LMFDB - Embedded newform 1028.1.l.a.15.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1028,1,Mod(15,1028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1028, base_ring=CyclotomicField(32))

chi = DirichletCharacter(H, H._module([16, 7]))

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

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

chi := DirichletCharacter("1028.15");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1028 = 2^{2} \cdot 257 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1028.l (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:	$0.513038832987$
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:	$\Z[a_1, \ldots, a_{9}]$
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			15.1
Root			$-0.980785 + 0.195090i$ of defining polynomial
Character	$\chi$	$=$	1028.15
Dual form			1028.1.l.a.891.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +(1.75535 + 0.172887i) q^{5} -1.00000i q^{8} +(0.195090 + 0.980785i) q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} +(1.75535 + 0.172887i) q^{5} -1.00000i q^{8} +(0.195090 + 0.980785i) q^{9} +(-0.172887 + 1.75535i) q^{10} +(-1.63099 - 1.08979i) q^{13} +1.00000 q^{16} +(1.17588 + 1.17588i) q^{17} +(-0.980785 + 0.195090i) q^{18} +(-1.75535 - 0.172887i) q^{20} +(2.07058 + 0.411863i) q^{25} +(1.08979 - 1.63099i) q^{26} +(0.275899 - 1.38704i) q^{29} +1.00000i q^{32} +(-1.17588 + 1.17588i) q^{34} +(-0.195090 - 0.980785i) q^{36} +(-1.68789 + 0.512016i) q^{37} +(0.172887 - 1.75535i) q^{40} +(-0.172887 - 0.0924099i) q^{41} +(0.172887 + 1.75535i) q^{45} +(-0.831470 - 0.555570i) q^{49} +(-0.411863 + 2.07058i) q^{50} +(1.63099 + 1.08979i) q^{52} +(0.0924099 + 0.938254i) q^{53} +(1.38704 + 0.275899i) q^{58} +(-0.636379 - 0.425215i) q^{61} -1.00000 q^{64} +(-2.67454 - 2.19494i) q^{65} +(-1.17588 - 1.17588i) q^{68} +(0.980785 - 0.195090i) q^{72} +(-0.149316 + 0.360480i) q^{73} +(-0.512016 - 1.68789i) q^{74} +(1.75535 + 0.172887i) q^{80} +(-0.923880 + 0.382683i) q^{81} +(0.0924099 - 0.172887i) q^{82} +(1.86078 + 2.26737i) q^{85} +(0.216773 + 0.324423i) q^{89} +(-1.75535 + 0.172887i) q^{90} +(0.124363 - 1.26268i) q^{97} +(0.555570 - 0.831470i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4}+O(q^{10}) $ 16 * q - 16 * q^4	$ 16 q - 16 q^{4} + 16 q^{16} - 16 q^{64} - 16 q^{65}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^16 - 16 * q^64 - 16 * q^65

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1028\mathbb{Z}\right)^\times$.

$n$	$515$	$517$
$\chi(n)$	$-1$	$e\left(\frac{7}{32}\right)$

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 $

$2$			1.00000i			1.00000i
$2$			1.00000i			1.00000i
$3$		0			0		−0.773010	−	0.634393i	$-0.781250\pi$
$3$		0			0		0.773010	+	0.634393i	$0.218750\pi$
$4$	−1.00000			−1.00000
$5$	1.75535	+	0.172887i	1.75535	+	0.172887i	0.923880	−	0.382683i	$-0.125000\pi$
$5$	1.75535	+	0.172887i	1.75535	+	0.172887i	0.831470	+	0.555570i	$0.187500\pi$
$6$		0			0
$7$		0			0		0.290285	−	0.956940i	$-0.406250\pi$
$7$		0			0		−0.290285	+	0.956940i	$0.593750\pi$
$8$		−	1.00000i		−	1.00000i
$9$	0.195090	+	0.980785i	0.195090	+	0.980785i
$10$	−0.172887	+	1.75535i	−0.172887	+	1.75535i
$11$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$11$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$12$		0			0
$13$	−1.63099	−	1.08979i	−1.63099	−	1.08979i	−0.923880	−	0.382683i	$-0.875000\pi$
$13$	−1.63099	−	1.08979i	−1.63099	−	1.08979i	−0.707107	−	0.707107i	$-0.750000\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			1.00000
$17$	1.17588	+	1.17588i	1.17588	+	1.17588i	0.980785	+	0.195090i	$0.0625000\pi$
$17$	1.17588	+	1.17588i	1.17588	+	1.17588i	0.195090	+	0.980785i	$0.437500\pi$
$18$	−0.980785	+	0.195090i	−0.980785	+	0.195090i
$19$		0			0		−0.471397	−	0.881921i	$-0.656250\pi$
$19$		0			0		0.471397	+	0.881921i	$0.343750\pi$
$20$	−1.75535	−	0.172887i	−1.75535	−	0.172887i
$21$		0			0
$22$		0			0
$23$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$23$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$24$		0			0
$25$	2.07058	+	0.411863i	2.07058	+	0.411863i
$26$	1.08979	−	1.63099i	1.08979	−	1.63099i
$27$		0			0
$28$		0			0
$29$	0.275899	−	1.38704i	0.275899	−	1.38704i	−0.555570	−	0.831470i	$-0.687500\pi$
$29$	0.275899	−	1.38704i	0.275899	−	1.38704i	0.831470	−	0.555570i	$-0.187500\pi$
$30$		0			0
$31$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$31$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$32$			1.00000i			1.00000i
$33$		0			0
$34$	−1.17588	+	1.17588i	−1.17588	+	1.17588i
$35$		0			0
$36$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$37$	−1.68789	+	0.512016i	−1.68789	+	0.512016i	−0.980785	−	0.195090i	$-0.937500\pi$
$37$	−1.68789	+	0.512016i	−1.68789	+	0.512016i	−0.707107	+	0.707107i	$0.750000\pi$
$38$		0			0
$39$		0			0
$40$	0.172887	−	1.75535i	0.172887	−	1.75535i
$41$	−0.172887	−	0.0924099i	−0.172887	−	0.0924099i	0.382683	−	0.923880i	$-0.375000\pi$
$41$	−0.172887	−	0.0924099i	−0.172887	−	0.0924099i	−0.555570	+	0.831470i	$0.687500\pi$
$42$		0			0
$43$		0			0		−0.634393	−	0.773010i	$-0.718750\pi$
$43$		0			0		0.634393	+	0.773010i	$0.281250\pi$
$44$		0			0
$45$	0.172887	+	1.75535i	0.172887	+	1.75535i
$46$		0			0
$47$		0			0		−0.471397	−	0.881921i	$-0.656250\pi$
$47$		0			0		0.471397	+	0.881921i	$0.343750\pi$
$48$		0			0
$49$	−0.831470	−	0.555570i	−0.831470	−	0.555570i
$50$	−0.411863	+	2.07058i	−0.411863	+	2.07058i
$51$		0			0
$52$	1.63099	+	1.08979i	1.63099	+	1.08979i
$53$	0.0924099	+	0.938254i	0.0924099	+	0.938254i	0.923880	+	0.382683i	$0.125000\pi$
$53$	0.0924099	+	0.938254i	0.0924099	+	0.938254i	−0.831470	+	0.555570i	$0.812500\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	1.38704	+	0.275899i	1.38704	+	0.275899i
$59$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$59$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$60$		0			0
$61$	−0.636379	−	0.425215i	−0.636379	−	0.425215i	0.195090	−	0.980785i	$-0.437500\pi$
$61$	−0.636379	−	0.425215i	−0.636379	−	0.425215i	−0.831470	+	0.555570i	$0.812500\pi$
$62$		0			0
$63$		0			0
$64$	−1.00000			−1.00000
$65$	−2.67454	−	2.19494i	−2.67454	−	2.19494i
$66$		0			0
$67$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$67$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$68$	−1.17588	−	1.17588i	−1.17588	−	1.17588i
$69$		0			0
$70$		0			0
$71$		0			0		0.471397	−	0.881921i	$-0.343750\pi$
$71$		0			0		−0.471397	+	0.881921i	$0.656250\pi$
$72$	0.980785	−	0.195090i	0.980785	−	0.195090i
$73$	−0.149316	+	0.360480i	−0.149316	+	0.360480i	−0.980785	−	0.195090i	$-0.937500\pi$
$73$	−0.149316	+	0.360480i	−0.149316	+	0.360480i	0.831470	+	0.555570i	$0.187500\pi$
$74$	−0.512016	−	1.68789i	−0.512016	−	1.68789i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$79$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$80$	1.75535	+	0.172887i	1.75535	+	0.172887i
$81$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$82$	0.0924099	−	0.172887i	0.0924099	−	0.172887i
$83$		0			0		−0.634393	−	0.773010i	$-0.718750\pi$
$83$		0			0		0.634393	+	0.773010i	$0.281250\pi$
$84$		0			0
$85$	1.86078	+	2.26737i	1.86078	+	2.26737i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.216773	+	0.324423i	0.216773	+	0.324423i	0.923880	−	0.382683i	$-0.125000\pi$
$89$	0.216773	+	0.324423i	0.216773	+	0.324423i	−0.707107	+	0.707107i	$0.750000\pi$
$90$	−1.75535	+	0.172887i	−1.75535	+	0.172887i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.124363	−	1.26268i	0.124363	−	1.26268i	−0.707107	−	0.707107i	$-0.750000\pi$
$97$	0.124363	−	1.26268i	0.124363	−	1.26268i	0.831470	−	0.555570i	$-0.187500\pi$
$98$	0.555570	−	0.831470i	0.555570	−	0.831470i
$99$		0			0
$100$	−2.07058	−	0.411863i	−2.07058	−	0.411863i
$101$	−0.577774	−	1.90466i	−0.577774	−	1.90466i	−0.382683	−	0.923880i	$-0.625000\pi$
$101$	−0.577774	−	1.90466i	−0.577774	−	1.90466i	−0.195090	−	0.980785i	$-0.562500\pi$
$102$		0			0
$103$		0			0		0.995185	−	0.0980171i	$-0.0312500\pi$
$103$		0			0		−0.995185	+	0.0980171i	$0.968750\pi$
$104$	−1.08979	+	1.63099i	−1.08979	+	1.63099i
$105$		0			0
$106$	−0.938254	+	0.0924099i	−0.938254	+	0.0924099i
$107$		0			0		0.773010	−	0.634393i	$-0.218750\pi$
$107$		0			0		−0.773010	+	0.634393i	$0.781250\pi$
$108$		0			0
$109$	1.47945	−	0.448786i	1.47945	−	0.448786i	0.555570	−	0.831470i	$-0.312500\pi$
$109$	1.47945	−	0.448786i	1.47945	−	0.448786i	0.923880	+	0.382683i	$0.125000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.382683	−	1.92388i	−0.382683	−	1.92388i	−0.382683	−	0.923880i	$-0.625000\pi$
$113$	−0.382683	−	1.92388i	−0.382683	−	1.92388i		−	1.00000i	$-0.5\pi$
$114$		0			0
$115$		0			0
$116$	−0.275899	+	1.38704i	−0.275899	+	1.38704i
$117$	0.750661	−	1.81225i	0.750661	−	1.81225i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.707107	−	0.707107i	0.707107	−	0.707107i
$122$	0.425215	−	0.636379i	0.425215	−	0.636379i
$123$		0			0
$124$		0			0
$125$	1.87549	+	0.568922i	1.87549	+	0.568922i
$126$		0			0
$127$		0			0		0.634393	−	0.773010i	$-0.281250\pi$
$127$		0			0		−0.634393	+	0.773010i	$0.718750\pi$
$128$		−	1.00000i		−	1.00000i
$129$		0			0
$130$	2.19494	−	2.67454i	2.19494	−	2.67454i
$131$		0			0		0.0980171	−	0.995185i	$-0.468750\pi$
$131$		0			0		−0.0980171	+	0.995185i	$0.531250\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	1.17588	−	1.17588i	1.17588	−	1.17588i
$137$	−1.17588	+	1.17588i	−1.17588	+	1.17588i	−0.195090	+	0.980785i	$0.562500\pi$
$137$	−1.17588	+	1.17588i	−1.17588	+	1.17588i	−0.980785	+	0.195090i	$0.937500\pi$
$138$		0			0
$139$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$139$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.195090	+	0.980785i	0.195090	+	0.980785i
$145$	0.724101	−	2.38704i	0.724101	−	2.38704i
$146$	−0.360480	−	0.149316i	−0.360480	−	0.149316i
$147$		0			0
$148$	1.68789	−	0.512016i	1.68789	−	0.512016i
$149$	0.598102	−	1.11897i	0.598102	−	1.11897i	−0.382683	−	0.923880i	$-0.625000\pi$
$149$	0.598102	−	1.11897i	0.598102	−	1.11897i	0.980785	−	0.195090i	$-0.0625000\pi$
$150$		0			0
$151$		0			0		0.995185	−	0.0980171i	$-0.0312500\pi$
$151$		0			0		−0.995185	+	0.0980171i	$0.968750\pi$
$152$		0			0
$153$	−0.923880	+	1.38268i	−0.923880	+	1.38268i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$157$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$158$		0			0
$159$		0			0
$160$	−0.172887	+	1.75535i	−0.172887	+	1.75535i
$161$		0			0
$162$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$163$		0			0		0.881921	−	0.471397i	$-0.156250\pi$
$163$		0			0		−0.881921	+	0.471397i	$0.843750\pi$
$164$	0.172887	+	0.0924099i	0.172887	+	0.0924099i
$165$		0			0
$166$		0			0
$167$		0			0		0.995185	−	0.0980171i	$-0.0312500\pi$
$167$		0			0		−0.995185	+	0.0980171i	$0.968750\pi$
$168$		0			0
$169$	1.08979	+	2.63099i	1.08979	+	2.63099i
$170$	−2.26737	+	1.86078i	−2.26737	+	1.86078i
$171$		0			0
$172$		0			0
$173$	−1.53636	+	1.02656i	−1.53636	+	1.02656i	−0.555570	+	0.831470i	$0.687500\pi$
$173$	−1.53636	+	1.02656i	−1.53636	+	1.02656i	−0.980785	+	0.195090i	$0.937500\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−0.324423	+	0.216773i	−0.324423	+	0.216773i
$179$		0			0		0.956940	−	0.290285i	$-0.0937500\pi$
$179$		0			0		−0.956940	+	0.290285i	$0.906250\pi$
$180$	−0.172887	−	1.75535i	−0.172887	−	1.75535i
$181$	0.273678	+	0.512016i	0.273678	+	0.512016i	0.980785	−	0.195090i	$-0.0625000\pi$
$181$	0.273678	+	0.512016i	0.273678	+	0.512016i	−0.707107	+	0.707107i	$0.750000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.05136	+	0.606953i	−3.05136	+	0.606953i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.290285	−	0.956940i	$-0.406250\pi$
$191$		0			0		−0.290285	+	0.956940i	$0.593750\pi$
$192$		0			0
$193$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$193$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$194$	1.26268	+	0.124363i	1.26268	+	0.124363i
$195$		0			0
$196$	0.831470	+	0.555570i	0.831470	+	0.555570i
$197$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$197$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$198$		0			0
$199$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$199$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$200$	0.411863	−	2.07058i	0.411863	−	2.07058i
$201$		0			0
$202$	1.90466	−	0.577774i	1.90466	−	0.577774i
$203$		0			0
$204$		0			0
$205$	−0.287500	−	0.192102i	−0.287500	−	0.192102i
$206$		0			0
$207$		0			0
$208$	−1.63099	−	1.08979i	−1.63099	−	1.08979i
$209$		0			0
$210$		0			0
$211$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$211$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$212$	−0.0924099	−	0.938254i	−0.0924099	−	0.938254i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.448786	+	1.47945i	0.448786	+	1.47945i
$219$		0			0
$220$		0			0
$221$	−0.636379	−	3.19929i	−0.636379	−	3.19929i
$222$		0			0
$223$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$223$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$224$		0			0
$225$			2.11114i			2.11114i
$226$	1.92388	−	0.382683i	1.92388	−	0.382683i
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$	−0.555570	+	1.83147i	−0.555570	+	1.83147i			1.00000i	$0.5\pi$
$229$	−0.555570	+	1.83147i	−0.555570	+	1.83147i	−0.555570	+	0.831470i	$0.687500\pi$
$230$		0			0
$231$		0			0
$232$	−1.38704	−	0.275899i	−1.38704	−	0.275899i
$233$	−0.368309	+	0.448786i	−0.368309	+	0.448786i	−0.923880	−	0.382683i	$-0.875000\pi$
$233$	−0.368309	+	0.448786i	−0.368309	+	0.448786i	0.555570	+	0.831470i	$0.312500\pi$
$234$	1.81225	+	0.750661i	1.81225	+	0.750661i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$239$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$240$		0			0
$241$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$241$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$242$	0.707107	+	0.707107i	0.707107	+	0.707107i
$243$		0			0
$244$	0.636379	+	0.425215i	0.636379	+	0.425215i
$245$	−1.36347	−	1.11897i	−1.36347	−	1.11897i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−0.568922	+	1.87549i	−0.568922	+	1.87549i
$251$		0			0		0.634393	−	0.773010i	$-0.281250\pi$
$251$		0			0		−0.634393	+	0.773010i	$0.718750\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1.00000			1.00000
$257$	−1.00000			−1.00000
$257$	−1.00000			−1.00000
$258$		0			0
$259$		0			0
$260$	2.67454	+	2.19494i	2.67454	+	2.19494i
$261$	1.41421			1.41421
$262$		0			0
$263$		0			0		0.634393	−	0.773010i	$-0.281250\pi$
$263$		0			0		−0.634393	+	0.773010i	$0.718750\pi$
$264$		0			0
$265$			1.66294i			1.66294i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.448786	−	0.368309i	−0.448786	−	0.368309i	0.382683	−	0.923880i	$-0.375000\pi$
$269$	−0.448786	−	0.368309i	−0.448786	−	0.368309i	−0.831470	+	0.555570i	$0.812500\pi$
$270$		0			0
$271$		0			0		−0.956940	−	0.290285i	$-0.906250\pi$
$271$		0			0		0.956940	+	0.290285i	$0.0937500\pi$
$272$	1.17588	+	1.17588i	1.17588	+	1.17588i
$273$		0			0
$274$	−1.17588	−	1.17588i	−1.17588	−	1.17588i
$275$		0			0
$276$		0			0
$277$	−0.577774	−	0.0569057i	−0.577774	−	0.0569057i	−0.195090	−	0.980785i	$-0.562500\pi$
$277$	−0.577774	−	0.0569057i	−0.577774	−	0.0569057i	−0.382683	+	0.923880i	$0.625000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.804910	−	0.980785i	0.804910	−	0.980785i	−0.195090	−	0.980785i	$-0.562500\pi$
$281$	0.804910	−	0.980785i	0.804910	−	0.980785i	1.00000			$0$
$282$		0			0
$283$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$283$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.980785	+	0.195090i	−0.980785	+	0.195090i
$289$			1.76537i			1.76537i
$290$	2.38704	+	0.724101i	2.38704	+	0.724101i
$291$		0			0
$292$	0.149316	−	0.360480i	0.149316	−	0.360480i
$293$	0.382683	+	1.92388i	0.382683	+	1.92388i	0.382683	+	0.923880i	$0.375000\pi$
$293$	0.382683	+	1.92388i	0.382683	+	1.92388i			1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$	0.512016	+	1.68789i	0.512016	+	1.68789i
$297$		0			0
$298$	1.11897	+	0.598102i	1.11897	+	0.598102i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.04355	−	0.856422i	−1.04355	−	0.856422i
$306$	−1.38268	−	0.923880i	−1.38268	−	0.923880i
$307$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$307$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.881921	−	0.471397i	$-0.843750\pi$
$311$		0			0		0.881921	+	0.471397i	$0.156250\pi$
$312$		0			0
$313$	1.21415	+	0.368309i	1.21415	+	0.368309i	0.831470	−	0.555570i	$-0.187500\pi$
$313$	1.21415	+	0.368309i	1.21415	+	0.368309i	0.382683	+	0.923880i	$0.375000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$	−1.75535	−	0.172887i	−1.75535	−	0.172887i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.923880	−	0.382683i	0.923880	−	0.382683i
$325$	−2.92824	−	2.92824i	−2.92824	−	2.92824i
$326$		0			0
$327$		0			0
$328$	−0.0924099	+	0.172887i	−0.0924099	+	0.172887i
$329$		0			0
$330$		0			0
$331$		0			0		−0.290285	−	0.956940i	$-0.593750\pi$
$331$		0			0		0.290285	+	0.956940i	$0.406250\pi$
$332$		0			0
$333$	−0.831470	−	1.55557i	−0.831470	−	1.55557i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.98079	−	0.195090i	−1.98079	−	0.195090i	−0.980785	−	0.195090i	$-0.937500\pi$
$337$	−1.98079	−	0.195090i	−1.98079	−	0.195090i	−1.00000			$\pi$
$338$	−2.63099	+	1.08979i	−2.63099	+	1.08979i
$339$		0			0
$340$	−1.86078	−	2.26737i	−1.86078	−	2.26737i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−1.02656	−	1.53636i	−1.02656	−	1.53636i
$347$		0			0		0.995185	−	0.0980171i	$-0.0312500\pi$
$347$		0			0		−0.995185	+	0.0980171i	$0.968750\pi$
$348$		0			0
$349$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$349$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.124363	+	0.151537i	−0.124363	+	0.151537i	−0.831470	−	0.555570i	$-0.812500\pi$
$353$	−0.124363	+	0.151537i	−0.124363	+	0.151537i	0.707107	+	0.707107i	$0.250000\pi$
$354$		0			0
$355$		0			0
$356$	−0.216773	−	0.324423i	−0.216773	−	0.324423i
$357$		0			0
$358$		0			0
$359$		0			0		0.995185	−	0.0980171i	$-0.0312500\pi$
$359$		0			0		−0.995185	+	0.0980171i	$0.968750\pi$
$360$	1.75535	−	0.172887i	1.75535	−	0.172887i
$361$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$362$	−0.512016	+	0.273678i	−0.512016	+	0.273678i
$363$		0			0
$364$		0			0
$365$	−0.324423	+	0.606953i	−0.324423	+	0.606953i
$366$		0			0
$367$		0			0		−0.290285	−	0.956940i	$-0.593750\pi$
$367$		0			0		0.290285	+	0.956940i	$0.406250\pi$
$368$		0			0
$369$	0.0569057	−	0.187593i	0.0569057	−	0.187593i
$370$	−0.606953	−	3.05136i	−0.606953	−	3.05136i
$371$		0			0
$372$		0			0
$373$	0.360480	−	1.81225i	0.360480	−	1.81225i	−0.195090	−	0.980785i	$-0.562500\pi$
$373$	0.360480	−	1.81225i	0.360480	−	1.81225i	0.555570	−	0.831470i	$-0.312500\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.96157	+	1.96157i	−1.96157	+	1.96157i
$378$		0			0
$379$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$379$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.0980171	−	0.995185i	$-0.468750\pi$
$383$		0			0		−0.0980171	+	0.995185i	$0.531250\pi$
$384$		0			0
$385$		0			0
$386$			1.84776i			1.84776i
$387$		0			0
$388$	−0.124363	+	1.26268i	−0.124363	+	1.26268i
$389$	1.90466	+	0.577774i	1.90466	+	0.577774i	0.980785	+	0.195090i	$0.0625000\pi$
$389$	1.90466	+	0.577774i	1.90466	+	0.577774i	0.923880	+	0.382683i	$0.125000\pi$
$390$		0			0
$391$		0			0
$392$	−0.555570	+	0.831470i	−0.555570	+	0.831470i
$393$		0			0
$394$	1.00000	−	1.00000i	1.00000	−	1.00000i
$395$		0			0
$396$		0			0
$397$	−0.425215	+	1.02656i	−0.425215	+	1.02656i	0.555570	+	0.831470i	$0.312500\pi$
$397$	−0.425215	+	1.02656i	−0.425215	+	1.02656i	−0.980785	+	0.195090i	$0.937500\pi$
$398$		0			0
$399$		0			0
$400$	2.07058	+	0.411863i	2.07058	+	0.411863i
$401$	0.0761205	+	0.382683i	0.0761205	+	0.382683i	1.00000			$0$
$401$	0.0761205	+	0.382683i	0.0761205	+	0.382683i	−0.923880	+	0.382683i	$0.875000\pi$
$402$		0			0
$403$		0			0
$404$	0.577774	+	1.90466i	0.577774	+	1.90466i
$405$	−1.68789	+	0.512016i	−1.68789	+	0.512016i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.831470	+	0.444430i	−0.831470	+	0.444430i	−0.831470	−	0.555570i	$-0.812500\pi$
$409$	−0.831470	+	0.444430i	−0.831470	+	0.444430i			1.00000i	$0.5\pi$
$410$	0.192102	−	0.287500i	0.192102	−	0.287500i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	1.08979	−	1.63099i	1.08979	−	1.63099i
$417$		0			0
$418$		0			0
$419$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$419$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$420$		0			0
$421$	1.11897	+	0.598102i	1.11897	+	0.598102i	0.923880	−	0.382683i	$-0.125000\pi$
$421$	1.11897	+	0.598102i	1.11897	+	0.598102i	0.195090	+	0.980785i	$0.437500\pi$
$422$		0			0
$423$		0			0
$424$	0.938254	−	0.0924099i	0.938254	−	0.0924099i
$425$	1.95044	+	2.91904i	1.95044	+	2.91904i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.634393	−	0.773010i	$-0.718750\pi$
$431$		0			0		0.634393	+	0.773010i	$0.281250\pi$
$432$		0			0
$433$	−0.360480	+	0.149316i	−0.360480	+	0.149316i	−0.555570	−	0.831470i	$-0.687500\pi$
$433$	−0.360480	+	0.149316i	−0.360480	+	0.149316i	0.195090	+	0.980785i	$0.437500\pi$
$434$		0			0
$435$		0			0
$436$	−1.47945	+	0.448786i	−1.47945	+	0.448786i
$437$		0			0
$438$		0			0
$439$		0			0		−0.634393	−	0.773010i	$-0.718750\pi$
$439$		0			0		0.634393	+	0.773010i	$0.281250\pi$
$440$		0			0
$441$	0.382683	−	0.923880i	0.382683	−	0.923880i
$442$	3.19929	−	0.636379i	3.19929	−	0.636379i
$443$		0			0		0.471397	−	0.881921i	$-0.343750\pi$
$443$		0			0		−0.471397	+	0.881921i	$0.656250\pi$
$444$		0			0
$445$	0.324423	+	0.606953i	0.324423	+	0.606953i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.53858	+	1.26268i	1.53858	+	1.26268i	0.831470	+	0.555570i	$0.187500\pi$
$449$	1.53858	+	1.26268i	1.53858	+	1.26268i	0.707107	+	0.707107i	$0.250000\pi$
$450$	−2.11114			−2.11114
$451$		0			0
$452$	0.382683	+	1.92388i	0.382683	+	1.92388i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.390181	+	1.96157i	−0.390181	+	1.96157i	−0.195090	+	0.980785i	$0.562500\pi$
$457$	−0.390181	+	1.96157i	−0.390181	+	1.96157i	−0.195090	+	0.980785i	$0.562500\pi$
$458$	−1.83147	−	0.555570i	−1.83147	−	0.555570i
$459$		0			0
$460$		0			0
$461$	−0.0924099	−	0.938254i	−0.0924099	−	0.938254i	−0.923880	−	0.382683i	$-0.875000\pi$
$461$	−0.0924099	−	0.938254i	−0.0924099	−	0.938254i	0.831470	−	0.555570i	$-0.187500\pi$
$462$		0			0
$463$		0			0		−0.0980171	−	0.995185i	$-0.531250\pi$
$463$		0			0		0.0980171	+	0.995185i	$0.468750\pi$
$464$	0.275899	−	1.38704i	0.275899	−	1.38704i
$465$		0			0
$466$	−0.448786	−	0.368309i	−0.448786	−	0.368309i
$467$		0			0		−0.471397	−	0.881921i	$-0.656250\pi$
$467$		0			0		0.471397	+	0.881921i	$0.343750\pi$
$468$	−0.750661	+	1.81225i	−0.750661	+	1.81225i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.902197	+	0.273678i	−0.902197	+	0.273678i
$478$		0			0
$479$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$479$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$480$		0			0
$481$	3.31092	+	1.00436i	3.31092	+	1.00436i
$482$			0.765367i			0.765367i
$483$		0			0
$484$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$485$	0.436600	−	2.19494i	0.436600	−	2.19494i
$486$		0			0
$487$		0			0		0.471397	−	0.881921i	$-0.343750\pi$
$487$		0			0		−0.471397	+	0.881921i	$0.656250\pi$
$488$	−0.425215	+	0.636379i	−0.425215	+	0.636379i
$489$		0			0
$490$	1.11897	−	1.36347i	1.11897	−	1.36347i
$491$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$491$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$492$		0			0
$493$	1.95541	−	1.30656i	1.95541	−	1.30656i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$499$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$500$	−1.87549	−	0.568922i	−1.87549	−	0.568922i
$501$		0			0
$502$		0			0
$503$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$503$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$504$		0			0
$505$	−0.684903	−	3.44324i	−0.684903	−	3.44324i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.90466	+	0.187593i	1.90466	+	0.187593i	0.980785	−	0.195090i	$-0.0625000\pi$
$509$	1.90466	+	0.187593i	1.90466	+	0.187593i	0.923880	+	0.382683i	$0.125000\pi$
$510$		0			0
$511$		0			0
$512$			1.00000i			1.00000i
$513$		0			0
$514$		−	1.00000i		−	1.00000i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−2.19494	+	2.67454i	−2.19494	+	2.67454i
$521$	−0.448786	+	1.47945i	−0.448786	+	1.47945i	0.382683	+	0.923880i	$0.375000\pi$
$521$	−0.448786	+	1.47945i	−0.448786	+	1.47945i	−0.831470	+	0.555570i	$0.812500\pi$
$522$			1.41421i			1.41421i
$523$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$523$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.707107	+	0.707107i	0.707107	+	0.707107i
$530$	−1.66294			−1.66294
$531$		0			0
$532$		0			0
$533$	0.181269	+	0.339130i	0.181269	+	0.339130i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	0.368309	−	0.448786i	0.368309	−	0.448786i
$539$		0			0
$540$		0			0
$541$	−0.938254	+	1.75535i	−0.938254	+	1.75535i	−0.382683	+	0.923880i	$0.625000\pi$
$541$	−0.938254	+	1.75535i	−0.938254	+	1.75535i	−0.555570	+	0.831470i	$0.687500\pi$
$542$		0			0
$543$		0			0
$544$	−1.17588	+	1.17588i	−1.17588	+	1.17588i
$545$	2.67454	−	0.531999i	2.67454	−	0.531999i
$546$		0			0
$547$		0			0		−0.956940	−	0.290285i	$-0.906250\pi$
$547$		0			0		0.956940	+	0.290285i	$0.0937500\pi$
$548$	1.17588	−	1.17588i	1.17588	−	1.17588i
$549$	0.292893	−	0.707107i	0.292893	−	0.707107i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.0569057	−	0.577774i	0.0569057	−	0.577774i
$555$		0			0
$556$		0			0
$557$	1.21415	+	1.47945i	1.21415	+	1.47945i	0.831470	+	0.555570i	$0.187500\pi$
$557$	1.21415	+	1.47945i	1.21415	+	1.47945i	0.382683	+	0.923880i	$0.375000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.980785	+	0.804910i	0.980785	+	0.804910i
$563$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$563$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$564$		0			0
$565$	−0.339130	−	3.44324i	−0.339130	−	3.44324i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.47945	+	0.448786i	−1.47945	+	0.448786i	−0.923880	−	0.382683i	$-0.875000\pi$
$569$	−1.47945	+	0.448786i	−1.47945	+	0.448786i	−0.555570	+	0.831470i	$0.687500\pi$
$570$		0			0
$571$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$571$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.195090	−	0.980785i	−0.195090	−	0.980785i
$577$	−1.98079	−	0.195090i	−1.98079	−	0.195090i	−0.980785	−	0.195090i	$-0.937500\pi$
$577$	−1.98079	−	0.195090i	−1.98079	−	0.195090i	−1.00000			$\pi$
$578$	−1.76537			−1.76537
$579$		0			0
$580$	−0.724101	+	2.38704i	−0.724101	+	2.38704i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.360480	+	0.149316i	0.360480	+	0.149316i
$585$	1.63099	−	3.05136i	1.63099	−	3.05136i
$586$	−1.92388	+	0.382683i	−1.92388	+	0.382683i
$587$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$587$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.68789	+	0.512016i	−1.68789	+	0.512016i
$593$	1.53636	−	1.02656i	1.53636	−	1.02656i	0.555570	−	0.831470i	$-0.312500\pi$
$593$	1.53636	−	1.02656i	1.53636	−	1.02656i	0.980785	−	0.195090i	$-0.0625000\pi$
$594$		0			0
$595$		0			0
$596$	−0.598102	+	1.11897i	−0.598102	+	1.11897i
$597$		0			0
$598$		0			0
$599$		0			0		−0.634393	−	0.773010i	$-0.718750\pi$
$599$		0			0		0.634393	+	0.773010i	$0.281250\pi$
$600$		0			0
$601$	0.448786	−	0.368309i	0.448786	−	0.368309i	−0.382683	−	0.923880i	$-0.625000\pi$
$601$	0.448786	−	0.368309i	0.448786	−	0.368309i	0.831470	+	0.555570i	$0.187500\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.36347	−	1.11897i	1.36347	−	1.11897i
$606$		0			0
$607$		0			0		−0.881921	−	0.471397i	$-0.843750\pi$
$607$		0			0		0.881921	+	0.471397i	$0.156250\pi$
$608$		0			0
$609$		0			0
$610$	0.856422	−	1.04355i	0.856422	−	1.04355i
$611$		0			0
$612$	0.923880	−	1.38268i	0.923880	−	1.38268i
$613$	1.02656	+	1.53636i	1.02656	+	1.53636i	0.831470	+	0.555570i	$0.187500\pi$
$613$	1.02656	+	1.53636i	1.02656	+	1.53636i	0.195090	+	0.980785i	$0.437500\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.90466	−	0.187593i	1.90466	−	0.187593i	0.923880	−	0.382683i	$-0.125000\pi$
$617$	1.90466	−	0.187593i	1.90466	−	0.187593i	0.980785	+	0.195090i	$0.0625000\pi$
$618$		0			0
$619$		0			0		0.881921	−	0.471397i	$-0.156250\pi$
$619$		0			0		−0.881921	+	0.471397i	$0.843750\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.24333	+	0.515005i	1.24333	+	0.515005i
$626$	−0.368309	+	1.21415i	−0.368309	+	1.21415i
$627$		0			0
$628$		0			0
$629$	−2.58682	−	1.38268i	−2.58682	−	1.38268i
$630$		0			0
$631$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$631$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.750661	+	1.81225i	0.750661	+	1.81225i
$638$		0			0
$639$		0			0
$640$	0.172887	−	1.75535i	0.172887	−	1.75535i
$641$	−1.21415	+	1.47945i	−1.21415	+	1.47945i	−0.382683	+	0.923880i	$0.625000\pi$
$641$	−1.21415	+	1.47945i	−1.21415	+	1.47945i	−0.831470	+	0.555570i	$0.812500\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$647$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$648$	0.382683	+	0.923880i	0.382683	+	0.923880i
$649$		0			0
$650$	2.92824	−	2.92824i	2.92824	−	2.92824i
$651$		0			0
$652$		0			0
$653$	−1.11114	−	1.66294i	−1.11114	−	1.66294i	−0.555570	−	0.831470i	$-0.687500\pi$
$653$	−1.11114	−	1.66294i	−1.11114	−	1.66294i	−0.555570	−	0.831470i	$-0.687500\pi$
$654$		0			0
$655$		0			0
$656$	−0.172887	−	0.0924099i	−0.172887	−	0.0924099i
$657$	−0.382683	−	0.0761205i	−0.382683	−	0.0761205i
$658$		0			0
$659$		0			0		0.290285	−	0.956940i	$-0.406250\pi$
$659$		0			0		−0.290285	+	0.956940i	$0.593750\pi$
$660$		0			0
$661$	−0.555570	−	1.83147i	−0.555570	−	1.83147i	−0.555570	−	0.831470i	$-0.687500\pi$
$661$	−0.555570	−	1.83147i	−0.555570	−	1.83147i		−	1.00000i	$-0.5\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	1.55557	−	0.831470i	1.55557	−	0.831470i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.08979	+	1.63099i	−1.08979	+	1.63099i	−0.382683	+	0.923880i	$0.625000\pi$
$673$	−1.08979	+	1.63099i	−1.08979	+	1.63099i	−0.707107	+	0.707107i	$0.750000\pi$
$674$	0.195090	−	1.98079i	0.195090	−	1.98079i
$675$		0			0
$676$	−1.08979	−	2.63099i	−1.08979	−	2.63099i
$677$	−1.36347	+	0.728789i	−1.36347	+	0.728789i	−0.980785	−	0.195090i	$-0.937500\pi$
$677$	−1.36347	+	0.728789i	−1.36347	+	0.728789i	−0.382683	+	0.923880i	$0.625000\pi$
$678$		0			0
$679$		0			0
$680$	2.26737	−	1.86078i	2.26737	−	1.86078i
$681$		0			0
$682$		0			0
$683$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$683$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$684$		0			0
$685$	−2.26737	+	1.86078i	−2.26737	+	1.86078i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.871780	−	1.63099i	0.871780	−	1.63099i
$690$		0			0
$691$		0			0		−0.995185	−	0.0980171i	$-0.968750\pi$
$691$		0			0		0.995185	+	0.0980171i	$0.0312500\pi$
$692$	1.53636	−	1.02656i	1.53636	−	1.02656i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.0946308	−	0.311956i	−0.0946308	−	0.311956i
$698$	0.541196	−	1.30656i	0.541196	−	1.30656i
$699$		0			0
$700$		0			0
$701$	1.53636	+	0.636379i	1.53636	+	0.636379i	0.980785	−	0.195090i	$-0.0625000\pi$
$701$	1.53636	+	0.636379i	1.53636	+	0.636379i	0.555570	+	0.831470i	$0.312500\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.151537	−	0.124363i	−0.151537	−	0.124363i
$707$		0			0
$708$		0			0
$709$	0.149316	+	0.750661i	0.149316	+	0.750661i	0.980785	+	0.195090i	$0.0625000\pi$
$709$	0.149316	+	0.750661i	0.149316	+	0.750661i	−0.831470	+	0.555570i	$0.812500\pi$
$710$		0			0
$711$		0			0
$712$	0.324423	−	0.216773i	0.324423	−	0.216773i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$719$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$720$	0.172887	+	1.75535i	0.172887	+	1.75535i
$721$		0			0
$722$	−0.831470	−	0.555570i	−0.831470	−	0.555570i
$723$		0			0
$724$	−0.273678	−	0.512016i	−0.273678	−	0.512016i
$725$	1.14254	−	2.75834i	1.14254	−	2.75834i
$726$		0			0
$727$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$727$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$728$		0			0
$729$	−0.555570	−	0.831470i	−0.555570	−	0.831470i
$730$	−0.606953	−	0.324423i	−0.606953	−	0.324423i
$731$		0			0
$732$		0			0
$733$	1.36347	−	0.728789i	1.36347	−	0.728789i	0.382683	−	0.923880i	$-0.375000\pi$
$733$	1.36347	−	0.728789i	1.36347	−	0.728789i	0.980785	+	0.195090i	$0.0625000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	0.187593	+	0.0569057i	0.187593	+	0.0569057i
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$	3.05136	−	0.606953i	3.05136	−	0.606953i
$741$		0			0
$742$		0			0
$743$		0			0		0.290285	−	0.956940i	$-0.406250\pi$
$743$		0			0		−0.290285	+	0.956940i	$0.593750\pi$
$744$		0			0
$745$	1.24333	−	1.86078i	1.24333	−	1.86078i
$746$	1.81225	+	0.360480i	1.81225	+	0.360480i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.995185	−	0.0980171i	$-0.968750\pi$
$751$		0			0		0.995185	+	0.0980171i	$0.0312500\pi$
$752$		0			0
$753$		0			0
$754$	−1.96157	−	1.96157i	−1.96157	−	1.96157i
$755$		0			0
$756$		0			0
$757$	−0.187593	−	0.0569057i	−0.187593	−	0.0569057i	0.195090	−	0.980785i	$-0.437500\pi$
$757$	−0.187593	−	0.0569057i	−0.187593	−	0.0569057i	−0.382683	+	0.923880i	$0.625000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.151537	−	1.53858i	0.151537	−	1.53858i	−0.555570	−	0.831470i	$-0.687500\pi$
$761$	0.151537	−	1.53858i	0.151537	−	1.53858i	0.707107	−	0.707107i	$-0.250000\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−1.86078	+	2.26737i	−1.86078	+	2.26737i
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	0.390181i		−	0.390181i	−0.980785	−	0.195090i	$-0.937500\pi$
$769$		−	0.390181i		−	0.390181i	0.980785	−	0.195090i	$-0.0625000\pi$
$770$		0			0
$771$		0			0
$772$	−1.84776			−1.84776
$773$		−	1.11114i		−	1.11114i	−0.831470	−	0.555570i	$-0.812500\pi$
$773$		−	1.11114i		−	1.11114i	0.831470	−	0.555570i	$-0.187500\pi$
$774$		0			0
$775$		0			0
$776$	−1.26268	−	0.124363i	−1.26268	−	0.124363i
$777$		0			0
$778$	−0.577774	+	1.90466i	−0.577774	+	1.90466i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.831470	−	0.555570i	−0.831470	−	0.555570i
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	1.00000	+	1.00000i	1.00000	+	1.00000i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.574531	+	1.38704i	0.574531	+	1.38704i
$794$	−1.02656	−	0.425215i	−1.02656	−	0.425215i
$795$		0			0
$796$		0			0
$797$	1.02656	−	1.53636i	1.02656	−	1.53636i	0.195090	−	0.980785i	$-0.437500\pi$
$797$	1.02656	−	1.53636i	1.02656	−	1.53636i	0.831470	−	0.555570i	$-0.187500\pi$
$798$		0			0
$799$		0			0
$800$	−0.411863	+	2.07058i	−0.411863	+	2.07058i
$801$	−0.275899	+	0.275899i	−0.275899	+	0.275899i
$802$	−0.382683	+	0.0761205i	−0.382683	+	0.0761205i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−1.90466	+	0.577774i	−1.90466	+	0.577774i
$809$	−0.172887	+	0.0924099i	−0.172887	+	0.0924099i	−0.555570	−	0.831470i	$-0.687500\pi$
$809$	−0.172887	+	0.0924099i	−0.172887	+	0.0924099i	0.382683	+	0.923880i	$0.375000\pi$
$810$	−0.512016	−	1.68789i	−0.512016	−	1.68789i
$811$		0			0		0.0980171	−	0.995185i	$-0.468750\pi$
$811$		0			0		−0.0980171	+	0.995185i	$0.531250\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.444430	−	0.831470i	−0.444430	−	0.831470i
$819$		0			0
$820$	0.287500	+	0.192102i	0.287500	+	0.192102i
$821$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i	0.923880	+	0.382683i	$0.125000\pi$
$821$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i	−1.00000			$\pi$
$822$		0			0
$823$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$823$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.956940	−	0.290285i	$-0.906250\pi$
$827$		0			0		0.956940	+	0.290285i	$0.0937500\pi$
$828$		0			0
$829$	1.63099	+	0.324423i	1.63099	+	0.324423i	0.923880	−	0.382683i	$-0.125000\pi$
$829$	1.63099	+	0.324423i	1.63099	+	0.324423i	0.707107	+	0.707107i	$0.250000\pi$
$830$		0			0
$831$		0			0
$832$	1.63099	+	1.08979i	1.63099	+	1.08979i
$833$	−0.324423	−	1.63099i	−0.324423	−	1.63099i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$839$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$840$		0			0
$841$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$842$	−0.598102	+	1.11897i	−0.598102	+	1.11897i
$843$		0			0
$844$		0			0
$845$	1.45810	+	4.80671i	1.45810	+	4.80671i
$846$		0			0
$847$		0			0
$848$	0.0924099	+	0.938254i	0.0924099	+	0.938254i
$849$		0			0
$850$	−2.91904	+	1.95044i	−2.91904	+	1.95044i
$851$		0			0
$852$		0			0
$853$	−0.444430	+	0.831470i	−0.444430	+	0.831470i	0.555570	+	0.831470i	$0.312500\pi$
$853$	−0.444430	+	0.831470i	−0.444430	+	0.831470i	−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.151537	+	0.124363i	−0.151537	+	0.124363i	−0.707107	−	0.707107i	$-0.750000\pi$
$857$	−0.151537	+	0.124363i	−0.151537	+	0.124363i	0.555570	+	0.831470i	$0.312500\pi$
$858$		0			0
$859$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$859$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$863$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$864$		0			0
$865$	−2.87432	+	1.53636i	−2.87432	+	1.53636i
$866$	−0.149316	−	0.360480i	−0.149316	−	0.360480i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−0.448786	−	1.47945i	−0.448786	−	1.47945i
$873$	1.26268	−	0.124363i	1.26268	−	0.124363i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.53858	−	0.151537i	1.53858	−	0.151537i	0.707107	−	0.707107i	$-0.250000\pi$
$877$	1.53858	−	0.151537i	1.53858	−	0.151537i	0.831470	+	0.555570i	$0.187500\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.577774	−	1.90466i	−0.577774	−	1.90466i	−0.382683	−	0.923880i	$-0.625000\pi$
$881$	−0.577774	−	1.90466i	−0.577774	−	1.90466i	−0.195090	−	0.980785i	$-0.562500\pi$
$882$	0.923880	+	0.382683i	0.923880	+	0.382683i
$883$		0			0		0.290285	−	0.956940i	$-0.406250\pi$
$883$		0			0		−0.290285	+	0.956940i	$0.593750\pi$
$884$	0.636379	+	3.19929i	0.636379	+	3.19929i
$885$		0			0
$886$		0			0
$887$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$887$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$888$		0			0
$889$		0			0
$890$	−0.606953	+	0.324423i	−0.606953	+	0.324423i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−1.26268	+	1.53858i	−1.26268	+	1.53858i
$899$		0			0
$900$		−	2.11114i		−	2.11114i
$901$	−0.994607	+	1.21193i	−0.994607	+	1.21193i
$902$		0			0
$903$		0			0
$904$	−1.92388	+	0.382683i	−1.92388	+	0.382683i
$905$	0.391880	+	0.946083i	0.391880	+	0.946083i
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$	1.75535	−	0.938254i	1.75535	−	0.938254i
$910$		0			0
$911$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$911$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$912$		0			0
$913$		0			0
$914$	−1.96157	−	0.390181i	−1.96157	−	0.390181i
$915$		0			0
$916$	0.555570	−	1.83147i	0.555570	−	1.83147i
$917$		0			0
$918$		0			0
$919$		0			0		0.956940	−	0.290285i	$-0.0937500\pi$
$919$		0			0		−0.956940	+	0.290285i	$0.906250\pi$
$920$		0			0
$921$		0			0
$922$	0.938254	−	0.0924099i	0.938254	−	0.0924099i
$923$		0			0
$924$		0			0
$925$	−3.70579	+	0.364988i	−3.70579	+	0.364988i
$926$		0			0
$927$		0			0
$928$	1.38704	+	0.275899i	1.38704	+	0.275899i
$929$	−0.785695	−	1.17588i	−0.785695	−	1.17588i	−0.980785	−	0.195090i	$-0.937500\pi$
$929$	−0.785695	−	1.17588i	−0.785695	−	1.17588i	0.195090	−	0.980785i	$-0.437500\pi$
$930$		0			0
$931$		0			0
$932$	0.368309	−	0.448786i	0.368309	−	0.448786i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−1.81225	−	0.750661i	−1.81225	−	0.750661i
$937$	0.151537	−	0.124363i	0.151537	−	0.124363i	−0.555570	−	0.831470i	$-0.687500\pi$
$937$	0.151537	−	0.124363i	0.151537	−	0.124363i	0.707107	+	0.707107i	$0.250000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.980785	+	0.804910i	−0.980785	+	0.804910i	−0.980785	−	0.195090i	$-0.937500\pi$
$941$	−0.980785	+	0.804910i	−0.980785	+	0.804910i			1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$947$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$948$		0			0
$949$	0.636379	−	0.425215i	0.636379	−	0.425215i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.598102	−	0.728789i	−0.598102	−	0.728789i	0.382683	−	0.923880i	$-0.375000\pi$
$953$	−0.598102	−	0.728789i	−0.598102	−	0.728789i	−0.980785	+	0.195090i	$0.937500\pi$
$954$	−0.273678	−	0.902197i	−0.273678	−	0.902197i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.923880	−	0.382683i	0.923880	−	0.382683i
$962$	−1.00436	+	3.31092i	−1.00436	+	3.31092i
$963$		0			0
$964$	−0.765367			−0.765367
$965$	3.24346	+	0.319453i	3.24346	+	0.319453i
$966$		0			0
$967$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$967$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$968$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$969$		0			0
$970$	2.19494	+	0.436600i	2.19494	+	0.436600i
$971$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$971$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−0.636379	−	0.425215i	−0.636379	−	0.425215i
$977$	0.124363	+	1.26268i	0.124363	+	1.26268i	0.831470	+	0.555570i	$0.187500\pi$
$977$	0.124363	+	1.26268i	0.124363	+	1.26268i	−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		0			0
$980$	1.36347	+	1.11897i	1.36347	+	1.11897i
$981$	0.728789	+	1.36347i	0.728789	+	1.36347i
$982$		0			0
$983$		0			0		−0.0980171	−	0.995185i	$-0.531250\pi$
$983$		0			0		0.0980171	+	0.995185i	$0.468750\pi$
$984$		0			0
$985$	−1.58246	−	1.92824i	−1.58246	−	1.92824i
$986$	1.30656	+	1.95541i	1.30656	+	1.95541i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.956940	−	0.290285i	$-0.0937500\pi$
$991$		0			0		−0.956940	+	0.290285i	$0.906250\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$997$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.1.l.a.15.1	✓	16
4.3	odd	2	CM	1028.1.l.a.15.1	✓	16
257.120	even	32	inner	1028.1.l.a.891.1	yes	16
1028.891	odd	32	inner	1028.1.l.a.891.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.1.l.a.15.1	✓	16	1.1	even	1	trivial
1028.1.l.a.15.1	✓	16	4.3	odd	2	CM
1028.1.l.a.891.1	yes	16	257.120	even	32	inner
1028.1.l.a.891.1	yes	16	1028.891	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:	\( N \)	\(=\)	\( 1028 = 2^{2} \cdot 257 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1028.l (of order \(32\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(1.75535 + 0.172887i) q^{5} -1.00000i q^{8} +(0.195090 + 0.980785i) q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(1.75535 + 0.172887i) q^{5} -1.00000i q^{8} +(0.195090 + 0.980785i) q^{9} +(-0.172887 + 1.75535i) q^{10} +(-1.63099 - 1.08979i) q^{13} +1.00000 q^{16} +(1.17588 + 1.17588i) q^{17} +(-0.980785 + 0.195090i) q^{18} +(-1.75535 - 0.172887i) q^{20} +(2.07058 + 0.411863i) q^{25} +(1.08979 - 1.63099i) q^{26} +(0.275899 - 1.38704i) q^{29} +1.00000i q^{32} +(-1.17588 + 1.17588i) q^{34} +(-0.195090 - 0.980785i) q^{36} +(-1.68789 + 0.512016i) q^{37} +(0.172887 - 1.75535i) q^{40} +(-0.172887 - 0.0924099i) q^{41} +(0.172887 + 1.75535i) q^{45} +(-0.831470 - 0.555570i) q^{49} +(-0.411863 + 2.07058i) q^{50} +(1.63099 + 1.08979i) q^{52} +(0.0924099 + 0.938254i) q^{53} +(1.38704 + 0.275899i) q^{58} +(-0.636379 - 0.425215i) q^{61} -1.00000 q^{64} +(-2.67454 - 2.19494i) q^{65} +(-1.17588 - 1.17588i) q^{68} +(0.980785 - 0.195090i) q^{72} +(-0.149316 + 0.360480i) q^{73} +(-0.512016 - 1.68789i) q^{74} +(1.75535 + 0.172887i) q^{80} +(-0.923880 + 0.382683i) q^{81} +(0.0924099 - 0.172887i) q^{82} +(1.86078 + 2.26737i) q^{85} +(0.216773 + 0.324423i) q^{89} +(-1.75535 + 0.172887i) q^{90} +(0.124363 - 1.26268i) q^{97} +(0.555570 - 0.831470i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 16 * q - 16 * q^4	\( 16 q - 16 q^{4} + 16 q^{16} - 16 q^{64} - 16 q^{65}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^16 - 16 * q^64 - 16 * q^65

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