Properties

 Label 243.2.a.e.1.2 Level $243$ Weight $2$ Character 243.1 Self dual yes Analytic conductor $1.940$ Analytic rank $1$ 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] = [243,2,Mod(1,243)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(243, 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("243.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$243 = 3^{5}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 243.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: $$1.94036476912$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 3x - 1$$ x^3 - 3*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.2 Root $$-0.347296$$ of defining polynomial Character $$\chi$$ $$=$$ 243.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.34730 q^{2} -0.184793 q^{4} -1.65270 q^{5} +2.41147 q^{7} +2.94356 q^{8} +O(q^{10})$$ $$q-1.34730 q^{2} -0.184793 q^{4} -1.65270 q^{5} +2.41147 q^{7} +2.94356 q^{8} +2.22668 q^{10} -5.94356 q^{11} -3.22668 q^{13} -3.24897 q^{14} -3.59627 q^{16} -3.00000 q^{17} -6.63816 q^{19} +0.305407 q^{20} +8.00774 q^{22} +2.94356 q^{23} -2.26857 q^{25} +4.34730 q^{26} -0.445622 q^{28} +1.29086 q^{29} -0.588526 q^{31} -1.04189 q^{32} +4.04189 q^{34} -3.98545 q^{35} +0.0418891 q^{37} +8.94356 q^{38} -4.86484 q^{40} +4.90167 q^{41} -5.18479 q^{43} +1.09833 q^{44} -3.96585 q^{46} +3.73648 q^{47} -1.18479 q^{49} +3.05644 q^{50} +0.596267 q^{52} -11.6382 q^{53} +9.82295 q^{55} +7.09833 q^{56} -1.73917 q^{58} +7.34730 q^{59} +11.0496 q^{61} +0.792919 q^{62} +8.59627 q^{64} +5.33275 q^{65} +1.85710 q^{67} +0.554378 q^{68} +5.36959 q^{70} +5.51249 q^{71} +5.55438 q^{73} -0.0564370 q^{74} +1.22668 q^{76} -14.3327 q^{77} -3.78106 q^{79} +5.94356 q^{80} -6.60401 q^{82} -3.98545 q^{83} +4.95811 q^{85} +6.98545 q^{86} -17.4953 q^{88} -8.15064 q^{89} -7.78106 q^{91} -0.543948 q^{92} -5.03415 q^{94} +10.9709 q^{95} +0.260830 q^{97} +1.59627 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 3 q^{7} - 6 q^{8}+O(q^{10})$$ 3 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 - 3 * q^7 - 6 * q^8 $$3 q - 3 q^{2} + 3 q^{4} - 6 q^{5} - 3 q^{7} - 6 q^{8} - 3 q^{11} - 3 q^{13} + 3 q^{14} + 3 q^{16} - 9 q^{17} - 3 q^{19} + 3 q^{20} - 6 q^{23} + 3 q^{25} + 12 q^{26} - 12 q^{28} - 12 q^{29} - 12 q^{31} + 9 q^{34} + 6 q^{35} - 3 q^{37} + 12 q^{38} + 9 q^{40} + 3 q^{41} - 12 q^{43} + 15 q^{44} + 9 q^{46} + 6 q^{47} + 24 q^{50} - 12 q^{52} - 18 q^{53} + 9 q^{55} + 33 q^{56} + 9 q^{58} + 21 q^{59} + 6 q^{61} + 12 q^{62} + 12 q^{64} - 3 q^{65} + 6 q^{67} - 9 q^{68} + 9 q^{70} + 9 q^{71} + 6 q^{73} - 15 q^{74} - 3 q^{76} - 24 q^{77} + 6 q^{79} + 3 q^{80} + 18 q^{82} + 6 q^{83} + 18 q^{85} + 3 q^{86} - 36 q^{88} - 6 q^{91} - 24 q^{92} - 36 q^{94} - 3 q^{95} + 15 q^{97} - 9 q^{98}+O(q^{100})$$ 3 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 - 3 * q^7 - 6 * q^8 - 3 * q^11 - 3 * q^13 + 3 * q^14 + 3 * q^16 - 9 * q^17 - 3 * q^19 + 3 * q^20 - 6 * q^23 + 3 * q^25 + 12 * q^26 - 12 * q^28 - 12 * q^29 - 12 * q^31 + 9 * q^34 + 6 * q^35 - 3 * q^37 + 12 * q^38 + 9 * q^40 + 3 * q^41 - 12 * q^43 + 15 * q^44 + 9 * q^46 + 6 * q^47 + 24 * q^50 - 12 * q^52 - 18 * q^53 + 9 * q^55 + 33 * q^56 + 9 * q^58 + 21 * q^59 + 6 * q^61 + 12 * q^62 + 12 * q^64 - 3 * q^65 + 6 * q^67 - 9 * q^68 + 9 * q^70 + 9 * q^71 + 6 * q^73 - 15 * q^74 - 3 * q^76 - 24 * q^77 + 6 * q^79 + 3 * q^80 + 18 * q^82 + 6 * q^83 + 18 * q^85 + 3 * q^86 - 36 * q^88 - 6 * q^91 - 24 * q^92 - 36 * q^94 - 3 * q^95 + 15 * q^97 - 9 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.34730 −0.952682 −0.476341 0.879261i $$-0.658037\pi$$
−0.476341 + 0.879261i $$0.658037\pi$$
$$3$$ 0 0
$$4$$ −0.184793 −0.0923963
$$5$$ −1.65270 −0.739112 −0.369556 0.929209i $$-0.620490\pi$$
−0.369556 + 0.929209i $$0.620490\pi$$
$$6$$ 0 0
$$7$$ 2.41147 0.911452 0.455726 0.890120i $$-0.349380\pi$$
0.455726 + 0.890120i $$0.349380\pi$$
$$8$$ 2.94356 1.04071
$$9$$ 0 0
$$10$$ 2.22668 0.704139
$$11$$ −5.94356 −1.79205 −0.896026 0.444002i $$-0.853558\pi$$
−0.896026 + 0.444002i $$0.853558\pi$$
$$12$$ 0 0
$$13$$ −3.22668 −0.894920 −0.447460 0.894304i $$-0.647671\pi$$
−0.447460 + 0.894304i $$0.647671\pi$$
$$14$$ −3.24897 −0.868324
$$15$$ 0 0
$$16$$ −3.59627 −0.899067
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ −6.63816 −1.52290 −0.761449 0.648225i $$-0.775512\pi$$
−0.761449 + 0.648225i $$0.775512\pi$$
$$20$$ 0.305407 0.0682911
$$21$$ 0 0
$$22$$ 8.00774 1.70726
$$23$$ 2.94356 0.613775 0.306888 0.951746i $$-0.400712\pi$$
0.306888 + 0.951746i $$0.400712\pi$$
$$24$$ 0 0
$$25$$ −2.26857 −0.453714
$$26$$ 4.34730 0.852575
$$27$$ 0 0
$$28$$ −0.445622 −0.0842147
$$29$$ 1.29086 0.239707 0.119853 0.992792i $$-0.461758\pi$$
0.119853 + 0.992792i $$0.461758\pi$$
$$30$$ 0 0
$$31$$ −0.588526 −0.105702 −0.0528512 0.998602i $$-0.516831\pi$$
−0.0528512 + 0.998602i $$0.516831\pi$$
$$32$$ −1.04189 −0.184182
$$33$$ 0 0
$$34$$ 4.04189 0.693178
$$35$$ −3.98545 −0.673664
$$36$$ 0 0
$$37$$ 0.0418891 0.00688652 0.00344326 0.999994i $$-0.498904\pi$$
0.00344326 + 0.999994i $$0.498904\pi$$
$$38$$ 8.94356 1.45084
$$39$$ 0 0
$$40$$ −4.86484 −0.769198
$$41$$ 4.90167 0.765513 0.382756 0.923849i $$-0.374975\pi$$
0.382756 + 0.923849i $$0.374975\pi$$
$$42$$ 0 0
$$43$$ −5.18479 −0.790673 −0.395337 0.918536i $$-0.629372\pi$$
−0.395337 + 0.918536i $$0.629372\pi$$
$$44$$ 1.09833 0.165579
$$45$$ 0 0
$$46$$ −3.96585 −0.584733
$$47$$ 3.73648 0.545022 0.272511 0.962153i $$-0.412146\pi$$
0.272511 + 0.962153i $$0.412146\pi$$
$$48$$ 0 0
$$49$$ −1.18479 −0.169256
$$50$$ 3.05644 0.432245
$$51$$ 0 0
$$52$$ 0.596267 0.0826873
$$53$$ −11.6382 −1.59862 −0.799312 0.600916i $$-0.794802\pi$$
−0.799312 + 0.600916i $$0.794802\pi$$
$$54$$ 0 0
$$55$$ 9.82295 1.32453
$$56$$ 7.09833 0.948554
$$57$$ 0 0
$$58$$ −1.73917 −0.228364
$$59$$ 7.34730 0.956537 0.478268 0.878214i $$-0.341265\pi$$
0.478268 + 0.878214i $$0.341265\pi$$
$$60$$ 0 0
$$61$$ 11.0496 1.41476 0.707380 0.706833i $$-0.249877\pi$$
0.707380 + 0.706833i $$0.249877\pi$$
$$62$$ 0.792919 0.100701
$$63$$ 0 0
$$64$$ 8.59627 1.07453
$$65$$ 5.33275 0.661446
$$66$$ 0 0
$$67$$ 1.85710 0.226880 0.113440 0.993545i $$-0.463813\pi$$
0.113440 + 0.993545i $$0.463813\pi$$
$$68$$ 0.554378 0.0672282
$$69$$ 0 0
$$70$$ 5.36959 0.641788
$$71$$ 5.51249 0.654212 0.327106 0.944988i $$-0.393927\pi$$
0.327106 + 0.944988i $$0.393927\pi$$
$$72$$ 0 0
$$73$$ 5.55438 0.650091 0.325045 0.945698i $$-0.394620\pi$$
0.325045 + 0.945698i $$0.394620\pi$$
$$74$$ −0.0564370 −0.00656067
$$75$$ 0 0
$$76$$ 1.22668 0.140710
$$77$$ −14.3327 −1.63337
$$78$$ 0 0
$$79$$ −3.78106 −0.425402 −0.212701 0.977117i $$-0.568226\pi$$
−0.212701 + 0.977117i $$0.568226\pi$$
$$80$$ 5.94356 0.664511
$$81$$ 0 0
$$82$$ −6.60401 −0.729291
$$83$$ −3.98545 −0.437460 −0.218730 0.975785i $$-0.570191\pi$$
−0.218730 + 0.975785i $$0.570191\pi$$
$$84$$ 0 0
$$85$$ 4.95811 0.537783
$$86$$ 6.98545 0.753261
$$87$$ 0 0
$$88$$ −17.4953 −1.86500
$$89$$ −8.15064 −0.863967 −0.431983 0.901882i $$-0.642186\pi$$
−0.431983 + 0.901882i $$0.642186\pi$$
$$90$$ 0 0
$$91$$ −7.78106 −0.815677
$$92$$ −0.543948 −0.0567105
$$93$$ 0 0
$$94$$ −5.03415 −0.519233
$$95$$ 10.9709 1.12559
$$96$$ 0 0
$$97$$ 0.260830 0.0264833 0.0132416 0.999912i $$-0.495785\pi$$
0.0132416 + 0.999912i $$0.495785\pi$$
$$98$$ 1.59627 0.161247
$$99$$ 0 0
$$100$$ 0.419215 0.0419215
$$101$$ −11.0273 −1.09726 −0.548631 0.836065i $$-0.684851\pi$$
−0.548631 + 0.836065i $$0.684851\pi$$
$$102$$ 0 0
$$103$$ −3.90673 −0.384941 −0.192471 0.981303i $$-0.561650\pi$$
−0.192471 + 0.981303i $$0.561650\pi$$
$$104$$ −9.49794 −0.931350
$$105$$ 0 0
$$106$$ 15.6800 1.52298
$$107$$ 2.63816 0.255040 0.127520 0.991836i $$-0.459298\pi$$
0.127520 + 0.991836i $$0.459298\pi$$
$$108$$ 0 0
$$109$$ −8.95811 −0.858031 −0.429016 0.903297i $$-0.641140\pi$$
−0.429016 + 0.903297i $$0.641140\pi$$
$$110$$ −13.2344 −1.26185
$$111$$ 0 0
$$112$$ −8.67230 −0.819456
$$113$$ −15.9290 −1.49848 −0.749238 0.662301i $$-0.769580\pi$$
−0.749238 + 0.662301i $$0.769580\pi$$
$$114$$ 0 0
$$115$$ −4.86484 −0.453648
$$116$$ −0.238541 −0.0221480
$$117$$ 0 0
$$118$$ −9.89899 −0.911275
$$119$$ −7.23442 −0.663178
$$120$$ 0 0
$$121$$ 24.3259 2.21145
$$122$$ −14.8871 −1.34782
$$123$$ 0 0
$$124$$ 0.108755 0.00976650
$$125$$ 12.0128 1.07446
$$126$$ 0 0
$$127$$ 3.59627 0.319117 0.159559 0.987188i $$-0.448993\pi$$
0.159559 + 0.987188i $$0.448993\pi$$
$$128$$ −9.49794 −0.839507
$$129$$ 0 0
$$130$$ −7.18479 −0.630148
$$131$$ 17.7074 1.54710 0.773551 0.633734i $$-0.218479\pi$$
0.773551 + 0.633734i $$0.218479\pi$$
$$132$$ 0 0
$$133$$ −16.0077 −1.38805
$$134$$ −2.50206 −0.216145
$$135$$ 0 0
$$136$$ −8.83069 −0.757225
$$137$$ 3.92902 0.335678 0.167839 0.985814i $$-0.446321\pi$$
0.167839 + 0.985814i $$0.446321\pi$$
$$138$$ 0 0
$$139$$ 11.9659 1.01493 0.507465 0.861672i $$-0.330583\pi$$
0.507465 + 0.861672i $$0.330583\pi$$
$$140$$ 0.736482 0.0622441
$$141$$ 0 0
$$142$$ −7.42696 −0.623256
$$143$$ 19.1780 1.60374
$$144$$ 0 0
$$145$$ −2.13341 −0.177170
$$146$$ −7.48339 −0.619330
$$147$$ 0 0
$$148$$ −0.00774079 −0.000636289 0
$$149$$ −20.5253 −1.68150 −0.840748 0.541426i $$-0.817885\pi$$
−0.840748 + 0.541426i $$0.817885\pi$$
$$150$$ 0 0
$$151$$ 16.0077 1.30269 0.651346 0.758781i $$-0.274205\pi$$
0.651346 + 0.758781i $$0.274205\pi$$
$$152$$ −19.5398 −1.58489
$$153$$ 0 0
$$154$$ 19.3105 1.55608
$$155$$ 0.972659 0.0781258
$$156$$ 0 0
$$157$$ −21.9736 −1.75368 −0.876842 0.480779i $$-0.840354\pi$$
−0.876842 + 0.480779i $$0.840354\pi$$
$$158$$ 5.09421 0.405273
$$159$$ 0 0
$$160$$ 1.72193 0.136131
$$161$$ 7.09833 0.559426
$$162$$ 0 0
$$163$$ 20.5107 1.60652 0.803262 0.595625i $$-0.203096\pi$$
0.803262 + 0.595625i $$0.203096\pi$$
$$164$$ −0.905793 −0.0707305
$$165$$ 0 0
$$166$$ 5.36959 0.416761
$$167$$ −4.29086 −0.332037 −0.166018 0.986123i $$-0.553091\pi$$
−0.166018 + 0.986123i $$0.553091\pi$$
$$168$$ 0 0
$$169$$ −2.58853 −0.199117
$$170$$ −6.68004 −0.512336
$$171$$ 0 0
$$172$$ 0.958111 0.0730553
$$173$$ −3.79292 −0.288370 −0.144185 0.989551i $$-0.546056\pi$$
−0.144185 + 0.989551i $$0.546056\pi$$
$$174$$ 0 0
$$175$$ −5.47060 −0.413538
$$176$$ 21.3746 1.61117
$$177$$ 0 0
$$178$$ 10.9813 0.823086
$$179$$ −8.27631 −0.618601 −0.309300 0.950964i $$-0.600095\pi$$
−0.309300 + 0.950964i $$0.600095\pi$$
$$180$$ 0 0
$$181$$ 6.72193 0.499637 0.249819 0.968293i $$-0.419629\pi$$
0.249819 + 0.968293i $$0.419629\pi$$
$$182$$ 10.4834 0.777081
$$183$$ 0 0
$$184$$ 8.66456 0.638760
$$185$$ −0.0692302 −0.00508991
$$186$$ 0 0
$$187$$ 17.8307 1.30391
$$188$$ −0.690474 −0.0503580
$$189$$ 0 0
$$190$$ −14.7811 −1.07233
$$191$$ 5.01455 0.362840 0.181420 0.983406i $$-0.441931\pi$$
0.181420 + 0.983406i $$0.441931\pi$$
$$192$$ 0 0
$$193$$ −17.8648 −1.28594 −0.642970 0.765892i $$-0.722298\pi$$
−0.642970 + 0.765892i $$0.722298\pi$$
$$194$$ −0.351415 −0.0252301
$$195$$ 0 0
$$196$$ 0.218941 0.0156386
$$197$$ −0.723689 −0.0515607 −0.0257803 0.999668i $$-0.508207\pi$$
−0.0257803 + 0.999668i $$0.508207\pi$$
$$198$$ 0 0
$$199$$ −10.1925 −0.722530 −0.361265 0.932463i $$-0.617655\pi$$
−0.361265 + 0.932463i $$0.617655\pi$$
$$200$$ −6.67768 −0.472183
$$201$$ 0 0
$$202$$ 14.8571 1.04534
$$203$$ 3.11287 0.218481
$$204$$ 0 0
$$205$$ −8.10101 −0.565799
$$206$$ 5.26352 0.366727
$$207$$ 0 0
$$208$$ 11.6040 0.804593
$$209$$ 39.4543 2.72911
$$210$$ 0 0
$$211$$ −14.8648 −1.02334 −0.511669 0.859183i $$-0.670973\pi$$
−0.511669 + 0.859183i $$0.670973\pi$$
$$212$$ 2.15064 0.147707
$$213$$ 0 0
$$214$$ −3.55438 −0.242972
$$215$$ 8.56893 0.584396
$$216$$ 0 0
$$217$$ −1.41921 −0.0963426
$$218$$ 12.0692 0.817431
$$219$$ 0 0
$$220$$ −1.81521 −0.122381
$$221$$ 9.68004 0.651150
$$222$$ 0 0
$$223$$ −10.9486 −0.733174 −0.366587 0.930384i $$-0.619474\pi$$
−0.366587 + 0.930384i $$0.619474\pi$$
$$224$$ −2.51249 −0.167873
$$225$$ 0 0
$$226$$ 21.4611 1.42757
$$227$$ 17.3327 1.15041 0.575207 0.818008i $$-0.304921\pi$$
0.575207 + 0.818008i $$0.304921\pi$$
$$228$$ 0 0
$$229$$ 1.56212 0.103228 0.0516138 0.998667i $$-0.483563\pi$$
0.0516138 + 0.998667i $$0.483563\pi$$
$$230$$ 6.55438 0.432183
$$231$$ 0 0
$$232$$ 3.79973 0.249464
$$233$$ −16.7888 −1.09987 −0.549935 0.835207i $$-0.685348\pi$$
−0.549935 + 0.835207i $$0.685348\pi$$
$$234$$ 0 0
$$235$$ −6.17530 −0.402832
$$236$$ −1.35773 −0.0883804
$$237$$ 0 0
$$238$$ 9.74691 0.631798
$$239$$ 4.02910 0.260621 0.130310 0.991473i $$-0.458403\pi$$
0.130310 + 0.991473i $$0.458403\pi$$
$$240$$ 0 0
$$241$$ −3.35235 −0.215944 −0.107972 0.994154i $$-0.534436\pi$$
−0.107972 + 0.994154i $$0.534436\pi$$
$$242$$ −32.7743 −2.10681
$$243$$ 0 0
$$244$$ −2.04189 −0.130719
$$245$$ 1.95811 0.125099
$$246$$ 0 0
$$247$$ 21.4192 1.36287
$$248$$ −1.73236 −0.110005
$$249$$ 0 0
$$250$$ −16.1848 −1.02362
$$251$$ 23.1506 1.46126 0.730628 0.682776i $$-0.239227\pi$$
0.730628 + 0.682776i $$0.239227\pi$$
$$252$$ 0 0
$$253$$ −17.4953 −1.09992
$$254$$ −4.84524 −0.304017
$$255$$ 0 0
$$256$$ −4.39599 −0.274750
$$257$$ −12.0128 −0.749337 −0.374669 0.927159i $$-0.622244\pi$$
−0.374669 + 0.927159i $$0.622244\pi$$
$$258$$ 0 0
$$259$$ 0.101014 0.00627673
$$260$$ −0.985452 −0.0611151
$$261$$ 0 0
$$262$$ −23.8571 −1.47390
$$263$$ −16.9017 −1.04220 −0.521101 0.853495i $$-0.674478\pi$$
−0.521101 + 0.853495i $$0.674478\pi$$
$$264$$ 0 0
$$265$$ 19.2344 1.18156
$$266$$ 21.5672 1.32237
$$267$$ 0 0
$$268$$ −0.343178 −0.0209629
$$269$$ 7.91447 0.482554 0.241277 0.970456i $$-0.422434\pi$$
0.241277 + 0.970456i $$0.422434\pi$$
$$270$$ 0 0
$$271$$ −17.2344 −1.04692 −0.523458 0.852051i $$-0.675358\pi$$
−0.523458 + 0.852051i $$0.675358\pi$$
$$272$$ 10.7888 0.654167
$$273$$ 0 0
$$274$$ −5.29355 −0.319795
$$275$$ 13.4834 0.813079
$$276$$ 0 0
$$277$$ 26.4347 1.58831 0.794153 0.607717i $$-0.207915\pi$$
0.794153 + 0.607717i $$0.207915\pi$$
$$278$$ −16.1215 −0.966906
$$279$$ 0 0
$$280$$ −11.7314 −0.701087
$$281$$ 18.9959 1.13320 0.566600 0.823993i $$-0.308259\pi$$
0.566600 + 0.823993i $$0.308259\pi$$
$$282$$ 0 0
$$283$$ −16.5868 −0.985981 −0.492991 0.870035i $$-0.664096\pi$$
−0.492991 + 0.870035i $$0.664096\pi$$
$$284$$ −1.01867 −0.0604467
$$285$$ 0 0
$$286$$ −25.8384 −1.52786
$$287$$ 11.8203 0.697728
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 2.87433 0.168787
$$291$$ 0 0
$$292$$ −1.02641 −0.0600660
$$293$$ −19.3577 −1.13089 −0.565445 0.824786i $$-0.691296\pi$$
−0.565445 + 0.824786i $$0.691296\pi$$
$$294$$ 0 0
$$295$$ −12.1429 −0.706987
$$296$$ 0.123303 0.00716685
$$297$$ 0 0
$$298$$ 27.6536 1.60193
$$299$$ −9.49794 −0.549280
$$300$$ 0 0
$$301$$ −12.5030 −0.720661
$$302$$ −21.5672 −1.24105
$$303$$ 0 0
$$304$$ 23.8726 1.36919
$$305$$ −18.2618 −1.04567
$$306$$ 0 0
$$307$$ 20.8057 1.18744 0.593722 0.804670i $$-0.297658\pi$$
0.593722 + 0.804670i $$0.297658\pi$$
$$308$$ 2.64858 0.150917
$$309$$ 0 0
$$310$$ −1.31046 −0.0744291
$$311$$ 10.6655 0.604785 0.302392 0.953184i $$-0.402215\pi$$
0.302392 + 0.953184i $$0.402215\pi$$
$$312$$ 0 0
$$313$$ 3.81521 0.215648 0.107824 0.994170i $$-0.465612\pi$$
0.107824 + 0.994170i $$0.465612\pi$$
$$314$$ 29.6049 1.67070
$$315$$ 0 0
$$316$$ 0.698711 0.0393056
$$317$$ −26.3892 −1.48216 −0.741082 0.671414i $$-0.765687\pi$$
−0.741082 + 0.671414i $$0.765687\pi$$
$$318$$ 0 0
$$319$$ −7.67230 −0.429567
$$320$$ −14.2071 −0.794200
$$321$$ 0 0
$$322$$ −9.56355 −0.532956
$$323$$ 19.9145 1.10807
$$324$$ 0 0
$$325$$ 7.31996 0.406038
$$326$$ −27.6340 −1.53051
$$327$$ 0 0
$$328$$ 14.4284 0.796674
$$329$$ 9.01043 0.496761
$$330$$ 0 0
$$331$$ −1.57161 −0.0863837 −0.0431919 0.999067i $$-0.513753\pi$$
−0.0431919 + 0.999067i $$0.513753\pi$$
$$332$$ 0.736482 0.0404197
$$333$$ 0 0
$$334$$ 5.78106 0.316325
$$335$$ −3.06923 −0.167690
$$336$$ 0 0
$$337$$ −8.01548 −0.436631 −0.218316 0.975878i $$-0.570056\pi$$
−0.218316 + 0.975878i $$0.570056\pi$$
$$338$$ 3.48751 0.189696
$$339$$ 0 0
$$340$$ −0.916222 −0.0496891
$$341$$ 3.49794 0.189424
$$342$$ 0 0
$$343$$ −19.7374 −1.06572
$$344$$ −15.2618 −0.822859
$$345$$ 0 0
$$346$$ 5.11019 0.274725
$$347$$ −19.8452 −1.06535 −0.532674 0.846320i $$-0.678813\pi$$
−0.532674 + 0.846320i $$0.678813\pi$$
$$348$$ 0 0
$$349$$ 11.0933 0.593809 0.296905 0.954907i $$-0.404046\pi$$
0.296905 + 0.954907i $$0.404046\pi$$
$$350$$ 7.37052 0.393971
$$351$$ 0 0
$$352$$ 6.19253 0.330063
$$353$$ 2.86390 0.152430 0.0762151 0.997091i $$-0.475716\pi$$
0.0762151 + 0.997091i $$0.475716\pi$$
$$354$$ 0 0
$$355$$ −9.11051 −0.483536
$$356$$ 1.50618 0.0798273
$$357$$ 0 0
$$358$$ 11.1506 0.589330
$$359$$ −28.7888 −1.51941 −0.759707 0.650265i $$-0.774658\pi$$
−0.759707 + 0.650265i $$0.774658\pi$$
$$360$$ 0 0
$$361$$ 25.0651 1.31922
$$362$$ −9.05644 −0.475996
$$363$$ 0 0
$$364$$ 1.43788 0.0753655
$$365$$ −9.17974 −0.480490
$$366$$ 0 0
$$367$$ −10.9923 −0.573791 −0.286896 0.957962i $$-0.592623\pi$$
−0.286896 + 0.957962i $$0.592623\pi$$
$$368$$ −10.5858 −0.551825
$$369$$ 0 0
$$370$$ 0.0932736 0.00484906
$$371$$ −28.0651 −1.45707
$$372$$ 0 0
$$373$$ 33.4097 1.72989 0.864945 0.501867i $$-0.167353\pi$$
0.864945 + 0.501867i $$0.167353\pi$$
$$374$$ −24.0232 −1.24221
$$375$$ 0 0
$$376$$ 10.9986 0.567208
$$377$$ −4.16519 −0.214518
$$378$$ 0 0
$$379$$ 20.9394 1.07559 0.537794 0.843077i $$-0.319258\pi$$
0.537794 + 0.843077i $$0.319258\pi$$
$$380$$ −2.02734 −0.104000
$$381$$ 0 0
$$382$$ −6.75608 −0.345671
$$383$$ 4.11112 0.210068 0.105034 0.994469i $$-0.466505\pi$$
0.105034 + 0.994469i $$0.466505\pi$$
$$384$$ 0 0
$$385$$ 23.6878 1.20724
$$386$$ 24.0692 1.22509
$$387$$ 0 0
$$388$$ −0.0481994 −0.00244695
$$389$$ −17.0942 −0.866711 −0.433355 0.901223i $$-0.642670\pi$$
−0.433355 + 0.901223i $$0.642670\pi$$
$$390$$ 0 0
$$391$$ −8.83069 −0.446587
$$392$$ −3.48751 −0.176146
$$393$$ 0 0
$$394$$ 0.975023 0.0491209
$$395$$ 6.24897 0.314420
$$396$$ 0 0
$$397$$ 22.4020 1.12432 0.562162 0.827027i $$-0.309970\pi$$
0.562162 + 0.827027i $$0.309970\pi$$
$$398$$ 13.7324 0.688341
$$399$$ 0 0
$$400$$ 8.15839 0.407919
$$401$$ 14.5817 0.728176 0.364088 0.931364i $$-0.381381\pi$$
0.364088 + 0.931364i $$0.381381\pi$$
$$402$$ 0 0
$$403$$ 1.89899 0.0945952
$$404$$ 2.03777 0.101383
$$405$$ 0 0
$$406$$ −4.19396 −0.208143
$$407$$ −0.248970 −0.0123410
$$408$$ 0 0
$$409$$ −17.5030 −0.865467 −0.432734 0.901522i $$-0.642451\pi$$
−0.432734 + 0.901522i $$0.642451\pi$$
$$410$$ 10.9145 0.539027
$$411$$ 0 0
$$412$$ 0.721934 0.0355671
$$413$$ 17.7178 0.871837
$$414$$ 0 0
$$415$$ 6.58677 0.323332
$$416$$ 3.36184 0.164828
$$417$$ 0 0
$$418$$ −53.1566 −2.59998
$$419$$ −18.8621 −0.921476 −0.460738 0.887536i $$-0.652415\pi$$
−0.460738 + 0.887536i $$0.652415\pi$$
$$420$$ 0 0
$$421$$ −32.3337 −1.57585 −0.787924 0.615773i $$-0.788844\pi$$
−0.787924 + 0.615773i $$0.788844\pi$$
$$422$$ 20.0273 0.974916
$$423$$ 0 0
$$424$$ −34.2576 −1.66370
$$425$$ 6.80571 0.330126
$$426$$ 0 0
$$427$$ 26.6459 1.28949
$$428$$ −0.487511 −0.0235648
$$429$$ 0 0
$$430$$ −11.5449 −0.556744
$$431$$ −34.3164 −1.65297 −0.826483 0.562962i $$-0.809662\pi$$
−0.826483 + 0.562962i $$0.809662\pi$$
$$432$$ 0 0
$$433$$ −25.0669 −1.20464 −0.602318 0.798256i $$-0.705756\pi$$
−0.602318 + 0.798256i $$0.705756\pi$$
$$434$$ 1.91210 0.0917839
$$435$$ 0 0
$$436$$ 1.65539 0.0792789
$$437$$ −19.5398 −0.934717
$$438$$ 0 0
$$439$$ −23.2080 −1.10766 −0.553829 0.832630i $$-0.686834\pi$$
−0.553829 + 0.832630i $$0.686834\pi$$
$$440$$ 28.9145 1.37844
$$441$$ 0 0
$$442$$ −13.0419 −0.620339
$$443$$ 4.12155 0.195821 0.0979103 0.995195i $$-0.468784\pi$$
0.0979103 + 0.995195i $$0.468784\pi$$
$$444$$ 0 0
$$445$$ 13.4706 0.638568
$$446$$ 14.7510 0.698482
$$447$$ 0 0
$$448$$ 20.7297 0.979385
$$449$$ −18.3414 −0.865585 −0.432793 0.901494i $$-0.642472\pi$$
−0.432793 + 0.901494i $$0.642472\pi$$
$$450$$ 0 0
$$451$$ −29.1334 −1.37184
$$452$$ 2.94356 0.138454
$$453$$ 0 0
$$454$$ −23.3523 −1.09598
$$455$$ 12.8598 0.602876
$$456$$ 0 0
$$457$$ −19.4611 −0.910352 −0.455176 0.890401i $$-0.650424\pi$$
−0.455176 + 0.890401i $$0.650424\pi$$
$$458$$ −2.10464 −0.0983432
$$459$$ 0 0
$$460$$ 0.898986 0.0419154
$$461$$ −27.7493 −1.29241 −0.646206 0.763163i $$-0.723645\pi$$
−0.646206 + 0.763163i $$0.723645\pi$$
$$462$$ 0 0
$$463$$ 38.6860 1.79789 0.898946 0.438059i $$-0.144334\pi$$
0.898946 + 0.438059i $$0.144334\pi$$
$$464$$ −4.64227 −0.215512
$$465$$ 0 0
$$466$$ 22.6195 1.04783
$$467$$ −29.7638 −1.37731 −0.688653 0.725091i $$-0.741798\pi$$
−0.688653 + 0.725091i $$0.741798\pi$$
$$468$$ 0 0
$$469$$ 4.47834 0.206791
$$470$$ 8.31996 0.383771
$$471$$ 0 0
$$472$$ 21.6272 0.995474
$$473$$ 30.8161 1.41693
$$474$$ 0 0
$$475$$ 15.0591 0.690960
$$476$$ 1.33687 0.0612752
$$477$$ 0 0
$$478$$ −5.42839 −0.248289
$$479$$ 37.6759 1.72146 0.860728 0.509064i $$-0.170008\pi$$
0.860728 + 0.509064i $$0.170008\pi$$
$$480$$ 0 0
$$481$$ −0.135163 −0.00616289
$$482$$ 4.51661 0.205726
$$483$$ 0 0
$$484$$ −4.49525 −0.204330
$$485$$ −0.431074 −0.0195741
$$486$$ 0 0
$$487$$ −0.763823 −0.0346121 −0.0173061 0.999850i $$-0.505509\pi$$
−0.0173061 + 0.999850i $$0.505509\pi$$
$$488$$ 32.5253 1.47235
$$489$$ 0 0
$$490$$ −2.63816 −0.119180
$$491$$ 0.497941 0.0224717 0.0112359 0.999937i $$-0.496423\pi$$
0.0112359 + 0.999937i $$0.496423\pi$$
$$492$$ 0 0
$$493$$ −3.87258 −0.174412
$$494$$ −28.8580 −1.29838
$$495$$ 0 0
$$496$$ 2.11650 0.0950335
$$497$$ 13.2932 0.596283
$$498$$ 0 0
$$499$$ 8.96585 0.401367 0.200683 0.979656i $$-0.435684\pi$$
0.200683 + 0.979656i $$0.435684\pi$$
$$500$$ −2.21987 −0.0992758
$$501$$ 0 0
$$502$$ −31.1908 −1.39211
$$503$$ −18.3618 −0.818714 −0.409357 0.912374i $$-0.634247\pi$$
−0.409357 + 0.912374i $$0.634247\pi$$
$$504$$ 0 0
$$505$$ 18.2249 0.810999
$$506$$ 23.5713 1.04787
$$507$$ 0 0
$$508$$ −0.664563 −0.0294852
$$509$$ 28.3705 1.25750 0.628751 0.777607i $$-0.283567\pi$$
0.628751 + 0.777607i $$0.283567\pi$$
$$510$$ 0 0
$$511$$ 13.3942 0.592526
$$512$$ 24.9186 1.10126
$$513$$ 0 0
$$514$$ 16.1848 0.713881
$$515$$ 6.45666 0.284514
$$516$$ 0 0
$$517$$ −22.2080 −0.976707
$$518$$ −0.136096 −0.00597973
$$519$$ 0 0
$$520$$ 15.6973 0.688371
$$521$$ 32.6382 1.42990 0.714952 0.699174i $$-0.246449\pi$$
0.714952 + 0.699174i $$0.246449\pi$$
$$522$$ 0 0
$$523$$ −22.0232 −0.963008 −0.481504 0.876444i $$-0.659909\pi$$
−0.481504 + 0.876444i $$0.659909\pi$$
$$524$$ −3.27219 −0.142946
$$525$$ 0 0
$$526$$ 22.7716 0.992887
$$527$$ 1.76558 0.0769098
$$528$$ 0 0
$$529$$ −14.3354 −0.623280
$$530$$ −25.9145 −1.12565
$$531$$ 0 0
$$532$$ 2.95811 0.128250
$$533$$ −15.8161 −0.685073
$$534$$ 0 0
$$535$$ −4.36009 −0.188503
$$536$$ 5.46648 0.236116
$$537$$ 0 0
$$538$$ −10.6631 −0.459720
$$539$$ 7.04189 0.303316
$$540$$ 0 0
$$541$$ −15.7870 −0.678738 −0.339369 0.940653i $$-0.610214\pi$$
−0.339369 + 0.940653i $$0.610214\pi$$
$$542$$ 23.2199 0.997379
$$543$$ 0 0
$$544$$ 3.12567 0.134012
$$545$$ 14.8051 0.634181
$$546$$ 0 0
$$547$$ −27.4192 −1.17236 −0.586180 0.810180i $$-0.699369\pi$$
−0.586180 + 0.810180i $$0.699369\pi$$
$$548$$ −0.726053 −0.0310154
$$549$$ 0 0
$$550$$ −18.1661 −0.774606
$$551$$ −8.56893 −0.365048
$$552$$ 0 0
$$553$$ −9.11793 −0.387734
$$554$$ −35.6154 −1.51315
$$555$$ 0 0
$$556$$ −2.21120 −0.0937758
$$557$$ −29.4020 −1.24580 −0.622901 0.782301i $$-0.714046\pi$$
−0.622901 + 0.782301i $$0.714046\pi$$
$$558$$ 0 0
$$559$$ 16.7297 0.707590
$$560$$ 14.3327 0.605669
$$561$$ 0 0
$$562$$ −25.5931 −1.07958
$$563$$ 10.3705 0.437065 0.218533 0.975830i $$-0.429873\pi$$
0.218533 + 0.975830i $$0.429873\pi$$
$$564$$ 0 0
$$565$$ 26.3259 1.10754
$$566$$ 22.3473 0.939327
$$567$$ 0 0
$$568$$ 16.2264 0.680843
$$569$$ 32.9691 1.38214 0.691069 0.722788i $$-0.257140\pi$$
0.691069 + 0.722788i $$0.257140\pi$$
$$570$$ 0 0
$$571$$ −0.737415 −0.0308599 −0.0154299 0.999881i $$-0.504912\pi$$
−0.0154299 + 0.999881i $$0.504912\pi$$
$$572$$ −3.54395 −0.148180
$$573$$ 0 0
$$574$$ −15.9254 −0.664713
$$575$$ −6.67768 −0.278479
$$576$$ 0 0
$$577$$ 19.3432 0.805267 0.402634 0.915361i $$-0.368095\pi$$
0.402634 + 0.915361i $$0.368095\pi$$
$$578$$ 10.7784 0.448321
$$579$$ 0 0
$$580$$ 0.394238 0.0163698
$$581$$ −9.61081 −0.398724
$$582$$ 0 0
$$583$$ 69.1721 2.86482
$$584$$ 16.3497 0.676554
$$585$$ 0 0
$$586$$ 26.0806 1.07738
$$587$$ 31.9121 1.31715 0.658577 0.752514i $$-0.271159\pi$$
0.658577 + 0.752514i $$0.271159\pi$$
$$588$$ 0 0
$$589$$ 3.90673 0.160974
$$590$$ 16.3601 0.673534
$$591$$ 0 0
$$592$$ −0.150644 −0.00619144
$$593$$ −31.6783 −1.30087 −0.650436 0.759561i $$-0.725414\pi$$
−0.650436 + 0.759561i $$0.725414\pi$$
$$594$$ 0 0
$$595$$ 11.9564 0.490163
$$596$$ 3.79292 0.155364
$$597$$ 0 0
$$598$$ 12.7965 0.523289
$$599$$ 12.6236 0.515787 0.257893 0.966173i $$-0.416972\pi$$
0.257893 + 0.966173i $$0.416972\pi$$
$$600$$ 0 0
$$601$$ −8.90848 −0.363385 −0.181692 0.983355i $$-0.558157\pi$$
−0.181692 + 0.983355i $$0.558157\pi$$
$$602$$ 16.8452 0.686561
$$603$$ 0 0
$$604$$ −2.95811 −0.120364
$$605$$ −40.2036 −1.63451
$$606$$ 0 0
$$607$$ −33.1242 −1.34447 −0.672236 0.740337i $$-0.734666\pi$$
−0.672236 + 0.740337i $$0.734666\pi$$
$$608$$ 6.91622 0.280490
$$609$$ 0 0
$$610$$ 24.6040 0.996187
$$611$$ −12.0564 −0.487751
$$612$$ 0 0
$$613$$ 17.6800 0.714090 0.357045 0.934087i $$-0.383784\pi$$
0.357045 + 0.934087i $$0.383784\pi$$
$$614$$ −28.0315 −1.13126
$$615$$ 0 0
$$616$$ −42.1893 −1.69986
$$617$$ 25.7324 1.03595 0.517973 0.855397i $$-0.326687\pi$$
0.517973 + 0.855397i $$0.326687\pi$$
$$618$$ 0 0
$$619$$ 27.7948 1.11717 0.558583 0.829448i $$-0.311345\pi$$
0.558583 + 0.829448i $$0.311345\pi$$
$$620$$ −0.179740 −0.00721854
$$621$$ 0 0
$$622$$ −14.3696 −0.576168
$$623$$ −19.6551 −0.787464
$$624$$ 0 0
$$625$$ −8.51073 −0.340429
$$626$$ −5.14022 −0.205444
$$627$$ 0 0
$$628$$ 4.06056 0.162034
$$629$$ −0.125667 −0.00501068
$$630$$ 0 0
$$631$$ 26.8138 1.06744 0.533720 0.845661i $$-0.320794\pi$$
0.533720 + 0.845661i $$0.320794\pi$$
$$632$$ −11.1298 −0.442719
$$633$$ 0 0
$$634$$ 35.5541 1.41203
$$635$$ −5.94356 −0.235863
$$636$$ 0 0
$$637$$ 3.82295 0.151471
$$638$$ 10.3369 0.409240
$$639$$ 0 0
$$640$$ 15.6973 0.620490
$$641$$ −12.6905 −0.501244 −0.250622 0.968085i $$-0.580635\pi$$
−0.250622 + 0.968085i $$0.580635\pi$$
$$642$$ 0 0
$$643$$ 15.4766 0.610337 0.305168 0.952298i $$-0.401287\pi$$
0.305168 + 0.952298i $$0.401287\pi$$
$$644$$ −1.31172 −0.0516889
$$645$$ 0 0
$$646$$ −26.8307 −1.05564
$$647$$ −11.1506 −0.438377 −0.219189 0.975683i $$-0.570341\pi$$
−0.219189 + 0.975683i $$0.570341\pi$$
$$648$$ 0 0
$$649$$ −43.6691 −1.71416
$$650$$ −9.86215 −0.386825
$$651$$ 0 0
$$652$$ −3.79023 −0.148437
$$653$$ −44.5921 −1.74503 −0.872513 0.488591i $$-0.837511\pi$$
−0.872513 + 0.488591i $$0.837511\pi$$
$$654$$ 0 0
$$655$$ −29.2651 −1.14348
$$656$$ −17.6277 −0.688247
$$657$$ 0 0
$$658$$ −12.1397 −0.473255
$$659$$ 14.0966 0.549124 0.274562 0.961569i $$-0.411467\pi$$
0.274562 + 0.961569i $$0.411467\pi$$
$$660$$ 0 0
$$661$$ 36.1147 1.40470 0.702350 0.711831i $$-0.252134\pi$$
0.702350 + 0.711831i $$0.252134\pi$$
$$662$$ 2.11743 0.0822962
$$663$$ 0 0
$$664$$ −11.7314 −0.455268
$$665$$ 26.4561 1.02592
$$666$$ 0 0
$$667$$ 3.79973 0.147126
$$668$$ 0.792919 0.0306789
$$669$$ 0 0
$$670$$ 4.13516 0.159755
$$671$$ −65.6742 −2.53532
$$672$$ 0 0
$$673$$ 2.24216 0.0864290 0.0432145 0.999066i $$-0.486240\pi$$
0.0432145 + 0.999066i $$0.486240\pi$$
$$674$$ 10.7992 0.415971
$$675$$ 0 0
$$676$$ 0.478340 0.0183977
$$677$$ −35.1762 −1.35193 −0.675966 0.736933i $$-0.736273\pi$$
−0.675966 + 0.736933i $$0.736273\pi$$
$$678$$ 0 0
$$679$$ 0.628984 0.0241382
$$680$$ 14.5945 0.559674
$$681$$ 0 0
$$682$$ −4.71276 −0.180461
$$683$$ 17.7638 0.679714 0.339857 0.940477i $$-0.389621\pi$$
0.339857 + 0.940477i $$0.389621\pi$$
$$684$$ 0 0
$$685$$ −6.49350 −0.248104
$$686$$ 26.5921 1.01529
$$687$$ 0 0
$$688$$ 18.6459 0.710868
$$689$$ 37.5526 1.43064
$$690$$ 0 0
$$691$$ −44.0306 −1.67500 −0.837502 0.546434i $$-0.815985\pi$$
−0.837502 + 0.546434i $$0.815985\pi$$
$$692$$ 0.700903 0.0266443
$$693$$ 0 0
$$694$$ 26.7374 1.01494
$$695$$ −19.7760 −0.750147
$$696$$ 0 0
$$697$$ −14.7050 −0.556992
$$698$$ −14.9459 −0.565712
$$699$$ 0 0
$$700$$ 1.01093 0.0382094
$$701$$ 30.1052 1.13706 0.568530 0.822663i $$-0.307512\pi$$
0.568530 + 0.822663i $$0.307512\pi$$
$$702$$ 0 0
$$703$$ −0.278066 −0.0104875
$$704$$ −51.0925 −1.92562
$$705$$ 0 0
$$706$$ −3.85853 −0.145218
$$707$$ −26.5921 −1.00010
$$708$$ 0 0
$$709$$ −24.7374 −0.929033 −0.464517 0.885564i $$-0.653772\pi$$
−0.464517 + 0.885564i $$0.653772\pi$$
$$710$$ 12.2746 0.460656
$$711$$ 0 0
$$712$$ −23.9919 −0.899136
$$713$$ −1.73236 −0.0648775
$$714$$ 0 0
$$715$$ −31.6955 −1.18535
$$716$$ 1.52940 0.0571564
$$717$$ 0 0
$$718$$ 38.7870 1.44752
$$719$$ 43.5526 1.62424 0.812119 0.583491i $$-0.198314\pi$$
0.812119 + 0.583491i $$0.198314\pi$$
$$720$$ 0 0
$$721$$ −9.42097 −0.350855
$$722$$ −33.7701 −1.25679
$$723$$ 0 0
$$724$$ −1.24216 −0.0461646
$$725$$ −2.92841 −0.108758
$$726$$ 0 0
$$727$$ 20.5371 0.761680 0.380840 0.924641i $$-0.375635\pi$$
0.380840 + 0.924641i $$0.375635\pi$$
$$728$$ −22.9040 −0.848880
$$729$$ 0 0
$$730$$ 12.3678 0.457754
$$731$$ 15.5544 0.575299
$$732$$ 0 0
$$733$$ 14.0060 0.517323 0.258661 0.965968i $$-0.416719\pi$$
0.258661 + 0.965968i $$0.416719\pi$$
$$734$$ 14.8098 0.546641
$$735$$ 0 0
$$736$$ −3.06687 −0.113046
$$737$$ −11.0378 −0.406581
$$738$$ 0 0
$$739$$ −41.9813 −1.54431 −0.772154 0.635435i $$-0.780821\pi$$
−0.772154 + 0.635435i $$0.780821\pi$$
$$740$$ 0.0127932 0.000470288 0
$$741$$ 0 0
$$742$$ 37.8120 1.38812
$$743$$ −27.8621 −1.02216 −0.511082 0.859532i $$-0.670755\pi$$
−0.511082 + 0.859532i $$0.670755\pi$$
$$744$$ 0 0
$$745$$ 33.9222 1.24281
$$746$$ −45.0128 −1.64804
$$747$$ 0 0
$$748$$ −3.29498 −0.120476
$$749$$ 6.36184 0.232457
$$750$$ 0 0
$$751$$ −52.9050 −1.93053 −0.965265 0.261273i $$-0.915858\pi$$
−0.965265 + 0.261273i $$0.915858\pi$$
$$752$$ −13.4374 −0.490011
$$753$$ 0 0
$$754$$ 5.61175 0.204368
$$755$$ −26.4561 −0.962834
$$756$$ 0 0
$$757$$ −41.4858 −1.50783 −0.753913 0.656975i $$-0.771836\pi$$
−0.753913 + 0.656975i $$0.771836\pi$$
$$758$$ −28.2116 −1.02469
$$759$$ 0 0
$$760$$ 32.2935 1.17141
$$761$$ 45.3874 1.64529 0.822647 0.568553i $$-0.192497\pi$$
0.822647 + 0.568553i $$0.192497\pi$$
$$762$$ 0 0
$$763$$ −21.6023 −0.782054
$$764$$ −0.926651 −0.0335251
$$765$$ 0 0
$$766$$ −5.53890 −0.200128
$$767$$ −23.7074 −0.856024
$$768$$ 0 0
$$769$$ 5.11650 0.184506 0.0922528 0.995736i $$-0.470593\pi$$
0.0922528 + 0.995736i $$0.470593\pi$$
$$770$$ −31.9145 −1.15012
$$771$$ 0 0
$$772$$ 3.30129 0.118816
$$773$$ 52.6427 1.89343 0.946713 0.322077i $$-0.104381\pi$$
0.946713 + 0.322077i $$0.104381\pi$$
$$774$$ 0 0
$$775$$ 1.33511 0.0479587
$$776$$ 0.767769 0.0275613
$$777$$ 0 0
$$778$$ 23.0310 0.825700
$$779$$ −32.5381 −1.16580
$$780$$ 0 0
$$781$$ −32.7638 −1.17238
$$782$$ 11.8976 0.425456
$$783$$ 0 0
$$784$$ 4.26083 0.152172
$$785$$ 36.3158 1.29617
$$786$$ 0 0
$$787$$ −20.7624 −0.740099 −0.370050 0.929012i $$-0.620659\pi$$
−0.370050 + 0.929012i $$0.620659\pi$$
$$788$$ 0.133732 0.00476401
$$789$$ 0 0
$$790$$ −8.41921 −0.299542
$$791$$ −38.4124 −1.36579
$$792$$ 0 0
$$793$$ −35.6536 −1.26610
$$794$$ −30.1821 −1.07112
$$795$$ 0 0
$$796$$ 1.88350 0.0667590
$$797$$ 45.5800 1.61453 0.807263 0.590192i $$-0.200948\pi$$
0.807263 + 0.590192i $$0.200948\pi$$
$$798$$ 0 0
$$799$$ −11.2094 −0.396562
$$800$$ 2.36360 0.0835658
$$801$$ 0 0
$$802$$ −19.6459 −0.693721
$$803$$ −33.0128 −1.16500
$$804$$ 0 0
$$805$$ −11.7314 −0.413479
$$806$$ −2.55850 −0.0901192
$$807$$ 0 0
$$808$$ −32.4597 −1.14193
$$809$$ 4.21120 0.148058 0.0740290 0.997256i $$-0.476414\pi$$
0.0740290 + 0.997256i $$0.476414\pi$$
$$810$$ 0 0
$$811$$ 11.3618 0.398968 0.199484 0.979901i $$-0.436073\pi$$
0.199484 + 0.979901i $$0.436073\pi$$
$$812$$ −0.575236 −0.0201868
$$813$$ 0 0
$$814$$ 0.335437 0.0117571
$$815$$ −33.8982 −1.18740
$$816$$ 0 0
$$817$$ 34.4175 1.20411
$$818$$ 23.5817 0.824515
$$819$$ 0 0
$$820$$ 1.49701 0.0522778
$$821$$ −1.64227 −0.0573158 −0.0286579 0.999589i $$-0.509123\pi$$
−0.0286579 + 0.999589i $$0.509123\pi$$
$$822$$ 0 0
$$823$$ 11.2189 0.391068 0.195534 0.980697i $$-0.437356\pi$$
0.195534 + 0.980697i $$0.437356\pi$$
$$824$$ −11.4997 −0.400611
$$825$$ 0 0
$$826$$ −23.8711 −0.830583
$$827$$ 9.61318 0.334283 0.167141 0.985933i $$-0.446546\pi$$
0.167141 + 0.985933i $$0.446546\pi$$
$$828$$ 0 0
$$829$$ 33.4938 1.16329 0.581644 0.813443i $$-0.302410\pi$$
0.581644 + 0.813443i $$0.302410\pi$$
$$830$$ −8.87433 −0.308033
$$831$$ 0 0
$$832$$ −27.7374 −0.961622
$$833$$ 3.55438 0.123152
$$834$$ 0 0
$$835$$ 7.09152 0.245412
$$836$$ −7.29086 −0.252160
$$837$$ 0 0
$$838$$ 25.4129 0.877874
$$839$$ 31.9941 1.10456 0.552280 0.833659i $$-0.313758\pi$$
0.552280 + 0.833659i $$0.313758\pi$$
$$840$$ 0 0
$$841$$ −27.3337 −0.942541
$$842$$ 43.5631 1.50128
$$843$$ 0 0
$$844$$ 2.74691 0.0945526
$$845$$ 4.27807 0.147170
$$846$$ 0 0
$$847$$ 58.6614 2.01563
$$848$$ 41.8539 1.43727
$$849$$ 0 0
$$850$$ −9.16931 −0.314505
$$851$$ 0.123303 0.00422678
$$852$$ 0 0
$$853$$ 35.2422 1.20667 0.603334 0.797488i $$-0.293838\pi$$
0.603334 + 0.797488i $$0.293838\pi$$
$$854$$ −35.8999 −1.22847
$$855$$ 0 0
$$856$$ 7.76558 0.265422
$$857$$ 21.2490 0.725851 0.362925 0.931818i $$-0.381778\pi$$
0.362925 + 0.931818i $$0.381778\pi$$
$$858$$ 0 0
$$859$$ 51.8384 1.76870 0.884352 0.466820i $$-0.154601\pi$$
0.884352 + 0.466820i $$0.154601\pi$$
$$860$$ −1.58347 −0.0539960
$$861$$ 0 0
$$862$$ 46.2344 1.57475
$$863$$ 22.6783 0.771978 0.385989 0.922503i $$-0.373860\pi$$
0.385989 + 0.922503i $$0.373860\pi$$
$$864$$ 0 0
$$865$$ 6.26857 0.213138
$$866$$ 33.7725 1.14764
$$867$$ 0 0
$$868$$ 0.262260 0.00890170
$$869$$ 22.4730 0.762343
$$870$$ 0 0
$$871$$ −5.99226 −0.203040
$$872$$ −26.3688 −0.892959
$$873$$ 0 0
$$874$$ 26.3259 0.890488
$$875$$ 28.9685 0.979315
$$876$$ 0 0
$$877$$ 1.13341 0.0382725 0.0191362 0.999817i $$-0.493908\pi$$
0.0191362 + 0.999817i $$0.493908\pi$$
$$878$$ 31.2681 1.05525
$$879$$ 0 0
$$880$$ −35.3259 −1.19084
$$881$$ 30.8289 1.03865 0.519327 0.854576i $$-0.326183\pi$$
0.519327 + 0.854576i $$0.326183\pi$$
$$882$$ 0 0
$$883$$ −9.33511 −0.314152 −0.157076 0.987587i $$-0.550207\pi$$
−0.157076 + 0.987587i $$0.550207\pi$$
$$884$$ −1.78880 −0.0601639
$$885$$ 0 0
$$886$$ −5.55295 −0.186555
$$887$$ 14.0942 0.473237 0.236619 0.971603i $$-0.423961\pi$$
0.236619 + 0.971603i $$0.423961\pi$$
$$888$$ 0 0
$$889$$ 8.67230 0.290860
$$890$$ −18.1489 −0.608352
$$891$$ 0 0
$$892$$ 2.02322 0.0677425
$$893$$ −24.8033 −0.830012
$$894$$ 0 0
$$895$$ 13.6783 0.457215
$$896$$ −22.9040 −0.765170
$$897$$ 0 0
$$898$$ 24.7113 0.824628
$$899$$ −0.759704 −0.0253376
$$900$$ 0 0
$$901$$ 34.9145 1.16317
$$902$$ 39.2513 1.30693
$$903$$ 0 0
$$904$$ −46.8881 −1.55947
$$905$$ −11.1094 −0.369288
$$906$$ 0 0
$$907$$ −8.73917 −0.290179 −0.145090 0.989419i $$-0.546347\pi$$
−0.145090 + 0.989419i $$0.546347\pi$$
$$908$$ −3.20296 −0.106294
$$909$$ 0 0
$$910$$ −17.3259 −0.574349
$$911$$ 20.6509 0.684196 0.342098 0.939664i $$-0.388862\pi$$
0.342098 + 0.939664i $$0.388862\pi$$
$$912$$ 0 0
$$913$$ 23.6878 0.783951
$$914$$ 26.2199 0.867276
$$915$$ 0 0
$$916$$ −0.288668 −0.00953785
$$917$$ 42.7009 1.41011
$$918$$ 0 0
$$919$$ −2.36009 −0.0778522 −0.0389261 0.999242i $$-0.512394\pi$$
−0.0389261 + 0.999242i $$0.512394\pi$$
$$920$$ −14.3200 −0.472115
$$921$$ 0 0
$$922$$ 37.3865 1.23126
$$923$$ −17.7870 −0.585468
$$924$$ 0 0
$$925$$ −0.0950283 −0.00312451
$$926$$ −52.1215 −1.71282
$$927$$ 0 0
$$928$$ −1.34493 −0.0441496
$$929$$ 26.8203 0.879944 0.439972 0.898011i $$-0.354988\pi$$
0.439972 + 0.898011i $$0.354988\pi$$
$$930$$ 0 0
$$931$$ 7.86484 0.257760
$$932$$ 3.10244 0.101624
$$933$$ 0 0
$$934$$ 40.1007 1.31213
$$935$$ −29.4688 −0.963734
$$936$$ 0 0
$$937$$ 1.93313 0.0631527 0.0315764 0.999501i $$-0.489947\pi$$
0.0315764 + 0.999501i $$0.489947\pi$$
$$938$$ −6.03365 −0.197006
$$939$$ 0 0
$$940$$ 1.14115 0.0372202
$$941$$ −11.9103 −0.388266 −0.194133 0.980975i $$-0.562189\pi$$
−0.194133 + 0.980975i $$0.562189\pi$$
$$942$$ 0 0
$$943$$ 14.4284 0.469853
$$944$$ −26.4228 −0.859990
$$945$$ 0 0
$$946$$ −41.5185 −1.34988
$$947$$ 0.315836 0.0102633 0.00513165 0.999987i $$-0.498367\pi$$
0.00513165 + 0.999987i $$0.498367\pi$$
$$948$$ 0 0
$$949$$ −17.9222 −0.581779
$$950$$ −20.2891 −0.658265
$$951$$ 0 0
$$952$$ −21.2950 −0.690174
$$953$$ −3.25133 −0.105321 −0.0526605 0.998612i $$-0.516770\pi$$
−0.0526605 + 0.998612i $$0.516770\pi$$
$$954$$ 0 0
$$955$$ −8.28756 −0.268179
$$956$$ −0.744547 −0.0240804
$$957$$ 0 0
$$958$$ −50.7606 −1.64000
$$959$$ 9.47472 0.305955
$$960$$ 0 0
$$961$$ −30.6536 −0.988827
$$962$$ 0.182104 0.00587127
$$963$$ 0 0
$$964$$ 0.619489 0.0199524
$$965$$ 29.5253 0.950452
$$966$$ 0 0
$$967$$ 10.9676 0.352694 0.176347 0.984328i $$-0.443572\pi$$
0.176347 + 0.984328i $$0.443572\pi$$
$$968$$ 71.6049 2.30147
$$969$$ 0 0
$$970$$ 0.580785 0.0186479
$$971$$ −23.3868 −0.750519 −0.375259 0.926920i $$-0.622446\pi$$
−0.375259 + 0.926920i $$0.622446\pi$$
$$972$$ 0 0
$$973$$ 28.8553 0.925060
$$974$$ 1.02910 0.0329744
$$975$$ 0 0
$$976$$ −39.7374 −1.27196
$$977$$ 50.1762 1.60528 0.802640 0.596464i $$-0.203428\pi$$
0.802640 + 0.596464i $$0.203428\pi$$
$$978$$ 0 0
$$979$$ 48.4439 1.54827
$$980$$ −0.361844 −0.0115587
$$981$$ 0 0
$$982$$ −0.670874 −0.0214084
$$983$$ 14.6946 0.468685 0.234342 0.972154i $$-0.424706\pi$$
0.234342 + 0.972154i $$0.424706\pi$$
$$984$$ 0 0
$$985$$ 1.19604 0.0381091
$$986$$ 5.21751 0.166159
$$987$$ 0 0
$$988$$ −3.95811 −0.125924
$$989$$ −15.2618 −0.485296
$$990$$ 0 0
$$991$$ 2.00000 0.0635321 0.0317660 0.999495i $$-0.489887\pi$$
0.0317660 + 0.999495i $$0.489887\pi$$
$$992$$ 0.613179 0.0194684
$$993$$ 0 0
$$994$$ −17.9099 −0.568068
$$995$$ 16.8452 0.534030
$$996$$ 0 0
$$997$$ −45.8863 −1.45323 −0.726617 0.687043i $$-0.758908\pi$$
−0.726617 + 0.687043i $$0.758908\pi$$
$$998$$ −12.0797 −0.382375
$$999$$ 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 243.2.a.e.1.2 3
3.2 odd 2 243.2.a.f.1.2 yes 3
4.3 odd 2 3888.2.a.bd.1.2 3
5.4 even 2 6075.2.a.bv.1.2 3
9.2 odd 6 243.2.c.e.82.2 6
9.4 even 3 243.2.c.f.163.2 6
9.5 odd 6 243.2.c.e.163.2 6
9.7 even 3 243.2.c.f.82.2 6
12.11 even 2 3888.2.a.bk.1.2 3
15.14 odd 2 6075.2.a.bq.1.2 3
27.2 odd 18 729.2.e.b.568.1 6
27.4 even 9 729.2.e.h.406.1 6
27.5 odd 18 729.2.e.i.649.1 6
27.7 even 9 729.2.e.h.325.1 6
27.11 odd 18 729.2.e.i.82.1 6
27.13 even 9 729.2.e.g.163.1 6
27.14 odd 18 729.2.e.b.163.1 6
27.16 even 9 729.2.e.a.82.1 6
27.20 odd 18 729.2.e.c.325.1 6
27.22 even 9 729.2.e.a.649.1 6
27.23 odd 18 729.2.e.c.406.1 6
27.25 even 9 729.2.e.g.568.1 6

By twisted newform
Twist Min Dim Char Parity Ord Type
243.2.a.e.1.2 3 1.1 even 1 trivial
243.2.a.f.1.2 yes 3 3.2 odd 2
243.2.c.e.82.2 6 9.2 odd 6
243.2.c.e.163.2 6 9.5 odd 6
243.2.c.f.82.2 6 9.7 even 3
243.2.c.f.163.2 6 9.4 even 3
729.2.e.a.82.1 6 27.16 even 9
729.2.e.a.649.1 6 27.22 even 9
729.2.e.b.163.1 6 27.14 odd 18
729.2.e.b.568.1 6 27.2 odd 18
729.2.e.c.325.1 6 27.20 odd 18
729.2.e.c.406.1 6 27.23 odd 18
729.2.e.g.163.1 6 27.13 even 9
729.2.e.g.568.1 6 27.25 even 9
729.2.e.h.325.1 6 27.7 even 9
729.2.e.h.406.1 6 27.4 even 9
729.2.e.i.82.1 6 27.11 odd 18
729.2.e.i.649.1 6 27.5 odd 18
3888.2.a.bd.1.2 3 4.3 odd 2
3888.2.a.bk.1.2 3 12.11 even 2
6075.2.a.bq.1.2 3 15.14 odd 2
6075.2.a.bv.1.2 3 5.4 even 2