LMFDB - Embedded newform 572.2.b.b.571.11

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			571.11
Root			$-0.356257 - 1.36861i$ of defining polynomial
Character	$\chi$	$=$	572.571
Dual form			572.2.b.b.571.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.356257 - 1.36861i) q^{2} +3.05807i q^{3} +(-1.74616 - 0.975150i) q^{4} +(4.18530 + 1.08946i) q^{6} -5.22251i q^{7} +(-1.95668 + 2.04240i) q^{8} -6.35182 q^{9} +O(q^{10})$	\(q+(0.356257 - 1.36861i) q^{2} +3.05807i q^{3} +(-1.74616 - 0.975150i) q^{4} +(4.18530 + 1.08946i) q^{6} -5.22251i q^{7} +(-1.95668 + 2.04240i) q^{8} -6.35182 q^{9} -3.31662i q^{11} +(2.98208 - 5.33989i) q^{12} -3.60555 q^{13} +(-7.14756 - 1.86056i) q^{14} +(2.09816 + 3.40554i) q^{16} +(-2.26288 + 8.69313i) q^{18} -0.0771978i q^{19} +15.9708 q^{21} +(-4.53915 - 1.18157i) q^{22} -8.87708i q^{23} +(-6.24582 - 5.98367i) q^{24} +5.00000 q^{25} +(-1.28450 + 4.93458i) q^{26} -10.2501i q^{27} +(-5.09273 + 9.11935i) q^{28} +(5.40833 - 1.65831i) q^{32} +10.1425 q^{33} +(11.0913 + 6.19397i) q^{36} +(-0.105653 - 0.0275022i) q^{38} -11.0260i q^{39} -10.3818 q^{41} +(5.68972 - 21.8578i) q^{42} +(-3.23421 + 5.79136i) q^{44} +(-12.1492 - 3.16252i) q^{46} +(-10.4144 + 6.41634i) q^{48} -20.2746 q^{49} +(1.78128 - 6.84303i) q^{50} +(6.29588 + 3.51595i) q^{52} +10.0200 q^{53} +(-14.0283 - 3.65167i) q^{54} +(10.6665 + 10.2188i) q^{56} +0.236076 q^{57} +33.1724i q^{63} +(-0.342822 - 7.99265i) q^{64} +(3.61333 - 13.8811i) q^{66} +27.1468 q^{69} +(12.4285 - 12.9730i) q^{72} +4.44237 q^{73} +15.2904i q^{75} +(-0.0752794 + 0.134800i) q^{76} -17.3211 q^{77} +(-15.0903 - 3.92810i) q^{78} +12.2901 q^{81} +(-3.69858 + 14.2085i) q^{82} -0.671690i q^{83} +(-27.8877 - 15.5740i) q^{84} +(6.77389 + 6.48957i) q^{88} +18.8300i q^{91} +(-8.65648 + 15.5008i) q^{92} +(5.07124 + 16.5391i) q^{96} +(-7.22298 + 27.7480i) q^{98} +21.0666i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.356257	−	1.36861i	0.251912	−	0.967750i
$2$	0.356257	−	1.36861i	0.251912	−	0.967750i
$3$			3.05807i			1.76558i	0.469768	+	0.882790i	$0.344338\pi$
$3$			3.05807i			1.76558i	−0.469768	+	0.882790i	$0.655662\pi$
$4$	−1.74616	−	0.975150i	−0.873081	−	0.487575i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$	4.18530	+	1.08946i	1.70864	+	0.444770i
$7$		−	5.22251i		−	1.97392i	−0.160954	−	0.986962i	$-0.551457\pi$
$7$		−	5.22251i		−	1.97392i	0.160954	−	0.986962i	$-0.448543\pi$
$8$	−1.95668	+	2.04240i	−0.691790	+	0.722099i
$9$	−6.35182			−2.11727
$10$		0			0
$11$		−	3.31662i		−	1.00000i
$11$		−	3.31662i		−	1.00000i
$12$	2.98208	−	5.33989i	0.860853	−	1.54149i
$13$	−3.60555			−1.00000
$13$	−3.60555			−1.00000
$14$	−7.14756	−	1.86056i	−1.91027	−	0.497254i
$15$		0			0
$16$	2.09816	+	3.40554i	0.524541	+	0.851385i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	−2.26288	+	8.69313i	−0.533365	+	2.04899i
$19$		−	0.0771978i		−	0.0177104i	−0.999961	−	0.00885519i	$-0.997181\pi$
$19$		−	0.0771978i		−	0.0177104i	0.999961	−	0.00885519i	$-0.00281873\pi$
$20$		0			0
$21$	15.9708			3.48512
$22$	−4.53915	−	1.18157i	−0.967750	−	0.251912i
$23$		−	8.87708i		−	1.85100i	−0.378749	−	0.925499i	$-0.623646\pi$
$23$		−	8.87708i		−	1.85100i	0.378749	−	0.925499i	$-0.376354\pi$
$24$	−6.24582	−	5.98367i	−1.27492	−	1.22141i
$25$	5.00000			1.00000
$26$	−1.28450	+	4.93458i	−0.251912	+	0.967750i
$27$		−	10.2501i		−	1.97263i
$28$	−5.09273	+	9.11935i	−0.962436	+	1.72340i
$29$		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
$33$	10.1425			1.76558
$34$		0			0
$35$		0			0
$36$	11.0913	+	6.19397i	1.84855	+	1.03233i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−0.105653	−	0.0275022i	−0.0171392	−	0.00446145i
$39$		−	11.0260i		−	1.76558i
$40$		0			0
$41$	−10.3818			−1.62136			−0.810680	−	0.585489i	$-0.800902\pi$
$41$	−10.3818			−1.62136			−0.810680	+	0.585489i	$0.800902\pi$
$42$	5.68972	−	21.8578i	0.877942	−	3.37273i
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−3.23421	+	5.79136i	−0.487575	+	0.873081i
$45$		0			0
$46$	−12.1492	−	3.16252i	−1.79130	−	0.466288i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	−10.4144	+	6.41634i	−1.50319	+	0.926119i
$49$	−20.2746			−2.89638
$50$	1.78128	−	6.84303i	0.251912	−	0.967750i
$51$		0			0
$52$	6.29588	+	3.51595i	0.873081	+	0.487575i
$53$	10.0200			1.37635			0.688175	−	0.725545i	$-0.258412\pi$
$53$	10.0200			1.37635			0.688175	+	0.725545i	$0.258412\pi$
$54$	−14.0283	−	3.65167i	−1.90902	−	0.496929i
$55$		0			0
$56$	10.6665	+	10.2188i	1.42537	+	1.36554i
$57$	0.236076			0.0312691
$58$		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
$63$			33.1724i			4.17933i
$64$	−0.342822	−	7.99265i	−0.0428528	−	0.999081i
$65$		0			0
$66$	3.61333	−	13.8811i	0.444770	−	1.70864i
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$	27.1468			3.26809
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	12.4285	−	12.9730i	1.46471	−	1.52888i
$73$	4.44237			0.519941			0.259970	−	0.965617i	$-0.416287\pi$
$73$	4.44237			0.519941			0.259970	+	0.965617i	$0.416287\pi$
$74$		0			0
$75$			15.2904i			1.76558i
$76$	−0.0752794	+	0.134800i	−0.00863514	+	0.0154626i
$77$	−17.3211			−1.97392
$78$	−15.0903	−	3.92810i	−1.70864	−	0.444770i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	12.2901			1.36557
$82$	−3.69858	+	14.2085i	−0.408439	+	1.56907i
$83$		−	0.671690i		−	0.0737276i	−0.999320	−	0.0368638i	$-0.988263\pi$
$83$		−	0.671690i		−	0.0737276i	0.999320	−	0.0368638i	$-0.0117368\pi$
$84$	−27.8877	−	15.5740i	−3.04279	−	1.69926i
$85$		0			0
$86$		0			0
$87$		0			0
$88$	6.77389	+	6.48957i	0.722099	+	0.691790i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$			18.8300i			1.97392i
$92$	−8.65648	+	15.5008i	−0.902501	+	1.61607i
$93$		0			0
$94$		0			0
$95$		0			0
$96$	5.07124	+	16.5391i	0.517581	+	1.68801i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−7.22298	+	27.7480i	−0.729631	+	2.80297i
$99$			21.0666i			2.11727i
$100$	−8.73081	−	4.87575i	−0.873081	−	0.487575i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$			11.0288i			1.08670i	0.839505	+	0.543352i	$0.182845\pi$
$103$			11.0288i			1.08670i	−0.839505	+	0.543352i	$0.817155\pi$
$104$	7.05490	−	7.36399i	0.691790	−	0.722099i
$105$		0			0
$106$	3.56969	−	13.7134i	0.346719	−	1.33196i
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−9.99539	+	17.8983i	−0.961806	+	1.72227i
$109$	12.9925			1.24446			0.622230	−	0.782835i	$-0.286227\pi$
$109$	12.9925			1.24446			0.622230	+	0.782835i	$0.286227\pi$
$110$		0			0
$111$		0			0
$112$	17.7855	−	10.9577i	1.68057	−	1.03540i
$113$	−19.7839			−1.86112			−0.930558	−	0.366145i	$-0.880677\pi$
$113$	−19.7839			−1.86112			−0.930558	+	0.366145i	$0.880677\pi$
$114$	0.0841038	−	0.323096i	0.00787705	−	0.0302607i
$115$		0			0
$116$		0			0
$117$	22.9018			2.11727
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		−	31.7482i		−	2.86264i
$124$		0			0
$125$		0			0
$126$	45.4000	+	11.8179i	4.04455	+	1.05282i
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	−11.0609	−	2.37825i	−0.977656	−	0.210209i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	−17.7104	−	9.89044i	−1.54149	−	0.860853i
$133$	−0.403166			−0.0349589
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$	9.67122	−	37.1532i	0.823269	−	3.16269i
$139$		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
$144$	−13.3272	−	21.6314i	−1.11060	−	1.80261i
$145$		0			0
$146$	1.58263	−	6.07986i	0.130979	−	0.503173i
$147$		−	62.0013i		−	5.11378i
$148$		0			0
$149$	5.27165			0.431871			0.215935	−	0.976408i	$-0.430720\pi$
$149$	5.27165			0.431871			0.215935	+	0.976408i	$0.430720\pi$
$150$	20.9265	+	5.44730i	1.70864	+	0.444770i
$151$		−	19.8997i		−	1.61942i	−0.586831	−	0.809709i	$-0.699625\pi$
$151$		−	19.8997i		−	1.61942i	0.586831	−	0.809709i	$-0.300375\pi$
$152$	0.157669	+	0.151051i	0.0127886	+	0.0122519i
$153$		0			0
$154$	−6.17076	+	23.7058i	−0.497254	+	1.91027i
$155$		0			0
$156$	−10.7520	+	19.2533i	−0.860853	+	1.54149i
$157$	24.0247			1.91738			0.958692	−	0.284447i	$-0.0918101\pi$
$157$	24.0247			1.91738			0.958692	+	0.284447i	$0.0918101\pi$
$158$		0			0
$159$			30.6418i			2.43006i
$160$		0			0
$161$	−46.3606			−3.65373
$162$	4.37844	−	16.8203i	0.344002	−	1.32153i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$	18.1283	+	10.1238i	1.41558	+	0.790535i
$165$		0			0
$166$	−0.919279	−	0.239294i	−0.0713499	−	0.0185728i
$167$			11.1804i			0.865168i	0.901594	+	0.432584i	$0.142398\pi$
$167$			11.1804i			0.865168i	−0.901594	+	0.432584i	$0.857602\pi$
$168$	−31.2498	+	32.6189i	−2.41097	+	2.51660i
$169$	13.0000			1.00000
$170$		0			0
$171$			0.490346i			0.0374977i
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		−	26.1126i		−	1.97392i
$176$	11.2949	−	6.95882i	0.851385	−	0.524541i
$177$		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
$181$	23.0386			1.71244			0.856222	−	0.516608i	$-0.172806\pi$
$181$	23.0386			1.71244			0.856222	+	0.516608i	$0.172806\pi$
$182$	25.7709	+	6.70833i	1.91027	+	0.497254i
$183$		0			0
$184$	18.1306	+	17.3696i	1.33660	+	1.28050i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−53.5313			−3.89383
$190$		0			0
$191$		−	11.9540i		−	0.864958i	−0.901644	−	0.432479i	$-0.857639\pi$
$191$		−	11.9540i		−	0.864958i	0.901644	−	0.432479i	$-0.142361\pi$
$192$	24.4421	−	1.04838i	1.76396	−	0.0756600i
$193$	27.7386			1.99667			0.998333	−	0.0577192i	$-0.0183828\pi$
$193$	27.7386			1.99667			0.998333	+	0.0577192i	$0.0183828\pi$
$194$		0			0
$195$		0			0
$196$	35.4028	+	19.7708i	2.52877	+	1.41220i
$197$	−27.5774			−1.96481			−0.982404	−	0.186768i	$-0.940199\pi$
$197$	−27.5774			−1.96481			−0.982404	+	0.186768i	$0.940199\pi$
$198$	28.8318	+	7.50512i	2.04899	+	0.533365i
$199$			18.0701i			1.28096i	0.767977	+	0.640478i	$0.221264\pi$
$199$			18.0701i			1.28096i	−0.767977	+	0.640478i	$0.778736\pi$
$200$	−9.78339	+	10.2120i	−0.691790	+	0.722099i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	15.0941	+	3.92910i	1.05166	+	0.273753i
$207$			56.3856i			3.91907i
$208$	−7.56504	−	12.2789i	−0.524541	−	0.851385i
$209$	−0.256036			−0.0177104
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$	−17.4965	−	9.77099i	−1.20167	−	0.671074i
$213$		0			0
$214$		0			0
$215$		0			0
$216$	20.9348	+	20.0561i	1.42444	+	1.36465i
$217$		0			0
$218$	4.62868	−	17.7817i	0.313494	−	1.20433i
$219$			13.5851i			0.917997i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−8.66056	−	28.2451i	−0.578658	−	1.88720i
$225$	−31.7591			−2.11727
$226$	−7.04816	+	27.0764i	−0.468837	+	1.80109i
$227$			22.1293i			1.46877i	0.678732	+	0.734386i	$0.262530\pi$
$227$			22.1293i			1.46877i	−0.678732	+	0.734386i	$0.737470\pi$
$228$	−0.412228	−	0.230210i	−0.0273004	−	0.0152460i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		−	52.9692i		−	3.48512i
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$	8.15892	−	31.3435i	0.533365	−	2.04899i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	10.7173i		−	0.693241i	−0.938005	−	0.346621i	$-0.887329\pi$
$239$		−	10.7173i		−	0.693241i	0.938005	−	0.346621i	$-0.112671\pi$
$240$		0			0
$241$	24.3318			1.56735			0.783675	−	0.621171i	$-0.213343\pi$
$241$	24.3318			1.56735			0.783675	+	0.621171i	$0.213343\pi$
$242$	−3.91883	+	15.0547i	−0.251912	+	0.967750i
$243$			6.83378i			0.438387i
$244$		0			0
$245$		0			0
$246$	−43.4508	−	11.3105i	−2.77032	−	0.721132i
$247$			0.278340i			0.0177104i
$248$		0			0
$249$	2.05408			0.130172
$250$		0			0
$251$		−	20.9818i		−	1.32436i	−0.749345	−	0.662179i	$-0.769632\pi$
$251$		−	20.9818i		−	1.32436i	0.749345	−	0.662179i	$-0.230368\pi$
$252$	32.3481	−	57.9244i	2.03774	−	3.64890i
$253$	−29.4419			−1.85100
$254$		0			0
$255$		0			0
$256$	−7.19541	+	14.2908i	−0.449713	+	0.893173i
$257$	24.5868			1.53368			0.766841	−	0.641838i	$-0.221828\pi$
$257$	24.5868			1.53368			0.766841	+	0.641838i	$0.221828\pi$
$258$		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	$0.5\pi$
$264$	−19.8456	+	20.7150i	−1.22141	+	1.27492i
$265$		0			0
$266$	−0.143631	+	0.551775i	−0.00880656	+	0.0338315i
$267$		0			0
$268$		0			0
$269$	−2.78978			−0.170096			−0.0850479	−	0.996377i	$-0.527104\pi$
$269$	−2.78978			−0.170096			−0.0850479	+	0.996377i	$0.527104\pi$
$270$		0			0
$271$		−	32.9237i		−	1.99997i	−0.00508114	−	0.999987i	$-0.501617\pi$
$271$		−	32.9237i		−	1.99997i	0.00508114	−	0.999987i	$-0.498383\pi$
$272$		0			0
$273$	−57.5836			−3.48512
$274$		0			0
$275$		−	16.5831i		−	1.00000i
$276$	−47.4026	−	26.4722i	−2.85330	−	1.59344i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−19.0272			−1.13507			−0.567535	−	0.823349i	$-0.692103\pi$
$281$	−19.0272			−1.13507			−0.567535	+	0.823349i	$0.692103\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$		0			0
$286$	16.3661	+	4.26021i	0.967750	+	0.251912i
$287$			54.2189i			3.20044i
$288$	−34.3527	+	10.5333i	−2.02425	+	0.620680i
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$	−7.75711	−	4.33198i	−0.453950	−	0.253510i
$293$	21.6333			1.26383			0.631916	−	0.775037i	$-0.282269\pi$
$293$	21.6333			1.26383			0.631916	+	0.775037i	$0.282269\pi$
$294$	−84.8553	−	22.0884i	−4.94886	−	1.28822i
$295$		0			0
$296$		0			0
$297$	−33.9957			−1.97263
$298$	1.87806	−	7.21481i	0.108793	−	0.417943i
$299$			32.0068i			1.85100i
$300$	14.9104	−	26.6995i	0.860853	−	1.54149i
$301$		0			0
$302$	−27.2349	−	7.08942i	−1.56719	−	0.407950i
$303$		0			0
$304$	0.262900	−	0.161974i	0.0150784	−	0.00928982i
$305$		0			0
$306$		0			0
$307$			19.8997i			1.13574i	0.823119	+	0.567869i	$0.192232\pi$
$307$			19.8997i			1.13574i	−0.823119	+	0.567869i	$0.807768\pi$
$308$	30.2455	+	16.8907i	1.72340	+	0.962436i
$309$	−33.7270			−1.91866
$310$		0			0
$311$		−	6.39449i		−	0.362598i	−0.983428	−	0.181299i	$-0.941970\pi$
$311$		−	6.39449i		−	0.362598i	0.983428	−	0.181299i	$-0.0580302\pi$
$312$	22.5196	+	21.5744i	1.27492	+	1.22141i
$313$	−17.8831			−1.01082			−0.505408	−	0.862881i	$-0.668658\pi$
$313$	−17.8831			−1.01082			−0.505408	+	0.862881i	$0.668658\pi$
$314$	8.55898	−	32.8804i	0.483011	−	1.85555i
$315$		0			0
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$	41.9366	+	10.9164i	2.35169	+	0.612159i
$319$		0			0
$320$		0			0
$321$		0			0
$322$	−16.5163	+	63.4494i	−0.920417	+	3.53590i
$323$		0			0
$324$	−21.4605	−	11.9847i	−1.19225	−	0.665817i
$325$	−18.0278			−1.00000
$326$		0			0
$327$			39.7321i			2.19719i
$328$	20.3138	−	21.2038i	1.12164	−	1.17078i
$329$		0			0
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$	−0.654999	+	1.17288i	−0.0359477	+	0.0643702i
$333$		0			0
$334$	15.3016	+	3.98311i	0.837267	+	0.217946i
$335$		0			0
$336$	33.5094	+	54.3893i	1.82809	+	2.96718i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	4.63134	−	17.7919i	0.251912	−	0.967750i
$339$		−	60.5007i		−	3.28595i
$340$		0			0
$341$		0			0
$342$	0.671090	+	0.174689i	0.0362884	+	0.00944610i
$343$			69.3269i			3.74330i
$344$		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
$349$	−22.6285			−1.21127			−0.605637	−	0.795741i	$-0.707081\pi$
$349$	−22.6285			−1.21127			−0.605637	+	0.795741i	$0.707081\pi$
$350$	−35.7378	−	9.30278i	−1.91027	−	0.497254i
$351$			36.9573i			1.97263i
$352$	−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
$358$	−32.7323	−	8.52042i	−1.72995	−	0.450318i
$359$		−	17.0109i		−	0.897802i	−0.893582	−	0.448901i	$-0.851816\pi$
$359$		−	17.0109i		−	0.897802i	0.893582	−	0.448901i	$-0.148184\pi$
$360$		0			0
$361$	18.9940			0.999686
$362$	8.20765	−	31.5307i	0.431385	−	1.65722i
$363$		−	33.6388i		−	1.76558i
$364$	18.3621	−	32.8803i	0.962436	−	1.72340i
$365$		0			0
$366$		0			0
$367$		−	36.4185i		−	1.90103i	−0.310675	−	0.950516i	$-0.600555\pi$
$367$		−	36.4185i		−	1.90103i	0.310675	−	0.950516i	$-0.399445\pi$
$368$	30.2313	−	18.6256i	1.57591	−	0.970925i
$369$	65.9431			3.43286
$370$		0			0
$371$		−	52.3295i		−	2.71681i
$372$		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
$378$	−19.0709	+	73.2632i	−0.980900	+	3.76825i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$	−16.3602	−	4.25868i	−0.837063	−	0.217893i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	7.27286	−	33.8251i	0.371142	−	1.72613i
$385$		0			0
$386$	9.88205	−	37.9632i	0.502983	−	1.93227i
$387$		0			0
$388$		0			0
$389$	29.5479			1.49814			0.749068	−	0.662493i	$-0.230501\pi$
$389$	29.5479			1.49814			0.749068	+	0.662493i	$0.230501\pi$
$390$		0			0
$391$		0			0
$392$	39.6709	−	41.4090i	2.00368	−	2.09147i
$393$		0			0
$394$	−9.82464	+	37.7426i	−0.494958	+	1.90144i
$395$		0			0
$396$	20.5431	−	36.7857i	1.03233	−	1.84855i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	24.7308	+	6.43760i	1.23965	+	0.322688i
$399$		−	1.23291i		−	0.0617228i
$400$	10.4908	+	17.0277i	0.524541	+	0.851385i
$401$		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
$412$	10.7548	−	19.2581i	0.529849	−	0.948780i
$413$		0			0
$414$	77.1696	+	20.0877i	3.79268	+	0.987259i
$415$		0			0
$416$	−19.5000	+	5.97913i	−0.956066	+	0.293151i
$417$		0			0
$418$	−0.0912146	+	0.350412i	−0.00446145	+	0.0171392i
$419$			14.5265i			0.709667i	0.934929	+	0.354834i	$0.115462\pi$
$419$			14.5265i			0.709667i	−0.934929	+	0.354834i	$0.884538\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	−19.6059	+	20.4648i	−0.952146	+	0.993861i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−36.5692			−1.76558
$430$		0			0
$431$		−	14.3242i		−	0.689970i	−0.938608	−	0.344985i	$-0.887884\pi$
$431$		−	14.3242i		−	0.689970i	0.938608	−	0.344985i	$-0.112116\pi$
$432$	34.9071	−	21.5064i	1.67947	−	1.03473i
$433$	6.25326			0.300513			0.150256	−	0.988647i	$-0.451990\pi$
$433$	6.25326			0.300513			0.150256	+	0.988647i	$0.451990\pi$
$434$		0			0
$435$		0			0
$436$	−22.6871	−	12.6697i	−1.08651	−	0.606767i
$437$	−0.685291			−0.0327819
$438$	18.5927	+	4.83979i	0.888391	+	0.231254i
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	128.781			6.13241
$442$		0			0
$443$			38.7359i			1.84040i	0.391448	+	0.920200i	$0.371974\pi$
$443$			38.7359i			1.84040i	−0.391448	+	0.920200i	$0.628026\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			16.1211i			0.762502i
$448$	−41.7417	+	1.79039i	−1.97211	+	0.0845881i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−11.3144	+	43.4656i	−0.533365	+	2.04899i
$451$			34.4324i			1.62136i
$452$	34.5459	+	19.2923i	1.62490	+	0.907433i
$453$	60.8549			2.85921
$454$	30.2863	+	7.88371i	1.42140	+	0.370001i
$455$		0			0
$456$	−0.461926	+	0.482163i	−0.0216316	+	0.0225794i
$457$	39.3122			1.83895			0.919474	−	0.393152i	$-0.128615\pi$
$457$	39.3122			1.83895			0.919474	+	0.393152i	$0.128615\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−33.2775			−1.54989			−0.774944	−	0.632030i	$-0.782222\pi$
$461$	−33.2775			−1.54989			−0.774944	+	0.632030i	$0.782222\pi$
$462$	−72.4940	−	18.8707i	−3.37273	−	0.877942i
$463$		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	$-0.186655\pi$
$468$	−39.9902	−	22.3327i	−1.84855	−	1.03233i
$469$		0			0
$470$		0			0
$471$			73.4694i			3.38529i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	0.385989i		−	0.0177104i
$476$		0			0
$477$	−63.6451			−2.91411
$478$	−14.6677	−	3.81809i	−0.670884	−	0.174636i
$479$			6.63325i			0.303081i	0.988451	+	0.151540i	$0.0484234\pi$
$479$			6.63325i			0.303081i	−0.988451	+	0.151540i	$0.951577\pi$
$480$		0			0
$481$		0			0
$482$	8.66838	−	33.3007i	0.394834	−	1.51680i
$483$		−	141.774i		−	6.45095i
$484$	19.2078	+	10.7267i	0.873081	+	0.487575i
$485$		0			0
$486$	9.35274	+	2.43458i	0.424249	+	0.110435i
$487$		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	$0.5\pi$
$492$	−30.9593	+	55.4375i	−1.39575	+	2.49932i
$493$		0			0
$494$	0.380938	+	0.0991607i	0.0171392	+	0.00446145i
$495$		0			0
$496$		0			0
$497$		0			0
$498$	0.731779	−	2.81122i	0.0327918	−	0.125974i
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	−34.1906			−1.52752
$502$	−28.7158	−	7.47491i	−1.28165	−	0.333621i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	−67.7515	−	64.9078i	−3.01789	−	2.89122i
$505$		0			0
$506$	−10.4889	+	40.2944i	−0.466288	+	1.79130i
$507$			39.7550i			1.76558i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		−	23.2004i		−	1.02632i
$512$	16.9950	+	14.9389i	0.751080	+	0.660211i
$513$	−0.791285			−0.0349361
$514$	8.75921	−	33.6496i	0.386352	−	1.48422i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−44.6977			−1.95824			−0.979120	−	0.203282i	$-0.934839\pi$
$521$	−44.6977			−1.95824			−0.979120	+	0.203282i	$0.934839\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$	79.8541			3.48512
$526$		0			0
$527$		0			0
$528$	21.2806	+	34.5406i	0.926119	+	1.50319i
$529$	−55.8025			−2.42620
$530$		0			0
$531$		0			0
$532$	0.703994	+	0.393148i	0.0305220	+	0.0170451i
$533$	37.4320			1.62136
$534$		0			0
$535$		0			0
$536$		0			0
$537$	73.1385			3.15616
$538$	−0.993878	+	3.81811i	−0.0428491	+	0.164610i
$539$			67.2433i			2.89638i
$540$		0			0
$541$	−32.8487			−1.41228			−0.706138	−	0.708075i	$-0.749564\pi$
$541$	−32.8487			−1.41228			−0.706138	+	0.708075i	$0.749564\pi$
$542$	−45.0596	−	11.7293i	−1.93548	−	0.503817i
$543$			70.4537i			3.02346i
$544$		0			0
$545$		0			0
$546$	−20.5146	+	78.8093i	−0.877942	+	3.37273i
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		0			0
$549$		0			0
$550$	−22.6958	−	5.90785i	−0.967750	−	0.251912i
$551$		0			0
$552$	−53.1175	+	55.4446i	−2.26083	+	2.35988i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	46.7987			1.98293			0.991463	−	0.130388i	$-0.0416224\pi$
$557$	46.7987			1.98293			0.991463	+	0.130388i	$0.0416224\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−6.77858	+	26.0408i	−0.285937	+	1.09846i
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	64.1853i		−	2.69553i
$568$		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	$0.5\pi$
$572$	11.6611	−	20.8811i	0.487575	−	0.873081i
$573$	36.5561			1.52715
$574$	74.2043	+	19.3159i	3.09723	+	0.806228i
$575$		−	44.3854i		−	1.85100i
$576$	2.17754	+	50.7678i	0.0907309	+	2.11533i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	6.05637	−	23.2663i	0.251912	−	0.967750i
$579$			84.8266i			3.52527i
$580$		0			0
$581$	−3.50791			−0.145533
$582$		0			0
$583$		−	33.2325i		−	1.37635i
$584$	−8.69230	+	9.07312i	−0.359690	+	0.375448i
$585$		0			0
$586$	7.70701	−	29.6075i	0.318374	−	1.22307i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−60.4606	+	108.264i	−2.49335	+	4.46475i
$589$		0			0
$590$		0			0
$591$		−	84.3337i		−	3.46903i
$592$		0			0
$593$	−41.6886			−1.71195			−0.855973	−	0.517020i	$-0.827041\pi$
$593$	−41.6886			−1.71195			−0.855973	+	0.517020i	$0.827041\pi$
$594$	−12.1112	+	46.5267i	−0.496929	+	1.90902i
$595$		0			0
$596$	−9.20516	−	5.14065i	−0.377058	−	0.210569i
$597$	−55.2597			−2.26163
$598$	43.8046	+	11.4026i	1.79130	+	0.466288i
$599$		−	48.6508i		−	1.98782i	−0.110197	−	0.993910i	$-0.535148\pi$
$599$		−	48.6508i		−	1.98782i	0.110197	−	0.993910i	$-0.464852\pi$
$600$	−31.2291	−	29.9183i	−1.27492	−	1.22141i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	−19.4052	+	34.7482i	−0.789588	+	1.41388i
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	−0.128018	−	0.417511i	−0.00519182	−	0.0169323i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−4.10779			−0.165912			−0.0829560	−	0.996553i	$-0.526436\pi$
$613$	−4.10779			−0.165912			−0.0829560	+	0.996553i	$0.526436\pi$
$614$	27.2349	+	7.08942i	1.09911	+	0.286106i
$615$		0			0
$616$	33.8918	−	35.3767i	1.36554	−	1.42537i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$	−12.0155	+	46.1589i	−0.483333	+	1.85678i
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$	−90.9909			−3.65134
$622$	−8.75153	−	2.27808i	−0.350904	−	0.0913427i
$623$		0			0
$624$	37.5496	−	23.1344i	1.50319	−	0.926119i
$625$	25.0000			1.00000
$626$	−6.37099	+	24.4750i	−0.254636	+	0.978217i
$627$		−	0.782977i		−	0.0312691i
$628$	−41.9511	−	23.4277i	−1.67403	−	0.934868i
$629$		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
$636$	29.8804	−	53.5056i	1.18483	−	2.12164i
$637$	73.1012			2.89638
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−4.47589			−0.176787			−0.0883934	−	0.996086i	$-0.528173\pi$
$641$	−4.47589			−0.176787			−0.0883934	+	0.996086i	$0.528173\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$	80.9532	+	45.2086i	3.19000	+	1.78147i
$645$		0			0
$646$		0			0
$647$			24.7429i			0.972745i	0.873752	+	0.486372i	$0.161680\pi$
$647$			24.7429i			0.972745i	−0.873752	+	0.486372i	$0.838320\pi$
$648$	−24.0478	+	25.1014i	−0.944687	+	0.986075i
$649$		0			0
$650$	−6.42251	+	24.6729i	−0.251912	+	0.967750i
$651$		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	$0.614565\pi$
$654$	54.3776	+	14.1548i	2.12633	+	0.553498i
$655$		0			0
$656$	−21.7827	−	35.3555i	−0.850470	−	1.38040i
$657$	−28.2171			−1.10086
$658$		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
$664$	1.37186	+	1.31428i	0.0532386	+	0.0510040i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	10.9026	−	19.5229i	0.421834	−	0.755362i
$669$		0			0
$670$		0			0
$671$		0			0
$672$	86.3755	−	26.4846i	3.33201	−	1.02167i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$		−	51.2505i		−	1.97263i
$676$	−22.7001	−	12.6770i	−0.873081	−	0.487575i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$	−82.8016	−	21.5538i	−3.17998	−	0.827768i
$679$		0			0
$680$		0			0
$681$	−67.6730			−2.59323
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	0.478161	−	0.856223i	0.0182829	−	0.0327385i
$685$		0			0
$686$	94.8812	+	24.6982i	3.62258	+	0.942981i
$687$		0			0
$688$		0			0
$689$	−36.1276			−1.37635
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$	110.021			4.17933
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−8.06154	+	30.9694i	−0.305134	+	1.17221i
$699$		0			0
$700$	−25.4637	+	45.5968i	−0.962436	+	1.72340i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	50.5799	+	13.1663i	1.90902	+	0.496929i
$703$		0			0
$704$	−26.5086	+	1.13701i	−0.999081	+	0.0428528i
$705$		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
$716$	−23.3222	+	41.7621i	−0.871591	+	1.56072i
$717$	32.7741			1.22397
$718$	−23.2812	−	6.06026i	−0.868848	−	0.226167i
$719$			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
$721$	57.5982			2.14507
$722$	6.76676	−	25.9953i	0.251833	−	0.967447i
$723$			74.4085i			2.76728i
$724$	−40.2291	−	22.4661i	−1.49510	−	0.834945i
$725$		0			0
$726$	−46.0383	−	11.9841i	−1.70864	−	0.444770i
$727$		−	48.6889i		−	1.80577i	−0.429881	−	0.902886i	$-0.641444\pi$
$727$		−	48.6889i		−	1.80577i	0.429881	−	0.902886i	$-0.358556\pi$
$728$	−38.4585	−	36.8443i	−1.42537	−	1.36554i
$729$	15.9721			0.591561
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−19.2217			−0.709970			−0.354985	−	0.934872i	$-0.615514\pi$
$733$	−19.2217			−0.709970			−0.354985	+	0.934872i	$0.615514\pi$
$734$	−49.8426	−	12.9744i	−1.83972	−	0.478892i
$735$		0			0
$736$	−14.7210	−	48.0101i	−0.542622	−	1.76968i
$737$		0			0
$738$	23.4927	−	90.2501i	0.864777	−	3.32215i
$739$		−	12.4601i		−	0.458352i	−0.973385	−	0.229176i	$-0.926397\pi$
$739$		−	12.4601i		−	0.458352i	0.973385	−	0.229176i	$-0.0736031\pi$
$740$		0			0
$741$	−0.851186			−0.0312691
$742$	−71.6184	−	18.6427i	−2.62919	−	0.684396i
$743$			33.1662i			1.21675i	0.793649	+	0.608376i	$0.208179\pi$
$743$			33.1662i			1.21675i	−0.793649	+	0.608376i	$0.791821\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			4.26645i			0.156101i
$748$		0			0
$749$		0			0
$750$		0			0
$751$			54.7670i			1.99848i	0.0390338	+	0.999238i	$0.487572\pi$
$751$			54.7670i			1.99848i	−0.0390338	+	0.999238i	$0.512428\pi$
$752$		0			0
$753$	64.1639			2.33826
$754$		0			0
$755$		0			0
$756$	93.4743	+	52.2010i	3.39963	+	1.89853i
$757$	−16.1970			−0.588691			−0.294346	−	0.955699i	$-0.595102\pi$
$757$	−16.1970			−0.588691			−0.294346	+	0.955699i	$0.595102\pi$
$758$		0			0
$759$		−	90.0356i		−	3.26809i
$760$		0			0
$761$	−0.161538			−0.00585574			−0.00292787	−	0.999996i	$-0.500932\pi$
$761$	−0.161538			−0.00585574			−0.00292787	+	0.999996i	$0.500932\pi$
$762$		0			0
$763$		−	67.8537i		−	2.45647i
$764$	−11.6569	+	20.8735i	−0.421732	+	0.755178i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−43.7022	−	22.0041i	−1.57697	−	0.794005i
$769$	34.5521			1.24598			0.622989	−	0.782230i	$-0.285918\pi$
$769$	34.5521			1.24598			0.622989	+	0.782230i	$0.285918\pi$
$770$		0			0
$771$			75.1882i			2.70784i
$772$	−48.4360	−	27.0493i	−1.74325	−	0.973524i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	10.5266	−	40.4394i	0.377398	−	1.44982i
$779$			0.801449i			0.0287149i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−42.5395	−	69.0461i	−1.51927	−	2.46593i
$785$		0			0
$786$		0			0
$787$			22.9051i			0.816479i	0.912875	+	0.408240i	$0.133857\pi$
$787$			22.9051i			0.816479i	−0.912875	+	0.408240i	$0.866143\pi$
$788$	48.1546	+	26.8921i	1.71544	+	0.957991i
$789$		0			0
$790$		0			0
$791$			103.322i			3.67370i
$792$	−43.0265	−	41.2205i	−1.52888	−	1.46471i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	17.6211	−	31.5533i	0.624562	−	1.11838i
$797$	30.0000			1.06265			0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000			1.06265			0.531327	+	0.847167i	$0.321693\pi$
$798$	−1.68737	−	0.439233i	−0.0597323	−	0.0155487i
$799$		0			0
$800$	27.0416	−	8.29156i	0.956066	−	0.293151i
$801$		0			0
$802$		0			0
$803$		−	14.7337i		−	0.519941i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	8.53135i		−	0.300318i
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$			8.42995i			0.296016i	0.988986	+	0.148008i	$0.0472861\pi$
$811$			8.42995i			0.296016i	−0.988986	+	0.148008i	$0.952714\pi$
$812$		0			0
$813$	100.683			3.53111
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	2.56900	−	9.86915i	0.0898231	−	0.345067i
$819$		−	119.605i		−	4.17933i
$820$		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
$823$			42.2336i			1.47217i	0.676888	+	0.736086i	$0.263328\pi$
$823$			42.2336i			1.47217i	−0.676888	+	0.736086i	$0.736672\pi$
$824$	−22.5253	−	21.5799i	−0.784707	−	0.751771i
$825$	50.7124			1.76558
$826$		0			0
$827$		−	43.5638i		−	1.51486i	−0.652916	−	0.757431i	$-0.726454\pi$
$827$		−	43.5638i		−	1.51486i	0.652916	−	0.757431i	$-0.273546\pi$
$828$	54.9844	−	98.4583i	1.91084	−	3.42166i
$829$	−44.1356			−1.53289			−0.766447	−	0.642308i	$-0.777977\pi$
$829$	−44.1356			−1.53289			−0.766447	+	0.642308i	$0.777977\pi$
$830$		0			0
$831$		0			0
$832$	1.23606	+	28.8179i	0.0428528	+	0.999081i
$833$		0			0
$834$		0			0
$835$		0			0
$836$	0.447080	+	0.249674i	0.0154626	+	0.00863514i
$837$		0			0
$838$	19.8811	+	5.17517i	0.686781	+	0.178773i
$839$		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
$843$		−	58.1867i		−	2.00406i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			57.4476i			1.97392i
$848$	21.0236	+	34.1235i	0.721952	+	1.17180i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	30.7620			1.05327			0.526636	−	0.850091i	$-0.323453\pi$
$853$	30.7620			1.05327			0.526636	+	0.850091i	$0.323453\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$	−13.0280	+	50.0489i	−0.444770	+	1.70864i
$859$			8.61754i			0.294027i	0.989135	+	0.147013i	$0.0469660\pi$
$859$			8.61754i			0.294027i	−0.989135	+	0.147013i	$0.953034\pi$
$860$		0			0
$861$	−165.805			−5.65063
$862$	−19.6041	−	5.10308i	−0.667719	−	0.173812i
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	−16.9979	−	55.4359i	−0.578279	−	1.88597i
$865$		0			0
$866$	2.22777	−	8.55825i	0.0757026	−	0.290821i
$867$			51.9873i			1.76558i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−25.4222	+	26.5360i	−0.860905	+	0.898622i
$873$		0			0
$874$	−0.244139	+	0.937892i	−0.00825814	+	0.0317247i
$875$		0			0
$876$	13.2475	−	23.7218i	0.447592	−	0.801485i
$877$	−56.7471			−1.91621			−0.958107	−	0.286411i	$-0.907538\pi$
$877$	−56.7471			−1.91621			−0.958107	+	0.286411i	$0.907538\pi$
$878$		0			0
$879$			66.1563i			2.23139i
$880$		0			0
$881$	−39.3118			−1.32445			−0.662224	−	0.749306i	$-0.730387\pi$
$881$	−39.3118			−1.32445			−0.662224	+	0.749306i	$0.730387\pi$
$882$	45.8790	−	176.250i	1.54483	−	5.93465i
$883$			45.3144i			1.52495i	0.647017	+	0.762476i	$0.276016\pi$
$883$			45.3144i			1.52495i	−0.647017	+	0.762476i	$0.723984\pi$
$884$		0			0
$885$		0			0
$886$	53.0142	+	13.7999i	1.78105	+	0.463618i
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		−	40.7617i		−	1.36557i
$892$		0			0
$893$		0			0
$894$	22.0634	+	5.74325i	0.737912	+	0.192083i
$895$		0			0
$896$	−12.4204	+	57.7658i	−0.414937	+	1.92982i
$897$	−97.8790			−3.26809
$898$		0			0
$899$		0			0
$900$	55.4565	+	30.9699i	1.84855	+	1.03233i
$901$		0			0
$902$	47.1244	+	12.2668i	1.56907	+	0.408439i
$903$		0			0
$904$	38.7108	−	40.4068i	1.28750	−	1.34391i
$905$		0			0
$906$	21.6800	−	83.2864i	0.720269	−	2.76700i
$907$		−	12.3748i		−	0.410897i	−0.978668	−	0.205449i	$-0.934135\pi$
$907$		−	12.3748i		−	0.410897i	0.978668	−	0.205449i	$-0.0658654\pi$
$908$	21.5794	−	38.6413i	0.716137	−	1.28236i
$909$		0			0
$910$		0			0
$911$			15.3323i			0.507983i	0.967206	+	0.253992i	$0.0817436\pi$
$911$			15.3323i			0.507983i	−0.967206	+	0.253992i	$0.918256\pi$
$912$	0.495327	+	0.803968i	0.0164019	+	0.0266220i
$913$	−2.22774			−0.0737276
$914$	14.0052	−	53.8029i	0.463252	−	1.77964i
$915$		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
$921$	−60.8549			−2.00524
$922$	−11.8553	+	45.5438i	−0.390435	+	1.49990i
$923$		0			0
$924$	−51.6530	+	92.4929i	−1.69926	+	3.04279i
$925$		0			0
$926$		0			0
$927$		−	70.0531i		−	2.30085i
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$			1.56516i			0.0512959i
$932$		0			0
$933$	19.5548			0.640196
$934$	−32.7323	−	8.52042i	−1.07103	−	0.278797i
$935$		0			0
$936$	−44.8114	+	46.7747i	−1.46471	+	1.52888i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		−	54.6880i		−	1.78467i
$940$		0			0
$941$	−10.4771			−0.341543			−0.170771	−	0.985311i	$-0.554626\pi$
$941$	−10.4771			−0.341543			−0.170771	+	0.985311i	$0.554626\pi$
$942$	100.551	+	26.1740i	3.27612	+	0.852795i
$943$			92.1598i			3.00114i
$944$		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
$949$	−16.0172			−0.519941
$950$	−0.528266	−	0.137511i	−0.0171392	−	0.00446145i
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$	−22.6740	+	87.1050i	−0.734098	+	2.82013i
$955$		0			0
$956$	−10.4509	+	18.7141i	−0.338007	+	0.605256i
$957$		0			0
$958$	9.07830	+	2.36314i	0.293307	+	0.0763496i
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$	−42.4873	−	23.7272i	−1.36842	−	0.764201i
$965$		0			0
$966$	−194.033	−	50.5081i	−6.24291	−	1.62507i
$967$			33.3869i			1.07365i	0.843693	+	0.536825i	$0.180377\pi$
$967$			33.3869i			1.07365i	−0.843693	+	0.536825i	$0.819623\pi$
$968$	21.5235	−	22.4664i	0.691790	−	0.722099i
$969$		0			0
$970$		0			0
$971$		−	51.4306i		−	1.65049i	−0.564778	−	0.825243i	$-0.691038\pi$
$971$		−	51.4306i		−	1.65049i	0.564778	−	0.825243i	$-0.308962\pi$
$972$	6.66396	−	11.9329i	0.213747	−	0.382747i
$973$		0			0
$974$		0			0
$975$		−	55.1302i		−	1.76558i
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−82.5262			−2.63486
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	64.8427	+	62.1210i	2.06711	+	1.98035i
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.271424	−	0.486028i	0.00863514	−	0.0154626i
$989$		0			0
$990$		0			0
$991$			36.9752i			1.17456i	0.809385	+	0.587278i	$0.199801\pi$
$991$			36.9752i			1.17456i	−0.809385	+	0.587278i	$0.800199\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	−3.58675	−	2.00303i	−0.113651	−	0.0634686i
$997$		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.571.11	yes	20
4.3	odd	2	inner	572.2.b.b.571.12	yes	20
11.10	odd	2	inner	572.2.b.b.571.10	yes	20
13.12	even	2	inner	572.2.b.b.571.10	yes	20
44.43	even	2	inner	572.2.b.b.571.9	✓	20
52.51	odd	2	inner	572.2.b.b.571.9	✓	20
143.142	odd	2	CM	572.2.b.b.571.11	yes	20
572.571	even	2	inner	572.2.b.b.571.12	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.b.b.571.9	✓	20	44.43	even	2	inner
572.2.b.b.571.9	✓	20	52.51	odd	2	inner
572.2.b.b.571.10	yes	20	11.10	odd	2	inner
572.2.b.b.571.10	yes	20	13.12	even	2	inner
572.2.b.b.571.11	yes	20	1.1	even	1	trivial
572.2.b.b.571.11	yes	20	143.142	odd	2	CM
572.2.b.b.571.12	yes	20	4.3	odd	2	inner
572.2.b.b.571.12	yes	20	572.571	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)

\(f(q)\)	\(=\)	\(q+(0.356257 - 1.36861i) q^{2} +3.05807i q^{3} +(-1.74616 - 0.975150i) q^{4} +(4.18530 + 1.08946i) q^{6} -5.22251i q^{7} +(-1.95668 + 2.04240i) q^{8} -6.35182 q^{9} +O(q^{10})\)	\(q+(0.356257 - 1.36861i) q^{2} +3.05807i q^{3} +(-1.74616 - 0.975150i) q^{4} +(4.18530 + 1.08946i) q^{6} -5.22251i q^{7} +(-1.95668 + 2.04240i) q^{8} -6.35182 q^{9} -3.31662i q^{11} +(2.98208 - 5.33989i) q^{12} -3.60555 q^{13} +(-7.14756 - 1.86056i) q^{14} +(2.09816 + 3.40554i) q^{16} +(-2.26288 + 8.69313i) q^{18} -0.0771978i q^{19} +15.9708 q^{21} +(-4.53915 - 1.18157i) q^{22} -8.87708i q^{23} +(-6.24582 - 5.98367i) q^{24} +5.00000 q^{25} +(-1.28450 + 4.93458i) q^{26} -10.2501i q^{27} +(-5.09273 + 9.11935i) q^{28} +(5.40833 - 1.65831i) q^{32} +10.1425 q^{33} +(11.0913 + 6.19397i) q^{36} +(-0.105653 - 0.0275022i) q^{38} -11.0260i q^{39} -10.3818 q^{41} +(5.68972 - 21.8578i) q^{42} +(-3.23421 + 5.79136i) q^{44} +(-12.1492 - 3.16252i) q^{46} +(-10.4144 + 6.41634i) q^{48} -20.2746 q^{49} +(1.78128 - 6.84303i) q^{50} +(6.29588 + 3.51595i) q^{52} +10.0200 q^{53} +(-14.0283 - 3.65167i) q^{54} +(10.6665 + 10.2188i) q^{56} +0.236076 q^{57} +33.1724i q^{63} +(-0.342822 - 7.99265i) q^{64} +(3.61333 - 13.8811i) q^{66} +27.1468 q^{69} +(12.4285 - 12.9730i) q^{72} +4.44237 q^{73} +15.2904i q^{75} +(-0.0752794 + 0.134800i) q^{76} -17.3211 q^{77} +(-15.0903 - 3.92810i) q^{78} +12.2901 q^{81} +(-3.69858 + 14.2085i) q^{82} -0.671690i q^{83} +(-27.8877 - 15.5740i) q^{84} +(6.77389 + 6.48957i) q^{88} +18.8300i q^{91} +(-8.65648 + 15.5008i) q^{92} +(5.07124 + 16.5391i) q^{96} +(-7.22298 + 27.7480i) q^{98} +21.0666i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 60 q^{9}+O(q^{10}) \) 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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