LMFDB - Embedded newform 1155.1.e.d.1154.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1155,1,Mod(1154,1155)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1155, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))

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

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

chi := DirichletCharacter("1155.1154");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1155 = 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1155.e (of order $2$, degree $1$, 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:	$0.576420089591$
Analytic rank:	$0$
Dimension:	$4$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.0.46690875.1

Embedding invariants

Embedding label			1154.4
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1155.1154
Dual form			1155.1.e.d.1154.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{2} +1.00000i q^{3} -2.00000 q^{4} +(0.866025 - 0.500000i) q^{5} -1.73205 q^{6} +1.00000i q^{7} -1.73205i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.73205i q^{2} +1.00000i q^{3} -2.00000 q^{4} +(0.866025 - 0.500000i) q^{5} -1.73205 q^{6} +1.00000i q^{7} -1.73205i q^{8} -1.00000 q^{9} +(0.866025 + 1.50000i) q^{10} +1.00000 q^{11} -2.00000i q^{12} +1.00000i q^{13} -1.73205 q^{14} +(0.500000 + 0.866025i) q^{15} +1.00000 q^{16} -1.73205i q^{18} -1.73205 q^{19} +(-1.73205 + 1.00000i) q^{20} -1.00000 q^{21} +1.73205i q^{22} +1.73205 q^{24} +(0.500000 - 0.866025i) q^{25} -1.73205 q^{26} -1.00000i q^{27} -2.00000i q^{28} +1.00000 q^{29} +(-1.50000 + 0.866025i) q^{30} +1.00000i q^{33} +(0.500000 + 0.866025i) q^{35} +2.00000 q^{36} -1.73205i q^{37} -3.00000i q^{38} -1.00000 q^{39} +(-0.866025 - 1.50000i) q^{40} -1.73205i q^{42} -2.00000 q^{44} +(-0.866025 + 0.500000i) q^{45} +1.00000i q^{47} +1.00000i q^{48} -1.00000 q^{49} +(1.50000 + 0.866025i) q^{50} -2.00000i q^{52} +1.73205 q^{54} +(0.866025 - 0.500000i) q^{55} +1.73205 q^{56} -1.73205i q^{57} +1.73205i q^{58} +1.73205 q^{59} +(-1.00000 - 1.73205i) q^{60} -1.00000i q^{63} +1.00000 q^{64} +(0.500000 + 0.866025i) q^{65} -1.73205 q^{66} +1.73205i q^{67} +(-1.50000 + 0.866025i) q^{70} +1.73205i q^{72} -1.00000i q^{73} +3.00000 q^{74} +(0.866025 + 0.500000i) q^{75} +3.46410 q^{76} +1.00000i q^{77} -1.73205i q^{78} +(0.866025 - 0.500000i) q^{80} +1.00000 q^{81} +2.00000 q^{84} +1.00000i q^{87} -1.73205i q^{88} +(-0.866025 - 1.50000i) q^{90} -1.00000 q^{91} -1.73205 q^{94} +(-1.50000 + 0.866025i) q^{95} -1.73205i q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 4 q^{9}+O(q^{10}) $ 4 * q - 8 * q^4 - 4 * q^9	$ 4 q - 8 q^{4} - 4 q^{9} + 4 q^{11} + 2 q^{15} + 4 q^{16} - 4 q^{21} + 2 q^{25} + 4 q^{29} - 6 q^{30} + 2 q^{35} + 8 q^{36} - 4 q^{39} - 8 q^{44} - 4 q^{49} + 6 q^{50} - 4 q^{60} + 4 q^{64} + 2 q^{65} - 6 q^{70} + 12 q^{74} + 4 q^{81} + 8 q^{84} - 4 q^{91} - 6 q^{95} - 4 q^{99}+O(q^{100}) $ 4 * q - 8 * q^4 - 4 * q^9 + 4 * q^11 + 2 * q^15 + 4 * q^16 - 4 * q^21 + 2 * q^25 + 4 * q^29 - 6 * q^30 + 2 * q^35 + 8 * q^36 - 4 * q^39 - 8 * q^44 - 4 * q^49 + 6 * q^50 - 4 * q^60 + 4 * q^64 + 2 * q^65 - 6 * q^70 + 12 * q^74 + 4 * q^81 + 8 * q^84 - 4 * q^91 - 6 * q^95 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times$.

$n$	$211$	$232$	$386$	$661$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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 $

$2$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$2$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$3$			1.00000i			1.00000i
$3$			1.00000i			1.00000i
$4$	−2.00000			−2.00000
$5$	0.866025	−	0.500000i	0.866025	−	0.500000i
$5$	0.866025	−	0.500000i	0.866025	−	0.500000i
$6$	−1.73205			−1.73205
$7$			1.00000i			1.00000i
$7$			1.00000i			1.00000i
$8$		−	1.73205i		−	1.73205i
$9$	−1.00000			−1.00000
$10$	0.866025	+	1.50000i	0.866025	+	1.50000i
$11$	1.00000			1.00000
$11$	1.00000			1.00000
$12$		−	2.00000i		−	2.00000i
$13$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$13$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$14$	−1.73205			−1.73205
$15$	0.500000	+	0.866025i	0.500000	+	0.866025i
$16$	1.00000			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		−	1.73205i		−	1.73205i
$19$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$19$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$20$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$21$	−1.00000			−1.00000
$22$			1.73205i			1.73205i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	1.73205			1.73205
$25$	0.500000	−	0.866025i	0.500000	−	0.866025i
$26$	−1.73205			−1.73205
$27$		−	1.00000i		−	1.00000i
$28$		−	2.00000i		−	2.00000i
$29$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$29$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$30$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$			1.00000i			1.00000i
$34$		0			0
$35$	0.500000	+	0.866025i	0.500000	+	0.866025i
$36$	2.00000			2.00000
$37$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$37$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$38$		−	3.00000i		−	3.00000i
$39$	−1.00000			−1.00000
$40$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		−	1.73205i		−	1.73205i
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−2.00000			−2.00000
$45$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$46$		0			0
$47$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$47$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$48$			1.00000i			1.00000i
$49$	−1.00000			−1.00000
$50$	1.50000	+	0.866025i	1.50000	+	0.866025i
$51$		0			0
$52$		−	2.00000i		−	2.00000i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	1.73205			1.73205
$55$	0.866025	−	0.500000i	0.866025	−	0.500000i
$56$	1.73205			1.73205
$57$		−	1.73205i		−	1.73205i
$58$			1.73205i			1.73205i
$59$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$59$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$60$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		−	1.00000i		−	1.00000i
$64$	1.00000			1.00000
$65$	0.500000	+	0.866025i	0.500000	+	0.866025i
$66$	−1.73205			−1.73205
$67$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$67$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$		0			0
$70$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$			1.73205i			1.73205i
$73$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$73$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$74$	3.00000			3.00000
$75$	0.866025	+	0.500000i	0.866025	+	0.500000i
$76$	3.46410			3.46410
$77$			1.00000i			1.00000i
$78$		−	1.73205i		−	1.73205i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	0.866025	−	0.500000i	0.866025	−	0.500000i
$81$	1.00000			1.00000
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	2.00000			2.00000
$85$		0			0
$86$		0			0
$87$			1.00000i			1.00000i
$88$		−	1.73205i		−	1.73205i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$91$	−1.00000			−1.00000
$92$		0			0
$93$		0			0
$94$	−1.73205			−1.73205
$95$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		−	1.73205i		−	1.73205i
$99$	−1.00000			−1.00000
$100$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$	1.73205			1.73205
$105$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$106$		0			0
$107$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$107$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$108$			2.00000i			2.00000i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	0.866025	+	1.50000i	0.866025	+	1.50000i
$111$	1.73205			1.73205
$112$			1.00000i			1.00000i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$	3.00000			3.00000
$115$		0			0
$116$	−2.00000			−2.00000
$117$		−	1.00000i		−	1.00000i
$118$			3.00000i			3.00000i
$119$		0			0
$120$	1.50000	−	0.866025i	1.50000	−	0.866025i
$121$	1.00000			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	1.00000i		−	1.00000i
$126$	1.73205			1.73205
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$			1.73205i			1.73205i
$129$		0			0
$130$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		−	2.00000i		−	2.00000i
$133$		−	1.73205i		−	1.73205i
$134$	−3.00000			−3.00000
$135$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$141$	−1.00000			−1.00000
$142$		0			0
$143$			1.00000i			1.00000i
$144$	−1.00000			−1.00000
$145$	0.866025	−	0.500000i	0.866025	−	0.500000i
$146$	1.73205			1.73205
$147$		−	1.00000i		−	1.00000i
$148$			3.46410i			3.46410i
$149$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$149$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$150$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$			3.00000i			3.00000i
$153$		0			0
$154$	−1.73205			−1.73205
$155$		0			0
$156$	2.00000			2.00000
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$			1.73205i			1.73205i
$163$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$163$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$164$		0			0
$165$	0.500000	+	0.866025i	0.500000	+	0.866025i
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$			1.73205i			1.73205i
$169$		0			0
$170$		0			0
$171$	1.73205			1.73205
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$	−1.73205			−1.73205
$175$	0.866025	+	0.500000i	0.866025	+	0.500000i
$176$	1.00000			1.00000
$177$			1.73205i			1.73205i
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	1.73205	−	1.00000i	1.73205	−	1.00000i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		−	1.73205i		−	1.73205i
$183$		0			0
$184$		0			0
$185$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$186$		0			0
$187$		0			0
$188$		−	2.00000i		−	2.00000i
$189$	1.00000			1.00000
$190$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$			1.00000i			1.00000i
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$196$	2.00000			2.00000
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		−	1.73205i		−	1.73205i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$201$	−1.73205			−1.73205
$202$		0			0
$203$			1.00000i			1.00000i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$			1.00000i			1.00000i
$209$	−1.73205			−1.73205
$210$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$	3.00000			3.00000
$215$		0			0
$216$	−1.73205			−1.73205
$217$		0			0
$218$		0			0
$219$	1.00000			1.00000
$220$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$221$		0			0
$222$			3.00000i			3.00000i
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$			3.46410i			3.46410i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$	−1.00000			−1.00000
$232$		−	1.73205i		−	1.73205i
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$	1.73205			1.73205
$235$	0.500000	+	0.866025i	0.500000	+	0.866025i
$236$	−3.46410			−3.46410
$237$		0			0
$238$		0			0
$239$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$239$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$240$	0.500000	+	0.866025i	0.500000	+	0.866025i
$241$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$241$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$242$			1.73205i			1.73205i
$243$			1.00000i			1.00000i
$244$		0			0
$245$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$246$		0			0
$247$		−	1.73205i		−	1.73205i
$248$		0			0
$249$		0			0
$250$	1.73205			1.73205
$251$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$251$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$252$			2.00000i			2.00000i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−2.00000			−2.00000
$257$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$257$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$258$		0			0
$259$	1.73205			1.73205
$260$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$261$	−1.00000			−1.00000
$262$		0			0
$263$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$263$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$264$	1.73205			1.73205
$265$		0			0
$266$	3.00000			3.00000
$267$		0			0
$268$		−	3.46410i		−	3.46410i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$	1.50000	−	0.866025i	1.50000	−	0.866025i
$271$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$271$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$272$		0			0
$273$		−	1.00000i		−	1.00000i
$274$		0			0
$275$	0.500000	−	0.866025i	0.500000	−	0.866025i
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$	1.50000	−	0.866025i	1.50000	−	0.866025i
$281$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$281$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$282$		−	1.73205i		−	1.73205i
$283$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$283$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$284$		0			0
$285$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$286$	−1.73205			−1.73205
$287$		0			0
$288$		0			0
$289$	−1.00000			−1.00000
$290$	0.866025	+	1.50000i	0.866025	+	1.50000i
$291$		0			0
$292$			2.00000i			2.00000i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	1.73205			1.73205
$295$	1.50000	−	0.866025i	1.50000	−	0.866025i
$296$	−3.00000			−3.00000
$297$		−	1.00000i		−	1.00000i
$298$		−	1.73205i		−	1.73205i
$299$		0			0
$300$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−1.73205			−1.73205
$305$		0			0
$306$		0			0
$307$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$307$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$308$		−	2.00000i		−	2.00000i
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$			1.73205i			1.73205i
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$	1.00000			1.00000
$320$	0.866025	−	0.500000i	0.866025	−	0.500000i
$321$	1.73205			1.73205
$322$		0			0
$323$		0			0
$324$	−2.00000			−2.00000
$325$	0.866025	+	0.500000i	0.866025	+	0.500000i
$326$	−3.00000			−3.00000
$327$		0			0
$328$		0			0
$329$	−1.00000			−1.00000
$330$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$331$	−2.00000			−2.00000			−1.00000			$\pi$
$331$	−2.00000			−2.00000			−1.00000			$\pi$
$332$		0			0
$333$			1.73205i			1.73205i
$334$		0			0
$335$	0.866025	+	1.50000i	0.866025	+	1.50000i
$336$	−1.00000			−1.00000
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$			3.00000i			3.00000i
$343$		−	1.00000i		−	1.00000i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		−	2.00000i		−	2.00000i
$349$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$349$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$350$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$351$	1.00000			1.00000
$352$		0			0
$353$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$353$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$354$	−3.00000			−3.00000
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2.00000			2.00000			1.00000			$0$
$359$	2.00000			2.00000			1.00000			$0$
$360$	0.866025	+	1.50000i	0.866025	+	1.50000i
$361$	2.00000			2.00000
$362$		0			0
$363$			1.00000i			1.00000i
$364$	2.00000			2.00000
$365$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$		0			0
$370$	2.59808	−	1.50000i	2.59808	−	1.50000i
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$	1.00000			1.00000
$376$	1.73205			1.73205
$377$			1.00000i			1.00000i
$378$			1.73205i			1.73205i
$379$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$379$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$380$	3.00000	−	1.73205i	3.00000	−	1.73205i
$381$		0			0
$382$		0			0
$383$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$383$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$384$	−1.73205			−1.73205
$385$	0.500000	+	0.866025i	0.500000	+	0.866025i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$391$		0			0
$392$			1.73205i			1.73205i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	2.00000			2.00000
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$	1.73205			1.73205
$400$	0.500000	−	0.866025i	0.500000	−	0.866025i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		−	3.00000i		−	3.00000i
$403$		0			0
$404$		0			0
$405$	0.866025	−	0.500000i	0.866025	−	0.500000i
$406$	−1.73205			−1.73205
$407$		−	1.73205i		−	1.73205i
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			1.73205i			1.73205i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		−	3.00000i		−	3.00000i
$419$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$419$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$420$	1.73205	−	1.00000i	1.73205	−	1.00000i
$421$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$421$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$422$		0			0
$423$		−	1.00000i		−	1.00000i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$			3.46410i			3.46410i
$429$	−1.00000			−1.00000
$430$		0			0
$431$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$431$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$432$		−	1.00000i		−	1.00000i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$	0.500000	+	0.866025i	0.500000	+	0.866025i
$436$		0			0
$437$		0			0
$438$			1.73205i			1.73205i
$439$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$439$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$440$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$441$	1.00000			1.00000
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$	−3.46410			−3.46410
$445$		0			0
$446$		0			0
$447$		−	1.00000i		−	1.00000i
$448$			1.00000i			1.00000i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$456$	−3.00000			−3.00000
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		−	1.73205i		−	1.73205i
$463$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$463$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$464$	1.00000			1.00000
$465$		0			0
$466$		0			0
$467$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$467$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$468$			2.00000i			2.00000i
$469$	−1.73205			−1.73205
$470$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$471$		0			0
$472$		−	3.00000i		−	3.00000i
$473$		0			0
$474$		0			0
$475$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$476$		0			0
$477$		0			0
$478$			1.73205i			1.73205i
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$	1.73205			1.73205
$482$		−	3.00000i		−	3.00000i
$483$		0			0
$484$	−2.00000			−2.00000
$485$		0			0
$486$	−1.73205			−1.73205
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	−1.73205			−1.73205
$490$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$491$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$491$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$	3.00000			3.00000
$495$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$499$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$500$			2.00000i			2.00000i
$501$		0			0
$502$			3.00000i			3.00000i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	−1.73205			−1.73205
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	1.00000			1.00000
$512$		−	1.73205i		−	1.73205i
$513$			1.73205i			1.73205i
$514$	1.73205			1.73205
$515$		0			0
$516$		0			0
$517$			1.00000i			1.00000i
$518$			3.00000i			3.00000i
$519$		0			0
$520$	1.50000	−	0.866025i	1.50000	−	0.866025i
$521$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$521$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$522$		−	1.73205i		−	1.73205i
$523$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$523$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$524$		0			0
$525$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$526$	−3.00000			−3.00000
$527$		0			0
$528$			1.00000i			1.00000i
$529$	−1.00000			−1.00000
$530$		0			0
$531$	−1.73205			−1.73205
$532$			3.46410i			3.46410i
$533$		0			0
$534$		0			0
$535$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$536$	3.00000			3.00000
$537$		0			0
$538$		0			0
$539$	−1.00000			−1.00000
$540$	1.00000	+	1.73205i	1.00000	+	1.73205i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		−	3.00000i		−	3.00000i
$543$		0			0
$544$		0			0
$545$		0			0
$546$	1.73205			1.73205
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		0			0
$549$		0			0
$550$	1.50000	+	0.866025i	1.50000	+	0.866025i
$551$	−1.73205			−1.73205
$552$		0			0
$553$		0			0
$554$		0			0
$555$	1.50000	−	0.866025i	1.50000	−	0.866025i
$556$		0			0
$557$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$557$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$558$		0			0
$559$		0			0
$560$	0.500000	+	0.866025i	0.500000	+	0.866025i
$561$		0			0
$562$			1.73205i			1.73205i
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	2.00000			2.00000
$565$		0			0
$566$	1.73205			1.73205
$567$			1.00000i			1.00000i
$568$		0			0
$569$	−2.00000			−2.00000			−1.00000			$\pi$
$569$	−2.00000			−2.00000			−1.00000			$\pi$
$570$	2.59808	−	1.50000i	2.59808	−	1.50000i
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		−	2.00000i		−	2.00000i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−1.00000			−1.00000
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$		−	1.73205i		−	1.73205i
$579$		0			0
$580$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−1.73205			−1.73205
$585$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$586$		0			0
$587$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$587$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$588$			2.00000i			2.00000i
$589$		0			0
$590$	1.50000	+	2.59808i	1.50000	+	2.59808i
$591$		0			0
$592$		−	1.73205i		−	1.73205i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$	1.73205			1.73205
$595$		0			0
$596$	2.00000			2.00000
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$	0.866025	−	1.50000i	0.866025	−	1.50000i
$601$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$601$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$602$		0			0
$603$		−	1.73205i		−	1.73205i
$604$		0			0
$605$	0.866025	−	0.500000i	0.866025	−	0.500000i
$606$		0			0
$607$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$607$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$608$		0			0
$609$	−1.00000			−1.00000
$610$		0			0
$611$	−1.00000			−1.00000
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$	3.46410			3.46410
$615$		0			0
$616$	1.73205			1.73205
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−1.00000			−1.00000
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$		−	1.73205i		−	1.73205i
$628$		0			0
$629$		0			0
$630$	1.50000	−	0.866025i	1.50000	−	0.866025i
$631$	−2.00000			−2.00000			−1.00000			$\pi$
$631$	−2.00000			−2.00000			−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	1.00000i		−	1.00000i
$638$			1.73205i			1.73205i
$639$		0			0
$640$	0.866025	+	1.50000i	0.866025	+	1.50000i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$			3.00000i			3.00000i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$647$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$648$		−	1.73205i		−	1.73205i
$649$	1.73205			1.73205
$650$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$651$		0			0
$652$		−	3.46410i		−	3.46410i
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			1.00000i			1.00000i
$658$		−	1.73205i		−	1.73205i
$659$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$659$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$660$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		−	3.46410i		−	3.46410i
$663$		0			0
$664$		0			0
$665$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$666$	−3.00000			−3.00000
$667$		0			0
$668$		0			0
$669$		0			0
$670$	−2.59808	+	1.50000i	−2.59808	+	1.50000i
$671$		0			0
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$676$		0			0
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	−3.46410			−3.46410
$685$		0			0
$686$	1.73205			1.73205
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		−	1.00000i		−	1.00000i
$694$		0			0
$695$		0			0
$696$	1.73205			1.73205
$697$		0			0
$698$			3.00000i			3.00000i
$699$		0			0
$700$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$701$	2.00000			2.00000			1.00000			$0$
$701$	2.00000			2.00000			1.00000			$0$
$702$			1.73205i			1.73205i
$703$			3.00000i			3.00000i
$704$	1.00000			1.00000
$705$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$706$	1.73205			1.73205
$707$		0			0
$708$		−	3.46410i		−	3.46410i
$709$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$709$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	0.500000	+	0.866025i	0.500000	+	0.866025i
$716$		0			0
$717$			1.00000i			1.00000i
$718$			3.46410i			3.46410i
$719$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$719$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$720$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$721$		0			0
$722$			3.46410i			3.46410i
$723$		−	1.73205i		−	1.73205i
$724$		0			0
$725$	0.500000	−	0.866025i	0.500000	−	0.866025i
$726$	−1.73205			−1.73205
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$			1.73205i			1.73205i
$729$	−1.00000			−1.00000
$730$	1.50000	−	0.866025i	1.50000	−	0.866025i
$731$		0			0
$732$		0			0
$733$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$733$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$734$		0			0
$735$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$736$		0			0
$737$			1.73205i			1.73205i
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$	1.73205	+	3.00000i	1.73205	+	3.00000i
$741$	1.73205			1.73205
$742$		0			0
$743$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$743$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$744$		0			0
$745$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.73205			1.73205
$750$			1.73205i			1.73205i
$751$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$751$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$752$			1.00000i			1.00000i
$753$			1.73205i			1.73205i
$754$	−1.73205			−1.73205
$755$		0			0
$756$	−2.00000			−2.00000
$757$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$757$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$758$		−	1.73205i		−	1.73205i
$759$		0			0
$760$	1.50000	+	2.59808i	1.50000	+	2.59808i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	3.46410			3.46410
$767$			1.73205i			1.73205i
$768$		−	2.00000i		−	2.00000i
$769$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$769$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$770$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$771$	1.00000			1.00000
$772$		0			0
$773$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$773$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$			1.73205i			1.73205i
$778$		0			0
$779$		0			0
$780$	1.73205	−	1.00000i	1.73205	−	1.00000i
$781$		0			0
$782$		0			0
$783$		−	1.00000i		−	1.00000i
$784$	−1.00000			−1.00000
$785$		0			0
$786$		0			0
$787$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$787$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$788$		0			0
$789$	−1.73205			−1.73205
$790$		0			0
$791$		0			0
$792$			1.73205i			1.73205i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$797$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$798$			3.00000i			3.00000i
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	1.00000i		−	1.00000i
$804$	3.46410			3.46410
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$809$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$810$	0.866025	+	1.50000i	0.866025	+	1.50000i
$811$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$811$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$812$		−	2.00000i		−	2.00000i
$813$		−	1.73205i		−	1.73205i
$814$	3.00000			3.00000
$815$	0.866025	+	1.50000i	0.866025	+	1.50000i
$816$		0			0
$817$		0			0
$818$		0			0
$819$	1.00000			1.00000
$820$		0			0
$821$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$821$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$823$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$824$		0			0
$825$	0.866025	+	0.500000i	0.866025	+	0.500000i
$826$	−3.00000			−3.00000
$827$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$827$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$			1.00000i			1.00000i
$833$		0			0
$834$		0			0
$835$		0			0
$836$	3.46410			3.46410
$837$		0			0
$838$		−	3.00000i		−	3.00000i
$839$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$839$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$840$	0.866025	+	1.50000i	0.866025	+	1.50000i
$841$		0			0
$842$			1.73205i			1.73205i
$843$			1.00000i			1.00000i
$844$		0			0
$845$		0			0
$846$	1.73205			1.73205
$847$			1.00000i			1.00000i
$848$		0			0
$849$	1.00000			1.00000
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$853$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$854$		0			0
$855$	1.50000	−	0.866025i	1.50000	−	0.866025i
$856$	−3.00000			−3.00000
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		−	1.73205i		−	1.73205i
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		−	1.73205i		−	1.73205i
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	1.00000i		−	1.00000i
$868$		0			0
$869$		0			0
$870$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$871$	−1.73205			−1.73205
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.00000			1.00000
$876$	−2.00000			−2.00000
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$			3.00000i			3.00000i
$879$		0			0
$880$	0.866025	−	0.500000i	0.866025	−	0.500000i
$881$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$881$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$882$			1.73205i			1.73205i
$883$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$883$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$884$		0			0
$885$	0.866025	+	1.50000i	0.866025	+	1.50000i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		−	3.00000i		−	3.00000i
$889$		0			0
$890$		0			0
$891$	1.00000			1.00000
$892$		0			0
$893$		−	1.73205i		−	1.73205i
$894$	1.73205			1.73205
$895$		0			0
$896$	−1.73205			−1.73205
$897$		0			0
$898$		0			0
$899$		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			0
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$		0			0
$910$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		−	1.73205i		−	1.73205i
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	2.00000			2.00000
$922$		0			0
$923$		0			0
$924$	2.00000			2.00000
$925$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$926$	3.00000			3.00000
$927$		0			0
$928$		0			0
$929$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$929$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	1.73205			1.73205
$932$		0			0
$933$		0			0
$934$	1.73205			1.73205
$935$		0			0
$936$	−1.73205			−1.73205
$937$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$937$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$938$		−	3.00000i		−	3.00000i
$939$		0			0
$940$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$	1.73205			1.73205
$945$	0.866025	−	0.500000i	0.866025	−	0.500000i
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$	1.00000			1.00000
$950$	−2.59808	−	1.50000i	−2.59808	−	1.50000i
$951$		0			0
$952$		0			0
$953$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$953$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$954$		0			0
$955$		0			0
$956$	−2.00000			−2.00000
$957$			1.00000i			1.00000i
$958$		0			0
$959$		0			0
$960$	0.500000	+	0.866025i	0.500000	+	0.866025i
$961$	1.00000			1.00000
$962$			3.00000i			3.00000i
$963$			1.73205i			1.73205i
$964$	3.46410			3.46410
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		−	1.73205i		−	1.73205i
$969$		0			0
$970$		0			0
$971$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$971$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$972$		−	2.00000i		−	2.00000i
$973$		0			0
$974$		0			0
$975$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		−	3.00000i		−	3.00000i
$979$		0			0
$980$	1.73205	−	1.00000i	1.73205	−	1.00000i
$981$		0			0
$982$		−	1.73205i		−	1.73205i
$983$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$983$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	1.00000i		−	1.00000i
$988$			3.46410i			3.46410i
$989$		0			0
$990$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$991$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$991$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$		−	2.00000i		−	2.00000i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$997$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$998$		−	1.73205i		−	1.73205i
$999$	−1.73205			−1.73205

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1155.1.e.d.1154.4	yes	4
3.2	odd	2		1155.1.e.c.1154.1	✓	4
5.4	even	2	inner	1155.1.e.d.1154.1	yes	4
7.6	odd	2	inner	1155.1.e.d.1154.3	yes	4
11.10	odd	2		1155.1.e.c.1154.2	yes	4
15.14	odd	2		1155.1.e.c.1154.4	yes	4
21.20	even	2		1155.1.e.c.1154.2	yes	4
33.32	even	2	inner	1155.1.e.d.1154.3	yes	4
35.34	odd	2	inner	1155.1.e.d.1154.2	yes	4
55.54	odd	2		1155.1.e.c.1154.3	yes	4
77.76	even	2		1155.1.e.c.1154.1	✓	4
105.104	even	2		1155.1.e.c.1154.3	yes	4
165.164	even	2	inner	1155.1.e.d.1154.2	yes	4
231.230	odd	2	CM	1155.1.e.d.1154.4	yes	4
385.384	even	2		1155.1.e.c.1154.4	yes	4
1155.1154	odd	2	inner	1155.1.e.d.1154.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.1.e.c.1154.1	✓	4	3.2	odd	2
1155.1.e.c.1154.1	✓	4	77.76	even	2
1155.1.e.c.1154.2	yes	4	11.10	odd	2
1155.1.e.c.1154.2	yes	4	21.20	even	2
1155.1.e.c.1154.3	yes	4	55.54	odd	2
1155.1.e.c.1154.3	yes	4	105.104	even	2
1155.1.e.c.1154.4	yes	4	15.14	odd	2
1155.1.e.c.1154.4	yes	4	385.384	even	2
1155.1.e.d.1154.1	yes	4	5.4	even	2	inner
1155.1.e.d.1154.1	yes	4	1155.1154	odd	2	inner
1155.1.e.d.1154.2	yes	4	35.34	odd	2	inner
1155.1.e.d.1154.2	yes	4	165.164	even	2	inner
1155.1.e.d.1154.3	yes	4	7.6	odd	2	inner
1155.1.e.d.1154.3	yes	4	33.32	even	2	inner
1155.1.e.d.1154.4	yes	4	1.1	even	1	trivial
1155.1.e.d.1154.4	yes	4	231.230	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1155.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{2} +1.00000i q^{3} -2.00000 q^{4} +(0.866025 - 0.500000i) q^{5} -1.73205 q^{6} +1.00000i q^{7} -1.73205i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{2} +1.00000i q^{3} -2.00000 q^{4} +(0.866025 - 0.500000i) q^{5} -1.73205 q^{6} +1.00000i q^{7} -1.73205i q^{8} -1.00000 q^{9} +(0.866025 + 1.50000i) q^{10} +1.00000 q^{11} -2.00000i q^{12} +1.00000i q^{13} -1.73205 q^{14} +(0.500000 + 0.866025i) q^{15} +1.00000 q^{16} -1.73205i q^{18} -1.73205 q^{19} +(-1.73205 + 1.00000i) q^{20} -1.00000 q^{21} +1.73205i q^{22} +1.73205 q^{24} +(0.500000 - 0.866025i) q^{25} -1.73205 q^{26} -1.00000i q^{27} -2.00000i q^{28} +1.00000 q^{29} +(-1.50000 + 0.866025i) q^{30} +1.00000i q^{33} +(0.500000 + 0.866025i) q^{35} +2.00000 q^{36} -1.73205i q^{37} -3.00000i q^{38} -1.00000 q^{39} +(-0.866025 - 1.50000i) q^{40} -1.73205i q^{42} -2.00000 q^{44} +(-0.866025 + 0.500000i) q^{45} +1.00000i q^{47} +1.00000i q^{48} -1.00000 q^{49} +(1.50000 + 0.866025i) q^{50} -2.00000i q^{52} +1.73205 q^{54} +(0.866025 - 0.500000i) q^{55} +1.73205 q^{56} -1.73205i q^{57} +1.73205i q^{58} +1.73205 q^{59} +(-1.00000 - 1.73205i) q^{60} -1.00000i q^{63} +1.00000 q^{64} +(0.500000 + 0.866025i) q^{65} -1.73205 q^{66} +1.73205i q^{67} +(-1.50000 + 0.866025i) q^{70} +1.73205i q^{72} -1.00000i q^{73} +3.00000 q^{74} +(0.866025 + 0.500000i) q^{75} +3.46410 q^{76} +1.00000i q^{77} -1.73205i q^{78} +(0.866025 - 0.500000i) q^{80} +1.00000 q^{81} +2.00000 q^{84} +1.00000i q^{87} -1.73205i q^{88} +(-0.866025 - 1.50000i) q^{90} -1.00000 q^{91} -1.73205 q^{94} +(-1.50000 + 0.866025i) q^{95} -1.73205i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 4 q^{9}+O(q^{10}) \) 4 * q - 8 * q^4 - 4 * q^9	\( 4 q - 8 q^{4} - 4 q^{9} + 4 q^{11} + 2 q^{15} + 4 q^{16} - 4 q^{21} + 2 q^{25} + 4 q^{29} - 6 q^{30} + 2 q^{35} + 8 q^{36} - 4 q^{39} - 8 q^{44} - 4 q^{49} + 6 q^{50} - 4 q^{60} + 4 q^{64} + 2 q^{65} - 6 q^{70} + 12 q^{74} + 4 q^{81} + 8 q^{84} - 4 q^{91} - 6 q^{95} - 4 q^{99}+O(q^{100}) \) 4 * q - 8 * q^4 - 4 * q^9 + 4 * q^11 + 2 * q^15 + 4 * q^16 - 4 * q^21 + 2 * q^25 + 4 * q^29 - 6 * q^30 + 2 * q^35 + 8 * q^36 - 4 * q^39 - 8 * q^44 - 4 * q^49 + 6 * q^50 - 4 * q^60 + 4 * q^64 + 2 * q^65 - 6 * q^70 + 12 * q^74 + 4 * q^81 + 8 * q^84 - 4 * q^91 - 6 * q^95 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more