LMFDB - Embedded newform 1755.1.e.e.1754.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1755,1,Mod(1754,1755)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1755, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1755.1754");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1755 = 3^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1755.e (of order $2$, degree $1$, 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:	$0.875859097171$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1754.4
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1755.1754
Dual form			1755.1.e.e.1754.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.517638i q^{2} +0.732051 q^{4} +(0.258819 - 0.965926i) q^{5} -0.896575i q^{8} +O(q^{10})$	$q-0.517638i q^{2} +0.732051 q^{4} +(0.258819 - 0.965926i) q^{5} -0.896575i q^{8} +(-0.500000 - 0.133975i) q^{10} -1.41421 q^{11} -1.00000i q^{13} +0.267949 q^{16} +(0.189469 - 0.707107i) q^{20} +0.732051i q^{22} +(-0.866025 - 0.500000i) q^{25} -0.517638 q^{26} -1.03528i q^{32} +(-0.866025 - 0.232051i) q^{40} +1.41421 q^{41} +1.73205i q^{43} -1.03528 q^{44} +0.517638i q^{47} -1.00000 q^{49} +(-0.258819 + 0.448288i) q^{50} -0.732051i q^{52} +(-0.366025 + 1.36603i) q^{55} +1.93185 q^{59} +1.73205 q^{61} -0.267949 q^{64} +(-0.965926 - 0.258819i) q^{65} +0.517638 q^{71} +(0.0693504 - 0.258819i) q^{80} -0.732051i q^{82} +1.93185i q^{83} +0.896575 q^{86} +1.26795i q^{88} -1.93185 q^{89} +0.267949 q^{94} +0.517638i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} - 4 q^{10} + 16 q^{16} - 8 q^{49} + 4 q^{55} - 16 q^{64} + 16 q^{94}+O(q^{100}) $ 8 * q - 8 * q^4 - 4 * q^10 + 16 * q^16 - 8 * q^49 + 4 * q^55 - 16 * q^64 + 16 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1755\mathbb{Z}\right)^\times$.

$n$	$326$	$352$	$1081$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$		−	0.517638i		−	0.517638i	−0.965926	−	0.258819i	$-0.916667\pi$
$2$		−	0.517638i		−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$3$		0			0
$3$		0			0
$4$	0.732051			0.732051
$5$	0.258819	−	0.965926i	0.258819	−	0.965926i
$5$	0.258819	−	0.965926i	0.258819	−	0.965926i
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$		−	0.896575i		−	0.896575i
$9$		0			0
$10$	−0.500000	−	0.133975i	−0.500000	−	0.133975i
$11$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$11$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$12$		0			0
$13$		−	1.00000i		−	1.00000i
$13$		−	1.00000i		−	1.00000i
$14$		0			0
$15$		0			0
$16$	0.267949			0.267949
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	0.189469	−	0.707107i	0.189469	−	0.707107i
$21$		0			0
$22$			0.732051i			0.732051i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$26$	−0.517638			−0.517638
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		−	1.03528i		−	1.03528i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$	−0.866025	−	0.232051i	−0.866025	−	0.232051i
$41$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$41$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$42$		0			0
$43$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$43$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$44$	−1.03528			−1.03528
$45$		0			0
$46$		0			0
$47$			0.517638i			0.517638i	0.965926	+	0.258819i	$0.0833333\pi$
$47$			0.517638i			0.517638i	−0.965926	+	0.258819i	$0.916667\pi$
$48$		0			0
$49$	−1.00000			−1.00000
$50$	−0.258819	+	0.448288i	−0.258819	+	0.448288i
$51$		0			0
$52$		−	0.732051i		−	0.732051i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.93185			1.93185			0.965926	−	0.258819i	$-0.0833333\pi$
$59$	1.93185			1.93185			0.965926	+	0.258819i	$0.0833333\pi$
$60$		0			0
$61$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$61$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$62$		0			0
$63$		0			0
$64$	−0.267949			−0.267949
$65$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.517638			0.517638			0.258819	−	0.965926i	$-0.416667\pi$
$71$	0.517638			0.517638			0.258819	+	0.965926i	$0.416667\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	0.0693504	−	0.258819i	0.0693504	−	0.258819i
$81$		0			0
$82$		−	0.732051i		−	0.732051i
$83$			1.93185i			1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$83$			1.93185i			1.93185i	−0.258819	+	0.965926i	$0.583333\pi$
$84$		0			0
$85$		0			0
$86$	0.896575			0.896575
$87$		0			0
$88$			1.26795i			1.26795i
$89$	−1.93185			−1.93185			−0.965926	−	0.258819i	$-0.916667\pi$
$89$	−1.93185			−1.93185			−0.965926	+	0.258819i	$0.916667\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	0.267949			0.267949
$95$		0			0
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$			0.517638i			0.517638i
$99$		0			0
$100$	−0.633975	−	0.366025i	−0.633975	−	0.366025i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$103$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$104$	−0.896575			−0.896575
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	0.707107	+	0.189469i	0.707107	+	0.189469i
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		−	1.00000i		−	1.00000i
$119$		0			0
$120$		0			0
$121$	1.00000			1.00000
$122$		−	0.896575i		−	0.896575i
$123$		0			0
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$126$		0			0
$127$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$127$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$128$		−	0.896575i		−	0.896575i
$129$		0			0
$130$	−0.133975	+	0.500000i	−0.133975	+	0.500000i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$137$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$138$		0			0
$139$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$139$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$140$		0			0
$141$		0			0
$142$		−	0.267949i		−	0.267949i
$143$			1.41421i			1.41421i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.93185			1.93185			0.965926	−	0.258819i	$-0.0833333\pi$
$149$	1.93185			1.93185			0.965926	+	0.258819i	$0.0833333\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$157$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$		0			0
$160$	−1.00000	−	0.267949i	−1.00000	−	0.267949i
$161$		0			0
$162$		0			0
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$	1.03528			1.03528
$165$		0			0
$166$	1.00000			1.00000
$167$			1.93185i			1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$167$			1.93185i			1.93185i	−0.258819	+	0.965926i	$0.583333\pi$
$168$		0			0
$169$	−1.00000			−1.00000
$170$		0			0
$171$		0			0
$172$			1.26795i			1.26795i
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$	−0.378937			−0.378937
$177$		0			0
$178$			1.00000i			1.00000i
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$181$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$			0.378937i			0.378937i
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$	−0.732051			−0.732051
$197$			0.517638i			0.517638i	0.965926	+	0.258819i	$0.0833333\pi$
$197$			0.517638i			0.517638i	−0.965926	+	0.258819i	$0.916667\pi$
$198$		0			0
$199$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$199$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$200$	−0.448288	+	0.776457i	−0.448288	+	0.776457i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	0.366025	−	1.36603i	0.366025	−	1.36603i
$206$	−0.896575			−0.896575
$207$		0			0
$208$		−	0.267949i		−	0.267949i
$209$		0			0
$210$		0			0
$211$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$211$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.67303	+	0.448288i	1.67303	+	0.448288i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.267949	+	1.00000i	−0.267949	+	1.00000i
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			1.93185i			1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$227$			1.93185i			1.93185i	−0.258819	+	0.965926i	$0.583333\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$	0.500000	+	0.133975i	0.500000	+	0.133975i
$236$	1.41421			1.41421
$237$		0			0
$238$		0			0
$239$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$239$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		−	0.517638i		−	0.517638i
$243$		0			0
$244$	1.26795			1.26795
$245$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.366025	+	0.366025i	0.366025	+	0.366025i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$	−0.517638			−0.517638
$255$		0			0
$256$	−0.732051			−0.732051
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$	−0.707107	−	0.189469i	−0.707107	−	0.189469i
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$	−0.732051			−0.732051
$275$	1.22474	+	0.707107i	1.22474	+	0.707107i
$276$		0			0
$277$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$277$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$278$			0.517638i			0.517638i
$279$		0			0
$280$		0			0
$281$	−0.517638			−0.517638			−0.258819	−	0.965926i	$-0.583333\pi$
$281$	−0.517638			−0.517638			−0.258819	+	0.965926i	$0.583333\pi$
$282$		0			0
$283$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$283$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$284$	0.378937			0.378937
$285$		0			0
$286$	0.732051			0.732051
$287$		0			0
$288$		0			0
$289$	−1.00000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$293$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$	0.500000	−	1.86603i	0.500000	−	1.86603i
$296$		0			0
$297$		0			0
$298$		−	1.00000i		−	1.00000i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.448288	−	1.67303i	0.448288	−	1.67303i
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$313$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$314$	0.896575			0.896575
$315$		0			0
$316$		0			0
$317$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$317$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$	−0.0693504	+	0.258819i	−0.0693504	+	0.258819i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$326$		0			0
$327$		0			0
$328$		−	1.26795i		−	1.26795i
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$			1.41421i			1.41421i
$333$		0			0
$334$	1.00000			1.00000
$335$		0			0
$336$		0			0
$337$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$337$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$338$			0.517638i			0.517638i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	1.55291			1.55291
$345$		0			0
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$			1.46410i			1.46410i
$353$		−	1.93185i		−	1.93185i	−0.258819	−	0.965926i	$-0.583333\pi$
$353$		−	1.93185i		−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$354$		0			0
$355$	0.133975	−	0.500000i	0.133975	−	0.500000i
$356$	−1.41421			−1.41421
$357$		0			0
$358$		0			0
$359$	0.517638			0.517638			0.258819	−	0.965926i	$-0.416667\pi$
$359$	0.517638			0.517638			0.258819	+	0.965926i	$0.416667\pi$
$360$		0			0
$361$	1.00000			1.00000
$362$		−	0.517638i		−	0.517638i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$373$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$374$		0			0
$375$		0			0
$376$	0.464102			0.464102
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$383$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$			0.896575i			0.896575i
$393$		0			0
$394$	0.267949			0.267949
$395$		0			0
$396$		0			0
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$			0.896575i			0.896575i
$399$		0			0
$400$	−0.232051	−	0.133975i	−0.232051	−	0.133975i
$401$	−1.93185			−1.93185			−0.965926	−	0.258819i	$-0.916667\pi$
$401$	−1.93185			−1.93185			−0.965926	+	0.258819i	$0.916667\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$	−0.707107	−	0.189469i	−0.707107	−	0.189469i
$411$		0			0
$412$		−	1.26795i		−	1.26795i
$413$		0			0
$414$		0			0
$415$	1.86603	+	0.500000i	1.86603	+	0.500000i
$416$	−1.03528			−1.03528
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		−	0.896575i		−	0.896575i
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$	0.232051	−	0.866025i	0.232051	−	0.866025i
$431$	1.93185			1.93185			0.965926	−	0.258819i	$-0.0833333\pi$
$431$	1.93185			1.93185			0.965926	+	0.258819i	$0.0833333\pi$
$432$		0			0
$433$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$433$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$439$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$440$	1.22474	+	0.328169i	1.22474	+	0.328169i
$441$		0			0
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.517638			−0.517638			−0.258819	−	0.965926i	$-0.583333\pi$
$449$	−0.517638			−0.517638			−0.258819	+	0.965926i	$0.583333\pi$
$450$		0			0
$451$	−2.00000			−2.00000
$452$		0			0
$453$		0			0
$454$	1.00000			1.00000
$455$		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
$461$	1.93185			1.93185			0.965926	−	0.258819i	$-0.0833333\pi$
$461$	1.93185			1.93185			0.965926	+	0.258819i	$0.0833333\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$	0.0693504	−	0.258819i	0.0693504	−	0.258819i
$471$		0			0
$472$		−	1.73205i		−	1.73205i
$473$		−	2.44949i		−	2.44949i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$			0.732051i			0.732051i
$479$	−1.93185			−1.93185			−0.965926	−	0.258819i	$-0.916667\pi$
$479$	−1.93185			−1.93185			−0.965926	+	0.258819i	$0.916667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.732051			0.732051
$485$		0			0
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		−	1.55291i		−	1.55291i
$489$		0			0
$490$	0.500000	+	0.133975i	0.500000	+	0.133975i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$	−0.517638	+	0.517638i	−0.517638	+	0.517638i
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		−	0.732051i		−	0.732051i
$509$	−1.93185			−1.93185			−0.965926	−	0.258819i	$-0.916667\pi$
$509$	−1.93185			−1.93185			−0.965926	+	0.258819i	$0.916667\pi$
$510$		0			0
$511$		0			0
$512$		−	0.517638i		−	0.517638i
$513$		0			0
$514$		0			0
$515$	−1.67303	−	0.448288i	−1.67303	−	0.448288i
$516$		0			0
$517$		−	0.732051i		−	0.732051i
$518$		0			0
$519$		0			0
$520$	−0.232051	+	0.866025i	−0.232051	+	0.866025i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$523$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−1.00000			−1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	1.41421i		−	1.41421i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.41421			1.41421
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		−	1.03528i		−	1.03528i
$549$		0			0
$550$	0.366025	−	0.633975i	0.366025	−	0.633975i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−0.896575			−0.896575
$555$		0			0
$556$	−0.732051			−0.732051
$557$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$557$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$558$		0			0
$559$	1.73205			1.73205
$560$		0			0
$561$		0			0
$562$			0.267949i			0.267949i
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$	0.517638			0.517638
$567$		0			0
$568$		−	0.464102i		−	0.464102i
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$571$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$572$			1.03528i			1.03528i
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$			0.517638i			0.517638i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	0.732051			0.732051
$587$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$587$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$588$		0			0
$589$		0			0
$590$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$591$		0			0
$592$		0			0
$593$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$593$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$594$		0			0
$595$		0			0
$596$	1.41421			1.41421
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$601$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.258819	−	0.965926i	0.258819	−	0.965926i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$	−0.866025	−	0.232051i	−0.866025	−	0.232051i
$611$	0.517638			0.517638
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1.93185i		−	1.93185i	−0.258819	−	0.965926i	$-0.583333\pi$
$617$		−	1.93185i		−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.500000	+	0.866025i	0.500000	+	0.866025i
$626$	0.517638			0.517638
$627$		0			0
$628$			1.26795i			1.26795i
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	0.732051			0.732051
$635$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$636$		0			0
$637$			1.00000i			1.00000i
$638$		0			0
$639$		0			0
$640$	−0.866025	−	0.232051i	−0.866025	−	0.232051i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		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
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	−2.73205			−2.73205
$650$	0.448288	+	0.258819i	0.448288	+	0.258819i
$651$		0			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$	0.378937			0.378937
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$	1.73205			1.73205
$665$		0			0
$666$		0			0
$667$		0			0
$668$			1.41421i			1.41421i
$669$		0			0
$670$		0			0
$671$	−2.44949			−2.44949
$672$		0			0
$673$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$673$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$674$	0.517638			0.517638
$675$		0			0
$676$	−0.732051			−0.732051
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1.93185i		−	1.93185i	−0.258819	−	0.965926i	$-0.583333\pi$
$683$		−	1.93185i		−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$684$		0			0
$685$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$686$		0			0
$687$		0			0
$688$			0.464102i			0.464102i
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	0.378937			0.378937
$705$		0			0
$706$	−1.00000			−1.00000
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$	−0.258819	−	0.0693504i	−0.258819	−	0.0693504i
$711$		0			0
$712$			1.73205i			1.73205i
$713$		0			0
$714$		0			0
$715$	1.36603	+	0.366025i	1.36603	+	0.366025i
$716$		0			0
$717$		0			0
$718$		−	0.267949i		−	0.267949i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$		−	0.517638i		−	0.517638i
$723$		0			0
$724$	0.732051			0.732051
$725$		0			0
$726$		0			0
$727$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$727$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1.93185i			1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$743$			1.93185i			1.93185i	−0.258819	+	0.965926i	$0.583333\pi$
$744$		0			0
$745$	0.500000	−	1.86603i	0.500000	−	1.86603i
$746$	−0.517638			−0.517638
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−2.00000			−2.00000			−1.00000			$\pi$
$751$	−2.00000			−2.00000			−1.00000			$\pi$
$752$			0.138701i			0.138701i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$757$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.517638			−0.517638			−0.258819	−	0.965926i	$-0.583333\pi$
$761$	−0.517638			−0.517638			−0.258819	+	0.965926i	$0.583333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−0.732051			−0.732051
$767$		−	1.93185i		−	1.93185i
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	0.517638i		−	0.517638i	−0.965926	−	0.258819i	$-0.916667\pi$
$773$		−	0.517638i		−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−0.732051			−0.732051
$782$		0			0
$783$		0			0
$784$	−0.267949			−0.267949
$785$	1.67303	+	0.448288i	1.67303	+	0.448288i
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$			0.378937i			0.378937i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	1.73205i		−	1.73205i
$794$		0			0
$795$		0			0
$796$	−1.26795			−1.26795
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	−0.517638	+	0.896575i	−0.517638	+	0.896575i
$801$		0			0
$802$			1.00000i			1.00000i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	0.267949	−	1.00000i	0.267949	−	1.00000i
$821$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$821$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$822$		0			0
$823$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$823$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$824$	−1.55291			−1.55291
$825$		0			0
$826$		0			0
$827$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$827$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$829$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$830$	0.258819	−	0.965926i	0.258819	−	0.965926i
$831$		0			0
$832$			0.267949i			0.267949i
$833$		0			0
$834$		0			0
$835$	1.86603	+	0.500000i	1.86603	+	0.500000i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	0.517638			0.517638			0.258819	−	0.965926i	$-0.416667\pi$
$839$	0.517638			0.517638			0.258819	+	0.965926i	$0.416667\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$		0			0
$843$		0			0
$844$	1.26795			1.26795
$845$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$	1.22474	+	0.328169i	1.22474	+	0.328169i
$861$		0			0
$862$		−	1.00000i		−	1.00000i
$863$		−	1.93185i		−	1.93185i	−0.258819	−	0.965926i	$-0.583333\pi$
$863$		−	1.93185i		−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$864$		0			0
$865$		0			0
$866$	1.03528			1.03528
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$			0.517638i			0.517638i
$879$		0			0
$880$	−0.0980762	+	0.366025i	−0.0980762	+	0.366025i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		−	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
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$	0.965926	+	0.258819i	0.965926	+	0.258819i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$			0.267949i			0.267949i
$899$		0			0
$900$		0			0
$901$		0			0
$902$			1.03528i			1.03528i
$903$		0			0
$904$		0			0
$905$	0.258819	−	0.965926i	0.258819	−	0.965926i
$906$		0			0
$907$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$907$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$908$			1.41421i			1.41421i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		−	2.73205i		−	2.73205i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$919$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$920$		0			0
$921$		0			0
$922$		−	1.00000i		−	1.00000i
$923$		−	0.517638i		−	0.517638i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$929$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$937$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$938$		0			0
$939$		0			0
$940$	0.366025	+	0.0980762i	0.366025	+	0.0980762i
$941$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$941$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$942$		0			0
$943$		0			0
$944$	0.517638			0.517638
$945$		0			0
$946$	−1.26795			−1.26795
$947$			0.517638i			0.517638i	0.965926	+	0.258819i	$0.0833333\pi$
$947$			0.517638i			0.517638i	−0.965926	+	0.258819i	$0.916667\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$	−1.03528			−1.03528
$957$		0			0
$958$			1.00000i			1.00000i
$959$		0			0
$960$		0			0
$961$	1.00000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		−	0.896575i		−	0.896575i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.464102			0.464102
$977$		−	1.93185i		−	1.93185i	−0.258819	−	0.965926i	$-0.583333\pi$
$977$		−	1.93185i		−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$978$		0			0
$979$	2.73205			2.73205
$980$	−0.189469	+	0.707107i	−0.189469	+	0.707107i
$981$		0			0
$982$		0			0
$983$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$983$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$	0.500000	+	0.133975i	0.500000	+	0.133975i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$991$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.448288	+	1.67303i	−0.448288	+	1.67303i
$996$		0			0
$997$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$997$			1.73205i			1.73205i	−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	1755.1.e.e.1754.4	yes	8
3.2	odd	2	inner	1755.1.e.e.1754.5	yes	8
5.4	even	2	inner	1755.1.e.e.1754.6	yes	8
13.12	even	2	inner	1755.1.e.e.1754.5	yes	8
15.14	odd	2	inner	1755.1.e.e.1754.3	✓	8
39.38	odd	2	CM	1755.1.e.e.1754.4	yes	8
65.64	even	2	inner	1755.1.e.e.1754.3	✓	8
195.194	odd	2	inner	1755.1.e.e.1754.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.1.e.e.1754.3	✓	8	15.14	odd	2	inner
1755.1.e.e.1754.3	✓	8	65.64	even	2	inner
1755.1.e.e.1754.4	yes	8	1.1	even	1	trivial
1755.1.e.e.1754.4	yes	8	39.38	odd	2	CM
1755.1.e.e.1754.5	yes	8	3.2	odd	2	inner
1755.1.e.e.1754.5	yes	8	13.12	even	2	inner
1755.1.e.e.1754.6	yes	8	5.4	even	2	inner
1755.1.e.e.1754.6	yes	8	195.194	odd	2	inner

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:	\( N \)	\(=\)	\( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1755.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.517638i q^{2} +0.732051 q^{4} +(0.258819 - 0.965926i) q^{5} -0.896575i q^{8} +O(q^{10})\)	\(q-0.517638i q^{2} +0.732051 q^{4} +(0.258819 - 0.965926i) q^{5} -0.896575i q^{8} +(-0.500000 - 0.133975i) q^{10} -1.41421 q^{11} -1.00000i q^{13} +0.267949 q^{16} +(0.189469 - 0.707107i) q^{20} +0.732051i q^{22} +(-0.866025 - 0.500000i) q^{25} -0.517638 q^{26} -1.03528i q^{32} +(-0.866025 - 0.232051i) q^{40} +1.41421 q^{41} +1.73205i q^{43} -1.03528 q^{44} +0.517638i q^{47} -1.00000 q^{49} +(-0.258819 + 0.448288i) q^{50} -0.732051i q^{52} +(-0.366025 + 1.36603i) q^{55} +1.93185 q^{59} +1.73205 q^{61} -0.267949 q^{64} +(-0.965926 - 0.258819i) q^{65} +0.517638 q^{71} +(0.0693504 - 0.258819i) q^{80} -0.732051i q^{82} +1.93185i q^{83} +0.896575 q^{86} +1.26795i q^{88} -1.93185 q^{89} +0.267949 q^{94} +0.517638i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} - 4 q^{10} + 16 q^{16} - 8 q^{49} + 4 q^{55} - 16 q^{64} + 16 q^{94}+O(q^{100}) \) 8 * q - 8 * q^4 - 4 * q^10 + 16 * q^16 - 8 * q^49 + 4 * q^55 - 16 * q^64 + 16 * q^94

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