LMFDB - Embedded newform 1728.2.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,2,Mod(1,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, 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("1728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1728.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:	$13.7981494693$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 864)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	1728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.60555 q^{5} -3.60555 q^{7} +O(q^{10})$	$q+3.60555 q^{5} -3.60555 q^{7} -1.00000 q^{11} -4.00000 q^{13} -7.21110 q^{19} -6.00000 q^{23} +8.00000 q^{25} -7.21110 q^{29} +3.60555 q^{31} -13.0000 q^{35} -10.0000 q^{37} +7.21110 q^{41} +7.21110 q^{43} -10.0000 q^{47} +6.00000 q^{49} -3.60555 q^{53} -3.60555 q^{55} +4.00000 q^{59} -14.4222 q^{65} +7.21110 q^{67} +8.00000 q^{71} -3.00000 q^{73} +3.60555 q^{77} +14.4222 q^{79} -9.00000 q^{83} -7.21110 q^{89} +14.4222 q^{91} -26.0000 q^{95} +7.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 2 q^{11} - 8 q^{13} - 12 q^{23} + 16 q^{25} - 26 q^{35} - 20 q^{37} - 20 q^{47} + 12 q^{49} + 8 q^{59} + 16 q^{71} - 6 q^{73} - 18 q^{83} - 52 q^{95} + 14 q^{97}+O(q^{100}) $ 2 * q - 2 * q^11 - 8 * q^13 - 12 * q^23 + 16 * q^25 - 26 * q^35 - 20 * q^37 - 20 * q^47 + 12 * q^49 + 8 * q^59 + 16 * q^71 - 6 * q^73 - 18 * q^83 - 52 * q^95 + 14 * q^97

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
$5$	3.60555	1.61245	0.806226	−	0.591608i	$-0.201507\pi$
$5$	3.60555	1.61245	0.806226	+	0.591608i	$0.201507\pi$
$6$	0	0
$7$	−3.60555	−1.36277	−0.681385	−	0.731925i	$-0.738622\pi$
$7$	−3.60555	−1.36277	−0.681385	+	0.731925i	$0.738622\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−7.21110	−1.65434	−0.827170	−	0.561951i	$-0.810051\pi$
$19$	−7.21110	−1.65434	−0.827170	+	0.561951i	$0.810051\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	8.00000	1.60000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.21110	−1.33907	−0.669534	−	0.742781i	$-0.733506\pi$
$29$	−7.21110	−1.33907	−0.669534	+	0.742781i	$0.733506\pi$
$30$	0	0
$31$	3.60555	0.647576	0.323788	−	0.946130i	$-0.395044\pi$
$31$	3.60555	0.647576	0.323788	+	0.946130i	$0.395044\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−13.0000	−2.19740
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.21110	1.12619	0.563093	−	0.826394i	$-0.309611\pi$
$41$	7.21110	1.12619	0.563093	+	0.826394i	$0.309611\pi$
$42$	0	0
$43$	7.21110	1.09968	0.549841	−	0.835269i	$-0.314688\pi$
$43$	7.21110	1.09968	0.549841	+	0.835269i	$0.314688\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.0000	−1.45865	−0.729325	−	0.684167i	$-0.760166\pi$
$47$	−10.0000	−1.45865	−0.729325	+	0.684167i	$0.760166\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.60555	−0.495261	−0.247630	−	0.968855i	$-0.579652\pi$
$53$	−3.60555	−0.495261	−0.247630	+	0.968855i	$0.579652\pi$
$54$	0	0
$55$	−3.60555	−0.486172
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	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
$65$	−14.4222	−1.78885
$66$	0	0
$67$	7.21110	0.880976	0.440488	−	0.897758i	$-0.354805\pi$
$67$	7.21110	0.880976	0.440488	+	0.897758i	$0.354805\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	−3.00000	−0.351123	−0.175562	−	0.984468i	$-0.556174\pi$
$73$	−3.00000	−0.351123	−0.175562	+	0.984468i	$0.556174\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.60555	0.410891
$78$	0	0
$79$	14.4222	1.62262	0.811312	−	0.584613i	$-0.198754\pi$
$79$	14.4222	1.62262	0.811312	+	0.584613i	$0.198754\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.21110	−0.764375	−0.382188	−	0.924085i	$-0.624829\pi$
$89$	−7.21110	−0.764375	−0.382188	+	0.924085i	$0.624829\pi$
$90$	0	0
$91$	14.4222	1.51186
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−26.0000	−2.66754
$96$	0	0
$97$	7.00000	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.8167	1.07630	0.538149	−	0.842850i	$-0.319124\pi$
$101$	10.8167	1.07630	0.538149	+	0.842850i	$0.319124\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.0000	−1.64345	−0.821726	−	0.569883i	$-0.806989\pi$
$107$	−17.0000	−1.64345	−0.821726	+	0.569883i	$0.806989\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.21110	0.678363	0.339182	−	0.940721i	$-0.389850\pi$
$113$	7.21110	0.678363	0.339182	+	0.940721i	$0.389850\pi$
$114$	0	0
$115$	−21.6333	−2.01732
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.8167	0.967471
$126$	0	0
$127$	3.60555	0.319941	0.159970	−	0.987122i	$-0.448860\pi$
$127$	3.60555	0.319941	0.159970	+	0.987122i	$0.448860\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.00000	−0.0873704	−0.0436852	−	0.999045i	$-0.513910\pi$
$131$	−1.00000	−0.0873704	−0.0436852	+	0.999045i	$0.513910\pi$
$132$	0	0
$133$	26.0000	2.25449
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.6333	1.84826	0.924129	−	0.382080i	$-0.124792\pi$
$137$	21.6333	1.84826	0.924129	+	0.382080i	$0.124792\pi$
$138$	0	0
$139$	−14.4222	−1.22328	−0.611638	−	0.791138i	$-0.709489\pi$
$139$	−14.4222	−1.22328	−0.611638	+	0.791138i	$0.709489\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	−26.0000	−2.15918
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.8167	−0.886135	−0.443067	−	0.896488i	$-0.646110\pi$
$149$	−10.8167	−0.886135	−0.443067	+	0.896488i	$0.646110\pi$
$150$	0	0
$151$	−10.8167	−0.880247	−0.440123	−	0.897937i	$-0.645065\pi$
$151$	−10.8167	−0.880247	−0.440123	+	0.897937i	$0.645065\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	13.0000	1.04419
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	21.6333	1.70494
$162$	0	0
$163$	14.4222	1.12963	0.564817	−	0.825216i	$-0.308947\pi$
$163$	14.4222	1.12963	0.564817	+	0.825216i	$0.308947\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−10.8167	−0.822375	−0.411187	−	0.911551i	$-0.634886\pi$
$173$	−10.8167	−0.822375	−0.411187	+	0.911551i	$0.634886\pi$
$174$	0	0
$175$	−28.8444	−2.18043
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−15.0000	−1.12115	−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000	−1.12115	−0.560576	+	0.828103i	$0.689420\pi$
$180$	0	0
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−36.0555	−2.65085
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	−15.0000	−1.07972	−0.539862	−	0.841754i	$-0.681524\pi$
$193$	−15.0000	−1.07972	−0.539862	+	0.841754i	$0.681524\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.0278	−1.28442	−0.642212	−	0.766527i	$-0.721983\pi$
$197$	−18.0278	−1.28442	−0.642212	+	0.766527i	$0.721983\pi$
$198$	0	0
$199$	3.60555	0.255591	0.127795	−	0.991801i	$-0.459210\pi$
$199$	3.60555	0.255591	0.127795	+	0.991801i	$0.459210\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	26.0000	1.82484
$204$	0	0
$205$	26.0000	1.81592
$206$	0	0
$207$	0	0
$208$	0	0
$209$	7.21110	0.498802
$210$	0	0
$211$	−7.21110	−0.496433	−0.248216	−	0.968705i	$-0.579844\pi$
$211$	−7.21110	−0.496433	−0.248216	+	0.968705i	$0.579844\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	26.0000	1.77319
$216$	0	0
$217$	−13.0000	−0.882498
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−14.4222	−0.965782	−0.482891	−	0.875680i	$-0.660413\pi$
$223$	−14.4222	−0.965782	−0.482891	+	0.875680i	$0.660413\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−21.6333	−1.41725	−0.708623	−	0.705588i	$-0.750683\pi$
$233$	−21.6333	−1.41725	−0.708623	+	0.705588i	$0.750683\pi$
$234$	0	0
$235$	−36.0555	−2.35200
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	21.6333	1.38210
$246$	0	0
$247$	28.8444	1.83533
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.4222	−0.899632	−0.449816	−	0.893121i	$-0.648511\pi$
$257$	−14.4222	−0.899632	−0.449816	+	0.893121i	$0.648511\pi$
$258$	0	0
$259$	36.0555	2.24038
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	−13.0000	−0.798584
$266$	0	0
$267$	0	0
$268$	0	0
$269$	21.6333	1.31901	0.659503	−	0.751702i	$-0.270767\pi$
$269$	21.6333	1.31901	0.659503	+	0.751702i	$0.270767\pi$
$270$	0	0
$271$	10.8167	0.657065	0.328532	−	0.944493i	$-0.393446\pi$
$271$	10.8167	0.657065	0.328532	+	0.944493i	$0.393446\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.00000	−0.482418
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	7.21110	0.428656	0.214328	−	0.976762i	$-0.431244\pi$
$283$	7.21110	0.428656	0.214328	+	0.976762i	$0.431244\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−26.0000	−1.53473
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−21.6333	−1.26383	−0.631916	−	0.775037i	$-0.717731\pi$
$293$	−21.6333	−1.26383	−0.631916	+	0.775037i	$0.717731\pi$
$294$	0	0
$295$	14.4222	0.839693
$296$	0	0
$297$	0	0
$298$	0	0
$299$	24.0000	1.38796
$300$	0	0
$301$	−26.0000	−1.49862
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	1.00000	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.8167	0.607524	0.303762	−	0.952748i	$-0.401757\pi$
$317$	10.8167	0.607524	0.303762	+	0.952748i	$0.401757\pi$
$318$	0	0
$319$	7.21110	0.403744
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−32.0000	−1.77504
$326$	0	0
$327$	0	0
$328$	0	0
$329$	36.0555	1.98780
$330$	0	0
$331$	7.21110	0.396358	0.198179	−	0.980166i	$-0.436497\pi$
$331$	7.21110	0.396358	0.198179	+	0.980166i	$0.436497\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	26.0000	1.42053
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.60555	−0.195252
$342$	0	0
$343$	3.60555	0.194681
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.00000	0.0536828	0.0268414	−	0.999640i	$-0.491455\pi$
$347$	1.00000	0.0536828	0.0268414	+	0.999640i	$0.491455\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7.21110	−0.383808	−0.191904	−	0.981414i	$-0.561466\pi$
$353$	−7.21110	−0.383808	−0.191904	+	0.981414i	$0.561466\pi$
$354$	0	0
$355$	28.8444	1.53090
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	33.0000	1.73684
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−10.8167	−0.566170
$366$	0	0
$367$	−10.8167	−0.564625	−0.282312	−	0.959323i	$-0.591101\pi$
$367$	−10.8167	−0.564625	−0.282312	+	0.959323i	$0.591101\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	13.0000	0.674926
$372$	0	0
$373$	16.0000	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$373$	16.0000	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	28.8444	1.48556
$378$	0	0
$379$	14.4222	0.740819	0.370409	−	0.928869i	$-0.379217\pi$
$379$	14.4222	0.740819	0.370409	+	0.928869i	$0.379217\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	0	0
$385$	13.0000	0.662541
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10.8167	−0.548426	−0.274213	−	0.961669i	$-0.588417\pi$
$389$	−10.8167	−0.548426	−0.274213	+	0.961669i	$0.588417\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	52.0000	2.61640
$396$	0	0
$397$	−4.00000	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.4222	0.720211	0.360105	−	0.932912i	$-0.382741\pi$
$401$	14.4222	0.720211	0.360105	+	0.932912i	$0.382741\pi$
$402$	0	0
$403$	−14.4222	−0.718421
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.0000	0.495682
$408$	0	0
$409$	7.00000	0.346128	0.173064	−	0.984911i	$-0.444633\pi$
$409$	7.00000	0.346128	0.173064	+	0.984911i	$0.444633\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−14.4222	−0.709670
$414$	0	0
$415$	−32.4500	−1.59291
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	−32.0000	−1.55958	−0.779792	−	0.626038i	$-0.784675\pi$
$421$	−32.0000	−1.55958	−0.779792	+	0.626038i	$0.784675\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	0	0
$433$	−35.0000	−1.68199	−0.840996	−	0.541041i	$-0.818030\pi$
$433$	−35.0000	−1.68199	−0.840996	+	0.541041i	$0.818030\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	43.2666	2.06972
$438$	0	0
$439$	18.0278	0.860418	0.430209	−	0.902729i	$-0.358440\pi$
$439$	18.0278	0.860418	0.430209	+	0.902729i	$0.358440\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	−26.0000	−1.23252
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−7.21110	−0.340313	−0.170156	−	0.985417i	$-0.554427\pi$
$449$	−7.21110	−0.340313	−0.170156	+	0.985417i	$0.554427\pi$
$450$	0	0
$451$	−7.21110	−0.339558
$452$	0	0
$453$	0	0
$454$	0	0
$455$	52.0000	2.43780
$456$	0	0
$457$	23.0000	1.07589	0.537947	−	0.842978i	$-0.319200\pi$
$457$	23.0000	1.07589	0.537947	+	0.842978i	$0.319200\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−10.8167	−0.503782	−0.251891	−	0.967756i	$-0.581052\pi$
$461$	−10.8167	−0.503782	−0.251891	+	0.967756i	$0.581052\pi$
$462$	0	0
$463$	25.2389	1.17295	0.586475	−	0.809968i	$-0.300515\pi$
$463$	25.2389	1.17295	0.586475	+	0.809968i	$0.300515\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	23.0000	1.06431	0.532157	−	0.846646i	$-0.321382\pi$
$467$	23.0000	1.06431	0.532157	+	0.846646i	$0.321382\pi$
$468$	0	0
$469$	−26.0000	−1.20057
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−7.21110	−0.331567
$474$	0	0
$475$	−57.6888	−2.64694
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−42.0000	−1.91903	−0.959514	−	0.281659i	$-0.909115\pi$
$479$	−42.0000	−1.91903	−0.959514	+	0.281659i	$0.909115\pi$
$480$	0	0
$481$	40.0000	1.82384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	25.2389	1.14604
$486$	0	0
$487$	−14.4222	−0.653532	−0.326766	−	0.945105i	$-0.605959\pi$
$487$	−14.4222	−0.653532	−0.326766	+	0.945105i	$0.605959\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7.00000	−0.315906	−0.157953	−	0.987447i	$-0.550489\pi$
$491$	−7.00000	−0.315906	−0.157953	+	0.987447i	$0.550489\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−28.8444	−1.29385
$498$	0	0
$499$	−21.6333	−0.968440	−0.484220	−	0.874946i	$-0.660897\pi$
$499$	−21.6333	−0.968440	−0.484220	+	0.874946i	$0.660897\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−42.0000	−1.87269	−0.936344	−	0.351085i	$-0.885813\pi$
$503$	−42.0000	−1.87269	−0.936344	+	0.351085i	$0.885813\pi$
$504$	0	0
$505$	39.0000	1.73548
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10.8167	0.479440	0.239720	−	0.970842i	$-0.422944\pi$
$509$	10.8167	0.479440	0.239720	+	0.970842i	$0.422944\pi$
$510$	0	0
$511$	10.8167	0.478501
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	10.0000	0.439799
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−43.2666	−1.89554	−0.947772	−	0.318947i	$-0.896671\pi$
$521$	−43.2666	−1.89554	−0.947772	+	0.318947i	$0.896671\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−28.8444	−1.24939
$534$	0	0
$535$	−61.2944	−2.64999
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−6.00000	−0.258438
$540$	0	0
$541$	−4.00000	−0.171973	−0.0859867	−	0.996296i	$-0.527404\pi$
$541$	−4.00000	−0.171973	−0.0859867	+	0.996296i	$0.527404\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7.21110	−0.308890
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	52.0000	2.21527
$552$	0	0
$553$	−52.0000	−2.21126
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−39.6611	−1.68049	−0.840247	−	0.542204i	$-0.817590\pi$
$557$	−39.6611	−1.68049	−0.840247	+	0.542204i	$0.817590\pi$
$558$	0	0
$559$	−28.8444	−1.21999
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−39.0000	−1.64365	−0.821827	−	0.569737i	$-0.807045\pi$
$563$	−39.0000	−1.64365	−0.821827	+	0.569737i	$0.807045\pi$
$564$	0	0
$565$	26.0000	1.09383
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−14.4222	−0.604610	−0.302305	−	0.953211i	$-0.597756\pi$
$569$	−14.4222	−0.604610	−0.302305	+	0.953211i	$0.597756\pi$
$570$	0	0
$571$	14.4222	0.603550	0.301775	−	0.953379i	$-0.402421\pi$
$571$	14.4222	0.603550	0.301775	+	0.953379i	$0.402421\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−48.0000	−2.00174
$576$	0	0
$577$	6.00000	0.249783	0.124892	−	0.992170i	$-0.460142\pi$
$577$	6.00000	0.249783	0.124892	+	0.992170i	$0.460142\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	32.4500	1.34625
$582$	0	0
$583$	3.60555	0.149327
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.00000	0.123823	0.0619116	−	0.998082i	$-0.480280\pi$
$587$	3.00000	0.123823	0.0619116	+	0.998082i	$0.480280\pi$
$588$	0	0
$589$	−26.0000	−1.07131
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−21.6333	−0.888373	−0.444187	−	0.895934i	$-0.646507\pi$
$593$	−21.6333	−0.888373	−0.444187	+	0.895934i	$0.646507\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	14.0000	0.572024	0.286012	−	0.958226i	$-0.407670\pi$
$599$	14.0000	0.572024	0.286012	+	0.958226i	$0.407670\pi$
$600$	0	0
$601$	−17.0000	−0.693444	−0.346722	−	0.937968i	$-0.612705\pi$
$601$	−17.0000	−0.693444	−0.346722	+	0.937968i	$0.612705\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−36.0555	−1.46587
$606$	0	0
$607$	14.4222	0.585379	0.292690	−	0.956208i	$-0.405450\pi$
$607$	14.4222	0.585379	0.292690	+	0.956208i	$0.405450\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	40.0000	1.61823
$612$	0	0
$613$	18.0000	0.727013	0.363507	−	0.931592i	$-0.381579\pi$
$613$	18.0000	0.727013	0.363507	+	0.931592i	$0.381579\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.6333	0.870924	0.435462	−	0.900207i	$-0.356585\pi$
$617$	21.6333	0.870924	0.435462	+	0.900207i	$0.356585\pi$
$618$	0	0
$619$	43.2666	1.73903	0.869516	−	0.493905i	$-0.164431\pi$
$619$	43.2666	1.73903	0.869516	+	0.493905i	$0.164431\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	26.0000	1.04167
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	25.2389	1.00474	0.502372	−	0.864652i	$-0.332461\pi$
$631$	25.2389	1.00474	0.502372	+	0.864652i	$0.332461\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	13.0000	0.515889
$636$	0	0
$637$	−24.0000	−0.950915
$638$	0	0
$639$	0	0
$640$	0	0
$641$	36.0555	1.42411	0.712054	−	0.702125i	$-0.247765\pi$
$641$	36.0555	1.42411	0.712054	+	0.702125i	$0.247765\pi$
$642$	0	0
$643$	−43.2666	−1.70627	−0.853134	−	0.521691i	$-0.825301\pi$
$643$	−43.2666	−1.70627	−0.853134	+	0.521691i	$0.825301\pi$
$644$	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
$649$	−4.00000	−0.157014
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18.0278	0.705481	0.352740	−	0.935721i	$-0.385250\pi$
$653$	18.0278	0.705481	0.352740	+	0.935721i	$0.385250\pi$
$654$	0	0
$655$	−3.60555	−0.140881
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−27.0000	−1.05177	−0.525885	−	0.850555i	$-0.676266\pi$
$659$	−27.0000	−1.05177	−0.525885	+	0.850555i	$0.676266\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	93.7443	3.63525
$666$	0	0
$667$	43.2666	1.67529
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.21110	0.277145	0.138573	−	0.990352i	$-0.455749\pi$
$677$	7.21110	0.277145	0.138573	+	0.990352i	$0.455749\pi$
$678$	0	0
$679$	−25.2389	−0.968579
$680$	0	0
$681$	0	0
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	78.0000	2.98023
$686$	0	0
$687$	0	0
$688$	0	0
$689$	14.4222	0.549442
$690$	0	0
$691$	14.4222	0.548647	0.274323	−	0.961638i	$-0.411546\pi$
$691$	14.4222	0.548647	0.274323	+	0.961638i	$0.411546\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−52.0000	−1.97247
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	10.8167	0.408539	0.204270	−	0.978915i	$-0.434518\pi$
$701$	10.8167	0.408539	0.204270	+	0.978915i	$0.434518\pi$
$702$	0	0
$703$	72.1110	2.71972
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−39.0000	−1.46675
$708$	0	0
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−21.6333	−0.810174
$714$	0	0
$715$	14.4222	0.539360
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4.00000	−0.149175	−0.0745874	−	0.997214i	$-0.523764\pi$
$719$	−4.00000	−0.149175	−0.0745874	+	0.997214i	$0.523764\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−57.6888	−2.14251
$726$	0	0
$727$	−32.4500	−1.20350	−0.601751	−	0.798684i	$-0.705530\pi$
$727$	−32.4500	−1.20350	−0.601751	+	0.798684i	$0.705530\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	46.0000	1.69905	0.849524	−	0.527549i	$-0.176889\pi$
$733$	46.0000	1.69905	0.849524	+	0.527549i	$0.176889\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.21110	−0.265624
$738$	0	0
$739$	−28.8444	−1.06106	−0.530529	−	0.847667i	$-0.678007\pi$
$739$	−28.8444	−1.06106	−0.530529	+	0.847667i	$0.678007\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	−39.0000	−1.42885
$746$	0	0
$747$	0	0
$748$	0	0
$749$	61.2944	2.23965
$750$	0	0
$751$	46.8722	1.71039	0.855195	−	0.518307i	$-0.173437\pi$
$751$	46.8722	1.71039	0.855195	+	0.518307i	$0.173437\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−39.0000	−1.41936
$756$	0	0
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	7.21110	0.261059
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−16.0000	−0.577727
$768$	0	0
$769$	−11.0000	−0.396670	−0.198335	−	0.980134i	$-0.563553\pi$
$769$	−11.0000	−0.396670	−0.198335	+	0.980134i	$0.563553\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	21.6333	0.778096	0.389048	−	0.921217i	$-0.372804\pi$
$773$	21.6333	0.778096	0.389048	+	0.921217i	$0.372804\pi$
$774$	0	0
$775$	28.8444	1.03612
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−52.0000	−1.86309
$780$	0	0
$781$	−8.00000	−0.286263
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.4222	0.514751
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−26.0000	−0.924454
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−10.8167	−0.383146	−0.191573	−	0.981478i	$-0.561359\pi$
$797$	−10.8167	−0.383146	−0.191573	+	0.981478i	$0.561359\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3.00000	0.105868
$804$	0	0
$805$	78.0000	2.74914
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−50.4777	−1.77470	−0.887351	−	0.461095i	$-0.847457\pi$
$809$	−50.4777	−1.77470	−0.887351	+	0.461095i	$0.847457\pi$
$810$	0	0
$811$	−7.21110	−0.253216	−0.126608	−	0.991953i	$-0.540409\pi$
$811$	−7.21110	−0.253216	−0.126608	+	0.991953i	$0.540409\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	52.0000	1.82148
$816$	0	0
$817$	−52.0000	−1.81925
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−7.21110	−0.251669	−0.125835	−	0.992051i	$-0.540161\pi$
$821$	−7.21110	−0.251669	−0.125835	+	0.992051i	$0.540161\pi$
$822$	0	0
$823$	18.0278	0.628408	0.314204	−	0.949355i	$-0.398262\pi$
$823$	18.0278	0.628408	0.314204	+	0.949355i	$0.398262\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−16.0000	−0.556375	−0.278187	−	0.960527i	$-0.589734\pi$
$827$	−16.0000	−0.556375	−0.278187	+	0.960527i	$0.589734\pi$
$828$	0	0
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	7.21110	0.249550
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	23.0000	0.793103
$842$	0	0
$843$	0	0
$844$	0	0
$845$	10.8167	0.372104
$846$	0	0
$847$	36.0555	1.23888
$848$	0	0
$849$	0	0
$850$	0	0
$851$	60.0000	2.05677
$852$	0	0
$853$	42.0000	1.43805	0.719026	−	0.694983i	$-0.244588\pi$
$853$	42.0000	1.43805	0.719026	+	0.694983i	$0.244588\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−28.8444	−0.985306	−0.492653	−	0.870226i	$-0.663973\pi$
$857$	−28.8444	−0.985306	−0.492653	+	0.870226i	$0.663973\pi$
$858$	0	0
$859$	28.8444	0.984159	0.492079	−	0.870550i	$-0.336237\pi$
$859$	28.8444	0.984159	0.492079	+	0.870550i	$0.336237\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−28.0000	−0.953131	−0.476566	−	0.879139i	$-0.658119\pi$
$863$	−28.0000	−0.953131	−0.476566	+	0.879139i	$0.658119\pi$
$864$	0	0
$865$	−39.0000	−1.32604
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−14.4222	−0.489240
$870$	0	0
$871$	−28.8444	−0.977356
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−39.0000	−1.31844
$876$	0	0
$877$	−34.0000	−1.14810	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−34.0000	−1.14810	−0.574049	+	0.818821i	$0.694628\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−43.2666	−1.45769	−0.728845	−	0.684679i	$-0.759942\pi$
$881$	−43.2666	−1.45769	−0.728845	+	0.684679i	$0.759942\pi$
$882$	0	0
$883$	50.4777	1.69871	0.849355	−	0.527822i	$-0.176991\pi$
$883$	50.4777	1.69871	0.849355	+	0.527822i	$0.176991\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	0	0
$889$	−13.0000	−0.436006
$890$	0	0
$891$	0	0
$892$	0	0
$893$	72.1110	2.41310
$894$	0	0
$895$	−54.0833	−1.80780
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−26.0000	−0.867149
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−28.8444	−0.958821
$906$	0	0
$907$	−7.21110	−0.239441	−0.119720	−	0.992808i	$-0.538200\pi$
$907$	−7.21110	−0.239441	−0.119720	+	0.992808i	$0.538200\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	0	0
$913$	9.00000	0.297857
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.60555	0.119066
$918$	0	0
$919$	−54.0833	−1.78404	−0.892021	−	0.451994i	$-0.850713\pi$
$919$	−54.0833	−1.78404	−0.892021	+	0.451994i	$0.850713\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−32.0000	−1.05329
$924$	0	0
$925$	−80.0000	−2.63038
$926$	0	0
$927$	0	0
$928$	0	0
$929$	43.2666	1.41953	0.709766	−	0.704438i	$-0.248801\pi$
$929$	43.2666	1.41953	0.709766	+	0.704438i	$0.248801\pi$
$930$	0	0
$931$	−43.2666	−1.41801
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	35.0000	1.14340	0.571700	−	0.820463i	$-0.306284\pi$
$937$	35.0000	1.14340	0.571700	+	0.820463i	$0.306284\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−25.2389	−0.822763	−0.411382	−	0.911463i	$-0.634954\pi$
$941$	−25.2389	−0.822763	−0.411382	+	0.911463i	$0.634954\pi$
$942$	0	0
$943$	−43.2666	−1.40895
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−27.0000	−0.877382	−0.438691	−	0.898638i	$-0.644558\pi$
$947$	−27.0000	−0.877382	−0.438691	+	0.898638i	$0.644558\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	0	0
$951$	0	0
$952$	0	0
$953$	43.2666	1.40154	0.700772	−	0.713386i	$-0.252839\pi$
$953$	43.2666	1.40154	0.700772	+	0.713386i	$0.252839\pi$
$954$	0	0
$955$	−43.2666	−1.40007
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−78.0000	−2.51875
$960$	0	0
$961$	−18.0000	−0.580645
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−54.0833	−1.74100
$966$	0	0
$967$	10.8167	0.347840	0.173920	−	0.984760i	$-0.444357\pi$
$967$	10.8167	0.347840	0.173920	+	0.984760i	$0.444357\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−25.0000	−0.802288	−0.401144	−	0.916015i	$-0.631387\pi$
$971$	−25.0000	−0.802288	−0.401144	+	0.916015i	$0.631387\pi$
$972$	0	0
$973$	52.0000	1.66704
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−36.0555	−1.15352	−0.576759	−	0.816914i	$-0.695683\pi$
$977$	−36.0555	−1.15352	−0.576759	+	0.816914i	$0.695683\pi$
$978$	0	0
$979$	7.21110	0.230468
$980$	0	0
$981$	0	0
$982$	0	0
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	0	0
$985$	−65.0000	−2.07107
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−43.2666	−1.37580
$990$	0	0
$991$	−61.2944	−1.94708	−0.973540	−	0.228517i	$-0.926612\pi$
$991$	−61.2944	−1.94708	−0.973540	+	0.228517i	$0.926612\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	13.0000	0.412128
$996$	0	0
$997$	−36.0000	−1.14013	−0.570066	−	0.821599i	$-0.693082\pi$
$997$	−36.0000	−1.14013	−0.570066	+	0.821599i	$0.693082\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.2.a.bc.1.2		2
3.2	odd	2		1728.2.a.bd.1.1		2
4.3	odd	2		1728.2.a.bd.1.2		2
8.3	odd	2		864.2.a.m.1.1	✓	2
8.5	even	2		864.2.a.n.1.1	yes	2
12.11	even	2	inner	1728.2.a.bc.1.1		2
24.5	odd	2		864.2.a.m.1.2	yes	2
24.11	even	2		864.2.a.n.1.2	yes	2
72.5	odd	6		2592.2.i.bc.865.1		4
72.11	even	6		2592.2.i.bb.1729.1		4
72.13	even	6		2592.2.i.bb.865.2		4
72.29	odd	6		2592.2.i.bc.1729.1		4
72.43	odd	6		2592.2.i.bc.1729.2		4
72.59	even	6		2592.2.i.bb.865.1		4
72.61	even	6		2592.2.i.bb.1729.2		4
72.67	odd	6		2592.2.i.bc.865.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.a.m.1.1	✓	2	8.3	odd	2
864.2.a.m.1.2	yes	2	24.5	odd	2
864.2.a.n.1.1	yes	2	8.5	even	2
864.2.a.n.1.2	yes	2	24.11	even	2
1728.2.a.bc.1.1		2	12.11	even	2	inner
1728.2.a.bc.1.2		2	1.1	even	1	trivial
1728.2.a.bd.1.1		2	3.2	odd	2
1728.2.a.bd.1.2		2	4.3	odd	2
2592.2.i.bb.865.1		4	72.59	even	6
2592.2.i.bb.865.2		4	72.13	even	6
2592.2.i.bb.1729.1		4	72.11	even	6
2592.2.i.bb.1729.2		4	72.61	even	6
2592.2.i.bc.865.1		4	72.5	odd	6
2592.2.i.bc.865.2		4	72.67	odd	6
2592.2.i.bc.1729.1		4	72.29	odd	6
2592.2.i.bc.1729.2		4	72.43	odd	6

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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.a (trivial)

\(f(q)\)	\(=\)	\(q+3.60555 q^{5} -3.60555 q^{7} +O(q^{10})\)	\(q+3.60555 q^{5} -3.60555 q^{7} -1.00000 q^{11} -4.00000 q^{13} -7.21110 q^{19} -6.00000 q^{23} +8.00000 q^{25} -7.21110 q^{29} +3.60555 q^{31} -13.0000 q^{35} -10.0000 q^{37} +7.21110 q^{41} +7.21110 q^{43} -10.0000 q^{47} +6.00000 q^{49} -3.60555 q^{53} -3.60555 q^{55} +4.00000 q^{59} -14.4222 q^{65} +7.21110 q^{67} +8.00000 q^{71} -3.00000 q^{73} +3.60555 q^{77} +14.4222 q^{79} -9.00000 q^{83} -7.21110 q^{89} +14.4222 q^{91} -26.0000 q^{95} +7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 2 q^{11} - 8 q^{13} - 12 q^{23} + 16 q^{25} - 26 q^{35} - 20 q^{37} - 20 q^{47} + 12 q^{49} + 8 q^{59} + 16 q^{71} - 6 q^{73} - 18 q^{83} - 52 q^{95} + 14 q^{97}+O(q^{100}) \) 2 * q - 2 * q^11 - 8 * q^13 - 12 * q^23 + 16 * q^25 - 26 * q^35 - 20 * q^37 - 20 * q^47 + 12 * q^49 + 8 * q^59 + 16 * q^71 - 6 * q^73 - 18 * q^83 - 52 * q^95 + 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more