LMFDB - Embedded newform 1001.1.y.c.571.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1001,1,Mod(142,1001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1001, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3, 3]))

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

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

chi := DirichletCharacter("1001.142");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1001 = 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1001.y (of order $6$, degree $2$, 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:	$0.499564077646$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{15})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 $ x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{15}$
Projective field:	Galois closure of 15.1.345408215017012205401343.1

Embedding invariants

Embedding label			571.4
Root			$-0.978148 - 0.207912i$ of defining polynomial
Character	$\chi$	$=$	1001.571
Dual form			1001.1.y.c.142.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.978148 + 1.69420i) q^{2} +(-0.669131 + 1.15897i) q^{3} +(-1.41355 + 2.44833i) q^{4} -2.61803 q^{6} +(-0.104528 - 0.994522i) q^{7} -3.57433 q^{8} +(-0.395472 - 0.684977i) q^{9} +O(q^{10})$	\(q+(0.978148 + 1.69420i) q^{2} +(-0.669131 + 1.15897i) q^{3} +(-1.41355 + 2.44833i) q^{4} -2.61803 q^{6} +(-0.104528 - 0.994522i) q^{7} -3.57433 q^{8} +(-0.395472 - 0.684977i) q^{9} +(-0.500000 + 0.866025i) q^{11} +(-1.89169 - 3.27651i) q^{12} +1.00000 q^{13} +(1.58268 - 1.14988i) q^{14} +(-2.08268 - 3.60730i) q^{16} +(0.773659 - 1.34002i) q^{18} +(-0.309017 - 0.535233i) q^{19} +(1.22256 + 0.544320i) q^{21} -1.95630 q^{22} +(0.809017 + 1.40126i) q^{23} +(2.39169 - 4.14253i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(0.978148 + 1.69420i) q^{26} -0.279773 q^{27} +(2.58268 + 1.14988i) q^{28} +(2.28716 - 3.96149i) q^{32} +(-0.669131 - 1.15897i) q^{33} +2.23607 q^{36} +(0.604528 - 1.04707i) q^{38} +(-0.669131 + 1.15897i) q^{39} +1.82709 q^{41} +(0.273659 + 2.60369i) q^{42} +(-1.41355 - 2.44833i) q^{44} +(-1.58268 + 2.74128i) q^{46} +5.57433 q^{48} +(-0.978148 + 0.207912i) q^{49} -1.95630 q^{50} +(-1.41355 + 2.44833i) q^{52} +(-0.309017 + 0.535233i) q^{53} +(-0.273659 - 0.473991i) q^{54} +(0.373619 + 3.55475i) q^{56} +0.827091 q^{57} +(-0.639886 + 0.464905i) q^{63} +4.78339 q^{64} +(1.30902 - 2.26728i) q^{66} -2.16535 q^{69} +(1.41355 + 2.44833i) q^{72} +(0.978148 - 1.69420i) q^{73} +(-0.669131 - 1.15897i) q^{75} +1.74724 q^{76} +(0.913545 + 0.406737i) q^{77} -2.61803 q^{78} +(0.582676 - 1.00922i) q^{81} +(1.78716 + 3.09546i) q^{82} -0.209057 q^{83} +(-3.06082 + 2.22382i) q^{84} +(1.78716 - 3.09546i) q^{88} +(-0.104528 - 0.994522i) q^{91} -4.57433 q^{92} +(3.06082 + 5.30150i) q^{96} +(-1.30902 - 1.45381i) q^{98} +0.790943 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} - q^{3} - 5 q^{4} - 12 q^{6} + q^{7} - 2 q^{8} - 5 q^{9}+O(q^{10}) $ 8 * q - q^2 - q^3 - 5 * q^4 - 12 * q^6 + q^7 - 2 * q^8 - 5 * q^9	$ 8 q - q^{2} - q^{3} - 5 q^{4} - 12 q^{6} + q^{7} - 2 q^{8} - 5 q^{9} - 4 q^{11} + 8 q^{13} + 2 q^{14} - 6 q^{16} + 2 q^{19} - q^{21} + 2 q^{22} + 2 q^{23} + 4 q^{24} - 4 q^{25} - q^{26} - 2 q^{27} + 10 q^{28} + 5 q^{32} - q^{33} + 3 q^{38} - q^{39} + 2 q^{41} - 4 q^{42} - 5 q^{44} - 2 q^{46} + 18 q^{48} + q^{49} + 2 q^{50} - 5 q^{52} + 2 q^{53} + 4 q^{54} - 4 q^{56} - 6 q^{57} - 5 q^{63} + 8 q^{64} + 6 q^{66} + 4 q^{69} + 5 q^{72} - q^{73} - q^{75} + q^{77} - 12 q^{78} - 6 q^{81} + q^{82} + 2 q^{83} - 5 q^{84} + q^{88} + q^{91} - 10 q^{92} + 5 q^{96} - 6 q^{98} + 10 q^{99}+O(q^{100}) $ 8 * q - q^2 - q^3 - 5 * q^4 - 12 * q^6 + q^7 - 2 * q^8 - 5 * q^9 - 4 * q^11 + 8 * q^13 + 2 * q^14 - 6 * q^16 + 2 * q^19 - q^21 + 2 * q^22 + 2 * q^23 + 4 * q^24 - 4 * q^25 - q^26 - 2 * q^27 + 10 * q^28 + 5 * q^32 - q^33 + 3 * q^38 - q^39 + 2 * q^41 - 4 * q^42 - 5 * q^44 - 2 * q^46 + 18 * q^48 + q^49 + 2 * q^50 - 5 * q^52 + 2 * q^53 + 4 * q^54 - 4 * q^56 - 6 * q^57 - 5 * q^63 + 8 * q^64 + 6 * q^66 + 4 * q^69 + 5 * q^72 - q^73 - q^75 + q^77 - 12 * q^78 - 6 * q^81 + q^82 + 2 * q^83 - 5 * q^84 + q^88 + q^91 - 10 * q^92 + 5 * q^96 - 6 * q^98 + 10 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1001\mathbb{Z}\right)^\times$.

$n$	$365$	$430$	$925$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$-1$

Coefficient data

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

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

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

$2$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.669131	+	0.743145i	$0.266667\pi$
$2$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.309017	+	0.951057i	$0.400000\pi$
$3$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$3$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$4$	−1.41355	+	2.44833i	−1.41355	+	2.44833i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$	−2.61803			−2.61803
$7$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$7$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$8$	−3.57433			−3.57433
$9$	−0.395472	−	0.684977i	−0.395472	−	0.684977i
$10$		0			0
$11$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$11$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$12$	−1.89169	−	3.27651i	−1.89169	−	3.27651i
$13$	1.00000			1.00000
$13$	1.00000			1.00000
$14$	1.58268	−	1.14988i	1.58268	−	1.14988i
$15$		0			0
$16$	−2.08268	−	3.60730i	−2.08268	−	3.60730i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$	0.773659	−	1.34002i	0.773659	−	1.34002i
$19$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$19$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$20$		0			0
$21$	1.22256	+	0.544320i	1.22256	+	0.544320i
$22$	−1.95630			−1.95630
$23$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$23$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$24$	2.39169	−	4.14253i	2.39169	−	4.14253i
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$	0.978148	+	1.69420i	0.978148	+	1.69420i
$27$	−0.279773			−0.279773
$28$	2.58268	+	1.14988i	2.58268	+	1.14988i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	2.28716	−	3.96149i	2.28716	−	3.96149i
$33$	−0.669131	−	1.15897i	−0.669131	−	1.15897i
$34$		0			0
$35$		0			0
$36$	2.23607			2.23607
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$	0.604528	−	1.04707i	0.604528	−	1.04707i
$39$	−0.669131	+	1.15897i	−0.669131	+	1.15897i
$40$		0			0
$41$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$41$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$42$	0.273659	+	2.60369i	0.273659	+	2.60369i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	−1.41355	−	2.44833i	−1.41355	−	2.44833i
$45$		0			0
$46$	−1.58268	+	2.74128i	−1.58268	+	2.74128i
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$	5.57433			5.57433
$49$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$50$	−1.95630			−1.95630
$51$		0			0
$52$	−1.41355	+	2.44833i	−1.41355	+	2.44833i
$53$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$53$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$54$	−0.273659	−	0.473991i	−0.273659	−	0.473991i
$55$		0			0
$56$	0.373619	+	3.55475i	0.373619	+	3.55475i
$57$	0.827091			0.827091
$58$		0			0
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		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
$63$	−0.639886	+	0.464905i	−0.639886	+	0.464905i
$64$	4.78339			4.78339
$65$		0			0
$66$	1.30902	−	2.26728i	1.30902	−	2.26728i
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$	−2.16535			−2.16535
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	1.41355	+	2.44833i	1.41355	+	2.44833i
$73$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$73$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$74$		0			0
$75$	−0.669131	−	1.15897i	−0.669131	−	1.15897i
$76$	1.74724			1.74724
$77$	0.913545	+	0.406737i	0.913545	+	0.406737i
$78$	−2.61803			−2.61803
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$		0			0
$81$	0.582676	−	1.00922i	0.582676	−	1.00922i
$82$	1.78716	+	3.09546i	1.78716	+	3.09546i
$83$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$83$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$84$	−3.06082	+	2.22382i	−3.06082	+	2.22382i
$85$		0			0
$86$		0			0
$87$		0			0
$88$	1.78716	−	3.09546i	1.78716	−	3.09546i
$89$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$89$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$90$		0			0
$91$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$92$	−4.57433			−4.57433
$93$		0			0
$94$		0			0
$95$		0			0
$96$	3.06082	+	5.30150i	3.06082	+	5.30150i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−1.30902	−	1.45381i	−1.30902	−	1.45381i
$99$	0.790943			0.790943
$100$	−1.41355	−	2.44833i	−1.41355	−	2.44833i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$103$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$104$	−3.57433			−3.57433
$105$		0			0
$106$	−1.20906			−1.20906
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$	0.395472	−	0.684977i	0.395472	−	0.684977i
$109$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$109$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$110$		0			0
$111$		0			0
$112$	−3.36984	+	2.44833i	−3.36984	+	2.44833i
$113$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$113$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$114$	0.809017	+	1.40126i	0.809017	+	1.40126i
$115$		0			0
$116$		0			0
$117$	−0.395472	−	0.684977i	−0.395472	−	0.684977i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$122$		0			0
$123$	−1.22256	+	2.11754i	−1.22256	+	2.11754i
$124$		0			0
$125$		0			0
$126$	−1.41355	−	0.629351i	−1.41355	−	0.629351i
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	2.39169	+	4.14253i	2.39169	+	4.14253i
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$	3.78339			3.78339
$133$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$	−2.11803	−	3.66854i	−2.11803	−	3.66854i
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$144$	−1.64728	+	2.85317i	−1.64728	+	2.85317i
$145$		0			0
$146$	3.82709			3.82709
$147$	0.413545	−	1.27276i	0.413545	−	1.27276i
$148$		0			0
$149$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$149$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$150$	1.30902	−	2.26728i	1.30902	−	2.26728i
$151$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$151$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$152$	1.10453	+	1.91310i	1.10453	+	1.91310i
$153$		0			0
$154$	0.204489	+	1.94558i	0.204489	+	1.94558i
$155$		0			0
$156$	−1.89169	−	3.27651i	−1.89169	−	3.27651i
$157$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.104528	+	0.994522i	$0.533333\pi$
$157$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.809017	+	0.587785i	$0.800000\pi$
$158$		0			0
$159$	−0.413545	−	0.716282i	−0.413545	−	0.716282i
$160$		0			0
$161$	1.30902	−	0.951057i	1.30902	−	0.951057i
$162$	2.27977			2.27977
$163$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$163$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$164$	−2.58268	+	4.47333i	−2.58268	+	4.47333i
$165$		0			0
$166$	−0.204489	−	0.354185i	−0.204489	−	0.354185i
$167$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$167$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$168$	−4.36984	−	1.94558i	−4.36984	−	1.94558i
$169$	1.00000			1.00000
$170$		0			0
$171$	−0.244415	+	0.423339i	−0.244415	+	0.423339i
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	0.913545	+	0.406737i	0.913545	+	0.406737i
$176$	4.16535			4.16535
$177$		0			0
$178$		0			0
$179$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$179$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$180$		0			0
$181$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$181$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$182$	1.58268	−	1.14988i	1.58268	−	1.14988i
$183$		0			0
$184$	−2.89169	−	5.00856i	−2.89169	−	5.00856i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	0.0292442	+	0.278240i	0.0292442	+	0.278240i
$190$		0			0
$191$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.669131	+	0.743145i	$0.266667\pi$
$191$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.309017	+	0.951057i	$0.400000\pi$
$192$	−3.20071	+	5.54379i	−3.20071	+	5.54379i
$193$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$193$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$194$		0			0
$195$		0			0
$196$	0.873619	−	2.68872i	0.873619	−	2.68872i
$197$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$197$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$198$	0.773659	+	1.34002i	0.773659	+	1.34002i
$199$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$199$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$200$	1.78716	−	3.09546i	1.78716	−	3.09546i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−0.204489	+	0.354185i	−0.204489	+	0.354185i
$207$	0.639886	−	1.10832i	0.639886	−	1.10832i
$208$	−2.08268	−	3.60730i	−2.08268	−	3.60730i
$209$	0.618034			0.618034
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	−0.873619	−	1.51315i	−0.873619	−	1.51315i
$213$		0			0
$214$		0			0
$215$		0			0
$216$	1.00000			1.00000
$217$		0			0
$218$	−1.20906			−1.20906
$219$	1.30902	+	2.26728i	1.30902	+	2.26728i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	−4.17886	−	1.86055i	−4.17886	−	1.86055i
$225$	0.790943			0.790943
$226$	1.30902	+	2.26728i	1.30902	+	2.26728i
$227$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$227$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$228$	−1.16913	+	2.02499i	−1.16913	+	2.02499i
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$	−1.08268	+	0.786610i	−1.08268	+	0.786610i
$232$		0			0
$233$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$233$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$234$	0.773659	−	1.34002i	0.773659	−	1.34002i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$239$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$240$		0			0
$241$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$241$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$242$	0.978148	−	1.69420i	0.978148	−	1.69420i
$243$	0.639886	+	1.10832i	0.639886	+	1.10832i
$244$		0			0
$245$		0			0
$246$	−4.78339			−4.78339
$247$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$248$		0			0
$249$	0.139886	−	0.242290i	0.139886	−	0.242290i
$250$		0			0
$251$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$251$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$252$	−0.233733	−	2.22382i	−0.233733	−	2.22382i
$253$	−1.61803			−1.61803
$254$		0			0
$255$		0			0
$256$	−2.28716	+	3.96149i	−2.28716	+	3.96149i
$257$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$257$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$	2.39169	+	4.14253i	2.39169	+	4.14253i
$265$		0			0
$266$	−1.10453	−	0.491768i	−1.10453	−	0.491768i
$267$		0			0
$268$		0			0
$269$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$269$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$270$		0			0
$271$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	−0.978148	−	0.207912i	$-0.933333\pi$
$271$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	0.309017	−	0.951057i	$-0.400000\pi$
$272$		0			0
$273$	1.22256	+	0.544320i	1.22256	+	0.544320i
$274$		0			0
$275$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$276$	3.06082	−	5.30150i	3.06082	−	5.30150i
$277$		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
$280$		0			0
$281$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$281$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$282$		0			0
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$		0			0
$285$		0			0
$286$	−1.95630			−1.95630
$287$	−0.190983	−	1.81708i	−0.190983	−	1.81708i
$288$	−3.61803			−3.61803
$289$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$	2.76531	+	4.78966i	2.76531	+	4.78966i
$293$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$293$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$294$	2.56082	−	0.544320i	2.56082	−	0.544320i
$295$		0			0
$296$		0			0
$297$	0.139886	−	0.242290i	0.139886	−	0.242290i
$298$	−0.204489	+	0.354185i	−0.204489	+	0.354185i
$299$	0.809017	+	1.40126i	0.809017	+	1.40126i
$300$	3.78339			3.78339
$301$		0			0
$302$	1.95630			1.95630
$303$		0			0
$304$	−1.28716	+	2.22943i	−1.28716	+	2.22943i
$305$		0			0
$306$		0			0
$307$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$307$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$308$	−2.28716	+	1.66172i	−2.28716	+	1.66172i
$309$	−0.279773			−0.279773
$310$		0			0
$311$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$311$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$312$	2.39169	−	4.14253i	2.39169	−	4.14253i
$313$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$313$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$314$	−3.57433			−3.57433
$315$		0			0
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$	0.809017	−	1.40126i	0.809017	−	1.40126i
$319$		0			0
$320$		0			0
$321$		0			0
$322$	2.89169	+	1.28746i	2.89169	+	1.28746i
$323$		0			0
$324$	1.64728	+	2.85317i	1.64728	+	2.85317i
$325$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$326$		0			0
$327$	−0.413545	−	0.716282i	−0.413545	−	0.716282i
$328$	−6.53062			−6.53062
$329$		0			0
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$	0.295511	−	0.511841i	0.295511	−	0.511841i
$333$		0			0
$334$	−1.91355	−	3.31436i	−1.91355	−	3.31436i
$335$		0			0
$336$	−0.582676	−	5.54379i	−0.582676	−	5.54379i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	0.978148	+	1.69420i	0.978148	+	1.69420i
$339$	−0.895472	+	1.55100i	−0.895472	+	1.55100i
$340$		0			0
$341$		0			0
$342$	−0.956295			−0.956295
$343$	0.309017	+	0.951057i	0.309017	+	0.951057i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$349$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$350$	0.204489	+	1.94558i	0.204489	+	1.94558i
$351$	−0.279773			−0.279773
$352$	2.28716	+	3.96149i	2.28716	+	3.96149i
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	1.95630			1.95630
$359$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$359$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$360$		0			0
$361$	0.309017	−	0.535233i	0.309017	−	0.535233i
$362$	0.604528	+	1.04707i	0.604528	+	1.04707i
$363$	1.33826			1.33826
$364$	2.58268	+	1.14988i	2.58268	+	1.14988i
$365$		0			0
$366$		0			0
$367$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$367$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$368$	3.36984	−	5.83674i	3.36984	−	5.83674i
$369$	−0.722562	−	1.25151i	−0.722562	−	1.25151i
$370$		0			0
$371$	0.564602	+	0.251377i	0.564602	+	0.251377i
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$	−0.442790	+	0.321706i	−0.442790	+	0.321706i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$	−1.91355	+	3.31436i	−1.91355	+	3.31436i
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$	−6.40142			−6.40142
$385$		0			0
$386$	0.408977			0.408977
$387$		0			0
$388$		0			0
$389$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$389$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$390$		0			0
$391$		0			0
$392$	3.49622	−	0.743145i	3.49622	−	0.743145i
$393$		0			0
$394$	1.30902	+	2.26728i	1.30902	+	2.26728i
$395$		0			0
$396$	−1.11803	+	1.93649i	−1.11803	+	1.93649i
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$	3.82709			3.82709
$399$	−0.0864545	−	0.822560i	−0.0864545	−	0.822560i
$400$	4.16535			4.16535
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$409$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$410$		0			0
$411$		0			0
$412$	−0.591023			−0.591023
$413$		0			0
$414$	2.50361			2.50361
$415$		0			0
$416$	2.28716	−	3.96149i	2.28716	−	3.96149i
$417$		0			0
$418$	0.604528	+	1.04707i	0.604528	+	1.04707i
$419$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$419$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	1.10453	−	1.91310i	1.10453	−	1.91310i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−0.669131	−	1.15897i	−0.669131	−	1.15897i
$430$		0			0
$431$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$431$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$432$	0.582676	+	1.00922i	0.582676	+	1.00922i
$433$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$433$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$434$		0			0
$435$		0			0
$436$	−0.873619	−	1.51315i	−0.873619	−	1.51315i
$437$	0.500000	−	0.866025i	0.500000	−	0.866025i
$438$	−2.56082	+	4.43548i	−2.56082	+	4.43548i
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	0.529244	+	0.587785i	0.529244	+	0.587785i
$442$		0			0
$443$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$443$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−0.279773			−0.279773
$448$	−0.500000	−	4.75718i	−0.500000	−	4.75718i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	0.773659	+	1.34002i	0.773659	+	1.34002i
$451$	−0.913545	+	1.58231i	−0.913545	+	1.58231i
$452$	−1.89169	+	3.27651i	−1.89169	+	3.27651i
$453$	0.669131	+	1.15897i	0.669131	+	1.15897i
$454$	−2.61803			−2.61803
$455$		0			0
$456$	−2.95630			−2.95630
$457$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	−0.978148	−	0.207912i	$-0.933333\pi$
$457$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	0.309017	−	0.951057i	$-0.400000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$461$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$462$	−2.39169	−	1.06485i	−2.39169	−	1.06485i
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$467$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$468$	2.23607			2.23607
$469$		0			0
$470$		0			0
$471$	−1.22256	−	2.11754i	−1.22256	−	2.11754i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.618034			0.618034
$476$		0			0
$477$	0.488830			0.488830
$478$	0.604528	+	1.04707i	0.604528	+	1.04707i
$479$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$479$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$480$		0			0
$481$		0			0
$482$	0.408977			0.408977
$483$	0.226341	+	2.15349i	0.226341	+	2.15349i
$484$	2.82709			2.82709
$485$		0			0
$486$	−1.25181	+	2.16819i	−1.25181	+	2.16819i
$487$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$487$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$	−3.45630	−	5.98648i	−3.45630	−	5.98648i
$493$		0			0
$494$	0.604528	−	1.04707i	0.604528	−	1.04707i
$495$		0			0
$496$		0			0
$497$		0			0
$498$	0.547318			0.547318
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$		0			0
$501$	1.30902	−	2.26728i	1.30902	−	2.26728i
$502$	1.78716	+	3.09546i	1.78716	+	3.09546i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$	2.28716	−	1.66172i	2.28716	−	1.66172i
$505$		0			0
$506$	−1.58268	−	2.74128i	−1.58268	−	2.74128i
$507$	−0.669131	+	1.15897i	−0.669131	+	1.15897i
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$	−1.78716	−	0.795697i	−1.78716	−	0.795697i
$512$	−4.16535			−4.16535
$513$	0.0864545	+	0.149744i	0.0864545	+	0.149744i
$514$	1.78716	−	3.09546i	1.78716	−	3.09546i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$521$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$522$		0			0
$523$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$523$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$524$		0			0
$525$	−1.08268	+	0.786610i	−1.08268	+	0.786610i
$526$		0			0
$527$		0			0
$528$	−2.78716	+	4.82751i	−2.78716	+	4.82751i
$529$	−0.809017	+	1.40126i	−0.809017	+	1.40126i
$530$		0			0
$531$		0			0
$532$	−0.182636	−	1.73767i	−0.182636	−	1.73767i
$533$	1.82709			1.82709
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.669131	+	1.15897i	0.669131	+	1.15897i
$538$	3.16535			3.16535
$539$	0.309017	−	0.951057i	0.309017	−	0.951057i
$540$		0			0
$541$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$541$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$542$	1.30902	−	2.26728i	1.30902	−	2.26728i
$543$	−0.413545	+	0.716282i	−0.413545	+	0.716282i
$544$		0			0
$545$		0			0
$546$	0.273659	+	2.60369i	0.273659	+	2.60369i
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$	0.978148	−	1.69420i	0.978148	−	1.69420i
$551$		0			0
$552$	7.73968			7.73968
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.104528	+	0.994522i	$0.533333\pi$
$557$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.809017	+	0.587785i	$0.800000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.604528	+	1.04707i	0.604528	+	1.04707i
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1.06460	−	0.473991i	−1.06460	−	0.473991i
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$	−1.41355	−	2.44833i	−1.41355	−	2.44833i
$573$	−2.61803			−2.61803
$574$	2.89169	−	2.10094i	2.89169	−	2.10094i
$575$	−1.61803			−1.61803
$576$	−1.89169	−	3.27651i	−1.89169	−	3.27651i
$577$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$577$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$578$	0.978148	−	1.69420i	0.978148	−	1.69420i
$579$	0.139886	+	0.242290i	0.139886	+	0.242290i
$580$		0			0
$581$	0.0218524	+	0.207912i	0.0218524	+	0.207912i
$582$		0			0
$583$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$584$	−3.49622	+	6.05563i	−3.49622	+	6.05563i
$585$		0			0
$586$	−0.978148	−	1.69420i	−0.978148	−	1.69420i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	2.53158	+	2.81160i	2.53158	+	2.81160i
$589$		0			0
$590$		0			0
$591$	−0.895472	+	1.55100i	−0.895472	+	1.55100i
$592$		0			0
$593$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$593$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$594$	0.547318			0.547318
$595$		0			0
$596$	−0.591023			−0.591023
$597$	1.30902	+	2.26728i	1.30902	+	2.26728i
$598$	−1.58268	+	2.74128i	−1.58268	+	2.74128i
$599$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$599$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$600$	2.39169	+	4.14253i	2.39169	+	4.14253i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	1.41355	+	2.44833i	1.41355	+	2.44833i
$605$		0			0
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$	−2.82709			−2.82709
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$613$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$614$	−0.978148	−	1.69420i	−0.978148	−	1.69420i
$615$		0			0
$616$	−3.26531	−	1.45381i	−3.26531	−	1.45381i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$	−0.273659	−	0.473991i	−0.273659	−	0.473991i
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$	−0.226341	−	0.392034i	−0.226341	−	0.392034i
$622$	−1.20906			−1.20906
$623$		0			0
$624$	5.57433			5.57433
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$	−1.58268	+	2.74128i	−1.58268	+	2.74128i
$627$	−0.413545	+	0.716282i	−0.413545	+	0.716282i
$628$	−2.58268	−	4.47333i	−2.58268	−	4.47333i
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	2.33826			2.33826
$637$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.104528	+	0.994522i	$0.533333\pi$
$641$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.809017	+	0.587785i	$0.800000\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$	0.478148	+	4.54927i	0.478148	+	4.54927i
$645$		0			0
$646$		0			0
$647$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$647$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$648$	−2.08268	+	3.60730i	−2.08268	+	3.60730i
$649$		0			0
$650$	−1.95630			−1.95630
$651$		0			0
$652$		0			0
$653$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$653$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$654$	0.809017	−	1.40126i	0.809017	−	1.40126i
$655$		0			0
$656$	−3.80524	−	6.59087i	−3.80524	−	6.59087i
$657$	−1.54732			−1.54732
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$		0			0
$664$	0.747238			0.747238
$665$		0			0
$666$		0			0
$667$		0			0
$668$	2.76531	−	4.78966i	2.76531	−	4.78966i
$669$		0			0
$670$		0			0
$671$		0			0
$672$	4.95252	−	3.59821i	4.95252	−	3.59821i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	0.139886	−	0.242290i	0.139886	−	0.242290i
$676$	−1.41355	+	2.44833i	−1.41355	+	2.44833i
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$	−3.50361			−3.50361
$679$		0			0
$680$		0			0
$681$	−0.895472	−	1.55100i	−0.895472	−	1.55100i
$682$		0			0
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$	−0.690983	−	1.19682i	−0.690983	−	1.19682i
$685$		0			0
$686$	−1.30902	+	1.45381i	−1.30902	+	1.45381i
$687$		0			0
$688$		0			0
$689$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$		0			0
$693$	−0.0826761	−	0.786610i	−0.0826761	−	0.786610i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−1.58268	−	2.74128i	−1.58268	−	2.74128i
$699$		0			0
$700$	−2.28716	+	1.66172i	−2.28716	+	1.66172i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	−0.273659	−	0.473991i	−0.273659	−	0.473991i
$703$		0			0
$704$	−2.39169	+	4.14253i	−2.39169	+	4.14253i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	1.41355	+	2.44833i	1.41355	+	2.44833i
$717$	−0.413545	+	0.716282i	−0.413545	+	0.716282i
$718$	1.78716	−	3.09546i	1.78716	−	3.09546i
$719$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$719$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$720$		0			0
$721$	0.169131	−	0.122881i	0.169131	−	0.122881i
$722$	1.20906			1.20906
$723$	0.139886	+	0.242290i	0.139886	+	0.242290i
$724$	−0.873619	+	1.51315i	−0.873619	+	1.51315i
$725$		0			0
$726$	1.30902	+	2.26728i	1.30902	+	2.26728i
$727$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$727$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$728$	0.373619	+	3.55475i	0.373619	+	3.55475i
$729$	−0.547318			−0.547318
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$733$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$734$	−1.20906			−1.20906
$735$		0			0
$736$	7.40142			7.40142
$737$		0			0
$738$	1.41355	−	2.44833i	1.41355	−	2.44833i
$739$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$739$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$740$		0			0
$741$	0.827091			0.827091
$742$	0.126381	+	1.20243i	0.126381	+	1.20243i
$743$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$743$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.0826761	+	0.143199i	0.0826761	+	0.143199i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.669131	+	0.743145i	$0.266667\pi$
$751$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.309017	+	0.951057i	$0.400000\pi$
$752$		0			0
$753$	−1.22256	+	2.11754i	−1.22256	+	2.11754i
$754$		0			0
$755$		0			0
$756$	−0.722562	−	0.321706i	−0.722562	−	0.321706i
$757$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$757$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$758$		0			0
$759$	1.08268	−	1.87525i	1.08268	−	1.87525i
$760$		0			0
$761$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$761$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$762$		0			0
$763$	0.564602	+	0.251377i	0.564602	+	0.251377i
$764$	−5.53062			−5.53062
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−3.06082	−	5.30150i	−3.06082	−	5.30150i
$769$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$769$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$770$		0			0
$771$	2.44512			2.44512
$772$	0.295511	+	0.511841i	0.295511	+	0.511841i
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	3.82709			3.82709
$779$	−0.564602	−	0.977920i	−0.564602	−	0.977920i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	2.78716	+	3.09546i	2.78716	+	3.09546i
$785$		0			0
$786$		0			0
$787$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$787$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$788$	−1.89169	+	3.27651i	−1.89169	+	3.27651i
$789$		0			0
$790$		0			0
$791$	−0.139886	−	1.33093i	−0.139886	−	1.33093i
$792$	−2.82709			−2.82709
$793$		0			0
$794$		0			0
$795$		0			0
$796$	2.76531	+	4.78966i	2.76531	+	4.78966i
$797$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$797$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$798$	1.30902	−	0.951057i	1.30902	−	0.951057i
$799$		0			0
$800$	2.28716	+	3.96149i	2.28716	+	3.96149i
$801$		0			0
$802$		0			0
$803$	0.978148	+	1.69420i	0.978148	+	1.69420i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	1.08268	+	1.87525i	1.08268	+	1.87525i
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$811$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$812$		0			0
$813$	1.79094			1.79094
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.95630			1.95630
$819$	−0.639886	+	0.464905i	−0.639886	+	0.464905i
$820$		0			0
$821$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$821$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$823$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$824$	−0.373619	−	0.647127i	−0.373619	−	0.647127i
$825$	1.33826			1.33826
$826$		0			0
$827$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$827$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$828$	1.80902	+	3.13331i	1.80902	+	3.13331i
$829$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$829$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$830$		0			0
$831$		0			0
$832$	4.78339			4.78339
$833$		0			0
$834$		0			0
$835$		0			0
$836$	−0.873619	+	1.51315i	−0.873619	+	1.51315i
$837$		0			0
$838$	−0.204489	−	0.354185i	−0.204489	−	0.354185i
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$		0			0
$843$	−0.413545	+	0.716282i	−0.413545	+	0.716282i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$848$	2.57433			2.57433
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$853$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$	1.30902	−	2.26728i	1.30902	−	2.26728i
$859$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$859$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$860$		0			0
$861$	2.23373	+	0.994522i	2.23373	+	0.994522i
$862$	0.408977			0.408977
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$	−0.639886	+	1.10832i	−0.639886	+	1.10832i
$865$		0			0
$866$	−1.91355	−	3.31436i	−1.91355	−	3.31436i
$867$	1.33826			1.33826
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	1.10453	−	1.91310i	1.10453	−	1.91310i
$873$		0			0
$874$	1.95630			1.95630
$875$		0			0
$876$	−7.40142			−7.40142
$877$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.669131	+	0.743145i	$0.266667\pi$
$877$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.309017	+	0.951057i	$0.400000\pi$
$878$		0			0
$879$	0.669131	−	1.15897i	0.669131	−	1.15897i
$880$		0			0
$881$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$881$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$882$	−0.478148	+	1.47159i	−0.478148	+	1.47159i
$883$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$883$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$884$		0			0
$885$		0			0
$886$	1.78716	−	3.09546i	1.78716	−	3.09546i
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.582676	+	1.00922i	0.582676	+	1.00922i
$892$		0			0
$893$		0			0
$894$	−0.273659	−	0.473991i	−0.273659	−	0.473991i
$895$		0			0
$896$	3.86984	−	2.81160i	3.86984	−	2.81160i
$897$	−2.16535			−2.16535
$898$		0			0
$899$		0			0
$900$	−1.11803	+	1.93649i	−1.11803	+	1.93649i
$901$		0			0
$902$	−3.57433			−3.57433
$903$		0			0
$904$	−4.78339			−4.78339
$905$		0			0
$906$	−1.30902	+	2.26728i	−1.30902	+	2.26728i
$907$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$907$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$908$	−1.89169	−	3.27651i	−1.89169	−	3.27651i
$909$		0			0
$910$		0			0
$911$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$911$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$912$	−1.72256	−	2.98357i	−1.72256	−	2.98357i
$913$	0.104528	−	0.181049i	0.104528	−	0.181049i
$914$	1.30902	−	2.26728i	1.30902	−	2.26728i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$919$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$920$		0			0
$921$	0.669131	−	1.15897i	0.669131	−	1.15897i
$922$	−1.91355	−	3.31436i	−1.91355	−	3.31436i
$923$		0			0
$924$	−0.395472	−	3.76266i	−0.395472	−	3.76266i
$925$		0			0
$926$		0			0
$927$	0.0826761	−	0.143199i	0.0826761	−	0.143199i
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$	0.413545	+	0.459289i	0.413545	+	0.459289i
$932$		0			0
$933$	−0.413545	−	0.716282i	−0.413545	−	0.716282i
$934$	−0.978148	+	1.69420i	−0.978148	+	1.69420i
$935$		0			0
$936$	1.41355	+	2.44833i	1.41355	+	2.44833i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$	−2.16535			−2.16535
$940$		0			0
$941$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$941$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$942$	2.39169	−	4.14253i	2.39169	−	4.14253i
$943$	1.47815	+	2.56023i	1.47815	+	2.56023i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$	0.978148	−	1.69420i	0.978148	−	1.69420i
$950$	0.604528	+	1.04707i	0.604528	+	1.04707i
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$	0.478148	+	0.828176i	0.478148	+	0.828176i
$955$		0			0
$956$	−0.873619	+	1.51315i	−0.873619	+	1.51315i
$957$		0			0
$958$	1.95630			1.95630
$959$		0			0
$960$		0			0
$961$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$	0.295511	+	0.511841i	0.295511	+	0.511841i
$965$		0			0
$966$	−3.42705	+	2.48990i	−3.42705	+	2.48990i
$967$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$967$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$968$	1.78716	+	3.09546i	1.78716	+	3.09546i
$969$		0			0
$970$		0			0
$971$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$971$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$972$	−3.61803			−3.61803
$973$		0			0
$974$		0			0
$975$	−0.669131	−	1.15897i	−0.669131	−	1.15897i
$976$		0			0
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.488830			0.488830
$982$		0			0
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$	4.36984	−	7.56879i	4.36984	−	7.56879i
$985$		0			0
$986$		0			0
$987$		0			0
$988$	1.74724			1.74724
$989$		0			0
$990$		0			0
$991$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$991$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	0.395472	+	0.684977i	0.395472	+	0.684977i
$997$		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	1001.1.y.c.571.4	yes	8
7.2	even	3	inner	1001.1.y.c.142.4	✓	8
11.10	odd	2		1001.1.y.d.571.1	yes	8
13.12	even	2		1001.1.y.d.571.1	yes	8
77.65	odd	6		1001.1.y.d.142.1	yes	8
91.51	even	6		1001.1.y.d.142.1	yes	8
143.142	odd	2	CM	1001.1.y.c.571.4	yes	8
1001.142	odd	6	inner	1001.1.y.c.142.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1001.1.y.c.142.4	✓	8	7.2	even	3	inner
1001.1.y.c.142.4	✓	8	1001.142	odd	6	inner
1001.1.y.c.571.4	yes	8	1.1	even	1	trivial
1001.1.y.c.571.4	yes	8	143.142	odd	2	CM
1001.1.y.d.142.1	yes	8	77.65	odd	6
1001.1.y.d.142.1	yes	8	91.51	even	6
1001.1.y.d.571.1	yes	8	11.10	odd	2
1001.1.y.d.571.1	yes	8	13.12	even	2

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:	\( N \)	\(=\)	\( 1001 = 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1001.y (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.978148 + 1.69420i) q^{2} +(-0.669131 + 1.15897i) q^{3} +(-1.41355 + 2.44833i) q^{4} -2.61803 q^{6} +(-0.104528 - 0.994522i) q^{7} -3.57433 q^{8} +(-0.395472 - 0.684977i) q^{9} +O(q^{10})\)	\(q+(0.978148 + 1.69420i) q^{2} +(-0.669131 + 1.15897i) q^{3} +(-1.41355 + 2.44833i) q^{4} -2.61803 q^{6} +(-0.104528 - 0.994522i) q^{7} -3.57433 q^{8} +(-0.395472 - 0.684977i) q^{9} +(-0.500000 + 0.866025i) q^{11} +(-1.89169 - 3.27651i) q^{12} +1.00000 q^{13} +(1.58268 - 1.14988i) q^{14} +(-2.08268 - 3.60730i) q^{16} +(0.773659 - 1.34002i) q^{18} +(-0.309017 - 0.535233i) q^{19} +(1.22256 + 0.544320i) q^{21} -1.95630 q^{22} +(0.809017 + 1.40126i) q^{23} +(2.39169 - 4.14253i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(0.978148 + 1.69420i) q^{26} -0.279773 q^{27} +(2.58268 + 1.14988i) q^{28} +(2.28716 - 3.96149i) q^{32} +(-0.669131 - 1.15897i) q^{33} +2.23607 q^{36} +(0.604528 - 1.04707i) q^{38} +(-0.669131 + 1.15897i) q^{39} +1.82709 q^{41} +(0.273659 + 2.60369i) q^{42} +(-1.41355 - 2.44833i) q^{44} +(-1.58268 + 2.74128i) q^{46} +5.57433 q^{48} +(-0.978148 + 0.207912i) q^{49} -1.95630 q^{50} +(-1.41355 + 2.44833i) q^{52} +(-0.309017 + 0.535233i) q^{53} +(-0.273659 - 0.473991i) q^{54} +(0.373619 + 3.55475i) q^{56} +0.827091 q^{57} +(-0.639886 + 0.464905i) q^{63} +4.78339 q^{64} +(1.30902 - 2.26728i) q^{66} -2.16535 q^{69} +(1.41355 + 2.44833i) q^{72} +(0.978148 - 1.69420i) q^{73} +(-0.669131 - 1.15897i) q^{75} +1.74724 q^{76} +(0.913545 + 0.406737i) q^{77} -2.61803 q^{78} +(0.582676 - 1.00922i) q^{81} +(1.78716 + 3.09546i) q^{82} -0.209057 q^{83} +(-3.06082 + 2.22382i) q^{84} +(1.78716 - 3.09546i) q^{88} +(-0.104528 - 0.994522i) q^{91} -4.57433 q^{92} +(3.06082 + 5.30150i) q^{96} +(-1.30902 - 1.45381i) q^{98} +0.790943 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} - q^{3} - 5 q^{4} - 12 q^{6} + q^{7} - 2 q^{8} - 5 q^{9}+O(q^{10}) \) 8 * q - q^2 - q^3 - 5 * q^4 - 12 * q^6 + q^7 - 2 * q^8 - 5 * q^9	\( 8 q - q^{2} - q^{3} - 5 q^{4} - 12 q^{6} + q^{7} - 2 q^{8} - 5 q^{9} - 4 q^{11} + 8 q^{13} + 2 q^{14} - 6 q^{16} + 2 q^{19} - q^{21} + 2 q^{22} + 2 q^{23} + 4 q^{24} - 4 q^{25} - q^{26} - 2 q^{27} + 10 q^{28} + 5 q^{32} - q^{33} + 3 q^{38} - q^{39} + 2 q^{41} - 4 q^{42} - 5 q^{44} - 2 q^{46} + 18 q^{48} + q^{49} + 2 q^{50} - 5 q^{52} + 2 q^{53} + 4 q^{54} - 4 q^{56} - 6 q^{57} - 5 q^{63} + 8 q^{64} + 6 q^{66} + 4 q^{69} + 5 q^{72} - q^{73} - q^{75} + q^{77} - 12 q^{78} - 6 q^{81} + q^{82} + 2 q^{83} - 5 q^{84} + q^{88} + q^{91} - 10 q^{92} + 5 q^{96} - 6 q^{98} + 10 q^{99}+O(q^{100}) \) 8 * q - q^2 - q^3 - 5 * q^4 - 12 * q^6 + q^7 - 2 * q^8 - 5 * q^9 - 4 * q^11 + 8 * q^13 + 2 * q^14 - 6 * q^16 + 2 * q^19 - q^21 + 2 * q^22 + 2 * q^23 + 4 * q^24 - 4 * q^25 - q^26 - 2 * q^27 + 10 * q^28 + 5 * q^32 - q^33 + 3 * q^38 - q^39 + 2 * q^41 - 4 * q^42 - 5 * q^44 - 2 * q^46 + 18 * q^48 + q^49 + 2 * q^50 - 5 * q^52 + 2 * q^53 + 4 * q^54 - 4 * q^56 - 6 * q^57 - 5 * q^63 + 8 * q^64 + 6 * q^66 + 4 * q^69 + 5 * q^72 - q^73 - q^75 + q^77 - 12 * q^78 - 6 * q^81 + q^82 + 2 * q^83 - 5 * q^84 + q^88 + q^91 - 10 * q^92 + 5 * q^96 - 6 * q^98 + 10 * q^99

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