Properties

 Label 243.2.a.d.1.1 Level $243$ Weight $2$ Character 243.1 Self dual yes Analytic conductor $1.940$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 6$$ x^2 - 6 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 1.1 Root $$-2.44949$$ 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)

 $$f(q)$$ $$=$$ $$q-2.44949 q^{2} +4.00000 q^{4} +2.44949 q^{5} +2.00000 q^{7} -4.89898 q^{8} +O(q^{10})$$ $$q-2.44949 q^{2} +4.00000 q^{4} +2.44949 q^{5} +2.00000 q^{7} -4.89898 q^{8} -6.00000 q^{10} -2.44949 q^{11} -1.00000 q^{13} -4.89898 q^{14} +4.00000 q^{16} +7.34847 q^{17} -1.00000 q^{19} +9.79796 q^{20} +6.00000 q^{22} +2.44949 q^{23} +1.00000 q^{25} +2.44949 q^{26} +8.00000 q^{28} +4.89898 q^{29} -1.00000 q^{31} -18.0000 q^{34} +4.89898 q^{35} +8.00000 q^{37} +2.44949 q^{38} -12.0000 q^{40} -4.89898 q^{41} +11.0000 q^{43} -9.79796 q^{44} -6.00000 q^{46} -9.79796 q^{47} -3.00000 q^{49} -2.44949 q^{50} -4.00000 q^{52} -7.34847 q^{53} -6.00000 q^{55} -9.79796 q^{56} -12.0000 q^{58} +2.44949 q^{59} +5.00000 q^{61} +2.44949 q^{62} -8.00000 q^{64} -2.44949 q^{65} -7.00000 q^{67} +29.3939 q^{68} -12.0000 q^{70} -7.34847 q^{71} +11.0000 q^{73} -19.5959 q^{74} -4.00000 q^{76} -4.89898 q^{77} -7.00000 q^{79} +9.79796 q^{80} +12.0000 q^{82} +12.2474 q^{83} +18.0000 q^{85} -26.9444 q^{86} +12.0000 q^{88} -2.00000 q^{91} +9.79796 q^{92} +24.0000 q^{94} -2.44949 q^{95} -7.00000 q^{97} +7.34847 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{4} + 4 q^{7}+O(q^{10})$$ 2 * q + 8 * q^4 + 4 * q^7 $$2 q + 8 q^{4} + 4 q^{7} - 12 q^{10} - 2 q^{13} + 8 q^{16} - 2 q^{19} + 12 q^{22} + 2 q^{25} + 16 q^{28} - 2 q^{31} - 36 q^{34} + 16 q^{37} - 24 q^{40} + 22 q^{43} - 12 q^{46} - 6 q^{49} - 8 q^{52} - 12 q^{55} - 24 q^{58} + 10 q^{61} - 16 q^{64} - 14 q^{67} - 24 q^{70} + 22 q^{73} - 8 q^{76} - 14 q^{79} + 24 q^{82} + 36 q^{85} + 24 q^{88} - 4 q^{91} + 48 q^{94} - 14 q^{97}+O(q^{100})$$ 2 * q + 8 * q^4 + 4 * q^7 - 12 * q^10 - 2 * q^13 + 8 * q^16 - 2 * q^19 + 12 * q^22 + 2 * q^25 + 16 * q^28 - 2 * q^31 - 36 * q^34 + 16 * q^37 - 24 * q^40 + 22 * q^43 - 12 * q^46 - 6 * q^49 - 8 * q^52 - 12 * q^55 - 24 * q^58 + 10 * q^61 - 16 * q^64 - 14 * q^67 - 24 * q^70 + 22 * q^73 - 8 * q^76 - 14 * q^79 + 24 * q^82 + 36 * q^85 + 24 * q^88 - 4 * q^91 + 48 * q^94 - 14 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.44949 −1.73205 −0.866025 0.500000i $$-0.833333\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$3$$ 0 0
$$4$$ 4.00000 2.00000
$$5$$ 2.44949 1.09545 0.547723 0.836660i $$-0.315495\pi$$
0.547723 + 0.836660i $$0.315495\pi$$
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ −4.89898 −1.73205
$$9$$ 0 0
$$10$$ −6.00000 −1.89737
$$11$$ −2.44949 −0.738549 −0.369274 0.929320i $$-0.620394\pi$$
−0.369274 + 0.929320i $$0.620394\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ −4.89898 −1.30931
$$15$$ 0 0
$$16$$ 4.00000 1.00000
$$17$$ 7.34847 1.78227 0.891133 0.453743i $$-0.149911\pi$$
0.891133 + 0.453743i $$0.149911\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 9.79796 2.19089
$$21$$ 0 0
$$22$$ 6.00000 1.27920
$$23$$ 2.44949 0.510754 0.255377 0.966842i $$-0.417800\pi$$
0.255377 + 0.966842i $$0.417800\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 2.44949 0.480384
$$27$$ 0 0
$$28$$ 8.00000 1.51186
$$29$$ 4.89898 0.909718 0.454859 0.890564i $$-0.349690\pi$$
0.454859 + 0.890564i $$0.349690\pi$$
$$30$$ 0 0
$$31$$ −1.00000 −0.179605 −0.0898027 0.995960i $$-0.528624\pi$$
−0.0898027 + 0.995960i $$0.528624\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ −18.0000 −3.08697
$$35$$ 4.89898 0.828079
$$36$$ 0 0
$$37$$ 8.00000 1.31519 0.657596 0.753371i $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ 2.44949 0.397360
$$39$$ 0 0
$$40$$ −12.0000 −1.89737
$$41$$ −4.89898 −0.765092 −0.382546 0.923936i $$-0.624953\pi$$
−0.382546 + 0.923936i $$0.624953\pi$$
$$42$$ 0 0
$$43$$ 11.0000 1.67748 0.838742 0.544529i $$-0.183292\pi$$
0.838742 + 0.544529i $$0.183292\pi$$
$$44$$ −9.79796 −1.47710
$$45$$ 0 0
$$46$$ −6.00000 −0.884652
$$47$$ −9.79796 −1.42918 −0.714590 0.699544i $$-0.753387\pi$$
−0.714590 + 0.699544i $$0.753387\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ −2.44949 −0.346410
$$51$$ 0 0
$$52$$ −4.00000 −0.554700
$$53$$ −7.34847 −1.00939 −0.504695 0.863298i $$-0.668395\pi$$
−0.504695 + 0.863298i $$0.668395\pi$$
$$54$$ 0 0
$$55$$ −6.00000 −0.809040
$$56$$ −9.79796 −1.30931
$$57$$ 0 0
$$58$$ −12.0000 −1.57568
$$59$$ 2.44949 0.318896 0.159448 0.987206i $$-0.449029\pi$$
0.159448 + 0.987206i $$0.449029\pi$$
$$60$$ 0 0
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ 2.44949 0.311086
$$63$$ 0 0
$$64$$ −8.00000 −1.00000
$$65$$ −2.44949 −0.303822
$$66$$ 0 0
$$67$$ −7.00000 −0.855186 −0.427593 0.903971i $$-0.640638\pi$$
−0.427593 + 0.903971i $$0.640638\pi$$
$$68$$ 29.3939 3.56453
$$69$$ 0 0
$$70$$ −12.0000 −1.43427
$$71$$ −7.34847 −0.872103 −0.436051 0.899922i $$-0.643623\pi$$
−0.436051 + 0.899922i $$0.643623\pi$$
$$72$$ 0 0
$$73$$ 11.0000 1.28745 0.643726 0.765256i $$-0.277388\pi$$
0.643726 + 0.765256i $$0.277388\pi$$
$$74$$ −19.5959 −2.27798
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ −4.89898 −0.558291
$$78$$ 0 0
$$79$$ −7.00000 −0.787562 −0.393781 0.919204i $$-0.628833\pi$$
−0.393781 + 0.919204i $$0.628833\pi$$
$$80$$ 9.79796 1.09545
$$81$$ 0 0
$$82$$ 12.0000 1.32518
$$83$$ 12.2474 1.34433 0.672166 0.740400i $$-0.265364\pi$$
0.672166 + 0.740400i $$0.265364\pi$$
$$84$$ 0 0
$$85$$ 18.0000 1.95237
$$86$$ −26.9444 −2.90549
$$87$$ 0 0
$$88$$ 12.0000 1.27920
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ 9.79796 1.02151
$$93$$ 0 0
$$94$$ 24.0000 2.47541
$$95$$ −2.44949 −0.251312
$$96$$ 0 0
$$97$$ −7.00000 −0.710742 −0.355371 0.934725i $$-0.615646\pi$$
−0.355371 + 0.934725i $$0.615646\pi$$
$$98$$ 7.34847 0.742307
$$99$$ 0 0
$$100$$ 4.00000 0.400000
$$101$$ 4.89898 0.487467 0.243733 0.969842i $$-0.421628\pi$$
0.243733 + 0.969842i $$0.421628\pi$$
$$102$$ 0 0
$$103$$ −7.00000 −0.689730 −0.344865 0.938652i $$-0.612075\pi$$
−0.344865 + 0.938652i $$0.612075\pi$$
$$104$$ 4.89898 0.480384
$$105$$ 0 0
$$106$$ 18.0000 1.74831
$$107$$ −14.6969 −1.42081 −0.710403 0.703795i $$-0.751487\pi$$
−0.710403 + 0.703795i $$0.751487\pi$$
$$108$$ 0 0
$$109$$ −1.00000 −0.0957826 −0.0478913 0.998853i $$-0.515250\pi$$
−0.0478913 + 0.998853i $$0.515250\pi$$
$$110$$ 14.6969 1.40130
$$111$$ 0 0
$$112$$ 8.00000 0.755929
$$113$$ 9.79796 0.921714 0.460857 0.887474i $$-0.347542\pi$$
0.460857 + 0.887474i $$0.347542\pi$$
$$114$$ 0 0
$$115$$ 6.00000 0.559503
$$116$$ 19.5959 1.81944
$$117$$ 0 0
$$118$$ −6.00000 −0.552345
$$119$$ 14.6969 1.34727
$$120$$ 0 0
$$121$$ −5.00000 −0.454545
$$122$$ −12.2474 −1.10883
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ −9.79796 −0.876356
$$126$$ 0 0
$$127$$ −19.0000 −1.68598 −0.842989 0.537931i $$-0.819206\pi$$
−0.842989 + 0.537931i $$0.819206\pi$$
$$128$$ 19.5959 1.73205
$$129$$ 0 0
$$130$$ 6.00000 0.526235
$$131$$ −12.2474 −1.07006 −0.535032 0.844832i $$-0.679701\pi$$
−0.535032 + 0.844832i $$0.679701\pi$$
$$132$$ 0 0
$$133$$ −2.00000 −0.173422
$$134$$ 17.1464 1.48123
$$135$$ 0 0
$$136$$ −36.0000 −3.08697
$$137$$ −9.79796 −0.837096 −0.418548 0.908195i $$-0.637461\pi$$
−0.418548 + 0.908195i $$0.637461\pi$$
$$138$$ 0 0
$$139$$ −10.0000 −0.848189 −0.424094 0.905618i $$-0.639408\pi$$
−0.424094 + 0.905618i $$0.639408\pi$$
$$140$$ 19.5959 1.65616
$$141$$ 0 0
$$142$$ 18.0000 1.51053
$$143$$ 2.44949 0.204837
$$144$$ 0 0
$$145$$ 12.0000 0.996546
$$146$$ −26.9444 −2.22993
$$147$$ 0 0
$$148$$ 32.0000 2.63038
$$149$$ −12.2474 −1.00335 −0.501675 0.865056i $$-0.667283\pi$$
−0.501675 + 0.865056i $$0.667283\pi$$
$$150$$ 0 0
$$151$$ 5.00000 0.406894 0.203447 0.979086i $$-0.434786\pi$$
0.203447 + 0.979086i $$0.434786\pi$$
$$152$$ 4.89898 0.397360
$$153$$ 0 0
$$154$$ 12.0000 0.966988
$$155$$ −2.44949 −0.196748
$$156$$ 0 0
$$157$$ 17.0000 1.35675 0.678374 0.734717i $$-0.262685\pi$$
0.678374 + 0.734717i $$0.262685\pi$$
$$158$$ 17.1464 1.36410
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 4.89898 0.386094
$$162$$ 0 0
$$163$$ −10.0000 −0.783260 −0.391630 0.920123i $$-0.628089\pi$$
−0.391630 + 0.920123i $$0.628089\pi$$
$$164$$ −19.5959 −1.53018
$$165$$ 0 0
$$166$$ −30.0000 −2.32845
$$167$$ −4.89898 −0.379094 −0.189547 0.981872i $$-0.560702\pi$$
−0.189547 + 0.981872i $$0.560702\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ −44.0908 −3.38161
$$171$$ 0 0
$$172$$ 44.0000 3.35497
$$173$$ −9.79796 −0.744925 −0.372463 0.928047i $$-0.621486\pi$$
−0.372463 + 0.928047i $$0.621486\pi$$
$$174$$ 0 0
$$175$$ 2.00000 0.151186
$$176$$ −9.79796 −0.738549
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 14.6969 1.09850 0.549250 0.835658i $$-0.314913\pi$$
0.549250 + 0.835658i $$0.314913\pi$$
$$180$$ 0 0
$$181$$ 8.00000 0.594635 0.297318 0.954779i $$-0.403908\pi$$
0.297318 + 0.954779i $$0.403908\pi$$
$$182$$ 4.89898 0.363137
$$183$$ 0 0
$$184$$ −12.0000 −0.884652
$$185$$ 19.5959 1.44072
$$186$$ 0 0
$$187$$ −18.0000 −1.31629
$$188$$ −39.1918 −2.85836
$$189$$ 0 0
$$190$$ 6.00000 0.435286
$$191$$ −9.79796 −0.708955 −0.354478 0.935064i $$-0.615341\pi$$
−0.354478 + 0.935064i $$0.615341\pi$$
$$192$$ 0 0
$$193$$ 11.0000 0.791797 0.395899 0.918294i $$-0.370433\pi$$
0.395899 + 0.918294i $$0.370433\pi$$
$$194$$ 17.1464 1.23104
$$195$$ 0 0
$$196$$ −12.0000 −0.857143
$$197$$ −14.6969 −1.04711 −0.523557 0.851991i $$-0.675395\pi$$
−0.523557 + 0.851991i $$0.675395\pi$$
$$198$$ 0 0
$$199$$ −1.00000 −0.0708881 −0.0354441 0.999372i $$-0.511285\pi$$
−0.0354441 + 0.999372i $$0.511285\pi$$
$$200$$ −4.89898 −0.346410
$$201$$ 0 0
$$202$$ −12.0000 −0.844317
$$203$$ 9.79796 0.687682
$$204$$ 0 0
$$205$$ −12.0000 −0.838116
$$206$$ 17.1464 1.19465
$$207$$ 0 0
$$208$$ −4.00000 −0.277350
$$209$$ 2.44949 0.169435
$$210$$ 0 0
$$211$$ −1.00000 −0.0688428 −0.0344214 0.999407i $$-0.510959\pi$$
−0.0344214 + 0.999407i $$0.510959\pi$$
$$212$$ −29.3939 −2.01878
$$213$$ 0 0
$$214$$ 36.0000 2.46091
$$215$$ 26.9444 1.83759
$$216$$ 0 0
$$217$$ −2.00000 −0.135769
$$218$$ 2.44949 0.165900
$$219$$ 0 0
$$220$$ −24.0000 −1.61808
$$221$$ −7.34847 −0.494312
$$222$$ 0 0
$$223$$ −7.00000 −0.468755 −0.234377 0.972146i $$-0.575305\pi$$
−0.234377 + 0.972146i $$0.575305\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −24.0000 −1.59646
$$227$$ −9.79796 −0.650313 −0.325157 0.945660i $$-0.605417\pi$$
−0.325157 + 0.945660i $$0.605417\pi$$
$$228$$ 0 0
$$229$$ −1.00000 −0.0660819 −0.0330409 0.999454i $$-0.510519\pi$$
−0.0330409 + 0.999454i $$0.510519\pi$$
$$230$$ −14.6969 −0.969087
$$231$$ 0 0
$$232$$ −24.0000 −1.57568
$$233$$ 7.34847 0.481414 0.240707 0.970598i $$-0.422621\pi$$
0.240707 + 0.970598i $$0.422621\pi$$
$$234$$ 0 0
$$235$$ −24.0000 −1.56559
$$236$$ 9.79796 0.637793
$$237$$ 0 0
$$238$$ −36.0000 −2.33353
$$239$$ 2.44949 0.158444 0.0792222 0.996857i $$-0.474756\pi$$
0.0792222 + 0.996857i $$0.474756\pi$$
$$240$$ 0 0
$$241$$ −16.0000 −1.03065 −0.515325 0.856995i $$-0.672329\pi$$
−0.515325 + 0.856995i $$0.672329\pi$$
$$242$$ 12.2474 0.787296
$$243$$ 0 0
$$244$$ 20.0000 1.28037
$$245$$ −7.34847 −0.469476
$$246$$ 0 0
$$247$$ 1.00000 0.0636285
$$248$$ 4.89898 0.311086
$$249$$ 0 0
$$250$$ 24.0000 1.51789
$$251$$ 7.34847 0.463831 0.231916 0.972736i $$-0.425501\pi$$
0.231916 + 0.972736i $$0.425501\pi$$
$$252$$ 0 0
$$253$$ −6.00000 −0.377217
$$254$$ 46.5403 2.92020
$$255$$ 0 0
$$256$$ −32.0000 −2.00000
$$257$$ 17.1464 1.06956 0.534782 0.844990i $$-0.320394\pi$$
0.534782 + 0.844990i $$0.320394\pi$$
$$258$$ 0 0
$$259$$ 16.0000 0.994192
$$260$$ −9.79796 −0.607644
$$261$$ 0 0
$$262$$ 30.0000 1.85341
$$263$$ 26.9444 1.66146 0.830731 0.556674i $$-0.187923\pi$$
0.830731 + 0.556674i $$0.187923\pi$$
$$264$$ 0 0
$$265$$ −18.0000 −1.10573
$$266$$ 4.89898 0.300376
$$267$$ 0 0
$$268$$ −28.0000 −1.71037
$$269$$ 22.0454 1.34413 0.672066 0.740491i $$-0.265407\pi$$
0.672066 + 0.740491i $$0.265407\pi$$
$$270$$ 0 0
$$271$$ −7.00000 −0.425220 −0.212610 0.977137i $$-0.568196\pi$$
−0.212610 + 0.977137i $$0.568196\pi$$
$$272$$ 29.3939 1.78227
$$273$$ 0 0
$$274$$ 24.0000 1.44989
$$275$$ −2.44949 −0.147710
$$276$$ 0 0
$$277$$ 11.0000 0.660926 0.330463 0.943819i $$-0.392795\pi$$
0.330463 + 0.943819i $$0.392795\pi$$
$$278$$ 24.4949 1.46911
$$279$$ 0 0
$$280$$ −24.0000 −1.43427
$$281$$ 12.2474 0.730622 0.365311 0.930886i $$-0.380963\pi$$
0.365311 + 0.930886i $$0.380963\pi$$
$$282$$ 0 0
$$283$$ 17.0000 1.01055 0.505273 0.862960i $$-0.331392\pi$$
0.505273 + 0.862960i $$0.331392\pi$$
$$284$$ −29.3939 −1.74421
$$285$$ 0 0
$$286$$ −6.00000 −0.354787
$$287$$ −9.79796 −0.578355
$$288$$ 0 0
$$289$$ 37.0000 2.17647
$$290$$ −29.3939 −1.72607
$$291$$ 0 0
$$292$$ 44.0000 2.57491
$$293$$ −4.89898 −0.286201 −0.143101 0.989708i $$-0.545707\pi$$
−0.143101 + 0.989708i $$0.545707\pi$$
$$294$$ 0 0
$$295$$ 6.00000 0.349334
$$296$$ −39.1918 −2.27798
$$297$$ 0 0
$$298$$ 30.0000 1.73785
$$299$$ −2.44949 −0.141658
$$300$$ 0 0
$$301$$ 22.0000 1.26806
$$302$$ −12.2474 −0.704761
$$303$$ 0 0
$$304$$ −4.00000 −0.229416
$$305$$ 12.2474 0.701287
$$306$$ 0 0
$$307$$ 2.00000 0.114146 0.0570730 0.998370i $$-0.481823\pi$$
0.0570730 + 0.998370i $$0.481823\pi$$
$$308$$ −19.5959 −1.11658
$$309$$ 0 0
$$310$$ 6.00000 0.340777
$$311$$ 24.4949 1.38898 0.694489 0.719503i $$-0.255630\pi$$
0.694489 + 0.719503i $$0.255630\pi$$
$$312$$ 0 0
$$313$$ −16.0000 −0.904373 −0.452187 0.891923i $$-0.649356\pi$$
−0.452187 + 0.891923i $$0.649356\pi$$
$$314$$ −41.6413 −2.34996
$$315$$ 0 0
$$316$$ −28.0000 −1.57512
$$317$$ −9.79796 −0.550308 −0.275154 0.961400i $$-0.588729\pi$$
−0.275154 + 0.961400i $$0.588729\pi$$
$$318$$ 0 0
$$319$$ −12.0000 −0.671871
$$320$$ −19.5959 −1.09545
$$321$$ 0 0
$$322$$ −12.0000 −0.668734
$$323$$ −7.34847 −0.408880
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ 24.4949 1.35665
$$327$$ 0 0
$$328$$ 24.0000 1.32518
$$329$$ −19.5959 −1.08036
$$330$$ 0 0
$$331$$ −7.00000 −0.384755 −0.192377 0.981321i $$-0.561620\pi$$
−0.192377 + 0.981321i $$0.561620\pi$$
$$332$$ 48.9898 2.68866
$$333$$ 0 0
$$334$$ 12.0000 0.656611
$$335$$ −17.1464 −0.936809
$$336$$ 0 0
$$337$$ −28.0000 −1.52526 −0.762629 0.646837i $$-0.776092\pi$$
−0.762629 + 0.646837i $$0.776092\pi$$
$$338$$ 29.3939 1.59882
$$339$$ 0 0
$$340$$ 72.0000 3.90475
$$341$$ 2.44949 0.132647
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ −53.8888 −2.90549
$$345$$ 0 0
$$346$$ 24.0000 1.29025
$$347$$ 24.4949 1.31495 0.657477 0.753474i $$-0.271623\pi$$
0.657477 + 0.753474i $$0.271623\pi$$
$$348$$ 0 0
$$349$$ 20.0000 1.07058 0.535288 0.844670i $$-0.320203\pi$$
0.535288 + 0.844670i $$0.320203\pi$$
$$350$$ −4.89898 −0.261861
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −2.44949 −0.130373 −0.0651866 0.997873i $$-0.520764\pi$$
−0.0651866 + 0.997873i $$0.520764\pi$$
$$354$$ 0 0
$$355$$ −18.0000 −0.955341
$$356$$ 0 0
$$357$$ 0 0
$$358$$ −36.0000 −1.90266
$$359$$ −29.3939 −1.55135 −0.775675 0.631133i $$-0.782590\pi$$
−0.775675 + 0.631133i $$0.782590\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ −19.5959 −1.02994
$$363$$ 0 0
$$364$$ −8.00000 −0.419314
$$365$$ 26.9444 1.41033
$$366$$ 0 0
$$367$$ 5.00000 0.260998 0.130499 0.991448i $$-0.458342\pi$$
0.130499 + 0.991448i $$0.458342\pi$$
$$368$$ 9.79796 0.510754
$$369$$ 0 0
$$370$$ −48.0000 −2.49540
$$371$$ −14.6969 −0.763027
$$372$$ 0 0
$$373$$ 35.0000 1.81223 0.906116 0.423030i $$-0.139034\pi$$
0.906116 + 0.423030i $$0.139034\pi$$
$$374$$ 44.0908 2.27988
$$375$$ 0 0
$$376$$ 48.0000 2.47541
$$377$$ −4.89898 −0.252310
$$378$$ 0 0
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ −9.79796 −0.502625
$$381$$ 0 0
$$382$$ 24.0000 1.22795
$$383$$ −34.2929 −1.75228 −0.876142 0.482054i $$-0.839891\pi$$
−0.876142 + 0.482054i $$0.839891\pi$$
$$384$$ 0 0
$$385$$ −12.0000 −0.611577
$$386$$ −26.9444 −1.37143
$$387$$ 0 0
$$388$$ −28.0000 −1.42148
$$389$$ 26.9444 1.36613 0.683067 0.730355i $$-0.260646\pi$$
0.683067 + 0.730355i $$0.260646\pi$$
$$390$$ 0 0
$$391$$ 18.0000 0.910299
$$392$$ 14.6969 0.742307
$$393$$ 0 0
$$394$$ 36.0000 1.81365
$$395$$ −17.1464 −0.862730
$$396$$ 0 0
$$397$$ −1.00000 −0.0501886 −0.0250943 0.999685i $$-0.507989\pi$$
−0.0250943 + 0.999685i $$0.507989\pi$$
$$398$$ 2.44949 0.122782
$$399$$ 0 0
$$400$$ 4.00000 0.200000
$$401$$ −34.2929 −1.71250 −0.856252 0.516559i $$-0.827213\pi$$
−0.856252 + 0.516559i $$0.827213\pi$$
$$402$$ 0 0
$$403$$ 1.00000 0.0498135
$$404$$ 19.5959 0.974933
$$405$$ 0 0
$$406$$ −24.0000 −1.19110
$$407$$ −19.5959 −0.971334
$$408$$ 0 0
$$409$$ −28.0000 −1.38451 −0.692255 0.721653i $$-0.743383\pi$$
−0.692255 + 0.721653i $$0.743383\pi$$
$$410$$ 29.3939 1.45166
$$411$$ 0 0
$$412$$ −28.0000 −1.37946
$$413$$ 4.89898 0.241063
$$414$$ 0 0
$$415$$ 30.0000 1.47264
$$416$$ 0 0
$$417$$ 0 0
$$418$$ −6.00000 −0.293470
$$419$$ −34.2929 −1.67532 −0.837658 0.546195i $$-0.816076\pi$$
−0.837658 + 0.546195i $$0.816076\pi$$
$$420$$ 0 0
$$421$$ 2.00000 0.0974740 0.0487370 0.998812i $$-0.484480\pi$$
0.0487370 + 0.998812i $$0.484480\pi$$
$$422$$ 2.44949 0.119239
$$423$$ 0 0
$$424$$ 36.0000 1.74831
$$425$$ 7.34847 0.356453
$$426$$ 0 0
$$427$$ 10.0000 0.483934
$$428$$ −58.7878 −2.84161
$$429$$ 0 0
$$430$$ −66.0000 −3.18280
$$431$$ 7.34847 0.353963 0.176982 0.984214i $$-0.443367\pi$$
0.176982 + 0.984214i $$0.443367\pi$$
$$432$$ 0 0
$$433$$ 17.0000 0.816968 0.408484 0.912766i $$-0.366058\pi$$
0.408484 + 0.912766i $$0.366058\pi$$
$$434$$ 4.89898 0.235159
$$435$$ 0 0
$$436$$ −4.00000 −0.191565
$$437$$ −2.44949 −0.117175
$$438$$ 0 0
$$439$$ 14.0000 0.668184 0.334092 0.942541i $$-0.391570\pi$$
0.334092 + 0.942541i $$0.391570\pi$$
$$440$$ 29.3939 1.40130
$$441$$ 0 0
$$442$$ 18.0000 0.856173
$$443$$ 12.2474 0.581894 0.290947 0.956739i $$-0.406030\pi$$
0.290947 + 0.956739i $$0.406030\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 17.1464 0.811907
$$447$$ 0 0
$$448$$ −16.0000 −0.755929
$$449$$ −22.0454 −1.04039 −0.520194 0.854048i $$-0.674140\pi$$
−0.520194 + 0.854048i $$0.674140\pi$$
$$450$$ 0 0
$$451$$ 12.0000 0.565058
$$452$$ 39.1918 1.84343
$$453$$ 0 0
$$454$$ 24.0000 1.12638
$$455$$ −4.89898 −0.229668
$$456$$ 0 0
$$457$$ 29.0000 1.35656 0.678281 0.734802i $$-0.262725\pi$$
0.678281 + 0.734802i $$0.262725\pi$$
$$458$$ 2.44949 0.114457
$$459$$ 0 0
$$460$$ 24.0000 1.11901
$$461$$ 26.9444 1.25493 0.627463 0.778647i $$-0.284094\pi$$
0.627463 + 0.778647i $$0.284094\pi$$
$$462$$ 0 0
$$463$$ −19.0000 −0.883005 −0.441502 0.897260i $$-0.645554\pi$$
−0.441502 + 0.897260i $$0.645554\pi$$
$$464$$ 19.5959 0.909718
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ −14.6969 −0.680093 −0.340047 0.940409i $$-0.610443\pi$$
−0.340047 + 0.940409i $$0.610443\pi$$
$$468$$ 0 0
$$469$$ −14.0000 −0.646460
$$470$$ 58.7878 2.71168
$$471$$ 0 0
$$472$$ −12.0000 −0.552345
$$473$$ −26.9444 −1.23890
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 58.7878 2.69453
$$477$$ 0 0
$$478$$ −6.00000 −0.274434
$$479$$ 26.9444 1.23112 0.615560 0.788090i $$-0.288930\pi$$
0.615560 + 0.788090i $$0.288930\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ 39.1918 1.78514
$$483$$ 0 0
$$484$$ −20.0000 −0.909091
$$485$$ −17.1464 −0.778579
$$486$$ 0 0
$$487$$ 35.0000 1.58600 0.793001 0.609221i $$-0.208518\pi$$
0.793001 + 0.609221i $$0.208518\pi$$
$$488$$ −24.4949 −1.10883
$$489$$ 0 0
$$490$$ 18.0000 0.813157
$$491$$ 39.1918 1.76870 0.884351 0.466822i $$-0.154601\pi$$
0.884351 + 0.466822i $$0.154601\pi$$
$$492$$ 0 0
$$493$$ 36.0000 1.62136
$$494$$ −2.44949 −0.110208
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ −14.6969 −0.659248
$$498$$ 0 0
$$499$$ 2.00000 0.0895323 0.0447661 0.998997i $$-0.485746\pi$$
0.0447661 + 0.998997i $$0.485746\pi$$
$$500$$ −39.1918 −1.75271
$$501$$ 0 0
$$502$$ −18.0000 −0.803379
$$503$$ 14.6969 0.655304 0.327652 0.944798i $$-0.393743\pi$$
0.327652 + 0.944798i $$0.393743\pi$$
$$504$$ 0 0
$$505$$ 12.0000 0.533993
$$506$$ 14.6969 0.653359
$$507$$ 0 0
$$508$$ −76.0000 −3.37195
$$509$$ 9.79796 0.434287 0.217143 0.976140i $$-0.430326\pi$$
0.217143 + 0.976140i $$0.430326\pi$$
$$510$$ 0 0
$$511$$ 22.0000 0.973223
$$512$$ 39.1918 1.73205
$$513$$ 0 0
$$514$$ −42.0000 −1.85254
$$515$$ −17.1464 −0.755562
$$516$$ 0 0
$$517$$ 24.0000 1.05552
$$518$$ −39.1918 −1.72199
$$519$$ 0 0
$$520$$ 12.0000 0.526235
$$521$$ −22.0454 −0.965827 −0.482913 0.875668i $$-0.660421\pi$$
−0.482913 + 0.875668i $$0.660421\pi$$
$$522$$ 0 0
$$523$$ −25.0000 −1.09317 −0.546587 0.837402i $$-0.684073\pi$$
−0.546587 + 0.837402i $$0.684073\pi$$
$$524$$ −48.9898 −2.14013
$$525$$ 0 0
$$526$$ −66.0000 −2.87774
$$527$$ −7.34847 −0.320104
$$528$$ 0 0
$$529$$ −17.0000 −0.739130
$$530$$ 44.0908 1.91518
$$531$$ 0 0
$$532$$ −8.00000 −0.346844
$$533$$ 4.89898 0.212198
$$534$$ 0 0
$$535$$ −36.0000 −1.55642
$$536$$ 34.2929 1.48123
$$537$$ 0 0
$$538$$ −54.0000 −2.32811
$$539$$ 7.34847 0.316521
$$540$$ 0 0
$$541$$ −28.0000 −1.20381 −0.601907 0.798566i $$-0.705592\pi$$
−0.601907 + 0.798566i $$0.705592\pi$$
$$542$$ 17.1464 0.736502
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −2.44949 −0.104925
$$546$$ 0 0
$$547$$ −13.0000 −0.555840 −0.277920 0.960604i $$-0.589645\pi$$
−0.277920 + 0.960604i $$0.589645\pi$$
$$548$$ −39.1918 −1.67419
$$549$$ 0 0
$$550$$ 6.00000 0.255841
$$551$$ −4.89898 −0.208704
$$552$$ 0 0
$$553$$ −14.0000 −0.595341
$$554$$ −26.9444 −1.14476
$$555$$ 0 0
$$556$$ −40.0000 −1.69638
$$557$$ −7.34847 −0.311365 −0.155682 0.987807i $$-0.549758\pi$$
−0.155682 + 0.987807i $$0.549758\pi$$
$$558$$ 0 0
$$559$$ −11.0000 −0.465250
$$560$$ 19.5959 0.828079
$$561$$ 0 0
$$562$$ −30.0000 −1.26547
$$563$$ −12.2474 −0.516168 −0.258084 0.966122i $$-0.583091\pi$$
−0.258084 + 0.966122i $$0.583091\pi$$
$$564$$ 0 0
$$565$$ 24.0000 1.00969
$$566$$ −41.6413 −1.75032
$$567$$ 0 0
$$568$$ 36.0000 1.51053
$$569$$ 12.2474 0.513440 0.256720 0.966486i $$-0.417358\pi$$
0.256720 + 0.966486i $$0.417358\pi$$
$$570$$ 0 0
$$571$$ −10.0000 −0.418487 −0.209243 0.977864i $$-0.567100\pi$$
−0.209243 + 0.977864i $$0.567100\pi$$
$$572$$ 9.79796 0.409673
$$573$$ 0 0
$$574$$ 24.0000 1.00174
$$575$$ 2.44949 0.102151
$$576$$ 0 0
$$577$$ −25.0000 −1.04076 −0.520382 0.853934i $$-0.674210\pi$$
−0.520382 + 0.853934i $$0.674210\pi$$
$$578$$ −90.6311 −3.76976
$$579$$ 0 0
$$580$$ 48.0000 1.99309
$$581$$ 24.4949 1.01622
$$582$$ 0 0
$$583$$ 18.0000 0.745484
$$584$$ −53.8888 −2.22993
$$585$$ 0 0
$$586$$ 12.0000 0.495715
$$587$$ −2.44949 −0.101101 −0.0505506 0.998721i $$-0.516098\pi$$
−0.0505506 + 0.998721i $$0.516098\pi$$
$$588$$ 0 0
$$589$$ 1.00000 0.0412043
$$590$$ −14.6969 −0.605063
$$591$$ 0 0
$$592$$ 32.0000 1.31519
$$593$$ −7.34847 −0.301765 −0.150883 0.988552i $$-0.548212\pi$$
−0.150883 + 0.988552i $$0.548212\pi$$
$$594$$ 0 0
$$595$$ 36.0000 1.47586
$$596$$ −48.9898 −2.00670
$$597$$ 0 0
$$598$$ 6.00000 0.245358
$$599$$ 39.1918 1.60134 0.800668 0.599109i $$-0.204478\pi$$
0.800668 + 0.599109i $$0.204478\pi$$
$$600$$ 0 0
$$601$$ −7.00000 −0.285536 −0.142768 0.989756i $$-0.545600\pi$$
−0.142768 + 0.989756i $$0.545600\pi$$
$$602$$ −53.8888 −2.19634
$$603$$ 0 0
$$604$$ 20.0000 0.813788
$$605$$ −12.2474 −0.497930
$$606$$ 0 0
$$607$$ 44.0000 1.78590 0.892952 0.450151i $$-0.148630\pi$$
0.892952 + 0.450151i $$0.148630\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ −30.0000 −1.21466
$$611$$ 9.79796 0.396383
$$612$$ 0 0
$$613$$ 11.0000 0.444286 0.222143 0.975014i $$-0.428695\pi$$
0.222143 + 0.975014i $$0.428695\pi$$
$$614$$ −4.89898 −0.197707
$$615$$ 0 0
$$616$$ 24.0000 0.966988
$$617$$ 24.4949 0.986127 0.493064 0.869993i $$-0.335877\pi$$
0.493064 + 0.869993i $$0.335877\pi$$
$$618$$ 0 0
$$619$$ −49.0000 −1.96948 −0.984738 0.174042i $$-0.944317\pi$$
−0.984738 + 0.174042i $$0.944317\pi$$
$$620$$ −9.79796 −0.393496
$$621$$ 0 0
$$622$$ −60.0000 −2.40578
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 39.1918 1.56642
$$627$$ 0 0
$$628$$ 68.0000 2.71350
$$629$$ 58.7878 2.34402
$$630$$ 0 0
$$631$$ 44.0000 1.75161 0.875806 0.482663i $$-0.160330\pi$$
0.875806 + 0.482663i $$0.160330\pi$$
$$632$$ 34.2929 1.36410
$$633$$ 0 0
$$634$$ 24.0000 0.953162
$$635$$ −46.5403 −1.84690
$$636$$ 0 0
$$637$$ 3.00000 0.118864
$$638$$ 29.3939 1.16371
$$639$$ 0 0
$$640$$ 48.0000 1.89737
$$641$$ −17.1464 −0.677243 −0.338622 0.940923i $$-0.609961\pi$$
−0.338622 + 0.940923i $$0.609961\pi$$
$$642$$ 0 0
$$643$$ 38.0000 1.49857 0.749287 0.662246i $$-0.230396\pi$$
0.749287 + 0.662246i $$0.230396\pi$$
$$644$$ 19.5959 0.772187
$$645$$ 0 0
$$646$$ 18.0000 0.708201
$$647$$ −36.7423 −1.44449 −0.722245 0.691637i $$-0.756890\pi$$
−0.722245 + 0.691637i $$0.756890\pi$$
$$648$$ 0 0
$$649$$ −6.00000 −0.235521
$$650$$ 2.44949 0.0960769
$$651$$ 0 0
$$652$$ −40.0000 −1.56652
$$653$$ 9.79796 0.383424 0.191712 0.981451i $$-0.438596\pi$$
0.191712 + 0.981451i $$0.438596\pi$$
$$654$$ 0 0
$$655$$ −30.0000 −1.17220
$$656$$ −19.5959 −0.765092
$$657$$ 0 0
$$658$$ 48.0000 1.87123
$$659$$ 19.5959 0.763349 0.381674 0.924297i $$-0.375348\pi$$
0.381674 + 0.924297i $$0.375348\pi$$
$$660$$ 0 0
$$661$$ 11.0000 0.427850 0.213925 0.976850i $$-0.431375\pi$$
0.213925 + 0.976850i $$0.431375\pi$$
$$662$$ 17.1464 0.666415
$$663$$ 0 0
$$664$$ −60.0000 −2.32845
$$665$$ −4.89898 −0.189974
$$666$$ 0 0
$$667$$ 12.0000 0.464642
$$668$$ −19.5959 −0.758189
$$669$$ 0 0
$$670$$ 42.0000 1.62260
$$671$$ −12.2474 −0.472808
$$672$$ 0 0
$$673$$ 29.0000 1.11787 0.558934 0.829212i $$-0.311211\pi$$
0.558934 + 0.829212i $$0.311211\pi$$
$$674$$ 68.5857 2.64182
$$675$$ 0 0
$$676$$ −48.0000 −1.84615
$$677$$ −46.5403 −1.78869 −0.894345 0.447379i $$-0.852358\pi$$
−0.894345 + 0.447379i $$0.852358\pi$$
$$678$$ 0 0
$$679$$ −14.0000 −0.537271
$$680$$ −88.1816 −3.38161
$$681$$ 0 0
$$682$$ −6.00000 −0.229752
$$683$$ 22.0454 0.843544 0.421772 0.906702i $$-0.361408\pi$$
0.421772 + 0.906702i $$0.361408\pi$$
$$684$$ 0 0
$$685$$ −24.0000 −0.916993
$$686$$ 48.9898 1.87044
$$687$$ 0 0
$$688$$ 44.0000 1.67748
$$689$$ 7.34847 0.279954
$$690$$ 0 0
$$691$$ 47.0000 1.78796 0.893982 0.448103i $$-0.147900\pi$$
0.893982 + 0.448103i $$0.147900\pi$$
$$692$$ −39.1918 −1.48985
$$693$$ 0 0
$$694$$ −60.0000 −2.27757
$$695$$ −24.4949 −0.929144
$$696$$ 0 0
$$697$$ −36.0000 −1.36360
$$698$$ −48.9898 −1.85429
$$699$$ 0 0
$$700$$ 8.00000 0.302372
$$701$$ 14.6969 0.555096 0.277548 0.960712i $$-0.410478\pi$$
0.277548 + 0.960712i $$0.410478\pi$$
$$702$$ 0 0
$$703$$ −8.00000 −0.301726
$$704$$ 19.5959 0.738549
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ 9.79796 0.368490
$$708$$ 0 0
$$709$$ −7.00000 −0.262891 −0.131445 0.991323i $$-0.541962\pi$$
−0.131445 + 0.991323i $$0.541962\pi$$
$$710$$ 44.0908 1.65470
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −2.44949 −0.0917341
$$714$$ 0 0
$$715$$ 6.00000 0.224387
$$716$$ 58.7878 2.19700
$$717$$ 0 0
$$718$$ 72.0000 2.68702
$$719$$ 36.7423 1.37026 0.685129 0.728422i $$-0.259746\pi$$
0.685129 + 0.728422i $$0.259746\pi$$
$$720$$ 0 0
$$721$$ −14.0000 −0.521387
$$722$$ 44.0908 1.64089
$$723$$ 0 0
$$724$$ 32.0000 1.18927
$$725$$ 4.89898 0.181944
$$726$$ 0 0
$$727$$ 14.0000 0.519231 0.259616 0.965712i $$-0.416404\pi$$
0.259616 + 0.965712i $$0.416404\pi$$
$$728$$ 9.79796 0.363137
$$729$$ 0 0
$$730$$ −66.0000 −2.44277
$$731$$ 80.8332 2.98972
$$732$$ 0 0
$$733$$ 17.0000 0.627909 0.313955 0.949438i $$-0.398346\pi$$
0.313955 + 0.949438i $$0.398346\pi$$
$$734$$ −12.2474 −0.452062
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 17.1464 0.631597
$$738$$ 0 0
$$739$$ −1.00000 −0.0367856 −0.0183928 0.999831i $$-0.505855\pi$$
−0.0183928 + 0.999831i $$0.505855\pi$$
$$740$$ 78.3837 2.88144
$$741$$ 0 0
$$742$$ 36.0000 1.32160
$$743$$ 31.8434 1.16822 0.584110 0.811675i $$-0.301444\pi$$
0.584110 + 0.811675i $$0.301444\pi$$
$$744$$ 0 0
$$745$$ −30.0000 −1.09911
$$746$$ −85.7321 −3.13888
$$747$$ 0 0
$$748$$ −72.0000 −2.63258
$$749$$ −29.3939 −1.07403
$$750$$ 0 0
$$751$$ 26.0000 0.948753 0.474377 0.880322i $$-0.342673\pi$$
0.474377 + 0.880322i $$0.342673\pi$$
$$752$$ −39.1918 −1.42918
$$753$$ 0 0
$$754$$ 12.0000 0.437014
$$755$$ 12.2474 0.445730
$$756$$ 0 0
$$757$$ −7.00000 −0.254419 −0.127210 0.991876i $$-0.540602\pi$$
−0.127210 + 0.991876i $$0.540602\pi$$
$$758$$ −19.5959 −0.711756
$$759$$ 0 0
$$760$$ 12.0000 0.435286
$$761$$ 2.44949 0.0887939 0.0443970 0.999014i $$-0.485863\pi$$
0.0443970 + 0.999014i $$0.485863\pi$$
$$762$$ 0 0
$$763$$ −2.00000 −0.0724049
$$764$$ −39.1918 −1.41791
$$765$$ 0 0
$$766$$ 84.0000 3.03504
$$767$$ −2.44949 −0.0884459
$$768$$ 0 0
$$769$$ −37.0000 −1.33425 −0.667127 0.744944i $$-0.732476\pi$$
−0.667127 + 0.744944i $$0.732476\pi$$
$$770$$ 29.3939 1.05928
$$771$$ 0 0
$$772$$ 44.0000 1.58359
$$773$$ 44.0908 1.58584 0.792918 0.609328i $$-0.208561\pi$$
0.792918 + 0.609328i $$0.208561\pi$$
$$774$$ 0 0
$$775$$ −1.00000 −0.0359211
$$776$$ 34.2929 1.23104
$$777$$ 0 0
$$778$$ −66.0000 −2.36621
$$779$$ 4.89898 0.175524
$$780$$ 0 0
$$781$$ 18.0000 0.644091
$$782$$ −44.0908 −1.57668
$$783$$ 0 0
$$784$$ −12.0000 −0.428571
$$785$$ 41.6413 1.48624
$$786$$ 0 0
$$787$$ −25.0000 −0.891154 −0.445577 0.895244i $$-0.647001\pi$$
−0.445577 + 0.895244i $$0.647001\pi$$
$$788$$ −58.7878 −2.09423
$$789$$ 0 0
$$790$$ 42.0000 1.49429
$$791$$ 19.5959 0.696751
$$792$$ 0 0
$$793$$ −5.00000 −0.177555
$$794$$ 2.44949 0.0869291
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ −41.6413 −1.47501 −0.737506 0.675341i $$-0.763997\pi$$
−0.737506 + 0.675341i $$0.763997\pi$$
$$798$$ 0 0
$$799$$ −72.0000 −2.54718
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 84.0000 2.96614
$$803$$ −26.9444 −0.950847
$$804$$ 0 0
$$805$$ 12.0000 0.422944
$$806$$ −2.44949 −0.0862796
$$807$$ 0 0
$$808$$ −24.0000 −0.844317
$$809$$ −22.0454 −0.775075 −0.387538 0.921854i $$-0.626674\pi$$
−0.387538 + 0.921854i $$0.626674\pi$$
$$810$$ 0 0
$$811$$ 35.0000 1.22902 0.614508 0.788911i $$-0.289355\pi$$
0.614508 + 0.788911i $$0.289355\pi$$
$$812$$ 39.1918 1.37536
$$813$$ 0 0
$$814$$ 48.0000 1.68240
$$815$$ −24.4949 −0.858019
$$816$$ 0 0
$$817$$ −11.0000 −0.384841
$$818$$ 68.5857 2.39804
$$819$$ 0 0
$$820$$ −48.0000 −1.67623
$$821$$ −39.1918 −1.36780 −0.683902 0.729574i $$-0.739719\pi$$
−0.683902 + 0.729574i $$0.739719\pi$$
$$822$$ 0 0
$$823$$ 35.0000 1.22002 0.610012 0.792392i $$-0.291165\pi$$
0.610012 + 0.792392i $$0.291165\pi$$
$$824$$ 34.2929 1.19465
$$825$$ 0 0
$$826$$ −12.0000 −0.417533
$$827$$ 22.0454 0.766594 0.383297 0.923625i $$-0.374789\pi$$
0.383297 + 0.923625i $$0.374789\pi$$
$$828$$ 0 0
$$829$$ −37.0000 −1.28506 −0.642532 0.766259i $$-0.722116\pi$$
−0.642532 + 0.766259i $$0.722116\pi$$
$$830$$ −73.4847 −2.55069
$$831$$ 0 0
$$832$$ 8.00000 0.277350
$$833$$ −22.0454 −0.763828
$$834$$ 0 0
$$835$$ −12.0000 −0.415277
$$836$$ 9.79796 0.338869
$$837$$ 0 0
$$838$$ 84.0000 2.90173
$$839$$ 4.89898 0.169132 0.0845658 0.996418i $$-0.473050\pi$$
0.0845658 + 0.996418i $$0.473050\pi$$
$$840$$ 0 0
$$841$$ −5.00000 −0.172414
$$842$$ −4.89898 −0.168830
$$843$$ 0 0
$$844$$ −4.00000 −0.137686
$$845$$ −29.3939 −1.01118
$$846$$ 0 0
$$847$$ −10.0000 −0.343604
$$848$$ −29.3939 −1.00939
$$849$$ 0 0
$$850$$ −18.0000 −0.617395
$$851$$ 19.5959 0.671739
$$852$$ 0 0
$$853$$ −13.0000 −0.445112 −0.222556 0.974920i $$-0.571440\pi$$
−0.222556 + 0.974920i $$0.571440\pi$$
$$854$$ −24.4949 −0.838198
$$855$$ 0 0
$$856$$ 72.0000 2.46091
$$857$$ −24.4949 −0.836730 −0.418365 0.908279i $$-0.637397\pi$$
−0.418365 + 0.908279i $$0.637397\pi$$
$$858$$ 0 0
$$859$$ 26.0000 0.887109 0.443554 0.896248i $$-0.353717\pi$$
0.443554 + 0.896248i $$0.353717\pi$$
$$860$$ 107.778 3.67518
$$861$$ 0 0
$$862$$ −18.0000 −0.613082
$$863$$ 7.34847 0.250145 0.125072 0.992148i $$-0.460084\pi$$
0.125072 + 0.992148i $$0.460084\pi$$
$$864$$ 0 0
$$865$$ −24.0000 −0.816024
$$866$$ −41.6413 −1.41503
$$867$$ 0 0
$$868$$ −8.00000 −0.271538
$$869$$ 17.1464 0.581653
$$870$$ 0 0
$$871$$ 7.00000 0.237186
$$872$$ 4.89898 0.165900
$$873$$ 0 0
$$874$$ 6.00000 0.202953
$$875$$ −19.5959 −0.662463
$$876$$ 0 0
$$877$$ 8.00000 0.270141 0.135070 0.990836i $$-0.456874\pi$$
0.135070 + 0.990836i $$0.456874\pi$$
$$878$$ −34.2929 −1.15733
$$879$$ 0 0
$$880$$ −24.0000 −0.809040
$$881$$ −36.7423 −1.23788 −0.618941 0.785438i $$-0.712438\pi$$
−0.618941 + 0.785438i $$0.712438\pi$$
$$882$$ 0 0
$$883$$ 17.0000 0.572096 0.286048 0.958215i $$-0.407658\pi$$
0.286048 + 0.958215i $$0.407658\pi$$
$$884$$ −29.3939 −0.988623
$$885$$ 0 0
$$886$$ −30.0000 −1.00787
$$887$$ 17.1464 0.575721 0.287860 0.957672i $$-0.407056\pi$$
0.287860 + 0.957672i $$0.407056\pi$$
$$888$$ 0 0
$$889$$ −38.0000 −1.27448
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −28.0000 −0.937509
$$893$$ 9.79796 0.327876
$$894$$ 0 0
$$895$$ 36.0000 1.20335
$$896$$ 39.1918 1.30931
$$897$$ 0 0
$$898$$ 54.0000 1.80200
$$899$$ −4.89898 −0.163390
$$900$$ 0 0
$$901$$ −54.0000 −1.79900
$$902$$ −29.3939 −0.978709
$$903$$ 0 0
$$904$$ −48.0000 −1.59646
$$905$$ 19.5959 0.651390
$$906$$ 0 0
$$907$$ −7.00000 −0.232431 −0.116216 0.993224i $$-0.537076\pi$$
−0.116216 + 0.993224i $$0.537076\pi$$
$$908$$ −39.1918 −1.30063
$$909$$ 0 0
$$910$$ 12.0000 0.397796
$$911$$ 12.2474 0.405776 0.202888 0.979202i $$-0.434967\pi$$
0.202888 + 0.979202i $$0.434967\pi$$
$$912$$ 0 0
$$913$$ −30.0000 −0.992855
$$914$$ −71.0352 −2.34964
$$915$$ 0 0
$$916$$ −4.00000 −0.132164
$$917$$ −24.4949 −0.808893
$$918$$ 0 0
$$919$$ 20.0000 0.659739 0.329870 0.944027i $$-0.392995\pi$$
0.329870 + 0.944027i $$0.392995\pi$$
$$920$$ −29.3939 −0.969087
$$921$$ 0 0
$$922$$ −66.0000 −2.17359
$$923$$ 7.34847 0.241878
$$924$$ 0 0
$$925$$ 8.00000 0.263038
$$926$$ 46.5403 1.52941
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 26.9444 0.884017 0.442008 0.897011i $$-0.354266\pi$$
0.442008 + 0.897011i $$0.354266\pi$$
$$930$$ 0 0
$$931$$ 3.00000 0.0983210
$$932$$ 29.3939 0.962828
$$933$$ 0 0
$$934$$ 36.0000 1.17796
$$935$$ −44.0908 −1.44192
$$936$$ 0 0
$$937$$ 8.00000 0.261349 0.130674 0.991425i $$-0.458286\pi$$
0.130674 + 0.991425i $$0.458286\pi$$
$$938$$ 34.2929 1.11970
$$939$$ 0 0
$$940$$ −96.0000 −3.13117
$$941$$ 9.79796 0.319404 0.159702 0.987165i $$-0.448947\pi$$
0.159702 + 0.987165i $$0.448947\pi$$
$$942$$ 0 0
$$943$$ −12.0000 −0.390774
$$944$$ 9.79796 0.318896
$$945$$ 0 0
$$946$$ 66.0000 2.14585
$$947$$ −24.4949 −0.795977 −0.397989 0.917390i $$-0.630292\pi$$
−0.397989 + 0.917390i $$0.630292\pi$$
$$948$$ 0 0
$$949$$ −11.0000 −0.357075
$$950$$ 2.44949 0.0794719
$$951$$ 0 0
$$952$$ −72.0000 −2.33353
$$953$$ −29.3939 −0.952161 −0.476081 0.879402i $$-0.657943\pi$$
−0.476081 + 0.879402i $$0.657943\pi$$
$$954$$ 0 0
$$955$$ −24.0000 −0.776622
$$956$$ 9.79796 0.316889
$$957$$ 0 0
$$958$$ −66.0000 −2.13236
$$959$$ −19.5959 −0.632785
$$960$$ 0 0
$$961$$ −30.0000 −0.967742
$$962$$ 19.5959 0.631798
$$963$$ 0 0
$$964$$ −64.0000 −2.06130
$$965$$ 26.9444 0.867371
$$966$$ 0 0
$$967$$ −7.00000 −0.225105 −0.112552 0.993646i $$-0.535903\pi$$
−0.112552 + 0.993646i $$0.535903\pi$$
$$968$$ 24.4949 0.787296
$$969$$ 0 0
$$970$$ 42.0000 1.34854
$$971$$ −29.3939 −0.943294 −0.471647 0.881787i $$-0.656340\pi$$
−0.471647 + 0.881787i $$0.656340\pi$$
$$972$$ 0 0
$$973$$ −20.0000 −0.641171
$$974$$ −85.7321 −2.74703
$$975$$ 0 0
$$976$$ 20.0000 0.640184
$$977$$ 24.4949 0.783661 0.391831 0.920037i $$-0.371842\pi$$
0.391831 + 0.920037i $$0.371842\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ −29.3939 −0.938953
$$981$$ 0 0
$$982$$ −96.0000 −3.06348
$$983$$ 4.89898 0.156253 0.0781266 0.996943i $$-0.475106\pi$$
0.0781266 + 0.996943i $$0.475106\pi$$
$$984$$ 0 0
$$985$$ −36.0000 −1.14706
$$986$$ −88.1816 −2.80828
$$987$$ 0 0
$$988$$ 4.00000 0.127257
$$989$$ 26.9444 0.856782
$$990$$ 0 0
$$991$$ −7.00000 −0.222362 −0.111181 0.993800i $$-0.535463\pi$$
−0.111181 + 0.993800i $$0.535463\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 36.0000 1.14185
$$995$$ −2.44949 −0.0776540
$$996$$ 0 0
$$997$$ 50.0000 1.58352 0.791758 0.610835i $$-0.209166\pi$$
0.791758 + 0.610835i $$0.209166\pi$$
$$998$$ −4.89898 −0.155074
$$999$$ 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.d.1.1 2
3.2 odd 2 inner 243.2.a.d.1.2 yes 2
4.3 odd 2 3888.2.a.z.1.2 2
5.4 even 2 6075.2.a.bn.1.2 2
9.2 odd 6 243.2.c.c.82.1 4
9.4 even 3 243.2.c.c.163.2 4
9.5 odd 6 243.2.c.c.163.1 4
9.7 even 3 243.2.c.c.82.2 4
12.11 even 2 3888.2.a.z.1.1 2
15.14 odd 2 6075.2.a.bn.1.1 2
27.2 odd 18 729.2.e.p.568.1 12
27.4 even 9 729.2.e.p.406.1 12
27.5 odd 18 729.2.e.p.649.2 12
27.7 even 9 729.2.e.p.325.1 12
27.11 odd 18 729.2.e.p.82.2 12
27.13 even 9 729.2.e.p.163.2 12
27.14 odd 18 729.2.e.p.163.1 12
27.16 even 9 729.2.e.p.82.1 12
27.20 odd 18 729.2.e.p.325.2 12
27.22 even 9 729.2.e.p.649.1 12
27.23 odd 18 729.2.e.p.406.2 12
27.25 even 9 729.2.e.p.568.2 12

By twisted newform
Twist Min Dim Char Parity Ord Type
243.2.a.d.1.1 2 1.1 even 1 trivial
243.2.a.d.1.2 yes 2 3.2 odd 2 inner
243.2.c.c.82.1 4 9.2 odd 6
243.2.c.c.82.2 4 9.7 even 3
243.2.c.c.163.1 4 9.5 odd 6
243.2.c.c.163.2 4 9.4 even 3
729.2.e.p.82.1 12 27.16 even 9
729.2.e.p.82.2 12 27.11 odd 18
729.2.e.p.163.1 12 27.14 odd 18
729.2.e.p.163.2 12 27.13 even 9
729.2.e.p.325.1 12 27.7 even 9
729.2.e.p.325.2 12 27.20 odd 18
729.2.e.p.406.1 12 27.4 even 9
729.2.e.p.406.2 12 27.23 odd 18
729.2.e.p.568.1 12 27.2 odd 18
729.2.e.p.568.2 12 27.25 even 9
729.2.e.p.649.1 12 27.22 even 9
729.2.e.p.649.2 12 27.5 odd 18
3888.2.a.z.1.1 2 12.11 even 2
3888.2.a.z.1.2 2 4.3 odd 2
6075.2.a.bn.1.1 2 15.14 odd 2
6075.2.a.bn.1.2 2 5.4 even 2