LMFDB - Embedded newform 572.2.b.b.571.18

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 53x^{10} + 1024 $ x^20 - 53*x^10 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			571.18
Root			$-1.19153 + 0.761743i$ of defining polynomial
Character	$\chi$	$=$	572.571
Dual form			572.2.b.b.571.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.19153 + 0.761743i) q^{2} -0.602681i q^{3} +(0.839496 + 1.81528i) q^{4} +(0.459088 - 0.718113i) q^{6} -3.72449i q^{7} +(-0.382491 + 2.80245i) q^{8} +2.63678 q^{9} +O(q^{10})$	\(q+(1.19153 + 0.761743i) q^{2} -0.602681i q^{3} +(0.839496 + 1.81528i) q^{4} +(0.459088 - 0.718113i) q^{6} -3.72449i q^{7} +(-0.382491 + 2.80245i) q^{8} +2.63678 q^{9} +3.31662i q^{11} +(1.09403 - 0.505948i) q^{12} +3.60555 q^{13} +(2.83711 - 4.43785i) q^{14} +(-2.59049 + 3.04784i) q^{16} +(3.14180 + 2.00854i) q^{18} -8.26694i q^{19} -2.24468 q^{21} +(-2.52641 + 3.95186i) q^{22} +9.31703i q^{23} +(1.68898 + 0.230520i) q^{24} +5.00000 q^{25} +(4.29613 + 2.74650i) q^{26} -3.39718i q^{27} +(6.76100 - 3.12670i) q^{28} +(-5.40833 + 1.65831i) q^{32} +1.99887 q^{33} +(2.21356 + 4.78649i) q^{36} +(6.29728 - 9.85032i) q^{38} -2.17300i q^{39} -12.8062 q^{41} +(-2.67461 - 1.70987i) q^{42} +(-6.02061 + 2.78429i) q^{44} +(-7.09718 + 11.1015i) q^{46} +(1.83688 + 1.56124i) q^{48} -6.87184 q^{49} +(5.95766 + 3.80871i) q^{50} +(3.02685 + 6.54509i) q^{52} -6.95070 q^{53} +(2.58777 - 4.04784i) q^{54} +(10.4377 + 1.42459i) q^{56} -4.98232 q^{57} -9.82065i q^{63} +(-7.70740 - 2.14382i) q^{64} +(2.38171 + 1.52262i) q^{66} +5.61520 q^{69} +(-1.00854 + 7.38942i) q^{72} -17.0656 q^{73} -3.01340i q^{75} +(15.0068 - 6.94006i) q^{76} +12.3527 q^{77} +(1.65526 - 2.58919i) q^{78} +5.86292 q^{81} +(-15.2590 - 9.75503i) q^{82} -11.2461i q^{83} +(-1.88440 - 4.07473i) q^{84} +(-9.29466 - 1.26858i) q^{88} -13.4288i q^{91} +(-16.9130 + 7.82161i) q^{92} +(0.999433 + 3.25949i) q^{96} +(-8.18802 - 5.23458i) q^{98} +8.74520i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 60 q^{9}+O(q^{10}) $ 20 * q - 60 * q^9	$ 20 q - 60 q^{9} + 100 q^{25} + 10 q^{36} + 30 q^{38} - 50 q^{42} - 70 q^{48} - 140 q^{49} + 90 q^{56} + 110 q^{66} - 130 q^{78} + 180 q^{81} - 150 q^{92}+O(q^{100}) $ 20 * q - 60 * q^9 + 100 * q^25 + 10 * q^36 + 30 * q^38 - 50 * q^42 - 70 * q^48 - 140 * q^49 + 90 * q^56 + 110 * q^66 - 130 * q^78 + 180 * q^81 - 150 * q^92

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 $

$2$	1.19153	+	0.761743i	0.842540	+	0.538633i
$2$	1.19153	+	0.761743i	0.842540	+	0.538633i
$3$		−	0.602681i		−	0.347958i	−0.984749	−	0.173979i	$-0.944338\pi$
$3$		−	0.602681i		−	0.347958i	0.984749	−	0.173979i	$-0.0556625\pi$
$4$	0.839496	+	1.81528i	0.419748	+	0.907641i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$	0.459088	−	0.718113i	0.187422	−	0.293168i
$7$		−	3.72449i		−	1.40773i	−0.710336	−	0.703863i	$-0.751457\pi$
$7$		−	3.72449i		−	1.40773i	0.710336	−	0.703863i	$-0.248543\pi$
$8$	−0.382491	+	2.80245i	−0.135231	+	0.990814i
$9$	2.63678			0.878925
$10$		0			0
$11$			3.31662i			1.00000i
$11$			3.31662i			1.00000i
$12$	1.09403	−	0.505948i	0.315821	−	0.146055i
$13$	3.60555			1.00000
$13$	3.60555			1.00000
$14$	2.83711	−	4.43785i	0.758248	−	1.18607i
$15$		0			0
$16$	−2.59049	+	3.04784i	−0.647623	+	0.761961i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	3.14180	+	2.00854i	0.740530	+	0.473419i
$19$		−	8.26694i		−	1.89657i	−0.317427	−	0.948283i	$-0.602819\pi$
$19$		−	8.26694i		−	1.89657i	0.317427	−	0.948283i	$-0.397181\pi$
$20$		0			0
$21$	−2.24468			−0.489829
$22$	−2.52641	+	3.95186i	−0.538633	+	0.842540i
$23$			9.31703i			1.94274i	0.237580	+	0.971368i	$0.423646\pi$
$23$			9.31703i			1.94274i	−0.237580	+	0.971368i	$0.576354\pi$
$24$	1.68898	+	0.230520i	0.344762	+	0.0470547i
$25$	5.00000			1.00000
$26$	4.29613	+	2.74650i	0.842540	+	0.538633i
$27$		−	3.39718i		−	0.653787i
$28$	6.76100	−	3.12670i	1.27771	−	0.590890i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	−5.40833	+	1.65831i	−0.956066	+	0.293151i
$33$	1.99887			0.347958
$34$		0			0
$35$		0			0
$36$	2.21356	+	4.78649i	0.368927	+	0.797748i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	6.29728	−	9.85032i	1.02155	−	1.59793i
$39$		−	2.17300i		−	0.347958i
$40$		0			0
$41$	−12.8062			−1.99999			−0.999996	−	0.00283490i	$-0.999098\pi$
$41$	−12.8062			−1.99999			−0.999996	+	0.00283490i	$0.999098\pi$
$42$	−2.67461	−	1.70987i	−0.412701	−	0.263838i
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−6.02061	+	2.78429i	−0.907641	+	0.419748i
$45$		0			0
$46$	−7.09718	+	11.1015i	−1.04642	+	1.63683i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	1.83688	+	1.56124i	0.265130	+	0.225346i
$49$	−6.87184			−0.981692
$50$	5.95766	+	3.80871i	0.842540	+	0.538633i
$51$		0			0
$52$	3.02685	+	6.54509i	0.419748	+	0.907641i
$53$	−6.95070			−0.954752			−0.477376	−	0.878699i	$-0.658412\pi$
$53$	−6.95070			−0.954752			−0.477376	+	0.878699i	$0.658412\pi$
$54$	2.58777	−	4.04784i	0.352151	−	0.550842i
$55$		0			0
$56$	10.4377	+	1.42459i	1.39479	+	0.190368i
$57$	−4.98232			−0.659925
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		−	9.82065i		−	1.23729i
$64$	−7.70740	−	2.14382i	−0.963425	−	0.267978i
$65$		0			0
$66$	2.38171	+	1.52262i	0.293168	+	0.187422i
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$	5.61520			0.675990
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−1.00854	+	7.38942i	−0.118858	+	0.870852i
$73$	−17.0656			−1.99738			−0.998691	−	0.0511455i	$-0.983713\pi$
$73$	−17.0656			−1.99738			−0.998691	+	0.0511455i	$0.983713\pi$
$74$		0			0
$75$		−	3.01340i		−	0.347958i
$76$	15.0068	−	6.94006i	1.72140	−	0.796080i
$77$	12.3527			1.40773
$78$	1.65526	−	2.58919i	0.187422	−	0.293168i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	5.86292			0.651435
$82$	−15.2590	−	9.75503i	−1.68507	−	1.07726i
$83$		−	11.2461i		−	1.23442i	−0.786799	−	0.617209i	$-0.788263\pi$
$83$		−	11.2461i		−	1.23442i	0.786799	−	0.617209i	$-0.211737\pi$
$84$	−1.88440	−	4.07473i	−0.205605	−	0.444589i
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−9.29466	−	1.26858i	−0.990814	−	0.135231i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		−	13.4288i		−	1.40773i
$92$	−16.9130	+	7.82161i	−1.76331	+	0.815459i
$93$		0			0
$94$		0			0
$95$		0			0
$96$	0.999433	+	3.25949i	0.102004	+	0.332671i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−8.18802	−	5.23458i	−0.827115	−	0.528772i
$99$			8.74520i			0.878925i
$100$	4.19748	+	9.07641i	0.419748	+	0.907641i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$			1.09340i			0.107736i	0.998548	+	0.0538681i	$0.0171550\pi$
$103$			1.09340i			0.107736i	−0.998548	+	0.0538681i	$0.982845\pi$
$104$	−1.37909	+	10.1044i	−0.135231	+	0.990814i
$105$		0			0
$106$	−8.28198	−	5.29465i	−0.804417	−	0.514261i
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	6.16683	−	2.85192i	0.593404	−	0.274426i
$109$	11.5311			1.10448			0.552240	−	0.833685i	$-0.313773\pi$
$109$	11.5311			1.10448			0.552240	+	0.833685i	$0.313773\pi$
$110$		0			0
$111$		0			0
$112$	11.3517	+	9.64827i	1.07263	+	0.911676i
$113$	−13.5169			−1.27157			−0.635783	−	0.771868i	$-0.719323\pi$
$113$	−13.5169			−1.27157			−0.635783	+	0.771868i	$0.719323\pi$
$114$	−5.93660	−	3.79525i	−0.556013	−	0.355458i
$115$		0			0
$116$		0			0
$117$	9.50703			0.878925
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$			7.71805i			0.695913i
$124$		0			0
$125$		0			0
$126$	7.48081	−	11.7016i	0.666444	−	1.04246i
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	−7.55057	−	8.42549i	−0.667383	−	0.744715i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	1.67804	+	3.62850i	0.146055	+	0.315821i
$133$	−30.7901			−2.66984
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$	6.69068	+	4.27733i	0.569549	+	0.364111i
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			11.9583i			1.00000i
$144$	−6.83055	+	8.03648i	−0.569212	+	0.669707i
$145$		0			0
$146$	−20.3343	−	12.9996i	−1.68287	−	1.07586i
$147$			4.14153i			0.341587i
$148$		0			0
$149$	−9.74627			−0.798445			−0.399223	−	0.916854i	$-0.630720\pi$
$149$	−9.74627			−0.798445			−0.399223	+	0.916854i	$0.630720\pi$
$150$	2.29544	−	3.59057i	0.187422	−	0.293168i
$151$			19.8997i			1.61942i	0.586831	+	0.809709i	$0.300375\pi$
$151$			19.8997i			1.61942i	−0.586831	+	0.809709i	$0.699625\pi$
$152$	23.1676	+	3.16203i	1.87914	+	0.256475i
$153$		0			0
$154$	14.7187	+	9.40961i	1.18607	+	0.758248i
$155$		0			0
$156$	3.94460	−	1.82422i	0.315821	−	0.146055i
$157$	−23.6263			−1.88558			−0.942792	−	0.333382i	$-0.891810\pi$
$157$	−23.6263			−1.88558			−0.942792	+	0.333382i	$0.891810\pi$
$158$		0			0
$159$			4.18905i			0.332213i
$160$		0			0
$161$	34.7012			2.73484
$162$	6.98585	+	4.46603i	0.548860	+	0.350885i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$	−10.7508	−	23.2469i	−0.839493	−	1.81527i
$165$		0			0
$166$	8.56662	−	13.4001i	0.664899	−	1.04005i
$167$			18.7069i			1.44758i	0.690020	+	0.723791i	$0.257602\pi$
$167$			18.7069i			1.44758i	−0.690020	+	0.723791i	$0.742398\pi$
$168$	0.858570	−	6.29059i	0.0662401	−	0.485330i
$169$	13.0000			1.00000
$170$		0			0
$171$		−	21.7981i		−	1.66694i
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		−	18.6225i		−	1.40773i
$176$	−10.1086	−	8.59169i	−0.761961	−	0.647623i
$177$		0			0
$178$		0			0
$179$		−	23.9165i		−	1.78760i	−0.448461	−	0.893802i	$-0.648028\pi$
$179$		−	23.9165i		−	1.78760i	0.448461	−	0.893802i	$-0.351972\pi$
$180$		0			0
$181$	20.3395			1.51182			0.755911	−	0.654675i	$-0.227194\pi$
$181$	20.3395			1.51182			0.755911	+	0.654675i	$0.227194\pi$
$182$	10.2293	−	16.0009i	0.758248	−	1.18607i
$183$		0			0
$184$	−26.1105	−	3.56368i	−1.92489	−	0.262718i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−12.6528			−0.920353
$190$		0			0
$191$		−	27.3961i		−	1.98232i	−0.132689	−	0.991158i	$-0.542361\pi$
$191$		−	27.3961i		−	1.98232i	0.132689	−	0.991158i	$-0.457639\pi$
$192$	−1.29204	+	4.64510i	−0.0932450	+	0.335231i
$193$	23.3836			1.68319			0.841595	−	0.540110i	$-0.181617\pi$
$193$	23.3836			1.68319			0.841595	+	0.540110i	$0.181617\pi$
$194$		0			0
$195$		0			0
$196$	−5.76889	−	12.4743i	−0.412063	−	0.891024i
$197$	3.53566			0.251905			0.125953	−	0.992036i	$-0.459801\pi$
$197$	3.53566			0.251905			0.125953	+	0.992036i	$0.459801\pi$
$198$	−6.66159	+	10.4202i	−0.473419	+	0.740530i
$199$			26.1908i			1.85662i	0.371813	+	0.928308i	$0.378736\pi$
$199$			26.1908i			1.85662i	−0.371813	+	0.928308i	$0.621264\pi$
$200$	−1.91246	+	14.0122i	−0.135231	+	0.990814i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−0.832892	+	1.30282i	−0.0580303	+	0.0907721i
$207$			24.5669i			1.70752i
$208$	−9.34016	+	10.9892i	−0.647623	+	0.761961i
$209$	27.4183			1.89657
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$	−5.83508	−	12.6175i	−0.400755	−	0.866572i
$213$		0			0
$214$		0			0
$215$		0			0
$216$	9.52040	+	1.29939i	0.647781	+	0.0884123i
$217$		0			0
$218$	13.7397	+	8.78375i	0.930570	+	0.594910i
$219$			10.2851i			0.695005i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	6.17637	+	20.1433i	0.412676	+	1.34588i
$225$	13.1839			0.878925
$226$	−16.1059	−	10.2964i	−1.07135	−	0.684908i
$227$			12.6129i			0.837150i	0.908182	+	0.418575i	$0.137470\pi$
$227$			12.6129i			0.837150i	−0.908182	+	0.418575i	$0.862530\pi$
$228$	−4.18264	−	9.04432i	−0.277002	−	0.598975i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		−	7.44476i		−	0.489829i
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$	11.3279	+	7.24191i	0.740530	+	0.473419i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			30.8948i			1.99842i	0.0397962	+	0.999208i	$0.487329\pi$
$239$			30.8948i			1.99842i	−0.0397962	+	0.999208i	$0.512671\pi$
$240$		0			0
$241$	8.34864			0.537783			0.268892	−	0.963170i	$-0.413343\pi$
$241$	8.34864			0.537783			0.268892	+	0.963170i	$0.413343\pi$
$242$	−13.1068	−	8.37917i	−0.842540	−	0.538633i
$243$		−	13.7250i		−	0.880459i
$244$		0			0
$245$		0			0
$246$	−5.87917	+	9.19630i	−0.374842	+	0.586335i
$247$		−	29.8069i		−	1.89657i
$248$		0			0
$249$	−6.77780			−0.429526
$250$		0			0
$251$			3.01841i			0.190521i	0.995452	+	0.0952603i	$0.0303683\pi$
$251$			3.01841i			0.190521i	−0.995452	+	0.0952603i	$0.969632\pi$
$252$	17.8272	−	8.24440i	1.12301	−	0.519348i
$253$	−30.9011			−1.94274
$254$		0			0
$255$		0			0
$256$	−2.57869	−	15.7908i	−0.161168	−	0.986927i
$257$	−7.79516			−0.486249			−0.243124	−	0.969995i	$-0.578172\pi$
$257$	−7.79516			−0.486249			−0.243124	+	0.969995i	$0.578172\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$	−0.764549	+	5.60171i	−0.0470547	+	0.344762i
$265$		0			0
$266$	−36.6874	−	23.4542i	−2.24945	−	1.43807i
$267$		0			0
$268$		0			0
$269$	21.4679			1.30892			0.654461	−	0.756096i	$-0.272896\pi$
$269$	21.4679			1.30892			0.654461	+	0.756096i	$0.272896\pi$
$270$		0			0
$271$			10.0149i			0.608361i	0.952614	+	0.304181i	$0.0983826\pi$
$271$			10.0149i			0.608361i	−0.952614	+	0.304181i	$0.901617\pi$
$272$		0			0
$273$	−8.09331			−0.489829
$274$		0			0
$275$			16.5831i			1.00000i
$276$	4.71393	+	10.1932i	0.283745	+	0.613556i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	32.1324			1.91686			0.958430	−	0.285329i	$-0.0921027\pi$
$281$	32.1324			1.91686			0.958430	+	0.285329i	$0.0921027\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$		0			0
$286$	−9.10912	+	14.2486i	−0.538633	+	0.842540i
$287$			47.6966i			2.81544i
$288$	−14.2605	+	4.37260i	−0.840311	+	0.257658i
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$	−14.3265	−	30.9789i	−0.838397	−	1.81291i
$293$	−21.6333			−1.26383			−0.631916	−	0.775037i	$-0.717731\pi$
$293$	−21.6333			−1.26383			−0.631916	+	0.775037i	$0.717731\pi$
$294$	−3.15478	+	4.93476i	−0.183990	+	0.287801i
$295$		0			0
$296$		0			0
$297$	11.2672			0.653787
$298$	−11.6130	−	7.42415i	−0.672722	−	0.430069i
$299$			33.5930i			1.94274i
$300$	5.47017	−	2.52974i	0.315821	−	0.146055i
$301$		0			0
$302$	−15.1585	+	23.7112i	−0.872273	+	1.36443i
$303$		0			0
$304$	25.1963	+	21.4154i	1.44511	+	1.22826i
$305$		0			0
$306$		0			0
$307$		−	19.8997i		−	1.13574i	−0.823119	−	0.567869i	$-0.807768\pi$
$307$		−	19.8997i		−	1.13574i	0.823119	−	0.567869i	$-0.192232\pi$
$308$	10.3701	+	22.4237i	0.590890	+	1.27771i
$309$	0.658973			0.0374876
$310$		0			0
$311$			31.0122i			1.75854i	0.476322	+	0.879271i	$0.341970\pi$
$311$			31.0122i			1.75854i	−0.476322	+	0.879271i	$0.658030\pi$
$312$	6.08970	+	0.831152i	0.344762	+	0.0470547i
$313$	−3.47839			−0.196610			−0.0983052	−	0.995156i	$-0.531342\pi$
$313$	−3.47839			−0.196610			−0.0983052	+	0.995156i	$0.531342\pi$
$314$	−28.1515	−	17.9972i	−1.58868	−	1.01564i
$315$		0			0
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$	−3.19098	+	4.99139i	−0.178941	+	0.279903i
$319$		0			0
$320$		0			0
$321$		0			0
$322$	41.3476	+	26.4334i	2.30421	+	1.47308i
$323$		0			0
$324$	4.92189	+	10.6428i	0.273439	+	0.591269i
$325$	18.0278			1.00000
$326$		0			0
$327$		−	6.94958i		−	0.384313i
$328$	4.89826	−	35.8887i	0.270461	−	1.98162i
$329$		0			0
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$	20.4148	−	9.44104i	1.12041	−	0.518145i
$333$		0			0
$334$	−14.2498	+	22.2898i	−0.779716	+	1.21965i
$335$		0			0
$336$	5.81483	−	6.84143i	0.317225	−	0.373231i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	15.4899	+	9.90266i	0.842540	+	0.538633i
$339$			8.14639i			0.442451i
$340$		0			0
$341$		0			0
$342$	16.6045	−	25.9731i	0.897869	−	1.40446i
$343$		−	0.477317i		−	0.0257727i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$	−0.831151			−0.0444905			−0.0222453	−	0.999753i	$-0.507081\pi$
$349$	−0.831151			−0.0444905			−0.0222453	+	0.999753i	$0.507081\pi$
$350$	14.1855	−	22.1893i	0.758248	−	1.18607i
$351$		−	12.2487i		−	0.653787i
$352$	−5.50000	−	17.9374i	−0.293151	−	0.956066i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	18.2182	−	28.4973i	0.962864	−	1.50613i
$359$		−	33.6656i		−	1.77681i	−0.459065	−	0.888403i	$-0.651816\pi$
$359$		−	33.6656i		−	1.77681i	0.459065	−	0.888403i	$-0.348184\pi$
$360$		0			0
$361$	−49.3423			−2.59696
$362$	24.2351	+	15.4934i	1.27377	+	0.814318i
$363$			6.62949i			0.347958i
$364$	24.3771	−	11.2735i	1.27771	−	0.590890i
$365$		0			0
$366$		0			0
$367$		−	22.5747i		−	1.17839i	−0.807991	−	0.589195i	$-0.799445\pi$
$367$		−	22.5747i		−	1.17839i	0.807991	−	0.589195i	$-0.200555\pi$
$368$	−28.3969	−	24.1357i	−1.48029	−	1.25816i
$369$	−33.7671			−1.75784
$370$		0			0
$371$			25.8878i			1.34403i
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$	−15.0762	−	9.63815i	−0.775434	−	0.495733i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$	20.8688	−	32.6434i	1.06774	−	1.67018i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	−5.07788	+	4.55058i	−0.259129	+	0.232221i
$385$		0			0
$386$	27.8623	+	17.8123i	1.41815	+	0.906622i
$387$		0			0
$388$		0			0
$389$	33.9846			1.72309			0.861543	−	0.507685i	$-0.169499\pi$
$389$	33.9846			1.72309			0.861543	+	0.507685i	$0.169499\pi$
$390$		0			0
$391$		0			0
$392$	2.62842	−	19.2580i	0.132755	−	0.972674i
$393$		0			0
$394$	4.21285	+	2.69326i	0.212240	+	0.135685i
$395$		0			0
$396$	−15.8750	+	7.34156i	−0.797748	+	0.368927i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	−19.9506	+	31.2072i	−1.00004	+	1.56427i
$399$			18.5566i			0.928993i
$400$	−12.9525	+	15.2392i	−0.647623	+	0.761961i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−7.21110			−0.356566			−0.178283	−	0.983979i	$-0.557054\pi$
$409$	−7.21110			−0.356566			−0.178283	+	0.983979i	$0.557054\pi$
$410$		0			0
$411$		0			0
$412$	−1.98483	+	0.917907i	−0.0977857	+	0.0452220i
$413$		0			0
$414$	−18.7137	+	29.2723i	−0.919727	+	1.43865i
$415$		0			0
$416$	−19.5000	+	5.97913i	−0.956066	+	0.293151i
$417$		0			0
$418$	32.6698	+	20.8857i	1.59793	+	1.02155i
$419$		−	34.2497i		−	1.67321i	−0.547808	−	0.836604i	$-0.684538\pi$
$419$		−	34.2497i		−	1.67321i	0.547808	−	0.836604i	$-0.315462\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	2.65858	−	19.4790i	0.129112	−	0.945982i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	7.20701			0.347958
$430$		0			0
$431$			11.3187i			0.545202i	0.962127	+	0.272601i	$0.0878840\pi$
$431$			11.3187i			0.545202i	−0.962127	+	0.272601i	$0.912116\pi$
$432$	10.3541	+	8.80036i	0.498160	+	0.423408i
$433$	41.0634			1.97338			0.986691	−	0.162607i	$-0.0519901\pi$
$433$	41.0634			1.97338			0.986691	+	0.162607i	$0.0519901\pi$
$434$		0			0
$435$		0			0
$436$	9.68033	+	20.9322i	0.463604	+	1.00247i
$437$	77.0233			3.68453
$438$	−7.83463	+	12.2551i	−0.374353	+	0.585570i
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−18.1195			−0.862834
$442$		0			0
$443$		−	21.6525i		−	1.02874i	−0.857568	−	0.514370i	$-0.828026\pi$
$443$		−	21.6525i		−	1.02874i	0.857568	−	0.514370i	$-0.171974\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			5.87389i			0.277825i
$448$	−7.98465	+	28.7062i	−0.377239	+	1.35624i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	15.7090	+	10.0427i	0.740530	+	0.473419i
$451$		−	42.4734i		−	1.99999i
$452$	−11.3474	−	24.5370i	−0.533737	−	1.15412i
$453$	11.9932			0.563489
$454$	−9.60781	+	15.0287i	−0.450917	+	0.705332i
$455$		0			0
$456$	1.90570	−	13.9627i	0.0892424	−	0.653863i
$457$	−28.1347			−1.31609			−0.658043	−	0.752981i	$-0.728615\pi$
$457$	−28.1347			−1.31609			−0.658043	+	0.752981i	$0.728615\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−15.5289			−0.723251			−0.361625	−	0.932323i	$-0.617778\pi$
$461$	−15.5289			−0.723251			−0.361625	+	0.932323i	$0.617778\pi$
$462$	5.67099	−	8.87067i	0.263838	−	0.412701i
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	23.9165i		−	1.10672i	−0.832941	−	0.553362i	$-0.813345\pi$
$467$		−	23.9165i		−	1.10672i	0.832941	−	0.553362i	$-0.186655\pi$
$468$	7.98111	+	17.2579i	0.368927	+	0.797748i
$469$		0			0
$470$		0			0
$471$			14.2391i			0.656104i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	41.3347i		−	1.89657i
$476$		0			0
$477$	−18.3274			−0.839156
$478$	−23.5339	+	36.8121i	−1.07641	+	1.68375i
$479$		−	6.63325i		−	0.303081i	−0.988451	−	0.151540i	$-0.951577\pi$
$479$		−	6.63325i		−	0.303081i	0.988451	−	0.151540i	$-0.0484234\pi$
$480$		0			0
$481$		0			0
$482$	9.94767	+	6.35952i	0.453104	+	0.289668i
$483$		−	20.9138i		−	0.951609i
$484$	−9.23446	−	19.9681i	−0.419748	−	0.907641i
$485$		0			0
$486$	10.4549	−	16.3538i	0.474245	−	0.741822i
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$	−14.0104	+	6.47927i	−0.631639	+	0.292108i
$493$		0			0
$494$	22.7052	−	35.5158i	1.02155	−	1.59793i
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−8.07596	−	5.16294i	−0.361893	−	0.231357i
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	11.2743			0.503697
$502$	−2.29925	+	3.59653i	−0.102621	+	0.160521i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	27.5218	+	3.75631i	1.22592	+	0.167320i
$505$		0			0
$506$	−36.8196	−	23.5387i	−1.63683	−	1.04642i
$507$		−	7.83485i		−	0.347958i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$			63.5609i			2.81177i
$512$	8.95596	−	20.7796i	0.395801	−	0.918336i
$513$	−28.0842			−1.23995
$514$	−9.28818	−	5.93791i	−0.409684	−	0.261910i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	41.6158			1.82322			0.911611	−	0.411054i	$-0.134839\pi$
$521$	41.6158			1.82322			0.911611	+	0.411054i	$0.134839\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$	−11.2234			−0.489829
$526$		0			0
$527$		0			0
$528$	−5.17805	+	6.09223i	−0.225346	+	0.265130i
$529$	−63.8071			−2.77422
$530$		0			0
$531$		0			0
$532$	−25.8482	−	55.8928i	−1.12066	−	2.42326i
$533$	−46.1734			−1.99999
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−14.4140			−0.622011
$538$	25.5797	+	16.3530i	1.10282	+	0.705029i
$539$		−	22.7913i		−	0.981692i
$540$		0			0
$541$	−45.9361			−1.97495			−0.987473	−	0.157787i	$-0.949564\pi$
$541$	−45.9361			−1.97495			−0.987473	+	0.157787i	$0.949564\pi$
$542$	−7.62877	+	11.9331i	−0.327684	+	0.512569i
$543$		−	12.2582i		−	0.526050i
$544$		0			0
$545$		0			0
$546$	−9.64343	−	6.16502i	−0.412701	−	0.263838i
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		0			0
$549$		0			0
$550$	−12.6321	+	19.7593i	−0.538633	+	0.842540i
$551$		0			0
$552$	−2.14776	+	15.7363i	−0.0914149	+	0.669781i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	41.4785			1.75750			0.878751	−	0.477281i	$-0.158378\pi$
$557$	41.4785			1.75750			0.878751	+	0.477281i	$0.158378\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	38.2868	+	24.4766i	1.61503	+	1.03248i
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	21.8364i		−	0.917042i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$	−21.7076	+	10.0389i	−0.907641	+	0.419748i
$573$	−16.5111			−0.689762
$574$	−36.3325	+	56.8320i	−1.51649	+	2.37212i
$575$			46.5852i			1.94274i
$576$	−20.3227	−	5.65278i	−0.846779	−	0.235532i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	20.2560	+	12.9496i	0.842540	+	0.538633i
$579$		−	14.0929i		−	0.585679i
$580$		0			0
$581$	−41.8860			−1.73772
$582$		0			0
$583$		−	23.0529i		−	0.954752i
$584$	6.52746	−	47.8255i	0.270108	−	1.97903i
$585$		0			0
$586$	−25.7768	−	16.4790i	−1.06483	−	0.680742i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−7.51804	+	3.47680i	−0.310039	+	0.143381i
$589$		0			0
$590$		0			0
$591$		−	2.13087i		−	0.0876524i
$592$		0			0
$593$	−18.9261			−0.777200			−0.388600	−	0.921407i	$-0.627041\pi$
$593$	−18.9261			−0.777200			−0.388600	+	0.921407i	$0.627041\pi$
$594$	13.4252	+	8.58268i	0.550842	+	0.352151i
$595$		0			0
$596$	−8.18195	−	17.6922i	−0.335146	−	0.724701i
$597$	15.7847			0.646024
$598$	−25.5893	+	40.0272i	−1.04642	+	1.63683i
$599$		−	20.1640i		−	0.823878i	−0.911211	−	0.411939i	$-0.864852\pi$
$599$		−	20.1640i		−	0.823878i	0.911211	−	0.411939i	$-0.135148\pi$
$600$	8.44490	+	1.15260i	0.344762	+	0.0470547i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	−36.1236	+	16.7058i	−1.46985	+	0.679748i
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	13.7092	+	44.7103i	0.555980	+	1.81324i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−45.6624			−1.84429			−0.922144	−	0.386848i	$-0.873564\pi$
$613$	−45.6624			−1.84429			−0.922144	+	0.386848i	$0.873564\pi$
$614$	15.1585	−	23.7112i	0.611747	−	0.956905i
$615$		0			0
$616$	−4.72482	+	34.6179i	−0.190368	+	1.39479i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$	0.785187	+	0.501968i	0.0315848	+	0.0201921i
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$	31.6516			1.27013
$622$	−23.6233	+	36.9521i	−0.947210	+	1.48164i
$623$		0			0
$624$	6.62295	+	5.62913i	0.265130	+	0.225346i
$625$	25.0000			1.00000
$626$	−4.14462	−	2.64964i	−0.165652	−	0.105901i
$627$		−	16.5245i		−	0.659925i
$628$	−19.8342	−	42.8884i	−0.791470	−	1.71143i
$629$		0			0
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−7.60431	+	3.51669i	−0.301530	+	0.139446i
$637$	−24.7768			−0.981692
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−26.0255			−1.02795			−0.513973	−	0.857807i	$-0.671827\pi$
$641$	−26.0255			−1.02795			−0.513973	+	0.857807i	$0.671827\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$	29.1315	+	62.9925i	1.14794	+	2.48225i
$645$		0			0
$646$		0			0
$647$		−	34.6283i		−	1.36138i	−0.732572	−	0.680690i	$-0.761680\pi$
$647$		−	34.6283i		−	1.36138i	0.732572	−	0.680690i	$-0.238320\pi$
$648$	−2.24251	+	16.4305i	−0.0880943	+	0.645451i
$649$		0			0
$650$	21.4806	+	13.7325i	0.842540	+	0.538633i
$651$		0			0
$652$		0			0
$653$	−18.0000			−0.704394			−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000			−0.704394			−0.352197	+	0.935926i	$0.614565\pi$
$654$	5.29379	−	8.28065i	0.207004	−	0.323799i
$655$		0			0
$656$	33.1744	−	39.0313i	1.29524	−	1.52392i
$657$	−44.9983			−1.75555
$658$		0			0
$659$		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			$\pi$
$662$		0			0
$663$		0			0
$664$	31.5165	+	4.30153i	1.22308	+	0.166932i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−33.9582	+	15.7043i	−1.31388	+	0.607619i
$669$		0			0
$670$		0			0
$671$		0			0
$672$	12.1400	−	3.72238i	0.468309	−	0.143594i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$		−	16.9859i		−	0.653787i
$676$	10.9134	+	23.5987i	0.419748	+	0.907641i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$	−6.20546	+	9.70669i	−0.238319	+	0.372783i
$679$		0			0
$680$		0			0
$681$	7.60157			0.291293
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	39.5696	−	18.2994i	1.51298	−	0.699695i
$685$		0			0
$686$	0.363593	−	0.568738i	0.0138820	−	0.0217145i
$687$		0			0
$688$		0			0
$689$	−25.0611			−0.954752
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$	32.5714			1.23729
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−0.990343	−	0.633123i	−0.0374850	−	0.0239641i
$699$		0			0
$700$	33.8050	−	15.6335i	1.27771	−	0.590890i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	9.33035	−	14.5947i	0.352151	−	0.550842i
$703$		0			0
$704$	7.11025	−	25.5626i	0.267978	−	0.963425i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		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
$716$	43.4152	−	20.0778i	1.62250	−	0.750344i
$717$	18.6197			0.695364
$718$	25.6446	−	40.1137i	0.957047	−	1.49703i
$719$			23.9165i			0.891936i	0.895049	+	0.445968i	$0.147140\pi$
$719$			23.9165i			0.891936i	−0.895049	+	0.445968i	$0.852860\pi$
$720$		0			0
$721$	4.07237			0.151663
$722$	−58.7929	−	37.5861i	−2.18804	−	1.39881i
$723$		−	5.03156i		−	0.187126i
$724$	17.0749	+	36.9219i	0.634584	+	1.37219i
$725$		0			0
$726$	−5.04996	+	7.89924i	−0.187422	+	0.293168i
$727$			25.7643i			0.955545i	0.878484	+	0.477772i	$0.158556\pi$
$727$			25.7643i			0.955545i	−0.878484	+	0.477772i	$0.841444\pi$
$728$	37.6336	+	5.13642i	1.39479	+	0.190368i
$729$	9.31696			0.345073
$730$		0			0
$731$		0			0
$732$		0			0
$733$	14.2038			0.524630			0.262315	−	0.964982i	$-0.415514\pi$
$733$	14.2038			0.524630			0.262315	+	0.964982i	$0.415514\pi$
$734$	17.1961	−	26.8985i	0.634720	−	0.992841i
$735$		0			0
$736$	−15.4506	−	50.3896i	−0.569515	−	1.85738i
$737$		0			0
$738$	−40.2345	−	25.7218i	−1.48105	−	0.946833i
$739$		−	41.1872i		−	1.51510i	−0.652779	−	0.757549i	$-0.726397\pi$
$739$		−	41.1872i		−	1.51510i	0.652779	−	0.757549i	$-0.273603\pi$
$740$		0			0
$741$	−17.9640			−0.659925
$742$	−19.7199	+	30.8462i	−0.723939	+	1.13240i
$743$		−	33.1662i		−	1.21675i	−0.793649	−	0.608376i	$-0.791821\pi$
$743$		−	33.1662i		−	1.21675i	0.793649	−	0.608376i	$-0.208179\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	29.6534i		−	1.08496i
$748$		0			0
$749$		0			0
$750$		0			0
$751$			18.9586i			0.691810i	0.938270	+	0.345905i	$0.112428\pi$
$751$			18.9586i			0.691810i	−0.938270	+	0.345905i	$0.887572\pi$
$752$		0			0
$753$	1.81914			0.0662931
$754$		0			0
$755$		0			0
$756$	−10.6219	−	22.9683i	−0.386316	−	0.835349i
$757$	44.0150			1.59975			0.799876	−	0.600165i	$-0.204898\pi$
$757$	44.0150			1.59975			0.799876	+	0.600165i	$0.204898\pi$
$758$		0			0
$759$			18.6235i			0.675990i
$760$		0			0
$761$	32.2987			1.17083			0.585414	−	0.810734i	$-0.300932\pi$
$761$	32.2987			1.17083			0.585414	+	0.810734i	$0.300932\pi$
$762$		0			0
$763$		−	42.9476i		−	1.55481i
$764$	49.7317	−	22.9990i	1.79923	−	0.832073i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−9.51683	+	1.55413i	−0.343409	+	0.0560798i
$769$	53.4536			1.92758			0.963792	−	0.266654i	$-0.0859179\pi$
$769$	53.4536			1.92758			0.963792	+	0.266654i	$0.0859179\pi$
$770$		0			0
$771$			4.69799i			0.169194i
$772$	19.6305	+	42.4478i	0.706515	+	1.52773i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	40.4937	+	25.8875i	1.45177	+	0.928112i
$779$			105.868i			3.79312i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	17.8015	−	20.9443i	0.635767	−	0.748011i
$785$		0			0
$786$		0			0
$787$			48.6362i			1.73369i	0.498574	+	0.866847i	$0.333857\pi$
$787$			48.6362i			1.73369i	−0.498574	+	0.866847i	$0.666143\pi$
$788$	2.96817	+	6.41821i	0.105737	+	0.228639i
$789$		0			0
$790$		0			0
$791$			50.3437i			1.79002i
$792$	−24.5079	−	3.34496i	−0.870852	−	0.118858i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−47.5436	+	21.9871i	−1.68514	+	0.779311i
$797$	30.0000			1.06265			0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000			1.06265			0.531327	+	0.847167i	$0.321693\pi$
$798$	−14.1354	+	22.1108i	−0.500387	+	0.782714i
$799$		0			0
$800$	−27.0416	+	8.29156i	−0.956066	+	0.293151i
$801$		0			0
$802$		0			0
$803$		−	56.6003i		−	1.99738i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	12.9383i		−	0.455450i
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$		−	26.2893i		−	0.923141i	−0.887103	−	0.461571i	$-0.847286\pi$
$811$		−	26.2893i		−	0.923141i	0.887103	−	0.461571i	$-0.152714\pi$
$812$		0			0
$813$	6.03578			0.211684
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−8.59226	−	5.49301i	−0.300421	−	0.192058i
$819$		−	35.4089i		−	1.23729i
$820$		0			0
$821$	−21.6333			−0.755008			−0.377504	−	0.926008i	$-0.623217\pi$
$821$	−21.6333			−0.755008			−0.377504	+	0.926008i	$0.623217\pi$
$822$		0			0
$823$		−	56.9956i		−	1.98674i	−0.114953	−	0.993371i	$-0.536672\pi$
$823$		−	56.9956i		−	1.98674i	0.114953	−	0.993371i	$-0.463328\pi$
$824$	−3.06420	−	0.418217i	−0.106747	−	0.0145693i
$825$	9.99433			0.347958
$826$		0			0
$827$			49.1766i			1.71004i	0.518597	+	0.855019i	$0.326454\pi$
$827$			49.1766i			1.71004i	−0.518597	+	0.855019i	$0.673546\pi$
$828$	−44.5959	+	20.6238i	−1.54981	+	0.716728i
$829$	57.4470			1.99522			0.997608	−	0.0691315i	$-0.0220228\pi$
$829$	57.4470			1.99522			0.997608	+	0.0691315i	$0.0220228\pi$
$830$		0			0
$831$		0			0
$832$	−27.7894	−	7.72966i	−0.963425	−	0.267978i
$833$		0			0
$834$		0			0
$835$		0			0
$836$	23.0176	+	49.7720i	0.796080	+	1.72140i
$837$		0			0
$838$	26.0895	−	40.8096i	0.901246	−	1.40975i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		−	19.3656i		−	0.666986i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			40.9694i			1.40773i
$848$	18.0057	−	21.1846i	0.618320	−	0.727484i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−56.7315			−1.94245			−0.971224	−	0.238168i	$-0.923453\pi$
$853$	−56.7315			−1.94245			−0.971224	+	0.238168i	$0.923453\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$	8.58738	+	5.48989i	0.293168	+	0.187422i
$859$			57.8057i			1.97230i	0.165841	+	0.986152i	$0.446966\pi$
$859$			57.8057i			1.97230i	−0.165841	+	0.986152i	$0.553034\pi$
$860$		0			0
$861$	28.7458			0.979655
$862$	−8.62193	+	13.4866i	−0.293664	+	0.459355i
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	5.63358	+	18.3730i	0.191658	+	0.625063i
$865$		0			0
$866$	48.9284	+	31.2798i	1.66265	+	1.06293i
$867$		−	10.2456i		−	0.347958i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−4.41055	+	32.3153i	−0.149360	+	1.09434i
$873$		0			0
$874$	91.7758	+	58.6720i	3.10436	+	1.98461i
$875$		0			0
$876$	−18.6704	+	8.63433i	−0.630815	+	0.291727i
$877$	33.6692			1.13693			0.568464	−	0.822708i	$-0.307538\pi$
$877$	33.6692			1.13693			0.568464	+	0.822708i	$0.307538\pi$
$878$		0			0
$879$			13.0380i			0.439760i
$880$		0			0
$881$	−54.4522			−1.83454			−0.917271	−	0.398265i	$-0.869613\pi$
$881$	−54.4522			−1.83454			−0.917271	+	0.398265i	$0.869613\pi$
$882$	−21.5900	−	13.8024i	−0.726972	−	0.464751i
$883$			50.5735i			1.70193i	0.525219	+	0.850967i	$0.323984\pi$
$883$			50.5735i			1.70193i	−0.525219	+	0.850967i	$0.676016\pi$
$884$		0			0
$885$		0			0
$886$	16.4936	−	25.7996i	0.554114	−	0.866755i
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$			19.4451i			0.651435i
$892$		0			0
$893$		0			0
$894$	−4.47439	+	6.99892i	−0.149646	+	0.234079i
$895$		0			0
$896$	−31.3807	+	28.1220i	−1.04835	+	0.939492i
$897$	20.2459			0.675990
$898$		0			0
$899$		0			0
$900$	11.0678	+	23.9325i	0.368927	+	0.797748i
$901$		0			0
$902$	32.3538	−	50.6083i	1.07726	−	1.68507i
$903$		0			0
$904$	5.17011	−	37.8805i	0.171955	−	1.25989i
$905$		0			0
$906$	14.2903	+	9.13573i	0.474762	+	0.303514i
$907$			44.6602i			1.48292i	0.670999	+	0.741458i	$0.265865\pi$
$907$			44.6602i			1.48292i	−0.670999	+	0.741458i	$0.734135\pi$
$908$	−22.8960	+	10.5885i	−0.759831	+	0.351392i
$909$		0			0
$910$		0			0
$911$			21.9143i			0.726052i	0.931779	+	0.363026i	$0.118256\pi$
$911$			21.9143i			0.726052i	−0.931779	+	0.363026i	$0.881744\pi$
$912$	12.9067	−	15.1853i	0.427383	−	0.502837i
$913$	37.2990			1.23442
$914$	−33.5234	−	21.4314i	−1.10885	−	0.708888i
$915$		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
$921$	−11.9932			−0.395189
$922$	−18.5031	−	11.8290i	−0.609368	−	0.389567i
$923$		0			0
$924$	13.5143	−	6.24985i	0.444589	−	0.205605i
$925$		0			0
$926$		0			0
$927$			2.88306i			0.0946920i
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$			56.8091i			1.86184i
$932$		0			0
$933$	18.6905			0.611899
$934$	18.2182	−	28.4973i	0.596119	−	0.932460i
$935$		0			0
$936$	−3.63636	+	26.6429i	−0.118858	+	0.870852i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$			2.09636i			0.0684121i
$940$		0			0
$941$	60.7292			1.97971			0.989857	−	0.142065i	$-0.0453741\pi$
$941$	60.7292			1.97971			0.989857	+	0.142065i	$0.0453741\pi$
$942$	−10.8465	+	16.9664i	−0.353399	+	0.552794i
$943$		−	119.316i		−	3.88546i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$	−61.5311			−1.99738
$950$	31.4864	−	49.2516i	1.02155	−	1.59793i
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$	−21.8377	−	13.9608i	−0.707022	−	0.451997i
$955$		0			0
$956$	−56.0827	+	25.9360i	−1.81384	+	0.838831i
$957$		0			0
$958$	5.05283	−	7.90373i	0.163250	−	0.255358i
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$	7.00865	+	15.1551i	0.225733	+	0.488114i
$965$		0			0
$966$	15.9309	−	24.9194i	0.512568	−	0.801769i
$967$			39.5867i			1.27302i	0.771267	+	0.636512i	$0.219623\pi$
$967$			39.5867i			1.27302i	−0.771267	+	0.636512i	$0.780377\pi$
$968$	4.20740	−	30.8269i	0.135231	−	0.990814i
$969$		0			0
$970$		0			0
$971$		−	49.3682i		−	1.58430i	−0.610327	−	0.792150i	$-0.708962\pi$
$971$		−	49.3682i		−	1.58430i	0.610327	−	0.792150i	$-0.291038\pi$
$972$	24.9147	−	11.5221i	0.799140	−	0.369571i
$973$		0			0
$974$		0			0
$975$		−	10.8650i		−	0.347958i
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	30.4050			0.970756
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	−21.6294	−	2.95209i	−0.689520	−	0.0941091i
$985$		0			0
$986$		0			0
$987$		0			0
$988$	54.1079	−	25.0227i	1.72140	−	0.796080i
$989$		0			0
$990$		0			0
$991$		−	37.0390i		−	1.17658i	−0.808649	−	0.588292i	$-0.799801\pi$
$991$		−	37.0390i		−	1.17658i	0.808649	−	0.588292i	$-0.200199\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	−5.68993	−	12.3036i	−0.180292	−	0.389855i
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.b.b.571.18	yes	20
4.3	odd	2	inner	572.2.b.b.571.17	yes	20
11.10	odd	2	inner	572.2.b.b.571.3	✓	20
13.12	even	2	inner	572.2.b.b.571.3	✓	20
44.43	even	2	inner	572.2.b.b.571.4	yes	20
52.51	odd	2	inner	572.2.b.b.571.4	yes	20
143.142	odd	2	CM	572.2.b.b.571.18	yes	20
572.571	even	2	inner	572.2.b.b.571.17	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.b.b.571.3	✓	20	11.10	odd	2	inner
572.2.b.b.571.3	✓	20	13.12	even	2	inner
572.2.b.b.571.4	yes	20	44.43	even	2	inner
572.2.b.b.571.4	yes	20	52.51	odd	2	inner
572.2.b.b.571.17	yes	20	4.3	odd	2	inner
572.2.b.b.571.17	yes	20	572.571	even	2	inner
572.2.b.b.571.18	yes	20	1.1	even	1	trivial
572.2.b.b.571.18	yes	20	143.142	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q+(1.19153 + 0.761743i) q^{2} -0.602681i q^{3} +(0.839496 + 1.81528i) q^{4} +(0.459088 - 0.718113i) q^{6} -3.72449i q^{7} +(-0.382491 + 2.80245i) q^{8} +2.63678 q^{9} +O(q^{10})\)	\(q+(1.19153 + 0.761743i) q^{2} -0.602681i q^{3} +(0.839496 + 1.81528i) q^{4} +(0.459088 - 0.718113i) q^{6} -3.72449i q^{7} +(-0.382491 + 2.80245i) q^{8} +2.63678 q^{9} +3.31662i q^{11} +(1.09403 - 0.505948i) q^{12} +3.60555 q^{13} +(2.83711 - 4.43785i) q^{14} +(-2.59049 + 3.04784i) q^{16} +(3.14180 + 2.00854i) q^{18} -8.26694i q^{19} -2.24468 q^{21} +(-2.52641 + 3.95186i) q^{22} +9.31703i q^{23} +(1.68898 + 0.230520i) q^{24} +5.00000 q^{25} +(4.29613 + 2.74650i) q^{26} -3.39718i q^{27} +(6.76100 - 3.12670i) q^{28} +(-5.40833 + 1.65831i) q^{32} +1.99887 q^{33} +(2.21356 + 4.78649i) q^{36} +(6.29728 - 9.85032i) q^{38} -2.17300i q^{39} -12.8062 q^{41} +(-2.67461 - 1.70987i) q^{42} +(-6.02061 + 2.78429i) q^{44} +(-7.09718 + 11.1015i) q^{46} +(1.83688 + 1.56124i) q^{48} -6.87184 q^{49} +(5.95766 + 3.80871i) q^{50} +(3.02685 + 6.54509i) q^{52} -6.95070 q^{53} +(2.58777 - 4.04784i) q^{54} +(10.4377 + 1.42459i) q^{56} -4.98232 q^{57} -9.82065i q^{63} +(-7.70740 - 2.14382i) q^{64} +(2.38171 + 1.52262i) q^{66} +5.61520 q^{69} +(-1.00854 + 7.38942i) q^{72} -17.0656 q^{73} -3.01340i q^{75} +(15.0068 - 6.94006i) q^{76} +12.3527 q^{77} +(1.65526 - 2.58919i) q^{78} +5.86292 q^{81} +(-15.2590 - 9.75503i) q^{82} -11.2461i q^{83} +(-1.88440 - 4.07473i) q^{84} +(-9.29466 - 1.26858i) q^{88} -13.4288i q^{91} +(-16.9130 + 7.82161i) q^{92} +(0.999433 + 3.25949i) q^{96} +(-8.18802 - 5.23458i) q^{98} +8.74520i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 60 q^{9}+O(q^{10}) \) 20 * q - 60 * q^9	\( 20 q - 60 q^{9} + 100 q^{25} + 10 q^{36} + 30 q^{38} - 50 q^{42} - 70 q^{48} - 140 q^{49} + 90 q^{56} + 110 q^{66} - 130 q^{78} + 180 q^{81} - 150 q^{92}+O(q^{100}) \) 20 * q - 60 * q^9 + 100 * q^25 + 10 * q^36 + 30 * q^38 - 50 * q^42 - 70 * q^48 - 140 * q^49 + 90 * q^56 + 110 * q^66 - 130 * q^78 + 180 * q^81 - 150 * q^92

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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