LMFDB - Embedded newform 1904.1.bt.a.1087.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1904,1,Mod(543,1904)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1904, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 4, 3]))

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

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

chi := DirichletCharacter("1904.543");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1904 = 2^{4} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1904.bt (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:	$0.950219784053$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
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:	$\Z[a_1, \ldots, a_{7}]$
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			1087.1
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1904.1087
Dual form			1904.1.bt.a.543.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.965926 - 1.67303i) q^{3} +(0.707107 + 0.707107i) q^{7} +(-1.36603 + 2.36603i) q^{9} +O(q^{10})$	$q+(-0.965926 - 1.67303i) q^{3} +(0.707107 + 0.707107i) q^{7} +(-1.36603 + 2.36603i) q^{9} +(-0.258819 - 0.448288i) q^{11} +1.73205 q^{13} +(0.500000 + 0.866025i) q^{17} +(0.500000 - 1.86603i) q^{21} +(0.707107 - 1.22474i) q^{23} +(-0.500000 - 0.866025i) q^{25} +3.34607 q^{27} +(0.707107 + 1.22474i) q^{31} +(-0.500000 + 0.866025i) q^{33} +(-1.67303 - 2.89778i) q^{39} +1.00000i q^{49} +(0.965926 - 1.67303i) q^{51} +(-0.500000 - 0.866025i) q^{53} +(-2.63896 + 0.707107i) q^{63} -2.73205 q^{69} -1.93185 q^{71} +(-0.965926 + 1.67303i) q^{75} +(0.133975 - 0.500000i) q^{77} +(0.258819 - 0.448288i) q^{79} +(-1.86603 - 3.23205i) q^{81} +(0.866025 - 1.50000i) q^{89} +(1.22474 + 1.22474i) q^{91} +(1.36603 - 2.36603i) q^{93} +1.41421 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{9}+O(q^{10}) $ 8 * q - 4 * q^9	$ 8 q - 4 q^{9} + 4 q^{17} + 4 q^{21} - 4 q^{25} - 4 q^{33} - 4 q^{53} - 8 q^{69} + 8 q^{77} - 8 q^{81} + 4 q^{93}+O(q^{100}) $ 8 * q - 4 * q^9 + 4 * q^17 + 4 * q^21 - 4 * q^25 - 4 * q^33 - 4 * q^53 - 8 * q^69 + 8 * q^77 - 8 * q^81 + 4 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1904\mathbb{Z}\right)^\times$.

$n$	$239$	$785$	$1361$	$1429$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{3}\right)$	$1$

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			0
$2$		0			0
$3$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.707107	−	0.707107i	$-0.750000\pi$
$3$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.258819	−	0.965926i	$-0.583333\pi$
$4$		0			0
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$		0			0
$7$	0.707107	+	0.707107i	0.707107	+	0.707107i
$7$	0.707107	+	0.707107i	0.707107	+	0.707107i
$8$		0			0
$9$	−1.36603	+	2.36603i	−1.36603	+	2.36603i
$10$		0			0
$11$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$11$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$12$		0			0
$13$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$13$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.500000	+	0.866025i	0.500000	+	0.866025i
$17$	0.500000	+	0.866025i	0.500000	+	0.866025i
$18$		0			0
$19$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$		0			0
$21$	0.500000	−	1.86603i	0.500000	−	1.86603i
$22$		0			0
$23$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$23$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$26$		0			0
$27$	3.34607			3.34607
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$31$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$32$		0			0
$33$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$	−1.67303	−	2.89778i	−1.67303	−	2.89778i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$		0			0
$49$			1.00000i			1.00000i
$50$		0			0
$51$	0.965926	−	1.67303i	0.965926	−	1.67303i
$52$		0			0
$53$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$53$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$	−2.63896	+	0.707107i	−2.63896	+	0.707107i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$	−2.73205			−2.73205
$70$		0			0
$71$	−1.93185			−1.93185			−0.965926	−	0.258819i	$-0.916667\pi$
$71$	−1.93185			−1.93185			−0.965926	+	0.258819i	$0.916667\pi$
$72$		0			0
$73$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$73$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$74$		0			0
$75$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$76$		0			0
$77$	0.133975	−	0.500000i	0.133975	−	0.500000i
$78$		0			0
$79$	0.258819	−	0.448288i	0.258819	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$79$	0.258819	−	0.448288i	0.258819	−	0.448288i	0.965926	+	0.258819i	$0.0833333\pi$
$80$		0			0
$81$	−1.86603	−	3.23205i	−1.86603	−	3.23205i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$89$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$90$		0			0
$91$	1.22474	+	1.22474i	1.22474	+	1.22474i
$92$		0			0
$93$	1.36603	−	2.36603i	1.36603	−	2.36603i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$	1.41421			1.41421
$100$		0			0
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.258819	+	0.965926i	$0.583333\pi$
$107$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−2.36603	+	4.09808i	−2.36603	+	4.09808i
$118$		0			0
$119$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$120$		0			0
$121$	0.366025	−	0.633975i	0.366025	−	0.633975i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$131$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$137$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$138$		0			0
$139$	0.517638			0.517638			0.258819	−	0.965926i	$-0.416667\pi$
$139$	0.517638			0.517638			0.258819	+	0.965926i	$0.416667\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.448288	−	0.776457i	−0.448288	−	0.776457i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	1.67303	−	0.965926i	1.67303	−	0.965926i
$148$		0			0
$149$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$149$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$150$		0			0
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$		0			0
$153$	−2.73205			−2.73205
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$157$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$158$		0			0
$159$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$160$		0			0
$161$	1.36603	−	0.366025i	1.36603	−	0.366025i
$162$		0			0
$163$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$163$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.517638			0.517638			0.258819	−	0.965926i	$-0.416667\pi$
$167$	0.517638			0.517638			0.258819	+	0.965926i	$0.416667\pi$
$168$		0			0
$169$	2.00000			2.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$	0.258819	−	0.965926i	0.258819	−	0.965926i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	0.258819	−	0.448288i	0.258819	−	0.448288i
$188$		0			0
$189$	2.36603	+	2.36603i	2.36603	+	2.36603i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	0.258819	+	0.448288i	0.258819	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$199$	0.258819	+	0.448288i	0.258819	+	0.448288i	−0.707107	+	0.707107i	$0.750000\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.93185	+	3.34607i	1.93185	+	3.34607i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$211$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$212$		0			0
$213$	1.86603	+	3.23205i	1.86603	+	3.23205i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	0.866025	+	1.50000i	0.866025	+	1.50000i
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	2.73205			2.73205
$226$		0			0
$227$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$227$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$228$		0			0
$229$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$229$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$230$		0			0
$231$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.00000			−1.00000
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$		0			0
$243$	−1.93185	+	3.34607i	−1.93185	+	3.34607i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$	−0.732051			−0.732051
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$257$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−3.34607			−3.34607
$268$		0			0
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$		0			0
$273$	0.866025	−	3.23205i	0.866025	−	3.23205i
$274$		0			0
$275$	−0.258819	+	0.448288i	−0.258819	+	0.448288i
$276$		0			0
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$		0			0
$279$	−3.86370			−3.86370
$280$		0			0
$281$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$281$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$282$		0			0
$283$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.707107	+	0.707107i	$0.250000\pi$
$283$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.258819	+	0.965926i	$0.416667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$293$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$298$		0			0
$299$	1.22474	−	2.12132i	1.22474	−	2.12132i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.707107	+	0.707107i	$0.250000\pi$
$311$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.258819	+	0.965926i	$0.416667\pi$
$312$		0			0
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	3.73205			3.73205
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	0.366025	−	0.633975i	0.366025	−	0.633975i
$342$		0			0
$343$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$347$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$348$		0			0
$349$	−2.00000			−2.00000			−1.00000			$\pi$
$349$	−2.00000			−2.00000			−1.00000			$\pi$
$350$		0			0
$351$	5.79555			5.79555
$352$		0			0
$353$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$353$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	1.86603	−	0.500000i	1.86603	−	0.500000i
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$362$		0			0
$363$	−1.41421			−1.41421
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$367$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.258819	−	0.965926i	0.258819	−	0.965926i
$372$		0			0
$373$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$373$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.93185			1.93185			0.965926	−	0.258819i	$-0.0833333\pi$
$379$	1.93185			1.93185			0.965926	+	0.258819i	$0.0833333\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$389$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$390$		0			0
$391$	1.41421			1.41421
$392$		0			0
$393$	−2.73205			−2.73205
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	1.22474	+	2.12132i	1.22474	+	2.12132i
$404$		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	$-0.333333\pi$
$409$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$410$		0			0
$411$	1.67303	−	2.89778i	1.67303	−	2.89778i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$418$		0			0
$419$	−0.517638			−0.517638			−0.258819	−	0.965926i	$-0.583333\pi$
$419$	−0.517638			−0.517638			−0.258819	+	0.965926i	$0.583333\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.500000	−	0.866025i	0.500000	−	0.866025i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$430$		0			0
$431$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.707107	−	0.707107i	$-0.750000\pi$
$431$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.258819	−	0.965926i	$-0.583333\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	0.965926	−	1.67303i	0.965926	−	1.67303i	0.258819	−	0.965926i	$-0.416667\pi$
$439$	0.965926	−	1.67303i	0.965926	−	1.67303i	0.707107	−	0.707107i	$-0.250000\pi$
$440$		0			0
$441$	−2.36603	−	1.36603i	−2.36603	−	1.36603i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−1.93185			−1.93185
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$457$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$458$		0			0
$459$	1.67303	+	2.89778i	1.67303	+	2.89778i
$460$		0			0
$461$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$461$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	2.73205			2.73205
$478$		0			0
$479$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$479$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−1.93185	−	1.93185i	−1.93185	−	1.93185i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$487$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$488$		0			0
$489$	2.73205			2.73205
$490$		0			0
$491$		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
$497$	−1.36603	−	1.36603i	−1.36603	−	1.36603i
$498$		0			0
$499$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.258819	+	0.965926i	$0.583333\pi$
$499$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.707107	+	0.707107i	$0.750000\pi$
$500$		0			0
$501$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$502$		0			0
$503$	−1.93185			−1.93185			−0.965926	−	0.258819i	$-0.916667\pi$
$503$	−1.93185			−1.93185			−0.965926	+	0.258819i	$0.916667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.93185	−	3.34607i	−1.93185	−	3.34607i
$508$		0			0
$509$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$509$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$	−1.86603	+	0.500000i	−1.86603	+	0.500000i
$526$		0			0
$527$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$528$		0			0
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.448288	−	0.258819i	0.448288	−	0.258819i
$540$		0			0
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.517638			−0.517638			−0.258819	−	0.965926i	$-0.583333\pi$
$547$	−0.517638			−0.517638			−0.258819	+	0.965926i	$0.583333\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.500000	−	0.133975i	0.500000	−	0.133975i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$557$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−1.00000			−1.00000
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.965926	−	3.60488i	0.965926	−	3.60488i
$568$		0			0
$569$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$569$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$570$		0			0
$571$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$571$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.41421			−1.41421
$576$		0			0
$577$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$577$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.258819	+	0.448288i	−0.258819	+	0.448288i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$593$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.500000	−	0.866025i	0.500000	−	0.866025i
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	0.258819	−	0.448288i	0.258819	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$607$	0.258819	−	0.448288i	0.258819	−	0.448288i	0.965926	+	0.258819i	$0.0833333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$613$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.707107	+	0.707107i	$0.250000\pi$
$619$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.258819	+	0.965926i	$0.416667\pi$
$620$		0			0
$621$	2.36603	−	4.09808i	2.36603	−	4.09808i
$622$		0			0
$623$	1.67303	−	0.448288i	1.67303	−	0.448288i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	1.36603	+	2.36603i	1.36603	+	2.36603i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			1.73205i			1.73205i
$638$		0			0
$639$	2.63896	−	4.57081i	2.63896	−	4.57081i
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$	−0.517638			−0.517638			−0.258819	−	0.965926i	$-0.583333\pi$
$643$	−0.517638			−0.517638			−0.258819	+	0.965926i	$0.583333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	2.63896	−	0.707107i	2.63896	−	0.707107i
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$661$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$662$		0			0
$663$	1.67303	−	2.89778i	1.67303	−	2.89778i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−1.67303	−	2.89778i	−1.67303	−	2.89778i
$676$		0			0
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$682$		0			0
$683$	0.258819	+	0.448288i	0.258819	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$683$	0.258819	+	0.448288i	0.258819	+	0.448288i	−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$690$		0			0
$691$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$691$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$692$		0			0
$693$	1.00000	+	1.00000i	1.00000	+	1.00000i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$701$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$	0.707107	+	1.22474i	0.707107	+	1.22474i
$712$		0			0
$713$	2.00000			2.00000
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	−0.965926	−	0.258819i	$-0.916667\pi$
$719$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	0.707107	+	0.707107i	$0.250000\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	3.73205			3.73205
$730$		0			0
$731$		0			0
$732$		0			0
$733$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$733$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$739$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−0.517638			−0.517638			−0.258819	−	0.965926i	$-0.583333\pi$
$743$	−0.517638			−0.517638			−0.258819	+	0.965926i	$0.583333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.86603	+	0.500000i	−1.86603	+	0.500000i
$750$		0			0
$751$	0.965926	−	1.67303i	0.965926	−	1.67303i	0.258819	−	0.965926i	$-0.416667\pi$
$751$	0.965926	−	1.67303i	0.965926	−	1.67303i	0.707107	−	0.707107i	$-0.250000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$757$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$758$		0			0
$759$	0.707107	+	1.22474i	0.707107	+	1.22474i
$760$		0			0
$761$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$761$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$769$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$770$		0			0
$771$	3.34607			3.34607
$772$		0			0
$773$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$773$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$774$		0			0
$775$	0.707107	−	1.22474i	0.707107	−	1.22474i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	0.500000	+	0.866025i	0.500000	+	0.866025i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$787$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$797$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	2.36603	+	4.09808i	2.36603	+	4.09808i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$	−1.93185			−1.93185			−0.965926	−	0.258819i	$-0.916667\pi$
$811$	−1.93185			−1.93185			−0.965926	+	0.258819i	$0.916667\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−4.57081	+	1.22474i	−4.57081	+	1.22474i
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.707107	−	0.707107i	$-0.750000\pi$
$823$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.258819	−	0.965926i	$-0.583333\pi$
$824$		0			0
$825$	1.00000			1.00000
$826$		0			0
$827$	−0.517638			−0.517638			−0.258819	−	0.965926i	$-0.583333\pi$
$827$	−0.517638			−0.517638			−0.258819	+	0.965926i	$0.583333\pi$
$828$		0			0
$829$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$829$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	2.36603	+	4.09808i	2.36603	+	4.09808i
$838$		0			0
$839$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$839$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$		0			0
$843$	0.965926	+	1.67303i	0.965926	+	1.67303i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	0.707107	−	0.189469i	0.707107	−	0.189469i
$848$		0			0
$849$	1.86603	−	3.23205i	1.86603	−	3.23205i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$859$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	1.93185			1.93185
$868$		0			0
$869$	−0.267949			−0.267949
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$		0			0
$879$	0.965926	+	1.67303i	0.965926	+	1.67303i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.258819	−	0.448288i	0.258819	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$887$	0.258819	−	0.448288i	0.258819	−	0.448288i	0.965926	+	0.258819i	$0.0833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−4.73205			−4.73205
$898$		0			0
$899$		0			0
$900$		0			0
$901$	0.500000	−	0.866025i	0.500000	−	0.866025i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$907$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$911$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.36603	−	0.366025i	1.36603	−	0.366025i
$918$		0			0
$919$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$919$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−3.34607			−3.34607
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	1.86603	−	3.23205i	1.86603	−	3.23205i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−2.00000			−2.00000			−1.00000			$\pi$
$937$	−2.00000			−2.00000			−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.965926	−	1.67303i	0.965926	−	1.67303i	0.258819	−	0.965926i	$-0.416667\pi$
$947$	0.965926	−	1.67303i	0.965926	−	1.67303i	0.707107	−	0.707107i	$-0.250000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$953$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−0.448288	+	1.67303i	−0.448288	+	1.67303i
$960$		0			0
$961$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$962$		0			0
$963$	−2.63896	−	4.57081i	−2.63896	−	4.57081i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$		0			0
$973$	0.366025	+	0.366025i	0.366025	+	0.366025i
$974$		0			0
$975$	−1.67303	+	2.89778i	−1.67303	+	2.89778i
$976$		0			0
$977$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$977$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$	−0.896575			−0.896575
$980$		0			0
$981$		0			0
$982$		0			0
$983$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.707107	+	0.707107i	$0.250000\pi$
$983$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.258819	+	0.965926i	$0.416667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$991$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1904.1.bt.a.1087.1	yes	8
4.3	odd	2	inner	1904.1.bt.a.1087.4	yes	8
7.4	even	3	inner	1904.1.bt.a.543.1	✓	8
17.16	even	2	inner	1904.1.bt.a.1087.4	yes	8
28.11	odd	6	inner	1904.1.bt.a.543.4	yes	8
68.67	odd	2	CM	1904.1.bt.a.1087.1	yes	8
119.67	even	6	inner	1904.1.bt.a.543.4	yes	8
476.67	odd	6	inner	1904.1.bt.a.543.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1904.1.bt.a.543.1	✓	8	7.4	even	3	inner
1904.1.bt.a.543.1	✓	8	476.67	odd	6	inner
1904.1.bt.a.543.4	yes	8	28.11	odd	6	inner
1904.1.bt.a.543.4	yes	8	119.67	even	6	inner
1904.1.bt.a.1087.1	yes	8	1.1	even	1	trivial
1904.1.bt.a.1087.1	yes	8	68.67	odd	2	CM
1904.1.bt.a.1087.4	yes	8	4.3	odd	2	inner
1904.1.bt.a.1087.4	yes	8	17.16	even	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:	\( N \)	\(=\)	\( 1904 = 2^{4} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1904.bt (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.965926 - 1.67303i) q^{3} +(0.707107 + 0.707107i) q^{7} +(-1.36603 + 2.36603i) q^{9} +O(q^{10})\)	\(q+(-0.965926 - 1.67303i) q^{3} +(0.707107 + 0.707107i) q^{7} +(-1.36603 + 2.36603i) q^{9} +(-0.258819 - 0.448288i) q^{11} +1.73205 q^{13} +(0.500000 + 0.866025i) q^{17} +(0.500000 - 1.86603i) q^{21} +(0.707107 - 1.22474i) q^{23} +(-0.500000 - 0.866025i) q^{25} +3.34607 q^{27} +(0.707107 + 1.22474i) q^{31} +(-0.500000 + 0.866025i) q^{33} +(-1.67303 - 2.89778i) q^{39} +1.00000i q^{49} +(0.965926 - 1.67303i) q^{51} +(-0.500000 - 0.866025i) q^{53} +(-2.63896 + 0.707107i) q^{63} -2.73205 q^{69} -1.93185 q^{71} +(-0.965926 + 1.67303i) q^{75} +(0.133975 - 0.500000i) q^{77} +(0.258819 - 0.448288i) q^{79} +(-1.86603 - 3.23205i) q^{81} +(0.866025 - 1.50000i) q^{89} +(1.22474 + 1.22474i) q^{91} +(1.36603 - 2.36603i) q^{93} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{9}+O(q^{10}) \) 8 * q - 4 * q^9	\( 8 q - 4 q^{9} + 4 q^{17} + 4 q^{21} - 4 q^{25} - 4 q^{33} - 4 q^{53} - 8 q^{69} + 8 q^{77} - 8 q^{81} + 4 q^{93}+O(q^{100}) \) 8 * q - 4 * q^9 + 4 * q^17 + 4 * q^21 - 4 * q^25 - 4 * q^33 - 4 * q^53 - 8 * q^69 + 8 * q^77 - 8 * q^81 + 4 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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