LMFDB - Embedded newform 1056.3.p.d.527.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1056,3,Mod(527,1056)] 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("1056.527"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1056, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1056 = 2^{5} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1056.p (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,0,0,0,0,-14,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$28.7739159164$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 264)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			527.1
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	1056.527
Dual form			1056.3.p.d.527.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 2.82843i) q^{3} +(-7.00000 + 5.65685i) q^{9} +(7.00000 - 8.48528i) q^{11} -2.00000 q^{17} -16.9706i q^{19} -25.0000 q^{25} +(23.0000 + 14.1421i) q^{27} +(-31.0000 - 11.3137i) q^{33} -46.0000 q^{41} -84.8528i q^{43} -49.0000 q^{49} +(2.00000 + 5.65685i) q^{51} +(-48.0000 + 16.9706i) q^{57} +84.8528i q^{59} -62.0000 q^{67} +33.9411i q^{73} +(25.0000 + 70.7107i) q^{75} +(17.0000 - 79.1960i) q^{81} -158.000 q^{83} +101.823i q^{89} +94.0000 q^{97} +(-1.00000 + 98.9949i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 14 q^{9} + 14 q^{11} - 4 q^{17} - 50 q^{25} + 46 q^{27} - 62 q^{33} - 92 q^{41} - 98 q^{49} + 4 q^{51} - 96 q^{57} - 124 q^{67} + 50 q^{75} + 34 q^{81} - 316 q^{83} + 188 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 14 * q^9 + 14 * q^11 - 4 * q^17 - 50 * q^25 + 46 * q^27 - 62 * q^33 - 92 * q^41 - 98 * q^49 + 4 * q^51 - 96 * q^57 - 124 * q^67 + 50 * q^75 + 34 * q^81 - 316 * q^83 + 188 * q^97 - 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1056\mathbb{Z}\right)^\times$.

$n$	$133$	$353$	$673$	$991$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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	−	2.82843i	−0.333333	−	0.942809i
$3$	−1.00000	−	2.82843i	−0.333333	−	0.942809i
$4$		0			0
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$		0			0
$9$	−7.00000	+	5.65685i	−0.777778	+	0.628539i
$10$		0			0
$11$	7.00000	−	8.48528i	0.636364	−	0.771389i
$11$	7.00000	−	8.48528i	0.636364	−	0.771389i
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.00000			−0.117647			−0.0588235	−	0.998268i	$-0.518735\pi$
$17$	−2.00000			−0.117647			−0.0588235	+	0.998268i	$0.518735\pi$
$18$		0			0
$19$		−	16.9706i		−	0.893188i	−0.894737	−	0.446594i	$-0.852637\pi$
$19$		−	16.9706i		−	0.893188i	0.894737	−	0.446594i	$-0.147363\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	−25.0000			−1.00000
$26$		0			0
$27$	23.0000	+	14.1421i	0.851852	+	0.523783i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$	−31.0000	−	11.3137i	−0.939394	−	0.342840i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−46.0000			−1.12195			−0.560976	−	0.827832i	$-0.689574\pi$
$41$	−46.0000			−1.12195			−0.560976	+	0.827832i	$0.689574\pi$
$42$		0			0
$43$		−	84.8528i		−	1.97332i	−0.162791	−	0.986661i	$-0.552050\pi$
$43$		−	84.8528i		−	1.97332i	0.162791	−	0.986661i	$-0.447950\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−49.0000			−1.00000
$50$		0			0
$51$	2.00000	+	5.65685i	0.0392157	+	0.110919i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−48.0000	+	16.9706i	−0.842105	+	0.297729i
$58$		0			0
$59$			84.8528i			1.43818i	0.694915	+	0.719092i	$0.255442\pi$
$59$			84.8528i			1.43818i	−0.694915	+	0.719092i	$0.744558\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−62.0000			−0.925373			−0.462687	−	0.886522i	$-0.653114\pi$
$67$	−62.0000			−0.925373			−0.462687	+	0.886522i	$0.653114\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$			33.9411i			0.464947i	0.972603	+	0.232473i	$0.0746819\pi$
$73$			33.9411i			0.464947i	−0.972603	+	0.232473i	$0.925318\pi$
$74$		0			0
$75$	25.0000	+	70.7107i	0.333333	+	0.942809i
$76$		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
$81$	17.0000	−	79.1960i	0.209877	−	0.977728i
$82$		0			0
$83$	−158.000			−1.90361			−0.951807	−	0.306697i	$-0.900776\pi$
$83$	−158.000			−1.90361			−0.951807	+	0.306697i	$0.900776\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			101.823i			1.14408i	0.820225	+	0.572041i	$0.193848\pi$
$89$			101.823i			1.14408i	−0.820225	+	0.572041i	$0.806152\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	94.0000			0.969072			0.484536	−	0.874771i	$-0.338988\pi$
$97$	94.0000			0.969072			0.484536	+	0.874771i	$0.338988\pi$
$98$		0			0
$99$	−1.00000	+	98.9949i	−0.0101010	+	0.999949i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−178.000			−1.66355			−0.831776	−	0.555112i	$-0.812675\pi$
$107$	−178.000			−1.66355			−0.831776	+	0.555112i	$0.812675\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			203.647i			1.80218i	0.433628	+	0.901092i	$0.357233\pi$
$113$			203.647i			1.80218i	−0.433628	+	0.901092i	$0.642767\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−23.0000	−	118.794i	−0.190083	−	0.981768i
$122$		0			0
$123$	46.0000	+	130.108i	0.373984	+	1.05779i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$	−240.000	+	84.8528i	−1.86047	+	0.657774i
$130$		0			0
$131$	−62.0000			−0.473282			−0.236641	−	0.971597i	$-0.576047\pi$
$131$	−62.0000			−0.473282			−0.236641	+	0.971597i	$0.576047\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$			135.765i			0.990982i	0.868613	+	0.495491i	$0.165012\pi$
$137$			135.765i			0.990982i	−0.868613	+	0.495491i	$0.834988\pi$
$138$		0			0
$139$		−	186.676i		−	1.34299i	−0.741007	−	0.671497i	$-0.765652\pi$
$139$		−	186.676i		−	1.34299i	0.741007	−	0.671497i	$-0.234348\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	49.0000	+	138.593i	0.333333	+	0.942809i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	14.0000	−	11.3137i	0.0915033	−	0.0739458i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−322.000			−1.97546			−0.987730	−	0.156171i	$-0.950085\pi$
$163$	−322.000			−1.97546			−0.987730	+	0.156171i	$0.950085\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−169.000			−1.00000
$170$		0			0
$171$	96.0000	+	118.794i	0.561404	+	0.694701i
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	240.000	−	84.8528i	1.35593	−	0.479394i
$178$		0			0
$179$		−	356.382i		−	1.99096i	−0.0949721	−	0.995480i	$-0.530276\pi$
$179$		−	356.382i		−	1.99096i	0.0949721	−	0.995480i	$-0.469724\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−14.0000	+	16.9706i	−0.0748663	+	0.0907517i
$188$		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
$193$		−	373.352i		−	1.93447i	−0.253886	−	0.967234i	$-0.581709\pi$
$193$		−	373.352i		−	1.93447i	0.253886	−	0.967234i	$-0.418291\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$	62.0000	+	175.362i	0.308458	+	0.872450i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−144.000	−	118.794i	−0.688995	−	0.568392i
$210$		0			0
$211$			356.382i			1.68901i	0.535545	+	0.844507i	$0.320106\pi$
$211$			356.382i			1.68901i	−0.535545	+	0.844507i	$0.679894\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	96.0000	−	33.9411i	0.438356	−	0.154982i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	175.000	−	141.421i	0.777778	−	0.628539i
$226$		0			0
$227$	446.000			1.96476			0.982379	−	0.186900i	$-0.0598442\pi$
$227$	446.000			1.96476			0.982379	+	0.186900i	$0.0598442\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	434.000			1.86266			0.931330	−	0.364175i	$-0.118649\pi$
$233$	434.000			1.86266			0.931330	+	0.364175i	$0.118649\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		−	441.235i		−	1.83085i	−0.402490	−	0.915425i	$-0.631855\pi$
$241$		−	441.235i		−	1.83085i	0.402490	−	0.915425i	$-0.368145\pi$
$242$		0			0
$243$	−241.000	+	31.1127i	−0.991770	+	0.128036i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	158.000	+	446.891i	0.634538	+	1.79474i
$250$		0			0
$251$		−	186.676i		−	0.743730i	−0.928287	−	0.371865i	$-0.878718\pi$
$251$		−	186.676i		−	0.743730i	0.928287	−	0.371865i	$-0.121282\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	339.411i		−	1.32067i	−0.750973	−	0.660333i	$-0.770415\pi$
$257$		−	339.411i		−	1.32067i	0.750973	−	0.660333i	$-0.229585\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	288.000	−	101.823i	1.07865	−	0.381361i
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		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
$275$	−175.000	+	212.132i	−0.636364	+	0.771389i
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	238.000			0.846975			0.423488	−	0.905902i	$-0.360806\pi$
$281$	238.000			0.846975			0.423488	+	0.905902i	$0.360806\pi$
$282$		0			0
$283$		−	560.029i		−	1.97890i	−0.144876	−	0.989450i	$-0.546278\pi$
$283$		−	560.029i		−	1.97890i	0.144876	−	0.989450i	$-0.453722\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−285.000			−0.986159
$290$		0			0
$291$	−94.0000	−	265.872i	−0.323024	−	0.913650i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	281.000	−	96.1665i	0.946128	−	0.323793i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	288.500i		−	0.939738i	−0.882736	−	0.469869i	$-0.844301\pi$
$307$		−	288.500i		−	0.939738i	0.882736	−	0.469869i	$-0.155699\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−526.000			−1.68051			−0.840256	−	0.542191i	$-0.817595\pi$
$313$	−526.000			−1.68051			−0.840256	+	0.542191i	$0.817595\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	178.000	+	503.460i	0.554517	+	1.56841i
$322$		0			0
$323$			33.9411i			0.105081i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	14.0000			0.0422961			0.0211480	−	0.999776i	$-0.493268\pi$
$331$	14.0000			0.0422961			0.0211480	+	0.999776i	$0.493268\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$			475.176i			1.41002i	0.709199	+	0.705009i	$0.249057\pi$
$337$			475.176i			1.41002i	−0.709199	+	0.705009i	$0.750943\pi$
$338$		0			0
$339$	576.000	−	203.647i	1.69912	−	0.600728i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	658.000			1.89625			0.948127	−	0.317892i	$-0.102975\pi$
$347$	658.000			1.89625			0.948127	+	0.317892i	$0.102975\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	678.823i		−	1.92301i	−0.274788	−	0.961505i	$-0.588608\pi$
$353$		−	678.823i		−	1.92301i	0.274788	−	0.961505i	$-0.411392\pi$
$354$		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
$361$	73.0000			0.202216
$362$		0			0
$363$	−313.000	+	183.848i	−0.862259	+	0.506468i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	322.000	−	260.215i	0.872629	−	0.705191i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	686.000			1.81003			0.905013	−	0.425383i	$-0.139861\pi$
$379$	686.000			1.81003			0.905013	+	0.425383i	$0.139861\pi$
$380$		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
$387$	480.000	+	593.970i	1.24031	+	1.53481i
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	62.0000	+	175.362i	0.157761	+	0.446215i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			237.588i			0.592488i	0.955112	+	0.296244i	$0.0957342\pi$
$401$			237.588i			0.592488i	−0.955112	+	0.296244i	$0.904266\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			746.705i			1.82568i	0.408313	+	0.912842i	$0.366117\pi$
$409$			746.705i			1.82568i	−0.408313	+	0.912842i	$0.633883\pi$
$410$		0			0
$411$	384.000	−	135.765i	0.934307	−	0.330327i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−528.000	+	186.676i	−1.26619	+	0.447665i
$418$		0			0
$419$			661.852i			1.57960i	0.613365	+	0.789799i	$0.289815\pi$
$419$			661.852i			1.57960i	−0.613365	+	0.789799i	$0.710185\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	50.0000			0.117647
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	578.000			1.33487			0.667436	−	0.744667i	$-0.267392\pi$
$433$	578.000			1.33487			0.667436	+	0.744667i	$0.267392\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	343.000	−	277.186i	0.777778	−	0.628539i
$442$		0			0
$443$		−	118.794i		−	0.268158i	−0.990971	−	0.134079i	$-0.957192\pi$
$443$		−	118.794i		−	0.268158i	0.990971	−	0.134079i	$-0.0428076\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			237.588i			0.529149i	0.964365	+	0.264574i	$0.0852315\pi$
$449$			237.588i			0.529149i	−0.964365	+	0.264574i	$0.914769\pi$
$450$		0			0
$451$	−322.000	+	390.323i	−0.713969	+	0.865461i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	882.469i		−	1.93100i	−0.260394	−	0.965502i	$-0.583852\pi$
$457$		−	882.469i		−	1.93100i	0.260394	−	0.965502i	$-0.416148\pi$
$458$		0			0
$459$	−46.0000	−	28.2843i	−0.100218	−	0.0616215i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	933.381i		−	1.99867i	−0.0364026	−	0.999337i	$-0.511590\pi$
$467$		−	933.381i		−	1.99867i	0.0364026	−	0.999337i	$-0.488410\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−720.000	−	593.970i	−1.52220	−	1.25575i
$474$		0			0
$475$			424.264i			0.893188i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	322.000	+	910.754i	0.658487	+	1.86248i
$490$		0			0
$491$	782.000			1.59267			0.796334	−	0.604857i	$-0.206770\pi$
$491$	782.000			1.59267			0.796334	+	0.604857i	$0.206770\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−802.000			−1.60721			−0.803607	−	0.595160i	$-0.797089\pi$
$499$	−802.000			−1.60721			−0.803607	+	0.595160i	$0.797089\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	169.000	+	478.004i	0.333333	+	0.942809i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	240.000	−	390.323i	0.467836	−	0.760863i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			271.529i			0.521169i	0.965451	+	0.260584i	$0.0839152\pi$
$521$			271.529i			0.521169i	−0.965451	+	0.260584i	$0.916085\pi$
$522$		0			0
$523$		−	967.322i		−	1.84956i	−0.380497	−	0.924782i	$-0.624247\pi$
$523$		−	967.322i		−	1.84956i	0.380497	−	0.924782i	$-0.375753\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−529.000			−1.00000
$530$		0			0
$531$	−480.000	−	593.970i	−0.903955	−	1.11859i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−1008.00	+	356.382i	−1.87709	+	0.663653i
$538$		0			0
$539$	−343.000	+	415.779i	−0.636364	+	0.771389i
$540$		0			0
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$			390.323i			0.713570i	0.934186	+	0.356785i	$0.116127\pi$
$547$			390.323i			0.713570i	−0.934186	+	0.356785i	$0.883873\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	62.0000	+	22.6274i	0.110517	+	0.0403341i
$562$		0			0
$563$	−226.000			−0.401421			−0.200710	−	0.979651i	$-0.564325\pi$
$563$	−226.000			−0.401421			−0.200710	+	0.979651i	$0.564325\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−626.000			−1.10018			−0.550088	−	0.835107i	$-0.685406\pi$
$569$	−626.000			−1.10018			−0.550088	+	0.835107i	$0.685406\pi$
$570$		0			0
$571$		−	933.381i		−	1.63464i	−0.576182	−	0.817321i	$-0.695458\pi$
$571$		−	933.381i		−	1.63464i	0.576182	−	0.817321i	$-0.304542\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−2.00000			−0.00346620			−0.00173310	−	0.999998i	$-0.500552\pi$
$577$	−2.00000			−0.00346620			−0.00173310	+	0.999998i	$0.500552\pi$
$578$		0			0
$579$	−1056.00	+	373.352i	−1.82383	+	0.644823i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$			288.500i			0.491481i	0.969336	+	0.245741i	$0.0790312\pi$
$587$			288.500i			0.491481i	−0.969336	+	0.245741i	$0.920969\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	862.000			1.45363			0.726813	−	0.686836i	$-0.241001\pi$
$593$	862.000			1.45363			0.726813	+	0.686836i	$0.241001\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$		−	780.646i		−	1.29891i	−0.760399	−	0.649456i	$-0.774997\pi$
$601$		−	780.646i		−	1.29891i	0.760399	−	0.649456i	$-0.225003\pi$
$602$		0			0
$603$	434.000	−	350.725i	0.719735	−	0.581633i
$604$		0			0
$605$		0			0
$606$		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
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1187.94i		−	1.92535i	−0.270665	−	0.962674i	$-0.587243\pi$
$617$		−	1187.94i		−	1.92535i	0.270665	−	0.962674i	$-0.412757\pi$
$618$		0			0
$619$	562.000			0.907916			0.453958	−	0.891023i	$-0.350012\pi$
$619$	562.000			0.907916			0.453958	+	0.891023i	$0.350012\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	625.000			1.00000
$626$		0			0
$627$	−192.000	+	526.087i	−0.306220	+	0.839055i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	1008.00	−	356.382i	1.59242	−	0.563004i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$			1187.94i			1.85326i	0.375975	+	0.926630i	$0.377308\pi$
$641$			1187.94i			1.85326i	−0.375975	+	0.926630i	$0.622692\pi$
$642$		0			0
$643$	−1214.00			−1.88802			−0.944012	−	0.329910i	$-0.892982\pi$
$643$	−1214.00			−1.88802			−0.944012	+	0.329910i	$0.892982\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	720.000	+	593.970i	1.10940	+	0.915208i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−192.000	−	237.588i	−0.292237	−	0.361625i
$658$		0			0
$659$	−994.000			−1.50835			−0.754173	−	0.656676i	$-0.771962\pi$
$659$	−994.000			−1.50835			−0.754173	+	0.656676i	$0.771962\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		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
$673$			509.117i			0.756489i	0.925706	+	0.378244i	$0.123472\pi$
$673$			509.117i			0.756489i	−0.925706	+	0.378244i	$0.876528\pi$
$674$		0			0
$675$	−575.000	−	353.553i	−0.851852	−	0.523783i
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−446.000	−	1261.48i	−0.654919	−	1.85239i
$682$		0			0
$683$			1306.73i			1.91323i	0.291362	+	0.956613i	$0.405892\pi$
$683$			1306.73i			1.91323i	−0.291362	+	0.956613i	$0.594108\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−734.000			−1.06223			−0.531114	−	0.847300i	$-0.678227\pi$
$691$	−734.000			−1.06223			−0.531114	+	0.847300i	$0.678227\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	92.0000			0.131994
$698$		0			0
$699$	−434.000	−	1227.54i	−0.620887	−	1.75613i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−1248.00	+	441.235i	−1.72614	+	0.610283i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	329.000	+	650.538i	0.451303	+	0.892371i
$730$		0			0
$731$			169.706i			0.232155i
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−434.000	+	526.087i	−0.588874	+	0.713823i
$738$		0			0
$739$			1442.50i			1.95196i	0.217862	+	0.975980i	$0.430092\pi$
$739$			1442.50i			1.95196i	−0.217862	+	0.975980i	$0.569908\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1106.00	−	893.783i	1.48059	−	1.19650i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	−528.000	+	186.676i	−0.701195	+	0.247910i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1394.00			−1.83180			−0.915900	−	0.401406i	$-0.868522\pi$
$761$	−1394.00			−1.83180			−0.915900	+	0.401406i	$0.868522\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	1120.06i		−	1.45651i	−0.685306	−	0.728256i	$-0.740331\pi$
$769$		−	1120.06i		−	1.45651i	0.685306	−	0.728256i	$-0.259669\pi$
$770$		0			0
$771$	−960.000	+	339.411i	−1.24514	+	0.440222i
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$			780.646i			1.00211i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	1272.79i		−	1.61727i	−0.588310	−	0.808635i	$-0.700207\pi$
$787$		−	1272.79i		−	1.61727i	0.588310	−	0.808635i	$-0.299793\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−576.000	−	712.764i	−0.719101	−	0.889842i
$802$		0			0
$803$	288.000	+	237.588i	0.358655	+	0.295875i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1582.00			−1.95550			−0.977750	−	0.209772i	$-0.932728\pi$
$809$	−1582.00			−1.95550			−0.977750	+	0.209772i	$0.932728\pi$
$810$		0			0
$811$			1612.20i			1.98792i	0.109741	+	0.993960i	$0.464998\pi$
$811$			1612.20i			1.98792i	−0.109741	+	0.993960i	$0.535002\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1440.00			−1.76255
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$	775.000	+	282.843i	0.939394	+	0.342840i
$826$		0			0
$827$	−1262.00			−1.52600			−0.762999	−	0.646400i	$-0.776274\pi$
$827$	−1262.00			−1.52600			−0.762999	+	0.646400i	$0.776274\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	98.0000			0.117647
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	841.000			1.00000
$842$		0			0
$843$	−238.000	−	673.166i	−0.282325	−	0.798536i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1584.00	+	560.029i	−1.86572	+	0.659633i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1202.00			1.40257			0.701284	−	0.712882i	$-0.252611\pi$
$857$	1202.00			1.40257			0.701284	+	0.712882i	$0.252611\pi$
$858$		0			0
$859$	1646.00			1.91618			0.958091	−	0.286465i	$-0.0924801\pi$
$859$	1646.00			1.91618			0.958091	+	0.286465i	$0.0924801\pi$
$860$		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
$865$		0			0
$866$		0			0
$867$	285.000	+	806.102i	0.328720	+	0.929760i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−658.000	+	531.744i	−0.753723	+	0.609100i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1018.23i			1.15577i	0.816118	+	0.577885i	$0.196122\pi$
$881$			1018.23i			1.15577i	−0.816118	+	0.577885i	$0.803878\pi$
$882$		0			0
$883$	1762.00			1.99547			0.997735	−	0.0672672i	$-0.0214280\pi$
$883$	1762.00			1.99547			0.997735	+	0.0672672i	$0.0214280\pi$
$884$		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
$891$	−553.000	−	698.621i	−0.620651	−	0.784087i
$892$		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
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1714.00			1.88975			0.944873	−	0.327436i	$-0.106185\pi$
$907$	1714.00			1.88975			0.944873	+	0.327436i	$0.106185\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	−1106.00	+	1340.67i	−1.21139	+	1.46843i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	−816.000	+	288.500i	−0.885993	+	0.313246i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1527.35i			1.64408i	0.569429	+	0.822040i	$0.307164\pi$
$929$			1527.35i			1.64408i	−0.569429	+	0.822040i	$0.692836\pi$
$930$		0			0
$931$			831.558i			0.893188i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			1731.00i			1.84738i	0.383138	+	0.923691i	$0.374843\pi$
$937$			1731.00i			1.84738i	−0.383138	+	0.923691i	$0.625157\pi$
$938$		0			0
$939$	526.000	+	1487.75i	0.560170	+	1.58440i
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1612.20i		−	1.70243i	−0.524815	−	0.851216i	$-0.675866\pi$
$947$		−	1612.20i		−	1.70243i	0.524815	−	0.851216i	$-0.324134\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−142.000			−0.149003			−0.0745016	−	0.997221i	$-0.523737\pi$
$953$	−142.000			−0.149003			−0.0745016	+	0.997221i	$0.523737\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	961.000			1.00000
$962$		0			0
$963$	1246.00	−	1006.92i	1.29387	−	1.04561i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		0			0
$969$	96.0000	−	33.9411i	0.0990712	−	0.0350270i
$970$		0			0
$971$		−	1680.09i		−	1.73026i	−0.501545	−	0.865132i	$-0.667235\pi$
$971$		−	1680.09i		−	1.73026i	0.501545	−	0.865132i	$-0.332765\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	373.352i		−	0.382142i	−0.981576	−	0.191071i	$-0.938804\pi$
$977$		−	373.352i		−	0.382142i	0.981576	−	0.191071i	$-0.0611960\pi$
$978$		0			0
$979$	864.000	+	712.764i	0.882533	+	0.728053i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	−14.0000	−	39.5980i	−0.0140987	−	0.0398771i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1056.3.p.d.527.1		2
3.2	odd	2		1056.3.p.c.527.2		2
4.3	odd	2		264.3.p.b.131.2	yes	2
8.3	odd	2	CM	1056.3.p.d.527.1		2
8.5	even	2		264.3.p.b.131.2	yes	2
11.10	odd	2		1056.3.p.c.527.1		2
12.11	even	2		264.3.p.e.131.1	yes	2
24.5	odd	2		264.3.p.e.131.1	yes	2
24.11	even	2		1056.3.p.c.527.2		2
33.32	even	2	inner	1056.3.p.d.527.2		2
44.43	even	2		264.3.p.e.131.2	yes	2
88.21	odd	2		264.3.p.e.131.2	yes	2
88.43	even	2		1056.3.p.c.527.1		2
132.131	odd	2		264.3.p.b.131.1	✓	2
264.131	odd	2	inner	1056.3.p.d.527.2		2
264.197	even	2		264.3.p.b.131.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
264.3.p.b.131.1	✓	2	132.131	odd	2
264.3.p.b.131.1	✓	2	264.197	even	2
264.3.p.b.131.2	yes	2	4.3	odd	2
264.3.p.b.131.2	yes	2	8.5	even	2
264.3.p.e.131.1	yes	2	12.11	even	2
264.3.p.e.131.1	yes	2	24.5	odd	2
264.3.p.e.131.2	yes	2	44.43	even	2
264.3.p.e.131.2	yes	2	88.21	odd	2
1056.3.p.c.527.1		2	11.10	odd	2
1056.3.p.c.527.1		2	88.43	even	2
1056.3.p.c.527.2		2	3.2	odd	2
1056.3.p.c.527.2		2	24.11	even	2
1056.3.p.d.527.1		2	1.1	even	1	trivial
1056.3.p.d.527.1		2	8.3	odd	2	CM
1056.3.p.d.527.2		2	33.32	even	2	inner
1056.3.p.d.527.2		2	264.131	odd	2	inner

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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:	\( N \)	\(=\)	\( 1056 = 2^{5} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1056.p (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 2.82843i) q^{3} +(-7.00000 + 5.65685i) q^{9} +(7.00000 - 8.48528i) q^{11} -2.00000 q^{17} -16.9706i q^{19} -25.0000 q^{25} +(23.0000 + 14.1421i) q^{27} +(-31.0000 - 11.3137i) q^{33} -46.0000 q^{41} -84.8528i q^{43} -49.0000 q^{49} +(2.00000 + 5.65685i) q^{51} +(-48.0000 + 16.9706i) q^{57} +84.8528i q^{59} -62.0000 q^{67} +33.9411i q^{73} +(25.0000 + 70.7107i) q^{75} +(17.0000 - 79.1960i) q^{81} -158.000 q^{83} +101.823i q^{89} +94.0000 q^{97} +(-1.00000 + 98.9949i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 14 q^{9} + 14 q^{11} - 4 q^{17} - 50 q^{25} + 46 q^{27} - 62 q^{33} - 92 q^{41} - 98 q^{49} + 4 q^{51} - 96 q^{57} - 124 q^{67} + 50 q^{75} + 34 q^{81} - 316 q^{83} + 188 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 14 * q^9 + 14 * q^11 - 4 * q^17 - 50 * q^25 + 46 * q^27 - 62 * q^33 - 92 * q^41 - 98 * q^49 + 4 * q^51 - 96 * q^57 - 124 * q^67 + 50 * q^75 + 34 * q^81 - 316 * q^83 + 188 * q^97 - 2 * q^99

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