LMFDB - Embedded newform 2160.4.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,4,Mod(1,2160)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$127.444125612$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.5637.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 23x + 6 $ x^3 - x^2 - 23*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.20067$ 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)

$f(q)$	$=$	$q+5.00000 q^{5} -24.4013 q^{7} +O(q^{10})$	$q+5.00000 q^{5} -24.4013 q^{7} -28.9839 q^{11} -65.3919 q^{13} -68.1718 q^{17} -104.424 q^{19} -154.807 q^{23} +25.0000 q^{25} +205.658 q^{29} +18.2497 q^{31} -122.007 q^{35} -337.613 q^{37} +195.969 q^{41} -334.882 q^{43} -5.00398 q^{47} +252.425 q^{49} +319.965 q^{53} -144.919 q^{55} +430.611 q^{59} +594.581 q^{61} -326.960 q^{65} -195.876 q^{67} +425.955 q^{71} +929.193 q^{73} +707.245 q^{77} -24.4296 q^{79} -545.859 q^{83} -340.859 q^{85} +84.1332 q^{89} +1595.65 q^{91} -522.121 q^{95} +827.613 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 15 q^{5} - 44 q^{7}+O(q^{10}) $ 3 * q + 15 * q^5 - 44 * q^7	$ 3 q + 15 q^{5} - 44 q^{7} - 38 q^{11} + 28 q^{13} - 19 q^{17} - 187 q^{19} + 81 q^{23} + 75 q^{25} + 160 q^{29} - 227 q^{31} - 220 q^{35} + 78 q^{37} - 338 q^{41} - 22 q^{43} + 472 q^{47} - 197 q^{49} + 521 q^{53} - 190 q^{55} - 140 q^{59} + 595 q^{61} + 140 q^{65} - 878 q^{67} + 602 q^{71} + 1294 q^{73} + 288 q^{77} - 629 q^{79} + 1287 q^{83} - 95 q^{85} + 2154 q^{89} + 440 q^{91} - 935 q^{95} + 1392 q^{97}+O(q^{100}) $ 3 * q + 15 * q^5 - 44 * q^7 - 38 * q^11 + 28 * q^13 - 19 * q^17 - 187 * q^19 + 81 * q^23 + 75 * q^25 + 160 * q^29 - 227 * q^31 - 220 * q^35 + 78 * q^37 - 338 * q^41 - 22 * q^43 + 472 * q^47 - 197 * q^49 + 521 * q^53 - 190 * q^55 - 140 * q^59 + 595 * q^61 + 140 * q^65 - 878 * q^67 + 602 * q^71 + 1294 * q^73 + 288 * q^77 - 629 * q^79 + 1287 * q^83 - 95 * q^85 + 2154 * q^89 + 440 * q^91 - 935 * q^95 + 1392 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−24.4013	−1.31755	−0.658774	−	0.752341i	$-0.728925\pi$
$7$	−24.4013	−1.31755	−0.658774	+	0.752341i	$0.728925\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−28.9839	−0.794451	−0.397226	−	0.917721i	$-0.630027\pi$
$11$	−28.9839	−0.794451	−0.397226	+	0.917721i	$0.630027\pi$
$12$	0	0
$13$	−65.3919	−1.39511	−0.697556	−	0.716530i	$-0.745729\pi$
$13$	−65.3919	−1.39511	−0.697556	+	0.716530i	$0.745729\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−68.1718	−0.972593	−0.486296	−	0.873794i	$-0.661653\pi$
$17$	−68.1718	−0.972593	−0.486296	+	0.873794i	$0.661653\pi$
$18$	0	0
$19$	−104.424	−1.26087	−0.630435	−	0.776242i	$-0.717124\pi$
$19$	−104.424	−1.26087	−0.630435	+	0.776242i	$0.717124\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−154.807	−1.40345	−0.701727	−	0.712446i	$-0.747587\pi$
$23$	−154.807	−1.40345	−0.701727	+	0.712446i	$0.747587\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	205.658	1.31689	0.658443	−	0.752631i	$-0.271215\pi$
$29$	205.658	1.31689	0.658443	+	0.752631i	$0.271215\pi$
$30$	0	0
$31$	18.2497	0.105734	0.0528668	−	0.998602i	$-0.483164\pi$
$31$	18.2497	0.105734	0.0528668	+	0.998602i	$0.483164\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−122.007	−0.589226
$36$	0	0
$37$	−337.613	−1.50009	−0.750044	−	0.661387i	$-0.769968\pi$
$37$	−337.613	−1.50009	−0.750044	+	0.661387i	$0.769968\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	195.969	0.746469	0.373234	−	0.927737i	$-0.378249\pi$
$41$	195.969	0.746469	0.373234	+	0.927737i	$0.378249\pi$
$42$	0	0
$43$	−334.882	−1.18765	−0.593826	−	0.804594i	$-0.702383\pi$
$43$	−334.882	−1.18765	−0.593826	+	0.804594i	$0.702383\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.00398	−0.0155299	−0.00776496	−	0.999970i	$-0.502472\pi$
$47$	−5.00398	−0.0155299	−0.00776496	+	0.999970i	$0.502472\pi$
$48$	0	0
$49$	252.425	0.735934
$50$	0	0
$51$	0	0
$52$	0	0
$53$	319.965	0.829256	0.414628	−	0.909991i	$-0.363912\pi$
$53$	319.965	0.829256	0.414628	+	0.909991i	$0.363912\pi$
$54$	0	0
$55$	−144.919	−0.355289
$56$	0	0
$57$	0	0
$58$	0	0
$59$	430.611	0.950182	0.475091	−	0.879937i	$-0.342415\pi$
$59$	430.611	0.950182	0.475091	+	0.879937i	$0.342415\pi$
$60$	0	0
$61$	594.581	1.24800	0.624002	−	0.781422i	$-0.285505\pi$
$61$	594.581	1.24800	0.624002	+	0.781422i	$0.285505\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−326.960	−0.623913
$66$	0	0
$67$	−195.876	−0.357166	−0.178583	−	0.983925i	$-0.557151\pi$
$67$	−195.876	−0.357166	−0.178583	+	0.983925i	$0.557151\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	425.955	0.711994	0.355997	−	0.934487i	$-0.384141\pi$
$71$	425.955	0.711994	0.355997	+	0.934487i	$0.384141\pi$
$72$	0	0
$73$	929.193	1.48978	0.744889	−	0.667188i	$-0.232502\pi$
$73$	929.193	1.48978	0.744889	+	0.667188i	$0.232502\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	707.245	1.04673
$78$	0	0
$79$	−24.4296	−0.0347917	−0.0173959	−	0.999849i	$-0.505538\pi$
$79$	−24.4296	−0.0347917	−0.0173959	+	0.999849i	$0.505538\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−545.859	−0.721877	−0.360938	−	0.932590i	$-0.617544\pi$
$83$	−545.859	−0.721877	−0.360938	+	0.932590i	$0.617544\pi$
$84$	0	0
$85$	−340.859	−0.434957
$86$	0	0
$87$	0	0
$88$	0	0
$89$	84.1332	0.100203	0.0501017	−	0.998744i	$-0.484045\pi$
$89$	84.1332	0.100203	0.0501017	+	0.998744i	$0.484045\pi$
$90$	0	0
$91$	1595.65	1.83813
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−522.121	−0.563879
$96$	0	0
$97$	827.613	0.866303	0.433152	−	0.901321i	$-0.357401\pi$
$97$	827.613	0.866303	0.433152	+	0.901321i	$0.357401\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−823.576	−0.811375	−0.405688	−	0.914012i	$-0.632968\pi$
$101$	−823.576	−0.811375	−0.405688	+	0.914012i	$0.632968\pi$
$102$	0	0
$103$	−1171.19	−1.12040	−0.560198	−	0.828359i	$-0.689275\pi$
$103$	−1171.19	−1.12040	−0.560198	+	0.828359i	$0.689275\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1023.21	0.924460	0.462230	−	0.886760i	$-0.347049\pi$
$107$	1023.21	0.924460	0.462230	+	0.886760i	$0.347049\pi$
$108$	0	0
$109$	−403.647	−0.354700	−0.177350	−	0.984148i	$-0.556753\pi$
$109$	−403.647	−0.354700	−0.177350	+	0.984148i	$0.556753\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1082.20	−0.900931	−0.450465	−	0.892794i	$-0.648742\pi$
$113$	−1082.20	−0.900931	−0.450465	+	0.892794i	$0.648742\pi$
$114$	0	0
$115$	−774.033	−0.627643
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1663.48	1.28144
$120$	0	0
$121$	−490.935	−0.368847
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	774.132	0.540890	0.270445	−	0.962735i	$-0.412829\pi$
$127$	774.132	0.540890	0.270445	+	0.962735i	$0.412829\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1214.04	−0.809702	−0.404851	−	0.914383i	$-0.632677\pi$
$131$	−1214.04	−0.809702	−0.404851	+	0.914383i	$0.632677\pi$
$132$	0	0
$133$	2548.09	1.66126
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2300.15	−1.43441	−0.717207	−	0.696860i	$-0.754580\pi$
$137$	−2300.15	−1.43441	−0.717207	+	0.696860i	$0.754580\pi$
$138$	0	0
$139$	1355.93	0.827396	0.413698	−	0.910414i	$-0.364237\pi$
$139$	1355.93	0.827396	0.413698	+	0.910414i	$0.364237\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1895.31	1.10835
$144$	0	0
$145$	1028.29	0.588929
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−259.845	−0.142868	−0.0714340	−	0.997445i	$-0.522758\pi$
$149$	−259.845	−0.142868	−0.0714340	+	0.997445i	$0.522758\pi$
$150$	0	0
$151$	508.304	0.273941	0.136971	−	0.990575i	$-0.456263\pi$
$151$	508.304	0.273941	0.136971	+	0.990575i	$0.456263\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	91.2485	0.0472855
$156$	0	0
$157$	−23.3052	−0.0118468	−0.00592342	−	0.999982i	$-0.501885\pi$
$157$	−23.3052	−0.0118468	−0.00592342	+	0.999982i	$0.501885\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3777.49	1.84912
$162$	0	0
$163$	4032.10	1.93754	0.968769	−	0.247964i	$-0.0797616\pi$
$163$	4032.10	1.93754	0.968769	+	0.247964i	$0.0797616\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−671.911	−0.311341	−0.155671	−	0.987809i	$-0.549754\pi$
$167$	−671.911	−0.311341	−0.155671	+	0.987809i	$0.549754\pi$
$168$	0	0
$169$	2079.10	0.946337
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1633.53	−0.717889	−0.358944	−	0.933359i	$-0.616863\pi$
$173$	−1633.53	−0.717889	−0.358944	+	0.933359i	$0.616863\pi$
$174$	0	0
$175$	−610.034	−0.263510
$176$	0	0
$177$	0	0
$178$	0	0
$179$	341.260	0.142497	0.0712485	−	0.997459i	$-0.477302\pi$
$179$	341.260	0.142497	0.0712485	+	0.997459i	$0.477302\pi$
$180$	0	0
$181$	−1695.92	−0.696447	−0.348223	−	0.937412i	$-0.613215\pi$
$181$	−1695.92	−0.696447	−0.348223	+	0.937412i	$0.613215\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1688.07	−0.670860
$186$	0	0
$187$	1975.88	0.772678
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−726.451	−0.275205	−0.137603	−	0.990488i	$-0.543940\pi$
$191$	−726.451	−0.275205	−0.137603	+	0.990488i	$0.543940\pi$
$192$	0	0
$193$	4247.26	1.58406	0.792032	−	0.610479i	$-0.209023\pi$
$193$	4247.26	1.58406	0.792032	+	0.610479i	$0.209023\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2678.52	−0.968713	−0.484357	−	0.874871i	$-0.660946\pi$
$197$	−2678.52	−0.968713	−0.484357	+	0.874871i	$0.660946\pi$
$198$	0	0
$199$	−1486.48	−0.529517	−0.264759	−	0.964315i	$-0.585292\pi$
$199$	−1486.48	−0.529517	−0.264759	+	0.964315i	$0.585292\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5018.32	−1.73506
$204$	0	0
$205$	979.845	0.333831
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3026.62	1.00170
$210$	0	0
$211$	4827.41	1.57504	0.787519	−	0.616291i	$-0.211365\pi$
$211$	4827.41	1.57504	0.787519	+	0.616291i	$0.211365\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1674.41	−0.531134
$216$	0	0
$217$	−445.317	−0.139309
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4457.88	1.35688
$222$	0	0
$223$	−2774.48	−0.833153	−0.416576	−	0.909101i	$-0.636770\pi$
$223$	−2774.48	−0.833153	−0.416576	+	0.909101i	$0.636770\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5101.34	−1.49158	−0.745788	−	0.666184i	$-0.767927\pi$
$227$	−5101.34	−1.49158	−0.745788	+	0.666184i	$0.767927\pi$
$228$	0	0
$229$	−4097.83	−1.18250	−0.591249	−	0.806489i	$-0.701365\pi$
$229$	−4097.83	−1.18250	−0.591249	+	0.806489i	$0.701365\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−357.613	−0.100549	−0.0502747	−	0.998735i	$-0.516010\pi$
$233$	−357.613	−0.100549	−0.0502747	+	0.998735i	$0.516010\pi$
$234$	0	0
$235$	−25.0199	−0.00694519
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−351.682	−0.0951818	−0.0475909	−	0.998867i	$-0.515154\pi$
$239$	−351.682	−0.0951818	−0.0475909	+	0.998867i	$0.515154\pi$
$240$	0	0
$241$	−6165.53	−1.64795	−0.823976	−	0.566624i	$-0.808249\pi$
$241$	−6165.53	−1.64795	−0.823976	+	0.566624i	$0.808249\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1262.13	0.329120
$246$	0	0
$247$	6828.50	1.75906
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3245.53	0.816160	0.408080	−	0.912946i	$-0.366198\pi$
$251$	3245.53	0.816160	0.408080	+	0.912946i	$0.366198\pi$
$252$	0	0
$253$	4486.90	1.11498
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3552.19	−0.862178	−0.431089	−	0.902309i	$-0.641871\pi$
$257$	−3552.19	−0.862178	−0.431089	+	0.902309i	$0.641871\pi$
$258$	0	0
$259$	8238.22	1.97644
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4416.59	1.03551	0.517754	−	0.855530i	$-0.326768\pi$
$263$	4416.59	1.03551	0.517754	+	0.855530i	$0.326768\pi$
$264$	0	0
$265$	1599.83	0.370855
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3419.93	0.775155	0.387578	−	0.921837i	$-0.373312\pi$
$269$	3419.93	0.775155	0.387578	+	0.921837i	$0.373312\pi$
$270$	0	0
$271$	−716.407	−0.160585	−0.0802927	−	0.996771i	$-0.525585\pi$
$271$	−716.407	−0.160585	−0.0802927	+	0.996771i	$0.525585\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−724.597	−0.158890
$276$	0	0
$277$	657.529	0.142625	0.0713124	−	0.997454i	$-0.477281\pi$
$277$	657.529	0.142625	0.0713124	+	0.997454i	$0.477281\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1513.91	−0.321397	−0.160698	−	0.987004i	$-0.551375\pi$
$281$	−1513.91	−0.321397	−0.160698	+	0.987004i	$0.551375\pi$
$282$	0	0
$283$	−3906.38	−0.820532	−0.410266	−	0.911966i	$-0.634564\pi$
$283$	−3906.38	−0.820532	−0.410266	+	0.911966i	$0.634564\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4781.91	−0.983509
$288$	0	0
$289$	−265.611	−0.0540629
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8048.76	1.60483	0.802413	−	0.596770i	$-0.203549\pi$
$293$	8048.76	1.60483	0.802413	+	0.596770i	$0.203549\pi$
$294$	0	0
$295$	2153.05	0.424934
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10123.1	1.95797
$300$	0	0
$301$	8171.57	1.56479
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2972.91	0.558125
$306$	0	0
$307$	−101.564	−0.0188814	−0.00944068	−	0.999955i	$-0.503005\pi$
$307$	−101.564	−0.0188814	−0.00944068	+	0.999955i	$0.503005\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7684.59	−1.40113	−0.700567	−	0.713586i	$-0.747070\pi$
$311$	−7684.59	−1.40113	−0.700567	+	0.713586i	$0.747070\pi$
$312$	0	0
$313$	−1345.15	−0.242915	−0.121457	−	0.992597i	$-0.538757\pi$
$313$	−1345.15	−0.242915	−0.121457	+	0.992597i	$0.538757\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7622.33	−1.35051	−0.675257	−	0.737583i	$-0.735967\pi$
$317$	−7622.33	−1.35051	−0.675257	+	0.737583i	$0.735967\pi$
$318$	0	0
$319$	−5960.76	−1.04620
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7118.78	1.22631
$324$	0	0
$325$	−1634.80	−0.279022
$326$	0	0
$327$	0	0
$328$	0	0
$329$	122.104	0.0204614
$330$	0	0
$331$	6585.09	1.09350	0.546751	−	0.837295i	$-0.315864\pi$
$331$	6585.09	1.09350	0.546751	+	0.837295i	$0.315864\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−979.382	−0.159729
$336$	0	0
$337$	−2946.94	−0.476351	−0.238175	−	0.971222i	$-0.576549\pi$
$337$	−2946.94	−0.476351	−0.238175	+	0.971222i	$0.576549\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−528.947	−0.0840002
$342$	0	0
$343$	2210.14	0.347919
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8493.48	1.31399	0.656994	−	0.753896i	$-0.271828\pi$
$347$	8493.48	1.31399	0.656994	+	0.753896i	$0.271828\pi$
$348$	0	0
$349$	5646.54	0.866053	0.433027	−	0.901381i	$-0.357446\pi$
$349$	5646.54	0.866053	0.433027	+	0.901381i	$0.357446\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1221.93	0.184240	0.0921202	−	0.995748i	$-0.470636\pi$
$353$	1221.93	0.184240	0.0921202	+	0.995748i	$0.470636\pi$
$354$	0	0
$355$	2129.78	0.318414
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4151.44	−0.610319	−0.305160	−	0.952301i	$-0.598710\pi$
$359$	−4151.44	−0.610319	−0.305160	+	0.952301i	$0.598710\pi$
$360$	0	0
$361$	4045.41	0.589796
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4645.96	0.666249
$366$	0	0
$367$	7038.71	1.00114	0.500569	−	0.865696i	$-0.333124\pi$
$367$	7038.71	1.00114	0.500569	+	0.865696i	$0.333124\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−7807.58	−1.09259
$372$	0	0
$373$	−7119.57	−0.988303	−0.494152	−	0.869376i	$-0.664521\pi$
$373$	−7119.57	−0.988303	−0.494152	+	0.869376i	$0.664521\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−13448.4	−1.83720
$378$	0	0
$379$	−3372.29	−0.457053	−0.228526	−	0.973538i	$-0.573391\pi$
$379$	−3372.29	−0.457053	−0.228526	+	0.973538i	$0.573391\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3958.63	0.528138	0.264069	−	0.964504i	$-0.414935\pi$
$383$	3958.63	0.528138	0.264069	+	0.964504i	$0.414935\pi$
$384$	0	0
$385$	3536.23	0.468111
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9654.01	−1.25830	−0.629148	−	0.777285i	$-0.716596\pi$
$389$	−9654.01	−1.25830	−0.629148	+	0.777285i	$0.716596\pi$
$390$	0	0
$391$	10553.4	1.36499
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−122.148	−0.0155593
$396$	0	0
$397$	10928.3	1.38155	0.690776	−	0.723068i	$-0.257269\pi$
$397$	10928.3	1.38155	0.690776	+	0.723068i	$0.257269\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4085.57	−0.508787	−0.254393	−	0.967101i	$-0.581876\pi$
$401$	−4085.57	−0.508787	−0.254393	+	0.967101i	$0.581876\pi$
$402$	0	0
$403$	−1193.38	−0.147510
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9785.34	1.19175
$408$	0	0
$409$	10156.3	1.22786	0.613930	−	0.789361i	$-0.289588\pi$
$409$	10156.3	1.22786	0.613930	+	0.789361i	$0.289588\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−10507.5	−1.25191
$414$	0	0
$415$	−2729.29	−0.322833
$416$	0	0
$417$	0	0
$418$	0	0
$419$	15878.8	1.85139	0.925693	−	0.378275i	$-0.123483\pi$
$419$	15878.8	1.85139	0.925693	+	0.378275i	$0.123483\pi$
$420$	0	0
$421$	−2279.85	−0.263926	−0.131963	−	0.991255i	$-0.542128\pi$
$421$	−2279.85	−0.263926	−0.131963	+	0.991255i	$0.542128\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1704.29	−0.194519
$426$	0	0
$427$	−14508.6	−1.64431
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1947.38	−0.217638	−0.108819	−	0.994062i	$-0.534707\pi$
$431$	−1947.38	−0.217638	−0.108819	+	0.994062i	$0.534707\pi$
$432$	0	0
$433$	12636.2	1.40244	0.701219	−	0.712946i	$-0.252639\pi$
$433$	12636.2	1.40244	0.701219	+	0.712946i	$0.252639\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	16165.6	1.76957
$438$	0	0
$439$	15849.8	1.72317	0.861585	−	0.507614i	$-0.169472\pi$
$439$	15849.8	1.72317	0.861585	+	0.507614i	$0.169472\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−17455.6	−1.87210	−0.936048	−	0.351872i	$-0.885545\pi$
$443$	−17455.6	−1.87210	−0.936048	+	0.351872i	$0.885545\pi$
$444$	0	0
$445$	420.666	0.0448123
$446$	0	0
$447$	0	0
$448$	0	0
$449$	16068.1	1.68887	0.844435	−	0.535658i	$-0.179936\pi$
$449$	16068.1	1.68887	0.844435	+	0.535658i	$0.179936\pi$
$450$	0	0
$451$	−5679.94	−0.593033
$452$	0	0
$453$	0	0
$454$	0	0
$455$	7978.25	0.822036
$456$	0	0
$457$	11891.7	1.21722	0.608612	−	0.793468i	$-0.291727\pi$
$457$	11891.7	1.21722	0.608612	+	0.793468i	$0.291727\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2802.23	0.283108	0.141554	−	0.989931i	$-0.454790\pi$
$461$	2802.23	0.283108	0.141554	+	0.989931i	$0.454790\pi$
$462$	0	0
$463$	12933.3	1.29819	0.649096	−	0.760707i	$-0.275147\pi$
$463$	12933.3	1.29819	0.649096	+	0.760707i	$0.275147\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5748.11	0.569573	0.284787	−	0.958591i	$-0.408077\pi$
$467$	5748.11	0.569573	0.284787	+	0.958591i	$0.408077\pi$
$468$	0	0
$469$	4779.65	0.470583
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9706.17	0.943531
$474$	0	0
$475$	−2610.60	−0.252174
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11217.3	−1.07000	−0.535002	−	0.844851i	$-0.679689\pi$
$479$	−11217.3	−1.07000	−0.535002	+	0.844851i	$0.679689\pi$
$480$	0	0
$481$	22077.2	2.09279
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4138.07	0.387423
$486$	0	0
$487$	8905.12	0.828603	0.414301	−	0.910140i	$-0.364026\pi$
$487$	8905.12	0.828603	0.414301	+	0.910140i	$0.364026\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6553.10	−0.602316	−0.301158	−	0.953574i	$-0.597373\pi$
$491$	−6553.10	−0.602316	−0.301158	+	0.953574i	$0.597373\pi$
$492$	0	0
$493$	−14020.1	−1.28079
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10393.9	−0.938087
$498$	0	0
$499$	4610.09	0.413579	0.206789	−	0.978385i	$-0.433698\pi$
$499$	4610.09	0.413579	0.206789	+	0.978385i	$0.433698\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	13069.1	1.15850	0.579249	−	0.815151i	$-0.303346\pi$
$503$	13069.1	1.15850	0.579249	+	0.815151i	$0.303346\pi$
$504$	0	0
$505$	−4117.88	−0.362858
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−15930.8	−1.38727	−0.693635	−	0.720327i	$-0.743992\pi$
$509$	−15930.8	−1.38727	−0.693635	+	0.720327i	$0.743992\pi$
$510$	0	0
$511$	−22673.6	−1.96286
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5855.95	−0.501056
$516$	0	0
$517$	145.035	0.0123378
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3654.38	−0.307296	−0.153648	−	0.988126i	$-0.549102\pi$
$521$	−3654.38	−0.307296	−0.153648	+	0.988126i	$0.549102\pi$
$522$	0	0
$523$	5138.66	0.429633	0.214816	−	0.976654i	$-0.431085\pi$
$523$	5138.66	0.429633	0.214816	+	0.976654i	$0.431085\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1244.11	−0.102836
$528$	0	0
$529$	11798.1	0.969681
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−12814.8	−1.04141
$534$	0	0
$535$	5116.04	0.413431
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−7316.27	−0.584664
$540$	0	0
$541$	6932.06	0.550892	0.275446	−	0.961317i	$-0.411175\pi$
$541$	6932.06	0.550892	0.275446	+	0.961317i	$0.411175\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2018.23	−0.158627
$546$	0	0
$547$	3423.11	0.267572	0.133786	−	0.991010i	$-0.457287\pi$
$547$	3423.11	0.267572	0.133786	+	0.991010i	$0.457287\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−21475.6	−1.66042
$552$	0	0
$553$	596.115	0.0458398
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24489.2	1.86291	0.931455	−	0.363856i	$-0.118540\pi$
$557$	24489.2	1.86291	0.931455	+	0.363856i	$0.118540\pi$
$558$	0	0
$559$	21898.6	1.65691
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10053.1	−0.752552	−0.376276	−	0.926508i	$-0.622796\pi$
$563$	−10053.1	−0.752552	−0.376276	+	0.926508i	$0.622796\pi$
$564$	0	0
$565$	−5411.02	−0.402908
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6670.45	0.491459	0.245729	−	0.969338i	$-0.420973\pi$
$569$	6670.45	0.491459	0.245729	+	0.969338i	$0.420973\pi$
$570$	0	0
$571$	−4633.55	−0.339594	−0.169797	−	0.985479i	$-0.554311\pi$
$571$	−4633.55	−0.339594	−0.169797	+	0.985479i	$0.554311\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3870.17	−0.280691
$576$	0	0
$577$	7045.15	0.508307	0.254154	−	0.967164i	$-0.418203\pi$
$577$	7045.15	0.508307	0.254154	+	0.967164i	$0.418203\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	13319.7	0.951108
$582$	0	0
$583$	−9273.82	−0.658804
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8001.06	−0.562588	−0.281294	−	0.959622i	$-0.590764\pi$
$587$	−8001.06	−0.562588	−0.281294	+	0.959622i	$0.590764\pi$
$588$	0	0
$589$	−1905.71	−0.133316
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6747.53	−0.467265	−0.233632	−	0.972325i	$-0.575061\pi$
$593$	−6747.53	−0.467265	−0.233632	+	0.972325i	$0.575061\pi$
$594$	0	0
$595$	8317.41	0.573077
$596$	0	0
$597$	0	0
$598$	0	0
$599$	21547.3	1.46978	0.734890	−	0.678186i	$-0.237234\pi$
$599$	21547.3	1.46978	0.734890	+	0.678186i	$0.237234\pi$
$600$	0	0
$601$	−12155.1	−0.824983	−0.412492	−	0.910961i	$-0.635341\pi$
$601$	−12155.1	−0.824983	−0.412492	+	0.910961i	$0.635341\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2454.68	−0.164953
$606$	0	0
$607$	−16348.9	−1.09322	−0.546608	−	0.837388i	$-0.684081\pi$
$607$	−16348.9	−1.09322	−0.546608	+	0.837388i	$0.684081\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	327.220	0.0216660
$612$	0	0
$613$	−29955.5	−1.97372	−0.986859	−	0.161581i	$-0.948341\pi$
$613$	−29955.5	−1.97372	−0.986859	+	0.161581i	$0.948341\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2159.74	−0.140921	−0.0704603	−	0.997515i	$-0.522447\pi$
$617$	−2159.74	−0.140921	−0.0704603	+	0.997515i	$0.522447\pi$
$618$	0	0
$619$	−22100.8	−1.43507	−0.717535	−	0.696523i	$-0.754730\pi$
$619$	−22100.8	−1.43507	−0.717535	+	0.696523i	$0.754730\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2052.96	−0.132023
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	23015.7	1.45898
$630$	0	0
$631$	−18360.1	−1.15833	−0.579164	−	0.815211i	$-0.696621\pi$
$631$	−18360.1	−1.15833	−0.579164	+	0.815211i	$0.696621\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3870.66	0.241893
$636$	0	0
$637$	−16506.6	−1.02671
$638$	0	0
$639$	0	0
$640$	0	0
$641$	21064.5	1.29797	0.648985	−	0.760802i	$-0.275194\pi$
$641$	21064.5	1.29797	0.648985	+	0.760802i	$0.275194\pi$
$642$	0	0
$643$	10539.1	0.646381	0.323190	−	0.946334i	$-0.395245\pi$
$643$	10539.1	0.646381	0.323190	+	0.946334i	$0.395245\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	22553.4	1.37043	0.685213	−	0.728343i	$-0.259709\pi$
$647$	22553.4	1.37043	0.685213	+	0.728343i	$0.259709\pi$
$648$	0	0
$649$	−12480.8	−0.754873
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−22624.0	−1.35582	−0.677908	−	0.735147i	$-0.737113\pi$
$653$	−22624.0	−1.35582	−0.677908	+	0.735147i	$0.737113\pi$
$654$	0	0
$655$	−6070.19	−0.362110
$656$	0	0
$657$	0	0
$658$	0	0
$659$	6376.60	0.376930	0.188465	−	0.982080i	$-0.439649\pi$
$659$	6376.60	0.376930	0.188465	+	0.982080i	$0.439649\pi$
$660$	0	0
$661$	−22097.5	−1.30029	−0.650146	−	0.759809i	$-0.725292\pi$
$661$	−22097.5	−1.30029	−0.650146	+	0.759809i	$0.725292\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	12740.4	0.742938
$666$	0	0
$667$	−31837.2	−1.84819
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−17233.3	−0.991479
$672$	0	0
$673$	24033.0	1.37653	0.688264	−	0.725461i	$-0.258373\pi$
$673$	24033.0	1.37653	0.688264	+	0.725461i	$0.258373\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−179.638	−0.0101980	−0.00509901	−	0.999987i	$-0.501623\pi$
$677$	−179.638	−0.0101980	−0.00509901	+	0.999987i	$0.501623\pi$
$678$	0	0
$679$	−20194.9	−1.14140
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−30434.2	−1.70502	−0.852511	−	0.522709i	$-0.824921\pi$
$683$	−30434.2	−1.70502	−0.852511	+	0.522709i	$0.824921\pi$
$684$	0	0
$685$	−11500.7	−0.641490
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−20923.1	−1.15691
$690$	0	0
$691$	−9792.73	−0.539122	−0.269561	−	0.962983i	$-0.586879\pi$
$691$	−9792.73	−0.539122	−0.269561	+	0.962983i	$0.586879\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6779.63	0.370023
$696$	0	0
$697$	−13359.6	−0.726010
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−8130.47	−0.438065	−0.219032	−	0.975718i	$-0.570290\pi$
$701$	−8130.47	−0.438065	−0.219032	+	0.975718i	$0.570290\pi$
$702$	0	0
$703$	35255.0	1.89142
$704$	0	0
$705$	0	0
$706$	0	0
$707$	20096.4	1.06903
$708$	0	0
$709$	−4859.95	−0.257432	−0.128716	−	0.991681i	$-0.541086\pi$
$709$	−4859.95	−0.257432	−0.128716	+	0.991681i	$0.541086\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2825.17	−0.148392
$714$	0	0
$715$	9476.55	0.495669
$716$	0	0
$717$	0	0
$718$	0	0
$719$	19463.0	1.00952	0.504762	−	0.863259i	$-0.331580\pi$
$719$	19463.0	1.00952	0.504762	+	0.863259i	$0.331580\pi$
$720$	0	0
$721$	28578.6	1.47618
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5141.44	0.263377
$726$	0	0
$727$	−2432.66	−0.124102	−0.0620512	−	0.998073i	$-0.519764\pi$
$727$	−2432.66	−0.124102	−0.0620512	+	0.998073i	$0.519764\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	22829.5	1.15510
$732$	0	0
$733$	−17967.6	−0.905386	−0.452693	−	0.891666i	$-0.649537\pi$
$733$	−17967.6	−0.905386	−0.452693	+	0.891666i	$0.649537\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5677.26	0.283751
$738$	0	0
$739$	−23473.0	−1.16843	−0.584214	−	0.811599i	$-0.698597\pi$
$739$	−23473.0	−1.16843	−0.584214	+	0.811599i	$0.698597\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−33559.2	−1.65702	−0.828512	−	0.559971i	$-0.810812\pi$
$743$	−33559.2	−1.65702	−0.828512	+	0.559971i	$0.810812\pi$
$744$	0	0
$745$	−1299.23	−0.0638925
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−24967.6	−1.21802
$750$	0	0
$751$	7782.75	0.378158	0.189079	−	0.981962i	$-0.439450\pi$
$751$	7782.75	0.378158	0.189079	+	0.981962i	$0.439450\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2541.52	0.122510
$756$	0	0
$757$	−38154.3	−1.83189	−0.915946	−	0.401300i	$-0.868558\pi$
$757$	−38154.3	−1.83189	−0.915946	+	0.401300i	$0.868558\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−19867.1	−0.946363	−0.473182	−	0.880965i	$-0.656895\pi$
$761$	−19867.1	−0.946363	−0.473182	+	0.880965i	$0.656895\pi$
$762$	0	0
$763$	9849.52	0.467335
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−28158.5	−1.32561
$768$	0	0
$769$	−15710.8	−0.736730	−0.368365	−	0.929681i	$-0.620082\pi$
$769$	−15710.8	−0.736730	−0.368365	+	0.929681i	$0.620082\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−25811.9	−1.20102	−0.600510	−	0.799617i	$-0.705036\pi$
$773$	−25811.9	−1.20102	−0.600510	+	0.799617i	$0.705036\pi$
$774$	0	0
$775$	456.242	0.0211467
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−20463.9	−0.941201
$780$	0	0
$781$	−12345.8	−0.565645
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−116.526	−0.00529807
$786$	0	0
$787$	29242.6	1.32450	0.662252	−	0.749281i	$-0.269601\pi$
$787$	29242.6	1.32450	0.662252	+	0.749281i	$0.269601\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	26407.2	1.18702
$792$	0	0
$793$	−38880.8	−1.74111
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−32573.7	−1.44771	−0.723853	−	0.689955i	$-0.757630\pi$
$797$	−32573.7	−1.44771	−0.723853	+	0.689955i	$0.757630\pi$
$798$	0	0
$799$	341.130	0.0151043
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−26931.6	−1.18356
$804$	0	0
$805$	18887.5	0.826951
$806$	0	0
$807$	0	0
$808$	0	0
$809$	34644.9	1.50562	0.752812	−	0.658236i	$-0.228697\pi$
$809$	34644.9	1.50562	0.752812	+	0.658236i	$0.228697\pi$
$810$	0	0
$811$	29057.9	1.25815	0.629077	−	0.777343i	$-0.283433\pi$
$811$	29057.9	1.25815	0.629077	+	0.777343i	$0.283433\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	20160.5	0.866493
$816$	0	0
$817$	34969.8	1.49748
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−46709.5	−1.98560	−0.992798	−	0.119802i	$-0.961774\pi$
$821$	−46709.5	−1.98560	−0.992798	+	0.119802i	$0.961774\pi$
$822$	0	0
$823$	3468.10	0.146890	0.0734450	−	0.997299i	$-0.476601\pi$
$823$	3468.10	0.146890	0.0734450	+	0.997299i	$0.476601\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	42454.9	1.78513	0.892564	−	0.450920i	$-0.148904\pi$
$827$	42454.9	1.78513	0.892564	+	0.450920i	$0.148904\pi$
$828$	0	0
$829$	−3933.47	−0.164795	−0.0823975	−	0.996600i	$-0.526258\pi$
$829$	−3933.47	−0.164795	−0.0823975	+	0.996600i	$0.526258\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−17208.3	−0.715764
$834$	0	0
$835$	−3359.55	−0.139236
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−32959.6	−1.35625	−0.678123	−	0.734948i	$-0.737206\pi$
$839$	−32959.6	−1.35625	−0.678123	+	0.734948i	$0.737206\pi$
$840$	0	0
$841$	17906.1	0.734188
$842$	0	0
$843$	0	0
$844$	0	0
$845$	10395.5	0.423215
$846$	0	0
$847$	11979.5	0.485974
$848$	0	0
$849$	0	0
$850$	0	0
$851$	52264.8	2.10530
$852$	0	0
$853$	38845.8	1.55927	0.779634	−	0.626235i	$-0.215405\pi$
$853$	38845.8	1.55927	0.779634	+	0.626235i	$0.215405\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	29305.3	1.16809	0.584043	−	0.811723i	$-0.301470\pi$
$857$	29305.3	1.16809	0.584043	+	0.811723i	$0.301470\pi$
$858$	0	0
$859$	909.659	0.0361318	0.0180659	−	0.999837i	$-0.494249\pi$
$859$	909.659	0.0361318	0.0180659	+	0.999837i	$0.494249\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	47998.0	1.89324	0.946622	−	0.322346i	$-0.104471\pi$
$863$	47998.0	1.89324	0.946622	+	0.322346i	$0.104471\pi$
$864$	0	0
$865$	−8167.64	−0.321050
$866$	0	0
$867$	0	0
$868$	0	0
$869$	708.065	0.0276403
$870$	0	0
$871$	12808.7	0.498286
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3050.17	−0.117845
$876$	0	0
$877$	3258.01	0.125445	0.0627225	−	0.998031i	$-0.480022\pi$
$877$	3258.01	0.125445	0.0627225	+	0.998031i	$0.480022\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	33380.2	1.27651	0.638256	−	0.769824i	$-0.279656\pi$
$881$	33380.2	1.27651	0.638256	+	0.769824i	$0.279656\pi$
$882$	0	0
$883$	−33714.6	−1.28492	−0.642460	−	0.766319i	$-0.722086\pi$
$883$	−33714.6	−1.28492	−0.642460	+	0.766319i	$0.722086\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6218.80	−0.235408	−0.117704	−	0.993049i	$-0.537553\pi$
$887$	−6218.80	−0.235408	−0.117704	+	0.993049i	$0.537553\pi$
$888$	0	0
$889$	−18889.8	−0.712649
$890$	0	0
$891$	0	0
$892$	0	0
$893$	522.537	0.0195812
$894$	0	0
$895$	1706.30	0.0637266
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3753.19	0.139239
$900$	0	0
$901$	−21812.6	−0.806529
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8479.61	−0.311460
$906$	0	0
$907$	22878.6	0.837566	0.418783	−	0.908086i	$-0.362457\pi$
$907$	22878.6	0.837566	0.418783	+	0.908086i	$0.362457\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−30144.8	−1.09631	−0.548157	−	0.836376i	$-0.684670\pi$
$911$	−30144.8	−1.09631	−0.548157	+	0.836376i	$0.684670\pi$
$912$	0	0
$913$	15821.1	0.573496
$914$	0	0
$915$	0	0
$916$	0	0
$917$	29624.1	1.06682
$918$	0	0
$919$	−3803.52	−0.136525	−0.0682625	−	0.997667i	$-0.521746\pi$
$919$	−3803.52	−0.136525	−0.0682625	+	0.997667i	$0.521746\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−27854.0	−0.993312
$924$	0	0
$925$	−8440.33	−0.300018
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−125.985	−0.00444934	−0.00222467	−	0.999998i	$-0.500708\pi$
$929$	−125.985	−0.00444934	−0.00222467	+	0.999998i	$0.500708\pi$
$930$	0	0
$931$	−26359.3	−0.927918
$932$	0	0
$933$	0	0
$934$	0	0
$935$	9879.41	0.345552
$936$	0	0
$937$	28107.9	0.979984	0.489992	−	0.871727i	$-0.337000\pi$
$937$	28107.9	0.979984	0.489992	+	0.871727i	$0.337000\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	49194.9	1.70426	0.852130	−	0.523330i	$-0.175311\pi$
$941$	49194.9	1.70426	0.852130	+	0.523330i	$0.175311\pi$
$942$	0	0
$943$	−30337.3	−1.04763
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−14498.0	−0.497490	−0.248745	−	0.968569i	$-0.580018\pi$
$947$	−14498.0	−0.497490	−0.248745	+	0.968569i	$0.580018\pi$
$948$	0	0
$949$	−60761.7	−2.07841
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3201.79	−0.108831	−0.0544155	−	0.998518i	$-0.517330\pi$
$953$	−3201.79	−0.108831	−0.0544155	+	0.998518i	$0.517330\pi$
$954$	0	0
$955$	−3632.26	−0.123075
$956$	0	0
$957$	0	0
$958$	0	0
$959$	56126.7	1.88991
$960$	0	0
$961$	−29457.9	−0.988820
$962$	0	0
$963$	0	0
$964$	0	0
$965$	21236.3	0.708415
$966$	0	0
$967$	−57781.9	−1.92155	−0.960776	−	0.277326i	$-0.910552\pi$
$967$	−57781.9	−1.92155	−0.960776	+	0.277326i	$0.910552\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2611.60	0.0863133	0.0431566	−	0.999068i	$-0.486259\pi$
$971$	2611.60	0.0863133	0.0431566	+	0.999068i	$0.486259\pi$
$972$	0	0
$973$	−33086.4	−1.09013
$974$	0	0
$975$	0	0
$976$	0	0
$977$	33598.3	1.10021	0.550104	−	0.835096i	$-0.314588\pi$
$977$	33598.3	1.10021	0.550104	+	0.835096i	$0.314588\pi$
$978$	0	0
$979$	−2438.51	−0.0796067
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−39484.8	−1.28115	−0.640575	−	0.767895i	$-0.721304\pi$
$983$	−39484.8	−1.28115	−0.640575	+	0.767895i	$0.721304\pi$
$984$	0	0
$985$	−13392.6	−0.433222
$986$	0	0
$987$	0	0
$988$	0	0
$989$	51841.9	1.66681
$990$	0	0
$991$	−39918.6	−1.27957	−0.639786	−	0.768553i	$-0.720977\pi$
$991$	−39918.6	−1.27957	−0.639786	+	0.768553i	$0.720977\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−7432.41	−0.236807
$996$	0	0
$997$	−25670.3	−0.815432	−0.407716	−	0.913109i	$-0.633675\pi$
$997$	−25670.3	−0.815432	−0.407716	+	0.913109i	$0.633675\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bm.1.1		3
3.2	odd	2		2160.4.a.be.1.1		3
4.3	odd	2		135.4.a.f.1.1	✓	3
12.11	even	2		135.4.a.g.1.3	yes	3
20.3	even	4		675.4.b.l.649.6		6
20.7	even	4		675.4.b.l.649.1		6
20.19	odd	2		675.4.a.r.1.3		3
36.7	odd	6		405.4.e.t.271.3		6
36.11	even	6		405.4.e.r.271.1		6
36.23	even	6		405.4.e.r.136.1		6
36.31	odd	6		405.4.e.t.136.3		6
60.23	odd	4		675.4.b.k.649.1		6
60.47	odd	4		675.4.b.k.649.6		6
60.59	even	2		675.4.a.q.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.f.1.1	✓	3	4.3	odd	2
135.4.a.g.1.3	yes	3	12.11	even	2
405.4.e.r.136.1		6	36.23	even	6
405.4.e.r.271.1		6	36.11	even	6
405.4.e.t.136.3		6	36.31	odd	6
405.4.e.t.271.3		6	36.7	odd	6
675.4.a.q.1.1		3	60.59	even	2
675.4.a.r.1.3		3	20.19	odd	2
675.4.b.k.649.1		6	60.23	odd	4
675.4.b.k.649.6		6	60.47	odd	4
675.4.b.l.649.1		6	20.7	even	4
675.4.b.l.649.6		6	20.3	even	4
2160.4.a.be.1.1		3	3.2	odd	2
2160.4.a.bm.1.1		3	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} -24.4013 q^{7} +O(q^{10})\)	\(q+5.00000 q^{5} -24.4013 q^{7} -28.9839 q^{11} -65.3919 q^{13} -68.1718 q^{17} -104.424 q^{19} -154.807 q^{23} +25.0000 q^{25} +205.658 q^{29} +18.2497 q^{31} -122.007 q^{35} -337.613 q^{37} +195.969 q^{41} -334.882 q^{43} -5.00398 q^{47} +252.425 q^{49} +319.965 q^{53} -144.919 q^{55} +430.611 q^{59} +594.581 q^{61} -326.960 q^{65} -195.876 q^{67} +425.955 q^{71} +929.193 q^{73} +707.245 q^{77} -24.4296 q^{79} -545.859 q^{83} -340.859 q^{85} +84.1332 q^{89} +1595.65 q^{91} -522.121 q^{95} +827.613 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 15 q^{5} - 44 q^{7}+O(q^{10}) \) 3 * q + 15 * q^5 - 44 * q^7	\( 3 q + 15 q^{5} - 44 q^{7} - 38 q^{11} + 28 q^{13} - 19 q^{17} - 187 q^{19} + 81 q^{23} + 75 q^{25} + 160 q^{29} - 227 q^{31} - 220 q^{35} + 78 q^{37} - 338 q^{41} - 22 q^{43} + 472 q^{47} - 197 q^{49} + 521 q^{53} - 190 q^{55} - 140 q^{59} + 595 q^{61} + 140 q^{65} - 878 q^{67} + 602 q^{71} + 1294 q^{73} + 288 q^{77} - 629 q^{79} + 1287 q^{83} - 95 q^{85} + 2154 q^{89} + 440 q^{91} - 935 q^{95} + 1392 q^{97}+O(q^{100}) \) 3 * q + 15 * q^5 - 44 * q^7 - 38 * q^11 + 28 * q^13 - 19 * q^17 - 187 * q^19 + 81 * q^23 + 75 * q^25 + 160 * q^29 - 227 * q^31 - 220 * q^35 + 78 * q^37 - 338 * q^41 - 22 * q^43 + 472 * q^47 - 197 * q^49 + 521 * q^53 - 190 * q^55 - 140 * q^59 + 595 * q^61 + 140 * q^65 - 878 * q^67 + 602 * q^71 + 1294 * q^73 + 288 * q^77 - 629 * q^79 + 1287 * q^83 - 95 * q^85 + 2154 * q^89 + 440 * q^91 - 935 * q^95 + 1392 * q^97

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