LMFDB - Embedded newform 252.2.bf.a.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,2,Mod(19,252)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 5]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("252.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	252.bf (of order $6$, degree $2$, 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:	$2.01223013094$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - x^{2} - 2x + 4 $ x^4 - x^3 - x^2 - 2*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			199.2
Root			$-0.895644 - 1.09445i$ of defining polynomial
Character	$\chi$	$=$	252.199
Dual form			252.2.bf.a.19.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.895644 - 1.09445i) q^{2} +(-0.395644 - 1.96048i) q^{4} +(1.50000 - 0.866025i) q^{5} +(-2.29129 - 1.32288i) q^{7} +(-2.50000 - 1.32288i) q^{8} +O(q^{10})$	\(q+(0.895644 - 1.09445i) q^{2} +(-0.395644 - 1.96048i) q^{4} +(1.50000 - 0.866025i) q^{5} +(-2.29129 - 1.32288i) q^{7} +(-2.50000 - 1.32288i) q^{8} +(0.395644 - 2.41733i) q^{10} +(2.29129 + 1.32288i) q^{11} -3.46410i q^{13} +(-3.50000 + 1.32288i) q^{14} +(-3.68693 + 1.55130i) q^{16} +(6.00000 + 3.46410i) q^{17} +(-2.29129 - 2.59808i) q^{20} +(3.50000 - 1.32288i) q^{22} +(-4.58258 + 2.64575i) q^{23} +(-1.00000 + 1.73205i) q^{25} +(-3.79129 - 3.10260i) q^{26} +(-1.68693 + 5.01540i) q^{28} +5.00000 q^{29} +(2.29129 - 3.96863i) q^{31} +(-1.60436 + 5.42458i) q^{32} +(9.16515 - 3.46410i) q^{34} -4.58258 q^{35} +(-4.89564 + 0.180750i) q^{40} -3.46410i q^{41} +10.5830i q^{43} +(1.68693 - 5.01540i) q^{44} +(-1.20871 + 7.38505i) q^{46} +(4.58258 + 7.93725i) q^{47} +(3.50000 + 6.06218i) q^{49} +(1.00000 + 2.64575i) q^{50} +(-6.79129 + 1.37055i) q^{52} +(-3.50000 + 6.06218i) q^{53} +4.58258 q^{55} +(3.97822 + 6.33828i) q^{56} +(4.47822 - 5.47225i) q^{58} +(6.87386 - 11.9059i) q^{59} +(-9.00000 + 5.19615i) q^{61} +(-2.29129 - 6.06218i) q^{62} +(4.50000 + 6.61438i) q^{64} +(-3.00000 - 5.19615i) q^{65} +(4.41742 - 13.1334i) q^{68} +(-4.10436 + 5.01540i) q^{70} -5.29150i q^{71} +(-6.00000 - 3.46410i) q^{73} +(-3.50000 - 6.06218i) q^{77} +(6.87386 - 3.96863i) q^{79} +(-4.18693 + 5.51993i) q^{80} +(-3.79129 - 3.10260i) q^{82} +4.58258 q^{83} +12.0000 q^{85} +(11.5826 + 9.47860i) q^{86} +(-3.97822 - 6.33828i) q^{88} +(-9.00000 + 5.19615i) q^{89} +(-4.58258 + 7.93725i) q^{91} +(7.00000 + 7.93725i) q^{92} +(12.7913 + 2.09355i) q^{94} -8.66025i q^{97} +(9.76951 + 1.59898i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 3 q^{4} + 6 q^{5} - 10 q^{8}+O(q^{10}) $ 4 * q - q^2 + 3 * q^4 + 6 * q^5 - 10 * q^8	$ 4 q - q^{2} + 3 q^{4} + 6 q^{5} - 10 q^{8} - 3 q^{10} - 14 q^{14} - q^{16} + 24 q^{17} + 14 q^{22} - 4 q^{25} - 6 q^{26} + 7 q^{28} + 20 q^{29} - 11 q^{32} - 15 q^{40} - 7 q^{44} - 14 q^{46} + 14 q^{49} + 4 q^{50} - 18 q^{52} - 14 q^{53} - 7 q^{56} - 5 q^{58} - 36 q^{61} + 18 q^{64} - 12 q^{65} + 36 q^{68} - 21 q^{70} - 24 q^{73} - 14 q^{77} - 3 q^{80} - 6 q^{82} + 48 q^{85} + 28 q^{86} + 7 q^{88} - 36 q^{89} + 28 q^{92} + 42 q^{94} + 7 q^{98}+O(q^{100}) $ 4 * q - q^2 + 3 * q^4 + 6 * q^5 - 10 * q^8 - 3 * q^10 - 14 * q^14 - q^16 + 24 * q^17 + 14 * q^22 - 4 * q^25 - 6 * q^26 + 7 * q^28 + 20 * q^29 - 11 * q^32 - 15 * q^40 - 7 * q^44 - 14 * q^46 + 14 * q^49 + 4 * q^50 - 18 * q^52 - 14 * q^53 - 7 * q^56 - 5 * q^58 - 36 * q^61 + 18 * q^64 - 12 * q^65 + 36 * q^68 - 21 * q^70 - 24 * q^73 - 14 * q^77 - 3 * q^80 - 6 * q^82 + 48 * q^85 + 28 * q^86 + 7 * q^88 - 36 * q^89 + 28 * q^92 + 42 * q^94 + 7 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$	$-1$

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 $

$2$	0.895644	−	1.09445i	0.633316	−	0.773893i
$2$	0.895644	−	1.09445i	0.633316	−	0.773893i
$3$		0			0
$3$		0			0
$4$	−0.395644	−	1.96048i	−0.197822	−	0.980238i
$5$	1.50000	−	0.866025i	0.670820	−	0.387298i	−0.125567	−	0.992085i	$-0.540075\pi$
$5$	1.50000	−	0.866025i	0.670820	−	0.387298i	0.796387	+	0.604787i	$0.206742\pi$
$6$		0			0
$7$	−2.29129	−	1.32288i	−0.866025	−	0.500000i
$7$	−2.29129	−	1.32288i	−0.866025	−	0.500000i
$8$	−2.50000	−	1.32288i	−0.883883	−	0.467707i
$9$		0			0
$10$	0.395644	−	2.41733i	0.125114	−	0.764426i
$11$	2.29129	+	1.32288i	0.690849	+	0.398862i	0.803930	−	0.594724i	$-0.202739\pi$
$11$	2.29129	+	1.32288i	0.690849	+	0.398862i	−0.113081	+	0.993586i	$0.536072\pi$
$12$		0			0
$13$		−	3.46410i		−	0.960769i	−0.877058	−	0.480384i	$-0.840497\pi$
$13$		−	3.46410i		−	0.960769i	0.877058	−	0.480384i	$-0.159503\pi$
$14$	−3.50000	+	1.32288i	−0.935414	+	0.353553i
$15$		0			0
$16$	−3.68693	+	1.55130i	−0.921733	+	0.387825i
$17$	6.00000	+	3.46410i	1.45521	+	0.840168i	0.998770	−	0.0495842i	$-0.0157896\pi$
$17$	6.00000	+	3.46410i	1.45521	+	0.840168i	0.456444	+	0.889752i	$0.349123\pi$
$18$		0			0
$19$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$19$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$20$	−2.29129	−	2.59808i	−0.512348	−	0.580948i
$21$		0			0
$22$	3.50000	−	1.32288i	0.746203	−	0.282038i
$23$	−4.58258	+	2.64575i	−0.955533	+	0.551677i	−0.894795	−	0.446476i	$-0.852679\pi$
$23$	−4.58258	+	2.64575i	−0.955533	+	0.551677i	−0.0607377	+	0.998154i	$0.519345\pi$
$24$		0			0
$25$	−1.00000	+	1.73205i	−0.200000	+	0.346410i
$26$	−3.79129	−	3.10260i	−0.743533	−	0.608470i
$27$		0			0
$28$	−1.68693	+	5.01540i	−0.318800	+	0.947822i
$29$	5.00000			0.928477			0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000			0.928477			0.464238	+	0.885710i	$0.346328\pi$
$30$		0			0
$31$	2.29129	−	3.96863i	0.411527	−	0.712786i	−0.583530	−	0.812092i	$-0.698329\pi$
$31$	2.29129	−	3.96863i	0.411527	−	0.712786i	0.995057	+	0.0993055i	$0.0316621\pi$
$32$	−1.60436	+	5.42458i	−0.283613	+	0.958939i
$33$		0			0
$34$	9.16515	−	3.46410i	1.57181	−	0.594089i
$35$	−4.58258			−0.774597
$36$		0			0
$37$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$37$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$38$		0			0
$39$		0			0
$40$	−4.89564	+	0.180750i	−0.774069	+	0.0285791i
$41$		−	3.46410i		−	0.541002i	−0.962720	−	0.270501i	$-0.912811\pi$
$41$		−	3.46410i		−	0.541002i	0.962720	−	0.270501i	$-0.0871893\pi$
$42$		0			0
$43$			10.5830i			1.61389i	0.590624	+	0.806947i	$0.298881\pi$
$43$			10.5830i			1.61389i	−0.590624	+	0.806947i	$0.701119\pi$
$44$	1.68693	−	5.01540i	0.254315	−	0.756100i
$45$		0			0
$46$	−1.20871	+	7.38505i	−0.178215	+	1.08887i
$47$	4.58258	+	7.93725i	0.668437	+	1.15777i	0.978341	+	0.207000i	$0.0663699\pi$
$47$	4.58258	+	7.93725i	0.668437	+	1.15777i	−0.309904	+	0.950768i	$0.600297\pi$
$48$		0			0
$49$	3.50000	+	6.06218i	0.500000	+	0.866025i
$50$	1.00000	+	2.64575i	0.141421	+	0.374166i
$51$		0			0
$52$	−6.79129	+	1.37055i	−0.941782	+	0.190061i
$53$	−3.50000	+	6.06218i	−0.480762	+	0.832704i	−0.999756	−	0.0220735i	$-0.992973\pi$
$53$	−3.50000	+	6.06218i	−0.480762	+	0.832704i	0.518994	+	0.854778i	$0.326307\pi$
$54$		0			0
$55$	4.58258			0.617914
$56$	3.97822	+	6.33828i	0.531612	+	0.846988i
$57$		0			0
$58$	4.47822	−	5.47225i	0.588019	−	0.718542i
$59$	6.87386	−	11.9059i	0.894901	−	1.55001i	0.0609735	−	0.998139i	$-0.480579\pi$
$59$	6.87386	−	11.9059i	0.894901	−	1.55001i	0.833927	−	0.551874i	$-0.186087\pi$
$60$		0			0
$61$	−9.00000	+	5.19615i	−1.15233	+	0.665299i	−0.949454	−	0.313905i	$-0.898363\pi$
$61$	−9.00000	+	5.19615i	−1.15233	+	0.665299i	−0.202878	+	0.979204i	$0.565029\pi$
$62$	−2.29129	−	6.06218i	−0.290994	−	0.769897i
$63$		0			0
$64$	4.50000	+	6.61438i	0.562500	+	0.826797i
$65$	−3.00000	−	5.19615i	−0.372104	−	0.644503i
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$	4.41742	−	13.1334i	0.535691	−	1.59266i
$69$		0			0
$70$	−4.10436	+	5.01540i	−0.490564	+	0.599455i
$71$		−	5.29150i		−	0.627986i	−0.949425	−	0.313993i	$-0.898333\pi$
$71$		−	5.29150i		−	0.627986i	0.949425	−	0.313993i	$-0.101667\pi$
$72$		0			0
$73$	−6.00000	−	3.46410i	−0.702247	−	0.405442i	0.105937	−	0.994373i	$-0.466216\pi$
$73$	−6.00000	−	3.46410i	−0.702247	−	0.405442i	−0.808184	+	0.588930i	$0.799549\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.50000	−	6.06218i	−0.398862	−	0.690849i
$78$		0			0
$79$	6.87386	−	3.96863i	0.773370	−	0.446505i	−0.0607054	−	0.998156i	$-0.519335\pi$
$79$	6.87386	−	3.96863i	0.773370	−	0.446505i	0.834075	+	0.551650i	$0.186002\pi$
$80$	−4.18693	+	5.51993i	−0.468113	+	0.617147i
$81$		0			0
$82$	−3.79129	−	3.10260i	−0.418678	−	0.342625i
$83$	4.58258			0.503003			0.251502	−	0.967857i	$-0.419076\pi$
$83$	4.58258			0.503003			0.251502	+	0.967857i	$0.419076\pi$
$84$		0			0
$85$	12.0000			1.30158
$86$	11.5826	+	9.47860i	1.24898	+	1.02210i
$87$		0			0
$88$	−3.97822	−	6.33828i	−0.424080	−	0.675663i
$89$	−9.00000	+	5.19615i	−0.953998	+	0.550791i	−0.894321	−	0.447427i	$-0.852341\pi$
$89$	−9.00000	+	5.19615i	−0.953998	+	0.550791i	−0.0596775	+	0.998218i	$0.519007\pi$
$90$		0			0
$91$	−4.58258	+	7.93725i	−0.480384	+	0.832050i
$92$	7.00000	+	7.93725i	0.729800	+	0.827516i
$93$		0			0
$94$	12.7913	+	2.09355i	1.31932	+	0.215933i
$95$		0			0
$96$		0			0
$97$		−	8.66025i		−	0.879316i	−0.898165	−	0.439658i	$-0.855100\pi$
$97$		−	8.66025i		−	0.879316i	0.898165	−	0.439658i	$-0.144900\pi$
$98$	9.76951	+	1.59898i	0.986869	+	0.161521i
$99$		0			0
$100$	3.79129	+	1.27520i	0.379129	+	0.127520i
$101$	−6.00000	−	3.46410i	−0.597022	−	0.344691i	0.170847	−	0.985298i	$-0.445350\pi$
$101$	−6.00000	−	3.46410i	−0.597022	−	0.344691i	−0.767869	+	0.640607i	$0.778683\pi$
$102$		0			0
$103$	−9.16515	−	15.8745i	−0.903069	−	1.56416i	−0.823488	−	0.567333i	$-0.807975\pi$
$103$	−9.16515	−	15.8745i	−0.903069	−	1.56416i	−0.0795810	−	0.996828i	$-0.525358\pi$
$104$	−4.58258	+	8.66025i	−0.449359	+	0.849208i
$105$		0			0
$106$	3.50000	+	9.26013i	0.339950	+	0.899423i
$107$	−16.0390	+	9.26013i	−1.55055	+	0.895211i	−0.552453	+	0.833544i	$0.686308\pi$
$107$	−16.0390	+	9.26013i	−1.55055	+	0.895211i	−0.998097	+	0.0616667i	$0.980358\pi$
$108$		0			0
$109$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$109$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$110$	4.10436	−	5.01540i	0.391335	−	0.478200i
$111$		0			0
$112$	10.5000	+	1.32288i	0.992157	+	0.125000i
$113$	−14.0000			−1.31701			−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000			−1.31701			−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$	−4.58258	+	7.93725i	−0.427327	+	0.740153i
$116$	−1.97822	−	9.80238i	−0.183673	−	0.910128i
$117$		0			0
$118$	−6.87386	−	18.1865i	−0.632790	−	1.67421i
$119$	−9.16515	−	15.8745i	−0.840168	−	1.45521i
$120$		0			0
$121$	−2.00000	−	3.46410i	−0.181818	−	0.314918i
$122$	−2.37386	+	14.5040i	−0.214920	+	1.31313i
$123$		0			0
$124$	−8.68693	−	2.92185i	−0.780110	−	0.262390i
$125$			12.1244i			1.08444i
$126$		0			0
$127$		−	7.93725i		−	0.704317i	−0.935940	−	0.352159i	$-0.885448\pi$
$127$		−	7.93725i		−	0.704317i	0.935940	−	0.352159i	$-0.114552\pi$
$128$	11.2695	+	0.999100i	0.996093	+	0.0883088i
$129$		0			0
$130$	−8.37386	−	1.37055i	−0.734436	−	0.120205i
$131$	6.87386	+	11.9059i	0.600572	+	1.04022i	0.992734	+	0.120326i	$0.0383938\pi$
$131$	6.87386	+	11.9059i	0.600572	+	1.04022i	−0.392162	+	0.919896i	$0.628273\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−10.4174	−	16.5975i	−0.893287	−	1.42322i
$137$	2.00000	−	3.46410i	0.170872	−	0.295958i	−0.767853	−	0.640626i	$-0.778675\pi$
$137$	2.00000	−	3.46410i	0.170872	−	0.295958i	0.938725	+	0.344668i	$0.112008\pi$
$138$		0			0
$139$	9.16515			0.777378			0.388689	−	0.921369i	$-0.372928\pi$
$139$	9.16515			0.777378			0.388689	+	0.921369i	$0.372928\pi$
$140$	1.81307	+	8.98403i	0.153232	+	0.759289i
$141$		0			0
$142$	−5.79129	−	4.73930i	−0.485994	−	0.397713i
$143$	4.58258	−	7.93725i	0.383214	−	0.663747i
$144$		0			0
$145$	7.50000	−	4.33013i	0.622841	−	0.359597i
$146$	−9.16515	+	3.46410i	−0.758513	+	0.286691i
$147$		0			0
$148$		0			0
$149$	−7.00000	−	12.1244i	−0.573462	−	0.993266i	−0.996207	−	0.0870170i	$-0.972267\pi$
$149$	−7.00000	−	12.1244i	−0.573462	−	0.993266i	0.422744	−	0.906249i	$-0.361067\pi$
$150$		0			0
$151$	−11.4564	−	6.61438i	−0.932312	−	0.538270i	−0.0447698	−	0.998997i	$-0.514255\pi$
$151$	−11.4564	−	6.61438i	−0.932312	−	0.538270i	−0.887542	+	0.460727i	$0.847589\pi$
$152$		0			0
$153$		0			0
$154$	−9.76951	−	1.59898i	−0.787249	−	0.128849i
$155$		−	7.93725i		−	0.637536i
$156$		0			0
$157$	6.00000	+	3.46410i	0.478852	+	0.276465i	0.719938	−	0.694038i	$-0.244170\pi$
$157$	6.00000	+	3.46410i	0.478852	+	0.276465i	−0.241086	+	0.970504i	$0.577504\pi$
$158$	1.81307	−	11.0776i	0.144240	−	0.881285i
$159$		0			0
$160$	2.29129	+	9.52628i	0.181142	+	0.753119i
$161$	14.0000			1.10335
$162$		0			0
$163$	−13.7477	+	7.93725i	−1.07681	+	0.621694i	−0.930033	−	0.367477i	$-0.880222\pi$
$163$	−13.7477	+	7.93725i	−1.07681	+	0.621694i	−0.146772	+	0.989170i	$0.546888\pi$
$164$	−6.79129	+	1.37055i	−0.530310	+	0.107022i
$165$		0			0
$166$	4.10436	−	5.01540i	0.318560	−	0.389271i
$167$	18.3303			1.41844			0.709221	−	0.704987i	$-0.249047\pi$
$167$	18.3303			1.41844			0.709221	+	0.704987i	$0.249047\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$	10.7477	−	13.1334i	0.824313	−	1.00729i
$171$		0			0
$172$	20.7477	−	4.18710i	1.58200	−	0.319264i
$173$	−12.0000	+	6.92820i	−0.912343	+	0.526742i	−0.881184	−	0.472773i	$-0.843253\pi$
$173$	−12.0000	+	6.92820i	−0.912343	+	0.526742i	−0.0311588	+	0.999514i	$0.509920\pi$
$174$		0			0
$175$	4.58258	−	2.64575i	0.346410	−	0.200000i
$176$	−10.5000	−	1.32288i	−0.791467	−	0.0997155i
$177$		0			0
$178$	−2.37386	+	14.5040i	−0.177929	+	1.08712i
$179$	4.58258	+	2.64575i	0.342518	+	0.197753i	0.661385	−	0.750047i	$-0.269969\pi$
$179$	4.58258	+	2.64575i	0.342518	+	0.197753i	−0.318867	+	0.947799i	$0.603302\pi$
$180$		0			0
$181$			3.46410i			0.257485i	0.991678	+	0.128742i	$0.0410940\pi$
$181$			3.46410i			0.257485i	−0.991678	+	0.128742i	$0.958906\pi$
$182$	4.58258	+	12.1244i	0.339683	+	0.898717i
$183$		0			0
$184$	14.9564	−	0.552200i	1.10260	−	0.0407088i
$185$		0			0
$186$		0			0
$187$	9.16515	+	15.8745i	0.670222	+	1.16086i
$188$	13.7477	−	12.1244i	1.00266	−	0.884260i
$189$		0			0
$190$		0			0
$191$	−9.16515	+	5.29150i	−0.663167	+	0.382880i	−0.793483	−	0.608593i	$-0.791734\pi$
$191$	−9.16515	+	5.29150i	−0.663167	+	0.382880i	0.130316	+	0.991473i	$0.458401\pi$
$192$		0			0
$193$	−1.50000	+	2.59808i	−0.107972	+	0.187014i	−0.914949	−	0.403570i	$-0.867769\pi$
$193$	−1.50000	+	2.59808i	−0.107972	+	0.187014i	0.806976	+	0.590584i	$0.201102\pi$
$194$	−9.47822	−	7.75650i	−0.680497	−	0.556885i
$195$		0			0
$196$	10.5000	−	9.26013i	0.750000	−	0.661438i
$197$	14.0000			0.997459			0.498729	−	0.866758i	$-0.333800\pi$
$197$	14.0000			0.997459			0.498729	+	0.866758i	$0.333800\pi$
$198$		0			0
$199$	9.16515	−	15.8745i	0.649700	−	1.12531i	−0.333494	−	0.942752i	$-0.608228\pi$
$199$	9.16515	−	15.8745i	0.649700	−	1.12531i	0.983194	−	0.182562i	$-0.0584390\pi$
$200$	4.79129	−	3.00725i	0.338795	−	0.212645i
$201$		0			0
$202$	−9.16515	+	3.46410i	−0.644858	+	0.243733i
$203$	−11.4564	−	6.61438i	−0.804084	−	0.464238i
$204$		0			0
$205$	−3.00000	−	5.19615i	−0.209529	−	0.362915i
$206$	−25.5826	−	4.18710i	−1.78242	−	0.291729i
$207$		0			0
$208$	5.37386	+	12.7719i	0.372610	+	0.885572i
$209$		0			0
$210$		0			0
$211$			5.29150i			0.364282i	0.983272	+	0.182141i	$0.0583027\pi$
$211$			5.29150i			0.364282i	−0.983272	+	0.182141i	$0.941697\pi$
$212$	13.2695	+	4.46320i	0.911354	+	0.306534i
$213$		0			0
$214$	−4.23049	+	25.8477i	−0.289191	+	1.76691i
$215$	9.16515	+	15.8745i	0.625058	+	1.08263i
$216$		0			0
$217$	−10.5000	+	6.06218i	−0.712786	+	0.411527i
$218$		0			0
$219$		0			0
$220$	−1.81307	−	8.98403i	−0.122237	−	0.605703i
$221$	12.0000	−	20.7846i	0.807207	−	1.39812i
$222$		0			0
$223$	4.58258			0.306872			0.153436	−	0.988159i	$-0.450966\pi$
$223$	4.58258			0.306872			0.153436	+	0.988159i	$0.450966\pi$
$224$	10.8521	−	10.3069i	0.725085	−	0.688659i
$225$		0			0
$226$	−12.5390	+	15.3223i	−0.834083	+	1.01922i
$227$	−2.29129	+	3.96863i	−0.152078	+	0.263407i	−0.931991	−	0.362481i	$-0.881930\pi$
$227$	−2.29129	+	3.96863i	−0.152078	+	0.263407i	0.779913	+	0.625888i	$0.215263\pi$
$228$		0			0
$229$	−9.00000	+	5.19615i	−0.594737	+	0.343371i	−0.766968	−	0.641685i	$-0.778236\pi$
$229$	−9.00000	+	5.19615i	−0.594737	+	0.343371i	0.172231	+	0.985057i	$0.444902\pi$
$230$	4.58258	+	12.1244i	0.302166	+	0.799456i
$231$		0			0
$232$	−12.5000	−	6.61438i	−0.820665	−	0.434255i
$233$	10.0000	+	17.3205i	0.655122	+	1.13470i	0.981863	+	0.189590i	$0.0607160\pi$
$233$	10.0000	+	17.3205i	0.655122	+	1.13470i	−0.326741	+	0.945114i	$0.605951\pi$
$234$		0			0
$235$	13.7477	+	7.93725i	0.896803	+	0.517769i
$236$	−26.0608	−	8.76555i	−1.69641	−	0.570589i
$237$		0			0
$238$	−25.5826	−	4.18710i	−1.65827	−	0.271409i
$239$			5.29150i			0.342279i	0.985247	+	0.171139i	$0.0547449\pi$
$239$			5.29150i			0.342279i	−0.985247	+	0.171139i	$0.945255\pi$
$240$		0			0
$241$	−4.50000	−	2.59808i	−0.289870	−	0.167357i	0.348013	−	0.937490i	$-0.386857\pi$
$241$	−4.50000	−	2.59808i	−0.289870	−	0.167357i	−0.637883	+	0.770133i	$0.720190\pi$
$242$	−5.58258	−	0.913701i	−0.358862	−	0.0587349i
$243$		0			0
$244$	13.7477	+	15.5885i	0.880108	+	0.997949i
$245$	10.5000	+	6.06218i	0.670820	+	0.387298i
$246$		0			0
$247$		0			0
$248$	−10.9782	+	6.89048i	−0.697118	+	0.437546i
$249$		0			0
$250$	13.2695	+	10.8591i	0.839237	+	0.686790i
$251$	−4.58258			−0.289250			−0.144625	−	0.989487i	$-0.546198\pi$
$251$	−4.58258			−0.289250			−0.144625	+	0.989487i	$0.546198\pi$
$252$		0			0
$253$	−14.0000			−0.880172
$254$	−8.68693	−	7.10895i	−0.545067	−	0.446055i
$255$		0			0
$256$	11.1869	−	11.4391i	0.699183	−	0.714943i
$257$	9.00000	−	5.19615i	0.561405	−	0.324127i	−0.192304	−	0.981335i	$-0.561596\pi$
$257$	9.00000	−	5.19615i	0.561405	−	0.324127i	0.753709	+	0.657208i	$0.228263\pi$
$258$		0			0
$259$		0			0
$260$	−9.00000	+	7.93725i	−0.558156	+	0.492248i
$261$		0			0
$262$	19.1869	+	3.14033i	1.18537	+	0.194010i
$263$	−9.16515	−	5.29150i	−0.565147	−	0.326288i	0.190061	−	0.981772i	$-0.439131\pi$
$263$	−9.16515	−	5.29150i	−0.565147	−	0.326288i	−0.755209	+	0.655484i	$0.772465\pi$
$264$		0			0
$265$			12.1244i			0.744793i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	16.5000	+	9.52628i	1.00602	+	0.580828i	0.910025	−	0.414554i	$-0.136062\pi$
$269$	16.5000	+	9.52628i	1.00602	+	0.580828i	0.0959980	+	0.995382i	$0.469396\pi$
$270$		0			0
$271$	6.87386	+	11.9059i	0.417557	+	0.723231i	0.995693	−	0.0927099i	$-0.0295529\pi$
$271$	6.87386	+	11.9059i	0.417557	+	0.723231i	−0.578136	+	0.815941i	$0.696220\pi$
$272$	−27.4955	−	3.46410i	−1.66716	−	0.210042i
$273$		0			0
$274$	−2.00000	−	5.29150i	−0.120824	−	0.319671i
$275$	−4.58258	+	2.64575i	−0.276340	+	0.159545i
$276$		0			0
$277$	2.00000	−	3.46410i	0.120168	−	0.208138i	−0.799666	−	0.600446i	$-0.794990\pi$
$277$	2.00000	−	3.46410i	0.120168	−	0.208138i	0.919834	+	0.392308i	$0.128323\pi$
$278$	8.20871	−	10.0308i	0.492326	−	0.601608i
$279$		0			0
$280$	11.4564	+	6.06218i	0.684653	+	0.362284i
$281$	16.0000			0.954480			0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000			0.954480			0.477240	+	0.878773i	$0.341637\pi$
$282$		0			0
$283$	−9.16515	+	15.8745i	−0.544812	+	0.943642i	0.453807	+	0.891100i	$0.350066\pi$
$283$	−9.16515	+	15.8745i	−0.544812	+	0.943642i	−0.998619	+	0.0525416i	$0.983268\pi$
$284$	−10.3739	+	2.09355i	−0.615576	+	0.124229i
$285$		0			0
$286$	−4.58258	−	12.1244i	−0.270973	−	0.716928i
$287$	−4.58258	+	7.93725i	−0.270501	+	0.468521i
$288$		0			0
$289$	15.5000	+	26.8468i	0.911765	+	1.57922i
$290$	1.97822	−	12.0866i	0.116165	−	0.709751i
$291$		0			0
$292$	−4.41742	+	13.1334i	−0.258510	+	0.768574i
$293$			15.5885i			0.910687i	0.890316	+	0.455344i	$0.150484\pi$
$293$			15.5885i			0.910687i	−0.890316	+	0.455344i	$0.849516\pi$
$294$		0			0
$295$		−	23.8118i		−	1.38637i
$296$		0			0
$297$		0			0
$298$	−19.5390	−	3.19795i	−1.13186	−	0.185252i
$299$	9.16515	+	15.8745i	0.530034	+	0.918046i
$300$		0			0
$301$	14.0000	−	24.2487i	0.806947	−	1.39767i
$302$	−17.5000	+	6.61438i	−1.00701	+	0.380615i
$303$		0			0
$304$		0			0
$305$	−9.00000	+	15.5885i	−0.515339	+	0.892592i
$306$		0			0
$307$	−27.4955			−1.56925			−0.784624	−	0.619972i	$-0.787144\pi$
$307$	−27.4955			−1.56925			−0.784624	+	0.619972i	$0.787144\pi$
$308$	−10.5000	+	9.26013i	−0.598293	+	0.527645i
$309$		0			0
$310$	−8.68693	−	7.10895i	−0.493385	−	0.403761i
$311$	4.58258	−	7.93725i	0.259854	−	0.450080i	−0.706349	−	0.707864i	$-0.749659\pi$
$311$	4.58258	−	7.93725i	0.259854	−	0.450080i	0.966203	+	0.257784i	$0.0829922\pi$
$312$		0			0
$313$	19.5000	−	11.2583i	1.10221	−	0.636358i	0.165406	−	0.986226i	$-0.447107\pi$
$313$	19.5000	−	11.2583i	1.10221	−	0.636358i	0.936799	+	0.349867i	$0.113773\pi$
$314$	9.16515	−	3.46410i	0.517219	−	0.195491i
$315$		0			0
$316$	−10.5000	−	11.9059i	−0.590671	−	0.669758i
$317$	−3.50000	−	6.06218i	−0.196580	−	0.340486i	0.750838	−	0.660487i	$-0.229650\pi$
$317$	−3.50000	−	6.06218i	−0.196580	−	0.340486i	−0.947417	+	0.320001i	$0.896317\pi$
$318$		0			0
$319$	11.4564	+	6.61438i	0.641437	+	0.370334i
$320$	12.4782	+	6.02445i	0.697554	+	0.336777i
$321$		0			0
$322$	12.5390	−	15.3223i	0.698772	−	0.853879i
$323$		0			0
$324$		0			0
$325$	6.00000	+	3.46410i	0.332820	+	0.192154i
$326$	−3.62614	+	22.1552i	−0.200833	+	1.22706i
$327$		0			0
$328$	−4.58258	+	8.66025i	−0.253030	+	0.478183i
$329$		−	24.2487i		−	1.33687i
$330$		0			0
$331$	9.16515	−	5.29150i	0.503762	−	0.290847i	−0.226504	−	0.974010i	$-0.572730\pi$
$331$	9.16515	−	5.29150i	0.503762	−	0.290847i	0.730266	+	0.683163i	$0.239396\pi$
$332$	−1.81307	−	8.98403i	−0.0995050	−	0.493063i
$333$		0			0
$334$	16.4174	−	20.0616i	0.898321	−	1.09772i
$335$		0			0
$336$		0			0
$337$	−21.0000			−1.14394			−0.571971	−	0.820274i	$-0.693821\pi$
$337$	−21.0000			−1.14394			−0.571971	+	0.820274i	$0.693821\pi$
$338$	0.895644	−	1.09445i	0.0487166	−	0.0595303i
$339$		0			0
$340$	−4.74773	−	23.5257i	−0.257482	−	1.27586i
$341$	10.5000	−	6.06218i	0.568607	−	0.328285i
$342$		0			0
$343$		−	18.5203i		−	1.00000i
$344$	14.0000	−	26.4575i	0.754829	−	1.42649i
$345$		0			0
$346$	−3.16515	+	19.3386i	−0.170160	+	1.03965i
$347$	4.58258	+	2.64575i	0.246006	+	0.142031i	0.617934	−	0.786230i	$-0.287970\pi$
$347$	4.58258	+	2.64575i	0.246006	+	0.142031i	−0.371928	+	0.928261i	$0.621303\pi$
$348$		0			0
$349$			20.7846i			1.11257i	0.830990	+	0.556287i	$0.187775\pi$
$349$			20.7846i			1.11257i	−0.830990	+	0.556287i	$0.812225\pi$
$350$	1.20871	−	7.38505i	0.0646084	−	0.394748i
$351$		0			0
$352$	−10.8521	+	10.3069i	−0.578418	+	0.549360i
$353$	−6.00000	−	3.46410i	−0.319348	−	0.184376i	0.331754	−	0.943366i	$-0.392360\pi$
$353$	−6.00000	−	3.46410i	−0.319348	−	0.184376i	−0.651102	+	0.758990i	$0.725693\pi$
$354$		0			0
$355$	−4.58258	−	7.93725i	−0.243218	−	0.421266i
$356$	13.7477	+	15.5885i	0.728628	+	0.826187i
$357$		0			0
$358$	7.00000	−	2.64575i	0.369961	−	0.139832i
$359$	−9.16515	+	5.29150i	−0.483718	+	0.279275i	−0.721965	−	0.691930i	$-0.756761\pi$
$359$	−9.16515	+	5.29150i	−0.483718	+	0.279275i	0.238247	+	0.971205i	$0.423427\pi$
$360$		0			0
$361$	9.50000	−	16.4545i	0.500000	−	0.866025i
$362$	3.79129	+	3.10260i	0.199266	+	0.163069i
$363$		0			0
$364$	17.3739	+	5.84370i	0.910638	+	0.306293i
$365$	−12.0000			−0.628109
$366$		0			0
$367$	−2.29129	+	3.96863i	−0.119604	+	0.207161i	−0.919611	−	0.392831i	$-0.871496\pi$
$367$	−2.29129	+	3.96863i	−0.119604	+	0.207161i	0.800007	+	0.599991i	$0.204829\pi$
$368$	12.7913	−	16.8637i	0.666792	−	0.879079i
$369$		0			0
$370$		0			0
$371$	16.0390	−	9.26013i	0.832704	−	0.480762i
$372$		0			0
$373$	−17.0000	−	29.4449i	−0.880227	−	1.52460i	−0.851089	−	0.525022i	$-0.824057\pi$
$373$	−17.0000	−	29.4449i	−0.880227	−	1.52460i	−0.0291379	−	0.999575i	$-0.509276\pi$
$374$	25.5826	+	4.18710i	1.32284	+	0.216510i
$375$		0			0
$376$	−0.956439	−	25.9053i	−0.0493246	−	1.33596i
$377$		−	17.3205i		−	0.892052i
$378$		0			0
$379$			5.29150i			0.271806i	0.990722	+	0.135903i	$0.0433936\pi$
$379$			5.29150i			0.271806i	−0.990722	+	0.135903i	$0.956606\pi$
$380$		0			0
$381$		0			0
$382$	−2.41742	+	14.7701i	−0.123686	+	0.755704i
$383$	−13.7477	−	23.8118i	−0.702476	−	1.21672i	−0.967595	−	0.252508i	$-0.918744\pi$
$383$	−13.7477	−	23.8118i	−0.702476	−	1.21672i	0.265119	−	0.964216i	$-0.414589\pi$
$384$		0			0
$385$	−10.5000	−	6.06218i	−0.535130	−	0.308957i
$386$	1.50000	+	3.96863i	0.0763480	+	0.201998i
$387$		0			0
$388$	−16.9782	+	3.42638i	−0.861939	+	0.173948i
$389$	−5.00000	+	8.66025i	−0.253510	+	0.439092i	−0.964490	−	0.264120i	$-0.914918\pi$
$389$	−5.00000	+	8.66025i	−0.253510	+	0.439092i	0.710980	+	0.703213i	$0.248252\pi$
$390$		0			0
$391$	−36.6606			−1.85401
$392$	−0.730493	−	19.7855i	−0.0368954	−	0.999319i
$393$		0			0
$394$	12.5390	−	15.3223i	0.631706	−	0.771927i
$395$	6.87386	−	11.9059i	0.345862	−	0.599050i
$396$		0			0
$397$	12.0000	−	6.92820i	0.602263	−	0.347717i	−0.167668	−	0.985843i	$-0.553624\pi$
$397$	12.0000	−	6.92820i	0.602263	−	0.347717i	0.769931	+	0.638127i	$0.220290\pi$
$398$	−9.16515	−	24.2487i	−0.459408	−	1.21548i
$399$		0			0
$400$	1.00000	−	7.93725i	0.0500000	−	0.396863i
$401$	−4.00000	−	6.92820i	−0.199750	−	0.345978i	0.748697	−	0.662912i	$-0.230680\pi$
$401$	−4.00000	−	6.92820i	−0.199750	−	0.345978i	−0.948447	+	0.316934i	$0.897346\pi$
$402$		0			0
$403$	−13.7477	−	7.93725i	−0.684823	−	0.395383i
$404$	−4.41742	+	13.1334i	−0.219775	+	0.653411i
$405$		0			0
$406$	−17.5000	+	6.61438i	−0.868510	+	0.328266i
$407$		0			0
$408$		0			0
$409$	4.50000	+	2.59808i	0.222511	+	0.128467i	0.607112	−	0.794616i	$-0.292328\pi$
$409$	4.50000	+	2.59808i	0.222511	+	0.128467i	−0.384602	+	0.923083i	$0.625661\pi$
$410$	−8.37386	−	1.37055i	−0.413556	−	0.0676867i
$411$		0			0
$412$	−27.4955	+	24.2487i	−1.35460	+	1.19465i
$413$	−31.5000	+	18.1865i	−1.55001	+	0.894901i
$414$		0			0
$415$	6.87386	−	3.96863i	0.337425	−	0.194812i
$416$	18.7913	+	5.55765i	0.921319	+	0.272486i
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−2.00000			−0.0974740			−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000			−0.0974740			−0.0487370	+	0.998812i	$0.515520\pi$
$422$	5.79129	+	4.73930i	0.281915	+	0.230706i
$423$		0			0
$424$	16.7695	−	10.5254i	0.814399	−	0.511158i
$425$	−12.0000	+	6.92820i	−0.582086	+	0.336067i
$426$		0			0
$427$	27.4955			1.33060
$428$	24.5000	+	27.7804i	1.18425	+	1.34282i
$429$		0			0
$430$	25.5826	+	4.18710i	1.23370	+	0.201920i
$431$	22.9129	+	13.2288i	1.10367	+	0.637207i	0.937184	−	0.348836i	$-0.113423\pi$
$431$	22.9129	+	13.2288i	1.10367	+	0.637207i	0.166491	+	0.986043i	$0.446756\pi$
$432$		0			0
$433$		−	20.7846i		−	0.998845i	−0.866359	−	0.499422i	$-0.833546\pi$
$433$		−	20.7846i		−	0.998845i	0.866359	−	0.499422i	$-0.166454\pi$
$434$	−2.76951	+	16.9213i	−0.132941	+	0.812248i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−2.29129	−	3.96863i	−0.109357	−	0.189412i	0.806153	−	0.591707i	$-0.201546\pi$
$439$	−2.29129	−	3.96863i	−0.109357	−	0.189412i	−0.915510	+	0.402295i	$0.868213\pi$
$440$	−11.4564	−	6.06218i	−0.546164	−	0.289003i
$441$		0			0
$442$	−12.0000	−	31.7490i	−0.570782	−	1.51015i
$443$	16.0390	−	9.26013i	0.762037	−	0.439962i	−0.0679899	−	0.997686i	$-0.521659\pi$
$443$	16.0390	−	9.26013i	0.762037	−	0.439962i	0.830026	+	0.557724i	$0.188325\pi$
$444$		0			0
$445$	−9.00000	+	15.5885i	−0.426641	+	0.738964i
$446$	4.10436	−	5.01540i	0.194347	−	0.237486i
$447$		0			0
$448$	−1.56080	−	21.1084i	−0.0737406	−	0.997277i
$449$	−14.0000			−0.660701			−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000			−0.660701			−0.330350	+	0.943858i	$0.607167\pi$
$450$		0			0
$451$	4.58258	−	7.93725i	0.215785	−	0.373751i
$452$	5.53901	+	27.4467i	0.260533	+	1.29098i
$453$		0			0
$454$	2.29129	+	6.06218i	0.107535	+	0.284512i
$455$			15.8745i			0.744208i
$456$		0			0
$457$	13.5000	+	23.3827i	0.631503	+	1.09380i	0.987245	+	0.159211i	$0.0508951\pi$
$457$	13.5000	+	23.3827i	0.631503	+	1.09380i	−0.355741	+	0.934585i	$0.615772\pi$
$458$	−2.37386	+	14.5040i	−0.110923	+	0.677725i
$459$		0			0
$460$	17.3739	+	5.84370i	0.810061	+	0.272464i
$461$		−	27.7128i		−	1.29071i	−0.763881	−	0.645357i	$-0.776709\pi$
$461$		−	27.7128i		−	1.29071i	0.763881	−	0.645357i	$-0.223291\pi$
$462$		0			0
$463$			15.8745i			0.737751i	0.929479	+	0.368875i	$0.120257\pi$
$463$			15.8745i			0.737751i	−0.929479	+	0.368875i	$0.879743\pi$
$464$	−18.4347	+	7.75650i	−0.855808	+	0.360087i
$465$		0			0
$466$	27.9129	+	4.56850i	1.29304	+	0.211632i
$467$	−18.3303	−	31.7490i	−0.848225	−	1.46917i	−0.882791	−	0.469767i	$-0.844338\pi$
$467$	−18.3303	−	31.7490i	−0.848225	−	1.46917i	0.0345653	−	0.999402i	$-0.488995\pi$
$468$		0			0
$469$		0			0
$470$	21.0000	−	7.93725i	0.968658	−	0.366118i
$471$		0			0
$472$	−32.9347	+	20.6714i	−1.51594	+	0.951480i
$473$	−14.0000	+	24.2487i	−0.643721	+	1.11496i
$474$		0			0
$475$		0			0
$476$	−27.4955	+	24.2487i	−1.26025	+	1.11144i
$477$		0			0
$478$	5.79129	+	4.73930i	0.264887	+	0.216771i
$479$	−9.16515	+	15.8745i	−0.418766	+	0.725325i	−0.995816	−	0.0913846i	$-0.970871\pi$
$479$	−9.16515	+	15.8745i	−0.418766	+	0.725325i	0.577049	+	0.816709i	$0.304204\pi$
$480$		0			0
$481$		0			0
$482$	−6.87386	+	2.59808i	−0.313096	+	0.118339i
$483$		0			0
$484$	−6.00000	+	5.29150i	−0.272727	+	0.240523i
$485$	−7.50000	−	12.9904i	−0.340557	−	0.589863i
$486$		0			0
$487$	−2.29129	−	1.32288i	−0.103828	−	0.0599452i	0.447187	−	0.894441i	$-0.352426\pi$
$487$	−2.29129	−	1.32288i	−0.103828	−	0.0599452i	−0.551015	+	0.834495i	$0.685759\pi$
$488$	29.3739	−	1.08450i	1.32969	−	0.0490930i
$489$		0			0
$490$	16.0390	−	6.06218i	0.724569	−	0.273861i
$491$		−	2.64575i		−	0.119401i	−0.998216	−	0.0597005i	$-0.980985\pi$
$491$		−	2.64575i		−	0.119401i	0.998216	−	0.0597005i	$-0.0190146\pi$
$492$		0			0
$493$	30.0000	+	17.3205i	1.35113	+	0.780076i
$494$		0			0
$495$		0			0
$496$	−2.29129	+	18.1865i	−0.102882	+	0.816599i
$497$	−7.00000	+	12.1244i	−0.313993	+	0.543852i
$498$		0			0
$499$	4.58258	−	2.64575i	0.205144	−	0.118440i	−0.393908	−	0.919150i	$-0.628877\pi$
$499$	4.58258	−	2.64575i	0.205144	−	0.118440i	0.599053	+	0.800710i	$0.295544\pi$
$500$	23.7695	−	4.79693i	1.06300	−	0.214525i
$501$		0			0
$502$	−4.10436	+	5.01540i	−0.183186	+	0.223848i
$503$	−36.6606			−1.63462			−0.817308	−	0.576201i	$-0.804534\pi$
$503$	−36.6606			−1.63462			−0.817308	+	0.576201i	$0.804534\pi$
$504$		0			0
$505$	−12.0000			−0.533993
$506$	−12.5390	+	15.3223i	−0.557427	+	0.681160i
$507$		0			0
$508$	−15.5608	+	3.14033i	−0.690399	+	0.139329i
$509$	−1.50000	+	0.866025i	−0.0664863	+	0.0383859i	−0.532875	−	0.846194i	$-0.678888\pi$
$509$	−1.50000	+	0.866025i	−0.0664863	+	0.0383859i	0.466388	+	0.884580i	$0.345555\pi$
$510$		0			0
$511$	9.16515	+	15.8745i	0.405442	+	0.702247i
$512$	−2.50000	−	22.4889i	−0.110485	−	0.993878i
$513$		0			0
$514$	2.37386	−	14.5040i	0.104707	−	0.639742i
$515$	−27.4955	−	15.8745i	−1.21159	−	0.699514i
$516$		0			0
$517$			24.2487i			1.06646i
$518$		0			0
$519$		0			0
$520$	0.626136	+	16.9590i	0.0274579	+	0.743702i
$521$	−6.00000	−	3.46410i	−0.262865	−	0.151765i	0.362776	−	0.931876i	$-0.381829\pi$
$521$	−6.00000	−	3.46410i	−0.262865	−	0.151765i	−0.625641	+	0.780111i	$0.715162\pi$
$522$		0			0
$523$	13.7477	+	23.8118i	0.601146	+	1.04122i	0.992648	+	0.121038i	$0.0386224\pi$
$523$	13.7477	+	23.8118i	0.601146	+	1.04122i	−0.391502	+	0.920177i	$0.628044\pi$
$524$	20.6216	−	18.1865i	0.900858	−	0.794482i
$525$		0			0
$526$	−14.0000	+	5.29150i	−0.610429	+	0.230720i
$527$	27.4955	−	15.8745i	1.19772	−	0.691504i
$528$		0			0
$529$	2.50000	−	4.33013i	0.108696	−	0.188266i
$530$	13.2695	+	10.8591i	0.576391	+	0.471689i
$531$		0			0
$532$		0			0
$533$	−12.0000			−0.519778
$534$		0			0
$535$	−16.0390	+	27.7804i	−0.693427	+	1.20105i
$536$		0			0
$537$		0			0
$538$	25.2042	−	9.52628i	1.08663	−	0.410707i
$539$			18.5203i			0.797724i
$540$		0			0
$541$	−4.00000	−	6.92820i	−0.171973	−	0.297867i	0.767136	−	0.641484i	$-0.221681\pi$
$541$	−4.00000	−	6.92820i	−0.171973	−	0.297867i	−0.939110	+	0.343617i	$0.888348\pi$
$542$	19.1869	+	3.14033i	0.824149	+	0.134889i
$543$		0			0
$544$	−28.4174	+	26.9898i	−1.21839	+	1.15718i
$545$		0			0
$546$		0			0
$547$		−	31.7490i		−	1.35749i	−0.734374	−	0.678745i	$-0.762524\pi$
$547$		−	31.7490i		−	1.35749i	0.734374	−	0.678745i	$-0.237476\pi$
$548$	−7.58258	−	2.55040i	−0.323912	−	0.108948i
$549$		0			0
$550$	−1.20871	+	7.38505i	−0.0515397	+	0.314900i
$551$		0			0
$552$		0			0
$553$	−21.0000			−0.893011
$554$	−2.00000	−	5.29150i	−0.0849719	−	0.224814i
$555$		0			0
$556$	−3.62614	−	17.9681i	−0.153782	−	0.762015i
$557$	3.50000	−	6.06218i	0.148300	−	0.256863i	−0.782299	−	0.622903i	$-0.785953\pi$
$557$	3.50000	−	6.06218i	0.148300	−	0.256863i	0.930599	+	0.366040i	$0.119287\pi$
$558$		0			0
$559$	36.6606			1.55058
$560$	16.8956	−	7.10895i	0.713971	−	0.300408i
$561$		0			0
$562$	14.3303	−	17.5112i	0.604487	−	0.738666i
$563$	−11.4564	+	19.8431i	−0.482831	+	0.836288i	−0.999806	−	0.0197125i	$-0.993725\pi$
$563$	−11.4564	+	19.8431i	−0.482831	+	0.836288i	0.516974	+	0.856001i	$0.327058\pi$
$564$		0			0
$565$	−21.0000	+	12.1244i	−0.883477	+	0.510075i
$566$	9.16515	+	24.2487i	0.385240	+	1.01925i
$567$		0			0
$568$	−7.00000	+	13.2288i	−0.293713	+	0.555066i
$569$	4.00000	+	6.92820i	0.167689	+	0.290445i	0.937607	−	0.347697i	$-0.113036\pi$
$569$	4.00000	+	6.92820i	0.167689	+	0.290445i	−0.769918	+	0.638143i	$0.779703\pi$
$570$		0			0
$571$	13.7477	+	7.93725i	0.575324	+	0.332164i	0.759273	−	0.650772i	$-0.225555\pi$
$571$	13.7477	+	7.93725i	0.575324	+	0.332164i	−0.183949	+	0.982936i	$0.558888\pi$
$572$	−17.3739	−	5.84370i	−0.726438	−	0.244338i
$573$		0			0
$574$	4.58258	+	12.1244i	0.191273	+	0.506061i
$575$		−	10.5830i		−	0.441342i
$576$		0			0
$577$	−37.5000	−	21.6506i	−1.56115	−	0.901328i	−0.997142	−	0.0755556i	$-0.975927\pi$
$577$	−37.5000	−	21.6506i	−1.56115	−	0.901328i	−0.564004	−	0.825772i	$-0.690740\pi$
$578$	43.2650	+	7.08118i	1.79959	+	0.294538i
$579$		0			0
$580$	−11.4564	−	12.9904i	−0.475703	−	0.539396i
$581$	−10.5000	−	6.06218i	−0.435613	−	0.251502i
$582$		0			0
$583$	−16.0390	+	9.26013i	−0.664268	+	0.383515i
$584$	10.4174	+	16.5975i	0.431076	+	0.686810i
$585$		0			0
$586$	17.0608	+	13.9617i	0.704775	+	0.576753i
$587$	22.9129			0.945716			0.472858	−	0.881139i	$-0.343222\pi$
$587$	22.9129			0.945716			0.472858	+	0.881139i	$0.343222\pi$
$588$		0			0
$589$		0			0
$590$	−26.0608	−	21.3269i	−1.07291	−	0.878013i
$591$		0			0
$592$		0			0
$593$	9.00000	−	5.19615i	0.369586	−	0.213380i	−0.303692	−	0.952770i	$-0.598219\pi$
$593$	9.00000	−	5.19615i	0.369586	−	0.213380i	0.673277	+	0.739390i	$0.264886\pi$
$594$		0			0
$595$	−27.4955	−	15.8745i	−1.12720	−	0.650791i
$596$	−21.0000	+	18.5203i	−0.860194	+	0.758619i
$597$		0			0
$598$	25.5826	+	4.18710i	1.04615	+	0.171223i
$599$	9.16515	+	5.29150i	0.374478	+	0.216205i	0.675413	−	0.737440i	$-0.263965\pi$
$599$	9.16515	+	5.29150i	0.374478	+	0.216205i	−0.300935	+	0.953645i	$0.597299\pi$
$600$		0			0
$601$			8.66025i			0.353259i	0.984277	+	0.176630i	$0.0565195\pi$
$601$			8.66025i			0.353259i	−0.984277	+	0.176630i	$0.943481\pi$
$602$	−14.0000	−	37.0405i	−0.570597	−	1.50966i
$603$		0			0
$604$	−8.43466	+	25.0770i	−0.343201	+	1.02037i
$605$	−6.00000	−	3.46410i	−0.243935	−	0.140836i
$606$		0			0
$607$	6.87386	+	11.9059i	0.279002	+	0.483245i	0.971137	−	0.238523i	$-0.0766632\pi$
$607$	6.87386	+	11.9059i	0.279002	+	0.483245i	−0.692135	+	0.721768i	$0.743330\pi$
$608$		0			0
$609$		0			0
$610$	9.00000	+	23.8118i	0.364399	+	0.964110i
$611$	27.4955	−	15.8745i	1.11235	−	0.642214i
$612$		0			0
$613$	−7.00000	+	12.1244i	−0.282727	+	0.489698i	−0.972056	−	0.234751i	$-0.924572\pi$
$613$	−7.00000	+	12.1244i	−0.282727	+	0.489698i	0.689328	+	0.724449i	$0.257906\pi$
$614$	−24.6261	+	30.0924i	−0.993830	+	1.21443i
$615$		0			0
$616$	0.730493	+	19.7855i	0.0294324	+	0.797181i
$617$	−2.00000			−0.0805170			−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000			−0.0805170			−0.0402585	+	0.999189i	$0.512818\pi$
$618$		0			0
$619$	13.7477	−	23.8118i	0.552568	−	0.957076i	−0.445521	−	0.895272i	$-0.646981\pi$
$619$	13.7477	−	23.8118i	0.552568	−	0.957076i	0.998088	−	0.0618038i	$-0.0196853\pi$
$620$	−15.5608	+	3.14033i	−0.624937	+	0.126119i
$621$		0			0
$622$	−4.58258	−	12.1244i	−0.183745	−	0.486142i
$623$	27.4955			1.10158
$624$		0			0
$625$	5.50000	+	9.52628i	0.220000	+	0.381051i
$626$	5.14337	−	31.4252i	0.205570	−	1.25601i
$627$		0			0
$628$	4.41742	−	13.1334i	0.176274	−	0.524080i
$629$		0			0
$630$		0			0
$631$		−	7.93725i		−	0.315977i	−0.987441	−	0.157989i	$-0.949499\pi$
$631$		−	7.93725i		−	0.315977i	0.987441	−	0.157989i	$-0.0505009\pi$
$632$	−22.4347	+	0.828301i	−0.892403	+	0.0329480i
$633$		0			0
$634$	−9.76951	−	1.59898i	−0.387997	−	0.0635034i
$635$	−6.87386	−	11.9059i	−0.272781	−	0.472470i
$636$		0			0
$637$	21.0000	−	12.1244i	0.832050	−	0.480384i
$638$	17.5000	−	6.61438i	0.692832	−	0.261866i
$639$		0			0
$640$	17.7695	−	8.26103i	0.702401	−	0.326546i
$641$	19.0000	−	32.9090i	0.750455	−	1.29983i	−0.197148	−	0.980374i	$-0.563168\pi$
$641$	19.0000	−	32.9090i	0.750455	−	1.29983i	0.947602	−	0.319452i	$-0.103499\pi$
$642$		0			0
$643$	18.3303			0.722877			0.361438	−	0.932396i	$-0.382286\pi$
$643$	18.3303			0.722877			0.361438	+	0.932396i	$0.382286\pi$
$644$	−5.53901	−	27.4467i	−0.218268	−	1.08155i
$645$		0			0
$646$		0			0
$647$	4.58258	−	7.93725i	0.180160	−	0.312046i	−0.761775	−	0.647842i	$-0.775672\pi$
$647$	4.58258	−	7.93725i	0.180160	−	0.312046i	0.941935	+	0.335796i	$0.109005\pi$
$648$		0			0
$649$	31.5000	−	18.1865i	1.23648	−	0.713884i
$650$	9.16515	−	3.46410i	0.359487	−	0.135873i
$651$		0			0
$652$	21.0000	+	23.8118i	0.822423	+	0.932541i
$653$	21.5000	+	37.2391i	0.841360	+	1.45728i	0.888745	+	0.458402i	$0.151578\pi$
$653$	21.5000	+	37.2391i	0.841360	+	1.45728i	−0.0473852	+	0.998877i	$0.515089\pi$
$654$		0			0
$655$	20.6216	+	11.9059i	0.805752	+	0.465201i
$656$	5.37386	+	12.7719i	0.209814	+	0.498659i
$657$		0			0
$658$	−26.5390	−	21.7182i	−1.03460	−	0.846664i
$659$			26.4575i			1.03064i	0.856998	+	0.515319i	$0.172327\pi$
$659$			26.4575i			1.03064i	−0.856998	+	0.515319i	$0.827673\pi$
$660$		0			0
$661$	6.00000	+	3.46410i	0.233373	+	0.134738i	0.612127	−	0.790759i	$-0.290314\pi$
$661$	6.00000	+	3.46410i	0.233373	+	0.134738i	−0.378754	+	0.925497i	$0.623647\pi$
$662$	2.41742	−	14.7701i	0.0939559	−	0.574057i
$663$		0			0
$664$	−11.4564	−	6.06218i	−0.444596	−	0.235258i
$665$		0			0
$666$		0			0
$667$	−22.9129	+	13.2288i	−0.887190	+	0.512219i
$668$	−7.25227	−	35.9361i	−0.280599	−	1.39041i
$669$		0			0
$670$		0			0
$671$	−27.4955			−1.06145
$672$		0			0
$673$	21.0000			0.809491			0.404745	−	0.914429i	$-0.367360\pi$
$673$	21.0000			0.809491			0.404745	+	0.914429i	$0.367360\pi$
$674$	−18.8085	+	22.9835i	−0.724477	+	0.885290i
$675$		0			0
$676$	−0.395644	−	1.96048i	−0.0152171	−	0.0754029i
$677$	19.5000	−	11.2583i	0.749446	−	0.432693i	−0.0760478	−	0.997104i	$-0.524230\pi$
$677$	19.5000	−	11.2583i	0.749446	−	0.432693i	0.825494	+	0.564411i	$0.190897\pi$
$678$		0			0
$679$	−11.4564	+	19.8431i	−0.439658	+	0.761510i
$680$	−30.0000	−	15.8745i	−1.15045	−	0.608760i
$681$		0			0
$682$	2.76951	−	16.9213i	0.106050	−	0.647949i
$683$	2.29129	+	1.32288i	0.0876737	+	0.0506184i	0.543196	−	0.839606i	$-0.317214\pi$
$683$	2.29129	+	1.32288i	0.0876737	+	0.0506184i	−0.455522	+	0.890224i	$0.650547\pi$
$684$		0			0
$685$		−	6.92820i		−	0.264713i
$686$	−20.2695	−	16.5876i	−0.773893	−	0.633316i
$687$		0			0
$688$	−16.4174	−	39.0188i	−0.625908	−	1.48758i
$689$	21.0000	+	12.1244i	0.800036	+	0.461901i
$690$		0			0
$691$	−22.9129	−	39.6863i	−0.871647	−	1.50974i	−0.860292	−	0.509801i	$-0.829719\pi$
$691$	−22.9129	−	39.6863i	−0.871647	−	1.50974i	−0.0113548	−	0.999936i	$-0.503614\pi$
$692$	18.3303	+	20.7846i	0.696814	+	0.790112i
$693$		0			0
$694$	7.00000	−	2.64575i	0.265716	−	0.100431i
$695$	13.7477	−	7.93725i	0.521481	−	0.301077i
$696$		0			0
$697$	12.0000	−	20.7846i	0.454532	−	0.787273i
$698$	22.7477	+	18.6156i	0.861014	+	0.704611i
$699$		0			0
$700$	−7.00000	−	7.93725i	−0.264575	−	0.300000i
$701$	23.0000			0.868698			0.434349	−	0.900745i	$-0.356978\pi$
$701$	23.0000			0.868698			0.434349	+	0.900745i	$0.356978\pi$
$702$		0			0
$703$		0			0
$704$	1.56080	+	21.1084i	0.0588247	+	0.795552i
$705$		0			0
$706$	−9.16515	+	3.46410i	−0.344935	+	0.130373i
$707$	9.16515	+	15.8745i	0.344691	+	0.597022i
$708$		0			0
$709$	−21.0000	−	36.3731i	−0.788672	−	1.36602i	−0.926781	−	0.375602i	$-0.877436\pi$
$709$	−21.0000	−	36.3731i	−0.788672	−	1.36602i	0.138109	−	0.990417i	$-0.455897\pi$
$710$	−12.7913	−	2.09355i	−0.480048	−	0.0785696i
$711$		0			0
$712$	29.3739	−	1.08450i	1.10083	−	0.0406434i
$713$			24.2487i			0.908121i
$714$		0			0
$715$		−	15.8745i		−	0.593673i
$716$	3.37386	−	10.0308i	0.126087	−	0.374869i
$717$		0			0
$718$	−2.41742	+	14.7701i	−0.0902175	+	0.551215i
$719$	−13.7477	−	23.8118i	−0.512704	−	0.888029i	−0.999891	−	0.0147316i	$-0.995311\pi$
$719$	−13.7477	−	23.8118i	−0.512704	−	0.888029i	0.487188	−	0.873297i	$-0.338023\pi$
$720$		0			0
$721$			48.4974i			1.80614i
$722$	−9.50000	−	25.1346i	−0.353553	−	0.935414i
$723$		0			0
$724$	6.79129	−	1.37055i	0.252396	−	0.0509361i
$725$	−5.00000	+	8.66025i	−0.185695	+	0.321634i
$726$		0			0
$727$	−13.7477			−0.509875			−0.254937	−	0.966958i	$-0.582055\pi$
$727$	−13.7477			−0.509875			−0.254937	+	0.966958i	$0.582055\pi$
$728$	21.9564	−	13.7810i	0.813760	−	0.510756i
$729$		0			0
$730$	−10.7477	+	13.1334i	−0.397791	+	0.486089i
$731$	−36.6606	+	63.4980i	−1.35594	+	2.34856i
$732$		0			0
$733$	−30.0000	+	17.3205i	−1.10808	+	0.639748i	−0.938330	−	0.345740i	$-0.887628\pi$
$733$	−30.0000	+	17.3205i	−1.10808	+	0.639748i	−0.169745	+	0.985488i	$0.554294\pi$
$734$	2.29129	+	6.06218i	0.0845730	+	0.223759i
$735$		0			0
$736$	−7.00000	−	29.1033i	−0.258023	−	1.07276i
$737$		0			0
$738$		0			0
$739$	−41.2432	−	23.8118i	−1.51715	−	0.875930i	−0.999797	−	0.0201609i	$-0.993582\pi$
$739$	−41.2432	−	23.8118i	−1.51715	−	0.875930i	−0.517358	−	0.855769i	$-0.673085\pi$
$740$		0			0
$741$		0			0
$742$	4.23049	−	25.8477i	0.155306	−	0.948898i
$743$			21.1660i			0.776506i	0.921553	+	0.388253i	$0.126921\pi$
$743$			21.1660i			0.776506i	−0.921553	+	0.388253i	$0.873079\pi$
$744$		0			0
$745$	−21.0000	−	12.1244i	−0.769380	−	0.444202i
$746$	−47.4519	−	7.76645i	−1.73734	−	0.284350i
$747$		0			0
$748$	27.4955	−	24.2487i	1.00533	−	0.886621i
$749$	49.0000			1.79042
$750$		0			0
$751$	−11.4564	+	6.61438i	−0.418051	+	0.241362i	−0.694243	−	0.719740i	$-0.744261\pi$
$751$	−11.4564	+	6.61438i	−0.418051	+	0.241362i	0.276192	+	0.961103i	$0.410927\pi$
$752$	−29.2087	−	22.1552i	−1.06513	−	0.807916i
$753$		0			0
$754$	−18.9564	−	15.5130i	−0.690353	−	0.564950i
$755$	−22.9129			−0.833885
$756$		0			0
$757$	42.0000			1.52652			0.763258	−	0.646094i	$-0.223599\pi$
$757$	42.0000			1.52652			0.763258	+	0.646094i	$0.223599\pi$
$758$	5.79129	+	4.73930i	0.210349	+	0.172139i
$759$		0			0
$760$		0			0
$761$	33.0000	−	19.0526i	1.19625	−	0.690655i	0.236532	−	0.971624i	$-0.423989\pi$
$761$	33.0000	−	19.0526i	1.19625	−	0.690655i	0.959717	+	0.280969i	$0.0906558\pi$
$762$		0			0
$763$		0			0
$764$	14.0000	+	15.8745i	0.506502	+	0.574320i
$765$		0			0
$766$	−38.3739	−	6.28065i	−1.38650	−	0.226929i
$767$	−41.2432	−	23.8118i	−1.48920	−	0.859793i
$768$		0			0
$769$		−	32.9090i		−	1.18673i	−0.804934	−	0.593364i	$-0.797800\pi$
$769$		−	32.9090i		−	1.18673i	0.804934	−	0.593364i	$-0.202200\pi$
$770$	−16.0390	+	6.06218i	−0.578006	+	0.218466i
$771$		0			0
$772$	5.68693	+	1.91280i	0.204677	+	0.0688432i
$773$	−36.0000	−	20.7846i	−1.29483	−	0.747570i	−0.315324	−	0.948984i	$-0.602113\pi$
$773$	−36.0000	−	20.7846i	−1.29483	−	0.747570i	−0.979506	+	0.201414i	$0.935446\pi$
$774$		0			0
$775$	4.58258	+	7.93725i	0.164611	+	0.285115i
$776$	−11.4564	+	21.6506i	−0.411262	+	0.777213i
$777$		0			0
$778$	5.00000	+	13.2288i	0.179259	+	0.474274i
$779$		0			0
$780$		0			0
$781$	7.00000	−	12.1244i	0.250480	−	0.433844i
$782$	−32.8348	+	40.1232i	−1.17417	+	1.43480i
$783$		0			0
$784$	−22.3085	−	16.9213i	−0.796733	−	0.604332i
$785$	12.0000			0.428298
$786$		0			0
$787$	13.7477	−	23.8118i	0.490054	−	0.848798i	−0.509881	−	0.860245i	$-0.670311\pi$
$787$	13.7477	−	23.8118i	0.490054	−	0.848798i	0.999934	+	0.0114473i	$0.00364388\pi$
$788$	−5.53901	−	27.4467i	−0.197319	−	0.977747i
$789$		0			0
$790$	−6.87386	−	18.1865i	−0.244561	−	0.647048i
$791$	32.0780	+	18.5203i	1.14056	+	0.658505i
$792$		0			0
$793$	18.0000	+	31.1769i	0.639199	+	1.10712i
$794$	3.16515	−	19.3386i	0.112327	−	0.686302i
$795$		0			0
$796$	−34.7477	−	11.6874i	−1.23160	−	0.414249i
$797$			15.5885i			0.552171i	0.961133	+	0.276086i	$0.0890374\pi$
$797$			15.5885i			0.552171i	−0.961133	+	0.276086i	$0.910963\pi$
$798$		0			0
$799$			63.4980i			2.24640i
$800$	−7.79129	−	8.20340i	−0.275464	−	0.290034i
$801$		0			0
$802$	−11.1652	−	1.82740i	−0.394255	−	0.0645278i
$803$	−9.16515	−	15.8745i	−0.323431	−	0.560199i
$804$		0			0
$805$	21.0000	−	12.1244i	0.740153	−	0.427327i
$806$	−21.0000	+	7.93725i	−0.739693	+	0.279578i
$807$		0			0
$808$	10.4174	+	16.5975i	0.366484	+	0.583898i
$809$	14.0000	−	24.2487i	0.492214	−	0.852539i	−0.507746	−	0.861507i	$-0.669521\pi$
$809$	14.0000	−	24.2487i	0.492214	−	0.852539i	0.999960	+	0.00896753i	$0.00285449\pi$
$810$		0			0
$811$	36.6606			1.28733			0.643664	−	0.765308i	$-0.277413\pi$
$811$	36.6606			1.28733			0.643664	+	0.765308i	$0.277413\pi$
$812$	−8.43466	+	25.0770i	−0.295998	+	0.880031i
$813$		0			0
$814$		0			0
$815$	−13.7477	+	23.8118i	−0.481562	+	0.834090i
$816$		0			0
$817$		0			0
$818$	6.87386	−	2.59808i	0.240339	−	0.0908396i
$819$		0			0
$820$	−9.00000	+	7.93725i	−0.314294	+	0.277181i
$821$	24.5000	+	42.4352i	0.855056	+	1.48100i	0.876593	+	0.481232i	$0.159811\pi$
$821$	24.5000	+	42.4352i	0.855056	+	1.48100i	−0.0215373	+	0.999768i	$0.506856\pi$
$822$		0			0
$823$	−41.2432	−	23.8118i	−1.43765	−	0.830026i	−0.439961	−	0.898017i	$-0.645008\pi$
$823$	−41.2432	−	23.8118i	−1.43765	−	0.830026i	−0.997686	+	0.0679910i	$0.978341\pi$
$824$	1.91288	+	51.8106i	0.0666383	+	1.80491i
$825$		0			0
$826$	−8.30852	+	50.7638i	−0.289090	+	1.76630i
$827$			34.3948i			1.19602i	0.801487	+	0.598012i	$0.204042\pi$
$827$			34.3948i			1.19602i	−0.801487	+	0.598012i	$0.795958\pi$
$828$		0			0
$829$	6.00000	+	3.46410i	0.208389	+	0.120313i	0.600562	−	0.799578i	$-0.294943\pi$
$829$	6.00000	+	3.46410i	0.208389	+	0.120313i	−0.392174	+	0.919891i	$0.628277\pi$
$830$	1.81307	−	11.0776i	0.0629325	−	0.384508i
$831$		0			0
$832$	22.9129	−	15.5885i	0.794361	−	0.540433i
$833$			48.4974i			1.68034i
$834$		0			0
$835$	27.4955	−	15.8745i	0.951519	−	0.549360i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	36.6606			1.26566			0.632832	−	0.774289i	$-0.281892\pi$
$839$	36.6606			1.26566			0.632832	+	0.774289i	$0.281892\pi$
$840$		0			0
$841$	−4.00000			−0.137931
$842$	−1.79129	+	2.18890i	−0.0617319	+	0.0754345i
$843$		0			0
$844$	10.3739	−	2.09355i	0.357083	−	0.0720630i
$845$	1.50000	−	0.866025i	0.0516016	−	0.0297922i
$846$		0			0
$847$			10.5830i			0.363636i
$848$	3.50000	−	27.7804i	0.120190	−	0.953982i
$849$		0			0
$850$	−3.16515	+	19.3386i	−0.108564	+	0.663309i
$851$		0			0
$852$		0			0
$853$		−	27.7128i		−	0.948869i	−0.880291	−	0.474434i	$-0.842653\pi$
$853$		−	27.7128i		−	0.948869i	0.880291	−	0.474434i	$-0.157347\pi$
$854$	24.6261	−	30.0924i	0.842689	−	1.02974i
$855$		0			0
$856$	52.3475	−	1.93270i	1.78920	−	0.0660584i
$857$	−27.0000	−	15.5885i	−0.922302	−	0.532492i	−0.0379336	−	0.999280i	$-0.512078\pi$
$857$	−27.0000	−	15.5885i	−0.922302	−	0.532492i	−0.884369	+	0.466789i	$0.845411\pi$
$858$		0			0
$859$	−18.3303	−	31.7490i	−0.625422	−	1.08326i	−0.988459	−	0.151488i	$-0.951594\pi$
$859$	−18.3303	−	31.7490i	−0.625422	−	1.08326i	0.363037	−	0.931775i	$-0.381740\pi$
$860$	27.4955	−	24.2487i	0.937587	−	0.826874i
$861$		0			0
$862$	35.0000	−	13.2288i	1.19210	−	0.450573i
$863$	22.9129	−	13.2288i	0.779963	−	0.450312i	−0.0564539	−	0.998405i	$-0.517979\pi$
$863$	22.9129	−	13.2288i	0.779963	−	0.450312i	0.836417	+	0.548093i	$0.184646\pi$
$864$		0			0
$865$	−12.0000	+	20.7846i	−0.408012	+	0.706698i
$866$	−22.7477	−	18.6156i	−0.772999	−	0.632584i
$867$		0			0
$868$	16.0390	+	18.1865i	0.544400	+	0.617291i
$869$	21.0000			0.712376
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	16.0390	−	27.7804i	0.542218	−	0.939149i
$876$		0			0
$877$	−14.0000	−	24.2487i	−0.472746	−	0.818821i	0.526767	−	0.850010i	$-0.323404\pi$
$877$	−14.0000	−	24.2487i	−0.472746	−	0.818821i	−0.999514	+	0.0311889i	$0.990071\pi$
$878$	−6.39564	−	1.04678i	−0.215843	−	0.0353270i
$879$		0			0
$880$	−16.8956	+	7.10895i	−0.569552	+	0.239643i
$881$		−	27.7128i		−	0.933668i	−0.884345	−	0.466834i	$-0.845394\pi$
$881$		−	27.7128i		−	0.933668i	0.884345	−	0.466834i	$-0.154606\pi$
$882$		0			0
$883$		−	5.29150i		−	0.178073i	−0.996028	−	0.0890366i	$-0.971621\pi$
$883$		−	5.29150i		−	0.178073i	0.996028	−	0.0890366i	$-0.0283788\pi$
$884$	−45.4955	−	15.3024i	−1.53018	−	0.514676i
$885$		0			0
$886$	4.23049	−	25.8477i	0.142126	−	0.868370i
$887$	−13.7477	−	23.8118i	−0.461603	−	0.799521i	0.537438	−	0.843304i	$-0.319392\pi$
$887$	−13.7477	−	23.8118i	−0.461603	−	0.799521i	−0.999041	+	0.0437828i	$0.986059\pi$
$888$		0			0
$889$	−10.5000	+	18.1865i	−0.352159	+	0.609957i
$890$	9.00000	+	23.8118i	0.301681	+	0.798172i
$891$		0			0
$892$	−1.81307	−	8.98403i	−0.0607060	−	0.300808i
$893$		0			0
$894$		0			0
$895$	9.16515			0.306357
$896$	−24.5000	−	17.1974i	−0.818488	−	0.574524i
$897$		0			0
$898$	−12.5390	+	15.3223i	−0.418432	+	0.511312i
$899$	11.4564	−	19.8431i	0.382094	−	0.661806i
$900$		0			0
$901$	−42.0000	+	24.2487i	−1.39922	+	0.807842i
$902$	−4.58258	−	12.1244i	−0.152583	−	0.403697i
$903$		0			0
$904$	35.0000	+	18.5203i	1.16408	+	0.615975i
$905$	3.00000	+	5.19615i	0.0997234	+	0.172726i
$906$		0			0
$907$	32.0780	+	18.5203i	1.06513	+	0.614955i	0.926848	−	0.375437i	$-0.122507\pi$
$907$	32.0780	+	18.5203i	1.06513	+	0.614955i	0.138286	+	0.990392i	$0.455841\pi$
$908$	8.68693	+	2.92185i	0.288286	+	0.0969650i
$909$		0			0
$910$	17.3739	+	14.2179i	0.575938	+	0.471319i
$911$		−	5.29150i		−	0.175315i	−0.996151	−	0.0876577i	$-0.972062\pi$
$911$		−	5.29150i		−	0.175315i	0.996151	−	0.0876577i	$-0.0279382\pi$
$912$		0			0
$913$	10.5000	+	6.06218i	0.347499	+	0.200629i
$914$	37.6824	+	6.16748i	1.24642	+	0.204002i
$915$		0			0
$916$	13.7477	+	15.5885i	0.454238	+	0.515057i
$917$		−	36.3731i		−	1.20114i
$918$		0			0
$919$	−4.58258	+	2.64575i	−0.151165	+	0.0872753i	−0.573675	−	0.819083i	$-0.694483\pi$
$919$	−4.58258	+	2.64575i	−0.151165	+	0.0872753i	0.422510	+	0.906358i	$0.361149\pi$
$920$	21.9564	−	13.7810i	0.723882	−	0.454345i
$921$		0			0
$922$	−30.3303	−	24.8208i	−0.998875	−	0.817430i
$923$	−18.3303			−0.603349
$924$		0			0
$925$		0			0
$926$	17.3739	+	14.2179i	0.570941	+	0.467229i
$927$		0			0
$928$	−8.02178	+	27.1229i	−0.263328	+	0.890352i
$929$	−30.0000	+	17.3205i	−0.984268	+	0.568267i	−0.903556	−	0.428470i	$-0.859053\pi$
$929$	−30.0000	+	17.3205i	−0.984268	+	0.568267i	−0.0807121	+	0.996737i	$0.525719\pi$
$930$		0			0
$931$		0			0
$932$	30.0000	−	26.4575i	0.982683	−	0.866645i
$933$		0			0
$934$	−51.1652	−	8.37420i	−1.67417	−	0.274012i
$935$	27.4955	+	15.8745i	0.899198	+	0.519152i
$936$		0			0
$937$		−	15.5885i		−	0.509253i	−0.967040	−	0.254626i	$-0.918048\pi$
$937$		−	15.5885i		−	0.509253i	0.967040	−	0.254626i	$-0.0819525\pi$
$938$		0			0
$939$		0			0
$940$	10.1216	−	30.0924i	0.330130	−	0.981506i
$941$	−4.50000	−	2.59808i	−0.146696	−	0.0846949i	0.424856	−	0.905261i	$-0.360325\pi$
$941$	−4.50000	−	2.59808i	−0.146696	−	0.0846949i	−0.571551	+	0.820566i	$0.693658\pi$
$942$		0			0
$943$	9.16515	+	15.8745i	0.298458	+	0.516945i
$944$	−6.87386	+	54.5596i	−0.223725	+	1.77576i
$945$		0			0
$946$	14.0000	+	37.0405i	0.455179	+	1.20429i
$947$	32.0780	−	18.5203i	1.04240	−	0.601828i	0.121885	−	0.992544i	$-0.461106\pi$
$947$	32.0780	−	18.5203i	1.04240	−	0.601828i	0.920511	+	0.390717i	$0.127773\pi$
$948$		0			0
$949$	−12.0000	+	20.7846i	−0.389536	+	0.674697i
$950$		0			0
$951$		0			0
$952$	1.91288	+	51.8106i	0.0619967	+	1.67919i
$953$	−28.0000			−0.907009			−0.453504	−	0.891254i	$-0.649826\pi$
$953$	−28.0000			−0.907009			−0.453504	+	0.891254i	$0.649826\pi$
$954$		0			0
$955$	−9.16515	+	15.8745i	−0.296577	+	0.513687i
$956$	10.3739	−	2.09355i	0.335515	−	0.0677103i
$957$		0			0
$958$	9.16515	+	24.2487i	0.296113	+	0.783440i
$959$	−9.16515	+	5.29150i	−0.295958	+	0.170872i
$960$		0			0
$961$	5.00000	+	8.66025i	0.161290	+	0.279363i
$962$		0			0
$963$		0			0
$964$	−3.31307	+	9.85005i	−0.106707	+	0.317249i
$965$			5.19615i			0.167270i
$966$		0			0
$967$			39.6863i			1.27622i	0.769943	+	0.638112i	$0.220284\pi$
$967$			39.6863i			1.27622i	−0.769943	+	0.638112i	$0.779716\pi$
$968$	0.417424	+	11.3060i	0.0134165	+	0.363389i
$969$		0			0
$970$	−20.9347	−	3.42638i	−0.672171	−	0.110014i
$971$	−2.29129	−	3.96863i	−0.0735309	−	0.127359i	0.826916	−	0.562326i	$-0.190093\pi$
$971$	−2.29129	−	3.96863i	−0.0735309	−	0.127359i	−0.900446	+	0.434967i	$0.856760\pi$
$972$		0			0
$973$	−21.0000	−	12.1244i	−0.673229	−	0.388689i
$974$	−3.50000	+	1.32288i	−0.112147	+	0.0423877i
$975$		0			0
$976$	25.1216	−	33.1196i	0.804123	−	1.06013i
$977$	16.0000	−	27.7128i	0.511885	−	0.886611i	−0.488020	−	0.872833i	$-0.662281\pi$
$977$	16.0000	−	27.7128i	0.511885	−	0.886611i	0.999905	−	0.0137788i	$-0.00438608\pi$
$978$		0			0
$979$	−27.4955			−0.878759
$980$	7.73049	−	22.9835i	0.246942	−	0.734180i
$981$		0			0
$982$	−2.89564	−	2.36965i	−0.0924037	−	0.0756186i
$983$	−22.9129	+	39.6863i	−0.730807	+	1.26580i	0.225731	+	0.974190i	$0.427523\pi$
$983$	−22.9129	+	39.6863i	−0.730807	+	1.26580i	−0.956539	+	0.291606i	$0.905810\pi$
$984$		0			0
$985$	21.0000	−	12.1244i	0.669116	−	0.386314i
$986$	45.8258	−	17.3205i	1.45939	−	0.551597i
$987$		0			0
$988$		0			0
$989$	−28.0000	−	48.4974i	−0.890348	−	1.54213i
$990$		0			0
$991$	43.5345	+	25.1346i	1.38292	+	0.798428i	0.992504	−	0.122211i	$-0.0389986\pi$
$991$	43.5345	+	25.1346i	1.38292	+	0.798428i	0.390414	+	0.920639i	$0.372332\pi$
$992$	17.8521	+	18.7964i	0.566804	+	0.596785i
$993$		0			0
$994$	7.00000	+	18.5203i	0.222027	+	0.587427i
$995$		−	31.7490i		−	1.00651i
$996$		0			0
$997$	6.00000	+	3.46410i	0.190022	+	0.109709i	0.591993	−	0.805943i	$-0.298341\pi$
$997$	6.00000	+	3.46410i	0.190022	+	0.109709i	−0.401971	+	0.915652i	$0.631675\pi$
$998$	1.20871	−	7.38505i	0.0382611	−	0.233770i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.2.bf.a.199.2	yes	4
3.2	odd	2		252.2.bf.d.199.1	yes	4
4.3	odd	2	inner	252.2.bf.a.199.1	yes	4
7.3	odd	6		1764.2.b.h.1567.3		4
7.4	even	3		1764.2.b.h.1567.4		4
7.5	odd	6	inner	252.2.bf.a.19.1	✓	4
12.11	even	2		252.2.bf.d.199.2	yes	4
21.5	even	6		252.2.bf.d.19.2	yes	4
21.11	odd	6		1764.2.b.b.1567.1		4
21.17	even	6		1764.2.b.b.1567.2		4
28.3	even	6		1764.2.b.h.1567.1		4
28.11	odd	6		1764.2.b.h.1567.2		4
28.19	even	6	inner	252.2.bf.a.19.2	yes	4
84.11	even	6		1764.2.b.b.1567.3		4
84.47	odd	6		252.2.bf.d.19.1	yes	4
84.59	odd	6		1764.2.b.b.1567.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.bf.a.19.1	✓	4	7.5	odd	6	inner
252.2.bf.a.19.2	yes	4	28.19	even	6	inner
252.2.bf.a.199.1	yes	4	4.3	odd	2	inner
252.2.bf.a.199.2	yes	4	1.1	even	1	trivial
252.2.bf.d.19.1	yes	4	84.47	odd	6
252.2.bf.d.19.2	yes	4	21.5	even	6
252.2.bf.d.199.1	yes	4	3.2	odd	2
252.2.bf.d.199.2	yes	4	12.11	even	2
1764.2.b.b.1567.1		4	21.11	odd	6
1764.2.b.b.1567.2		4	21.17	even	6
1764.2.b.b.1567.3		4	84.11	even	6
1764.2.b.b.1567.4		4	84.59	odd	6
1764.2.b.h.1567.1		4	28.3	even	6
1764.2.b.h.1567.2		4	28.11	odd	6
1764.2.b.h.1567.3		4	7.3	odd	6
1764.2.b.h.1567.4		4	7.4	even	3

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	252.bf (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.895644 - 1.09445i) q^{2} +(-0.395644 - 1.96048i) q^{4} +(1.50000 - 0.866025i) q^{5} +(-2.29129 - 1.32288i) q^{7} +(-2.50000 - 1.32288i) q^{8} +O(q^{10})\)	\(q+(0.895644 - 1.09445i) q^{2} +(-0.395644 - 1.96048i) q^{4} +(1.50000 - 0.866025i) q^{5} +(-2.29129 - 1.32288i) q^{7} +(-2.50000 - 1.32288i) q^{8} +(0.395644 - 2.41733i) q^{10} +(2.29129 + 1.32288i) q^{11} -3.46410i q^{13} +(-3.50000 + 1.32288i) q^{14} +(-3.68693 + 1.55130i) q^{16} +(6.00000 + 3.46410i) q^{17} +(-2.29129 - 2.59808i) q^{20} +(3.50000 - 1.32288i) q^{22} +(-4.58258 + 2.64575i) q^{23} +(-1.00000 + 1.73205i) q^{25} +(-3.79129 - 3.10260i) q^{26} +(-1.68693 + 5.01540i) q^{28} +5.00000 q^{29} +(2.29129 - 3.96863i) q^{31} +(-1.60436 + 5.42458i) q^{32} +(9.16515 - 3.46410i) q^{34} -4.58258 q^{35} +(-4.89564 + 0.180750i) q^{40} -3.46410i q^{41} +10.5830i q^{43} +(1.68693 - 5.01540i) q^{44} +(-1.20871 + 7.38505i) q^{46} +(4.58258 + 7.93725i) q^{47} +(3.50000 + 6.06218i) q^{49} +(1.00000 + 2.64575i) q^{50} +(-6.79129 + 1.37055i) q^{52} +(-3.50000 + 6.06218i) q^{53} +4.58258 q^{55} +(3.97822 + 6.33828i) q^{56} +(4.47822 - 5.47225i) q^{58} +(6.87386 - 11.9059i) q^{59} +(-9.00000 + 5.19615i) q^{61} +(-2.29129 - 6.06218i) q^{62} +(4.50000 + 6.61438i) q^{64} +(-3.00000 - 5.19615i) q^{65} +(4.41742 - 13.1334i) q^{68} +(-4.10436 + 5.01540i) q^{70} -5.29150i q^{71} +(-6.00000 - 3.46410i) q^{73} +(-3.50000 - 6.06218i) q^{77} +(6.87386 - 3.96863i) q^{79} +(-4.18693 + 5.51993i) q^{80} +(-3.79129 - 3.10260i) q^{82} +4.58258 q^{83} +12.0000 q^{85} +(11.5826 + 9.47860i) q^{86} +(-3.97822 - 6.33828i) q^{88} +(-9.00000 + 5.19615i) q^{89} +(-4.58258 + 7.93725i) q^{91} +(7.00000 + 7.93725i) q^{92} +(12.7913 + 2.09355i) q^{94} -8.66025i q^{97} +(9.76951 + 1.59898i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 3 q^{4} + 6 q^{5} - 10 q^{8}+O(q^{10}) \) 4 * q - q^2 + 3 * q^4 + 6 * q^5 - 10 * q^8	\( 4 q - q^{2} + 3 q^{4} + 6 q^{5} - 10 q^{8} - 3 q^{10} - 14 q^{14} - q^{16} + 24 q^{17} + 14 q^{22} - 4 q^{25} - 6 q^{26} + 7 q^{28} + 20 q^{29} - 11 q^{32} - 15 q^{40} - 7 q^{44} - 14 q^{46} + 14 q^{49} + 4 q^{50} - 18 q^{52} - 14 q^{53} - 7 q^{56} - 5 q^{58} - 36 q^{61} + 18 q^{64} - 12 q^{65} + 36 q^{68} - 21 q^{70} - 24 q^{73} - 14 q^{77} - 3 q^{80} - 6 q^{82} + 48 q^{85} + 28 q^{86} + 7 q^{88} - 36 q^{89} + 28 q^{92} + 42 q^{94} + 7 q^{98}+O(q^{100}) \) 4 * q - q^2 + 3 * q^4 + 6 * q^5 - 10 * q^8 - 3 * q^10 - 14 * q^14 - q^16 + 24 * q^17 + 14 * q^22 - 4 * q^25 - 6 * q^26 + 7 * q^28 + 20 * q^29 - 11 * q^32 - 15 * q^40 - 7 * q^44 - 14 * q^46 + 14 * q^49 + 4 * q^50 - 18 * q^52 - 14 * q^53 - 7 * q^56 - 5 * q^58 - 36 * q^61 + 18 * q^64 - 12 * q^65 + 36 * q^68 - 21 * q^70 - 24 * q^73 - 14 * q^77 - 3 * q^80 - 6 * q^82 + 48 * q^85 + 28 * q^86 + 7 * q^88 - 36 * q^89 + 28 * q^92 + 42 * q^94 + 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more