LMFDB - Embedded newform 1700.1.ct.a.147.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1700,1,Mod(3,1700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1700, base_ring=CyclotomicField(80))

chi = DirichletCharacter(H, H._module([40, 28, 5]))

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

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

chi := DirichletCharacter("1700.3");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1700 = 2^{2} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1700.ct (of order $80$, degree $32$, 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:	$0.848410521476$
Analytic rank:	$0$
Dimension:	$32$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{80}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{80} - \cdots)$

Embedding invariants

Embedding label			147.1
Root			$0.522499 - 0.852640i$ of defining polynomial
Character	$\chi$	$=$	1700.147
Dual form			1700.1.ct.a.983.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.996917 + 0.0784591i) q^{2} +(0.987688 - 0.156434i) q^{4} +(-0.987688 - 0.156434i) q^{5} +(-0.972370 + 0.233445i) q^{8} +(-0.760406 + 0.649448i) q^{9} +(0.996917 + 0.0784591i) q^{10} +(0.444039 - 0.144277i) q^{13} +(0.951057 - 0.309017i) q^{16} +(0.453990 - 0.891007i) q^{17} +(0.707107 - 0.707107i) q^{18} -1.00000 q^{20} +(0.951057 + 0.309017i) q^{25} +(-0.431351 + 0.178671i) q^{26} +(0.666244 + 1.80593i) q^{29} +(-0.923880 + 0.382683i) q^{32} +(-0.382683 + 0.923880i) q^{34} +(-0.649448 + 0.760406i) q^{36} +(0.226249 + 0.0638088i) q^{37} +(0.996917 - 0.0784591i) q^{40} +(1.56942 + 1.23723i) q^{41} +(0.852640 - 0.522499i) q^{45} +(0.923880 + 0.382683i) q^{49} +(-0.972370 - 0.233445i) q^{50} +(0.416003 - 0.211964i) q^{52} +(-1.01612 - 0.243950i) q^{53} +(-0.805883 - 1.74809i) q^{58} +(1.41351 - 0.398650i) q^{61} +(0.891007 - 0.453990i) q^{64} +(-0.461143 + 0.0730378i) q^{65} +(0.309017 - 0.951057i) q^{68} +(0.587785 - 0.809017i) q^{72} +(-0.0984164 + 0.831516i) q^{73} +(-0.230558 - 0.0458608i) q^{74} +(-0.987688 + 0.156434i) q^{80} +(0.156434 - 0.987688i) q^{81} +(-1.66166 - 1.11028i) q^{82} +(-0.587785 + 0.809017i) q^{85} +(0.0712394 + 0.139815i) q^{89} +(-0.809017 + 0.587785i) q^{90} +(1.42636 + 0.657561i) q^{97} +(-0.951057 - 0.309017i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 32 q^{20} + 8 q^{41} + 8 q^{53} - 8 q^{68} + 8 q^{74} - 8 q^{82} - 8 q^{90}+O(q^{100}) $ 32 * q - 32 * q^20 + 8 * q^41 + 8 * q^53 - 8 * q^68 + 8 * q^74 - 8 * q^82 - 8 * q^90

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1700\mathbb{Z}\right)^\times$.

$n$	$477$	$851$	$1601$
$\chi(n)$	$e\left(\frac{17}{20}\right)$	$-1$	$e\left(\frac{7}{16}\right)$

Coefficient data

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

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

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

$2$	−0.996917	+	0.0784591i	−0.996917	+	0.0784591i
$2$	−0.996917	+	0.0784591i	−0.996917	+	0.0784591i
$3$		0			0		−0.346117	−	0.938191i	$-0.612500\pi$
$3$		0			0		0.346117	+	0.938191i	$0.387500\pi$
$4$	0.987688	−	0.156434i	0.987688	−	0.156434i
$5$	−0.987688	−	0.156434i	−0.987688	−	0.156434i
$5$	−0.987688	−	0.156434i	−0.987688	−	0.156434i
$6$		0			0
$7$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$7$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$8$	−0.972370	+	0.233445i	−0.972370	+	0.233445i
$9$	−0.760406	+	0.649448i	−0.760406	+	0.649448i
$10$	0.996917	+	0.0784591i	0.996917	+	0.0784591i
$11$		0			0		0.488621	−	0.872496i	$-0.337500\pi$
$11$		0			0		−0.488621	+	0.872496i	$0.662500\pi$
$12$		0			0
$13$	0.444039	−	0.144277i	0.444039	−	0.144277i	−0.0784591	−	0.996917i	$-0.525000\pi$
$13$	0.444039	−	0.144277i	0.444039	−	0.144277i	0.522499	+	0.852640i	$0.325000\pi$
$14$		0			0
$15$		0			0
$16$	0.951057	−	0.309017i	0.951057	−	0.309017i
$17$	0.453990	−	0.891007i	0.453990	−	0.891007i
$17$	0.453990	−	0.891007i	0.453990	−	0.891007i
$18$	0.707107	−	0.707107i	0.707107	−	0.707107i
$19$		0			0		−0.233445	−	0.972370i	$-0.575000\pi$
$19$		0			0		0.233445	+	0.972370i	$0.425000\pi$
$20$	−1.00000			−1.00000
$21$		0			0
$22$		0			0
$23$		0			0		0.962455	−	0.271440i	$-0.0875000\pi$
$23$		0			0		−0.962455	+	0.271440i	$0.912500\pi$
$24$		0			0
$25$	0.951057	+	0.309017i	0.951057	+	0.309017i
$26$	−0.431351	+	0.178671i	−0.431351	+	0.178671i
$27$		0			0
$28$		0			0
$29$	0.666244	+	1.80593i	0.666244	+	1.80593i	0.587785	+	0.809017i	$0.300000\pi$
$29$	0.666244	+	1.80593i	0.666244	+	1.80593i	0.0784591	+	0.996917i	$0.475000\pi$
$30$		0			0
$31$		0			0		0.678801	−	0.734323i	$-0.262500\pi$
$31$		0			0		−0.678801	+	0.734323i	$0.737500\pi$
$32$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$33$		0			0
$34$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$35$		0			0
$36$	−0.649448	+	0.760406i	−0.649448	+	0.760406i
$37$	0.226249	+	0.0638088i	0.226249	+	0.0638088i	0.382683	−	0.923880i	$-0.375000\pi$
$37$	0.226249	+	0.0638088i	0.226249	+	0.0638088i	−0.156434	+	0.987688i	$0.550000\pi$
$38$		0			0
$39$		0			0
$40$	0.996917	−	0.0784591i	0.996917	−	0.0784591i
$41$	1.56942	+	1.23723i	1.56942	+	1.23723i	0.809017	+	0.587785i	$0.200000\pi$
$41$	1.56942	+	1.23723i	1.56942	+	1.23723i	0.760406	+	0.649448i	$0.225000\pi$
$42$		0			0
$43$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$43$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$44$		0			0
$45$	0.852640	−	0.522499i	0.852640	−	0.522499i
$46$		0			0
$47$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$47$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$48$		0			0
$49$	0.923880	+	0.382683i	0.923880	+	0.382683i
$50$	−0.972370	−	0.233445i	−0.972370	−	0.233445i
$51$		0			0
$52$	0.416003	−	0.211964i	0.416003	−	0.211964i
$53$	−1.01612	−	0.243950i	−1.01612	−	0.243950i	−0.309017	−	0.951057i	$-0.600000\pi$
$53$	−1.01612	−	0.243950i	−1.01612	−	0.243950i	−0.707107	+	0.707107i	$0.750000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−0.805883	−	1.74809i	−0.805883	−	1.74809i
$59$		0			0		−0.649448	−	0.760406i	$-0.725000\pi$
$59$		0			0		0.649448	+	0.760406i	$0.275000\pi$
$60$		0			0
$61$	1.41351	−	0.398650i	1.41351	−	0.398650i	0.522499	−	0.852640i	$-0.325000\pi$
$61$	1.41351	−	0.398650i	1.41351	−	0.398650i	0.891007	+	0.453990i	$0.150000\pi$
$62$		0			0
$63$		0			0
$64$	0.891007	−	0.453990i	0.891007	−	0.453990i
$65$	−0.461143	+	0.0730378i	−0.461143	+	0.0730378i
$66$		0			0
$67$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$67$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$68$	0.309017	−	0.951057i	0.309017	−	0.951057i
$69$		0			0
$70$		0			0
$71$		0			0		0.418660	−	0.908143i	$-0.362500\pi$
$71$		0			0		−0.418660	+	0.908143i	$0.637500\pi$
$72$	0.587785	−	0.809017i	0.587785	−	0.809017i
$73$	−0.0984164	+	0.831516i	−0.0984164	+	0.831516i	0.852640	+	0.522499i	$0.175000\pi$
$73$	−0.0984164	+	0.831516i	−0.0984164	+	0.831516i	−0.951057	+	0.309017i	$0.900000\pi$
$74$	−0.230558	−	0.0458608i	−0.230558	−	0.0458608i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.734323	−	0.678801i	$-0.237500\pi$
$79$		0			0		−0.734323	+	0.678801i	$0.762500\pi$
$80$	−0.987688	+	0.156434i	−0.987688	+	0.156434i
$81$	0.156434	−	0.987688i	0.156434	−	0.987688i
$82$	−1.66166	−	1.11028i	−1.66166	−	1.11028i
$83$		0			0		−0.852640	−	0.522499i	$-0.825000\pi$
$83$		0			0		0.852640	+	0.522499i	$0.175000\pi$
$84$		0			0
$85$	−0.587785	+	0.809017i	−0.587785	+	0.809017i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.0712394	+	0.139815i	0.0712394	+	0.139815i	0.923880	−	0.382683i	$-0.125000\pi$
$89$	0.0712394	+	0.139815i	0.0712394	+	0.139815i	−0.852640	+	0.522499i	$0.825000\pi$
$90$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.42636	+	0.657561i	1.42636	+	0.657561i	0.972370	−	0.233445i	$-0.0750000\pi$
$97$	1.42636	+	0.657561i	1.42636	+	0.657561i	0.453990	+	0.891007i	$0.350000\pi$
$98$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$99$		0			0
$100$	0.987688	+	0.156434i	0.987688	+	0.156434i
$101$		−	1.29890i		−	1.29890i	−0.760406	−	0.649448i	$-0.775000\pi$
$101$		−	1.29890i		−	1.29890i	0.760406	−	0.649448i	$-0.225000\pi$
$102$		0			0
$103$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$103$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$104$	−0.398090	+	0.243950i	−0.398090	+	0.243950i
$105$		0			0
$106$	1.03213	+	0.163474i	1.03213	+	0.163474i
$107$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$107$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$108$		0			0
$109$	−1.12445	+	1.42636i	−1.12445	+	1.42636i	−0.233445	+	0.972370i	$0.575000\pi$
$109$	−1.12445	+	1.42636i	−1.12445	+	1.42636i	−0.891007	+	0.453990i	$0.850000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.37037	+	1.08031i	1.37037	+	1.08031i	0.987688	+	0.156434i	$0.0500000\pi$
$113$	1.37037	+	1.08031i	1.37037	+	1.08031i	0.382683	+	0.923880i	$0.375000\pi$
$114$		0			0
$115$		0			0
$116$	0.940552	+	1.67948i	0.940552	+	1.67948i
$117$	−0.243950	+	0.398090i	−0.243950	+	0.398090i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.522499	−	0.852640i	−0.522499	−	0.852640i
$122$	−1.37787	+	0.508323i	−1.37787	+	0.508323i
$123$		0			0
$124$		0			0
$125$	−0.891007	−	0.453990i	−0.891007	−	0.453990i
$126$		0			0
$127$		0			0		−0.760406	−	0.649448i	$-0.775000\pi$
$127$		0			0		0.760406	+	0.649448i	$0.225000\pi$
$128$	−0.852640	+	0.522499i	−0.852640	+	0.522499i
$129$		0			0
$130$	0.453990	−	0.108993i	0.453990	−	0.108993i
$131$		0			0		−0.0392598	−	0.999229i	$-0.512500\pi$
$131$		0			0		0.0392598	+	0.999229i	$0.487500\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.233445	+	0.972370i	−0.233445	+	0.972370i
$137$	0.882893	−	1.73278i	0.882893	−	1.73278i	0.233445	−	0.972370i	$-0.425000\pi$
$137$	0.882893	−	1.73278i	0.882893	−	1.73278i	0.649448	−	0.760406i	$-0.275000\pi$
$138$		0			0
$139$		0			0		0.117537	−	0.993068i	$-0.462500\pi$
$139$		0			0		−0.117537	+	0.993068i	$0.537500\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.522499	+	0.852640i	−0.522499	+	0.852640i
$145$	−0.375531	−	1.88792i	−0.375531	−	1.88792i
$146$	0.0328730	−	0.836674i	0.0328730	−	0.836674i
$147$		0			0
$148$	0.233445	+	0.0276301i	0.233445	+	0.0276301i
$149$	−0.221232	−	0.221232i	−0.221232	−	0.221232i	0.587785	−	0.809017i	$-0.300000\pi$
$149$	−0.221232	−	0.221232i	−0.221232	−	0.221232i	−0.809017	+	0.587785i	$0.800000\pi$
$150$		0			0
$151$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$151$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$152$		0			0
$153$	0.233445	+	0.972370i	0.233445	+	0.972370i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			1.90211i			1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$157$			1.90211i			1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$158$		0			0
$159$		0			0
$160$	0.972370	−	0.233445i	0.972370	−	0.233445i
$161$		0			0
$162$	−0.0784591	+	0.996917i	−0.0784591	+	0.996917i
$163$		0			0		−0.117537	−	0.993068i	$-0.537500\pi$
$163$		0			0		0.117537	+	0.993068i	$0.462500\pi$
$164$	1.74365	+	0.976489i	1.74365	+	0.976489i
$165$		0			0
$166$		0			0
$167$		0			0		0.999229	−	0.0392598i	$-0.0125000\pi$
$167$		0			0		−0.999229	+	0.0392598i	$0.987500\pi$
$168$		0			0
$169$	−0.632662	+	0.459656i	−0.632662	+	0.459656i
$170$	0.522499	−	0.852640i	0.522499	−	0.852640i
$171$		0			0
$172$		0			0
$173$	0.972370	−	0.766555i	0.972370	−	0.766555i		−	1.00000i	$-0.5\pi$
$173$	0.972370	−	0.766555i	0.972370	−	0.766555i	0.972370	+	0.233445i	$0.0750000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−0.0819895	−	0.133795i	−0.0819895	−	0.133795i
$179$		0			0		0.233445	−	0.972370i	$-0.425000\pi$
$179$		0			0		−0.233445	+	0.972370i	$0.575000\pi$
$180$	0.760406	−	0.649448i	0.760406	−	0.649448i
$181$	0.493014	−	0.227282i	0.493014	−	0.227282i	−0.156434	−	0.987688i	$-0.550000\pi$
$181$	0.493014	−	0.227282i	0.493014	−	0.227282i	0.649448	+	0.760406i	$0.275000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.213482	−	0.0984164i	−0.213482	−	0.0984164i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$191$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$192$		0			0
$193$	−1.04246	−	1.56016i	−1.04246	−	1.56016i	−0.809017	−	0.587785i	$-0.800000\pi$
$193$	−1.04246	−	1.56016i	−1.04246	−	1.56016i	−0.233445	−	0.972370i	$-0.575000\pi$
$194$	−1.47356	−	0.543623i	−1.47356	−	0.543623i
$195$		0			0
$196$	0.972370	+	0.233445i	0.972370	+	0.233445i
$197$	−0.996917	+	0.921541i	−0.996917	+	0.921541i	−0.996917	−	0.0784591i	$-0.975000\pi$
$197$	−0.996917	+	0.921541i	−0.996917	+	0.921541i			1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$199$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$200$	−0.996917	−	0.0784591i	−0.996917	−	0.0784591i
$201$		0			0
$202$	0.101910	+	1.29489i	0.101910	+	1.29489i
$203$		0			0
$204$		0			0
$205$	−1.35655	−	1.46751i	−1.35655	−	1.46751i
$206$		0			0
$207$		0			0
$208$	0.377723	−	0.274431i	0.377723	−	0.274431i
$209$		0			0
$210$		0			0
$211$		0			0		−0.271440	−	0.962455i	$-0.587500\pi$
$211$		0			0		0.271440	+	0.962455i	$0.412500\pi$
$212$	−1.04178	−	0.0819895i	−1.04178	−	0.0819895i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	1.00907	−	1.51019i	1.00907	−	1.51019i
$219$		0			0
$220$		0			0
$221$	0.0730378	−	0.461143i	0.0730378	−	0.461143i
$222$		0			0
$223$		0			0		−0.0784591	−	0.996917i	$-0.525000\pi$
$223$		0			0		0.0784591	+	0.996917i	$0.475000\pi$
$224$		0			0
$225$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$226$	−1.45091	−	0.969466i	−1.45091	−	0.969466i
$227$		0			0		−0.271440	−	0.962455i	$-0.587500\pi$
$227$		0			0		0.271440	+	0.962455i	$0.412500\pi$
$228$		0			0
$229$	−1.92080	−	0.461143i	−1.92080	−	0.461143i	−0.996917	−	0.0784591i	$-0.975000\pi$
$229$	−1.92080	−	0.461143i	−1.92080	−	0.461143i	−0.923880	−	0.382683i	$-0.875000\pi$
$230$		0			0
$231$		0			0
$232$	−1.06942	−	1.60050i	−1.06942	−	1.60050i
$233$	0.508323	+	0.469889i	0.508323	+	0.469889i	0.891007	−	0.453990i	$-0.150000\pi$
$233$	0.508323	+	0.469889i	0.508323	+	0.469889i	−0.382683	+	0.923880i	$0.625000\pi$
$234$	0.211964	−	0.416003i	0.211964	−	0.416003i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$239$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$240$		0			0
$241$	−0.852640	+	0.477501i	−0.852640	+	0.477501i	−0.852640	−	0.522499i	$-0.825000\pi$
$241$	−0.852640	+	0.477501i	−0.852640	+	0.477501i			1.00000i	$0.5\pi$
$242$	0.587785	+	0.809017i	0.587785	+	0.809017i
$243$		0			0
$244$	1.33374	−	0.614863i	1.33374	−	0.614863i
$245$	−0.852640	−	0.522499i	−0.852640	−	0.522499i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.923880	+	0.382683i	0.923880	+	0.382683i
$251$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$251$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.809017	−	0.587785i	0.809017	−	0.587785i
$257$	0.449871	+	1.08609i	0.449871	+	1.08609i	0.972370	+	0.233445i	$0.0750000\pi$
$257$	0.449871	+	1.08609i	0.449871	+	1.08609i	−0.522499	+	0.852640i	$0.675000\pi$
$258$		0			0
$259$		0			0
$260$	−0.444039	+	0.144277i	−0.444039	+	0.144277i
$261$	−1.67948	−	0.940552i	−1.67948	−	0.940552i
$262$		0			0
$263$		0			0		0.0784591	−	0.996917i	$-0.475000\pi$
$263$		0			0		−0.0784591	+	0.996917i	$0.525000\pi$
$264$		0			0
$265$	0.965451	+	0.399903i	0.965451	+	0.399903i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.0576587	+	0.0532992i	0.0576587	+	0.0532992i	0.707107	−	0.707107i	$-0.250000\pi$
$269$	0.0576587	+	0.0532992i	0.0576587	+	0.0532992i	−0.649448	+	0.760406i	$0.725000\pi$
$270$		0			0
$271$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$271$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$272$	0.156434	−	0.987688i	0.156434	−	0.987688i
$273$		0			0
$274$	−0.744220	+	1.79671i	−0.744220	+	1.79671i
$275$		0			0
$276$		0			0
$277$	−1.97237	−	0.233445i	−1.97237	−	0.233445i	−0.972370	−	0.233445i	$-0.925000\pi$
$277$	−1.97237	−	0.233445i	−1.97237	−	0.233445i	−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.891007	−	1.45399i	0.891007	−	1.45399i		−	1.00000i	$-0.5\pi$
$281$	0.891007	−	1.45399i	0.891007	−	1.45399i	0.891007	−	0.453990i	$-0.150000\pi$
$282$		0			0
$283$		0			0		0.346117	−	0.938191i	$-0.387500\pi$
$283$		0			0		−0.346117	+	0.938191i	$0.612500\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.453990	−	0.891007i	0.453990	−	0.891007i
$289$	−0.587785	−	0.809017i	−0.587785	−	0.809017i
$290$	0.522499	+	1.85264i	0.522499	+	1.85264i
$291$		0			0
$292$	0.0328730	+	0.836674i	0.0328730	+	0.836674i
$293$	−0.907981			−0.907981			−0.453990	−	0.891007i	$-0.650000\pi$
$293$	−0.907981			−0.907981			−0.453990	+	0.891007i	$0.650000\pi$
$294$		0			0
$295$		0			0
$296$	−0.234894	−	0.00922899i	−0.234894	−	0.00922899i
$297$		0			0
$298$	0.237907	+	0.203192i	0.237907	+	0.203192i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.45847	+	0.172621i	−1.45847	+	0.172621i
$306$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$307$		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		0.785317	−	0.619094i	$-0.212500\pi$
$311$		0			0		−0.785317	+	0.619094i	$0.787500\pi$
$312$		0			0
$313$	1.94797	−	0.230558i	1.94797	−	0.230558i	0.951057	−	0.309017i	$-0.100000\pi$
$313$	1.94797	−	0.230558i	1.94797	−	0.230558i	0.996917	+	0.0784591i	$0.0250000\pi$
$314$	−0.149238	−	1.89625i	−0.149238	−	1.89625i
$315$		0			0
$316$		0			0
$317$	−1.56016	−	0.575572i	−1.56016	−	0.575572i	−0.587785	−	0.809017i	$-0.700000\pi$
$317$	−1.56016	−	0.575572i	−1.56016	−	0.575572i	−0.972370	+	0.233445i	$0.925000\pi$
$318$		0			0
$319$		0			0
$320$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		−	1.00000i		−	1.00000i
$325$	0.466891			0.466891
$326$		0			0
$327$		0			0
$328$	−1.81489	−	0.836674i	−1.81489	−	0.836674i
$329$		0			0
$330$		0			0
$331$		0			0		−0.852640	−	0.522499i	$-0.825000\pi$
$331$		0			0		0.852640	+	0.522499i	$0.175000\pi$
$332$		0			0
$333$	−0.213482	+	0.0984164i	−0.213482	+	0.0984164i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.301608	−	1.06942i	0.301608	−	1.06942i	−0.649448	−	0.760406i	$-0.725000\pi$
$337$	0.301608	−	1.06942i	0.301608	−	1.06942i	0.951057	−	0.309017i	$-0.100000\pi$
$338$	0.594647	−	0.507877i	0.594647	−	0.507877i
$339$		0			0
$340$	−0.453990	+	0.891007i	−0.453990	+	0.891007i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.909229	+	0.840483i	−0.909229	+	0.840483i
$347$		0			0		−0.678801	−	0.734323i	$-0.737500\pi$
$347$		0			0		0.678801	+	0.734323i	$0.262500\pi$
$348$		0			0
$349$	−0.581990	−	1.40505i	−0.581990	−	1.40505i	−0.891007	−	0.453990i	$-0.850000\pi$
$349$	−0.581990	−	1.40505i	−0.581990	−	1.40505i	0.309017	−	0.951057i	$-0.400000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	0.156434	+	0.987688i	$0.450000\pi$
$353$	−0.831254	+	1.14412i	−0.831254	+	1.14412i	−0.987688	+	0.156434i	$0.950000\pi$
$354$		0			0
$355$		0			0
$356$	0.0922342	+	0.126949i	0.0922342	+	0.126949i
$357$		0			0
$358$		0			0
$359$		0			0		−0.996917	−	0.0784591i	$-0.975000\pi$
$359$		0			0		0.996917	+	0.0784591i	$0.0250000\pi$
$360$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$361$	−0.891007	+	0.453990i	−0.891007	+	0.453990i
$362$	−0.473661	+	0.265263i	−0.473661	+	0.265263i
$363$		0			0
$364$		0			0
$365$	0.227282	−	0.805883i	0.227282	−	0.805883i
$366$		0			0
$367$		0			0		−0.418660	−	0.908143i	$-0.637500\pi$
$367$		0			0		0.418660	+	0.908143i	$0.362500\pi$
$368$		0			0
$369$	−1.99692	+	0.0784591i	−1.99692	+	0.0784591i
$370$	0.220545	+	0.0813634i	0.220545	+	0.0813634i
$371$		0			0
$372$		0			0
$373$	1.64637	−	0.838865i	1.64637	−	0.838865i	0.649448	−	0.760406i	$-0.275000\pi$
$373$	1.64637	−	0.838865i	1.64637	−	0.838865i	0.996917	−	0.0784591i	$-0.0250000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.556394	+	0.705782i	0.556394	+	0.705782i
$378$		0			0
$379$		0			0		0.938191	−	0.346117i	$-0.112500\pi$
$379$		0			0		−0.938191	+	0.346117i	$0.887500\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.233445	−	0.972370i	$-0.575000\pi$
$383$		0			0		0.233445	+	0.972370i	$0.425000\pi$
$384$		0			0
$385$		0			0
$386$	1.16166	+	1.47356i	1.16166	+	1.47356i
$387$		0			0
$388$	1.51166	+	0.426334i	1.51166	+	0.426334i
$389$	0.763472	−	0.893911i	0.763472	−	0.893911i	−0.233445	−	0.972370i	$-0.575000\pi$
$389$	0.763472	−	0.893911i	0.763472	−	0.893911i	0.996917	+	0.0784591i	$0.0250000\pi$
$390$		0			0
$391$		0			0
$392$	−0.987688	−	0.156434i	−0.987688	−	0.156434i
$393$		0			0
$394$	0.921541	−	0.996917i	0.921541	−	0.996917i
$395$		0			0
$396$		0			0
$397$	−0.389880	−	0.0153184i	−0.389880	−	0.0153184i	−0.156434	−	0.987688i	$-0.550000\pi$
$397$	−0.389880	−	0.0153184i	−0.389880	−	0.0153184i	−0.233445	+	0.972370i	$0.575000\pi$
$398$		0			0
$399$		0			0
$400$	1.00000			1.00000
$401$	0.384585	+	0.575572i	0.384585	+	0.575572i	0.972370	−	0.233445i	$-0.0750000\pi$
$401$	0.384585	+	0.575572i	0.384585	+	0.575572i	−0.587785	+	0.809017i	$0.700000\pi$
$402$		0			0
$403$		0			0
$404$	−0.203192	−	1.28290i	−0.203192	−	1.28290i
$405$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.863541	−	0.280582i	0.863541	−	0.280582i	0.156434	−	0.987688i	$-0.450000\pi$
$409$	0.863541	−	0.280582i	0.863541	−	0.280582i	0.707107	+	0.707107i	$0.250000\pi$
$410$	1.46751	+	1.35655i	1.46751	+	1.35655i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−0.355026	+	0.303221i	−0.355026	+	0.303221i
$417$		0			0
$418$		0			0
$419$		0			0		−0.418660	−	0.908143i	$-0.637500\pi$
$419$		0			0		0.418660	+	0.908143i	$0.362500\pi$
$420$		0			0
$421$	−1.96929	+	0.311904i	−1.96929	+	0.311904i	−0.972370	+	0.233445i	$0.925000\pi$
$421$	−1.96929	+	0.311904i	−1.96929	+	0.311904i	−0.996917	+	0.0784591i	$0.975000\pi$
$422$		0			0
$423$		0			0
$424$	1.04500			1.04500
$425$	0.707107	−	0.707107i	0.707107	−	0.707107i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.418660	−	0.908143i	$-0.637500\pi$
$431$		0			0		0.418660	+	0.908143i	$0.362500\pi$
$432$		0			0
$433$	−1.92080	+	0.461143i	−1.92080	+	0.461143i	−0.923880	+	0.382683i	$0.875000\pi$
$433$	−1.92080	+	0.461143i	−1.92080	+	0.461143i	−0.996917	+	0.0784591i	$0.975000\pi$
$434$		0			0
$435$		0			0
$436$	−0.887476	+	1.58470i	−0.887476	+	1.58470i
$437$		0			0
$438$		0			0
$439$		0			0		−0.993068	−	0.117537i	$-0.962500\pi$
$439$		0			0		0.993068	+	0.117537i	$0.0375000\pi$
$440$		0			0
$441$	−0.951057	+	0.309017i	−0.951057	+	0.309017i
$442$	−0.0366318	+	0.465451i	−0.0366318	+	0.465451i
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$	−0.0484904	−	0.149238i	−0.0484904	−	0.149238i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.10344	+	1.65141i	1.10344	+	1.65141i	0.649448	+	0.760406i	$0.275000\pi$
$449$	1.10344	+	1.65141i	1.10344	+	1.65141i	0.453990	+	0.891007i	$0.350000\pi$
$450$	0.891007	−	0.453990i	0.891007	−	0.453990i
$451$		0			0
$452$	1.52250	+	0.852640i	1.52250	+	0.852640i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$457$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$458$	1.95106	+	0.309017i	1.95106	+	0.309017i
$459$		0			0
$460$		0			0
$461$	1.29489	−	1.51612i	1.29489	−	1.51612i	0.587785	−	0.809017i	$-0.300000\pi$
$461$	1.29489	−	1.51612i	1.29489	−	1.51612i	0.707107	−	0.707107i	$-0.250000\pi$
$462$		0			0
$463$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$463$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$464$	1.19170	+	1.51166i	1.19170	+	1.51166i
$465$		0			0
$466$	−0.543623	−	0.428558i	−0.543623	−	0.428558i
$467$		0			0		−0.233445	−	0.972370i	$-0.575000\pi$
$467$		0			0		0.233445	+	0.972370i	$0.425000\pi$
$468$	−0.178671	+	0.431351i	−0.178671	+	0.431351i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.931099	−	0.474419i	0.931099	−	0.474419i
$478$		0			0
$479$		0			0		−0.908143	−	0.418660i	$-0.862500\pi$
$479$		0			0		0.908143	+	0.418660i	$0.137500\pi$
$480$		0			0
$481$	0.109670	−	0.00430893i	0.109670	−	0.00430893i
$482$	0.812547	−	0.542927i	0.812547	−	0.542927i
$483$		0			0
$484$	−0.649448	−	0.760406i	−0.649448	−	0.760406i
$485$	−1.30593	−	0.872597i	−1.30593	−	0.872597i
$486$		0			0
$487$		0			0		0.619094	−	0.785317i	$-0.287500\pi$
$487$		0			0		−0.619094	+	0.785317i	$0.712500\pi$
$488$	−1.28139	+	0.717611i	−1.28139	+	0.717611i
$489$		0			0
$490$	0.891007	+	0.453990i	0.891007	+	0.453990i
$491$		0			0		−0.996917	−	0.0784591i	$-0.975000\pi$
$491$		0			0		0.996917	+	0.0784591i	$0.0250000\pi$
$492$		0			0
$493$	1.91157	+	0.226249i	1.91157	+	0.226249i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$499$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$500$	−0.951057	−	0.309017i	−0.951057	−	0.309017i
$501$		0			0
$502$		0			0
$503$		0			0		−0.678801	−	0.734323i	$-0.737500\pi$
$503$		0			0		0.678801	+	0.734323i	$0.262500\pi$
$504$		0			0
$505$	−0.203192	+	1.28290i	−0.203192	+	1.28290i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.280582	+	0.863541i	0.280582	+	0.863541i	0.987688	+	0.156434i	$0.0500000\pi$
$509$	0.280582	+	0.863541i	0.280582	+	0.863541i	−0.707107	+	0.707107i	$0.750000\pi$
$510$		0			0
$511$		0			0
$512$	−0.760406	+	0.649448i	−0.760406	+	0.649448i
$513$		0			0
$514$	−0.533698	−	1.04744i	−0.533698	−	1.04744i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	0.431351	−	0.178671i	0.431351	−	0.178671i
$521$	−0.921541	−	0.996917i	−0.921541	−	0.996917i	0.0784591	−	0.996917i	$-0.475000\pi$
$521$	−0.921541	−	0.996917i	−0.921541	−	0.996917i	−1.00000			$\pi$
$522$	1.74809	+	0.805883i	1.74809	+	0.805883i
$523$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$523$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.852640	−	0.522499i	0.852640	−	0.522499i
$530$	−0.993851	−	0.322922i	−0.993851	−	0.322922i
$531$		0			0
$532$		0			0
$533$	0.875390	+	0.322949i	0.875390	+	0.322949i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−0.0616628	−	0.0486110i	−0.0616628	−	0.0486110i
$539$		0			0
$540$		0			0
$541$	0.605005	+	1.08031i	0.605005	+	1.08031i	0.987688	+	0.156434i	$0.0500000\pi$
$541$	0.605005	+	1.08031i	0.605005	+	1.08031i	−0.382683	+	0.923880i	$0.625000\pi$
$542$		0			0
$543$		0			0
$544$	−0.0784591	+	0.996917i	−0.0784591	+	0.996917i
$545$	1.33374	−	1.23290i	1.33374	−	1.23290i
$546$		0			0
$547$		0			0		0.938191	−	0.346117i	$-0.112500\pi$
$547$		0			0		−0.938191	+	0.346117i	$0.887500\pi$
$548$	0.600958	−	1.84956i	0.600958	−	1.84956i
$549$	−0.815935	+	1.22113i	−0.815935	+	1.22113i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	1.98461	+	0.0779754i	1.98461	+	0.0779754i
$555$		0			0
$556$		0			0
$557$	1.99383			1.99383			0.996917	−	0.0784591i	$-0.0250000\pi$
$557$	1.99383			1.99383			0.996917	+	0.0784591i	$0.0250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.774181	+	1.51942i	−0.774181	+	1.51942i
$563$		0			0		−0.649448	−	0.760406i	$-0.725000\pi$
$563$		0			0		0.649448	+	0.760406i	$0.275000\pi$
$564$		0			0
$565$	−1.18450	−	1.28139i	−1.18450	−	1.28139i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.794622	−	1.29671i	0.794622	−	1.29671i	−0.156434	−	0.987688i	$-0.550000\pi$
$569$	0.794622	−	1.29671i	0.794622	−	1.29671i	0.951057	−	0.309017i	$-0.100000\pi$
$570$		0			0
$571$		0			0		0.0392598	−	0.999229i	$-0.487500\pi$
$571$		0			0		−0.0392598	+	0.999229i	$0.512500\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$577$	0.838865	+	1.64637i	0.838865	+	1.64637i	0.760406	+	0.649448i	$0.225000\pi$
$577$	0.838865	+	1.64637i	0.838865	+	1.64637i	0.0784591	+	0.996917i	$0.475000\pi$
$578$	0.649448	+	0.760406i	0.649448	+	0.760406i
$579$		0			0
$580$	−0.666244	−	1.80593i	−0.666244	−	1.80593i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.0984164	−	0.831516i	−0.0984164	−	0.831516i
$585$	0.303221	−	0.355026i	0.303221	−	0.355026i
$586$	0.905182	−	0.0712394i	0.905182	−	0.0712394i
$587$		0			0		0.0784591	−	0.996917i	$-0.475000\pi$
$587$		0			0		−0.0784591	+	0.996917i	$0.525000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.234894	−	0.00922899i	0.234894	−	0.00922899i
$593$	0.581990	+	1.40505i	0.581990	+	1.40505i	0.891007	+	0.453990i	$0.150000\pi$
$593$	0.581990	+	1.40505i	0.581990	+	1.40505i	−0.309017	+	0.951057i	$0.600000\pi$
$594$		0			0
$595$		0			0
$596$	−0.253116	−	0.183900i	−0.253116	−	0.183900i
$597$		0			0
$598$		0			0
$599$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$599$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$600$		0			0
$601$	−0.264855	+	1.33152i	−0.264855	+	1.33152i	0.587785	+	0.809017i	$0.300000\pi$
$601$	−0.264855	+	1.33152i	−0.264855	+	1.33152i	−0.852640	+	0.522499i	$0.825000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.382683	+	0.923880i	0.382683	+	0.923880i
$606$		0			0
$607$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$607$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$608$		0			0
$609$		0			0
$610$	1.44043	−	0.286518i	1.44043	−	0.286518i
$611$		0			0
$612$	0.382683	+	0.923880i	0.382683	+	0.923880i
$613$	−1.58779	−	0.809017i	−1.58779	−	0.809017i	−0.587785	−	0.809017i	$-0.700000\pi$
$613$	−1.58779	−	0.809017i	−1.58779	−	0.809017i	−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.0576587	+	0.0532992i	0.0576587	+	0.0532992i	0.707107	−	0.707107i	$-0.250000\pi$
$617$	0.0576587	+	0.0532992i	0.0576587	+	0.0532992i	−0.649448	+	0.760406i	$0.725000\pi$
$618$		0			0
$619$		0			0		−0.938191	−	0.346117i	$-0.887500\pi$
$619$		0			0		0.938191	+	0.346117i	$0.112500\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.809017	+	0.587785i	0.809017	+	0.587785i
$626$	−1.92388	+	0.382683i	−1.92388	+	0.382683i
$627$		0			0
$628$	0.297556	+	1.87869i	0.297556	+	1.87869i
$629$	0.159569	−	0.172621i	0.159569	−	0.172621i
$630$		0			0
$631$		0			0		0.972370	−	0.233445i	$-0.0750000\pi$
$631$		0			0		−0.972370	+	0.233445i	$0.925000\pi$
$632$		0			0
$633$		0			0
$634$	1.60050	+	0.451389i	1.60050	+	0.451389i
$635$		0			0
$636$		0			0
$637$	0.465451	+	0.0366318i	0.465451	+	0.0366318i
$638$		0			0
$639$		0			0
$640$	0.923880	−	0.382683i	0.923880	−	0.382683i
$641$	0.375531	+	0.105911i	0.375531	+	0.105911i	0.453990	−	0.891007i	$-0.350000\pi$
$641$	0.375531	+	0.105911i	0.375531	+	0.105911i	−0.0784591	+	0.996917i	$0.525000\pi$
$642$		0			0
$643$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$643$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$647$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$648$	0.0784591	+	0.996917i	0.0784591	+	0.996917i
$649$		0			0
$650$	−0.465451	+	0.0366318i	−0.465451	+	0.0366318i
$651$		0			0
$652$		0			0
$653$	−1.45847	+	1.34819i	−1.45847	+	1.34819i	−0.649448	+	0.760406i	$0.725000\pi$
$653$	−1.45847	+	1.34819i	−1.45847	+	1.34819i	−0.809017	+	0.587785i	$0.800000\pi$
$654$		0			0
$655$		0			0
$656$	1.87494	+	0.691700i	1.87494	+	0.691700i
$657$	−0.465190	−	0.696206i	−0.465190	−	0.696206i
$658$		0			0
$659$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$659$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$660$		0			0
$661$	1.40505	+	1.20002i	1.40505	+	1.20002i	0.951057	+	0.309017i	$0.100000\pi$
$661$	1.40505	+	1.20002i	1.40505	+	1.20002i	0.453990	+	0.891007i	$0.350000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	0.205102	−	0.114863i	0.205102	−	0.114863i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.265263	+	0.473661i	−0.265263	+	0.473661i	−0.972370	−	0.233445i	$-0.925000\pi$
$673$	−0.265263	+	0.473661i	−0.265263	+	0.473661i	0.707107	+	0.707107i	$0.250000\pi$
$674$	−0.216773	+	1.08979i	−0.216773	+	1.08979i
$675$		0			0
$676$	−0.552967	+	0.552967i	−0.552967	+	0.552967i
$677$	0.872597	−	0.687900i	0.872597	−	0.687900i	−0.0784591	−	0.996917i	$-0.525000\pi$
$677$	0.872597	−	0.687900i	0.872597	−	0.687900i	0.951057	+	0.309017i	$0.100000\pi$
$678$		0			0
$679$		0			0
$680$	0.382683	−	0.923880i	0.382683	−	0.923880i
$681$		0			0
$682$		0			0
$683$		0			0		0.999229	−	0.0392598i	$-0.0125000\pi$
$683$		0			0		−0.999229	+	0.0392598i	$0.987500\pi$
$684$		0			0
$685$	−1.14309	+	1.57333i	−1.14309	+	1.57333i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.486395	+	0.0382802i	−0.486395	+	0.0382802i
$690$		0			0
$691$		0			0		−0.117537	−	0.993068i	$-0.537500\pi$
$691$		0			0		0.117537	+	0.993068i	$0.462500\pi$
$692$	0.840483	−	0.909229i	0.840483	−	0.909229i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1.81489	−	0.836674i	1.81489	−	0.836674i
$698$	0.690434	+	1.35505i	0.690434	+	1.35505i
$699$		0			0
$700$		0			0
$701$	1.37514	+	1.37514i	1.37514	+	1.37514i	0.852640	+	0.522499i	$0.175000\pi$
$701$	1.37514	+	1.37514i	1.37514	+	1.37514i	0.522499	+	0.852640i	$0.325000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	0.738925	−	1.20582i	0.738925	−	1.20582i
$707$		0			0
$708$		0			0
$709$	0.187900	−	0.666244i	0.187900	−	0.666244i	−0.809017	−	0.587785i	$-0.800000\pi$
$709$	0.187900	−	0.666244i	0.187900	−	0.666244i	0.996917	−	0.0784591i	$-0.0250000\pi$
$710$		0			0
$711$		0			0
$712$	−0.101910	−	0.119322i	−0.101910	−	0.119322i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.0392598	−	0.999229i	$-0.512500\pi$
$719$		0			0		0.0392598	+	0.999229i	$0.487500\pi$
$720$	0.649448	−	0.760406i	0.649448	−	0.760406i
$721$		0			0
$722$	0.852640	−	0.522499i	0.852640	−	0.522499i
$723$		0			0
$724$	0.451389	−	0.301608i	0.451389	−	0.301608i
$725$	0.0755716	+	1.92343i	0.0755716	+	1.92343i
$726$		0			0
$727$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$727$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$728$		0			0
$729$	0.522499	+	0.852640i	0.522499	+	0.852640i
$730$	−0.163353	+	0.821231i	−0.163353	+	0.821231i
$731$		0			0
$732$		0			0
$733$	−0.399903	+	0.652583i	−0.399903	+	0.652583i	−0.987688	−	0.156434i	$-0.950000\pi$
$733$	−0.399903	+	0.652583i	−0.399903	+	0.652583i	0.587785	+	0.809017i	$0.300000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	1.98461	−	0.234894i	1.98461	−	0.234894i
$739$		0			0		−0.0784591	−	0.996917i	$-0.525000\pi$
$739$		0			0		0.0784591	+	0.996917i	$0.475000\pi$
$740$	−0.226249	−	0.0638088i	−0.226249	−	0.0638088i
$741$		0			0
$742$		0			0
$743$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$743$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$744$		0			0
$745$	0.183900	+	0.253116i	0.183900	+	0.253116i
$746$	−1.57547	+	0.965451i	−1.57547	+	0.965451i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$751$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$752$		0			0
$753$		0			0
$754$	−0.610054	−	0.659952i	−0.610054	−	0.659952i
$755$		0			0
$756$		0			0
$757$	1.08609	+	0.449871i	1.08609	+	0.449871i	0.852640	−	0.522499i	$-0.175000\pi$
$757$	1.08609	+	0.449871i	1.08609	+	0.449871i	0.233445	+	0.972370i	$0.425000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.882893	−	1.73278i	−0.882893	−	1.73278i	−0.649448	−	0.760406i	$-0.725000\pi$
$761$	−0.882893	−	1.73278i	−0.882893	−	1.73278i	−0.233445	−	0.972370i	$-0.575000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−0.0784591	−	0.996917i	−0.0784591	−	0.996917i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.119730	−	0.755944i	0.119730	−	0.755944i	−0.852640	−	0.522499i	$-0.825000\pi$
$769$	0.119730	−	0.755944i	0.119730	−	0.755944i	0.972370	−	0.233445i	$-0.0750000\pi$
$770$		0			0
$771$		0			0
$772$	−1.27369	−	1.37787i	−1.27369	−	1.37787i
$773$	−0.203192	+	0.237907i	−0.203192	+	0.237907i	−0.852640	−	0.522499i	$-0.825000\pi$
$773$	−0.203192	+	0.237907i	−0.203192	+	0.237907i	0.649448	+	0.760406i	$0.275000\pi$
$774$		0			0
$775$		0			0
$776$	−1.54045	−	0.306415i	−1.54045	−	0.306415i
$777$		0			0
$778$	−0.690983	+	0.951057i	−0.690983	+	0.951057i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.996917	+	0.0784591i	0.996917	+	0.0784591i
$785$	0.297556	−	1.87869i	0.297556	−	1.87869i
$786$		0			0
$787$		0			0		0.872496	−	0.488621i	$-0.162500\pi$
$787$		0			0		−0.872496	+	0.488621i	$0.837500\pi$
$788$	−0.840483	+	1.06615i	−0.840483	+	1.06615i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.570136	−	0.380953i	0.570136	−	0.380953i
$794$	0.389880	−	0.0153184i	0.389880	−	0.0153184i
$795$		0			0
$796$		0			0
$797$	−1.65816	−	0.398090i	−1.65816	−	0.398090i	−0.707107	−	0.707107i	$-0.750000\pi$
$797$	−1.65816	−	0.398090i	−1.65816	−	0.398090i	−0.951057	+	0.309017i	$0.900000\pi$
$798$		0			0
$799$		0			0
$800$	−0.996917	+	0.0784591i	−0.996917	+	0.0784591i
$801$	−0.144974	−	0.0600500i	−0.144974	−	0.0600500i
$802$	−0.428558	−	0.543623i	−0.428558	−	0.543623i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.303221	+	1.26301i	0.303221	+	1.26301i
$809$	−1.37037	−	1.08031i	−1.37037	−	1.08031i	−0.987688	−	0.156434i	$-0.950000\pi$
$809$	−1.37037	−	1.08031i	−1.37037	−	1.08031i	−0.382683	−	0.923880i	$-0.625000\pi$
$810$	0.233445	−	0.972370i	0.233445	−	0.972370i
$811$		0			0		−0.619094	−	0.785317i	$-0.712500\pi$
$811$		0			0		0.619094	+	0.785317i	$0.287500\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.838865	+	0.347469i	−0.838865	+	0.347469i
$819$		0			0
$820$	−1.56942	−	1.23723i	−1.56942	−	1.23723i
$821$	−0.575572	−	1.56016i	−0.575572	−	1.56016i	−0.809017	−	0.587785i	$-0.800000\pi$
$821$	−0.575572	−	1.56016i	−0.575572	−	1.56016i	0.233445	−	0.972370i	$-0.425000\pi$
$822$		0			0
$823$		0			0		−0.872496	−	0.488621i	$-0.837500\pi$
$823$		0			0		0.872496	+	0.488621i	$0.162500\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.962455	−	0.271440i	$-0.0875000\pi$
$827$		0			0		−0.962455	+	0.271440i	$0.912500\pi$
$828$		0			0
$829$	−0.309017	−	1.95106i	−0.309017	−	1.95106i	−0.309017	−	0.951057i	$-0.600000\pi$
$829$	−0.309017	−	1.95106i	−0.309017	−	1.95106i		−	1.00000i	$-0.5\pi$
$830$		0			0
$831$		0			0
$832$	0.330142	−	0.330142i	0.330142	−	0.330142i
$833$	0.760406	−	0.649448i	0.760406	−	0.649448i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.488621	−	0.872496i	$-0.337500\pi$
$839$		0			0		−0.488621	+	0.872496i	$0.662500\pi$
$840$		0			0
$841$	−2.05711	+	1.75694i	−2.05711	+	1.75694i
$842$	1.93874	−	0.465451i	1.93874	−	0.465451i
$843$		0			0
$844$		0			0
$845$	0.696779	−	0.355026i	0.696779	−	0.355026i
$846$		0			0
$847$		0			0
$848$	−1.04178	+	0.0819895i	−1.04178	+	0.0819895i
$849$		0			0
$850$	−0.649448	+	0.760406i	−0.649448	+	0.760406i
$851$		0			0
$852$		0			0
$853$	−0.628648	−	1.70402i	−0.628648	−	1.70402i	−0.707107	−	0.707107i	$-0.750000\pi$
$853$	−0.628648	−	1.70402i	−0.628648	−	1.70402i	0.0784591	−	0.996917i	$-0.475000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.92388	+	0.382683i	1.92388	+	0.382683i	1.00000			$0$
$857$	1.92388	+	0.382683i	1.92388	+	0.382683i	0.923880	+	0.382683i	$0.125000\pi$
$858$		0			0
$859$		0			0		0.760406	−	0.649448i	$-0.225000\pi$
$859$		0			0		−0.760406	+	0.649448i	$0.775000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$863$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$864$		0			0
$865$	−1.08031	+	0.605005i	−1.08031	+	0.605005i
$866$	1.87869	−	0.610425i	1.87869	−	0.610425i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.760406	−	1.64945i	0.760406	−	1.64945i
$873$	−1.51166	+	0.426334i	−1.51166	+	0.426334i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.603972	−	0.338240i	−0.603972	−	0.338240i	0.156434	−	0.987688i	$-0.450000\pi$
$877$	−0.603972	−	0.338240i	−0.603972	−	0.338240i	−0.760406	+	0.649448i	$0.775000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.33152	+	1.44043i	−1.33152	+	1.44043i	−0.522499	+	0.852640i	$0.675000\pi$
$881$	−1.33152	+	1.44043i	−1.33152	+	1.44043i	−0.809017	+	0.587785i	$0.800000\pi$
$882$	0.923880	−	0.382683i	0.923880	−	0.382683i
$883$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$883$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$884$		−	0.466891i		−	0.466891i
$885$		0			0
$886$		0			0
$887$		0			0		−0.962455	−	0.271440i	$-0.912500\pi$
$887$		0			0		0.962455	+	0.271440i	$0.0875000\pi$
$888$		0			0
$889$		0			0
$890$	0.0600500	+	0.144974i	0.0600500	+	0.144974i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−1.22961	−	1.55975i	−1.22961	−	1.55975i
$899$		0			0
$900$	−0.852640	+	0.522499i	−0.852640	+	0.522499i
$901$	−0.678671	+	0.794622i	−0.678671	+	0.794622i
$902$		0			0
$903$		0			0
$904$	−1.58470	−	0.730558i	−1.58470	−	0.730558i
$905$	−0.522499	+	0.147360i	−0.522499	+	0.147360i
$906$		0			0
$907$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$907$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$908$		0			0
$909$	0.843566	+	0.987688i	0.843566	+	0.987688i
$910$		0			0
$911$		0			0		0.962455	−	0.271440i	$-0.0875000\pi$
$911$		0			0		−0.962455	+	0.271440i	$0.912500\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.96929	−	0.154986i	−1.96929	−	0.154986i
$917$		0			0
$918$		0			0
$919$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$919$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$920$		0			0
$921$		0			0
$922$	−1.17195	+	1.61305i	−1.17195	+	1.61305i
$923$		0			0
$924$		0			0
$925$	0.195458	+	0.130601i	0.195458	+	0.130601i
$926$		0			0
$927$		0			0
$928$	−1.30663	−	1.41351i	−1.30663	−	1.41351i
$929$	−0.717611	+	0.663353i	−0.717611	+	0.663353i	−0.951057	−	0.309017i	$-0.900000\pi$
$929$	−0.717611	+	0.663353i	−0.717611	+	0.663353i	0.233445	+	0.972370i	$0.425000\pi$
$930$		0			0
$931$		0			0
$932$	0.575572	+	0.384585i	0.575572	+	0.384585i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	0.144277	−	0.444039i	0.144277	−	0.444039i
$937$	0.987688	−	0.843566i	0.987688	−	0.843566i		−	1.00000i	$-0.5\pi$
$937$	0.987688	−	0.843566i	0.987688	−	0.843566i	0.987688	+	0.156434i	$0.0500000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.73290	−	0.205102i	1.73290	−	0.205102i	0.809017	−	0.587785i	$-0.200000\pi$
$941$	1.73290	−	0.205102i	1.73290	−	0.205102i	0.923880	+	0.382683i	$0.125000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.908143	−	0.418660i	$-0.862500\pi$
$947$		0			0		0.908143	+	0.418660i	$0.137500\pi$
$948$		0			0
$949$	0.0762680	+	0.383425i	0.0762680	+	0.383425i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.28290	+	0.203192i	−1.28290	+	0.203192i	−0.760406	−	0.649448i	$-0.775000\pi$
$953$	−1.28290	+	0.203192i	−1.28290	+	0.203192i	−0.522499	+	0.852640i	$0.675000\pi$
$954$	−0.891007	+	0.546010i	−0.891007	+	0.546010i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.0784591	−	0.996917i	−0.0784591	−	0.996917i
$962$	−0.108993	+	0.0129002i	−0.108993	+	0.0129002i
$963$		0			0
$964$	−0.767445	+	0.605005i	−0.767445	+	0.605005i
$965$	0.785566	+	1.70402i	0.785566	+	1.70402i
$966$		0			0
$967$		0			0		0.522499	−	0.852640i	$-0.325000\pi$
$967$		0			0		−0.522499	+	0.852640i	$0.675000\pi$
$968$	0.707107	+	0.707107i	0.707107	+	0.707107i
$969$		0			0
$970$	1.37037	+	0.767445i	1.37037	+	0.767445i
$971$		0			0		−0.522499	−	0.852640i	$-0.675000\pi$
$971$		0			0		0.522499	+	0.852640i	$0.325000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	1.22113	−	0.815935i	1.22113	−	0.815935i
$977$	−1.29671	−	1.10749i	−1.29671	−	1.10749i	−0.987688	−	0.156434i	$-0.950000\pi$
$977$	−1.29671	−	1.10749i	−1.29671	−	1.10749i	−0.309017	−	0.951057i	$-0.600000\pi$
$978$		0			0
$979$		0			0
$980$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$981$	−0.0713071	−	1.81489i	−0.0713071	−	1.81489i
$982$		0			0
$983$		0			0		−0.0392598	−	0.999229i	$-0.512500\pi$
$983$		0			0		0.0392598	+	0.999229i	$0.487500\pi$
$984$		0			0
$985$	1.12880	−	0.754243i	1.12880	−	0.754243i
$986$	−1.92343	−	0.0755716i	−1.92343	−	0.0755716i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.271440	−	0.962455i	$-0.412500\pi$
$991$		0			0		−0.271440	+	0.962455i	$0.587500\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.0652867	+	1.66166i	−0.0652867	+	1.66166i	0.522499	+	0.852640i	$0.325000\pi$
$997$	−0.0652867	+	1.66166i	−0.0652867	+	1.66166i	−0.587785	+	0.809017i	$0.700000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1700.1.ct.a.147.1	yes	32
4.3	odd	2	CM	1700.1.ct.a.147.1	yes	32
17.14	odd	16		1700.1.cm.a.847.1	yes	32
25.8	odd	20		1700.1.cm.a.283.1	✓	32
68.31	even	16		1700.1.cm.a.847.1	yes	32
100.83	even	20		1700.1.cm.a.283.1	✓	32
425.133	even	80	inner	1700.1.ct.a.983.1	yes	32
1700.983	odd	80	inner	1700.1.ct.a.983.1	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1700.1.cm.a.283.1	✓	32	25.8	odd	20
1700.1.cm.a.283.1	✓	32	100.83	even	20
1700.1.cm.a.847.1	yes	32	17.14	odd	16
1700.1.cm.a.847.1	yes	32	68.31	even	16
1700.1.ct.a.147.1	yes	32	1.1	even	1	trivial
1700.1.ct.a.147.1	yes	32	4.3	odd	2	CM
1700.1.ct.a.983.1	yes	32	425.133	even	80	inner
1700.1.ct.a.983.1	yes	32	1700.983	odd	80	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:	\( N \)	\(=\)	\( 1700 = 2^{2} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1700.ct (of order \(80\), degree \(32\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.996917 + 0.0784591i) q^{2} +(0.987688 - 0.156434i) q^{4} +(-0.987688 - 0.156434i) q^{5} +(-0.972370 + 0.233445i) q^{8} +(-0.760406 + 0.649448i) q^{9} +(0.996917 + 0.0784591i) q^{10} +(0.444039 - 0.144277i) q^{13} +(0.951057 - 0.309017i) q^{16} +(0.453990 - 0.891007i) q^{17} +(0.707107 - 0.707107i) q^{18} -1.00000 q^{20} +(0.951057 + 0.309017i) q^{25} +(-0.431351 + 0.178671i) q^{26} +(0.666244 + 1.80593i) q^{29} +(-0.923880 + 0.382683i) q^{32} +(-0.382683 + 0.923880i) q^{34} +(-0.649448 + 0.760406i) q^{36} +(0.226249 + 0.0638088i) q^{37} +(0.996917 - 0.0784591i) q^{40} +(1.56942 + 1.23723i) q^{41} +(0.852640 - 0.522499i) q^{45} +(0.923880 + 0.382683i) q^{49} +(-0.972370 - 0.233445i) q^{50} +(0.416003 - 0.211964i) q^{52} +(-1.01612 - 0.243950i) q^{53} +(-0.805883 - 1.74809i) q^{58} +(1.41351 - 0.398650i) q^{61} +(0.891007 - 0.453990i) q^{64} +(-0.461143 + 0.0730378i) q^{65} +(0.309017 - 0.951057i) q^{68} +(0.587785 - 0.809017i) q^{72} +(-0.0984164 + 0.831516i) q^{73} +(-0.230558 - 0.0458608i) q^{74} +(-0.987688 + 0.156434i) q^{80} +(0.156434 - 0.987688i) q^{81} +(-1.66166 - 1.11028i) q^{82} +(-0.587785 + 0.809017i) q^{85} +(0.0712394 + 0.139815i) q^{89} +(-0.809017 + 0.587785i) q^{90} +(1.42636 + 0.657561i) q^{97} +(-0.951057 - 0.309017i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{20} + 8 q^{41} + 8 q^{53} - 8 q^{68} + 8 q^{74} - 8 q^{82} - 8 q^{90}+O(q^{100}) \) 32 * q - 32 * q^20 + 8 * q^41 + 8 * q^53 - 8 * q^68 + 8 * q^74 - 8 * q^82 - 8 * q^90

Introduction

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