LMFDB - Embedded newform 4608.2.a.j.1.2

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1536)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+1.41421 q^{5} +1.41421 q^{7} +O(q^{10})$	$q+1.41421 q^{5} +1.41421 q^{7} +2.00000 q^{11} -2.00000 q^{17} -4.00000 q^{19} -2.82843 q^{23} -3.00000 q^{25} -9.89949 q^{29} -7.07107 q^{31} +2.00000 q^{35} -8.48528 q^{37} -6.00000 q^{41} -8.00000 q^{43} -2.82843 q^{47} -5.00000 q^{49} +1.41421 q^{53} +2.82843 q^{55} +12.0000 q^{59} +14.1421 q^{61} -8.00000 q^{67} +14.1421 q^{71} -8.00000 q^{73} +2.82843 q^{77} -4.24264 q^{79} +6.00000 q^{83} -2.82843 q^{85} -2.00000 q^{89} -5.65685 q^{95} -14.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 4 q^{11} - 4 q^{17} - 8 q^{19} - 6 q^{25} + 4 q^{35} - 12 q^{41} - 16 q^{43} - 10 q^{49} + 24 q^{59} - 16 q^{67} - 16 q^{73} + 12 q^{83} - 4 q^{89} - 28 q^{97}+O(q^{100}) $ 2 * q + 4 * q^11 - 4 * q^17 - 8 * q^19 - 6 * q^25 + 4 * q^35 - 12 * q^41 - 16 * q^43 - 10 * q^49 + 24 * q^59 - 16 * q^67 - 16 * q^73 + 12 * q^83 - 4 * q^89 - 28 * 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
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	0	0
$7$	1.41421	0.534522	0.267261	−	0.963624i	$-0.413881\pi$
$7$	1.41421	0.534522	0.267261	+	0.963624i	$0.413881\pi$
$8$	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
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−9.89949	−1.83829	−0.919145	−	0.393919i	$-0.871119\pi$
$29$	−9.89949	−1.83829	−0.919145	+	0.393919i	$0.871119\pi$
$30$	0	0
$31$	−7.07107	−1.27000	−0.635001	−	0.772512i	$-0.719000\pi$
$31$	−7.07107	−1.27000	−0.635001	+	0.772512i	$0.719000\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	−8.48528	−1.39497	−0.697486	−	0.716599i	$-0.745698\pi$
$37$	−8.48528	−1.39497	−0.697486	+	0.716599i	$0.745698\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.41421	0.194257	0.0971286	−	0.995272i	$-0.469034\pi$
$53$	1.41421	0.194257	0.0971286	+	0.995272i	$0.469034\pi$
$54$	0	0
$55$	2.82843	0.381385
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	14.1421	1.81071	0.905357	−	0.424650i	$-0.139603\pi$
$61$	14.1421	1.81071	0.905357	+	0.424650i	$0.139603\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	14.1421	1.67836	0.839181	−	0.543852i	$-0.183035\pi$
$71$	14.1421	1.67836	0.839181	+	0.543852i	$0.183035\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.82843	0.322329
$78$	0	0
$79$	−4.24264	−0.477334	−0.238667	−	0.971101i	$-0.576710\pi$
$79$	−4.24264	−0.477334	−0.238667	+	0.971101i	$0.576710\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−2.82843	−0.306786
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.65685	−0.580381
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.89949	0.985037	0.492518	−	0.870302i	$-0.336076\pi$
$101$	9.89949	0.985037	0.492518	+	0.870302i	$0.336076\pi$
$102$	0	0
$103$	12.7279	1.25412	0.627060	−	0.778971i	$-0.284258\pi$
$103$	12.7279	1.25412	0.627060	+	0.778971i	$0.284258\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	−11.3137	−1.08366	−0.541828	−	0.840489i	$-0.682268\pi$
$109$	−11.3137	−1.08366	−0.541828	+	0.840489i	$0.682268\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.82843	−0.259281
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.3137	−1.01193
$126$	0	0
$127$	−4.24264	−0.376473	−0.188237	−	0.982124i	$-0.560277\pi$
$127$	−4.24264	−0.376473	−0.188237	+	0.982124i	$0.560277\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	−5.65685	−0.490511
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−14.0000	−1.16264
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.24264	−0.347571	−0.173785	−	0.984784i	$-0.555600\pi$
$149$	−4.24264	−0.347571	−0.173785	+	0.984784i	$0.555600\pi$
$150$	0	0
$151$	4.24264	0.345261	0.172631	−	0.984987i	$-0.444773\pi$
$151$	4.24264	0.345261	0.172631	+	0.984987i	$0.444773\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−10.0000	−0.803219
$156$	0	0
$157$	14.1421	1.12867	0.564333	−	0.825547i	$-0.309134\pi$
$157$	14.1421	1.12867	0.564333	+	0.825547i	$0.309134\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	16.0000	1.25322	0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000	1.25322	0.626608	+	0.779334i	$0.284443\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.3137	0.875481	0.437741	−	0.899101i	$-0.355779\pi$
$167$	11.3137	0.875481	0.437741	+	0.899101i	$0.355779\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.5563	1.18273	0.591364	−	0.806405i	$-0.298590\pi$
$173$	15.5563	1.18273	0.591364	+	0.806405i	$0.298590\pi$
$174$	0	0
$175$	−4.24264	−0.320713
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	11.3137	0.840941	0.420471	−	0.907306i	$-0.361865\pi$
$181$	11.3137	0.840941	0.420471	+	0.907306i	$0.361865\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−22.6274	−1.63726	−0.818631	−	0.574320i	$-0.805267\pi$
$191$	−22.6274	−1.63726	−0.818631	+	0.574320i	$0.805267\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	7.07107	0.503793	0.251896	−	0.967754i	$-0.418946\pi$
$197$	7.07107	0.503793	0.251896	+	0.967754i	$0.418946\pi$
$198$	0	0
$199$	24.0416	1.70427	0.852133	−	0.523325i	$-0.175309\pi$
$199$	24.0416	1.70427	0.852133	+	0.523325i	$0.175309\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−14.0000	−0.982607
$204$	0	0
$205$	−8.48528	−0.592638
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8.00000	−0.553372
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−11.3137	−0.771589
$216$	0	0
$217$	−10.0000	−0.678844
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−12.7279	−0.852325	−0.426162	−	0.904647i	$-0.640135\pi$
$223$	−12.7279	−0.852325	−0.426162	+	0.904647i	$0.640135\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−22.0000	−1.46019	−0.730096	−	0.683345i	$-0.760525\pi$
$227$	−22.0000	−1.46019	−0.730096	+	0.683345i	$0.760525\pi$
$228$	0	0
$229$	16.9706	1.12145	0.560723	−	0.828003i	$-0.310523\pi$
$229$	16.9706	1.12145	0.560723	+	0.828003i	$0.310523\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.6274	1.46365	0.731823	−	0.681495i	$-0.238670\pi$
$239$	22.6274	1.46365	0.731823	+	0.681495i	$0.238670\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−7.07107	−0.451754
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6.00000	−0.378717	−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000	−0.378717	−0.189358	+	0.981908i	$0.560641\pi$
$252$	0	0
$253$	−5.65685	−0.355643
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−12.0000	−0.745644
$260$	0	0
$261$	0	0
$262$	0	0
$263$	28.2843	1.74408	0.872041	−	0.489432i	$-0.162796\pi$
$263$	28.2843	1.74408	0.872041	+	0.489432i	$0.162796\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	0	0
$268$	0	0
$269$	21.2132	1.29339	0.646696	−	0.762748i	$-0.276150\pi$
$269$	21.2132	1.29339	0.646696	+	0.762748i	$0.276150\pi$
$270$	0	0
$271$	−24.0416	−1.46043	−0.730213	−	0.683220i	$-0.760579\pi$
$271$	−24.0416	−1.46043	−0.730213	+	0.683220i	$0.760579\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.00000	−0.361814
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.48528	−0.500870
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9.89949	−0.578335	−0.289167	−	0.957279i	$-0.593378\pi$
$293$	−9.89949	−0.578335	−0.289167	+	0.957279i	$0.593378\pi$
$294$	0	0
$295$	16.9706	0.988064
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−11.3137	−0.652111
$302$	0	0
$303$	0	0
$304$	0	0
$305$	20.0000	1.14520
$306$	0	0
$307$	32.0000	1.82634	0.913168	−	0.407583i	$-0.133628\pi$
$307$	32.0000	1.82634	0.913168	+	0.407583i	$0.133628\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−22.6274	−1.28308	−0.641542	−	0.767088i	$-0.721705\pi$
$311$	−22.6274	−1.28308	−0.641542	+	0.767088i	$0.721705\pi$
$312$	0	0
$313$	−4.00000	−0.226093	−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000	−0.226093	−0.113047	+	0.993590i	$0.536061\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.89949	0.556011	0.278006	−	0.960579i	$-0.410327\pi$
$317$	9.89949	0.556011	0.278006	+	0.960579i	$0.410327\pi$
$318$	0	0
$319$	−19.7990	−1.10853
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.00000	0.445132
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.3137	−0.618134
$336$	0	0
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−14.1421	−0.765840
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−26.0000	−1.39575	−0.697877	−	0.716218i	$-0.745872\pi$
$347$	−26.0000	−1.39575	−0.697877	+	0.716218i	$0.745872\pi$
$348$	0	0
$349$	−25.4558	−1.36262	−0.681310	−	0.731995i	$-0.738589\pi$
$349$	−25.4558	−1.36262	−0.681310	+	0.731995i	$0.738589\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	20.0000	1.06149
$356$	0	0
$357$	0	0
$358$	0	0
$359$	19.7990	1.04495	0.522475	−	0.852654i	$-0.325009\pi$
$359$	19.7990	1.04495	0.522475	+	0.852654i	$0.325009\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−11.3137	−0.592187
$366$	0	0
$367$	4.24264	0.221464	0.110732	−	0.993850i	$-0.464680\pi$
$367$	4.24264	0.221464	0.110732	+	0.993850i	$0.464680\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.00000	0.103835
$372$	0	0
$373$	19.7990	1.02515	0.512576	−	0.858642i	$-0.328691\pi$
$373$	19.7990	1.02515	0.512576	+	0.858642i	$0.328691\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.65685	0.289052	0.144526	−	0.989501i	$-0.453834\pi$
$383$	5.65685	0.289052	0.144526	+	0.989501i	$0.453834\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−24.0416	−1.21896	−0.609480	−	0.792802i	$-0.708622\pi$
$389$	−24.0416	−1.21896	−0.609480	+	0.792802i	$0.708622\pi$
$390$	0	0
$391$	5.65685	0.286079
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6.00000	−0.301893
$396$	0	0
$397$	2.82843	0.141955	0.0709773	−	0.997478i	$-0.477388\pi$
$397$	2.82843	0.141955	0.0709773	+	0.997478i	$0.477388\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−16.9706	−0.841200
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	16.9706	0.835067
$414$	0	0
$415$	8.48528	0.416526
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	5.65685	0.275698	0.137849	−	0.990453i	$-0.455981\pi$
$421$	5.65685	0.275698	0.137849	+	0.990453i	$0.455981\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	20.0000	0.967868
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.82843	−0.136241	−0.0681203	−	0.997677i	$-0.521700\pi$
$431$	−2.82843	−0.136241	−0.0681203	+	0.997677i	$0.521700\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	11.3137	0.541208
$438$	0	0
$439$	−4.24264	−0.202490	−0.101245	−	0.994862i	$-0.532283\pi$
$439$	−4.24264	−0.202490	−0.101245	+	0.994862i	$0.532283\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	30.0000	1.42534	0.712672	−	0.701498i	$-0.247485\pi$
$443$	30.0000	1.42534	0.712672	+	0.701498i	$0.247485\pi$
$444$	0	0
$445$	−2.82843	−0.134080
$446$	0	0
$447$	0	0
$448$	0	0
$449$	22.0000	1.03824	0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000	1.03824	0.519122	+	0.854700i	$0.326259\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.00000	−0.280668	−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000	−0.280668	−0.140334	+	0.990104i	$0.544818\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12.7279	−0.592798	−0.296399	−	0.955064i	$-0.595786\pi$
$461$	−12.7279	−0.592798	−0.296399	+	0.955064i	$0.595786\pi$
$462$	0	0
$463$	35.3553	1.64310	0.821551	−	0.570135i	$-0.193109\pi$
$463$	35.3553	1.64310	0.821551	+	0.570135i	$0.193109\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.0000	−1.01804	−0.509019	−	0.860755i	$-0.669992\pi$
$467$	−22.0000	−1.01804	−0.509019	+	0.860755i	$0.669992\pi$
$468$	0	0
$469$	−11.3137	−0.522419
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	12.0000	0.550598
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−25.4558	−1.16311	−0.581554	−	0.813508i	$-0.697555\pi$
$479$	−25.4558	−1.16311	−0.581554	+	0.813508i	$0.697555\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−19.7990	−0.899026
$486$	0	0
$487$	−4.24264	−0.192252	−0.0961262	−	0.995369i	$-0.530645\pi$
$487$	−4.24264	−0.192252	−0.0961262	+	0.995369i	$0.530645\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	0	0
$493$	19.7990	0.891702
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.0000	0.897123
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.82843	−0.126113	−0.0630567	−	0.998010i	$-0.520085\pi$
$503$	−2.82843	−0.126113	−0.0630567	+	0.998010i	$0.520085\pi$
$504$	0	0
$505$	14.0000	0.622992
$506$	0	0
$507$	0	0
$508$	0	0
$509$	24.0416	1.06563	0.532813	−	0.846233i	$-0.321135\pi$
$509$	24.0416	1.06563	0.532813	+	0.846233i	$0.321135\pi$
$510$	0	0
$511$	−11.3137	−0.500489
$512$	0	0
$513$	0	0
$514$	0	0
$515$	18.0000	0.793175
$516$	0	0
$517$	−5.65685	−0.248788
$518$	0	0
$519$	0	0
$520$	0	0
$521$	38.0000	1.66481	0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000	1.66481	0.832405	+	0.554168i	$0.186963\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.1421	0.616041
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	5.65685	0.244567
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−10.0000	−0.430730
$540$	0	0
$541$	−28.2843	−1.21604	−0.608018	−	0.793923i	$-0.708035\pi$
$541$	−28.2843	−1.21604	−0.608018	+	0.793923i	$0.708035\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−16.0000	−0.685365
$546$	0	0
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	39.5980	1.68693
$552$	0	0
$553$	−6.00000	−0.255146
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−38.1838	−1.61790	−0.808949	−	0.587879i	$-0.799963\pi$
$557$	−38.1838	−1.61790	−0.808949	+	0.587879i	$0.799963\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	0	0
$565$	−8.48528	−0.356978
$566$	0	0
$567$	0	0
$568$	0	0
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.48528	0.353861
$576$	0	0
$577$	−4.00000	−0.166522	−0.0832611	−	0.996528i	$-0.526534\pi$
$577$	−4.00000	−0.166522	−0.0832611	+	0.996528i	$0.526534\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	8.48528	0.352029
$582$	0	0
$583$	2.82843	0.117141
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	28.2843	1.16543
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−36.7696	−1.50236	−0.751182	−	0.660096i	$-0.770516\pi$
$599$	−36.7696	−1.50236	−0.751182	+	0.660096i	$0.770516\pi$
$600$	0	0
$601$	44.0000	1.79480	0.897399	−	0.441221i	$-0.145454\pi$
$601$	44.0000	1.79480	0.897399	+	0.441221i	$0.145454\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.89949	−0.402472
$606$	0	0
$607$	−4.24264	−0.172203	−0.0861017	−	0.996286i	$-0.527441\pi$
$607$	−4.24264	−0.172203	−0.0861017	+	0.996286i	$0.527441\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−8.48528	−0.342717	−0.171359	−	0.985209i	$-0.554816\pi$
$613$	−8.48528	−0.342717	−0.171359	+	0.985209i	$0.554816\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−46.0000	−1.85189	−0.925945	−	0.377658i	$-0.876729\pi$
$617$	−46.0000	−1.85189	−0.925945	+	0.377658i	$0.876729\pi$
$618$	0	0
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.82843	−0.113319
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	16.9706	0.676661
$630$	0	0
$631$	−15.5563	−0.619288	−0.309644	−	0.950852i	$-0.600210\pi$
$631$	−15.5563	−0.619288	−0.309644	+	0.950852i	$0.600210\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6.00000	−0.238103
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	0	0
$643$	−40.0000	−1.57745	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−40.0000	−1.57745	−0.788723	+	0.614749i	$0.789257\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.1421	0.555985	0.277992	−	0.960583i	$-0.410331\pi$
$647$	14.1421	0.555985	0.277992	+	0.960583i	$0.410331\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−29.6985	−1.16219	−0.581096	−	0.813835i	$-0.697376\pi$
$653$	−29.6985	−1.16219	−0.581096	+	0.813835i	$0.697376\pi$
$654$	0	0
$655$	16.9706	0.663095
$656$	0	0
$657$	0	0
$658$	0	0
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	0	0
$661$	−19.7990	−0.770091	−0.385046	−	0.922897i	$-0.625814\pi$
$661$	−19.7990	−0.770091	−0.385046	+	0.922897i	$0.625814\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	28.0000	1.08416
$668$	0	0
$669$	0	0
$670$	0	0
$671$	28.2843	1.09190
$672$	0	0
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	24.0416	0.923995	0.461997	−	0.886881i	$-0.347133\pi$
$677$	24.0416	0.923995	0.461997	+	0.886881i	$0.347133\pi$
$678$	0	0
$679$	−19.7990	−0.759815
$680$	0	0
$681$	0	0
$682$	0	0
$683$	26.0000	0.994862	0.497431	−	0.867503i	$-0.334277\pi$
$683$	26.0000	0.994862	0.497431	+	0.867503i	$0.334277\pi$
$684$	0	0
$685$	14.1421	0.540343
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	0	0
$700$	0	0
$701$	4.24264	0.160242	0.0801212	−	0.996785i	$-0.474469\pi$
$701$	4.24264	0.160242	0.0801212	+	0.996785i	$0.474469\pi$
$702$	0	0
$703$	33.9411	1.28011
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.0000	0.526524
$708$	0	0
$709$	33.9411	1.27469	0.637343	−	0.770580i	$-0.280034\pi$
$709$	33.9411	1.27469	0.637343	+	0.770580i	$0.280034\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	20.0000	0.749006
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	25.4558	0.949343	0.474671	−	0.880163i	$-0.342567\pi$
$719$	25.4558	0.949343	0.474671	+	0.880163i	$0.342567\pi$
$720$	0	0
$721$	18.0000	0.670355
$722$	0	0
$723$	0	0
$724$	0	0
$725$	29.6985	1.10297
$726$	0	0
$727$	−24.0416	−0.891655	−0.445827	−	0.895119i	$-0.647090\pi$
$727$	−24.0416	−0.891655	−0.445827	+	0.895119i	$0.647090\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	16.0000	0.591781
$732$	0	0
$733$	−11.3137	−0.417881	−0.208941	−	0.977928i	$-0.567002\pi$
$733$	−11.3137	−0.417881	−0.208941	+	0.977928i	$0.567002\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.0000	−0.589368
$738$	0	0
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−28.2843	−1.03765	−0.518825	−	0.854881i	$-0.673630\pi$
$743$	−28.2843	−1.03765	−0.518825	+	0.854881i	$0.673630\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5.65685	0.206697
$750$	0	0
$751$	12.7279	0.464448	0.232224	−	0.972662i	$-0.425400\pi$
$751$	12.7279	0.464448	0.232224	+	0.972662i	$0.425400\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.00000	0.218362
$756$	0	0
$757$	−50.9117	−1.85042	−0.925208	−	0.379459i	$-0.876110\pi$
$757$	−50.9117	−1.85042	−0.925208	+	0.379459i	$0.876110\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−10.0000	−0.362500	−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000	−0.362500	−0.181250	+	0.983437i	$0.558014\pi$
$762$	0	0
$763$	−16.0000	−0.579239
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	20.0000	0.721218	0.360609	−	0.932717i	$-0.382569\pi$
$769$	20.0000	0.721218	0.360609	+	0.932717i	$0.382569\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	15.5563	0.559523	0.279761	−	0.960070i	$-0.409745\pi$
$773$	15.5563	0.559523	0.279761	+	0.960070i	$0.409745\pi$
$774$	0	0
$775$	21.2132	0.762001
$776$	0	0
$777$	0	0
$778$	0	0
$779$	24.0000	0.859889
$780$	0	0
$781$	28.2843	1.01209
$782$	0	0
$783$	0	0
$784$	0	0
$785$	20.0000	0.713831
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−8.48528	−0.301702
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.41421	0.0500940	0.0250470	−	0.999686i	$-0.492026\pi$
$797$	1.41421	0.0500940	0.0250470	+	0.999686i	$0.492026\pi$
$798$	0	0
$799$	5.65685	0.200125
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−16.0000	−0.564628
$804$	0	0
$805$	−5.65685	−0.199378
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	22.6274	0.792604
$816$	0	0
$817$	32.0000	1.11954
$818$	0	0
$819$	0	0
$820$	0	0
$821$	9.89949	0.345495	0.172747	−	0.984966i	$-0.444736\pi$
$821$	9.89949	0.345495	0.172747	+	0.984966i	$0.444736\pi$
$822$	0	0
$823$	−21.2132	−0.739446	−0.369723	−	0.929142i	$-0.620547\pi$
$823$	−21.2132	−0.739446	−0.369723	+	0.929142i	$0.620547\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	0	0
$829$	11.3137	0.392941	0.196471	−	0.980510i	$-0.437052\pi$
$829$	11.3137	0.392941	0.196471	+	0.980510i	$0.437052\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	10.0000	0.346479
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−14.1421	−0.488241	−0.244120	−	0.969745i	$-0.578499\pi$
$839$	−14.1421	−0.488241	−0.244120	+	0.969745i	$0.578499\pi$
$840$	0	0
$841$	69.0000	2.37931
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−18.3848	−0.632456
$846$	0	0
$847$	−9.89949	−0.340151
$848$	0	0
$849$	0	0
$850$	0	0
$851$	24.0000	0.822709
$852$	0	0
$853$	14.1421	0.484218	0.242109	−	0.970249i	$-0.422161\pi$
$853$	14.1421	0.484218	0.242109	+	0.970249i	$0.422161\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	26.0000	0.888143	0.444072	−	0.895991i	$-0.353534\pi$
$857$	26.0000	0.888143	0.444072	+	0.895991i	$0.353534\pi$
$858$	0	0
$859$	−8.00000	−0.272956	−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000	−0.272956	−0.136478	+	0.990643i	$0.543578\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−22.6274	−0.770246	−0.385123	−	0.922865i	$-0.625841\pi$
$863$	−22.6274	−0.770246	−0.385123	+	0.922865i	$0.625841\pi$
$864$	0	0
$865$	22.0000	0.748022
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8.48528	−0.287843
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−16.0000	−0.540899
$876$	0	0
$877$	42.4264	1.43264	0.716319	−	0.697773i	$-0.245826\pi$
$877$	42.4264	1.43264	0.716319	+	0.697773i	$0.245826\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	22.0000	0.741199	0.370599	−	0.928793i	$-0.379152\pi$
$881$	22.0000	0.741199	0.370599	+	0.928793i	$0.379152\pi$
$882$	0	0
$883$	−48.0000	−1.61533	−0.807664	−	0.589643i	$-0.799269\pi$
$883$	−48.0000	−1.61533	−0.807664	+	0.589643i	$0.799269\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−50.9117	−1.70945	−0.854724	−	0.519083i	$-0.826273\pi$
$887$	−50.9117	−1.70945	−0.854724	+	0.519083i	$0.826273\pi$
$888$	0	0
$889$	−6.00000	−0.201234
$890$	0	0
$891$	0	0
$892$	0	0
$893$	11.3137	0.378599
$894$	0	0
$895$	−5.65685	−0.189088
$896$	0	0
$897$	0	0
$898$	0	0
$899$	70.0000	2.33463
$900$	0	0
$901$	−2.82843	−0.0942286
$902$	0	0
$903$	0	0
$904$	0	0
$905$	16.0000	0.531858
$906$	0	0
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.65685	−0.187420	−0.0937100	−	0.995600i	$-0.529873\pi$
$911$	−5.65685	−0.187420	−0.0937100	+	0.995600i	$0.529873\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.9706	0.560417
$918$	0	0
$919$	−21.2132	−0.699759	−0.349880	−	0.936795i	$-0.613777\pi$
$919$	−21.2132	−0.699759	−0.349880	+	0.936795i	$0.613777\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	25.4558	0.836983
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	20.0000	0.655474
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.65685	−0.184999
$936$	0	0
$937$	−18.0000	−0.588034	−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000	−0.588034	−0.294017	+	0.955800i	$0.594992\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.41421	−0.0461020	−0.0230510	−	0.999734i	$-0.507338\pi$
$941$	−1.41421	−0.0461020	−0.0230510	+	0.999734i	$0.507338\pi$
$942$	0	0
$943$	16.9706	0.552638
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	0	0
$955$	−32.0000	−1.03550
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14.1421	0.456673
$960$	0	0
$961$	19.0000	0.612903
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−5.65685	−0.182101
$966$	0	0
$967$	−52.3259	−1.68269	−0.841344	−	0.540500i	$-0.818235\pi$
$967$	−52.3259	−1.68269	−0.841344	+	0.540500i	$0.818235\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−22.0000	−0.706014	−0.353007	−	0.935621i	$-0.614841\pi$
$971$	−22.0000	−0.706014	−0.353007	+	0.935621i	$0.614841\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−50.0000	−1.59964	−0.799821	−	0.600239i	$-0.795072\pi$
$977$	−50.0000	−1.59964	−0.799821	+	0.600239i	$0.795072\pi$
$978$	0	0
$979$	−4.00000	−0.127841
$980$	0	0
$981$	0	0
$982$	0	0
$983$	50.9117	1.62383	0.811915	−	0.583775i	$-0.198425\pi$
$983$	50.9117	1.62383	0.811915	+	0.583775i	$0.198425\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	0	0
$987$	0	0
$988$	0	0
$989$	22.6274	0.719510
$990$	0	0
$991$	46.6690	1.48249	0.741246	−	0.671234i	$-0.234235\pi$
$991$	46.6690	1.48249	0.741246	+	0.671234i	$0.234235\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	34.0000	1.07787
$996$	0	0
$997$	−31.1127	−0.985349	−0.492675	−	0.870214i	$-0.663981\pi$
$997$	−31.1127	−0.985349	−0.492675	+	0.870214i	$0.663981\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.a.j.1.2		2
3.2	odd	2		1536.2.a.c.1.1	✓	2
4.3	odd	2		4608.2.a.h.1.2		2
8.3	odd	2	inner	4608.2.a.j.1.1		2
8.5	even	2		4608.2.a.h.1.1		2
12.11	even	2		1536.2.a.j.1.1	yes	2
16.3	odd	4		4608.2.d.g.2305.2		4
16.5	even	4		4608.2.d.g.2305.3		4
16.11	odd	4		4608.2.d.g.2305.4		4
16.13	even	4		4608.2.d.g.2305.1		4
24.5	odd	2		1536.2.a.j.1.2	yes	2
24.11	even	2		1536.2.a.c.1.2	yes	2
48.5	odd	4		1536.2.d.d.769.3		4
48.11	even	4		1536.2.d.d.769.1		4
48.29	odd	4		1536.2.d.d.769.2		4
48.35	even	4		1536.2.d.d.769.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.a.c.1.1	✓	2	3.2	odd	2
1536.2.a.c.1.2	yes	2	24.11	even	2
1536.2.a.j.1.1	yes	2	12.11	even	2
1536.2.a.j.1.2	yes	2	24.5	odd	2
1536.2.d.d.769.1		4	48.11	even	4
1536.2.d.d.769.2		4	48.29	odd	4
1536.2.d.d.769.3		4	48.5	odd	4
1536.2.d.d.769.4		4	48.35	even	4
4608.2.a.h.1.1		2	8.5	even	2
4608.2.a.h.1.2		2	4.3	odd	2
4608.2.a.j.1.1		2	8.3	odd	2	inner
4608.2.a.j.1.2		2	1.1	even	1	trivial
4608.2.d.g.2305.1		4	16.13	even	4
4608.2.d.g.2305.2		4	16.3	odd	4
4608.2.d.g.2305.3		4	16.5	even	4
4608.2.d.g.2305.4		4	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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{5} +1.41421 q^{7} +O(q^{10})\)	\(q+1.41421 q^{5} +1.41421 q^{7} +2.00000 q^{11} -2.00000 q^{17} -4.00000 q^{19} -2.82843 q^{23} -3.00000 q^{25} -9.89949 q^{29} -7.07107 q^{31} +2.00000 q^{35} -8.48528 q^{37} -6.00000 q^{41} -8.00000 q^{43} -2.82843 q^{47} -5.00000 q^{49} +1.41421 q^{53} +2.82843 q^{55} +12.0000 q^{59} +14.1421 q^{61} -8.00000 q^{67} +14.1421 q^{71} -8.00000 q^{73} +2.82843 q^{77} -4.24264 q^{79} +6.00000 q^{83} -2.82843 q^{85} -2.00000 q^{89} -5.65685 q^{95} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 4 q^{11} - 4 q^{17} - 8 q^{19} - 6 q^{25} + 4 q^{35} - 12 q^{41} - 16 q^{43} - 10 q^{49} + 24 q^{59} - 16 q^{67} - 16 q^{73} + 12 q^{83} - 4 q^{89} - 28 q^{97}+O(q^{100}) \) 2 * q + 4 * q^11 - 4 * q^17 - 8 * q^19 - 6 * q^25 + 4 * q^35 - 12 * q^41 - 16 * q^43 - 10 * q^49 + 24 * q^59 - 16 * q^67 - 16 * q^73 + 12 * q^83 - 4 * q^89 - 28 * q^97

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