589$	1.52884			0.0629946
$590$			0.514319i			0.0211742i
$591$		0			0
$592$			7.13342i			0.293182i
$593$			8.07841i			0.331740i	0.986148	+	0.165870i	$0.0530433\pi$
$593$			8.07841i			0.331740i	−0.986148	+	0.165870i	$0.946957\pi$
$594$		0			0
$595$	−14.2880			−0.585750
$596$		−	18.9793i		−	0.777421i
$597$		0			0
$598$	4.50641	+	2.18319i	0.184281	+	0.0892773i
$599$	−19.1440			−0.782202			−0.391101	−	0.920348i	$-0.627906\pi$
$599$	−19.1440			−0.782202			−0.391101	+	0.920348i	$0.627906\pi$
$600$		0			0
$601$	−4.28797			−0.174910			−0.0874550	−	0.996168i	$-0.527873\pi$
$601$	−4.28797			−0.174910			−0.0874550	+	0.996168i	$0.527873\pi$
$602$	16.6913			0.680286
$603$		0			0
$604$			35.1668i			1.43092i
$605$			15.0185i			0.610588i
$606$		0			0
$607$	29.2426			1.18692			0.593459	−	0.804864i	$-0.297762\pi$
$607$	29.2426			1.18692			0.593459	+	0.804864i	$0.297762\pi$
$608$	33.8969			1.37470
$609$		0			0
$610$	−3.32718			−0.134713
$611$	8.67282	+	4.20166i	0.350865	+	0.169981i
$612$		0			0
$613$		−	1.42405i		−	0.0575170i	−0.999586	−	0.0287585i	$-0.990845\pi$
$613$		−	1.42405i		−	0.0575170i	0.999586	−	0.0287585i	$-0.00915538\pi$
$614$	1.36864			0.0552338
$615$		0			0
$616$		−	28.6419i		−	1.15402i
$617$		−	46.5266i		−	1.87309i	−0.350548	−	0.936545i	$-0.614005\pi$
$617$		−	46.5266i		−	1.87309i	0.350548	−	0.936545i	$-0.385995\pi$
$618$		0			0
$619$		−	34.2818i		−	1.37790i	−0.724809	−	0.688950i	$-0.758072\pi$
$619$		−	34.2818i		−	1.37790i	0.724809	−	0.688950i	$-0.241928\pi$
$620$	0.409536			0.0164473
$621$		0			0
$622$		−	2.02904i		−	0.0813569i
$623$	−20.4033			−0.817442
$624$		0			0
$625$	1.00000			0.0400000
$626$			2.84811i			0.113833i
$627$		0			0
$628$	−16.4527			−0.656534
$629$			17.7859i			0.709171i
$630$		0			0
$631$			2.32322i			0.0924861i	0.998930	+	0.0462430i	$0.0147249\pi$
$631$			2.32322i			0.0924861i	−0.998930	+	0.0462430i	$0.985275\pi$
$632$		−	7.63362i		−	0.303649i
$633$		0			0
$634$	10.6359			0.422405
$635$			12.0616i			0.478651i
$636$		0			0
$637$	0.226346	−	0.467210i	0.00896815	−	0.0185115i
$638$	−7.79834			−0.308739
$639$		0			0
$640$	11.5328			0.455874
$641$	−28.1312			−1.11111			−0.555557	−	0.831478i	$-0.687495\pi$
$641$	−28.1312			−1.11111			−0.555557	+	0.831478i	$0.687495\pi$
$642$		0			0
$643$			14.1338i			0.557383i	0.960381	+	0.278692i	$0.0899008\pi$
$643$			14.1338i			0.557383i	−0.960381	+	0.278692i	$0.910099\pi$
$644$			10.8560i			0.427787i
$645$		0			0
$646$	19.0946			0.751268
$647$	−45.8969			−1.80439			−0.902197	−	0.431325i	$-0.858046\pi$
$647$	−45.8969			−1.80439			−0.902197	+	0.431325i	$0.858046\pi$
$648$		0			0
$649$	4.58651			0.180036
$650$	1.85601	+	0.899170i	0.0727988	+	0.0352683i
$651$		0			0
$652$			20.7177i			0.811367i
$653$	−50.4482			−1.97419			−0.987095	−	0.160137i	$-0.948806\pi$
$653$	−50.4482			−1.97419			−0.987095	+	0.160137i	$0.948806\pi$
$654$		0			0
$655$			7.63362i			0.298270i
$656$			13.9137i			0.543238i
$657$		0			0
$658$		−	4.08631i		−	0.159301i
$659$	45.8722			1.78693			0.893464	−	0.449135i	$-0.148268\pi$
$659$	45.8722			1.78693			0.893464	+	0.449135i	$0.148268\pi$
$660$		0			0
$661$		−	25.8432i		−	1.00518i	−0.864524	−	0.502592i	$-0.832380\pi$
$661$		−	25.8432i		−	1.00518i	0.864524	−	0.502592i	$-0.167620\pi$
$662$	3.38614			0.131606
$663$		0			0
$664$	−20.6235			−0.800345
$665$			16.6913i			0.647261i
$666$		0			0
$667$	6.48963			0.251280
$668$			14.2326i			0.550674i
$669$		0			0
$670$		−	1.24877i		−	0.0482442i
$671$			29.6706i			1.14542i
$672$		0			0
$673$	−4.97927			−0.191937			−0.0959683	−	0.995384i	$-0.530595\pi$
$673$	−4.97927			−0.191937			−0.0959683	+	0.995384i	$0.530595\pi$
$674$			7.96306i			0.306726i
$675$		0			0
$676$	−13.4791	−	17.0656i	−0.518426	−	0.656368i
$677$	35.8353			1.37726			0.688631	−	0.725112i	$-0.258212\pi$
$677$	35.8353			1.37726			0.688631	+	0.725112i	$0.258212\pi$
$678$		0			0
$679$	30.3249			1.16376
$680$	11.2303			0.430662
$681$		0			0
$682$			0.714288i			0.0273515i
$683$		−	40.4712i		−	1.54859i	−0.632827	−	0.774293i	$-0.718106\pi$
$683$		−	40.4712i		−	1.54859i	0.632827	−	0.774293i	$-0.281894\pi$
$684$		0			0
$685$	3.43196			0.131128
$686$	10.4817			0.400194
$687$		0			0
$688$	23.4073			0.892394
$689$	−6.60498	+	13.6336i	−0.251630	+	0.519400i
$690$		0			0
$691$		−	12.2448i		−	0.465815i	−0.972499	−	0.232907i	$-0.925176\pi$
$691$		−	12.2448i		−	0.465815i	0.972499	−	0.232907i	$-0.0748239\pi$
$692$	−36.0448			−1.37022
$693$		0			0
$694$		−	4.38050i		−	0.166281i
$695$			0.942326i			0.0357445i
$696$		0			0
$697$			34.6913i			1.31403i
$698$	−6.39542			−0.242070
$699$		0			0
$700$			4.47116i			0.168994i
$701$	11.8353			0.447012			0.223506	−	0.974703i	$-0.428250\pi$
$701$	11.8353			0.447012			0.223506	+	0.974703i	$0.428250\pi$
$702$		0			0
$703$	20.7776			0.783642
$704$		−	6.03525i		−	0.227462i
$705$		0			0
$706$	5.52093			0.207783
$707$		−	29.0162i		−	1.09127i
$708$		0			0
$709$			6.37429i			0.239391i	0.992811	+	0.119696i	$0.0381919\pi$
$709$			6.37429i			0.239391i	−0.992811	+	0.119696i	$0.961808\pi$
$710$			3.57199i			0.134055i
$711$		0			0
$712$	16.0369			0.601010
$713$		−	0.594417i		−	0.0222611i
$714$		0			0
$715$	8.01847	−	16.5513i	0.299874	−	0.618982i
$716$	4.29588			0.160545
$717$		0			0
$718$	5.44083			0.203050
$719$	−34.9009			−1.30158			−0.650791	−	0.759257i	$-0.725563\pi$
$719$	−34.9009			−1.30158			−0.650791	+	0.759257i	$0.725563\pi$
$720$		0			0
$721$			28.4112i			1.05809i
$722$		−	11.4386i		−	0.425700i
$723$		0			0
$724$	22.9687			0.853625
$725$	2.67282			0.0992662
$726$		0			0
$727$	−19.8106			−0.734734			−0.367367	−	0.930076i	$-0.619741\pi$
$727$	−19.8106			−0.734734			−0.367367	+	0.930076i	$0.619741\pi$
$728$	−8.82698	+	18.2201i	−0.327150	+	0.675283i
$729$		0			0
$730$			6.26950i			0.232045i
$731$	58.3619			2.15859
$732$		0			0
$733$		−	12.6050i		−	0.465576i	−0.972528	−	0.232788i	$-0.925215\pi$
$733$		−	12.6050i		−	0.465576i	0.972528	−	0.232788i	$-0.0747848\pi$
$734$		−	9.72615i		−	0.358999i
$735$		0			0
$736$		−	13.1792i		−	0.485793i
$737$	−11.1361			−0.410203
$738$		0			0
$739$		−	37.1747i		−	1.36749i	−0.729719	−	0.683747i	$-0.760349\pi$
$739$		−	37.1747i		−	1.36749i	0.729719	−	0.683747i	$-0.239651\pi$
$740$	5.56578			0.204602
$741$		0			0
$742$	6.42366			0.235820
$743$		−	27.9322i		−	1.02473i	−0.858767	−	0.512366i	$-0.828769\pi$
$743$		−	27.9322i		−	1.02473i	0.858767	−	0.512366i	$-0.171231\pi$
$744$		0			0
$745$	−11.3456			−0.415672
$746$		−	1.04977i		−	0.0384348i
$747$		0			0
$748$		−	45.6133i		−	1.66779i
$749$			0.374285i			0.0136761i
$750$		0			0
$751$	13.6257			0.497209			0.248605	−	0.968605i	$-0.420028\pi$
$751$	13.6257			0.497209			0.248605	+	0.968605i	$0.420028\pi$
$752$		−	5.73050i		−	0.208970i
$753$		0			0
$754$	4.96080	+	2.40332i	0.180662	+	0.0875238i
$755$	21.0224			0.765084
$756$		0			0
$757$	−30.8560			−1.12148			−0.560740	−	0.827992i	$-0.689483\pi$
$757$	−30.8560			−1.12148			−0.560740	+	0.827992i	$0.689483\pi$
$758$	18.5143			0.672470
$759$		0			0
$760$		−	13.1193i		−	0.475887i
$761$		−	17.0162i		−	0.616837i	−0.951251	−	0.308419i	$-0.900200\pi$
$761$		−	17.0162i		−	0.616837i	0.951251	−	0.308419i	$-0.0997997\pi$
$762$		0			0
$763$	−24.2096			−0.876445
$764$	17.8847			0.647044
$765$		0			0
$766$	−16.0969			−0.581604
$767$	−2.91764	−	1.41349i	−0.105350	−	0.0510381i
$768$		0			0
$769$			29.0162i			1.04635i	0.852225	+	0.523176i	$0.175253\pi$
$769$			29.0162i			1.04635i	−0.852225	+	0.523176i	$0.824747\pi$
$770$	−7.79834			−0.281033
$771$		0			0
$772$		−	32.2175i		−	1.15953i
$773$		−	40.5160i		−	1.45726i	−0.684908	−	0.728630i	$-0.740158\pi$
$773$		−	40.5160i		−	1.45726i	0.684908	−	0.728630i	$-0.259842\pi$
$774$		0			0
$775$		−	0.244817i		−	0.00879409i
$776$	−23.8353			−0.855637
$777$		0			0
$778$			18.2801i			0.655372i
$779$	40.5266			1.45202
$780$		0			0
$781$	31.8538			1.13982
$782$		−	7.42405i		−	0.265484i
$783$		0			0
$784$	−0.308705			−0.0110252
$785$			9.83528i			0.351036i
$786$		0			0
$787$			34.0264i			1.21291i	0.795118	+	0.606455i	$0.207409\pi$
$787$			34.0264i			1.21291i	−0.795118	+	0.606455i	$0.792591\pi$
$788$		−	28.7407i		−	1.02384i
$789$		0			0
$790$	−2.07841			−0.0739464
$791$			34.6913i			1.23348i
$792$		0			0
$793$	9.14399	−	18.8745i	0.324712	−	0.670253i
$794$	−7.26990			−0.257999
$795$		0			0
$796$	−19.3166			−0.684659
$797$	−14.7776			−0.523450			−0.261725	−	0.965143i	$-0.584291\pi$
$797$	−14.7776			−0.523450			−0.261725	+	0.965143i	$0.584291\pi$
$798$		0			0
$799$		−	14.2880i		−	0.505472i
$800$		−	5.42801i		−	0.191909i
$801$		0			0
$802$	−0.560135			−0.0197790
$803$	55.9092			1.97299
$804$		0			0
$805$	6.48963			0.228730
$806$	0.220132	−	0.454384i	0.00775382	−	0.0160050i
$807$		0			0
$808$			22.8066i			0.802335i
$809$	36.0554			1.26764			0.633820	−	0.773480i	$-0.281486\pi$
$809$	36.0554			1.26764			0.633820	+	0.773480i	$0.281486\pi$
$810$		0			0
$811$			5.31040i			0.186473i	0.995644	+	0.0932366i	$0.0297213\pi$
$811$			5.31040i			0.186473i	−0.995644	+	0.0932366i	$0.970279\pi$
$812$			11.9506i			0.419385i
$813$		0			0
$814$			9.70750i			0.340248i
$815$	12.3849			0.433822
$816$		0			0
$817$		−	68.1787i		−	2.38527i
$818$	−4.46060			−0.155961
$819$		0			0
$820$	10.8560			0.379108
$821$		−	13.6336i		−	0.475817i	−0.971288	−	0.237908i	$-0.923538\pi$
$821$		−	13.6336i		−	0.475817i	0.971288	−	0.237908i	$-0.0764618\pi$
$822$		0			0
$823$	−39.6459			−1.38197			−0.690984	−	0.722870i	$-0.742823\pi$
$823$	−39.6459			−1.38197			−0.690984	+	0.722870i	$0.742823\pi$
$824$		−	22.3311i		−	0.777942i
$825$		0			0
$826$			1.37468i			0.0478314i
$827$			23.8952i			0.830918i	0.909612	+	0.415459i	$0.136379\pi$
$827$			23.8952i			0.830918i	−0.909612	+	0.415459i	$0.863621\pi$
$828$		0			0
$829$	27.3562			0.950121			0.475060	−	0.879953i	$-0.342426\pi$
$829$	27.3562			0.950121			0.475060	+	0.879953i	$0.342426\pi$
$830$			5.61515i			0.194905i
$831$		0			0
$832$	−1.85997	+	3.83923i	−0.0644827	+	0.133101i
$833$	−0.769701			−0.0266686
$834$		0			0
$835$	8.50811			0.294435
$836$	−53.2857			−1.84292
$837$		0			0
$838$		−	5.83528i		−	0.201576i
$839$		−	29.5411i		−	1.01987i	−0.860212	−	0.509936i	$-0.829669\pi$
$839$		−	29.5411i		−	1.01987i	0.860212	−	0.509936i	$-0.170331\pi$
$840$		0			0
$841$	−21.8560			−0.753656
$842$	−17.8398			−0.614800
$843$		0			0
$844$	40.3540			1.38904
$845$	−10.2017	+	8.05767i	−0.350948	+	0.277192i
$846$		0			0
$847$			40.1417i			1.37929i
$848$	9.00830			0.309346
$849$		0			0
$850$		−	3.05767i		−	0.104877i
$851$		−	8.07841i		−	0.276924i
$852$		0			0
$853$		−	5.61515i		−	0.192259i	−0.995369	−	0.0961295i	$-0.969354\pi$
$853$		−	5.61515i		−	0.192259i	0.995369	−	0.0961295i	$-0.0306463\pi$
$854$	−8.89296			−0.304311
$855$		0			0
$856$		−	0.294187i		−	0.0100551i
$857$	−39.9216			−1.36370			−0.681848	−	0.731494i	$-0.738823\pi$
$857$	−39.9216			−1.36370			−0.681848	+	0.731494i	$0.738823\pi$
$858$		0			0
$859$	28.6498			0.977520			0.488760	−	0.872418i	$-0.337449\pi$
$859$	28.6498			0.977520			0.488760	+	0.872418i	$0.337449\pi$
$860$		−	18.2633i		−	0.622773i
$861$		0			0
$862$	13.1193			0.446845
$863$			38.5530i			1.31236i	0.754605	+	0.656179i	$0.227828\pi$
$863$			38.5530i			1.31236i	−0.754605	+	0.656179i	$0.772172\pi$
$864$		0			0
$865$			21.5473i			0.732630i
$866$		−	9.31661i		−	0.316591i
$867$		0			0
$868$	1.09462			0.0371537
$869$			18.5345i			0.628739i
$870$		0			0
$871$	7.08405	+	3.43196i	0.240034	+	0.116288i
$872$	19.0286			0.644391
$873$		0			0
$874$	−8.67282			−0.293363
$875$	2.67282			0.0903579
$876$		0			0
$877$		−	27.3826i		−	0.924644i	−0.886712	−	0.462322i	$-0.847016\pi$
$877$		−	27.3826i		−	0.924644i	0.886712	−	0.462322i	$-0.152984\pi$
$878$			13.5888i			0.458599i
$879$		0			0
$880$	−10.9361			−0.368656
$881$	−25.0841			−0.845103			−0.422552	−	0.906339i	$-0.638865\pi$
$881$	−25.0841			−0.845103			−0.422552	+	0.906339i	$0.638865\pi$
$882$		0			0
$883$	50.5513			1.70119			0.850593	−	0.525825i	$-0.176243\pi$
$883$	50.5513			1.70119			0.850593	+	0.525825i	$0.176243\pi$
$884$	−14.0573	+	29.0162i	−0.472797	+	0.975921i
$885$		0			0
$886$			1.57465i			0.0529015i
$887$	19.6583			0.660061			0.330031	−	0.943970i	$-0.392941\pi$
$887$	19.6583			0.660061			0.330031	+	0.943970i	$0.392941\pi$
$888$		0			0
$889$			32.2386i			1.08125i
$890$		−	4.36638i		−	0.146361i
$891$		0			0
$892$		−	9.58611i		−	0.320967i
$893$	−16.6913			−0.558553
$894$		0			0
$895$		−	2.56804i		−	0.0858401i
$896$	30.8251			1.02979
$897$		0			0
$898$	−15.5025			−0.517324
$899$		−	0.654353i		−	0.0218239i
$900$		0			0
$901$	22.4606			0.748271
$902$			18.9344i			0.630447i
$903$		0			0
$904$		−	27.2672i		−	0.906895i
$905$		−	13.7305i		−	0.456417i
$906$		0			0
$907$	−22.0537			−0.732282			−0.366141	−	0.930559i	$-0.619321\pi$
$907$	−22.0537			−0.732282			−0.366141	+	0.930559i	$0.619321\pi$
$908$			4.93967i			0.163929i
$909$		0			0
$910$	4.96080	+	2.40332i	0.164449	+	0.0796693i
$911$	14.0079			0.464103			0.232051	−	0.972704i	$-0.425456\pi$
$911$	14.0079			0.464103			0.232051	+	0.972704i	$0.425456\pi$
$912$		0			0
$913$	50.0739			1.65720
$914$	22.9668			0.759676
$915$		0			0
$916$			42.2544i			1.39613i
$917$			20.4033i			0.673777i
$918$		0			0
$919$	21.8353			0.720279			0.360140	−	0.932898i	$-0.382729\pi$
$919$	21.8353			0.720279			0.360140	+	0.932898i	$0.382729\pi$
$920$	−5.10083			−0.168169
$921$		0			0
$922$	−1.14851			−0.0378241
$923$	−20.2633	−	9.81681i	−0.666974	−	0.323124i
$924$		0			0
$925$		−	3.32718i		−	0.109397i
$926$	−1.52884			−0.0502407
$927$		0			0
$928$		−	14.5081i		−	0.476252i
$929$			15.2224i			0.499431i	0.968319	+	0.249715i	$0.0803370\pi$
$929$			15.2224i			0.499431i	−0.968319	+	0.249715i	$0.919663\pi$
$930$		0			0
$931$			0.899170i			0.0294691i
$932$	39.0532			1.27923
$933$		0			0
$934$		−	18.2403i		−	0.596841i
$935$	−27.2672			−0.891734
$936$		0			0
$937$	−25.4610			−0.831774			−0.415887	−	0.909416i	$-0.636529\pi$
$937$	−25.4610			−0.831774			−0.415887	+	0.909416i	$0.636529\pi$
$938$		−	3.33774i		−	0.108981i
$939$		0			0
$940$	−4.47116			−0.145833
$941$		−	24.9299i		−	0.812691i	−0.913719	−	0.406346i	$-0.866803\pi$
$941$		−	24.9299i		−	0.812691i	0.913719	−	0.406346i	$-0.133197\pi$
$942$		0			0
$943$		−	15.7569i		−	0.513114i
$944$			1.92781i			0.0627448i
$945$		0			0
$946$	31.8538			1.03565
$947$		−	16.6807i		−	0.542051i	−0.962572	−	0.271025i	$-0.912637\pi$
$947$		−	16.6807i		−	0.542051i	0.962572	−	0.271025i	$-0.0873628\pi$
$948$		0			0
$949$	−35.5658	−	17.2303i	−1.15451	−	0.559319i
$950$	−3.57199			−0.115891
$951$		0			0
$952$	30.0166			0.972844
$953$	27.1730			0.880221			0.440110	−	0.897944i	$-0.354939\pi$
$953$	27.1730			0.880221			0.440110	+	0.897944i	$0.354939\pi$
$954$		0			0
$955$		−	10.6913i		−	0.345962i
$956$		−	6.34356i		−	0.205165i
$957$		0			0
$958$	−3.85206			−0.124454
$959$	9.17302			0.296212
$960$		0			0
$961$	30.9401			0.998067
$962$	2.99170	−	6.17528i	0.0964561	−	0.199099i
$963$		0			0
$964$		−	5.74106i		−	0.184907i
$965$	−19.2593			−0.619980
$966$		0			0
$967$			6.75914i			0.217359i	0.994077	+	0.108680i	$0.0346622\pi$
$967$			6.75914i			0.217359i	−0.994077	+	0.108680i	$0.965338\pi$
$968$		−	31.5513i		−	1.01410i
$969$		0			0
$970$			6.48963i			0.208370i
$971$	−46.6419			−1.49681			−0.748405	−	0.663242i	$-0.769180\pi$
$971$	−46.6419			−1.49681			−0.748405	+	0.663242i	$0.769180\pi$
$972$		0			0
$973$			2.51867i			0.0807449i
$974$	22.6807			0.726737
$975$		0			0
$976$	−12.4712			−0.399192
$977$			52.7467i			1.68752i	0.536723	+	0.843758i	$0.319662\pi$
$977$			52.7467i			1.68752i	−0.536723	+	0.843758i	$0.680338\pi$
$978$		0			0
$979$	−38.9378			−1.24446
$980$			0.240864i			0.00769412i
$981$		0			0
$982$		−	17.7859i		−	0.567571i
$983$			20.9977i			0.669724i	0.942267	+	0.334862i	$0.108690\pi$
$983$			20.9977i			0.669724i	−0.942267	+	0.334862i	$0.891310\pi$
$984$		0			0
$985$	−17.1809			−0.547430
$986$		−	8.17262i		−	0.260269i
$987$		0			0
$988$	33.8969	+	16.4218i	1.07840	+	0.522447i
$989$	−26.5081			−0.842909
$990$		0			0
$991$	−32.3170			−1.02658			−0.513292	−	0.858214i	$-0.671574\pi$
$991$	−32.3170			−1.02658			−0.513292	+	0.858214i	$0.671574\pi$
$992$	−1.32887			−0.0421916
$993$		0			0
$994$			9.54731i			0.302822i
$995$			11.5473i			0.366074i
$996$		0			0
$997$	−21.3905			−0.677444			−0.338722	−	0.940887i	$-0.609995\pi$
$997$	−21.3905			−0.677444			−0.338722	+	0.940887i	$0.609995\pi$
$998$	7.49811			0.237348
$999$		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	585.2.b.g.181.4		6
3.2	odd	2		65.2.c.a.51.3	✓	6
12.11	even	2		1040.2.k.d.961.4		6
13.5	odd	4		7605.2.a.cc.1.2		3
13.8	odd	4		7605.2.a.bs.1.2		3
13.12	even	2	inner	585.2.b.g.181.3		6
15.2	even	4		325.2.d.f.324.4		6
15.8	even	4		325.2.d.e.324.3		6
15.14	odd	2		325.2.c.g.51.4		6
39.2	even	12		845.2.e.k.191.2		6
39.5	even	4		845.2.a.i.1.2		3
39.8	even	4		845.2.a.k.1.2		3
39.11	even	12		845.2.e.i.191.2		6
39.17	odd	6		845.2.m.h.361.4		12
39.20	even	12		845.2.e.i.146.2		6
39.23	odd	6		845.2.m.h.316.3		12
39.29	odd	6		845.2.m.h.316.4		12
39.32	even	12		845.2.e.k.146.2		6
39.35	odd	6		845.2.m.h.361.3		12
39.38	odd	2		65.2.c.a.51.4	yes	6
156.155	even	2		1040.2.k.d.961.3		6
195.38	even	4		325.2.d.f.324.3		6
195.44	even	4		4225.2.a.be.1.2		3
195.77	even	4		325.2.d.e.324.4		6
195.164	even	4		4225.2.a.bc.1.2		3
195.194	odd	2		325.2.c.g.51.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.c.a.51.3	✓	6	3.2	odd	2
65.2.c.a.51.4	yes	6	39.38	odd	2
325.2.c.g.51.3		6	195.194	odd	2
325.2.c.g.51.4		6	15.14	odd	2
325.2.d.e.324.3		6	15.8	even	4
325.2.d.e.324.4		6	195.77	even	4
325.2.d.f.324.3		6	195.38	even	4
325.2.d.f.324.4		6	15.2	even	4
585.2.b.g.181.3		6	13.12	even	2	inner
585.2.b.g.181.4		6	1.1	even	1	trivial
845.2.a.i.1.2		3	39.5	even	4
845.2.a.k.1.2		3	39.8	even	4
845.2.e.i.146.2		6	39.20	even	12
845.2.e.i.191.2		6	39.11	even	12
845.2.e.k.146.2		6	39.32	even	12
845.2.e.k.191.2		6	39.2	even	12
845.2.m.h.316.3		12	39.23	odd	6
845.2.m.h.316.4		12	39.29	odd	6
845.2.m.h.361.3		12	39.35	odd	6
845.2.m.h.361.4		12	39.17	odd	6
1040.2.k.d.961.3		6	156.155	even	2
1040.2.k.d.961.4		6	12.11	even	2
4225.2.a.bc.1.2		3	195.164	even	4
4225.2.a.be.1.2		3	195.44	even	4
7605.2.a.bs.1.2		3	13.8	odd	4
7605.2.a.cc.1.2		3	13.5	odd	4

Overview	Random
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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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

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	585.2.b.g.181.4

Level	$585$
Weight	$2$
Character	585.181
Analytic conductor	$4.671$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$