LMFDB - Embedded newform 299.1.j.a.114.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [299,1,Mod(114,299)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(299, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("299.114");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 299 = 13 \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	299.j (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:	$0.149220438777$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{18}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{18} - \cdots)$

Embedding invariants

Embedding label			114.3
Root			$0.939693 - 0.342020i$ of defining polynomial
Character	$\chi$	$=$	299.114
Dual form			299.1.j.a.160.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.11334 - 0.642788i) q^{2} +(-0.939693 - 1.62760i) q^{3} +(0.326352 - 0.565258i) q^{4} +(-2.09240 - 1.20805i) q^{6} +0.446476i q^{8} +(-1.26604 + 2.19285i) q^{9} +O(q^{10})$	\(q+(1.11334 - 0.642788i) q^{2} +(-0.939693 - 1.62760i) q^{3} +(0.326352 - 0.565258i) q^{4} +(-2.09240 - 1.20805i) q^{6} +0.446476i q^{8} +(-1.26604 + 2.19285i) q^{9} -1.22668 q^{12} +(0.766044 - 0.642788i) q^{13} +(0.613341 + 1.06234i) q^{16} +3.25519i q^{18} +(-0.500000 - 0.866025i) q^{23} +(0.726682 - 0.419550i) q^{24} -1.00000 q^{25} +(0.439693 - 1.20805i) q^{26} +2.87939 q^{27} +(0.766044 + 1.32683i) q^{29} -0.684040i q^{31} +(0.979055 + 0.565258i) q^{32} +(0.826352 + 1.43128i) q^{36} +(-1.76604 - 0.642788i) q^{39} +(-1.70574 + 0.984808i) q^{41} +(-1.11334 - 0.642788i) q^{46} +1.28558i q^{47} +(1.15270 - 1.99654i) q^{48} +(0.500000 + 0.866025i) q^{49} +(-1.11334 + 0.642788i) q^{50} +(-0.113341 - 0.642788i) q^{52} +(3.20574 - 1.85083i) q^{54} +(1.70574 + 0.984808i) q^{58} +(-0.439693 - 0.761570i) q^{62} +0.226682 q^{64} +(-0.939693 + 1.62760i) q^{69} +(-1.70574 - 0.984808i) q^{71} +(-0.979055 - 0.565258i) q^{72} -1.96962i q^{73} +(0.939693 + 1.62760i) q^{75} +(-2.37939 + 0.419550i) q^{78} +(-1.43969 - 2.49362i) q^{81} +(-1.26604 + 2.19285i) q^{82} +(1.43969 - 2.49362i) q^{87} -0.652704 q^{92} +(-1.11334 + 0.642788i) q^{93} +(0.826352 + 1.43128i) q^{94} -2.12467i q^{96} +(1.11334 + 0.642788i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{4} - 9 q^{6} - 3 q^{9}+O(q^{10}) $ 6 * q + 3 * q^4 - 9 * q^6 - 3 * q^9	$ 6 q + 3 q^{4} - 9 q^{6} - 3 q^{9} + 6 q^{12} - 3 q^{16} - 3 q^{23} - 9 q^{24} - 6 q^{25} - 3 q^{26} + 6 q^{27} + 9 q^{32} + 6 q^{36} - 6 q^{39} + 9 q^{48} + 3 q^{49} + 6 q^{52} + 9 q^{54} + 3 q^{62} - 12 q^{64} - 9 q^{72} - 3 q^{78} - 3 q^{81} - 3 q^{82} + 3 q^{87} - 6 q^{92} + 6 q^{94}+O(q^{100}) $ 6 * q + 3 * q^4 - 9 * q^6 - 3 * q^9 + 6 * q^12 - 3 * q^16 - 3 * q^23 - 9 * q^24 - 6 * q^25 - 3 * q^26 + 6 * q^27 + 9 * q^32 + 6 * q^36 - 6 * q^39 + 9 * q^48 + 3 * q^49 + 6 * q^52 + 9 * q^54 + 3 * q^62 - 12 * q^64 - 9 * q^72 - 3 * q^78 - 3 * q^81 - 3 * q^82 + 3 * q^87 - 6 * q^92 + 6 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/299\mathbb{Z}\right)^\times$.

$n$	$93$	$235$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$

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 $

$2$	1.11334	−	0.642788i	1.11334	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$2$	1.11334	−	0.642788i	1.11334	−	0.642788i	0.939693	+	0.342020i	$0.111111\pi$
$3$	−0.939693	−	1.62760i	−0.939693	−	1.62760i	−0.766044	−	0.642788i	$-0.777778\pi$
$3$	−0.939693	−	1.62760i	−0.939693	−	1.62760i	−0.173648	−	0.984808i	$-0.555556\pi$
$4$	0.326352	−	0.565258i	0.326352	−	0.565258i
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	−2.09240	−	1.20805i	−2.09240	−	1.20805i
$7$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$7$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$8$			0.446476i			0.446476i
$9$	−1.26604	+	2.19285i	−1.26604	+	2.19285i
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$	−1.22668			−1.22668
$13$	0.766044	−	0.642788i	0.766044	−	0.642788i
$13$	0.766044	−	0.642788i	0.766044	−	0.642788i
$14$		0			0
$15$		0			0
$16$	0.613341	+	1.06234i	0.613341	+	1.06234i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$			3.25519i			3.25519i
$19$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$19$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$23$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$24$	0.726682	−	0.419550i	0.726682	−	0.419550i
$25$	−1.00000			−1.00000
$26$	0.439693	−	1.20805i	0.439693	−	1.20805i
$27$	2.87939			2.87939
$28$		0			0
$29$	0.766044	+	1.32683i	0.766044	+	1.32683i	0.939693	+	0.342020i	$0.111111\pi$
$29$	0.766044	+	1.32683i	0.766044	+	1.32683i	−0.173648	+	0.984808i	$0.555556\pi$
$30$		0			0
$31$		−	0.684040i		−	0.684040i	−0.939693	−	0.342020i	$-0.888889\pi$
$31$		−	0.684040i		−	0.684040i	0.939693	−	0.342020i	$-0.111111\pi$
$32$	0.979055	+	0.565258i	0.979055	+	0.565258i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.826352	+	1.43128i	0.826352	+	1.43128i
$37$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$37$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$38$		0			0
$39$	−1.76604	−	0.642788i	−1.76604	−	0.642788i
$40$		0			0
$41$	−1.70574	+	0.984808i	−1.70574	+	0.984808i	−0.766044	+	0.642788i	$0.777778\pi$
$41$	−1.70574	+	0.984808i	−1.70574	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$42$		0			0
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$		0			0
$45$		0			0
$46$	−1.11334	−	0.642788i	−1.11334	−	0.642788i
$47$			1.28558i			1.28558i	0.766044	+	0.642788i	$0.222222\pi$
$47$			1.28558i			1.28558i	−0.766044	+	0.642788i	$0.777778\pi$
$48$	1.15270	−	1.99654i	1.15270	−	1.99654i
$49$	0.500000	+	0.866025i	0.500000	+	0.866025i
$50$	−1.11334	+	0.642788i	−1.11334	+	0.642788i
$51$		0			0
$52$	−0.113341	−	0.642788i	−0.113341	−	0.642788i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	3.20574	−	1.85083i	3.20574	−	1.85083i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	1.70574	+	0.984808i	1.70574	+	0.984808i
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$	−0.439693	−	0.761570i	−0.439693	−	0.761570i
$63$		0			0
$64$	0.226682			0.226682
$65$		0			0
$66$		0			0
$67$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$67$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$68$		0			0
$69$	−0.939693	+	1.62760i	−0.939693	+	1.62760i
$70$		0			0
$71$	−1.70574	−	0.984808i	−1.70574	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$71$	−1.70574	−	0.984808i	−1.70574	−	0.984808i	−0.766044	−	0.642788i	$-0.777778\pi$
$72$	−0.979055	−	0.565258i	−0.979055	−	0.565258i
$73$		−	1.96962i		−	1.96962i	−0.173648	−	0.984808i	$-0.555556\pi$
$73$		−	1.96962i		−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$74$		0			0
$75$	0.939693	+	1.62760i	0.939693	+	1.62760i
$76$		0			0
$77$		0			0
$78$	−2.37939	+	0.419550i	−2.37939	+	0.419550i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−1.43969	−	2.49362i	−1.43969	−	2.49362i
$82$	−1.26604	+	2.19285i	−1.26604	+	2.19285i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	1.43969	−	2.49362i	1.43969	−	2.49362i
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$		0			0
$92$	−0.652704			−0.652704
$93$	−1.11334	+	0.642788i	−1.11334	+	0.642788i
$94$	0.826352	+	1.43128i	0.826352	+	1.43128i
$95$		0			0
$96$		−	2.12467i		−	2.12467i
$97$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$97$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$98$	1.11334	+	0.642788i	1.11334	+	0.642788i
$99$		0			0
$100$	−0.326352	+	0.565258i	−0.326352	+	0.565258i
$101$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$101$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	0.286989	+	0.342020i	0.286989	+	0.342020i
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$	0.939693	−	1.62760i	0.939693	−	1.62760i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$		0			0
$115$		0			0
$116$	1.00000			1.00000
$117$	0.439693	+	2.49362i	0.439693	+	2.49362i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.500000	−	0.866025i	0.500000	−	0.866025i
$122$		0			0
$123$	3.20574	+	1.85083i	3.20574	+	1.85083i
$124$	−0.386659	−	0.223238i	−0.386659	−	0.223238i
$125$		0			0
$126$		0			0
$127$	0.173648	+	0.300767i	0.173648	+	0.300767i	0.939693	−	0.342020i	$-0.111111\pi$
$127$	0.173648	+	0.300767i	0.173648	+	0.300767i	−0.766044	+	0.642788i	$0.777778\pi$
$128$	−0.726682	+	0.419550i	−0.726682	+	0.419550i
$129$		0			0
$130$		0			0
$131$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$131$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$			2.41609i			2.41609i
$139$	0.766044	−	1.32683i	0.766044	−	1.32683i	−0.173648	−	0.984808i	$-0.555556\pi$
$139$	0.766044	−	1.32683i	0.766044	−	1.32683i	0.939693	−	0.342020i	$-0.111111\pi$
$140$		0			0
$141$	2.09240	−	1.20805i	2.09240	−	1.20805i
$142$	−2.53209			−2.53209
$143$		0			0
$144$	−3.10607			−3.10607
$145$		0			0
$146$	−1.26604	−	2.19285i	−1.26604	−	2.19285i
$147$	0.939693	−	1.62760i	0.939693	−	1.62760i
$148$		0			0
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$	2.09240	+	1.20805i	2.09240	+	1.20805i
$151$			1.96962i			1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$151$			1.96962i			1.96962i	−0.173648	+	0.984808i	$0.555556\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−0.939693	+	0.788496i	−0.939693	+	0.788496i
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−3.20574	−	1.85083i	−3.20574	−	1.85083i
$163$	0.592396	+	0.342020i	0.592396	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$163$	0.592396	+	0.342020i	0.592396	+	0.342020i	−0.173648	+	0.984808i	$0.555556\pi$
$164$			1.28558i			1.28558i
$165$		0			0
$166$		0			0
$167$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$167$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$168$		0			0
$169$	0.173648	−	0.984808i	0.173648	−	0.984808i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$173$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$174$		−	3.70167i		−	3.70167i
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−0.939693	−	1.62760i	−0.939693	−	1.62760i	−0.766044	−	0.642788i	$-0.777778\pi$
$179$	−0.939693	−	1.62760i	−0.939693	−	1.62760i	−0.173648	−	0.984808i	$-0.555556\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	0.386659	−	0.223238i	0.386659	−	0.223238i
$185$		0			0
$186$	−0.826352	+	1.43128i	−0.826352	+	1.43128i
$187$		0			0
$188$	0.726682	+	0.419550i	0.726682	+	0.419550i
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	−0.213011	−	0.368946i	−0.213011	−	0.368946i
$193$	1.11334	−	0.642788i	1.11334	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$193$	1.11334	−	0.642788i	1.11334	−	0.642788i	0.939693	+	0.342020i	$0.111111\pi$
$194$		0			0
$195$		0			0
$196$	0.652704			0.652704
$197$	0.592396	−	0.342020i	0.592396	−	0.342020i	−0.173648	−	0.984808i	$-0.555556\pi$
$197$	0.592396	−	0.342020i	0.592396	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$198$		0			0
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$		−	0.446476i		−	0.446476i
$201$		0			0
$202$	−1.11334	−	0.642788i	−1.11334	−	0.642788i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	2.53209			2.53209
$208$	1.15270	+	0.419550i	1.15270	+	0.419550i
$209$		0			0
$210$		0			0
$211$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$211$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$212$		0			0
$213$			3.70167i			3.70167i
$214$		0			0
$215$		0			0
$216$			1.28558i			1.28558i
$217$		0			0
$218$		0			0
$219$	−3.20574	+	1.85083i	−3.20574	+	1.85083i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$223$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$224$		0			0
$225$	1.26604	−	2.19285i	1.26604	−	2.19285i
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$	−0.592396	+	0.342020i	−0.592396	+	0.342020i
$233$	1.87939			1.87939			0.939693	−	0.342020i	$-0.111111\pi$
$233$	1.87939			1.87939			0.939693	+	0.342020i	$0.111111\pi$
$234$	2.09240	+	2.49362i	2.09240	+	2.49362i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			0.684040i			0.684040i	0.939693	+	0.342020i	$0.111111\pi$
$239$			0.684040i			0.684040i	−0.939693	+	0.342020i	$0.888889\pi$
$240$		0			0
$241$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$241$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$242$		−	1.28558i		−	1.28558i
$243$	−1.26604	+	2.19285i	−1.26604	+	2.19285i
$244$		0			0
$245$		0			0
$246$	4.75877			4.75877
$247$		0			0
$248$	0.305407			0.305407
$249$		0			0
$250$		0			0
$251$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$251$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$252$		0			0
$253$		0			0
$254$	0.386659	+	0.223238i	0.386659	+	0.223238i
$255$		0			0
$256$	−0.652704	+	1.13052i	−0.652704	+	1.13052i
$257$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$257$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−3.87939			−3.87939
$262$	0.386659	−	0.223238i	0.386659	−	0.223238i
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$269$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$270$		0			0
$271$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$271$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	0.613341	+	1.06234i	0.613341	+	1.06234i
$277$	0.173648	−	0.300767i	0.173648	−	0.300767i	−0.766044	−	0.642788i	$-0.777778\pi$
$277$	0.173648	−	0.300767i	0.173648	−	0.300767i	0.939693	+	0.342020i	$0.111111\pi$
$278$		−	1.96962i		−	1.96962i
$279$	1.50000	+	0.866025i	1.50000	+	0.866025i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	1.55303	−	2.68993i	1.55303	−	2.68993i
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$	−1.11334	+	0.642788i	−1.11334	+	0.642788i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−2.47906	+	1.43128i	−2.47906	+	1.43128i
$289$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$	−1.11334	−	0.642788i	−1.11334	−	0.642788i
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$		−	2.41609i		−	2.41609i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$300$	1.22668			1.22668
$301$		0			0
$302$	1.26604	+	2.19285i	1.26604	+	2.19285i
$303$	−0.939693	+	1.62760i	−0.939693	+	1.62760i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$307$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$311$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$312$	0.286989	−	0.788496i	0.286989	−	0.788496i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$317$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−1.87939			−1.87939
$325$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$326$	0.879385			0.879385
$327$		0			0
$328$	−0.439693	−	0.761570i	−0.439693	−	0.761570i
$329$		0			0
$330$		0			0
$331$	1.11334	+	0.642788i	1.11334	+	0.642788i	0.939693	−	0.342020i	$-0.111111\pi$
$331$	1.11334	+	0.642788i	1.11334	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$332$		0			0
$333$		0			0
$334$	−1.11334	+	1.92836i	−1.11334	+	1.92836i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−0.439693	−	1.20805i	−0.439693	−	1.20805i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		−	1.28558i		−	1.28558i
$347$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$347$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$348$	−0.939693	−	1.62760i	−0.939693	−	1.62760i
$349$	−0.592396	+	0.342020i	−0.592396	+	0.342020i	−0.766044	−	0.642788i	$-0.777778\pi$
$349$	−0.592396	+	0.342020i	−0.592396	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$350$		0			0
$351$	2.20574	−	1.85083i	2.20574	−	1.85083i
$352$		0			0
$353$	−0.592396	+	0.342020i	−0.592396	+	0.342020i	−0.766044	−	0.642788i	$-0.777778\pi$
$353$	−0.592396	+	0.342020i	−0.592396	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−2.09240	−	1.20805i	−2.09240	−	1.20805i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	0.500000	+	0.866025i	0.500000	+	0.866025i
$362$		0			0
$363$	−1.87939			−1.87939
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$	0.613341	−	1.06234i	0.613341	−	1.06234i
$369$		−	4.98724i		−	4.98724i
$370$		0			0
$371$		0			0
$372$			0.839100i			0.839100i
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$		0			0
$376$	−0.573978			−0.573978
$377$	1.43969	+	0.524005i	1.43969	+	0.524005i
$378$		0			0
$379$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$379$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$380$		0			0
$381$	0.326352	−	0.565258i	0.326352	−	0.565258i
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$	1.36571	+	0.788496i	1.36571	+	0.788496i
$385$		0			0
$386$	0.826352	−	1.43128i	0.826352	−	1.43128i
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	−0.386659	+	0.223238i	−0.386659	+	0.223238i
$393$	−0.326352	−	0.565258i	−0.326352	−	0.565258i
$394$	0.439693	−	0.761570i	0.439693	−	0.761570i
$395$		0			0
$396$		0			0
$397$	−1.70574	−	0.984808i	−1.70574	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$397$	−1.70574	−	0.984808i	−1.70574	−	0.984808i	−0.766044	−	0.642788i	$-0.777778\pi$
$398$		0			0
$399$		0			0
$400$	−0.613341	−	1.06234i	−0.613341	−	1.06234i
$401$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$401$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$402$		0			0
$403$	−0.439693	−	0.524005i	−0.439693	−	0.524005i
$404$	−0.652704			−0.652704
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.592396	−	0.342020i	−0.592396	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$409$	−0.592396	−	0.342020i	−0.592396	−	0.342020i	−0.766044	+	0.642788i	$0.777778\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	2.81908	−	1.62760i	2.81908	−	1.62760i
$415$		0			0
$416$	1.11334	−	0.196312i	1.11334	−	0.196312i
$417$	−2.87939			−2.87939
$418$		0			0
$419$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$419$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$	1.11334	+	0.642788i	1.11334	+	0.642788i
$423$	−2.81908	−	1.62760i	−2.81908	−	1.62760i
$424$		0			0
$425$		0			0
$426$	2.37939	+	4.12122i	2.37939	+	4.12122i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$431$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$432$	1.76604	+	3.05888i	1.76604	+	3.05888i
$433$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$433$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−2.37939	+	4.12122i	−2.37939	+	4.12122i
$439$	0.766044	+	1.32683i	0.766044	+	1.32683i	0.939693	+	0.342020i	$0.111111\pi$
$439$	0.766044	+	1.32683i	0.766044	+	1.32683i	−0.173648	+	0.984808i	$0.555556\pi$
$440$		0			0
$441$	−2.53209			−2.53209
$442$		0			0
$443$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$443$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$444$		0			0
$445$		0			0
$446$	−1.11334	+	1.92836i	−1.11334	+	1.92836i
$447$		0			0
$448$		0			0
$449$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$449$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$450$		−	3.25519i		−	3.25519i
$451$		0			0
$452$		0			0
$453$	3.20574	−	1.85083i	3.20574	−	1.85083i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$457$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	0.592396	+	0.342020i	0.592396	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$461$	0.592396	+	0.342020i	0.592396	+	0.342020i	−0.173648	+	0.984808i	$0.555556\pi$
$462$		0			0
$463$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$463$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$464$	−0.939693	+	1.62760i	−0.939693	+	1.62760i
$465$		0			0
$466$	2.09240	−	1.20805i	2.09240	−	1.20805i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	1.55303	+	0.565258i	1.55303	+	0.565258i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	0.439693	+	0.761570i	0.439693	+	0.761570i
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.326352	−	0.565258i	−0.326352	−	0.565258i
$485$		0			0
$486$			3.25519i			3.25519i
$487$	−1.11334	−	0.642788i	−1.11334	−	0.642788i	−0.173648	−	0.984808i	$-0.555556\pi$
$487$	−1.11334	−	0.642788i	−1.11334	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$488$		0			0
$489$		−	1.28558i		−	1.28558i
$490$		0			0
$491$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$491$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$492$	2.09240	−	1.20805i	2.09240	−	1.20805i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.726682	−	0.419550i	0.726682	−	0.419550i
$497$		0			0
$498$		0			0
$499$			1.96962i			1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$499$			1.96962i			1.96962i	−0.173648	+	0.984808i	$0.555556\pi$
$500$		0			0
$501$	2.81908	+	1.62760i	2.81908	+	1.62760i
$502$		0			0
$503$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$503$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.76604	+	0.642788i	−1.76604	+	0.642788i
$508$	0.226682			0.226682
$509$	1.70574	−	0.984808i	1.70574	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$509$	1.70574	−	0.984808i	1.70574	−	0.984808i	0.939693	−	0.342020i	$-0.111111\pi$
$510$		0			0
$511$		0			0
$512$			0.839100i			0.839100i
$513$		0			0
$514$	−0.386659	−	0.223238i	−0.386659	−	0.223238i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−1.87939			−1.87939
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	−4.31908	+	2.49362i	−4.31908	+	2.49362i
$523$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$523$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$524$	0.113341	−	0.196312i	0.113341	−	0.196312i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.673648	+	1.85083i	−0.673648	+	1.85083i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−1.76604	+	3.05888i	−1.76604	+	3.05888i
$538$			0.446476i			0.446476i
$539$		0			0
$540$		0			0
$541$		−	1.28558i		−	1.28558i	−0.766044	−	0.642788i	$-0.777778\pi$
$541$		−	1.28558i		−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$547$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	−0.726682	−	0.419550i	−0.726682	−	0.419550i
$553$		0			0
$554$		−	0.446476i		−	0.446476i
$555$		0			0
$556$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$557$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$557$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$558$	2.22668			2.22668
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		−	1.57699i		−	1.57699i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	0.439693	−	0.761570i	0.439693	−	0.761570i
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.500000	+	0.866025i	0.500000	+	0.866025i
$576$	−0.286989	+	0.497079i	−0.286989	+	0.497079i
$577$			1.28558i			1.28558i	0.766044	+	0.642788i	$0.222222\pi$
$577$			1.28558i			1.28558i	−0.766044	+	0.642788i	$0.777778\pi$
$578$	−1.11334	−	0.642788i	−1.11334	−	0.642788i
$579$	−2.09240	−	1.20805i	−2.09240	−	1.20805i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.879385			0.879385
$585$		0			0
$586$		0			0
$587$	0.592396	−	0.342020i	0.592396	−	0.342020i	−0.173648	−	0.984808i	$-0.555556\pi$
$587$	0.592396	−	0.342020i	0.592396	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$588$	−0.613341	−	1.06234i	−0.613341	−	1.06234i
$589$		0			0
$590$		0			0
$591$	−1.11334	−	0.642788i	−1.11334	−	0.642788i
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−1.26604	+	0.223238i	−1.26604	+	0.223238i
$599$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$599$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$600$	−0.726682	+	0.419550i	−0.726682	+	0.419550i
$601$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$601$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$602$		0			0
$603$		0			0
$604$	1.11334	+	0.642788i	1.11334	+	0.642788i
$605$		0			0
$606$			2.41609i			2.41609i
$607$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$607$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	0.826352	+	0.984808i	0.826352	+	0.984808i
$612$		0			0
$613$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$613$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$614$	−1.11334	−	1.92836i	−1.11334	−	1.92836i
$615$		0			0
$616$		0			0
$617$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$617$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$	−1.43969	−	2.49362i	−1.43969	−	2.49362i
$622$	0.386659	−	0.223238i	0.386659	−	0.223238i
$623$		0			0
$624$	−0.400330	−	2.27038i	−0.400330	−	2.27038i
$625$	1.00000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$631$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$632$		0			0
$633$	0.939693	−	1.62760i	0.939693	−	1.62760i
$634$	1.11334	+	1.92836i	1.11334	+	1.92836i
$635$		0			0
$636$		0			0
$637$	0.939693	+	0.342020i	0.939693	+	0.342020i
$638$		0			0
$639$	4.31908	−	2.49362i	4.31908	−	2.49362i
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$643$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.766044	+	1.32683i	0.766044	+	1.32683i	0.939693	+	0.342020i	$0.111111\pi$
$647$	0.766044	+	1.32683i	0.766044	+	1.32683i	−0.173648	+	0.984808i	$0.555556\pi$
$648$	1.11334	−	0.642788i	1.11334	−	0.642788i
$649$		0			0
$650$	−0.439693	+	1.20805i	−0.439693	+	1.20805i
$651$		0			0
$652$	0.386659	−	0.223238i	0.386659	−	0.223238i
$653$	0.173648	+	0.300767i	0.173648	+	0.300767i	0.939693	−	0.342020i	$-0.111111\pi$
$653$	0.173648	+	0.300767i	0.173648	+	0.300767i	−0.766044	+	0.642788i	$0.777778\pi$
$654$		0			0
$655$		0			0
$656$	−2.09240	−	1.20805i	−2.09240	−	1.20805i
$657$	4.31908	+	2.49362i	4.31908	+	2.49362i
$658$		0			0
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$661$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$662$	1.65270			1.65270
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	0.766044	−	1.32683i	0.766044	−	1.32683i
$668$			1.13052i			1.13052i
$669$	2.81908	+	1.62760i	2.81908	+	1.62760i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$673$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$674$		0			0
$675$	−2.87939			−2.87939
$676$	−0.500000	−	0.419550i	−0.500000	−	0.419550i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.592396	−	0.342020i	−0.592396	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$683$	−0.592396	−	0.342020i	−0.592396	−	0.342020i	−0.766044	+	0.642788i	$0.777778\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$691$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$692$	−0.326352	−	0.565258i	−0.326352	−	0.565258i
$693$		0			0
$694$		−	1.28558i		−	1.28558i
$695$		0			0
$696$	1.11334	+	0.642788i	1.11334	+	0.642788i
$697$		0			0
$698$	−0.439693	+	0.761570i	−0.439693	+	0.761570i
$699$	−1.76604	−	3.05888i	−1.76604	−	3.05888i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	1.26604	−	3.47843i	1.26604	−	3.47843i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.439693	+	0.761570i	−0.439693	+	0.761570i
$707$		0			0
$708$		0			0
$709$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$709$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.592396	+	0.342020i	−0.592396	+	0.342020i
$714$		0			0
$715$		0			0
$716$	−1.22668			−1.22668
$717$	1.11334	−	0.642788i	1.11334	−	0.642788i
$718$		0			0
$719$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$719$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	1.11334	+	0.642788i	1.11334	+	0.642788i
$723$		0			0
$724$		0			0
$725$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$726$	−2.09240	+	1.20805i	−2.09240	+	1.20805i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	1.87939			1.87939
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$		−	1.13052i		−	1.13052i
$737$		0			0
$738$	−3.20574	−	5.55250i	−3.20574	−	5.55250i
$739$	−1.11334	+	0.642788i	−1.11334	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$739$	−1.11334	+	0.642788i	−1.11334	+	0.642788i	−0.173648	+	0.984808i	$0.555556\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$743$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$744$	−0.286989	−	0.497079i	−0.286989	−	0.497079i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$	−1.36571	+	0.788496i	−1.36571	+	0.788496i
$753$		0			0
$754$	1.93969	−	0.342020i	1.93969	−	0.342020i
$755$		0			0
$756$		0			0
$757$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$757$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.11334	−	0.642788i	−1.11334	−	0.642788i	−0.173648	−	0.984808i	$-0.555556\pi$
$761$	−1.11334	−	0.642788i	−1.11334	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$762$		−	0.839100i		−	0.839100i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	2.45336			2.45336
$769$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$769$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$770$		0			0
$771$	−0.326352	+	0.565258i	−0.326352	+	0.565258i
$772$		−	0.839100i		−	0.839100i
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$			0.684040i			0.684040i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	2.20574	+	3.82045i	2.20574	+	3.82045i
$784$	−0.613341	+	1.06234i	−0.613341	+	1.06234i
$785$		0			0
$786$	−0.726682	−	0.419550i	−0.726682	−	0.419550i
$787$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$787$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$788$		−	0.446476i		−	0.446476i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−2.53209			−2.53209
$795$		0			0
$796$		0			0
$797$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$797$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$798$		0			0
$799$		0			0
$800$	−0.979055	−	0.565258i	−0.979055	−	0.565258i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	−0.826352	−	0.300767i	−0.826352	−	0.300767i
$807$	0.652704			0.652704
$808$	0.386659	−	0.223238i	0.386659	−	0.223238i
$809$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$809$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$810$		0			0
$811$			1.28558i			1.28558i	0.766044	+	0.642788i	$0.222222\pi$
$811$			1.28558i			1.28558i	−0.766044	+	0.642788i	$0.777778\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.879385			−0.879385
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	−0.939693	+	1.62760i	−0.939693	+	1.62760i	−0.173648	+	0.984808i	$0.555556\pi$
$823$	−0.939693	+	1.62760i	−0.939693	+	1.62760i	−0.766044	+	0.642788i	$0.777778\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	0.826352	−	1.43128i	0.826352	−	1.43128i
$829$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$829$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$	−0.652704			−0.652704
$832$	0.173648	−	0.145708i	0.173648	−	0.145708i
$833$		0			0
$834$	−3.20574	+	1.85083i	−3.20574	+	1.85083i
$835$		0			0
$836$		0			0
$837$		−	1.96962i		−	1.96962i
$838$		0			0
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	−0.673648	+	1.16679i	−0.673648	+	1.16679i
$842$		0			0
$843$		0			0
$844$	0.652704			0.652704
$845$		0			0
$846$	−4.18479			−4.18479
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	2.09240	+	1.20805i	2.09240	+	1.20805i
$853$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$853$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.347296			−0.347296			−0.173648	−	0.984808i	$-0.555556\pi$
$857$	−0.347296			−0.347296			−0.173648	+	0.984808i	$0.555556\pi$
$858$		0			0
$859$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$859$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1.96962i		−	1.96962i	−0.173648	−	0.984808i	$-0.555556\pi$
$863$		−	1.96962i		−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$864$	2.81908	+	1.62760i	2.81908	+	1.62760i
$865$		0			0
$866$		0			0
$867$	−0.939693	+	1.62760i	−0.939693	+	1.62760i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$			2.41609i			2.41609i
$877$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$877$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$878$	1.70574	+	0.984808i	1.70574	+	0.984808i
$879$		0			0
$880$		0			0
$881$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$881$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$882$	−2.81908	+	1.62760i	−2.81908	+	1.62760i
$883$	2.00000			2.00000			1.00000			$0$
$883$	2.00000			2.00000			1.00000			$0$
$884$		0			0
$885$		0			0
$886$	1.70574	−	0.984808i	1.70574	−	0.984808i
$887$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$887$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$			1.13052i			1.13052i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	0.326352	+	1.85083i	0.326352	+	1.85083i
$898$	2.22668			2.22668
$899$	0.907604	−	0.524005i	0.907604	−	0.524005i
$900$	−0.826352	−	1.43128i	−0.826352	−	1.43128i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	2.37939	−	4.12122i	2.37939	−	4.12122i
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$		0			0
$909$	2.53209			2.53209
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$919$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$	−2.81908	+	1.62760i	−2.81908	+	1.62760i
$922$	0.879385			0.879385
$923$	−1.93969	+	0.342020i	−1.93969	+	0.342020i
$924$		0			0
$925$		0			0
$926$	−1.11334	−	1.92836i	−1.11334	−	1.92836i
$927$		0			0
$928$			1.73205i			1.73205i
$929$	−1.70574	−	0.984808i	−1.70574	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$929$	−1.70574	−	0.984808i	−1.70574	−	0.984808i	−0.766044	−	0.642788i	$-0.777778\pi$
$930$		0			0
$931$		0			0
$932$	0.613341	−	1.06234i	0.613341	−	1.06234i
$933$	−0.326352	−	0.565258i	−0.326352	−	0.565258i
$934$		0			0
$935$		0			0
$936$	−1.11334	+	0.196312i	−1.11334	+	0.196312i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$	1.70574	+	0.984808i	1.70574	+	0.984808i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.70574	−	0.984808i	1.70574	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$947$	1.70574	−	0.984808i	1.70574	−	0.984808i	0.939693	−	0.342020i	$-0.111111\pi$
$948$		0			0
$949$	−1.26604	−	1.50881i	−1.26604	−	1.50881i
$950$		0			0
$951$	2.81908	−	1.62760i	2.81908	−	1.62760i
$952$		0			0
$953$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$953$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$954$		0			0
$955$		0			0
$956$	0.386659	+	0.223238i	0.386659	+	0.223238i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.532089			0.532089
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$			0.684040i			0.684040i	0.939693	+	0.342020i	$0.111111\pi$
$967$			0.684040i			0.684040i	−0.939693	+	0.342020i	$0.888889\pi$
$968$	0.386659	+	0.223238i	0.386659	+	0.223238i
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$	0.826352	+	1.43128i	0.826352	+	1.43128i
$973$		0			0
$974$	−1.65270			−1.65270
$975$	1.76604	+	0.642788i	1.76604	+	0.642788i
$976$		0			0
$977$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$977$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$978$	−0.826352	−	1.43128i	−0.826352	−	1.43128i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−0.386659	−	0.223238i	−0.386659	−	0.223238i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	−0.826352	+	1.43128i	−0.826352	+	1.43128i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$991$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$992$	0.386659	−	0.669713i	0.386659	−	0.669713i
$993$		−	2.41609i		−	2.41609i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$997$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$998$	1.26604	+	2.19285i	1.26604	+	2.19285i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	299.1.j.a.114.3	✓	6
3.2	odd	2		2691.1.bn.a.712.1		6
13.2	odd	12		3887.1.d.g.2874.5		6
13.3	even	3		3887.1.c.c.3886.5		6
13.4	even	6	inner	299.1.j.a.160.3	yes	6
13.5	odd	4		3887.1.h.g.3357.2		12
13.6	odd	12		3887.1.h.g.22.2		12
13.7	odd	12		3887.1.h.g.22.5		12
13.8	odd	4		3887.1.h.g.3357.5		12
13.9	even	3		3887.1.j.f.2851.1		6
13.10	even	6		3887.1.c.c.3886.2		6
13.11	odd	12		3887.1.d.g.2874.2		6
13.12	even	2		3887.1.j.f.3403.1		6
23.22	odd	2	CM	299.1.j.a.114.3	✓	6
39.17	odd	6		2691.1.bn.a.1954.1		6
69.68	even	2		2691.1.bn.a.712.1		6
299.22	odd	6		3887.1.j.f.2851.1		6
299.45	even	12		3887.1.h.g.22.2		12
299.68	odd	6		3887.1.c.c.3886.5		6
299.114	odd	6		3887.1.c.c.3886.2		6
299.137	even	12		3887.1.h.g.22.5		12
299.160	odd	6	inner	299.1.j.a.160.3	yes	6
299.206	even	12		3887.1.d.g.2874.2		6
299.229	even	4		3887.1.h.g.3357.5		12
299.252	even	4		3887.1.h.g.3357.2		12
299.275	even	12		3887.1.d.g.2874.5		6
299.298	odd	2		3887.1.j.f.3403.1		6
897.758	even	6		2691.1.bn.a.1954.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
299.1.j.a.114.3	✓	6	1.1	even	1	trivial
299.1.j.a.114.3	✓	6	23.22	odd	2	CM
299.1.j.a.160.3	yes	6	13.4	even	6	inner
299.1.j.a.160.3	yes	6	299.160	odd	6	inner
2691.1.bn.a.712.1		6	3.2	odd	2
2691.1.bn.a.712.1		6	69.68	even	2
2691.1.bn.a.1954.1		6	39.17	odd	6
2691.1.bn.a.1954.1		6	897.758	even	6
3887.1.c.c.3886.2		6	13.10	even	6
3887.1.c.c.3886.2		6	299.114	odd	6
3887.1.c.c.3886.5		6	13.3	even	3
3887.1.c.c.3886.5		6	299.68	odd	6
3887.1.d.g.2874.2		6	13.11	odd	12
3887.1.d.g.2874.2		6	299.206	even	12
3887.1.d.g.2874.5		6	13.2	odd	12
3887.1.d.g.2874.5		6	299.275	even	12
3887.1.h.g.22.2		12	13.6	odd	12
3887.1.h.g.22.2		12	299.45	even	12
3887.1.h.g.22.5		12	13.7	odd	12
3887.1.h.g.22.5		12	299.137	even	12
3887.1.h.g.3357.2		12	13.5	odd	4
3887.1.h.g.3357.2		12	299.252	even	4
3887.1.h.g.3357.5		12	13.8	odd	4
3887.1.h.g.3357.5		12	299.229	even	4
3887.1.j.f.2851.1		6	13.9	even	3
3887.1.j.f.2851.1		6	299.22	odd	6
3887.1.j.f.3403.1		6	13.12	even	2
3887.1.j.f.3403.1		6	299.298	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 299 = 13 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	299.j (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.11334 - 0.642788i) q^{2} +(-0.939693 - 1.62760i) q^{3} +(0.326352 - 0.565258i) q^{4} +(-2.09240 - 1.20805i) q^{6} +0.446476i q^{8} +(-1.26604 + 2.19285i) q^{9} +O(q^{10})\)	\(q+(1.11334 - 0.642788i) q^{2} +(-0.939693 - 1.62760i) q^{3} +(0.326352 - 0.565258i) q^{4} +(-2.09240 - 1.20805i) q^{6} +0.446476i q^{8} +(-1.26604 + 2.19285i) q^{9} -1.22668 q^{12} +(0.766044 - 0.642788i) q^{13} +(0.613341 + 1.06234i) q^{16} +3.25519i q^{18} +(-0.500000 - 0.866025i) q^{23} +(0.726682 - 0.419550i) q^{24} -1.00000 q^{25} +(0.439693 - 1.20805i) q^{26} +2.87939 q^{27} +(0.766044 + 1.32683i) q^{29} -0.684040i q^{31} +(0.979055 + 0.565258i) q^{32} +(0.826352 + 1.43128i) q^{36} +(-1.76604 - 0.642788i) q^{39} +(-1.70574 + 0.984808i) q^{41} +(-1.11334 - 0.642788i) q^{46} +1.28558i q^{47} +(1.15270 - 1.99654i) q^{48} +(0.500000 + 0.866025i) q^{49} +(-1.11334 + 0.642788i) q^{50} +(-0.113341 - 0.642788i) q^{52} +(3.20574 - 1.85083i) q^{54} +(1.70574 + 0.984808i) q^{58} +(-0.439693 - 0.761570i) q^{62} +0.226682 q^{64} +(-0.939693 + 1.62760i) q^{69} +(-1.70574 - 0.984808i) q^{71} +(-0.979055 - 0.565258i) q^{72} -1.96962i q^{73} +(0.939693 + 1.62760i) q^{75} +(-2.37939 + 0.419550i) q^{78} +(-1.43969 - 2.49362i) q^{81} +(-1.26604 + 2.19285i) q^{82} +(1.43969 - 2.49362i) q^{87} -0.652704 q^{92} +(-1.11334 + 0.642788i) q^{93} +(0.826352 + 1.43128i) q^{94} -2.12467i q^{96} +(1.11334 + 0.642788i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{4} - 9 q^{6} - 3 q^{9}+O(q^{10}) \) 6 * q + 3 * q^4 - 9 * q^6 - 3 * q^9	\( 6 q + 3 q^{4} - 9 q^{6} - 3 q^{9} + 6 q^{12} - 3 q^{16} - 3 q^{23} - 9 q^{24} - 6 q^{25} - 3 q^{26} + 6 q^{27} + 9 q^{32} + 6 q^{36} - 6 q^{39} + 9 q^{48} + 3 q^{49} + 6 q^{52} + 9 q^{54} + 3 q^{62} - 12 q^{64} - 9 q^{72} - 3 q^{78} - 3 q^{81} - 3 q^{82} + 3 q^{87} - 6 q^{92} + 6 q^{94}+O(q^{100}) \) 6 * q + 3 * q^4 - 9 * q^6 - 3 * q^9 + 6 * q^12 - 3 * q^16 - 3 * q^23 - 9 * q^24 - 6 * q^25 - 3 * q^26 + 6 * q^27 + 9 * q^32 + 6 * q^36 - 6 * q^39 + 9 * q^48 + 3 * q^49 + 6 * q^52 + 9 * q^54 + 3 * q^62 - 12 * q^64 - 9 * q^72 - 3 * q^78 - 3 * q^81 - 3 * q^82 + 3 * q^87 - 6 * q^92 + 6 * q^94

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