LMFDB - Embedded newform 2496.4.a.bp.1.3

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 = [3,0,9,0,-4,0,-30,0,27,0,-16] 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:	$3$
Coefficient field:	3.3.3144.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 8 $ x^3 - x^2 - 16*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.20905$ of defining polynomial
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)

$f(q)$	$=$	$q+3.00000 q^{3} +11.4322 q^{5} +11.2543 q^{7} +9.00000 q^{9} +25.8785 q^{11} -13.0000 q^{13} +34.2966 q^{15} -20.3276 q^{17} +154.712 q^{19} +33.7629 q^{21} +180.418 q^{23} +5.69520 q^{25} +27.0000 q^{27} +20.4522 q^{29} -266.424 q^{31} +77.6355 q^{33} +128.661 q^{35} -115.984 q^{37} -39.0000 q^{39} +391.184 q^{41} +151.407 q^{43} +102.890 q^{45} +467.365 q^{47} -216.341 q^{49} -60.9828 q^{51} -79.9842 q^{53} +295.848 q^{55} +464.136 q^{57} -873.710 q^{59} +187.068 q^{61} +101.289 q^{63} -148.619 q^{65} -609.204 q^{67} +541.255 q^{69} -248.038 q^{71} +852.765 q^{73} +17.0856 q^{75} +291.244 q^{77} +331.221 q^{79} +81.0000 q^{81} -435.432 q^{83} -232.389 q^{85} +61.3566 q^{87} +259.233 q^{89} -146.306 q^{91} -799.273 q^{93} +1768.70 q^{95} +1225.17 q^{97} +232.907 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 9 q^{3} - 4 q^{5} - 30 q^{7} + 27 q^{9} - 16 q^{11} - 39 q^{13} - 12 q^{15} - 146 q^{17} + 94 q^{19} - 90 q^{21} + 48 q^{23} + 145 q^{25} + 81 q^{27} + 2 q^{29} - 302 q^{31} - 48 q^{33} + 80 q^{35}+ \cdots - 144 q^{99}+O(q^{100}) $ 3 * q + 9 * q^3 - 4 * q^5 - 30 * q^7 + 27 * q^9 - 16 * q^11 - 39 * q^13 - 12 * q^15 - 146 * q^17 + 94 * q^19 - 90 * q^21 + 48 * q^23 + 145 * q^25 + 81 * q^27 + 2 * q^29 - 302 * q^31 - 48 * q^33 + 80 * q^35 - 374 * q^37 - 117 * q^39 + 480 * q^41 - 260 * q^43 - 36 * q^45 + 24 * q^47 + 447 * q^49 - 438 * q^51 + 678 * q^53 + 1552 * q^55 + 282 * q^57 - 1788 * q^59 - 230 * q^61 - 270 * q^63 + 52 * q^65 + 74 * q^67 + 144 * q^69 + 948 * q^71 - 222 * q^73 + 435 * q^75 - 112 * q^77 + 24 * q^79 + 243 * q^81 - 796 * q^83 + 248 * q^85 + 6 * q^87 + 1436 * q^89 + 390 * q^91 - 906 * q^93 + 4032 * q^95 + 3242 * q^97 - 144 * q^99

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
$5$	11.4322	1.02253	0.511264	−	0.859424i	$-0.329178\pi$
$5$	11.4322	1.02253	0.511264	+	0.859424i	$0.329178\pi$
$6$	0	0
$7$	11.2543	0.607675	0.303838	−	0.952724i	$-0.401732\pi$
$7$	11.2543	0.607675	0.303838	+	0.952724i	$0.401732\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	25.8785	0.709333	0.354666	−	0.934993i	$-0.384594\pi$
$11$	25.8785	0.709333	0.354666	+	0.934993i	$0.384594\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	34.2966	0.590356
$16$	0	0
$17$	−20.3276	−0.290010	−0.145005	−	0.989431i	$-0.546320\pi$
$17$	−20.3276	−0.290010	−0.145005	+	0.989431i	$0.546320\pi$
$18$	0	0
$19$	154.712	1.86807	0.934035	−	0.357181i	$-0.116262\pi$
$19$	154.712	1.86807	0.934035	+	0.357181i	$0.116262\pi$
$20$	0	0
$21$	33.7629	0.350841
$22$	0	0
$23$	180.418	1.63565	0.817823	−	0.575471i	$-0.195181\pi$
$23$	180.418	1.63565	0.817823	+	0.575471i	$0.195181\pi$
$24$	0	0
$25$	5.69520	0.0455616
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	20.4522	0.130961	0.0654806	−	0.997854i	$-0.479142\pi$
$29$	20.4522	0.130961	0.0654806	+	0.997854i	$0.479142\pi$
$30$	0	0
$31$	−266.424	−1.54359	−0.771794	−	0.635873i	$-0.780640\pi$
$31$	−266.424	−1.54359	−0.771794	+	0.635873i	$0.780640\pi$
$32$	0	0
$33$	77.6355	0.409534
$34$	0	0
$35$	128.661	0.621364
$36$	0	0
$37$	−115.984	−0.515340	−0.257670	−	0.966233i	$-0.582955\pi$
$37$	−115.984	−0.515340	−0.257670	+	0.966233i	$0.582955\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	391.184	1.49006	0.745032	−	0.667029i	$-0.232434\pi$
$41$	391.184	1.49006	0.745032	+	0.667029i	$0.232434\pi$
$42$	0	0
$43$	151.407	0.536963	0.268482	−	0.963285i	$-0.413478\pi$
$43$	151.407	0.536963	0.268482	+	0.963285i	$0.413478\pi$
$44$	0	0
$45$	102.890	0.340842
$46$	0	0
$47$	467.365	1.45047	0.725236	−	0.688500i	$-0.241731\pi$
$47$	467.365	1.45047	0.725236	+	0.688500i	$0.241731\pi$
$48$	0	0
$49$	−216.341	−0.630731
$50$	0	0
$51$	−60.9828	−0.167437
$52$	0	0
$53$	−79.9842	−0.207296	−0.103648	−	0.994614i	$-0.533051\pi$
$53$	−79.9842	−0.207296	−0.103648	+	0.994614i	$0.533051\pi$
$54$	0	0
$55$	295.848	0.725312
$56$	0	0
$57$	464.136	1.07853
$58$	0	0
$59$	−873.710	−1.92792	−0.963960	−	0.266045i	$-0.914283\pi$
$59$	−873.710	−1.92792	−0.963960	+	0.266045i	$0.914283\pi$
$60$	0	0
$61$	187.068	0.392649	0.196325	−	0.980539i	$-0.437099\pi$
$61$	187.068	0.392649	0.196325	+	0.980539i	$0.437099\pi$
$62$	0	0
$63$	101.289	0.202558
$64$	0	0
$65$	−148.619	−0.283598
$66$	0	0
$67$	−609.204	−1.11084	−0.555418	−	0.831571i	$-0.687442\pi$
$67$	−609.204	−1.11084	−0.555418	+	0.831571i	$0.687442\pi$
$68$	0	0
$69$	541.255	0.944340
$70$	0	0
$71$	−248.038	−0.414601	−0.207301	−	0.978277i	$-0.566468\pi$
$71$	−248.038	−0.414601	−0.207301	+	0.978277i	$0.566468\pi$
$72$	0	0
$73$	852.765	1.36724	0.683621	−	0.729838i	$-0.260404\pi$
$73$	852.765	1.36724	0.683621	+	0.729838i	$0.260404\pi$
$74$	0	0
$75$	17.0856	0.0263050
$76$	0	0
$77$	291.244	0.431044
$78$	0	0
$79$	331.221	0.471712	0.235856	−	0.971788i	$-0.424211\pi$
$79$	331.221	0.471712	0.235856	+	0.971788i	$0.424211\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−435.432	−0.575842	−0.287921	−	0.957654i	$-0.592964\pi$
$83$	−435.432	−0.575842	−0.287921	+	0.957654i	$0.592964\pi$
$84$	0	0
$85$	−232.389	−0.296543
$86$	0	0
$87$	61.3566	0.0756105
$88$	0	0
$89$	259.233	0.308749	0.154375	−	0.988012i	$-0.450664\pi$
$89$	259.233	0.308749	0.154375	+	0.988012i	$0.450664\pi$
$90$	0	0
$91$	−146.306	−0.168539
$92$	0	0
$93$	−799.273	−0.891191
$94$	0	0
$95$	1768.70	1.91015
$96$	0	0
$97$	1225.17	1.28245	0.641223	−	0.767355i	$-0.278428\pi$
$97$	1225.17	1.28245	0.641223	+	0.767355i	$0.278428\pi$
$98$	0	0
$99$	232.907	0.236444
$100$	0	0
$101$	−645.416	−0.635855	−0.317927	−	0.948115i	$-0.602987\pi$
$101$	−645.416	−0.635855	−0.317927	+	0.948115i	$0.602987\pi$
$102$	0	0
$103$	511.137	0.488969	0.244484	−	0.969653i	$-0.421381\pi$
$103$	511.137	0.488969	0.244484	+	0.969653i	$0.421381\pi$
$104$	0	0
$105$	385.984	0.358745
$106$	0	0
$107$	608.195	0.549499	0.274750	−	0.961516i	$-0.411405\pi$
$107$	608.195	0.549499	0.274750	+	0.961516i	$0.411405\pi$
$108$	0	0
$109$	1300.04	1.14239	0.571197	−	0.820813i	$-0.306479\pi$
$109$	1300.04	1.14239	0.571197	+	0.820813i	$0.306479\pi$
$110$	0	0
$111$	−347.951	−0.297532
$112$	0	0
$113$	42.1953	0.0351274	0.0175637	−	0.999846i	$-0.494409\pi$
$113$	42.1953	0.0351274	0.0175637	+	0.999846i	$0.494409\pi$
$114$	0	0
$115$	2062.58	1.67249
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	−228.773	−0.176232
$120$	0	0
$121$	−661.303	−0.496847
$122$	0	0
$123$	1173.55	0.860289
$124$	0	0
$125$	−1363.92	−0.975939
$126$	0	0
$127$	311.018	0.217310	0.108655	−	0.994080i	$-0.465346\pi$
$127$	311.018	0.217310	0.108655	+	0.994080i	$0.465346\pi$
$128$	0	0
$129$	454.222	0.310016
$130$	0	0
$131$	2000.98	1.33456	0.667278	−	0.744809i	$-0.267459\pi$
$131$	2000.98	1.33456	0.667278	+	0.744809i	$0.267459\pi$
$132$	0	0
$133$	1741.17	1.13518
$134$	0	0
$135$	308.669	0.196785
$136$	0	0
$137$	1038.53	0.647644	0.323822	−	0.946118i	$-0.395032\pi$
$137$	1038.53	0.647644	0.323822	+	0.946118i	$0.395032\pi$
$138$	0	0
$139$	−2858.46	−1.74426	−0.872128	−	0.489277i	$-0.837261\pi$
$139$	−2858.46	−1.74426	−0.872128	+	0.489277i	$0.837261\pi$
$140$	0	0
$141$	1402.09	0.837430
$142$	0	0
$143$	−336.421	−0.196734
$144$	0	0
$145$	233.814	0.133911
$146$	0	0
$147$	−649.022	−0.364153
$148$	0	0
$149$	−743.479	−0.408780	−0.204390	−	0.978890i	$-0.565521\pi$
$149$	−743.479	−0.408780	−0.204390	+	0.978890i	$0.565521\pi$
$150$	0	0
$151$	−2277.24	−1.22728	−0.613640	−	0.789586i	$-0.710295\pi$
$151$	−2277.24	−1.22728	−0.613640	+	0.789586i	$0.710295\pi$
$152$	0	0
$153$	−182.948	−0.0966700
$154$	0	0
$155$	−3045.82	−1.57836
$156$	0	0
$157$	−3173.51	−1.61321	−0.806605	−	0.591091i	$-0.798697\pi$
$157$	−3173.51	−1.61321	−0.806605	+	0.591091i	$0.798697\pi$
$158$	0	0
$159$	−239.953	−0.119682
$160$	0	0
$161$	2030.48	0.993941
$162$	0	0
$163$	−2314.65	−1.11225	−0.556126	−	0.831098i	$-0.687713\pi$
$163$	−2314.65	−1.11225	−0.556126	+	0.831098i	$0.687713\pi$
$164$	0	0
$165$	887.545	0.418759
$166$	0	0
$167$	2665.65	1.23517	0.617587	−	0.786502i	$-0.288110\pi$
$167$	2665.65	1.23517	0.617587	+	0.786502i	$0.288110\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	1392.41	0.622690
$172$	0	0
$173$	165.243	0.0726198	0.0363099	−	0.999341i	$-0.488440\pi$
$173$	165.243	0.0726198	0.0363099	+	0.999341i	$0.488440\pi$
$174$	0	0
$175$	64.0954	0.0276866
$176$	0	0
$177$	−2621.13	−1.11309
$178$	0	0
$179$	712.339	0.297446	0.148723	−	0.988879i	$-0.452484\pi$
$179$	712.339	0.297446	0.148723	+	0.988879i	$0.452484\pi$
$180$	0	0
$181$	−2206.53	−0.906133	−0.453066	−	0.891477i	$-0.649670\pi$
$181$	−2206.53	−0.906133	−0.453066	+	0.891477i	$0.649670\pi$
$182$	0	0
$183$	561.204	0.226696
$184$	0	0
$185$	−1325.95	−0.526949
$186$	0	0
$187$	−526.048	−0.205714
$188$	0	0
$189$	303.866	0.116947
$190$	0	0
$191$	−1470.64	−0.557129	−0.278565	−	0.960417i	$-0.589859\pi$
$191$	−1470.64	−0.557129	−0.278565	+	0.960417i	$0.589859\pi$
$192$	0	0
$193$	369.560	0.137832	0.0689158	−	0.997622i	$-0.478046\pi$
$193$	369.560	0.137832	0.0689158	+	0.997622i	$0.478046\pi$
$194$	0	0
$195$	−445.856	−0.163735
$196$	0	0
$197$	4273.41	1.54552	0.772761	−	0.634697i	$-0.218875\pi$
$197$	4273.41	1.54552	0.772761	+	0.634697i	$0.218875\pi$
$198$	0	0
$199$	−4154.31	−1.47985	−0.739927	−	0.672687i	$-0.765140\pi$
$199$	−4154.31	−1.47985	−0.739927	+	0.672687i	$0.765140\pi$
$200$	0	0
$201$	−1827.61	−0.641342
$202$	0	0
$203$	230.175	0.0795819
$204$	0	0
$205$	4472.09	1.52363
$206$	0	0
$207$	1623.77	0.545215
$208$	0	0
$209$	4003.71	1.32508
$210$	0	0
$211$	1231.59	0.401830	0.200915	−	0.979609i	$-0.435608\pi$
$211$	1231.59	0.401830	0.200915	+	0.979609i	$0.435608\pi$
$212$	0	0
$213$	−744.114	−0.239370
$214$	0	0
$215$	1730.92	0.549059
$216$	0	0
$217$	−2998.42	−0.938000
$218$	0	0
$219$	2558.30	0.789377
$220$	0	0
$221$	264.259	0.0804343
$222$	0	0
$223$	2187.24	0.656809	0.328404	−	0.944537i	$-0.393489\pi$
$223$	2187.24	0.656809	0.328404	+	0.944537i	$0.393489\pi$
$224$	0	0
$225$	51.2568	0.0151872
$226$	0	0
$227$	4138.67	1.21010	0.605051	−	0.796187i	$-0.293153\pi$
$227$	4138.67	1.21010	0.605051	+	0.796187i	$0.293153\pi$
$228$	0	0
$229$	835.354	0.241056	0.120528	−	0.992710i	$-0.461541\pi$
$229$	835.354	0.241056	0.120528	+	0.992710i	$0.461541\pi$
$230$	0	0
$231$	873.733	0.248863
$232$	0	0
$233$	3685.51	1.03625	0.518124	−	0.855305i	$-0.326630\pi$
$233$	3685.51	1.03625	0.518124	+	0.855305i	$0.326630\pi$
$234$	0	0
$235$	5343.01	1.48315
$236$	0	0
$237$	993.662	0.272343
$238$	0	0
$239$	−3026.21	−0.819034	−0.409517	−	0.912303i	$-0.634303\pi$
$239$	−3026.21	−0.819034	−0.409517	+	0.912303i	$0.634303\pi$
$240$	0	0
$241$	3265.58	0.872839	0.436420	−	0.899743i	$-0.356246\pi$
$241$	3265.58	0.872839	0.436420	+	0.899743i	$0.356246\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−2473.25	−0.644940
$246$	0	0
$247$	−2011.25	−0.518110
$248$	0	0
$249$	−1306.30	−0.332463
$250$	0	0
$251$	−6363.16	−1.60016	−0.800078	−	0.599897i	$-0.795208\pi$
$251$	−6363.16	−1.60016	−0.800078	+	0.599897i	$0.795208\pi$
$252$	0	0
$253$	4668.96	1.16022
$254$	0	0
$255$	−697.168	−0.171209
$256$	0	0
$257$	−6085.36	−1.47702	−0.738511	−	0.674242i	$-0.764471\pi$
$257$	−6085.36	−1.47702	−0.738511	+	0.674242i	$0.764471\pi$
$258$	0	0
$259$	−1305.31	−0.313159
$260$	0	0
$261$	184.070	0.0436538
$262$	0	0
$263$	−123.227	−0.0288916	−0.0144458	−	0.999896i	$-0.504598\pi$
$263$	−123.227	−0.0288916	−0.0144458	+	0.999896i	$0.504598\pi$
$264$	0	0
$265$	−914.395	−0.211965
$266$	0	0
$267$	777.700	0.178256
$268$	0	0
$269$	1935.79	0.438763	0.219381	−	0.975639i	$-0.429596\pi$
$269$	1935.79	0.438763	0.219381	+	0.975639i	$0.429596\pi$
$270$	0	0
$271$	4612.69	1.03395	0.516976	−	0.856000i	$-0.327058\pi$
$271$	4612.69	1.03395	0.516976	+	0.856000i	$0.327058\pi$
$272$	0	0
$273$	−438.918	−0.0973059
$274$	0	0
$275$	147.383	0.0323183
$276$	0	0
$277$	5834.30	1.26552	0.632761	−	0.774347i	$-0.281922\pi$
$277$	5834.30	1.26552	0.632761	+	0.774347i	$0.281922\pi$
$278$	0	0
$279$	−2397.82	−0.514529
$280$	0	0
$281$	4691.91	0.996071	0.498036	−	0.867157i	$-0.334055\pi$
$281$	4691.91	0.996071	0.498036	+	0.867157i	$0.334055\pi$
$282$	0	0
$283$	3465.60	0.727945	0.363973	−	0.931410i	$-0.381420\pi$
$283$	3465.60	0.727945	0.363973	+	0.931410i	$0.381420\pi$
$284$	0	0
$285$	5306.09	1.10283
$286$	0	0
$287$	4402.50	0.905475
$288$	0	0
$289$	−4499.79	−0.915894
$290$	0	0
$291$	3675.51	0.740420
$292$	0	0
$293$	−2677.31	−0.533822	−0.266911	−	0.963721i	$-0.586003\pi$
$293$	−2677.31	−0.533822	−0.266911	+	0.963721i	$0.586003\pi$
$294$	0	0
$295$	−9988.43	−1.97135
$296$	0	0
$297$	698.720	0.136511
$298$	0	0
$299$	−2345.44	−0.453646
$300$	0	0
$301$	1703.98	0.326299
$302$	0	0
$303$	−1936.25	−0.367111
$304$	0	0
$305$	2138.60	0.401494
$306$	0	0
$307$	471.915	0.0877316	0.0438658	−	0.999037i	$-0.486033\pi$
$307$	471.915	0.0877316	0.0438658	+	0.999037i	$0.486033\pi$
$308$	0	0
$309$	1533.41	0.282306
$310$	0	0
$311$	1518.52	0.276872	0.138436	−	0.990371i	$-0.455793\pi$
$311$	1518.52	0.276872	0.138436	+	0.990371i	$0.455793\pi$
$312$	0	0
$313$	4049.86	0.731348	0.365674	−	0.930743i	$-0.380839\pi$
$313$	4049.86	0.731348	0.365674	+	0.930743i	$0.380839\pi$
$314$	0	0
$315$	1157.95	0.207121
$316$	0	0
$317$	−3253.96	−0.576532	−0.288266	−	0.957550i	$-0.593079\pi$
$317$	−3253.96	−0.576532	−0.288266	+	0.957550i	$0.593079\pi$
$318$	0	0
$319$	529.272	0.0928951
$320$	0	0
$321$	1824.59	0.317254
$322$	0	0
$323$	−3144.92	−0.541759
$324$	0	0
$325$	−74.0375	−0.0126365
$326$	0	0
$327$	3900.11	0.659561
$328$	0	0
$329$	5259.86	0.881415
$330$	0	0
$331$	−3422.45	−0.568322	−0.284161	−	0.958777i	$-0.591715\pi$
$331$	−3422.45	−0.568322	−0.284161	+	0.958777i	$0.591715\pi$
$332$	0	0
$333$	−1043.85	−0.171780
$334$	0	0
$335$	−6964.54	−1.13586
$336$	0	0
$337$	−9301.67	−1.50354	−0.751772	−	0.659423i	$-0.770801\pi$
$337$	−9301.67	−1.50354	−0.751772	+	0.659423i	$0.770801\pi$
$338$	0	0
$339$	126.586	0.0202808
$340$	0	0
$341$	−6894.66	−1.09492
$342$	0	0
$343$	−6294.99	−0.990955
$344$	0	0
$345$	6187.74	0.965613
$346$	0	0
$347$	216.898	0.0335554	0.0167777	−	0.999859i	$-0.494659\pi$
$347$	216.898	0.0335554	0.0167777	+	0.999859i	$0.494659\pi$
$348$	0	0
$349$	4809.84	0.737721	0.368861	−	0.929485i	$-0.379748\pi$
$349$	4809.84	0.737721	0.368861	+	0.929485i	$0.379748\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	−2859.64	−0.431170	−0.215585	−	0.976485i	$-0.569166\pi$
$353$	−2859.64	−0.431170	−0.215585	+	0.976485i	$0.569166\pi$
$354$	0	0
$355$	−2835.62	−0.423941
$356$	0	0
$357$	−686.319	−0.101747
$358$	0	0
$359$	−3686.04	−0.541899	−0.270949	−	0.962594i	$-0.587338\pi$
$359$	−3686.04	−0.541899	−0.270949	+	0.962594i	$0.587338\pi$
$360$	0	0
$361$	17076.8	2.48969
$362$	0	0
$363$	−1983.91	−0.286855
$364$	0	0
$365$	9748.98	1.39804
$366$	0	0
$367$	3470.59	0.493633	0.246816	−	0.969062i	$-0.420616\pi$
$367$	3470.59	0.493633	0.246816	+	0.969062i	$0.420616\pi$
$368$	0	0
$369$	3520.65	0.496688
$370$	0	0
$371$	−900.166	−0.125968
$372$	0	0
$373$	11963.4	1.66070	0.830352	−	0.557240i	$-0.188140\pi$
$373$	11963.4	1.66070	0.830352	+	0.557240i	$0.188140\pi$
$374$	0	0
$375$	−4091.75	−0.563459
$376$	0	0
$377$	−265.879	−0.0363221
$378$	0	0
$379$	345.604	0.0468403	0.0234202	−	0.999726i	$-0.492544\pi$
$379$	345.604	0.0468403	0.0234202	+	0.999726i	$0.492544\pi$
$380$	0	0
$381$	933.055	0.125464
$382$	0	0
$383$	3386.40	0.451793	0.225897	−	0.974151i	$-0.427469\pi$
$383$	3386.40	0.451793	0.225897	+	0.974151i	$0.427469\pi$
$384$	0	0
$385$	3329.56	0.440754
$386$	0	0
$387$	1362.67	0.178988
$388$	0	0
$389$	1629.88	0.212438	0.106219	−	0.994343i	$-0.466126\pi$
$389$	1629.88	0.212438	0.106219	+	0.994343i	$0.466126\pi$
$390$	0	0
$391$	−3667.47	−0.474353
$392$	0	0
$393$	6002.95	0.770506
$394$	0	0
$395$	3786.58	0.482338
$396$	0	0
$397$	−7938.94	−1.00364	−0.501819	−	0.864973i	$-0.667336\pi$
$397$	−7938.94	−1.00364	−0.501819	+	0.864973i	$0.667336\pi$
$398$	0	0
$399$	5223.52	0.655396
$400$	0	0
$401$	214.402	0.0267001	0.0133500	−	0.999911i	$-0.495750\pi$
$401$	214.402	0.0267001	0.0133500	+	0.999911i	$0.495750\pi$
$402$	0	0
$403$	3463.52	0.428114
$404$	0	0
$405$	926.008	0.113614
$406$	0	0
$407$	−3001.48	−0.365548
$408$	0	0
$409$	−4783.73	−0.578338	−0.289169	−	0.957278i	$-0.593379\pi$
$409$	−4783.73	−0.578338	−0.289169	+	0.957278i	$0.593379\pi$
$410$	0	0
$411$	3115.58	0.373917
$412$	0	0
$413$	−9832.99	−1.17155
$414$	0	0
$415$	−4977.95	−0.588815
$416$	0	0
$417$	−8575.39	−1.00705
$418$	0	0
$419$	−9903.67	−1.15472	−0.577358	−	0.816491i	$-0.695916\pi$
$419$	−9903.67	−1.15472	−0.577358	+	0.816491i	$0.695916\pi$
$420$	0	0
$421$	12120.6	1.40314	0.701572	−	0.712598i	$-0.252482\pi$
$421$	12120.6	1.40314	0.701572	+	0.712598i	$0.252482\pi$
$422$	0	0
$423$	4206.28	0.483491
$424$	0	0
$425$	−115.770	−0.0132133
$426$	0	0
$427$	2105.32	0.238603
$428$	0	0
$429$	−1009.26	−0.113584
$430$	0	0
$431$	13672.6	1.52805	0.764023	−	0.645189i	$-0.223221\pi$
$431$	13672.6	1.52805	0.764023	+	0.645189i	$0.223221\pi$
$432$	0	0
$433$	7113.10	0.789455	0.394727	−	0.918798i	$-0.370839\pi$
$433$	7113.10	0.789455	0.394727	+	0.918798i	$0.370839\pi$
$434$	0	0
$435$	701.441	0.0773138
$436$	0	0
$437$	27912.9	3.05550
$438$	0	0
$439$	6022.04	0.654707	0.327353	−	0.944902i	$-0.393843\pi$
$439$	6022.04	0.654707	0.327353	+	0.944902i	$0.393843\pi$
$440$	0	0
$441$	−1947.07	−0.210244
$442$	0	0
$443$	−12994.4	−1.39364	−0.696821	−	0.717245i	$-0.745403\pi$
$443$	−12994.4	−1.39364	−0.696821	+	0.717245i	$0.745403\pi$
$444$	0	0
$445$	2963.61	0.315704
$446$	0	0
$447$	−2230.44	−0.236009
$448$	0	0
$449$	10984.3	1.15452	0.577260	−	0.816560i	$-0.304122\pi$
$449$	10984.3	1.15452	0.577260	+	0.816560i	$0.304122\pi$
$450$	0	0
$451$	10123.2	1.05695
$452$	0	0
$453$	−6831.72	−0.708570
$454$	0	0
$455$	−1672.60	−0.172335
$456$	0	0
$457$	9834.10	1.00661	0.503304	−	0.864109i	$-0.332118\pi$
$457$	9834.10	1.00661	0.503304	+	0.864109i	$0.332118\pi$
$458$	0	0
$459$	−548.845	−0.0558124
$460$	0	0
$461$	−3401.42	−0.343644	−0.171822	−	0.985128i	$-0.554965\pi$
$461$	−3401.42	−0.343644	−0.171822	+	0.985128i	$0.554965\pi$
$462$	0	0
$463$	−1739.42	−0.174596	−0.0872979	−	0.996182i	$-0.527823\pi$
$463$	−1739.42	−0.174596	−0.0872979	+	0.996182i	$0.527823\pi$
$464$	0	0
$465$	−9137.45	−0.911267
$466$	0	0
$467$	−7958.82	−0.788630	−0.394315	−	0.918975i	$-0.629018\pi$
$467$	−7958.82	−0.788630	−0.394315	+	0.918975i	$0.629018\pi$
$468$	0	0
$469$	−6856.16	−0.675028
$470$	0	0
$471$	−9520.54	−0.931387
$472$	0	0
$473$	3918.20	0.380886
$474$	0	0
$475$	881.114	0.0851122
$476$	0	0
$477$	−719.858	−0.0690986
$478$	0	0
$479$	−8431.98	−0.804315	−0.402158	−	0.915570i	$-0.631740\pi$
$479$	−8431.98	−0.804315	−0.402158	+	0.915570i	$0.631740\pi$
$480$	0	0
$481$	1507.79	0.142930
$482$	0	0
$483$	6091.45	0.573852
$484$	0	0
$485$	14006.4	1.31133
$486$	0	0
$487$	11684.7	1.08723	0.543617	−	0.839334i	$-0.317055\pi$
$487$	11684.7	1.08723	0.543617	+	0.839334i	$0.317055\pi$
$488$	0	0
$489$	−6943.94	−0.642159
$490$	0	0
$491$	−3954.70	−0.363489	−0.181745	−	0.983346i	$-0.558174\pi$
$491$	−3954.70	−0.363489	−0.181745	+	0.983346i	$0.558174\pi$
$492$	0	0
$493$	−415.744	−0.0379801
$494$	0	0
$495$	2662.63	0.241771
$496$	0	0
$497$	−2791.49	−0.251943
$498$	0	0
$499$	−5690.37	−0.510493	−0.255246	−	0.966876i	$-0.582157\pi$
$499$	−5690.37	−0.510493	−0.255246	+	0.966876i	$0.582157\pi$
$500$	0	0
$501$	7996.95	0.713128
$502$	0	0
$503$	−10859.1	−0.962595	−0.481298	−	0.876557i	$-0.659834\pi$
$503$	−10859.1	−0.962595	−0.481298	+	0.876557i	$0.659834\pi$
$504$	0	0
$505$	−7378.53	−0.650178
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	18558.6	1.61610	0.808049	−	0.589115i	$-0.200524\pi$
$509$	18558.6	1.61610	0.808049	+	0.589115i	$0.200524\pi$
$510$	0	0
$511$	9597.27	0.830838
$512$	0	0
$513$	4177.22	0.359510
$514$	0	0
$515$	5843.42	0.499984
$516$	0	0
$517$	12094.7	1.02887
$518$	0	0
$519$	495.730	0.0419271
$520$	0	0
$521$	17297.5	1.45454	0.727271	−	0.686350i	$-0.240788\pi$
$521$	17297.5	1.45454	0.727271	+	0.686350i	$0.240788\pi$
$522$	0	0
$523$	−5016.11	−0.419386	−0.209693	−	0.977767i	$-0.567247\pi$
$523$	−5016.11	−0.419386	−0.209693	+	0.977767i	$0.567247\pi$
$524$	0	0
$525$	192.286	0.0159849
$526$	0	0
$527$	5415.77	0.447656
$528$	0	0
$529$	20383.8	1.67533
$530$	0	0
$531$	−7863.39	−0.642640
$532$	0	0
$533$	−5085.39	−0.413269
$534$	0	0
$535$	6953.01	0.561878
$536$	0	0
$537$	2137.02	0.171730
$538$	0	0
$539$	−5598.57	−0.447398
$540$	0	0
$541$	−17642.3	−1.40204	−0.701018	−	0.713144i	$-0.747271\pi$
$541$	−17642.3	−1.40204	−0.701018	+	0.713144i	$0.747271\pi$
$542$	0	0
$543$	−6619.59	−0.523156
$544$	0	0
$545$	14862.3	1.16813
$546$	0	0
$547$	−18414.9	−1.43943	−0.719713	−	0.694271i	$-0.755727\pi$
$547$	−18414.9	−1.43943	−0.719713	+	0.694271i	$0.755727\pi$
$548$	0	0
$549$	1683.61	0.130883
$550$	0	0
$551$	3164.20	0.244645
$552$	0	0
$553$	3727.66	0.286648
$554$	0	0
$555$	−3977.84	−0.304234
$556$	0	0
$557$	−8179.15	−0.622193	−0.311096	−	0.950378i	$-0.600696\pi$
$557$	−8179.15	−0.622193	−0.311096	+	0.950378i	$0.600696\pi$
$558$	0	0
$559$	−1968.30	−0.148927
$560$	0	0
$561$	−1578.14	−0.118769
$562$	0	0
$563$	−1880.07	−0.140738	−0.0703690	−	0.997521i	$-0.522418\pi$
$563$	−1880.07	−0.140738	−0.0703690	+	0.997521i	$0.522418\pi$
$564$	0	0
$565$	482.385	0.0359187
$566$	0	0
$567$	911.598	0.0675194
$568$	0	0
$569$	10118.3	0.745485	0.372743	−	0.927935i	$-0.378417\pi$
$569$	10118.3	0.745485	0.372743	+	0.927935i	$0.378417\pi$
$570$	0	0
$571$	23428.9	1.71711	0.858555	−	0.512721i	$-0.171362\pi$
$571$	23428.9	1.71711	0.858555	+	0.512721i	$0.171362\pi$
$572$	0	0
$573$	−4411.92	−0.321659
$574$	0	0
$575$	1027.52	0.0745225
$576$	0	0
$577$	20508.1	1.47966	0.739831	−	0.672793i	$-0.234906\pi$
$577$	20508.1	1.47966	0.739831	+	0.672793i	$0.234906\pi$
$578$	0	0
$579$	1108.68	0.0795771
$580$	0	0
$581$	−4900.49	−0.349925
$582$	0	0
$583$	−2069.87	−0.147042
$584$	0	0
$585$	−1337.57	−0.0945327
$586$	0	0
$587$	−5968.43	−0.419665	−0.209833	−	0.977737i	$-0.567292\pi$
$587$	−5968.43	−0.419665	−0.209833	+	0.977737i	$0.567292\pi$
$588$	0	0
$589$	−41219.0	−2.88353
$590$	0	0
$591$	12820.2	0.892308
$592$	0	0
$593$	−14659.5	−1.01517	−0.507584	−	0.861602i	$-0.669461\pi$
$593$	−14659.5	−1.01517	−0.507584	+	0.861602i	$0.669461\pi$
$594$	0	0
$595$	−2615.38	−0.180202
$596$	0	0
$597$	−12462.9	−0.854394
$598$	0	0
$599$	−23635.9	−1.61225	−0.806125	−	0.591746i	$-0.798439\pi$
$599$	−23635.9	−1.61225	−0.806125	+	0.591746i	$0.798439\pi$
$600$	0	0
$601$	−11527.0	−0.782356	−0.391178	−	0.920315i	$-0.627932\pi$
$601$	−11527.0	−0.782356	−0.391178	+	0.920315i	$0.627932\pi$
$602$	0	0
$603$	−5482.83	−0.370279
$604$	0	0
$605$	−7560.15	−0.508039
$606$	0	0
$607$	5098.56	0.340930	0.170465	−	0.985364i	$-0.445473\pi$
$607$	5098.56	0.340930	0.170465	+	0.985364i	$0.445473\pi$
$608$	0	0
$609$	690.525	0.0459466
$610$	0	0
$611$	−6075.74	−0.402288
$612$	0	0
$613$	−1516.39	−0.0999128	−0.0499564	−	0.998751i	$-0.515908\pi$
$613$	−1516.39	−0.0999128	−0.0499564	+	0.998751i	$0.515908\pi$
$614$	0	0
$615$	13416.3	0.879668
$616$	0	0
$617$	18539.3	1.20966	0.604832	−	0.796353i	$-0.293240\pi$
$617$	18539.3	1.20966	0.604832	+	0.796353i	$0.293240\pi$
$618$	0	0
$619$	25684.9	1.66779	0.833897	−	0.551920i	$-0.186105\pi$
$619$	25684.9	1.66779	0.833897	+	0.551920i	$0.186105\pi$
$620$	0	0
$621$	4871.30	0.314780
$622$	0	0
$623$	2917.49	0.187619
$624$	0	0
$625$	−16304.5	−1.04349
$626$	0	0
$627$	12011.1	0.765038
$628$	0	0
$629$	2357.67	0.149454
$630$	0	0
$631$	22410.9	1.41389	0.706945	−	0.707269i	$-0.250073\pi$
$631$	22410.9	1.41389	0.706945	+	0.707269i	$0.250073\pi$
$632$	0	0
$633$	3694.77	0.231997
$634$	0	0
$635$	3555.62	0.222206
$636$	0	0
$637$	2812.43	0.174933
$638$	0	0
$639$	−2232.34	−0.138200
$640$	0	0
$641$	6827.81	0.420721	0.210361	−	0.977624i	$-0.432536\pi$
$641$	6827.81	0.420721	0.210361	+	0.977624i	$0.432536\pi$
$642$	0	0
$643$	−23264.3	−1.42684	−0.713418	−	0.700738i	$-0.752854\pi$
$643$	−23264.3	−1.42684	−0.713418	+	0.700738i	$0.752854\pi$
$644$	0	0
$645$	5192.76	0.316999
$646$	0	0
$647$	−14745.9	−0.896014	−0.448007	−	0.894030i	$-0.647866\pi$
$647$	−14745.9	−0.896014	−0.448007	+	0.894030i	$0.647866\pi$
$648$	0	0
$649$	−22610.3	−1.36754
$650$	0	0
$651$	−8995.26	−0.541554
$652$	0	0
$653$	−10909.0	−0.653755	−0.326878	−	0.945067i	$-0.605997\pi$
$653$	−10909.0	−0.653755	−0.326878	+	0.945067i	$0.605997\pi$
$654$	0	0
$655$	22875.7	1.36462
$656$	0	0
$657$	7674.89	0.455747
$658$	0	0
$659$	−4182.99	−0.247263	−0.123631	−	0.992328i	$-0.539454\pi$
$659$	−4182.99	−0.247263	−0.123631	+	0.992328i	$0.539454\pi$
$660$	0	0
$661$	−2224.23	−0.130881	−0.0654406	−	0.997856i	$-0.520845\pi$
$661$	−2224.23	−0.130881	−0.0654406	+	0.997856i	$0.520845\pi$
$662$	0	0
$663$	792.776	0.0464387
$664$	0	0
$665$	19905.4	1.16075
$666$	0	0
$667$	3689.95	0.214206
$668$	0	0
$669$	6561.72	0.379209
$670$	0	0
$671$	4841.04	0.278519
$672$	0	0
$673$	−24152.5	−1.38337	−0.691687	−	0.722197i	$-0.743132\pi$
$673$	−24152.5	−1.38337	−0.691687	+	0.722197i	$0.743132\pi$
$674$	0	0
$675$	153.770	0.00876833
$676$	0	0
$677$	15310.7	0.869187	0.434593	−	0.900627i	$-0.356892\pi$
$677$	15310.7	0.869187	0.434593	+	0.900627i	$0.356892\pi$
$678$	0	0
$679$	13788.4	0.779310
$680$	0	0
$681$	12416.0	0.698652
$682$	0	0
$683$	11399.6	0.638646	0.319323	−	0.947646i	$-0.396545\pi$
$683$	11399.6	0.638646	0.319323	+	0.947646i	$0.396545\pi$
$684$	0	0
$685$	11872.6	0.662233
$686$	0	0
$687$	2506.06	0.139174
$688$	0	0
$689$	1039.79	0.0574935
$690$	0	0
$691$	−3323.23	−0.182955	−0.0914773	−	0.995807i	$-0.529159\pi$
$691$	−3323.23	−0.182955	−0.0914773	+	0.995807i	$0.529159\pi$
$692$	0	0
$693$	2621.20	0.143681
$694$	0	0
$695$	−32678.5	−1.78355
$696$	0	0
$697$	−7951.82	−0.432133
$698$	0	0
$699$	11056.5	0.598279
$700$	0	0
$701$	12670.4	0.682673	0.341336	−	0.939941i	$-0.389120\pi$
$701$	12670.4	0.682673	0.341336	+	0.939941i	$0.389120\pi$
$702$	0	0
$703$	−17944.0	−0.962692
$704$	0	0
$705$	16029.0	0.856295
$706$	0	0
$707$	−7263.71	−0.386393
$708$	0	0
$709$	−13075.2	−0.692594	−0.346297	−	0.938125i	$-0.612561\pi$
$709$	−13075.2	−0.692594	−0.346297	+	0.938125i	$0.612561\pi$
$710$	0	0
$711$	2980.99	0.157237
$712$	0	0
$713$	−48067.8	−2.52476
$714$	0	0
$715$	−3846.03	−0.201165
$716$	0	0
$717$	−9078.62	−0.472869
$718$	0	0
$719$	2988.41	0.155005	0.0775026	−	0.996992i	$-0.475305\pi$
$719$	2988.41	0.155005	0.0775026	+	0.996992i	$0.475305\pi$
$720$	0	0
$721$	5752.48	0.297134
$722$	0	0
$723$	9796.73	0.503934
$724$	0	0
$725$	116.479	0.00596680
$726$	0	0
$727$	5507.46	0.280963	0.140482	−	0.990083i	$-0.455135\pi$
$727$	5507.46	0.280963	0.140482	+	0.990083i	$0.455135\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−3077.75	−0.155725
$732$	0	0
$733$	36585.2	1.84353	0.921764	−	0.387751i	$-0.126748\pi$
$733$	36585.2	1.84353	0.921764	+	0.387751i	$0.126748\pi$
$734$	0	0
$735$	−7419.75	−0.372356
$736$	0	0
$737$	−15765.3	−0.787953
$738$	0	0
$739$	6425.89	0.319865	0.159933	−	0.987128i	$-0.448872\pi$
$739$	6425.89	0.319865	0.159933	+	0.987128i	$0.448872\pi$
$740$	0	0
$741$	−6033.76	−0.299131
$742$	0	0
$743$	−20411.0	−1.00782	−0.503908	−	0.863757i	$-0.668105\pi$
$743$	−20411.0	−1.00782	−0.503908	+	0.863757i	$0.668105\pi$
$744$	0	0
$745$	−8499.60	−0.417988
$746$	0	0
$747$	−3918.89	−0.191947
$748$	0	0
$749$	6844.81	0.333917
$750$	0	0
$751$	24259.5	1.17875	0.589375	−	0.807860i	$-0.299374\pi$
$751$	24259.5	1.17875	0.589375	+	0.807860i	$0.299374\pi$
$752$	0	0
$753$	−19089.5	−0.923850
$754$	0	0
$755$	−26033.9	−1.25493
$756$	0	0
$757$	−9295.39	−0.446297	−0.223148	−	0.974785i	$-0.571633\pi$
$757$	−9295.39	−0.446297	−0.223148	+	0.974785i	$0.571633\pi$
$758$	0	0
$759$	14006.9	0.669851
$760$	0	0
$761$	−21974.7	−1.04676	−0.523378	−	0.852101i	$-0.675328\pi$
$761$	−21974.7	−1.04676	−0.523378	+	0.852101i	$0.675328\pi$
$762$	0	0
$763$	14631.0	0.694204
$764$	0	0
$765$	−2091.50	−0.0988476
$766$	0	0
$767$	11358.2	0.534709
$768$	0	0
$769$	22987.4	1.07795	0.538977	−	0.842320i	$-0.318811\pi$
$769$	22987.4	1.07795	0.538977	+	0.842320i	$0.318811\pi$
$770$	0	0
$771$	−18256.1	−0.852759
$772$	0	0
$773$	−31970.9	−1.48760	−0.743799	−	0.668404i	$-0.766978\pi$
$773$	−31970.9	−1.48760	−0.743799	+	0.668404i	$0.766978\pi$
$774$	0	0
$775$	−1517.34	−0.0703283
$776$	0	0
$777$	−3915.94	−0.180803
$778$	0	0
$779$	60520.8	2.78354
$780$	0	0
$781$	−6418.85	−0.294090
$782$	0	0
$783$	552.209	0.0252035
$784$	0	0
$785$	−36280.2	−1.64955
$786$	0	0
$787$	6087.26	0.275715	0.137857	−	0.990452i	$-0.455978\pi$
$787$	6087.26	0.275715	0.137857	+	0.990452i	$0.455978\pi$
$788$	0	0
$789$	−369.680	−0.0166805
$790$	0	0
$791$	474.878	0.0213460
$792$	0	0
$793$	−2431.88	−0.108901
$794$	0	0
$795$	−2743.19	−0.122378
$796$	0	0
$797$	−23080.0	−1.02577	−0.512883	−	0.858458i	$-0.671423\pi$
$797$	−23080.0	−1.02577	−0.512883	+	0.858458i	$0.671423\pi$
$798$	0	0
$799$	−9500.41	−0.420651
$800$	0	0
$801$	2333.10	0.102916
$802$	0	0
$803$	22068.3	0.969829
$804$	0	0
$805$	23212.9	1.01633
$806$	0	0
$807$	5807.37	0.253320
$808$	0	0
$809$	−32377.8	−1.40710	−0.703550	−	0.710646i	$-0.748403\pi$
$809$	−32377.8	−1.40710	−0.703550	+	0.710646i	$0.748403\pi$
$810$	0	0
$811$	26352.8	1.14103	0.570513	−	0.821288i	$-0.306744\pi$
$811$	26352.8	1.14103	0.570513	+	0.821288i	$0.306744\pi$
$812$	0	0
$813$	13838.1	0.596952
$814$	0	0
$815$	−26461.5	−1.13731
$816$	0	0
$817$	23424.5	1.00308
$818$	0	0
$819$	−1316.75	−0.0561796
$820$	0	0
$821$	−35355.3	−1.50294	−0.751468	−	0.659770i	$-0.770654\pi$
$821$	−35355.3	−1.50294	−0.751468	+	0.659770i	$0.770654\pi$
$822$	0	0
$823$	12663.3	0.536347	0.268173	−	0.963371i	$-0.413580\pi$
$823$	12663.3	0.536347	0.268173	+	0.963371i	$0.413580\pi$
$824$	0	0
$825$	442.149	0.0186590
$826$	0	0
$827$	−16295.2	−0.685176	−0.342588	−	0.939486i	$-0.611303\pi$
$827$	−16295.2	−0.685176	−0.342588	+	0.939486i	$0.611303\pi$
$828$	0	0
$829$	−13638.9	−0.571411	−0.285705	−	0.958318i	$-0.592228\pi$
$829$	−13638.9	−0.571411	−0.285705	+	0.958318i	$0.592228\pi$
$830$	0	0
$831$	17502.9	0.730649
$832$	0	0
$833$	4397.69	0.182918
$834$	0	0
$835$	30474.2	1.26300
$836$	0	0
$837$	−7193.46	−0.297064
$838$	0	0
$839$	1890.31	0.0777838	0.0388919	−	0.999243i	$-0.487617\pi$
$839$	1890.31	0.0777838	0.0388919	+	0.999243i	$0.487617\pi$
$840$	0	0
$841$	−23970.7	−0.982849
$842$	0	0
$843$	14075.7	0.575082
$844$	0	0
$845$	1932.04	0.0786559
$846$	0	0
$847$	−7442.50	−0.301921
$848$	0	0
$849$	10396.8	0.420279
$850$	0	0
$851$	−20925.6	−0.842914
$852$	0	0
$853$	−1620.21	−0.0650351	−0.0325175	−	0.999471i	$-0.510352\pi$
$853$	−1620.21	−0.0650351	−0.0325175	+	0.999471i	$0.510352\pi$
$854$	0	0
$855$	15918.3	0.636718
$856$	0	0
$857$	−14508.4	−0.578292	−0.289146	−	0.957285i	$-0.593371\pi$
$857$	−14508.4	−0.578292	−0.289146	+	0.957285i	$0.593371\pi$
$858$	0	0
$859$	29639.8	1.17730	0.588648	−	0.808389i	$-0.299660\pi$
$859$	29639.8	1.17730	0.588648	+	0.808389i	$0.299660\pi$
$860$	0	0
$861$	13207.5	0.522776
$862$	0	0
$863$	−21528.8	−0.849186	−0.424593	−	0.905384i	$-0.639583\pi$
$863$	−21528.8	−0.849186	−0.424593	+	0.905384i	$0.639583\pi$
$864$	0	0
$865$	1889.10	0.0742557
$866$	0	0
$867$	−13499.4	−0.528792
$868$	0	0
$869$	8571.50	0.334601
$870$	0	0
$871$	7919.65	0.308091
$872$	0	0
$873$	11026.5	0.427482
$874$	0	0
$875$	−15349.9	−0.593054
$876$	0	0
$877$	−14865.3	−0.572366	−0.286183	−	0.958175i	$-0.592387\pi$
$877$	−14865.3	−0.572366	−0.286183	+	0.958175i	$0.592387\pi$
$878$	0	0
$879$	−8031.92	−0.308202
$880$	0	0
$881$	−21336.0	−0.815921	−0.407961	−	0.913000i	$-0.633760\pi$
$881$	−21336.0	−0.815921	−0.407961	+	0.913000i	$0.633760\pi$
$882$	0	0
$883$	37538.2	1.43065	0.715323	−	0.698794i	$-0.246280\pi$
$883$	37538.2	1.43065	0.715323	+	0.698794i	$0.246280\pi$
$884$	0	0
$885$	−29965.3	−1.13816
$886$	0	0
$887$	−34575.0	−1.30881	−0.654406	−	0.756144i	$-0.727081\pi$
$887$	−34575.0	−1.30881	−0.654406	+	0.756144i	$0.727081\pi$
$888$	0	0
$889$	3500.29	0.132054
$890$	0	0
$891$	2096.16	0.0788148
$892$	0	0
$893$	72306.9	2.70958
$894$	0	0
$895$	8143.61	0.304146
$896$	0	0
$897$	−7036.32	−0.261913
$898$	0	0
$899$	−5448.96	−0.202150
$900$	0	0
$901$	1625.89	0.0601178
$902$	0	0
$903$	5111.95	0.188389
$904$	0	0
$905$	−25225.5	−0.926545
$906$	0	0
$907$	−10424.8	−0.381641	−0.190820	−	0.981625i	$-0.561115\pi$
$907$	−10424.8	−0.381641	−0.190820	+	0.981625i	$0.561115\pi$
$908$	0	0
$909$	−5808.75	−0.211952
$910$	0	0
$911$	−10961.8	−0.398661	−0.199331	−	0.979932i	$-0.563877\pi$
$911$	−10961.8	−0.398661	−0.199331	+	0.979932i	$0.563877\pi$
$912$	0	0
$913$	−11268.3	−0.408464
$914$	0	0
$915$	6415.80	0.231803
$916$	0	0
$917$	22519.7	0.810976
$918$	0	0
$919$	10779.2	0.386914	0.193457	−	0.981109i	$-0.438030\pi$
$919$	10779.2	0.386914	0.193457	+	0.981109i	$0.438030\pi$
$920$	0	0
$921$	1415.74	0.0506519
$922$	0	0
$923$	3224.49	0.114990
$924$	0	0
$925$	−660.549	−0.0234797
$926$	0	0
$927$	4600.23	0.162990
$928$	0	0
$929$	5429.07	0.191735	0.0958675	−	0.995394i	$-0.469437\pi$
$929$	5429.07	0.191735	0.0958675	+	0.995394i	$0.469437\pi$
$930$	0	0
$931$	−33470.5	−1.17825
$932$	0	0
$933$	4555.55	0.159852
$934$	0	0
$935$	−6013.88	−0.210348
$936$	0	0
$937$	−21300.1	−0.742631	−0.371315	−	0.928507i	$-0.621093\pi$
$937$	−21300.1	−0.742631	−0.371315	+	0.928507i	$0.621093\pi$
$938$	0	0
$939$	12149.6	0.422244
$940$	0	0
$941$	−26851.2	−0.930207	−0.465103	−	0.885256i	$-0.653983\pi$
$941$	−26851.2	−0.930207	−0.465103	+	0.885256i	$0.653983\pi$
$942$	0	0
$943$	70576.7	2.43722
$944$	0	0
$945$	3473.86	0.119582
$946$	0	0
$947$	−8021.68	−0.275258	−0.137629	−	0.990484i	$-0.543948\pi$
$947$	−8021.68	−0.275258	−0.137629	+	0.990484i	$0.543948\pi$
$948$	0	0
$949$	−11085.9	−0.379204
$950$	0	0
$951$	−9761.88	−0.332861
$952$	0	0
$953$	35715.0	1.21398	0.606990	−	0.794709i	$-0.292377\pi$
$953$	35715.0	1.21398	0.606990	+	0.794709i	$0.292377\pi$
$954$	0	0
$955$	−16812.6	−0.569680
$956$	0	0
$957$	1587.82	0.0536330
$958$	0	0
$959$	11687.9	0.393557
$960$	0	0
$961$	41190.9	1.38266
$962$	0	0
$963$	5473.76	0.183166
$964$	0	0
$965$	4224.88	0.140936
$966$	0	0
$967$	−53338.8	−1.77380	−0.886898	−	0.461965i	$-0.847145\pi$
$967$	−53338.8	−1.77380	−0.886898	+	0.461965i	$0.847145\pi$
$968$	0	0
$969$	−9434.77	−0.312785
$970$	0	0
$971$	23112.9	0.763882	0.381941	−	0.924187i	$-0.375256\pi$
$971$	23112.9	0.763882	0.381941	+	0.924187i	$0.375256\pi$
$972$	0	0
$973$	−32170.0	−1.05994
$974$	0	0
$975$	−222.113	−0.00729569
$976$	0	0
$977$	−52874.6	−1.73143	−0.865715	−	0.500538i	$-0.833136\pi$
$977$	−52874.6	−1.73143	−0.865715	+	0.500538i	$0.833136\pi$
$978$	0	0
$979$	6708.57	0.219006
$980$	0	0
$981$	11700.3	0.380798
$982$	0	0
$983$	−45173.1	−1.46572	−0.732858	−	0.680381i	$-0.761814\pi$
$983$	−45173.1	−1.46572	−0.732858	+	0.680381i	$0.761814\pi$
$984$	0	0
$985$	48854.5	1.58034
$986$	0	0
$987$	15779.6	0.508885
$988$	0	0
$989$	27316.7	0.878281
$990$	0	0
$991$	−60485.6	−1.93884	−0.969418	−	0.245414i	$-0.921076\pi$
$991$	−60485.6	−1.93884	−0.969418	+	0.245414i	$0.921076\pi$
$992$	0	0
$993$	−10267.3	−0.328121
$994$	0	0
$995$	−47492.8	−1.51319
$996$	0	0
$997$	18108.1	0.575214	0.287607	−	0.957749i	$-0.407140\pi$
$997$	18108.1	0.575214	0.287607	+	0.957749i	$0.407140\pi$
$998$	0	0
$999$	−3131.56	−0.0991773

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.4.a.bp.1.3		3
4.3	odd	2		2496.4.a.bl.1.3		3
8.3	odd	2		39.4.a.c.1.3	✓	3
8.5	even	2		624.4.a.t.1.1		3
24.5	odd	2		1872.4.a.bk.1.3		3
24.11	even	2		117.4.a.f.1.1		3
40.19	odd	2		975.4.a.l.1.1		3
56.27	even	2		1911.4.a.k.1.3		3
104.51	odd	2		507.4.a.h.1.1		3
104.83	even	4		507.4.b.g.337.1		6
104.99	even	4		507.4.b.g.337.6		6
312.155	even	2		1521.4.a.u.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.c.1.3	✓	3	8.3	odd	2
117.4.a.f.1.1		3	24.11	even	2
507.4.a.h.1.1		3	104.51	odd	2
507.4.b.g.337.1		6	104.83	even	4
507.4.b.g.337.6		6	104.99	even	4
624.4.a.t.1.1		3	8.5	even	2
975.4.a.l.1.1		3	40.19	odd	2
1521.4.a.u.1.3		3	312.155	even	2
1872.4.a.bk.1.3		3	24.5	odd	2
1911.4.a.k.1.3		3	56.27	even	2
2496.4.a.bl.1.3		3	4.3	odd	2
2496.4.a.bp.1.3		3	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +11.4322 q^{5} +11.2543 q^{7} +9.00000 q^{9} +25.8785 q^{11} -13.0000 q^{13} +34.2966 q^{15} -20.3276 q^{17} +154.712 q^{19} +33.7629 q^{21} +180.418 q^{23} +5.69520 q^{25} +27.0000 q^{27} +20.4522 q^{29} -266.424 q^{31} +77.6355 q^{33} +128.661 q^{35} -115.984 q^{37} -39.0000 q^{39} +391.184 q^{41} +151.407 q^{43} +102.890 q^{45} +467.365 q^{47} -216.341 q^{49} -60.9828 q^{51} -79.9842 q^{53} +295.848 q^{55} +464.136 q^{57} -873.710 q^{59} +187.068 q^{61} +101.289 q^{63} -148.619 q^{65} -609.204 q^{67} +541.255 q^{69} -248.038 q^{71} +852.765 q^{73} +17.0856 q^{75} +291.244 q^{77} +331.221 q^{79} +81.0000 q^{81} -435.432 q^{83} -232.389 q^{85} +61.3566 q^{87} +259.233 q^{89} -146.306 q^{91} -799.273 q^{93} +1768.70 q^{95} +1225.17 q^{97} +232.907 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 9 q^{3} - 4 q^{5} - 30 q^{7} + 27 q^{9} - 16 q^{11} - 39 q^{13} - 12 q^{15} - 146 q^{17} + 94 q^{19} - 90 q^{21} + 48 q^{23} + 145 q^{25} + 81 q^{27} + 2 q^{29} - 302 q^{31} - 48 q^{33} + 80 q^{35}+ \cdots - 144 q^{99}+O(q^{100}) \) 3 * q + 9 * q^3 - 4 * q^5 - 30 * q^7 + 27 * q^9 - 16 * q^11 - 39 * q^13 - 12 * q^15 - 146 * q^17 + 94 * q^19 - 90 * q^21 + 48 * q^23 + 145 * q^25 + 81 * q^27 + 2 * q^29 - 302 * q^31 - 48 * q^33 + 80 * q^35 - 374 * q^37 - 117 * q^39 + 480 * q^41 - 260 * q^43 - 36 * q^45 + 24 * q^47 + 447 * q^49 - 438 * q^51 + 678 * q^53 + 1552 * q^55 + 282 * q^57 - 1788 * q^59 - 230 * q^61 - 270 * q^63 + 52 * q^65 + 74 * q^67 + 144 * q^69 + 948 * q^71 - 222 * q^73 + 435 * q^75 - 112 * q^77 + 24 * q^79 + 243 * q^81 - 796 * q^83 + 248 * q^85 + 6 * q^87 + 1436 * q^89 + 390 * q^91 - 906 * q^93 + 4032 * q^95 + 3242 * q^97 - 144 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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