LMFDB - Embedded newform 3380.1.w.e.699.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,1,Mod(699,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 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("3380.699");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3380.w (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:	$1.68683974270$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.38614472000.1

Embedding invariants

Embedding label			699.3
Root			$-0.385418 - 0.222521i$ of defining polynomial
Character	$\chi$	$=$	3380.699
Dual form			3380.1.w.e.1499.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.866025 + 0.500000i) q^{2} +(0.623490 + 1.07992i) q^{3} +(0.500000 - 0.866025i) q^{4} +1.00000i q^{5} +(-1.07992 - 0.623490i) q^{6} +(1.56052 + 0.900969i) q^{7} +1.00000i q^{8} +(-0.277479 + 0.480608i) q^{9} +(-0.500000 - 0.866025i) q^{10} +1.24698 q^{12} -1.80194 q^{14} +(-1.07992 + 0.623490i) q^{15} +(-0.500000 - 0.866025i) q^{16} -0.554958i q^{18} +(0.866025 + 0.500000i) q^{20} +2.24698i q^{21} +(0.222521 + 0.385418i) q^{23} +(-1.07992 + 0.623490i) q^{24} -1.00000 q^{25} +0.554958 q^{27} +(1.56052 - 0.900969i) q^{28} +(-0.623490 - 1.07992i) q^{29} +(0.623490 - 1.07992i) q^{30} +(0.866025 + 0.500000i) q^{32} +(-0.900969 + 1.56052i) q^{35} +(0.277479 + 0.480608i) q^{36} -1.00000 q^{40} +(-1.56052 + 0.900969i) q^{41} +(-1.12349 - 1.94594i) q^{42} +(0.222521 - 0.385418i) q^{43} +(-0.480608 - 0.277479i) q^{45} +(-0.385418 - 0.222521i) q^{46} -0.445042i q^{47} +(0.623490 - 1.07992i) q^{48} +(1.12349 + 1.94594i) q^{49} +(0.866025 - 0.500000i) q^{50} +(-0.480608 + 0.277479i) q^{54} +(-0.900969 + 1.56052i) q^{56} +(1.07992 + 0.623490i) q^{58} +1.24698i q^{60} +(0.900969 - 1.56052i) q^{61} +(-0.866025 + 0.500000i) q^{63} -1.00000 q^{64} +(0.385418 - 0.222521i) q^{67} +(-0.277479 + 0.480608i) q^{69} -1.80194i q^{70} +(-0.480608 - 0.277479i) q^{72} +(-0.623490 - 1.07992i) q^{75} +(0.866025 - 0.500000i) q^{80} +(0.623490 + 1.07992i) q^{81} +(0.900969 - 1.56052i) q^{82} -1.24698i q^{83} +(1.94594 + 1.12349i) q^{84} +0.445042i q^{86} +(0.777479 - 1.34663i) q^{87} +(0.385418 - 0.222521i) q^{89} +0.554958 q^{90} +0.445042 q^{92} +(0.222521 + 0.385418i) q^{94} +1.24698i q^{96} +(-1.94594 - 1.12349i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 2 q^{3} + 6 q^{4} - 4 q^{9} - 6 q^{10} - 4 q^{12} - 4 q^{14} - 6 q^{16} + 2 q^{23} - 12 q^{25} + 8 q^{27} + 2 q^{29} - 2 q^{30} - 2 q^{35} + 4 q^{36} - 12 q^{40} - 4 q^{42} + 2 q^{43} - 2 q^{48}+ \cdots + 2 q^{94}+O(q^{100}) $ 12 * q - 2 * q^3 + 6 * q^4 - 4 * q^9 - 6 * q^10 - 4 * q^12 - 4 * q^14 - 6 * q^16 + 2 * q^23 - 12 * q^25 + 8 * q^27 + 2 * q^29 - 2 * q^30 - 2 * q^35 + 4 * q^36 - 12 * q^40 - 4 * q^42 + 2 * q^43 - 2 * q^48 + 4 * q^49 - 2 * q^56 + 2 * q^61 - 12 * q^64 - 4 * q^69 + 2 * q^75 - 2 * q^81 + 2 * q^82 + 10 * q^87 + 8 * q^90 + 4 * q^92 + 2 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times$.

$n$	$677$	$1691$	$1861$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{5}{6}\right)$

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 $

$2$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$2$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$3$	0.623490	+	1.07992i	0.623490	+	1.07992i	0.988831	+	0.149042i	$0.0476190\pi$
$3$	0.623490	+	1.07992i	0.623490	+	1.07992i	−0.365341	+	0.930874i	$0.619048\pi$
$4$	0.500000	−	0.866025i	0.500000	−	0.866025i
$5$			1.00000i			1.00000i
$5$			1.00000i			1.00000i
$6$	−1.07992	−	0.623490i	−1.07992	−	0.623490i
$7$	1.56052	+	0.900969i	1.56052	+	0.900969i	0.997204	+	0.0747301i	$0.0238095\pi$
$7$	1.56052	+	0.900969i	1.56052	+	0.900969i	0.563320	+	0.826239i	$0.309524\pi$
$8$			1.00000i			1.00000i
$9$	−0.277479	+	0.480608i	−0.277479	+	0.480608i
$10$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$	1.24698			1.24698
$13$		0			0
$13$		0			0
$14$	−1.80194			−1.80194
$15$	−1.07992	+	0.623490i	−1.07992	+	0.623490i
$16$	−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	$0.666667\pi$
$18$		−	0.554958i		−	0.554958i
$19$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$19$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$20$	0.866025	+	0.500000i	0.866025	+	0.500000i
$21$			2.24698i			2.24698i
$22$		0			0
$23$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$23$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$24$	−1.07992	+	0.623490i	−1.07992	+	0.623490i
$25$	−1.00000			−1.00000
$26$		0			0
$27$	0.554958			0.554958
$28$	1.56052	−	0.900969i	1.56052	−	0.900969i
$29$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$29$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$30$	0.623490	−	1.07992i	0.623490	−	1.07992i
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	0.866025	+	0.500000i	0.866025	+	0.500000i
$33$		0			0
$34$		0			0
$35$	−0.900969	+	1.56052i	−0.900969	+	1.56052i
$36$	0.277479	+	0.480608i	0.277479	+	0.480608i
$37$		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
$40$	−1.00000			−1.00000
$41$	−1.56052	+	0.900969i	−1.56052	+	0.900969i	−0.563320	+	0.826239i	$0.690476\pi$
$41$	−1.56052	+	0.900969i	−1.56052	+	0.900969i	−0.997204	+	0.0747301i	$0.976190\pi$
$42$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$43$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$43$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$44$		0			0
$45$	−0.480608	−	0.277479i	−0.480608	−	0.277479i
$46$	−0.385418	−	0.222521i	−0.385418	−	0.222521i
$47$		−	0.445042i		−	0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$47$		−	0.445042i		−	0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$48$	0.623490	−	1.07992i	0.623490	−	1.07992i
$49$	1.12349	+	1.94594i	1.12349	+	1.94594i
$50$	0.866025	−	0.500000i	0.866025	−	0.500000i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−0.480608	+	0.277479i	−0.480608	+	0.277479i
$55$		0			0
$56$	−0.900969	+	1.56052i	−0.900969	+	1.56052i
$57$		0			0
$58$	1.07992	+	0.623490i	1.07992	+	0.623490i
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$			1.24698i			1.24698i
$61$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$61$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$62$		0			0
$63$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$64$	−1.00000			−1.00000
$65$		0			0
$66$		0			0
$67$	0.385418	−	0.222521i	0.385418	−	0.222521i	−0.294755	−	0.955573i	$-0.595238\pi$
$67$	0.385418	−	0.222521i	0.385418	−	0.222521i	0.680173	+	0.733052i	$0.261905\pi$
$68$		0			0
$69$	−0.277479	+	0.480608i	−0.277479	+	0.480608i
$70$		−	1.80194i		−	1.80194i
$71$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$71$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$72$	−0.480608	−	0.277479i	−0.480608	−	0.277479i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	0.866025	−	0.500000i	0.866025	−	0.500000i
$81$	0.623490	+	1.07992i	0.623490	+	1.07992i
$82$	0.900969	−	1.56052i	0.900969	−	1.56052i
$83$		−	1.24698i		−	1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$83$		−	1.24698i		−	1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$84$	1.94594	+	1.12349i	1.94594	+	1.12349i
$85$		0			0
$86$			0.445042i			0.445042i
$87$	0.777479	−	1.34663i	0.777479	−	1.34663i
$88$		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	$0.261905\pi$
$90$	0.554958			0.554958
$91$		0			0
$92$	0.445042			0.445042
$93$		0			0
$94$	0.222521	+	0.385418i	0.222521	+	0.385418i
$95$		0			0
$96$			1.24698i			1.24698i
$97$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$97$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$98$	−1.94594	−	1.12349i	−1.94594	−	1.12349i
$99$		0			0
$100$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$101$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	0.733052	−	0.680173i	$-0.238095\pi$
$101$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	−0.955573	+	0.294755i	$0.904762\pi$
$102$		0			0
$103$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$103$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$104$		0			0
$105$	−2.24698			−2.24698
$106$		0			0
$107$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$107$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$108$	0.277479	−	0.480608i	0.277479	−	0.480608i
$109$		−	1.80194i		−	1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$109$		−	1.80194i		−	1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$110$		0			0
$111$		0			0
$112$		−	1.80194i		−	1.80194i
$113$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$		0			0
$115$	−0.385418	+	0.222521i	−0.385418	+	0.222521i
$116$	−1.24698			−1.24698
$117$		0			0
$118$		0			0
$119$		0			0
$120$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$121$	0.500000	−	0.866025i	0.500000	−	0.866025i
$122$			1.80194i			1.80194i
$123$	−1.94594	−	1.12349i	−1.94594	−	1.12349i
$124$		0			0
$125$		−	1.00000i		−	1.00000i
$126$	0.500000	−	0.866025i	0.500000	−	0.866025i
$127$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$127$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$128$	0.866025	−	0.500000i	0.866025	−	0.500000i
$129$	0.554958			0.554958
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$	−0.222521	+	0.385418i	−0.222521	+	0.385418i
$135$			0.554958i			0.554958i
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		−	0.554958i		−	0.554958i
$139$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$139$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$140$	0.900969	+	1.56052i	0.900969	+	1.56052i
$141$	0.480608	−	0.277479i	0.480608	−	0.277479i
$142$		0			0
$143$		0			0
$144$	0.554958			0.554958
$145$	1.07992	−	0.623490i	1.07992	−	0.623490i
$146$		0			0
$147$	−1.40097	+	2.42655i	−1.40097	+	2.42655i
$148$		0			0
$149$	−1.07992	−	0.623490i	−1.07992	−	0.623490i	−0.149042	−	0.988831i	$-0.547619\pi$
$149$	−1.07992	−	0.623490i	−1.07992	−	0.623490i	−0.930874	+	0.365341i	$0.880952\pi$
$150$	1.07992	+	0.623490i	1.07992	+	0.623490i
$151$		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			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$161$			0.801938i			0.801938i
$162$	−1.07992	−	0.623490i	−1.07992	−	0.623490i
$163$	−1.07992	−	0.623490i	−1.07992	−	0.623490i	−0.149042	−	0.988831i	$-0.547619\pi$
$163$	−1.07992	−	0.623490i	−1.07992	−	0.623490i	−0.930874	+	0.365341i	$0.880952\pi$
$164$			1.80194i			1.80194i
$165$		0			0
$166$	0.623490	+	1.07992i	0.623490	+	1.07992i
$167$	−1.56052	+	0.900969i	−1.56052	+	0.900969i	−0.563320	+	0.826239i	$0.690476\pi$
$167$	−1.56052	+	0.900969i	−1.56052	+	0.900969i	−0.997204	+	0.0747301i	$0.976190\pi$
$168$	−2.24698			−2.24698
$169$		0			0
$170$		0			0
$171$		0			0
$172$	−0.222521	−	0.385418i	−0.222521	−	0.385418i
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$			1.55496i			1.55496i
$175$	−1.56052	−	0.900969i	−1.56052	−	0.900969i
$176$		0			0
$177$		0			0
$178$	−0.222521	+	0.385418i	−0.222521	+	0.385418i
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$	−0.480608	+	0.277479i	−0.480608	+	0.277479i
$181$	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
$183$	2.24698			2.24698
$184$	−0.385418	+	0.222521i	−0.385418	+	0.222521i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−0.385418	−	0.222521i	−0.385418	−	0.222521i
$189$	0.866025	+	0.500000i	0.866025	+	0.500000i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$193$		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
$196$	2.24698			2.24698
$197$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$197$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$198$		0			0
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$		−	1.00000i		−	1.00000i
$201$	0.480608	+	0.277479i	0.480608	+	0.277479i
$202$	0.385418	+	0.222521i	0.385418	+	0.222521i
$203$		−	2.24698i		−	2.24698i
$204$		0			0
$205$	−0.900969	−	1.56052i	−0.900969	−	1.56052i
$206$	1.56052	−	0.900969i	1.56052	−	0.900969i
$207$	−0.246980			−0.246980
$208$		0			0
$209$		0			0
$210$	1.94594	−	1.12349i	1.94594	−	1.12349i
$211$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$211$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$212$		0			0
$213$		0			0
$214$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$215$	0.385418	+	0.222521i	0.385418	+	0.222521i
$216$			0.554958i			0.554958i
$217$		0			0
$218$	0.900969	+	1.56052i	0.900969	+	1.56052i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.07992	+	0.623490i	−1.07992	+	0.623490i	−0.930874	−	0.365341i	$-0.880952\pi$
$223$	−1.07992	+	0.623490i	−1.07992	+	0.623490i	−0.149042	+	0.988831i	$0.547619\pi$
$224$	0.900969	+	1.56052i	0.900969	+	1.56052i
$225$	0.277479	−	0.480608i	0.277479	−	0.480608i
$226$		0			0
$227$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.930874	−	0.365341i	$-0.119048\pi$
$227$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.149042	+	0.988831i	$0.452381\pi$
$228$		0			0
$229$		−	1.24698i		−	1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$229$		−	1.24698i		−	1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$230$	0.222521	−	0.385418i	0.222521	−	0.385418i
$231$		0			0
$232$	1.07992	−	0.623490i	1.07992	−	0.623490i
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$	0.445042			0.445042
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	1.07992	+	0.623490i	1.07992	+	0.623490i
$241$	−0.385418	−	0.222521i	−0.385418	−	0.222521i	0.294755	−	0.955573i	$-0.404762\pi$
$241$	−0.385418	−	0.222521i	−0.385418	−	0.222521i	−0.680173	+	0.733052i	$0.738095\pi$
$242$			1.00000i			1.00000i
$243$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$244$	−0.900969	−	1.56052i	−0.900969	−	1.56052i
$245$	−1.94594	+	1.12349i	−1.94594	+	1.12349i
$246$	2.24698			2.24698
$247$		0			0
$248$		0			0
$249$	1.34663	−	0.777479i	1.34663	−	0.777479i
$250$	0.500000	+	0.866025i	0.500000	+	0.866025i
$251$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$251$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$252$			1.00000i			1.00000i
$253$		0			0
$254$	−1.56052	−	0.900969i	−1.56052	−	0.900969i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$	−0.480608	+	0.277479i	−0.480608	+	0.277479i
$259$		0			0
$260$		0			0
$261$	0.692021			0.692021
$262$		0			0
$263$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.826239	−	0.563320i	$-0.809524\pi$
$263$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.0747301	−	0.997204i	$-0.523810\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	0.480608	+	0.277479i	0.480608	+	0.277479i
$268$		−	0.445042i		−	0.445042i
$269$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$269$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$270$	−0.277479	−	0.480608i	−0.277479	−	0.480608i
$271$		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			0
$276$	0.277479	+	0.480608i	0.277479	+	0.480608i
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$		0			0
$280$	−1.56052	−	0.900969i	−1.56052	−	0.900969i
$281$		−	1.24698i		−	1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$281$		−	1.24698i		−	1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$282$	−0.277479	+	0.480608i	−0.277479	+	0.480608i
$283$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$283$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3.24698			−3.24698
$288$	−0.480608	+	0.277479i	−0.480608	+	0.277479i
$289$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$290$	−0.623490	+	1.07992i	−0.623490	+	1.07992i
$291$		0			0
$292$		0			0
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$		−	2.80194i		−	2.80194i
$295$		0			0
$296$		0			0
$297$		0			0
$298$	1.24698			1.24698
$299$		0			0
$300$	−1.24698			−1.24698
$301$	0.694498	−	0.400969i	0.694498	−	0.400969i
$302$		0			0
$303$	0.277479	−	0.480608i	0.277479	−	0.480608i
$304$		0			0
$305$	1.56052	+	0.900969i	1.56052	+	0.900969i
$306$		0			0
$307$			1.24698i			1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$307$			1.24698i			1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$308$		0			0
$309$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		−	1.00000i		−	1.00000i
$321$	−1.24698	+	2.15983i	−1.24698	+	2.15983i
$322$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$323$		0			0
$324$	1.24698			1.24698
$325$		0			0
$326$	1.24698			1.24698
$327$	1.94594	−	1.12349i	1.94594	−	1.12349i
$328$	−0.900969	−	1.56052i	−0.900969	−	1.56052i
$329$	0.400969	−	0.694498i	0.400969	−	0.694498i
$330$		0			0
$331$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$331$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$332$	−1.07992	−	0.623490i	−1.07992	−	0.623490i
$333$		0			0
$334$	0.900969	−	1.56052i	0.900969	−	1.56052i
$335$	0.222521	+	0.385418i	0.222521	+	0.385418i
$336$	1.94594	−	1.12349i	1.94594	−	1.12349i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			2.24698i			2.24698i
$344$	0.385418	+	0.222521i	0.385418	+	0.222521i
$345$	−0.480608	−	0.277479i	−0.480608	−	0.277479i
$346$		0			0
$347$	0.623490	−	1.07992i	0.623490	−	1.07992i	−0.365341	−	0.930874i	$-0.619048\pi$
$347$	0.623490	−	1.07992i	0.623490	−	1.07992i	0.988831	−	0.149042i	$-0.0476190\pi$
$348$	−0.777479	−	1.34663i	−0.777479	−	1.34663i
$349$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$349$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$350$	1.80194			1.80194
$351$		0			0
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		−	0.445042i		−	0.445042i
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	0.277479	−	0.480608i	0.277479	−	0.480608i
$361$	0.500000	+	0.866025i	0.500000	+	0.866025i
$362$	−0.385418	+	0.222521i	−0.385418	+	0.222521i
$363$	1.24698			1.24698
$364$		0			0
$365$		0			0
$366$	−1.94594	+	1.12349i	−1.94594	+	1.12349i
$367$	0.623490	+	1.07992i	0.623490	+	1.07992i	0.988831	+	0.149042i	$0.0476190\pi$
$367$	0.623490	+	1.07992i	0.623490	+	1.07992i	−0.365341	+	0.930874i	$0.619048\pi$
$368$	0.222521	−	0.385418i	0.222521	−	0.385418i
$369$		−	1.00000i		−	1.00000i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$	1.07992	−	0.623490i	1.07992	−	0.623490i
$376$	0.445042			0.445042
$377$		0			0
$378$	−1.00000			−1.00000
$379$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$379$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$380$		0			0
$381$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$382$		0			0
$383$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.930874	−	0.365341i	$-0.119048\pi$
$383$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.149042	+	0.988831i	$0.452381\pi$
$384$	1.07992	+	0.623490i	1.07992	+	0.623490i
$385$		0			0
$386$		0			0
$387$	0.123490	+	0.213891i	0.123490	+	0.213891i
$388$		0			0
$389$	0.445042			0.445042			0.222521	−	0.974928i	$-0.428571\pi$
$389$	0.445042			0.445042			0.222521	+	0.974928i	$0.428571\pi$
$390$		0			0
$391$		0			0
$392$	−1.94594	+	1.12349i	−1.94594	+	1.12349i
$393$		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
$400$	0.500000	+	0.866025i	0.500000	+	0.866025i
$401$	1.56052	−	0.900969i	1.56052	−	0.900969i	0.563320	−	0.826239i	$-0.309524\pi$
$401$	1.56052	−	0.900969i	1.56052	−	0.900969i	0.997204	−	0.0747301i	$-0.0238095\pi$
$402$	−0.554958			−0.554958
$403$		0			0
$404$	−0.445042			−0.445042
$405$	−1.07992	+	0.623490i	−1.07992	+	0.623490i
$406$	1.12349	+	1.94594i	1.12349	+	1.94594i
$407$		0			0
$408$		0			0
$409$	1.56052	+	0.900969i	1.56052	+	0.900969i	0.997204	+	0.0747301i	$0.0238095\pi$
$409$	1.56052	+	0.900969i	1.56052	+	0.900969i	0.563320	+	0.826239i	$0.309524\pi$
$410$	1.56052	+	0.900969i	1.56052	+	0.900969i
$411$		0			0
$412$	−0.900969	+	1.56052i	−0.900969	+	1.56052i
$413$		0			0
$414$	0.213891	−	0.123490i	0.213891	−	0.123490i
$415$	1.24698			1.24698
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$419$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$420$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$421$			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
$423$	0.213891	+	0.123490i	0.213891	+	0.123490i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	2.81197	−	1.62349i	2.81197	−	1.62349i
$428$	2.00000			2.00000
$429$		0			0
$430$	−0.445042			−0.445042
$431$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$431$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$432$	−0.277479	−	0.480608i	−0.277479	−	0.480608i
$433$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$433$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$434$		0			0
$435$	1.34663	+	0.777479i	1.34663	+	0.777479i
$436$	−1.56052	−	0.900969i	−1.56052	−	0.900969i
$437$		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
$441$	−1.24698			−1.24698
$442$		0			0
$443$	−1.24698			−1.24698			−0.623490	−	0.781831i	$-0.714286\pi$
$443$	−1.24698			−1.24698			−0.623490	+	0.781831i	$0.714286\pi$
$444$		0			0
$445$	0.222521	+	0.385418i	0.222521	+	0.385418i
$446$	0.623490	−	1.07992i	0.623490	−	1.07992i
$447$		−	1.55496i		−	1.55496i
$448$	−1.56052	−	0.900969i	−1.56052	−	0.900969i
$449$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.930874	−	0.365341i	$-0.119048\pi$
$449$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.149042	+	0.988831i	$0.452381\pi$
$450$			0.554958i			0.554958i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	−1.24698			−1.24698
$455$		0			0
$456$		0			0
$457$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$457$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$458$	0.623490	+	1.07992i	0.623490	+	1.07992i
$459$		0			0
$460$			0.445042i			0.445042i
$461$	0.385418	+	0.222521i	0.385418	+	0.222521i	0.680173	−	0.733052i	$-0.261905\pi$
$461$	0.385418	+	0.222521i	0.385418	+	0.222521i	−0.294755	+	0.955573i	$0.595238\pi$
$462$		0			0
$463$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$463$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$464$	−0.623490	+	1.07992i	−0.623490	+	1.07992i
$465$		0			0
$466$		0			0
$467$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$467$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$468$		0			0
$469$	0.801938			0.801938
$470$	−0.385418	+	0.222521i	−0.385418	+	0.222521i
$471$		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		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$	−1.24698			−1.24698
$481$		0			0
$482$	0.445042			0.445042
$483$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$484$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$485$		0			0
$486$		−	1.00000i		−	1.00000i
$487$	−0.385418	−	0.222521i	−0.385418	−	0.222521i	0.294755	−	0.955573i	$-0.404762\pi$
$487$	−0.385418	−	0.222521i	−0.385418	−	0.222521i	−0.680173	+	0.733052i	$0.738095\pi$
$488$	1.56052	+	0.900969i	1.56052	+	0.900969i
$489$		−	1.55496i		−	1.55496i
$490$	1.12349	−	1.94594i	1.12349	−	1.94594i
$491$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$491$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$492$	−1.94594	+	1.12349i	−1.94594	+	1.12349i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−0.777479	+	1.34663i	−0.777479	+	1.34663i
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$501$	−1.94594	−	1.12349i	−1.94594	−	1.12349i
$502$		0			0
$503$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.0747301	+	0.997204i	$0.523810\pi$
$503$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.826239	+	0.563320i	$0.809524\pi$
$504$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$505$	0.385418	−	0.222521i	0.385418	−	0.222521i
$506$		0			0
$507$		0			0
$508$	1.80194			1.80194
$509$	1.07992	−	0.623490i	1.07992	−	0.623490i	0.149042	−	0.988831i	$-0.452381\pi$
$509$	1.07992	−	0.623490i	1.07992	−	0.623490i	0.930874	+	0.365341i	$0.119048\pi$
$510$		0			0
$511$		0			0
$512$		−	1.00000i		−	1.00000i
$513$		0			0
$514$		0			0
$515$		−	1.80194i		−	1.80194i
$516$	0.277479	−	0.480608i	0.277479	−	0.480608i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$521$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$522$	−0.599308	+	0.346011i	−0.599308	+	0.346011i
$523$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	0.733052	−	0.680173i	$-0.238095\pi$
$523$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	−0.955573	+	0.294755i	$0.904762\pi$
$524$		0			0
$525$		−	2.24698i		−	2.24698i
$526$	1.56052	+	0.900969i	1.56052	+	0.900969i
$527$		0			0
$528$		0			0
$529$	0.400969	−	0.694498i	0.400969	−	0.694498i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	−0.554958			−0.554958
$535$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$536$	0.222521	+	0.385418i	0.222521	+	0.385418i
$537$		0			0
$538$		−	1.24698i		−	1.24698i
$539$		0			0
$540$	0.480608	+	0.277479i	0.480608	+	0.277479i
$541$		−	1.24698i		−	1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$541$		−	1.24698i		−	1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$542$		0			0
$543$	0.277479	+	0.480608i	0.277479	+	0.480608i
$544$		0			0
$545$	1.80194			1.80194
$546$		0			0
$547$	1.80194			1.80194			0.900969	−	0.433884i	$-0.142857\pi$
$547$	1.80194			1.80194			0.900969	+	0.433884i	$0.142857\pi$
$548$		0			0
$549$	0.500000	+	0.866025i	0.500000	+	0.866025i
$550$		0			0
$551$		0			0
$552$	−0.480608	−	0.277479i	−0.480608	−	0.277479i
$553$		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
$560$	1.80194			1.80194
$561$		0			0
$562$	0.623490	+	1.07992i	0.623490	+	1.07992i
$563$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$563$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$564$		−	0.554958i		−	0.554958i
$565$		0			0
$566$	1.07992	+	0.623490i	1.07992	+	0.623490i
$567$			2.24698i			2.24698i
$568$		0			0
$569$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.826239	−	0.563320i	$-0.809524\pi$
$569$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.0747301	−	0.997204i	$-0.523810\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$	2.81197	−	1.62349i	2.81197	−	1.62349i
$575$	−0.222521	−	0.385418i	−0.222521	−	0.385418i
$576$	0.277479	−	0.480608i	0.277479	−	0.480608i
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	0.866025	+	0.500000i	0.866025	+	0.500000i
$579$		0			0
$580$		−	1.24698i		−	1.24698i
$581$	1.12349	−	1.94594i	1.12349	−	1.94594i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.56052	−	0.900969i	1.56052	−	0.900969i	0.563320	−	0.826239i	$-0.309524\pi$
$587$	1.56052	−	0.900969i	1.56052	−	0.900969i	0.997204	−	0.0747301i	$-0.0238095\pi$
$588$	1.40097	+	2.42655i	1.40097	+	2.42655i
$589$		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
$596$	−1.07992	+	0.623490i	−1.07992	+	0.623490i
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$	1.07992	−	0.623490i	1.07992	−	0.623490i
$601$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$601$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$602$	−0.400969	+	0.694498i	−0.400969	+	0.694498i
$603$			0.246980i			0.246980i
$604$		0			0
$605$	0.866025	+	0.500000i	0.866025	+	0.500000i
$606$			0.554958i			0.554958i
$607$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	−0.955573	−	0.294755i	$-0.904762\pi$
$607$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	0.733052	+	0.680173i	$0.238095\pi$
$608$		0			0
$609$	2.42655	−	1.40097i	2.42655	−	1.40097i
$610$	−1.80194			−1.80194
$611$		0			0
$612$		0			0
$613$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$613$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$614$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$615$	1.12349	−	1.94594i	1.12349	−	1.94594i
$616$		0			0
$617$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$617$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$618$	1.94594	+	1.12349i	1.94594	+	1.12349i
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$	0.123490	+	0.213891i	0.123490	+	0.213891i
$622$		0			0
$623$	0.801938			0.801938
$624$		0			0
$625$	1.00000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$	0.866025	+	0.500000i	0.866025	+	0.500000i
$631$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$631$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.56052	+	0.900969i	−1.56052	+	0.900969i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	0.500000	+	0.866025i	0.500000	+	0.866025i
$641$	−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	$0.809524\pi$
$642$		−	2.49396i		−	2.49396i
$643$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$643$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$644$	0.694498	+	0.400969i	0.694498	+	0.400969i
$645$			0.554958i			0.554958i
$646$		0			0
$647$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$647$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$648$	−1.07992	+	0.623490i	−1.07992	+	0.623490i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−1.07992	+	0.623490i	−1.07992	+	0.623490i
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$655$		0			0
$656$	1.56052	+	0.900969i	1.56052	+	0.900969i
$657$		0			0
$658$			0.801938i			0.801938i
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$	0.385418	−	0.222521i	0.385418	−	0.222521i	−0.294755	−	0.955573i	$-0.595238\pi$
$661$	0.385418	−	0.222521i	0.385418	−	0.222521i	0.680173	+	0.733052i	$0.261905\pi$
$662$		0			0
$663$		0			0
$664$	1.24698			1.24698
$665$		0			0
$666$		0			0
$667$	0.277479	−	0.480608i	0.277479	−	0.480608i
$668$			1.80194i			1.80194i
$669$	−1.34663	−	0.777479i	−1.34663	−	0.777479i
$670$	−0.385418	−	0.222521i	−0.385418	−	0.222521i
$671$		0			0
$672$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$673$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$673$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$674$		0			0
$675$	−0.554958			−0.554958
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$			1.55496i			1.55496i
$682$		0			0
$683$	0.385418	+	0.222521i	0.385418	+	0.222521i	0.680173	−	0.733052i	$-0.261905\pi$
$683$	0.385418	+	0.222521i	0.385418	+	0.222521i	−0.294755	+	0.955573i	$0.595238\pi$
$684$		0			0
$685$		0			0
$686$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$687$	1.34663	−	0.777479i	1.34663	−	0.777479i
$688$	−0.445042			−0.445042
$689$		0			0
$690$	0.554958			0.554958
$691$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$691$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$692$		0			0
$693$		0			0
$694$			1.24698i			1.24698i
$695$		0			0
$696$	1.34663	+	0.777479i	1.34663	+	0.777479i
$697$		0			0
$698$	1.00000	−	1.73205i	1.00000	−	1.73205i
$699$		0			0
$700$	−1.56052	+	0.900969i	−1.56052	+	0.900969i
$701$	1.80194			1.80194			0.900969	−	0.433884i	$-0.142857\pi$
$701$	1.80194			1.80194			0.900969	+	0.433884i	$0.142857\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	0.277479	+	0.480608i	0.277479	+	0.480608i
$706$		0			0
$707$		−	0.801938i		−	0.801938i
$708$		0			0
$709$	−1.56052	−	0.900969i	−1.56052	−	0.900969i	−0.997204	−	0.0747301i	$-0.976190\pi$
$709$	−1.56052	−	0.900969i	−1.56052	−	0.900969i	−0.563320	−	0.826239i	$-0.690476\pi$
$710$		0			0
$711$		0			0
$712$	0.222521	+	0.385418i	0.222521	+	0.385418i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$			0.554958i			0.554958i
$721$	−2.81197	−	1.62349i	−2.81197	−	1.62349i
$722$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$723$		−	0.554958i		−	0.554958i
$724$	0.222521	−	0.385418i	0.222521	−	0.385418i
$725$	0.623490	+	1.07992i	0.623490	+	1.07992i
$726$	−1.07992	+	0.623490i	−1.07992	+	0.623490i
$727$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$727$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$	1.12349	−	1.94594i	1.12349	−	1.94594i
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$	−1.07992	−	0.623490i	−1.07992	−	0.623490i
$735$	−2.42655	−	1.40097i	−2.42655	−	1.40097i
$736$			0.445042i			0.445042i
$737$		0			0
$738$	0.500000	+	0.866025i	0.500000	+	0.866025i
$739$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$739$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.56052	−	0.900969i	1.56052	−	0.900969i	0.563320	−	0.826239i	$-0.309524\pi$
$743$	1.56052	−	0.900969i	1.56052	−	0.900969i	0.997204	−	0.0747301i	$-0.0238095\pi$
$744$		0			0
$745$	0.623490	−	1.07992i	0.623490	−	1.07992i
$746$		0			0
$747$	0.599308	+	0.346011i	0.599308	+	0.346011i
$748$		0			0
$749$			3.60388i			3.60388i
$750$	−0.623490	+	1.07992i	−0.623490	+	1.07992i
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$	−0.385418	+	0.222521i	−0.385418	+	0.222521i
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.866025	−	0.500000i	0.866025	−	0.500000i
$757$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$757$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.930874	−	0.365341i	$-0.119048\pi$
$761$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.149042	+	0.988831i	$0.452381\pi$
$762$		−	2.24698i		−	2.24698i
$763$	1.62349	−	2.81197i	1.62349	−	2.81197i
$764$		0			0
$765$		0			0
$766$	−1.24698			−1.24698
$767$		0			0
$768$	−1.24698			−1.24698
$769$	−1.56052	+	0.900969i	−1.56052	+	0.900969i	−0.563320	+	0.826239i	$0.690476\pi$
$769$	−1.56052	+	0.900969i	−1.56052	+	0.900969i	−0.997204	+	0.0747301i	$0.976190\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$	−0.213891	−	0.123490i	−0.213891	−	0.123490i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−0.385418	+	0.222521i	−0.385418	+	0.222521i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−0.346011	−	0.599308i	−0.346011	−	0.599308i
$784$	1.12349	−	1.94594i	1.12349	−	1.94594i
$785$		0			0
$786$		0			0
$787$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$787$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$788$		0			0
$789$	1.12349	−	1.94594i	1.12349	−	1.94594i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$797$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$798$		0			0
$799$		0			0
$800$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$801$			0.246980i			0.246980i
$802$	−0.900969	+	1.56052i	−0.900969	+	1.56052i
$803$		0			0
$804$	0.480608	−	0.277479i	0.480608	−	0.277479i
$805$	−0.801938			−0.801938
$806$		0			0
$807$	−1.55496			−1.55496
$808$	0.385418	−	0.222521i	0.385418	−	0.222521i
$809$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$809$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$810$	0.623490	−	1.07992i	0.623490	−	1.07992i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	−1.94594	−	1.12349i	−1.94594	−	1.12349i
$813$		0			0
$814$		0			0
$815$	0.623490	−	1.07992i	0.623490	−	1.07992i
$816$		0			0
$817$		0			0
$818$	−1.80194			−1.80194
$819$		0			0
$820$	−1.80194			−1.80194
$821$	−0.385418	+	0.222521i	−0.385418	+	0.222521i	−0.680173	−	0.733052i	$-0.738095\pi$
$821$	−0.385418	+	0.222521i	−0.385418	+	0.222521i	0.294755	+	0.955573i	$0.404762\pi$
$822$		0			0
$823$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$823$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$824$		−	1.80194i		−	1.80194i
$825$		0			0
$826$		0			0
$827$			1.24698i			1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$827$			1.24698i			1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$828$	−0.123490	+	0.213891i	−0.123490	+	0.213891i
$829$	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	$0.619048\pi$
$830$	−1.07992	+	0.623490i	−1.07992	+	0.623490i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−0.900969	−	1.56052i	−0.900969	−	1.56052i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		−	2.24698i		−	2.24698i
$841$	−0.277479	+	0.480608i	−0.277479	+	0.480608i
$842$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$843$	1.34663	−	0.777479i	1.34663	−	0.777479i
$844$		0			0
$845$		0			0
$846$	−0.246980			−0.246980
$847$	1.56052	−	0.900969i	1.56052	−	0.900969i
$848$		0			0
$849$	0.777479	−	1.34663i	0.777479	−	1.34663i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$	−1.62349	+	2.81197i	−1.62349	+	2.81197i
$855$		0			0
$856$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$	0.385418	−	0.222521i	0.385418	−	0.222521i
$861$	−2.02446	−	3.50647i	−2.02446	−	3.50647i
$862$		0			0
$863$		−	1.24698i		−	1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$863$		−	1.24698i		−	1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$864$	0.480608	+	0.277479i	0.480608	+	0.277479i
$865$		0			0
$866$		0			0
$867$	0.623490	−	1.07992i	0.623490	−	1.07992i
$868$		0			0
$869$		0			0
$870$	−1.55496			−1.55496
$871$		0			0
$872$	1.80194			1.80194
$873$		0			0
$874$		0			0
$875$	0.900969	−	1.56052i	0.900969	−	1.56052i
$876$		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			0
$881$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$881$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$882$	1.07992	−	0.623490i	1.07992	−	0.623490i
$883$	2.00000			2.00000			1.00000			$0$
$883$	2.00000			2.00000			1.00000			$0$
$884$		0			0
$885$		0			0
$886$	1.07992	−	0.623490i	1.07992	−	0.623490i
$887$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.826239	−	0.563320i	$-0.809524\pi$
$887$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.0747301	−	0.997204i	$-0.523810\pi$
$888$		0			0
$889$			3.24698i			3.24698i
$890$	−0.385418	−	0.222521i	−0.385418	−	0.222521i
$891$		0			0
$892$			1.24698i			1.24698i
$893$		0			0
$894$	0.777479	+	1.34663i	0.777479	+	1.34663i
$895$		0			0
$896$	1.80194			1.80194
$897$		0			0
$898$	−1.24698			−1.24698
$899$		0			0
$900$	−0.277479	−	0.480608i	−0.277479	−	0.480608i
$901$		0			0
$902$		0			0
$903$	0.866025	+	0.500000i	0.866025	+	0.500000i
$904$		0			0
$905$			0.445042i			0.445042i
$906$		0			0
$907$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$907$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$908$	1.07992	−	0.623490i	1.07992	−	0.623490i
$909$	0.246980			0.246980
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$			2.24698i			2.24698i
$916$	−1.07992	−	0.623490i	−1.07992	−	0.623490i
$917$		0			0
$918$		0			0
$919$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$919$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$920$	−0.222521	−	0.385418i	−0.222521	−	0.385418i
$921$	−1.34663	+	0.777479i	−1.34663	+	0.777479i
$922$	−0.445042			−0.445042
$923$		0			0
$924$		0			0
$925$		0			0
$926$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$927$	0.500000	−	0.866025i	0.500000	−	0.866025i
$928$		−	1.24698i		−	1.24698i
$929$	0.385418	+	0.222521i	0.385418	+	0.222521i	0.680173	−	0.733052i	$-0.261905\pi$
$929$	0.385418	+	0.222521i	0.385418	+	0.222521i	−0.294755	+	0.955573i	$0.595238\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	0.385418	−	0.222521i	0.385418	−	0.222521i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	−0.694498	+	0.400969i	−0.694498	+	0.400969i
$939$		0			0
$940$	0.222521	−	0.385418i	0.222521	−	0.385418i
$941$		−	1.80194i		−	1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$941$		−	1.80194i		−	1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$942$		0			0
$943$	−0.694498	−	0.400969i	−0.694498	−	0.400969i
$944$		0			0
$945$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$946$		0			0
$947$	−1.56052	+	0.900969i	−1.56052	+	0.900969i	−0.563320	+	0.826239i	$0.690476\pi$
$947$	−1.56052	+	0.900969i	−1.56052	+	0.900969i	−0.997204	+	0.0747301i	$0.976190\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$953$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	1.07992	−	0.623490i	1.07992	−	0.623490i
$961$	−1.00000			−1.00000
$962$		0			0
$963$	−1.10992			−1.10992
$964$	−0.385418	+	0.222521i	−0.385418	+	0.222521i
$965$		0			0
$966$	0.500000	−	0.866025i	0.500000	−	0.866025i
$967$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$967$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$968$	0.866025	+	0.500000i	0.866025	+	0.500000i
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$	0.500000	+	0.866025i	0.500000	+	0.866025i
$973$		0			0
$974$	0.445042			0.445042
$975$		0			0
$976$	−1.80194			−1.80194
$977$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$977$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$978$	0.777479	+	1.34663i	0.777479	+	1.34663i
$979$		0			0
$980$			2.24698i			2.24698i
$981$	0.866025	+	0.500000i	0.866025	+	0.500000i
$982$		0			0
$983$			1.24698i			1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$983$			1.24698i			1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$984$	1.12349	−	1.94594i	1.12349	−	1.94594i
$985$		0			0
$986$		0			0
$987$	1.00000			1.00000
$988$		0			0
$989$	0.198062			0.198062
$990$		0			0
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		−	1.55496i		−	1.55496i
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.1.w.e.699.3		12
4.3	odd	2		3380.1.w.f.699.4		12
5.4	even	2		3380.1.w.f.699.4		12
13.2	odd	12		3380.1.h.c.339.1	yes	3
13.3	even	3		3380.1.g.d.3379.1		6
13.4	even	6	inner	3380.1.w.e.1499.3		12
13.5	odd	4		3380.1.v.f.3019.3		6
13.6	odd	12		3380.1.v.f.2219.3		6
13.7	odd	12		3380.1.v.d.2219.3		6
13.8	odd	4		3380.1.v.d.3019.3		6
13.9	even	3	inner	3380.1.w.e.1499.6		12
13.10	even	6		3380.1.g.d.3379.4		6
13.11	odd	12		3380.1.h.e.339.1	yes	3
13.12	even	2	inner	3380.1.w.e.699.6		12
20.19	odd	2	CM	3380.1.w.e.699.3		12
52.3	odd	6		3380.1.g.c.3379.6		6
52.7	even	12		3380.1.v.g.2219.1		6
52.11	even	12		3380.1.h.b.339.3	✓	3
52.15	even	12		3380.1.h.d.339.3	yes	3
52.19	even	12		3380.1.v.e.2219.1		6
52.23	odd	6		3380.1.g.c.3379.3		6
52.31	even	4		3380.1.v.e.3019.1		6
52.35	odd	6		3380.1.w.f.1499.1		12
52.43	odd	6		3380.1.w.f.1499.4		12
52.47	even	4		3380.1.v.g.3019.1		6
52.51	odd	2		3380.1.w.f.699.1		12
65.4	even	6		3380.1.w.f.1499.4		12
65.9	even	6		3380.1.w.f.1499.1		12
65.19	odd	12		3380.1.v.e.2219.1		6
65.24	odd	12		3380.1.h.b.339.3	✓	3
65.29	even	6		3380.1.g.c.3379.6		6
65.34	odd	4		3380.1.v.g.3019.1		6
65.44	odd	4		3380.1.v.e.3019.1		6
65.49	even	6		3380.1.g.c.3379.3		6
65.54	odd	12		3380.1.h.d.339.3	yes	3
65.59	odd	12		3380.1.v.g.2219.1		6
65.64	even	2		3380.1.w.f.699.1		12
260.19	even	12		3380.1.v.f.2219.3		6
260.59	even	12		3380.1.v.d.2219.3		6
260.99	even	4		3380.1.v.d.3019.3		6
260.119	even	12		3380.1.h.c.339.1	yes	3
260.139	odd	6	inner	3380.1.w.e.1499.6		12
260.159	odd	6		3380.1.g.d.3379.1		6
260.179	odd	6		3380.1.g.d.3379.4		6
260.199	odd	6	inner	3380.1.w.e.1499.3		12
260.219	even	12		3380.1.h.e.339.1	yes	3
260.239	even	4		3380.1.v.f.3019.3		6
260.259	odd	2	inner	3380.1.w.e.699.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.1.g.c.3379.3		6	52.23	odd	6
3380.1.g.c.3379.3		6	65.49	even	6
3380.1.g.c.3379.6		6	52.3	odd	6
3380.1.g.c.3379.6		6	65.29	even	6
3380.1.g.d.3379.1		6	13.3	even	3
3380.1.g.d.3379.1		6	260.159	odd	6
3380.1.g.d.3379.4		6	13.10	even	6
3380.1.g.d.3379.4		6	260.179	odd	6
3380.1.h.b.339.3	✓	3	52.11	even	12
3380.1.h.b.339.3	✓	3	65.24	odd	12
3380.1.h.c.339.1	yes	3	13.2	odd	12
3380.1.h.c.339.1	yes	3	260.119	even	12
3380.1.h.d.339.3	yes	3	52.15	even	12
3380.1.h.d.339.3	yes	3	65.54	odd	12
3380.1.h.e.339.1	yes	3	13.11	odd	12
3380.1.h.e.339.1	yes	3	260.219	even	12
3380.1.v.d.2219.3		6	13.7	odd	12
3380.1.v.d.2219.3		6	260.59	even	12
3380.1.v.d.3019.3		6	13.8	odd	4
3380.1.v.d.3019.3		6	260.99	even	4
3380.1.v.e.2219.1		6	52.19	even	12
3380.1.v.e.2219.1		6	65.19	odd	12
3380.1.v.e.3019.1		6	52.31	even	4
3380.1.v.e.3019.1		6	65.44	odd	4
3380.1.v.f.2219.3		6	13.6	odd	12
3380.1.v.f.2219.3		6	260.19	even	12
3380.1.v.f.3019.3		6	13.5	odd	4
3380.1.v.f.3019.3		6	260.239	even	4
3380.1.v.g.2219.1		6	52.7	even	12
3380.1.v.g.2219.1		6	65.59	odd	12
3380.1.v.g.3019.1		6	52.47	even	4
3380.1.v.g.3019.1		6	65.34	odd	4
3380.1.w.e.699.3		12	1.1	even	1	trivial
3380.1.w.e.699.3		12	20.19	odd	2	CM
3380.1.w.e.699.6		12	13.12	even	2	inner
3380.1.w.e.699.6		12	260.259	odd	2	inner
3380.1.w.e.1499.3		12	13.4	even	6	inner
3380.1.w.e.1499.3		12	260.199	odd	6	inner
3380.1.w.e.1499.6		12	13.9	even	3	inner
3380.1.w.e.1499.6		12	260.139	odd	6	inner
3380.1.w.f.699.1		12	52.51	odd	2
3380.1.w.f.699.1		12	65.64	even	2
3380.1.w.f.699.4		12	4.3	odd	2
3380.1.w.f.699.4		12	5.4	even	2
3380.1.w.f.1499.1		12	52.35	odd	6
3380.1.w.f.1499.1		12	65.9	even	6
3380.1.w.f.1499.4		12	52.43	odd	6
3380.1.w.f.1499.4		12	65.4	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3380.w (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 + 0.500000i) q^{2} +(0.623490 + 1.07992i) q^{3} +(0.500000 - 0.866025i) q^{4} +1.00000i q^{5} +(-1.07992 - 0.623490i) q^{6} +(1.56052 + 0.900969i) q^{7} +1.00000i q^{8} +(-0.277479 + 0.480608i) q^{9} +(-0.500000 - 0.866025i) q^{10} +1.24698 q^{12} -1.80194 q^{14} +(-1.07992 + 0.623490i) q^{15} +(-0.500000 - 0.866025i) q^{16} -0.554958i q^{18} +(0.866025 + 0.500000i) q^{20} +2.24698i q^{21} +(0.222521 + 0.385418i) q^{23} +(-1.07992 + 0.623490i) q^{24} -1.00000 q^{25} +0.554958 q^{27} +(1.56052 - 0.900969i) q^{28} +(-0.623490 - 1.07992i) q^{29} +(0.623490 - 1.07992i) q^{30} +(0.866025 + 0.500000i) q^{32} +(-0.900969 + 1.56052i) q^{35} +(0.277479 + 0.480608i) q^{36} -1.00000 q^{40} +(-1.56052 + 0.900969i) q^{41} +(-1.12349 - 1.94594i) q^{42} +(0.222521 - 0.385418i) q^{43} +(-0.480608 - 0.277479i) q^{45} +(-0.385418 - 0.222521i) q^{46} -0.445042i q^{47} +(0.623490 - 1.07992i) q^{48} +(1.12349 + 1.94594i) q^{49} +(0.866025 - 0.500000i) q^{50} +(-0.480608 + 0.277479i) q^{54} +(-0.900969 + 1.56052i) q^{56} +(1.07992 + 0.623490i) q^{58} +1.24698i q^{60} +(0.900969 - 1.56052i) q^{61} +(-0.866025 + 0.500000i) q^{63} -1.00000 q^{64} +(0.385418 - 0.222521i) q^{67} +(-0.277479 + 0.480608i) q^{69} -1.80194i q^{70} +(-0.480608 - 0.277479i) q^{72} +(-0.623490 - 1.07992i) q^{75} +(0.866025 - 0.500000i) q^{80} +(0.623490 + 1.07992i) q^{81} +(0.900969 - 1.56052i) q^{82} -1.24698i q^{83} +(1.94594 + 1.12349i) q^{84} +0.445042i q^{86} +(0.777479 - 1.34663i) q^{87} +(0.385418 - 0.222521i) q^{89} +0.554958 q^{90} +0.445042 q^{92} +(0.222521 + 0.385418i) q^{94} +1.24698i q^{96} +(-1.94594 - 1.12349i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{3} + 6 q^{4} - 4 q^{9} - 6 q^{10} - 4 q^{12} - 4 q^{14} - 6 q^{16} + 2 q^{23} - 12 q^{25} + 8 q^{27} + 2 q^{29} - 2 q^{30} - 2 q^{35} + 4 q^{36} - 12 q^{40} - 4 q^{42} + 2 q^{43} - 2 q^{48}+ \cdots + 2 q^{94}+O(q^{100}) \) 12 * q - 2 * q^3 + 6 * q^4 - 4 * q^9 - 6 * q^10 - 4 * q^12 - 4 * q^14 - 6 * q^16 + 2 * q^23 - 12 * q^25 + 8 * q^27 + 2 * q^29 - 2 * q^30 - 2 * q^35 + 4 * q^36 - 12 * q^40 - 4 * q^42 + 2 * q^43 - 2 * q^48 + 4 * q^49 - 2 * q^56 + 2 * q^61 - 12 * q^64 - 4 * q^69 + 2 * q^75 - 2 * q^81 + 2 * q^82 + 10 * q^87 + 8 * q^90 + 4 * q^92 + 2 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more