# Properties

 Label 1080.4.a.h.1.1 Level $1080$ Weight $4$ Character 1080.1 Self dual yes Analytic conductor $63.722$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

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: $$3$$ Coefficient field: 3.3.47977.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 60x - 44$$ x^3 - x^2 - 60*x - 44 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2\cdot 3^{2}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.749725$$ 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)

 $$f(q)$$ $$=$$ $$q-5.00000 q^{5} -24.3916 q^{7} +O(q^{10})$$ $$q-5.00000 q^{5} -24.3916 q^{7} -26.8932 q^{11} -68.2815 q^{13} -61.8899 q^{17} -75.1747 q^{19} -43.3916 q^{23} +25.0000 q^{25} +174.951 q^{29} +222.666 q^{31} +121.958 q^{35} +67.1813 q^{37} -22.2070 q^{41} -84.7442 q^{43} -585.650 q^{47} +251.948 q^{49} +38.3625 q^{53} +134.466 q^{55} +92.0000 q^{59} -226.300 q^{61} +341.407 q^{65} +858.035 q^{67} +116.912 q^{71} +911.769 q^{73} +655.967 q^{77} -285.129 q^{79} -999.099 q^{83} +309.450 q^{85} -374.848 q^{89} +1665.49 q^{91} +375.873 q^{95} -227.413 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 15 q^{5} + 9 q^{7}+O(q^{10})$$ 3 * q - 15 * q^5 + 9 * q^7 $$3 q - 15 q^{5} + 9 q^{7} - 18 q^{11} - 21 q^{13} - 84 q^{17} + 21 q^{19} - 48 q^{23} + 75 q^{25} + 36 q^{29} + 324 q^{31} - 45 q^{35} + 33 q^{37} - 114 q^{41} + 282 q^{43} - 282 q^{47} + 228 q^{49} - 222 q^{53} + 90 q^{55} + 276 q^{59} + 303 q^{61} + 105 q^{65} + 1035 q^{67} - 510 q^{71} + 447 q^{73} + 1578 q^{77} + 777 q^{79} - 78 q^{83} + 420 q^{85} + 324 q^{89} + 1995 q^{91} - 105 q^{95} + 1191 q^{97}+O(q^{100})$$ 3 * q - 15 * q^5 + 9 * q^7 - 18 * q^11 - 21 * q^13 - 84 * q^17 + 21 * q^19 - 48 * q^23 + 75 * q^25 + 36 * q^29 + 324 * q^31 - 45 * q^35 + 33 * q^37 - 114 * q^41 + 282 * q^43 - 282 * q^47 + 228 * q^49 - 222 * q^53 + 90 * q^55 + 276 * q^59 + 303 * q^61 + 105 * q^65 + 1035 * q^67 - 510 * q^71 + 447 * q^73 + 1578 * q^77 + 777 * q^79 - 78 * q^83 + 420 * q^85 + 324 * q^89 + 1995 * q^91 - 105 * q^95 + 1191 * q^97

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ −24.3916 −1.31702 −0.658510 0.752572i $$-0.728813\pi$$
−0.658510 + 0.752572i $$0.728813\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −26.8932 −0.737146 −0.368573 0.929599i $$-0.620154\pi$$
−0.368573 + 0.929599i $$0.620154\pi$$
$$12$$ 0 0
$$13$$ −68.2815 −1.45676 −0.728380 0.685174i $$-0.759726\pi$$
−0.728380 + 0.685174i $$0.759726\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −61.8899 −0.882971 −0.441485 0.897268i $$-0.645548\pi$$
−0.441485 + 0.897268i $$0.645548\pi$$
$$18$$ 0 0
$$19$$ −75.1747 −0.907697 −0.453849 0.891079i $$-0.649949\pi$$
−0.453849 + 0.891079i $$0.649949\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −43.3916 −0.393381 −0.196691 0.980466i $$-0.563019\pi$$
−0.196691 + 0.980466i $$0.563019\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 174.951 1.12026 0.560131 0.828404i $$-0.310751\pi$$
0.560131 + 0.828404i $$0.310751\pi$$
$$30$$ 0 0
$$31$$ 222.666 1.29007 0.645033 0.764154i $$-0.276843\pi$$
0.645033 + 0.764154i $$0.276843\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 121.958 0.588989
$$36$$ 0 0
$$37$$ 67.1813 0.298501 0.149250 0.988799i $$-0.452314\pi$$
0.149250 + 0.988799i $$0.452314\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −22.2070 −0.0845890 −0.0422945 0.999105i $$-0.513467\pi$$
−0.0422945 + 0.999105i $$0.513467\pi$$
$$42$$ 0 0
$$43$$ −84.7442 −0.300543 −0.150272 0.988645i $$-0.548015\pi$$
−0.150272 + 0.988645i $$0.548015\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −585.650 −1.81757 −0.908785 0.417264i $$-0.862989\pi$$
−0.908785 + 0.417264i $$0.862989\pi$$
$$48$$ 0 0
$$49$$ 251.948 0.734542
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 38.3625 0.0994245 0.0497122 0.998764i $$-0.484170\pi$$
0.0497122 + 0.998764i $$0.484170\pi$$
$$54$$ 0 0
$$55$$ 134.466 0.329662
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 92.0000 0.203006 0.101503 0.994835i $$-0.467635\pi$$
0.101503 + 0.994835i $$0.467635\pi$$
$$60$$ 0 0
$$61$$ −226.300 −0.474997 −0.237498 0.971388i $$-0.576327\pi$$
−0.237498 + 0.971388i $$0.576327\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 341.407 0.651482
$$66$$ 0 0
$$67$$ 858.035 1.56456 0.782281 0.622925i $$-0.214056\pi$$
0.782281 + 0.622925i $$0.214056\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 116.912 0.195422 0.0977108 0.995215i $$-0.468848\pi$$
0.0977108 + 0.995215i $$0.468848\pi$$
$$72$$ 0 0
$$73$$ 911.769 1.46184 0.730921 0.682462i $$-0.239091\pi$$
0.730921 + 0.682462i $$0.239091\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 655.967 0.970836
$$78$$ 0 0
$$79$$ −285.129 −0.406070 −0.203035 0.979171i $$-0.565081\pi$$
−0.203035 + 0.979171i $$0.565081\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −999.099 −1.32127 −0.660635 0.750707i $$-0.729713\pi$$
−0.660635 + 0.750707i $$0.729713\pi$$
$$84$$ 0 0
$$85$$ 309.450 0.394877
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −374.848 −0.446447 −0.223223 0.974767i $$-0.571658\pi$$
−0.223223 + 0.974767i $$0.571658\pi$$
$$90$$ 0 0
$$91$$ 1665.49 1.91858
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 375.873 0.405935
$$96$$ 0 0
$$97$$ −227.413 −0.238044 −0.119022 0.992892i $$-0.537976\pi$$
−0.119022 + 0.992892i $$0.537976\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1857.75 1.83023 0.915115 0.403193i $$-0.132100\pi$$
0.915115 + 0.403193i $$0.132100\pi$$
$$102$$ 0 0
$$103$$ 172.787 0.165294 0.0826468 0.996579i $$-0.473663\pi$$
0.0826468 + 0.996579i $$0.473663\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −1806.79 −1.63242 −0.816209 0.577756i $$-0.803928\pi$$
−0.816209 + 0.577756i $$0.803928\pi$$
$$108$$ 0 0
$$109$$ 287.644 0.252764 0.126382 0.991982i $$-0.459663\pi$$
0.126382 + 0.991982i $$0.459663\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −671.875 −0.559333 −0.279667 0.960097i $$-0.590224\pi$$
−0.279667 + 0.960097i $$0.590224\pi$$
$$114$$ 0 0
$$115$$ 216.958 0.175925
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 1509.59 1.16289
$$120$$ 0 0
$$121$$ −607.756 −0.456616
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ 1818.41 1.27053 0.635265 0.772294i $$-0.280891\pi$$
0.635265 + 0.772294i $$0.280891\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1768.99 1.17983 0.589915 0.807465i $$-0.299161\pi$$
0.589915 + 0.807465i $$0.299161\pi$$
$$132$$ 0 0
$$133$$ 1833.63 1.19546
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −1842.85 −1.14924 −0.574618 0.818422i $$-0.694849\pi$$
−0.574618 + 0.818422i $$0.694849\pi$$
$$138$$ 0 0
$$139$$ 1371.14 0.836678 0.418339 0.908291i $$-0.362612\pi$$
0.418339 + 0.908291i $$0.362612\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 1836.31 1.07384
$$144$$ 0 0
$$145$$ −874.756 −0.500997
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −2139.65 −1.17642 −0.588212 0.808707i $$-0.700168\pi$$
−0.588212 + 0.808707i $$0.700168\pi$$
$$150$$ 0 0
$$151$$ −1199.81 −0.646616 −0.323308 0.946294i $$-0.604795\pi$$
−0.323308 + 0.946294i $$0.604795\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −1113.33 −0.576935
$$156$$ 0 0
$$157$$ 1847.13 0.938960 0.469480 0.882943i $$-0.344441\pi$$
0.469480 + 0.882943i $$0.344441\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1058.39 0.518091
$$162$$ 0 0
$$163$$ −3203.12 −1.53919 −0.769595 0.638532i $$-0.779542\pi$$
−0.769595 + 0.638532i $$0.779542\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 590.884 0.273796 0.136898 0.990585i $$-0.456287\pi$$
0.136898 + 0.990585i $$0.456287\pi$$
$$168$$ 0 0
$$169$$ 2465.36 1.12215
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 3110.68 1.36705 0.683527 0.729925i $$-0.260445\pi$$
0.683527 + 0.729925i $$0.260445\pi$$
$$174$$ 0 0
$$175$$ −609.789 −0.263404
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −2448.94 −1.02258 −0.511291 0.859408i $$-0.670832\pi$$
−0.511291 + 0.859408i $$0.670832\pi$$
$$180$$ 0 0
$$181$$ 2416.13 0.992206 0.496103 0.868264i $$-0.334764\pi$$
0.496103 + 0.868264i $$0.334764\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −335.906 −0.133494
$$186$$ 0 0
$$187$$ 1664.42 0.650879
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −1652.25 −0.625929 −0.312964 0.949765i $$-0.601322\pi$$
−0.312964 + 0.949765i $$0.601322\pi$$
$$192$$ 0 0
$$193$$ 4736.65 1.76659 0.883295 0.468818i $$-0.155320\pi$$
0.883295 + 0.468818i $$0.155320\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −2106.79 −0.761944 −0.380972 0.924587i $$-0.624411\pi$$
−0.380972 + 0.924587i $$0.624411\pi$$
$$198$$ 0 0
$$199$$ 891.149 0.317447 0.158723 0.987323i $$-0.449262\pi$$
0.158723 + 0.987323i $$0.449262\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −4267.33 −1.47541
$$204$$ 0 0
$$205$$ 111.035 0.0378294
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 2021.69 0.669105
$$210$$ 0 0
$$211$$ 3607.11 1.17689 0.588444 0.808538i $$-0.299741\pi$$
0.588444 + 0.808538i $$0.299741\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 423.721 0.134407
$$216$$ 0 0
$$217$$ −5431.18 −1.69904
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 4225.93 1.28628
$$222$$ 0 0
$$223$$ 2507.12 0.752865 0.376432 0.926444i $$-0.377151\pi$$
0.376432 + 0.926444i $$0.377151\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −3970.82 −1.16102 −0.580512 0.814252i $$-0.697147\pi$$
−0.580512 + 0.814252i $$0.697147\pi$$
$$228$$ 0 0
$$229$$ 5494.88 1.58564 0.792820 0.609455i $$-0.208612\pi$$
0.792820 + 0.609455i $$0.208612\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 6895.27 1.93873 0.969365 0.245625i $$-0.0789931\pi$$
0.969365 + 0.245625i $$0.0789931\pi$$
$$234$$ 0 0
$$235$$ 2928.25 0.812842
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −4308.62 −1.16612 −0.583058 0.812431i $$-0.698144\pi$$
−0.583058 + 0.812431i $$0.698144\pi$$
$$240$$ 0 0
$$241$$ 3448.92 0.921843 0.460922 0.887441i $$-0.347519\pi$$
0.460922 + 0.887441i $$0.347519\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −1259.74 −0.328497
$$246$$ 0 0
$$247$$ 5133.04 1.32230
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −6184.80 −1.55530 −0.777652 0.628695i $$-0.783589\pi$$
−0.777652 + 0.628695i $$0.783589\pi$$
$$252$$ 0 0
$$253$$ 1166.94 0.289979
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 4616.98 1.12062 0.560310 0.828283i $$-0.310682\pi$$
0.560310 + 0.828283i $$0.310682\pi$$
$$258$$ 0 0
$$259$$ −1638.66 −0.393132
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −728.557 −0.170816 −0.0854082 0.996346i $$-0.527219\pi$$
−0.0854082 + 0.996346i $$0.527219\pi$$
$$264$$ 0 0
$$265$$ −191.813 −0.0444640
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 6718.17 1.52273 0.761364 0.648325i $$-0.224530\pi$$
0.761364 + 0.648325i $$0.224530\pi$$
$$270$$ 0 0
$$271$$ −2435.89 −0.546015 −0.273007 0.962012i $$-0.588018\pi$$
−0.273007 + 0.962012i $$0.588018\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −672.330 −0.147429
$$276$$ 0 0
$$277$$ −4808.18 −1.04295 −0.521473 0.853268i $$-0.674617\pi$$
−0.521473 + 0.853268i $$0.674617\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −3535.67 −0.750606 −0.375303 0.926902i $$-0.622461\pi$$
−0.375303 + 0.926902i $$0.622461\pi$$
$$282$$ 0 0
$$283$$ 3070.64 0.644984 0.322492 0.946572i $$-0.395479\pi$$
0.322492 + 0.946572i $$0.395479\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 541.663 0.111405
$$288$$ 0 0
$$289$$ −1082.64 −0.220362
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 1245.18 0.248273 0.124136 0.992265i $$-0.460384\pi$$
0.124136 + 0.992265i $$0.460384\pi$$
$$294$$ 0 0
$$295$$ −460.000 −0.0907872
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 2962.84 0.573061
$$300$$ 0 0
$$301$$ 2067.04 0.395822
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 1131.50 0.212425
$$306$$ 0 0
$$307$$ −4269.79 −0.793778 −0.396889 0.917867i $$-0.629910\pi$$
−0.396889 + 0.917867i $$0.629910\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −2584.39 −0.471213 −0.235607 0.971849i $$-0.575708\pi$$
−0.235607 + 0.971849i $$0.575708\pi$$
$$312$$ 0 0
$$313$$ −8899.44 −1.60711 −0.803556 0.595229i $$-0.797061\pi$$
−0.803556 + 0.595229i $$0.797061\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 6168.58 1.09294 0.546470 0.837479i $$-0.315971\pi$$
0.546470 + 0.837479i $$0.315971\pi$$
$$318$$ 0 0
$$319$$ −4705.00 −0.825797
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 4652.55 0.801470
$$324$$ 0 0
$$325$$ −1707.04 −0.291352
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 14284.9 2.39378
$$330$$ 0 0
$$331$$ −9996.33 −1.65996 −0.829982 0.557791i $$-0.811649\pi$$
−0.829982 + 0.557791i $$0.811649\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −4290.18 −0.699694
$$336$$ 0 0
$$337$$ −2497.40 −0.403685 −0.201843 0.979418i $$-0.564693\pi$$
−0.201843 + 0.979418i $$0.564693\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −5988.21 −0.950967
$$342$$ 0 0
$$343$$ 2220.90 0.349614
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −9001.22 −1.39254 −0.696269 0.717780i $$-0.745158\pi$$
−0.696269 + 0.717780i $$0.745158\pi$$
$$348$$ 0 0
$$349$$ −10707.4 −1.64227 −0.821135 0.570733i $$-0.806659\pi$$
−0.821135 + 0.570733i $$0.806659\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −2843.35 −0.428714 −0.214357 0.976755i $$-0.568766\pi$$
−0.214357 + 0.976755i $$0.568766\pi$$
$$354$$ 0 0
$$355$$ −584.561 −0.0873952
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 9328.89 1.37148 0.685738 0.727849i $$-0.259480\pi$$
0.685738 + 0.727849i $$0.259480\pi$$
$$360$$ 0 0
$$361$$ −1207.77 −0.176086
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −4558.85 −0.653756
$$366$$ 0 0
$$367$$ 46.7367 0.00664751 0.00332376 0.999994i $$-0.498942\pi$$
0.00332376 + 0.999994i $$0.498942\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −935.721 −0.130944
$$372$$ 0 0
$$373$$ −3791.98 −0.526384 −0.263192 0.964743i $$-0.584775\pi$$
−0.263192 + 0.964743i $$0.584775\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −11945.9 −1.63195
$$378$$ 0 0
$$379$$ 1881.74 0.255035 0.127518 0.991836i $$-0.459299\pi$$
0.127518 + 0.991836i $$0.459299\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −12011.9 −1.60256 −0.801280 0.598289i $$-0.795847\pi$$
−0.801280 + 0.598289i $$0.795847\pi$$
$$384$$ 0 0
$$385$$ −3279.83 −0.434171
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 7758.91 1.01129 0.505645 0.862741i $$-0.331254\pi$$
0.505645 + 0.862741i $$0.331254\pi$$
$$390$$ 0 0
$$391$$ 2685.50 0.347344
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 1425.65 0.181600
$$396$$ 0 0
$$397$$ 14841.9 1.87630 0.938152 0.346223i $$-0.112536\pi$$
0.938152 + 0.346223i $$0.112536\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −435.446 −0.0542273 −0.0271136 0.999632i $$-0.508632\pi$$
−0.0271136 + 0.999632i $$0.508632\pi$$
$$402$$ 0 0
$$403$$ −15204.0 −1.87932
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −1806.72 −0.220039
$$408$$ 0 0
$$409$$ 10550.6 1.27554 0.637770 0.770227i $$-0.279857\pi$$
0.637770 + 0.770227i $$0.279857\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −2244.02 −0.267364
$$414$$ 0 0
$$415$$ 4995.50 0.590890
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 4161.11 0.485164 0.242582 0.970131i $$-0.422006\pi$$
0.242582 + 0.970131i $$0.422006\pi$$
$$420$$ 0 0
$$421$$ 154.745 0.0179140 0.00895701 0.999960i $$-0.497149\pi$$
0.00895701 + 0.999960i $$0.497149\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −1547.25 −0.176594
$$426$$ 0 0
$$427$$ 5519.82 0.625580
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −4393.72 −0.491040 −0.245520 0.969392i $$-0.578959\pi$$
−0.245520 + 0.969392i $$0.578959\pi$$
$$432$$ 0 0
$$433$$ 4437.08 0.492454 0.246227 0.969212i $$-0.420809\pi$$
0.246227 + 0.969212i $$0.420809\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 3261.95 0.357071
$$438$$ 0 0
$$439$$ −7046.68 −0.766104 −0.383052 0.923727i $$-0.625127\pi$$
−0.383052 + 0.923727i $$0.625127\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 7335.09 0.786683 0.393341 0.919392i $$-0.371319\pi$$
0.393341 + 0.919392i $$0.371319\pi$$
$$444$$ 0 0
$$445$$ 1874.24 0.199657
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 731.914 0.0769290 0.0384645 0.999260i $$-0.487753\pi$$
0.0384645 + 0.999260i $$0.487753\pi$$
$$450$$ 0 0
$$451$$ 597.217 0.0623545
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −8327.45 −0.858016
$$456$$ 0 0
$$457$$ −12611.7 −1.29092 −0.645462 0.763792i $$-0.723335\pi$$
−0.645462 + 0.763792i $$0.723335\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 12971.9 1.31055 0.655275 0.755391i $$-0.272553\pi$$
0.655275 + 0.755391i $$0.272553\pi$$
$$462$$ 0 0
$$463$$ 9899.48 0.993667 0.496833 0.867846i $$-0.334496\pi$$
0.496833 + 0.867846i $$0.334496\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −2180.60 −0.216073 −0.108037 0.994147i $$-0.534456\pi$$
−0.108037 + 0.994147i $$0.534456\pi$$
$$468$$ 0 0
$$469$$ −20928.8 −2.06056
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 2279.04 0.221544
$$474$$ 0 0
$$475$$ −1879.37 −0.181539
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −4132.54 −0.394197 −0.197099 0.980384i $$-0.563152\pi$$
−0.197099 + 0.980384i $$0.563152\pi$$
$$480$$ 0 0
$$481$$ −4587.23 −0.434844
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 1137.07 0.106457
$$486$$ 0 0
$$487$$ −19277.8 −1.79376 −0.896880 0.442275i $$-0.854172\pi$$
−0.896880 + 0.442275i $$0.854172\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 1315.88 0.120946 0.0604732 0.998170i $$-0.480739\pi$$
0.0604732 + 0.998170i $$0.480739\pi$$
$$492$$ 0 0
$$493$$ −10827.7 −0.989159
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −2851.67 −0.257374
$$498$$ 0 0
$$499$$ 10477.9 0.939988 0.469994 0.882670i $$-0.344256\pi$$
0.469994 + 0.882670i $$0.344256\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 9312.29 0.825476 0.412738 0.910850i $$-0.364572\pi$$
0.412738 + 0.910850i $$0.364572\pi$$
$$504$$ 0 0
$$505$$ −9288.76 −0.818504
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 21595.3 1.88054 0.940270 0.340429i $$-0.110572\pi$$
0.940270 + 0.340429i $$0.110572\pi$$
$$510$$ 0 0
$$511$$ −22239.5 −1.92528
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −863.936 −0.0739215
$$516$$ 0 0
$$517$$ 15750.0 1.33981
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −9386.88 −0.789341 −0.394670 0.918823i $$-0.629141\pi$$
−0.394670 + 0.918823i $$0.629141\pi$$
$$522$$ 0 0
$$523$$ −19068.9 −1.59431 −0.797156 0.603773i $$-0.793663\pi$$
−0.797156 + 0.603773i $$0.793663\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −13780.8 −1.13909
$$528$$ 0 0
$$529$$ −10284.2 −0.845251
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 1516.33 0.123226
$$534$$ 0 0
$$535$$ 9033.94 0.730040
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −6775.69 −0.541465
$$540$$ 0 0
$$541$$ −22209.4 −1.76499 −0.882493 0.470325i $$-0.844137\pi$$
−0.882493 + 0.470325i $$0.844137\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −1438.22 −0.113040
$$546$$ 0 0
$$547$$ −2233.14 −0.174556 −0.0872782 0.996184i $$-0.527817\pi$$
−0.0872782 + 0.996184i $$0.527817\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −13151.9 −1.01686
$$552$$ 0 0
$$553$$ 6954.74 0.534802
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −19727.7 −1.50070 −0.750351 0.661039i $$-0.770116\pi$$
−0.750351 + 0.661039i $$0.770116\pi$$
$$558$$ 0 0
$$559$$ 5786.46 0.437819
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 12025.7 0.900216 0.450108 0.892974i $$-0.351385\pi$$
0.450108 + 0.892974i $$0.351385\pi$$
$$564$$ 0 0
$$565$$ 3359.37 0.250142
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −19708.8 −1.45209 −0.726044 0.687648i $$-0.758643\pi$$
−0.726044 + 0.687648i $$0.758643\pi$$
$$570$$ 0 0
$$571$$ 3965.65 0.290643 0.145322 0.989384i $$-0.453578\pi$$
0.145322 + 0.989384i $$0.453578\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −1084.79 −0.0786762
$$576$$ 0 0
$$577$$ 4134.92 0.298334 0.149167 0.988812i $$-0.452341\pi$$
0.149167 + 0.988812i $$0.452341\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 24369.6 1.74014
$$582$$ 0 0
$$583$$ −1031.69 −0.0732904
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −10949.0 −0.769873 −0.384937 0.922943i $$-0.625777\pi$$
−0.384937 + 0.922943i $$0.625777\pi$$
$$588$$ 0 0
$$589$$ −16738.9 −1.17099
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 17834.7 1.23505 0.617523 0.786553i $$-0.288136\pi$$
0.617523 + 0.786553i $$0.288136\pi$$
$$594$$ 0 0
$$595$$ −7547.95 −0.520060
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 6052.79 0.412872 0.206436 0.978460i $$-0.433814\pi$$
0.206436 + 0.978460i $$0.433814\pi$$
$$600$$ 0 0
$$601$$ 5233.02 0.355174 0.177587 0.984105i $$-0.443171\pi$$
0.177587 + 0.984105i $$0.443171\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 3038.78 0.204205
$$606$$ 0 0
$$607$$ −24209.4 −1.61883 −0.809413 0.587240i $$-0.800215\pi$$
−0.809413 + 0.587240i $$0.800215\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 39989.0 2.64776
$$612$$ 0 0
$$613$$ 7255.45 0.478050 0.239025 0.971013i $$-0.423172\pi$$
0.239025 + 0.971013i $$0.423172\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −29694.7 −1.93754 −0.968772 0.247951i $$-0.920243\pi$$
−0.968772 + 0.247951i $$0.920243\pi$$
$$618$$ 0 0
$$619$$ 10775.2 0.699665 0.349833 0.936812i $$-0.386238\pi$$
0.349833 + 0.936812i $$0.386238\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 9143.11 0.587979
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −4157.84 −0.263568
$$630$$ 0 0
$$631$$ −3661.18 −0.230981 −0.115491 0.993309i $$-0.536844\pi$$
−0.115491 + 0.993309i $$0.536844\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −9092.03 −0.568199
$$636$$ 0 0
$$637$$ −17203.4 −1.07005
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −5627.43 −0.346755 −0.173378 0.984855i $$-0.555468\pi$$
−0.173378 + 0.984855i $$0.555468\pi$$
$$642$$ 0 0
$$643$$ −10732.5 −0.658243 −0.329122 0.944288i $$-0.606753\pi$$
−0.329122 + 0.944288i $$0.606753\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −28753.1 −1.74714 −0.873571 0.486697i $$-0.838202\pi$$
−0.873571 + 0.486697i $$0.838202\pi$$
$$648$$ 0 0
$$649$$ −2474.17 −0.149645
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 27398.6 1.64195 0.820974 0.570966i $$-0.193431\pi$$
0.820974 + 0.570966i $$0.193431\pi$$
$$654$$ 0 0
$$655$$ −8844.97 −0.527636
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 26446.7 1.56331 0.781653 0.623714i $$-0.214377\pi$$
0.781653 + 0.623714i $$0.214377\pi$$
$$660$$ 0 0
$$661$$ 25643.7 1.50897 0.754483 0.656320i $$-0.227888\pi$$
0.754483 + 0.656320i $$0.227888\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −9168.13 −0.534624
$$666$$ 0 0
$$667$$ −7591.40 −0.440690
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 6085.94 0.350142
$$672$$ 0 0
$$673$$ 10796.0 0.618357 0.309179 0.951004i $$-0.399946\pi$$
0.309179 + 0.951004i $$0.399946\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 13183.2 0.748406 0.374203 0.927347i $$-0.377916\pi$$
0.374203 + 0.927347i $$0.377916\pi$$
$$678$$ 0 0
$$679$$ 5546.96 0.313509
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 1349.06 0.0755786 0.0377893 0.999286i $$-0.487968\pi$$
0.0377893 + 0.999286i $$0.487968\pi$$
$$684$$ 0 0
$$685$$ 9214.26 0.513954
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −2619.45 −0.144838
$$690$$ 0 0
$$691$$ 25133.8 1.38370 0.691849 0.722042i $$-0.256796\pi$$
0.691849 + 0.722042i $$0.256796\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −6855.68 −0.374174
$$696$$ 0 0
$$697$$ 1374.39 0.0746896
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 14808.9 0.797895 0.398948 0.916974i $$-0.369375\pi$$
0.398948 + 0.916974i $$0.369375\pi$$
$$702$$ 0 0
$$703$$ −5050.33 −0.270948
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −45313.5 −2.41045
$$708$$ 0 0
$$709$$ −13857.2 −0.734019 −0.367009 0.930217i $$-0.619618\pi$$
−0.367009 + 0.930217i $$0.619618\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −9661.84 −0.507488
$$714$$ 0 0
$$715$$ −9181.54 −0.480238
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 21853.5 1.13352 0.566759 0.823884i $$-0.308197\pi$$
0.566759 + 0.823884i $$0.308197\pi$$
$$720$$ 0 0
$$721$$ −4214.55 −0.217695
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 4373.78 0.224053
$$726$$ 0 0
$$727$$ −1442.23 −0.0735757 −0.0367878 0.999323i $$-0.511713\pi$$
−0.0367878 + 0.999323i $$0.511713\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 5244.81 0.265371
$$732$$ 0 0
$$733$$ 2606.50 0.131342 0.0656708 0.997841i $$-0.479081\pi$$
0.0656708 + 0.997841i $$0.479081\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −23075.3 −1.15331
$$738$$ 0 0
$$739$$ 9072.37 0.451600 0.225800 0.974174i $$-0.427500\pi$$
0.225800 + 0.974174i $$0.427500\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 10093.2 0.498364 0.249182 0.968457i $$-0.419838\pi$$
0.249182 + 0.968457i $$0.419838\pi$$
$$744$$ 0 0
$$745$$ 10698.3 0.526113
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 44070.4 2.14993
$$750$$ 0 0
$$751$$ 22033.4 1.07059 0.535294 0.844666i $$-0.320201\pi$$
0.535294 + 0.844666i $$0.320201\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 5999.04 0.289175
$$756$$ 0 0
$$757$$ 8362.75 0.401518 0.200759 0.979641i $$-0.435659\pi$$
0.200759 + 0.979641i $$0.435659\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 11405.1 0.543280 0.271640 0.962399i $$-0.412434\pi$$
0.271640 + 0.962399i $$0.412434\pi$$
$$762$$ 0 0
$$763$$ −7016.09 −0.332896
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −6281.89 −0.295731
$$768$$ 0 0
$$769$$ 1447.34 0.0678704 0.0339352 0.999424i $$-0.489196\pi$$
0.0339352 + 0.999424i $$0.489196\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −25835.9 −1.20214 −0.601070 0.799196i $$-0.705259\pi$$
−0.601070 + 0.799196i $$0.705259\pi$$
$$774$$ 0 0
$$775$$ 5566.66 0.258013
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 1669.40 0.0767812
$$780$$ 0 0
$$781$$ −3144.14 −0.144054
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −9235.63 −0.419916
$$786$$ 0 0
$$787$$ 15859.9 0.718355 0.359178 0.933269i $$-0.383057\pi$$
0.359178 + 0.933269i $$0.383057\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 16388.1 0.736653
$$792$$ 0 0
$$793$$ 15452.1 0.691956
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −32805.2 −1.45799 −0.728997 0.684517i $$-0.760013\pi$$
−0.728997 + 0.684517i $$0.760013\pi$$
$$798$$ 0 0
$$799$$ 36245.8 1.60486
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −24520.4 −1.07759
$$804$$ 0 0
$$805$$ −5291.94 −0.231697
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −17068.4 −0.741772 −0.370886 0.928678i $$-0.620946\pi$$
−0.370886 + 0.928678i $$0.620946\pi$$
$$810$$ 0 0
$$811$$ 12079.5 0.523018 0.261509 0.965201i $$-0.415780\pi$$
0.261509 + 0.965201i $$0.415780\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 16015.6 0.688347
$$816$$ 0 0
$$817$$ 6370.61 0.272802
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −34364.3 −1.46081 −0.730403 0.683016i $$-0.760668\pi$$
−0.730403 + 0.683016i $$0.760668\pi$$
$$822$$ 0 0
$$823$$ 12983.0 0.549891 0.274946 0.961460i $$-0.411340\pi$$
0.274946 + 0.961460i $$0.411340\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 25503.7 1.07237 0.536186 0.844100i $$-0.319865\pi$$
0.536186 + 0.844100i $$0.319865\pi$$
$$828$$ 0 0
$$829$$ 26016.9 1.08999 0.544997 0.838438i $$-0.316531\pi$$
0.544997 + 0.838438i $$0.316531\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −15593.0 −0.648579
$$834$$ 0 0
$$835$$ −2954.42 −0.122445
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 11090.3 0.456353 0.228177 0.973620i $$-0.426724\pi$$
0.228177 + 0.973620i $$0.426724\pi$$
$$840$$ 0 0
$$841$$ 6218.91 0.254988
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −12326.8 −0.501839
$$846$$ 0 0
$$847$$ 14824.1 0.601372
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −2915.10 −0.117425
$$852$$ 0 0
$$853$$ 41006.8 1.64601 0.823004 0.568036i $$-0.192296\pi$$
0.823004 + 0.568036i $$0.192296\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −38890.1 −1.55013 −0.775065 0.631881i $$-0.782283\pi$$
−0.775065 + 0.631881i $$0.782283\pi$$
$$858$$ 0 0
$$859$$ 43062.9 1.71046 0.855232 0.518246i $$-0.173415\pi$$
0.855232 + 0.518246i $$0.173415\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 2294.97 0.0905232 0.0452616 0.998975i $$-0.485588\pi$$
0.0452616 + 0.998975i $$0.485588\pi$$
$$864$$ 0 0
$$865$$ −15553.4 −0.611365
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 7668.03 0.299333
$$870$$ 0 0
$$871$$ −58587.9 −2.27919
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 3048.94 0.117798
$$876$$ 0 0
$$877$$ 33278.9 1.28136 0.640678 0.767810i $$-0.278653\pi$$
0.640678 + 0.767810i $$0.278653\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 9821.12 0.375575 0.187788 0.982210i $$-0.439868\pi$$
0.187788 + 0.982210i $$0.439868\pi$$
$$882$$ 0 0
$$883$$ −13745.6 −0.523869 −0.261934 0.965086i $$-0.584360\pi$$
−0.261934 + 0.965086i $$0.584360\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 24565.8 0.929922 0.464961 0.885331i $$-0.346068\pi$$
0.464961 + 0.885331i $$0.346068\pi$$
$$888$$ 0 0
$$889$$ −44353.7 −1.67331
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 44026.0 1.64980
$$894$$ 0 0
$$895$$ 12244.7 0.457312
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 38955.7 1.44521
$$900$$ 0 0
$$901$$ −2374.25 −0.0877889
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −12080.6 −0.443728
$$906$$ 0 0
$$907$$ 20218.5 0.740181 0.370091 0.928996i $$-0.379327\pi$$
0.370091 + 0.928996i $$0.379327\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 4552.96 0.165583 0.0827915 0.996567i $$-0.473616\pi$$
0.0827915 + 0.996567i $$0.473616\pi$$
$$912$$ 0 0
$$913$$ 26869.0 0.973969
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −43148.5 −1.55386
$$918$$ 0 0
$$919$$ 19814.0 0.711211 0.355605 0.934636i $$-0.384275\pi$$
0.355605 + 0.934636i $$0.384275\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −7982.94 −0.284682
$$924$$ 0 0
$$925$$ 1679.53 0.0597002
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 40202.9 1.41982 0.709911 0.704291i $$-0.248735\pi$$
0.709911 + 0.704291i $$0.248735\pi$$
$$930$$ 0 0
$$931$$ −18940.1 −0.666742
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −8322.09 −0.291082
$$936$$ 0 0
$$937$$ 24587.9 0.857260 0.428630 0.903480i $$-0.358996\pi$$
0.428630 + 0.903480i $$0.358996\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −33462.1 −1.15923 −0.579614 0.814891i $$-0.696797\pi$$
−0.579614 + 0.814891i $$0.696797\pi$$
$$942$$ 0 0
$$943$$ 963.596 0.0332757
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −8385.17 −0.287731 −0.143866 0.989597i $$-0.545953\pi$$
−0.143866 + 0.989597i $$0.545953\pi$$
$$948$$ 0 0
$$949$$ −62256.9 −2.12955
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −55532.0 −1.88758 −0.943788 0.330552i $$-0.892765\pi$$
−0.943788 + 0.330552i $$0.892765\pi$$
$$954$$ 0 0
$$955$$ 8261.23 0.279924
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 44950.0 1.51357
$$960$$ 0 0
$$961$$ 19789.3 0.664272
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −23683.3 −0.790043
$$966$$ 0 0
$$967$$ −44080.8 −1.46592 −0.732960 0.680272i $$-0.761862\pi$$
−0.732960 + 0.680272i $$0.761862\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 29721.4 0.982291 0.491146 0.871077i $$-0.336578\pi$$
0.491146 + 0.871077i $$0.336578\pi$$
$$972$$ 0 0
$$973$$ −33444.2 −1.10192
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −462.881 −0.0151575 −0.00757876 0.999971i $$-0.502412\pi$$
−0.00757876 + 0.999971i $$0.502412\pi$$
$$978$$ 0 0
$$979$$ 10080.9 0.329096
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 27389.7 0.888703 0.444351 0.895853i $$-0.353434\pi$$
0.444351 + 0.895853i $$0.353434\pi$$
$$984$$ 0 0
$$985$$ 10534.0 0.340752
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 3677.18 0.118228
$$990$$ 0 0
$$991$$ 32596.6 1.04487 0.522435 0.852679i $$-0.325024\pi$$
0.522435 + 0.852679i $$0.325024\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −4455.75 −0.141966
$$996$$ 0 0
$$997$$ −26630.6 −0.845936 −0.422968 0.906145i $$-0.639012\pi$$
−0.422968 + 0.906145i $$0.639012\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

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

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