LMFDB - Embedded newform 4608.2.a.t.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:	$1$
Dimension:	$4$
Coefficient field:	4.4.4352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1536)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.74912$ 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)

$f(q)$	$=$	$q+2.08402 q^{5} -5.03127 q^{7} +O(q^{10})$	$q+2.08402 q^{5} -5.03127 q^{7} +0.828427 q^{11} +2.94725 q^{13} -4.82843 q^{17} +2.82843 q^{19} -4.16804 q^{23} -0.656854 q^{25} +7.97852 q^{29} +5.03127 q^{31} -10.4853 q^{35} +7.11529 q^{37} -8.82843 q^{41} -12.4853 q^{43} -4.16804 q^{47} +18.3137 q^{49} -12.1466 q^{53} +1.72646 q^{55} +1.65685 q^{59} +7.11529 q^{61} +6.14214 q^{65} -2.34315 q^{67} -10.0625 q^{71} +4.00000 q^{73} -4.16804 q^{77} -5.03127 q^{79} -3.17157 q^{83} -10.0625 q^{85} -10.0000 q^{89} -14.8284 q^{91} +5.89450 q^{95} +0.343146 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 8 q^{11} - 8 q^{17} + 20 q^{25} - 8 q^{35} - 24 q^{41} - 16 q^{43} + 28 q^{49} - 16 q^{59} - 32 q^{65} - 32 q^{67} + 16 q^{73} - 24 q^{83} - 40 q^{89} - 48 q^{91} + 24 q^{97}+O(q^{100}) $ 4 * q - 8 * q^11 - 8 * q^17 + 20 * q^25 - 8 * q^35 - 24 * q^41 - 16 * q^43 + 28 * q^49 - 16 * q^59 - 32 * q^65 - 32 * q^67 + 16 * q^73 - 24 * q^83 - 40 * q^89 - 48 * q^91 + 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
$5$	2.08402	0.932003	0.466001	−	0.884784i	$-0.345694\pi$
$5$	2.08402	0.932003	0.466001	+	0.884784i	$0.345694\pi$
$6$	0	0
$7$	−5.03127	−1.90164	−0.950821	−	0.309740i	$-0.899758\pi$
$7$	−5.03127	−1.90164	−0.950821	+	0.309740i	$0.899758\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.828427	0.249780	0.124890	−	0.992171i	$-0.460142\pi$
$11$	0.828427	0.249780	0.124890	+	0.992171i	$0.460142\pi$
$12$	0	0
$13$	2.94725	0.817420	0.408710	−	0.912664i	$-0.365979\pi$
$13$	2.94725	0.817420	0.408710	+	0.912664i	$0.365979\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.82843	−1.17107	−0.585533	−	0.810649i	$-0.699115\pi$
$17$	−4.82843	−1.17107	−0.585533	+	0.810649i	$0.699115\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.16804	−0.869097	−0.434549	−	0.900648i	$-0.643092\pi$
$23$	−4.16804	−0.869097	−0.434549	+	0.900648i	$0.643092\pi$
$24$	0	0
$25$	−0.656854	−0.131371
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.97852	1.48157	0.740787	−	0.671740i	$-0.234453\pi$
$29$	7.97852	1.48157	0.740787	+	0.671740i	$0.234453\pi$
$30$	0	0
$31$	5.03127	0.903643	0.451822	−	0.892108i	$-0.350774\pi$
$31$	5.03127	0.903643	0.451822	+	0.892108i	$0.350774\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−10.4853	−1.77234
$36$	0	0
$37$	7.11529	1.16975	0.584874	−	0.811124i	$-0.301144\pi$
$37$	7.11529	1.16975	0.584874	+	0.811124i	$0.301144\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.82843	−1.37877	−0.689384	−	0.724396i	$-0.742119\pi$
$41$	−8.82843	−1.37877	−0.689384	+	0.724396i	$0.742119\pi$
$42$	0	0
$43$	−12.4853	−1.90399	−0.951994	−	0.306117i	$-0.900970\pi$
$43$	−12.4853	−1.90399	−0.951994	+	0.306117i	$0.900970\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.16804	−0.607972	−0.303986	−	0.952677i	$-0.598318\pi$
$47$	−4.16804	−0.607972	−0.303986	+	0.952677i	$0.598318\pi$
$48$	0	0
$49$	18.3137	2.61624
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−12.1466	−1.66846	−0.834230	−	0.551417i	$-0.814087\pi$
$53$	−12.1466	−1.66846	−0.834230	+	0.551417i	$0.814087\pi$
$54$	0	0
$55$	1.72646	0.232796
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.65685	0.215704	0.107852	−	0.994167i	$-0.465603\pi$
$59$	1.65685	0.215704	0.107852	+	0.994167i	$0.465603\pi$
$60$	0	0
$61$	7.11529	0.911020	0.455510	−	0.890231i	$-0.349457\pi$
$61$	7.11529	0.911020	0.455510	+	0.890231i	$0.349457\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.14214	0.761838
$66$	0	0
$67$	−2.34315	−0.286261	−0.143130	−	0.989704i	$-0.545717\pi$
$67$	−2.34315	−0.286261	−0.143130	+	0.989704i	$0.545717\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.0625	−1.19420	−0.597102	−	0.802165i	$-0.703681\pi$
$71$	−10.0625	−1.19420	−0.597102	+	0.802165i	$0.703681\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.16804	−0.474993
$78$	0	0
$79$	−5.03127	−0.566062	−0.283031	−	0.959111i	$-0.591340\pi$
$79$	−5.03127	−0.566062	−0.283031	+	0.959111i	$0.591340\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.17157	−0.348125	−0.174063	−	0.984735i	$-0.555690\pi$
$83$	−3.17157	−0.348125	−0.174063	+	0.984735i	$0.555690\pi$
$84$	0	0
$85$	−10.0625	−1.09144
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−14.8284	−1.55444
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.89450	0.604763
$96$	0	0
$97$	0.343146	0.0348412	0.0174206	−	0.999848i	$-0.494455\pi$
$97$	0.343146	0.0348412	0.0174206	+	0.999848i	$0.494455\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.1466	1.20863	0.604314	−	0.796746i	$-0.293447\pi$
$101$	12.1466	1.20863	0.604314	+	0.796746i	$0.293447\pi$
$102$	0	0
$103$	3.30481	0.325633	0.162816	−	0.986656i	$-0.447942\pi$
$103$	3.30481	0.325633	0.162816	+	0.986656i	$0.447942\pi$
$104$	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
$109$	−8.84175	−0.846886	−0.423443	−	0.905923i	$-0.639179\pi$
$109$	−8.84175	−0.846886	−0.423443	+	0.905923i	$0.639179\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−11.6569	−1.09658	−0.548292	−	0.836287i	$-0.684722\pi$
$113$	−11.6569	−1.09658	−0.548292	+	0.836287i	$0.684722\pi$
$114$	0	0
$115$	−8.68629	−0.810001
$116$	0	0
$117$	0	0
$118$	0	0
$119$	24.2931	2.22695
$120$	0	0
$121$	−10.3137	−0.937610
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.7890	−1.05444
$126$	0	0
$127$	−13.3674	−1.18616	−0.593081	−	0.805143i	$-0.702088\pi$
$127$	−13.3674	−1.18616	−0.593081	+	0.805143i	$0.702088\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.3137	−1.33796	−0.668982	−	0.743278i	$-0.733270\pi$
$131$	−15.3137	−1.33796	−0.668982	+	0.743278i	$0.733270\pi$
$132$	0	0
$133$	−14.2306	−1.23395
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.1421	−1.03737	−0.518686	−	0.854965i	$-0.673579\pi$
$137$	−12.1421	−1.03737	−0.518686	+	0.854965i	$0.673579\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.44158	0.204175
$144$	0	0
$145$	16.6274	1.38083
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.08402	0.170730	0.0853648	−	0.996350i	$-0.472794\pi$
$149$	2.08402	0.170730	0.0853648	+	0.996350i	$0.472794\pi$
$150$	0	0
$151$	13.3674	1.08782	0.543910	−	0.839143i	$-0.316943\pi$
$151$	13.3674	1.08782	0.543910	+	0.839143i	$0.316943\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	10.4853	0.842198
$156$	0	0
$157$	−15.4514	−1.23315	−0.616577	−	0.787294i	$-0.711481\pi$
$157$	−15.4514	−1.23315	−0.616577	+	0.787294i	$0.711481\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	20.9706	1.65271
$162$	0	0
$163$	−18.1421	−1.42100	−0.710501	−	0.703696i	$-0.751532\pi$
$163$	−18.1421	−1.42100	−0.710501	+	0.703696i	$0.751532\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	14.2306	1.10120	0.550598	−	0.834771i	$-0.314400\pi$
$167$	14.2306	1.10120	0.550598	+	0.834771i	$0.314400\pi$
$168$	0	0
$169$	−4.31371	−0.331824
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.25206	0.475336	0.237668	−	0.971346i	$-0.423617\pi$
$173$	6.25206	0.475336	0.237668	+	0.971346i	$0.423617\pi$
$174$	0	0
$175$	3.30481	0.249820
$176$	0	0
$177$	0	0
$178$	0	0
$179$	20.9706	1.56741	0.783707	−	0.621131i	$-0.213327\pi$
$179$	20.9706	1.56741	0.783707	+	0.621131i	$0.213327\pi$
$180$	0	0
$181$	8.84175	0.657202	0.328601	−	0.944469i	$-0.393423\pi$
$181$	8.84175	0.657202	0.328601	+	0.944469i	$0.393423\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	14.8284	1.09021
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−26.0196	−1.88271	−0.941356	−	0.337415i	$-0.890447\pi$
$191$	−26.0196	−1.88271	−0.941356	+	0.337415i	$0.890447\pi$
$192$	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
$197$	−20.4827	−1.45933	−0.729664	−	0.683806i	$-0.760324\pi$
$197$	−20.4827	−1.45933	−0.729664	+	0.683806i	$0.760324\pi$
$198$	0	0
$199$	−5.03127	−0.356657	−0.178329	−	0.983971i	$-0.557069\pi$
$199$	−5.03127	−0.356657	−0.178329	+	0.983971i	$0.557069\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−40.1421	−2.81743
$204$	0	0
$205$	−18.3986	−1.28502
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.34315	0.162079
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−26.0196	−1.77452
$216$	0	0
$217$	−25.3137	−1.71841
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−14.2306	−0.957253
$222$	0	0
$223$	13.3674	0.895145	0.447572	−	0.894248i	$-0.352289\pi$
$223$	13.3674	0.895145	0.447572	+	0.894248i	$0.352289\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−16.1421	−1.07139	−0.535696	−	0.844411i	$-0.679951\pi$
$227$	−16.1421	−1.07139	−0.535696	+	0.844411i	$0.679951\pi$
$228$	0	0
$229$	5.38883	0.356104	0.178052	−	0.984021i	$-0.443020\pi$
$229$	5.38883	0.356104	0.178052	+	0.984021i	$0.443020\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.31371	−0.0860639	−0.0430320	−	0.999074i	$-0.513702\pi$
$233$	−1.31371	−0.0860639	−0.0430320	+	0.999074i	$0.513702\pi$
$234$	0	0
$235$	−8.68629	−0.566631
$236$	0	0
$237$	0	0
$238$	0	0
$239$	14.2306	0.920500	0.460250	−	0.887789i	$-0.347760\pi$
$239$	14.2306	0.920500	0.460250	+	0.887789i	$0.347760\pi$
$240$	0	0
$241$	16.9706	1.09317	0.546585	−	0.837404i	$-0.315928\pi$
$241$	16.9706	1.09317	0.546585	+	0.837404i	$0.315928\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	38.1662	2.43835
$246$	0	0
$247$	8.33609	0.530412
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−15.1716	−0.957621	−0.478811	−	0.877918i	$-0.658932\pi$
$251$	−15.1716	−0.957621	−0.478811	+	0.877918i	$0.658932\pi$
$252$	0	0
$253$	−3.45292	−0.217083
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.34315	0.520431	0.260216	−	0.965551i	$-0.416206\pi$
$257$	8.34315	0.520431	0.260216	+	0.965551i	$0.416206\pi$
$258$	0	0
$259$	−35.7990	−2.22444
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.33609	0.514025	0.257013	−	0.966408i	$-0.417262\pi$
$263$	8.33609	0.514025	0.257013	+	0.966408i	$0.417262\pi$
$264$	0	0
$265$	−25.3137	−1.55501
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.97852	−0.486459	−0.243230	−	0.969969i	$-0.578207\pi$
$269$	−7.97852	−0.486459	−0.243230	+	0.969969i	$0.578207\pi$
$270$	0	0
$271$	−3.30481	−0.200753	−0.100377	−	0.994950i	$-0.532005\pi$
$271$	−3.30481	−0.200753	−0.100377	+	0.994950i	$0.532005\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.544156	−0.0328138
$276$	0	0
$277$	25.5139	1.53298	0.766492	−	0.642254i	$-0.222001\pi$
$277$	25.5139	1.53298	0.766492	+	0.642254i	$0.222001\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−11.6569	−0.695390	−0.347695	−	0.937608i	$-0.613035\pi$
$281$	−11.6569	−0.695390	−0.347695	+	0.937608i	$0.613035\pi$
$282$	0	0
$283$	4.97056	0.295469	0.147735	−	0.989027i	$-0.452802\pi$
$283$	4.97056	0.295469	0.147735	+	0.989027i	$0.452802\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	44.4182	2.62193
$288$	0	0
$289$	6.31371	0.371395
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−12.1466	−0.709610	−0.354805	−	0.934940i	$-0.615453\pi$
$293$	−12.1466	−0.709610	−0.354805	+	0.934940i	$0.615453\pi$
$294$	0	0
$295$	3.45292	0.201037
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−12.2843	−0.710418
$300$	0	0
$301$	62.8169	3.62070
$302$	0	0
$303$	0	0
$304$	0	0
$305$	14.8284	0.849073
$306$	0	0
$307$	−10.3431	−0.590315	−0.295157	−	0.955449i	$-0.595372\pi$
$307$	−10.3431	−0.590315	−0.295157	+	0.955449i	$0.595372\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−15.3137	−0.865582	−0.432791	−	0.901494i	$-0.642471\pi$
$313$	−15.3137	−0.865582	−0.432791	+	0.901494i	$0.642471\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.9356	1.34436	0.672178	−	0.740390i	$-0.265359\pi$
$317$	23.9356	1.34436	0.672178	+	0.740390i	$0.265359\pi$
$318$	0	0
$319$	6.60963	0.370068
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−13.6569	−0.759888
$324$	0	0
$325$	−1.93591	−0.107385
$326$	0	0
$327$	0	0
$328$	0	0
$329$	20.9706	1.15614
$330$	0	0
$331$	21.6569	1.19037	0.595184	−	0.803589i	$-0.297079\pi$
$331$	21.6569	1.19037	0.595184	+	0.803589i	$0.297079\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.88317	−0.266796
$336$	0	0
$337$	−20.0000	−1.08947	−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000	−1.08947	−0.544735	+	0.838608i	$0.683370\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.16804	0.225712
$342$	0	0
$343$	−56.9224	−3.07352
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.82843	0.259204	0.129602	−	0.991566i	$-0.458630\pi$
$347$	4.82843	0.259204	0.129602	+	0.991566i	$0.458630\pi$
$348$	0	0
$349$	27.2404	1.45814	0.729072	−	0.684437i	$-0.239952\pi$
$349$	27.2404	1.45814	0.729072	+	0.684437i	$0.239952\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	21.3137	1.13441	0.567207	−	0.823575i	$-0.308024\pi$
$353$	21.3137	1.13441	0.567207	+	0.823575i	$0.308024\pi$
$354$	0	0
$355$	−20.9706	−1.11300
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.60963	−0.348843	−0.174421	−	0.984671i	$-0.555805\pi$
$359$	−6.60963	−0.348843	−0.174421	+	0.984671i	$0.555805\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8.33609	0.436331
$366$	0	0
$367$	16.8203	0.878011	0.439006	−	0.898484i	$-0.355331\pi$
$367$	16.8203	0.878011	0.439006	+	0.898484i	$0.355331\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	61.1127	3.17281
$372$	0	0
$373$	−18.9043	−0.978828	−0.489414	−	0.872052i	$-0.662789\pi$
$373$	−18.9043	−0.978828	−0.489414	+	0.872052i	$0.662789\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	23.5147	1.21107
$378$	0	0
$379$	16.4853	0.846792	0.423396	−	0.905945i	$-0.360838\pi$
$379$	16.4853	0.846792	0.423396	+	0.905945i	$0.360838\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5.89450	−0.301195	−0.150598	−	0.988595i	$-0.548120\pi$
$383$	−5.89450	−0.301195	−0.150598	+	0.988595i	$0.548120\pi$
$384$	0	0
$385$	−8.68629	−0.442694
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.08402	−0.105664	−0.0528320	−	0.998603i	$-0.516825\pi$
$389$	−2.08402	−0.105664	−0.0528320	+	0.998603i	$0.516825\pi$
$390$	0	0
$391$	20.1251	1.01777
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−10.4853	−0.527572
$396$	0	0
$397$	−15.4514	−0.775483	−0.387741	−	0.921768i	$-0.626745\pi$
$397$	−15.4514	−0.775483	−0.387741	+	0.921768i	$0.626745\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.14214	0.206848	0.103424	−	0.994637i	$-0.467020\pi$
$401$	4.14214	0.206848	0.103424	+	0.994637i	$0.467020\pi$
$402$	0	0
$403$	14.8284	0.738657
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.89450	0.292180
$408$	0	0
$409$	7.65685	0.378607	0.189304	−	0.981919i	$-0.439377\pi$
$409$	7.65685	0.378607	0.189304	+	0.981919i	$0.439377\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−8.33609	−0.410192
$414$	0	0
$415$	−6.60963	−0.324454
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−20.8284	−1.01754	−0.508768	−	0.860904i	$-0.669899\pi$
$419$	−20.8284	−1.01754	−0.508768	+	0.860904i	$0.669899\pi$
$420$	0	0
$421$	5.38883	0.262636	0.131318	−	0.991340i	$-0.458079\pi$
$421$	5.38883	0.262636	0.131318	+	0.991340i	$0.458079\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.17157	0.153844
$426$	0	0
$427$	−35.7990	−1.73243
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−15.9570	−0.768624	−0.384312	−	0.923203i	$-0.625561\pi$
$431$	−15.9570	−0.768624	−0.384312	+	0.923203i	$0.625561\pi$
$432$	0	0
$433$	−20.6274	−0.991290	−0.495645	−	0.868525i	$-0.665068\pi$
$433$	−20.6274	−0.991290	−0.495645	+	0.868525i	$0.665068\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−11.7890	−0.563945
$438$	0	0
$439$	−8.48419	−0.404928	−0.202464	−	0.979290i	$-0.564895\pi$
$439$	−8.48419	−0.404928	−0.202464	+	0.979290i	$0.564895\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.51472	0.0719665	0.0359832	−	0.999352i	$-0.488544\pi$
$443$	1.51472	0.0719665	0.0359832	+	0.999352i	$0.488544\pi$
$444$	0	0
$445$	−20.8402	−0.987921
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−34.4853	−1.62746	−0.813731	−	0.581242i	$-0.802567\pi$
$449$	−34.4853	−1.62746	−0.813731	+	0.581242i	$0.802567\pi$
$450$	0	0
$451$	−7.31371	−0.344389
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−30.9028	−1.44874
$456$	0	0
$457$	−12.3431	−0.577388	−0.288694	−	0.957421i	$-0.593221\pi$
$457$	−12.3431	−0.577388	−0.288694	+	0.957421i	$0.593221\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	29.8301	1.38933	0.694663	−	0.719336i	$-0.255554\pi$
$461$	29.8301	1.38933	0.694663	+	0.719336i	$0.255554\pi$
$462$	0	0
$463$	23.4299	1.08888	0.544440	−	0.838800i	$-0.316742\pi$
$463$	23.4299	1.08888	0.544440	+	0.838800i	$0.316742\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.79899	0.0832473	0.0416237	−	0.999133i	$-0.486747\pi$
$467$	1.79899	0.0832473	0.0416237	+	0.999133i	$0.486747\pi$
$468$	0	0
$469$	11.7890	0.544366
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−10.3431	−0.475578
$474$	0	0
$475$	−1.85786	−0.0852447
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−41.9766	−1.91796	−0.958981	−	0.283471i	$-0.908514\pi$
$479$	−41.9766	−1.91796	−0.958981	+	0.283471i	$0.908514\pi$
$480$	0	0
$481$	20.9706	0.956175
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.715123	0.0324721
$486$	0	0
$487$	6.75773	0.306222	0.153111	−	0.988209i	$-0.451071\pi$
$487$	6.75773	0.306222	0.153111	+	0.988209i	$0.451071\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9.65685	0.435808	0.217904	−	0.975970i	$-0.430078\pi$
$491$	9.65685	0.435808	0.217904	+	0.975970i	$0.430078\pi$
$492$	0	0
$493$	−38.5237	−1.73502
$494$	0	0
$495$	0	0
$496$	0	0
$497$	50.6274	2.27095
$498$	0	0
$499$	−41.6569	−1.86482	−0.932408	−	0.361406i	$-0.882297\pi$
$499$	−41.6569	−1.86482	−0.932408	+	0.361406i	$0.882297\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−18.3986	−0.820354	−0.410177	−	0.912006i	$-0.634533\pi$
$503$	−18.3986	−0.820354	−0.410177	+	0.912006i	$0.634533\pi$
$504$	0	0
$505$	25.3137	1.12645
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16.3146	0.723132	0.361566	−	0.932346i	$-0.382242\pi$
$509$	16.3146	0.723132	0.361566	+	0.932346i	$0.382242\pi$
$510$	0	0
$511$	−20.1251	−0.890282
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.88730	0.303491
$516$	0	0
$517$	−3.45292	−0.151859
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−20.8284	−0.912510	−0.456255	−	0.889849i	$-0.650810\pi$
$521$	−20.8284	−0.912510	−0.456255	+	0.889849i	$0.650810\pi$
$522$	0	0
$523$	18.8284	0.823310	0.411655	−	0.911340i	$-0.364951\pi$
$523$	18.8284	0.823310	0.411655	+	0.911340i	$0.364951\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−24.2931	−1.05823
$528$	0	0
$529$	−5.62742	−0.244670
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−26.0196	−1.12703
$534$	0	0
$535$	8.33609	0.360400
$536$	0	0
$537$	0	0
$538$	0	0
$539$	15.1716	0.653486
$540$	0	0
$541$	−11.2833	−0.485109	−0.242554	−	0.970138i	$-0.577985\pi$
$541$	−11.2833	−0.485109	−0.242554	+	0.970138i	$0.577985\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−18.4264	−0.789301
$546$	0	0
$547$	31.7990	1.35963	0.679813	−	0.733385i	$-0.262061\pi$
$547$	31.7990	1.35963	0.679813	+	0.733385i	$0.262061\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	22.5667	0.961373
$552$	0	0
$553$	25.3137	1.07645
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14.5882	−0.618120	−0.309060	−	0.951043i	$-0.600014\pi$
$557$	−14.5882	−0.618120	−0.309060	+	0.951043i	$0.600014\pi$
$558$	0	0
$559$	−36.7973	−1.55636
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0.828427	0.0349140	0.0174570	−	0.999848i	$-0.494443\pi$
$563$	0.828427	0.0349140	0.0174570	+	0.999848i	$0.494443\pi$
$564$	0	0
$565$	−24.2931	−1.02202
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−23.4558	−0.983320	−0.491660	−	0.870787i	$-0.663610\pi$
$569$	−23.4558	−0.983320	−0.491660	+	0.870787i	$0.663610\pi$
$570$	0	0
$571$	23.3137	0.975648	0.487824	−	0.872942i	$-0.337791\pi$
$571$	23.3137	0.975648	0.487824	+	0.872942i	$0.337791\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.73780	0.114174
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	15.9570	0.662010
$582$	0	0
$583$	−10.0625	−0.416748
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.97056	0.205157	0.102579	−	0.994725i	$-0.467291\pi$
$587$	4.97056	0.205157	0.102579	+	0.994725i	$0.467291\pi$
$588$	0	0
$589$	14.2306	0.586361
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.97056	0.121986	0.0609932	−	0.998138i	$-0.480573\pi$
$593$	2.97056	0.121986	0.0609932	+	0.998138i	$0.480573\pi$
$594$	0	0
$595$	50.6274	2.07552
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0.715123	0.0292191	0.0146096	−	0.999893i	$-0.495349\pi$
$599$	0.715123	0.0292191	0.0146096	+	0.999893i	$0.495349\pi$
$600$	0	0
$601$	−4.68629	−0.191158	−0.0955789	−	0.995422i	$-0.530470\pi$
$601$	−4.68629	−0.191158	−0.0955789	+	0.995422i	$0.530470\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−21.4940	−0.873855
$606$	0	0
$607$	−25.1564	−1.02107	−0.510533	−	0.859858i	$-0.670552\pi$
$607$	−25.1564	−1.02107	−0.510533	+	0.859858i	$0.670552\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−12.2843	−0.496968
$612$	0	0
$613$	−10.5682	−0.426846	−0.213423	−	0.976960i	$-0.568461\pi$
$613$	−10.5682	−0.426846	−0.213423	+	0.976960i	$0.568461\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	43.9411	1.76900	0.884502	−	0.466537i	$-0.154499\pi$
$617$	43.9411	1.76900	0.884502	+	0.466537i	$0.154499\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	50.3127	2.01574
$624$	0	0
$625$	−21.2843	−0.851371
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−34.3557	−1.36985
$630$	0	0
$631$	35.2189	1.40204	0.701021	−	0.713140i	$-0.252728\pi$
$631$	35.2189	1.40204	0.701021	+	0.713140i	$0.252728\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−27.8579	−1.10551
$636$	0	0
$637$	53.9751	2.13857
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−32.8284	−1.29664	−0.648322	−	0.761366i	$-0.724529\pi$
$641$	−32.8284	−1.29664	−0.648322	+	0.761366i	$0.724529\pi$
$642$	0	0
$643$	3.51472	0.138607	0.0693035	−	0.997596i	$-0.477922\pi$
$643$	3.51472	0.138607	0.0693035	+	0.997596i	$0.477922\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	44.4182	1.74626	0.873130	−	0.487487i	$-0.162086\pi$
$647$	44.4182	1.74626	0.873130	+	0.487487i	$0.162086\pi$
$648$	0	0
$649$	1.37258	0.0538786
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30.5452	−1.19533	−0.597663	−	0.801747i	$-0.703904\pi$
$653$	−30.5452	−1.19533	−0.597663	+	0.801747i	$0.703904\pi$
$654$	0	0
$655$	−31.9141	−1.24699
$656$	0	0
$657$	0	0
$658$	0	0
$659$	9.65685	0.376178	0.188089	−	0.982152i	$-0.439771\pi$
$659$	9.65685	0.376178	0.188089	+	0.982152i	$0.439771\pi$
$660$	0	0
$661$	−27.2404	−1.05953	−0.529764	−	0.848145i	$-0.677720\pi$
$661$	−27.2404	−1.05953	−0.529764	+	0.848145i	$0.677720\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−29.6569	−1.15004
$666$	0	0
$667$	−33.2548	−1.28763
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.89450	0.227555
$672$	0	0
$673$	37.3137	1.43834	0.719169	−	0.694835i	$-0.244523\pi$
$673$	37.3137	1.43834	0.719169	+	0.694835i	$0.244523\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.2091	0.853566	0.426783	−	0.904354i	$-0.359647\pi$
$677$	22.2091	0.853566	0.426783	+	0.904354i	$0.359647\pi$
$678$	0	0
$679$	−1.72646	−0.0662555
$680$	0	0
$681$	0	0
$682$	0	0
$683$	22.4853	0.860375	0.430188	−	0.902739i	$-0.358447\pi$
$683$	22.4853	0.860375	0.430188	+	0.902739i	$0.358447\pi$
$684$	0	0
$685$	−25.3045	−0.966834
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−35.7990	−1.36383
$690$	0	0
$691$	33.1716	1.26191	0.630953	−	0.775821i	$-0.282664\pi$
$691$	33.1716	1.26191	0.630953	+	0.775821i	$0.282664\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−16.6722	−0.632412
$696$	0	0
$697$	42.6274	1.61463
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−4.52560	−0.170930	−0.0854649	−	0.996341i	$-0.527238\pi$
$701$	−4.52560	−0.170930	−0.0854649	+	0.996341i	$0.527238\pi$
$702$	0	0
$703$	20.1251	0.759032
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−61.1127	−2.29838
$708$	0	0
$709$	−19.6194	−0.736823	−0.368411	−	0.929663i	$-0.620098\pi$
$709$	−19.6194	−0.736823	−0.368411	+	0.929663i	$0.620098\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−20.9706	−0.785354
$714$	0	0
$715$	5.08831	0.190292
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6.60963	0.246497	0.123249	−	0.992376i	$-0.460669\pi$
$719$	6.60963	0.246497	0.123249	+	0.992376i	$0.460669\pi$
$720$	0	0
$721$	−16.6274	−0.619237
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5.24073	−0.194636
$726$	0	0
$727$	36.9454	1.37023	0.685114	−	0.728436i	$-0.259752\pi$
$727$	36.9454	1.37023	0.685114	+	0.728436i	$0.259752\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	60.2843	2.22969
$732$	0	0
$733$	−43.1974	−1.59553	−0.797767	−	0.602966i	$-0.793985\pi$
$733$	−43.1974	−1.59553	−0.797767	+	0.602966i	$0.793985\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.94113	−0.0715023
$738$	0	0
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−17.6835	−0.648745	−0.324373	−	0.945929i	$-0.605153\pi$
$743$	−17.6835	−0.648745	−0.324373	+	0.945929i	$0.605153\pi$
$744$	0	0
$745$	4.34315	0.159121
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−20.1251	−0.735355
$750$	0	0
$751$	−5.03127	−0.183594	−0.0917969	−	0.995778i	$-0.529261\pi$
$751$	−5.03127	−0.183594	−0.0917969	+	0.995778i	$0.529261\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	27.8579	1.01385
$756$	0	0
$757$	17.1778	0.624339	0.312170	−	0.950026i	$-0.398944\pi$
$757$	17.1778	0.624339	0.312170	+	0.950026i	$0.398944\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	42.7696	1.55040	0.775198	−	0.631719i	$-0.217650\pi$
$761$	42.7696	1.55040	0.775198	+	0.631719i	$0.217650\pi$
$762$	0	0
$763$	44.4853	1.61048
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.88317	0.176321
$768$	0	0
$769$	12.0000	0.432731	0.216366	−	0.976312i	$-0.430580\pi$
$769$	12.0000	0.432731	0.216366	+	0.976312i	$0.430580\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	32.2717	1.16073	0.580365	−	0.814356i	$-0.302910\pi$
$773$	32.2717	1.16073	0.580365	+	0.814356i	$0.302910\pi$
$774$	0	0
$775$	−3.30481	−0.118712
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−24.9706	−0.894663
$780$	0	0
$781$	−8.33609	−0.298289
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−32.2010	−1.14930
$786$	0	0
$787$	52.7696	1.88103	0.940516	−	0.339750i	$-0.110343\pi$
$787$	52.7696	1.88103	0.940516	+	0.339750i	$0.110343\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	58.6488	2.08531
$792$	0	0
$793$	20.9706	0.744687
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.5452	1.08197	0.540983	−	0.841033i	$-0.318052\pi$
$797$	30.5452	1.08197	0.540983	+	0.841033i	$0.318052\pi$
$798$	0	0
$799$	20.1251	0.711975
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3.31371	0.116938
$804$	0	0
$805$	43.7031	1.54033
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3.45584	0.121501	0.0607505	−	0.998153i	$-0.480651\pi$
$809$	3.45584	0.121501	0.0607505	+	0.998153i	$0.480651\pi$
$810$	0	0
$811$	49.4558	1.73663	0.868315	−	0.496014i	$-0.165203\pi$
$811$	49.4558	1.73663	0.868315	+	0.496014i	$0.165203\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−37.8086	−1.32438
$816$	0	0
$817$	−35.3137	−1.23547
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−36.4397	−1.27175	−0.635877	−	0.771790i	$-0.719362\pi$
$821$	−36.4397	−1.27175	−0.635877	+	0.771790i	$0.719362\pi$
$822$	0	0
$823$	−13.3674	−0.465957	−0.232978	−	0.972482i	$-0.574847\pi$
$823$	−13.3674	−0.465957	−0.232978	+	0.972482i	$0.574847\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	53.9411	1.87572	0.937858	−	0.347018i	$-0.112806\pi$
$827$	53.9411	1.87572	0.937858	+	0.347018i	$0.112806\pi$
$828$	0	0
$829$	2.94725	0.102362	0.0511811	−	0.998689i	$-0.483701\pi$
$829$	2.94725	0.102362	0.0511811	+	0.998689i	$0.483701\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−88.4264	−3.06379
$834$	0	0
$835$	29.6569	1.02632
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−41.9766	−1.44919	−0.724597	−	0.689172i	$-0.757974\pi$
$839$	−41.9766	−1.44919	−0.724597	+	0.689172i	$0.757974\pi$
$840$	0	0
$841$	34.6569	1.19506
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−8.98986	−0.309261
$846$	0	0
$847$	51.8911	1.78300
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−29.6569	−1.01662
$852$	0	0
$853$	−15.4514	−0.529045	−0.264523	−	0.964379i	$-0.585214\pi$
$853$	−15.4514	−0.529045	−0.264523	+	0.964379i	$0.585214\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−30.4853	−1.04136	−0.520679	−	0.853753i	$-0.674321\pi$
$857$	−30.4853	−1.04136	−0.520679	+	0.853753i	$0.674321\pi$
$858$	0	0
$859$	21.4558	0.732064	0.366032	−	0.930602i	$-0.380716\pi$
$859$	21.4558	0.732064	0.366032	+	0.930602i	$0.380716\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−14.2306	−0.484415	−0.242207	−	0.970224i	$-0.577871\pi$
$863$	−14.2306	−0.484415	−0.242207	+	0.970224i	$0.577871\pi$
$864$	0	0
$865$	13.0294	0.443014
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4.16804	−0.141391
$870$	0	0
$871$	−6.90584	−0.233995
$872$	0	0
$873$	0	0
$874$	0	0
$875$	59.3137	2.00517
$876$	0	0
$877$	22.3572	0.754950	0.377475	−	0.926020i	$-0.376792\pi$
$877$	22.3572	0.754950	0.377475	+	0.926020i	$0.376792\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	38.9706	1.31295	0.656476	−	0.754347i	$-0.272046\pi$
$881$	38.9706	1.31295	0.656476	+	0.754347i	$0.272046\pi$
$882$	0	0
$883$	24.7696	0.833562	0.416781	−	0.909007i	$-0.363158\pi$
$883$	24.7696	0.833562	0.416781	+	0.909007i	$0.363158\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.33609	0.279898	0.139949	−	0.990159i	$-0.455306\pi$
$887$	8.33609	0.279898	0.139949	+	0.990159i	$0.455306\pi$
$888$	0	0
$889$	67.2548	2.25565
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−11.7890	−0.394504
$894$	0	0
$895$	43.7031	1.46083
$896$	0	0
$897$	0	0
$898$	0	0
$899$	40.1421	1.33882
$900$	0	0
$901$	58.6488	1.95388
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.4264	0.612514
$906$	0	0
$907$	−23.7990	−0.790232	−0.395116	−	0.918631i	$-0.629296\pi$
$907$	−23.7990	−0.790232	−0.395116	+	0.918631i	$0.629296\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	34.3557	1.13825	0.569127	−	0.822249i	$-0.307281\pi$
$911$	34.3557	1.13825	0.569127	+	0.822249i	$0.307281\pi$
$912$	0	0
$913$	−2.62742	−0.0869548
$914$	0	0
$915$	0	0
$916$	0	0
$917$	77.0474	2.54433
$918$	0	0
$919$	15.0938	0.497899	0.248950	−	0.968516i	$-0.419915\pi$
$919$	15.0938	0.497899	0.248950	+	0.968516i	$0.419915\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−29.6569	−0.976167
$924$	0	0
$925$	−4.67371	−0.153671
$926$	0	0
$927$	0	0
$928$	0	0
$929$	17.7990	0.583966	0.291983	−	0.956424i	$-0.405685\pi$
$929$	17.7990	0.583966	0.291983	+	0.956424i	$0.405685\pi$
$930$	0	0
$931$	51.7990	1.69764
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.33609	−0.272619
$936$	0	0
$937$	4.62742	0.151171	0.0755856	−	0.997139i	$-0.475917\pi$
$937$	4.62742	0.151171	0.0755856	+	0.997139i	$0.475917\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	51.3854	1.67512	0.837558	−	0.546348i	$-0.183982\pi$
$941$	51.3854	1.67512	0.837558	+	0.546348i	$0.183982\pi$
$942$	0	0
$943$	36.7973	1.19828
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.97056	−0.161522	−0.0807608	−	0.996734i	$-0.525735\pi$
$947$	−4.97056	−0.161522	−0.0807608	+	0.996734i	$0.525735\pi$
$948$	0	0
$949$	11.7890	0.382687
$950$	0	0
$951$	0	0
$952$	0	0
$953$	9.79899	0.317420	0.158710	−	0.987325i	$-0.449266\pi$
$953$	9.79899	0.317420	0.158710	+	0.987325i	$0.449266\pi$
$954$	0	0
$955$	−54.2254	−1.75469
$956$	0	0
$957$	0	0
$958$	0	0
$959$	61.0904	1.97271
$960$	0	0
$961$	−5.68629	−0.183429
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−8.33609	−0.268348
$966$	0	0
$967$	−18.5467	−0.596423	−0.298211	−	0.954500i	$-0.596390\pi$
$967$	−18.5467	−0.596423	−0.298211	+	0.954500i	$0.596390\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−35.4558	−1.13783	−0.568916	−	0.822396i	$-0.692637\pi$
$971$	−35.4558	−1.13783	−0.568916	+	0.822396i	$0.692637\pi$
$972$	0	0
$973$	40.2502	1.29036
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30.7696	−0.984405	−0.492203	−	0.870481i	$-0.663808\pi$
$977$	−30.7696	−0.984405	−0.492203	+	0.870481i	$0.663808\pi$
$978$	0	0
$979$	−8.28427	−0.264766
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.45292	0.110131	0.0550655	−	0.998483i	$-0.482463\pi$
$983$	3.45292	0.110131	0.0550655	+	0.998483i	$0.482463\pi$
$984$	0	0
$985$	−42.6863	−1.36010
$986$	0	0
$987$	0	0
$988$	0	0
$989$	52.0392	1.65475
$990$	0	0
$991$	−28.6093	−0.908804	−0.454402	−	0.890797i	$-0.650147\pi$
$991$	−28.6093	−0.908804	−0.454402	+	0.890797i	$0.650147\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−10.4853	−0.332406
$996$	0	0
$997$	−9.55688	−0.302669	−0.151335	−	0.988483i	$-0.548357\pi$
$997$	−9.55688	−0.302669	−0.151335	+	0.988483i	$0.548357\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.a.t.1.3		4
3.2	odd	2		1536.2.a.n.1.2	yes	4
4.3	odd	2		4608.2.a.ba.1.3		4
8.3	odd	2	inner	4608.2.a.t.1.2		4
8.5	even	2		4608.2.a.ba.1.2		4
12.11	even	2		1536.2.a.m.1.2	✓	4
16.3	odd	4		4608.2.d.p.2305.3		8
16.5	even	4		4608.2.d.p.2305.6		8
16.11	odd	4		4608.2.d.p.2305.5		8
16.13	even	4		4608.2.d.p.2305.4		8
24.5	odd	2		1536.2.a.m.1.3	yes	4
24.11	even	2		1536.2.a.n.1.3	yes	4
48.5	odd	4		1536.2.d.g.769.2		8
48.11	even	4		1536.2.d.g.769.6		8
48.29	odd	4		1536.2.d.g.769.7		8
48.35	even	4		1536.2.d.g.769.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.a.m.1.2	✓	4	12.11	even	2
1536.2.a.m.1.3	yes	4	24.5	odd	2
1536.2.a.n.1.2	yes	4	3.2	odd	2
1536.2.a.n.1.3	yes	4	24.11	even	2
1536.2.d.g.769.2		8	48.5	odd	4
1536.2.d.g.769.3		8	48.35	even	4
1536.2.d.g.769.6		8	48.11	even	4
1536.2.d.g.769.7		8	48.29	odd	4
4608.2.a.t.1.2		4	8.3	odd	2	inner
4608.2.a.t.1.3		4	1.1	even	1	trivial
4608.2.a.ba.1.2		4	8.5	even	2
4608.2.a.ba.1.3		4	4.3	odd	2
4608.2.d.p.2305.3		8	16.3	odd	4
4608.2.d.p.2305.4		8	16.13	even	4
4608.2.d.p.2305.5		8	16.11	odd	4
4608.2.d.p.2305.6		8	16.5	even	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)

\(f(q)\)	\(=\)	\(q+2.08402 q^{5} -5.03127 q^{7} +O(q^{10})\)	\(q+2.08402 q^{5} -5.03127 q^{7} +0.828427 q^{11} +2.94725 q^{13} -4.82843 q^{17} +2.82843 q^{19} -4.16804 q^{23} -0.656854 q^{25} +7.97852 q^{29} +5.03127 q^{31} -10.4853 q^{35} +7.11529 q^{37} -8.82843 q^{41} -12.4853 q^{43} -4.16804 q^{47} +18.3137 q^{49} -12.1466 q^{53} +1.72646 q^{55} +1.65685 q^{59} +7.11529 q^{61} +6.14214 q^{65} -2.34315 q^{67} -10.0625 q^{71} +4.00000 q^{73} -4.16804 q^{77} -5.03127 q^{79} -3.17157 q^{83} -10.0625 q^{85} -10.0000 q^{89} -14.8284 q^{91} +5.89450 q^{95} +0.343146 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 8 q^{11} - 8 q^{17} + 20 q^{25} - 8 q^{35} - 24 q^{41} - 16 q^{43} + 28 q^{49} - 16 q^{59} - 32 q^{65} - 32 q^{67} + 16 q^{73} - 24 q^{83} - 40 q^{89} - 48 q^{91} + 24 q^{97}+O(q^{100}) \) 4 * q - 8 * q^11 - 8 * q^17 + 20 * q^25 - 8 * q^35 - 24 * q^41 - 16 * q^43 + 28 * q^49 - 16 * q^59 - 32 * q^65 - 32 * q^67 + 16 * q^73 - 24 * q^83 - 40 * q^89 - 48 * q^91 + 24 * q^97

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