LMFDB - Embedded newform 1024.2.e.n.769.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,2,Mod(257,1024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1024.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1024.e (of order $4$, degree $2$, not 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:	$8.17668116698$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			769.2
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1024.769
Dual form			1024.2.e.n.257.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 - 1.00000i) q^{3} +(1.41421 + 1.41421i) q^{5} -2.82843i q^{7} +1.00000i q^{9} +(-3.00000 - 3.00000i) q^{11} +(4.24264 - 4.24264i) q^{13} +2.82843 q^{15} +(-3.00000 + 3.00000i) q^{19} +(-2.82843 - 2.82843i) q^{21} -8.48528i q^{23} -1.00000i q^{25} +(4.00000 + 4.00000i) q^{27} +(-1.41421 + 1.41421i) q^{29} +5.65685 q^{31} -6.00000 q^{33} +(4.00000 - 4.00000i) q^{35} +(4.24264 + 4.24264i) q^{37} -8.48528i q^{39} -6.00000i q^{41} +(3.00000 + 3.00000i) q^{43} +(-1.41421 + 1.41421i) q^{45} -1.00000 q^{49} +(1.41421 + 1.41421i) q^{53} -8.48528i q^{55} +6.00000i q^{57} +(1.00000 + 1.00000i) q^{59} +(-4.24264 + 4.24264i) q^{61} +2.82843 q^{63} +12.0000 q^{65} +(-9.00000 + 9.00000i) q^{67} +(-8.48528 - 8.48528i) q^{69} -8.48528i q^{71} +12.0000i q^{73} +(-1.00000 - 1.00000i) q^{75} +(-8.48528 + 8.48528i) q^{77} +5.65685 q^{79} +5.00000 q^{81} +(3.00000 - 3.00000i) q^{83} +2.82843i q^{87} -12.0000i q^{89} +(-12.0000 - 12.0000i) q^{91} +(5.65685 - 5.65685i) q^{93} -8.48528 q^{95} -8.00000 q^{97} +(3.00000 - 3.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 12 q^{11} - 12 q^{19} + 16 q^{27} - 24 q^{33} + 16 q^{35} + 12 q^{43} - 4 q^{49} + 4 q^{59} + 48 q^{65} - 36 q^{67} - 4 q^{75} + 20 q^{81} + 12 q^{83} - 48 q^{91} - 32 q^{97} + 12 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 12 * q^11 - 12 * q^19 + 16 * q^27 - 24 * q^33 + 16 * q^35 + 12 * q^43 - 4 * q^49 + 4 * q^59 + 48 * q^65 - 36 * q^67 - 4 * q^75 + 20 * q^81 + 12 * q^83 - 48 * q^91 - 32 * q^97 + 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times$.

$n$	$5$	$1023$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$

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
$2$		0			0
$3$	1.00000	−	1.00000i	0.577350	−	0.577350i	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	1.00000	−	1.00000i	0.577350	−	0.577350i	0.934172	+	0.356822i	$0.116140\pi$
$4$		0			0
$5$	1.41421	+	1.41421i	0.632456	+	0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$5$	1.41421	+	1.41421i	0.632456	+	0.632456i	−0.316228	+	0.948683i	$0.602416\pi$
$6$		0			0
$7$		−	2.82843i		−	1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$		−	2.82843i		−	1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$	−3.00000	−	3.00000i	−0.904534	−	0.904534i	0.0912903	−	0.995824i	$-0.470901\pi$
$11$	−3.00000	−	3.00000i	−0.904534	−	0.904534i	−0.995824	+	0.0912903i	$0.970901\pi$
$12$		0			0
$13$	4.24264	−	4.24264i	1.17670	−	1.17670i	0.196116	−	0.980581i	$-0.437167\pi$
$13$	4.24264	−	4.24264i	1.17670	−	1.17670i	0.980581	−	0.196116i	$-0.0628330\pi$
$14$		0			0
$15$	2.82843			0.730297
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	−3.00000	+	3.00000i	−0.688247	+	0.688247i	−0.961844	−	0.273597i	$-0.911786\pi$
$19$	−3.00000	+	3.00000i	−0.688247	+	0.688247i	0.273597	+	0.961844i	$0.411786\pi$
$20$		0			0
$21$	−2.82843	−	2.82843i	−0.617213	−	0.617213i
$22$		0			0
$23$		−	8.48528i		−	1.76930i	−0.466252	−	0.884652i	$-0.654396\pi$
$23$		−	8.48528i		−	1.76930i	0.466252	−	0.884652i	$-0.345604\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$	4.00000	+	4.00000i	0.769800	+	0.769800i
$28$		0			0
$29$	−1.41421	+	1.41421i	−0.262613	+	0.262613i	−0.826115	−	0.563502i	$-0.809454\pi$
$29$	−1.41421	+	1.41421i	−0.262613	+	0.262613i	0.563502	+	0.826115i	$0.309454\pi$
$30$		0			0
$31$	5.65685			1.01600			0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685			1.01600			0.508001	+	0.861357i	$0.330385\pi$
$32$		0			0
$33$	−6.00000			−1.04447
$34$		0			0
$35$	4.00000	−	4.00000i	0.676123	−	0.676123i
$36$		0			0
$37$	4.24264	+	4.24264i	0.697486	+	0.697486i	0.963868	−	0.266382i	$-0.0858282\pi$
$37$	4.24264	+	4.24264i	0.697486	+	0.697486i	−0.266382	+	0.963868i	$0.585828\pi$
$38$		0			0
$39$		−	8.48528i		−	1.35873i
$40$		0			0
$41$		−	6.00000i		−	0.937043i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$		−	6.00000i		−	0.937043i	0.883452	−	0.468521i	$-0.155213\pi$
$42$		0			0
$43$	3.00000	+	3.00000i	0.457496	+	0.457496i	0.897833	−	0.440337i	$-0.145141\pi$
$43$	3.00000	+	3.00000i	0.457496	+	0.457496i	−0.440337	+	0.897833i	$0.645141\pi$
$44$		0			0
$45$	−1.41421	+	1.41421i	−0.210819	+	0.210819i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	1.41421	+	1.41421i	0.194257	+	0.194257i	0.797533	−	0.603276i	$-0.206138\pi$
$53$	1.41421	+	1.41421i	0.194257	+	0.194257i	−0.603276	+	0.797533i	$0.706138\pi$
$54$		0			0
$55$		−	8.48528i		−	1.14416i
$56$		0			0
$57$			6.00000i			0.794719i
$58$		0			0
$59$	1.00000	+	1.00000i	0.130189	+	0.130189i	0.769199	−	0.639010i	$-0.220656\pi$
$59$	1.00000	+	1.00000i	0.130189	+	0.130189i	−0.639010	+	0.769199i	$0.720656\pi$
$60$		0			0
$61$	−4.24264	+	4.24264i	−0.543214	+	0.543214i	−0.924470	−	0.381255i	$-0.875492\pi$
$61$	−4.24264	+	4.24264i	−0.543214	+	0.543214i	0.381255	+	0.924470i	$0.375492\pi$
$62$		0			0
$63$	2.82843			0.356348
$64$		0			0
$65$	12.0000			1.48842
$66$		0			0
$67$	−9.00000	+	9.00000i	−1.09952	+	1.09952i	−0.105059	+	0.994466i	$0.533503\pi$
$67$	−9.00000	+	9.00000i	−1.09952	+	1.09952i	−0.994466	+	0.105059i	$0.966497\pi$
$68$		0			0
$69$	−8.48528	−	8.48528i	−1.02151	−	1.02151i
$70$		0			0
$71$		−	8.48528i		−	1.00702i	−0.863990	−	0.503509i	$-0.832042\pi$
$71$		−	8.48528i		−	1.00702i	0.863990	−	0.503509i	$-0.167958\pi$
$72$		0			0
$73$			12.0000i			1.40449i	0.711934	+	0.702247i	$0.247820\pi$
$73$			12.0000i			1.40449i	−0.711934	+	0.702247i	$0.752180\pi$
$74$		0			0
$75$	−1.00000	−	1.00000i	−0.115470	−	0.115470i
$76$		0			0
$77$	−8.48528	+	8.48528i	−0.966988	+	0.966988i
$78$		0			0
$79$	5.65685			0.636446			0.318223	−	0.948016i	$-0.396914\pi$
$79$	5.65685			0.636446			0.318223	+	0.948016i	$0.396914\pi$
$80$		0			0
$81$	5.00000			0.555556
$82$		0			0
$83$	3.00000	−	3.00000i	0.329293	−	0.329293i	−0.523025	−	0.852318i	$-0.675196\pi$
$83$	3.00000	−	3.00000i	0.329293	−	0.329293i	0.852318	+	0.523025i	$0.175196\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			2.82843i			0.303239i
$88$		0			0
$89$		−	12.0000i		−	1.27200i	−0.771690	−	0.635999i	$-0.780588\pi$
$89$		−	12.0000i		−	1.27200i	0.771690	−	0.635999i	$-0.219412\pi$
$90$		0			0
$91$	−12.0000	−	12.0000i	−1.25794	−	1.25794i
$92$		0			0
$93$	5.65685	−	5.65685i	0.586588	−	0.586588i
$94$		0			0
$95$	−8.48528			−0.870572
$96$		0			0
$97$	−8.00000			−0.812277			−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000			−0.812277			−0.406138	+	0.913812i	$0.633125\pi$
$98$		0			0
$99$	3.00000	−	3.00000i	0.301511	−	0.301511i
$100$		0			0
$101$	−7.07107	−	7.07107i	−0.703598	−	0.703598i	0.261583	−	0.965181i	$-0.415755\pi$
$101$	−7.07107	−	7.07107i	−0.703598	−	0.703598i	−0.965181	+	0.261583i	$0.915755\pi$
$102$		0			0
$103$			2.82843i			0.278693i	0.990244	+	0.139347i	$0.0445002\pi$
$103$			2.82843i			0.278693i	−0.990244	+	0.139347i	$0.955500\pi$
$104$		0			0
$105$		−	8.00000i		−	0.780720i
$106$		0			0
$107$	5.00000	+	5.00000i	0.483368	+	0.483368i	0.906206	−	0.422837i	$-0.138966\pi$
$107$	5.00000	+	5.00000i	0.483368	+	0.483368i	−0.422837	+	0.906206i	$0.638966\pi$
$108$		0			0
$109$	−4.24264	+	4.24264i	−0.406371	+	0.406371i	−0.880471	−	0.474100i	$-0.842774\pi$
$109$	−4.24264	+	4.24264i	−0.406371	+	0.406371i	0.474100	+	0.880471i	$0.342774\pi$
$110$		0			0
$111$	8.48528			0.805387
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$	12.0000	−	12.0000i	1.11901	−	1.11901i
$116$		0			0
$117$	4.24264	+	4.24264i	0.392232	+	0.392232i
$118$		0			0
$119$		0			0
$120$		0			0
$121$			7.00000i			0.636364i
$122$		0			0
$123$	−6.00000	−	6.00000i	−0.541002	−	0.541002i
$124$		0			0
$125$	8.48528	−	8.48528i	0.758947	−	0.758947i
$126$		0			0
$127$	5.65685			0.501965			0.250982	−	0.967992i	$-0.419246\pi$
$127$	5.65685			0.501965			0.250982	+	0.967992i	$0.419246\pi$
$128$		0			0
$129$	6.00000			0.528271
$130$		0			0
$131$	−1.00000	+	1.00000i	−0.0873704	+	0.0873704i	−0.749441	−	0.662071i	$-0.769678\pi$
$131$	−1.00000	+	1.00000i	−0.0873704	+	0.0873704i	0.662071	+	0.749441i	$0.269678\pi$
$132$		0			0
$133$	8.48528	+	8.48528i	0.735767	+	0.735767i
$134$		0			0
$135$			11.3137i			0.973729i
$136$		0			0
$137$			6.00000i			0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$			6.00000i			0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$		0			0
$139$	3.00000	+	3.00000i	0.254457	+	0.254457i	0.822795	−	0.568338i	$-0.192414\pi$
$139$	3.00000	+	3.00000i	0.254457	+	0.254457i	−0.568338	+	0.822795i	$0.692414\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−25.4558			−2.12872
$144$		0			0
$145$	−4.00000			−0.332182
$146$		0			0
$147$	−1.00000	+	1.00000i	−0.0824786	+	0.0824786i
$148$		0			0
$149$	15.5563	+	15.5563i	1.27443	+	1.27443i	0.943741	+	0.330684i	$0.107280\pi$
$149$	15.5563	+	15.5563i	1.27443	+	1.27443i	0.330684	+	0.943741i	$0.392720\pi$
$150$		0			0
$151$			14.1421i			1.15087i	0.817847	+	0.575435i	$0.195167\pi$
$151$			14.1421i			1.15087i	−0.817847	+	0.575435i	$0.804833\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.00000	+	8.00000i	0.642575	+	0.642575i
$156$		0			0
$157$	−4.24264	+	4.24264i	−0.338600	+	0.338600i	−0.855840	−	0.517241i	$-0.826959\pi$
$157$	−4.24264	+	4.24264i	−0.338600	+	0.338600i	0.517241	+	0.855840i	$0.326959\pi$
$158$		0			0
$159$	2.82843			0.224309
$160$		0			0
$161$	−24.0000			−1.89146
$162$		0			0
$163$	−15.0000	+	15.0000i	−1.17489	+	1.17489i	−0.193862	+	0.981029i	$0.562101\pi$
$163$	−15.0000	+	15.0000i	−1.17489	+	1.17489i	−0.981029	+	0.193862i	$0.937899\pi$
$164$		0			0
$165$	−8.48528	−	8.48528i	−0.660578	−	0.660578i
$166$		0			0
$167$			8.48528i			0.656611i	0.944572	+	0.328305i	$0.106478\pi$
$167$			8.48528i			0.656611i	−0.944572	+	0.328305i	$0.893522\pi$
$168$		0			0
$169$		−	23.0000i		−	1.76923i
$170$		0			0
$171$	−3.00000	−	3.00000i	−0.229416	−	0.229416i
$172$		0			0
$173$	9.89949	−	9.89949i	0.752645	−	0.752645i	−0.222327	−	0.974972i	$-0.571365\pi$
$173$	9.89949	−	9.89949i	0.752645	−	0.752645i	0.974972	+	0.222327i	$0.0713654\pi$
$174$		0			0
$175$	−2.82843			−0.213809
$176$		0			0
$177$	2.00000			0.150329
$178$		0			0
$179$	−13.0000	+	13.0000i	−0.971666	+	0.971666i	−0.999609	−	0.0279439i	$-0.991104\pi$
$179$	−13.0000	+	13.0000i	−0.971666	+	0.971666i	0.0279439	+	0.999609i	$0.491104\pi$
$180$		0			0
$181$	−12.7279	−	12.7279i	−0.946059	−	0.946059i	0.0525588	−	0.998618i	$-0.483262\pi$
$181$	−12.7279	−	12.7279i	−0.946059	−	0.946059i	−0.998618	+	0.0525588i	$0.983262\pi$
$182$		0			0
$183$			8.48528i			0.627250i
$184$		0			0
$185$			12.0000i			0.882258i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	11.3137	−	11.3137i	0.822951	−	0.822951i
$190$		0			0
$191$	−16.9706			−1.22795			−0.613973	−	0.789327i	$-0.710430\pi$
$191$	−16.9706			−1.22795			−0.613973	+	0.789327i	$0.710430\pi$
$192$		0			0
$193$	24.0000			1.72756			0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000			1.72756			0.863779	+	0.503871i	$0.168091\pi$
$194$		0			0
$195$	12.0000	−	12.0000i	0.859338	−	0.859338i
$196$		0			0
$197$	7.07107	+	7.07107i	0.503793	+	0.503793i	0.912614	−	0.408822i	$-0.134060\pi$
$197$	7.07107	+	7.07107i	0.503793	+	0.503793i	−0.408822	+	0.912614i	$0.634060\pi$
$198$		0			0
$199$			19.7990i			1.40351i	0.712417	+	0.701757i	$0.247601\pi$
$199$			19.7990i			1.40351i	−0.712417	+	0.701757i	$0.752399\pi$
$200$		0			0
$201$			18.0000i			1.26962i
$202$		0			0
$203$	4.00000	+	4.00000i	0.280745	+	0.280745i
$204$		0			0
$205$	8.48528	−	8.48528i	0.592638	−	0.592638i
$206$		0			0
$207$	8.48528			0.589768
$208$		0			0
$209$	18.0000			1.24509
$210$		0			0
$211$	3.00000	−	3.00000i	0.206529	−	0.206529i	−0.596262	−	0.802790i	$-0.703348\pi$
$211$	3.00000	−	3.00000i	0.206529	−	0.206529i	0.802790	+	0.596262i	$0.203348\pi$
$212$		0			0
$213$	−8.48528	−	8.48528i	−0.581402	−	0.581402i
$214$		0			0
$215$			8.48528i			0.578691i
$216$		0			0
$217$		−	16.0000i		−	1.08615i
$218$		0			0
$219$	12.0000	+	12.0000i	0.810885	+	0.810885i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−5.65685			−0.378811			−0.189405	−	0.981899i	$-0.560656\pi$
$223$	−5.65685			−0.378811			−0.189405	+	0.981899i	$0.560656\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	9.00000	−	9.00000i	0.597351	−	0.597351i	−0.342256	−	0.939607i	$-0.611191\pi$
$227$	9.00000	−	9.00000i	0.597351	−	0.597351i	0.939607	+	0.342256i	$0.111191\pi$
$228$		0			0
$229$	12.7279	+	12.7279i	0.841085	+	0.841085i	0.989000	−	0.147915i	$-0.0472563\pi$
$229$	12.7279	+	12.7279i	0.841085	+	0.841085i	−0.147915	+	0.989000i	$0.547256\pi$
$230$		0			0
$231$			16.9706i			1.11658i
$232$		0			0
$233$			12.0000i			0.786146i	0.919507	+	0.393073i	$0.128588\pi$
$233$			12.0000i			0.786146i	−0.919507	+	0.393073i	$0.871412\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	5.65685	−	5.65685i	0.367452	−	0.367452i
$238$		0			0
$239$	−16.9706			−1.09773			−0.548867	−	0.835910i	$-0.684941\pi$
$239$	−16.9706			−1.09773			−0.548867	+	0.835910i	$0.684941\pi$
$240$		0			0
$241$		0			0			−	1.00000i	$-0.5\pi$
$241$		0			0				1.00000i	$0.5\pi$
$242$		0			0
$243$	−7.00000	+	7.00000i	−0.449050	+	0.449050i
$244$		0			0
$245$	−1.41421	−	1.41421i	−0.0903508	−	0.0903508i
$246$		0			0
$247$			25.4558i			1.61972i
$248$		0			0
$249$		−	6.00000i		−	0.380235i
$250$		0			0
$251$	9.00000	+	9.00000i	0.568075	+	0.568075i	0.931589	−	0.363514i	$-0.118423\pi$
$251$	9.00000	+	9.00000i	0.568075	+	0.568075i	−0.363514	+	0.931589i	$0.618423\pi$
$252$		0			0
$253$	−25.4558	+	25.4558i	−1.60040	+	1.60040i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.00000			0.374270			0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000			0.374270			0.187135	+	0.982334i	$0.440080\pi$
$258$		0			0
$259$	12.0000	−	12.0000i	0.745644	−	0.745644i
$260$		0			0
$261$	−1.41421	−	1.41421i	−0.0875376	−	0.0875376i
$262$		0			0
$263$		−	8.48528i		−	0.523225i	−0.965173	−	0.261612i	$-0.915746\pi$
$263$		−	8.48528i		−	0.523225i	0.965173	−	0.261612i	$-0.0842542\pi$
$264$		0			0
$265$			4.00000i			0.245718i
$266$		0			0
$267$	−12.0000	−	12.0000i	−0.734388	−	0.734388i
$268$		0			0
$269$	−9.89949	+	9.89949i	−0.603583	+	0.603583i	−0.941261	−	0.337679i	$-0.890358\pi$
$269$	−9.89949	+	9.89949i	−0.603583	+	0.603583i	0.337679	+	0.941261i	$0.390358\pi$
$270$		0			0
$271$	22.6274			1.37452			0.687259	−	0.726413i	$-0.258814\pi$
$271$	22.6274			1.37452			0.687259	+	0.726413i	$0.258814\pi$
$272$		0			0
$273$	−24.0000			−1.45255
$274$		0			0
$275$	−3.00000	+	3.00000i	−0.180907	+	0.180907i
$276$		0			0
$277$	−4.24264	−	4.24264i	−0.254916	−	0.254916i	0.568067	−	0.822982i	$-0.307691\pi$
$277$	−4.24264	−	4.24264i	−0.254916	−	0.254916i	−0.822982	+	0.568067i	$0.807691\pi$
$278$		0			0
$279$			5.65685i			0.338667i
$280$		0			0
$281$			12.0000i			0.715860i	0.933748	+	0.357930i	$0.116517\pi$
$281$			12.0000i			0.715860i	−0.933748	+	0.357930i	$0.883483\pi$
$282$		0			0
$283$	−9.00000	−	9.00000i	−0.534994	−	0.534994i	0.387060	−	0.922055i	$-0.373491\pi$
$283$	−9.00000	−	9.00000i	−0.534994	−	0.534994i	−0.922055	+	0.387060i	$0.873491\pi$
$284$		0			0
$285$	−8.48528	+	8.48528i	−0.502625	+	0.502625i
$286$		0			0
$287$	−16.9706			−1.00174
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$	−8.00000	+	8.00000i	−0.468968	+	0.468968i
$292$		0			0
$293$	7.07107	+	7.07107i	0.413096	+	0.413096i	0.882816	−	0.469720i	$-0.155645\pi$
$293$	7.07107	+	7.07107i	0.413096	+	0.413096i	−0.469720	+	0.882816i	$0.655645\pi$
$294$		0			0
$295$			2.82843i			0.164677i
$296$		0			0
$297$		−	24.0000i		−	1.39262i
$298$		0			0
$299$	−36.0000	−	36.0000i	−2.08193	−	2.08193i
$300$		0			0
$301$	8.48528	−	8.48528i	0.489083	−	0.489083i
$302$		0			0
$303$	−14.1421			−0.812444
$304$		0			0
$305$	−12.0000			−0.687118
$306$		0			0
$307$	−3.00000	+	3.00000i	−0.171219	+	0.171219i	−0.787515	−	0.616296i	$-0.788633\pi$
$307$	−3.00000	+	3.00000i	−0.171219	+	0.171219i	0.616296	+	0.787515i	$0.288633\pi$
$308$		0			0
$309$	2.82843	+	2.82843i	0.160904	+	0.160904i
$310$		0			0
$311$			8.48528i			0.481156i	0.970630	+	0.240578i	$0.0773370\pi$
$311$			8.48528i			0.481156i	−0.970630	+	0.240578i	$0.922663\pi$
$312$		0			0
$313$			6.00000i			0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$			6.00000i			0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$		0			0
$315$	4.00000	+	4.00000i	0.225374	+	0.225374i
$316$		0			0
$317$	−7.07107	+	7.07107i	−0.397151	+	0.397151i	−0.877227	−	0.480076i	$-0.840609\pi$
$317$	−7.07107	+	7.07107i	−0.397151	+	0.397151i	0.480076	+	0.877227i	$0.340609\pi$
$318$		0			0
$319$	8.48528			0.475085
$320$		0			0
$321$	10.0000			0.558146
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−4.24264	−	4.24264i	−0.235339	−	0.235339i
$326$		0			0
$327$			8.48528i			0.469237i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	21.0000	+	21.0000i	1.15426	+	1.15426i	0.985689	+	0.168576i	$0.0539168\pi$
$331$	21.0000	+	21.0000i	1.15426	+	1.15426i	0.168576	+	0.985689i	$0.446083\pi$
$332$		0			0
$333$	−4.24264	+	4.24264i	−0.232495	+	0.232495i
$334$		0			0
$335$	−25.4558			−1.39080
$336$		0			0
$337$	18.0000			0.980522			0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000			0.980522			0.490261	+	0.871576i	$0.336901\pi$
$338$		0			0
$339$	−6.00000	+	6.00000i	−0.325875	+	0.325875i
$340$		0			0
$341$	−16.9706	−	16.9706i	−0.919007	−	0.919007i
$342$		0			0
$343$		−	16.9706i		−	0.916324i
$344$		0			0
$345$		−	24.0000i		−	1.29212i
$346$		0			0
$347$	−15.0000	−	15.0000i	−0.805242	−	0.805242i	0.178667	−	0.983910i	$-0.442821\pi$
$347$	−15.0000	−	15.0000i	−0.805242	−	0.805242i	−0.983910	+	0.178667i	$0.942821\pi$
$348$		0			0
$349$	4.24264	−	4.24264i	0.227103	−	0.227103i	−0.584378	−	0.811481i	$-0.698662\pi$
$349$	4.24264	−	4.24264i	0.227103	−	0.227103i	0.811481	+	0.584378i	$0.198662\pi$
$350$		0			0
$351$	33.9411			1.81164
$352$		0			0
$353$	−6.00000			−0.319348			−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000			−0.319348			−0.159674	+	0.987170i	$0.551044\pi$
$354$		0			0
$355$	12.0000	−	12.0000i	0.636894	−	0.636894i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	8.48528i		−	0.447836i	−0.974608	−	0.223918i	$-0.928115\pi$
$359$		−	8.48528i		−	0.447836i	0.974608	−	0.223918i	$-0.0718848\pi$
$360$		0			0
$361$			1.00000i			0.0526316i
$362$		0			0
$363$	7.00000	+	7.00000i	0.367405	+	0.367405i
$364$		0			0
$365$	−16.9706	+	16.9706i	−0.888280	+	0.888280i
$366$		0			0
$367$	11.3137			0.590571			0.295285	−	0.955409i	$-0.404585\pi$
$367$	11.3137			0.590571			0.295285	+	0.955409i	$0.404585\pi$
$368$		0			0
$369$	6.00000			0.312348
$370$		0			0
$371$	4.00000	−	4.00000i	0.207670	−	0.207670i
$372$		0			0
$373$	−4.24264	−	4.24264i	−0.219676	−	0.219676i	0.588686	−	0.808362i	$-0.299645\pi$
$373$	−4.24264	−	4.24264i	−0.219676	−	0.219676i	−0.808362	+	0.588686i	$0.799645\pi$
$374$		0			0
$375$		−	16.9706i		−	0.876356i
$376$		0			0
$377$			12.0000i			0.618031i
$378$		0			0
$379$	9.00000	+	9.00000i	0.462299	+	0.462299i	0.899408	−	0.437109i	$-0.143998\pi$
$379$	9.00000	+	9.00000i	0.462299	+	0.462299i	−0.437109	+	0.899408i	$0.643998\pi$
$380$		0			0
$381$	5.65685	−	5.65685i	0.289809	−	0.289809i
$382$		0			0
$383$	−33.9411			−1.73431			−0.867155	−	0.498038i	$-0.834054\pi$
$383$	−33.9411			−1.73431			−0.867155	+	0.498038i	$0.834054\pi$
$384$		0			0
$385$	−24.0000			−1.22315
$386$		0			0
$387$	−3.00000	+	3.00000i	−0.152499	+	0.152499i
$388$		0			0
$389$	−18.3848	−	18.3848i	−0.932145	−	0.932145i	0.0656946	−	0.997840i	$-0.479074\pi$
$389$	−18.3848	−	18.3848i	−0.932145	−	0.932145i	−0.997840	+	0.0656946i	$0.979074\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			2.00000i			0.100887i
$394$		0			0
$395$	8.00000	+	8.00000i	0.402524	+	0.402524i
$396$		0			0
$397$	12.7279	−	12.7279i	0.638796	−	0.638796i	−0.311462	−	0.950258i	$-0.600819\pi$
$397$	12.7279	−	12.7279i	0.638796	−	0.638796i	0.950258	+	0.311462i	$0.100819\pi$
$398$		0			0
$399$	16.9706			0.849591
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	24.0000	−	24.0000i	1.19553	−	1.19553i
$404$		0			0
$405$	7.07107	+	7.07107i	0.351364	+	0.351364i
$406$		0			0
$407$		−	25.4558i		−	1.26180i
$408$		0			0
$409$			22.0000i			1.08783i	0.839140	+	0.543915i	$0.183059\pi$
$409$			22.0000i			1.08783i	−0.839140	+	0.543915i	$0.816941\pi$
$410$		0			0
$411$	6.00000	+	6.00000i	0.295958	+	0.295958i
$412$		0			0
$413$	2.82843	−	2.82843i	0.139178	−	0.139178i
$414$		0			0
$415$	8.48528			0.416526
$416$		0			0
$417$	6.00000			0.293821
$418$		0			0
$419$	−9.00000	+	9.00000i	−0.439679	+	0.439679i	−0.891904	−	0.452225i	$-0.850630\pi$
$419$	−9.00000	+	9.00000i	−0.439679	+	0.439679i	0.452225	+	0.891904i	$0.350630\pi$
$420$		0			0
$421$	−4.24264	−	4.24264i	−0.206774	−	0.206774i	0.596121	−	0.802895i	$-0.296708\pi$
$421$	−4.24264	−	4.24264i	−0.206774	−	0.206774i	−0.802895	+	0.596121i	$0.796708\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	12.0000	+	12.0000i	0.580721	+	0.580721i
$428$		0			0
$429$	−25.4558	+	25.4558i	−1.22902	+	1.22902i
$430$		0			0
$431$	16.9706			0.817443			0.408722	−	0.912659i	$-0.365975\pi$
$431$	16.9706			0.817443			0.408722	+	0.912659i	$0.365975\pi$
$432$		0			0
$433$	16.0000			0.768911			0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000			0.768911			0.384455	+	0.923144i	$0.374389\pi$
$434$		0			0
$435$	−4.00000	+	4.00000i	−0.191785	+	0.191785i
$436$		0			0
$437$	25.4558	+	25.4558i	1.21772	+	1.21772i
$438$		0			0
$439$		−	2.82843i		−	0.134993i	−0.997719	−	0.0674967i	$-0.978499\pi$
$439$		−	2.82843i		−	0.134993i	0.997719	−	0.0674967i	$-0.0215012\pi$
$440$		0			0
$441$		−	1.00000i		−	0.0476190i
$442$		0			0
$443$	−9.00000	−	9.00000i	−0.427603	−	0.427603i	0.460208	−	0.887811i	$-0.347775\pi$
$443$	−9.00000	−	9.00000i	−0.427603	−	0.427603i	−0.887811	+	0.460208i	$0.847775\pi$
$444$		0			0
$445$	16.9706	−	16.9706i	0.804482	−	0.804482i
$446$		0			0
$447$	31.1127			1.47158
$448$		0			0
$449$	24.0000			1.13263			0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000			1.13263			0.566315	+	0.824189i	$0.308369\pi$
$450$		0			0
$451$	−18.0000	+	18.0000i	−0.847587	+	0.847587i
$452$		0			0
$453$	14.1421	+	14.1421i	0.664455	+	0.664455i
$454$		0			0
$455$		−	33.9411i		−	1.59118i
$456$		0			0
$457$			10.0000i			0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$			10.0000i			0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	18.3848	−	18.3848i	0.856264	−	0.856264i	−0.134631	−	0.990896i	$-0.542985\pi$
$461$	18.3848	−	18.3848i	0.856264	−	0.856264i	0.990896	+	0.134631i	$0.0429850\pi$
$462$		0			0
$463$	−5.65685			−0.262896			−0.131448	−	0.991323i	$-0.541963\pi$
$463$	−5.65685			−0.262896			−0.131448	+	0.991323i	$0.541963\pi$
$464$		0			0
$465$	16.0000			0.741982
$466$		0			0
$467$	−3.00000	+	3.00000i	−0.138823	+	0.138823i	−0.773103	−	0.634280i	$-0.781297\pi$
$467$	−3.00000	+	3.00000i	−0.138823	+	0.138823i	0.634280	+	0.773103i	$0.281297\pi$
$468$		0			0
$469$	25.4558	+	25.4558i	1.17544	+	1.17544i
$470$		0			0
$471$			8.48528i			0.390981i
$472$		0			0
$473$		−	18.0000i		−	0.827641i
$474$		0			0
$475$	3.00000	+	3.00000i	0.137649	+	0.137649i
$476$		0			0
$477$	−1.41421	+	1.41421i	−0.0647524	+	0.0647524i
$478$		0			0
$479$	−16.9706			−0.775405			−0.387702	−	0.921785i	$-0.626731\pi$
$479$	−16.9706			−0.775405			−0.387702	+	0.921785i	$0.626731\pi$
$480$		0			0
$481$	36.0000			1.64146
$482$		0			0
$483$	−24.0000	+	24.0000i	−1.09204	+	1.09204i
$484$		0			0
$485$	−11.3137	−	11.3137i	−0.513729	−	0.513729i
$486$		0			0
$487$		−	36.7696i		−	1.66619i	−0.553132	−	0.833094i	$-0.686567\pi$
$487$		−	36.7696i		−	1.66619i	0.553132	−	0.833094i	$-0.313433\pi$
$488$		0			0
$489$			30.0000i			1.35665i
$490$		0			0
$491$	19.0000	+	19.0000i	0.857458	+	0.857458i	0.991038	−	0.133580i	$-0.0426473\pi$
$491$	19.0000	+	19.0000i	0.857458	+	0.857458i	−0.133580	+	0.991038i	$0.542647\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	8.48528			0.381385
$496$		0			0
$497$	−24.0000			−1.07655
$498$		0			0
$499$	3.00000	−	3.00000i	0.134298	−	0.134298i	−0.636762	−	0.771060i	$-0.719727\pi$
$499$	3.00000	−	3.00000i	0.134298	−	0.134298i	0.771060	+	0.636762i	$0.219727\pi$
$500$		0			0
$501$	8.48528	+	8.48528i	0.379094	+	0.379094i
$502$		0			0
$503$		−	42.4264i		−	1.89170i	−0.324604	−	0.945850i	$-0.605231\pi$
$503$		−	42.4264i		−	1.89170i	0.324604	−	0.945850i	$-0.394769\pi$
$504$		0			0
$505$		−	20.0000i		−	0.889988i
$506$		0			0
$507$	−23.0000	−	23.0000i	−1.02147	−	1.02147i
$508$		0			0
$509$	24.0416	−	24.0416i	1.06563	−	1.06563i	0.0679369	−	0.997690i	$-0.478358\pi$
$509$	24.0416	−	24.0416i	1.06563	−	1.06563i	0.997690	−	0.0679369i	$-0.0216417\pi$
$510$		0			0
$511$	33.9411			1.50147
$512$		0			0
$513$	−24.0000			−1.05963
$514$		0			0
$515$	−4.00000	+	4.00000i	−0.176261	+	0.176261i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		−	19.7990i		−	0.869079i
$520$		0			0
$521$		−	42.0000i		−	1.84005i	−0.391856	−	0.920027i	$-0.628167\pi$
$521$		−	42.0000i		−	1.84005i	0.391856	−	0.920027i	$-0.371833\pi$
$522$		0			0
$523$	−3.00000	−	3.00000i	−0.131181	−	0.131181i	0.638468	−	0.769649i	$-0.279569\pi$
$523$	−3.00000	−	3.00000i	−0.131181	−	0.131181i	−0.769649	+	0.638468i	$0.779569\pi$
$524$		0			0
$525$	−2.82843	+	2.82843i	−0.123443	+	0.123443i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−49.0000			−2.13043
$530$		0			0
$531$	−1.00000	+	1.00000i	−0.0433963	+	0.0433963i
$532$		0			0
$533$	−25.4558	−	25.4558i	−1.10262	−	1.10262i
$534$		0			0
$535$			14.1421i			0.611418i
$536$		0			0
$537$			26.0000i			1.12198i
$538$		0			0
$539$	3.00000	+	3.00000i	0.129219	+	0.129219i
$540$		0			0
$541$	29.6985	−	29.6985i	1.27684	−	1.27684i	0.334410	−	0.942428i	$-0.391463\pi$
$541$	29.6985	−	29.6985i	1.27684	−	1.27684i	0.942428	−	0.334410i	$-0.108537\pi$
$542$		0			0
$543$	−25.4558			−1.09241
$544$		0			0
$545$	−12.0000			−0.514024
$546$		0			0
$547$	15.0000	−	15.0000i	0.641354	−	0.641354i	−0.309535	−	0.950888i	$-0.600173\pi$
$547$	15.0000	−	15.0000i	0.641354	−	0.641354i	0.950888	+	0.309535i	$0.100173\pi$
$548$		0			0
$549$	−4.24264	−	4.24264i	−0.181071	−	0.181071i
$550$		0			0
$551$		−	8.48528i		−	0.361485i
$552$		0			0
$553$		−	16.0000i		−	0.680389i
$554$		0			0
$555$	12.0000	+	12.0000i	0.509372	+	0.509372i
$556$		0			0
$557$	−1.41421	+	1.41421i	−0.0599222	+	0.0599222i	−0.736433	−	0.676511i	$-0.763491\pi$
$557$	−1.41421	+	1.41421i	−0.0599222	+	0.0599222i	0.676511	+	0.736433i	$0.263491\pi$
$558$		0			0
$559$	25.4558			1.07667
$560$		0			0
$561$		0			0
$562$		0			0
$563$	21.0000	−	21.0000i	0.885044	−	0.885044i	−0.108998	−	0.994042i	$-0.534764\pi$
$563$	21.0000	−	21.0000i	0.885044	−	0.885044i	0.994042	+	0.108998i	$0.0347641\pi$
$564$		0			0
$565$	−8.48528	−	8.48528i	−0.356978	−	0.356978i
$566$		0			0
$567$		−	14.1421i		−	0.593914i
$568$		0			0
$569$		−	6.00000i		−	0.251533i	−0.992060	−	0.125767i	$-0.959861\pi$
$569$		−	6.00000i		−	0.251533i	0.992060	−	0.125767i	$-0.0401390\pi$
$570$		0			0
$571$	−9.00000	−	9.00000i	−0.376638	−	0.376638i	0.493250	−	0.869888i	$-0.335809\pi$
$571$	−9.00000	−	9.00000i	−0.376638	−	0.376638i	−0.869888	+	0.493250i	$0.835809\pi$
$572$		0			0
$573$	−16.9706	+	16.9706i	−0.708955	+	0.708955i
$574$		0			0
$575$	−8.48528			−0.353861
$576$		0			0
$577$	−18.0000			−0.749350			−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000			−0.749350			−0.374675	+	0.927156i	$0.622246\pi$
$578$		0			0
$579$	24.0000	−	24.0000i	0.997406	−	0.997406i
$580$		0			0
$581$	−8.48528	−	8.48528i	−0.352029	−	0.352029i
$582$		0			0
$583$		−	8.48528i		−	0.351424i
$584$		0			0
$585$			12.0000i			0.496139i
$586$		0			0
$587$	−11.0000	−	11.0000i	−0.454019	−	0.454019i	0.442667	−	0.896686i	$-0.354032\pi$
$587$	−11.0000	−	11.0000i	−0.454019	−	0.454019i	−0.896686	+	0.442667i	$0.854032\pi$
$588$		0			0
$589$	−16.9706	+	16.9706i	−0.699260	+	0.699260i
$590$		0			0
$591$	14.1421			0.581730
$592$		0			0
$593$	30.0000			1.23195			0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000			1.23195			0.615976	+	0.787765i	$0.288762\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	19.7990	+	19.7990i	0.810319	+	0.810319i
$598$		0			0
$599$		−	25.4558i		−	1.04010i	−0.854137	−	0.520049i	$-0.825914\pi$
$599$		−	25.4558i		−	1.04010i	0.854137	−	0.520049i	$-0.174086\pi$
$600$		0			0
$601$		−	12.0000i		−	0.489490i	−0.969587	−	0.244745i	$-0.921296\pi$
$601$		−	12.0000i		−	0.489490i	0.969587	−	0.244745i	$-0.0787043\pi$
$602$		0			0
$603$	−9.00000	−	9.00000i	−0.366508	−	0.366508i
$604$		0			0
$605$	−9.89949	+	9.89949i	−0.402472	+	0.402472i
$606$		0			0
$607$	22.6274			0.918419			0.459209	−	0.888328i	$-0.348133\pi$
$607$	22.6274			0.918419			0.459209	+	0.888328i	$0.348133\pi$
$608$		0			0
$609$	8.00000			0.324176
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−4.24264	−	4.24264i	−0.171359	−	0.171359i	0.616217	−	0.787576i	$-0.288664\pi$
$613$	−4.24264	−	4.24264i	−0.171359	−	0.171359i	−0.787576	+	0.616217i	$0.788664\pi$
$614$		0			0
$615$		−	16.9706i		−	0.684319i
$616$		0			0
$617$			36.0000i			1.44931i	0.689114	+	0.724653i	$0.258000\pi$
$617$			36.0000i			1.44931i	−0.689114	+	0.724653i	$0.742000\pi$
$618$		0			0
$619$	−27.0000	−	27.0000i	−1.08522	−	1.08522i	−0.996013	−	0.0892087i	$-0.971566\pi$
$619$	−27.0000	−	27.0000i	−1.08522	−	1.08522i	−0.0892087	−	0.996013i	$-0.528434\pi$
$620$		0			0
$621$	33.9411	−	33.9411i	1.36201	−	1.36201i
$622$		0			0
$623$	−33.9411			−1.35982
$624$		0			0
$625$	19.0000			0.760000
$626$		0			0
$627$	18.0000	−	18.0000i	0.718851	−	0.718851i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	31.1127i		−	1.23858i	−0.785164	−	0.619288i	$-0.787421\pi$
$631$		−	31.1127i		−	1.23858i	0.785164	−	0.619288i	$-0.212579\pi$
$632$		0			0
$633$		−	6.00000i		−	0.238479i
$634$		0			0
$635$	8.00000	+	8.00000i	0.317470	+	0.317470i
$636$		0			0
$637$	−4.24264	+	4.24264i	−0.168100	+	0.168100i
$638$		0			0
$639$	8.48528			0.335673
$640$		0			0
$641$	−24.0000			−0.947943			−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000			−0.947943			−0.473972	+	0.880540i	$0.657180\pi$
$642$		0			0
$643$	−9.00000	+	9.00000i	−0.354925	+	0.354925i	−0.861938	−	0.507013i	$-0.830750\pi$
$643$	−9.00000	+	9.00000i	−0.354925	+	0.354925i	0.507013	+	0.861938i	$0.330750\pi$
$644$		0			0
$645$	8.48528	+	8.48528i	0.334108	+	0.334108i
$646$		0			0
$647$			25.4558i			1.00077i	0.865802	+	0.500386i	$0.166809\pi$
$647$			25.4558i			1.00077i	−0.865802	+	0.500386i	$0.833191\pi$
$648$		0			0
$649$		−	6.00000i		−	0.235521i
$650$		0			0
$651$	−16.0000	−	16.0000i	−0.627089	−	0.627089i
$652$		0			0
$653$	1.41421	−	1.41421i	0.0553425	−	0.0553425i	−0.678894	−	0.734236i	$-0.737540\pi$
$653$	1.41421	−	1.41421i	0.0553425	−	0.0553425i	0.734236	+	0.678894i	$0.237540\pi$
$654$		0			0
$655$	−2.82843			−0.110516
$656$		0			0
$657$	−12.0000			−0.468165
$658$		0			0
$659$	−29.0000	+	29.0000i	−1.12968	+	1.12968i	−0.139451	+	0.990229i	$0.544534\pi$
$659$	−29.0000	+	29.0000i	−1.12968	+	1.12968i	−0.990229	+	0.139451i	$0.955466\pi$
$660$		0			0
$661$	−12.7279	−	12.7279i	−0.495059	−	0.495059i	0.414837	−	0.909896i	$-0.363839\pi$
$661$	−12.7279	−	12.7279i	−0.495059	−	0.495059i	−0.909896	+	0.414837i	$0.863839\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			24.0000i			0.930680i
$666$		0			0
$667$	12.0000	+	12.0000i	0.464642	+	0.464642i
$668$		0			0
$669$	−5.65685	+	5.65685i	−0.218707	+	0.218707i
$670$		0			0
$671$	25.4558			0.982712
$672$		0			0
$673$	8.00000			0.308377			0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000			0.308377			0.154189	+	0.988041i	$0.450724\pi$
$674$		0			0
$675$	4.00000	−	4.00000i	0.153960	−	0.153960i
$676$		0			0
$677$	26.8701	+	26.8701i	1.03270	+	1.03270i	0.999447	+	0.0332533i	$0.0105868\pi$
$677$	26.8701	+	26.8701i	1.03270	+	1.03270i	0.0332533	+	0.999447i	$0.489413\pi$
$678$		0			0
$679$			22.6274i			0.868361i
$680$		0			0
$681$		−	18.0000i		−	0.689761i
$682$		0			0
$683$	27.0000	+	27.0000i	1.03313	+	1.03313i	0.999432	+	0.0336941i	$0.0107272\pi$
$683$	27.0000	+	27.0000i	1.03313	+	1.03313i	0.0336941	+	0.999432i	$0.489273\pi$
$684$		0			0
$685$	−8.48528	+	8.48528i	−0.324206	+	0.324206i
$686$		0			0
$687$	25.4558			0.971201
$688$		0			0
$689$	12.0000			0.457164
$690$		0			0
$691$	−21.0000	+	21.0000i	−0.798878	+	0.798878i	−0.982919	−	0.184041i	$-0.941082\pi$
$691$	−21.0000	+	21.0000i	−0.798878	+	0.798878i	0.184041	+	0.982919i	$0.441082\pi$
$692$		0			0
$693$	−8.48528	−	8.48528i	−0.322329	−	0.322329i
$694$		0			0
$695$			8.48528i			0.321865i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	12.0000	+	12.0000i	0.453882	+	0.453882i
$700$		0			0
$701$	−32.5269	+	32.5269i	−1.22852	+	1.22852i	−0.264003	+	0.964522i	$0.585043\pi$
$701$	−32.5269	+	32.5269i	−1.22852	+	1.22852i	−0.964522	+	0.264003i	$0.914957\pi$
$702$		0			0
$703$	−25.4558			−0.960085
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−20.0000	+	20.0000i	−0.752177	+	0.752177i
$708$		0			0
$709$	−12.7279	−	12.7279i	−0.478007	−	0.478007i	0.426487	−	0.904494i	$-0.359751\pi$
$709$	−12.7279	−	12.7279i	−0.478007	−	0.478007i	−0.904494	+	0.426487i	$0.859751\pi$
$710$		0			0
$711$			5.65685i			0.212149i
$712$		0			0
$713$		−	48.0000i		−	1.79761i
$714$		0			0
$715$	−36.0000	−	36.0000i	−1.34632	−	1.34632i
$716$		0			0
$717$	−16.9706	+	16.9706i	−0.633777	+	0.633777i
$718$		0			0
$719$	−16.9706			−0.632895			−0.316448	−	0.948610i	$-0.602490\pi$
$719$	−16.9706			−0.632895			−0.316448	+	0.948610i	$0.602490\pi$
$720$		0			0
$721$	8.00000			0.297936
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.41421	+	1.41421i	0.0525226	+	0.0525226i
$726$		0			0
$727$		−	2.82843i		−	0.104901i	−0.998624	−	0.0524503i	$-0.983297\pi$
$727$		−	2.82843i		−	0.104901i	0.998624	−	0.0524503i	$-0.0167031\pi$
$728$		0			0
$729$			29.0000i			1.07407i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−29.6985	+	29.6985i	−1.09694	+	1.09694i	−0.102172	+	0.994767i	$0.532579\pi$
$733$	−29.6985	+	29.6985i	−1.09694	+	1.09694i	−0.994767	+	0.102172i	$0.967421\pi$
$734$		0			0
$735$	−2.82843			−0.104328
$736$		0			0
$737$	54.0000			1.98912
$738$		0			0
$739$	33.0000	−	33.0000i	1.21392	−	1.21392i	0.244200	−	0.969725i	$-0.421475\pi$
$739$	33.0000	−	33.0000i	1.21392	−	1.21392i	0.969725	−	0.244200i	$-0.0785252\pi$
$740$		0			0
$741$	25.4558	+	25.4558i	0.935144	+	0.935144i
$742$		0			0
$743$		−	25.4558i		−	0.933884i	−0.884288	−	0.466942i	$-0.845356\pi$
$743$		−	25.4558i		−	0.933884i	0.884288	−	0.466942i	$-0.154644\pi$
$744$		0			0
$745$			44.0000i			1.61204i
$746$		0			0
$747$	3.00000	+	3.00000i	0.109764	+	0.109764i
$748$		0			0
$749$	14.1421	−	14.1421i	0.516742	−	0.516742i
$750$		0			0
$751$	5.65685			0.206422			0.103211	−	0.994660i	$-0.467088\pi$
$751$	5.65685			0.206422			0.103211	+	0.994660i	$0.467088\pi$
$752$		0			0
$753$	18.0000			0.655956
$754$		0			0
$755$	−20.0000	+	20.0000i	−0.727875	+	0.727875i
$756$		0			0
$757$	−29.6985	−	29.6985i	−1.07941	−	1.07941i	−0.996562	−	0.0828476i	$-0.973599\pi$
$757$	−29.6985	−	29.6985i	−1.07941	−	1.07941i	−0.0828476	−	0.996562i	$-0.526401\pi$
$758$		0			0
$759$			50.9117i			1.84798i
$760$		0			0
$761$		−	42.0000i		−	1.52250i	−0.648459	−	0.761249i	$-0.724586\pi$
$761$		−	42.0000i		−	1.52250i	0.648459	−	0.761249i	$-0.275414\pi$
$762$		0			0
$763$	12.0000	+	12.0000i	0.434429	+	0.434429i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	8.48528			0.306386
$768$		0			0
$769$	−24.0000			−0.865462			−0.432731	−	0.901523i	$-0.642450\pi$
$769$	−24.0000			−0.865462			−0.432731	+	0.901523i	$0.642450\pi$
$770$		0			0
$771$	6.00000	−	6.00000i	0.216085	−	0.216085i
$772$		0			0
$773$	9.89949	+	9.89949i	0.356060	+	0.356060i	0.862358	−	0.506298i	$-0.168987\pi$
$773$	9.89949	+	9.89949i	0.356060	+	0.356060i	−0.506298	+	0.862358i	$0.668987\pi$
$774$		0			0
$775$		−	5.65685i		−	0.203200i
$776$		0			0
$777$		−	24.0000i		−	0.860995i
$778$		0			0
$779$	18.0000	+	18.0000i	0.644917	+	0.644917i
$780$		0			0
$781$	−25.4558	+	25.4558i	−0.910882	+	0.910882i
$782$		0			0
$783$	−11.3137			−0.404319
$784$		0			0
$785$	−12.0000			−0.428298
$786$		0			0
$787$	−21.0000	+	21.0000i	−0.748569	+	0.748569i	−0.974210	−	0.225641i	$-0.927552\pi$
$787$	−21.0000	+	21.0000i	−0.748569	+	0.748569i	0.225641	+	0.974210i	$0.427552\pi$
$788$		0			0
$789$	−8.48528	−	8.48528i	−0.302084	−	0.302084i
$790$		0			0
$791$			16.9706i			0.603404i
$792$		0			0
$793$			36.0000i			1.27840i
$794$		0			0
$795$	4.00000	+	4.00000i	0.141865	+	0.141865i
$796$		0			0
$797$	1.41421	−	1.41421i	0.0500940	−	0.0500940i	−0.681616	−	0.731710i	$-0.738723\pi$
$797$	1.41421	−	1.41421i	0.0500940	−	0.0500940i	0.731710	+	0.681616i	$0.238723\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	12.0000			0.423999
$802$		0			0
$803$	36.0000	−	36.0000i	1.27041	−	1.27041i
$804$		0			0
$805$	−33.9411	−	33.9411i	−1.19627	−	1.19627i
$806$		0			0
$807$			19.7990i			0.696957i
$808$		0			0
$809$			42.0000i			1.47664i	0.674450	+	0.738321i	$0.264381\pi$
$809$			42.0000i			1.47664i	−0.674450	+	0.738321i	$0.735619\pi$
$810$		0			0
$811$	−3.00000	−	3.00000i	−0.105344	−	0.105344i	0.652470	−	0.757814i	$-0.273733\pi$
$811$	−3.00000	−	3.00000i	−0.105344	−	0.105344i	−0.757814	+	0.652470i	$0.773733\pi$
$812$		0			0
$813$	22.6274	−	22.6274i	0.793578	−	0.793578i
$814$		0			0
$815$	−42.4264			−1.48613
$816$		0			0
$817$	−18.0000			−0.629740
$818$		0			0
$819$	12.0000	−	12.0000i	0.419314	−	0.419314i
$820$		0			0
$821$	−26.8701	−	26.8701i	−0.937771	−	0.937771i	0.0604026	−	0.998174i	$-0.480762\pi$
$821$	−26.8701	−	26.8701i	−0.937771	−	0.937771i	−0.998174	+	0.0604026i	$0.980762\pi$
$822$		0			0
$823$			19.7990i			0.690149i	0.938575	+	0.345075i	$0.112146\pi$
$823$			19.7990i			0.690149i	−0.938575	+	0.345075i	$0.887854\pi$
$824$		0			0
$825$			6.00000i			0.208893i
$826$		0			0
$827$	31.0000	+	31.0000i	1.07798	+	1.07798i	0.996691	+	0.0812847i	$0.0259023\pi$
$827$	31.0000	+	31.0000i	1.07798	+	1.07798i	0.0812847	+	0.996691i	$0.474098\pi$
$828$		0			0
$829$	12.7279	−	12.7279i	0.442059	−	0.442059i	−0.450644	−	0.892704i	$-0.648806\pi$
$829$	12.7279	−	12.7279i	0.442059	−	0.442059i	0.892704	+	0.450644i	$0.148806\pi$
$830$		0			0
$831$	−8.48528			−0.294351
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−12.0000	+	12.0000i	−0.415277	+	0.415277i
$836$		0			0
$837$	22.6274	+	22.6274i	0.782118	+	0.782118i
$838$		0			0
$839$			8.48528i			0.292944i	0.989215	+	0.146472i	$0.0467919\pi$
$839$			8.48528i			0.292944i	−0.989215	+	0.146472i	$0.953208\pi$
$840$		0			0
$841$			25.0000i			0.862069i
$842$		0			0
$843$	12.0000	+	12.0000i	0.413302	+	0.413302i
$844$		0			0
$845$	32.5269	−	32.5269i	1.11896	−	1.11896i
$846$		0			0
$847$	19.7990			0.680301
$848$		0			0
$849$	−18.0000			−0.617758
$850$		0			0
$851$	36.0000	−	36.0000i	1.23406	−	1.23406i
$852$		0			0
$853$	12.7279	+	12.7279i	0.435796	+	0.435796i	0.890594	−	0.454799i	$-0.150289\pi$
$853$	12.7279	+	12.7279i	0.435796	+	0.435796i	−0.454799	+	0.890594i	$0.650289\pi$
$854$		0			0
$855$		−	8.48528i		−	0.290191i
$856$		0			0
$857$		−	54.0000i		−	1.84460i	−0.386469	−	0.922302i	$-0.626305\pi$
$857$		−	54.0000i		−	1.84460i	0.386469	−	0.922302i	$-0.373695\pi$
$858$		0			0
$859$	33.0000	+	33.0000i	1.12595	+	1.12595i	0.990830	+	0.135116i	$0.0431406\pi$
$859$	33.0000	+	33.0000i	1.12595	+	1.12595i	0.135116	+	0.990830i	$0.456859\pi$
$860$		0			0
$861$	−16.9706	+	16.9706i	−0.578355	+	0.578355i
$862$		0			0
$863$	−50.9117			−1.73305			−0.866527	−	0.499130i	$-0.833653\pi$
$863$	−50.9117			−1.73305			−0.866527	+	0.499130i	$0.833653\pi$
$864$		0			0
$865$	28.0000			0.952029
$866$		0			0
$867$	−17.0000	+	17.0000i	−0.577350	+	0.577350i
$868$		0			0
$869$	−16.9706	−	16.9706i	−0.575687	−	0.575687i
$870$		0			0
$871$			76.3675i			2.58762i
$872$		0			0
$873$		−	8.00000i		−	0.270759i
$874$		0			0
$875$	−24.0000	−	24.0000i	−0.811348	−	0.811348i
$876$		0			0
$877$	−4.24264	+	4.24264i	−0.143264	+	0.143264i	−0.775101	−	0.631837i	$-0.782301\pi$
$877$	−4.24264	+	4.24264i	−0.143264	+	0.143264i	0.631837	+	0.775101i	$0.282301\pi$
$878$		0			0
$879$	14.1421			0.477002
$880$		0			0
$881$	−30.0000			−1.01073			−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000			−1.01073			−0.505363	+	0.862907i	$0.668641\pi$
$882$		0			0
$883$	−27.0000	+	27.0000i	−0.908622	+	0.908622i	−0.996161	−	0.0875388i	$-0.972100\pi$
$883$	−27.0000	+	27.0000i	−0.908622	+	0.908622i	0.0875388	+	0.996161i	$0.472100\pi$
$884$		0			0
$885$	2.82843	+	2.82843i	0.0950765	+	0.0950765i
$886$		0			0
$887$			42.4264i			1.42454i	0.701906	+	0.712270i	$0.252333\pi$
$887$			42.4264i			1.42454i	−0.701906	+	0.712270i	$0.747667\pi$
$888$		0			0
$889$		−	16.0000i		−	0.536623i
$890$		0			0
$891$	−15.0000	−	15.0000i	−0.502519	−	0.502519i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−36.7696			−1.22907
$896$		0			0
$897$	−72.0000			−2.40401
$898$		0			0
$899$	−8.00000	+	8.00000i	−0.266815	+	0.266815i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		−	16.9706i		−	0.564745i
$904$		0			0
$905$		−	36.0000i		−	1.19668i
$906$		0			0
$907$	3.00000	+	3.00000i	0.0996134	+	0.0996134i	0.755157	−	0.655544i	$-0.227561\pi$
$907$	3.00000	+	3.00000i	0.0996134	+	0.0996134i	−0.655544	+	0.755157i	$0.727561\pi$
$908$		0			0
$909$	7.07107	−	7.07107i	0.234533	−	0.234533i
$910$		0			0
$911$	33.9411			1.12452			0.562260	−	0.826961i	$-0.309932\pi$
$911$	33.9411			1.12452			0.562260	+	0.826961i	$0.309932\pi$
$912$		0			0
$913$	−18.0000			−0.595713
$914$		0			0
$915$	−12.0000	+	12.0000i	−0.396708	+	0.396708i
$916$		0			0
$917$	2.82843	+	2.82843i	0.0934029	+	0.0934029i
$918$		0			0
$919$			2.82843i			0.0933012i	0.998911	+	0.0466506i	$0.0148547\pi$
$919$			2.82843i			0.0933012i	−0.998911	+	0.0466506i	$0.985145\pi$
$920$		0			0
$921$			6.00000i			0.197707i
$922$		0			0
$923$	−36.0000	−	36.0000i	−1.18495	−	1.18495i
$924$		0			0
$925$	4.24264	−	4.24264i	0.139497	−	0.139497i
$926$		0			0
$927$	−2.82843			−0.0928977
$928$		0			0
$929$	−24.0000			−0.787414			−0.393707	−	0.919236i	$-0.628808\pi$
$929$	−24.0000			−0.787414			−0.393707	+	0.919236i	$0.628808\pi$
$930$		0			0
$931$	3.00000	−	3.00000i	0.0983210	−	0.0983210i
$932$		0			0
$933$	8.48528	+	8.48528i	0.277796	+	0.277796i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	20.0000i		−	0.653372i	−0.945133	−	0.326686i	$-0.894068\pi$
$937$		−	20.0000i		−	0.653372i	0.945133	−	0.326686i	$-0.105932\pi$
$938$		0			0
$939$	6.00000	+	6.00000i	0.195803	+	0.195803i
$940$		0			0
$941$	−15.5563	+	15.5563i	−0.507122	+	0.507122i	−0.913642	−	0.406520i	$-0.866742\pi$
$941$	−15.5563	+	15.5563i	−0.507122	+	0.507122i	0.406520	+	0.913642i	$0.366742\pi$
$942$		0			0
$943$	−50.9117			−1.65791
$944$		0			0
$945$	32.0000			1.04096
$946$		0			0
$947$	29.0000	−	29.0000i	0.942373	−	0.942373i	−0.0560543	−	0.998428i	$-0.517852\pi$
$947$	29.0000	−	29.0000i	0.942373	−	0.942373i	0.998428	+	0.0560543i	$0.0178520\pi$
$948$		0			0
$949$	50.9117	+	50.9117i	1.65266	+	1.65266i
$950$		0			0
$951$			14.1421i			0.458590i
$952$		0			0
$953$		−	6.00000i		−	0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$		−	6.00000i		−	0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$		0			0
$955$	−24.0000	−	24.0000i	−0.776622	−	0.776622i
$956$		0			0
$957$	8.48528	−	8.48528i	0.274290	−	0.274290i
$958$		0			0
$959$	16.9706			0.548008
$960$		0			0
$961$	1.00000			0.0322581
$962$		0			0
$963$	−5.00000	+	5.00000i	−0.161123	+	0.161123i
$964$		0			0
$965$	33.9411	+	33.9411i	1.09260	+	1.09260i
$966$		0			0
$967$			31.1127i			1.00052i	0.865876	+	0.500258i	$0.166762\pi$
$967$			31.1127i			1.00052i	−0.865876	+	0.500258i	$0.833238\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−21.0000	−	21.0000i	−0.673922	−	0.673922i	0.284696	−	0.958618i	$-0.408107\pi$
$971$	−21.0000	−	21.0000i	−0.673922	−	0.673922i	−0.958618	+	0.284696i	$0.908107\pi$
$972$		0			0
$973$	8.48528	−	8.48528i	0.272026	−	0.272026i
$974$		0			0
$975$	−8.48528			−0.271746
$976$		0			0
$977$	48.0000			1.53566			0.767828	−	0.640656i	$-0.221338\pi$
$977$	48.0000			1.53566			0.767828	+	0.640656i	$0.221338\pi$
$978$		0			0
$979$	−36.0000	+	36.0000i	−1.15056	+	1.15056i
$980$		0			0
$981$	−4.24264	−	4.24264i	−0.135457	−	0.135457i
$982$		0			0
$983$			42.4264i			1.35319i	0.736354	+	0.676596i	$0.236546\pi$
$983$			42.4264i			1.35319i	−0.736354	+	0.676596i	$0.763454\pi$
$984$		0			0
$985$			20.0000i			0.637253i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	25.4558	−	25.4558i	0.809449	−	0.809449i
$990$		0			0
$991$	45.2548			1.43757			0.718784	−	0.695234i	$-0.244699\pi$
$991$	45.2548			1.43757			0.718784	+	0.695234i	$0.244699\pi$
$992$		0			0
$993$	42.0000			1.33283
$994$		0			0
$995$	−28.0000	+	28.0000i	−0.887660	+	0.887660i
$996$		0			0
$997$	−12.7279	−	12.7279i	−0.403097	−	0.403097i	0.476226	−	0.879323i	$-0.342005\pi$
$997$	−12.7279	−	12.7279i	−0.403097	−	0.403097i	−0.879323	+	0.476226i	$0.842005\pi$
$998$		0			0
$999$			33.9411i			1.07385i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.2.e.n.769.2		4
4.3	odd	2		1024.2.e.h.769.2		4
8.3	odd	2	inner	1024.2.e.n.769.1		4
8.5	even	2		1024.2.e.h.769.1		4
16.3	odd	4		1024.2.e.h.257.2		4
16.5	even	4		1024.2.e.h.257.1		4
16.11	odd	4	inner	1024.2.e.n.257.1		4
16.13	even	4	inner	1024.2.e.n.257.2		4
32.3	odd	8		512.2.a.b.1.1	✓	2
32.5	even	8		512.2.b.e.257.4		4
32.11	odd	8		512.2.b.e.257.3		4
32.13	even	8		512.2.a.e.1.1	yes	2
32.19	odd	8		512.2.a.e.1.2	yes	2
32.21	even	8		512.2.b.e.257.1		4
32.27	odd	8		512.2.b.e.257.2		4
32.29	even	8		512.2.a.b.1.2	yes	2
96.5	odd	8		4608.2.d.j.2305.2		4
96.11	even	8		4608.2.d.j.2305.3		4
96.29	odd	8		4608.2.a.p.1.1		2
96.35	even	8		4608.2.a.p.1.2		2
96.53	odd	8		4608.2.d.j.2305.4		4
96.59	even	8		4608.2.d.j.2305.1		4
96.77	odd	8		4608.2.a.c.1.1		2
96.83	even	8		4608.2.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.a.b.1.1	✓	2	32.3	odd	8
512.2.a.b.1.2	yes	2	32.29	even	8
512.2.a.e.1.1	yes	2	32.13	even	8
512.2.a.e.1.2	yes	2	32.19	odd	8
512.2.b.e.257.1		4	32.21	even	8
512.2.b.e.257.2		4	32.27	odd	8
512.2.b.e.257.3		4	32.11	odd	8
512.2.b.e.257.4		4	32.5	even	8
1024.2.e.h.257.1		4	16.5	even	4
1024.2.e.h.257.2		4	16.3	odd	4
1024.2.e.h.769.1		4	8.5	even	2
1024.2.e.h.769.2		4	4.3	odd	2
1024.2.e.n.257.1		4	16.11	odd	4	inner
1024.2.e.n.257.2		4	16.13	even	4	inner
1024.2.e.n.769.1		4	8.3	odd	2	inner
1024.2.e.n.769.2		4	1.1	even	1	trivial
4608.2.a.c.1.1		2	96.77	odd	8
4608.2.a.c.1.2		2	96.83	even	8
4608.2.a.p.1.1		2	96.29	odd	8
4608.2.a.p.1.2		2	96.35	even	8
4608.2.d.j.2305.1		4	96.59	even	8
4608.2.d.j.2305.2		4	96.5	odd	8
4608.2.d.j.2305.3		4	96.11	even	8
4608.2.d.j.2305.4		4	96.53	odd	8

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:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1024.e (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{3} +(1.41421 + 1.41421i) q^{5} -2.82843i q^{7} +1.00000i q^{9} +(-3.00000 - 3.00000i) q^{11} +(4.24264 - 4.24264i) q^{13} +2.82843 q^{15} +(-3.00000 + 3.00000i) q^{19} +(-2.82843 - 2.82843i) q^{21} -8.48528i q^{23} -1.00000i q^{25} +(4.00000 + 4.00000i) q^{27} +(-1.41421 + 1.41421i) q^{29} +5.65685 q^{31} -6.00000 q^{33} +(4.00000 - 4.00000i) q^{35} +(4.24264 + 4.24264i) q^{37} -8.48528i q^{39} -6.00000i q^{41} +(3.00000 + 3.00000i) q^{43} +(-1.41421 + 1.41421i) q^{45} -1.00000 q^{49} +(1.41421 + 1.41421i) q^{53} -8.48528i q^{55} +6.00000i q^{57} +(1.00000 + 1.00000i) q^{59} +(-4.24264 + 4.24264i) q^{61} +2.82843 q^{63} +12.0000 q^{65} +(-9.00000 + 9.00000i) q^{67} +(-8.48528 - 8.48528i) q^{69} -8.48528i q^{71} +12.0000i q^{73} +(-1.00000 - 1.00000i) q^{75} +(-8.48528 + 8.48528i) q^{77} +5.65685 q^{79} +5.00000 q^{81} +(3.00000 - 3.00000i) q^{83} +2.82843i q^{87} -12.0000i q^{89} +(-12.0000 - 12.0000i) q^{91} +(5.65685 - 5.65685i) q^{93} -8.48528 q^{95} -8.00000 q^{97} +(3.00000 - 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 12 q^{11} - 12 q^{19} + 16 q^{27} - 24 q^{33} + 16 q^{35} + 12 q^{43} - 4 q^{49} + 4 q^{59} + 48 q^{65} - 36 q^{67} - 4 q^{75} + 20 q^{81} + 12 q^{83} - 48 q^{91} - 32 q^{97} + 12 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 12 * q^11 - 12 * q^19 + 16 * q^27 - 24 * q^33 + 16 * q^35 + 12 * q^43 - 4 * q^49 + 4 * q^59 + 48 * q^65 - 36 * q^67 - 4 * q^75 + 20 * q^81 + 12 * q^83 - 48 * q^91 - 32 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more