LMFDB - Embedded newform 9216.2.a.bq.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9216,2,Mod(1,9216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9216, 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("9216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9216 = 2^{10} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9216.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:	$73.5901305028$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.10070523904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 8x^{6} + 24x^{5} + 30x^{4} - 16x^{3} - 20x^{2} + 2 $ x^8 - 4x^7 - 8x^6 + 24x^5 + 30x^4 - 16x^3 - 20x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 144)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.305093$ of defining polynomial
Character	$\chi$	$=$	9216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.841723 q^{5} +1.64575 q^{7} +O(q^{10})$	$q-0.841723 q^{5} +1.64575 q^{7} +4.75216 q^{11} -3.74166 q^{13} -5.53019 q^{17} +5.15587 q^{19} +4.33981 q^{23} -4.29150 q^{25} -8.66259 q^{29} -5.64575 q^{31} -1.38527 q^{35} +0.913230 q^{37} +7.91094 q^{41} -0.500983 q^{43} +9.10132 q^{47} -4.29150 q^{49} -6.97915 q^{53} -4.00000 q^{55} -6.13742 q^{59} -0.913230 q^{61} +3.14944 q^{65} +5.65685 q^{67} -13.4411 q^{71} -3.29150 q^{73} +7.82087 q^{77} -9.64575 q^{79} +4.75216 q^{83} +4.65489 q^{85} -2.38075 q^{89} -6.15784 q^{91} -4.33981 q^{95} -10.5830 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7}+O(q^{10}) $ 8 * q - 8 * q^7	$ 8 q - 8 q^{7} + 8 q^{25} - 24 q^{31} + 8 q^{49} - 32 q^{55} + 16 q^{73} - 56 q^{79}+O(q^{100}) $ 8 * q - 8 * q^7 + 8 * q^25 - 24 * q^31 + 8 * q^49 - 32 * q^55 + 16 * q^73 - 56 * q^79

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
$5$	−0.841723	−0.376430	−0.188215	−	0.982128i	$-0.560270\pi$
$5$	−0.841723	−0.376430	−0.188215	+	0.982128i	$0.560270\pi$
$6$	0	0
$7$	1.64575	0.622036	0.311018	−	0.950404i	$-0.399330\pi$
$7$	1.64575	0.622036	0.311018	+	0.950404i	$0.399330\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.75216	1.43283	0.716415	−	0.697675i	$-0.245782\pi$
$11$	4.75216	1.43283	0.716415	+	0.697675i	$0.245782\pi$
$12$	0	0
$13$	−3.74166	−1.03775	−0.518875	−	0.854850i	$-0.673649\pi$
$13$	−3.74166	−1.03775	−0.518875	+	0.854850i	$0.673649\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.53019	−1.34127	−0.670634	−	0.741788i	$-0.733978\pi$
$17$	−5.53019	−1.34127	−0.670634	+	0.741788i	$0.733978\pi$
$18$	0	0
$19$	5.15587	1.18284	0.591419	−	0.806364i	$-0.298568\pi$
$19$	5.15587	1.18284	0.591419	+	0.806364i	$0.298568\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.33981	0.904914	0.452457	−	0.891786i	$-0.350548\pi$
$23$	4.33981	0.904914	0.452457	+	0.891786i	$0.350548\pi$
$24$	0	0
$25$	−4.29150	−0.858301
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.66259	−1.60860	−0.804302	−	0.594221i	$-0.797460\pi$
$29$	−8.66259	−1.60860	−0.804302	+	0.594221i	$0.797460\pi$
$30$	0	0
$31$	−5.64575	−1.01401	−0.507003	−	0.861944i	$-0.669247\pi$
$31$	−5.64575	−1.01401	−0.507003	+	0.861944i	$0.669247\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.38527	−0.234153
$36$	0	0
$37$	0.913230	0.150134	0.0750671	−	0.997178i	$-0.476083\pi$
$37$	0.913230	0.150134	0.0750671	+	0.997178i	$0.476083\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.91094	1.23548	0.617741	−	0.786382i	$-0.288048\pi$
$41$	7.91094	1.23548	0.617741	+	0.786382i	$0.288048\pi$
$42$	0	0
$43$	−0.500983	−0.0763992	−0.0381996	−	0.999270i	$-0.512162\pi$
$43$	−0.500983	−0.0763992	−0.0381996	+	0.999270i	$0.512162\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.10132	1.32756	0.663782	−	0.747926i	$-0.268950\pi$
$47$	9.10132	1.32756	0.663782	+	0.747926i	$0.268950\pi$
$48$	0	0
$49$	−4.29150	−0.613072
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.97915	−0.958660	−0.479330	−	0.877635i	$-0.659120\pi$
$53$	−6.97915	−0.958660	−0.479330	+	0.877635i	$0.659120\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.13742	−0.799025	−0.399512	−	0.916728i	$-0.630821\pi$
$59$	−6.13742	−0.799025	−0.399512	+	0.916728i	$0.630821\pi$
$60$	0	0
$61$	−0.913230	−0.116927	−0.0584636	−	0.998290i	$-0.518620\pi$
$61$	−0.913230	−0.116927	−0.0584636	+	0.998290i	$0.518620\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.14944	0.390640
$66$	0	0
$67$	5.65685	0.691095	0.345547	−	0.938401i	$-0.387693\pi$
$67$	5.65685	0.691095	0.345547	+	0.938401i	$0.387693\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.4411	−1.59517	−0.797584	−	0.603207i	$-0.793889\pi$
$71$	−13.4411	−1.59517	−0.797584	+	0.603207i	$0.793889\pi$
$72$	0	0
$73$	−3.29150	−0.385241	−0.192621	−	0.981273i	$-0.561699\pi$
$73$	−3.29150	−0.385241	−0.192621	+	0.981273i	$0.561699\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.82087	0.891271
$78$	0	0
$79$	−9.64575	−1.08523	−0.542616	−	0.839981i	$-0.682566\pi$
$79$	−9.64575	−1.08523	−0.542616	+	0.839981i	$0.682566\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.75216	0.521617	0.260809	−	0.965391i	$-0.416011\pi$
$83$	4.75216	0.521617	0.260809	+	0.965391i	$0.416011\pi$
$84$	0	0
$85$	4.65489	0.504893
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.38075	−0.252359	−0.126180	−	0.992007i	$-0.540272\pi$
$89$	−2.38075	−0.252359	−0.126180	+	0.992007i	$0.540272\pi$
$90$	0	0
$91$	−6.15784	−0.645517
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.33981	−0.445256
$96$	0	0
$97$	−10.5830	−1.07454	−0.537271	−	0.843410i	$-0.680545\pi$
$97$	−10.5830	−1.07454	−0.537271	+	0.843410i	$0.680545\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−0.841723	−0.0837546	−0.0418773	−	0.999123i	$-0.513334\pi$
$101$	−0.841723	−0.0837546	−0.0418773	+	0.999123i	$0.513334\pi$
$102$	0	0
$103$	−16.9373	−1.66888	−0.834439	−	0.551101i	$-0.814208\pi$
$103$	−16.9373	−1.66888	−0.834439	+	0.551101i	$0.814208\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.36689	0.325490	0.162745	−	0.986668i	$-0.447965\pi$
$107$	3.36689	0.325490	0.162745	+	0.986668i	$0.447965\pi$
$108$	0	0
$109$	9.39851	0.900214	0.450107	−	0.892975i	$-0.351386\pi$
$109$	9.39851	0.900214	0.450107	+	0.892975i	$0.351386\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	−3.65292	−0.340637
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−9.10132	−0.834316
$120$	0	0
$121$	11.5830	1.05300
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.82087	0.699520
$126$	0	0
$127$	0.937254	0.0831678	0.0415839	−	0.999135i	$-0.486760\pi$
$127$	0.937254	0.0831678	0.0415839	+	0.999135i	$0.486760\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.50432	0.830396	0.415198	−	0.909731i	$-0.363712\pi$
$131$	9.50432	0.830396	0.415198	+	0.909731i	$0.363712\pi$
$132$	0	0
$133$	8.48528	0.735767
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.2917	−0.879279	−0.439639	−	0.898174i	$-0.644894\pi$
$137$	−10.2917	−0.879279	−0.439639	+	0.898174i	$0.644894\pi$
$138$	0	0
$139$	−20.6235	−1.74926	−0.874631	−	0.484790i	$-0.838896\pi$
$139$	−20.6235	−1.74926	−0.874631	+	0.484790i	$0.838896\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−17.7809	−1.48692
$144$	0	0
$145$	7.29150	0.605526
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.4835	1.35038	0.675189	−	0.737645i	$-0.264062\pi$
$149$	16.4835	1.35038	0.675189	+	0.737645i	$0.264062\pi$
$150$	0	0
$151$	−10.3542	−0.842617	−0.421308	−	0.906917i	$-0.638429\pi$
$151$	−10.3542	−0.842617	−0.421308	+	0.906917i	$0.638429\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.75216	0.381703
$156$	0	0
$157$	−9.39851	−0.750083	−0.375041	−	0.927008i	$-0.622372\pi$
$157$	−9.39851	−0.750083	−0.375041	+	0.927008i	$0.622372\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.14226	0.562889
$162$	0	0
$163$	16.4696	1.29000	0.644999	−	0.764184i	$-0.276858\pi$
$163$	16.4696	1.29000	0.644999	+	0.764184i	$0.276858\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.7809	1.37593	0.687965	−	0.725743i	$-0.258504\pi$
$167$	17.7809	1.37593	0.687965	+	0.725743i	$0.258504\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.97915	−0.530615	−0.265307	−	0.964164i	$-0.585473\pi$
$173$	−6.97915	−0.530615	−0.265307	+	0.964164i	$0.585473\pi$
$174$	0	0
$175$	−7.06275	−0.533893
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.36689	0.251653	0.125827	−	0.992052i	$-0.459842\pi$
$179$	3.36689	0.251653	0.125827	+	0.992052i	$0.459842\pi$
$180$	0	0
$181$	−0.913230	−0.0678799	−0.0339399	−	0.999424i	$-0.510805\pi$
$181$	−0.913230	−0.0678799	−0.0339399	+	0.999424i	$0.510805\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.768687	−0.0565150
$186$	0	0
$187$	−26.2803	−1.92181
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.10132	0.658548	0.329274	−	0.944234i	$-0.393196\pi$
$191$	9.10132	0.658548	0.329274	+	0.944234i	$0.393196\pi$
$192$	0	0
$193$	11.8745	0.854746	0.427373	−	0.904075i	$-0.359439\pi$
$193$	11.8745	0.854746	0.427373	+	0.904075i	$0.359439\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.0295	0.857065	0.428533	−	0.903526i	$-0.359031\pi$
$197$	12.0295	0.857065	0.428533	+	0.903526i	$0.359031\pi$
$198$	0	0
$199$	−13.6458	−0.967322	−0.483661	−	0.875256i	$-0.660693\pi$
$199$	−13.6458	−0.967322	−0.483661	+	0.875256i	$0.660693\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−14.2565	−1.00061
$204$	0	0
$205$	−6.65882	−0.465072
$206$	0	0
$207$	0	0
$208$	0	0
$209$	24.5015	1.69481
$210$	0	0
$211$	20.6235	1.41978	0.709890	−	0.704313i	$-0.248745\pi$
$211$	20.6235	1.41978	0.709890	+	0.704313i	$0.248745\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.421689	0.0287590
$216$	0	0
$217$	−9.29150	−0.630748
$218$	0	0
$219$	0	0
$220$	0	0
$221$	20.6921	1.39190
$222$	0	0
$223$	−14.3542	−0.961232	−0.480616	−	0.876931i	$-0.659587\pi$
$223$	−14.3542	−0.961232	−0.480616	+	0.876931i	$0.659587\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.98162	−0.131525	−0.0657625	−	0.997835i	$-0.520948\pi$
$227$	−1.98162	−0.131525	−0.0657625	+	0.997835i	$0.520948\pi$
$228$	0	0
$229$	−25.3671	−1.67631	−0.838153	−	0.545435i	$-0.816364\pi$
$229$	−25.3671	−1.67631	−0.838153	+	0.545435i	$0.816364\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−19.7400	−1.29321	−0.646606	−	0.762825i	$-0.723812\pi$
$233$	−19.7400	−1.29321	−0.646606	+	0.762825i	$0.723812\pi$
$234$	0	0
$235$	−7.66079	−0.499735
$236$	0	0
$237$	0	0
$238$	0	0
$239$	13.0194	0.842158	0.421079	−	0.907024i	$-0.361652\pi$
$239$	13.0194	0.842158	0.421079	+	0.907024i	$0.361652\pi$
$240$	0	0
$241$	1.29150	0.0831930	0.0415965	−	0.999134i	$-0.486756\pi$
$241$	1.29150	0.0831930	0.0415965	+	0.999134i	$0.486756\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.61226	0.230779
$246$	0	0
$247$	−19.2915	−1.22749
$248$	0	0
$249$	0	0
$250$	0	0
$251$	26.5313	1.67464	0.837321	−	0.546711i	$-0.184121\pi$
$251$	26.5313	1.67464	0.837321	+	0.546711i	$0.184121\pi$
$252$	0	0
$253$	20.6235	1.29659
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−29.2630	−1.82538	−0.912688	−	0.408656i	$-0.865998\pi$
$257$	−29.2630	−1.82538	−0.912688	+	0.408656i	$0.865998\pi$
$258$	0	0
$259$	1.50295	0.0933888
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.67963	0.535209	0.267604	−	0.963529i	$-0.413768\pi$
$263$	8.67963	0.535209	0.267604	+	0.963529i	$0.413768\pi$
$264$	0	0
$265$	5.87451	0.360868
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−14.8000	−0.902373	−0.451186	−	0.892430i	$-0.648999\pi$
$269$	−14.8000	−0.902373	−0.451186	+	0.892430i	$0.648999\pi$
$270$	0	0
$271$	−6.35425	−0.385993	−0.192997	−	0.981199i	$-0.561821\pi$
$271$	−6.35425	−0.385993	−0.192997	+	0.981199i	$0.561821\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−20.3939	−1.22980
$276$	0	0
$277$	−23.3632	−1.40376	−0.701879	−	0.712297i	$-0.747655\pi$
$277$	−23.3632	−1.40376	−0.701879	+	0.712297i	$0.747655\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.38075	−0.142024	−0.0710119	−	0.997475i	$-0.522623\pi$
$281$	−2.38075	−0.142024	−0.0710119	+	0.997475i	$0.522623\pi$
$282$	0	0
$283$	3.65292	0.217144	0.108572	−	0.994089i	$-0.465372\pi$
$283$	3.65292	0.217144	0.108572	+	0.994089i	$0.465372\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	13.0194	0.768513
$288$	0	0
$289$	13.5830	0.799000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3.61226	−0.211030	−0.105515	−	0.994418i	$-0.533649\pi$
$293$	−3.61226	−0.211030	−0.105515	+	0.994418i	$0.533649\pi$
$294$	0	0
$295$	5.16601	0.300777
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−16.2381	−0.939074
$300$	0	0
$301$	−0.824494	−0.0475230
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.768687	0.0440149
$306$	0	0
$307$	28.2843	1.61427	0.807134	−	0.590368i	$-0.201017\pi$
$307$	28.2843	1.61427	0.807134	+	0.590368i	$0.201017\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8.67963	−0.492177	−0.246088	−	0.969247i	$-0.579145\pi$
$311$	−8.67963	−0.492177	−0.246088	+	0.969247i	$0.579145\pi$
$312$	0	0
$313$	−9.29150	−0.525187	−0.262593	−	0.964907i	$-0.584578\pi$
$313$	−9.29150	−0.525187	−0.262593	+	0.964907i	$0.584578\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.841723	−0.0472759	−0.0236379	−	0.999721i	$-0.507525\pi$
$317$	−0.841723	−0.0472759	−0.0236379	+	0.999721i	$0.507525\pi$
$318$	0	0
$319$	−41.1660	−2.30485
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−28.5129	−1.58650
$324$	0	0
$325$	16.0573	0.890701
$326$	0	0
$327$	0	0
$328$	0	0
$329$	14.9785	0.825792
$330$	0	0
$331$	−11.3137	−0.621858	−0.310929	−	0.950433i	$-0.600640\pi$
$331$	−11.3137	−0.621858	−0.310929	+	0.950433i	$0.600640\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.76150	−0.260149
$336$	0	0
$337$	−4.70850	−0.256488	−0.128244	−	0.991743i	$-0.540934\pi$
$337$	−4.70850	−0.256488	−0.128244	+	0.991743i	$0.540934\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−26.8295	−1.45290
$342$	0	0
$343$	−18.5830	−1.00339
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.75216	−0.255109	−0.127555	−	0.991832i	$-0.540713\pi$
$347$	−4.75216	−0.255109	−0.127555	+	0.991832i	$0.540713\pi$
$348$	0	0
$349$	4.56615	0.244420	0.122210	−	0.992504i	$-0.461002\pi$
$349$	4.56615	0.244420	0.122210	+	0.992504i	$0.461002\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7.14226	0.380144	0.190072	−	0.981770i	$-0.439128\pi$
$353$	7.14226	0.380144	0.190072	+	0.981770i	$0.439128\pi$
$354$	0	0
$355$	11.3137	0.600469
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.76150	0.251303	0.125651	−	0.992074i	$-0.459898\pi$
$359$	4.76150	0.251303	0.125651	+	0.992074i	$0.459898\pi$
$360$	0	0
$361$	7.58301	0.399106
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.77053	0.145016
$366$	0	0
$367$	−34.8118	−1.81716	−0.908580	−	0.417712i	$-0.862832\pi$
$367$	−34.8118	−1.81716	−0.908580	+	0.417712i	$0.862832\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.4859	−0.596320
$372$	0	0
$373$	16.8818	0.874108	0.437054	−	0.899435i	$-0.356022\pi$
$373$	16.8818	0.874108	0.437054	+	0.899435i	$0.356022\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	32.4125	1.66933
$378$	0	0
$379$	−11.8147	−0.606880	−0.303440	−	0.952851i	$-0.598135\pi$
$379$	−11.8147	−0.606880	−0.303440	+	0.952851i	$0.598135\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9.10132	−0.465056	−0.232528	−	0.972590i	$-0.574700\pi$
$383$	−9.10132	−0.465056	−0.232528	+	0.972590i	$0.574700\pi$
$384$	0	0
$385$	−6.58301	−0.335501
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−29.3547	−1.48834	−0.744170	−	0.667990i	$-0.767155\pi$
$389$	−29.3547	−1.48834	−0.744170	+	0.667990i	$0.767155\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.11905	0.408514
$396$	0	0
$397$	−11.4024	−0.572272	−0.286136	−	0.958189i	$-0.592371\pi$
$397$	−11.4024	−0.572272	−0.286136	+	0.958189i	$0.592371\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.5906	−0.828494	−0.414247	−	0.910165i	$-0.635955\pi$
$401$	−16.5906	−0.828494	−0.414247	+	0.910165i	$0.635955\pi$
$402$	0	0
$403$	21.1245	1.05228
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.33981	0.215117
$408$	0	0
$409$	25.1660	1.24438	0.622190	−	0.782867i	$-0.286243\pi$
$409$	25.1660	1.24438	0.622190	+	0.782867i	$0.286243\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−10.1007	−0.497022
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	0	0
$418$	0	0
$419$	26.5313	1.29614	0.648070	−	0.761581i	$-0.275576\pi$
$419$	26.5313	1.29614	0.648070	+	0.761581i	$0.275576\pi$
$420$	0	0
$421$	−16.0573	−0.782587	−0.391293	−	0.920266i	$-0.627972\pi$
$421$	−16.0573	−0.782587	−0.391293	+	0.920266i	$0.627972\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	23.7328	1.15121
$426$	0	0
$427$	−1.50295	−0.0727328
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−31.2221	−1.50391	−0.751957	−	0.659212i	$-0.770890\pi$
$431$	−31.2221	−1.50391	−0.751957	+	0.659212i	$0.770890\pi$
$432$	0	0
$433$	4.00000	0.192228	0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000	0.192228	0.0961139	+	0.995370i	$0.469359\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	22.3755	1.07037
$438$	0	0
$439$	−0.479741	−0.0228968	−0.0114484	−	0.999934i	$-0.503644\pi$
$439$	−0.479741	−0.0228968	−0.0114484	+	0.999934i	$0.503644\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−30.4946	−1.44884	−0.724420	−	0.689358i	$-0.757893\pi$
$443$	−30.4946	−1.44884	−0.724420	+	0.689358i	$0.757893\pi$
$444$	0	0
$445$	2.00393	0.0949955
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9.44832	0.445894	0.222947	−	0.974831i	$-0.428432\pi$
$449$	9.44832	0.445894	0.222947	+	0.974831i	$0.428432\pi$
$450$	0	0
$451$	37.5940	1.77023
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.18319	0.242992
$456$	0	0
$457$	5.41699	0.253396	0.126698	−	0.991941i	$-0.459562\pi$
$457$	5.41699	0.253396	0.126698	+	0.991941i	$0.459562\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12.0295	−0.560269	−0.280134	−	0.959961i	$-0.590379\pi$
$461$	−12.0295	−0.560269	−0.280134	+	0.959961i	$0.590379\pi$
$462$	0	0
$463$	−15.0627	−0.700025	−0.350013	−	0.936745i	$-0.613823\pi$
$463$	−15.0627	−0.700025	−0.350013	+	0.936745i	$0.613823\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	36.0356	1.66753	0.833765	−	0.552119i	$-0.186181\pi$
$467$	36.0356	1.66753	0.833765	+	0.552119i	$0.186181\pi$
$468$	0	0
$469$	9.30978	0.429885
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.38075	−0.109467
$474$	0	0
$475$	−22.1264	−1.01523
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−22.1208	−1.01072	−0.505362	−	0.862908i	$-0.668641\pi$
$479$	−22.1208	−1.01072	−0.505362	+	0.862908i	$0.668641\pi$
$480$	0	0
$481$	−3.41699	−0.155802
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.90796	0.404490
$486$	0	0
$487$	−8.22876	−0.372881	−0.186440	−	0.982466i	$-0.559695\pi$
$487$	−8.22876	−0.372881	−0.186440	+	0.982466i	$0.559695\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	21.7792	0.982880	0.491440	−	0.870911i	$-0.336471\pi$
$491$	21.7792	0.982880	0.491440	+	0.870911i	$0.336471\pi$
$492$	0	0
$493$	47.9058	2.15757
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−22.1208	−0.992252
$498$	0	0
$499$	−14.9666	−0.669998	−0.334999	−	0.942218i	$-0.608736\pi$
$499$	−14.9666	−0.669998	−0.334999	+	0.942218i	$0.608736\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	31.6438	1.41093	0.705463	−	0.708747i	$-0.250739\pi$
$503$	31.6438	1.41093	0.705463	+	0.708747i	$0.250739\pi$
$504$	0	0
$505$	0.708497	0.0315277
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−34.8957	−1.54673	−0.773363	−	0.633963i	$-0.781427\pi$
$509$	−34.8957	−1.54673	−0.773363	+	0.633963i	$0.781427\pi$
$510$	0	0
$511$	−5.41699	−0.239634
$512$	0	0
$513$	0	0
$514$	0	0
$515$	14.2565	0.628215
$516$	0	0
$517$	43.2509	1.90217
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14.2098	−0.622543	−0.311272	−	0.950321i	$-0.600755\pi$
$521$	−14.2098	−0.622543	−0.311272	+	0.950321i	$0.600755\pi$
$522$	0	0
$523$	−26.7813	−1.17107	−0.585533	−	0.810649i	$-0.699115\pi$
$523$	−26.7813	−1.17107	−0.585533	+	0.810649i	$0.699115\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	31.2221	1.36006
$528$	0	0
$529$	−4.16601	−0.181131
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−29.6000	−1.28212
$534$	0	0
$535$	−2.83399	−0.122524
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−20.3939	−0.878427
$540$	0	0
$541$	−40.3337	−1.73408	−0.867041	−	0.498236i	$-0.833981\pi$
$541$	−40.3337	−1.73408	−0.867041	+	0.498236i	$0.833981\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7.91094	−0.338868
$546$	0	0
$547$	21.1245	0.903217	0.451608	−	0.892216i	$-0.350850\pi$
$547$	21.1245	0.903217	0.451608	+	0.892216i	$0.350850\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−44.6632	−1.90272
$552$	0	0
$553$	−15.8745	−0.675053
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26.5841	1.12641	0.563203	−	0.826319i	$-0.309569\pi$
$557$	26.5841	1.12641	0.563203	+	0.826319i	$0.309569\pi$
$558$	0	0
$559$	1.87451	0.0792832
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7.52269	0.317044	0.158522	−	0.987355i	$-0.449327\pi$
$563$	7.52269	0.317044	0.158522	+	0.987355i	$0.449327\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−41.0921	−1.72267	−0.861335	−	0.508038i	$-0.830371\pi$
$569$	−41.0921	−1.72267	−0.861335	+	0.508038i	$0.830371\pi$
$570$	0	0
$571$	−5.65685	−0.236732	−0.118366	−	0.992970i	$-0.537766\pi$
$571$	−5.65685	−0.236732	−0.118366	+	0.992970i	$0.537766\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−18.6243	−0.776688
$576$	0	0
$577$	−39.0405	−1.62528	−0.812639	−	0.582767i	$-0.801970\pi$
$577$	−39.0405	−1.62528	−0.812639	+	0.582767i	$0.801970\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.82087	0.324464
$582$	0	0
$583$	−33.1660	−1.37360
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.73378	−0.277933	−0.138966	−	0.990297i	$-0.544378\pi$
$587$	−6.73378	−0.277933	−0.138966	+	0.990297i	$0.544378\pi$
$588$	0	0
$589$	−29.1088	−1.19941
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−22.1208	−0.908391	−0.454195	−	0.890902i	$-0.650073\pi$
$593$	−22.1208	−0.908391	−0.454195	+	0.890902i	$0.650073\pi$
$594$	0	0
$595$	7.66079	0.314062
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−13.8628	−0.566420	−0.283210	−	0.959058i	$-0.591399\pi$
$599$	−13.8628	−0.566420	−0.283210	+	0.959058i	$0.591399\pi$
$600$	0	0
$601$	3.29150	0.134263	0.0671316	−	0.997744i	$-0.478615\pi$
$601$	3.29150	0.134263	0.0671316	+	0.997744i	$0.478615\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.74968	−0.396381
$606$	0	0
$607$	14.1033	0.572434	0.286217	−	0.958165i	$-0.407602\pi$
$607$	14.1033	0.572434	0.286217	+	0.958165i	$0.407602\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−34.0540	−1.37768
$612$	0	0
$613$	−37.6828	−1.52199	−0.760997	−	0.648756i	$-0.775290\pi$
$613$	−37.6828	−1.52199	−0.760997	+	0.648756i	$0.775290\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.6438	1.27393	0.636965	−	0.770893i	$-0.280190\pi$
$617$	31.6438	1.27393	0.636965	+	0.770893i	$0.280190\pi$
$618$	0	0
$619$	−29.9333	−1.20312	−0.601560	−	0.798828i	$-0.705454\pi$
$619$	−29.9333	−1.20312	−0.601560	+	0.798828i	$0.705454\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.91813	−0.156976
$624$	0	0
$625$	14.8745	0.594980
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.05034	−0.201370
$630$	0	0
$631$	−8.22876	−0.327582	−0.163791	−	0.986495i	$-0.552372\pi$
$631$	−8.22876	−0.327582	−0.163791	+	0.986495i	$0.552372\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−0.788908	−0.0313069
$636$	0	0
$637$	16.0573	0.636215
$638$	0	0
$639$	0	0
$640$	0	0
$641$	16.5906	0.655288	0.327644	−	0.944801i	$-0.393745\pi$
$641$	16.5906	0.655288	0.327644	+	0.944801i	$0.393745\pi$
$642$	0	0
$643$	−5.15587	−0.203328	−0.101664	−	0.994819i	$-0.532417\pi$
$643$	−5.15587	−0.203328	−0.101664	+	0.994819i	$0.532417\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4.33981	−0.170616	−0.0853079	−	0.996355i	$-0.527187\pi$
$647$	−4.33981	−0.170616	−0.0853079	+	0.996355i	$0.527187\pi$
$648$	0	0
$649$	−29.1660	−1.14487
$650$	0	0
$651$	0	0
$652$	0	0
$653$	29.3547	1.14874	0.574369	−	0.818597i	$-0.305248\pi$
$653$	29.3547	1.14874	0.574369	+	0.818597i	$0.305248\pi$
$654$	0	0
$655$	−8.00000	−0.312586
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−35.2467	−1.37302	−0.686509	−	0.727121i	$-0.740858\pi$
$659$	−35.2467	−1.37302	−0.686509	+	0.727121i	$0.740858\pi$
$660$	0	0
$661$	−3.91913	−0.152436	−0.0762182	−	0.997091i	$-0.524285\pi$
$661$	−3.91913	−0.152436	−0.0762182	+	0.997091i	$0.524285\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−7.14226	−0.276965
$666$	0	0
$667$	−37.5940	−1.45565
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.33981	−0.167537
$672$	0	0
$673$	20.0000	0.770943	0.385472	−	0.922720i	$-0.374039\pi$
$673$	20.0000	0.770943	0.385472	+	0.922720i	$0.374039\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	19.8504	0.762911	0.381456	−	0.924387i	$-0.375423\pi$
$677$	19.8504	0.762911	0.381456	+	0.924387i	$0.375423\pi$
$678$	0	0
$679$	−17.4170	−0.668403
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−0.788908	−0.0301867	−0.0150934	−	0.999886i	$-0.504805\pi$
$683$	−0.788908	−0.0301867	−0.0150934	+	0.999886i	$0.504805\pi$
$684$	0	0
$685$	8.66275	0.330987
$686$	0	0
$687$	0	0
$688$	0	0
$689$	26.1136	0.994848
$690$	0	0
$691$	5.15587	0.196139	0.0980693	−	0.995180i	$-0.468733\pi$
$691$	5.15587	0.196139	0.0980693	+	0.995180i	$0.468733\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17.3593	0.658474
$696$	0	0
$697$	−43.7490	−1.65711
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−5.89206	−0.222540	−0.111270	−	0.993790i	$-0.535492\pi$
$701$	−5.89206	−0.222540	−0.111270	+	0.993790i	$0.535492\pi$
$702$	0	0
$703$	4.70850	0.177584
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.38527	−0.0520983
$708$	0	0
$709$	12.4044	0.465857	0.232929	−	0.972494i	$-0.425169\pi$
$709$	12.4044	0.465857	0.232929	+	0.972494i	$0.425169\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−24.5015	−0.917589
$714$	0	0
$715$	14.9666	0.559720
$716$	0	0
$717$	0	0
$718$	0	0
$719$	40.3234	1.50381	0.751904	−	0.659272i	$-0.229135\pi$
$719$	40.3234	1.50381	0.751904	+	0.659272i	$0.229135\pi$
$720$	0	0
$721$	−27.8745	−1.03810
$722$	0	0
$723$	0	0
$724$	0	0
$725$	37.1755	1.38066
$726$	0	0
$727$	−33.3948	−1.23854	−0.619272	−	0.785177i	$-0.712572\pi$
$727$	−33.3948	−1.23854	−0.619272	+	0.785177i	$0.712572\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.77053	0.102472
$732$	0	0
$733$	−17.8838	−0.660553	−0.330276	−	0.943884i	$-0.607142\pi$
$733$	−17.8838	−0.660553	−0.330276	+	0.943884i	$0.607142\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	26.8823	0.990221
$738$	0	0
$739$	−12.9627	−0.476840	−0.238420	−	0.971162i	$-0.576630\pi$
$739$	−12.9627	−0.476840	−0.238420	+	0.971162i	$0.576630\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−22.5425	−0.827002	−0.413501	−	0.910504i	$-0.635694\pi$
$743$	−22.5425	−0.827002	−0.413501	+	0.910504i	$0.635694\pi$
$744$	0	0
$745$	−13.8745	−0.508323
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5.54107	0.202466
$750$	0	0
$751$	−12.9373	−0.472087	−0.236044	−	0.971742i	$-0.575851\pi$
$751$	−12.9373	−0.472087	−0.236044	+	0.971742i	$0.575851\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8.71541	0.317186
$756$	0	0
$757$	−10.2230	−0.371561	−0.185781	−	0.982591i	$-0.559481\pi$
$757$	−10.2230	−0.371561	−0.185781	+	0.982591i	$0.559481\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7.91094	−0.286771	−0.143386	−	0.989667i	$-0.545799\pi$
$761$	−7.91094	−0.286771	−0.143386	+	0.989667i	$0.545799\pi$
$762$	0	0
$763$	15.4676	0.559965
$764$	0	0
$765$	0	0
$766$	0	0
$767$	22.9641	0.829187
$768$	0	0
$769$	17.2915	0.623548	0.311774	−	0.950156i	$-0.399077\pi$
$769$	17.2915	0.623548	0.311774	+	0.950156i	$0.399077\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	30.4418	1.09491	0.547457	−	0.836834i	$-0.315596\pi$
$773$	30.4418	1.09491	0.547457	+	0.836834i	$0.315596\pi$
$774$	0	0
$775$	24.2288	0.870323
$776$	0	0
$777$	0	0
$778$	0	0
$779$	40.7878	1.46137
$780$	0	0
$781$	−63.8744	−2.28561
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7.91094	0.282354
$786$	0	0
$787$	−8.80879	−0.314000	−0.157000	−	0.987599i	$-0.550182\pi$
$787$	−8.80879	−0.314000	−0.157000	+	0.987599i	$0.550182\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	3.41699	0.121341
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.4418	1.07830	0.539151	−	0.842209i	$-0.318745\pi$
$797$	30.4418	1.07830	0.539151	+	0.842209i	$0.318745\pi$
$798$	0	0
$799$	−50.3320	−1.78062
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−15.6417	−0.551985
$804$	0	0
$805$	−6.01180	−0.211888
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−17.4339	−0.612945	−0.306473	−	0.951880i	$-0.599149\pi$
$809$	−17.4339	−0.612945	−0.306473	+	0.951880i	$0.599149\pi$
$810$	0	0
$811$	−22.1264	−0.776964	−0.388482	−	0.921456i	$-0.627000\pi$
$811$	−22.1264	−0.776964	−0.388482	+	0.921456i	$0.627000\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−13.8628	−0.485593
$816$	0	0
$817$	−2.58301	−0.0903679
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10.9424	0.381892	0.190946	−	0.981601i	$-0.438844\pi$
$821$	10.9424	0.381892	0.190946	+	0.981601i	$0.438844\pi$
$822$	0	0
$823$	−32.2288	−1.12342	−0.561712	−	0.827333i	$-0.689857\pi$
$823$	−32.2288	−1.12342	−0.561712	+	0.827333i	$0.689857\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−6.13742	−0.213419	−0.106710	−	0.994290i	$-0.534032\pi$
$827$	−6.13742	−0.213419	−0.106710	+	0.994290i	$0.534032\pi$
$828$	0	0
$829$	20.7122	0.719365	0.359683	−	0.933075i	$-0.382885\pi$
$829$	20.7122	0.719365	0.359683	+	0.933075i	$0.382885\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	23.7328	0.822294
$834$	0	0
$835$	−14.9666	−0.517942
$836$	0	0
$837$	0	0
$838$	0	0
$839$	52.9212	1.82704	0.913521	−	0.406793i	$-0.133353\pi$
$839$	52.9212	1.82704	0.913521	+	0.406793i	$0.133353\pi$
$840$	0	0
$841$	46.0405	1.58760
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−0.841723	−0.0289561
$846$	0	0
$847$	19.0627	0.655004
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.96325	0.135858
$852$	0	0
$853$	16.8818	0.578023	0.289011	−	0.957326i	$-0.406673\pi$
$853$	16.8818	0.578023	0.289011	+	0.957326i	$0.406673\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.2917	0.351558	0.175779	−	0.984430i	$-0.443756\pi$
$857$	10.2917	0.351558	0.175779	+	0.984430i	$0.443756\pi$
$858$	0	0
$859$	8.16177	0.278476	0.139238	−	0.990259i	$-0.455535\pi$
$859$	8.16177	0.278476	0.139238	+	0.990259i	$0.455535\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	53.3428	1.81581	0.907906	−	0.419174i	$-0.137680\pi$
$863$	53.3428	1.81581	0.907906	+	0.419174i	$0.137680\pi$
$864$	0	0
$865$	5.87451	0.199739
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−45.8381	−1.55495
$870$	0	0
$871$	−21.1660	−0.717183
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12.8712	0.435126
$876$	0	0
$877$	31.0240	1.04760	0.523802	−	0.851840i	$-0.324513\pi$
$877$	31.0240	1.04760	0.523802	+	0.851840i	$0.324513\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	14.9785	0.504639	0.252319	−	0.967644i	$-0.418807\pi$
$881$	14.9785	0.504639	0.252319	+	0.967644i	$0.418807\pi$
$882$	0	0
$883$	50.0558	1.68451	0.842255	−	0.539079i	$-0.181228\pi$
$883$	50.0558	1.68451	0.842255	+	0.539079i	$0.181228\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.76150	−0.159876	−0.0799378	−	0.996800i	$-0.525472\pi$
$887$	−4.76150	−0.159876	−0.0799378	+	0.996800i	$0.525472\pi$
$888$	0	0
$889$	1.54249	0.0517333
$890$	0	0
$891$	0	0
$892$	0	0
$893$	46.9252	1.57029
$894$	0	0
$895$	−2.83399	−0.0947298
$896$	0	0
$897$	0	0
$898$	0	0
$899$	48.9068	1.63113
$900$	0	0
$901$	38.5960	1.28582
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0.768687	0.0255520
$906$	0	0
$907$	22.7735	0.756180	0.378090	−	0.925769i	$-0.376581\pi$
$907$	22.7735	0.756180	0.378090	+	0.925769i	$0.376581\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5.18319	0.171727	0.0858634	−	0.996307i	$-0.472635\pi$
$911$	5.18319	0.171727	0.0858634	+	0.996307i	$0.472635\pi$
$912$	0	0
$913$	22.5830	0.747388
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15.6417	0.516536
$918$	0	0
$919$	42.1033	1.38886	0.694429	−	0.719561i	$-0.255657\pi$
$919$	42.1033	1.38886	0.694429	+	0.719561i	$0.255657\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	50.2921	1.65539
$924$	0	0
$925$	−3.91913	−0.128860
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−23.7328	−0.778649	−0.389324	−	0.921101i	$-0.627291\pi$
$929$	−23.7328	−0.778649	−0.389324	+	0.921101i	$0.627291\pi$
$930$	0	0
$931$	−22.1264	−0.725165
$932$	0	0
$933$	0	0
$934$	0	0
$935$	22.1208	0.723426
$936$	0	0
$937$	31.1660	1.01815	0.509075	−	0.860722i	$-0.329988\pi$
$937$	31.1660	1.01815	0.509075	+	0.860722i	$0.329988\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−17.5706	−0.572784	−0.286392	−	0.958113i	$-0.592456\pi$
$941$	−17.5706	−0.572784	−0.286392	+	0.958113i	$0.592456\pi$
$942$	0	0
$943$	34.3320	1.11800
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.36689	0.109409	0.0547046	−	0.998503i	$-0.482578\pi$
$947$	3.36689	0.109409	0.0547046	+	0.998503i	$0.482578\pi$
$948$	0	0
$949$	12.3157	0.399784
$950$	0	0
$951$	0	0
$952$	0	0
$953$	25.2702	0.818582	0.409291	−	0.912404i	$-0.365776\pi$
$953$	25.2702	0.818582	0.409291	+	0.912404i	$0.365776\pi$
$954$	0	0
$955$	−7.66079	−0.247897
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−16.9376	−0.546943
$960$	0	0
$961$	0.874508	0.0282099
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−9.99504	−0.321752
$966$	0	0
$967$	45.3948	1.45980	0.729899	−	0.683555i	$-0.239567\pi$
$967$	45.3948	1.45980	0.729899	+	0.683555i	$0.239567\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−29.3019	−0.940341	−0.470171	−	0.882576i	$-0.655808\pi$
$971$	−29.3019	−0.940341	−0.470171	+	0.882576i	$0.655808\pi$
$972$	0	0
$973$	−33.9411	−1.08810
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20.5087	0.656131	0.328066	−	0.944655i	$-0.393603\pi$
$977$	20.5087	0.656131	0.328066	+	0.944655i	$0.393603\pi$
$978$	0	0
$979$	−11.3137	−0.361588
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−26.8823	−0.857411	−0.428706	−	0.903444i	$-0.641030\pi$
$983$	−26.8823	−0.857411	−0.428706	+	0.903444i	$0.641030\pi$
$984$	0	0
$985$	−10.1255	−0.322625
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.17417	−0.0691347
$990$	0	0
$991$	3.06275	0.0972913	0.0486457	−	0.998816i	$-0.484509\pi$
$991$	3.06275	0.0972913	0.0486457	+	0.998816i	$0.484509\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	11.4859	0.364129
$996$	0	0
$997$	−51.6474	−1.63569	−0.817846	−	0.575438i	$-0.804832\pi$
$997$	−51.6474	−1.63569	−0.817846	+	0.575438i	$0.804832\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9216.2.a.bq.1.3		8
3.2	odd	2	inner	9216.2.a.bq.1.5		8
4.3	odd	2		9216.2.a.bt.1.3		8
8.3	odd	2		9216.2.a.bt.1.6		8
8.5	even	2	inner	9216.2.a.bq.1.6		8
12.11	even	2		9216.2.a.bt.1.5		8
24.5	odd	2	inner	9216.2.a.bq.1.4		8
24.11	even	2		9216.2.a.bt.1.4		8
32.3	odd	8		576.2.k.c.145.3		8
32.5	even	8		1152.2.k.e.865.2		8
32.11	odd	8		576.2.k.c.433.3		8
32.13	even	8		1152.2.k.e.289.2		8
32.19	odd	8		1152.2.k.d.289.2		8
32.21	even	8		144.2.k.c.37.1	✓	8
32.27	odd	8		1152.2.k.d.865.2		8
32.29	even	8		144.2.k.c.109.1	yes	8
96.5	odd	8		1152.2.k.e.865.3		8
96.11	even	8		576.2.k.c.433.2		8
96.29	odd	8		144.2.k.c.109.4	yes	8
96.35	even	8		576.2.k.c.145.2		8
96.53	odd	8		144.2.k.c.37.4	yes	8
96.59	even	8		1152.2.k.d.865.3		8
96.77	odd	8		1152.2.k.e.289.3		8
96.83	even	8		1152.2.k.d.289.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.k.c.37.1	✓	8	32.21	even	8
144.2.k.c.37.4	yes	8	96.53	odd	8
144.2.k.c.109.1	yes	8	32.29	even	8
144.2.k.c.109.4	yes	8	96.29	odd	8
576.2.k.c.145.2		8	96.35	even	8
576.2.k.c.145.3		8	32.3	odd	8
576.2.k.c.433.2		8	96.11	even	8
576.2.k.c.433.3		8	32.11	odd	8
1152.2.k.d.289.2		8	32.19	odd	8
1152.2.k.d.289.3		8	96.83	even	8
1152.2.k.d.865.2		8	32.27	odd	8
1152.2.k.d.865.3		8	96.59	even	8
1152.2.k.e.289.2		8	32.13	even	8
1152.2.k.e.289.3		8	96.77	odd	8
1152.2.k.e.865.2		8	32.5	even	8
1152.2.k.e.865.3		8	96.5	odd	8
9216.2.a.bq.1.3		8	1.1	even	1	trivial
9216.2.a.bq.1.4		8	24.5	odd	2	inner
9216.2.a.bq.1.5		8	3.2	odd	2	inner
9216.2.a.bq.1.6		8	8.5	even	2	inner
9216.2.a.bt.1.3		8	4.3	odd	2
9216.2.a.bt.1.4		8	24.11	even	2
9216.2.a.bt.1.5		8	12.11	even	2
9216.2.a.bt.1.6		8	8.3	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:	\( N \)	\(=\)	\( 9216 = 2^{10} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9216.a (trivial)

\(f(q)\)	\(=\)	\(q-0.841723 q^{5} +1.64575 q^{7} +O(q^{10})\)	\(q-0.841723 q^{5} +1.64575 q^{7} +4.75216 q^{11} -3.74166 q^{13} -5.53019 q^{17} +5.15587 q^{19} +4.33981 q^{23} -4.29150 q^{25} -8.66259 q^{29} -5.64575 q^{31} -1.38527 q^{35} +0.913230 q^{37} +7.91094 q^{41} -0.500983 q^{43} +9.10132 q^{47} -4.29150 q^{49} -6.97915 q^{53} -4.00000 q^{55} -6.13742 q^{59} -0.913230 q^{61} +3.14944 q^{65} +5.65685 q^{67} -13.4411 q^{71} -3.29150 q^{73} +7.82087 q^{77} -9.64575 q^{79} +4.75216 q^{83} +4.65489 q^{85} -2.38075 q^{89} -6.15784 q^{91} -4.33981 q^{95} -10.5830 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7}+O(q^{10}) \) 8 * q - 8 * q^7	\( 8 q - 8 q^{7} + 8 q^{25} - 24 q^{31} + 8 q^{49} - 32 q^{55} + 16 q^{73} - 56 q^{79}+O(q^{100}) \) 8 * q - 8 * q^7 + 8 * q^25 - 24 * q^31 + 8 * q^49 - 32 * q^55 + 16 * q^73 - 56 * q^79

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