LMFDB - Embedded newform 1575.1.cn.a.293.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,1,Mod(293,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([10, 9, 6]))

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

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

chi := DirichletCharacter("1575.293");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1575.cn (of order $12$, degree $4$, 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:	$0.786027394897$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.2.843908625.2

Embedding invariants

Embedding label			293.1
Root			$-0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1575.293
Dual form			1575.1.cn.a.1532.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{3} +(0.866025 + 0.500000i) q^{4} +(-0.258819 + 0.965926i) q^{7} -1.00000i q^{9} +(-1.50000 + 0.866025i) q^{11} +(-0.965926 + 0.258819i) q^{12} +(-0.258819 - 0.965926i) q^{13} +(0.500000 + 0.866025i) q^{16} +(-1.22474 + 1.22474i) q^{17} +(-0.500000 - 0.866025i) q^{21} +(0.707107 + 0.707107i) q^{27} +(-0.707107 + 0.707107i) q^{28} +(0.448288 - 1.67303i) q^{33} +(0.500000 - 0.866025i) q^{36} +(0.866025 + 0.500000i) q^{39} -1.73205 q^{44} +(1.67303 + 0.448288i) q^{47} +(-0.965926 - 0.258819i) q^{48} +(-0.866025 - 0.500000i) q^{49} -1.73205i q^{51} +(0.258819 - 0.965926i) q^{52} +(0.965926 + 0.258819i) q^{63} +1.00000i q^{64} +(-1.67303 + 0.448288i) q^{68} +1.73205i q^{71} +(0.707107 - 0.707107i) q^{73} +(-0.448288 - 1.67303i) q^{77} +(-0.866025 + 0.500000i) q^{79} -1.00000 q^{81} +(0.448288 - 1.67303i) q^{83} -1.00000i q^{84} +1.00000 q^{91} +(-0.258819 + 0.965926i) q^{97} +(0.866025 + 1.50000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{11} + 4 q^{16} - 4 q^{21} + 4 q^{36} - 8 q^{81} + 8 q^{91}+O(q^{100}) $ 8 * q - 12 * q^11 + 4 * q^16 - 4 * q^21 + 4 * q^36 - 8 * q^81 + 8 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$1226$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$e\left(\frac{5}{6}\right)$

Coefficient data

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

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

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

$2$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$2$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$3$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$3$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$4$	0.866025	+	0.500000i	0.866025	+	0.500000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$7$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$8$		0			0
$9$		−	1.00000i		−	1.00000i
$10$		0			0
$11$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$11$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$12$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$13$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	−0.965926	−	0.258819i	$-0.916667\pi$
$13$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	0.707107	−	0.707107i	$-0.250000\pi$
$14$		0			0
$15$		0			0
$16$	0.500000	+	0.866025i	0.500000	+	0.866025i
$17$	−1.22474	+	1.22474i	−1.22474	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$17$	−1.22474	+	1.22474i	−1.22474	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$22$		0			0
$23$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$23$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	0.707107	+	0.707107i	0.707107	+	0.707107i
$28$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$		0			0
$33$	0.448288	−	1.67303i	0.448288	−	1.67303i
$34$		0			0
$35$		0			0
$36$	0.500000	−	0.866025i	0.500000	−	0.866025i
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$	0.866025	+	0.500000i	0.866025	+	0.500000i
$40$		0			0
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$43$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$44$	−1.73205			−1.73205
$45$		0			0
$46$		0			0
$47$	1.67303	+	0.448288i	1.67303	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$47$	1.67303	+	0.448288i	1.67303	+	0.448288i	0.707107	+	0.707107i	$0.250000\pi$
$48$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$49$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$50$		0			0
$51$		−	1.73205i		−	1.73205i
$52$	0.258819	−	0.965926i	0.258819	−	0.965926i
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$	0.965926	+	0.258819i	0.965926	+	0.258819i
$64$			1.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$67$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$68$	−1.67303	+	0.448288i	−1.67303	+	0.448288i
$69$		0			0
$70$		0			0
$71$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$71$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$72$		0			0
$73$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$73$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.448288	−	1.67303i	−0.448288	−	1.67303i
$78$		0			0
$79$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$79$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$80$		0			0
$81$	−1.00000			−1.00000
$82$		0			0
$83$	0.448288	−	1.67303i	0.448288	−	1.67303i	−0.258819	−	0.965926i	$-0.583333\pi$
$83$	0.448288	−	1.67303i	0.448288	−	1.67303i	0.707107	−	0.707107i	$-0.250000\pi$
$84$		−	1.00000i		−	1.00000i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	1.00000			1.00000
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$97$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$98$		0			0
$99$	0.866025	+	1.50000i	0.866025	+	1.50000i
$100$		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
$103$	0.517638	+	1.93185i	0.517638	+	1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$103$	0.517638	+	1.93185i	0.517638	+	1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$	0.258819	+	0.965926i	0.258819	+	0.965926i
$109$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$109$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$110$		0			0
$111$		0			0
$112$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$113$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$113$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$118$		0			0
$119$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$120$		0			0
$121$	1.00000	−	1.73205i	1.00000	−	1.73205i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$127$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$	1.22474	−	1.22474i	1.22474	−	1.22474i
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$137$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$138$		0			0
$139$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$139$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$140$		0			0
$141$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$142$		0			0
$143$	1.22474	+	1.22474i	1.22474	+	1.22474i
$144$	0.866025	−	0.500000i	0.866025	−	0.500000i
$145$		0			0
$146$		0			0
$147$	0.965926	−	0.258819i	0.965926	−	0.258819i
$148$		0			0
$149$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$149$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$150$		0			0
$151$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$151$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$	1.22474	+	1.22474i	1.22474	+	1.22474i
$154$		0			0
$155$		0			0
$156$	0.500000	+	0.866025i	0.500000	+	0.866025i
$157$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.258819	−	0.965926i	$-0.416667\pi$
$157$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−0.448288	−	1.67303i	−0.448288	−	1.67303i	−0.707107	−	0.707107i	$-0.750000\pi$
$167$	−0.448288	−	1.67303i	−0.448288	−	1.67303i	0.258819	−	0.965926i	$-0.416667\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$173$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$174$		0			0
$175$		0			0
$176$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$177$		0			0
$178$		0			0
$179$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$179$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$180$		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
$187$	0.776457	−	2.89778i	0.776457	−	2.89778i
$188$	1.22474	+	1.22474i	1.22474	+	1.22474i
$189$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$193$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$193$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$194$		0			0
$195$		0			0
$196$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$	0.866025	−	1.50000i	0.866025	−	1.50000i
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.707107	−	0.707107i	0.707107	−	0.707107i
$209$		0			0
$210$		0			0
$211$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$211$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$212$		0			0
$213$	−1.22474	−	1.22474i	−1.22474	−	1.22474i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$			1.00000i			1.00000i
$220$		0			0
$221$	1.50000	+	0.866025i	1.50000	+	0.866025i
$222$		0			0
$223$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$223$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.67303	+	0.448288i	1.67303	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$227$	1.67303	+	0.448288i	1.67303	+	0.448288i	0.707107	+	0.707107i	$0.250000\pi$
$228$		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
$231$	1.50000	+	0.866025i	1.50000	+	0.866025i
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	0.258819	−	0.965926i	0.258819	−	0.965926i
$238$		0			0
$239$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$		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
$243$	0.707107	−	0.707107i	0.707107	−	0.707107i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	0.866025	+	1.50000i	0.866025	+	1.50000i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	0.707107	+	0.707107i	0.707107	+	0.707107i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$	0.448288	+	1.67303i	0.448288	+	1.67303i	0.707107	+	0.707107i	$0.250000\pi$
$257$	0.448288	+	1.67303i	0.448288	+	1.67303i	−0.258819	+	0.965926i	$0.583333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$263$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$264$		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			$\pi$
$272$	−1.67303	−	0.448288i	−1.67303	−	0.448288i
$273$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$277$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$281$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$282$		0			0
$283$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	−0.965926	−	0.258819i	$-0.916667\pi$
$283$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	0.707107	−	0.707107i	$-0.250000\pi$
$284$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	2.00000i		−	2.00000i
$290$		0			0
$291$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$292$	0.965926	−	0.258819i	0.965926	−	0.258819i
$293$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$293$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−1.67303	−	0.448288i	−1.67303	−	0.448288i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$307$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$308$	0.448288	−	1.67303i	0.448288	−	1.67303i
$309$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	1.93185	+	0.517638i	1.93185	+	0.517638i	0.965926	+	0.258819i	$0.0833333\pi$
$313$	1.93185	+	0.517638i	1.93185	+	0.517638i	0.965926	+	0.258819i	$0.0833333\pi$
$314$		0			0
$315$		0			0
$316$	−1.00000			−1.00000
$317$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$317$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$325$		0			0
$326$		0			0
$327$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$328$		0			0
$329$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$330$		0			0
$331$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$331$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$332$	1.22474	−	1.22474i	1.22474	−	1.22474i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	0.500000	−	0.866025i	0.500000	−	0.866025i
$337$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$337$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.707107	−	0.707107i	0.707107	−	0.707107i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$347$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$348$		0			0
$349$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$349$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$350$		0			0
$351$	0.500000	−	0.866025i	0.500000	−	0.866025i
$352$		0			0
$353$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$353$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	1.67303	+	0.448288i	1.67303	+	0.448288i
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−1.00000			−1.00000
$362$		0			0
$363$	0.517638	+	1.93185i	0.517638	+	1.93185i
$364$	0.866025	+	0.500000i	0.866025	+	0.500000i
$365$		0			0
$366$		0			0
$367$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$367$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$373$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$379$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.67303	+	0.448288i	−1.67303	+	0.448288i	−0.965926	−	0.258819i	$-0.916667\pi$
$383$	−1.67303	+	0.448288i	−1.67303	+	0.448288i	−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$389$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$389$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$			1.73205i			1.73205i
$397$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$397$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$398$		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
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$409$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$410$		0			0
$411$		0			0
$412$	−0.517638	+	1.93185i	−0.517638	+	1.93185i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$421$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$422$		0			0
$423$	0.448288	−	1.67303i	0.448288	−	1.67303i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−1.73205			−1.73205
$430$		0			0
$431$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$431$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$432$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$433$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$433$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$437$		0			0
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$442$		0			0
$443$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$443$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	0.448288	+	1.67303i	0.448288	+	1.67303i
$448$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$457$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$458$		0			0
$459$	−1.73205			−1.73205
$460$		0			0
$461$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$461$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$462$		0			0
$463$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$463$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.22474	+	1.22474i	−1.22474	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$467$	−1.22474	+	1.22474i	−1.22474	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$468$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$469$		0			0
$470$		0			0
$471$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		−	1.73205i		−	1.73205i
$477$		0			0
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	1.73205	−	1.00000i	1.73205	−	1.00000i
$485$		0			0
$486$		0			0
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$491$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.67303	−	0.448288i	−1.67303	−	0.448288i
$498$		0			0
$499$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$499$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$500$		0			0
$501$	1.50000	+	0.866025i	1.50000	+	0.866025i
$502$		0			0
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	0.500000	+	0.866025i	0.500000	+	0.866025i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−2.89778	+	0.776457i	−2.89778	+	0.776457i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$523$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	1.67303	−	0.448288i	1.67303	−	0.448288i
$529$	0.866025	−	0.500000i	0.866025	−	0.500000i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$538$		0			0
$539$	1.73205			1.73205
$540$		0			0
$541$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$541$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$547$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	1.50000	+	2.59808i	1.50000	+	2.59808i
$562$		0			0
$563$	−1.67303	+	0.448288i	−1.67303	+	0.448288i	−0.965926	−	0.258819i	$-0.916667\pi$
$563$	−1.67303	+	0.448288i	−1.67303	+	0.448288i	−0.707107	+	0.707107i	$0.750000\pi$
$564$	−1.73205			−1.73205
$565$		0			0
$566$		0			0
$567$	0.258819	−	0.965926i	0.258819	−	0.965926i
$568$		0			0
$569$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$569$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$570$		0			0
$571$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$571$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$572$	0.448288	+	1.67303i	0.448288	+	1.67303i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	1.00000			1.00000
$577$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$577$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.50000	+	0.866025i	1.50000	+	0.866025i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$587$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$588$	0.965926	+	0.258819i	0.965926	+	0.258819i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$	1.50000	−	0.866025i	1.50000	−	0.866025i
$597$		0			0
$598$		0			0
$599$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$599$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$600$		0			0
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$		0			0
$604$			1.00000i			1.00000i
$605$		0			0
$606$		0			0
$607$	1.93185	−	0.517638i	1.93185	−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$607$	1.93185	−	0.517638i	1.93185	−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	1.73205i		−	1.73205i
$612$	0.448288	+	1.67303i	0.448288	+	1.67303i
$613$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$613$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$617$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$618$		0			0
$619$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$619$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$			1.00000i			1.00000i
$625$		0			0
$626$		0			0
$627$		0			0
$628$	0.965926	+	0.258819i	0.965926	+	0.258819i
$629$		0			0
$630$		0			0
$631$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$631$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$632$		0			0
$633$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$638$		0			0
$639$	1.73205			1.73205
$640$		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
$643$	0.258819	+	0.965926i	0.258819	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$643$	0.258819	+	0.965926i	0.258819	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$653$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$658$		0			0
$659$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$659$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$	−1.67303	+	0.448288i	−1.67303	+	0.448288i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.448288	−	1.67303i	0.448288	−	1.67303i
$669$	0.866025	−	0.500000i	0.866025	−	0.500000i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$673$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.67303	+	0.448288i	1.67303	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$677$	1.67303	+	0.448288i	1.67303	+	0.448288i	0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$680$		0			0
$681$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$		0			0
$693$	−1.67303	+	0.448288i	−1.67303	+	0.448288i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$701$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$702$		0			0
$703$		0			0
$704$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$709$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$710$		0			0
$711$	0.500000	+	0.866025i	0.500000	+	0.866025i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	1.50000	+	0.866025i	1.50000	+	0.866025i
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$	−2.00000			−2.00000
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	0.258819	−	0.965926i	0.258819	−	0.965926i	−0.707107	−	0.707107i	$-0.750000\pi$
$727$	0.258819	−	0.965926i	0.258819	−	0.965926i	0.965926	−	0.258819i	$-0.0833333\pi$
$728$		0			0
$729$			1.00000i			1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−0.517638	−	1.93185i	−0.517638	−	1.93185i	−0.258819	−	0.965926i	$-0.583333\pi$
$733$	−0.517638	−	1.93185i	−0.517638	−	1.93185i	−0.258819	−	0.965926i	$-0.583333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$739$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$743$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.67303	−	0.448288i	−1.67303	−	0.448288i
$748$	2.12132	−	2.12132i	2.12132	−	2.12132i
$749$		0			0
$750$		0			0
$751$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$751$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$752$	0.448288	+	1.67303i	0.448288	+	1.67303i
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−1.00000			−1.00000
$757$		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
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$769$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$769$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$770$		0			0
$771$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$772$		0			0
$773$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$773$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$782$		0			0
$783$		0			0
$784$		−	1.00000i		−	1.00000i
$785$		0			0
$786$		0			0
$787$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.707107	−	0.707107i	$-0.750000\pi$
$787$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$788$		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
$797$	−0.448288	−	1.67303i	−0.448288	−	1.67303i	−0.707107	−	0.707107i	$-0.750000\pi$
$797$	−0.448288	−	1.67303i	−0.448288	−	1.67303i	0.258819	−	0.965926i	$-0.416667\pi$
$798$		0			0
$799$	−2.59808	+	1.50000i	−2.59808	+	1.50000i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.448288	+	1.67303i	−0.448288	+	1.67303i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$809$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$810$		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
$816$	1.50000	−	0.866025i	1.50000	−	0.866025i
$817$		0			0
$818$		0			0
$819$		−	1.00000i		−	1.00000i
$820$		0			0
$821$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$821$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$822$		0			0
$823$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$823$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$	0.965926	−	0.258819i	0.965926	−	0.258819i
$833$	1.67303	−	0.448288i	1.67303	−	0.448288i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	0.500000	−	0.866025i	0.500000	−	0.866025i
$842$		0			0
$843$	−0.448288	+	1.67303i	−0.448288	+	1.67303i
$844$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$845$		0			0
$846$		0			0
$847$	1.41421	+	1.41421i	1.41421	+	1.41421i
$848$		0			0
$849$	0.866025	+	0.500000i	0.866025	+	0.500000i
$850$		0			0
$851$		0			0
$852$	−0.448288	−	1.67303i	−0.448288	−	1.67303i
$853$	−1.93185	−	0.517638i	−1.93185	−	0.517638i	−0.965926	−	0.258819i	$-0.916667\pi$
$853$	−1.93185	−	0.517638i	−1.93185	−	0.517638i	−0.965926	−	0.258819i	$-0.916667\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.67303	−	0.448288i	−1.67303	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$857$	−1.67303	−	0.448288i	−1.67303	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$858$		0			0
$859$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$859$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	1.41421	+	1.41421i	1.41421	+	1.41421i
$868$		0			0
$869$	0.866025	−	1.50000i	0.866025	−	1.50000i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.965926	+	0.258819i	0.965926	+	0.258819i
$874$		0			0
$875$		0			0
$876$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$877$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$877$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$	0.866025	+	1.50000i	0.866025	+	1.50000i
$885$		0			0
$886$		0			0
$887$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$887$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1.50000	−	0.866025i	1.50000	−	0.866025i
$892$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$907$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$908$	1.22474	+	1.22474i	1.22474	+	1.22474i
$909$		0			0
$910$		0			0
$911$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$911$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$912$		0			0
$913$	0.776457	+	2.89778i	0.776457	+	2.89778i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$919$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$920$		0			0
$921$	−1.00000			−1.00000
$922$		0			0
$923$	1.67303	−	0.448288i	1.67303	−	0.448288i
$924$	0.866025	+	1.50000i	0.866025	+	1.50000i
$925$		0			0
$926$		0			0
$927$	1.93185	−	0.517638i	1.93185	−	0.517638i
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$937$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$938$		0			0
$939$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$947$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$948$	0.707107	−	0.707107i	0.707107	−	0.707107i
$949$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$953$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$967$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	0.965926	−	0.258819i	0.965926	−	0.258819i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$977$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	1.00000			1.00000
$982$		0			0
$983$	−0.448288	+	1.67303i	−0.448288	+	1.67303i	0.258819	+	0.965926i	$0.416667\pi$
$983$	−0.448288	+	1.67303i	−0.448288	+	1.67303i	−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−0.448288	−	1.67303i	−0.448288	−	1.67303i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$991$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$	0.965926	+	0.258819i	0.965926	+	0.258819i
$994$		0			0
$995$		0			0
$996$			1.73205i			1.73205i
$997$	0.517638	−	1.93185i	0.517638	−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$997$	0.517638	−	1.93185i	0.517638	−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.1.cn.a.293.1	✓	8
5.2	odd	4	inner	1575.1.cn.a.482.2	yes	8
5.3	odd	4	inner	1575.1.cn.a.482.1	yes	8
5.4	even	2	inner	1575.1.cn.a.293.2	yes	8
7.6	odd	2	inner	1575.1.cn.a.293.2	yes	8
9.2	odd	6	inner	1575.1.cn.a.1343.1	yes	8
35.13	even	4	inner	1575.1.cn.a.482.2	yes	8
35.27	even	4	inner	1575.1.cn.a.482.1	yes	8
35.34	odd	2	CM	1575.1.cn.a.293.1	✓	8
45.2	even	12	inner	1575.1.cn.a.1532.2	yes	8
45.29	odd	6	inner	1575.1.cn.a.1343.2	yes	8
45.38	even	12	inner	1575.1.cn.a.1532.1	yes	8
63.20	even	6	inner	1575.1.cn.a.1343.2	yes	8
315.83	odd	12	inner	1575.1.cn.a.1532.2	yes	8
315.209	even	6	inner	1575.1.cn.a.1343.1	yes	8
315.272	odd	12	inner	1575.1.cn.a.1532.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1575.1.cn.a.293.1	✓	8	1.1	even	1	trivial
1575.1.cn.a.293.1	✓	8	35.34	odd	2	CM
1575.1.cn.a.293.2	yes	8	5.4	even	2	inner
1575.1.cn.a.293.2	yes	8	7.6	odd	2	inner
1575.1.cn.a.482.1	yes	8	5.3	odd	4	inner
1575.1.cn.a.482.1	yes	8	35.27	even	4	inner
1575.1.cn.a.482.2	yes	8	5.2	odd	4	inner
1575.1.cn.a.482.2	yes	8	35.13	even	4	inner
1575.1.cn.a.1343.1	yes	8	9.2	odd	6	inner
1575.1.cn.a.1343.1	yes	8	315.209	even	6	inner
1575.1.cn.a.1343.2	yes	8	45.29	odd	6	inner
1575.1.cn.a.1343.2	yes	8	63.20	even	6	inner
1575.1.cn.a.1532.1	yes	8	45.38	even	12	inner
1575.1.cn.a.1532.1	yes	8	315.272	odd	12	inner
1575.1.cn.a.1532.2	yes	8	45.2	even	12	inner
1575.1.cn.a.1532.2	yes	8	315.83	odd	12	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1575.cn (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{3} +(0.866025 + 0.500000i) q^{4} +(-0.258819 + 0.965926i) q^{7} -1.00000i q^{9} +(-1.50000 + 0.866025i) q^{11} +(-0.965926 + 0.258819i) q^{12} +(-0.258819 - 0.965926i) q^{13} +(0.500000 + 0.866025i) q^{16} +(-1.22474 + 1.22474i) q^{17} +(-0.500000 - 0.866025i) q^{21} +(0.707107 + 0.707107i) q^{27} +(-0.707107 + 0.707107i) q^{28} +(0.448288 - 1.67303i) q^{33} +(0.500000 - 0.866025i) q^{36} +(0.866025 + 0.500000i) q^{39} -1.73205 q^{44} +(1.67303 + 0.448288i) q^{47} +(-0.965926 - 0.258819i) q^{48} +(-0.866025 - 0.500000i) q^{49} -1.73205i q^{51} +(0.258819 - 0.965926i) q^{52} +(0.965926 + 0.258819i) q^{63} +1.00000i q^{64} +(-1.67303 + 0.448288i) q^{68} +1.73205i q^{71} +(0.707107 - 0.707107i) q^{73} +(-0.448288 - 1.67303i) q^{77} +(-0.866025 + 0.500000i) q^{79} -1.00000 q^{81} +(0.448288 - 1.67303i) q^{83} -1.00000i q^{84} +1.00000 q^{91} +(-0.258819 + 0.965926i) q^{97} +(0.866025 + 1.50000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{11} + 4 q^{16} - 4 q^{21} + 4 q^{36} - 8 q^{81} + 8 q^{91}+O(q^{100}) \) 8 * q - 12 * q^11 + 4 * q^16 - 4 * q^21 + 4 * q^36 - 8 * q^81 + 8 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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