LMFDB - Embedded newform 2169.1.de.a.298.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2169,1,Mod(28,2169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2169, base_ring=CyclotomicField(80))

chi = DirichletCharacter(H, H._module([0, 47]))

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

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

chi := DirichletCharacter("2169.28");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2169 = 3^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2169.de (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:	$1.08247201240$
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:	$\Z[a_1, \ldots, a_{13}]$
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			298.1
Root			$0.972370 - 0.233445i$ of defining polynomial
Character	$\chi$	$=$	2169.298
Dual form			2169.1.de.a.1594.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{4} +(-0.220545 - 0.0813634i) q^{7} +O(q^{10})$	$q+(0.707107 - 0.707107i) q^{4} +(-0.220545 - 0.0813634i) q^{7} +(1.08031 + 1.37037i) q^{13} -1.00000i q^{16} +(0.241558 - 1.21440i) q^{19} +(0.987688 + 0.156434i) q^{25} +(-0.213482 + 0.0984164i) q^{28} +(-1.66166 + 0.0652867i) q^{31} +(1.42636 + 1.12445i) q^{37} +(0.568373 - 1.23290i) q^{43} +(-0.718386 - 0.613559i) q^{49} +(1.73290 + 0.205102i) q^{52} +(1.28290 - 1.50209i) q^{61} +(-0.707107 - 0.707107i) q^{64} +(-0.931099 + 1.51942i) q^{67} +(-1.56942 - 1.23723i) q^{73} +(-0.687900 - 1.02952i) q^{76} +(-1.44638 + 1.23532i) q^{79} +(-0.126760 - 0.390127i) q^{91} +(0.154986 - 0.0245474i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 8 q^{28} - 8 q^{31} + 8 q^{52} - 8 q^{73} - 8 q^{76}+O(q^{100}) $ 32 * q - 8 * q^28 - 8 * q^31 + 8 * q^52 - 8 * q^73 - 8 * q^76

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2169\mathbb{Z}\right)^\times$.

$n$	$730$	$965$
$\chi(n)$	$e\left(\frac{9}{80}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$2$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$3$		0			0
$3$		0			0
$4$	0.707107	−	0.707107i	0.707107	−	0.707107i
$5$		0			0		−0.996917	−	0.0784591i	$-0.975000\pi$
$5$		0			0		0.996917	+	0.0784591i	$0.0250000\pi$
$6$		0			0
$7$	−0.220545	−	0.0813634i	−0.220545	−	0.0813634i	0.233445	−	0.972370i	$-0.425000\pi$
$7$	−0.220545	−	0.0813634i	−0.220545	−	0.0813634i	−0.453990	+	0.891007i	$0.650000\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$11$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$12$		0			0
$13$	1.08031	+	1.37037i	1.08031	+	1.37037i	0.923880	+	0.382683i	$0.125000\pi$
$13$	1.08031	+	1.37037i	1.08031	+	1.37037i	0.156434	+	0.987688i	$0.450000\pi$
$14$		0			0
$15$		0			0
$16$		−	1.00000i		−	1.00000i
$17$		0			0		0.999229	−	0.0392598i	$-0.0125000\pi$
$17$		0			0		−0.999229	+	0.0392598i	$0.987500\pi$
$18$		0			0
$19$	0.241558	−	1.21440i	0.241558	−	1.21440i	−0.649448	−	0.760406i	$-0.725000\pi$
$19$	0.241558	−	1.21440i	0.241558	−	1.21440i	0.891007	−	0.453990i	$-0.150000\pi$
$20$		0			0
$21$		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
$25$	0.987688	+	0.156434i	0.987688	+	0.156434i
$26$		0			0
$27$		0			0
$28$	−0.213482	+	0.0984164i	−0.213482	+	0.0984164i
$29$		0			0		0.852640	−	0.522499i	$-0.175000\pi$
$29$		0			0		−0.852640	+	0.522499i	$0.825000\pi$
$30$		0			0
$31$	−1.66166	+	0.0652867i	−1.66166	+	0.0652867i	−0.852640	−	0.522499i	$-0.825000\pi$
$31$	−1.66166	+	0.0652867i	−1.66166	+	0.0652867i	−0.809017	+	0.587785i	$0.800000\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.42636	+	1.12445i	1.42636	+	1.12445i	0.972370	+	0.233445i	$0.0750000\pi$
$37$	1.42636	+	1.12445i	1.42636	+	1.12445i	0.453990	+	0.891007i	$0.350000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.852640	−	0.522499i	$-0.825000\pi$
$41$		0			0		0.852640	+	0.522499i	$0.175000\pi$
$42$		0			0
$43$	0.568373	−	1.23290i	0.568373	−	1.23290i	−0.382683	−	0.923880i	$-0.625000\pi$
$43$	0.568373	−	1.23290i	0.568373	−	1.23290i	0.951057	−	0.309017i	$-0.100000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.522499	−	0.852640i	$-0.675000\pi$
$47$		0			0		0.522499	+	0.852640i	$0.325000\pi$
$48$		0			0
$49$	−0.718386	−	0.613559i	−0.718386	−	0.613559i
$50$		0			0
$51$		0			0
$52$	1.73290	+	0.205102i	1.73290	+	0.205102i
$53$		0			0		0.233445	−	0.972370i	$-0.425000\pi$
$53$		0			0		−0.233445	+	0.972370i	$0.575000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.760406	−	0.649448i	$-0.225000\pi$
$59$		0			0		−0.760406	+	0.649448i	$0.775000\pi$
$60$		0			0
$61$	1.28290	−	1.50209i	1.28290	−	1.50209i	0.522499	−	0.852640i	$-0.325000\pi$
$61$	1.28290	−	1.50209i	1.28290	−	1.50209i	0.760406	−	0.649448i	$-0.225000\pi$
$62$		0			0
$63$		0			0
$64$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$65$		0			0
$66$		0			0
$67$	−0.931099	+	1.51942i	−0.931099	+	1.51942i	−0.0784591	+	0.996917i	$0.525000\pi$
$67$	−0.931099	+	1.51942i	−0.931099	+	1.51942i	−0.852640	+	0.522499i	$0.825000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.346117	−	0.938191i	$-0.612500\pi$
$71$		0			0		0.346117	+	0.938191i	$0.387500\pi$
$72$		0			0
$73$	−1.56942	−	1.23723i	−1.56942	−	1.23723i	−0.809017	−	0.587785i	$-0.800000\pi$
$73$	−1.56942	−	1.23723i	−1.56942	−	1.23723i	−0.760406	−	0.649448i	$-0.775000\pi$
$74$		0			0
$75$		0			0
$76$	−0.687900	−	1.02952i	−0.687900	−	1.02952i
$77$		0			0
$78$		0			0
$79$	−1.44638	+	1.23532i	−1.44638	+	1.23532i	−0.522499	+	0.852640i	$0.675000\pi$
$79$	−1.44638	+	1.23532i	−1.44638	+	1.23532i	−0.923880	+	0.382683i	$0.875000\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$83$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$89$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$90$		0			0
$91$	−0.126760	−	0.390127i	−0.126760	−	0.390127i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.154986	−	0.0245474i	0.154986	−	0.0245474i	−0.0784591	−	0.996917i	$-0.525000\pi$
$97$	0.154986	−	0.0245474i	0.154986	−	0.0245474i	0.233445	+	0.972370i	$0.425000\pi$
$98$		0			0
$99$		0			0
$100$	0.809017	−	0.587785i	0.809017	−	0.587785i
$101$		0			0		−0.346117	−	0.938191i	$-0.612500\pi$
$101$		0			0		0.346117	+	0.938191i	$0.387500\pi$
$102$		0			0
$103$	0.398650	−	0.368508i	0.398650	−	0.368508i	−0.453990	−	0.891007i	$-0.650000\pi$
$103$	0.398650	−	0.368508i	0.398650	−	0.368508i	0.852640	+	0.522499i	$0.175000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$107$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$108$		0			0
$109$	−0.230558	+	1.94797i	−0.230558	+	1.94797i	0.0784591	+	0.996917i	$0.475000\pi$
$109$	−0.230558	+	1.94797i	−0.230558	+	1.94797i	−0.309017	+	0.951057i	$0.600000\pi$
$110$		0			0
$111$		0			0
$112$	−0.0813634	+	0.220545i	−0.0813634	+	0.220545i
$113$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$113$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$122$		0			0
$123$		0			0
$124$	−1.12880	+	1.22113i	−1.12880	+	1.22113i
$125$		0			0
$126$		0			0
$127$	0.666244	−	0.187900i	0.666244	−	0.187900i	0.0784591	−	0.996917i	$-0.475000\pi$
$127$	0.666244	−	0.187900i	0.666244	−	0.187900i	0.587785	+	0.809017i	$0.300000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.619094	−	0.785317i	$-0.287500\pi$
$131$		0			0		−0.619094	+	0.785317i	$0.712500\pi$
$132$		0			0
$133$	−0.152082	+	0.248175i	−0.152082	+	0.248175i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.271440	−	0.962455i	$-0.587500\pi$
$137$		0			0		0.271440	+	0.962455i	$0.412500\pi$
$138$		0			0
$139$	0.340431	−	0.190651i	0.340431	−	0.190651i	−0.309017	−	0.951057i	$-0.600000\pi$
$139$	0.340431	−	0.190651i	0.340431	−	0.190651i	0.649448	+	0.760406i	$0.275000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	1.80370	−	0.213482i	1.80370	−	0.213482i
$149$		0			0		−0.872496	−	0.488621i	$-0.837500\pi$
$149$		0			0		0.872496	+	0.488621i	$0.162500\pi$
$150$		0			0
$151$	0.905182	+	1.77652i	0.905182	+	1.77652i	0.522499	+	0.852640i	$0.325000\pi$
$151$	0.905182	+	1.77652i	0.905182	+	1.77652i	0.382683	+	0.923880i	$0.375000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.63714	+	0.916840i	−1.63714	+	0.916840i	−0.649448	+	0.760406i	$0.725000\pi$
$157$	−1.63714	+	0.916840i	−1.63714	+	0.916840i	−0.987688	+	0.156434i	$0.950000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.805883	−	1.74809i	0.805883	−	1.74809i	0.156434	−	0.987688i	$-0.450000\pi$
$163$	0.805883	−	1.74809i	0.805883	−	1.74809i	0.649448	−	0.760406i	$-0.275000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.271440	−	0.962455i	$-0.412500\pi$
$167$		0			0		−0.271440	+	0.962455i	$0.587500\pi$
$168$		0			0
$169$	−0.477395	+	1.98849i	−0.477395	+	1.98849i
$170$		0			0
$171$		0			0
$172$	−0.469889	−	1.27369i	−0.469889	−	1.27369i
$173$		0			0		−0.734323	−	0.678801i	$-0.762500\pi$
$173$		0			0		0.734323	+	0.678801i	$0.237500\pi$
$174$		0			0
$175$	−0.205102	−	0.114863i	−0.205102	−	0.114863i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.418660	−	0.908143i	$-0.637500\pi$
$179$		0			0		0.418660	+	0.908143i	$0.362500\pi$
$180$		0			0
$181$	−1.14412	−	1.14412i	−1.14412	−	1.14412i	−0.987688	−	0.156434i	$-0.950000\pi$
$181$	−1.14412	−	1.14412i	−1.14412	−	1.14412i	−0.156434	−	0.987688i	$-0.550000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.233445	−	0.972370i	$-0.575000\pi$
$191$		0			0		0.233445	+	0.972370i	$0.425000\pi$
$192$		0			0
$193$	0.119322	+	1.51612i	0.119322	+	1.51612i	0.707107	+	0.707107i	$0.250000\pi$
$193$	0.119322	+	1.51612i	0.119322	+	1.51612i	−0.587785	+	0.809017i	$0.700000\pi$
$194$		0			0
$195$		0			0
$196$	−0.941828	+	0.0741234i	−0.941828	+	0.0741234i
$197$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$197$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$198$		0			0
$199$	−0.00922899	−	0.0779754i	−0.00922899	−	0.0779754i	0.987688	−	0.156434i	$-0.0500000\pi$
$199$	−0.00922899	−	0.0779754i	−0.00922899	−	0.0779754i	−0.996917	+	0.0784591i	$0.975000\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	1.37037	−	1.08031i	1.37037	−	1.08031i
$209$		0			0
$210$		0			0
$211$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$211$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.371782	+	0.120799i	0.371782	+	0.120799i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1.10749	−	0.678671i	1.10749	−	0.678671i	0.156434	−	0.987688i	$-0.450000\pi$
$223$	1.10749	−	0.678671i	1.10749	−	0.678671i	0.951057	+	0.309017i	$0.100000\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.0392598	−	0.999229i	$-0.512500\pi$
$227$		0			0		0.0392598	+	0.999229i	$0.487500\pi$
$228$		0			0
$229$	−0.401381	+	0.469957i	−0.401381	+	0.469957i	−0.923880	−	0.382683i	$-0.875000\pi$
$229$	−0.401381	+	0.469957i	−0.401381	+	0.469957i	0.522499	+	0.852640i	$0.325000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$233$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$239$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$240$		0			0
$241$	−0.996917	+	0.0784591i	−0.996917	+	0.0784591i
$241$	−0.996917	+	0.0784591i	−0.996917	+	0.0784591i
$242$		0			0
$243$		0			0
$244$	−0.154986	−	1.96929i	−0.154986	−	1.96929i
$245$		0			0
$246$		0			0
$247$	1.92513	−	0.980905i	1.92513	−	0.980905i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$251$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−1.00000			−1.00000
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$	−0.223088	−	0.364046i	−0.223088	−	0.364046i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$263$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.416003	+	1.73278i	0.416003	+	1.73278i
$269$		0			0		−0.418660	−	0.908143i	$-0.637500\pi$
$269$		0			0		0.418660	+	0.908143i	$0.362500\pi$
$270$		0			0
$271$	0.581990	+	1.40505i	0.581990	+	1.40505i	0.891007	+	0.453990i	$0.150000\pi$
$271$	0.581990	+	1.40505i	0.581990	+	1.40505i	−0.309017	+	0.951057i	$0.600000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.893911	+	1.23036i	−0.893911	+	1.23036i	0.0784591	+	0.996917i	$0.475000\pi$
$277$	−0.893911	+	1.23036i	−0.893911	+	1.23036i	−0.972370	+	0.233445i	$0.925000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$281$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$282$		0			0
$283$	−1.86338	+	0.220545i	−1.86338	+	0.220545i	−0.972370	−	0.233445i	$-0.925000\pi$
$283$	−1.86338	+	0.220545i	−1.86338	+	0.220545i	−0.891007	+	0.453990i	$0.850000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.996917	−	0.0784591i	0.996917	−	0.0784591i
$290$		0			0
$291$		0			0
$292$	−1.98461	+	0.234894i	−1.98461	+	0.234894i
$293$		0			0		0.117537	−	0.993068i	$-0.462500\pi$
$293$		0			0		−0.117537	+	0.993068i	$0.537500\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−0.225665	+	0.225665i	−0.225665	+	0.225665i
$302$		0			0
$303$		0			0
$304$	−1.21440	−	0.241558i	−1.21440	−	0.241558i
$305$		0			0
$306$		0			0
$307$	−0.940552	+	1.67948i	−0.940552	+	1.67948i	−0.233445	+	0.972370i	$0.575000\pi$
$307$	−0.940552	+	1.67948i	−0.940552	+	1.67948i	−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.0392598	−	0.999229i	$-0.487500\pi$
$311$		0			0		−0.0392598	+	0.999229i	$0.512500\pi$
$312$		0			0
$313$	0.453990	+	0.108993i	0.453990	+	0.108993i	0.453990	−	0.891007i	$-0.350000\pi$
$313$	0.453990	+	0.108993i	0.453990	+	0.108993i			1.00000i	$0.5\pi$
$314$		0			0
$315$		0			0
$316$	−0.149238	+	1.89625i	−0.149238	+	1.89625i
$317$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$317$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	0.852640	+	1.52250i	0.852640	+	1.52250i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.51942	+	0.774181i	−1.51942	+	0.774181i	−0.996917	−	0.0784591i	$-0.975000\pi$
$331$	−1.51942	+	0.774181i	−1.51942	+	0.774181i	−0.522499	+	0.852640i	$0.675000\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.642040	−	1.26007i	−0.642040	−	1.26007i	−0.951057	−	0.309017i	$-0.900000\pi$
$337$	−0.642040	−	1.26007i	−0.642040	−	1.26007i	0.309017	−	0.951057i	$-0.400000\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.223378	+	0.398870i	0.223378	+	0.398870i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.987688	−	0.156434i	$-0.0500000\pi$
$347$		0			0		−0.987688	+	0.156434i	$0.950000\pi$
$348$		0			0
$349$	0.266765	+	0.163474i	0.266765	+	0.163474i	0.649448	−	0.760406i	$-0.275000\pi$
$349$	0.266765	+	0.163474i	0.266765	+	0.163474i	−0.382683	+	0.923880i	$0.625000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.938191	−	0.346117i	$-0.887500\pi$
$353$		0			0		0.938191	+	0.346117i	$0.112500\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.891007	−	0.453990i	$-0.850000\pi$
$359$		0			0		0.891007	+	0.453990i	$0.150000\pi$
$360$		0			0
$361$	−0.492529	−	0.204012i	−0.492529	−	0.204012i
$362$		0			0
$363$		0			0
$364$	−0.365494	−	0.186229i	−0.365494	−	0.186229i
$365$		0			0
$366$		0			0
$367$	−0.286518	−	1.44043i	−0.286518	−	1.44043i	−0.809017	−	0.587785i	$-0.800000\pi$
$367$	−0.286518	−	1.44043i	−0.286518	−	1.44043i	0.522499	−	0.852640i	$-0.325000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.45847	+	0.172621i	1.45847	+	0.172621i	0.809017	−	0.587785i	$-0.200000\pi$
$373$	1.45847	+	0.172621i	1.45847	+	0.172621i	0.649448	+	0.760406i	$0.275000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.264855	−	0.286518i	−0.264855	−	0.286518i	0.587785	−	0.809017i	$-0.300000\pi$
$379$	−0.264855	−	0.286518i	−0.264855	−	0.286518i	−0.852640	+	0.522499i	$0.825000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.734323	−	0.678801i	$-0.237500\pi$
$383$		0			0		−0.734323	+	0.678801i	$0.762500\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.0922342	−	0.126949i	0.0922342	−	0.126949i
$389$		0			0		−0.117537	−	0.993068i	$-0.537500\pi$
$389$		0			0		0.117537	+	0.993068i	$0.462500\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.0328730	−	0.836674i	0.0328730	−	0.836674i	−0.891007	−	0.453990i	$-0.850000\pi$
$397$	0.0328730	−	0.836674i	0.0328730	−	0.836674i	0.923880	−	0.382683i	$-0.125000\pi$
$398$		0			0
$399$		0			0
$400$	0.156434	−	0.987688i	0.156434	−	0.987688i
$401$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$401$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$402$		0			0
$403$	−1.88458	−	2.20656i	−1.88458	−	2.20656i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.0486110	−	0.0616628i	0.0486110	−	0.0616628i	−0.760406	−	0.649448i	$-0.775000\pi$
$409$	0.0486110	−	0.0616628i	0.0486110	−	0.0616628i	0.809017	+	0.587785i	$0.200000\pi$
$410$		0			0
$411$		0			0
$412$	0.0213134	−	0.542462i	0.0213134	−	0.542462i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$419$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$420$		0			0
$421$	−1.29671	−	1.10749i	−1.29671	−	1.10749i	−0.987688	−	0.156434i	$-0.950000\pi$
$421$	−1.29671	−	1.10749i	−1.29671	−	1.10749i	−0.309017	−	0.951057i	$-0.600000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.405153	+	0.226897i	−0.405153	+	0.226897i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.993068	−	0.117537i	$-0.0375000\pi$
$431$		0			0		−0.993068	+	0.117537i	$0.962500\pi$
$432$		0			0
$433$	0.589686	−	0.690434i	0.589686	−	0.690434i	−0.382683	−	0.923880i	$-0.625000\pi$
$433$	0.589686	−	0.690434i	0.589686	−	0.690434i	0.972370	+	0.233445i	$0.0750000\pi$
$434$		0			0
$435$		0			0
$436$	1.21440	+	1.54045i	1.21440	+	1.54045i
$437$		0			0
$438$		0			0
$439$	−1.78139	−	0.821231i	−1.78139	−	0.821231i	−0.972370	−	0.233445i	$-0.925000\pi$
$439$	−1.78139	−	0.821231i	−1.78139	−	0.821231i	−0.809017	−	0.587785i	$-0.800000\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.734323	−	0.678801i	$-0.237500\pi$
$443$		0			0		−0.734323	+	0.678801i	$0.762500\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.0984164	+	0.213482i	0.0984164	+	0.213482i
$449$		0			0		−0.619094	−	0.785317i	$-0.712500\pi$
$449$		0			0		0.619094	+	0.785317i	$0.287500\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.163474	+	1.03213i	−0.163474	+	1.03213i	0.760406	+	0.649448i	$0.225000\pi$
$457$	−0.163474	+	1.03213i	−0.163474	+	1.03213i	−0.923880	+	0.382683i	$0.875000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.271440	−	0.962455i	$-0.412500\pi$
$461$		0			0		−0.271440	+	0.962455i	$0.587500\pi$
$462$		0			0
$463$	−0.821231	−	0.163353i	−0.821231	−	0.163353i	−0.233445	−	0.972370i	$-0.575000\pi$
$463$	−0.821231	−	0.163353i	−0.821231	−	0.163353i	−0.587785	+	0.809017i	$0.700000\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	0.328974	−	0.259342i	0.328974	−	0.259342i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.428558	−	1.16166i	0.428558	−	1.16166i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.0784591	−	0.996917i	$-0.525000\pi$
$479$		0			0		0.0784591	+	0.996917i	$0.475000\pi$
$480$		0			0
$481$			3.16941i			3.16941i
$482$		0			0
$483$		0			0
$484$	0.382683	+	0.923880i	0.382683	+	0.923880i
$485$		0			0
$486$		0			0
$487$	−0.0600500	+	0.763007i	−0.0600500	+	0.763007i	0.891007	+	0.453990i	$0.150000\pi$
$487$	−0.0600500	+	0.763007i	−0.0600500	+	0.763007i	−0.951057	+	0.309017i	$0.900000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$491$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.0652867	+	1.66166i	0.0652867	+	1.66166i
$497$		0			0
$498$		0			0
$499$	−0.0736664	−	1.87494i	−0.0736664	−	1.87494i	−0.382683	−	0.923880i	$-0.625000\pi$
$499$	−0.0736664	−	1.87494i	−0.0736664	−	1.87494i	0.309017	−	0.951057i	$-0.400000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.962455	−	0.271440i	$-0.912500\pi$
$503$		0			0		0.962455	+	0.271440i	$0.0875000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	0.338240	−	0.603972i	0.338240	−	0.603972i
$509$		0			0		0.972370	−	0.233445i	$-0.0750000\pi$
$509$		0			0		−0.972370	+	0.233445i	$0.925000\pi$
$510$		0			0
$511$	0.245463	+	0.400559i	0.245463	+	0.400559i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.678801	−	0.734323i	$-0.737500\pi$
$521$		0			0		0.678801	+	0.734323i	$0.262500\pi$
$522$		0			0
$523$	0.399903	−	0.652583i	0.399903	−	0.652583i	−0.587785	−	0.809017i	$-0.700000\pi$
$523$	0.399903	−	0.652583i	0.399903	−	0.652583i	0.987688	+	0.156434i	$0.0500000\pi$
$524$		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			0
$531$		0			0
$532$	0.0679482	+	0.283025i	0.0679482	+	0.283025i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.918458	−	1.07538i	−0.918458	−	1.07538i	−0.996917	−	0.0784591i	$-0.975000\pi$
$541$	−0.918458	−	1.07538i	−0.918458	−	1.07538i	0.0784591	−	0.996917i	$-0.475000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.923880	−	0.617317i	0.923880	−	0.617317i		−	1.00000i	$-0.5\pi$
$547$	0.923880	−	0.617317i	0.923880	−	0.617317i	0.923880	+	0.382683i	$0.125000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.419502	−	0.154762i	0.419502	−	0.154762i
$554$		0			0
$555$		0			0
$556$	0.105911	−	0.375531i	0.105911	−	0.375531i
$557$		0			0		−0.996917	−	0.0784591i	$-0.975000\pi$
$557$		0			0		0.996917	+	0.0784591i	$0.0250000\pi$
$558$		0			0
$559$	2.30355	−	0.553033i	2.30355	−	0.553033i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$563$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$569$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$570$		0			0
$571$	0.451389	+	0.301608i	0.451389	+	0.301608i	0.760406	−	0.649448i	$-0.225000\pi$
$571$	0.451389	+	0.301608i	0.451389	+	0.301608i	−0.309017	+	0.951057i	$0.600000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.06942	+	0.301608i	1.06942	+	0.301608i	0.760406	−	0.649448i	$-0.225000\pi$
$577$	1.06942	+	0.301608i	1.06942	+	0.301608i	0.309017	+	0.951057i	$0.400000\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.346117	−	0.938191i	$-0.387500\pi$
$587$		0			0		−0.346117	+	0.938191i	$0.612500\pi$
$588$		0			0
$589$	−0.322103	+	2.03368i	−0.322103	+	2.03368i
$590$		0			0
$591$		0			0
$592$	1.12445	−	1.42636i	1.12445	−	1.42636i
$593$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$593$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.678801	−	0.734323i	$-0.262500\pi$
$599$		0			0		−0.678801	+	0.734323i	$0.737500\pi$
$600$		0			0
$601$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	0.587785	+	0.809017i	$0.300000\pi$
$601$	−0.363271	+	1.11803i	−0.363271	+	1.11803i	−0.951057	+	0.309017i	$0.900000\pi$
$602$		0			0
$603$		0			0
$604$	1.89625	+	0.616129i	1.89625	+	0.616129i
$605$		0			0
$606$		0			0
$607$	−0.377723	+	1.57333i	−0.377723	+	1.57333i	0.382683	+	0.923880i	$0.375000\pi$
$607$	−0.377723	+	1.57333i	−0.377723	+	1.57333i	−0.760406	+	0.649448i	$0.775000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.54045	−	1.21440i	−1.54045	−	1.21440i	−0.891007	−	0.453990i	$-0.850000\pi$
$613$	−1.54045	−	1.21440i	−1.54045	−	1.21440i	−0.649448	−	0.760406i	$-0.725000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.156434	−	0.987688i	$-0.550000\pi$
$617$		0			0		0.156434	+	0.987688i	$0.450000\pi$
$618$		0			0
$619$	1.41351	−	0.398650i	1.41351	−	0.398650i	0.522499	−	0.852640i	$-0.325000\pi$
$619$	1.41351	−	0.398650i	1.41351	−	0.398650i	0.891007	+	0.453990i	$0.150000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.951057	+	0.309017i	0.951057	+	0.309017i
$626$		0			0
$627$		0			0
$628$	−0.509326	+	1.80593i	−0.509326	+	1.80593i
$629$		0			0
$630$		0			0
$631$	0.940552	−	1.67948i	0.940552	−	1.67948i	0.233445	−	0.972370i	$-0.425000\pi$
$631$	0.940552	−	1.67948i	0.940552	−	1.67948i	0.707107	−	0.707107i	$-0.250000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.0647223	−	1.64729i	0.0647223	−	1.64729i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.987688	−	0.156434i	$-0.950000\pi$
$641$		0			0		0.987688	+	0.156434i	$0.0500000\pi$
$642$		0			0
$643$	−0.144974	+	1.84206i	−0.144974	+	1.84206i	0.309017	+	0.951057i	$0.400000\pi$
$643$	−0.144974	+	1.84206i	−0.144974	+	1.84206i	−0.453990	+	0.891007i	$0.650000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$647$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−0.666244	−	1.80593i	−0.666244	−	1.80593i
$653$		0			0		−0.999229	−	0.0392598i	$-0.987500\pi$
$653$		0			0		0.999229	+	0.0392598i	$0.0125000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$659$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$660$		0			0
$661$	1.81489	−	0.836674i	1.81489	−	0.836674i	0.891007	−	0.453990i	$-0.150000\pi$
$661$	1.81489	−	0.836674i	1.81489	−	0.836674i	0.923880	−	0.382683i	$-0.125000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.600958	+	0.144277i	−0.600958	+	0.144277i	−0.522499	−	0.852640i	$-0.675000\pi$
$673$	−0.600958	+	0.144277i	−0.600958	+	0.144277i	−0.0784591	+	0.996917i	$0.525000\pi$
$674$		0			0
$675$		0			0
$676$	1.06851	+	1.74365i	1.06851	+	1.74365i
$677$		0			0		0.785317	−	0.619094i	$-0.212500\pi$
$677$		0			0		−0.785317	+	0.619094i	$0.787500\pi$
$678$		0			0
$679$	−0.0361787	−	0.00719640i	−0.0361787	−	0.00719640i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$683$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−1.23290	−	0.568373i	−1.23290	−	0.568373i
$689$		0			0
$690$		0			0
$691$	−0.619195	−	1.49487i	−0.619195	−	1.49487i	−0.852640	−	0.522499i	$-0.825000\pi$
$691$	−0.619195	−	1.49487i	−0.619195	−	1.49487i	0.233445	−	0.972370i	$-0.425000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.226249	+	0.0638088i	−0.226249	+	0.0638088i
$701$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$701$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$702$		0			0
$703$	1.71008	−	1.46055i	1.71008	−	1.46055i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.389880	−	0.0153184i	0.389880	−	0.0153184i	0.156434	−	0.987688i	$-0.450000\pi$
$709$	0.389880	−	0.0153184i	0.389880	−	0.0153184i	0.233445	+	0.972370i	$0.425000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$719$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$720$		0			0
$721$	−0.117903	+	0.0488371i	−0.117903	+	0.0488371i
$722$		0			0
$723$		0			0
$724$	−1.61803			−1.61803
$725$		0			0
$726$		0			0
$727$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$727$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	1.69480	−	0.550672i	1.69480	−	0.550672i	0.707107	−	0.707107i	$-0.250000\pi$
$733$	1.69480	−	0.550672i	1.69480	−	0.550672i	0.987688	+	0.156434i	$0.0500000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	0.618034i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$739$		−	0.618034i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.760406	−	0.649448i	$-0.225000\pi$
$743$		0			0		−0.760406	+	0.649448i	$0.775000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.760406	−	0.350552i	0.760406	−	0.350552i		−	1.00000i	$-0.5\pi$
$751$	0.760406	−	0.350552i	0.760406	−	0.350552i	0.760406	+	0.649448i	$0.225000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.493014	−	0.227282i	0.493014	−	0.227282i	−0.156434	−	0.987688i	$-0.550000\pi$
$757$	0.493014	−	0.227282i	0.493014	−	0.227282i	0.649448	+	0.760406i	$0.275000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$761$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$762$		0			0
$763$	0.209342	−	0.410857i	0.209342	−	0.410857i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.23345	−	0.972370i	1.23345	−	0.972370i	0.233445	−	0.972370i	$-0.425000\pi$
$769$	1.23345	−	0.972370i	1.23345	−	0.972370i	1.00000			$0$
$770$		0			0
$771$		0			0
$772$	1.15643	+	0.987688i	1.15643	+	0.987688i
$773$		0			0		0.972370	−	0.233445i	$-0.0750000\pi$
$773$		0			0		−0.972370	+	0.233445i	$0.925000\pi$
$774$		0			0
$775$	−1.65141	−	0.195458i	−1.65141	−	0.195458i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.613559	+	0.718386i	−0.613559	+	0.718386i
$785$		0			0
$786$		0			0
$787$	0.831254	+	0.831254i	0.831254	+	0.831254i	0.987688	−	0.156434i	$-0.0500000\pi$
$787$	0.831254	+	0.831254i	0.831254	+	0.831254i	−0.156434	+	0.987688i	$0.550000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	3.44436	+	0.135329i	3.44436	+	0.135329i
$794$		0			0
$795$		0			0
$796$	−0.0616628	−	0.0486110i	−0.0616628	−	0.0486110i
$797$		0			0		−0.962455	−	0.271440i	$-0.912500\pi$
$797$		0			0		0.962455	+	0.271440i	$0.0875000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.0392598	−	0.999229i	$-0.487500\pi$
$809$		0			0		−0.0392598	+	0.999229i	$0.512500\pi$
$810$		0			0
$811$	−1.08979	+	0.216773i	−1.08979	+	0.216773i	−0.707107	−	0.707107i	$-0.750000\pi$
$811$	−1.08979	+	0.216773i	−1.08979	+	0.216773i	−0.382683	+	0.923880i	$0.625000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.35993	−	0.988047i	−1.35993	−	0.988047i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$821$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$822$		0			0
$823$	1.59811	−	1.16110i	1.59811	−	1.16110i	0.707107	−	0.707107i	$-0.250000\pi$
$823$	1.59811	−	1.16110i	1.59811	−	1.16110i	0.891007	−	0.453990i	$-0.150000\pi$
$824$		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
$829$	−0.289053	−	1.82501i	−0.289053	−	1.82501i	−0.522499	−	0.852640i	$-0.675000\pi$
$829$	−0.289053	−	1.82501i	−0.289053	−	1.82501i	0.233445	−	0.972370i	$-0.425000\pi$
$830$		0			0
$831$		0			0
$832$	0.205102	−	1.73290i	0.205102	−	1.73290i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.233445	−	0.972370i	$-0.425000\pi$
$839$		0			0		−0.233445	+	0.972370i	$0.575000\pi$
$840$		0			0
$841$	0.453990	−	0.891007i	0.453990	−	0.891007i
$842$		0			0
$843$		0			0
$844$	−0.707107	+	1.70711i	−0.707107	+	1.70711i
$845$		0			0
$846$		0			0
$847$	0.159569	−	0.172621i	0.159569	−	0.172621i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.575572	+	0.384585i	0.575572	+	0.384585i	0.809017	−	0.587785i	$-0.200000\pi$
$853$	0.575572	+	0.384585i	0.575572	+	0.384585i	−0.233445	+	0.972370i	$0.575000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.156434	−	0.987688i	$-0.450000\pi$
$857$		0			0		−0.156434	+	0.987688i	$0.550000\pi$
$858$		0			0
$859$	−0.543623	+	1.47356i	−0.543623	+	1.47356i	0.309017	+	0.951057i	$0.400000\pi$
$859$	−0.543623	+	1.47356i	−0.543623	+	1.47356i	−0.852640	+	0.522499i	$0.825000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.938191	−	0.346117i	$-0.112500\pi$
$863$		0			0		−0.938191	+	0.346117i	$0.887500\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	0.348308	−	0.177472i	0.348308	−	0.177472i
$869$		0			0
$870$		0			0
$871$	−3.08804	+	0.365494i	−3.08804	+	0.365494i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.614234	+	0.845420i	0.614234	+	0.845420i	0.996917	−	0.0784591i	$-0.0250000\pi$
$877$	0.614234	+	0.845420i	0.614234	+	0.845420i	−0.382683	+	0.923880i	$0.625000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.453990	−	0.891007i	$-0.350000\pi$
$881$		0			0		−0.453990	+	0.891007i	$0.650000\pi$
$882$		0			0
$883$	0.610425	+	1.87869i	0.610425	+	1.87869i	0.453990	+	0.891007i	$0.350000\pi$
$883$	0.610425	+	1.87869i	0.610425	+	1.87869i	0.156434	+	0.987688i	$0.450000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.972370	−	0.233445i	$-0.0750000\pi$
$887$		0			0		−0.972370	+	0.233445i	$0.925000\pi$
$888$		0			0
$889$	−0.162225	−	0.0127674i	−0.162225	−	0.0127674i
$890$		0			0
$891$		0			0
$892$	0.303221	−	1.26301i	0.303221	−	1.26301i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.234894	+	1.98461i	−0.234894	+	1.98461i	−0.0784591	+	0.996917i	$0.525000\pi$
$907$	−0.234894	+	1.98461i	−0.234894	+	1.98461i	−0.156434	+	0.987688i	$0.550000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.972370	−	0.233445i	$-0.925000\pi$
$911$		0			0		0.972370	+	0.233445i	$0.0750000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.0484904	+	0.616129i	0.0484904	+	0.616129i
$917$		0			0
$918$		0			0
$919$	0.156434	−	0.0123117i	0.156434	−	0.0123117i		−	1.00000i	$-0.5\pi$
$919$	0.156434	−	0.0123117i	0.156434	−	0.0123117i	0.156434	+	0.987688i	$0.450000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.23290	+	1.33374i	1.23290	+	1.33374i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.418660	−	0.908143i	$-0.362500\pi$
$929$		0			0		−0.418660	+	0.908143i	$0.637500\pi$
$930$		0			0
$931$	−0.918637	+	0.724195i	−0.918637	+	0.724195i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.89101	+	0.453990i	−1.89101	+	0.453990i	−0.891007	+	0.453990i	$0.850000\pi$
$937$	−1.89101	+	0.453990i	−1.89101	+	0.453990i	−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.271440	−	0.962455i	$-0.587500\pi$
$941$		0			0		0.271440	+	0.962455i	$0.412500\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.0392598	−	0.999229i	$-0.512500\pi$
$947$		0			0		0.0392598	+	0.999229i	$0.487500\pi$
$948$		0			0
$949$		−	3.48729i		−	3.48729i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$953$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1.75992	−	0.138509i	1.75992	−	0.138509i
$962$		0			0
$963$		0			0
$964$	−0.649448	+	0.760406i	−0.649448	+	0.760406i
$965$		0			0
$966$		0			0
$967$	0.126949	+	1.61305i	0.126949	+	1.61305i	0.649448	+	0.760406i	$0.275000\pi$
$967$	0.126949	+	1.61305i	0.126949	+	1.61305i	−0.522499	+	0.852640i	$0.675000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.346117	−	0.938191i	$-0.387500\pi$
$971$		0			0		−0.346117	+	0.938191i	$0.612500\pi$
$972$		0			0
$973$	−0.0905924	+	0.0143484i	−0.0905924	+	0.0143484i
$974$		0			0
$975$		0			0
$976$	−1.50209	−	1.28290i	−1.50209	−	1.28290i
$977$		0			0		0.785317	−	0.619094i	$-0.212500\pi$
$977$		0			0		−0.785317	+	0.619094i	$0.787500\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$983$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.667671	−	2.05488i	0.667671	−	2.05488i
$989$		0			0
$990$		0			0
$991$	−0.211964	−	0.882893i	−0.211964	−	0.882893i	−0.972370	−	0.233445i	$-0.925000\pi$
$991$	−0.211964	−	0.882893i	−0.211964	−	0.882893i	0.760406	−	0.649448i	$-0.225000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.0486110	−	0.0616628i	−0.0486110	−	0.0616628i	0.760406	−	0.649448i	$-0.225000\pi$
$997$	−0.0486110	−	0.0616628i	−0.0486110	−	0.0616628i	−0.809017	+	0.587785i	$0.800000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2169.1.de.a.298.1	✓	32
3.2	odd	2	CM	2169.1.de.a.298.1	✓	32
241.148	odd	80	inner	2169.1.de.a.1594.1	yes	32
723.389	even	80	inner	2169.1.de.a.1594.1	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2169.1.de.a.298.1	✓	32	1.1	even	1	trivial
2169.1.de.a.298.1	✓	32	3.2	odd	2	CM
2169.1.de.a.1594.1	yes	32	241.148	odd	80	inner
2169.1.de.a.1594.1	yes	32	723.389	even	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:	\( N \)	\(=\)	\( 2169 = 3^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2169.de (of order \(80\), degree \(32\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{4} +(-0.220545 - 0.0813634i) q^{7} +O(q^{10})\)	\(q+(0.707107 - 0.707107i) q^{4} +(-0.220545 - 0.0813634i) q^{7} +(1.08031 + 1.37037i) q^{13} -1.00000i q^{16} +(0.241558 - 1.21440i) q^{19} +(0.987688 + 0.156434i) q^{25} +(-0.213482 + 0.0984164i) q^{28} +(-1.66166 + 0.0652867i) q^{31} +(1.42636 + 1.12445i) q^{37} +(0.568373 - 1.23290i) q^{43} +(-0.718386 - 0.613559i) q^{49} +(1.73290 + 0.205102i) q^{52} +(1.28290 - 1.50209i) q^{61} +(-0.707107 - 0.707107i) q^{64} +(-0.931099 + 1.51942i) q^{67} +(-1.56942 - 1.23723i) q^{73} +(-0.687900 - 1.02952i) q^{76} +(-1.44638 + 1.23532i) q^{79} +(-0.126760 - 0.390127i) q^{91} +(0.154986 - 0.0245474i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 8 q^{28} - 8 q^{31} + 8 q^{52} - 8 q^{73} - 8 q^{76}+O(q^{100}) \) 32 * q - 8 * q^28 - 8 * q^31 + 8 * q^52 - 8 * q^73 - 8 * q^76

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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