# Properties

 Label 1521.4.a.t.1.1 Level $1521$ Weight $4$ Character 1521.1 Self dual yes Analytic conductor $89.742$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1521.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: $$89.7419051187$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.438447 q^{2} -7.80776 q^{4} +17.8078 q^{5} +5.43845 q^{7} -6.93087 q^{8} +O(q^{10})$$ $$q+0.438447 q^{2} -7.80776 q^{4} +17.8078 q^{5} +5.43845 q^{7} -6.93087 q^{8} +7.80776 q^{10} +22.4233 q^{11} +2.38447 q^{14} +59.4233 q^{16} -67.9848 q^{17} -80.8078 q^{19} -139.039 q^{20} +9.83143 q^{22} -140.531 q^{23} +192.116 q^{25} -42.4621 q^{28} +106.693 q^{29} -276.155 q^{31} +81.5009 q^{32} -29.8078 q^{34} +96.8466 q^{35} -4.29168 q^{37} -35.4299 q^{38} -123.423 q^{40} -227.769 q^{41} +27.5294 q^{43} -175.076 q^{44} -61.6155 q^{46} -318.617 q^{47} -313.423 q^{49} +84.2329 q^{50} +67.6562 q^{53} +399.309 q^{55} -37.6932 q^{56} +46.7793 q^{58} +291.115 q^{59} +663.311 q^{61} -121.080 q^{62} -439.652 q^{64} -425.101 q^{67} +530.810 q^{68} +42.4621 q^{70} +152.963 q^{71} +117.268 q^{73} -1.88167 q^{74} +630.928 q^{76} +121.948 q^{77} +202.462 q^{79} +1058.20 q^{80} -99.8647 q^{82} -336.155 q^{83} -1210.66 q^{85} +12.0702 q^{86} -155.413 q^{88} -718.194 q^{89} +1097.23 q^{92} -139.697 q^{94} -1439.01 q^{95} +759.368 q^{97} -137.420 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{2} + 5 q^{4} + 15 q^{5} + 15 q^{7} + 15 q^{8}+O(q^{10})$$ 2 * q + 5 * q^2 + 5 * q^4 + 15 * q^5 + 15 * q^7 + 15 * q^8 $$2 q + 5 q^{2} + 5 q^{4} + 15 q^{5} + 15 q^{7} + 15 q^{8} - 5 q^{10} - 17 q^{11} + 46 q^{14} + 57 q^{16} - 70 q^{17} - 141 q^{19} - 175 q^{20} - 170 q^{22} - 145 q^{23} + 75 q^{25} + 80 q^{28} - 34 q^{29} - 140 q^{31} - 105 q^{32} - 39 q^{34} + 70 q^{35} - 190 q^{37} - 310 q^{38} - 185 q^{40} - 538 q^{41} + 455 q^{43} - 680 q^{44} - 82 q^{46} - 60 q^{47} - 565 q^{49} - 450 q^{50} - 545 q^{53} + 510 q^{55} + 172 q^{56} - 595 q^{58} + 809 q^{59} + 502 q^{61} + 500 q^{62} - 1271 q^{64} - 475 q^{67} + 505 q^{68} - 80 q^{70} - 127 q^{71} + 585 q^{73} - 849 q^{74} - 140 q^{76} - 255 q^{77} + 240 q^{79} + 1065 q^{80} - 1515 q^{82} - 260 q^{83} - 1205 q^{85} + 1962 q^{86} - 1020 q^{88} - 921 q^{89} + 1040 q^{92} + 1040 q^{94} - 1270 q^{95} - 415 q^{97} - 1285 q^{98}+O(q^{100})$$ 2 * q + 5 * q^2 + 5 * q^4 + 15 * q^5 + 15 * q^7 + 15 * q^8 - 5 * q^10 - 17 * q^11 + 46 * q^14 + 57 * q^16 - 70 * q^17 - 141 * q^19 - 175 * q^20 - 170 * q^22 - 145 * q^23 + 75 * q^25 + 80 * q^28 - 34 * q^29 - 140 * q^31 - 105 * q^32 - 39 * q^34 + 70 * q^35 - 190 * q^37 - 310 * q^38 - 185 * q^40 - 538 * q^41 + 455 * q^43 - 680 * q^44 - 82 * q^46 - 60 * q^47 - 565 * q^49 - 450 * q^50 - 545 * q^53 + 510 * q^55 + 172 * q^56 - 595 * q^58 + 809 * q^59 + 502 * q^61 + 500 * q^62 - 1271 * q^64 - 475 * q^67 + 505 * q^68 - 80 * q^70 - 127 * q^71 + 585 * q^73 - 849 * q^74 - 140 * q^76 - 255 * q^77 + 240 * q^79 + 1065 * q^80 - 1515 * q^82 - 260 * q^83 - 1205 * q^85 + 1962 * q^86 - 1020 * q^88 - 921 * q^89 + 1040 * q^92 + 1040 * q^94 - 1270 * q^95 - 415 * q^97 - 1285 * q^98

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.438447 0.155014 0.0775072 0.996992i $$-0.475304\pi$$
0.0775072 + 0.996992i $$0.475304\pi$$
$$3$$ 0 0
$$4$$ −7.80776 −0.975971
$$5$$ 17.8078 1.59277 0.796387 0.604787i $$-0.206742\pi$$
0.796387 + 0.604787i $$0.206742\pi$$
$$6$$ 0 0
$$7$$ 5.43845 0.293649 0.146824 0.989163i $$-0.453095\pi$$
0.146824 + 0.989163i $$0.453095\pi$$
$$8$$ −6.93087 −0.306304
$$9$$ 0 0
$$10$$ 7.80776 0.246903
$$11$$ 22.4233 0.614625 0.307313 0.951609i $$-0.400570\pi$$
0.307313 + 0.951609i $$0.400570\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 2.38447 0.0455198
$$15$$ 0 0
$$16$$ 59.4233 0.928489
$$17$$ −67.9848 −0.969926 −0.484963 0.874535i $$-0.661167\pi$$
−0.484963 + 0.874535i $$0.661167\pi$$
$$18$$ 0 0
$$19$$ −80.8078 −0.975714 −0.487857 0.872923i $$-0.662221\pi$$
−0.487857 + 0.872923i $$0.662221\pi$$
$$20$$ −139.039 −1.55450
$$21$$ 0 0
$$22$$ 9.83143 0.0952758
$$23$$ −140.531 −1.27403 −0.637017 0.770850i $$-0.719832\pi$$
−0.637017 + 0.770850i $$0.719832\pi$$
$$24$$ 0 0
$$25$$ 192.116 1.53693
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −42.4621 −0.286592
$$29$$ 106.693 0.683187 0.341594 0.939848i $$-0.389033\pi$$
0.341594 + 0.939848i $$0.389033\pi$$
$$30$$ 0 0
$$31$$ −276.155 −1.59997 −0.799983 0.600023i $$-0.795158\pi$$
−0.799983 + 0.600023i $$0.795158\pi$$
$$32$$ 81.5009 0.450233
$$33$$ 0 0
$$34$$ −29.8078 −0.150353
$$35$$ 96.8466 0.467716
$$36$$ 0 0
$$37$$ −4.29168 −0.0190688 −0.00953442 0.999955i $$-0.503035\pi$$
−0.00953442 + 0.999955i $$0.503035\pi$$
$$38$$ −35.4299 −0.151250
$$39$$ 0 0
$$40$$ −123.423 −0.487873
$$41$$ −227.769 −0.867598 −0.433799 0.901010i $$-0.642827\pi$$
−0.433799 + 0.901010i $$0.642827\pi$$
$$42$$ 0 0
$$43$$ 27.5294 0.0976323 0.0488162 0.998808i $$-0.484455\pi$$
0.0488162 + 0.998808i $$0.484455\pi$$
$$44$$ −175.076 −0.599856
$$45$$ 0 0
$$46$$ −61.6155 −0.197494
$$47$$ −318.617 −0.988832 −0.494416 0.869225i $$-0.664618\pi$$
−0.494416 + 0.869225i $$0.664618\pi$$
$$48$$ 0 0
$$49$$ −313.423 −0.913771
$$50$$ 84.2329 0.238247
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 67.6562 0.175345 0.0876726 0.996149i $$-0.472057\pi$$
0.0876726 + 0.996149i $$0.472057\pi$$
$$54$$ 0 0
$$55$$ 399.309 0.978960
$$56$$ −37.6932 −0.0899457
$$57$$ 0 0
$$58$$ 46.7793 0.105904
$$59$$ 291.115 0.642371 0.321186 0.947016i $$-0.395919\pi$$
0.321186 + 0.947016i $$0.395919\pi$$
$$60$$ 0 0
$$61$$ 663.311 1.39227 0.696133 0.717913i $$-0.254902\pi$$
0.696133 + 0.717913i $$0.254902\pi$$
$$62$$ −121.080 −0.248018
$$63$$ 0 0
$$64$$ −439.652 −0.858696
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −425.101 −0.775140 −0.387570 0.921840i $$-0.626685\pi$$
−0.387570 + 0.921840i $$0.626685\pi$$
$$68$$ 530.810 0.946619
$$69$$ 0 0
$$70$$ 42.4621 0.0725028
$$71$$ 152.963 0.255681 0.127841 0.991795i $$-0.459195\pi$$
0.127841 + 0.991795i $$0.459195\pi$$
$$72$$ 0 0
$$73$$ 117.268 0.188016 0.0940081 0.995571i $$-0.470032\pi$$
0.0940081 + 0.995571i $$0.470032\pi$$
$$74$$ −1.88167 −0.00295595
$$75$$ 0 0
$$76$$ 630.928 0.952268
$$77$$ 121.948 0.180484
$$78$$ 0 0
$$79$$ 202.462 0.288339 0.144169 0.989553i $$-0.453949\pi$$
0.144169 + 0.989553i $$0.453949\pi$$
$$80$$ 1058.20 1.47887
$$81$$ 0 0
$$82$$ −99.8647 −0.134490
$$83$$ −336.155 −0.444552 −0.222276 0.974984i $$-0.571349\pi$$
−0.222276 + 0.974984i $$0.571349\pi$$
$$84$$ 0 0
$$85$$ −1210.66 −1.54487
$$86$$ 12.0702 0.0151344
$$87$$ 0 0
$$88$$ −155.413 −0.188262
$$89$$ −718.194 −0.855376 −0.427688 0.903927i $$-0.640672\pi$$
−0.427688 + 0.903927i $$0.640672\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 1097.23 1.24342
$$93$$ 0 0
$$94$$ −139.697 −0.153283
$$95$$ −1439.01 −1.55409
$$96$$ 0 0
$$97$$ 759.368 0.794868 0.397434 0.917631i $$-0.369901\pi$$
0.397434 + 0.917631i $$0.369901\pi$$
$$98$$ −137.420 −0.141648
$$99$$ 0 0
$$100$$ −1500.00 −1.50000
$$101$$ 348.697 0.343531 0.171766 0.985138i $$-0.445053\pi$$
0.171766 + 0.985138i $$0.445053\pi$$
$$102$$ 0 0
$$103$$ −580.303 −0.555136 −0.277568 0.960706i $$-0.589528\pi$$
−0.277568 + 0.960706i $$0.589528\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 29.6637 0.0271810
$$107$$ −571.493 −0.516340 −0.258170 0.966100i $$-0.583119\pi$$
−0.258170 + 0.966100i $$0.583119\pi$$
$$108$$ 0 0
$$109$$ 176.004 0.154661 0.0773307 0.997005i $$-0.475360\pi$$
0.0773307 + 0.997005i $$0.475360\pi$$
$$110$$ 175.076 0.151753
$$111$$ 0 0
$$112$$ 323.170 0.272649
$$113$$ −1264.88 −1.05301 −0.526505 0.850172i $$-0.676498\pi$$
−0.526505 + 0.850172i $$0.676498\pi$$
$$114$$ 0 0
$$115$$ −2502.55 −2.02925
$$116$$ −833.035 −0.666770
$$117$$ 0 0
$$118$$ 127.638 0.0995768
$$119$$ −369.732 −0.284817
$$120$$ 0 0
$$121$$ −828.196 −0.622236
$$122$$ 290.827 0.215821
$$123$$ 0 0
$$124$$ 2156.16 1.56152
$$125$$ 1195.19 0.855211
$$126$$ 0 0
$$127$$ 2604.11 1.81950 0.909752 0.415151i $$-0.136271\pi$$
0.909752 + 0.415151i $$0.136271\pi$$
$$128$$ −844.772 −0.583344
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −2131.70 −1.42174 −0.710870 0.703324i $$-0.751698\pi$$
−0.710870 + 0.703324i $$0.751698\pi$$
$$132$$ 0 0
$$133$$ −439.469 −0.286517
$$134$$ −186.384 −0.120158
$$135$$ 0 0
$$136$$ 471.194 0.297092
$$137$$ −687.985 −0.429040 −0.214520 0.976720i $$-0.568819\pi$$
−0.214520 + 0.976720i $$0.568819\pi$$
$$138$$ 0 0
$$139$$ −679.580 −0.414685 −0.207343 0.978268i $$-0.566482\pi$$
−0.207343 + 0.978268i $$0.566482\pi$$
$$140$$ −756.155 −0.456477
$$141$$ 0 0
$$142$$ 67.0662 0.0396343
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 1899.97 1.08816
$$146$$ 51.4158 0.0291452
$$147$$ 0 0
$$148$$ 33.5084 0.0186106
$$149$$ 1975.46 1.08615 0.543074 0.839685i $$-0.317260\pi$$
0.543074 + 0.839685i $$0.317260\pi$$
$$150$$ 0 0
$$151$$ 1803.24 0.971824 0.485912 0.874008i $$-0.338487\pi$$
0.485912 + 0.874008i $$0.338487\pi$$
$$152$$ 560.068 0.298865
$$153$$ 0 0
$$154$$ 53.4677 0.0279776
$$155$$ −4917.71 −2.54839
$$156$$ 0 0
$$157$$ −397.168 −0.201894 −0.100947 0.994892i $$-0.532187\pi$$
−0.100947 + 0.994892i $$0.532187\pi$$
$$158$$ 88.7689 0.0446967
$$159$$ 0 0
$$160$$ 1451.35 0.717120
$$161$$ −764.272 −0.374118
$$162$$ 0 0
$$163$$ 941.393 0.452365 0.226183 0.974085i $$-0.427375\pi$$
0.226183 + 0.974085i $$0.427375\pi$$
$$164$$ 1778.37 0.846750
$$165$$ 0 0
$$166$$ −147.386 −0.0689120
$$167$$ −3680.43 −1.70539 −0.852696 0.522408i $$-0.825034\pi$$
−0.852696 + 0.522408i $$0.825034\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −530.810 −0.239478
$$171$$ 0 0
$$172$$ −214.943 −0.0952863
$$173$$ −1422.77 −0.625269 −0.312634 0.949874i $$-0.601211\pi$$
−0.312634 + 0.949874i $$0.601211\pi$$
$$174$$ 0 0
$$175$$ 1044.82 0.451318
$$176$$ 1332.47 0.570673
$$177$$ 0 0
$$178$$ −314.890 −0.132596
$$179$$ −1167.89 −0.487666 −0.243833 0.969817i $$-0.578405\pi$$
−0.243833 + 0.969817i $$0.578405\pi$$
$$180$$ 0 0
$$181$$ −1133.96 −0.465673 −0.232836 0.972516i $$-0.574801\pi$$
−0.232836 + 0.972516i $$0.574801\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 974.004 0.390242
$$185$$ −76.4252 −0.0303724
$$186$$ 0 0
$$187$$ −1524.44 −0.596141
$$188$$ 2487.69 0.965071
$$189$$ 0 0
$$190$$ −630.928 −0.240907
$$191$$ −2682.12 −1.01608 −0.508040 0.861333i $$-0.669630\pi$$
−0.508040 + 0.861333i $$0.669630\pi$$
$$192$$ 0 0
$$193$$ −1970.67 −0.734983 −0.367491 0.930027i $$-0.619783\pi$$
−0.367491 + 0.930027i $$0.619783\pi$$
$$194$$ 332.943 0.123216
$$195$$ 0 0
$$196$$ 2447.14 0.891813
$$197$$ 4016.05 1.45244 0.726222 0.687460i $$-0.241274\pi$$
0.726222 + 0.687460i $$0.241274\pi$$
$$198$$ 0 0
$$199$$ −4226.06 −1.50541 −0.752707 0.658356i $$-0.771252\pi$$
−0.752707 + 0.658356i $$0.771252\pi$$
$$200$$ −1331.53 −0.470768
$$201$$ 0 0
$$202$$ 152.885 0.0532523
$$203$$ 580.245 0.200617
$$204$$ 0 0
$$205$$ −4056.06 −1.38189
$$206$$ −254.432 −0.0860541
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −1811.98 −0.599699
$$210$$ 0 0
$$211$$ 1364.67 0.445249 0.222625 0.974904i $$-0.428538\pi$$
0.222625 + 0.974904i $$0.428538\pi$$
$$212$$ −528.244 −0.171132
$$213$$ 0 0
$$214$$ −250.570 −0.0800401
$$215$$ 490.237 0.155506
$$216$$ 0 0
$$217$$ −1501.86 −0.469828
$$218$$ 77.1683 0.0239748
$$219$$ 0 0
$$220$$ −3117.71 −0.955436
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −1059.47 −0.318149 −0.159075 0.987267i $$-0.550851\pi$$
−0.159075 + 0.987267i $$0.550851\pi$$
$$224$$ 443.239 0.132210
$$225$$ 0 0
$$226$$ −554.584 −0.163232
$$227$$ −3464.19 −1.01289 −0.506446 0.862272i $$-0.669041\pi$$
−0.506446 + 0.862272i $$0.669041\pi$$
$$228$$ 0 0
$$229$$ −2324.64 −0.670815 −0.335407 0.942073i $$-0.608874\pi$$
−0.335407 + 0.942073i $$0.608874\pi$$
$$230$$ −1097.23 −0.314563
$$231$$ 0 0
$$232$$ −739.476 −0.209263
$$233$$ 3731.01 1.04904 0.524521 0.851398i $$-0.324245\pi$$
0.524521 + 0.851398i $$0.324245\pi$$
$$234$$ 0 0
$$235$$ −5673.86 −1.57499
$$236$$ −2272.95 −0.626935
$$237$$ 0 0
$$238$$ −162.108 −0.0441508
$$239$$ −6044.47 −1.63592 −0.817958 0.575278i $$-0.804894\pi$$
−0.817958 + 0.575278i $$0.804894\pi$$
$$240$$ 0 0
$$241$$ −5173.96 −1.38292 −0.691461 0.722414i $$-0.743033\pi$$
−0.691461 + 0.722414i $$0.743033\pi$$
$$242$$ −363.120 −0.0964556
$$243$$ 0 0
$$244$$ −5178.97 −1.35881
$$245$$ −5581.37 −1.45543
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 1914.00 0.490076
$$249$$ 0 0
$$250$$ 524.029 0.132570
$$251$$ −5620.73 −1.41346 −0.706728 0.707486i $$-0.749829\pi$$
−0.706728 + 0.707486i $$0.749829\pi$$
$$252$$ 0 0
$$253$$ −3151.17 −0.783054
$$254$$ 1141.76 0.282050
$$255$$ 0 0
$$256$$ 3146.83 0.768270
$$257$$ 1674.14 0.406342 0.203171 0.979143i $$-0.434875\pi$$
0.203171 + 0.979143i $$0.434875\pi$$
$$258$$ 0 0
$$259$$ −23.3401 −0.00559954
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −934.640 −0.220390
$$263$$ 6309.18 1.47924 0.739622 0.673023i $$-0.235004\pi$$
0.739622 + 0.673023i $$0.235004\pi$$
$$264$$ 0 0
$$265$$ 1204.81 0.279285
$$266$$ −192.684 −0.0444143
$$267$$ 0 0
$$268$$ 3319.09 0.756514
$$269$$ 2482.73 0.562731 0.281366 0.959601i $$-0.409213\pi$$
0.281366 + 0.959601i $$0.409213\pi$$
$$270$$ 0 0
$$271$$ 2835.72 0.635638 0.317819 0.948151i $$-0.397050\pi$$
0.317819 + 0.948151i $$0.397050\pi$$
$$272$$ −4039.88 −0.900566
$$273$$ 0 0
$$274$$ −301.645 −0.0665075
$$275$$ 4307.88 0.944637
$$276$$ 0 0
$$277$$ −3837.51 −0.832396 −0.416198 0.909274i $$-0.636638\pi$$
−0.416198 + 0.909274i $$0.636638\pi$$
$$278$$ −297.960 −0.0642822
$$279$$ 0 0
$$280$$ −671.231 −0.143263
$$281$$ 9122.13 1.93659 0.968293 0.249819i $$-0.0803712\pi$$
0.968293 + 0.249819i $$0.0803712\pi$$
$$282$$ 0 0
$$283$$ 2127.85 0.446952 0.223476 0.974709i $$-0.428260\pi$$
0.223476 + 0.974709i $$0.428260\pi$$
$$284$$ −1194.30 −0.249537
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −1238.71 −0.254769
$$288$$ 0 0
$$289$$ −291.061 −0.0592430
$$290$$ 833.035 0.168681
$$291$$ 0 0
$$292$$ −915.601 −0.183498
$$293$$ −8274.77 −1.64989 −0.824944 0.565215i $$-0.808793\pi$$
−0.824944 + 0.565215i $$0.808793\pi$$
$$294$$ 0 0
$$295$$ 5184.10 1.02315
$$296$$ 29.7450 0.00584086
$$297$$ 0 0
$$298$$ 866.136 0.168369
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 149.717 0.0286696
$$302$$ 790.625 0.150647
$$303$$ 0 0
$$304$$ −4801.86 −0.905940
$$305$$ 11812.1 2.21757
$$306$$ 0 0
$$307$$ −3610.49 −0.671211 −0.335605 0.942003i $$-0.608941\pi$$
−0.335605 + 0.942003i $$0.608941\pi$$
$$308$$ −952.140 −0.176147
$$309$$ 0 0
$$310$$ −2156.16 −0.395037
$$311$$ −3331.06 −0.607354 −0.303677 0.952775i $$-0.598214\pi$$
−0.303677 + 0.952775i $$0.598214\pi$$
$$312$$ 0 0
$$313$$ −358.125 −0.0646724 −0.0323362 0.999477i $$-0.510295\pi$$
−0.0323362 + 0.999477i $$0.510295\pi$$
$$314$$ −174.137 −0.0312966
$$315$$ 0 0
$$316$$ −1580.78 −0.281410
$$317$$ −3047.46 −0.539944 −0.269972 0.962868i $$-0.587014\pi$$
−0.269972 + 0.962868i $$0.587014\pi$$
$$318$$ 0 0
$$319$$ 2392.41 0.419904
$$320$$ −7829.23 −1.36771
$$321$$ 0 0
$$322$$ −335.093 −0.0579938
$$323$$ 5493.70 0.946371
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 412.751 0.0701232
$$327$$ 0 0
$$328$$ 1578.64 0.265749
$$329$$ −1732.78 −0.290369
$$330$$ 0 0
$$331$$ 7694.77 1.27777 0.638887 0.769301i $$-0.279395\pi$$
0.638887 + 0.769301i $$0.279395\pi$$
$$332$$ 2624.62 0.433870
$$333$$ 0 0
$$334$$ −1613.68 −0.264360
$$335$$ −7570.10 −1.23462
$$336$$ 0 0
$$337$$ 4712.21 0.761693 0.380846 0.924638i $$-0.375633\pi$$
0.380846 + 0.924638i $$0.375633\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 9452.53 1.50775
$$341$$ −6192.31 −0.983380
$$342$$ 0 0
$$343$$ −3569.92 −0.561976
$$344$$ −190.803 −0.0299052
$$345$$ 0 0
$$346$$ −623.811 −0.0969257
$$347$$ 5261.98 0.814058 0.407029 0.913415i $$-0.366565\pi$$
0.407029 + 0.913415i $$0.366565\pi$$
$$348$$ 0 0
$$349$$ 50.3345 0.00772018 0.00386009 0.999993i $$-0.498771\pi$$
0.00386009 + 0.999993i $$0.498771\pi$$
$$350$$ 458.096 0.0699608
$$351$$ 0 0
$$352$$ 1827.52 0.276725
$$353$$ −9057.64 −1.36569 −0.682846 0.730562i $$-0.739258\pi$$
−0.682846 + 0.730562i $$0.739258\pi$$
$$354$$ 0 0
$$355$$ 2723.93 0.407243
$$356$$ 5607.49 0.834821
$$357$$ 0 0
$$358$$ −512.059 −0.0755953
$$359$$ −7177.86 −1.05525 −0.527623 0.849479i $$-0.676917\pi$$
−0.527623 + 0.849479i $$0.676917\pi$$
$$360$$ 0 0
$$361$$ −329.105 −0.0479815
$$362$$ −497.183 −0.0721861
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 2088.28 0.299467
$$366$$ 0 0
$$367$$ 4004.14 0.569522 0.284761 0.958599i $$-0.408086\pi$$
0.284761 + 0.958599i $$0.408086\pi$$
$$368$$ −8350.83 −1.18293
$$369$$ 0 0
$$370$$ −33.5084 −0.00470816
$$371$$ 367.945 0.0514899
$$372$$ 0 0
$$373$$ −10014.2 −1.39012 −0.695060 0.718952i $$-0.744622\pi$$
−0.695060 + 0.718952i $$0.744622\pi$$
$$374$$ −668.388 −0.0924105
$$375$$ 0 0
$$376$$ 2208.30 0.302883
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −8169.12 −1.10717 −0.553587 0.832791i $$-0.686742\pi$$
−0.553587 + 0.832791i $$0.686742\pi$$
$$380$$ 11235.4 1.51675
$$381$$ 0 0
$$382$$ −1175.97 −0.157507
$$383$$ 7310.25 0.975290 0.487645 0.873042i $$-0.337856\pi$$
0.487645 + 0.873042i $$0.337856\pi$$
$$384$$ 0 0
$$385$$ 2171.62 0.287470
$$386$$ −864.033 −0.113933
$$387$$ 0 0
$$388$$ −5928.97 −0.775767
$$389$$ −8785.47 −1.14509 −0.572546 0.819872i $$-0.694044\pi$$
−0.572546 + 0.819872i $$0.694044\pi$$
$$390$$ 0 0
$$391$$ 9553.99 1.23572
$$392$$ 2172.30 0.279892
$$393$$ 0 0
$$394$$ 1760.82 0.225150
$$395$$ 3605.40 0.459259
$$396$$ 0 0
$$397$$ 11266.8 1.42434 0.712171 0.702006i $$-0.247712\pi$$
0.712171 + 0.702006i $$0.247712\pi$$
$$398$$ −1852.90 −0.233361
$$399$$ 0 0
$$400$$ 11416.2 1.42702
$$401$$ −1576.23 −0.196293 −0.0981464 0.995172i $$-0.531291\pi$$
−0.0981464 + 0.995172i $$0.531291\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −2722.54 −0.335276
$$405$$ 0 0
$$406$$ 254.407 0.0310985
$$407$$ −96.2335 −0.0117202
$$408$$ 0 0
$$409$$ 6755.78 0.816753 0.408377 0.912814i $$-0.366095\pi$$
0.408377 + 0.912814i $$0.366095\pi$$
$$410$$ −1778.37 −0.214213
$$411$$ 0 0
$$412$$ 4530.87 0.541796
$$413$$ 1583.21 0.188631
$$414$$ 0 0
$$415$$ −5986.17 −0.708072
$$416$$ 0 0
$$417$$ 0 0
$$418$$ −794.456 −0.0929620
$$419$$ −10756.2 −1.25411 −0.627057 0.778973i $$-0.715741\pi$$
−0.627057 + 0.778973i $$0.715741\pi$$
$$420$$ 0 0
$$421$$ 7886.03 0.912925 0.456463 0.889743i $$-0.349116\pi$$
0.456463 + 0.889743i $$0.349116\pi$$
$$422$$ 598.335 0.0690201
$$423$$ 0 0
$$424$$ −468.916 −0.0537089
$$425$$ −13061.0 −1.49071
$$426$$ 0 0
$$427$$ 3607.38 0.408837
$$428$$ 4462.08 0.503932
$$429$$ 0 0
$$430$$ 214.943 0.0241057
$$431$$ 14084.6 1.57409 0.787044 0.616897i $$-0.211610\pi$$
0.787044 + 0.616897i $$0.211610\pi$$
$$432$$ 0 0
$$433$$ 1864.14 0.206894 0.103447 0.994635i $$-0.467013\pi$$
0.103447 + 0.994635i $$0.467013\pi$$
$$434$$ −658.485 −0.0728301
$$435$$ 0 0
$$436$$ −1374.20 −0.150945
$$437$$ 11356.0 1.24309
$$438$$ 0 0
$$439$$ 6154.49 0.669106 0.334553 0.942377i $$-0.391415\pi$$
0.334553 + 0.942377i $$0.391415\pi$$
$$440$$ −2767.56 −0.299859
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 14539.3 1.55933 0.779663 0.626200i $$-0.215391\pi$$
0.779663 + 0.626200i $$0.215391\pi$$
$$444$$ 0 0
$$445$$ −12789.4 −1.36242
$$446$$ −464.521 −0.0493177
$$447$$ 0 0
$$448$$ −2391.03 −0.252155
$$449$$ 7043.87 0.740358 0.370179 0.928960i $$-0.379296\pi$$
0.370179 + 0.928960i $$0.379296\pi$$
$$450$$ 0 0
$$451$$ −5107.33 −0.533248
$$452$$ 9875.90 1.02771
$$453$$ 0 0
$$454$$ −1518.87 −0.157013
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 14098.9 1.44314 0.721572 0.692340i $$-0.243420\pi$$
0.721572 + 0.692340i $$0.243420\pi$$
$$458$$ −1019.23 −0.103986
$$459$$ 0 0
$$460$$ 19539.3 1.98049
$$461$$ −14449.7 −1.45985 −0.729924 0.683529i $$-0.760444\pi$$
−0.729924 + 0.683529i $$0.760444\pi$$
$$462$$ 0 0
$$463$$ 15806.5 1.58659 0.793293 0.608840i $$-0.208365\pi$$
0.793293 + 0.608840i $$0.208365\pi$$
$$464$$ 6340.06 0.634332
$$465$$ 0 0
$$466$$ 1635.85 0.162617
$$467$$ 15071.3 1.49340 0.746699 0.665162i $$-0.231638\pi$$
0.746699 + 0.665162i $$0.231638\pi$$
$$468$$ 0 0
$$469$$ −2311.89 −0.227619
$$470$$ −2487.69 −0.244146
$$471$$ 0 0
$$472$$ −2017.68 −0.196761
$$473$$ 617.299 0.0600073
$$474$$ 0 0
$$475$$ −15524.5 −1.49961
$$476$$ 2886.78 0.277973
$$477$$ 0 0
$$478$$ −2650.18 −0.253591
$$479$$ 392.545 0.0374443 0.0187222 0.999825i $$-0.494040\pi$$
0.0187222 + 0.999825i $$0.494040\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −2268.51 −0.214373
$$483$$ 0 0
$$484$$ 6466.36 0.607284
$$485$$ 13522.7 1.26605
$$486$$ 0 0
$$487$$ 9497.89 0.883758 0.441879 0.897075i $$-0.354312\pi$$
0.441879 + 0.897075i $$0.354312\pi$$
$$488$$ −4597.32 −0.426457
$$489$$ 0 0
$$490$$ −2447.14 −0.225613
$$491$$ 1893.82 0.174067 0.0870337 0.996205i $$-0.472261\pi$$
0.0870337 + 0.996205i $$0.472261\pi$$
$$492$$ 0 0
$$493$$ −7253.52 −0.662641
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −16410.1 −1.48555
$$497$$ 831.881 0.0750804
$$498$$ 0 0
$$499$$ −13370.1 −1.19945 −0.599727 0.800205i $$-0.704724\pi$$
−0.599727 + 0.800205i $$0.704724\pi$$
$$500$$ −9331.79 −0.834661
$$501$$ 0 0
$$502$$ −2464.39 −0.219106
$$503$$ −5554.71 −0.492391 −0.246195 0.969220i $$-0.579180\pi$$
−0.246195 + 0.969220i $$0.579180\pi$$
$$504$$ 0 0
$$505$$ 6209.51 0.547168
$$506$$ −1381.62 −0.121385
$$507$$ 0 0
$$508$$ −20332.3 −1.77578
$$509$$ −2197.55 −0.191365 −0.0956824 0.995412i $$-0.530503\pi$$
−0.0956824 + 0.995412i $$0.530503\pi$$
$$510$$ 0 0
$$511$$ 637.756 0.0552107
$$512$$ 8137.89 0.702437
$$513$$ 0 0
$$514$$ 734.022 0.0629890
$$515$$ −10333.9 −0.884206
$$516$$ 0 0
$$517$$ −7144.45 −0.607761
$$518$$ −10.2334 −0.000868010 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −17005.2 −1.42997 −0.714983 0.699142i $$-0.753565\pi$$
−0.714983 + 0.699142i $$0.753565\pi$$
$$522$$ 0 0
$$523$$ −14486.2 −1.21116 −0.605581 0.795783i $$-0.707059\pi$$
−0.605581 + 0.795783i $$0.707059\pi$$
$$524$$ 16643.8 1.38758
$$525$$ 0 0
$$526$$ 2766.24 0.229304
$$527$$ 18774.4 1.55185
$$528$$ 0 0
$$529$$ 7582.03 0.623163
$$530$$ 528.244 0.0432933
$$531$$ 0 0
$$532$$ 3431.27 0.279632
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −10177.0 −0.822413
$$536$$ 2946.32 0.237429
$$537$$ 0 0
$$538$$ 1088.55 0.0872315
$$539$$ −7027.98 −0.561626
$$540$$ 0 0
$$541$$ −15266.7 −1.21325 −0.606623 0.794990i $$-0.707476\pi$$
−0.606623 + 0.794990i $$0.707476\pi$$
$$542$$ 1243.31 0.0985330
$$543$$ 0 0
$$544$$ −5540.83 −0.436693
$$545$$ 3134.23 0.246341
$$546$$ 0 0
$$547$$ 15260.5 1.19286 0.596430 0.802665i $$-0.296586\pi$$
0.596430 + 0.802665i $$0.296586\pi$$
$$548$$ 5371.62 0.418731
$$549$$ 0 0
$$550$$ 1888.78 0.146432
$$551$$ −8621.64 −0.666595
$$552$$ 0 0
$$553$$ 1101.08 0.0846703
$$554$$ −1682.55 −0.129033
$$555$$ 0 0
$$556$$ 5306.00 0.404721
$$557$$ 10442.1 0.794337 0.397169 0.917746i $$-0.369993\pi$$
0.397169 + 0.917746i $$0.369993\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 5754.94 0.434269
$$561$$ 0 0
$$562$$ 3999.57 0.300199
$$563$$ −7145.26 −0.534879 −0.267440 0.963575i $$-0.586178\pi$$
−0.267440 + 0.963575i $$0.586178\pi$$
$$564$$ 0 0
$$565$$ −22524.7 −1.67721
$$566$$ 932.950 0.0692841
$$567$$ 0 0
$$568$$ −1060.17 −0.0783162
$$569$$ −4438.86 −0.327042 −0.163521 0.986540i $$-0.552285\pi$$
−0.163521 + 0.986540i $$0.552285\pi$$
$$570$$ 0 0
$$571$$ 10117.3 0.741497 0.370748 0.928733i $$-0.379101\pi$$
0.370748 + 0.928733i $$0.379101\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −543.109 −0.0394929
$$575$$ −26998.4 −1.95810
$$576$$ 0 0
$$577$$ 3105.60 0.224069 0.112035 0.993704i $$-0.464263\pi$$
0.112035 + 0.993704i $$0.464263\pi$$
$$578$$ −127.615 −0.00918352
$$579$$ 0 0
$$580$$ −14834.5 −1.06202
$$581$$ −1828.16 −0.130542
$$582$$ 0 0
$$583$$ 1517.08 0.107772
$$584$$ −812.769 −0.0575901
$$585$$ 0 0
$$586$$ −3628.05 −0.255757
$$587$$ −19662.3 −1.38254 −0.691270 0.722597i $$-0.742948\pi$$
−0.691270 + 0.722597i $$0.742948\pi$$
$$588$$ 0 0
$$589$$ 22315.5 1.56111
$$590$$ 2272.95 0.158603
$$591$$ 0 0
$$592$$ −255.026 −0.0177052
$$593$$ −6395.51 −0.442888 −0.221444 0.975173i $$-0.571077\pi$$
−0.221444 + 0.975173i $$0.571077\pi$$
$$594$$ 0 0
$$595$$ −6584.10 −0.453650
$$596$$ −15423.9 −1.06005
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −8878.48 −0.605618 −0.302809 0.953051i $$-0.597924\pi$$
−0.302809 + 0.953051i $$0.597924\pi$$
$$600$$ 0 0
$$601$$ 19100.6 1.29639 0.648194 0.761475i $$-0.275525\pi$$
0.648194 + 0.761475i $$0.275525\pi$$
$$602$$ 65.6430 0.00444420
$$603$$ 0 0
$$604$$ −14079.3 −0.948472
$$605$$ −14748.3 −0.991082
$$606$$ 0 0
$$607$$ 16595.8 1.10972 0.554861 0.831943i $$-0.312771\pi$$
0.554861 + 0.831943i $$0.312771\pi$$
$$608$$ −6585.91 −0.439299
$$609$$ 0 0
$$610$$ 5178.97 0.343755
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 16469.2 1.08513 0.542564 0.840015i $$-0.317454\pi$$
0.542564 + 0.840015i $$0.317454\pi$$
$$614$$ −1583.01 −0.104047
$$615$$ 0 0
$$616$$ −845.205 −0.0552829
$$617$$ −10116.0 −0.660055 −0.330027 0.943971i $$-0.607058\pi$$
−0.330027 + 0.943971i $$0.607058\pi$$
$$618$$ 0 0
$$619$$ 18854.8 1.22430 0.612148 0.790743i $$-0.290306\pi$$
0.612148 + 0.790743i $$0.290306\pi$$
$$620$$ 38396.3 2.48715
$$621$$ 0 0
$$622$$ −1460.49 −0.0941487
$$623$$ −3905.86 −0.251180
$$624$$ 0 0
$$625$$ −2730.82 −0.174773
$$626$$ −157.019 −0.0100252
$$627$$ 0 0
$$628$$ 3100.99 0.197043
$$629$$ 291.769 0.0184954
$$630$$ 0 0
$$631$$ 18946.2 1.19531 0.597653 0.801755i $$-0.296100\pi$$
0.597653 + 0.801755i $$0.296100\pi$$
$$632$$ −1403.24 −0.0883194
$$633$$ 0 0
$$634$$ −1336.15 −0.0836991
$$635$$ 46373.3 2.89806
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 1048.95 0.0650912
$$639$$ 0 0
$$640$$ −15043.5 −0.929135
$$641$$ 23586.9 1.45340 0.726698 0.686957i $$-0.241054\pi$$
0.726698 + 0.686957i $$0.241054\pi$$
$$642$$ 0 0
$$643$$ −27153.0 −1.66534 −0.832669 0.553772i $$-0.813188\pi$$
−0.832669 + 0.553772i $$0.813188\pi$$
$$644$$ 5967.25 0.365128
$$645$$ 0 0
$$646$$ 2408.70 0.146701
$$647$$ −6856.72 −0.416639 −0.208319 0.978061i $$-0.566799\pi$$
−0.208319 + 0.978061i $$0.566799\pi$$
$$648$$ 0 0
$$649$$ 6527.75 0.394817
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −7350.17 −0.441495
$$653$$ 8073.89 0.483853 0.241926 0.970295i $$-0.422221\pi$$
0.241926 + 0.970295i $$0.422221\pi$$
$$654$$ 0 0
$$655$$ −37960.9 −2.26451
$$656$$ −13534.8 −0.805555
$$657$$ 0 0
$$658$$ −759.734 −0.0450114
$$659$$ −5305.73 −0.313629 −0.156815 0.987628i $$-0.550123\pi$$
−0.156815 + 0.987628i $$0.550123\pi$$
$$660$$ 0 0
$$661$$ 25848.3 1.52100 0.760502 0.649336i $$-0.224953\pi$$
0.760502 + 0.649336i $$0.224953\pi$$
$$662$$ 3373.75 0.198073
$$663$$ 0 0
$$664$$ 2329.85 0.136168
$$665$$ −7825.96 −0.456357
$$666$$ 0 0
$$667$$ −14993.7 −0.870404
$$668$$ 28735.9 1.66441
$$669$$ 0 0
$$670$$ −3319.09 −0.191385
$$671$$ 14873.6 0.855722
$$672$$ 0 0
$$673$$ −14529.1 −0.832177 −0.416089 0.909324i $$-0.636599\pi$$
−0.416089 + 0.909324i $$0.636599\pi$$
$$674$$ 2066.06 0.118073
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −12058.1 −0.684535 −0.342267 0.939603i $$-0.611195\pi$$
−0.342267 + 0.939603i $$0.611195\pi$$
$$678$$ 0 0
$$679$$ 4129.78 0.233412
$$680$$ 8390.91 0.473201
$$681$$ 0 0
$$682$$ −2715.00 −0.152438
$$683$$ −30028.8 −1.68231 −0.841156 0.540792i $$-0.818125\pi$$
−0.841156 + 0.540792i $$0.818125\pi$$
$$684$$ 0 0
$$685$$ −12251.5 −0.683364
$$686$$ −1565.22 −0.0871144
$$687$$ 0 0
$$688$$ 1635.89 0.0906505
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 449.696 0.0247572 0.0123786 0.999923i $$-0.496060\pi$$
0.0123786 + 0.999923i $$0.496060\pi$$
$$692$$ 11108.7 0.610244
$$693$$ 0 0
$$694$$ 2307.10 0.126191
$$695$$ −12101.8 −0.660500
$$696$$ 0 0
$$697$$ 15484.8 0.841506
$$698$$ 22.0690 0.00119674
$$699$$ 0 0
$$700$$ −8157.67 −0.440473
$$701$$ 26986.0 1.45399 0.726994 0.686644i $$-0.240917\pi$$
0.726994 + 0.686644i $$0.240917\pi$$
$$702$$ 0 0
$$703$$ 346.801 0.0186057
$$704$$ −9858.46 −0.527776
$$705$$ 0 0
$$706$$ −3971.29 −0.211702
$$707$$ 1896.37 0.100877
$$708$$ 0 0
$$709$$ −9098.87 −0.481968 −0.240984 0.970529i $$-0.577470\pi$$
−0.240984 + 0.970529i $$0.577470\pi$$
$$710$$ 1194.30 0.0631285
$$711$$ 0 0
$$712$$ 4977.71 0.262005
$$713$$ 38808.4 2.03841
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 9118.62 0.475948
$$717$$ 0 0
$$718$$ −3147.11 −0.163578
$$719$$ 6293.55 0.326439 0.163220 0.986590i $$-0.447812\pi$$
0.163220 + 0.986590i $$0.447812\pi$$
$$720$$ 0 0
$$721$$ −3155.95 −0.163015
$$722$$ −144.295 −0.00743783
$$723$$ 0 0
$$724$$ 8853.72 0.454483
$$725$$ 20497.5 1.05001
$$726$$ 0 0
$$727$$ 18070.7 0.921878 0.460939 0.887432i $$-0.347513\pi$$
0.460939 + 0.887432i $$0.347513\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 915.601 0.0464218
$$731$$ −1871.58 −0.0946962
$$732$$ 0 0
$$733$$ 34771.5 1.75214 0.876068 0.482188i $$-0.160158\pi$$
0.876068 + 0.482188i $$0.160158\pi$$
$$734$$ 1755.61 0.0882842
$$735$$ 0 0
$$736$$ −11453.4 −0.573613
$$737$$ −9532.17 −0.476421
$$738$$ 0 0
$$739$$ 23631.5 1.17632 0.588158 0.808746i $$-0.299853\pi$$
0.588158 + 0.808746i $$0.299853\pi$$
$$740$$ 596.710 0.0296425
$$741$$ 0 0
$$742$$ 161.324 0.00798167
$$743$$ −32502.8 −1.60486 −0.802431 0.596745i $$-0.796460\pi$$
−0.802431 + 0.596745i $$0.796460\pi$$
$$744$$ 0 0
$$745$$ 35178.6 1.72999
$$746$$ −4390.69 −0.215489
$$747$$ 0 0
$$748$$ 11902.5 0.581816
$$749$$ −3108.04 −0.151622
$$750$$ 0 0
$$751$$ 2020.86 0.0981920 0.0490960 0.998794i $$-0.484366\pi$$
0.0490960 + 0.998794i $$0.484366\pi$$
$$752$$ −18933.3 −0.918120
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 32111.6 1.54790
$$756$$ 0 0
$$757$$ 12568.2 0.603434 0.301717 0.953398i $$-0.402440\pi$$
0.301717 + 0.953398i $$0.402440\pi$$
$$758$$ −3581.73 −0.171628
$$759$$ 0 0
$$760$$ 9973.56 0.476025
$$761$$ 8704.81 0.414651 0.207325 0.978272i $$-0.433524\pi$$
0.207325 + 0.978272i $$0.433524\pi$$
$$762$$ 0 0
$$763$$ 957.187 0.0454161
$$764$$ 20941.4 0.991665
$$765$$ 0 0
$$766$$ 3205.16 0.151184
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −21915.9 −1.02771 −0.513853 0.857878i $$-0.671782\pi$$
−0.513853 + 0.857878i $$0.671782\pi$$
$$770$$ 952.140 0.0445620
$$771$$ 0 0
$$772$$ 15386.5 0.717322
$$773$$ 23077.5 1.07379 0.536896 0.843649i $$-0.319597\pi$$
0.536896 + 0.843649i $$0.319597\pi$$
$$774$$ 0 0
$$775$$ −53054.0 −2.45904
$$776$$ −5263.08 −0.243471
$$777$$ 0 0
$$778$$ −3851.96 −0.177506
$$779$$ 18405.5 0.846528
$$780$$ 0 0
$$781$$ 3429.94 0.157148
$$782$$ 4188.92 0.191554
$$783$$ 0 0
$$784$$ −18624.6 −0.848426
$$785$$ −7072.67 −0.321572
$$786$$ 0 0
$$787$$ −16522.4 −0.748362 −0.374181 0.927356i $$-0.622076\pi$$
−0.374181 + 0.927356i $$0.622076\pi$$
$$788$$ −31356.4 −1.41754
$$789$$ 0 0
$$790$$ 1580.78 0.0711918
$$791$$ −6879.00 −0.309215
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 4939.89 0.220794
$$795$$ 0 0
$$796$$ 32996.1 1.46924
$$797$$ 11719.4 0.520855 0.260427 0.965493i $$-0.416137\pi$$
0.260427 + 0.965493i $$0.416137\pi$$
$$798$$ 0 0
$$799$$ 21661.2 0.959095
$$800$$ 15657.7 0.691978
$$801$$ 0 0
$$802$$ −691.096 −0.0304282
$$803$$ 2629.53 0.115559
$$804$$ 0 0
$$805$$ −13610.0 −0.595886
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −2416.77 −0.105225
$$809$$ 24096.0 1.04718 0.523592 0.851969i $$-0.324592\pi$$
0.523592 + 0.851969i $$0.324592\pi$$
$$810$$ 0 0
$$811$$ 16622.6 0.719729 0.359864 0.933005i $$-0.382823\pi$$
0.359864 + 0.933005i $$0.382823\pi$$
$$812$$ −4530.42 −0.195796
$$813$$ 0 0
$$814$$ −42.1933 −0.00181680
$$815$$ 16764.1 0.720516
$$816$$ 0 0
$$817$$ −2224.59 −0.0952613
$$818$$ 2962.05 0.126609
$$819$$ 0 0
$$820$$ 31668.7 1.34868
$$821$$ 38005.5 1.61559 0.807797 0.589461i $$-0.200660\pi$$
0.807797 + 0.589461i $$0.200660\pi$$
$$822$$ 0 0
$$823$$ −15859.5 −0.671722 −0.335861 0.941912i $$-0.609027\pi$$
−0.335861 + 0.941912i $$0.609027\pi$$
$$824$$ 4022.01 0.170040
$$825$$ 0 0
$$826$$ 694.155 0.0292406
$$827$$ −12201.0 −0.513023 −0.256512 0.966541i $$-0.582573\pi$$
−0.256512 + 0.966541i $$0.582573\pi$$
$$828$$ 0 0
$$829$$ −5431.41 −0.227552 −0.113776 0.993506i $$-0.536295\pi$$
−0.113776 + 0.993506i $$0.536295\pi$$
$$830$$ −2624.62 −0.109761
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 21308.0 0.886290
$$834$$ 0 0
$$835$$ −65540.3 −2.71630
$$836$$ 14147.5 0.585288
$$837$$ 0 0
$$838$$ −4716.02 −0.194406
$$839$$ 7960.90 0.327582 0.163791 0.986495i $$-0.447628\pi$$
0.163791 + 0.986495i $$0.447628\pi$$
$$840$$ 0 0
$$841$$ −13005.6 −0.533255
$$842$$ 3457.61 0.141517
$$843$$ 0 0
$$844$$ −10655.0 −0.434550
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −4504.10 −0.182719
$$848$$ 4020.35 0.162806
$$849$$ 0 0
$$850$$ −5726.56 −0.231082
$$851$$ 603.115 0.0242944
$$852$$ 0 0
$$853$$ 13576.7 0.544969 0.272485 0.962160i $$-0.412155\pi$$
0.272485 + 0.962160i $$0.412155\pi$$
$$854$$ 1581.65 0.0633756
$$855$$ 0 0
$$856$$ 3960.95 0.158157
$$857$$ 31223.9 1.24456 0.622281 0.782794i $$-0.286206\pi$$
0.622281 + 0.782794i $$0.286206\pi$$
$$858$$ 0 0
$$859$$ −11815.8 −0.469323 −0.234661 0.972077i $$-0.575398\pi$$
−0.234661 + 0.972077i $$0.575398\pi$$
$$860$$ −3827.65 −0.151770
$$861$$ 0 0
$$862$$ 6175.36 0.244006
$$863$$ 1790.84 0.0706384 0.0353192 0.999376i $$-0.488755\pi$$
0.0353192 + 0.999376i $$0.488755\pi$$
$$864$$ 0 0
$$865$$ −25336.4 −0.995912
$$866$$ 817.328 0.0320715
$$867$$ 0 0
$$868$$ 11726.1 0.458538
$$869$$ 4539.87 0.177220
$$870$$ 0 0
$$871$$ 0 0
$$872$$ −1219.86 −0.0473734
$$873$$ 0 0
$$874$$ 4979.01 0.192698
$$875$$ 6500.00 0.251132
$$876$$ 0 0
$$877$$ 43542.5 1.67654 0.838270 0.545255i $$-0.183567\pi$$
0.838270 + 0.545255i $$0.183567\pi$$
$$878$$ 2698.42 0.103721
$$879$$ 0 0
$$880$$ 23728.2 0.908953
$$881$$ 1020.04 0.0390080 0.0195040 0.999810i $$-0.493791\pi$$
0.0195040 + 0.999810i $$0.493791\pi$$
$$882$$ 0 0
$$883$$ −34781.9 −1.32560 −0.662800 0.748797i $$-0.730632\pi$$
−0.662800 + 0.748797i $$0.730632\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 6374.70 0.241718
$$887$$ −49785.1 −1.88458 −0.942288 0.334802i $$-0.891330\pi$$
−0.942288 + 0.334802i $$0.891330\pi$$
$$888$$ 0 0
$$889$$ 14162.3 0.534295
$$890$$ −5607.49 −0.211195
$$891$$ 0 0
$$892$$ 8272.08 0.310504
$$893$$ 25746.8 0.964818
$$894$$ 0 0
$$895$$ −20797.5 −0.776743
$$896$$ −4594.25 −0.171298
$$897$$ 0 0
$$898$$ 3088.36 0.114766
$$899$$ −29463.9 −1.09308
$$900$$ 0 0
$$901$$ −4599.60 −0.170072
$$902$$ −2239.29 −0.0826611
$$903$$ 0 0
$$904$$ 8766.74 0.322541
$$905$$ −20193.3 −0.741712
$$906$$ 0 0
$$907$$ 17389.9 0.636627 0.318314 0.947985i $$-0.396884\pi$$
0.318314 + 0.947985i $$0.396884\pi$$
$$908$$ 27047.6 0.988553
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −20419.5 −0.742621 −0.371311 0.928509i $$-0.621091\pi$$
−0.371311 + 0.928509i $$0.621091\pi$$
$$912$$ 0 0
$$913$$ −7537.71 −0.273233
$$914$$ 6181.60 0.223708
$$915$$ 0 0
$$916$$ 18150.2 0.654695
$$917$$ −11593.2 −0.417492
$$918$$ 0 0
$$919$$ −33231.8 −1.19283 −0.596417 0.802674i $$-0.703410\pi$$
−0.596417 + 0.802674i $$0.703410\pi$$
$$920$$ 17344.8 0.621567
$$921$$ 0 0
$$922$$ −6335.43 −0.226297
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −824.502 −0.0293075
$$926$$ 6930.31 0.245944
$$927$$ 0 0
$$928$$ 8695.59 0.307594
$$929$$ 25222.8 0.890780 0.445390 0.895337i $$-0.353065\pi$$
0.445390 + 0.895337i $$0.353065\pi$$
$$930$$ 0 0
$$931$$ 25327.0 0.891579
$$932$$ −29130.9 −1.02383
$$933$$ 0 0
$$934$$ 6607.97 0.231498
$$935$$ −27146.9 −0.949519
$$936$$ 0 0
$$937$$ −26979.4 −0.940639 −0.470319 0.882496i $$-0.655861\pi$$
−0.470319 + 0.882496i $$0.655861\pi$$
$$938$$ −1013.64 −0.0352842
$$939$$ 0 0
$$940$$ 44300.2 1.53714
$$941$$ −7641.67 −0.264730 −0.132365 0.991201i $$-0.542257\pi$$
−0.132365 + 0.991201i $$0.542257\pi$$
$$942$$ 0 0
$$943$$ 32008.7 1.10535
$$944$$ 17299.0 0.596434
$$945$$ 0 0
$$946$$ 270.653 0.00930200
$$947$$ 2869.32 0.0984587 0.0492293 0.998788i $$-0.484323\pi$$
0.0492293 + 0.998788i $$0.484323\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ −6806.67 −0.232461
$$951$$ 0 0
$$952$$ 2562.56 0.0872407
$$953$$ −12313.6 −0.418548 −0.209274 0.977857i $$-0.567110\pi$$
−0.209274 + 0.977857i $$0.567110\pi$$
$$954$$ 0 0
$$955$$ −47762.6 −1.61839
$$956$$ 47193.8 1.59661
$$957$$ 0 0
$$958$$ 172.110 0.00580442
$$959$$ −3741.57 −0.125987
$$960$$ 0 0
$$961$$ 46470.7 1.55989
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 40397.1 1.34969
$$965$$ −35093.2 −1.17066
$$966$$ 0 0
$$967$$ −17838.0 −0.593207 −0.296603 0.955001i $$-0.595854\pi$$
−0.296603 + 0.955001i $$0.595854\pi$$
$$968$$ 5740.12 0.190593
$$969$$ 0 0
$$970$$ 5928.97 0.196255
$$971$$ −41525.3 −1.37241 −0.686206 0.727408i $$-0.740725\pi$$
−0.686206 + 0.727408i $$0.740725\pi$$
$$972$$ 0 0
$$973$$ −3695.86 −0.121772
$$974$$ 4164.32 0.136995
$$975$$ 0 0
$$976$$ 39416.1 1.29270
$$977$$ 31654.4 1.03655 0.518277 0.855213i $$-0.326574\pi$$
0.518277 + 0.855213i $$0.326574\pi$$
$$978$$ 0 0
$$979$$ −16104.3 −0.525735
$$980$$ 43578.0 1.42046
$$981$$ 0 0
$$982$$ 830.342 0.0269830
$$983$$ 39913.2 1.29505 0.647525 0.762045i $$-0.275804\pi$$
0.647525 + 0.762045i $$0.275804\pi$$
$$984$$ 0 0
$$985$$ 71516.8 2.31342
$$986$$ −3180.28 −0.102719
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −3868.74 −0.124387
$$990$$ 0 0
$$991$$ −2700.94 −0.0865773 −0.0432887 0.999063i $$-0.513784\pi$$
−0.0432887 + 0.999063i $$0.513784\pi$$
$$992$$ −22506.9 −0.720358
$$993$$ 0 0
$$994$$ 364.736 0.0116386
$$995$$ −75256.6 −2.39778
$$996$$ 0 0
$$997$$ −9729.08 −0.309050 −0.154525 0.987989i $$-0.549385\pi$$
−0.154525 + 0.987989i $$0.549385\pi$$
$$998$$ −5862.08 −0.185933
$$999$$ 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 1521.4.a.t.1.1 2
3.2 odd 2 169.4.a.f.1.2 2
13.3 even 3 117.4.g.d.100.2 4
13.9 even 3 117.4.g.d.55.2 4
13.12 even 2 1521.4.a.l.1.2 2
39.2 even 12 169.4.e.g.147.2 8
39.5 even 4 169.4.b.e.168.3 4
39.8 even 4 169.4.b.e.168.2 4
39.11 even 12 169.4.e.g.147.3 8
39.17 odd 6 169.4.c.f.146.2 4
39.20 even 12 169.4.e.g.23.2 8
39.23 odd 6 169.4.c.f.22.2 4
39.29 odd 6 13.4.c.b.9.1 yes 4
39.32 even 12 169.4.e.g.23.3 8
39.35 odd 6 13.4.c.b.3.1 4
39.38 odd 2 169.4.a.j.1.1 2
156.35 even 6 208.4.i.e.81.1 4
156.107 even 6 208.4.i.e.113.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
13.4.c.b.3.1 4 39.35 odd 6
13.4.c.b.9.1 yes 4 39.29 odd 6
117.4.g.d.55.2 4 13.9 even 3
117.4.g.d.100.2 4 13.3 even 3
169.4.a.f.1.2 2 3.2 odd 2
169.4.a.j.1.1 2 39.38 odd 2
169.4.b.e.168.2 4 39.8 even 4
169.4.b.e.168.3 4 39.5 even 4
169.4.c.f.22.2 4 39.23 odd 6
169.4.c.f.146.2 4 39.17 odd 6
169.4.e.g.23.2 8 39.20 even 12
169.4.e.g.23.3 8 39.32 even 12
169.4.e.g.147.2 8 39.2 even 12
169.4.e.g.147.3 8 39.11 even 12
208.4.i.e.81.1 4 156.35 even 6
208.4.i.e.113.1 4 156.107 even 6
1521.4.a.l.1.2 2 13.12 even 2
1521.4.a.t.1.1 2 1.1 even 1 trivial