LMFDB - Embedded newform 1080.4.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,4,Mod(1,1080)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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

Self dual:	yes
Analytic conductor:	$63.7220628062$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 141x^{2} + 200x + 3500 $ x^4 - x^3 - 141x^2 + 200x + 3500
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$9.55367$ 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)

$f(q)$	$=$	$q-5.00000 q^{5} -2.40438 q^{7} +O(q^{10})$	$q-5.00000 q^{5} -2.40438 q^{7} -17.3282 q^{11} +71.2940 q^{13} -111.164 q^{17} -86.2397 q^{19} +148.266 q^{23} +25.0000 q^{25} -71.1182 q^{29} +212.038 q^{31} +12.0219 q^{35} -132.980 q^{37} +29.6830 q^{41} -126.879 q^{43} +103.317 q^{47} -337.219 q^{49} +298.464 q^{53} +86.6410 q^{55} -893.675 q^{59} +752.720 q^{61} -356.470 q^{65} +477.167 q^{67} +58.7339 q^{71} +873.851 q^{73} +41.6635 q^{77} +819.768 q^{79} +959.392 q^{83} +555.818 q^{85} +406.735 q^{89} -171.418 q^{91} +431.199 q^{95} +623.748 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{5} + 14 q^{7}+O(q^{10}) $ 4 * q - 20 * q^5 + 14 * q^7	$ 4 q - 20 q^{5} + 14 q^{7} + 4 q^{11} + 30 q^{13} + 28 q^{17} + 78 q^{19} - 182 q^{23} + 100 q^{25} - 202 q^{29} - 76 q^{31} - 70 q^{35} + 302 q^{37} - 380 q^{41} + 178 q^{43} - 114 q^{47} + 958 q^{49} + 256 q^{53} - 20 q^{55} + 204 q^{59} + 766 q^{61} - 150 q^{65} + 330 q^{67} + 1060 q^{71} + 1442 q^{73} - 216 q^{77} + 742 q^{79} + 768 q^{83} - 140 q^{85} + 400 q^{89} + 3066 q^{91} - 390 q^{95} + 3338 q^{97}+O(q^{100}) $ 4 * q - 20 * q^5 + 14 * q^7 + 4 * q^11 + 30 * q^13 + 28 * q^17 + 78 * q^19 - 182 * q^23 + 100 * q^25 - 202 * q^29 - 76 * q^31 - 70 * q^35 + 302 * q^37 - 380 * q^41 + 178 * q^43 - 114 * q^47 + 958 * q^49 + 256 * q^53 - 20 * q^55 + 204 * q^59 + 766 * q^61 - 150 * q^65 + 330 * q^67 + 1060 * q^71 + 1442 * q^73 - 216 * q^77 + 742 * q^79 + 768 * q^83 - 140 * q^85 + 400 * q^89 + 3066 * q^91 - 390 * q^95 + 3338 * 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
$7$	−2.40438	−0.129824	−0.0649121	−	0.997891i	$-0.520677\pi$
$7$	−2.40438	−0.129824	−0.0649121	+	0.997891i	$0.520677\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−17.3282	−0.474968	−0.237484	−	0.971391i	$-0.576323\pi$
$11$	−17.3282	−0.474968	−0.237484	+	0.971391i	$0.576323\pi$
$12$	0	0
$13$	71.2940	1.52103	0.760515	−	0.649320i	$-0.224946\pi$
$13$	71.2940	1.52103	0.760515	+	0.649320i	$0.224946\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−111.164	−1.58595	−0.792974	−	0.609255i	$-0.791468\pi$
$17$	−111.164	−1.58595	−0.792974	+	0.609255i	$0.791468\pi$
$18$	0	0
$19$	−86.2397	−1.04130	−0.520651	−	0.853769i	$-0.674311\pi$
$19$	−86.2397	−1.04130	−0.520651	+	0.853769i	$0.674311\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	148.266	1.34416	0.672080	−	0.740479i	$-0.265401\pi$
$23$	148.266	1.34416	0.672080	+	0.740479i	$0.265401\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−71.1182	−0.455390	−0.227695	−	0.973732i	$-0.573119\pi$
$29$	−71.1182	−0.455390	−0.227695	+	0.973732i	$0.573119\pi$
$30$	0	0
$31$	212.038	1.22849	0.614244	−	0.789116i	$-0.289461\pi$
$31$	212.038	1.22849	0.614244	+	0.789116i	$0.289461\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	12.0219	0.0580591
$36$	0	0
$37$	−132.980	−0.590859	−0.295430	−	0.955365i	$-0.595463\pi$
$37$	−132.980	−0.590859	−0.295430	+	0.955365i	$0.595463\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	29.6830	0.113066	0.0565330	−	0.998401i	$-0.481995\pi$
$41$	29.6830	0.113066	0.0565330	+	0.998401i	$0.481995\pi$
$42$	0	0
$43$	−126.879	−0.449973	−0.224987	−	0.974362i	$-0.572234\pi$
$43$	−126.879	−0.449973	−0.224987	+	0.974362i	$0.572234\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	103.317	0.320647	0.160323	−	0.987065i	$-0.448746\pi$
$47$	103.317	0.320647	0.160323	+	0.987065i	$0.448746\pi$
$48$	0	0
$49$	−337.219	−0.983146
$50$	0	0
$51$	0	0
$52$	0	0
$53$	298.464	0.773531	0.386765	−	0.922178i	$-0.373592\pi$
$53$	298.464	0.773531	0.386765	+	0.922178i	$0.373592\pi$
$54$	0	0
$55$	86.6410	0.212412
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−893.675	−1.97198	−0.985988	−	0.166817i	$-0.946651\pi$
$59$	−893.675	−1.97198	−0.985988	+	0.166817i	$0.946651\pi$
$60$	0	0
$61$	752.720	1.57993	0.789966	−	0.613150i	$-0.210098\pi$
$61$	752.720	1.57993	0.789966	+	0.613150i	$0.210098\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−356.470	−0.680226
$66$	0	0
$67$	477.167	0.870078	0.435039	−	0.900412i	$-0.356735\pi$
$67$	477.167	0.870078	0.435039	+	0.900412i	$0.356735\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	58.7339	0.0981751	0.0490875	−	0.998794i	$-0.484369\pi$
$71$	58.7339	0.0981751	0.0490875	+	0.998794i	$0.484369\pi$
$72$	0	0
$73$	873.851	1.40105	0.700524	−	0.713628i	$-0.252949\pi$
$73$	873.851	1.40105	0.700524	+	0.713628i	$0.252949\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	41.6635	0.0616623
$78$	0	0
$79$	819.768	1.16748	0.583741	−	0.811940i	$-0.301588\pi$
$79$	819.768	1.16748	0.583741	+	0.811940i	$0.301588\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	959.392	1.26876	0.634379	−	0.773022i	$-0.281256\pi$
$83$	959.392	1.26876	0.634379	+	0.773022i	$0.281256\pi$
$84$	0	0
$85$	555.818	0.709258
$86$	0	0
$87$	0	0
$88$	0	0
$89$	406.735	0.484425	0.242212	−	0.970223i	$-0.422127\pi$
$89$	406.735	0.484425	0.242212	+	0.970223i	$0.422127\pi$
$90$	0	0
$91$	−171.418	−0.197467
$92$	0	0
$93$	0	0
$94$	0	0
$95$	431.199	0.465685
$96$	0	0
$97$	623.748	0.652908	0.326454	−	0.945213i	$-0.394146\pi$
$97$	623.748	0.652908	0.326454	+	0.945213i	$0.394146\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	468.202	0.461266	0.230633	−	0.973041i	$-0.425920\pi$
$101$	468.202	0.461266	0.230633	+	0.973041i	$0.425920\pi$
$102$	0	0
$103$	−949.097	−0.907935	−0.453967	−	0.891018i	$-0.649992\pi$
$103$	−949.097	−0.907935	−0.453967	+	0.891018i	$0.649992\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	649.086	0.586444	0.293222	−	0.956044i	$-0.405273\pi$
$107$	649.086	0.586444	0.293222	+	0.956044i	$0.405273\pi$
$108$	0	0
$109$	1152.44	1.01269	0.506346	−	0.862330i	$-0.330996\pi$
$109$	1152.44	1.01269	0.506346	+	0.862330i	$0.330996\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1537.75	−1.28017	−0.640087	−	0.768302i	$-0.721102\pi$
$113$	−1537.75	−1.28017	−0.640087	+	0.768302i	$0.721102\pi$
$114$	0	0
$115$	−741.331	−0.601126
$116$	0	0
$117$	0	0
$118$	0	0
$119$	267.279	0.205894
$120$	0	0
$121$	−1030.73	−0.774405
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1950.33	1.36271	0.681353	−	0.731955i	$-0.261392\pi$
$127$	1950.33	1.36271	0.681353	+	0.731955i	$0.261392\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1219.32	0.813222	0.406611	−	0.913601i	$-0.366710\pi$
$131$	1219.32	0.813222	0.406611	+	0.913601i	$0.366710\pi$
$132$	0	0
$133$	207.353	0.135186
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2632.77	−1.64185	−0.820924	−	0.571038i	$-0.806541\pi$
$137$	−2632.77	−1.64185	−0.820924	+	0.571038i	$0.806541\pi$
$138$	0	0
$139$	943.117	0.575498	0.287749	−	0.957706i	$-0.407093\pi$
$139$	943.117	0.575498	0.287749	+	0.957706i	$0.407093\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1235.40	−0.722441
$144$	0	0
$145$	355.591	0.203657
$146$	0	0
$147$	0	0
$148$	0	0
$149$	833.371	0.458204	0.229102	−	0.973402i	$-0.426421\pi$
$149$	833.371	0.458204	0.229102	+	0.973402i	$0.426421\pi$
$150$	0	0
$151$	2413.13	1.30052	0.650258	−	0.759714i	$-0.274661\pi$
$151$	2413.13	1.30052	0.650258	+	0.759714i	$0.274661\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1060.19	−0.549396
$156$	0	0
$157$	1085.41	0.551755	0.275877	−	0.961193i	$-0.411032\pi$
$157$	1085.41	0.551755	0.275877	+	0.961193i	$0.411032\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−356.488	−0.174504
$162$	0	0
$163$	3657.89	1.75772	0.878859	−	0.477082i	$-0.158306\pi$
$163$	3657.89	1.75772	0.878859	+	0.477082i	$0.158306\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1506.46	0.698043	0.349022	−	0.937115i	$-0.386514\pi$
$167$	1506.46	0.698043	0.349022	+	0.937115i	$0.386514\pi$
$168$	0	0
$169$	2885.84	1.31353
$170$	0	0
$171$	0	0
$172$	0	0
$173$	83.8876	0.0368662	0.0184331	−	0.999830i	$-0.494132\pi$
$173$	83.8876	0.0368662	0.0184331	+	0.999830i	$0.494132\pi$
$174$	0	0
$175$	−60.1094	−0.0259648
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4698.85	1.96206	0.981030	−	0.193855i	$-0.0620990\pi$
$179$	4698.85	1.96206	0.981030	+	0.193855i	$0.0620990\pi$
$180$	0	0
$181$	−2160.95	−0.887413	−0.443707	−	0.896172i	$-0.646337\pi$
$181$	−2160.95	−0.887413	−0.443707	+	0.896172i	$0.646337\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	664.900	0.264240
$186$	0	0
$187$	1926.26	0.753275
$188$	0	0
$189$	0	0
$190$	0	0
$191$	160.758	0.0609009	0.0304504	−	0.999536i	$-0.490306\pi$
$191$	160.758	0.0609009	0.0304504	+	0.999536i	$0.490306\pi$
$192$	0	0
$193$	3743.78	1.39628	0.698142	−	0.715959i	$-0.254010\pi$
$193$	3743.78	1.39628	0.698142	+	0.715959i	$0.254010\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3812.48	−1.37882	−0.689411	−	0.724371i	$-0.742130\pi$
$197$	−3812.48	−1.37882	−0.689411	+	0.724371i	$0.742130\pi$
$198$	0	0
$199$	−52.1965	−0.0185935	−0.00929676	−	0.999957i	$-0.502959\pi$
$199$	−52.1965	−0.0185935	−0.00929676	+	0.999957i	$0.502959\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	170.995	0.0591206
$204$	0	0
$205$	−148.415	−0.0505646
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1494.38	0.494585
$210$	0	0
$211$	2781.76	0.907603	0.453802	−	0.891103i	$-0.350067\pi$
$211$	2781.76	0.907603	0.453802	+	0.891103i	$0.350067\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	634.395	0.201234
$216$	0	0
$217$	−509.819	−0.159487
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−7925.29	−2.41228
$222$	0	0
$223$	1310.27	0.393462	0.196731	−	0.980458i	$-0.436968\pi$
$223$	1310.27	0.393462	0.196731	+	0.980458i	$0.436968\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3413.82	−0.998163	−0.499082	−	0.866555i	$-0.666329\pi$
$227$	−3413.82	−0.998163	−0.499082	+	0.866555i	$0.666329\pi$
$228$	0	0
$229$	−4317.15	−1.24579	−0.622894	−	0.782306i	$-0.714043\pi$
$229$	−4317.15	−1.24579	−0.622894	+	0.782306i	$0.714043\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2244.87	−0.631185	−0.315593	−	0.948895i	$-0.602203\pi$
$233$	−2244.87	−0.631185	−0.315593	+	0.948895i	$0.602203\pi$
$234$	0	0
$235$	−516.587	−0.143398
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−924.054	−0.250092	−0.125046	−	0.992151i	$-0.539908\pi$
$239$	−924.054	−0.250092	−0.125046	+	0.992151i	$0.539908\pi$
$240$	0	0
$241$	5505.32	1.47149	0.735744	−	0.677260i	$-0.236833\pi$
$241$	5505.32	1.47149	0.735744	+	0.677260i	$0.236833\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1686.09	0.439676
$246$	0	0
$247$	−6148.37	−1.58385
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1654.48	0.416056	0.208028	−	0.978123i	$-0.433295\pi$
$251$	1654.48	0.416056	0.208028	+	0.978123i	$0.433295\pi$
$252$	0	0
$253$	−2569.19	−0.638433
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5099.06	1.23763	0.618814	−	0.785538i	$-0.287613\pi$
$257$	5099.06	1.23763	0.618814	+	0.785538i	$0.287613\pi$
$258$	0	0
$259$	319.734	0.0767078
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4300.91	1.00838	0.504192	−	0.863591i	$-0.331790\pi$
$263$	4300.91	1.00838	0.504192	+	0.863591i	$0.331790\pi$
$264$	0	0
$265$	−1492.32	−0.345933
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5589.55	−1.26692	−0.633459	−	0.773776i	$-0.718366\pi$
$269$	−5589.55	−1.26692	−0.633459	+	0.773776i	$0.718366\pi$
$270$	0	0
$271$	3435.85	0.770158	0.385079	−	0.922884i	$-0.374174\pi$
$271$	3435.85	0.770158	0.385079	+	0.922884i	$0.374174\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−433.205	−0.0949936
$276$	0	0
$277$	4652.36	1.00914	0.504572	−	0.863369i	$-0.331650\pi$
$277$	4652.36	1.00914	0.504572	+	0.863369i	$0.331650\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3173.95	0.673814	0.336907	−	0.941538i	$-0.390619\pi$
$281$	3173.95	0.673814	0.336907	+	0.941538i	$0.390619\pi$
$282$	0	0
$283$	−4344.90	−0.912641	−0.456321	−	0.889815i	$-0.650833\pi$
$283$	−4344.90	−0.912641	−0.456321	+	0.889815i	$0.650833\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−71.3691	−0.0146787
$288$	0	0
$289$	7444.33	1.51523
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3010.86	0.600329	0.300165	−	0.953887i	$-0.402958\pi$
$293$	3010.86	0.600329	0.300165	+	0.953887i	$0.402958\pi$
$294$	0	0
$295$	4468.37	0.881894
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10570.5	2.04451
$300$	0	0
$301$	305.065	0.0584174
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3763.60	−0.706567
$306$	0	0
$307$	−1083.82	−0.201487	−0.100744	−	0.994912i	$-0.532122\pi$
$307$	−1083.82	−0.201487	−0.100744	+	0.994912i	$0.532122\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7202.17	1.31318	0.656588	−	0.754249i	$-0.271999\pi$
$311$	7202.17	1.31318	0.656588	+	0.754249i	$0.271999\pi$
$312$	0	0
$313$	55.5588	0.0100331	0.00501656	−	0.999987i	$-0.498403\pi$
$313$	55.5588	0.0100331	0.00501656	+	0.999987i	$0.498403\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6762.76	−1.19822	−0.599108	−	0.800668i	$-0.704478\pi$
$317$	−6762.76	−1.19822	−0.599108	+	0.800668i	$0.704478\pi$
$318$	0	0
$319$	1232.35	0.216296
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9586.71	1.65145
$324$	0	0
$325$	1782.35	0.304206
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−248.414	−0.0416277
$330$	0	0
$331$	−4689.11	−0.778660	−0.389330	−	0.921098i	$-0.627293\pi$
$331$	−4689.11	−0.778660	−0.389330	+	0.921098i	$0.627293\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2385.84	−0.389111
$336$	0	0
$337$	−11110.1	−1.79587	−0.897935	−	0.440128i	$-0.854933\pi$
$337$	−11110.1	−1.79587	−0.897935	+	0.440128i	$0.854933\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3674.23	−0.583492
$342$	0	0
$343$	1635.50	0.257460
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8190.20	−1.26707	−0.633535	−	0.773714i	$-0.718397\pi$
$347$	−8190.20	−1.26707	−0.633535	+	0.773714i	$0.718397\pi$
$348$	0	0
$349$	−2804.03	−0.430075	−0.215037	−	0.976606i	$-0.568987\pi$
$349$	−2804.03	−0.430075	−0.215037	+	0.976606i	$0.568987\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5285.46	−0.796931	−0.398466	−	0.917183i	$-0.630457\pi$
$353$	−5285.46	−0.796931	−0.398466	+	0.917183i	$0.630457\pi$
$354$	0	0
$355$	−293.669	−0.0439052
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3078.37	0.452563	0.226281	−	0.974062i	$-0.427343\pi$
$359$	3078.37	0.452563	0.226281	+	0.974062i	$0.427343\pi$
$360$	0	0
$361$	578.288	0.0843109
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4369.26	−0.626568
$366$	0	0
$367$	−81.5000	−0.0115920	−0.00579600	−	0.999983i	$-0.501845\pi$
$367$	−81.5000	−0.0115920	−0.00579600	+	0.999983i	$0.501845\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−717.619	−0.100423
$372$	0	0
$373$	−11888.6	−1.65032	−0.825160	−	0.564899i	$-0.808915\pi$
$373$	−11888.6	−1.65032	−0.825160	+	0.564899i	$0.808915\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5070.30	−0.692662
$378$	0	0
$379$	−10663.7	−1.44527	−0.722636	−	0.691229i	$-0.757070\pi$
$379$	−10663.7	−1.44527	−0.722636	+	0.691229i	$0.757070\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6178.97	0.824362	0.412181	−	0.911102i	$-0.364767\pi$
$383$	6178.97	0.824362	0.412181	+	0.911102i	$0.364767\pi$
$384$	0	0
$385$	−208.318	−0.0275762
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7948.36	−1.03598	−0.517992	−	0.855386i	$-0.673320\pi$
$389$	−7948.36	−1.03598	−0.517992	+	0.855386i	$0.673320\pi$
$390$	0	0
$391$	−16481.8	−2.13177
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4098.84	−0.522114
$396$	0	0
$397$	11191.2	1.41478	0.707392	−	0.706822i	$-0.249872\pi$
$397$	11191.2	1.41478	0.707392	+	0.706822i	$0.249872\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9357.36	1.16530	0.582649	−	0.812724i	$-0.302016\pi$
$401$	9357.36	1.16530	0.582649	+	0.812724i	$0.302016\pi$
$402$	0	0
$403$	15117.0	1.86857
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2304.31	0.280639
$408$	0	0
$409$	−3229.59	−0.390447	−0.195224	−	0.980759i	$-0.562543\pi$
$409$	−3229.59	−0.390447	−0.195224	+	0.980759i	$0.562543\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2148.73	0.256010
$414$	0	0
$415$	−4796.96	−0.567406
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4305.10	0.501951	0.250976	−	0.967993i	$-0.419249\pi$
$419$	4305.10	0.501951	0.250976	+	0.967993i	$0.419249\pi$
$420$	0	0
$421$	−4330.36	−0.501304	−0.250652	−	0.968077i	$-0.580645\pi$
$421$	−4330.36	−0.501304	−0.250652	+	0.968077i	$0.580645\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2779.09	−0.317190
$426$	0	0
$427$	−1809.82	−0.205113
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2774.39	0.310064	0.155032	−	0.987909i	$-0.450452\pi$
$431$	2774.39	0.310064	0.155032	+	0.987909i	$0.450452\pi$
$432$	0	0
$433$	−16958.3	−1.88213	−0.941065	−	0.338225i	$-0.890173\pi$
$433$	−16958.3	−1.88213	−0.941065	+	0.338225i	$0.890173\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−12786.4	−1.39968
$438$	0	0
$439$	1334.39	0.145072	0.0725362	−	0.997366i	$-0.476891\pi$
$439$	1334.39	0.145072	0.0725362	+	0.997366i	$0.476891\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1990.19	−0.213447	−0.106723	−	0.994289i	$-0.534036\pi$
$443$	−1990.19	−0.213447	−0.106723	+	0.994289i	$0.534036\pi$
$444$	0	0
$445$	−2033.67	−0.216641
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5858.34	0.615751	0.307876	−	0.951427i	$-0.400382\pi$
$449$	5858.34	0.615751	0.307876	+	0.951427i	$0.400382\pi$
$450$	0	0
$451$	−514.353	−0.0537027
$452$	0	0
$453$	0	0
$454$	0	0
$455$	857.088	0.0883097
$456$	0	0
$457$	1838.53	0.188190	0.0940952	−	0.995563i	$-0.470004\pi$
$457$	1838.53	0.188190	0.0940952	+	0.995563i	$0.470004\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2805.80	−0.283469	−0.141735	−	0.989905i	$-0.545268\pi$
$461$	−2805.80	−0.283469	−0.141735	+	0.989905i	$0.545268\pi$
$462$	0	0
$463$	−9647.34	−0.968359	−0.484179	−	0.874969i	$-0.660882\pi$
$463$	−9647.34	−0.968359	−0.484179	+	0.874969i	$0.660882\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8339.39	0.826340	0.413170	−	0.910654i	$-0.364421\pi$
$467$	8339.39	0.826340	0.413170	+	0.910654i	$0.364421\pi$
$468$	0	0
$469$	−1147.29	−0.112957
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2198.58	0.213723
$474$	0	0
$475$	−2155.99	−0.208260
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2842.91	0.271181	0.135591	−	0.990765i	$-0.456707\pi$
$479$	2842.91	0.271181	0.135591	+	0.990765i	$0.456707\pi$
$480$	0	0
$481$	−9480.68	−0.898715
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3118.74	−0.291989
$486$	0	0
$487$	55.8309	0.00519495	0.00259747	−	0.999997i	$-0.499173\pi$
$487$	55.8309	0.00519495	0.00259747	+	0.999997i	$0.499173\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−10131.2	−0.931188	−0.465594	−	0.884998i	$-0.654159\pi$
$491$	−10131.2	−0.931188	−0.465594	+	0.884998i	$0.654159\pi$
$492$	0	0
$493$	7905.75	0.722225
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−141.218	−0.0127455
$498$	0	0
$499$	−18760.6	−1.68305	−0.841524	−	0.540219i	$-0.818341\pi$
$499$	−18760.6	−1.68305	−0.841524	+	0.540219i	$0.818341\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1263.21	−0.111975	−0.0559877	−	0.998431i	$-0.517831\pi$
$503$	−1263.21	−0.111975	−0.0559877	+	0.998431i	$0.517831\pi$
$504$	0	0
$505$	−2341.01	−0.206284
$506$	0	0
$507$	0	0
$508$	0	0
$509$	20105.1	1.75077	0.875386	−	0.483424i	$-0.160607\pi$
$509$	20105.1	1.75077	0.875386	+	0.483424i	$0.160607\pi$
$510$	0	0
$511$	−2101.07	−0.181890
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4745.48	0.406041
$516$	0	0
$517$	−1790.31	−0.152297
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1644.53	−0.138288	−0.0691440	−	0.997607i	$-0.522027\pi$
$521$	−1644.53	−0.138288	−0.0691440	+	0.997607i	$0.522027\pi$
$522$	0	0
$523$	4830.92	0.403903	0.201952	−	0.979395i	$-0.435272\pi$
$523$	4830.92	0.403903	0.201952	+	0.979395i	$0.435272\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−23570.9	−1.94832
$528$	0	0
$529$	9815.89	0.806764
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2116.22	0.171977
$534$	0	0
$535$	−3245.43	−0.262266
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5843.40	0.466963
$540$	0	0
$541$	159.541	0.0126787	0.00633936	−	0.999980i	$-0.497982\pi$
$541$	159.541	0.0126787	0.00633936	+	0.999980i	$0.497982\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5762.18	−0.452890
$546$	0	0
$547$	3217.83	0.251526	0.125763	−	0.992060i	$-0.459862\pi$
$547$	3217.83	0.251526	0.125763	+	0.992060i	$0.459862\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6133.21	0.474199
$552$	0	0
$553$	−1971.03	−0.151567
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−21394.0	−1.62745	−0.813727	−	0.581247i	$-0.802565\pi$
$557$	−21394.0	−1.62745	−0.813727	+	0.581247i	$0.802565\pi$
$558$	0	0
$559$	−9045.71	−0.684423
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−3427.87	−0.256603	−0.128302	−	0.991735i	$-0.540953\pi$
$563$	−3427.87	−0.256603	−0.128302	+	0.991735i	$0.540953\pi$
$564$	0	0
$565$	7688.77	0.572511
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12420.8	0.915128	0.457564	−	0.889177i	$-0.348722\pi$
$569$	12420.8	0.915128	0.457564	+	0.889177i	$0.348722\pi$
$570$	0	0
$571$	−3954.37	−0.289816	−0.144908	−	0.989445i	$-0.546289\pi$
$571$	−3954.37	−0.289816	−0.144908	+	0.989445i	$0.546289\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3706.66	0.268832
$576$	0	0
$577$	−24982.7	−1.80250	−0.901250	−	0.433299i	$-0.857349\pi$
$577$	−24982.7	−1.80250	−0.901250	+	0.433299i	$0.857349\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2306.74	−0.164715
$582$	0	0
$583$	−5171.84	−0.367402
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7981.44	−0.561208	−0.280604	−	0.959824i	$-0.590535\pi$
$587$	−7981.44	−0.561208	−0.280604	+	0.959824i	$0.590535\pi$
$588$	0	0
$589$	−18286.1	−1.27923
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2465.27	−0.170719	−0.0853597	−	0.996350i	$-0.527204\pi$
$593$	−2465.27	−0.170719	−0.0853597	+	0.996350i	$0.527204\pi$
$594$	0	0
$595$	−1336.40	−0.0920788
$596$	0	0
$597$	0	0
$598$	0	0
$599$	21413.6	1.46066	0.730331	−	0.683093i	$-0.239366\pi$
$599$	21413.6	1.46066	0.730331	+	0.683093i	$0.239366\pi$
$600$	0	0
$601$	−307.813	−0.0208918	−0.0104459	−	0.999945i	$-0.503325\pi$
$601$	−307.813	−0.0208918	−0.0104459	+	0.999945i	$0.503325\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5153.67	0.346325
$606$	0	0
$607$	3953.10	0.264335	0.132168	−	0.991227i	$-0.457806\pi$
$607$	3953.10	0.264335	0.132168	+	0.991227i	$0.457806\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7365.92	0.487714
$612$	0	0
$613$	−8356.90	−0.550623	−0.275311	−	0.961355i	$-0.588781\pi$
$613$	−8356.90	−0.550623	−0.275311	+	0.961355i	$0.588781\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−25301.3	−1.65088	−0.825439	−	0.564491i	$-0.809072\pi$
$617$	−25301.3	−1.65088	−0.825439	+	0.564491i	$0.809072\pi$
$618$	0	0
$619$	1542.86	0.100182	0.0500912	−	0.998745i	$-0.484049\pi$
$619$	1542.86	0.100182	0.0500912	+	0.998745i	$0.484049\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−977.944	−0.0628900
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14782.5	0.937072
$630$	0	0
$631$	12888.9	0.813152	0.406576	−	0.913617i	$-0.366723\pi$
$631$	12888.9	0.813152	0.406576	+	0.913617i	$0.366723\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−9751.64	−0.609420
$636$	0	0
$637$	−24041.7	−1.49539
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4590.77	0.282878	0.141439	−	0.989947i	$-0.454827\pi$
$641$	4590.77	0.282878	0.141439	+	0.989947i	$0.454827\pi$
$642$	0	0
$643$	−12085.3	−0.741209	−0.370605	−	0.928791i	$-0.620850\pi$
$643$	−12085.3	−0.741209	−0.370605	+	0.928791i	$0.620850\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6233.43	0.378766	0.189383	−	0.981903i	$-0.439351\pi$
$647$	6233.43	0.378766	0.189383	+	0.981903i	$0.439351\pi$
$648$	0	0
$649$	15485.8	0.936625
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24307.8	1.45672	0.728361	−	0.685194i	$-0.240282\pi$
$653$	24307.8	1.45672	0.728361	+	0.685194i	$0.240282\pi$
$654$	0	0
$655$	−6096.58	−0.363684
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1445.94	−0.0854719	−0.0427359	−	0.999086i	$-0.513607\pi$
$659$	−1445.94	−0.0854719	−0.0427359	+	0.999086i	$0.513607\pi$
$660$	0	0
$661$	−11159.3	−0.656654	−0.328327	−	0.944564i	$-0.606485\pi$
$661$	−11159.3	−0.656654	−0.328327	+	0.944564i	$0.606485\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1036.76	−0.0604571
$666$	0	0
$667$	−10544.4	−0.612117
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−13043.3	−0.750418
$672$	0	0
$673$	−4473.63	−0.256234	−0.128117	−	0.991759i	$-0.540893\pi$
$673$	−4473.63	−0.256234	−0.128117	+	0.991759i	$0.540893\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−19997.2	−1.13523	−0.567617	−	0.823292i	$-0.692135\pi$
$677$	−19997.2	−1.13523	−0.567617	+	0.823292i	$0.692135\pi$
$678$	0	0
$679$	−1499.73	−0.0847632
$680$	0	0
$681$	0	0
$682$	0	0
$683$	21261.5	1.19114	0.595571	−	0.803303i	$-0.296926\pi$
$683$	21261.5	1.19114	0.595571	+	0.803303i	$0.296926\pi$
$684$	0	0
$685$	13163.9	0.734256
$686$	0	0
$687$	0	0
$688$	0	0
$689$	21278.7	1.17656
$690$	0	0
$691$	31303.9	1.72338	0.861691	−	0.507434i	$-0.169406\pi$
$691$	31303.9	1.72338	0.861691	+	0.507434i	$0.169406\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4715.59	−0.257370
$696$	0	0
$697$	−3299.67	−0.179317
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−30288.1	−1.63190	−0.815952	−	0.578120i	$-0.803786\pi$
$701$	−30288.1	−1.63190	−0.815952	+	0.578120i	$0.803786\pi$
$702$	0	0
$703$	11468.2	0.615263
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1125.73	−0.0598835
$708$	0	0
$709$	26181.8	1.38685	0.693426	−	0.720528i	$-0.256101\pi$
$709$	26181.8	1.38685	0.693426	+	0.720528i	$0.256101\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	31438.1	1.65128
$714$	0	0
$715$	6176.98	0.323085
$716$	0	0
$717$	0	0
$718$	0	0
$719$	32762.6	1.69936	0.849679	−	0.527301i	$-0.176796\pi$
$719$	32762.6	1.69936	0.849679	+	0.527301i	$0.176796\pi$
$720$	0	0
$721$	2281.99	0.117872
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1777.95	−0.0910780
$726$	0	0
$727$	19439.8	0.991721	0.495861	−	0.868402i	$-0.334853\pi$
$727$	19439.8	0.991721	0.495861	+	0.868402i	$0.334853\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	14104.3	0.713634
$732$	0	0
$733$	−16886.4	−0.850906	−0.425453	−	0.904981i	$-0.639885\pi$
$733$	−16886.4	−0.850906	−0.425453	+	0.904981i	$0.639885\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8268.45	−0.413259
$738$	0	0
$739$	9048.46	0.450410	0.225205	−	0.974311i	$-0.427695\pi$
$739$	9048.46	0.450410	0.225205	+	0.974311i	$0.427695\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−22272.8	−1.09975	−0.549873	−	0.835249i	$-0.685324\pi$
$743$	−22272.8	−1.09975	−0.549873	+	0.835249i	$0.685324\pi$
$744$	0	0
$745$	−4166.85	−0.204915
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1560.65	−0.0761346
$750$	0	0
$751$	2223.89	0.108057	0.0540284	−	0.998539i	$-0.482794\pi$
$751$	2223.89	0.108057	0.0540284	+	0.998539i	$0.482794\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−12065.7	−0.581608
$756$	0	0
$757$	21253.7	1.02045	0.510225	−	0.860041i	$-0.329562\pi$
$757$	21253.7	1.02045	0.510225	+	0.860041i	$0.329562\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	36585.2	1.74272	0.871362	−	0.490640i	$-0.163237\pi$
$761$	36585.2	1.74272	0.871362	+	0.490640i	$0.163237\pi$
$762$	0	0
$763$	−2770.89	−0.131472
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−63713.7	−2.99944
$768$	0	0
$769$	14952.5	0.701171	0.350585	−	0.936531i	$-0.385983\pi$
$769$	14952.5	0.701171	0.350585	+	0.936531i	$0.385983\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−41504.0	−1.93117	−0.965586	−	0.260082i	$-0.916250\pi$
$773$	−41504.0	−1.93117	−0.965586	+	0.260082i	$0.916250\pi$
$774$	0	0
$775$	5300.94	0.245697
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2559.85	−0.117736
$780$	0	0
$781$	−1017.75	−0.0466300
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5427.07	−0.246752
$786$	0	0
$787$	−34867.0	−1.57926	−0.789628	−	0.613585i	$-0.789727\pi$
$787$	−34867.0	−1.57926	−0.789628	+	0.613585i	$0.789727\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3697.34	0.166198
$792$	0	0
$793$	53664.4	2.40313
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14744.4	0.655299	0.327649	−	0.944799i	$-0.393743\pi$
$797$	14744.4	0.655299	0.327649	+	0.944799i	$0.393743\pi$
$798$	0	0
$799$	−11485.1	−0.508529
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−15142.3	−0.665454
$804$	0	0
$805$	1782.44	0.0780407
$806$	0	0
$807$	0	0
$808$	0	0
$809$	26179.5	1.13773	0.568864	−	0.822431i	$-0.307383\pi$
$809$	26179.5	1.13773	0.568864	+	0.822431i	$0.307383\pi$
$810$	0	0
$811$	−1162.53	−0.0503354	−0.0251677	−	0.999683i	$-0.508012\pi$
$811$	−1162.53	−0.0503354	−0.0251677	+	0.999683i	$0.508012\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−18289.4	−0.786075
$816$	0	0
$817$	10942.0	0.468558
$818$	0	0
$819$	0	0
$820$	0	0
$821$	35989.7	1.52990	0.764951	−	0.644088i	$-0.222763\pi$
$821$	35989.7	1.52990	0.764951	+	0.644088i	$0.222763\pi$
$822$	0	0
$823$	36003.6	1.52492	0.762459	−	0.647036i	$-0.223992\pi$
$823$	36003.6	1.52492	0.762459	+	0.647036i	$0.223992\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−22096.7	−0.929114	−0.464557	−	0.885543i	$-0.653786\pi$
$827$	−22096.7	−0.929114	−0.464557	+	0.885543i	$0.653786\pi$
$828$	0	0
$829$	14.9350	0.000625710 0	0.000312855	−	1.00000i	$-0.499900\pi$
$829$	14.9350	0.000625710 0	0.000312855		1.00000i	$0.499900\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	37486.5	1.55922
$834$	0	0
$835$	−7532.29	−0.312174
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−41995.0	−1.72804	−0.864022	−	0.503453i	$-0.832063\pi$
$839$	−41995.0	−1.72804	−0.864022	+	0.503453i	$0.832063\pi$
$840$	0	0
$841$	−19331.2	−0.792620
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−14429.2	−0.587430
$846$	0	0
$847$	2478.27	0.100537
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−19716.5	−0.794209
$852$	0	0
$853$	9809.11	0.393737	0.196868	−	0.980430i	$-0.436923\pi$
$853$	9809.11	0.393737	0.196868	+	0.980430i	$0.436923\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31063.8	1.23818	0.619089	−	0.785321i	$-0.287502\pi$
$857$	31063.8	1.23818	0.619089	+	0.785321i	$0.287502\pi$
$858$	0	0
$859$	−5957.06	−0.236615	−0.118307	−	0.992977i	$-0.537747\pi$
$859$	−5957.06	−0.236615	−0.118307	+	0.992977i	$0.537747\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31548.8	1.24442	0.622210	−	0.782850i	$-0.286235\pi$
$863$	31548.8	1.24442	0.622210	+	0.782850i	$0.286235\pi$
$864$	0	0
$865$	−419.438	−0.0164871
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−14205.1	−0.554517
$870$	0	0
$871$	34019.2	1.32342
$872$	0	0
$873$	0	0
$874$	0	0
$875$	300.547	0.0116118
$876$	0	0
$877$	44007.7	1.69445	0.847226	−	0.531233i	$-0.178271\pi$
$877$	44007.7	1.69445	0.847226	+	0.531233i	$0.178271\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−8635.87	−0.330250	−0.165125	−	0.986273i	$-0.552803\pi$
$881$	−8635.87	−0.330250	−0.165125	+	0.986273i	$0.552803\pi$
$882$	0	0
$883$	−25587.5	−0.975184	−0.487592	−	0.873072i	$-0.662125\pi$
$883$	−25587.5	−0.975184	−0.487592	+	0.873072i	$0.662125\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−22534.0	−0.853007	−0.426503	−	0.904486i	$-0.640255\pi$
$887$	−22534.0	−0.853007	−0.426503	+	0.904486i	$0.640255\pi$
$888$	0	0
$889$	−4689.32	−0.176912
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8910.07	−0.333890
$894$	0	0
$895$	−23494.3	−0.877460
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−15079.7	−0.559441
$900$	0	0
$901$	−33178.3	−1.22678
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10804.7	0.396863
$906$	0	0
$907$	44792.3	1.63980	0.819902	−	0.572504i	$-0.194028\pi$
$907$	44792.3	1.63980	0.819902	+	0.572504i	$0.194028\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	44378.6	1.61397	0.806986	−	0.590571i	$-0.201097\pi$
$911$	44378.6	1.61397	0.806986	+	0.590571i	$0.201097\pi$
$912$	0	0
$913$	−16624.5	−0.602620
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2931.69	−0.105576
$918$	0	0
$919$	−19542.4	−0.701462	−0.350731	−	0.936476i	$-0.614067\pi$
$919$	−19542.4	−0.701462	−0.350731	+	0.936476i	$0.614067\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4187.37	0.149327
$924$	0	0
$925$	−3324.50	−0.118172
$926$	0	0
$927$	0	0
$928$	0	0
$929$	23231.5	0.820453	0.410227	−	0.911984i	$-0.365450\pi$
$929$	23231.5	0.820453	0.410227	+	0.911984i	$0.365450\pi$
$930$	0	0
$931$	29081.7	1.02375
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9631.32	−0.336875
$936$	0	0
$937$	−14026.4	−0.489032	−0.244516	−	0.969645i	$-0.578629\pi$
$937$	−14026.4	−0.489032	−0.244516	+	0.969645i	$0.578629\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	986.880	0.0341885	0.0170943	−	0.999854i	$-0.494558\pi$
$941$	986.880	0.0341885	0.0170943	+	0.999854i	$0.494558\pi$
$942$	0	0
$943$	4400.99	0.151979
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−34679.9	−1.19002	−0.595008	−	0.803720i	$-0.702851\pi$
$947$	−34679.9	−1.19002	−0.595008	+	0.803720i	$0.702851\pi$
$948$	0	0
$949$	62300.4	2.13104
$950$	0	0
$951$	0	0
$952$	0	0
$953$	52384.1	1.78058	0.890288	−	0.455398i	$-0.150503\pi$
$953$	52384.1	1.78058	0.890288	+	0.455398i	$0.150503\pi$
$954$	0	0
$955$	−803.792	−0.0272357
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6330.18	0.213151
$960$	0	0
$961$	15169.0	0.509181
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−18718.9	−0.624437
$966$	0	0
$967$	29842.4	0.992417	0.496208	−	0.868203i	$-0.334725\pi$
$967$	29842.4	0.992417	0.496208	+	0.868203i	$0.334725\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−21134.5	−0.698494	−0.349247	−	0.937031i	$-0.613563\pi$
$971$	−21134.5	−0.698494	−0.349247	+	0.937031i	$0.613563\pi$
$972$	0	0
$973$	−2267.61	−0.0747135
$974$	0	0
$975$	0	0
$976$	0	0
$977$	11583.6	0.379318	0.189659	−	0.981850i	$-0.439262\pi$
$977$	11583.6	0.379318	0.189659	+	0.981850i	$0.439262\pi$
$978$	0	0
$979$	−7047.98	−0.230086
$980$	0	0
$981$	0	0
$982$	0	0
$983$	54382.8	1.76454	0.882269	−	0.470745i	$-0.156015\pi$
$983$	54382.8	1.76454	0.882269	+	0.470745i	$0.156015\pi$
$984$	0	0
$985$	19062.4	0.616628
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−18811.9	−0.604836
$990$	0	0
$991$	8319.58	0.266680	0.133340	−	0.991070i	$-0.457430\pi$
$991$	8319.58	0.266680	0.133340	+	0.991070i	$0.457430\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	260.982	0.00831527
$996$	0	0
$997$	33697.8	1.07043	0.535216	−	0.844715i	$-0.320230\pi$
$997$	33697.8	1.07043	0.535216	+	0.844715i	$0.320230\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.4.a.o.1.2	✓	4
3.2	odd	2		1080.4.a.p.1.2	yes	4
4.3	odd	2		2160.4.a.bu.1.3		4
12.11	even	2		2160.4.a.bv.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.o.1.2	✓	4	1.1	even	1	trivial
1080.4.a.p.1.2	yes	4	3.2	odd	2
2160.4.a.bu.1.3		4	4.3	odd	2
2160.4.a.bv.1.3		4	12.11	even	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:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1080.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} -2.40438 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} -2.40438 q^{7} -17.3282 q^{11} +71.2940 q^{13} -111.164 q^{17} -86.2397 q^{19} +148.266 q^{23} +25.0000 q^{25} -71.1182 q^{29} +212.038 q^{31} +12.0219 q^{35} -132.980 q^{37} +29.6830 q^{41} -126.879 q^{43} +103.317 q^{47} -337.219 q^{49} +298.464 q^{53} +86.6410 q^{55} -893.675 q^{59} +752.720 q^{61} -356.470 q^{65} +477.167 q^{67} +58.7339 q^{71} +873.851 q^{73} +41.6635 q^{77} +819.768 q^{79} +959.392 q^{83} +555.818 q^{85} +406.735 q^{89} -171.418 q^{91} +431.199 q^{95} +623.748 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{5} + 14 q^{7}+O(q^{10}) \) 4 * q - 20 * q^5 + 14 * q^7	\( 4 q - 20 q^{5} + 14 q^{7} + 4 q^{11} + 30 q^{13} + 28 q^{17} + 78 q^{19} - 182 q^{23} + 100 q^{25} - 202 q^{29} - 76 q^{31} - 70 q^{35} + 302 q^{37} - 380 q^{41} + 178 q^{43} - 114 q^{47} + 958 q^{49} + 256 q^{53} - 20 q^{55} + 204 q^{59} + 766 q^{61} - 150 q^{65} + 330 q^{67} + 1060 q^{71} + 1442 q^{73} - 216 q^{77} + 742 q^{79} + 768 q^{83} - 140 q^{85} + 400 q^{89} + 3066 q^{91} - 390 q^{95} + 3338 q^{97}+O(q^{100}) \) 4 * q - 20 * q^5 + 14 * q^7 + 4 * q^11 + 30 * q^13 + 28 * q^17 + 78 * q^19 - 182 * q^23 + 100 * q^25 - 202 * q^29 - 76 * q^31 - 70 * q^35 + 302 * q^37 - 380 * q^41 + 178 * q^43 - 114 * q^47 + 958 * q^49 + 256 * q^53 - 20 * q^55 + 204 * q^59 + 766 * q^61 - 150 * q^65 + 330 * q^67 + 1060 * q^71 + 1442 * q^73 - 216 * q^77 + 742 * q^79 + 768 * q^83 - 140 * q^85 + 400 * q^89 + 3066 * q^91 - 390 * q^95 + 3338 * q^97

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