LMFDB - Embedded newform 572.2.b.b.571.8

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			571.8
Root			$0.516228 + 1.31663i$ of defining polynomial
Character	$\chi$	$=$	572.571
Dual form			572.2.b.b.571.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.516228 + 1.31663i) q^{2} +1.51752i q^{3} +(-1.46702 - 1.35936i) q^{4} +(-1.99800 - 0.783385i) q^{6} -2.42385i q^{7} +(2.54709 - 1.22977i) q^{8} +0.697144 q^{9} +O(q^{10})$	\(q+(-0.516228 + 1.31663i) q^{2} +1.51752i q^{3} +(-1.46702 - 1.35936i) q^{4} +(-1.99800 - 0.783385i) q^{6} -2.42385i q^{7} +(2.54709 - 1.22977i) q^{8} +0.697144 q^{9} -3.31662i q^{11} +(2.06285 - 2.22622i) q^{12} +3.60555 q^{13} +(3.19130 + 1.25126i) q^{14} +(0.304276 + 3.98841i) q^{16} +(-0.359885 + 0.917879i) q^{18} -5.06154i q^{19} +3.67823 q^{21} +(4.36676 + 1.71214i) q^{22} -0.711862i q^{23} +(1.86620 + 3.86525i) q^{24} +5.00000 q^{25} +(-1.86129 + 4.74717i) q^{26} +5.61048i q^{27} +(-3.29488 + 3.55582i) q^{28} +(-5.40833 - 1.65831i) q^{32} +5.03303 q^{33} +(-1.02272 - 0.947670i) q^{36} +(6.66416 + 2.61291i) q^{38} +5.47148i q^{39} -3.92280 q^{41} +(-1.89880 + 4.84286i) q^{42} +(-4.50849 + 4.86554i) q^{44} +(0.937257 + 0.367483i) q^{46} +(-6.05248 + 0.461743i) q^{48} +1.12497 q^{49} +(-2.58114 + 6.58314i) q^{50} +(-5.28940 - 4.90125i) q^{52} -1.89692 q^{53} +(-7.38691 - 2.89629i) q^{54} +(-2.98078 - 6.17375i) q^{56} +7.68097 q^{57} -1.68977i q^{63} +(4.97531 - 6.26468i) q^{64} +(-2.59819 + 6.62663i) q^{66} +1.08026 q^{69} +(1.77569 - 0.857330i) q^{72} +13.2927 q^{73} +7.58758i q^{75} +(-6.88046 + 7.42536i) q^{76} -8.03899 q^{77} +(-7.20391 - 2.82453i) q^{78} -6.42256 q^{81} +(2.02506 - 5.16487i) q^{82} +17.1097i q^{83} +(-5.39602 - 5.00004i) q^{84} +(-4.07870 - 8.44773i) q^{88} -8.73930i q^{91} +(-0.967677 + 1.04431i) q^{92} +(2.51652 - 8.20722i) q^{96} +(-0.580740 + 1.48116i) q^{98} -2.31216i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.516228	+	1.31663i	−0.365029	+	0.930996i
$2$	−0.516228	+	1.31663i	−0.365029	+	0.930996i
$3$			1.51752i			0.876139i	0.898941	+	0.438069i	$0.144338\pi$
$3$			1.51752i			0.876139i	−0.898941	+	0.438069i	$0.855662\pi$
$4$	−1.46702	−	1.35936i	−0.733508	−	0.679680i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$	−1.99800	−	0.783385i	−0.815682	−	0.319816i
$7$		−	2.42385i		−	0.916128i	−0.888919	−	0.458064i	$-0.848543\pi$
$7$		−	2.42385i		−	0.916128i	0.888919	−	0.458064i	$-0.151457\pi$
$8$	2.54709	−	1.22977i	0.900531	−	0.434791i
$9$	0.697144			0.232381
$10$		0			0
$11$		−	3.31662i		−	1.00000i
$11$		−	3.31662i		−	1.00000i
$12$	2.06285	−	2.22622i	0.595494	−	0.642655i
$13$	3.60555			1.00000
$13$	3.60555			1.00000
$14$	3.19130	+	1.25126i	0.852912	+	0.334413i
$15$		0			0
$16$	0.304276	+	3.98841i	0.0760689	+	0.997103i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	−0.359885	+	0.917879i	−0.0848258	+	0.216346i
$19$		−	5.06154i		−	1.16120i	−0.814190	−	0.580598i	$-0.802819\pi$
$19$		−	5.06154i		−	1.16120i	0.814190	−	0.580598i	$-0.197181\pi$
$20$		0			0
$21$	3.67823			0.802655
$22$	4.36676	+	1.71214i	0.930996	+	0.365029i
$23$		−	0.711862i		−	0.148433i	−0.997242	−	0.0742167i	$-0.976354\pi$
$23$		−	0.711862i		−	0.148433i	0.997242	−	0.0742167i	$-0.0236456\pi$
$24$	1.86620	+	3.86525i	0.380937	+	0.788990i
$25$	5.00000			1.00000
$26$	−1.86129	+	4.74717i	−0.365029	+	0.930996i
$27$			5.61048i			1.07974i
$28$	−3.29488	+	3.55582i	−0.622674	+	0.671987i
$29$		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
$33$	5.03303			0.876139
$34$		0			0
$35$		0			0
$36$	−1.02272	−	0.947670i	−0.170454	−	0.157945i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	6.66416	+	2.61291i	1.08107	+	0.423870i
$39$			5.47148i			0.876139i
$40$		0			0
$41$	−3.92280			−0.612639			−0.306320	−	0.951929i	$-0.599098\pi$
$41$	−3.92280			−0.612639			−0.306320	+	0.951929i	$0.599098\pi$
$42$	−1.89880	+	4.84286i	−0.292992	+	0.747269i
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−4.50849	+	4.86554i	−0.679680	+	0.733508i
$45$		0			0
$46$	0.937257	+	0.367483i	0.138191	+	0.0541824i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	−6.05248	+	0.461743i	−0.873600	+	0.0666469i
$49$	1.12497			0.160710
$50$	−2.58114	+	6.58314i	−0.365029	+	0.930996i
$51$		0			0
$52$	−5.28940	−	4.90125i	−0.733508	−	0.679680i
$53$	−1.89692			−0.260562			−0.130281	−	0.991477i	$-0.541588\pi$
$53$	−1.89692			−0.260562			−0.130281	+	0.991477i	$0.541588\pi$
$54$	−7.38691	−	2.89629i	−1.00523	−	0.394135i
$55$		0			0
$56$	−2.98078	−	6.17375i	−0.398324	−	0.825002i
$57$	7.68097			1.01737
$58$		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
$63$		−	1.68977i		−	0.212891i
$64$	4.97531	−	6.26468i	0.621914	−	0.783086i
$65$		0			0
$66$	−2.59819	+	6.62663i	−0.319816	+	0.815682i
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$	1.08026			0.130048
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	1.77569	−	0.857330i	0.209267	−	0.101037i
$73$	13.2927			1.55579			0.777896	−	0.628394i	$-0.216287\pi$
$73$	13.2927			1.55579			0.777896	+	0.628394i	$0.216287\pi$
$74$		0			0
$75$			7.58758i			0.876139i
$76$	−6.88046	+	7.42536i	−0.789243	+	0.851747i
$77$	−8.03899			−0.916128
$78$	−7.20391	−	2.82453i	−0.815682	−	0.319816i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−6.42256			−0.713618
$82$	2.02506	−	5.16487i	0.223631	−	0.570365i
$83$			17.1097i			1.87804i	0.343867	+	0.939019i	$0.388263\pi$
$83$			17.1097i			1.87804i	−0.343867	+	0.939019i	$0.611737\pi$
$84$	−5.39602	−	5.00004i	−0.588754	−	0.545549i
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−4.07870	−	8.44773i	−0.434791	−	0.900531i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		−	8.73930i		−	0.916128i
$92$	−0.967677	+	1.04431i	−0.100887	+	0.108877i
$93$		0			0
$94$		0			0
$95$		0			0
$96$	2.51652	−	8.20722i	0.256841	−	0.837646i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−0.580740	+	1.48116i	−0.0586636	+	0.149620i
$99$		−	2.31216i		−	0.232381i
$100$	−7.33508	−	6.79680i	−0.733508	−	0.679680i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		−	19.6142i		−	1.93264i	−0.257337	−	0.966322i	$-0.582845\pi$
$103$		−	19.6142i		−	1.93264i	0.257337	−	0.966322i	$-0.417155\pi$
$104$	9.18366	−	4.43401i	0.900531	−	0.434791i
$105$		0			0
$106$	0.979244	−	2.49754i	0.0951126	−	0.242582i
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	7.62666	−	8.23066i	0.733876	−	0.791996i
$109$	0.903206			0.0865115			0.0432557	−	0.999064i	$-0.486227\pi$
$109$	0.903206			0.0865115			0.0432557	+	0.999064i	$0.486227\pi$
$110$		0			0
$111$		0			0
$112$	9.66729	−	0.737518i	0.913473	−	0.0696889i
$113$	20.5811			1.93610			0.968052	−	0.250750i	$-0.0806773\pi$
$113$	20.5811			1.93610			0.968052	+	0.250750i	$0.0806773\pi$
$114$	−3.96513	+	10.1130i	−0.371369	+	0.947167i
$115$		0			0
$116$		0			0
$117$	2.51359			0.232381
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		−	5.95292i		−	0.536757i
$124$		0			0
$125$		0			0
$126$	2.22480	+	0.872307i	0.198201	+	0.0777113i
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	5.67986	+	9.78464i	0.502033	+	0.864848i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	−7.38354	−	6.84171i	−0.642655	−	0.595494i
$133$	−12.2684			−1.06380
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$	−0.557662	+	1.42230i	−0.0474713	+	0.121074i
$139$		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
$144$	0.212124	+	2.78050i	0.0176770	+	0.231708i
$145$		0			0
$146$	−6.86206	+	17.5015i	−0.567908	+	1.44844i
$147$			1.70716i			0.140804i
$148$		0			0
$149$	−24.2995			−1.99069			−0.995347	−	0.0963598i	$-0.969280\pi$
$149$	−24.2995			−1.99069			−0.995347	+	0.0963598i	$0.969280\pi$
$150$	−9.99002	−	3.91692i	−0.815682	−	0.319816i
$151$		−	19.8997i		−	1.61942i	−0.586831	−	0.809709i	$-0.699625\pi$
$151$		−	19.8997i		−	1.61942i	0.586831	−	0.809709i	$-0.300375\pi$
$152$	−6.22455	−	12.8922i	−0.504878	−	1.04569i
$153$		0			0
$154$	4.14995	−	10.5844i	0.334413	−	0.852912i
$155$		0			0
$156$	7.43772	−	8.02676i	0.595494	−	0.642655i
$157$	0.644702			0.0514528			0.0257264	−	0.999669i	$-0.491810\pi$
$157$	0.644702			0.0514528			0.0257264	+	0.999669i	$0.491810\pi$
$158$		0			0
$159$		−	2.87861i		−	0.228288i
$160$		0			0
$161$	−1.72544			−0.135984
$162$	3.31551	−	8.45612i	0.260491	−	0.664375i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$	5.75482	+	5.33251i	0.449376	+	0.416399i
$165$		0			0
$166$	−22.5271	−	8.83253i	−1.74845	−	0.685537i
$167$			4.65159i			0.359951i	0.983671	+	0.179976i	$0.0576019\pi$
$167$			4.65159i			0.359951i	−0.983671	+	0.179976i	$0.942398\pi$
$168$	9.36877	−	4.52339i	0.722816	−	0.348987i
$169$	13.0000			1.00000
$170$		0			0
$171$		−	3.52862i		−	0.269840i
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		−	12.1192i		−	0.916128i
$176$	13.2281	−	1.00917i	0.997103	−	0.0760689i
$177$		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
$181$	−26.8091			−1.99271			−0.996353	−	0.0853299i	$-0.972806\pi$
$181$	−26.8091			−1.99271			−0.996353	+	0.0853299i	$0.972806\pi$
$182$	11.5064	+	4.51148i	0.852912	+	0.334413i
$183$		0			0
$184$	−0.875429	−	1.81317i	−0.0645375	−	0.133669i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	13.5989			0.989177
$190$		0			0
$191$		−	24.3197i		−	1.75971i	−0.475241	−	0.879856i	$-0.657639\pi$
$191$		−	24.3197i		−	1.75971i	0.475241	−	0.879856i	$-0.342361\pi$
$192$	9.50676	+	7.55012i	0.686091	+	0.544883i
$193$	−7.04646			−0.507215			−0.253608	−	0.967307i	$-0.581617\pi$
$193$	−7.04646			−0.507215			−0.253608	+	0.967307i	$0.581617\pi$
$194$		0			0
$195$		0			0
$196$	−1.65035	−	1.52924i	−0.117882	−	0.109231i
$197$	−19.2289			−1.37000			−0.685002	−	0.728541i	$-0.740199\pi$
$197$	−19.2289			−1.37000			−0.685002	+	0.728541i	$0.740199\pi$
$198$	3.04426	+	1.19360i	0.216346	+	0.0848258i
$199$			27.3547i			1.93913i	0.244843	+	0.969563i	$0.421264\pi$
$199$			27.3547i			1.93913i	−0.244843	+	0.969563i	$0.578736\pi$
$200$	12.7354	−	6.14887i	0.900531	−	0.434791i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	25.8246	+	10.1254i	1.79928	+	0.705470i
$207$		−	0.496270i		−	0.0344931i
$208$	1.09708	+	14.3804i	0.0760689	+	0.997103i
$209$	−16.7872			−1.16120
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$	2.78281	+	2.57860i	0.191124	+	0.177099i
$213$		0			0
$214$		0			0
$215$		0			0
$216$	6.89962	+	14.2904i	0.469460	+	0.972337i
$217$		0			0
$218$	−0.466261	+	1.18919i	−0.0315791	+	0.0805418i
$219$			20.1719i			1.36309i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−4.01949	+	13.1090i	−0.268564	+	0.875879i
$225$	3.48572			0.232381
$226$	−10.6245	+	27.0976i	−0.706733	+	1.80251i
$227$		−	5.88143i		−	0.390364i	−0.980767	−	0.195182i	$-0.937470\pi$
$227$		−	5.88143i		−	0.390364i	0.980767	−	0.195182i	$-0.0625298\pi$
$228$	−11.2681	−	10.4412i	−0.746249	−	0.691486i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		−	12.1993i		−	0.802655i
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$	−1.29759	+	3.30946i	−0.0848258	+	0.216346i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			25.7176i			1.66354i	0.555124	+	0.831768i	$0.312671\pi$
$239$			25.7176i			1.66354i	−0.555124	+	0.831768i	$0.687329\pi$
$240$		0			0
$241$	−25.8613			−1.66587			−0.832937	−	0.553367i	$-0.813343\pi$
$241$	−25.8613			−1.66587			−0.832937	+	0.553367i	$0.813343\pi$
$242$	5.67851	−	14.4829i	0.365029	−	0.930996i
$243$			7.08509i			0.454509i
$244$		0			0
$245$		0			0
$246$	7.83778	+	3.07307i	0.499719	+	0.195932i
$247$		−	18.2496i		−	1.16120i
$248$		0			0
$249$	−25.9643			−1.64542
$250$		0			0
$251$			29.0654i			1.83459i	0.398210	+	0.917294i	$0.369632\pi$
$251$			29.0654i			1.83459i	−0.398210	+	0.917294i	$0.630368\pi$
$252$	−2.29701	+	2.47892i	−0.144698	+	0.156157i
$253$	−2.36098			−0.148433
$254$		0			0
$255$		0			0
$256$	−15.8148	+	2.42715i	−0.988427	+	0.151697i
$257$	27.1694			1.69478			0.847391	−	0.530970i	$-0.178172\pi$
$257$	27.1694			1.69478			0.847391	+	0.530970i	$0.178172\pi$
$258$		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	$0.5\pi$
$264$	12.8196	−	6.18949i	0.788990	−	0.380937i
$265$		0			0
$266$	6.33329	−	16.1529i	0.388319	−	0.990398i
$267$		0			0
$268$		0			0
$269$	30.2219			1.84266			0.921329	−	0.388783i	$-0.127104\pi$
$269$	30.2219			1.84266			0.921329	+	0.388783i	$0.127104\pi$
$270$		0			0
$271$			26.5375i			1.61204i	0.591888	+	0.806020i	$0.298383\pi$
$271$			26.5375i			1.61204i	−0.591888	+	0.806020i	$0.701617\pi$
$272$		0			0
$273$	13.2620			0.802655
$274$		0			0
$275$		−	16.5831i		−	1.00000i
$276$	−1.58476	−	1.46847i	−0.0953914	−	0.0883912i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−31.6184			−1.88620			−0.943098	−	0.332515i	$-0.892103\pi$
$281$	−31.6184			−1.88620			−0.943098	+	0.332515i	$0.892103\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$		0			0
$286$	15.7446	+	6.17319i	0.930996	+	0.365029i
$287$			9.50828i			0.561256i
$288$	−3.77038	−	1.15608i	−0.222172	−	0.0681228i
$289$	17.0000			1.00000
$290$		0			0
$291$		0			0
$292$	−19.5006	−	18.0696i	−1.14119	−	1.05744i
$293$	−21.6333			−1.26383			−0.631916	−	0.775037i	$-0.717731\pi$
$293$	−21.6333			−1.26383			−0.631916	+	0.775037i	$0.717731\pi$
$294$	−2.24769	−	0.881283i	−0.131088	−	0.0513975i
$295$		0			0
$296$		0			0
$297$	18.6078			1.07974
$298$	12.5441	−	31.9934i	0.726660	−	1.85333i
$299$		−	2.56665i		−	0.148433i
$300$	10.3143	−	11.1311i	0.595494	−	0.642655i
$301$		0			0
$302$	26.2006	+	10.2728i	1.50767	+	0.591134i
$303$		0			0
$304$	20.1875	−	1.54010i	1.15783	−	0.0883310i
$305$		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	$0.807768\pi$
$308$	11.7933	+	10.9279i	0.671987	+	0.622674i
$309$	29.7649			1.69326
$310$		0			0
$311$			15.2146i			0.862741i	0.902175	+	0.431370i	$0.141970\pi$
$311$			15.2146i			0.862741i	−0.902175	+	0.431370i	$0.858030\pi$
$312$	6.72869	+	13.9363i	0.380937	+	0.788990i
$313$	−34.5637			−1.95366			−0.976828	−	0.214026i	$-0.931342\pi$
$313$	−34.5637			−1.95366			−0.976828	+	0.214026i	$0.931342\pi$
$314$	−0.332813	+	0.848832i	−0.0187817	+	0.0479024i
$315$		0			0
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$	3.79006	+	1.48602i	0.212536	+	0.0833318i
$319$		0			0
$320$		0			0
$321$		0			0
$322$	0.890723	−	2.27177i	0.0496380	−	0.126601i
$323$		0			0
$324$	9.42200	+	8.73058i	0.523444	+	0.485032i
$325$	18.0278			1.00000
$326$		0			0
$327$			1.37063i			0.0757960i
$328$	−9.99173	+	4.82416i	−0.551701	+	0.266370i
$329$		0			0
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$	23.2583	−	25.1003i	1.27647	−	1.37756i
$333$		0			0
$334$	−6.12442	−	2.40128i	−0.335113	−	0.131392i
$335$		0			0
$336$	1.11920	+	14.6703i	0.0610571	+	0.800329i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−6.71097	+	17.1162i	−0.365029	+	0.930996i
$339$			31.2321i			1.69629i
$340$		0			0
$341$		0			0
$342$	4.64588	+	1.82157i	0.251220	+	0.0984994i
$343$		−	19.6937i		−	1.06336i
$344$		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
$349$	35.2688			1.88789			0.943947	−	0.330097i	$-0.107081\pi$
$349$	35.2688			1.88789			0.943947	+	0.330097i	$0.107081\pi$
$350$	15.9565	+	6.25629i	0.852912	+	0.334413i
$351$			20.2289i			1.07974i
$352$	−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
$358$	−31.4892	−	12.3464i	−1.66425	−	0.652527i
$359$			26.9479i			1.42226i	0.703062	+	0.711129i	$0.251816\pi$
$359$			26.9479i			1.42226i	−0.703062	+	0.711129i	$0.748184\pi$
$360$		0			0
$361$	−6.61917			−0.348377
$362$	13.8396	−	35.2976i	0.727394	−	1.85520i
$363$		−	16.6927i		−	0.876139i
$364$	−11.8799	+	12.8207i	−0.622674	+	0.671987i
$365$		0			0
$366$		0			0
$367$		−	36.4598i		−	1.90319i	−0.307358	−	0.951594i	$-0.599445\pi$
$367$		−	36.4598i		−	1.90319i	0.307358	−	0.951594i	$-0.400555\pi$
$368$	2.83920	−	0.216602i	0.148003	−	0.0112912i
$369$	−2.73476			−0.142366
$370$		0			0
$371$			4.59784i			0.238708i
$372$		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
$378$	−7.02015	+	17.9047i	−0.361078	+	0.920920i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$	32.0200	+	12.5545i	1.63829	+	0.642345i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	−14.8484	+	8.61928i	−0.757727	+	0.439851i
$385$		0			0
$386$	3.63758	−	9.27756i	0.185148	−	0.472216i
$387$		0			0
$388$		0			0
$389$	−39.2652			−1.99082			−0.995412	−	0.0956765i	$-0.969499\pi$
$389$	−39.2652			−1.99082			−0.995412	+	0.0956765i	$0.969499\pi$
$390$		0			0
$391$		0			0
$392$	2.86539	−	1.38346i	0.144724	−	0.0698751i
$393$		0			0
$394$	9.92652	−	25.3173i	0.500091	−	1.27547i
$395$		0			0
$396$	−3.14307	+	3.39198i	−0.157945	+	0.170454i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	−36.0160	−	14.1213i	−1.80532	−	0.707836i
$399$		−	18.6175i		−	0.932040i
$400$	1.52138	+	19.9421i	0.0760689	+	0.997103i
$401$		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
$412$	−26.6628	+	28.7743i	−1.31358	+	1.41761i
$413$		0			0
$414$	0.653403	+	0.256189i	0.0321130	+	0.0125910i
$415$		0			0
$416$	−19.5000	−	5.97913i	−0.956066	−	0.293151i
$417$		0			0
$418$	8.66604	−	22.1025i	0.423870	−	1.08107i
$419$			31.9128i			1.55904i	0.626376	+	0.779521i	$0.284538\pi$
$419$			31.9128i			1.55904i	−0.626376	+	0.779521i	$0.715462\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	−4.83162	+	2.33278i	−0.234644	+	0.113290i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	18.1469			0.876139
$430$		0			0
$431$		−	41.4910i		−	1.99855i	−0.0380545	−	0.999276i	$-0.512116\pi$
$431$		−	41.4910i		−	1.99855i	0.0380545	−	0.999276i	$-0.487884\pi$
$432$	−22.3769	+	1.70713i	−1.07661	+	0.0821344i
$433$	−29.2433			−1.40534			−0.702672	−	0.711514i	$-0.748010\pi$
$433$	−29.2433			−1.40534			−0.702672	+	0.711514i	$0.748010\pi$
$434$		0			0
$435$		0			0
$436$	−1.32502	−	1.22778i	−0.0634569	−	0.0588001i
$437$	−3.60311			−0.172360
$438$	−26.5588	−	10.4133i	−1.26903	−	0.497566i
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	0.784265			0.0373459
$442$		0			0
$443$		−	27.6416i		−	1.31329i	−0.754198	−	0.656647i	$-0.771974\pi$
$443$		−	27.6416i		−	1.31329i	0.754198	−	0.656647i	$-0.228026\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	36.8749i		−	1.74412i
$448$	−15.1846	−	12.0594i	−0.717407	−	0.569753i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−1.79943	+	4.58939i	−0.0848258	+	0.216346i
$451$			13.0105i			0.612639i
$452$	−30.1928	−	27.9771i	−1.42015	−	1.31593i
$453$	30.1982			1.41883
$454$	7.74365	+	3.03616i	0.363428	+	0.142494i
$455$		0			0
$456$	19.5641	−	9.44585i	0.916173	−	0.442343i
$457$	41.6845			1.94992			0.974959	−	0.222386i	$-0.0713847\pi$
$457$	41.6845			1.94992			0.974959	+	0.222386i	$0.0713847\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−10.9693			−0.510890			−0.255445	−	0.966824i	$-0.582222\pi$
$461$	−10.9693			−0.510890			−0.255445	+	0.966824i	$0.582222\pi$
$462$	16.0619	+	6.29762i	0.747269	+	0.292992i
$463$		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	$0.813345\pi$
$468$	−3.68748	−	3.41687i	−0.170454	−	0.157945i
$469$		0			0
$470$		0			0
$471$			0.978345i			0.0450798i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	25.3077i		−	1.16120i
$476$		0			0
$477$	−1.32243			−0.0605498
$478$	−33.8605	−	13.2762i	−1.54875	−	0.607238i
$479$			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
$482$	13.3504	−	34.0497i	0.608092	−	1.55092i
$483$		−	2.61839i		−	0.119141i
$484$	16.1372	+	14.9530i	0.733508	+	0.679680i
$485$		0			0
$486$	−9.32843	−	3.65752i	−0.423146	−	0.165909i
$487$		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	$0.5\pi$
$492$	−8.09217	+	8.73303i	−0.364823	+	0.393716i
$493$		0			0
$494$	24.0280	+	9.42098i	1.08107	+	0.423870i
$495$		0			0
$496$		0			0
$497$		0			0
$498$	13.4035	−	34.1853i	0.600625	−	1.53188i
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	−7.05887			−0.315367
$502$	−38.2682	−	15.0044i	−1.70800	−	0.669677i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	−2.07804	−	4.30399i	−0.0925630	−	0.191715i
$505$		0			0
$506$	1.21880	−	3.10853i	0.0541824	−	0.138191i
$507$			19.7277i			0.876139i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		−	32.2194i		−	1.42530i
$512$	4.96841	−	22.0752i	0.219575	−	0.975596i
$513$	28.3976			1.25379
$514$	−14.0256	+	35.7720i	−0.618644	+	1.57784i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−4.98655			−0.218465			−0.109232	−	0.994016i	$-0.534839\pi$
$521$	−4.98655			−0.218465			−0.109232	+	0.994016i	$0.534839\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$	18.3911			0.802655
$526$		0			0
$527$		0			0
$528$	1.53143	+	20.0738i	0.0666469	+	0.873600i
$529$	22.4933			0.977968
$530$		0			0
$531$		0			0
$532$	17.9979	+	16.6772i	0.780309	+	0.723047i
$533$	−14.1439			−0.612639
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−36.2937			−1.56619
$538$	−15.6014	+	39.7909i	−0.672623	+	1.71551i
$539$		−	3.73110i		−	0.160710i
$540$		0			0
$541$	−21.1759			−0.910421			−0.455210	−	0.890384i	$-0.650436\pi$
$541$	−21.1759			−0.910421			−0.455210	+	0.890384i	$0.650436\pi$
$542$	−34.9400	−	13.6994i	−1.50080	−	0.588441i
$543$		−	40.6833i		−	1.74589i
$544$		0			0
$545$		0			0
$546$	−6.84624	+	17.4612i	−0.292992	+	0.747269i
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		0			0
$549$		0			0
$550$	21.8338	+	8.56068i	0.930996	+	0.365029i
$551$		0			0
$552$	2.75152	−	1.32848i	0.117113	−	0.0565438i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8.60829			−0.364745			−0.182372	−	0.983230i	$-0.558378\pi$
$557$	−8.60829			−0.364745			−0.182372	+	0.983230i	$0.558378\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	16.3223	−	41.6297i	0.688515	−	1.75604i
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$			15.5673i			0.653765i
$568$		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	$0.5\pi$
$572$	−16.2556	+	17.5430i	−0.679680	+	0.733508i
$573$	36.9055			1.54175
$574$	−12.5189	−	4.90844i	−0.522527	−	0.204874i
$575$		−	3.55931i		−	0.148433i
$576$	3.46851	−	4.36739i	0.144521	−	0.181974i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−8.77588	+	22.3827i	−0.365029	+	0.930996i
$579$		−	10.6931i		−	0.444391i
$580$		0			0
$581$	41.4714			1.72052
$582$		0			0
$583$			6.29137i			0.260562i
$584$	33.8576	−	16.3470i	1.40104	−	0.676444i
$585$		0			0
$586$	11.1677	−	28.4830i	0.461335	−	1.17662i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	2.32064	−	2.50443i	0.0957017	−	0.103281i
$589$		0			0
$590$		0			0
$591$		−	29.1802i		−	1.20031i
$592$		0			0
$593$	36.8306			1.51245			0.756226	−	0.654311i	$-0.227041\pi$
$593$	36.8306			1.51245			0.756226	+	0.654311i	$0.227041\pi$
$594$	−9.60590	+	24.4996i	−0.394135	+	1.00523i
$595$		0			0
$596$	35.6478	+	33.0318i	1.46019	+	1.35304i
$597$	−41.5113			−1.69894
$598$	3.37933	+	1.32498i	0.138191	+	0.0541824i
$599$		−	42.5299i		−	1.73772i	−0.495054	−	0.868862i	$-0.664852\pi$
$599$		−	42.5299i		−	1.73772i	0.495054	−	0.868862i	$-0.335148\pi$
$600$	9.33101	+	19.3262i	0.380937	+	0.788990i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	−27.0509	+	29.1933i	−1.10069	+	1.18786i
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	−8.39361	+	27.3745i	−0.340406	+	1.11018i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	25.6822			1.03729			0.518646	−	0.854989i	$-0.326436\pi$
$613$	25.6822			1.03729			0.518646	+	0.854989i	$0.326436\pi$
$614$	−26.2006	−	10.2728i	−1.05737	−	0.414577i
$615$		0			0
$616$	−20.4760	+	9.88614i	−0.825002	+	0.398324i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$	−15.3655	+	39.1892i	−0.618089	+	1.57642i
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$	3.99388			0.160269
$622$	−20.0320	−	7.85421i	−0.803209	−	0.314925i
$623$		0			0
$624$	−21.8225	+	1.66484i	−0.873600	+	0.0666469i
$625$	25.0000			1.00000
$626$	17.8428	−	45.5075i	0.713140	−	1.81885i
$627$		−	25.4749i		−	1.01737i
$628$	−0.945788	−	0.876382i	−0.0377410	−	0.0349715i
$629$		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
$636$	−3.91307	+	4.22297i	−0.155163	+	0.167452i
$637$	4.05613			0.160710
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−49.3523			−1.94930			−0.974649	−	0.223740i	$-0.928173\pi$
$641$	−49.3523			−1.94930			−0.974649	+	0.223740i	$0.928173\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$	2.53125	+	2.34550i	0.0997454	+	0.0924256i
$645$		0			0
$646$		0			0
$647$		−	6.10950i		−	0.240189i	−0.992762	−	0.120095i	$-0.961680\pi$
$647$		−	6.10950i		−	0.240189i	0.992762	−	0.120095i	$-0.0383198\pi$
$648$	−16.3588	+	7.89830i	−0.642635	+	0.310274i
$649$		0			0
$650$	−9.30644	+	23.7358i	−0.365029	+	0.930996i
$651$		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	$0.614565\pi$
$654$	−1.80461	−	0.707558i	−0.0705658	−	0.0276677i
$655$		0			0
$656$	−1.19361	−	15.6458i	−0.0466028	−	0.610864i
$657$	9.26691			0.361537
$658$		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
$664$	21.0411	+	43.5800i	0.816553	+	1.69123i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	6.32319	−	6.82396i	0.244652	−	0.264027i
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−19.8931	−	6.09965i	−0.767391	−	0.235299i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$			28.0524i			1.07974i
$676$	−19.0712	−	17.6717i	−0.733508	−	0.679680i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$	−41.1210	−	16.1229i	−1.57924	−	0.619196i
$679$		0			0
$680$		0			0
$681$	8.92516			0.342013
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	−4.79667	+	5.17654i	−0.183405	+	0.197930i
$685$		0			0
$686$	25.9292	+	10.1664i	0.989983	+	0.388156i
$687$		0			0
$688$		0			0
$689$	−6.83944			−0.260562
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$	−5.60433			−0.212891
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−18.2067	+	46.4358i	−0.689135	+	1.75762i
$699$		0			0
$700$	−16.4744	+	17.7791i	−0.622674	+	0.671987i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	−26.6339	−	10.4427i	−1.00523	−	0.394135i
$703$		0			0
$704$	−20.7776	−	16.5012i	−0.783086	−	0.621914i
$705$		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
$716$	32.5112	−	35.0859i	1.21500	−	1.31122i
$717$	−39.0269			−1.45749
$718$	−35.4804	−	13.9113i	−1.32412	−	0.519164i
$719$		−	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
$721$	−47.5418			−1.77055
$722$	3.41700	−	8.71498i	0.127168	−	0.324338i
$723$		−	39.2450i		−	1.45954i
$724$	39.3294	+	36.4433i	1.46167	+	1.35440i
$725$		0			0
$726$	21.9780	+	8.61723i	0.815682	+	0.319816i
$727$			37.0928i			1.37570i	0.725855	+	0.687848i	$0.241444\pi$
$727$			37.0928i			1.37570i	−0.725855	+	0.687848i	$0.758556\pi$
$728$	−10.7474	−	22.2598i	−0.398324	−	0.825002i
$729$	−30.0194			−1.11183
$730$		0			0
$731$		0			0
$732$		0			0
$733$	54.0836			1.99763			0.998813	−	0.0487194i	$-0.0155140\pi$
$733$	54.0836			1.99763			0.998813	+	0.0487194i	$0.0155140\pi$
$734$	48.0040	+	18.8216i	1.77186	+	0.694718i
$735$		0			0
$736$	−1.18049	+	3.84998i	−0.0435134	+	0.141912i
$737$		0			0
$738$	1.41176	−	3.60066i	0.0519676	−	0.132542i
$739$			46.4815i			1.70985i	0.518752	+	0.854925i	$0.326397\pi$
$739$			46.4815i			1.70985i	−0.518752	+	0.854925i	$0.673603\pi$
$740$		0			0
$741$	27.6941			1.01737
$742$	−6.05365	−	2.37354i	−0.222236	−	0.0871353i
$743$			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
$747$			11.9279i			0.436421i
$748$		0			0
$749$		0			0
$750$		0			0
$751$			45.5649i			1.66269i	0.555758	+	0.831344i	$0.312428\pi$
$751$			45.5649i			1.66269i	−0.555758	+	0.831344i	$0.687572\pi$
$752$		0			0
$753$	−44.1072			−1.60735
$754$		0			0
$755$		0			0
$756$	−19.9499	−	18.4859i	−0.725569	−	0.672324i
$757$	45.0104			1.63593			0.817966	−	0.575267i	$-0.195102\pi$
$757$	45.0104			1.63593			0.817966	+	0.575267i	$0.195102\pi$
$758$		0			0
$759$		−	3.58282i		−	0.130048i
$760$		0			0
$761$	52.5218			1.90391			0.951957	−	0.306231i	$-0.0990680\pi$
$761$	52.5218			1.90391			0.951957	+	0.306231i	$0.0990680\pi$
$762$		0			0
$763$		−	2.18923i		−	0.0792556i
$764$	−33.0592	+	35.6774i	−1.19604	+	1.29076i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−3.68324	−	23.9993i	−0.132908	−	0.865999i
$769$	30.5833			1.10286			0.551431	−	0.834221i	$-0.314082\pi$
$769$	30.5833			1.10286			0.551431	+	0.834221i	$0.314082\pi$
$770$		0			0
$771$			41.2300i			1.48486i
$772$	10.3373	+	9.57868i	0.372047	+	0.344744i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	20.2698	−	51.6977i	0.726708	−	1.85345i
$779$			19.8554i			0.711394i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.342301	+	4.48683i	0.0122250	+	0.160244i
$785$		0			0
$786$		0			0
$787$		−	41.6338i		−	1.48409i	−0.670353	−	0.742043i	$-0.733857\pi$
$787$		−	41.6338i		−	1.48409i	0.670353	−	0.742043i	$-0.266143\pi$
$788$	28.2092	+	26.1390i	1.00491	+	0.931165i
$789$		0			0
$790$		0			0
$791$		−	49.8853i		−	1.77372i
$792$	−2.84344	−	5.88929i	−0.101037	−	0.209267i
$793$		0			0
$794$		0			0
$795$		0			0
$796$	37.1850	−	40.1298i	1.31799	−	1.42236i
$797$	30.0000			1.06265			0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000			1.06265			0.531327	+	0.847167i	$0.321693\pi$
$798$	24.5123	+	9.61087i	0.867726	+	0.340221i
$799$		0			0
$800$	−27.0416	−	8.29156i	−0.956066	−	0.293151i
$801$		0			0
$802$		0			0
$803$		−	44.0869i		−	1.55579i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			45.8622i			1.61442i
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$			56.1769i			1.97264i	0.164850	+	0.986319i	$0.447286\pi$
$811$			56.1769i			1.97264i	−0.164850	+	0.986319i	$0.552714\pi$
$812$		0			0
$813$	−40.2711			−1.41237
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	3.72258	−	9.49434i	0.130157	−	0.331962i
$819$		−	6.09255i		−	0.212891i
$820$		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
$823$			23.8854i			0.832591i	0.909229	+	0.416296i	$0.136672\pi$
$823$			23.8854i			0.832591i	−0.909229	+	0.416296i	$0.863328\pi$
$824$	−24.1210	−	49.9591i	−0.840296	−	1.74041i
$825$	25.1652			0.876139
$826$		0			0
$827$			57.3167i			1.99310i	0.0830134	+	0.996548i	$0.473546\pi$
$827$			57.3167i			1.99310i	−0.0830134	+	0.996548i	$0.526454\pi$
$828$	−0.674610	+	0.728036i	−0.0234443	+	0.0253010i
$829$	21.5382			0.748051			0.374026	−	0.927418i	$-0.377977\pi$
$829$	21.5382			0.748051			0.374026	+	0.927418i	$0.377977\pi$
$830$		0			0
$831$		0			0
$832$	17.9387	−	22.5876i	0.621914	−	0.783086i
$833$		0			0
$834$		0			0
$835$		0			0
$836$	24.6271	+	22.8199i	0.851747	+	0.789243i
$837$		0			0
$838$	−42.0173	−	16.4743i	−1.45146	−	0.569095i
$839$		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
$843$		−	47.9814i		−	1.65257i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			26.6623i			0.916128i
$848$	−0.577187	−	7.56570i	−0.0198207	−	0.259807i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	54.0739			1.85146			0.925728	−	0.378189i	$-0.123453\pi$
$853$	54.0739			1.85146			0.925728	+	0.378189i	$0.123453\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$	−9.36792	+	23.8927i	−0.319816	+	0.815682i
$859$			41.0518i			1.40067i	0.713814	+	0.700335i	$0.246966\pi$
$859$			41.0518i			1.40067i	−0.713814	+	0.700335i	$0.753034\pi$
$860$		0			0
$861$	−14.4290			−0.491738
$862$	54.6282	+	21.4188i	1.86064	+	0.729528i
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	9.30392	−	30.3433i	0.316526	−	1.03230i
$865$		0			0
$866$	15.0962	−	38.5026i	0.512991	−	1.30837i
$867$			25.7978i			0.876139i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	2.30055	−	1.11074i	0.0779063	−	0.0376144i
$873$		0			0
$874$	1.86003	−	4.74396i	0.0629164	−	0.160467i
$875$		0			0
$876$	27.4208	−	29.5925i	0.926465	−	0.999837i
$877$	−55.8804			−1.88695			−0.943473	−	0.331450i	$-0.892462\pi$
$877$	−55.8804			−1.88695			−0.943473	+	0.331450i	$0.892462\pi$
$878$		0			0
$879$		−	32.8289i		−	1.10729i
$880$		0			0
$881$	57.9493			1.95236			0.976182	−	0.216955i	$-0.0696125\pi$
$881$	57.9493			1.95236			0.976182	+	0.216955i	$0.0696125\pi$
$882$	−0.404860	+	1.03258i	−0.0136323	+	0.0347689i
$883$			59.2620i			1.99433i	0.0752753	+	0.997163i	$0.476016\pi$
$883$			59.2620i			1.99433i	−0.0752753	+	0.997163i	$0.523984\pi$
$884$		0			0
$885$		0			0
$886$	36.3937	+	14.2694i	1.22267	+	0.479390i
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$			21.3012i			0.713618i
$892$		0			0
$893$		0			0
$894$	48.5505	+	19.0359i	1.62377	+	0.636655i
$895$		0			0
$896$	23.7165	−	13.7671i	0.792312	−	0.459927i
$897$	3.89494			0.130048
$898$		0			0
$899$		0			0
$900$	−5.11361	−	4.73835i	−0.170454	−	0.157945i
$901$		0			0
$902$	−17.1299	−	6.71637i	−0.570365	−	0.223631i
$903$		0			0
$904$	52.4218	−	25.3101i	1.74352	−	0.841800i
$905$		0			0
$906$	−15.5892	+	39.7598i	−0.517915	+	1.32093i
$907$		−	52.2389i		−	1.73456i	−0.497818	−	0.867281i	$-0.665865\pi$
$907$		−	52.2389i		−	1.73456i	0.497818	−	0.867281i	$-0.334135\pi$
$908$	−7.99498	+	8.62815i	−0.265323	+	0.286335i
$909$		0			0
$910$		0			0
$911$		−	60.2663i		−	1.99671i	−0.0573229	−	0.998356i	$-0.518256\pi$
$911$		−	60.2663i		−	1.99671i	0.0573229	−	0.998356i	$-0.481744\pi$
$912$	2.33713	+	30.6348i	0.0773902	+	1.01442i
$913$	56.7466			1.87804
$914$	−21.5187	+	54.8829i	−0.711775	+	1.81537i
$915$		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
$921$	−30.1982			−0.995064
$922$	5.66265	−	14.4424i	0.186489	−	0.475637i
$923$		0			0
$924$	−16.5832	+	17.8966i	−0.545549	+	0.588754i
$925$		0			0
$926$		0			0
$927$		−	13.6739i		−	0.449110i
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		−	5.69407i		−	0.186616i
$932$		0			0
$933$	−23.0884			−0.755881
$934$	−31.4892	−	12.3464i	−1.03036	−	0.403986i
$935$		0			0
$936$	6.40233	−	3.09115i	0.209267	−	0.101037i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		−	52.4510i		−	1.71167i
$940$		0			0
$941$	−44.0079			−1.43462			−0.717308	−	0.696756i	$-0.754626\pi$
$941$	−44.0079			−1.43462			−0.717308	+	0.696756i	$0.754626\pi$
$942$	−1.28812	−	0.505049i	−0.0419691	−	0.0164554i
$943$			2.79249i			0.0909361i
$944$		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
$949$	47.9275			1.55579
$950$	33.3208	+	13.0645i	1.08107	+	0.423870i
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$	0.682674	−	1.74114i	0.0221024	−	0.0563716i
$955$		0			0
$956$	34.9595	−	37.7282i	1.13067	−	1.22022i
$957$		0			0
$958$	−8.73352	−	3.42427i	−0.282167	−	0.110633i
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$	37.9390	+	35.1549i	1.22193	+	1.13226i
$965$		0			0
$966$	3.44744	+	1.35169i	0.110920	+	0.0434898i
$967$			3.83170i			0.123219i	0.998100	+	0.0616096i	$0.0196234\pi$
$967$			3.83170i			0.123219i	−0.998100	+	0.0616096i	$0.980377\pi$
$968$	−28.0180	+	13.5275i	−0.900531	+	0.434791i
$969$		0			0
$970$		0			0
$971$		−	62.2970i		−	1.99921i	−0.0281505	−	0.999604i	$-0.508962\pi$
$971$		−	62.2970i		−	1.99921i	0.0281505	−	0.999604i	$-0.491038\pi$
$972$	9.63120	−	10.3939i	0.308921	−	0.333386i
$973$		0			0
$974$		0			0
$975$			27.3574i			0.876139i
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.629665			0.0201036
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	−7.32075	−	15.1626i	−0.233377	−	0.483366i
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−24.8078	+	26.7725i	−0.789243	+	0.851747i
$989$		0			0
$990$		0			0
$991$		−	0.0394371i		−	0.00125276i	−1.00000		0.000626381i	$-0.999801\pi$
$991$		−	0.0394371i		−	0.00125276i	1.00000		0.000626381i	$-0.000199383\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	38.0901	+	35.2949i	1.20693	+	1.11836i
$997$		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.571.8	yes	20
4.3	odd	2	inner	572.2.b.b.571.7	✓	20
11.10	odd	2	inner	572.2.b.b.571.13	yes	20
13.12	even	2	inner	572.2.b.b.571.13	yes	20
44.43	even	2	inner	572.2.b.b.571.14	yes	20
52.51	odd	2	inner	572.2.b.b.571.14	yes	20
143.142	odd	2	CM	572.2.b.b.571.8	yes	20
572.571	even	2	inner	572.2.b.b.571.7	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.b.b.571.7	✓	20	4.3	odd	2	inner
572.2.b.b.571.7	✓	20	572.571	even	2	inner
572.2.b.b.571.8	yes	20	1.1	even	1	trivial
572.2.b.b.571.8	yes	20	143.142	odd	2	CM
572.2.b.b.571.13	yes	20	11.10	odd	2	inner
572.2.b.b.571.13	yes	20	13.12	even	2	inner
572.2.b.b.571.14	yes	20	44.43	even	2	inner
572.2.b.b.571.14	yes	20	52.51	odd	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)

\(f(q)\)	\(=\)	\(q+(-0.516228 + 1.31663i) q^{2} +1.51752i q^{3} +(-1.46702 - 1.35936i) q^{4} +(-1.99800 - 0.783385i) q^{6} -2.42385i q^{7} +(2.54709 - 1.22977i) q^{8} +0.697144 q^{9} +O(q^{10})\)	\(q+(-0.516228 + 1.31663i) q^{2} +1.51752i q^{3} +(-1.46702 - 1.35936i) q^{4} +(-1.99800 - 0.783385i) q^{6} -2.42385i q^{7} +(2.54709 - 1.22977i) q^{8} +0.697144 q^{9} -3.31662i q^{11} +(2.06285 - 2.22622i) q^{12} +3.60555 q^{13} +(3.19130 + 1.25126i) q^{14} +(0.304276 + 3.98841i) q^{16} +(-0.359885 + 0.917879i) q^{18} -5.06154i q^{19} +3.67823 q^{21} +(4.36676 + 1.71214i) q^{22} -0.711862i q^{23} +(1.86620 + 3.86525i) q^{24} +5.00000 q^{25} +(-1.86129 + 4.74717i) q^{26} +5.61048i q^{27} +(-3.29488 + 3.55582i) q^{28} +(-5.40833 - 1.65831i) q^{32} +5.03303 q^{33} +(-1.02272 - 0.947670i) q^{36} +(6.66416 + 2.61291i) q^{38} +5.47148i q^{39} -3.92280 q^{41} +(-1.89880 + 4.84286i) q^{42} +(-4.50849 + 4.86554i) q^{44} +(0.937257 + 0.367483i) q^{46} +(-6.05248 + 0.461743i) q^{48} +1.12497 q^{49} +(-2.58114 + 6.58314i) q^{50} +(-5.28940 - 4.90125i) q^{52} -1.89692 q^{53} +(-7.38691 - 2.89629i) q^{54} +(-2.98078 - 6.17375i) q^{56} +7.68097 q^{57} -1.68977i q^{63} +(4.97531 - 6.26468i) q^{64} +(-2.59819 + 6.62663i) q^{66} +1.08026 q^{69} +(1.77569 - 0.857330i) q^{72} +13.2927 q^{73} +7.58758i q^{75} +(-6.88046 + 7.42536i) q^{76} -8.03899 q^{77} +(-7.20391 - 2.82453i) q^{78} -6.42256 q^{81} +(2.02506 - 5.16487i) q^{82} +17.1097i q^{83} +(-5.39602 - 5.00004i) q^{84} +(-4.07870 - 8.44773i) q^{88} -8.73930i q^{91} +(-0.967677 + 1.04431i) q^{92} +(2.51652 - 8.20722i) q^{96} +(-0.580740 + 1.48116i) q^{98} -2.31216i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 60 q^{9}+O(q^{10}) \) 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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