LMFDB - Embedded newform 2496.4.a.bl.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2496,4,Mod(1,2496)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2496.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:	$147.268767374$
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)

$f(q)$	$=$	$q-3.00000 q^{3} +11.4322 q^{5} -11.2543 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +11.4322 q^{5} -11.2543 q^{7} +9.00000 q^{9} -25.8785 q^{11} -13.0000 q^{13} -34.2966 q^{15} -20.3276 q^{17} -154.712 q^{19} +33.7629 q^{21} -180.418 q^{23} +5.69520 q^{25} -27.0000 q^{27} +20.4522 q^{29} +266.424 q^{31} +77.6355 q^{33} -128.661 q^{35} -115.984 q^{37} +39.0000 q^{39} +391.184 q^{41} -151.407 q^{43} +102.890 q^{45} -467.365 q^{47} -216.341 q^{49} +60.9828 q^{51} -79.9842 q^{53} -295.848 q^{55} +464.136 q^{57} +873.710 q^{59} +187.068 q^{61} -101.289 q^{63} -148.619 q^{65} +609.204 q^{67} +541.255 q^{69} +248.038 q^{71} +852.765 q^{73} -17.0856 q^{75} +291.244 q^{77} -331.221 q^{79} +81.0000 q^{81} +435.432 q^{83} -232.389 q^{85} -61.3566 q^{87} +259.233 q^{89} +146.306 q^{91} -799.273 q^{93} -1768.70 q^{95} +1225.17 q^{97} -232.907 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} - 4 q^{5} + 30 q^{7} + 27 q^{9}+O(q^{10}) $ 3 * q - 9 * q^3 - 4 * q^5 + 30 * q^7 + 27 * q^9	$ 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}+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

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
$3$	−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	$-0.598268\pi$
$7$	−11.2543	−0.607675	−0.303838	+	0.952724i	$0.598268\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−25.8785	−0.709333	−0.354666	−	0.934993i	$-0.615406\pi$
$11$	−25.8785	−0.709333	−0.354666	+	0.934993i	$0.615406\pi$
$12$	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	$-0.883738\pi$
$19$	−154.712	−1.86807	−0.934035	+	0.357181i	$0.883738\pi$
$20$	0	0
$21$	33.7629	0.350841
$22$	0	0
$23$	−180.418	−1.63565	−0.817823	−	0.575471i	$-0.804819\pi$
$23$	−180.418	−1.63565	−0.817823	+	0.575471i	$0.804819\pi$
$24$	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	$-0.219360\pi$
$31$	266.424	1.54359	0.771794	+	0.635873i	$0.219360\pi$
$32$	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	$-0.586522\pi$
$43$	−151.407	−0.536963	−0.268482	+	0.963285i	$0.586522\pi$
$44$	0	0
$45$	102.890	0.340842
$46$	0	0
$47$	−467.365	−1.45047	−0.725236	−	0.688500i	$-0.758269\pi$
$47$	−467.365	−1.45047	−0.725236	+	0.688500i	$0.758269\pi$
$48$	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	$-0.0857171\pi$
$59$	873.710	1.92792	0.963960	+	0.266045i	$0.0857171\pi$
$60$	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	$-0.312558\pi$
$67$	609.204	1.11084	0.555418	+	0.831571i	$0.312558\pi$
$68$	0	0
$69$	541.255	0.944340
$70$	0	0
$71$	248.038	0.414601	0.207301	−	0.978277i	$-0.433532\pi$
$71$	248.038	0.414601	0.207301	+	0.978277i	$0.433532\pi$
$72$	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	$-0.575789\pi$
$79$	−331.221	−0.471712	−0.235856	+	0.971788i	$0.575789\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	435.432	0.575842	0.287921	−	0.957654i	$-0.407036\pi$
$83$	435.432	0.575842	0.287921	+	0.957654i	$0.407036\pi$
$84$	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	$-0.578619\pi$
$103$	−511.137	−0.488969	−0.244484	+	0.969653i	$0.578619\pi$
$104$	0	0
$105$	385.984	0.358745
$106$	0	0
$107$	−608.195	−0.549499	−0.274750	−	0.961516i	$-0.588595\pi$
$107$	−608.195	−0.549499	−0.274750	+	0.961516i	$0.588595\pi$
$108$	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	$-0.534654\pi$
$127$	−311.018	−0.217310	−0.108655	+	0.994080i	$0.534654\pi$
$128$	0	0
$129$	454.222	0.310016
$130$	0	0
$131$	−2000.98	−1.33456	−0.667278	−	0.744809i	$-0.732541\pi$
$131$	−2000.98	−1.33456	−0.667278	+	0.744809i	$0.732541\pi$
$132$	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	$-0.162739\pi$
$139$	2858.46	1.74426	0.872128	+	0.489277i	$0.162739\pi$
$140$	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	$-0.289705\pi$
$151$	2277.24	1.22728	0.613640	+	0.789586i	$0.289705\pi$
$152$	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	$-0.312287\pi$
$163$	2314.65	1.11225	0.556126	+	0.831098i	$0.312287\pi$
$164$	0	0
$165$	887.545	0.418759
$166$	0	0
$167$	−2665.65	−1.23517	−0.617587	−	0.786502i	$-0.711890\pi$
$167$	−2665.65	−1.23517	−0.617587	+	0.786502i	$0.711890\pi$
$168$	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	$-0.547516\pi$
$179$	−712.339	−0.297446	−0.148723	+	0.988879i	$0.547516\pi$
$180$	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	$-0.410141\pi$
$191$	1470.64	0.557129	0.278565	+	0.960417i	$0.410141\pi$
$192$	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	$-0.234860\pi$
$199$	4154.31	1.47985	0.739927	+	0.672687i	$0.234860\pi$
$200$	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	$-0.564392\pi$
$211$	−1231.59	−0.401830	−0.200915	+	0.979609i	$0.564392\pi$
$212$	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	$-0.606511\pi$
$223$	−2187.24	−0.656809	−0.328404	+	0.944537i	$0.606511\pi$
$224$	0	0
$225$	51.2568	0.0151872
$226$	0	0
$227$	−4138.67	−1.21010	−0.605051	−	0.796187i	$-0.706847\pi$
$227$	−4138.67	−1.21010	−0.605051	+	0.796187i	$0.706847\pi$
$228$	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	$-0.365697\pi$
$239$	3026.21	0.819034	0.409517	+	0.912303i	$0.365697\pi$
$240$	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	$-0.204792\pi$
$251$	6363.16	1.60016	0.800078	+	0.599897i	$0.204792\pi$
$252$	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	$-0.495402\pi$
$263$	123.227	0.0288916	0.0144458	+	0.999896i	$0.495402\pi$
$264$	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	$-0.672942\pi$
$271$	−4612.69	−1.03395	−0.516976	+	0.856000i	$0.672942\pi$
$272$	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	$-0.618580\pi$
$283$	−3465.60	−0.727945	−0.363973	+	0.931410i	$0.618580\pi$
$284$	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	$-0.513967\pi$
$307$	−471.915	−0.0877316	−0.0438658	+	0.999037i	$0.513967\pi$
$308$	0	0
$309$	1533.41	0.282306
$310$	0	0
$311$	−1518.52	−0.276872	−0.138436	−	0.990371i	$-0.544207\pi$
$311$	−1518.52	−0.276872	−0.138436	+	0.990371i	$0.544207\pi$
$312$	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	$-0.408285\pi$
$331$	3422.45	0.568322	0.284161	+	0.958777i	$0.408285\pi$
$332$	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	$-0.505341\pi$
$347$	−216.898	−0.0335554	−0.0167777	+	0.999859i	$0.505341\pi$
$348$	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	$-0.412662\pi$
$359$	3686.04	0.541899	0.270949	+	0.962594i	$0.412662\pi$
$360$	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	$-0.579384\pi$
$367$	−3470.59	−0.493633	−0.246816	+	0.969062i	$0.579384\pi$
$368$	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	$-0.507456\pi$
$379$	−345.604	−0.0468403	−0.0234202	+	0.999726i	$0.507456\pi$
$380$	0	0
$381$	933.055	0.125464
$382$	0	0
$383$	−3386.40	−0.451793	−0.225897	−	0.974151i	$-0.572531\pi$
$383$	−3386.40	−0.451793	−0.225897	+	0.974151i	$0.572531\pi$
$384$	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	$-0.304084\pi$
$419$	9903.67	1.15472	0.577358	+	0.816491i	$0.304084\pi$
$420$	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	$-0.776779\pi$
$431$	−13672.6	−1.52805	−0.764023	+	0.645189i	$0.776779\pi$
$432$	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	$-0.606157\pi$
$439$	−6022.04	−0.654707	−0.327353	+	0.944902i	$0.606157\pi$
$440$	0	0
$441$	−1947.07	−0.210244
$442$	0	0
$443$	12994.4	1.39364	0.696821	−	0.717245i	$-0.254597\pi$
$443$	12994.4	1.39364	0.696821	+	0.717245i	$0.254597\pi$
$444$	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	$-0.472177\pi$
$463$	1739.42	0.174596	0.0872979	+	0.996182i	$0.472177\pi$
$464$	0	0
$465$	−9137.45	−0.911267
$466$	0	0
$467$	7958.82	0.788630	0.394315	−	0.918975i	$-0.370982\pi$
$467$	7958.82	0.788630	0.394315	+	0.918975i	$0.370982\pi$
$468$	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	$-0.368260\pi$
$479$	8431.98	0.804315	0.402158	+	0.915570i	$0.368260\pi$
$480$	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	$-0.682945\pi$
$487$	−11684.7	−1.08723	−0.543617	+	0.839334i	$0.682945\pi$
$488$	0	0
$489$	−6943.94	−0.642159
$490$	0	0
$491$	3954.70	0.363489	0.181745	−	0.983346i	$-0.441826\pi$
$491$	3954.70	0.363489	0.181745	+	0.983346i	$0.441826\pi$
$492$	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	$-0.417843\pi$
$499$	5690.37	0.510493	0.255246	+	0.966876i	$0.417843\pi$
$500$	0	0
$501$	7996.95	0.713128
$502$	0	0
$503$	10859.1	0.962595	0.481298	−	0.876557i	$-0.340166\pi$
$503$	10859.1	0.962595	0.481298	+	0.876557i	$0.340166\pi$
$504$	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	$-0.432753\pi$
$523$	5016.11	0.419386	0.209693	+	0.977767i	$0.432753\pi$
$524$	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	$-0.244273\pi$
$547$	18414.9	1.43943	0.719713	+	0.694271i	$0.244273\pi$
$548$	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	$-0.477582\pi$
$563$	1880.07	0.140738	0.0703690	+	0.997521i	$0.477582\pi$
$564$	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	$-0.828638\pi$
$571$	−23428.9	−1.71711	−0.858555	+	0.512721i	$0.828638\pi$
$572$	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	$-0.432708\pi$
$587$	5968.43	0.419665	0.209833	+	0.977737i	$0.432708\pi$
$588$	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	$-0.201561\pi$
$599$	23635.9	1.61225	0.806125	+	0.591746i	$0.201561\pi$
$600$	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	$-0.554527\pi$
$607$	−5098.56	−0.340930	−0.170465	+	0.985364i	$0.554527\pi$
$608$	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	$-0.813895\pi$
$619$	−25684.9	−1.66779	−0.833897	+	0.551920i	$0.813895\pi$
$620$	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	$-0.749927\pi$
$631$	−22410.9	−1.41389	−0.706945	+	0.707269i	$0.749927\pi$
$632$	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	$-0.247146\pi$
$643$	23264.3	1.42684	0.713418	+	0.700738i	$0.247146\pi$
$644$	0	0
$645$	5192.76	0.316999
$646$	0	0
$647$	14745.9	0.896014	0.448007	−	0.894030i	$-0.352134\pi$
$647$	14745.9	0.896014	0.448007	+	0.894030i	$0.352134\pi$
$648$	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	$-0.460546\pi$
$659$	4182.99	0.247263	0.123631	+	0.992328i	$0.460546\pi$
$660$	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	$-0.603455\pi$
$683$	−11399.6	−0.638646	−0.319323	+	0.947646i	$0.603455\pi$
$684$	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	$-0.470841\pi$
$691$	3323.23	0.182955	0.0914773	+	0.995807i	$0.470841\pi$
$692$	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	$-0.524695\pi$
$719$	−2988.41	−0.155005	−0.0775026	+	0.996992i	$0.524695\pi$
$720$	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	$-0.544865\pi$
$727$	−5507.46	−0.280963	−0.140482	+	0.990083i	$0.544865\pi$
$728$	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	$-0.551128\pi$
$739$	−6425.89	−0.319865	−0.159933	+	0.987128i	$0.551128\pi$
$740$	0	0
$741$	−6033.76	−0.299131
$742$	0	0
$743$	20411.0	1.00782	0.503908	−	0.863757i	$-0.331895\pi$
$743$	20411.0	1.00782	0.503908	+	0.863757i	$0.331895\pi$
$744$	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	$-0.700626\pi$
$751$	−24259.5	−1.17875	−0.589375	+	0.807860i	$0.700626\pi$
$752$	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	$-0.544022\pi$
$787$	−6087.26	−0.275715	−0.137857	+	0.990452i	$0.544022\pi$
$788$	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	$-0.693256\pi$
$811$	−26352.8	−1.14103	−0.570513	+	0.821288i	$0.693256\pi$
$812$	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	$-0.586420\pi$
$823$	−12663.3	−0.536347	−0.268173	+	0.963371i	$0.586420\pi$
$824$	0	0
$825$	442.149	0.0186590
$826$	0	0
$827$	16295.2	0.685176	0.342588	−	0.939486i	$-0.388697\pi$
$827$	16295.2	0.685176	0.342588	+	0.939486i	$0.388697\pi$
$828$	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	$-0.512383\pi$
$839$	−1890.31	−0.0777838	−0.0388919	+	0.999243i	$0.512383\pi$
$840$	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	$-0.700340\pi$
$859$	−29639.8	−1.17730	−0.588648	+	0.808389i	$0.700340\pi$
$860$	0	0
$861$	13207.5	0.522776
$862$	0	0
$863$	21528.8	0.849186	0.424593	−	0.905384i	$-0.360417\pi$
$863$	21528.8	0.849186	0.424593	+	0.905384i	$0.360417\pi$
$864$	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	$-0.753720\pi$
$883$	−37538.2	−1.43065	−0.715323	+	0.698794i	$0.753720\pi$
$884$	0	0
$885$	−29965.3	−1.13816
$886$	0	0
$887$	34575.0	1.30881	0.654406	−	0.756144i	$-0.272919\pi$
$887$	34575.0	1.30881	0.654406	+	0.756144i	$0.272919\pi$
$888$	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	$-0.438885\pi$
$907$	10424.8	0.381641	0.190820	+	0.981625i	$0.438885\pi$
$908$	0	0
$909$	−5808.75	−0.211952
$910$	0	0
$911$	10961.8	0.398661	0.199331	−	0.979932i	$-0.436123\pi$
$911$	10961.8	0.398661	0.199331	+	0.979932i	$0.436123\pi$
$912$	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	$-0.561970\pi$
$919$	−10779.2	−0.386914	−0.193457	+	0.981109i	$0.561970\pi$
$920$	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	$-0.456052\pi$
$947$	8021.68	0.275258	0.137629	+	0.990484i	$0.456052\pi$
$948$	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	$-0.152855\pi$
$967$	53338.8	1.77380	0.886898	+	0.461965i	$0.152855\pi$
$968$	0	0
$969$	−9434.77	−0.312785
$970$	0	0
$971$	−23112.9	−0.763882	−0.381941	−	0.924187i	$-0.624744\pi$
$971$	−23112.9	−0.763882	−0.381941	+	0.924187i	$0.624744\pi$
$972$	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	$-0.238186\pi$
$983$	45173.1	1.46572	0.732858	+	0.680381i	$0.238186\pi$
$984$	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	$-0.0789239\pi$
$991$	60485.6	1.93884	0.969418	+	0.245414i	$0.0789239\pi$
$992$	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.bl.1.3		3
4.3	odd	2		2496.4.a.bp.1.3		3
8.3	odd	2		624.4.a.t.1.1		3
8.5	even	2		39.4.a.c.1.3	✓	3
24.5	odd	2		117.4.a.f.1.1		3
24.11	even	2		1872.4.a.bk.1.3		3
40.29	even	2		975.4.a.l.1.1		3
56.13	odd	2		1911.4.a.k.1.3		3
104.5	odd	4		507.4.b.g.337.1		6
104.21	odd	4		507.4.b.g.337.6		6
104.77	even	2		507.4.a.h.1.1		3
312.77	odd	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	8.5	even	2
117.4.a.f.1.1		3	24.5	odd	2
507.4.a.h.1.1		3	104.77	even	2
507.4.b.g.337.1		6	104.5	odd	4
507.4.b.g.337.6		6	104.21	odd	4
624.4.a.t.1.1		3	8.3	odd	2
975.4.a.l.1.1		3	40.29	even	2
1521.4.a.u.1.3		3	312.77	odd	2
1872.4.a.bk.1.3		3	24.11	even	2
1911.4.a.k.1.3		3	56.13	odd	2
2496.4.a.bl.1.3		3	1.1	even	1	trivial
2496.4.a.bp.1.3		3	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +11.4322 q^{5} -11.2543 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +11.4322 q^{5} -11.2543 q^{7} +9.00000 q^{9} -25.8785 q^{11} -13.0000 q^{13} -34.2966 q^{15} -20.3276 q^{17} -154.712 q^{19} +33.7629 q^{21} -180.418 q^{23} +5.69520 q^{25} -27.0000 q^{27} +20.4522 q^{29} +266.424 q^{31} +77.6355 q^{33} -128.661 q^{35} -115.984 q^{37} +39.0000 q^{39} +391.184 q^{41} -151.407 q^{43} +102.890 q^{45} -467.365 q^{47} -216.341 q^{49} +60.9828 q^{51} -79.9842 q^{53} -295.848 q^{55} +464.136 q^{57} +873.710 q^{59} +187.068 q^{61} -101.289 q^{63} -148.619 q^{65} +609.204 q^{67} +541.255 q^{69} +248.038 q^{71} +852.765 q^{73} -17.0856 q^{75} +291.244 q^{77} -331.221 q^{79} +81.0000 q^{81} +435.432 q^{83} -232.389 q^{85} -61.3566 q^{87} +259.233 q^{89} +146.306 q^{91} -799.273 q^{93} -1768.70 q^{95} +1225.17 q^{97} -232.907 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} - 4 q^{5} + 30 q^{7} + 27 q^{9}+O(q^{10}) \) 3 * q - 9 * q^3 - 4 * q^5 + 30 * q^7 + 27 * q^9	\( 3 q - 9 q^{3} - 4 q^{5} + 30 q^{7} + 27 q^{9} + 16 q^{11} - 39 q^{13} + 12 q^{15} - 146 q^{17} - 94 q^{19} - 90 q^{21} - 48 q^{23} + 145 q^{25} - 81 q^{27} + 2 q^{29} + 302 q^{31} - 48 q^{33} - 80 q^{35} - 374 q^{37} + 117 q^{39} + 480 q^{41} + 260 q^{43} - 36 q^{45} - 24 q^{47} + 447 q^{49} + 438 q^{51} + 678 q^{53} - 1552 q^{55} + 282 q^{57} + 1788 q^{59} - 230 q^{61} + 270 q^{63} + 52 q^{65} - 74 q^{67} + 144 q^{69} - 948 q^{71} - 222 q^{73} - 435 q^{75} - 112 q^{77} - 24 q^{79} + 243 q^{81} + 796 q^{83} + 248 q^{85} - 6 q^{87} + 1436 q^{89} - 390 q^{91} - 906 q^{93} - 4032 q^{95} + 3242 q^{97} + 144 q^{99}+O(q^{100}) \) 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

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