LMFDB - Embedded newform 2700.1.bj.a.151.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,1,Mod(151,2700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2700, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 4, 0]))

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

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

chi := DirichletCharacter("2700.151");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$2700 = 2^{2} \cdot 3^{3} \cdot 5^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	2700.bj (of order $18$ , degree $6$ , 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:	$1.34747553411$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{18})$
Coefficient field:	$\Q(\zeta_{36})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} - x^{6} + 1$ x^12 - x^6 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 540)
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.5020969537440000.1

Embedding invariants

Embedding label			151.1
Root			$-0.342020 + 0.939693i$ of defining polynomial
Character	$\chi$	$=$	2700.151
Dual form			2700.1.bj.a.751.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.642788 + 0.766044i) q^{2} +(-0.342020 + 0.939693i) q^{3} +(-0.173648 - 0.984808i) q^{4} +(-0.500000 - 0.866025i) q^{6} +(-1.85083 - 0.326352i) q^{7} +(0.866025 + 0.500000i) q^{8} +(-0.766044 - 0.642788i) q^{9} +(0.984808 + 0.173648i) q^{12} +(1.43969 - 1.20805i) q^{14} +(-0.939693 + 0.342020i) q^{16} +(0.984808 - 0.173648i) q^{18} +(0.939693 - 1.62760i) q^{21} +(0.342020 - 0.0603074i) q^{23} +(-0.766044 + 0.642788i) q^{24} +(0.866025 - 0.500000i) q^{27} +1.87939i q^{28} +(-0.266044 - 0.223238i) q^{29} +(0.342020 - 0.939693i) q^{32} +(-0.500000 + 0.866025i) q^{36} +(1.17365 - 0.984808i) q^{41} +(0.642788 + 1.76604i) q^{42} +(-0.342020 - 0.939693i) q^{43} +(-0.173648 + 0.300767i) q^{46} +(1.50881 + 0.266044i) q^{47} -1.00000i q^{48} +(2.37939 + 0.866025i) q^{49} +(-0.173648 + 0.984808i) q^{54} +(-1.43969 - 1.20805i) q^{56} +(0.342020 - 0.0603074i) q^{58} +(-0.326352 + 1.85083i) q^{61} +(1.20805 + 1.43969i) q^{63} +(0.500000 + 0.866025i) q^{64} +(0.984808 + 1.17365i) q^{67} +(-0.0603074 + 0.342020i) q^{69} +(-0.342020 - 0.939693i) q^{72} +(0.173648 + 0.984808i) q^{81} +1.53209i q^{82} +(0.984808 - 1.17365i) q^{83} +(-1.76604 - 0.642788i) q^{84} +(0.939693 + 0.342020i) q^{86} +(0.300767 - 0.173648i) q^{87} +(0.766044 - 1.32683i) q^{89} +(-0.118782 - 0.326352i) q^{92} +(-1.17365 + 0.984808i) q^{94} +(0.766044 + 0.642788i) q^{96} +(-2.19285 + 1.26604i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 6 q^{6} + 6 q^{14} + 6 q^{29} - 6 q^{36} + 12 q^{41} + 6 q^{49} - 6 q^{56} - 6 q^{61} + 6 q^{64} - 12 q^{69} - 12 q^{84} - 12 q^{94}+O(q^{100})$ 12 * q - 6 * q^6 + 6 * q^14 + 6 * q^29 - 6 * q^36 + 12 * q^41 + 6 * q^49 - 6 * q^56 - 6 * q^61 + 6 * q^64 - 12 * q^69 - 12 * q^84 - 12 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$ .

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$e\left(\frac{2}{9}\right)$	$-1$	$1$

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

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$	−0.642788	+	0.766044i	−0.642788	+	0.766044i
$2$	−0.642788	+	0.766044i	−0.642788	+	0.766044i
$3$	−0.342020	+	0.939693i	−0.342020	+	0.939693i
$3$	−0.342020	+	0.939693i	−0.342020	+	0.939693i
$4$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$5$		0			0
$5$		0			0
$6$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$7$	−1.85083	−	0.326352i	−1.85083	−	0.326352i	−0.866025	−	0.500000i	$-0.833333\pi$
$7$	−1.85083	−	0.326352i	−1.85083	−	0.326352i	−0.984808	+	0.173648i	$0.944444\pi$
$8$	0.866025	+	0.500000i	0.866025	+	0.500000i
$9$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$10$		0			0
$11$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$11$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$12$	0.984808	+	0.173648i	0.984808	+	0.173648i
$13$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$13$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$14$	1.43969	−	1.20805i	1.43969	−	1.20805i
$15$		0			0
$16$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$	0.984808	−	0.173648i	0.984808	−	0.173648i
$19$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$		0			0
$21$	0.939693	−	1.62760i	0.939693	−	1.62760i
$22$		0			0
$23$	0.342020	−	0.0603074i	0.342020	−	0.0603074i		−	1.00000i	$-0.5\pi$
$23$	0.342020	−	0.0603074i	0.342020	−	0.0603074i	0.342020	+	0.939693i	$0.388889\pi$
$24$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$25$		0			0
$26$		0			0
$27$	0.866025	−	0.500000i	0.866025	−	0.500000i
$28$			1.87939i			1.87939i
$29$	−0.266044	−	0.223238i	−0.266044	−	0.223238i	0.500000	−	0.866025i	$-0.333333\pi$
$29$	−0.266044	−	0.223238i	−0.266044	−	0.223238i	−0.766044	+	0.642788i	$0.777778\pi$
$30$		0			0
$31$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$31$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$32$	0.342020	−	0.939693i	0.342020	−	0.939693i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$37$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$37$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.17365	−	0.984808i	1.17365	−	0.984808i	0.173648	−	0.984808i	$-0.444444\pi$
$41$	1.17365	−	0.984808i	1.17365	−	0.984808i	1.00000			$0$
$42$	0.642788	+	1.76604i	0.642788	+	1.76604i
$43$	−0.342020	−	0.939693i	−0.342020	−	0.939693i	−0.984808	−	0.173648i	$-0.944444\pi$
$43$	−0.342020	−	0.939693i	−0.342020	−	0.939693i	0.642788	−	0.766044i	$-0.277778\pi$
$44$		0			0
$45$		0			0
$46$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$47$	1.50881	+	0.266044i	1.50881	+	0.266044i	0.866025	−	0.500000i	$-0.166667\pi$
$47$	1.50881	+	0.266044i	1.50881	+	0.266044i	0.642788	+	0.766044i	$0.277778\pi$
$48$		−	1.00000i		−	1.00000i
$49$	2.37939	+	0.866025i	2.37939	+	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$55$		0			0
$56$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$57$		0			0
$58$	0.342020	−	0.0603074i	0.342020	−	0.0603074i
$59$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$59$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$60$		0			0
$61$	−0.326352	+	1.85083i	−0.326352	+	1.85083i	0.173648	+	0.984808i	$0.444444\pi$
$61$	−0.326352	+	1.85083i	−0.326352	+	1.85083i	−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$	1.20805	+	1.43969i	1.20805	+	1.43969i
$64$	0.500000	+	0.866025i	0.500000	+	0.866025i
$65$		0			0
$66$		0			0
$67$	0.984808	+	1.17365i	0.984808	+	1.17365i	0.984808	+	0.173648i	$0.0555556\pi$
$67$	0.984808	+	1.17365i	0.984808	+	1.17365i			1.00000i	$0.5\pi$
$68$		0			0
$69$	−0.0603074	+	0.342020i	−0.0603074	+	0.342020i
$70$		0			0
$71$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$71$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$72$	−0.342020	−	0.939693i	−0.342020	−	0.939693i
$73$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$73$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$79$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$80$		0			0
$81$	0.173648	+	0.984808i	0.173648	+	0.984808i
$82$			1.53209i			1.53209i
$83$	0.984808	−	1.17365i	0.984808	−	1.17365i		−	1.00000i	$-0.5\pi$
$83$	0.984808	−	1.17365i	0.984808	−	1.17365i	0.984808	−	0.173648i	$-0.0555556\pi$
$84$	−1.76604	−	0.642788i	−1.76604	−	0.642788i
$85$		0			0
$86$	0.939693	+	0.342020i	0.939693	+	0.342020i
$87$	0.300767	−	0.173648i	0.300767	−	0.173648i
$88$		0			0
$89$	0.766044	−	1.32683i	0.766044	−	1.32683i	−0.173648	−	0.984808i	$-0.555556\pi$
$89$	0.766044	−	1.32683i	0.766044	−	1.32683i	0.939693	−	0.342020i	$-0.111111\pi$
$90$		0			0
$91$		0			0
$92$	−0.118782	−	0.326352i	−0.118782	−	0.326352i
$93$		0			0
$94$	−1.17365	+	0.984808i	−1.17365	+	0.984808i
$95$		0			0
$96$	0.766044	+	0.642788i	0.766044	+	0.642788i
$97$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$97$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$98$	−2.19285	+	1.26604i	−2.19285	+	1.26604i
$99$		0			0
$100$		0			0
$101$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$101$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$102$		0			0
$103$	0.342020	−	0.939693i	0.342020	−	0.939693i	−0.642788	−	0.766044i	$-0.722222\pi$
$103$	0.342020	−	0.939693i	0.342020	−	0.939693i	0.984808	−	0.173648i	$-0.0555556\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1.87939i		−	1.87939i	−0.342020	−	0.939693i	$-0.611111\pi$
$107$		−	1.87939i		−	1.87939i	0.342020	−	0.939693i	$-0.388889\pi$
$108$	−0.642788	−	0.766044i	−0.642788	−	0.766044i
$109$	−0.347296			−0.347296			−0.173648	−	0.984808i	$-0.555556\pi$
$109$	−0.347296			−0.347296			−0.173648	+	0.984808i	$0.555556\pi$
$110$		0			0
$111$		0			0
$112$	1.85083	−	0.326352i	1.85083	−	0.326352i
$113$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$113$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$114$		0			0
$115$		0			0
$116$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.766044	−	0.642788i	0.766044	−	0.642788i
$122$	−1.20805	−	1.43969i	−1.20805	−	1.43969i
$123$	0.524005	+	1.43969i	0.524005	+	1.43969i
$124$		0			0
$125$		0			0
$126$	−1.87939			−1.87939
$127$	−0.300767	−	0.173648i	−0.300767	−	0.173648i	0.342020	−	0.939693i	$-0.388889\pi$
$127$	−0.300767	−	0.173648i	−0.300767	−	0.173648i	−0.642788	+	0.766044i	$0.722222\pi$
$128$	−0.984808	−	0.173648i	−0.984808	−	0.173648i
$129$	1.00000			1.00000
$130$		0			0
$131$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$131$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$132$		0			0
$133$		0			0
$134$	−1.53209			−1.53209
$135$		0			0
$136$		0			0
$137$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$137$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$138$	−0.223238	−	0.266044i	−0.223238	−	0.266044i
$139$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$139$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$140$		0			0
$141$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$142$		0			0
$143$		0			0
$144$	0.939693	+	0.342020i	0.939693	+	0.342020i
$145$		0			0
$146$		0			0
$147$	−1.62760	+	1.93969i	−1.62760	+	1.93969i
$148$		0			0
$149$	−0.266044	+	0.223238i	−0.266044	+	0.223238i	−0.766044	−	0.642788i	$-0.777778\pi$
$149$	−0.266044	+	0.223238i	−0.266044	+	0.223238i	0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$151$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$157$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.652704			−0.652704
$162$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$163$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$163$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$164$	−1.17365	−	0.984808i	−1.17365	−	0.984808i
$165$		0			0
$166$	0.266044	+	1.50881i	0.266044	+	1.50881i
$167$	0.118782	−	0.326352i	0.118782	−	0.326352i	−0.866025	−	0.500000i	$-0.833333\pi$
$167$	0.118782	−	0.326352i	0.118782	−	0.326352i	0.984808	+	0.173648i	$0.0555556\pi$
$168$	1.62760	−	0.939693i	1.62760	−	0.939693i
$169$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$170$		0			0
$171$		0			0
$172$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$173$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$173$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$174$	−0.0603074	+	0.342020i	−0.0603074	+	0.342020i
$175$		0			0
$176$		0			0
$177$		0			0
$178$	0.524005	+	1.43969i	0.524005	+	1.43969i
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$181$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$182$		0			0
$183$	−1.62760	−	0.939693i	−1.62760	−	0.939693i
$184$	0.326352	+	0.118782i	0.326352	+	0.118782i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		−	1.53209i		−	1.53209i
$189$	−1.76604	+	0.642788i	−1.76604	+	0.642788i
$190$		0			0
$191$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$191$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$192$	−0.984808	+	0.173648i	−0.984808	+	0.173648i
$193$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$193$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$194$		0			0
$195$		0			0
$196$	0.439693	−	2.49362i	0.439693	−	2.49362i
$197$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$197$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$198$		0			0
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$		0			0
$201$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$202$	−0.642788	−	0.766044i	−0.642788	−	0.766044i
$203$	0.419550	+	0.500000i	0.419550	+	0.500000i
$204$		0			0
$205$		0			0
$206$	0.500000	+	0.866025i	0.500000	+	0.866025i
$207$	−0.300767	−	0.173648i	−0.300767	−	0.173648i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$211$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$212$		0			0
$213$		0			0
$214$	1.43969	+	1.20805i	1.43969	+	1.20805i
$215$		0			0
$216$	1.00000			1.00000
$217$		0			0
$218$	0.223238	−	0.266044i	0.223238	−	0.266044i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.50881	−	0.266044i	−1.50881	−	0.266044i	−0.642788	−	0.766044i	$-0.722222\pi$
$223$	−1.50881	−	0.266044i	−1.50881	−	0.266044i	−0.866025	+	0.500000i	$0.833333\pi$
$224$	−0.939693	+	1.62760i	−0.939693	+	1.62760i
$225$		0			0
$226$		0			0
$227$	0.342020	+	0.939693i	0.342020	+	0.939693i	0.984808	+	0.173648i	$0.0555556\pi$
$227$	0.342020	+	0.939693i	0.342020	+	0.939693i	−0.642788	+	0.766044i	$0.722222\pi$
$228$		0			0
$229$	−0.266044	+	0.223238i	−0.266044	+	0.223238i	−0.766044	−	0.642788i	$-0.777778\pi$
$229$	−0.266044	+	0.223238i	−0.266044	+	0.223238i	0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$	−0.118782	−	0.326352i	−0.118782	−	0.326352i
$233$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$233$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$239$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$240$		0			0
$241$	−1.43969	−	1.20805i	−1.43969	−	1.20805i	−0.939693	−	0.342020i	$-0.888889\pi$
$241$	−1.43969	−	1.20805i	−1.43969	−	1.20805i	−0.500000	−	0.866025i	$-0.666667\pi$
$242$			1.00000i			1.00000i
$243$	−0.984808	−	0.173648i	−0.984808	−	0.173648i
$244$	1.87939			1.87939
$245$		0			0
$246$	−1.43969	−	0.524005i	−1.43969	−	0.524005i
$247$		0			0
$248$		0			0
$249$	0.766044	+	1.32683i	0.766044	+	1.32683i
$250$		0			0
$251$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$251$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$252$	1.20805	−	1.43969i	1.20805	−	1.43969i
$253$		0			0
$254$	0.326352	−	0.118782i	0.326352	−	0.118782i
$255$		0			0
$256$	0.766044	−	0.642788i	0.766044	−	0.642788i
$257$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$257$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$258$	−0.642788	+	0.766044i	−0.642788	+	0.766044i
$259$		0			0
$260$		0			0
$261$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$262$		0			0
$263$	0.984808	+	0.173648i	0.984808	+	0.173648i	0.642788	−	0.766044i	$-0.277778\pi$
$263$	0.984808	+	0.173648i	0.984808	+	0.173648i	0.342020	+	0.939693i	$0.388889\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	0.984808	+	1.17365i	0.984808	+	1.17365i
$268$	0.984808	−	1.17365i	0.984808	−	1.17365i
$269$	1.87939			1.87939			0.939693	−	0.342020i	$-0.111111\pi$
$269$	1.87939			1.87939			0.939693	+	0.342020i	$0.111111\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	0.347296			0.347296
$277$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$277$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.76604	−	0.642788i	1.76604	−	0.642788i	0.766044	−	0.642788i	$-0.222222\pi$
$281$	1.76604	−	0.642788i	1.76604	−	0.642788i	1.00000			$0$
$282$	−0.524005	−	1.43969i	−0.524005	−	1.43969i
$283$	−0.223238	−	0.266044i	−0.223238	−	0.266044i	0.642788	−	0.766044i	$-0.277778\pi$
$283$	−0.223238	−	0.266044i	−0.223238	−	0.266044i	−0.866025	+	0.500000i	$0.833333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.49362	+	1.43969i	−2.49362	+	1.43969i
$288$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$289$	0.500000	−	0.866025i	0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$293$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$294$	−0.439693	−	2.49362i	−0.439693	−	2.49362i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		−	0.347296i		−	0.347296i
$299$		0			0
$300$		0			0
$301$	0.326352	+	1.85083i	0.326352	+	1.85083i
$302$		0			0
$303$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	0.300767	−	0.173648i	0.300767	−	0.173648i	−0.342020	−	0.939693i	$-0.611111\pi$
$307$	0.300767	−	0.173648i	0.300767	−	0.173648i	0.642788	+	0.766044i	$0.277778\pi$
$308$		0			0
$309$	0.766044	+	0.642788i	0.766044	+	0.642788i
$310$		0			0
$311$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$311$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$312$		0			0
$313$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$313$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$317$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	1.76604	+	0.642788i	1.76604	+	0.642788i
$322$	0.419550	−	0.500000i	0.419550	−	0.500000i
$323$		0			0
$324$	0.939693	−	0.342020i	0.939693	−	0.342020i
$325$		0			0
$326$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$327$	0.118782	−	0.326352i	0.118782	−	0.326352i
$328$	1.50881	−	0.266044i	1.50881	−	0.266044i
$329$	−2.70574	−	0.984808i	−2.70574	−	0.984808i
$330$		0			0
$331$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$331$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$332$	−1.32683	−	0.766044i	−1.32683	−	0.766044i
$333$		0			0
$334$	0.173648	+	0.300767i	0.173648	+	0.300767i
$335$		0			0
$336$	−0.326352	+	1.85083i	−0.326352	+	1.85083i
$337$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$337$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$338$	−0.642788	−	0.766044i	−0.642788	−	0.766044i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−2.49362	−	1.43969i	−2.49362	−	1.43969i
$344$	0.173648	−	0.984808i	0.173648	−	0.984808i
$345$		0			0
$346$		0			0
$347$	0.984808	−	0.173648i	0.984808	−	0.173648i	0.342020	−	0.939693i	$-0.388889\pi$
$347$	0.984808	−	0.173648i	0.984808	−	0.173648i	0.642788	+	0.766044i	$0.277778\pi$
$348$	−0.223238	−	0.266044i	−0.223238	−	0.266044i
$349$	−1.17365	−	0.984808i	−1.17365	−	0.984808i	−0.173648	−	0.984808i	$-0.555556\pi$
$349$	−1.17365	−	0.984808i	−1.17365	−	0.984808i	−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$353$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$354$		0			0
$355$		0			0
$356$	−1.43969	−	0.524005i	−1.43969	−	0.524005i
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$362$	0.642788	+	1.76604i	0.642788	+	1.76604i
$363$	0.342020	+	0.939693i	0.342020	+	0.939693i
$364$		0			0
$365$		0			0
$366$	1.76604	−	0.642788i	1.76604	−	0.642788i
$367$	−0.684040	−	1.87939i	−0.684040	−	1.87939i	−0.342020	−	0.939693i	$-0.611111\pi$
$367$	−0.684040	−	1.87939i	−0.684040	−	1.87939i	−0.342020	−	0.939693i	$-0.611111\pi$
$368$	−0.300767	+	0.173648i	−0.300767	+	0.173648i
$369$	−1.53209			−1.53209
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$373$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$374$		0			0
$375$		0			0
$376$	1.17365	+	0.984808i	1.17365	+	0.984808i
$377$		0			0
$378$	0.642788	−	1.76604i	0.642788	−	1.76604i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	0.266044	−	0.223238i	0.266044	−	0.223238i
$382$		0			0
$383$	0.342020	−	0.939693i	0.342020	−	0.939693i	−0.642788	−	0.766044i	$-0.722222\pi$
$383$	0.342020	−	0.939693i	0.342020	−	0.939693i	0.984808	−	0.173648i	$-0.0555556\pi$
$384$	0.500000	−	0.866025i	0.500000	−	0.866025i
$385$		0			0
$386$		0			0
$387$	−0.342020	+	0.939693i	−0.342020	+	0.939693i
$388$		0			0
$389$	1.43969	−	0.524005i	1.43969	−	0.524005i	0.500000	−	0.866025i	$-0.333333\pi$
$389$	1.43969	−	0.524005i	1.43969	−	0.524005i	0.939693	+	0.342020i	$0.111111\pi$
$390$		0			0
$391$		0			0
$392$	1.62760	+	1.93969i	1.62760	+	1.93969i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$397$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$401$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$402$	0.524005	−	1.43969i	0.524005	−	1.43969i
$403$		0			0
$404$	1.00000			1.00000
$405$		0			0
$406$	−0.652704			−0.652704
$407$		0			0
$408$		0			0
$409$	0.173648	+	0.984808i	0.173648	+	0.984808i	0.939693	+	0.342020i	$0.111111\pi$
$409$	0.173648	+	0.984808i	0.173648	+	0.984808i	−0.766044	+	0.642788i	$0.777778\pi$
$410$		0			0
$411$		0			0
$412$	−0.984808	−	0.173648i	−0.984808	−	0.173648i
$413$		0			0
$414$	0.326352	−	0.118782i	0.326352	−	0.118782i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$419$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$420$		0			0
$421$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$421$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$422$		0			0
$423$	−0.984808	−	1.17365i	−0.984808	−	1.17365i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.20805	−	3.31908i	1.20805	−	3.31908i
$428$	−1.85083	+	0.326352i	−1.85083	+	0.326352i
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	−0.642788	+	0.766044i	−0.642788	+	0.766044i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$437$		0			0
$438$		0			0
$439$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$439$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$440$		0			0
$441$	−1.26604	−	2.19285i	−1.26604	−	2.19285i
$442$		0			0
$443$	0.524005	+	1.43969i	0.524005	+	1.43969i	0.866025	+	0.500000i	$0.166667\pi$
$443$	0.524005	+	1.43969i	0.524005	+	1.43969i	−0.342020	+	0.939693i	$0.611111\pi$
$444$		0			0
$445$		0			0
$446$	1.17365	−	0.984808i	1.17365	−	0.984808i
$447$	−0.118782	−	0.326352i	−0.118782	−	0.326352i
$448$	−0.642788	−	1.76604i	−0.642788	−	1.76604i
$449$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$449$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$455$		0			0
$456$		0			0
$457$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$457$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$458$		−	0.347296i		−	0.347296i
$459$		0			0
$460$		0			0
$461$	−1.43969	−	1.20805i	−1.43969	−	1.20805i	−0.939693	−	0.342020i	$-0.888889\pi$
$461$	−1.43969	−	1.20805i	−1.43969	−	1.20805i	−0.500000	−	0.866025i	$-0.666667\pi$
$462$		0			0
$463$	−0.984808	+	0.173648i	−0.984808	+	0.173648i	−0.642788	−	0.766044i	$-0.722222\pi$
$463$	−0.984808	+	0.173648i	−0.984808	+	0.173648i	−0.342020	+	0.939693i	$0.611111\pi$
$464$	0.326352	+	0.118782i	0.326352	+	0.118782i
$465$		0			0
$466$		0			0
$467$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$467$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$468$		0			0
$469$	−1.43969	−	2.49362i	−1.43969	−	2.49362i
$470$		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			0
$479$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$479$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$480$		0			0
$481$		0			0
$482$	1.85083	−	0.326352i	1.85083	−	0.326352i
$483$	0.223238	−	0.613341i	0.223238	−	0.613341i
$484$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$485$		0			0
$486$	0.766044	−	0.642788i	0.766044	−	0.642788i
$487$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$487$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$488$	−1.20805	+	1.43969i	−1.20805	+	1.43969i
$489$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$490$		0			0
$491$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$491$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$492$	1.32683	−	0.766044i	1.32683	−	0.766044i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−1.50881	−	0.266044i	−1.50881	−	0.266044i
$499$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$499$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$500$		0			0
$501$	0.266044	+	0.223238i	0.266044	+	0.223238i
$502$		0			0
$503$	1.62760	−	0.939693i	1.62760	−	0.939693i	0.642788	−	0.766044i	$-0.277778\pi$
$503$	1.62760	−	0.939693i	1.62760	−	0.939693i	0.984808	−	0.173648i	$-0.0555556\pi$
$504$	0.326352	+	1.85083i	0.326352	+	1.85083i
$505$		0			0
$506$		0			0
$507$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$508$	−0.118782	+	0.326352i	−0.118782	+	0.326352i
$509$	0.326352	+	1.85083i	0.326352	+	1.85083i	0.500000	+	0.866025i	$0.333333\pi$
$509$	0.326352	+	1.85083i	0.326352	+	1.85083i	−0.173648	+	0.984808i	$0.555556\pi$
$510$		0			0
$511$		0			0
$512$			1.00000i			1.00000i
$513$		0			0
$514$		0			0
$515$		0			0
$516$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$521$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$522$	−0.300767	−	0.173648i	−0.300767	−	0.173648i
$523$	1.62760	−	0.939693i	1.62760	−	0.939693i	0.642788	−	0.766044i	$-0.277778\pi$
$523$	1.62760	−	0.939693i	1.62760	−	0.939693i	0.984808	−	0.173648i	$-0.0555556\pi$
$524$		0			0
$525$		0			0
$526$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$527$		0			0
$528$		0			0
$529$	−0.826352	+	0.300767i	−0.826352	+	0.300767i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	−1.53209			−1.53209
$535$		0			0
$536$	0.266044	+	1.50881i	0.266044	+	1.50881i
$537$		0			0
$538$	−1.20805	+	1.43969i	−1.20805	+	1.43969i
$539$		0			0
$540$		0			0
$541$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$541$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$542$		0			0
$543$	1.20805	+	1.43969i	1.20805	+	1.43969i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.50881	+	0.266044i	1.50881	+	0.266044i	0.866025	−	0.500000i	$-0.166667\pi$
$547$	1.50881	+	0.266044i	1.50881	+	0.266044i	0.642788	+	0.766044i	$0.277778\pi$
$548$		0			0
$549$	1.43969	−	1.20805i	1.43969	−	1.20805i
$550$		0			0
$551$		0			0
$552$	−0.223238	+	0.266044i	−0.223238	+	0.266044i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$557$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.642788	+	1.76604i	−0.642788	+	1.76604i
$563$	1.50881	−	0.266044i	1.50881	−	0.266044i	0.642788	−	0.766044i	$-0.277778\pi$
$563$	1.50881	−	0.266044i	1.50881	−	0.266044i	0.866025	+	0.500000i	$0.166667\pi$
$564$	1.43969	+	0.524005i	1.43969	+	0.524005i
$565$		0			0
$566$	0.347296			0.347296
$567$		−	1.87939i		−	1.87939i
$568$		0			0
$569$	0.766044	+	0.642788i	0.766044	+	0.642788i	0.939693	−	0.342020i	$-0.111111\pi$
$569$	0.766044	+	0.642788i	0.766044	+	0.642788i	−0.173648	+	0.984808i	$0.555556\pi$
$570$		0			0
$571$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$571$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$572$		0			0
$573$		0			0
$574$	0.500000	−	2.83564i	0.500000	−	2.83564i
$575$		0			0
$576$	0.173648	−	0.984808i	0.173648	−	0.984808i
$577$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$577$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$578$	0.342020	+	0.939693i	0.342020	+	0.939693i
$579$		0			0
$580$		0			0
$581$	−2.20574	+	1.85083i	−2.20574	+	1.85083i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.85083	−	0.326352i	−1.85083	−	0.326352i	−0.866025	−	0.500000i	$-0.833333\pi$
$587$	−1.85083	−	0.326352i	−1.85083	−	0.326352i	−0.984808	+	0.173648i	$0.944444\pi$
$588$	2.19285	+	1.26604i	2.19285	+	1.26604i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	0.266044	+	0.223238i	0.266044	+	0.223238i
$597$		0			0
$598$		0			0
$599$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$599$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$600$		0			0
$601$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$601$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$602$	−1.62760	−	0.939693i	−1.62760	−	0.939693i
$603$		−	1.53209i		−	1.53209i
$604$		0			0
$605$		0			0
$606$	0.939693	−	0.342020i	0.939693	−	0.342020i
$607$	0.223238	+	0.266044i	0.223238	+	0.266044i	0.866025	−	0.500000i	$-0.166667\pi$
$607$	0.223238	+	0.266044i	0.223238	+	0.266044i	−0.642788	+	0.766044i	$0.722222\pi$
$608$		0			0
$609$	−0.613341	+	0.223238i	−0.613341	+	0.223238i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$613$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$614$	−0.0603074	+	0.342020i	−0.0603074	+	0.342020i
$615$		0			0
$616$		0			0
$617$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$617$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$618$	−0.984808	+	0.173648i	−0.984808	+	0.173648i
$619$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$619$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$620$		0			0
$621$	0.266044	−	0.223238i	0.266044	−	0.223238i
$622$		0			0
$623$	−1.85083	+	2.20574i	−1.85083	+	2.20574i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$631$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.326352	+	1.85083i	−0.326352	+	1.85083i	0.173648	+	0.984808i	$0.444444\pi$
$641$	−0.326352	+	1.85083i	−0.326352	+	1.85083i	−0.500000	+	0.866025i	$0.666667\pi$
$642$	−1.62760	+	0.939693i	−1.62760	+	0.939693i
$643$	−0.118782	+	0.326352i	−0.118782	+	0.326352i	−0.984808	−	0.173648i	$-0.944444\pi$
$643$	−0.118782	+	0.326352i	−0.118782	+	0.326352i	0.866025	+	0.500000i	$0.166667\pi$
$644$	0.113341	+	0.642788i	0.113341	+	0.642788i
$645$		0			0
$646$		0			0
$647$			0.347296i			0.347296i	0.984808	+	0.173648i	$0.0555556\pi$
$647$			0.347296i			0.347296i	−0.984808	+	0.173648i	$0.944444\pi$
$648$	−0.342020	+	0.939693i	−0.342020	+	0.939693i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.984808	−	0.173648i	0.984808	−	0.173648i
$653$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$653$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$654$	0.173648	+	0.300767i	0.173648	+	0.300767i
$655$		0			0
$656$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$657$		0			0
$658$	2.49362	−	1.43969i	2.49362	−	1.43969i
$659$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$659$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$660$		0			0
$661$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$661$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$662$		0			0
$663$		0			0
$664$	1.43969	−	0.524005i	1.43969	−	0.524005i
$665$		0			0
$666$		0			0
$667$	−0.104455	−	0.0603074i	−0.104455	−	0.0603074i
$668$	−0.342020	−	0.0603074i	−0.342020	−	0.0603074i
$669$	0.766044	−	1.32683i	0.766044	−	1.32683i
$670$		0			0
$671$		0			0
$672$	−1.20805	−	1.43969i	−1.20805	−	1.43969i
$673$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$673$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$674$		0			0
$675$		0			0
$676$	1.00000			1.00000
$677$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$677$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−1.00000			−1.00000
$682$		0			0
$683$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$683$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$684$		0			0
$685$		0			0
$686$	2.70574	−	0.984808i	2.70574	−	0.984808i
$687$	−0.118782	−	0.326352i	−0.118782	−	0.326352i
$688$	0.642788	+	0.766044i	0.642788	+	0.766044i
$689$		0			0
$690$		0			0
$691$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$691$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$692$		0			0
$693$		0			0
$694$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$695$		0			0
$696$	0.347296			0.347296
$697$		0			0
$698$	1.50881	−	0.266044i	1.50881	−	0.266044i
$699$		0			0
$700$		0			0
$701$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$701$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.642788	−	1.76604i	0.642788	−	1.76604i
$708$		0			0
$709$	−0.266044	+	1.50881i	−0.266044	+	1.50881i	0.500000	+	0.866025i	$0.333333\pi$
$709$	−0.266044	+	1.50881i	−0.266044	+	1.50881i	−0.766044	+	0.642788i	$0.777778\pi$
$710$		0			0
$711$		0			0
$712$	1.32683	−	0.766044i	1.32683	−	0.766044i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$	−0.939693	+	1.62760i	−0.939693	+	1.62760i
$722$	0.984808	+	0.173648i	0.984808	+	0.173648i
$723$	1.62760	−	0.939693i	1.62760	−	0.939693i
$724$	−1.76604	−	0.642788i	−1.76604	−	0.642788i
$725$		0			0
$726$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$727$	1.20805	−	1.43969i	1.20805	−	1.43969i	0.342020	−	0.939693i	$-0.388889\pi$
$727$	1.20805	−	1.43969i	1.20805	−	1.43969i	0.866025	−	0.500000i	$-0.166667\pi$
$728$		0			0
$729$	0.500000	−	0.866025i	0.500000	−	0.866025i
$730$		0			0
$731$		0			0
$732$	−0.642788	+	1.76604i	−0.642788	+	1.76604i
$733$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$733$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$734$	1.87939	+	0.684040i	1.87939	+	0.684040i
$735$		0			0
$736$	0.0603074	−	0.342020i	0.0603074	−	0.342020i
$737$		0			0
$738$	0.984808	−	1.17365i	0.984808	−	1.17365i
$739$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$739$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−0.984808	−	1.17365i	−0.984808	−	1.17365i	−0.984808	−	0.173648i	$-0.944444\pi$
$743$	−0.984808	−	1.17365i	−0.984808	−	1.17365i		−	1.00000i	$-0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.50881	+	0.266044i	−1.50881	+	0.266044i
$748$		0			0
$749$	−0.613341	+	3.47843i	−0.613341	+	3.47843i
$750$		0			0
$751$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$751$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$752$	−1.50881	+	0.266044i	−1.50881	+	0.266044i
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.939693	+	1.62760i	0.939693	+	1.62760i
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.43969	−	0.524005i	−1.43969	−	0.524005i	−0.500000	−	0.866025i	$-0.666667\pi$
$761$	−1.43969	−	0.524005i	−1.43969	−	0.524005i	−0.939693	+	0.342020i	$0.888889\pi$
$762$			0.347296i			0.347296i
$763$	0.642788	+	0.113341i	0.642788	+	0.113341i
$764$		0			0
$765$		0			0
$766$	0.500000	+	0.866025i	0.500000	+	0.866025i
$767$		0			0
$768$	0.342020	+	0.939693i	0.342020	+	0.939693i
$769$	1.43969	−	1.20805i	1.43969	−	1.20805i	0.500000	−	0.866025i	$-0.333333\pi$
$769$	1.43969	−	1.20805i	1.43969	−	1.20805i	0.939693	−	0.342020i	$-0.111111\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−0.524005	+	1.43969i	−0.524005	+	1.43969i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−0.342020	−	0.0603074i	−0.342020	−	0.0603074i
$784$	−2.53209			−2.53209
$785$		0			0
$786$		0			0
$787$	−1.96962	+	0.347296i	−1.96962	+	0.347296i	−0.984808	+	0.173648i	$0.944444\pi$
$787$	−1.96962	+	0.347296i	−1.96962	+	0.347296i	−0.984808	+	0.173648i	$0.944444\pi$
$788$		0			0
$789$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$797$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$802$	0.866025	+	0.500000i	0.866025	+	0.500000i
$803$		0			0
$804$	0.766044	+	1.32683i	0.766044	+	1.32683i
$805$		0			0
$806$		0			0
$807$	−0.642788	+	1.76604i	−0.642788	+	1.76604i
$808$	−0.642788	+	0.766044i	−0.642788	+	0.766044i
$809$	−2.00000			−2.00000			−1.00000			$\pi$
$809$	−2.00000			−2.00000			−1.00000			$\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	0.419550	−	0.500000i	0.419550	−	0.500000i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$819$		0			0
$820$		0			0
$821$	−0.326352	+	0.118782i	−0.326352	+	0.118782i	−0.500000	−	0.866025i	$-0.666667\pi$
$821$	−0.326352	+	0.118782i	−0.326352	+	0.118782i	0.173648	+	0.984808i	$0.444444\pi$
$822$		0			0
$823$	1.20805	+	1.43969i	1.20805	+	1.43969i	0.866025	+	0.500000i	$0.166667\pi$
$823$	1.20805	+	1.43969i	1.20805	+	1.43969i	0.342020	+	0.939693i	$0.388889\pi$
$824$	0.766044	−	0.642788i	0.766044	−	0.642788i
$825$		0			0
$826$		0			0
$827$	0.300767	−	0.173648i	0.300767	−	0.173648i	−0.342020	−	0.939693i	$-0.611111\pi$
$827$	0.300767	−	0.173648i	0.300767	−	0.173648i	0.642788	+	0.766044i	$0.277778\pi$
$828$	−0.118782	+	0.326352i	−0.118782	+	0.326352i
$829$	0.173648	−	0.300767i	0.173648	−	0.300767i	−0.766044	−	0.642788i	$-0.777778\pi$
$829$	0.173648	−	0.300767i	0.173648	−	0.300767i	0.939693	+	0.342020i	$0.111111\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$839$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$840$		0			0
$841$	−0.152704	−	0.866025i	−0.152704	−	0.866025i
$842$	−0.342020	+	0.939693i	−0.342020	+	0.939693i
$843$			1.87939i			1.87939i
$844$		0			0
$845$		0			0
$846$	1.53209			1.53209
$847$	−1.62760	+	0.939693i	−1.62760	+	0.939693i
$848$		0			0
$849$	0.326352	−	0.118782i	0.326352	−	0.118782i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$853$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$854$	1.76604	+	3.05888i	1.76604	+	3.05888i
$855$		0			0
$856$	0.939693	−	1.62760i	0.939693	−	1.62760i
$857$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$857$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$858$		0			0
$859$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$859$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$860$		0			0
$861$	−0.500000	−	2.83564i	−0.500000	−	2.83564i
$862$		0			0
$863$			1.87939i			1.87939i	0.342020	+	0.939693i	$0.388889\pi$
$863$			1.87939i			1.87939i	−0.342020	+	0.939693i	$0.611111\pi$
$864$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$865$		0			0
$866$		0			0
$867$	0.642788	+	0.766044i	0.642788	+	0.766044i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−0.300767	−	0.173648i	−0.300767	−	0.173648i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$877$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$881$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$882$	2.49362	+	0.439693i	2.49362	+	0.439693i
$883$	1.32683	+	0.766044i	1.32683	+	0.766044i	0.984808	−	0.173648i	$-0.0555556\pi$
$883$	1.32683	+	0.766044i	1.32683	+	0.766044i	0.342020	+	0.939693i	$0.388889\pi$
$884$		0			0
$885$		0			0
$886$	−1.43969	−	0.524005i	−1.43969	−	0.524005i
$887$	0.984808	−	0.173648i	0.984808	−	0.173648i	0.342020	−	0.939693i	$-0.388889\pi$
$887$	0.984808	−	0.173648i	0.984808	−	0.173648i	0.642788	+	0.766044i	$0.277778\pi$
$888$		0			0
$889$	0.500000	+	0.419550i	0.500000	+	0.419550i
$890$		0			0
$891$		0			0
$892$			1.53209i			1.53209i
$893$		0			0
$894$	0.326352	+	0.118782i	0.326352	+	0.118782i
$895$		0			0
$896$	1.76604	+	0.642788i	1.76604	+	0.642788i
$897$		0			0
$898$	0.984808	+	0.173648i	0.984808	+	0.173648i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−1.85083	−	0.326352i	−1.85083	−	0.326352i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.642788	+	1.76604i	0.642788	+	1.76604i	0.642788	+	0.766044i	$0.277778\pi$
$907$	0.642788	+	1.76604i	0.642788	+	1.76604i			1.00000i	$0.5\pi$
$908$	0.866025	−	0.500000i	0.866025	−	0.500000i
$909$	0.766044	−	0.642788i	0.766044	−	0.642788i
$910$		0			0
$911$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$911$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.266044	+	0.223238i	0.266044	+	0.223238i
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$922$	1.85083	−	0.326352i	1.85083	−	0.326352i
$923$		0			0
$924$		0			0
$925$		0			0
$926$	0.500000	−	0.866025i	0.500000	−	0.866025i
$927$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$928$	−0.300767	+	0.173648i	−0.300767	+	0.173648i
$929$	−0.939693	+	0.342020i	−0.939693	+	0.342020i	−0.766044	−	0.642788i	$-0.777778\pi$
$929$	−0.939693	+	0.342020i	−0.939693	+	0.342020i	−0.173648	+	0.984808i	$0.555556\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$935$		0			0
$936$		0			0
$937$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$937$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$938$	2.83564	+	0.500000i	2.83564	+	0.500000i
$939$		0			0
$940$		0			0
$941$	0.266044	+	1.50881i	0.266044	+	1.50881i	0.766044	+	0.642788i	$0.222222\pi$
$941$	0.266044	+	1.50881i	0.266044	+	1.50881i	−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$	0.342020	−	0.407604i	0.342020	−	0.407604i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.984808	+	1.17365i	−0.984808	+	1.17365i			1.00000i	$0.5\pi$
$947$	−0.984808	+	1.17365i	−0.984808	+	1.17365i	−0.984808	+	0.173648i	$0.944444\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$953$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$962$		0			0
$963$	−1.20805	+	1.43969i	−1.20805	+	1.43969i
$964$	−0.939693	+	1.62760i	−0.939693	+	1.62760i
$965$		0			0
$966$	0.326352	+	0.565258i	0.326352	+	0.565258i
$967$	0.118782	−	0.326352i	0.118782	−	0.326352i	−0.866025	−	0.500000i	$-0.833333\pi$
$967$	0.118782	−	0.326352i	0.118782	−	0.326352i	0.984808	+	0.173648i	$0.0555556\pi$
$968$	0.984808	−	0.173648i	0.984808	−	0.173648i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$			1.00000i			1.00000i
$973$		0			0
$974$	−1.53209	−	1.28558i	−1.53209	−	1.28558i
$975$		0			0
$976$	−0.326352	−	1.85083i	−0.326352	−	1.85083i
$977$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$977$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$978$	0.866025	−	0.500000i	0.866025	−	0.500000i
$979$		0			0
$980$		0			0
$981$	0.266044	+	0.223238i	0.266044	+	0.223238i
$982$		0			0
$983$	−0.642788	−	1.76604i	−0.642788	−	1.76604i	−0.642788	−	0.766044i	$-0.722222\pi$
$983$	−0.642788	−	1.76604i	−0.642788	−	1.76604i		−	1.00000i	$-0.5\pi$
$984$	−0.266044	+	1.50881i	−0.266044	+	1.50881i
$985$		0			0
$986$		0			0
$987$	1.85083	−	2.20574i	1.85083	−	2.20574i
$988$		0			0
$989$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	1.17365	−	0.984808i	1.17365	−	0.984808i
$997$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$997$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.1.bj.a.151.1		12
4.3	odd	2	inner	2700.1.bj.a.151.2		12
5.2	odd	4		540.1.bf.b.259.1	yes	6
5.3	odd	4		540.1.bf.a.259.1	✓	6
5.4	even	2	inner	2700.1.bj.a.151.2		12
15.2	even	4		1620.1.bf.b.1099.1		6
15.8	even	4		1620.1.bf.a.1099.1		6
20.3	even	4		540.1.bf.b.259.1	yes	6
20.7	even	4		540.1.bf.a.259.1	✓	6
20.19	odd	2	CM	2700.1.bj.a.151.1		12
27.22	even	9	inner	2700.1.bj.a.751.2		12
60.23	odd	4		1620.1.bf.b.1099.1		6
60.47	odd	4		1620.1.bf.a.1099.1		6
108.103	odd	18	inner	2700.1.bj.a.751.1		12
135.22	odd	36		540.1.bf.b.319.1	yes	6
135.32	even	36		1620.1.bf.b.199.1		6
135.49	even	18	inner	2700.1.bj.a.751.1		12
135.103	odd	36		540.1.bf.a.319.1	yes	6
135.113	even	36		1620.1.bf.a.199.1		6
540.103	even	36		540.1.bf.b.319.1	yes	6
540.167	odd	36		1620.1.bf.a.199.1		6
540.319	odd	18	inner	2700.1.bj.a.751.2		12
540.383	odd	36		1620.1.bf.b.199.1		6
540.427	even	36		540.1.bf.a.319.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.1.bf.a.259.1	✓	6	5.3	odd	4
540.1.bf.a.259.1	✓	6	20.7	even	4
540.1.bf.a.319.1	yes	6	135.103	odd	36
540.1.bf.a.319.1	yes	6	540.427	even	36
540.1.bf.b.259.1	yes	6	5.2	odd	4
540.1.bf.b.259.1	yes	6	20.3	even	4
540.1.bf.b.319.1	yes	6	135.22	odd	36
540.1.bf.b.319.1	yes	6	540.103	even	36
1620.1.bf.a.199.1		6	135.113	even	36
1620.1.bf.a.199.1		6	540.167	odd	36
1620.1.bf.a.1099.1		6	15.8	even	4
1620.1.bf.a.1099.1		6	60.47	odd	4
1620.1.bf.b.199.1		6	135.32	even	36
1620.1.bf.b.199.1		6	540.383	odd	36
1620.1.bf.b.1099.1		6	15.2	even	4
1620.1.bf.b.1099.1		6	60.23	odd	4
2700.1.bj.a.151.1		12	1.1	even	1	trivial
2700.1.bj.a.151.1		12	20.19	odd	2	CM
2700.1.bj.a.151.2		12	4.3	odd	2	inner
2700.1.bj.a.151.2		12	5.4	even	2	inner
2700.1.bj.a.751.1		12	108.103	odd	18	inner
2700.1.bj.a.751.1		12	135.49	even	18	inner
2700.1.bj.a.751.2		12	27.22	even	9	inner
2700.1.bj.a.751.2		12	540.319	odd	18	inner

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2700.1.bj.a.151.1

Level	$2700$
Weight	$1$
Character	2700.151
Analytic conductor	$1.347$
Analytic rank	$0$
Dimension	$12$
Projective image	$D_{9}$
CM discriminant	-20
Inner twists	$8$