LMFDB - Embedded newform 1800.1.cr.a.1333.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,1,Mod(37,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([0, 10, 0, 9]))

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

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

chi := DirichletCharacter("1800.37");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1800.cr (of order $20$, degree $8$, 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:	$0.898317022739$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{20})$
Coefficient field:	$\Q(\zeta_{40})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{20}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{20} - \cdots)$

Embedding invariants

Embedding label			1333.1
Root			$-0.453990 - 0.891007i$ of defining polynomial
Character	$\chi$	$=$	1800.1333
Dual form			1800.1.cr.a.397.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.453990 + 0.891007i) q^{2} +(-0.587785 - 0.809017i) q^{4} +(0.891007 + 0.453990i) q^{5} +(1.39680 - 1.39680i) q^{7} +(0.987688 - 0.156434i) q^{8} +O(q^{10})$	\(q+(-0.453990 + 0.891007i) q^{2} +(-0.587785 - 0.809017i) q^{4} +(0.891007 + 0.453990i) q^{5} +(1.39680 - 1.39680i) q^{7} +(0.987688 - 0.156434i) q^{8} +(-0.809017 + 0.587785i) q^{10} +(-0.863541 + 0.280582i) q^{11} +(0.610425 + 1.87869i) q^{14} +(-0.309017 + 0.951057i) q^{16} +(-0.156434 - 0.987688i) q^{20} +(0.142040 - 0.896802i) q^{22} +(0.587785 + 0.809017i) q^{25} +(-1.95106 - 0.309017i) q^{28} +(1.14412 - 0.831254i) q^{29} +(-0.951057 - 0.690983i) q^{31} +(-0.707107 - 0.707107i) q^{32} +(1.87869 - 0.610425i) q^{35} +(0.951057 + 0.309017i) q^{40} +(0.734572 + 0.533698i) q^{44} -2.90211i q^{49} +(-0.987688 + 0.156434i) q^{50} +(-0.297556 + 1.87869i) q^{53} +(-0.896802 - 0.142040i) q^{55} +(1.16110 - 1.59811i) q^{56} +(0.221232 + 1.39680i) q^{58} +(0.550672 - 1.69480i) q^{59} +(1.04744 - 0.533698i) q^{62} +(0.951057 - 0.309017i) q^{64} +(-0.309017 + 1.95106i) q^{70} +(-0.642040 + 1.26007i) q^{73} +(-0.814279 + 1.59811i) q^{77} +(1.11803 + 1.53884i) q^{79} +(-0.707107 + 0.707107i) q^{80} +(-1.59811 + 0.253116i) q^{83} +(-0.809017 + 0.412215i) q^{88} +(-0.0489435 + 0.309017i) q^{97} +(2.58580 + 1.31753i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{7}+O(q^{10}) $ 16 * q + 4 * q^7	$ 16 q + 4 q^{7} - 4 q^{10} + 4 q^{16} - 4 q^{22} - 16 q^{28} + 4 q^{55} + 4 q^{58} + 4 q^{70} - 4 q^{73} - 4 q^{88} - 16 q^{97}+O(q^{100}) $ 16 * q + 4 * q^7 - 4 * q^10 + 4 * q^16 - 4 * q^22 - 16 * q^28 + 4 * q^55 + 4 * q^58 + 4 * q^70 - 4 * q^73 - 4 * q^88 - 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$e\left(\frac{3}{20}\right)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	−0.453990	+	0.891007i	−0.453990	+	0.891007i
$2$	−0.453990	+	0.891007i	−0.453990	+	0.891007i
$3$		0			0
$3$		0			0
$4$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$5$	0.891007	+	0.453990i	0.891007	+	0.453990i
$5$	0.891007	+	0.453990i	0.891007	+	0.453990i
$6$		0			0
$7$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.587785	−	0.809017i	$-0.300000\pi$
$7$	1.39680	−	1.39680i	1.39680	−	1.39680i	0.809017	−	0.587785i	$-0.200000\pi$
$8$	0.987688	−	0.156434i	0.987688	−	0.156434i
$9$		0			0
$10$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$11$	−0.863541	+	0.280582i	−0.863541	+	0.280582i	−0.707107	−	0.707107i	$-0.750000\pi$
$11$	−0.863541	+	0.280582i	−0.863541	+	0.280582i	−0.156434	+	0.987688i	$0.550000\pi$
$12$		0			0
$13$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$13$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$14$	0.610425	+	1.87869i	0.610425	+	1.87869i
$15$		0			0
$16$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$17$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$17$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$18$		0			0
$19$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$19$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$20$	−0.156434	−	0.987688i	−0.156434	−	0.987688i
$21$		0			0
$22$	0.142040	−	0.896802i	0.142040	−	0.896802i
$23$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$23$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$24$		0			0
$25$	0.587785	+	0.809017i	0.587785	+	0.809017i
$26$		0			0
$27$		0			0
$28$	−1.95106	−	0.309017i	−1.95106	−	0.309017i
$29$	1.14412	−	0.831254i	1.14412	−	0.831254i	0.156434	−	0.987688i	$-0.450000\pi$
$29$	1.14412	−	0.831254i	1.14412	−	0.831254i	0.987688	+	0.156434i	$0.0500000\pi$
$30$		0			0
$31$	−0.951057	−	0.690983i	−0.951057	−	0.690983i		−	1.00000i	$-0.5\pi$
$31$	−0.951057	−	0.690983i	−0.951057	−	0.690983i	−0.951057	+	0.309017i	$0.900000\pi$
$32$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$33$		0			0
$34$		0			0
$35$	1.87869	−	0.610425i	1.87869	−	0.610425i
$36$		0			0
$37$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$37$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$38$		0			0
$39$		0			0
$40$	0.951057	+	0.309017i	0.951057	+	0.309017i
$41$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$41$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$	0.734572	+	0.533698i	0.734572	+	0.533698i
$45$		0			0
$46$		0			0
$47$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$47$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$48$		0			0
$49$		−	2.90211i		−	2.90211i
$50$	−0.987688	+	0.156434i	−0.987688	+	0.156434i
$51$		0			0
$52$		0			0
$53$	−0.297556	+	1.87869i	−0.297556	+	1.87869i	0.156434	+	0.987688i	$0.450000\pi$
$53$	−0.297556	+	1.87869i	−0.297556	+	1.87869i	−0.453990	+	0.891007i	$0.650000\pi$
$54$		0			0
$55$	−0.896802	−	0.142040i	−0.896802	−	0.142040i
$56$	1.16110	−	1.59811i	1.16110	−	1.59811i
$57$		0			0
$58$	0.221232	+	1.39680i	0.221232	+	1.39680i
$59$	0.550672	−	1.69480i	0.550672	−	1.69480i	−0.156434	−	0.987688i	$-0.550000\pi$
$59$	0.550672	−	1.69480i	0.550672	−	1.69480i	0.707107	−	0.707107i	$-0.250000\pi$
$60$		0			0
$61$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$61$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$62$	1.04744	−	0.533698i	1.04744	−	0.533698i
$63$		0			0
$64$	0.951057	−	0.309017i	0.951057	−	0.309017i
$65$		0			0
$66$		0			0
$67$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$67$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$68$		0			0
$69$		0			0
$70$	−0.309017	+	1.95106i	−0.309017	+	1.95106i
$71$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$71$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$72$		0			0
$73$	−0.642040	+	1.26007i	−0.642040	+	1.26007i	0.309017	+	0.951057i	$0.400000\pi$
$73$	−0.642040	+	1.26007i	−0.642040	+	1.26007i	−0.951057	+	0.309017i	$0.900000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.814279	+	1.59811i	−0.814279	+	1.59811i
$78$		0			0
$79$	1.11803	+	1.53884i	1.11803	+	1.53884i	0.809017	+	0.587785i	$0.200000\pi$
$79$	1.11803	+	1.53884i	1.11803	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$80$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$81$		0			0
$82$		0			0
$83$	−1.59811	+	0.253116i	−1.59811	+	0.253116i	−0.891007	−	0.453990i	$-0.850000\pi$
$83$	−1.59811	+	0.253116i	−1.59811	+	0.253116i	−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−0.809017	+	0.412215i	−0.809017	+	0.412215i
$89$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$89$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.0489435	+	0.309017i	−0.0489435	+	0.309017i	0.951057	+	0.309017i	$0.100000\pi$
$97$	−0.0489435	+	0.309017i	−0.0489435	+	0.309017i	−1.00000			$\pi$
$98$	2.58580	+	1.31753i	2.58580	+	1.31753i
$99$		0			0
$100$	0.309017	−	0.951057i	0.309017	−	0.951057i
$101$			1.78201i			1.78201i	0.453990	+	0.891007i	$0.350000\pi$
$101$			1.78201i			1.78201i	−0.453990	+	0.891007i	$0.650000\pi$
$102$		0			0
$103$	0.309017	+	0.0489435i	0.309017	+	0.0489435i	0.309017	−	0.951057i	$-0.400000\pi$
$103$	0.309017	+	0.0489435i	0.309017	+	0.0489435i			1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$	−1.53884	−	1.11803i	−1.53884	−	1.11803i
$107$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	0.453990	−	0.891007i	$-0.350000\pi$
$107$	−0.437016	−	0.437016i	−0.437016	−	0.437016i	−0.891007	+	0.453990i	$0.850000\pi$
$108$		0			0
$109$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$109$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$110$	0.533698	−	0.734572i	0.533698	−	0.734572i
$111$		0			0
$112$	0.896802	+	1.76007i	0.896802	+	1.76007i
$113$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$113$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$114$		0			0
$115$		0			0
$116$	−1.34500	−	0.437016i	−1.34500	−	0.437016i
$117$		0			0
$118$	1.26007	+	1.26007i	1.26007	+	1.26007i
$119$		0			0
$120$		0			0
$121$	−0.142040	+	0.103198i	−0.142040	+	0.103198i
$122$		0			0
$123$		0			0
$124$			1.17557i			1.17557i
$125$	0.156434	+	0.987688i	0.156434	+	0.987688i
$126$		0			0
$127$	1.58779	+	0.809017i	1.58779	+	0.809017i	1.00000			$0$
$127$	1.58779	+	0.809017i	1.58779	+	0.809017i	0.587785	+	0.809017i	$0.300000\pi$
$128$	−0.156434	+	0.987688i	−0.156434	+	0.987688i
$129$		0			0
$130$		0			0
$131$	0.831254	−	1.14412i	0.831254	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$131$	0.831254	−	1.14412i	0.831254	−	1.14412i	0.987688	−	0.156434i	$-0.0500000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$137$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$138$		0			0
$139$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$139$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$140$	−1.59811	−	1.16110i	−1.59811	−	1.16110i
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	1.39680	−	0.221232i	1.39680	−	0.221232i
$146$	−0.831254	−	1.14412i	−0.831254	−	1.14412i
$147$		0			0
$148$		0			0
$149$	−1.78201			−1.78201			−0.891007	−	0.453990i	$-0.850000\pi$
$149$	−1.78201			−1.78201			−0.891007	+	0.453990i	$0.850000\pi$
$150$		0			0
$151$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$151$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$152$		0			0
$153$		0			0
$154$	−1.05425	−	1.45106i	−1.05425	−	1.45106i
$155$	−0.533698	−	1.04744i	−0.533698	−	1.04744i
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$	−1.87869	+	0.297556i	−1.87869	+	0.297556i
$159$		0			0
$160$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$161$		0			0
$162$		0			0
$163$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$163$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$164$		0			0
$165$		0			0
$166$	0.500000	−	1.53884i	0.500000	−	1.53884i
$167$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$167$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$168$		0			0
$169$	0.587785	−	0.809017i	0.587785	−	0.809017i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.44168	−	0.734572i	−1.44168	−	0.734572i	−0.453990	−	0.891007i	$-0.650000\pi$
$173$	−1.44168	−	0.734572i	−1.44168	−	0.734572i	−0.987688	+	0.156434i	$0.950000\pi$
$174$		0			0
$175$	1.95106	+	0.309017i	1.95106	+	0.309017i
$176$		−	0.907981i		−	0.907981i
$177$		0			0
$178$		0			0
$179$	0.253116	−	0.183900i	0.253116	−	0.183900i	−0.453990	−	0.891007i	$-0.650000\pi$
$179$	0.253116	−	0.183900i	0.253116	−	0.183900i	0.707107	+	0.707107i	$0.250000\pi$
$180$		0			0
$181$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$181$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$191$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$192$		0			0
$193$	0.221232	+	0.221232i	0.221232	+	0.221232i	0.809017	−	0.587785i	$-0.200000\pi$
$193$	0.221232	+	0.221232i	0.221232	+	0.221232i	−0.587785	+	0.809017i	$0.700000\pi$
$194$	−0.253116	−	0.183900i	−0.253116	−	0.183900i
$195$		0			0
$196$	−2.34786	+	1.70582i	−2.34786	+	1.70582i
$197$	−1.16110	−	0.183900i	−1.16110	−	0.183900i	−0.453990	−	0.891007i	$-0.650000\pi$
$197$	−1.16110	−	0.183900i	−1.16110	−	0.183900i	−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$199$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$200$	0.707107	+	0.707107i	0.707107	+	0.707107i
$201$		0			0
$202$	−1.58779	−	0.809017i	−1.58779	−	0.809017i
$203$	0.437016	−	2.75921i	0.437016	−	2.75921i
$204$		0			0
$205$		0			0
$206$	−0.183900	+	0.253116i	−0.183900	+	0.253116i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$211$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$212$	1.69480	−	0.863541i	1.69480	−	0.863541i
$213$		0			0
$214$	0.587785	−	0.190983i	0.587785	−	0.190983i
$215$		0			0
$216$		0			0
$217$	−2.29360	+	0.363271i	−2.29360	+	0.363271i
$218$		0			0
$219$		0			0
$220$	0.412215	+	0.809017i	0.412215	+	0.809017i
$221$		0			0
$222$		0			0
$223$	−0.412215	+	0.809017i	−0.412215	+	0.809017i	0.587785	+	0.809017i	$0.300000\pi$
$223$	−0.412215	+	0.809017i	−0.412215	+	0.809017i	−1.00000			$\pi$
$224$	−1.97538			−1.97538
$225$		0			0
$226$		0			0
$227$	−0.863541	+	1.69480i	−0.863541	+	1.69480i	−0.156434	+	0.987688i	$0.550000\pi$
$227$	−0.863541	+	1.69480i	−0.863541	+	1.69480i	−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$229$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$230$		0			0
$231$		0			0
$232$	1.00000	−	1.00000i	1.00000	−	1.00000i
$233$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$233$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$234$		0			0
$235$		0			0
$236$	−1.69480	+	0.550672i	−1.69480	+	0.550672i
$237$		0			0
$238$		0			0
$239$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$239$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$240$		0			0
$241$	0.363271	−	1.11803i	0.363271	−	1.11803i	−0.587785	−	0.809017i	$-0.700000\pi$
$241$	0.363271	−	1.11803i	0.363271	−	1.11803i	0.951057	−	0.309017i	$-0.100000\pi$
$242$	−0.0274653	−	0.173409i	−0.0274653	−	0.173409i
$243$		0			0
$244$		0			0
$245$	1.31753	−	2.58580i	1.31753	−	2.58580i
$246$		0			0
$247$		0			0
$248$	−1.04744	−	0.533698i	−1.04744	−	0.533698i
$249$		0			0
$250$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$251$			0.312869i			0.312869i	0.987688	+	0.156434i	$0.0500000\pi$
$251$			0.312869i			0.312869i	−0.987688	+	0.156434i	$0.950000\pi$
$252$		0			0
$253$		0			0
$254$	−1.44168	+	1.04744i	−1.44168	+	1.04744i
$255$		0			0
$256$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$257$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$257$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	0.642040	+	1.26007i	0.642040	+	1.26007i
$263$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$263$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$264$		0			0
$265$	−1.11803	+	1.53884i	−1.11803	+	1.53884i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.59811	+	1.16110i	1.59811	+	1.16110i	0.891007	+	0.453990i	$0.150000\pi$
$269$	1.59811	+	1.16110i	1.59811	+	1.16110i	0.707107	+	0.707107i	$0.250000\pi$
$270$		0			0
$271$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−0.309017	+	0.951057i	$0.600000\pi$
$271$	−1.30902	+	0.951057i	−1.30902	+	0.951057i	−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.734572	−	0.533698i	−0.734572	−	0.533698i
$276$		0			0
$277$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$277$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$278$		0			0
$279$		0			0
$280$	1.76007	−	0.896802i	1.76007	−	0.896802i
$281$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$281$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$282$		0			0
$283$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$283$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$290$	−0.437016	+	1.34500i	−0.437016	+	1.34500i
$291$		0			0
$292$	1.39680	−	0.221232i	1.39680	−	0.221232i
$293$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.453990	−	0.891007i	$-0.350000\pi$
$293$	1.34500	−	1.34500i	1.34500	−	1.34500i	0.891007	−	0.453990i	$-0.150000\pi$
$294$		0			0
$295$	1.26007	−	1.26007i	1.26007	−	1.26007i
$296$		0			0
$297$		0			0
$298$	0.809017	−	1.58779i	0.809017	−	1.58779i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	0.734572	−	1.44168i	0.734572	−	1.44168i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$	1.77152	−	0.280582i	1.77152	−	0.280582i
$309$		0			0
$310$	1.17557			1.17557
$311$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$311$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$312$		0			0
$313$	0.809017	−	0.412215i	0.809017	−	0.412215i		−	1.00000i	$-0.5\pi$
$313$	0.809017	−	0.412215i	0.809017	−	0.412215i	0.809017	+	0.587785i	$0.200000\pi$
$314$		0			0
$315$		0			0
$316$	0.587785	−	1.80902i	0.587785	−	1.80902i
$317$	0.0966818	+	0.610425i	0.0966818	+	0.610425i	0.987688	+	0.156434i	$0.0500000\pi$
$317$	0.0966818	+	0.610425i	0.0966818	+	0.610425i	−0.891007	+	0.453990i	$0.850000\pi$
$318$		0			0
$319$	−0.754763	+	1.03884i	−0.754763	+	1.03884i
$320$	0.987688	+	0.156434i	0.987688	+	0.156434i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$331$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$332$	1.14412	+	1.14412i	1.14412	+	1.14412i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.142040	+	0.278768i	0.142040	+	0.278768i	0.951057	−	0.309017i	$-0.100000\pi$
$337$	0.142040	+	0.278768i	0.142040	+	0.278768i	−0.809017	+	0.587785i	$0.800000\pi$
$338$	0.453990	+	0.891007i	0.453990	+	0.891007i
$339$		0			0
$340$		0			0
$341$	1.01515	+	0.329843i	1.01515	+	0.329843i
$342$		0			0
$343$	−2.65688	−	2.65688i	−2.65688	−	2.65688i
$344$		0			0
$345$		0			0
$346$	1.30902	−	0.951057i	1.30902	−	0.951057i
$347$	−1.87869	−	0.297556i	−1.87869	−	0.297556i	−0.891007	−	0.453990i	$-0.850000\pi$
$347$	−1.87869	−	0.297556i	−1.87869	−	0.297556i	−0.987688	+	0.156434i	$0.950000\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	−1.16110	+	1.59811i	−1.16110	+	1.59811i
$351$		0			0
$352$	0.809017	+	0.412215i	0.809017	+	0.412215i
$353$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$353$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	0.0489435	+	0.309017i	0.0489435	+	0.309017i
$359$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$359$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$360$		0			0
$361$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.14412	+	0.831254i	−1.14412	+	0.831254i
$366$		0			0
$367$	0.309017	−	0.0489435i	0.309017	−	0.0489435i		−	1.00000i	$-0.5\pi$
$367$	0.309017	−	0.0489435i	0.309017	−	0.0489435i	0.309017	+	0.951057i	$0.400000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.20854	+	3.03979i	2.20854	+	3.03979i
$372$		0			0
$373$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$373$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$379$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$383$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$384$		0			0
$385$	−1.45106	+	1.05425i	−1.45106	+	1.05425i
$386$	−0.297556	+	0.0966818i	−0.297556	+	0.0966818i
$387$		0			0
$388$	0.278768	−	0.142040i	0.278768	−	0.142040i
$389$	−0.280582	−	0.863541i	−0.280582	−	0.863541i	−0.987688	−	0.156434i	$-0.950000\pi$
$389$	−0.280582	−	0.863541i	−0.280582	−	0.863541i	0.707107	−	0.707107i	$-0.250000\pi$
$390$		0			0
$391$		0			0
$392$	−0.453990	−	2.86638i	−0.453990	−	2.86638i
$393$		0			0
$394$	0.690983	−	0.951057i	0.690983	−	0.951057i
$395$	0.297556	+	1.87869i	0.297556	+	1.87869i
$396$		0			0
$397$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$397$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$398$	0.550672	+	0.280582i	0.550672	+	0.280582i
$399$		0			0
$400$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$	1.44168	−	1.04744i	1.44168	−	1.04744i
$405$		0			0
$406$	2.26007	+	1.64204i	2.26007	+	1.64204i
$407$		0			0
$408$		0			0
$409$	−1.80902	−	0.587785i	−1.80902	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$409$	−1.80902	−	0.587785i	−1.80902	−	0.587785i	−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$	−0.142040	−	0.278768i	−0.142040	−	0.278768i
$413$	−1.59811	−	3.13647i	−1.59811	−	3.13647i
$414$		0			0
$415$	−1.53884	−	0.500000i	−1.53884	−	0.500000i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1.59811	−	1.16110i	−1.59811	−	1.16110i	−0.891007	−	0.453990i	$-0.850000\pi$
$419$	−1.59811	−	1.16110i	−1.59811	−	1.16110i	−0.707107	−	0.707107i	$-0.750000\pi$
$420$		0			0
$421$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$421$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$422$		0			0
$423$		0			0
$424$			1.90211i			1.90211i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.0966818	+	0.610425i	−0.0966818	+	0.610425i
$429$		0			0
$430$		0			0
$431$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$431$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$432$		0			0
$433$	0.142040	+	0.896802i	0.142040	+	0.896802i	0.951057	+	0.309017i	$0.100000\pi$
$433$	0.142040	+	0.896802i	0.142040	+	0.896802i	−0.809017	+	0.587785i	$0.800000\pi$
$434$	0.717598	−	2.20854i	0.717598	−	2.20854i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−0.587785	+	0.190983i	−0.587785	+	0.190983i	−0.587785	−	0.809017i	$-0.700000\pi$
$439$	−0.587785	+	0.190983i	−0.587785	+	0.190983i			1.00000i	$0.5\pi$
$440$	−0.907981			−0.907981
$441$		0			0
$442$		0			0
$443$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	−0.987688	−	0.156434i	$-0.950000\pi$
$443$	−0.831254	+	0.831254i	−0.831254	+	0.831254i	0.156434	+	0.987688i	$0.450000\pi$
$444$		0			0
$445$		0			0
$446$	−0.533698	−	0.734572i	−0.533698	−	0.734572i
$447$		0			0
$448$	0.896802	−	1.76007i	0.896802	−	1.76007i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$	−1.11803	−	1.53884i	−1.11803	−	1.53884i
$455$		0			0
$456$		0			0
$457$	1.26007	−	1.26007i	1.26007	−	1.26007i	0.309017	−	0.951057i	$-0.400000\pi$
$457$	1.26007	−	1.26007i	1.26007	−	1.26007i	0.951057	−	0.309017i	$-0.100000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.87869	−	0.610425i	1.87869	−	0.610425i	0.891007	−	0.453990i	$-0.150000\pi$
$461$	1.87869	−	0.610425i	1.87869	−	0.610425i	0.987688	−	0.156434i	$-0.0500000\pi$
$462$		0			0
$463$	−1.26007	+	0.642040i	−1.26007	+	0.642040i	−0.951057	−	0.309017i	$-0.900000\pi$
$463$	−1.26007	+	0.642040i	−1.26007	+	0.642040i	−0.309017	+	0.951057i	$0.600000\pi$
$464$	0.437016	+	1.34500i	0.437016	+	1.34500i
$465$		0			0
$466$		0			0
$467$	−0.0966818	−	0.610425i	−0.0966818	−	0.610425i	−0.987688	−	0.156434i	$-0.950000\pi$
$467$	−0.0966818	−	0.610425i	−0.0966818	−	0.610425i	0.891007	−	0.453990i	$-0.150000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	0.278768	−	1.76007i	0.278768	−	1.76007i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$479$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$480$		0			0
$481$		0			0
$482$	0.831254	+	0.831254i	0.831254	+	0.831254i
$483$		0			0
$484$	0.166977	+	0.0542543i	0.166977	+	0.0542543i
$485$	−0.183900	+	0.253116i	−0.183900	+	0.253116i
$486$		0			0
$487$	0.809017	+	1.58779i	0.809017	+	1.58779i	0.809017	+	0.587785i	$0.200000\pi$
$487$	0.809017	+	1.58779i	0.809017	+	1.58779i			1.00000i	$0.500000\pi$
$488$		0			0
$489$		0			0
$490$	1.70582	+	2.34786i	1.70582	+	2.34786i
$491$	−0.297556	−	0.0966818i	−0.297556	−	0.0966818i	0.156434	−	0.987688i	$-0.450000\pi$
$491$	−0.297556	−	0.0966818i	−0.297556	−	0.0966818i	−0.453990	+	0.891007i	$0.650000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.951057	−	0.690983i	0.951057	−	0.690983i
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	0.707107	−	0.707107i	0.707107	−	0.707107i
$501$		0			0
$502$	−0.278768	−	0.142040i	−0.278768	−	0.142040i
$503$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$503$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$504$		0			0
$505$	−0.809017	+	1.58779i	−0.809017	+	1.58779i
$506$		0			0
$507$		0			0
$508$	−0.278768	−	1.76007i	−0.278768	−	1.76007i
$509$	0.280582	−	0.863541i	0.280582	−	0.863541i	−0.707107	−	0.707107i	$-0.750000\pi$
$509$	0.280582	−	0.863541i	0.280582	−	0.863541i	0.987688	−	0.156434i	$-0.0500000\pi$
$510$		0			0
$511$	0.863271	+	2.65688i	0.863271	+	2.65688i
$512$	0.891007	−	0.453990i	0.891007	−	0.453990i
$513$		0			0
$514$		0			0
$515$	0.253116	+	0.183900i	0.253116	+	0.183900i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$521$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$522$		0			0
$523$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$523$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$524$	−1.41421			−1.41421
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.587785	+	0.809017i	0.587785	+	0.809017i
$530$	−0.863541	−	1.69480i	−0.863541	−	1.69480i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$536$		0			0
$537$		0			0
$538$	−1.76007	+	0.896802i	−1.76007	+	0.896802i
$539$	0.814279	+	2.50609i	0.814279	+	2.50609i
$540$		0			0
$541$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$541$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$542$	−0.253116	−	1.59811i	−0.253116	−	1.59811i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$547$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$548$		0			0
$549$		0			0
$550$	0.809017	−	0.412215i	0.809017	−	0.412215i
$551$		0			0
$552$		0			0
$553$	3.71113	+	0.587785i	3.71113	+	0.587785i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	0.831254	+	0.831254i	0.831254	+	0.831254i	0.987688	−	0.156434i	$-0.0500000\pi$
$557$	0.831254	+	0.831254i	0.831254	+	0.831254i	−0.156434	+	0.987688i	$0.550000\pi$
$558$		0			0
$559$		0			0
$560$			1.97538i			1.97538i
$561$		0			0
$562$		0			0
$563$	0.533698	+	1.04744i	0.533698	+	1.04744i	0.987688	+	0.156434i	$0.0500000\pi$
$563$	0.533698	+	1.04744i	0.533698	+	1.04744i	−0.453990	+	0.891007i	$0.650000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$569$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$570$		0			0
$571$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$571$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1.76007	−	0.896802i	−1.76007	−	0.896802i	−0.951057	−	0.309017i	$-0.900000\pi$
$577$	−1.76007	−	0.896802i	−1.76007	−	0.896802i	−0.809017	−	0.587785i	$-0.800000\pi$
$578$	0.156434	−	0.987688i	0.156434	−	0.987688i
$579$		0			0
$580$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$581$	−1.87869	+	2.58580i	−1.87869	+	2.58580i
$582$		0			0
$583$	−0.270175	−	1.70582i	−0.270175	−	1.70582i
$584$	−0.437016	+	1.34500i	−0.437016	+	1.34500i
$585$		0			0
$586$	0.587785	+	1.80902i	0.587785	+	1.80902i
$587$	0.550672	−	0.280582i	0.550672	−	0.280582i	−0.156434	−	0.987688i	$-0.550000\pi$
$587$	0.550672	−	0.280582i	0.550672	−	0.280582i	0.707107	+	0.707107i	$0.250000\pi$
$588$		0			0
$589$		0			0
$590$	0.550672	+	1.69480i	0.550672	+	1.69480i
$591$		0			0
$592$		0			0
$593$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$593$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$594$		0			0
$595$		0			0
$596$	1.04744	+	1.44168i	1.04744	+	1.44168i
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$601$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$602$		0			0
$603$		0			0
$604$	0.951057	+	1.30902i	0.951057	+	1.30902i
$605$	−0.173409	+	0.0274653i	−0.173409	+	0.0274653i
$606$		0			0
$607$	−0.642040	+	0.642040i	−0.642040	+	0.642040i	−0.951057	−	0.309017i	$-0.900000\pi$
$607$	−0.642040	+	0.642040i	−0.642040	+	0.642040i	0.309017	+	0.951057i	$0.400000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$613$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$614$		0			0
$615$		0			0
$616$	−0.554254	+	1.70582i	−0.554254	+	1.70582i
$617$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$617$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$618$		0			0
$619$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$619$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$620$	−0.533698	+	1.04744i	−0.533698	+	1.04744i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$626$			0.907981i			0.907981i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$631$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$632$	1.34500	+	1.34500i	1.34500	+	1.34500i
$633$		0			0
$634$	−0.587785	−	0.190983i	−0.587785	−	0.190983i
$635$	1.04744	+	1.44168i	1.04744	+	1.44168i
$636$		0			0
$637$		0			0
$638$	−0.582960	−	1.14412i	−0.582960	−	1.14412i
$639$		0			0
$640$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$641$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$641$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$647$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$648$		0			0
$649$			1.61803i			1.61803i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.253116	+	1.59811i	−0.253116	+	1.59811i	0.453990	+	0.891007i	$0.350000\pi$
$653$	−0.253116	+	1.59811i	−0.253116	+	1.59811i	−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$	1.26007	−	0.642040i	1.26007	−	0.642040i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.610425	−	1.87869i	0.610425	−	1.87869i	0.156434	−	0.987688i	$-0.450000\pi$
$659$	0.610425	−	1.87869i	0.610425	−	1.87869i	0.453990	−	0.891007i	$-0.350000\pi$
$660$		0			0
$661$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$661$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$662$		0			0
$663$		0			0
$664$	−1.53884	+	0.500000i	−1.53884	+	0.500000i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.896802	+	1.76007i	−0.896802	+	1.76007i	−0.309017	+	0.951057i	$0.600000\pi$
$673$	−0.896802	+	1.76007i	−0.896802	+	1.76007i	−0.587785	+	0.809017i	$0.700000\pi$
$674$	−0.312869			−0.312869
$675$		0			0
$676$	−1.00000			−1.00000
$677$	0.734572	−	1.44168i	0.734572	−	1.44168i	−0.156434	−	0.987688i	$-0.550000\pi$
$677$	0.734572	−	1.44168i	0.734572	−	1.44168i	0.891007	−	0.453990i	$-0.150000\pi$
$678$		0			0
$679$	0.363271	+	0.500000i	0.363271	+	0.500000i
$680$		0			0
$681$		0			0
$682$	−0.754763	+	0.754763i	−0.754763	+	0.754763i
$683$	0.610425	−	0.0966818i	0.610425	−	0.0966818i	0.156434	−	0.987688i	$-0.450000\pi$
$683$	0.610425	−	0.0966818i	0.610425	−	0.0966818i	0.453990	+	0.891007i	$0.350000\pi$
$684$		0			0
$685$		0			0
$686$	3.57349	−	1.16110i	3.57349	−	1.16110i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$691$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$692$	0.253116	+	1.59811i	0.253116	+	1.59811i
$693$		0			0
$694$	1.11803	−	1.53884i	1.11803	−	1.53884i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.896802	−	1.76007i	−0.896802	−	1.76007i
$701$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$701$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$702$		0			0
$703$		0			0
$704$	−0.734572	+	0.533698i	−0.734572	+	0.533698i
$705$		0			0
$706$		0			0
$707$	2.48912	+	2.48912i	2.48912	+	2.48912i
$708$		0			0
$709$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$709$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.297556	−	0.0966818i	−0.297556	−	0.0966818i
$717$		0			0
$718$		0			0
$719$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$719$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$720$		0			0
$721$	0.500000	−	0.363271i	0.500000	−	0.363271i
$722$	0.987688	+	0.156434i	0.987688	+	0.156434i
$723$		0			0
$724$		0			0
$725$	1.34500	+	0.437016i	1.34500	+	0.437016i
$726$		0			0
$727$	−1.26007	−	0.642040i	−1.26007	−	0.642040i	−0.309017	−	0.951057i	$-0.600000\pi$
$727$	−1.26007	−	0.642040i	−1.26007	−	0.642040i	−0.951057	+	0.309017i	$0.900000\pi$
$728$		0			0
$729$		0			0
$730$	−0.221232	−	1.39680i	−0.221232	−	1.39680i
$731$		0			0
$732$		0			0
$733$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$733$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$734$	−0.0966818	+	0.297556i	−0.0966818	+	0.297556i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$739$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$740$		0			0
$741$		0			0
$742$	−3.71113	+	0.587785i	−3.71113	+	0.587785i
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$	−1.58779	−	0.809017i	−1.58779	−	0.809017i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.22085			−1.22085
$750$		0			0
$751$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$751$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.44168	−	0.734572i	−1.44168	−	0.734572i
$756$		0			0
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$761$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.951057	+	1.30902i	−0.951057	+	1.30902i			1.00000i	$0.5\pi$
$769$	−0.951057	+	1.30902i	−0.951057	+	1.30902i	−0.951057	+	0.309017i	$0.900000\pi$
$770$	−0.280582	−	1.77152i	−0.280582	−	1.77152i
$771$		0			0
$772$	0.0489435	−	0.309017i	0.0489435	−	0.309017i
$773$	0.550672	+	0.280582i	0.550672	+	0.280582i	0.707107	−	0.707107i	$-0.250000\pi$
$773$	0.550672	+	0.280582i	0.550672	+	0.280582i	−0.156434	+	0.987688i	$0.550000\pi$
$774$		0			0
$775$		−	1.17557i		−	1.17557i
$776$			0.312869i			0.312869i
$777$		0			0
$778$	0.896802	+	0.142040i	0.896802	+	0.142040i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	2.76007	+	0.896802i	2.76007	+	0.896802i
$785$		0			0
$786$		0			0
$787$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$787$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$788$	0.533698	+	1.04744i	0.533698	+	1.04744i
$789$		0			0
$790$	−1.80902	−	0.587785i	−1.80902	−	0.587785i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$797$	0.610425	+	0.0966818i	0.610425	+	0.0966818i	0.453990	−	0.891007i	$-0.350000\pi$
$797$	0.610425	+	0.0966818i	0.610425	+	0.0966818i	0.156434	+	0.987688i	$0.450000\pi$
$798$		0			0
$799$		0			0
$800$	0.156434	−	0.987688i	0.156434	−	0.987688i
$801$		0			0
$802$		0			0
$803$	0.200874	−	1.26827i	0.200874	−	1.26827i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.278768	+	1.76007i	0.278768	+	1.76007i
$809$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$809$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$810$		0			0
$811$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$811$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$812$	−2.48912	+	1.26827i	−2.48912	+	1.26827i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.34500	−	1.34500i	1.34500	−	1.34500i
$819$		0			0
$820$		0			0
$821$	−1.16110	−	1.59811i	−1.16110	−	1.59811i	−0.707107	−	0.707107i	$-0.750000\pi$
$821$	−1.16110	−	1.59811i	−1.16110	−	1.59811i	−0.453990	−	0.891007i	$-0.650000\pi$
$822$		0			0
$823$	−0.809017	+	1.58779i	−0.809017	+	1.58779i			1.00000i	$0.5\pi$
$823$	−0.809017	+	1.58779i	−0.809017	+	1.58779i	−0.809017	+	0.587785i	$0.800000\pi$
$824$	0.312869			0.312869
$825$		0			0
$826$	3.52015			3.52015
$827$	0.533698	−	1.04744i	0.533698	−	1.04744i	−0.453990	−	0.891007i	$-0.650000\pi$
$827$	0.533698	−	1.04744i	0.533698	−	1.04744i	0.987688	−	0.156434i	$-0.0500000\pi$
$828$		0			0
$829$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$829$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$830$	1.14412	−	1.14412i	1.14412	−	1.14412i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	1.76007	−	0.896802i	1.76007	−	0.896802i
$839$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$839$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$840$		0			0
$841$	0.309017	−	0.951057i	0.309017	−	0.951057i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.891007	−	0.453990i	0.891007	−	0.453990i
$846$		0			0
$847$	−0.0542543	+	0.342548i	−0.0542543	+	0.342548i
$848$	−1.69480	−	0.863541i	−1.69480	−	0.863541i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$853$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$854$		0			0
$855$		0			0
$856$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$859$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$863$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$864$		0			0
$865$	−0.951057	−	1.30902i	−0.951057	−	1.30902i
$866$	−0.863541	−	0.280582i	−0.863541	−	0.280582i
$867$		0			0
$868$	1.64204	+	1.64204i	1.64204	+	1.64204i
$869$	−1.39724	−	1.01515i	−1.39724	−	1.01515i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.59811	+	1.16110i	1.59811	+	1.16110i
$876$		0			0
$877$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$877$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$878$	0.0966818	−	0.610425i	0.0966818	−	0.610425i
$879$		0			0
$880$	0.412215	−	0.809017i	0.412215	−	0.809017i
$881$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$881$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$882$		0			0
$883$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$883$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$884$		0			0
$885$		0			0
$886$	−0.363271	−	1.11803i	−0.363271	−	1.11803i
$887$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$887$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$888$		0			0
$889$	3.34786	−	1.08779i	3.34786	−	1.08779i
$890$		0			0
$891$		0			0
$892$	0.896802	−	0.142040i	0.896802	−	0.142040i
$893$		0			0
$894$		0			0
$895$	0.309017	−	0.0489435i	0.309017	−	0.0489435i
$896$	1.16110	+	1.59811i	1.16110	+	1.59811i
$897$		0			0
$898$		0			0
$899$	−1.66251			−1.66251
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$	1.87869	−	0.297556i	1.87869	−	0.297556i
$909$		0			0
$910$		0			0
$911$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$911$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$912$		0			0
$913$	1.30902	−	0.666977i	1.30902	−	0.666977i
$914$	0.550672	+	1.69480i	0.550672	+	1.69480i
$915$		0			0
$916$		0			0
$917$	−0.437016	−	2.75921i	−0.437016	−	2.75921i
$918$		0			0
$919$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$919$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$920$		0			0
$921$		0			0
$922$	−0.309017	+	1.95106i	−0.309017	+	1.95106i
$923$		0			0
$924$		0			0
$925$		0			0
$926$		−	1.41421i		−	1.41421i
$927$		0			0
$928$	−1.39680	−	0.221232i	−1.39680	−	0.221232i
$929$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$929$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	0.587785	+	0.190983i	0.587785	+	0.190983i
$935$		0			0
$936$		0			0
$937$	−0.896802	−	1.76007i	−0.896802	−	1.76007i	−0.587785	−	0.809017i	$-0.700000\pi$
$937$	−0.896802	−	1.76007i	−0.896802	−	1.76007i	−0.309017	−	0.951057i	$-0.600000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.297556	+	0.0966818i	0.297556	+	0.0966818i	0.453990	−	0.891007i	$-0.350000\pi$
$941$	0.297556	+	0.0966818i	0.297556	+	0.0966818i	−0.156434	+	0.987688i	$0.550000\pi$
$942$		0			0
$943$		0			0
$944$	1.44168	+	1.04744i	1.44168	+	1.04744i
$945$		0			0
$946$		0			0
$947$	1.16110	+	0.183900i	1.16110	+	0.183900i	0.707107	−	0.707107i	$-0.250000\pi$
$947$	1.16110	+	0.183900i	1.16110	+	0.183900i	0.453990	+	0.891007i	$0.350000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$953$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.118034	+	0.363271i	0.118034	+	0.363271i
$962$		0			0
$963$		0			0
$964$	−1.11803	+	0.363271i	−1.11803	+	0.363271i
$965$	0.0966818	+	0.297556i	0.0966818	+	0.297556i
$966$		0			0
$967$	−0.309017	+	0.0489435i	−0.309017	+	0.0489435i	−0.309017	−	0.951057i	$-0.600000\pi$
$967$	−0.309017	+	0.0489435i	−0.309017	+	0.0489435i			1.00000i	$0.5\pi$
$968$	−0.124147	+	0.124147i	−0.124147	+	0.124147i
$969$		0			0
$970$	−0.142040	−	0.278768i	−0.142040	−	0.278768i
$971$	0.183900	+	0.253116i	0.183900	+	0.253116i	0.891007	−	0.453990i	$-0.150000\pi$
$971$	0.183900	+	0.253116i	0.183900	+	0.253116i	−0.707107	+	0.707107i	$0.750000\pi$
$972$		0			0
$973$		0			0
$974$	−1.78201			−1.78201
$975$		0			0
$976$		0			0
$977$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$977$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$978$		0			0
$979$		0			0
$980$	−2.86638	+	0.453990i	−2.86638	+	0.453990i
$981$		0			0
$982$	0.221232	−	0.221232i	0.221232	−	0.221232i
$983$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$983$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$984$		0			0
$985$	−0.951057	−	0.690983i	−0.951057	−	0.690983i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$991$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$992$	0.183900	+	1.16110i	0.183900	+	1.16110i
$993$		0			0
$994$		0			0
$995$	0.280582	−	0.550672i	0.280582	−	0.550672i
$996$		0			0
$997$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$997$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.1.cr.a.1333.1	yes	16
3.2	odd	2	inner	1800.1.cr.a.1333.2	yes	16
8.5	even	2	inner	1800.1.cr.a.1333.2	yes	16
24.5	odd	2	CM	1800.1.cr.a.1333.1	yes	16
25.22	odd	20	inner	1800.1.cr.a.397.2	yes	16
75.47	even	20	inner	1800.1.cr.a.397.1	✓	16
200.197	odd	20	inner	1800.1.cr.a.397.1	✓	16
600.197	even	20	inner	1800.1.cr.a.397.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1800.1.cr.a.397.1	✓	16	75.47	even	20	inner
1800.1.cr.a.397.1	✓	16	200.197	odd	20	inner
1800.1.cr.a.397.2	yes	16	25.22	odd	20	inner
1800.1.cr.a.397.2	yes	16	600.197	even	20	inner
1800.1.cr.a.1333.1	yes	16	1.1	even	1	trivial
1800.1.cr.a.1333.1	yes	16	24.5	odd	2	CM
1800.1.cr.a.1333.2	yes	16	3.2	odd	2	inner
1800.1.cr.a.1333.2	yes	16	8.5	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1800.cr (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.453990 + 0.891007i) q^{2} +(-0.587785 - 0.809017i) q^{4} +(0.891007 + 0.453990i) q^{5} +(1.39680 - 1.39680i) q^{7} +(0.987688 - 0.156434i) q^{8} +O(q^{10})\)	\(q+(-0.453990 + 0.891007i) q^{2} +(-0.587785 - 0.809017i) q^{4} +(0.891007 + 0.453990i) q^{5} +(1.39680 - 1.39680i) q^{7} +(0.987688 - 0.156434i) q^{8} +(-0.809017 + 0.587785i) q^{10} +(-0.863541 + 0.280582i) q^{11} +(0.610425 + 1.87869i) q^{14} +(-0.309017 + 0.951057i) q^{16} +(-0.156434 - 0.987688i) q^{20} +(0.142040 - 0.896802i) q^{22} +(0.587785 + 0.809017i) q^{25} +(-1.95106 - 0.309017i) q^{28} +(1.14412 - 0.831254i) q^{29} +(-0.951057 - 0.690983i) q^{31} +(-0.707107 - 0.707107i) q^{32} +(1.87869 - 0.610425i) q^{35} +(0.951057 + 0.309017i) q^{40} +(0.734572 + 0.533698i) q^{44} -2.90211i q^{49} +(-0.987688 + 0.156434i) q^{50} +(-0.297556 + 1.87869i) q^{53} +(-0.896802 - 0.142040i) q^{55} +(1.16110 - 1.59811i) q^{56} +(0.221232 + 1.39680i) q^{58} +(0.550672 - 1.69480i) q^{59} +(1.04744 - 0.533698i) q^{62} +(0.951057 - 0.309017i) q^{64} +(-0.309017 + 1.95106i) q^{70} +(-0.642040 + 1.26007i) q^{73} +(-0.814279 + 1.59811i) q^{77} +(1.11803 + 1.53884i) q^{79} +(-0.707107 + 0.707107i) q^{80} +(-1.59811 + 0.253116i) q^{83} +(-0.809017 + 0.412215i) q^{88} +(-0.0489435 + 0.309017i) q^{97} +(2.58580 + 1.31753i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{7}+O(q^{10}) \) 16 * q + 4 * q^7	\( 16 q + 4 q^{7} - 4 q^{10} + 4 q^{16} - 4 q^{22} - 16 q^{28} + 4 q^{55} + 4 q^{58} + 4 q^{70} - 4 q^{73} - 4 q^{88} - 16 q^{97}+O(q^{100}) \) 16 * q + 4 * q^7 - 4 * q^10 + 4 * q^16 - 4 * q^22 - 16 * q^28 + 4 * q^55 + 4 * q^58 + 4 * q^70 - 4 * q^73 - 4 * q^88 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more