LMFDB - Embedded newform 1080.4.a.p.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	$-0.423677\pi$
$11$	17.3282	0.474968	0.237484	+	0.971391i	$0.423677\pi$
$12$	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	$-0.208532\pi$
$17$	111.164	1.58595	0.792974	+	0.609255i	$0.208532\pi$
$18$	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	$-0.734599\pi$
$23$	−148.266	−1.34416	−0.672080	+	0.740479i	$0.734599\pi$
$24$	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	$-0.426881\pi$
$29$	71.1182	0.455390	0.227695	+	0.973732i	$0.426881\pi$
$30$	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	$-0.518005\pi$
$41$	−29.6830	−0.113066	−0.0565330	+	0.998401i	$0.518005\pi$
$42$	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	$-0.551254\pi$
$47$	−103.317	−0.320647	−0.160323	+	0.987065i	$0.551254\pi$
$48$	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	$-0.626408\pi$
$53$	−298.464	−0.773531	−0.386765	+	0.922178i	$0.626408\pi$
$54$	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	$-0.0533490\pi$
$59$	893.675	1.97198	0.985988	+	0.166817i	$0.0533490\pi$
$60$	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	$-0.515631\pi$
$71$	−58.7339	−0.0981751	−0.0490875	+	0.998794i	$0.515631\pi$
$72$	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	$-0.718744\pi$
$83$	−959.392	−1.26876	−0.634379	+	0.773022i	$0.718744\pi$
$84$	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	$-0.577873\pi$
$89$	−406.735	−0.484425	−0.242212	+	0.970223i	$0.577873\pi$
$90$	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	$-0.574080\pi$
$101$	−468.202	−0.461266	−0.230633	+	0.973041i	$0.574080\pi$
$102$	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	$-0.594727\pi$
$107$	−649.086	−0.586444	−0.293222	+	0.956044i	$0.594727\pi$
$108$	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	$-0.278898\pi$
$113$	1537.75	1.28017	0.640087	+	0.768302i	$0.278898\pi$
$114$	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	$-0.633290\pi$
$131$	−1219.32	−0.813222	−0.406611	+	0.913601i	$0.633290\pi$
$132$	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	$-0.193459\pi$
$137$	2632.77	1.64185	0.820924	+	0.571038i	$0.193459\pi$
$138$	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	$-0.573579\pi$
$149$	−833.371	−0.458204	−0.229102	+	0.973402i	$0.573579\pi$
$150$	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	$-0.613486\pi$
$167$	−1506.46	−0.698043	−0.349022	+	0.937115i	$0.613486\pi$
$168$	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	$-0.505868\pi$
$173$	−83.8876	−0.0368662	−0.0184331	+	0.999830i	$0.505868\pi$
$174$	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	$-0.937901\pi$
$179$	−4698.85	−1.96206	−0.981030	+	0.193855i	$0.937901\pi$
$180$	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	$-0.509694\pi$
$191$	−160.758	−0.0609009	−0.0304504	+	0.999536i	$0.509694\pi$
$192$	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	$-0.257870\pi$
$197$	3812.48	1.37882	0.689411	+	0.724371i	$0.257870\pi$
$198$	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	$-0.333671\pi$
$227$	3413.82	0.998163	0.499082	+	0.866555i	$0.333671\pi$
$228$	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	$-0.397797\pi$
$233$	2244.87	0.631185	0.315593	+	0.948895i	$0.397797\pi$
$234$	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	$-0.460092\pi$
$239$	924.054	0.250092	0.125046	+	0.992151i	$0.460092\pi$
$240$	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	$-0.566705\pi$
$251$	−1654.48	−0.416056	−0.208028	+	0.978123i	$0.566705\pi$
$252$	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	$-0.712387\pi$
$257$	−5099.06	−1.23763	−0.618814	+	0.785538i	$0.712387\pi$
$258$	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	$-0.668210\pi$
$263$	−4300.91	−1.00838	−0.504192	+	0.863591i	$0.668210\pi$
$264$	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	$-0.281634\pi$
$269$	5589.55	1.26692	0.633459	+	0.773776i	$0.281634\pi$
$270$	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	$-0.609381\pi$
$281$	−3173.95	−0.673814	−0.336907	+	0.941538i	$0.609381\pi$
$282$	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	$-0.597042\pi$
$293$	−3010.86	−0.600329	−0.300165	+	0.953887i	$0.597042\pi$
$294$	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	$-0.728001\pi$
$311$	−7202.17	−1.31318	−0.656588	+	0.754249i	$0.728001\pi$
$312$	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	$-0.295522\pi$
$317$	6762.76	1.19822	0.599108	+	0.800668i	$0.295522\pi$
$318$	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	$-0.281603\pi$
$347$	8190.20	1.26707	0.633535	+	0.773714i	$0.281603\pi$
$348$	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	$-0.369543\pi$
$353$	5285.46	0.796931	0.398466	+	0.917183i	$0.369543\pi$
$354$	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	$-0.572657\pi$
$359$	−3078.37	−0.452563	−0.226281	+	0.974062i	$0.572657\pi$
$360$	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	$-0.635233\pi$
$383$	−6178.97	−0.824362	−0.412181	+	0.911102i	$0.635233\pi$
$384$	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	$-0.326680\pi$
$389$	7948.36	1.03598	0.517992	+	0.855386i	$0.326680\pi$
$390$	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	$-0.697984\pi$
$401$	−9357.36	−1.16530	−0.582649	+	0.812724i	$0.697984\pi$
$402$	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	$-0.580751\pi$
$419$	−4305.10	−0.501951	−0.250976	+	0.967993i	$0.580751\pi$
$420$	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	$-0.549548\pi$
$431$	−2774.39	−0.310064	−0.155032	+	0.987909i	$0.549548\pi$
$432$	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	$-0.465964\pi$
$443$	1990.19	0.213447	0.106723	+	0.994289i	$0.465964\pi$
$444$	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	$-0.599618\pi$
$449$	−5858.34	−0.615751	−0.307876	+	0.951427i	$0.599618\pi$
$450$	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	$-0.454732\pi$
$461$	2805.80	0.283469	0.141735	+	0.989905i	$0.454732\pi$
$462$	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	$-0.635579\pi$
$467$	−8339.39	−0.826340	−0.413170	+	0.910654i	$0.635579\pi$
$468$	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	$-0.543293\pi$
$479$	−2842.91	−0.271181	−0.135591	+	0.990765i	$0.543293\pi$
$480$	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	$-0.345841\pi$
$491$	10131.2	0.931188	0.465594	+	0.884998i	$0.345841\pi$
$492$	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	$-0.482169\pi$
$503$	1263.21	0.111975	0.0559877	+	0.998431i	$0.482169\pi$
$504$	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	$-0.839393\pi$
$509$	−20105.1	−1.75077	−0.875386	+	0.483424i	$0.839393\pi$
$510$	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	$-0.477973\pi$
$521$	1644.53	0.138288	0.0691440	+	0.997607i	$0.477973\pi$
$522$	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	$-0.197435\pi$
$557$	21394.0	1.62745	0.813727	+	0.581247i	$0.197435\pi$
$558$	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	$-0.459047\pi$
$563$	3427.87	0.256603	0.128302	+	0.991735i	$0.459047\pi$
$564$	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	$-0.651278\pi$
$569$	−12420.8	−0.915128	−0.457564	+	0.889177i	$0.651278\pi$
$570$	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	$-0.409465\pi$
$587$	7981.44	0.561208	0.280604	+	0.959824i	$0.409465\pi$
$588$	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	$-0.472796\pi$
$593$	2465.27	0.170719	0.0853597	+	0.996350i	$0.472796\pi$
$594$	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	$-0.760634\pi$
$599$	−21413.6	−1.46066	−0.730331	+	0.683093i	$0.760634\pi$
$600$	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	$-0.190928\pi$
$617$	25301.3	1.65088	0.825439	+	0.564491i	$0.190928\pi$
$618$	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	$-0.545173\pi$
$641$	−4590.77	−0.282878	−0.141439	+	0.989947i	$0.545173\pi$
$642$	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	$-0.560649\pi$
$647$	−6233.43	−0.378766	−0.189383	+	0.981903i	$0.560649\pi$
$648$	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	$-0.759718\pi$
$653$	−24307.8	−1.45672	−0.728361	+	0.685194i	$0.759718\pi$
$654$	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	$-0.486393\pi$
$659$	1445.94	0.0854719	0.0427359	+	0.999086i	$0.486393\pi$
$660$	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	$-0.307865\pi$
$677$	19997.2	1.13523	0.567617	+	0.823292i	$0.307865\pi$
$678$	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	$-0.703074\pi$
$683$	−21261.5	−1.19114	−0.595571	+	0.803303i	$0.703074\pi$
$684$	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	$-0.196214\pi$
$701$	30288.1	1.63190	0.815952	+	0.578120i	$0.196214\pi$
$702$	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	$-0.823204\pi$
$719$	−32762.6	−1.69936	−0.849679	+	0.527301i	$0.823204\pi$
$720$	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	$-0.314676\pi$
$743$	22272.8	1.09975	0.549873	+	0.835249i	$0.314676\pi$
$744$	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	$-0.836763\pi$
$761$	−36585.2	−1.74272	−0.871362	+	0.490640i	$0.836763\pi$
$762$	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	$-0.0837497\pi$
$773$	41504.0	1.93117	0.965586	+	0.260082i	$0.0837497\pi$
$774$	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	$-0.606257\pi$
$797$	−14744.4	−0.655299	−0.327649	+	0.944799i	$0.606257\pi$
$798$	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	$-0.692617\pi$
$809$	−26179.5	−1.13773	−0.568864	+	0.822431i	$0.692617\pi$
$810$	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	$-0.777237\pi$
$821$	−35989.7	−1.52990	−0.764951	+	0.644088i	$0.777237\pi$
$822$	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	$-0.346214\pi$
$827$	22096.7	0.929114	0.464557	+	0.885543i	$0.346214\pi$
$828$	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	$-0.167937\pi$
$839$	41995.0	1.72804	0.864022	+	0.503453i	$0.167937\pi$
$840$	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	$-0.712498\pi$
$857$	−31063.8	−1.23818	−0.619089	+	0.785321i	$0.712498\pi$
$858$	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	$-0.713765\pi$
$863$	−31548.8	−1.24442	−0.622210	+	0.782850i	$0.713765\pi$
$864$	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	$-0.447197\pi$
$881$	8635.87	0.330250	0.165125	+	0.986273i	$0.447197\pi$
$882$	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	$-0.359745\pi$
$887$	22534.0	0.853007	0.426503	+	0.904486i	$0.359745\pi$
$888$	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	$-0.798903\pi$
$911$	−44378.6	−1.61397	−0.806986	+	0.590571i	$0.798903\pi$
$912$	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	$-0.634550\pi$
$929$	−23231.5	−0.820453	−0.410227	+	0.911984i	$0.634550\pi$
$930$	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	$-0.505442\pi$
$941$	−986.880	−0.0341885	−0.0170943	+	0.999854i	$0.505442\pi$
$942$	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	$-0.297149\pi$
$947$	34679.9	1.19002	0.595008	+	0.803720i	$0.297149\pi$
$948$	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	$-0.849497\pi$
$953$	−52384.1	−1.78058	−0.890288	+	0.455398i	$0.849497\pi$
$954$	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	$-0.386437\pi$
$971$	21134.5	0.698494	0.349247	+	0.937031i	$0.386437\pi$
$972$	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	$-0.560738\pi$
$977$	−11583.6	−0.379318	−0.189659	+	0.981850i	$0.560738\pi$
$978$	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	$-0.843985\pi$
$983$	−54382.8	−1.76454	−0.882269	+	0.470745i	$0.843985\pi$
$984$	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.p.1.2	yes	4
3.2	odd	2		1080.4.a.o.1.2	✓	4
4.3	odd	2		2160.4.a.bv.1.3		4
12.11	even	2		2160.4.a.bu.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.o.1.2	✓	4	3.2	odd	2
1080.4.a.p.1.2	yes	4	1.1	even	1	trivial
2160.4.a.bu.1.3		4	12.11	even	2
2160.4.a.bv.1.3		4	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 $\F_{q}$

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more