LMFDB - Embedded newform 385.1.q.c.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [385,1,Mod(109,385)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(385, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4, 3]))

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

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

chi := DirichletCharacter("385.109");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 385 = 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	385.q (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:	$0.192140029864$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.0.79893275.1

Embedding invariants

Embedding label			109.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	385.109
Dual form			385.1.q.c.219.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 - 1.50000i) q^{2} +(-1.00000 + 1.73205i) q^{4} +(0.500000 + 0.866025i) q^{5} +1.00000i q^{7} +1.73205 q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(-0.866025 - 1.50000i) q^{2} +(-1.00000 + 1.73205i) q^{4} +(0.500000 + 0.866025i) q^{5} +1.00000i q^{7} +1.73205 q^{8} +(-0.500000 - 0.866025i) q^{9} +(0.866025 - 1.50000i) q^{10} +(0.500000 - 0.866025i) q^{11} +1.73205 q^{13} +(1.50000 - 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-0.866025 + 1.50000i) q^{18} -2.00000 q^{20} -1.73205 q^{22} +(-0.500000 + 0.866025i) q^{25} +(-1.50000 - 2.59808i) q^{26} +(-1.73205 - 1.00000i) q^{28} +(-0.500000 + 0.866025i) q^{31} +(-0.866025 + 0.500000i) q^{35} +2.00000 q^{36} +(0.866025 + 1.50000i) q^{40} -1.73205 q^{43} +(1.00000 + 1.73205i) q^{44} +(0.500000 - 0.866025i) q^{45} -1.00000 q^{49} +1.73205 q^{50} +(-1.73205 + 3.00000i) q^{52} +1.00000 q^{55} +1.73205i q^{56} +(0.500000 - 0.866025i) q^{59} +1.73205 q^{62} +(0.866025 - 0.500000i) q^{63} -1.00000 q^{64} +(0.866025 + 1.50000i) q^{65} +(1.50000 + 0.866025i) q^{70} -1.00000 q^{71} +(-0.866025 - 1.50000i) q^{72} +(0.866025 - 1.50000i) q^{73} +(0.866025 + 0.500000i) q^{77} +(0.500000 - 0.866025i) q^{80} +(-0.500000 + 0.866025i) q^{81} -1.73205 q^{83} +(1.50000 + 2.59808i) q^{86} +(0.866025 - 1.50000i) q^{88} +(-0.500000 - 0.866025i) q^{89} -1.73205 q^{90} +1.73205i q^{91} +(0.866025 + 1.50000i) q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 2 q^{5} - 2 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 + 2 * q^5 - 2 * q^9	$ 4 q - 4 q^{4} + 2 q^{5} - 2 q^{9} + 2 q^{11} + 6 q^{14} - 2 q^{16} - 8 q^{20} - 2 q^{25} - 6 q^{26} - 2 q^{31} + 8 q^{36} + 4 q^{44} + 2 q^{45} - 4 q^{49} + 4 q^{55} + 2 q^{59} - 4 q^{64} + 6 q^{70} - 4 q^{71} + 2 q^{80} - 2 q^{81} + 6 q^{86} - 2 q^{89} - 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 + 2 * q^5 - 2 * q^9 + 2 * q^11 + 6 * q^14 - 2 * q^16 - 8 * q^20 - 2 * q^25 - 6 * q^26 - 2 * q^31 + 8 * q^36 + 4 * q^44 + 2 * q^45 - 4 * q^49 + 4 * q^55 + 2 * q^59 - 4 * q^64 + 6 * q^70 - 4 * q^71 + 2 * q^80 - 2 * q^81 + 6 * q^86 - 2 * q^89 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/385\mathbb{Z}\right)^\times$.

$n$	$211$	$232$	$276$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$2$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$3$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$3$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$4$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$5$	0.500000	+	0.866025i	0.500000	+	0.866025i
$5$	0.500000	+	0.866025i	0.500000	+	0.866025i
$6$		0			0
$7$			1.00000i			1.00000i
$7$			1.00000i			1.00000i
$8$	1.73205			1.73205
$9$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$10$	0.866025	−	1.50000i	0.866025	−	1.50000i
$11$	0.500000	−	0.866025i	0.500000	−	0.866025i
$11$	0.500000	−	0.866025i	0.500000	−	0.866025i
$12$		0			0
$13$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$13$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$14$	1.50000	−	0.866025i	1.50000	−	0.866025i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$19$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$19$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$20$	−2.00000			−2.00000
$21$		0			0
$22$	−1.73205			−1.73205
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$27$		0			0
$28$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$31$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$36$	2.00000			2.00000
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$		0			0
$39$		0			0
$40$	0.866025	+	1.50000i	0.866025	+	1.50000i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$43$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$44$	1.00000	+	1.73205i	1.00000	+	1.73205i
$45$	0.500000	−	0.866025i	0.500000	−	0.866025i
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$		0			0
$49$	−1.00000			−1.00000
$50$	1.73205			1.73205
$51$		0			0
$52$	−1.73205	+	3.00000i	−1.73205	+	3.00000i
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$	1.00000			1.00000
$56$			1.73205i			1.73205i
$57$		0			0
$58$		0			0
$59$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$59$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$	1.73205			1.73205
$63$	0.866025	−	0.500000i	0.866025	−	0.500000i
$64$	−1.00000			−1.00000
$65$	0.866025	+	1.50000i	0.866025	+	1.50000i
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$		0			0
$70$	1.50000	+	0.866025i	1.50000	+	0.866025i
$71$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$71$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$72$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$73$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$73$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.866025	+	0.500000i	0.866025	+	0.500000i
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$	0.500000	−	0.866025i	0.500000	−	0.866025i
$81$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$82$		0			0
$83$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$83$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$84$		0			0
$85$		0			0
$86$	1.50000	+	2.59808i	1.50000	+	2.59808i
$87$		0			0
$88$	0.866025	−	1.50000i	0.866025	−	1.50000i
$89$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$89$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$90$	−1.73205			−1.73205
$91$			1.73205i			1.73205i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	0.866025	+	1.50000i	0.866025	+	1.50000i
$99$	−1.00000			−1.00000
$100$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$	3.00000			3.00000
$105$		0			0
$106$		0			0
$107$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$107$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$108$		0			0
$109$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$109$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$110$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$111$		0			0
$112$	0.866025	−	0.500000i	0.866025	−	0.500000i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$118$	−1.73205			−1.73205
$119$		0			0
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$125$	−1.00000			−1.00000
$126$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	0.866025	+	1.50000i	0.866025	+	1.50000i
$129$		0			0
$130$	1.50000	−	2.59808i	1.50000	−	2.59808i
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		−	2.00000i		−	2.00000i
$141$		0			0
$142$	0.866025	+	1.50000i	0.866025	+	1.50000i
$143$	0.866025	−	1.50000i	0.866025	−	1.50000i
$144$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$145$		0			0
$146$	−3.00000			−3.00000
$147$		0			0
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$		0			0
$154$		−	1.73205i		−	1.73205i
$155$	−1.00000			−1.00000
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	1.73205			1.73205
$163$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$163$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$164$		0			0
$165$		0			0
$166$	1.50000	+	2.59808i	1.50000	+	2.59808i
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	2.00000			2.00000
$170$		0			0
$171$		0			0
$172$	1.73205	−	3.00000i	1.73205	−	3.00000i
$173$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$173$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$174$		0			0
$175$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$176$	−1.00000			−1.00000
$177$		0			0
$178$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$179$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$179$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$180$	1.00000	+	1.73205i	1.00000	+	1.73205i
$181$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$181$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$182$	2.59808	−	1.50000i	2.59808	−	1.50000i
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$191$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$192$		0			0
$193$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$193$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$194$		0			0
$195$		0			0
$196$	1.00000	−	1.73205i	1.00000	−	1.73205i
$197$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$197$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$198$	0.866025	+	1.50000i	0.866025	+	1.50000i
$199$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$199$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$200$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$216$		0			0
$217$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$218$		0			0
$219$		0			0
$220$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	1.00000			1.00000
$226$		0			0
$227$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$227$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$229$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$233$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$234$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$235$		0			0
$236$	1.00000	+	1.73205i	1.00000	+	1.73205i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$243$		0			0
$244$		0			0
$245$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$246$		0			0
$247$		0			0
$248$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$249$		0			0
$250$	0.866025	+	1.50000i	0.866025	+	1.50000i
$251$	2.00000			2.00000			1.00000			$0$
$251$	2.00000			2.00000			1.00000			$0$
$252$			2.00000i			2.00000i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1.00000	−	1.73205i	1.00000	−	1.73205i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$		0			0
$260$	−3.46410			−3.46410
$261$		0			0
$262$		0			0
$263$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$263$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$269$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$270$		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
$275$	0.500000	+	0.866025i	0.500000	+	0.866025i
$276$		0			0
$277$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$277$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$278$		0			0
$279$	1.00000			1.00000
$280$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$283$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$284$	1.00000	−	1.73205i	1.00000	−	1.73205i
$285$		0			0
$286$	−3.00000			−3.00000
$287$		0			0
$288$		0			0
$289$	0.500000	+	0.866025i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$	1.73205	+	3.00000i	1.73205	+	3.00000i
$293$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$293$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$294$		0			0
$295$	1.00000			1.00000
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		−	1.73205i		−	1.73205i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$307$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$308$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$309$		0			0
$310$	0.866025	+	1.50000i	0.866025	+	1.50000i
$311$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$311$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$		0			0
$315$	0.866025	+	0.500000i	0.866025	+	0.500000i
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$		0			0
$320$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$325$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$331$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$332$	1.73205	−	3.00000i	1.73205	−	3.00000i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$337$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$338$	−1.73205	−	3.00000i	−1.73205	−	3.00000i
$339$		0			0
$340$		0			0
$341$	0.500000	+	0.866025i	0.500000	+	0.866025i
$342$		0			0
$343$		−	1.00000i		−	1.00000i
$344$	−3.00000			−3.00000
$345$		0			0
$346$	1.50000	−	2.59808i	1.50000	−	2.59808i
$347$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$347$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$			1.73205i			1.73205i
$351$		0			0
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$356$	2.00000			2.00000
$357$		0			0
$358$	1.73205			1.73205
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$	0.866025	−	1.50000i	0.866025	−	1.50000i
$361$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$362$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$363$		0			0
$364$	−3.00000	−	1.73205i	−3.00000	−	1.73205i
$365$	1.73205			1.73205
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$373$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$379$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$380$		0			0
$381$		0			0
$382$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$			1.00000i			1.00000i
$386$		0			0
$387$	0.866025	+	1.50000i	0.866025	+	1.50000i
$388$		0			0
$389$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$389$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$390$		0			0
$391$		0			0
$392$	−1.73205			−1.73205
$393$		0			0
$394$	1.50000	+	2.59808i	1.50000	+	2.59808i
$395$		0			0
$396$	1.00000	−	1.73205i	1.00000	−	1.73205i
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$	−1.73205			−1.73205
$399$		0			0
$400$	1.00000			1.00000
$401$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$401$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$402$		0			0
$403$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$404$		0			0
$405$	−1.00000			−1.00000
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	0.866025	+	0.500000i	0.866025	+	0.500000i
$414$		0			0
$415$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$419$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$420$		0			0
$421$	−2.00000			−2.00000			−1.00000			$\pi$
$421$	−2.00000			−2.00000			−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$			1.73205i			1.73205i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$	1.73205			1.73205
$441$	0.500000	+	0.866025i	0.500000	+	0.866025i
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$	0.500000	−	0.866025i	0.500000	−	0.866025i
$446$		0			0
$447$		0			0
$448$		−	1.00000i		−	1.00000i
$449$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$449$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$450$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	3.00000			3.00000
$455$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$456$		0			0
$457$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$457$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$458$	0.866025	−	1.50000i	0.866025	−	1.50000i
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$467$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$468$	3.46410			3.46410
$469$		0			0
$470$		0			0
$471$		0			0
$472$	0.866025	−	1.50000i	0.866025	−	1.50000i
$473$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	2.00000			2.00000
$485$		0			0
$486$		0			0
$487$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$487$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$488$		0			0
$489$		0			0
$490$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$496$	1.00000			1.00000
$497$		−	1.00000i		−	1.00000i
$498$		0			0
$499$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$499$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$500$	1.00000	−	1.73205i	1.00000	−	1.73205i
$501$		0			0
$502$	−1.73205	−	3.00000i	−1.73205	−	3.00000i
$503$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$503$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$504$	1.50000	−	0.866025i	1.50000	−	0.866025i
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$509$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	1.50000	+	0.866025i	1.50000	+	0.866025i
$512$	−1.73205			−1.73205
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	1.50000	+	2.59808i	1.50000	+	2.59808i
$521$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$521$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$522$		0			0
$523$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$523$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$	−3.00000			−3.00000
$527$		0			0
$528$		0			0
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$		0			0
$531$	−1.00000			−1.00000
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	3.46410			3.46410
$539$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$540$		0			0
$541$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$541$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$547$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$548$		0			0
$549$		0			0
$550$	0.866025	−	1.50000i	0.866025	−	1.50000i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−3.00000			−3.00000
$555$		0			0
$556$		0			0
$557$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$557$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$558$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$559$	−3.00000			−3.00000
$560$	0.866025	+	0.500000i	0.866025	+	0.500000i
$561$		0			0
$562$		0			0
$563$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$563$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$568$	−1.73205			−1.73205
$569$		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		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$	1.73205	+	3.00000i	1.73205	+	3.00000i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.500000	+	0.866025i	0.500000	+	0.866025i
$577$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$577$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$578$	0.866025	−	1.50000i	0.866025	−	1.50000i
$579$		0			0
$580$		0			0
$581$		−	1.73205i		−	1.73205i
$582$		0			0
$583$		0			0
$584$	1.50000	−	2.59808i	1.50000	−	2.59808i
$585$	0.866025	−	1.50000i	0.866025	−	1.50000i
$586$	1.50000	+	2.59808i	1.50000	+	2.59808i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$591$		0			0
$592$		0			0
$593$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$593$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$599$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	−2.59808	+	1.50000i	−2.59808	+	1.50000i
$603$		0			0
$604$		0			0
$605$	0.500000	−	0.866025i	0.500000	−	0.866025i
$606$		0			0
$607$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$607$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$613$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$614$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$615$		0			0
$616$	1.50000	+	0.866025i	1.50000	+	0.866025i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$619$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$620$	1.00000	−	1.73205i	1.00000	−	1.73205i
$621$		0			0
$622$	−1.73205			−1.73205
$623$	0.866025	−	0.500000i	0.866025	−	0.500000i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		−	1.73205i		−	1.73205i
$631$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$631$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1.73205			−1.73205
$638$		0			0
$639$	0.500000	+	0.866025i	0.500000	+	0.866025i
$640$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$641$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$641$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$649$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$650$	3.00000			3.00000
$651$		0			0
$652$		0			0
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1.73205			−1.73205
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$661$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$662$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$663$		0			0
$664$	−3.00000			−3.00000
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$673$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$674$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$675$		0			0
$676$	−2.00000	+	3.46410i	−2.00000	+	3.46410i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	0.866025	−	1.50000i	0.866025	−	1.50000i
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$		0			0
$686$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$687$		0			0
$688$	0.866025	+	1.50000i	0.866025	+	1.50000i
$689$		0			0
$690$		0			0
$691$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$691$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$692$	−3.46410			−3.46410
$693$		−	1.00000i		−	1.00000i
$694$	3.00000			3.00000
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.73205	−	1.00000i	1.73205	−	1.00000i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$709$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$710$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$711$		0			0
$712$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$713$		0			0
$714$		0			0
$715$	1.73205			1.73205
$716$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$717$		0			0
$718$		0			0
$719$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$719$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$720$	−1.00000			−1.00000
$721$		0			0
$722$	1.73205			1.73205
$723$		0			0
$724$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$			3.00000i			3.00000i
$729$	1.00000			1.00000
$730$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$731$		0			0
$732$		0			0
$733$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$733$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$739$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$743$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$		0			0
$746$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$747$	0.866025	+	1.50000i	0.866025	+	1.50000i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$751$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	0.866025	+	1.50000i	0.866025	+	1.50000i
$759$		0			0
$760$		0			0
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$		0			0
$764$	2.00000			2.00000
$765$		0			0
$766$		0			0
$767$	0.866025	−	1.50000i	0.866025	−	1.50000i
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$	1.50000	−	0.866025i	1.50000	−	0.866025i
$771$		0			0
$772$		0			0
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$	1.50000	−	2.59808i	1.50000	−	2.59808i
$775$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$776$		0			0
$777$		0			0
$778$	−1.73205			−1.73205
$779$		0			0
$780$		0			0
$781$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$782$		0			0
$783$		0			0
$784$	0.500000	+	0.866025i	0.500000	+	0.866025i
$785$		0			0
$786$		0			0
$787$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$787$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$788$	1.73205	−	3.00000i	1.73205	−	3.00000i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−1.73205			−1.73205
$793$		0			0
$794$		0			0
$795$		0			0
$796$	1.00000	+	1.73205i	1.00000	+	1.73205i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$802$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$803$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$804$		0			0
$805$		0			0
$806$	3.00000			3.00000
$807$		0			0
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$	0.866025	+	1.50000i	0.866025	+	1.50000i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	1.50000	−	0.866025i	1.50000	−	0.866025i
$820$		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			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$		0			0
$826$		−	1.73205i		−	1.73205i
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$829$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$830$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$831$		0			0
$832$	−1.73205			−1.73205
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$839$	2.00000			2.00000			1.00000			$0$
$839$	2.00000			2.00000			1.00000			$0$
$840$		0			0
$841$	1.00000			1.00000
$842$	1.73205	+	3.00000i	1.73205	+	3.00000i
$843$		0			0
$844$		0			0
$845$	1.00000	+	1.73205i	1.00000	+	1.73205i
$846$		0			0
$847$	0.866025	−	0.500000i	0.866025	−	0.500000i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$857$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$859$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$860$	3.46410			3.46410
$861$		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
$865$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$866$		0			0
$867$		0			0
$868$	1.73205	−	1.00000i	1.73205	−	1.00000i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	1.00000i		−	1.00000i
$876$		0			0
$877$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$877$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$881$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$881$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$882$	0.866025	−	1.50000i	0.866025	−	1.50000i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$887$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$888$		0			0
$889$		0			0
$890$	−1.73205			−1.73205
$891$	0.500000	+	0.866025i	0.500000	+	0.866025i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−1.00000			−1.00000
$896$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$897$		0			0
$898$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$899$		0			0
$900$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.500000	+	0.866025i	0.500000	+	0.866025i
$906$		0			0
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$	−1.73205	−	3.00000i	−1.73205	−	3.00000i
$909$		0			0
$910$	2.59808	+	1.50000i	2.59808	+	1.50000i
$911$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$911$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$912$		0			0
$913$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$914$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$915$		0			0
$916$	−2.00000			−2.00000
$917$		0			0
$918$		0			0
$919$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$919$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.73205			−1.73205
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$929$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$937$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$937$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$	−1.00000			−1.00000
$945$		0			0
$946$	3.00000			3.00000
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$	1.50000	−	2.59808i	1.50000	−	2.59808i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	0.500000	−	0.866025i	0.500000	−	0.866025i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		0			0
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$967$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$968$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$969$		0			0
$970$		0			0
$971$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$971$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$	−1.00000			−1.00000
$980$	2.00000			2.00000
$981$		0			0
$982$		0			0
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$991$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$991$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$992$		0			0
$993$		0			0
$994$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$995$	1.00000			1.00000
$996$		0			0
$997$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$997$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$998$	0.866025	−	1.50000i	0.866025	−	1.50000i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	385.1.q.c.109.1	✓	4
3.2	odd	2		3465.1.cd.c.109.2		4
5.2	odd	4		1925.1.w.b.1726.1		2
5.3	odd	4		1925.1.w.a.1726.1		2
5.4	even	2	inner	385.1.q.c.109.2	yes	4
7.2	even	3	inner	385.1.q.c.219.1	yes	4
7.3	odd	6		2695.1.g.h.1814.2		2
7.4	even	3		2695.1.g.g.1814.2		2
7.5	odd	6		2695.1.q.e.2529.1		4
7.6	odd	2		2695.1.q.e.2419.1		4
11.10	odd	2	inner	385.1.q.c.109.2	yes	4
15.14	odd	2		3465.1.cd.c.109.1		4
21.2	odd	6		3465.1.cd.c.604.2		4
33.32	even	2		3465.1.cd.c.109.1		4
35.2	odd	12		1925.1.w.a.1451.1		2
35.4	even	6		2695.1.g.g.1814.1		2
35.9	even	6	inner	385.1.q.c.219.2	yes	4
35.19	odd	6		2695.1.q.e.2529.2		4
35.23	odd	12		1925.1.w.b.1451.1		2
35.24	odd	6		2695.1.g.h.1814.1		2
35.34	odd	2		2695.1.q.e.2419.2		4
55.32	even	4		1925.1.w.a.1726.1		2
55.43	even	4		1925.1.w.b.1726.1		2
55.54	odd	2	CM	385.1.q.c.109.1	✓	4
77.10	even	6		2695.1.g.h.1814.1		2
77.32	odd	6		2695.1.g.g.1814.1		2
77.54	even	6		2695.1.q.e.2529.2		4
77.65	odd	6	inner	385.1.q.c.219.2	yes	4
77.76	even	2		2695.1.q.e.2419.2		4
105.44	odd	6		3465.1.cd.c.604.1		4
165.164	even	2		3465.1.cd.c.109.2		4
231.65	even	6		3465.1.cd.c.604.1		4
385.54	even	6		2695.1.q.e.2529.1		4
385.109	odd	6		2695.1.g.g.1814.2		2
385.142	even	12		1925.1.w.b.1451.1		2
385.164	even	6		2695.1.g.h.1814.2		2
385.219	odd	6	inner	385.1.q.c.219.1	yes	4
385.373	even	12		1925.1.w.a.1451.1		2
385.384	even	2		2695.1.q.e.2419.1		4
1155.989	even	6		3465.1.cd.c.604.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.1.q.c.109.1	✓	4	1.1	even	1	trivial
385.1.q.c.109.1	✓	4	55.54	odd	2	CM
385.1.q.c.109.2	yes	4	5.4	even	2	inner
385.1.q.c.109.2	yes	4	11.10	odd	2	inner
385.1.q.c.219.1	yes	4	7.2	even	3	inner
385.1.q.c.219.1	yes	4	385.219	odd	6	inner
385.1.q.c.219.2	yes	4	35.9	even	6	inner
385.1.q.c.219.2	yes	4	77.65	odd	6	inner
1925.1.w.a.1451.1		2	35.2	odd	12
1925.1.w.a.1451.1		2	385.373	even	12
1925.1.w.a.1726.1		2	5.3	odd	4
1925.1.w.a.1726.1		2	55.32	even	4
1925.1.w.b.1451.1		2	35.23	odd	12
1925.1.w.b.1451.1		2	385.142	even	12
1925.1.w.b.1726.1		2	5.2	odd	4
1925.1.w.b.1726.1		2	55.43	even	4
2695.1.g.g.1814.1		2	35.4	even	6
2695.1.g.g.1814.1		2	77.32	odd	6
2695.1.g.g.1814.2		2	7.4	even	3
2695.1.g.g.1814.2		2	385.109	odd	6
2695.1.g.h.1814.1		2	35.24	odd	6
2695.1.g.h.1814.1		2	77.10	even	6
2695.1.g.h.1814.2		2	7.3	odd	6
2695.1.g.h.1814.2		2	385.164	even	6
2695.1.q.e.2419.1		4	7.6	odd	2
2695.1.q.e.2419.1		4	385.384	even	2
2695.1.q.e.2419.2		4	35.34	odd	2
2695.1.q.e.2419.2		4	77.76	even	2
2695.1.q.e.2529.1		4	7.5	odd	6
2695.1.q.e.2529.1		4	385.54	even	6
2695.1.q.e.2529.2		4	35.19	odd	6
2695.1.q.e.2529.2		4	77.54	even	6
3465.1.cd.c.109.1		4	15.14	odd	2
3465.1.cd.c.109.1		4	33.32	even	2
3465.1.cd.c.109.2		4	3.2	odd	2
3465.1.cd.c.109.2		4	165.164	even	2
3465.1.cd.c.604.1		4	105.44	odd	6
3465.1.cd.c.604.1		4	231.65	even	6
3465.1.cd.c.604.2		4	21.2	odd	6
3465.1.cd.c.604.2		4	1155.989	even	6

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:	\( N \)	\(=\)	\( 385 = 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	385.q (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 - 1.50000i) q^{2} +(-1.00000 + 1.73205i) q^{4} +(0.500000 + 0.866025i) q^{5} +1.00000i q^{7} +1.73205 q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.866025 - 1.50000i) q^{2} +(-1.00000 + 1.73205i) q^{4} +(0.500000 + 0.866025i) q^{5} +1.00000i q^{7} +1.73205 q^{8} +(-0.500000 - 0.866025i) q^{9} +(0.866025 - 1.50000i) q^{10} +(0.500000 - 0.866025i) q^{11} +1.73205 q^{13} +(1.50000 - 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-0.866025 + 1.50000i) q^{18} -2.00000 q^{20} -1.73205 q^{22} +(-0.500000 + 0.866025i) q^{25} +(-1.50000 - 2.59808i) q^{26} +(-1.73205 - 1.00000i) q^{28} +(-0.500000 + 0.866025i) q^{31} +(-0.866025 + 0.500000i) q^{35} +2.00000 q^{36} +(0.866025 + 1.50000i) q^{40} -1.73205 q^{43} +(1.00000 + 1.73205i) q^{44} +(0.500000 - 0.866025i) q^{45} -1.00000 q^{49} +1.73205 q^{50} +(-1.73205 + 3.00000i) q^{52} +1.00000 q^{55} +1.73205i q^{56} +(0.500000 - 0.866025i) q^{59} +1.73205 q^{62} +(0.866025 - 0.500000i) q^{63} -1.00000 q^{64} +(0.866025 + 1.50000i) q^{65} +(1.50000 + 0.866025i) q^{70} -1.00000 q^{71} +(-0.866025 - 1.50000i) q^{72} +(0.866025 - 1.50000i) q^{73} +(0.866025 + 0.500000i) q^{77} +(0.500000 - 0.866025i) q^{80} +(-0.500000 + 0.866025i) q^{81} -1.73205 q^{83} +(1.50000 + 2.59808i) q^{86} +(0.866025 - 1.50000i) q^{88} +(-0.500000 - 0.866025i) q^{89} -1.73205 q^{90} +1.73205i q^{91} +(0.866025 + 1.50000i) q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 2 q^{5} - 2 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 + 2 * q^5 - 2 * q^9	\( 4 q - 4 q^{4} + 2 q^{5} - 2 q^{9} + 2 q^{11} + 6 q^{14} - 2 q^{16} - 8 q^{20} - 2 q^{25} - 6 q^{26} - 2 q^{31} + 8 q^{36} + 4 q^{44} + 2 q^{45} - 4 q^{49} + 4 q^{55} + 2 q^{59} - 4 q^{64} + 6 q^{70} - 4 q^{71} + 2 q^{80} - 2 q^{81} + 6 q^{86} - 2 q^{89} - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 + 2 * q^5 - 2 * q^9 + 2 * q^11 + 6 * q^14 - 2 * q^16 - 8 * q^20 - 2 * q^25 - 6 * q^26 - 2 * q^31 + 8 * q^36 + 4 * q^44 + 2 * q^45 - 4 * q^49 + 4 * q^55 + 2 * q^59 - 4 * q^64 + 6 * q^70 - 4 * q^71 + 2 * q^80 - 2 * q^81 + 6 * q^86 - 2 * q^89 - 4 * q^99

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