LMFDB - Embedded newform 2001.1.i.c.1172.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2001,1,Mod(1172,2001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2001, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 1]))

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

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

chi := DirichletCharacter("2001.1172");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2001 = 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2001.i (of order $4$, 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:	$0.998629090279$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{12}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{12} - \cdots)$

Embedding invariants

Embedding label			1172.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2001.1172
Dual form			2001.1.i.c.1931.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.36603 + 1.36603i) q^{2} +(0.866025 - 0.500000i) q^{3} -2.73205i q^{4} +(-0.500000 + 1.86603i) q^{6} +(2.36603 + 2.36603i) q^{8} +(0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(-1.36603 + 1.36603i) q^{2} +(0.866025 - 0.500000i) q^{3} -2.73205i q^{4} +(-0.500000 + 1.86603i) q^{6} +(2.36603 + 2.36603i) q^{8} +(0.500000 - 0.866025i) q^{9} +(-1.36603 - 2.36603i) q^{12} -1.00000i q^{13} -3.73205 q^{16} +(0.500000 + 1.86603i) q^{18} +1.00000i q^{23} +(3.23205 + 0.866025i) q^{24} +1.00000 q^{25} +(1.36603 + 1.36603i) q^{26} -1.00000i q^{27} +(0.866025 + 0.500000i) q^{29} +(-0.366025 - 0.366025i) q^{31} +(2.73205 - 2.73205i) q^{32} +(-2.36603 - 1.36603i) q^{36} +(-0.500000 - 0.866025i) q^{39} +(-0.366025 - 0.366025i) q^{41} +(-1.36603 - 1.36603i) q^{46} +(-1.36603 - 1.36603i) q^{47} +(-3.23205 + 1.86603i) q^{48} +1.00000 q^{49} +(-1.36603 + 1.36603i) q^{50} -2.73205 q^{52} +(1.36603 + 1.36603i) q^{54} +(-1.86603 + 0.500000i) q^{58} -2.00000i q^{59} +1.00000 q^{62} +3.73205i q^{64} +(0.500000 + 0.866025i) q^{69} +1.73205 q^{71} +(3.23205 - 0.866025i) q^{72} +(-0.366025 + 0.366025i) q^{73} +(0.866025 - 0.500000i) q^{75} +(1.86603 + 0.500000i) q^{78} +(-0.500000 - 0.866025i) q^{81} +1.00000 q^{82} +1.00000 q^{87} +2.73205 q^{92} +(-0.500000 - 0.133975i) q^{93} +3.73205 q^{94} +(1.00000 - 3.73205i) q^{96} +(-1.36603 + 1.36603i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 2 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10}) $ 4 * q - 2 * q^2 - 2 * q^6 + 6 * q^8 + 2 * q^9	$ 4 q - 2 q^{2} - 2 q^{6} + 6 q^{8} + 2 q^{9} - 2 q^{12} - 8 q^{16} + 2 q^{18} + 6 q^{24} + 4 q^{25} + 2 q^{26} + 2 q^{31} + 4 q^{32} - 6 q^{36} - 2 q^{39} + 2 q^{41} - 2 q^{46} - 2 q^{47} - 6 q^{48} + 4 q^{49} - 2 q^{50} - 4 q^{52} + 2 q^{54} - 4 q^{58} + 4 q^{62} + 2 q^{69} + 6 q^{72} + 2 q^{73} + 4 q^{78} - 2 q^{81} + 4 q^{82} + 4 q^{87} + 4 q^{92} - 2 q^{93} + 8 q^{94} + 4 q^{96} - 2 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 - 2 * q^6 + 6 * q^8 + 2 * q^9 - 2 * q^12 - 8 * q^16 + 2 * q^18 + 6 * q^24 + 4 * q^25 + 2 * q^26 + 2 * q^31 + 4 * q^32 - 6 * q^36 - 2 * q^39 + 2 * q^41 - 2 * q^46 - 2 * q^47 - 6 * q^48 + 4 * q^49 - 2 * q^50 - 4 * q^52 + 2 * q^54 - 4 * q^58 + 4 * q^62 + 2 * q^69 + 6 * q^72 + 2 * q^73 + 4 * q^78 - 2 * q^81 + 4 * q^82 + 4 * q^87 + 4 * q^92 - 2 * q^93 + 8 * q^94 + 4 * q^96 - 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2001\mathbb{Z}\right)^\times$.

$n$	$553$	$668$	$1132$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$-1$

Coefficient data

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

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

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

$2$	−1.36603	+	1.36603i	−1.36603	+	1.36603i	−0.500000	+	0.866025i	$0.666667\pi$
$2$	−1.36603	+	1.36603i	−1.36603	+	1.36603i	−0.866025	+	0.500000i	$0.833333\pi$
$3$	0.866025	−	0.500000i	0.866025	−	0.500000i
$3$	0.866025	−	0.500000i	0.866025	−	0.500000i
$4$		−	2.73205i		−	2.73205i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	2.36603	+	2.36603i	2.36603	+	2.36603i
$9$	0.500000	−	0.866025i	0.500000	−	0.866025i
$10$		0			0
$11$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$11$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$12$	−1.36603	−	2.36603i	−1.36603	−	2.36603i
$13$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$13$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$14$		0			0
$15$		0			0
$16$	−3.73205			−3.73205
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$	0.500000	+	1.86603i	0.500000	+	1.86603i
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.00000i			1.00000i
$23$			1.00000i			1.00000i
$24$	3.23205	+	0.866025i	3.23205	+	0.866025i
$25$	1.00000			1.00000
$26$	1.36603	+	1.36603i	1.36603	+	1.36603i
$27$		−	1.00000i		−	1.00000i
$28$		0			0
$29$	0.866025	+	0.500000i	0.866025	+	0.500000i
$29$	0.866025	+	0.500000i	0.866025	+	0.500000i
$30$		0			0
$31$	−0.366025	−	0.366025i	−0.366025	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$31$	−0.366025	−	0.366025i	−0.366025	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$32$	2.73205	−	2.73205i	2.73205	−	2.73205i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−2.36603	−	1.36603i	−2.36603	−	1.36603i
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$		0			0
$39$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$40$		0			0
$41$	−0.366025	−	0.366025i	−0.366025	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$41$	−0.366025	−	0.366025i	−0.366025	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		0			0
$45$		0			0
$46$	−1.36603	−	1.36603i	−1.36603	−	1.36603i
$47$	−1.36603	−	1.36603i	−1.36603	−	1.36603i	−0.866025	−	0.500000i	$-0.833333\pi$
$47$	−1.36603	−	1.36603i	−1.36603	−	1.36603i	−0.500000	−	0.866025i	$-0.666667\pi$
$48$	−3.23205	+	1.86603i	−3.23205	+	1.86603i
$49$	1.00000			1.00000
$50$	−1.36603	+	1.36603i	−1.36603	+	1.36603i
$51$		0			0
$52$	−2.73205			−2.73205
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	1.36603	+	1.36603i	1.36603	+	1.36603i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−1.86603	+	0.500000i	−1.86603	+	0.500000i
$59$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$59$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$60$		0			0
$61$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$61$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$62$	1.00000			1.00000
$63$		0			0
$64$			3.73205i			3.73205i
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$	0.500000	+	0.866025i	0.500000	+	0.866025i
$70$		0			0
$71$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$71$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$72$	3.23205	−	0.866025i	3.23205	−	0.866025i
$73$	−0.366025	+	0.366025i	−0.366025	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$73$	−0.366025	+	0.366025i	−0.366025	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$74$		0			0
$75$	0.866025	−	0.500000i	0.866025	−	0.500000i
$76$		0			0
$77$		0			0
$78$	1.86603	+	0.500000i	1.86603	+	0.500000i
$79$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$79$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$82$	1.00000			1.00000
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	1.00000			1.00000
$88$		0			0
$89$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$89$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$90$		0			0
$91$		0			0
$92$	2.73205			2.73205
$93$	−0.500000	−	0.133975i	−0.500000	−	0.133975i
$94$	3.73205			3.73205
$95$		0			0
$96$	1.00000	−	3.73205i	1.00000	−	3.73205i
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$	−1.36603	+	1.36603i	−1.36603	+	1.36603i
$99$		0			0
$100$		−	2.73205i		−	2.73205i
$101$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$101$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	2.36603	−	2.36603i	2.36603	−	2.36603i
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−2.73205			−2.73205
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$		0			0
$115$		0			0
$116$	1.36603	−	2.36603i	1.36603	−	2.36603i
$117$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$118$	2.73205	+	2.73205i	2.73205	+	2.73205i
$119$		0			0
$120$		0			0
$121$		−	1.00000i		−	1.00000i
$122$		0			0
$123$	−0.500000	−	0.133975i	−0.500000	−	0.133975i
$124$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$125$		0			0
$126$		0			0
$127$	1.36603	+	1.36603i	1.36603	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$127$	1.36603	+	1.36603i	1.36603	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$128$	−2.36603	−	2.36603i	−2.36603	−	2.36603i
$129$		0			0
$130$		0			0
$131$	−0.366025	−	0.366025i	−0.366025	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$131$	−0.366025	−	0.366025i	−0.366025	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$	−1.86603	−	0.500000i	−1.86603	−	0.500000i
$139$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$139$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$140$		0			0
$141$	−1.86603	−	0.500000i	−1.86603	−	0.500000i
$142$	−2.36603	+	2.36603i	−2.36603	+	2.36603i
$143$		0			0
$144$	−1.86603	+	3.23205i	−1.86603	+	3.23205i
$145$		0			0
$146$		−	1.00000i		−	1.00000i
$147$	0.866025	−	0.500000i	0.866025	−	0.500000i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$151$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$151$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−2.36603	+	1.36603i	−2.36603	+	1.36603i
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	1.86603	+	0.500000i	1.86603	+	0.500000i
$163$	−1.36603	+	1.36603i	−1.36603	+	1.36603i	−0.500000	+	0.866025i	$0.666667\pi$
$163$	−1.36603	+	1.36603i	−1.36603	+	1.36603i	−0.866025	+	0.500000i	$0.833333\pi$
$164$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$165$		0			0
$166$		0			0
$167$	−2.00000			−2.00000			−1.00000			$\pi$
$167$	−2.00000			−2.00000			−1.00000			$\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2.00000			2.00000			1.00000			$0$
$173$	2.00000			2.00000			1.00000			$0$
$174$	−1.36603	+	1.36603i	−1.36603	+	1.36603i
$175$		0			0
$176$		0			0
$177$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$178$		0			0
$179$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$179$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	−2.36603	+	2.36603i	−2.36603	+	2.36603i
$185$		0			0
$186$	0.866025	−	0.500000i	0.866025	−	0.500000i
$187$		0			0
$188$	−3.73205	+	3.73205i	−3.73205	+	3.73205i
$189$		0			0
$190$		0			0
$191$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$191$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$192$	1.86603	+	3.23205i	1.86603	+	3.23205i
$193$	1.36603	+	1.36603i	1.36603	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$193$	1.36603	+	1.36603i	1.36603	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$		0			0
$196$		−	2.73205i		−	2.73205i
$197$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$197$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	2.36603	+	2.36603i	2.36603	+	2.36603i
$201$		0			0
$202$		−	2.73205i		−	2.73205i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.866025	+	0.500000i	0.866025	+	0.500000i
$208$			3.73205i			3.73205i
$209$		0			0
$210$		0			0
$211$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$211$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$212$		0			0
$213$	1.50000	−	0.866025i	1.50000	−	0.866025i
$214$		0			0
$215$		0			0
$216$	2.36603	−	2.36603i	2.36603	−	2.36603i
$217$		0			0
$218$		0			0
$219$	−0.133975	+	0.500000i	−0.133975	+	0.500000i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−2.00000			−2.00000			−1.00000			$\pi$
$223$	−2.00000			−2.00000			−1.00000			$\pi$
$224$		0			0
$225$	0.500000	−	0.866025i	0.500000	−	0.866025i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$229$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$230$		0			0
$231$		0			0
$232$	0.866025	+	3.23205i	0.866025	+	3.23205i
$233$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$233$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$234$	1.86603	−	0.500000i	1.86603	−	0.500000i
$235$		0			0
$236$	−5.46410			−5.46410
$237$		0			0
$238$		0			0
$239$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$239$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$	1.36603	+	1.36603i	1.36603	+	1.36603i
$243$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$244$		0			0
$245$		0			0
$246$	0.866025	−	0.500000i	0.866025	−	0.500000i
$247$		0			0
$248$		−	1.73205i		−	1.73205i
$249$		0			0
$250$		0			0
$251$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$251$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$252$		0			0
$253$		0			0
$254$	−3.73205			−3.73205
$255$		0			0
$256$	2.73205			2.73205
$257$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$257$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.866025	−	0.500000i	0.866025	−	0.500000i
$262$	1.00000			1.00000
$263$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$263$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.36603	−	1.36603i	−1.36603	−	1.36603i	−0.866025	−	0.500000i	$-0.833333\pi$
$269$	−1.36603	−	1.36603i	−1.36603	−	1.36603i	−0.500000	−	0.866025i	$-0.666667\pi$
$270$		0			0
$271$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$271$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	2.36603	−	1.36603i	2.36603	−	1.36603i
$277$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$277$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$278$	−2.36603	+	2.36603i	−2.36603	+	2.36603i
$279$	−0.500000	+	0.133975i	−0.500000	+	0.133975i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	3.23205	−	1.86603i	3.23205	−	1.86603i
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		−	4.73205i		−	4.73205i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−1.00000	−	3.73205i	−1.00000	−	3.73205i
$289$		−	1.00000i		−	1.00000i
$290$		0			0
$291$		0			0
$292$	1.00000	+	1.00000i	1.00000	+	1.00000i
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.00000			1.00000
$300$	−1.36603	−	2.36603i	−1.36603	−	2.36603i
$301$		0			0
$302$	−1.36603	−	1.36603i	−1.36603	−	1.36603i
$303$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$307$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−0.366025	+	0.366025i	−0.366025	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$311$	−0.366025	+	0.366025i	−0.366025	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$312$	0.866025	−	3.23205i	0.866025	−	3.23205i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$317$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−2.36603	+	1.36603i	−2.36603	+	1.36603i
$325$		−	1.00000i		−	1.00000i
$326$		−	3.73205i		−	3.73205i
$327$		0			0
$328$		−	1.73205i		−	1.73205i
$329$		0			0
$330$		0			0
$331$	0.366025	−	0.366025i	0.366025	−	0.366025i	−0.500000	−	0.866025i	$-0.666667\pi$
$331$	0.366025	−	0.366025i	0.366025	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$332$		0			0
$333$		0			0
$334$	2.73205	−	2.73205i	2.73205	−	2.73205i
$335$		0			0
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−2.73205	+	2.73205i	−2.73205	+	2.73205i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		−	2.73205i		−	2.73205i
$349$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$349$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$	−1.00000			−1.00000
$352$		0			0
$353$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$353$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$354$	3.73205	+	1.00000i	3.73205	+	1.00000i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	1.36603	−	1.36603i	1.36603	−	1.36603i
$359$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$359$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$360$		0			0
$361$			1.00000i			1.00000i
$362$		0			0
$363$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$		−	3.73205i		−	3.73205i
$369$	−0.500000	+	0.133975i	−0.500000	+	0.133975i
$370$		0			0
$371$		0			0
$372$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		−	6.46410i		−	6.46410i
$377$	0.500000	−	0.866025i	0.500000	−	0.866025i
$378$		0			0
$379$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$379$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$380$		0			0
$381$	1.86603	+	0.500000i	1.86603	+	0.500000i
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$	−3.23205	−	0.866025i	−3.23205	−	0.866025i
$385$		0			0
$386$	−3.73205			−3.73205
$387$		0			0
$388$		0			0
$389$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$389$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$390$		0			0
$391$		0			0
$392$	2.36603	+	2.36603i	2.36603	+	2.36603i
$393$	−0.500000	−	0.133975i	−0.500000	−	0.133975i
$394$	−2.36603	−	2.36603i	−2.36603	−	2.36603i
$395$		0			0
$396$		0			0
$397$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$397$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$398$		0			0
$399$		0			0
$400$	−3.73205			−3.73205
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−0.366025	+	0.366025i	−0.366025	+	0.366025i
$404$	2.73205	+	2.73205i	2.73205	+	2.73205i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.366025	+	0.366025i	0.366025	+	0.366025i	0.866025	−	0.500000i	$-0.166667\pi$
$409$	0.366025	+	0.366025i	0.366025	+	0.366025i	−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−1.86603	+	0.500000i	−1.86603	+	0.500000i
$415$		0			0
$416$	−2.73205	−	2.73205i	−2.73205	−	2.73205i
$417$	1.50000	−	0.866025i	1.50000	−	0.866025i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$421$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$422$		−	2.73205i		−	2.73205i
$423$	−1.86603	+	0.500000i	−1.86603	+	0.500000i
$424$		0			0
$425$		0			0
$426$	−0.866025	+	3.23205i	−0.866025	+	3.23205i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$			3.73205i			3.73205i
$433$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$439$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$439$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	0.500000	−	0.866025i	0.500000	−	0.866025i
$442$		0			0
$443$	1.36603	+	1.36603i	1.36603	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$443$	1.36603	+	1.36603i	1.36603	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$	2.73205	−	2.73205i	2.73205	−	2.73205i
$447$		0			0
$448$		0			0
$449$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$449$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$450$	0.500000	+	1.86603i	0.500000	+	1.86603i
$451$		0			0
$452$		0			0
$453$	0.500000	+	0.866025i	0.500000	+	0.866025i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	0.366025	+	0.366025i	0.366025	+	0.366025i	0.866025	−	0.500000i	$-0.166667\pi$
$461$	0.366025	+	0.366025i	0.366025	+	0.366025i	−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$463$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$464$	−3.23205	−	1.86603i	−3.23205	−	1.86603i
$465$		0			0
$466$	1.36603	+	1.36603i	1.36603	+	1.36603i
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	−1.36603	+	2.36603i	−1.36603	+	2.36603i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	4.73205	−	4.73205i	4.73205	−	4.73205i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−2.36603	−	2.36603i	−2.36603	−	2.36603i
$479$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$479$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−2.73205			−2.73205
$485$		0			0
$486$	1.86603	−	0.500000i	1.86603	−	0.500000i
$487$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$487$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$488$		0			0
$489$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$490$		0			0
$491$	−1.36603	−	1.36603i	−1.36603	−	1.36603i	−0.866025	−	0.500000i	$-0.833333\pi$
$491$	−1.36603	−	1.36603i	−1.36603	−	1.36603i	−0.500000	−	0.866025i	$-0.666667\pi$
$492$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1.36603	+	1.36603i	1.36603	+	1.36603i
$497$		0			0
$498$		0			0
$499$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$499$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$500$		0			0
$501$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$502$		0			0
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	3.73205	−	3.73205i	3.73205	−	3.73205i
$509$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$509$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$510$		0			0
$511$		0			0
$512$	−1.36603	+	1.36603i	−1.36603	+	1.36603i
$513$		0			0
$514$	−2.36603	−	2.36603i	−2.36603	−	2.36603i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	1.73205	−	1.00000i	1.73205	−	1.00000i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−1.00000			−1.00000
$530$		0			0
$531$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$532$		0			0
$533$	−0.366025	+	0.366025i	−0.366025	+	0.366025i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$538$	3.73205			3.73205
$539$		0			0
$540$		0			0
$541$	−0.366025	−	0.366025i	−0.366025	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$541$	−0.366025	−	0.366025i	−0.366025	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$542$			2.73205i			2.73205i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$547$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	−0.866025	+	3.23205i	−0.866025	+	3.23205i
$553$		0			0
$554$	1.36603	−	1.36603i	1.36603	−	1.36603i
$555$		0			0
$556$		−	4.73205i		−	4.73205i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$	0.500000	−	0.866025i	0.500000	−	0.866025i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$	−1.36603	+	5.09808i	−1.36603	+	5.09808i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	4.09808	+	4.09808i	4.09808	+	4.09808i
$569$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$569$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			1.00000i			1.00000i
$576$	3.23205	+	1.86603i	3.23205	+	1.86603i
$577$	−0.366025	+	0.366025i	−0.366025	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$577$	−0.366025	+	0.366025i	−0.366025	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$578$	1.36603	+	1.36603i	1.36603	+	1.36603i
$579$	1.86603	+	0.500000i	1.86603	+	0.500000i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−1.73205			−1.73205
$585$		0			0
$586$		0			0
$587$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$587$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$588$	−1.36603	−	2.36603i	−1.36603	−	2.36603i
$589$		0			0
$590$		0			0
$591$	0.866025	+	1.50000i	0.866025	+	1.50000i
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−1.36603	+	1.36603i	−1.36603	+	1.36603i
$599$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$599$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$600$	3.23205	+	0.866025i	3.23205	+	0.866025i
$601$	1.36603	+	1.36603i	1.36603	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$601$	1.36603	+	1.36603i	1.36603	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$		0			0
$604$	2.73205			2.73205
$605$		0			0
$606$	−1.36603	−	2.36603i	−1.36603	−	2.36603i
$607$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$607$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.36603	+	1.36603i	−1.36603	+	1.36603i
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$	2.73205			2.73205
$615$		0			0
$616$		0			0
$617$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$617$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$618$		0			0
$619$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$619$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$620$		0			0
$621$	1.00000			1.00000
$622$		−	1.00000i		−	1.00000i
$623$		0			0
$624$	1.86603	+	3.23205i	1.86603	+	3.23205i
$625$	1.00000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$634$	−2.73205			−2.73205
$635$		0			0
$636$		0			0
$637$		−	1.00000i		−	1.00000i
$638$		0			0
$639$	0.866025	−	1.50000i	0.866025	−	1.50000i
$640$		0			0
$641$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$641$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$647$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$648$	0.866025	−	3.23205i	0.866025	−	3.23205i
$649$		0			0
$650$	1.36603	+	1.36603i	1.36603	+	1.36603i
$651$		0			0
$652$	3.73205	+	3.73205i	3.73205	+	3.73205i
$653$	1.36603	+	1.36603i	1.36603	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$653$	1.36603	+	1.36603i	1.36603	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$	1.36603	+	1.36603i	1.36603	+	1.36603i
$657$	0.133975	+	0.500000i	0.133975	+	0.500000i
$658$		0			0
$659$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$659$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$			1.00000i			1.00000i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$668$			5.46410i			5.46410i
$669$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$673$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$674$		0			0
$675$		−	1.00000i		−	1.00000i
$676$		0			0
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$683$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		−	5.46410i		−	5.46410i
$693$		0			0
$694$		0			0
$695$		0			0
$696$	2.36603	+	2.36603i	2.36603	+	2.36603i
$697$		0			0
$698$	1.36603	−	1.36603i	1.36603	−	1.36603i
$699$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	1.36603	−	1.36603i	1.36603	−	1.36603i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−2.36603	+	2.36603i	−2.36603	+	2.36603i
$707$		0			0
$708$	−4.73205	+	2.73205i	−4.73205	+	2.73205i
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.366025	−	0.366025i	0.366025	−	0.366025i
$714$		0			0
$715$		0			0
$716$			2.73205i			2.73205i
$717$	0.866025	+	1.50000i	0.866025	+	1.50000i
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	−1.36603	−	1.36603i	−1.36603	−	1.36603i
$723$		0			0
$724$		0			0
$725$	0.866025	+	0.500000i	0.866025	+	0.500000i
$726$	1.86603	+	0.500000i	1.86603	+	0.500000i
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$	−1.00000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$		0			0
$735$		0			0
$736$	2.73205	+	2.73205i	2.73205	+	2.73205i
$737$		0			0
$738$	0.500000	−	0.866025i	0.500000	−	0.866025i
$739$	−0.366025	−	0.366025i	−0.366025	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$739$	−0.366025	−	0.366025i	−0.366025	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$751$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$752$	5.09808	+	5.09808i	5.09808	+	5.09808i
$753$		0			0
$754$	0.500000	+	1.86603i	0.500000	+	1.86603i
$755$		0			0
$756$		0			0
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$761$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$762$	−3.23205	+	1.86603i	−3.23205	+	1.86603i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.00000			−2.00000
$768$	2.36603	−	1.36603i	2.36603	−	1.36603i
$769$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$769$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$770$		0			0
$771$	0.866025	+	1.50000i	0.866025	+	1.50000i
$772$	3.73205	−	3.73205i	3.73205	−	3.73205i
$773$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$773$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$774$		0			0
$775$	−0.366025	−	0.366025i	−0.366025	−	0.366025i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	0.500000	−	0.866025i	0.500000	−	0.866025i
$784$	−3.73205			−3.73205
$785$		0			0
$786$	0.866025	−	0.500000i	0.866025	−	0.500000i
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	4.73205			4.73205
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	2.36603	−	2.36603i	2.36603	−	2.36603i
$795$		0			0
$796$		0			0
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		0			0
$800$	2.73205	−	2.73205i	2.73205	−	2.73205i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		−	1.00000i		−	1.00000i
$807$	−1.86603	−	0.500000i	−1.86603	−	0.500000i
$808$	−4.73205			−4.73205
$809$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$809$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$810$		0			0
$811$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$811$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$812$		0			0
$813$	0.366025	−	1.36603i	0.366025	−	1.36603i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−1.00000			−1.00000
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$	−1.36603	−	1.36603i	−1.36603	−	1.36603i	−0.866025	−	0.500000i	$-0.833333\pi$
$823$	−1.36603	−	1.36603i	−1.36603	−	1.36603i	−0.500000	−	0.866025i	$-0.666667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$	1.36603	−	2.36603i	1.36603	−	2.36603i
$829$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$829$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$830$		0			0
$831$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$832$	3.73205			3.73205
$833$		0			0
$834$	−0.866025	+	3.23205i	−0.866025	+	3.23205i
$835$		0			0
$836$		0			0
$837$	−0.366025	+	0.366025i	−0.366025	+	0.366025i
$838$		0			0
$839$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$839$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$840$		0			0
$841$	0.500000	+	0.866025i	0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$	2.73205	+	2.73205i	2.73205	+	2.73205i
$845$		0			0
$846$	1.86603	−	3.23205i	1.86603	−	3.23205i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	−2.36603	−	4.09808i	−2.36603	−	4.09808i
$853$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$853$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$857$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	−1.36603	+	1.36603i	−1.36603	+	1.36603i	−0.500000	+	0.866025i	$0.666667\pi$
$859$	−1.36603	+	1.36603i	−1.36603	+	1.36603i	−0.866025	+	0.500000i	$0.833333\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$863$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$864$	−2.73205	−	2.73205i	−2.73205	−	2.73205i
$865$		0			0
$866$		0			0
$867$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	1.36603	+	0.366025i	1.36603	+	0.366025i
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$	−2.36603	−	2.36603i	−2.36603	−	2.36603i
$879$		0			0
$880$		0			0
$881$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$881$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$882$	0.500000	+	1.86603i	0.500000	+	1.86603i
$883$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$883$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$884$		0			0
$885$		0			0
$886$	−3.73205			−3.73205
$887$	−1.36603	+	1.36603i	−1.36603	+	1.36603i	−0.500000	+	0.866025i	$0.666667\pi$
$887$	−1.36603	+	1.36603i	−1.36603	+	1.36603i	−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$			5.46410i			5.46410i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	0.866025	−	0.500000i	0.866025	−	0.500000i
$898$		−	2.73205i		−	2.73205i
$899$	−0.133975	−	0.500000i	−0.133975	−	0.500000i
$900$	−2.36603	−	1.36603i	−2.36603	−	1.36603i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	−1.86603	−	0.500000i	−1.86603	−	0.500000i
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$	0.366025	+	1.36603i	0.366025	+	1.36603i
$910$		0			0
$911$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$911$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$922$	−1.00000			−1.00000
$923$		−	1.73205i		−	1.73205i
$924$		0			0
$925$		0			0
$926$	−2.73205	−	2.73205i	−2.73205	−	2.73205i
$927$		0			0
$928$	3.73205	−	1.00000i	3.73205	−	1.00000i
$929$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$929$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$		0			0
$932$	−2.73205			−2.73205
$933$	−0.133975	+	0.500000i	−0.133975	+	0.500000i
$934$		0			0
$935$		0			0
$936$	−0.866025	−	3.23205i	−0.866025	−	3.23205i
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$	0.366025	−	0.366025i	0.366025	−	0.366025i
$944$			7.46410i			7.46410i
$945$		0			0
$946$		0			0
$947$	1.36603	−	1.36603i	1.36603	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$947$	1.36603	−	1.36603i	1.36603	−	1.36603i	0.866025	−	0.500000i	$-0.166667\pi$
$948$		0			0
$949$	0.366025	+	0.366025i	0.366025	+	0.366025i
$950$		0			0
$951$	1.36603	+	0.366025i	1.36603	+	0.366025i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$	4.73205			4.73205
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		−	0.732051i		−	0.732051i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1.36603	−	1.36603i	1.36603	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$967$	1.36603	−	1.36603i	1.36603	−	1.36603i	0.866025	−	0.500000i	$-0.166667\pi$
$968$	2.36603	−	2.36603i	2.36603	−	2.36603i
$969$		0			0
$970$		0			0
$971$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$971$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$972$	−1.36603	+	2.36603i	−1.36603	+	2.36603i
$973$		0			0
$974$	2.36603	−	2.36603i	2.36603	−	2.36603i
$975$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$	−1.86603	−	3.23205i	−1.86603	−	3.23205i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	3.73205			3.73205
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−2.00000			−2.00000
$993$	0.133975	−	0.500000i	0.133975	−	0.500000i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$997$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$998$	1.36603	+	1.36603i	1.36603	+	1.36603i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2001.1.i.c.1172.1	✓	4
3.2	odd	2		2001.1.i.d.1172.2	yes	4
23.22	odd	2	CM	2001.1.i.c.1172.1	✓	4
29.17	odd	4		2001.1.i.d.1931.2	yes	4
69.68	even	2		2001.1.i.d.1172.2	yes	4
87.17	even	4	inner	2001.1.i.c.1931.1	yes	4
667.597	even	4		2001.1.i.d.1931.2	yes	4
2001.1931	odd	4	inner	2001.1.i.c.1931.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.1.i.c.1172.1	✓	4	1.1	even	1	trivial
2001.1.i.c.1172.1	✓	4	23.22	odd	2	CM
2001.1.i.c.1931.1	yes	4	87.17	even	4	inner
2001.1.i.c.1931.1	yes	4	2001.1931	odd	4	inner
2001.1.i.d.1172.2	yes	4	3.2	odd	2
2001.1.i.d.1172.2	yes	4	69.68	even	2
2001.1.i.d.1931.2	yes	4	29.17	odd	4
2001.1.i.d.1931.2	yes	4	667.597	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2001 = 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2001.i (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.36603 + 1.36603i) q^{2} +(0.866025 - 0.500000i) q^{3} -2.73205i q^{4} +(-0.500000 + 1.86603i) q^{6} +(2.36603 + 2.36603i) q^{8} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(-1.36603 + 1.36603i) q^{2} +(0.866025 - 0.500000i) q^{3} -2.73205i q^{4} +(-0.500000 + 1.86603i) q^{6} +(2.36603 + 2.36603i) q^{8} +(0.500000 - 0.866025i) q^{9} +(-1.36603 - 2.36603i) q^{12} -1.00000i q^{13} -3.73205 q^{16} +(0.500000 + 1.86603i) q^{18} +1.00000i q^{23} +(3.23205 + 0.866025i) q^{24} +1.00000 q^{25} +(1.36603 + 1.36603i) q^{26} -1.00000i q^{27} +(0.866025 + 0.500000i) q^{29} +(-0.366025 - 0.366025i) q^{31} +(2.73205 - 2.73205i) q^{32} +(-2.36603 - 1.36603i) q^{36} +(-0.500000 - 0.866025i) q^{39} +(-0.366025 - 0.366025i) q^{41} +(-1.36603 - 1.36603i) q^{46} +(-1.36603 - 1.36603i) q^{47} +(-3.23205 + 1.86603i) q^{48} +1.00000 q^{49} +(-1.36603 + 1.36603i) q^{50} -2.73205 q^{52} +(1.36603 + 1.36603i) q^{54} +(-1.86603 + 0.500000i) q^{58} -2.00000i q^{59} +1.00000 q^{62} +3.73205i q^{64} +(0.500000 + 0.866025i) q^{69} +1.73205 q^{71} +(3.23205 - 0.866025i) q^{72} +(-0.366025 + 0.366025i) q^{73} +(0.866025 - 0.500000i) q^{75} +(1.86603 + 0.500000i) q^{78} +(-0.500000 - 0.866025i) q^{81} +1.00000 q^{82} +1.00000 q^{87} +2.73205 q^{92} +(-0.500000 - 0.133975i) q^{93} +3.73205 q^{94} +(1.00000 - 3.73205i) q^{96} +(-1.36603 + 1.36603i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 2 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) 4 * q - 2 * q^2 - 2 * q^6 + 6 * q^8 + 2 * q^9	\( 4 q - 2 q^{2} - 2 q^{6} + 6 q^{8} + 2 q^{9} - 2 q^{12} - 8 q^{16} + 2 q^{18} + 6 q^{24} + 4 q^{25} + 2 q^{26} + 2 q^{31} + 4 q^{32} - 6 q^{36} - 2 q^{39} + 2 q^{41} - 2 q^{46} - 2 q^{47} - 6 q^{48} + 4 q^{49} - 2 q^{50} - 4 q^{52} + 2 q^{54} - 4 q^{58} + 4 q^{62} + 2 q^{69} + 6 q^{72} + 2 q^{73} + 4 q^{78} - 2 q^{81} + 4 q^{82} + 4 q^{87} + 4 q^{92} - 2 q^{93} + 8 q^{94} + 4 q^{96} - 2 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 - 2 * q^6 + 6 * q^8 + 2 * q^9 - 2 * q^12 - 8 * q^16 + 2 * q^18 + 6 * q^24 + 4 * q^25 + 2 * q^26 + 2 * q^31 + 4 * q^32 - 6 * q^36 - 2 * q^39 + 2 * q^41 - 2 * q^46 - 2 * q^47 - 6 * q^48 + 4 * q^49 - 2 * q^50 - 4 * q^52 + 2 * q^54 - 4 * q^58 + 4 * q^62 + 2 * q^69 + 6 * q^72 + 2 * q^73 + 4 * q^78 - 2 * q^81 + 4 * q^82 + 4 * q^87 + 4 * q^92 - 2 * q^93 + 8 * q^94 + 4 * q^96 - 2 * q^98

Introduction

L-functions

Modular forms

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