LMFDB - Embedded newform 1024.2.e.k.769.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1024,2,Mod(257,1024)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			769.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1024.769
Dual form			1024.2.e.k.257.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$		0			0
$5$	−2.82843	−	2.82843i	−1.26491	−	1.26491i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$	−2.82843	−	2.82843i	−1.26491	−	1.26491i	−0.316228	−	0.948683i	$-0.602416\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$			3.00000i			1.00000i
$10$		0			0
$11$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$11$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$12$		0			0
$13$	−2.82843	+	2.82843i	−0.784465	+	0.784465i	−0.980581	−	0.196116i	$-0.937167\pi$
$13$	−2.82843	+	2.82843i	−0.784465	+	0.784465i	0.196116	+	0.980581i	$0.437167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$19$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$			11.0000i			2.20000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.82843	−	2.82843i	0.525226	−	0.525226i	−0.393919	−	0.919145i	$-0.628881\pi$
$29$	2.82843	−	2.82843i	0.525226	−	0.525226i	0.919145	+	0.393919i	$0.128881\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	8.48528	+	8.48528i	1.39497	+	1.39497i	0.813733	+	0.581238i	$0.197432\pi$
$37$	8.48528	+	8.48528i	1.39497	+	1.39497i	0.581238	+	0.813733i	$0.302568\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			10.0000i			1.56174i	0.624695	+	0.780869i	$0.285223\pi$
$41$			10.0000i			1.56174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	8.48528	−	8.48528i	1.26491	−	1.26491i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.82843	+	2.82843i	0.388514	+	0.388514i	0.874157	−	0.485643i	$-0.161414\pi$
$53$	2.82843	+	2.82843i	0.388514	+	0.388514i	−0.485643	+	0.874157i	$0.661414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$59$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$60$		0			0
$61$	−8.48528	+	8.48528i	−1.08643	+	1.08643i	−0.0905357	+	0.995893i	$0.528858\pi$
$61$	−8.48528	+	8.48528i	−1.08643	+	1.08643i	−0.995893	+	0.0905357i	$0.971142\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	16.0000			1.98456
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$			6.00000i			0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	−5.65685	−	5.65685i	−0.613572	−	0.613572i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			10.0000i			1.06000i	0.847998	+	0.529999i	$0.177808\pi$
$89$			10.0000i			1.06000i	−0.847998	+	0.529999i	$0.822192\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−18.0000			−1.82762			−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−18.0000			−1.82762			−0.913812	+	0.406138i	$0.866875\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−14.1421	−	14.1421i	−1.40720	−	1.40720i	−0.773957	−	0.633238i	$-0.781726\pi$
$101$	−14.1421	−	14.1421i	−1.40720	−	1.40720i	−0.633238	−	0.773957i	$-0.718274\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$	−14.1421	+	14.1421i	−1.35457	+	1.35457i	−0.474100	+	0.880471i	$0.657226\pi$
$109$	−14.1421	+	14.1421i	−1.35457	+	1.35457i	−0.880471	+	0.474100i	$0.842774\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	14.0000			1.31701			0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000			1.31701			0.658505	+	0.752577i	$0.271189\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−8.48528	−	8.48528i	−0.784465	−	0.784465i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	11.0000i		−	1.00000i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	16.9706	−	16.9706i	1.51789	−	1.51789i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$131$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	22.0000i		−	1.87959i	−0.341743	−	0.939793i	$-0.611017\pi$
$137$		−	22.0000i		−	1.87959i	0.341743	−	0.939793i	$-0.388983\pi$
$138$		0			0
$139$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$139$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−16.0000			−1.32873
$146$		0			0
$147$		0			0
$148$		0			0
$149$	14.1421	+	14.1421i	1.15857	+	1.15857i	0.984784	+	0.173785i	$0.0555999\pi$
$149$	14.1421	+	14.1421i	1.15857	+	1.15857i	0.173785	+	0.984784i	$0.444400\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$			6.00000i			0.485071i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−8.48528	+	8.48528i	−0.677199	+	0.677199i	−0.959366	−	0.282166i	$-0.908947\pi$
$157$	−8.48528	+	8.48528i	−0.677199	+	0.677199i	0.282166	+	0.959366i	$0.408947\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$163$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$		−	3.00000i		−	0.230769i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.82843	+	2.82843i	−0.215041	+	0.215041i	−0.806405	−	0.591364i	$-0.798590\pi$
$173$	−2.82843	+	2.82843i	−0.215041	+	0.215041i	0.591364	+	0.806405i	$0.298590\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$179$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$180$		0			0
$181$	14.1421	+	14.1421i	1.05118	+	1.05118i	0.998618	+	0.0525588i	$0.0167377\pi$
$181$	14.1421	+	14.1421i	1.05118	+	1.05118i	0.0525588	+	0.998618i	$0.483262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	48.0000i		−	3.52903i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	14.0000			1.00774			0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000			1.00774			0.503871	+	0.863779i	$0.331909\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	19.7990	+	19.7990i	1.41062	+	1.41062i	0.755689	+	0.654931i	$0.227302\pi$
$197$	19.7990	+	19.7990i	1.41062	+	1.41062i	0.654931	+	0.755689i	$0.272698\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	28.2843	−	28.2843i	1.97546	−	1.97546i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$211$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−5.65685	+	5.65685i	−0.380521	+	0.380521i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	−33.0000			−2.20000
$226$		0			0
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$	−2.82843	−	2.82843i	−0.186908	−	0.186908i	0.607450	−	0.794358i	$-0.292192\pi$
$229$	−2.82843	−	2.82843i	−0.186908	−	0.186908i	−0.794358	+	0.607450i	$0.792192\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	26.0000i		−	1.70332i	−0.524097	−	0.851658i	$-0.675597\pi$
$233$		−	26.0000i		−	1.70332i	0.524097	−	0.851658i	$-0.324403\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−30.0000			−1.93247			−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000			−1.93247			−0.966235	+	0.257663i	$0.917048\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−19.7990	−	19.7990i	−1.26491	−	1.26491i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$251$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.00000			0.124757			0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000			0.124757			0.0623783	+	0.998053i	$0.480131\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	8.48528	+	8.48528i	0.525226	+	0.525226i
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		−	16.0000i		−	0.982872i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−14.1421	+	14.1421i	−0.862261	+	0.862261i	−0.991600	−	0.129339i	$-0.958714\pi$
$269$	−14.1421	+	14.1421i	−0.862261	+	0.862261i	0.129339	+	0.991600i	$0.458714\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−19.7990	−	19.7990i	−1.18961	−	1.18961i	−0.977176	−	0.212430i	$-0.931862\pi$
$277$	−19.7990	−	19.7990i	−1.18961	−	1.18961i	−0.212430	−	0.977176i	$-0.568138\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			10.0000i			0.596550i	0.954480	+	0.298275i	$0.0964112\pi$
$281$			10.0000i			0.596550i	−0.954480	+	0.298275i	$0.903589\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−2.82843	−	2.82843i	−0.165238	−	0.165238i	0.619644	−	0.784883i	$-0.287277\pi$
$293$	−2.82843	−	2.82843i	−0.165238	−	0.165238i	−0.784883	+	0.619644i	$0.787277\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	48.0000			2.74847
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		−	26.0000i		−	1.46961i	−0.678280	−	0.734803i	$-0.737274\pi$
$313$		−	26.0000i		−	1.46961i	0.678280	−	0.734803i	$-0.262726\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−19.7990	+	19.7990i	−1.11202	+	1.11202i	−0.119145	+	0.992877i	$0.538015\pi$
$317$	−19.7990	+	19.7990i	−1.11202	+	1.11202i	−0.992877	+	0.119145i	$0.961985\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−31.1127	−	31.1127i	−1.72582	−	1.72582i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$331$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$332$		0			0
$333$	−25.4558	+	25.4558i	−1.39497	+	1.39497i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−18.0000			−0.980522			−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000			−0.980522			−0.490261	+	0.871576i	$0.663099\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$	25.4558	−	25.4558i	1.36262	−	1.36262i	0.492057	−	0.870563i	$-0.336245\pi$
$349$	25.4558	−	25.4558i	1.36262	−	1.36262i	0.870563	−	0.492057i	$-0.163755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	34.0000			1.80964			0.904819	−	0.425797i	$-0.140006\pi$
$353$	34.0000			1.80964			0.904819	+	0.425797i	$0.140006\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$			19.0000i			1.00000i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	16.9706	−	16.9706i	0.888280	−	0.888280i
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$	−30.0000			−1.56174
$370$		0			0
$371$		0			0
$372$		0			0
$373$	25.4558	+	25.4558i	1.31805	+	1.31805i	0.915315	+	0.402739i	$0.131942\pi$
$373$	25.4558	+	25.4558i	1.31805	+	1.31805i	0.402739	+	0.915315i	$0.368058\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			16.0000i			0.824042i
$378$		0			0
$379$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$379$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−14.1421	−	14.1421i	−0.717035	−	0.717035i	0.250962	−	0.967997i	$-0.419253\pi$
$389$	−14.1421	−	14.1421i	−0.717035	−	0.717035i	−0.967997	+	0.250962i	$0.919253\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	8.48528	−	8.48528i	0.425864	−	0.425864i	−0.461353	−	0.887217i	$-0.652636\pi$
$397$	8.48528	−	8.48528i	0.425864	−	0.425864i	0.887217	+	0.461353i	$0.152636\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.00000			0.0998752			0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000			0.0998752			0.0499376	+	0.998752i	$0.484098\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	25.4558	+	25.4558i	1.26491	+	1.26491i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			6.00000i			0.296681i	0.988936	+	0.148340i	$0.0473931\pi$
$409$			6.00000i			0.296681i	−0.988936	+	0.148340i	$0.952607\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$419$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$420$		0			0
$421$	19.7990	+	19.7990i	0.964944	+	0.964944i	0.999406	−	0.0344623i	$-0.0109719\pi$
$421$	19.7990	+	19.7990i	0.964944	+	0.964944i	−0.0344623	+	0.999406i	$0.510972\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			22.0000i			1.06716i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	34.0000			1.63394			0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000			1.63394			0.816968	+	0.576683i	$0.195653\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$			21.0000i			1.00000i
$442$		0			0
$443$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$443$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$444$		0			0
$445$	28.2843	−	28.2843i	1.34080	−	1.34080i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	14.0000			0.660701			0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000			0.660701			0.330350	+	0.943858i	$0.392833\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			42.0000i			1.96468i	0.187112	+	0.982339i	$0.440087\pi$
$457$			42.0000i			1.96468i	−0.187112	+	0.982339i	$0.559913\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−14.1421	+	14.1421i	−0.658665	+	0.658665i	−0.955064	−	0.296399i	$-0.904214\pi$
$461$	−14.1421	+	14.1421i	−0.658665	+	0.658665i	0.296399	+	0.955064i	$0.404214\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$467$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−8.48528	+	8.48528i	−0.388514	+	0.388514i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$	−48.0000			−2.18861
$482$		0			0
$483$		0			0
$484$		0			0
$485$	50.9117	+	50.9117i	2.31178	+	2.31178i
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$491$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$492$		0			0
$493$	5.65685	−	5.65685i	0.254772	−	0.254772i
$494$		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	$0.250000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$			80.0000i			3.55995i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−31.1127	+	31.1127i	−1.37905	+	1.37905i	−0.532813	+	0.846233i	$0.678865\pi$
$509$	−31.1127	+	31.1127i	−1.37905	+	1.37905i	−0.846233	+	0.532813i	$0.821135\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	22.0000i		−	0.963837i	−0.876216	−	0.481919i	$-0.839940\pi$
$521$		−	22.0000i		−	0.963837i	0.876216	−	0.481919i	$-0.160060\pi$
$522$		0			0
$523$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$523$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−28.2843	−	28.2843i	−1.22513	−	1.22513i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	14.1421	−	14.1421i	0.608018	−	0.608018i	−0.334410	−	0.942428i	$-0.608537\pi$
$541$	14.1421	−	14.1421i	0.608018	−	0.608018i	0.942428	+	0.334410i	$0.108537\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	80.0000			3.42682
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$		0			0
$549$	−25.4558	−	25.4558i	−1.08643	−	1.08643i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	19.7990	−	19.7990i	0.838910	−	0.838910i	−0.149805	−	0.988716i	$-0.547865\pi$
$557$	19.7990	−	19.7990i	0.838910	−	0.838910i	0.988716	+	0.149805i	$0.0478647\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$	−39.5980	−	39.5980i	−1.66590	−	1.66590i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	26.0000i		−	1.08998i	−0.838444	−	0.544988i	$-0.816534\pi$
$569$		−	26.0000i		−	1.08998i	0.838444	−	0.544988i	$-0.183466\pi$
$570$		0			0
$571$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$571$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	2.00000			0.0832611			0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000			0.0832611			0.0416305	+	0.999133i	$0.486745\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$			48.0000i			1.98456i
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	46.0000			1.88899			0.944497	−	0.328521i	$-0.106550\pi$
$593$	46.0000			1.88899			0.944497	+	0.328521i	$0.106550\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$			10.0000i			0.407909i	0.978980	+	0.203954i	$0.0653794\pi$
$601$			10.0000i			0.407909i	−0.978980	+	0.203954i	$0.934621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−31.1127	+	31.1127i	−1.26491	+	1.26491i
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−25.4558	−	25.4558i	−1.02815	−	1.02815i	−0.999592	−	0.0285598i	$-0.990908\pi$
$613$	−25.4558	−	25.4558i	−1.02815	−	1.02815i	−0.0285598	−	0.999592i	$-0.509092\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			38.0000i			1.52982i	0.644136	+	0.764911i	$0.277217\pi$
$617$			38.0000i			1.52982i	−0.644136	+	0.764911i	$0.722783\pi$
$618$		0			0
$619$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$619$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−41.0000			−1.64000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	16.9706	+	16.9706i	0.676661	+	0.676661i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−19.7990	+	19.7990i	−0.784465	+	0.784465i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−50.0000			−1.97488			−0.987441	−	0.157991i	$-0.949498\pi$
$641$	−50.0000			−1.97488			−0.987441	+	0.157991i	$0.949498\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	31.1127	−	31.1127i	1.21753	−	1.21753i	0.249041	−	0.968493i	$-0.419885\pi$
$653$	31.1127	−	31.1127i	1.21753	−	1.21753i	0.968493	−	0.249041i	$-0.0801154\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−18.0000			−0.702247
$658$		0			0
$659$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$659$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$660$		0			0
$661$	−8.48528	−	8.48528i	−0.330039	−	0.330039i	0.522562	−	0.852601i	$-0.324976\pi$
$661$	−8.48528	−	8.48528i	−0.330039	−	0.330039i	−0.852601	+	0.522562i	$0.824976\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	46.0000			1.77317			0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000			1.77317			0.886585	+	0.462566i	$0.153071\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−36.7696	−	36.7696i	−1.41317	−	1.41317i	−0.733761	−	0.679408i	$-0.762237\pi$
$677$	−36.7696	−	36.7696i	−1.41317	−	1.41317i	−0.679408	−	0.733761i	$-0.737763\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$	−62.2254	+	62.2254i	−2.37751	+	2.37751i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−16.0000			−0.609551
$690$		0			0
$691$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$691$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$			20.0000i			0.757554i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	36.7696	−	36.7696i	1.38877	−	1.38877i	0.560848	−	0.827919i	$-0.310475\pi$
$701$	36.7696	−	36.7696i	1.38877	−	1.38877i	0.827919	−	0.560848i	$-0.189525\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	31.1127	+	31.1127i	1.16846	+	1.16846i	0.982570	+	0.185892i	$0.0595174\pi$
$709$	31.1127	+	31.1127i	1.16846	+	1.16846i	0.185892	+	0.982570i	$0.440483\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	31.1127	+	31.1127i	1.15550	+	1.15550i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	2.82843	−	2.82843i	0.104470	−	0.104470i	−0.652940	−	0.757410i	$-0.726464\pi$
$733$	2.82843	−	2.82843i	0.104470	−	0.104470i	0.757410	+	0.652940i	$0.226464\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$739$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		−	80.0000i		−	2.93097i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	36.7696	+	36.7696i	1.33641	+	1.33641i	0.899508	+	0.436904i	$0.143925\pi$
$757$	36.7696	+	36.7696i	1.33641	+	1.33641i	0.436904	+	0.899508i	$0.356075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			38.0000i			1.37750i	0.724999	+	0.688749i	$0.241840\pi$
$761$			38.0000i			1.37750i	−0.724999	+	0.688749i	$0.758160\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	16.9706	−	16.9706i	0.613572	−	0.613572i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−50.0000			−1.80305			−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000			−1.80305			−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	31.1127	+	31.1127i	1.11905	+	1.11905i	0.991882	+	0.127164i	$0.0405876\pi$
$773$	31.1127	+	31.1127i	1.11905	+	1.11905i	0.127164	+	0.991882i	$0.459412\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	48.0000			1.71319
$786$		0			0
$787$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$787$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	48.0000i		−	1.70453i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	36.7696	−	36.7696i	1.30244	−	1.30244i	0.375705	−	0.926739i	$-0.377401\pi$
$797$	36.7696	−	36.7696i	1.30244	−	1.30244i	0.926739	−	0.375705i	$-0.122599\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−30.0000			−1.06000
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			10.0000i			0.351581i	0.984428	+	0.175791i	$0.0562482\pi$
$809$			10.0000i			0.351581i	−0.984428	+	0.175791i	$0.943752\pi$
$810$		0			0
$811$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$811$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−19.7990	−	19.7990i	−0.690990	−	0.690990i	0.271460	−	0.962450i	$-0.412493\pi$
$821$	−19.7990	−	19.7990i	−0.690990	−	0.690990i	−0.962450	+	0.271460i	$0.912493\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$	14.1421	−	14.1421i	0.491177	−	0.491177i	−0.417500	−	0.908677i	$-0.637094\pi$
$829$	14.1421	−	14.1421i	0.491177	−	0.491177i	0.908677	+	0.417500i	$0.137094\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	14.0000			0.485071
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$			13.0000i			0.448276i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−8.48528	+	8.48528i	−0.291903	+	0.291903i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	25.4558	+	25.4558i	0.871592	+	0.871592i	0.992646	−	0.121054i	$-0.0386275\pi$
$853$	25.4558	+	25.4558i	0.871592	+	0.871592i	−0.121054	+	0.992646i	$0.538628\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	58.0000i		−	1.98124i	−0.136637	−	0.990621i	$-0.543630\pi$
$857$		−	58.0000i		−	1.98124i	0.136637	−	0.990621i	$-0.456370\pi$
$858$		0			0
$859$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$859$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	16.0000			0.544016
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		−	54.0000i		−	1.82762i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	8.48528	−	8.48528i	0.286528	−	0.286528i	−0.549178	−	0.835705i	$-0.685059\pi$
$877$	8.48528	−	8.48528i	0.286528	−	0.286528i	0.835705	+	0.549178i	$0.185059\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−50.0000			−1.68454			−0.842271	−	0.539054i	$-0.818782\pi$
$881$	−50.0000			−1.68454			−0.842271	+	0.539054i	$0.818782\pi$
$882$		0			0
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	5.65685	+	5.65685i	0.188457	+	0.188457i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	80.0000i		−	2.65929i
$906$		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
$909$	42.4264	−	42.4264i	1.40720	−	1.40720i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−93.3381	+	93.3381i	−3.06894	+	3.06894i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	46.0000			1.50921			0.754606	−	0.656179i	$-0.227828\pi$
$929$	46.0000			1.50921			0.754606	+	0.656179i	$0.227828\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			38.0000i			1.24141i	0.784046	+	0.620703i	$0.213153\pi$
$937$			38.0000i			1.24141i	−0.784046	+	0.620703i	$0.786847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−14.1421	+	14.1421i	−0.461020	+	0.461020i	−0.898990	−	0.437969i	$-0.855698\pi$
$941$	−14.1421	+	14.1421i	−0.461020	+	0.461020i	0.437969	+	0.898990i	$0.355698\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$947$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$948$		0			0
$949$	−16.9706	−	16.9706i	−0.550888	−	0.550888i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	26.0000i		−	0.842223i	−0.907009	−	0.421111i	$-0.861640\pi$
$953$		−	26.0000i		−	0.842223i	0.907009	−	0.421111i	$-0.138360\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−39.5980	−	39.5980i	−1.27470	−	1.27470i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$971$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−62.0000			−1.98356			−0.991778	−	0.127971i	$-0.959153\pi$
$977$	−62.0000			−1.98356			−0.991778	+	0.127971i	$0.959153\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−42.4264	−	42.4264i	−1.35457	−	1.35457i
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		−	112.000i		−	3.56862i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	8.48528	+	8.48528i	0.268732	+	0.268732i	0.828589	−	0.559857i	$-0.189144\pi$
$997$	8.48528	+	8.48528i	0.268732	+	0.268732i	−0.559857	+	0.828589i	$0.689144\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.2.e.k.769.1		4
4.3	odd	2	CM	1024.2.e.k.769.1		4
8.3	odd	2	inner	1024.2.e.k.769.2		4
8.5	even	2	inner	1024.2.e.k.769.2		4
16.3	odd	4	inner	1024.2.e.k.257.1		4
16.5	even	4	inner	1024.2.e.k.257.2		4
16.11	odd	4	inner	1024.2.e.k.257.2		4
16.13	even	4	inner	1024.2.e.k.257.1		4
32.3	odd	8		256.2.a.c.1.1		1
32.5	even	8		128.2.b.b.65.1	✓	2
32.11	odd	8		128.2.b.b.65.2	yes	2
32.13	even	8		256.2.a.b.1.1		1
32.19	odd	8		256.2.a.b.1.1		1
32.21	even	8		128.2.b.b.65.2	yes	2
32.27	odd	8		128.2.b.b.65.1	✓	2
32.29	even	8		256.2.a.c.1.1		1
96.5	odd	8		1152.2.d.d.577.2		2
96.11	even	8		1152.2.d.d.577.1		2
96.29	odd	8		2304.2.a.a.1.1		1
96.35	even	8		2304.2.a.a.1.1		1
96.53	odd	8		1152.2.d.d.577.1		2
96.59	even	8		1152.2.d.d.577.2		2
96.77	odd	8		2304.2.a.p.1.1		1
96.83	even	8		2304.2.a.p.1.1		1
160.19	odd	8		6400.2.a.m.1.1		1
160.27	even	8		3200.2.f.d.449.1		2
160.29	even	8		6400.2.a.l.1.1		1
160.37	odd	8		3200.2.f.d.449.1		2
160.43	even	8		3200.2.f.d.449.2		2
160.53	odd	8		3200.2.f.d.449.2		2
160.59	odd	8		3200.2.d.e.1601.1		2
160.69	even	8		3200.2.d.e.1601.1		2
160.99	odd	8		6400.2.a.l.1.1		1
160.107	even	8		3200.2.f.c.449.1		2
160.109	even	8		6400.2.a.m.1.1		1
160.117	odd	8		3200.2.f.c.449.1		2
160.123	even	8		3200.2.f.c.449.2		2
160.133	odd	8		3200.2.f.c.449.2		2
160.139	odd	8		3200.2.d.e.1601.2		2
160.149	even	8		3200.2.d.e.1601.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.2.b.b.65.1	✓	2	32.5	even	8
128.2.b.b.65.1	✓	2	32.27	odd	8
128.2.b.b.65.2	yes	2	32.11	odd	8
128.2.b.b.65.2	yes	2	32.21	even	8
256.2.a.b.1.1		1	32.13	even	8
256.2.a.b.1.1		1	32.19	odd	8
256.2.a.c.1.1		1	32.3	odd	8
256.2.a.c.1.1		1	32.29	even	8
1024.2.e.k.257.1		4	16.3	odd	4	inner
1024.2.e.k.257.1		4	16.13	even	4	inner
1024.2.e.k.257.2		4	16.5	even	4	inner
1024.2.e.k.257.2		4	16.11	odd	4	inner
1024.2.e.k.769.1		4	1.1	even	1	trivial
1024.2.e.k.769.1		4	4.3	odd	2	CM
1024.2.e.k.769.2		4	8.3	odd	2	inner
1024.2.e.k.769.2		4	8.5	even	2	inner
1152.2.d.d.577.1		2	96.11	even	8
1152.2.d.d.577.1		2	96.53	odd	8
1152.2.d.d.577.2		2	96.5	odd	8
1152.2.d.d.577.2		2	96.59	even	8
2304.2.a.a.1.1		1	96.29	odd	8
2304.2.a.a.1.1		1	96.35	even	8
2304.2.a.p.1.1		1	96.77	odd	8
2304.2.a.p.1.1		1	96.83	even	8
3200.2.d.e.1601.1		2	160.59	odd	8
3200.2.d.e.1601.1		2	160.69	even	8
3200.2.d.e.1601.2		2	160.139	odd	8
3200.2.d.e.1601.2		2	160.149	even	8
3200.2.f.c.449.1		2	160.107	even	8
3200.2.f.c.449.1		2	160.117	odd	8
3200.2.f.c.449.2		2	160.123	even	8
3200.2.f.c.449.2		2	160.133	odd	8
3200.2.f.d.449.1		2	160.27	even	8
3200.2.f.d.449.1		2	160.37	odd	8
3200.2.f.d.449.2		2	160.43	even	8
3200.2.f.d.449.2		2	160.53	odd	8
6400.2.a.l.1.1		1	160.29	even	8
6400.2.a.l.1.1		1	160.99	odd	8
6400.2.a.m.1.1		1	160.19	odd	8
6400.2.a.m.1.1		1	160.109	even	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(-2.82843 - 2.82843i) q^{5} +3.00000i q^{9} +(-2.82843 + 2.82843i) q^{13} +2.00000 q^{17} +11.0000i q^{25} +(2.82843 - 2.82843i) q^{29} +(8.48528 + 8.48528i) q^{37} +10.0000i q^{41} +(8.48528 - 8.48528i) q^{45} +7.00000 q^{49} +(2.82843 + 2.82843i) q^{53} +(-8.48528 + 8.48528i) q^{61} +16.0000 q^{65} +6.00000i q^{73} -9.00000 q^{81} +(-5.65685 - 5.65685i) q^{85} +10.0000i q^{89} -18.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{17} + 28 q^{49} + 64 q^{65} - 36 q^{81} - 72 q^{97}+O(q^{100}) \) 4 * q + 8 * q^17 + 28 * q^49 + 64 * q^65 - 36 * q^81 - 72 * q^97

Introduction

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