LMFDB - Embedded newform 2160.4.a.bf.1.2

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,-9,0,0,0,18] 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:	3.3.47977.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 60x - 44 $ x^3 - x^2 - 60*x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1080)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$8.58548$ 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)

$f(q)$	$=$	$q-5.00000 q^{5} -9.46549 q^{7} +49.0474 q^{11} +55.4438 q^{13} +27.9783 q^{17} -26.3965 q^{19} +9.53451 q^{23} +25.0000 q^{25} -218.379 q^{29} +158.961 q^{31} +47.3274 q^{35} +189.655 q^{37} +246.146 q^{41} -40.2327 q^{43} -213.965 q^{47} -253.405 q^{49} +283.310 q^{53} -245.237 q^{55} -92.0000 q^{59} +370.172 q^{61} -277.219 q^{65} -1087.53 q^{67} +333.681 q^{71} -606.791 q^{73} -464.257 q^{77} +44.3751 q^{79} -107.732 q^{83} -139.892 q^{85} -257.819 q^{89} -524.803 q^{91} +131.982 q^{95} +1312.05 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 15 q^{5} - 9 q^{7} + 18 q^{11} - 21 q^{13} - 84 q^{17} - 21 q^{19} + 48 q^{23} + 75 q^{25} + 36 q^{29} - 324 q^{31} + 45 q^{35} + 33 q^{37} - 114 q^{41} - 282 q^{43} + 282 q^{47} + 228 q^{49} - 222 q^{53}+ \cdots + 1191 q^{97}+O(q^{100}) $ 3 * q - 15 * q^5 - 9 * q^7 + 18 * q^11 - 21 * q^13 - 84 * q^17 - 21 * q^19 + 48 * q^23 + 75 * q^25 + 36 * q^29 - 324 * q^31 + 45 * q^35 + 33 * q^37 - 114 * q^41 - 282 * q^43 + 282 * q^47 + 228 * q^49 - 222 * q^53 - 90 * q^55 - 276 * q^59 + 303 * q^61 + 105 * q^65 - 1035 * q^67 + 510 * q^71 + 447 * q^73 + 1578 * q^77 - 777 * q^79 + 78 * q^83 + 420 * q^85 + 324 * q^89 - 1995 * q^91 + 105 * q^95 + 1191 * 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
$7$	−9.46549	−0.511088	−0.255544	−	0.966797i	$-0.582255\pi$
$7$	−9.46549	−0.511088	−0.255544	+	0.966797i	$0.582255\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	49.0474	1.34439	0.672197	−	0.740372i	$-0.265351\pi$
$11$	49.0474	1.34439	0.672197	+	0.740372i	$0.265351\pi$
$12$	0	0
$13$	55.4438	1.18287	0.591437	−	0.806351i	$-0.298561\pi$
$13$	55.4438	1.18287	0.591437	+	0.806351i	$0.298561\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	27.9783	0.399162	0.199581	−	0.979881i	$-0.436042\pi$
$17$	27.9783	0.399162	0.199581	+	0.979881i	$0.436042\pi$
$18$	0	0
$19$	−26.3965	−0.318724	−0.159362	−	0.987220i	$-0.550944\pi$
$19$	−26.3965	−0.318724	−0.159362	+	0.987220i	$0.550944\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	9.53451	0.0864384	0.0432192	−	0.999066i	$-0.486239\pi$
$23$	9.53451	0.0864384	0.0432192	+	0.999066i	$0.486239\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−218.379	−1.39834	−0.699171	−	0.714954i	$-0.746448\pi$
$29$	−218.379	−1.39834	−0.699171	+	0.714954i	$0.746448\pi$
$30$	0	0
$31$	158.961	0.920974	0.460487	−	0.887666i	$-0.347675\pi$
$31$	158.961	0.920974	0.460487	+	0.887666i	$0.347675\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	47.3274	0.228566
$36$	0	0
$37$	189.655	0.842678	0.421339	−	0.906903i	$-0.361560\pi$
$37$	189.655	0.842678	0.421339	+	0.906903i	$0.361560\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	246.146	0.937599	0.468800	−	0.883304i	$-0.344687\pi$
$41$	246.146	0.937599	0.468800	+	0.883304i	$0.344687\pi$
$42$	0	0
$43$	−40.2327	−0.142684	−0.0713422	−	0.997452i	$-0.522728\pi$
$43$	−40.2327	−0.142684	−0.0713422	+	0.997452i	$0.522728\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−213.965	−0.664042	−0.332021	−	0.943272i	$-0.607730\pi$
$47$	−213.965	−0.664042	−0.332021	+	0.943272i	$0.607730\pi$
$48$	0	0
$49$	−253.405	−0.738789
$50$	0	0
$51$	0	0
$52$	0	0
$53$	283.310	0.734257	0.367128	−	0.930170i	$-0.380341\pi$
$53$	283.310	0.734257	0.367128	+	0.930170i	$0.380341\pi$
$54$	0	0
$55$	−245.237	−0.601232
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−92.0000	−0.203006	−0.101503	−	0.994835i	$-0.532365\pi$
$59$	−92.0000	−0.203006	−0.101503	+	0.994835i	$0.532365\pi$
$60$	0	0
$61$	370.172	0.776978	0.388489	−	0.921453i	$-0.372997\pi$
$61$	370.172	0.776978	0.388489	+	0.921453i	$0.372997\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−277.219	−0.528997
$66$	0	0
$67$	−1087.53	−1.98302	−0.991510	−	0.130029i	$-0.958493\pi$
$67$	−1087.53	−1.98302	−0.991510	+	0.130029i	$0.958493\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	333.681	0.557755	0.278877	−	0.960327i	$-0.410038\pi$
$71$	333.681	0.557755	0.278877	+	0.960327i	$0.410038\pi$
$72$	0	0
$73$	−606.791	−0.972871	−0.486435	−	0.873717i	$-0.661703\pi$
$73$	−606.791	−0.972871	−0.486435	+	0.873717i	$0.661703\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−464.257	−0.687104
$78$	0	0
$79$	44.3751	0.0631973	0.0315986	−	0.999501i	$-0.489940\pi$
$79$	44.3751	0.0631973	0.0315986	+	0.999501i	$0.489940\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−107.732	−0.142471	−0.0712355	−	0.997460i	$-0.522694\pi$
$83$	−107.732	−0.142471	−0.0712355	+	0.997460i	$0.522694\pi$
$84$	0	0
$85$	−139.892	−0.178510
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−257.819	−0.307065	−0.153532	−	0.988144i	$-0.549065\pi$
$89$	−257.819	−0.307065	−0.153532	+	0.988144i	$0.549065\pi$
$90$	0	0
$91$	−524.803	−0.604553
$92$	0	0
$93$	0	0
$94$	0	0
$95$	131.982	0.142538
$96$	0	0
$97$	1312.05	1.37339	0.686693	−	0.726947i	$-0.259062\pi$
$97$	1312.05	1.37339	0.686693	+	0.726947i	$0.259062\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	63.3293	0.0623911	0.0311955	−	0.999513i	$-0.490069\pi$
$101$	63.3293	0.0623911	0.0311955	+	0.999513i	$0.490069\pi$
$102$	0	0
$103$	−1219.85	−1.16695	−0.583474	−	0.812132i	$-0.698307\pi$
$103$	−1219.85	−1.16695	−0.583474	+	0.812132i	$0.698307\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−42.3947	−0.0383033	−0.0191517	−	0.999817i	$-0.506097\pi$
$107$	−42.3947	−0.0383033	−0.0191517	+	0.999817i	$0.506097\pi$
$108$	0	0
$109$	266.741	0.234396	0.117198	−	0.993109i	$-0.462609\pi$
$109$	266.741	0.234396	0.117198	+	0.993109i	$0.462609\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1871.60	1.55810	0.779049	−	0.626964i	$-0.215703\pi$
$113$	1871.60	1.55810	0.779049	+	0.626964i	$0.215703\pi$
$114$	0	0
$115$	−47.6726	−0.0386564
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−264.829	−0.204007
$120$	0	0
$121$	1074.64	0.807397
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1232.11	0.860881	0.430441	−	0.902619i	$-0.358358\pi$
$127$	1232.11	0.860881	0.430441	+	0.902619i	$0.358358\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1209.17	0.806458	0.403229	−	0.915099i	$-0.367888\pi$
$131$	1209.17	0.806458	0.403229	+	0.915099i	$0.367888\pi$
$132$	0	0
$133$	249.855	0.162896
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2629.66	1.63990	0.819952	−	0.572432i	$-0.194000\pi$
$137$	2629.66	1.63990	0.819952	+	0.572432i	$0.194000\pi$
$138$	0	0
$139$	−2215.06	−1.35165	−0.675823	−	0.737064i	$-0.736212\pi$
$139$	−2215.06	−1.35165	−0.675823	+	0.737064i	$0.736212\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2719.37	1.59025
$144$	0	0
$145$	1091.89	0.625358
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2446.94	1.34538	0.672689	−	0.739925i	$-0.265139\pi$
$149$	2446.94	1.34538	0.672689	+	0.739925i	$0.265139\pi$
$150$	0	0
$151$	−119.365	−0.0643298	−0.0321649	−	0.999483i	$-0.510240\pi$
$151$	−119.365	−0.0643298	−0.0321649	+	0.999483i	$0.510240\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−794.804	−0.411872
$156$	0	0
$157$	2212.86	1.12487	0.562437	−	0.826840i	$-0.309864\pi$
$157$	2212.86	1.12487	0.562437	+	0.826840i	$0.309864\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−90.2488	−0.0441777
$162$	0	0
$163$	−3480.54	−1.67250	−0.836249	−	0.548349i	$-0.815256\pi$
$163$	−3480.54	−1.67250	−0.836249	+	0.548349i	$0.815256\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2477.15	1.14783	0.573916	−	0.818914i	$-0.305424\pi$
$167$	2477.15	1.14783	0.573916	+	0.818914i	$0.305424\pi$
$168$	0	0
$169$	877.018	0.399189
$170$	0	0
$171$	0	0
$172$	0	0
$173$	26.7446	0.0117535	0.00587676	−	0.999983i	$-0.498129\pi$
$173$	26.7446	0.0117535	0.00587676	+	0.999983i	$0.498129\pi$
$174$	0	0
$175$	−236.637	−0.102218
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1400.75	−0.584901	−0.292451	−	0.956281i	$-0.594471\pi$
$179$	−1400.75	−0.584901	−0.292451	+	0.956281i	$0.594471\pi$
$180$	0	0
$181$	3492.48	1.43422	0.717111	−	0.696959i	$-0.245464\pi$
$181$	3492.48	1.43422	0.717111	+	0.696959i	$0.245464\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−948.275	−0.376857
$186$	0	0
$187$	1372.26	0.536631
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2728.16	1.03352	0.516762	−	0.856129i	$-0.327137\pi$
$191$	2728.16	1.03352	0.516762	+	0.856129i	$0.327137\pi$
$192$	0	0
$193$	2439.82	0.909959	0.454980	−	0.890502i	$-0.349647\pi$
$193$	2439.82	0.909959	0.454980	+	0.890502i	$0.349647\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−765.906	−0.276998	−0.138499	−	0.990363i	$-0.544228\pi$
$197$	−765.906	−0.276998	−0.138499	+	0.990363i	$0.544228\pi$
$198$	0	0
$199$	−754.031	−0.268602	−0.134301	−	0.990941i	$-0.542879\pi$
$199$	−754.031	−0.268602	−0.134301	+	0.990941i	$0.542879\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2067.06	0.714676
$204$	0	0
$205$	−1230.73	−0.419307
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1294.68	−0.428491
$210$	0	0
$211$	1024.68	0.334320	0.167160	−	0.985930i	$-0.446540\pi$
$211$	1024.68	0.334320	0.167160	+	0.985930i	$0.446540\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	201.163	0.0638104
$216$	0	0
$217$	−1504.64	−0.470699
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1551.23	0.472157
$222$	0	0
$223$	1938.61	0.582147	0.291074	−	0.956701i	$-0.405988\pi$
$223$	1938.61	0.582147	0.291074	+	0.956701i	$0.405988\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5876.07	1.71810	0.859050	−	0.511892i	$-0.171055\pi$
$227$	5876.07	1.71810	0.859050	+	0.511892i	$0.171055\pi$
$228$	0	0
$229$	5061.06	1.46046	0.730228	−	0.683204i	$-0.239414\pi$
$229$	5061.06	1.46046	0.730228	+	0.683204i	$0.239414\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1241.58	0.349093	0.174547	−	0.984649i	$-0.444154\pi$
$233$	1241.58	0.349093	0.174547	+	0.984649i	$0.444154\pi$
$234$	0	0
$235$	1069.82	0.296968
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3141.68	−0.850286	−0.425143	−	0.905126i	$-0.639776\pi$
$239$	−3141.68	−0.850286	−0.425143	+	0.905126i	$0.639776\pi$
$240$	0	0
$241$	−2682.15	−0.716898	−0.358449	−	0.933549i	$-0.616694\pi$
$241$	−2682.15	−0.716898	−0.358449	+	0.933549i	$0.616694\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1267.02	0.330396
$246$	0	0
$247$	−1463.52	−0.377010
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4159.18	−1.04592	−0.522959	−	0.852358i	$-0.675172\pi$
$251$	−4159.18	−1.04592	−0.522959	+	0.852358i	$0.675172\pi$
$252$	0	0
$253$	467.643	0.116207
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2820.31	−0.684537	−0.342268	−	0.939602i	$-0.611195\pi$
$257$	−2820.31	−0.684537	−0.342268	+	0.939602i	$0.611195\pi$
$258$	0	0
$259$	−1795.18	−0.430683
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−27.1884	−0.00637457	−0.00318728	−	0.999995i	$-0.501015\pi$
$263$	−27.1884	−0.00637457	−0.00318728	+	0.999995i	$0.501015\pi$
$264$	0	0
$265$	−1416.55	−0.328370
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4955.54	1.12321	0.561607	−	0.827404i	$-0.310183\pi$
$269$	4955.54	1.12321	0.561607	+	0.827404i	$0.310183\pi$
$270$	0	0
$271$	−6619.08	−1.48369	−0.741846	−	0.670570i	$-0.766050\pi$
$271$	−6619.08	−1.48369	−0.741846	+	0.670570i	$0.766050\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1226.18	0.268879
$276$	0	0
$277$	−2999.18	−0.650554	−0.325277	−	0.945619i	$-0.605457\pi$
$277$	−2999.18	−0.650554	−0.325277	+	0.945619i	$0.605457\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3387.06	0.719058	0.359529	−	0.933134i	$-0.382937\pi$
$281$	3387.06	0.719058	0.359529	+	0.933134i	$0.382937\pi$
$282$	0	0
$283$	−79.4525	−0.0166889	−0.00834446	−	0.999965i	$-0.502656\pi$
$283$	−79.4525	−0.0166889	−0.00834446	+	0.999965i	$0.502656\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2329.89	−0.479196
$288$	0	0
$289$	−4130.21	−0.840670
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6045.37	1.20537	0.602686	−	0.797978i	$-0.294097\pi$
$293$	6045.37	1.20537	0.602686	+	0.797978i	$0.294097\pi$
$294$	0	0
$295$	460.000	0.0907872
$296$	0	0
$297$	0	0
$298$	0	0
$299$	528.630	0.102246
$300$	0	0
$301$	380.822	0.0729243
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1850.86	−0.347475
$306$	0	0
$307$	1529.87	0.284412	0.142206	−	0.989837i	$-0.454580\pi$
$307$	1529.87	0.284412	0.142206	+	0.989837i	$0.454580\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6583.78	1.20042	0.600212	−	0.799841i	$-0.295083\pi$
$311$	6583.78	1.20042	0.600212	+	0.799841i	$0.295083\pi$
$312$	0	0
$313$	110.352	0.0199281	0.00996403	−	0.999950i	$-0.496828\pi$
$313$	110.352	0.0199281	0.00996403	+	0.999950i	$0.496828\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4003.48	0.709331	0.354666	−	0.934993i	$-0.384595\pi$
$317$	4003.48	0.709331	0.354666	+	0.934993i	$0.384595\pi$
$318$	0	0
$319$	−10710.9	−1.87992
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−738.529	−0.127222
$324$	0	0
$325$	1386.10	0.236575
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2025.28	0.339384
$330$	0	0
$331$	3066.52	0.509219	0.254609	−	0.967044i	$-0.418053\pi$
$331$	3066.52	0.509219	0.254609	+	0.967044i	$0.418053\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5437.63	0.886834
$336$	0	0
$337$	10714.9	1.73199	0.865993	−	0.500057i	$-0.166687\pi$
$337$	10714.9	1.73199	0.865993	+	0.500057i	$0.166687\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7796.61	1.23815
$342$	0	0
$343$	5645.26	0.888674
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9008.73	1.39370	0.696850	−	0.717217i	$-0.254584\pi$
$347$	9008.73	1.39370	0.696850	+	0.717217i	$0.254584\pi$
$348$	0	0
$349$	−9529.14	−1.46156	−0.730778	−	0.682615i	$-0.760843\pi$
$349$	−9529.14	−1.46156	−0.730778	+	0.682615i	$0.760843\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	982.433	0.148129	0.0740646	−	0.997253i	$-0.476403\pi$
$353$	982.433	0.148129	0.0740646	+	0.997253i	$0.476403\pi$
$354$	0	0
$355$	−1668.40	−0.249436
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9887.90	1.45366	0.726829	−	0.686818i	$-0.240993\pi$
$359$	9887.90	1.45366	0.726829	+	0.686818i	$0.240993\pi$
$360$	0	0
$361$	−6162.23	−0.898415
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3033.96	0.435081
$366$	0	0
$367$	3348.54	0.476274	0.238137	−	0.971232i	$-0.423463\pi$
$367$	3348.54	0.476274	0.238137	+	0.971232i	$0.423463\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2681.67	−0.375270
$372$	0	0
$373$	9572.04	1.32874	0.664372	−	0.747402i	$-0.268699\pi$
$373$	9572.04	1.32874	0.664372	+	0.747402i	$0.268699\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−12107.8	−1.65406
$378$	0	0
$379$	9976.30	1.35211	0.676053	−	0.736853i	$-0.263689\pi$
$379$	9976.30	1.35211	0.676053	+	0.736853i	$0.263689\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7619.84	−1.01659	−0.508297	−	0.861182i	$-0.669725\pi$
$383$	−7619.84	−1.01659	−0.508297	+	0.861182i	$0.669725\pi$
$384$	0	0
$385$	2321.29	0.307282
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7802.78	1.01701	0.508504	−	0.861059i	$-0.330199\pi$
$389$	7802.78	1.01701	0.508504	+	0.861059i	$0.330199\pi$
$390$	0	0
$391$	266.760	0.0345029
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−221.875	−0.0282627
$396$	0	0
$397$	−8073.51	−1.02065	−0.510325	−	0.859982i	$-0.670475\pi$
$397$	−8073.51	−1.02065	−0.510325	+	0.859982i	$0.670475\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6306.80	0.785403	0.392702	−	0.919666i	$-0.371541\pi$
$401$	6306.80	0.785403	0.392702	+	0.919666i	$0.371541\pi$
$402$	0	0
$403$	8813.39	1.08940
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9302.08	1.13289
$408$	0	0
$409$	−15362.3	−1.85726	−0.928629	−	0.371010i	$-0.879012\pi$
$409$	−15362.3	−1.85726	−0.928629	+	0.371010i	$0.879012\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	870.825	0.103754
$414$	0	0
$415$	538.659	0.0637150
$416$	0	0
$417$	0	0
$418$	0	0
$419$	13791.3	1.60799	0.803994	−	0.594637i	$-0.202704\pi$
$419$	13791.3	1.60799	0.803994	+	0.594637i	$0.202704\pi$
$420$	0	0
$421$	−9093.23	−1.05268	−0.526339	−	0.850275i	$-0.676436\pi$
$421$	−9093.23	−1.05268	−0.526339	+	0.850275i	$0.676436\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	699.459	0.0798323
$426$	0	0
$427$	−3503.86	−0.397104
$428$	0	0
$429$	0	0
$430$	0	0
$431$	400.347	0.0447425	0.0223713	−	0.999750i	$-0.492878\pi$
$431$	400.347	0.0447425	0.0223713	+	0.999750i	$0.492878\pi$
$432$	0	0
$433$	−10063.9	−1.11695	−0.558474	−	0.829522i	$-0.688613\pi$
$433$	−10063.9	−1.11695	−0.558474	+	0.829522i	$0.688613\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−251.677	−0.0275500
$438$	0	0
$439$	11684.8	1.27035	0.635176	−	0.772368i	$-0.280928\pi$
$439$	11684.8	1.27035	0.635176	+	0.772368i	$0.280928\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3698.68	−0.396681	−0.198341	−	0.980133i	$-0.563555\pi$
$443$	−3698.68	−0.396681	−0.198341	+	0.980133i	$0.563555\pi$
$444$	0	0
$445$	1289.09	0.137323
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11318.1	1.18961	0.594804	−	0.803871i	$-0.297230\pi$
$449$	11318.1	1.18961	0.594804	+	0.803871i	$0.297230\pi$
$450$	0	0
$451$	12072.8	1.26050
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2624.01	0.270364
$456$	0	0
$457$	−2436.98	−0.249446	−0.124723	−	0.992192i	$-0.539804\pi$
$457$	−2436.98	−0.249446	−0.124723	+	0.992192i	$0.539804\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−8575.54	−0.866384	−0.433192	−	0.901302i	$-0.642613\pi$
$461$	−8575.54	−0.866384	−0.433192	+	0.901302i	$0.642613\pi$
$462$	0	0
$463$	4711.30	0.472900	0.236450	−	0.971644i	$-0.424016\pi$
$463$	4711.30	0.472900	0.236450	+	0.971644i	$0.424016\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11075.4	1.09745	0.548726	−	0.836002i	$-0.315113\pi$
$467$	11075.4	1.09745	0.548726	+	0.836002i	$0.315113\pi$
$468$	0	0
$469$	10294.0	1.01350
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1973.31	−0.191824
$474$	0	0
$475$	−659.911	−0.0637449
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17455.1	−1.66502	−0.832510	−	0.554010i	$-0.813097\pi$
$479$	−17455.1	−1.66502	−0.832510	+	0.554010i	$0.813097\pi$
$480$	0	0
$481$	10515.2	0.996781
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6560.25	−0.614197
$486$	0	0
$487$	−8285.77	−0.770974	−0.385487	−	0.922713i	$-0.625966\pi$
$487$	−8285.77	−0.770974	−0.385487	+	0.922713i	$0.625966\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7055.06	−0.648453	−0.324226	−	0.945980i	$-0.605104\pi$
$491$	−7055.06	−0.648453	−0.324226	+	0.945980i	$0.605104\pi$
$492$	0	0
$493$	−6109.88	−0.558165
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3158.45	−0.285062
$498$	0	0
$499$	−2778.49	−0.249263	−0.124632	−	0.992203i	$-0.539775\pi$
$499$	−2778.49	−0.249263	−0.124632	+	0.992203i	$0.539775\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−5445.52	−0.482711	−0.241356	−	0.970437i	$-0.577592\pi$
$503$	−5445.52	−0.482711	−0.241356	+	0.970437i	$0.577592\pi$
$504$	0	0
$505$	−316.646	−0.0279021
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16567.4	1.44270	0.721352	−	0.692569i	$-0.243521\pi$
$509$	16567.4	1.44270	0.721352	+	0.692569i	$0.243521\pi$
$510$	0	0
$511$	5743.58	0.497223
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6099.26	0.521875
$516$	0	0
$517$	−10494.4	−0.892734
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3805.03	−0.319964	−0.159982	−	0.987120i	$-0.551144\pi$
$521$	−3805.03	−0.319964	−0.159982	+	0.987120i	$0.551144\pi$
$522$	0	0
$523$	7489.80	0.626207	0.313104	−	0.949719i	$-0.398631\pi$
$523$	7489.80	0.626207	0.313104	+	0.949719i	$0.398631\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4447.46	0.367617
$528$	0	0
$529$	−12076.1	−0.992528
$530$	0	0
$531$	0	0
$532$	0	0
$533$	13647.3	1.10906
$534$	0	0
$535$	211.974	0.0171298
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−12428.8	−0.993224
$540$	0	0
$541$	−2063.73	−0.164005	−0.0820026	−	0.996632i	$-0.526132\pi$
$541$	−2063.73	−0.164005	−0.0820026	+	0.996632i	$0.526132\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1333.71	−0.104825
$546$	0	0
$547$	12954.6	1.01261	0.506307	−	0.862353i	$-0.331010\pi$
$547$	12954.6	1.01261	0.506307	+	0.862353i	$0.331010\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5764.43	0.445686
$552$	0	0
$553$	−420.032	−0.0322994
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14340.8	−1.09091	−0.545457	−	0.838139i	$-0.683644\pi$
$557$	−14340.8	−1.09091	−0.545457	+	0.838139i	$0.683644\pi$
$558$	0	0
$559$	−2230.65	−0.168777
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12827.3	0.960222	0.480111	−	0.877208i	$-0.340596\pi$
$563$	12827.3	0.960222	0.480111	+	0.877208i	$0.340596\pi$
$564$	0	0
$565$	−9357.98	−0.696802
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15102.9	1.11274	0.556370	−	0.830935i	$-0.312194\pi$
$569$	15102.9	1.11274	0.556370	+	0.830935i	$0.312194\pi$
$570$	0	0
$571$	−22553.6	−1.65295	−0.826477	−	0.562970i	$-0.809658\pi$
$571$	−22553.6	−1.65295	−0.826477	+	0.562970i	$0.809658\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	238.363	0.0172877
$576$	0	0
$577$	5007.38	0.361283	0.180641	−	0.983549i	$-0.442183\pi$
$577$	5007.38	0.361283	0.180641	+	0.983549i	$0.442183\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1019.73	0.0728153
$582$	0	0
$583$	13895.6	0.987131
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24975.4	1.75612	0.878062	−	0.478547i	$-0.158837\pi$
$587$	24975.4	1.75612	0.878062	+	0.478547i	$0.158837\pi$
$588$	0	0
$589$	−4196.00	−0.293537
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−19247.2	−1.33287	−0.666433	−	0.745565i	$-0.732180\pi$
$593$	−19247.2	−1.33287	−0.666433	+	0.745565i	$0.732180\pi$
$594$	0	0
$595$	1324.14	0.0912346
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6000.84	−0.409328	−0.204664	−	0.978832i	$-0.565610\pi$
$599$	−6000.84	−0.409328	−0.204664	+	0.978832i	$0.565610\pi$
$600$	0	0
$601$	−8576.08	−0.582073	−0.291036	−	0.956712i	$-0.594000\pi$
$601$	−8576.08	−0.582073	−0.291036	+	0.956712i	$0.594000\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−5373.22	−0.361079
$606$	0	0
$607$	2960.22	0.197943	0.0989715	−	0.995090i	$-0.468445\pi$
$607$	2960.22	0.197943	0.0989715	+	0.995090i	$0.468445\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−11863.0	−0.785477
$612$	0	0
$613$	6483.88	0.427212	0.213606	−	0.976920i	$-0.431479\pi$
$613$	6483.88	0.427212	0.213606	+	0.976920i	$0.431479\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1169.84	−0.0763304	−0.0381652	−	0.999271i	$-0.512151\pi$
$617$	−1169.84	−0.0763304	−0.0381652	+	0.999271i	$0.512151\pi$
$618$	0	0
$619$	−2930.97	−0.190316	−0.0951580	−	0.995462i	$-0.530336\pi$
$619$	−2930.97	−0.190316	−0.0951580	+	0.995462i	$0.530336\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2440.38	0.156937
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5306.23	0.336365
$630$	0	0
$631$	29359.9	1.85230	0.926150	−	0.377156i	$-0.123098\pi$
$631$	29359.9	1.85230	0.926150	+	0.377156i	$0.123098\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6160.54	−0.384998
$636$	0	0
$637$	−14049.7	−0.873894
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−14661.8	−0.903443	−0.451721	−	0.892159i	$-0.649190\pi$
$641$	−14661.8	−0.903443	−0.451721	+	0.892159i	$0.649190\pi$
$642$	0	0
$643$	−11141.3	−0.683312	−0.341656	−	0.939825i	$-0.610988\pi$
$643$	−11141.3	−0.683312	−0.341656	+	0.939825i	$0.610988\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−17044.5	−1.03569	−0.517844	−	0.855475i	$-0.673265\pi$
$647$	−17044.5	−1.03569	−0.517844	+	0.855475i	$0.673265\pi$
$648$	0	0
$649$	−4512.36	−0.272921
$650$	0	0
$651$	0	0
$652$	0	0
$653$	19960.2	1.19618	0.598089	−	0.801430i	$-0.295927\pi$
$653$	19960.2	1.19618	0.598089	+	0.801430i	$0.295927\pi$
$654$	0	0
$655$	−6045.87	−0.360659
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−13422.1	−0.793399	−0.396700	−	0.917948i	$-0.629845\pi$
$659$	−13422.1	−0.793399	−0.396700	+	0.917948i	$0.629845\pi$
$660$	0	0
$661$	13349.5	0.785531	0.392765	−	0.919639i	$-0.371518\pi$
$661$	13349.5	0.785531	0.392765	+	0.919639i	$0.371518\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1249.28	−0.0728494
$666$	0	0
$667$	−2082.14	−0.120871
$668$	0	0
$669$	0	0
$670$	0	0
$671$	18156.0	1.04456
$672$	0	0
$673$	6927.58	0.396788	0.198394	−	0.980122i	$-0.436427\pi$
$673$	6927.58	0.396788	0.198394	+	0.980122i	$0.436427\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6607.86	−0.375126	−0.187563	−	0.982253i	$-0.560059\pi$
$677$	−6607.86	−0.375126	−0.187563	+	0.982253i	$0.560059\pi$
$678$	0	0
$679$	−12419.2	−0.701922
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−15422.7	−0.864031	−0.432015	−	0.901866i	$-0.642197\pi$
$683$	−15422.7	−0.864031	−0.432015	+	0.901866i	$0.642197\pi$
$684$	0	0
$685$	−13148.3	−0.733388
$686$	0	0
$687$	0	0
$688$	0	0
$689$	15707.8	0.868533
$690$	0	0
$691$	16254.6	0.894870	0.447435	−	0.894317i	$-0.352338\pi$
$691$	16254.6	0.894870	0.447435	+	0.894317i	$0.352338\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11075.3	0.604475
$696$	0	0
$697$	6886.76	0.374254
$698$	0	0
$699$	0	0
$700$	0	0
$701$	10389.8	0.559796	0.279898	−	0.960030i	$-0.409699\pi$
$701$	10389.8	0.559796	0.279898	+	0.960030i	$0.409699\pi$
$702$	0	0
$703$	−5006.22	−0.268582
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−599.442	−0.0318873
$708$	0	0
$709$	7530.22	0.398877	0.199438	−	0.979910i	$-0.436088\pi$
$709$	7530.22	0.398877	0.199438	+	0.979910i	$0.436088\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1515.61	0.0796075
$714$	0	0
$715$	−13596.9	−0.711181
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−33888.4	−1.75775	−0.878875	−	0.477051i	$-0.841706\pi$
$719$	−33888.4	−1.75775	−0.878875	+	0.477051i	$0.841706\pi$
$720$	0	0
$721$	11546.5	0.596413
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5459.47	−0.279669
$726$	0	0
$727$	13863.3	0.707239	0.353619	−	0.935389i	$-0.384951\pi$
$727$	13863.3	0.707239	0.353619	+	0.935389i	$0.384951\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1125.64	−0.0569541
$732$	0	0
$733$	24497.7	1.23444	0.617219	−	0.786792i	$-0.288259\pi$
$733$	24497.7	1.23444	0.617219	+	0.786792i	$0.288259\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−53340.3	−2.66596
$738$	0	0
$739$	−19029.1	−0.947221	−0.473611	−	0.880734i	$-0.657050\pi$
$739$	−19029.1	−0.947221	−0.473611	+	0.880734i	$0.657050\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−5639.67	−0.278465	−0.139233	−	0.990260i	$-0.544464\pi$
$743$	−5639.67	−0.278465	−0.139233	+	0.990260i	$0.544464\pi$
$744$	0	0
$745$	−12234.7	−0.601672
$746$	0	0
$747$	0	0
$748$	0	0
$749$	401.287	0.0195764
$750$	0	0
$751$	14566.0	0.707753	0.353876	−	0.935292i	$-0.384863\pi$
$751$	14566.0	0.707753	0.353876	+	0.935292i	$0.384863\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	596.825	0.0287691
$756$	0	0
$757$	−2958.83	−0.142061	−0.0710307	−	0.997474i	$-0.522629\pi$
$757$	−2958.83	−0.142061	−0.0710307	+	0.997474i	$0.522629\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	281.640	0.0134158	0.00670791	−	0.999978i	$-0.497865\pi$
$761$	281.640	0.0134158	0.00670791	+	0.999978i	$0.497865\pi$
$762$	0	0
$763$	−2524.84	−0.119797
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5100.83	−0.240131
$768$	0	0
$769$	29799.2	1.39738	0.698690	−	0.715425i	$-0.253767\pi$
$769$	29799.2	1.39738	0.698690	+	0.715425i	$0.253767\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8377.18	0.389788	0.194894	−	0.980824i	$-0.437564\pi$
$773$	8377.18	0.389788	0.194894	+	0.980824i	$0.437564\pi$
$774$	0	0
$775$	3974.02	0.184195
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6497.39	−0.298836
$780$	0	0
$781$	16366.2	0.749843
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−11064.3	−0.503059
$786$	0	0
$787$	17368.8	0.786697	0.393348	−	0.919389i	$-0.371317\pi$
$787$	17368.8	0.786697	0.393348	+	0.919389i	$0.371317\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−17715.6	−0.796325
$792$	0	0
$793$	20523.7	0.919066
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−6003.86	−0.266835	−0.133418	−	0.991060i	$-0.542595\pi$
$797$	−6003.86	−0.266835	−0.133418	+	0.991060i	$0.542595\pi$
$798$	0	0
$799$	−5986.38	−0.265060
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−29761.5	−1.30792
$804$	0	0
$805$	451.244	0.0197568
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−42392.2	−1.84231	−0.921157	−	0.389191i	$-0.872754\pi$
$809$	−42392.2	−1.84231	−0.921157	+	0.389191i	$0.872754\pi$
$810$	0	0
$811$	7148.82	0.309530	0.154765	−	0.987951i	$-0.450538\pi$
$811$	7148.82	0.309530	0.154765	+	0.987951i	$0.450538\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	17402.7	0.747964
$816$	0	0
$817$	1062.00	0.0454770
$818$	0	0
$819$	0	0
$820$	0	0
$821$	25700.4	1.09251	0.546254	−	0.837619i	$-0.316053\pi$
$821$	25700.4	1.09251	0.546254	+	0.837619i	$0.316053\pi$
$822$	0	0
$823$	19064.7	0.807476	0.403738	−	0.914875i	$-0.367711\pi$
$823$	19064.7	0.807476	0.403738	+	0.914875i	$0.367711\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	41947.6	1.76380	0.881900	−	0.471437i	$-0.156264\pi$
$827$	41947.6	1.76380	0.881900	+	0.471437i	$0.156264\pi$
$828$	0	0
$829$	−31403.8	−1.31568	−0.657841	−	0.753157i	$-0.728530\pi$
$829$	−31403.8	−1.31568	−0.657841	+	0.753157i	$0.728530\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7089.84	−0.294896
$834$	0	0
$835$	−12385.8	−0.513326
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−25495.3	−1.04910	−0.524550	−	0.851380i	$-0.675766\pi$
$839$	−25495.3	−1.04910	−0.524550	+	0.851380i	$0.675766\pi$
$840$	0	0
$841$	23300.3	0.955362
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−4385.09	−0.178523
$846$	0	0
$847$	−10172.0	−0.412651
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1808.27	0.0728398
$852$	0	0
$853$	−13537.1	−0.543377	−0.271689	−	0.962385i	$-0.587582\pi$
$853$	−13537.1	−0.543377	−0.271689	+	0.962385i	$0.587582\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−15337.3	−0.611332	−0.305666	−	0.952139i	$-0.598879\pi$
$857$	−15337.3	−0.611332	−0.305666	+	0.952139i	$0.598879\pi$
$858$	0	0
$859$	−19720.1	−0.783286	−0.391643	−	0.920117i	$-0.628093\pi$
$859$	−19720.1	−0.783286	−0.391643	+	0.920117i	$0.628093\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−21464.2	−0.846641	−0.423321	−	0.905980i	$-0.639136\pi$
$863$	−21464.2	−0.846641	−0.423321	+	0.905980i	$0.639136\pi$
$864$	0	0
$865$	−133.723	−0.00525633
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2176.48	0.0849621
$870$	0	0
$871$	−60296.6	−2.34566
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1183.19	0.0457131
$876$	0	0
$877$	3671.90	0.141381	0.0706905	−	0.997498i	$-0.477480\pi$
$877$	3671.90	0.141381	0.0706905	+	0.997498i	$0.477480\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−48518.9	−1.85544	−0.927720	−	0.373276i	$-0.878234\pi$
$881$	−48518.9	−1.85544	−0.927720	+	0.373276i	$0.878234\pi$
$882$	0	0
$883$	−13413.5	−0.511210	−0.255605	−	0.966781i	$-0.582275\pi$
$883$	−13413.5	−0.511210	−0.255605	+	0.966781i	$0.582275\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	50035.0	1.89404	0.947018	−	0.321180i	$-0.104079\pi$
$887$	50035.0	1.89404	0.947018	+	0.321180i	$0.104079\pi$
$888$	0	0
$889$	−11662.5	−0.439986
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5647.91	0.211646
$894$	0	0
$895$	7003.77	0.261576
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−34713.7	−1.28784
$900$	0	0
$901$	7926.54	0.293087
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−17462.4	−0.641403
$906$	0	0
$907$	−35905.1	−1.31446	−0.657228	−	0.753692i	$-0.728271\pi$
$907$	−35905.1	−1.31446	−0.657228	+	0.753692i	$0.728271\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	36938.6	1.34339	0.671697	−	0.740826i	$-0.265566\pi$
$911$	36938.6	1.34339	0.671697	+	0.740826i	$0.265566\pi$
$912$	0	0
$913$	−5283.96	−0.191537
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−11445.4	−0.412171
$918$	0	0
$919$	−35867.0	−1.28742	−0.643712	−	0.765268i	$-0.722606\pi$
$919$	−35867.0	−1.28742	−0.643712	+	0.765268i	$0.722606\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	18500.5	0.659753
$924$	0	0
$925$	4741.37	0.168536
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−11391.2	−0.402297	−0.201148	−	0.979561i	$-0.564467\pi$
$929$	−11391.2	−0.402297	−0.201148	+	0.979561i	$0.564467\pi$
$930$	0	0
$931$	6688.98	0.235470
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−6861.32	−0.239988
$936$	0	0
$937$	30928.7	1.07833	0.539166	−	0.842199i	$-0.318740\pi$
$937$	30928.7	1.07833	0.539166	+	0.842199i	$0.318740\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	36684.3	1.27086	0.635428	−	0.772160i	$-0.280824\pi$
$941$	36684.3	1.27086	0.635428	+	0.772160i	$0.280824\pi$
$942$	0	0
$943$	2346.88	0.0810446
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−39211.7	−1.34552	−0.672761	−	0.739860i	$-0.734892\pi$
$947$	−39211.7	−1.34552	−0.672761	+	0.739860i	$0.734892\pi$
$948$	0	0
$949$	−33642.8	−1.15078
$950$	0	0
$951$	0	0
$952$	0	0
$953$	30476.1	1.03591	0.517953	−	0.855409i	$-0.326694\pi$
$953$	30476.1	1.03591	0.517953	+	0.855409i	$0.326694\pi$
$954$	0	0
$955$	−13640.8	−0.462206
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−24891.0	−0.838136
$960$	0	0
$961$	−4522.48	−0.151807
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−12199.1	−0.406946
$966$	0	0
$967$	−48866.8	−1.62508	−0.812538	−	0.582908i	$-0.801915\pi$
$967$	−48866.8	−1.62508	−0.812538	+	0.582908i	$0.801915\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−54060.7	−1.78670	−0.893352	−	0.449358i	$-0.851653\pi$
$971$	−54060.7	−1.78670	−0.893352	+	0.449358i	$0.851653\pi$
$972$	0	0
$973$	20966.6	0.690811
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−27233.7	−0.891795	−0.445898	−	0.895084i	$-0.647115\pi$
$977$	−27233.7	−0.891795	−0.445898	+	0.895084i	$0.647115\pi$
$978$	0	0
$979$	−12645.3	−0.412816
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−949.700	−0.0308146	−0.0154073	−	0.999881i	$-0.504904\pi$
$983$	−949.700	−0.0308146	−0.0154073	+	0.999881i	$0.504904\pi$
$984$	0	0
$985$	3829.53	0.123877
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−383.599	−0.0123334
$990$	0	0
$991$	−37146.4	−1.19071	−0.595356	−	0.803462i	$-0.702989\pi$
$991$	−37146.4	−1.19071	−0.595356	+	0.803462i	$0.702989\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3770.15	0.120123
$996$	0	0
$997$	45360.4	1.44090	0.720451	−	0.693506i	$-0.243935\pi$
$997$	45360.4	1.44090	0.720451	+	0.693506i	$0.243935\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bf.1.2		3
3.2	odd	2		2160.4.a.bn.1.2		3
4.3	odd	2		1080.4.a.h.1.2	✓	3
12.11	even	2		1080.4.a.n.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.h.1.2	✓	3	4.3	odd	2
1080.4.a.n.1.2	yes	3	12.11	even	2
2160.4.a.bf.1.2		3	1.1	even	1	trivial
2160.4.a.bn.1.2		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)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} -9.46549 q^{7} +49.0474 q^{11} +55.4438 q^{13} +27.9783 q^{17} -26.3965 q^{19} +9.53451 q^{23} +25.0000 q^{25} -218.379 q^{29} +158.961 q^{31} +47.3274 q^{35} +189.655 q^{37} +246.146 q^{41} -40.2327 q^{43} -213.965 q^{47} -253.405 q^{49} +283.310 q^{53} -245.237 q^{55} -92.0000 q^{59} +370.172 q^{61} -277.219 q^{65} -1087.53 q^{67} +333.681 q^{71} -606.791 q^{73} -464.257 q^{77} +44.3751 q^{79} -107.732 q^{83} -139.892 q^{85} -257.819 q^{89} -524.803 q^{91} +131.982 q^{95} +1312.05 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 15 q^{5} - 9 q^{7} + 18 q^{11} - 21 q^{13} - 84 q^{17} - 21 q^{19} + 48 q^{23} + 75 q^{25} + 36 q^{29} - 324 q^{31} + 45 q^{35} + 33 q^{37} - 114 q^{41} - 282 q^{43} + 282 q^{47} + 228 q^{49} - 222 q^{53}+ \cdots + 1191 q^{97}+O(q^{100}) \) 3 * q - 15 * q^5 - 9 * q^7 + 18 * q^11 - 21 * q^13 - 84 * q^17 - 21 * q^19 + 48 * q^23 + 75 * q^25 + 36 * q^29 - 324 * q^31 + 45 * q^35 + 33 * q^37 - 114 * q^41 - 282 * q^43 + 282 * q^47 + 228 * q^49 - 222 * q^53 - 90 * q^55 - 276 * q^59 + 303 * q^61 + 105 * q^65 - 1035 * q^67 + 510 * q^71 + 447 * q^73 + 1578 * q^77 - 777 * q^79 + 78 * q^83 + 420 * q^85 + 324 * q^89 - 1995 * q^91 + 105 * q^95 + 1191 * q^97

Introduction

L-functions

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