LMFDB - Embedded newform 873.1.br.a.451.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [873,1,Mod(19,873)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(873, base_ring=CyclotomicField(32))

chi = DirichletCharacter(H, H._module([0, 27]))

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

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

chi := DirichletCharacter("873.19");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 873 = 3^{2} \cdot 97 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	873.br (of order $32$, degree $16$, 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:	$0.435683756029$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{32})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 1 $ x^16 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{32}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{32} - \cdots)$

Embedding invariants

Embedding label			451.1
Root			$0.555570 - 0.831470i$ of defining polynomial
Character	$\chi$	$=$	873.451
Dual form			873.1.br.a.271.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.923880 + 0.382683i) q^{4} +(-0.151537 - 0.124363i) q^{7} +O(q^{10})$	$q+(0.923880 + 0.382683i) q^{4} +(-0.151537 - 0.124363i) q^{7} +(0.187593 - 1.90466i) q^{13} +(0.707107 + 0.707107i) q^{16} +(0.980785 + 1.19509i) q^{19} +(-0.195090 + 0.980785i) q^{25} +(-0.0924099 - 0.172887i) q^{28} +(-1.81225 + 0.360480i) q^{31} +(-1.68789 - 0.512016i) q^{37} +(1.53636 + 0.636379i) q^{43} +(-0.187593 - 0.943094i) q^{49} +(0.902197 - 1.68789i) q^{52} -1.11114 q^{61} +(0.382683 + 0.923880i) q^{64} +(0.0924099 - 0.938254i) q^{67} +(-1.30656 + 0.541196i) q^{73} +(0.448786 + 1.47945i) q^{76} +(-0.0761205 - 0.382683i) q^{79} +(-0.265297 + 0.265297i) q^{91} +(0.195090 - 0.980785i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 16 q^{79}+O(q^{100}) $ 16 * q - 16 * q^79

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/873\mathbb{Z}\right)^\times$.

$n$	$199$	$389$
$\chi(n)$	$e\left(\frac{25}{32}\right)$	$1$

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		−0.980785	−	0.195090i	$-0.937500\pi$
$2$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$3$		0			0
$3$		0			0
$4$	0.923880	+	0.382683i	0.923880	+	0.382683i
$5$		0			0		−0.634393	−	0.773010i	$-0.718750\pi$
$5$		0			0		0.634393	+	0.773010i	$0.281250\pi$
$6$		0			0
$7$	−0.151537	−	0.124363i	−0.151537	−	0.124363i	0.555570	−	0.831470i	$-0.312500\pi$
$7$	−0.151537	−	0.124363i	−0.151537	−	0.124363i	−0.707107	+	0.707107i	$0.750000\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$11$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$12$		0			0
$13$	0.187593	−	1.90466i	0.187593	−	1.90466i	−0.195090	−	0.980785i	$-0.562500\pi$
$13$	0.187593	−	1.90466i	0.187593	−	1.90466i	0.382683	−	0.923880i	$-0.375000\pi$
$14$		0			0
$15$		0			0
$16$	0.707107	+	0.707107i	0.707107	+	0.707107i
$17$		0			0		−0.995185	−	0.0980171i	$-0.968750\pi$
$17$		0			0		0.995185	+	0.0980171i	$0.0312500\pi$
$18$		0			0
$19$	0.980785	+	1.19509i	0.980785	+	1.19509i	0.980785	+	0.195090i	$0.0625000\pi$
$19$	0.980785	+	1.19509i	0.980785	+	1.19509i			1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.471397	−	0.881921i	$-0.343750\pi$
$23$		0			0		−0.471397	+	0.881921i	$0.656250\pi$
$24$		0			0
$25$	−0.195090	+	0.980785i	−0.195090	+	0.980785i
$26$		0			0
$27$		0			0
$28$	−0.0924099	−	0.172887i	−0.0924099	−	0.172887i
$29$		0			0		0.471397	−	0.881921i	$-0.343750\pi$
$29$		0			0		−0.471397	+	0.881921i	$0.656250\pi$
$30$		0			0
$31$	−1.81225	+	0.360480i	−1.81225	+	0.360480i	−0.980785	−	0.195090i	$-0.937500\pi$
$31$	−1.81225	+	0.360480i	−1.81225	+	0.360480i	−0.831470	+	0.555570i	$0.812500\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.68789	−	0.512016i	−1.68789	−	0.512016i	−0.707107	−	0.707107i	$-0.750000\pi$
$37$	−1.68789	−	0.512016i	−1.68789	−	0.512016i	−0.980785	+	0.195090i	$0.937500\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.956940	−	0.290285i	$-0.0937500\pi$
$41$		0			0		−0.956940	+	0.290285i	$0.906250\pi$
$42$		0			0
$43$	1.53636	+	0.636379i	1.53636	+	0.636379i	0.980785	−	0.195090i	$-0.0625000\pi$
$43$	1.53636	+	0.636379i	1.53636	+	0.636379i	0.555570	+	0.831470i	$0.312500\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$47$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$48$		0			0
$49$	−0.187593	−	0.943094i	−0.187593	−	0.943094i
$50$		0			0
$51$		0			0
$52$	0.902197	−	1.68789i	0.902197	−	1.68789i
$53$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$53$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.881921	−	0.471397i	$-0.156250\pi$
$59$		0			0		−0.881921	+	0.471397i	$0.843750\pi$
$60$		0			0
$61$	−1.11114			−1.11114			−0.555570	−	0.831470i	$-0.687500\pi$
$61$	−1.11114			−1.11114			−0.555570	+	0.831470i	$0.687500\pi$
$62$		0			0
$63$		0			0
$64$	0.382683	+	0.923880i	0.382683	+	0.923880i
$65$		0			0
$66$		0			0
$67$	0.0924099	−	0.938254i	0.0924099	−	0.938254i	−0.831470	−	0.555570i	$-0.812500\pi$
$67$	0.0924099	−	0.938254i	0.0924099	−	0.938254i	0.923880	−	0.382683i	$-0.125000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.956940	−	0.290285i	$-0.906250\pi$
$71$		0			0		0.956940	+	0.290285i	$0.0937500\pi$
$72$		0			0
$73$	−1.30656	+	0.541196i	−1.30656	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$73$	−1.30656	+	0.541196i	−1.30656	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$74$		0			0
$75$		0			0
$76$	0.448786	+	1.47945i	0.448786	+	1.47945i
$77$		0			0
$78$		0			0
$79$	−0.0761205	−	0.382683i	−0.0761205	−	0.382683i	0.923880	−	0.382683i	$-0.125000\pi$
$79$	−0.0761205	−	0.382683i	−0.0761205	−	0.382683i	−1.00000			$\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.773010	−	0.634393i	$-0.218750\pi$
$83$		0			0		−0.773010	+	0.634393i	$0.781250\pi$
$84$		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
$91$	−0.265297	+	0.265297i	−0.265297	+	0.265297i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.195090	−	0.980785i	0.195090	−	0.980785i
$97$	0.195090	−	0.980785i	0.195090	−	0.980785i
$98$		0			0
$99$		0			0
$100$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$101$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$101$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$102$		0			0
$103$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$103$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.881921	−	0.471397i	$-0.156250\pi$
$107$		0			0		−0.881921	+	0.471397i	$0.843750\pi$
$108$		0			0
$109$	−0.636379	+	0.425215i	−0.636379	+	0.425215i	−0.831470	−	0.555570i	$-0.812500\pi$
$109$	−0.636379	+	0.425215i	−0.636379	+	0.425215i	0.195090	+	0.980785i	$0.437500\pi$
$110$		0			0
$111$		0			0
$112$	−0.0192147	−	0.195090i	−0.0192147	−	0.195090i
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		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
$124$	−1.81225	−	0.360480i	−1.81225	−	0.360480i
$125$		0			0
$126$		0			0
$127$	1.26268	+	0.124363i	1.26268	+	0.124363i	0.707107	−	0.707107i	$-0.250000\pi$
$127$	1.26268	+	0.124363i	1.26268	+	0.124363i	0.555570	+	0.831470i	$0.312500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.290285	−	0.956940i	$-0.406250\pi$
$131$		0			0		−0.290285	+	0.956940i	$0.593750\pi$
$132$		0			0
$133$		−	0.303073i		−	0.303073i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.995185	−	0.0980171i	$-0.0312500\pi$
$137$		0			0		−0.995185	+	0.0980171i	$0.968750\pi$
$138$		0			0
$139$	−0.0924099	−	0.938254i	−0.0924099	−	0.938254i	−0.923880	−	0.382683i	$-0.875000\pi$
$139$	−0.0924099	−	0.938254i	−0.0924099	−	0.938254i	0.831470	−	0.555570i	$-0.187500\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−1.36347	−	1.11897i	−1.36347	−	1.11897i
$149$		0			0		−0.881921	−	0.471397i	$-0.843750\pi$
$149$		0			0		0.881921	+	0.471397i	$0.156250\pi$
$150$		0			0
$151$	−0.750661	+	1.81225i	−0.750661	+	1.81225i	−0.195090	+	0.980785i	$0.562500\pi$
$151$	−0.750661	+	1.81225i	−0.750661	+	1.81225i	−0.555570	+	0.831470i	$0.687500\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.368309	+	1.21415i	−0.368309	+	1.21415i	0.555570	+	0.831470i	$0.312500\pi$
$157$	−0.368309	+	1.21415i	−0.368309	+	1.21415i	−0.923880	+	0.382683i	$0.875000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.382683	+	1.92388i	0.382683	+	1.92388i	0.382683	+	0.923880i	$0.375000\pi$
$163$	0.382683	+	1.92388i	0.382683	+	1.92388i			1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$167$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$168$		0			0
$169$	−2.61177	−	0.519514i	−2.61177	−	0.519514i
$170$		0			0
$171$		0			0
$172$	1.17588	+	1.17588i	1.17588	+	1.17588i
$173$		0			0		0.956940	−	0.290285i	$-0.0937500\pi$
$173$		0			0		−0.956940	+	0.290285i	$0.906250\pi$
$174$		0			0
$175$	0.151537	−	0.124363i	0.151537	−	0.124363i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.0980171	−	0.995185i	$-0.531250\pi$
$179$		0			0		0.0980171	+	0.995185i	$0.468750\pi$
$180$		0			0
$181$	−0.577774	−	0.0569057i	−0.577774	−	0.0569057i	−0.195090	−	0.980785i	$-0.562500\pi$
$181$	−0.577774	−	0.0569057i	−0.577774	−	0.0569057i	−0.382683	+	0.923880i	$0.625000\pi$
$182$		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		0.555570	−	0.831470i	$-0.312500\pi$
$191$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$192$		0			0
$193$			0.390181i			0.390181i	0.980785	+	0.195090i	$0.0625000\pi$
$193$			0.390181i			0.390181i	−0.980785	+	0.195090i	$0.937500\pi$
$194$		0			0
$195$		0			0
$196$	0.187593	−	0.943094i	0.187593	−	0.943094i
$197$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$197$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$198$		0			0
$199$	1.36347	−	1.11897i	1.36347	−	1.11897i	0.382683	−	0.923880i	$-0.375000\pi$
$199$	1.36347	−	1.11897i	1.36347	−	1.11897i	0.980785	−	0.195090i	$-0.0625000\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	1.47945	−	1.21415i	1.47945	−	1.21415i
$209$		0			0
$210$		0			0
$211$	−0.124363	+	1.26268i	−0.124363	+	1.26268i	0.707107	+	0.707107i	$0.250000\pi$
$211$	−0.124363	+	1.26268i	−0.124363	+	1.26268i	−0.831470	+	0.555570i	$0.812500\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.319453	+	0.170751i	0.319453	+	0.170751i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.512016	−	0.273678i	−0.512016	−	0.273678i	0.195090	−	0.980785i	$-0.437500\pi$
$223$	−0.512016	−	0.273678i	−0.512016	−	0.273678i	−0.707107	+	0.707107i	$0.750000\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$227$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$228$		0			0
$229$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$229$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.634393	−	0.773010i	$-0.281250\pi$
$233$		0			0		−0.634393	+	0.773010i	$0.718750\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.471397	−	0.881921i	$-0.343750\pi$
$239$		0			0		−0.471397	+	0.881921i	$0.656250\pi$
$240$		0			0
$241$	0.636379	−	1.53636i	0.636379	−	1.53636i	−0.195090	−	0.980785i	$-0.562500\pi$
$241$	0.636379	−	1.53636i	0.636379	−	1.53636i	0.831470	−	0.555570i	$-0.187500\pi$
$242$		0			0
$243$		0			0
$244$	−1.02656	−	0.425215i	−1.02656	−	0.425215i
$245$		0			0
$246$		0			0
$247$	2.46024	−	1.64388i	2.46024	−	1.64388i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.0980171	−	0.995185i	$-0.531250\pi$
$251$		0			0		0.0980171	+	0.995185i	$0.468750\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$			1.00000i			1.00000i
$257$		0			0		−0.956940	−	0.290285i	$-0.906250\pi$
$257$		0			0		0.956940	+	0.290285i	$0.0937500\pi$
$258$		0			0
$259$	0.192102	+	0.287500i	0.192102	+	0.287500i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.471397	−	0.881921i	$-0.656250\pi$
$263$		0			0		0.471397	+	0.881921i	$0.343750\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.444430	−	0.831470i	0.444430	−	0.831470i
$269$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$269$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$270$		0			0
$271$	0.512016	+	1.68789i	0.512016	+	1.68789i	0.707107	+	0.707107i	$0.250000\pi$
$271$	0.512016	+	1.68789i	0.512016	+	1.68789i	−0.195090	+	0.980785i	$0.562500\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.11897	+	1.36347i	1.11897	+	1.36347i	0.923880	+	0.382683i	$0.125000\pi$
$277$	1.11897	+	1.36347i	1.11897	+	1.36347i	0.195090	+	0.980785i	$0.437500\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.471397	−	0.881921i	$-0.656250\pi$
$281$		0			0		0.471397	+	0.881921i	$0.343750\pi$
$282$		0			0
$283$	1.38268	+	0.923880i	1.38268	+	0.923880i	1.00000			$0$
$283$	1.38268	+	0.923880i	1.38268	+	0.923880i	0.382683	+	0.923880i	$0.375000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.980785	+	0.195090i	0.980785	+	0.195090i
$290$		0			0
$291$		0			0
$292$	−1.41421			−1.41421
$293$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$293$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−0.153672	−	0.287500i	−0.153672	−	0.287500i
$302$		0			0
$303$		0			0
$304$	−0.151537	+	1.53858i	−0.151537	+	1.53858i
$305$		0			0
$306$		0			0
$307$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$307$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.290285	−	0.956940i	$-0.593750\pi$
$311$		0			0		0.290285	+	0.956940i	$0.406250\pi$
$312$		0			0
$313$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$313$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$314$		0			0
$315$		0			0
$316$	0.0761205	−	0.382683i	0.0761205	−	0.382683i
$317$		0			0		0.290285	−	0.956940i	$-0.406250\pi$
$317$		0			0		−0.290285	+	0.956940i	$0.593750\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	1.83147	+	0.555570i	1.83147	+	0.555570i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.0569057	+	0.577774i	0.0569057	+	0.577774i	0.980785	+	0.195090i	$0.0625000\pi$
$331$	0.0569057	+	0.577774i	0.0569057	+	0.577774i	−0.923880	+	0.382683i	$0.875000\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.26268	−	1.53858i	1.26268	−	1.53858i	0.555570	−	0.831470i	$-0.312500\pi$
$337$	1.26268	−	1.53858i	1.26268	−	1.53858i	0.707107	−	0.707107i	$-0.250000\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.181269	+	0.339130i	−0.181269	+	0.339130i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.290285	−	0.956940i	$-0.593750\pi$
$347$		0			0		0.290285	+	0.956940i	$0.406250\pi$
$348$		0			0
$349$	1.26268	−	1.53858i	1.26268	−	1.53858i	0.555570	−	0.831470i	$-0.312500\pi$
$349$	1.26268	−	1.53858i	1.26268	−	1.53858i	0.707107	−	0.707107i	$-0.250000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.881921	−	0.471397i	$-0.843750\pi$
$359$		0			0		0.881921	+	0.471397i	$0.156250\pi$
$360$		0			0
$361$	−0.271211	+	1.36347i	−0.271211	+	1.36347i
$362$		0			0
$363$		0			0
$364$	−0.346627	+	0.143578i	−0.346627	+	0.143578i
$365$		0			0
$366$		0			0
$367$	−0.577774	−	1.90466i	−0.577774	−	1.90466i	−0.382683	−	0.923880i	$-0.625000\pi$
$367$	−0.577774	−	1.90466i	−0.577774	−	1.90466i	−0.195090	−	0.980785i	$-0.562500\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1.53858	+	0.151537i	−1.53858	+	0.151537i	−0.831470	−	0.555570i	$-0.812500\pi$
$373$	−1.53858	+	0.151537i	−1.53858	+	0.151537i	−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$379$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.773010	−	0.634393i	$-0.218750\pi$
$383$		0			0		−0.773010	+	0.634393i	$0.781250\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.555570	−	0.831470i	0.555570	−	0.831470i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.149316	−	0.360480i	−0.149316	−	0.360480i	0.831470	−	0.555570i	$-0.187500\pi$
$397$	−0.149316	−	0.360480i	−0.149316	−	0.360480i	−0.980785	+	0.195090i	$0.937500\pi$
$398$		0			0
$399$		0			0
$400$	−0.831470	+	0.555570i	−0.831470	+	0.555570i
$401$		0			0		−0.995185	−	0.0980171i	$-0.968750\pi$
$401$		0			0		0.995185	+	0.0980171i	$0.0312500\pi$
$402$		0			0
$403$	0.346627	+	3.51936i	0.346627	+	3.51936i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1.21415	−	0.368309i	1.21415	−	0.368309i	0.382683	−	0.923880i	$-0.375000\pi$
$409$	1.21415	−	0.368309i	1.21415	−	0.368309i	0.831470	+	0.555570i	$0.187500\pi$
$410$		0			0
$411$		0			0
$412$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$419$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$420$		0			0
$421$	1.81225	−	0.750661i	1.81225	−	0.750661i	0.831470	−	0.555570i	$-0.187500\pi$
$421$	1.81225	−	0.750661i	1.81225	−	0.750661i	0.980785	−	0.195090i	$-0.0625000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.168378	+	0.138185i	0.168378	+	0.138185i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$431$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$432$		0			0
$433$	−0.831470	−	0.444430i	−0.831470	−	0.444430i		−	1.00000i	$-0.5\pi$
$433$	−0.831470	−	0.444430i	−0.831470	−	0.444430i	−0.831470	+	0.555570i	$0.812500\pi$
$434$		0			0
$435$		0			0
$436$	−0.750661	+	0.149316i	−0.750661	+	0.149316i
$437$		0			0
$438$		0			0
$439$	1.53858	+	1.26268i	1.53858	+	1.26268i	0.831470	+	0.555570i	$0.187500\pi$
$439$	1.53858	+	1.26268i	1.53858	+	1.26268i	0.707107	+	0.707107i	$0.250000\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.0980171	−	0.995185i	$-0.531250\pi$
$443$		0			0		0.0980171	+	0.995185i	$0.468750\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.0569057	−	0.187593i	0.0569057	−	0.187593i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.831470	+	0.444430i	−0.831470	+	0.444430i	−0.831470	−	0.555570i	$-0.812500\pi$
$457$	−0.831470	+	0.444430i	−0.831470	+	0.444430i			1.00000i	$0.5\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$461$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$462$		0			0
$463$	−0.785695	−	0.785695i	−0.785695	−	0.785695i	0.195090	−	0.980785i	$-0.437500\pi$
$463$	−0.785695	−	0.785695i	−0.785695	−	0.785695i	−0.980785	+	0.195090i	$0.937500\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$467$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$468$		0			0
$469$	−0.130687	+	0.130687i	−0.130687	+	0.130687i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.36347	+	0.728789i	−1.36347	+	0.728789i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$479$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$480$		0			0
$481$	−1.29186	+	3.11882i	−1.29186	+	3.11882i
$482$		0			0
$483$		0			0
$484$		−	1.00000i		−	1.00000i
$485$		0			0
$486$		0			0
$487$	0.216773	−	1.08979i	0.216773	−	1.08979i	−0.707107	−	0.707107i	$-0.750000\pi$
$487$	0.216773	−	1.08979i	0.216773	−	1.08979i	0.923880	−	0.382683i	$-0.125000\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$491$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.53636	−	1.02656i	−1.53636	−	1.02656i
$497$		0			0
$498$		0			0
$499$	1.53858	−	1.26268i	1.53858	−	1.26268i	0.707107	−	0.707107i	$-0.250000\pi$
$499$	1.53858	−	1.26268i	1.53858	−	1.26268i	0.831470	−	0.555570i	$-0.187500\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$503$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	1.11897	+	0.598102i	1.11897	+	0.598102i
$509$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$509$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$510$		0			0
$511$	0.265297	+	0.0804769i	0.265297	+	0.0804769i
$512$		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		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−0.512016	+	0.273678i	−0.512016	+	0.273678i	−0.707107	−	0.707107i	$-0.750000\pi$
$523$	−0.512016	+	0.273678i	−0.512016	+	0.273678i	0.195090	+	0.980785i	$0.437500\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.555570	−	0.831470i	−0.555570	−	0.831470i
$530$		0			0
$531$		0			0
$532$	0.115981	−	0.280003i	0.115981	−	0.280003i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.555570	−	0.168530i	0.555570	−	0.168530i		−	1.00000i	$-0.5\pi$
$541$	0.555570	−	0.168530i	0.555570	−	0.168530i	0.555570	+	0.831470i	$0.312500\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$547$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.0360566	+	0.0674571i	−0.0360566	+	0.0674571i
$554$		0			0
$555$		0			0
$556$	0.273678	−	0.902197i	0.273678	−	0.902197i
$557$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$557$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$558$		0			0
$559$	1.50030	−	2.80686i	1.50030	−	2.80686i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.634393	−	0.773010i	$-0.718750\pi$
$563$		0			0		0.634393	+	0.773010i	$0.281250\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.0980171	−	0.995185i	$-0.468750\pi$
$569$		0			0		−0.0980171	+	0.995185i	$0.531250\pi$
$570$		0			0
$571$	−0.785695	+	1.17588i	−0.785695	+	1.17588i	0.195090	+	0.980785i	$0.437500\pi$
$571$	−0.785695	+	1.17588i	−0.785695	+	1.17588i	−0.980785	+	0.195090i	$0.937500\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−0.980785	−	1.19509i	−0.980785	−	1.19509i	−0.980785	−	0.195090i	$-0.937500\pi$
$577$	−0.980785	−	1.19509i	−0.980785	−	1.19509i		−	1.00000i	$-0.5\pi$
$578$		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		−0.634393	−	0.773010i	$-0.718750\pi$
$587$		0			0		0.634393	+	0.773010i	$0.281250\pi$
$588$		0			0
$589$	−2.20824	−	1.81225i	−2.20824	−	1.81225i
$590$		0			0
$591$		0			0
$592$	−0.831470	−	1.55557i	−0.831470	−	1.55557i
$593$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$593$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.995185	−	0.0980171i	$-0.968750\pi$
$599$		0			0		0.995185	+	0.0980171i	$0.0312500\pi$
$600$		0			0
$601$	1.21415	+	1.47945i	1.21415	+	1.47945i	0.831470	+	0.555570i	$0.187500\pi$
$601$	1.21415	+	1.47945i	1.21415	+	1.47945i	0.382683	+	0.923880i	$0.375000\pi$
$602$		0			0
$603$		0			0
$604$	−1.38704	+	1.38704i	−1.38704	+	1.38704i
$605$		0			0
$606$		0			0
$607$	0.149316	−	0.750661i	0.149316	−	0.750661i	−0.831470	−	0.555570i	$-0.812500\pi$
$607$	0.149316	−	0.750661i	0.149316	−	0.750661i	0.980785	−	0.195090i	$-0.0625000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.750661	+	0.149316i	−0.750661	+	0.149316i	−0.555570	−	0.831470i	$-0.687500\pi$
$613$	−0.750661	+	0.149316i	−0.750661	+	0.149316i	−0.195090	+	0.980785i	$0.562500\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$	1.21415	+	0.368309i	1.21415	+	0.368309i	0.831470	−	0.555570i	$-0.187500\pi$
$619$	1.21415	+	0.368309i	1.21415	+	0.368309i	0.382683	+	0.923880i	$0.375000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$626$		0			0
$627$		0			0
$628$	−0.804910	+	0.980785i	−0.804910	+	0.980785i
$629$		0			0
$630$		0			0
$631$	0.275899	+	1.38704i	0.275899	+	1.38704i	0.831470	+	0.555570i	$0.187500\pi$
$631$	0.275899	+	1.38704i	0.275899	+	1.38704i	−0.555570	+	0.831470i	$0.687500\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1.83147	+	0.180384i	−1.83147	+	0.180384i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.881921	−	0.471397i	$-0.156250\pi$
$641$		0			0		−0.881921	+	0.471397i	$0.843750\pi$
$642$		0			0
$643$	1.66294			1.66294			0.831470	−	0.555570i	$-0.187500\pi$
$643$	1.66294			1.66294			0.831470	+	0.555570i	$0.187500\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$647$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−0.382683	+	1.92388i	−0.382683	+	1.92388i
$653$		0			0		−0.956940	−	0.290285i	$-0.906250\pi$
$653$		0			0		0.956940	+	0.290285i	$0.0937500\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.956940	−	0.290285i	$-0.0937500\pi$
$659$		0			0		−0.956940	+	0.290285i	$0.906250\pi$
$660$		0			0
$661$	0.324423	+	1.63099i	0.324423	+	1.63099i	0.707107	+	0.707107i	$0.250000\pi$
$661$	0.324423	+	1.63099i	0.324423	+	1.63099i	−0.382683	+	0.923880i	$0.625000\pi$
$662$		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
$673$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$673$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$674$		0			0
$675$		0			0
$676$	−2.21415	−	1.47945i	−2.21415	−	1.47945i
$677$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$677$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$678$		0			0
$679$	−0.151537	+	0.124363i	−0.151537	+	0.124363i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$683$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	0.636379	+	1.53636i	0.636379	+	1.53636i
$689$		0			0
$690$		0			0
$691$	−0.923880	+	0.617317i	−0.923880	+	0.617317i	−0.923880	−	0.382683i	$-0.875000\pi$
$691$	−0.923880	+	0.617317i	−0.923880	+	0.617317i			1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.187593	−	0.0569057i	0.187593	−	0.0569057i
$701$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$701$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$702$		0			0
$703$	−1.04355	−	2.51936i	−1.04355	−	2.51936i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.98079	−	0.195090i	−1.98079	−	0.195090i	−0.980785	−	0.195090i	$-0.937500\pi$
$709$	−1.98079	−	0.195090i	−1.98079	−	0.195090i	−1.00000			$\pi$
$710$		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		0.995185	−	0.0980171i	$-0.0312500\pi$
$719$		0			0		−0.995185	+	0.0980171i	$0.968750\pi$
$720$		0			0
$721$	0.0271737	+	0.275899i	0.0271737	+	0.275899i
$722$		0			0
$723$		0			0
$724$	−0.512016	−	0.273678i	−0.512016	−	0.273678i
$725$		0			0
$726$		0			0
$727$	−1.92388	+	0.382683i	−1.92388	+	0.382683i	−0.923880	+	0.382683i	$0.875000\pi$
$727$	−1.92388	+	0.382683i	−1.92388	+	0.382683i	−1.00000			$\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	0.149316	−	0.360480i	0.149316	−	0.360480i	−0.831470	−	0.555570i	$-0.812500\pi$
$733$	0.149316	−	0.360480i	0.149316	−	0.360480i	0.980785	+	0.195090i	$0.0625000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.555570	−	1.83147i	0.555570	−	1.83147i		−	1.00000i	$-0.5\pi$
$739$	0.555570	−	1.83147i	0.555570	−	1.83147i	0.555570	−	0.831470i	$-0.312500\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$743$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.750661	+	0.149316i	0.750661	+	0.149316i	0.555570	−	0.831470i	$-0.312500\pi$
$751$	0.750661	+	0.149316i	0.750661	+	0.149316i	0.195090	+	0.980785i	$0.437500\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.448786	−	0.368309i	0.448786	−	0.368309i	−0.382683	−	0.923880i	$-0.625000\pi$
$757$	0.448786	−	0.368309i	0.448786	−	0.368309i	0.831470	+	0.555570i	$0.187500\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.0980171	−	0.995185i	$-0.531250\pi$
$761$		0			0		0.0980171	+	0.995185i	$0.468750\pi$
$762$		0			0
$763$	0.149316	+	0.0147063i	0.149316	+	0.0147063i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.368309	+	0.448786i	−0.368309	+	0.448786i	−0.923880	−	0.382683i	$-0.875000\pi$
$769$	−0.368309	+	0.448786i	−0.368309	+	0.448786i	0.555570	+	0.831470i	$0.312500\pi$
$770$		0			0
$771$		0			0
$772$	−0.149316	+	0.360480i	−0.149316	+	0.360480i
$773$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$773$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$774$		0			0
$775$		−	1.84776i		−	1.84776i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.534220	−	0.799517i	0.534220	−	0.799517i
$785$		0			0
$786$		0			0
$787$	−1.63099	−	1.08979i	−1.63099	−	1.08979i	−0.923880	−	0.382683i	$-0.875000\pi$
$787$	−1.63099	−	1.08979i	−1.63099	−	1.08979i	−0.707107	−	0.707107i	$-0.750000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.208442	+	2.11635i	−0.208442	+	2.11635i
$794$		0			0
$795$		0			0
$796$	1.68789	−	0.512016i	1.68789	−	0.512016i
$797$		0			0		−0.290285	−	0.956940i	$-0.593750\pi$
$797$		0			0		0.290285	+	0.956940i	$0.406250\pi$
$798$		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		−0.382683	−	0.923880i	$-0.625000\pi$
$809$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$810$		0			0
$811$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$811$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.746304	+	2.46024i	0.746304	+	2.46024i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.471397	−	0.881921i	$-0.343750\pi$
$821$		0			0		−0.471397	+	0.881921i	$0.656250\pi$
$822$		0			0
$823$	−0.750661	+	1.81225i	−0.750661	+	1.81225i	−0.195090	+	0.980785i	$0.562500\pi$
$823$	−0.750661	+	1.81225i	−0.750661	+	1.81225i	−0.555570	+	0.831470i	$0.687500\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.634393	−	0.773010i	$-0.281250\pi$
$827$		0			0		−0.634393	+	0.773010i	$0.718750\pi$
$828$		0			0
$829$	0.324423	−	0.216773i	0.324423	−	0.216773i	−0.382683	−	0.923880i	$-0.625000\pi$
$829$	0.324423	−	0.216773i	0.324423	−	0.216773i	0.707107	+	0.707107i	$0.250000\pi$
$830$		0			0
$831$		0			0
$832$	1.83147	−	0.555570i	1.83147	−	0.555570i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.956940	−	0.290285i	$-0.906250\pi$
$839$		0			0		0.956940	+	0.290285i	$0.0937500\pi$
$840$		0			0
$841$	−0.555570	−	0.831470i	−0.555570	−	0.831470i
$842$		0			0
$843$		0			0
$844$	−0.598102	+	1.11897i	−0.598102	+	1.11897i
$845$		0			0
$846$		0			0
$847$	−0.0569057	+	0.187593i	−0.0569057	+	0.187593i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.555570	+	1.83147i	0.555570	+	1.83147i	0.555570	+	0.831470i	$0.312500\pi$
$853$	0.555570	+	1.83147i	0.555570	+	1.83147i			1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$857$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$	0.804910	+	0.980785i	0.804910	+	0.980785i	1.00000			$0$
$859$	0.804910	+	0.980785i	0.804910	+	0.980785i	−0.195090	+	0.980785i	$0.562500\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.471397	−	0.881921i	$-0.656250\pi$
$863$		0			0		0.471397	+	0.881921i	$0.343750\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	0.229793	+	0.280003i	0.229793	+	0.280003i
$869$		0			0
$870$		0			0
$871$	−1.76972	−	0.352020i	−1.76972	−	0.352020i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.02656	+	0.425215i	1.02656	+	0.425215i	0.831470	−	0.555570i	$-0.187500\pi$
$877$	1.02656	+	0.425215i	1.02656	+	0.425215i	0.195090	+	0.980785i	$0.437500\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$881$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$882$		0			0
$883$	−0.902197	−	1.68789i	−0.902197	−	1.68789i	−0.707107	−	0.707107i	$-0.750000\pi$
$883$	−0.902197	−	1.68789i	−0.902197	−	1.68789i	−0.195090	−	0.980785i	$-0.562500\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.634393	−	0.773010i	$-0.718750\pi$
$887$		0			0		0.634393	+	0.773010i	$0.281250\pi$
$888$		0			0
$889$	−0.175876	−	0.175876i	−0.175876	−	0.175876i
$890$		0			0
$891$		0			0
$892$	−0.368309	−	0.448786i	−0.368309	−	0.448786i
$893$		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
$907$	1.90466	+	0.577774i	1.90466	+	0.577774i	0.980785	+	0.195090i	$0.0625000\pi$
$907$	1.90466	+	0.577774i	1.90466	+	0.577774i	0.923880	+	0.382683i	$0.125000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.471397	−	0.881921i	$-0.656250\pi$
$911$		0			0		0.471397	+	0.881921i	$0.343750\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.70711	+	0.707107i	1.70711	+	0.707107i
$917$		0			0
$918$		0			0
$919$	1.11897	−	1.36347i	1.11897	−	1.36347i	0.195090	−	0.980785i	$-0.437500\pi$
$919$	1.11897	−	1.36347i	1.11897	−	1.36347i	0.923880	−	0.382683i	$-0.125000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.831470	−	1.55557i	0.831470	−	1.55557i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.290285	−	0.956940i	$-0.593750\pi$
$929$		0			0		0.290285	+	0.956940i	$0.406250\pi$
$930$		0			0
$931$	0.943094	−	1.14916i	0.943094	−	1.14916i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.149316	+	0.360480i	0.149316	+	0.360480i	0.980785	−	0.195090i	$-0.0625000\pi$
$937$	0.149316	+	0.360480i	0.149316	+	0.360480i	−0.831470	+	0.555570i	$0.812500\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.881921	−	0.471397i	$-0.843750\pi$
$941$		0			0		0.881921	+	0.471397i	$0.156250\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.881921	−	0.471397i	$-0.843750\pi$
$947$		0			0		0.881921	+	0.471397i	$0.156250\pi$
$948$		0			0
$949$	0.785695	+	2.59009i	0.785695	+	2.59009i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.0980171	−	0.995185i	$-0.468750\pi$
$953$		0			0		−0.0980171	+	0.995185i	$0.531250\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	2.23044	−	0.923880i	2.23044	−	0.923880i
$962$		0			0
$963$		0			0
$964$	1.17588	−	1.17588i	1.17588	−	1.17588i
$965$		0			0
$966$		0			0
$967$	−0.324423	−	0.216773i	−0.324423	−	0.216773i	0.382683	−	0.923880i	$-0.375000\pi$
$967$	−0.324423	−	0.216773i	−0.324423	−	0.216773i	−0.707107	+	0.707107i	$0.750000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$	−0.102680	+	0.153672i	−0.102680	+	0.153672i
$974$		0			0
$975$		0			0
$976$	−0.785695	−	0.785695i	−0.785695	−	0.785695i
$977$		0			0		0.634393	−	0.773010i	$-0.281250\pi$
$977$		0			0		−0.634393	+	0.773010i	$0.718750\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.995185	−	0.0980171i	$-0.968750\pi$
$983$		0			0		0.995185	+	0.0980171i	$0.0312500\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	2.90205	−	0.577253i	2.90205	−	0.577253i
$989$		0			0
$990$		0			0
$991$	1.47945	−	0.448786i	1.47945	−	0.448786i	0.555570	−	0.831470i	$-0.312500\pi$
$991$	1.47945	−	0.448786i	1.47945	−	0.448786i	0.923880	+	0.382683i	$0.125000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.382683	+	0.0761205i	0.382683	+	0.0761205i	0.382683	−	0.923880i	$-0.375000\pi$
$997$	0.382683	+	0.0761205i	0.382683	+	0.0761205i			1.00000i	$0.5\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	873.1.br.a.451.1	yes	16
3.2	odd	2	CM	873.1.br.a.451.1	yes	16
97.77	odd	32	inner	873.1.br.a.271.1	✓	16
291.77	even	32	inner	873.1.br.a.271.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
873.1.br.a.271.1	✓	16	97.77	odd	32	inner
873.1.br.a.271.1	✓	16	291.77	even	32	inner
873.1.br.a.451.1	yes	16	1.1	even	1	trivial
873.1.br.a.451.1	yes	16	3.2	odd	2	CM

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:	\( N \)	\(=\)	\( 873 = 3^{2} \cdot 97 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	873.br (of order \(32\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\(q+(0.923880 + 0.382683i) q^{4} +(-0.151537 - 0.124363i) q^{7} +O(q^{10})\)	\(q+(0.923880 + 0.382683i) q^{4} +(-0.151537 - 0.124363i) q^{7} +(0.187593 - 1.90466i) q^{13} +(0.707107 + 0.707107i) q^{16} +(0.980785 + 1.19509i) q^{19} +(-0.195090 + 0.980785i) q^{25} +(-0.0924099 - 0.172887i) q^{28} +(-1.81225 + 0.360480i) q^{31} +(-1.68789 - 0.512016i) q^{37} +(1.53636 + 0.636379i) q^{43} +(-0.187593 - 0.943094i) q^{49} +(0.902197 - 1.68789i) q^{52} -1.11114 q^{61} +(0.382683 + 0.923880i) q^{64} +(0.0924099 - 0.938254i) q^{67} +(-1.30656 + 0.541196i) q^{73} +(0.448786 + 1.47945i) q^{76} +(-0.0761205 - 0.382683i) q^{79} +(-0.265297 + 0.265297i) q^{91} +(0.195090 - 0.980785i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 16 q^{79}+O(q^{100}) \) 16 * q - 16 * q^79

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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