# Properties

 Label 729.2.a.d.1.2 Level $729$ Weight $2$ Character 729.1 Self dual yes Analytic conductor $5.821$ Analytic rank $0$ 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: $$0$$ 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 1.2 Root $$-1.40162$$ 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)

 $$f(q)$$ $$=$$ $$q-1.05432 q^{2} -0.888399 q^{4} +1.74579 q^{5} +2.45925 q^{7} +3.04531 q^{8} +O(q^{10})$$ $$q-1.05432 q^{2} -0.888399 q^{4} +1.74579 q^{5} +2.45925 q^{7} +3.04531 q^{8} -1.84063 q^{10} +1.25421 q^{11} -4.54903 q^{13} -2.59284 q^{14} -1.43395 q^{16} +6.64717 q^{17} +0.249156 q^{19} -1.55096 q^{20} -1.32235 q^{22} -0.842001 q^{23} -1.95223 q^{25} +4.79615 q^{26} -2.18479 q^{28} +0.512383 q^{29} -0.820004 q^{31} -4.57877 q^{32} -7.00828 q^{34} +4.29332 q^{35} +2.60806 q^{37} -0.262692 q^{38} +5.31646 q^{40} +8.15281 q^{41} +4.32714 q^{43} -1.11424 q^{44} +0.887743 q^{46} +5.30233 q^{47} -0.952106 q^{49} +2.05828 q^{50} +4.04135 q^{52} +10.4841 q^{53} +2.18959 q^{55} +7.48917 q^{56} -0.540218 q^{58} +3.00620 q^{59} +2.88317 q^{61} +0.864550 q^{62} +7.69541 q^{64} -7.94164 q^{65} +10.0863 q^{67} -5.90534 q^{68} -4.52655 q^{70} -0.0894756 q^{71} -5.32114 q^{73} -2.74974 q^{74} -0.221350 q^{76} +3.08442 q^{77} +4.77692 q^{79} -2.50337 q^{80} -8.59571 q^{82} -8.04066 q^{83} +11.6045 q^{85} -4.56222 q^{86} +3.81947 q^{88} -6.70377 q^{89} -11.1872 q^{91} +0.748033 q^{92} -5.59038 q^{94} +0.434974 q^{95} +5.49058 q^{97} +1.00383 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 6 q^{8}+O(q^{10})$$ 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.05432 −0.745520 −0.372760 0.927928i $$-0.621589\pi$$
−0.372760 + 0.927928i $$0.621589\pi$$
$$3$$ 0 0
$$4$$ −0.888399 −0.444200
$$5$$ 1.74579 0.780740 0.390370 0.920658i $$-0.372347\pi$$
0.390370 + 0.920658i $$0.372347\pi$$
$$6$$ 0 0
$$7$$ 2.45925 0.929508 0.464754 0.885440i $$-0.346143\pi$$
0.464754 + 0.885440i $$0.346143\pi$$
$$8$$ 3.04531 1.07668
$$9$$ 0 0
$$10$$ −1.84063 −0.582057
$$11$$ 1.25421 0.378159 0.189080 0.981962i $$-0.439450\pi$$
0.189080 + 0.981962i $$0.439450\pi$$
$$12$$ 0 0
$$13$$ −4.54903 −1.26167 −0.630837 0.775915i $$-0.717288\pi$$
−0.630837 + 0.775915i $$0.717288\pi$$
$$14$$ −2.59284 −0.692967
$$15$$ 0 0
$$16$$ −1.43395 −0.358487
$$17$$ 6.64717 1.61218 0.806088 0.591796i $$-0.201581\pi$$
0.806088 + 0.591796i $$0.201581\pi$$
$$18$$ 0 0
$$19$$ 0.249156 0.0571604 0.0285802 0.999592i $$-0.490901\pi$$
0.0285802 + 0.999592i $$0.490901\pi$$
$$20$$ −1.55096 −0.346804
$$21$$ 0 0
$$22$$ −1.32235 −0.281926
$$23$$ −0.842001 −0.175569 −0.0877847 0.996139i $$-0.527979\pi$$
−0.0877847 + 0.996139i $$0.527979\pi$$
$$24$$ 0 0
$$25$$ −1.95223 −0.390446
$$26$$ 4.79615 0.940603
$$27$$ 0 0
$$28$$ −2.18479 −0.412887
$$29$$ 0.512383 0.0951471 0.0475736 0.998868i $$-0.484851\pi$$
0.0475736 + 0.998868i $$0.484851\pi$$
$$30$$ 0 0
$$31$$ −0.820004 −0.147277 −0.0736385 0.997285i $$-0.523461\pi$$
−0.0736385 + 0.997285i $$0.523461\pi$$
$$32$$ −4.57877 −0.809421
$$33$$ 0 0
$$34$$ −7.00828 −1.20191
$$35$$ 4.29332 0.725704
$$36$$ 0 0
$$37$$ 2.60806 0.428763 0.214381 0.976750i $$-0.431226\pi$$
0.214381 + 0.976750i $$0.431226\pi$$
$$38$$ −0.262692 −0.0426142
$$39$$ 0 0
$$40$$ 5.31646 0.840607
$$41$$ 8.15281 1.27326 0.636628 0.771171i $$-0.280329\pi$$
0.636628 + 0.771171i $$0.280329\pi$$
$$42$$ 0 0
$$43$$ 4.32714 0.659883 0.329942 0.944001i $$-0.392971\pi$$
0.329942 + 0.944001i $$0.392971\pi$$
$$44$$ −1.11424 −0.167978
$$45$$ 0 0
$$46$$ 0.887743 0.130890
$$47$$ 5.30233 0.773425 0.386713 0.922200i $$-0.373611\pi$$
0.386713 + 0.922200i $$0.373611\pi$$
$$48$$ 0 0
$$49$$ −0.952106 −0.136015
$$50$$ 2.05828 0.291085
$$51$$ 0 0
$$52$$ 4.04135 0.560435
$$53$$ 10.4841 1.44010 0.720052 0.693920i $$-0.244118\pi$$
0.720052 + 0.693920i $$0.244118\pi$$
$$54$$ 0 0
$$55$$ 2.18959 0.295244
$$56$$ 7.48917 1.00078
$$57$$ 0 0
$$58$$ −0.540218 −0.0709341
$$59$$ 3.00620 0.391374 0.195687 0.980666i $$-0.437306\pi$$
0.195687 + 0.980666i $$0.437306\pi$$
$$60$$ 0 0
$$61$$ 2.88317 0.369152 0.184576 0.982818i $$-0.440909\pi$$
0.184576 + 0.982818i $$0.440909\pi$$
$$62$$ 0.864550 0.109798
$$63$$ 0 0
$$64$$ 7.69541 0.961927
$$65$$ −7.94164 −0.985039
$$66$$ 0 0
$$67$$ 10.0863 1.23223 0.616117 0.787655i $$-0.288705\pi$$
0.616117 + 0.787655i $$0.288705\pi$$
$$68$$ −5.90534 −0.716128
$$69$$ 0 0
$$70$$ −4.52655 −0.541027
$$71$$ −0.0894756 −0.0106188 −0.00530940 0.999986i $$-0.501690\pi$$
−0.00530940 + 0.999986i $$0.501690\pi$$
$$72$$ 0 0
$$73$$ −5.32114 −0.622792 −0.311396 0.950280i $$-0.600797\pi$$
−0.311396 + 0.950280i $$0.600797\pi$$
$$74$$ −2.74974 −0.319651
$$75$$ 0 0
$$76$$ −0.221350 −0.0253906
$$77$$ 3.08442 0.351502
$$78$$ 0 0
$$79$$ 4.77692 0.537445 0.268723 0.963218i $$-0.413398\pi$$
0.268723 + 0.963218i $$0.413398\pi$$
$$80$$ −2.50337 −0.279885
$$81$$ 0 0
$$82$$ −8.59571 −0.949238
$$83$$ −8.04066 −0.882577 −0.441289 0.897365i $$-0.645478\pi$$
−0.441289 + 0.897365i $$0.645478\pi$$
$$84$$ 0 0
$$85$$ 11.6045 1.25869
$$86$$ −4.56222 −0.491956
$$87$$ 0 0
$$88$$ 3.81947 0.407157
$$89$$ −6.70377 −0.710598 −0.355299 0.934753i $$-0.615621\pi$$
−0.355299 + 0.934753i $$0.615621\pi$$
$$90$$ 0 0
$$91$$ −11.1872 −1.17274
$$92$$ 0.748033 0.0779878
$$93$$ 0 0
$$94$$ −5.59038 −0.576604
$$95$$ 0.434974 0.0446274
$$96$$ 0 0
$$97$$ 5.49058 0.557484 0.278742 0.960366i $$-0.410083\pi$$
0.278742 + 0.960366i $$0.410083\pi$$
$$98$$ 1.00383 0.101402
$$99$$ 0 0
$$100$$ 1.73436 0.173436
$$101$$ −5.00546 −0.498062 −0.249031 0.968495i $$-0.580112\pi$$
−0.249031 + 0.968495i $$0.580112\pi$$
$$102$$ 0 0
$$103$$ 11.6192 1.14487 0.572435 0.819950i $$-0.305999\pi$$
0.572435 + 0.819950i $$0.305999\pi$$
$$104$$ −13.8532 −1.35842
$$105$$ 0 0
$$106$$ −11.0537 −1.07363
$$107$$ −19.4581 −1.88109 −0.940544 0.339673i $$-0.889684\pi$$
−0.940544 + 0.339673i $$0.889684\pi$$
$$108$$ 0 0
$$109$$ 6.31515 0.604881 0.302441 0.953168i $$-0.402199\pi$$
0.302441 + 0.953168i $$0.402199\pi$$
$$110$$ −2.30854 −0.220110
$$111$$ 0 0
$$112$$ −3.52643 −0.333217
$$113$$ 6.91572 0.650577 0.325288 0.945615i $$-0.394539\pi$$
0.325288 + 0.945615i $$0.394539\pi$$
$$114$$ 0 0
$$115$$ −1.46995 −0.137074
$$116$$ −0.455201 −0.0422643
$$117$$ 0 0
$$118$$ −3.16951 −0.291777
$$119$$ 16.3470 1.49853
$$120$$ 0 0
$$121$$ −9.42695 −0.856995
$$122$$ −3.03980 −0.275211
$$123$$ 0 0
$$124$$ 0.728491 0.0654204
$$125$$ −12.1371 −1.08558
$$126$$ 0 0
$$127$$ 12.0232 1.06689 0.533445 0.845835i $$-0.320897\pi$$
0.533445 + 0.845835i $$0.320897\pi$$
$$128$$ 1.04408 0.0922848
$$129$$ 0 0
$$130$$ 8.37306 0.734366
$$131$$ 14.0848 1.23059 0.615297 0.788295i $$-0.289036\pi$$
0.615297 + 0.788295i $$0.289036\pi$$
$$132$$ 0 0
$$133$$ 0.612737 0.0531310
$$134$$ −10.6342 −0.918655
$$135$$ 0 0
$$136$$ 20.2427 1.73580
$$137$$ −2.25814 −0.192926 −0.0964630 0.995337i $$-0.530753\pi$$
−0.0964630 + 0.995337i $$0.530753\pi$$
$$138$$ 0 0
$$139$$ −7.97509 −0.676439 −0.338219 0.941067i $$-0.609825\pi$$
−0.338219 + 0.941067i $$0.609825\pi$$
$$140$$ −3.81418 −0.322357
$$141$$ 0 0
$$142$$ 0.0943364 0.00791653
$$143$$ −5.70545 −0.477114
$$144$$ 0 0
$$145$$ 0.894512 0.0742851
$$146$$ 5.61021 0.464304
$$147$$ 0 0
$$148$$ −2.31700 −0.190456
$$149$$ 0.106938 0.00876074 0.00438037 0.999990i $$-0.498606\pi$$
0.00438037 + 0.999990i $$0.498606\pi$$
$$150$$ 0 0
$$151$$ −20.2594 −1.64869 −0.824344 0.566090i $$-0.808456\pi$$
−0.824344 + 0.566090i $$0.808456\pi$$
$$152$$ 0.758758 0.0615434
$$153$$ 0 0
$$154$$ −3.25198 −0.262052
$$155$$ −1.43155 −0.114985
$$156$$ 0 0
$$157$$ −20.7498 −1.65602 −0.828009 0.560715i $$-0.810526\pi$$
−0.828009 + 0.560715i $$0.810526\pi$$
$$158$$ −5.03642 −0.400676
$$159$$ 0 0
$$160$$ −7.99356 −0.631947
$$161$$ −2.07069 −0.163193
$$162$$ 0 0
$$163$$ −20.1346 −1.57706 −0.788531 0.614995i $$-0.789158\pi$$
−0.788531 + 0.614995i $$0.789158\pi$$
$$164$$ −7.24295 −0.565580
$$165$$ 0 0
$$166$$ 8.47747 0.657979
$$167$$ 19.8626 1.53701 0.768507 0.639841i $$-0.221000\pi$$
0.768507 + 0.639841i $$0.221000\pi$$
$$168$$ 0 0
$$169$$ 7.69367 0.591821
$$170$$ −12.2350 −0.938378
$$171$$ 0 0
$$172$$ −3.84423 −0.293120
$$173$$ −18.9251 −1.43885 −0.719425 0.694570i $$-0.755595\pi$$
−0.719425 + 0.694570i $$0.755595\pi$$
$$174$$ 0 0
$$175$$ −4.80101 −0.362922
$$176$$ −1.79848 −0.135565
$$177$$ 0 0
$$178$$ 7.06795 0.529765
$$179$$ −10.9137 −0.815725 −0.407863 0.913043i $$-0.633726\pi$$
−0.407863 + 0.913043i $$0.633726\pi$$
$$180$$ 0 0
$$181$$ −17.9479 −1.33405 −0.667027 0.745033i $$-0.732433\pi$$
−0.667027 + 0.745033i $$0.732433\pi$$
$$182$$ 11.7949 0.874298
$$183$$ 0 0
$$184$$ −2.56415 −0.189032
$$185$$ 4.55312 0.334752
$$186$$ 0 0
$$187$$ 8.33697 0.609659
$$188$$ −4.71059 −0.343555
$$189$$ 0 0
$$190$$ −0.458604 −0.0332706
$$191$$ 26.9661 1.95120 0.975598 0.219566i $$-0.0704640\pi$$
0.975598 + 0.219566i $$0.0704640\pi$$
$$192$$ 0 0
$$193$$ −17.1548 −1.23483 −0.617415 0.786638i $$-0.711820\pi$$
−0.617415 + 0.786638i $$0.711820\pi$$
$$194$$ −5.78885 −0.415615
$$195$$ 0 0
$$196$$ 0.845850 0.0604179
$$197$$ −2.51225 −0.178990 −0.0894951 0.995987i $$-0.528525\pi$$
−0.0894951 + 0.995987i $$0.528525\pi$$
$$198$$ 0 0
$$199$$ 18.5388 1.31418 0.657092 0.753810i $$-0.271786\pi$$
0.657092 + 0.753810i $$0.271786\pi$$
$$200$$ −5.94514 −0.420385
$$201$$ 0 0
$$202$$ 5.27739 0.371316
$$203$$ 1.26008 0.0884400
$$204$$ 0 0
$$205$$ 14.2331 0.994081
$$206$$ −12.2504 −0.853524
$$207$$ 0 0
$$208$$ 6.52307 0.452294
$$209$$ 0.312495 0.0216157
$$210$$ 0 0
$$211$$ −3.69118 −0.254111 −0.127056 0.991896i $$-0.540553\pi$$
−0.127056 + 0.991896i $$0.540553\pi$$
$$212$$ −9.31408 −0.639693
$$213$$ 0 0
$$214$$ 20.5152 1.40239
$$215$$ 7.55427 0.515197
$$216$$ 0 0
$$217$$ −2.01659 −0.136895
$$218$$ −6.65822 −0.450951
$$219$$ 0 0
$$220$$ −1.94523 −0.131147
$$221$$ −30.2382 −2.03404
$$222$$ 0 0
$$223$$ −21.2410 −1.42241 −0.711203 0.702987i $$-0.751849\pi$$
−0.711203 + 0.702987i $$0.751849\pi$$
$$224$$ −11.2603 −0.752363
$$225$$ 0 0
$$226$$ −7.29142 −0.485018
$$227$$ 14.3400 0.951783 0.475891 0.879504i $$-0.342126\pi$$
0.475891 + 0.879504i $$0.342126\pi$$
$$228$$ 0 0
$$229$$ 16.8858 1.11585 0.557923 0.829893i $$-0.311598\pi$$
0.557923 + 0.829893i $$0.311598\pi$$
$$230$$ 1.54981 0.102191
$$231$$ 0 0
$$232$$ 1.56037 0.102443
$$233$$ −5.59945 −0.366832 −0.183416 0.983035i $$-0.558716\pi$$
−0.183416 + 0.983035i $$0.558716\pi$$
$$234$$ 0 0
$$235$$ 9.25675 0.603844
$$236$$ −2.67071 −0.173848
$$237$$ 0 0
$$238$$ −17.2351 −1.11718
$$239$$ 5.27427 0.341164 0.170582 0.985343i $$-0.445435\pi$$
0.170582 + 0.985343i $$0.445435\pi$$
$$240$$ 0 0
$$241$$ −8.90248 −0.573459 −0.286730 0.958012i $$-0.592568\pi$$
−0.286730 + 0.958012i $$0.592568\pi$$
$$242$$ 9.93907 0.638907
$$243$$ 0 0
$$244$$ −2.56141 −0.163977
$$245$$ −1.66217 −0.106192
$$246$$ 0 0
$$247$$ −1.13342 −0.0721177
$$248$$ −2.49717 −0.158570
$$249$$ 0 0
$$250$$ 12.7965 0.809319
$$251$$ 7.78021 0.491082 0.245541 0.969386i $$-0.421034\pi$$
0.245541 + 0.969386i $$0.421034\pi$$
$$252$$ 0 0
$$253$$ −1.05605 −0.0663932
$$254$$ −12.6764 −0.795388
$$255$$ 0 0
$$256$$ −16.4916 −1.03073
$$257$$ 20.4366 1.27480 0.637399 0.770534i $$-0.280011\pi$$
0.637399 + 0.770534i $$0.280011\pi$$
$$258$$ 0 0
$$259$$ 6.41387 0.398538
$$260$$ 7.05534 0.437554
$$261$$ 0 0
$$262$$ −14.8500 −0.917433
$$263$$ −11.2798 −0.695543 −0.347771 0.937579i $$-0.613061\pi$$
−0.347771 + 0.937579i $$0.613061\pi$$
$$264$$ 0 0
$$265$$ 18.3030 1.12435
$$266$$ −0.646023 −0.0396102
$$267$$ 0 0
$$268$$ −8.96063 −0.547357
$$269$$ 0.307761 0.0187645 0.00938226 0.999956i $$-0.497013\pi$$
0.00938226 + 0.999956i $$0.497013\pi$$
$$270$$ 0 0
$$271$$ −2.22251 −0.135008 −0.0675040 0.997719i $$-0.521504\pi$$
−0.0675040 + 0.997719i $$0.521504\pi$$
$$272$$ −9.53170 −0.577944
$$273$$ 0 0
$$274$$ 2.38081 0.143830
$$275$$ −2.44851 −0.147651
$$276$$ 0 0
$$277$$ −23.3297 −1.40175 −0.700874 0.713285i $$-0.747206\pi$$
−0.700874 + 0.713285i $$0.747206\pi$$
$$278$$ 8.40834 0.504299
$$279$$ 0 0
$$280$$ 13.0745 0.781351
$$281$$ −7.22546 −0.431035 −0.215517 0.976500i $$-0.569144\pi$$
−0.215517 + 0.976500i $$0.569144\pi$$
$$282$$ 0 0
$$283$$ 7.12029 0.423257 0.211629 0.977350i $$-0.432123\pi$$
0.211629 + 0.977350i $$0.432123\pi$$
$$284$$ 0.0794901 0.00471687
$$285$$ 0 0
$$286$$ 6.01540 0.355698
$$287$$ 20.0498 1.18350
$$288$$ 0 0
$$289$$ 27.1849 1.59911
$$290$$ −0.943106 −0.0553811
$$291$$ 0 0
$$292$$ 4.72730 0.276644
$$293$$ −0.552485 −0.0322765 −0.0161383 0.999870i $$-0.505137\pi$$
−0.0161383 + 0.999870i $$0.505137\pi$$
$$294$$ 0 0
$$295$$ 5.24819 0.305561
$$296$$ 7.94236 0.461640
$$297$$ 0 0
$$298$$ −0.112748 −0.00653131
$$299$$ 3.83029 0.221511
$$300$$ 0 0
$$301$$ 10.6415 0.613367
$$302$$ 21.3600 1.22913
$$303$$ 0 0
$$304$$ −0.357277 −0.0204913
$$305$$ 5.03340 0.288212
$$306$$ 0 0
$$307$$ 6.72876 0.384031 0.192015 0.981392i $$-0.438498\pi$$
0.192015 + 0.981392i $$0.438498\pi$$
$$308$$ −2.74020 −0.156137
$$309$$ 0 0
$$310$$ 1.50932 0.0857236
$$311$$ −15.4235 −0.874584 −0.437292 0.899320i $$-0.644062\pi$$
−0.437292 + 0.899320i $$0.644062\pi$$
$$312$$ 0 0
$$313$$ −23.5609 −1.33174 −0.665870 0.746068i $$-0.731939\pi$$
−0.665870 + 0.746068i $$0.731939\pi$$
$$314$$ 21.8771 1.23459
$$315$$ 0 0
$$316$$ −4.24381 −0.238733
$$317$$ −7.25204 −0.407315 −0.203658 0.979042i $$-0.565283\pi$$
−0.203658 + 0.979042i $$0.565283\pi$$
$$318$$ 0 0
$$319$$ 0.642637 0.0359808
$$320$$ 13.4346 0.751014
$$321$$ 0 0
$$322$$ 2.18318 0.121664
$$323$$ 1.65618 0.0921525
$$324$$ 0 0
$$325$$ 8.88074 0.492615
$$326$$ 21.2284 1.17573
$$327$$ 0 0
$$328$$ 24.8279 1.37089
$$329$$ 13.0397 0.718905
$$330$$ 0 0
$$331$$ 29.0345 1.59588 0.797939 0.602738i $$-0.205924\pi$$
0.797939 + 0.602738i $$0.205924\pi$$
$$332$$ 7.14332 0.392040
$$333$$ 0 0
$$334$$ −20.9416 −1.14588
$$335$$ 17.6085 0.962053
$$336$$ 0 0
$$337$$ −1.15910 −0.0631400 −0.0315700 0.999502i $$-0.510051\pi$$
−0.0315700 + 0.999502i $$0.510051\pi$$
$$338$$ −8.11163 −0.441214
$$339$$ 0 0
$$340$$ −10.3095 −0.559109
$$341$$ −1.02846 −0.0556942
$$342$$ 0 0
$$343$$ −19.5562 −1.05594
$$344$$ 13.1775 0.710483
$$345$$ 0 0
$$346$$ 19.9532 1.07269
$$347$$ −5.88971 −0.316176 −0.158088 0.987425i $$-0.550533\pi$$
−0.158088 + 0.987425i $$0.550533\pi$$
$$348$$ 0 0
$$349$$ −30.5927 −1.63759 −0.818795 0.574087i $$-0.805357\pi$$
−0.818795 + 0.574087i $$0.805357\pi$$
$$350$$ 5.06182 0.270566
$$351$$ 0 0
$$352$$ −5.74276 −0.306090
$$353$$ −36.9613 −1.96725 −0.983625 0.180227i $$-0.942317\pi$$
−0.983625 + 0.180227i $$0.942317\pi$$
$$354$$ 0 0
$$355$$ −0.156205 −0.00829052
$$356$$ 5.95562 0.315647
$$357$$ 0 0
$$358$$ 11.5065 0.608140
$$359$$ 26.3761 1.39207 0.696037 0.718006i $$-0.254945\pi$$
0.696037 + 0.718006i $$0.254945\pi$$
$$360$$ 0 0
$$361$$ −18.9379 −0.996733
$$362$$ 18.9229 0.994564
$$363$$ 0 0
$$364$$ 9.93869 0.520929
$$365$$ −9.28958 −0.486239
$$366$$ 0 0
$$367$$ −11.3131 −0.590541 −0.295270 0.955414i $$-0.595410\pi$$
−0.295270 + 0.955414i $$0.595410\pi$$
$$368$$ 1.20739 0.0629394
$$369$$ 0 0
$$370$$ −4.80047 −0.249564
$$371$$ 25.7830 1.33859
$$372$$ 0 0
$$373$$ 5.84408 0.302595 0.151297 0.988488i $$-0.451655\pi$$
0.151297 + 0.988488i $$0.451655\pi$$
$$374$$ −8.78987 −0.454513
$$375$$ 0 0
$$376$$ 16.1473 0.832731
$$377$$ −2.33085 −0.120045
$$378$$ 0 0
$$379$$ 24.3265 1.24957 0.624783 0.780798i $$-0.285187\pi$$
0.624783 + 0.780798i $$0.285187\pi$$
$$380$$ −0.386430 −0.0198235
$$381$$ 0 0
$$382$$ −28.4310 −1.45466
$$383$$ 3.81605 0.194991 0.0974955 0.995236i $$-0.468917\pi$$
0.0974955 + 0.995236i $$0.468917\pi$$
$$384$$ 0 0
$$385$$ 5.38474 0.274432
$$386$$ 18.0867 0.920591
$$387$$ 0 0
$$388$$ −4.87782 −0.247634
$$389$$ −10.8418 −0.549704 −0.274852 0.961487i $$-0.588629\pi$$
−0.274852 + 0.961487i $$0.588629\pi$$
$$390$$ 0 0
$$391$$ −5.59692 −0.283049
$$392$$ −2.89946 −0.146445
$$393$$ 0 0
$$394$$ 2.64872 0.133441
$$395$$ 8.33948 0.419605
$$396$$ 0 0
$$397$$ −10.5092 −0.527442 −0.263721 0.964599i $$-0.584950\pi$$
−0.263721 + 0.964599i $$0.584950\pi$$
$$398$$ −19.5460 −0.979751
$$399$$ 0 0
$$400$$ 2.79939 0.139970
$$401$$ −14.3656 −0.717383 −0.358691 0.933456i $$-0.616777\pi$$
−0.358691 + 0.933456i $$0.616777\pi$$
$$402$$ 0 0
$$403$$ 3.73022 0.185816
$$404$$ 4.44685 0.221239
$$405$$ 0 0
$$406$$ −1.32853 −0.0659338
$$407$$ 3.27106 0.162141
$$408$$ 0 0
$$409$$ 17.6823 0.874332 0.437166 0.899381i $$-0.355982\pi$$
0.437166 + 0.899381i $$0.355982\pi$$
$$410$$ −15.0063 −0.741108
$$411$$ 0 0
$$412$$ −10.3225 −0.508551
$$413$$ 7.39299 0.363785
$$414$$ 0 0
$$415$$ −14.0373 −0.689063
$$416$$ 20.8290 1.02122
$$417$$ 0 0
$$418$$ −0.329471 −0.0161150
$$419$$ −9.13376 −0.446214 −0.223107 0.974794i $$-0.571620\pi$$
−0.223107 + 0.974794i $$0.571620\pi$$
$$420$$ 0 0
$$421$$ 24.1949 1.17919 0.589594 0.807700i $$-0.299288\pi$$
0.589594 + 0.807700i $$0.299288\pi$$
$$422$$ 3.89170 0.189445
$$423$$ 0 0
$$424$$ 31.9274 1.55053
$$425$$ −12.9768 −0.629467
$$426$$ 0 0
$$427$$ 7.09043 0.343130
$$428$$ 17.2866 0.835578
$$429$$ 0 0
$$430$$ −7.96466 −0.384090
$$431$$ 29.5332 1.42256 0.711282 0.702907i $$-0.248115\pi$$
0.711282 + 0.702907i $$0.248115\pi$$
$$432$$ 0 0
$$433$$ 0.669754 0.0321863 0.0160932 0.999870i $$-0.494877\pi$$
0.0160932 + 0.999870i $$0.494877\pi$$
$$434$$ 2.12614 0.102058
$$435$$ 0 0
$$436$$ −5.61037 −0.268688
$$437$$ −0.209790 −0.0100356
$$438$$ 0 0
$$439$$ −6.34887 −0.303015 −0.151507 0.988456i $$-0.548413\pi$$
−0.151507 + 0.988456i $$0.548413\pi$$
$$440$$ 6.66798 0.317883
$$441$$ 0 0
$$442$$ 31.8809 1.51642
$$443$$ −15.3539 −0.729487 −0.364743 0.931108i $$-0.618843\pi$$
−0.364743 + 0.931108i $$0.618843\pi$$
$$444$$ 0 0
$$445$$ −11.7034 −0.554792
$$446$$ 22.3950 1.06043
$$447$$ 0 0
$$448$$ 18.9249 0.894118
$$449$$ 32.0398 1.51205 0.756027 0.654541i $$-0.227138\pi$$
0.756027 + 0.654541i $$0.227138\pi$$
$$450$$ 0 0
$$451$$ 10.2254 0.481494
$$452$$ −6.14392 −0.288986
$$453$$ 0 0
$$454$$ −15.1191 −0.709573
$$455$$ −19.5304 −0.915601
$$456$$ 0 0
$$457$$ 19.1680 0.896643 0.448321 0.893872i $$-0.352022\pi$$
0.448321 + 0.893872i $$0.352022\pi$$
$$458$$ −17.8031 −0.831886
$$459$$ 0 0
$$460$$ 1.30591 0.0608882
$$461$$ 5.22029 0.243133 0.121567 0.992583i $$-0.461208\pi$$
0.121567 + 0.992583i $$0.461208\pi$$
$$462$$ 0 0
$$463$$ 1.69739 0.0788844 0.0394422 0.999222i $$-0.487442\pi$$
0.0394422 + 0.999222i $$0.487442\pi$$
$$464$$ −0.734731 −0.0341090
$$465$$ 0 0
$$466$$ 5.90364 0.273481
$$467$$ −19.6827 −0.910808 −0.455404 0.890285i $$-0.650505\pi$$
−0.455404 + 0.890285i $$0.650505\pi$$
$$468$$ 0 0
$$469$$ 24.8046 1.14537
$$470$$ −9.75962 −0.450178
$$471$$ 0 0
$$472$$ 9.15482 0.421385
$$473$$ 5.42716 0.249541
$$474$$ 0 0
$$475$$ −0.486410 −0.0223180
$$476$$ −14.5227 −0.665646
$$477$$ 0 0
$$478$$ −5.56080 −0.254345
$$479$$ 29.2534 1.33662 0.668311 0.743882i $$-0.267017\pi$$
0.668311 + 0.743882i $$0.267017\pi$$
$$480$$ 0 0
$$481$$ −11.8641 −0.540959
$$482$$ 9.38610 0.427525
$$483$$ 0 0
$$484$$ 8.37489 0.380677
$$485$$ 9.58538 0.435250
$$486$$ 0 0
$$487$$ −20.5056 −0.929199 −0.464600 0.885521i $$-0.653802\pi$$
−0.464600 + 0.885521i $$0.653802\pi$$
$$488$$ 8.78016 0.397459
$$489$$ 0 0
$$490$$ 1.75247 0.0791686
$$491$$ −17.5270 −0.790982 −0.395491 0.918470i $$-0.629426\pi$$
−0.395491 + 0.918470i $$0.629426\pi$$
$$492$$ 0 0
$$493$$ 3.40590 0.153394
$$494$$ 1.19499 0.0537652
$$495$$ 0 0
$$496$$ 1.17584 0.0527969
$$497$$ −0.220043 −0.00987026
$$498$$ 0 0
$$499$$ −19.1060 −0.855301 −0.427651 0.903944i $$-0.640659\pi$$
−0.427651 + 0.903944i $$0.640659\pi$$
$$500$$ 10.7826 0.482212
$$501$$ 0 0
$$502$$ −8.20287 −0.366112
$$503$$ 10.9676 0.489022 0.244511 0.969646i $$-0.421373\pi$$
0.244511 + 0.969646i $$0.421373\pi$$
$$504$$ 0 0
$$505$$ −8.73847 −0.388857
$$506$$ 1.11342 0.0494975
$$507$$ 0 0
$$508$$ −10.6814 −0.473912
$$509$$ 19.8994 0.882023 0.441012 0.897501i $$-0.354620\pi$$
0.441012 + 0.897501i $$0.354620\pi$$
$$510$$ 0 0
$$511$$ −13.0860 −0.578890
$$512$$ 15.2994 0.676143
$$513$$ 0 0
$$514$$ −21.5468 −0.950387
$$515$$ 20.2846 0.893845
$$516$$ 0 0
$$517$$ 6.65026 0.292478
$$518$$ −6.76230 −0.297118
$$519$$ 0 0
$$520$$ −24.1848 −1.06057
$$521$$ −35.1167 −1.53849 −0.769244 0.638955i $$-0.779367\pi$$
−0.769244 + 0.638955i $$0.779367\pi$$
$$522$$ 0 0
$$523$$ −14.2454 −0.622907 −0.311453 0.950261i $$-0.600816\pi$$
−0.311453 + 0.950261i $$0.600816\pi$$
$$524$$ −12.5129 −0.546629
$$525$$ 0 0
$$526$$ 11.8926 0.518541
$$527$$ −5.45070 −0.237436
$$528$$ 0 0
$$529$$ −22.2910 −0.969175
$$530$$ −19.2973 −0.838223
$$531$$ 0 0
$$532$$ −0.544355 −0.0236008
$$533$$ −37.0874 −1.60643
$$534$$ 0 0
$$535$$ −33.9697 −1.46864
$$536$$ 30.7158 1.32672
$$537$$ 0 0
$$538$$ −0.324480 −0.0139893
$$539$$ −1.19414 −0.0514354
$$540$$ 0 0
$$541$$ −13.2368 −0.569094 −0.284547 0.958662i $$-0.591843\pi$$
−0.284547 + 0.958662i $$0.591843\pi$$
$$542$$ 2.34325 0.100651
$$543$$ 0 0
$$544$$ −30.4359 −1.30493
$$545$$ 11.0249 0.472255
$$546$$ 0 0
$$547$$ −16.3049 −0.697148 −0.348574 0.937281i $$-0.613334\pi$$
−0.348574 + 0.937281i $$0.613334\pi$$
$$548$$ 2.00613 0.0856976
$$549$$ 0 0
$$550$$ 2.58152 0.110077
$$551$$ 0.127663 0.00543864
$$552$$ 0 0
$$553$$ 11.7476 0.499560
$$554$$ 24.5971 1.04503
$$555$$ 0 0
$$556$$ 7.08507 0.300474
$$557$$ −30.8972 −1.30915 −0.654577 0.755995i $$-0.727153\pi$$
−0.654577 + 0.755995i $$0.727153\pi$$
$$558$$ 0 0
$$559$$ −19.6843 −0.832557
$$560$$ −6.15640 −0.260155
$$561$$ 0 0
$$562$$ 7.61798 0.321345
$$563$$ −26.7759 −1.12847 −0.564236 0.825614i $$-0.690829\pi$$
−0.564236 + 0.825614i $$0.690829\pi$$
$$564$$ 0 0
$$565$$ 12.0734 0.507931
$$566$$ −7.50710 −0.315547
$$567$$ 0 0
$$568$$ −0.272481 −0.0114331
$$569$$ −19.0606 −0.799064 −0.399532 0.916719i $$-0.630827\pi$$
−0.399532 + 0.916719i $$0.630827\pi$$
$$570$$ 0 0
$$571$$ −18.7742 −0.785676 −0.392838 0.919608i $$-0.628507\pi$$
−0.392838 + 0.919608i $$0.628507\pi$$
$$572$$ 5.06872 0.211934
$$573$$ 0 0
$$574$$ −21.1390 −0.882324
$$575$$ 1.64378 0.0685503
$$576$$ 0 0
$$577$$ −4.85962 −0.202309 −0.101154 0.994871i $$-0.532254\pi$$
−0.101154 + 0.994871i $$0.532254\pi$$
$$578$$ −28.6617 −1.19217
$$579$$ 0 0
$$580$$ −0.794683 −0.0329974
$$581$$ −19.7740 −0.820362
$$582$$ 0 0
$$583$$ 13.1493 0.544589
$$584$$ −16.2045 −0.670548
$$585$$ 0 0
$$586$$ 0.582499 0.0240628
$$587$$ 32.7973 1.35369 0.676846 0.736125i $$-0.263346\pi$$
0.676846 + 0.736125i $$0.263346\pi$$
$$588$$ 0 0
$$589$$ −0.204309 −0.00841841
$$590$$ −5.53330 −0.227802
$$591$$ 0 0
$$592$$ −3.73983 −0.153706
$$593$$ −17.3446 −0.712258 −0.356129 0.934437i $$-0.615904\pi$$
−0.356129 + 0.934437i $$0.615904\pi$$
$$594$$ 0 0
$$595$$ 28.5384 1.16996
$$596$$ −0.0950040 −0.00389151
$$597$$ 0 0
$$598$$ −4.03837 −0.165141
$$599$$ −24.4079 −0.997280 −0.498640 0.866809i $$-0.666167\pi$$
−0.498640 + 0.866809i $$0.666167\pi$$
$$600$$ 0 0
$$601$$ 7.87027 0.321035 0.160517 0.987033i $$-0.448684\pi$$
0.160517 + 0.987033i $$0.448684\pi$$
$$602$$ −11.2196 −0.457277
$$603$$ 0 0
$$604$$ 17.9984 0.732346
$$605$$ −16.4574 −0.669090
$$606$$ 0 0
$$607$$ −10.2375 −0.415526 −0.207763 0.978179i $$-0.566618\pi$$
−0.207763 + 0.978179i $$0.566618\pi$$
$$608$$ −1.14083 −0.0462668
$$609$$ 0 0
$$610$$ −5.30684 −0.214868
$$611$$ −24.1205 −0.975810
$$612$$ 0 0
$$613$$ 2.23507 0.0902736 0.0451368 0.998981i $$-0.485628\pi$$
0.0451368 + 0.998981i $$0.485628\pi$$
$$614$$ −7.09430 −0.286303
$$615$$ 0 0
$$616$$ 9.39301 0.378455
$$617$$ −33.9757 −1.36781 −0.683905 0.729571i $$-0.739720\pi$$
−0.683905 + 0.729571i $$0.739720\pi$$
$$618$$ 0 0
$$619$$ −28.8560 −1.15982 −0.579910 0.814681i $$-0.696912\pi$$
−0.579910 + 0.814681i $$0.696912\pi$$
$$620$$ 1.27179 0.0510763
$$621$$ 0 0
$$622$$ 16.2613 0.652020
$$623$$ −16.4862 −0.660507
$$624$$ 0 0
$$625$$ −11.4277 −0.457107
$$626$$ 24.8408 0.992838
$$627$$ 0 0
$$628$$ 18.4341 0.735602
$$629$$ 17.3362 0.691241
$$630$$ 0 0
$$631$$ −3.14078 −0.125032 −0.0625162 0.998044i $$-0.519913\pi$$
−0.0625162 + 0.998044i $$0.519913\pi$$
$$632$$ 14.5472 0.578657
$$633$$ 0 0
$$634$$ 7.64601 0.303662
$$635$$ 20.9900 0.832964
$$636$$ 0 0
$$637$$ 4.33116 0.171607
$$638$$ −0.677549 −0.0268244
$$639$$ 0 0
$$640$$ 1.82275 0.0720504
$$641$$ 31.8225 1.25691 0.628457 0.777844i $$-0.283687\pi$$
0.628457 + 0.777844i $$0.283687\pi$$
$$642$$ 0 0
$$643$$ −12.8471 −0.506639 −0.253319 0.967383i $$-0.581522\pi$$
−0.253319 + 0.967383i $$0.581522\pi$$
$$644$$ 1.83960 0.0724903
$$645$$ 0 0
$$646$$ −1.74616 −0.0687016
$$647$$ −28.2444 −1.11040 −0.555200 0.831717i $$-0.687358\pi$$
−0.555200 + 0.831717i $$0.687358\pi$$
$$648$$ 0 0
$$649$$ 3.77042 0.148002
$$650$$ −9.36319 −0.367254
$$651$$ 0 0
$$652$$ 17.8875 0.700530
$$653$$ −25.9905 −1.01709 −0.508543 0.861036i $$-0.669816\pi$$
−0.508543 + 0.861036i $$0.669816\pi$$
$$654$$ 0 0
$$655$$ 24.5891 0.960774
$$656$$ −11.6907 −0.456446
$$657$$ 0 0
$$658$$ −13.7481 −0.535958
$$659$$ 47.6178 1.85493 0.927463 0.373914i $$-0.121985\pi$$
0.927463 + 0.373914i $$0.121985\pi$$
$$660$$ 0 0
$$661$$ −0.876508 −0.0340922 −0.0170461 0.999855i $$-0.505426\pi$$
−0.0170461 + 0.999855i $$0.505426\pi$$
$$662$$ −30.6118 −1.18976
$$663$$ 0 0
$$664$$ −24.4863 −0.950253
$$665$$ 1.06971 0.0414815
$$666$$ 0 0
$$667$$ −0.431427 −0.0167049
$$668$$ −17.6459 −0.682741
$$669$$ 0 0
$$670$$ −18.5650 −0.717230
$$671$$ 3.61611 0.139598
$$672$$ 0 0
$$673$$ 37.3394 1.43933 0.719665 0.694321i $$-0.244295\pi$$
0.719665 + 0.694321i $$0.244295\pi$$
$$674$$ 1.22206 0.0470721
$$675$$ 0 0
$$676$$ −6.83505 −0.262886
$$677$$ 13.7192 0.527270 0.263635 0.964622i $$-0.415078\pi$$
0.263635 + 0.964622i $$0.415078\pi$$
$$678$$ 0 0
$$679$$ 13.5027 0.518185
$$680$$ 35.3394 1.35521
$$681$$ 0 0
$$682$$ 1.08433 0.0415211
$$683$$ −49.9887 −1.91276 −0.956381 0.292121i $$-0.905639\pi$$
−0.956381 + 0.292121i $$0.905639\pi$$
$$684$$ 0 0
$$685$$ −3.94223 −0.150625
$$686$$ 20.6186 0.787221
$$687$$ 0 0
$$688$$ −6.20490 −0.236560
$$689$$ −47.6925 −1.81694
$$690$$ 0 0
$$691$$ 23.8151 0.905967 0.452984 0.891519i $$-0.350360\pi$$
0.452984 + 0.891519i $$0.350360\pi$$
$$692$$ 16.8131 0.639137
$$693$$ 0 0
$$694$$ 6.20967 0.235716
$$695$$ −13.9228 −0.528122
$$696$$ 0 0
$$697$$ 54.1931 2.05271
$$698$$ 32.2546 1.22086
$$699$$ 0 0
$$700$$ 4.26521 0.161210
$$701$$ −34.4493 −1.30113 −0.650565 0.759450i $$-0.725468\pi$$
−0.650565 + 0.759450i $$0.725468\pi$$
$$702$$ 0 0
$$703$$ 0.649815 0.0245082
$$704$$ 9.65169 0.363762
$$705$$ 0 0
$$706$$ 38.9692 1.46662
$$707$$ −12.3097 −0.462953
$$708$$ 0 0
$$709$$ 15.5233 0.582989 0.291494 0.956573i $$-0.405848\pi$$
0.291494 + 0.956573i $$0.405848\pi$$
$$710$$ 0.164691 0.00618075
$$711$$ 0 0
$$712$$ −20.4151 −0.765087
$$713$$ 0.690444 0.0258573
$$714$$ 0 0
$$715$$ −9.96050 −0.372502
$$716$$ 9.69569 0.362345
$$717$$ 0 0
$$718$$ −27.8089 −1.03782
$$719$$ −12.0537 −0.449528 −0.224764 0.974413i $$-0.572161\pi$$
−0.224764 + 0.974413i $$0.572161\pi$$
$$720$$ 0 0
$$721$$ 28.5744 1.06417
$$722$$ 19.9667 0.743084
$$723$$ 0 0
$$724$$ 15.9449 0.592586
$$725$$ −1.00029 −0.0371498
$$726$$ 0 0
$$727$$ −31.6302 −1.17310 −0.586550 0.809913i $$-0.699514\pi$$
−0.586550 + 0.809913i $$0.699514\pi$$
$$728$$ −34.0685 −1.26266
$$729$$ 0 0
$$730$$ 9.79423 0.362501
$$731$$ 28.7633 1.06385
$$732$$ 0 0
$$733$$ −19.0308 −0.702918 −0.351459 0.936203i $$-0.614314\pi$$
−0.351459 + 0.936203i $$0.614314\pi$$
$$734$$ 11.9277 0.440260
$$735$$ 0 0
$$736$$ 3.85533 0.142109
$$737$$ 12.6503 0.465981
$$738$$ 0 0
$$739$$ 16.6007 0.610668 0.305334 0.952245i $$-0.401232\pi$$
0.305334 + 0.952245i $$0.401232\pi$$
$$740$$ −4.04499 −0.148697
$$741$$ 0 0
$$742$$ −27.1837 −0.997944
$$743$$ 33.3035 1.22179 0.610894 0.791712i $$-0.290810\pi$$
0.610894 + 0.791712i $$0.290810\pi$$
$$744$$ 0 0
$$745$$ 0.186692 0.00683985
$$746$$ −6.16156 −0.225591
$$747$$ 0 0
$$748$$ −7.40655 −0.270810
$$749$$ −47.8523 −1.74849
$$750$$ 0 0
$$751$$ 27.7816 1.01376 0.506882 0.862015i $$-0.330798\pi$$
0.506882 + 0.862015i $$0.330798\pi$$
$$752$$ −7.60328 −0.277263
$$753$$ 0 0
$$754$$ 2.45747 0.0894957
$$755$$ −35.3686 −1.28720
$$756$$ 0 0
$$757$$ −3.12036 −0.113411 −0.0567057 0.998391i $$-0.518060\pi$$
−0.0567057 + 0.998391i $$0.518060\pi$$
$$758$$ −25.6480 −0.931577
$$759$$ 0 0
$$760$$ 1.32463 0.0480494
$$761$$ 43.9592 1.59352 0.796760 0.604296i $$-0.206546\pi$$
0.796760 + 0.604296i $$0.206546\pi$$
$$762$$ 0 0
$$763$$ 15.5305 0.562242
$$764$$ −23.9566 −0.866720
$$765$$ 0 0
$$766$$ −4.02336 −0.145370
$$767$$ −13.6753 −0.493787
$$768$$ 0 0
$$769$$ 3.71286 0.133889 0.0669445 0.997757i $$-0.478675\pi$$
0.0669445 + 0.997757i $$0.478675\pi$$
$$770$$ −5.67726 −0.204594
$$771$$ 0 0
$$772$$ 15.2403 0.548511
$$773$$ 8.96903 0.322594 0.161297 0.986906i $$-0.448432\pi$$
0.161297 + 0.986906i $$0.448432\pi$$
$$774$$ 0 0
$$775$$ 1.60083 0.0575036
$$776$$ 16.7205 0.600232
$$777$$ 0 0
$$778$$ 11.4308 0.409815
$$779$$ 2.03132 0.0727797
$$780$$ 0 0
$$781$$ −0.112222 −0.00401560
$$782$$ 5.90098 0.211018
$$783$$ 0 0
$$784$$ 1.36527 0.0487597
$$785$$ −36.2248 −1.29292
$$786$$ 0 0
$$787$$ 43.8514 1.56313 0.781567 0.623821i $$-0.214421\pi$$
0.781567 + 0.623821i $$0.214421\pi$$
$$788$$ 2.23188 0.0795074
$$789$$ 0 0
$$790$$ −8.79253 −0.312824
$$791$$ 17.0075 0.604716
$$792$$ 0 0
$$793$$ −13.1156 −0.465750
$$794$$ 11.0801 0.393219
$$795$$ 0 0
$$796$$ −16.4699 −0.583760
$$797$$ −12.0160 −0.425629 −0.212815 0.977093i $$-0.568263\pi$$
−0.212815 + 0.977093i $$0.568263\pi$$
$$798$$ 0 0
$$799$$ 35.2455 1.24690
$$800$$ 8.93881 0.316035
$$801$$ 0 0
$$802$$ 15.1460 0.534823
$$803$$ −6.67384 −0.235515
$$804$$ 0 0
$$805$$ −3.61498 −0.127411
$$806$$ −3.93286 −0.138529
$$807$$ 0 0
$$808$$ −15.2432 −0.536254
$$809$$ 8.02937 0.282298 0.141149 0.989988i $$-0.454920\pi$$
0.141149 + 0.989988i $$0.454920\pi$$
$$810$$ 0 0
$$811$$ −12.8345 −0.450681 −0.225341 0.974280i $$-0.572349\pi$$
−0.225341 + 0.974280i $$0.572349\pi$$
$$812$$ −1.11945 −0.0392850
$$813$$ 0 0
$$814$$ −3.44876 −0.120879
$$815$$ −35.1507 −1.23127
$$816$$ 0 0
$$817$$ 1.07813 0.0377192
$$818$$ −18.6428 −0.651832
$$819$$ 0 0
$$820$$ −12.6447 −0.441570
$$821$$ 29.6654 1.03533 0.517665 0.855584i $$-0.326802\pi$$
0.517665 + 0.855584i $$0.326802\pi$$
$$822$$ 0 0
$$823$$ 49.5407 1.72688 0.863441 0.504451i $$-0.168305\pi$$
0.863441 + 0.504451i $$0.168305\pi$$
$$824$$ 35.3840 1.23266
$$825$$ 0 0
$$826$$ −7.79462 −0.271209
$$827$$ −40.8431 −1.42025 −0.710126 0.704074i $$-0.751362\pi$$
−0.710126 + 0.704074i $$0.751362\pi$$
$$828$$ 0 0
$$829$$ −9.45276 −0.328308 −0.164154 0.986435i $$-0.552489\pi$$
−0.164154 + 0.986435i $$0.552489\pi$$
$$830$$ 14.7999 0.513711
$$831$$ 0 0
$$832$$ −35.0067 −1.21364
$$833$$ −6.32881 −0.219280
$$834$$ 0 0
$$835$$ 34.6759 1.20001
$$836$$ −0.277620 −0.00960170
$$837$$ 0 0
$$838$$ 9.62995 0.332661
$$839$$ 12.2869 0.424191 0.212095 0.977249i $$-0.431971\pi$$
0.212095 + 0.977249i $$0.431971\pi$$
$$840$$ 0 0
$$841$$ −28.7375 −0.990947
$$842$$ −25.5093 −0.879108
$$843$$ 0 0
$$844$$ 3.27924 0.112876
$$845$$ 13.4315 0.462058
$$846$$ 0 0
$$847$$ −23.1832 −0.796584
$$848$$ −15.0337 −0.516259
$$849$$ 0 0
$$850$$ 13.6817 0.469280
$$851$$ −2.19599 −0.0752776
$$852$$ 0 0
$$853$$ −30.8659 −1.05683 −0.528413 0.848987i $$-0.677213\pi$$
−0.528413 + 0.848987i $$0.677213\pi$$
$$854$$ −7.47562 −0.255810
$$855$$ 0 0
$$856$$ −59.2560 −2.02533
$$857$$ −11.1460 −0.380741 −0.190371 0.981712i $$-0.560969\pi$$
−0.190371 + 0.981712i $$0.560969\pi$$
$$858$$ 0 0
$$859$$ 4.14868 0.141551 0.0707755 0.997492i $$-0.477453\pi$$
0.0707755 + 0.997492i $$0.477453\pi$$
$$860$$ −6.71121 −0.228850
$$861$$ 0 0
$$862$$ −31.1376 −1.06055
$$863$$ 47.2534 1.60852 0.804262 0.594275i $$-0.202561\pi$$
0.804262 + 0.594275i $$0.202561\pi$$
$$864$$ 0 0
$$865$$ −33.0392 −1.12337
$$866$$ −0.706138 −0.0239956
$$867$$ 0 0
$$868$$ 1.79154 0.0608088
$$869$$ 5.99127 0.203240
$$870$$ 0 0
$$871$$ −45.8827 −1.55468
$$872$$ 19.2316 0.651264
$$873$$ 0 0
$$874$$ 0.221187 0.00748175
$$875$$ −29.8481 −1.00905
$$876$$ 0 0
$$877$$ 35.1504 1.18695 0.593473 0.804854i $$-0.297756\pi$$
0.593473 + 0.804854i $$0.297756\pi$$
$$878$$ 6.69377 0.225904
$$879$$ 0 0
$$880$$ −3.13976 −0.105841
$$881$$ 19.3596 0.652242 0.326121 0.945328i $$-0.394258\pi$$
0.326121 + 0.945328i $$0.394258\pi$$
$$882$$ 0 0
$$883$$ −13.7860 −0.463937 −0.231969 0.972723i $$-0.574517\pi$$
−0.231969 + 0.972723i $$0.574517\pi$$
$$884$$ 26.8636 0.903519
$$885$$ 0 0
$$886$$ 16.1880 0.543847
$$887$$ 29.8175 1.00117 0.500586 0.865687i $$-0.333118\pi$$
0.500586 + 0.865687i $$0.333118\pi$$
$$888$$ 0 0
$$889$$ 29.5681 0.991683
$$890$$ 12.3391 0.413609
$$891$$ 0 0
$$892$$ 18.8705 0.631832
$$893$$ 1.32111 0.0442092
$$894$$ 0 0
$$895$$ −19.0529 −0.636869
$$896$$ 2.56766 0.0857795
$$897$$ 0 0
$$898$$ −33.7804 −1.12727
$$899$$ −0.420156 −0.0140130
$$900$$ 0 0
$$901$$ 69.6897 2.32170
$$902$$ −10.7809 −0.358963
$$903$$ 0 0
$$904$$ 21.0605 0.700463
$$905$$ −31.3331 −1.04155
$$906$$ 0 0
$$907$$ −6.23157 −0.206916 −0.103458 0.994634i $$-0.532991\pi$$
−0.103458 + 0.994634i $$0.532991\pi$$
$$908$$ −12.7397 −0.422781
$$909$$ 0 0
$$910$$ 20.5914 0.682599
$$911$$ 31.6395 1.04826 0.524131 0.851638i $$-0.324390\pi$$
0.524131 + 0.851638i $$0.324390\pi$$
$$912$$ 0 0
$$913$$ −10.0847 −0.333755
$$914$$ −20.2093 −0.668465
$$915$$ 0 0
$$916$$ −15.0013 −0.495658
$$917$$ 34.6380 1.14385
$$918$$ 0 0
$$919$$ 36.0031 1.18763 0.593816 0.804601i $$-0.297621\pi$$
0.593816 + 0.804601i $$0.297621\pi$$
$$920$$ −4.47647 −0.147585
$$921$$ 0 0
$$922$$ −5.50388 −0.181261
$$923$$ 0.407027 0.0133975
$$924$$ 0 0
$$925$$ −5.09153 −0.167408
$$926$$ −1.78960 −0.0588099
$$927$$ 0 0
$$928$$ −2.34609 −0.0770141
$$929$$ −25.4477 −0.834913 −0.417456 0.908697i $$-0.637078\pi$$
−0.417456 + 0.908697i $$0.637078\pi$$
$$930$$ 0 0
$$931$$ −0.237223 −0.00777468
$$932$$ 4.97454 0.162947
$$933$$ 0 0
$$934$$ 20.7520 0.679026
$$935$$ 14.5546 0.475985
$$936$$ 0 0
$$937$$ 28.3048 0.924677 0.462338 0.886704i $$-0.347011\pi$$
0.462338 + 0.886704i $$0.347011\pi$$
$$938$$ −26.1521 −0.853897
$$939$$ 0 0
$$940$$ −8.22369 −0.268227
$$941$$ −8.29444 −0.270391 −0.135196 0.990819i $$-0.543166\pi$$
−0.135196 + 0.990819i $$0.543166\pi$$
$$942$$ 0 0
$$943$$ −6.86468 −0.223545
$$944$$ −4.31074 −0.140303
$$945$$ 0 0
$$946$$ −5.72199 −0.186038
$$947$$ 44.4506 1.44445 0.722225 0.691658i $$-0.243119\pi$$
0.722225 + 0.691658i $$0.243119\pi$$
$$948$$ 0 0
$$949$$ 24.2060 0.785761
$$950$$ 0.512834 0.0166385
$$951$$ 0 0
$$952$$ 49.7818 1.61344
$$953$$ 9.67149 0.313290 0.156645 0.987655i $$-0.449932\pi$$
0.156645 + 0.987655i $$0.449932\pi$$
$$954$$ 0 0
$$955$$ 47.0770 1.52338
$$956$$ −4.68566 −0.151545
$$957$$ 0 0
$$958$$ −30.8426 −0.996479
$$959$$ −5.55332 −0.179326
$$960$$ 0 0
$$961$$ −30.3276 −0.978309
$$962$$ 12.5087 0.403296
$$963$$ 0 0
$$964$$ 7.90895 0.254730
$$965$$ −29.9486 −0.964081
$$966$$ 0 0
$$967$$ −33.1151 −1.06491 −0.532455 0.846459i $$-0.678730\pi$$
−0.532455 + 0.846459i $$0.678730\pi$$
$$968$$ −28.7080 −0.922710
$$969$$ 0 0
$$970$$ −10.1061 −0.324487
$$971$$ 27.4309 0.880298 0.440149 0.897925i $$-0.354926\pi$$
0.440149 + 0.897925i $$0.354926\pi$$
$$972$$ 0 0
$$973$$ −19.6127 −0.628755
$$974$$ 21.6196 0.692737
$$975$$ 0 0
$$976$$ −4.13432 −0.132336
$$977$$ −36.3776 −1.16382 −0.581911 0.813253i $$-0.697695\pi$$
−0.581911 + 0.813253i $$0.697695\pi$$
$$978$$ 0 0
$$979$$ −8.40796 −0.268719
$$980$$ 1.47667 0.0471706
$$981$$ 0 0
$$982$$ 18.4791 0.589693
$$983$$ −42.1372 −1.34397 −0.671984 0.740566i $$-0.734558\pi$$
−0.671984 + 0.740566i $$0.734558\pi$$
$$984$$ 0 0
$$985$$ −4.38585 −0.139745
$$986$$ −3.59092 −0.114358
$$987$$ 0 0
$$988$$ 1.00693 0.0320347
$$989$$ −3.64346 −0.115855
$$990$$ 0 0
$$991$$ 25.5409 0.811333 0.405667 0.914021i $$-0.367039\pi$$
0.405667 + 0.914021i $$0.367039\pi$$
$$992$$ 3.75461 0.119209
$$993$$ 0 0
$$994$$ 0.231996 0.00735848
$$995$$ 32.3649 1.02604
$$996$$ 0 0
$$997$$ 23.5329 0.745294 0.372647 0.927973i $$-0.378450\pi$$
0.372647 + 0.927973i $$0.378450\pi$$
$$998$$ 20.1439 0.637644
$$999$$ 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.d.1.2 6
3.2 odd 2 729.2.a.a.1.5 6
9.2 odd 6 729.2.c.e.244.2 12
9.4 even 3 729.2.c.b.487.5 12
9.5 odd 6 729.2.c.e.487.2 12
9.7 even 3 729.2.c.b.244.5 12
27.2 odd 18 243.2.e.d.190.1 12
27.4 even 9 243.2.e.b.136.1 12
27.5 odd 18 27.2.e.a.25.2 yes 12
27.7 even 9 243.2.e.b.109.1 12
27.11 odd 18 27.2.e.a.13.2 12
27.13 even 9 243.2.e.a.55.2 12
27.14 odd 18 243.2.e.d.55.1 12
27.16 even 9 81.2.e.a.10.1 12
27.20 odd 18 243.2.e.c.109.2 12
27.22 even 9 81.2.e.a.73.1 12
27.23 odd 18 243.2.e.c.136.2 12
27.25 even 9 243.2.e.a.190.2 12
108.11 even 18 432.2.u.c.337.2 12
108.59 even 18 432.2.u.c.241.2 12
135.32 even 36 675.2.u.b.349.3 24
135.38 even 36 675.2.u.b.499.3 24
135.59 odd 18 675.2.l.c.376.1 12
135.92 even 36 675.2.u.b.499.2 24
135.113 even 36 675.2.u.b.349.2 24
135.119 odd 18 675.2.l.c.526.1 12

By twisted newform
Twist Min Dim Char Parity Ord Type
27.2.e.a.13.2 12 27.11 odd 18
27.2.e.a.25.2 yes 12 27.5 odd 18
81.2.e.a.10.1 12 27.16 even 9
81.2.e.a.73.1 12 27.22 even 9
243.2.e.a.55.2 12 27.13 even 9
243.2.e.a.190.2 12 27.25 even 9
243.2.e.b.109.1 12 27.7 even 9
243.2.e.b.136.1 12 27.4 even 9
243.2.e.c.109.2 12 27.20 odd 18
243.2.e.c.136.2 12 27.23 odd 18
243.2.e.d.55.1 12 27.14 odd 18
243.2.e.d.190.1 12 27.2 odd 18
432.2.u.c.241.2 12 108.59 even 18
432.2.u.c.337.2 12 108.11 even 18
675.2.l.c.376.1 12 135.59 odd 18
675.2.l.c.526.1 12 135.119 odd 18
675.2.u.b.349.2 24 135.113 even 36
675.2.u.b.349.3 24 135.32 even 36
675.2.u.b.499.2 24 135.92 even 36
675.2.u.b.499.3 24 135.38 even 36
729.2.a.a.1.5 6 3.2 odd 2
729.2.a.d.1.2 6 1.1 even 1 trivial
729.2.c.b.244.5 12 9.7 even 3
729.2.c.b.487.5 12 9.4 even 3
729.2.c.e.244.2 12 9.2 odd 6
729.2.c.e.487.2 12 9.5 odd 6