# Properties

 Label 9.22.a.a.1.1 Level $9$ Weight $22$ Character 9.1 Self dual yes Analytic conductor $25.153$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9,22,Mod(1,9)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 22, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9 = 3^{2}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 9.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: $$25.1529609858$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 9.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1728.00 q^{2} +888832. q^{4} +4.15128e7 q^{5} +5.38430e8 q^{7} +2.08798e9 q^{8} +O(q^{10})$$ $$q-1728.00 q^{2} +888832. q^{4} +4.15128e7 q^{5} +5.38430e8 q^{7} +2.08798e9 q^{8} -7.17341e10 q^{10} +6.41130e10 q^{11} -1.30980e11 q^{13} -9.30407e11 q^{14} -5.47204e12 q^{16} -8.24203e12 q^{17} +1.34921e13 q^{19} +3.68979e13 q^{20} -1.10787e14 q^{22} +2.33185e14 q^{23} +1.24647e15 q^{25} +2.26334e14 q^{26} +4.78574e14 q^{28} +2.02456e15 q^{29} -6.86919e15 q^{31} +5.07688e15 q^{32} +1.42422e16 q^{34} +2.23517e16 q^{35} +3.44400e15 q^{37} -2.33144e16 q^{38} +8.66777e16 q^{40} +2.18424e16 q^{41} -7.17928e16 q^{43} +5.69857e16 q^{44} -4.02943e17 q^{46} -2.83545e17 q^{47} -2.68639e17 q^{49} -2.15391e18 q^{50} -1.16419e17 q^{52} +2.17229e18 q^{53} +2.66151e18 q^{55} +1.12423e18 q^{56} -3.49844e18 q^{58} -1.53483e18 q^{59} +4.31159e18 q^{61} +1.18700e19 q^{62} +2.70285e18 q^{64} -5.43735e18 q^{65} +9.24391e18 q^{67} -7.32578e18 q^{68} -3.86238e19 q^{70} +2.03874e19 q^{71} +1.66178e19 q^{73} -5.95123e18 q^{74} +1.19922e19 q^{76} +3.45204e19 q^{77} +6.79403e19 q^{79} -2.27160e20 q^{80} -3.77437e19 q^{82} -3.95037e19 q^{83} -3.42149e20 q^{85} +1.24058e20 q^{86} +1.33867e20 q^{88} -4.16117e19 q^{89} -7.05236e19 q^{91} +2.07262e20 q^{92} +4.89965e20 q^{94} +5.60095e20 q^{95} +5.71815e19 q^{97} +4.64209e20 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1728.00 −1.19324 −0.596621 0.802523i $$-0.703491\pi$$
−0.596621 + 0.802523i $$0.703491\pi$$
$$3$$ 0 0
$$4$$ 888832. 0.423828
$$5$$ 4.15128e7 1.90106 0.950532 0.310627i $$-0.100539\pi$$
0.950532 + 0.310627i $$0.100539\pi$$
$$6$$ 0 0
$$7$$ 5.38430e8 0.720443 0.360222 0.932867i $$-0.382701\pi$$
0.360222 + 0.932867i $$0.382701\pi$$
$$8$$ 2.08798e9 0.687513
$$9$$ 0 0
$$10$$ −7.17341e10 −2.26843
$$11$$ 6.41130e10 0.745286 0.372643 0.927975i $$-0.378452\pi$$
0.372643 + 0.927975i $$0.378452\pi$$
$$12$$ 0 0
$$13$$ −1.30980e11 −0.263512 −0.131756 0.991282i $$-0.542062\pi$$
−0.131756 + 0.991282i $$0.542062\pi$$
$$14$$ −9.30407e11 −0.859663
$$15$$ 0 0
$$16$$ −5.47204e12 −1.24420
$$17$$ −8.24203e12 −0.991563 −0.495782 0.868447i $$-0.665118\pi$$
−0.495782 + 0.868447i $$0.665118\pi$$
$$18$$ 0 0
$$19$$ 1.34921e13 0.504855 0.252428 0.967616i $$-0.418771\pi$$
0.252428 + 0.967616i $$0.418771\pi$$
$$20$$ 3.68979e13 0.805724
$$21$$ 0 0
$$22$$ −1.10787e14 −0.889308
$$23$$ 2.33185e14 1.17370 0.586851 0.809695i $$-0.300367\pi$$
0.586851 + 0.809695i $$0.300367\pi$$
$$24$$ 0 0
$$25$$ 1.24647e15 2.61404
$$26$$ 2.26334e14 0.314434
$$27$$ 0 0
$$28$$ 4.78574e14 0.305344
$$29$$ 2.02456e15 0.893618 0.446809 0.894629i $$-0.352560\pi$$
0.446809 + 0.894629i $$0.352560\pi$$
$$30$$ 0 0
$$31$$ −6.86919e15 −1.50525 −0.752624 0.658451i $$-0.771212\pi$$
−0.752624 + 0.658451i $$0.771212\pi$$
$$32$$ 5.07688e15 0.797117
$$33$$ 0 0
$$34$$ 1.42422e16 1.18318
$$35$$ 2.23517e16 1.36961
$$36$$ 0 0
$$37$$ 3.44400e15 0.117746 0.0588728 0.998265i $$-0.481249\pi$$
0.0588728 + 0.998265i $$0.481249\pi$$
$$38$$ −2.33144e16 −0.602415
$$39$$ 0 0
$$40$$ 8.66777e16 1.30701
$$41$$ 2.18424e16 0.254138 0.127069 0.991894i $$-0.459443\pi$$
0.127069 + 0.991894i $$0.459443\pi$$
$$42$$ 0 0
$$43$$ −7.17928e16 −0.506597 −0.253298 0.967388i $$-0.581515\pi$$
−0.253298 + 0.967388i $$0.581515\pi$$
$$44$$ 5.69857e16 0.315873
$$45$$ 0 0
$$46$$ −4.02943e17 −1.40051
$$47$$ −2.83545e17 −0.786310 −0.393155 0.919472i $$-0.628617\pi$$
−0.393155 + 0.919472i $$0.628617\pi$$
$$48$$ 0 0
$$49$$ −2.68639e17 −0.480962
$$50$$ −2.15391e18 −3.11919
$$51$$ 0 0
$$52$$ −1.16419e17 −0.111684
$$53$$ 2.17229e18 1.70616 0.853081 0.521779i $$-0.174731\pi$$
0.853081 + 0.521779i $$0.174731\pi$$
$$54$$ 0 0
$$55$$ 2.66151e18 1.41684
$$56$$ 1.12423e18 0.495314
$$57$$ 0 0
$$58$$ −3.49844e18 −1.06630
$$59$$ −1.53483e18 −0.390944 −0.195472 0.980709i $$-0.562624\pi$$
−0.195472 + 0.980709i $$0.562624\pi$$
$$60$$ 0 0
$$61$$ 4.31159e18 0.773881 0.386940 0.922105i $$-0.373532\pi$$
0.386940 + 0.922105i $$0.373532\pi$$
$$62$$ 1.18700e19 1.79613
$$63$$ 0 0
$$64$$ 2.70285e18 0.293044
$$65$$ −5.43735e18 −0.500953
$$66$$ 0 0
$$67$$ 9.24391e18 0.619541 0.309771 0.950811i $$-0.399748\pi$$
0.309771 + 0.950811i $$0.399748\pi$$
$$68$$ −7.32578e18 −0.420252
$$69$$ 0 0
$$70$$ −3.86238e19 −1.63427
$$71$$ 2.03874e19 0.743273 0.371636 0.928378i $$-0.378797\pi$$
0.371636 + 0.928378i $$0.378797\pi$$
$$72$$ 0 0
$$73$$ 1.66178e19 0.452566 0.226283 0.974062i $$-0.427343\pi$$
0.226283 + 0.974062i $$0.427343\pi$$
$$74$$ −5.95123e18 −0.140499
$$75$$ 0 0
$$76$$ 1.19922e19 0.213972
$$77$$ 3.45204e19 0.536936
$$78$$ 0 0
$$79$$ 6.79403e19 0.807315 0.403658 0.914910i $$-0.367739\pi$$
0.403658 + 0.914910i $$0.367739\pi$$
$$80$$ −2.27160e20 −2.36530
$$81$$ 0 0
$$82$$ −3.77437e19 −0.303249
$$83$$ −3.95037e19 −0.279459 −0.139730 0.990190i $$-0.544623\pi$$
−0.139730 + 0.990190i $$0.544623\pi$$
$$84$$ 0 0
$$85$$ −3.42149e20 −1.88503
$$86$$ 1.24058e20 0.604493
$$87$$ 0 0
$$88$$ 1.33867e20 0.512394
$$89$$ −4.16117e19 −0.141456 −0.0707278 0.997496i $$-0.522532\pi$$
−0.0707278 + 0.997496i $$0.522532\pi$$
$$90$$ 0 0
$$91$$ −7.05236e19 −0.189845
$$92$$ 2.07262e20 0.497448
$$93$$ 0 0
$$94$$ 4.89965e20 0.938259
$$95$$ 5.60095e20 0.959762
$$96$$ 0 0
$$97$$ 5.71815e19 0.0787322 0.0393661 0.999225i $$-0.487466\pi$$
0.0393661 + 0.999225i $$0.487466\pi$$
$$98$$ 4.64209e20 0.573904
$$99$$ 0 0
$$100$$ 1.10791e21 1.10791
$$101$$ −4.32417e20 −0.389518 −0.194759 0.980851i $$-0.562393\pi$$
−0.194759 + 0.980851i $$0.562393\pi$$
$$102$$ 0 0
$$103$$ 1.84123e21 1.34995 0.674974 0.737841i $$-0.264155\pi$$
0.674974 + 0.737841i $$0.264155\pi$$
$$104$$ −2.73483e20 −0.181168
$$105$$ 0 0
$$106$$ −3.75371e21 −2.03587
$$107$$ 2.43805e21 1.19815 0.599077 0.800691i $$-0.295534\pi$$
0.599077 + 0.800691i $$0.295534\pi$$
$$108$$ 0 0
$$109$$ −4.13676e21 −1.67372 −0.836859 0.547418i $$-0.815611\pi$$
−0.836859 + 0.547418i $$0.815611\pi$$
$$110$$ −4.59909e21 −1.69063
$$111$$ 0 0
$$112$$ −2.94631e21 −0.896374
$$113$$ −3.47910e21 −0.964146 −0.482073 0.876131i $$-0.660116\pi$$
−0.482073 + 0.876131i $$0.660116\pi$$
$$114$$ 0 0
$$115$$ 9.68015e21 2.23128
$$116$$ 1.79950e21 0.378741
$$117$$ 0 0
$$118$$ 2.65219e21 0.466491
$$119$$ −4.43775e21 −0.714365
$$120$$ 0 0
$$121$$ −3.28977e21 −0.444548
$$122$$ −7.45043e21 −0.923428
$$123$$ 0 0
$$124$$ −6.10556e21 −0.637966
$$125$$ 3.19497e22 3.06840
$$126$$ 0 0
$$127$$ 1.37141e21 0.111488 0.0557438 0.998445i $$-0.482247\pi$$
0.0557438 + 0.998445i $$0.482247\pi$$
$$128$$ −1.53175e22 −1.14679
$$129$$ 0 0
$$130$$ 9.39574e21 0.597758
$$131$$ 2.45276e22 1.43981 0.719907 0.694071i $$-0.244185\pi$$
0.719907 + 0.694071i $$0.244185\pi$$
$$132$$ 0 0
$$133$$ 7.26455e21 0.363719
$$134$$ −1.59735e22 −0.739263
$$135$$ 0 0
$$136$$ −1.72092e22 −0.681713
$$137$$ −1.02835e22 −0.377204 −0.188602 0.982054i $$-0.560396\pi$$
−0.188602 + 0.982054i $$0.560396\pi$$
$$138$$ 0 0
$$139$$ 8.70692e21 0.274289 0.137145 0.990551i $$-0.456207\pi$$
0.137145 + 0.990551i $$0.456207\pi$$
$$140$$ 1.98669e22 0.580478
$$141$$ 0 0
$$142$$ −3.52294e22 −0.886905
$$143$$ −8.39753e21 −0.196392
$$144$$ 0 0
$$145$$ 8.40452e22 1.69883
$$146$$ −2.87155e22 −0.540022
$$147$$ 0 0
$$148$$ 3.06114e21 0.0499039
$$149$$ 9.03997e22 1.37313 0.686564 0.727069i $$-0.259118\pi$$
0.686564 + 0.727069i $$0.259118\pi$$
$$150$$ 0 0
$$151$$ −4.75206e22 −0.627514 −0.313757 0.949503i $$-0.601588\pi$$
−0.313757 + 0.949503i $$0.601588\pi$$
$$152$$ 2.81712e22 0.347094
$$153$$ 0 0
$$154$$ −5.96512e22 −0.640695
$$155$$ −2.85159e23 −2.86157
$$156$$ 0 0
$$157$$ −1.50901e23 −1.32356 −0.661781 0.749697i $$-0.730199\pi$$
−0.661781 + 0.749697i $$0.730199\pi$$
$$158$$ −1.17401e23 −0.963323
$$159$$ 0 0
$$160$$ 2.10755e23 1.51537
$$161$$ 1.25554e23 0.845586
$$162$$ 0 0
$$163$$ −4.83503e22 −0.286042 −0.143021 0.989720i $$-0.545682\pi$$
−0.143021 + 0.989720i $$0.545682\pi$$
$$164$$ 1.94142e22 0.107711
$$165$$ 0 0
$$166$$ 6.82624e22 0.333462
$$167$$ −4.78731e20 −0.00219568 −0.00109784 0.999999i $$-0.500349\pi$$
−0.00109784 + 0.999999i $$0.500349\pi$$
$$168$$ 0 0
$$169$$ −2.29909e23 −0.930562
$$170$$ 5.91234e23 2.24929
$$171$$ 0 0
$$172$$ −6.38118e22 −0.214710
$$173$$ 1.61804e23 0.512277 0.256139 0.966640i $$-0.417550\pi$$
0.256139 + 0.966640i $$0.417550\pi$$
$$174$$ 0 0
$$175$$ 6.71138e23 1.88327
$$176$$ −3.50829e23 −0.927284
$$177$$ 0 0
$$178$$ 7.19050e22 0.168791
$$179$$ 8.76377e22 0.193970 0.0969849 0.995286i $$-0.469080\pi$$
0.0969849 + 0.995286i $$0.469080\pi$$
$$180$$ 0 0
$$181$$ 9.36624e22 0.184476 0.0922381 0.995737i $$-0.470598\pi$$
0.0922381 + 0.995737i $$0.470598\pi$$
$$182$$ 1.21865e23 0.226532
$$183$$ 0 0
$$184$$ 4.86885e23 0.806936
$$185$$ 1.42970e23 0.223842
$$186$$ 0 0
$$187$$ −5.28422e23 −0.738999
$$188$$ −2.52024e23 −0.333260
$$189$$ 0 0
$$190$$ −9.67843e23 −1.14523
$$191$$ −1.20858e24 −1.35340 −0.676699 0.736260i $$-0.736590\pi$$
−0.676699 + 0.736260i $$0.736590\pi$$
$$192$$ 0 0
$$193$$ −1.78822e24 −1.79502 −0.897509 0.440997i $$-0.854625\pi$$
−0.897509 + 0.440997i $$0.854625\pi$$
$$194$$ −9.88096e22 −0.0939466
$$195$$ 0 0
$$196$$ −2.38775e23 −0.203845
$$197$$ −1.90963e24 −1.54545 −0.772723 0.634743i $$-0.781106\pi$$
−0.772723 + 0.634743i $$0.781106\pi$$
$$198$$ 0 0
$$199$$ 1.44254e24 1.04995 0.524977 0.851116i $$-0.324074\pi$$
0.524977 + 0.851116i $$0.324074\pi$$
$$200$$ 2.60261e24 1.79719
$$201$$ 0 0
$$202$$ 7.47216e23 0.464790
$$203$$ 1.09008e24 0.643801
$$204$$ 0 0
$$205$$ 9.06739e23 0.483133
$$206$$ −3.18165e24 −1.61082
$$207$$ 0 0
$$208$$ 7.16728e23 0.327861
$$209$$ 8.65020e23 0.376262
$$210$$ 0 0
$$211$$ 3.98848e24 1.56979 0.784895 0.619629i $$-0.212717\pi$$
0.784895 + 0.619629i $$0.212717\pi$$
$$212$$ 1.93080e24 0.723119
$$213$$ 0 0
$$214$$ −4.21295e24 −1.42969
$$215$$ −2.98032e24 −0.963072
$$216$$ 0 0
$$217$$ −3.69858e24 −1.08445
$$218$$ 7.14833e24 1.99715
$$219$$ 0 0
$$220$$ 2.36564e24 0.600495
$$221$$ 1.07954e24 0.261289
$$222$$ 0 0
$$223$$ −4.62963e24 −1.01940 −0.509700 0.860352i $$-0.670244\pi$$
−0.509700 + 0.860352i $$0.670244\pi$$
$$224$$ 2.73354e24 0.574278
$$225$$ 0 0
$$226$$ 6.01188e24 1.15046
$$227$$ 3.43010e24 0.626664 0.313332 0.949644i $$-0.398555\pi$$
0.313332 + 0.949644i $$0.398555\pi$$
$$228$$ 0 0
$$229$$ 8.11792e23 0.135261 0.0676304 0.997710i $$-0.478456\pi$$
0.0676304 + 0.997710i $$0.478456\pi$$
$$230$$ −1.67273e25 −2.66246
$$231$$ 0 0
$$232$$ 4.22724e24 0.614374
$$233$$ −8.22188e23 −0.114218 −0.0571089 0.998368i $$-0.518188\pi$$
−0.0571089 + 0.998368i $$0.518188\pi$$
$$234$$ 0 0
$$235$$ −1.17707e25 −1.49483
$$236$$ −1.36421e24 −0.165693
$$237$$ 0 0
$$238$$ 7.66844e24 0.852411
$$239$$ 8.85525e24 0.941940 0.470970 0.882149i $$-0.343904\pi$$
0.470970 + 0.882149i $$0.343904\pi$$
$$240$$ 0 0
$$241$$ 7.46934e24 0.727953 0.363977 0.931408i $$-0.381419\pi$$
0.363977 + 0.931408i $$0.381419\pi$$
$$242$$ 5.68472e24 0.530454
$$243$$ 0 0
$$244$$ 3.83228e24 0.327992
$$245$$ −1.11520e25 −0.914339
$$246$$ 0 0
$$247$$ −1.76720e24 −0.133035
$$248$$ −1.43427e25 −1.03488
$$249$$ 0 0
$$250$$ −5.52091e25 −3.66134
$$251$$ −9.46474e23 −0.0601914 −0.0300957 0.999547i $$-0.509581\pi$$
−0.0300957 + 0.999547i $$0.509581\pi$$
$$252$$ 0 0
$$253$$ 1.49502e25 0.874744
$$254$$ −2.36979e24 −0.133032
$$255$$ 0 0
$$256$$ 2.08004e25 1.07535
$$257$$ 1.91825e25 0.951936 0.475968 0.879463i $$-0.342098\pi$$
0.475968 + 0.879463i $$0.342098\pi$$
$$258$$ 0 0
$$259$$ 1.85435e24 0.0848290
$$260$$ −4.83289e24 −0.212318
$$261$$ 0 0
$$262$$ −4.23837e25 −1.71805
$$263$$ −8.88429e23 −0.0346009 −0.0173004 0.999850i $$-0.505507\pi$$
−0.0173004 + 0.999850i $$0.505507\pi$$
$$264$$ 0 0
$$265$$ 9.01776e25 3.24352
$$266$$ −1.25531e25 −0.434006
$$267$$ 0 0
$$268$$ 8.21628e24 0.262579
$$269$$ 2.13847e25 0.657211 0.328605 0.944467i $$-0.393421\pi$$
0.328605 + 0.944467i $$0.393421\pi$$
$$270$$ 0 0
$$271$$ −1.56435e25 −0.444791 −0.222395 0.974957i $$-0.571388\pi$$
−0.222395 + 0.974957i $$0.571388\pi$$
$$272$$ 4.51007e25 1.23370
$$273$$ 0 0
$$274$$ 1.77699e25 0.450095
$$275$$ 7.99152e25 1.94821
$$276$$ 0 0
$$277$$ −8.04973e25 −1.81863 −0.909313 0.416112i $$-0.863392\pi$$
−0.909313 + 0.416112i $$0.863392\pi$$
$$278$$ −1.50456e25 −0.327294
$$279$$ 0 0
$$280$$ 4.66699e25 0.941623
$$281$$ −8.33171e25 −1.61926 −0.809632 0.586938i $$-0.800333\pi$$
−0.809632 + 0.586938i $$0.800333\pi$$
$$282$$ 0 0
$$283$$ −4.46130e24 −0.0804829 −0.0402415 0.999190i $$-0.512813\pi$$
−0.0402415 + 0.999190i $$0.512813\pi$$
$$284$$ 1.81209e25 0.315020
$$285$$ 0 0
$$286$$ 1.45109e25 0.234343
$$287$$ 1.17606e25 0.183092
$$288$$ 0 0
$$289$$ −1.16088e24 −0.0168020
$$290$$ −1.45230e26 −2.02711
$$291$$ 0 0
$$292$$ 1.47704e25 0.191810
$$293$$ −9.67128e25 −1.21164 −0.605820 0.795602i $$-0.707155\pi$$
−0.605820 + 0.795602i $$0.707155\pi$$
$$294$$ 0 0
$$295$$ −6.37151e25 −0.743209
$$296$$ 7.19099e24 0.0809516
$$297$$ 0 0
$$298$$ −1.56211e26 −1.63848
$$299$$ −3.05426e25 −0.309284
$$300$$ 0 0
$$301$$ −3.86554e25 −0.364974
$$302$$ 8.21155e25 0.748777
$$303$$ 0 0
$$304$$ −7.38293e25 −0.628140
$$305$$ 1.78986e26 1.47120
$$306$$ 0 0
$$307$$ 1.68163e26 1.29056 0.645278 0.763948i $$-0.276742\pi$$
0.645278 + 0.763948i $$0.276742\pi$$
$$308$$ 3.06828e25 0.227569
$$309$$ 0 0
$$310$$ 4.92755e26 3.41455
$$311$$ −2.30370e26 −1.54327 −0.771636 0.636065i $$-0.780561\pi$$
−0.771636 + 0.636065i $$0.780561\pi$$
$$312$$ 0 0
$$313$$ −2.79658e26 −1.75151 −0.875753 0.482759i $$-0.839635\pi$$
−0.875753 + 0.482759i $$0.839635\pi$$
$$314$$ 2.60756e26 1.57933
$$315$$ 0 0
$$316$$ 6.03875e25 0.342163
$$317$$ −2.98501e25 −0.163615 −0.0818075 0.996648i $$-0.526069\pi$$
−0.0818075 + 0.996648i $$0.526069\pi$$
$$318$$ 0 0
$$319$$ 1.29801e26 0.666002
$$320$$ 1.12203e26 0.557095
$$321$$ 0 0
$$322$$ −2.16957e26 −1.00899
$$323$$ −1.11202e26 −0.500596
$$324$$ 0 0
$$325$$ −1.63263e26 −0.688831
$$326$$ 8.35494e25 0.341317
$$327$$ 0 0
$$328$$ 4.56064e25 0.174723
$$329$$ −1.52669e26 −0.566492
$$330$$ 0 0
$$331$$ 2.55594e26 0.889933 0.444967 0.895547i $$-0.353216\pi$$
0.444967 + 0.895547i $$0.353216\pi$$
$$332$$ −3.51122e25 −0.118443
$$333$$ 0 0
$$334$$ 8.27248e23 0.00261998
$$335$$ 3.83740e26 1.17779
$$336$$ 0 0
$$337$$ −4.91931e25 −0.141837 −0.0709187 0.997482i $$-0.522593\pi$$
−0.0709187 + 0.997482i $$0.522593\pi$$
$$338$$ 3.97282e26 1.11039
$$339$$ 0 0
$$340$$ −3.04113e26 −0.798927
$$341$$ −4.40405e26 −1.12184
$$342$$ 0 0
$$343$$ −4.45381e26 −1.06695
$$344$$ −1.49902e26 −0.348292
$$345$$ 0 0
$$346$$ −2.79597e26 −0.611271
$$347$$ −2.98136e26 −0.632345 −0.316173 0.948702i $$-0.602398\pi$$
−0.316173 + 0.948702i $$0.602398\pi$$
$$348$$ 0 0
$$349$$ 7.72834e26 1.54319 0.771595 0.636115i $$-0.219459\pi$$
0.771595 + 0.636115i $$0.219459\pi$$
$$350$$ −1.15973e27 −2.24720
$$351$$ 0 0
$$352$$ 3.25494e26 0.594081
$$353$$ 7.30755e26 1.29461 0.647303 0.762233i $$-0.275897\pi$$
0.647303 + 0.762233i $$0.275897\pi$$
$$354$$ 0 0
$$355$$ 8.46336e26 1.41301
$$356$$ −3.69858e25 −0.0599529
$$357$$ 0 0
$$358$$ −1.51438e26 −0.231453
$$359$$ 1.58936e25 0.0235901 0.0117951 0.999930i $$-0.496245\pi$$
0.0117951 + 0.999930i $$0.496245\pi$$
$$360$$ 0 0
$$361$$ −5.32173e26 −0.745121
$$362$$ −1.61849e26 −0.220125
$$363$$ 0 0
$$364$$ −6.26836e25 −0.0804618
$$365$$ 6.89849e26 0.860358
$$366$$ 0 0
$$367$$ −1.40734e27 −1.65732 −0.828660 0.559752i $$-0.810896\pi$$
−0.828660 + 0.559752i $$0.810896\pi$$
$$368$$ −1.27600e27 −1.46032
$$369$$ 0 0
$$370$$ −2.47052e26 −0.267098
$$371$$ 1.16962e27 1.22919
$$372$$ 0 0
$$373$$ −9.30077e26 −0.923797 −0.461898 0.886933i $$-0.652831\pi$$
−0.461898 + 0.886933i $$0.652831\pi$$
$$374$$ 9.13112e26 0.881805
$$375$$ 0 0
$$376$$ −5.92035e26 −0.540599
$$377$$ −2.65177e26 −0.235479
$$378$$ 0 0
$$379$$ 2.18541e27 1.83578 0.917892 0.396830i $$-0.129890\pi$$
0.917892 + 0.396830i $$0.129890\pi$$
$$380$$ 4.97830e26 0.406774
$$381$$ 0 0
$$382$$ 2.08843e27 1.61493
$$383$$ −2.10347e27 −1.58252 −0.791258 0.611482i $$-0.790574\pi$$
−0.791258 + 0.611482i $$0.790574\pi$$
$$384$$ 0 0
$$385$$ 1.43304e27 1.02075
$$386$$ 3.09004e27 2.14189
$$387$$ 0 0
$$388$$ 5.08247e25 0.0333689
$$389$$ 2.97815e26 0.190316 0.0951582 0.995462i $$-0.469664\pi$$
0.0951582 + 0.995462i $$0.469664\pi$$
$$390$$ 0 0
$$391$$ −1.92192e27 −1.16380
$$392$$ −5.60912e26 −0.330667
$$393$$ 0 0
$$394$$ 3.29984e27 1.84409
$$395$$ 2.82039e27 1.53476
$$396$$ 0 0
$$397$$ 6.36504e26 0.328474 0.164237 0.986421i $$-0.447484\pi$$
0.164237 + 0.986421i $$0.447484\pi$$
$$398$$ −2.49271e27 −1.25285
$$399$$ 0 0
$$400$$ −6.82075e27 −3.25239
$$401$$ −2.43888e27 −1.13286 −0.566428 0.824111i $$-0.691675\pi$$
−0.566428 + 0.824111i $$0.691675\pi$$
$$402$$ 0 0
$$403$$ 8.99728e26 0.396651
$$404$$ −3.84346e26 −0.165089
$$405$$ 0 0
$$406$$ −1.88367e27 −0.768211
$$407$$ 2.20805e26 0.0877542
$$408$$ 0 0
$$409$$ 5.48032e26 0.206876 0.103438 0.994636i $$-0.467016\pi$$
0.103438 + 0.994636i $$0.467016\pi$$
$$410$$ −1.56684e27 −0.576495
$$411$$ 0 0
$$412$$ 1.63654e27 0.572146
$$413$$ −8.26399e26 −0.281653
$$414$$ 0 0
$$415$$ −1.63991e27 −0.531269
$$416$$ −6.64970e26 −0.210050
$$417$$ 0 0
$$418$$ −1.49475e27 −0.448971
$$419$$ 6.08246e27 1.78169 0.890844 0.454309i $$-0.150114\pi$$
0.890844 + 0.454309i $$0.150114\pi$$
$$420$$ 0 0
$$421$$ −4.05990e27 −1.13124 −0.565618 0.824667i $$-0.691362\pi$$
−0.565618 + 0.824667i $$0.691362\pi$$
$$422$$ −6.89209e27 −1.87314
$$423$$ 0 0
$$424$$ 4.53568e27 1.17301
$$425$$ −1.02735e28 −2.59199
$$426$$ 0 0
$$427$$ 2.32149e27 0.557537
$$428$$ 2.16702e27 0.507811
$$429$$ 0 0
$$430$$ 5.14999e27 1.14918
$$431$$ 7.87214e27 1.71428 0.857140 0.515084i $$-0.172239\pi$$
0.857140 + 0.515084i $$0.172239\pi$$
$$432$$ 0 0
$$433$$ −1.73785e27 −0.360486 −0.180243 0.983622i $$-0.557688\pi$$
−0.180243 + 0.983622i $$0.557688\pi$$
$$434$$ 6.39115e27 1.29401
$$435$$ 0 0
$$436$$ −3.67689e27 −0.709369
$$437$$ 3.14615e27 0.592550
$$438$$ 0 0
$$439$$ 8.37416e27 1.50336 0.751681 0.659526i $$-0.229243\pi$$
0.751681 + 0.659526i $$0.229243\pi$$
$$440$$ 5.55717e27 0.974094
$$441$$ 0 0
$$442$$ −1.86545e27 −0.311781
$$443$$ 3.30286e25 0.00539077 0.00269539 0.999996i $$-0.499142\pi$$
0.00269539 + 0.999996i $$0.499142\pi$$
$$444$$ 0 0
$$445$$ −1.72742e27 −0.268916
$$446$$ 7.99999e27 1.21639
$$447$$ 0 0
$$448$$ 1.45530e27 0.211121
$$449$$ −5.21713e27 −0.739341 −0.369670 0.929163i $$-0.620529\pi$$
−0.369670 + 0.929163i $$0.620529\pi$$
$$450$$ 0 0
$$451$$ 1.40038e27 0.189406
$$452$$ −3.09233e27 −0.408632
$$453$$ 0 0
$$454$$ −5.92721e27 −0.747763
$$455$$ −2.92763e27 −0.360908
$$456$$ 0 0
$$457$$ 2.15211e26 0.0253363 0.0126682 0.999920i $$-0.495967\pi$$
0.0126682 + 0.999920i $$0.495967\pi$$
$$458$$ −1.40278e27 −0.161399
$$459$$ 0 0
$$460$$ 8.60403e27 0.945681
$$461$$ −1.68699e28 −1.81239 −0.906197 0.422855i $$-0.861028\pi$$
−0.906197 + 0.422855i $$0.861028\pi$$
$$462$$ 0 0
$$463$$ −1.90352e28 −1.95415 −0.977074 0.212898i $$-0.931710\pi$$
−0.977074 + 0.212898i $$0.931710\pi$$
$$464$$ −1.10785e28 −1.11184
$$465$$ 0 0
$$466$$ 1.42074e27 0.136290
$$467$$ 1.21027e28 1.13515 0.567576 0.823321i $$-0.307881\pi$$
0.567576 + 0.823321i $$0.307881\pi$$
$$468$$ 0 0
$$469$$ 4.97720e27 0.446344
$$470$$ 2.03398e28 1.78369
$$471$$ 0 0
$$472$$ −3.20469e27 −0.268779
$$473$$ −4.60286e27 −0.377559
$$474$$ 0 0
$$475$$ 1.68175e28 1.31971
$$476$$ −3.94442e27 −0.302768
$$477$$ 0 0
$$478$$ −1.53019e28 −1.12396
$$479$$ −6.95253e27 −0.499597 −0.249798 0.968298i $$-0.580364\pi$$
−0.249798 + 0.968298i $$0.580364\pi$$
$$480$$ 0 0
$$481$$ −4.51095e26 −0.0310274
$$482$$ −1.29070e28 −0.868625
$$483$$ 0 0
$$484$$ −2.92405e27 −0.188412
$$485$$ 2.37376e27 0.149675
$$486$$ 0 0
$$487$$ −1.06412e28 −0.642596 −0.321298 0.946978i $$-0.604119\pi$$
−0.321298 + 0.946978i $$0.604119\pi$$
$$488$$ 9.00250e27 0.532053
$$489$$ 0 0
$$490$$ 1.92706e28 1.09103
$$491$$ −1.68064e28 −0.931361 −0.465681 0.884953i $$-0.654190\pi$$
−0.465681 + 0.884953i $$0.654190\pi$$
$$492$$ 0 0
$$493$$ −1.66865e28 −0.886079
$$494$$ 3.05372e27 0.158743
$$495$$ 0 0
$$496$$ 3.75885e28 1.87283
$$497$$ 1.09772e28 0.535486
$$498$$ 0 0
$$499$$ −5.12285e27 −0.239583 −0.119792 0.992799i $$-0.538223\pi$$
−0.119792 + 0.992799i $$0.538223\pi$$
$$500$$ 2.83979e28 1.30047
$$501$$ 0 0
$$502$$ 1.63551e27 0.0718229
$$503$$ −1.99606e28 −0.858442 −0.429221 0.903200i $$-0.641212\pi$$
−0.429221 + 0.903200i $$0.641212\pi$$
$$504$$ 0 0
$$505$$ −1.79508e28 −0.740499
$$506$$ −2.58339e28 −1.04378
$$507$$ 0 0
$$508$$ 1.21895e27 0.0472516
$$509$$ 2.57966e27 0.0979550 0.0489775 0.998800i $$-0.484404\pi$$
0.0489775 + 0.998800i $$0.484404\pi$$
$$510$$ 0 0
$$511$$ 8.94749e27 0.326048
$$512$$ −3.81989e27 −0.136369
$$513$$ 0 0
$$514$$ −3.31474e28 −1.13589
$$515$$ 7.64346e28 2.56634
$$516$$ 0 0
$$517$$ −1.81789e28 −0.586026
$$518$$ −3.20432e27 −0.101222
$$519$$ 0 0
$$520$$ −1.13531e28 −0.344412
$$521$$ 2.61230e28 0.776652 0.388326 0.921522i $$-0.373053\pi$$
0.388326 + 0.921522i $$0.373053\pi$$
$$522$$ 0 0
$$523$$ 6.70750e28 1.91555 0.957774 0.287523i $$-0.0928317\pi$$
0.957774 + 0.287523i $$0.0928317\pi$$
$$524$$ 2.18009e28 0.610233
$$525$$ 0 0
$$526$$ 1.53520e27 0.0412873
$$527$$ 5.66161e28 1.49255
$$528$$ 0 0
$$529$$ 1.49036e28 0.377577
$$530$$ −1.55827e29 −3.87031
$$531$$ 0 0
$$532$$ 6.45696e27 0.154155
$$533$$ −2.86092e27 −0.0669684
$$534$$ 0 0
$$535$$ 1.01210e29 2.27777
$$536$$ 1.93011e28 0.425943
$$537$$ 0 0
$$538$$ −3.69528e28 −0.784212
$$539$$ −1.72233e28 −0.358454
$$540$$ 0 0
$$541$$ −2.15196e28 −0.430787 −0.215394 0.976527i $$-0.569103\pi$$
−0.215394 + 0.976527i $$0.569103\pi$$
$$542$$ 2.70319e28 0.530743
$$543$$ 0 0
$$544$$ −4.18438e28 −0.790392
$$545$$ −1.71728e29 −3.18185
$$546$$ 0 0
$$547$$ −7.46789e28 −1.33147 −0.665734 0.746189i $$-0.731882\pi$$
−0.665734 + 0.746189i $$0.731882\pi$$
$$548$$ −9.14032e27 −0.159870
$$549$$ 0 0
$$550$$ −1.38093e29 −2.32469
$$551$$ 2.73156e28 0.451148
$$552$$ 0 0
$$553$$ 3.65811e28 0.581625
$$554$$ 1.39099e29 2.17006
$$555$$ 0 0
$$556$$ 7.73899e27 0.116252
$$557$$ 7.95166e28 1.17214 0.586068 0.810262i $$-0.300675\pi$$
0.586068 + 0.810262i $$0.300675\pi$$
$$558$$ 0 0
$$559$$ 9.40343e27 0.133494
$$560$$ −1.22309e29 −1.70406
$$561$$ 0 0
$$562$$ 1.43972e29 1.93217
$$563$$ −5.46305e28 −0.719609 −0.359805 0.933028i $$-0.617157\pi$$
−0.359805 + 0.933028i $$0.617157\pi$$
$$564$$ 0 0
$$565$$ −1.44427e29 −1.83290
$$566$$ 7.70912e27 0.0960356
$$567$$ 0 0
$$568$$ 4.25683e28 0.511010
$$569$$ −9.43478e28 −1.11187 −0.555933 0.831227i $$-0.687639\pi$$
−0.555933 + 0.831227i $$0.687639\pi$$
$$570$$ 0 0
$$571$$ 8.05027e28 0.914390 0.457195 0.889367i $$-0.348854\pi$$
0.457195 + 0.889367i $$0.348854\pi$$
$$572$$ −7.46400e27 −0.0832364
$$573$$ 0 0
$$574$$ −2.03223e28 −0.218473
$$575$$ 2.90659e29 3.06811
$$576$$ 0 0
$$577$$ −1.67132e28 −0.170104 −0.0850519 0.996377i $$-0.527106\pi$$
−0.0850519 + 0.996377i $$0.527106\pi$$
$$578$$ 2.00600e27 0.0200488
$$579$$ 0 0
$$580$$ 7.47020e28 0.720010
$$581$$ −2.12700e28 −0.201334
$$582$$ 0 0
$$583$$ 1.39272e29 1.27158
$$584$$ 3.46975e28 0.311145
$$585$$ 0 0
$$586$$ 1.67120e29 1.44578
$$587$$ −3.15730e28 −0.268297 −0.134149 0.990961i $$-0.542830\pi$$
−0.134149 + 0.990961i $$0.542830\pi$$
$$588$$ 0 0
$$589$$ −9.26799e28 −0.759932
$$590$$ 1.10100e29 0.886829
$$591$$ 0 0
$$592$$ −1.88457e28 −0.146499
$$593$$ 5.48493e27 0.0418887 0.0209443 0.999781i $$-0.493333\pi$$
0.0209443 + 0.999781i $$0.493333\pi$$
$$594$$ 0 0
$$595$$ −1.84223e29 −1.35805
$$596$$ 8.03502e28 0.581971
$$597$$ 0 0
$$598$$ 5.27776e28 0.369051
$$599$$ −1.25621e29 −0.863136 −0.431568 0.902080i $$-0.642040\pi$$
−0.431568 + 0.902080i $$0.642040\pi$$
$$600$$ 0 0
$$601$$ 3.99325e28 0.264938 0.132469 0.991187i $$-0.457709\pi$$
0.132469 + 0.991187i $$0.457709\pi$$
$$602$$ 6.67965e28 0.435503
$$603$$ 0 0
$$604$$ −4.22378e28 −0.265958
$$605$$ −1.36567e29 −0.845115
$$606$$ 0 0
$$607$$ −2.46990e29 −1.47638 −0.738189 0.674594i $$-0.764319\pi$$
−0.738189 + 0.674594i $$0.764319\pi$$
$$608$$ 6.84978e28 0.402429
$$609$$ 0 0
$$610$$ −3.09288e29 −1.75549
$$611$$ 3.71387e28 0.207202
$$612$$ 0 0
$$613$$ 2.63911e28 0.142273 0.0711364 0.997467i $$-0.477337\pi$$
0.0711364 + 0.997467i $$0.477337\pi$$
$$614$$ −2.90585e29 −1.53995
$$615$$ 0 0
$$616$$ 7.20777e28 0.369151
$$617$$ −3.09820e29 −1.55997 −0.779984 0.625800i $$-0.784773\pi$$
−0.779984 + 0.625800i $$0.784773\pi$$
$$618$$ 0 0
$$619$$ −2.50758e29 −1.22040 −0.610202 0.792246i $$-0.708912\pi$$
−0.610202 + 0.792246i $$0.708912\pi$$
$$620$$ −2.53459e29 −1.21281
$$621$$ 0 0
$$622$$ 3.98080e29 1.84150
$$623$$ −2.24050e28 −0.101911
$$624$$ 0 0
$$625$$ 7.31956e29 3.21918
$$626$$ 4.83249e29 2.08997
$$627$$ 0 0
$$628$$ −1.34125e29 −0.560963
$$629$$ −2.83855e28 −0.116752
$$630$$ 0 0
$$631$$ −4.32770e28 −0.172167 −0.0860833 0.996288i $$-0.527435\pi$$
−0.0860833 + 0.996288i $$0.527435\pi$$
$$632$$ 1.41858e29 0.555039
$$633$$ 0 0
$$634$$ 5.15809e28 0.195232
$$635$$ 5.69309e28 0.211945
$$636$$ 0 0
$$637$$ 3.51864e28 0.126739
$$638$$ −2.24296e29 −0.794702
$$639$$ 0 0
$$640$$ −6.35872e29 −2.18012
$$641$$ 8.73381e28 0.294574 0.147287 0.989094i $$-0.452946\pi$$
0.147287 + 0.989094i $$0.452946\pi$$
$$642$$ 0 0
$$643$$ −4.72013e29 −1.54077 −0.770386 0.637578i $$-0.779937\pi$$
−0.770386 + 0.637578i $$0.779937\pi$$
$$644$$ 1.11596e29 0.358383
$$645$$ 0 0
$$646$$ 1.92158e29 0.597332
$$647$$ −1.26799e28 −0.0387812 −0.0193906 0.999812i $$-0.506173\pi$$
−0.0193906 + 0.999812i $$0.506173\pi$$
$$648$$ 0 0
$$649$$ −9.84027e28 −0.291365
$$650$$ 2.82119e29 0.821943
$$651$$ 0 0
$$652$$ −4.29753e28 −0.121233
$$653$$ 2.76226e29 0.766790 0.383395 0.923584i $$-0.374755\pi$$
0.383395 + 0.923584i $$0.374755\pi$$
$$654$$ 0 0
$$655$$ 1.01821e30 2.73718
$$656$$ −1.19523e29 −0.316198
$$657$$ 0 0
$$658$$ 2.63812e29 0.675962
$$659$$ −6.46511e28 −0.163034 −0.0815172 0.996672i $$-0.525977\pi$$
−0.0815172 + 0.996672i $$0.525977\pi$$
$$660$$ 0 0
$$661$$ −2.27730e29 −0.556295 −0.278147 0.960538i $$-0.589720\pi$$
−0.278147 + 0.960538i $$0.589720\pi$$
$$662$$ −4.41667e29 −1.06191
$$663$$ 0 0
$$664$$ −8.24829e28 −0.192132
$$665$$ 3.01572e29 0.691454
$$666$$ 0 0
$$667$$ 4.72097e29 1.04884
$$668$$ −4.25512e26 −0.000930591 0
$$669$$ 0 0
$$670$$ −6.63103e29 −1.40539
$$671$$ 2.76429e29 0.576763
$$672$$ 0 0
$$673$$ −3.79243e29 −0.766936 −0.383468 0.923554i $$-0.625270\pi$$
−0.383468 + 0.923554i $$0.625270\pi$$
$$674$$ 8.50057e28 0.169246
$$675$$ 0 0
$$676$$ −2.04350e29 −0.394398
$$677$$ 3.39717e29 0.645559 0.322780 0.946474i $$-0.395383\pi$$
0.322780 + 0.946474i $$0.395383\pi$$
$$678$$ 0 0
$$679$$ 3.07882e28 0.0567220
$$680$$ −7.14400e29 −1.29598
$$681$$ 0 0
$$682$$ 7.61020e29 1.33863
$$683$$ −4.54742e29 −0.787677 −0.393838 0.919180i $$-0.628853\pi$$
−0.393838 + 0.919180i $$0.628853\pi$$
$$684$$ 0 0
$$685$$ −4.26897e29 −0.717088
$$686$$ 7.69619e29 1.27313
$$687$$ 0 0
$$688$$ 3.92853e29 0.630306
$$689$$ −2.84526e29 −0.449594
$$690$$ 0 0
$$691$$ −3.71838e29 −0.569947 −0.284974 0.958535i $$-0.591985\pi$$
−0.284974 + 0.958535i $$0.591985\pi$$
$$692$$ 1.43817e29 0.217118
$$693$$ 0 0
$$694$$ 5.15178e29 0.754542
$$695$$ 3.61448e29 0.521442
$$696$$ 0 0
$$697$$ −1.80026e29 −0.251994
$$698$$ −1.33546e30 −1.84140
$$699$$ 0 0
$$700$$ 5.96529e29 0.798183
$$701$$ 9.37969e29 1.23637 0.618186 0.786032i $$-0.287868\pi$$
0.618186 + 0.786032i $$0.287868\pi$$
$$702$$ 0 0
$$703$$ 4.64668e28 0.0594445
$$704$$ 1.73288e29 0.218401
$$705$$ 0 0
$$706$$ −1.26275e30 −1.54478
$$707$$ −2.32826e29 −0.280626
$$708$$ 0 0
$$709$$ −7.54578e29 −0.882914 −0.441457 0.897282i $$-0.645538\pi$$
−0.441457 + 0.897282i $$0.645538\pi$$
$$710$$ −1.46247e30 −1.68606
$$711$$ 0 0
$$712$$ −8.68842e28 −0.0972525
$$713$$ −1.60179e30 −1.76671
$$714$$ 0 0
$$715$$ −3.48605e29 −0.373353
$$716$$ 7.78952e28 0.0822098
$$717$$ 0 0
$$718$$ −2.74641e28 −0.0281488
$$719$$ −1.22754e30 −1.23989 −0.619944 0.784646i $$-0.712845\pi$$
−0.619944 + 0.784646i $$0.712845\pi$$
$$720$$ 0 0
$$721$$ 9.91374e29 0.972561
$$722$$ 9.19594e29 0.889111
$$723$$ 0 0
$$724$$ 8.32502e28 0.0781862
$$725$$ 2.52356e30 2.33596
$$726$$ 0 0
$$727$$ 9.04407e29 0.813303 0.406652 0.913583i $$-0.366696\pi$$
0.406652 + 0.913583i $$0.366696\pi$$
$$728$$ −1.47252e29 −0.130521
$$729$$ 0 0
$$730$$ −1.19206e30 −1.02662
$$731$$ 5.91719e29 0.502323
$$732$$ 0 0
$$733$$ −1.38874e30 −1.14559 −0.572795 0.819699i $$-0.694141\pi$$
−0.572795 + 0.819699i $$0.694141\pi$$
$$734$$ 2.43189e30 1.97759
$$735$$ 0 0
$$736$$ 1.18385e30 0.935578
$$737$$ 5.92655e29 0.461736
$$738$$ 0 0
$$739$$ 2.11506e30 1.60161 0.800804 0.598927i $$-0.204406\pi$$
0.800804 + 0.598927i $$0.204406\pi$$
$$740$$ 1.27076e29 0.0948705
$$741$$ 0 0
$$742$$ −2.02111e30 −1.46673
$$743$$ −2.26805e30 −1.62282 −0.811411 0.584476i $$-0.801300\pi$$
−0.811411 + 0.584476i $$0.801300\pi$$
$$744$$ 0 0
$$745$$ 3.75274e30 2.61040
$$746$$ 1.60717e30 1.10231
$$747$$ 0 0
$$748$$ −4.69678e29 −0.313208
$$749$$ 1.31272e30 0.863202
$$750$$ 0 0
$$751$$ −6.98349e29 −0.446533 −0.223266 0.974757i $$-0.571672\pi$$
−0.223266 + 0.974757i $$0.571672\pi$$
$$752$$ 1.55157e30 0.978326
$$753$$ 0 0
$$754$$ 4.58226e29 0.280984
$$755$$ −1.97271e30 −1.19294
$$756$$ 0 0
$$757$$ −3.79778e29 −0.223369 −0.111685 0.993744i $$-0.535625\pi$$
−0.111685 + 0.993744i $$0.535625\pi$$
$$758$$ −3.77639e30 −2.19054
$$759$$ 0 0
$$760$$ 1.16946e30 0.659849
$$761$$ 7.50371e29 0.417577 0.208789 0.977961i $$-0.433048\pi$$
0.208789 + 0.977961i $$0.433048\pi$$
$$762$$ 0 0
$$763$$ −2.22736e30 −1.20582
$$764$$ −1.07423e30 −0.573608
$$765$$ 0 0
$$766$$ 3.63479e30 1.88833
$$767$$ 2.01032e29 0.103018
$$768$$ 0 0
$$769$$ 2.16884e29 0.108144 0.0540719 0.998537i $$-0.482780\pi$$
0.0540719 + 0.998537i $$0.482780\pi$$
$$770$$ −2.47629e30 −1.21800
$$771$$ 0 0
$$772$$ −1.58943e30 −0.760779
$$773$$ 3.08128e30 1.45495 0.727473 0.686136i $$-0.240695\pi$$
0.727473 + 0.686136i $$0.240695\pi$$
$$774$$ 0 0
$$775$$ −8.56227e30 −3.93478
$$776$$ 1.19394e29 0.0541294
$$777$$ 0 0
$$778$$ −5.14625e29 −0.227094
$$779$$ 2.94700e29 0.128303
$$780$$ 0 0
$$781$$ 1.30710e30 0.553951
$$782$$ 3.32107e30 1.38870
$$783$$ 0 0
$$784$$ 1.47000e30 0.598412
$$785$$ −6.26430e30 −2.51618
$$786$$ 0 0
$$787$$ 1.46460e30 0.572775 0.286387 0.958114i $$-0.407546\pi$$
0.286387 + 0.958114i $$0.407546\pi$$
$$788$$ −1.69734e30 −0.655004
$$789$$ 0 0
$$790$$ −4.87363e30 −1.83134
$$791$$ −1.87325e30 −0.694612
$$792$$ 0 0
$$793$$ −5.64732e29 −0.203927
$$794$$ −1.09988e30 −0.391949
$$795$$ 0 0
$$796$$ 1.28217e30 0.445000
$$797$$ −1.27731e30 −0.437505 −0.218752 0.975780i $$-0.570199\pi$$
−0.218752 + 0.975780i $$0.570199\pi$$
$$798$$ 0 0
$$799$$ 2.33698e30 0.779677
$$800$$ 6.32819e30 2.08370
$$801$$ 0 0
$$802$$ 4.21439e30 1.35177
$$803$$ 1.06541e30 0.337292
$$804$$ 0 0
$$805$$ 5.21208e30 1.60751
$$806$$ −1.55473e30 −0.473301
$$807$$ 0 0
$$808$$ −9.02876e29 −0.267799
$$809$$ 4.24975e30 1.24424 0.622120 0.782922i $$-0.286272\pi$$
0.622120 + 0.782922i $$0.286272\pi$$
$$810$$ 0 0
$$811$$ 2.05863e30 0.587298 0.293649 0.955913i $$-0.405130\pi$$
0.293649 + 0.955913i $$0.405130\pi$$
$$812$$ 9.68902e29 0.272861
$$813$$ 0 0
$$814$$ −3.81551e29 −0.104712
$$815$$ −2.00716e30 −0.543784
$$816$$ 0 0
$$817$$ −9.68636e29 −0.255758
$$818$$ −9.46999e29 −0.246854
$$819$$ 0 0
$$820$$ 8.05938e29 0.204765
$$821$$ 1.80717e29 0.0453309 0.0226655 0.999743i $$-0.492785\pi$$
0.0226655 + 0.999743i $$0.492785\pi$$
$$822$$ 0 0
$$823$$ −6.23532e30 −1.52462 −0.762308 0.647214i $$-0.775934\pi$$
−0.762308 + 0.647214i $$0.775934\pi$$
$$824$$ 3.84445e30 0.928107
$$825$$ 0 0
$$826$$ 1.42802e30 0.336080
$$827$$ −3.37179e30 −0.783524 −0.391762 0.920067i $$-0.628134\pi$$
−0.391762 + 0.920067i $$0.628134\pi$$
$$828$$ 0 0
$$829$$ −2.70711e29 −0.0613315 −0.0306658 0.999530i $$-0.509763\pi$$
−0.0306658 + 0.999530i $$0.509763\pi$$
$$830$$ 2.83376e30 0.633933
$$831$$ 0 0
$$832$$ −3.54020e29 −0.0772205
$$833$$ 2.21413e30 0.476904
$$834$$ 0 0
$$835$$ −1.98735e28 −0.00417413
$$836$$ 7.68857e29 0.159470
$$837$$ 0 0
$$838$$ −1.05105e31 −2.12599
$$839$$ 7.45457e30 1.48909 0.744547 0.667570i $$-0.232666\pi$$
0.744547 + 0.667570i $$0.232666\pi$$
$$840$$ 0 0
$$841$$ −1.03399e30 −0.201446
$$842$$ 7.01551e30 1.34984
$$843$$ 0 0
$$844$$ 3.54509e30 0.665321
$$845$$ −9.54415e30 −1.76906
$$846$$ 0 0
$$847$$ −1.77131e30 −0.320272
$$848$$ −1.18868e31 −2.12280
$$849$$ 0 0
$$850$$ 1.77526e31 3.09287
$$851$$ 8.03088e29 0.138198
$$852$$ 0 0
$$853$$ −6.10653e30 −1.02525 −0.512625 0.858613i $$-0.671327\pi$$
−0.512625 + 0.858613i $$0.671327\pi$$
$$854$$ −4.01153e30 −0.665277
$$855$$ 0 0
$$856$$ 5.09059e30 0.823746
$$857$$ 4.08307e30 0.652662 0.326331 0.945256i $$-0.394188\pi$$
0.326331 + 0.945256i $$0.394188\pi$$
$$858$$ 0 0
$$859$$ 5.27189e30 0.822316 0.411158 0.911564i $$-0.365125\pi$$
0.411158 + 0.911564i $$0.365125\pi$$
$$860$$ −2.64900e30 −0.408177
$$861$$ 0 0
$$862$$ −1.36031e31 −2.04555
$$863$$ −1.05537e31 −1.56780 −0.783902 0.620884i $$-0.786774\pi$$
−0.783902 + 0.620884i $$0.786774\pi$$
$$864$$ 0 0
$$865$$ 6.71693e30 0.973872
$$866$$ 3.00300e30 0.430147
$$867$$ 0 0
$$868$$ −3.28742e30 −0.459619
$$869$$ 4.35586e30 0.601681
$$870$$ 0 0
$$871$$ −1.21077e30 −0.163256
$$872$$ −8.63747e30 −1.15070
$$873$$ 0 0
$$874$$ −5.43655e30 −0.707056
$$875$$ 1.72027e31 2.21061
$$876$$ 0 0
$$877$$ 7.16069e30 0.898378 0.449189 0.893437i $$-0.351713\pi$$
0.449189 + 0.893437i $$0.351713\pi$$
$$878$$ −1.44706e31 −1.79388
$$879$$ 0 0
$$880$$ −1.45639e31 −1.76283
$$881$$ 1.05943e31 1.26714 0.633568 0.773687i $$-0.281590\pi$$
0.633568 + 0.773687i $$0.281590\pi$$
$$882$$ 0 0
$$883$$ 6.60744e30 0.771695 0.385848 0.922562i $$-0.373909\pi$$
0.385848 + 0.922562i $$0.373909\pi$$
$$884$$ 9.59531e29 0.110742
$$885$$ 0 0
$$886$$ −5.70734e28 −0.00643250
$$887$$ 2.74467e30 0.305698 0.152849 0.988250i $$-0.451155\pi$$
0.152849 + 0.988250i $$0.451155\pi$$
$$888$$ 0 0
$$889$$ 7.38406e29 0.0803205
$$890$$ 2.98497e30 0.320882
$$891$$ 0 0
$$892$$ −4.11496e30 −0.432051
$$893$$ −3.82561e30 −0.396973
$$894$$ 0 0
$$895$$ 3.63808e30 0.368749
$$896$$ −8.24741e30 −0.826196
$$897$$ 0 0
$$898$$ 9.01519e30 0.882213
$$899$$ −1.39071e31 −1.34512
$$900$$ 0 0
$$901$$ −1.79040e31 −1.69177
$$902$$ −2.41986e30 −0.226007
$$903$$ 0 0
$$904$$ −7.26427e30 −0.662863
$$905$$ 3.88819e30 0.350701
$$906$$ 0 0
$$907$$ 5.45568e30 0.480809 0.240404 0.970673i $$-0.422720\pi$$
0.240404 + 0.970673i $$0.422720\pi$$
$$908$$ 3.04878e30 0.265598
$$909$$ 0 0
$$910$$ 5.05894e30 0.430651
$$911$$ 8.75739e30 0.736940 0.368470 0.929640i $$-0.379882\pi$$
0.368470 + 0.929640i $$0.379882\pi$$
$$912$$ 0 0
$$913$$ −2.53270e30 −0.208277
$$914$$ −3.71884e29 −0.0302324
$$915$$ 0 0
$$916$$ 7.21547e29 0.0573274
$$917$$ 1.32064e31 1.03730
$$918$$ 0 0
$$919$$ 1.24295e31 0.954206 0.477103 0.878847i $$-0.341687\pi$$
0.477103 + 0.878847i $$0.341687\pi$$
$$920$$ 2.02119e31 1.53404
$$921$$ 0 0
$$922$$ 2.91512e31 2.16263
$$923$$ −2.67034e30 −0.195861
$$924$$ 0 0
$$925$$ 4.29285e30 0.307792
$$926$$ 3.28929e31 2.33177
$$927$$ 0 0
$$928$$ 1.02785e31 0.712319
$$929$$ −2.24310e31 −1.53703 −0.768517 0.639829i $$-0.779005\pi$$
−0.768517 + 0.639829i $$0.779005\pi$$
$$930$$ 0 0
$$931$$ −3.62451e30 −0.242816
$$932$$ −7.30787e29 −0.0484087
$$933$$ 0 0
$$934$$ −2.09134e31 −1.35451
$$935$$ −2.19362e31 −1.40488
$$936$$ 0 0
$$937$$ −1.10052e31 −0.689176 −0.344588 0.938754i $$-0.611981\pi$$
−0.344588 + 0.938754i $$0.611981\pi$$
$$938$$ −8.60060e30 −0.532597
$$939$$ 0 0
$$940$$ −1.04622e31 −0.633549
$$941$$ 2.38036e31 1.42545 0.712725 0.701444i $$-0.247461\pi$$
0.712725 + 0.701444i $$0.247461\pi$$
$$942$$ 0 0
$$943$$ 5.09332e30 0.298283
$$944$$ 8.39866e30 0.486412
$$945$$ 0 0
$$946$$ 7.95373e30 0.450520
$$947$$ −8.09762e30 −0.453610 −0.226805 0.973940i $$-0.572828\pi$$
−0.226805 + 0.973940i $$0.572828\pi$$
$$948$$ 0 0
$$949$$ −2.17660e30 −0.119257
$$950$$ −2.90607e31 −1.57474
$$951$$ 0 0
$$952$$ −9.26593e30 −0.491135
$$953$$ 3.42232e30 0.179410 0.0897048 0.995968i $$-0.471408\pi$$
0.0897048 + 0.995968i $$0.471408\pi$$
$$954$$ 0 0
$$955$$ −5.01715e31 −2.57290
$$956$$ 7.87083e30 0.399220
$$957$$ 0 0
$$958$$ 1.20140e31 0.596140
$$959$$ −5.53695e30 −0.271754
$$960$$ 0 0
$$961$$ 2.63603e31 1.26577
$$962$$ 7.79493e29 0.0370232
$$963$$ 0 0
$$964$$ 6.63899e30 0.308527
$$965$$ −7.42339e31 −3.41244
$$966$$ 0 0
$$967$$ 1.33121e31 0.598780 0.299390 0.954131i $$-0.403217\pi$$
0.299390 + 0.954131i $$0.403217\pi$$
$$968$$ −6.86896e30 −0.305633
$$969$$ 0 0
$$970$$ −4.10186e30 −0.178598
$$971$$ −9.71774e30 −0.418565 −0.209283 0.977855i $$-0.567113\pi$$
−0.209283 + 0.977855i $$0.567113\pi$$
$$972$$ 0 0
$$973$$ 4.68806e30 0.197610
$$974$$ 1.83881e31 0.766773
$$975$$ 0 0
$$976$$ −2.35932e31 −0.962861
$$977$$ 1.40637e31 0.567816 0.283908 0.958851i $$-0.408369\pi$$
0.283908 + 0.958851i $$0.408369\pi$$
$$978$$ 0 0
$$979$$ −2.66785e30 −0.105425
$$980$$ −9.91222e30 −0.387523
$$981$$ 0 0
$$982$$ 2.90414e31 1.11134
$$983$$ 4.87291e31 1.84492 0.922458 0.386098i $$-0.126177\pi$$
0.922458 + 0.386098i $$0.126177\pi$$
$$984$$ 0 0
$$985$$ −7.92741e31 −2.93799
$$986$$ 2.88343e31 1.05731
$$987$$ 0 0
$$988$$ −1.57074e30 −0.0563841
$$989$$ −1.67410e31 −0.594594
$$990$$ 0 0
$$991$$ 1.17847e30 0.0409773 0.0204887 0.999790i $$-0.493478\pi$$
0.0204887 + 0.999790i $$0.493478\pi$$
$$992$$ −3.48741e31 −1.19986
$$993$$ 0 0
$$994$$ −1.89685e31 −0.638965
$$995$$ 5.98838e31 1.99603
$$996$$ 0 0
$$997$$ 5.05438e31 1.64956 0.824781 0.565453i $$-0.191299\pi$$
0.824781 + 0.565453i $$0.191299\pi$$
$$998$$ 8.85229e30 0.285881
$$999$$ 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 9.22.a.a.1.1 1
3.2 odd 2 3.22.a.b.1.1 1
12.11 even 2 48.22.a.d.1.1 1
15.2 even 4 75.22.b.b.49.2 2
15.8 even 4 75.22.b.b.49.1 2
15.14 odd 2 75.22.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.b.1.1 1 3.2 odd 2
9.22.a.a.1.1 1 1.1 even 1 trivial
48.22.a.d.1.1 1 12.11 even 2
75.22.a.a.1.1 1 15.14 odd 2
75.22.b.b.49.1 2 15.8 even 4
75.22.b.b.49.2 2 15.2 even 4