LMFDB - Embedded newform 572.2.b.a.571.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [572,2,Mod(571,572)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(572, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

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

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

chi := DirichletCharacter("572.571");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 572 = 2^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	572.b (of order $2$, degree $1$, 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:	$4.56744299562$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 12x^{2} + 49 $ x^4 - 12*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			571.1
Root			$-2.54951 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	572.571
Dual form			572.2.b.a.571.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} +1.41421i q^{7} +2.82843i q^{8} +3.00000 q^{9} +O(q^{10})$	$q-1.41421i q^{2} -2.00000 q^{4} +1.41421i q^{7} +2.82843i q^{8} +3.00000 q^{9} +(-2.54951 - 2.12132i) q^{11} -3.60555i q^{13} +2.00000 q^{14} +4.00000 q^{16} -7.21110i q^{17} -4.24264i q^{18} -7.07107i q^{19} +(-3.00000 + 3.60555i) q^{22} +5.00000 q^{25} -5.09902 q^{26} -2.82843i q^{28} +7.21110i q^{29} -5.09902 q^{31} -5.65685i q^{32} -10.1980 q^{34} -6.00000 q^{36} -10.0000 q^{38} +(5.09902 + 4.24264i) q^{44} +5.09902 q^{47} +5.00000 q^{49} -7.07107i q^{50} +7.21110i q^{52} -2.00000 q^{53} -4.00000 q^{56} +10.1980 q^{58} -15.2971 q^{59} -14.4222i q^{61} +7.21110i q^{62} +4.24264i q^{63} -8.00000 q^{64} +5.09902 q^{67} +14.4222i q^{68} +15.2971 q^{71} +8.48528i q^{72} +14.1421i q^{76} +(3.00000 - 3.60555i) q^{77} +9.00000 q^{81} +9.89949i q^{83} +(6.00000 - 7.21110i) q^{88} +5.09902 q^{91} -7.21110i q^{94} -7.07107i q^{98} +(-7.64853 - 6.36396i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} + 12 q^{9}+O(q^{10}) $ 4 * q - 8 * q^4 + 12 * q^9	$ 4 q - 8 q^{4} + 12 q^{9} + 8 q^{14} + 16 q^{16} - 12 q^{22} + 20 q^{25} - 24 q^{36} - 40 q^{38} + 20 q^{49} - 8 q^{53} - 16 q^{56} - 32 q^{64} + 12 q^{77} + 36 q^{81} + 24 q^{88}+O(q^{100}) $ 4 * q - 8 * q^4 + 12 * q^9 + 8 * q^14 + 16 * q^16 - 12 * q^22 + 20 * q^25 - 24 * q^36 - 40 * q^38 + 20 * q^49 - 8 * q^53 - 16 * q^56 - 32 * q^64 + 12 * q^77 + 36 * q^81 + 24 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/572\mathbb{Z}\right)^\times$.

$n$	$287$	$353$	$365$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$		−	1.41421i		−	1.00000i
$2$		−	1.41421i		−	1.00000i
$3$		0			0		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$	−2.00000			−1.00000
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$			1.41421i			0.534522i	0.963624	+	0.267261i	$0.0861187\pi$
$7$			1.41421i			0.534522i	−0.963624	+	0.267261i	$0.913881\pi$
$8$			2.82843i			1.00000i
$9$	3.00000			1.00000
$10$		0			0
$11$	−2.54951	−	2.12132i	−0.768706	−	0.639602i
$11$	−2.54951	−	2.12132i	−0.768706	−	0.639602i
$12$		0			0
$13$		−	3.60555i		−	1.00000i
$13$		−	3.60555i		−	1.00000i
$14$	2.00000			0.534522
$15$		0			0
$16$	4.00000			1.00000
$17$		−	7.21110i		−	1.74895i	−0.485071	−	0.874475i	$-0.661206\pi$
$17$		−	7.21110i		−	1.74895i	0.485071	−	0.874475i	$-0.338794\pi$
$18$		−	4.24264i		−	1.00000i
$19$		−	7.07107i		−	1.62221i	−0.584898	−	0.811107i	$-0.698865\pi$
$19$		−	7.07107i		−	1.62221i	0.584898	−	0.811107i	$-0.301135\pi$
$20$		0			0
$21$		0			0
$22$	−3.00000	+	3.60555i	−0.639602	+	0.768706i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$	−5.09902			−1.00000
$27$		0			0
$28$		−	2.82843i		−	0.534522i
$29$			7.21110i			1.33907i	0.742781	+	0.669534i	$0.233506\pi$
$29$			7.21110i			1.33907i	−0.742781	+	0.669534i	$0.766494\pi$
$30$		0			0
$31$	−5.09902			−0.915811			−0.457905	−	0.889001i	$-0.651400\pi$
$31$	−5.09902			−0.915811			−0.457905	+	0.889001i	$0.651400\pi$
$32$		−	5.65685i		−	1.00000i
$33$		0			0
$34$	−10.1980			−1.74895
$35$		0			0
$36$	−6.00000			−1.00000
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−10.0000			−1.62221
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	5.09902	+	4.24264i	0.768706	+	0.639602i
$45$		0			0
$46$		0			0
$47$	5.09902			0.743768			0.371884	−	0.928279i	$-0.378712\pi$
$47$	5.09902			0.743768			0.371884	+	0.928279i	$0.378712\pi$
$48$		0			0
$49$	5.00000			0.714286
$50$		−	7.07107i		−	1.00000i
$51$		0			0
$52$			7.21110i			1.00000i
$53$	−2.00000			−0.274721			−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000			−0.274721			−0.137361	+	0.990521i	$0.543862\pi$
$54$		0			0
$55$		0			0
$56$	−4.00000			−0.534522
$57$		0			0
$58$	10.1980			1.33907
$59$	−15.2971			−1.99151			−0.995754	−	0.0920575i	$-0.970656\pi$
$59$	−15.2971			−1.99151			−0.995754	+	0.0920575i	$0.970656\pi$
$60$		0			0
$61$		−	14.4222i		−	1.84657i	−0.384111	−	0.923287i	$-0.625492\pi$
$61$		−	14.4222i		−	1.84657i	0.384111	−	0.923287i	$-0.374508\pi$
$62$			7.21110i			0.915811i
$63$			4.24264i			0.534522i
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$	5.09902			0.622944			0.311472	−	0.950255i	$-0.399178\pi$
$67$	5.09902			0.622944			0.311472	+	0.950255i	$0.399178\pi$
$68$			14.4222i			1.74895i
$69$		0			0
$70$		0			0
$71$	15.2971			1.81543			0.907713	−	0.419591i	$-0.137826\pi$
$71$	15.2971			1.81543			0.907713	+	0.419591i	$0.137826\pi$
$72$			8.48528i			1.00000i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$		0			0
$76$			14.1421i			1.62221i
$77$	3.00000	−	3.60555i	0.341882	−	0.410891i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$			9.89949i			1.08661i	0.839535	+	0.543305i	$0.182827\pi$
$83$			9.89949i			1.08661i	−0.839535	+	0.543305i	$0.817173\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	6.00000	−	7.21110i	0.639602	−	0.768706i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	5.09902			0.534522
$92$		0			0
$93$		0			0
$94$		−	7.21110i		−	0.743768i
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		−	7.07107i		−	0.714286i
$99$	−7.64853	−	6.36396i	−0.768706	−	0.639602i
$100$	−10.0000			−1.00000
$101$			14.4222i			1.43506i	0.696526	+	0.717532i	$0.254728\pi$
$101$			14.4222i			1.43506i	−0.696526	+	0.717532i	$0.745272\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	10.1980			1.00000
$105$		0			0
$106$			2.82843i			0.274721i
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$			5.65685i			0.534522i
$113$	−20.0000			−1.88144			−0.940721	−	0.339182i	$-0.889850\pi$
$113$	−20.0000			−1.88144			−0.940721	+	0.339182i	$0.889850\pi$
$114$		0			0
$115$		0			0
$116$		−	14.4222i		−	1.33907i
$117$		−	10.8167i		−	1.00000i
$118$			21.6333i			1.99151i
$119$	10.1980			0.934853
$120$		0			0
$121$	2.00000	+	10.8167i	0.181818	+	0.983332i
$122$	−20.3961			−1.84657
$123$		0			0
$124$	10.1980			0.915811
$125$		0			0
$126$	6.00000			0.534522
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$			11.3137i			1.00000i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	10.0000			0.867110
$134$		−	7.21110i		−	0.622944i
$135$		0			0
$136$	20.3961			1.74895
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		−	21.6333i		−	1.81543i
$143$	−7.64853	+	9.19239i	−0.639602	+	0.768706i
$144$	12.0000			1.00000
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$			24.0416i			1.95648i	0.207476	+	0.978240i	$0.433475\pi$
$151$			24.0416i			1.95648i	−0.207476	+	0.978240i	$0.566525\pi$
$152$	20.0000			1.62221
$153$		−	21.6333i		−	1.74895i
$154$	−5.09902	−	4.24264i	−0.410891	−	0.341882i
$155$		0			0
$156$		0			0
$157$	24.0000			1.91541			0.957704	−	0.287754i	$-0.0929087\pi$
$157$	24.0000			1.91541			0.957704	+	0.287754i	$0.0929087\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		−	12.7279i		−	1.00000i
$163$	25.4951			1.99693			0.998465	−	0.0553849i	$-0.0176386\pi$
$163$	25.4951			1.99693			0.998465	+	0.0553849i	$0.0176386\pi$
$164$		0			0
$165$		0			0
$166$	14.0000			1.08661
$167$		−	4.24264i		−	0.328305i	−0.986435	−	0.164153i	$-0.947511\pi$
$167$		−	4.24264i		−	0.328305i	0.986435	−	0.164153i	$-0.0524890\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		−	21.2132i		−	1.62221i
$172$		0			0
$173$			14.4222i			1.09650i	0.836315	+	0.548250i	$0.184706\pi$
$173$			14.4222i			1.09650i	−0.836315	+	0.548250i	$0.815294\pi$
$174$		0			0
$175$			7.07107i			0.534522i
$176$	−10.1980	−	8.48528i	−0.768706	−	0.639602i
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	−16.0000			−1.18927			−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000			−1.18927			−0.594635	+	0.803996i	$0.702704\pi$
$182$		−	7.21110i		−	0.534522i
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−15.2971	+	18.3848i	−1.11863	+	1.34443i
$188$	−10.1980			−0.743768
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$	−10.0000			−0.714286
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	−9.00000	+	10.8167i	−0.639602	+	0.768706i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$			14.1421i			1.00000i
$201$		0			0
$202$	20.3961			1.43506
$203$	−10.1980			−0.715762
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		−	14.4222i		−	1.00000i
$209$	−15.0000	+	18.0278i	−1.03757	+	1.24701i
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$	4.00000			0.274721
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	7.21110i		−	0.489522i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−26.0000			−1.74895
$222$		0			0
$223$	25.4951			1.70728			0.853639	−	0.520865i	$-0.174391\pi$
$223$	25.4951			1.70728			0.853639	+	0.520865i	$0.174391\pi$
$224$	8.00000			0.534522
$225$	15.0000			1.00000
$226$			28.2843i			1.88144i
$227$		−	29.6985i		−	1.97116i	−0.169217	−	0.985579i	$-0.554124\pi$
$227$		−	29.6985i		−	1.97116i	0.169217	−	0.985579i	$-0.445876\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$	−20.3961			−1.33907
$233$			28.8444i			1.88966i	0.327561	+	0.944830i	$0.393773\pi$
$233$			28.8444i			1.88966i	−0.327561	+	0.944830i	$0.606227\pi$
$234$	−15.2971			−1.00000
$235$		0			0
$236$	30.5941			1.99151
$237$		0			0
$238$		−	14.4222i		−	0.934853i
$239$		−	26.8701i		−	1.73808i	−0.494742	−	0.869040i	$-0.664738\pi$
$239$		−	26.8701i		−	1.73808i	0.494742	−	0.869040i	$-0.335262\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	15.2971	−	2.82843i	0.983332	−	0.181818i
$243$		0			0
$244$			28.8444i			1.84657i
$245$		0			0
$246$		0			0
$247$	−25.4951			−1.62221
$248$		−	14.4222i		−	0.915811i
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		−	8.48528i		−	0.534522i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	−14.0000			−0.873296			−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000			−0.873296			−0.436648	+	0.899632i	$0.643834\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$			21.6333i			1.33907i
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		−	14.1421i		−	0.867110i
$267$		0			0
$268$	−10.1980			−0.622944
$269$	−32.0000			−1.95107			−0.975537	−	0.219834i	$-0.929448\pi$
$269$	−32.0000			−1.95107			−0.975537	+	0.219834i	$0.929448\pi$
$270$		0			0
$271$			32.5269i			1.97587i	0.154872	+	0.987935i	$0.450504\pi$
$271$			32.5269i			1.97587i	−0.154872	+	0.987935i	$0.549496\pi$
$272$		−	28.8444i		−	1.74895i
$273$		0			0
$274$		0			0
$275$	−12.7475	−	10.6066i	−0.768706	−	0.639602i
$276$		0			0
$277$			14.4222i			0.866546i	0.901263	+	0.433273i	$0.142641\pi$
$277$			14.4222i			0.866546i	−0.901263	+	0.433273i	$0.857359\pi$
$278$		0			0
$279$	−15.2971			−0.915811
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$	−30.5941			−1.81543
$285$		0			0
$286$	13.0000	+	10.8167i	0.768706	+	0.639602i
$287$		0			0
$288$		−	16.9706i		−	1.00000i
$289$	−35.0000			−2.05882
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	34.0000			1.95648
$303$		0			0
$304$		−	28.2843i		−	1.62221i
$305$		0			0
$306$	−30.5941			−1.74895
$307$		−	24.0416i		−	1.37213i	−0.727541	−	0.686064i	$-0.759337\pi$
$307$		−	24.0416i		−	1.37213i	0.727541	−	0.686064i	$-0.240663\pi$
$308$	−6.00000	+	7.21110i	−0.341882	+	0.410891i
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	28.0000			1.58265			0.791327	−	0.611393i	$-0.209391\pi$
$313$	28.0000			1.58265			0.791327	+	0.611393i	$0.209391\pi$
$314$		−	33.9411i		−	1.91541i
$315$		0			0
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$	15.2971	−	18.3848i	0.856471	−	1.02935i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−50.9902			−2.83717
$324$	−18.0000			−1.00000
$325$		−	18.0278i		−	1.00000i
$326$		−	36.0555i		−	1.99693i
$327$		0			0
$328$		0			0
$329$			7.21110i			0.397561i
$330$		0			0
$331$	35.6931			1.96187			0.980936	−	0.194331i	$-0.0622534\pi$
$331$	35.6931			1.96187			0.980936	+	0.194331i	$0.0622534\pi$
$332$		−	19.7990i		−	1.08661i
$333$		0			0
$334$	−6.00000			−0.328305
$335$		0			0
$336$		0			0
$337$			7.21110i			0.392814i	0.980522	+	0.196407i	$0.0629273\pi$
$337$			7.21110i			0.392814i	−0.980522	+	0.196407i	$0.937073\pi$
$338$			18.3848i			1.00000i
$339$		0			0
$340$		0			0
$341$	13.0000	+	10.8167i	0.703989	+	0.585755i
$342$	−30.0000			−1.62221
$343$			16.9706i			0.916324i
$344$		0			0
$345$		0			0
$346$	20.3961			1.09650
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	10.0000			0.534522
$351$		0			0
$352$	−12.0000	+	14.4222i	−0.639602	+	0.768706i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	12.7279i		−	0.671754i	−0.941906	−	0.335877i	$-0.890967\pi$
$359$		−	12.7279i		−	0.671754i	0.941906	−	0.335877i	$-0.109033\pi$
$360$		0			0
$361$	−31.0000			−1.63158
$362$			22.6274i			1.18927i
$363$		0			0
$364$	−10.1980			−0.534522
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	2.82843i		−	0.146845i
$372$		0			0
$373$		−	21.6333i		−	1.12013i	−0.828449	−	0.560065i	$-0.810776\pi$
$373$		−	21.6333i		−	1.12013i	0.828449	−	0.560065i	$-0.189224\pi$
$374$	26.0000	+	21.6333i	1.34443	+	1.11863i
$375$		0			0
$376$			14.4222i			0.743768i
$377$	26.0000			1.33907
$378$		0			0
$379$	−35.6931			−1.83343			−0.916717	−	0.399538i	$-0.869171\pi$
$379$	−35.6931			−1.83343			−0.916717	+	0.399538i	$0.869171\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	25.4951			1.30274			0.651369	−	0.758761i	$-0.274195\pi$
$383$	25.4951			1.30274			0.651369	+	0.758761i	$0.274195\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	16.0000			0.811232			0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000			0.811232			0.405616	+	0.914044i	$0.367057\pi$
$390$		0			0
$391$		0			0
$392$			14.1421i			0.714286i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	15.2971	+	12.7279i	0.768706	+	0.639602i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	20.0000			1.00000
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$			18.3848i			0.915811i
$404$		−	28.8444i		−	1.43506i
$405$		0			0
$406$			14.4222i			0.715762i
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	21.6333i		−	1.06451i
$414$		0			0
$415$		0			0
$416$	−20.3961			−1.00000
$417$		0			0
$418$	25.4951	+	21.2132i	1.24701	+	1.03757i
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$	15.2971			0.743768
$424$		−	5.65685i		−	0.274721i
$425$		−	36.0555i		−	1.74895i
$426$		0			0
$427$	20.3961			0.987035
$428$		0			0
$429$		0			0
$430$		0			0
$431$			21.2132i			1.02180i	0.859639	+	0.510902i	$0.170689\pi$
$431$			21.2132i			1.02180i	−0.859639	+	0.510902i	$0.829311\pi$
$432$		0			0
$433$	30.0000			1.44171			0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000			1.44171			0.720854	+	0.693087i	$0.243750\pi$
$434$	−10.1980			−0.489522
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	15.0000			0.714286
$442$			36.7696i			1.74895i
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		−	36.0555i		−	1.70728i
$447$		0			0
$448$		−	11.3137i		−	0.534522i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		−	21.2132i		−	1.00000i
$451$		0			0
$452$	40.0000			1.88144
$453$		0			0
$454$	−42.0000			−1.97116
$455$		0			0
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	35.6931			1.65880			0.829400	−	0.558655i	$-0.188682\pi$
$463$	35.6931			1.65880			0.829400	+	0.558655i	$0.188682\pi$
$464$			28.8444i			1.33907i
$465$		0			0
$466$	40.7922			1.88966
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$			21.6333i			1.00000i
$469$			7.21110i			0.332978i
$470$		0			0
$471$		0			0
$472$		−	43.2666i		−	1.99151i
$473$		0			0
$474$		0			0
$475$		−	35.3553i		−	1.62221i
$476$	−20.3961			−0.934853
$477$	−6.00000			−0.274721
$478$	−38.0000			−1.73808
$479$			41.0122i			1.87389i	0.349470	+	0.936947i	$0.386362\pi$
$479$			41.0122i			1.87389i	−0.349470	+	0.936947i	$0.613638\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−4.00000	−	21.6333i	−0.181818	−	0.983332i
$485$		0			0
$486$		0			0
$487$	5.09902			0.231059			0.115529	−	0.993304i	$-0.463144\pi$
$487$	5.09902			0.231059			0.115529	+	0.993304i	$0.463144\pi$
$488$	40.7922			1.84657
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$	52.0000			2.34196
$494$			36.0555i			1.62221i
$495$		0			0
$496$	−20.3961			−0.915811
$497$			21.6333i			0.970386i
$498$		0			0
$499$	−35.6931			−1.59784			−0.798922	−	0.601434i	$-0.794596\pi$
$499$	−35.6931			−1.59784			−0.798922	+	0.601434i	$0.794596\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	−12.0000			−0.534522
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$		−	22.6274i		−	1.00000i
$513$		0			0
$514$			19.7990i			0.873296i
$515$		0			0
$516$		0			0
$517$	−13.0000	−	10.8167i	−0.571739	−	0.475716i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−28.0000			−1.22670			−0.613351	−	0.789810i	$-0.710179\pi$
$521$	−28.0000			−1.22670			−0.613351	+	0.789810i	$0.710179\pi$
$522$	30.5941			1.33907
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			36.7696i			1.60171i
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$	−45.8912			−1.99151
$532$	−20.0000			−0.867110
$533$		0			0
$534$		0			0
$535$		0			0
$536$			14.4222i			0.622944i
$537$		0			0
$538$			45.2548i			1.95107i
$539$	−12.7475	−	10.6066i	−0.549076	−	0.456859i
$540$		0			0
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$	46.0000			1.97587
$543$		0			0
$544$	−40.7922			−1.74895
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		0			0
$549$		−	43.2666i		−	1.84657i
$550$	−15.0000	+	18.0278i	−0.639602	+	0.768706i
$551$	50.9902			2.17226
$552$		0			0
$553$		0			0
$554$	20.3961			0.866546
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$			21.6333i			0.915811i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$			12.7279i			0.534522i
$568$			43.2666i			1.81543i
$569$		−	28.8444i		−	1.20922i	−0.796521	−	0.604610i	$-0.793329\pi$
$569$		−	28.8444i		−	1.20922i	0.796521	−	0.604610i	$-0.206671\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$	15.2971	−	18.3848i	0.639602	−	0.768706i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−24.0000			−1.00000
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$			49.4975i			2.05882i
$579$		0			0
$580$		0			0
$581$	−14.0000			−0.580818
$582$		0			0
$583$	5.09902	+	4.24264i	0.211180	+	0.175712i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	45.8912			1.89413			0.947066	−	0.321040i	$-0.104032\pi$
$587$	45.8912			1.89413			0.947066	+	0.321040i	$0.104032\pi$
$588$		0			0
$589$			36.0555i			1.48564i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$		−	21.6333i		−	0.882441i	−0.897399	−	0.441221i	$-0.854546\pi$
$601$		−	21.6333i		−	0.882441i	0.897399	−	0.441221i	$-0.145454\pi$
$602$		0			0
$603$	15.2971			0.622944
$604$		−	48.0833i		−	1.95648i
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	−40.0000			−1.62221
$609$		0			0
$610$		0			0
$611$		−	18.3848i		−	0.743768i
$612$			43.2666i			1.74895i
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$	−34.0000			−1.37213
$615$		0			0
$616$	10.1980	+	8.48528i	0.410891	+	0.341882i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	5.09902			0.204947			0.102473	−	0.994736i	$-0.467324\pi$
$619$	5.09902			0.204947			0.102473	+	0.994736i	$0.467324\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	25.0000			1.00000
$626$		−	39.5980i		−	1.58265i
$627$		0			0
$628$	−48.0000			−1.91541
$629$		0			0
$630$		0			0
$631$	−35.6931			−1.42092			−0.710461	−	0.703737i	$-0.751513\pi$
$631$	−35.6931			−1.42092			−0.710461	+	0.703737i	$0.751513\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	18.0278i		−	0.714286i
$638$	−26.0000	−	21.6333i	−1.02935	−	0.856471i
$639$	45.8912			1.81543
$640$		0			0
$641$	4.00000			0.157991			0.0789953	−	0.996875i	$-0.474829\pi$
$641$	4.00000			0.157991			0.0789953	+	0.996875i	$0.474829\pi$
$642$		0			0
$643$	−25.4951			−1.00543			−0.502714	−	0.864453i	$-0.667665\pi$
$643$	−25.4951			−1.00543			−0.502714	+	0.864453i	$0.667665\pi$
$644$		0			0
$645$		0			0
$646$			72.1110i			2.83717i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$			25.4558i			1.00000i
$649$	39.0000	+	32.4500i	1.53088	+	1.27377i
$650$	−25.4951			−1.00000
$651$		0			0
$652$	−50.9902			−1.99693
$653$	8.00000			0.313064			0.156532	−	0.987673i	$-0.449969\pi$
$653$	8.00000			0.313064			0.156532	+	0.987673i	$0.449969\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	10.1980			0.397561
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		−	50.4777i		−	1.96187i
$663$		0			0
$664$	−28.0000			−1.08661
$665$		0			0
$666$		0			0
$667$		0			0
$668$			8.48528i			0.328305i
$669$		0			0
$670$		0			0
$671$	−30.5941	+	36.7696i	−1.18107	+	1.41947i
$672$		0			0
$673$		−	50.4777i		−	1.94577i	−0.231283	−	0.972887i	$-0.574292\pi$
$673$		−	50.4777i		−	1.94577i	0.231283	−	0.972887i	$-0.425708\pi$
$674$	10.1980			0.392814
$675$		0			0
$676$	26.0000			1.00000
$677$		−	14.4222i		−	0.554290i	−0.960828	−	0.277145i	$-0.910612\pi$
$677$		−	14.4222i		−	0.554290i	0.960828	−	0.277145i	$-0.0893883\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	15.2971	−	18.3848i	0.585755	−	0.703989i
$683$	35.6931			1.36576			0.682880	−	0.730531i	$-0.260727\pi$
$683$	35.6931			1.36576			0.682880	+	0.730531i	$0.260727\pi$
$684$			42.4264i			1.62221i
$685$		0			0
$686$	24.0000			0.916324
$687$		0			0
$688$		0			0
$689$			7.21110i			0.274721i
$690$		0			0
$691$	−5.09902			−0.193976			−0.0969879	−	0.995286i	$-0.530921\pi$
$691$	−5.09902			−0.193976			−0.0969879	+	0.995286i	$0.530921\pi$
$692$		−	28.8444i		−	1.09650i
$693$	9.00000	−	10.8167i	0.341882	−	0.410891i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		−	14.1421i		−	0.534522i
$701$			50.4777i			1.90652i	0.302156	+	0.953259i	$0.402294\pi$
$701$			50.4777i			1.90652i	−0.302156	+	0.953259i	$0.597706\pi$
$702$		0			0
$703$		0			0
$704$	20.3961	+	16.9706i	0.768706	+	0.639602i
$705$		0			0
$706$		0			0
$707$	−20.3961			−0.767073
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	−18.0000			−0.671754
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$			43.8406i			1.63158i
$723$		0			0
$724$	32.0000			1.18927
$725$			36.0555i			1.33907i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$			14.4222i			0.534522i
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−13.0000	−	10.8167i	−0.478861	−	0.398437i
$738$		0			0
$739$		−	41.0122i		−	1.50866i	−0.656497	−	0.754329i	$-0.727962\pi$
$739$		−	41.0122i		−	1.50866i	0.656497	−	0.754329i	$-0.272038\pi$
$740$		0			0
$741$		0			0
$742$	−4.00000			−0.146845
$743$		−	52.3259i		−	1.91965i	−0.280597	−	0.959826i	$-0.590533\pi$
$743$		−	52.3259i		−	1.91965i	0.280597	−	0.959826i	$-0.409467\pi$
$744$		0			0
$745$		0			0
$746$	−30.5941			−1.12013
$747$			29.6985i			1.08661i
$748$	30.5941	−	36.7696i	1.11863	−	1.34443i
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$	20.3961			0.743768
$753$		0			0
$754$		−	36.7696i		−	1.33907i
$755$		0			0
$756$		0			0
$757$	−34.0000			−1.23575			−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000			−1.23575			−0.617876	+	0.786276i	$0.712006\pi$
$758$			50.4777i			1.83343i
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		−	36.0555i		−	1.30274i
$767$			55.1543i			1.99151i
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	−25.4951			−0.915811
$776$		0			0
$777$		0			0
$778$		−	22.6274i		−	0.811232i
$779$		0			0
$780$		0			0
$781$	−39.0000	−	32.4500i	−1.39553	−	1.16115i
$782$		0			0
$783$		0			0
$784$	20.0000			0.714286
$785$		0			0
$786$		0			0
$787$			1.41421i			0.0504113i	0.999682	+	0.0252056i	$0.00802405\pi$
$787$			1.41421i			0.0504113i	−0.999682	+	0.0252056i	$0.991976\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	28.2843i		−	1.00567i
$792$	18.0000	−	21.6333i	0.639602	−	0.768706i
$793$	−52.0000			−1.84657
$794$		0			0
$795$		0			0
$796$		0			0
$797$	56.0000			1.98362			0.991811	−	0.127715i	$-0.0407643\pi$
$797$	56.0000			1.98362			0.991811	+	0.127715i	$0.0407643\pi$
$798$		0			0
$799$		−	36.7696i		−	1.30081i
$800$		−	28.2843i		−	1.00000i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	26.0000			0.915811
$807$		0			0
$808$	−40.7922			−1.43506
$809$		−	36.0555i		−	1.26764i	−0.773479	−	0.633822i	$-0.781485\pi$
$809$		−	36.0555i		−	1.26764i	0.773479	−	0.633822i	$-0.218515\pi$
$810$		0			0
$811$		−	9.89949i		−	0.347618i	−0.984779	−	0.173809i	$-0.944392\pi$
$811$		−	9.89949i		−	0.347618i	0.984779	−	0.173809i	$-0.0556076\pi$
$812$	20.3961			0.715762
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	15.2971			0.534522
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$	−30.5941			−1.06451
$827$			12.7279i			0.442593i	0.975207	+	0.221297i	$0.0710289\pi$
$827$			12.7279i			0.442593i	−0.975207	+	0.221297i	$0.928971\pi$
$828$		0			0
$829$	38.0000			1.31979			0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000			1.31979			0.659897	+	0.751356i	$0.270600\pi$
$830$		0			0
$831$		0			0
$832$			28.8444i			1.00000i
$833$		−	36.0555i		−	1.24925i
$834$		0			0
$835$		0			0
$836$	30.0000	−	36.0555i	1.03757	−	1.24701i
$837$		0			0
$838$		0			0
$839$	−45.8912			−1.58434			−0.792170	−	0.610301i	$-0.791049\pi$
$839$	−45.8912			−1.58434			−0.792170	+	0.610301i	$0.791049\pi$
$840$		0			0
$841$	−23.0000			−0.793103
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		−	21.6333i		−	0.743768i
$847$	−15.2971	+	2.82843i	−0.525613	+	0.0971859i
$848$	−8.00000			−0.274721
$849$		0			0
$850$	−50.9902			−1.74895
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		−	28.8444i		−	0.987035i
$855$		0			0
$856$		0			0
$857$			57.6888i			1.97061i	0.170797	+	0.985306i	$0.445366\pi$
$857$			57.6888i			1.97061i	−0.170797	+	0.985306i	$0.554634\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	30.0000			1.02180
$863$	−35.6931			−1.21501			−0.607504	−	0.794316i	$-0.707829\pi$
$863$	−35.6931			−1.21501			−0.607504	+	0.794316i	$0.707829\pi$
$864$		0			0
$865$		0			0
$866$		−	42.4264i		−	1.44171i
$867$		0			0
$868$			14.4222i			0.489522i
$869$		0			0
$870$		0			0
$871$		−	18.3848i		−	0.622944i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	14.0000			0.471672			0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000			0.471672			0.235836	+	0.971793i	$0.424217\pi$
$882$		−	21.2132i		−	0.714286i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	52.0000			1.74895
$885$		0			0
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−22.9456	−	19.0919i	−0.768706	−	0.639602i
$892$	−50.9902			−1.70728
$893$		−	36.0555i		−	1.20655i
$894$		0			0
$895$		0			0
$896$	−16.0000			−0.534522
$897$		0			0
$898$		0			0
$899$		−	36.7696i		−	1.22633i
$900$	−30.0000			−1.00000
$901$			14.4222i			0.480473i
$902$		0			0
$903$		0			0
$904$		−	56.5685i		−	1.88144i
$905$		0			0
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$			59.3970i			1.97116i
$909$			43.2666i			1.43506i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$	21.0000	−	25.2389i	0.694999	−	0.835284i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	55.1543i		−	1.81543i
$924$		0			0
$925$		0			0
$926$		−	50.4777i		−	1.65880i
$927$		0			0
$928$	40.7922			1.33907
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		−	35.3553i		−	1.15872i
$932$		−	57.6888i		−	1.88966i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	30.5941			1.00000
$937$			28.8444i			0.942306i	0.882052	+	0.471153i	$0.156162\pi$
$937$			28.8444i			0.942306i	−0.882052	+	0.471153i	$0.843838\pi$
$938$	10.1980			0.332978
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$	−61.1882			−1.99151
$945$		0			0
$946$		0			0
$947$	45.8912			1.49126			0.745631	−	0.666359i	$-0.232148\pi$
$947$	45.8912			1.49126			0.745631	+	0.666359i	$0.232148\pi$
$948$		0			0
$949$		0			0
$950$	−50.0000			−1.62221
$951$		0			0
$952$			28.8444i			0.934853i
$953$			57.6888i			1.86872i	0.356325	+	0.934362i	$0.384030\pi$
$953$			57.6888i			1.86872i	−0.356325	+	0.934362i	$0.615970\pi$
$954$			8.48528i			0.274721i
$955$		0			0
$956$			53.7401i			1.73808i
$957$		0			0
$958$	58.0000			1.87389
$959$		0			0
$960$		0			0
$961$	−5.00000			−0.161290
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	26.8701i		−	0.864083i	−0.901854	−	0.432041i	$-0.857793\pi$
$967$		−	26.8701i		−	0.864083i	0.901854	−	0.432041i	$-0.142207\pi$
$968$	−30.5941	+	5.65685i	−0.983332	+	0.181818i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		−	7.21110i		−	0.231059i
$975$		0			0
$976$		−	57.6888i		−	1.84657i
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−15.2971			−0.487900			−0.243950	−	0.969788i	$-0.578443\pi$
$983$	−15.2971			−0.487900			−0.243950	+	0.969788i	$0.578443\pi$
$984$		0			0
$985$		0			0
$986$		−	73.5391i		−	2.34196i
$987$		0			0
$988$	50.9902			1.62221
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$			28.8444i			0.915811i
$993$		0			0
$994$	30.5941			0.970386
$995$		0			0
$996$		0			0
$997$			43.2666i			1.37027i	0.728417	+	0.685134i	$0.240256\pi$
$997$			43.2666i			1.37027i	−0.728417	+	0.685134i	$0.759744\pi$
$998$			50.4777i			1.59784i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.b.a.571.1	✓	4
4.3	odd	2	inner	572.2.b.a.571.4	yes	4
11.10	odd	2	inner	572.2.b.a.571.3	yes	4
13.12	even	2	inner	572.2.b.a.571.4	yes	4
44.43	even	2	inner	572.2.b.a.571.2	yes	4
52.51	odd	2	CM	572.2.b.a.571.1	✓	4
143.142	odd	2	inner	572.2.b.a.571.2	yes	4
572.571	even	2	inner	572.2.b.a.571.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.b.a.571.1	✓	4	1.1	even	1	trivial
572.2.b.a.571.1	✓	4	52.51	odd	2	CM
572.2.b.a.571.2	yes	4	44.43	even	2	inner
572.2.b.a.571.2	yes	4	143.142	odd	2	inner
572.2.b.a.571.3	yes	4	11.10	odd	2	inner
572.2.b.a.571.3	yes	4	572.571	even	2	inner
572.2.b.a.571.4	yes	4	4.3	odd	2	inner
572.2.b.a.571.4	yes	4	13.12	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +1.41421i q^{7} +2.82843i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} +1.41421i q^{7} +2.82843i q^{8} +3.00000 q^{9} +(-2.54951 - 2.12132i) q^{11} -3.60555i q^{13} +2.00000 q^{14} +4.00000 q^{16} -7.21110i q^{17} -4.24264i q^{18} -7.07107i q^{19} +(-3.00000 + 3.60555i) q^{22} +5.00000 q^{25} -5.09902 q^{26} -2.82843i q^{28} +7.21110i q^{29} -5.09902 q^{31} -5.65685i q^{32} -10.1980 q^{34} -6.00000 q^{36} -10.0000 q^{38} +(5.09902 + 4.24264i) q^{44} +5.09902 q^{47} +5.00000 q^{49} -7.07107i q^{50} +7.21110i q^{52} -2.00000 q^{53} -4.00000 q^{56} +10.1980 q^{58} -15.2971 q^{59} -14.4222i q^{61} +7.21110i q^{62} +4.24264i q^{63} -8.00000 q^{64} +5.09902 q^{67} +14.4222i q^{68} +15.2971 q^{71} +8.48528i q^{72} +14.1421i q^{76} +(3.00000 - 3.60555i) q^{77} +9.00000 q^{81} +9.89949i q^{83} +(6.00000 - 7.21110i) q^{88} +5.09902 q^{91} -7.21110i q^{94} -7.07107i q^{98} +(-7.64853 - 6.36396i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} + 12 q^{9}+O(q^{10}) \) 4 * q - 8 * q^4 + 12 * q^9	\( 4 q - 8 q^{4} + 12 q^{9} + 8 q^{14} + 16 q^{16} - 12 q^{22} + 20 q^{25} - 24 q^{36} - 40 q^{38} + 20 q^{49} - 8 q^{53} - 16 q^{56} - 32 q^{64} + 12 q^{77} + 36 q^{81} + 24 q^{88}+O(q^{100}) \) 4 * q - 8 * q^4 + 12 * q^9 + 8 * q^14 + 16 * q^16 - 12 * q^22 + 20 * q^25 - 24 * q^36 - 40 * q^38 + 20 * q^49 - 8 * q^53 - 16 * q^56 - 32 * q^64 + 12 * q^77 + 36 * q^81 + 24 * q^88

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