LMFDB - Embedded newform 572.2.b.b.571.20

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			571.20
Root			$-1.41171 + 0.0841020i$ of defining polynomial
Character	$\chi$	$=$	572.571
Dual form			572.2.b.b.571.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.41171 + 0.0841020i) q^{2} -2.49267i q^{3} +(1.98585 + 0.237455i) q^{4} +(0.209639 - 3.51893i) q^{6} -4.72571i q^{7} +(2.78348 + 0.502232i) q^{8} -3.21342 q^{9} +O(q^{10})$	\(q+(1.41171 + 0.0841020i) q^{2} -2.49267i q^{3} +(1.98585 + 0.237455i) q^{4} +(0.209639 - 3.51893i) q^{6} -4.72571i q^{7} +(2.78348 + 0.502232i) q^{8} -3.21342 q^{9} +3.31662i q^{11} +(0.591899 - 4.95009i) q^{12} -3.60555 q^{13} +(0.397442 - 6.67133i) q^{14} +(3.88723 + 0.943103i) q^{16} +(-4.53643 - 0.270255i) q^{18} +8.31465i q^{19} -11.7797 q^{21} +(-0.278935 + 4.68211i) q^{22} -5.04638i q^{23} +(1.25190 - 6.93831i) q^{24} +5.00000 q^{25} +(-5.09000 - 0.303234i) q^{26} +0.531998i q^{27} +(1.12214 - 9.38457i) q^{28} +(5.40833 + 1.65831i) q^{32} +8.26727 q^{33} +(-6.38139 - 0.763045i) q^{36} +(-0.699278 + 11.7379i) q^{38} +8.98746i q^{39} +3.99186 q^{41} +(-16.6295 - 0.990692i) q^{42} +(-0.787550 + 6.58633i) q^{44} +(0.424411 - 7.12403i) q^{46} +(2.35085 - 9.68960i) q^{48} -15.3323 q^{49} +(7.05855 + 0.420510i) q^{50} +(-7.16010 - 0.856157i) q^{52} +13.1434 q^{53} +(-0.0447421 + 0.751028i) q^{54} +(2.37340 - 13.1539i) q^{56} +20.7257 q^{57} +15.1857i q^{63} +(7.49553 + 2.79591i) q^{64} +(11.6710 + 0.695293i) q^{66} -12.5790 q^{69} +(-8.94450 - 1.61389i) q^{72} -14.3201 q^{73} -12.4634i q^{75} +(-1.97436 + 16.5117i) q^{76} +15.6734 q^{77} +(-0.755864 + 12.6877i) q^{78} -8.31418 q^{81} +(5.63535 + 0.335723i) q^{82} +10.1593i q^{83} +(-23.3927 - 2.79714i) q^{84} +(-1.66572 + 9.23176i) q^{88} +17.0388i q^{91} +(1.19829 - 10.0214i) q^{92} +(4.13363 - 13.4812i) q^{96} +(-21.6448 - 1.28948i) q^{98} -10.6577i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.41171	+	0.0841020i	0.998230	+	0.0594691i
$2$	1.41171	+	0.0841020i	0.998230	+	0.0594691i
$3$		−	2.49267i		−	1.43915i	−0.694417	−	0.719573i	$-0.744338\pi$
$3$		−	2.49267i		−	1.43915i	0.694417	−	0.719573i	$-0.255662\pi$
$4$	1.98585	+	0.237455i	0.992927	+	0.118728i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$	0.209639	−	3.51893i	0.0855847	−	1.43660i
$7$		−	4.72571i		−	1.78615i	−0.449908	−	0.893075i	$-0.648543\pi$
$7$		−	4.72571i		−	1.78615i	0.449908	−	0.893075i	$-0.351457\pi$
$8$	2.78348	+	0.502232i	0.984109	+	0.177566i
$9$	−3.21342			−1.07114
$10$		0			0
$11$			3.31662i			1.00000i
$11$			3.31662i			1.00000i
$12$	0.591899	−	4.95009i	0.170866	−	1.42897i
$13$	−3.60555			−1.00000
$13$	−3.60555			−1.00000
$14$	0.397442	−	6.67133i	0.106221	−	1.78299i
$15$		0			0
$16$	3.88723	+	0.943103i	0.971807	+	0.235776i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	−4.53643	−	0.270255i	−1.06925	−	0.0636998i
$19$			8.31465i			1.90751i	0.300583	+	0.953756i	$0.402819\pi$
$19$			8.31465i			1.90751i	−0.300583	+	0.953756i	$0.597181\pi$
$20$		0			0
$21$	−11.7797			−2.57053
$22$	−0.278935	+	4.68211i	−0.0594691	+	0.998230i
$23$		−	5.04638i		−	1.05224i	−0.850409	−	0.526122i	$-0.823646\pi$
$23$		−	5.04638i		−	1.05224i	0.850409	−	0.526122i	$-0.176354\pi$
$24$	1.25190	−	6.93831i	0.255543	−	1.41628i
$25$	5.00000			1.00000
$26$	−5.09000	−	0.303234i	−0.998230	−	0.0594691i
$27$			0.531998i			0.102383i
$28$	1.12214	−	9.38457i	0.212065	−	1.77352i
$29$		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
$33$	8.26727			1.43915
$34$		0			0
$35$		0			0
$36$	−6.38139	−	0.763045i	−1.06357	−	0.127174i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−0.699278	+	11.7379i	−0.113438	+	1.90414i
$39$			8.98746i			1.43915i
$40$		0			0
$41$	3.99186			0.623424			0.311712	−	0.950177i	$-0.399098\pi$
$41$	3.99186			0.623424			0.311712	+	0.950177i	$0.399098\pi$
$42$	−16.6295	−	0.990692i	−2.56598	−	0.152867i
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−0.787550	+	6.58633i	−0.118728	+	0.992927i
$45$		0			0
$46$	0.424411	−	7.12403i	0.0625759	−	1.05038i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	2.35085	−	9.68960i	0.339316	−	1.39857i
$49$	−15.3323			−2.19033
$50$	7.05855	+	0.420510i	0.998230	+	0.0594691i
$51$		0			0
$52$	−7.16010	−	0.856157i	−0.992927	−	0.118728i
$53$	13.1434			1.80538			0.902692	−	0.430288i	$-0.141588\pi$
$53$	13.1434			1.80538			0.902692	+	0.430288i	$0.141588\pi$
$54$	−0.0447421	+	0.751028i	−0.00608863	+	0.102202i
$55$		0			0
$56$	2.37340	−	13.1539i	0.317160	−	1.75777i
$57$	20.7257			2.74519
$58$		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
$63$			15.1857i			1.91322i
$64$	7.49553	+	2.79591i	0.936941	+	0.349489i
$65$		0			0
$66$	11.6710	+	0.695293i	1.43660	+	0.0855847i
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$	−12.5790			−1.51433
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−8.94450	−	1.61389i	−1.05412	−	0.190198i
$73$	−14.3201			−1.67604			−0.838021	−	0.545638i	$-0.816287\pi$
$73$	−14.3201			−1.67604			−0.838021	+	0.545638i	$0.816287\pi$
$74$		0			0
$75$		−	12.4634i		−	1.43915i
$76$	−1.97436	+	16.5117i	−0.226474	+	1.89402i
$77$	15.6734			1.78615
$78$	−0.755864	+	12.6877i	−0.0855847	+	1.43660i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−8.31418			−0.923797
$82$	5.63535	+	0.335723i	0.622320	+	0.0370744i
$83$			10.1593i			1.11512i	0.830135	+	0.557562i	$0.188263\pi$
$83$			10.1593i			1.11512i	−0.830135	+	0.557562i	$0.811737\pi$
$84$	−23.3927	−	2.79714i	−2.55235	−	0.305193i
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−1.66572	+	9.23176i	−0.177566	+	0.984109i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$			17.0388i			1.78615i
$92$	1.19829	−	10.0214i	0.124930	−	1.04480i
$93$		0			0
$94$		0			0
$95$		0			0
$96$	4.13363	−	13.4812i	0.421887	−	1.37592i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−21.6448	−	1.28948i	−2.18646	−	0.130257i
$99$		−	10.6577i		−	1.07114i
$100$	9.92927	+	1.18728i	0.992927	+	0.118728i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$			18.9384i			1.86606i	0.359800	+	0.933029i	$0.382845\pi$
$103$			18.9384i			1.86606i	−0.359800	+	0.933029i	$0.617155\pi$
$104$	−10.0360	−	1.81082i	−0.984109	−	0.177566i
$105$		0			0
$106$	18.5547	+	1.10539i	1.80219	+	0.107364i
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−0.126326	+	1.05647i	−0.0121557	+	0.101659i
$109$	19.5610			1.87360			0.936800	−	0.349866i	$-0.113773\pi$
$109$	19.5610			1.87360			0.936800	+	0.349866i	$0.113773\pi$
$110$		0			0
$111$		0			0
$112$	4.45683	−	18.3699i	0.421131	−	1.73579i
$113$	1.28979			0.121333			0.0606667	−	0.998158i	$-0.480677\pi$
$113$	1.28979			0.121333			0.0606667	+	0.998158i	$0.480677\pi$
$114$	29.2587	+	1.74307i	2.74033	+	0.163254i
$115$		0			0
$116$		0			0
$117$	11.5862			1.07114
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		−	9.95041i		−	0.897198i
$124$		0			0
$125$		0			0
$126$	−1.27715	+	21.4378i	−0.113777	+	1.90983i
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	10.3464	+	4.57740i	0.914499	+	0.404589i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	16.4176	+	1.96311i	1.42897	+	0.170866i
$133$	39.2926			3.40710
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$	−17.7579	−	1.05792i	−1.51165	−	0.0900559i
$139$		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
$144$	−12.4913	−	3.03059i	−1.04094	−	0.252549i
$145$		0			0
$146$	−20.2158	−	1.20435i	−1.67308	−	0.0996727i
$147$			38.2185i			3.15221i
$148$		0			0
$149$	−18.2760			−1.49723			−0.748613	−	0.663007i	$-0.769280\pi$
$149$	−18.2760			−1.49723			−0.748613	+	0.663007i	$0.769280\pi$
$150$	1.04819	−	17.5947i	0.0855847	−	1.43660i
$151$			19.8997i			1.61942i	0.586831	+	0.809709i	$0.300375\pi$
$151$			19.8997i			1.61942i	−0.586831	+	0.809709i	$0.699625\pi$
$152$	−4.17589	+	23.1437i	−0.338709	+	1.87720i
$153$		0			0
$154$	22.1263	+	1.31816i	1.78299	+	0.106221i
$155$		0			0
$156$	−2.13412	+	17.8478i	−0.170866	+	1.42897i
$157$	−15.2466			−1.21681			−0.608404	−	0.793628i	$-0.708190\pi$
$157$	−15.2466			−1.21681			−0.608404	+	0.793628i	$0.708190\pi$
$158$		0			0
$159$		−	32.7622i		−	2.59821i
$160$		0			0
$161$	−23.8477			−1.87946
$162$	−11.7372	−	0.699239i	−0.922162	−	0.0549374i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$	7.92725	+	0.947888i	0.619014	+	0.0740177i
$165$		0			0
$166$	−0.854415	+	14.3419i	−0.0663154	+	1.11315i
$167$		−	25.6168i		−	1.98228i	−0.132804	−	0.991142i	$-0.542398\pi$
$167$		−	25.6168i		−	1.98228i	0.132804	−	0.991142i	$-0.457602\pi$
$168$	−32.7884	−	5.91612i	−2.52968	−	0.456439i
$169$	13.0000			1.00000
$170$		0			0
$171$		−	26.7185i		−	2.04321i
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		−	23.6285i		−	1.78615i
$176$	−3.12792	+	12.8925i	−0.235776	+	0.971807i
$177$		0			0
$178$		0			0
$179$			23.9165i			1.78760i	0.448461	+	0.893802i	$0.351972\pi$
$179$			23.9165i			1.78760i	−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	−6.10085			−0.453473			−0.226736	−	0.973956i	$-0.572806\pi$
$181$	−6.10085			−0.453473			−0.226736	+	0.973956i	$0.572806\pi$
$182$	−1.43300	+	24.0538i	−0.106221	+	1.78299i
$183$		0			0
$184$	2.53446	−	14.0465i	0.186843	−	1.03552i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	2.51407			0.182872
$190$		0			0
$191$		−	20.0082i		−	1.44774i	−0.689935	−	0.723871i	$-0.742361\pi$
$191$		−	20.0082i		−	1.44774i	0.689935	−	0.723871i	$-0.257639\pi$
$192$	6.96929	−	18.6839i	0.502965	−	1.34839i
$193$	−21.4983			−1.54748			−0.773742	−	0.633501i	$-0.781617\pi$
$193$	−21.4983			−1.54748			−0.773742	+	0.633501i	$0.781617\pi$
$194$		0			0
$195$		0			0
$196$	−30.4478	−	3.64074i	−2.17484	−	0.260053i
$197$	−13.5081			−0.962413			−0.481207	−	0.876607i	$-0.659801\pi$
$197$	−13.5081			−0.962413			−0.481207	+	0.876607i	$0.659801\pi$
$198$	0.896336	−	15.0456i	0.0636998	−	1.06925i
$199$			15.0228i			1.06494i	0.846448	+	0.532471i	$0.178736\pi$
$199$			15.0228i			1.06494i	−0.846448	+	0.532471i	$0.821264\pi$
$200$	13.9174	+	2.51116i	0.984109	+	0.177566i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−1.59276	+	26.7356i	−0.110973	+	1.86276i
$207$			16.2162i			1.12710i
$208$	−14.0156	−	3.40041i	−0.971807	−	0.235776i
$209$	−27.5766			−1.90751
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$	26.1008	+	3.12097i	1.79261	+	0.214349i
$213$		0			0
$214$		0			0
$215$		0			0
$216$	−0.267187	+	1.48081i	−0.0181798	+	0.100756i
$217$		0			0
$218$	27.6144	+	1.64511i	1.87028	+	0.111421i
$219$			35.6954i			2.41207i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	7.83670	−	25.5582i	0.523612	−	1.70768i
$225$	−16.0671			−1.07114
$226$	1.82081	+	0.108474i	0.121119	+	0.00721558i
$227$		−	26.2896i		−	1.74490i	−0.488703	−	0.872450i	$-0.662530\pi$
$227$		−	26.2896i		−	1.74490i	0.488703	−	0.872450i	$-0.337470\pi$
$228$	41.1582	+	4.92143i	2.72577	+	0.325930i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		−	39.0687i		−	2.57053i
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$	16.3563	+	0.974420i	1.06925	+	0.0636998i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	24.2711i		−	1.56997i	−0.619515	−	0.784984i	$-0.712671\pi$
$239$		−	24.2711i		−	1.56997i	0.619515	−	0.784984i	$-0.287329\pi$
$240$		0			0
$241$	−31.0211			−1.99824			−0.999122	−	0.0419050i	$-0.986657\pi$
$241$	−31.0211			−1.99824			−0.999122	+	0.0419050i	$0.986657\pi$
$242$	−15.5288	−	0.925122i	−0.998230	−	0.0594691i
$243$			22.3205i			1.43186i
$244$		0			0
$245$		0			0
$246$	0.836849	−	14.0471i	0.0533555	−	0.895610i
$247$		−	29.9789i		−	1.90751i
$248$		0			0
$249$	25.3237			1.60483
$250$		0			0
$251$		−	30.9308i		−	1.95234i	−0.217014	−	0.976169i	$-0.569632\pi$
$251$		−	30.9308i		−	1.95234i	0.217014	−	0.976169i	$-0.430368\pi$
$252$	−3.60593	+	30.1566i	−0.227152	+	1.89969i
$253$	16.7370			1.05224
$254$		0			0
$255$		0			0
$256$	14.2211	+	7.33212i	0.888820	+	0.458257i
$257$	−31.9871			−1.99530			−0.997650	−	0.0685200i	$-0.978172\pi$
$257$	−31.9871			−1.99530			−0.997650	+	0.0685200i	$0.978172\pi$
$258$		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	$0.5\pi$
$264$	23.0118	+	4.15209i	1.41628	+	0.255543i
$265$		0			0
$266$	55.4698	+	3.30459i	3.40107	+	0.202617i
$267$		0			0
$268$		0			0
$269$	−16.9540			−1.03370			−0.516850	−	0.856076i	$-0.672896\pi$
$269$	−16.9540			−1.03370			−0.516850	+	0.856076i	$0.672896\pi$
$270$		0			0
$271$			10.3331i			0.627691i	0.949474	+	0.313845i	$0.101617\pi$
$271$			10.3331i			0.627691i	−0.949474	+	0.313845i	$0.898383\pi$
$272$		0			0
$273$	42.4721			2.57053
$274$		0			0
$275$			16.5831i			1.00000i
$276$	−24.9800	−	2.98695i	−1.50362	−	0.179793i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	20.3729			1.21535			0.607674	−	0.794187i	$-0.292103\pi$
$281$	20.3729			1.21535			0.607674	+	0.794187i	$0.292103\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$		0			0
$286$	1.00571	−	16.8816i	0.0594691	−	0.998230i
$287$		−	18.8644i		−	1.11353i
$288$	−17.3793	−	5.32886i	−1.02408	−	0.314006i
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$	−28.4376	−	3.40039i	−1.66419	−	0.198992i
$293$	21.6333			1.26383			0.631916	−	0.775037i	$-0.282269\pi$
$293$	21.6333			1.26383			0.631916	+	0.775037i	$0.282269\pi$
$294$	−3.21425	+	53.9535i	−0.187459	+	3.14663i
$295$		0			0
$296$		0			0
$297$	−1.76444			−0.102383
$298$	−25.8004	−	1.53705i	−1.49458	−	0.0890387i
$299$			18.1950i			1.05224i
$300$	2.95949	−	24.7504i	0.170866	−	1.42897i
$301$		0			0
$302$	−1.67361	+	28.0927i	−0.0963053	+	1.61655i
$303$		0			0
$304$	−7.84157	+	32.3210i	−0.449745	+	1.85373i
$305$		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	$-0.192232\pi$
$308$	31.1251	+	3.72173i	1.77352	+	0.212065i
$309$	47.2073			2.68553
$310$		0			0
$311$			34.9642i			1.98264i	0.131470	+	0.991320i	$0.458030\pi$
$311$			34.9642i			1.98264i	−0.131470	+	0.991320i	$0.541970\pi$
$312$	−4.51380	+	25.0164i	−0.255543	+	1.41628i
$313$	32.4139			1.83214			0.916072	−	0.401014i	$-0.131342\pi$
$313$	32.4139			1.83214			0.916072	+	0.401014i	$0.131342\pi$
$314$	−21.5237	−	1.28227i	−1.21465	−	0.0723624i
$315$		0			0
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$	2.75537	−	46.2507i	0.154513	−	2.59361i
$319$		0			0
$320$		0			0
$321$		0			0
$322$	−33.6661	−	2.00564i	−1.87614	−	0.111770i
$323$		0			0
$324$	−16.5107	−	1.97425i	−0.917263	−	0.109680i
$325$	−18.0278			−1.00000
$326$		0			0
$327$		−	48.7591i		−	2.69638i
$328$	11.1113	+	2.00484i	0.613517	+	0.110699i
$329$		0			0
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$	−2.41237	+	20.1748i	−0.132396	+	1.10724i
$333$		0			0
$334$	2.15442	−	36.1635i	0.117885	−	1.97878i
$335$		0			0
$336$	−45.7902	−	11.1094i	−2.49806	−	0.606069i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	18.3522	+	1.09333i	0.998230	+	0.0594691i
$339$		−	3.21503i		−	0.174616i
$340$		0			0
$341$		0			0
$342$	2.24708	−	37.7188i	0.121508	−	2.03960i
$343$			39.3762i			2.12611i
$344$		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
$349$	35.7825			1.91539			0.957695	−	0.287784i	$-0.0929185\pi$
$349$	35.7825			1.91539			0.957695	+	0.287784i	$0.0929185\pi$
$350$	1.98721	−	33.3567i	0.106221	−	1.78299i
$351$		−	1.91815i		−	0.102383i
$352$	−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
$358$	−2.01143	+	33.7632i	−0.106307	+	1.78444i
$359$			6.14141i			0.324131i	0.986780	+	0.162066i	$0.0518156\pi$
$359$			6.14141i			0.324131i	−0.986780	+	0.162066i	$0.948184\pi$
$360$		0			0
$361$	−50.1334			−2.63860
$362$	−8.61264	−	0.513094i	−0.452670	−	0.0269676i
$363$			27.4194i			1.43915i
$364$	−4.04595	+	33.8365i	−0.212065	+	1.77352i
$365$		0			0
$366$		0			0
$367$		−	0.0668021i		−	0.00348704i	−0.999998	−	0.00174352i	$-0.999445\pi$
$367$		−	0.0668021i		−	0.00348704i	0.999998	−	0.00174352i	$-0.000554980\pi$
$368$	4.75926	−	19.6164i	0.248093	−	1.02258i
$369$	−12.8275			−0.667775
$370$		0			0
$371$		−	62.1118i		−	3.22469i
$372$		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
$378$	3.54914	+	0.211438i	0.182548	+	0.0108752i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$	1.68273	−	28.2458i	0.0860959	−	1.44518i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	11.4100	−	25.7901i	0.582263	−	1.31610i
$385$		0			0
$386$	−30.3494	−	1.80805i	−1.54474	−	0.0920274i
$387$		0			0
$388$		0			0
$389$	−15.7230			−0.797186			−0.398593	−	0.917128i	$-0.630501\pi$
$389$	−15.7230			−0.797186			−0.398593	+	0.917128i	$0.630501\pi$
$390$		0			0
$391$		0			0
$392$	−42.6772	−	7.70039i	−2.15553	−	0.388929i
$393$		0			0
$394$	−19.0695	−	1.13606i	−0.960710	−	0.0572338i
$395$		0			0
$396$	2.53073	−	21.1647i	0.127174	−	1.06357i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	−1.26345	+	21.2079i	−0.0633311	+	1.06306i
$399$		−	97.9437i		−	4.90332i
$400$	19.4361	+	4.71552i	0.971807	+	0.235776i
$401$		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	$-0.442946\pi$
$409$	7.21110			0.356566			0.178283	+	0.983979i	$0.442946\pi$
$410$		0			0
$411$		0			0
$412$	−4.49703	+	37.6090i	−0.221553	+	1.85286i
$413$		0			0
$414$	−1.36381	+	22.8925i	−0.0670277	+	1.12511i
$415$		0			0
$416$	−19.5000	−	5.97913i	−0.956066	−	0.293151i
$417$		0			0
$418$	−38.9301	−	2.31924i	−1.90414	−	0.113438i
$419$		−	10.7453i		−	0.524943i	−0.964940	−	0.262471i	$-0.915462\pi$
$419$		−	10.7453i		−	0.524943i	0.964940	−	0.262471i	$-0.0845375\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	36.5844	+	6.60104i	1.77669	+	0.320575i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−29.8080			−1.43915
$430$		0			0
$431$		−	34.4957i		−	1.66160i	−0.556573	−	0.830799i	$-0.687884\pi$
$431$		−	34.4957i		−	1.66160i	0.556573	−	0.830799i	$-0.312116\pi$
$432$	−0.501729	+	2.06800i	−0.0241395	+	0.0994967i
$433$	−37.1987			−1.78766			−0.893828	−	0.448411i	$-0.851990\pi$
$433$	−37.1987			−1.78766			−0.893828	+	0.448411i	$0.851990\pi$
$434$		0			0
$435$		0			0
$436$	38.8452	+	4.64485i	1.86035	+	0.222448i
$437$	41.9589			2.00717
$438$	−3.00205	+	50.3915i	−0.143444	+	2.40780i
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	49.2693			2.34616
$442$		0			0
$443$			41.0236i			1.94909i	0.224192	+	0.974545i	$0.428026\pi$
$443$			41.0236i			1.94909i	−0.224192	+	0.974545i	$0.571974\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			45.5561i			2.15473i
$448$	13.2126	−	35.4217i	0.624239	−	1.67352i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−22.6821	−	1.35128i	−1.06925	−	0.0636998i
$451$			13.2395i			0.623424i
$452$	2.56134	+	0.306268i	0.120475	+	0.0144056i
$453$	49.6036			2.33058
$454$	2.21101	−	37.1133i	0.103768	−	1.74181i
$455$		0			0
$456$	57.6896	+	10.4091i	2.70156	+	0.487452i
$457$	−3.83841			−0.179553			−0.0897767	−	0.995962i	$-0.528615\pi$
$457$	−3.83841			−0.179553			−0.0897767	+	0.995962i	$0.528615\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−36.0955			−1.68113			−0.840567	−	0.541708i	$-0.817778\pi$
$461$	−36.0955			−1.68113			−0.840567	+	0.541708i	$0.817778\pi$
$462$	3.28575	−	55.1537i	0.152867	−	2.56598i
$463$		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	$0.186655\pi$
$467$			23.9165i			1.10672i	−0.832941	+	0.553362i	$0.813345\pi$
$468$	23.0084	+	2.75120i	1.06357	+	0.127174i
$469$		0			0
$470$		0			0
$471$			38.0047i			1.75116i
$472$		0			0
$473$		0			0
$474$		0			0
$475$			41.5732i			1.90751i
$476$		0			0
$477$	−42.2353			−1.93382
$478$	2.04125	−	34.2638i	0.0933646	−	1.56719i
$479$		−	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
$482$	−43.7928	−	2.60893i	−1.99471	−	0.118834i
$483$			59.4446i			2.70482i
$484$	−21.8444	−	2.61201i	−0.992927	−	0.118728i
$485$		0			0
$486$	−1.87720	+	31.5101i	−0.0851515	+	1.42933i
$487$		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	$0.5\pi$
$492$	2.36278	−	19.7601i	0.106522	−	0.890852i
$493$		0			0
$494$	2.52128	−	42.3215i	0.113438	−	1.90414i
$495$		0			0
$496$		0			0
$497$		0			0
$498$	35.7498	+	2.12978i	1.60199	+	0.0954376i
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	−63.8542			−2.85280
$502$	2.60134	−	43.6654i	0.116104	−	1.94888i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	−7.62676	+	42.2691i	−0.339723	+	1.88282i
$505$		0			0
$506$	23.6277	+	1.40761i	1.05038	+	0.0625759i
$507$		−	32.4048i		−	1.43915i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$			67.6727i			2.99366i
$512$	19.4595	+	11.5469i	0.859994	+	0.510304i
$513$	−4.42338			−0.195297
$514$	−45.1565	−	2.69018i	−1.99177	−	0.118659i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	30.7065			1.34528			0.672639	−	0.739971i	$-0.265161\pi$
$521$	30.7065			1.34528			0.672639	+	0.739971i	$0.265161\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$	−58.8983			−2.57053
$526$		0			0
$527$		0			0
$528$	32.1368	+	7.79688i	1.39857	+	0.339316i
$529$	−2.46597			−0.107216
$530$		0			0
$531$		0			0
$532$	78.0294	+	9.33024i	3.38300	+	0.404517i
$533$	−14.3929			−0.623424
$534$		0			0
$535$		0			0
$536$		0			0
$537$	59.6161			2.57262
$538$	−23.9341	−	1.42586i	−1.03187	−	0.0614732i
$539$		−	50.8516i		−	2.19033i
$540$		0			0
$541$	7.21421			0.310163			0.155081	−	0.987902i	$-0.450436\pi$
$541$	7.21421			0.310163			0.155081	+	0.987902i	$0.450436\pi$
$542$	−0.869034	+	14.5873i	−0.0373282	+	0.626580i
$543$			15.2074i			0.652614i
$544$		0			0
$545$		0			0
$546$	59.9584	+	3.57199i	2.56598	+	0.152867i
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		0			0
$549$		0			0
$550$	−1.39467	+	23.4106i	−0.0594691	+	0.998230i
$551$		0			0
$552$	−35.0134	−	6.31757i	−1.49027	−	0.268894i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−34.2434			−1.45094			−0.725470	−	0.688254i	$-0.758378\pi$
$557$	−34.2434			−1.45094			−0.725470	+	0.688254i	$0.758378\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	28.7607	+	1.71341i	1.21320	+	0.0722756i
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$			39.2904i			1.65004i
$568$		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	$0.5\pi$
$572$	2.83955	−	23.7474i	0.118728	−	0.992927i
$573$	−49.8739			−2.08351
$574$	1.58653	−	26.6310i	0.0662205	−	1.11156i
$575$		−	25.2319i		−	1.05224i
$576$	−24.0863	−	8.98444i	−1.00360	−	0.374352i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	23.9991	+	1.42973i	0.998230	+	0.0594691i
$579$			53.5883i			2.22705i
$580$		0			0
$581$	48.0098			1.99178
$582$		0			0
$583$			43.5917i			1.80538i
$584$	−39.8597	−	7.19202i	−1.64941	−	0.297608i
$585$		0			0
$586$	30.5400	+	1.81940i	1.26159	+	0.0751589i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−9.07519	+	75.8964i	−0.374254	+	3.12991i
$589$		0			0
$590$		0			0
$591$			33.6713i			1.38505i
$592$		0			0
$593$	48.5275			1.99279			0.996394	−	0.0848503i	$-0.0270412\pi$
$593$	48.5275			1.99279			0.996394	+	0.0848503i	$0.0270412\pi$
$594$	−2.49088	−	0.148393i	−0.102202	−	0.00608863i
$595$		0			0
$596$	−36.2934	−	4.33973i	−1.48664	−	0.177762i
$597$	37.4471			1.53261
$598$	−1.53023	+	25.6861i	−0.0625759	+	1.05038i
$599$			9.90389i			0.404662i	0.979317	+	0.202331i	$0.0648517\pi$
$599$			9.90389i			0.404662i	−0.979317	+	0.202331i	$0.935148\pi$
$600$	6.25951	−	34.6915i	0.255543	−	1.41628i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	−4.72530	+	39.5180i	−0.192270	+	1.60796i
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	−13.7883	+	44.9683i	−0.559189	+	1.82371i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−48.2012			−1.94683			−0.973413	−	0.229056i	$-0.926436\pi$
$613$	−48.2012			−1.94683			−0.973413	+	0.229056i	$0.926436\pi$
$614$	1.67361	−	28.0927i	0.0675413	−	1.13373i
$615$		0			0
$616$	43.6266	+	7.87169i	1.75777	+	0.317160i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$	66.6431	+	3.97023i	2.68078	+	0.159706i
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$	2.68467			0.107732
$622$	−2.94056	+	49.3594i	−0.117906	+	1.97913i
$623$		0			0
$624$	−8.47611	+	34.9363i	−0.339316	+	1.39857i
$625$	25.0000			1.00000
$626$	45.7591	+	2.72608i	1.82890	+	0.108956i
$627$			68.7394i			2.74519i
$628$	−30.2774	−	3.62038i	−1.20820	−	0.144469i
$629$		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
$636$	7.77956	−	65.0609i	0.308479	−	2.57983i
$637$	55.2815			2.19033
$638$		0			0
$639$		0			0
$640$		0			0
$641$	33.2676			1.31399			0.656996	−	0.753894i	$-0.271827\pi$
$641$	33.2676			1.31399			0.656996	+	0.753894i	$0.271827\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$	−47.3581	−	5.66277i	−1.86617	−	0.223144i
$645$		0			0
$646$		0			0
$647$		−	49.9203i		−	1.96257i	−0.192564	−	0.981284i	$-0.561680\pi$
$647$		−	49.9203i		−	1.96257i	0.192564	−	0.981284i	$-0.438320\pi$
$648$	−23.1423	−	4.17565i	−0.909117	−	0.164035i
$649$		0			0
$650$	−25.4500	−	1.51617i	−0.998230	−	0.0594691i
$651$		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	$0.614565\pi$
$654$	4.10074	−	68.8337i	0.160351	−	2.69161i
$655$		0			0
$656$	15.5173	+	3.76474i	0.605848	+	0.146988i
$657$	46.0166			1.79528
$658$		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
$664$	−5.10231	+	28.2781i	−0.198008	+	1.09740i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	6.08284	−	50.8711i	0.235352	−	1.96826i
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−63.7082	−	19.5343i	−2.45760	−	0.753554i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$			2.65999i			0.102383i
$676$	25.8161	+	3.08692i	0.992927	+	0.118728i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$	0.270390	−	4.53869i	0.0103843	−	0.174307i
$679$		0			0
$680$		0			0
$681$	−65.5314			−2.51117
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	6.34445	−	53.0590i	0.242586	−	2.02876i
$685$		0			0
$686$	−3.31161	+	55.5878i	−0.126438	+	2.12235i
$687$		0			0
$688$		0			0
$689$	−47.3892			−1.80538
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$	−50.3653			−1.91322
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	50.5145	+	3.00938i	1.91200	+	0.113907i
$699$		0			0
$700$	5.61072	−	46.9228i	0.212065	−	1.77352i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	0.161320	−	2.70787i	0.00608863	−	0.102202i
$703$		0			0
$704$	−9.27298	+	24.8598i	−0.349489	+	0.936941i
$705$		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
$716$	−5.67911	+	47.4947i	−0.212238	+	1.77496i
$717$	−60.5000			−2.25941
$718$	−0.516505	+	8.66989i	−0.0192758	+	0.323558i
$719$		−	23.9165i		−	0.891936i	−0.895049	−	0.445968i	$-0.852860\pi$
$719$		−	23.9165i		−	0.891936i	0.895049	−	0.445968i	$-0.147140\pi$
$720$		0			0
$721$	89.4975			3.33306
$722$	−70.7738	−	4.21632i	−2.63393	−	0.156915i
$723$			77.3254i			2.87576i
$724$	−12.1154	−	1.44868i	−0.450265	−	0.0538398i
$725$		0			0
$726$	−2.30603	+	38.7083i	−0.0855847	+	1.43660i
$727$		−	53.0160i		−	1.96625i	−0.182922	−	0.983127i	$-0.558556\pi$
$727$		−	53.0160i		−	1.96625i	0.182922	−	0.983127i	$-0.441444\pi$
$728$	−8.55743	+	47.4271i	−0.317160	+	1.75777i
$729$	30.6953			1.13686
$730$		0			0
$731$		0			0
$732$		0			0
$733$	45.3052			1.67339			0.836693	−	0.547672i	$-0.184486\pi$
$733$	45.3052			1.67339			0.836693	+	0.547672i	$0.184486\pi$
$734$	0.00561819	−	0.0943052i	0.000207371	−	0.00348087i
$735$		0			0
$736$	8.36848	−	27.2925i	0.308466	−	1.00601i
$737$		0			0
$738$	−18.1088	−	1.07882i	−0.666593	−	0.0397120i
$739$			21.0264i			0.773468i	0.922191	+	0.386734i	$0.126397\pi$
$739$			21.0264i			0.773468i	−0.922191	+	0.386734i	$0.873603\pi$
$740$		0			0
$741$	−74.7276			−2.74519
$742$	5.22373	−	87.6839i	0.191769	−	3.21898i
$743$		−	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
$747$		−	32.6460i		−	1.19446i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	14.8892i		−	0.543316i	−0.962394	−	0.271658i	$-0.912428\pi$
$751$		−	14.8892i		−	0.543316i	0.962394	−	0.271658i	$-0.0875720\pi$
$752$		0			0
$753$	−77.1005			−2.80970
$754$		0			0
$755$		0			0
$756$	4.99257	+	0.596979i	0.181578	+	0.0217119i
$757$	−17.8077			−0.647230			−0.323615	−	0.946189i	$-0.604898\pi$
$757$	−17.8077			−0.647230			−0.323615	+	0.946189i	$0.604898\pi$
$758$		0			0
$759$		−	41.7198i		−	1.51433i
$760$		0			0
$761$	32.5601			1.18030			0.590151	−	0.807293i	$-0.299068\pi$
$761$	32.5601			1.18030			0.590151	+	0.807293i	$0.299068\pi$
$762$		0			0
$763$		−	92.4394i		−	3.34653i
$764$	4.75105	−	39.7333i	0.171887	−	1.43750i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	18.2766	−	35.4486i	0.659499	−	1.27914i
$769$	−2.45283			−0.0884514			−0.0442257	−	0.999022i	$-0.514082\pi$
$769$	−2.45283			−0.0884514			−0.0442257	+	0.999022i	$0.514082\pi$
$770$		0			0
$771$			79.7334i			2.87153i
$772$	−42.6925	−	5.10489i	−1.53654	−	0.183729i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−22.1963	−	1.32233i	−0.795775	−	0.0474079i
$779$			33.1909i			1.18919i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−59.6003	−	14.4600i	−2.12858	−	0.516427i
$785$		0			0
$786$		0			0
$787$		−	11.5750i		−	0.412603i	−0.978488	−	0.206302i	$-0.933857\pi$
$787$		−	11.5750i		−	0.412603i	0.978488	−	0.206302i	$-0.0661428\pi$
$788$	−26.8251	−	3.20757i	−0.955606	−	0.114265i
$789$		0			0
$790$		0			0
$791$		−	6.09518i		−	0.216720i
$792$	5.35265	−	29.6656i	0.190198	−	1.05412i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−3.56726	+	29.8332i	−0.126438	+	1.05741i
$797$	30.0000			1.06265			0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000			1.06265			0.531327	+	0.847167i	$0.321693\pi$
$798$	8.23726	−	138.268i	0.291596	−	4.89464i
$799$		0			0
$800$	27.0416	+	8.29156i	0.956066	+	0.293151i
$801$		0			0
$802$		0			0
$803$		−	47.4944i		−	1.67604i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			42.2607i			1.48765i
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$			39.9292i			1.40210i	0.713110	+	0.701052i	$0.247286\pi$
$811$			39.9292i			1.40210i	−0.713110	+	0.701052i	$0.752714\pi$
$812$		0			0
$813$	25.7570			0.903339
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	10.1800	+	0.606468i	0.355935	+	0.0212047i
$819$		−	54.7529i		−	1.91322i
$820$		0			0
$821$	21.6333			0.755008			0.377504	−	0.926008i	$-0.376783\pi$
$821$	21.6333			0.755008			0.377504	+	0.926008i	$0.376783\pi$
$822$		0			0
$823$			11.3399i			0.395283i	0.980274	+	0.197641i	$0.0633281\pi$
$823$			11.3399i			0.395283i	−0.980274	+	0.197641i	$0.936672\pi$
$824$	−9.51149	+	52.7147i	−0.331349	+	1.83641i
$825$	41.3363			1.43915
$826$		0			0
$827$		−	22.2527i		−	0.773802i	−0.922121	−	0.386901i	$-0.873546\pi$
$827$		−	22.2527i		−	0.773802i	0.922121	−	0.386901i	$-0.126454\pi$
$828$	−3.85062	+	32.2029i	−0.133818	+	1.11913i
$829$	13.9660			0.485059			0.242530	−	0.970144i	$-0.422023\pi$
$829$	13.9660			0.485059			0.242530	+	0.970144i	$0.422023\pi$
$830$		0			0
$831$		0			0
$832$	−27.0255	−	10.0808i	−0.936941	−	0.349489i
$833$		0			0
$834$		0			0
$835$		0			0
$836$	−54.7630	−	6.54820i	−1.89402	−	0.226474i
$837$		0			0
$838$	0.903702	−	15.1693i	0.0312179	−	0.524014i
$839$		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
$843$		−	50.7831i		−	1.74906i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			51.9828i			1.78615i
$848$	51.0914	+	12.3956i	1.75449	+	0.425666i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−37.7195			−1.29149			−0.645745	−	0.763553i	$-0.723453\pi$
$853$	−37.7195			−1.29149			−0.645745	+	0.763553i	$0.723453\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$	−42.0803	−	2.50692i	−1.43660	−	0.0855847i
$859$			52.4798i			1.79059i	0.445477	+	0.895293i	$0.353034\pi$
$859$			52.4798i			1.79059i	−0.445477	+	0.895293i	$0.646966\pi$
$860$		0			0
$861$	−47.0227			−1.60253
$862$	2.90115	−	48.6979i	0.0988137	−	1.65866i
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	−0.882220	+	2.87722i	−0.0300137	+	0.0978851i
$865$		0			0
$866$	−52.5138	−	3.12848i	−1.78449	−	0.106310i
$867$		−	42.3755i		−	1.43915i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	54.4475	+	9.82414i	1.84383	+	0.332687i
$873$		0			0
$874$	59.2338	+	3.52883i	2.00361	+	0.119364i
$875$		0			0
$876$	−8.47605	+	70.8858i	−0.286379	+	2.39501i
$877$	−1.40243			−0.0473567			−0.0236784	−	0.999720i	$-0.507538\pi$
$877$	−1.40243			−0.0473567			−0.0236784	+	0.999720i	$0.507538\pi$
$878$		0			0
$879$		−	53.9248i		−	1.81884i
$880$		0			0
$881$	30.1562			1.01599			0.507993	−	0.861361i	$-0.330387\pi$
$881$	30.1562			1.01599			0.507993	+	0.861361i	$0.330387\pi$
$882$	69.5540	+	4.14364i	2.34200	+	0.139524i
$883$			22.5677i			0.759463i	0.925097	+	0.379731i	$0.123984\pi$
$883$			22.5677i			0.759463i	−0.925097	+	0.379731i	$0.876016\pi$
$884$		0			0
$885$		0			0
$886$	−3.45017	+	57.9135i	−0.115911	+	1.94564i
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		−	27.5750i		−	0.923797i
$892$		0			0
$893$		0			0
$894$	−3.83136	+	64.3120i	−0.128140	+	2.15091i
$895$		0			0
$896$	21.6315	−	48.8939i	0.722657	−	1.63343i
$897$	45.3542			1.51433
$898$		0			0
$899$		0			0
$900$	−31.9070	−	3.81522i	−1.06357	−	0.127174i
$901$		0			0
$902$	−1.11347	+	18.6903i	−0.0370744	+	0.622320i
$903$		0			0
$904$	3.59011	+	0.647775i	0.119405	+	0.0215447i
$905$		0			0
$906$	70.0259	+	4.17176i	2.32646	+	0.138597i
$907$			24.6374i			0.818070i	0.912519	+	0.409035i	$0.134135\pi$
$907$			24.6374i			0.818070i	−0.912519	+	0.409035i	$0.865865\pi$
$908$	6.24260	−	52.2073i	0.207168	−	1.73256i
$909$		0			0
$910$		0			0
$911$			46.7225i			1.54799i	0.633194	+	0.773993i	$0.281744\pi$
$911$			46.7225i			1.54799i	−0.633194	+	0.773993i	$0.718256\pi$
$912$	80.5656	+	19.5465i	2.66779	+	0.647249i
$913$	−33.6945			−1.11512
$914$	−5.41873	−	0.322818i	−0.179236	−	0.0106779i
$915$		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
$921$	−49.6036			−1.63449
$922$	−50.9564	−	3.03570i	−1.67816	−	0.0999755i
$923$		0			0
$924$	9.27707	−	77.5847i	0.305193	−	2.55235i
$925$		0			0
$926$		0			0
$927$		−	60.8572i		−	1.99881i
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		−	127.483i		−	4.17808i
$932$		0			0
$933$	87.1545			2.85331
$934$	−2.01143	+	33.7632i	−0.0658159	+	1.10477i
$935$		0			0
$936$	32.2499	+	5.81895i	1.05412	+	0.190198i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		−	80.7974i		−	2.63672i
$940$		0			0
$941$	54.2540			1.76863			0.884315	−	0.466891i	$-0.154626\pi$
$941$	54.2540			1.76863			0.884315	+	0.466891i	$0.154626\pi$
$942$	−3.19627	+	53.6516i	−0.104140	+	1.74806i
$943$		−	20.1444i		−	0.655994i
$944$		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
$949$	51.6319			1.67604
$950$	−3.49639	+	58.6894i	−0.113438	+	1.90414i
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$	−59.6240	−	3.55207i	−1.93040	−	0.115003i
$955$		0			0
$956$	5.76331	−	48.1989i	0.186399	−	1.55886i
$957$		0			0
$958$	0.557869	−	9.36423i	0.0180239	−	0.302545i
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$	−61.6033	−	7.36612i	−1.98411	−	0.237247i
$965$		0			0
$966$	−4.99941	+	83.9186i	−0.160853	+	2.70004i
$967$		−	60.2210i		−	1.93658i	−0.249836	−	0.968288i	$-0.580377\pi$
$967$		−	60.2210i		−	1.93658i	0.249836	−	0.968288i	$-0.419623\pi$
$968$	−30.6183	−	5.52456i	−0.984109	−	0.177566i
$969$		0			0
$970$		0			0
$971$		−	17.5823i		−	0.564244i	−0.959379	−	0.282122i	$-0.908962\pi$
$971$		−	17.5823i		−	0.564244i	0.959379	−	0.282122i	$-0.0910382\pi$
$972$	−5.30013	+	44.3253i	−0.170002	+	1.42173i
$973$		0			0
$974$		0			0
$975$			44.9373i			1.43915i
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−62.8576			−2.00689
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	4.99742	−	27.6968i	0.159312	−	0.882941i
$985$		0			0
$986$		0			0
$987$		0			0
$988$	7.11865	−	59.5337i	0.226474	−	1.89402i
$989$		0			0
$990$		0			0
$991$		−	59.8910i		−	1.90250i	−0.308421	−	0.951250i	$-0.599801\pi$
$991$		−	59.8910i		−	1.90250i	0.308421	−	0.951250i	$-0.400199\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	50.2893	+	6.01326i	1.59348	+	0.190537i
$997$		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.571.20	yes	20
4.3	odd	2	inner	572.2.b.b.571.19	yes	20
11.10	odd	2	inner	572.2.b.b.571.1	✓	20
13.12	even	2	inner	572.2.b.b.571.1	✓	20
44.43	even	2	inner	572.2.b.b.571.2	yes	20
52.51	odd	2	inner	572.2.b.b.571.2	yes	20
143.142	odd	2	CM	572.2.b.b.571.20	yes	20
572.571	even	2	inner	572.2.b.b.571.19	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.b.b.571.1	✓	20	11.10	odd	2	inner
572.2.b.b.571.1	✓	20	13.12	even	2	inner
572.2.b.b.571.2	yes	20	44.43	even	2	inner
572.2.b.b.571.2	yes	20	52.51	odd	2	inner
572.2.b.b.571.19	yes	20	4.3	odd	2	inner
572.2.b.b.571.19	yes	20	572.571	even	2	inner
572.2.b.b.571.20	yes	20	1.1	even	1	trivial
572.2.b.b.571.20	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)

\(f(q)\)	\(=\)	\(q+(1.41171 + 0.0841020i) q^{2} -2.49267i q^{3} +(1.98585 + 0.237455i) q^{4} +(0.209639 - 3.51893i) q^{6} -4.72571i q^{7} +(2.78348 + 0.502232i) q^{8} -3.21342 q^{9} +O(q^{10})\)	\(q+(1.41171 + 0.0841020i) q^{2} -2.49267i q^{3} +(1.98585 + 0.237455i) q^{4} +(0.209639 - 3.51893i) q^{6} -4.72571i q^{7} +(2.78348 + 0.502232i) q^{8} -3.21342 q^{9} +3.31662i q^{11} +(0.591899 - 4.95009i) q^{12} -3.60555 q^{13} +(0.397442 - 6.67133i) q^{14} +(3.88723 + 0.943103i) q^{16} +(-4.53643 - 0.270255i) q^{18} +8.31465i q^{19} -11.7797 q^{21} +(-0.278935 + 4.68211i) q^{22} -5.04638i q^{23} +(1.25190 - 6.93831i) q^{24} +5.00000 q^{25} +(-5.09000 - 0.303234i) q^{26} +0.531998i q^{27} +(1.12214 - 9.38457i) q^{28} +(5.40833 + 1.65831i) q^{32} +8.26727 q^{33} +(-6.38139 - 0.763045i) q^{36} +(-0.699278 + 11.7379i) q^{38} +8.98746i q^{39} +3.99186 q^{41} +(-16.6295 - 0.990692i) q^{42} +(-0.787550 + 6.58633i) q^{44} +(0.424411 - 7.12403i) q^{46} +(2.35085 - 9.68960i) q^{48} -15.3323 q^{49} +(7.05855 + 0.420510i) q^{50} +(-7.16010 - 0.856157i) q^{52} +13.1434 q^{53} +(-0.0447421 + 0.751028i) q^{54} +(2.37340 - 13.1539i) q^{56} +20.7257 q^{57} +15.1857i q^{63} +(7.49553 + 2.79591i) q^{64} +(11.6710 + 0.695293i) q^{66} -12.5790 q^{69} +(-8.94450 - 1.61389i) q^{72} -14.3201 q^{73} -12.4634i q^{75} +(-1.97436 + 16.5117i) q^{76} +15.6734 q^{77} +(-0.755864 + 12.6877i) q^{78} -8.31418 q^{81} +(5.63535 + 0.335723i) q^{82} +10.1593i q^{83} +(-23.3927 - 2.79714i) q^{84} +(-1.66572 + 9.23176i) q^{88} +17.0388i q^{91} +(1.19829 - 10.0214i) q^{92} +(4.13363 - 13.4812i) q^{96} +(-21.6448 - 1.28948i) q^{98} -10.6577i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 60 q^{9}+O(q^{10}) \) 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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