LMFDB - Embedded newform 2592.1.bz.a.175.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,1,Mod(79,2592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2592, base_ring=CyclotomicField(54))

chi = DirichletCharacter(H, H._module([27, 27, 28]))

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

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

chi := DirichletCharacter("2592.79");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2592.bz (of order $54$, degree $18$, not 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:	$1.29357651274$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\Q(\zeta_{54})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{9} + 1 $ x^18 - x^9 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 648)
Projective image:	$D_{27}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{27} + \cdots)$

Embedding invariants

Embedding label			175.1
Root			$0.686242 - 0.727374i$ of defining polynomial
Character	$\chi$	$=$	2592.175
Dual form			2592.1.bz.a.1807.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.686242 + 0.727374i) q^{3} +(-0.0581448 + 0.998308i) q^{9} +O(q^{10})$	$q+(0.686242 + 0.727374i) q^{3} +(-0.0581448 + 0.998308i) q^{9} +(-1.16212 + 0.275428i) q^{11} +(1.57020 - 0.571507i) q^{17} +(1.82873 + 0.665602i) q^{19} +(-0.835488 + 0.549509i) q^{25} +(-0.766044 + 0.642788i) q^{27} +(-0.997837 - 0.656288i) q^{33} +(-1.22650 + 0.615969i) q^{41} +(1.22650 + 1.30001i) q^{43} +(-0.686242 + 0.727374i) q^{49} +(1.49324 + 0.749932i) q^{51} +(0.770807 + 1.78693i) q^{57} +(0.558145 + 0.132283i) q^{59} +(0.0460600 - 0.790819i) q^{67} +(-0.0201935 - 0.114523i) q^{73} +(-0.973045 - 0.230616i) q^{75} +(-0.993238 - 0.116093i) q^{81} +(-1.36912 - 0.687600i) q^{83} +(-0.326352 - 1.85083i) q^{89} +(0.569728 - 1.90302i) q^{97} +(-0.207391 - 1.17617i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q+O(q^{10}) $ 18 * q	$ 18 q + 9 q^{51} + 9 q^{59} - 9 q^{89}+O(q^{100}) $ 18 * q + 9 * q^51 + 9 * q^59 - 9 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$-1$	$e\left(\frac{4}{27}\right)$	$-1$

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
$3$	0.686242	+	0.727374i	0.686242	+	0.727374i
$3$	0.686242	+	0.727374i	0.686242	+	0.727374i
$4$		0			0
$5$		0			0		−0.286803	−	0.957990i	$-0.592593\pi$
$5$		0			0		0.286803	+	0.957990i	$0.407407\pi$
$6$		0			0
$7$		0			0		−0.396080	−	0.918216i	$-0.629630\pi$
$7$		0			0		0.396080	+	0.918216i	$0.370370\pi$
$8$		0			0
$9$	−0.0581448	+	0.998308i	−0.0581448	+	0.998308i
$10$		0			0
$11$	−1.16212	+	0.275428i	−1.16212	+	0.275428i	−0.766044	−	0.642788i	$-0.777778\pi$
$11$	−1.16212	+	0.275428i	−1.16212	+	0.275428i	−0.396080	+	0.918216i	$0.629630\pi$
$12$		0			0
$13$		0			0		−0.835488	−	0.549509i	$-0.814815\pi$
$13$		0			0		0.835488	+	0.549509i	$0.185185\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.57020	−	0.571507i	1.57020	−	0.571507i	0.597159	−	0.802123i	$-0.296296\pi$
$17$	1.57020	−	0.571507i	1.57020	−	0.571507i	0.973045	+	0.230616i	$0.0740741\pi$
$18$		0			0
$19$	1.82873	+	0.665602i	1.82873	+	0.665602i	0.993238	+	0.116093i	$0.0370370\pi$
$19$	1.82873	+	0.665602i	1.82873	+	0.665602i	0.835488	+	0.549509i	$0.185185\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.396080	−	0.918216i	$-0.370370\pi$
$23$		0			0		−0.396080	+	0.918216i	$0.629630\pi$
$24$		0			0
$25$	−0.835488	+	0.549509i	−0.835488	+	0.549509i
$26$		0			0
$27$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$28$		0			0
$29$		0			0		−0.0581448	−	0.998308i	$-0.518519\pi$
$29$		0			0		0.0581448	+	0.998308i	$0.481481\pi$
$30$		0			0
$31$		0			0		0.993238	−	0.116093i	$-0.0370370\pi$
$31$		0			0		−0.993238	+	0.116093i	$0.962963\pi$
$32$		0			0
$33$	−0.997837	−	0.656288i	−0.997837	−	0.656288i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$37$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.22650	+	0.615969i	−1.22650	+	0.615969i	−0.939693	−	0.342020i	$-0.888889\pi$
$41$	−1.22650	+	0.615969i	−1.22650	+	0.615969i	−0.286803	+	0.957990i	$0.592593\pi$
$42$		0			0
$43$	1.22650	+	1.30001i	1.22650	+	1.30001i	0.939693	+	0.342020i	$0.111111\pi$
$43$	1.22650	+	1.30001i	1.22650	+	1.30001i	0.286803	+	0.957990i	$0.407407\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.993238	−	0.116093i	$-0.962963\pi$
$47$		0			0		0.993238	+	0.116093i	$0.0370370\pi$
$48$		0			0
$49$	−0.686242	+	0.727374i	−0.686242	+	0.727374i
$50$		0			0
$51$	1.49324	+	0.749932i	1.49324	+	0.749932i
$52$		0			0
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.770807	+	1.78693i	0.770807	+	1.78693i
$58$		0			0
$59$	0.558145	+	0.132283i	0.558145	+	0.132283i	0.500000	−	0.866025i	$-0.333333\pi$
$59$	0.558145	+	0.132283i	0.558145	+	0.132283i	0.0581448	+	0.998308i	$0.481481\pi$
$60$		0			0
$61$		0			0		0.597159	−	0.802123i	$-0.296296\pi$
$61$		0			0		−0.597159	+	0.802123i	$0.703704\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	0.0460600	−	0.790819i	0.0460600	−	0.790819i	−0.893633	−	0.448799i	$-0.851852\pi$
$67$	0.0460600	−	0.790819i	0.0460600	−	0.790819i	0.939693	−	0.342020i	$-0.111111\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$71$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$72$		0			0
$73$	−0.0201935	−	0.114523i	−0.0201935	−	0.114523i	0.973045	−	0.230616i	$-0.0740741\pi$
$73$	−0.0201935	−	0.114523i	−0.0201935	−	0.114523i	−0.993238	+	0.116093i	$0.962963\pi$
$74$		0			0
$75$	−0.973045	−	0.230616i	−0.973045	−	0.230616i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.893633	−	0.448799i	$-0.851852\pi$
$79$		0			0		0.893633	+	0.448799i	$0.148148\pi$
$80$		0			0
$81$	−0.993238	−	0.116093i	−0.993238	−	0.116093i
$82$		0			0
$83$	−1.36912	−	0.687600i	−1.36912	−	0.687600i	−0.396080	−	0.918216i	$-0.629630\pi$
$83$	−1.36912	−	0.687600i	−1.36912	−	0.687600i	−0.973045	+	0.230616i	$0.925926\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.326352	−	1.85083i	−0.326352	−	1.85083i	−0.500000	−	0.866025i	$-0.666667\pi$
$89$	−0.326352	−	1.85083i	−0.326352	−	1.85083i	0.173648	−	0.984808i	$-0.444444\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.569728	−	1.90302i	0.569728	−	1.90302i	0.173648	−	0.984808i	$-0.444444\pi$
$97$	0.569728	−	1.90302i	0.569728	−	1.90302i	0.396080	−	0.918216i	$-0.370370\pi$
$98$		0			0
$99$	−0.207391	−	1.17617i	−0.207391	−	1.17617i
$100$		0			0
$101$		0			0		0.597159	−	0.802123i	$-0.296296\pi$
$101$		0			0		−0.597159	+	0.802123i	$0.703704\pi$
$102$		0			0
$103$		0			0		−0.973045	−	0.230616i	$-0.925926\pi$
$103$		0			0		0.973045	+	0.230616i	$0.0740741\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.0581448	−	0.100710i	−0.0581448	−	0.100710i	0.835488	−	0.549509i	$-0.185185\pi$
$107$	−0.0581448	−	0.100710i	−0.0581448	−	0.100710i	−0.893633	+	0.448799i	$0.851852\pi$
$108$		0			0
$109$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$109$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.238329	+	0.252614i	−0.238329	+	0.252614i	−0.835488	−	0.549509i	$-0.814815\pi$
$113$	−0.238329	+	0.252614i	−0.238329	+	0.252614i	0.597159	+	0.802123i	$0.296296\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.381039	−	0.191365i	0.381039	−	0.191365i
$122$		0			0
$123$	−1.28971	−	0.469417i	−1.28971	−	0.469417i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$127$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$128$		0			0
$129$	−0.103920	+	1.78424i	−0.103920	+	1.78424i
$130$		0			0
$131$	0.344948	−	0.0403186i	0.344948	−	0.0403186i	0.0581448	−	0.998308i	$-0.481481\pi$
$131$	0.344948	−	0.0403186i	0.344948	−	0.0403186i	0.286803	+	0.957990i	$0.407407\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.65968	−	1.09159i	1.65968	−	1.09159i	0.766044	−	0.642788i	$-0.222222\pi$
$137$	1.65968	−	1.09159i	1.65968	−	1.09159i	0.893633	−	0.448799i	$-0.148148\pi$
$138$		0			0
$139$	0.786803	−	1.82401i	0.786803	−	1.82401i	0.286803	−	0.957990i	$-0.407407\pi$
$139$	0.786803	−	1.82401i	0.786803	−	1.82401i	0.500000	−	0.866025i	$-0.333333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−1.00000			−1.00000
$148$		0			0
$149$		0			0		−0.835488	−	0.549509i	$-0.814815\pi$
$149$		0			0		0.835488	+	0.549509i	$0.185185\pi$
$150$		0			0
$151$		0			0		0.973045	−	0.230616i	$-0.0740741\pi$
$151$		0			0		−0.973045	+	0.230616i	$0.925926\pi$
$152$		0			0
$153$	0.479241	+	1.60078i	0.479241	+	1.60078i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.286803	−	0.957990i	$-0.592593\pi$
$157$		0			0		0.286803	+	0.957990i	$0.407407\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.347296			−0.347296			−0.173648	−	0.984808i	$-0.555556\pi$
$163$	−0.347296			−0.347296			−0.173648	+	0.984808i	$0.555556\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.286803	−	0.957990i	$-0.592593\pi$
$167$		0			0		0.286803	+	0.957990i	$0.407407\pi$
$168$		0			0
$169$	0.396080	+	0.918216i	0.396080	+	0.918216i
$170$		0			0
$171$	−0.770807	+	1.78693i	−0.770807	+	1.78693i
$172$		0			0
$173$		0			0		0.973045	−	0.230616i	$-0.0740741\pi$
$173$		0			0		−0.973045	+	0.230616i	$0.925926\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	0.286803	+	0.496758i	0.286803	+	0.496758i
$178$		0			0
$179$	−1.76604	+	0.642788i	−1.76604	+	0.642788i	−0.766044	+	0.642788i	$0.777778\pi$
$179$	−1.76604	+	0.642788i	−1.76604	+	0.642788i	−1.00000			$1.00000\pi$
$180$		0			0
$181$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$181$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−1.66736	+	1.09664i	−1.66736	+	1.09664i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.0581448	−	0.998308i	$-0.518519\pi$
$191$		0			0		0.0581448	+	0.998308i	$0.481481\pi$
$192$		0			0
$193$	1.36320	−	0.159336i	1.36320	−	0.159336i	0.597159	−	0.802123i	$-0.296296\pi$
$193$	1.36320	−	0.159336i	1.36320	−	0.159336i	0.766044	+	0.642788i	$0.222222\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$197$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$198$		0			0
$199$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$199$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$200$		0			0
$201$	0.606829	−	0.509190i	0.606829	−	0.509190i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.30853	−	0.269829i	−2.30853	−	0.269829i
$210$		0			0
$211$	1.05138	−	1.11440i	1.05138	−	1.11440i	0.0581448	−	0.998308i	$-0.481481\pi$
$211$	1.05138	−	1.11440i	1.05138	−	1.11440i	0.993238	−	0.116093i	$-0.0370370\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	0.0694434	−	0.0932786i	0.0694434	−	0.0932786i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.597159	−	0.802123i	$-0.296296\pi$
$223$		0			0		−0.597159	+	0.802123i	$0.703704\pi$
$224$		0			0
$225$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$226$		0			0
$227$	0.512593	−	1.71218i	0.512593	−	1.71218i	−0.173648	−	0.984808i	$-0.555556\pi$
$227$	0.512593	−	1.71218i	0.512593	−	1.71218i	0.686242	−	0.727374i	$-0.259259\pi$
$228$		0			0
$229$		0			0		0.0581448	−	0.998308i	$-0.481481\pi$
$229$		0			0		−0.0581448	+	0.998308i	$0.518519\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.0996057	+	0.564892i	−0.0996057	+	0.564892i	0.893633	+	0.448799i	$0.148148\pi$
$233$	−0.0996057	+	0.564892i	−0.0996057	+	0.564892i	−0.993238	+	0.116093i	$0.962963\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.597159	−	0.802123i	$-0.703704\pi$
$239$		0			0		0.597159	+	0.802123i	$0.296296\pi$
$240$		0			0
$241$	0.707900	+	0.355521i	0.707900	+	0.355521i	0.766044	−	0.642788i	$-0.222222\pi$
$241$	0.707900	+	0.355521i	0.707900	+	0.355521i	−0.0581448	+	0.998308i	$0.518519\pi$
$242$		0			0
$243$	−0.597159	−	0.802123i	−0.597159	−	0.802123i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−0.439408	−	1.46773i	−0.439408	−	1.46773i
$250$		0			0
$251$	−0.310355	−	1.76011i	−0.310355	−	1.76011i	−0.597159	−	0.802123i	$-0.703704\pi$
$251$	−0.310355	−	1.76011i	−0.310355	−	1.76011i	0.286803	−	0.957990i	$-0.407407\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−0.0460600	+	0.790819i	−0.0460600	+	0.790819i	0.893633	+	0.448799i	$0.148148\pi$
$257$	−0.0460600	+	0.790819i	−0.0460600	+	0.790819i	−0.939693	+	0.342020i	$0.888889\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.597159	−	0.802123i	$-0.296296\pi$
$263$		0			0		−0.597159	+	0.802123i	$0.703704\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	1.12229	−	1.50750i	1.12229	−	1.50750i
$268$		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			0
$274$		0			0
$275$	0.819590	−	0.868715i	0.819590	−	0.868715i
$276$		0			0
$277$		0			0		−0.993238	−	0.116093i	$-0.962963\pi$
$277$		0			0		0.993238	+	0.116093i	$0.0370370\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.05138	−	1.11440i	−1.05138	−	1.11440i	−0.993238	−	0.116093i	$-0.962963\pi$
$281$	−1.05138	−	1.11440i	−1.05138	−	1.11440i	−0.0581448	−	0.998308i	$-0.518519\pi$
$282$		0			0
$283$	−1.36912	+	0.687600i	−1.36912	+	0.687600i	−0.973045	−	0.230616i	$-0.925926\pi$
$283$	−1.36912	+	0.687600i	−1.36912	+	0.687600i	−0.396080	+	0.918216i	$0.629630\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	1.37287	−	1.15198i	1.37287	−	1.15198i
$290$		0			0
$291$	1.77518	−	0.891529i	1.77518	−	0.891529i
$292$		0			0
$293$		0			0		0.993238	−	0.116093i	$-0.0370370\pi$
$293$		0			0		−0.993238	+	0.116093i	$0.962963\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	0.713197	−	0.957990i	0.713197	−	0.957990i
$298$		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
$307$	−0.539014	+	0.196185i	−0.539014	+	0.196185i	−0.597159	−	0.802123i	$-0.703704\pi$
$307$	−0.539014	+	0.196185i	−0.539014	+	0.196185i	0.0581448	+	0.998308i	$0.481481\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.835488	−	0.549509i	$-0.814815\pi$
$311$		0			0		0.835488	+	0.549509i	$0.185185\pi$
$312$		0			0
$313$	−1.33549	+	0.316516i	−1.33549	+	0.316516i	−0.835488	−	0.549509i	$-0.814815\pi$
$313$	−1.33549	+	0.316516i	−1.33549	+	0.316516i	−0.500000	+	0.866025i	$0.666667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.396080	−	0.918216i	$-0.629630\pi$
$317$		0			0		0.396080	+	0.918216i	$0.370370\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	0.0333522	−	0.111404i	0.0333522	−	0.111404i
$322$		0			0
$323$	3.25187			3.25187
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.744386	+	1.72568i	0.744386	+	1.72568i	0.686242	+	0.727374i	$0.259259\pi$
$331$	0.744386	+	1.72568i	0.744386	+	1.72568i	0.0581448	+	0.998308i	$0.481481\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.49324	−	0.982118i	−1.49324	−	0.982118i	−0.993238	−	0.116093i	$-0.962963\pi$
$337$	−1.49324	−	0.982118i	−1.49324	−	0.982118i	−0.500000	−	0.866025i	$-0.666667\pi$
$338$		0			0
$339$	−0.347296			−0.347296
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.770807	+	1.78693i	−0.770807	+	1.78693i	−0.173648	+	0.984808i	$0.555556\pi$
$347$	−0.770807	+	1.78693i	−0.770807	+	1.78693i	−0.597159	+	0.802123i	$0.703704\pi$
$348$		0			0
$349$		0			0		0.835488	−	0.549509i	$-0.185185\pi$
$349$		0			0		−0.835488	+	0.549509i	$0.814815\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.00676164	+	0.116093i	0.00676164	+	0.116093i	1.00000			$0$
$353$	0.00676164	+	0.116093i	0.00676164	+	0.116093i	−0.993238	+	0.116093i	$0.962963\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$359$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$360$		0			0
$361$	2.13517	+	1.79162i	2.13517	+	1.79162i
$362$		0			0
$363$	0.400679	+	0.145835i	0.400679	+	0.145835i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.686242	−	0.727374i	$-0.740741\pi$
$367$		0			0		0.686242	+	0.727374i	$0.259259\pi$
$368$		0			0
$369$	−0.543613	−	1.26024i	−0.543613	−	1.26024i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.686242	−	0.727374i	$-0.259259\pi$
$373$		0			0		−0.686242	+	0.727374i	$0.740741\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.686242	−	1.18861i	−0.686242	−	1.18861i	−0.973045	−	0.230616i	$-0.925926\pi$
$379$	−0.686242	−	1.18861i	−0.686242	−	1.18861i	0.286803	−	0.957990i	$-0.407407\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.973045	−	0.230616i	$-0.925926\pi$
$383$		0			0		0.973045	+	0.230616i	$0.0740741\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.36912	+	1.14883i	−1.36912	+	1.14883i
$388$		0			0
$389$		0			0		0.286803	−	0.957990i	$-0.407407\pi$
$389$		0			0		−0.286803	+	0.957990i	$0.592593\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	0.266044	+	0.223238i	0.266044	+	0.223238i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$397$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.16212	+	1.56100i	1.16212	+	1.56100i	0.766044	+	0.642788i	$0.222222\pi$
$401$	1.16212	+	1.56100i	1.16212	+	1.56100i	0.396080	+	0.918216i	$0.370370\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.997837	−	1.34033i	−0.997837	−	1.34033i	−0.939693	−	0.342020i	$-0.888889\pi$
$409$	−0.997837	−	1.34033i	−0.997837	−	1.34033i	−0.0581448	−	0.998308i	$-0.518519\pi$
$410$		0			0
$411$	1.93293	+	0.458113i	1.93293	+	0.458113i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.86668	−	0.679415i	1.86668	−	0.679415i
$418$		0			0
$419$	0.0201935	−	0.346709i	0.0201935	−	0.346709i	−0.973045	−	0.230616i	$-0.925926\pi$
$419$	0.0201935	−	0.346709i	0.0201935	−	0.346709i	0.993238	−	0.116093i	$-0.0370370\pi$
$420$		0			0
$421$		0			0		0.286803	−	0.957990i	$-0.407407\pi$
$421$		0			0		−0.286803	+	0.957990i	$0.592593\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.997837	+	1.34033i	−0.997837	+	1.34033i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$		0			0
$433$	0.835488	−	1.44711i	0.835488	−	1.44711i	−0.0581448	−	0.998308i	$-0.518519\pi$
$433$	0.835488	−	1.44711i	0.835488	−	1.44711i	0.893633	−	0.448799i	$-0.148148\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.993238	−	0.116093i	$-0.962963\pi$
$439$		0			0		0.993238	+	0.116093i	$0.0370370\pi$
$440$		0			0
$441$	−0.686242	−	0.727374i	−0.686242	−	0.727374i
$442$		0			0
$443$	−0.941855	−	0.998308i	−0.941855	−	0.998308i	0.0581448	−	0.998308i	$-0.481481\pi$
$443$	−0.941855	−	0.998308i	−0.941855	−	0.998308i	−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.52173	−	1.27688i	−1.52173	−	1.27688i	−0.835488	−	0.549509i	$-0.814815\pi$
$449$	−1.52173	−	1.27688i	−1.52173	−	1.27688i	−0.686242	−	0.727374i	$-0.740741\pi$
$450$		0			0
$451$	1.25569	−	1.05364i	1.25569	−	1.05364i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.0333522	+	0.572636i	0.0333522	+	0.572636i	0.973045	+	0.230616i	$0.0740741\pi$
$457$	0.0333522	+	0.572636i	0.0333522	+	0.572636i	−0.939693	+	0.342020i	$0.888889\pi$
$458$		0			0
$459$	−0.835488	+	1.44711i	−0.835488	+	1.44711i
$460$		0			0
$461$		0			0		0.835488	−	0.549509i	$-0.185185\pi$
$461$		0			0		−0.835488	+	0.549509i	$0.814815\pi$
$462$		0			0
$463$		0			0		0.396080	−	0.918216i	$-0.370370\pi$
$463$		0			0		−0.396080	+	0.918216i	$0.629630\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.82873	+	0.665602i	1.82873	+	0.665602i	0.993238	+	0.116093i	$0.0370370\pi$
$467$	1.82873	+	0.665602i	1.82873	+	0.665602i	0.835488	+	0.549509i	$0.185185\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.78340	−	1.17296i	−1.78340	−	1.17296i
$474$		0			0
$475$	−1.89363	+	0.448799i	−1.89363	+	0.448799i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.396080	−	0.918216i	$-0.629630\pi$
$479$		0			0		0.396080	+	0.918216i	$0.370370\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	−0.238329	−	0.252614i	−0.238329	−	0.252614i
$490$		0			0
$491$	−0.0333522	−	0.111404i	−0.0333522	−	0.111404i	0.939693	−	0.342020i	$-0.111111\pi$
$491$	−0.0333522	−	0.111404i	−0.0333522	−	0.111404i	−0.973045	+	0.230616i	$0.925926\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−1.39608	−	0.918216i	−1.39608	−	0.918216i	−0.396080	−	0.918216i	$-0.629630\pi$
$499$	−1.39608	−	0.918216i	−1.39608	−	0.918216i	−1.00000			$\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$503$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.396080	+	0.918216i	−0.396080	+	0.918216i
$508$		0			0
$509$		0			0		0.396080	−	0.918216i	$-0.370370\pi$
$509$		0			0		−0.396080	+	0.918216i	$0.629630\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−1.82873	+	0.665602i	−1.82873	+	0.665602i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.36912	−	1.14883i	1.36912	−	1.14883i	0.396080	−	0.918216i	$-0.370370\pi$
$521$	1.36912	−	1.14883i	1.36912	−	1.14883i	0.973045	−	0.230616i	$-0.0740741\pi$
$522$		0			0
$523$	−0.266044	−	0.223238i	−0.266044	−	0.223238i	0.500000	−	0.866025i	$-0.333333\pi$
$523$	−0.266044	−	0.223238i	−0.266044	−	0.223238i	−0.766044	+	0.642788i	$0.777778\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.686242	−	0.727374i	−0.686242	−	0.727374i
$530$		0			0
$531$	−0.164512	+	0.549509i	−0.164512	+	0.549509i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−1.67948	−	0.843467i	−1.67948	−	0.843467i
$538$		0			0
$539$	0.597159	−	1.03431i	0.597159	−	1.03431i
$540$		0			0
$541$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$541$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.997837	−	1.34033i	0.997837	−	1.34033i	0.0581448	−	0.998308i	$-0.481481\pi$
$547$	0.997837	−	1.34033i	0.997837	−	1.34033i	0.939693	−	0.342020i	$-0.111111\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$557$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−1.94188	−	0.460234i	−1.94188	−	0.460234i
$562$		0			0
$563$	1.18624	+	1.59340i	1.18624	+	1.59340i	0.686242	+	0.727374i	$0.259259\pi$
$563$	1.18624	+	1.59340i	1.18624	+	1.59340i	0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.512593	−	0.257434i	−0.512593	−	0.257434i	0.173648	−	0.984808i	$-0.444444\pi$
$569$	−0.512593	−	0.257434i	−0.512593	−	0.257434i	−0.686242	+	0.727374i	$0.740741\pi$
$570$		0			0
$571$	0.0694434	+	0.0932786i	0.0694434	+	0.0932786i	0.835488	−	0.549509i	$-0.185185\pi$
$571$	0.0694434	+	0.0932786i	0.0694434	+	0.0932786i	−0.766044	+	0.642788i	$0.777778\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−0.344948	+	1.95630i	−0.344948	+	1.95630i	−0.0581448	+	0.998308i	$0.518519\pi$
$577$	−0.344948	+	1.95630i	−0.344948	+	1.95630i	−0.286803	+	0.957990i	$0.592593\pi$
$578$		0			0
$579$	1.05138	+	0.882215i	1.05138	+	0.882215i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.473045	+	0.635410i	−0.473045	+	0.635410i	−0.973045	−	0.230616i	$-0.925926\pi$
$587$	−0.473045	+	0.635410i	−0.473045	+	0.635410i	0.500000	+	0.866025i	$0.333333\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$593$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.686242	−	0.727374i	$-0.259259\pi$
$599$		0			0		−0.686242	+	0.727374i	$0.740741\pi$
$600$		0			0
$601$	−1.18624	−	0.138652i	−1.18624	−	0.138652i	−0.500000	−	0.866025i	$-0.666667\pi$
$601$	−1.18624	−	0.138652i	−1.18624	−	0.138652i	−0.686242	+	0.727374i	$0.740741\pi$
$602$		0			0
$603$	0.786803	+	0.0919641i	0.786803	+	0.0919641i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.893633	−	0.448799i	$-0.148148\pi$
$607$		0			0		−0.893633	+	0.448799i	$0.851852\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$613$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.97304	−	0.230616i	1.97304	−	0.230616i	0.973045	−	0.230616i	$-0.0740741\pi$
$617$	1.97304	−	0.230616i	1.97304	−	0.230616i	1.00000			$0$
$618$		0			0
$619$	−0.0798028	−	1.37016i	−0.0798028	−	1.37016i	−0.766044	−	0.642788i	$-0.777778\pi$
$619$	−0.0798028	−	1.37016i	−0.0798028	−	1.37016i	0.686242	−	0.727374i	$-0.259259\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.396080	−	0.918216i	0.396080	−	0.918216i
$626$		0			0
$627$	−1.38794	−	1.86433i	−1.38794	−	1.86433i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$631$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$632$		0			0
$633$	1.53209			1.53209
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.770807	+	1.78693i	0.770807	+	1.78693i	0.597159	+	0.802123i	$0.296296\pi$
$641$	0.770807	+	1.78693i	0.770807	+	1.78693i	0.173648	+	0.984808i	$0.444444\pi$
$642$		0			0
$643$	0.342534	+	1.14414i	0.342534	+	1.14414i	0.939693	+	0.342020i	$0.111111\pi$
$643$	0.342534	+	1.14414i	0.342534	+	1.14414i	−0.597159	+	0.802123i	$0.703704\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	−0.685068			−0.685068
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.286803	−	0.957990i	$-0.592593\pi$
$653$		0			0		0.286803	+	0.957990i	$0.407407\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	0.115503	−	0.0135004i	0.115503	−	0.0135004i
$658$		0			0
$659$	−1.49079	+	0.353324i	−1.49079	+	0.353324i	−0.893633	−	0.448799i	$-0.851852\pi$
$659$	−1.49079	+	0.353324i	−1.49079	+	0.353324i	−0.597159	+	0.802123i	$0.703704\pi$
$660$		0			0
$661$		0			0		−0.835488	−	0.549509i	$-0.814815\pi$
$661$		0			0		0.835488	+	0.549509i	$0.185185\pi$
$662$		0			0
$663$		0			0
$664$		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
$673$	−1.28004	+	0.841897i	−1.28004	+	0.841897i	−0.993238	−	0.116093i	$-0.962963\pi$
$673$	−1.28004	+	0.841897i	−1.28004	+	0.841897i	−0.286803	+	0.957990i	$0.592593\pi$
$674$		0			0
$675$	0.286803	−	0.957990i	0.286803	−	0.957990i
$676$		0			0
$677$		0			0		−0.0581448	−	0.998308i	$-0.518519\pi$
$677$		0			0		0.0581448	+	0.998308i	$0.481481\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	1.59716	−	0.802123i	1.59716	−	0.802123i
$682$		0			0
$683$	−0.606829	+	0.509190i	−0.606829	+	0.509190i	−0.893633	−	0.448799i	$-0.851852\pi$
$683$	−0.606829	+	0.509190i	−0.606829	+	0.509190i	0.286803	+	0.957990i	$0.407407\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−1.28971	−	1.36702i	−1.28971	−	1.36702i	−0.893633	−	0.448799i	$-0.851852\pi$
$691$	−1.28971	−	1.36702i	−1.28971	−	1.36702i	−0.396080	−	0.918216i	$-0.629630\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1.57382	+	1.66815i	−1.57382	+	1.66815i
$698$		0			0
$699$	−0.479241	+	0.315202i	−0.479241	+	0.315202i
$700$		0			0
$701$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$701$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.597159	−	0.802123i	$-0.296296\pi$
$709$		0			0		−0.597159	+	0.802123i	$0.703704\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$719$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	0.227194	+	0.758881i	0.227194	+	0.758881i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		−0.893633	−	0.448799i	$-0.851852\pi$
$727$		0			0		0.893633	+	0.448799i	$0.148148\pi$
$728$		0			0
$729$	0.173648	−	0.984808i	0.173648	−	0.984808i
$730$		0			0
$731$	2.66881	+	1.34033i	2.66881	+	1.34033i
$732$		0			0
$733$		0			0		−0.597159	−	0.802123i	$-0.703704\pi$
$733$		0			0		0.597159	+	0.802123i	$0.296296\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	0.164287	+	0.931717i	0.164287	+	0.931717i
$738$		0			0
$739$	−0.207391	+	1.17617i	−0.207391	+	1.17617i	0.686242	+	0.727374i	$0.259259\pi$
$739$	−0.207391	+	1.17617i	−0.207391	+	1.17617i	−0.893633	+	0.448799i	$0.851852\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.0581448	−	0.998308i	$-0.481481\pi$
$743$		0			0		−0.0581448	+	0.998308i	$0.518519\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.766044	−	1.32683i	0.766044	−	1.32683i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.973045	−	0.230616i	$-0.925926\pi$
$751$		0			0		0.973045	+	0.230616i	$0.0740741\pi$
$752$		0			0
$753$	1.06728	−	1.43361i	1.06728	−	1.43361i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$757$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.686242	−	0.727374i	0.686242	−	0.727374i	−0.286803	−	0.957990i	$-0.592593\pi$
$761$	0.686242	−	0.727374i	0.686242	−	0.727374i	0.973045	+	0.230616i	$0.0740741\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.310355	−	0.155866i	0.310355	−	0.155866i	−0.286803	−	0.957990i	$-0.592593\pi$
$769$	0.310355	−	0.155866i	0.310355	−	0.155866i	0.597159	+	0.802123i	$0.296296\pi$
$770$		0			0
$771$	−0.606829	+	0.509190i	−0.606829	+	0.509190i
$772$		0			0
$773$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$773$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.65292	+	0.310081i	−2.65292	+	0.310081i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−0.606829	+	1.40679i	−0.606829	+	1.40679i	0.286803	+	0.957990i	$0.407407\pi$
$787$	−0.606829	+	1.40679i	−0.606829	+	1.40679i	−0.893633	+	0.448799i	$0.851852\pi$
$788$		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			0		−0.835488	−	0.549509i	$-0.814815\pi$
$797$		0			0		0.835488	+	0.549509i	$0.185185\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	1.86668	−	0.218183i	1.86668	−	0.218183i
$802$		0			0
$803$	0.0550102	+	0.127528i	0.0550102	+	0.127528i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.116290			−0.116290			−0.0581448	−	0.998308i	$-0.518519\pi$
$809$	−0.116290			−0.116290			−0.0581448	+	0.998308i	$0.518519\pi$
$810$		0			0
$811$	−0.792160			−0.792160			−0.396080	−	0.918216i	$-0.629630\pi$
$811$	−0.792160			−0.792160			−0.396080	+	0.918216i	$0.629630\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1.37764	+	3.19372i	1.37764	+	3.19372i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.973045	−	0.230616i	$-0.0740741\pi$
$821$		0			0		−0.973045	+	0.230616i	$0.925926\pi$
$822$		0			0
$823$		0			0		−0.835488	−	0.549509i	$-0.814815\pi$
$823$		0			0		0.835488	+	0.549509i	$0.185185\pi$
$824$		0			0
$825$	1.19432			1.19432
$826$		0			0
$827$	−0.939693	+	0.342020i	−0.939693	+	0.342020i	−0.766044	−	0.642788i	$-0.777778\pi$
$827$	−0.939693	+	0.342020i	−0.939693	+	0.342020i	−0.173648	+	0.984808i	$0.555556\pi$
$828$		0			0
$829$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$829$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.661840	+	1.53432i	−0.661840	+	1.53432i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.0581448	−	0.998308i	$-0.518519\pi$
$839$		0			0		0.0581448	+	0.998308i	$0.481481\pi$
$840$		0			0
$841$	−0.993238	+	0.116093i	−0.993238	+	0.116093i
$842$		0			0
$843$	0.0890830	−	1.52950i	0.0890830	−	1.52950i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1.43969	−	0.524005i	−1.43969	−	0.524005i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.686242	−	0.727374i	$-0.740741\pi$
$853$		0			0		0.686242	+	0.727374i	$0.259259\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.993238	+	0.116093i	0.993238	+	0.116093i	0.597159	−	0.802123i	$-0.296296\pi$
$857$	0.993238	+	0.116093i	0.993238	+	0.116093i	0.396080	+	0.918216i	$0.370370\pi$
$858$		0			0
$859$	0.819590	−	0.868715i	0.819590	−	0.868715i	−0.173648	−	0.984808i	$-0.555556\pi$
$859$	0.819590	−	0.868715i	0.819590	−	0.868715i	0.993238	+	0.116093i	$0.0370370\pi$
$860$		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
$867$	1.78004	+	0.208057i	1.78004	+	0.208057i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1.86668	+	0.679415i	1.86668	+	0.679415i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.0581448	−	0.998308i	$-0.481481\pi$
$877$		0			0		−0.0581448	+	0.998308i	$0.518519\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.326352	+	1.85083i	−0.326352	+	1.85083i	0.173648	+	0.984808i	$0.444444\pi$
$881$	−0.326352	+	1.85083i	−0.326352	+	1.85083i	−0.500000	+	0.866025i	$0.666667\pi$
$882$		0			0
$883$	0.290162	+	1.64559i	0.290162	+	1.64559i	0.686242	+	0.727374i	$0.259259\pi$
$883$	0.290162	+	1.64559i	0.290162	+	1.64559i	−0.396080	+	0.918216i	$0.629630\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.597159	−	0.802123i	$-0.703704\pi$
$887$		0			0		0.597159	+	0.802123i	$0.296296\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1.18624	−	0.138652i	1.18624	−	0.138652i
$892$		0			0
$893$		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
$907$	0.227194	−	0.758881i	0.227194	−	0.758881i	−0.766044	−	0.642788i	$-0.777778\pi$
$907$	0.227194	−	0.758881i	0.227194	−	0.758881i	0.993238	−	0.116093i	$-0.0370370\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.597159	−	0.802123i	$-0.296296\pi$
$911$		0			0		−0.597159	+	0.802123i	$0.703704\pi$
$912$		0			0
$913$	1.78048	+	0.421981i	1.78048	+	0.421981i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		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
$921$	−0.512593	−	0.257434i	−0.512593	−	0.257434i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.686242	+	0.727374i	0.686242	+	0.727374i	0.973045	−	0.230616i	$-0.0740741\pi$
$929$	0.686242	+	0.727374i	0.686242	+	0.727374i	−0.286803	+	0.957990i	$0.592593\pi$
$930$		0			0
$931$	−1.73909	+	0.873403i	−1.73909	+	0.873403i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.43969	+	1.20805i	−1.43969	+	1.20805i	−0.500000	+	0.866025i	$0.666667\pi$
$937$	−1.43969	+	1.20805i	−1.43969	+	1.20805i	−0.939693	+	0.342020i	$0.888889\pi$
$938$		0			0
$939$	−1.14669	−	0.754192i	−1.14669	−	0.754192i
$940$		0			0
$941$		0			0		0.993238	−	0.116093i	$-0.0370370\pi$
$941$		0			0		−0.993238	+	0.116093i	$0.962963\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.0971586	+	0.0639022i	−0.0971586	+	0.0639022i	−0.597159	−	0.802123i	$-0.703704\pi$
$947$	−0.0971586	+	0.0639022i	−0.0971586	+	0.0639022i	0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.744386	−	0.270935i	−0.744386	−	0.270935i	−0.0581448	−	0.998308i	$-0.518519\pi$
$953$	−0.744386	−	0.270935i	−0.744386	−	0.270935i	−0.686242	+	0.727374i	$0.740741\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.973045	−	0.230616i	0.973045	−	0.230616i
$962$		0			0
$963$	0.103920	−	0.0521907i	0.103920	−	0.0521907i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.286803	−	0.957990i	$-0.592593\pi$
$967$		0			0		0.286803	+	0.957990i	$0.407407\pi$
$968$		0			0
$969$	2.23157	+	2.36532i	2.23157	+	2.36532i
$970$		0			0
$971$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$971$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.164512	+	0.549509i	0.164512	+	0.549509i	1.00000			$0$
$977$	0.164512	+	0.549509i	0.164512	+	0.549509i	−0.835488	+	0.549509i	$0.814815\pi$
$978$		0			0
$979$	0.889034	+	2.06101i	0.889034	+	2.06101i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.973045	−	0.230616i	$-0.0740741\pi$
$983$		0			0		−0.973045	+	0.230616i	$0.925926\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$991$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$992$		0			0
$993$	−0.744386	+	1.72568i	−0.744386	+	1.72568i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.835488	−	0.549509i	$-0.185185\pi$
$997$		0			0		−0.835488	+	0.549509i	$0.814815\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.1.bz.a.175.1		18
4.3	odd	2		648.1.bf.a.499.1	yes	18
8.3	odd	2	CM	2592.1.bz.a.175.1		18
8.5	even	2		648.1.bf.a.499.1	yes	18
12.11	even	2		1944.1.bf.a.739.1		18
24.5	odd	2		1944.1.bf.a.739.1		18
81.25	even	27	inner	2592.1.bz.a.1807.1		18
324.187	odd	54		648.1.bf.a.187.1	✓	18
324.299	even	54		1944.1.bf.a.1531.1		18
648.187	odd	54	inner	2592.1.bz.a.1807.1		18
648.349	even	54		648.1.bf.a.187.1	✓	18
648.461	odd	54		1944.1.bf.a.1531.1		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.1.bf.a.187.1	✓	18	324.187	odd	54
648.1.bf.a.187.1	✓	18	648.349	even	54
648.1.bf.a.499.1	yes	18	4.3	odd	2
648.1.bf.a.499.1	yes	18	8.5	even	2
1944.1.bf.a.739.1		18	12.11	even	2
1944.1.bf.a.739.1		18	24.5	odd	2
1944.1.bf.a.1531.1		18	324.299	even	54
1944.1.bf.a.1531.1		18	648.461	odd	54
2592.1.bz.a.175.1		18	1.1	even	1	trivial
2592.1.bz.a.175.1		18	8.3	odd	2	CM
2592.1.bz.a.1807.1		18	81.25	even	27	inner
2592.1.bz.a.1807.1		18	648.187	odd	54	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:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2592.bz (of order \(54\), degree \(18\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.686242 + 0.727374i) q^{3} +(-0.0581448 + 0.998308i) q^{9} +O(q^{10})\)	\(q+(0.686242 + 0.727374i) q^{3} +(-0.0581448 + 0.998308i) q^{9} +(-1.16212 + 0.275428i) q^{11} +(1.57020 - 0.571507i) q^{17} +(1.82873 + 0.665602i) q^{19} +(-0.835488 + 0.549509i) q^{25} +(-0.766044 + 0.642788i) q^{27} +(-0.997837 - 0.656288i) q^{33} +(-1.22650 + 0.615969i) q^{41} +(1.22650 + 1.30001i) q^{43} +(-0.686242 + 0.727374i) q^{49} +(1.49324 + 0.749932i) q^{51} +(0.770807 + 1.78693i) q^{57} +(0.558145 + 0.132283i) q^{59} +(0.0460600 - 0.790819i) q^{67} +(-0.0201935 - 0.114523i) q^{73} +(-0.973045 - 0.230616i) q^{75} +(-0.993238 - 0.116093i) q^{81} +(-1.36912 - 0.687600i) q^{83} +(-0.326352 - 1.85083i) q^{89} +(0.569728 - 1.90302i) q^{97} +(-0.207391 - 1.17617i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q+O(q^{10}) \) 18 * q	\( 18 q + 9 q^{51} + 9 q^{59} - 9 q^{89}+O(q^{100}) \) 18 * q + 9 * q^51 + 9 * q^59 - 9 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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