LMFDB - Embedded newform 600.2.w.a.557.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [600,2,Mod(293,600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("600.293"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 2, 3])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	600.w (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,-4,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.79102412128$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			557.2
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	600.557
Dual form			600.2.w.a.293.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 + 1.00000i) q^{2} +(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} -2.44949 q^{6} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} -4.89898 q^{11} +(2.44949 - 2.44949i) q^{12} +(2.44949 + 2.44949i) q^{13} -4.00000 q^{16} +(-2.00000 + 2.00000i) q^{17} +(-3.00000 - 3.00000i) q^{18} +(4.89898 - 4.89898i) q^{22} +(4.00000 + 4.00000i) q^{23} +4.89898i q^{24} -4.89898 q^{26} +(-3.67423 + 3.67423i) q^{27} +9.79796i q^{29} +2.00000 q^{31} +(4.00000 - 4.00000i) q^{32} +(-6.00000 - 6.00000i) q^{33} -4.00000i q^{34} +6.00000 q^{36} +(-7.34847 + 7.34847i) q^{37} +6.00000i q^{39} +(2.44949 + 2.44949i) q^{43} +9.79796i q^{44} -8.00000 q^{46} +(8.00000 - 8.00000i) q^{47} +(-4.89898 - 4.89898i) q^{48} -7.00000i q^{49} -4.89898 q^{51} +(4.89898 - 4.89898i) q^{52} -7.34847i q^{54} +(-9.79796 - 9.79796i) q^{58} -14.6969i q^{59} +(-2.00000 + 2.00000i) q^{62} +8.00000i q^{64} +12.0000 q^{66} +(-7.34847 + 7.34847i) q^{67} +(4.00000 + 4.00000i) q^{68} +9.79796i q^{69} +(-6.00000 + 6.00000i) q^{72} -14.6969i q^{74} +(-6.00000 - 6.00000i) q^{78} -14.0000i q^{79} -9.00000 q^{81} -4.89898 q^{86} +(-12.0000 + 12.0000i) q^{87} +(-9.79796 - 9.79796i) q^{88} +(8.00000 - 8.00000i) q^{92} +(2.44949 + 2.44949i) q^{93} +16.0000i q^{94} +9.79796 q^{96} +(7.00000 + 7.00000i) q^{98} -14.6969i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 8 q^{8} - 16 q^{16} - 8 q^{17} - 12 q^{18} + 16 q^{23} + 8 q^{31} + 16 q^{32} - 24 q^{33} + 24 q^{36} - 32 q^{46} + 32 q^{47} - 8 q^{62} + 48 q^{66} + 16 q^{68} - 24 q^{72} - 24 q^{78} - 36 q^{81}+ \cdots + 28 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 8 * q^8 - 16 * q^16 - 8 * q^17 - 12 * q^18 + 16 * q^23 + 8 * q^31 + 16 * q^32 - 24 * q^33 + 24 * q^36 - 32 * q^46 + 32 * q^47 - 8 * q^62 + 48 * q^66 + 16 * q^68 - 24 * q^72 - 24 * q^78 - 36 * q^81 - 48 * q^87 + 32 * q^92 + 28 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	−1.00000	+	1.00000i	−0.707107	+	0.707107i
$2$	−1.00000	+	1.00000i	−0.707107	+	0.707107i
$3$	1.22474	+	1.22474i	0.707107	+	0.707107i
$3$	1.22474	+	1.22474i	0.707107	+	0.707107i
$4$		−	2.00000i		−	1.00000i
$5$		0			0
$5$		0			0
$6$	−2.44949			−1.00000
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	2.00000	+	2.00000i	0.707107	+	0.707107i
$9$			3.00000i			1.00000i
$10$		0			0
$11$	−4.89898			−1.47710			−0.738549	−	0.674200i	$-0.764489\pi$
$11$	−4.89898			−1.47710			−0.738549	+	0.674200i	$0.764489\pi$
$12$	2.44949	−	2.44949i	0.707107	−	0.707107i
$13$	2.44949	+	2.44949i	0.679366	+	0.679366i	0.959857	−	0.280491i	$-0.0904971\pi$
$13$	2.44949	+	2.44949i	0.679366	+	0.679366i	−0.280491	+	0.959857i	$0.590497\pi$
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$	−2.00000	+	2.00000i	−0.485071	+	0.485071i	−0.906747	−	0.421676i	$-0.861442\pi$
$17$	−2.00000	+	2.00000i	−0.485071	+	0.485071i	0.421676	+	0.906747i	$0.361442\pi$
$18$	−3.00000	−	3.00000i	−0.707107	−	0.707107i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$	4.89898	−	4.89898i	1.04447	−	1.04447i
$23$	4.00000	+	4.00000i	0.834058	+	0.834058i	0.988069	−	0.154011i	$-0.0492193\pi$
$23$	4.00000	+	4.00000i	0.834058	+	0.834058i	−0.154011	+	0.988069i	$0.549219\pi$
$24$			4.89898i			1.00000i
$25$		0			0
$26$	−4.89898			−0.960769
$27$	−3.67423	+	3.67423i	−0.707107	+	0.707107i
$28$		0			0
$29$			9.79796i			1.81944i	0.415227	+	0.909718i	$0.363702\pi$
$29$			9.79796i			1.81944i	−0.415227	+	0.909718i	$0.636298\pi$
$30$		0			0
$31$	2.00000			0.359211			0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000			0.359211			0.179605	+	0.983739i	$0.442518\pi$
$32$	4.00000	−	4.00000i	0.707107	−	0.707107i
$33$	−6.00000	−	6.00000i	−1.04447	−	1.04447i
$34$		−	4.00000i		−	0.685994i
$35$		0			0
$36$	6.00000			1.00000
$37$	−7.34847	+	7.34847i	−1.20808	+	1.20808i	−0.236433	+	0.971648i	$0.575978\pi$
$37$	−7.34847	+	7.34847i	−1.20808	+	1.20808i	−0.971648	+	0.236433i	$0.924022\pi$
$38$		0			0
$39$			6.00000i			0.960769i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	2.44949	+	2.44949i	0.373544	+	0.373544i	0.868766	−	0.495222i	$-0.164913\pi$
$43$	2.44949	+	2.44949i	0.373544	+	0.373544i	−0.495222	+	0.868766i	$0.664913\pi$
$44$			9.79796i			1.47710i
$45$		0			0
$46$	−8.00000			−1.17954
$47$	8.00000	−	8.00000i	1.16692	−	1.16692i	0.183992	−	0.982928i	$-0.441098\pi$
$47$	8.00000	−	8.00000i	1.16692	−	1.16692i	0.982928	−	0.183992i	$-0.0589021\pi$
$48$	−4.89898	−	4.89898i	−0.707107	−	0.707107i
$49$		−	7.00000i		−	1.00000i
$50$		0			0
$51$	−4.89898			−0.685994
$52$	4.89898	−	4.89898i	0.679366	−	0.679366i
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		−	7.34847i		−	1.00000i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−9.79796	−	9.79796i	−1.28654	−	1.28654i
$59$		−	14.6969i		−	1.91338i	−0.291111	−	0.956689i	$-0.594025\pi$
$59$		−	14.6969i		−	1.91338i	0.291111	−	0.956689i	$-0.405975\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	−2.00000	+	2.00000i	−0.254000	+	0.254000i
$63$		0			0
$64$			8.00000i			1.00000i
$65$		0			0
$66$	12.0000			1.47710
$67$	−7.34847	+	7.34847i	−0.897758	+	0.897758i	−0.995238	−	0.0974792i	$-0.968922\pi$
$67$	−7.34847	+	7.34847i	−0.897758	+	0.897758i	0.0974792	+	0.995238i	$0.468922\pi$
$68$	4.00000	+	4.00000i	0.485071	+	0.485071i
$69$			9.79796i			1.17954i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−6.00000	+	6.00000i	−0.707107	+	0.707107i
$73$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$73$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$74$		−	14.6969i		−	1.70848i
$75$		0			0
$76$		0			0
$77$		0			0
$78$	−6.00000	−	6.00000i	−0.679366	−	0.679366i
$79$		−	14.0000i		−	1.57512i	−0.616236	−	0.787562i	$-0.711343\pi$
$79$		−	14.0000i		−	1.57512i	0.616236	−	0.787562i	$-0.288657\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$		0			0
$86$	−4.89898			−0.528271
$87$	−12.0000	+	12.0000i	−1.28654	+	1.28654i
$88$	−9.79796	−	9.79796i	−1.04447	−	1.04447i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	8.00000	−	8.00000i	0.834058	−	0.834058i
$93$	2.44949	+	2.44949i	0.254000	+	0.254000i
$94$			16.0000i			1.65027i
$95$		0			0
$96$	9.79796			1.00000
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$	7.00000	+	7.00000i	0.707107	+	0.707107i
$99$		−	14.6969i		−	1.47710i
$100$		0			0
$101$	19.5959			1.94987			0.974933	−	0.222497i	$-0.0714208\pi$
$101$	19.5959			1.94987			0.974933	+	0.222497i	$0.0714208\pi$
$102$	4.89898	−	4.89898i	0.485071	−	0.485071i
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$			9.79796i			0.960769i
$105$		0			0
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$	7.34847	+	7.34847i	0.707107	+	0.707107i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	−18.0000			−1.70848
$112$		0			0
$113$	14.0000	+	14.0000i	1.31701	+	1.31701i	0.916132	+	0.400878i	$0.131295\pi$
$113$	14.0000	+	14.0000i	1.31701	+	1.31701i	0.400878	+	0.916132i	$0.368705\pi$
$114$		0			0
$115$		0			0
$116$	19.5959			1.81944
$117$	−7.34847	+	7.34847i	−0.679366	+	0.679366i
$118$	14.6969	+	14.6969i	1.35296	+	1.35296i
$119$		0			0
$120$		0			0
$121$	13.0000			1.18182
$122$		0			0
$123$		0			0
$124$		−	4.00000i		−	0.359211i
$125$		0			0
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$	−8.00000	−	8.00000i	−0.707107	−	0.707107i
$129$			6.00000i			0.528271i
$130$		0			0
$131$	−4.89898			−0.428026			−0.214013	−	0.976831i	$-0.568653\pi$
$131$	−4.89898			−0.428026			−0.214013	+	0.976831i	$0.568653\pi$
$132$	−12.0000	+	12.0000i	−1.04447	+	1.04447i
$133$		0			0
$134$		−	14.6969i		−	1.26962i
$135$		0			0
$136$	−8.00000			−0.685994
$137$	−2.00000	+	2.00000i	−0.170872	+	0.170872i	−0.787362	−	0.616491i	$-0.788554\pi$
$137$	−2.00000	+	2.00000i	−0.170872	+	0.170872i	0.616491	+	0.787362i	$0.288554\pi$
$138$	−9.79796	−	9.79796i	−0.834058	−	0.834058i
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	19.5959			1.65027
$142$		0			0
$143$	−12.0000	−	12.0000i	−1.00349	−	1.00349i
$144$		−	12.0000i		−	1.00000i
$145$		0			0
$146$		0			0
$147$	8.57321	−	8.57321i	0.707107	−	0.707107i
$148$	14.6969	+	14.6969i	1.20808	+	1.20808i
$149$			9.79796i			0.802680i	0.915929	+	0.401340i	$0.131455\pi$
$149$			9.79796i			0.802680i	−0.915929	+	0.401340i	$0.868545\pi$
$150$		0			0
$151$	22.0000			1.79033			0.895167	−	0.445730i	$-0.147056\pi$
$151$	22.0000			1.79033			0.895167	+	0.445730i	$0.147056\pi$
$152$		0			0
$153$	−6.00000	−	6.00000i	−0.485071	−	0.485071i
$154$		0			0
$155$		0			0
$156$	12.0000			0.960769
$157$	17.1464	−	17.1464i	1.36843	−	1.36843i	0.505759	−	0.862675i	$-0.331212\pi$
$157$	17.1464	−	17.1464i	1.36843	−	1.36843i	0.862675	−	0.505759i	$-0.168788\pi$
$158$	14.0000	+	14.0000i	1.11378	+	1.11378i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	9.00000	−	9.00000i	0.707107	−	0.707107i
$163$	2.44949	+	2.44949i	0.191859	+	0.191859i	0.796499	−	0.604640i	$-0.206683\pi$
$163$	2.44949	+	2.44949i	0.191859	+	0.191859i	−0.604640	+	0.796499i	$0.706683\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	8.00000	−	8.00000i	0.619059	−	0.619059i	−0.326231	−	0.945290i	$-0.605779\pi$
$167$	8.00000	−	8.00000i	0.619059	−	0.619059i	0.945290	+	0.326231i	$0.105779\pi$
$168$		0			0
$169$		−	1.00000i		−	0.0769231i
$170$		0			0
$171$		0			0
$172$	4.89898	−	4.89898i	0.373544	−	0.373544i
$173$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$173$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$174$		−	24.0000i		−	1.81944i
$175$		0			0
$176$	19.5959			1.47710
$177$	18.0000	−	18.0000i	1.35296	−	1.35296i
$178$		0			0
$179$		−	14.6969i		−	1.09850i	−0.835658	−	0.549250i	$-0.814913\pi$
$179$		−	14.6969i		−	1.09850i	0.835658	−	0.549250i	$-0.185087\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$			16.0000i			1.17954i
$185$		0			0
$186$	−4.89898			−0.359211
$187$	9.79796	−	9.79796i	0.716498	−	0.716498i
$188$	−16.0000	−	16.0000i	−1.16692	−	1.16692i
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−9.79796	+	9.79796i	−0.707107	+	0.707107i
$193$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$193$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$194$		0			0
$195$		0			0
$196$	−14.0000			−1.00000
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$	14.6969	+	14.6969i	1.04447	+	1.04447i
$199$			26.0000i			1.84309i	0.388270	+	0.921546i	$0.373073\pi$
$199$			26.0000i			1.84309i	−0.388270	+	0.921546i	$0.626927\pi$
$200$		0			0
$201$	−18.0000			−1.26962
$202$	−19.5959	+	19.5959i	−1.37876	+	1.37876i
$203$		0			0
$204$			9.79796i			0.685994i
$205$		0			0
$206$		0			0
$207$	−12.0000	+	12.0000i	−0.834058	+	0.834058i
$208$	−9.79796	−	9.79796i	−0.679366	−	0.679366i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$	−14.6969			−1.00000
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−9.79796			−0.659082
$222$	18.0000	−	18.0000i	1.20808	−	1.20808i
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$		0			0
$225$		0			0
$226$	−28.0000			−1.86253
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$	−19.5959	+	19.5959i	−1.28654	+	1.28654i
$233$	14.0000	+	14.0000i	0.917170	+	0.917170i	0.996823	−	0.0796522i	$-0.0253810\pi$
$233$	14.0000	+	14.0000i	0.917170	+	0.917170i	−0.0796522	+	0.996823i	$0.525381\pi$
$234$		−	14.6969i		−	0.960769i
$235$		0			0
$236$	−29.3939			−1.91338
$237$	17.1464	−	17.1464i	1.11378	−	1.11378i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	22.0000			1.41714			0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000			1.41714			0.708572	+	0.705638i	$0.249340\pi$
$242$	−13.0000	+	13.0000i	−0.835672	+	0.835672i
$243$	−11.0227	−	11.0227i	−0.707107	−	0.707107i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$	4.00000	+	4.00000i	0.254000	+	0.254000i
$249$		0			0
$250$		0			0
$251$	−4.89898			−0.309221			−0.154610	−	0.987976i	$-0.549412\pi$
$251$	−4.89898			−0.309221			−0.154610	+	0.987976i	$0.549412\pi$
$252$		0			0
$253$	−19.5959	−	19.5959i	−1.23198	−	1.23198i
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	−22.0000	+	22.0000i	−1.37232	+	1.37232i	−0.515331	+	0.856991i	$0.672331\pi$
$257$	−22.0000	+	22.0000i	−1.37232	+	1.37232i	−0.856991	+	0.515331i	$0.827669\pi$
$258$	−6.00000	−	6.00000i	−0.373544	−	0.373544i
$259$		0			0
$260$		0			0
$261$	−29.3939			−1.81944
$262$	4.89898	−	4.89898i	0.302660	−	0.302660i
$263$	−16.0000	−	16.0000i	−0.986602	−	0.986602i	0.0133092	−	0.999911i	$-0.495763\pi$
$263$	−16.0000	−	16.0000i	−0.986602	−	0.986602i	−0.999911	+	0.0133092i	$0.995763\pi$
$264$		−	24.0000i		−	1.47710i
$265$		0			0
$266$		0			0
$267$		0			0
$268$	14.6969	+	14.6969i	0.897758	+	0.897758i
$269$			9.79796i			0.597392i	0.954348	+	0.298696i	$0.0965517\pi$
$269$			9.79796i			0.597392i	−0.954348	+	0.298696i	$0.903448\pi$
$270$		0			0
$271$	2.00000			0.121491			0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000			0.121491			0.0607457	+	0.998153i	$0.480652\pi$
$272$	8.00000	−	8.00000i	0.485071	−	0.485071i
$273$		0			0
$274$		−	4.00000i		−	0.241649i
$275$		0			0
$276$	19.5959			1.17954
$277$	−7.34847	+	7.34847i	−0.441527	+	0.441527i	−0.892525	−	0.450998i	$-0.851068\pi$
$277$	−7.34847	+	7.34847i	−0.441527	+	0.441527i	0.450998	+	0.892525i	$0.351068\pi$
$278$		0			0
$279$			6.00000i			0.359211i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	−19.5959	+	19.5959i	−1.16692	+	1.16692i
$283$	−22.0454	−	22.0454i	−1.31046	−	1.31046i	−0.921073	−	0.389391i	$-0.872686\pi$
$283$	−22.0454	−	22.0454i	−1.31046	−	1.31046i	−0.389391	−	0.921073i	$-0.627314\pi$
$284$		0			0
$285$		0			0
$286$	24.0000			1.41915
$287$		0			0
$288$	12.0000	+	12.0000i	0.707107	+	0.707107i
$289$			9.00000i			0.529412i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$			17.1464i			1.00000i
$295$		0			0
$296$	−29.3939			−1.70848
$297$	18.0000	−	18.0000i	1.04447	−	1.04447i
$298$	−9.79796	−	9.79796i	−0.567581	−	0.567581i
$299$			19.5959i			1.13326i
$300$		0			0
$301$		0			0
$302$	−22.0000	+	22.0000i	−1.26596	+	1.26596i
$303$	24.0000	+	24.0000i	1.37876	+	1.37876i
$304$		0			0
$305$		0			0
$306$	12.0000			0.685994
$307$	17.1464	−	17.1464i	0.978598	−	0.978598i	−0.0211774	−	0.999776i	$-0.506741\pi$
$307$	17.1464	−	17.1464i	0.978598	−	0.978598i	0.999776	+	0.0211774i	$0.00674148\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$	−12.0000	+	12.0000i	−0.679366	+	0.679366i
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$			34.2929i			1.93526i
$315$		0			0
$316$	−28.0000			−1.57512
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		−	48.0000i		−	2.68748i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$			18.0000i			1.00000i
$325$		0			0
$326$	−4.89898			−0.271329
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	−22.0454	−	22.0454i	−1.20808	−	1.20808i
$334$			16.0000i			0.875481i
$335$		0			0
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$	1.00000	+	1.00000i	0.0543928	+	0.0543928i
$339$			34.2929i			1.86253i
$340$		0			0
$341$	−9.79796			−0.530589
$342$		0			0
$343$		0			0
$344$			9.79796i			0.528271i
$345$		0			0
$346$		0			0
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$	24.0000	+	24.0000i	1.28654	+	1.28654i
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$	−18.0000			−0.960769
$352$	−19.5959	+	19.5959i	−1.04447	+	1.04447i
$353$	−26.0000	−	26.0000i	−1.38384	−	1.38384i	−0.837682	−	0.546159i	$-0.816090\pi$
$353$	−26.0000	−	26.0000i	−1.38384	−	1.38384i	−0.546159	−	0.837682i	$-0.683910\pi$
$354$			36.0000i			1.91338i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	14.6969	+	14.6969i	0.776757	+	0.776757i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$	15.9217	+	15.9217i	0.835672	+	0.835672i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$	−16.0000	−	16.0000i	−0.834058	−	0.834058i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	4.89898	−	4.89898i	0.254000	−	0.254000i
$373$	26.9444	+	26.9444i	1.39513	+	1.39513i	0.813343	+	0.581785i	$0.197645\pi$
$373$	26.9444	+	26.9444i	1.39513	+	1.39513i	0.581785	+	0.813343i	$0.302355\pi$
$374$			19.5959i			1.01328i
$375$		0			0
$376$	32.0000			1.65027
$377$	−24.0000	+	24.0000i	−1.23606	+	1.23606i
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4.00000	+	4.00000i	0.204390	+	0.204390i	0.801878	−	0.597488i	$-0.203834\pi$
$383$	4.00000	+	4.00000i	0.204390	+	0.204390i	−0.597488	+	0.801878i	$0.703834\pi$
$384$		−	19.5959i		−	1.00000i
$385$		0			0
$386$		0			0
$387$	−7.34847	+	7.34847i	−0.373544	+	0.373544i
$388$		0			0
$389$		−	39.1918i		−	1.98710i	−0.113373	−	0.993552i	$-0.536166\pi$
$389$		−	39.1918i		−	1.98710i	0.113373	−	0.993552i	$-0.463834\pi$
$390$		0			0
$391$	−16.0000			−0.809155
$392$	14.0000	−	14.0000i	0.707107	−	0.707107i
$393$	−6.00000	−	6.00000i	−0.302660	−	0.302660i
$394$		0			0
$395$		0			0
$396$	−29.3939			−1.47710
$397$	17.1464	−	17.1464i	0.860555	−	0.860555i	−0.130848	−	0.991402i	$-0.541770\pi$
$397$	17.1464	−	17.1464i	0.860555	−	0.860555i	0.991402	+	0.130848i	$0.0417700\pi$
$398$	−26.0000	−	26.0000i	−1.30326	−	1.30326i
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$	18.0000	−	18.0000i	0.897758	−	0.897758i
$403$	4.89898	+	4.89898i	0.244036	+	0.244036i
$404$		−	39.1918i		−	1.94987i
$405$		0			0
$406$		0			0
$407$	36.0000	−	36.0000i	1.78445	−	1.78445i
$408$	−9.79796	−	9.79796i	−0.485071	−	0.485071i
$409$		−	34.0000i		−	1.68119i	−0.541663	−	0.840596i	$-0.682205\pi$
$409$		−	34.0000i		−	1.68119i	0.541663	−	0.840596i	$-0.317795\pi$
$410$		0			0
$411$	−4.89898			−0.241649
$412$		0			0
$413$		0			0
$414$		−	24.0000i		−	1.17954i
$415$		0			0
$416$	19.5959			0.960769
$417$		0			0
$418$		0			0
$419$			34.2929i			1.67532i	0.546195	+	0.837658i	$0.316076\pi$
$419$			34.2929i			1.67532i	−0.546195	+	0.837658i	$0.683924\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$	24.0000	+	24.0000i	1.16692	+	1.16692i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		−	29.3939i		−	1.41915i
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	14.6969	−	14.6969i	0.707107	−	0.707107i
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$			26.0000i			1.24091i	0.784241	+	0.620456i	$0.213053\pi$
$439$			26.0000i			1.24091i	−0.784241	+	0.620456i	$0.786947\pi$
$440$		0			0
$441$	21.0000			1.00000
$442$	9.79796	−	9.79796i	0.466041	−	0.466041i
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$			36.0000i			1.70848i
$445$		0			0
$446$		0			0
$447$	−12.0000	+	12.0000i	−0.567581	+	0.567581i
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$	28.0000	−	28.0000i	1.31701	−	1.31701i
$453$	26.9444	+	26.9444i	1.26596	+	1.26596i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$457$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$458$		0			0
$459$		−	14.6969i		−	0.685994i
$460$		0			0
$461$	−29.3939			−1.36901			−0.684505	−	0.729008i	$-0.739981\pi$
$461$	−29.3939			−1.36901			−0.684505	+	0.729008i	$0.739981\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$		−	39.1918i		−	1.81944i
$465$		0			0
$466$	−28.0000			−1.29707
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	14.6969	+	14.6969i	0.679366	+	0.679366i
$469$		0			0
$470$		0			0
$471$	42.0000			1.93526
$472$	29.3939	−	29.3939i	1.35296	−	1.35296i
$473$	−12.0000	−	12.0000i	−0.551761	−	0.551761i
$474$			34.2929i			1.57512i
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−36.0000			−1.64146
$482$	−22.0000	+	22.0000i	−1.00207	+	1.00207i
$483$		0			0
$484$		−	26.0000i		−	1.18182i
$485$		0			0
$486$	22.0454			1.00000
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$		0			0
$489$			6.00000i			0.271329i
$490$		0			0
$491$	44.0908			1.98979			0.994895	−	0.100912i	$-0.0321762\pi$
$491$	44.0908			1.98979			0.994895	+	0.100912i	$0.0321762\pi$
$492$		0			0
$493$	−19.5959	−	19.5959i	−0.882556	−	0.882556i
$494$		0			0
$495$		0			0
$496$	−8.00000			−0.359211
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	19.5959			0.875481
$502$	4.89898	−	4.89898i	0.218652	−	0.218652i
$503$	−16.0000	−	16.0000i	−0.713405	−	0.713405i	0.253841	−	0.967246i	$-0.418306\pi$
$503$	−16.0000	−	16.0000i	−0.713405	−	0.713405i	−0.967246	+	0.253841i	$0.918306\pi$
$504$		0			0
$505$		0			0
$506$	39.1918			1.74229
$507$	1.22474	−	1.22474i	0.0543928	−	0.0543928i
$508$		0			0
$509$		−	39.1918i		−	1.73715i	−0.495560	−	0.868574i	$-0.665037\pi$
$509$		−	39.1918i		−	1.73715i	0.495560	−	0.868574i	$-0.334963\pi$
$510$		0			0
$511$		0			0
$512$	−16.0000	+	16.0000i	−0.707107	+	0.707107i
$513$		0			0
$514$		−	44.0000i		−	1.94076i
$515$		0			0
$516$	12.0000			0.528271
$517$	−39.1918	+	39.1918i	−1.72365	+	1.72365i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	29.3939	−	29.3939i	1.28654	−	1.28654i
$523$	26.9444	+	26.9444i	1.17820	+	1.17820i	0.980204	+	0.197992i	$0.0634421\pi$
$523$	26.9444	+	26.9444i	1.17820	+	1.17820i	0.197992	+	0.980204i	$0.436558\pi$
$524$			9.79796i			0.428026i
$525$		0			0
$526$	32.0000			1.39527
$527$	−4.00000	+	4.00000i	−0.174243	+	0.174243i
$528$	24.0000	+	24.0000i	1.04447	+	1.04447i
$529$			9.00000i			0.391304i
$530$		0			0
$531$	44.0908			1.91338
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−29.3939			−1.26962
$537$	18.0000	−	18.0000i	0.776757	−	0.776757i
$538$	−9.79796	−	9.79796i	−0.422420	−	0.422420i
$539$			34.2929i			1.47710i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$	−2.00000	+	2.00000i	−0.0859074	+	0.0859074i
$543$		0			0
$544$			16.0000i			0.685994i
$545$		0			0
$546$		0			0
$547$	−31.8434	+	31.8434i	−1.36152	+	1.36152i	−0.489547	+	0.871977i	$0.662838\pi$
$547$	−31.8434	+	31.8434i	−1.36152	+	1.36152i	−0.871977	+	0.489547i	$0.837162\pi$
$548$	4.00000	+	4.00000i	0.170872	+	0.170872i
$549$		0			0
$550$		0			0
$551$		0			0
$552$	−19.5959	+	19.5959i	−0.834058	+	0.834058i
$553$		0			0
$554$		−	14.6969i		−	0.624413i
$555$		0			0
$556$		0			0
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$	−6.00000	−	6.00000i	−0.254000	−	0.254000i
$559$			12.0000i			0.507546i
$560$		0			0
$561$	24.0000			1.01328
$562$		0			0
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		−	39.1918i		−	1.65027i
$565$		0			0
$566$	44.0908			1.85328
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$	−24.0000	+	24.0000i	−1.00349	+	1.00349i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−24.0000			−1.00000
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	−9.00000	−	9.00000i	−0.374351	−	0.374351i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$	−17.1464	−	17.1464i	−0.707107	−	0.707107i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	29.3939	−	29.3939i	1.20808	−	1.20808i
$593$	34.0000	+	34.0000i	1.39621	+	1.39621i	0.810568	+	0.585645i	$0.199159\pi$
$593$	34.0000	+	34.0000i	1.39621	+	1.39621i	0.585645	+	0.810568i	$0.300841\pi$
$594$			36.0000i			1.47710i
$595$		0			0
$596$	19.5959			0.802680
$597$	−31.8434	+	31.8434i	−1.30326	+	1.30326i
$598$	−19.5959	−	19.5959i	−0.801337	−	0.801337i
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	22.0000			0.897399			0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000			0.897399			0.448699	+	0.893683i	$0.351887\pi$
$602$		0			0
$603$	−22.0454	−	22.0454i	−0.897758	−	0.897758i
$604$		−	44.0000i		−	1.79033i
$605$		0			0
$606$	−48.0000			−1.94987
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	39.1918			1.58553
$612$	−12.0000	+	12.0000i	−0.485071	+	0.485071i
$613$	26.9444	+	26.9444i	1.08827	+	1.08827i	0.995706	+	0.0925671i	$0.0295073\pi$
$613$	26.9444	+	26.9444i	1.08827	+	1.08827i	0.0925671	+	0.995706i	$0.470493\pi$
$614$			34.2929i			1.38395i
$615$		0			0
$616$		0			0
$617$	−22.0000	+	22.0000i	−0.885687	+	0.885687i	−0.994105	−	0.108419i	$-0.965421\pi$
$617$	−22.0000	+	22.0000i	−0.885687	+	0.885687i	0.108419	+	0.994105i	$0.465421\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$	−29.3939			−1.17954
$622$		0			0
$623$		0			0
$624$		−	24.0000i		−	0.960769i
$625$		0			0
$626$		0			0
$627$		0			0
$628$	−34.2929	−	34.2929i	−1.36843	−	1.36843i
$629$		−	29.3939i		−	1.17201i
$630$		0			0
$631$	−38.0000			−1.51276			−0.756378	−	0.654135i	$-0.773033\pi$
$631$	−38.0000			−1.51276			−0.756378	+	0.654135i	$0.773033\pi$
$632$	28.0000	−	28.0000i	1.11378	−	1.11378i
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	17.1464	−	17.1464i	0.679366	−	0.679366i
$638$	48.0000	+	48.0000i	1.90034	+	1.90034i
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$	2.44949	+	2.44949i	0.0965984	+	0.0965984i	0.753755	−	0.657156i	$-0.228241\pi$
$643$	2.44949	+	2.44949i	0.0965984	+	0.0965984i	−0.657156	+	0.753755i	$0.728241\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−32.0000	+	32.0000i	−1.25805	+	1.25805i	−0.306027	+	0.952023i	$0.599000\pi$
$647$	−32.0000	+	32.0000i	−1.25805	+	1.25805i	−0.952023	+	0.306027i	$0.901000\pi$
$648$	−18.0000	−	18.0000i	−0.707107	−	0.707107i
$649$			72.0000i			2.82625i
$650$		0			0
$651$		0			0
$652$	4.89898	−	4.89898i	0.191859	−	0.191859i
$653$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$653$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	14.6969i		−	0.572511i	−0.958153	−	0.286256i	$-0.907589\pi$
$659$		−	14.6969i		−	0.572511i	0.958153	−	0.286256i	$-0.0924107\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$	−12.0000	−	12.0000i	−0.466041	−	0.466041i
$664$		0			0
$665$		0			0
$666$	44.0908			1.70848
$667$	−39.1918	+	39.1918i	−1.51751	+	1.51751i
$668$	−16.0000	−	16.0000i	−0.619059	−	0.619059i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$673$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$674$		0			0
$675$		0			0
$676$	−2.00000			−0.0769231
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$	−34.2929	−	34.2929i	−1.31701	−	1.31701i
$679$		0			0
$680$		0			0
$681$		0			0
$682$	9.79796	−	9.79796i	0.375183	−	0.375183i
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−9.79796	−	9.79796i	−0.373544	−	0.373544i
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$	−48.0000			−1.81944
$697$		0			0
$698$		0			0
$699$			34.2929i			1.29707i
$700$		0			0
$701$	19.5959			0.740128			0.370064	−	0.929006i	$-0.379336\pi$
$701$	19.5959			0.740128			0.370064	+	0.929006i	$0.379336\pi$
$702$	18.0000	−	18.0000i	0.679366	−	0.679366i
$703$		0			0
$704$		−	39.1918i		−	1.47710i
$705$		0			0
$706$	52.0000			1.95705
$707$		0			0
$708$	−36.0000	−	36.0000i	−1.35296	−	1.35296i
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$	42.0000			1.57512
$712$		0			0
$713$	8.00000	+	8.00000i	0.299602	+	0.299602i
$714$		0			0
$715$		0			0
$716$	−29.3939			−1.09850
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	19.0000	−	19.0000i	0.707107	−	0.707107i
$723$	26.9444	+	26.9444i	1.00207	+	1.00207i
$724$		0			0
$725$		0			0
$726$	−31.8434			−1.18182
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$	−9.79796			−0.362391
$732$		0			0
$733$	−22.0454	−	22.0454i	−0.814266	−	0.814266i	0.171005	−	0.985270i	$-0.445299\pi$
$733$	−22.0454	−	22.0454i	−0.814266	−	0.814266i	−0.985270	+	0.171005i	$0.945299\pi$
$734$		0			0
$735$		0			0
$736$	32.0000			1.17954
$737$	36.0000	−	36.0000i	1.32608	−	1.32608i
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	4.00000	+	4.00000i	0.146746	+	0.146746i	0.776663	−	0.629917i	$-0.216911\pi$
$743$	4.00000	+	4.00000i	0.146746	+	0.146746i	−0.629917	+	0.776663i	$0.716911\pi$
$744$			9.79796i			0.359211i
$745$		0			0
$746$	−53.8888			−1.97301
$747$		0			0
$748$	−19.5959	−	19.5959i	−0.716498	−	0.716498i
$749$		0			0
$750$		0			0
$751$	2.00000			0.0729810			0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000			0.0729810			0.0364905	+	0.999334i	$0.488382\pi$
$752$	−32.0000	+	32.0000i	−1.16692	+	1.16692i
$753$	−6.00000	−	6.00000i	−0.218652	−	0.218652i
$754$		−	48.0000i		−	1.74806i
$755$		0			0
$756$		0			0
$757$	−31.8434	+	31.8434i	−1.15737	+	1.15737i	−0.172327	+	0.985040i	$0.555129\pi$
$757$	−31.8434	+	31.8434i	−1.15737	+	1.15737i	−0.985040	+	0.172327i	$0.944871\pi$
$758$		0			0
$759$		−	48.0000i		−	1.74229i
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−8.00000			−0.289052
$767$	36.0000	−	36.0000i	1.29988	−	1.29988i
$768$	19.5959	+	19.5959i	0.707107	+	0.707107i
$769$		−	34.0000i		−	1.22607i	−0.790055	−	0.613036i	$-0.789948\pi$
$769$		−	34.0000i		−	1.22607i	0.790055	−	0.613036i	$-0.210052\pi$
$770$		0			0
$771$	−53.8888			−1.94076
$772$		0			0
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$		−	14.6969i		−	0.528271i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	39.1918	+	39.1918i	1.40510	+	1.40510i
$779$		0			0
$780$		0			0
$781$		0			0
$782$	16.0000	−	16.0000i	0.572159	−	0.572159i
$783$	−36.0000	−	36.0000i	−1.28654	−	1.28654i
$784$			28.0000i			1.00000i
$785$		0			0
$786$	12.0000			0.428026
$787$	17.1464	−	17.1464i	0.611204	−	0.611204i	−0.332056	−	0.943260i	$-0.607742\pi$
$787$	17.1464	−	17.1464i	0.611204	−	0.611204i	0.943260	+	0.332056i	$0.107742\pi$
$788$		0			0
$789$		−	39.1918i		−	1.39527i
$790$		0			0
$791$		0			0
$792$	29.3939	−	29.3939i	1.04447	−	1.04447i
$793$		0			0
$794$			34.2929i			1.21701i
$795$		0			0
$796$	52.0000			1.84309
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$			32.0000i			1.13208i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$			36.0000i			1.26962i
$805$		0			0
$806$	−9.79796			−0.345118
$807$	−12.0000	+	12.0000i	−0.422420	+	0.422420i
$808$	39.1918	+	39.1918i	1.37876	+	1.37876i
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$	2.44949	+	2.44949i	0.0859074	+	0.0859074i
$814$			72.0000i			2.52360i
$815$		0			0
$816$	19.5959			0.685994
$817$		0			0
$818$	34.0000	+	34.0000i	1.18878	+	1.18878i
$819$		0			0
$820$		0			0
$821$	−29.3939			−1.02585			−0.512927	−	0.858432i	$-0.671439\pi$
$821$	−29.3939			−1.02585			−0.512927	+	0.858432i	$0.671439\pi$
$822$	4.89898	−	4.89898i	0.170872	−	0.170872i
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$	24.0000	+	24.0000i	0.834058	+	0.834058i
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$	−18.0000			−0.624413
$832$	−19.5959	+	19.5959i	−0.679366	+	0.679366i
$833$	14.0000	+	14.0000i	0.485071	+	0.485071i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−7.34847	+	7.34847i	−0.254000	+	0.254000i
$838$	−34.2929	−	34.2929i	−1.18463	−	1.18463i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−67.0000			−2.31034
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$	−48.0000			−1.65027
$847$		0			0
$848$		0			0
$849$		−	54.0000i		−	1.85328i
$850$		0			0
$851$	−58.7878			−2.01522
$852$		0			0
$853$	26.9444	+	26.9444i	0.922558	+	0.922558i	0.997210	−	0.0746514i	$-0.0237844\pi$
$853$	26.9444	+	26.9444i	0.922558	+	0.922558i	−0.0746514	+	0.997210i	$0.523784\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	38.0000	−	38.0000i	1.29806	−	1.29806i	0.368380	−	0.929675i	$-0.379913\pi$
$857$	38.0000	−	38.0000i	1.29806	−	1.29806i	0.929675	−	0.368380i	$-0.120087\pi$
$858$	29.3939	+	29.3939i	1.00349	+	1.00349i
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−16.0000	−	16.0000i	−0.544646	−	0.544646i	0.380241	−	0.924887i	$-0.375841\pi$
$863$	−16.0000	−	16.0000i	−0.544646	−	0.544646i	−0.924887	+	0.380241i	$0.875841\pi$
$864$			29.3939i			1.00000i
$865$		0			0
$866$		0			0
$867$	−11.0227	+	11.0227i	−0.374351	+	0.374351i
$868$		0			0
$869$			68.5857i			2.32661i
$870$		0			0
$871$	−36.0000			−1.21981
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	41.6413	−	41.6413i	1.40613	−	1.40613i	0.627557	−	0.778570i	$-0.284055\pi$
$877$	41.6413	−	41.6413i	1.40613	−	1.40613i	0.778570	−	0.627557i	$-0.215945\pi$
$878$	−26.0000	−	26.0000i	−0.877457	−	0.877457i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−21.0000	+	21.0000i	−0.707107	+	0.707107i
$883$	−22.0454	−	22.0454i	−0.741887	−	0.741887i	0.231054	−	0.972941i	$-0.425783\pi$
$883$	−22.0454	−	22.0454i	−0.741887	−	0.741887i	−0.972941	+	0.231054i	$0.925783\pi$
$884$			19.5959i			0.659082i
$885$		0			0
$886$		0			0
$887$	−32.0000	+	32.0000i	−1.07445	+	1.07445i	−0.0774593	+	0.996996i	$0.524681\pi$
$887$	−32.0000	+	32.0000i	−1.07445	+	1.07445i	−0.996996	+	0.0774593i	$0.975319\pi$
$888$	−36.0000	−	36.0000i	−1.20808	−	1.20808i
$889$		0			0
$890$		0			0
$891$	44.0908			1.47710
$892$		0			0
$893$		0			0
$894$		−	24.0000i		−	0.802680i
$895$		0			0
$896$		0			0
$897$	−24.0000	+	24.0000i	−0.801337	+	0.801337i
$898$		0			0
$899$			19.5959i			0.653560i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$			56.0000i			1.86253i
$905$		0			0
$906$	−53.8888			−1.79033
$907$	41.6413	−	41.6413i	1.38268	−	1.38268i	0.542844	−	0.839834i	$-0.317348\pi$
$907$	41.6413	−	41.6413i	1.38268	−	1.38268i	0.839834	−	0.542844i	$-0.182652\pi$
$908$		0			0
$909$			58.7878i			1.94987i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$	14.6969	+	14.6969i	0.485071	+	0.485071i
$919$			26.0000i			0.857661i	0.903385	+	0.428830i	$0.141074\pi$
$919$			26.0000i			0.857661i	−0.903385	+	0.428830i	$0.858926\pi$
$920$		0			0
$921$	42.0000			1.38395
$922$	29.3939	−	29.3939i	0.968036	−	0.968036i
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	39.1918	+	39.1918i	1.28654	+	1.28654i
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$		0			0
$932$	28.0000	−	28.0000i	0.917170	−	0.917170i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−29.3939			−0.960769
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	19.5959			0.638809			0.319404	−	0.947619i	$-0.396517\pi$
$941$	19.5959			0.638809			0.319404	+	0.947619i	$0.396517\pi$
$942$	−42.0000	+	42.0000i	−1.36843	+	1.36843i
$943$		0			0
$944$			58.7878i			1.91338i
$945$		0			0
$946$	24.0000			0.780307
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$	−34.2929	−	34.2929i	−1.11378	−	1.11378i
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	34.0000	+	34.0000i	1.10137	+	1.10137i	0.994246	+	0.107122i	$0.0341636\pi$
$953$	34.0000	+	34.0000i	1.10137	+	1.10137i	0.107122	+	0.994246i	$0.465836\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	58.7878	−	58.7878i	1.90034	−	1.90034i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−27.0000			−0.870968
$962$	36.0000	−	36.0000i	1.16069	−	1.16069i
$963$		0			0
$964$		−	44.0000i		−	1.41714i
$965$		0			0
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$	26.0000	+	26.0000i	0.835672	+	0.835672i
$969$		0			0
$970$		0			0
$971$	−53.8888			−1.72937			−0.864687	−	0.502312i	$-0.832483\pi$
$971$	−53.8888			−1.72937			−0.864687	+	0.502312i	$0.832483\pi$
$972$	−22.0454	+	22.0454i	−0.707107	+	0.707107i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−22.0000	+	22.0000i	−0.703842	+	0.703842i	−0.965233	−	0.261391i	$-0.915819\pi$
$977$	−22.0000	+	22.0000i	−0.703842	+	0.703842i	0.261391	+	0.965233i	$0.415819\pi$
$978$	−6.00000	−	6.00000i	−0.191859	−	0.191859i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−44.0908	+	44.0908i	−1.40699	+	1.40699i
$983$	44.0000	+	44.0000i	1.40338	+	1.40338i	0.789039	+	0.614343i	$0.210579\pi$
$983$	44.0000	+	44.0000i	1.40338	+	1.40338i	0.614343	+	0.789039i	$0.289421\pi$
$984$		0			0
$985$		0			0
$986$	39.1918			1.24812
$987$		0			0
$988$		0			0
$989$			19.5959i			0.623114i
$990$		0			0
$991$	62.0000			1.96949			0.984747	−	0.173990i	$-0.0556660\pi$
$991$	62.0000			1.96949			0.984747	+	0.173990i	$0.0556660\pi$
$992$	8.00000	−	8.00000i	0.254000	−	0.254000i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−31.8434	+	31.8434i	−1.00849	+	1.00849i	−0.00852589	+	0.999964i	$0.502714\pi$
$997$	−31.8434	+	31.8434i	−1.00849	+	1.00849i	−0.999964	+	0.00852589i	$0.997286\pi$
$998$		0			0
$999$		−	54.0000i		−	1.70848i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.2.w.a.557.2	yes	4
3.2	odd	2		600.2.w.g.557.2	yes	4
5.2	odd	4	inner	600.2.w.a.293.2	yes	4
5.3	odd	4		600.2.w.g.293.1	yes	4
5.4	even	2		600.2.w.g.557.1	yes	4
8.5	even	2	inner	600.2.w.a.557.1	yes	4
15.2	even	4		600.2.w.g.293.2	yes	4
15.8	even	4	inner	600.2.w.a.293.1	✓	4
15.14	odd	2	inner	600.2.w.a.557.1	yes	4
24.5	odd	2		600.2.w.g.557.1	yes	4
40.13	odd	4		600.2.w.g.293.2	yes	4
40.29	even	2		600.2.w.g.557.2	yes	4
40.37	odd	4	inner	600.2.w.a.293.1	✓	4
120.29	odd	2	CM	600.2.w.a.557.2	yes	4
120.53	even	4	inner	600.2.w.a.293.2	yes	4
120.77	even	4		600.2.w.g.293.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.2.w.a.293.1	✓	4	15.8	even	4	inner
600.2.w.a.293.1	✓	4	40.37	odd	4	inner
600.2.w.a.293.2	yes	4	5.2	odd	4	inner
600.2.w.a.293.2	yes	4	120.53	even	4	inner
600.2.w.a.557.1	yes	4	8.5	even	2	inner
600.2.w.a.557.1	yes	4	15.14	odd	2	inner
600.2.w.a.557.2	yes	4	1.1	even	1	trivial
600.2.w.a.557.2	yes	4	120.29	odd	2	CM
600.2.w.g.293.1	yes	4	5.3	odd	4
600.2.w.g.293.1	yes	4	120.77	even	4
600.2.w.g.293.2	yes	4	15.2	even	4
600.2.w.g.293.2	yes	4	40.13	odd	4
600.2.w.g.557.1	yes	4	5.4	even	2
600.2.w.g.557.1	yes	4	24.5	odd	2
600.2.w.g.557.2	yes	4	3.2	odd	2
600.2.w.g.557.2	yes	4	40.29	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	600.w (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} +(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} -2.44949 q^{6} +(2.00000 + 2.00000i) q^{8} +3.00000i q^{9} -4.89898 q^{11} +(2.44949 - 2.44949i) q^{12} +(2.44949 + 2.44949i) q^{13} -4.00000 q^{16} +(-2.00000 + 2.00000i) q^{17} +(-3.00000 - 3.00000i) q^{18} +(4.89898 - 4.89898i) q^{22} +(4.00000 + 4.00000i) q^{23} +4.89898i q^{24} -4.89898 q^{26} +(-3.67423 + 3.67423i) q^{27} +9.79796i q^{29} +2.00000 q^{31} +(4.00000 - 4.00000i) q^{32} +(-6.00000 - 6.00000i) q^{33} -4.00000i q^{34} +6.00000 q^{36} +(-7.34847 + 7.34847i) q^{37} +6.00000i q^{39} +(2.44949 + 2.44949i) q^{43} +9.79796i q^{44} -8.00000 q^{46} +(8.00000 - 8.00000i) q^{47} +(-4.89898 - 4.89898i) q^{48} -7.00000i q^{49} -4.89898 q^{51} +(4.89898 - 4.89898i) q^{52} -7.34847i q^{54} +(-9.79796 - 9.79796i) q^{58} -14.6969i q^{59} +(-2.00000 + 2.00000i) q^{62} +8.00000i q^{64} +12.0000 q^{66} +(-7.34847 + 7.34847i) q^{67} +(4.00000 + 4.00000i) q^{68} +9.79796i q^{69} +(-6.00000 + 6.00000i) q^{72} -14.6969i q^{74} +(-6.00000 - 6.00000i) q^{78} -14.0000i q^{79} -9.00000 q^{81} -4.89898 q^{86} +(-12.0000 + 12.0000i) q^{87} +(-9.79796 - 9.79796i) q^{88} +(8.00000 - 8.00000i) q^{92} +(2.44949 + 2.44949i) q^{93} +16.0000i q^{94} +9.79796 q^{96} +(7.00000 + 7.00000i) q^{98} -14.6969i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 8 q^{8} - 16 q^{16} - 8 q^{17} - 12 q^{18} + 16 q^{23} + 8 q^{31} + 16 q^{32} - 24 q^{33} + 24 q^{36} - 32 q^{46} + 32 q^{47} - 8 q^{62} + 48 q^{66} + 16 q^{68} - 24 q^{72} - 24 q^{78} - 36 q^{81}+ \cdots + 28 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 8 * q^8 - 16 * q^16 - 8 * q^17 - 12 * q^18 + 16 * q^23 + 8 * q^31 + 16 * q^32 - 24 * q^33 + 24 * q^36 - 32 * q^46 + 32 * q^47 - 8 * q^62 + 48 * q^66 + 16 * q^68 - 24 * q^72 - 24 * q^78 - 36 * q^81 - 48 * q^87 + 32 * q^92 + 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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