LMFDB - Embedded newform 2925.1.er.a.1117.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,1,Mod(298,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([0, 11, 10]))

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

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

chi := DirichletCharacter("2925.298");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2925.er (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:	$1.45976516195$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$4$ over $\Q(\zeta_{20})$
Coefficient field:	$\Q(\zeta_{80})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{32} - x^{24} + x^{16} - x^{8} + 1 $ x^32 - x^24 + x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{40}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{40} + \cdots)$

Embedding invariants

Embedding label			1117.2
Root			$-0.996917 - 0.0784591i$ of defining polynomial
Character	$\chi$	$=$	2925.1117
Dual form			2925.1.er.a.2053.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.15732 - 0.589686i) q^{2} +(0.403886 + 0.555901i) q^{4} +(-0.522499 - 0.852640i) q^{5} +(0.0635725 + 0.401381i) q^{8} +O(q^{10})$	\(q+(-1.15732 - 0.589686i) q^{2} +(0.403886 + 0.555901i) q^{4} +(-0.522499 - 0.852640i) q^{5} +(0.0635725 + 0.401381i) q^{8} +(0.101910 + 1.29489i) q^{10} +(-1.84956 + 0.600958i) q^{11} +(-0.453990 - 0.891007i) q^{13} +(0.375450 - 1.15552i) q^{16} +(0.262954 - 0.634826i) q^{20} +(2.49492 + 0.395156i) q^{22} +(-0.453990 + 0.891007i) q^{25} +1.29890i q^{26} +(-0.828553 + 0.828553i) q^{32} +(0.309017 - 0.263925i) q^{40} +(1.89625 + 0.616129i) q^{41} +(-1.26007 + 1.26007i) q^{43} +(-1.08108 - 0.785452i) q^{44} +(0.311904 - 1.96929i) q^{47} +1.00000i q^{49} +(1.05083 - 0.763472i) q^{50} +(0.311951 - 0.612238i) q^{52} +(1.47879 + 1.26301i) q^{55} +(-0.526961 + 1.62182i) q^{59} +(0.610425 + 1.87869i) q^{61} +(0.291975 - 0.0948685i) q^{64} +(-0.522499 + 0.852640i) q^{65} +(0.274431 + 0.377723i) q^{71} +(-0.831254 - 1.14412i) q^{79} +(-1.18141 + 0.283632i) q^{80} +(-1.83125 - 1.83125i) q^{82} +(-0.237907 - 1.50209i) q^{83} +(2.20136 - 0.715266i) q^{86} +(-0.358794 - 0.704173i) q^{88} +(0.570989 + 1.75732i) q^{89} +(-1.52224 + 2.09518i) q^{94} +(0.589686 - 1.15732i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 8 q^{10} + 8 q^{16} - 8 q^{22} - 8 q^{40} + 8 q^{43} + 8 q^{52} + 40 q^{64} - 32 q^{82} + 24 q^{88} - 40 q^{94}+O(q^{100}) $ 32 * q + 8 * q^10 + 8 * q^16 - 8 * q^22 - 8 * q^40 + 8 * q^43 + 8 * q^52 + 40 * q^64 - 32 * q^82 + 24 * q^88 - 40 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times$.

$n$	$326$	$352$	$2251$
$\chi(n)$	$1$	$e\left(\frac{13}{20}\right)$	$-1$

Coefficient data

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

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

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

$2$	−1.15732	−	0.589686i	−1.15732	−	0.589686i	−0.233445	−	0.972370i	$-0.575000\pi$
$2$	−1.15732	−	0.589686i	−1.15732	−	0.589686i	−0.923880	+	0.382683i	$0.875000\pi$
$3$		0			0
$3$		0			0
$4$	0.403886	+	0.555901i	0.403886	+	0.555901i
$5$	−0.522499	−	0.852640i	−0.522499	−	0.852640i
$5$	−0.522499	−	0.852640i	−0.522499	−	0.852640i
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$	0.0635725	+	0.401381i	0.0635725	+	0.401381i
$9$		0			0
$10$	0.101910	+	1.29489i	0.101910	+	1.29489i
$11$	−1.84956	+	0.600958i	−1.84956	+	0.600958i	−0.852640	+	0.522499i	$0.825000\pi$
$11$	−1.84956	+	0.600958i	−1.84956	+	0.600958i	−0.996917	+	0.0784591i	$0.975000\pi$
$12$		0			0
$13$	−0.453990	−	0.891007i	−0.453990	−	0.891007i
$13$	−0.453990	−	0.891007i	−0.453990	−	0.891007i
$14$		0			0
$15$		0			0
$16$	0.375450	−	1.15552i	0.375450	−	1.15552i
$17$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$17$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$18$		0			0
$19$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$19$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$20$	0.262954	−	0.634826i	0.262954	−	0.634826i
$21$		0			0
$22$	2.49492	+	0.395156i	2.49492	+	0.395156i
$23$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$23$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$24$		0			0
$25$	−0.453990	+	0.891007i	−0.453990	+	0.891007i
$26$			1.29890i			1.29890i
$27$		0			0
$28$		0			0
$29$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$29$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$30$		0			0
$31$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$31$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$32$	−0.828553	+	0.828553i	−0.828553	+	0.828553i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$37$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$38$		0			0
$39$		0			0
$40$	0.309017	−	0.263925i	0.309017	−	0.263925i
$41$	1.89625	+	0.616129i	1.89625	+	0.616129i	0.972370	+	0.233445i	$0.0750000\pi$
$41$	1.89625	+	0.616129i	1.89625	+	0.616129i	0.923880	+	0.382683i	$0.125000\pi$
$42$		0			0
$43$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.309017	+	0.951057i	$0.600000\pi$
$43$	−1.26007	+	1.26007i	−1.26007	+	1.26007i	−0.951057	+	0.309017i	$0.900000\pi$
$44$	−1.08108	−	0.785452i	−1.08108	−	0.785452i
$45$		0			0
$46$		0			0
$47$	0.311904	−	1.96929i	0.311904	−	1.96929i	0.0784591	−	0.996917i	$-0.475000\pi$
$47$	0.311904	−	1.96929i	0.311904	−	1.96929i	0.233445	−	0.972370i	$-0.425000\pi$
$48$		0			0
$49$			1.00000i			1.00000i
$50$	1.05083	−	0.763472i	1.05083	−	0.763472i
$51$		0			0
$52$	0.311951	−	0.612238i	0.311951	−	0.612238i
$53$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$53$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$54$		0			0
$55$	1.47879	+	1.26301i	1.47879	+	1.26301i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.526961	+	1.62182i	−0.526961	+	1.62182i	0.233445	+	0.972370i	$0.425000\pi$
$59$	−0.526961	+	1.62182i	−0.526961	+	1.62182i	−0.760406	+	0.649448i	$0.775000\pi$
$60$		0			0
$61$	0.610425	+	1.87869i	0.610425	+	1.87869i	0.453990	+	0.891007i	$0.350000\pi$
$61$	0.610425	+	1.87869i	0.610425	+	1.87869i	0.156434	+	0.987688i	$0.450000\pi$
$62$		0			0
$63$		0			0
$64$	0.291975	−	0.0948685i	0.291975	−	0.0948685i
$65$	−0.522499	+	0.852640i	−0.522499	+	0.852640i
$66$		0			0
$67$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$67$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.274431	+	0.377723i	0.274431	+	0.377723i	0.923880	−	0.382683i	$-0.125000\pi$
$71$	0.274431	+	0.377723i	0.274431	+	0.377723i	−0.649448	+	0.760406i	$0.725000\pi$
$72$		0			0
$73$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$73$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$79$	−0.831254	−	1.14412i	−0.831254	−	1.14412i	0.156434	−	0.987688i	$-0.450000\pi$
$80$	−1.18141	+	0.283632i	−1.18141	+	0.283632i
$81$		0			0
$82$	−1.83125	−	1.83125i	−1.83125	−	1.83125i
$83$	−0.237907	−	1.50209i	−0.237907	−	1.50209i	−0.760406	−	0.649448i	$-0.775000\pi$
$83$	−0.237907	−	1.50209i	−0.237907	−	1.50209i	0.522499	−	0.852640i	$-0.325000\pi$
$84$		0			0
$85$		0			0
$86$	2.20136	−	0.715266i	2.20136	−	0.715266i
$87$		0			0
$88$	−0.358794	−	0.704173i	−0.358794	−	0.704173i
$89$	0.570989	+	1.75732i	0.570989	+	1.75732i	0.649448	+	0.760406i	$0.275000\pi$
$89$	0.570989	+	1.75732i	0.570989	+	1.75732i	−0.0784591	+	0.996917i	$0.525000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	−1.52224	+	2.09518i	−1.52224	+	2.09518i
$95$		0			0
$96$		0			0
$97$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$97$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$98$	0.589686	−	1.15732i	0.589686	−	1.15732i
$99$		0			0
$100$	−0.678671	+	0.107491i	−0.678671	+	0.107491i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$	−0.221232	+	1.39680i	−0.221232	+	1.39680i	0.587785	+	0.809017i	$0.300000\pi$
$103$	−0.221232	+	1.39680i	−0.221232	+	1.39680i	−0.809017	+	0.587785i	$0.800000\pi$
$104$	0.328772	−	0.238867i	0.328772	−	0.238867i
$105$		0			0
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$109$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$110$	−0.966664	−	2.33373i	−0.966664	−	2.33373i
$111$		0			0
$112$		0			0
$113$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$113$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	1.56623	−	1.56623i	1.56623	−	1.56623i
$119$		0			0
$120$		0			0
$121$	2.25070	−	1.63523i	2.25070	−	1.63523i
$122$	0.401381	−	2.53422i	0.401381	−	2.53422i
$123$		0			0
$124$		0			0
$125$	0.996917	−	0.0784591i	0.996917	−	0.0784591i
$126$		0			0
$127$	0.280582	−	0.550672i	0.280582	−	0.550672i	−0.707107	−	0.707107i	$-0.750000\pi$
$127$	0.280582	−	0.550672i	0.280582	−	0.550672i	0.987688	+	0.156434i	$0.0500000\pi$
$128$	0.763472	+	0.120922i	0.763472	+	0.120922i
$129$		0			0
$130$	1.10749	−	0.678671i	1.10749	−	0.678671i
$131$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$131$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.838865	+	1.64637i	0.838865	+	1.64637i	0.760406	+	0.649448i	$0.225000\pi$
$137$	0.838865	+	1.64637i	0.838865	+	1.64637i	0.0784591	+	0.996917i	$0.475000\pi$
$138$		0			0
$139$	−1.11803	+	0.363271i	−1.11803	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$139$	−1.11803	+	0.363271i	−1.11803	+	0.363271i	−0.309017	+	0.951057i	$0.600000\pi$
$140$		0			0
$141$		0			0
$142$	−0.0948685	−	0.598976i	−0.0948685	−	0.598976i
$143$	1.37514	+	1.37514i	1.37514	+	1.37514i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$149$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.221232	+	0.221232i	0.221232	+	0.221232i	0.809017	−	0.587785i	$-0.200000\pi$
$157$	0.221232	+	0.221232i	0.221232	+	0.221232i	−0.587785	+	0.809017i	$0.700000\pi$
$158$	0.287357	+	1.81430i	0.287357	+	1.81430i
$159$		0			0
$160$	1.13938	+	0.273540i	1.13938	+	0.273540i
$161$		0			0
$162$		0			0
$163$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$163$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$164$	0.423361	+	1.30297i	0.423361	+	1.30297i
$165$		0			0
$166$	−0.610425	+	1.87869i	−0.610425	+	1.87869i
$167$	−1.03213	+	0.163474i	−1.03213	+	0.163474i	−0.649448	−	0.760406i	$-0.725000\pi$
$167$	−1.03213	+	0.163474i	−1.03213	+	0.163474i	−0.382683	+	0.923880i	$0.625000\pi$
$168$		0			0
$169$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$170$		0			0
$171$		0			0
$172$	−1.20940	−	0.191550i	−1.20940	−	0.191550i
$173$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$173$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$174$		0			0
$175$		0			0
$176$			2.36282i			2.36282i
$177$		0			0
$178$	0.375450	−	2.37050i	0.375450	−	2.37050i
$179$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$179$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$180$		0			0
$181$	0.951057	+	0.690983i	0.951057	+	0.690983i	0.951057	−	0.309017i	$-0.100000\pi$
$181$	0.951057	+	0.690983i	0.951057	+	0.690983i			1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	1.22070	−	0.621979i	1.22070	−	0.621979i
$189$		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			0		0.707107	−	0.707107i	$-0.250000\pi$
$193$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$194$		0			0
$195$		0			0
$196$	−0.555901	+	0.403886i	−0.555901	+	0.403886i
$197$	−0.119730	+	0.755944i	−0.119730	+	0.755944i	0.852640	+	0.522499i	$0.175000\pi$
$197$	−0.119730	+	0.755944i	−0.119730	+	0.755944i	−0.972370	+	0.233445i	$0.925000\pi$
$198$		0			0
$199$		−	0.907981i		−	0.907981i	−0.891007	−	0.453990i	$-0.850000\pi$
$199$		−	0.907981i		−	0.907981i	0.891007	−	0.453990i	$-0.150000\pi$
$200$	−0.386494	−	0.125580i	−0.386494	−	0.125580i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−0.465451	−	1.93874i	−0.465451	−	1.93874i
$206$	1.07971	−	1.48610i	1.07971	−	1.48610i
$207$		0			0
$208$	−1.20002	+	0.190065i	−1.20002	+	0.190065i
$209$		0			0
$210$		0			0
$211$	0.550672	+	1.69480i	0.550672	+	1.69480i	0.707107	+	0.707107i	$0.250000\pi$
$211$	0.550672	+	1.69480i	0.550672	+	1.69480i	−0.156434	+	0.987688i	$0.550000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.73278	+	0.416003i	1.73278	+	0.416003i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.104844	+	1.33217i	−0.104844	+	1.33217i
$221$		0			0
$222$		0			0
$223$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$223$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.64637	+	0.838865i	1.64637	+	0.838865i	0.996917	+	0.0784591i	$0.0250000\pi$
$227$	1.64637	+	0.838865i	1.64637	+	0.838865i	0.649448	+	0.760406i	$0.275000\pi$
$228$		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
$232$		0			0
$233$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$233$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$234$		0			0
$235$	−1.84206	+	0.763007i	−1.84206	+	0.763007i
$236$	−1.11440	+	0.362091i	−1.11440	+	0.362091i
$237$		0			0
$238$		0			0
$239$	−0.236511	−	0.727907i	−0.236511	−	0.727907i	−0.996917	−	0.0784591i	$-0.975000\pi$
$239$	−0.236511	−	0.727907i	−0.236511	−	0.727907i	0.760406	−	0.649448i	$-0.225000\pi$
$240$		0			0
$241$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$241$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$242$	−3.56906	+	0.565283i	−3.56906	+	0.565283i
$243$		0			0
$244$	−0.797826	+	1.09811i	−0.797826	+	1.09811i
$245$	0.852640	−	0.522499i	0.852640	−	0.522499i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−1.20002	−	0.497066i	−1.20002	−	0.497066i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$	−0.649448	+	0.471852i	−0.649448	+	0.471852i
$255$		0			0
$256$	−1.06065	−	0.770606i	−1.06065	−	0.770606i
$257$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$257$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$		0			0
$260$	−0.685013	+	0.0539117i	−0.685013	+	0.0539117i
$261$		0			0
$262$		0			0
$263$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$263$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$269$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$270$		0			0
$271$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$271$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$272$		0			0
$273$		0			0
$274$		−	2.40005i		−	2.40005i
$275$	0.304224	−	1.92080i	0.304224	−	1.92080i
$276$		0			0
$277$	0.412215	−	0.809017i	0.412215	−	0.809017i	−0.587785	−	0.809017i	$-0.700000\pi$
$277$	0.412215	−	0.809017i	0.412215	−	0.809017i	1.00000			$0$
$278$	1.50814	+	0.238867i	1.50814	+	0.238867i
$279$		0			0
$280$		0			0
$281$	−0.614234	+	0.845420i	−0.614234	+	0.845420i	−0.996917	−	0.0784591i	$-0.975000\pi$
$281$	−0.614234	+	0.845420i	−0.614234	+	0.845420i	0.382683	+	0.923880i	$0.375000\pi$
$282$		0			0
$283$	−1.59811	+	0.253116i	−1.59811	+	0.253116i	−0.891007	−	0.453990i	$-0.850000\pi$
$283$	−1.59811	+	0.253116i	−1.59811	+	0.253116i	−0.707107	+	0.707107i	$0.750000\pi$
$284$	−0.0991373	+	0.305113i	−0.0991373	+	0.305113i
$285$		0			0
$286$	−0.780582	−	2.40238i	−0.780582	−	2.40238i
$287$		0			0
$288$		0			0
$289$	0.951057	−	0.309017i	0.951057	−	0.309017i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.110958	−	0.110958i	−0.110958	−	0.110958i	0.649448	−	0.760406i	$-0.275000\pi$
$293$	−0.110958	−	0.110958i	−0.110958	−	0.110958i	−0.760406	+	0.649448i	$0.775000\pi$
$294$		0			0
$295$	1.65816	−	0.398090i	1.65816	−	0.398090i
$296$		0			0
$297$		0			0
$298$	0.885778	+	0.451326i	0.885778	+	0.451326i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.28290	−	1.50209i	1.28290	−	1.50209i
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$311$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$312$		0			0
$313$	−0.280582	−	0.550672i	−0.280582	−	0.550672i	0.707107	−	0.707107i	$-0.250000\pi$
$313$	−0.280582	−	0.550672i	−0.280582	−	0.550672i	−0.987688	+	0.156434i	$0.950000\pi$
$314$	−0.125580	−	0.386494i	−0.125580	−	0.386494i
$315$		0			0
$316$	0.300287	−	0.924189i	0.300287	−	0.924189i
$317$	−1.50209	+	0.237907i	−1.50209	+	0.237907i	−0.852640	−	0.522499i	$-0.825000\pi$
$317$	−1.50209	+	0.237907i	−1.50209	+	0.237907i	−0.649448	+	0.760406i	$0.725000\pi$
$318$		0			0
$319$		0			0
$320$	−0.233445	−	0.199381i	−0.233445	−	0.199381i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	1.00000			1.00000
$326$		0			0
$327$		0			0
$328$	−0.126753	+	0.800287i	−0.126753	+	0.800287i
$329$		0			0
$330$		0			0
$331$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$331$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$332$	0.738925	−	0.738925i	0.738925	−	0.738925i
$333$		0			0
$334$	1.29091	+	0.419442i	1.29091	+	0.419442i
$335$		0			0
$336$		0			0
$337$	−1.69480	+	0.863541i	−1.69480	+	0.863541i	−0.707107	+	0.707107i	$0.750000\pi$
$337$	−1.69480	+	0.863541i	−1.69480	+	0.863541i	−0.987688	+	0.156434i	$0.950000\pi$
$338$	1.15732	−	0.589686i	1.15732	−	0.589686i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−0.585876	−	0.425664i	−0.585876	−	0.425664i
$345$		0			0
$346$		0			0
$347$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$347$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	1.03453	−	2.03038i	1.03453	−	2.03038i
$353$	−0.154986	−	0.0245474i	−0.154986	−	0.0245474i	0.0784591	−	0.996917i	$-0.475000\pi$
$353$	−0.154986	−	0.0245474i	−0.154986	−	0.0245474i	−0.233445	+	0.972370i	$0.575000\pi$
$354$		0			0
$355$	0.178671	−	0.431351i	0.178671	−	0.431351i
$356$	−0.746283	+	1.02717i	−0.746283	+	1.02717i
$357$		0			0
$358$		0			0
$359$	0.0484904	−	0.149238i	0.0484904	−	0.149238i	−0.923880	−	0.382683i	$-0.875000\pi$
$359$	0.0484904	−	0.149238i	0.0484904	−	0.149238i	0.972370	+	0.233445i	$0.0750000\pi$
$360$		0			0
$361$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$362$	−0.693218	−	1.36052i	−0.693218	−	1.36052i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	0.142040	+	0.896802i	0.142040	+	0.896802i	0.951057	+	0.309017i	$0.100000\pi$
$367$	0.142040	+	0.896802i	0.142040	+	0.896802i	−0.809017	+	0.587785i	$0.800000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1.04744	−	0.533698i	−1.04744	−	0.533698i	−0.156434	−	0.987688i	$-0.550000\pi$
$373$	−1.04744	−	0.533698i	−1.04744	−	0.533698i	−0.891007	+	0.453990i	$0.850000\pi$
$374$		0			0
$375$		0			0
$376$	0.810263			0.810263
$377$		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
$383$	−0.119730	−	0.755944i	−0.119730	−	0.755944i	−0.972370	−	0.233445i	$-0.925000\pi$
$383$	−0.119730	−	0.755944i	−0.119730	−	0.755944i	0.852640	−	0.522499i	$-0.175000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$389$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$390$		0			0
$391$		0			0
$392$	−0.401381	+	0.0635725i	−0.401381	+	0.0635725i
$393$		0			0
$394$	0.584336	−	0.804270i	0.584336	−	0.804270i
$395$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$396$		0			0
$397$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$397$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$398$	−0.535424	+	1.05083i	−0.535424	+	1.05083i
$399$		0			0
$400$	0.859122	+	0.859122i	0.859122	+	0.859122i
$401$		−	0.466891i		−	0.466891i	−0.972370	−	0.233445i	$-0.925000\pi$
$401$		−	0.466891i		−	0.466891i	0.972370	−	0.233445i	$-0.0750000\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$409$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$410$	−0.604573	+	2.51823i	−0.604573	+	2.51823i
$411$		0			0
$412$	−0.865836	+	0.441165i	−0.865836	+	0.441165i
$413$		0			0
$414$		0			0
$415$	−1.15643	+	0.987688i	−1.15643	+	0.987688i
$416$	1.11440	+	0.362091i	1.11440	+	0.362091i
$417$		0			0
$418$		0			0
$419$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$419$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$420$		0			0
$421$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$421$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$422$	0.362091	−	2.28615i	0.362091	−	2.28615i
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$	−1.76007	−	1.50324i	−1.76007	−	1.50324i
$431$	0.763472	−	1.05083i	0.763472	−	1.05083i	−0.233445	−	0.972370i	$-0.575000\pi$
$431$	0.763472	−	1.05083i	0.763472	−	1.05083i	0.996917	−	0.0784591i	$-0.0250000\pi$
$432$		0			0
$433$	1.16110	−	0.183900i	1.16110	−	0.183900i	0.453990	−	0.891007i	$-0.350000\pi$
$433$	1.16110	−	0.183900i	1.16110	−	0.183900i	0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−1.80902	+	0.587785i	−1.80902	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$439$	−1.80902	+	0.587785i	−1.80902	+	0.587785i	−1.00000			$\pi$
$440$	−0.412937	+	0.673851i	−0.412937	+	0.673851i
$441$		0			0
$442$		0			0
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$	1.20002	−	1.40505i	1.20002	−	1.40505i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.70528			−1.70528			−0.852640	−	0.522499i	$-0.825000\pi$
$449$	−1.70528			−1.70528			−0.852640	+	0.522499i	$0.825000\pi$
$450$		0			0
$451$	−3.87749			−3.87749
$452$		0			0
$453$		0			0
$454$	−1.41071	−	1.94168i	−1.41071	−	1.94168i
$455$		0			0
$456$		0			0
$457$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$457$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−0.993851	+	0.322922i	−0.993851	+	0.322922i	−0.760406	−	0.649448i	$-0.775000\pi$
$461$	−0.993851	+	0.322922i	−0.993851	+	0.322922i	−0.233445	+	0.972370i	$0.575000\pi$
$462$		0			0
$463$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$463$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$467$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$468$		0			0
$469$		0			0
$470$	2.58180	+	0.203192i	2.58180	+	0.203192i
$471$		0			0
$472$	−0.684467	−	0.108409i	−0.684467	−	0.108409i
$473$	1.57333	−	3.08783i	1.57333	−	3.08783i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−0.155517	+	0.981893i	−0.155517	+	0.981893i
$479$	−0.845420	+	0.614234i	−0.845420	+	0.614234i	−0.923880	−	0.382683i	$-0.875000\pi$
$479$	−0.845420	+	0.614234i	−0.845420	+	0.614234i	0.0784591	+	0.996917i	$0.475000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	1.81805	+	0.590719i	1.81805	+	0.590719i
$485$		0			0
$486$		0			0
$487$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$487$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$488$	−0.715266	+	0.364446i	−0.715266	+	0.364446i
$489$		0			0
$490$	−1.29489	+	0.101910i	−1.29489	+	0.101910i
$491$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$491$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	0.446256	+	0.522499i	0.446256	+	0.522499i
$501$		0			0
$502$		0			0
$503$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$503$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	0.419442	−	0.0664331i	0.419442	−	0.0664331i
$509$	−0.401381	+	1.23532i	−0.401381	+	1.23532i	0.522499	+	0.852640i	$0.325000\pi$
$509$	−0.401381	+	1.23532i	−0.401381	+	1.23532i	−0.923880	+	0.382683i	$0.875000\pi$
$510$		0			0
$511$		0			0
$512$	0.422169	+	0.828553i	0.422169	+	0.828553i
$513$		0			0
$514$		0			0
$515$	1.30656	−	0.541196i	1.30656	−	0.541196i
$516$		0			0
$517$	0.606573	+	3.82975i	0.606573	+	3.82975i
$518$		0			0
$519$		0			0
$520$	−0.375450	−	0.155517i	−0.375450	−	0.155517i
$521$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$521$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$522$		0			0
$523$	−1.76007	−	0.896802i	−1.76007	−	0.896802i	−0.951057	−	0.309017i	$-0.900000\pi$
$523$	−1.76007	−	0.896802i	−1.76007	−	0.896802i	−0.809017	−	0.587785i	$-0.800000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.311904	−	1.96929i	−0.311904	−	1.96929i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.600958	−	1.84956i	−0.600958	−	1.84956i
$540$		0			0
$541$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$541$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.309017	−	0.0489435i	−0.309017	−	0.0489435i		−	1.00000i	$-0.5\pi$
$547$	−0.309017	−	0.0489435i	−0.309017	−	0.0489435i	−0.309017	+	0.951057i	$0.600000\pi$
$548$	−0.576410	+	1.13127i	−0.576410	+	1.13127i
$549$		0			0
$550$	−1.48475	+	2.04359i	−1.48475	+	2.04359i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−0.954133	+	0.693218i	−0.954133	+	0.693218i
$555$		0			0
$556$	−0.653500	−	0.474796i	−0.653500	−	0.474796i
$557$	−0.738925	+	0.738925i	−0.738925	+	0.738925i	−0.972370	−	0.233445i	$-0.925000\pi$
$557$	−0.738925	+	0.738925i	−0.738925	+	0.738925i	0.233445	+	0.972370i	$0.425000\pi$
$558$		0			0
$559$	1.69480	+	0.550672i	1.69480	+	0.550672i
$560$		0			0
$561$		0			0
$562$	1.20940	−	0.616221i	1.20940	−	0.616221i
$563$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$563$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$564$		0			0
$565$		0			0
$566$	1.99880	+	0.649448i	1.99880	+	0.649448i
$567$		0			0
$568$	−0.134164	+	0.134164i	−0.134164	+	0.134164i
$569$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$569$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$570$		0			0
$571$	0.734572	−	0.533698i	0.734572	−	0.533698i	−0.156434	−	0.987688i	$-0.550000\pi$
$571$	0.734572	−	0.533698i	0.734572	−	0.533698i	0.891007	+	0.453990i	$0.150000\pi$
$572$	−0.209042	+	1.31984i	−0.209042	+	1.31984i
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$577$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$578$	−1.28290	−	0.203192i	−1.28290	−	0.203192i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	0.0629840	+	0.193845i	0.0629840	+	0.193845i
$587$	0.474419	+	0.931099i	0.474419	+	0.931099i	0.996917	+	0.0784591i	$0.0250000\pi$
$587$	0.474419	+	0.931099i	0.474419	+	0.931099i	−0.522499	+	0.852640i	$0.675000\pi$
$588$		0			0
$589$		0			0
$590$	−2.15378	−	0.517077i	−2.15378	−	0.517077i
$591$		0			0
$592$		0			0
$593$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$593$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$594$		0			0
$595$		0			0
$596$	−0.309121	−	0.425468i	−0.309121	−	0.425468i
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$601$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.57024	−	1.06463i	−2.57024	−	1.06463i
$606$		0			0
$607$	1.39680	+	1.39680i	1.39680	+	1.39680i	0.809017	+	0.587785i	$0.200000\pi$
$607$	1.39680	+	1.39680i	1.39680	+	1.39680i	0.587785	+	0.809017i	$0.300000\pi$
$608$		0			0
$609$		0			0
$610$	−2.37050	+	0.981893i	−2.37050	+	0.981893i
$611$	−1.89625	+	0.616129i	−1.89625	+	0.616129i
$612$		0			0
$613$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$613$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.96929	+	0.311904i	−1.96929	+	0.311904i	−0.972370	+	0.233445i	$0.925000\pi$
$617$	−1.96929	+	0.311904i	−1.96929	+	0.311904i	−0.996917	+	0.0784591i	$0.975000\pi$
$618$		0			0
$619$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$619$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$626$			0.802762i			0.802762i
$627$		0			0
$628$	−0.0336306	+	0.212335i	−0.0336306	+	0.212335i
$629$		0			0
$630$		0			0
$631$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$631$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$632$	0.406384	−	0.406384i	0.406384	−	0.406384i
$633$		0			0
$634$	1.87869	+	0.610425i	1.87869	+	0.610425i
$635$	−0.616129	+	0.0484904i	−0.616129	+	0.0484904i
$636$		0			0
$637$	0.891007	−	0.453990i	0.891007	−	0.453990i
$638$		0			0
$639$		0			0
$640$	−0.295810	−	0.714148i	−0.295810	−	0.714148i
$641$		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	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$647$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$648$		0			0
$649$		−	3.31633i		−	3.31633i
$650$	−1.15732	−	0.589686i	−1.15732	−	0.589686i
$651$		0			0
$652$		0			0
$653$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$653$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$654$		0			0
$655$		0			0
$656$	1.42389	−	1.95982i	1.42389	−	1.95982i
$657$		0			0
$658$		0			0
$659$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$659$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$660$		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
$664$	0.587785	−	0.190983i	0.587785	−	0.190983i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−0.507738	−	0.507738i	−0.507738	−	0.507738i
$669$		0			0
$670$		0			0
$671$	−2.25803	−	3.10791i	−2.25803	−	3.10791i
$672$		0			0
$673$	1.26007	+	0.642040i	1.26007	+	0.642040i	0.951057	−	0.309017i	$-0.100000\pi$
$673$	1.26007	+	0.642040i	1.26007	+	0.642040i	0.309017	+	0.951057i	$0.400000\pi$
$674$	2.47065			2.47065
$675$		0			0
$676$	−0.687131			−0.687131
$677$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$677$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.304224	+	1.92080i	0.304224	+	1.92080i	0.382683	+	0.923880i	$0.375000\pi$
$683$	0.304224	+	1.92080i	0.304224	+	1.92080i	−0.0784591	+	0.996917i	$0.525000\pi$
$684$		0			0
$685$	0.965451	−	1.57547i	0.965451	−	1.57547i
$686$		0			0
$687$		0			0
$688$	0.982941	+	1.92913i	0.982941	+	1.92913i
$689$		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			0
$693$		0			0
$694$		0			0
$695$	0.893911	+	0.763472i	0.893911	+	0.763472i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	−0.483013	+	0.350930i	−0.483013	+	0.350930i
$705$		0			0
$706$	0.164894	+	0.119803i	0.164894	+	0.119803i
$707$		0			0
$708$		0			0
$709$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$709$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$710$	−0.461143	+	0.393853i	−0.461143	+	0.393853i
$711$		0			0
$712$	−0.669057	+	0.340902i	−0.669057	+	0.340902i
$713$		0			0
$714$		0			0
$715$	0.453990	−	1.89101i	0.453990	−	1.89101i
$716$		0			0
$717$		0			0
$718$	−0.144123	+	0.144123i	−0.144123	+	0.144123i
$719$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$719$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$720$		0			0
$721$		0			0
$722$	−0.203192	+	1.28290i	−0.203192	+	1.28290i
$723$		0			0
$724$			0.807771i			0.807771i
$725$		0			0
$726$		0			0
$727$	−0.734572	+	1.44168i	−0.734572	+	1.44168i	0.156434	+	0.987688i	$0.450000\pi$
$727$	−0.734572	+	1.44168i	−0.734572	+	1.44168i	−0.891007	+	0.453990i	$0.850000\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$733$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$734$	0.364446	−	1.12165i	0.364446	−	1.12165i
$735$		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
$742$		0			0
$743$	0.330142	+	0.330142i	0.330142	+	0.330142i	0.852640	−	0.522499i	$-0.175000\pi$
$743$	0.330142	+	0.330142i	0.330142	+	0.330142i	−0.522499	+	0.852640i	$0.675000\pi$
$744$		0			0
$745$	0.399903	+	0.652583i	0.399903	+	0.652583i
$746$	0.897515	+	1.23532i	0.897515	+	1.23532i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.17557			1.17557			0.587785	−	0.809017i	$-0.300000\pi$
$751$	1.17557			1.17557			0.587785	+	0.809017i	$0.300000\pi$
$752$	−2.15844	−	1.09978i	−2.15844	−	1.09978i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.891007	−	0.453990i	$-0.850000\pi$
$757$	−1.34500	−	1.34500i	−1.34500	−	1.34500i	−0.453990	−	0.891007i	$-0.650000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.149238	+	0.0484904i	−0.149238	+	0.0484904i	−0.382683	−	0.923880i	$-0.625000\pi$
$761$	−0.149238	+	0.0484904i	−0.149238	+	0.0484904i	0.233445	+	0.972370i	$0.425000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−0.307204	+	0.945476i	−0.307204	+	0.945476i
$767$	1.68429	−	0.266765i	1.68429	−	0.266765i
$768$		0			0
$769$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$769$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−0.774181	+	1.51942i	−0.774181	+	1.51942i	0.0784591	+	0.996917i	$0.475000\pi$
$773$	−0.774181	+	1.51942i	−0.774181	+	1.51942i	−0.852640	+	0.522499i	$0.825000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−0.734572	−	0.533698i	−0.734572	−	0.533698i
$782$		0			0
$783$		0			0
$784$	1.15552	+	0.375450i	1.15552	+	0.375450i
$785$	0.0730378	−	0.304224i	0.0730378	−	0.304224i
$786$		0			0
$787$		0			0		0.891007	−	0.453990i	$-0.150000\pi$
$787$		0			0		−0.891007	+	0.453990i	$0.850000\pi$
$788$	−0.468587	+	0.238757i	−0.468587	+	0.238757i
$789$		0			0
$790$	1.39680	−	1.19298i	1.39680	−	1.19298i
$791$		0			0
$792$		0			0
$793$	1.39680	−	1.39680i	1.39680	−	1.39680i
$794$		0			0
$795$		0			0
$796$	0.504747	−	0.366720i	0.504747	−	0.366720i
$797$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$797$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$798$		0			0
$799$		0			0
$800$	−0.362091	−	1.11440i	−0.362091	−	1.11440i
$801$		0			0
$802$	−0.275319	+	0.540344i	−0.275319	+	0.540344i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		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	$0.400000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	0.889761	−	1.04178i	0.889761	−	1.04178i
$821$	−0.763472	−	1.05083i	−0.763472	−	1.05083i	−0.996917	−	0.0784591i	$-0.975000\pi$
$821$	−0.763472	−	1.05083i	−0.763472	−	1.05083i	0.233445	−	0.972370i	$-0.425000\pi$
$822$		0			0
$823$	1.76007	+	0.896802i	1.76007	+	0.896802i	0.951057	+	0.309017i	$0.100000\pi$
$823$	1.76007	+	0.896802i	1.76007	+	0.896802i	0.809017	+	0.587785i	$0.200000\pi$
$824$	−0.574714			−0.574714
$825$		0			0
$826$		0			0
$827$	0.416003	+	0.211964i	0.416003	+	0.211964i	0.649448	−	0.760406i	$-0.275000\pi$
$827$	0.416003	+	0.211964i	0.416003	+	0.211964i	−0.233445	+	0.972370i	$0.575000\pi$
$828$		0			0
$829$	1.04744	+	1.44168i	1.04744	+	1.44168i	0.891007	+	0.453990i	$0.150000\pi$
$829$	1.04744	+	1.44168i	1.04744	+	1.44168i	0.156434	+	0.987688i	$0.450000\pi$
$830$	1.92080	−	0.461143i	1.92080	−	0.461143i
$831$		0			0
$832$	−0.217082	−	0.217082i	−0.217082	−	0.217082i
$833$		0			0
$834$		0			0
$835$	0.678671	+	0.794622i	0.678671	+	0.794622i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−0.322922	−	0.993851i	−0.322922	−	0.993851i	−0.972370	−	0.233445i	$-0.925000\pi$
$839$	−0.322922	−	0.993851i	−0.322922	−	0.993851i	0.649448	−	0.760406i	$-0.275000\pi$
$840$		0			0
$841$	0.309017	−	0.951057i	0.309017	−	0.951057i
$842$		0			0
$843$		0			0
$844$	−0.719729	+	0.990622i	−0.719729	+	0.990622i
$845$	0.996917	+	0.0784591i	0.996917	+	0.0784591i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$853$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$857$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$	0.863541	+	0.280582i	0.863541	+	0.280582i	0.707107	−	0.707107i	$-0.250000\pi$
$859$	0.863541	+	0.280582i	0.863541	+	0.280582i	0.156434	+	0.987688i	$0.450000\pi$
$860$	0.468587	+	1.13127i	0.468587	+	1.13127i
$861$		0			0
$862$	−1.50324	+	0.765941i	−1.50324	+	0.765941i
$863$	0.139815	−	0.0712394i	0.139815	−	0.0712394i	−0.382683	−	0.923880i	$-0.625000\pi$
$863$	0.139815	−	0.0712394i	0.139815	−	0.0712394i	0.522499	+	0.852640i	$0.325000\pi$
$864$		0			0
$865$		0			0
$866$	−1.45221	−	0.471852i	−1.45221	−	0.471852i
$867$		0			0
$868$		0			0
$869$	2.22502	+	1.61657i	2.22502	+	1.61657i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$877$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$878$	2.44023	+	0.386494i	2.44023	+	0.386494i
$879$		0			0
$880$	2.01464	−	1.23457i	2.01464	−	1.23457i
$881$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$881$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$882$		0			0
$883$	0.610425	−	0.0966818i	0.610425	−	0.0966818i	0.156434	−	0.987688i	$-0.450000\pi$
$883$	0.610425	−	0.0966818i	0.610425	−	0.0966818i	0.453990	+	0.891007i	$0.350000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.453990	−	0.891007i	$-0.650000\pi$
$887$		0			0		0.453990	+	0.891007i	$0.350000\pi$
$888$		0			0
$889$		0			0
$890$	−2.21735	+	0.918458i	−2.21735	+	0.918458i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	1.97356	+	1.00558i	1.97356	+	1.00558i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	4.48752	+	2.28650i	4.48752	+	2.28650i
$903$		0			0
$904$		0			0
$905$	0.0922342	−	1.17195i	0.0922342	−	1.17195i
$906$		0			0
$907$	0.437016	+	0.437016i	0.437016	+	0.437016i	0.891007	−	0.453990i	$-0.150000\pi$
$907$	0.437016	+	0.437016i	0.437016	+	0.437016i	−0.453990	+	0.891007i	$0.650000\pi$
$908$	0.198617	+	1.25402i	0.198617	+	1.25402i
$909$		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
$913$	1.34271	+	2.63523i	1.34271	+	2.63523i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1.16110	+	1.59811i	−1.16110	+	1.59811i	−0.453990	+	0.891007i	$0.650000\pi$
$919$	−1.16110	+	1.59811i	−1.16110	+	1.59811i	−0.707107	+	0.707107i	$0.750000\pi$
$920$		0			0
$921$		0			0
$922$	1.34063	+	0.212335i	1.34063	+	0.212335i
$923$	0.211964	−	0.416003i	0.211964	−	0.416003i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−0.126949	+	0.0922342i	−0.126949	+	0.0922342i	−0.649448	−	0.760406i	$-0.725000\pi$
$929$	−0.126949	+	0.0922342i	−0.126949	+	0.0922342i	0.522499	+	0.852640i	$0.325000\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.44168	+	0.734572i	−1.44168	+	0.734572i	−0.987688	−	0.156434i	$-0.950000\pi$
$937$	−1.44168	+	0.734572i	−1.44168	+	0.734572i	−0.453990	+	0.891007i	$0.650000\pi$
$938$		0			0
$939$		0			0
$940$	−1.16814	−	0.715836i	−1.16814	−	0.715836i
$941$	−0.444039	−	0.144277i	−0.444039	−	0.144277i	0.0784591	−	0.996917i	$-0.475000\pi$
$941$	−0.444039	−	0.144277i	−0.444039	−	0.144277i	−0.522499	+	0.852640i	$0.675000\pi$
$942$		0			0
$943$		0			0
$944$	1.67619	+	1.21782i	1.67619	+	1.21782i
$945$		0			0
$946$	−3.64170	+	2.64585i	−3.64170	+	2.64585i
$947$	0.266765	−	1.68429i	0.266765	−	1.68429i	−0.382683	−	0.923880i	$-0.625000\pi$
$947$	0.266765	−	1.68429i	0.266765	−	1.68429i	0.649448	−	0.760406i	$-0.275000\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$953$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$954$		0			0
$955$		0			0
$956$	0.309121	−	0.425468i	0.309121	−	0.425468i
$957$		0			0
$958$	1.34063	−	0.212335i	1.34063	−	0.212335i
$959$		0			0
$960$		0			0
$961$	0.309017	+	0.951057i	0.309017	+	0.951057i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$967$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$968$	0.799431	+	0.799431i	0.799431	+	0.799431i
$969$		0			0
$970$		0			0
$971$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$971$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	2.40005			2.40005
$977$	1.51942	+	0.774181i	1.51942	+	0.774181i	0.996917	−	0.0784591i	$-0.0250000\pi$
$977$	1.51942	+	0.774181i	1.51942	+	0.774181i	0.522499	+	0.852640i	$0.325000\pi$
$978$		0			0
$979$	−2.11215	−	2.90713i	−2.11215	−	2.90713i
$980$	0.634826	+	0.262954i	0.634826	+	0.262954i
$981$		0			0
$982$		0			0
$983$	0.0245474	+	0.154986i	0.0245474	+	0.154986i	0.996917	−	0.0784591i	$-0.0250000\pi$
$983$	0.0245474	+	0.154986i	0.0245474	+	0.154986i	−0.972370	+	0.233445i	$0.925000\pi$
$984$		0			0
$985$	0.707107	−	0.292893i	0.707107	−	0.292893i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	0.809017	+	0.587785i	$0.200000\pi$
$991$	−0.190983	+	0.587785i	−0.190983	+	0.587785i	−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.774181	+	0.474419i	−0.774181	+	0.474419i
$996$		0			0
$997$	1.76007	+	0.278768i	1.76007	+	0.278768i	0.951057	−	0.309017i	$-0.100000\pi$
$997$	1.76007	+	0.278768i	1.76007	+	0.278768i	0.809017	+	0.587785i	$0.200000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.1.er.a.1117.2	✓	32
3.2	odd	2	inner	2925.1.er.a.1117.3	yes	32
13.12	even	2	inner	2925.1.er.a.1117.3	yes	32
25.3	odd	20	inner	2925.1.er.a.2053.3	yes	32
39.38	odd	2	CM	2925.1.er.a.1117.2	✓	32
75.53	even	20	inner	2925.1.er.a.2053.2	yes	32
325.103	odd	20	inner	2925.1.er.a.2053.2	yes	32
975.428	even	20	inner	2925.1.er.a.2053.3	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2925.1.er.a.1117.2	✓	32	1.1	even	1	trivial
2925.1.er.a.1117.2	✓	32	39.38	odd	2	CM
2925.1.er.a.1117.3	yes	32	3.2	odd	2	inner
2925.1.er.a.1117.3	yes	32	13.12	even	2	inner
2925.1.er.a.2053.2	yes	32	75.53	even	20	inner
2925.1.er.a.2053.2	yes	32	325.103	odd	20	inner
2925.1.er.a.2053.3	yes	32	25.3	odd	20	inner
2925.1.er.a.2053.3	yes	32	975.428	even	20	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:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2925.er (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.15732 - 0.589686i) q^{2} +(0.403886 + 0.555901i) q^{4} +(-0.522499 - 0.852640i) q^{5} +(0.0635725 + 0.401381i) q^{8} +O(q^{10})\)	\(q+(-1.15732 - 0.589686i) q^{2} +(0.403886 + 0.555901i) q^{4} +(-0.522499 - 0.852640i) q^{5} +(0.0635725 + 0.401381i) q^{8} +(0.101910 + 1.29489i) q^{10} +(-1.84956 + 0.600958i) q^{11} +(-0.453990 - 0.891007i) q^{13} +(0.375450 - 1.15552i) q^{16} +(0.262954 - 0.634826i) q^{20} +(2.49492 + 0.395156i) q^{22} +(-0.453990 + 0.891007i) q^{25} +1.29890i q^{26} +(-0.828553 + 0.828553i) q^{32} +(0.309017 - 0.263925i) q^{40} +(1.89625 + 0.616129i) q^{41} +(-1.26007 + 1.26007i) q^{43} +(-1.08108 - 0.785452i) q^{44} +(0.311904 - 1.96929i) q^{47} +1.00000i q^{49} +(1.05083 - 0.763472i) q^{50} +(0.311951 - 0.612238i) q^{52} +(1.47879 + 1.26301i) q^{55} +(-0.526961 + 1.62182i) q^{59} +(0.610425 + 1.87869i) q^{61} +(0.291975 - 0.0948685i) q^{64} +(-0.522499 + 0.852640i) q^{65} +(0.274431 + 0.377723i) q^{71} +(-0.831254 - 1.14412i) q^{79} +(-1.18141 + 0.283632i) q^{80} +(-1.83125 - 1.83125i) q^{82} +(-0.237907 - 1.50209i) q^{83} +(2.20136 - 0.715266i) q^{86} +(-0.358794 - 0.704173i) q^{88} +(0.570989 + 1.75732i) q^{89} +(-1.52224 + 2.09518i) q^{94} +(0.589686 - 1.15732i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 8 q^{10} + 8 q^{16} - 8 q^{22} - 8 q^{40} + 8 q^{43} + 8 q^{52} + 40 q^{64} - 32 q^{82} + 24 q^{88} - 40 q^{94}+O(q^{100}) \) 32 * q + 8 * q^10 + 8 * q^16 - 8 * q^22 - 8 * q^40 + 8 * q^43 + 8 * q^52 + 40 * q^64 - 32 * q^82 + 24 * q^88 - 40 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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