LMFDB - Embedded newform 1859.1.i.c.868.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1859,1,Mod(868,1859)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1859, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1859.868");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1859 = 11 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1859.i (of order $6$, degree $2$, not 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:	$0.927761858485$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	12.0.17213603549184.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 $ x^12 - 5x^10 + 19x^8 - 28x^6 + 31x^4 - 6*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.6424482779.1
Artin image:	$C_{12}\times D_7$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{84} - \cdots)$

Embedding invariants

Embedding label			868.6
Root			$1.56052 + 0.900969i$ of defining polynomial
Character	$\chi$	$=$	1859.868
Dual form			1859.1.i.c.1330.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.900969 + 1.56052i) q^{3} +(0.500000 - 0.866025i) q^{4} +1.24698i q^{5} +(-1.12349 + 1.94594i) q^{9} +O(q^{10})$	$q+(0.900969 + 1.56052i) q^{3} +(0.500000 - 0.866025i) q^{4} +1.24698i q^{5} +(-1.12349 + 1.94594i) q^{9} +(-0.866025 + 0.500000i) q^{11} +1.80194 q^{12} +(-1.94594 + 1.12349i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(1.07992 + 0.623490i) q^{20} +(-0.222521 - 0.385418i) q^{23} -0.554958 q^{25} -2.24698 q^{27} -0.445042i q^{31} +(-1.56052 - 0.900969i) q^{33} +(1.12349 + 1.94594i) q^{36} +(1.56052 - 0.900969i) q^{37} +1.00000i q^{44} +(-2.42655 - 1.40097i) q^{45} +0.445042i q^{47} +(0.900969 - 1.56052i) q^{48} +(0.500000 + 0.866025i) q^{49} +1.24698 q^{53} +(-0.623490 - 1.07992i) q^{55} +(-1.56052 - 0.900969i) q^{59} +2.24698i q^{60} -1.00000 q^{64} +(1.73205 - 1.00000i) q^{67} +(0.400969 - 0.694498i) q^{69} +(1.56052 + 0.900969i) q^{71} +(-0.500000 - 0.866025i) q^{75} +(1.07992 - 0.623490i) q^{80} +(-0.900969 - 1.56052i) q^{81} +(0.385418 - 0.222521i) q^{89} -0.445042 q^{92} +(0.694498 - 0.400969i) q^{93} +(-1.07992 - 0.623490i) q^{97} -2.24698i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{3} + 6 q^{4} - 4 q^{9}+O(q^{10}) $ 12 * q + 2 * q^3 + 6 * q^4 - 4 * q^9	$ 12 q + 2 q^{3} + 6 q^{4} - 4 q^{9} + 4 q^{12} - 6 q^{16} - 2 q^{23} - 8 q^{25} - 8 q^{27} + 4 q^{36} + 2 q^{48} + 6 q^{49} - 4 q^{53} + 2 q^{55} - 12 q^{64} - 4 q^{69} - 6 q^{75} - 2 q^{81} - 4 q^{92}+O(q^{100}) $ 12 * q + 2 * q^3 + 6 * q^4 - 4 * q^9 + 4 * q^12 - 6 * q^16 - 2 * q^23 - 8 * q^25 - 8 * q^27 + 4 * q^36 + 2 * q^48 + 6 * q^49 - 4 * q^53 + 2 * q^55 - 12 * q^64 - 4 * q^69 - 6 * q^75 - 2 * q^81 - 4 * q^92

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1859\mathbb{Z}\right)^\times$.

$n$	$508$	$1354$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$2$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$3$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$3$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$4$	0.500000	−	0.866025i	0.500000	−	0.866025i
$5$			1.24698i			1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$5$			1.24698i			1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$6$		0			0
$7$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$7$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$8$		0			0
$9$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$10$		0			0
$11$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$11$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$12$	1.80194			1.80194
$13$		0			0
$13$		0			0
$14$		0			0
$15$	−1.94594	+	1.12349i	−1.94594	+	1.12349i
$16$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$		0			0
$19$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$19$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$20$	1.07992	+	0.623490i	1.07992	+	0.623490i
$21$		0			0
$22$		0			0
$23$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	0.733052	−	0.680173i	$-0.238095\pi$
$23$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	−0.955573	+	0.294755i	$0.904762\pi$
$24$		0			0
$25$	−0.554958			−0.554958
$26$		0			0
$27$	−2.24698			−2.24698
$28$		0			0
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$		−	0.445042i		−	0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$31$		−	0.445042i		−	0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$32$		0			0
$33$	−1.56052	−	0.900969i	−1.56052	−	0.900969i
$34$		0			0
$35$		0			0
$36$	1.12349	+	1.94594i	1.12349	+	1.94594i
$37$	1.56052	−	0.900969i	1.56052	−	0.900969i	0.563320	−	0.826239i	$-0.309524\pi$
$37$	1.56052	−	0.900969i	1.56052	−	0.900969i	0.997204	−	0.0747301i	$-0.0238095\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$			1.00000i			1.00000i
$45$	−2.42655	−	1.40097i	−2.42655	−	1.40097i
$46$		0			0
$47$			0.445042i			0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$47$			0.445042i			0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$48$	0.900969	−	1.56052i	0.900969	−	1.56052i
$49$	0.500000	+	0.866025i	0.500000	+	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$53$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$54$		0			0
$55$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.56052	−	0.900969i	−1.56052	−	0.900969i	−0.997204	−	0.0747301i	$-0.976190\pi$
$59$	−1.56052	−	0.900969i	−1.56052	−	0.900969i	−0.563320	−	0.826239i	$-0.690476\pi$
$60$			2.24698i			2.24698i
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$	−1.00000			−1.00000
$65$		0			0
$66$		0			0
$67$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$67$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$68$		0			0
$69$	0.400969	−	0.694498i	0.400969	−	0.694498i
$70$		0			0
$71$	1.56052	+	0.900969i	1.56052	+	0.900969i	0.997204	+	0.0747301i	$0.0238095\pi$
$71$	1.56052	+	0.900969i	1.56052	+	0.900969i	0.563320	+	0.826239i	$0.309524\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	1.07992	−	0.623490i	1.07992	−	0.623490i
$81$	−0.900969	−	1.56052i	−0.900969	−	1.56052i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.385418	−	0.222521i	0.385418	−	0.222521i	−0.294755	−	0.955573i	$-0.595238\pi$
$89$	0.385418	−	0.222521i	0.385418	−	0.222521i	0.680173	+	0.733052i	$0.261905\pi$
$90$		0			0
$91$		0			0
$92$	−0.445042			−0.445042
$93$	0.694498	−	0.400969i	0.694498	−	0.400969i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.07992	−	0.623490i	−1.07992	−	0.623490i	−0.149042	−	0.988831i	$-0.547619\pi$
$97$	−1.07992	−	0.623490i	−1.07992	−	0.623490i	−0.930874	+	0.365341i	$0.880952\pi$
$98$		0			0
$99$		−	2.24698i		−	2.24698i
$100$	−0.277479	+	0.480608i	−0.277479	+	0.480608i
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$	−1.24698			−1.24698			−0.623490	−	0.781831i	$-0.714286\pi$
$103$	−1.24698			−1.24698			−0.623490	+	0.781831i	$0.714286\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	2.81197	+	1.62349i	2.81197	+	1.62349i
$112$		0			0
$113$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$113$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$114$		0			0
$115$	0.480608	−	0.277479i	0.480608	−	0.277479i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.500000	−	0.866025i	0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	−0.385418	−	0.222521i	−0.385418	−	0.222521i
$125$			0.554958i			0.554958i
$126$		0			0
$127$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$127$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	−1.56052	+	0.900969i	−1.56052	+	0.900969i
$133$		0			0
$134$		0			0
$135$		−	2.80194i		−	2.80194i
$136$		0			0
$137$	−0.385418	−	0.222521i	−0.385418	−	0.222521i	0.294755	−	0.955573i	$-0.404762\pi$
$137$	−0.385418	−	0.222521i	−0.385418	−	0.222521i	−0.680173	+	0.733052i	$0.738095\pi$
$138$		0			0
$139$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$139$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$140$		0			0
$141$	−0.694498	+	0.400969i	−0.694498	+	0.400969i
$142$		0			0
$143$		0			0
$144$	2.24698			2.24698
$145$		0			0
$146$		0			0
$147$	−0.900969	+	1.56052i	−0.900969	+	1.56052i
$148$		−	1.80194i		−	1.80194i
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.554958			0.554958
$156$		0			0
$157$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$157$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$158$		0			0
$159$	1.12349	+	1.94594i	1.12349	+	1.94594i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.385418	−	0.222521i	−0.385418	−	0.222521i	0.294755	−	0.955573i	$-0.404762\pi$
$163$	−0.385418	−	0.222521i	−0.385418	−	0.222521i	−0.680173	+	0.733052i	$0.738095\pi$
$164$		0			0
$165$	1.12349	−	1.94594i	1.12349	−	1.94594i
$166$		0			0
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$		0			0
$176$	0.866025	+	0.500000i	0.866025	+	0.500000i
$177$		−	3.24698i		−	3.24698i
$178$		0			0
$179$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.826239	−	0.563320i	$-0.809524\pi$
$179$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.0747301	−	0.997204i	$-0.523810\pi$
$180$	−2.42655	+	1.40097i	−2.42655	+	1.40097i
$181$	0.445042			0.445042			0.222521	−	0.974928i	$-0.428571\pi$
$181$	0.445042			0.445042			0.222521	+	0.974928i	$0.428571\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.12349	+	1.94594i	1.12349	+	1.94594i
$186$		0			0
$187$		0			0
$188$	0.385418	+	0.222521i	0.385418	+	0.222521i
$189$		0			0
$190$		0			0
$191$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$191$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$192$	−0.900969	−	1.56052i	−0.900969	−	1.56052i
$193$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$193$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$194$		0			0
$195$		0			0
$196$	1.00000			1.00000
$197$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$197$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$198$		0			0
$199$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.0747301	+	0.997204i	$0.523810\pi$
$199$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.826239	+	0.563320i	$0.809524\pi$
$200$		0			0
$201$	3.12105	+	1.80194i	3.12105	+	1.80194i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.00000			1.00000
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$211$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$212$	0.623490	−	1.07992i	0.623490	−	1.07992i
$213$			3.24698i			3.24698i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−1.24698			−1.24698
$221$		0			0
$222$		0			0
$223$	1.07992	−	0.623490i	1.07992	−	0.623490i	0.149042	−	0.988831i	$-0.452381\pi$
$223$	1.07992	−	0.623490i	1.07992	−	0.623490i	0.930874	+	0.365341i	$0.119048\pi$
$224$		0			0
$225$	0.623490	−	1.07992i	0.623490	−	1.07992i
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$			1.80194i			1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$229$			1.80194i			1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$	−0.554958			−0.554958
$236$	−1.56052	+	0.900969i	−1.56052	+	0.900969i
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	1.94594	+	1.12349i	1.94594	+	1.12349i
$241$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$241$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$242$		0			0
$243$	0.500000	−	0.866025i	0.500000	−	0.866025i
$244$		0			0
$245$	−1.07992	+	0.623490i	−1.07992	+	0.623490i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	0.623490	−	1.07992i	0.623490	−	1.07992i	−0.365341	−	0.930874i	$-0.619048\pi$
$251$	0.623490	−	1.07992i	0.623490	−	1.07992i	0.988831	−	0.149042i	$-0.0476190\pi$
$252$		0			0
$253$	0.385418	+	0.222521i	0.385418	+	0.222521i
$254$		0			0
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.826239	−	0.563320i	$-0.809524\pi$
$257$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.0747301	−	0.997204i	$-0.523810\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$			1.55496i			1.55496i
$266$		0			0
$267$	0.694498	+	0.400969i	0.694498	+	0.400969i
$268$		−	2.00000i		−	2.00000i
$269$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$269$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$270$		0			0
$271$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.480608	−	0.277479i	0.480608	−	0.277479i
$276$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$	0.866025	+	0.500000i	0.866025	+	0.500000i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$	1.56052	−	0.900969i	1.56052	−	0.900969i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$290$		0			0
$291$		−	2.24698i		−	2.24698i
$292$		0			0
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$		0			0
$295$	1.12349	−	1.94594i	1.12349	−	1.94594i
$296$		0			0
$297$	1.94594	−	1.12349i	1.94594	−	1.12349i
$298$		0			0
$299$		0			0
$300$	−1.00000			−1.00000
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$310$		0			0
$311$	0.445042			0.445042			0.222521	−	0.974928i	$-0.428571\pi$
$311$	0.445042			0.445042			0.222521	+	0.974928i	$0.428571\pi$
$312$		0			0
$313$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$313$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	1.80194i		−	1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$317$		−	1.80194i		−	1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$318$		0			0
$319$		0			0
$320$		−	1.24698i		−	1.24698i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−1.80194			−1.80194
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.07992	−	0.623490i	−1.07992	−	0.623490i	−0.149042	−	0.988831i	$-0.547619\pi$
$331$	−1.07992	−	0.623490i	−1.07992	−	0.623490i	−0.930874	+	0.365341i	$0.880952\pi$
$332$		0			0
$333$			4.04892i			4.04892i
$334$		0			0
$335$	1.24698	+	2.15983i	1.24698	+	2.15983i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$	−2.24698			−2.24698
$340$		0			0
$341$	0.222521	+	0.385418i	0.222521	+	0.385418i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	0.866025	+	0.500000i	0.866025	+	0.500000i
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$349$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.56052	+	0.900969i	−1.56052	+	0.900969i	−0.563320	+	0.826239i	$0.690476\pi$
$353$	−1.56052	+	0.900969i	−1.56052	+	0.900969i	−0.997204	+	0.0747301i	$0.976190\pi$
$354$		0			0
$355$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$356$		−	0.445042i		−	0.445042i
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	0.500000	+	0.866025i	0.500000	+	0.866025i
$362$		0			0
$363$	1.80194			1.80194
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$367$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$368$	−0.222521	+	0.385418i	−0.222521	+	0.385418i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		−	0.801938i		−	0.801938i
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.07992	−	0.623490i	1.07992	−	0.623490i	0.149042	−	0.988831i	$-0.452381\pi$
$379$	1.07992	−	0.623490i	1.07992	−	0.623490i	0.930874	+	0.365341i	$0.119048\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$383$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−1.07992	+	0.623490i	−1.07992	+	0.623490i
$389$	1.80194			1.80194			0.900969	−	0.433884i	$-0.142857\pi$
$389$	1.80194			1.80194			0.900969	+	0.433884i	$0.142857\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−1.94594	−	1.12349i	−1.94594	−	1.12349i
$397$	−1.56052	−	0.900969i	−1.56052	−	0.900969i	−0.997204	−	0.0747301i	$-0.976190\pi$
$397$	−1.56052	−	0.900969i	−1.56052	−	0.900969i	−0.563320	−	0.826239i	$-0.690476\pi$
$398$		0			0
$399$		0			0
$400$	0.277479	+	0.480608i	0.277479	+	0.480608i
$401$	0.385418	−	0.222521i	0.385418	−	0.222521i	−0.294755	−	0.955573i	$-0.595238\pi$
$401$	0.385418	−	0.222521i	0.385418	−	0.222521i	0.680173	+	0.733052i	$0.261905\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	1.94594	−	1.12349i	1.94594	−	1.12349i
$406$		0			0
$407$	−0.900969	+	1.56052i	−0.900969	+	1.56052i
$408$		0			0
$409$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$409$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$410$		0			0
$411$		−	0.801938i		−	0.801938i
$412$	−0.623490	+	1.07992i	−0.623490	+	1.07992i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$419$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$420$		0			0
$421$			1.24698i			1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$421$			1.24698i			1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$422$		0			0
$423$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$431$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$432$	1.12349	+	1.94594i	1.12349	+	1.94594i
$433$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	−0.955573	−	0.294755i	$-0.904762\pi$
$433$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	0.733052	+	0.680173i	$0.238095\pi$
$434$		0			0
$435$		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	$0.333333\pi$
$440$		0			0
$441$	−2.24698			−2.24698
$442$		0			0
$443$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$443$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$444$	2.81197	−	1.62349i	2.81197	−	1.62349i
$445$	0.277479	+	0.480608i	0.277479	+	0.480608i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$449$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$450$		0			0
$451$		0			0
$452$	0.623490	+	1.07992i	0.623490	+	1.07992i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$457$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$458$		0			0
$459$		0			0
$460$		−	0.554958i		−	0.554958i
$461$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$461$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$462$		0			0
$463$			1.80194i			1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$463$			1.80194i			1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$464$		0			0
$465$	0.500000	+	0.866025i	0.500000	+	0.866025i
$466$		0			0
$467$	−2.00000			−2.00000			−1.00000			$\pi$
$467$	−2.00000			−2.00000			−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−1.62349	−	2.81197i	−1.62349	−	2.81197i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.40097	+	2.42655i	−1.40097	+	2.42655i
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$485$	0.777479	−	1.34663i	0.777479	−	1.34663i
$486$		0			0
$487$	−1.07992	−	0.623490i	−1.07992	−	0.623490i	−0.149042	−	0.988831i	$-0.547619\pi$
$487$	−1.07992	−	0.623490i	−1.07992	−	0.623490i	−0.930874	+	0.365341i	$0.880952\pi$
$488$		0			0
$489$		−	0.801938i		−	0.801938i
$490$		0			0
$491$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$491$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	2.80194			2.80194
$496$	−0.385418	+	0.222521i	−0.385418	+	0.222521i
$497$		0			0
$498$		0			0
$499$		−	1.80194i		−	1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$499$		−	1.80194i		−	1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$500$	0.480608	+	0.277479i	0.480608	+	0.277479i
$501$		0			0
$502$		0			0
$503$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$503$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.385418	+	0.222521i	−0.385418	+	0.222521i	−0.680173	−	0.733052i	$-0.738095\pi$
$509$	−0.385418	+	0.222521i	−0.385418	+	0.222521i	0.294755	+	0.955573i	$0.404762\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	1.55496i		−	1.55496i
$516$		0			0
$517$	−0.222521	−	0.385418i	−0.222521	−	0.385418i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	2.00000			2.00000			1.00000			$0$
$521$	2.00000			2.00000			1.00000			$0$
$522$		0			0
$523$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$523$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$			1.80194i			1.80194i
$529$	0.400969	−	0.694498i	0.400969	−	0.694498i
$530$		0			0
$531$	3.50647	−	2.02446i	3.50647	−	2.02446i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	1.62349	−	2.81197i	1.62349	−	2.81197i
$538$		0			0
$539$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$540$	−2.42655	−	1.40097i	−2.42655	−	1.40097i
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$	0.400969	+	0.694498i	0.400969	+	0.694498i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	−0.385418	+	0.222521i	−0.385418	+	0.222521i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−2.02446	+	3.50647i	−2.02446	+	3.50647i
$556$		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			0
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$			0.801938i			0.801938i
$565$	−1.34663	−	0.777479i	−1.34663	−	0.777479i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	0.801938			0.801938
$574$		0			0
$575$	0.123490	+	0.213891i	0.123490	+	0.213891i
$576$	1.12349	−	1.94594i	1.12349	−	1.94594i
$577$		−	0.445042i		−	0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$577$		−	0.445042i		−	0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.07992	+	0.623490i	−1.07992	+	0.623490i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.56052	+	0.900969i	−1.56052	+	0.900969i	−0.563320	+	0.826239i	$0.690476\pi$
$587$	−1.56052	+	0.900969i	−1.56052	+	0.900969i	−0.997204	+	0.0747301i	$0.976190\pi$
$588$	0.900969	+	1.56052i	0.900969	+	1.56052i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.56052	−	0.900969i	−1.56052	−	0.900969i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−3.24698			−3.24698
$598$		0			0
$599$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$599$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$600$		0			0
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$			4.49396i			4.49396i
$604$		0			0
$605$	1.07992	+	0.623490i	1.07992	+	0.623490i
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$613$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.56052	+	0.900969i	1.56052	+	0.900969i	0.997204	+	0.0747301i	$0.0238095\pi$
$617$	1.56052	+	0.900969i	1.56052	+	0.900969i	0.563320	+	0.826239i	$0.309524\pi$
$618$		0			0
$619$			0.445042i			0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$619$			0.445042i			0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$620$	0.277479	−	0.480608i	0.277479	−	0.480608i
$621$	0.500000	+	0.866025i	0.500000	+	0.866025i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.24698			−1.24698
$626$		0			0
$627$		0			0
$628$	−0.900969	+	1.56052i	−0.900969	+	1.56052i
$629$		0			0
$630$		0			0
$631$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.930874	−	0.365341i	$-0.119048\pi$
$631$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.149042	+	0.988831i	$0.452381\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	2.24698			2.24698
$637$		0			0
$638$		0			0
$639$	−3.50647	+	2.02446i	−3.50647	+	2.02446i
$640$		0			0
$641$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.0747301	+	0.997204i	$0.523810\pi$
$641$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.826239	+	0.563320i	$0.809524\pi$
$642$		0			0
$643$	0.385418	+	0.222521i	0.385418	+	0.222521i	0.680173	−	0.733052i	$-0.261905\pi$
$643$	0.385418	+	0.222521i	0.385418	+	0.222521i	−0.294755	+	0.955573i	$0.595238\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$647$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$648$		0			0
$649$	1.80194			1.80194
$650$		0			0
$651$		0			0
$652$	−0.385418	+	0.222521i	−0.385418	+	0.222521i
$653$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$653$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$661$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$661$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	1.94594	+	1.12349i	1.94594	+	1.12349i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$673$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$674$		0			0
$675$	1.24698			1.24698
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.56052	−	0.900969i	−1.56052	−	0.900969i	−0.997204	−	0.0747301i	$-0.976190\pi$
$683$	−1.56052	−	0.900969i	−1.56052	−	0.900969i	−0.563320	−	0.826239i	$-0.690476\pi$
$684$		0			0
$685$	0.277479	−	0.480608i	0.277479	−	0.480608i
$686$		0			0
$687$	−2.81197	+	1.62349i	−2.81197	+	1.62349i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−1.56052	+	0.900969i	−1.56052	+	0.900969i	−0.563320	+	0.826239i	$0.690476\pi$
$691$	−1.56052	+	0.900969i	−1.56052	+	0.900969i	−0.997204	+	0.0747301i	$0.976190\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	0.866025	−	0.500000i	0.866025	−	0.500000i
$705$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$706$		0			0
$707$		0			0
$708$	−2.81197	−	1.62349i	−2.81197	−	1.62349i
$709$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.930874	−	0.365341i	$-0.119048\pi$
$709$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.149042	+	0.988831i	$0.452381\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.171527	+	0.0990311i	−0.171527	+	0.0990311i
$714$		0			0
$715$		0			0
$716$	−1.80194			−1.80194
$717$		0			0
$718$		0			0
$719$	0.623490	−	1.07992i	0.623490	−	1.07992i	−0.365341	−	0.930874i	$-0.619048\pi$
$719$	0.623490	−	1.07992i	0.623490	−	1.07992i	0.988831	−	0.149042i	$-0.0476190\pi$
$720$			2.80194i			2.80194i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	0.222521	−	0.385418i	0.222521	−	0.385418i
$725$		0			0
$726$		0			0
$727$	0.445042			0.445042			0.222521	−	0.974928i	$-0.428571\pi$
$727$	0.445042			0.445042			0.222521	+	0.974928i	$0.428571\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$	−1.94594	−	1.12349i	−1.94594	−	1.12349i
$736$		0			0
$737$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$738$		0			0
$739$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$739$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$740$	2.24698			2.24698
$741$		0			0
$742$		0			0
$743$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$743$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.623490	+	1.07992i	0.623490	+	1.07992i	0.988831	+	0.149042i	$0.0476190\pi$
$751$	0.623490	+	1.07992i	0.623490	+	1.07992i	−0.365341	+	0.930874i	$0.619048\pi$
$752$	0.385418	−	0.222521i	0.385418	−	0.222521i
$753$	2.24698			2.24698
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$757$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$758$		0			0
$759$			0.801938i			0.801938i
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$		0			0
$764$	−0.222521	−	0.385418i	−0.222521	−	0.385418i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−1.80194			−1.80194
$769$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$769$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$770$		0			0
$771$	1.62349	−	2.81197i	1.62349	−	2.81197i
$772$		0			0
$773$	0.385418	+	0.222521i	0.385418	+	0.222521i	0.680173	−	0.733052i	$-0.261905\pi$
$773$	0.385418	+	0.222521i	0.385418	+	0.222521i	−0.294755	+	0.955573i	$0.595238\pi$
$774$		0			0
$775$			0.246980i			0.246980i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−1.80194			−1.80194
$782$		0			0
$783$		0			0
$784$	0.500000	−	0.866025i	0.500000	−	0.866025i
$785$		−	2.24698i		−	2.24698i
$786$		0			0
$787$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$787$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−2.42655	+	1.40097i	−2.42655	+	1.40097i
$796$	0.900969	+	1.56052i	0.900969	+	1.56052i
$797$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	−0.955573	−	0.294755i	$-0.904762\pi$
$797$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	0.733052	+	0.680173i	$0.238095\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$			1.00000i			1.00000i
$802$		0			0
$803$		0			0
$804$	3.12105	−	1.80194i	3.12105	−	1.80194i
$805$		0			0
$806$		0			0
$807$	3.24698			3.24698
$808$		0			0
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.277479	−	0.480608i	0.277479	−	0.480608i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$821$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$822$		0			0
$823$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	−0.955573	−	0.294755i	$-0.904762\pi$
$823$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	0.733052	+	0.680173i	$0.238095\pi$
$824$		0			0
$825$	0.866025	+	0.500000i	0.866025	+	0.500000i
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	0.500000	−	0.866025i	0.500000	−	0.866025i
$829$	0.623490	+	1.07992i	0.623490	+	1.07992i	0.988831	+	0.149042i	$0.0476190\pi$
$829$	0.623490	+	1.07992i	0.623490	+	1.07992i	−0.365341	+	0.930874i	$0.619048\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$			1.00000i			1.00000i
$838$		0			0
$839$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.930874	−	0.365341i	$-0.119048\pi$
$839$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.149042	+	0.988831i	$0.452381\pi$
$840$		0			0
$841$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$849$		0			0
$850$		0			0
$851$	−0.694498	−	0.400969i	−0.694498	−	0.400969i
$852$	2.81197	+	1.62349i	2.81197	+	1.62349i
$853$		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			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$859$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1.24698i			1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$863$			1.24698i			1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.900969	−	1.56052i	0.900969	−	1.56052i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	2.42655	−	1.40097i	2.42655	−	1.40097i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$877$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$878$		0			0
$879$		0			0
$880$	−0.623490	+	1.07992i	−0.623490	+	1.07992i
$881$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.826239	−	0.563320i	$-0.809524\pi$
$881$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.0747301	−	0.997204i	$-0.523810\pi$
$882$		0			0
$883$	−1.24698			−1.24698			−0.623490	−	0.781831i	$-0.714286\pi$
$883$	−1.24698			−1.24698			−0.623490	+	0.781831i	$0.714286\pi$
$884$		0			0
$885$	4.04892			4.04892
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1.56052	+	0.900969i	1.56052	+	0.900969i
$892$		−	1.24698i		−	1.24698i
$893$		0			0
$894$		0			0
$895$	1.94594	−	1.12349i	1.94594	−	1.12349i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$			0.554958i			0.554958i
$906$		0			0
$907$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.826239	−	0.563320i	$-0.809524\pi$
$907$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.0747301	−	0.997204i	$-0.523810\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$911$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.56052	+	0.900969i	1.56052	+	0.900969i
$917$		0			0
$918$		0			0
$919$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$919$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$926$		0			0
$927$	1.40097	−	2.42655i	1.40097	−	2.42655i
$928$		0			0
$929$	−1.07992	−	0.623490i	−1.07992	−	0.623490i	−0.149042	−	0.988831i	$-0.547619\pi$
$929$	−1.07992	−	0.623490i	−1.07992	−	0.623490i	−0.930874	+	0.365341i	$0.880952\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	0.400969	+	0.694498i	0.400969	+	0.694498i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$	1.12349	+	1.94594i	1.12349	+	1.94594i
$940$	−0.277479	+	0.480608i	−0.277479	+	0.480608i
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$			1.80194i			1.80194i
$945$		0			0
$946$		0			0
$947$	−1.07992	+	0.623490i	−1.07992	+	0.623490i	−0.930874	−	0.365341i	$-0.880952\pi$
$947$	−1.07992	+	0.623490i	−1.07992	+	0.623490i	−0.149042	+	0.988831i	$0.547619\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	2.81197	−	1.62349i	2.81197	−	1.62349i
$952$		0			0
$953$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$953$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$954$		0			0
$955$	0.480608	+	0.277479i	0.480608	+	0.277479i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	1.94594	−	1.12349i	1.94594	−	1.12349i
$961$	0.801938			0.801938
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$971$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$972$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.07992	−	0.623490i	1.07992	−	0.623490i	0.149042	−	0.988831i	$-0.452381\pi$
$977$	1.07992	−	0.623490i	1.07992	−	0.623490i	0.930874	+	0.365341i	$0.119048\pi$
$978$		0			0
$979$	−0.222521	+	0.385418i	−0.222521	+	0.385418i
$980$			1.24698i			1.24698i
$981$		0			0
$982$		0			0
$983$			0.445042i			0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$983$			0.445042i			0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$991$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$992$		0			0
$993$		−	2.24698i		−	2.24698i
$994$		0			0
$995$	−1.94594	−	1.12349i	−1.94594	−	1.12349i
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$		0			0
$999$	−3.50647	+	2.02446i	−3.50647	+	2.02446i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1859.1.i.c.868.6		12
11.10	odd	2	CM	1859.1.i.c.868.6		12
13.2	odd	12		1859.1.c.a.846.1	✓	3
13.3	even	3		1859.1.d.a.1858.2		6
13.4	even	6	inner	1859.1.i.c.1330.5		12
13.5	odd	4		1859.1.k.a.1836.3		6
13.6	odd	12		1859.1.k.a.1374.3		6
13.7	odd	12		1859.1.k.b.1374.3		6
13.8	odd	4		1859.1.k.b.1836.3		6
13.9	even	3	inner	1859.1.i.c.1330.6		12
13.10	even	6		1859.1.d.a.1858.1		6
13.11	odd	12		1859.1.c.b.846.1	yes	3
13.12	even	2	inner	1859.1.i.c.868.5		12
143.10	odd	6		1859.1.d.a.1858.1		6
143.21	even	4		1859.1.k.b.1836.3		6
143.32	even	12		1859.1.k.a.1374.3		6
143.43	odd	6	inner	1859.1.i.c.1330.5		12
143.54	even	12		1859.1.c.a.846.1	✓	3
143.76	even	12		1859.1.c.b.846.1	yes	3
143.87	odd	6	inner	1859.1.i.c.1330.6		12
143.98	even	12		1859.1.k.b.1374.3		6
143.109	even	4		1859.1.k.a.1836.3		6
143.120	odd	6		1859.1.d.a.1858.2		6
143.142	odd	2	inner	1859.1.i.c.868.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1859.1.c.a.846.1	✓	3	13.2	odd	12
1859.1.c.a.846.1	✓	3	143.54	even	12
1859.1.c.b.846.1	yes	3	13.11	odd	12
1859.1.c.b.846.1	yes	3	143.76	even	12
1859.1.d.a.1858.1		6	13.10	even	6
1859.1.d.a.1858.1		6	143.10	odd	6
1859.1.d.a.1858.2		6	13.3	even	3
1859.1.d.a.1858.2		6	143.120	odd	6
1859.1.i.c.868.5		12	13.12	even	2	inner
1859.1.i.c.868.5		12	143.142	odd	2	inner
1859.1.i.c.868.6		12	1.1	even	1	trivial
1859.1.i.c.868.6		12	11.10	odd	2	CM
1859.1.i.c.1330.5		12	13.4	even	6	inner
1859.1.i.c.1330.5		12	143.43	odd	6	inner
1859.1.i.c.1330.6		12	13.9	even	3	inner
1859.1.i.c.1330.6		12	143.87	odd	6	inner
1859.1.k.a.1374.3		6	13.6	odd	12
1859.1.k.a.1374.3		6	143.32	even	12
1859.1.k.a.1836.3		6	13.5	odd	4
1859.1.k.a.1836.3		6	143.109	even	4
1859.1.k.b.1374.3		6	13.7	odd	12
1859.1.k.b.1374.3		6	143.98	even	12
1859.1.k.b.1836.3		6	13.8	odd	4
1859.1.k.b.1836.3		6	143.21	even	4

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:	\( N \)	\(=\)	\( 1859 = 11 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1859.i (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.900969 + 1.56052i) q^{3} +(0.500000 - 0.866025i) q^{4} +1.24698i q^{5} +(-1.12349 + 1.94594i) q^{9} +O(q^{10})\)	\(q+(0.900969 + 1.56052i) q^{3} +(0.500000 - 0.866025i) q^{4} +1.24698i q^{5} +(-1.12349 + 1.94594i) q^{9} +(-0.866025 + 0.500000i) q^{11} +1.80194 q^{12} +(-1.94594 + 1.12349i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(1.07992 + 0.623490i) q^{20} +(-0.222521 - 0.385418i) q^{23} -0.554958 q^{25} -2.24698 q^{27} -0.445042i q^{31} +(-1.56052 - 0.900969i) q^{33} +(1.12349 + 1.94594i) q^{36} +(1.56052 - 0.900969i) q^{37} +1.00000i q^{44} +(-2.42655 - 1.40097i) q^{45} +0.445042i q^{47} +(0.900969 - 1.56052i) q^{48} +(0.500000 + 0.866025i) q^{49} +1.24698 q^{53} +(-0.623490 - 1.07992i) q^{55} +(-1.56052 - 0.900969i) q^{59} +2.24698i q^{60} -1.00000 q^{64} +(1.73205 - 1.00000i) q^{67} +(0.400969 - 0.694498i) q^{69} +(1.56052 + 0.900969i) q^{71} +(-0.500000 - 0.866025i) q^{75} +(1.07992 - 0.623490i) q^{80} +(-0.900969 - 1.56052i) q^{81} +(0.385418 - 0.222521i) q^{89} -0.445042 q^{92} +(0.694498 - 0.400969i) q^{93} +(-1.07992 - 0.623490i) q^{97} -2.24698i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{3} + 6 q^{4} - 4 q^{9}+O(q^{10}) \) 12 * q + 2 * q^3 + 6 * q^4 - 4 * q^9	\( 12 q + 2 q^{3} + 6 q^{4} - 4 q^{9} + 4 q^{12} - 6 q^{16} - 2 q^{23} - 8 q^{25} - 8 q^{27} + 4 q^{36} + 2 q^{48} + 6 q^{49} - 4 q^{53} + 2 q^{55} - 12 q^{64} - 4 q^{69} - 6 q^{75} - 2 q^{81} - 4 q^{92}+O(q^{100}) \) 12 * q + 2 * q^3 + 6 * q^4 - 4 * q^9 + 4 * q^12 - 6 * q^16 - 2 * q^23 - 8 * q^25 - 8 * q^27 + 4 * q^36 + 2 * q^48 + 6 * q^49 - 4 * q^53 + 2 * q^55 - 12 * q^64 - 4 * q^69 - 6 * q^75 - 2 * q^81 - 4 * q^92

Introduction

L-functions

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Database

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