LMFDB - Embedded newform 5292.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(1,5292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5292, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5292.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5292.a (trivial)

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:	yes
Analytic conductor:	$42.2568327497$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.7168.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 7 $ x^4 - 6*x^2 + 7
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.10100$ of defining polynomial
Character	$\chi$	$=$	5292.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.870264 q^{5} +O(q^{10})$	$q-0.870264 q^{5} +5.07227 q^{11} -7.17327 q^{17} -1.58579 q^{19} +3.84153 q^{23} -4.24264 q^{25} -2.46148 q^{29} -3.00000 q^{31} +5.24264 q^{37} -4.56248 q^{41} +0.242641 q^{43} +5.43275 q^{47} -10.1445 q^{53} -4.41421 q^{55} +10.6543 q^{59} -11.6569 q^{61} -8.48528 q^{67} +2.61079 q^{71} -7.41421 q^{73} +0.242641 q^{79} -1.74053 q^{83} +6.24264 q^{85} +4.56248 q^{89} +1.38005 q^{95} +5.65685 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 12 q^{19} - 12 q^{31} + 4 q^{37} - 16 q^{43} - 12 q^{55} - 24 q^{61} - 24 q^{73} - 16 q^{79} + 8 q^{85}+O(q^{100}) $ 4 * q - 12 * q^19 - 12 * q^31 + 4 * q^37 - 16 * q^43 - 12 * q^55 - 24 * q^61 - 24 * q^73 - 16 * q^79 + 8 * q^85

Display 100 coefficients 10 coefficients

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
$3$	0	0
$4$	0	0
$5$	−0.870264	−0.389194	−0.194597	−	0.980883i	$-0.562340\pi$
$5$	−0.870264	−0.389194	−0.194597	+	0.980883i	$0.562340\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.07227	1.52935	0.764673	−	0.644418i	$-0.222900\pi$
$11$	5.07227	1.52935	0.764673	+	0.644418i	$0.222900\pi$
$12$	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
$17$	−7.17327	−1.73977	−0.869887	−	0.493251i	$-0.835808\pi$
$17$	−7.17327	−1.73977	−0.869887	+	0.493251i	$0.835808\pi$
$18$	0	0
$19$	−1.58579	−0.363804	−0.181902	−	0.983317i	$-0.558225\pi$
$19$	−1.58579	−0.363804	−0.181902	+	0.983317i	$0.558225\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.84153	0.801015	0.400507	−	0.916294i	$-0.368834\pi$
$23$	3.84153	0.801015	0.400507	+	0.916294i	$0.368834\pi$
$24$	0	0
$25$	−4.24264	−0.848528
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.46148	−0.457085	−0.228543	−	0.973534i	$-0.573396\pi$
$29$	−2.46148	−0.457085	−0.228543	+	0.973534i	$0.573396\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.24264	0.861885	0.430942	−	0.902379i	$-0.358181\pi$
$37$	5.24264	0.861885	0.430942	+	0.902379i	$0.358181\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.56248	−0.712540	−0.356270	−	0.934383i	$-0.615952\pi$
$41$	−4.56248	−0.712540	−0.356270	+	0.934383i	$0.615952\pi$
$42$	0	0
$43$	0.242641	0.0370024	0.0185012	−	0.999829i	$-0.494111\pi$
$43$	0.242641	0.0370024	0.0185012	+	0.999829i	$0.494111\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.43275	0.792447	0.396224	−	0.918154i	$-0.370320\pi$
$47$	5.43275	0.792447	0.396224	+	0.918154i	$0.370320\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.1445	−1.39346	−0.696730	−	0.717334i	$-0.745362\pi$
$53$	−10.1445	−1.39346	−0.696730	+	0.717334i	$0.745362\pi$
$54$	0	0
$55$	−4.41421	−0.595212
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.6543	1.38708	0.693538	−	0.720420i	$-0.256051\pi$
$59$	10.6543	1.38708	0.693538	+	0.720420i	$0.256051\pi$
$60$	0	0
$61$	−11.6569	−1.49251	−0.746254	−	0.665662i	$-0.768149\pi$
$61$	−11.6569	−1.49251	−0.746254	+	0.665662i	$0.768149\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.48528	−1.03664	−0.518321	−	0.855186i	$-0.673443\pi$
$67$	−8.48528	−1.03664	−0.518321	+	0.855186i	$0.673443\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.61079	0.309844	0.154922	−	0.987927i	$-0.450487\pi$
$71$	2.61079	0.309844	0.154922	+	0.987927i	$0.450487\pi$
$72$	0	0
$73$	−7.41421	−0.867768	−0.433884	−	0.900969i	$-0.642857\pi$
$73$	−7.41421	−0.867768	−0.433884	+	0.900969i	$0.642857\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.242641	0.0272992	0.0136496	−	0.999907i	$-0.495655\pi$
$79$	0.242641	0.0272992	0.0136496	+	0.999907i	$0.495655\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.74053	−0.191048	−0.0955239	−	0.995427i	$-0.530453\pi$
$83$	−1.74053	−0.191048	−0.0955239	+	0.995427i	$0.530453\pi$
$84$	0	0
$85$	6.24264	0.677109
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.56248	0.483622	0.241811	−	0.970323i	$-0.422259\pi$
$89$	4.56248	0.483622	0.241811	+	0.970323i	$0.422259\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.38005	0.141590
$96$	0	0
$97$	5.65685	0.574367	0.287183	−	0.957876i	$-0.407281\pi$
$97$	5.65685	0.574367	0.287183	+	0.957876i	$0.407281\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.69222	0.367389	0.183695	−	0.982983i	$-0.441194\pi$
$101$	3.69222	0.367389	0.183695	+	0.982983i	$0.441194\pi$
$102$	0	0
$103$	−12.1716	−1.19930	−0.599650	−	0.800262i	$-0.704694\pi$
$103$	−12.1716	−1.19930	−0.599650	+	0.800262i	$0.704694\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.3753	−1.09969	−0.549845	−	0.835267i	$-0.685313\pi$
$107$	−11.3753	−1.09969	−0.549845	+	0.835267i	$0.685313\pi$
$108$	0	0
$109$	−15.9706	−1.52970	−0.764851	−	0.644207i	$-0.777188\pi$
$109$	−15.9706	−1.52970	−0.764851	+	0.644207i	$0.777188\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.2982	1.53321	0.766604	−	0.642120i	$-0.221945\pi$
$113$	16.2982	1.53321	0.766604	+	0.642120i	$0.221945\pi$
$114$	0	0
$115$	−3.34315	−0.311750
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	14.7279	1.33890
$122$	0	0
$123$	0	0
$124$	0	0
$125$	8.04354	0.719436
$126$	0	0
$127$	18.4853	1.64030	0.820152	−	0.572146i	$-0.193889\pi$
$127$	18.4853	1.64030	0.820152	+	0.572146i	$0.193889\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.69222	−0.322591	−0.161295	−	0.986906i	$-0.551567\pi$
$131$	−3.69222	−0.322591	−0.161295	+	0.986906i	$0.551567\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.22158	−0.446110	−0.223055	−	0.974806i	$-0.571603\pi$
$137$	−5.22158	−0.446110	−0.223055	+	0.974806i	$0.571603\pi$
$138$	0	0
$139$	−8.82843	−0.748817	−0.374409	−	0.927264i	$-0.622154\pi$
$139$	−8.82843	−0.748817	−0.374409	+	0.927264i	$0.622154\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.14214	0.177895
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.23074	−0.100826	−0.0504130	−	0.998728i	$-0.516054\pi$
$149$	−1.23074	−0.100826	−0.0504130	+	0.998728i	$0.516054\pi$
$150$	0	0
$151$	−16.7279	−1.36130	−0.680649	−	0.732609i	$-0.738302\pi$
$151$	−16.7279	−1.36130	−0.680649	+	0.732609i	$0.738302\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.61079	0.209704
$156$	0	0
$157$	−8.82843	−0.704585	−0.352293	−	0.935890i	$-0.614598\pi$
$157$	−8.82843	−0.704585	−0.352293	+	0.935890i	$0.614598\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.7793	1.53057	0.765284	−	0.643693i	$-0.222599\pi$
$167$	19.7793	1.53057	0.765284	+	0.643693i	$0.222599\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−18.9090	−1.43763	−0.718813	−	0.695203i	$-0.755314\pi$
$173$	−18.9090	−1.43763	−0.718813	+	0.695203i	$0.755314\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	21.5198	1.60847	0.804233	−	0.594314i	$-0.202576\pi$
$179$	21.5198	1.60847	0.804233	+	0.594314i	$0.202576\pi$
$180$	0	0
$181$	−6.72792	−0.500083	−0.250041	−	0.968235i	$-0.580444\pi$
$181$	−6.72792	−0.500083	−0.250041	+	0.968235i	$0.580444\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.56248	−0.335440
$186$	0	0
$187$	−36.3848	−2.66072
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.2260	0.812282	0.406141	−	0.913810i	$-0.366874\pi$
$191$	11.2260	0.812282	0.406141	+	0.913810i	$0.366874\pi$
$192$	0	0
$193$	−14.9706	−1.07760	−0.538802	−	0.842432i	$-0.681123\pi$
$193$	−14.9706	−1.07760	−0.538802	+	0.842432i	$0.681123\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−16.5969	−1.18248	−0.591239	−	0.806497i	$-0.701361\pi$
$197$	−16.5969	−1.18248	−0.591239	+	0.806497i	$0.701361\pi$
$198$	0	0
$199$	−21.7279	−1.54025	−0.770126	−	0.637892i	$-0.779807\pi$
$199$	−21.7279	−1.54025	−0.770126	+	0.637892i	$0.779807\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3.97056	0.277316
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8.04354	−0.556383
$210$	0	0
$211$	12.7279	0.876226	0.438113	−	0.898920i	$-0.355647\pi$
$211$	12.7279	0.876226	0.438113	+	0.898920i	$0.355647\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.211161	−0.0144011
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−2.27208	−0.152150	−0.0760748	−	0.997102i	$-0.524239\pi$
$223$	−2.27208	−0.152150	−0.0760748	+	0.997102i	$0.524239\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−26.7414	−1.77489	−0.887445	−	0.460914i	$-0.847522\pi$
$227$	−26.7414	−1.77489	−0.887445	+	0.460914i	$0.847522\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.2982	1.06773	0.533866	−	0.845569i	$-0.320739\pi$
$233$	16.2982	1.06773	0.533866	+	0.845569i	$0.320739\pi$
$234$	0	0
$235$	−4.72792	−0.308416
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−26.7414	−1.72976	−0.864879	−	0.501981i	$-0.832605\pi$
$239$	−26.7414	−1.72976	−0.864879	+	0.501981i	$0.832605\pi$
$240$	0	0
$241$	0.727922	0.0468896	0.0234448	−	0.999725i	$-0.492537\pi$
$241$	0.727922	0.0468896	0.0234448	+	0.999725i	$0.492537\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−19.5681	−1.23513	−0.617565	−	0.786520i	$-0.711881\pi$
$251$	−19.5681	−1.23513	−0.617565	+	0.786520i	$0.711881\pi$
$252$	0	0
$253$	19.4853	1.22503
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.2168	0.949199	0.474599	−	0.880202i	$-0.342593\pi$
$257$	15.2168	0.949199	0.474599	+	0.880202i	$0.342593\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−18.9090	−1.16598	−0.582990	−	0.812479i	$-0.698117\pi$
$263$	−18.9090	−1.16598	−0.582990	+	0.812479i	$0.698117\pi$
$264$	0	0
$265$	8.82843	0.542326
$266$	0	0
$267$	0	0
$268$	0	0
$269$	29.7745	1.81538	0.907692	−	0.419637i	$-0.137843\pi$
$269$	29.7745	1.81538	0.907692	+	0.419637i	$0.137843\pi$
$270$	0	0
$271$	−3.85786	−0.234349	−0.117174	−	0.993111i	$-0.537384\pi$
$271$	−3.85786	−0.234349	−0.117174	+	0.993111i	$0.537384\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−21.5198	−1.29769
$276$	0	0
$277$	−26.2132	−1.57500	−0.787499	−	0.616315i	$-0.788625\pi$
$277$	−26.2132	−1.57500	−0.787499	+	0.616315i	$0.788625\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−19.0583	−1.13693	−0.568463	−	0.822709i	$-0.692462\pi$
$281$	−19.0583	−1.13693	−0.568463	+	0.822709i	$0.692462\pi$
$282$	0	0
$283$	−9.17157	−0.545193	−0.272597	−	0.962128i	$-0.587882\pi$
$283$	−9.17157	−0.545193	−0.272597	+	0.962128i	$0.587882\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	34.4558	2.02681
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−18.0388	−1.05384	−0.526918	−	0.849916i	$-0.676652\pi$
$293$	−18.0388	−1.05384	−0.526918	+	0.849916i	$0.676652\pi$
$294$	0	0
$295$	−9.27208	−0.539841
$296$	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
$305$	10.1445	0.580875
$306$	0	0
$307$	−16.4142	−0.936809	−0.468404	−	0.883514i	$-0.655171\pi$
$307$	−16.4142	−0.936809	−0.468404	+	0.883514i	$0.655171\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−14.3465	−0.813518	−0.406759	−	0.913536i	$-0.633341\pi$
$311$	−14.3465	−0.813518	−0.406759	+	0.913536i	$0.633341\pi$
$312$	0	0
$313$	−31.4558	−1.77799	−0.888995	−	0.457917i	$-0.848596\pi$
$313$	−31.4558	−1.77799	−0.888995	+	0.457917i	$0.848596\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−13.8368	−0.777150	−0.388575	−	0.921417i	$-0.627033\pi$
$317$	−13.8368	−0.777150	−0.388575	+	0.921417i	$0.627033\pi$
$318$	0	0
$319$	−12.4853	−0.699042
$320$	0	0
$321$	0	0
$322$	0	0
$323$	11.3753	0.632937
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−22.4853	−1.23590	−0.617951	−	0.786216i	$-0.712037\pi$
$331$	−22.4853	−1.23590	−0.617951	+	0.786216i	$0.712037\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7.38443	0.403455
$336$	0	0
$337$	2.51472	0.136985	0.0684927	−	0.997652i	$-0.478181\pi$
$337$	2.51472	0.136985	0.0684927	+	0.997652i	$0.478181\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−15.2168	−0.824036
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−24.1306	−1.29540	−0.647700	−	0.761896i	$-0.724269\pi$
$347$	−24.1306	−1.29540	−0.647700	+	0.761896i	$0.724269\pi$
$348$	0	0
$349$	30.7279	1.64483	0.822414	−	0.568889i	$-0.192627\pi$
$349$	30.7279	1.64483	0.822414	+	0.568889i	$0.192627\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	13.2651	0.706031	0.353016	−	0.935617i	$-0.385156\pi$
$353$	13.2651	0.706031	0.353016	+	0.935617i	$0.385156\pi$
$354$	0	0
$355$	−2.27208	−0.120589
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.5969	0.875949	0.437974	−	0.898987i	$-0.355696\pi$
$359$	16.5969	0.875949	0.437974	+	0.898987i	$0.355696\pi$
$360$	0	0
$361$	−16.4853	−0.867646
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.45232	0.337730
$366$	0	0
$367$	4.07107	0.212508	0.106254	−	0.994339i	$-0.466114\pi$
$367$	4.07107	0.212508	0.106254	+	0.994339i	$0.466114\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	15.4853	0.801797	0.400899	−	0.916122i	$-0.368698\pi$
$373$	15.4853	0.801797	0.400899	+	0.916122i	$0.368698\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	32.2426	1.65619	0.828097	−	0.560585i	$-0.189424\pi$
$379$	32.2426	1.65619	0.828097	+	0.560585i	$0.189424\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10.6543	−0.544411	−0.272205	−	0.962239i	$-0.587753\pi$
$383$	−10.6543	−0.544411	−0.272205	+	0.962239i	$0.587753\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	35.3566	1.79265	0.896325	−	0.443398i	$-0.146227\pi$
$389$	35.3566	1.79265	0.896325	+	0.443398i	$0.146227\pi$
$390$	0	0
$391$	−27.5563	−1.39358
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−0.211161	−0.0106247
$396$	0	0
$397$	26.8701	1.34857	0.674285	−	0.738471i	$-0.264452\pi$
$397$	26.8701	1.34857	0.674285	+	0.738471i	$0.264452\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5.22158	−0.260753	−0.130377	−	0.991465i	$-0.541619\pi$
$401$	−5.22158	−0.260753	−0.130377	+	0.991465i	$0.541619\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	26.5921	1.31812
$408$	0	0
$409$	−18.7279	−0.926036	−0.463018	−	0.886349i	$-0.653234\pi$
$409$	−18.7279	−0.926036	−0.463018	+	0.886349i	$0.653234\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1.51472	0.0743546
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10.6543	0.520498	0.260249	−	0.965542i	$-0.416195\pi$
$419$	10.6543	0.520498	0.260249	+	0.965542i	$0.416195\pi$
$420$	0	0
$421$	−22.7574	−1.10913	−0.554563	−	0.832142i	$-0.687115\pi$
$421$	−22.7574	−1.10913	−0.554563	+	0.832142i	$0.687115\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	30.4336	1.47625
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.1306	−1.16233	−0.581165	−	0.813786i	$-0.697403\pi$
$431$	−24.1306	−1.16233	−0.581165	+	0.813786i	$0.697403\pi$
$432$	0	0
$433$	−25.4558	−1.22333	−0.611665	−	0.791117i	$-0.709500\pi$
$433$	−25.4558	−1.22333	−0.611665	+	0.791117i	$0.709500\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.09185	−0.291413
$438$	0	0
$439$	−37.4558	−1.78767	−0.893835	−	0.448396i	$-0.851995\pi$
$439$	−37.4558	−1.78767	−0.893835	+	0.448396i	$0.851995\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.149314	0.00709411	0.00354705	−	0.999994i	$-0.498871\pi$
$443$	0.149314	0.00709411	0.00354705	+	0.999994i	$0.498871\pi$
$444$	0	0
$445$	−3.97056	−0.188223
$446$	0	0
$447$	0	0
$448$	0	0
$449$	26.7414	1.26200	0.631002	−	0.775781i	$-0.282644\pi$
$449$	26.7414	1.26200	0.631002	+	0.775781i	$0.282644\pi$
$450$	0	0
$451$	−23.1421	−1.08972
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.97056	0.372847	0.186424	−	0.982469i	$-0.440310\pi$
$457$	7.97056	0.372847	0.186424	+	0.982469i	$0.440310\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	32.8332	1.52920	0.764598	−	0.644507i	$-0.222937\pi$
$461$	32.8332	1.52920	0.764598	+	0.644507i	$0.222937\pi$
$462$	0	0
$463$	10.7279	0.498569	0.249284	−	0.968430i	$-0.419805\pi$
$463$	10.7279	0.498569	0.249284	+	0.968430i	$0.419805\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12.3949	−0.573566	−0.286783	−	0.957996i	$-0.592586\pi$
$467$	−12.3949	−0.573566	−0.286783	+	0.957996i	$0.592586\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.23074	0.0565894
$474$	0	0
$475$	6.72792	0.308698
$476$	0	0
$477$	0	0
$478$	0	0
$479$	32.1741	1.47007	0.735037	−	0.678027i	$-0.237165\pi$
$479$	32.1741	1.47007	0.735037	+	0.678027i	$0.237165\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.92296	−0.223540
$486$	0	0
$487$	22.2426	1.00791	0.503955	−	0.863730i	$-0.331878\pi$
$487$	22.2426	1.00791	0.503955	+	0.863730i	$0.331878\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	29.3522	1.32465	0.662323	−	0.749218i	$-0.269571\pi$
$491$	29.3522	1.32465	0.662323	+	0.749218i	$0.269571\pi$
$492$	0	0
$493$	17.6569	0.795225
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−2.24264	−0.100394	−0.0501972	−	0.998739i	$-0.515985\pi$
$499$	−2.24264	−0.100394	−0.0501972	+	0.998739i	$0.515985\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	26.5302	1.18292	0.591462	−	0.806333i	$-0.298551\pi$
$503$	26.5302	1.18292	0.591462	+	0.806333i	$0.298551\pi$
$504$	0	0
$505$	−3.21320	−0.142986
$506$	0	0
$507$	0	0
$508$	0	0
$509$	19.7793	0.876702	0.438351	−	0.898804i	$-0.355563\pi$
$509$	19.7793	0.876702	0.438351	+	0.898804i	$0.355563\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	10.5925	0.466760
$516$	0	0
$517$	27.5563	1.21193
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−26.0823	−1.14269	−0.571343	−	0.820711i	$-0.693577\pi$
$521$	−26.0823	−1.14269	−0.571343	+	0.820711i	$0.693577\pi$
$522$	0	0
$523$	16.4558	0.719564	0.359782	−	0.933036i	$-0.382851\pi$
$523$	16.4558	0.719564	0.359782	+	0.933036i	$0.382851\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	21.5198	0.937418
$528$	0	0
$529$	−8.24264	−0.358376
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	9.89949	0.427992
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	23.2426	0.999279	0.499640	−	0.866233i	$-0.333466\pi$
$541$	23.2426	0.999279	0.499640	+	0.866233i	$0.333466\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	13.8986	0.595351
$546$	0	0
$547$	−23.5147	−1.00542	−0.502708	−	0.864456i	$-0.667663\pi$
$547$	−23.5147	−1.00542	−0.502708	+	0.864456i	$0.667663\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.90338	0.166290
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−11.0767	−0.469333	−0.234666	−	0.972076i	$-0.575400\pi$
$557$	−11.0767	−0.469333	−0.234666	+	0.972076i	$0.575400\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.12496	0.384571	0.192286	−	0.981339i	$-0.438410\pi$
$563$	9.12496	0.384571	0.192286	+	0.981339i	$0.438410\pi$
$564$	0	0
$565$	−14.1838	−0.596716
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−19.0583	−0.798967	−0.399484	−	0.916740i	$-0.630811\pi$
$569$	−19.0583	−0.798967	−0.399484	+	0.916740i	$0.630811\pi$
$570$	0	0
$571$	1.21320	0.0507710	0.0253855	−	0.999678i	$-0.491919\pi$
$571$	1.21320	0.0507710	0.0253855	+	0.999678i	$0.491919\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−16.2982	−0.679683
$576$	0	0
$577$	−18.7279	−0.779654	−0.389827	−	0.920888i	$-0.627465\pi$
$577$	−18.7279	−0.779654	−0.389827	+	0.920888i	$0.627465\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−51.4558	−2.13108
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−41.5103	−1.71331	−0.856656	−	0.515888i	$-0.827462\pi$
$587$	−41.5103	−1.71331	−0.856656	+	0.515888i	$0.827462\pi$
$588$	0	0
$589$	4.75736	0.196024
$590$	0	0
$591$	0	0
$592$	0	0
$593$	3.03311	0.124555	0.0622775	−	0.998059i	$-0.480164\pi$
$593$	3.03311	0.124555	0.0622775	+	0.998059i	$0.480164\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	36.4380	1.48882	0.744408	−	0.667725i	$-0.232732\pi$
$599$	36.4380	1.48882	0.744408	+	0.667725i	$0.232732\pi$
$600$	0	0
$601$	25.4558	1.03837	0.519183	−	0.854663i	$-0.326236\pi$
$601$	25.4558	1.03837	0.519183	+	0.854663i	$0.326236\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−12.8172	−0.521092
$606$	0	0
$607$	6.00000	0.243532	0.121766	−	0.992559i	$-0.461144\pi$
$607$	6.00000	0.243532	0.121766	+	0.992559i	$0.461144\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−9.00000	−0.363507	−0.181753	−	0.983344i	$-0.558177\pi$
$613$	−9.00000	−0.363507	−0.181753	+	0.983344i	$0.558177\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−33.8272	−1.36183	−0.680916	−	0.732361i	$-0.738418\pi$
$617$	−33.8272	−1.36183	−0.680916	+	0.732361i	$0.738418\pi$
$618$	0	0
$619$	34.1127	1.37111	0.685553	−	0.728023i	$-0.259561\pi$
$619$	34.1127	1.37111	0.685553	+	0.728023i	$0.259561\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	14.2132	0.568528
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−37.6069	−1.49948
$630$	0	0
$631$	38.7279	1.54173	0.770867	−	0.636996i	$-0.219823\pi$
$631$	38.7279	1.54173	0.770867	+	0.636996i	$0.219823\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−16.0871	−0.638396
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−34.4245	−1.35968	−0.679842	−	0.733358i	$-0.737952\pi$
$641$	−34.4245	−1.35968	−0.679842	+	0.733358i	$0.737952\pi$
$642$	0	0
$643$	−36.5563	−1.44164	−0.720821	−	0.693121i	$-0.756235\pi$
$643$	−36.5563	−1.44164	−0.720821	+	0.693121i	$0.756235\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3.69222	−0.145156	−0.0725780	−	0.997363i	$-0.523123\pi$
$647$	−3.69222	−0.145156	−0.0725780	+	0.997363i	$0.523123\pi$
$648$	0	0
$649$	54.0416	2.12132
$650$	0	0
$651$	0	0
$652$	0	0
$653$	31.9630	1.25081	0.625404	−	0.780301i	$-0.284934\pi$
$653$	31.9630	1.25081	0.625404	+	0.780301i	$0.284934\pi$
$654$	0	0
$655$	3.21320	0.125550
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−13.0540	−0.508510	−0.254255	−	0.967137i	$-0.581830\pi$
$659$	−13.0540	−0.508510	−0.254255	+	0.967137i	$0.581830\pi$
$660$	0	0
$661$	24.0000	0.933492	0.466746	−	0.884391i	$-0.345426\pi$
$661$	24.0000	0.933492	0.466746	+	0.884391i	$0.345426\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−9.45584	−0.366132
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−59.1267	−2.28256
$672$	0	0
$673$	−43.4558	−1.67510	−0.837550	−	0.546361i	$-0.816013\pi$
$673$	−43.4558	−1.67510	−0.837550	+	0.546361i	$0.816013\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.83238	0.301023	0.150511	−	0.988608i	$-0.451908\pi$
$677$	7.83238	0.301023	0.150511	+	0.988608i	$0.451908\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−13.0540	−0.499496	−0.249748	−	0.968311i	$-0.580348\pi$
$683$	−13.0540	−0.499496	−0.249748	+	0.968311i	$0.580348\pi$
$684$	0	0
$685$	4.54416	0.173623
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−31.4558	−1.19664	−0.598318	−	0.801258i	$-0.704164\pi$
$691$	−31.4558	−1.19664	−0.598318	+	0.801258i	$0.704164\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.68306	0.291435
$696$	0	0
$697$	32.7279	1.23966
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−5.22158	−0.197216	−0.0986082	−	0.995126i	$-0.531439\pi$
$701$	−5.22158	−0.197216	−0.0986082	+	0.995126i	$0.531439\pi$
$702$	0	0
$703$	−8.31371	−0.313557
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−6.02944	−0.226440	−0.113220	−	0.993570i	$-0.536117\pi$
$709$	−6.02944	−0.226440	−0.113220	+	0.993570i	$0.536117\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−11.5246	−0.431599
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	46.3095	1.72705	0.863527	−	0.504303i	$-0.168250\pi$
$719$	46.3095	1.72705	0.863527	+	0.504303i	$0.168250\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	10.4432	0.387849
$726$	0	0
$727$	−12.3431	−0.457782	−0.228891	−	0.973452i	$-0.573510\pi$
$727$	−12.3431	−0.457782	−0.228891	+	0.973452i	$0.573510\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.74053	−0.0643757
$732$	0	0
$733$	24.7279	0.913347	0.456673	−	0.889634i	$-0.349041\pi$
$733$	24.7279	0.913347	0.456673	+	0.889634i	$0.349041\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−43.0396	−1.58539
$738$	0	0
$739$	30.7279	1.13034	0.565172	−	0.824973i	$-0.308810\pi$
$739$	30.7279	1.13034	0.565172	+	0.824973i	$0.308810\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14.9182	−0.547295	−0.273648	−	0.961830i	$-0.588230\pi$
$743$	−14.9182	−0.547295	−0.273648	+	0.961830i	$0.588230\pi$
$744$	0	0
$745$	1.07107	0.0392409
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−24.1838	−0.882478	−0.441239	−	0.897390i	$-0.645461\pi$
$751$	−24.1838	−0.882478	−0.441239	+	0.897390i	$0.645461\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14.5577	0.529809
$756$	0	0
$757$	8.00000	0.290765	0.145382	−	0.989376i	$-0.453559\pi$
$757$	8.00000	0.290765	0.145382	+	0.989376i	$0.453559\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−41.0879	−1.48944	−0.744718	−	0.667379i	$-0.767416\pi$
$761$	−41.0879	−1.48944	−0.744718	+	0.667379i	$0.767416\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−34.7849	−1.25113	−0.625564	−	0.780173i	$-0.715131\pi$
$773$	−34.7849	−1.25113	−0.625564	+	0.780173i	$0.715131\pi$
$774$	0	0
$775$	12.7279	0.457200
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.23512	0.259225
$780$	0	0
$781$	13.2426	0.473859
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7.68306	0.274220
$786$	0	0
$787$	−10.6274	−0.378827	−0.189413	−	0.981897i	$-0.560659\pi$
$787$	−10.6274	−0.378827	−0.189413	+	0.981897i	$0.560659\pi$
$788$	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
$797$	15.2168	0.539007	0.269504	−	0.962999i	$-0.413140\pi$
$797$	15.2168	0.539007	0.269504	+	0.962999i	$0.413140\pi$
$798$	0	0
$799$	−38.9706	−1.37868
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−37.6069	−1.32712
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	15.3661	0.540244	0.270122	−	0.962826i	$-0.412936\pi$
$809$	15.3661	0.540244	0.270122	+	0.962826i	$0.412936\pi$
$810$	0	0
$811$	−21.0000	−0.737410	−0.368705	−	0.929547i	$-0.620199\pi$
$811$	−21.0000	−0.737410	−0.368705	+	0.929547i	$0.620199\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−8.70264	−0.304840
$816$	0	0
$817$	−0.384776	−0.0134616
$818$	0	0
$819$	0	0
$820$	0	0
$821$	37.1846	1.29775	0.648875	−	0.760895i	$-0.275240\pi$
$821$	37.1846	1.29775	0.648875	+	0.760895i	$0.275240\pi$
$822$	0	0
$823$	27.6985	0.965508	0.482754	−	0.875756i	$-0.339636\pi$
$823$	27.6985	0.965508	0.482754	+	0.875756i	$0.339636\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−29.3522	−1.02068	−0.510338	−	0.859974i	$-0.670480\pi$
$827$	−29.3522	−1.02068	−0.510338	+	0.859974i	$0.670480\pi$
$828$	0	0
$829$	40.5858	1.40960	0.704801	−	0.709405i	$-0.251036\pi$
$829$	40.5858	1.40960	0.704801	+	0.709405i	$0.251036\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−17.2132	−0.595687
$836$	0	0
$837$	0	0
$838$	0	0
$839$	24.7897	0.855836	0.427918	−	0.903818i	$-0.359247\pi$
$839$	24.7897	0.855836	0.427918	+	0.903818i	$0.359247\pi$
$840$	0	0
$841$	−22.9411	−0.791073
$842$	0	0
$843$	0	0
$844$	0	0
$845$	11.3134	0.389194
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	20.1398	0.690382
$852$	0	0
$853$	49.4558	1.69334	0.846668	−	0.532122i	$-0.178605\pi$
$853$	49.4558	1.69334	0.846668	+	0.532122i	$0.178605\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.39963	0.0819698	0.0409849	−	0.999160i	$-0.486950\pi$
$857$	2.39963	0.0819698	0.0409849	+	0.999160i	$0.486950\pi$
$858$	0	0
$859$	42.5563	1.45200	0.726002	−	0.687693i	$-0.241376\pi$
$859$	42.5563	1.45200	0.726002	+	0.687693i	$0.241376\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−21.8184	−0.742709	−0.371354	−	0.928491i	$-0.621106\pi$
$863$	−21.8184	−0.742709	−0.371354	+	0.928491i	$0.621106\pi$
$864$	0	0
$865$	16.4558	0.559515
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.23074	0.0417500
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	11.5147	0.388824	0.194412	−	0.980920i	$-0.437720\pi$
$877$	11.5147	0.388824	0.194412	+	0.980920i	$0.437720\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	49.3426	1.66240	0.831198	−	0.555977i	$-0.187656\pi$
$881$	49.3426	1.66240	0.831198	+	0.555977i	$0.187656\pi$
$882$	0	0
$883$	34.4853	1.16052	0.580261	−	0.814431i	$-0.302951\pi$
$883$	34.4853	1.16052	0.580261	+	0.814431i	$0.302951\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.0871	0.540151	0.270076	−	0.962839i	$-0.412951\pi$
$887$	16.0871	0.540151	0.270076	+	0.962839i	$0.412951\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.61517	−0.288296
$894$	0	0
$895$	−18.7279	−0.626005
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.38443	0.246285
$900$	0	0
$901$	72.7696	2.42431
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.85507	0.194629
$906$	0	0
$907$	24.1838	0.803009	0.401504	−	0.915857i	$-0.368487\pi$
$907$	24.1838	0.803009	0.401504	+	0.915857i	$0.368487\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.39359	0.112435	0.0562173	−	0.998419i	$-0.482096\pi$
$911$	3.39359	0.112435	0.0562173	+	0.998419i	$0.482096\pi$
$912$	0	0
$913$	−8.82843	−0.292178
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	19.5147	0.643731	0.321866	−	0.946785i	$-0.395690\pi$
$919$	19.5147	0.643731	0.321866	+	0.946785i	$0.395690\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−22.2426	−0.731334
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−32.3853	−1.06253	−0.531264	−	0.847206i	$-0.678283\pi$
$929$	−32.3853	−1.06253	−0.531264	+	0.847206i	$0.678283\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	31.6644	1.03554
$936$	0	0
$937$	−30.7696	−1.00520	−0.502599	−	0.864520i	$-0.667623\pi$
$937$	−30.7696	−1.00520	−0.502599	+	0.864520i	$0.667623\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	36.9478	1.20446	0.602232	−	0.798321i	$-0.294278\pi$
$941$	36.9478	1.20446	0.602232	+	0.798321i	$0.294278\pi$
$942$	0	0
$943$	−17.5269	−0.570755
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.46586	0.275103	0.137552	−	0.990495i	$-0.456077\pi$
$947$	8.46586	0.275103	0.137552	+	0.990495i	$0.456077\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22.4519	0.727289	0.363645	−	0.931538i	$-0.381532\pi$
$953$	22.4519	0.727289	0.363645	+	0.931538i	$0.381532\pi$
$954$	0	0
$955$	−9.76955	−0.316135
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	0	0
$964$	0	0
$965$	13.0283	0.419397
$966$	0	0
$967$	−4.24264	−0.136434	−0.0682171	−	0.997671i	$-0.521731\pi$
$967$	−4.24264	−0.136434	−0.0682171	+	0.997671i	$0.521731\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	26.9526	0.864949	0.432474	−	0.901646i	$-0.357641\pi$
$971$	26.9526	0.864949	0.432474	+	0.901646i	$0.357641\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.4432	−0.334107	−0.167053	−	0.985948i	$-0.553425\pi$
$977$	−10.4432	−0.334107	−0.167053	+	0.985948i	$0.553425\pi$
$978$	0	0
$979$	23.1421	0.739626
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.95169	0.0622492	0.0311246	−	0.999516i	$-0.490091\pi$
$983$	1.95169	0.0622492	0.0311246	+	0.999516i	$0.490091\pi$
$984$	0	0
$985$	14.4437	0.460213
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.932112	0.0296394
$990$	0	0
$991$	59.9411	1.90409	0.952046	−	0.305954i	$-0.0989754\pi$
$991$	59.9411	1.90409	0.952046	+	0.305954i	$0.0989754\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	18.9090	0.599456
$996$	0	0
$997$	62.5269	1.98025	0.990124	−	0.140197i	$-0.0447737\pi$
$997$	62.5269	1.98025	0.990124	+	0.140197i	$0.0447737\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.a.z.1.2	✓	4
3.2	odd	2	inner	5292.2.a.z.1.3	yes	4
7.6	odd	2		5292.2.a.ba.1.3	yes	4
21.20	even	2		5292.2.a.ba.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5292.2.a.z.1.2	✓	4	1.1	even	1	trivial
5292.2.a.z.1.3	yes	4	3.2	odd	2	inner
5292.2.a.ba.1.2	yes	4	21.20	even	2
5292.2.a.ba.1.3	yes	4	7.6	odd	2

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:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.a (trivial)

\(f(q)\)	\(=\)	\(q-0.870264 q^{5} +O(q^{10})\)	\(q-0.870264 q^{5} +5.07227 q^{11} -7.17327 q^{17} -1.58579 q^{19} +3.84153 q^{23} -4.24264 q^{25} -2.46148 q^{29} -3.00000 q^{31} +5.24264 q^{37} -4.56248 q^{41} +0.242641 q^{43} +5.43275 q^{47} -10.1445 q^{53} -4.41421 q^{55} +10.6543 q^{59} -11.6569 q^{61} -8.48528 q^{67} +2.61079 q^{71} -7.41421 q^{73} +0.242641 q^{79} -1.74053 q^{83} +6.24264 q^{85} +4.56248 q^{89} +1.38005 q^{95} +5.65685 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 12 q^{19} - 12 q^{31} + 4 q^{37} - 16 q^{43} - 12 q^{55} - 24 q^{61} - 24 q^{73} - 16 q^{79} + 8 q^{85}+O(q^{100}) \) 4 * q - 12 * q^19 - 12 * q^31 + 4 * q^37 - 16 * q^43 - 12 * q^55 - 24 * q^61 - 24 * q^73 - 16 * q^79 + 8 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more