# Properties

 Label 729.2.a.c.1.2 Level $729$ Weight $2$ Character 729.1 Self dual yes Analytic conductor $5.821$ Analytic rank $1$ Dimension $6$ CM no Inner twists $2$

# 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: $$\Q(\zeta_{36})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 6x^{4} + 9x^{2} - 3$$ x^6 - 6*x^4 + 9*x^2 - 3 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 $$-1.28558$$ 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)

 $$f(q)$$ $$=$$ $$q-1.28558 q^{2} -0.347296 q^{4} -0.446476 q^{5} -3.53209 q^{7} +3.01763 q^{8} +O(q^{10})$$ $$q-1.28558 q^{2} -0.347296 q^{4} -0.446476 q^{5} -3.53209 q^{7} +3.01763 q^{8} +0.573978 q^{10} +2.78006 q^{11} +3.29086 q^{13} +4.54077 q^{14} -3.18479 q^{16} +7.03936 q^{17} -5.18479 q^{19} +0.155059 q^{20} -3.57398 q^{22} -7.27693 q^{23} -4.80066 q^{25} -4.23065 q^{26} +1.22668 q^{28} -3.61916 q^{29} -1.93582 q^{31} -1.94096 q^{32} -9.04963 q^{34} +1.57699 q^{35} -3.22668 q^{37} +6.66544 q^{38} -1.34730 q^{40} +4.86084 q^{41} -5.75877 q^{43} -0.965505 q^{44} +9.35504 q^{46} -3.01763 q^{47} +5.47565 q^{49} +6.17161 q^{50} -1.14290 q^{52} -8.77141 q^{53} -1.24123 q^{55} -10.6585 q^{56} +4.65270 q^{58} -2.96377 q^{59} +7.88713 q^{61} +2.48865 q^{62} +8.86484 q^{64} -1.46929 q^{65} -9.43882 q^{67} -2.44474 q^{68} -2.02734 q^{70} -5.30731 q^{71} -1.55438 q^{73} +4.14814 q^{74} +1.80066 q^{76} -9.81942 q^{77} -11.9017 q^{79} +1.42193 q^{80} -6.24897 q^{82} +16.2573 q^{83} -3.14290 q^{85} +7.40333 q^{86} +8.38919 q^{88} -18.4258 q^{89} -11.6236 q^{91} +2.52725 q^{92} +3.87939 q^{94} +2.31488 q^{95} +10.1138 q^{97} -7.03936 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 12 q^{7}+O(q^{10})$$ 6 * q - 12 * q^7 $$6 q - 12 q^{7} - 12 q^{10} - 12 q^{13} - 12 q^{16} - 24 q^{19} - 6 q^{22} - 6 q^{28} - 30 q^{31} - 6 q^{37} - 6 q^{40} - 12 q^{43} + 6 q^{46} - 6 q^{49} - 6 q^{52} - 30 q^{55} + 30 q^{58} - 12 q^{61} + 6 q^{64} + 6 q^{67} + 30 q^{70} + 12 q^{73} - 18 q^{76} - 48 q^{79} - 12 q^{82} - 18 q^{85} + 42 q^{88} + 12 q^{94} - 12 q^{97}+O(q^{100})$$ 6 * q - 12 * q^7 - 12 * q^10 - 12 * q^13 - 12 * q^16 - 24 * q^19 - 6 * q^22 - 6 * q^28 - 30 * q^31 - 6 * q^37 - 6 * q^40 - 12 * q^43 + 6 * q^46 - 6 * q^49 - 6 * q^52 - 30 * q^55 + 30 * q^58 - 12 * q^61 + 6 * q^64 + 6 * q^67 + 30 * q^70 + 12 * q^73 - 18 * q^76 - 48 * q^79 - 12 * q^82 - 18 * q^85 + 42 * q^88 + 12 * q^94 - 12 * q^97

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.28558 −0.909039 −0.454519 0.890737i $$-0.650189\pi$$
−0.454519 + 0.890737i $$0.650189\pi$$
$$3$$ 0 0
$$4$$ −0.347296 −0.173648
$$5$$ −0.446476 −0.199670 −0.0998350 0.995004i $$-0.531831\pi$$
−0.0998350 + 0.995004i $$0.531831\pi$$
$$6$$ 0 0
$$7$$ −3.53209 −1.33500 −0.667502 0.744608i $$-0.732636\pi$$
−0.667502 + 0.744608i $$0.732636\pi$$
$$8$$ 3.01763 1.06689
$$9$$ 0 0
$$10$$ 0.573978 0.181508
$$11$$ 2.78006 0.838220 0.419110 0.907935i $$-0.362342\pi$$
0.419110 + 0.907935i $$0.362342\pi$$
$$12$$ 0 0
$$13$$ 3.29086 0.912720 0.456360 0.889795i $$-0.349153\pi$$
0.456360 + 0.889795i $$0.349153\pi$$
$$14$$ 4.54077 1.21357
$$15$$ 0 0
$$16$$ −3.18479 −0.796198
$$17$$ 7.03936 1.70730 0.853648 0.520850i $$-0.174385\pi$$
0.853648 + 0.520850i $$0.174385\pi$$
$$18$$ 0 0
$$19$$ −5.18479 −1.18947 −0.594736 0.803921i $$-0.702744\pi$$
−0.594736 + 0.803921i $$0.702744\pi$$
$$20$$ 0.155059 0.0346723
$$21$$ 0 0
$$22$$ −3.57398 −0.761975
$$23$$ −7.27693 −1.51734 −0.758672 0.651473i $$-0.774152\pi$$
−0.758672 + 0.651473i $$0.774152\pi$$
$$24$$ 0 0
$$25$$ −4.80066 −0.960132
$$26$$ −4.23065 −0.829698
$$27$$ 0 0
$$28$$ 1.22668 0.231821
$$29$$ −3.61916 −0.672061 −0.336031 0.941851i $$-0.609085\pi$$
−0.336031 + 0.941851i $$0.609085\pi$$
$$30$$ 0 0
$$31$$ −1.93582 −0.347684 −0.173842 0.984774i $$-0.555618\pi$$
−0.173842 + 0.984774i $$0.555618\pi$$
$$32$$ −1.94096 −0.343117
$$33$$ 0 0
$$34$$ −9.04963 −1.55200
$$35$$ 1.57699 0.266560
$$36$$ 0 0
$$37$$ −3.22668 −0.530463 −0.265232 0.964185i $$-0.585448\pi$$
−0.265232 + 0.964185i $$0.585448\pi$$
$$38$$ 6.66544 1.08128
$$39$$ 0 0
$$40$$ −1.34730 −0.213026
$$41$$ 4.86084 0.759135 0.379568 0.925164i $$-0.376073\pi$$
0.379568 + 0.925164i $$0.376073\pi$$
$$42$$ 0 0
$$43$$ −5.75877 −0.878204 −0.439102 0.898437i $$-0.644703\pi$$
−0.439102 + 0.898437i $$0.644703\pi$$
$$44$$ −0.965505 −0.145555
$$45$$ 0 0
$$46$$ 9.35504 1.37932
$$47$$ −3.01763 −0.440166 −0.220083 0.975481i $$-0.570633\pi$$
−0.220083 + 0.975481i $$0.570633\pi$$
$$48$$ 0 0
$$49$$ 5.47565 0.782236
$$50$$ 6.17161 0.872797
$$51$$ 0 0
$$52$$ −1.14290 −0.158492
$$53$$ −8.77141 −1.20485 −0.602423 0.798177i $$-0.705798\pi$$
−0.602423 + 0.798177i $$0.705798\pi$$
$$54$$ 0 0
$$55$$ −1.24123 −0.167367
$$56$$ −10.6585 −1.42431
$$57$$ 0 0
$$58$$ 4.65270 0.610930
$$59$$ −2.96377 −0.385851 −0.192925 0.981213i $$-0.561797\pi$$
−0.192925 + 0.981213i $$0.561797\pi$$
$$60$$ 0 0
$$61$$ 7.88713 1.00984 0.504922 0.863165i $$-0.331521\pi$$
0.504922 + 0.863165i $$0.331521\pi$$
$$62$$ 2.48865 0.316058
$$63$$ 0 0
$$64$$ 8.86484 1.10810
$$65$$ −1.46929 −0.182243
$$66$$ 0 0
$$67$$ −9.43882 −1.15313 −0.576567 0.817050i $$-0.695608\pi$$
−0.576567 + 0.817050i $$0.695608\pi$$
$$68$$ −2.44474 −0.296469
$$69$$ 0 0
$$70$$ −2.02734 −0.242314
$$71$$ −5.30731 −0.629862 −0.314931 0.949115i $$-0.601981\pi$$
−0.314931 + 0.949115i $$0.601981\pi$$
$$72$$ 0 0
$$73$$ −1.55438 −0.181926 −0.0909631 0.995854i $$-0.528995\pi$$
−0.0909631 + 0.995854i $$0.528995\pi$$
$$74$$ 4.14814 0.482212
$$75$$ 0 0
$$76$$ 1.80066 0.206550
$$77$$ −9.81942 −1.11903
$$78$$ 0 0
$$79$$ −11.9017 −1.33904 −0.669521 0.742793i $$-0.733501\pi$$
−0.669521 + 0.742793i $$0.733501\pi$$
$$80$$ 1.42193 0.158977
$$81$$ 0 0
$$82$$ −6.24897 −0.690083
$$83$$ 16.2573 1.78447 0.892233 0.451576i $$-0.149138\pi$$
0.892233 + 0.451576i $$0.149138\pi$$
$$84$$ 0 0
$$85$$ −3.14290 −0.340896
$$86$$ 7.40333 0.798322
$$87$$ 0 0
$$88$$ 8.38919 0.894290
$$89$$ −18.4258 −1.95313 −0.976567 0.215214i $$-0.930955\pi$$
−0.976567 + 0.215214i $$0.930955\pi$$
$$90$$ 0 0
$$91$$ −11.6236 −1.21849
$$92$$ 2.52725 0.263484
$$93$$ 0 0
$$94$$ 3.87939 0.400128
$$95$$ 2.31488 0.237502
$$96$$ 0 0
$$97$$ 10.1138 1.02690 0.513451 0.858119i $$-0.328367\pi$$
0.513451 + 0.858119i $$0.328367\pi$$
$$98$$ −7.03936 −0.711083
$$99$$ 0 0
$$100$$ 1.66725 0.166725
$$101$$ 2.08077 0.207045 0.103522 0.994627i $$-0.466989\pi$$
0.103522 + 0.994627i $$0.466989\pi$$
$$102$$ 0 0
$$103$$ −1.44831 −0.142706 −0.0713531 0.997451i $$-0.522732\pi$$
−0.0713531 + 0.997451i $$0.522732\pi$$
$$104$$ 9.93058 0.973774
$$105$$ 0 0
$$106$$ 11.2763 1.09525
$$107$$ 2.23583 0.216146 0.108073 0.994143i $$-0.465532\pi$$
0.108073 + 0.994143i $$0.465532\pi$$
$$108$$ 0 0
$$109$$ −11.5030 −1.10179 −0.550893 0.834576i $$-0.685713\pi$$
−0.550893 + 0.834576i $$0.685713\pi$$
$$110$$ 1.59569 0.152143
$$111$$ 0 0
$$112$$ 11.2490 1.06293
$$113$$ −1.68815 −0.158808 −0.0794039 0.996843i $$-0.525302\pi$$
−0.0794039 + 0.996843i $$0.525302\pi$$
$$114$$ 0 0
$$115$$ 3.24897 0.302968
$$116$$ 1.25692 0.116702
$$117$$ 0 0
$$118$$ 3.81016 0.350753
$$119$$ −24.8637 −2.27925
$$120$$ 0 0
$$121$$ −3.27126 −0.297387
$$122$$ −10.1395 −0.917987
$$123$$ 0 0
$$124$$ 0.672304 0.0603747
$$125$$ 4.37576 0.391379
$$126$$ 0 0
$$127$$ −2.67230 −0.237129 −0.118564 0.992946i $$-0.537829\pi$$
−0.118564 + 0.992946i $$0.537829\pi$$
$$128$$ −7.51449 −0.664194
$$129$$ 0 0
$$130$$ 1.88888 0.165666
$$131$$ 2.91987 0.255111 0.127555 0.991831i $$-0.459287\pi$$
0.127555 + 0.991831i $$0.459287\pi$$
$$132$$ 0 0
$$133$$ 18.3131 1.58795
$$134$$ 12.1343 1.04824
$$135$$ 0 0
$$136$$ 21.2422 1.82150
$$137$$ 4.65722 0.397893 0.198947 0.980010i $$-0.436248\pi$$
0.198947 + 0.980010i $$0.436248\pi$$
$$138$$ 0 0
$$139$$ −8.00269 −0.678779 −0.339390 0.940646i $$-0.610220\pi$$
−0.339390 + 0.940646i $$0.610220\pi$$
$$140$$ −0.547683 −0.0462877
$$141$$ 0 0
$$142$$ 6.82295 0.572569
$$143$$ 9.14879 0.765060
$$144$$ 0 0
$$145$$ 1.61587 0.134190
$$146$$ 1.99827 0.165378
$$147$$ 0 0
$$148$$ 1.12061 0.0921140
$$149$$ −20.2117 −1.65581 −0.827905 0.560869i $$-0.810467\pi$$
−0.827905 + 0.560869i $$0.810467\pi$$
$$150$$ 0 0
$$151$$ 6.70233 0.545428 0.272714 0.962095i $$-0.412079\pi$$
0.272714 + 0.962095i $$0.412079\pi$$
$$152$$ −15.6458 −1.26904
$$153$$ 0 0
$$154$$ 12.6236 1.01724
$$155$$ 0.864297 0.0694220
$$156$$ 0 0
$$157$$ 5.32501 0.424982 0.212491 0.977163i $$-0.431842\pi$$
0.212491 + 0.977163i $$0.431842\pi$$
$$158$$ 15.3005 1.21724
$$159$$ 0 0
$$160$$ 0.866592 0.0685101
$$161$$ 25.7028 2.02566
$$162$$ 0 0
$$163$$ 3.81521 0.298830 0.149415 0.988775i $$-0.452261\pi$$
0.149415 + 0.988775i $$0.452261\pi$$
$$164$$ −1.68815 −0.131822
$$165$$ 0 0
$$166$$ −20.8999 −1.62215
$$167$$ 9.13538 0.706917 0.353459 0.935450i $$-0.385006\pi$$
0.353459 + 0.935450i $$0.385006\pi$$
$$168$$ 0 0
$$169$$ −2.17024 −0.166942
$$170$$ 4.04044 0.309888
$$171$$ 0 0
$$172$$ 2.00000 0.152499
$$173$$ 6.07386 0.461787 0.230893 0.972979i $$-0.425835\pi$$
0.230893 + 0.972979i $$0.425835\pi$$
$$174$$ 0 0
$$175$$ 16.9564 1.28178
$$176$$ −8.85392 −0.667389
$$177$$ 0 0
$$178$$ 23.6878 1.77547
$$179$$ 10.2811 0.768449 0.384224 0.923240i $$-0.374469\pi$$
0.384224 + 0.923240i $$0.374469\pi$$
$$180$$ 0 0
$$181$$ −23.1411 −1.72007 −0.860034 0.510237i $$-0.829558\pi$$
−0.860034 + 0.510237i $$0.829558\pi$$
$$182$$ 14.9430 1.10765
$$183$$ 0 0
$$184$$ −21.9590 −1.61884
$$185$$ 1.44063 0.105918
$$186$$ 0 0
$$187$$ 19.5699 1.43109
$$188$$ 1.04801 0.0764340
$$189$$ 0 0
$$190$$ −2.97596 −0.215899
$$191$$ 14.6703 1.06151 0.530753 0.847526i $$-0.321909\pi$$
0.530753 + 0.847526i $$0.321909\pi$$
$$192$$ 0 0
$$193$$ 23.4415 1.68736 0.843678 0.536849i $$-0.180386\pi$$
0.843678 + 0.536849i $$0.180386\pi$$
$$194$$ −13.0021 −0.933494
$$195$$ 0 0
$$196$$ −1.90167 −0.135834
$$197$$ −9.02768 −0.643196 −0.321598 0.946876i $$-0.604220\pi$$
−0.321598 + 0.946876i $$0.604220\pi$$
$$198$$ 0 0
$$199$$ 2.60401 0.184593 0.0922966 0.995732i $$-0.470579\pi$$
0.0922966 + 0.995732i $$0.470579\pi$$
$$200$$ −14.4866 −1.02436
$$201$$ 0 0
$$202$$ −2.67499 −0.188212
$$203$$ 12.7832 0.897205
$$204$$ 0 0
$$205$$ −2.17024 −0.151576
$$206$$ 1.86191 0.129726
$$207$$ 0 0
$$208$$ −10.4807 −0.726706
$$209$$ −14.4140 −0.997040
$$210$$ 0 0
$$211$$ −16.3550 −1.12593 −0.562964 0.826482i $$-0.690339\pi$$
−0.562964 + 0.826482i $$0.690339\pi$$
$$212$$ 3.04628 0.209219
$$213$$ 0 0
$$214$$ −2.87433 −0.196485
$$215$$ 2.57115 0.175351
$$216$$ 0 0
$$217$$ 6.83750 0.464159
$$218$$ 14.7880 1.00157
$$219$$ 0 0
$$220$$ 0.431074 0.0290630
$$221$$ 23.1656 1.55828
$$222$$ 0 0
$$223$$ 3.70233 0.247927 0.123963 0.992287i $$-0.460439\pi$$
0.123963 + 0.992287i $$0.460439\pi$$
$$224$$ 6.85565 0.458062
$$225$$ 0 0
$$226$$ 2.17024 0.144363
$$227$$ −10.5608 −0.700943 −0.350472 0.936573i $$-0.613979\pi$$
−0.350472 + 0.936573i $$0.613979\pi$$
$$228$$ 0 0
$$229$$ 13.5253 0.893776 0.446888 0.894590i $$-0.352532\pi$$
0.446888 + 0.894590i $$0.352532\pi$$
$$230$$ −4.17680 −0.275410
$$231$$ 0 0
$$232$$ −10.9213 −0.717017
$$233$$ 12.7007 0.832050 0.416025 0.909353i $$-0.363423\pi$$
0.416025 + 0.909353i $$0.363423\pi$$
$$234$$ 0 0
$$235$$ 1.34730 0.0878879
$$236$$ 1.02931 0.0670022
$$237$$ 0 0
$$238$$ 31.9641 2.07192
$$239$$ −5.85154 −0.378505 −0.189252 0.981928i $$-0.560606\pi$$
−0.189252 + 0.981928i $$0.560606\pi$$
$$240$$ 0 0
$$241$$ −8.53983 −0.550099 −0.275049 0.961430i $$-0.588694\pi$$
−0.275049 + 0.961430i $$0.588694\pi$$
$$242$$ 4.20545 0.270337
$$243$$ 0 0
$$244$$ −2.73917 −0.175357
$$245$$ −2.44474 −0.156189
$$246$$ 0 0
$$247$$ −17.0624 −1.08566
$$248$$ −5.84159 −0.370941
$$249$$ 0 0
$$250$$ −5.62536 −0.355779
$$251$$ −6.75790 −0.426555 −0.213277 0.976992i $$-0.568414\pi$$
−0.213277 + 0.976992i $$0.568414\pi$$
$$252$$ 0 0
$$253$$ −20.2303 −1.27187
$$254$$ 3.43545 0.215559
$$255$$ 0 0
$$256$$ −8.06923 −0.504327
$$257$$ 3.68296 0.229737 0.114868 0.993381i $$-0.463355\pi$$
0.114868 + 0.993381i $$0.463355\pi$$
$$258$$ 0 0
$$259$$ 11.3969 0.708171
$$260$$ 0.510278 0.0316461
$$261$$ 0 0
$$262$$ −3.75372 −0.231905
$$263$$ 3.63786 0.224320 0.112160 0.993690i $$-0.464223\pi$$
0.112160 + 0.993690i $$0.464223\pi$$
$$264$$ 0 0
$$265$$ 3.91622 0.240572
$$266$$ −23.5429 −1.44351
$$267$$ 0 0
$$268$$ 3.27807 0.200240
$$269$$ −7.08672 −0.432085 −0.216042 0.976384i $$-0.569315\pi$$
−0.216042 + 0.976384i $$0.569315\pi$$
$$270$$ 0 0
$$271$$ −19.0000 −1.15417 −0.577084 0.816685i $$-0.695809\pi$$
−0.577084 + 0.816685i $$0.695809\pi$$
$$272$$ −22.4189 −1.35935
$$273$$ 0 0
$$274$$ −5.98721 −0.361700
$$275$$ −13.3461 −0.804802
$$276$$ 0 0
$$277$$ −13.8922 −0.834700 −0.417350 0.908746i $$-0.637041\pi$$
−0.417350 + 0.908746i $$0.637041\pi$$
$$278$$ 10.2881 0.617037
$$279$$ 0 0
$$280$$ 4.75877 0.284391
$$281$$ 22.1275 1.32002 0.660008 0.751259i $$-0.270553\pi$$
0.660008 + 0.751259i $$0.270553\pi$$
$$282$$ 0 0
$$283$$ −23.7324 −1.41074 −0.705371 0.708838i $$-0.749220\pi$$
−0.705371 + 0.708838i $$0.749220\pi$$
$$284$$ 1.84321 0.109374
$$285$$ 0 0
$$286$$ −11.7615 −0.695470
$$287$$ −17.1689 −1.01345
$$288$$ 0 0
$$289$$ 32.5526 1.91486
$$290$$ −2.07732 −0.121984
$$291$$ 0 0
$$292$$ 0.539830 0.0315911
$$293$$ 18.6061 1.08698 0.543489 0.839416i $$-0.317103\pi$$
0.543489 + 0.839416i $$0.317103\pi$$
$$294$$ 0 0
$$295$$ 1.32325 0.0770428
$$296$$ −9.73692 −0.565947
$$297$$ 0 0
$$298$$ 25.9837 1.50520
$$299$$ −23.9473 −1.38491
$$300$$ 0 0
$$301$$ 20.3405 1.17241
$$302$$ −8.61635 −0.495815
$$303$$ 0 0
$$304$$ 16.5125 0.947056
$$305$$ −3.52141 −0.201635
$$306$$ 0 0
$$307$$ 20.7469 1.18409 0.592044 0.805905i $$-0.298321\pi$$
0.592044 + 0.805905i $$0.298321\pi$$
$$308$$ 3.41025 0.194317
$$309$$ 0 0
$$310$$ −1.11112 −0.0631073
$$311$$ 20.3855 1.15596 0.577978 0.816053i $$-0.303842\pi$$
0.577978 + 0.816053i $$0.303842\pi$$
$$312$$ 0 0
$$313$$ −29.8408 −1.68670 −0.843351 0.537364i $$-0.819420\pi$$
−0.843351 + 0.537364i $$0.819420\pi$$
$$314$$ −6.84570 −0.386325
$$315$$ 0 0
$$316$$ 4.13341 0.232522
$$317$$ 4.31661 0.242445 0.121222 0.992625i $$-0.461319\pi$$
0.121222 + 0.992625i $$0.461319\pi$$
$$318$$ 0 0
$$319$$ −10.0615 −0.563335
$$320$$ −3.95793 −0.221255
$$321$$ 0 0
$$322$$ −33.0428 −1.84140
$$323$$ −36.4976 −2.03078
$$324$$ 0 0
$$325$$ −15.7983 −0.876332
$$326$$ −4.90474 −0.271648
$$327$$ 0 0
$$328$$ 14.6682 0.809915
$$329$$ 10.6585 0.587623
$$330$$ 0 0
$$331$$ −2.91891 −0.160438 −0.0802189 0.996777i $$-0.525562\pi$$
−0.0802189 + 0.996777i $$0.525562\pi$$
$$332$$ −5.64608 −0.309869
$$333$$ 0 0
$$334$$ −11.7442 −0.642615
$$335$$ 4.21420 0.230246
$$336$$ 0 0
$$337$$ 14.4415 0.786679 0.393339 0.919393i $$-0.371320\pi$$
0.393339 + 0.919393i $$0.371320\pi$$
$$338$$ 2.79001 0.151757
$$339$$ 0 0
$$340$$ 1.09152 0.0591959
$$341$$ −5.38170 −0.291436
$$342$$ 0 0
$$343$$ 5.38413 0.290716
$$344$$ −17.3778 −0.936949
$$345$$ 0 0
$$346$$ −7.80840 −0.419782
$$347$$ −1.45404 −0.0780571 −0.0390285 0.999238i $$-0.512426\pi$$
−0.0390285 + 0.999238i $$0.512426\pi$$
$$348$$ 0 0
$$349$$ −27.5817 −1.47642 −0.738208 0.674573i $$-0.764328\pi$$
−0.738208 + 0.674573i $$0.764328\pi$$
$$350$$ −21.7987 −1.16519
$$351$$ 0 0
$$352$$ −5.39599 −0.287607
$$353$$ 10.5321 0.560568 0.280284 0.959917i $$-0.409571\pi$$
0.280284 + 0.959917i $$0.409571\pi$$
$$354$$ 0 0
$$355$$ 2.36959 0.125765
$$356$$ 6.39922 0.339158
$$357$$ 0 0
$$358$$ −13.2172 −0.698550
$$359$$ 14.7055 0.776124 0.388062 0.921633i $$-0.373145\pi$$
0.388062 + 0.921633i $$0.373145\pi$$
$$360$$ 0 0
$$361$$ 7.88207 0.414846
$$362$$ 29.7497 1.56361
$$363$$ 0 0
$$364$$ 4.03684 0.211588
$$365$$ 0.693992 0.0363252
$$366$$ 0 0
$$367$$ −21.3628 −1.11513 −0.557564 0.830134i $$-0.688264\pi$$
−0.557564 + 0.830134i $$0.688264\pi$$
$$368$$ 23.1755 1.20811
$$369$$ 0 0
$$370$$ −1.85204 −0.0962832
$$371$$ 30.9814 1.60847
$$372$$ 0 0
$$373$$ 22.6245 1.17145 0.585727 0.810508i $$-0.300809\pi$$
0.585727 + 0.810508i $$0.300809\pi$$
$$374$$ −25.1585 −1.30092
$$375$$ 0 0
$$376$$ −9.10607 −0.469610
$$377$$ −11.9101 −0.613404
$$378$$ 0 0
$$379$$ −17.0743 −0.877047 −0.438523 0.898720i $$-0.644498\pi$$
−0.438523 + 0.898720i $$0.644498\pi$$
$$380$$ −0.803951 −0.0412418
$$381$$ 0 0
$$382$$ −18.8598 −0.964951
$$383$$ −38.9916 −1.99238 −0.996188 0.0872303i $$-0.972198\pi$$
−0.996188 + 0.0872303i $$0.972198\pi$$
$$384$$ 0 0
$$385$$ 4.38413 0.223436
$$386$$ −30.1358 −1.53387
$$387$$ 0 0
$$388$$ −3.51249 −0.178320
$$389$$ 10.2120 0.517771 0.258886 0.965908i $$-0.416645\pi$$
0.258886 + 0.965908i $$0.416645\pi$$
$$390$$ 0 0
$$391$$ −51.2249 −2.59056
$$392$$ 16.5235 0.834561
$$393$$ 0 0
$$394$$ 11.6058 0.584690
$$395$$ 5.31381 0.267367
$$396$$ 0 0
$$397$$ −1.14290 −0.0573607 −0.0286803 0.999589i $$-0.509130\pi$$
−0.0286803 + 0.999589i $$0.509130\pi$$
$$398$$ −3.34765 −0.167802
$$399$$ 0 0
$$400$$ 15.2891 0.764455
$$401$$ −20.5540 −1.02642 −0.513208 0.858264i $$-0.671543\pi$$
−0.513208 + 0.858264i $$0.671543\pi$$
$$402$$ 0 0
$$403$$ −6.37052 −0.317338
$$404$$ −0.722645 −0.0359530
$$405$$ 0 0
$$406$$ −16.4338 −0.815594
$$407$$ −8.97037 −0.444645
$$408$$ 0 0
$$409$$ −8.12836 −0.401921 −0.200961 0.979599i $$-0.564406\pi$$
−0.200961 + 0.979599i $$0.564406\pi$$
$$410$$ 2.79001 0.137789
$$411$$ 0 0
$$412$$ 0.502993 0.0247807
$$413$$ 10.4683 0.515112
$$414$$ 0 0
$$415$$ −7.25847 −0.356304
$$416$$ −6.38743 −0.313170
$$417$$ 0 0
$$418$$ 18.5303 0.906348
$$419$$ 35.7784 1.74789 0.873946 0.486024i $$-0.161553\pi$$
0.873946 + 0.486024i $$0.161553\pi$$
$$420$$ 0 0
$$421$$ −13.1189 −0.639374 −0.319687 0.947523i $$-0.603578\pi$$
−0.319687 + 0.947523i $$0.603578\pi$$
$$422$$ 21.0256 1.02351
$$423$$ 0 0
$$424$$ −26.4688 −1.28544
$$425$$ −33.7936 −1.63923
$$426$$ 0 0
$$427$$ −27.8580 −1.34814
$$428$$ −0.776497 −0.0375334
$$429$$ 0 0
$$430$$ −3.30541 −0.159401
$$431$$ 9.48411 0.456833 0.228417 0.973563i $$-0.426645\pi$$
0.228417 + 0.973563i $$0.426645\pi$$
$$432$$ 0 0
$$433$$ 17.6628 0.848820 0.424410 0.905470i $$-0.360481\pi$$
0.424410 + 0.905470i $$0.360481\pi$$
$$434$$ −8.79012 −0.421939
$$435$$ 0 0
$$436$$ 3.99495 0.191323
$$437$$ 37.7294 1.80484
$$438$$ 0 0
$$439$$ −17.6655 −0.843128 −0.421564 0.906799i $$-0.638519\pi$$
−0.421564 + 0.906799i $$0.638519\pi$$
$$440$$ −3.74557 −0.178563
$$441$$ 0 0
$$442$$ −29.7811 −1.41654
$$443$$ −12.9921 −0.617274 −0.308637 0.951180i $$-0.599873\pi$$
−0.308637 + 0.951180i $$0.599873\pi$$
$$444$$ 0 0
$$445$$ 8.22668 0.389982
$$446$$ −4.75963 −0.225375
$$447$$ 0 0
$$448$$ −31.3114 −1.47932
$$449$$ −4.62857 −0.218436 −0.109218 0.994018i $$-0.534835\pi$$
−0.109218 + 0.994018i $$0.534835\pi$$
$$450$$ 0 0
$$451$$ 13.5134 0.636322
$$452$$ 0.586289 0.0275767
$$453$$ 0 0
$$454$$ 13.5767 0.637185
$$455$$ 5.18966 0.243295
$$456$$ 0 0
$$457$$ 20.5672 0.962092 0.481046 0.876695i $$-0.340257\pi$$
0.481046 + 0.876695i $$0.340257\pi$$
$$458$$ −17.3878 −0.812477
$$459$$ 0 0
$$460$$ −1.12836 −0.0526098
$$461$$ −33.5725 −1.56363 −0.781813 0.623513i $$-0.785705\pi$$
−0.781813 + 0.623513i $$0.785705\pi$$
$$462$$ 0 0
$$463$$ 13.1310 0.610251 0.305126 0.952312i $$-0.401302\pi$$
0.305126 + 0.952312i $$0.401302\pi$$
$$464$$ 11.5263 0.535094
$$465$$ 0 0
$$466$$ −16.3277 −0.756366
$$467$$ 23.6307 1.09350 0.546750 0.837296i $$-0.315865\pi$$
0.546750 + 0.837296i $$0.315865\pi$$
$$468$$ 0 0
$$469$$ 33.3387 1.53944
$$470$$ −1.73205 −0.0798935
$$471$$ 0 0
$$472$$ −8.94356 −0.411661
$$473$$ −16.0097 −0.736128
$$474$$ 0 0
$$475$$ 24.8904 1.14205
$$476$$ 8.63506 0.395787
$$477$$ 0 0
$$478$$ 7.52259 0.344075
$$479$$ −5.88019 −0.268673 −0.134336 0.990936i $$-0.542890\pi$$
−0.134336 + 0.990936i $$0.542890\pi$$
$$480$$ 0 0
$$481$$ −10.6186 −0.484164
$$482$$ 10.9786 0.500061
$$483$$ 0 0
$$484$$ 1.13610 0.0516407
$$485$$ −4.51557 −0.205041
$$486$$ 0 0
$$487$$ 38.7965 1.75804 0.879020 0.476786i $$-0.158198\pi$$
0.879020 + 0.476786i $$0.158198\pi$$
$$488$$ 23.8004 1.07739
$$489$$ 0 0
$$490$$ 3.14290 0.141982
$$491$$ −37.5842 −1.69615 −0.848077 0.529873i $$-0.822239\pi$$
−0.848077 + 0.529873i $$0.822239\pi$$
$$492$$ 0 0
$$493$$ −25.4766 −1.14741
$$494$$ 21.9350 0.986904
$$495$$ 0 0
$$496$$ 6.16519 0.276825
$$497$$ 18.7459 0.840868
$$498$$ 0 0
$$499$$ 33.7520 1.51095 0.755473 0.655180i $$-0.227407\pi$$
0.755473 + 0.655180i $$0.227407\pi$$
$$500$$ −1.51968 −0.0679623
$$501$$ 0 0
$$502$$ 8.68779 0.387755
$$503$$ −18.7119 −0.834324 −0.417162 0.908832i $$-0.636975\pi$$
−0.417162 + 0.908832i $$0.636975\pi$$
$$504$$ 0 0
$$505$$ −0.929015 −0.0413406
$$506$$ 26.0076 1.15618
$$507$$ 0 0
$$508$$ 0.928081 0.0411770
$$509$$ 21.7682 0.964858 0.482429 0.875935i $$-0.339755\pi$$
0.482429 + 0.875935i $$0.339755\pi$$
$$510$$ 0 0
$$511$$ 5.49020 0.242872
$$512$$ 25.4026 1.12265
$$513$$ 0 0
$$514$$ −4.73473 −0.208840
$$515$$ 0.646635 0.0284942
$$516$$ 0 0
$$517$$ −8.38919 −0.368956
$$518$$ −14.6516 −0.643755
$$519$$ 0 0
$$520$$ −4.43376 −0.194433
$$521$$ −6.47643 −0.283738 −0.141869 0.989885i $$-0.545311\pi$$
−0.141869 + 0.989885i $$0.545311\pi$$
$$522$$ 0 0
$$523$$ −10.8726 −0.475425 −0.237712 0.971336i $$-0.576398\pi$$
−0.237712 + 0.971336i $$0.576398\pi$$
$$524$$ −1.01406 −0.0442995
$$525$$ 0 0
$$526$$ −4.67675 −0.203916
$$527$$ −13.6270 −0.593599
$$528$$ 0 0
$$529$$ 29.9537 1.30233
$$530$$ −5.03460 −0.218689
$$531$$ 0 0
$$532$$ −6.36009 −0.275745
$$533$$ 15.9963 0.692878
$$534$$ 0 0
$$535$$ −0.998245 −0.0431579
$$536$$ −28.4828 −1.23027
$$537$$ 0 0
$$538$$ 9.11051 0.392782
$$539$$ 15.2226 0.655686
$$540$$ 0 0
$$541$$ 24.6459 1.05961 0.529805 0.848120i $$-0.322265\pi$$
0.529805 + 0.848120i $$0.322265\pi$$
$$542$$ 24.4259 1.04918
$$543$$ 0 0
$$544$$ −13.6631 −0.585802
$$545$$ 5.13581 0.219994
$$546$$ 0 0
$$547$$ 30.9273 1.32235 0.661177 0.750230i $$-0.270057\pi$$
0.661177 + 0.750230i $$0.270057\pi$$
$$548$$ −1.61744 −0.0690934
$$549$$ 0 0
$$550$$ 17.1575 0.731596
$$551$$ 18.7646 0.799399
$$552$$ 0 0
$$553$$ 42.0378 1.78763
$$554$$ 17.8594 0.758775
$$555$$ 0 0
$$556$$ 2.77930 0.117869
$$557$$ 23.3627 0.989908 0.494954 0.868919i $$-0.335185\pi$$
0.494954 + 0.868919i $$0.335185\pi$$
$$558$$ 0 0
$$559$$ −18.9513 −0.801555
$$560$$ −5.02239 −0.212235
$$561$$ 0 0
$$562$$ −28.4466 −1.19995
$$563$$ 33.6877 1.41977 0.709884 0.704319i $$-0.248747\pi$$
0.709884 + 0.704319i $$0.248747\pi$$
$$564$$ 0 0
$$565$$ 0.753718 0.0317092
$$566$$ 30.5097 1.28242
$$567$$ 0 0
$$568$$ −16.0155 −0.671995
$$569$$ −30.7959 −1.29103 −0.645515 0.763748i $$-0.723357\pi$$
−0.645515 + 0.763748i $$0.723357\pi$$
$$570$$ 0 0
$$571$$ −29.8753 −1.25024 −0.625120 0.780528i $$-0.714950\pi$$
−0.625120 + 0.780528i $$0.714950\pi$$
$$572$$ −3.17734 −0.132851
$$573$$ 0 0
$$574$$ 22.0719 0.921264
$$575$$ 34.9340 1.45685
$$576$$ 0 0
$$577$$ 4.80747 0.200137 0.100069 0.994981i $$-0.468094\pi$$
0.100069 + 0.994981i $$0.468094\pi$$
$$578$$ −41.8488 −1.74068
$$579$$ 0 0
$$580$$ −0.561185 −0.0233019
$$581$$ −57.4221 −2.38227
$$582$$ 0 0
$$583$$ −24.3851 −1.00993
$$584$$ −4.69053 −0.194096
$$585$$ 0 0
$$586$$ −23.9195 −0.988106
$$587$$ 7.93761 0.327620 0.163810 0.986492i $$-0.447622\pi$$
0.163810 + 0.986492i $$0.447622\pi$$
$$588$$ 0 0
$$589$$ 10.0368 0.413561
$$590$$ −1.70114 −0.0700349
$$591$$ 0 0
$$592$$ 10.2763 0.422354
$$593$$ 36.2753 1.48965 0.744824 0.667261i $$-0.232533\pi$$
0.744824 + 0.667261i $$0.232533\pi$$
$$594$$ 0 0
$$595$$ 11.1010 0.455097
$$596$$ 7.01946 0.287528
$$597$$ 0 0
$$598$$ 30.7861 1.25894
$$599$$ 33.6450 1.37470 0.687349 0.726327i $$-0.258774\pi$$
0.687349 + 0.726327i $$0.258774\pi$$
$$600$$ 0 0
$$601$$ −2.97771 −0.121463 −0.0607317 0.998154i $$-0.519343\pi$$
−0.0607317 + 0.998154i $$0.519343\pi$$
$$602$$ −26.1492 −1.06576
$$603$$ 0 0
$$604$$ −2.32770 −0.0947126
$$605$$ 1.46054 0.0593793
$$606$$ 0 0
$$607$$ −15.6486 −0.635156 −0.317578 0.948232i $$-0.602870\pi$$
−0.317578 + 0.948232i $$0.602870\pi$$
$$608$$ 10.0635 0.408128
$$609$$ 0 0
$$610$$ 4.52704 0.183294
$$611$$ −9.93058 −0.401748
$$612$$ 0 0
$$613$$ −1.06687 −0.0430903 −0.0215452 0.999768i $$-0.506859\pi$$
−0.0215452 + 0.999768i $$0.506859\pi$$
$$614$$ −26.6717 −1.07638
$$615$$ 0 0
$$616$$ −29.6313 −1.19388
$$617$$ −13.0700 −0.526177 −0.263088 0.964772i $$-0.584741\pi$$
−0.263088 + 0.964772i $$0.584741\pi$$
$$618$$ 0 0
$$619$$ 20.5175 0.824670 0.412335 0.911032i $$-0.364713\pi$$
0.412335 + 0.911032i $$0.364713\pi$$
$$620$$ −0.300167 −0.0120550
$$621$$ 0 0
$$622$$ −26.2071 −1.05081
$$623$$ 65.0817 2.60744
$$624$$ 0 0
$$625$$ 22.0496 0.881985
$$626$$ 38.3626 1.53328
$$627$$ 0 0
$$628$$ −1.84936 −0.0737973
$$629$$ −22.7138 −0.905658
$$630$$ 0 0
$$631$$ −10.3122 −0.410523 −0.205261 0.978707i $$-0.565804\pi$$
−0.205261 + 0.978707i $$0.565804\pi$$
$$632$$ −35.9148 −1.42861
$$633$$ 0 0
$$634$$ −5.54933 −0.220392
$$635$$ 1.19312 0.0473475
$$636$$ 0 0
$$637$$ 18.0196 0.713963
$$638$$ 12.9348 0.512094
$$639$$ 0 0
$$640$$ 3.35504 0.132619
$$641$$ 3.50616 0.138485 0.0692426 0.997600i $$-0.477942\pi$$
0.0692426 + 0.997600i $$0.477942\pi$$
$$642$$ 0 0
$$643$$ 31.5517 1.24428 0.622139 0.782907i $$-0.286264\pi$$
0.622139 + 0.782907i $$0.286264\pi$$
$$644$$ −8.92647 −0.351752
$$645$$ 0 0
$$646$$ 46.9205 1.84606
$$647$$ −3.04628 −0.119762 −0.0598808 0.998206i $$-0.519072\pi$$
−0.0598808 + 0.998206i $$0.519072\pi$$
$$648$$ 0 0
$$649$$ −8.23947 −0.323428
$$650$$ 20.3099 0.796620
$$651$$ 0 0
$$652$$ −1.32501 −0.0518913
$$653$$ −30.7374 −1.20285 −0.601423 0.798931i $$-0.705399\pi$$
−0.601423 + 0.798931i $$0.705399\pi$$
$$654$$ 0 0
$$655$$ −1.30365 −0.0509379
$$656$$ −15.4808 −0.604422
$$657$$ 0 0
$$658$$ −13.7023 −0.534173
$$659$$ −33.1839 −1.29266 −0.646331 0.763057i $$-0.723698\pi$$
−0.646331 + 0.763057i $$0.723698\pi$$
$$660$$ 0 0
$$661$$ −14.9077 −0.579841 −0.289920 0.957051i $$-0.593629\pi$$
−0.289920 + 0.957051i $$0.593629\pi$$
$$662$$ 3.75248 0.145844
$$663$$ 0 0
$$664$$ 49.0583 1.90383
$$665$$ −8.17637 −0.317066
$$666$$ 0 0
$$667$$ 26.3364 1.01975
$$668$$ −3.17269 −0.122755
$$669$$ 0 0
$$670$$ −5.41767 −0.209303
$$671$$ 21.9267 0.846471
$$672$$ 0 0
$$673$$ 6.88888 0.265547 0.132773 0.991146i $$-0.457612\pi$$
0.132773 + 0.991146i $$0.457612\pi$$
$$674$$ −18.5656 −0.715122
$$675$$ 0 0
$$676$$ 0.753718 0.0289892
$$677$$ −13.1892 −0.506903 −0.253452 0.967348i $$-0.581566\pi$$
−0.253452 + 0.967348i $$0.581566\pi$$
$$678$$ 0 0
$$679$$ −35.7229 −1.37092
$$680$$ −9.48411 −0.363699
$$681$$ 0 0
$$682$$ 6.91859 0.264926
$$683$$ −3.37814 −0.129261 −0.0646305 0.997909i $$-0.520587\pi$$
−0.0646305 + 0.997909i $$0.520587\pi$$
$$684$$ 0 0
$$685$$ −2.07934 −0.0794473
$$686$$ −6.92171 −0.264272
$$687$$ 0 0
$$688$$ 18.3405 0.699225
$$689$$ −28.8655 −1.09969
$$690$$ 0 0
$$691$$ 23.4151 0.890752 0.445376 0.895344i $$-0.353070\pi$$
0.445376 + 0.895344i $$0.353070\pi$$
$$692$$ −2.10943 −0.0801884
$$693$$ 0 0
$$694$$ 1.86928 0.0709569
$$695$$ 3.57300 0.135532
$$696$$ 0 0
$$697$$ 34.2172 1.29607
$$698$$ 35.4584 1.34212
$$699$$ 0 0
$$700$$ −5.88888 −0.222579
$$701$$ 45.5001 1.71852 0.859258 0.511543i $$-0.170926\pi$$
0.859258 + 0.511543i $$0.170926\pi$$
$$702$$ 0 0
$$703$$ 16.7297 0.630972
$$704$$ 24.6448 0.928836
$$705$$ 0 0
$$706$$ −13.5398 −0.509578
$$707$$ −7.34948 −0.276406
$$708$$ 0 0
$$709$$ 38.6168 1.45028 0.725142 0.688599i $$-0.241774\pi$$
0.725142 + 0.688599i $$0.241774\pi$$
$$710$$ −3.04628 −0.114325
$$711$$ 0 0
$$712$$ −55.6023 −2.08378
$$713$$ 14.0868 0.527556
$$714$$ 0 0
$$715$$ −4.08471 −0.152760
$$716$$ −3.57060 −0.133440
$$717$$ 0 0
$$718$$ −18.9050 −0.705527
$$719$$ −49.3182 −1.83926 −0.919630 0.392786i $$-0.871511\pi$$
−0.919630 + 0.392786i $$0.871511\pi$$
$$720$$ 0 0
$$721$$ 5.11556 0.190513
$$722$$ −10.1330 −0.377111
$$723$$ 0 0
$$724$$ 8.03684 0.298687
$$725$$ 17.3744 0.645268
$$726$$ 0 0
$$727$$ −32.2945 −1.19774 −0.598868 0.800848i $$-0.704383\pi$$
−0.598868 + 0.800848i $$0.704383\pi$$
$$728$$ −35.0757 −1.29999
$$729$$ 0 0
$$730$$ −0.892178 −0.0330210
$$731$$ −40.5381 −1.49935
$$732$$ 0 0
$$733$$ 39.4662 1.45772 0.728858 0.684665i $$-0.240051\pi$$
0.728858 + 0.684665i $$0.240051\pi$$
$$734$$ 27.4635 1.01369
$$735$$ 0 0
$$736$$ 14.1242 0.520626
$$737$$ −26.2405 −0.966581
$$738$$ 0 0
$$739$$ 35.3090 1.29886 0.649432 0.760420i $$-0.275007\pi$$
0.649432 + 0.760420i $$0.275007\pi$$
$$740$$ −0.500327 −0.0183924
$$741$$ 0 0
$$742$$ −39.8289 −1.46217
$$743$$ −47.4106 −1.73933 −0.869663 0.493646i $$-0.835664\pi$$
−0.869663 + 0.493646i $$0.835664\pi$$
$$744$$ 0 0
$$745$$ 9.02404 0.330615
$$746$$ −29.0855 −1.06490
$$747$$ 0 0
$$748$$ −6.79654 −0.248506
$$749$$ −7.89716 −0.288556
$$750$$ 0 0
$$751$$ −8.92808 −0.325790 −0.162895 0.986643i $$-0.552083\pi$$
−0.162895 + 0.986643i $$0.552083\pi$$
$$752$$ 9.61051 0.350459
$$753$$ 0 0
$$754$$ 15.3114 0.557608
$$755$$ −2.99243 −0.108906
$$756$$ 0 0
$$757$$ −3.63816 −0.132231 −0.0661155 0.997812i $$-0.521061\pi$$
−0.0661155 + 0.997812i $$0.521061\pi$$
$$758$$ 21.9503 0.797270
$$759$$ 0 0
$$760$$ 6.98545 0.253389
$$761$$ −6.68880 −0.242469 −0.121234 0.992624i $$-0.538685\pi$$
−0.121234 + 0.992624i $$0.538685\pi$$
$$762$$ 0 0
$$763$$ 40.6296 1.47089
$$764$$ −5.09494 −0.184329
$$765$$ 0 0
$$766$$ 50.1266 1.81115
$$767$$ −9.75337 −0.352174
$$768$$ 0 0
$$769$$ −21.4712 −0.774272 −0.387136 0.922023i $$-0.626536\pi$$
−0.387136 + 0.922023i $$0.626536\pi$$
$$770$$ −5.63613 −0.203112
$$771$$ 0 0
$$772$$ −8.14115 −0.293006
$$773$$ −10.2442 −0.368457 −0.184228 0.982883i $$-0.558979\pi$$
−0.184228 + 0.982883i $$0.558979\pi$$
$$774$$ 0 0
$$775$$ 9.29322 0.333822
$$776$$ 30.5197 1.09559
$$777$$ 0 0
$$778$$ −13.1284 −0.470674
$$779$$ −25.2024 −0.902971
$$780$$ 0 0
$$781$$ −14.7547 −0.527963
$$782$$ 65.8535 2.35492
$$783$$ 0 0
$$784$$ −17.4388 −0.622815
$$785$$ −2.37749 −0.0848561
$$786$$ 0 0
$$787$$ 0.947682 0.0337812 0.0168906 0.999857i $$-0.494623\pi$$
0.0168906 + 0.999857i $$0.494623\pi$$
$$788$$ 3.13528 0.111690
$$789$$ 0 0
$$790$$ −6.83130 −0.243047
$$791$$ 5.96270 0.212009
$$792$$ 0 0
$$793$$ 25.9554 0.921704
$$794$$ 1.46929 0.0521431
$$795$$ 0 0
$$796$$ −0.904362 −0.0320543
$$797$$ 7.01071 0.248332 0.124166 0.992261i $$-0.460374\pi$$
0.124166 + 0.992261i $$0.460374\pi$$
$$798$$ 0 0
$$799$$ −21.2422 −0.751494
$$800$$ 9.31790 0.329437
$$801$$ 0 0
$$802$$ 26.4237 0.933052
$$803$$ −4.32127 −0.152494
$$804$$ 0 0
$$805$$ −11.4757 −0.404464
$$806$$ 8.18978 0.288473
$$807$$ 0 0
$$808$$ 6.27900 0.220894
$$809$$ −45.1028 −1.58573 −0.792866 0.609396i $$-0.791412\pi$$
−0.792866 + 0.609396i $$0.791412\pi$$
$$810$$ 0 0
$$811$$ 8.07285 0.283476 0.141738 0.989904i $$-0.454731\pi$$
0.141738 + 0.989904i $$0.454731\pi$$
$$812$$ −4.43956 −0.155798
$$813$$ 0 0
$$814$$ 11.5321 0.404200
$$815$$ −1.70340 −0.0596674
$$816$$ 0 0
$$817$$ 29.8580 1.04460
$$818$$ 10.4496 0.365362
$$819$$ 0 0
$$820$$ 0.753718 0.0263210
$$821$$ 6.26936 0.218802 0.109401 0.993998i $$-0.465107\pi$$
0.109401 + 0.993998i $$0.465107\pi$$
$$822$$ 0 0
$$823$$ −19.7929 −0.689938 −0.344969 0.938614i $$-0.612111\pi$$
−0.344969 + 0.938614i $$0.612111\pi$$
$$824$$ −4.37046 −0.152252
$$825$$ 0 0
$$826$$ −13.4578 −0.468257
$$827$$ −41.8003 −1.45354 −0.726769 0.686882i $$-0.758979\pi$$
−0.726769 + 0.686882i $$0.758979\pi$$
$$828$$ 0 0
$$829$$ 33.7279 1.17142 0.585710 0.810521i $$-0.300816\pi$$
0.585710 + 0.810521i $$0.300816\pi$$
$$830$$ 9.33130 0.323894
$$831$$ 0 0
$$832$$ 29.1729 1.01139
$$833$$ 38.5451 1.33551
$$834$$ 0 0
$$835$$ −4.07873 −0.141150
$$836$$ 5.00594 0.173134
$$837$$ 0 0
$$838$$ −45.9959 −1.58890
$$839$$ 29.2868 1.01109 0.505546 0.862800i $$-0.331291\pi$$
0.505546 + 0.862800i $$0.331291\pi$$
$$840$$ 0 0
$$841$$ −15.9017 −0.548334
$$842$$ 16.8653 0.581216
$$843$$ 0 0
$$844$$ 5.68004 0.195515
$$845$$ 0.968961 0.0333333
$$846$$ 0 0
$$847$$ 11.5544 0.397013
$$848$$ 27.9351 0.959296
$$849$$ 0 0
$$850$$ 43.4442 1.49012
$$851$$ 23.4803 0.804895
$$852$$ 0 0
$$853$$ 37.5981 1.28734 0.643668 0.765305i $$-0.277412\pi$$
0.643668 + 0.765305i $$0.277412\pi$$
$$854$$ 35.8136 1.22552
$$855$$ 0 0
$$856$$ 6.74691 0.230605
$$857$$ 29.0100 0.990961 0.495481 0.868619i $$-0.334992\pi$$
0.495481 + 0.868619i $$0.334992\pi$$
$$858$$ 0 0
$$859$$ −24.8675 −0.848469 −0.424235 0.905552i $$-0.639457\pi$$
−0.424235 + 0.905552i $$0.639457\pi$$
$$860$$ −0.892951 −0.0304494
$$861$$ 0 0
$$862$$ −12.1925 −0.415279
$$863$$ −42.4018 −1.44337 −0.721687 0.692219i $$-0.756633\pi$$
−0.721687 + 0.692219i $$0.756633\pi$$
$$864$$ 0 0
$$865$$ −2.71183 −0.0922050
$$866$$ −22.7069 −0.771611
$$867$$ 0 0
$$868$$ −2.37464 −0.0806004
$$869$$ −33.0874 −1.12241
$$870$$ 0 0
$$871$$ −31.0618 −1.05249
$$872$$ −34.7117 −1.17549
$$873$$ 0 0
$$874$$ −48.5039 −1.64067
$$875$$ −15.4556 −0.522493
$$876$$ 0 0
$$877$$ 12.0547 0.407058 0.203529 0.979069i $$-0.434759\pi$$
0.203529 + 0.979069i $$0.434759\pi$$
$$878$$ 22.7103 0.766436
$$879$$ 0 0
$$880$$ 3.95306 0.133258
$$881$$ 14.7827 0.498041 0.249020 0.968498i $$-0.419891\pi$$
0.249020 + 0.968498i $$0.419891\pi$$
$$882$$ 0 0
$$883$$ −25.8462 −0.869793 −0.434896 0.900480i $$-0.643215\pi$$
−0.434896 + 0.900480i $$0.643215\pi$$
$$884$$ −8.04531 −0.270593
$$885$$ 0 0
$$886$$ 16.7023 0.561126
$$887$$ −45.8560 −1.53969 −0.769846 0.638229i $$-0.779667\pi$$
−0.769846 + 0.638229i $$0.779667\pi$$
$$888$$ 0 0
$$889$$ 9.43882 0.316568
$$890$$ −10.5760 −0.354509
$$891$$ 0 0
$$892$$ −1.28581 −0.0430520
$$893$$ 15.6458 0.523566
$$894$$ 0 0
$$895$$ −4.59028 −0.153436
$$896$$ 26.5419 0.886701
$$897$$ 0 0
$$898$$ 5.95037 0.198566
$$899$$ 7.00605 0.233665
$$900$$ 0 0
$$901$$ −61.7452 −2.05703
$$902$$ −17.3725 −0.578442
$$903$$ 0 0
$$904$$ −5.09421 −0.169431
$$905$$ 10.3320 0.343446
$$906$$ 0 0
$$907$$ −11.7000 −0.388491 −0.194246 0.980953i $$-0.562226\pi$$
−0.194246 + 0.980953i $$0.562226\pi$$
$$908$$ 3.66772 0.121717
$$909$$ 0 0
$$910$$ −6.67169 −0.221165
$$911$$ −28.6753 −0.950054 −0.475027 0.879971i $$-0.657562\pi$$
−0.475027 + 0.879971i $$0.657562\pi$$
$$912$$ 0 0
$$913$$ 45.1962 1.49577
$$914$$ −26.4406 −0.874579
$$915$$ 0 0
$$916$$ −4.69728 −0.155203
$$917$$ −10.3133 −0.340574
$$918$$ 0 0
$$919$$ 4.33511 0.143002 0.0715011 0.997441i $$-0.477221\pi$$
0.0715011 + 0.997441i $$0.477221\pi$$
$$920$$ 9.80418 0.323234
$$921$$ 0 0
$$922$$ 43.1599 1.42140
$$923$$ −17.4656 −0.574888
$$924$$ 0 0
$$925$$ 15.4902 0.509315
$$926$$ −16.8809 −0.554742
$$927$$ 0 0
$$928$$ 7.02465 0.230596
$$929$$ 12.8909 0.422937 0.211468 0.977385i $$-0.432175\pi$$
0.211468 + 0.977385i $$0.432175\pi$$
$$930$$ 0 0
$$931$$ −28.3901 −0.930449
$$932$$ −4.41090 −0.144484
$$933$$ 0 0
$$934$$ −30.3791 −0.994034
$$935$$ −8.73746 −0.285746
$$936$$ 0 0
$$937$$ −53.2080 −1.73823 −0.869115 0.494610i $$-0.835311\pi$$
−0.869115 + 0.494610i $$0.835311\pi$$
$$938$$ −42.8595 −1.39941
$$939$$ 0 0
$$940$$ −0.467911 −0.0152616
$$941$$ −33.4075 −1.08905 −0.544526 0.838744i $$-0.683290\pi$$
−0.544526 + 0.838744i $$0.683290\pi$$
$$942$$ 0 0
$$943$$ −35.3719 −1.15187
$$944$$ 9.43901 0.307214
$$945$$ 0 0
$$946$$ 20.5817 0.669169
$$947$$ −15.5048 −0.503837 −0.251918 0.967748i $$-0.581061\pi$$
−0.251918 + 0.967748i $$0.581061\pi$$
$$948$$ 0 0
$$949$$ −5.11524 −0.166048
$$950$$ −31.9985 −1.03817
$$951$$ 0 0
$$952$$ −75.0292 −2.43171
$$953$$ 22.5049 0.729004 0.364502 0.931203i $$-0.381239\pi$$
0.364502 + 0.931203i $$0.381239\pi$$
$$954$$ 0 0
$$955$$ −6.54993 −0.211951
$$956$$ 2.03222 0.0657266
$$957$$ 0 0
$$958$$ 7.55943 0.244234
$$959$$ −16.4497 −0.531189
$$960$$ 0 0
$$961$$ −27.2526 −0.879116
$$962$$ 13.6510 0.440124
$$963$$ 0 0
$$964$$ 2.96585 0.0955237
$$965$$ −10.4661 −0.336914
$$966$$ 0 0
$$967$$ −41.8084 −1.34447 −0.672234 0.740339i $$-0.734665\pi$$
−0.672234 + 0.740339i $$0.734665\pi$$
$$968$$ −9.87144 −0.317280
$$969$$ 0 0
$$970$$ 5.80510 0.186391
$$971$$ 35.8662 1.15100 0.575501 0.817801i $$-0.304807\pi$$
0.575501 + 0.817801i $$0.304807\pi$$
$$972$$ 0 0
$$973$$ 28.2662 0.906173
$$974$$ −49.8759 −1.59813
$$975$$ 0 0
$$976$$ −25.1189 −0.804035
$$977$$ 14.0501 0.449502 0.224751 0.974416i $$-0.427843\pi$$
0.224751 + 0.974416i $$0.427843\pi$$
$$978$$ 0 0
$$979$$ −51.2249 −1.63716
$$980$$ 0.849051 0.0271219
$$981$$ 0 0
$$982$$ 48.3174 1.54187
$$983$$ −17.6891 −0.564196 −0.282098 0.959386i $$-0.591030\pi$$
−0.282098 + 0.959386i $$0.591030\pi$$
$$984$$ 0 0
$$985$$ 4.03064 0.128427
$$986$$ 32.7521 1.04304
$$987$$ 0 0
$$988$$ 5.92572 0.188522
$$989$$ 41.9062 1.33254
$$990$$ 0 0
$$991$$ 32.8958 1.04497 0.522485 0.852649i $$-0.325005\pi$$
0.522485 + 0.852649i $$0.325005\pi$$
$$992$$ 3.75736 0.119296
$$993$$ 0 0
$$994$$ −24.0993 −0.764382
$$995$$ −1.16263 −0.0368577
$$996$$ 0 0
$$997$$ 20.0068 0.633622 0.316811 0.948489i $$-0.397388\pi$$
0.316811 + 0.948489i $$0.397388\pi$$
$$998$$ −43.3907 −1.37351
$$999$$ 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.c.1.2 6
3.2 odd 2 inner 729.2.a.c.1.5 yes 6
9.2 odd 6 729.2.c.c.244.2 12
9.4 even 3 729.2.c.c.487.5 12
9.5 odd 6 729.2.c.c.487.2 12
9.7 even 3 729.2.c.c.244.5 12
27.2 odd 18 729.2.e.q.568.1 12
27.4 even 9 729.2.e.m.406.1 12
27.5 odd 18 729.2.e.r.649.2 12
27.7 even 9 729.2.e.m.325.1 12
27.11 odd 18 729.2.e.r.82.2 12
27.13 even 9 729.2.e.q.163.2 12
27.14 odd 18 729.2.e.q.163.1 12
27.16 even 9 729.2.e.r.82.1 12
27.20 odd 18 729.2.e.m.325.2 12
27.22 even 9 729.2.e.r.649.1 12
27.23 odd 18 729.2.e.m.406.2 12
27.25 even 9 729.2.e.q.568.2 12

By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.c.1.2 6 1.1 even 1 trivial
729.2.a.c.1.5 yes 6 3.2 odd 2 inner
729.2.c.c.244.2 12 9.2 odd 6
729.2.c.c.244.5 12 9.7 even 3
729.2.c.c.487.2 12 9.5 odd 6
729.2.c.c.487.5 12 9.4 even 3
729.2.e.m.325.1 12 27.7 even 9
729.2.e.m.325.2 12 27.20 odd 18
729.2.e.m.406.1 12 27.4 even 9
729.2.e.m.406.2 12 27.23 odd 18
729.2.e.q.163.1 12 27.14 odd 18
729.2.e.q.163.2 12 27.13 even 9
729.2.e.q.568.1 12 27.2 odd 18
729.2.e.q.568.2 12 27.25 even 9
729.2.e.r.82.1 12 27.16 even 9
729.2.e.r.82.2 12 27.11 odd 18
729.2.e.r.649.1 12 27.22 even 9
729.2.e.r.649.2 12 27.5 odd 18