# Properties

 Label 729.2.a.a.1.4 Level $729$ Weight $2$ Character 729.1 Self dual yes Analytic conductor $5.821$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,2,Mod(1,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("729.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$729 = 3^{6}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 729.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: $$5.82109430735$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.1397493.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1$$ x^6 - 3*x^5 - 3*x^4 + 10*x^3 + 3*x^2 - 6*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 27) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-1.11662$$ of defining polynomial Character $$\chi$$ $$=$$ 729.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.415466 q^{2} -1.82739 q^{4} +2.21519 q^{5} -1.31963 q^{7} +1.59015 q^{8} +O(q^{10})$$ $$q-0.415466 q^{2} -1.82739 q^{4} +2.21519 q^{5} -1.31963 q^{7} +1.59015 q^{8} -0.920335 q^{10} -5.21519 q^{11} -0.0180585 q^{13} +0.548261 q^{14} +2.99412 q^{16} -3.13280 q^{17} +0.417352 q^{19} -4.04801 q^{20} +2.16673 q^{22} -1.03439 q^{23} -0.0929475 q^{25} +0.00750270 q^{26} +2.41147 q^{28} -7.80722 q^{29} +3.72966 q^{31} -4.42426 q^{32} +1.30157 q^{34} -2.92322 q^{35} +4.42476 q^{37} -0.173396 q^{38} +3.52248 q^{40} -3.67494 q^{41} -8.30787 q^{43} +9.53017 q^{44} +0.429753 q^{46} -7.09791 q^{47} -5.25858 q^{49} +0.0386165 q^{50} +0.0329999 q^{52} -1.30057 q^{53} -11.5526 q^{55} -2.09841 q^{56} +3.24364 q^{58} -3.70181 q^{59} +6.91424 q^{61} -1.54955 q^{62} -4.15011 q^{64} -0.0400030 q^{65} -11.0268 q^{67} +5.72483 q^{68} +1.21450 q^{70} -6.08428 q^{71} -0.546973 q^{73} -1.83834 q^{74} -0.762665 q^{76} +6.88211 q^{77} +0.489144 q^{79} +6.63254 q^{80} +1.52681 q^{82} +4.61367 q^{83} -6.93973 q^{85} +3.45164 q^{86} -8.29293 q^{88} -3.37307 q^{89} +0.0238305 q^{91} +1.89023 q^{92} +2.94894 q^{94} +0.924513 q^{95} +9.94136 q^{97} +2.18476 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8}+O(q^{10})$$ 6 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 - 6 * q^8 $$6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8} + 3 q^{10} - 12 q^{11} - 6 q^{14} - 3 q^{16} - 9 q^{17} + 3 q^{19} - 6 q^{20} + 6 q^{22} - 15 q^{23} - 6 q^{25} - 15 q^{26} - 6 q^{28} - 12 q^{29} - 12 q^{35} + 3 q^{37} + 3 q^{38} + 6 q^{40} - 15 q^{41} - 3 q^{44} + 3 q^{46} - 21 q^{47} - 12 q^{49} - 3 q^{50} + 12 q^{52} - 9 q^{53} - 6 q^{55} + 6 q^{56} - 12 q^{58} - 24 q^{59} - 9 q^{61} + 12 q^{62} - 12 q^{64} + 6 q^{65} - 9 q^{67} + 9 q^{68} + 15 q^{70} - 27 q^{71} - 6 q^{73} + 12 q^{74} + 6 q^{76} + 12 q^{77} + 21 q^{80} - 6 q^{82} - 12 q^{83} + 21 q^{86} + 12 q^{88} - 9 q^{89} - 6 q^{91} - 6 q^{92} + 6 q^{94} - 12 q^{95} + 45 q^{98}+O(q^{100})$$ 6 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 - 6 * q^8 + 3 * q^10 - 12 * q^11 - 6 * q^14 - 3 * q^16 - 9 * q^17 + 3 * q^19 - 6 * q^20 + 6 * q^22 - 15 * q^23 - 6 * q^25 - 15 * q^26 - 6 * q^28 - 12 * q^29 - 12 * q^35 + 3 * q^37 + 3 * q^38 + 6 * q^40 - 15 * q^41 - 3 * q^44 + 3 * q^46 - 21 * q^47 - 12 * q^49 - 3 * q^50 + 12 * q^52 - 9 * q^53 - 6 * q^55 + 6 * q^56 - 12 * q^58 - 24 * q^59 - 9 * q^61 + 12 * q^62 - 12 * q^64 + 6 * q^65 - 9 * q^67 + 9 * q^68 + 15 * q^70 - 27 * q^71 - 6 * q^73 + 12 * q^74 + 6 * q^76 + 12 * q^77 + 21 * q^80 - 6 * q^82 - 12 * q^83 + 21 * q^86 + 12 * q^88 - 9 * q^89 - 6 * q^91 - 6 * q^92 + 6 * q^94 - 12 * q^95 + 45 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.415466 −0.293779 −0.146889 0.989153i $$-0.546926\pi$$
−0.146889 + 0.989153i $$0.546926\pi$$
$$3$$ 0 0
$$4$$ −1.82739 −0.913694
$$5$$ 2.21519 0.990662 0.495331 0.868704i $$-0.335047\pi$$
0.495331 + 0.868704i $$0.335047\pi$$
$$6$$ 0 0
$$7$$ −1.31963 −0.498773 −0.249386 0.968404i $$-0.580229\pi$$
−0.249386 + 0.968404i $$0.580229\pi$$
$$8$$ 1.59015 0.562203
$$9$$ 0 0
$$10$$ −0.920335 −0.291036
$$11$$ −5.21519 −1.57244 −0.786219 0.617948i $$-0.787964\pi$$
−0.786219 + 0.617948i $$0.787964\pi$$
$$12$$ 0 0
$$13$$ −0.0180585 −0.00500853 −0.00250427 0.999997i $$-0.500797\pi$$
−0.00250427 + 0.999997i $$0.500797\pi$$
$$14$$ 0.548261 0.146529
$$15$$ 0 0
$$16$$ 2.99412 0.748530
$$17$$ −3.13280 −0.759814 −0.379907 0.925025i $$-0.624044\pi$$
−0.379907 + 0.925025i $$0.624044\pi$$
$$18$$ 0 0
$$19$$ 0.417352 0.0957472 0.0478736 0.998853i $$-0.484756\pi$$
0.0478736 + 0.998853i $$0.484756\pi$$
$$20$$ −4.04801 −0.905162
$$21$$ 0 0
$$22$$ 2.16673 0.461949
$$23$$ −1.03439 −0.215684 −0.107842 0.994168i $$-0.534394\pi$$
−0.107842 + 0.994168i $$0.534394\pi$$
$$24$$ 0 0
$$25$$ −0.0929475 −0.0185895
$$26$$ 0.00750270 0.00147140
$$27$$ 0 0
$$28$$ 2.41147 0.455726
$$29$$ −7.80722 −1.44976 −0.724882 0.688873i $$-0.758106\pi$$
−0.724882 + 0.688873i $$0.758106\pi$$
$$30$$ 0 0
$$31$$ 3.72966 0.669868 0.334934 0.942242i $$-0.391286\pi$$
0.334934 + 0.942242i $$0.391286\pi$$
$$32$$ −4.42426 −0.782106
$$33$$ 0 0
$$34$$ 1.30157 0.223218
$$35$$ −2.92322 −0.494115
$$36$$ 0 0
$$37$$ 4.42476 0.727426 0.363713 0.931511i $$-0.381509\pi$$
0.363713 + 0.931511i $$0.381509\pi$$
$$38$$ −0.173396 −0.0281285
$$39$$ 0 0
$$40$$ 3.52248 0.556953
$$41$$ −3.67494 −0.573929 −0.286965 0.957941i $$-0.592646\pi$$
−0.286965 + 0.957941i $$0.592646\pi$$
$$42$$ 0 0
$$43$$ −8.30787 −1.26694 −0.633469 0.773768i $$-0.718370\pi$$
−0.633469 + 0.773768i $$0.718370\pi$$
$$44$$ 9.53017 1.43673
$$45$$ 0 0
$$46$$ 0.429753 0.0633636
$$47$$ −7.09791 −1.03534 −0.517668 0.855581i $$-0.673200\pi$$
−0.517668 + 0.855581i $$0.673200\pi$$
$$48$$ 0 0
$$49$$ −5.25858 −0.751226
$$50$$ 0.0386165 0.00546120
$$51$$ 0 0
$$52$$ 0.0329999 0.00457627
$$53$$ −1.30057 −0.178648 −0.0893238 0.996003i $$-0.528471\pi$$
−0.0893238 + 0.996003i $$0.528471\pi$$
$$54$$ 0 0
$$55$$ −11.5526 −1.55775
$$56$$ −2.09841 −0.280412
$$57$$ 0 0
$$58$$ 3.24364 0.425910
$$59$$ −3.70181 −0.481935 −0.240967 0.970533i $$-0.577465\pi$$
−0.240967 + 0.970533i $$0.577465\pi$$
$$60$$ 0 0
$$61$$ 6.91424 0.885277 0.442639 0.896700i $$-0.354042\pi$$
0.442639 + 0.896700i $$0.354042\pi$$
$$62$$ −1.54955 −0.196793
$$63$$ 0 0
$$64$$ −4.15011 −0.518764
$$65$$ −0.0400030 −0.00496176
$$66$$ 0 0
$$67$$ −11.0268 −1.34714 −0.673569 0.739125i $$-0.735239\pi$$
−0.673569 + 0.739125i $$0.735239\pi$$
$$68$$ 5.72483 0.694238
$$69$$ 0 0
$$70$$ 1.21450 0.145161
$$71$$ −6.08428 −0.722071 −0.361035 0.932552i $$-0.617577\pi$$
−0.361035 + 0.932552i $$0.617577\pi$$
$$72$$ 0 0
$$73$$ −0.546973 −0.0640183 −0.0320092 0.999488i $$-0.510191\pi$$
−0.0320092 + 0.999488i $$0.510191\pi$$
$$74$$ −1.83834 −0.213702
$$75$$ 0 0
$$76$$ −0.762665 −0.0874836
$$77$$ 6.88211 0.784289
$$78$$ 0 0
$$79$$ 0.489144 0.0550330 0.0275165 0.999621i $$-0.491240\pi$$
0.0275165 + 0.999621i $$0.491240\pi$$
$$80$$ 6.63254 0.741540
$$81$$ 0 0
$$82$$ 1.52681 0.168608
$$83$$ 4.61367 0.506416 0.253208 0.967412i $$-0.418514\pi$$
0.253208 + 0.967412i $$0.418514\pi$$
$$84$$ 0 0
$$85$$ −6.93973 −0.752719
$$86$$ 3.45164 0.372200
$$87$$ 0 0
$$88$$ −8.29293 −0.884029
$$89$$ −3.37307 −0.357544 −0.178772 0.983891i $$-0.557212\pi$$
−0.178772 + 0.983891i $$0.557212\pi$$
$$90$$ 0 0
$$91$$ 0.0238305 0.00249812
$$92$$ 1.89023 0.197070
$$93$$ 0 0
$$94$$ 2.94894 0.304160
$$95$$ 0.924513 0.0948531
$$96$$ 0 0
$$97$$ 9.94136 1.00939 0.504696 0.863297i $$-0.331605\pi$$
0.504696 + 0.863297i $$0.331605\pi$$
$$98$$ 2.18476 0.220694
$$99$$ 0 0
$$100$$ 0.169851 0.0169851
$$101$$ 13.7995 1.37310 0.686550 0.727082i $$-0.259124\pi$$
0.686550 + 0.727082i $$0.259124\pi$$
$$102$$ 0 0
$$103$$ 4.56512 0.449815 0.224907 0.974380i $$-0.427792\pi$$
0.224907 + 0.974380i $$0.427792\pi$$
$$104$$ −0.0287158 −0.00281581
$$105$$ 0 0
$$106$$ 0.540345 0.0524829
$$107$$ −11.2965 −1.09207 −0.546035 0.837762i $$-0.683864\pi$$
−0.546035 + 0.837762i $$0.683864\pi$$
$$108$$ 0 0
$$109$$ 14.5032 1.38915 0.694577 0.719419i $$-0.255592\pi$$
0.694577 + 0.719419i $$0.255592\pi$$
$$110$$ 4.79972 0.457635
$$111$$ 0 0
$$112$$ −3.95113 −0.373347
$$113$$ 12.5584 1.18140 0.590699 0.806892i $$-0.298852\pi$$
0.590699 + 0.806892i $$0.298852\pi$$
$$114$$ 0 0
$$115$$ −2.29136 −0.213670
$$116$$ 14.2668 1.32464
$$117$$ 0 0
$$118$$ 1.53798 0.141582
$$119$$ 4.13413 0.378975
$$120$$ 0 0
$$121$$ 16.1982 1.47256
$$122$$ −2.87263 −0.260076
$$123$$ 0 0
$$124$$ −6.81554 −0.612054
$$125$$ −11.2818 −1.00908
$$126$$ 0 0
$$127$$ −8.39499 −0.744935 −0.372467 0.928045i $$-0.621488\pi$$
−0.372467 + 0.928045i $$0.621488\pi$$
$$128$$ 10.5727 0.934508
$$129$$ 0 0
$$130$$ 0.0166199 0.00145766
$$131$$ −15.5349 −1.35729 −0.678645 0.734466i $$-0.737433\pi$$
−0.678645 + 0.734466i $$0.737433\pi$$
$$132$$ 0 0
$$133$$ −0.550750 −0.0477561
$$134$$ 4.58126 0.395761
$$135$$ 0 0
$$136$$ −4.98162 −0.427170
$$137$$ 12.0074 1.02586 0.512930 0.858430i $$-0.328560\pi$$
0.512930 + 0.858430i $$0.328560\pi$$
$$138$$ 0 0
$$139$$ 6.14512 0.521222 0.260611 0.965444i $$-0.416076\pi$$
0.260611 + 0.965444i $$0.416076\pi$$
$$140$$ 5.34187 0.451470
$$141$$ 0 0
$$142$$ 2.52781 0.212129
$$143$$ 0.0941785 0.00787561
$$144$$ 0 0
$$145$$ −17.2945 −1.43623
$$146$$ 0.227249 0.0188072
$$147$$ 0 0
$$148$$ −8.08575 −0.664644
$$149$$ 0.882820 0.0723235 0.0361617 0.999346i $$-0.488487\pi$$
0.0361617 + 0.999346i $$0.488487\pi$$
$$150$$ 0 0
$$151$$ 8.22547 0.669379 0.334690 0.942328i $$-0.391368\pi$$
0.334690 + 0.942328i $$0.391368\pi$$
$$152$$ 0.663653 0.0538294
$$153$$ 0 0
$$154$$ −2.85929 −0.230408
$$155$$ 8.26190 0.663612
$$156$$ 0 0
$$157$$ 12.5598 1.00238 0.501192 0.865336i $$-0.332895\pi$$
0.501192 + 0.865336i $$0.332895\pi$$
$$158$$ −0.203223 −0.0161675
$$159$$ 0 0
$$160$$ −9.80056 −0.774802
$$161$$ 1.36501 0.107578
$$162$$ 0 0
$$163$$ 3.31466 0.259624 0.129812 0.991539i $$-0.458563\pi$$
0.129812 + 0.991539i $$0.458563\pi$$
$$164$$ 6.71554 0.524396
$$165$$ 0 0
$$166$$ −1.91682 −0.148774
$$167$$ −20.5630 −1.59121 −0.795606 0.605815i $$-0.792847\pi$$
−0.795606 + 0.605815i $$0.792847\pi$$
$$168$$ 0 0
$$169$$ −12.9997 −0.999975
$$170$$ 2.88322 0.221133
$$171$$ 0 0
$$172$$ 15.1817 1.15759
$$173$$ 14.0333 1.06693 0.533465 0.845822i $$-0.320890\pi$$
0.533465 + 0.845822i $$0.320890\pi$$
$$174$$ 0 0
$$175$$ 0.122656 0.00927194
$$176$$ −15.6149 −1.17702
$$177$$ 0 0
$$178$$ 1.40139 0.105039
$$179$$ 10.1900 0.761636 0.380818 0.924650i $$-0.375642\pi$$
0.380818 + 0.924650i $$0.375642\pi$$
$$180$$ 0 0
$$181$$ 24.0547 1.78797 0.893987 0.448093i $$-0.147897\pi$$
0.893987 + 0.448093i $$0.147897\pi$$
$$182$$ −0.00990079 −0.000733895 0
$$183$$ 0 0
$$184$$ −1.64483 −0.121258
$$185$$ 9.80166 0.720633
$$186$$ 0 0
$$187$$ 16.3381 1.19476
$$188$$ 12.9706 0.945981
$$189$$ 0 0
$$190$$ −0.384104 −0.0278658
$$191$$ 10.9464 0.792052 0.396026 0.918239i $$-0.370389\pi$$
0.396026 + 0.918239i $$0.370389\pi$$
$$192$$ 0 0
$$193$$ −10.8060 −0.777830 −0.388915 0.921274i $$-0.627150\pi$$
−0.388915 + 0.921274i $$0.627150\pi$$
$$194$$ −4.13030 −0.296538
$$195$$ 0 0
$$196$$ 9.60946 0.686390
$$197$$ −22.0734 −1.57266 −0.786331 0.617806i $$-0.788022\pi$$
−0.786331 + 0.617806i $$0.788022\pi$$
$$198$$ 0 0
$$199$$ 12.8868 0.913518 0.456759 0.889590i $$-0.349010\pi$$
0.456759 + 0.889590i $$0.349010\pi$$
$$200$$ −0.147801 −0.0104511
$$201$$ 0 0
$$202$$ −5.73322 −0.403388
$$203$$ 10.3026 0.723103
$$204$$ 0 0
$$205$$ −8.14068 −0.568570
$$206$$ −1.89665 −0.132146
$$207$$ 0 0
$$208$$ −0.0540694 −0.00374904
$$209$$ −2.17657 −0.150557
$$210$$ 0 0
$$211$$ −23.9956 −1.65193 −0.825964 0.563723i $$-0.809368\pi$$
−0.825964 + 0.563723i $$0.809368\pi$$
$$212$$ 2.37665 0.163229
$$213$$ 0 0
$$214$$ 4.69330 0.320827
$$215$$ −18.4035 −1.25511
$$216$$ 0 0
$$217$$ −4.92177 −0.334112
$$218$$ −6.02558 −0.408104
$$219$$ 0 0
$$220$$ 21.1111 1.42331
$$221$$ 0.0565736 0.00380555
$$222$$ 0 0
$$223$$ 21.6622 1.45061 0.725303 0.688430i $$-0.241700\pi$$
0.725303 + 0.688430i $$0.241700\pi$$
$$224$$ 5.83838 0.390093
$$225$$ 0 0
$$226$$ −5.21760 −0.347070
$$227$$ 21.6419 1.43642 0.718211 0.695826i $$-0.244961\pi$$
0.718211 + 0.695826i $$0.244961\pi$$
$$228$$ 0 0
$$229$$ −10.8054 −0.714038 −0.357019 0.934097i $$-0.616207\pi$$
−0.357019 + 0.934097i $$0.616207\pi$$
$$230$$ 0.951982 0.0627719
$$231$$ 0 0
$$232$$ −12.4147 −0.815062
$$233$$ 7.63900 0.500447 0.250224 0.968188i $$-0.419496\pi$$
0.250224 + 0.968188i $$0.419496\pi$$
$$234$$ 0 0
$$235$$ −15.7232 −1.02567
$$236$$ 6.76465 0.440341
$$237$$ 0 0
$$238$$ −1.71759 −0.111335
$$239$$ −3.23149 −0.209028 −0.104514 0.994523i $$-0.533329\pi$$
−0.104514 + 0.994523i $$0.533329\pi$$
$$240$$ 0 0
$$241$$ −26.5449 −1.70991 −0.854955 0.518702i $$-0.826415\pi$$
−0.854955 + 0.518702i $$0.826415\pi$$
$$242$$ −6.72979 −0.432608
$$243$$ 0 0
$$244$$ −12.6350 −0.808873
$$245$$ −11.6487 −0.744210
$$246$$ 0 0
$$247$$ −0.00753676 −0.000479553 0
$$248$$ 5.93073 0.376602
$$249$$ 0 0
$$250$$ 4.68722 0.296446
$$251$$ −4.49930 −0.283993 −0.141997 0.989867i $$-0.545352\pi$$
−0.141997 + 0.989867i $$0.545352\pi$$
$$252$$ 0 0
$$253$$ 5.39452 0.339150
$$254$$ 3.48783 0.218846
$$255$$ 0 0
$$256$$ 3.90761 0.244226
$$257$$ 13.7354 0.856792 0.428396 0.903591i $$-0.359079\pi$$
0.428396 + 0.903591i $$0.359079\pi$$
$$258$$ 0 0
$$259$$ −5.83904 −0.362820
$$260$$ 0.0731010 0.00453353
$$261$$ 0 0
$$262$$ 6.45423 0.398743
$$263$$ −24.2026 −1.49239 −0.746197 0.665725i $$-0.768122\pi$$
−0.746197 + 0.665725i $$0.768122\pi$$
$$264$$ 0 0
$$265$$ −2.88101 −0.176979
$$266$$ 0.228818 0.0140297
$$267$$ 0 0
$$268$$ 20.1502 1.23087
$$269$$ 12.0062 0.732032 0.366016 0.930609i $$-0.380722\pi$$
0.366016 + 0.930609i $$0.380722\pi$$
$$270$$ 0 0
$$271$$ 3.71777 0.225839 0.112919 0.993604i $$-0.463980\pi$$
0.112919 + 0.993604i $$0.463980\pi$$
$$272$$ −9.37997 −0.568744
$$273$$ 0 0
$$274$$ −4.98867 −0.301376
$$275$$ 0.484739 0.0292308
$$276$$ 0 0
$$277$$ −23.4831 −1.41096 −0.705482 0.708728i $$-0.749269\pi$$
−0.705482 + 0.708728i $$0.749269\pi$$
$$278$$ −2.55309 −0.153124
$$279$$ 0 0
$$280$$ −4.64837 −0.277793
$$281$$ −20.3717 −1.21528 −0.607638 0.794214i $$-0.707883\pi$$
−0.607638 + 0.794214i $$0.707883\pi$$
$$282$$ 0 0
$$283$$ 11.5999 0.689545 0.344772 0.938686i $$-0.387956\pi$$
0.344772 + 0.938686i $$0.387956\pi$$
$$284$$ 11.1183 0.659752
$$285$$ 0 0
$$286$$ −0.0391280 −0.00231369
$$287$$ 4.84956 0.286260
$$288$$ 0 0
$$289$$ −7.18559 −0.422682
$$290$$ 7.18526 0.421933
$$291$$ 0 0
$$292$$ 0.999532 0.0584932
$$293$$ 31.5742 1.84458 0.922291 0.386496i $$-0.126315\pi$$
0.922291 + 0.386496i $$0.126315\pi$$
$$294$$ 0 0
$$295$$ −8.20020 −0.477434
$$296$$ 7.03603 0.408961
$$297$$ 0 0
$$298$$ −0.366782 −0.0212471
$$299$$ 0.0186795 0.00108026
$$300$$ 0 0
$$301$$ 10.9633 0.631915
$$302$$ −3.41741 −0.196650
$$303$$ 0 0
$$304$$ 1.24960 0.0716697
$$305$$ 15.3163 0.877010
$$306$$ 0 0
$$307$$ −8.12054 −0.463464 −0.231732 0.972780i $$-0.574439\pi$$
−0.231732 + 0.972780i $$0.574439\pi$$
$$308$$ −12.5763 −0.716601
$$309$$ 0 0
$$310$$ −3.43254 −0.194955
$$311$$ −23.8486 −1.35233 −0.676164 0.736751i $$-0.736359\pi$$
−0.676164 + 0.736751i $$0.736359\pi$$
$$312$$ 0 0
$$313$$ 26.9105 1.52107 0.760535 0.649297i $$-0.224937\pi$$
0.760535 + 0.649297i $$0.224937\pi$$
$$314$$ −5.21818 −0.294479
$$315$$ 0 0
$$316$$ −0.893856 −0.0502833
$$317$$ −8.33233 −0.467990 −0.233995 0.972238i $$-0.575180\pi$$
−0.233995 + 0.972238i $$0.575180\pi$$
$$318$$ 0 0
$$319$$ 40.7161 2.27967
$$320$$ −9.19328 −0.513920
$$321$$ 0 0
$$322$$ −0.567114 −0.0316040
$$323$$ −1.30748 −0.0727501
$$324$$ 0 0
$$325$$ 0.00167849 9.31061e−5 0
$$326$$ −1.37713 −0.0762721
$$327$$ 0 0
$$328$$ −5.84371 −0.322665
$$329$$ 9.36661 0.516398
$$330$$ 0 0
$$331$$ −6.42026 −0.352889 −0.176445 0.984311i $$-0.556460\pi$$
−0.176445 + 0.984311i $$0.556460\pi$$
$$332$$ −8.43096 −0.462709
$$333$$ 0 0
$$334$$ 8.54322 0.467464
$$335$$ −24.4264 −1.33456
$$336$$ 0 0
$$337$$ 7.47489 0.407183 0.203592 0.979056i $$-0.434739\pi$$
0.203592 + 0.979056i $$0.434739\pi$$
$$338$$ 5.40093 0.293772
$$339$$ 0 0
$$340$$ 12.6816 0.687755
$$341$$ −19.4509 −1.05333
$$342$$ 0 0
$$343$$ 16.1768 0.873464
$$344$$ −13.2108 −0.712277
$$345$$ 0 0
$$346$$ −5.83035 −0.313442
$$347$$ −31.4545 −1.68857 −0.844283 0.535898i $$-0.819973\pi$$
−0.844283 + 0.535898i $$0.819973\pi$$
$$348$$ 0 0
$$349$$ 11.8529 0.634474 0.317237 0.948346i $$-0.397245\pi$$
0.317237 + 0.948346i $$0.397245\pi$$
$$350$$ −0.0509595 −0.00272390
$$351$$ 0 0
$$352$$ 23.0733 1.22981
$$353$$ −8.20708 −0.436819 −0.218409 0.975857i $$-0.570087\pi$$
−0.218409 + 0.975857i $$0.570087\pi$$
$$354$$ 0 0
$$355$$ −13.4778 −0.715328
$$356$$ 6.16390 0.326686
$$357$$ 0 0
$$358$$ −4.23360 −0.223753
$$359$$ 17.7273 0.935611 0.467806 0.883831i $$-0.345045\pi$$
0.467806 + 0.883831i $$0.345045\pi$$
$$360$$ 0 0
$$361$$ −18.8258 −0.990832
$$362$$ −9.99393 −0.525269
$$363$$ 0 0
$$364$$ −0.0435477 −0.00228252
$$365$$ −1.21165 −0.0634205
$$366$$ 0 0
$$367$$ −20.3195 −1.06067 −0.530335 0.847788i $$-0.677934\pi$$
−0.530335 + 0.847788i $$0.677934\pi$$
$$368$$ −3.09708 −0.161446
$$369$$ 0 0
$$370$$ −4.07226 −0.211707
$$371$$ 1.71628 0.0891046
$$372$$ 0 0
$$373$$ −9.68144 −0.501286 −0.250643 0.968080i $$-0.580642\pi$$
−0.250643 + 0.968080i $$0.580642\pi$$
$$374$$ −6.78793 −0.350996
$$375$$ 0 0
$$376$$ −11.2867 −0.582069
$$377$$ 0.140987 0.00726119
$$378$$ 0 0
$$379$$ −4.12905 −0.212095 −0.106048 0.994361i $$-0.533820\pi$$
−0.106048 + 0.994361i $$0.533820\pi$$
$$380$$ −1.68944 −0.0866667
$$381$$ 0 0
$$382$$ −4.54785 −0.232688
$$383$$ 4.75018 0.242723 0.121362 0.992608i $$-0.461274\pi$$
0.121362 + 0.992608i $$0.461274\pi$$
$$384$$ 0 0
$$385$$ 15.2452 0.776966
$$386$$ 4.48951 0.228510
$$387$$ 0 0
$$388$$ −18.1667 −0.922275
$$389$$ −21.8133 −1.10598 −0.552990 0.833188i $$-0.686513\pi$$
−0.552990 + 0.833188i $$0.686513\pi$$
$$390$$ 0 0
$$391$$ 3.24052 0.163880
$$392$$ −8.36193 −0.422341
$$393$$ 0 0
$$394$$ 9.17074 0.462015
$$395$$ 1.08355 0.0545191
$$396$$ 0 0
$$397$$ −34.8490 −1.74902 −0.874512 0.485005i $$-0.838818\pi$$
−0.874512 + 0.485005i $$0.838818\pi$$
$$398$$ −5.35401 −0.268373
$$399$$ 0 0
$$400$$ −0.278296 −0.0139148
$$401$$ −18.8261 −0.940130 −0.470065 0.882632i $$-0.655769\pi$$
−0.470065 + 0.882632i $$0.655769\pi$$
$$402$$ 0 0
$$403$$ −0.0673522 −0.00335505
$$404$$ −25.2170 −1.25459
$$405$$ 0 0
$$406$$ −4.28040 −0.212433
$$407$$ −23.0759 −1.14383
$$408$$ 0 0
$$409$$ −6.35996 −0.314480 −0.157240 0.987560i $$-0.550260\pi$$
−0.157240 + 0.987560i $$0.550260\pi$$
$$410$$ 3.38218 0.167034
$$411$$ 0 0
$$412$$ −8.34224 −0.410993
$$413$$ 4.88502 0.240376
$$414$$ 0 0
$$415$$ 10.2201 0.501687
$$416$$ 0.0798955 0.00391720
$$417$$ 0 0
$$418$$ 0.904291 0.0442303
$$419$$ −24.3180 −1.18801 −0.594005 0.804461i $$-0.702454\pi$$
−0.594005 + 0.804461i $$0.702454\pi$$
$$420$$ 0 0
$$421$$ −7.99004 −0.389411 −0.194705 0.980862i $$-0.562375\pi$$
−0.194705 + 0.980862i $$0.562375\pi$$
$$422$$ 9.96937 0.485302
$$423$$ 0 0
$$424$$ −2.06811 −0.100436
$$425$$ 0.291185 0.0141246
$$426$$ 0 0
$$427$$ −9.12423 −0.441552
$$428$$ 20.6430 0.997818
$$429$$ 0 0
$$430$$ 7.64603 0.368724
$$431$$ 9.87124 0.475481 0.237740 0.971329i $$-0.423593\pi$$
0.237740 + 0.971329i $$0.423593\pi$$
$$432$$ 0 0
$$433$$ −6.10369 −0.293325 −0.146662 0.989187i $$-0.546853\pi$$
−0.146662 + 0.989187i $$0.546853\pi$$
$$434$$ 2.04483 0.0981550
$$435$$ 0 0
$$436$$ −26.5029 −1.26926
$$437$$ −0.431704 −0.0206512
$$438$$ 0 0
$$439$$ −15.1340 −0.722308 −0.361154 0.932506i $$-0.617617\pi$$
−0.361154 + 0.932506i $$0.617617\pi$$
$$440$$ −18.3704 −0.875774
$$441$$ 0 0
$$442$$ −0.0235044 −0.00111799
$$443$$ −0.722793 −0.0343410 −0.0171705 0.999853i $$-0.505466\pi$$
−0.0171705 + 0.999853i $$0.505466\pi$$
$$444$$ 0 0
$$445$$ −7.47197 −0.354205
$$446$$ −8.99990 −0.426158
$$447$$ 0 0
$$448$$ 5.47661 0.258746
$$449$$ 1.66845 0.0787389 0.0393695 0.999225i $$-0.487465\pi$$
0.0393695 + 0.999225i $$0.487465\pi$$
$$450$$ 0 0
$$451$$ 19.1655 0.902468
$$452$$ −22.9491 −1.07944
$$453$$ 0 0
$$454$$ −8.99147 −0.421991
$$455$$ 0.0527891 0.00247479
$$456$$ 0 0
$$457$$ 11.0834 0.518462 0.259231 0.965815i $$-0.416531\pi$$
0.259231 + 0.965815i $$0.416531\pi$$
$$458$$ 4.48926 0.209769
$$459$$ 0 0
$$460$$ 4.18720 0.195229
$$461$$ −21.8844 −1.01926 −0.509629 0.860394i $$-0.670217\pi$$
−0.509629 + 0.860394i $$0.670217\pi$$
$$462$$ 0 0
$$463$$ 24.8517 1.15496 0.577479 0.816406i $$-0.304037\pi$$
0.577479 + 0.816406i $$0.304037\pi$$
$$464$$ −23.3758 −1.08519
$$465$$ 0 0
$$466$$ −3.17375 −0.147021
$$467$$ −11.8355 −0.547683 −0.273842 0.961775i $$-0.588294\pi$$
−0.273842 + 0.961775i $$0.588294\pi$$
$$468$$ 0 0
$$469$$ 14.5513 0.671916
$$470$$ 6.53246 0.301320
$$471$$ 0 0
$$472$$ −5.88644 −0.270945
$$473$$ 43.3271 1.99218
$$474$$ 0 0
$$475$$ −0.0387919 −0.00177989
$$476$$ −7.55465 −0.346267
$$477$$ 0 0
$$478$$ 1.34258 0.0614080
$$479$$ 2.88735 0.131926 0.0659632 0.997822i $$-0.478988\pi$$
0.0659632 + 0.997822i $$0.478988\pi$$
$$480$$ 0 0
$$481$$ −0.0799046 −0.00364333
$$482$$ 11.0285 0.502336
$$483$$ 0 0
$$484$$ −29.6003 −1.34547
$$485$$ 22.0220 0.999966
$$486$$ 0 0
$$487$$ 8.75903 0.396910 0.198455 0.980110i $$-0.436408\pi$$
0.198455 + 0.980110i $$0.436408\pi$$
$$488$$ 10.9947 0.497706
$$489$$ 0 0
$$490$$ 4.83966 0.218633
$$491$$ −22.5730 −1.01871 −0.509354 0.860557i $$-0.670115\pi$$
−0.509354 + 0.860557i $$0.670115\pi$$
$$492$$ 0 0
$$493$$ 24.4584 1.10155
$$494$$ 0.00313127 0.000140883 0
$$495$$ 0 0
$$496$$ 11.1671 0.501416
$$497$$ 8.02899 0.360149
$$498$$ 0 0
$$499$$ −25.3328 −1.13405 −0.567026 0.823700i $$-0.691906\pi$$
−0.567026 + 0.823700i $$0.691906\pi$$
$$500$$ 20.6163 0.921988
$$501$$ 0 0
$$502$$ 1.86931 0.0834312
$$503$$ 3.74414 0.166943 0.0834714 0.996510i $$-0.473399\pi$$
0.0834714 + 0.996510i $$0.473399\pi$$
$$504$$ 0 0
$$505$$ 30.5684 1.36028
$$506$$ −2.24124 −0.0996353
$$507$$ 0 0
$$508$$ 15.3409 0.680642
$$509$$ 24.3499 1.07929 0.539645 0.841893i $$-0.318559\pi$$
0.539645 + 0.841893i $$0.318559\pi$$
$$510$$ 0 0
$$511$$ 0.721801 0.0319306
$$512$$ −22.7690 −1.00626
$$513$$ 0 0
$$514$$ −5.70660 −0.251707
$$515$$ 10.1126 0.445614
$$516$$ 0 0
$$517$$ 37.0169 1.62800
$$518$$ 2.42592 0.106589
$$519$$ 0 0
$$520$$ −0.0636108 −0.00278952
$$521$$ 19.6209 0.859608 0.429804 0.902922i $$-0.358583\pi$$
0.429804 + 0.902922i $$0.358583\pi$$
$$522$$ 0 0
$$523$$ 20.8154 0.910194 0.455097 0.890442i $$-0.349605\pi$$
0.455097 + 0.890442i $$0.349605\pi$$
$$524$$ 28.3883 1.24015
$$525$$ 0 0
$$526$$ 10.0553 0.438434
$$527$$ −11.6843 −0.508975
$$528$$ 0 0
$$529$$ −21.9300 −0.953480
$$530$$ 1.19696 0.0519928
$$531$$ 0 0
$$532$$ 1.00643 0.0436345
$$533$$ 0.0663640 0.00287454
$$534$$ 0 0
$$535$$ −25.0238 −1.08187
$$536$$ −17.5343 −0.757365
$$537$$ 0 0
$$538$$ −4.98818 −0.215056
$$539$$ 27.4245 1.18126
$$540$$ 0 0
$$541$$ −30.6272 −1.31676 −0.658382 0.752684i $$-0.728759\pi$$
−0.658382 + 0.752684i $$0.728759\pi$$
$$542$$ −1.54461 −0.0663467
$$543$$ 0 0
$$544$$ 13.8603 0.594255
$$545$$ 32.1273 1.37618
$$546$$ 0 0
$$547$$ 22.6477 0.968345 0.484172 0.874973i $$-0.339121\pi$$
0.484172 + 0.874973i $$0.339121\pi$$
$$548$$ −21.9422 −0.937323
$$549$$ 0 0
$$550$$ −0.201393 −0.00858741
$$551$$ −3.25836 −0.138811
$$552$$ 0 0
$$553$$ −0.645489 −0.0274490
$$554$$ 9.75644 0.414512
$$555$$ 0 0
$$556$$ −11.2295 −0.476237
$$557$$ 36.4518 1.54451 0.772256 0.635311i $$-0.219128\pi$$
0.772256 + 0.635311i $$0.219128\pi$$
$$558$$ 0 0
$$559$$ 0.150028 0.00634550
$$560$$ −8.75249 −0.369860
$$561$$ 0 0
$$562$$ 8.46377 0.357023
$$563$$ 26.5162 1.11753 0.558763 0.829327i $$-0.311276\pi$$
0.558763 + 0.829327i $$0.311276\pi$$
$$564$$ 0 0
$$565$$ 27.8193 1.17037
$$566$$ −4.81938 −0.202574
$$567$$ 0 0
$$568$$ −9.67492 −0.405950
$$569$$ 22.9674 0.962844 0.481422 0.876489i $$-0.340121\pi$$
0.481422 + 0.876489i $$0.340121\pi$$
$$570$$ 0 0
$$571$$ −4.79801 −0.200790 −0.100395 0.994948i $$-0.532011\pi$$
−0.100395 + 0.994948i $$0.532011\pi$$
$$572$$ −0.172101 −0.00719589
$$573$$ 0 0
$$574$$ −2.01483 −0.0840973
$$575$$ 0.0961436 0.00400947
$$576$$ 0 0
$$577$$ −4.31333 −0.179566 −0.0897831 0.995961i $$-0.528617\pi$$
−0.0897831 + 0.995961i $$0.528617\pi$$
$$578$$ 2.98537 0.124175
$$579$$ 0 0
$$580$$ 31.6037 1.31227
$$581$$ −6.08833 −0.252587
$$582$$ 0 0
$$583$$ 6.78274 0.280912
$$584$$ −0.869769 −0.0359913
$$585$$ 0 0
$$586$$ −13.1180 −0.541899
$$587$$ 41.8222 1.72619 0.863094 0.505044i $$-0.168524\pi$$
0.863094 + 0.505044i $$0.168524\pi$$
$$588$$ 0 0
$$589$$ 1.55658 0.0641379
$$590$$ 3.40691 0.140260
$$591$$ 0 0
$$592$$ 13.2483 0.544500
$$593$$ −31.5370 −1.29507 −0.647536 0.762035i $$-0.724200\pi$$
−0.647536 + 0.762035i $$0.724200\pi$$
$$594$$ 0 0
$$595$$ 9.15786 0.375436
$$596$$ −1.61326 −0.0660815
$$597$$ 0 0
$$598$$ −0.00776070 −0.000317358 0
$$599$$ −12.6303 −0.516060 −0.258030 0.966137i $$-0.583073\pi$$
−0.258030 + 0.966137i $$0.583073\pi$$
$$600$$ 0 0
$$601$$ 20.5430 0.837965 0.418983 0.907994i $$-0.362387\pi$$
0.418983 + 0.907994i $$0.362387\pi$$
$$602$$ −4.55489 −0.185643
$$603$$ 0 0
$$604$$ −15.0311 −0.611608
$$605$$ 35.8820 1.45881
$$606$$ 0 0
$$607$$ 12.9126 0.524105 0.262052 0.965054i $$-0.415601\pi$$
0.262052 + 0.965054i $$0.415601\pi$$
$$608$$ −1.84647 −0.0748844
$$609$$ 0 0
$$610$$ −6.36342 −0.257647
$$611$$ 0.128178 0.00518552
$$612$$ 0 0
$$613$$ −31.1598 −1.25853 −0.629265 0.777191i $$-0.716644\pi$$
−0.629265 + 0.777191i $$0.716644\pi$$
$$614$$ 3.37381 0.136156
$$615$$ 0 0
$$616$$ 10.9436 0.440930
$$617$$ −7.14078 −0.287477 −0.143739 0.989616i $$-0.545912\pi$$
−0.143739 + 0.989616i $$0.545912\pi$$
$$618$$ 0 0
$$619$$ −10.0309 −0.403176 −0.201588 0.979470i $$-0.564610\pi$$
−0.201588 + 0.979470i $$0.564610\pi$$
$$620$$ −15.0977 −0.606338
$$621$$ 0 0
$$622$$ 9.90827 0.397285
$$623$$ 4.45119 0.178333
$$624$$ 0 0
$$625$$ −24.5266 −0.981065
$$626$$ −11.1804 −0.446858
$$627$$ 0 0
$$628$$ −22.9517 −0.915871
$$629$$ −13.8619 −0.552708
$$630$$ 0 0
$$631$$ −7.07560 −0.281675 −0.140838 0.990033i $$-0.544980\pi$$
−0.140838 + 0.990033i $$0.544980\pi$$
$$632$$ 0.777813 0.0309397
$$633$$ 0 0
$$634$$ 3.46180 0.137486
$$635$$ −18.5965 −0.737978
$$636$$ 0 0
$$637$$ 0.0949621 0.00376254
$$638$$ −16.9162 −0.669718
$$639$$ 0 0
$$640$$ 23.4206 0.925781
$$641$$ −5.01121 −0.197931 −0.0989655 0.995091i $$-0.531553\pi$$
−0.0989655 + 0.995091i $$0.531553\pi$$
$$642$$ 0 0
$$643$$ 1.63840 0.0646123 0.0323062 0.999478i $$-0.489715\pi$$
0.0323062 + 0.999478i $$0.489715\pi$$
$$644$$ −2.49440 −0.0982930
$$645$$ 0 0
$$646$$ 0.543213 0.0213724
$$647$$ 34.4927 1.35605 0.678024 0.735040i $$-0.262836\pi$$
0.678024 + 0.735040i $$0.262836\pi$$
$$648$$ 0 0
$$649$$ 19.3056 0.757813
$$650$$ −0.000697358 0 −2.73526e−5 0
$$651$$ 0 0
$$652$$ −6.05717 −0.237217
$$653$$ 38.7606 1.51682 0.758410 0.651778i $$-0.225977\pi$$
0.758410 + 0.651778i $$0.225977\pi$$
$$654$$ 0 0
$$655$$ −34.4127 −1.34462
$$656$$ −11.0032 −0.429604
$$657$$ 0 0
$$658$$ −3.89151 −0.151707
$$659$$ 9.39192 0.365857 0.182929 0.983126i $$-0.441442\pi$$
0.182929 + 0.983126i $$0.441442\pi$$
$$660$$ 0 0
$$661$$ −24.1474 −0.939226 −0.469613 0.882872i $$-0.655607\pi$$
−0.469613 + 0.882872i $$0.655607\pi$$
$$662$$ 2.66740 0.103671
$$663$$ 0 0
$$664$$ 7.33643 0.284709
$$665$$ −1.22001 −0.0473101
$$666$$ 0 0
$$667$$ 8.07569 0.312692
$$668$$ 37.5765 1.45388
$$669$$ 0 0
$$670$$ 10.1483 0.392065
$$671$$ −36.0590 −1.39204
$$672$$ 0 0
$$673$$ −26.4661 −1.02019 −0.510097 0.860117i $$-0.670391\pi$$
−0.510097 + 0.860117i $$0.670391\pi$$
$$674$$ −3.10557 −0.119622
$$675$$ 0 0
$$676$$ 23.7554 0.913671
$$677$$ 31.0668 1.19400 0.596998 0.802243i $$-0.296360\pi$$
0.596998 + 0.802243i $$0.296360\pi$$
$$678$$ 0 0
$$679$$ −13.1189 −0.503457
$$680$$ −11.0352 −0.423181
$$681$$ 0 0
$$682$$ 8.08119 0.309445
$$683$$ 38.1361 1.45924 0.729619 0.683854i $$-0.239697\pi$$
0.729619 + 0.683854i $$0.239697\pi$$
$$684$$ 0 0
$$685$$ 26.5986 1.01628
$$686$$ −6.72090 −0.256605
$$687$$ 0 0
$$688$$ −24.8748 −0.948342
$$689$$ 0.0234864 0.000894762 0
$$690$$ 0 0
$$691$$ 32.9295 1.25270 0.626349 0.779543i $$-0.284548\pi$$
0.626349 + 0.779543i $$0.284548\pi$$
$$692$$ −25.6442 −0.974847
$$693$$ 0 0
$$694$$ 13.0683 0.496065
$$695$$ 13.6126 0.516355
$$696$$ 0 0
$$697$$ 11.5128 0.436080
$$698$$ −4.92450 −0.186395
$$699$$ 0 0
$$700$$ −0.224140 −0.00847171
$$701$$ −2.30710 −0.0871381 −0.0435690 0.999050i $$-0.513873\pi$$
−0.0435690 + 0.999050i $$0.513873\pi$$
$$702$$ 0 0
$$703$$ 1.84668 0.0696490
$$704$$ 21.6436 0.815725
$$705$$ 0 0
$$706$$ 3.40977 0.128328
$$707$$ −18.2102 −0.684865
$$708$$ 0 0
$$709$$ −11.1521 −0.418825 −0.209412 0.977827i $$-0.567155\pi$$
−0.209412 + 0.977827i $$0.567155\pi$$
$$710$$ 5.59958 0.210148
$$711$$ 0 0
$$712$$ −5.36368 −0.201012
$$713$$ −3.85791 −0.144480
$$714$$ 0 0
$$715$$ 0.208623 0.00780206
$$716$$ −18.6211 −0.695902
$$717$$ 0 0
$$718$$ −7.36510 −0.274863
$$719$$ −32.1700 −1.19974 −0.599869 0.800098i $$-0.704781\pi$$
−0.599869 + 0.800098i $$0.704781\pi$$
$$720$$ 0 0
$$721$$ −6.02426 −0.224355
$$722$$ 7.82149 0.291086
$$723$$ 0 0
$$724$$ −43.9573 −1.63366
$$725$$ 0.725662 0.0269504
$$726$$ 0 0
$$727$$ −5.36551 −0.198996 −0.0994979 0.995038i $$-0.531724\pi$$
−0.0994979 + 0.995038i $$0.531724\pi$$
$$728$$ 0.0378942 0.00140445
$$729$$ 0 0
$$730$$ 0.503399 0.0186316
$$731$$ 26.0269 0.962638
$$732$$ 0 0
$$733$$ −14.5964 −0.539129 −0.269564 0.962982i $$-0.586880\pi$$
−0.269564 + 0.962982i $$0.586880\pi$$
$$734$$ 8.44208 0.311603
$$735$$ 0 0
$$736$$ 4.57639 0.168688
$$737$$ 57.5068 2.11829
$$738$$ 0 0
$$739$$ −43.2165 −1.58975 −0.794873 0.606776i $$-0.792463\pi$$
−0.794873 + 0.606776i $$0.792463\pi$$
$$740$$ −17.9114 −0.658438
$$741$$ 0 0
$$742$$ −0.713055 −0.0261771
$$743$$ 8.11221 0.297608 0.148804 0.988867i $$-0.452458\pi$$
0.148804 + 0.988867i $$0.452458\pi$$
$$744$$ 0 0
$$745$$ 1.95561 0.0716481
$$746$$ 4.02231 0.147267
$$747$$ 0 0
$$748$$ −29.8561 −1.09165
$$749$$ 14.9071 0.544695
$$750$$ 0 0
$$751$$ 8.75545 0.319491 0.159746 0.987158i $$-0.448933\pi$$
0.159746 + 0.987158i $$0.448933\pi$$
$$752$$ −21.2520 −0.774981
$$753$$ 0 0
$$754$$ −0.0585753 −0.00213319
$$755$$ 18.2210 0.663129
$$756$$ 0 0
$$757$$ −32.1511 −1.16855 −0.584276 0.811555i $$-0.698622\pi$$
−0.584276 + 0.811555i $$0.698622\pi$$
$$758$$ 1.71548 0.0623091
$$759$$ 0 0
$$760$$ 1.47012 0.0533267
$$761$$ −24.5459 −0.889789 −0.444894 0.895583i $$-0.646759\pi$$
−0.444894 + 0.895583i $$0.646759\pi$$
$$762$$ 0 0
$$763$$ −19.1388 −0.692872
$$764$$ −20.0033 −0.723693
$$765$$ 0 0
$$766$$ −1.97354 −0.0713069
$$767$$ 0.0668492 0.00241379
$$768$$ 0 0
$$769$$ 31.4144 1.13283 0.566416 0.824119i $$-0.308330\pi$$
0.566416 + 0.824119i $$0.308330\pi$$
$$770$$ −6.33385 −0.228256
$$771$$ 0 0
$$772$$ 19.7467 0.710699
$$773$$ 28.7145 1.03279 0.516395 0.856351i $$-0.327274\pi$$
0.516395 + 0.856351i $$0.327274\pi$$
$$774$$ 0 0
$$775$$ −0.346663 −0.0124525
$$776$$ 15.8083 0.567483
$$777$$ 0 0
$$778$$ 9.06270 0.324913
$$779$$ −1.53374 −0.0549521
$$780$$ 0 0
$$781$$ 31.7306 1.13541
$$782$$ −1.34633 −0.0481446
$$783$$ 0 0
$$784$$ −15.7448 −0.562315
$$785$$ 27.8224 0.993022
$$786$$ 0 0
$$787$$ −38.8160 −1.38364 −0.691821 0.722069i $$-0.743191\pi$$
−0.691821 + 0.722069i $$0.743191\pi$$
$$788$$ 40.3366 1.43693
$$789$$ 0 0
$$790$$ −0.450177 −0.0160166
$$791$$ −16.5725 −0.589249
$$792$$ 0 0
$$793$$ −0.124861 −0.00443394
$$794$$ 14.4786 0.513826
$$795$$ 0 0
$$796$$ −23.5491 −0.834676
$$797$$ −4.03410 −0.142895 −0.0714476 0.997444i $$-0.522762\pi$$
−0.0714476 + 0.997444i $$0.522762\pi$$
$$798$$ 0 0
$$799$$ 22.2363 0.786664
$$800$$ 0.411224 0.0145390
$$801$$ 0 0
$$802$$ 7.82160 0.276190
$$803$$ 2.85257 0.100665
$$804$$ 0 0
$$805$$ 3.02374 0.106573
$$806$$ 0.0279826 0.000985644 0
$$807$$ 0 0
$$808$$ 21.9433 0.771961
$$809$$ −29.9454 −1.05283 −0.526413 0.850229i $$-0.676463\pi$$
−0.526413 + 0.850229i $$0.676463\pi$$
$$810$$ 0 0
$$811$$ 20.2173 0.709927 0.354963 0.934880i $$-0.384493\pi$$
0.354963 + 0.934880i $$0.384493\pi$$
$$812$$ −18.8269 −0.660695
$$813$$ 0 0
$$814$$ 9.58727 0.336034
$$815$$ 7.34259 0.257200
$$816$$ 0 0
$$817$$ −3.46731 −0.121306
$$818$$ 2.64235 0.0923875
$$819$$ 0 0
$$820$$ 14.8762 0.519499
$$821$$ 26.8736 0.937896 0.468948 0.883226i $$-0.344633\pi$$
0.468948 + 0.883226i $$0.344633\pi$$
$$822$$ 0 0
$$823$$ 23.0543 0.803623 0.401812 0.915722i $$-0.368381\pi$$
0.401812 + 0.915722i $$0.368381\pi$$
$$824$$ 7.25923 0.252887
$$825$$ 0 0
$$826$$ −2.02956 −0.0706174
$$827$$ −5.10953 −0.177676 −0.0888378 0.996046i $$-0.528315\pi$$
−0.0888378 + 0.996046i $$0.528315\pi$$
$$828$$ 0 0
$$829$$ −30.5982 −1.06272 −0.531360 0.847146i $$-0.678319\pi$$
−0.531360 + 0.847146i $$0.678319\pi$$
$$830$$ −4.24612 −0.147385
$$831$$ 0 0
$$832$$ 0.0749449 0.00259825
$$833$$ 16.4741 0.570792
$$834$$ 0 0
$$835$$ −45.5508 −1.57635
$$836$$ 3.97744 0.137563
$$837$$ 0 0
$$838$$ 10.1033 0.349013
$$839$$ −56.3087 −1.94399 −0.971996 0.234997i $$-0.924492\pi$$
−0.971996 + 0.234997i $$0.924492\pi$$
$$840$$ 0 0
$$841$$ 31.9527 1.10182
$$842$$ 3.31959 0.114401
$$843$$ 0 0
$$844$$ 43.8493 1.50936
$$845$$ −28.7967 −0.990637
$$846$$ 0 0
$$847$$ −21.3756 −0.734474
$$848$$ −3.89408 −0.133723
$$849$$ 0 0
$$850$$ −0.120978 −0.00414950
$$851$$ −4.57691 −0.156894
$$852$$ 0 0
$$853$$ −45.5450 −1.55943 −0.779715 0.626134i $$-0.784636\pi$$
−0.779715 + 0.626134i $$0.784636\pi$$
$$854$$ 3.79081 0.129719
$$855$$ 0 0
$$856$$ −17.9631 −0.613966
$$857$$ −17.4892 −0.597419 −0.298709 0.954344i $$-0.596556\pi$$
−0.298709 + 0.954344i $$0.596556\pi$$
$$858$$ 0 0
$$859$$ 18.3460 0.625958 0.312979 0.949760i $$-0.398673\pi$$
0.312979 + 0.949760i $$0.398673\pi$$
$$860$$ 33.6303 1.14678
$$861$$ 0 0
$$862$$ −4.10117 −0.139686
$$863$$ 4.65373 0.158415 0.0792073 0.996858i $$-0.474761\pi$$
0.0792073 + 0.996858i $$0.474761\pi$$
$$864$$ 0 0
$$865$$ 31.0863 1.05697
$$866$$ 2.53588 0.0861726
$$867$$ 0 0
$$868$$ 8.99399 0.305276
$$869$$ −2.55098 −0.0865360
$$870$$ 0 0
$$871$$ 0.199128 0.00674718
$$872$$ 23.0622 0.780986
$$873$$ 0 0
$$874$$ 0.179358 0.00606688
$$875$$ 14.8878 0.503301
$$876$$ 0 0
$$877$$ −3.66710 −0.123829 −0.0619145 0.998081i $$-0.519721\pi$$
−0.0619145 + 0.998081i $$0.519721\pi$$
$$878$$ 6.28768 0.212199
$$879$$ 0 0
$$880$$ −34.5899 −1.16603
$$881$$ 38.3008 1.29039 0.645193 0.764020i $$-0.276777\pi$$
0.645193 + 0.764020i $$0.276777\pi$$
$$882$$ 0 0
$$883$$ 22.6142 0.761027 0.380513 0.924775i $$-0.375747\pi$$
0.380513 + 0.924775i $$0.375747\pi$$
$$884$$ −0.103382 −0.00347711
$$885$$ 0 0
$$886$$ 0.300296 0.0100887
$$887$$ −1.89656 −0.0636802 −0.0318401 0.999493i $$-0.510137\pi$$
−0.0318401 + 0.999493i $$0.510137\pi$$
$$888$$ 0 0
$$889$$ 11.0783 0.371553
$$890$$ 3.10435 0.104058
$$891$$ 0 0
$$892$$ −39.5852 −1.32541
$$893$$ −2.96233 −0.0991306
$$894$$ 0 0
$$895$$ 22.5727 0.754523
$$896$$ −13.9521 −0.466107
$$897$$ 0 0
$$898$$ −0.693184 −0.0231318
$$899$$ −29.1183 −0.971150
$$900$$ 0 0
$$901$$ 4.07443 0.135739
$$902$$ −7.96262 −0.265126
$$903$$ 0 0
$$904$$ 19.9698 0.664185
$$905$$ 53.2857 1.77128
$$906$$ 0 0
$$907$$ 6.53094 0.216856 0.108428 0.994104i $$-0.465418\pi$$
0.108428 + 0.994104i $$0.465418\pi$$
$$908$$ −39.5481 −1.31245
$$909$$ 0 0
$$910$$ −0.0219321 −0.000727042 0
$$911$$ −43.0371 −1.42588 −0.712942 0.701223i $$-0.752638\pi$$
−0.712942 + 0.701223i $$0.752638\pi$$
$$912$$ 0 0
$$913$$ −24.0612 −0.796308
$$914$$ −4.60480 −0.152313
$$915$$ 0 0
$$916$$ 19.7456 0.652412
$$917$$ 20.5003 0.676980
$$918$$ 0 0
$$919$$ −16.7911 −0.553887 −0.276943 0.960886i $$-0.589321\pi$$
−0.276943 + 0.960886i $$0.589321\pi$$
$$920$$ −3.64361 −0.120126
$$921$$ 0 0
$$922$$ 9.09222 0.299436
$$923$$ 0.109873 0.00361652
$$924$$ 0 0
$$925$$ −0.411270 −0.0135225
$$926$$ −10.3251 −0.339302
$$927$$ 0 0
$$928$$ 34.5412 1.13387
$$929$$ −11.6000 −0.380584 −0.190292 0.981728i $$-0.560943\pi$$
−0.190292 + 0.981728i $$0.560943\pi$$
$$930$$ 0 0
$$931$$ −2.19468 −0.0719277
$$932$$ −13.9594 −0.457256
$$933$$ 0 0
$$934$$ 4.91727 0.160898
$$935$$ 36.1920 1.18360
$$936$$ 0 0
$$937$$ 47.7953 1.56140 0.780702 0.624904i $$-0.214862\pi$$
0.780702 + 0.624904i $$0.214862\pi$$
$$938$$ −6.04557 −0.197395
$$939$$ 0 0
$$940$$ 28.7324 0.937147
$$941$$ 11.2604 0.367077 0.183538 0.983013i $$-0.441245\pi$$
0.183538 + 0.983013i $$0.441245\pi$$
$$942$$ 0 0
$$943$$ 3.80131 0.123788
$$944$$ −11.0837 −0.360743
$$945$$ 0 0
$$946$$ −18.0010 −0.585261
$$947$$ 7.33189 0.238255 0.119127 0.992879i $$-0.461990\pi$$
0.119127 + 0.992879i $$0.461990\pi$$
$$948$$ 0 0
$$949$$ 0.00987752 0.000320638 0
$$950$$ 0.0161167 0.000522895 0
$$951$$ 0 0
$$952$$ 6.57388 0.213061
$$953$$ −24.8753 −0.805791 −0.402895 0.915246i $$-0.631996\pi$$
−0.402895 + 0.915246i $$0.631996\pi$$
$$954$$ 0 0
$$955$$ 24.2483 0.784655
$$956$$ 5.90519 0.190987
$$957$$ 0 0
$$958$$ −1.19960 −0.0387572
$$959$$ −15.8453 −0.511672
$$960$$ 0 0
$$961$$ −17.0896 −0.551277
$$962$$ 0.0331976 0.00107034
$$963$$ 0 0
$$964$$ 48.5079 1.56233
$$965$$ −23.9372 −0.770567
$$966$$ 0 0
$$967$$ −34.0300 −1.09433 −0.547165 0.837025i $$-0.684293\pi$$
−0.547165 + 0.837025i $$0.684293\pi$$
$$968$$ 25.7575 0.827878
$$969$$ 0 0
$$970$$ −9.14938 −0.293769
$$971$$ −34.2476 −1.09906 −0.549530 0.835474i $$-0.685193\pi$$
−0.549530 + 0.835474i $$0.685193\pi$$
$$972$$ 0 0
$$973$$ −8.10928 −0.259971
$$974$$ −3.63908 −0.116604
$$975$$ 0 0
$$976$$ 20.7021 0.662657
$$977$$ −23.4173 −0.749186 −0.374593 0.927189i $$-0.622218\pi$$
−0.374593 + 0.927189i $$0.622218\pi$$
$$978$$ 0 0
$$979$$ 17.5912 0.562216
$$980$$ 21.2868 0.679980
$$981$$ 0 0
$$982$$ 9.37834 0.299275
$$983$$ −33.2031 −1.05902 −0.529508 0.848305i $$-0.677623\pi$$
−0.529508 + 0.848305i $$0.677623\pi$$
$$984$$ 0 0
$$985$$ −48.8966 −1.55798
$$986$$ −10.1617 −0.323613
$$987$$ 0 0
$$988$$ 0.0137726 0.000438165 0
$$989$$ 8.59355 0.273259
$$990$$ 0 0
$$991$$ −28.1806 −0.895187 −0.447594 0.894237i $$-0.647719\pi$$
−0.447594 + 0.894237i $$0.647719\pi$$
$$992$$ −16.5010 −0.523907
$$993$$ 0 0
$$994$$ −3.33577 −0.105804
$$995$$ 28.5466 0.904988
$$996$$ 0 0
$$997$$ −44.9507 −1.42360 −0.711802 0.702381i $$-0.752120\pi$$
−0.711802 + 0.702381i $$0.752120\pi$$
$$998$$ 10.5249 0.333161
$$999$$ 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 729.2.a.a.1.4 6
3.2 odd 2 729.2.a.d.1.3 6
9.2 odd 6 729.2.c.b.244.4 12
9.4 even 3 729.2.c.e.487.3 12
9.5 odd 6 729.2.c.b.487.4 12
9.7 even 3 729.2.c.e.244.3 12
27.2 odd 18 243.2.e.b.190.2 12
27.4 even 9 27.2.e.a.16.2 12
27.5 odd 18 243.2.e.a.217.1 12
27.7 even 9 27.2.e.a.22.2 yes 12
27.11 odd 18 243.2.e.a.28.1 12
27.13 even 9 243.2.e.c.55.1 12
27.14 odd 18 243.2.e.b.55.2 12
27.16 even 9 243.2.e.d.28.2 12
27.20 odd 18 81.2.e.a.37.1 12
27.22 even 9 243.2.e.d.217.2 12
27.23 odd 18 81.2.e.a.46.1 12
27.25 even 9 243.2.e.c.190.1 12
108.7 odd 18 432.2.u.c.49.1 12
108.31 odd 18 432.2.u.c.97.1 12
135.4 even 18 675.2.l.c.151.1 12
135.7 odd 36 675.2.u.b.49.2 24
135.34 even 18 675.2.l.c.76.1 12
135.58 odd 36 675.2.u.b.124.2 24
135.88 odd 36 675.2.u.b.49.3 24
135.112 odd 36 675.2.u.b.124.3 24

By twisted newform
Twist Min Dim Char Parity Ord Type
27.2.e.a.16.2 12 27.4 even 9
27.2.e.a.22.2 yes 12 27.7 even 9
81.2.e.a.37.1 12 27.20 odd 18
81.2.e.a.46.1 12 27.23 odd 18
243.2.e.a.28.1 12 27.11 odd 18
243.2.e.a.217.1 12 27.5 odd 18
243.2.e.b.55.2 12 27.14 odd 18
243.2.e.b.190.2 12 27.2 odd 18
243.2.e.c.55.1 12 27.13 even 9
243.2.e.c.190.1 12 27.25 even 9
243.2.e.d.28.2 12 27.16 even 9
243.2.e.d.217.2 12 27.22 even 9
432.2.u.c.49.1 12 108.7 odd 18
432.2.u.c.97.1 12 108.31 odd 18
675.2.l.c.76.1 12 135.34 even 18
675.2.l.c.151.1 12 135.4 even 18
675.2.u.b.49.2 24 135.7 odd 36
675.2.u.b.49.3 24 135.88 odd 36
675.2.u.b.124.2 24 135.58 odd 36
675.2.u.b.124.3 24 135.112 odd 36
729.2.a.a.1.4 6 1.1 even 1 trivial
729.2.a.d.1.3 6 3.2 odd 2
729.2.c.b.244.4 12 9.2 odd 6
729.2.c.b.487.4 12 9.5 odd 6
729.2.c.e.244.3 12 9.7 even 3
729.2.c.e.487.3 12 9.4 even 3