LMFDB - Embedded newform 2592.2.s.d.863.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2592,2,Mod(863,2592)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2592, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2592.863"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.s (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,-16,0,0,0,0,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$20.6972242039$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			863.1
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2592.863
Dual form			2592.2.s.d.1727.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 0.707107i) q^{5} +(-3.46410 + 2.00000i) q^{7} +(-2.82843 - 4.89898i) q^{11} +(-2.00000 + 3.46410i) q^{13} +4.24264i q^{17} +(2.82843 - 4.89898i) q^{23} +(-1.50000 - 2.59808i) q^{25} +(-1.22474 + 0.707107i) q^{29} +(3.46410 + 2.00000i) q^{31} +5.65685 q^{35} -6.00000 q^{37} +(8.57321 + 4.94975i) q^{41} +(6.92820 - 4.00000i) q^{43} +(2.82843 + 4.89898i) q^{47} +(4.50000 - 7.79423i) q^{49} +4.24264i q^{53} +8.00000i q^{55} +(5.65685 - 9.79796i) q^{59} +(1.00000 + 1.73205i) q^{61} +(4.89898 - 2.82843i) q^{65} +(-6.92820 - 4.00000i) q^{67} +5.65685 q^{71} +(19.5959 + 11.3137i) q^{77} +(3.46410 - 2.00000i) q^{79} +(2.82843 + 4.89898i) q^{83} +(3.00000 - 5.19615i) q^{85} -4.24264i q^{89} -16.0000i q^{91} +(4.00000 + 6.92820i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{13} - 12 q^{25} - 48 q^{37} + 36 q^{49} + 8 q^{61} + 24 q^{85} + 32 q^{97}+O(q^{100}) $ 8 * q - 16 * q^13 - 12 * q^25 - 48 * q^37 + 36 * q^49 + 8 * q^61 + 24 * q^85 + 32 * q^97

Character values

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

$n$	$325$	$1217$	$2431$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$-1$

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
$5$	−1.22474	−	0.707107i	−0.547723	−	0.316228i	0.200480	−	0.979698i	$-0.435750\pi$
$5$	−1.22474	−	0.707107i	−0.547723	−	0.316228i	−0.748203	+	0.663470i	$0.769083\pi$
$6$		0			0
$7$	−3.46410	+	2.00000i	−1.30931	+	0.755929i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	−3.46410	+	2.00000i	−1.30931	+	0.755929i	−0.327327	+	0.944911i	$0.606148\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.82843	−	4.89898i	−0.852803	−	1.47710i	−0.878668	−	0.477432i	$-0.841568\pi$
$11$	−2.82843	−	4.89898i	−0.852803	−	1.47710i	0.0258656	−	0.999665i	$-0.491766\pi$
$12$		0			0
$13$	−2.00000	+	3.46410i	−0.554700	+	0.960769i	0.443227	+	0.896410i	$0.353834\pi$
$13$	−2.00000	+	3.46410i	−0.554700	+	0.960769i	−0.997927	+	0.0643593i	$0.979500\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			4.24264i			1.02899i	0.857493	+	0.514496i	$0.172021\pi$
$17$			4.24264i			1.02899i	−0.857493	+	0.514496i	$0.827979\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.82843	−	4.89898i	0.589768	−	1.02151i	−0.404495	−	0.914540i	$-0.632553\pi$
$23$	2.82843	−	4.89898i	0.589768	−	1.02151i	0.994263	−	0.106967i	$-0.0341141\pi$
$24$		0			0
$25$	−1.50000	−	2.59808i	−0.300000	−	0.519615i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.22474	+	0.707107i	−0.227429	+	0.131306i	−0.609386	−	0.792874i	$-0.708584\pi$
$29$	−1.22474	+	0.707107i	−0.227429	+	0.131306i	0.381956	+	0.924180i	$0.375251\pi$
$30$		0			0
$31$	3.46410	+	2.00000i	0.622171	+	0.359211i	0.777714	−	0.628619i	$-0.216379\pi$
$31$	3.46410	+	2.00000i	0.622171	+	0.359211i	−0.155543	+	0.987829i	$0.549713\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	5.65685			0.956183
$36$		0			0
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.57321	+	4.94975i	1.33891	+	0.773021i	0.986646	−	0.162880i	$-0.0520782\pi$
$41$	8.57321	+	4.94975i	1.33891	+	0.773021i	0.352265	+	0.935900i	$0.385412\pi$
$42$		0			0
$43$	6.92820	−	4.00000i	1.05654	−	0.609994i	0.132068	−	0.991241i	$-0.457838\pi$
$43$	6.92820	−	4.00000i	1.05654	−	0.609994i	0.924473	+	0.381246i	$0.124505\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.82843	+	4.89898i	0.412568	+	0.714590i	0.995170	−	0.0981685i	$-0.0312984\pi$
$47$	2.82843	+	4.89898i	0.412568	+	0.714590i	−0.582601	+	0.812758i	$0.697965\pi$
$48$		0			0
$49$	4.50000	−	7.79423i	0.642857	−	1.11346i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			4.24264i			0.582772i	0.956606	+	0.291386i	$0.0941163\pi$
$53$			4.24264i			0.582772i	−0.956606	+	0.291386i	$0.905884\pi$
$54$		0			0
$55$			8.00000i			1.07872i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.65685	−	9.79796i	0.736460	−	1.27559i	−0.217620	−	0.976034i	$-0.569829\pi$
$59$	5.65685	−	9.79796i	0.736460	−	1.27559i	0.954080	−	0.299552i	$-0.0968372\pi$
$60$		0			0
$61$	1.00000	+	1.73205i	0.128037	+	0.221766i	0.922916	−	0.385002i	$-0.125799\pi$
$61$	1.00000	+	1.73205i	0.128037	+	0.221766i	−0.794879	+	0.606768i	$0.792466\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.89898	−	2.82843i	0.607644	−	0.350823i
$66$		0			0
$67$	−6.92820	−	4.00000i	−0.846415	−	0.488678i	0.0130248	−	0.999915i	$-0.495854\pi$
$67$	−6.92820	−	4.00000i	−0.846415	−	0.488678i	−0.859440	+	0.511237i	$0.829187\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	5.65685			0.671345			0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685			0.671345			0.335673	+	0.941979i	$0.391036\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	19.5959	+	11.3137i	2.23316	+	1.28932i
$78$		0			0
$79$	3.46410	−	2.00000i	0.389742	−	0.225018i	−0.292306	−	0.956325i	$-0.594423\pi$
$79$	3.46410	−	2.00000i	0.389742	−	0.225018i	0.682048	+	0.731307i	$0.261089\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.82843	+	4.89898i	0.310460	+	0.537733i	0.978462	−	0.206427i	$-0.0661835\pi$
$83$	2.82843	+	4.89898i	0.310460	+	0.537733i	−0.668002	+	0.744160i	$0.732850\pi$
$84$		0			0
$85$	3.00000	−	5.19615i	0.325396	−	0.563602i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	4.24264i		−	0.449719i	−0.974391	−	0.224860i	$-0.927808\pi$
$89$		−	4.24264i		−	0.449719i	0.974391	−	0.224860i	$-0.0721923\pi$
$90$		0			0
$91$		−	16.0000i		−	1.67726i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	4.00000	+	6.92820i	0.406138	+	0.703452i	0.994453	−	0.105180i	$-0.0335417\pi$
$97$	4.00000	+	6.92820i	0.406138	+	0.703452i	−0.588315	+	0.808632i	$0.700208\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	15.9217	−	9.19239i	1.58427	−	0.914677i	0.590040	−	0.807374i	$-0.299112\pi$
$101$	15.9217	−	9.19239i	1.58427	−	0.914677i	0.994226	−	0.107303i	$-0.0342215\pi$
$102$		0			0
$103$	−3.46410	−	2.00000i	−0.341328	−	0.197066i	0.319531	−	0.947576i	$-0.396475\pi$
$103$	−3.46410	−	2.00000i	−0.341328	−	0.197066i	−0.660859	+	0.750510i	$0.729808\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	12.0000			1.14939			0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000			1.14939			0.574696	+	0.818367i	$0.305120\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.22474	−	0.707107i	−0.115214	−	0.0665190i	0.441285	−	0.897367i	$-0.354523\pi$
$113$	−1.22474	−	0.707107i	−0.115214	−	0.0665190i	−0.556500	+	0.830848i	$0.687856\pi$
$114$		0			0
$115$	−6.92820	+	4.00000i	−0.646058	+	0.373002i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−8.48528	−	14.6969i	−0.777844	−	1.34727i
$120$		0			0
$121$	−10.5000	+	18.1865i	−0.954545	+	1.65332i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.3137i			1.01193i
$126$		0			0
$127$			12.0000i			1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$			12.0000i			1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.65685	−	9.79796i	0.494242	−	0.856052i	−0.505736	−	0.862688i	$-0.668779\pi$
$131$	5.65685	−	9.79796i	0.494242	−	0.856052i	0.999978	+	0.00663646i	$0.00211246\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.22474	+	0.707107i	−0.104637	+	0.0604122i	−0.551405	−	0.834238i	$-0.685908\pi$
$137$	−1.22474	+	0.707107i	−0.104637	+	0.0604122i	0.446768	+	0.894650i	$0.352575\pi$
$138$		0			0
$139$	6.92820	+	4.00000i	0.587643	+	0.339276i	0.764165	−	0.645021i	$-0.223151\pi$
$139$	6.92820	+	4.00000i	0.587643	+	0.339276i	−0.176522	+	0.984297i	$0.556485\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	22.6274			1.89220
$144$		0			0
$145$	2.00000			0.166091
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.22474	+	0.707107i	0.100335	+	0.0579284i	0.549328	−	0.835607i	$-0.314884\pi$
$149$	1.22474	+	0.707107i	0.100335	+	0.0579284i	−0.448993	+	0.893535i	$0.648217\pi$
$150$		0			0
$151$	−17.3205	+	10.0000i	−1.40952	+	0.813788i	−0.995342	−	0.0964061i	$-0.969265\pi$
$151$	−17.3205	+	10.0000i	−1.40952	+	0.813788i	−0.414181	+	0.910195i	$0.635932\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.82843	−	4.89898i	−0.227185	−	0.393496i
$156$		0			0
$157$	−1.00000	+	1.73205i	−0.0798087	+	0.138233i	−0.903167	−	0.429289i	$-0.858764\pi$
$157$	−1.00000	+	1.73205i	−0.0798087	+	0.138233i	0.823359	+	0.567521i	$0.192098\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			22.6274i			1.78329i
$162$		0			0
$163$		−	24.0000i		−	1.87983i	−0.341415	−	0.939913i	$-0.610906\pi$
$163$		−	24.0000i		−	1.87983i	0.341415	−	0.939913i	$-0.389094\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.65685	−	9.79796i	0.437741	−	0.758189i	−0.559774	−	0.828645i	$-0.689112\pi$
$167$	5.65685	−	9.79796i	0.437741	−	0.758189i	0.997515	+	0.0704563i	$0.0224455\pi$
$168$		0			0
$169$	−1.50000	−	2.59808i	−0.115385	−	0.199852i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−13.4722	+	7.77817i	−1.02427	+	0.591364i	−0.915338	−	0.402685i	$-0.868077\pi$
$173$	−13.4722	+	7.77817i	−1.02427	+	0.591364i	−0.108933	+	0.994049i	$0.534744\pi$
$174$		0			0
$175$	10.3923	+	6.00000i	0.785584	+	0.453557i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	12.0000			0.891953			0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000			0.891953			0.445976	+	0.895045i	$0.352856\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	7.34847	+	4.24264i	0.540270	+	0.311925i
$186$		0			0
$187$	20.7846	−	12.0000i	1.51992	−	0.877527i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.65685	−	9.79796i	−0.409316	−	0.708955i	0.585498	−	0.810674i	$-0.300899\pi$
$191$	−5.65685	−	9.79796i	−0.409316	−	0.708955i	−0.994813	+	0.101719i	$0.967566\pi$
$192$		0			0
$193$	1.00000	−	1.73205i	0.0719816	−	0.124676i	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	1.00000	−	1.73205i	0.0719816	−	0.124676i	0.899770	+	0.436365i	$0.143734\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			7.07107i			0.503793i	0.967754	+	0.251896i	$0.0810542\pi$
$197$			7.07107i			0.503793i	−0.967754	+	0.251896i	$0.918946\pi$
$198$		0			0
$199$			20.0000i			1.41776i	0.705328	+	0.708881i	$0.250800\pi$
$199$			20.0000i			1.41776i	−0.705328	+	0.708881i	$0.749200\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.82843	−	4.89898i	0.198517	−	0.343841i
$204$		0			0
$205$	−7.00000	−	12.1244i	−0.488901	−	0.846802i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−6.92820	−	4.00000i	−0.476957	−	0.275371i	0.242190	−	0.970229i	$-0.422134\pi$
$211$	−6.92820	−	4.00000i	−0.476957	−	0.275371i	−0.719148	+	0.694857i	$0.755467\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−11.3137			−0.771589
$216$		0			0
$217$	−16.0000			−1.08615
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−14.6969	−	8.48528i	−0.988623	−	0.570782i
$222$		0			0
$223$	−3.46410	+	2.00000i	−0.231973	+	0.133930i	−0.611482	−	0.791258i	$-0.709426\pi$
$223$	−3.46410	+	2.00000i	−0.231973	+	0.133930i	0.379509	+	0.925188i	$0.376093\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.82843	−	4.89898i	−0.187729	−	0.325157i	0.756764	−	0.653689i	$-0.226779\pi$
$227$	−2.82843	−	4.89898i	−0.187729	−	0.325157i	−0.944493	+	0.328532i	$0.893446\pi$
$228$		0			0
$229$	14.0000	−	24.2487i	0.925146	−	1.60240i	0.133820	−	0.991006i	$-0.457276\pi$
$229$	14.0000	−	24.2487i	0.925146	−	1.60240i	0.791326	−	0.611394i	$-0.209391\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	7.07107i		−	0.463241i	−0.972806	−	0.231621i	$-0.925597\pi$
$233$		−	7.07107i		−	0.463241i	0.972806	−	0.231621i	$-0.0744028\pi$
$234$		0			0
$235$		−	8.00000i		−	0.521862i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	11.3137	−	19.5959i	0.731823	−	1.26755i	−0.224280	−	0.974525i	$-0.572003\pi$
$239$	11.3137	−	19.5959i	0.731823	−	1.26755i	0.956103	−	0.293030i	$-0.0946635\pi$
$240$		0			0
$241$	−4.00000	−	6.92820i	−0.257663	−	0.446285i	0.707953	−	0.706260i	$-0.249619\pi$
$241$	−4.00000	−	6.92820i	−0.257663	−	0.446285i	−0.965615	+	0.259975i	$0.916286\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−11.0227	+	6.36396i	−0.704215	+	0.406579i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	16.9706			1.07117			0.535586	−	0.844481i	$-0.320091\pi$
$251$	16.9706			1.07117			0.535586	+	0.844481i	$0.320091\pi$
$252$		0			0
$253$	−32.0000			−2.01182
$254$		0			0
$255$		0			0
$256$		0			0
$257$	20.8207	+	12.0208i	1.29876	+	0.749838i	0.980189	−	0.198062i	$-0.0634648\pi$
$257$	20.8207	+	12.0208i	1.29876	+	0.749838i	0.318568	+	0.947900i	$0.396798\pi$
$258$		0			0
$259$	20.7846	−	12.0000i	1.29149	−	0.745644i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	5.65685	+	9.79796i	0.348817	+	0.604168i	0.986040	−	0.166511i	$-0.0532503\pi$
$263$	5.65685	+	9.79796i	0.348817	+	0.604168i	−0.637223	+	0.770680i	$0.719917\pi$
$264$		0			0
$265$	3.00000	−	5.19615i	0.184289	−	0.319197i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	21.2132i		−	1.29339i	−0.762748	−	0.646696i	$-0.776150\pi$
$269$		−	21.2132i		−	1.29339i	0.762748	−	0.646696i	$-0.223850\pi$
$270$		0			0
$271$			12.0000i			0.728948i	0.931214	+	0.364474i	$0.118751\pi$
$271$			12.0000i			0.728948i	−0.931214	+	0.364474i	$0.881249\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.48528	+	14.6969i	−0.511682	+	0.886259i
$276$		0			0
$277$	2.00000	+	3.46410i	0.120168	+	0.208138i	0.919834	−	0.392308i	$-0.128323\pi$
$277$	2.00000	+	3.46410i	0.120168	+	0.208138i	−0.799666	+	0.600446i	$0.794990\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.12372	−	3.53553i	0.365311	−	0.210912i	−0.306097	−	0.952000i	$-0.599023\pi$
$281$	6.12372	−	3.53553i	0.365311	−	0.210912i	0.671408	+	0.741088i	$0.265690\pi$
$282$		0			0
$283$	−13.8564	−	8.00000i	−0.823678	−	0.475551i	0.0280052	−	0.999608i	$-0.491084\pi$
$283$	−13.8564	−	8.00000i	−0.823678	−	0.475551i	−0.851683	+	0.524057i	$0.824418\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−39.5980			−2.33739
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6.12372	+	3.53553i	0.357752	+	0.206548i	0.668094	−	0.744077i	$-0.267110\pi$
$293$	6.12372	+	3.53553i	0.357752	+	0.206548i	−0.310342	+	0.950625i	$0.600444\pi$
$294$		0			0
$295$	−13.8564	+	8.00000i	−0.806751	+	0.465778i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	11.3137	+	19.5959i	0.654289	+	1.13326i
$300$		0			0
$301$	−16.0000	+	27.7128i	−0.922225	+	1.59734i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	2.82843i		−	0.161955i
$306$		0			0
$307$			24.0000i			1.36975i	0.728659	+	0.684876i	$0.240144\pi$
$307$			24.0000i			1.36975i	−0.728659	+	0.684876i	$0.759856\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5.65685	+	9.79796i	−0.320771	+	0.555591i	−0.980647	−	0.195783i	$-0.937275\pi$
$311$	−5.65685	+	9.79796i	−0.320771	+	0.555591i	0.659877	+	0.751374i	$0.270609\pi$
$312$		0			0
$313$	11.0000	+	19.0526i	0.621757	+	1.07691i	0.989158	+	0.146852i	$0.0469141\pi$
$313$	11.0000	+	19.0526i	0.621757	+	1.07691i	−0.367402	+	0.930062i	$0.619753\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	20.8207	−	12.0208i	1.16940	−	0.675156i	0.215865	−	0.976423i	$-0.430743\pi$
$317$	20.8207	−	12.0208i	1.16940	−	0.675156i	0.953540	+	0.301267i	$0.0974095\pi$
$318$		0			0
$319$	6.92820	+	4.00000i	0.387905	+	0.223957i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	12.0000			0.665640
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−19.5959	−	11.3137i	−1.08036	−	0.623745i
$330$		0			0
$331$	−6.92820	+	4.00000i	−0.380808	+	0.219860i	−0.678170	−	0.734905i	$-0.737227\pi$
$331$	−6.92820	+	4.00000i	−0.380808	+	0.219860i	0.297361	+	0.954765i	$0.403893\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.65685	+	9.79796i	0.309067	+	0.535320i
$336$		0			0
$337$	8.00000	−	13.8564i	0.435788	−	0.754807i	−0.561572	−	0.827428i	$-0.689803\pi$
$337$	8.00000	−	13.8564i	0.435788	−	0.754807i	0.997360	+	0.0726214i	$0.0231365\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	22.6274i		−	1.22534i
$342$		0			0
$343$			8.00000i			0.431959i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−14.1421	+	24.4949i	−0.759190	+	1.31495i	0.184075	+	0.982912i	$0.441071\pi$
$347$	−14.1421	+	24.4949i	−0.759190	+	1.31495i	−0.943264	+	0.332043i	$0.892262\pi$
$348$		0			0
$349$	17.0000	+	29.4449i	0.909989	+	1.57615i	0.814076	+	0.580758i	$0.197244\pi$
$349$	17.0000	+	29.4449i	0.909989	+	1.57615i	0.0959126	+	0.995390i	$0.469423\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−8.57321	+	4.94975i	−0.456306	+	0.263448i	−0.710490	−	0.703707i	$-0.751527\pi$
$353$	−8.57321	+	4.94975i	−0.456306	+	0.263448i	0.254184	+	0.967156i	$0.418193\pi$
$354$		0			0
$355$	−6.92820	−	4.00000i	−0.367711	−	0.212298i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	16.9706			0.895672			0.447836	−	0.894116i	$-0.352195\pi$
$359$	16.9706			0.895672			0.447836	+	0.894116i	$0.352195\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−17.3205	+	10.0000i	−0.904123	+	0.521996i	−0.878536	−	0.477677i	$-0.841479\pi$
$367$	−17.3205	+	10.0000i	−0.904123	+	0.521996i	−0.0255875	+	0.999673i	$0.508146\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−8.48528	−	14.6969i	−0.440534	−	0.763027i
$372$		0			0
$373$	−11.0000	+	19.0526i	−0.569558	+	0.986504i	0.427051	+	0.904227i	$0.359552\pi$
$373$	−11.0000	+	19.0526i	−0.569558	+	0.986504i	−0.996610	+	0.0822766i	$0.973781\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	5.65685i		−	0.291343i
$378$		0			0
$379$		−	16.0000i		−	0.821865i	−0.911666	−	0.410932i	$-0.865203\pi$
$379$		−	16.0000i		−	0.821865i	0.911666	−	0.410932i	$-0.134797\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−11.3137	+	19.5959i	−0.578103	+	1.00130i	0.417593	+	0.908634i	$0.362874\pi$
$383$	−11.3137	+	19.5959i	−0.578103	+	1.00130i	−0.995697	+	0.0926706i	$0.970460\pi$
$384$		0			0
$385$	−16.0000	−	27.7128i	−0.815436	−	1.41238i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8.57321	−	4.94975i	0.434679	−	0.250962i	−0.266659	−	0.963791i	$-0.585920\pi$
$389$	8.57321	−	4.94975i	0.434679	−	0.250962i	0.701338	+	0.712829i	$0.252586\pi$
$390$		0			0
$391$	20.7846	+	12.0000i	1.05112	+	0.606866i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−5.65685			−0.284627
$396$		0			0
$397$	−18.0000			−0.903394			−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000			−0.903394			−0.451697	+	0.892171i	$0.649181\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−13.4722	−	7.77817i	−0.672769	−	0.388424i	0.124356	−	0.992238i	$-0.460314\pi$
$401$	−13.4722	−	7.77817i	−0.672769	−	0.388424i	−0.797125	+	0.603814i	$0.793647\pi$
$402$		0			0
$403$	−13.8564	+	8.00000i	−0.690237	+	0.398508i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	16.9706	+	29.3939i	0.841200	+	1.45700i
$408$		0			0
$409$	−4.00000	+	6.92820i	−0.197787	+	0.342578i	−0.947811	−	0.318834i	$-0.896709\pi$
$409$	−4.00000	+	6.92820i	−0.197787	+	0.342578i	0.750023	+	0.661411i	$0.230042\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			45.2548i			2.22684i
$414$		0			0
$415$		−	8.00000i		−	0.392705i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2.82843	−	4.89898i	0.138178	−	0.239331i	−0.788629	−	0.614869i	$-0.789209\pi$
$419$	2.82843	−	4.89898i	0.138178	−	0.239331i	0.926807	+	0.375538i	$0.122542\pi$
$420$		0			0
$421$	−14.0000	−	24.2487i	−0.682318	−	1.18181i	−0.974272	−	0.225377i	$-0.927639\pi$
$421$	−14.0000	−	24.2487i	−0.682318	−	1.18181i	0.291953	−	0.956433i	$-0.405695\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	11.0227	−	6.36396i	0.534680	−	0.308697i
$426$		0			0
$427$	−6.92820	−	4.00000i	−0.335279	−	0.193574i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	16.9706			0.817443			0.408722	−	0.912659i	$-0.365975\pi$
$431$	16.9706			0.817443			0.408722	+	0.912659i	$0.365975\pi$
$432$		0			0
$433$	30.0000			1.44171			0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000			1.44171			0.720854	+	0.693087i	$0.243750\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	24.2487	−	14.0000i	1.15733	−	0.668184i	0.206666	−	0.978412i	$-0.433739\pi$
$439$	24.2487	−	14.0000i	1.15733	−	0.668184i	0.950662	+	0.310228i	$0.100405\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.82843	+	4.89898i	0.134383	+	0.232758i	0.925361	−	0.379086i	$-0.123762\pi$
$443$	2.82843	+	4.89898i	0.134383	+	0.232758i	−0.790979	+	0.611844i	$0.790428\pi$
$444$		0			0
$445$	−3.00000	+	5.19615i	−0.142214	+	0.246321i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	38.1838i		−	1.80200i	−0.433816	−	0.901002i	$-0.642833\pi$
$449$		−	38.1838i		−	1.80200i	0.433816	−	0.901002i	$-0.357167\pi$
$450$		0			0
$451$		−	56.0000i		−	2.63694i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−11.3137	+	19.5959i	−0.530395	+	0.918671i
$456$		0			0
$457$	4.00000	+	6.92820i	0.187112	+	0.324088i	0.944286	−	0.329125i	$-0.106754\pi$
$457$	4.00000	+	6.92820i	0.187112	+	0.324088i	−0.757174	+	0.653213i	$0.773421\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−15.9217	+	9.19239i	−0.741547	+	0.428132i	−0.822631	−	0.568575i	$-0.807495\pi$
$461$	−15.9217	+	9.19239i	−0.741547	+	0.428132i	0.0810847	+	0.996707i	$0.474162\pi$
$462$		0			0
$463$	−17.3205	−	10.0000i	−0.804952	−	0.464739i	0.0402476	−	0.999190i	$-0.487185\pi$
$463$	−17.3205	−	10.0000i	−0.804952	−	0.464739i	−0.845200	+	0.534450i	$0.820519\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−16.9706			−0.785304			−0.392652	−	0.919687i	$-0.628442\pi$
$467$	−16.9706			−0.785304			−0.392652	+	0.919687i	$0.628442\pi$
$468$		0			0
$469$	32.0000			1.47762
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−39.1918	−	22.6274i	−1.80204	−	1.04041i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−19.7990	−	34.2929i	−0.904639	−	1.56688i	−0.821401	−	0.570351i	$-0.806807\pi$
$479$	−19.7990	−	34.2929i	−0.904639	−	1.56688i	−0.0832378	−	0.996530i	$-0.526526\pi$
$480$		0			0
$481$	12.0000	−	20.7846i	0.547153	−	0.947697i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	11.3137i		−	0.513729i
$486$		0			0
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−11.3137	+	19.5959i	−0.510581	+	0.884351i	0.489344	+	0.872091i	$0.337236\pi$
$491$	−11.3137	+	19.5959i	−0.510581	+	0.884351i	−0.999925	+	0.0122607i	$0.996097\pi$
$492$		0			0
$493$	−3.00000	−	5.19615i	−0.135113	−	0.234023i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−19.5959	+	11.3137i	−0.878997	+	0.507489i
$498$		0			0
$499$	27.7128	+	16.0000i	1.24060	+	0.716258i	0.969216	−	0.246214i	$-0.0791865\pi$
$499$	27.7128	+	16.0000i	1.24060	+	0.716258i	0.271380	+	0.962472i	$0.412520\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−5.65685			−0.252227			−0.126113	−	0.992016i	$-0.540250\pi$
$503$	−5.65685			−0.252227			−0.126113	+	0.992016i	$0.540250\pi$
$504$		0			0
$505$	−26.0000			−1.15698
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−28.1691	−	16.2635i	−1.24857	−	0.720865i	−0.277750	−	0.960653i	$-0.589589\pi$
$509$	−28.1691	−	16.2635i	−1.24857	−	0.720865i	−0.970825	+	0.239788i	$0.922922\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.82843	+	4.89898i	0.124635	+	0.215875i
$516$		0			0
$517$	16.0000	−	27.7128i	0.703679	−	1.21881i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			32.5269i			1.42503i	0.701657	+	0.712515i	$0.252444\pi$
$521$			32.5269i			1.42503i	−0.701657	+	0.712515i	$0.747556\pi$
$522$		0			0
$523$			16.0000i			0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$			16.0000i			0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−8.48528	+	14.6969i	−0.369625	+	0.640209i
$528$		0			0
$529$	−4.50000	−	7.79423i	−0.195652	−	0.338880i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−34.2929	+	19.7990i	−1.48539	+	0.857589i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−50.9117			−2.19292
$540$		0			0
$541$	−20.0000			−0.859867			−0.429934	−	0.902861i	$-0.641463\pi$
$541$	−20.0000			−0.859867			−0.429934	+	0.902861i	$0.641463\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−14.6969	−	8.48528i	−0.629548	−	0.363470i
$546$		0			0
$547$	34.6410	−	20.0000i	1.48114	−	0.855138i	0.481371	−	0.876517i	$-0.340139\pi$
$547$	34.6410	−	20.0000i	1.48114	−	0.855138i	0.999771	+	0.0213785i	$0.00680549\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−8.00000	+	13.8564i	−0.340195	+	0.589234i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	29.6985i		−	1.25837i	−0.777258	−	0.629183i	$-0.783390\pi$
$557$		−	29.6985i		−	1.25837i	0.777258	−	0.629183i	$-0.216610\pi$
$558$		0			0
$559$			32.0000i			1.35346i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	19.7990	−	34.2929i	0.834428	−	1.44527i	−0.0600674	−	0.998194i	$-0.519132\pi$
$563$	19.7990	−	34.2929i	0.834428	−	1.44527i	0.894495	−	0.447077i	$-0.147535\pi$
$564$		0			0
$565$	1.00000	+	1.73205i	0.0420703	+	0.0728679i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−20.8207	+	12.0208i	−0.872848	+	0.503939i	−0.868293	−	0.496051i	$-0.834783\pi$
$569$	−20.8207	+	12.0208i	−0.872848	+	0.503939i	−0.00455411	+	0.999990i	$0.501450\pi$
$570$		0			0
$571$	13.8564	+	8.00000i	0.579873	+	0.334790i	0.761083	−	0.648655i	$-0.224668\pi$
$571$	13.8564	+	8.00000i	0.579873	+	0.334790i	−0.181210	+	0.983444i	$0.558001\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−16.9706			−0.707721
$576$		0			0
$577$	18.0000			0.749350			0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000			0.749350			0.374675	+	0.927156i	$0.377754\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−19.5959	−	11.3137i	−0.812976	−	0.469372i
$582$		0			0
$583$	20.7846	−	12.0000i	0.860811	−	0.496989i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−5.65685	−	9.79796i	−0.233483	−	0.404405i	0.725347	−	0.688383i	$-0.241679\pi$
$587$	−5.65685	−	9.79796i	−0.233483	−	0.404405i	−0.958831	+	0.283978i	$0.908346\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	46.6690i		−	1.91647i	−0.285985	−	0.958234i	$-0.592321\pi$
$593$		−	46.6690i		−	1.91647i	0.285985	−	0.958234i	$-0.407679\pi$
$594$		0			0
$595$			24.0000i			0.983904i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−19.7990	+	34.2929i	−0.808965	+	1.40117i	0.104617	+	0.994513i	$0.466638\pi$
$599$	−19.7990	+	34.2929i	−0.808965	+	1.40117i	−0.913582	+	0.406656i	$0.866695\pi$
$600$		0			0
$601$	13.0000	+	22.5167i	0.530281	+	0.918474i	0.999376	+	0.0353259i	$0.0112469\pi$
$601$	13.0000	+	22.5167i	0.530281	+	0.918474i	−0.469095	+	0.883148i	$0.655420\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	25.7196	−	14.8492i	1.04565	−	0.603708i
$606$		0			0
$607$	−24.2487	−	14.0000i	−0.984225	−	0.568242i	−0.0806818	−	0.996740i	$-0.525710\pi$
$607$	−24.2487	−	14.0000i	−0.984225	−	0.568242i	−0.903543	+	0.428497i	$0.859043\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−22.6274			−0.915407
$612$		0			0
$613$	6.00000			0.242338			0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000			0.242338			0.121169	+	0.992632i	$0.461336\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	23.2702	+	13.4350i	0.936821	+	0.540874i	0.888962	−	0.457980i	$-0.151427\pi$
$617$	23.2702	+	13.4350i	0.936821	+	0.540874i	0.0478587	+	0.998854i	$0.484760\pi$
$618$		0			0
$619$	−27.7128	+	16.0000i	−1.11387	+	0.643094i	−0.939829	−	0.341644i	$-0.889016\pi$
$619$	−27.7128	+	16.0000i	−1.11387	+	0.643094i	−0.174042	+	0.984738i	$0.555683\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	8.48528	+	14.6969i	0.339956	+	0.588820i
$624$		0			0
$625$	0.500000	−	0.866025i	0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	25.4558i		−	1.01499i
$630$		0			0
$631$		−	12.0000i		−	0.477712i	−0.971055	−	0.238856i	$-0.923228\pi$
$631$		−	12.0000i		−	0.477712i	0.971055	−	0.238856i	$-0.0767725\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.48528	−	14.6969i	0.336728	−	0.583230i
$636$		0			0
$637$	18.0000	+	31.1769i	0.713186	+	1.23527i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.12372	+	3.53553i	−0.241873	+	0.139645i	−0.616037	−	0.787717i	$-0.711263\pi$
$641$	−6.12372	+	3.53553i	−0.241873	+	0.139645i	0.374165	+	0.927362i	$0.377930\pi$
$642$		0			0
$643$	−6.92820	−	4.00000i	−0.273222	−	0.157745i	0.357129	−	0.934055i	$-0.383756\pi$
$643$	−6.92820	−	4.00000i	−0.273222	−	0.157745i	−0.630351	+	0.776310i	$0.717089\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−16.9706			−0.667182			−0.333591	−	0.942718i	$-0.608260\pi$
$647$	−16.9706			−0.667182			−0.333591	+	0.942718i	$0.608260\pi$
$648$		0			0
$649$	−64.0000			−2.51222
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15.9217	−	9.19239i	−0.623064	−	0.359726i	0.154997	−	0.987915i	$-0.450463\pi$
$653$	−15.9217	−	9.19239i	−0.623064	−	0.359726i	−0.778061	+	0.628189i	$0.783796\pi$
$654$		0			0
$655$	−13.8564	+	8.00000i	−0.541415	+	0.312586i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	5.65685	+	9.79796i	0.220360	+	0.381674i	0.954917	−	0.296872i	$-0.0959435\pi$
$659$	5.65685	+	9.79796i	0.220360	+	0.381674i	−0.734557	+	0.678546i	$0.762610\pi$
$660$		0			0
$661$	11.0000	−	19.0526i	0.427850	−	0.741059i	−0.568831	−	0.822454i	$-0.692604\pi$
$661$	11.0000	−	19.0526i	0.427850	−	0.741059i	0.996682	+	0.0813955i	$0.0259377\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			8.00000i			0.309761i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.65685	−	9.79796i	0.218380	−	0.378246i
$672$		0			0
$673$	7.00000	+	12.1244i	0.269830	+	0.467360i	0.968818	−	0.247774i	$-0.0796991\pi$
$673$	7.00000	+	12.1244i	0.269830	+	0.467360i	−0.698988	+	0.715134i	$0.746366\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−28.1691	+	16.2635i	−1.08263	+	0.625055i	−0.931604	−	0.363475i	$-0.881590\pi$
$677$	−28.1691	+	16.2635i	−1.08263	+	0.625055i	−0.151024	+	0.988530i	$0.548257\pi$
$678$		0			0
$679$	−27.7128	−	16.0000i	−1.06352	−	0.614024i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	16.9706			0.649361			0.324680	−	0.945824i	$-0.394743\pi$
$683$	16.9706			0.649361			0.324680	+	0.945824i	$0.394743\pi$
$684$		0			0
$685$	2.00000			0.0764161
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−14.6969	−	8.48528i	−0.559909	−	0.323263i
$690$		0			0
$691$	6.92820	−	4.00000i	0.263561	−	0.152167i	−0.362397	−	0.932024i	$-0.618041\pi$
$691$	6.92820	−	4.00000i	0.263561	−	0.152167i	0.625958	+	0.779857i	$0.284708\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−5.65685	−	9.79796i	−0.214577	−	0.371658i
$696$		0			0
$697$	−21.0000	+	36.3731i	−0.795432	+	1.37773i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	12.7279i		−	0.480727i	−0.970683	−	0.240363i	$-0.922733\pi$
$701$		−	12.7279i		−	0.480727i	0.970683	−	0.240363i	$-0.0772666\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−36.7696	+	63.6867i	−1.38286	+	2.39519i
$708$		0			0
$709$	−10.0000	−	17.3205i	−0.375558	−	0.650485i	0.614852	−	0.788642i	$-0.289216\pi$
$709$	−10.0000	−	17.3205i	−0.375558	−	0.650485i	−0.990410	+	0.138157i	$0.955882\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	19.5959	−	11.3137i	0.733873	−	0.423702i
$714$		0			0
$715$	−27.7128	−	16.0000i	−1.03640	−	0.598366i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16.9706			−0.632895			−0.316448	−	0.948610i	$-0.602490\pi$
$719$	−16.9706			−0.632895			−0.316448	+	0.948610i	$0.602490\pi$
$720$		0			0
$721$	16.0000			0.595871
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.67423	+	2.12132i	0.136458	+	0.0787839i
$726$		0			0
$727$	3.46410	−	2.00000i	0.128476	−	0.0741759i	−0.434384	−	0.900728i	$-0.643034\pi$
$727$	3.46410	−	2.00000i	0.128476	−	0.0741759i	0.562861	+	0.826552i	$0.309701\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	16.9706	+	29.3939i	0.627679	+	1.08717i
$732$		0			0
$733$	−2.00000	+	3.46410i	−0.0738717	+	0.127950i	−0.900595	−	0.434659i	$-0.856869\pi$
$733$	−2.00000	+	3.46410i	−0.0738717	+	0.127950i	0.826723	+	0.562609i	$0.190202\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			45.2548i			1.66698i
$738$		0			0
$739$			16.0000i			0.588570i	0.955718	+	0.294285i	$0.0950814\pi$
$739$			16.0000i			0.588570i	−0.955718	+	0.294285i	$0.904919\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	5.65685	−	9.79796i	0.207530	−	0.359452i	−0.743406	−	0.668840i	$-0.766791\pi$
$743$	5.65685	−	9.79796i	0.207530	−	0.359452i	0.950936	+	0.309388i	$0.100124\pi$
$744$		0			0
$745$	−1.00000	−	1.73205i	−0.0366372	−	0.0634574i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	17.3205	+	10.0000i	0.632034	+	0.364905i	0.781540	−	0.623856i	$-0.214435\pi$
$751$	17.3205	+	10.0000i	0.632034	+	0.364905i	−0.149505	+	0.988761i	$0.547768\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	28.2843			1.02937
$756$		0			0
$757$	20.0000			0.726912			0.363456	−	0.931611i	$-0.381597\pi$
$757$	20.0000			0.726912			0.363456	+	0.931611i	$0.381597\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	8.57321	+	4.94975i	0.310779	+	0.179428i	0.647275	−	0.762257i	$-0.275909\pi$
$761$	8.57321	+	4.94975i	0.310779	+	0.179428i	−0.336496	+	0.941685i	$0.609242\pi$
$762$		0			0
$763$	−41.5692	+	24.0000i	−1.50491	+	0.868858i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	22.6274	+	39.1918i	0.817029	+	1.41514i
$768$		0			0
$769$	−1.00000	+	1.73205i	−0.0360609	+	0.0624593i	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	−1.00000	+	1.73205i	−0.0360609	+	0.0624593i	0.847432	+	0.530904i	$0.178148\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			4.24264i			0.152597i	0.997085	+	0.0762986i	$0.0243102\pi$
$773$			4.24264i			0.152597i	−0.997085	+	0.0762986i	$0.975690\pi$
$774$		0			0
$775$		−	12.0000i		−	0.431053i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−16.0000	−	27.7128i	−0.572525	−	0.991642i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2.44949	−	1.41421i	0.0874260	−	0.0504754i
$786$		0			0
$787$	27.7128	+	16.0000i	0.987855	+	0.570338i	0.904632	−	0.426193i	$-0.140145\pi$
$787$	27.7128	+	16.0000i	0.987855	+	0.570338i	0.0832226	+	0.996531i	$0.473479\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	5.65685			0.201135
$792$		0			0
$793$	−8.00000			−0.284088
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−6.12372	−	3.53553i	−0.216913	−	0.125235i	0.387607	−	0.921825i	$-0.373302\pi$
$797$	−6.12372	−	3.53553i	−0.216913	−	0.125235i	−0.604520	+	0.796590i	$0.706635\pi$
$798$		0			0
$799$	−20.7846	+	12.0000i	−0.735307	+	0.424529i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	16.0000	−	27.7128i	0.563926	−	0.976748i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	21.2132i		−	0.745817i	−0.927868	−	0.372908i	$-0.878361\pi$
$809$		−	21.2132i		−	0.745817i	0.927868	−	0.372908i	$-0.121639\pi$
$810$		0			0
$811$			48.0000i			1.68551i	0.538299	+	0.842754i	$0.319067\pi$
$811$			48.0000i			1.68551i	−0.538299	+	0.842754i	$0.680933\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−16.9706	+	29.3939i	−0.594453	+	1.02962i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	35.5176	−	20.5061i	1.23957	−	0.715668i	0.270566	−	0.962701i	$-0.412789\pi$
$821$	35.5176	−	20.5061i	1.23957	−	0.715668i	0.969007	+	0.247034i	$0.0794559\pi$
$822$		0			0
$823$	45.0333	+	26.0000i	1.56976	+	0.906303i	0.996196	+	0.0871445i	$0.0277742\pi$
$823$	45.0333	+	26.0000i	1.56976	+	0.906303i	0.573567	+	0.819159i	$0.305559\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	33.9411			1.18025			0.590124	−	0.807312i	$-0.299079\pi$
$827$	33.9411			1.18025			0.590124	+	0.807312i	$0.299079\pi$
$828$		0			0
$829$	12.0000			0.416777			0.208389	−	0.978046i	$-0.433178\pi$
$829$	12.0000			0.416777			0.208389	+	0.978046i	$0.433178\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	33.0681	+	19.0919i	1.14574	+	0.661495i
$834$		0			0
$835$	−13.8564	+	8.00000i	−0.479521	+	0.276851i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2.82843	+	4.89898i	0.0976481	+	0.169132i	0.910711	−	0.413045i	$-0.135535\pi$
$839$	2.82843	+	4.89898i	0.0976481	+	0.169132i	−0.813063	+	0.582176i	$0.802201\pi$
$840$		0			0
$841$	−13.5000	+	23.3827i	−0.465517	+	0.806300i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			4.24264i			0.145951i
$846$		0			0
$847$		−	84.0000i		−	2.88627i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−16.9706	+	29.3939i	−0.581743	+	1.00761i
$852$		0			0
$853$	−5.00000	−	8.66025i	−0.171197	−	0.296521i	0.767642	−	0.640879i	$-0.221430\pi$
$853$	−5.00000	−	8.66025i	−0.171197	−	0.296521i	−0.938839	+	0.344358i	$0.888097\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	20.8207	−	12.0208i	0.711220	−	0.410623i	−0.100292	−	0.994958i	$-0.531978\pi$
$857$	20.8207	−	12.0208i	0.711220	−	0.410623i	0.811513	+	0.584335i	$0.198644\pi$
$858$		0			0
$859$	−6.92820	−	4.00000i	−0.236387	−	0.136478i	0.377128	−	0.926161i	$-0.376912\pi$
$859$	−6.92820	−	4.00000i	−0.236387	−	0.136478i	−0.613515	+	0.789683i	$0.710245\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	33.9411			1.15537			0.577685	−	0.816260i	$-0.303956\pi$
$863$	33.9411			1.15537			0.577685	+	0.816260i	$0.303956\pi$
$864$		0			0
$865$	22.0000			0.748022
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−19.5959	−	11.3137i	−0.664746	−	0.383791i
$870$		0			0
$871$	27.7128	−	16.0000i	0.939013	−	0.542139i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−22.6274	−	39.1918i	−0.764946	−	1.32493i
$876$		0			0
$877$	−23.0000	+	39.8372i	−0.776655	+	1.34521i	0.157205	+	0.987566i	$0.449752\pi$
$877$	−23.0000	+	39.8372i	−0.776655	+	1.34521i	−0.933860	+	0.357640i	$0.883582\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	12.7279i		−	0.428815i	−0.976744	−	0.214407i	$-0.931218\pi$
$881$		−	12.7279i		−	0.428815i	0.976744	−	0.214407i	$-0.0687820\pi$
$882$		0			0
$883$			24.0000i			0.807664i	0.914833	+	0.403832i	$0.132322\pi$
$883$			24.0000i			0.807664i	−0.914833	+	0.403832i	$0.867678\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	28.2843	−	48.9898i	0.949693	−	1.64492i	0.203622	−	0.979050i	$-0.434729\pi$
$887$	28.2843	−	48.9898i	0.949693	−	1.64492i	0.746071	−	0.665867i	$-0.231938\pi$
$888$		0			0
$889$	−24.0000	−	41.5692i	−0.804934	−	1.39419i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−5.65685			−0.188667
$900$		0			0
$901$	−18.0000			−0.599667
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−14.6969	−	8.48528i	−0.488543	−	0.282060i
$906$		0			0
$907$	6.92820	−	4.00000i	0.230047	−	0.132818i	−0.380547	−	0.924762i	$-0.624264\pi$
$907$	6.92820	−	4.00000i	0.230047	−	0.132818i	0.610594	+	0.791944i	$0.290931\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	5.65685	+	9.79796i	0.187420	+	0.324621i	0.944389	−	0.328830i	$-0.106654\pi$
$911$	5.65685	+	9.79796i	0.187420	+	0.324621i	−0.756969	+	0.653450i	$0.773321\pi$
$912$		0			0
$913$	16.0000	−	27.7128i	0.529523	−	0.917160i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			45.2548i			1.49445i
$918$		0			0
$919$		−	36.0000i		−	1.18753i	−0.804638	−	0.593765i	$-0.797641\pi$
$919$		−	36.0000i		−	1.18753i	0.804638	−	0.593765i	$-0.202359\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−11.3137	+	19.5959i	−0.372395	+	0.645007i
$924$		0			0
$925$	9.00000	+	15.5885i	0.295918	+	0.512545i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	42.8661	−	24.7487i	1.40639	−	0.811980i	0.411352	−	0.911476i	$-0.365057\pi$
$929$	42.8661	−	24.7487i	1.40639	−	0.811980i	0.995038	+	0.0994967i	$0.0317233\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−33.9411			−1.10999
$936$		0			0
$937$	6.00000			0.196011			0.0980057	−	0.995186i	$-0.468754\pi$
$937$	6.00000			0.196011			0.0980057	+	0.995186i	$0.468754\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	35.5176	+	20.5061i	1.15784	+	0.668480i	0.950786	−	0.309850i	$-0.100279\pi$
$941$	35.5176	+	20.5061i	1.15784	+	0.668480i	0.207055	+	0.978329i	$0.433612\pi$
$942$		0			0
$943$	48.4974	−	28.0000i	1.57929	−	0.911805i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5.65685	+	9.79796i	0.183823	+	0.318391i	0.943179	−	0.332284i	$-0.107819\pi$
$947$	5.65685	+	9.79796i	0.183823	+	0.318391i	−0.759356	+	0.650675i	$0.774486\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			21.2132i			0.687163i	0.939123	+	0.343582i	$0.111640\pi$
$953$			21.2132i			0.687163i	−0.939123	+	0.343582i	$0.888360\pi$
$954$		0			0
$955$			16.0000i			0.517748i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2.82843	−	4.89898i	0.0913347	−	0.158196i
$960$		0			0
$961$	−7.50000	−	12.9904i	−0.241935	−	0.419045i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.44949	+	1.41421i	−0.0788519	+	0.0455251i
$966$		0			0
$967$	24.2487	+	14.0000i	0.779786	+	0.450210i	0.836354	−	0.548189i	$-0.184683\pi$
$967$	24.2487	+	14.0000i	0.779786	+	0.450210i	−0.0565684	+	0.998399i	$0.518016\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−5.65685			−0.181537			−0.0907685	−	0.995872i	$-0.528932\pi$
$971$	−5.65685			−0.181537			−0.0907685	+	0.995872i	$0.528932\pi$
$972$		0			0
$973$	−32.0000			−1.02587
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−45.3156	−	26.1630i	−1.44977	−	0.837027i	−0.451306	−	0.892369i	$-0.649042\pi$
$977$	−45.3156	−	26.1630i	−1.44977	−	0.837027i	−0.998467	+	0.0553424i	$0.982375\pi$
$978$		0			0
$979$	−20.7846	+	12.0000i	−0.664279	+	0.383522i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−28.2843	−	48.9898i	−0.902128	−	1.56253i	−0.824726	−	0.565532i	$-0.808671\pi$
$983$	−28.2843	−	48.9898i	−0.902128	−	1.56253i	−0.0774017	−	0.997000i	$-0.524662\pi$
$984$		0			0
$985$	5.00000	−	8.66025i	0.159313	−	0.275939i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	45.2548i		−	1.43902i
$990$		0			0
$991$			12.0000i			0.381193i	0.981669	+	0.190596i	$0.0610421\pi$
$991$			12.0000i			0.381193i	−0.981669	+	0.190596i	$0.938958\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	14.1421	−	24.4949i	0.448336	−	0.776540i
$996$		0			0
$997$	−19.0000	−	32.9090i	−0.601736	−	1.04224i	−0.992558	−	0.121771i	$-0.961143\pi$
$997$	−19.0000	−	32.9090i	−0.601736	−	1.04224i	0.390822	−	0.920466i	$-0.372191\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.s.d.863.1		8
3.2	odd	2	inner	2592.2.s.d.863.3		8
4.3	odd	2	inner	2592.2.s.d.863.2		8
9.2	odd	6	inner	2592.2.s.d.1727.2		8
9.4	even	3		288.2.c.a.287.3	yes	4
9.5	odd	6		288.2.c.a.287.1	✓	4
9.7	even	3	inner	2592.2.s.d.1727.4		8
12.11	even	2	inner	2592.2.s.d.863.4		8
36.7	odd	6	inner	2592.2.s.d.1727.3		8
36.11	even	6	inner	2592.2.s.d.1727.1		8
36.23	even	6		288.2.c.a.287.2	yes	4
36.31	odd	6		288.2.c.a.287.4	yes	4
45.4	even	6		7200.2.h.d.1151.4		4
45.13	odd	12		7200.2.o.a.7199.4		4
45.14	odd	6		7200.2.h.d.1151.3		4
45.22	odd	12		7200.2.o.n.7199.3		4
45.23	even	12		7200.2.o.a.7199.2		4
45.32	even	12		7200.2.o.n.7199.1		4
72.5	odd	6		576.2.c.c.575.3		4
72.13	even	6		576.2.c.c.575.1		4
72.59	even	6		576.2.c.c.575.4		4
72.67	odd	6		576.2.c.c.575.2		4
144.5	odd	12		2304.2.f.c.1151.4		4
144.13	even	12		2304.2.f.e.1151.4		4
144.59	even	12		2304.2.f.e.1151.3		4
144.67	odd	12		2304.2.f.c.1151.3		4
144.77	odd	12		2304.2.f.e.1151.2		4
144.85	even	12		2304.2.f.c.1151.2		4
144.131	even	12		2304.2.f.c.1151.1		4
144.139	odd	12		2304.2.f.e.1151.1		4
180.23	odd	12		7200.2.o.n.7199.4		4
180.59	even	6		7200.2.h.d.1151.2		4
180.67	even	12		7200.2.o.a.7199.1		4
180.103	even	12		7200.2.o.n.7199.2		4
180.139	odd	6		7200.2.h.d.1151.1		4
180.167	odd	12		7200.2.o.a.7199.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.2.c.a.287.1	✓	4	9.5	odd	6
288.2.c.a.287.2	yes	4	36.23	even	6
288.2.c.a.287.3	yes	4	9.4	even	3
288.2.c.a.287.4	yes	4	36.31	odd	6
576.2.c.c.575.1		4	72.13	even	6
576.2.c.c.575.2		4	72.67	odd	6
576.2.c.c.575.3		4	72.5	odd	6
576.2.c.c.575.4		4	72.59	even	6
2304.2.f.c.1151.1		4	144.131	even	12
2304.2.f.c.1151.2		4	144.85	even	12
2304.2.f.c.1151.3		4	144.67	odd	12
2304.2.f.c.1151.4		4	144.5	odd	12
2304.2.f.e.1151.1		4	144.139	odd	12
2304.2.f.e.1151.2		4	144.77	odd	12
2304.2.f.e.1151.3		4	144.59	even	12
2304.2.f.e.1151.4		4	144.13	even	12
2592.2.s.d.863.1		8	1.1	even	1	trivial
2592.2.s.d.863.2		8	4.3	odd	2	inner
2592.2.s.d.863.3		8	3.2	odd	2	inner
2592.2.s.d.863.4		8	12.11	even	2	inner
2592.2.s.d.1727.1		8	36.11	even	6	inner
2592.2.s.d.1727.2		8	9.2	odd	6	inner
2592.2.s.d.1727.3		8	36.7	odd	6	inner
2592.2.s.d.1727.4		8	9.7	even	3	inner
7200.2.h.d.1151.1		4	180.139	odd	6
7200.2.h.d.1151.2		4	180.59	even	6
7200.2.h.d.1151.3		4	45.14	odd	6
7200.2.h.d.1151.4		4	45.4	even	6
7200.2.o.a.7199.1		4	180.67	even	12
7200.2.o.a.7199.2		4	45.23	even	12
7200.2.o.a.7199.3		4	180.167	odd	12
7200.2.o.a.7199.4		4	45.13	odd	12
7200.2.o.n.7199.1		4	45.32	even	12
7200.2.o.n.7199.2		4	180.103	even	12
7200.2.o.n.7199.3		4	45.22	odd	12
7200.2.o.n.7199.4		4	180.23	odd	12

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

Level:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 0.707107i) q^{5} +(-3.46410 + 2.00000i) q^{7} +(-2.82843 - 4.89898i) q^{11} +(-2.00000 + 3.46410i) q^{13} +4.24264i q^{17} +(2.82843 - 4.89898i) q^{23} +(-1.50000 - 2.59808i) q^{25} +(-1.22474 + 0.707107i) q^{29} +(3.46410 + 2.00000i) q^{31} +5.65685 q^{35} -6.00000 q^{37} +(8.57321 + 4.94975i) q^{41} +(6.92820 - 4.00000i) q^{43} +(2.82843 + 4.89898i) q^{47} +(4.50000 - 7.79423i) q^{49} +4.24264i q^{53} +8.00000i q^{55} +(5.65685 - 9.79796i) q^{59} +(1.00000 + 1.73205i) q^{61} +(4.89898 - 2.82843i) q^{65} +(-6.92820 - 4.00000i) q^{67} +5.65685 q^{71} +(19.5959 + 11.3137i) q^{77} +(3.46410 - 2.00000i) q^{79} +(2.82843 + 4.89898i) q^{83} +(3.00000 - 5.19615i) q^{85} -4.24264i q^{89} -16.0000i q^{91} +(4.00000 + 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{13} - 12 q^{25} - 48 q^{37} + 36 q^{49} + 8 q^{61} + 24 q^{85} + 32 q^{97}+O(q^{100}) \) 8 * q - 16 * q^13 - 12 * q^25 - 48 * q^37 + 36 * q^49 + 8 * q^61 + 24 * q^85 + 32 * q^97

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