LMFDB - Embedded newform 1080.6.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,6,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, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1080.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
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:	$173.214525398$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 100x^{3} - 199x^{2} + 871x + 1455 $ x^5 - 2x^4 - 100x^3 - 199x^2 + 871x + 1455
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{8} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.57454$ 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)

$f(q)$	$=$	$q-25.0000 q^{5} -77.0356 q^{7} +O(q^{10})$	$q-25.0000 q^{5} -77.0356 q^{7} -403.188 q^{11} +858.883 q^{13} -1396.30 q^{17} +1288.71 q^{19} -794.472 q^{23} +625.000 q^{25} +3191.08 q^{29} +6561.95 q^{31} +1925.89 q^{35} +2929.24 q^{37} -8093.84 q^{41} -4964.94 q^{43} +1684.77 q^{47} -10872.5 q^{49} +33651.4 q^{53} +10079.7 q^{55} +24735.4 q^{59} -40656.6 q^{61} -21472.1 q^{65} +48458.7 q^{67} +3153.41 q^{71} +12251.6 q^{73} +31059.8 q^{77} +48200.5 q^{79} -13826.8 q^{83} +34907.4 q^{85} -73867.1 q^{89} -66164.6 q^{91} -32217.7 q^{95} -135749. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 125 q^{5} + 88 q^{7}+O(q^{10}) $ 5 * q - 125 * q^5 + 88 * q^7	$ 5 q - 125 q^{5} + 88 q^{7} - 134 q^{11} - 88 q^{13} - 211 q^{17} + 857 q^{19} - 461 q^{23} + 3125 q^{25} - 68 q^{29} + 1409 q^{31} - 2200 q^{35} + 5346 q^{37} - 750 q^{41} + 14486 q^{43} - 6704 q^{47} + 19005 q^{49} - 8583 q^{53} + 3350 q^{55} - 25384 q^{59} + 38885 q^{61} + 2200 q^{65} + 23146 q^{67} - 62910 q^{71} + 54342 q^{73} - 5368 q^{77} + 24207 q^{79} - 20179 q^{83} + 5275 q^{85} + 10230 q^{89} + 41752 q^{91} - 21425 q^{95} + 43184 q^{97}+O(q^{100}) $ 5 * q - 125 * q^5 + 88 * q^7 - 134 * q^11 - 88 * q^13 - 211 * q^17 + 857 * q^19 - 461 * q^23 + 3125 * q^25 - 68 * q^29 + 1409 * q^31 - 2200 * q^35 + 5346 * q^37 - 750 * q^41 + 14486 * q^43 - 6704 * q^47 + 19005 * q^49 - 8583 * q^53 + 3350 * q^55 - 25384 * q^59 + 38885 * q^61 + 2200 * q^65 + 23146 * q^67 - 62910 * q^71 + 54342 * q^73 - 5368 * q^77 + 24207 * q^79 - 20179 * q^83 + 5275 * q^85 + 10230 * q^89 + 41752 * q^91 - 21425 * q^95 + 43184 * 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
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−77.0356	−0.594219	−0.297110	−	0.954843i	$-0.596023\pi$
$7$	−77.0356	−0.594219	−0.297110	+	0.954843i	$0.596023\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−403.188	−1.00467	−0.502337	−	0.864672i	$-0.667526\pi$
$11$	−403.188	−1.00467	−0.502337	+	0.864672i	$0.667526\pi$
$12$	0	0
$13$	858.883	1.40953	0.704767	−	0.709439i	$-0.251052\pi$
$13$	858.883	1.40953	0.704767	+	0.709439i	$0.251052\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1396.30	−1.17181	−0.585903	−	0.810382i	$-0.699260\pi$
$17$	−1396.30	−1.17181	−0.585903	+	0.810382i	$0.699260\pi$
$18$	0	0
$19$	1288.71	0.818974	0.409487	−	0.912316i	$-0.365708\pi$
$19$	1288.71	0.818974	0.409487	+	0.912316i	$0.365708\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−794.472	−0.313155	−0.156577	−	0.987666i	$-0.550046\pi$
$23$	−794.472	−0.313155	−0.156577	+	0.987666i	$0.550046\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3191.08	0.704600	0.352300	−	0.935887i	$-0.385400\pi$
$29$	3191.08	0.704600	0.352300	+	0.935887i	$0.385400\pi$
$30$	0	0
$31$	6561.95	1.22639	0.613195	−	0.789932i	$-0.289884\pi$
$31$	6561.95	1.22639	0.613195	+	0.789932i	$0.289884\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1925.89	0.265743
$36$	0	0
$37$	2929.24	0.351764	0.175882	−	0.984411i	$-0.443722\pi$
$37$	2929.24	0.351764	0.175882	+	0.984411i	$0.443722\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8093.84	−0.751960	−0.375980	−	0.926628i	$-0.622694\pi$
$41$	−8093.84	−0.751960	−0.375980	+	0.926628i	$0.622694\pi$
$42$	0	0
$43$	−4964.94	−0.409489	−0.204745	−	0.978815i	$-0.565636\pi$
$43$	−4964.94	−0.409489	−0.204745	+	0.978815i	$0.565636\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1684.77	0.111249	0.0556245	−	0.998452i	$-0.482285\pi$
$47$	1684.77	0.111249	0.0556245	+	0.998452i	$0.482285\pi$
$48$	0	0
$49$	−10872.5	−0.646904
$50$	0	0
$51$	0	0
$52$	0	0
$53$	33651.4	1.64556	0.822780	−	0.568360i	$-0.192422\pi$
$53$	33651.4	1.64556	0.822780	+	0.568360i	$0.192422\pi$
$54$	0	0
$55$	10079.7	0.449304
$56$	0	0
$57$	0	0
$58$	0	0
$59$	24735.4	0.925100	0.462550	−	0.886593i	$-0.346935\pi$
$59$	24735.4	0.925100	0.462550	+	0.886593i	$0.346935\pi$
$60$	0	0
$61$	−40656.6	−1.39896	−0.699481	−	0.714651i	$-0.746586\pi$
$61$	−40656.6	−1.39896	−0.699481	+	0.714651i	$0.746586\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−21472.1	−0.630363
$66$	0	0
$67$	48458.7	1.31882	0.659408	−	0.751785i	$-0.270807\pi$
$67$	48458.7	1.31882	0.659408	+	0.751785i	$0.270807\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3153.41	0.0742395	0.0371198	−	0.999311i	$-0.488182\pi$
$71$	3153.41	0.0742395	0.0371198	+	0.999311i	$0.488182\pi$
$72$	0	0
$73$	12251.6	0.269083	0.134542	−	0.990908i	$-0.457044\pi$
$73$	12251.6	0.269083	0.134542	+	0.990908i	$0.457044\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	31059.8	0.596997
$78$	0	0
$79$	48200.5	0.868929	0.434464	−	0.900689i	$-0.356938\pi$
$79$	48200.5	0.868929	0.434464	+	0.900689i	$0.356938\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13826.8	−0.220306	−0.110153	−	0.993915i	$-0.535134\pi$
$83$	−13826.8	−0.220306	−0.110153	+	0.993915i	$0.535134\pi$
$84$	0	0
$85$	34907.4	0.524047
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−73867.1	−0.988499	−0.494250	−	0.869320i	$-0.664557\pi$
$89$	−73867.1	−0.988499	−0.494250	+	0.869320i	$0.664557\pi$
$90$	0	0
$91$	−66164.6	−0.837572
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−32217.7	−0.366256
$96$	0	0
$97$	−135749.	−1.46490	−0.732450	−	0.680821i	$-0.761623\pi$
$97$	−135749.	−1.46490	−0.732450	+	0.680821i	$0.761623\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−143102.	−1.39586	−0.697930	−	0.716166i	$-0.745895\pi$
$101$	−143102.	−1.39586	−0.697930	+	0.716166i	$0.745895\pi$
$102$	0	0
$103$	−30845.4	−0.286482	−0.143241	−	0.989688i	$-0.545752\pi$
$103$	−30845.4	−0.286482	−0.143241	+	0.989688i	$0.545752\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−102319.	−0.863971	−0.431985	−	0.901881i	$-0.642187\pi$
$107$	−102319.	−0.863971	−0.431985	+	0.901881i	$0.642187\pi$
$108$	0	0
$109$	209177.	1.68635	0.843175	−	0.537640i	$-0.180684\pi$
$109$	209177.	1.68635	0.843175	+	0.537640i	$0.180684\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	23946.7	0.176421	0.0882105	−	0.996102i	$-0.471885\pi$
$113$	23946.7	0.176421	0.0882105	+	0.996102i	$0.471885\pi$
$114$	0	0
$115$	19861.8	0.140047
$116$	0	0
$117$	0	0
$118$	0	0
$119$	107565.	0.696309
$120$	0	0
$121$	1509.23	0.00937112
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	17849.6	0.0982016	0.0491008	−	0.998794i	$-0.484364\pi$
$127$	17849.6	0.0982016	0.0491008	+	0.998794i	$0.484364\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−177874.	−0.905597	−0.452798	−	0.891613i	$-0.649574\pi$
$131$	−177874.	−0.905597	−0.452798	+	0.891613i	$0.649574\pi$
$132$	0	0
$133$	−99276.3	−0.486650
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−82021.0	−0.373357	−0.186678	−	0.982421i	$-0.559772\pi$
$137$	−82021.0	−0.373357	−0.186678	+	0.982421i	$0.559772\pi$
$138$	0	0
$139$	176000.	0.772637	0.386319	−	0.922365i	$-0.373747\pi$
$139$	176000.	0.772637	0.386319	+	0.922365i	$0.373747\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−346291.	−1.41612
$144$	0	0
$145$	−79777.0	−0.315107
$146$	0	0
$147$	0	0
$148$	0	0
$149$	424796.	1.56752	0.783762	−	0.621061i	$-0.213298\pi$
$149$	424796.	1.56752	0.783762	+	0.621061i	$0.213298\pi$
$150$	0	0
$151$	−330404.	−1.17924	−0.589621	−	0.807680i	$-0.700723\pi$
$151$	−330404.	−1.17924	−0.589621	+	0.807680i	$0.700723\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−164049.	−0.548458
$156$	0	0
$157$	37513.0	0.121460	0.0607300	−	0.998154i	$-0.480657\pi$
$157$	37513.0	0.121460	0.0607300	+	0.998154i	$0.480657\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	61202.6	0.186082
$162$	0	0
$163$	−121730.	−0.358862	−0.179431	−	0.983771i	$-0.557426\pi$
$163$	−121730.	−0.358862	−0.179431	+	0.983771i	$0.557426\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−456744.	−1.26731	−0.633654	−	0.773617i	$-0.718446\pi$
$167$	−456744.	−1.26731	−0.633654	+	0.773617i	$0.718446\pi$
$168$	0	0
$169$	366386.	0.986785
$170$	0	0
$171$	0	0
$172$	0	0
$173$	306205.	0.777851	0.388926	−	0.921269i	$-0.372846\pi$
$173$	306205.	0.777851	0.388926	+	0.921269i	$0.372846\pi$
$174$	0	0
$175$	−48147.3	−0.118844
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−259843.	−0.606148	−0.303074	−	0.952967i	$-0.598013\pi$
$179$	−259843.	−0.606148	−0.303074	+	0.952967i	$0.598013\pi$
$180$	0	0
$181$	−208135.	−0.472224	−0.236112	−	0.971726i	$-0.575873\pi$
$181$	−208135.	−0.472224	−0.236112	+	0.971726i	$0.575873\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−73231.0	−0.157313
$186$	0	0
$187$	562969.	1.17728
$188$	0	0
$189$	0	0
$190$	0	0
$191$	870514.	1.72660	0.863302	−	0.504688i	$-0.168393\pi$
$191$	870514.	1.72660	0.863302	+	0.504688i	$0.168393\pi$
$192$	0	0
$193$	−692613.	−1.33844	−0.669218	−	0.743066i	$-0.733371\pi$
$193$	−692613.	−1.33844	−0.669218	+	0.743066i	$0.733371\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−356273.	−0.654060	−0.327030	−	0.945014i	$-0.606048\pi$
$197$	−356273.	−0.654060	−0.327030	+	0.945014i	$0.606048\pi$
$198$	0	0
$199$	−943916.	−1.68967	−0.844833	−	0.535030i	$-0.820300\pi$
$199$	−943916.	−1.68967	−0.844833	+	0.535030i	$0.820300\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−245827.	−0.418687
$204$	0	0
$205$	202346.	0.336287
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−519590.	−0.822802
$210$	0	0
$211$	547138.	0.846040	0.423020	−	0.906120i	$-0.360970\pi$
$211$	547138.	0.846040	0.423020	+	0.906120i	$0.360970\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	124123.	0.183129
$216$	0	0
$217$	−505504.	−0.728744
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.19925e6	−1.65170
$222$	0	0
$223$	252624.	0.340183	0.170091	−	0.985428i	$-0.445594\pi$
$223$	252624.	0.340183	0.170091	+	0.985428i	$0.445594\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−703008.	−0.905515	−0.452758	−	0.891634i	$-0.649560\pi$
$227$	−703008.	−0.905515	−0.452758	+	0.891634i	$0.649560\pi$
$228$	0	0
$229$	386011.	0.486419	0.243210	−	0.969974i	$-0.421800\pi$
$229$	386011.	0.486419	0.243210	+	0.969974i	$0.421800\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	613.158	0.000739916 0	0.000369958	−	1.00000i	$-0.499882\pi$
$233$	613.158	0.000739916 0	0.000369958		1.00000i	$0.499882\pi$
$234$	0	0
$235$	−42119.3	−0.0497520
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−254445.	−0.288137	−0.144069	−	0.989568i	$-0.546019\pi$
$239$	−254445.	−0.288137	−0.144069	+	0.989568i	$0.546019\pi$
$240$	0	0
$241$	−1.16674e6	−1.29399	−0.646994	−	0.762495i	$-0.723974\pi$
$241$	−1.16674e6	−1.29399	−0.646994	+	0.762495i	$0.723974\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	271813.	0.289304
$246$	0	0
$247$	1.10685e6	1.15437
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−502349.	−0.503294	−0.251647	−	0.967819i	$-0.580972\pi$
$251$	−502349.	−0.503294	−0.251647	+	0.967819i	$0.580972\pi$
$252$	0	0
$253$	320321.	0.314619
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.75208e6	−1.65471	−0.827353	−	0.561682i	$-0.810154\pi$
$257$	−1.75208e6	−1.65471	−0.827353	+	0.561682i	$0.810154\pi$
$258$	0	0
$259$	−225656.	−0.209025
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−761538.	−0.678894	−0.339447	−	0.940625i	$-0.610240\pi$
$263$	−761538.	−0.678894	−0.339447	+	0.940625i	$0.610240\pi$
$264$	0	0
$265$	−841285.	−0.735917
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.77390e6	−1.49468	−0.747342	−	0.664440i	$-0.768670\pi$
$269$	−1.77390e6	−1.49468	−0.747342	+	0.664440i	$0.768670\pi$
$270$	0	0
$271$	1.11039e6	0.918446	0.459223	−	0.888321i	$-0.348128\pi$
$271$	1.11039e6	0.918446	0.459223	+	0.888321i	$0.348128\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−251992.	−0.200935
$276$	0	0
$277$	1.43729e6	1.12550	0.562749	−	0.826628i	$-0.309744\pi$
$277$	1.43729e6	1.12550	0.562749	+	0.826628i	$0.309744\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−997894.	−0.753909	−0.376954	−	0.926232i	$-0.623029\pi$
$281$	−997894.	−0.753909	−0.376954	+	0.926232i	$0.623029\pi$
$282$	0	0
$283$	747075.	0.554496	0.277248	−	0.960798i	$-0.410578\pi$
$283$	747075.	0.554496	0.277248	+	0.960798i	$0.410578\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	623514.	0.446829
$288$	0	0
$289$	529787.	0.373127
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.65822e6	−1.12843	−0.564213	−	0.825629i	$-0.690820\pi$
$293$	−1.65822e6	−1.12843	−0.564213	+	0.825629i	$0.690820\pi$
$294$	0	0
$295$	−618384.	−0.413717
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−682358.	−0.441402
$300$	0	0
$301$	382477.	0.243326
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.01641e6	0.625635
$306$	0	0
$307$	614029.	0.371829	0.185914	−	0.982566i	$-0.440475\pi$
$307$	614029.	0.371829	0.185914	+	0.982566i	$0.440475\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.73373e6	1.01644	0.508219	−	0.861228i	$-0.330304\pi$
$311$	1.73373e6	1.01644	0.508219	+	0.861228i	$0.330304\pi$
$312$	0	0
$313$	1.89669e6	1.09430	0.547150	−	0.837034i	$-0.315713\pi$
$313$	1.89669e6	1.09430	0.547150	+	0.837034i	$0.315713\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	593893.	0.331941	0.165970	−	0.986131i	$-0.446924\pi$
$317$	593893.	0.331941	0.165970	+	0.986131i	$0.446924\pi$
$318$	0	0
$319$	−1.28660e6	−0.707894
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.79942e6	−0.959678
$324$	0	0
$325$	536802.	0.281907
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−129787.	−0.0661063
$330$	0	0
$331$	−1.43992e6	−0.722387	−0.361193	−	0.932491i	$-0.617631\pi$
$331$	−1.43992e6	−0.722387	−0.361193	+	0.932491i	$0.617631\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.21147e6	−0.589793
$336$	0	0
$337$	8665.31	0.00415632	0.00207816	−	0.999998i	$-0.499339\pi$
$337$	8665.31	0.00415632	0.00207816	+	0.999998i	$0.499339\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.64570e6	−1.23212
$342$	0	0
$343$	2.13231e6	0.978622
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.59465e6	−0.710953	−0.355477	−	0.934685i	$-0.615681\pi$
$347$	−1.59465e6	−0.710953	−0.355477	+	0.934685i	$0.615681\pi$
$348$	0	0
$349$	−2.22383e6	−0.977322	−0.488661	−	0.872474i	$-0.662515\pi$
$349$	−2.22383e6	−0.977322	−0.488661	+	0.872474i	$0.662515\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−414863.	−0.177202	−0.0886009	−	0.996067i	$-0.528240\pi$
$353$	−414863.	−0.177202	−0.0886009	+	0.996067i	$0.528240\pi$
$354$	0	0
$355$	−78835.4	−0.0332009
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.94165e6	−1.61414	−0.807072	−	0.590453i	$-0.798949\pi$
$359$	−3.94165e6	−1.61414	−0.807072	+	0.590453i	$0.798949\pi$
$360$	0	0
$361$	−815335.	−0.329282
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−306291.	−0.120338
$366$	0	0
$367$	−2.01448e6	−0.780723	−0.390362	−	0.920662i	$-0.627650\pi$
$367$	−2.01448e6	−0.780723	−0.390362	+	0.920662i	$0.627650\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.59236e6	−0.977823
$372$	0	0
$373$	−2.81680e6	−1.04830	−0.524148	−	0.851627i	$-0.675616\pi$
$373$	−2.81680e6	−1.04830	−0.524148	+	0.851627i	$0.675616\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.74076e6	0.993158
$378$	0	0
$379$	−1.47128e6	−0.526136	−0.263068	−	0.964777i	$-0.584734\pi$
$379$	−1.47128e6	−0.526136	−0.263068	+	0.964777i	$0.584734\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−453535.	−0.157984	−0.0789921	−	0.996875i	$-0.525170\pi$
$383$	−453535.	−0.157984	−0.0789921	+	0.996875i	$0.525170\pi$
$384$	0	0
$385$	−776495.	−0.266985
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4.31465e6	−1.44568	−0.722840	−	0.691016i	$-0.757164\pi$
$389$	−4.31465e6	−1.44568	−0.722840	+	0.691016i	$0.757164\pi$
$390$	0	0
$391$	1.10932e6	0.366956
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.20501e6	−0.388597
$396$	0	0
$397$	−5.53431e6	−1.76233	−0.881166	−	0.472808i	$-0.843240\pi$
$397$	−5.53431e6	−1.76233	−0.881166	+	0.472808i	$0.843240\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−551779.	−0.171358	−0.0856790	−	0.996323i	$-0.527306\pi$
$401$	−551779.	−0.171358	−0.0856790	+	0.996323i	$0.527306\pi$
$402$	0	0
$403$	5.63594e6	1.72864
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.18103e6	−0.353408
$408$	0	0
$409$	−1.73292e6	−0.512236	−0.256118	−	0.966646i	$-0.582444\pi$
$409$	−1.73292e6	−0.512236	−0.256118	+	0.966646i	$0.582444\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.90551e6	−0.549712
$414$	0	0
$415$	345670.	0.0985238
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.84537e6	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$419$	4.84537e6	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$420$	0	0
$421$	−3.35095e6	−0.921430	−0.460715	−	0.887548i	$-0.652407\pi$
$421$	−3.35095e6	−0.921430	−0.460715	+	0.887548i	$0.652407\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−872685.	−0.234361
$426$	0	0
$427$	3.13201e6	0.831290
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6.08329e6	−1.57741	−0.788706	−	0.614770i	$-0.789249\pi$
$431$	−6.08329e6	−1.57741	−0.788706	+	0.614770i	$0.789249\pi$
$432$	0	0
$433$	−6.70936e6	−1.71973	−0.859867	−	0.510518i	$-0.829454\pi$
$433$	−6.70936e6	−1.71973	−0.859867	+	0.510518i	$0.829454\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.02384e6	−0.256465
$438$	0	0
$439$	6.54083e6	1.61984	0.809918	−	0.586543i	$-0.199511\pi$
$439$	6.54083e6	1.61984	0.809918	+	0.586543i	$0.199511\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.56672e6	−1.10559	−0.552796	−	0.833317i	$-0.686439\pi$
$443$	−4.56672e6	−1.10559	−0.552796	+	0.833317i	$0.686439\pi$
$444$	0	0
$445$	1.84668e6	0.442070
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.95800e6	−0.692441	−0.346221	−	0.938153i	$-0.612535\pi$
$449$	−2.95800e6	−0.692441	−0.346221	+	0.938153i	$0.612535\pi$
$450$	0	0
$451$	3.26334e6	0.755476
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.65411e6	0.374573
$456$	0	0
$457$	5.88839e6	1.31888	0.659442	−	0.751756i	$-0.270793\pi$
$457$	5.88839e6	1.31888	0.659442	+	0.751756i	$0.270793\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.63633e6	0.358608	0.179304	−	0.983794i	$-0.442615\pi$
$461$	1.63633e6	0.358608	0.179304	+	0.983794i	$0.442615\pi$
$462$	0	0
$463$	2.52074e6	0.546481	0.273240	−	0.961946i	$-0.411905\pi$
$463$	2.52074e6	0.546481	0.273240	+	0.961946i	$0.411905\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.08209e6	1.71487	0.857435	−	0.514592i	$-0.172057\pi$
$467$	8.08209e6	1.71487	0.857435	+	0.514592i	$0.172057\pi$
$468$	0	0
$469$	−3.73305e6	−0.783666
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.00180e6	0.411403
$474$	0	0
$475$	805442.	0.163795
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.74241e6	0.546127	0.273064	−	0.961996i	$-0.411963\pi$
$479$	2.74241e6	0.546127	0.273064	+	0.961996i	$0.411963\pi$
$480$	0	0
$481$	2.51587e6	0.495823
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.39373e6	0.655123
$486$	0	0
$487$	233803.	0.0446712	0.0223356	−	0.999751i	$-0.492890\pi$
$487$	233803.	0.0446712	0.0223356	+	0.999751i	$0.492890\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5.52223e6	−1.03374	−0.516869	−	0.856064i	$-0.672903\pi$
$491$	−5.52223e6	−1.03374	−0.516869	+	0.856064i	$0.672903\pi$
$492$	0	0
$493$	−4.45570e6	−0.825654
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−242925.	−0.0441146
$498$	0	0
$499$	4.57328e6	0.822198	0.411099	−	0.911591i	$-0.365145\pi$
$499$	4.57328e6	0.822198	0.411099	+	0.911591i	$0.365145\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	5.29964e6	0.933956	0.466978	−	0.884269i	$-0.345343\pi$
$503$	5.29964e6	0.933956	0.466978	+	0.884269i	$0.345343\pi$
$504$	0	0
$505$	3.57755e6	0.624248
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.43744e6	−0.417004	−0.208502	−	0.978022i	$-0.566859\pi$
$509$	−2.43744e6	−0.417004	−0.208502	+	0.978022i	$0.566859\pi$
$510$	0	0
$511$	−943812.	−0.159894
$512$	0	0
$513$	0	0
$514$	0	0
$515$	771135.	0.128119
$516$	0	0
$517$	−679278.	−0.111769
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3.40265e6	0.549190	0.274595	−	0.961560i	$-0.411456\pi$
$521$	3.40265e6	0.549190	0.274595	+	0.961560i	$0.411456\pi$
$522$	0	0
$523$	−8.86584e6	−1.41731	−0.708657	−	0.705554i	$-0.750698\pi$
$523$	−8.86584e6	−1.41731	−0.708657	+	0.705554i	$0.750698\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.16242e6	−1.43709
$528$	0	0
$529$	−5.80516e6	−0.901934
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−6.95166e6	−1.05991
$534$	0	0
$535$	2.55799e6	0.386379
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.38366e6	0.649928
$540$	0	0
$541$	−9.73366e6	−1.42983	−0.714913	−	0.699214i	$-0.753534\pi$
$541$	−9.73366e6	−1.42983	−0.714913	+	0.699214i	$0.753534\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.22942e6	−0.754158
$546$	0	0
$547$	8.68670e6	1.24133	0.620664	−	0.784077i	$-0.286863\pi$
$547$	8.68670e6	1.24133	0.620664	+	0.784077i	$0.286863\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.11237e6	0.577049
$552$	0	0
$553$	−3.71316e6	−0.516334
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.34194e6	−0.183271	−0.0916357	−	0.995793i	$-0.529210\pi$
$557$	−1.34194e6	−0.183271	−0.0916357	+	0.995793i	$0.529210\pi$
$558$	0	0
$559$	−4.26430e6	−0.577189
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.61322e6	0.746347	0.373173	−	0.927762i	$-0.378270\pi$
$563$	5.61322e6	0.746347	0.373173	+	0.927762i	$0.378270\pi$
$564$	0	0
$565$	−598668.	−0.0788978
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6.92406e6	0.896562	0.448281	−	0.893893i	$-0.352036\pi$
$569$	6.92406e6	0.896562	0.448281	+	0.893893i	$0.352036\pi$
$570$	0	0
$571$	1.03713e7	1.33120	0.665599	−	0.746309i	$-0.268176\pi$
$571$	1.03713e7	1.33120	0.665599	+	0.746309i	$0.268176\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−496545.	−0.0626309
$576$	0	0
$577$	−2.73261e6	−0.341694	−0.170847	−	0.985298i	$-0.554650\pi$
$577$	−2.73261e6	−0.341694	−0.170847	+	0.985298i	$0.554650\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.06516e6	0.130910
$582$	0	0
$583$	−1.35678e7	−1.65325
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.29208e7	−1.54773	−0.773864	−	0.633352i	$-0.781678\pi$
$587$	−1.29208e7	−1.54773	−0.773864	+	0.633352i	$0.781678\pi$
$588$	0	0
$589$	8.45642e6	1.00438
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4.38986e6	−0.512642	−0.256321	−	0.966592i	$-0.582510\pi$
$593$	−4.38986e6	−0.512642	−0.256321	+	0.966592i	$0.582510\pi$
$594$	0	0
$595$	−2.68912e6	−0.311399
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.59554e7	−1.81694	−0.908472	−	0.417946i	$-0.862750\pi$
$599$	−1.59554e7	−1.81694	−0.908472	+	0.417946i	$0.862750\pi$
$600$	0	0
$601$	1.04833e7	1.18389	0.591947	−	0.805977i	$-0.298360\pi$
$601$	1.04833e7	1.18389	0.591947	+	0.805977i	$0.298360\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−37730.7	−0.00419089
$606$	0	0
$607$	−6.93883e6	−0.764389	−0.382194	−	0.924082i	$-0.624832\pi$
$607$	−6.93883e6	−0.764389	−0.382194	+	0.924082i	$0.624832\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.44702e6	0.156809
$612$	0	0
$613$	−441717.	−0.0474781	−0.0237390	−	0.999718i	$-0.507557\pi$
$613$	−441717.	−0.0474781	−0.0237390	+	0.999718i	$0.507557\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	949536.	0.100415	0.0502075	−	0.998739i	$-0.484012\pi$
$617$	949536.	0.100415	0.0502075	+	0.998739i	$0.484012\pi$
$618$	0	0
$619$	1.07175e7	1.12426	0.562131	−	0.827048i	$-0.309982\pi$
$619$	1.07175e7	1.12426	0.562131	+	0.827048i	$0.309982\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.69040e6	0.587385
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.09009e6	−0.412198
$630$	0	0
$631$	1.01034e7	1.01017	0.505086	−	0.863069i	$-0.331461\pi$
$631$	1.01034e7	1.01017	0.505086	+	0.863069i	$0.331461\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−446239.	−0.0439171
$636$	0	0
$637$	−9.33821e6	−0.911832
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.82373e6	0.655959	0.327979	−	0.944685i	$-0.393632\pi$
$641$	6.82373e6	0.655959	0.327979	+	0.944685i	$0.393632\pi$
$642$	0	0
$643$	1.71706e7	1.63779	0.818893	−	0.573946i	$-0.194588\pi$
$643$	1.71706e7	1.63779	0.818893	+	0.573946i	$0.194588\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.56448e7	−1.46929	−0.734646	−	0.678450i	$-0.762652\pi$
$647$	−1.56448e7	−1.46929	−0.734646	+	0.678450i	$0.762652\pi$
$648$	0	0
$649$	−9.97300e6	−0.929424
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.16942e6	0.566189	0.283094	−	0.959092i	$-0.408639\pi$
$653$	6.16942e6	0.566189	0.283094	+	0.959092i	$0.408639\pi$
$654$	0	0
$655$	4.44686e6	0.404995
$656$	0	0
$657$	0	0
$658$	0	0
$659$	7.24146e6	0.649550	0.324775	−	0.945791i	$-0.394711\pi$
$659$	7.24146e6	0.649550	0.324775	+	0.945791i	$0.394711\pi$
$660$	0	0
$661$	−7.65330e6	−0.681311	−0.340655	−	0.940188i	$-0.610649\pi$
$661$	−7.65330e6	−0.681311	−0.340655	+	0.940188i	$0.610649\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.48191e6	0.217636
$666$	0	0
$667$	−2.53522e6	−0.220649
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.63922e7	1.40550
$672$	0	0
$673$	1.79900e6	0.153106	0.0765531	−	0.997066i	$-0.475609\pi$
$673$	1.79900e6	0.153106	0.0765531	+	0.997066i	$0.475609\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.92362e7	1.61305	0.806523	−	0.591203i	$-0.201347\pi$
$677$	1.92362e7	1.61305	0.806523	+	0.591203i	$0.201347\pi$
$678$	0	0
$679$	1.04575e7	0.870471
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.14323e7	1.75799	0.878996	−	0.476830i	$-0.158214\pi$
$683$	2.14323e7	1.75799	0.878996	+	0.476830i	$0.158214\pi$
$684$	0	0
$685$	2.05053e6	0.166970
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.89026e7	2.31947
$690$	0	0
$691$	−1.31513e7	−1.04779	−0.523895	−	0.851783i	$-0.675522\pi$
$691$	−1.31513e7	−1.04779	−0.523895	+	0.851783i	$0.675522\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.40000e6	−0.345534
$696$	0	0
$697$	1.13014e7	0.881151
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.17904e7	0.906222	0.453111	−	0.891454i	$-0.350314\pi$
$701$	1.17904e7	0.906222	0.453111	+	0.891454i	$0.350314\pi$
$702$	0	0
$703$	3.77493e6	0.288085
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.10239e7	0.829447
$708$	0	0
$709$	−1.81192e7	−1.35370	−0.676851	−	0.736120i	$-0.736656\pi$
$709$	−1.81192e7	−1.35370	−0.676851	+	0.736120i	$0.736656\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5.21328e6	−0.384050
$714$	0	0
$715$	8.65727e6	0.633309
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.72847e7	1.24692	0.623461	−	0.781854i	$-0.285726\pi$
$719$	1.72847e7	1.24692	0.623461	+	0.781854i	$0.285726\pi$
$720$	0	0
$721$	2.37619e6	0.170233
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.99443e6	0.140920
$726$	0	0
$727$	2.34011e7	1.64210	0.821052	−	0.570853i	$-0.193387\pi$
$727$	2.34011e7	1.64210	0.821052	+	0.570853i	$0.193387\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6.93252e6	0.479842
$732$	0	0
$733$	1.48055e7	1.01780	0.508901	−	0.860825i	$-0.330052\pi$
$733$	1.48055e7	1.01780	0.508901	+	0.860825i	$0.330052\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.95379e7	−1.32498
$738$	0	0
$739$	−1.16381e7	−0.783916	−0.391958	−	0.919983i	$-0.628202\pi$
$739$	−1.16381e7	−0.783916	−0.391958	+	0.919983i	$0.628202\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	751074.	0.0499126	0.0249563	−	0.999689i	$-0.492055\pi$
$743$	751074.	0.0499126	0.0249563	+	0.999689i	$0.492055\pi$
$744$	0	0
$745$	−1.06199e7	−0.701018
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7.88225e6	0.513388
$750$	0	0
$751$	8.65354e6	0.559878	0.279939	−	0.960018i	$-0.409686\pi$
$751$	8.65354e6	0.559878	0.279939	+	0.960018i	$0.409686\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	8.26011e6	0.527373
$756$	0	0
$757$	−1.01514e7	−0.643854	−0.321927	−	0.946765i	$-0.604331\pi$
$757$	−1.01514e7	−0.643854	−0.321927	+	0.946765i	$0.604331\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.15451e6	−0.0722663	−0.0361332	−	0.999347i	$-0.511504\pi$
$761$	−1.15451e6	−0.0722663	−0.0361332	+	0.999347i	$0.511504\pi$
$762$	0	0
$763$	−1.61141e7	−1.00206
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.12448e7	1.30396
$768$	0	0
$769$	−2.86350e7	−1.74615	−0.873075	−	0.487586i	$-0.837878\pi$
$769$	−2.86350e7	−1.74615	−0.873075	+	0.487586i	$0.837878\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−2.52306e6	−0.151872	−0.0759362	−	0.997113i	$-0.524195\pi$
$773$	−2.52306e6	−0.151872	−0.0759362	+	0.997113i	$0.524195\pi$
$774$	0	0
$775$	4.10122e6	0.245278
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.04306e7	−0.615836
$780$	0	0
$781$	−1.27142e6	−0.0745866
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−937826.	−0.0543185
$786$	0	0
$787$	3.00605e7	1.73005	0.865025	−	0.501728i	$-0.167302\pi$
$787$	3.00605e7	1.73005	0.865025	+	0.501728i	$0.167302\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.84475e6	−0.104833
$792$	0	0
$793$	−3.49192e7	−1.97188
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.16978e7	1.20996	0.604979	−	0.796241i	$-0.293182\pi$
$797$	2.16978e7	1.20996	0.604979	+	0.796241i	$0.293182\pi$
$798$	0	0
$799$	−2.35244e6	−0.130362
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4.93970e6	−0.270341
$804$	0	0
$805$	−1.53007e6	−0.0832186
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6.42278e6	0.345026	0.172513	−	0.985007i	$-0.444811\pi$
$809$	6.42278e6	0.345026	0.172513	+	0.985007i	$0.444811\pi$
$810$	0	0
$811$	−1.53609e7	−0.820096	−0.410048	−	0.912064i	$-0.634488\pi$
$811$	−1.53609e7	−0.820096	−0.410048	+	0.912064i	$0.634488\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.04324e6	0.160488
$816$	0	0
$817$	−6.39835e6	−0.335361
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.02681e6	0.104943	0.0524717	−	0.998622i	$-0.483290\pi$
$821$	2.02681e6	0.104943	0.0524717	+	0.998622i	$0.483290\pi$
$822$	0	0
$823$	2.24714e7	1.15646	0.578229	−	0.815875i	$-0.303744\pi$
$823$	2.24714e7	1.15646	0.578229	+	0.815875i	$0.303744\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.16101e6	0.109874	0.0549368	−	0.998490i	$-0.482504\pi$
$827$	2.16101e6	0.109874	0.0549368	+	0.998490i	$0.482504\pi$
$828$	0	0
$829$	−2.98092e7	−1.50648	−0.753242	−	0.657744i	$-0.771511\pi$
$829$	−2.98092e7	−1.50648	−0.753242	+	0.657744i	$0.771511\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.51812e7	0.758045
$834$	0	0
$835$	1.14186e7	0.566757
$836$	0	0
$837$	0	0
$838$	0	0
$839$	557320.	0.0273338	0.0136669	−	0.999907i	$-0.495650\pi$
$839$	557320.	0.0273338	0.0136669	+	0.999907i	$0.495650\pi$
$840$	0	0
$841$	−1.03281e7	−0.503538
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−9.15966e6	−0.441304
$846$	0	0
$847$	−116264.	−0.00556850
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.32720e6	−0.110156
$852$	0	0
$853$	2.30074e7	1.08267	0.541334	−	0.840808i	$-0.317919\pi$
$853$	2.30074e7	1.08267	0.541334	+	0.840808i	$0.317919\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.78607e7	0.830702	0.415351	−	0.909661i	$-0.363659\pi$
$857$	1.78607e7	0.830702	0.415351	+	0.909661i	$0.363659\pi$
$858$	0	0
$859$	−2.21472e7	−1.02408	−0.512041	−	0.858961i	$-0.671111\pi$
$859$	−2.21472e7	−1.02408	−0.512041	+	0.858961i	$0.671111\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.39641e7	0.638244	0.319122	−	0.947714i	$-0.396612\pi$
$863$	1.39641e7	0.638244	0.319122	+	0.947714i	$0.396612\pi$
$864$	0	0
$865$	−7.65512e6	−0.347866
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.94339e7	−0.872991
$870$	0	0
$871$	4.16203e7	1.85892
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.20368e6	0.0531486
$876$	0	0
$877$	2.45796e7	1.07914	0.539568	−	0.841942i	$-0.318588\pi$
$877$	2.45796e7	1.07914	0.539568	+	0.841942i	$0.318588\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.41282e7	−0.613265	−0.306633	−	0.951828i	$-0.599202\pi$
$881$	−1.41282e7	−0.613265	−0.306633	+	0.951828i	$0.599202\pi$
$882$	0	0
$883$	7.34339e6	0.316953	0.158477	−	0.987363i	$-0.449342\pi$
$883$	7.34339e6	0.316953	0.158477	+	0.987363i	$0.449342\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.73707e7	1.59486	0.797430	−	0.603411i	$-0.206192\pi$
$887$	3.73707e7	1.59486	0.797430	+	0.603411i	$0.206192\pi$
$888$	0	0
$889$	−1.37505e6	−0.0583533
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.17117e6	0.0911100
$894$	0	0
$895$	6.49608e6	0.271078
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.09397e7	0.864114
$900$	0	0
$901$	−4.69874e7	−1.92828
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.20337e6	0.211185
$906$	0	0
$907$	2.58057e7	1.04159	0.520796	−	0.853681i	$-0.325635\pi$
$907$	2.58057e7	1.04159	0.520796	+	0.853681i	$0.325635\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.48871e6	−0.219116	−0.109558	−	0.993980i	$-0.534944\pi$
$911$	−5.48871e6	−0.219116	−0.109558	+	0.993980i	$0.534944\pi$
$912$	0	0
$913$	5.57479e6	0.221336
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.37027e7	0.538123
$918$	0	0
$919$	−3.97336e7	−1.55192	−0.775959	−	0.630784i	$-0.782734\pi$
$919$	−3.97336e7	−1.55192	−0.775959	+	0.630784i	$0.782734\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.70841e6	0.104643
$924$	0	0
$925$	1.83078e6	0.0703527
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−111003.	−0.00421982	−0.00210991	−	0.999998i	$-0.500672\pi$
$929$	−111003.	−0.00421982	−0.00210991	+	0.999998i	$0.500672\pi$
$930$	0	0
$931$	−1.40115e7	−0.529797
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.40742e7	−0.526497
$936$	0	0
$937$	1.92423e6	0.0715993	0.0357996	−	0.999359i	$-0.488602\pi$
$937$	1.92423e6	0.0715993	0.0357996	+	0.999359i	$0.488602\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−2.15993e7	−0.795180	−0.397590	−	0.917563i	$-0.630153\pi$
$941$	−2.15993e7	−0.795180	−0.397590	+	0.917563i	$0.630153\pi$
$942$	0	0
$943$	6.43033e6	0.235480
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.85084e7	−1.03299	−0.516496	−	0.856289i	$-0.672764\pi$
$947$	−2.85084e7	−1.03299	−0.516496	+	0.856289i	$0.672764\pi$
$948$	0	0
$949$	1.05227e7	0.379282
$950$	0	0
$951$	0	0
$952$	0	0
$953$	4.37241e7	1.55951	0.779756	−	0.626083i	$-0.215343\pi$
$953$	4.37241e7	1.55951	0.779756	+	0.626083i	$0.215343\pi$
$954$	0	0
$955$	−2.17629e7	−0.772160
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.31854e6	0.221856
$960$	0	0
$961$	1.44300e7	0.504031
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.73153e7	0.598567
$966$	0	0
$967$	3.44157e7	1.18356	0.591779	−	0.806100i	$-0.298426\pi$
$967$	3.44157e7	1.18356	0.591779	+	0.806100i	$0.298426\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.50051e6	0.153184	0.0765920	−	0.997063i	$-0.475596\pi$
$971$	4.50051e6	0.153184	0.0765920	+	0.997063i	$0.475596\pi$
$972$	0	0
$973$	−1.35583e7	−0.459116
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.38444e7	−1.13436	−0.567180	−	0.823594i	$-0.691966\pi$
$977$	−3.38444e7	−1.13436	−0.567180	+	0.823594i	$0.691966\pi$
$978$	0	0
$979$	2.97823e7	0.993120
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.02242e7	0.337479	0.168740	−	0.985661i	$-0.446030\pi$
$983$	1.02242e7	0.337479	0.168740	+	0.985661i	$0.446030\pi$
$984$	0	0
$985$	8.90683e6	0.292504
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.94450e6	0.128233
$990$	0	0
$991$	−3.43962e7	−1.11257	−0.556284	−	0.830992i	$-0.687773\pi$
$991$	−3.43962e7	−1.11257	−0.556284	+	0.830992i	$0.687773\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2.35979e7	0.755642
$996$	0	0
$997$	1.49667e7	0.476857	0.238428	−	0.971160i	$-0.423368\pi$
$997$	1.49667e7	0.476857	0.238428	+	0.971160i	$0.423368\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.6.a.m.1.2	✓	5
3.2	odd	2		1080.6.a.r.1.2	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.6.a.m.1.2	✓	5	1.1	even	1	trivial
1080.6.a.r.1.2	yes	5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1080.a (trivial)

\(f(q)\)	\(=\)	\(q-25.0000 q^{5} -77.0356 q^{7} +O(q^{10})\)	\(q-25.0000 q^{5} -77.0356 q^{7} -403.188 q^{11} +858.883 q^{13} -1396.30 q^{17} +1288.71 q^{19} -794.472 q^{23} +625.000 q^{25} +3191.08 q^{29} +6561.95 q^{31} +1925.89 q^{35} +2929.24 q^{37} -8093.84 q^{41} -4964.94 q^{43} +1684.77 q^{47} -10872.5 q^{49} +33651.4 q^{53} +10079.7 q^{55} +24735.4 q^{59} -40656.6 q^{61} -21472.1 q^{65} +48458.7 q^{67} +3153.41 q^{71} +12251.6 q^{73} +31059.8 q^{77} +48200.5 q^{79} -13826.8 q^{83} +34907.4 q^{85} -73867.1 q^{89} -66164.6 q^{91} -32217.7 q^{95} -135749. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 125 q^{5} + 88 q^{7}+O(q^{10}) \) 5 * q - 125 * q^5 + 88 * q^7	\( 5 q - 125 q^{5} + 88 q^{7} - 134 q^{11} - 88 q^{13} - 211 q^{17} + 857 q^{19} - 461 q^{23} + 3125 q^{25} - 68 q^{29} + 1409 q^{31} - 2200 q^{35} + 5346 q^{37} - 750 q^{41} + 14486 q^{43} - 6704 q^{47} + 19005 q^{49} - 8583 q^{53} + 3350 q^{55} - 25384 q^{59} + 38885 q^{61} + 2200 q^{65} + 23146 q^{67} - 62910 q^{71} + 54342 q^{73} - 5368 q^{77} + 24207 q^{79} - 20179 q^{83} + 5275 q^{85} + 10230 q^{89} + 41752 q^{91} - 21425 q^{95} + 43184 q^{97}+O(q^{100}) \) 5 * q - 125 * q^5 + 88 * q^7 - 134 * q^11 - 88 * q^13 - 211 * q^17 + 857 * q^19 - 461 * q^23 + 3125 * q^25 - 68 * q^29 + 1409 * q^31 - 2200 * q^35 + 5346 * q^37 - 750 * q^41 + 14486 * q^43 - 6704 * q^47 + 19005 * q^49 - 8583 * q^53 + 3350 * q^55 - 25384 * q^59 + 38885 * q^61 + 2200 * q^65 + 23146 * q^67 - 62910 * q^71 + 54342 * q^73 - 5368 * q^77 + 24207 * q^79 - 20179 * q^83 + 5275 * q^85 + 10230 * q^89 + 41752 * q^91 - 21425 * q^95 + 43184 * q^97

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