LMFDB - Embedded newform 1080.4.a.n.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1080,4,Mod(1,1080)] 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("1080.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1080, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1080.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,15,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$63.7220628062$
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:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-6.83575$ of defining polynomial
Character	$\chi$	$=$	1080.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000 q^{5} +23.9261 q^{7} -57.9406 q^{11} -8.16237 q^{13} +50.0884 q^{17} +69.7782 q^{19} -4.92607 q^{23} +25.0000 q^{25} -79.4277 q^{29} +260.294 q^{31} +119.630 q^{35} -223.836 q^{37} +337.939 q^{41} +326.511 q^{43} -89.6851 q^{47} +229.457 q^{49} +543.672 q^{53} -289.703 q^{55} -92.0000 q^{59} +159.129 q^{61} -40.8119 q^{65} -910.561 q^{67} +293.232 q^{71} +142.022 q^{73} -1386.29 q^{77} +1106.50 q^{79} -813.367 q^{83} +250.442 q^{85} -956.666 q^{89} -195.294 q^{91} +348.891 q^{95} +106.363 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	23.9261	1.29189	0.645943	−	0.763386i	$-0.276464\pi$
$7$	23.9261	1.29189	0.645943	+	0.763386i	$0.276464\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−57.9406	−1.58816	−0.794079	−	0.607814i	$-0.792046\pi$
$11$	−57.9406	−1.58816	−0.794079	+	0.607814i	$0.792046\pi$
$12$	0	0
$13$	−8.16237	−0.174141	−0.0870706	−	0.996202i	$-0.527751\pi$
$13$	−8.16237	−0.174141	−0.0870706	+	0.996202i	$0.527751\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	50.0884	0.714602	0.357301	−	0.933989i	$-0.383697\pi$
$17$	50.0884	0.714602	0.357301	+	0.933989i	$0.383697\pi$
$18$	0	0
$19$	69.7782	0.842538	0.421269	−	0.906936i	$-0.361585\pi$
$19$	69.7782	0.842538	0.421269	+	0.906936i	$0.361585\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.92607	−0.0446590	−0.0223295	−	0.999751i	$-0.507108\pi$
$23$	−4.92607	−0.0446590	−0.0223295	+	0.999751i	$0.507108\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−79.4277	−0.508598	−0.254299	−	0.967126i	$-0.581845\pi$
$29$	−79.4277	−0.508598	−0.254299	+	0.967126i	$0.581845\pi$
$30$	0	0
$31$	260.294	1.50807	0.754036	−	0.656833i	$-0.228104\pi$
$31$	260.294	1.50807	0.754036	+	0.656833i	$0.228104\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	119.630	0.577749
$36$	0	0
$37$	−223.836	−0.994553	−0.497276	−	0.867592i	$-0.665666\pi$
$37$	−223.836	−0.994553	−0.497276	+	0.867592i	$0.665666\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	337.939	1.28725	0.643625	−	0.765341i	$-0.277430\pi$
$41$	337.939	1.28725	0.643625	+	0.765341i	$0.277430\pi$
$42$	0	0
$43$	326.511	1.15797	0.578983	−	0.815340i	$-0.303450\pi$
$43$	326.511	1.15797	0.578983	+	0.815340i	$0.303450\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−89.6851	−0.278339	−0.139169	−	0.990269i	$-0.544443\pi$
$47$	−89.6851	−0.278339	−0.139169	+	0.990269i	$0.544443\pi$
$48$	0	0
$49$	229.457	0.668970
$50$	0	0
$51$	0	0
$52$	0	0
$53$	543.672	1.40904	0.704520	−	0.709684i	$-0.251162\pi$
$53$	543.672	1.40904	0.704520	+	0.709684i	$0.251162\pi$
$54$	0	0
$55$	−289.703	−0.710246
$56$	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
$61$	159.129	0.334006	0.167003	−	0.985956i	$-0.446591\pi$
$61$	159.129	0.334006	0.167003	+	0.985956i	$0.446591\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−40.8119	−0.0778783
$66$	0	0
$67$	−910.561	−1.66034	−0.830169	−	0.557511i	$-0.811756\pi$
$67$	−910.561	−1.66034	−0.830169	+	0.557511i	$0.811756\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	293.232	0.490144	0.245072	−	0.969505i	$-0.421188\pi$
$71$	293.232	0.490144	0.245072	+	0.969505i	$0.421188\pi$
$72$	0	0
$73$	142.022	0.227705	0.113853	−	0.993498i	$-0.463681\pi$
$73$	142.022	0.227705	0.113853	+	0.993498i	$0.463681\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1386.29	−2.05172
$78$	0	0
$79$	1106.50	1.57584	0.787920	−	0.615777i	$-0.211158\pi$
$79$	1106.50	1.57584	0.787920	+	0.615777i	$0.211158\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−813.367	−1.07565	−0.537823	−	0.843058i	$-0.680753\pi$
$83$	−813.367	−1.07565	−0.537823	+	0.843058i	$0.680753\pi$
$84$	0	0
$85$	250.442	0.319580
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−956.666	−1.13940	−0.569699	−	0.821853i	$-0.692940\pi$
$89$	−956.666	−1.13940	−0.569699	+	0.821853i	$0.692940\pi$
$90$	0	0
$91$	−195.294	−0.224971
$92$	0	0
$93$	0	0
$94$	0	0
$95$	348.891	0.376794
$96$	0	0
$97$	106.363	0.111335	0.0556677	−	0.998449i	$-0.482271\pi$
$97$	106.363	0.111335	0.0556677	+	0.998449i	$0.482271\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	703.082	0.692666	0.346333	−	0.938112i	$-0.387427\pi$
$101$	703.082	0.692666	0.346333	+	0.938112i	$0.387427\pi$
$102$	0	0
$103$	968.360	0.926363	0.463181	−	0.886263i	$-0.346708\pi$
$103$	968.360	0.926363	0.463181	+	0.886263i	$0.346708\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−366.393	−0.331033	−0.165517	−	0.986207i	$-0.552929\pi$
$107$	−366.393	−0.331033	−0.165517	+	0.986207i	$0.552929\pi$
$108$	0	0
$109$	723.614	0.635869	0.317934	−	0.948113i	$-0.397011\pi$
$109$	723.614	0.635869	0.317934	+	0.948113i	$0.397011\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−978.279	−0.814413	−0.407207	−	0.913336i	$-0.633497\pi$
$113$	−978.279	−0.814413	−0.407207	+	0.913336i	$0.633497\pi$
$114$	0	0
$115$	−24.6303	−0.0199721
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1198.42	0.923184
$120$	0	0
$121$	2026.11	1.52225
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	2821.70	1.97154	0.985770	−	0.168099i	$-0.0537628\pi$
$127$	2821.70	1.97154	0.985770	+	0.168099i	$0.0537628\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2131.82	1.42182	0.710908	−	0.703285i	$-0.248284\pi$
$131$	2131.82	1.42182	0.710908	+	0.703285i	$0.248284\pi$
$132$	0	0
$133$	1669.52	1.08846
$134$	0	0
$135$	0	0
$136$	0	0
$137$	816.808	0.509377	0.254688	−	0.967023i	$-0.418027\pi$
$137$	816.808	0.509377	0.254688	+	0.967023i	$0.418027\pi$
$138$	0	0
$139$	1876.80	1.14524	0.572620	−	0.819821i	$-0.305927\pi$
$139$	1876.80	1.14524	0.572620	+	0.819821i	$0.305927\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	472.933	0.276564
$144$	0	0
$145$	−397.139	−0.227452
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2542.71	−1.39803	−0.699016	−	0.715106i	$-0.746379\pi$
$149$	−2542.71	−1.39803	−0.699016	+	0.715106i	$0.746379\pi$
$150$	0	0
$151$	333.443	0.179703	0.0898516	−	0.995955i	$-0.471361\pi$
$151$	333.443	0.179703	0.0898516	+	0.995955i	$0.471361\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1301.47	0.674430
$156$	0	0
$157$	3650.02	1.85543	0.927716	−	0.373286i	$-0.121769\pi$
$157$	3650.02	1.85543	0.927716	+	0.373286i	$0.121769\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−117.861	−0.0576943
$162$	0	0
$163$	745.580	0.358272	0.179136	−	0.983824i	$-0.442670\pi$
$163$	745.580	0.358272	0.179136	+	0.983824i	$0.442670\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3321.73	1.53918	0.769591	−	0.638537i	$-0.220460\pi$
$167$	3321.73	1.53918	0.769591	+	0.638537i	$0.220460\pi$
$168$	0	0
$169$	−2130.38	−0.969675
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1038.58	−0.456426	−0.228213	−	0.973611i	$-0.573288\pi$
$173$	−1038.58	−0.456426	−0.228213	+	0.973611i	$0.573288\pi$
$174$	0	0
$175$	598.152	0.258377
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−250.181	−0.104466	−0.0522329	−	0.998635i	$-0.516634\pi$
$179$	−250.181	−0.104466	−0.0522329	+	0.998635i	$0.516634\pi$
$180$	0	0
$181$	−1171.61	−0.481132	−0.240566	−	0.970633i	$-0.577333\pi$
$181$	−1171.61	−0.481132	−0.240566	+	0.970633i	$0.577333\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1119.18	−0.444778
$186$	0	0
$187$	−2902.15	−1.13490
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1091.59	0.413533	0.206766	−	0.978390i	$-0.433706\pi$
$191$	1091.59	0.413533	0.206766	+	0.978390i	$0.433706\pi$
$192$	0	0
$193$	−171.475	−0.0639537	−0.0319768	−	0.999489i	$-0.510180\pi$
$193$	−171.475	−0.0639537	−0.0319768	+	0.999489i	$0.510180\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2368.70	−0.856665	−0.428332	−	0.903621i	$-0.640899\pi$
$197$	−2368.70	−0.856665	−0.428332	+	0.903621i	$0.640899\pi$
$198$	0	0
$199$	3373.82	1.20183	0.600914	−	0.799314i	$-0.294803\pi$
$199$	3373.82	1.20183	0.600914	+	0.799314i	$0.294803\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1900.39	−0.657051
$204$	0	0
$205$	1689.70	0.575676
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4042.99	−1.33808
$210$	0	0
$211$	5148.57	1.67982	0.839910	−	0.542726i	$-0.182608\pi$
$211$	5148.57	1.67982	0.839910	+	0.542726i	$0.182608\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1632.56	0.517858
$216$	0	0
$217$	6227.82	1.94826
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−408.841	−0.124442
$222$	0	0
$223$	−3604.51	−1.08240	−0.541201	−	0.840893i	$-0.682030\pi$
$223$	−3604.51	−1.08240	−0.541201	+	0.840893i	$0.682030\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3324.89	−0.972162	−0.486081	−	0.873914i	$-0.661574\pi$
$227$	−3324.89	−0.972162	−0.486081	+	0.873914i	$0.661574\pi$
$228$	0	0
$229$	−2809.94	−0.810856	−0.405428	−	0.914127i	$-0.632877\pi$
$229$	−2809.94	−0.810856	−0.405428	+	0.914127i	$0.632877\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4200.85	1.18115	0.590573	−	0.806984i	$-0.298902\pi$
$233$	4200.85	1.18115	0.590573	+	0.806984i	$0.298902\pi$
$234$	0	0
$235$	−448.426	−0.124477
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5787.06	1.56625	0.783125	−	0.621865i	$-0.213625\pi$
$239$	5787.06	1.56625	0.783125	+	0.621865i	$0.213625\pi$
$240$	0	0
$241$	−2971.77	−0.794309	−0.397154	−	0.917752i	$-0.630002\pi$
$241$	−2971.77	−0.794309	−0.397154	+	0.917752i	$0.630002\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1147.28	0.299172
$246$	0	0
$247$	−569.556	−0.146721
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4035.62	−1.01484	−0.507422	−	0.861698i	$-0.669401\pi$
$251$	−4035.62	−1.01484	−0.507422	+	0.861698i	$0.669401\pi$
$252$	0	0
$253$	285.419	0.0709255
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2852.67	0.692392	0.346196	−	0.938162i	$-0.387473\pi$
$257$	2852.67	0.692392	0.346196	+	0.938162i	$0.387473\pi$
$258$	0	0
$259$	−5355.52	−1.28485
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2910.63	0.682423	0.341212	−	0.939987i	$-0.389163\pi$
$263$	2910.63	0.682423	0.341212	+	0.939987i	$0.389163\pi$
$264$	0	0
$265$	2718.36	0.630142
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−476.299	−0.107957	−0.0539785	−	0.998542i	$-0.517190\pi$
$269$	−476.299	−0.107957	−0.0539785	+	0.998542i	$0.517190\pi$
$270$	0	0
$271$	−4186.19	−0.938350	−0.469175	−	0.883105i	$-0.655449\pi$
$271$	−4186.19	−0.938350	−0.469175	+	0.883105i	$0.655449\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1448.51	−0.317632
$276$	0	0
$277$	4993.37	1.08311	0.541556	−	0.840664i	$-0.317835\pi$
$277$	4993.37	1.08311	0.541556	+	0.840664i	$0.317835\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4480.61	−0.951212	−0.475606	−	0.879658i	$-0.657771\pi$
$281$	−4480.61	−0.951212	−0.475606	+	0.879658i	$0.657771\pi$
$282$	0	0
$283$	8117.91	1.70516	0.852579	−	0.522599i	$-0.175037\pi$
$283$	8117.91	1.70516	0.852579	+	0.522599i	$0.175037\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8085.56	1.66298
$288$	0	0
$289$	−2404.15	−0.489344
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−5999.46	−1.19622	−0.598109	−	0.801414i	$-0.704081\pi$
$293$	−5999.46	−1.19622	−0.598109	+	0.801414i	$0.704081\pi$
$294$	0	0
$295$	−460.000	−0.0907872
$296$	0	0
$297$	0	0
$298$	0	0
$299$	40.2084	0.00777696
$300$	0	0
$301$	7812.14	1.49596
$302$	0	0
$303$	0	0
$304$	0	0
$305$	795.644	0.149372
$306$	0	0
$307$	−1058.34	−0.196751	−0.0983753	−	0.995149i	$-0.531365\pi$
$307$	−1058.34	−0.196751	−0.0983753	+	0.995149i	$0.531365\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3120.17	−0.568902	−0.284451	−	0.958691i	$-0.591811\pi$
$311$	−3120.17	−0.568902	−0.284451	+	0.958691i	$0.591811\pi$
$312$	0	0
$313$	−4425.91	−0.799257	−0.399628	−	0.916677i	$-0.630861\pi$
$313$	−4425.91	−0.799257	−0.399628	+	0.916677i	$0.630861\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9032.06	1.60029	0.800144	−	0.599808i	$-0.204757\pi$
$317$	9032.06	1.60029	0.800144	+	0.599808i	$0.204757\pi$
$318$	0	0
$319$	4602.09	0.807735
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3495.08	0.602079
$324$	0	0
$325$	−204.059	−0.0348282
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2145.81	−0.359582
$330$	0	0
$331$	3153.85	0.523720	0.261860	−	0.965106i	$-0.415664\pi$
$331$	3153.85	0.523720	0.261860	+	0.965106i	$0.415664\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4552.80	−0.742526
$336$	0	0
$337$	−3438.53	−0.555811	−0.277906	−	0.960608i	$-0.589640\pi$
$337$	−3438.53	−0.555811	−0.277906	+	0.960608i	$0.589640\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−15081.6	−2.39506
$342$	0	0
$343$	−2716.64	−0.427653
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11108.0	1.71848	0.859238	−	0.511576i	$-0.170938\pi$
$347$	11108.0	1.71848	0.859238	+	0.511576i	$0.170938\pi$
$348$	0	0
$349$	1687.51	0.258826	0.129413	−	0.991591i	$-0.458691\pi$
$349$	1687.51	0.258826	0.129413	+	0.991591i	$0.458691\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−12048.9	−1.81671	−0.908356	−	0.418199i	$-0.862662\pi$
$353$	−12048.9	−1.81671	−0.908356	+	0.418199i	$0.862662\pi$
$354$	0	0
$355$	1466.16	0.219199
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11017.0	−1.61965	−0.809827	−	0.586669i	$-0.800439\pi$
$359$	−11017.0	−1.61965	−0.809827	+	0.586669i	$0.800439\pi$
$360$	0	0
$361$	−1990.00	−0.290130
$362$	0	0
$363$	0	0
$364$	0	0
$365$	710.112	0.101833
$366$	0	0
$367$	−10621.2	−1.51069	−0.755343	−	0.655329i	$-0.772530\pi$
$367$	−10621.2	−1.51069	−0.755343	+	0.655329i	$0.772530\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	13007.9	1.82032
$372$	0	0
$373$	24.9392	0.00346194	0.00173097	−	0.999999i	$-0.499449\pi$
$373$	24.9392	0.00346194	0.00173097	+	0.999999i	$0.499449\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	648.319	0.0885679
$378$	0	0
$379$	4755.57	0.644531	0.322265	−	0.946649i	$-0.395556\pi$
$379$	4755.57	0.644531	0.322265	+	0.946649i	$0.395556\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1229.91	0.164088	0.0820440	−	0.996629i	$-0.473855\pi$
$383$	1229.91	0.164088	0.0820440	+	0.996629i	$0.473855\pi$
$384$	0	0
$385$	−6931.45	−0.917557
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10616.3	−1.38372	−0.691862	−	0.722030i	$-0.743209\pi$
$389$	−10616.3	−1.38372	−0.691862	+	0.722030i	$0.743209\pi$
$390$	0	0
$391$	−246.739	−0.0319134
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5532.52	0.704738
$396$	0	0
$397$	−13086.4	−1.65437	−0.827187	−	0.561927i	$-0.810060\pi$
$397$	−13086.4	−1.65437	−0.827187	+	0.561927i	$0.810060\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5253.36	0.654215	0.327107	−	0.944987i	$-0.393926\pi$
$401$	5253.36	0.654215	0.327107	+	0.944987i	$0.393926\pi$
$402$	0	0
$403$	−2124.62	−0.262618
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12969.2	1.57951
$408$	0	0
$409$	4778.69	0.577729	0.288865	−	0.957370i	$-0.406722\pi$
$409$	4778.69	0.577729	0.288865	+	0.957370i	$0.406722\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2201.20	−0.262261
$414$	0	0
$415$	−4066.84	−0.481044
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−7770.14	−0.905958	−0.452979	−	0.891521i	$-0.649639\pi$
$419$	−7770.14	−0.905958	−0.452979	+	0.891521i	$0.649639\pi$
$420$	0	0
$421$	−14440.5	−1.67170	−0.835852	−	0.548955i	$-0.815026\pi$
$421$	−14440.5	−1.67170	−0.835852	+	0.548955i	$0.815026\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1252.21	0.142920
$426$	0	0
$427$	3807.32	0.431497
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13416.1	−1.49937	−0.749686	−	0.661794i	$-0.769795\pi$
$431$	−13416.1	−1.49937	−0.749686	+	0.661794i	$0.769795\pi$
$432$	0	0
$433$	−187.211	−0.0207778	−0.0103889	−	0.999946i	$-0.503307\pi$
$433$	−187.211	−0.0207778	−0.0103889	+	0.999946i	$0.503307\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−343.732	−0.0376269
$438$	0	0
$439$	2171.47	0.236079	0.118039	−	0.993009i	$-0.462339\pi$
$439$	2171.47	0.236079	0.118039	+	0.993009i	$0.462339\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11129.8	1.19366	0.596830	−	0.802368i	$-0.296427\pi$
$443$	11129.8	1.19366	0.596830	+	0.802368i	$0.296427\pi$
$444$	0	0
$445$	−4783.33	−0.509554
$446$	0	0
$447$	0	0
$448$	0	0
$449$	15602.0	1.63988	0.819938	−	0.572452i	$-0.194008\pi$
$449$	15602.0	1.63988	0.819938	+	0.572452i	$0.194008\pi$
$450$	0	0
$451$	−19580.4	−2.04436
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−976.468	−0.100610
$456$	0	0
$457$	−12065.3	−1.23499	−0.617495	−	0.786575i	$-0.711852\pi$
$457$	−12065.3	−1.23499	−0.617495	+	0.786575i	$0.711852\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−9703.61	−0.980352	−0.490176	−	0.871623i	$-0.663067\pi$
$461$	−9703.61	−0.980352	−0.490176	+	0.871623i	$0.663067\pi$
$462$	0	0
$463$	−9787.17	−0.982394	−0.491197	−	0.871048i	$-0.663441\pi$
$463$	−9787.17	−0.982394	−0.491197	+	0.871048i	$0.663441\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9051.97	0.896949	0.448474	−	0.893796i	$-0.351968\pi$
$467$	9051.97	0.896949	0.448474	+	0.893796i	$0.351968\pi$
$468$	0	0
$469$	−21786.1	−2.14497
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−18918.3	−1.83903
$474$	0	0
$475$	1744.46	0.168508
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8492.58	0.810096	0.405048	−	0.914295i	$-0.367255\pi$
$479$	8492.58	0.810096	0.405048	+	0.914295i	$0.367255\pi$
$480$	0	0
$481$	1827.04	0.173193
$482$	0	0
$483$	0	0
$484$	0	0
$485$	531.816	0.0497907
$486$	0	0
$487$	−7190.96	−0.669104	−0.334552	−	0.942377i	$-0.608585\pi$
$487$	−7190.96	−0.669104	−0.334552	+	0.942377i	$0.608585\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	20304.9	1.86629	0.933146	−	0.359499i	$-0.117052\pi$
$491$	20304.9	1.86629	0.933146	+	0.359499i	$0.117052\pi$
$492$	0	0
$493$	−3978.41	−0.363445
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7015.88	0.633210
$498$	0	0
$499$	5079.64	0.455703	0.227852	−	0.973696i	$-0.426830\pi$
$499$	5079.64	0.455703	0.227852	+	0.973696i	$0.426830\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1329.81	0.117880	0.0589398	−	0.998262i	$-0.481228\pi$
$503$	1329.81	0.117880	0.0589398	+	0.998262i	$0.481228\pi$
$504$	0	0
$505$	3515.41	0.309770
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−6501.31	−0.566140	−0.283070	−	0.959099i	$-0.591353\pi$
$509$	−6501.31	−0.566140	−0.283070	+	0.959099i	$0.591353\pi$
$510$	0	0
$511$	3398.04	0.294169
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4841.80	0.414282
$516$	0	0
$517$	5196.41	0.442046
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5846.09	0.491597	0.245798	−	0.969321i	$-0.420950\pi$
$521$	5846.09	0.491597	0.245798	+	0.969321i	$0.420950\pi$
$522$	0	0
$523$	−2544.28	−0.212722	−0.106361	−	0.994328i	$-0.533920\pi$
$523$	−2544.28	−0.212722	−0.106361	+	0.994328i	$0.533920\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13037.7	1.07767
$528$	0	0
$529$	−12142.7	−0.998006
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2758.39	−0.224163
$534$	0	0
$535$	−1831.97	−0.148043
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−13294.9	−1.06243
$540$	0	0
$541$	19350.2	1.53776	0.768880	−	0.639393i	$-0.220814\pi$
$541$	19350.2	1.53776	0.768880	+	0.639393i	$0.220814\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3618.07	0.284369
$546$	0	0
$547$	−4957.23	−0.387488	−0.193744	−	0.981052i	$-0.562063\pi$
$547$	−4957.23	−0.387488	−0.193744	+	0.981052i	$0.562063\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5542.32	−0.428513
$552$	0	0
$553$	26474.3	2.03581
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1559.47	0.118630	0.0593148	−	0.998239i	$-0.481108\pi$
$557$	1559.47	0.118630	0.0593148	+	0.998239i	$0.481108\pi$
$558$	0	0
$559$	−2665.11	−0.201650
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7533.60	−0.563949	−0.281974	−	0.959422i	$-0.590989\pi$
$563$	−7533.60	−0.563949	−0.281974	+	0.959422i	$0.590989\pi$
$564$	0	0
$565$	−4891.39	−0.364217
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−19257.9	−1.41886	−0.709432	−	0.704774i	$-0.751048\pi$
$569$	−19257.9	−1.41886	−0.709432	+	0.704774i	$0.751048\pi$
$570$	0	0
$571$	−26102.2	−1.91304	−0.956518	−	0.291675i	$-0.905788\pi$
$571$	−26102.2	−1.91304	−0.956518	+	0.291675i	$0.905788\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−123.152	−0.00893179
$576$	0	0
$577$	−12553.3	−0.905721	−0.452860	−	0.891581i	$-0.649596\pi$
$577$	−12553.3	−0.905721	−0.452860	+	0.891581i	$0.649596\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−19460.7	−1.38961
$582$	0	0
$583$	−31500.7	−2.23778
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−16778.4	−1.17976	−0.589881	−	0.807490i	$-0.700825\pi$
$587$	−16778.4	−1.17976	−0.589881	+	0.807490i	$0.700825\pi$
$588$	0	0
$589$	18162.9	1.27061
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21693.4	1.50226	0.751132	−	0.660152i	$-0.229508\pi$
$593$	21693.4	1.50226	0.751132	+	0.660152i	$0.229508\pi$
$594$	0	0
$595$	5992.10	0.412861
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−22980.4	−1.56753	−0.783767	−	0.621055i	$-0.786704\pi$
$599$	−22980.4	−1.56753	−0.783767	+	0.621055i	$0.786704\pi$
$600$	0	0
$601$	−10048.9	−0.682038	−0.341019	−	0.940056i	$-0.610772\pi$
$601$	−10048.9	−0.682038	−0.341019	+	0.940056i	$0.610772\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	10130.6	0.680770
$606$	0	0
$607$	24514.6	1.63923	0.819617	−	0.572911i	$-0.194186\pi$
$607$	24514.6	1.63923	0.819617	+	0.572911i	$0.194186\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	732.043	0.0484702
$612$	0	0
$613$	1845.67	0.121609	0.0608043	−	0.998150i	$-0.480633\pi$
$613$	1845.67	0.121609	0.0608043	+	0.998150i	$0.480633\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9365.42	0.611082	0.305541	−	0.952179i	$-0.401163\pi$
$617$	9365.42	0.611082	0.305541	+	0.952179i	$0.401163\pi$
$618$	0	0
$619$	−19517.2	−1.26731	−0.633653	−	0.773618i	$-0.718445\pi$
$619$	−19517.2	−1.26731	−0.633653	+	0.773618i	$0.718445\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−22889.3	−1.47197
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−11211.6	−0.710709
$630$	0	0
$631$	22314.1	1.40778	0.703892	−	0.710307i	$-0.251444\pi$
$631$	22314.1	1.40778	0.703892	+	0.710307i	$0.251444\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	14108.5	0.881700
$636$	0	0
$637$	−1872.91	−0.116495
$638$	0	0
$639$	0	0
$640$	0	0
$641$	20150.8	1.24167	0.620833	−	0.783943i	$-0.286795\pi$
$641$	20150.8	1.24167	0.620833	+	0.783943i	$0.286795\pi$
$642$	0	0
$643$	5345.25	0.327832	0.163916	−	0.986474i	$-0.447587\pi$
$643$	5345.25	0.327832	0.163916	+	0.986474i	$0.447587\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	13701.4	0.832549	0.416274	−	0.909239i	$-0.363336\pi$
$647$	13701.4	0.832549	0.416274	+	0.909239i	$0.363336\pi$
$648$	0	0
$649$	5330.53	0.322406
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−713.119	−0.0427358	−0.0213679	−	0.999772i	$-0.506802\pi$
$653$	−713.119	−0.0427358	−0.0213679	+	0.999772i	$0.506802\pi$
$654$	0	0
$655$	10659.1	0.635856
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−26491.2	−1.56593	−0.782967	−	0.622063i	$-0.786295\pi$
$659$	−26491.2	−1.56593	−0.782967	+	0.622063i	$0.786295\pi$
$660$	0	0
$661$	645.746	0.0379979	0.0189989	−	0.999820i	$-0.493952\pi$
$661$	645.746	0.0379979	0.0189989	+	0.999820i	$0.493952\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8347.59	0.486775
$666$	0	0
$667$	391.266	0.0227135
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−9220.01	−0.530454
$672$	0	0
$673$	−6602.56	−0.378172	−0.189086	−	0.981961i	$-0.560553\pi$
$673$	−6602.56	−0.378172	−0.189086	+	0.981961i	$0.560553\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4968.68	−0.282070	−0.141035	−	0.990005i	$-0.545043\pi$
$677$	−4968.68	−0.282070	−0.141035	+	0.990005i	$0.545043\pi$
$678$	0	0
$679$	2544.85	0.143833
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24694.2	−1.38345	−0.691727	−	0.722159i	$-0.743150\pi$
$683$	−24694.2	−1.38345	−0.691727	+	0.722159i	$0.743150\pi$
$684$	0	0
$685$	4084.04	0.227800
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4437.66	−0.245372
$690$	0	0
$691$	−19259.2	−1.06028	−0.530141	−	0.847910i	$-0.677861\pi$
$691$	−19259.2	−1.06028	−0.530141	+	0.847910i	$0.677861\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9384.02	0.512167
$696$	0	0
$697$	16926.8	0.919871
$698$	0	0
$699$	0	0
$700$	0	0
$701$	31090.7	1.67515	0.837575	−	0.546323i	$-0.183973\pi$
$701$	31090.7	1.67515	0.837575	+	0.546323i	$0.183973\pi$
$702$	0	0
$703$	−15618.9	−0.837948
$704$	0	0
$705$	0	0
$706$	0	0
$707$	16822.0	0.894845
$708$	0	0
$709$	−20832.0	−1.10347	−0.551736	−	0.834019i	$-0.686034\pi$
$709$	−20832.0	−1.10347	−0.551736	+	0.834019i	$0.686034\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1282.23	−0.0673490
$714$	0	0
$715$	2364.66	0.123683
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−21328.1	−1.10626	−0.553132	−	0.833093i	$-0.686568\pi$
$719$	−21328.1	−1.10626	−0.553132	+	0.833093i	$0.686568\pi$
$720$	0	0
$721$	23169.1	1.19676
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1985.69	−0.101720
$726$	0	0
$727$	4121.56	0.210262	0.105131	−	0.994458i	$-0.466474\pi$
$727$	4121.56	0.210262	0.105131	+	0.994458i	$0.466474\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	16354.5	0.827485
$732$	0	0
$733$	−11102.2	−0.559439	−0.279720	−	0.960082i	$-0.590241\pi$
$733$	−11102.2	−0.559439	−0.279720	+	0.960082i	$0.590241\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	52758.4	2.63688
$738$	0	0
$739$	−13749.5	−0.684414	−0.342207	−	0.939625i	$-0.611174\pi$
$739$	−13749.5	−0.684414	−0.342207	+	0.939625i	$0.611174\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	13962.9	0.689433	0.344717	−	0.938707i	$-0.387975\pi$
$743$	13962.9	0.689433	0.344717	+	0.938707i	$0.387975\pi$
$744$	0	0
$745$	−12713.5	−0.625219
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−8766.35	−0.427657
$750$	0	0
$751$	32801.6	1.59380	0.796902	−	0.604108i	$-0.206470\pi$
$751$	32801.6	1.59380	0.796902	+	0.604108i	$0.206470\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1667.21	0.0803657
$756$	0	0
$757$	−14664.9	−0.704102	−0.352051	−	0.935981i	$-0.614516\pi$
$757$	−14664.9	−0.704102	−0.352051	+	0.935981i	$0.614516\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3304.79	0.157422	0.0787112	−	0.996897i	$-0.474919\pi$
$761$	3304.79	0.157422	0.0787112	+	0.996897i	$0.474919\pi$
$762$	0	0
$763$	17313.2	0.821470
$764$	0	0
$765$	0	0
$766$	0	0
$767$	750.938	0.0353518
$768$	0	0
$769$	−33085.5	−1.55149	−0.775743	−	0.631049i	$-0.782625\pi$
$769$	−33085.5	−1.55149	−0.775743	+	0.631049i	$0.782625\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−27880.8	−1.29729	−0.648643	−	0.761093i	$-0.724663\pi$
$773$	−27880.8	−1.29729	−0.648643	+	0.761093i	$0.724663\pi$
$774$	0	0
$775$	6507.36	0.301614
$776$	0	0
$777$	0	0
$778$	0	0
$779$	23580.8	1.08456
$780$	0	0
$781$	−16990.0	−0.778426
$782$	0	0
$783$	0	0
$784$	0	0
$785$	18250.1	0.829775
$786$	0	0
$787$	17813.9	0.806856	0.403428	−	0.915011i	$-0.367819\pi$
$787$	17813.9	0.806856	0.403428	+	0.915011i	$0.367819\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−23406.4	−1.05213
$792$	0	0
$793$	−1298.87	−0.0581641
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−8005.10	−0.355778	−0.177889	−	0.984051i	$-0.556927\pi$
$797$	−8005.10	−0.355778	−0.177889	+	0.984051i	$0.556927\pi$
$798$	0	0
$799$	−4492.19	−0.198901
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−8228.86	−0.361632
$804$	0	0
$805$	−589.307	−0.0258017
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−10140.7	−0.440700	−0.220350	−	0.975421i	$-0.570720\pi$
$809$	−10140.7	−0.440700	−0.220350	+	0.975421i	$0.570720\pi$
$810$	0	0
$811$	−14662.7	−0.634865	−0.317433	−	0.948281i	$-0.602821\pi$
$811$	−14662.7	−0.634865	−0.317433	+	0.948281i	$0.602821\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3727.90	0.160224
$816$	0	0
$817$	22783.4	0.975630
$818$	0	0
$819$	0	0
$820$	0	0
$821$	36216.1	1.53953	0.769763	−	0.638330i	$-0.220374\pi$
$821$	36216.1	1.53953	0.769763	+	0.638330i	$0.220374\pi$
$822$	0	0
$823$	−9695.36	−0.410643	−0.205321	−	0.978695i	$-0.565824\pi$
$823$	−9695.36	−0.410643	−0.205321	+	0.978695i	$0.565824\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−21471.9	−0.902844	−0.451422	−	0.892311i	$-0.649083\pi$
$827$	−21471.9	−0.902844	−0.451422	+	0.892311i	$0.649083\pi$
$828$	0	0
$829$	10447.9	0.437723	0.218861	−	0.975756i	$-0.429766\pi$
$829$	10447.9	0.437723	0.218861	+	0.975756i	$0.429766\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	11493.1	0.478047
$834$	0	0
$835$	16608.7	0.688343
$836$	0	0
$837$	0	0
$838$	0	0
$839$	4917.61	0.202354	0.101177	−	0.994868i	$-0.467739\pi$
$839$	4917.61	0.202354	0.101177	+	0.994868i	$0.467739\pi$
$840$	0	0
$841$	−18080.2	−0.741328
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10651.9	−0.433652
$846$	0	0
$847$	48476.9	1.96657
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1102.63	0.0444157
$852$	0	0
$853$	45031.3	1.80755	0.903777	−	0.428004i	$-0.140783\pi$
$853$	45031.3	1.80755	0.903777	+	0.428004i	$0.140783\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14826.6	0.590976	0.295488	−	0.955346i	$-0.404518\pi$
$857$	14826.6	0.590976	0.295488	+	0.955346i	$0.404518\pi$
$858$	0	0
$859$	43287.9	1.71940	0.859700	−	0.510799i	$-0.170650\pi$
$859$	43287.9	1.71940	0.859700	+	0.510799i	$0.170650\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−40830.8	−1.61054	−0.805270	−	0.592908i	$-0.797980\pi$
$863$	−40830.8	−1.61054	−0.805270	+	0.592908i	$0.797980\pi$
$864$	0	0
$865$	−5192.89	−0.204120
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−64111.5	−2.50268
$870$	0	0
$871$	7432.34	0.289133
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2990.76	0.115550
$876$	0	0
$877$	2322.20	0.0894127	0.0447064	−	0.999000i	$-0.485765\pi$
$877$	2322.20	0.0894127	0.0447064	+	0.999000i	$0.485765\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−10407.8	−0.398010	−0.199005	−	0.979998i	$-0.563771\pi$
$881$	−10407.8	−0.398010	−0.199005	+	0.979998i	$0.563771\pi$
$882$	0	0
$883$	−14346.9	−0.546784	−0.273392	−	0.961903i	$-0.588146\pi$
$883$	−14346.9	−0.546784	−0.273392	+	0.961903i	$0.588146\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22740.9	0.860838	0.430419	−	0.902629i	$-0.358366\pi$
$887$	22740.9	0.860838	0.430419	+	0.902629i	$0.358366\pi$
$888$	0	0
$889$	67512.2	2.54701
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−6258.07	−0.234511
$894$	0	0
$895$	−1250.90	−0.0467186
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−20674.6	−0.767003
$900$	0	0
$901$	27231.7	1.00690
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5858.04	−0.215169
$906$	0	0
$907$	−14222.6	−0.520678	−0.260339	−	0.965517i	$-0.583834\pi$
$907$	−14222.6	−0.520678	−0.260339	+	0.965517i	$0.583834\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−6825.69	−0.248238	−0.124119	−	0.992267i	$-0.539610\pi$
$911$	−6825.69	−0.248238	−0.124119	+	0.992267i	$0.539610\pi$
$912$	0	0
$913$	47127.0	1.70830
$914$	0	0
$915$	0	0
$916$	0	0
$917$	51006.1	1.83683
$918$	0	0
$919$	18851.0	0.676647	0.338323	−	0.941030i	$-0.390140\pi$
$919$	18851.0	0.676647	0.338323	+	0.941030i	$0.390140\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2393.47	−0.0853542
$924$	0	0
$925$	−5595.91	−0.198911
$926$	0	0
$927$	0	0
$928$	0	0
$929$	7109.70	0.251089	0.125544	−	0.992088i	$-0.459932\pi$
$929$	7109.70	0.251089	0.125544	+	0.992088i	$0.459932\pi$
$930$	0	0
$931$	16011.1	0.563633
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−14510.8	−0.507543
$936$	0	0
$937$	−14107.6	−0.491864	−0.245932	−	0.969287i	$-0.579094\pi$
$937$	−14107.6	−0.491864	−0.245932	+	0.969287i	$0.579094\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−49409.8	−1.71170	−0.855852	−	0.517221i	$-0.826966\pi$
$941$	−49409.8	−1.71170	−0.855852	+	0.517221i	$0.826966\pi$
$942$	0	0
$943$	−1664.71	−0.0574872
$944$	0	0
$945$	0	0
$946$	0	0
$947$	24070.5	0.825963	0.412982	−	0.910739i	$-0.364487\pi$
$947$	24070.5	0.825963	0.412982	+	0.910739i	$0.364487\pi$
$948$	0	0
$949$	−1159.24	−0.0396528
$950$	0	0
$951$	0	0
$952$	0	0
$953$	55644.1	1.89138	0.945692	−	0.325064i	$-0.105386\pi$
$953$	55644.1	1.89138	0.945692	+	0.325064i	$0.105386\pi$
$954$	0	0
$955$	5457.95	0.184937
$956$	0	0
$957$	0	0
$958$	0	0
$959$	19543.0	0.658057
$960$	0	0
$961$	37962.1	1.27428
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−857.377	−0.0286010
$966$	0	0
$967$	−34638.9	−1.15193	−0.575963	−	0.817476i	$-0.695373\pi$
$967$	−34638.9	−1.15193	−0.575963	+	0.817476i	$0.695373\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−28705.9	−0.948731	−0.474365	−	0.880328i	$-0.657322\pi$
$971$	−28705.9	−0.948731	−0.474365	+	0.880328i	$0.657322\pi$
$972$	0	0
$973$	44904.5	1.47952
$974$	0	0
$975$	0	0
$976$	0	0
$977$	45305.4	1.48357	0.741785	−	0.670638i	$-0.233979\pi$
$977$	45305.4	1.48357	0.741785	+	0.670638i	$0.233979\pi$
$978$	0	0
$979$	55429.8	1.80954
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−9220.63	−0.299178	−0.149589	−	0.988748i	$-0.547795\pi$
$983$	−9220.63	−0.299178	−0.149589	+	0.988748i	$0.547795\pi$
$984$	0	0
$985$	−11843.5	−0.383112
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1608.42	−0.0517136
$990$	0	0
$991$	−23444.0	−0.751488	−0.375744	−	0.926723i	$-0.622613\pi$
$991$	−23444.0	−0.751488	−0.375744	+	0.926723i	$0.622613\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16869.1	0.537474
$996$	0	0
$997$	9848.13	0.312832	0.156416	−	0.987691i	$-0.450006\pi$
$997$	9848.13	0.312832	0.156416	+	0.987691i	$0.450006\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.4.a.n.1.3	yes	3
3.2	odd	2		1080.4.a.h.1.3	✓	3
4.3	odd	2		2160.4.a.bn.1.1		3
12.11	even	2		2160.4.a.bf.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.h.1.3	✓	3	3.2	odd	2
1080.4.a.n.1.3	yes	3	1.1	even	1	trivial
2160.4.a.bf.1.1		3	12.11	even	2
2160.4.a.bn.1.1		3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1080.a (trivial)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} +23.9261 q^{7} -57.9406 q^{11} -8.16237 q^{13} +50.0884 q^{17} +69.7782 q^{19} -4.92607 q^{23} +25.0000 q^{25} -79.4277 q^{29} +260.294 q^{31} +119.630 q^{35} -223.836 q^{37} +337.939 q^{41} +326.511 q^{43} -89.6851 q^{47} +229.457 q^{49} +543.672 q^{53} -289.703 q^{55} -92.0000 q^{59} +159.129 q^{61} -40.8119 q^{65} -910.561 q^{67} +293.232 q^{71} +142.022 q^{73} -1386.29 q^{77} +1106.50 q^{79} -813.367 q^{83} +250.442 q^{85} -956.666 q^{89} -195.294 q^{91} +348.891 q^{95} +106.363 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 15 q^{5} + 9 q^{7} + 18 q^{11} - 21 q^{13} + 84 q^{17} + 21 q^{19} + 48 q^{23} + 75 q^{25} - 36 q^{29} + 324 q^{31} + 45 q^{35} + 33 q^{37} + 114 q^{41} + 282 q^{43} + 282 q^{47} + 228 q^{49} + 222 q^{53}+ \cdots + 1191 q^{97}+O(q^{100}) \) 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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