LMFDB - Embedded newform 2496.4.a.i.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2496,4,Mod(1,2496)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2496.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2496, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2496.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-3,0,20,0,32,0,9,0,50] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$147.268767374$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2496.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} +20.0000 q^{5} +32.0000 q^{7} +9.00000 q^{9} +50.0000 q^{11} +13.0000 q^{13} -60.0000 q^{15} -30.0000 q^{17} -120.000 q^{19} -96.0000 q^{21} +20.0000 q^{23} +275.000 q^{25} -27.0000 q^{27} -82.0000 q^{29} +44.0000 q^{31} -150.000 q^{33} +640.000 q^{35} +306.000 q^{37} -39.0000 q^{39} +108.000 q^{41} -356.000 q^{43} +180.000 q^{45} +178.000 q^{47} +681.000 q^{49} +90.0000 q^{51} -198.000 q^{53} +1000.00 q^{55} +360.000 q^{57} +94.0000 q^{59} +62.0000 q^{61} +288.000 q^{63} +260.000 q^{65} -140.000 q^{67} -60.0000 q^{69} +778.000 q^{71} +62.0000 q^{73} -825.000 q^{75} +1600.00 q^{77} +1096.00 q^{79} +81.0000 q^{81} -462.000 q^{83} -600.000 q^{85} +246.000 q^{87} +1224.00 q^{89} +416.000 q^{91} -132.000 q^{93} -2400.00 q^{95} +614.000 q^{97} +450.000 q^{99} +O(q^{100})$

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	20.0000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	20.0000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	0	0
$7$	32.0000	1.72784	0.863919	−	0.503631i	$-0.168003\pi$
$7$	32.0000	1.72784	0.863919	+	0.503631i	$0.168003\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	50.0000	1.37051	0.685253	−	0.728305i	$-0.259692\pi$
$11$	50.0000	1.37051	0.685253	+	0.728305i	$0.259692\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−60.0000	−1.03280
$16$	0	0
$17$	−30.0000	−0.428004	−0.214002	−	0.976833i	$-0.568650\pi$
$17$	−30.0000	−0.428004	−0.214002	+	0.976833i	$0.568650\pi$
$18$	0	0
$19$	−120.000	−1.44894	−0.724471	−	0.689306i	$-0.757916\pi$
$19$	−120.000	−1.44894	−0.724471	+	0.689306i	$0.757916\pi$
$20$	0	0
$21$	−96.0000	−0.997567
$22$	0	0
$23$	20.0000	0.181317	0.0906584	−	0.995882i	$-0.471103\pi$
$23$	20.0000	0.181317	0.0906584	+	0.995882i	$0.471103\pi$
$24$	0	0
$25$	275.000	2.20000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−82.0000	−0.525070	−0.262535	−	0.964923i	$-0.584558\pi$
$29$	−82.0000	−0.525070	−0.262535	+	0.964923i	$0.584558\pi$
$30$	0	0
$31$	44.0000	0.254924	0.127462	−	0.991843i	$-0.459317\pi$
$31$	44.0000	0.254924	0.127462	+	0.991843i	$0.459317\pi$
$32$	0	0
$33$	−150.000	−0.791262
$34$	0	0
$35$	640.000	3.09085
$36$	0	0
$37$	306.000	1.35962	0.679812	−	0.733386i	$-0.262061\pi$
$37$	306.000	1.35962	0.679812	+	0.733386i	$0.262061\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	108.000	0.411385	0.205692	−	0.978617i	$-0.434055\pi$
$41$	108.000	0.411385	0.205692	+	0.978617i	$0.434055\pi$
$42$	0	0
$43$	−356.000	−1.26255	−0.631273	−	0.775561i	$-0.717467\pi$
$43$	−356.000	−1.26255	−0.631273	+	0.775561i	$0.717467\pi$
$44$	0	0
$45$	180.000	0.596285
$46$	0	0
$47$	178.000	0.552425	0.276212	−	0.961097i	$-0.410921\pi$
$47$	178.000	0.552425	0.276212	+	0.961097i	$0.410921\pi$
$48$	0	0
$49$	681.000	1.98542
$50$	0	0
$51$	90.0000	0.247108
$52$	0	0
$53$	−198.000	−0.513158	−0.256579	−	0.966523i	$-0.582595\pi$
$53$	−198.000	−0.513158	−0.256579	+	0.966523i	$0.582595\pi$
$54$	0	0
$55$	1000.00	2.45164
$56$	0	0
$57$	360.000	0.836547
$58$	0	0
$59$	94.0000	0.207420	0.103710	−	0.994608i	$-0.466929\pi$
$59$	94.0000	0.207420	0.103710	+	0.994608i	$0.466929\pi$
$60$	0	0
$61$	62.0000	0.130136	0.0650679	−	0.997881i	$-0.479274\pi$
$61$	62.0000	0.130136	0.0650679	+	0.997881i	$0.479274\pi$
$62$	0	0
$63$	288.000	0.575946
$64$	0	0
$65$	260.000	0.496139
$66$	0	0
$67$	−140.000	−0.255279	−0.127640	−	0.991821i	$-0.540740\pi$
$67$	−140.000	−0.255279	−0.127640	+	0.991821i	$0.540740\pi$
$68$	0	0
$69$	−60.0000	−0.104683
$70$	0	0
$71$	778.000	1.30045	0.650223	−	0.759744i	$-0.274676\pi$
$71$	778.000	1.30045	0.650223	+	0.759744i	$0.274676\pi$
$72$	0	0
$73$	62.0000	0.0994048	0.0497024	−	0.998764i	$-0.484173\pi$
$73$	62.0000	0.0994048	0.0497024	+	0.998764i	$0.484173\pi$
$74$	0	0
$75$	−825.000	−1.27017
$76$	0	0
$77$	1600.00	2.36801
$78$	0	0
$79$	1096.00	1.56088	0.780441	−	0.625230i	$-0.214995\pi$
$79$	1096.00	1.56088	0.780441	+	0.625230i	$0.214995\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−462.000	−0.610977	−0.305488	−	0.952196i	$-0.598820\pi$
$83$	−462.000	−0.610977	−0.305488	+	0.952196i	$0.598820\pi$
$84$	0	0
$85$	−600.000	−0.765637
$86$	0	0
$87$	246.000	0.303149
$88$	0	0
$89$	1224.00	1.45779	0.728897	−	0.684623i	$-0.240033\pi$
$89$	1224.00	1.45779	0.728897	+	0.684623i	$0.240033\pi$
$90$	0	0
$91$	416.000	0.479216
$92$	0	0
$93$	−132.000	−0.147180
$94$	0	0
$95$	−2400.00	−2.59195
$96$	0	0
$97$	614.000	0.642704	0.321352	−	0.946960i	$-0.395863\pi$
$97$	614.000	0.642704	0.321352	+	0.946960i	$0.395863\pi$
$98$	0	0
$99$	450.000	0.456835
$100$	0	0
$101$	−1058.00	−1.04233	−0.521163	−	0.853457i	$-0.674502\pi$
$101$	−1058.00	−1.04233	−0.521163	+	0.853457i	$0.674502\pi$
$102$	0	0
$103$	−1768.00	−1.69132	−0.845661	−	0.533720i	$-0.820794\pi$
$103$	−1768.00	−1.69132	−0.845661	+	0.533720i	$0.820794\pi$
$104$	0	0
$105$	−1920.00	−1.78450
$106$	0	0
$107$	−1808.00	−1.63351	−0.816757	−	0.576982i	$-0.804230\pi$
$107$	−1808.00	−1.63351	−0.816757	+	0.576982i	$0.804230\pi$
$108$	0	0
$109$	1886.00	1.65730	0.828652	−	0.559765i	$-0.189109\pi$
$109$	1886.00	1.65730	0.828652	+	0.559765i	$0.189109\pi$
$110$	0	0
$111$	−918.000	−0.784979
$112$	0	0
$113$	1246.00	1.03729	0.518645	−	0.854990i	$-0.326437\pi$
$113$	1246.00	1.03729	0.518645	+	0.854990i	$0.326437\pi$
$114$	0	0
$115$	400.000	0.324349
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	−960.000	−0.739521
$120$	0	0
$121$	1169.00	0.878287
$122$	0	0
$123$	−324.000	−0.237513
$124$	0	0
$125$	3000.00	2.14663
$126$	0	0
$127$	−1624.00	−1.13470	−0.567349	−	0.823477i	$-0.692031\pi$
$127$	−1624.00	−1.13470	−0.567349	+	0.823477i	$0.692031\pi$
$128$	0	0
$129$	1068.00	0.728931
$130$	0	0
$131$	−2072.00	−1.38192	−0.690960	−	0.722893i	$-0.742812\pi$
$131$	−2072.00	−1.38192	−0.690960	+	0.722893i	$0.742812\pi$
$132$	0	0
$133$	−3840.00	−2.50354
$134$	0	0
$135$	−540.000	−0.344265
$136$	0	0
$137$	−756.000	−0.471456	−0.235728	−	0.971819i	$-0.575747\pi$
$137$	−756.000	−0.471456	−0.235728	+	0.971819i	$0.575747\pi$
$138$	0	0
$139$	172.000	0.104956	0.0524779	−	0.998622i	$-0.483288\pi$
$139$	172.000	0.104956	0.0524779	+	0.998622i	$0.483288\pi$
$140$	0	0
$141$	−534.000	−0.318943
$142$	0	0
$143$	650.000	0.380110
$144$	0	0
$145$	−1640.00	−0.939273
$146$	0	0
$147$	−2043.00	−1.14628
$148$	0	0
$149$	−1272.00	−0.699371	−0.349686	−	0.936867i	$-0.613712\pi$
$149$	−1272.00	−0.699371	−0.349686	+	0.936867i	$0.613712\pi$
$150$	0	0
$151$	−1404.00	−0.756662	−0.378331	−	0.925670i	$-0.623502\pi$
$151$	−1404.00	−0.756662	−0.378331	+	0.925670i	$0.623502\pi$
$152$	0	0
$153$	−270.000	−0.142668
$154$	0	0
$155$	880.000	0.456021
$156$	0	0
$157$	2170.00	1.10309	0.551544	−	0.834146i	$-0.314039\pi$
$157$	2170.00	1.10309	0.551544	+	0.834146i	$0.314039\pi$
$158$	0	0
$159$	594.000	0.296272
$160$	0	0
$161$	640.000	0.313286
$162$	0	0
$163$	248.000	0.119171	0.0595855	−	0.998223i	$-0.481022\pi$
$163$	248.000	0.119171	0.0595855	+	0.998223i	$0.481022\pi$
$164$	0	0
$165$	−3000.00	−1.41545
$166$	0	0
$167$	−102.000	−0.0472635	−0.0236317	−	0.999721i	$-0.507523\pi$
$167$	−102.000	−0.0472635	−0.0236317	+	0.999721i	$0.507523\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−1080.00	−0.482980
$172$	0	0
$173$	−682.000	−0.299720	−0.149860	−	0.988707i	$-0.547882\pi$
$173$	−682.000	−0.299720	−0.149860	+	0.988707i	$0.547882\pi$
$174$	0	0
$175$	8800.00	3.80124
$176$	0	0
$177$	−282.000	−0.119754
$178$	0	0
$179$	−612.000	−0.255548	−0.127774	−	0.991803i	$-0.540783\pi$
$179$	−612.000	−0.255548	−0.127774	+	0.991803i	$0.540783\pi$
$180$	0	0
$181$	66.0000	0.0271035	0.0135518	−	0.999908i	$-0.495686\pi$
$181$	66.0000	0.0271035	0.0135518	+	0.999908i	$0.495686\pi$
$182$	0	0
$183$	−186.000	−0.0751340
$184$	0	0
$185$	6120.00	2.43217
$186$	0	0
$187$	−1500.00	−0.586582
$188$	0	0
$189$	−864.000	−0.332522
$190$	0	0
$191$	−608.000	−0.230332	−0.115166	−	0.993346i	$-0.536740\pi$
$191$	−608.000	−0.230332	−0.115166	+	0.993346i	$0.536740\pi$
$192$	0	0
$193$	1370.00	0.510957	0.255479	−	0.966815i	$-0.417767\pi$
$193$	1370.00	0.510957	0.255479	+	0.966815i	$0.417767\pi$
$194$	0	0
$195$	−780.000	−0.286446
$196$	0	0
$197$	4908.00	1.77503	0.887514	−	0.460781i	$-0.152431\pi$
$197$	4908.00	1.77503	0.887514	+	0.460781i	$0.152431\pi$
$198$	0	0
$199$	328.000	0.116841	0.0584204	−	0.998292i	$-0.481394\pi$
$199$	328.000	0.116841	0.0584204	+	0.998292i	$0.481394\pi$
$200$	0	0
$201$	420.000	0.147386
$202$	0	0
$203$	−2624.00	−0.907235
$204$	0	0
$205$	2160.00	0.735907
$206$	0	0
$207$	180.000	0.0604390
$208$	0	0
$209$	−6000.00	−1.98578
$210$	0	0
$211$	1316.00	0.429371	0.214685	−	0.976683i	$-0.431127\pi$
$211$	1316.00	0.429371	0.214685	+	0.976683i	$0.431127\pi$
$212$	0	0
$213$	−2334.00	−0.750812
$214$	0	0
$215$	−7120.00	−2.25851
$216$	0	0
$217$	1408.00	0.440467
$218$	0	0
$219$	−186.000	−0.0573914
$220$	0	0
$221$	−390.000	−0.118707
$222$	0	0
$223$	1932.00	0.580163	0.290081	−	0.957002i	$-0.406318\pi$
$223$	1932.00	0.580163	0.290081	+	0.957002i	$0.406318\pi$
$224$	0	0
$225$	2475.00	0.733333
$226$	0	0
$227$	4998.00	1.46136	0.730680	−	0.682720i	$-0.239203\pi$
$227$	4998.00	1.46136	0.730680	+	0.682720i	$0.239203\pi$
$228$	0	0
$229$	78.0000	0.0225082	0.0112541	−	0.999937i	$-0.496418\pi$
$229$	78.0000	0.0225082	0.0112541	+	0.999937i	$0.496418\pi$
$230$	0	0
$231$	−4800.00	−1.36717
$232$	0	0
$233$	−1282.00	−0.360458	−0.180229	−	0.983625i	$-0.557684\pi$
$233$	−1282.00	−0.360458	−0.180229	+	0.983625i	$0.557684\pi$
$234$	0	0
$235$	3560.00	0.988208
$236$	0	0
$237$	−3288.00	−0.901175
$238$	0	0
$239$	−294.000	−0.0795702	−0.0397851	−	0.999208i	$-0.512667\pi$
$239$	−294.000	−0.0795702	−0.0397851	+	0.999208i	$0.512667\pi$
$240$	0	0
$241$	−4962.00	−1.32627	−0.663134	−	0.748501i	$-0.730774\pi$
$241$	−4962.00	−1.32627	−0.663134	+	0.748501i	$0.730774\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	13620.0	3.55163
$246$	0	0
$247$	−1560.00	−0.401864
$248$	0	0
$249$	1386.00	0.352748
$250$	0	0
$251$	744.000	0.187095	0.0935475	−	0.995615i	$-0.470179\pi$
$251$	744.000	0.187095	0.0935475	+	0.995615i	$0.470179\pi$
$252$	0	0
$253$	1000.00	0.248496
$254$	0	0
$255$	1800.00	0.442041
$256$	0	0
$257$	−1026.00	−0.249028	−0.124514	−	0.992218i	$-0.539737\pi$
$257$	−1026.00	−0.249028	−0.124514	+	0.992218i	$0.539737\pi$
$258$	0	0
$259$	9792.00	2.34921
$260$	0	0
$261$	−738.000	−0.175023
$262$	0	0
$263$	5532.00	1.29703	0.648513	−	0.761204i	$-0.275391\pi$
$263$	5532.00	1.29703	0.648513	+	0.761204i	$0.275391\pi$
$264$	0	0
$265$	−3960.00	−0.917966
$266$	0	0
$267$	−3672.00	−0.841658
$268$	0	0
$269$	3534.00	0.801010	0.400505	−	0.916294i	$-0.368835\pi$
$269$	3534.00	0.801010	0.400505	+	0.916294i	$0.368835\pi$
$270$	0	0
$271$	−2392.00	−0.536176	−0.268088	−	0.963394i	$-0.586392\pi$
$271$	−2392.00	−0.536176	−0.268088	+	0.963394i	$0.586392\pi$
$272$	0	0
$273$	−1248.00	−0.276675
$274$	0	0
$275$	13750.0	3.01511
$276$	0	0
$277$	−6102.00	−1.32359	−0.661794	−	0.749686i	$-0.730204\pi$
$277$	−6102.00	−1.32359	−0.661794	+	0.749686i	$0.730204\pi$
$278$	0	0
$279$	396.000	0.0849746
$280$	0	0
$281$	−7540.00	−1.60071	−0.800354	−	0.599528i	$-0.795355\pi$
$281$	−7540.00	−1.60071	−0.800354	+	0.599528i	$0.795355\pi$
$282$	0	0
$283$	−2756.00	−0.578895	−0.289447	−	0.957194i	$-0.593472\pi$
$283$	−2756.00	−0.578895	−0.289447	+	0.957194i	$0.593472\pi$
$284$	0	0
$285$	7200.00	1.49646
$286$	0	0
$287$	3456.00	0.710806
$288$	0	0
$289$	−4013.00	−0.816813
$290$	0	0
$291$	−1842.00	−0.371065
$292$	0	0
$293$	−968.000	−0.193007	−0.0965037	−	0.995333i	$-0.530766\pi$
$293$	−968.000	−0.193007	−0.0965037	+	0.995333i	$0.530766\pi$
$294$	0	0
$295$	1880.00	0.371043
$296$	0	0
$297$	−1350.00	−0.263754
$298$	0	0
$299$	260.000	0.0502883
$300$	0	0
$301$	−11392.0	−2.18147
$302$	0	0
$303$	3174.00	0.601787
$304$	0	0
$305$	1240.00	0.232794
$306$	0	0
$307$	−6436.00	−1.19649	−0.598244	−	0.801314i	$-0.704135\pi$
$307$	−6436.00	−1.19649	−0.598244	+	0.801314i	$0.704135\pi$
$308$	0	0
$309$	5304.00	0.976485
$310$	0	0
$311$	−7932.00	−1.44625	−0.723123	−	0.690719i	$-0.757294\pi$
$311$	−7932.00	−1.44625	−0.723123	+	0.690719i	$0.757294\pi$
$312$	0	0
$313$	10358.0	1.87051	0.935254	−	0.353978i	$-0.115171\pi$
$313$	10358.0	1.87051	0.935254	+	0.353978i	$0.115171\pi$
$314$	0	0
$315$	5760.00	1.03028
$316$	0	0
$317$	2820.00	0.499643	0.249822	−	0.968292i	$-0.419628\pi$
$317$	2820.00	0.499643	0.249822	+	0.968292i	$0.419628\pi$
$318$	0	0
$319$	−4100.00	−0.719611
$320$	0	0
$321$	5424.00	0.943110
$322$	0	0
$323$	3600.00	0.620153
$324$	0	0
$325$	3575.00	0.610170
$326$	0	0
$327$	−5658.00	−0.956844
$328$	0	0
$329$	5696.00	0.954500
$330$	0	0
$331$	−4180.00	−0.694120	−0.347060	−	0.937843i	$-0.612820\pi$
$331$	−4180.00	−0.694120	−0.347060	+	0.937843i	$0.612820\pi$
$332$	0	0
$333$	2754.00	0.453208
$334$	0	0
$335$	−2800.00	−0.456658
$336$	0	0
$337$	−5026.00	−0.812414	−0.406207	−	0.913781i	$-0.633149\pi$
$337$	−5026.00	−0.812414	−0.406207	+	0.913781i	$0.633149\pi$
$338$	0	0
$339$	−3738.00	−0.598880
$340$	0	0
$341$	2200.00	0.349374
$342$	0	0
$343$	10816.0	1.70265
$344$	0	0
$345$	−1200.00	−0.187263
$346$	0	0
$347$	−7332.00	−1.13430	−0.567150	−	0.823614i	$-0.691954\pi$
$347$	−7332.00	−1.13430	−0.567150	+	0.823614i	$0.691954\pi$
$348$	0	0
$349$	8162.00	1.25187	0.625934	−	0.779876i	$-0.284718\pi$
$349$	8162.00	1.25187	0.625934	+	0.779876i	$0.284718\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	1244.00	0.187568	0.0937839	−	0.995593i	$-0.470104\pi$
$353$	1244.00	0.187568	0.0937839	+	0.995593i	$0.470104\pi$
$354$	0	0
$355$	15560.0	2.32631
$356$	0	0
$357$	2880.00	0.426963
$358$	0	0
$359$	−9558.00	−1.40516	−0.702579	−	0.711605i	$-0.747968\pi$
$359$	−9558.00	−1.40516	−0.702579	+	0.711605i	$0.747968\pi$
$360$	0	0
$361$	7541.00	1.09943
$362$	0	0
$363$	−3507.00	−0.507079
$364$	0	0
$365$	1240.00	0.177821
$366$	0	0
$367$	11032.0	1.56912	0.784558	−	0.620055i	$-0.212890\pi$
$367$	11032.0	1.56912	0.784558	+	0.620055i	$0.212890\pi$
$368$	0	0
$369$	972.000	0.137128
$370$	0	0
$371$	−6336.00	−0.886654
$372$	0	0
$373$	−5474.00	−0.759874	−0.379937	−	0.925012i	$-0.624054\pi$
$373$	−5474.00	−0.759874	−0.379937	+	0.925012i	$0.624054\pi$
$374$	0	0
$375$	−9000.00	−1.23935
$376$	0	0
$377$	−1066.00	−0.145628
$378$	0	0
$379$	−7040.00	−0.954144	−0.477072	−	0.878864i	$-0.658302\pi$
$379$	−7040.00	−0.954144	−0.477072	+	0.878864i	$0.658302\pi$
$380$	0	0
$381$	4872.00	0.655118
$382$	0	0
$383$	1830.00	0.244148	0.122074	−	0.992521i	$-0.461045\pi$
$383$	1830.00	0.244148	0.122074	+	0.992521i	$0.461045\pi$
$384$	0	0
$385$	32000.0	4.23603
$386$	0	0
$387$	−3204.00	−0.420849
$388$	0	0
$389$	−10158.0	−1.32399	−0.661994	−	0.749509i	$-0.730289\pi$
$389$	−10158.0	−1.32399	−0.661994	+	0.749509i	$0.730289\pi$
$390$	0	0
$391$	−600.000	−0.0776044
$392$	0	0
$393$	6216.00	0.797852
$394$	0	0
$395$	21920.0	2.79219
$396$	0	0
$397$	12658.0	1.60022	0.800109	−	0.599854i	$-0.204775\pi$
$397$	12658.0	1.60022	0.800109	+	0.599854i	$0.204775\pi$
$398$	0	0
$399$	11520.0	1.44542
$400$	0	0
$401$	15720.0	1.95765	0.978827	−	0.204689i	$-0.0656182\pi$
$401$	15720.0	1.95765	0.978827	+	0.204689i	$0.0656182\pi$
$402$	0	0
$403$	572.000	0.0707031
$404$	0	0
$405$	1620.00	0.198762
$406$	0	0
$407$	15300.0	1.86337
$408$	0	0
$409$	7654.00	0.925345	0.462672	−	0.886529i	$-0.346891\pi$
$409$	7654.00	0.925345	0.462672	+	0.886529i	$0.346891\pi$
$410$	0	0
$411$	2268.00	0.272195
$412$	0	0
$413$	3008.00	0.358387
$414$	0	0
$415$	−9240.00	−1.09295
$416$	0	0
$417$	−516.000	−0.0605962
$418$	0	0
$419$	−1848.00	−0.215467	−0.107734	−	0.994180i	$-0.534359\pi$
$419$	−1848.00	−0.215467	−0.107734	+	0.994180i	$0.534359\pi$
$420$	0	0
$421$	12542.0	1.45192	0.725962	−	0.687735i	$-0.241395\pi$
$421$	12542.0	1.45192	0.725962	+	0.687735i	$0.241395\pi$
$422$	0	0
$423$	1602.00	0.184142
$424$	0	0
$425$	−8250.00	−0.941609
$426$	0	0
$427$	1984.00	0.224854
$428$	0	0
$429$	−1950.00	−0.219457
$430$	0	0
$431$	5238.00	0.585396	0.292698	−	0.956205i	$-0.405447\pi$
$431$	5238.00	0.585396	0.292698	+	0.956205i	$0.405447\pi$
$432$	0	0
$433$	−8258.00	−0.916522	−0.458261	−	0.888818i	$-0.651528\pi$
$433$	−8258.00	−0.916522	−0.458261	+	0.888818i	$0.651528\pi$
$434$	0	0
$435$	4920.00	0.542290
$436$	0	0
$437$	−2400.00	−0.262718
$438$	0	0
$439$	6304.00	0.685361	0.342681	−	0.939452i	$-0.388665\pi$
$439$	6304.00	0.685361	0.342681	+	0.939452i	$0.388665\pi$
$440$	0	0
$441$	6129.00	0.661808
$442$	0	0
$443$	12744.0	1.36678	0.683392	−	0.730051i	$-0.260504\pi$
$443$	12744.0	1.36678	0.683392	+	0.730051i	$0.260504\pi$
$444$	0	0
$445$	24480.0	2.60778
$446$	0	0
$447$	3816.00	0.403782
$448$	0	0
$449$	−11776.0	−1.23774	−0.618868	−	0.785495i	$-0.712409\pi$
$449$	−11776.0	−1.23774	−0.618868	+	0.785495i	$0.712409\pi$
$450$	0	0
$451$	5400.00	0.563805
$452$	0	0
$453$	4212.00	0.436859
$454$	0	0
$455$	8320.00	0.857248
$456$	0	0
$457$	2134.00	0.218434	0.109217	−	0.994018i	$-0.465166\pi$
$457$	2134.00	0.218434	0.109217	+	0.994018i	$0.465166\pi$
$458$	0	0
$459$	810.000	0.0823694
$460$	0	0
$461$	−2724.00	−0.275205	−0.137602	−	0.990488i	$-0.543940\pi$
$461$	−2724.00	−0.275205	−0.137602	+	0.990488i	$0.543940\pi$
$462$	0	0
$463$	5648.00	0.566922	0.283461	−	0.958984i	$-0.408517\pi$
$463$	5648.00	0.566922	0.283461	+	0.958984i	$0.408517\pi$
$464$	0	0
$465$	−2640.00	−0.263284
$466$	0	0
$467$	−18224.0	−1.80579	−0.902897	−	0.429856i	$-0.858564\pi$
$467$	−18224.0	−1.80579	−0.902897	+	0.429856i	$0.858564\pi$
$468$	0	0
$469$	−4480.00	−0.441081
$470$	0	0
$471$	−6510.00	−0.636868
$472$	0	0
$473$	−17800.0	−1.73033
$474$	0	0
$475$	−33000.0	−3.18767
$476$	0	0
$477$	−1782.00	−0.171053
$478$	0	0
$479$	−9066.00	−0.864794	−0.432397	−	0.901683i	$-0.642332\pi$
$479$	−9066.00	−0.864794	−0.432397	+	0.901683i	$0.642332\pi$
$480$	0	0
$481$	3978.00	0.377092
$482$	0	0
$483$	−1920.00	−0.180876
$484$	0	0
$485$	12280.0	1.14970
$486$	0	0
$487$	−8948.00	−0.832593	−0.416296	−	0.909229i	$-0.636672\pi$
$487$	−8948.00	−0.832593	−0.416296	+	0.909229i	$0.636672\pi$
$488$	0	0
$489$	−744.000	−0.0688034
$490$	0	0
$491$	8720.00	0.801483	0.400741	−	0.916191i	$-0.368753\pi$
$491$	8720.00	0.801483	0.400741	+	0.916191i	$0.368753\pi$
$492$	0	0
$493$	2460.00	0.224732
$494$	0	0
$495$	9000.00	0.817212
$496$	0	0
$497$	24896.0	2.24696
$498$	0	0
$499$	6604.00	0.592456	0.296228	−	0.955117i	$-0.404271\pi$
$499$	6604.00	0.592456	0.296228	+	0.955117i	$0.404271\pi$
$500$	0	0
$501$	306.000	0.0272876
$502$	0	0
$503$	−3404.00	−0.301743	−0.150872	−	0.988553i	$-0.548208\pi$
$503$	−3404.00	−0.301743	−0.150872	+	0.988553i	$0.548208\pi$
$504$	0	0
$505$	−21160.0	−1.86457
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	76.0000	0.00661815	0.00330908	−	0.999995i	$-0.498947\pi$
$509$	76.0000	0.00661815	0.00330908	+	0.999995i	$0.498947\pi$
$510$	0	0
$511$	1984.00	0.171755
$512$	0	0
$513$	3240.00	0.278849
$514$	0	0
$515$	−35360.0	−3.02553
$516$	0	0
$517$	8900.00	0.757102
$518$	0	0
$519$	2046.00	0.173043
$520$	0	0
$521$	12054.0	1.01362	0.506809	−	0.862058i	$-0.330825\pi$
$521$	12054.0	1.01362	0.506809	+	0.862058i	$0.330825\pi$
$522$	0	0
$523$	276.000	0.0230758	0.0115379	−	0.999933i	$-0.496327\pi$
$523$	276.000	0.0230758	0.0115379	+	0.999933i	$0.496327\pi$
$524$	0	0
$525$	−26400.0	−2.19465
$526$	0	0
$527$	−1320.00	−0.109108
$528$	0	0
$529$	−11767.0	−0.967124
$530$	0	0
$531$	846.000	0.0691399
$532$	0	0
$533$	1404.00	0.114098
$534$	0	0
$535$	−36160.0	−2.92212
$536$	0	0
$537$	1836.00	0.147540
$538$	0	0
$539$	34050.0	2.72103
$540$	0	0
$541$	−13778.0	−1.09494	−0.547470	−	0.836825i	$-0.684409\pi$
$541$	−13778.0	−1.09494	−0.547470	+	0.836825i	$0.684409\pi$
$542$	0	0
$543$	−198.000	−0.0156482
$544$	0	0
$545$	37720.0	2.96467
$546$	0	0
$547$	−10844.0	−0.847634	−0.423817	−	0.905748i	$-0.639310\pi$
$547$	−10844.0	−0.847634	−0.423817	+	0.905748i	$0.639310\pi$
$548$	0	0
$549$	558.000	0.0433786
$550$	0	0
$551$	9840.00	0.760795
$552$	0	0
$553$	35072.0	2.69695
$554$	0	0
$555$	−18360.0	−1.40421
$556$	0	0
$557$	−20544.0	−1.56280	−0.781398	−	0.624033i	$-0.785493\pi$
$557$	−20544.0	−1.56280	−0.781398	+	0.624033i	$0.785493\pi$
$558$	0	0
$559$	−4628.00	−0.350167
$560$	0	0
$561$	4500.00	0.338663
$562$	0	0
$563$	6988.00	0.523107	0.261553	−	0.965189i	$-0.415765\pi$
$563$	6988.00	0.523107	0.261553	+	0.965189i	$0.415765\pi$
$564$	0	0
$565$	24920.0	1.85556
$566$	0	0
$567$	2592.00	0.191982
$568$	0	0
$569$	−706.000	−0.0520159	−0.0260080	−	0.999662i	$-0.508280\pi$
$569$	−706.000	−0.0520159	−0.0260080	+	0.999662i	$0.508280\pi$
$570$	0	0
$571$	−17532.0	−1.28492	−0.642462	−	0.766318i	$-0.722087\pi$
$571$	−17532.0	−1.28492	−0.642462	+	0.766318i	$0.722087\pi$
$572$	0	0
$573$	1824.00	0.132982
$574$	0	0
$575$	5500.00	0.398897
$576$	0	0
$577$	−14814.0	−1.06883	−0.534415	−	0.845222i	$-0.679468\pi$
$577$	−14814.0	−1.06883	−0.534415	+	0.845222i	$0.679468\pi$
$578$	0	0
$579$	−4110.00	−0.295001
$580$	0	0
$581$	−14784.0	−1.05567
$582$	0	0
$583$	−9900.00	−0.703287
$584$	0	0
$585$	2340.00	0.165380
$586$	0	0
$587$	14170.0	0.996352	0.498176	−	0.867076i	$-0.334003\pi$
$587$	14170.0	0.996352	0.498176	+	0.867076i	$0.334003\pi$
$588$	0	0
$589$	−5280.00	−0.369369
$590$	0	0
$591$	−14724.0	−1.02481
$592$	0	0
$593$	−11744.0	−0.813269	−0.406634	−	0.913591i	$-0.633298\pi$
$593$	−11744.0	−0.813269	−0.406634	+	0.913591i	$0.633298\pi$
$594$	0	0
$595$	−19200.0	−1.32290
$596$	0	0
$597$	−984.000	−0.0674580
$598$	0	0
$599$	15076.0	1.02836	0.514181	−	0.857682i	$-0.328096\pi$
$599$	15076.0	1.02836	0.514181	+	0.857682i	$0.328096\pi$
$600$	0	0
$601$	20230.0	1.37304	0.686522	−	0.727109i	$-0.259137\pi$
$601$	20230.0	1.37304	0.686522	+	0.727109i	$0.259137\pi$
$602$	0	0
$603$	−1260.00	−0.0850931
$604$	0	0
$605$	23380.0	1.57113
$606$	0	0
$607$	28056.0	1.87604	0.938021	−	0.346577i	$-0.112656\pi$
$607$	28056.0	1.87604	0.938021	+	0.346577i	$0.112656\pi$
$608$	0	0
$609$	7872.00	0.523792
$610$	0	0
$611$	2314.00	0.153215
$612$	0	0
$613$	−27446.0	−1.80837	−0.904187	−	0.427136i	$-0.859522\pi$
$613$	−27446.0	−1.80837	−0.904187	+	0.427136i	$0.859522\pi$
$614$	0	0
$615$	−6480.00	−0.424876
$616$	0	0
$617$	8804.00	0.574450	0.287225	−	0.957863i	$-0.407267\pi$
$617$	8804.00	0.574450	0.287225	+	0.957863i	$0.407267\pi$
$618$	0	0
$619$	3508.00	0.227784	0.113892	−	0.993493i	$-0.463668\pi$
$619$	3508.00	0.227784	0.113892	+	0.993493i	$0.463668\pi$
$620$	0	0
$621$	−540.000	−0.0348945
$622$	0	0
$623$	39168.0	2.51883
$624$	0	0
$625$	25625.0	1.64000
$626$	0	0
$627$	18000.0	1.14649
$628$	0	0
$629$	−9180.00	−0.581925
$630$	0	0
$631$	−22084.0	−1.39326	−0.696632	−	0.717428i	$-0.745319\pi$
$631$	−22084.0	−1.39326	−0.696632	+	0.717428i	$0.745319\pi$
$632$	0	0
$633$	−3948.00	−0.247897
$634$	0	0
$635$	−32480.0	−2.02981
$636$	0	0
$637$	8853.00	0.550657
$638$	0	0
$639$	7002.00	0.433482
$640$	0	0
$641$	−7342.00	−0.452405	−0.226202	−	0.974080i	$-0.572631\pi$
$641$	−7342.00	−0.452405	−0.226202	+	0.974080i	$0.572631\pi$
$642$	0	0
$643$	2996.00	0.183749	0.0918746	−	0.995771i	$-0.470714\pi$
$643$	2996.00	0.183749	0.0918746	+	0.995771i	$0.470714\pi$
$644$	0	0
$645$	21360.0	1.30395
$646$	0	0
$647$	−9344.00	−0.567775	−0.283888	−	0.958858i	$-0.591624\pi$
$647$	−9344.00	−0.567775	−0.283888	+	0.958858i	$0.591624\pi$
$648$	0	0
$649$	4700.00	0.284270
$650$	0	0
$651$	−4224.00	−0.254304
$652$	0	0
$653$	16686.0	0.999960	0.499980	−	0.866037i	$-0.333341\pi$
$653$	16686.0	0.999960	0.499980	+	0.866037i	$0.333341\pi$
$654$	0	0
$655$	−41440.0	−2.47205
$656$	0	0
$657$	558.000	0.0331349
$658$	0	0
$659$	31356.0	1.85350	0.926750	−	0.375679i	$-0.122590\pi$
$659$	31356.0	1.85350	0.926750	+	0.375679i	$0.122590\pi$
$660$	0	0
$661$	−590.000	−0.0347176	−0.0173588	−	0.999849i	$-0.505526\pi$
$661$	−590.000	−0.0347176	−0.0173588	+	0.999849i	$0.505526\pi$
$662$	0	0
$663$	1170.00	0.0685355
$664$	0	0
$665$	−76800.0	−4.47846
$666$	0	0
$667$	−1640.00	−0.0952040
$668$	0	0
$669$	−5796.00	−0.334957
$670$	0	0
$671$	3100.00	0.178352
$672$	0	0
$673$	5938.00	0.340109	0.170054	−	0.985435i	$-0.445606\pi$
$673$	5938.00	0.340109	0.170054	+	0.985435i	$0.445606\pi$
$674$	0	0
$675$	−7425.00	−0.423390
$676$	0	0
$677$	−9486.00	−0.538518	−0.269259	−	0.963068i	$-0.586779\pi$
$677$	−9486.00	−0.538518	−0.269259	+	0.963068i	$0.586779\pi$
$678$	0	0
$679$	19648.0	1.11049
$680$	0	0
$681$	−14994.0	−0.843717
$682$	0	0
$683$	26162.0	1.46568	0.732841	−	0.680400i	$-0.238194\pi$
$683$	26162.0	1.46568	0.732841	+	0.680400i	$0.238194\pi$
$684$	0	0
$685$	−15120.0	−0.843366
$686$	0	0
$687$	−234.000	−0.0129951
$688$	0	0
$689$	−2574.00	−0.142325
$690$	0	0
$691$	−17348.0	−0.955064	−0.477532	−	0.878614i	$-0.658468\pi$
$691$	−17348.0	−0.955064	−0.477532	+	0.878614i	$0.658468\pi$
$692$	0	0
$693$	14400.0	0.789337
$694$	0	0
$695$	3440.00	0.187751
$696$	0	0
$697$	−3240.00	−0.176074
$698$	0	0
$699$	3846.00	0.208110
$700$	0	0
$701$	−30.0000	−0.00161638	−0.000808191	−	1.00000i	$-0.500257\pi$
$701$	−30.0000	−0.00161638	−0.000808191		1.00000i	$0.500257\pi$
$702$	0	0
$703$	−36720.0	−1.97002
$704$	0	0
$705$	−10680.0	−0.570542
$706$	0	0
$707$	−33856.0	−1.80097
$708$	0	0
$709$	−31466.0	−1.66676	−0.833378	−	0.552703i	$-0.813596\pi$
$709$	−31466.0	−1.66676	−0.833378	+	0.552703i	$0.813596\pi$
$710$	0	0
$711$	9864.00	0.520294
$712$	0	0
$713$	880.000	0.0462220
$714$	0	0
$715$	13000.0	0.679961
$716$	0	0
$717$	882.000	0.0459399
$718$	0	0
$719$	28892.0	1.49859	0.749297	−	0.662234i	$-0.230391\pi$
$719$	28892.0	1.49859	0.749297	+	0.662234i	$0.230391\pi$
$720$	0	0
$721$	−56576.0	−2.92233
$722$	0	0
$723$	14886.0	0.765721
$724$	0	0
$725$	−22550.0	−1.15515
$726$	0	0
$727$	−13384.0	−0.682786	−0.341393	−	0.939921i	$-0.610899\pi$
$727$	−13384.0	−0.682786	−0.341393	+	0.939921i	$0.610899\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	10680.0	0.540375
$732$	0	0
$733$	−7130.00	−0.359280	−0.179640	−	0.983732i	$-0.557493\pi$
$733$	−7130.00	−0.359280	−0.179640	+	0.983732i	$0.557493\pi$
$734$	0	0
$735$	−40860.0	−2.05054
$736$	0	0
$737$	−7000.00	−0.349862
$738$	0	0
$739$	−29268.0	−1.45689	−0.728444	−	0.685105i	$-0.759756\pi$
$739$	−29268.0	−1.45689	−0.728444	+	0.685105i	$0.759756\pi$
$740$	0	0
$741$	4680.00	0.232016
$742$	0	0
$743$	9898.00	0.488725	0.244362	−	0.969684i	$-0.421421\pi$
$743$	9898.00	0.488725	0.244362	+	0.969684i	$0.421421\pi$
$744$	0	0
$745$	−25440.0	−1.25107
$746$	0	0
$747$	−4158.00	−0.203659
$748$	0	0
$749$	−57856.0	−2.82245
$750$	0	0
$751$	15120.0	0.734669	0.367335	−	0.930089i	$-0.380270\pi$
$751$	15120.0	0.734669	0.367335	+	0.930089i	$0.380270\pi$
$752$	0	0
$753$	−2232.00	−0.108019
$754$	0	0
$755$	−28080.0	−1.35356
$756$	0	0
$757$	5454.00	0.261861	0.130931	−	0.991392i	$-0.458203\pi$
$757$	5454.00	0.261861	0.130931	+	0.991392i	$0.458203\pi$
$758$	0	0
$759$	−3000.00	−0.143469
$760$	0	0
$761$	−11988.0	−0.571044	−0.285522	−	0.958372i	$-0.592167\pi$
$761$	−11988.0	−0.571044	−0.285522	+	0.958372i	$0.592167\pi$
$762$	0	0
$763$	60352.0	2.86355
$764$	0	0
$765$	−5400.00	−0.255212
$766$	0	0
$767$	1222.00	0.0575279
$768$	0	0
$769$	1338.00	0.0627432	0.0313716	−	0.999508i	$-0.490012\pi$
$769$	1338.00	0.0627432	0.0313716	+	0.999508i	$0.490012\pi$
$770$	0	0
$771$	3078.00	0.143776
$772$	0	0
$773$	14408.0	0.670401	0.335200	−	0.942147i	$-0.391196\pi$
$773$	14408.0	0.670401	0.335200	+	0.942147i	$0.391196\pi$
$774$	0	0
$775$	12100.0	0.560832
$776$	0	0
$777$	−29376.0	−1.35632
$778$	0	0
$779$	−12960.0	−0.596072
$780$	0	0
$781$	38900.0	1.78227
$782$	0	0
$783$	2214.00	0.101050
$784$	0	0
$785$	43400.0	1.97326
$786$	0	0
$787$	−10660.0	−0.482831	−0.241415	−	0.970422i	$-0.577612\pi$
$787$	−10660.0	−0.482831	−0.241415	+	0.970422i	$0.577612\pi$
$788$	0	0
$789$	−16596.0	−0.748838
$790$	0	0
$791$	39872.0	1.79227
$792$	0	0
$793$	806.000	0.0360932
$794$	0	0
$795$	11880.0	0.529988
$796$	0	0
$797$	−1974.00	−0.0877323	−0.0438662	−	0.999037i	$-0.513968\pi$
$797$	−1974.00	−0.0877323	−0.0438662	+	0.999037i	$0.513968\pi$
$798$	0	0
$799$	−5340.00	−0.236440
$800$	0	0
$801$	11016.0	0.485932
$802$	0	0
$803$	3100.00	0.136235
$804$	0	0
$805$	12800.0	0.560423
$806$	0	0
$807$	−10602.0	−0.462464
$808$	0	0
$809$	31734.0	1.37912	0.689560	−	0.724229i	$-0.257804\pi$
$809$	31734.0	1.37912	0.689560	+	0.724229i	$0.257804\pi$
$810$	0	0
$811$	−38824.0	−1.68100	−0.840502	−	0.541808i	$-0.817740\pi$
$811$	−38824.0	−1.68100	−0.840502	+	0.541808i	$0.817740\pi$
$812$	0	0
$813$	7176.00	0.309561
$814$	0	0
$815$	4960.00	0.213179
$816$	0	0
$817$	42720.0	1.82936
$818$	0	0
$819$	3744.00	0.159739
$820$	0	0
$821$	16736.0	0.711438	0.355719	−	0.934593i	$-0.384236\pi$
$821$	16736.0	0.711438	0.355719	+	0.934593i	$0.384236\pi$
$822$	0	0
$823$	−42096.0	−1.78296	−0.891479	−	0.453062i	$-0.850332\pi$
$823$	−42096.0	−1.78296	−0.891479	+	0.453062i	$0.850332\pi$
$824$	0	0
$825$	−41250.0	−1.74078
$826$	0	0
$827$	24858.0	1.04522	0.522610	−	0.852572i	$-0.324958\pi$
$827$	24858.0	1.04522	0.522610	+	0.852572i	$0.324958\pi$
$828$	0	0
$829$	−922.000	−0.0386277	−0.0193139	−	0.999813i	$-0.506148\pi$
$829$	−922.000	−0.0386277	−0.0193139	+	0.999813i	$0.506148\pi$
$830$	0	0
$831$	18306.0	0.764173
$832$	0	0
$833$	−20430.0	−0.849769
$834$	0	0
$835$	−2040.00	−0.0845474
$836$	0	0
$837$	−1188.00	−0.0490601
$838$	0	0
$839$	14294.0	0.588181	0.294090	−	0.955778i	$-0.404983\pi$
$839$	14294.0	0.588181	0.294090	+	0.955778i	$0.404983\pi$
$840$	0	0
$841$	−17665.0	−0.724302
$842$	0	0
$843$	22620.0	0.924169
$844$	0	0
$845$	3380.00	0.137604
$846$	0	0
$847$	37408.0	1.51754
$848$	0	0
$849$	8268.00	0.334225
$850$	0	0
$851$	6120.00	0.246523
$852$	0	0
$853$	−37966.0	−1.52395	−0.761976	−	0.647605i	$-0.775771\pi$
$853$	−37966.0	−1.52395	−0.761976	+	0.647605i	$0.775771\pi$
$854$	0	0
$855$	−21600.0	−0.863982
$856$	0	0
$857$	39038.0	1.55602	0.778012	−	0.628249i	$-0.216228\pi$
$857$	39038.0	1.55602	0.778012	+	0.628249i	$0.216228\pi$
$858$	0	0
$859$	20564.0	0.816804	0.408402	−	0.912802i	$-0.366086\pi$
$859$	20564.0	0.816804	0.408402	+	0.912802i	$0.366086\pi$
$860$	0	0
$861$	−10368.0	−0.410384
$862$	0	0
$863$	−39866.0	−1.57248	−0.786242	−	0.617918i	$-0.787976\pi$
$863$	−39866.0	−1.57248	−0.786242	+	0.617918i	$0.787976\pi$
$864$	0	0
$865$	−13640.0	−0.536155
$866$	0	0
$867$	12039.0	0.471587
$868$	0	0
$869$	54800.0	2.13920
$870$	0	0
$871$	−1820.00	−0.0708018
$872$	0	0
$873$	5526.00	0.214235
$874$	0	0
$875$	96000.0	3.70902
$876$	0	0
$877$	−30990.0	−1.19322	−0.596612	−	0.802530i	$-0.703487\pi$
$877$	−30990.0	−1.19322	−0.596612	+	0.802530i	$0.703487\pi$
$878$	0	0
$879$	2904.00	0.111433
$880$	0	0
$881$	−4458.00	−0.170481	−0.0852405	−	0.996360i	$-0.527166\pi$
$881$	−4458.00	−0.170481	−0.0852405	+	0.996360i	$0.527166\pi$
$882$	0	0
$883$	−3164.00	−0.120586	−0.0602928	−	0.998181i	$-0.519203\pi$
$883$	−3164.00	−0.120586	−0.0602928	+	0.998181i	$0.519203\pi$
$884$	0	0
$885$	−5640.00	−0.214222
$886$	0	0
$887$	32512.0	1.23072	0.615359	−	0.788247i	$-0.289011\pi$
$887$	32512.0	1.23072	0.615359	+	0.788247i	$0.289011\pi$
$888$	0	0
$889$	−51968.0	−1.96057
$890$	0	0
$891$	4050.00	0.152278
$892$	0	0
$893$	−21360.0	−0.800431
$894$	0	0
$895$	−12240.0	−0.457138
$896$	0	0
$897$	−780.000	−0.0290339
$898$	0	0
$899$	−3608.00	−0.133853
$900$	0	0
$901$	5940.00	0.219634
$902$	0	0
$903$	34176.0	1.25948
$904$	0	0
$905$	1320.00	0.0484843
$906$	0	0
$907$	10500.0	0.384396	0.192198	−	0.981356i	$-0.438438\pi$
$907$	10500.0	0.384396	0.192198	+	0.981356i	$0.438438\pi$
$908$	0	0
$909$	−9522.00	−0.347442
$910$	0	0
$911$	−9840.00	−0.357864	−0.178932	−	0.983861i	$-0.557264\pi$
$911$	−9840.00	−0.357864	−0.178932	+	0.983861i	$0.557264\pi$
$912$	0	0
$913$	−23100.0	−0.837348
$914$	0	0
$915$	−3720.00	−0.134404
$916$	0	0
$917$	−66304.0	−2.38773
$918$	0	0
$919$	35040.0	1.25774	0.628870	−	0.777511i	$-0.283518\pi$
$919$	35040.0	1.25774	0.628870	+	0.777511i	$0.283518\pi$
$920$	0	0
$921$	19308.0	0.690793
$922$	0	0
$923$	10114.0	0.360679
$924$	0	0
$925$	84150.0	2.99117
$926$	0	0
$927$	−15912.0	−0.563774
$928$	0	0
$929$	44172.0	1.56000	0.779998	−	0.625782i	$-0.215220\pi$
$929$	44172.0	1.56000	0.779998	+	0.625782i	$0.215220\pi$
$930$	0	0
$931$	−81720.0	−2.87676
$932$	0	0
$933$	23796.0	0.834990
$934$	0	0
$935$	−30000.0	−1.04931
$936$	0	0
$937$	−54018.0	−1.88334	−0.941671	−	0.336535i	$-0.890745\pi$
$937$	−54018.0	−1.88334	−0.941671	+	0.336535i	$0.890745\pi$
$938$	0	0
$939$	−31074.0	−1.07994
$940$	0	0
$941$	1672.00	0.0579231	0.0289616	−	0.999581i	$-0.490780\pi$
$941$	1672.00	0.0579231	0.0289616	+	0.999581i	$0.490780\pi$
$942$	0	0
$943$	2160.00	0.0745910
$944$	0	0
$945$	−17280.0	−0.594834
$946$	0	0
$947$	5238.00	0.179738	0.0898691	−	0.995954i	$-0.471355\pi$
$947$	5238.00	0.179738	0.0898691	+	0.995954i	$0.471355\pi$
$948$	0	0
$949$	806.000	0.0275699
$950$	0	0
$951$	−8460.00	−0.288469
$952$	0	0
$953$	−50042.0	−1.70096	−0.850482	−	0.526004i	$-0.823690\pi$
$953$	−50042.0	−1.70096	−0.850482	+	0.526004i	$0.823690\pi$
$954$	0	0
$955$	−12160.0	−0.412030
$956$	0	0
$957$	12300.0	0.415468
$958$	0	0
$959$	−24192.0	−0.814599
$960$	0	0
$961$	−27855.0	−0.935014
$962$	0	0
$963$	−16272.0	−0.544505
$964$	0	0
$965$	27400.0	0.914028
$966$	0	0
$967$	−37676.0	−1.25293	−0.626463	−	0.779452i	$-0.715498\pi$
$967$	−37676.0	−1.25293	−0.626463	+	0.779452i	$0.715498\pi$
$968$	0	0
$969$	−10800.0	−0.358045
$970$	0	0
$971$	−17364.0	−0.573880	−0.286940	−	0.957949i	$-0.592638\pi$
$971$	−17364.0	−0.573880	−0.286940	+	0.957949i	$0.592638\pi$
$972$	0	0
$973$	5504.00	0.181346
$974$	0	0
$975$	−10725.0	−0.352282
$976$	0	0
$977$	−14904.0	−0.488046	−0.244023	−	0.969769i	$-0.578467\pi$
$977$	−14904.0	−0.488046	−0.244023	+	0.969769i	$0.578467\pi$
$978$	0	0
$979$	61200.0	1.99792
$980$	0	0
$981$	16974.0	0.552434
$982$	0	0
$983$	18038.0	0.585272	0.292636	−	0.956224i	$-0.405467\pi$
$983$	18038.0	0.585272	0.292636	+	0.956224i	$0.405467\pi$
$984$	0	0
$985$	98160.0	3.17527
$986$	0	0
$987$	−17088.0	−0.551081
$988$	0	0
$989$	−7120.00	−0.228921
$990$	0	0
$991$	−46176.0	−1.48015	−0.740075	−	0.672524i	$-0.765210\pi$
$991$	−46176.0	−1.48015	−0.740075	+	0.672524i	$0.765210\pi$
$992$	0	0
$993$	12540.0	0.400750
$994$	0	0
$995$	6560.00	0.209011
$996$	0	0
$997$	−55838.0	−1.77373	−0.886864	−	0.462030i	$-0.847121\pi$
$997$	−55838.0	−1.77373	−0.886864	+	0.462030i	$0.847121\pi$
$998$	0	0
$999$	−8262.00	−0.261660

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.4.a.i.1.1		1
4.3	odd	2		2496.4.a.r.1.1		1
8.3	odd	2		78.4.a.d.1.1	✓	1
8.5	even	2		624.4.a.e.1.1		1
24.5	odd	2		1872.4.a.r.1.1		1
24.11	even	2		234.4.a.f.1.1		1
40.19	odd	2		1950.4.a.h.1.1		1
104.51	odd	2		1014.4.a.d.1.1		1
104.83	even	4		1014.4.b.e.337.1		2
104.99	even	4		1014.4.b.e.337.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.4.a.d.1.1	✓	1	8.3	odd	2
234.4.a.f.1.1		1	24.11	even	2
624.4.a.e.1.1		1	8.5	even	2
1014.4.a.d.1.1		1	104.51	odd	2
1014.4.b.e.337.1		2	104.83	even	4
1014.4.b.e.337.2		2	104.99	even	4
1872.4.a.r.1.1		1	24.5	odd	2
1950.4.a.h.1.1		1	40.19	odd	2
2496.4.a.i.1.1		1	1.1	even	1	trivial
2496.4.a.r.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2496.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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