# Properties

 Label 729.2.a.a.1.3 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 1.3 Root $$2.68091$$ 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)

 $$f(q)$$ $$=$$ $$q-0.801527 q^{2} -1.35755 q^{4} -2.74984 q^{5} +2.37683 q^{7} +2.69117 q^{8} +O(q^{10})$$ $$q-0.801527 q^{2} -1.35755 q^{4} -2.74984 q^{5} +2.37683 q^{7} +2.69117 q^{8} +2.20407 q^{10} -0.250159 q^{11} +2.61198 q^{13} -1.90510 q^{14} +0.558064 q^{16} -0.293377 q^{17} -2.78475 q^{19} +3.73306 q^{20} +0.200509 q^{22} -6.68984 q^{23} +2.56163 q^{25} -2.09357 q^{26} -3.22668 q^{28} -0.355057 q^{29} +2.76547 q^{31} -5.82964 q^{32} +0.235149 q^{34} -6.53592 q^{35} -6.99238 q^{37} +2.23205 q^{38} -7.40029 q^{40} -9.71761 q^{41} +0.260706 q^{43} +0.339604 q^{44} +5.36209 q^{46} -11.4256 q^{47} -1.35066 q^{49} -2.05321 q^{50} -3.54591 q^{52} +5.43137 q^{53} +0.687897 q^{55} +6.39646 q^{56} +0.284588 q^{58} +5.97693 q^{59} -11.8468 q^{61} -2.21660 q^{62} +3.55649 q^{64} -7.18254 q^{65} +1.81030 q^{67} +0.398275 q^{68} +5.23871 q^{70} -0.370510 q^{71} +5.02679 q^{73} +5.60458 q^{74} +3.78044 q^{76} -0.594586 q^{77} +0.802822 q^{79} -1.53459 q^{80} +7.78892 q^{82} -2.75565 q^{83} +0.806740 q^{85} -0.208963 q^{86} -0.673220 q^{88} +10.4507 q^{89} +6.20825 q^{91} +9.08183 q^{92} +9.15789 q^{94} +7.65761 q^{95} -14.8346 q^{97} +1.08259 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 6 q^{8}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.801527 −0.566765 −0.283383 0.959007i $$-0.591457\pi$$
−0.283383 + 0.959007i $$0.591457\pi$$
$$3$$ 0 0
$$4$$ −1.35755 −0.678777
$$5$$ −2.74984 −1.22977 −0.614883 0.788618i $$-0.710797\pi$$
−0.614883 + 0.788618i $$0.710797\pi$$
$$6$$ 0 0
$$7$$ 2.37683 0.898359 0.449179 0.893442i $$-0.351716\pi$$
0.449179 + 0.893442i $$0.351716\pi$$
$$8$$ 2.69117 0.951472
$$9$$ 0 0
$$10$$ 2.20407 0.696989
$$11$$ −0.250159 −0.0754257 −0.0377129 0.999289i $$-0.512007\pi$$
−0.0377129 + 0.999289i $$0.512007\pi$$
$$12$$ 0 0
$$13$$ 2.61198 0.724434 0.362217 0.932094i $$-0.382020\pi$$
0.362217 + 0.932094i $$0.382020\pi$$
$$14$$ −1.90510 −0.509158
$$15$$ 0 0
$$16$$ 0.558064 0.139516
$$17$$ −0.293377 −0.0711543 −0.0355772 0.999367i $$-0.511327\pi$$
−0.0355772 + 0.999367i $$0.511327\pi$$
$$18$$ 0 0
$$19$$ −2.78475 −0.638864 −0.319432 0.947609i $$-0.603492\pi$$
−0.319432 + 0.947609i $$0.603492\pi$$
$$20$$ 3.73306 0.834737
$$21$$ 0 0
$$22$$ 0.200509 0.0427487
$$23$$ −6.68984 −1.39493 −0.697464 0.716620i $$-0.745688\pi$$
−0.697464 + 0.716620i $$0.745688\pi$$
$$24$$ 0 0
$$25$$ 2.56163 0.512325
$$26$$ −2.09357 −0.410584
$$27$$ 0 0
$$28$$ −3.22668 −0.609786
$$29$$ −0.355057 −0.0659324 −0.0329662 0.999456i $$-0.510495\pi$$
−0.0329662 + 0.999456i $$0.510495\pi$$
$$30$$ 0 0
$$31$$ 2.76547 0.496692 0.248346 0.968671i $$-0.420113\pi$$
0.248346 + 0.968671i $$0.420113\pi$$
$$32$$ −5.82964 −1.03055
$$33$$ 0 0
$$34$$ 0.235149 0.0403278
$$35$$ −6.53592 −1.10477
$$36$$ 0 0
$$37$$ −6.99238 −1.14954 −0.574770 0.818315i $$-0.694909\pi$$
−0.574770 + 0.818315i $$0.694909\pi$$
$$38$$ 2.23205 0.362086
$$39$$ 0 0
$$40$$ −7.40029 −1.17009
$$41$$ −9.71761 −1.51764 −0.758818 0.651303i $$-0.774223\pi$$
−0.758818 + 0.651303i $$0.774223\pi$$
$$42$$ 0 0
$$43$$ 0.260706 0.0397574 0.0198787 0.999802i $$-0.493672\pi$$
0.0198787 + 0.999802i $$0.493672\pi$$
$$44$$ 0.339604 0.0511973
$$45$$ 0 0
$$46$$ 5.36209 0.790597
$$47$$ −11.4256 −1.66659 −0.833295 0.552829i $$-0.813548\pi$$
−0.833295 + 0.552829i $$0.813548\pi$$
$$48$$ 0 0
$$49$$ −1.35066 −0.192952
$$50$$ −2.05321 −0.290368
$$51$$ 0 0
$$52$$ −3.54591 −0.491729
$$53$$ 5.43137 0.746056 0.373028 0.927820i $$-0.378320\pi$$
0.373028 + 0.927820i $$0.378320\pi$$
$$54$$ 0 0
$$55$$ 0.687897 0.0927560
$$56$$ 6.39646 0.854764
$$57$$ 0 0
$$58$$ 0.284588 0.0373682
$$59$$ 5.97693 0.778130 0.389065 0.921210i $$-0.372798\pi$$
0.389065 + 0.921210i $$0.372798\pi$$
$$60$$ 0 0
$$61$$ −11.8468 −1.51682 −0.758411 0.651776i $$-0.774024\pi$$
−0.758411 + 0.651776i $$0.774024\pi$$
$$62$$ −2.21660 −0.281508
$$63$$ 0 0
$$64$$ 3.55649 0.444561
$$65$$ −7.18254 −0.890884
$$66$$ 0 0
$$67$$ 1.81030 0.221164 0.110582 0.993867i $$-0.464729\pi$$
0.110582 + 0.993867i $$0.464729\pi$$
$$68$$ 0.398275 0.0482979
$$69$$ 0 0
$$70$$ 5.23871 0.626146
$$71$$ −0.370510 −0.0439714 −0.0219857 0.999758i $$-0.506999\pi$$
−0.0219857 + 0.999758i $$0.506999\pi$$
$$72$$ 0 0
$$73$$ 5.02679 0.588341 0.294171 0.955753i $$-0.404957\pi$$
0.294171 + 0.955753i $$0.404957\pi$$
$$74$$ 5.60458 0.651519
$$75$$ 0 0
$$76$$ 3.78044 0.433647
$$77$$ −0.594586 −0.0677594
$$78$$ 0 0
$$79$$ 0.802822 0.0903245 0.0451622 0.998980i $$-0.485620\pi$$
0.0451622 + 0.998980i $$0.485620\pi$$
$$80$$ −1.53459 −0.171572
$$81$$ 0 0
$$82$$ 7.78892 0.860143
$$83$$ −2.75565 −0.302472 −0.151236 0.988498i $$-0.548325\pi$$
−0.151236 + 0.988498i $$0.548325\pi$$
$$84$$ 0 0
$$85$$ 0.806740 0.0875032
$$86$$ −0.208963 −0.0225331
$$87$$ 0 0
$$88$$ −0.673220 −0.0717655
$$89$$ 10.4507 1.10777 0.553884 0.832594i $$-0.313145\pi$$
0.553884 + 0.832594i $$0.313145\pi$$
$$90$$ 0 0
$$91$$ 6.20825 0.650801
$$92$$ 9.08183 0.946846
$$93$$ 0 0
$$94$$ 9.15789 0.944565
$$95$$ 7.65761 0.785654
$$96$$ 0 0
$$97$$ −14.8346 −1.50623 −0.753113 0.657891i $$-0.771449\pi$$
−0.753113 + 0.657891i $$0.771449\pi$$
$$98$$ 1.08259 0.109358
$$99$$ 0 0
$$100$$ −3.47755 −0.347755
$$101$$ 4.00533 0.398545 0.199272 0.979944i $$-0.436142\pi$$
0.199272 + 0.979944i $$0.436142\pi$$
$$102$$ 0 0
$$103$$ −5.91936 −0.583252 −0.291626 0.956532i $$-0.594196\pi$$
−0.291626 + 0.956532i $$0.594196\pi$$
$$104$$ 7.02929 0.689279
$$105$$ 0 0
$$106$$ −4.35339 −0.422838
$$107$$ −0.258978 −0.0250364 −0.0125182 0.999922i $$-0.503985\pi$$
−0.0125182 + 0.999922i $$0.503985\pi$$
$$108$$ 0 0
$$109$$ −8.55787 −0.819695 −0.409848 0.912154i $$-0.634418\pi$$
−0.409848 + 0.912154i $$0.634418\pi$$
$$110$$ −0.551368 −0.0525709
$$111$$ 0 0
$$112$$ 1.32642 0.125335
$$113$$ −3.11918 −0.293428 −0.146714 0.989179i $$-0.546870\pi$$
−0.146714 + 0.989179i $$0.546870\pi$$
$$114$$ 0 0
$$115$$ 18.3960 1.71544
$$116$$ 0.482009 0.0447534
$$117$$ 0 0
$$118$$ −4.79067 −0.441017
$$119$$ −0.697308 −0.0639221
$$120$$ 0 0
$$121$$ −10.9374 −0.994311
$$122$$ 9.49550 0.859682
$$123$$ 0 0
$$124$$ −3.75427 −0.337144
$$125$$ 6.70514 0.599726
$$126$$ 0 0
$$127$$ −18.4545 −1.63757 −0.818787 0.574097i $$-0.805353\pi$$
−0.818787 + 0.574097i $$0.805353\pi$$
$$128$$ 8.80867 0.778583
$$129$$ 0 0
$$130$$ 5.75700 0.504922
$$131$$ 14.2255 1.24289 0.621443 0.783460i $$-0.286547\pi$$
0.621443 + 0.783460i $$0.286547\pi$$
$$132$$ 0 0
$$133$$ −6.61888 −0.573929
$$134$$ −1.45101 −0.125348
$$135$$ 0 0
$$136$$ −0.789527 −0.0677014
$$137$$ −19.6856 −1.68185 −0.840926 0.541149i $$-0.817989\pi$$
−0.840926 + 0.541149i $$0.817989\pi$$
$$138$$ 0 0
$$139$$ −17.9110 −1.51919 −0.759594 0.650398i $$-0.774602\pi$$
−0.759594 + 0.650398i $$0.774602\pi$$
$$140$$ 8.87286 0.749894
$$141$$ 0 0
$$142$$ 0.296974 0.0249215
$$143$$ −0.653411 −0.0546410
$$144$$ 0 0
$$145$$ 0.976351 0.0810815
$$146$$ −4.02911 −0.333451
$$147$$ 0 0
$$148$$ 9.49254 0.780282
$$149$$ 16.2895 1.33448 0.667242 0.744841i $$-0.267475\pi$$
0.667242 + 0.744841i $$0.267475\pi$$
$$150$$ 0 0
$$151$$ 14.2749 1.16167 0.580836 0.814021i $$-0.302726\pi$$
0.580836 + 0.814021i $$0.302726\pi$$
$$152$$ −7.49422 −0.607862
$$153$$ 0 0
$$154$$ 0.476577 0.0384036
$$155$$ −7.60459 −0.610816
$$156$$ 0 0
$$157$$ 0.763354 0.0609223 0.0304612 0.999536i $$-0.490302\pi$$
0.0304612 + 0.999536i $$0.490302\pi$$
$$158$$ −0.643483 −0.0511928
$$159$$ 0 0
$$160$$ 16.0306 1.26733
$$161$$ −15.9006 −1.25315
$$162$$ 0 0
$$163$$ 5.12834 0.401682 0.200841 0.979624i $$-0.435632\pi$$
0.200841 + 0.979624i $$0.435632\pi$$
$$164$$ 13.1922 1.03014
$$165$$ 0 0
$$166$$ 2.20873 0.171431
$$167$$ 8.90112 0.688790 0.344395 0.938825i $$-0.388084\pi$$
0.344395 + 0.938825i $$0.388084\pi$$
$$168$$ 0 0
$$169$$ −6.17754 −0.475196
$$170$$ −0.646624 −0.0495938
$$171$$ 0 0
$$172$$ −0.353923 −0.0269864
$$173$$ −6.81124 −0.517849 −0.258924 0.965898i $$-0.583368\pi$$
−0.258924 + 0.965898i $$0.583368\pi$$
$$174$$ 0 0
$$175$$ 6.08856 0.460252
$$176$$ −0.139605 −0.0105231
$$177$$ 0 0
$$178$$ −8.37649 −0.627844
$$179$$ −18.3476 −1.37137 −0.685684 0.727900i $$-0.740497\pi$$
−0.685684 + 0.727900i $$0.740497\pi$$
$$180$$ 0 0
$$181$$ 11.3256 0.841829 0.420914 0.907100i $$-0.361709\pi$$
0.420914 + 0.907100i $$0.361709\pi$$
$$182$$ −4.97608 −0.368852
$$183$$ 0 0
$$184$$ −18.0035 −1.32724
$$185$$ 19.2279 1.41367
$$186$$ 0 0
$$187$$ 0.0733908 0.00536687
$$188$$ 15.5108 1.13124
$$189$$ 0 0
$$190$$ −6.13778 −0.445281
$$191$$ 6.85841 0.496257 0.248129 0.968727i $$-0.420184\pi$$
0.248129 + 0.968727i $$0.420184\pi$$
$$192$$ 0 0
$$193$$ 20.4128 1.46935 0.734673 0.678422i $$-0.237336\pi$$
0.734673 + 0.678422i $$0.237336\pi$$
$$194$$ 11.8903 0.853676
$$195$$ 0 0
$$196$$ 1.83360 0.130971
$$197$$ 3.03573 0.216287 0.108143 0.994135i $$-0.465509\pi$$
0.108143 + 0.994135i $$0.465509\pi$$
$$198$$ 0 0
$$199$$ −2.26247 −0.160382 −0.0801912 0.996779i $$-0.525553\pi$$
−0.0801912 + 0.996779i $$0.525553\pi$$
$$200$$ 6.89377 0.487463
$$201$$ 0 0
$$202$$ −3.21038 −0.225881
$$203$$ −0.843912 −0.0592310
$$204$$ 0 0
$$205$$ 26.7219 1.86634
$$206$$ 4.74453 0.330567
$$207$$ 0 0
$$208$$ 1.45765 0.101070
$$209$$ 0.696629 0.0481868
$$210$$ 0 0
$$211$$ 25.4308 1.75073 0.875364 0.483464i $$-0.160621\pi$$
0.875364 + 0.483464i $$0.160621\pi$$
$$212$$ −7.37338 −0.506406
$$213$$ 0 0
$$214$$ 0.207578 0.0141898
$$215$$ −0.716901 −0.0488923
$$216$$ 0 0
$$217$$ 6.57305 0.446208
$$218$$ 6.85936 0.464575
$$219$$ 0 0
$$220$$ −0.933858 −0.0629607
$$221$$ −0.766295 −0.0515466
$$222$$ 0 0
$$223$$ −3.83134 −0.256565 −0.128283 0.991738i $$-0.540946\pi$$
−0.128283 + 0.991738i $$0.540946\pi$$
$$224$$ −13.8561 −0.925799
$$225$$ 0 0
$$226$$ 2.50011 0.166305
$$227$$ 2.51599 0.166992 0.0834961 0.996508i $$-0.473391\pi$$
0.0834961 + 0.996508i $$0.473391\pi$$
$$228$$ 0 0
$$229$$ −15.9396 −1.05332 −0.526660 0.850076i $$-0.676556\pi$$
−0.526660 + 0.850076i $$0.676556\pi$$
$$230$$ −14.7449 −0.972249
$$231$$ 0 0
$$232$$ −0.955519 −0.0627329
$$233$$ 28.1283 1.84274 0.921372 0.388682i $$-0.127070\pi$$
0.921372 + 0.388682i $$0.127070\pi$$
$$234$$ 0 0
$$235$$ 31.4185 2.04952
$$236$$ −8.11400 −0.528177
$$237$$ 0 0
$$238$$ 0.558911 0.0362288
$$239$$ −14.7058 −0.951238 −0.475619 0.879651i $$-0.657776\pi$$
−0.475619 + 0.879651i $$0.657776\pi$$
$$240$$ 0 0
$$241$$ 8.44295 0.543858 0.271929 0.962317i $$-0.412338\pi$$
0.271929 + 0.962317i $$0.412338\pi$$
$$242$$ 8.76664 0.563541
$$243$$ 0 0
$$244$$ 16.0826 1.02958
$$245$$ 3.71410 0.237285
$$246$$ 0 0
$$247$$ −7.27371 −0.462815
$$248$$ 7.44234 0.472589
$$249$$ 0 0
$$250$$ −5.37435 −0.339904
$$251$$ −23.2205 −1.46566 −0.732832 0.680409i $$-0.761802\pi$$
−0.732832 + 0.680409i $$0.761802\pi$$
$$252$$ 0 0
$$253$$ 1.67352 0.105213
$$254$$ 14.7918 0.928120
$$255$$ 0 0
$$256$$ −14.1734 −0.885835
$$257$$ 6.86520 0.428239 0.214120 0.976807i $$-0.431312\pi$$
0.214120 + 0.976807i $$0.431312\pi$$
$$258$$ 0 0
$$259$$ −16.6197 −1.03270
$$260$$ 9.75069 0.604712
$$261$$ 0 0
$$262$$ −11.4021 −0.704424
$$263$$ 3.35294 0.206751 0.103376 0.994642i $$-0.467036\pi$$
0.103376 + 0.994642i $$0.467036\pi$$
$$264$$ 0 0
$$265$$ −14.9354 −0.917474
$$266$$ 5.30521 0.325283
$$267$$ 0 0
$$268$$ −2.45758 −0.150121
$$269$$ 12.7416 0.776869 0.388434 0.921476i $$-0.373016\pi$$
0.388434 + 0.921476i $$0.373016\pi$$
$$270$$ 0 0
$$271$$ −23.5566 −1.43096 −0.715481 0.698632i $$-0.753792\pi$$
−0.715481 + 0.698632i $$0.753792\pi$$
$$272$$ −0.163723 −0.00992716
$$273$$ 0 0
$$274$$ 15.7785 0.953216
$$275$$ −0.640814 −0.0386425
$$276$$ 0 0
$$277$$ 4.18122 0.251225 0.125613 0.992079i $$-0.459910\pi$$
0.125613 + 0.992079i $$0.459910\pi$$
$$278$$ 14.3561 0.861023
$$279$$ 0 0
$$280$$ −17.5893 −1.05116
$$281$$ −21.6360 −1.29070 −0.645348 0.763888i $$-0.723288\pi$$
−0.645348 + 0.763888i $$0.723288\pi$$
$$282$$ 0 0
$$283$$ 5.22734 0.310733 0.155366 0.987857i $$-0.450344\pi$$
0.155366 + 0.987857i $$0.450344\pi$$
$$284$$ 0.502987 0.0298468
$$285$$ 0 0
$$286$$ 0.523726 0.0309686
$$287$$ −23.0971 −1.36338
$$288$$ 0 0
$$289$$ −16.9139 −0.994937
$$290$$ −0.782571 −0.0459542
$$291$$ 0 0
$$292$$ −6.82414 −0.399353
$$293$$ −6.14217 −0.358829 −0.179415 0.983774i $$-0.557420\pi$$
−0.179415 + 0.983774i $$0.557420\pi$$
$$294$$ 0 0
$$295$$ −16.4356 −0.956918
$$296$$ −18.8177 −1.09376
$$297$$ 0 0
$$298$$ −13.0564 −0.756339
$$299$$ −17.4738 −1.01053
$$300$$ 0 0
$$301$$ 0.619656 0.0357164
$$302$$ −11.4417 −0.658395
$$303$$ 0 0
$$304$$ −1.55407 −0.0891317
$$305$$ 32.5767 1.86534
$$306$$ 0 0
$$307$$ 19.0039 1.08461 0.542304 0.840182i $$-0.317552\pi$$
0.542304 + 0.840182i $$0.317552\pi$$
$$308$$ 0.807183 0.0459935
$$309$$ 0 0
$$310$$ 6.09529 0.346189
$$311$$ −21.5469 −1.22181 −0.610907 0.791703i $$-0.709195\pi$$
−0.610907 + 0.791703i $$0.709195\pi$$
$$312$$ 0 0
$$313$$ −3.81362 −0.215558 −0.107779 0.994175i $$-0.534374\pi$$
−0.107779 + 0.994175i $$0.534374\pi$$
$$314$$ −0.611849 −0.0345286
$$315$$ 0 0
$$316$$ −1.08987 −0.0613102
$$317$$ 4.25449 0.238956 0.119478 0.992837i $$-0.461878\pi$$
0.119478 + 0.992837i $$0.461878\pi$$
$$318$$ 0 0
$$319$$ 0.0888207 0.00497300
$$320$$ −9.77978 −0.546706
$$321$$ 0 0
$$322$$ 12.7448 0.710239
$$323$$ 0.816980 0.0454580
$$324$$ 0 0
$$325$$ 6.69092 0.371146
$$326$$ −4.11050 −0.227660
$$327$$ 0 0
$$328$$ −26.1517 −1.44399
$$329$$ −27.1567 −1.49719
$$330$$ 0 0
$$331$$ −14.2938 −0.785658 −0.392829 0.919612i $$-0.628504\pi$$
−0.392829 + 0.919612i $$0.628504\pi$$
$$332$$ 3.74095 0.205311
$$333$$ 0 0
$$334$$ −7.13449 −0.390382
$$335$$ −4.97804 −0.271980
$$336$$ 0 0
$$337$$ 35.7185 1.94571 0.972855 0.231414i $$-0.0743352\pi$$
0.972855 + 0.231414i $$0.0743352\pi$$
$$338$$ 4.95147 0.269324
$$339$$ 0 0
$$340$$ −1.09519 −0.0593952
$$341$$ −0.691806 −0.0374634
$$342$$ 0 0
$$343$$ −19.8481 −1.07170
$$344$$ 0.701606 0.0378280
$$345$$ 0 0
$$346$$ 5.45939 0.293499
$$347$$ −19.4415 −1.04368 −0.521838 0.853045i $$-0.674753\pi$$
−0.521838 + 0.853045i $$0.674753\pi$$
$$348$$ 0 0
$$349$$ −8.02070 −0.429338 −0.214669 0.976687i $$-0.568867\pi$$
−0.214669 + 0.976687i $$0.568867\pi$$
$$350$$ −4.88014 −0.260855
$$351$$ 0 0
$$352$$ 1.45834 0.0777296
$$353$$ 8.75381 0.465919 0.232959 0.972486i $$-0.425159\pi$$
0.232959 + 0.972486i $$0.425159\pi$$
$$354$$ 0 0
$$355$$ 1.01884 0.0540746
$$356$$ −14.1873 −0.751928
$$357$$ 0 0
$$358$$ 14.7061 0.777243
$$359$$ −8.27791 −0.436892 −0.218446 0.975849i $$-0.570099\pi$$
−0.218446 + 0.975849i $$0.570099\pi$$
$$360$$ 0 0
$$361$$ −11.2452 −0.591852
$$362$$ −9.07781 −0.477119
$$363$$ 0 0
$$364$$ −8.42804 −0.441749
$$365$$ −13.8229 −0.723522
$$366$$ 0 0
$$367$$ −14.7999 −0.772546 −0.386273 0.922384i $$-0.626238\pi$$
−0.386273 + 0.922384i $$0.626238\pi$$
$$368$$ −3.73336 −0.194615
$$369$$ 0 0
$$370$$ −15.4117 −0.801216
$$371$$ 12.9095 0.670226
$$372$$ 0 0
$$373$$ 25.5334 1.32207 0.661035 0.750355i $$-0.270118\pi$$
0.661035 + 0.750355i $$0.270118\pi$$
$$374$$ −0.0588247 −0.00304175
$$375$$ 0 0
$$376$$ −30.7481 −1.58571
$$377$$ −0.927403 −0.0477637
$$378$$ 0 0
$$379$$ 20.1244 1.03372 0.516861 0.856070i $$-0.327101\pi$$
0.516861 + 0.856070i $$0.327101\pi$$
$$380$$ −10.3956 −0.533284
$$381$$ 0 0
$$382$$ −5.49720 −0.281261
$$383$$ −23.8613 −1.21925 −0.609627 0.792689i $$-0.708681\pi$$
−0.609627 + 0.792689i $$0.708681\pi$$
$$384$$ 0 0
$$385$$ 1.63502 0.0833282
$$386$$ −16.3614 −0.832774
$$387$$ 0 0
$$388$$ 20.1388 1.02239
$$389$$ 37.9733 1.92532 0.962662 0.270708i $$-0.0872577\pi$$
0.962662 + 0.270708i $$0.0872577\pi$$
$$390$$ 0 0
$$391$$ 1.96264 0.0992552
$$392$$ −3.63486 −0.183588
$$393$$ 0 0
$$394$$ −2.43322 −0.122584
$$395$$ −2.20763 −0.111078
$$396$$ 0 0
$$397$$ 20.3493 1.02130 0.510651 0.859788i $$-0.329404\pi$$
0.510651 + 0.859788i $$0.329404\pi$$
$$398$$ 1.81343 0.0908992
$$399$$ 0 0
$$400$$ 1.42955 0.0714775
$$401$$ 6.94497 0.346815 0.173408 0.984850i $$-0.444522\pi$$
0.173408 + 0.984850i $$0.444522\pi$$
$$402$$ 0 0
$$403$$ 7.22335 0.359821
$$404$$ −5.43745 −0.270523
$$405$$ 0 0
$$406$$ 0.676418 0.0335701
$$407$$ 1.74921 0.0867049
$$408$$ 0 0
$$409$$ −10.9060 −0.539265 −0.269632 0.962963i $$-0.586902\pi$$
−0.269632 + 0.962963i $$0.586902\pi$$
$$410$$ −21.4183 −1.05777
$$411$$ 0 0
$$412$$ 8.03586 0.395898
$$413$$ 14.2062 0.699040
$$414$$ 0 0
$$415$$ 7.57760 0.371970
$$416$$ −15.2269 −0.746562
$$417$$ 0 0
$$418$$ −0.558367 −0.0273106
$$419$$ −10.0692 −0.491912 −0.245956 0.969281i $$-0.579102\pi$$
−0.245956 + 0.969281i $$0.579102\pi$$
$$420$$ 0 0
$$421$$ −3.10756 −0.151453 −0.0757267 0.997129i $$-0.524128\pi$$
−0.0757267 + 0.997129i $$0.524128\pi$$
$$422$$ −20.3835 −0.992252
$$423$$ 0 0
$$424$$ 14.6167 0.709852
$$425$$ −0.751522 −0.0364542
$$426$$ 0 0
$$427$$ −28.1578 −1.36265
$$428$$ 0.351577 0.0169941
$$429$$ 0 0
$$430$$ 0.574616 0.0277104
$$431$$ −28.0701 −1.35209 −0.676044 0.736862i $$-0.736307\pi$$
−0.676044 + 0.736862i $$0.736307\pi$$
$$432$$ 0 0
$$433$$ 19.5251 0.938317 0.469158 0.883114i $$-0.344557\pi$$
0.469158 + 0.883114i $$0.344557\pi$$
$$434$$ −5.26848 −0.252895
$$435$$ 0 0
$$436$$ 11.6178 0.556391
$$437$$ 18.6295 0.891170
$$438$$ 0 0
$$439$$ 14.6296 0.698232 0.349116 0.937080i $$-0.386482\pi$$
0.349116 + 0.937080i $$0.386482\pi$$
$$440$$ 1.85125 0.0882548
$$441$$ 0 0
$$442$$ 0.614206 0.0292148
$$443$$ 18.3559 0.872117 0.436059 0.899918i $$-0.356374\pi$$
0.436059 + 0.899918i $$0.356374\pi$$
$$444$$ 0 0
$$445$$ −28.7377 −1.36230
$$446$$ 3.07092 0.145412
$$447$$ 0 0
$$448$$ 8.45318 0.399375
$$449$$ 13.8594 0.654065 0.327032 0.945013i $$-0.393951\pi$$
0.327032 + 0.945013i $$0.393951\pi$$
$$450$$ 0 0
$$451$$ 2.43095 0.114469
$$452$$ 4.23445 0.199172
$$453$$ 0 0
$$454$$ −2.01663 −0.0946454
$$455$$ −17.0717 −0.800334
$$456$$ 0 0
$$457$$ 17.6481 0.825545 0.412772 0.910834i $$-0.364561\pi$$
0.412772 + 0.910834i $$0.364561\pi$$
$$458$$ 12.7760 0.596985
$$459$$ 0 0
$$460$$ −24.9736 −1.16440
$$461$$ 25.6380 1.19408 0.597041 0.802211i $$-0.296343\pi$$
0.597041 + 0.802211i $$0.296343\pi$$
$$462$$ 0 0
$$463$$ 18.3474 0.852677 0.426339 0.904564i $$-0.359803\pi$$
0.426339 + 0.904564i $$0.359803\pi$$
$$464$$ −0.198144 −0.00919863
$$465$$ 0 0
$$466$$ −22.5456 −1.04440
$$467$$ −16.2618 −0.752509 −0.376254 0.926516i $$-0.622788\pi$$
−0.376254 + 0.926516i $$0.622788\pi$$
$$468$$ 0 0
$$469$$ 4.30279 0.198684
$$470$$ −25.1828 −1.16159
$$471$$ 0 0
$$472$$ 16.0849 0.740369
$$473$$ −0.0652180 −0.00299873
$$474$$ 0 0
$$475$$ −7.13348 −0.327306
$$476$$ 0.946634 0.0433889
$$477$$ 0 0
$$478$$ 11.7871 0.539129
$$479$$ −9.47171 −0.432773 −0.216387 0.976308i $$-0.569427\pi$$
−0.216387 + 0.976308i $$0.569427\pi$$
$$480$$ 0 0
$$481$$ −18.2640 −0.832766
$$482$$ −6.76725 −0.308240
$$483$$ 0 0
$$484$$ 14.8481 0.674916
$$485$$ 40.7928 1.85231
$$486$$ 0 0
$$487$$ −0.467564 −0.0211874 −0.0105937 0.999944i $$-0.503372\pi$$
−0.0105937 + 0.999944i $$0.503372\pi$$
$$488$$ −31.8817 −1.44321
$$489$$ 0 0
$$490$$ −2.97695 −0.134485
$$491$$ 25.0470 1.13035 0.565177 0.824969i $$-0.308808\pi$$
0.565177 + 0.824969i $$0.308808\pi$$
$$492$$ 0 0
$$493$$ 0.104166 0.00469138
$$494$$ 5.83007 0.262307
$$495$$ 0 0
$$496$$ 1.54331 0.0692965
$$497$$ −0.880640 −0.0395021
$$498$$ 0 0
$$499$$ −14.0342 −0.628255 −0.314128 0.949381i $$-0.601712\pi$$
−0.314128 + 0.949381i $$0.601712\pi$$
$$500$$ −9.10259 −0.407080
$$501$$ 0 0
$$502$$ 18.6119 0.830688
$$503$$ 28.3116 1.26235 0.631176 0.775640i $$-0.282573\pi$$
0.631176 + 0.775640i $$0.282573\pi$$
$$504$$ 0 0
$$505$$ −11.0140 −0.490117
$$506$$ −1.34137 −0.0596313
$$507$$ 0 0
$$508$$ 25.0530 1.11155
$$509$$ −28.6875 −1.27155 −0.635774 0.771875i $$-0.719319\pi$$
−0.635774 + 0.771875i $$0.719319\pi$$
$$510$$ 0 0
$$511$$ 11.9478 0.528541
$$512$$ −6.25700 −0.276523
$$513$$ 0 0
$$514$$ −5.50264 −0.242711
$$515$$ 16.2773 0.717264
$$516$$ 0 0
$$517$$ 2.85820 0.125704
$$518$$ 13.3212 0.585298
$$519$$ 0 0
$$520$$ −19.3294 −0.847652
$$521$$ −24.9096 −1.09131 −0.545655 0.838010i $$-0.683719\pi$$
−0.545655 + 0.838010i $$0.683719\pi$$
$$522$$ 0 0
$$523$$ −25.8648 −1.13099 −0.565494 0.824753i $$-0.691314\pi$$
−0.565494 + 0.824753i $$0.691314\pi$$
$$524$$ −19.3119 −0.843642
$$525$$ 0 0
$$526$$ −2.68747 −0.117179
$$527$$ −0.811324 −0.0353418
$$528$$ 0 0
$$529$$ 21.7540 0.945825
$$530$$ 11.9711 0.519992
$$531$$ 0 0
$$532$$ 8.98549 0.389570
$$533$$ −25.3822 −1.09943
$$534$$ 0 0
$$535$$ 0.712150 0.0307889
$$536$$ 4.87183 0.210431
$$537$$ 0 0
$$538$$ −10.2127 −0.440302
$$539$$ 0.337880 0.0145535
$$540$$ 0 0
$$541$$ −21.9158 −0.942232 −0.471116 0.882071i $$-0.656149\pi$$
−0.471116 + 0.882071i $$0.656149\pi$$
$$542$$ 18.8813 0.811019
$$543$$ 0 0
$$544$$ 1.71028 0.0733278
$$545$$ 23.5328 1.00803
$$546$$ 0 0
$$547$$ −9.97605 −0.426545 −0.213273 0.976993i $$-0.568412\pi$$
−0.213273 + 0.976993i $$0.568412\pi$$
$$548$$ 26.7243 1.14160
$$549$$ 0 0
$$550$$ 0.513629 0.0219012
$$551$$ 0.988744 0.0421219
$$552$$ 0 0
$$553$$ 1.90817 0.0811438
$$554$$ −3.35136 −0.142386
$$555$$ 0 0
$$556$$ 24.3151 1.03119
$$557$$ −18.5330 −0.785268 −0.392634 0.919695i $$-0.628436\pi$$
−0.392634 + 0.919695i $$0.628436\pi$$
$$558$$ 0 0
$$559$$ 0.680961 0.0288016
$$560$$ −3.64746 −0.154133
$$561$$ 0 0
$$562$$ 17.3418 0.731522
$$563$$ 43.6831 1.84102 0.920511 0.390716i $$-0.127772\pi$$
0.920511 + 0.390716i $$0.127772\pi$$
$$564$$ 0 0
$$565$$ 8.57724 0.360847
$$566$$ −4.18985 −0.176113
$$567$$ 0 0
$$568$$ −0.997105 −0.0418376
$$569$$ 13.5667 0.568745 0.284373 0.958714i $$-0.408215\pi$$
0.284373 + 0.958714i $$0.408215\pi$$
$$570$$ 0 0
$$571$$ 23.7487 0.993853 0.496926 0.867793i $$-0.334462\pi$$
0.496926 + 0.867793i $$0.334462\pi$$
$$572$$ 0.887041 0.0370890
$$573$$ 0 0
$$574$$ 18.5130 0.772717
$$575$$ −17.1369 −0.714657
$$576$$ 0 0
$$577$$ −8.11902 −0.337999 −0.169000 0.985616i $$-0.554054\pi$$
−0.169000 + 0.985616i $$0.554054\pi$$
$$578$$ 13.5570 0.563896
$$579$$ 0 0
$$580$$ −1.32545 −0.0550363
$$581$$ −6.54972 −0.271728
$$582$$ 0 0
$$583$$ −1.35871 −0.0562718
$$584$$ 13.5279 0.559790
$$585$$ 0 0
$$586$$ 4.92311 0.203372
$$587$$ −3.69199 −0.152385 −0.0761925 0.997093i $$-0.524276\pi$$
−0.0761925 + 0.997093i $$0.524276\pi$$
$$588$$ 0 0
$$589$$ −7.70112 −0.317319
$$590$$ 13.1736 0.542348
$$591$$ 0 0
$$592$$ −3.90219 −0.160379
$$593$$ 29.4590 1.20974 0.604869 0.796325i $$-0.293226\pi$$
0.604869 + 0.796325i $$0.293226\pi$$
$$594$$ 0 0
$$595$$ 1.91749 0.0786093
$$596$$ −22.1138 −0.905818
$$597$$ 0 0
$$598$$ 14.0057 0.572735
$$599$$ −21.8754 −0.893804 −0.446902 0.894583i $$-0.647473\pi$$
−0.446902 + 0.894583i $$0.647473\pi$$
$$600$$ 0 0
$$601$$ 36.5207 1.48971 0.744854 0.667227i $$-0.232519\pi$$
0.744854 + 0.667227i $$0.232519\pi$$
$$602$$ −0.496671 −0.0202428
$$603$$ 0 0
$$604$$ −19.3789 −0.788517
$$605$$ 30.0762 1.22277
$$606$$ 0 0
$$607$$ −6.58082 −0.267107 −0.133554 0.991042i $$-0.542639\pi$$
−0.133554 + 0.991042i $$0.542639\pi$$
$$608$$ 16.2341 0.658379
$$609$$ 0 0
$$610$$ −26.1111 −1.05721
$$611$$ −29.8434 −1.20733
$$612$$ 0 0
$$613$$ −7.14867 −0.288732 −0.144366 0.989524i $$-0.546114\pi$$
−0.144366 + 0.989524i $$0.546114\pi$$
$$614$$ −15.2321 −0.614718
$$615$$ 0 0
$$616$$ −1.60013 −0.0644712
$$617$$ 16.5375 0.665774 0.332887 0.942967i $$-0.391977\pi$$
0.332887 + 0.942967i $$0.391977\pi$$
$$618$$ 0 0
$$619$$ −1.49935 −0.0602640 −0.0301320 0.999546i $$-0.509593\pi$$
−0.0301320 + 0.999546i $$0.509593\pi$$
$$620$$ 10.3236 0.414608
$$621$$ 0 0
$$622$$ 17.2704 0.692481
$$623$$ 24.8395 0.995173
$$624$$ 0 0
$$625$$ −31.2462 −1.24985
$$626$$ 3.05672 0.122171
$$627$$ 0 0
$$628$$ −1.03630 −0.0413527
$$629$$ 2.05140 0.0817948
$$630$$ 0 0
$$631$$ −35.8913 −1.42881 −0.714404 0.699733i $$-0.753302\pi$$
−0.714404 + 0.699733i $$0.753302\pi$$
$$632$$ 2.16053 0.0859413
$$633$$ 0 0
$$634$$ −3.41009 −0.135432
$$635$$ 50.7470 2.01383
$$636$$ 0 0
$$637$$ −3.52790 −0.139781
$$638$$ −0.0711922 −0.00281852
$$639$$ 0 0
$$640$$ −24.2224 −0.957476
$$641$$ 39.2279 1.54941 0.774705 0.632322i $$-0.217898\pi$$
0.774705 + 0.632322i $$0.217898\pi$$
$$642$$ 0 0
$$643$$ −10.4185 −0.410866 −0.205433 0.978671i $$-0.565860\pi$$
−0.205433 + 0.978671i $$0.565860\pi$$
$$644$$ 21.5860 0.850607
$$645$$ 0 0
$$646$$ −0.654831 −0.0257640
$$647$$ −39.1517 −1.53921 −0.769606 0.638519i $$-0.779547\pi$$
−0.769606 + 0.638519i $$0.779547\pi$$
$$648$$ 0 0
$$649$$ −1.49518 −0.0586910
$$650$$ −5.36296 −0.210352
$$651$$ 0 0
$$652$$ −6.96200 −0.272653
$$653$$ 32.9099 1.28786 0.643932 0.765083i $$-0.277302\pi$$
0.643932 + 0.765083i $$0.277302\pi$$
$$654$$ 0 0
$$655$$ −39.1178 −1.52846
$$656$$ −5.42304 −0.211734
$$657$$ 0 0
$$658$$ 21.7668 0.848558
$$659$$ −21.5684 −0.840186 −0.420093 0.907481i $$-0.638003\pi$$
−0.420093 + 0.907481i $$0.638003\pi$$
$$660$$ 0 0
$$661$$ 26.2964 1.02281 0.511405 0.859340i $$-0.329125\pi$$
0.511405 + 0.859340i $$0.329125\pi$$
$$662$$ 11.4569 0.445283
$$663$$ 0 0
$$664$$ −7.41593 −0.287794
$$665$$ 18.2009 0.705799
$$666$$ 0 0
$$667$$ 2.37528 0.0919710
$$668$$ −12.0838 −0.467535
$$669$$ 0 0
$$670$$ 3.99004 0.154149
$$671$$ 2.96357 0.114407
$$672$$ 0 0
$$673$$ 11.5221 0.444145 0.222073 0.975030i $$-0.428718\pi$$
0.222073 + 0.975030i $$0.428718\pi$$
$$674$$ −28.6293 −1.10276
$$675$$ 0 0
$$676$$ 8.38635 0.322552
$$677$$ −33.9250 −1.30384 −0.651922 0.758286i $$-0.726037\pi$$
−0.651922 + 0.758286i $$0.726037\pi$$
$$678$$ 0 0
$$679$$ −35.2594 −1.35313
$$680$$ 2.17107 0.0832569
$$681$$ 0 0
$$682$$ 0.554501 0.0212329
$$683$$ 36.7553 1.40640 0.703201 0.710991i $$-0.251753\pi$$
0.703201 + 0.710991i $$0.251753\pi$$
$$684$$ 0 0
$$685$$ 54.1322 2.06829
$$686$$ 15.9088 0.607401
$$687$$ 0 0
$$688$$ 0.145491 0.00554678
$$689$$ 14.1866 0.540468
$$690$$ 0 0
$$691$$ 13.3781 0.508928 0.254464 0.967082i $$-0.418101\pi$$
0.254464 + 0.967082i $$0.418101\pi$$
$$692$$ 9.24663 0.351504
$$693$$ 0 0
$$694$$ 15.5829 0.591519
$$695$$ 49.2523 1.86825
$$696$$ 0 0
$$697$$ 2.85092 0.107986
$$698$$ 6.42881 0.243334
$$699$$ 0 0
$$700$$ −8.26555 −0.312409
$$701$$ 5.00452 0.189018 0.0945091 0.995524i $$-0.469872\pi$$
0.0945091 + 0.995524i $$0.469872\pi$$
$$702$$ 0 0
$$703$$ 19.4720 0.734400
$$704$$ −0.889687 −0.0335314
$$705$$ 0 0
$$706$$ −7.01642 −0.264066
$$707$$ 9.52000 0.358036
$$708$$ 0 0
$$709$$ 17.1439 0.643851 0.321925 0.946765i $$-0.395670\pi$$
0.321925 + 0.946765i $$0.395670\pi$$
$$710$$ −0.816630 −0.0306476
$$711$$ 0 0
$$712$$ 28.1245 1.05401
$$713$$ −18.5005 −0.692850
$$714$$ 0 0
$$715$$ 1.79678 0.0671956
$$716$$ 24.9079 0.930853
$$717$$ 0 0
$$718$$ 6.63497 0.247615
$$719$$ −43.3519 −1.61675 −0.808377 0.588665i $$-0.799654\pi$$
−0.808377 + 0.588665i $$0.799654\pi$$
$$720$$ 0 0
$$721$$ −14.0693 −0.523969
$$722$$ 9.01333 0.335441
$$723$$ 0 0
$$724$$ −15.3752 −0.571414
$$725$$ −0.909524 −0.0337789
$$726$$ 0 0
$$727$$ −36.3439 −1.34792 −0.673960 0.738768i $$-0.735408\pi$$
−0.673960 + 0.738768i $$0.735408\pi$$
$$728$$ 16.7075 0.619220
$$729$$ 0 0
$$730$$ 11.0794 0.410067
$$731$$ −0.0764852 −0.00282891
$$732$$ 0 0
$$733$$ 3.87561 0.143149 0.0715744 0.997435i $$-0.477198\pi$$
0.0715744 + 0.997435i $$0.477198\pi$$
$$734$$ 11.8625 0.437852
$$735$$ 0 0
$$736$$ 38.9994 1.43754
$$737$$ −0.452863 −0.0166814
$$738$$ 0 0
$$739$$ 26.4482 0.972913 0.486456 0.873705i $$-0.338289\pi$$
0.486456 + 0.873705i $$0.338289\pi$$
$$740$$ −26.1030 −0.959564
$$741$$ 0 0
$$742$$ −10.3473 −0.379861
$$743$$ −13.4634 −0.493923 −0.246961 0.969025i $$-0.579432\pi$$
−0.246961 + 0.969025i $$0.579432\pi$$
$$744$$ 0 0
$$745$$ −44.7934 −1.64110
$$746$$ −20.4657 −0.749303
$$747$$ 0 0
$$748$$ −0.0996320 −0.00364291
$$749$$ −0.615549 −0.0224917
$$750$$ 0 0
$$751$$ 3.76830 0.137507 0.0687537 0.997634i $$-0.478098\pi$$
0.0687537 + 0.997634i $$0.478098\pi$$
$$752$$ −6.37619 −0.232516
$$753$$ 0 0
$$754$$ 0.743339 0.0270708
$$755$$ −39.2536 −1.42859
$$756$$ 0 0
$$757$$ −33.7073 −1.22511 −0.612556 0.790427i $$-0.709859\pi$$
−0.612556 + 0.790427i $$0.709859\pi$$
$$758$$ −16.1303 −0.585877
$$759$$ 0 0
$$760$$ 20.6079 0.747528
$$761$$ −9.65543 −0.350009 −0.175005 0.984568i $$-0.555994\pi$$
−0.175005 + 0.984568i $$0.555994\pi$$
$$762$$ 0 0
$$763$$ −20.3406 −0.736381
$$764$$ −9.31067 −0.336848
$$765$$ 0 0
$$766$$ 19.1254 0.691030
$$767$$ 15.6116 0.563703
$$768$$ 0 0
$$769$$ 38.7110 1.39595 0.697977 0.716120i $$-0.254084\pi$$
0.697977 + 0.716120i $$0.254084\pi$$
$$770$$ −1.31051 −0.0472275
$$771$$ 0 0
$$772$$ −27.7115 −0.997358
$$773$$ 24.3039 0.874150 0.437075 0.899425i $$-0.356014\pi$$
0.437075 + 0.899425i $$0.356014\pi$$
$$774$$ 0 0
$$775$$ 7.08409 0.254468
$$776$$ −39.9225 −1.43313
$$777$$ 0 0
$$778$$ −30.4366 −1.09121
$$779$$ 27.0611 0.969563
$$780$$ 0 0
$$781$$ 0.0926863 0.00331658
$$782$$ −1.57311 −0.0562544
$$783$$ 0 0
$$784$$ −0.753755 −0.0269198
$$785$$ −2.09910 −0.0749202
$$786$$ 0 0
$$787$$ 20.9406 0.746452 0.373226 0.927740i $$-0.378252\pi$$
0.373226 + 0.927740i $$0.378252\pi$$
$$788$$ −4.12117 −0.146810
$$789$$ 0 0
$$790$$ 1.76948 0.0629551
$$791$$ −7.41377 −0.263603
$$792$$ 0 0
$$793$$ −30.9435 −1.09884
$$794$$ −16.3105 −0.578839
$$795$$ 0 0
$$796$$ 3.07143 0.108864
$$797$$ 11.9368 0.422822 0.211411 0.977397i $$-0.432194\pi$$
0.211411 + 0.977397i $$0.432194\pi$$
$$798$$ 0 0
$$799$$ 3.35199 0.118585
$$800$$ −14.9334 −0.527974
$$801$$ 0 0
$$802$$ −5.56658 −0.196563
$$803$$ −1.25750 −0.0443761
$$804$$ 0 0
$$805$$ 43.7242 1.54108
$$806$$ −5.78971 −0.203934
$$807$$ 0 0
$$808$$ 10.7790 0.379205
$$809$$ 8.60808 0.302644 0.151322 0.988485i $$-0.451647\pi$$
0.151322 + 0.988485i $$0.451647\pi$$
$$810$$ 0 0
$$811$$ 1.53770 0.0539958 0.0269979 0.999635i $$-0.491405\pi$$
0.0269979 + 0.999635i $$0.491405\pi$$
$$812$$ 1.14566 0.0402046
$$813$$ 0 0
$$814$$ −1.40204 −0.0491413
$$815$$ −14.1021 −0.493975
$$816$$ 0 0
$$817$$ −0.726001 −0.0253996
$$818$$ 8.74141 0.305636
$$819$$ 0 0
$$820$$ −36.2764 −1.26683
$$821$$ 28.9763 1.01128 0.505639 0.862745i $$-0.331257\pi$$
0.505639 + 0.862745i $$0.331257\pi$$
$$822$$ 0 0
$$823$$ −11.2568 −0.392386 −0.196193 0.980565i $$-0.562858\pi$$
−0.196193 + 0.980565i $$0.562858\pi$$
$$824$$ −15.9300 −0.554948
$$825$$ 0 0
$$826$$ −11.3866 −0.396191
$$827$$ 30.9279 1.07547 0.537734 0.843114i $$-0.319280\pi$$
0.537734 + 0.843114i $$0.319280\pi$$
$$828$$ 0 0
$$829$$ −9.83524 −0.341592 −0.170796 0.985306i $$-0.554634\pi$$
−0.170796 + 0.985306i $$0.554634\pi$$
$$830$$ −6.07365 −0.210820
$$831$$ 0 0
$$832$$ 9.28949 0.322055
$$833$$ 0.396253 0.0137293
$$834$$ 0 0
$$835$$ −24.4767 −0.847050
$$836$$ −0.945711 −0.0327081
$$837$$ 0 0
$$838$$ 8.07072 0.278798
$$839$$ −13.1432 −0.453754 −0.226877 0.973923i $$-0.572852\pi$$
−0.226877 + 0.973923i $$0.572852\pi$$
$$840$$ 0 0
$$841$$ −28.8739 −0.995653
$$842$$ 2.49080 0.0858385
$$843$$ 0 0
$$844$$ −34.5237 −1.18835
$$845$$ 16.9873 0.584380
$$846$$ 0 0
$$847$$ −25.9964 −0.893248
$$848$$ 3.03105 0.104087
$$849$$ 0 0
$$850$$ 0.602365 0.0206609
$$851$$ 46.7779 1.60353
$$852$$ 0 0
$$853$$ 15.4423 0.528735 0.264368 0.964422i $$-0.414837\pi$$
0.264368 + 0.964422i $$0.414837\pi$$
$$854$$ 22.5692 0.772303
$$855$$ 0 0
$$856$$ −0.696955 −0.0238214
$$857$$ 21.9827 0.750916 0.375458 0.926839i $$-0.377485\pi$$
0.375458 + 0.926839i $$0.377485\pi$$
$$858$$ 0 0
$$859$$ 19.5667 0.667606 0.333803 0.942643i $$-0.391668\pi$$
0.333803 + 0.942643i $$0.391668\pi$$
$$860$$ 0.973233 0.0331870
$$861$$ 0 0
$$862$$ 22.4989 0.766316
$$863$$ 21.8676 0.744383 0.372191 0.928156i $$-0.378607\pi$$
0.372191 + 0.928156i $$0.378607\pi$$
$$864$$ 0 0
$$865$$ 18.7298 0.636833
$$866$$ −15.6499 −0.531805
$$867$$ 0 0
$$868$$ −8.92328 −0.302876
$$869$$ −0.200833 −0.00681279
$$870$$ 0 0
$$871$$ 4.72848 0.160218
$$872$$ −23.0307 −0.779918
$$873$$ 0 0
$$874$$ −14.9320 −0.505084
$$875$$ 15.9370 0.538769
$$876$$ 0 0
$$877$$ −39.0999 −1.32031 −0.660155 0.751130i $$-0.729509\pi$$
−0.660155 + 0.751130i $$0.729509\pi$$
$$878$$ −11.7260 −0.395733
$$879$$ 0 0
$$880$$ 0.383890 0.0129409
$$881$$ −7.30508 −0.246115 −0.123057 0.992400i $$-0.539270\pi$$
−0.123057 + 0.992400i $$0.539270\pi$$
$$882$$ 0 0
$$883$$ −3.49293 −0.117546 −0.0587732 0.998271i $$-0.518719\pi$$
−0.0587732 + 0.998271i $$0.518719\pi$$
$$884$$ 1.04029 0.0349887
$$885$$ 0 0
$$886$$ −14.7128 −0.494286
$$887$$ 28.4213 0.954293 0.477147 0.878824i $$-0.341671\pi$$
0.477147 + 0.878824i $$0.341671\pi$$
$$888$$ 0 0
$$889$$ −43.8633 −1.47113
$$890$$ 23.0340 0.772102
$$891$$ 0 0
$$892$$ 5.20125 0.174151
$$893$$ 31.8173 1.06472
$$894$$ 0 0
$$895$$ 50.4531 1.68646
$$896$$ 20.9367 0.699447
$$897$$ 0 0
$$898$$ −11.1087 −0.370701
$$899$$ −0.981898 −0.0327481
$$900$$ 0 0
$$901$$ −1.59344 −0.0530851
$$902$$ −1.94847 −0.0648769
$$903$$ 0 0
$$904$$ −8.39424 −0.279188
$$905$$ −31.1437 −1.03525
$$906$$ 0 0
$$907$$ 53.1167 1.76371 0.881855 0.471520i $$-0.156295\pi$$
0.881855 + 0.471520i $$0.156295\pi$$
$$908$$ −3.41560 −0.113351
$$909$$ 0 0
$$910$$ 13.6834 0.453601
$$911$$ −8.07589 −0.267566 −0.133783 0.991011i $$-0.542713\pi$$
−0.133783 + 0.991011i $$0.542713\pi$$
$$912$$ 0 0
$$913$$ 0.689351 0.0228142
$$914$$ −14.1455 −0.467890
$$915$$ 0 0
$$916$$ 21.6389 0.714970
$$917$$ 33.8116 1.11656
$$918$$ 0 0
$$919$$ −47.9961 −1.58325 −0.791623 0.611009i $$-0.790764\pi$$
−0.791623 + 0.611009i $$0.790764\pi$$
$$920$$ 49.5068 1.63219
$$921$$ 0 0
$$922$$ −20.5496 −0.676764
$$923$$ −0.967766 −0.0318544
$$924$$ 0 0
$$925$$ −17.9119 −0.588938
$$926$$ −14.7060 −0.483268
$$927$$ 0 0
$$928$$ 2.06986 0.0679464
$$929$$ −28.9939 −0.951260 −0.475630 0.879645i $$-0.657780\pi$$
−0.475630 + 0.879645i $$0.657780\pi$$
$$930$$ 0 0
$$931$$ 3.76125 0.123270
$$932$$ −38.1857 −1.25081
$$933$$ 0 0
$$934$$ 13.0343 0.426496
$$935$$ −0.201813 −0.00659999
$$936$$ 0 0
$$937$$ 5.02850 0.164274 0.0821369 0.996621i $$-0.473826\pi$$
0.0821369 + 0.996621i $$0.473826\pi$$
$$938$$ −3.44880 −0.112607
$$939$$ 0 0
$$940$$ −42.6523 −1.39116
$$941$$ −55.8411 −1.82037 −0.910183 0.414206i $$-0.864059\pi$$
−0.910183 + 0.414206i $$0.864059\pi$$
$$942$$ 0 0
$$943$$ 65.0092 2.11699
$$944$$ 3.33551 0.108561
$$945$$ 0 0
$$946$$ 0.0522740 0.00169957
$$947$$ 42.5294 1.38202 0.691009 0.722846i $$-0.257166\pi$$
0.691009 + 0.722846i $$0.257166\pi$$
$$948$$ 0 0
$$949$$ 13.1299 0.426214
$$950$$ 5.71767 0.185506
$$951$$ 0 0
$$952$$ −1.87657 −0.0608201
$$953$$ 21.8148 0.706651 0.353325 0.935501i $$-0.385051\pi$$
0.353325 + 0.935501i $$0.385051\pi$$
$$954$$ 0 0
$$955$$ −18.8595 −0.610280
$$956$$ 19.9639 0.645679
$$957$$ 0 0
$$958$$ 7.59183 0.245281
$$959$$ −46.7894 −1.51091
$$960$$ 0 0
$$961$$ −23.3522 −0.753297
$$962$$ 14.6391 0.471983
$$963$$ 0 0
$$964$$ −11.4618 −0.369158
$$965$$ −56.1319 −1.80695
$$966$$ 0 0
$$967$$ 4.62859 0.148845 0.0744227 0.997227i $$-0.476289\pi$$
0.0744227 + 0.997227i $$0.476289\pi$$
$$968$$ −29.4345 −0.946059
$$969$$ 0 0
$$970$$ −32.6965 −1.04982
$$971$$ −21.6509 −0.694809 −0.347405 0.937715i $$-0.612937\pi$$
−0.347405 + 0.937715i $$0.612937\pi$$
$$972$$ 0 0
$$973$$ −42.5714 −1.36478
$$974$$ 0.374766 0.0120083
$$975$$ 0 0
$$976$$ −6.61125 −0.211621
$$977$$ −22.0848 −0.706556 −0.353278 0.935518i $$-0.614933\pi$$
−0.353278 + 0.935518i $$0.614933\pi$$
$$978$$ 0 0
$$979$$ −2.61433 −0.0835542
$$980$$ −5.04210 −0.161064
$$981$$ 0 0
$$982$$ −20.0758 −0.640646
$$983$$ −13.8630 −0.442163 −0.221081 0.975255i $$-0.570959\pi$$
−0.221081 + 0.975255i $$0.570959\pi$$
$$984$$ 0 0
$$985$$ −8.34777 −0.265982
$$986$$ −0.0834915 −0.00265891
$$987$$ 0 0
$$988$$ 9.87446 0.314148
$$989$$ −1.74408 −0.0554587
$$990$$ 0 0
$$991$$ 34.8224 1.10617 0.553084 0.833125i $$-0.313451\pi$$
0.553084 + 0.833125i $$0.313451\pi$$
$$992$$ −16.1217 −0.511864
$$993$$ 0 0
$$994$$ 0.705857 0.0223884
$$995$$ 6.22144 0.197233
$$996$$ 0 0
$$997$$ −24.7498 −0.783833 −0.391916 0.920001i $$-0.628188\pi$$
−0.391916 + 0.920001i $$0.628188\pi$$
$$998$$ 11.2488 0.356073
$$999$$ 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.1.3 6
3.2 odd 2 729.2.a.d.1.4 6
9.2 odd 6 729.2.c.b.244.3 12
9.4 even 3 729.2.c.e.487.4 12
9.5 odd 6 729.2.c.b.487.3 12
9.7 even 3 729.2.c.e.244.4 12
27.2 odd 18 81.2.e.a.64.1 12
27.4 even 9 243.2.e.d.136.1 12
27.5 odd 18 243.2.e.b.217.2 12
27.7 even 9 243.2.e.d.109.1 12
27.11 odd 18 243.2.e.b.28.2 12
27.13 even 9 27.2.e.a.7.2 yes 12
27.14 odd 18 81.2.e.a.19.1 12
27.16 even 9 243.2.e.c.28.1 12
27.20 odd 18 243.2.e.a.109.2 12
27.22 even 9 243.2.e.c.217.1 12
27.23 odd 18 243.2.e.a.136.2 12
27.25 even 9 27.2.e.a.4.2 12
108.67 odd 18 432.2.u.c.385.2 12
108.79 odd 18 432.2.u.c.193.2 12
135.13 odd 36 675.2.u.b.574.2 24
135.52 odd 36 675.2.u.b.274.2 24
135.67 odd 36 675.2.u.b.574.3 24
135.79 even 18 675.2.l.c.301.1 12
135.94 even 18 675.2.l.c.601.1 12
135.133 odd 36 675.2.u.b.274.3 24

By twisted newform
Twist Min Dim Char Parity Ord Type
27.2.e.a.4.2 12 27.25 even 9
27.2.e.a.7.2 yes 12 27.13 even 9
81.2.e.a.19.1 12 27.14 odd 18
81.2.e.a.64.1 12 27.2 odd 18
243.2.e.a.109.2 12 27.20 odd 18
243.2.e.a.136.2 12 27.23 odd 18
243.2.e.b.28.2 12 27.11 odd 18
243.2.e.b.217.2 12 27.5 odd 18
243.2.e.c.28.1 12 27.16 even 9
243.2.e.c.217.1 12 27.22 even 9
243.2.e.d.109.1 12 27.7 even 9
243.2.e.d.136.1 12 27.4 even 9
432.2.u.c.193.2 12 108.79 odd 18
432.2.u.c.385.2 12 108.67 odd 18
675.2.l.c.301.1 12 135.79 even 18
675.2.l.c.601.1 12 135.94 even 18
675.2.u.b.274.2 24 135.52 odd 36
675.2.u.b.274.3 24 135.133 odd 36
675.2.u.b.574.2 24 135.13 odd 36
675.2.u.b.574.3 24 135.67 odd 36
729.2.a.a.1.3 6 1.1 even 1 trivial
729.2.a.d.1.4 6 3.2 odd 2
729.2.c.b.244.3 12 9.2 odd 6
729.2.c.b.487.3 12 9.5 odd 6
729.2.c.e.244.4 12 9.7 even 3
729.2.c.e.487.4 12 9.4 even 3