LMFDB - Embedded newform 1183.1.t.a.1161.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1183,1,Mod(699,1183)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1183, 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("1183.699");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1183 = 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1183.t (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:	$0.590393909945$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.1655595487.1

Embedding invariants

Embedding label			1161.5
Root			$1.56052 + 0.900969i$ of defining polynomial
Character	$\chi$	$=$	1183.1161
Dual form			1183.1.t.a.699.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.07992 + 0.623490i) q^{2} +(0.277479 + 0.480608i) q^{4} +(0.866025 - 0.500000i) q^{7} -0.554958i q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(1.07992 + 0.623490i) q^{2} +(0.277479 + 0.480608i) q^{4} +(0.866025 - 0.500000i) q^{7} -0.554958i q^{8} +(-0.500000 - 0.866025i) q^{9} +(0.385418 + 0.222521i) q^{11} +1.24698 q^{14} +(0.623490 - 1.07992i) q^{16} -1.24698i q^{18} +(0.277479 + 0.480608i) q^{22} +(-0.900969 + 1.56052i) q^{23} -1.00000 q^{25} +(0.480608 + 0.277479i) q^{28} +(-0.623490 + 1.07992i) q^{29} +(0.866025 - 0.500000i) q^{32} +(0.277479 - 0.480608i) q^{36} +(1.56052 + 0.900969i) q^{37} +(0.623490 + 1.07992i) q^{43} +0.246980i q^{44} +(-1.94594 + 1.12349i) q^{46} +(0.500000 - 0.866025i) q^{49} +(-1.07992 - 0.623490i) q^{50} -1.80194 q^{53} +(-0.277479 - 0.480608i) q^{56} +(-1.34663 + 0.777479i) q^{58} +(-0.866025 - 0.500000i) q^{63} +(-1.56052 - 0.900969i) q^{67} +(1.56052 - 0.900969i) q^{71} +(-0.480608 + 0.277479i) q^{72} +(1.12349 + 1.94594i) q^{74} +0.445042 q^{77} -0.445042 q^{79} +(-0.500000 + 0.866025i) q^{81} +1.55496i q^{86} +(0.123490 - 0.213891i) q^{88} -1.00000 q^{92} +(1.07992 - 0.623490i) q^{98} -0.445042i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{4} - 6 q^{9}+O(q^{10}) $ 12 * q + 4 * q^4 - 6 * q^9	$ 12 q + 4 q^{4} - 6 q^{9} - 4 q^{14} - 2 q^{16} + 4 q^{22} - 2 q^{23} - 12 q^{25} + 2 q^{29} + 4 q^{36} - 2 q^{43} + 6 q^{49} - 4 q^{53} - 4 q^{56} + 4 q^{74} + 4 q^{77} - 4 q^{79} - 6 q^{81} - 8 q^{88} - 12 q^{92}+O(q^{100}) $ 12 * q + 4 * q^4 - 6 * q^9 - 4 * q^14 - 2 * q^16 + 4 * q^22 - 2 * q^23 - 12 * q^25 + 2 * q^29 + 4 * q^36 - 2 * q^43 + 6 * q^49 - 4 * q^53 - 4 * q^56 + 4 * q^74 + 4 * q^77 - 4 * q^79 - 6 * q^81 - 8 * q^88 - 12 * q^92

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times$.

$n$	$339$	$1016$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

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 $

$2$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.930874	−	0.365341i	$-0.119048\pi$
$2$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.149042	+	0.988831i	$0.452381\pi$
$3$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$3$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$4$	0.277479	+	0.480608i	0.277479	+	0.480608i
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$	0.866025	−	0.500000i	0.866025	−	0.500000i
$7$	0.866025	−	0.500000i	0.866025	−	0.500000i
$8$		−	0.554958i		−	0.554958i
$9$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$10$		0			0
$11$	0.385418	+	0.222521i	0.385418	+	0.222521i	0.680173	−	0.733052i	$-0.261905\pi$
$11$	0.385418	+	0.222521i	0.385418	+	0.222521i	−0.294755	+	0.955573i	$0.595238\pi$
$12$		0			0
$13$		0			0
$13$		0			0
$14$	1.24698			1.24698
$15$		0			0
$16$	0.623490	−	1.07992i	0.623490	−	1.07992i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		−	1.24698i		−	1.24698i
$19$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$19$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$20$		0			0
$21$		0			0
$22$	0.277479	+	0.480608i	0.277479	+	0.480608i
$23$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.0747301	+	0.997204i	$0.523810\pi$
$23$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.826239	+	0.563320i	$0.809524\pi$
$24$		0			0
$25$	−1.00000			−1.00000
$26$		0			0
$27$		0			0
$28$	0.480608	+	0.277479i	0.480608	+	0.277479i
$29$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$29$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$30$		0			0
$31$		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			0
$36$	0.277479	−	0.480608i	0.277479	−	0.480608i
$37$	1.56052	+	0.900969i	1.56052	+	0.900969i	0.997204	+	0.0747301i	$0.0238095\pi$
$37$	1.56052	+	0.900969i	1.56052	+	0.900969i	0.563320	+	0.826239i	$0.309524\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$	0.623490	+	1.07992i	0.623490	+	1.07992i	0.988831	+	0.149042i	$0.0476190\pi$
$43$	0.623490	+	1.07992i	0.623490	+	1.07992i	−0.365341	+	0.930874i	$0.619048\pi$
$44$			0.246980i			0.246980i
$45$		0			0
$46$	−1.94594	+	1.12349i	−1.94594	+	1.12349i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	0.500000	−	0.866025i	0.500000	−	0.866025i
$50$	−1.07992	−	0.623490i	−1.07992	−	0.623490i
$51$		0			0
$52$		0			0
$53$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$53$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$54$		0			0
$55$		0			0
$56$	−0.277479	−	0.480608i	−0.277479	−	0.480608i
$57$		0			0
$58$	−1.34663	+	0.777479i	−1.34663	+	0.777479i
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−1.56052	−	0.900969i	−1.56052	−	0.900969i	−0.997204	−	0.0747301i	$-0.976190\pi$
$67$	−1.56052	−	0.900969i	−1.56052	−	0.900969i	−0.563320	−	0.826239i	$-0.690476\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.56052	−	0.900969i	1.56052	−	0.900969i	0.563320	−	0.826239i	$-0.309524\pi$
$71$	1.56052	−	0.900969i	1.56052	−	0.900969i	0.997204	−	0.0747301i	$-0.0238095\pi$
$72$	−0.480608	+	0.277479i	−0.480608	+	0.277479i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$	1.12349	+	1.94594i	1.12349	+	1.94594i
$75$		0			0
$76$		0			0
$77$	0.445042			0.445042
$78$		0			0
$79$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$79$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$			1.55496i			1.55496i
$87$		0			0
$88$	0.123490	−	0.213891i	0.123490	−	0.213891i
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$		0			0
$92$	−1.00000			−1.00000
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$97$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$98$	1.07992	−	0.623490i	1.07992	−	0.623490i
$99$		−	0.445042i		−	0.445042i
$100$	−0.277479	−	0.480608i	−0.277479	−	0.480608i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$	−1.94594	−	1.12349i	−1.94594	−	1.12349i
$107$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$107$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$108$		0			0
$109$			0.445042i			0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$109$			0.445042i			0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$110$		0			0
$111$		0			0
$112$		−	1.24698i		−	1.24698i
$113$	0.222521	+	0.385418i	0.222521	+	0.385418i	0.955573	−	0.294755i	$-0.0952381\pi$
$113$	0.222521	+	0.385418i	0.222521	+	0.385418i	−0.733052	+	0.680173i	$0.761905\pi$
$114$		0			0
$115$		0			0
$116$	−0.692021			−0.692021
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$127$	0.623490	−	1.07992i	0.623490	−	1.07992i	−0.365341	−	0.930874i	$-0.619048\pi$
$127$	0.623490	−	1.07992i	0.623490	−	1.07992i	0.988831	−	0.149042i	$-0.0476190\pi$
$128$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$135$		0			0
$136$		0			0
$137$	−0.385418	+	0.222521i	−0.385418	+	0.222521i	−0.680173	−	0.733052i	$-0.738095\pi$
$137$	−0.385418	+	0.222521i	−0.385418	+	0.222521i	0.294755	+	0.955573i	$0.404762\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$		0			0
$141$		0			0
$142$	2.24698			2.24698
$143$		0			0
$144$	−1.24698			−1.24698
$145$		0			0
$146$		0			0
$147$		0			0
$148$			1.00000i			1.00000i
$149$	−1.07992	+	0.623490i	−1.07992	+	0.623490i	−0.930874	−	0.365341i	$-0.880952\pi$
$149$	−1.07992	+	0.623490i	−1.07992	+	0.623490i	−0.149042	+	0.988831i	$0.547619\pi$
$150$		0			0
$151$			1.24698i			1.24698i	0.781831	+	0.623490i	$0.214286\pi$
$151$			1.24698i			1.24698i	−0.781831	+	0.623490i	$0.785714\pi$
$152$		0			0
$153$		0			0
$154$	0.480608	+	0.277479i	0.480608	+	0.277479i
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$	−0.480608	−	0.277479i	−0.480608	−	0.277479i
$159$		0			0
$160$		0			0
$161$			1.80194i			1.80194i
$162$	−1.07992	+	0.623490i	−1.07992	+	0.623490i
$163$	1.07992	−	0.623490i	1.07992	−	0.623490i	0.149042	−	0.988831i	$-0.452381\pi$
$163$	1.07992	−	0.623490i	1.07992	−	0.623490i	0.930874	+	0.365341i	$0.119048\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$		0			0
$172$	−0.346011	+	0.599308i	−0.346011	+	0.599308i
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$176$	0.480608	−	0.277479i	0.480608	−	0.277479i
$177$		0			0
$178$		0			0
$179$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	−0.955573	−	0.294755i	$-0.904762\pi$
$179$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	0.733052	+	0.680173i	$0.238095\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	0.866025	+	0.500000i	0.866025	+	0.500000i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$191$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$192$		0			0
$193$	−1.07992	−	0.623490i	−1.07992	−	0.623490i	−0.149042	−	0.988831i	$-0.547619\pi$
$193$	−1.07992	−	0.623490i	−1.07992	−	0.623490i	−0.930874	+	0.365341i	$0.880952\pi$
$194$		0			0
$195$		0			0
$196$	0.554958			0.554958
$197$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$197$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$198$	0.277479	−	0.480608i	0.277479	−	0.480608i
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$			0.554958i			0.554958i
$201$		0			0
$202$		0			0
$203$			1.24698i			1.24698i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.80194			1.80194
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$211$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$212$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$213$		0			0
$214$	0.480608	−	0.277479i	0.480608	−	0.277479i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−0.277479	+	0.480608i	−0.277479	+	0.480608i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$223$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$224$	0.500000	−	0.866025i	0.500000	−	0.866025i
$225$	0.500000	+	0.866025i	0.500000	+	0.866025i
$226$			0.554958i			0.554958i
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$	0.599308	+	0.346011i	0.599308	+	0.346011i
$233$	−1.24698			−1.24698			−0.623490	−	0.781831i	$-0.714286\pi$
$233$	−1.24698			−1.24698			−0.623490	+	0.781831i	$0.714286\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	1.24698i		−	1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$239$		−	1.24698i		−	1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$240$		0			0
$241$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$241$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$242$		−	1.00000i		−	1.00000i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$251$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$252$		−	0.554958i		−	0.554958i
$253$	−0.694498	+	0.400969i	−0.694498	+	0.400969i
$254$	1.34663	−	0.777479i	1.34663	−	0.777479i
$255$		0			0
$256$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$	1.80194			1.80194
$260$		0			0
$261$	1.24698			1.24698
$262$		0			0
$263$	0.222521	−	0.385418i	0.222521	−	0.385418i	−0.733052	−	0.680173i	$-0.761905\pi$
$263$	0.222521	−	0.385418i	0.222521	−	0.385418i	0.955573	+	0.294755i	$0.0952381\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		−	1.00000i		−	1.00000i
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$271$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$272$		0			0
$273$		0			0
$274$	−0.554958			−0.554958
$275$	−0.385418	−	0.222521i	−0.385418	−	0.222521i
$276$		0			0
$277$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	0.733052	−	0.680173i	$-0.238095\pi$
$277$	−0.222521	−	0.385418i	−0.222521	−	0.385418i	−0.955573	+	0.294755i	$0.904762\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$281$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$	0.866025	+	0.500000i	0.866025	+	0.500000i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$289$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$293$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$294$		0			0
$295$		0			0
$296$	0.500000	−	0.866025i	0.500000	−	0.866025i
$297$		0			0
$298$	−1.55496			−1.55496
$299$		0			0
$300$		0			0
$301$	1.07992	+	0.623490i	1.07992	+	0.623490i
$302$	−0.777479	+	1.34663i	−0.777479	+	1.34663i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	0.123490	+	0.213891i	0.123490	+	0.213891i
$309$		0			0
$310$		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			0
$316$	−0.123490	−	0.213891i	−0.123490	−	0.213891i
$317$			0.445042i			0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$317$			0.445042i			0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$318$		0			0
$319$	−0.480608	+	0.277479i	−0.480608	+	0.277479i
$320$		0			0
$321$		0			0
$322$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$323$		0			0
$324$	−0.554958			−0.554958
$325$		0			0
$326$	1.55496			1.55496
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.56052	−	0.900969i	1.56052	−	0.900969i	0.563320	−	0.826239i	$-0.309524\pi$
$331$	1.56052	−	0.900969i	1.56052	−	0.900969i	0.997204	−	0.0747301i	$-0.0238095\pi$
$332$		0			0
$333$		−	1.80194i		−	1.80194i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.24698			−1.24698			−0.623490	−	0.781831i	$-0.714286\pi$
$337$	−1.24698			−1.24698			−0.623490	+	0.781831i	$0.714286\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	1.00000i		−	1.00000i
$344$	0.599308	−	0.346011i	0.599308	−	0.346011i
$345$		0			0
$346$		0			0
$347$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$347$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$348$		0			0
$349$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$349$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$350$	−1.24698			−1.24698
$351$		0			0
$352$	0.445042			0.445042
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−0.480608	+	0.277479i	−0.480608	+	0.277479i
$359$		−	1.80194i		−	1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$359$		−	1.80194i		−	1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$360$		0			0
$361$	0.500000	−	0.866025i	0.500000	−	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$	1.12349	+	1.94594i	1.12349	+	1.94594i
$369$		0			0
$370$		0			0
$371$	−1.56052	+	0.900969i	−1.56052	+	0.900969i
$372$		0			0
$373$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$373$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.930874	−	0.365341i	$-0.119048\pi$
$379$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.149042	+	0.988831i	$0.452381\pi$
$380$		0			0
$381$		0			0
$382$		−	1.55496i		−	1.55496i
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$	−0.777479	−	1.34663i	−0.777479	−	1.34663i
$387$	0.623490	−	1.07992i	0.623490	−	1.07992i
$388$		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
$392$	−0.480608	−	0.277479i	−0.480608	−	0.277479i
$393$		0			0
$394$	1.24698	+	2.15983i	1.24698	+	2.15983i
$395$		0			0
$396$	0.213891	−	0.123490i	0.213891	−	0.123490i
$397$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$397$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$398$		0			0
$399$		0			0
$400$	−0.623490	+	1.07992i	−0.623490	+	1.07992i
$401$	−1.07992	−	0.623490i	−1.07992	−	0.623490i	−0.149042	−	0.988831i	$-0.547619\pi$
$401$	−1.07992	−	0.623490i	−1.07992	−	0.623490i	−0.930874	+	0.365341i	$0.880952\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−0.777479	+	1.34663i	−0.777479	+	1.34663i
$407$	0.400969	+	0.694498i	0.400969	+	0.694498i
$408$		0			0
$409$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$409$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	1.94594	+	1.12349i	1.94594	+	1.12349i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$421$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$422$	1.94594	−	1.12349i	1.94594	−	1.12349i
$423$		0			0
$424$			1.00000i			1.00000i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	0.246980			0.246980
$429$		0			0
$430$		0			0
$431$	−0.385418	−	0.222521i	−0.385418	−	0.222521i	0.294755	−	0.955573i	$-0.404762\pi$
$431$	−0.385418	−	0.222521i	−0.385418	−	0.222521i	−0.680173	+	0.733052i	$0.738095\pi$
$432$		0			0
$433$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$433$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$434$		0			0
$435$		0			0
$436$	−0.213891	+	0.123490i	−0.213891	+	0.123490i
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	−1.00000			−1.00000
$442$		0			0
$443$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$443$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.56052	+	0.900969i	−1.56052	+	0.900969i	−0.563320	+	0.826239i	$0.690476\pi$
$449$	−1.56052	+	0.900969i	−1.56052	+	0.900969i	−0.997204	+	0.0747301i	$0.976190\pi$
$450$			1.24698i			1.24698i
$451$		0			0
$452$	−0.123490	+	0.213891i	−0.123490	+	0.213891i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.385418	−	0.222521i	−0.385418	−	0.222521i	0.294755	−	0.955573i	$-0.404762\pi$
$457$	−0.385418	−	0.222521i	−0.385418	−	0.222521i	−0.680173	+	0.733052i	$0.738095\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$461$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$462$		0			0
$463$		−	0.445042i		−	0.445042i	−0.974928	−	0.222521i	$-0.928571\pi$
$463$		−	0.445042i		−	0.445042i	0.974928	−	0.222521i	$-0.0714286\pi$
$464$	0.777479	+	1.34663i	0.777479	+	1.34663i
$465$		0			0
$466$	−1.34663	−	0.777479i	−1.34663	−	0.777479i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	−1.80194			−1.80194
$470$		0			0
$471$		0			0
$472$		0			0
$473$			0.554958i			0.554958i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.900969	+	1.56052i	0.900969	+	1.56052i
$478$	0.777479	−	1.34663i	0.777479	−	1.34663i
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.222521	−	0.385418i	0.222521	−	0.385418i
$485$		0			0
$486$		0			0
$487$	0.385418	−	0.222521i	0.385418	−	0.222521i	−0.294755	−	0.955573i	$-0.595238\pi$
$487$	0.385418	−	0.222521i	0.385418	−	0.222521i	0.680173	+	0.733052i	$0.261905\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.0747301	+	0.997204i	$0.523810\pi$
$491$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.826239	+	0.563320i	$0.809524\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.900969	−	1.56052i	0.900969	−	1.56052i
$498$		0			0
$499$		−	1.24698i		−	1.24698i	−0.781831	−	0.623490i	$-0.785714\pi$
$499$		−	1.24698i		−	1.24698i	0.781831	−	0.623490i	$-0.214286\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$	−0.277479	+	0.480608i	−0.277479	+	0.480608i
$505$		0			0
$506$	−1.00000			−1.00000
$507$		0			0
$508$	0.692021			0.692021
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$		0			0
$512$		−	0.554958i		−	0.554958i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	1.94594	+	1.12349i	1.94594	+	1.12349i
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	1.34663	+	0.777479i	1.34663	+	0.777479i
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$		0			0
$526$	0.480608	−	0.277479i	0.480608	−	0.277479i
$527$		0			0
$528$		0			0
$529$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$537$		0			0
$538$		0			0
$539$	0.385418	−	0.222521i	0.385418	−	0.222521i
$540$		0			0
$541$		−	1.80194i		−	1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$541$		−	1.80194i		−	1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.445042			−0.445042			−0.222521	−	0.974928i	$-0.571429\pi$
$547$	−0.445042			−0.445042			−0.222521	+	0.974928i	$0.571429\pi$
$548$	−0.213891	−	0.123490i	−0.213891	−	0.123490i
$549$		0			0
$550$	−0.277479	−	0.480608i	−0.277479	−	0.480608i
$551$		0			0
$552$		0			0
$553$	−0.385418	+	0.222521i	−0.385418	+	0.222521i
$554$		−	0.554958i		−	0.554958i
$555$		0			0
$556$		0			0
$557$	1.56052	+	0.900969i	1.56052	+	0.900969i	0.997204	+	0.0747301i	$0.0238095\pi$
$557$	1.56052	+	0.900969i	1.56052	+	0.900969i	0.563320	+	0.826239i	$0.309524\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−1.24698	+	2.15983i	−1.24698	+	2.15983i
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$			1.00000i			1.00000i
$568$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$569$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	−0.955573	−	0.294755i	$-0.904762\pi$
$569$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	0.733052	+	0.680173i	$0.238095\pi$
$570$		0			0
$571$	−1.24698			−1.24698			−0.623490	−	0.781831i	$-0.714286\pi$
$571$	−1.24698			−1.24698			−0.623490	+	0.781831i	$0.714286\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.900969	−	1.56052i	0.900969	−	1.56052i
$576$		0			0
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	−1.07992	+	0.623490i	−1.07992	+	0.623490i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.694498	−	0.400969i	−0.694498	−	0.400969i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$587$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	1.94594	−	1.12349i	1.94594	−	1.12349i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	−0.599308	−	0.346011i	−0.599308	−	0.346011i
$597$		0			0
$598$		0			0
$599$	1.24698			1.24698			0.623490	−	0.781831i	$-0.285714\pi$
$599$	1.24698			1.24698			0.623490	+	0.781831i	$0.285714\pi$
$600$		0			0
$601$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$601$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$602$	0.777479	+	1.34663i	0.777479	+	1.34663i
$603$			1.80194i			1.80194i
$604$	−0.599308	+	0.346011i	−0.599308	+	0.346011i
$605$		0			0
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.930874	−	0.365341i	$-0.119048\pi$
$613$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.149042	+	0.988831i	$0.452381\pi$
$614$		0			0
$615$		0			0
$616$		−	0.246980i		−	0.246980i
$617$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$617$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.00000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.385418	+	0.222521i	−0.385418	+	0.222521i	−0.680173	−	0.733052i	$-0.738095\pi$
$631$	−0.385418	+	0.222521i	−0.385418	+	0.222521i	0.294755	+	0.955573i	$0.404762\pi$
$632$			0.246980i			0.246980i
$633$		0			0
$634$	−0.277479	+	0.480608i	−0.277479	+	0.480608i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−0.692021			−0.692021
$639$	−1.56052	−	0.900969i	−1.56052	−	0.900969i
$640$		0			0
$641$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.826239	−	0.563320i	$-0.809524\pi$
$641$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.0747301	−	0.997204i	$-0.523810\pi$
$642$		0			0
$643$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$643$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$644$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$	0.480608	+	0.277479i	0.480608	+	0.277479i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.599308	+	0.346011i	0.599308	+	0.346011i
$653$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$653$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	−0.988831	−	0.149042i	$-0.952381\pi$
$659$	−0.623490	−	1.07992i	−0.623490	−	1.07992i	0.365341	−	0.930874i	$-0.380952\pi$
$660$		0			0
$661$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$661$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$662$	2.24698			2.24698
$663$		0			0
$664$		0			0
$665$		0			0
$666$	1.12349	−	1.94594i	1.12349	−	1.94594i
$667$	−1.12349	−	1.94594i	−1.12349	−	1.94594i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	−0.955573	−	0.294755i	$-0.904762\pi$
$673$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	0.733052	+	0.680173i	$0.238095\pi$
$674$	−1.34663	−	0.777479i	−1.34663	−	0.777479i
$675$		0			0
$676$		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
$683$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$683$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$684$		0			0
$685$		0			0
$686$	0.623490	−	1.07992i	0.623490	−	1.07992i
$687$		0			0
$688$	1.55496			1.55496
$689$		0			0
$690$		0			0
$691$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$691$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$692$		0			0
$693$	−0.222521	−	0.385418i	−0.222521	−	0.385418i
$694$			2.24698i			2.24698i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.480608	−	0.277479i	−0.480608	−	0.277479i
$701$	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			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.07992	−	0.623490i	1.07992	−	0.623490i	0.149042	−	0.988831i	$-0.452381\pi$
$709$	1.07992	−	0.623490i	1.07992	−	0.623490i	0.930874	+	0.365341i	$0.119048\pi$
$710$		0			0
$711$	0.222521	+	0.385418i	0.222521	+	0.385418i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.246980			−0.246980
$717$		0			0
$718$	1.12349	−	1.94594i	1.12349	−	1.94594i
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$		0			0
$722$	1.07992	−	0.623490i	1.07992	−	0.623490i
$723$		0			0
$724$		0			0
$725$	0.623490	−	1.07992i	0.623490	−	1.07992i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	1.00000			1.00000
$730$		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
$735$		0			0
$736$			1.80194i			1.80194i
$737$	−0.400969	−	0.694498i	−0.400969	−	0.694498i
$738$		0			0
$739$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$739$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$740$		0			0
$741$		0			0
$742$	−2.24698			−2.24698
$743$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.930874	−	0.365341i	$-0.119048\pi$
$743$	1.07992	+	0.623490i	1.07992	+	0.623490i	0.149042	+	0.988831i	$0.452381\pi$
$744$		0			0
$745$		0			0
$746$			2.24698i			2.24698i
$747$		0			0
$748$		0			0
$749$		−	0.445042i		−	0.445042i
$750$		0			0
$751$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	−0.955573	−	0.294755i	$-0.904762\pi$
$751$	−0.222521	+	0.385418i	−0.222521	+	0.385418i	0.733052	+	0.680173i	$0.238095\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$757$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$758$	0.777479	+	1.34663i	0.777479	+	1.34663i
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$	0.222521	+	0.385418i	0.222521	+	0.385418i
$764$	0.346011	−	0.599308i	0.346011	−	0.599308i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$769$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$770$		0			0
$771$		0			0
$772$		−	0.692021i		−	0.692021i
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$	1.34663	−	0.777479i	1.34663	−	0.777479i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	0.480608	+	0.277479i	0.480608	+	0.277479i
$779$		0			0
$780$		0			0
$781$	0.801938			0.801938
$782$		0			0
$783$		0			0
$784$	−0.623490	−	1.07992i	−0.623490	−	1.07992i
$785$		0			0
$786$		0			0
$787$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$787$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$788$			1.10992i			1.10992i
$789$		0			0
$790$		0			0
$791$	0.385418	+	0.222521i	0.385418	+	0.222521i
$792$	−0.246980			−0.246980
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$797$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$798$		0			0
$799$		0			0
$800$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$801$		0			0
$802$	−0.777479	−	1.34663i	−0.777479	−	1.34663i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.0747301	−	0.997204i	$-0.476190\pi$
$809$	0.900969	−	1.56052i	0.900969	−	1.56052i	0.826239	−	0.563320i	$-0.190476\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	−0.599308	+	0.346011i	−0.599308	+	0.346011i
$813$		0			0
$814$			1.00000i			1.00000i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.56052	−	0.900969i	−1.56052	−	0.900969i	−0.997204	−	0.0747301i	$-0.976190\pi$
$821$	−1.56052	−	0.900969i	−1.56052	−	0.900969i	−0.563320	−	0.826239i	$-0.690476\pi$
$822$		0			0
$823$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.826239	−	0.563320i	$-0.809524\pi$
$823$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.0747301	−	0.997204i	$-0.523810\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1.80194i		−	1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$827$		−	1.80194i		−	1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$828$	0.500000	+	0.866025i	0.500000	+	0.866025i
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$		0			0
$841$	−0.277479	−	0.480608i	−0.277479	−	0.480608i
$842$	1.24698	−	2.15983i	1.24698	−	2.15983i
$843$		0			0
$844$	1.00000			1.00000
$845$		0			0
$846$		0			0
$847$	−0.694498	−	0.400969i	−0.694498	−	0.400969i
$848$	−1.12349	+	1.94594i	−1.12349	+	1.94594i
$849$		0			0
$850$		0			0
$851$	−2.81197	+	1.62349i	−2.81197	+	1.62349i
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	−0.213891	−	0.123490i	−0.213891	−	0.123490i
$857$		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			0
$861$		0			0
$862$	−0.277479	−	0.480608i	−0.277479	−	0.480608i
$863$			0.445042i			0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$863$			0.445042i			0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.171527	−	0.0990311i	−0.171527	−	0.0990311i
$870$		0			0
$871$		0			0
$872$	0.246980			0.246980
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.385418	−	0.222521i	0.385418	−	0.222521i	−0.294755	−	0.955573i	$-0.595238\pi$
$877$	0.385418	−	0.222521i	0.385418	−	0.222521i	0.680173	+	0.733052i	$0.261905\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$881$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$882$	−1.07992	−	0.623490i	−1.07992	−	0.623490i
$883$	0.445042			0.445042			0.222521	−	0.974928i	$-0.428571\pi$
$883$	0.445042			0.445042			0.222521	+	0.974928i	$0.428571\pi$
$884$		0			0
$885$		0			0
$886$	1.34663	+	0.777479i	1.34663	+	0.777479i
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$		−	1.24698i		−	1.24698i
$890$		0			0
$891$	−0.385418	+	0.222521i	−0.385418	+	0.222521i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−1.00000			−1.00000
$897$		0			0
$898$	−2.24698			−2.24698
$899$		0			0
$900$	−0.277479	+	0.480608i	−0.277479	+	0.480608i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	0.213891	−	0.123490i	0.213891	−	0.123490i
$905$		0			0
$906$		0			0
$907$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.0747301	+	0.997204i	$0.523810\pi$
$907$	−0.900969	+	1.56052i	−0.900969	+	1.56052i	−0.826239	+	0.563320i	$0.809524\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.80194			−1.80194			−0.900969	−	0.433884i	$-0.857143\pi$
$911$	−1.80194			−1.80194			−0.900969	+	0.433884i	$0.857143\pi$
$912$		0			0
$913$		0			0
$914$	−0.277479	−	0.480608i	−0.277479	−	0.480608i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.826239	+	0.563320i	$0.190476\pi$
$919$	0.900969	+	1.56052i	0.900969	+	1.56052i	0.0747301	+	0.997204i	$0.476190\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.56052	−	0.900969i	−1.56052	−	0.900969i
$926$	0.277479	−	0.480608i	0.277479	−	0.480608i
$927$		0			0
$928$			1.24698i			1.24698i
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$		0			0
$932$	−0.346011	−	0.599308i	−0.346011	−	0.599308i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	−1.94594	−	1.12349i	−1.94594	−	1.12349i
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−0.346011	+	0.599308i	−0.346011	+	0.599308i
$947$	1.56052	+	0.900969i	1.56052	+	0.900969i	0.997204	+	0.0747301i	$0.0238095\pi$
$947$	1.56052	+	0.900969i	1.56052	+	0.900969i	0.563320	+	0.826239i	$0.309524\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.826239	−	0.563320i	$-0.809524\pi$
$953$	−0.900969	−	1.56052i	−0.900969	−	1.56052i	−0.0747301	−	0.997204i	$-0.523810\pi$
$954$			2.24698i			2.24698i
$955$		0			0
$956$	0.599308	−	0.346011i	0.599308	−	0.346011i
$957$		0			0
$958$		0			0
$959$	−0.222521	+	0.385418i	−0.222521	+	0.385418i
$960$		0			0
$961$	−1.00000			−1.00000
$962$		0			0
$963$	−0.445042			−0.445042
$964$		0			0
$965$		0			0
$966$		0			0
$967$			1.80194i			1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$967$			1.80194i			1.80194i	−0.433884	+	0.900969i	$0.642857\pi$
$968$	−0.385418	+	0.222521i	−0.385418	+	0.222521i
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$		0			0
$973$		0			0
$974$	0.554958			0.554958
$975$		0			0
$976$		0			0
$977$	−0.385418	−	0.222521i	−0.385418	−	0.222521i	0.294755	−	0.955573i	$-0.404762\pi$
$977$	−0.385418	−	0.222521i	−0.385418	−	0.222521i	−0.680173	+	0.733052i	$0.738095\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.385418	−	0.222521i	0.385418	−	0.222521i
$982$	−1.94594	+	1.12349i	−1.94594	+	1.12349i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.24698			−2.24698
$990$		0			0
$991$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	0.365341	+	0.930874i	$0.380952\pi$
$991$	−0.623490	+	1.07992i	−0.623490	+	1.07992i	−0.988831	+	0.149042i	$0.952381\pi$
$992$		0			0
$993$		0			0
$994$	1.94594	−	1.12349i	1.94594	−	1.12349i
$995$		0			0
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$	0.777479	−	1.34663i	0.777479	−	1.34663i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1183.1.t.a.1161.5		12
7.6	odd	2	CM	1183.1.t.a.1161.5		12
13.2	odd	12		1183.1.n.a.146.3		6
13.3	even	3	inner	1183.1.t.a.699.2		12
13.4	even	6		1183.1.b.a.1182.5		6
13.5	odd	4		1183.1.n.a.867.3		6
13.6	odd	12		1183.1.d.b.846.1	yes	3
13.7	odd	12		1183.1.d.a.846.3	✓	3
13.8	odd	4		1183.1.n.b.867.1		6
13.9	even	3		1183.1.b.a.1182.2		6
13.10	even	6	inner	1183.1.t.a.699.5		12
13.11	odd	12		1183.1.n.b.146.1		6
13.12	even	2	inner	1183.1.t.a.1161.2		12
91.6	even	12		1183.1.d.b.846.1	yes	3
91.20	even	12		1183.1.d.a.846.3	✓	3
91.34	even	4		1183.1.n.b.867.1		6
91.41	even	12		1183.1.n.a.146.3		6
91.48	odd	6		1183.1.b.a.1182.2		6
91.55	odd	6	inner	1183.1.t.a.699.2		12
91.62	odd	6	inner	1183.1.t.a.699.5		12
91.69	odd	6		1183.1.b.a.1182.5		6
91.76	even	12		1183.1.n.b.146.1		6
91.83	even	4		1183.1.n.a.867.3		6
91.90	odd	2	inner	1183.1.t.a.1161.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1183.1.b.a.1182.2		6	13.9	even	3
1183.1.b.a.1182.2		6	91.48	odd	6
1183.1.b.a.1182.5		6	13.4	even	6
1183.1.b.a.1182.5		6	91.69	odd	6
1183.1.d.a.846.3	✓	3	13.7	odd	12
1183.1.d.a.846.3	✓	3	91.20	even	12
1183.1.d.b.846.1	yes	3	13.6	odd	12
1183.1.d.b.846.1	yes	3	91.6	even	12
1183.1.n.a.146.3		6	13.2	odd	12
1183.1.n.a.146.3		6	91.41	even	12
1183.1.n.a.867.3		6	13.5	odd	4
1183.1.n.a.867.3		6	91.83	even	4
1183.1.n.b.146.1		6	13.11	odd	12
1183.1.n.b.146.1		6	91.76	even	12
1183.1.n.b.867.1		6	13.8	odd	4
1183.1.n.b.867.1		6	91.34	even	4
1183.1.t.a.699.2		12	13.3	even	3	inner
1183.1.t.a.699.2		12	91.55	odd	6	inner
1183.1.t.a.699.5		12	13.10	even	6	inner
1183.1.t.a.699.5		12	91.62	odd	6	inner
1183.1.t.a.1161.2		12	13.12	even	2	inner
1183.1.t.a.1161.2		12	91.90	odd	2	inner
1183.1.t.a.1161.5		12	1.1	even	1	trivial
1183.1.t.a.1161.5		12	7.6	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1183 = 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1183.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.07992 + 0.623490i) q^{2} +(0.277479 + 0.480608i) q^{4} +(0.866025 - 0.500000i) q^{7} -0.554958i q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(1.07992 + 0.623490i) q^{2} +(0.277479 + 0.480608i) q^{4} +(0.866025 - 0.500000i) q^{7} -0.554958i q^{8} +(-0.500000 - 0.866025i) q^{9} +(0.385418 + 0.222521i) q^{11} +1.24698 q^{14} +(0.623490 - 1.07992i) q^{16} -1.24698i q^{18} +(0.277479 + 0.480608i) q^{22} +(-0.900969 + 1.56052i) q^{23} -1.00000 q^{25} +(0.480608 + 0.277479i) q^{28} +(-0.623490 + 1.07992i) q^{29} +(0.866025 - 0.500000i) q^{32} +(0.277479 - 0.480608i) q^{36} +(1.56052 + 0.900969i) q^{37} +(0.623490 + 1.07992i) q^{43} +0.246980i q^{44} +(-1.94594 + 1.12349i) q^{46} +(0.500000 - 0.866025i) q^{49} +(-1.07992 - 0.623490i) q^{50} -1.80194 q^{53} +(-0.277479 - 0.480608i) q^{56} +(-1.34663 + 0.777479i) q^{58} +(-0.866025 - 0.500000i) q^{63} +(-1.56052 - 0.900969i) q^{67} +(1.56052 - 0.900969i) q^{71} +(-0.480608 + 0.277479i) q^{72} +(1.12349 + 1.94594i) q^{74} +0.445042 q^{77} -0.445042 q^{79} +(-0.500000 + 0.866025i) q^{81} +1.55496i q^{86} +(0.123490 - 0.213891i) q^{88} -1.00000 q^{92} +(1.07992 - 0.623490i) q^{98} -0.445042i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{4} - 6 q^{9}+O(q^{10}) \) 12 * q + 4 * q^4 - 6 * q^9	\( 12 q + 4 q^{4} - 6 q^{9} - 4 q^{14} - 2 q^{16} + 4 q^{22} - 2 q^{23} - 12 q^{25} + 2 q^{29} + 4 q^{36} - 2 q^{43} + 6 q^{49} - 4 q^{53} - 4 q^{56} + 4 q^{74} + 4 q^{77} - 4 q^{79} - 6 q^{81} - 8 q^{88} - 12 q^{92}+O(q^{100}) \) 12 * q + 4 * q^4 - 6 * q^9 - 4 * q^14 - 2 * q^16 + 4 * q^22 - 2 * q^23 - 12 * q^25 + 2 * q^29 + 4 * q^36 - 2 * q^43 + 6 * q^49 - 4 * q^53 - 4 * q^56 + 4 * q^74 + 4 * q^77 - 4 * q^79 - 6 * q^81 - 8 * q^88 - 12 * q^92

Introduction

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