LMFDB - Embedded newform 1299.1.v.a.812.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1299,1,Mod(8,1299)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1299, base_ring=CyclotomicField(24))

chi = DirichletCharacter(H, H._module([12, 13]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1299.8");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1299 = 3 \cdot 433 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1299.v (of order $24$, degree $8$, 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:	$0.648285451410$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{24}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{24} + \cdots)$

Embedding invariants

Embedding label			812.1
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1299.812
Dual form			1299.1.v.a.8.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{3} -1.00000i q^{4} +(1.12484 + 1.46593i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})$	$q+(0.500000 + 0.866025i) q^{3} -1.00000i q^{4} +(1.12484 + 1.46593i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(0.866025 - 0.500000i) q^{12} +(-0.448288 + 1.67303i) q^{13} -1.00000 q^{16} +(-0.258819 - 1.96593i) q^{19} +(-0.707107 + 1.70711i) q^{21} +(-0.965926 + 0.258819i) q^{25} -1.00000 q^{27} +(1.46593 - 1.12484i) q^{28} +(0.758819 - 0.0999004i) q^{31} +(0.866025 + 0.500000i) q^{36} +(1.41421 - 1.41421i) q^{37} +(-1.67303 + 0.448288i) q^{39} +(0.241181 + 0.0999004i) q^{43} +(-0.500000 - 0.866025i) q^{48} +(-0.624844 + 2.33195i) q^{49} +(1.67303 + 0.448288i) q^{52} +(1.57313 - 1.20711i) q^{57} +(0.965926 - 0.741181i) q^{61} +(-1.83195 + 0.241181i) q^{63} +1.00000i q^{64} +(1.12484 - 0.465926i) q^{67} +(-0.758819 - 1.83195i) q^{73} +(-0.707107 - 0.707107i) q^{75} +(-1.96593 + 0.258819i) q^{76} +(-0.500000 - 0.866025i) q^{81} +(1.70711 + 0.707107i) q^{84} +(-2.95680 + 1.22474i) q^{91} +(0.465926 + 0.607206i) q^{93} +(0.292893 - 0.707107i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} - 4 q^{9}+O(q^{10}) $ 8 * q + 4 * q^3 - 4 * q^9	$ 8 q + 4 q^{3} - 4 q^{9} - 8 q^{16} - 8 q^{27} + 4 q^{28} + 4 q^{31} + 4 q^{43} - 4 q^{48} + 4 q^{49} - 4 q^{73} - 8 q^{76} - 4 q^{81} + 8 q^{84} - 4 q^{93} + 8 q^{97}+O(q^{100}) $ 8 * q + 4 * q^3 - 4 * q^9 - 8 * q^16 - 8 * q^27 + 4 * q^28 + 4 * q^31 + 4 * q^43 - 4 * q^48 + 4 * q^49 - 4 * q^73 - 8 * q^76 - 4 * q^81 + 8 * q^84 - 4 * q^93 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1299\mathbb{Z}\right)^\times$.

$n$	$434$	$871$
$\chi(n)$	$-1$	$e\left(\frac{11}{24}\right)$

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 $

$2$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$2$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$3$	0.500000	+	0.866025i	0.500000	+	0.866025i
$3$	0.500000	+	0.866025i	0.500000	+	0.866025i
$4$		−	1.00000i		−	1.00000i
$5$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$5$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$6$		0			0
$7$	1.12484	+	1.46593i	1.12484	+	1.46593i	0.866025	+	0.500000i	$0.166667\pi$
$7$	1.12484	+	1.46593i	1.12484	+	1.46593i	0.258819	+	0.965926i	$0.416667\pi$
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$10$		0			0
$11$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$11$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$12$	0.866025	−	0.500000i	0.866025	−	0.500000i
$13$	−0.448288	+	1.67303i	−0.448288	+	1.67303i	0.258819	+	0.965926i	$0.416667\pi$
$13$	−0.448288	+	1.67303i	−0.448288	+	1.67303i	−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$	−1.00000			−1.00000
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		0			0
$19$	−0.258819	−	1.96593i	−0.258819	−	1.96593i	−0.258819	−	0.965926i	$-0.583333\pi$
$19$	−0.258819	−	1.96593i	−0.258819	−	1.96593i		−	1.00000i	$-0.5\pi$
$20$		0			0
$21$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$22$		0			0
$23$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$23$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$24$		0			0
$25$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$26$		0			0
$27$	−1.00000			−1.00000
$28$	1.46593	−	1.12484i	1.46593	−	1.12484i
$29$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$29$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$30$		0			0
$31$	0.758819	−	0.0999004i	0.758819	−	0.0999004i	0.258819	−	0.965926i	$-0.416667\pi$
$31$	0.758819	−	0.0999004i	0.758819	−	0.0999004i	0.500000	+	0.866025i	$0.333333\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.866025	+	0.500000i	0.866025	+	0.500000i
$37$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$37$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$38$		0			0
$39$	−1.67303	+	0.448288i	−1.67303	+	0.448288i
$40$		0			0
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$	0.241181	+	0.0999004i	0.241181	+	0.0999004i	0.500000	−	0.866025i	$-0.333333\pi$
$43$	0.241181	+	0.0999004i	0.241181	+	0.0999004i	−0.258819	+	0.965926i	$0.583333\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$47$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$48$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$49$	−0.624844	+	2.33195i	−0.624844	+	2.33195i
$50$		0			0
$51$		0			0
$52$	1.67303	+	0.448288i	1.67303	+	0.448288i
$53$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$53$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.57313	−	1.20711i	1.57313	−	1.20711i
$58$		0			0
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$	0.965926	−	0.741181i	0.965926	−	0.741181i		−	1.00000i	$-0.5\pi$
$61$	0.965926	−	0.741181i	0.965926	−	0.741181i	0.965926	+	0.258819i	$0.0833333\pi$
$62$		0			0
$63$	−1.83195	+	0.241181i	−1.83195	+	0.241181i
$64$			1.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$	1.12484	−	0.465926i	1.12484	−	0.465926i	0.258819	−	0.965926i	$-0.416667\pi$
$67$	1.12484	−	0.465926i	1.12484	−	0.465926i	0.866025	+	0.500000i	$0.166667\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$71$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$72$		0			0
$73$	−0.758819	−	1.83195i	−0.758819	−	1.83195i	−0.500000	−	0.866025i	$-0.666667\pi$
$73$	−0.758819	−	1.83195i	−0.758819	−	1.83195i	−0.258819	−	0.965926i	$-0.583333\pi$
$74$		0			0
$75$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$76$	−1.96593	+	0.258819i	−1.96593	+	0.258819i
$77$		0			0
$78$		0			0
$79$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$79$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$83$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$84$	1.70711	+	0.707107i	1.70711	+	0.707107i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$89$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$90$		0			0
$91$	−2.95680	+	1.22474i	−2.95680	+	1.22474i
$92$		0			0
$93$	0.465926	+	0.607206i	0.465926	+	0.607206i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.292893	−	0.707107i	0.292893	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$97$	0.292893	−	0.707107i	0.292893	−	0.707107i	1.00000			$0$
$98$		0			0
$99$		0			0
$100$	0.258819	+	0.965926i	0.258819	+	0.965926i
$101$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$101$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$102$		0			0
$103$	−0.241181	−	0.0999004i	−0.241181	−	0.0999004i	0.258819	−	0.965926i	$-0.416667\pi$
$103$	−0.241181	−	0.0999004i	−0.241181	−	0.0999004i	−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$107$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$108$			1.00000i			1.00000i
$109$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$109$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$110$		0			0
$111$	1.93185	+	0.517638i	1.93185	+	0.517638i
$112$	−1.12484	−	1.46593i	−1.12484	−	1.46593i
$113$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$113$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.22474	−	1.22474i	−1.22474	−	1.22474i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$122$		0			0
$123$		0			0
$124$	−0.0999004	−	0.758819i	−0.0999004	−	0.758819i
$125$		0			0
$126$		0			0
$127$	−1.36603	+	1.36603i	−1.36603	+	1.36603i	−0.500000	+	0.866025i	$0.666667\pi$
$127$	−1.36603	+	1.36603i	−1.36603	+	1.36603i	−0.866025	+	0.500000i	$0.833333\pi$
$128$		0			0
$129$	0.0340742	+	0.258819i	0.0340742	+	0.258819i
$130$		0			0
$131$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$131$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$132$		0			0
$133$	2.59077	−	2.59077i	2.59077	−	2.59077i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$	0.258819	−	0.448288i	0.258819	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$139$	0.258819	−	0.448288i	0.258819	−	0.448288i	0.965926	+	0.258819i	$0.0833333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.500000	−	0.866025i	0.500000	−	0.866025i
$145$		0			0
$146$		0			0
$147$	−2.33195	+	0.624844i	−2.33195	+	0.624844i
$148$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$149$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$149$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$150$		0			0
$151$	−0.292893	+	0.707107i	−0.292893	+	0.707107i	0.707107	+	0.707107i	$0.250000\pi$
$151$	−0.292893	+	0.707107i	−0.292893	+	0.707107i	−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	0.448288	+	1.67303i	0.448288	+	1.67303i
$157$	1.46593	−	0.607206i	1.46593	−	0.607206i	0.500000	−	0.866025i	$-0.333333\pi$
$157$	1.46593	−	0.607206i	1.46593	−	0.607206i	0.965926	+	0.258819i	$0.0833333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.25882	+	0.965926i	1.25882	+	0.965926i	1.00000			$0$
$163$	1.25882	+	0.965926i	1.25882	+	0.965926i	0.258819	+	0.965926i	$0.416667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$170$		0			0
$171$	1.83195	+	0.758819i	1.83195	+	0.758819i
$172$	0.0999004	−	0.241181i	0.0999004	−	0.241181i
$173$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$173$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$174$		0			0
$175$	−1.46593	−	1.12484i	−1.46593	−	1.12484i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−0.241181	−	1.83195i	−0.241181	−	1.83195i	−0.500000	−	0.866025i	$-0.666667\pi$
$181$	−0.241181	−	1.83195i	−0.241181	−	1.83195i	0.258819	−	0.965926i	$-0.416667\pi$
$182$		0			0
$183$	1.12484	+	0.465926i	1.12484	+	0.465926i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−1.12484	−	1.46593i	−1.12484	−	1.46593i
$190$		0			0
$191$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$191$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$192$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$193$	0.207107	−	0.158919i	0.207107	−	0.158919i	−0.500000	−	0.866025i	$-0.666667\pi$
$193$	0.207107	−	0.158919i	0.207107	−	0.158919i	0.707107	+	0.707107i	$0.250000\pi$
$194$		0			0
$195$		0			0
$196$	2.33195	+	0.624844i	2.33195	+	0.624844i
$197$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$197$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$198$		0			0
$199$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$199$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$200$		0			0
$201$	0.965926	+	0.741181i	0.965926	+	0.741181i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.448288	−	1.67303i	0.448288	−	1.67303i
$209$		0			0
$210$		0			0
$211$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$211$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	1.00000	+	1.00000i	1.00000	+	1.00000i
$218$		0			0
$219$	1.20711	−	1.57313i	1.20711	−	1.57313i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$223$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$224$		0			0
$225$	0.258819	−	0.965926i	0.258819	−	0.965926i
$226$		0			0
$227$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$227$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$228$	−1.20711	−	1.57313i	−1.20711	−	1.57313i
$229$	0.448288	+	0.258819i	0.448288	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$229$	0.448288	+	0.258819i	0.448288	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$233$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$239$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$240$		0			0
$241$	0.448288	−	0.258819i	0.448288	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$241$	0.448288	−	0.258819i	0.448288	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$242$		0			0
$243$	0.500000	−	0.866025i	0.500000	−	0.866025i
$244$	−0.741181	−	0.965926i	−0.741181	−	0.965926i
$245$		0			0
$246$		0			0
$247$	3.40508	+	0.448288i	3.40508	+	0.448288i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$251$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$252$	0.241181	+	1.83195i	0.241181	+	1.83195i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1.00000			1.00000
$257$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$257$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$258$		0			0
$259$	3.66390	+	0.482362i	3.66390	+	0.482362i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$263$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.465926	−	1.12484i	−0.465926	−	1.12484i
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$271$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$272$		0			0
$273$	−2.53906	−	1.94829i	−2.53906	−	1.94829i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.500000	+	1.86603i	0.500000	+	1.86603i	0.500000	+	0.866025i	$0.333333\pi$
$277$	0.500000	+	1.86603i	0.500000	+	1.86603i			1.00000i	$0.5\pi$
$278$		0			0
$279$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$280$		0			0
$281$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$281$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$282$		0			0
$283$	−1.93185			−1.93185			−0.965926	−	0.258819i	$-0.916667\pi$
$283$	−1.93185			−1.93185			−0.965926	+	0.258819i	$0.916667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$290$		0			0
$291$	0.758819	−	0.0999004i	0.758819	−	0.0999004i
$292$	−1.83195	+	0.758819i	−1.83195	+	0.758819i
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$301$	0.124844	+	0.465926i	0.124844	+	0.465926i
$302$		0			0
$303$		0			0
$304$	0.258819	+	1.96593i	0.258819	+	1.96593i
$305$		0			0
$306$		0			0
$307$	−0.965926	+	1.25882i	−0.965926	+	1.25882i			1.00000i	$0.5\pi$
$307$	−0.965926	+	1.25882i	−0.965926	+	1.25882i	−0.965926	+	0.258819i	$0.916667\pi$
$308$		0			0
$309$	−0.0340742	−	0.258819i	−0.0340742	−	0.258819i
$310$		0			0
$311$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$311$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$312$		0			0
$313$	−1.57313	−	0.207107i	−1.57313	−	0.207107i	−0.707107	−	0.707107i	$-0.750000\pi$
$313$	−1.57313	−	0.207107i	−1.57313	−	0.207107i	−0.866025	+	0.500000i	$0.833333\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$317$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$325$		−	1.73205i		−	1.73205i
$326$		0			0
$327$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.366025	+	1.36603i	0.366025	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$331$	0.366025	+	1.36603i	0.366025	+	1.36603i	−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$	0.517638	+	1.93185i	0.517638	+	1.93185i
$334$		0			0
$335$		0			0
$336$	0.707107	−	1.70711i	0.707107	−	1.70711i
$337$	−0.500000	−	0.133975i	−0.500000	−	0.133975i		−	1.00000i	$-0.5\pi$
$337$	−0.500000	−	0.133975i	−0.500000	−	0.133975i	−0.500000	+	0.866025i	$0.666667\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−2.41421	+	1.00000i	−2.41421	+	1.00000i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$347$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$348$		0			0
$349$	−1.46593	−	0.607206i	−1.46593	−	0.607206i	−0.500000	−	0.866025i	$-0.666667\pi$
$349$	−1.46593	−	0.607206i	−1.46593	−	0.607206i	−0.965926	+	0.258819i	$0.916667\pi$
$350$		0			0
$351$	0.448288	−	1.67303i	0.448288	−	1.67303i
$352$		0			0
$353$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$353$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−2.83195	+	0.758819i	−2.83195	+	0.758819i
$362$		0			0
$363$		−	1.00000i		−	1.00000i
$364$	1.22474	+	2.95680i	1.22474	+	2.95680i
$365$		0			0
$366$		0			0
$367$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$367$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.607206	−	0.465926i	0.607206	−	0.465926i
$373$	−0.158919	−	0.207107i	−0.158919	−	0.207107i	0.707107	−	0.707107i	$-0.250000\pi$
$373$	−0.158919	−	0.207107i	−0.158919	−	0.207107i	−0.866025	+	0.500000i	$0.833333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.36603	−	1.36603i	1.36603	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$379$	1.36603	−	1.36603i	1.36603	−	1.36603i	0.866025	−	0.500000i	$-0.166667\pi$
$380$		0			0
$381$	−1.86603	−	0.500000i	−1.86603	−	0.500000i
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.207107	+	0.158919i	−0.207107	+	0.158919i
$388$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$389$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$389$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$397$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$398$		0			0
$399$	3.53906	+	0.948288i	3.53906	+	0.948288i
$400$	0.965926	−	0.258819i	0.965926	−	0.258819i
$401$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$401$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$402$		0			0
$403$	−0.173033	+	1.31431i	−0.173033	+	1.31431i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.448288	+	1.67303i	−0.448288	+	1.67303i	0.258819	+	0.965926i	$0.416667\pi$
$409$	−0.448288	+	1.67303i	−0.448288	+	1.67303i	−0.707107	+	0.707107i	$0.750000\pi$
$410$		0			0
$411$		0			0
$412$	−0.0999004	+	0.241181i	−0.0999004	+	0.241181i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.517638			0.517638
$418$		0			0
$419$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$419$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$420$		0			0
$421$	0.448288	−	0.258819i	0.448288	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$421$	0.448288	−	0.258819i	0.448288	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	2.17303	+	0.582262i	2.17303	+	0.582262i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$431$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$432$	1.00000			1.00000
$433$		−	1.00000i		−	1.00000i
$433$		−	1.00000i		−	1.00000i
$434$		0			0
$435$		0			0
$436$	0.500000	+	0.866025i	0.500000	+	0.866025i
$437$		0			0
$438$		0			0
$439$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.866025	−	0.500000i	$-0.166667\pi$
$439$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	−1.70711	−	1.70711i	−1.70711	−	1.70711i
$442$		0			0
$443$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$443$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$444$	0.517638	−	1.93185i	0.517638	−	1.93185i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−1.46593	+	1.12484i	−1.46593	+	1.12484i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−0.758819	+	0.0999004i	−0.758819	+	0.0999004i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.133975	+	0.500000i	−0.133975	+	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$457$	−0.133975	+	0.500000i	−0.133975	+	0.500000i	−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$461$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$462$		0			0
$463$	0.0340742	−	0.258819i	0.0340742	−	0.258819i	−0.965926	−	0.258819i	$-0.916667\pi$
$463$	0.0340742	−	0.258819i	0.0340742	−	0.258819i	1.00000			$0$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$467$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$468$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$469$	1.94829	+	1.12484i	1.94829	+	1.12484i
$470$		0			0
$471$	1.25882	+	0.965926i	1.25882	+	0.965926i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.758819	+	1.83195i	0.758819	+	1.83195i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$479$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$480$		0			0
$481$	1.73205	+	3.00000i	1.73205	+	3.00000i
$482$		0			0
$483$		0			0
$484$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$487$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$488$		0			0
$489$	−0.207107	+	1.57313i	−0.207107	+	1.57313i
$490$		0			0
$491$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$491$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.758819	+	0.0999004i	−0.758819	+	0.0999004i
$497$		0			0
$498$		0			0
$499$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.258819	+	0.965926i	$0.583333\pi$
$499$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.707107	+	0.707107i	$0.750000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	2.00000i		−	2.00000i
$508$	1.36603	+	1.36603i	1.36603	+	1.36603i
$509$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$509$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$510$		0			0
$511$	1.83195	−	3.17303i	1.83195	−	3.17303i
$512$		0			0
$513$	0.258819	+	1.96593i	0.258819	+	1.96593i
$514$		0			0
$515$		0			0
$516$	0.258819	−	0.0340742i	0.258819	−	0.0340742i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	1.83195	−	0.758819i	1.83195	−	0.758819i	0.866025	−	0.500000i	$-0.166667\pi$
$523$	1.83195	−	0.758819i	1.83195	−	0.758819i	0.965926	−	0.258819i	$-0.0833333\pi$
$524$		0			0
$525$	0.241181	−	1.83195i	0.241181	−	1.83195i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.965926	+	0.258819i	0.965926	+	0.258819i
$530$		0			0
$531$		0			0
$532$	−2.59077	−	2.59077i	−2.59077	−	2.59077i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$541$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$542$		0			0
$543$	1.46593	−	1.12484i	1.46593	−	1.12484i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.241181	+	1.83195i	−0.241181	+	1.83195i	0.258819	+	0.965926i	$0.416667\pi$
$547$	−0.241181	+	1.83195i	−0.241181	+	1.83195i	−0.500000	+	0.866025i	$0.666667\pi$
$548$		0			0
$549$	0.158919	+	1.20711i	0.158919	+	1.20711i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−0.448288	−	0.258819i	−0.448288	−	0.258819i
$557$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$557$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$558$		0			0
$559$	−0.275255	+	0.358719i	−0.275255	+	0.358719i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$563$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.707107	−	1.70711i	0.707107	−	1.70711i
$568$		0			0
$569$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$569$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$570$		0			0
$571$	1.70711	+	0.707107i	1.70711	+	0.707107i	1.00000			$0$
$571$	1.70711	+	0.707107i	1.70711	+	0.707107i	0.707107	+	0.707107i	$0.250000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$577$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$577$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$578$		0			0
$579$	0.241181	+	0.0999004i	0.241181	+	0.0999004i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$587$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$588$	0.624844	+	2.33195i	0.624844	+	2.33195i
$589$	−0.392794	−	1.46593i	−0.392794	−	1.46593i
$590$		0			0
$591$		0			0
$592$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$593$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$593$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$598$		0			0
$599$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$599$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$600$		0			0
$601$	1.46593	−	0.607206i	1.46593	−	0.607206i	0.500000	−	0.866025i	$-0.333333\pi$
$601$	1.46593	−	0.607206i	1.46593	−	0.607206i	0.965926	+	0.258819i	$0.0833333\pi$
$602$		0			0
$603$	−0.158919	+	1.20711i	−0.158919	+	1.20711i
$604$	0.707107	+	0.292893i	0.707107	+	0.292893i
$605$		0			0
$606$		0			0
$607$	1.57313	+	0.207107i	1.57313	+	0.207107i	0.866025	−	0.500000i	$-0.166667\pi$
$607$	1.57313	+	0.207107i	1.57313	+	0.207107i	0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.465926	−	1.12484i	0.465926	−	1.12484i	−0.500000	−	0.866025i	$-0.666667\pi$
$613$	0.465926	−	1.12484i	0.465926	−	1.12484i	0.965926	−	0.258819i	$-0.0833333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$617$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$618$		0			0
$619$	−1.57313	−	0.207107i	−1.57313	−	0.207107i	−0.707107	−	0.707107i	$-0.750000\pi$
$619$	−1.57313	−	0.207107i	−1.57313	−	0.207107i	−0.866025	+	0.500000i	$0.833333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	1.67303	−	0.448288i	1.67303	−	0.448288i
$625$	0.866025	−	0.500000i	0.866025	−	0.500000i
$626$		0			0
$627$		0			0
$628$	−0.607206	−	1.46593i	−0.607206	−	1.46593i
$629$		0			0
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−3.62132	−	2.09077i	−3.62132	−	2.09077i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$641$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$642$		0			0
$643$	−1.67303	+	0.965926i	−1.67303	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$643$	−1.67303	+	0.965926i	−1.67303	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$647$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$652$	0.965926	−	1.25882i	0.965926	−	1.25882i
$653$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$653$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.96593	+	0.258819i	1.96593	+	0.258819i
$658$		0			0
$659$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$659$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$660$		0			0
$661$	−0.965926	−	1.25882i	−0.965926	−	1.25882i	−0.965926	−	0.258819i	$-0.916667\pi$
$661$	−0.965926	−	1.25882i	−0.965926	−	1.25882i		−	1.00000i	$-0.5\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.207107	+	0.158919i	−0.207107	+	0.158919i	−0.707107	−	0.707107i	$-0.750000\pi$
$673$	−0.207107	+	0.158919i	−0.207107	+	0.158919i	0.500000	+	0.866025i	$0.333333\pi$
$674$		0			0
$675$	0.965926	−	0.258819i	0.965926	−	0.258819i
$676$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$677$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$677$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$678$		0			0
$679$	1.36603	−	0.366025i	1.36603	−	0.366025i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$683$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$684$	0.758819	−	1.83195i	0.758819	−	1.83195i
$685$		0			0
$686$		0			0
$687$			0.517638i			0.517638i
$688$	−0.241181	−	0.0999004i	−0.241181	−	0.0999004i
$689$		0			0
$690$		0			0
$691$	1.57313	+	1.20711i	1.57313	+	1.20711i	0.866025	+	0.500000i	$0.166667\pi$
$691$	1.57313	+	1.20711i	1.57313	+	1.20711i	0.707107	+	0.707107i	$0.250000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.12484	+	1.46593i	−1.12484	+	1.46593i
$701$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$701$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$702$		0			0
$703$	−3.14626	−	2.41421i	−3.14626	−	2.41421i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.12484	+	0.465926i	−1.12484	+	0.465926i	−0.866025	−	0.500000i	$-0.833333\pi$
$709$	−1.12484	+	0.465926i	−1.12484	+	0.465926i	−0.258819	+	0.965926i	$0.583333\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$719$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$720$		0			0
$721$	−0.124844	−	0.465926i	−0.124844	−	0.465926i
$722$		0			0
$723$	0.448288	+	0.258819i	0.448288	+	0.258819i
$724$	−1.83195	+	0.241181i	−1.83195	+	0.241181i
$725$		0			0
$726$		0			0
$727$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.258819	+	0.965926i	$0.583333\pi$
$727$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$	1.00000			1.00000
$730$		0			0
$731$		0			0
$732$	0.465926	−	1.12484i	0.465926	−	1.12484i
$733$	1.36603	−	1.36603i	1.36603	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$733$	1.36603	−	1.36603i	1.36603	−	1.36603i	0.866025	−	0.500000i	$-0.166667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$739$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$740$		0			0
$741$	1.31431	+	3.17303i	1.31431	+	3.17303i
$742$		0			0
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$751$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−1.46593	+	1.12484i	−1.46593	+	1.12484i
$757$	0.448288	−	0.258819i	0.448288	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$757$	0.448288	−	0.258819i	0.448288	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$761$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$762$		0			0
$763$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.500000	+	0.866025i	0.500000	+	0.866025i
$769$	0.465926	−	1.12484i	0.465926	−	1.12484i	−0.500000	−	0.866025i	$-0.666667\pi$
$769$	0.465926	−	1.12484i	0.465926	−	1.12484i	0.965926	−	0.258819i	$-0.0833333\pi$
$770$		0			0
$771$		0			0
$772$	−0.158919	−	0.207107i	−0.158919	−	0.207107i
$773$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$773$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$774$		0			0
$775$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$776$		0			0
$777$	1.41421	+	3.41421i	1.41421	+	3.41421i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.624844	−	2.33195i	0.624844	−	2.33195i
$785$		0			0
$786$		0			0
$787$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$787$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.807007	+	1.94829i	0.807007	+	1.94829i
$794$		0			0
$795$		0			0
$796$			1.00000i			1.00000i
$797$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$797$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	0.741181	−	0.965926i	0.741181	−	0.965926i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$809$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$810$		0			0
$811$	−1.83195	−	0.241181i	−1.83195	−	0.241181i	−0.866025	−	0.500000i	$-0.833333\pi$
$811$	−1.83195	−	0.241181i	−1.83195	−	0.241181i	−0.965926	+	0.258819i	$0.916667\pi$
$812$		0			0
$813$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.133975	−	0.500000i	0.133975	−	0.500000i
$818$		0			0
$819$	0.417738	−	3.17303i	0.417738	−	3.17303i
$820$		0			0
$821$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$821$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$822$		0			0
$823$	−1.12484	−	0.465926i	−1.12484	−	0.465926i	−0.258819	−	0.965926i	$-0.583333\pi$
$823$	−1.12484	−	0.465926i	−1.12484	−	0.465926i	−0.866025	+	0.500000i	$0.833333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$827$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$828$		0			0
$829$	0.366025	−	0.366025i	0.366025	−	0.366025i	−0.500000	−	0.866025i	$-0.666667\pi$
$829$	0.366025	−	0.366025i	0.366025	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$830$		0			0
$831$	−1.36603	+	1.36603i	−1.36603	+	1.36603i
$832$	−1.67303	−	0.448288i	−1.67303	−	0.448288i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−0.758819	+	0.0999004i	−0.758819	+	0.0999004i
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	0.965926	−	0.258819i	0.965926	−	0.258819i
$842$		0			0
$843$		0			0
$844$	0.707107	+	1.22474i	0.707107	+	1.22474i
$845$		0			0
$846$		0			0
$847$	−0.241181	−	1.83195i	−0.241181	−	1.83195i
$848$		0			0
$849$	−0.965926	−	1.67303i	−0.965926	−	1.67303i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$853$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$	−1.12484	−	1.46593i	−1.12484	−	1.46593i	−0.866025	−	0.500000i	$-0.833333\pi$
$859$	−1.12484	−	1.46593i	−1.12484	−	1.46593i	−0.258819	−	0.965926i	$-0.583333\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−1.00000			−1.00000
$868$	1.00000	−	1.00000i	1.00000	−	1.00000i
$869$		0			0
$870$		0			0
$871$	0.275255	+	2.09077i	0.275255	+	2.09077i
$872$		0			0
$873$	0.465926	+	0.607206i	0.465926	+	0.607206i
$874$		0			0
$875$		0			0
$876$	−1.57313	−	1.20711i	−1.57313	−	1.20711i
$877$	0.500000	−	1.86603i	0.500000	−	1.86603i		−	1.00000i	$-0.5\pi$
$877$	0.500000	−	1.86603i	0.500000	−	1.86603i	0.500000	−	0.866025i	$-0.333333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$881$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$882$		0			0
$883$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.707107	+	0.707107i	$0.250000\pi$
$883$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.258819	+	0.965926i	$0.416667\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$887$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$888$		0			0
$889$	−3.53906	−	0.465926i	−3.53906	−	0.465926i
$890$		0			0
$891$		0			0
$892$	0.707107	+	1.22474i	0.707107	+	1.22474i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$901$		0			0
$902$		0			0
$903$	−0.341081	+	0.341081i	−0.341081	+	0.341081i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$907$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$911$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$912$	−1.57313	+	1.20711i	−1.57313	+	1.20711i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.258819	−	0.448288i	0.258819	−	0.448288i
$917$		0			0
$918$		0			0
$919$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$919$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$920$		0			0
$921$	−1.57313	−	0.207107i	−1.57313	−	0.207107i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$926$		0			0
$927$	0.207107	−	0.158919i	0.207107	−	0.158919i
$928$		0			0
$929$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$929$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$930$		0			0
$931$	4.74616	+	0.624844i	4.74616	+	0.624844i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.607206	−	0.465926i	−0.607206	−	0.465926i	0.258819	−	0.965926i	$-0.416667\pi$
$937$	−0.607206	−	0.465926i	−0.607206	−	0.465926i	−0.866025	+	0.500000i	$0.833333\pi$
$938$		0			0
$939$	−0.607206	−	1.46593i	−0.607206	−	1.46593i
$940$		0			0
$941$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$941$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$	3.40508	−	0.448288i	3.40508	−	0.448288i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$953$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.400100	+	0.107206i	−0.400100	+	0.107206i
$962$		0			0
$963$		0			0
$964$	−0.258819	−	0.448288i	−0.258819	−	0.448288i
$965$		0			0
$966$		0			0
$967$	0.241181	+	1.83195i	0.241181	+	1.83195i	0.500000	+	0.866025i	$0.333333\pi$
$967$	0.241181	+	1.83195i	0.241181	+	1.83195i	−0.258819	+	0.965926i	$0.583333\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$971$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$972$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$973$	0.948288	−	0.124844i	0.948288	−	0.124844i
$974$		0			0
$975$	1.50000	−	0.866025i	1.50000	−	0.866025i
$976$	−0.965926	+	0.741181i	−0.965926	+	0.741181i
$977$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$977$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		−	1.00000i		−	1.00000i
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.448288	−	3.40508i	0.448288	−	3.40508i
$989$		0			0
$990$		0			0
$991$	−0.607206	−	1.46593i	−0.607206	−	1.46593i	−0.866025	−	0.500000i	$-0.833333\pi$
$991$	−0.607206	−	1.46593i	−0.607206	−	1.46593i	0.258819	−	0.965926i	$-0.416667\pi$
$992$		0			0
$993$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.465926	−	1.12484i	−0.465926	−	1.12484i	−0.965926	−	0.258819i	$-0.916667\pi$
$997$	−0.465926	−	1.12484i	−0.465926	−	1.12484i	0.500000	−	0.866025i	$-0.333333\pi$
$998$		0			0
$999$	−1.41421	+	1.41421i	−1.41421	+	1.41421i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1299.1.v.a.812.1	yes	8
3.2	odd	2	CM	1299.1.v.a.812.1	yes	8
433.8	even	24	inner	1299.1.v.a.8.1	✓	8
1299.8	odd	24	inner	1299.1.v.a.8.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1299.1.v.a.8.1	✓	8	433.8	even	24	inner
1299.1.v.a.8.1	✓	8	1299.8	odd	24	inner
1299.1.v.a.812.1	yes	8	1.1	even	1	trivial
1299.1.v.a.812.1	yes	8	3.2	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1299 = 3 \cdot 433 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1299.v (of order \(24\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{3} -1.00000i q^{4} +(1.12484 + 1.46593i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{3} -1.00000i q^{4} +(1.12484 + 1.46593i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(0.866025 - 0.500000i) q^{12} +(-0.448288 + 1.67303i) q^{13} -1.00000 q^{16} +(-0.258819 - 1.96593i) q^{19} +(-0.707107 + 1.70711i) q^{21} +(-0.965926 + 0.258819i) q^{25} -1.00000 q^{27} +(1.46593 - 1.12484i) q^{28} +(0.758819 - 0.0999004i) q^{31} +(0.866025 + 0.500000i) q^{36} +(1.41421 - 1.41421i) q^{37} +(-1.67303 + 0.448288i) q^{39} +(0.241181 + 0.0999004i) q^{43} +(-0.500000 - 0.866025i) q^{48} +(-0.624844 + 2.33195i) q^{49} +(1.67303 + 0.448288i) q^{52} +(1.57313 - 1.20711i) q^{57} +(0.965926 - 0.741181i) q^{61} +(-1.83195 + 0.241181i) q^{63} +1.00000i q^{64} +(1.12484 - 0.465926i) q^{67} +(-0.758819 - 1.83195i) q^{73} +(-0.707107 - 0.707107i) q^{75} +(-1.96593 + 0.258819i) q^{76} +(-0.500000 - 0.866025i) q^{81} +(1.70711 + 0.707107i) q^{84} +(-2.95680 + 1.22474i) q^{91} +(0.465926 + 0.607206i) q^{93} +(0.292893 - 0.707107i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 4 q^{9}+O(q^{10}) \) 8 * q + 4 * q^3 - 4 * q^9	\( 8 q + 4 q^{3} - 4 q^{9} - 8 q^{16} - 8 q^{27} + 4 q^{28} + 4 q^{31} + 4 q^{43} - 4 q^{48} + 4 q^{49} - 4 q^{73} - 8 q^{76} - 4 q^{81} + 8 q^{84} - 4 q^{93} + 8 q^{97}+O(q^{100}) \) 8 * q + 4 * q^3 - 4 * q^9 - 8 * q^16 - 8 * q^27 + 4 * q^28 + 4 * q^31 + 4 * q^43 - 4 * q^48 + 4 * q^49 - 4 * q^73 - 8 * q^76 - 4 * q^81 + 8 * q^84 - 4 * q^93 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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