LMFDB - Embedded newform 648.1.bf.a.139.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,1,Mod(43,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(54))

chi = DirichletCharacter(H, H._module([27, 27, 22]))

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

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

chi := DirichletCharacter("648.43");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	648.bf (of order $54$, degree $18$, 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:	$0.323394128186$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{27}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{27} + \cdots)$

Embedding invariants

Embedding label			139.1
Root			$0.835488 - 0.549509i$ of defining polynomial
Character	$\chi$	$=$	648.139
Dual form			648.1.bf.a.331.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.597159 - 0.802123i) q^{2} +(-0.0581448 - 0.998308i) q^{3} +(-0.286803 - 0.957990i) q^{4} +(-0.835488 - 0.549509i) q^{6} +(-0.939693 - 0.342020i) q^{8} +(-0.993238 + 0.116093i) q^{9} +O(q^{10})$	\(q+(0.597159 - 0.802123i) q^{2} +(-0.0581448 - 0.998308i) q^{3} +(-0.286803 - 0.957990i) q^{4} +(-0.835488 - 0.549509i) q^{6} +(-0.939693 - 0.342020i) q^{8} +(-0.993238 + 0.116093i) q^{9} +(-0.512593 - 0.257434i) q^{11} +(-0.939693 + 0.342020i) q^{12} +(-0.835488 + 0.549509i) q^{16} +(0.606829 + 0.509190i) q^{17} +(-0.500000 + 0.866025i) q^{18} +(1.36912 - 1.14883i) q^{19} +(-0.512593 + 0.257434i) q^{22} +(-0.286803 + 0.957990i) q^{24} +(0.396080 + 0.918216i) q^{25} +(0.173648 + 0.984808i) q^{27} +(-0.0581448 + 0.998308i) q^{32} +(-0.227194 + 0.526695i) q^{33} +(0.770807 - 0.182685i) q^{34} +(0.396080 + 0.918216i) q^{36} +(-0.103920 - 1.78424i) q^{38} +(-0.0694434 - 0.0932786i) q^{41} +(-0.0694434 - 1.19230i) q^{43} +(-0.0996057 + 0.564892i) q^{44} +(0.597159 + 0.802123i) q^{48} +(-0.0581448 + 0.998308i) q^{49} +(0.973045 + 0.230616i) q^{50} +(0.473045 - 0.635410i) q^{51} +(0.893633 + 0.448799i) q^{54} +(-1.22650 - 1.30001i) q^{57} +(-1.49324 + 0.749932i) q^{59} +(0.766044 + 0.642788i) q^{64} +(0.286803 + 0.496758i) q^{66} +(1.36320 - 0.159336i) q^{67} +(0.313758 - 0.727374i) q^{68} +(0.973045 + 0.230616i) q^{72} +(1.86668 + 0.679415i) q^{73} +(0.893633 - 0.448799i) q^{75} +(-1.49324 - 0.982118i) q^{76} +(0.973045 - 0.230616i) q^{81} -0.116290 q^{82} +(0.207391 - 0.278574i) q^{83} +(-0.997837 - 0.656288i) q^{86} +(0.393633 + 0.417226i) q^{88} +(-1.43969 - 0.524005i) q^{89} +1.00000 q^{96} +(-1.62593 + 1.06939i) q^{97} +(0.766044 + 0.642788i) q^{98} +(0.539014 + 0.196185i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q+O(q^{10}) $ 18 * q	$ 18 q - 9 q^{18} - 9 q^{38} - 9 q^{51} - 9 q^{59} + 18 q^{68} - 9 q^{76} - 9 q^{88} - 9 q^{89} + 18 q^{96}+O(q^{100}) $ 18 * q - 9 * q^18 - 9 * q^38 - 9 * q^51 - 9 * q^59 + 18 * q^68 - 9 * q^76 - 9 * q^88 - 9 * q^89 + 18 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{19}{27}\right)$

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 $

$2$	0.597159	−	0.802123i	0.597159	−	0.802123i
$2$	0.597159	−	0.802123i	0.597159	−	0.802123i
$3$	−0.0581448	−	0.998308i	−0.0581448	−	0.998308i
$3$	−0.0581448	−	0.998308i	−0.0581448	−	0.998308i
$4$	−0.286803	−	0.957990i	−0.286803	−	0.957990i
$5$		0			0		−0.835488	−	0.549509i	$-0.814815\pi$
$5$		0			0		0.835488	+	0.549509i	$0.185185\pi$
$6$	−0.835488	−	0.549509i	−0.835488	−	0.549509i
$7$		0			0		−0.686242	−	0.727374i	$-0.740741\pi$
$7$		0			0		0.686242	+	0.727374i	$0.259259\pi$
$8$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$9$	−0.993238	+	0.116093i	−0.993238	+	0.116093i
$10$		0			0
$11$	−0.512593	−	0.257434i	−0.512593	−	0.257434i	0.173648	−	0.984808i	$-0.444444\pi$
$11$	−0.512593	−	0.257434i	−0.512593	−	0.257434i	−0.686242	+	0.727374i	$0.740741\pi$
$12$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$13$		0			0		0.396080	−	0.918216i	$-0.370370\pi$
$13$		0			0		−0.396080	+	0.918216i	$0.629630\pi$
$14$		0			0
$15$		0			0
$16$	−0.835488	+	0.549509i	−0.835488	+	0.549509i
$17$	0.606829	+	0.509190i	0.606829	+	0.509190i	0.893633	−	0.448799i	$-0.148148\pi$
$17$	0.606829	+	0.509190i	0.606829	+	0.509190i	−0.286803	+	0.957990i	$0.592593\pi$
$18$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$19$	1.36912	−	1.14883i	1.36912	−	1.14883i	0.396080	−	0.918216i	$-0.370370\pi$
$19$	1.36912	−	1.14883i	1.36912	−	1.14883i	0.973045	−	0.230616i	$-0.0740741\pi$
$20$		0			0
$21$		0			0
$22$	−0.512593	+	0.257434i	−0.512593	+	0.257434i
$23$		0			0		0.686242	−	0.727374i	$-0.259259\pi$
$23$		0			0		−0.686242	+	0.727374i	$0.740741\pi$
$24$	−0.286803	+	0.957990i	−0.286803	+	0.957990i
$25$	0.396080	+	0.918216i	0.396080	+	0.918216i
$26$		0			0
$27$	0.173648	+	0.984808i	0.173648	+	0.984808i
$28$		0			0
$29$		0			0		−0.993238	−	0.116093i	$-0.962963\pi$
$29$		0			0		0.993238	+	0.116093i	$0.0370370\pi$
$30$		0			0
$31$		0			0		−0.973045	−	0.230616i	$-0.925926\pi$
$31$		0			0		0.973045	+	0.230616i	$0.0740741\pi$
$32$	−0.0581448	+	0.998308i	−0.0581448	+	0.998308i
$33$	−0.227194	+	0.526695i	−0.227194	+	0.526695i
$34$	0.770807	−	0.182685i	0.770807	−	0.182685i
$35$		0			0
$36$	0.396080	+	0.918216i	0.396080	+	0.918216i
$37$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$37$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$38$	−0.103920	−	1.78424i	−0.103920	−	1.78424i
$39$		0			0
$40$		0			0
$41$	−0.0694434	−	0.0932786i	−0.0694434	−	0.0932786i	0.766044	−	0.642788i	$-0.222222\pi$
$41$	−0.0694434	−	0.0932786i	−0.0694434	−	0.0932786i	−0.835488	+	0.549509i	$0.814815\pi$
$42$		0			0
$43$	−0.0694434	−	1.19230i	−0.0694434	−	1.19230i	−0.835488	−	0.549509i	$-0.814815\pi$
$43$	−0.0694434	−	1.19230i	−0.0694434	−	1.19230i	0.766044	−	0.642788i	$-0.222222\pi$
$44$	−0.0996057	+	0.564892i	−0.0996057	+	0.564892i
$45$		0			0
$46$		0			0
$47$		0			0		0.973045	−	0.230616i	$-0.0740741\pi$
$47$		0			0		−0.973045	+	0.230616i	$0.925926\pi$
$48$	0.597159	+	0.802123i	0.597159	+	0.802123i
$49$	−0.0581448	+	0.998308i	−0.0581448	+	0.998308i
$50$	0.973045	+	0.230616i	0.973045	+	0.230616i
$51$	0.473045	−	0.635410i	0.473045	−	0.635410i
$52$		0			0
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$	0.893633	+	0.448799i	0.893633	+	0.448799i
$55$		0			0
$56$		0			0
$57$	−1.22650	−	1.30001i	−1.22650	−	1.30001i
$58$		0			0
$59$	−1.49324	+	0.749932i	−1.49324	+	0.749932i	−0.993238	−	0.116093i	$-0.962963\pi$
$59$	−1.49324	+	0.749932i	−1.49324	+	0.749932i	−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$		0			0		0.286803	−	0.957990i	$-0.407407\pi$
$61$		0			0		−0.286803	+	0.957990i	$0.592593\pi$
$62$		0			0
$63$		0			0
$64$	0.766044	+	0.642788i	0.766044	+	0.642788i
$65$		0			0
$66$	0.286803	+	0.496758i	0.286803	+	0.496758i
$67$	1.36320	−	0.159336i	1.36320	−	0.159336i	0.597159	−	0.802123i	$-0.296296\pi$
$67$	1.36320	−	0.159336i	1.36320	−	0.159336i	0.766044	+	0.642788i	$0.222222\pi$
$68$	0.313758	−	0.727374i	0.313758	−	0.727374i
$69$		0			0
$70$		0			0
$71$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$71$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$72$	0.973045	+	0.230616i	0.973045	+	0.230616i
$73$	1.86668	+	0.679415i	1.86668	+	0.679415i	0.973045	+	0.230616i	$0.0740741\pi$
$73$	1.86668	+	0.679415i	1.86668	+	0.679415i	0.893633	+	0.448799i	$0.148148\pi$
$74$		0			0
$75$	0.893633	−	0.448799i	0.893633	−	0.448799i
$76$	−1.49324	−	0.982118i	−1.49324	−	0.982118i
$77$		0			0
$78$		0			0
$79$		0			0		0.597159	−	0.802123i	$-0.296296\pi$
$79$		0			0		−0.597159	+	0.802123i	$0.703704\pi$
$80$		0			0
$81$	0.973045	−	0.230616i	0.973045	−	0.230616i
$82$	−0.116290			−0.116290
$83$	0.207391	−	0.278574i	0.207391	−	0.278574i	−0.686242	−	0.727374i	$-0.740741\pi$
$83$	0.207391	−	0.278574i	0.207391	−	0.278574i	0.893633	+	0.448799i	$0.148148\pi$
$84$		0			0
$85$		0			0
$86$	−0.997837	−	0.656288i	−0.997837	−	0.656288i
$87$		0			0
$88$	0.393633	+	0.417226i	0.393633	+	0.417226i
$89$	−1.43969	−	0.524005i	−1.43969	−	0.524005i	−0.500000	−	0.866025i	$-0.666667\pi$
$89$	−1.43969	−	0.524005i	−1.43969	−	0.524005i	−0.939693	+	0.342020i	$0.888889\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	1.00000			1.00000
$97$	−1.62593	+	1.06939i	−1.62593	+	1.06939i	−0.686242	+	0.727374i	$0.740741\pi$
$97$	−1.62593	+	1.06939i	−1.62593	+	1.06939i	−0.939693	+	0.342020i	$0.888889\pi$
$98$	0.766044	+	0.642788i	0.766044	+	0.642788i
$99$	0.539014	+	0.196185i	0.539014	+	0.196185i
$100$	0.766044	−	0.642788i	0.766044	−	0.642788i
$101$		0			0		0.286803	−	0.957990i	$-0.407407\pi$
$101$		0			0		−0.286803	+	0.957990i	$0.592593\pi$
$102$	−0.227194	−	0.758881i	−0.227194	−	0.758881i
$103$		0			0		0.893633	−	0.448799i	$-0.148148\pi$
$103$		0			0		−0.893633	+	0.448799i	$0.851852\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.993238	+	1.72034i	0.993238	+	1.72034i	0.597159	+	0.802123i	$0.296296\pi$
$107$	0.993238	+	1.72034i	0.993238	+	1.72034i	0.396080	+	0.918216i	$0.370370\pi$
$108$	0.893633	−	0.448799i	0.893633	−	0.448799i
$109$		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
$113$	0.109277	−	1.87621i	0.109277	−	1.87621i	−0.286803	−	0.957990i	$-0.592593\pi$
$113$	0.109277	−	1.87621i	0.109277	−	1.87621i	0.396080	−	0.918216i	$-0.370370\pi$
$114$	−1.77518	+	0.207489i	−1.77518	+	0.207489i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−0.290162	+	1.64559i	−0.290162	+	1.64559i
$119$		0			0
$120$		0			0
$121$	−0.400679	−	0.538205i	−0.400679	−	0.538205i
$122$		0			0
$123$	−0.0890830	+	0.0747496i	−0.0890830	+	0.0747496i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$127$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$128$	0.973045	−	0.230616i	0.973045	−	0.230616i
$129$	−1.18624	+	0.138652i	−1.18624	+	0.138652i
$130$		0			0
$131$	−1.82873	−	0.433416i	−1.82873	−	0.433416i	−0.835488	−	0.549509i	$-0.814815\pi$
$131$	−1.82873	−	0.433416i	−1.82873	−	0.433416i	−0.993238	+	0.116093i	$0.962963\pi$
$132$	0.569728	+	0.0665916i	0.569728	+	0.0665916i
$133$		0			0
$134$	0.686242	−	1.18861i	0.686242	−	1.18861i
$135$		0			0
$136$	−0.396080	−	0.686030i	−0.396080	−	0.686030i
$137$	0.770807	+	1.78693i	0.770807	+	1.78693i	0.597159	+	0.802123i	$0.296296\pi$
$137$	0.770807	+	1.78693i	0.770807	+	1.78693i	0.173648	+	0.984808i	$0.444444\pi$
$138$		0			0
$139$	−1.33549	+	1.41553i	−1.33549	+	1.41553i	−0.500000	+	0.866025i	$0.666667\pi$
$139$	−1.33549	+	1.41553i	−1.33549	+	1.41553i	−0.835488	+	0.549509i	$0.814815\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.766044	−	0.642788i	0.766044	−	0.642788i
$145$		0			0
$146$	1.65968	−	1.09159i	1.65968	−	1.09159i
$147$	1.00000			1.00000
$148$		0			0
$149$		0			0		0.396080	−	0.918216i	$-0.370370\pi$
$149$		0			0		−0.396080	+	0.918216i	$0.629630\pi$
$150$	0.173648	−	0.984808i	0.173648	−	0.984808i
$151$		0			0		−0.893633	−	0.448799i	$-0.851852\pi$
$151$		0			0		0.893633	+	0.448799i	$0.148148\pi$
$152$	−1.67948	+	0.611281i	−1.67948	+	0.611281i
$153$	−0.661840	−	0.435299i	−0.661840	−	0.435299i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.835488	−	0.549509i	$-0.814815\pi$
$157$		0			0		0.835488	+	0.549509i	$0.185185\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.396080	−	0.918216i	0.396080	−	0.918216i
$163$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$163$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$164$	−0.0694434	+	0.0932786i	−0.0694434	+	0.0932786i
$165$		0			0
$166$	−0.0996057	−	0.332706i	−0.0996057	−	0.332706i
$167$		0			0		−0.835488	−	0.549509i	$-0.814815\pi$
$167$		0			0		0.835488	+	0.549509i	$0.185185\pi$
$168$		0			0
$169$	−0.686242	−	0.727374i	−0.686242	−	0.727374i
$170$		0			0
$171$	−1.22650	+	1.30001i	−1.22650	+	1.30001i
$172$	−1.12229	+	0.408481i	−1.12229	+	0.408481i
$173$		0			0		−0.893633	−	0.448799i	$-0.851852\pi$
$173$		0			0		0.893633	+	0.448799i	$0.148148\pi$
$174$		0			0
$175$		0			0
$176$	0.569728	−	0.0665916i	0.569728	−	0.0665916i
$177$	0.835488	+	1.44711i	0.835488	+	1.44711i
$178$	−1.28004	+	0.841897i	−1.28004	+	0.841897i
$179$	1.17365	+	0.984808i	1.17365	+	0.984808i	1.00000			$0$
$179$	1.17365	+	0.984808i	1.17365	+	0.984808i	0.173648	+	0.984808i	$0.444444\pi$
$180$		0			0
$181$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$181$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−0.179974	−	0.417226i	−0.179974	−	0.417226i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.993238	−	0.116093i	$-0.962963\pi$
$191$		0			0		0.993238	+	0.116093i	$0.0370370\pi$
$192$	0.597159	−	0.802123i	0.597159	−	0.802123i
$193$	−0.113155	−	0.0268182i	−0.113155	−	0.0268182i	0.173648	−	0.984808i	$-0.444444\pi$
$193$	−0.113155	−	0.0268182i	−0.113155	−	0.0268182i	−0.286803	+	0.957990i	$0.592593\pi$
$194$	−0.113155	+	1.94280i	−0.113155	+	1.94280i
$195$		0			0
$196$	0.973045	−	0.230616i	0.973045	−	0.230616i
$197$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$197$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$198$	0.479241	−	0.315202i	0.479241	−	0.315202i
$199$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$199$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$200$	−0.0581448	−	0.998308i	−0.0581448	−	0.998308i
$201$	−0.238329	−	1.35163i	−0.238329	−	1.35163i
$202$		0			0
$203$		0			0
$204$	−0.744386	−	0.270935i	−0.744386	−	0.270935i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.997553	+	0.236424i	−0.997553	+	0.236424i
$210$		0			0
$211$	−0.0201935	+	0.346709i	−0.0201935	+	0.346709i	0.973045	+	0.230616i	$0.0740741\pi$
$211$	−0.0201935	+	0.346709i	−0.0201935	+	0.346709i	−0.993238	+	0.116093i	$0.962963\pi$
$212$		0			0
$213$		0			0
$214$	1.97304	+	0.230616i	1.97304	+	0.230616i
$215$		0			0
$216$	0.173648	−	0.984808i	0.173648	−	0.984808i
$217$		0			0
$218$		0			0
$219$	0.569728	−	1.90302i	0.569728	−	1.90302i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.286803	−	0.957990i	$-0.407407\pi$
$223$		0			0		−0.286803	+	0.957990i	$0.592593\pi$
$224$		0			0
$225$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$226$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$227$	−0.997837	+	0.656288i	−0.997837	+	0.656288i	−0.939693	−	0.342020i	$-0.888889\pi$
$227$	−0.997837	+	0.656288i	−0.997837	+	0.656288i	−0.0581448	+	0.998308i	$0.518519\pi$
$228$	−0.893633	+	1.54782i	−0.893633	+	1.54782i
$229$		0			0		0.993238	−	0.116093i	$-0.0370370\pi$
$229$		0			0		−0.993238	+	0.116093i	$0.962963\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.57020	−	0.571507i	1.57020	−	0.571507i	0.597159	−	0.802123i	$-0.296296\pi$
$233$	1.57020	−	0.571507i	1.57020	−	0.571507i	0.973045	+	0.230616i	$0.0740741\pi$
$234$		0			0
$235$		0			0
$236$	1.14669	+	1.21542i	1.14669	+	1.21542i
$237$		0			0
$238$		0			0
$239$		0			0		−0.286803	−	0.957990i	$-0.592593\pi$
$239$		0			0		0.286803	+	0.957990i	$0.407407\pi$
$240$		0			0
$241$	−0.819590	+	1.10090i	−0.819590	+	1.10090i	0.173648	+	0.984808i	$0.444444\pi$
$241$	−0.819590	+	1.10090i	−0.819590	+	1.10090i	−0.993238	+	0.116093i	$0.962963\pi$
$242$	−0.670976			−0.670976
$243$	−0.286803	−	0.957990i	−0.286803	−	0.957990i
$244$		0			0
$245$		0			0
$246$	0.00676164	+	0.116093i	0.00676164	+	0.116093i
$247$		0			0
$248$		0			0
$249$	−0.290162	−	0.190842i	−0.290162	−	0.190842i
$250$		0			0
$251$	−1.12229	−	0.408481i	−1.12229	−	0.408481i	−0.286803	−	0.957990i	$-0.592593\pi$
$251$	−1.12229	−	0.408481i	−1.12229	−	0.408481i	−0.835488	+	0.549509i	$0.814815\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.396080	−	0.918216i	0.396080	−	0.918216i
$257$	1.36320	−	0.159336i	1.36320	−	0.159336i	0.597159	−	0.802123i	$-0.296296\pi$
$257$	1.36320	−	0.159336i	1.36320	−	0.159336i	0.766044	+	0.642788i	$0.222222\pi$
$258$	−0.597159	+	1.03431i	−0.597159	+	1.03431i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−1.43969	+	1.20805i	−1.43969	+	1.20805i
$263$		0			0		0.286803	−	0.957990i	$-0.407407\pi$
$263$		0			0		−0.286803	+	0.957990i	$0.592593\pi$
$264$	0.393633	−	0.417226i	0.393633	−	0.417226i
$265$		0			0
$266$		0			0
$267$	−0.439408	+	1.46773i	−0.439408	+	1.46773i
$268$	−0.543613	−	1.26024i	−0.543613	−	1.26024i
$269$		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	$0.666667\pi$
$272$	−0.786803	−	0.0919641i	−0.786803	−	0.0919641i
$273$		0			0
$274$	1.89363	+	0.448799i	1.89363	+	0.448799i
$275$	0.0333522	−	0.572636i	0.0333522	−	0.572636i
$276$		0			0
$277$		0			0		0.973045	−	0.230616i	$-0.0740741\pi$
$277$		0			0		−0.973045	+	0.230616i	$0.925926\pi$
$278$	0.337935	+	1.91652i	0.337935	+	1.91652i
$279$		0			0
$280$		0			0
$281$	−0.0201935	−	0.346709i	−0.0201935	−	0.346709i	−0.993238	−	0.116093i	$-0.962963\pi$
$281$	−0.0201935	−	0.346709i	−0.0201935	−	0.346709i	0.973045	−	0.230616i	$-0.0740741\pi$
$282$		0			0
$283$	0.207391	+	0.278574i	0.207391	+	0.278574i	0.893633	−	0.448799i	$-0.148148\pi$
$283$	0.207391	+	0.278574i	0.207391	+	0.278574i	−0.686242	+	0.727374i	$0.740741\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.0581448	−	0.998308i	−0.0581448	−	0.998308i
$289$	−0.0646810	−	0.366824i	−0.0646810	−	0.366824i
$290$		0			0
$291$	1.16212	+	1.56100i	1.16212	+	1.56100i
$292$	0.115503	−	1.98312i	0.115503	−	1.98312i
$293$		0			0		−0.973045	−	0.230616i	$-0.925926\pi$
$293$		0			0		0.973045	+	0.230616i	$0.0740741\pi$
$294$	0.597159	−	0.802123i	0.597159	−	0.802123i
$295$		0			0
$296$		0			0
$297$	0.164512	−	0.549509i	0.164512	−	0.549509i
$298$		0			0
$299$		0			0
$300$	−0.686242	−	0.727374i	−0.686242	−	0.727374i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−0.512593	+	1.71218i	−0.512593	+	1.71218i
$305$		0			0
$306$	−0.744386	+	0.270935i	−0.744386	+	0.270935i
$307$	−1.28004	−	1.07408i	−1.28004	−	1.07408i	−0.993238	−	0.116093i	$-0.962963\pi$
$307$	−1.28004	−	1.07408i	−1.28004	−	1.07408i	−0.286803	−	0.957990i	$-0.592593\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.396080	−	0.918216i	$-0.370370\pi$
$311$		0			0		−0.396080	+	0.918216i	$0.629630\pi$
$312$		0			0
$313$	−0.103920	−	0.0521907i	−0.103920	−	0.0521907i	0.396080	−	0.918216i	$-0.370370\pi$
$313$	−0.103920	−	0.0521907i	−0.103920	−	0.0521907i	−0.500000	+	0.866025i	$0.666667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.686242	−	0.727374i	$-0.740741\pi$
$317$		0			0		0.686242	+	0.727374i	$0.259259\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	1.65968	−	1.09159i	1.65968	−	1.09159i
$322$		0			0
$323$	1.41580			1.41580
$324$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$325$		0			0
$326$	−1.12229	+	1.50750i	−1.12229	+	1.50750i
$327$		0			0
$328$	0.0333522	+	0.111404i	0.0333522	+	0.111404i
$329$		0			0
$330$		0			0
$331$	−1.05138	−	1.11440i	−1.05138	−	1.11440i	−0.993238	−	0.116093i	$-0.962963\pi$
$331$	−1.05138	−	1.11440i	−1.05138	−	1.11440i	−0.0581448	−	0.998308i	$-0.518519\pi$
$332$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.473045	−	1.09664i	0.473045	−	1.09664i	−0.500000	−	0.866025i	$-0.666667\pi$
$337$	0.473045	−	1.09664i	0.473045	−	1.09664i	0.973045	−	0.230616i	$-0.0740741\pi$
$338$	−0.993238	+	0.116093i	−0.993238	+	0.116093i
$339$	−1.87939			−1.87939
$340$		0			0
$341$		0			0
$342$	0.310355	+	1.76011i	0.310355	+	1.76011i
$343$		0			0
$344$	−0.342534	+	1.14414i	−0.342534	+	1.14414i
$345$		0			0
$346$		0			0
$347$	−1.22650	+	1.30001i	−1.22650	+	1.30001i	−0.286803	+	0.957990i	$0.592593\pi$
$347$	−1.22650	+	1.30001i	−1.22650	+	1.30001i	−0.939693	+	0.342020i	$0.888889\pi$
$348$		0			0
$349$		0			0		−0.396080	−	0.918216i	$-0.629630\pi$
$349$		0			0		0.396080	+	0.918216i	$0.370370\pi$
$350$		0			0
$351$		0			0
$352$	0.286803	−	0.496758i	0.286803	−	0.496758i
$353$	1.97304	+	0.230616i	1.97304	+	0.230616i	1.00000			$0$
$353$	1.97304	+	0.230616i	1.97304	+	0.230616i	0.973045	+	0.230616i	$0.0740741\pi$
$354$	1.65968	+	0.193988i	1.65968	+	0.193988i
$355$		0			0
$356$	−0.0890830	+	1.52950i	−0.0890830	+	1.52950i
$357$		0			0
$358$	1.49079	−	0.353324i	1.49079	−	0.353324i
$359$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$359$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$360$		0			0
$361$	0.381039	−	2.16098i	0.381039	−	2.16098i
$362$		0			0
$363$	−0.513997	+	0.431295i	−0.513997	+	0.431295i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.0581448	−	0.998308i	$-0.518519\pi$
$367$		0			0		0.0581448	+	0.998308i	$0.481481\pi$
$368$		0			0
$369$	0.0798028	+	0.0845860i	0.0798028	+	0.0845860i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.0581448	−	0.998308i	$-0.481481\pi$
$373$		0			0		−0.0581448	+	0.998308i	$0.518519\pi$
$374$	−0.442140	−	0.104789i	−0.442140	−	0.104789i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.0581448	+	0.100710i	0.0581448	+	0.100710i	0.893633	−	0.448799i	$-0.148148\pi$
$379$	0.0581448	+	0.100710i	0.0581448	+	0.100710i	−0.835488	+	0.549509i	$0.814815\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.893633	−	0.448799i	$-0.148148\pi$
$383$		0			0		−0.893633	+	0.448799i	$0.851852\pi$
$384$	−0.286803	−	0.957990i	−0.286803	−	0.957990i
$385$		0			0
$386$	−0.0890830	+	0.0747496i	−0.0890830	+	0.0747496i
$387$	0.207391	+	1.17617i	0.207391	+	1.17617i
$388$	1.49079	+	1.25092i	1.49079	+	1.25092i
$389$		0			0		0.835488	−	0.549509i	$-0.185185\pi$
$389$		0			0		−0.835488	+	0.549509i	$0.814815\pi$
$390$		0			0
$391$		0			0
$392$	0.396080	−	0.918216i	0.396080	−	0.918216i
$393$	−0.326352	+	1.85083i	−0.326352	+	1.85083i
$394$		0			0
$395$		0			0
$396$	0.0333522	−	0.572636i	0.0333522	−	0.572636i
$397$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$397$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$398$		0			0
$399$		0			0
$400$	−0.835488	−	0.549509i	−0.835488	−	0.549509i
$401$	−0.512593	−	1.71218i	−0.512593	−	1.71218i	−0.686242	−	0.727374i	$-0.740741\pi$
$401$	−0.512593	−	1.71218i	−0.512593	−	1.71218i	0.173648	−	0.984808i	$-0.444444\pi$
$402$	−1.22650	−	0.615969i	−1.22650	−	0.615969i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−0.661840	+	0.435299i	−0.661840	+	0.435299i
$409$	−0.227194	−	0.758881i	−0.227194	−	0.758881i	−0.993238	−	0.116093i	$-0.962963\pi$
$409$	−0.227194	−	0.758881i	−0.227194	−	0.758881i	0.766044	−	0.642788i	$-0.222222\pi$
$410$		0			0
$411$	1.73909	−	0.873403i	1.73909	−	0.873403i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.49079	+	1.25092i	1.49079	+	1.25092i
$418$	−0.406056	+	0.941343i	−0.406056	+	0.941343i
$419$	1.86668	−	0.218183i	1.86668	−	0.218183i	0.893633	−	0.448799i	$-0.148148\pi$
$419$	1.86668	−	0.218183i	1.86668	−	0.218183i	0.973045	+	0.230616i	$0.0740741\pi$
$420$		0			0
$421$		0			0		0.835488	−	0.549509i	$-0.185185\pi$
$421$		0			0		−0.835488	+	0.549509i	$0.814815\pi$
$422$	0.266044	+	0.223238i	0.266044	+	0.223238i
$423$		0			0
$424$		0			0
$425$	−0.227194	+	0.758881i	−0.227194	+	0.758881i
$426$		0			0
$427$		0			0
$428$	1.36320	−	1.44491i	1.36320	−	1.44491i
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$	−0.686242	−	0.727374i	−0.686242	−	0.727374i
$433$	−0.396080	+	0.686030i	−0.396080	+	0.686030i	−0.993238	−	0.116093i	$-0.962963\pi$
$433$	−0.396080	+	0.686030i	−0.396080	+	0.686030i	0.597159	+	0.802123i	$0.296296\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−1.18624	−	1.59340i	−1.18624	−	1.59340i
$439$		0			0		0.973045	−	0.230616i	$-0.0740741\pi$
$439$		0			0		−0.973045	+	0.230616i	$0.925926\pi$
$440$		0			0
$441$	−0.0581448	−	0.998308i	−0.0581448	−	0.998308i
$442$		0			0
$443$	0.00676164	+	0.116093i	0.00676164	+	0.116093i	1.00000			$0$
$443$	0.00676164	+	0.116093i	0.00676164	+	0.116093i	−0.993238	+	0.116093i	$0.962963\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.337935	−	1.91652i	0.337935	−	1.91652i	−0.0581448	−	0.998308i	$-0.518519\pi$
$449$	0.337935	−	1.91652i	0.337935	−	1.91652i	0.396080	−	0.918216i	$-0.370370\pi$
$450$	−0.993238	−	0.116093i	−0.993238	−	0.116093i
$451$	0.0115831	+	0.0656911i	0.0115831	+	0.0656911i
$452$	−1.82873	+	0.433416i	−1.82873	+	0.433416i
$453$		0			0
$454$	−0.0694434	+	1.19230i	−0.0694434	+	1.19230i
$455$		0			0
$456$	0.707900	+	1.64110i	0.707900	+	1.64110i
$457$	1.65968	+	0.193988i	1.65968	+	0.193988i	0.893633	−	0.448799i	$-0.148148\pi$
$457$	1.65968	+	0.193988i	1.65968	+	0.193988i	0.766044	+	0.642788i	$0.222222\pi$
$458$		0			0
$459$	−0.396080	+	0.686030i	−0.396080	+	0.686030i
$460$		0			0
$461$		0			0		−0.396080	−	0.918216i	$-0.629630\pi$
$461$		0			0		0.396080	+	0.918216i	$0.370370\pi$
$462$		0			0
$463$		0			0		0.686242	−	0.727374i	$-0.259259\pi$
$463$		0			0		−0.686242	+	0.727374i	$0.740741\pi$
$464$		0			0
$465$		0			0
$466$	0.479241	−	1.60078i	0.479241	−	1.60078i
$467$	1.36912	−	1.14883i	1.36912	−	1.14883i	0.396080	−	0.918216i	$-0.370370\pi$
$467$	1.36912	−	1.14883i	1.36912	−	1.14883i	0.973045	−	0.230616i	$-0.0740741\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	1.65968	−	0.193988i	1.65968	−	0.193988i
$473$	−0.271342	+	0.629041i	−0.271342	+	0.629041i
$474$		0			0
$475$	1.59716	+	0.802123i	1.59716	+	0.802123i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.686242	−	0.727374i	$-0.740741\pi$
$479$		0			0		0.686242	+	0.727374i	$0.259259\pi$
$480$		0			0
$481$		0			0
$482$	0.393633	+	1.31482i	0.393633	+	1.31482i
$483$		0			0
$484$	−0.400679	+	0.538205i	−0.400679	+	0.538205i
$485$		0			0
$486$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	0.109277	+	1.87621i	0.109277	+	1.87621i
$490$		0			0
$491$	1.65968	+	1.09159i	1.65968	+	1.09159i	0.893633	+	0.448799i	$0.148148\pi$
$491$	1.65968	+	1.09159i	1.65968	+	1.09159i	0.766044	+	0.642788i	$0.222222\pi$
$492$	0.0971586	+	0.0639022i	0.0971586	+	0.0639022i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−0.326352	+	0.118782i	−0.326352	+	0.118782i
$499$	0.313758	−	0.727374i	0.313758	−	0.727374i	−0.686242	−	0.727374i	$-0.740741\pi$
$499$	0.313758	−	0.727374i	0.313758	−	0.727374i	1.00000			$0$
$500$		0			0
$501$		0			0
$502$	−0.997837	+	0.656288i	−0.997837	+	0.656288i
$503$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$503$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.686242	+	0.727374i	−0.686242	+	0.727374i
$508$		0			0
$509$		0			0		0.686242	−	0.727374i	$-0.259259\pi$
$509$		0			0		−0.686242	+	0.727374i	$0.740741\pi$
$510$		0			0
$511$		0			0
$512$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$513$	1.36912	+	1.14883i	1.36912	+	1.14883i
$514$	0.686242	−	1.18861i	0.686242	−	1.18861i
$515$		0			0
$516$	0.473045	+	1.09664i	0.473045	+	1.09664i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.207391	+	1.17617i	0.207391	+	1.17617i	0.893633	+	0.448799i	$0.148148\pi$
$521$	0.207391	+	1.17617i	0.207391	+	1.17617i	−0.686242	+	0.727374i	$0.740741\pi$
$522$		0			0
$523$	−0.326352	+	1.85083i	−0.326352	+	1.85083i	0.173648	+	0.984808i	$0.444444\pi$
$523$	−0.326352	+	1.85083i	−0.326352	+	1.85083i	−0.500000	+	0.866025i	$0.666667\pi$
$524$	0.109277	+	1.87621i	0.109277	+	1.87621i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−0.0996057	−	0.564892i	−0.0996057	−	0.564892i
$529$	−0.0581448	−	0.998308i	−0.0581448	−	0.998308i
$530$		0			0
$531$	1.39608	−	0.918216i	1.39608	−	0.918216i
$532$		0			0
$533$		0			0
$534$	0.914900	+	1.22892i	0.914900	+	1.22892i
$535$		0			0
$536$	−1.33549	−	0.316516i	−1.33549	−	0.316516i
$537$	0.914900	−	1.22892i	0.914900	−	1.22892i
$538$		0			0
$539$	0.286803	−	0.496758i	0.286803	−	0.496758i
$540$		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
$544$	−0.543613	+	0.576196i	−0.543613	+	0.576196i
$545$		0			0
$546$		0			0
$547$	−0.227194	+	0.758881i	−0.227194	+	0.758881i	0.766044	+	0.642788i	$0.222222\pi$
$547$	−0.227194	+	0.758881i	−0.227194	+	0.758881i	−0.993238	+	0.116093i	$0.962963\pi$
$548$	1.49079	−	1.25092i	1.49079	−	1.25092i
$549$		0			0
$550$	−0.439408	−	0.368707i	−0.439408	−	0.368707i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	1.73909	+	0.873403i	1.73909	+	0.873403i
$557$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$557$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−0.406056	+	0.203929i	−0.406056	+	0.203929i
$562$	−0.290162	−	0.190842i	−0.290162	−	0.190842i
$563$	−0.558145	−	1.86433i	−0.558145	−	1.86433i	−0.500000	−	0.866025i	$-0.666667\pi$
$563$	−0.558145	−	1.86433i	−0.558145	−	1.86433i	−0.0581448	−	0.998308i	$-0.518519\pi$
$564$		0			0
$565$		0			0
$566$	0.347296			0.347296
$567$		0			0
$568$		0			0
$569$	−0.997837	+	1.34033i	−0.997837	+	1.34033i	−0.0581448	+	0.998308i	$0.518519\pi$
$569$	−0.997837	+	1.34033i	−0.997837	+	1.34033i	−0.939693	+	0.342020i	$0.888889\pi$
$570$		0			0
$571$	0.569728	+	1.90302i	0.569728	+	1.90302i	0.396080	+	0.918216i	$0.370370\pi$
$571$	0.569728	+	1.90302i	0.569728	+	1.90302i	0.173648	+	0.984808i	$0.444444\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.835488	−	0.549509i	−0.835488	−	0.549509i
$577$	−1.82873	+	0.665602i	−1.82873	+	0.665602i	−0.835488	+	0.549509i	$0.814815\pi$
$577$	−1.82873	+	0.665602i	−1.82873	+	0.665602i	−0.993238	+	0.116093i	$0.962963\pi$
$578$	−0.332863	−	0.167170i	−0.332863	−	0.167170i
$579$	−0.0201935	+	0.114523i	−0.0201935	+	0.114523i
$580$		0			0
$581$		0			0
$582$	1.94609			1.94609
$583$		0			0
$584$	−1.52173	−	1.27688i	−1.52173	−	1.27688i
$585$		0			0
$586$		0			0
$587$	0.393633	−	1.31482i	0.393633	−	1.31482i	−0.500000	−	0.866025i	$-0.666667\pi$
$587$	0.393633	−	1.31482i	0.393633	−	1.31482i	0.893633	−	0.448799i	$-0.148148\pi$
$588$	−0.286803	−	0.957990i	−0.286803	−	0.957990i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$593$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$594$	−0.342534	−	0.460103i	−0.342534	−	0.460103i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.0581448	−	0.998308i	$-0.481481\pi$
$599$		0			0		−0.0581448	+	0.998308i	$0.518519\pi$
$600$	−0.993238	+	0.116093i	−0.993238	+	0.116093i
$601$	−0.558145	+	0.132283i	−0.558145	+	0.132283i	−0.500000	−	0.866025i	$-0.666667\pi$
$601$	−0.558145	+	0.132283i	−0.558145	+	0.132283i	−0.0581448	+	0.998308i	$0.518519\pi$
$602$		0			0
$603$	−1.33549	+	0.316516i	−1.33549	+	0.316516i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.597159	−	0.802123i	$-0.703704\pi$
$607$		0			0		0.597159	+	0.802123i	$0.296296\pi$
$608$	1.06728	+	1.43361i	1.06728	+	1.43361i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.227194	+	0.758881i	−0.227194	+	0.758881i
$613$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$613$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$614$	−1.62593	+	0.385353i	−1.62593	+	0.385353i
$615$		0			0
$616$		0			0
$617$	1.89363	+	0.448799i	1.89363	+	0.448799i	1.00000			$0$
$617$	1.89363	+	0.448799i	1.89363	+	0.448799i	0.893633	+	0.448799i	$0.148148\pi$
$618$		0			0
$619$	0.115503	+	0.0135004i	0.115503	+	0.0135004i	0.173648	−	0.984808i	$-0.444444\pi$
$619$	0.115503	+	0.0135004i	0.115503	+	0.0135004i	−0.0581448	+	0.998308i	$0.518519\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.686242	+	0.727374i	−0.686242	+	0.727374i
$626$	−0.103920	+	0.0521907i	−0.103920	+	0.0521907i
$627$	0.294027	+	0.982118i	0.294027	+	0.982118i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$631$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$632$		0			0
$633$	0.347296			0.347296
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.22650	−	1.30001i	−1.22650	−	1.30001i	−0.939693	−	0.342020i	$-0.888889\pi$
$641$	−1.22650	−	1.30001i	−1.22650	−	1.30001i	−0.286803	−	0.957990i	$-0.592593\pi$
$642$	0.115503	−	1.98312i	0.115503	−	1.98312i
$643$	0.479241	+	0.315202i	0.479241	+	0.315202i	0.766044	−	0.642788i	$-0.222222\pi$
$643$	0.479241	+	0.315202i	0.479241	+	0.315202i	−0.286803	+	0.957990i	$0.592593\pi$
$644$		0			0
$645$		0			0
$646$	0.845457	−	1.13565i	0.845457	−	1.13565i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−0.993238	−	0.116093i	−0.993238	−	0.116093i
$649$	0.958482			0.958482
$650$		0			0
$651$		0			0
$652$	0.539014	+	1.80043i	0.539014	+	1.80043i
$653$		0			0		−0.835488	−	0.549509i	$-0.814815\pi$
$653$		0			0		0.835488	+	0.549509i	$0.185185\pi$
$654$		0			0
$655$		0			0
$656$	0.109277	+	0.0397734i	0.109277	+	0.0397734i
$657$	−1.93293	−	0.458113i	−1.93293	−	0.458113i
$658$		0			0
$659$	0.310355	+	0.155866i	0.310355	+	0.155866i	0.597159	−	0.802123i	$-0.296296\pi$
$659$	0.310355	+	0.155866i	0.310355	+	0.155866i	−0.286803	+	0.957990i	$0.592593\pi$
$660$		0			0
$661$		0			0		0.396080	−	0.918216i	$-0.370370\pi$
$661$		0			0		−0.396080	+	0.918216i	$0.629630\pi$
$662$	−1.52173	+	0.177865i	−1.52173	+	0.177865i
$663$		0			0
$664$	−0.290162	+	0.190842i	−0.290162	+	0.190842i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.137557	+	0.318893i	0.137557	+	0.318893i	0.973045	−	0.230616i	$-0.0740741\pi$
$673$	0.137557	+	0.318893i	0.137557	+	0.318893i	−0.835488	+	0.549509i	$0.814815\pi$
$674$	−0.597159	−	1.03431i	−0.597159	−	1.03431i
$675$	−0.835488	+	0.549509i	−0.835488	+	0.549509i
$676$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$677$		0			0		−0.993238	−	0.116093i	$-0.962963\pi$
$677$		0			0		0.993238	+	0.116093i	$0.0370370\pi$
$678$	−1.12229	+	1.50750i	−1.12229	+	1.50750i
$679$		0			0
$680$		0			0
$681$	0.713197	+	0.957990i	0.713197	+	0.957990i
$682$		0			0
$683$	−0.238329	−	1.35163i	−0.238329	−	1.35163i	−0.835488	−	0.549509i	$-0.814815\pi$
$683$	−0.238329	−	1.35163i	−0.238329	−	1.35163i	0.597159	−	0.802123i	$-0.296296\pi$
$684$	1.59716	+	0.802123i	1.59716	+	0.802123i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	0.713197	+	0.957990i	0.713197	+	0.957990i
$689$		0			0
$690$		0			0
$691$	−0.0890830	−	1.52950i	−0.0890830	−	1.52950i	−0.686242	−	0.727374i	$-0.740741\pi$
$691$	−0.0890830	−	1.52950i	−0.0890830	−	1.52950i	0.597159	−	0.802123i	$-0.296296\pi$
$692$		0			0
$693$		0			0
$694$	0.310355	+	1.76011i	0.310355	+	1.76011i
$695$		0			0
$696$		0			0
$697$	0.00535630	−	0.0919641i	0.00535630	−	0.0919641i
$698$		0			0
$699$	−0.661840	−	1.53432i	−0.661840	−	1.53432i
$700$		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
$704$	−0.227194	−	0.526695i	−0.227194	−	0.526695i
$705$		0			0
$706$	1.36320	−	1.44491i	1.36320	−	1.44491i
$707$		0			0
$708$	1.14669	−	1.21542i	1.14669	−	1.21542i
$709$		0			0		0.286803	−	0.957990i	$-0.407407\pi$
$709$		0			0		−0.286803	+	0.957990i	$0.592593\pi$
$710$		0			0
$711$		0			0
$712$	1.17365	+	0.984808i	1.17365	+	0.984808i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.606829	−	1.40679i	0.606829	−	1.40679i
$717$		0			0
$718$		0			0
$719$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$719$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$720$		0			0
$721$		0			0
$722$	−1.50583	−	1.59609i	−1.50583	−	1.59609i
$723$	1.14669	+	0.754192i	1.14669	+	0.754192i
$724$		0			0
$725$		0			0
$726$	0.0390138	+	0.669840i	0.0390138	+	0.669840i
$727$		0			0		0.597159	−	0.802123i	$-0.296296\pi$
$727$		0			0		−0.597159	+	0.802123i	$0.703704\pi$
$728$		0			0
$729$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$730$		0			0
$731$	0.564966	−	0.758881i	0.564966	−	0.758881i
$732$		0			0
$733$		0			0		−0.286803	−	0.957990i	$-0.592593\pi$
$733$		0			0		0.286803	+	0.957990i	$0.407407\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.739787	−	0.269261i	−0.739787	−	0.269261i
$738$	0.115503	−	0.0135004i	0.115503	−	0.0135004i
$739$	0.539014	−	0.196185i	0.539014	−	0.196185i	−0.0581448	−	0.998308i	$-0.518519\pi$
$739$	0.539014	−	0.196185i	0.539014	−	0.196185i	0.597159	+	0.802123i	$0.296296\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.993238	−	0.116093i	$-0.0370370\pi$
$743$		0			0		−0.993238	+	0.116093i	$0.962963\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$748$	−0.348081	+	0.292075i	−0.348081	+	0.292075i
$749$		0			0
$750$		0			0
$751$		0			0		0.893633	−	0.448799i	$-0.148148\pi$
$751$		0			0		−0.893633	+	0.448799i	$0.851852\pi$
$752$		0			0
$753$	−0.342534	+	1.14414i	−0.342534	+	1.14414i
$754$		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	$0.666667\pi$
$758$	0.115503	+	0.0135004i	0.115503	+	0.0135004i
$759$		0			0
$760$		0			0
$761$	0.0581448	−	0.998308i	0.0581448	−	0.998308i	−0.835488	−	0.549509i	$-0.814815\pi$
$761$	0.0581448	−	0.998308i	0.0581448	−	0.998308i	0.893633	−	0.448799i	$-0.148148\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$769$	−1.12229	−	1.50750i	−1.12229	−	1.50750i	−0.835488	−	0.549509i	$-0.814815\pi$
$769$	−1.12229	−	1.50750i	−1.12229	−	1.50750i	−0.286803	−	0.957990i	$-0.592593\pi$
$770$		0			0
$771$	−0.238329	−	1.35163i	−0.238329	−	1.35163i
$772$	0.00676164	+	0.116093i	0.00676164	+	0.116093i
$773$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$773$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$774$	1.06728	+	0.536009i	1.06728	+	0.536009i
$775$		0			0
$776$	1.89363	−	0.448799i	1.89363	−	0.448799i
$777$		0			0
$778$		0			0
$779$	−0.202238	−	0.0479313i	−0.202238	−	0.0479313i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$785$		0			0
$786$	1.28971	+	1.36702i	1.28971	+	1.36702i
$787$	−0.238329	+	0.252614i	−0.238329	+	0.252614i	−0.835488	−	0.549509i	$-0.814815\pi$
$787$	−0.238329	+	0.252614i	−0.238329	+	0.252614i	0.597159	+	0.802123i	$0.296296\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−0.439408	−	0.368707i	−0.439408	−	0.368707i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.396080	−	0.918216i	$-0.370370\pi$
$797$		0			0		−0.396080	+	0.918216i	$0.629630\pi$
$798$		0			0
$799$		0			0
$800$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$801$	1.49079	+	0.353324i	1.49079	+	0.353324i
$802$	−1.67948	−	0.611281i	−1.67948	−	0.611281i
$803$	−0.781942	−	0.828810i	−0.781942	−	0.828810i
$804$	−1.22650	+	0.615969i	−1.22650	+	0.615969i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.98648			−1.98648			−0.993238	−	0.116093i	$-0.962963\pi$
$809$	−1.98648			−1.98648			−0.993238	+	0.116093i	$0.962963\pi$
$810$		0			0
$811$	−1.37248			−1.37248			−0.686242	−	0.727374i	$-0.740741\pi$
$811$	−1.37248			−1.37248			−0.686242	+	0.727374i	$0.740741\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−0.0460600	+	0.790819i	−0.0460600	+	0.790819i
$817$	−1.46483	−	1.55262i	−1.46483	−	1.55262i
$818$	−0.744386	−	0.270935i	−0.744386	−	0.270935i
$819$		0			0
$820$		0			0
$821$		0			0		−0.893633	−	0.448799i	$-0.851852\pi$
$821$		0			0		0.893633	+	0.448799i	$0.148148\pi$
$822$	0.337935	−	1.91652i	0.337935	−	1.91652i
$823$		0			0		0.396080	−	0.918216i	$-0.370370\pi$
$823$		0			0		−0.396080	+	0.918216i	$0.629630\pi$
$824$		0			0
$825$	−0.573606			−0.573606
$826$		0			0
$827$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$827$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$828$		0			0
$829$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$829$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.543613	+	0.576196i	−0.543613	+	0.576196i
$834$	1.89363	−	0.448799i	1.89363	−	0.448799i
$835$		0			0
$836$	0.512593	+	0.887838i	0.512593	+	0.887838i
$837$		0			0
$838$	0.939693	−	1.62760i	0.939693	−	1.62760i
$839$		0			0		−0.993238	−	0.116093i	$-0.962963\pi$
$839$		0			0		0.993238	+	0.116093i	$0.0370370\pi$
$840$		0			0
$841$	0.973045	+	0.230616i	0.973045	+	0.230616i
$842$		0			0
$843$	−0.344948	+	0.0403186i	−0.344948	+	0.0403186i
$844$	0.337935	−	0.0800921i	0.337935	−	0.0800921i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	0.266044	−	0.223238i	0.266044	−	0.223238i
$850$	0.473045	+	0.635410i	0.473045	+	0.635410i
$851$		0			0
$852$		0			0
$853$		0			0		−0.0581448	−	0.998308i	$-0.518519\pi$
$853$		0			0		0.0581448	+	0.998308i	$0.481481\pi$
$854$		0			0
$855$		0			0
$856$	−0.344948	−	1.95630i	−0.344948	−	1.95630i
$857$	−0.973045	+	0.230616i	−0.973045	+	0.230616i	−0.686242	−	0.727374i	$-0.740741\pi$
$857$	−0.973045	+	0.230616i	−0.973045	+	0.230616i	−0.286803	+	0.957990i	$0.592593\pi$
$858$		0			0
$859$	0.0333522	−	0.572636i	0.0333522	−	0.572636i	−0.939693	−	0.342020i	$-0.888889\pi$
$859$	0.0333522	−	0.572636i	0.0333522	−	0.572636i	0.973045	−	0.230616i	$-0.0740741\pi$
$860$		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	$0.666667\pi$
$864$	−0.993238	+	0.116093i	−0.993238	+	0.116093i
$865$		0			0
$866$	0.313758	+	0.727374i	0.313758	+	0.727374i
$867$	−0.362443	+	0.0859006i	−0.362443	+	0.0859006i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1.49079	−	1.25092i	1.49079	−	1.25092i
$874$		0			0
$875$		0			0
$876$	−1.98648			−1.98648
$877$		0			0		0.993238	−	0.116093i	$-0.0370370\pi$
$877$		0			0		−0.993238	+	0.116093i	$0.962963\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.43969	+	0.524005i	−1.43969	+	0.524005i	−0.939693	−	0.342020i	$-0.888889\pi$
$881$	−1.43969	+	0.524005i	−1.43969	+	0.524005i	−0.500000	+	0.866025i	$0.666667\pi$
$882$	−0.835488	−	0.549509i	−0.835488	−	0.549509i
$883$	−0.744386	−	0.270935i	−0.744386	−	0.270935i	−0.0581448	−	0.998308i	$-0.518519\pi$
$883$	−0.744386	−	0.270935i	−0.744386	−	0.270935i	−0.686242	+	0.727374i	$0.740741\pi$
$884$		0			0
$885$		0			0
$886$	0.0971586	+	0.0639022i	0.0971586	+	0.0639022i
$887$		0			0		−0.286803	−	0.957990i	$-0.592593\pi$
$887$		0			0		0.286803	+	0.957990i	$0.407407\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.558145	−	0.132283i	−0.558145	−	0.132283i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−1.33549	−	1.41553i	−1.33549	−	1.41553i
$899$		0			0
$900$	−0.686242	+	0.727374i	−0.686242	+	0.727374i
$901$		0			0
$902$	0.0596093	+	0.0299369i	0.0596093	+	0.0299369i
$903$		0			0
$904$	−0.744386	+	1.72568i	−0.744386	+	1.72568i
$905$		0			0
$906$		0			0
$907$	1.14669	−	0.754192i	1.14669	−	0.754192i	0.173648	−	0.984808i	$-0.444444\pi$
$907$	1.14669	−	0.754192i	1.14669	−	0.754192i	0.973045	+	0.230616i	$0.0740741\pi$
$908$	0.914900	+	0.767692i	0.914900	+	0.767692i
$909$		0			0
$910$		0			0
$911$		0			0		0.286803	−	0.957990i	$-0.407407\pi$
$911$		0			0		−0.286803	+	0.957990i	$0.592593\pi$
$912$	1.73909	+	0.412172i	1.73909	+	0.412172i
$913$	−0.178022	+	0.0894059i	−0.178022	+	0.0894059i
$914$	1.14669	−	1.21542i	1.14669	−	1.21542i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	0.313758	+	0.727374i	0.313758	+	0.727374i
$919$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$919$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$	−0.997837	+	1.34033i	−0.997837	+	1.34033i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.0581448	+	0.998308i	0.0581448	+	0.998308i	0.893633	+	0.448799i	$0.148148\pi$
$929$	0.0581448	+	0.998308i	0.0581448	+	0.998308i	−0.835488	+	0.549509i	$0.814815\pi$
$930$		0			0
$931$	1.06728	+	1.43361i	1.06728	+	1.43361i
$932$	−0.997837	−	1.34033i	−0.997837	−	1.34033i
$933$		0			0
$934$	−0.103920	−	1.78424i	−0.103920	−	1.78424i
$935$		0			0
$936$		0			0
$937$	0.266044	+	1.50881i	0.266044	+	1.50881i	0.766044	+	0.642788i	$0.222222\pi$
$937$	0.266044	+	1.50881i	0.266044	+	1.50881i	−0.500000	+	0.866025i	$0.666667\pi$
$938$		0			0
$939$	−0.0460600	+	0.106779i	−0.0460600	+	0.106779i
$940$		0			0
$941$		0			0		−0.973045	−	0.230616i	$-0.925926\pi$
$941$		0			0		0.973045	+	0.230616i	$0.0740741\pi$
$942$		0			0
$943$		0			0
$944$	0.835488	−	1.44711i	0.835488	−	1.44711i
$945$		0			0
$946$	0.342534	+	0.593286i	0.342534	+	0.593286i
$947$	−0.786803	−	1.82401i	−0.786803	−	1.82401i	−0.500000	−	0.866025i	$-0.666667\pi$
$947$	−0.786803	−	1.82401i	−0.786803	−	1.82401i	−0.286803	−	0.957990i	$-0.592593\pi$
$948$		0			0
$949$		0			0
$950$	1.59716	−	0.802123i	1.59716	−	0.802123i
$951$		0			0
$952$		0			0
$953$	−1.05138	+	0.882215i	−1.05138	+	0.882215i	−0.993238	−	0.116093i	$-0.962963\pi$
$953$	−1.05138	+	0.882215i	−1.05138	+	0.882215i	−0.0581448	+	0.998308i	$0.518519\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.893633	+	0.448799i	0.893633	+	0.448799i
$962$		0			0
$963$	−1.18624	−	1.59340i	−1.18624	−	1.59340i
$964$	1.28971	+	0.469417i	1.28971	+	0.469417i
$965$		0			0
$966$		0			0
$967$		0			0		−0.835488	−	0.549509i	$-0.814815\pi$
$967$		0			0		0.835488	+	0.549509i	$0.185185\pi$
$968$	0.192438	+	0.642788i	0.192438	+	0.642788i
$969$	−0.0823214	−	1.41340i	−0.0823214	−	1.41340i
$970$		0			0
$971$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$971$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$972$	−0.835488	+	0.549509i	−0.835488	+	0.549509i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.39608	+	0.918216i	1.39608	+	0.918216i	1.00000			$0$
$977$	1.39608	+	0.918216i	1.39608	+	0.918216i	0.396080	+	0.918216i	$0.370370\pi$
$978$	1.57020	+	1.03274i	1.57020	+	1.03274i
$979$	0.603080	+	0.639228i	0.603080	+	0.639228i
$980$		0			0
$981$		0			0
$982$	1.86668	−	0.679415i	1.86668	−	0.679415i
$983$		0			0		−0.893633	−	0.448799i	$-0.851852\pi$
$983$		0			0		0.893633	+	0.448799i	$0.148148\pi$
$984$	0.109277	−	0.0397734i	0.109277	−	0.0397734i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$991$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$992$		0			0
$993$	−1.05138	+	1.11440i	−1.05138	+	1.11440i
$994$		0			0
$995$		0			0
$996$	−0.0996057	+	0.332706i	−0.0996057	+	0.332706i
$997$		0			0		−0.396080	−	0.918216i	$-0.629630\pi$
$997$		0			0		0.396080	+	0.918216i	$0.370370\pi$
$998$	−0.396080	−	0.686030i	−0.396080	−	0.686030i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.1.bf.a.139.1	✓	18
3.2	odd	2		1944.1.bf.a.955.1		18
4.3	odd	2		2592.1.bz.a.463.1		18
8.3	odd	2	CM	648.1.bf.a.139.1	✓	18
8.5	even	2		2592.1.bz.a.463.1		18
24.11	even	2		1944.1.bf.a.955.1		18
81.7	even	27	inner	648.1.bf.a.331.1	yes	18
81.74	odd	54		1944.1.bf.a.1315.1		18
324.7	odd	54		2592.1.bz.a.655.1		18
648.155	even	54		1944.1.bf.a.1315.1		18
648.331	odd	54	inner	648.1.bf.a.331.1	yes	18
648.493	even	54		2592.1.bz.a.655.1		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.1.bf.a.139.1	✓	18	1.1	even	1	trivial
648.1.bf.a.139.1	✓	18	8.3	odd	2	CM
648.1.bf.a.331.1	yes	18	81.7	even	27	inner
648.1.bf.a.331.1	yes	18	648.331	odd	54	inner
1944.1.bf.a.955.1		18	3.2	odd	2
1944.1.bf.a.955.1		18	24.11	even	2
1944.1.bf.a.1315.1		18	81.74	odd	54
1944.1.bf.a.1315.1		18	648.155	even	54
2592.1.bz.a.463.1		18	4.3	odd	2
2592.1.bz.a.463.1		18	8.5	even	2
2592.1.bz.a.655.1		18	324.7	odd	54
2592.1.bz.a.655.1		18	648.493	even	54

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:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	648.bf (of order \(54\), degree \(18\), minimal)

\(f(q)\)	\(=\)	\(q+(0.597159 - 0.802123i) q^{2} +(-0.0581448 - 0.998308i) q^{3} +(-0.286803 - 0.957990i) q^{4} +(-0.835488 - 0.549509i) q^{6} +(-0.939693 - 0.342020i) q^{8} +(-0.993238 + 0.116093i) q^{9} +O(q^{10})\)	\(q+(0.597159 - 0.802123i) q^{2} +(-0.0581448 - 0.998308i) q^{3} +(-0.286803 - 0.957990i) q^{4} +(-0.835488 - 0.549509i) q^{6} +(-0.939693 - 0.342020i) q^{8} +(-0.993238 + 0.116093i) q^{9} +(-0.512593 - 0.257434i) q^{11} +(-0.939693 + 0.342020i) q^{12} +(-0.835488 + 0.549509i) q^{16} +(0.606829 + 0.509190i) q^{17} +(-0.500000 + 0.866025i) q^{18} +(1.36912 - 1.14883i) q^{19} +(-0.512593 + 0.257434i) q^{22} +(-0.286803 + 0.957990i) q^{24} +(0.396080 + 0.918216i) q^{25} +(0.173648 + 0.984808i) q^{27} +(-0.0581448 + 0.998308i) q^{32} +(-0.227194 + 0.526695i) q^{33} +(0.770807 - 0.182685i) q^{34} +(0.396080 + 0.918216i) q^{36} +(-0.103920 - 1.78424i) q^{38} +(-0.0694434 - 0.0932786i) q^{41} +(-0.0694434 - 1.19230i) q^{43} +(-0.0996057 + 0.564892i) q^{44} +(0.597159 + 0.802123i) q^{48} +(-0.0581448 + 0.998308i) q^{49} +(0.973045 + 0.230616i) q^{50} +(0.473045 - 0.635410i) q^{51} +(0.893633 + 0.448799i) q^{54} +(-1.22650 - 1.30001i) q^{57} +(-1.49324 + 0.749932i) q^{59} +(0.766044 + 0.642788i) q^{64} +(0.286803 + 0.496758i) q^{66} +(1.36320 - 0.159336i) q^{67} +(0.313758 - 0.727374i) q^{68} +(0.973045 + 0.230616i) q^{72} +(1.86668 + 0.679415i) q^{73} +(0.893633 - 0.448799i) q^{75} +(-1.49324 - 0.982118i) q^{76} +(0.973045 - 0.230616i) q^{81} -0.116290 q^{82} +(0.207391 - 0.278574i) q^{83} +(-0.997837 - 0.656288i) q^{86} +(0.393633 + 0.417226i) q^{88} +(-1.43969 - 0.524005i) q^{89} +1.00000 q^{96} +(-1.62593 + 1.06939i) q^{97} +(0.766044 + 0.642788i) q^{98} +(0.539014 + 0.196185i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q+O(q^{10}) \) 18 * q	\( 18 q - 9 q^{18} - 9 q^{38} - 9 q^{51} - 9 q^{59} + 18 q^{68} - 9 q^{76} - 9 q^{88} - 9 q^{89} + 18 q^{96}+O(q^{100}) \) 18 * q - 9 * q^18 - 9 * q^38 - 9 * q^51 - 9 * q^59 + 18 * q^68 - 9 * q^76 - 9 * q^88 - 9 * q^89 + 18 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

Properties

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