LMFDB - Embedded newform 4608.2.a.bb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(1,4608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4608, 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("4608.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.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:	$36.7950652514$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.93185$ of defining polynomial
Character	$\chi$	$=$	4608.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949 q^{5} +2.44949 q^{7} +O(q^{10})$	$q+2.44949 q^{5} +2.44949 q^{7} +2.00000 q^{11} -3.46410 q^{13} -2.82843 q^{17} -2.82843 q^{19} -6.92820 q^{23} +1.00000 q^{25} +2.44949 q^{29} +7.34847 q^{31} +6.00000 q^{35} +10.3923 q^{37} +8.48528 q^{41} +2.82843 q^{43} +6.92820 q^{47} -1.00000 q^{49} +2.44949 q^{53} +4.89898 q^{55} +8.00000 q^{59} -3.46410 q^{61} -8.48528 q^{65} +11.3137 q^{67} +13.8564 q^{71} +4.89898 q^{77} +2.44949 q^{79} +14.0000 q^{83} -6.92820 q^{85} +11.3137 q^{89} -8.48528 q^{91} -6.92820 q^{95} -6.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 8 q^{11} + 4 q^{25} + 24 q^{35} - 4 q^{49} + 32 q^{59} + 56 q^{83} - 24 q^{97}+O(q^{100}) $ 4 * q + 8 * q^11 + 4 * q^25 + 24 * q^35 - 4 * q^49 + 32 * q^59 + 56 * q^83 - 24 * 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
$5$	2.44949	1.09545	0.547723	−	0.836660i	$-0.315495\pi$
$5$	2.44949	1.09545	0.547723	+	0.836660i	$0.315495\pi$
$6$	0	0
$7$	2.44949	0.925820	0.462910	−	0.886405i	$-0.346805\pi$
$7$	2.44949	0.925820	0.462910	+	0.886405i	$0.346805\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−3.46410	−0.960769	−0.480384	−	0.877058i	$-0.659503\pi$
$13$	−3.46410	−0.960769	−0.480384	+	0.877058i	$0.659503\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.92820	−1.44463	−0.722315	−	0.691564i	$-0.756922\pi$
$23$	−6.92820	−1.44463	−0.722315	+	0.691564i	$0.756922\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.44949	0.454859	0.227429	−	0.973795i	$-0.426968\pi$
$29$	2.44949	0.454859	0.227429	+	0.973795i	$0.426968\pi$
$30$	0	0
$31$	7.34847	1.31982	0.659912	−	0.751343i	$-0.270594\pi$
$31$	7.34847	1.31982	0.659912	+	0.751343i	$0.270594\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	6.00000	1.01419
$36$	0	0
$37$	10.3923	1.70848	0.854242	−	0.519875i	$-0.174022\pi$
$37$	10.3923	1.70848	0.854242	+	0.519875i	$0.174022\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.48528	1.32518	0.662589	−	0.748983i	$-0.269458\pi$
$41$	8.48528	1.32518	0.662589	+	0.748983i	$0.269458\pi$
$42$	0	0
$43$	2.82843	0.431331	0.215666	−	0.976467i	$-0.430808\pi$
$43$	2.82843	0.431331	0.215666	+	0.976467i	$0.430808\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.92820	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.44949	0.336463	0.168232	−	0.985747i	$-0.446194\pi$
$53$	2.44949	0.336463	0.168232	+	0.985747i	$0.446194\pi$
$54$	0	0
$55$	4.89898	0.660578
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−3.46410	−0.443533	−0.221766	−	0.975100i	$-0.571182\pi$
$61$	−3.46410	−0.443533	−0.221766	+	0.975100i	$0.571182\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−8.48528	−1.05247
$66$	0	0
$67$	11.3137	1.38219	0.691095	−	0.722764i	$-0.257129\pi$
$67$	11.3137	1.38219	0.691095	+	0.722764i	$0.257129\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.8564	1.64445	0.822226	−	0.569160i	$-0.192732\pi$
$71$	13.8564	1.64445	0.822226	+	0.569160i	$0.192732\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.89898	0.558291
$78$	0	0
$79$	2.44949	0.275589	0.137795	−	0.990461i	$-0.455999\pi$
$79$	2.44949	0.275589	0.137795	+	0.990461i	$0.455999\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	14.0000	1.53670	0.768350	−	0.640030i	$-0.221078\pi$
$83$	14.0000	1.53670	0.768350	+	0.640030i	$0.221078\pi$
$84$	0	0
$85$	−6.92820	−0.751469
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.3137	1.19925	0.599625	−	0.800281i	$-0.295316\pi$
$89$	11.3137	1.19925	0.599625	+	0.800281i	$0.295316\pi$
$90$	0	0
$91$	−8.48528	−0.889499
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.92820	−0.710819
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.2474	−1.21867	−0.609333	−	0.792914i	$-0.708563\pi$
$101$	−12.2474	−1.21867	−0.609333	+	0.792914i	$0.708563\pi$
$102$	0	0
$103$	2.44949	0.241355	0.120678	−	0.992692i	$-0.461493\pi$
$103$	2.44949	0.241355	0.120678	+	0.992692i	$0.461493\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	0	0
$109$	−3.46410	−0.331801	−0.165900	−	0.986143i	$-0.553053\pi$
$109$	−3.46410	−0.331801	−0.165900	+	0.986143i	$0.553053\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−16.9706	−1.59646	−0.798228	−	0.602355i	$-0.794229\pi$
$113$	−16.9706	−1.59646	−0.798228	+	0.602355i	$0.794229\pi$
$114$	0	0
$115$	−16.9706	−1.58251
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.92820	−0.635107
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.79796	−0.876356
$126$	0	0
$127$	−17.1464	−1.52150	−0.760750	−	0.649045i	$-0.775169\pi$
$127$	−17.1464	−1.52150	−0.760750	+	0.649045i	$0.775169\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	−6.92820	−0.600751
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.48528	0.724947	0.362473	−	0.931994i	$-0.381932\pi$
$137$	8.48528	0.724947	0.362473	+	0.931994i	$0.381932\pi$
$138$	0	0
$139$	5.65685	0.479808	0.239904	−	0.970797i	$-0.422884\pi$
$139$	5.65685	0.479808	0.239904	+	0.970797i	$0.422884\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.92820	−0.579365
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.34847	−0.602010	−0.301005	−	0.953623i	$-0.597322\pi$
$149$	−7.34847	−0.602010	−0.301005	+	0.953623i	$0.597322\pi$
$150$	0	0
$151$	−12.2474	−0.996683	−0.498342	−	0.866981i	$-0.666057\pi$
$151$	−12.2474	−0.996683	−0.498342	+	0.866981i	$0.666057\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	18.0000	1.44579
$156$	0	0
$157$	17.3205	1.38233	0.691164	−	0.722698i	$-0.257098\pi$
$157$	17.3205	1.38233	0.691164	+	0.722698i	$0.257098\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−16.9706	−1.33747
$162$	0	0
$163$	19.7990	1.55078	0.775388	−	0.631485i	$-0.217554\pi$
$163$	19.7990	1.55078	0.775388	+	0.631485i	$0.217554\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.92820	0.536120	0.268060	−	0.963402i	$-0.413617\pi$
$167$	6.92820	0.536120	0.268060	+	0.963402i	$0.413617\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−22.0454	−1.67608	−0.838041	−	0.545608i	$-0.816299\pi$
$173$	−22.0454	−1.67608	−0.838041	+	0.545608i	$0.816299\pi$
$174$	0	0
$175$	2.44949	0.185164
$176$	0	0
$177$	0	0
$178$	0	0
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	3.46410	0.257485	0.128742	−	0.991678i	$-0.458906\pi$
$181$	3.46410	0.257485	0.128742	+	0.991678i	$0.458906\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	25.4558	1.87155
$186$	0	0
$187$	−5.65685	−0.413670
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.92820	−0.501307	−0.250654	−	0.968077i	$-0.580646\pi$
$191$	−6.92820	−0.501307	−0.250654	+	0.968077i	$0.580646\pi$
$192$	0	0
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.0454	1.57067	0.785335	−	0.619071i	$-0.212491\pi$
$197$	22.0454	1.57067	0.785335	+	0.619071i	$0.212491\pi$
$198$	0	0
$199$	−26.9444	−1.91004	−0.955018	−	0.296546i	$-0.904165\pi$
$199$	−26.9444	−1.91004	−0.955018	+	0.296546i	$0.904165\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.00000	0.421117
$204$	0	0
$205$	20.7846	1.45166
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.65685	−0.391293
$210$	0	0
$211$	−28.2843	−1.94717	−0.973585	−	0.228326i	$-0.926675\pi$
$211$	−28.2843	−1.94717	−0.973585	+	0.228326i	$0.926675\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.92820	0.472500
$216$	0	0
$217$	18.0000	1.22192
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9.79796	0.659082
$222$	0	0
$223$	−2.44949	−0.164030	−0.0820150	−	0.996631i	$-0.526136\pi$
$223$	−2.44949	−0.164030	−0.0820150	+	0.996631i	$0.526136\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.0000	1.72568	0.862840	−	0.505477i	$-0.168683\pi$
$227$	26.0000	1.72568	0.862840	+	0.505477i	$0.168683\pi$
$228$	0	0
$229$	−24.2487	−1.60240	−0.801200	−	0.598397i	$-0.795805\pi$
$229$	−24.2487	−1.60240	−0.801200	+	0.598397i	$0.795805\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.3137	0.741186	0.370593	−	0.928795i	$-0.379155\pi$
$233$	11.3137	0.741186	0.370593	+	0.928795i	$0.379155\pi$
$234$	0	0
$235$	16.9706	1.10704
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.92820	−0.448148	−0.224074	−	0.974572i	$-0.571936\pi$
$239$	−6.92820	−0.448148	−0.224074	+	0.974572i	$0.571936\pi$
$240$	0	0
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.44949	−0.156492
$246$	0	0
$247$	9.79796	0.623429
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.0000	0.631194	0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000	0.631194	0.315597	+	0.948893i	$0.397795\pi$
$252$	0	0
$253$	−13.8564	−0.871145
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.9706	−1.05859	−0.529297	−	0.848436i	$-0.677544\pi$
$257$	−16.9706	−1.05859	−0.529297	+	0.848436i	$0.677544\pi$
$258$	0	0
$259$	25.4558	1.58175
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−27.7128	−1.70885	−0.854423	−	0.519579i	$-0.826089\pi$
$263$	−27.7128	−1.70885	−0.854423	+	0.519579i	$0.826089\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	0	0
$268$	0	0
$269$	26.9444	1.64283	0.821414	−	0.570332i	$-0.193186\pi$
$269$	26.9444	1.64283	0.821414	+	0.570332i	$0.193186\pi$
$270$	0	0
$271$	−22.0454	−1.33916	−0.669582	−	0.742739i	$-0.733527\pi$
$271$	−22.0454	−1.33916	−0.669582	+	0.742739i	$0.733527\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.00000	0.120605
$276$	0	0
$277$	3.46410	0.208138	0.104069	−	0.994570i	$-0.466814\pi$
$277$	3.46410	0.208138	0.104069	+	0.994570i	$0.466814\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.65685	0.337460	0.168730	−	0.985662i	$-0.446033\pi$
$281$	5.65685	0.337460	0.168730	+	0.985662i	$0.446033\pi$
$282$	0	0
$283$	−5.65685	−0.336265	−0.168133	−	0.985764i	$-0.553774\pi$
$283$	−5.65685	−0.336265	−0.168133	+	0.985764i	$0.553774\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	20.7846	1.22688
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−17.1464	−1.00171	−0.500853	−	0.865533i	$-0.666980\pi$
$293$	−17.1464	−1.00171	−0.500853	+	0.865533i	$0.666980\pi$
$294$	0	0
$295$	19.5959	1.14092
$296$	0	0
$297$	0	0
$298$	0	0
$299$	24.0000	1.38796
$300$	0	0
$301$	6.92820	0.399335
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−8.48528	−0.485866
$306$	0	0
$307$	−22.6274	−1.29141	−0.645707	−	0.763585i	$-0.723437\pi$
$307$	−22.6274	−1.29141	−0.645707	+	0.763585i	$0.723437\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.8564	0.785725	0.392862	−	0.919597i	$-0.371485\pi$
$311$	13.8564	0.785725	0.392862	+	0.919597i	$0.371485\pi$
$312$	0	0
$313$	−20.0000	−1.13047	−0.565233	−	0.824931i	$-0.691214\pi$
$313$	−20.0000	−1.13047	−0.565233	+	0.824931i	$0.691214\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.44949	−0.137577	−0.0687885	−	0.997631i	$-0.521913\pi$
$317$	−2.44949	−0.137577	−0.0687885	+	0.997631i	$0.521913\pi$
$318$	0	0
$319$	4.89898	0.274290
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.00000	0.445132
$324$	0	0
$325$	−3.46410	−0.192154
$326$	0	0
$327$	0	0
$328$	0	0
$329$	16.9706	0.935617
$330$	0	0
$331$	11.3137	0.621858	0.310929	−	0.950433i	$-0.399360\pi$
$331$	11.3137	0.621858	0.310929	+	0.950433i	$0.399360\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	27.7128	1.51411
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	14.6969	0.795884
$342$	0	0
$343$	−19.5959	−1.05808
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.00000	−0.107366	−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000	−0.107366	−0.0536828	+	0.998558i	$0.517096\pi$
$348$	0	0
$349$	−17.3205	−0.927146	−0.463573	−	0.886059i	$-0.653433\pi$
$349$	−17.3205	−0.927146	−0.463573	+	0.886059i	$0.653433\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	33.9411	1.80141
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−13.8564	−0.731313	−0.365657	−	0.930750i	$-0.619156\pi$
$359$	−13.8564	−0.731313	−0.365657	+	0.930750i	$0.619156\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7.34847	0.383587	0.191793	−	0.981435i	$-0.438570\pi$
$367$	7.34847	0.383587	0.191793	+	0.981435i	$0.438570\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6.00000	0.311504
$372$	0	0
$373$	17.3205	0.896822	0.448411	−	0.893828i	$-0.351990\pi$
$373$	17.3205	0.896822	0.448411	+	0.893828i	$0.351990\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.48528	−0.437014
$378$	0	0
$379$	−31.1127	−1.59815	−0.799076	−	0.601230i	$-0.794678\pi$
$379$	−31.1127	−1.59815	−0.799076	+	0.601230i	$0.794678\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	20.7846	1.06204	0.531022	−	0.847358i	$-0.321808\pi$
$383$	20.7846	1.06204	0.531022	+	0.847358i	$0.321808\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7.34847	0.372582	0.186291	−	0.982495i	$-0.440353\pi$
$389$	7.34847	0.372582	0.186291	+	0.982495i	$0.440353\pi$
$390$	0	0
$391$	19.5959	0.991008
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.00000	0.301893
$396$	0	0
$397$	−24.2487	−1.21701	−0.608504	−	0.793551i	$-0.708230\pi$
$397$	−24.2487	−1.21701	−0.608504	+	0.793551i	$0.708230\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−31.1127	−1.55369	−0.776847	−	0.629689i	$-0.783182\pi$
$401$	−31.1127	−1.55369	−0.776847	+	0.629689i	$0.783182\pi$
$402$	0	0
$403$	−25.4558	−1.26805
$404$	0	0
$405$	0	0
$406$	0	0
$407$	20.7846	1.03025
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	19.5959	0.964252
$414$	0	0
$415$	34.2929	1.68337
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−14.0000	−0.683945	−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000	−0.683945	−0.341972	+	0.939710i	$0.611095\pi$
$420$	0	0
$421$	17.3205	0.844150	0.422075	−	0.906561i	$-0.361302\pi$
$421$	17.3205	0.844150	0.422075	+	0.906561i	$0.361302\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.82843	−0.137199
$426$	0	0
$427$	−8.48528	−0.410632
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−34.6410	−1.66860	−0.834300	−	0.551311i	$-0.814128\pi$
$431$	−34.6410	−1.66860	−0.834300	+	0.551311i	$0.814128\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	19.5959	0.937400
$438$	0	0
$439$	12.2474	0.584539	0.292269	−	0.956336i	$-0.405590\pi$
$439$	12.2474	0.584539	0.292269	+	0.956336i	$0.405590\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14.0000	0.665160	0.332580	−	0.943075i	$-0.392081\pi$
$443$	14.0000	0.665160	0.332580	+	0.943075i	$0.392081\pi$
$444$	0	0
$445$	27.7128	1.31371
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14.1421	0.667409	0.333704	−	0.942678i	$-0.391701\pi$
$449$	14.1421	0.667409	0.333704	+	0.942678i	$0.391701\pi$
$450$	0	0
$451$	16.9706	0.799113
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−20.7846	−0.974398
$456$	0	0
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7.34847	0.342252	0.171126	−	0.985249i	$-0.445259\pi$
$461$	7.34847	0.342252	0.171126	+	0.985249i	$0.445259\pi$
$462$	0	0
$463$	22.0454	1.02454	0.512268	−	0.858825i	$-0.328805\pi$
$463$	22.0454	1.02454	0.512268	+	0.858825i	$0.328805\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.0000	0.462745	0.231372	−	0.972865i	$-0.425678\pi$
$467$	10.0000	0.462745	0.231372	+	0.972865i	$0.425678\pi$
$468$	0	0
$469$	27.7128	1.27966
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.65685	0.260102
$474$	0	0
$475$	−2.82843	−0.129777
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−13.8564	−0.633115	−0.316558	−	0.948573i	$-0.602527\pi$
$479$	−13.8564	−0.633115	−0.316558	+	0.948573i	$0.602527\pi$
$480$	0	0
$481$	−36.0000	−1.64146
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−14.6969	−0.667354
$486$	0	0
$487$	−7.34847	−0.332991	−0.166495	−	0.986042i	$-0.553245\pi$
$487$	−7.34847	−0.332991	−0.166495	+	0.986042i	$0.553245\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−40.0000	−1.80517	−0.902587	−	0.430507i	$-0.858335\pi$
$491$	−40.0000	−1.80517	−0.902587	+	0.430507i	$0.858335\pi$
$492$	0	0
$493$	−6.92820	−0.312031
$494$	0	0
$495$	0	0
$496$	0	0
$497$	33.9411	1.52247
$498$	0	0
$499$	−28.2843	−1.26618	−0.633089	−	0.774079i	$-0.718213\pi$
$499$	−28.2843	−1.26618	−0.633089	+	0.774079i	$0.718213\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−30.0000	−1.33498
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−26.9444	−1.19429	−0.597144	−	0.802134i	$-0.703698\pi$
$509$	−26.9444	−1.19429	−0.597144	+	0.802134i	$0.703698\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.00000	0.264392
$516$	0	0
$517$	13.8564	0.609404
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.48528	0.371747	0.185873	−	0.982574i	$-0.440489\pi$
$521$	8.48528	0.371747	0.185873	+	0.982574i	$0.440489\pi$
$522$	0	0
$523$	31.1127	1.36046	0.680232	−	0.732997i	$-0.261879\pi$
$523$	31.1127	1.36046	0.680232	+	0.732997i	$0.261879\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−20.7846	−0.905392
$528$	0	0
$529$	25.0000	1.08696
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−29.3939	−1.27319
$534$	0	0
$535$	39.1918	1.69441
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−38.1051	−1.63827	−0.819133	−	0.573603i	$-0.805545\pi$
$541$	−38.1051	−1.63827	−0.819133	+	0.573603i	$0.805545\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−8.48528	−0.363470
$546$	0	0
$547$	31.1127	1.33028	0.665141	−	0.746717i	$-0.268371\pi$
$547$	31.1127	1.33028	0.665141	+	0.746717i	$0.268371\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.92820	−0.295151
$552$	0	0
$553$	6.00000	0.255146
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−36.7423	−1.55682	−0.778412	−	0.627754i	$-0.783974\pi$
$557$	−36.7423	−1.55682	−0.778412	+	0.627754i	$0.783974\pi$
$558$	0	0
$559$	−9.79796	−0.414410
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14.0000	−0.590030	−0.295015	−	0.955493i	$-0.595325\pi$
$563$	−14.0000	−0.590030	−0.295015	+	0.955493i	$0.595325\pi$
$564$	0	0
$565$	−41.5692	−1.74883
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−25.4558	−1.06716	−0.533582	−	0.845748i	$-0.679155\pi$
$569$	−25.4558	−1.06716	−0.533582	+	0.845748i	$0.679155\pi$
$570$	0	0
$571$	11.3137	0.473464	0.236732	−	0.971575i	$-0.423924\pi$
$571$	11.3137	0.473464	0.236732	+	0.971575i	$0.423924\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.92820	−0.288926
$576$	0	0
$577$	32.0000	1.33218	0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000	1.33218	0.666089	+	0.745873i	$0.267967\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	34.2929	1.42271
$582$	0	0
$583$	4.89898	0.202895
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.0000	0.660391	0.330195	−	0.943913i	$-0.392885\pi$
$587$	16.0000	0.660391	0.330195	+	0.943913i	$0.392885\pi$
$588$	0	0
$589$	−20.7846	−0.856415
$590$	0	0
$591$	0	0
$592$	0	0
$593$	16.9706	0.696897	0.348449	−	0.937328i	$-0.386709\pi$
$593$	16.9706	0.696897	0.348449	+	0.937328i	$0.386709\pi$
$594$	0	0
$595$	−16.9706	−0.695725
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6.92820	0.283079	0.141539	−	0.989933i	$-0.454795\pi$
$599$	6.92820	0.283079	0.141539	+	0.989933i	$0.454795\pi$
$600$	0	0
$601$	48.0000	1.95796	0.978980	−	0.203954i	$-0.0653794\pi$
$601$	48.0000	1.95796	0.978980	+	0.203954i	$0.0653794\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−17.1464	−0.697101
$606$	0	0
$607$	−17.1464	−0.695952	−0.347976	−	0.937503i	$-0.613131\pi$
$607$	−17.1464	−0.695952	−0.347976	+	0.937503i	$0.613131\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−24.0000	−0.970936
$612$	0	0
$613$	17.3205	0.699569	0.349784	−	0.936830i	$-0.386255\pi$
$613$	17.3205	0.699569	0.349784	+	0.936830i	$0.386255\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−11.3137	−0.455473	−0.227736	−	0.973723i	$-0.573132\pi$
$617$	−11.3137	−0.455473	−0.227736	+	0.973723i	$0.573132\pi$
$618$	0	0
$619$	11.3137	0.454736	0.227368	−	0.973809i	$-0.426988\pi$
$619$	11.3137	0.454736	0.227368	+	0.973809i	$0.426988\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	27.7128	1.11029
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−29.3939	−1.17201
$630$	0	0
$631$	−7.34847	−0.292538	−0.146269	−	0.989245i	$-0.546726\pi$
$631$	−7.34847	−0.292538	−0.146269	+	0.989245i	$0.546726\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−42.0000	−1.66672
$636$	0	0
$637$	3.46410	0.137253
$638$	0	0
$639$	0	0
$640$	0	0
$641$	19.7990	0.782013	0.391007	−	0.920388i	$-0.372127\pi$
$641$	19.7990	0.782013	0.391007	+	0.920388i	$0.372127\pi$
$642$	0	0
$643$	14.1421	0.557711	0.278856	−	0.960333i	$-0.410045\pi$
$643$	14.1421	0.557711	0.278856	+	0.960333i	$0.410045\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	34.6410	1.36188	0.680939	−	0.732340i	$-0.261572\pi$
$647$	34.6410	1.36188	0.680939	+	0.732340i	$0.261572\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12.2474	−0.479280	−0.239640	−	0.970862i	$-0.577029\pi$
$653$	−12.2474	−0.479280	−0.239640	+	0.970862i	$0.577029\pi$
$654$	0	0
$655$	19.5959	0.765676
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−8.00000	−0.311636	−0.155818	−	0.987786i	$-0.549801\pi$
$659$	−8.00000	−0.311636	−0.155818	+	0.987786i	$0.549801\pi$
$660$	0	0
$661$	−24.2487	−0.943166	−0.471583	−	0.881822i	$-0.656317\pi$
$661$	−24.2487	−0.943166	−0.471583	+	0.881822i	$0.656317\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−16.9706	−0.658090
$666$	0	0
$667$	−16.9706	−0.657103
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6.92820	−0.267460
$672$	0	0
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7.34847	−0.282425	−0.141212	−	0.989979i	$-0.545100\pi$
$677$	−7.34847	−0.282425	−0.141212	+	0.989979i	$0.545100\pi$
$678$	0	0
$679$	−14.6969	−0.564017
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−14.0000	−0.535695	−0.267848	−	0.963461i	$-0.586312\pi$
$683$	−14.0000	−0.535695	−0.267848	+	0.963461i	$0.586312\pi$
$684$	0	0
$685$	20.7846	0.794139
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−8.48528	−0.323263
$690$	0	0
$691$	−14.1421	−0.537992	−0.268996	−	0.963141i	$-0.586692\pi$
$691$	−14.1421	−0.537992	−0.268996	+	0.963141i	$0.586692\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.8564	0.525603
$696$	0	0
$697$	−24.0000	−0.909065
$698$	0	0
$699$	0	0
$700$	0	0
$701$	26.9444	1.01768	0.508838	−	0.860862i	$-0.330075\pi$
$701$	26.9444	1.01768	0.508838	+	0.860862i	$0.330075\pi$
$702$	0	0
$703$	−29.3939	−1.10861
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−30.0000	−1.12827
$708$	0	0
$709$	31.1769	1.17087	0.585437	−	0.810718i	$-0.300923\pi$
$709$	31.1769	1.17087	0.585437	+	0.810718i	$0.300923\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−50.9117	−1.90666
$714$	0	0
$715$	−16.9706	−0.634663
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−27.7128	−1.03351	−0.516757	−	0.856132i	$-0.672861\pi$
$719$	−27.7128	−1.03351	−0.516757	+	0.856132i	$0.672861\pi$
$720$	0	0
$721$	6.00000	0.223452
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.44949	0.0909718
$726$	0	0
$727$	−22.0454	−0.817619	−0.408809	−	0.912620i	$-0.634056\pi$
$727$	−22.0454	−0.817619	−0.408809	+	0.912620i	$0.634056\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	31.1769	1.15155	0.575773	−	0.817610i	$-0.304701\pi$
$733$	31.1769	1.15155	0.575773	+	0.817610i	$0.304701\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	22.6274	0.833492
$738$	0	0
$739$	−11.3137	−0.416181	−0.208091	−	0.978110i	$-0.566725\pi$
$739$	−11.3137	−0.416181	−0.208091	+	0.978110i	$0.566725\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	20.7846	0.762513	0.381257	−	0.924469i	$-0.375491\pi$
$743$	20.7846	0.762513	0.381257	+	0.924469i	$0.375491\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	0	0
$747$	0	0
$748$	0	0
$749$	39.1918	1.43204
$750$	0	0
$751$	12.2474	0.446916	0.223458	−	0.974714i	$-0.428265\pi$
$751$	12.2474	0.446916	0.223458	+	0.974714i	$0.428265\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−30.0000	−1.09181
$756$	0	0
$757$	−10.3923	−0.377715	−0.188857	−	0.982005i	$-0.560478\pi$
$757$	−10.3923	−0.377715	−0.188857	+	0.982005i	$0.560478\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	25.4558	0.922774	0.461387	−	0.887199i	$-0.347352\pi$
$761$	25.4558	0.922774	0.461387	+	0.887199i	$0.347352\pi$
$762$	0	0
$763$	−8.48528	−0.307188
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−27.7128	−1.00065
$768$	0	0
$769$	36.0000	1.29819	0.649097	−	0.760706i	$-0.275147\pi$
$769$	36.0000	1.29819	0.649097	+	0.760706i	$0.275147\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−51.4393	−1.85014	−0.925071	−	0.379794i	$-0.875995\pi$
$773$	−51.4393	−1.85014	−0.925071	+	0.379794i	$0.875995\pi$
$774$	0	0
$775$	7.34847	0.263965
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	27.7128	0.991642
$782$	0	0
$783$	0	0
$784$	0	0
$785$	42.4264	1.51426
$786$	0	0
$787$	−14.1421	−0.504113	−0.252056	−	0.967713i	$-0.581107\pi$
$787$	−14.1421	−0.504113	−0.252056	+	0.967713i	$0.581107\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−41.5692	−1.47803
$792$	0	0
$793$	12.0000	0.426132
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.2474	0.433827	0.216913	−	0.976191i	$-0.430401\pi$
$797$	12.2474	0.433827	0.216913	+	0.976191i	$0.430401\pi$
$798$	0	0
$799$	−19.5959	−0.693254
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−41.5692	−1.46512
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−42.4264	−1.49163	−0.745817	−	0.666151i	$-0.767940\pi$
$809$	−42.4264	−1.49163	−0.745817	+	0.666151i	$0.767940\pi$
$810$	0	0
$811$	31.1127	1.09251	0.546257	−	0.837617i	$-0.316052\pi$
$811$	31.1127	1.09251	0.546257	+	0.837617i	$0.316052\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	48.4974	1.69879
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	36.7423	1.28232	0.641158	−	0.767409i	$-0.278454\pi$
$821$	36.7423	1.28232	0.641158	+	0.767409i	$0.278454\pi$
$822$	0	0
$823$	−46.5403	−1.62229	−0.811147	−	0.584843i	$-0.801156\pi$
$823$	−46.5403	−1.62229	−0.811147	+	0.584843i	$0.801156\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	0	0
$829$	38.1051	1.32345	0.661723	−	0.749749i	$-0.269826\pi$
$829$	38.1051	1.32345	0.661723	+	0.749749i	$0.269826\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.82843	0.0979992
$834$	0	0
$835$	16.9706	0.587291
$836$	0	0
$837$	0	0
$838$	0	0
$839$	27.7128	0.956753	0.478376	−	0.878155i	$-0.341226\pi$
$839$	27.7128	0.956753	0.478376	+	0.878155i	$0.341226\pi$
$840$	0	0
$841$	−23.0000	−0.793103
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.44949	−0.0842650
$846$	0	0
$847$	−17.1464	−0.589158
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−72.0000	−2.46813
$852$	0	0
$853$	−10.3923	−0.355826	−0.177913	−	0.984046i	$-0.556935\pi$
$853$	−10.3923	−0.355826	−0.177913	+	0.984046i	$0.556935\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−42.4264	−1.44926	−0.724629	−	0.689139i	$-0.757989\pi$
$857$	−42.4264	−1.44926	−0.724629	+	0.689139i	$0.757989\pi$
$858$	0	0
$859$	2.82843	0.0965047	0.0482523	−	0.998835i	$-0.484635\pi$
$859$	2.82843	0.0965047	0.0482523	+	0.998835i	$0.484635\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	34.6410	1.17919	0.589597	−	0.807698i	$-0.299287\pi$
$863$	34.6410	1.17919	0.589597	+	0.807698i	$0.299287\pi$
$864$	0	0
$865$	−54.0000	−1.83606
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.89898	0.166186
$870$	0	0
$871$	−39.1918	−1.32796
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−24.0000	−0.811348
$876$	0	0
$877$	−31.1769	−1.05277	−0.526385	−	0.850246i	$-0.676453\pi$
$877$	−31.1769	−1.05277	−0.526385	+	0.850246i	$0.676453\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16.9706	0.571753	0.285876	−	0.958267i	$-0.407715\pi$
$881$	16.9706	0.571753	0.285876	+	0.958267i	$0.407715\pi$
$882$	0	0
$883$	14.1421	0.475921	0.237960	−	0.971275i	$-0.423521\pi$
$883$	14.1421	0.475921	0.237960	+	0.971275i	$0.423521\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−27.7128	−0.930505	−0.465253	−	0.885178i	$-0.654037\pi$
$887$	−27.7128	−0.930505	−0.465253	+	0.885178i	$0.654037\pi$
$888$	0	0
$889$	−42.0000	−1.40863
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−19.5959	−0.655752
$894$	0	0
$895$	39.1918	1.31004
$896$	0	0
$897$	0	0
$898$	0	0
$899$	18.0000	0.600334
$900$	0	0
$901$	−6.92820	−0.230812
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.48528	0.282060
$906$	0	0
$907$	−31.1127	−1.03308	−0.516540	−	0.856263i	$-0.672780\pi$
$907$	−31.1127	−1.03308	−0.516540	+	0.856263i	$0.672780\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20.7846	0.688625	0.344312	−	0.938855i	$-0.388112\pi$
$911$	20.7846	0.688625	0.344312	+	0.938855i	$0.388112\pi$
$912$	0	0
$913$	28.0000	0.926665
$914$	0	0
$915$	0	0
$916$	0	0
$917$	19.5959	0.647114
$918$	0	0
$919$	31.8434	1.05042	0.525208	−	0.850974i	$-0.323988\pi$
$919$	31.8434	1.05042	0.525208	+	0.850974i	$0.323988\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−48.0000	−1.57994
$924$	0	0
$925$	10.3923	0.341697
$926$	0	0
$927$	0	0
$928$	0	0
$929$	53.7401	1.76316	0.881578	−	0.472038i	$-0.156482\pi$
$929$	53.7401	1.76316	0.881578	+	0.472038i	$0.156482\pi$
$930$	0	0
$931$	2.82843	0.0926980
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−13.8564	−0.453153
$936$	0	0
$937$	6.00000	0.196011	0.0980057	−	0.995186i	$-0.468754\pi$
$937$	6.00000	0.196011	0.0980057	+	0.995186i	$0.468754\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	7.34847	0.239553	0.119777	−	0.992801i	$-0.461782\pi$
$941$	7.34847	0.239553	0.119777	+	0.992801i	$0.461782\pi$
$942$	0	0
$943$	−58.7878	−1.91439
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32.0000	1.03986	0.519930	−	0.854209i	$-0.325958\pi$
$947$	32.0000	1.03986	0.519930	+	0.854209i	$0.325958\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−25.4558	−0.824596	−0.412298	−	0.911049i	$-0.635274\pi$
$953$	−25.4558	−0.824596	−0.412298	+	0.911049i	$0.635274\pi$
$954$	0	0
$955$	−16.9706	−0.549155
$956$	0	0
$957$	0	0
$958$	0	0
$959$	20.7846	0.671170
$960$	0	0
$961$	23.0000	0.741935
$962$	0	0
$963$	0	0
$964$	0	0
$965$	29.3939	0.946222
$966$	0	0
$967$	−12.2474	−0.393851	−0.196926	−	0.980418i	$-0.563096\pi$
$967$	−12.2474	−0.393851	−0.196926	+	0.980418i	$0.563096\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−38.0000	−1.21948	−0.609739	−	0.792602i	$-0.708726\pi$
$971$	−38.0000	−1.21948	−0.609739	+	0.792602i	$0.708726\pi$
$972$	0	0
$973$	13.8564	0.444216
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.1127	0.995383	0.497692	−	0.867354i	$-0.334181\pi$
$977$	31.1127	0.995383	0.497692	+	0.867354i	$0.334181\pi$
$978$	0	0
$979$	22.6274	0.723175
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−41.5692	−1.32585	−0.662926	−	0.748685i	$-0.730686\pi$
$983$	−41.5692	−1.32585	−0.662926	+	0.748685i	$0.730686\pi$
$984$	0	0
$985$	54.0000	1.72058
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−19.5959	−0.623114
$990$	0	0
$991$	−46.5403	−1.47840	−0.739201	−	0.673485i	$-0.764797\pi$
$991$	−46.5403	−1.47840	−0.739201	+	0.673485i	$0.764797\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−66.0000	−2.09234
$996$	0	0
$997$	−31.1769	−0.987383	−0.493691	−	0.869637i	$-0.664353\pi$
$997$	−31.1769	−0.987383	−0.493691	+	0.869637i	$0.664353\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.a.bb.1.3	yes	4
3.2	odd	2		4608.2.a.u.1.1	✓	4
4.3	odd	2		4608.2.a.u.1.3	yes	4
8.3	odd	2	inner	4608.2.a.bb.1.2	yes	4
8.5	even	2		4608.2.a.u.1.2	yes	4
12.11	even	2	inner	4608.2.a.bb.1.1	yes	4
16.3	odd	4		4608.2.d.r.2305.3		8
16.5	even	4		4608.2.d.r.2305.6		8
16.11	odd	4		4608.2.d.r.2305.8		8
16.13	even	4		4608.2.d.r.2305.1		8
24.5	odd	2	inner	4608.2.a.bb.1.4	yes	4
24.11	even	2		4608.2.a.u.1.4	yes	4
48.5	odd	4		4608.2.d.r.2305.2		8
48.11	even	4		4608.2.d.r.2305.4		8
48.29	odd	4		4608.2.d.r.2305.5		8
48.35	even	4		4608.2.d.r.2305.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4608.2.a.u.1.1	✓	4	3.2	odd	2
4608.2.a.u.1.2	yes	4	8.5	even	2
4608.2.a.u.1.3	yes	4	4.3	odd	2
4608.2.a.u.1.4	yes	4	24.11	even	2
4608.2.a.bb.1.1	yes	4	12.11	even	2	inner
4608.2.a.bb.1.2	yes	4	8.3	odd	2	inner
4608.2.a.bb.1.3	yes	4	1.1	even	1	trivial
4608.2.a.bb.1.4	yes	4	24.5	odd	2	inner
4608.2.d.r.2305.1		8	16.13	even	4
4608.2.d.r.2305.2		8	48.5	odd	4
4608.2.d.r.2305.3		8	16.3	odd	4
4608.2.d.r.2305.4		8	48.11	even	4
4608.2.d.r.2305.5		8	48.29	odd	4
4608.2.d.r.2305.6		8	16.5	even	4
4608.2.d.r.2305.7		8	48.35	even	4
4608.2.d.r.2305.8		8	16.11	odd	4

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:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.a (trivial)

\(f(q)\)	\(=\)	\(q+2.44949 q^{5} +2.44949 q^{7} +O(q^{10})\)	\(q+2.44949 q^{5} +2.44949 q^{7} +2.00000 q^{11} -3.46410 q^{13} -2.82843 q^{17} -2.82843 q^{19} -6.92820 q^{23} +1.00000 q^{25} +2.44949 q^{29} +7.34847 q^{31} +6.00000 q^{35} +10.3923 q^{37} +8.48528 q^{41} +2.82843 q^{43} +6.92820 q^{47} -1.00000 q^{49} +2.44949 q^{53} +4.89898 q^{55} +8.00000 q^{59} -3.46410 q^{61} -8.48528 q^{65} +11.3137 q^{67} +13.8564 q^{71} +4.89898 q^{77} +2.44949 q^{79} +14.0000 q^{83} -6.92820 q^{85} +11.3137 q^{89} -8.48528 q^{91} -6.92820 q^{95} -6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 8 q^{11} + 4 q^{25} + 24 q^{35} - 4 q^{49} + 32 q^{59} + 56 q^{83} - 24 q^{97}+O(q^{100}) \) 4 * q + 8 * q^11 + 4 * q^25 + 24 * q^35 - 4 * q^49 + 32 * q^59 + 56 * q^83 - 24 * q^97

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