LMFDB - Embedded newform 2160.4.a.be.1.3

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			1.3
Root			$-4.45938$ 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)

$f(q)$	$=$	$q-5.00000 q^{5} -5.08123 q^{7} +O(q^{10})$	$q-5.00000 q^{5} -5.08123 q^{7} +58.3007 q^{11} +21.2119 q^{13} +68.8451 q^{17} +40.8133 q^{19} -144.318 q^{23} +25.0000 q^{25} +220.058 q^{29} -291.545 q^{31} +25.4062 q^{35} +260.637 q^{37} +169.766 q^{41} +438.596 q^{43} -255.481 q^{47} -317.181 q^{49} -214.714 q^{53} -291.503 q^{55} +331.524 q^{59} +54.9647 q^{61} -106.060 q^{65} -758.179 q^{67} -904.348 q^{71} +866.622 q^{73} -296.239 q^{77} -206.961 q^{79} -463.397 q^{83} -344.225 q^{85} -601.736 q^{89} -107.783 q^{91} -204.066 q^{95} +229.363 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−5.08123	−0.274361	−0.137180	−	0.990546i	$-0.543804\pi$
$7$	−5.08123	−0.274361	−0.137180	+	0.990546i	$0.543804\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	58.3007	1.59803	0.799014	−	0.601312i	$-0.205355\pi$
$11$	58.3007	1.59803	0.799014	+	0.601312i	$0.205355\pi$
$12$	0	0
$13$	21.2119	0.452548	0.226274	−	0.974064i	$-0.427345\pi$
$13$	21.2119	0.452548	0.226274	+	0.974064i	$0.427345\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	68.8451	0.982199	0.491099	−	0.871104i	$-0.336595\pi$
$17$	68.8451	0.982199	0.491099	+	0.871104i	$0.336595\pi$
$18$	0	0
$19$	40.8133	0.492800	0.246400	−	0.969168i	$-0.420752\pi$
$19$	40.8133	0.492800	0.246400	+	0.969168i	$0.420752\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−144.318	−1.30837	−0.654184	−	0.756336i	$-0.726988\pi$
$23$	−144.318	−1.30837	−0.654184	+	0.756336i	$0.726988\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	220.058	1.40909	0.704547	−	0.709657i	$-0.251150\pi$
$29$	220.058	1.40909	0.704547	+	0.709657i	$0.251150\pi$
$30$	0	0
$31$	−291.545	−1.68913	−0.844566	−	0.535452i	$-0.820141\pi$
$31$	−291.545	−1.68913	−0.844566	+	0.535452i	$0.820141\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	25.4062	0.122698
$36$	0	0
$37$	260.637	1.15807	0.579033	−	0.815304i	$-0.303430\pi$
$37$	260.637	1.15807	0.579033	+	0.815304i	$0.303430\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	169.766	0.646660	0.323330	−	0.946286i	$-0.395198\pi$
$41$	169.766	0.646660	0.323330	+	0.946286i	$0.395198\pi$
$42$	0	0
$43$	438.596	1.55547	0.777735	−	0.628592i	$-0.216369\pi$
$43$	438.596	1.55547	0.777735	+	0.628592i	$0.216369\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−255.481	−0.792887	−0.396444	−	0.918059i	$-0.629756\pi$
$47$	−255.481	−0.792887	−0.396444	+	0.918059i	$0.629756\pi$
$48$	0	0
$49$	−317.181	−0.924726
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−214.714	−0.556477	−0.278239	−	0.960512i	$-0.589751\pi$
$53$	−214.714	−0.556477	−0.278239	+	0.960512i	$0.589751\pi$
$54$	0	0
$55$	−291.503	−0.714660
$56$	0	0
$57$	0	0
$58$	0	0
$59$	331.524	0.731537	0.365769	−	0.930706i	$-0.380806\pi$
$59$	331.524	0.731537	0.365769	+	0.930706i	$0.380806\pi$
$60$	0	0
$61$	54.9647	0.115369	0.0576845	−	0.998335i	$-0.481628\pi$
$61$	54.9647	0.115369	0.0576845	+	0.998335i	$0.481628\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−106.060	−0.202386
$66$	0	0
$67$	−758.179	−1.38248	−0.691241	−	0.722624i	$-0.742936\pi$
$67$	−758.179	−1.38248	−0.691241	+	0.722624i	$0.742936\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−904.348	−1.51164	−0.755819	−	0.654780i	$-0.772761\pi$
$71$	−904.348	−1.51164	−0.755819	+	0.654780i	$0.772761\pi$
$72$	0	0
$73$	866.622	1.38946	0.694729	−	0.719271i	$-0.255524\pi$
$73$	866.622	1.38946	0.694729	+	0.719271i	$0.255524\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−296.239	−0.438437
$78$	0	0
$79$	−206.961	−0.294746	−0.147373	−	0.989081i	$-0.547082\pi$
$79$	−206.961	−0.294746	−0.147373	+	0.989081i	$0.547082\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−463.397	−0.612825	−0.306412	−	0.951899i	$-0.599129\pi$
$83$	−463.397	−0.612825	−0.306412	+	0.951899i	$0.599129\pi$
$84$	0	0
$85$	−344.225	−0.439253
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−601.736	−0.716673	−0.358337	−	0.933592i	$-0.616656\pi$
$89$	−601.736	−0.716673	−0.358337	+	0.933592i	$0.616656\pi$
$90$	0	0
$91$	−107.783	−0.124162
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−204.066	−0.220387
$96$	0	0
$97$	229.363	0.240086	0.120043	−	0.992769i	$-0.461697\pi$
$97$	229.363	0.240086	0.120043	+	0.992769i	$0.461697\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1345.66	−1.32573	−0.662863	−	0.748740i	$-0.730659\pi$
$101$	−1345.66	−1.32573	−0.662863	+	0.748740i	$0.730659\pi$
$102$	0	0
$103$	1596.30	1.52707	0.763534	−	0.645768i	$-0.223463\pi$
$103$	1596.30	1.52707	0.763534	+	0.645768i	$0.223463\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	958.786	0.866256	0.433128	−	0.901333i	$-0.357410\pi$
$107$	958.786	0.866256	0.433128	+	0.901333i	$0.357410\pi$
$108$	0	0
$109$	1690.23	1.48527	0.742635	−	0.669696i	$-0.233576\pi$
$109$	1690.23	1.48527	0.742635	+	0.669696i	$0.233576\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.6211	0.00967456	0.00483728	−	0.999988i	$-0.498460\pi$
$113$	11.6211	0.00967456	0.00483728	+	0.999988i	$0.498460\pi$
$114$	0	0
$115$	721.592	0.585120
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−349.818	−0.269477
$120$	0	0
$121$	2067.97	1.55370
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−309.141	−0.215999	−0.107999	−	0.994151i	$-0.534444\pi$
$127$	−309.141	−0.215999	−0.107999	+	0.994151i	$0.534444\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2785.03	1.85747	0.928736	−	0.370742i	$-0.120897\pi$
$131$	2785.03	1.85747	0.928736	+	0.370742i	$0.120897\pi$
$132$	0	0
$133$	−207.382	−0.135205
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2489.41	1.55244	0.776222	−	0.630460i	$-0.217134\pi$
$137$	2489.41	1.55244	0.776222	+	0.630460i	$0.217134\pi$
$138$	0	0
$139$	−1786.05	−1.08986	−0.544931	−	0.838481i	$-0.683444\pi$
$139$	−1786.05	−1.08986	−0.544931	+	0.838481i	$0.683444\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1236.67	0.723185
$144$	0	0
$145$	−1100.29	−0.630166
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1568.83	−0.862575	−0.431288	−	0.902214i	$-0.641941\pi$
$149$	−1568.83	−0.862575	−0.431288	+	0.902214i	$0.641941\pi$
$150$	0	0
$151$	438.327	0.236229	0.118114	−	0.993000i	$-0.462315\pi$
$151$	438.327	0.236229	0.118114	+	0.993000i	$0.462315\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1457.73	0.755403
$156$	0	0
$157$	−44.7479	−0.0227469	−0.0113735	−	0.999935i	$-0.503620\pi$
$157$	−44.7479	−0.0227469	−0.0113735	+	0.999935i	$0.503620\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	733.315	0.358965
$162$	0	0
$163$	2611.84	1.25506	0.627531	−	0.778591i	$-0.284065\pi$
$163$	2611.84	1.25506	0.627531	+	0.778591i	$0.284065\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−188.947	−0.0875516	−0.0437758	−	0.999041i	$-0.513939\pi$
$167$	−188.947	−0.0875516	−0.0437758	+	0.999041i	$0.513939\pi$
$168$	0	0
$169$	−1747.05	−0.795200
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1505.02	0.661413	0.330707	−	0.943734i	$-0.392713\pi$
$173$	1505.02	0.661413	0.330707	+	0.943734i	$0.392713\pi$
$174$	0	0
$175$	−127.031	−0.0548722
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3136.62	−1.30973	−0.654865	−	0.755746i	$-0.727275\pi$
$179$	−3136.62	−1.30973	−0.654865	+	0.755746i	$0.727275\pi$
$180$	0	0
$181$	4512.67	1.85317	0.926586	−	0.376084i	$-0.122730\pi$
$181$	4512.67	1.85317	0.926586	+	0.376084i	$0.122730\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1303.18	−0.517902
$186$	0	0
$187$	4013.71	1.56958
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1207.43	0.457418	0.228709	−	0.973495i	$-0.426550\pi$
$191$	1207.43	0.457418	0.228709	+	0.973495i	$0.426550\pi$
$192$	0	0
$193$	923.164	0.344305	0.172152	−	0.985070i	$-0.444928\pi$
$193$	923.164	0.344305	0.172152	+	0.985070i	$0.444928\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1180.87	0.427075	0.213537	−	0.976935i	$-0.431501\pi$
$197$	1180.87	0.427075	0.213537	+	0.976935i	$0.431501\pi$
$198$	0	0
$199$	839.805	0.299157	0.149578	−	0.988750i	$-0.452208\pi$
$199$	839.805	0.299157	0.149578	+	0.988750i	$0.452208\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1118.17	−0.386600
$204$	0	0
$205$	−848.832	−0.289195
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2379.44	0.787509
$210$	0	0
$211$	2589.65	0.844923	0.422461	−	0.906381i	$-0.361166\pi$
$211$	2589.65	0.844923	0.422461	+	0.906381i	$0.361166\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2192.98	−0.695627
$216$	0	0
$217$	1481.41	0.463432
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1460.34	0.444492
$222$	0	0
$223$	4180.76	1.25544	0.627722	−	0.778437i	$-0.283987\pi$
$223$	4180.76	1.25544	0.627722	+	0.778437i	$0.283987\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2602.67	−0.760993	−0.380497	−	0.924782i	$-0.624247\pi$
$227$	−2602.67	−0.760993	−0.380497	+	0.924782i	$0.624247\pi$
$228$	0	0
$229$	−1845.35	−0.532508	−0.266254	−	0.963903i	$-0.585786\pi$
$229$	−1845.35	−0.532508	−0.266254	+	0.963903i	$0.585786\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−240.637	−0.0676594	−0.0338297	−	0.999428i	$-0.510770\pi$
$233$	−240.637	−0.0676594	−0.0338297	+	0.999428i	$0.510770\pi$
$234$	0	0
$235$	1277.40	0.354590
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2567.64	0.694924	0.347462	−	0.937694i	$-0.387044\pi$
$239$	2567.64	0.694924	0.347462	+	0.937694i	$0.387044\pi$
$240$	0	0
$241$	3987.99	1.06593	0.532965	−	0.846137i	$-0.321078\pi$
$241$	3987.99	1.06593	0.532965	+	0.846137i	$0.321078\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1585.91	0.413550
$246$	0	0
$247$	865.728	0.223016
$248$	0	0
$249$	0	0
$250$	0	0
$251$	967.393	0.243272	0.121636	−	0.992575i	$-0.461186\pi$
$251$	967.393	0.243272	0.121636	+	0.992575i	$0.461186\pi$
$252$	0	0
$253$	−8413.86	−2.09081
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3743.03	−0.908498	−0.454249	−	0.890875i	$-0.650092\pi$
$257$	−3743.03	−0.908498	−0.454249	+	0.890875i	$0.650092\pi$
$258$	0	0
$259$	−1324.36	−0.317728
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2497.47	−0.585554	−0.292777	−	0.956181i	$-0.594579\pi$
$263$	−2497.47	−0.585554	−0.292777	+	0.956181i	$0.594579\pi$
$264$	0	0
$265$	1073.57	0.248864
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2142.91	0.485709	0.242855	−	0.970063i	$-0.421916\pi$
$269$	2142.91	0.485709	0.242855	+	0.970063i	$0.421916\pi$
$270$	0	0
$271$	−1540.64	−0.345341	−0.172671	−	0.984980i	$-0.555240\pi$
$271$	−1540.64	−0.345341	−0.172671	+	0.984980i	$0.555240\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1457.52	0.319606
$276$	0	0
$277$	6777.80	1.47018	0.735088	−	0.677972i	$-0.237141\pi$
$277$	6777.80	1.47018	0.735088	+	0.677972i	$0.237141\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−827.653	−0.175707	−0.0878535	−	0.996133i	$-0.528001\pi$
$281$	−827.653	−0.175707	−0.0878535	+	0.996133i	$0.528001\pi$
$282$	0	0
$283$	3171.98	0.666270	0.333135	−	0.942879i	$-0.391894\pi$
$283$	3171.98	0.666270	0.333135	+	0.942879i	$0.391894\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−862.623	−0.177418
$288$	0	0
$289$	−173.358	−0.0352856
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1376.02	−0.274362	−0.137181	−	0.990546i	$-0.543804\pi$
$293$	−1376.02	−0.274362	−0.137181	+	0.990546i	$0.543804\pi$
$294$	0	0
$295$	−1657.62	−0.327153
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3061.27	−0.592099
$300$	0	0
$301$	−2228.61	−0.426760
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−274.823	−0.0515946
$306$	0	0
$307$	119.504	0.0222165	0.0111083	−	0.999938i	$-0.496464\pi$
$307$	119.504	0.0222165	0.0111083	+	0.999938i	$0.496464\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2139.35	0.390069	0.195035	−	0.980796i	$-0.437518\pi$
$311$	2139.35	0.390069	0.195035	+	0.980796i	$0.437518\pi$
$312$	0	0
$313$	5163.50	0.932455	0.466227	−	0.884665i	$-0.345613\pi$
$313$	5163.50	0.932455	0.466227	+	0.884665i	$0.345613\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8631.69	1.52935	0.764675	−	0.644416i	$-0.222899\pi$
$317$	8631.69	1.52935	0.764675	+	0.644416i	$0.222899\pi$
$318$	0	0
$319$	12829.5	2.25177
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2809.79	0.484028
$324$	0	0
$325$	530.298	0.0905097
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1298.16	0.217537
$330$	0	0
$331$	2942.34	0.488597	0.244298	−	0.969700i	$-0.421442\pi$
$331$	2942.34	0.488597	0.244298	+	0.969700i	$0.421442\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3790.90	0.618265
$336$	0	0
$337$	9897.46	1.59985	0.799924	−	0.600101i	$-0.204873\pi$
$337$	9897.46	1.59985	0.799924	+	0.600101i	$0.204873\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−16997.3	−2.69928
$342$	0	0
$343$	3354.53	0.528070
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12015.7	−1.85890	−0.929451	−	0.368947i	$-0.879718\pi$
$347$	−12015.7	−1.85890	−0.929451	+	0.368947i	$0.879718\pi$
$348$	0	0
$349$	7894.62	1.21086	0.605428	−	0.795900i	$-0.293002\pi$
$349$	7894.62	1.21086	0.605428	+	0.795900i	$0.293002\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−741.154	−0.111750	−0.0558748	−	0.998438i	$-0.517795\pi$
$353$	−741.154	−0.111750	−0.0558748	+	0.998438i	$0.517795\pi$
$354$	0	0
$355$	4521.74	0.676025
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11564.4	−1.70012	−0.850060	−	0.526685i	$-0.823435\pi$
$359$	−11564.4	−1.70012	−0.850060	+	0.526685i	$0.823435\pi$
$360$	0	0
$361$	−5193.28	−0.757148
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4333.11	−0.621385
$366$	0	0
$367$	1148.57	0.163365	0.0816823	−	0.996658i	$-0.473971\pi$
$367$	1148.57	0.163365	0.0816823	+	0.996658i	$0.473971\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1091.01	0.152676
$372$	0	0
$373$	−4602.98	−0.638963	−0.319481	−	0.947593i	$-0.603509\pi$
$373$	−4602.98	−0.638963	−0.319481	+	0.947593i	$0.603509\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4667.85	0.637683
$378$	0	0
$379$	3988.46	0.540563	0.270282	−	0.962781i	$-0.412883\pi$
$379$	3988.46	0.540563	0.270282	+	0.962781i	$0.412883\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7788.20	1.03906	0.519528	−	0.854454i	$-0.326108\pi$
$383$	7788.20	1.03906	0.519528	+	0.854454i	$0.326108\pi$
$384$	0	0
$385$	1481.20	0.196075
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−8524.87	−1.11113	−0.555563	−	0.831474i	$-0.687497\pi$
$389$	−8524.87	−1.11113	−0.555563	+	0.831474i	$0.687497\pi$
$390$	0	0
$391$	−9935.60	−1.28508
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1034.80	0.131814
$396$	0	0
$397$	−155.729	−0.0196872	−0.00984361	−	0.999952i	$-0.503133\pi$
$397$	−155.729	−0.0196872	−0.00984361	+	0.999952i	$0.503133\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5933.96	−0.738972	−0.369486	−	0.929236i	$-0.620466\pi$
$401$	−5933.96	−0.738972	−0.369486	+	0.929236i	$0.620466\pi$
$402$	0	0
$403$	−6184.23	−0.764414
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15195.3	1.85062
$408$	0	0
$409$	14161.4	1.71207	0.856035	−	0.516917i	$-0.172921\pi$
$409$	14161.4	1.71207	0.856035	+	0.516917i	$0.172921\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1684.55	−0.200705
$414$	0	0
$415$	2316.99	0.274064
$416$	0	0
$417$	0	0
$418$	0	0
$419$	9624.24	1.12214	0.561068	−	0.827770i	$-0.310391\pi$
$419$	9624.24	1.12214	0.561068	+	0.827770i	$0.310391\pi$
$420$	0	0
$421$	−1536.26	−0.177845	−0.0889223	−	0.996039i	$-0.528342\pi$
$421$	−1536.26	−0.177845	−0.0889223	+	0.996039i	$0.528342\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1721.13	0.196440
$426$	0	0
$427$	−279.288	−0.0316527
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11582.2	1.29441	0.647207	−	0.762314i	$-0.275937\pi$
$431$	11582.2	1.29441	0.647207	+	0.762314i	$0.275937\pi$
$432$	0	0
$433$	14892.6	1.65287	0.826437	−	0.563029i	$-0.190364\pi$
$433$	14892.6	1.65287	0.826437	+	0.563029i	$0.190364\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5890.10	−0.644764
$438$	0	0
$439$	1642.51	0.178571	0.0892853	−	0.996006i	$-0.471542\pi$
$439$	1642.51	0.178571	0.0892853	+	0.996006i	$0.471542\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3916.31	0.420021	0.210011	−	0.977699i	$-0.432650\pi$
$443$	3916.31	0.420021	0.210011	+	0.977699i	$0.432650\pi$
$444$	0	0
$445$	3008.68	0.320506
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3985.25	0.418876	0.209438	−	0.977822i	$-0.432837\pi$
$449$	3985.25	0.418876	0.209438	+	0.977822i	$0.432837\pi$
$450$	0	0
$451$	9897.50	1.03338
$452$	0	0
$453$	0	0
$454$	0	0
$455$	538.914	0.0555267
$456$	0	0
$457$	−14177.8	−1.45122	−0.725611	−	0.688105i	$-0.758443\pi$
$457$	−14177.8	−1.45122	−0.725611	+	0.688105i	$0.758443\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	16394.3	1.65631	0.828154	−	0.560500i	$-0.189391\pi$
$461$	16394.3	1.65631	0.828154	+	0.560500i	$0.189391\pi$
$462$	0	0
$463$	−3319.60	−0.333207	−0.166603	−	0.986024i	$-0.553280\pi$
$463$	−3319.60	−0.333207	−0.166603	+	0.986024i	$0.553280\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2529.79	−0.250674	−0.125337	−	0.992114i	$-0.540001\pi$
$467$	−2529.79	−0.250674	−0.125337	+	0.992114i	$0.540001\pi$
$468$	0	0
$469$	3852.49	0.379299
$470$	0	0
$471$	0	0
$472$	0	0
$473$	25570.4	2.48569
$474$	0	0
$475$	1020.33	0.0985601
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8646.75	−0.824802	−0.412401	−	0.911002i	$-0.635310\pi$
$479$	−8646.75	−0.824802	−0.412401	+	0.911002i	$0.635310\pi$
$480$	0	0
$481$	5528.60	0.524080
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1146.82	−0.107370
$486$	0	0
$487$	15251.7	1.41914	0.709569	−	0.704636i	$-0.248890\pi$
$487$	15251.7	1.41914	0.709569	+	0.704636i	$0.248890\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1766.87	−0.162398	−0.0811992	−	0.996698i	$-0.525875\pi$
$491$	−1766.87	−0.162398	−0.0811992	+	0.996698i	$0.525875\pi$
$492$	0	0
$493$	15149.9	1.38401
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4595.20	0.414734
$498$	0	0
$499$	−11733.8	−1.05266	−0.526332	−	0.850279i	$-0.676433\pi$
$499$	−11733.8	−1.05266	−0.526332	+	0.850279i	$0.676433\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−977.608	−0.0866589	−0.0433294	−	0.999061i	$-0.513797\pi$
$503$	−977.608	−0.0866589	−0.0433294	+	0.999061i	$0.513797\pi$
$504$	0	0
$505$	6728.31	0.592883
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9674.72	−0.842484	−0.421242	−	0.906948i	$-0.638406\pi$
$509$	−9674.72	−0.842484	−0.421242	+	0.906948i	$0.638406\pi$
$510$	0	0
$511$	−4403.51	−0.381213
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−7981.49	−0.682926
$516$	0	0
$517$	−14894.7	−1.26706
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−5178.95	−0.435497	−0.217748	−	0.976005i	$-0.569871\pi$
$521$	−5178.95	−0.435497	−0.217748	+	0.976005i	$0.569871\pi$
$522$	0	0
$523$	−14280.7	−1.19398	−0.596992	−	0.802248i	$-0.703637\pi$
$523$	−14280.7	−1.19398	−0.596992	+	0.802248i	$0.703637\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−20071.5	−1.65906
$528$	0	0
$529$	8660.78	0.711826
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3601.07	0.292645
$534$	0	0
$535$	−4793.93	−0.387401
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−18491.9	−1.47774
$540$	0	0
$541$	12923.8	1.02706	0.513529	−	0.858072i	$-0.328338\pi$
$541$	12923.8	1.02706	0.513529	+	0.858072i	$0.328338\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−8451.14	−0.664233
$546$	0	0
$547$	−13653.3	−1.06723	−0.533614	−	0.845728i	$-0.679166\pi$
$547$	−13653.3	−1.06723	−0.533614	+	0.845728i	$0.679166\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8981.28	0.694402
$552$	0	0
$553$	1051.62	0.0808666
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9313.63	0.708494	0.354247	−	0.935152i	$-0.384737\pi$
$557$	9313.63	0.708494	0.354247	+	0.935152i	$0.384737\pi$
$558$	0	0
$559$	9303.46	0.703925
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10625.4	−0.795394	−0.397697	−	0.917517i	$-0.630190\pi$
$563$	−10625.4	−0.795394	−0.397697	+	0.917517i	$0.630190\pi$
$564$	0	0
$565$	−58.1057	−0.00432660
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−9060.50	−0.667550	−0.333775	−	0.942653i	$-0.608323\pi$
$569$	−9060.50	−0.667550	−0.333775	+	0.942653i	$0.608323\pi$
$570$	0	0
$571$	21379.1	1.56688	0.783440	−	0.621467i	$-0.213463\pi$
$571$	21379.1	1.56688	0.783440	+	0.621467i	$0.213463\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3607.96	−0.261674
$576$	0	0
$577$	6347.76	0.457991	0.228996	−	0.973427i	$-0.426456\pi$
$577$	6347.76	0.457991	0.228996	+	0.973427i	$0.426456\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2354.63	0.168135
$582$	0	0
$583$	−12518.0	−0.889266
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14773.7	1.03880	0.519401	−	0.854531i	$-0.326155\pi$
$587$	14773.7	1.03880	0.519401	+	0.854531i	$0.326155\pi$
$588$	0	0
$589$	−11898.9	−0.832405
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−26868.1	−1.86061	−0.930304	−	0.366790i	$-0.880457\pi$
$593$	−26868.1	−1.86061	−0.930304	+	0.366790i	$0.880457\pi$
$594$	0	0
$595$	1749.09	0.120514
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1749.83	−0.119359	−0.0596794	−	0.998218i	$-0.519008\pi$
$599$	−1749.83	−0.119359	−0.0596794	+	0.998218i	$0.519008\pi$
$600$	0	0
$601$	−17964.0	−1.21924	−0.609622	−	0.792692i	$-0.708679\pi$
$601$	−17964.0	−1.21924	−0.609622	+	0.792692i	$0.708679\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10339.8	−0.694834
$606$	0	0
$607$	4418.22	0.295437	0.147718	−	0.989029i	$-0.452807\pi$
$607$	4418.22	0.295437	0.147718	+	0.989029i	$0.452807\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5419.24	−0.358820
$612$	0	0
$613$	−20179.7	−1.32961	−0.664804	−	0.747018i	$-0.731485\pi$
$613$	−20179.7	−1.32961	−0.664804	+	0.747018i	$0.731485\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4673.01	0.304908	0.152454	−	0.988311i	$-0.451282\pi$
$617$	4673.01	0.304908	0.152454	+	0.988311i	$0.451282\pi$
$618$	0	0
$619$	19976.8	1.29715	0.648574	−	0.761151i	$-0.275366\pi$
$619$	19976.8	1.29715	0.648574	+	0.761151i	$0.275366\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3057.56	0.196627
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	17943.5	1.13745
$630$	0	0
$631$	−10457.5	−0.659757	−0.329879	−	0.944023i	$-0.607008\pi$
$631$	−10457.5	−0.659757	−0.329879	+	0.944023i	$0.607008\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1545.70	0.0965975
$636$	0	0
$637$	−6728.02	−0.418483
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−4395.86	−0.270867	−0.135434	−	0.990786i	$-0.543243\pi$
$641$	−4395.86	−0.270867	−0.135434	+	0.990786i	$0.543243\pi$
$642$	0	0
$643$	−5786.36	−0.354886	−0.177443	−	0.984131i	$-0.556783\pi$
$643$	−5786.36	−0.354886	−0.177443	+	0.984131i	$0.556783\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25367.7	−1.54143	−0.770717	−	0.637178i	$-0.780102\pi$
$647$	−25367.7	−1.54143	−0.770717	+	0.637178i	$0.780102\pi$
$648$	0	0
$649$	19328.1	1.16902
$650$	0	0
$651$	0	0
$652$	0	0
$653$	21633.2	1.29644	0.648218	−	0.761454i	$-0.275514\pi$
$653$	21633.2	1.29644	0.648218	+	0.761454i	$0.275514\pi$
$654$	0	0
$655$	−13925.1	−0.830687
$656$	0	0
$657$	0	0
$658$	0	0
$659$	18312.4	1.08248	0.541238	−	0.840870i	$-0.317956\pi$
$659$	18312.4	1.08248	0.541238	+	0.840870i	$0.317956\pi$
$660$	0	0
$661$	−5526.08	−0.325174	−0.162587	−	0.986694i	$-0.551984\pi$
$661$	−5526.08	−0.325174	−0.162587	+	0.986694i	$0.551984\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1036.91	0.0604656
$666$	0	0
$667$	−31758.4	−1.84361
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3204.48	0.184363
$672$	0	0
$673$	1437.24	0.0823204	0.0411602	−	0.999153i	$-0.486895\pi$
$673$	1437.24	0.0823204	0.0411602	+	0.999153i	$0.486895\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	23405.9	1.32875	0.664373	−	0.747401i	$-0.268699\pi$
$677$	23405.9	1.32875	0.664373	+	0.747401i	$0.268699\pi$
$678$	0	0
$679$	−1165.45	−0.0658701
$680$	0	0
$681$	0	0
$682$	0	0
$683$	19227.4	1.07719	0.538593	−	0.842566i	$-0.318956\pi$
$683$	19227.4	1.07719	0.538593	+	0.842566i	$0.318956\pi$
$684$	0	0
$685$	−12447.1	−0.694274
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4554.50	−0.251833
$690$	0	0
$691$	35284.8	1.94254	0.971271	−	0.237975i	$-0.0764835\pi$
$691$	35284.8	1.94254	0.971271	+	0.237975i	$0.0764835\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8930.26	0.487401
$696$	0	0
$697$	11687.6	0.635149
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−9173.00	−0.494236	−0.247118	−	0.968985i	$-0.579484\pi$
$701$	−9173.00	−0.494236	−0.247118	+	0.968985i	$0.579484\pi$
$702$	0	0
$703$	10637.4	0.570695
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6837.63	0.363728
$708$	0	0
$709$	−33951.6	−1.79842	−0.899210	−	0.437517i	$-0.855858\pi$
$709$	−33951.6	−1.79842	−0.899210	+	0.437517i	$0.855858\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	42075.3	2.21001
$714$	0	0
$715$	−6183.35	−0.323418
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6727.90	−0.348968	−0.174484	−	0.984660i	$-0.555826\pi$
$719$	−6727.90	−0.348968	−0.174484	+	0.984660i	$0.555826\pi$
$720$	0	0
$721$	−8111.17	−0.418968
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5501.45	0.281819
$726$	0	0
$727$	36726.1	1.87359	0.936793	−	0.349885i	$-0.113779\pi$
$727$	36726.1	1.87359	0.936793	+	0.349885i	$0.113779\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	30195.1	1.52778
$732$	0	0
$733$	26691.4	1.34498	0.672489	−	0.740108i	$-0.265225\pi$
$733$	26691.4	1.34498	0.672489	+	0.740108i	$0.265225\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−44202.4	−2.20925
$738$	0	0
$739$	−12207.0	−0.607634	−0.303817	−	0.952730i	$-0.598261\pi$
$739$	−12207.0	−0.607634	−0.303817	+	0.952730i	$0.598261\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−12473.3	−0.615882	−0.307941	−	0.951405i	$-0.599640\pi$
$743$	−12473.3	−0.615882	−0.307941	+	0.951405i	$0.599640\pi$
$744$	0	0
$745$	7844.16	0.385755
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−4871.82	−0.237667
$750$	0	0
$751$	15102.6	0.733825	0.366913	−	0.930255i	$-0.380415\pi$
$751$	15102.6	0.733825	0.366913	+	0.930255i	$0.380415\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−2191.63	−0.105645
$756$	0	0
$757$	−3418.34	−0.164124	−0.0820618	−	0.996627i	$-0.526150\pi$
$757$	−3418.34	−0.164124	−0.0820618	+	0.996627i	$0.526150\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	12684.5	0.604224	0.302112	−	0.953272i	$-0.402308\pi$
$761$	12684.5	0.604224	0.302112	+	0.953272i	$0.402308\pi$
$762$	0	0
$763$	−8588.45	−0.407500
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7032.25	0.331056
$768$	0	0
$769$	−27580.2	−1.29333	−0.646663	−	0.762776i	$-0.723836\pi$
$769$	−27580.2	−1.29333	−0.646663	+	0.762776i	$0.723836\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−17386.0	−0.808966	−0.404483	−	0.914546i	$-0.632548\pi$
$773$	−17386.0	−0.808966	−0.404483	+	0.914546i	$0.632548\pi$
$774$	0	0
$775$	−7288.63	−0.337826
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6928.72	0.318674
$780$	0	0
$781$	−52724.1	−2.41564
$782$	0	0
$783$	0	0
$784$	0	0
$785$	223.739	0.0101727
$786$	0	0
$787$	−4680.29	−0.211988	−0.105994	−	0.994367i	$-0.533802\pi$
$787$	−4680.29	−0.211988	−0.105994	+	0.994367i	$0.533802\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−59.0498	−0.00265432
$792$	0	0
$793$	1165.91	0.0522100
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−7278.62	−0.323490	−0.161745	−	0.986833i	$-0.551712\pi$
$797$	−7278.62	−0.323490	−0.161745	+	0.986833i	$0.551712\pi$
$798$	0	0
$799$	−17588.6	−0.778773
$800$	0	0
$801$	0	0
$802$	0	0
$803$	50524.7	2.22039
$804$	0	0
$805$	−3666.58	−0.160534
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−29212.5	−1.26954	−0.634770	−	0.772701i	$-0.718905\pi$
$809$	−29212.5	−1.26954	−0.634770	+	0.772701i	$0.718905\pi$
$810$	0	0
$811$	−41992.4	−1.81819	−0.909094	−	0.416590i	$-0.863225\pi$
$811$	−41992.4	−1.81819	−0.909094	+	0.416590i	$0.863225\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−13059.2	−0.561281
$816$	0	0
$817$	17900.5	0.766536
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8722.99	0.370809	0.185405	−	0.982662i	$-0.440640\pi$
$821$	8722.99	0.370809	0.185405	+	0.982662i	$0.440640\pi$
$822$	0	0
$823$	−13584.8	−0.575379	−0.287690	−	0.957724i	$-0.592887\pi$
$823$	−13584.8	−0.575379	−0.287690	+	0.957724i	$0.592887\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26573.6	1.11736	0.558679	−	0.829384i	$-0.311308\pi$
$827$	26573.6	1.11736	0.558679	+	0.829384i	$0.311308\pi$
$828$	0	0
$829$	−43238.4	−1.81150	−0.905748	−	0.423816i	$-0.860690\pi$
$829$	−43238.4	−1.81150	−0.905748	+	0.423816i	$0.860690\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−21836.3	−0.908265
$834$	0	0
$835$	944.733	0.0391543
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−4093.21	−0.168431	−0.0842154	−	0.996448i	$-0.526838\pi$
$839$	−4093.21	−0.168431	−0.0842154	+	0.996448i	$0.526838\pi$
$840$	0	0
$841$	24036.5	0.985546
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8735.27	0.355624
$846$	0	0
$847$	−10507.8	−0.426273
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−37614.7	−1.51517
$852$	0	0
$853$	−20201.5	−0.810886	−0.405443	−	0.914120i	$-0.632883\pi$
$853$	−20201.5	−0.810886	−0.405443	+	0.914120i	$0.632883\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4551.65	−0.181425	−0.0907126	−	0.995877i	$-0.528914\pi$
$857$	−4551.65	−0.181425	−0.0907126	+	0.995877i	$0.528914\pi$
$858$	0	0
$859$	−11962.6	−0.475154	−0.237577	−	0.971369i	$-0.576353\pi$
$859$	−11962.6	−0.475154	−0.237577	+	0.971369i	$0.576353\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−7164.61	−0.282603	−0.141301	−	0.989967i	$-0.545129\pi$
$863$	−7164.61	−0.282603	−0.141301	+	0.989967i	$0.545129\pi$
$864$	0	0
$865$	−7525.10	−0.295793
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−12065.9	−0.471012
$870$	0	0
$871$	−16082.4	−0.625640
$872$	0	0
$873$	0	0
$874$	0	0
$875$	635.154	0.0245396
$876$	0	0
$877$	−19218.7	−0.739987	−0.369994	−	0.929034i	$-0.620640\pi$
$877$	−19218.7	−0.739987	−0.369994	+	0.929034i	$0.620640\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	45914.5	1.75585	0.877923	−	0.478802i	$-0.158929\pi$
$881$	45914.5	1.75585	0.877923	+	0.478802i	$0.158929\pi$
$882$	0	0
$883$	44656.7	1.70194	0.850972	−	0.525211i	$-0.176014\pi$
$883$	44656.7	1.70194	0.850972	+	0.525211i	$0.176014\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	14975.2	0.566873	0.283437	−	0.958991i	$-0.408525\pi$
$887$	14975.2	0.566873	0.283437	+	0.958991i	$0.408525\pi$
$888$	0	0
$889$	1570.82	0.0592616
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−10427.0	−0.390735
$894$	0	0
$895$	15683.1	0.585729
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−64156.8	−2.38015
$900$	0	0
$901$	−14782.0	−0.546571
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−22563.3	−0.828763
$906$	0	0
$907$	14818.1	0.542479	0.271240	−	0.962512i	$-0.412566\pi$
$907$	14818.1	0.542479	0.271240	+	0.962512i	$0.412566\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	21846.5	0.794518	0.397259	−	0.917707i	$-0.369961\pi$
$911$	21846.5	0.794518	0.397259	+	0.917707i	$0.369961\pi$
$912$	0	0
$913$	−27016.4	−0.979312
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−14151.4	−0.509618
$918$	0	0
$919$	−28878.9	−1.03659	−0.518296	−	0.855201i	$-0.673434\pi$
$919$	−28878.9	−1.03659	−0.518296	+	0.855201i	$0.673434\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−19182.9	−0.684089
$924$	0	0
$925$	6515.92	0.231613
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−4608.79	−0.162766	−0.0813829	−	0.996683i	$-0.525934\pi$
$929$	−4608.79	−0.162766	−0.0813829	+	0.996683i	$0.525934\pi$
$930$	0	0
$931$	−12945.2	−0.455705
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−20068.6	−0.701938
$936$	0	0
$937$	−21063.8	−0.734391	−0.367195	−	0.930144i	$-0.619682\pi$
$937$	−21063.8	−0.734391	−0.367195	+	0.930144i	$0.619682\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	20244.8	0.701339	0.350670	−	0.936499i	$-0.385954\pi$
$941$	20244.8	0.701339	0.350670	+	0.936499i	$0.385954\pi$
$942$	0	0
$943$	−24500.4	−0.846069
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−29978.3	−1.02868	−0.514342	−	0.857585i	$-0.671964\pi$
$947$	−29978.3	−1.02868	−0.514342	+	0.857585i	$0.671964\pi$
$948$	0	0
$949$	18382.7	0.628797
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−5782.65	−0.196556	−0.0982782	−	0.995159i	$-0.531334\pi$
$953$	−5782.65	−0.196556	−0.0982782	+	0.995159i	$0.531334\pi$
$954$	0	0
$955$	−6037.17	−0.204564
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12649.3	−0.425930
$960$	0	0
$961$	55207.7	1.85317
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4615.82	−0.153978
$966$	0	0
$967$	26119.2	0.868600	0.434300	−	0.900768i	$-0.356996\pi$
$967$	26119.2	0.868600	0.434300	+	0.900768i	$0.356996\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5101.93	0.168619	0.0843093	−	0.996440i	$-0.473132\pi$
$971$	5101.93	0.168619	0.0843093	+	0.996440i	$0.473132\pi$
$972$	0	0
$973$	9075.34	0.299016
$974$	0	0
$975$	0	0
$976$	0	0
$977$	45902.4	1.50312	0.751559	−	0.659666i	$-0.229302\pi$
$977$	45902.4	1.50312	0.751559	+	0.659666i	$0.229302\pi$
$978$	0	0
$979$	−35081.6	−1.14526
$980$	0	0
$981$	0	0
$982$	0	0
$983$	13416.5	0.435321	0.217661	−	0.976025i	$-0.430157\pi$
$983$	13416.5	0.435321	0.217661	+	0.976025i	$0.430157\pi$
$984$	0	0
$985$	−5904.37	−0.190994
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−63297.4	−2.03513
$990$	0	0
$991$	−3806.66	−0.122021	−0.0610104	−	0.998137i	$-0.519432\pi$
$991$	−3806.66	−0.122021	−0.0610104	+	0.998137i	$0.519432\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−4199.02	−0.133787
$996$	0	0
$997$	−22523.8	−0.715483	−0.357742	−	0.933821i	$-0.616453\pi$
$997$	−22523.8	−0.715483	−0.357742	+	0.933821i	$0.616453\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.be.1.3		3
3.2	odd	2		2160.4.a.bm.1.3		3
4.3	odd	2		135.4.a.g.1.1	yes	3
12.11	even	2		135.4.a.f.1.3	✓	3
20.3	even	4		675.4.b.k.649.5		6
20.7	even	4		675.4.b.k.649.2		6
20.19	odd	2		675.4.a.q.1.3		3
36.7	odd	6		405.4.e.r.271.3		6
36.11	even	6		405.4.e.t.271.1		6
36.23	even	6		405.4.e.t.136.1		6
36.31	odd	6		405.4.e.r.136.3		6
60.23	odd	4		675.4.b.l.649.2		6
60.47	odd	4		675.4.b.l.649.5		6
60.59	even	2		675.4.a.r.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.f.1.3	✓	3	12.11	even	2
135.4.a.g.1.1	yes	3	4.3	odd	2
405.4.e.r.136.3		6	36.31	odd	6
405.4.e.r.271.3		6	36.7	odd	6
405.4.e.t.136.1		6	36.23	even	6
405.4.e.t.271.1		6	36.11	even	6
675.4.a.q.1.3		3	20.19	odd	2
675.4.a.r.1.1		3	60.59	even	2
675.4.b.k.649.2		6	20.7	even	4
675.4.b.k.649.5		6	20.3	even	4
675.4.b.l.649.2		6	60.23	odd	4
675.4.b.l.649.5		6	60.47	odd	4
2160.4.a.be.1.3		3	1.1	even	1	trivial
2160.4.a.bm.1.3		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} -5.08123 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} -5.08123 q^{7} +58.3007 q^{11} +21.2119 q^{13} +68.8451 q^{17} +40.8133 q^{19} -144.318 q^{23} +25.0000 q^{25} +220.058 q^{29} -291.545 q^{31} +25.4062 q^{35} +260.637 q^{37} +169.766 q^{41} +438.596 q^{43} -255.481 q^{47} -317.181 q^{49} -214.714 q^{53} -291.503 q^{55} +331.524 q^{59} +54.9647 q^{61} -106.060 q^{65} -758.179 q^{67} -904.348 q^{71} +866.622 q^{73} -296.239 q^{77} -206.961 q^{79} -463.397 q^{83} -344.225 q^{85} -601.736 q^{89} -107.783 q^{91} -204.066 q^{95} +229.363 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 15 q^{5} - 44 q^{7}+O(q^{10}) \) 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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