LMFDB - Embedded newform 2160.4.a.bj.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2160,4,Mod(1,2160)] 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("2160.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2160.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-15,0,8,0,0,0,-10] 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:	$127.444125612$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.4281.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 12x + 15 $ x^3 - x^2 - 12*x + 15
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1080)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.29027$ of defining polynomial
Character	$\chi$	$=$	2160.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000 q^{5} -16.4120 q^{7} +57.6104 q^{11} +38.7864 q^{13} -31.4120 q^{17} -70.6104 q^{19} -7.37433 q^{23} +25.0000 q^{25} +17.1607 q^{29} +50.9847 q^{31} +82.0601 q^{35} -159.693 q^{37} +63.2056 q^{41} +84.6580 q^{43} +434.434 q^{47} -73.6454 q^{49} -138.490 q^{53} -288.052 q^{55} -631.011 q^{59} +82.8743 q^{61} -193.932 q^{65} +431.813 q^{67} +450.284 q^{71} -33.5655 q^{73} -945.504 q^{77} -509.312 q^{79} -811.292 q^{83} +157.060 q^{85} -499.820 q^{89} -636.563 q^{91} +353.052 q^{95} +625.481 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 15 q^{5} + 8 q^{7} - 10 q^{11} + 48 q^{13} - 37 q^{17} - 29 q^{19} - 11 q^{23} + 75 q^{25} - 28 q^{29} - 41 q^{31} - 40 q^{35} + 230 q^{37} - 370 q^{41} + 130 q^{43} - 56 q^{47} + 547 q^{49} - 805 q^{53}+ \cdots + 576 q^{97}+O(q^{100}) $ 3 * q - 15 * q^5 + 8 * q^7 - 10 * q^11 + 48 * q^13 - 37 * q^17 - 29 * q^19 - 11 * q^23 + 75 * q^25 - 28 * q^29 - 41 * q^31 - 40 * q^35 + 230 * q^37 - 370 * q^41 + 130 * q^43 - 56 * q^47 + 547 * q^49 - 805 * q^53 + 50 * q^55 + 576 * q^59 - 257 * q^61 - 240 * q^65 + 14 * q^67 + 1238 * q^71 - 398 * q^73 - 1296 * q^77 + 321 * q^79 + 687 * q^83 + 185 * q^85 - 2358 * q^89 + 1968 * q^91 + 145 * q^95 + 576 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−16.4120	−0.886166	−0.443083	−	0.896481i	$-0.646115\pi$
$7$	−16.4120	−0.886166	−0.443083	+	0.896481i	$0.646115\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	57.6104	1.57911	0.789554	−	0.613681i	$-0.210312\pi$
$11$	57.6104	1.57911	0.789554	+	0.613681i	$0.210312\pi$
$12$	0	0
$13$	38.7864	0.827492	0.413746	−	0.910392i	$-0.364220\pi$
$13$	38.7864	0.827492	0.413746	+	0.910392i	$0.364220\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−31.4120	−0.448149	−0.224075	−	0.974572i	$-0.571936\pi$
$17$	−31.4120	−0.448149	−0.224075	+	0.974572i	$0.571936\pi$
$18$	0	0
$19$	−70.6104	−0.852586	−0.426293	−	0.904585i	$-0.640181\pi$
$19$	−70.6104	−0.852586	−0.426293	+	0.904585i	$0.640181\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.37433	−0.0668545	−0.0334273	−	0.999441i	$-0.510642\pi$
$23$	−7.37433	−0.0668545	−0.0334273	+	0.999441i	$0.510642\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	17.1607	0.109885	0.0549424	−	0.998490i	$-0.482502\pi$
$29$	17.1607	0.109885	0.0549424	+	0.998490i	$0.482502\pi$
$30$	0	0
$31$	50.9847	0.295391	0.147696	−	0.989033i	$-0.452814\pi$
$31$	50.9847	0.295391	0.147696	+	0.989033i	$0.452814\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	82.0601	0.396306
$36$	0	0
$37$	−159.693	−0.709550	−0.354775	−	0.934952i	$-0.615443\pi$
$37$	−159.693	−0.709550	−0.354775	+	0.934952i	$0.615443\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	63.2056	0.240757	0.120379	−	0.992728i	$-0.461589\pi$
$41$	63.2056	0.240757	0.120379	+	0.992728i	$0.461589\pi$
$42$	0	0
$43$	84.6580	0.300238	0.150119	−	0.988668i	$-0.452034\pi$
$43$	84.6580	0.300238	0.150119	+	0.988668i	$0.452034\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	434.434	1.34827	0.674135	−	0.738609i	$-0.264517\pi$
$47$	434.434	1.34827	0.674135	+	0.738609i	$0.264517\pi$
$48$	0	0
$49$	−73.6454	−0.214710
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−138.490	−0.358926	−0.179463	−	0.983765i	$-0.557436\pi$
$53$	−138.490	−0.358926	−0.179463	+	0.983765i	$0.557436\pi$
$54$	0	0
$55$	−288.052	−0.706199
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−631.011	−1.39238	−0.696192	−	0.717856i	$-0.745124\pi$
$59$	−631.011	−1.39238	−0.696192	+	0.717856i	$0.745124\pi$
$60$	0	0
$61$	82.8743	0.173950	0.0869751	−	0.996210i	$-0.472280\pi$
$61$	82.8743	0.173950	0.0869751	+	0.996210i	$0.472280\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−193.932	−0.370066
$66$	0	0
$67$	431.813	0.787379	0.393689	−	0.919244i	$-0.371199\pi$
$67$	431.813	0.787379	0.393689	+	0.919244i	$0.371199\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	450.284	0.752660	0.376330	−	0.926486i	$-0.377186\pi$
$71$	450.284	0.752660	0.376330	+	0.926486i	$0.377186\pi$
$72$	0	0
$73$	−33.5655	−0.0538157	−0.0269079	−	0.999638i	$-0.508566\pi$
$73$	−33.5655	−0.0538157	−0.0269079	+	0.999638i	$0.508566\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−945.504	−1.39935
$78$	0	0
$79$	−509.312	−0.725343	−0.362672	−	0.931917i	$-0.618135\pi$
$79$	−509.312	−0.725343	−0.362672	+	0.931917i	$0.618135\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−811.292	−1.07290	−0.536451	−	0.843932i	$-0.680235\pi$
$83$	−811.292	−1.07290	−0.536451	+	0.843932i	$0.680235\pi$
$84$	0	0
$85$	157.060	0.200418
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−499.820	−0.595290	−0.297645	−	0.954677i	$-0.596201\pi$
$89$	−499.820	−0.595290	−0.297645	+	0.954677i	$0.596201\pi$
$90$	0	0
$91$	−636.563	−0.733296
$92$	0	0
$93$	0	0
$94$	0	0
$95$	353.052	0.381288
$96$	0	0
$97$	625.481	0.654722	0.327361	−	0.944899i	$-0.393841\pi$
$97$	625.481	0.654722	0.327361	+	0.944899i	$0.393841\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−241.156	−0.237583	−0.118792	−	0.992919i	$-0.537902\pi$
$101$	−241.156	−0.237583	−0.118792	+	0.992919i	$0.537902\pi$
$102$	0	0
$103$	1436.46	1.37417	0.687083	−	0.726579i	$-0.258891\pi$
$103$	1436.46	1.37417	0.687083	+	0.726579i	$0.258891\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	801.329	0.723995	0.361997	−	0.932179i	$-0.382095\pi$
$107$	801.329	0.723995	0.361997	+	0.932179i	$0.382095\pi$
$108$	0	0
$109$	−1573.48	−1.38268	−0.691338	−	0.722531i	$-0.742979\pi$
$109$	−1573.48	−1.38268	−0.691338	+	0.722531i	$0.742979\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1777.62	−1.47986	−0.739930	−	0.672683i	$-0.765142\pi$
$113$	−1777.62	−1.47986	−0.739930	+	0.672683i	$0.765142\pi$
$114$	0	0
$115$	36.8717	0.0298983
$116$	0	0
$117$	0	0
$118$	0	0
$119$	515.535	0.397135
$120$	0	0
$121$	1987.96	1.49358
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	2006.78	1.40215	0.701075	−	0.713088i	$-0.252704\pi$
$127$	2006.78	1.40215	0.701075	+	0.713088i	$0.252704\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1262.69	0.842149	0.421075	−	0.907026i	$-0.361653\pi$
$131$	1262.69	0.842149	0.421075	+	0.907026i	$0.361653\pi$
$132$	0	0
$133$	1158.86	0.755533
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−678.275	−0.422985	−0.211492	−	0.977380i	$-0.567832\pi$
$137$	−678.275	−0.422985	−0.211492	+	0.977380i	$0.567832\pi$
$138$	0	0
$139$	136.614	0.0833630	0.0416815	−	0.999131i	$-0.486729\pi$
$139$	136.614	0.0833630	0.0416815	+	0.999131i	$0.486729\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2234.50	1.30670
$144$	0	0
$145$	−85.8034	−0.0491420
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1445.14	−0.794569	−0.397285	−	0.917695i	$-0.630047\pi$
$149$	−1445.14	−0.794569	−0.397285	+	0.917695i	$0.630047\pi$
$150$	0	0
$151$	2635.74	1.42048	0.710242	−	0.703958i	$-0.248586\pi$
$151$	2635.74	1.42048	0.710242	+	0.703958i	$0.248586\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−254.924	−0.132103
$156$	0	0
$157$	1523.92	0.774662	0.387331	−	0.921941i	$-0.373397\pi$
$157$	1523.92	0.774662	0.387331	+	0.921941i	$0.373397\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	121.028	0.0592442
$162$	0	0
$163$	2382.91	1.14506	0.572528	−	0.819885i	$-0.305963\pi$
$163$	2382.91	1.14506	0.572528	+	0.819885i	$0.305963\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3380.54	1.56643	0.783215	−	0.621751i	$-0.213578\pi$
$167$	3380.54	1.56643	0.783215	+	0.621751i	$0.213578\pi$
$168$	0	0
$169$	−692.618	−0.315256
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−99.6851	−0.0438088	−0.0219044	−	0.999760i	$-0.506973\pi$
$173$	−99.6851	−0.0438088	−0.0219044	+	0.999760i	$0.506973\pi$
$174$	0	0
$175$	−410.301	−0.177233
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3412.90	1.42510	0.712549	−	0.701622i	$-0.247541\pi$
$179$	3412.90	1.42510	0.712549	+	0.701622i	$0.247541\pi$
$180$	0	0
$181$	4063.94	1.66890	0.834448	−	0.551087i	$-0.185787\pi$
$181$	4063.94	1.66890	0.834448	+	0.551087i	$0.185787\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	798.465	0.317321
$186$	0	0
$187$	−1809.66	−0.707676
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2163.72	0.819693	0.409847	−	0.912154i	$-0.365582\pi$
$191$	2163.72	0.819693	0.409847	+	0.912154i	$0.365582\pi$
$192$	0	0
$193$	4432.06	1.65299	0.826493	−	0.562946i	$-0.190332\pi$
$193$	4432.06	1.65299	0.826493	+	0.562946i	$0.190332\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4233.88	1.53122	0.765612	−	0.643302i	$-0.222436\pi$
$197$	4233.88	1.53122	0.765612	+	0.643302i	$0.222436\pi$
$198$	0	0
$199$	2833.94	1.00951	0.504755	−	0.863263i	$-0.331583\pi$
$199$	2833.94	1.00951	0.504755	+	0.863263i	$0.331583\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−281.642	−0.0973762
$204$	0	0
$205$	−316.028	−0.107670
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4067.89	−1.34633
$210$	0	0
$211$	−2775.59	−0.905589	−0.452794	−	0.891615i	$-0.649573\pi$
$211$	−2775.59	−0.905589	−0.452794	+	0.891615i	$0.649573\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−423.290	−0.134270
$216$	0	0
$217$	−836.763	−0.261766
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1218.36	−0.370840
$222$	0	0
$223$	−317.706	−0.0954043	−0.0477021	−	0.998862i	$-0.515190\pi$
$223$	−317.706	−0.0954043	−0.0477021	+	0.998862i	$0.515190\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−855.603	−0.250169	−0.125085	−	0.992146i	$-0.539920\pi$
$227$	−855.603	−0.250169	−0.125085	+	0.992146i	$0.539920\pi$
$228$	0	0
$229$	687.782	0.198471	0.0992356	−	0.995064i	$-0.468360\pi$
$229$	687.782	0.198471	0.0992356	+	0.995064i	$0.468360\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4228.01	−1.18878	−0.594390	−	0.804177i	$-0.702607\pi$
$233$	−4228.01	−1.18878	−0.594390	+	0.804177i	$0.702607\pi$
$234$	0	0
$235$	−2172.17	−0.602964
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3113.46	−0.842648	−0.421324	−	0.906910i	$-0.638435\pi$
$239$	−3113.46	−0.842648	−0.421324	+	0.906910i	$0.638435\pi$
$240$	0	0
$241$	−4185.57	−1.11874	−0.559370	−	0.828918i	$-0.688957\pi$
$241$	−4185.57	−1.11874	−0.559370	+	0.828918i	$0.688957\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	368.227	0.0960210
$246$	0	0
$247$	−2738.72	−0.705509
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1137.16	−0.285963	−0.142981	−	0.989725i	$-0.545669\pi$
$251$	−1137.16	−0.285963	−0.142981	+	0.989725i	$0.545669\pi$
$252$	0	0
$253$	−424.838	−0.105571
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6957.03	1.68859	0.844295	−	0.535878i	$-0.180019\pi$
$257$	6957.03	1.68859	0.844295	+	0.535878i	$0.180019\pi$
$258$	0	0
$259$	2620.89	0.628780
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1187.16	0.278339	0.139169	−	0.990269i	$-0.455557\pi$
$263$	1187.16	0.278339	0.139169	+	0.990269i	$0.455557\pi$
$264$	0	0
$265$	692.451	0.160517
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1879.17	−0.425929	−0.212965	−	0.977060i	$-0.568312\pi$
$269$	−1879.17	−0.425929	−0.212965	+	0.977060i	$0.568312\pi$
$270$	0	0
$271$	8561.28	1.91904	0.959522	−	0.281635i	$-0.0908768\pi$
$271$	8561.28	1.91904	0.959522	+	0.281635i	$0.0908768\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1440.26	0.315822
$276$	0	0
$277$	7204.78	1.56279	0.781396	−	0.624036i	$-0.214508\pi$
$277$	7204.78	1.56279	0.781396	+	0.624036i	$0.214508\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7115.84	−1.51066	−0.755329	−	0.655345i	$-0.772523\pi$
$281$	−7115.84	−1.51066	−0.755329	+	0.655345i	$0.772523\pi$
$282$	0	0
$283$	1361.24	0.285926	0.142963	−	0.989728i	$-0.454337\pi$
$283$	1361.24	0.285926	0.142963	+	0.989728i	$0.454337\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1037.33	−0.213351
$288$	0	0
$289$	−3926.28	−0.799162
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1323.60	0.263910	0.131955	−	0.991256i	$-0.457875\pi$
$293$	1323.60	0.263910	0.131955	+	0.991256i	$0.457875\pi$
$294$	0	0
$295$	3155.05	0.622693
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−286.023	−0.0553216
$300$	0	0
$301$	−1389.41	−0.266061
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−414.371	−0.0777929
$306$	0	0
$307$	5864.61	1.09026	0.545132	−	0.838350i	$-0.316479\pi$
$307$	5864.61	1.09026	0.545132	+	0.838350i	$0.316479\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5084.53	−0.927065	−0.463533	−	0.886080i	$-0.653418\pi$
$311$	−5084.53	−0.927065	−0.463533	+	0.886080i	$0.653418\pi$
$312$	0	0
$313$	−2964.66	−0.535375	−0.267687	−	0.963506i	$-0.586259\pi$
$313$	−2964.66	−0.535375	−0.267687	+	0.963506i	$0.586259\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7443.17	−1.31877	−0.659385	−	0.751806i	$-0.729183\pi$
$317$	−7443.17	−1.31877	−0.659385	+	0.751806i	$0.729183\pi$
$318$	0	0
$319$	988.634	0.173520
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2218.02	0.382086
$324$	0	0
$325$	969.659	0.165498
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7129.94	−1.19479
$330$	0	0
$331$	3965.35	0.658475	0.329238	−	0.944247i	$-0.393208\pi$
$331$	3965.35	0.658475	0.329238	+	0.944247i	$0.393208\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2159.07	−0.352127
$336$	0	0
$337$	10319.1	1.66800	0.834000	−	0.551765i	$-0.186045\pi$
$337$	10319.1	1.66800	0.834000	+	0.551765i	$0.186045\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2937.25	0.466455
$342$	0	0
$343$	6838.00	1.07643
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7755.95	1.19989	0.599944	−	0.800042i	$-0.295189\pi$
$347$	7755.95	1.19989	0.599944	+	0.800042i	$0.295189\pi$
$348$	0	0
$349$	11579.1	1.77597	0.887987	−	0.459869i	$-0.152104\pi$
$349$	11579.1	1.77597	0.887987	+	0.459869i	$0.152104\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8495.91	1.28100	0.640498	−	0.767960i	$-0.278728\pi$
$353$	8495.91	1.28100	0.640498	+	0.767960i	$0.278728\pi$
$354$	0	0
$355$	−2251.42	−0.336600
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8260.19	−1.21436	−0.607181	−	0.794563i	$-0.707700\pi$
$359$	−8260.19	−1.21436	−0.607181	+	0.794563i	$0.707700\pi$
$360$	0	0
$361$	−1873.17	−0.273097
$362$	0	0
$363$	0	0
$364$	0	0
$365$	167.828	0.0240671
$366$	0	0
$367$	−13765.5	−1.95791	−0.978953	−	0.204085i	$-0.934578\pi$
$367$	−13765.5	−1.95791	−0.978953	+	0.204085i	$0.934578\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2272.90	0.318068
$372$	0	0
$373$	1048.30	0.145520	0.0727602	−	0.997349i	$-0.476819\pi$
$373$	1048.30	0.145520	0.0727602	+	0.997349i	$0.476819\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	665.601	0.0909289
$378$	0	0
$379$	10636.6	1.44160	0.720799	−	0.693144i	$-0.243775\pi$
$379$	10636.6	1.44160	0.720799	+	0.693144i	$0.243775\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1505.81	0.200896	0.100448	−	0.994942i	$-0.467972\pi$
$383$	1505.81	0.200896	0.100448	+	0.994942i	$0.467972\pi$
$384$	0	0
$385$	4727.52	0.625809
$386$	0	0
$387$	0	0
$388$	0	0
$389$	12411.4	1.61769	0.808846	−	0.588020i	$-0.200092\pi$
$389$	12411.4	1.61769	0.808846	+	0.588020i	$0.200092\pi$
$390$	0	0
$391$	231.643	0.0299608
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2546.56	0.324383
$396$	0	0
$397$	4807.65	0.607781	0.303891	−	0.952707i	$-0.401714\pi$
$397$	4807.65	0.607781	0.303891	+	0.952707i	$0.401714\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8688.87	1.08205	0.541024	−	0.841007i	$-0.318037\pi$
$401$	8688.87	1.08205	0.541024	+	0.841007i	$0.318037\pi$
$402$	0	0
$403$	1977.51	0.244434
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9199.98	−1.12046
$408$	0	0
$409$	10056.9	1.21584	0.607921	−	0.793997i	$-0.292004\pi$
$409$	10056.9	1.21584	0.607921	+	0.793997i	$0.292004\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	10356.2	1.23388
$414$	0	0
$415$	4056.46	0.479816
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−6643.15	−0.774556	−0.387278	−	0.921963i	$-0.626585\pi$
$419$	−6643.15	−0.774556	−0.387278	+	0.921963i	$0.626585\pi$
$420$	0	0
$421$	9771.06	1.13115	0.565573	−	0.824698i	$-0.308655\pi$
$421$	9771.06	1.13115	0.565573	+	0.824698i	$0.308655\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−785.301	−0.0896298
$426$	0	0
$427$	−1360.13	−0.154149
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7498.22	−0.837997	−0.418998	−	0.907987i	$-0.637619\pi$
$431$	−7498.22	−0.837997	−0.418998	+	0.907987i	$0.637619\pi$
$432$	0	0
$433$	−12325.1	−1.36792	−0.683959	−	0.729521i	$-0.739743\pi$
$433$	−12325.1	−1.36792	−0.683959	+	0.729521i	$0.739743\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	520.705	0.0569993
$438$	0	0
$439$	11635.6	1.26500	0.632500	−	0.774560i	$-0.282029\pi$
$439$	11635.6	1.26500	0.632500	+	0.774560i	$0.282029\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12038.6	−1.29113	−0.645567	−	0.763703i	$-0.723379\pi$
$443$	−12038.6	−1.29113	−0.645567	+	0.763703i	$0.723379\pi$
$444$	0	0
$445$	2499.10	0.266222
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10905.2	1.14621	0.573103	−	0.819483i	$-0.305739\pi$
$449$	10905.2	1.14621	0.573103	+	0.819483i	$0.305739\pi$
$450$	0	0
$451$	3641.30	0.380182
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3182.81	0.327940
$456$	0	0
$457$	831.136	0.0850742	0.0425371	−	0.999095i	$-0.486456\pi$
$457$	831.136	0.0850742	0.0425371	+	0.999095i	$0.486456\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3846.15	0.388575	0.194288	−	0.980945i	$-0.437760\pi$
$461$	3846.15	0.388575	0.194288	+	0.980945i	$0.437760\pi$
$462$	0	0
$463$	−2127.77	−0.213576	−0.106788	−	0.994282i	$-0.534057\pi$
$463$	−2127.77	−0.213576	−0.106788	+	0.994282i	$0.534057\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1169.95	0.115929	0.0579647	−	0.998319i	$-0.481539\pi$
$467$	1169.95	0.115929	0.0579647	+	0.998319i	$0.481539\pi$
$468$	0	0
$469$	−7086.93	−0.697749
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4877.18	0.474108
$474$	0	0
$475$	−1765.26	−0.170517
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11243.2	−1.07247	−0.536236	−	0.844068i	$-0.680154\pi$
$479$	−11243.2	−1.07247	−0.536236	+	0.844068i	$0.680154\pi$
$480$	0	0
$481$	−6193.91	−0.587148
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3127.41	−0.292800
$486$	0	0
$487$	−12169.5	−1.13234	−0.566172	−	0.824287i	$-0.691576\pi$
$487$	−12169.5	−1.13234	−0.566172	+	0.824287i	$0.691576\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1702.75	−0.156505	−0.0782524	−	0.996934i	$-0.524934\pi$
$491$	−1702.75	−0.156505	−0.0782524	+	0.996934i	$0.524934\pi$
$492$	0	0
$493$	−539.052	−0.0492448
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7390.07	−0.666982
$498$	0	0
$499$	16867.1	1.51318	0.756590	−	0.653890i	$-0.226864\pi$
$499$	16867.1	1.51318	0.756590	+	0.653890i	$0.226864\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	10396.8	0.921608	0.460804	−	0.887502i	$-0.347561\pi$
$503$	10396.8	0.921608	0.460804	+	0.887502i	$0.347561\pi$
$504$	0	0
$505$	1205.78	0.106251
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5656.96	−0.492614	−0.246307	−	0.969192i	$-0.579217\pi$
$509$	−5656.96	−0.492614	−0.246307	+	0.969192i	$0.579217\pi$
$510$	0	0
$511$	550.878	0.0476897
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−7182.32	−0.614546
$516$	0	0
$517$	25027.9	2.12906
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8808.99	0.740747	0.370373	−	0.928883i	$-0.379230\pi$
$521$	8808.99	0.740747	0.370373	+	0.928883i	$0.379230\pi$
$522$	0	0
$523$	6509.27	0.544227	0.272113	−	0.962265i	$-0.412277\pi$
$523$	6509.27	0.544227	0.272113	+	0.962265i	$0.412277\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1601.53	−0.132379
$528$	0	0
$529$	−12112.6	−0.995530
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2451.51	0.199225
$534$	0	0
$535$	−4006.65	−0.323780
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4242.74	−0.339050
$540$	0	0
$541$	−14119.6	−1.12208	−0.561042	−	0.827787i	$-0.689600\pi$
$541$	−14119.6	−1.12208	−0.561042	+	0.827787i	$0.689600\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7867.38	0.618352
$546$	0	0
$547$	7536.09	0.589068	0.294534	−	0.955641i	$-0.404836\pi$
$547$	7536.09	0.589068	0.294534	+	0.955641i	$0.404836\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1211.72	−0.0936863
$552$	0	0
$553$	8358.85	0.642775
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7113.19	−0.541105	−0.270553	−	0.962705i	$-0.587206\pi$
$557$	−7113.19	−0.541105	−0.270553	+	0.962705i	$0.587206\pi$
$558$	0	0
$559$	3283.58	0.248444
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8995.70	−0.673399	−0.336700	−	0.941612i	$-0.609311\pi$
$563$	−8995.70	−0.673399	−0.336700	+	0.941612i	$0.609311\pi$
$564$	0	0
$565$	8888.09	0.661814
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12326.8	−0.908202	−0.454101	−	0.890950i	$-0.650040\pi$
$569$	−12326.8	−0.908202	−0.454101	+	0.890950i	$0.650040\pi$
$570$	0	0
$571$	−20671.0	−1.51498	−0.757492	−	0.652844i	$-0.773576\pi$
$571$	−20671.0	−1.51498	−0.757492	+	0.652844i	$0.773576\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−184.358	−0.0133709
$576$	0	0
$577$	4732.48	0.341448	0.170724	−	0.985319i	$-0.445389\pi$
$577$	4732.48	0.341448	0.170724	+	0.985319i	$0.445389\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	13314.9	0.950769
$582$	0	0
$583$	−7978.47	−0.566783
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22571.9	−1.58713	−0.793564	−	0.608487i	$-0.791777\pi$
$587$	−22571.9	−1.58713	−0.793564	+	0.608487i	$0.791777\pi$
$588$	0	0
$589$	−3600.05	−0.251847
$590$	0	0
$591$	0	0
$592$	0	0
$593$	8442.07	0.584611	0.292306	−	0.956325i	$-0.405578\pi$
$593$	8442.07	0.584611	0.292306	+	0.956325i	$0.405578\pi$
$594$	0	0
$595$	−2577.68	−0.177604
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1168.01	−0.0796722	−0.0398361	−	0.999206i	$-0.512684\pi$
$599$	−1168.01	−0.0796722	−0.0398361	+	0.999206i	$0.512684\pi$
$600$	0	0
$601$	14952.9	1.01488	0.507439	−	0.861687i	$-0.330592\pi$
$601$	14952.9	1.01488	0.507439	+	0.861687i	$0.330592\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9939.80	−0.667951
$606$	0	0
$607$	−15021.3	−1.00444	−0.502221	−	0.864739i	$-0.667484\pi$
$607$	−15021.3	−1.00444	−0.502221	+	0.864739i	$0.667484\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	16850.1	1.11568
$612$	0	0
$613$	−3705.74	−0.244165	−0.122083	−	0.992520i	$-0.538957\pi$
$613$	−3705.74	−0.244165	−0.122083	+	0.992520i	$0.538957\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19883.5	1.29737	0.648685	−	0.761057i	$-0.275319\pi$
$617$	19883.5	1.29737	0.648685	+	0.761057i	$0.275319\pi$
$618$	0	0
$619$	18451.8	1.19813	0.599064	−	0.800701i	$-0.295539\pi$
$619$	18451.8	1.19813	0.599064	+	0.800701i	$0.295539\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8203.05	0.527525
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5016.28	0.317984
$630$	0	0
$631$	24501.2	1.54576	0.772882	−	0.634549i	$-0.218814\pi$
$631$	24501.2	1.54576	0.772882	+	0.634549i	$0.218814\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−10033.9	−0.627060
$636$	0	0
$637$	−2856.44	−0.177671
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−12596.9	−0.776205	−0.388102	−	0.921616i	$-0.626869\pi$
$641$	−12596.9	−0.776205	−0.388102	+	0.921616i	$0.626869\pi$
$642$	0	0
$643$	−11302.8	−0.693220	−0.346610	−	0.938009i	$-0.612667\pi$
$643$	−11302.8	−0.693220	−0.346610	+	0.938009i	$0.612667\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1011.85	−0.0614839	−0.0307419	−	0.999527i	$-0.509787\pi$
$647$	−1011.85	−0.0614839	−0.0307419	+	0.999527i	$0.509787\pi$
$648$	0	0
$649$	−36352.8	−2.19872
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4763.76	−0.285483	−0.142741	−	0.989760i	$-0.545592\pi$
$653$	−4763.76	−0.285483	−0.142741	+	0.989760i	$0.545592\pi$
$654$	0	0
$655$	−6313.44	−0.376621
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3887.39	0.229789	0.114895	−	0.993378i	$-0.463347\pi$
$659$	3887.39	0.229789	0.114895	+	0.993378i	$0.463347\pi$
$660$	0	0
$661$	13798.4	0.811943	0.405972	−	0.913886i	$-0.366933\pi$
$661$	13798.4	0.811943	0.405972	+	0.913886i	$0.366933\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5794.30	−0.337885
$666$	0	0
$667$	−126.549	−0.00734630
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4774.42	0.274686
$672$	0	0
$673$	−22080.3	−1.26469	−0.632343	−	0.774688i	$-0.717907\pi$
$673$	−22080.3	−1.26469	−0.632343	+	0.774688i	$0.717907\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8197.20	0.465353	0.232677	−	0.972554i	$-0.425252\pi$
$677$	8197.20	0.465353	0.232677	+	0.972554i	$0.425252\pi$
$678$	0	0
$679$	−10265.4	−0.580192
$680$	0	0
$681$	0	0
$682$	0	0
$683$	6789.28	0.380358	0.190179	−	0.981749i	$-0.439093\pi$
$683$	6789.28	0.380358	0.190179	+	0.981749i	$0.439093\pi$
$684$	0	0
$685$	3391.37	0.189165
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5371.53	−0.297009
$690$	0	0
$691$	−19512.4	−1.07422	−0.537109	−	0.843513i	$-0.680484\pi$
$691$	−19512.4	−1.07422	−0.537109	+	0.843513i	$0.680484\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−683.070	−0.0372810
$696$	0	0
$697$	−1985.41	−0.107895
$698$	0	0
$699$	0	0
$700$	0	0
$701$	10670.1	0.574898	0.287449	−	0.957796i	$-0.407193\pi$
$701$	10670.1	0.574898	0.287449	+	0.957796i	$0.407193\pi$
$702$	0	0
$703$	11276.0	0.604953
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3957.86	0.210538
$708$	0	0
$709$	−27297.4	−1.44595	−0.722973	−	0.690876i	$-0.757225\pi$
$709$	−27297.4	−1.44595	−0.722973	+	0.690876i	$0.757225\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−375.978	−0.0197482
$714$	0	0
$715$	−11172.5	−0.584374
$716$	0	0
$717$	0	0
$718$	0	0
$719$	23257.1	1.20632	0.603159	−	0.797621i	$-0.293908\pi$
$719$	23257.1	1.20632	0.603159	+	0.797621i	$0.293908\pi$
$720$	0	0
$721$	−23575.3	−1.21774
$722$	0	0
$723$	0	0
$724$	0	0
$725$	429.017	0.0219770
$726$	0	0
$727$	−18331.1	−0.935163	−0.467582	−	0.883950i	$-0.654875\pi$
$727$	−18331.1	−0.935163	−0.467582	+	0.883950i	$0.654875\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2659.28	−0.134551
$732$	0	0
$733$	−13078.2	−0.659011	−0.329505	−	0.944154i	$-0.606882\pi$
$733$	−13078.2	−0.659011	−0.329505	+	0.944154i	$0.606882\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24876.9	1.24336
$738$	0	0
$739$	12129.6	0.603782	0.301891	−	0.953342i	$-0.402382\pi$
$739$	12129.6	0.603782	0.301891	+	0.953342i	$0.402382\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	33145.5	1.63660	0.818298	−	0.574794i	$-0.194918\pi$
$743$	33145.5	1.63660	0.818298	+	0.574794i	$0.194918\pi$
$744$	0	0
$745$	7225.72	0.355342
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−13151.4	−0.641580
$750$	0	0
$751$	−2612.04	−0.126917	−0.0634585	−	0.997984i	$-0.520213\pi$
$751$	−2612.04	−0.126917	−0.0634585	+	0.997984i	$0.520213\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−13178.7	−0.635260
$756$	0	0
$757$	15839.7	0.760506	0.380253	−	0.924883i	$-0.375837\pi$
$757$	15839.7	0.760506	0.380253	+	0.924883i	$0.375837\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	742.568	0.0353720	0.0176860	−	0.999844i	$-0.494370\pi$
$761$	742.568	0.0353720	0.0176860	+	0.999844i	$0.494370\pi$
$762$	0	0
$763$	25823.9	1.22528
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−24474.6	−1.15219
$768$	0	0
$769$	−22719.0	−1.06537	−0.532683	−	0.846315i	$-0.678816\pi$
$769$	−22719.0	−1.06537	−0.532683	+	0.846315i	$0.678816\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	23215.6	1.08022	0.540108	−	0.841596i	$-0.318383\pi$
$773$	23215.6	1.08022	0.540108	+	0.841596i	$0.318383\pi$
$774$	0	0
$775$	1274.62	0.0590783
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4462.97	−0.205266
$780$	0	0
$781$	25941.0	1.18853
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7619.60	−0.346440
$786$	0	0
$787$	−25481.4	−1.15415	−0.577073	−	0.816692i	$-0.695805\pi$
$787$	−25481.4	−1.15415	−0.577073	+	0.816692i	$0.695805\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	29174.3	1.31140
$792$	0	0
$793$	3214.39	0.143942
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−22375.2	−0.994440	−0.497220	−	0.867624i	$-0.665646\pi$
$797$	−22375.2	−0.994440	−0.497220	+	0.867624i	$0.665646\pi$
$798$	0	0
$799$	−13646.4	−0.604226
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1933.72	−0.0849809
$804$	0	0
$805$	−605.139	−0.0264948
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−9293.69	−0.403892	−0.201946	−	0.979397i	$-0.564727\pi$
$809$	−9293.69	−0.403892	−0.201946	+	0.979397i	$0.564727\pi$
$810$	0	0
$811$	−26415.5	−1.14374	−0.571871	−	0.820344i	$-0.693782\pi$
$811$	−26415.5	−1.14374	−0.571871	+	0.820344i	$0.693782\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−11914.6	−0.512084
$816$	0	0
$817$	−5977.74	−0.255979
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−18148.6	−0.771486	−0.385743	−	0.922606i	$-0.626055\pi$
$821$	−18148.6	−0.771486	−0.385743	+	0.922606i	$0.626055\pi$
$822$	0	0
$823$	10438.4	0.442115	0.221058	−	0.975261i	$-0.429049\pi$
$823$	10438.4	0.442115	0.221058	+	0.975261i	$0.429049\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	18909.8	0.795112	0.397556	−	0.917578i	$-0.369858\pi$
$827$	18909.8	0.795112	0.397556	+	0.917578i	$0.369858\pi$
$828$	0	0
$829$	10162.8	0.425777	0.212888	−	0.977077i	$-0.431713\pi$
$829$	10162.8	0.425777	0.212888	+	0.977077i	$0.431713\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2313.35	0.0962219
$834$	0	0
$835$	−16902.7	−0.700529
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−16310.9	−0.671174	−0.335587	−	0.942009i	$-0.608935\pi$
$839$	−16310.9	−0.671174	−0.335587	+	0.942009i	$0.608935\pi$
$840$	0	0
$841$	−24094.5	−0.987925
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3463.09	0.140987
$846$	0	0
$847$	−32626.4	−1.32356
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1177.63	0.0474367
$852$	0	0
$853$	26423.7	1.06064	0.530322	−	0.847796i	$-0.322071\pi$
$853$	26423.7	1.06064	0.530322	+	0.847796i	$0.322071\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−19242.4	−0.766985	−0.383493	−	0.923544i	$-0.625279\pi$
$857$	−19242.4	−0.766985	−0.383493	+	0.923544i	$0.625279\pi$
$858$	0	0
$859$	−3598.56	−0.142935	−0.0714677	−	0.997443i	$-0.522768\pi$
$859$	−3598.56	−0.142935	−0.0714677	+	0.997443i	$0.522768\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	13195.7	0.520496	0.260248	−	0.965542i	$-0.416196\pi$
$863$	13195.7	0.520496	0.260248	+	0.965542i	$0.416196\pi$
$864$	0	0
$865$	498.425	0.0195919
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−29341.7	−1.14540
$870$	0	0
$871$	16748.5	0.651550
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2051.50	0.0792611
$876$	0	0
$877$	40460.0	1.55785	0.778926	−	0.627116i	$-0.215765\pi$
$877$	40460.0	1.55785	0.778926	+	0.627116i	$0.215765\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	23036.5	0.880955	0.440477	−	0.897764i	$-0.354809\pi$
$881$	23036.5	0.880955	0.440477	+	0.897764i	$0.354809\pi$
$882$	0	0
$883$	24591.6	0.937228	0.468614	−	0.883403i	$-0.344754\pi$
$883$	24591.6	0.937228	0.468614	+	0.883403i	$0.344754\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31117.5	1.17793	0.588966	−	0.808158i	$-0.299535\pi$
$887$	31117.5	1.17793	0.588966	+	0.808158i	$0.299535\pi$
$888$	0	0
$889$	−32935.3	−1.24254
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−30675.5	−1.14952
$894$	0	0
$895$	−17064.5	−0.637323
$896$	0	0
$897$	0	0
$898$	0	0
$899$	874.933	0.0324590
$900$	0	0
$901$	4350.26	0.160852
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−20319.7	−0.746353
$906$	0	0
$907$	−25694.5	−0.940651	−0.470326	−	0.882493i	$-0.655864\pi$
$907$	−25694.5	−0.940651	−0.470326	+	0.882493i	$0.655864\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−19061.3	−0.693225	−0.346612	−	0.938008i	$-0.612668\pi$
$911$	−19061.3	−0.693225	−0.346612	+	0.938008i	$0.612668\pi$
$912$	0	0
$913$	−46738.8	−1.69423
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−20723.3	−0.746284
$918$	0	0
$919$	−2322.58	−0.0833675	−0.0416838	−	0.999131i	$-0.513272\pi$
$919$	−2322.58	−0.0833675	−0.0416838	+	0.999131i	$0.513272\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	17464.9	0.622820
$924$	0	0
$925$	−3992.32	−0.141910
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−25814.2	−0.911664	−0.455832	−	0.890066i	$-0.650658\pi$
$929$	−25814.2	−0.911664	−0.455832	+	0.890066i	$0.650658\pi$
$930$	0	0
$931$	5200.13	0.183058
$932$	0	0
$933$	0	0
$934$	0	0
$935$	9048.30	0.316482
$936$	0	0
$937$	17757.8	0.619128	0.309564	−	0.950879i	$-0.399817\pi$
$937$	17757.8	0.619128	0.309564	+	0.950879i	$0.399817\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1857.32	−0.0643431	−0.0321715	−	0.999482i	$-0.510242\pi$
$941$	−1857.32	−0.0643431	−0.0321715	+	0.999482i	$0.510242\pi$
$942$	0	0
$943$	−466.099	−0.0160957
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10837.6	0.371885	0.185943	−	0.982561i	$-0.440466\pi$
$947$	10837.6	0.371885	0.185943	+	0.982561i	$0.440466\pi$
$948$	0	0
$949$	−1301.88	−0.0445321
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2386.31	0.0811125	0.0405562	−	0.999177i	$-0.487087\pi$
$953$	2386.31	0.0811125	0.0405562	+	0.999177i	$0.487087\pi$
$954$	0	0
$955$	−10818.6	−0.366578
$956$	0	0
$957$	0	0
$958$	0	0
$959$	11131.9	0.374835
$960$	0	0
$961$	−27191.6	−0.912744
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−22160.3	−0.739238
$966$	0	0
$967$	−12418.4	−0.412977	−0.206489	−	0.978449i	$-0.566204\pi$
$967$	−12418.4	−0.412977	−0.206489	+	0.978449i	$0.566204\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	50399.5	1.66570	0.832851	−	0.553498i	$-0.186707\pi$
$971$	50399.5	1.66570	0.832851	+	0.553498i	$0.186707\pi$
$972$	0	0
$973$	−2242.11	−0.0738734
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−56849.3	−1.86159	−0.930793	−	0.365547i	$-0.880882\pi$
$977$	−56849.3	−1.86159	−0.930793	+	0.365547i	$0.880882\pi$
$978$	0	0
$979$	−28794.8	−0.940027
$980$	0	0
$981$	0	0
$982$	0	0
$983$	13309.5	0.431850	0.215925	−	0.976410i	$-0.430723\pi$
$983$	13309.5	0.431850	0.215925	+	0.976410i	$0.430723\pi$
$984$	0	0
$985$	−21169.4	−0.684785
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−624.296	−0.0200723
$990$	0	0
$991$	19322.3	0.619369	0.309684	−	0.950839i	$-0.399777\pi$
$991$	19322.3	0.619369	0.309684	+	0.950839i	$0.399777\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−14169.7	−0.451466
$996$	0	0
$997$	8263.31	0.262489	0.131245	−	0.991350i	$-0.458103\pi$
$997$	8263.31	0.262489	0.131245	+	0.991350i	$0.458103\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bj.1.1		3
3.2	odd	2		2160.4.a.br.1.1		3
4.3	odd	2		1080.4.a.e.1.3	✓	3
12.11	even	2		1080.4.a.k.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.e.1.3	✓	3	4.3	odd	2
1080.4.a.k.1.3	yes	3	12.11	even	2
2160.4.a.bj.1.1		3	1.1	even	1	trivial
2160.4.a.br.1.1		3	3.2	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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2160.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} -16.4120 q^{7} +57.6104 q^{11} +38.7864 q^{13} -31.4120 q^{17} -70.6104 q^{19} -7.37433 q^{23} +25.0000 q^{25} +17.1607 q^{29} +50.9847 q^{31} +82.0601 q^{35} -159.693 q^{37} +63.2056 q^{41} +84.6580 q^{43} +434.434 q^{47} -73.6454 q^{49} -138.490 q^{53} -288.052 q^{55} -631.011 q^{59} +82.8743 q^{61} -193.932 q^{65} +431.813 q^{67} +450.284 q^{71} -33.5655 q^{73} -945.504 q^{77} -509.312 q^{79} -811.292 q^{83} +157.060 q^{85} -499.820 q^{89} -636.563 q^{91} +353.052 q^{95} +625.481 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 15 q^{5} + 8 q^{7} - 10 q^{11} + 48 q^{13} - 37 q^{17} - 29 q^{19} - 11 q^{23} + 75 q^{25} - 28 q^{29} - 41 q^{31} - 40 q^{35} + 230 q^{37} - 370 q^{41} + 130 q^{43} - 56 q^{47} + 547 q^{49} - 805 q^{53}+ \cdots + 576 q^{97}+O(q^{100}) \) 3 * q - 15 * q^5 + 8 * q^7 - 10 * q^11 + 48 * q^13 - 37 * q^17 - 29 * q^19 - 11 * q^23 + 75 * q^25 - 28 * q^29 - 41 * q^31 - 40 * q^35 + 230 * q^37 - 370 * q^41 + 130 * q^43 - 56 * q^47 + 547 * q^49 - 805 * q^53 + 50 * q^55 + 576 * q^59 - 257 * q^61 - 240 * q^65 + 14 * q^67 + 1238 * q^71 - 398 * q^73 - 1296 * q^77 + 321 * q^79 + 687 * q^83 + 185 * q^85 - 2358 * q^89 + 1968 * q^91 + 145 * q^95 + 576 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more