LMFDB - Embedded newform 1600.2.n.a.1407.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(1343,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

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

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

chi := DirichletCharacter("1600.1343");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.n (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1407.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1600.1407
Dual form			1600.2.n.a.1343.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 - 2.00000i) q^{3} +(2.00000 - 2.00000i) q^{7} +5.00000i q^{9} +O(q^{10})$	$q+(-2.00000 - 2.00000i) q^{3} +(2.00000 - 2.00000i) q^{7} +5.00000i q^{9} +(-1.00000 + 1.00000i) q^{13} +(5.00000 + 5.00000i) q^{17} +4.00000 q^{19} -8.00000 q^{21} +(2.00000 + 2.00000i) q^{23} +(4.00000 - 4.00000i) q^{27} +4.00000i q^{29} +4.00000i q^{31} +(1.00000 + 1.00000i) q^{37} +4.00000 q^{39} +(6.00000 + 6.00000i) q^{43} +(-2.00000 + 2.00000i) q^{47} -1.00000i q^{49} -20.0000i q^{51} +(-7.00000 + 7.00000i) q^{53} +(-8.00000 - 8.00000i) q^{57} +4.00000 q^{59} +4.00000 q^{61} +(10.0000 + 10.0000i) q^{63} +(10.0000 - 10.0000i) q^{67} -8.00000i q^{69} -12.0000i q^{71} +(3.00000 - 3.00000i) q^{73} -16.0000 q^{79} -1.00000 q^{81} +(2.00000 + 2.00000i) q^{83} +(8.00000 - 8.00000i) q^{87} +4.00000i q^{91} +(8.00000 - 8.00000i) q^{93} +(3.00000 + 3.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} + 4 q^{7}+O(q^{10}) $ 2 * q - 4 * q^3 + 4 * q^7	$ 2 q - 4 q^{3} + 4 q^{7} - 2 q^{13} + 10 q^{17} + 8 q^{19} - 16 q^{21} + 4 q^{23} + 8 q^{27} + 2 q^{37} + 8 q^{39} + 12 q^{43} - 4 q^{47} - 14 q^{53} - 16 q^{57} + 8 q^{59} + 8 q^{61} + 20 q^{63} + 20 q^{67} + 6 q^{73} - 32 q^{79} - 2 q^{81} + 4 q^{83} + 16 q^{87} + 16 q^{93} + 6 q^{97}+O(q^{100}) $ 2 * q - 4 * q^3 + 4 * q^7 - 2 * q^13 + 10 * q^17 + 8 * q^19 - 16 * q^21 + 4 * q^23 + 8 * q^27 + 2 * q^37 + 8 * q^39 + 12 * q^43 - 4 * q^47 - 14 * q^53 - 16 * q^57 + 8 * q^59 + 8 * q^61 + 20 * q^63 + 20 * q^67 + 6 * q^73 - 32 * q^79 - 2 * q^81 + 4 * q^83 + 16 * q^87 + 16 * q^93 + 6 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$-1$

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
$3$	−2.00000	−	2.00000i	−1.15470	−	1.15470i	−0.985599	−	0.169102i	$-0.945913\pi$
$3$	−2.00000	−	2.00000i	−1.15470	−	1.15470i	−0.169102	−	0.985599i	$-0.554087\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.00000	−	2.00000i	0.755929	−	0.755929i	−0.219650	−	0.975579i	$-0.570491\pi$
$7$	2.00000	−	2.00000i	0.755929	−	0.755929i	0.975579	+	0.219650i	$0.0704915\pi$
$8$		0			0
$9$			5.00000i			1.66667i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−1.00000	+	1.00000i	−0.277350	+	0.277350i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−1.00000	+	1.00000i	−0.277350	+	0.277350i	0.554700	+	0.832050i	$0.312833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.00000	+	5.00000i	1.21268	+	1.21268i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	5.00000	+	5.00000i	1.21268	+	1.21268i	0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$	−8.00000			−1.74574
$22$		0			0
$23$	2.00000	+	2.00000i	0.417029	+	0.417029i	0.884178	−	0.467150i	$-0.154719\pi$
$23$	2.00000	+	2.00000i	0.417029	+	0.417029i	−0.467150	+	0.884178i	$0.654719\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	4.00000	−	4.00000i	0.769800	−	0.769800i
$28$		0			0
$29$			4.00000i			0.742781i	0.928477	+	0.371391i	$0.121119\pi$
$29$			4.00000i			0.742781i	−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$			4.00000i			0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$			4.00000i			0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.00000	+	1.00000i	0.164399	+	0.164399i	0.784512	−	0.620113i	$-0.212913\pi$
$37$	1.00000	+	1.00000i	0.164399	+	0.164399i	−0.620113	+	0.784512i	$0.712913\pi$
$38$		0			0
$39$	4.00000			0.640513
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	6.00000	+	6.00000i	0.914991	+	0.914991i	0.996660	−	0.0816682i	$-0.0260248\pi$
$43$	6.00000	+	6.00000i	0.914991	+	0.914991i	−0.0816682	+	0.996660i	$0.526025\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.00000	+	2.00000i	−0.291730	+	0.291730i	−0.837763	−	0.546033i	$-0.816137\pi$
$47$	−2.00000	+	2.00000i	−0.291730	+	0.291730i	0.546033	+	0.837763i	$0.316137\pi$
$48$		0			0
$49$		−	1.00000i		−	0.142857i
$50$		0			0
$51$		−	20.0000i		−	2.80056i
$52$		0			0
$53$	−7.00000	+	7.00000i	−0.961524	+	0.961524i	−0.999287	−	0.0377628i	$-0.987977\pi$
$53$	−7.00000	+	7.00000i	−0.961524	+	0.961524i	0.0377628	+	0.999287i	$0.487977\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−8.00000	−	8.00000i	−1.05963	−	1.05963i
$58$		0			0
$59$	4.00000			0.520756			0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000			0.520756			0.260378	+	0.965507i	$0.416153\pi$
$60$		0			0
$61$	4.00000			0.512148			0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000			0.512148			0.256074	+	0.966657i	$0.417571\pi$
$62$		0			0
$63$	10.0000	+	10.0000i	1.25988	+	1.25988i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	10.0000	−	10.0000i	1.22169	−	1.22169i	0.254665	−	0.967029i	$-0.418035\pi$
$67$	10.0000	−	10.0000i	1.22169	−	1.22169i	0.967029	−	0.254665i	$-0.0819652\pi$
$68$		0			0
$69$		−	8.00000i		−	0.963087i
$70$		0			0
$71$		−	12.0000i		−	1.42414i	−0.702109	−	0.712069i	$-0.747758\pi$
$71$		−	12.0000i		−	1.42414i	0.702109	−	0.712069i	$-0.252242\pi$
$72$		0			0
$73$	3.00000	−	3.00000i	0.351123	−	0.351123i	−0.509404	−	0.860527i	$-0.670134\pi$
$73$	3.00000	−	3.00000i	0.351123	−	0.351123i	0.860527	+	0.509404i	$0.170134\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−16.0000			−1.80014			−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000			−1.80014			−0.900070	+	0.435745i	$0.856485\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	2.00000	+	2.00000i	0.219529	+	0.219529i	0.808300	−	0.588771i	$-0.200388\pi$
$83$	2.00000	+	2.00000i	0.219529	+	0.219529i	−0.588771	+	0.808300i	$0.700388\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	8.00000	−	8.00000i	0.857690	−	0.857690i
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$			4.00000i			0.419314i
$92$		0			0
$93$	8.00000	−	8.00000i	0.829561	−	0.829561i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	3.00000	+	3.00000i	0.304604	+	0.304604i	0.842812	−	0.538208i	$-0.180899\pi$
$97$	3.00000	+	3.00000i	0.304604	+	0.304604i	−0.538208	+	0.842812i	$0.680899\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$	−6.00000	−	6.00000i	−0.591198	−	0.591198i	0.346757	−	0.937955i	$-0.387283\pi$
$103$	−6.00000	−	6.00000i	−0.591198	−	0.591198i	−0.937955	+	0.346757i	$0.887283\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.00000	+	6.00000i	−0.580042	+	0.580042i	−0.934915	−	0.354873i	$-0.884524\pi$
$107$	−6.00000	+	6.00000i	−0.580042	+	0.580042i	0.354873	+	0.934915i	$0.384524\pi$
$108$		0			0
$109$		−	10.0000i		−	0.957826i	−0.877862	−	0.478913i	$-0.841031\pi$
$109$		−	10.0000i		−	0.957826i	0.877862	−	0.478913i	$-0.158969\pi$
$110$		0			0
$111$		−	4.00000i		−	0.379663i
$112$		0			0
$113$	9.00000	−	9.00000i	0.846649	−	0.846649i	−0.143065	−	0.989713i	$-0.545696\pi$
$113$	9.00000	−	9.00000i	0.846649	−	0.846649i	0.989713	+	0.143065i	$0.0456957\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−5.00000	−	5.00000i	−0.462250	−	0.462250i
$118$		0			0
$119$	20.0000			1.83340
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	10.0000	−	10.0000i	0.887357	−	0.887357i	−0.106912	−	0.994268i	$-0.534096\pi$
$127$	10.0000	−	10.0000i	0.887357	−	0.887357i	0.994268	+	0.106912i	$0.0340963\pi$
$128$		0			0
$129$		−	24.0000i		−	2.11308i
$130$		0			0
$131$			8.00000i			0.698963i	0.936943	+	0.349482i	$0.113642\pi$
$131$			8.00000i			0.698963i	−0.936943	+	0.349482i	$0.886358\pi$
$132$		0			0
$133$	8.00000	−	8.00000i	0.693688	−	0.693688i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.00000	−	1.00000i	−0.0854358	−	0.0854358i	0.663097	−	0.748533i	$-0.269242\pi$
$137$	−1.00000	−	1.00000i	−0.0854358	−	0.0854358i	−0.748533	+	0.663097i	$0.769242\pi$
$138$		0			0
$139$	−12.0000			−1.01783			−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000			−1.01783			−0.508913	+	0.860818i	$0.669953\pi$
$140$		0			0
$141$	8.00000			0.673722
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−2.00000	+	2.00000i	−0.164957	+	0.164957i
$148$		0			0
$149$			18.0000i			1.47462i	0.675556	+	0.737309i	$0.263904\pi$
$149$			18.0000i			1.47462i	−0.675556	+	0.737309i	$0.736096\pi$
$150$		0			0
$151$		−	12.0000i		−	0.976546i	−0.872691	−	0.488273i	$-0.837627\pi$
$151$		−	12.0000i		−	0.976546i	0.872691	−	0.488273i	$-0.162373\pi$
$152$		0			0
$153$	−25.0000	+	25.0000i	−2.02113	+	2.02113i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−9.00000	−	9.00000i	−0.718278	−	0.718278i	0.249974	−	0.968252i	$-0.419578\pi$
$157$	−9.00000	−	9.00000i	−0.718278	−	0.718278i	−0.968252	+	0.249974i	$0.919578\pi$
$158$		0			0
$159$	28.0000			2.22054
$160$		0			0
$161$	8.00000			0.630488
$162$		0			0
$163$	−2.00000	−	2.00000i	−0.156652	−	0.156652i	0.624429	−	0.781081i	$-0.285332\pi$
$163$	−2.00000	−	2.00000i	−0.156652	−	0.156652i	−0.781081	+	0.624429i	$0.785332\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.00000	+	2.00000i	−0.154765	+	0.154765i	−0.780242	−	0.625478i	$-0.784904\pi$
$167$	−2.00000	+	2.00000i	−0.154765	+	0.154765i	0.625478	+	0.780242i	$0.284904\pi$
$168$		0			0
$169$			11.0000i			0.846154i
$170$		0			0
$171$			20.0000i			1.52944i
$172$		0			0
$173$	13.0000	−	13.0000i	0.988372	−	0.988372i	−0.0115615	−	0.999933i	$-0.503680\pi$
$173$	13.0000	−	13.0000i	0.988372	−	0.988372i	0.999933	+	0.0115615i	$0.00368021\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−8.00000	−	8.00000i	−0.601317	−	0.601317i
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	10.0000			0.743294			0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000			0.743294			0.371647	+	0.928374i	$0.378793\pi$
$182$		0			0
$183$	−8.00000	−	8.00000i	−0.591377	−	0.591377i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		−	16.0000i		−	1.16383i
$190$		0			0
$191$			20.0000i			1.44715i	0.690246	+	0.723575i	$0.257502\pi$
$191$			20.0000i			1.44715i	−0.690246	+	0.723575i	$0.742498\pi$
$192$		0			0
$193$	5.00000	−	5.00000i	0.359908	−	0.359908i	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	5.00000	−	5.00000i	0.359908	−	0.359908i	0.863779	+	0.503871i	$0.168091\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.00000	+	5.00000i	0.356235	+	0.356235i	0.862423	−	0.506188i	$-0.168946\pi$
$197$	5.00000	+	5.00000i	0.356235	+	0.356235i	−0.506188	+	0.862423i	$0.668946\pi$
$198$		0			0
$199$	−24.0000			−1.70131			−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000			−1.70131			−0.850657	+	0.525720i	$0.823796\pi$
$200$		0			0
$201$	−40.0000			−2.82138
$202$		0			0
$203$	8.00000	+	8.00000i	0.561490	+	0.561490i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−10.0000	+	10.0000i	−0.695048	+	0.695048i
$208$		0			0
$209$		0			0
$210$		0			0
$211$			16.0000i			1.10149i	0.834675	+	0.550743i	$0.185655\pi$
$211$			16.0000i			1.10149i	−0.834675	+	0.550743i	$0.814345\pi$
$212$		0			0
$213$	−24.0000	+	24.0000i	−1.64445	+	1.64445i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	8.00000	+	8.00000i	0.543075	+	0.543075i
$218$		0			0
$219$	−12.0000			−0.810885
$220$		0			0
$221$	−10.0000			−0.672673
$222$		0			0
$223$	−10.0000	−	10.0000i	−0.669650	−	0.669650i	0.287985	−	0.957635i	$-0.407015\pi$
$223$	−10.0000	−	10.0000i	−0.669650	−	0.669650i	−0.957635	+	0.287985i	$0.907015\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.0000	+	10.0000i	−0.663723	+	0.663723i	−0.956256	−	0.292532i	$-0.905502\pi$
$227$	−10.0000	+	10.0000i	−0.663723	+	0.663723i	0.292532	+	0.956256i	$0.405502\pi$
$228$		0			0
$229$			20.0000i			1.32164i	0.750546	+	0.660819i	$0.229791\pi$
$229$			20.0000i			1.32164i	−0.750546	+	0.660819i	$0.770209\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5.00000	−	5.00000i	0.327561	−	0.327561i	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	5.00000	−	5.00000i	0.327561	−	0.327561i	0.851658	+	0.524097i	$0.175597\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	32.0000	+	32.0000i	2.07862	+	2.07862i
$238$		0			0
$239$	−8.00000			−0.517477			−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000			−0.517477			−0.258738	+	0.965947i	$0.583307\pi$
$240$		0			0
$241$	−16.0000			−1.03065			−0.515325	−	0.856995i	$-0.672329\pi$
$241$	−16.0000			−1.03065			−0.515325	+	0.856995i	$0.672329\pi$
$242$		0			0
$243$	−10.0000	−	10.0000i	−0.641500	−	0.641500i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4.00000	+	4.00000i	−0.254514	+	0.254514i
$248$		0			0
$249$		−	8.00000i		−	0.506979i
$250$		0			0
$251$		−	24.0000i		−	1.51487i	−0.652913	−	0.757433i	$-0.726453\pi$
$251$		−	24.0000i		−	1.51487i	0.652913	−	0.757433i	$-0.273547\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.00000	−	7.00000i	−0.436648	−	0.436648i	0.454234	−	0.890882i	$-0.349913\pi$
$257$	−7.00000	−	7.00000i	−0.436648	−	0.436648i	−0.890882	+	0.454234i	$0.849913\pi$
$258$		0			0
$259$	4.00000			0.248548
$260$		0			0
$261$	−20.0000			−1.23797
$262$		0			0
$263$	6.00000	+	6.00000i	0.369976	+	0.369976i	0.867468	−	0.497492i	$-0.165746\pi$
$263$	6.00000	+	6.00000i	0.369976	+	0.369976i	−0.497492	+	0.867468i	$0.665746\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			10.0000i			0.609711i	0.952399	+	0.304855i	$0.0986081\pi$
$269$			10.0000i			0.609711i	−0.952399	+	0.304855i	$0.901392\pi$
$270$		0			0
$271$		−	20.0000i		−	1.21491i	−0.794353	−	0.607457i	$-0.792190\pi$
$271$		−	20.0000i		−	1.21491i	0.794353	−	0.607457i	$-0.207810\pi$
$272$		0			0
$273$	8.00000	−	8.00000i	0.484182	−	0.484182i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−9.00000	−	9.00000i	−0.540758	−	0.540758i	0.382993	−	0.923751i	$-0.374893\pi$
$277$	−9.00000	−	9.00000i	−0.540758	−	0.540758i	−0.923751	+	0.382993i	$0.874893\pi$
$278$		0			0
$279$	−20.0000			−1.19737
$280$		0			0
$281$	8.00000			0.477240			0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000			0.477240			0.238620	+	0.971113i	$0.423305\pi$
$282$		0			0
$283$	−6.00000	−	6.00000i	−0.356663	−	0.356663i	0.505918	−	0.862581i	$-0.331154\pi$
$283$	−6.00000	−	6.00000i	−0.356663	−	0.356663i	−0.862581	+	0.505918i	$0.831154\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$			33.0000i			1.94118i
$290$		0			0
$291$		−	12.0000i		−	0.703452i
$292$		0			0
$293$	−5.00000	+	5.00000i	−0.292103	+	0.292103i	−0.837911	−	0.545807i	$-0.816223\pi$
$293$	−5.00000	+	5.00000i	−0.292103	+	0.292103i	0.545807	+	0.837911i	$0.316223\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.00000			−0.231326
$300$		0			0
$301$	24.0000			1.38334
$302$		0			0
$303$	−12.0000	−	12.0000i	−0.689382	−	0.689382i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	10.0000	−	10.0000i	0.570730	−	0.570730i	−0.361602	−	0.932332i	$-0.617770\pi$
$307$	10.0000	−	10.0000i	0.570730	−	0.570730i	0.932332	+	0.361602i	$0.117770\pi$
$308$		0			0
$309$			24.0000i			1.36531i
$310$		0			0
$311$			28.0000i			1.58773i	0.608091	+	0.793867i	$0.291935\pi$
$311$			28.0000i			1.58773i	−0.608091	+	0.793867i	$0.708065\pi$
$312$		0			0
$313$	−15.0000	+	15.0000i	−0.847850	+	0.847850i	−0.989865	−	0.142014i	$-0.954642\pi$
$313$	−15.0000	+	15.0000i	−0.847850	+	0.847850i	0.142014	+	0.989865i	$0.454642\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	11.0000	+	11.0000i	0.617822	+	0.617822i	0.944972	−	0.327151i	$-0.106088\pi$
$317$	11.0000	+	11.0000i	0.617822	+	0.617822i	−0.327151	+	0.944972i	$0.606088\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	24.0000			1.33955
$322$		0			0
$323$	20.0000	+	20.0000i	1.11283	+	1.11283i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−20.0000	+	20.0000i	−1.10600	+	1.10600i
$328$		0			0
$329$			8.00000i			0.441054i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	−5.00000	+	5.00000i	−0.273998	+	0.273998i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	23.0000	+	23.0000i	1.25289	+	1.25289i	0.954419	+	0.298471i	$0.0964767\pi$
$337$	23.0000	+	23.0000i	1.25289	+	1.25289i	0.298471	+	0.954419i	$0.403523\pi$
$338$		0			0
$339$	−36.0000			−1.95525
$340$		0			0
$341$		0			0
$342$		0			0
$343$	12.0000	+	12.0000i	0.647939	+	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−18.0000	+	18.0000i	−0.966291	+	0.966291i	−0.999450	−	0.0331594i	$-0.989443\pi$
$347$	−18.0000	+	18.0000i	−0.966291	+	0.966291i	0.0331594	+	0.999450i	$0.489443\pi$
$348$		0			0
$349$		−	20.0000i		−	1.07058i	−0.844670	−	0.535288i	$-0.820203\pi$
$349$		−	20.0000i		−	1.07058i	0.844670	−	0.535288i	$-0.179797\pi$
$350$		0			0
$351$			8.00000i			0.427008i
$352$		0			0
$353$	−9.00000	+	9.00000i	−0.479022	+	0.479022i	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−9.00000	+	9.00000i	−0.479022	+	0.479022i	0.425797	+	0.904819i	$0.359994\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−40.0000	−	40.0000i	−2.11702	−	2.11702i
$358$		0			0
$359$	16.0000			0.844448			0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000			0.844448			0.422224	+	0.906492i	$0.361250\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$	−22.0000	−	22.0000i	−1.15470	−	1.15470i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−22.0000	+	22.0000i	−1.14839	+	1.14839i	−0.161521	+	0.986869i	$0.551640\pi$
$367$	−22.0000	+	22.0000i	−1.14839	+	1.14839i	−0.986869	+	0.161521i	$0.948360\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			28.0000i			1.45369i
$372$		0			0
$373$	21.0000	−	21.0000i	1.08734	−	1.08734i	0.0915371	−	0.995802i	$-0.470822\pi$
$373$	21.0000	−	21.0000i	1.08734	−	1.08734i	0.995802	−	0.0915371i	$-0.0291780\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4.00000	−	4.00000i	−0.206010	−	0.206010i
$378$		0			0
$379$	28.0000			1.43826			0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000			1.43826			0.719132	+	0.694874i	$0.244540\pi$
$380$		0			0
$381$	−40.0000			−2.04926
$382$		0			0
$383$	22.0000	+	22.0000i	1.12415	+	1.12415i	0.991111	+	0.133036i	$0.0424727\pi$
$383$	22.0000	+	22.0000i	1.12415	+	1.12415i	0.133036	+	0.991111i	$0.457527\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−30.0000	+	30.0000i	−1.52499	+	1.52499i
$388$		0			0
$389$		−	18.0000i		−	0.912636i	−0.889817	−	0.456318i	$-0.849168\pi$
$389$		−	18.0000i		−	0.912636i	0.889817	−	0.456318i	$-0.150832\pi$
$390$		0			0
$391$			20.0000i			1.01144i
$392$		0			0
$393$	16.0000	−	16.0000i	0.807093	−	0.807093i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	13.0000	+	13.0000i	0.652451	+	0.652451i	0.953583	−	0.301131i	$-0.0973643\pi$
$397$	13.0000	+	13.0000i	0.652451	+	0.652451i	−0.301131	+	0.953583i	$0.597364\pi$
$398$		0			0
$399$	−32.0000			−1.60200
$400$		0			0
$401$	34.0000			1.69788			0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000			1.69788			0.848939	+	0.528490i	$0.177242\pi$
$402$		0			0
$403$	−4.00000	−	4.00000i	−0.199254	−	0.199254i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			2.00000i			0.0988936i	0.998777	+	0.0494468i	$0.0157458\pi$
$409$			2.00000i			0.0988936i	−0.998777	+	0.0494468i	$0.984254\pi$
$410$		0			0
$411$			4.00000i			0.197305i
$412$		0			0
$413$	8.00000	−	8.00000i	0.393654	−	0.393654i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	24.0000	+	24.0000i	1.17529	+	1.17529i
$418$		0			0
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$	−20.0000			−0.974740			−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000			−0.974740			−0.487370	+	0.873195i	$0.662044\pi$
$422$		0			0
$423$	−10.0000	−	10.0000i	−0.486217	−	0.486217i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	8.00000	−	8.00000i	0.387147	−	0.387147i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	4.00000i		−	0.192673i	−0.995349	−	0.0963366i	$-0.969287\pi$
$431$		−	4.00000i		−	0.192673i	0.995349	−	0.0963366i	$-0.0307125\pi$
$432$		0			0
$433$	19.0000	−	19.0000i	0.913082	−	0.913082i	−0.0834318	−	0.996513i	$-0.526588\pi$
$433$	19.0000	−	19.0000i	0.913082	−	0.913082i	0.996513	+	0.0834318i	$0.0265881\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	8.00000	+	8.00000i	0.382692	+	0.382692i
$438$		0			0
$439$	16.0000			0.763638			0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000			0.763638			0.381819	+	0.924237i	$0.375298\pi$
$440$		0			0
$441$	5.00000			0.238095
$442$		0			0
$443$	−22.0000	−	22.0000i	−1.04525	−	1.04525i	−0.998926	−	0.0463251i	$-0.985249\pi$
$443$	−22.0000	−	22.0000i	−1.04525	−	1.04525i	−0.0463251	−	0.998926i	$-0.514751\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	36.0000	−	36.0000i	1.70274	−	1.70274i
$448$		0			0
$449$		−	26.0000i		−	1.22702i	−0.789689	−	0.613508i	$-0.789758\pi$
$449$		−	26.0000i		−	1.22702i	0.789689	−	0.613508i	$-0.210242\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−24.0000	+	24.0000i	−1.12762	+	1.12762i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−15.0000	−	15.0000i	−0.701670	−	0.701670i	0.263099	−	0.964769i	$-0.415256\pi$
$457$	−15.0000	−	15.0000i	−0.701670	−	0.701670i	−0.964769	+	0.263099i	$0.915256\pi$
$458$		0			0
$459$	40.0000			1.86704
$460$		0			0
$461$	14.0000			0.652045			0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000			0.652045			0.326023	+	0.945362i	$0.394291\pi$
$462$		0			0
$463$	−22.0000	−	22.0000i	−1.02243	−	1.02243i	−0.999743	−	0.0226840i	$-0.992779\pi$
$463$	−22.0000	−	22.0000i	−1.02243	−	1.02243i	−0.0226840	−	0.999743i	$-0.507221\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2.00000	+	2.00000i	−0.0925490	+	0.0925490i	−0.751865	−	0.659317i	$-0.770846\pi$
$467$	−2.00000	+	2.00000i	−0.0925490	+	0.0925490i	0.659317	+	0.751865i	$0.270846\pi$
$468$		0			0
$469$		−	40.0000i		−	1.84703i
$470$		0			0
$471$			36.0000i			1.65879i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−35.0000	−	35.0000i	−1.60254	−	1.60254i
$478$		0			0
$479$	24.0000			1.09659			0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000			1.09659			0.548294	+	0.836286i	$0.315277\pi$
$480$		0			0
$481$	−2.00000			−0.0911922
$482$		0			0
$483$	−16.0000	−	16.0000i	−0.728025	−	0.728025i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−6.00000	+	6.00000i	−0.271886	+	0.271886i	−0.829859	−	0.557973i	$-0.811579\pi$
$487$	−6.00000	+	6.00000i	−0.271886	+	0.271886i	0.557973	+	0.829859i	$0.311579\pi$
$488$		0			0
$489$			8.00000i			0.361773i
$490$		0			0
$491$		−	16.0000i		−	0.722070i	−0.932552	−	0.361035i	$-0.882424\pi$
$491$		−	16.0000i		−	0.722070i	0.932552	−	0.361035i	$-0.117576\pi$
$492$		0			0
$493$	−20.0000	+	20.0000i	−0.900755	+	0.900755i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−24.0000	−	24.0000i	−1.07655	−	1.07655i
$498$		0			0
$499$	4.00000			0.179065			0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000			0.179065			0.0895323	+	0.995984i	$0.471463\pi$
$500$		0			0
$501$	8.00000			0.357414
$502$		0			0
$503$	10.0000	+	10.0000i	0.445878	+	0.445878i	0.893982	−	0.448104i	$-0.147900\pi$
$503$	10.0000	+	10.0000i	0.445878	+	0.445878i	−0.448104	+	0.893982i	$0.647900\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	22.0000	−	22.0000i	0.977054	−	0.977054i
$508$		0			0
$509$		−	36.0000i		−	1.59567i	−0.602875	−	0.797836i	$-0.705978\pi$
$509$		−	36.0000i		−	1.59567i	0.602875	−	0.797836i	$-0.294022\pi$
$510$		0			0
$511$		−	12.0000i		−	0.530849i
$512$		0			0
$513$	16.0000	−	16.0000i	0.706417	−	0.706417i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−52.0000			−2.28255
$520$		0			0
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$		0			0
$523$	−14.0000	−	14.0000i	−0.612177	−	0.612177i	0.331336	−	0.943513i	$-0.392501\pi$
$523$	−14.0000	−	14.0000i	−0.612177	−	0.612177i	−0.943513	+	0.331336i	$0.892501\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−20.0000	+	20.0000i	−0.871214	+	0.871214i
$528$		0			0
$529$		−	15.0000i		−	0.652174i
$530$		0			0
$531$			20.0000i			0.867926i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−24.0000	−	24.0000i	−1.03568	−	1.03568i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	30.0000			1.28980			0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000			1.28980			0.644900	+	0.764267i	$0.276899\pi$
$542$		0			0
$543$	−20.0000	−	20.0000i	−0.858282	−	0.858282i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	6.00000	−	6.00000i	0.256541	−	0.256541i	−0.567104	−	0.823646i	$-0.691936\pi$
$547$	6.00000	−	6.00000i	0.256541	−	0.256541i	0.823646	+	0.567104i	$0.191936\pi$
$548$		0			0
$549$			20.0000i			0.853579i
$550$		0			0
$551$			16.0000i			0.681623i
$552$		0			0
$553$	−32.0000	+	32.0000i	−1.36078	+	1.36078i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−15.0000	−	15.0000i	−0.635570	−	0.635570i	0.313889	−	0.949460i	$-0.398368\pi$
$557$	−15.0000	−	15.0000i	−0.635570	−	0.635570i	−0.949460	+	0.313889i	$0.898368\pi$
$558$		0			0
$559$	−12.0000			−0.507546
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6.00000	+	6.00000i	0.252870	+	0.252870i	0.822146	−	0.569276i	$-0.192777\pi$
$563$	6.00000	+	6.00000i	0.252870	+	0.252870i	−0.569276	+	0.822146i	$0.692777\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−2.00000	+	2.00000i	−0.0839921	+	0.0839921i
$568$		0			0
$569$		−	2.00000i		−	0.0838444i	−0.999121	−	0.0419222i	$-0.986652\pi$
$569$		−	2.00000i		−	0.0838444i	0.999121	−	0.0419222i	$-0.0133482\pi$
$570$		0			0
$571$		−	16.0000i		−	0.669579i	−0.942293	−	0.334790i	$-0.891335\pi$
$571$		−	16.0000i		−	0.669579i	0.942293	−	0.334790i	$-0.108665\pi$
$572$		0			0
$573$	40.0000	−	40.0000i	1.67102	−	1.67102i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−15.0000	−	15.0000i	−0.624458	−	0.624458i	0.322210	−	0.946668i	$-0.395574\pi$
$577$	−15.0000	−	15.0000i	−0.624458	−	0.624458i	−0.946668	+	0.322210i	$0.895574\pi$
$578$		0			0
$579$	−20.0000			−0.831172
$580$		0			0
$581$	8.00000			0.331896
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	14.0000	−	14.0000i	0.577842	−	0.577842i	−0.356466	−	0.934308i	$-0.616019\pi$
$587$	14.0000	−	14.0000i	0.577842	−	0.577842i	0.934308	+	0.356466i	$0.116019\pi$
$588$		0			0
$589$			16.0000i			0.659269i
$590$		0			0
$591$		−	20.0000i		−	0.822690i
$592$		0			0
$593$	−1.00000	+	1.00000i	−0.0410651	+	0.0410651i	−0.727341	−	0.686276i	$-0.759244\pi$
$593$	−1.00000	+	1.00000i	−0.0410651	+	0.0410651i	0.686276	+	0.727341i	$0.259244\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	48.0000	+	48.0000i	1.96451	+	1.96451i
$598$		0			0
$599$	8.00000			0.326871			0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000			0.326871			0.163436	+	0.986554i	$0.447742\pi$
$600$		0			0
$601$	−8.00000			−0.326327			−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000			−0.326327			−0.163163	+	0.986599i	$0.552170\pi$
$602$		0			0
$603$	50.0000	+	50.0000i	2.03616	+	2.03616i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−18.0000	+	18.0000i	−0.730597	+	0.730597i	−0.970738	−	0.240141i	$-0.922806\pi$
$607$	−18.0000	+	18.0000i	−0.730597	+	0.730597i	0.240141	+	0.970738i	$0.422806\pi$
$608$		0			0
$609$		−	32.0000i		−	1.29671i
$610$		0			0
$611$		−	4.00000i		−	0.161823i
$612$		0			0
$613$	−9.00000	+	9.00000i	−0.363507	+	0.363507i	−0.865102	−	0.501596i	$-0.832747\pi$
$613$	−9.00000	+	9.00000i	−0.363507	+	0.363507i	0.501596	+	0.865102i	$0.332747\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	29.0000	+	29.0000i	1.16750	+	1.16750i	0.982795	+	0.184701i	$0.0591318\pi$
$617$	29.0000	+	29.0000i	1.16750	+	1.16750i	0.184701	+	0.982795i	$0.440868\pi$
$618$		0			0
$619$	−28.0000			−1.12542			−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000			−1.12542			−0.562708	+	0.826656i	$0.690240\pi$
$620$		0			0
$621$	16.0000			0.642058
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$			10.0000i			0.398726i
$630$		0			0
$631$		−	4.00000i		−	0.159237i	−0.996825	−	0.0796187i	$-0.974630\pi$
$631$		−	4.00000i		−	0.159237i	0.996825	−	0.0796187i	$-0.0253703\pi$
$632$		0			0
$633$	32.0000	−	32.0000i	1.27189	−	1.27189i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.00000	+	1.00000i	0.0396214	+	0.0396214i
$638$		0			0
$639$	60.0000			2.37356
$640$		0			0
$641$	−48.0000			−1.89589			−0.947943	−	0.318440i	$-0.896841\pi$
$641$	−48.0000			−1.89589			−0.947943	+	0.318440i	$0.896841\pi$
$642$		0			0
$643$	10.0000	+	10.0000i	0.394362	+	0.394362i	0.876239	−	0.481877i	$-0.160045\pi$
$643$	10.0000	+	10.0000i	0.394362	+	0.394362i	−0.481877	+	0.876239i	$0.660045\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	10.0000	−	10.0000i	0.393141	−	0.393141i	−0.482665	−	0.875805i	$-0.660331\pi$
$647$	10.0000	−	10.0000i	0.393141	−	0.393141i	0.875805	+	0.482665i	$0.160331\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		−	32.0000i		−	1.25418i
$652$		0			0
$653$	1.00000	−	1.00000i	0.0391330	−	0.0391330i	−0.687270	−	0.726403i	$-0.741191\pi$
$653$	1.00000	−	1.00000i	0.0391330	−	0.0391330i	0.726403	+	0.687270i	$0.241191\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	15.0000	+	15.0000i	0.585206	+	0.585206i
$658$		0			0
$659$	20.0000			0.779089			0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000			0.779089			0.389545	+	0.921008i	$0.372632\pi$
$660$		0			0
$661$	−12.0000			−0.466746			−0.233373	−	0.972387i	$-0.574976\pi$
$661$	−12.0000			−0.466746			−0.233373	+	0.972387i	$0.574976\pi$
$662$		0			0
$663$	20.0000	+	20.0000i	0.776736	+	0.776736i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−8.00000	+	8.00000i	−0.309761	+	0.309761i
$668$		0			0
$669$			40.0000i			1.54649i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−5.00000	+	5.00000i	−0.192736	+	0.192736i	−0.796877	−	0.604141i	$-0.793516\pi$
$673$	−5.00000	+	5.00000i	−0.192736	+	0.192736i	0.604141	+	0.796877i	$0.293516\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3.00000	−	3.00000i	−0.115299	−	0.115299i	0.647103	−	0.762402i	$-0.275980\pi$
$677$	−3.00000	−	3.00000i	−0.115299	−	0.115299i	−0.762402	+	0.647103i	$0.775980\pi$
$678$		0			0
$679$	12.0000			0.460518
$680$		0			0
$681$	40.0000			1.53280
$682$		0			0
$683$	22.0000	+	22.0000i	0.841807	+	0.841807i	0.989094	−	0.147287i	$-0.0470541\pi$
$683$	22.0000	+	22.0000i	0.841807	+	0.841807i	−0.147287	+	0.989094i	$0.547054\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	40.0000	−	40.0000i	1.52610	−	1.52610i
$688$		0			0
$689$		−	14.0000i		−	0.533358i
$690$		0			0
$691$		−	32.0000i		−	1.21734i	−0.793424	−	0.608669i	$-0.791704\pi$
$691$		−	32.0000i		−	1.21734i	0.793424	−	0.608669i	$-0.208296\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−20.0000			−0.756469
$700$		0			0
$701$	−20.0000			−0.755390			−0.377695	−	0.925930i	$-0.623283\pi$
$701$	−20.0000			−0.755390			−0.377695	+	0.925930i	$0.623283\pi$
$702$		0			0
$703$	4.00000	+	4.00000i	0.150863	+	0.150863i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	12.0000	−	12.0000i	0.451306	−	0.451306i
$708$		0			0
$709$			12.0000i			0.450669i	0.974281	+	0.225335i	$0.0723476\pi$
$709$			12.0000i			0.450669i	−0.974281	+	0.225335i	$0.927652\pi$
$710$		0			0
$711$		−	80.0000i		−	3.00023i
$712$		0			0
$713$	−8.00000	+	8.00000i	−0.299602	+	0.299602i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	16.0000	+	16.0000i	0.597531	+	0.597531i
$718$		0			0
$719$	−8.00000			−0.298350			−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000			−0.298350			−0.149175	+	0.988811i	$0.547662\pi$
$720$		0			0
$721$	−24.0000			−0.893807
$722$		0			0
$723$	32.0000	+	32.0000i	1.19009	+	1.19009i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	18.0000	−	18.0000i	0.667583	−	0.667583i	−0.289573	−	0.957156i	$-0.593513\pi$
$727$	18.0000	−	18.0000i	0.667583	−	0.667583i	0.957156	+	0.289573i	$0.0935133\pi$
$728$		0			0
$729$			43.0000i			1.59259i
$730$		0			0
$731$			60.0000i			2.21918i
$732$		0			0
$733$	21.0000	−	21.0000i	0.775653	−	0.775653i	−0.203436	−	0.979088i	$-0.565211\pi$
$733$	21.0000	−	21.0000i	0.775653	−	0.775653i	0.979088	+	0.203436i	$0.0652108\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	44.0000			1.61857			0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000			1.61857			0.809283	+	0.587419i	$0.199856\pi$
$740$		0			0
$741$	16.0000			0.587775
$742$		0			0
$743$	30.0000	+	30.0000i	1.10059	+	1.10059i	0.994339	+	0.106254i	$0.0338857\pi$
$743$	30.0000	+	30.0000i	1.10059	+	1.10059i	0.106254	+	0.994339i	$0.466114\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−10.0000	+	10.0000i	−0.365881	+	0.365881i
$748$		0			0
$749$			24.0000i			0.876941i
$750$		0			0
$751$		−	44.0000i		−	1.60558i	−0.596260	−	0.802791i	$-0.703347\pi$
$751$		−	44.0000i		−	1.60558i	0.596260	−	0.802791i	$-0.296653\pi$
$752$		0			0
$753$	−48.0000	+	48.0000i	−1.74922	+	1.74922i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.00000	−	1.00000i	−0.0363456	−	0.0363456i	0.688700	−	0.725046i	$-0.258182\pi$
$757$	−1.00000	−	1.00000i	−0.0363456	−	0.0363456i	−0.725046	+	0.688700i	$0.758182\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−22.0000			−0.797499			−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000			−0.797499			−0.398750	+	0.917060i	$0.630556\pi$
$762$		0			0
$763$	−20.0000	−	20.0000i	−0.724049	−	0.724049i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.00000	+	4.00000i	−0.144432	+	0.144432i
$768$		0			0
$769$			40.0000i			1.44244i	0.692708	+	0.721218i	$0.256418\pi$
$769$			40.0000i			1.44244i	−0.692708	+	0.721218i	$0.743582\pi$
$770$		0			0
$771$			28.0000i			1.00840i
$772$		0			0
$773$	−1.00000	+	1.00000i	−0.0359675	+	0.0359675i	−0.724862	−	0.688894i	$-0.758096\pi$
$773$	−1.00000	+	1.00000i	−0.0359675	+	0.0359675i	0.688894	+	0.724862i	$0.258096\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−8.00000	−	8.00000i	−0.286998	−	0.286998i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	16.0000	+	16.0000i	0.571793	+	0.571793i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−30.0000	+	30.0000i	−1.06938	+	1.06938i	−0.0719783	+	0.997406i	$0.522931\pi$
$787$	−30.0000	+	30.0000i	−1.06938	+	1.06938i	−0.997406	+	0.0719783i	$0.977069\pi$
$788$		0			0
$789$		−	24.0000i		−	0.854423i
$790$		0			0
$791$		−	36.0000i		−	1.28001i
$792$		0			0
$793$	−4.00000	+	4.00000i	−0.142044	+	0.142044i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−29.0000	−	29.0000i	−1.02723	−	1.02723i	−0.999619	−	0.0276140i	$-0.991209\pi$
$797$	−29.0000	−	29.0000i	−1.02723	−	1.02723i	−0.0276140	−	0.999619i	$-0.508791\pi$
$798$		0			0
$799$	−20.0000			−0.707549
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	20.0000	−	20.0000i	0.704033	−	0.704033i
$808$		0			0
$809$			24.0000i			0.843795i	0.906644	+	0.421898i	$0.138636\pi$
$809$			24.0000i			0.843795i	−0.906644	+	0.421898i	$0.861364\pi$
$810$		0			0
$811$		−	24.0000i		−	0.842754i	−0.906886	−	0.421377i	$-0.861547\pi$
$811$		−	24.0000i		−	0.842754i	0.906886	−	0.421377i	$-0.138453\pi$
$812$		0			0
$813$	−40.0000	+	40.0000i	−1.40286	+	1.40286i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	24.0000	+	24.0000i	0.839654	+	0.839654i
$818$		0			0
$819$	−20.0000			−0.698857
$820$		0			0
$821$	28.0000			0.977207			0.488603	−	0.872506i	$-0.337507\pi$
$821$	28.0000			0.977207			0.488603	+	0.872506i	$0.337507\pi$
$822$		0			0
$823$	−30.0000	−	30.0000i	−1.04573	−	1.04573i	−0.998903	−	0.0468315i	$-0.985088\pi$
$823$	−30.0000	−	30.0000i	−1.04573	−	1.04573i	−0.0468315	−	0.998903i	$-0.514912\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−2.00000	+	2.00000i	−0.0695468	+	0.0695468i	−0.741025	−	0.671478i	$-0.765660\pi$
$827$	−2.00000	+	2.00000i	−0.0695468	+	0.0695468i	0.671478	+	0.741025i	$0.265660\pi$
$828$		0			0
$829$		−	6.00000i		−	0.208389i	−0.994557	−	0.104194i	$-0.966774\pi$
$829$		−	6.00000i		−	0.208389i	0.994557	−	0.104194i	$-0.0332264\pi$
$830$		0			0
$831$			36.0000i			1.24883i
$832$		0			0
$833$	5.00000	−	5.00000i	0.173240	−	0.173240i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	16.0000	+	16.0000i	0.553041	+	0.553041i
$838$		0			0
$839$	24.0000			0.828572			0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000			0.828572			0.414286	+	0.910147i	$0.364031\pi$
$840$		0			0
$841$	13.0000			0.448276
$842$		0			0
$843$	−16.0000	−	16.0000i	−0.551069	−	0.551069i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	22.0000	−	22.0000i	0.755929	−	0.755929i
$848$		0			0
$849$			24.0000i			0.823678i
$850$		0			0
$851$			4.00000i			0.137118i
$852$		0			0
$853$	15.0000	−	15.0000i	0.513590	−	0.513590i	−0.402034	−	0.915625i	$-0.631697\pi$
$853$	15.0000	−	15.0000i	0.513590	−	0.513590i	0.915625	+	0.402034i	$0.131697\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	7.00000	+	7.00000i	0.239115	+	0.239115i	0.816484	−	0.577368i	$-0.195920\pi$
$857$	7.00000	+	7.00000i	0.239115	+	0.239115i	−0.577368	+	0.816484i	$0.695920\pi$
$858$		0			0
$859$	−28.0000			−0.955348			−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000			−0.955348			−0.477674	+	0.878537i	$0.658520\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	34.0000	+	34.0000i	1.15737	+	1.15737i	0.985038	+	0.172335i	$0.0551312\pi$
$863$	34.0000	+	34.0000i	1.15737	+	1.15737i	0.172335	+	0.985038i	$0.444869\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	66.0000	−	66.0000i	2.24148	−	2.24148i
$868$		0			0
$869$		0			0
$870$		0			0
$871$			20.0000i			0.677674i
$872$		0			0
$873$	−15.0000	+	15.0000i	−0.507673	+	0.507673i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	15.0000	+	15.0000i	0.506514	+	0.506514i	0.913455	−	0.406941i	$-0.133404\pi$
$877$	15.0000	+	15.0000i	0.506514	+	0.506514i	−0.406941	+	0.913455i	$0.633404\pi$
$878$		0			0
$879$	20.0000			0.674583
$880$		0			0
$881$	−40.0000			−1.34763			−0.673817	−	0.738898i	$-0.735346\pi$
$881$	−40.0000			−1.34763			−0.673817	+	0.738898i	$0.735346\pi$
$882$		0			0
$883$	−10.0000	−	10.0000i	−0.336527	−	0.336527i	0.518532	−	0.855058i	$-0.326479\pi$
$883$	−10.0000	−	10.0000i	−0.336527	−	0.336527i	−0.855058	+	0.518532i	$0.826479\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	10.0000	−	10.0000i	0.335767	−	0.335767i	−0.519004	−	0.854772i	$-0.673697\pi$
$887$	10.0000	−	10.0000i	0.335767	−	0.335767i	0.854772	+	0.519004i	$0.173697\pi$
$888$		0			0
$889$		−	40.0000i		−	1.34156i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−8.00000	+	8.00000i	−0.267710	+	0.267710i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	8.00000	+	8.00000i	0.267112	+	0.267112i
$898$		0			0
$899$	−16.0000			−0.533630
$900$		0			0
$901$	−70.0000			−2.33204
$902$		0			0
$903$	−48.0000	−	48.0000i	−1.59734	−	1.59734i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−6.00000	+	6.00000i	−0.199227	+	0.199227i	−0.799668	−	0.600442i	$-0.794991\pi$
$907$	−6.00000	+	6.00000i	−0.199227	+	0.199227i	0.600442	+	0.799668i	$0.294991\pi$
$908$		0			0
$909$			30.0000i			0.995037i
$910$		0			0
$911$			4.00000i			0.132526i	0.997802	+	0.0662630i	$0.0211076\pi$
$911$			4.00000i			0.132526i	−0.997802	+	0.0662630i	$0.978892\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	16.0000	+	16.0000i	0.528367	+	0.528367i
$918$		0			0
$919$	24.0000			0.791687			0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000			0.791687			0.395843	+	0.918318i	$0.370452\pi$
$920$		0			0
$921$	−40.0000			−1.31804
$922$		0			0
$923$	12.0000	+	12.0000i	0.394985	+	0.394985i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	30.0000	−	30.0000i	0.985329	−	0.985329i
$928$		0			0
$929$		−	6.00000i		−	0.196854i	−0.995144	−	0.0984268i	$-0.968619\pi$
$929$		−	6.00000i		−	0.196854i	0.995144	−	0.0984268i	$-0.0313810\pi$
$930$		0			0
$931$		−	4.00000i		−	0.131095i
$932$		0			0
$933$	56.0000	−	56.0000i	1.83336	−	1.83336i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−27.0000	−	27.0000i	−0.882052	−	0.882052i	0.111691	−	0.993743i	$-0.464373\pi$
$937$	−27.0000	−	27.0000i	−0.882052	−	0.882052i	−0.993743	+	0.111691i	$0.964373\pi$
$938$		0			0
$939$	60.0000			1.95803
$940$		0			0
$941$	−46.0000			−1.49956			−0.749779	−	0.661689i	$-0.769840\pi$
$941$	−46.0000			−1.49956			−0.749779	+	0.661689i	$0.769840\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	10.0000	−	10.0000i	0.324956	−	0.324956i	−0.525708	−	0.850665i	$-0.676200\pi$
$947$	10.0000	−	10.0000i	0.324956	−	0.324956i	0.850665	+	0.525708i	$0.176200\pi$
$948$		0			0
$949$			6.00000i			0.194768i
$950$		0			0
$951$		−	44.0000i		−	1.42680i
$952$		0			0
$953$	9.00000	−	9.00000i	0.291539	−	0.291539i	−0.546149	−	0.837688i	$-0.683907\pi$
$953$	9.00000	−	9.00000i	0.291539	−	0.291539i	0.837688	+	0.546149i	$0.183907\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.00000			−0.129167
$960$		0			0
$961$	15.0000			0.483871
$962$		0			0
$963$	−30.0000	−	30.0000i	−0.966736	−	0.966736i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−10.0000	+	10.0000i	−0.321578	+	0.321578i	−0.849372	−	0.527794i	$-0.823019\pi$
$967$	−10.0000	+	10.0000i	−0.321578	+	0.321578i	0.527794	+	0.849372i	$0.323019\pi$
$968$		0			0
$969$		−	80.0000i		−	2.56997i
$970$		0			0
$971$			8.00000i			0.256732i	0.991727	+	0.128366i	$0.0409733\pi$
$971$			8.00000i			0.256732i	−0.991727	+	0.128366i	$0.959027\pi$
$972$		0			0
$973$	−24.0000	+	24.0000i	−0.769405	+	0.769405i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−21.0000	−	21.0000i	−0.671850	−	0.671850i	0.286293	−	0.958142i	$-0.407577\pi$
$977$	−21.0000	−	21.0000i	−0.671850	−	0.671850i	−0.958142	+	0.286293i	$0.907577\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	50.0000			1.59638
$982$		0			0
$983$	−34.0000	−	34.0000i	−1.08443	−	1.08443i	−0.996090	−	0.0883413i	$-0.971843\pi$
$983$	−34.0000	−	34.0000i	−1.08443	−	1.08443i	−0.0883413	−	0.996090i	$-0.528157\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	16.0000	−	16.0000i	0.509286	−	0.509286i
$988$		0			0
$989$			24.0000i			0.763156i
$990$		0			0
$991$		−	4.00000i		−	0.127064i	−0.997980	−	0.0635321i	$-0.979763\pi$
$991$		−	4.00000i		−	0.127064i	0.997980	−	0.0635321i	$-0.0202365\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−21.0000	−	21.0000i	−0.665077	−	0.665077i	0.291496	−	0.956572i	$-0.405847\pi$
$997$	−21.0000	−	21.0000i	−0.665077	−	0.665077i	−0.956572	+	0.291496i	$0.905847\pi$
$998$		0			0
$999$	8.00000			0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.n.a.1407.1		2
4.3	odd	2		1600.2.n.n.1407.1		2
5.2	odd	4		320.2.n.a.63.1		2
5.3	odd	4		1600.2.n.n.1343.1		2
5.4	even	2		320.2.n.h.127.1		2
8.3	odd	2		800.2.n.a.607.1		2
8.5	even	2		800.2.n.j.607.1		2
20.3	even	4	inner	1600.2.n.a.1343.1		2
20.7	even	4		320.2.n.h.63.1		2
20.19	odd	2		320.2.n.a.127.1		2
40.3	even	4		800.2.n.j.543.1		2
40.13	odd	4		800.2.n.a.543.1		2
40.19	odd	2		160.2.n.f.127.1	yes	2
40.27	even	4		160.2.n.a.63.1	✓	2
40.29	even	2		160.2.n.a.127.1	yes	2
40.37	odd	4		160.2.n.f.63.1	yes	2
80.19	odd	4		1280.2.o.o.127.1		2
80.27	even	4		1280.2.o.a.383.1		2
80.29	even	4		1280.2.o.a.127.1		2
80.37	odd	4		1280.2.o.o.383.1		2
80.59	odd	4		1280.2.o.b.127.1		2
80.67	even	4		1280.2.o.p.383.1		2
80.69	even	4		1280.2.o.p.127.1		2
80.77	odd	4		1280.2.o.b.383.1		2
120.29	odd	2		1440.2.x.i.127.1		2
120.59	even	2		1440.2.x.j.127.1		2
120.77	even	4		1440.2.x.j.703.1		2
120.107	odd	4		1440.2.x.i.703.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.2.n.a.63.1	✓	2	40.27	even	4
160.2.n.a.127.1	yes	2	40.29	even	2
160.2.n.f.63.1	yes	2	40.37	odd	4
160.2.n.f.127.1	yes	2	40.19	odd	2
320.2.n.a.63.1		2	5.2	odd	4
320.2.n.a.127.1		2	20.19	odd	2
320.2.n.h.63.1		2	20.7	even	4
320.2.n.h.127.1		2	5.4	even	2
800.2.n.a.543.1		2	40.13	odd	4
800.2.n.a.607.1		2	8.3	odd	2
800.2.n.j.543.1		2	40.3	even	4
800.2.n.j.607.1		2	8.5	even	2
1280.2.o.a.127.1		2	80.29	even	4
1280.2.o.a.383.1		2	80.27	even	4
1280.2.o.b.127.1		2	80.59	odd	4
1280.2.o.b.383.1		2	80.77	odd	4
1280.2.o.o.127.1		2	80.19	odd	4
1280.2.o.o.383.1		2	80.37	odd	4
1280.2.o.p.127.1		2	80.69	even	4
1280.2.o.p.383.1		2	80.67	even	4
1440.2.x.i.127.1		2	120.29	odd	2
1440.2.x.i.703.1		2	120.107	odd	4
1440.2.x.j.127.1		2	120.59	even	2
1440.2.x.j.703.1		2	120.77	even	4
1600.2.n.a.1343.1		2	20.3	even	4	inner
1600.2.n.a.1407.1		2	1.1	even	1	trivial
1600.2.n.n.1343.1		2	5.3	odd	4
1600.2.n.n.1407.1		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.n (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 2.00000i) q^{3} +(2.00000 - 2.00000i) q^{7} +5.00000i q^{9} +O(q^{10})\)	\(q+(-2.00000 - 2.00000i) q^{3} +(2.00000 - 2.00000i) q^{7} +5.00000i q^{9} +(-1.00000 + 1.00000i) q^{13} +(5.00000 + 5.00000i) q^{17} +4.00000 q^{19} -8.00000 q^{21} +(2.00000 + 2.00000i) q^{23} +(4.00000 - 4.00000i) q^{27} +4.00000i q^{29} +4.00000i q^{31} +(1.00000 + 1.00000i) q^{37} +4.00000 q^{39} +(6.00000 + 6.00000i) q^{43} +(-2.00000 + 2.00000i) q^{47} -1.00000i q^{49} -20.0000i q^{51} +(-7.00000 + 7.00000i) q^{53} +(-8.00000 - 8.00000i) q^{57} +4.00000 q^{59} +4.00000 q^{61} +(10.0000 + 10.0000i) q^{63} +(10.0000 - 10.0000i) q^{67} -8.00000i q^{69} -12.0000i q^{71} +(3.00000 - 3.00000i) q^{73} -16.0000 q^{79} -1.00000 q^{81} +(2.00000 + 2.00000i) q^{83} +(8.00000 - 8.00000i) q^{87} +4.00000i q^{91} +(8.00000 - 8.00000i) q^{93} +(3.00000 + 3.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} + 4 q^{7}+O(q^{10}) \) 2 * q - 4 * q^3 + 4 * q^7	\( 2 q - 4 q^{3} + 4 q^{7} - 2 q^{13} + 10 q^{17} + 8 q^{19} - 16 q^{21} + 4 q^{23} + 8 q^{27} + 2 q^{37} + 8 q^{39} + 12 q^{43} - 4 q^{47} - 14 q^{53} - 16 q^{57} + 8 q^{59} + 8 q^{61} + 20 q^{63} + 20 q^{67} + 6 q^{73} - 32 q^{79} - 2 q^{81} + 4 q^{83} + 16 q^{87} + 16 q^{93} + 6 q^{97}+O(q^{100}) \) 2 * q - 4 * q^3 + 4 * q^7 - 2 * q^13 + 10 * q^17 + 8 * q^19 - 16 * q^21 + 4 * q^23 + 8 * q^27 + 2 * q^37 + 8 * q^39 + 12 * q^43 - 4 * q^47 - 14 * q^53 - 16 * q^57 + 8 * q^59 + 8 * q^61 + 20 * q^63 + 20 * q^67 + 6 * q^73 - 32 * q^79 - 2 * q^81 + 4 * q^83 + 16 * q^87 + 16 * q^93 + 6 * q^97

Introduction

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