LMFDB - Embedded newform 2592.2.d.h.1297.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2592,2,Mod(1297,2592)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2592.1297"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2592, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$2592 = 2^{5} \cdot 3^{4}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2592.d (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-4,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$20.6972242039$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{2} + 1$ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 648)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1297.4
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2592.1297
Dual form			2592.2.d.h.1297.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.73205i q^{5} +0.732051 q^{7} -4.73205i q^{11} -2.46410i q^{13} +3.73205 q^{17} -3.26795i q^{19} +8.73205 q^{23} -8.92820 q^{25} -5.19615i q^{29} -2.00000 q^{31} +2.73205i q^{35} -5.00000i q^{37} -4.53590 q^{41} -9.66025i q^{43} +3.46410 q^{47} -6.46410 q^{49} -0.928203i q^{53} +17.6603 q^{55} -8.39230i q^{59} +9.00000i q^{61} +9.19615 q^{65} -4.73205i q^{67} +5.66025 q^{71} +9.00000 q^{73} -3.46410i q^{77} -5.26795 q^{79} +14.0000i q^{83} +13.9282i q^{85} +11.7321 q^{89} -1.80385i q^{91} +12.1962 q^{95} -8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{7} + 8 q^{17} + 28 q^{23} - 8 q^{25} - 8 q^{31} - 32 q^{41} - 12 q^{49} + 36 q^{55} + 16 q^{65} - 12 q^{71} + 36 q^{73} - 28 q^{79} + 40 q^{89} + 28 q^{95} - 32 q^{97}+O(q^{100})$ 4 * q - 4 * q^7 + 8 * q^17 + 28 * q^23 - 8 * q^25 - 8 * q^31 - 32 * q^41 - 12 * q^49 + 36 * q^55 + 16 * q^65 - 12 * q^71 + 36 * q^73 - 28 * q^79 + 40 * q^89 + 28 * q^95 - 32 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$ .

$n$	$325$	$1217$	$2431$
$\chi(n)$	$-1$	$1$	$1$

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	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.73205i	1.66902i	0.550990	+	0.834512i	$0.314250\pi$
$5$	3.73205i	1.66902i	−0.550990	+	0.834512i	$0.685750\pi$
$6$	0	0
$7$	0.732051	0.276689	0.138345	−	0.990384i	$-0.455822\pi$
$7$	0.732051	0.276689	0.138345	+	0.990384i	$0.455822\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 4.73205i	− 1.42677i	−0.700774	−	0.713384i	$-0.747162\pi$
$11$	− 4.73205i	− 1.42677i	0.700774	−	0.713384i	$-0.252838\pi$
$12$	0	0
$13$	− 2.46410i	− 0.683419i	−0.939806	−	0.341709i	$-0.888994\pi$
$13$	− 2.46410i	− 0.683419i	0.939806	−	0.341709i	$-0.111006\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.73205	0.905155	0.452578	−	0.891725i	$-0.350505\pi$
$17$	3.73205	0.905155	0.452578	+	0.891725i	$0.350505\pi$
$18$	0	0
$19$	− 3.26795i	− 0.749719i	−0.927082	−	0.374859i	$-0.877691\pi$
$19$	− 3.26795i	− 0.749719i	0.927082	−	0.374859i	$-0.122309\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.73205	1.82076	0.910379	−	0.413775i	$-0.135790\pi$
$23$	8.73205	1.82076	0.910379	+	0.413775i	$0.135790\pi$
$24$	0	0
$25$	−8.92820	−1.78564
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 5.19615i	− 0.964901i	−0.875923	−	0.482451i	$-0.839747\pi$
$29$	− 5.19615i	− 0.964901i	0.875923	−	0.482451i	$-0.160253\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.73205i	0.461801i
$36$	0	0
$37$	− 5.00000i	− 0.821995i	−0.911636	−	0.410997i	$-0.865181\pi$
$37$	− 5.00000i	− 0.821995i	0.911636	−	0.410997i	$-0.134819\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.53590	−0.708388	−0.354194	−	0.935172i	$-0.615245\pi$
$41$	−4.53590	−0.708388	−0.354194	+	0.935172i	$0.615245\pi$
$42$	0	0
$43$	− 9.66025i	− 1.47317i	−0.676342	−	0.736587i	$-0.736436\pi$
$43$	− 9.66025i	− 1.47317i	0.676342	−	0.736587i	$-0.263564\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.46410	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$48$	0	0
$49$	−6.46410	−0.923443
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 0.928203i	− 0.127499i	−0.997966	−	0.0637493i	$-0.979694\pi$
$53$	− 0.928203i	− 0.127499i	0.997966	−	0.0637493i	$-0.0203058\pi$
$54$	0	0
$55$	17.6603	2.38131
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 8.39230i	− 1.09259i	−0.837594	−	0.546293i	$-0.816039\pi$
$59$	− 8.39230i	− 1.09259i	0.837594	−	0.546293i	$-0.183961\pi$
$60$	0	0
$61$	9.00000i	1.15233i	0.817333	+	0.576166i	$0.195452\pi$
$61$	9.00000i	1.15233i	−0.817333	+	0.576166i	$0.804548\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.19615	1.14064
$66$	0	0
$67$	− 4.73205i	− 0.578112i	−0.957312	−	0.289056i	$-0.906659\pi$
$67$	− 4.73205i	− 0.578112i	0.957312	−	0.289056i	$-0.0933414\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.66025	0.671749	0.335874	−	0.941907i	$-0.390968\pi$
$71$	5.66025	0.671749	0.335874	+	0.941907i	$0.390968\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.46410i	− 0.394771i
$78$	0	0
$79$	−5.26795	−0.592691	−0.296345	−	0.955081i	$-0.595768\pi$
$79$	−5.26795	−0.592691	−0.296345	+	0.955081i	$0.595768\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	14.0000i	1.53670i	0.640030	+	0.768350i	$0.278922\pi$
$83$	14.0000i	1.53670i	−0.640030	+	0.768350i	$0.721078\pi$
$84$	0	0
$85$	13.9282i	1.51073i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.7321	1.24359	0.621797	−	0.783178i	$-0.286403\pi$
$89$	11.7321	1.24359	0.621797	+	0.783178i	$0.286403\pi$
$90$	0	0
$91$	− 1.80385i	− 0.189095i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	12.1962	1.25130
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.00000i	0.796030i	0.917379	+	0.398015i	$0.130301\pi$
$101$	8.00000i	0.796030i	−0.917379	+	0.398015i	$0.869699\pi$
$102$	0	0
$103$	3.46410	0.341328	0.170664	−	0.985329i	$-0.445409\pi$
$103$	3.46410	0.341328	0.170664	+	0.985329i	$0.445409\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 10.3923i	− 1.00466i	−0.864675	−	0.502331i	$-0.832476\pi$
$107$	− 10.3923i	− 1.00466i	0.864675	−	0.502331i	$-0.167524\pi$
$108$	0	0
$109$	12.8564i	1.23142i	0.787973	+	0.615710i	$0.211131\pi$
$109$	12.8564i	1.23142i	−0.787973	+	0.615710i	$0.788869\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.73205	0.727370	0.363685	−	0.931522i	$-0.381518\pi$
$113$	7.73205	0.727370	0.363685	+	0.931522i	$0.381518\pi$
$114$	0	0
$115$	32.5885i	3.03889i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.73205	0.250447
$120$	0	0
$121$	−11.3923	−1.03566
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 14.6603i	− 1.31125i
$126$	0	0
$127$	13.6603	1.21215	0.606076	−	0.795407i	$-0.292743\pi$
$127$	13.6603	1.21215	0.606076	+	0.795407i	$0.292743\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 0.196152i	− 0.0171379i	−0.999963	−	0.00856896i	$-0.997272\pi$
$131$	− 0.196152i	− 0.0171379i	0.999963	−	0.00856896i	$-0.00272762\pi$
$132$	0	0
$133$	− 2.39230i	− 0.207439i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.1962	1.46917	0.734583	−	0.678519i	$-0.237378\pi$
$137$	17.1962	1.46917	0.734583	+	0.678519i	$0.237378\pi$
$138$	0	0
$139$	19.4641i	1.65092i	0.564458	+	0.825462i	$0.309085\pi$
$139$	19.4641i	1.65092i	−0.564458	+	0.825462i	$0.690915\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−11.6603	−0.975079
$144$	0	0
$145$	19.3923	1.61044
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 5.19615i	− 0.425685i	−0.977086	−	0.212843i	$-0.931728\pi$
$149$	− 5.19615i	− 0.425685i	0.977086	−	0.212843i	$-0.0682722\pi$
$150$	0	0
$151$	−13.8564	−1.12762	−0.563809	−	0.825905i	$-0.690665\pi$
$151$	−13.8564	−1.12762	−0.563809	+	0.825905i	$0.690665\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 7.46410i	− 0.599531i
$156$	0	0
$157$	− 2.46410i	− 0.196657i	−0.995154	−	0.0983284i	$-0.968650\pi$
$157$	− 2.46410i	− 0.196657i	0.995154	−	0.0983284i	$-0.0313495\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.39230	0.503784
$162$	0	0
$163$	− 6.53590i	− 0.511931i	−0.966686	−	0.255966i	$-0.917607\pi$
$163$	− 6.53590i	− 0.511931i	0.966686	−	0.255966i	$-0.0823934\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.7321	−1.14000	−0.570000	−	0.821645i	$-0.693057\pi$
$167$	−14.7321	−1.14000	−0.570000	+	0.821645i	$0.693057\pi$
$168$	0	0
$169$	6.92820	0.532939
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.73205i	0.131685i	0.997830	+	0.0658427i	$0.0209736\pi$
$173$	1.73205i	0.131685i	−0.997830	+	0.0658427i	$0.979026\pi$
$174$	0	0
$175$	−6.53590	−0.494067
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.0000i	1.04641i	0.852207	+	0.523205i	$0.175264\pi$
$179$	14.0000i	1.04641i	−0.852207	+	0.523205i	$0.824736\pi$
$180$	0	0
$181$	− 6.53590i	− 0.485810i	−0.970050	−	0.242905i	$-0.921900\pi$
$181$	− 6.53590i	− 0.485810i	0.970050	−	0.242905i	$-0.0781002\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	18.6603	1.37193
$186$	0	0
$187$	− 17.6603i	− 1.29145i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.5885	1.20030	0.600149	−	0.799888i	$-0.295108\pi$
$191$	16.5885	1.20030	0.600149	+	0.799888i	$0.295108\pi$
$192$	0	0
$193$	10.8564	0.781461	0.390731	−	0.920505i	$-0.372222\pi$
$193$	10.8564	0.781461	0.390731	+	0.920505i	$0.372222\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.12436i	0.293848i	0.989148	+	0.146924i	$0.0469373\pi$
$197$	4.12436i	0.293848i	−0.989148	+	0.146924i	$0.953063\pi$
$198$	0	0
$199$	−2.39230	−0.169586	−0.0847930	−	0.996399i	$-0.527023\pi$
$199$	−2.39230	−0.169586	−0.0847930	+	0.996399i	$0.527023\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 3.80385i	− 0.266978i
$204$	0	0
$205$	− 16.9282i	− 1.18232i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−15.4641	−1.06967
$210$	0	0
$211$	− 26.0526i	− 1.79353i	−0.442505	−	0.896766i	$-0.645910\pi$
$211$	− 26.0526i	− 1.79353i	0.442505	−	0.896766i	$-0.354090\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	36.0526	2.45876
$216$	0	0
$217$	−1.46410	−0.0993897
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 9.19615i	− 0.618600i
$222$	0	0
$223$	−23.1244	−1.54852	−0.774261	−	0.632867i	$-0.781878\pi$
$223$	−23.1244	−1.54852	−0.774261	+	0.632867i	$0.781878\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 6.73205i	− 0.446822i	−0.974724	−	0.223411i	$-0.928281\pi$
$227$	− 6.73205i	− 0.446822i	0.974724	−	0.223411i	$-0.0717192\pi$
$228$	0	0
$229$	− 11.3923i	− 0.752825i	−0.926452	−	0.376412i	$-0.877158\pi$
$229$	− 11.3923i	− 0.752825i	0.926452	−	0.376412i	$-0.122842\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.73205	−0.244495	−0.122247	−	0.992500i	$-0.539010\pi$
$233$	−3.73205	−0.244495	−0.122247	+	0.992500i	$0.539010\pi$
$234$	0	0
$235$	12.9282i	0.843343i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	7.85641	0.508189	0.254094	−	0.967179i	$-0.418223\pi$
$239$	7.85641	0.508189	0.254094	+	0.967179i	$0.418223\pi$
$240$	0	0
$241$	−4.46410	−0.287558	−0.143779	−	0.989610i	$-0.545925\pi$
$241$	−4.46410	−0.287558	−0.143779	+	0.989610i	$0.545925\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 24.1244i	− 1.54125i
$246$	0	0
$247$	−8.05256	−0.512372
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 7.26795i	− 0.458749i	−0.973338	−	0.229374i	$-0.926332\pi$
$251$	− 7.26795i	− 0.458749i	0.973338	−	0.229374i	$-0.0736680\pi$
$252$	0	0
$253$	− 41.3205i	− 2.59780i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.33975	0.208328	0.104164	−	0.994560i	$-0.466783\pi$
$257$	3.33975	0.208328	0.104164	+	0.994560i	$0.466783\pi$
$258$	0	0
$259$	− 3.66025i	− 0.227437i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.2487	−1.00194	−0.500969	−	0.865465i	$-0.667023\pi$
$263$	−16.2487	−1.00194	−0.500969	+	0.865465i	$0.667023\pi$
$264$	0	0
$265$	3.46410	0.212798
$266$	0	0
$267$	0	0
$268$	0	0
$269$	25.7321i	1.56891i	0.620185	+	0.784455i	$0.287057\pi$
$269$	25.7321i	1.56891i	−0.620185	+	0.784455i	$0.712943\pi$
$270$	0	0
$271$	−10.7321	−0.651926	−0.325963	−	0.945383i	$-0.605688\pi$
$271$	−10.7321	−0.651926	−0.325963	+	0.945383i	$0.605688\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	42.2487i	2.54769i
$276$	0	0
$277$	9.85641i	0.592214i	0.955155	+	0.296107i	$0.0956885\pi$
$277$	9.85641i	0.592214i	−0.955155	+	0.296107i	$0.904311\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−13.7321	−0.819185	−0.409593	−	0.912268i	$-0.634329\pi$
$281$	−13.7321	−0.819185	−0.409593	+	0.912268i	$0.634329\pi$
$282$	0	0
$283$	6.39230i	0.379983i	0.981786	+	0.189992i	$0.0608461\pi$
$283$	6.39230i	0.379983i	−0.981786	+	0.189992i	$0.939154\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.32051	−0.196003
$288$	0	0
$289$	−3.07180	−0.180694
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.33975i	0.195110i	0.995230	+	0.0975550i	$0.0311022\pi$
$293$	3.33975i	0.195110i	−0.995230	+	0.0975550i	$0.968898\pi$
$294$	0	0
$295$	31.3205	1.82355
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 21.5167i	− 1.24434i
$300$	0	0
$301$	− 7.07180i	− 0.407612i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−33.5885	−1.92327
$306$	0	0
$307$	− 14.7846i	− 0.843802i	−0.906642	−	0.421901i	$-0.861363\pi$
$307$	− 14.7846i	− 0.843802i	0.906642	−	0.421901i	$-0.138637\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−19.3205	−1.09557	−0.547783	−	0.836621i	$-0.684528\pi$
$311$	−19.3205	−1.09557	−0.547783	+	0.836621i	$0.684528\pi$
$312$	0	0
$313$	2.46410	0.139279	0.0696396	−	0.997572i	$-0.477815\pi$
$313$	2.46410	0.139279	0.0696396	+	0.997572i	$0.477815\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.19615i	0.0671826i	0.999436	+	0.0335913i	$0.0106945\pi$
$317$	1.19615i	0.0671826i	−0.999436	+	0.0335913i	$0.989306\pi$
$318$	0	0
$319$	−24.5885	−1.37669
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 12.1962i	− 0.678612i
$324$	0	0
$325$	22.0000i	1.22034i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.53590	0.139809
$330$	0	0
$331$	− 13.1244i	− 0.721380i	−0.932686	−	0.360690i	$-0.882541\pi$
$331$	− 13.1244i	− 0.721380i	0.932686	−	0.360690i	$-0.117459\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	17.6603	0.964883
$336$	0	0
$337$	4.00000	0.217894	0.108947	−	0.994048i	$-0.465252\pi$
$337$	4.00000	0.217894	0.108947	+	0.994048i	$0.465252\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	9.46410i	0.512510i
$342$	0	0
$343$	−9.85641	−0.532196
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 6.73205i	− 0.361395i	−0.983539	−	0.180698i	$-0.942164\pi$
$347$	− 6.73205i	− 0.361395i	0.983539	−	0.180698i	$-0.0578356\pi$
$348$	0	0
$349$	11.8564i	0.634659i	0.948315	+	0.317329i	$0.102786\pi$
$349$	11.8564i	0.634659i	−0.948315	+	0.317329i	$0.897214\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.92820	0.368751	0.184376	−	0.982856i	$-0.440974\pi$
$353$	6.92820	0.368751	0.184376	+	0.982856i	$0.440974\pi$
$354$	0	0
$355$	21.1244i	1.12116i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.12436	0.0593412	0.0296706	−	0.999560i	$-0.490554\pi$
$359$	1.12436	0.0593412	0.0296706	+	0.999560i	$0.490554\pi$
$360$	0	0
$361$	8.32051	0.437921
$362$	0	0
$363$	0	0
$364$	0	0
$365$	33.5885i	1.75810i
$366$	0	0
$367$	−2.53590	−0.132373	−0.0661864	−	0.997807i	$-0.521083\pi$
$367$	−2.53590	−0.132373	−0.0661864	+	0.997807i	$0.521083\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 0.679492i	− 0.0352775i
$372$	0	0
$373$	− 22.9282i	− 1.18718i	−0.804769	−	0.593589i	$-0.797711\pi$
$373$	− 22.9282i	− 1.18718i	0.804769	−	0.593589i	$-0.202289\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−12.8038	−0.659432
$378$	0	0
$379$	10.0000i	0.513665i	0.966456	+	0.256833i	$0.0826790\pi$
$379$	10.0000i	0.513665i	−0.966456	+	0.256833i	$0.917321\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−20.1962	−1.03198	−0.515988	−	0.856596i	$-0.672575\pi$
$383$	−20.1962	−1.03198	−0.515988	+	0.856596i	$0.672575\pi$
$384$	0	0
$385$	12.9282	0.658882
$386$	0	0
$387$	0	0
$388$	0	0
$389$	16.9282i	0.858294i	0.903235	+	0.429147i	$0.141186\pi$
$389$	16.9282i	0.858294i	−0.903235	+	0.429147i	$0.858814\pi$
$390$	0	0
$391$	32.5885	1.64807
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 19.6603i	− 0.989215i
$396$	0	0
$397$	32.4641i	1.62933i	0.579934	+	0.814663i	$0.303078\pi$
$397$	32.4641i	1.62933i	−0.579934	+	0.814663i	$0.696922\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9.58846	0.478825	0.239412	−	0.970918i	$-0.423045\pi$
$401$	9.58846	0.478825	0.239412	+	0.970918i	$0.423045\pi$
$402$	0	0
$403$	4.92820i	0.245491i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−23.6603	−1.17280
$408$	0	0
$409$	28.7128	1.41976	0.709879	−	0.704324i	$-0.248750\pi$
$409$	28.7128	1.41976	0.709879	+	0.704324i	$0.248750\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 6.14359i	− 0.302306i
$414$	0	0
$415$	−52.2487	−2.56479
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 21.4641i	− 1.04859i	−0.851537	−	0.524295i	$-0.824329\pi$
$419$	− 21.4641i	− 1.04859i	0.851537	−	0.524295i	$-0.175671\pi$
$420$	0	0
$421$	− 13.0000i	− 0.633581i	−0.948495	−	0.316791i	$-0.897395\pi$
$421$	− 13.0000i	− 0.633581i	0.948495	−	0.316791i	$-0.102605\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−33.3205	−1.61628
$426$	0	0
$427$	6.58846i	0.318838i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.7128	−1.04587	−0.522935	−	0.852373i	$-0.675163\pi$
$431$	−21.7128	−1.04587	−0.522935	+	0.852373i	$0.675163\pi$
$432$	0	0
$433$	−21.7846	−1.04690	−0.523451	−	0.852056i	$-0.675356\pi$
$433$	−21.7846	−1.04690	−0.523451	+	0.852056i	$0.675356\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 28.5359i	− 1.36506i
$438$	0	0
$439$	30.0000	1.43182	0.715911	−	0.698192i	$-0.246012\pi$
$439$	30.0000	1.43182	0.715911	+	0.698192i	$0.246012\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	29.3205i	1.39306i	0.717528	+	0.696530i	$0.245274\pi$
$443$	29.3205i	1.39306i	−0.717528	+	0.696530i	$0.754726\pi$
$444$	0	0
$445$	43.7846i	2.07559i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	24.9282	1.17643	0.588217	−	0.808703i	$-0.299830\pi$
$449$	24.9282	1.17643	0.588217	+	0.808703i	$0.299830\pi$
$450$	0	0
$451$	21.4641i	1.01071i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.73205	0.315603
$456$	0	0
$457$	−6.32051	−0.295661	−0.147830	−	0.989013i	$-0.547229\pi$
$457$	−6.32051	−0.295661	−0.147830	+	0.989013i	$0.547229\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	41.3205i	1.92449i	0.272188	+	0.962244i	$0.412253\pi$
$461$	41.3205i	1.92449i	−0.272188	+	0.962244i	$0.587747\pi$
$462$	0	0
$463$	−13.4641	−0.625730	−0.312865	−	0.949798i	$-0.601289\pi$
$463$	−13.4641	−0.625730	−0.312865	+	0.949798i	$0.601289\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.7846i	1.14689i	0.819242	+	0.573447i	$0.194394\pi$
$467$	24.7846i	1.14689i	−0.819242	+	0.573447i	$0.805606\pi$
$468$	0	0
$469$	− 3.46410i	− 0.159957i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−45.7128	−2.10188
$474$	0	0
$475$	29.1769i	1.33873i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.66025	0.0758589	0.0379295	−	0.999280i	$-0.487924\pi$
$479$	1.66025	0.0758589	0.0379295	+	0.999280i	$0.487924\pi$
$480$	0	0
$481$	−12.3205	−0.561767
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 29.8564i	− 1.35571i
$486$	0	0
$487$	25.7128	1.16516	0.582579	−	0.812774i	$-0.302044\pi$
$487$	25.7128	1.16516	0.582579	+	0.812774i	$0.302044\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 13.8038i	− 0.622959i	−0.950253	−	0.311479i	$-0.899175\pi$
$491$	− 13.8038i	− 0.622959i	0.950253	−	0.311479i	$-0.100825\pi$
$492$	0	0
$493$	− 19.3923i	− 0.873385i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.14359	0.185866
$498$	0	0
$499$	− 8.19615i	− 0.366910i	−0.983028	−	0.183455i	$-0.941272\pi$
$499$	− 8.19615i	− 0.366910i	0.983028	−	0.183455i	$-0.0587282\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−0.679492	−0.0302970	−0.0151485	−	0.999885i	$-0.504822\pi$
$503$	−0.679492	−0.0302970	−0.0151485	+	0.999885i	$0.504822\pi$
$504$	0	0
$505$	−29.8564	−1.32859
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 29.3205i	− 1.29961i	−0.760102	−	0.649804i	$-0.774851\pi$
$509$	− 29.3205i	− 1.29961i	0.760102	−	0.649804i	$-0.225149\pi$
$510$	0	0
$511$	6.58846	0.291456
$512$	0	0
$513$	0	0
$514$	0	0
$515$	12.9282i	0.569685i
$516$	0	0
$517$	− 16.3923i	− 0.720933i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−26.7846	−1.17346	−0.586728	−	0.809784i	$-0.699584\pi$
$521$	−26.7846	−1.17346	−0.586728	+	0.809784i	$0.699584\pi$
$522$	0	0
$523$	13.2679i	0.580167i	0.957001	+	0.290083i	$0.0936831\pi$
$523$	13.2679i	0.580167i	−0.957001	+	0.290083i	$0.906317\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.46410	−0.325141
$528$	0	0
$529$	53.2487	2.31516
$530$	0	0
$531$	0	0
$532$	0	0
$533$	11.1769i	0.484126i
$534$	0	0
$535$	38.7846	1.67680
$536$	0	0
$537$	0	0
$538$	0	0
$539$	30.5885i	1.31754i
$540$	0	0
$541$	− 24.1769i	− 1.03945i	−0.854335	−	0.519723i	$-0.826035\pi$
$541$	− 24.1769i	− 1.03945i	0.854335	−	0.519723i	$-0.173965\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−47.9808	−2.05527
$546$	0	0
$547$	− 26.0000i	− 1.11168i	−0.831289	−	0.555840i	$-0.812397\pi$
$547$	− 26.0000i	− 1.11168i	0.831289	−	0.555840i	$-0.187603\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−16.9808	−0.723405
$552$	0	0
$553$	−3.85641	−0.163991
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.0526i	0.553055i	0.961006	+	0.276527i	$0.0891836\pi$
$557$	13.0526i	0.553055i	−0.961006	+	0.276527i	$0.910816\pi$
$558$	0	0
$559$	−23.8038	−1.00680
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 24.2487i	− 1.02196i	−0.859592	−	0.510981i	$-0.829282\pi$
$563$	− 24.2487i	− 1.02196i	0.859592	−	0.510981i	$-0.170718\pi$
$564$	0	0
$565$	28.8564i	1.21400i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	33.5885	1.40810	0.704051	−	0.710150i	$-0.251373\pi$
$569$	33.5885	1.40810	0.704051	+	0.710150i	$0.251373\pi$
$570$	0	0
$571$	− 26.2487i	− 1.09847i	−0.835667	−	0.549237i	$-0.814918\pi$
$571$	− 26.2487i	− 1.09847i	0.835667	−	0.549237i	$-0.185082\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−77.9615	−3.25122
$576$	0	0
$577$	−45.2487	−1.88373	−0.941864	−	0.335994i	$-0.890928\pi$
$577$	−45.2487	−1.88373	−0.941864	+	0.335994i	$0.890928\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10.2487i	0.425188i
$582$	0	0
$583$	−4.39230	−0.181911
$584$	0	0
$585$	0	0
$586$	0	0
$587$	42.3013i	1.74596i	0.487756	+	0.872980i	$0.337816\pi$
$587$	42.3013i	1.74596i	−0.487756	+	0.872980i	$0.662184\pi$
$588$	0	0
$589$	6.53590i	0.269307i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	14.4115	0.591811	0.295906	−	0.955217i	$-0.404379\pi$
$593$	14.4115	0.591811	0.295906	+	0.955217i	$0.404379\pi$
$594$	0	0
$595$	10.1962i	0.418001i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−8.24871	−0.337033	−0.168517	−	0.985699i	$-0.553898\pi$
$599$	−8.24871	−0.337033	−0.168517	+	0.985699i	$0.553898\pi$
$600$	0	0
$601$	17.2487	0.703590	0.351795	−	0.936077i	$-0.385571\pi$
$601$	17.2487	0.703590	0.351795	+	0.936077i	$0.385571\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 42.5167i	− 1.72855i
$606$	0	0
$607$	25.9090	1.05161	0.525806	−	0.850604i	$-0.323764\pi$
$607$	25.9090	1.05161	0.525806	+	0.850604i	$0.323764\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 8.53590i	− 0.345325i
$612$	0	0
$613$	24.3923i	0.985196i	0.870257	+	0.492598i	$0.163953\pi$
$613$	24.3923i	0.985196i	−0.870257	+	0.492598i	$0.836047\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26.1244	1.05173	0.525863	−	0.850569i	$-0.323742\pi$
$617$	26.1244	1.05173	0.525863	+	0.850569i	$0.323742\pi$
$618$	0	0
$619$	6.24871i	0.251157i	0.992084	+	0.125578i	$0.0400787\pi$
$619$	6.24871i	0.251157i	−0.992084	+	0.125578i	$0.959921\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.58846	0.344089
$624$	0	0
$625$	10.0718	0.402872
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 18.6603i	− 0.744033i
$630$	0	0
$631$	−0.784610	−0.0312348	−0.0156174	−	0.999878i	$-0.504971\pi$
$631$	−0.784610	−0.0312348	−0.0156174	+	0.999878i	$0.504971\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	50.9808i	2.02311i
$636$	0	0
$637$	15.9282i	0.631098i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−23.0526	−0.910521	−0.455261	−	0.890358i	$-0.650454\pi$
$641$	−23.0526	−0.910521	−0.455261	+	0.890358i	$0.650454\pi$
$642$	0	0
$643$	6.39230i	0.252088i	0.992025	+	0.126044i	$0.0402280\pi$
$643$	6.39230i	0.252088i	−0.992025	+	0.126044i	$0.959772\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	49.7654	1.95648	0.978239	−	0.207480i	$-0.0665261\pi$
$647$	49.7654	1.95648	0.978239	+	0.207480i	$0.0665261\pi$
$648$	0	0
$649$	−39.7128	−1.55886
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.5692i	1.39193i	0.718075	+	0.695966i	$0.245023\pi$
$653$	35.5692i	1.39193i	−0.718075	+	0.695966i	$0.754977\pi$
$654$	0	0
$655$	0.732051	0.0286036
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 10.0526i	− 0.391592i	−0.980645	−	0.195796i	$-0.937271\pi$
$659$	− 10.0526i	− 0.391592i	0.980645	−	0.195796i	$-0.0627291\pi$
$660$	0	0
$661$	32.1769i	1.25154i	0.780009	+	0.625768i	$0.215215\pi$
$661$	32.1769i	1.25154i	−0.780009	+	0.625768i	$0.784785\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8.92820	0.346221
$666$	0	0
$667$	− 45.3731i	− 1.75685i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	42.5885	1.64411
$672$	0	0
$673$	−37.2487	−1.43583	−0.717916	−	0.696130i	$-0.754904\pi$
$673$	−37.2487	−1.43583	−0.717916	+	0.696130i	$0.754904\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 29.3205i	− 1.12688i	−0.826157	−	0.563439i	$-0.809478\pi$
$677$	− 29.3205i	− 1.12688i	0.826157	−	0.563439i	$-0.190522\pi$
$678$	0	0
$679$	−5.85641	−0.224748
$680$	0	0
$681$	0	0
$682$	0	0
$683$	34.1051i	1.30500i	0.757790	+	0.652498i	$0.226279\pi$
$683$	34.1051i	1.30500i	−0.757790	+	0.652498i	$0.773721\pi$
$684$	0	0
$685$	64.1769i	2.45207i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.28719	−0.0871349
$690$	0	0
$691$	− 12.7321i	− 0.484350i	−0.970233	−	0.242175i	$-0.922139\pi$
$691$	− 12.7321i	− 0.484350i	0.970233	−	0.242175i	$-0.0778608\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−72.6410	−2.75543
$696$	0	0
$697$	−16.9282	−0.641201
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 10.1244i	− 0.382392i	−0.981552	−	0.191196i	$-0.938763\pi$
$701$	− 10.1244i	− 0.382392i	0.981552	−	0.191196i	$-0.0612365\pi$
$702$	0	0
$703$	−16.3397	−0.616265
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.85641i	0.220253i
$708$	0	0
$709$	− 47.3923i	− 1.77986i	−0.456102	−	0.889928i	$-0.650755\pi$
$709$	− 47.3923i	− 1.77986i	0.456102	−	0.889928i	$-0.349245\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−17.4641	−0.654036
$714$	0	0
$715$	− 43.5167i	− 1.62743i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−48.0526	−1.79206	−0.896029	−	0.443995i	$-0.853561\pi$
$719$	−48.0526	−1.79206	−0.896029	+	0.443995i	$0.853561\pi$
$720$	0	0
$721$	2.53590	0.0944418
$722$	0	0
$723$	0	0
$724$	0	0
$725$	46.3923i	1.72297i
$726$	0	0
$727$	49.5167	1.83647	0.918236	−	0.396034i	$-0.129614\pi$
$727$	49.5167	1.83647	0.918236	+	0.396034i	$0.129614\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 36.0526i	− 1.33345i
$732$	0	0
$733$	− 29.4641i	− 1.08828i	−0.838994	−	0.544141i	$-0.816856\pi$
$733$	− 29.4641i	− 1.08828i	0.838994	−	0.544141i	$-0.183144\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−22.3923	−0.824831
$738$	0	0
$739$	9.60770i	0.353425i	0.984263	+	0.176712i	$0.0565462\pi$
$739$	9.60770i	0.353425i	−0.984263	+	0.176712i	$0.943454\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−30.7846	−1.12938	−0.564689	−	0.825304i	$-0.691004\pi$
$743$	−30.7846	−1.12938	−0.564689	+	0.825304i	$0.691004\pi$
$744$	0	0
$745$	19.3923	0.710479
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 7.60770i	− 0.277979i
$750$	0	0
$751$	17.5167	0.639192	0.319596	−	0.947554i	$-0.396453\pi$
$751$	17.5167	0.639192	0.319596	+	0.947554i	$0.396453\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 51.7128i	− 1.88202i
$756$	0	0
$757$	− 18.2487i	− 0.663261i	−0.943409	−	0.331630i	$-0.892401\pi$
$757$	− 18.2487i	− 0.663261i	0.943409	−	0.331630i	$-0.107599\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−35.9808	−1.30430	−0.652151	−	0.758089i	$-0.726133\pi$
$761$	−35.9808	−1.30430	−0.652151	+	0.758089i	$0.726133\pi$
$762$	0	0
$763$	9.41154i	0.340721i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−20.6795	−0.746693
$768$	0	0
$769$	−39.3923	−1.42052	−0.710261	−	0.703938i	$-0.751423\pi$
$769$	−39.3923	−1.42052	−0.710261	+	0.703938i	$0.751423\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	40.6603i	1.46245i	0.682138	+	0.731224i	$0.261051\pi$
$773$	40.6603i	1.46245i	−0.682138	+	0.731224i	$0.738949\pi$
$774$	0	0
$775$	17.8564	0.641421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14.8231i	0.531092i
$780$	0	0
$781$	− 26.7846i	− 0.958429i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	9.19615	0.328225
$786$	0	0
$787$	14.4449i	0.514904i	0.966291	+	0.257452i	$0.0828829\pi$
$787$	14.4449i	0.514904i	−0.966291	+	0.257452i	$0.917117\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.66025	0.201255
$792$	0	0
$793$	22.1769	0.787525
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 12.2679i	− 0.434553i	−0.976110	−	0.217277i	$-0.930283\pi$
$797$	− 12.2679i	− 0.434553i	0.976110	−	0.217277i	$-0.0697173\pi$
$798$	0	0
$799$	12.9282	0.457367
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 42.5885i	− 1.50291i
$804$	0	0
$805$	23.8564i	0.840828i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	49.0526	1.72460	0.862298	−	0.506401i	$-0.169024\pi$
$809$	49.0526	1.72460	0.862298	+	0.506401i	$0.169024\pi$
$810$	0	0
$811$	− 48.7846i	− 1.71306i	−0.516098	−	0.856530i	$-0.672616\pi$
$811$	− 48.7846i	− 1.71306i	0.516098	−	0.856530i	$-0.327384\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	24.3923	0.854425
$816$	0	0
$817$	−31.5692	−1.10447
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1.58846i	− 0.0554375i	−0.999616	−	0.0277188i	$-0.991176\pi$
$821$	− 1.58846i	− 0.0554375i	0.999616	−	0.0277188i	$-0.00882429\pi$
$822$	0	0
$823$	17.6077	0.613766	0.306883	−	0.951747i	$-0.400714\pi$
$823$	17.6077	0.613766	0.306883	+	0.951747i	$0.400714\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.3397i	0.429095i	0.976714	+	0.214548i	$0.0688277\pi$
$827$	12.3397i	0.429095i	−0.976714	+	0.214548i	$0.931172\pi$
$828$	0	0
$829$	− 53.1769i	− 1.84691i	−0.383706	−	0.923455i	$-0.625352\pi$
$829$	− 53.1769i	− 1.84691i	0.383706	−	0.923455i	$-0.374648\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−24.1244	−0.835859
$834$	0	0
$835$	− 54.9808i	− 1.90269i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	41.8038	1.44323	0.721615	−	0.692295i	$-0.243400\pi$
$839$	41.8038	1.44323	0.721615	+	0.692295i	$0.243400\pi$
$840$	0	0
$841$	2.00000	0.0689655
$842$	0	0
$843$	0	0
$844$	0	0
$845$	25.8564i	0.889487i
$846$	0	0
$847$	−8.33975	−0.286557
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 43.6603i	− 1.49665i
$852$	0	0
$853$	5.07180i	0.173655i	0.996223	+	0.0868275i	$0.0276729\pi$
$853$	5.07180i	0.173655i	−0.996223	+	0.0868275i	$0.972327\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−0.124356	−0.00424791	−0.00212395	−	0.999998i	$-0.500676\pi$
$857$	−0.124356	−0.00424791	−0.00212395	+	0.999998i	$0.500676\pi$
$858$	0	0
$859$	2.00000i	0.0682391i	0.999418	+	0.0341196i	$0.0108627\pi$
$859$	2.00000i	0.0682391i	−0.999418	+	0.0341196i	$0.989137\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−25.5167	−0.868597	−0.434299	−	0.900769i	$-0.643004\pi$
$863$	−25.5167	−0.868597	−0.434299	+	0.900769i	$0.643004\pi$
$864$	0	0
$865$	−6.46410	−0.219786
$866$	0	0
$867$	0	0
$868$	0	0
$869$	24.9282i	0.845631i
$870$	0	0
$871$	−11.6603	−0.395093
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 10.7321i	− 0.362810i
$876$	0	0
$877$	− 57.1051i	− 1.92830i	−0.265353	−	0.964151i	$-0.585488\pi$
$877$	− 57.1051i	− 1.92830i	0.265353	−	0.964151i	$-0.414512\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−27.8564	−0.938506	−0.469253	−	0.883064i	$-0.655477\pi$
$881$	−27.8564	−0.938506	−0.469253	+	0.883064i	$0.655477\pi$
$882$	0	0
$883$	1.32051i	0.0444386i	0.999753	+	0.0222193i	$0.00707321\pi$
$883$	1.32051i	0.0444386i	−0.999753	+	0.0222193i	$0.992927\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.19615	0.0737396	0.0368698	−	0.999320i	$-0.488261\pi$
$887$	2.19615	0.0737396	0.0368698	+	0.999320i	$0.488261\pi$
$888$	0	0
$889$	10.0000	0.335389
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 11.3205i	− 0.378826i
$894$	0	0
$895$	−52.2487	−1.74648
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.3923i	0.346603i
$900$	0	0
$901$	− 3.46410i	− 0.115406i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	24.3923	0.810828
$906$	0	0
$907$	32.3923i	1.07557i	0.843082	+	0.537784i	$0.180739\pi$
$907$	32.3923i	1.07557i	−0.843082	+	0.537784i	$0.819261\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	16.7321	0.554358	0.277179	−	0.960818i	$-0.410601\pi$
$911$	16.7321	0.554358	0.277179	+	0.960818i	$0.410601\pi$
$912$	0	0
$913$	66.2487	2.19251
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 0.143594i	− 0.00474188i
$918$	0	0
$919$	−16.9808	−0.560144	−0.280072	−	0.959979i	$-0.590358\pi$
$919$	−16.9808	−0.560144	−0.280072	+	0.959979i	$0.590358\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 13.9474i	− 0.459086i
$924$	0	0
$925$	44.6410i	1.46779i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0.516660	0.0169511	0.00847554	−	0.999964i	$-0.497302\pi$
$929$	0.516660	0.0169511	0.00847554	+	0.999964i	$0.497302\pi$
$930$	0	0
$931$	21.1244i	0.692323i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	65.9090	2.15545
$936$	0	0
$937$	−7.24871	−0.236805	−0.118403	−	0.992966i	$-0.537777\pi$
$937$	−7.24871	−0.236805	−0.118403	+	0.992966i	$0.537777\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 18.3731i	− 0.598945i	−0.954105	−	0.299472i	$-0.903189\pi$
$941$	− 18.3731i	− 0.598945i	0.954105	−	0.299472i	$-0.0968107\pi$
$942$	0	0
$943$	−39.6077	−1.28980
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20.1962i	0.656287i	0.944628	+	0.328143i	$0.106423\pi$
$947$	20.1962i	0.656287i	−0.944628	+	0.328143i	$0.893577\pi$
$948$	0	0
$949$	− 22.1769i	− 0.719893i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−26.6603	−0.863610	−0.431805	−	0.901967i	$-0.642123\pi$
$953$	−26.6603	−0.863610	−0.431805	+	0.901967i	$0.642123\pi$
$954$	0	0
$955$	61.9090i	2.00333i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	12.5885	0.406502
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	40.5167i	1.30428i
$966$	0	0
$967$	−10.1962	−0.327886	−0.163943	−	0.986470i	$-0.552421\pi$
$967$	−10.1962	−0.327886	−0.163943	+	0.986470i	$0.552421\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	51.1244i	1.64066i	0.571891	+	0.820329i	$0.306210\pi$
$971$	51.1244i	1.64066i	−0.571891	+	0.820329i	$0.693790\pi$
$972$	0	0
$973$	14.2487i	0.456793i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22.7846	0.728944	0.364472	−	0.931214i	$-0.381249\pi$
$977$	22.7846	0.728944	0.364472	+	0.931214i	$0.381249\pi$
$978$	0	0
$979$	− 55.5167i	− 1.77432i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−1.46410	−0.0466976	−0.0233488	−	0.999727i	$-0.507433\pi$
$983$	−1.46410	−0.0466976	−0.0233488	+	0.999727i	$0.507433\pi$
$984$	0	0
$985$	−15.3923	−0.490440
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 84.3538i	− 2.68230i
$990$	0	0
$991$	−2.58846	−0.0822251	−0.0411125	−	0.999155i	$-0.513090\pi$
$991$	−2.58846	−0.0822251	−0.0411125	+	0.999155i	$0.513090\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 8.92820i	− 0.283043i
$996$	0	0
$997$	27.9282i	0.884495i	0.896893	+	0.442248i	$0.145819\pi$
$997$	27.9282i	0.884495i	−0.896893	+	0.442248i	$0.854181\pi$
$998$	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	2592.2.d.h.1297.4		4
3.2	odd	2		2592.2.d.g.1297.1		4
4.3	odd	2		648.2.d.e.325.1	✓	4
8.3	odd	2		648.2.d.e.325.2	yes	4
8.5	even	2	inner	2592.2.d.h.1297.1		4
9.2	odd	6		2592.2.r.k.433.2		4
9.4	even	3		2592.2.r.j.2161.2		4
9.5	odd	6		2592.2.r.a.2161.1		4
9.7	even	3		2592.2.r.b.433.1		4
12.11	even	2		648.2.d.i.325.4	yes	4
24.5	odd	2		2592.2.d.g.1297.4		4
24.11	even	2		648.2.d.i.325.3	yes	4
36.7	odd	6		648.2.n.b.109.2		4
36.11	even	6		648.2.n.l.109.1		4
36.23	even	6		648.2.n.a.541.2		4
36.31	odd	6		648.2.n.m.541.1		4
72.5	odd	6		2592.2.r.k.2161.2		4
72.11	even	6		648.2.n.a.109.1		4
72.13	even	6		2592.2.r.b.2161.1		4
72.29	odd	6		2592.2.r.a.433.1		4
72.43	odd	6		648.2.n.m.109.2		4
72.59	even	6		648.2.n.l.541.1		4
72.61	even	6		2592.2.r.j.433.2		4
72.67	odd	6		648.2.n.b.541.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.d.e.325.1	✓	4	4.3	odd	2
648.2.d.e.325.2	yes	4	8.3	odd	2
648.2.d.i.325.3	yes	4	24.11	even	2
648.2.d.i.325.4	yes	4	12.11	even	2
648.2.n.a.109.1		4	72.11	even	6
648.2.n.a.541.2		4	36.23	even	6
648.2.n.b.109.2		4	36.7	odd	6
648.2.n.b.541.2		4	72.67	odd	6
648.2.n.l.109.1		4	36.11	even	6
648.2.n.l.541.1		4	72.59	even	6
648.2.n.m.109.2		4	72.43	odd	6
648.2.n.m.541.1		4	36.31	odd	6
2592.2.d.g.1297.1		4	3.2	odd	2
2592.2.d.g.1297.4		4	24.5	odd	2
2592.2.d.h.1297.1		4	8.5	even	2	inner
2592.2.d.h.1297.4		4	1.1	even	1	trivial
2592.2.r.a.433.1		4	72.29	odd	6
2592.2.r.a.2161.1		4	9.5	odd	6
2592.2.r.b.433.1		4	9.7	even	3
2592.2.r.b.2161.1		4	72.13	even	6
2592.2.r.j.433.2		4	72.61	even	6
2592.2.r.j.2161.2		4	9.4	even	3
2592.2.r.k.433.2		4	9.2	odd	6
2592.2.r.k.2161.2		4	72.5	odd	6

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

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Fields

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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	2592.2.d.h.1297.4

Level	$2592$
Weight	$2$
Character	2592.1297
Analytic conductor	$20.697$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$