LMFDB - Embedded newform 579.1.s.a.179.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [579,1,Mod(8,579)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(579, base_ring=CyclotomicField(32))

chi = DirichletCharacter(H, H._module([16, 17]))

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

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

chi := DirichletCharacter("579.8");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 579 = 3 \cdot 193 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	579.s (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:	$0.288958642315$
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:	$\Z[a_1, \ldots, a_{13}]$
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			179.1
Root			$-0.980785 + 0.195090i$ of defining polynomial
Character	$\chi$	$=$	579.179
Dual form			579.1.s.a.317.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.382683 - 0.923880i) q^{3} +(-0.923880 + 0.382683i) q^{4} +(-1.17588 + 1.17588i) q^{7} +(-0.707107 + 0.707107i) q^{9} +O(q^{10})$	$q+(-0.382683 - 0.923880i) q^{3} +(-0.923880 + 0.382683i) q^{4} +(-1.17588 + 1.17588i) q^{7} +(-0.707107 + 0.707107i) q^{9} +(0.707107 + 0.707107i) q^{12} +(-1.55557 + 0.831470i) q^{13} +(0.707107 - 0.707107i) q^{16} +(0.124363 - 0.151537i) q^{19} +(1.53636 + 0.636379i) q^{21} +(0.195090 + 0.980785i) q^{25} +(0.923880 + 0.382683i) q^{27} +(0.636379 - 1.53636i) q^{28} +(-1.81225 + 0.360480i) q^{31} +(0.382683 - 0.923880i) q^{36} +(0.555570 + 0.168530i) q^{37} +(1.36347 + 1.11897i) q^{39} +(-1.38704 - 1.38704i) q^{43} +(-0.923880 - 0.382683i) q^{48} -1.76537i q^{49} +(1.11897 - 1.36347i) q^{52} +(-0.187593 - 0.0569057i) q^{57} +(0.980785 + 1.19509i) q^{61} -1.66294i q^{63} +(-0.382683 + 0.923880i) q^{64} +(0.382683 - 0.0761205i) q^{67} +(-1.21415 + 0.368309i) q^{73} +(0.831470 - 0.555570i) q^{75} +(-0.0569057 + 0.187593i) q^{76} +(0.187593 - 1.90466i) q^{79} -1.00000i q^{81} -1.66294 q^{84} +(0.851452 - 2.80686i) q^{91} +(1.02656 + 1.53636i) q^{93} +(-0.275899 + 1.38704i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 16 q^{13}+O(q^{100}) $ 16 * q - 16 * q^13

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/579\mathbb{Z}\right)^\times$.

$n$	$194$	$391$
$\chi(n)$	$-1$	$e\left(\frac{7}{32}\right)$

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			0		−0.195090	−	0.980785i	$-0.562500\pi$
$2$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$3$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$3$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$4$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$5$		0			0		−0.773010	−	0.634393i	$-0.781250\pi$
$5$		0			0		0.773010	+	0.634393i	$0.218750\pi$
$6$		0			0
$7$	−1.17588	+	1.17588i	−1.17588	+	1.17588i	−0.195090	+	0.980785i	$0.562500\pi$
$7$	−1.17588	+	1.17588i	−1.17588	+	1.17588i	−0.980785	+	0.195090i	$0.937500\pi$
$8$		0			0
$9$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$10$		0			0
$11$		0			0		−0.995185	−	0.0980171i	$-0.968750\pi$
$11$		0			0		0.995185	+	0.0980171i	$0.0312500\pi$
$12$	0.707107	+	0.707107i	0.707107	+	0.707107i
$13$	−1.55557	+	0.831470i	−1.55557	+	0.831470i	−0.555570	+	0.831470i	$0.687500\pi$
$13$	−1.55557	+	0.831470i	−1.55557	+	0.831470i	−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$	0.707107	−	0.707107i	0.707107	−	0.707107i
$17$		0			0		0.773010	−	0.634393i	$-0.218750\pi$
$17$		0			0		−0.773010	+	0.634393i	$0.781250\pi$
$18$		0			0
$19$	0.124363	−	0.151537i	0.124363	−	0.151537i	−0.707107	−	0.707107i	$-0.750000\pi$
$19$	0.124363	−	0.151537i	0.124363	−	0.151537i	0.831470	+	0.555570i	$0.187500\pi$
$20$		0			0
$21$	1.53636	+	0.636379i	1.53636	+	0.636379i
$22$		0			0
$23$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$23$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$24$		0			0
$25$	0.195090	+	0.980785i	0.195090	+	0.980785i
$26$		0			0
$27$	0.923880	+	0.382683i	0.923880	+	0.382683i
$28$	0.636379	−	1.53636i	0.636379	−	1.53636i
$29$		0			0		0.956940	−	0.290285i	$-0.0937500\pi$
$29$		0			0		−0.956940	+	0.290285i	$0.906250\pi$
$30$		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
$36$	0.382683	−	0.923880i	0.382683	−	0.923880i
$37$	0.555570	+	0.168530i	0.555570	+	0.168530i	0.555570	−	0.831470i	$-0.312500\pi$
$37$	0.555570	+	0.168530i	0.555570	+	0.168530i			1.00000i	$0.5\pi$
$38$		0			0
$39$	1.36347	+	1.11897i	1.36347	+	1.11897i
$40$		0			0
$41$		0			0		−0.995185	−	0.0980171i	$-0.968750\pi$
$41$		0			0		0.995185	+	0.0980171i	$0.0312500\pi$
$42$		0			0
$43$	−1.38704	−	1.38704i	−1.38704	−	1.38704i	−0.831470	−	0.555570i	$-0.812500\pi$
$43$	−1.38704	−	1.38704i	−1.38704	−	1.38704i	−0.555570	−	0.831470i	$-0.687500\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.471397	−	0.881921i	$-0.656250\pi$
$47$		0			0		0.471397	+	0.881921i	$0.343750\pi$
$48$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$49$		−	1.76537i		−	1.76537i
$50$		0			0
$51$		0			0
$52$	1.11897	−	1.36347i	1.11897	−	1.36347i
$53$		0			0		0.881921	−	0.471397i	$-0.156250\pi$
$53$		0			0		−0.881921	+	0.471397i	$0.843750\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.187593	−	0.0569057i	−0.187593	−	0.0569057i
$58$		0			0
$59$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$59$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$60$		0			0
$61$	0.980785	+	1.19509i	0.980785	+	1.19509i	0.980785	+	0.195090i	$0.0625000\pi$
$61$	0.980785	+	1.19509i	0.980785	+	1.19509i			1.00000i	$0.5\pi$
$62$		0			0
$63$		−	1.66294i		−	1.66294i
$64$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$65$		0			0
$66$		0			0
$67$	0.382683	−	0.0761205i	0.382683	−	0.0761205i		−	1.00000i	$-0.5\pi$
$67$	0.382683	−	0.0761205i	0.382683	−	0.0761205i	0.382683	+	0.923880i	$0.375000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.634393	−	0.773010i	$-0.281250\pi$
$71$		0			0		−0.634393	+	0.773010i	$0.718750\pi$
$72$		0			0
$73$	−1.21415	+	0.368309i	−1.21415	+	0.368309i	−0.831470	−	0.555570i	$-0.812500\pi$
$73$	−1.21415	+	0.368309i	−1.21415	+	0.368309i	−0.382683	+	0.923880i	$0.625000\pi$
$74$		0			0
$75$	0.831470	−	0.555570i	0.831470	−	0.555570i
$76$	−0.0569057	+	0.187593i	−0.0569057	+	0.187593i
$77$		0			0
$78$		0			0
$79$	0.187593	−	1.90466i	0.187593	−	1.90466i	−0.195090	−	0.980785i	$-0.562500\pi$
$79$	0.187593	−	1.90466i	0.187593	−	1.90466i	0.382683	−	0.923880i	$-0.375000\pi$
$80$		0			0
$81$		−	1.00000i		−	1.00000i
$82$		0			0
$83$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$83$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$84$	−1.66294			−1.66294
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.881921	−	0.471397i	$-0.843750\pi$
$89$		0			0		0.881921	+	0.471397i	$0.156250\pi$
$90$		0			0
$91$	0.851452	−	2.80686i	0.851452	−	2.80686i
$92$		0			0
$93$	1.02656	+	1.53636i	1.02656	+	1.53636i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.275899	+	1.38704i	−0.275899	+	1.38704i	0.555570	+	0.831470i	$0.312500\pi$
$97$	−0.275899	+	1.38704i	−0.275899	+	1.38704i	−0.831470	+	0.555570i	$0.812500\pi$
$98$		0			0
$99$		0			0
$100$	−0.555570	−	0.831470i	−0.555570	−	0.831470i
$101$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$101$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$102$		0			0
$103$	0.577774	+	1.90466i	0.577774	+	1.90466i	0.382683	+	0.923880i	$0.375000\pi$
$103$	0.577774	+	1.90466i	0.577774	+	1.90466i	0.195090	+	0.980785i	$0.437500\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$107$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$108$	−1.00000			−1.00000
$109$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$109$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$110$		0			0
$111$	−0.0569057	−	0.577774i	−0.0569057	−	0.577774i
$112$			1.66294i			1.66294i
$113$		0			0		0.995185	−	0.0980171i	$-0.0312500\pi$
$113$		0			0		−0.995185	+	0.0980171i	$0.968750\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.512016	−	1.68789i	0.512016	−	1.68789i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.980785	+	0.195090i	0.980785	+	0.195090i
$122$		0			0
$123$		0			0
$124$	1.53636	−	1.02656i	1.53636	−	1.02656i
$125$		0			0
$126$		0			0
$127$	−1.68789	+	0.902197i	−1.68789	+	0.902197i	−0.707107	+	0.707107i	$0.750000\pi$
$127$	−1.68789	+	0.902197i	−1.68789	+	0.902197i	−0.980785	+	0.195090i	$0.937500\pi$
$128$		0			0
$129$	−0.750661	+	1.81225i	−0.750661	+	1.81225i
$130$		0			0
$131$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$131$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$132$		0			0
$133$	0.0319529	+	0.324423i	0.0319529	+	0.324423i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$137$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$138$		0			0
$139$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i	0.923880	+	0.382683i	$0.125000\pi$
$139$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i	−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$			1.00000i			1.00000i
$145$		0			0
$146$		0			0
$147$	−1.63099	+	0.675577i	−1.63099	+	0.675577i
$148$	−0.577774	+	0.0569057i	−0.577774	+	0.0569057i
$149$		0			0		0.956940	−	0.290285i	$-0.0937500\pi$
$149$		0			0		−0.956940	+	0.290285i	$0.906250\pi$
$150$		0			0
$151$	−0.216773	+	1.08979i	−0.216773	+	1.08979i	0.707107	+	0.707107i	$0.250000\pi$
$151$	−0.216773	+	1.08979i	−0.216773	+	1.08979i	−0.923880	+	0.382683i	$0.875000\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.68789	−	0.512016i	−1.68789	−	0.512016i
$157$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$157$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.53858	+	0.151537i	1.53858	+	0.151537i	0.831470	−	0.555570i	$-0.187500\pi$
$163$	1.53858	+	0.151537i	1.53858	+	0.151537i	0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.634393	−	0.773010i	$-0.718750\pi$
$167$		0			0		0.634393	+	0.773010i	$0.281250\pi$
$168$		0			0
$169$	1.17289	−	1.75535i	1.17289	−	1.75535i
$170$		0			0
$171$	0.0192147	+	0.195090i	0.0192147	+	0.195090i
$172$	1.81225	+	0.750661i	1.81225	+	0.750661i
$173$		0			0		−0.956940	−	0.290285i	$-0.906250\pi$
$173$		0			0		0.956940	+	0.290285i	$0.0937500\pi$
$174$		0			0
$175$	−1.38268	−	0.923880i	−1.38268	−	0.923880i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$179$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$180$		0			0
$181$	0.785695	+	0.785695i	0.785695	+	0.785695i	0.980785	−	0.195090i	$-0.0625000\pi$
$181$	0.785695	+	0.785695i	0.785695	+	0.785695i	−0.195090	+	0.980785i	$0.562500\pi$
$182$		0			0
$183$	0.728789	−	1.36347i	0.728789	−	1.36347i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−1.53636	+	0.636379i	−1.53636	+	0.636379i
$190$		0			0
$191$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$191$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$192$	1.00000			1.00000
$193$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$193$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$194$		0			0
$195$		0			0
$196$	0.675577	+	1.63099i	0.675577	+	1.63099i
$197$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$197$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$198$		0			0
$199$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$199$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$200$		0			0
$201$	−0.216773	−	0.324423i	−0.216773	−	0.324423i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−0.512016	+	1.68789i	−0.512016	+	1.68789i
$209$		0			0
$210$		0			0
$211$	−0.636379	−	0.425215i	−0.636379	−	0.425215i	0.195090	−	0.980785i	$-0.437500\pi$
$211$	−0.636379	−	0.425215i	−0.636379	−	0.425215i	−0.831470	+	0.555570i	$0.812500\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	1.70711	−	2.55487i	1.70711	−	2.55487i
$218$		0			0
$219$	0.804910	+	0.980785i	0.804910	+	0.980785i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.938254	−	0.0924099i	−0.938254	−	0.0924099i	−0.382683	−	0.923880i	$-0.625000\pi$
$223$	−0.938254	−	0.0924099i	−0.938254	−	0.0924099i	−0.555570	+	0.831470i	$0.687500\pi$
$224$		0			0
$225$	−0.831470	−	0.555570i	−0.831470	−	0.555570i
$226$		0			0
$227$		0			0		−0.773010	−	0.634393i	$-0.781250\pi$
$227$		0			0		0.773010	+	0.634393i	$0.218750\pi$
$228$	0.195090	−	0.0192147i	0.195090	−	0.0192147i
$229$	−0.425215	+	1.02656i	−0.425215	+	1.02656i	0.555570	+	0.831470i	$0.312500\pi$
$229$	−0.425215	+	1.02656i	−0.425215	+	1.02656i	−0.980785	+	0.195090i	$0.937500\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.0980171	−	0.995185i	$-0.468750\pi$
$233$		0			0		−0.0980171	+	0.995185i	$0.531250\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.83147	+	0.555570i	−1.83147	+	0.555570i
$238$		0			0
$239$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$239$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$240$		0			0
$241$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$241$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$242$		0			0
$243$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$244$	−1.36347	−	0.728789i	−1.36347	−	0.728789i
$245$		0			0
$246$		0			0
$247$	−0.0674571	+	0.339130i	−0.0674571	+	0.339130i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.471397	−	0.881921i	$-0.656250\pi$
$251$		0			0		0.471397	+	0.881921i	$0.343750\pi$
$252$	0.636379	+	1.53636i	0.636379	+	1.53636i
$253$		0			0
$254$		0			0
$255$		0			0
$256$		−	1.00000i		−	1.00000i
$257$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$257$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$258$		0			0
$259$	−0.851452	+	0.455111i	−0.851452	+	0.455111i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.290285	−	0.956940i	$-0.593750\pi$
$263$		0			0		0.290285	+	0.956940i	$0.406250\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.324423	+	0.216773i	−0.324423	+	0.216773i
$269$		0			0		0.290285	−	0.956940i	$-0.406250\pi$
$269$		0			0		−0.290285	+	0.956940i	$0.593750\pi$
$270$		0			0
$271$	−0.902197	+	1.68789i	−0.902197	+	1.68789i	−0.195090	+	0.980785i	$0.562500\pi$
$271$	−0.902197	+	1.68789i	−0.902197	+	1.68789i	−0.707107	+	0.707107i	$0.750000\pi$
$272$		0			0
$273$	−2.91904	+	0.287500i	−2.91904	+	0.287500i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$277$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$278$		0			0
$279$	1.02656	−	1.53636i	1.02656	−	1.53636i
$280$		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
$283$	0.368309	+	1.21415i	0.368309	+	1.21415i	0.923880	+	0.382683i	$0.125000\pi$
$283$	0.368309	+	1.21415i	0.368309	+	1.21415i	−0.555570	+	0.831470i	$0.687500\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.195090	−	0.980785i	0.195090	−	0.980785i
$290$		0			0
$291$	1.38704	−	0.275899i	1.38704	−	0.275899i
$292$	0.980785	−	0.804910i	0.980785	−	0.804910i
$293$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$293$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$301$	3.26197			3.26197
$302$		0			0
$303$		0			0
$304$	−0.0192147	−	0.195090i	−0.0192147	−	0.195090i
$305$		0			0
$306$		0			0
$307$	0.0924099	−	0.938254i	0.0924099	−	0.938254i	−0.831470	−	0.555570i	$-0.812500\pi$
$307$	0.0924099	−	0.938254i	0.0924099	−	0.938254i	0.923880	−	0.382683i	$-0.125000\pi$
$308$		0			0
$309$	1.53858	−	1.26268i	1.53858	−	1.26268i
$310$		0			0
$311$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$311$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$312$		0			0
$313$	−0.555570	+	0.168530i	−0.555570	+	0.168530i	−0.555570	−	0.831470i	$-0.687500\pi$
$313$	−0.555570	+	0.168530i	−0.555570	+	0.168530i			1.00000i	$0.5\pi$
$314$		0			0
$315$		0			0
$316$	0.555570	+	1.83147i	0.555570	+	1.83147i
$317$		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
$324$	0.382683	+	0.923880i	0.382683	+	0.923880i
$325$	−1.11897	−	1.36347i	−1.11897	−	1.36347i
$326$		0			0
$327$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.38704	+	1.38704i	1.38704	+	1.38704i	0.831470	+	0.555570i	$0.187500\pi$
$331$	1.38704	+	1.38704i	1.38704	+	1.38704i	0.555570	+	0.831470i	$0.312500\pi$
$332$		0			0
$333$	−0.512016	+	0.273678i	−0.512016	+	0.273678i
$334$		0			0
$335$		0			0
$336$	1.53636	−	0.636379i	1.53636	−	0.636379i
$337$		−	0.390181i		−	0.390181i	−0.980785	−	0.195090i	$-0.937500\pi$
$337$		−	0.390181i		−	0.390181i	0.980785	−	0.195090i	$-0.0625000\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.899976	+	0.899976i	0.899976	+	0.899976i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.773010	−	0.634393i	$-0.781250\pi$
$347$		0			0		0.773010	+	0.634393i	$0.218750\pi$
$348$		0			0
$349$	−1.90466	−	0.577774i	−1.90466	−	0.577774i	−0.980785	−	0.195090i	$-0.937500\pi$
$349$	−1.90466	−	0.577774i	−1.90466	−	0.577774i	−0.923880	−	0.382683i	$-0.875000\pi$
$350$		0			0
$351$	−1.75535	+	0.172887i	−1.75535	+	0.172887i
$352$		0			0
$353$		0			0		−0.290285	−	0.956940i	$-0.593750\pi$
$353$		0			0		0.290285	+	0.956940i	$0.406250\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$359$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$360$		0			0
$361$	0.187593	+	0.943094i	0.187593	+	0.943094i
$362$		0			0
$363$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$364$	0.287500	+	2.91904i	0.287500	+	2.91904i
$365$		0			0
$366$		0			0
$367$	1.21415	−	1.47945i	1.21415	−	1.47945i	0.382683	−	0.923880i	$-0.375000\pi$
$367$	1.21415	−	1.47945i	1.21415	−	1.47945i	0.831470	−	0.555570i	$-0.187500\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−1.53636	−	1.02656i	−1.53636	−	1.02656i
$373$	1.75535	−	0.938254i	1.75535	−	0.938254i	0.831470	−	0.555570i	$-0.187500\pi$
$373$	1.75535	−	0.938254i	1.75535	−	0.938254i	0.923880	−	0.382683i	$-0.125000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$379$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$380$		0			0
$381$	1.47945	+	1.21415i	1.47945	+	1.21415i
$382$		0			0
$383$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$383$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1.96157			1.96157
$388$	−0.275899	−	1.38704i	−0.275899	−	1.38704i
$389$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$389$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.938254	+	0.0924099i	0.938254	+	0.0924099i	0.555570	−	0.831470i	$-0.312500\pi$
$397$	0.938254	+	0.0924099i	0.938254	+	0.0924099i	0.382683	+	0.923880i	$0.375000\pi$
$398$		0			0
$399$	0.287500	−	0.153672i	0.287500	−	0.153672i
$400$	0.831470	+	0.555570i	0.831470	+	0.555570i
$401$		0			0		0.290285	−	0.956940i	$-0.406250\pi$
$401$		0			0		−0.290285	+	0.956940i	$0.593750\pi$
$402$		0			0
$403$	2.51936	−	2.06759i	2.51936	−	2.06759i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.360480	−	1.81225i	−0.360480	−	1.81225i	−0.555570	−	0.831470i	$-0.687500\pi$
$409$	−0.360480	−	1.81225i	−0.360480	−	1.81225i	0.195090	−	0.980785i	$-0.437500\pi$
$410$		0			0
$411$		0			0
$412$	−1.26268	−	1.53858i	−1.26268	−	1.53858i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.382683	−	0.0761205i	0.382683	−	0.0761205i
$418$		0			0
$419$		0			0		−0.290285	−	0.956940i	$-0.593750\pi$
$419$		0			0		0.290285	+	0.956940i	$0.406250\pi$
$420$		0			0
$421$	−1.26268	+	0.124363i	−1.26268	+	0.124363i	−0.707107	−	0.707107i	$-0.750000\pi$
$421$	−1.26268	+	0.124363i	−1.26268	+	0.124363i	−0.555570	+	0.831470i	$0.687500\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−2.55856	−	0.251996i	−2.55856	−	0.251996i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.995185	−	0.0980171i	$-0.0312500\pi$
$431$		0			0		−0.995185	+	0.0980171i	$0.968750\pi$
$432$	0.923880	−	0.382683i	0.923880	−	0.382683i
$433$	−0.728789	−	1.36347i	−0.728789	−	1.36347i	−0.923880	−	0.382683i	$-0.875000\pi$
$433$	−0.728789	−	1.36347i	−0.728789	−	1.36347i	0.195090	−	0.980785i	$-0.437500\pi$
$434$		0			0
$435$		0			0
$436$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$437$		0			0
$438$		0			0
$439$	0.831470	−	0.444430i	0.831470	−	0.444430i		−	1.00000i	$-0.5\pi$
$439$	0.831470	−	0.444430i	0.831470	−	0.444430i	0.831470	+	0.555570i	$0.187500\pi$
$440$		0			0
$441$	1.24830	+	1.24830i	1.24830	+	1.24830i
$442$		0			0
$443$		0			0		−0.956940	−	0.290285i	$-0.906250\pi$
$443$		0			0		0.956940	+	0.290285i	$0.0937500\pi$
$444$	0.273678	+	0.512016i	0.273678	+	0.512016i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.636379	−	1.53636i	−0.636379	−	1.53636i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	1.08979	−	0.216773i	1.08979	−	0.216773i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.21415	+	1.47945i	−1.21415	+	1.47945i	−0.382683	+	0.923880i	$0.625000\pi$
$457$	−1.21415	+	1.47945i	−1.21415	+	1.47945i	−0.831470	+	0.555570i	$0.812500\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$461$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$462$		0			0
$463$	−1.19509	+	0.980785i	−1.19509	+	0.980785i	−0.195090	+	0.980785i	$0.562500\pi$
$463$	−1.19509	+	0.980785i	−1.19509	+	0.980785i	−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	0.172887	+	1.75535i	0.172887	+	1.75535i
$469$	−0.360480	+	0.539496i	−0.360480	+	0.539496i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.172887	+	0.0924099i	0.172887	+	0.0924099i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$479$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$480$		0			0
$481$	−1.00436	+	0.199779i	−1.00436	+	0.199779i
$482$		0			0
$483$		0			0
$484$	−0.980785	+	0.195090i	−0.980785	+	0.195090i
$485$		0			0
$486$		0			0
$487$	−0.216773	−	0.324423i	−0.216773	−	0.324423i	0.707107	−	0.707107i	$-0.250000\pi$
$487$	−0.216773	−	0.324423i	−0.216773	−	0.324423i	−0.923880	+	0.382683i	$0.875000\pi$
$488$		0			0
$489$	−0.448786	−	1.47945i	−0.448786	−	1.47945i
$490$		0			0
$491$		0			0		−0.471397	−	0.881921i	$-0.656250\pi$
$491$		0			0		0.471397	+	0.881921i	$0.343750\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.02656	+	1.53636i	−1.02656	+	1.53636i
$497$		0			0
$498$		0			0
$499$	0.195090	−	0.0192147i	0.195090	−	0.0192147i		−	1.00000i	$-0.5\pi$
$499$	0.195090	−	0.0192147i	0.195090	−	0.0192147i	0.195090	+	0.980785i	$0.437500\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.290285	−	0.956940i	$-0.406250\pi$
$503$		0			0		−0.290285	+	0.956940i	$0.593750\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−2.07058	−	0.411863i	−2.07058	−	0.411863i
$508$	1.21415	−	1.47945i	1.21415	−	1.47945i
$509$		0			0		−0.290285	−	0.956940i	$-0.593750\pi$
$509$		0			0		0.290285	+	0.956940i	$0.406250\pi$
$510$		0			0
$511$	0.994607	−	1.86078i	0.994607	−	1.86078i
$512$		0			0
$513$	0.172887	−	0.0924099i	0.172887	−	0.0924099i
$514$		0			0
$515$		0			0
$516$		−	1.96157i		−	1.96157i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.471397	−	0.881921i	$-0.656250\pi$
$521$		0			0		0.471397	+	0.881921i	$0.343750\pi$
$522$		0			0
$523$	−1.38704	−	0.275899i	−1.38704	−	0.275899i	−0.555570	−	0.831470i	$-0.687500\pi$
$523$	−1.38704	−	0.275899i	−1.38704	−	0.275899i	−0.831470	+	0.555570i	$0.812500\pi$
$524$		0			0
$525$	−0.324423	+	1.63099i	−0.324423	+	1.63099i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$530$		0			0
$531$		0			0
$532$	−0.153672	−	0.287500i	−0.153672	−	0.287500i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.11897	−	0.598102i	−1.11897	−	0.598102i	−0.195090	−	0.980785i	$-0.562500\pi$
$541$	−1.11897	−	0.598102i	−1.11897	−	0.598102i	−0.923880	+	0.382683i	$0.875000\pi$
$542$		0			0
$543$	0.425215	−	1.02656i	0.425215	−	1.02656i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.17588	+	0.785695i	1.17588	+	0.785695i	0.980785	−	0.195090i	$-0.0625000\pi$
$547$	1.17588	+	0.785695i	1.17588	+	0.785695i	0.195090	+	0.980785i	$0.437500\pi$
$548$		0			0
$549$	−1.53858	−	0.151537i	−1.53858	−	0.151537i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	2.01906	+	2.46024i	2.01906	+	2.46024i
$554$		0			0
$555$		0			0
$556$	−0.0761205	−	0.382683i	−0.0761205	−	0.382683i
$557$		0			0		−0.0980171	−	0.995185i	$-0.531250\pi$
$557$		0			0		0.0980171	+	0.995185i	$0.468750\pi$
$558$		0			0
$559$	3.31092	+	1.00436i	3.31092	+	1.00436i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	1.17588	+	1.17588i	1.17588	+	1.17588i
$568$		0			0
$569$		0			0		0.471397	−	0.881921i	$-0.343750\pi$
$569$		0			0		−0.471397	+	0.881921i	$0.656250\pi$
$570$		0			0
$571$	−0.425215	−	0.636379i	−0.425215	−	0.636379i	0.555570	−	0.831470i	$-0.312500\pi$
$571$	−0.425215	−	0.636379i	−0.425215	−	0.636379i	−0.980785	+	0.195090i	$0.937500\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$577$	−0.149316	−	0.750661i	−0.149316	−	0.750661i	−0.980785	−	0.195090i	$-0.937500\pi$
$577$	−0.149316	−	0.750661i	−0.149316	−	0.750661i	0.831470	−	0.555570i	$-0.187500\pi$
$578$		0			0
$579$	0.707107	+	0.707107i	0.707107	+	0.707107i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$587$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$588$	1.24830	−	1.24830i	1.24830	−	1.24830i
$589$	−0.170751	+	0.319453i	−0.170751	+	0.319453i
$590$		0			0
$591$		0			0
$592$	0.512016	−	0.273678i	0.512016	−	0.273678i
$593$		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		−0.956940	−	0.290285i	$-0.906250\pi$
$599$		0			0		0.956940	+	0.290285i	$0.0937500\pi$
$600$		0			0
$601$	0.0192147	+	0.195090i	0.0192147	+	0.195090i	1.00000			$0$
$601$	0.0192147	+	0.195090i	0.0192147	+	0.195090i	−0.980785	+	0.195090i	$0.937500\pi$
$602$		0			0
$603$	−0.216773	+	0.324423i	−0.216773	+	0.324423i
$604$	−0.216773	−	1.08979i	−0.216773	−	1.08979i
$605$		0			0
$606$		0			0
$607$	0.292893	−	0.707107i	0.292893	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$607$	0.292893	−	0.707107i	0.292893	−	0.707107i	1.00000			$0$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.980785	+	0.804910i	0.980785	+	0.804910i	0.980785	−	0.195090i	$-0.0625000\pi$
$613$	0.980785	+	0.804910i	0.980785	+	0.804910i			1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.881921	−	0.471397i	$-0.843750\pi$
$617$		0			0		0.881921	+	0.471397i	$0.156250\pi$
$618$		0			0
$619$	−0.195090	+	1.98079i	−0.195090	+	1.98079i			1.00000i	$0.5\pi$
$619$	−0.195090	+	1.98079i	−0.195090	+	1.98079i	−0.195090	+	0.980785i	$0.562500\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	1.75535	−	0.172887i	1.75535	−	0.172887i
$625$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.26268	+	1.53858i	−1.26268	+	1.53858i	−0.555570	+	0.831470i	$0.687500\pi$
$631$	−1.26268	+	1.53858i	−1.26268	+	1.53858i	−0.707107	+	0.707107i	$0.750000\pi$
$632$		0			0
$633$	−0.149316	+	0.750661i	−0.149316	+	0.750661i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.46785	+	2.74615i	1.46785	+	2.74615i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$641$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$642$		0			0
$643$	−0.707107	+	1.70711i	−0.707107	+	1.70711i			1.00000i	$0.5\pi$
$643$	−0.707107	+	1.70711i	−0.707107	+	1.70711i	−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.471397	−	0.881921i	$-0.343750\pi$
$647$		0			0		−0.471397	+	0.881921i	$0.656250\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−3.01367	−	0.599456i	−3.01367	−	0.599456i
$652$	−1.47945	+	0.448786i	−1.47945	+	0.448786i
$653$		0			0		0.0980171	−	0.995185i	$-0.468750\pi$
$653$		0			0		−0.0980171	+	0.995185i	$0.531250\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	0.598102	−	1.11897i	0.598102	−	1.11897i
$658$		0			0
$659$		0			0		0.995185	−	0.0980171i	$-0.0312500\pi$
$659$		0			0		−0.995185	+	0.0980171i	$0.968750\pi$
$660$		0			0
$661$	−0.151537	−	1.53858i	−0.151537	−	1.53858i	−0.707107	−	0.707107i	$-0.750000\pi$
$661$	−0.151537	−	1.53858i	−0.151537	−	1.53858i	0.555570	−	0.831470i	$-0.312500\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	0.273678	+	0.902197i	0.273678	+	0.902197i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.728789	−	0.598102i	0.728789	−	0.598102i	−0.195090	−	0.980785i	$-0.562500\pi$
$673$	0.728789	−	0.598102i	0.728789	−	0.598102i	0.923880	+	0.382683i	$0.125000\pi$
$674$		0			0
$675$	−0.195090	+	0.980785i	−0.195090	+	0.980785i
$676$	−0.411863	+	2.07058i	−0.411863	+	2.07058i
$677$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$677$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$678$		0			0
$679$	−1.30656	−	1.95541i	−1.30656	−	1.95541i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.881921	−	0.471397i	$-0.843750\pi$
$683$		0			0		0.881921	+	0.471397i	$0.156250\pi$
$684$	−0.0924099	−	0.172887i	−0.0924099	−	0.172887i
$685$		0			0
$686$		0			0
$687$	1.11114			1.11114
$688$	−1.96157			−1.96157
$689$		0			0
$690$		0			0
$691$			0.390181i			0.390181i	0.980785	+	0.195090i	$0.0625000\pi$
$691$			0.390181i			0.390181i	−0.980785	+	0.195090i	$0.937500\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.63099	+	0.324423i	1.63099	+	0.324423i
$701$		0			0		0.634393	−	0.773010i	$-0.281250\pi$
$701$		0			0		−0.634393	+	0.773010i	$0.718750\pi$
$702$		0			0
$703$	0.0946308	−	0.0632303i	0.0946308	−	0.0632303i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			1.96157i			1.96157i	0.195090	+	0.980785i	$0.437500\pi$
$709$			1.96157i			1.96157i	−0.195090	+	0.980785i	$0.562500\pi$
$710$		0			0
$711$	1.21415	+	1.47945i	1.21415	+	1.47945i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.881921	−	0.471397i	$-0.156250\pi$
$719$		0			0		−0.881921	+	0.471397i	$0.843750\pi$
$720$		0			0
$721$	−2.91904	−	1.56026i	−2.91904	−	1.56026i
$722$		0			0
$723$		0			0
$724$	−1.02656	−	0.425215i	−1.02656	−	0.425215i
$725$		0			0
$726$		0			0
$727$	1.98079	−	0.195090i	1.98079	−	0.195090i	0.980785	−	0.195090i	$-0.0625000\pi$
$727$	1.98079	−	0.195090i	1.98079	−	0.195090i	1.00000			$0$
$728$		0			0
$729$	0.707107	+	0.707107i	0.707107	+	0.707107i
$730$		0			0
$731$		0			0
$732$	−0.151537	+	1.53858i	−0.151537	+	1.53858i
$733$	−1.36347	−	1.11897i	−1.36347	−	1.11897i	−0.980785	−	0.195090i	$-0.937500\pi$
$733$	−1.36347	−	1.11897i	−1.36347	−	1.11897i	−0.382683	−	0.923880i	$-0.625000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.577774	+	1.90466i	0.577774	+	1.90466i	0.382683	+	0.923880i	$0.375000\pi$
$739$	0.577774	+	1.90466i	0.577774	+	1.90466i	0.195090	+	0.980785i	$0.437500\pi$
$740$		0			0
$741$	0.339130	−	0.0674571i	0.339130	−	0.0674571i
$742$		0			0
$743$		0			0		0.956940	−	0.290285i	$-0.0937500\pi$
$743$		0			0		−0.956940	+	0.290285i	$0.906250\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.360480	−	0.149316i	−0.360480	−	0.149316i	0.195090	−	0.980785i	$-0.437500\pi$
$751$	−0.360480	−	0.149316i	−0.360480	−	0.149316i	−0.555570	+	0.831470i	$0.687500\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	1.17588	−	1.17588i	1.17588	−	1.17588i
$757$	−0.168530	+	0.555570i	−0.168530	+	0.555570i	0.831470	+	0.555570i	$0.187500\pi$
$757$	−0.168530	+	0.555570i	−0.168530	+	0.555570i	−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.995185	−	0.0980171i	$-0.968750\pi$
$761$		0			0		0.995185	+	0.0980171i	$0.0312500\pi$
$762$		0			0
$763$	−1.66294	+	1.66294i	−1.66294	+	1.66294i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$769$	0.292893	+	0.707107i	0.292893	+	0.707107i	1.00000			$0$
$769$	0.292893	+	0.707107i	0.292893	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$770$		0			0
$771$		0			0
$772$	0.707107	−	0.707107i	0.707107	−	0.707107i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	−0.707107	−	1.70711i	−0.707107	−	1.70711i
$776$		0			0
$777$	0.746304	+	0.612476i	0.746304	+	0.612476i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1.24830	−	1.24830i	−1.24830	−	1.24830i
$785$		0			0
$786$		0			0
$787$	0.512016	−	1.68789i	0.512016	−	1.68789i	−0.195090	−	0.980785i	$-0.562500\pi$
$787$	0.512016	−	1.68789i	0.512016	−	1.68789i	0.707107	−	0.707107i	$-0.250000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−2.51936	−	1.04355i	−2.51936	−	1.04355i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$797$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	0.324423	+	0.216773i	0.324423	+	0.216773i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.956940	−	0.290285i	$-0.906250\pi$
$809$		0			0		0.956940	+	0.290285i	$0.0937500\pi$
$810$		0			0
$811$	0.448786	+	0.368309i	0.448786	+	0.368309i	0.831470	−	0.555570i	$-0.187500\pi$
$811$	0.448786	+	0.368309i	0.448786	+	0.368309i	−0.382683	+	0.923880i	$0.625000\pi$
$812$		0			0
$813$	1.90466	+	0.187593i	1.90466	+	0.187593i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.382683	+	0.0376910i	−0.382683	+	0.0376910i
$818$		0			0
$819$	1.38268	+	2.58682i	1.38268	+	2.58682i
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$	1.55557	+	0.831470i	1.55557	+	0.831470i	1.00000			$0$
$823$	1.55557	+	0.831470i	1.55557	+	0.831470i	0.555570	+	0.831470i	$0.312500\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$	−1.21415	−	0.368309i	−1.21415	−	0.368309i	−0.382683	−	0.923880i	$-0.625000\pi$
$829$	−1.21415	−	0.368309i	−1.21415	−	0.368309i	−0.831470	+	0.555570i	$0.812500\pi$
$830$		0			0
$831$	0.541196	+	1.30656i	0.541196	+	1.30656i
$832$	−0.172887	−	1.75535i	−0.172887	−	1.75535i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1.81225	−	0.360480i	−1.81225	−	0.360480i
$838$		0			0
$839$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$839$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$840$		0			0
$841$	0.831470	−	0.555570i	0.831470	−	0.555570i
$842$		0			0
$843$		0			0
$844$	0.750661	+	0.149316i	0.750661	+	0.149316i
$845$		0			0
$846$		0			0
$847$	−1.38268	+	0.923880i	−1.38268	+	0.923880i
$848$		0			0
$849$	0.980785	−	0.804910i	0.980785	−	0.804910i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	1.96157i		−	1.96157i	−0.195090	−	0.980785i	$-0.562500\pi$
$853$		−	1.96157i		−	1.96157i	0.195090	−	0.980785i	$-0.437500\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	0.598102	+	0.728789i	0.598102	+	0.728789i	0.980785	−	0.195090i	$-0.0625000\pi$
$859$	0.598102	+	0.728789i	0.598102	+	0.728789i	−0.382683	+	0.923880i	$0.625000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.290285	−	0.956940i	$-0.406250\pi$
$863$		0			0		−0.290285	+	0.956940i	$0.593750\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.980785	+	0.195090i	−0.980785	+	0.195090i
$868$	−0.599456	+	3.01367i	−0.599456	+	3.01367i
$869$		0			0
$870$		0			0
$871$	−0.531999	+	0.436600i	−0.531999	+	0.436600i
$872$		0			0
$873$	−0.785695	−	1.17588i	−0.785695	−	1.17588i
$874$		0			0
$875$		0			0
$876$	−1.11897	−	0.598102i	−1.11897	−	0.598102i
$877$	0.0924099	+	0.172887i	0.0924099	+	0.172887i	0.923880	−	0.382683i	$-0.125000\pi$
$877$	0.0924099	+	0.172887i	0.0924099	+	0.172887i	−0.831470	+	0.555570i	$0.812500\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	0.187593	+	1.90466i	0.187593	+	1.90466i	0.382683	+	0.923880i	$0.375000\pi$
$883$	0.187593	+	1.90466i	0.187593	+	1.90466i	−0.195090	+	0.980785i	$0.562500\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.471397	−	0.881921i	$-0.343750\pi$
$887$		0			0		−0.471397	+	0.881921i	$0.656250\pi$
$888$		0			0
$889$	0.923880	−	3.04562i	0.923880	−	3.04562i
$890$		0			0
$891$		0			0
$892$	0.902197	−	0.273678i	0.902197	−	0.273678i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.980785	+	0.195090i	0.980785	+	0.195090i
$901$		0			0
$902$		0			0
$903$	−1.24830	−	3.01367i	−1.24830	−	3.01367i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.728789	−	1.36347i	−0.728789	−	1.36347i	−0.923880	−	0.382683i	$-0.875000\pi$
$907$	−0.728789	−	1.36347i	−0.728789	−	1.36347i	0.195090	−	0.980785i	$-0.437500\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$911$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$912$	−0.172887	+	0.0924099i	−0.172887	+	0.0924099i
$913$		0			0
$914$		0			0
$915$		0			0
$916$		−	1.11114i		−	1.11114i
$917$		0			0
$918$		0			0
$919$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$919$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−1.00000			$\pi$
$920$		0			0
$921$	−0.902197	+	0.273678i	−0.902197	+	0.273678i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.0569057	+	0.577774i	−0.0569057	+	0.577774i
$926$		0			0
$927$	−1.75535	−	0.938254i	−1.75535	−	0.938254i
$928$		0			0
$929$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$929$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$930$		0			0
$931$	−0.267518	−	0.219546i	−0.267518	−	0.219546i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.149316	−	0.360480i	0.149316	−	0.360480i	−0.831470	−	0.555570i	$-0.812500\pi$
$937$	0.149316	−	0.360480i	0.149316	−	0.360480i	0.980785	+	0.195090i	$0.0625000\pi$
$938$		0			0
$939$	0.368309	+	0.448786i	0.368309	+	0.448786i
$940$		0			0
$941$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$941$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$947$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$948$	1.47945	−	1.21415i	1.47945	−	1.21415i
$949$	1.58246	−	1.58246i	1.58246	−	1.58246i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$953$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$954$		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
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.382683	+	1.92388i	0.382683	+	1.92388i	0.382683	+	0.923880i	$0.375000\pi$
$967$	0.382683	+	1.92388i	0.382683	+	1.92388i			1.00000i	$0.5\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$971$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$972$	0.707107	−	0.707107i	0.707107	−	0.707107i
$973$	−0.360480	−	0.539496i	−0.360480	−	0.539496i
$974$		0			0
$975$	−0.831470	+	1.55557i	−0.831470	+	1.55557i
$976$	1.53858	+	0.151537i	1.53858	+	0.151537i
$977$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$977$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$982$		0			0
$983$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$983$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−0.0674571	−	0.339130i	−0.0674571	−	0.339130i
$989$		0			0
$990$		0			0
$991$	−1.11897	−	1.36347i	−1.11897	−	1.36347i	−0.923880	−	0.382683i	$-0.875000\pi$
$991$	−1.11897	−	1.36347i	−1.11897	−	1.36347i	−0.195090	−	0.980785i	$-0.562500\pi$
$992$		0			0
$993$	0.750661	−	1.81225i	0.750661	−	1.81225i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.53636	+	1.02656i	1.53636	+	1.02656i	0.980785	+	0.195090i	$0.0625000\pi$
$997$	1.53636	+	1.02656i	1.53636	+	1.02656i	0.555570	+	0.831470i	$0.312500\pi$
$998$		0			0
$999$	0.448786	+	0.368309i	0.448786	+	0.368309i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	579.1.s.a.179.1	✓	16
3.2	odd	2	CM	579.1.s.a.179.1	✓	16
193.124	even	32	inner	579.1.s.a.317.1	yes	16
579.317	odd	32	inner	579.1.s.a.317.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
579.1.s.a.179.1	✓	16	1.1	even	1	trivial
579.1.s.a.179.1	✓	16	3.2	odd	2	CM
579.1.s.a.317.1	yes	16	193.124	even	32	inner
579.1.s.a.317.1	yes	16	579.317	odd	32	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:	\( N \)	\(=\)	\( 579 = 3 \cdot 193 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	579.s (of order \(32\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.382683 - 0.923880i) q^{3} +(-0.923880 + 0.382683i) q^{4} +(-1.17588 + 1.17588i) q^{7} +(-0.707107 + 0.707107i) q^{9} +O(q^{10})\)	\(q+(-0.382683 - 0.923880i) q^{3} +(-0.923880 + 0.382683i) q^{4} +(-1.17588 + 1.17588i) q^{7} +(-0.707107 + 0.707107i) q^{9} +(0.707107 + 0.707107i) q^{12} +(-1.55557 + 0.831470i) q^{13} +(0.707107 - 0.707107i) q^{16} +(0.124363 - 0.151537i) q^{19} +(1.53636 + 0.636379i) q^{21} +(0.195090 + 0.980785i) q^{25} +(0.923880 + 0.382683i) q^{27} +(0.636379 - 1.53636i) q^{28} +(-1.81225 + 0.360480i) q^{31} +(0.382683 - 0.923880i) q^{36} +(0.555570 + 0.168530i) q^{37} +(1.36347 + 1.11897i) q^{39} +(-1.38704 - 1.38704i) q^{43} +(-0.923880 - 0.382683i) q^{48} -1.76537i q^{49} +(1.11897 - 1.36347i) q^{52} +(-0.187593 - 0.0569057i) q^{57} +(0.980785 + 1.19509i) q^{61} -1.66294i q^{63} +(-0.382683 + 0.923880i) q^{64} +(0.382683 - 0.0761205i) q^{67} +(-1.21415 + 0.368309i) q^{73} +(0.831470 - 0.555570i) q^{75} +(-0.0569057 + 0.187593i) q^{76} +(0.187593 - 1.90466i) q^{79} -1.00000i q^{81} -1.66294 q^{84} +(0.851452 - 2.80686i) q^{91} +(1.02656 + 1.53636i) q^{93} +(-0.275899 + 1.38704i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 16 q^{13}+O(q^{100}) \) 16 * q - 16 * q^13

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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