# Properties

 Label 729.2.a.b.1.4 Level $729$ Weight $2$ Character 729.1 Self dual yes Analytic conductor $5.821$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,2,Mod(1,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("729.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$729 = 3^{6}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 729.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$5.82109430735$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.7459857.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} - 6x^{4} + 13x^{3} + 12x^{2} - 12x - 8$$ x^6 - 3*x^5 - 6*x^4 + 13*x^3 + 12*x^2 - 12*x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-1.70506$$ of defining polynomial Character $$\chi$$ $$=$$ 729.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.172976 q^{2} -1.97008 q^{4} -3.73656 q^{5} +3.03150 q^{7} -0.686728 q^{8} +O(q^{10})$$ $$q+0.172976 q^{2} -1.97008 q^{4} -3.73656 q^{5} +3.03150 q^{7} -0.686728 q^{8} -0.646335 q^{10} -2.49170 q^{11} -0.765139 q^{13} +0.524376 q^{14} +3.82137 q^{16} +4.62278 q^{17} -0.611844 q^{19} +7.36132 q^{20} -0.431003 q^{22} +6.52438 q^{23} +8.96190 q^{25} -0.132351 q^{26} -5.97229 q^{28} -6.55089 q^{29} +6.55043 q^{31} +2.03446 q^{32} +0.799630 q^{34} -11.3274 q^{35} +4.95969 q^{37} -0.105834 q^{38} +2.56600 q^{40} +5.26024 q^{41} +5.57057 q^{43} +4.90884 q^{44} +1.12856 q^{46} +1.10762 q^{47} +2.18998 q^{49} +1.55019 q^{50} +1.50738 q^{52} +8.84310 q^{53} +9.31038 q^{55} -2.08181 q^{56} -1.13315 q^{58} -11.8518 q^{59} +8.18700 q^{61} +1.13307 q^{62} -7.29083 q^{64} +2.85899 q^{65} -1.21234 q^{67} -9.10725 q^{68} -1.95936 q^{70} -4.91946 q^{71} +4.29945 q^{73} +0.857907 q^{74} +1.20538 q^{76} -7.55357 q^{77} -11.7946 q^{79} -14.2788 q^{80} +0.909895 q^{82} +9.01607 q^{83} -17.2733 q^{85} +0.963575 q^{86} +1.71112 q^{88} +7.53885 q^{89} -2.31952 q^{91} -12.8535 q^{92} +0.191591 q^{94} +2.28619 q^{95} -0.948354 q^{97} +0.378814 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 6 q^{7} - 6 q^{8}+O(q^{10})$$ 6 * q - 3 * q^2 + 9 * q^4 + 3 * q^5 + 6 * q^7 - 6 * q^8 $$6 q - 3 q^{2} + 9 q^{4} + 3 q^{5} + 6 q^{7} - 6 q^{8} + 6 q^{10} + 6 q^{11} + 6 q^{13} - 24 q^{14} + 15 q^{16} + 9 q^{17} + 12 q^{19} + 21 q^{20} + 3 q^{22} + 12 q^{23} + 9 q^{25} - 24 q^{26} + 3 q^{28} - 21 q^{29} + 15 q^{31} - 30 q^{35} + 3 q^{37} - 15 q^{38} + 3 q^{40} + 12 q^{41} + 6 q^{43} + 33 q^{44} - 3 q^{46} + 15 q^{47} + 12 q^{49} + 24 q^{50} + 3 q^{52} + 9 q^{53} + 15 q^{55} - 12 q^{56} - 15 q^{58} - 6 q^{59} + 24 q^{61} + 30 q^{62} + 6 q^{64} + 15 q^{65} + 15 q^{67} - 36 q^{68} - 15 q^{70} + 12 q^{73} - 24 q^{74} + 9 q^{76} - 15 q^{77} + 24 q^{79} + 21 q^{80} - 21 q^{82} + 6 q^{83} - 18 q^{85} + 30 q^{86} - 21 q^{88} + 9 q^{89} + 18 q^{91} - 6 q^{92} - 6 q^{94} + 33 q^{95} - 21 q^{97} - 18 q^{98}+O(q^{100})$$ 6 * q - 3 * q^2 + 9 * q^4 + 3 * q^5 + 6 * q^7 - 6 * q^8 + 6 * q^10 + 6 * q^11 + 6 * q^13 - 24 * q^14 + 15 * q^16 + 9 * q^17 + 12 * q^19 + 21 * q^20 + 3 * q^22 + 12 * q^23 + 9 * q^25 - 24 * q^26 + 3 * q^28 - 21 * q^29 + 15 * q^31 - 30 * q^35 + 3 * q^37 - 15 * q^38 + 3 * q^40 + 12 * q^41 + 6 * q^43 + 33 * q^44 - 3 * q^46 + 15 * q^47 + 12 * q^49 + 24 * q^50 + 3 * q^52 + 9 * q^53 + 15 * q^55 - 12 * q^56 - 15 * q^58 - 6 * q^59 + 24 * q^61 + 30 * q^62 + 6 * q^64 + 15 * q^65 + 15 * q^67 - 36 * q^68 - 15 * q^70 + 12 * q^73 - 24 * q^74 + 9 * q^76 - 15 * q^77 + 24 * q^79 + 21 * q^80 - 21 * q^82 + 6 * q^83 - 18 * q^85 + 30 * q^86 - 21 * q^88 + 9 * q^89 + 18 * q^91 - 6 * q^92 - 6 * q^94 + 33 * q^95 - 21 * q^97 - 18 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.172976 0.122312 0.0611562 0.998128i $$-0.480521\pi$$
0.0611562 + 0.998128i $$0.480521\pi$$
$$3$$ 0 0
$$4$$ −1.97008 −0.985040
$$5$$ −3.73656 −1.67104 −0.835521 0.549459i $$-0.814834\pi$$
−0.835521 + 0.549459i $$0.814834\pi$$
$$6$$ 0 0
$$7$$ 3.03150 1.14580 0.572899 0.819626i $$-0.305819\pi$$
0.572899 + 0.819626i $$0.305819\pi$$
$$8$$ −0.686728 −0.242795
$$9$$ 0 0
$$10$$ −0.646335 −0.204389
$$11$$ −2.49170 −0.751275 −0.375637 0.926767i $$-0.622576\pi$$
−0.375637 + 0.926767i $$0.622576\pi$$
$$12$$ 0 0
$$13$$ −0.765139 −0.212211 −0.106106 0.994355i $$-0.533838\pi$$
−0.106106 + 0.994355i $$0.533838\pi$$
$$14$$ 0.524376 0.140145
$$15$$ 0 0
$$16$$ 3.82137 0.955343
$$17$$ 4.62278 1.12119 0.560595 0.828090i $$-0.310573\pi$$
0.560595 + 0.828090i $$0.310573\pi$$
$$18$$ 0 0
$$19$$ −0.611844 −0.140367 −0.0701833 0.997534i $$-0.522358\pi$$
−0.0701833 + 0.997534i $$0.522358\pi$$
$$20$$ 7.36132 1.64604
$$21$$ 0 0
$$22$$ −0.431003 −0.0918902
$$23$$ 6.52438 1.36043 0.680213 0.733014i $$-0.261887\pi$$
0.680213 + 0.733014i $$0.261887\pi$$
$$24$$ 0 0
$$25$$ 8.96190 1.79238
$$26$$ −0.132351 −0.0259561
$$27$$ 0 0
$$28$$ −5.97229 −1.12866
$$29$$ −6.55089 −1.21647 −0.608235 0.793757i $$-0.708122\pi$$
−0.608235 + 0.793757i $$0.708122\pi$$
$$30$$ 0 0
$$31$$ 6.55043 1.17649 0.588246 0.808682i $$-0.299819\pi$$
0.588246 + 0.808682i $$0.299819\pi$$
$$32$$ 2.03446 0.359645
$$33$$ 0 0
$$34$$ 0.799630 0.137135
$$35$$ −11.3274 −1.91468
$$36$$ 0 0
$$37$$ 4.95969 0.815368 0.407684 0.913123i $$-0.366337\pi$$
0.407684 + 0.913123i $$0.366337\pi$$
$$38$$ −0.105834 −0.0171686
$$39$$ 0 0
$$40$$ 2.56600 0.405721
$$41$$ 5.26024 0.821511 0.410756 0.911745i $$-0.365265\pi$$
0.410756 + 0.911745i $$0.365265\pi$$
$$42$$ 0 0
$$43$$ 5.57057 0.849505 0.424752 0.905310i $$-0.360361\pi$$
0.424752 + 0.905310i $$0.360361\pi$$
$$44$$ 4.90884 0.740035
$$45$$ 0 0
$$46$$ 1.12856 0.166397
$$47$$ 1.10762 0.161562 0.0807812 0.996732i $$-0.474259\pi$$
0.0807812 + 0.996732i $$0.474259\pi$$
$$48$$ 0 0
$$49$$ 2.18998 0.312854
$$50$$ 1.55019 0.219230
$$51$$ 0 0
$$52$$ 1.50738 0.209037
$$53$$ 8.84310 1.21469 0.607346 0.794437i $$-0.292234\pi$$
0.607346 + 0.794437i $$0.292234\pi$$
$$54$$ 0 0
$$55$$ 9.31038 1.25541
$$56$$ −2.08181 −0.278194
$$57$$ 0 0
$$58$$ −1.13315 −0.148789
$$59$$ −11.8518 −1.54297 −0.771484 0.636249i $$-0.780485\pi$$
−0.771484 + 0.636249i $$0.780485\pi$$
$$60$$ 0 0
$$61$$ 8.18700 1.04824 0.524119 0.851645i $$-0.324395\pi$$
0.524119 + 0.851645i $$0.324395\pi$$
$$62$$ 1.13307 0.143900
$$63$$ 0 0
$$64$$ −7.29083 −0.911354
$$65$$ 2.85899 0.354614
$$66$$ 0 0
$$67$$ −1.21234 −0.148111 −0.0740553 0.997254i $$-0.523594\pi$$
−0.0740553 + 0.997254i $$0.523594\pi$$
$$68$$ −9.10725 −1.10442
$$69$$ 0 0
$$70$$ −1.95936 −0.234189
$$71$$ −4.91946 −0.583833 −0.291916 0.956444i $$-0.594293\pi$$
−0.291916 + 0.956444i $$0.594293\pi$$
$$72$$ 0 0
$$73$$ 4.29945 0.503213 0.251606 0.967830i $$-0.419041\pi$$
0.251606 + 0.967830i $$0.419041\pi$$
$$74$$ 0.857907 0.0997296
$$75$$ 0 0
$$76$$ 1.20538 0.138267
$$77$$ −7.55357 −0.860809
$$78$$ 0 0
$$79$$ −11.7946 −1.32700 −0.663498 0.748178i $$-0.730929\pi$$
−0.663498 + 0.748178i $$0.730929\pi$$
$$80$$ −14.2788 −1.59642
$$81$$ 0 0
$$82$$ 0.909895 0.100481
$$83$$ 9.01607 0.989642 0.494821 0.868995i $$-0.335234\pi$$
0.494821 + 0.868995i $$0.335234\pi$$
$$84$$ 0 0
$$85$$ −17.2733 −1.87355
$$86$$ 0.963575 0.103905
$$87$$ 0 0
$$88$$ 1.71112 0.182406
$$89$$ 7.53885 0.799117 0.399558 0.916708i $$-0.369163\pi$$
0.399558 + 0.916708i $$0.369163\pi$$
$$90$$ 0 0
$$91$$ −2.31952 −0.243151
$$92$$ −12.8535 −1.34007
$$93$$ 0 0
$$94$$ 0.191591 0.0197611
$$95$$ 2.28619 0.234558
$$96$$ 0 0
$$97$$ −0.948354 −0.0962908 −0.0481454 0.998840i $$-0.515331\pi$$
−0.0481454 + 0.998840i $$0.515331\pi$$
$$98$$ 0.378814 0.0382659
$$99$$ 0 0
$$100$$ −17.6557 −1.76557
$$101$$ −5.60815 −0.558032 −0.279016 0.960286i $$-0.590008\pi$$
−0.279016 + 0.960286i $$0.590008\pi$$
$$102$$ 0 0
$$103$$ −9.42502 −0.928675 −0.464337 0.885658i $$-0.653707\pi$$
−0.464337 + 0.885658i $$0.653707\pi$$
$$104$$ 0.525442 0.0515239
$$105$$ 0 0
$$106$$ 1.52964 0.148572
$$107$$ −1.27825 −0.123573 −0.0617864 0.998089i $$-0.519680\pi$$
−0.0617864 + 0.998089i $$0.519680\pi$$
$$108$$ 0 0
$$109$$ −7.40689 −0.709451 −0.354726 0.934970i $$-0.615426\pi$$
−0.354726 + 0.934970i $$0.615426\pi$$
$$110$$ 1.61047 0.153552
$$111$$ 0 0
$$112$$ 11.5845 1.09463
$$113$$ 9.35196 0.879759 0.439879 0.898057i $$-0.355021\pi$$
0.439879 + 0.898057i $$0.355021\pi$$
$$114$$ 0 0
$$115$$ −24.3787 −2.27333
$$116$$ 12.9058 1.19827
$$117$$ 0 0
$$118$$ −2.05007 −0.188724
$$119$$ 14.0140 1.28466
$$120$$ 0 0
$$121$$ −4.79145 −0.435586
$$122$$ 1.41615 0.128213
$$123$$ 0 0
$$124$$ −12.9049 −1.15889
$$125$$ −14.8039 −1.32410
$$126$$ 0 0
$$127$$ 20.7968 1.84542 0.922710 0.385496i $$-0.125970\pi$$
0.922710 + 0.385496i $$0.125970\pi$$
$$128$$ −5.33006 −0.471115
$$129$$ 0 0
$$130$$ 0.494536 0.0433737
$$131$$ −0.655830 −0.0573001 −0.0286501 0.999590i $$-0.509121\pi$$
−0.0286501 + 0.999590i $$0.509121\pi$$
$$132$$ 0 0
$$133$$ −1.85480 −0.160832
$$134$$ −0.209705 −0.0181158
$$135$$ 0 0
$$136$$ −3.17460 −0.272219
$$137$$ 8.58760 0.733689 0.366844 0.930282i $$-0.380438\pi$$
0.366844 + 0.930282i $$0.380438\pi$$
$$138$$ 0 0
$$139$$ 13.4461 1.14049 0.570243 0.821476i $$-0.306849\pi$$
0.570243 + 0.821476i $$0.306849\pi$$
$$140$$ 22.3158 1.88603
$$141$$ 0 0
$$142$$ −0.850949 −0.0714100
$$143$$ 1.90649 0.159429
$$144$$ 0 0
$$145$$ 24.4778 2.03277
$$146$$ 0.743701 0.0615492
$$147$$ 0 0
$$148$$ −9.77098 −0.803170
$$149$$ 9.62207 0.788270 0.394135 0.919052i $$-0.371044\pi$$
0.394135 + 0.919052i $$0.371044\pi$$
$$150$$ 0 0
$$151$$ −7.12820 −0.580085 −0.290042 0.957014i $$-0.593669\pi$$
−0.290042 + 0.957014i $$0.593669\pi$$
$$152$$ 0.420170 0.0340803
$$153$$ 0 0
$$154$$ −1.30659 −0.105288
$$155$$ −24.4761 −1.96597
$$156$$ 0 0
$$157$$ 7.68577 0.613391 0.306696 0.951808i $$-0.400777\pi$$
0.306696 + 0.951808i $$0.400777\pi$$
$$158$$ −2.04018 −0.162308
$$159$$ 0 0
$$160$$ −7.60189 −0.600982
$$161$$ 19.7786 1.55877
$$162$$ 0 0
$$163$$ 1.04750 0.0820465 0.0410232 0.999158i $$-0.486938\pi$$
0.0410232 + 0.999158i $$0.486938\pi$$
$$164$$ −10.3631 −0.809221
$$165$$ 0 0
$$166$$ 1.55956 0.121046
$$167$$ 8.35408 0.646458 0.323229 0.946321i $$-0.395232\pi$$
0.323229 + 0.946321i $$0.395232\pi$$
$$168$$ 0 0
$$169$$ −12.4146 −0.954966
$$170$$ −2.98787 −0.229159
$$171$$ 0 0
$$172$$ −10.9745 −0.836796
$$173$$ 21.8458 1.66090 0.830452 0.557090i $$-0.188082\pi$$
0.830452 + 0.557090i $$0.188082\pi$$
$$174$$ 0 0
$$175$$ 27.1680 2.05371
$$176$$ −9.52170 −0.717725
$$177$$ 0 0
$$178$$ 1.30404 0.0977419
$$179$$ −9.08866 −0.679319 −0.339659 0.940549i $$-0.610312\pi$$
−0.339659 + 0.940549i $$0.610312\pi$$
$$180$$ 0 0
$$181$$ −7.13077 −0.530026 −0.265013 0.964245i $$-0.585376\pi$$
−0.265013 + 0.964245i $$0.585376\pi$$
$$182$$ −0.401221 −0.0297404
$$183$$ 0 0
$$184$$ −4.48047 −0.330305
$$185$$ −18.5322 −1.36251
$$186$$ 0 0
$$187$$ −11.5186 −0.842321
$$188$$ −2.18209 −0.159145
$$189$$ 0 0
$$190$$ 0.395456 0.0286894
$$191$$ 11.9556 0.865079 0.432539 0.901615i $$-0.357618\pi$$
0.432539 + 0.901615i $$0.357618\pi$$
$$192$$ 0 0
$$193$$ −8.87364 −0.638738 −0.319369 0.947630i $$-0.603471\pi$$
−0.319369 + 0.947630i $$0.603471\pi$$
$$194$$ −0.164042 −0.0117776
$$195$$ 0 0
$$196$$ −4.31443 −0.308174
$$197$$ 7.39790 0.527079 0.263539 0.964649i $$-0.415110\pi$$
0.263539 + 0.964649i $$0.415110\pi$$
$$198$$ 0 0
$$199$$ −10.3837 −0.736084 −0.368042 0.929809i $$-0.619972\pi$$
−0.368042 + 0.929809i $$0.619972\pi$$
$$200$$ −6.15439 −0.435181
$$201$$ 0 0
$$202$$ −0.970076 −0.0682543
$$203$$ −19.8590 −1.39383
$$204$$ 0 0
$$205$$ −19.6552 −1.37278
$$206$$ −1.63030 −0.113588
$$207$$ 0 0
$$208$$ −2.92388 −0.202735
$$209$$ 1.52453 0.105454
$$210$$ 0 0
$$211$$ 20.8611 1.43614 0.718070 0.695971i $$-0.245026\pi$$
0.718070 + 0.695971i $$0.245026\pi$$
$$212$$ −17.4216 −1.19652
$$213$$ 0 0
$$214$$ −0.221106 −0.0151145
$$215$$ −20.8148 −1.41956
$$216$$ 0 0
$$217$$ 19.8576 1.34802
$$218$$ −1.28121 −0.0867747
$$219$$ 0 0
$$220$$ −18.3422 −1.23663
$$221$$ −3.53707 −0.237929
$$222$$ 0 0
$$223$$ 23.5785 1.57893 0.789466 0.613794i $$-0.210357\pi$$
0.789466 + 0.613794i $$0.210357\pi$$
$$224$$ 6.16747 0.412081
$$225$$ 0 0
$$226$$ 1.61766 0.107605
$$227$$ −10.4841 −0.695856 −0.347928 0.937521i $$-0.613115\pi$$
−0.347928 + 0.937521i $$0.613115\pi$$
$$228$$ 0 0
$$229$$ −13.8824 −0.917376 −0.458688 0.888597i $$-0.651680\pi$$
−0.458688 + 0.888597i $$0.651680\pi$$
$$230$$ −4.21694 −0.278056
$$231$$ 0 0
$$232$$ 4.49868 0.295353
$$233$$ 7.59964 0.497869 0.248935 0.968520i $$-0.419920\pi$$
0.248935 + 0.968520i $$0.419920\pi$$
$$234$$ 0 0
$$235$$ −4.13868 −0.269977
$$236$$ 23.3489 1.51988
$$237$$ 0 0
$$238$$ 2.42408 0.157130
$$239$$ 16.5587 1.07109 0.535546 0.844506i $$-0.320106\pi$$
0.535546 + 0.844506i $$0.320106\pi$$
$$240$$ 0 0
$$241$$ −14.5185 −0.935218 −0.467609 0.883935i $$-0.654884\pi$$
−0.467609 + 0.883935i $$0.654884\pi$$
$$242$$ −0.828806 −0.0532776
$$243$$ 0 0
$$244$$ −16.1290 −1.03256
$$245$$ −8.18299 −0.522792
$$246$$ 0 0
$$247$$ 0.468145 0.0297874
$$248$$ −4.49836 −0.285646
$$249$$ 0 0
$$250$$ −2.56072 −0.161954
$$251$$ 9.05181 0.571345 0.285673 0.958327i $$-0.407783\pi$$
0.285673 + 0.958327i $$0.407783\pi$$
$$252$$ 0 0
$$253$$ −16.2568 −1.02205
$$254$$ 3.59735 0.225718
$$255$$ 0 0
$$256$$ 13.6597 0.853730
$$257$$ 9.69988 0.605062 0.302531 0.953140i $$-0.402168\pi$$
0.302531 + 0.953140i $$0.402168\pi$$
$$258$$ 0 0
$$259$$ 15.0353 0.934247
$$260$$ −5.63243 −0.349309
$$261$$ 0 0
$$262$$ −0.113443 −0.00700852
$$263$$ −26.8552 −1.65597 −0.827983 0.560754i $$-0.810511\pi$$
−0.827983 + 0.560754i $$0.810511\pi$$
$$264$$ 0 0
$$265$$ −33.0428 −2.02980
$$266$$ −0.320836 −0.0196717
$$267$$ 0 0
$$268$$ 2.38840 0.145895
$$269$$ 11.7388 0.715729 0.357865 0.933774i $$-0.383505\pi$$
0.357865 + 0.933774i $$0.383505\pi$$
$$270$$ 0 0
$$271$$ 0.144576 0.00878238 0.00439119 0.999990i $$-0.498602\pi$$
0.00439119 + 0.999990i $$0.498602\pi$$
$$272$$ 17.6654 1.07112
$$273$$ 0 0
$$274$$ 1.48545 0.0897392
$$275$$ −22.3303 −1.34657
$$276$$ 0 0
$$277$$ −1.01442 −0.0609509 −0.0304754 0.999536i $$-0.509702\pi$$
−0.0304754 + 0.999536i $$0.509702\pi$$
$$278$$ 2.32586 0.139496
$$279$$ 0 0
$$280$$ 7.77883 0.464874
$$281$$ −27.5295 −1.64227 −0.821135 0.570734i $$-0.806659\pi$$
−0.821135 + 0.570734i $$0.806659\pi$$
$$282$$ 0 0
$$283$$ −26.8029 −1.59327 −0.796633 0.604463i $$-0.793388\pi$$
−0.796633 + 0.604463i $$0.793388\pi$$
$$284$$ 9.69173 0.575098
$$285$$ 0 0
$$286$$ 0.329777 0.0195001
$$287$$ 15.9464 0.941286
$$288$$ 0 0
$$289$$ 4.37012 0.257066
$$290$$ 4.23407 0.248633
$$291$$ 0 0
$$292$$ −8.47026 −0.495684
$$293$$ 18.6573 1.08997 0.544984 0.838446i $$-0.316536\pi$$
0.544984 + 0.838446i $$0.316536\pi$$
$$294$$ 0 0
$$295$$ 44.2848 2.57836
$$296$$ −3.40596 −0.197967
$$297$$ 0 0
$$298$$ 1.66439 0.0964153
$$299$$ −4.99205 −0.288698
$$300$$ 0 0
$$301$$ 16.8872 0.973361
$$302$$ −1.23301 −0.0709516
$$303$$ 0 0
$$304$$ −2.33808 −0.134098
$$305$$ −30.5913 −1.75165
$$306$$ 0 0
$$307$$ 33.7893 1.92845 0.964227 0.265077i $$-0.0853973\pi$$
0.964227 + 0.265077i $$0.0853973\pi$$
$$308$$ 14.8811 0.847931
$$309$$ 0 0
$$310$$ −4.23377 −0.240462
$$311$$ 34.6866 1.96690 0.983448 0.181193i $$-0.0579959\pi$$
0.983448 + 0.181193i $$0.0579959\pi$$
$$312$$ 0 0
$$313$$ 3.34038 0.188809 0.0944047 0.995534i $$-0.469905\pi$$
0.0944047 + 0.995534i $$0.469905\pi$$
$$314$$ 1.32945 0.0750254
$$315$$ 0 0
$$316$$ 23.2363 1.30714
$$317$$ 31.0328 1.74298 0.871488 0.490417i $$-0.163156\pi$$
0.871488 + 0.490417i $$0.163156\pi$$
$$318$$ 0 0
$$319$$ 16.3228 0.913903
$$320$$ 27.2426 1.52291
$$321$$ 0 0
$$322$$ 3.42123 0.190658
$$323$$ −2.82842 −0.157378
$$324$$ 0 0
$$325$$ −6.85710 −0.380363
$$326$$ 0.181192 0.0100353
$$327$$ 0 0
$$328$$ −3.61235 −0.199459
$$329$$ 3.35773 0.185118
$$330$$ 0 0
$$331$$ 3.27168 0.179828 0.0899138 0.995950i $$-0.471341\pi$$
0.0899138 + 0.995950i $$0.471341\pi$$
$$332$$ −17.7624 −0.974837
$$333$$ 0 0
$$334$$ 1.44505 0.0790699
$$335$$ 4.52998 0.247499
$$336$$ 0 0
$$337$$ 6.36581 0.346768 0.173384 0.984854i $$-0.444530\pi$$
0.173384 + 0.984854i $$0.444530\pi$$
$$338$$ −2.14742 −0.116804
$$339$$ 0 0
$$340$$ 34.0298 1.84553
$$341$$ −16.3217 −0.883868
$$342$$ 0 0
$$343$$ −14.5816 −0.787331
$$344$$ −3.82547 −0.206256
$$345$$ 0 0
$$346$$ 3.77880 0.203149
$$347$$ 8.79241 0.472001 0.236001 0.971753i $$-0.424163\pi$$
0.236001 + 0.971753i $$0.424163\pi$$
$$348$$ 0 0
$$349$$ 14.4002 0.770826 0.385413 0.922744i $$-0.374059\pi$$
0.385413 + 0.922744i $$0.374059\pi$$
$$350$$ 4.69941 0.251194
$$351$$ 0 0
$$352$$ −5.06926 −0.270192
$$353$$ −33.2005 −1.76708 −0.883542 0.468353i $$-0.844848\pi$$
−0.883542 + 0.468353i $$0.844848\pi$$
$$354$$ 0 0
$$355$$ 18.3819 0.975609
$$356$$ −14.8521 −0.787162
$$357$$ 0 0
$$358$$ −1.57212 −0.0830891
$$359$$ −4.94514 −0.260995 −0.130497 0.991449i $$-0.541657\pi$$
−0.130497 + 0.991449i $$0.541657\pi$$
$$360$$ 0 0
$$361$$ −18.6256 −0.980297
$$362$$ −1.23345 −0.0648288
$$363$$ 0 0
$$364$$ 4.56963 0.239514
$$365$$ −16.0652 −0.840889
$$366$$ 0 0
$$367$$ −2.49245 −0.130105 −0.0650525 0.997882i $$-0.520721\pi$$
−0.0650525 + 0.997882i $$0.520721\pi$$
$$368$$ 24.9321 1.29967
$$369$$ 0 0
$$370$$ −3.20562 −0.166652
$$371$$ 26.8078 1.39179
$$372$$ 0 0
$$373$$ −28.0476 −1.45225 −0.726124 0.687563i $$-0.758680\pi$$
−0.726124 + 0.687563i $$0.758680\pi$$
$$374$$ −1.99244 −0.103026
$$375$$ 0 0
$$376$$ −0.760631 −0.0392265
$$377$$ 5.01234 0.258149
$$378$$ 0 0
$$379$$ 5.13991 0.264019 0.132010 0.991248i $$-0.457857\pi$$
0.132010 + 0.991248i $$0.457857\pi$$
$$380$$ −4.50398 −0.231049
$$381$$ 0 0
$$382$$ 2.06804 0.105810
$$383$$ 0.0446729 0.00228268 0.00114134 0.999999i $$-0.499637\pi$$
0.00114134 + 0.999999i $$0.499637\pi$$
$$384$$ 0 0
$$385$$ 28.2244 1.43845
$$386$$ −1.53493 −0.0781256
$$387$$ 0 0
$$388$$ 1.86833 0.0948502
$$389$$ −20.9823 −1.06384 −0.531921 0.846794i $$-0.678530\pi$$
−0.531921 + 0.846794i $$0.678530\pi$$
$$390$$ 0 0
$$391$$ 30.1608 1.52530
$$392$$ −1.50392 −0.0759594
$$393$$ 0 0
$$394$$ 1.27966 0.0644683
$$395$$ 44.0713 2.21747
$$396$$ 0 0
$$397$$ −0.00245641 −0.000123284 0 −6.16419e−5 1.00000i $$-0.500020\pi$$
−6.16419e−5 1.00000i $$0.500020\pi$$
$$398$$ −1.79614 −0.0900323
$$399$$ 0 0
$$400$$ 34.2467 1.71234
$$401$$ −25.2563 −1.26124 −0.630620 0.776091i $$-0.717199\pi$$
−0.630620 + 0.776091i $$0.717199\pi$$
$$402$$ 0 0
$$403$$ −5.01198 −0.249665
$$404$$ 11.0485 0.549684
$$405$$ 0 0
$$406$$ −3.43513 −0.170483
$$407$$ −12.3580 −0.612565
$$408$$ 0 0
$$409$$ −23.2885 −1.15154 −0.575772 0.817610i $$-0.695298\pi$$
−0.575772 + 0.817610i $$0.695298\pi$$
$$410$$ −3.39988 −0.167908
$$411$$ 0 0
$$412$$ 18.5680 0.914781
$$413$$ −35.9286 −1.76793
$$414$$ 0 0
$$415$$ −33.6891 −1.65373
$$416$$ −1.55665 −0.0763208
$$417$$ 0 0
$$418$$ 0.263707 0.0128983
$$419$$ −31.2884 −1.52854 −0.764268 0.644898i $$-0.776900\pi$$
−0.764268 + 0.644898i $$0.776900\pi$$
$$420$$ 0 0
$$421$$ 30.5296 1.48792 0.743960 0.668224i $$-0.232945\pi$$
0.743960 + 0.668224i $$0.232945\pi$$
$$422$$ 3.60848 0.175658
$$423$$ 0 0
$$424$$ −6.07280 −0.294921
$$425$$ 41.4289 2.00960
$$426$$ 0 0
$$427$$ 24.8189 1.20107
$$428$$ 2.51825 0.121724
$$429$$ 0 0
$$430$$ −3.60046 −0.173630
$$431$$ −12.4246 −0.598474 −0.299237 0.954179i $$-0.596732\pi$$
−0.299237 + 0.954179i $$0.596732\pi$$
$$432$$ 0 0
$$433$$ −0.760649 −0.0365545 −0.0182772 0.999833i $$-0.505818\pi$$
−0.0182772 + 0.999833i $$0.505818\pi$$
$$434$$ 3.43489 0.164880
$$435$$ 0 0
$$436$$ 14.5922 0.698838
$$437$$ −3.99190 −0.190958
$$438$$ 0 0
$$439$$ 30.1094 1.43704 0.718521 0.695506i $$-0.244820\pi$$
0.718521 + 0.695506i $$0.244820\pi$$
$$440$$ −6.39370 −0.304808
$$441$$ 0 0
$$442$$ −0.611828 −0.0291017
$$443$$ 13.6616 0.649081 0.324541 0.945872i $$-0.394790\pi$$
0.324541 + 0.945872i $$0.394790\pi$$
$$444$$ 0 0
$$445$$ −28.1694 −1.33536
$$446$$ 4.07851 0.193123
$$447$$ 0 0
$$448$$ −22.1021 −1.04423
$$449$$ −21.9989 −1.03819 −0.519097 0.854715i $$-0.673732\pi$$
−0.519097 + 0.854715i $$0.673732\pi$$
$$450$$ 0 0
$$451$$ −13.1069 −0.617181
$$452$$ −18.4241 −0.866597
$$453$$ 0 0
$$454$$ −1.81350 −0.0851119
$$455$$ 8.66702 0.406316
$$456$$ 0 0
$$457$$ −1.48883 −0.0696444 −0.0348222 0.999394i $$-0.511086\pi$$
−0.0348222 + 0.999394i $$0.511086\pi$$
$$458$$ −2.40132 −0.112206
$$459$$ 0 0
$$460$$ 48.0281 2.23932
$$461$$ −7.11334 −0.331301 −0.165651 0.986185i $$-0.552972\pi$$
−0.165651 + 0.986185i $$0.552972\pi$$
$$462$$ 0 0
$$463$$ −26.5407 −1.23345 −0.616726 0.787178i $$-0.711541\pi$$
−0.616726 + 0.787178i $$0.711541\pi$$
$$464$$ −25.0334 −1.16215
$$465$$ 0 0
$$466$$ 1.31456 0.0608956
$$467$$ 26.1519 1.21017 0.605084 0.796162i $$-0.293140\pi$$
0.605084 + 0.796162i $$0.293140\pi$$
$$468$$ 0 0
$$469$$ −3.67520 −0.169705
$$470$$ −0.715891 −0.0330216
$$471$$ 0 0
$$472$$ 8.13894 0.374625
$$473$$ −13.8802 −0.638211
$$474$$ 0 0
$$475$$ −5.48328 −0.251590
$$476$$ −27.6086 −1.26544
$$477$$ 0 0
$$478$$ 2.86425 0.131008
$$479$$ −10.4065 −0.475487 −0.237744 0.971328i $$-0.576408\pi$$
−0.237744 + 0.971328i $$0.576408\pi$$
$$480$$ 0 0
$$481$$ −3.79485 −0.173030
$$482$$ −2.51135 −0.114389
$$483$$ 0 0
$$484$$ 9.43954 0.429070
$$485$$ 3.54358 0.160906
$$486$$ 0 0
$$487$$ −18.4664 −0.836791 −0.418396 0.908265i $$-0.637407\pi$$
−0.418396 + 0.908265i $$0.637407\pi$$
$$488$$ −5.62225 −0.254507
$$489$$ 0 0
$$490$$ −1.41546 −0.0639440
$$491$$ −16.9739 −0.766021 −0.383011 0.923744i $$-0.625113\pi$$
−0.383011 + 0.923744i $$0.625113\pi$$
$$492$$ 0 0
$$493$$ −30.2834 −1.36389
$$494$$ 0.0809779 0.00364337
$$495$$ 0 0
$$496$$ 25.0316 1.12395
$$497$$ −14.9133 −0.668955
$$498$$ 0 0
$$499$$ 24.6462 1.10331 0.551657 0.834071i $$-0.313996\pi$$
0.551657 + 0.834071i $$0.313996\pi$$
$$500$$ 29.1648 1.30429
$$501$$ 0 0
$$502$$ 1.56575 0.0698826
$$503$$ 40.1137 1.78858 0.894291 0.447485i $$-0.147680\pi$$
0.894291 + 0.447485i $$0.147680\pi$$
$$504$$ 0 0
$$505$$ 20.9552 0.932495
$$506$$ −2.81203 −0.125010
$$507$$ 0 0
$$508$$ −40.9714 −1.81781
$$509$$ 4.94852 0.219339 0.109670 0.993968i $$-0.465021\pi$$
0.109670 + 0.993968i $$0.465021\pi$$
$$510$$ 0 0
$$511$$ 13.0338 0.576580
$$512$$ 13.0229 0.575537
$$513$$ 0 0
$$514$$ 1.67785 0.0740066
$$515$$ 35.2172 1.55185
$$516$$ 0 0
$$517$$ −2.75984 −0.121378
$$518$$ 2.60074 0.114270
$$519$$ 0 0
$$520$$ −1.96335 −0.0860985
$$521$$ 7.73958 0.339077 0.169539 0.985524i $$-0.445772\pi$$
0.169539 + 0.985524i $$0.445772\pi$$
$$522$$ 0 0
$$523$$ 36.0140 1.57478 0.787391 0.616453i $$-0.211431\pi$$
0.787391 + 0.616453i $$0.211431\pi$$
$$524$$ 1.29204 0.0564429
$$525$$ 0 0
$$526$$ −4.64531 −0.202545
$$527$$ 30.2812 1.31907
$$528$$ 0 0
$$529$$ 19.5675 0.850760
$$530$$ −5.71561 −0.248270
$$531$$ 0 0
$$532$$ 3.65411 0.158426
$$533$$ −4.02481 −0.174334
$$534$$ 0 0
$$535$$ 4.77625 0.206495
$$536$$ 0.832547 0.0359605
$$537$$ 0 0
$$538$$ 2.03053 0.0875426
$$539$$ −5.45676 −0.235039
$$540$$ 0 0
$$541$$ −24.4147 −1.04967 −0.524834 0.851204i $$-0.675873\pi$$
−0.524834 + 0.851204i $$0.675873\pi$$
$$542$$ 0.0250082 0.00107419
$$543$$ 0 0
$$544$$ 9.40487 0.403231
$$545$$ 27.6763 1.18552
$$546$$ 0 0
$$547$$ −28.3618 −1.21266 −0.606331 0.795212i $$-0.707359\pi$$
−0.606331 + 0.795212i $$0.707359\pi$$
$$548$$ −16.9183 −0.722712
$$549$$ 0 0
$$550$$ −3.86261 −0.164702
$$551$$ 4.00812 0.170752
$$552$$ 0 0
$$553$$ −35.7553 −1.52047
$$554$$ −0.175471 −0.00745505
$$555$$ 0 0
$$556$$ −26.4900 −1.12342
$$557$$ −36.9373 −1.56508 −0.782542 0.622598i $$-0.786077\pi$$
−0.782542 + 0.622598i $$0.786077\pi$$
$$558$$ 0 0
$$559$$ −4.26226 −0.180274
$$560$$ −43.2861 −1.82917
$$561$$ 0 0
$$562$$ −4.76193 −0.200870
$$563$$ 22.7754 0.959869 0.479935 0.877304i $$-0.340660\pi$$
0.479935 + 0.877304i $$0.340660\pi$$
$$564$$ 0 0
$$565$$ −34.9442 −1.47011
$$566$$ −4.63625 −0.194876
$$567$$ 0 0
$$568$$ 3.37833 0.141752
$$569$$ 30.8688 1.29409 0.647043 0.762454i $$-0.276005\pi$$
0.647043 + 0.762454i $$0.276005\pi$$
$$570$$ 0 0
$$571$$ 12.8205 0.536521 0.268260 0.963346i $$-0.413551\pi$$
0.268260 + 0.963346i $$0.413551\pi$$
$$572$$ −3.75594 −0.157044
$$573$$ 0 0
$$574$$ 2.75834 0.115131
$$575$$ 58.4708 2.43840
$$576$$ 0 0
$$577$$ −23.5264 −0.979417 −0.489708 0.871886i $$-0.662897\pi$$
−0.489708 + 0.871886i $$0.662897\pi$$
$$578$$ 0.755926 0.0314424
$$579$$ 0 0
$$580$$ −48.2232 −2.00236
$$581$$ 27.3322 1.13393
$$582$$ 0 0
$$583$$ −22.0343 −0.912568
$$584$$ −2.95255 −0.122178
$$585$$ 0 0
$$586$$ 3.22726 0.133317
$$587$$ 11.3874 0.470010 0.235005 0.971994i $$-0.424489\pi$$
0.235005 + 0.971994i $$0.424489\pi$$
$$588$$ 0 0
$$589$$ −4.00784 −0.165140
$$590$$ 7.66021 0.315366
$$591$$ 0 0
$$592$$ 18.9528 0.778956
$$593$$ −37.7324 −1.54948 −0.774742 0.632277i $$-0.782120\pi$$
−0.774742 + 0.632277i $$0.782120\pi$$
$$594$$ 0 0
$$595$$ −52.3640 −2.14672
$$596$$ −18.9562 −0.776478
$$597$$ 0 0
$$598$$ −0.863505 −0.0353113
$$599$$ 47.3582 1.93500 0.967502 0.252865i $$-0.0813727\pi$$
0.967502 + 0.252865i $$0.0813727\pi$$
$$600$$ 0 0
$$601$$ −31.1074 −1.26890 −0.634449 0.772964i $$-0.718773\pi$$
−0.634449 + 0.772964i $$0.718773\pi$$
$$602$$ 2.92108 0.119054
$$603$$ 0 0
$$604$$ 14.0431 0.571407
$$605$$ 17.9036 0.727883
$$606$$ 0 0
$$607$$ 29.4864 1.19682 0.598409 0.801191i $$-0.295800\pi$$
0.598409 + 0.801191i $$0.295800\pi$$
$$608$$ −1.24477 −0.0504822
$$609$$ 0 0
$$610$$ −5.29155 −0.214249
$$611$$ −0.847480 −0.0342854
$$612$$ 0 0
$$613$$ −6.10428 −0.246550 −0.123275 0.992373i $$-0.539340\pi$$
−0.123275 + 0.992373i $$0.539340\pi$$
$$614$$ 5.84473 0.235874
$$615$$ 0 0
$$616$$ 5.18725 0.209000
$$617$$ −19.1201 −0.769747 −0.384873 0.922969i $$-0.625755\pi$$
−0.384873 + 0.922969i $$0.625755\pi$$
$$618$$ 0 0
$$619$$ −6.75385 −0.271460 −0.135730 0.990746i $$-0.543338\pi$$
−0.135730 + 0.990746i $$0.543338\pi$$
$$620$$ 48.2198 1.93655
$$621$$ 0 0
$$622$$ 5.99994 0.240576
$$623$$ 22.8540 0.915627
$$624$$ 0 0
$$625$$ 10.5061 0.420246
$$626$$ 0.577805 0.0230937
$$627$$ 0 0
$$628$$ −15.1416 −0.604215
$$629$$ 22.9276 0.914182
$$630$$ 0 0
$$631$$ −0.456907 −0.0181892 −0.00909458 0.999959i $$-0.502895\pi$$
−0.00909458 + 0.999959i $$0.502895\pi$$
$$632$$ 8.09969 0.322188
$$633$$ 0 0
$$634$$ 5.36793 0.213188
$$635$$ −77.7086 −3.08377
$$636$$ 0 0
$$637$$ −1.67564 −0.0663912
$$638$$ 2.82346 0.111782
$$639$$ 0 0
$$640$$ 19.9161 0.787253
$$641$$ 2.87103 0.113399 0.0566994 0.998391i $$-0.481942\pi$$
0.0566994 + 0.998391i $$0.481942\pi$$
$$642$$ 0 0
$$643$$ −1.70284 −0.0671536 −0.0335768 0.999436i $$-0.510690\pi$$
−0.0335768 + 0.999436i $$0.510690\pi$$
$$644$$ −38.9655 −1.53545
$$645$$ 0 0
$$646$$ −0.489249 −0.0192492
$$647$$ −36.1004 −1.41925 −0.709626 0.704579i $$-0.751136\pi$$
−0.709626 + 0.704579i $$0.751136\pi$$
$$648$$ 0 0
$$649$$ 29.5310 1.15919
$$650$$ −1.18611 −0.0465232
$$651$$ 0 0
$$652$$ −2.06366 −0.0808191
$$653$$ 43.5680 1.70495 0.852473 0.522771i $$-0.175102\pi$$
0.852473 + 0.522771i $$0.175102\pi$$
$$654$$ 0 0
$$655$$ 2.45055 0.0957509
$$656$$ 20.1013 0.784825
$$657$$ 0 0
$$658$$ 0.580807 0.0226422
$$659$$ −25.4810 −0.992598 −0.496299 0.868152i $$-0.665308\pi$$
−0.496299 + 0.868152i $$0.665308\pi$$
$$660$$ 0 0
$$661$$ 34.1672 1.32895 0.664475 0.747310i $$-0.268655\pi$$
0.664475 + 0.747310i $$0.268655\pi$$
$$662$$ 0.565921 0.0219952
$$663$$ 0 0
$$664$$ −6.19159 −0.240280
$$665$$ 6.93059 0.268757
$$666$$ 0 0
$$667$$ −42.7405 −1.65492
$$668$$ −16.4582 −0.636787
$$669$$ 0 0
$$670$$ 0.783577 0.0302722
$$671$$ −20.3995 −0.787515
$$672$$ 0 0
$$673$$ −29.5437 −1.13883 −0.569413 0.822051i $$-0.692830\pi$$
−0.569413 + 0.822051i $$0.692830\pi$$
$$674$$ 1.10113 0.0424140
$$675$$ 0 0
$$676$$ 24.4577 0.940680
$$677$$ −40.7802 −1.56731 −0.783656 0.621195i $$-0.786647\pi$$
−0.783656 + 0.621195i $$0.786647\pi$$
$$678$$ 0 0
$$679$$ −2.87493 −0.110330
$$680$$ 11.8621 0.454890
$$681$$ 0 0
$$682$$ −2.82326 −0.108108
$$683$$ −31.6426 −1.21077 −0.605384 0.795933i $$-0.706981\pi$$
−0.605384 + 0.795933i $$0.706981\pi$$
$$684$$ 0 0
$$685$$ −32.0881 −1.22602
$$686$$ −2.52226 −0.0963004
$$687$$ 0 0
$$688$$ 21.2872 0.811568
$$689$$ −6.76620 −0.257772
$$690$$ 0 0
$$691$$ −28.5848 −1.08742 −0.543708 0.839275i $$-0.682980\pi$$
−0.543708 + 0.839275i $$0.682980\pi$$
$$692$$ −43.0379 −1.63606
$$693$$ 0 0
$$694$$ 1.52088 0.0577316
$$695$$ −50.2423 −1.90580
$$696$$ 0 0
$$697$$ 24.3169 0.921070
$$698$$ 2.49089 0.0942817
$$699$$ 0 0
$$700$$ −53.5231 −2.02298
$$701$$ 7.52982 0.284397 0.142199 0.989838i $$-0.454583\pi$$
0.142199 + 0.989838i $$0.454583\pi$$
$$702$$ 0 0
$$703$$ −3.03455 −0.114450
$$704$$ 18.1665 0.684677
$$705$$ 0 0
$$706$$ −5.74288 −0.216136
$$707$$ −17.0011 −0.639392
$$708$$ 0 0
$$709$$ 8.01399 0.300972 0.150486 0.988612i $$-0.451916\pi$$
0.150486 + 0.988612i $$0.451916\pi$$
$$710$$ 3.17962 0.119329
$$711$$ 0 0
$$712$$ −5.17714 −0.194022
$$713$$ 42.7374 1.60053
$$714$$ 0 0
$$715$$ −7.12373 −0.266412
$$716$$ 17.9054 0.669156
$$717$$ 0 0
$$718$$ −0.855391 −0.0319229
$$719$$ −26.9826 −1.00628 −0.503140 0.864205i $$-0.667822\pi$$
−0.503140 + 0.864205i $$0.667822\pi$$
$$720$$ 0 0
$$721$$ −28.5719 −1.06407
$$722$$ −3.22179 −0.119903
$$723$$ 0 0
$$724$$ 14.0482 0.522097
$$725$$ −58.7084 −2.18038
$$726$$ 0 0
$$727$$ −14.6943 −0.544983 −0.272491 0.962158i $$-0.587848\pi$$
−0.272491 + 0.962158i $$0.587848\pi$$
$$728$$ 1.59288 0.0590360
$$729$$ 0 0
$$730$$ −2.77889 −0.102851
$$731$$ 25.7516 0.952456
$$732$$ 0 0
$$733$$ −31.3438 −1.15771 −0.578854 0.815431i $$-0.696500\pi$$
−0.578854 + 0.815431i $$0.696500\pi$$
$$734$$ −0.431134 −0.0159135
$$735$$ 0 0
$$736$$ 13.2736 0.489271
$$737$$ 3.02078 0.111272
$$738$$ 0 0
$$739$$ 0.482909 0.0177641 0.00888205 0.999961i $$-0.497173\pi$$
0.00888205 + 0.999961i $$0.497173\pi$$
$$740$$ 36.5099 1.34213
$$741$$ 0 0
$$742$$ 4.63711 0.170234
$$743$$ −43.0507 −1.57938 −0.789689 0.613507i $$-0.789758\pi$$
−0.789689 + 0.613507i $$0.789758\pi$$
$$744$$ 0 0
$$745$$ −35.9535 −1.31723
$$746$$ −4.85156 −0.177628
$$747$$ 0 0
$$748$$ 22.6925 0.829720
$$749$$ −3.87500 −0.141589
$$750$$ 0 0
$$751$$ 43.9216 1.60272 0.801361 0.598181i $$-0.204109\pi$$
0.801361 + 0.598181i $$0.204109\pi$$
$$752$$ 4.23261 0.154347
$$753$$ 0 0
$$754$$ 0.867014 0.0315748
$$755$$ 26.6350 0.969346
$$756$$ 0 0
$$757$$ 22.4143 0.814661 0.407331 0.913281i $$-0.366460\pi$$
0.407331 + 0.913281i $$0.366460\pi$$
$$758$$ 0.889081 0.0322929
$$759$$ 0 0
$$760$$ −1.56999 −0.0569496
$$761$$ 9.99674 0.362382 0.181191 0.983448i $$-0.442005\pi$$
0.181191 + 0.983448i $$0.442005\pi$$
$$762$$ 0 0
$$763$$ −22.4540 −0.812888
$$764$$ −23.5535 −0.852137
$$765$$ 0 0
$$766$$ 0.00772733 0.000279200 0
$$767$$ 9.06824 0.327435
$$768$$ 0 0
$$769$$ −7.49619 −0.270320 −0.135160 0.990824i $$-0.543155\pi$$
−0.135160 + 0.990824i $$0.543155\pi$$
$$770$$ 4.88214 0.175940
$$771$$ 0 0
$$772$$ 17.4818 0.629183
$$773$$ 19.8391 0.713562 0.356781 0.934188i $$-0.383874\pi$$
0.356781 + 0.934188i $$0.383874\pi$$
$$774$$ 0 0
$$775$$ 58.7043 2.10872
$$776$$ 0.651262 0.0233789
$$777$$ 0 0
$$778$$ −3.62943 −0.130121
$$779$$ −3.21845 −0.115313
$$780$$ 0 0
$$781$$ 12.2578 0.438619
$$782$$ 5.21709 0.186563
$$783$$ 0 0
$$784$$ 8.36872 0.298883
$$785$$ −28.7184 −1.02500
$$786$$ 0 0
$$787$$ 39.7283 1.41616 0.708080 0.706133i $$-0.249562\pi$$
0.708080 + 0.706133i $$0.249562\pi$$
$$788$$ −14.5745 −0.519193
$$789$$ 0 0
$$790$$ 7.62327 0.271224
$$791$$ 28.3505 1.00803
$$792$$ 0 0
$$793$$ −6.26419 −0.222448
$$794$$ −0.000424900 0 −1.50791e−5 0
$$795$$ 0 0
$$796$$ 20.4568 0.725072
$$797$$ 9.10595 0.322549 0.161275 0.986910i $$-0.448439\pi$$
0.161275 + 0.986910i $$0.448439\pi$$
$$798$$ 0 0
$$799$$ 5.12027 0.181142
$$800$$ 18.2326 0.644621
$$801$$ 0 0
$$802$$ −4.36874 −0.154265
$$803$$ −10.7129 −0.378051
$$804$$ 0 0
$$805$$ −73.9041 −2.60478
$$806$$ −0.866953 −0.0305371
$$807$$ 0 0
$$808$$ 3.85128 0.135487
$$809$$ −3.01910 −0.106146 −0.0530730 0.998591i $$-0.516902\pi$$
−0.0530730 + 0.998591i $$0.516902\pi$$
$$810$$ 0 0
$$811$$ 33.1722 1.16483 0.582416 0.812891i $$-0.302107\pi$$
0.582416 + 0.812891i $$0.302107\pi$$
$$812$$ 39.1238 1.37298
$$813$$ 0 0
$$814$$ −2.13764 −0.0749243
$$815$$ −3.91405 −0.137103
$$816$$ 0 0
$$817$$ −3.40832 −0.119242
$$818$$ −4.02836 −0.140848
$$819$$ 0 0
$$820$$ 38.7223 1.35224
$$821$$ −42.7620 −1.49240 −0.746201 0.665720i $$-0.768124\pi$$
−0.746201 + 0.665720i $$0.768124\pi$$
$$822$$ 0 0
$$823$$ −35.2289 −1.22800 −0.614000 0.789306i $$-0.710441\pi$$
−0.614000 + 0.789306i $$0.710441\pi$$
$$824$$ 6.47243 0.225478
$$825$$ 0 0
$$826$$ −6.21478 −0.216240
$$827$$ −26.0380 −0.905429 −0.452714 0.891656i $$-0.649544\pi$$
−0.452714 + 0.891656i $$0.649544\pi$$
$$828$$ 0 0
$$829$$ −7.90268 −0.274471 −0.137236 0.990538i $$-0.543822\pi$$
−0.137236 + 0.990538i $$0.543822\pi$$
$$830$$ −5.82741 −0.202272
$$831$$ 0 0
$$832$$ 5.57850 0.193400
$$833$$ 10.1238 0.350769
$$834$$ 0 0
$$835$$ −31.2155 −1.08026
$$836$$ −3.00344 −0.103876
$$837$$ 0 0
$$838$$ −5.41213 −0.186959
$$839$$ −5.14839 −0.177742 −0.0888711 0.996043i $$-0.528326\pi$$
−0.0888711 + 0.996043i $$0.528326\pi$$
$$840$$ 0 0
$$841$$ 13.9142 0.479800
$$842$$ 5.28088 0.181991
$$843$$ 0 0
$$844$$ −41.0981 −1.41466
$$845$$ 46.3878 1.59579
$$846$$ 0 0
$$847$$ −14.5253 −0.499094
$$848$$ 33.7928 1.16045
$$849$$ 0 0
$$850$$ 7.16621 0.245799
$$851$$ 32.3589 1.10925
$$852$$ 0 0
$$853$$ 25.9905 0.889896 0.444948 0.895556i $$-0.353222\pi$$
0.444948 + 0.895556i $$0.353222\pi$$
$$854$$ 4.29307 0.146906
$$855$$ 0 0
$$856$$ 0.877808 0.0300028
$$857$$ 4.11913 0.140707 0.0703535 0.997522i $$-0.477587\pi$$
0.0703535 + 0.997522i $$0.477587\pi$$
$$858$$ 0 0
$$859$$ −6.42943 −0.219369 −0.109685 0.993966i $$-0.534984\pi$$
−0.109685 + 0.993966i $$0.534984\pi$$
$$860$$ 41.0068 1.39832
$$861$$ 0 0
$$862$$ −2.14916 −0.0732008
$$863$$ −29.6195 −1.00826 −0.504129 0.863628i $$-0.668187\pi$$
−0.504129 + 0.863628i $$0.668187\pi$$
$$864$$ 0 0
$$865$$ −81.6282 −2.77544
$$866$$ −0.131574 −0.00447107
$$867$$ 0 0
$$868$$ −39.1210 −1.32785
$$869$$ 29.3886 0.996939
$$870$$ 0 0
$$871$$ 0.927607 0.0314308
$$872$$ 5.08652 0.172251
$$873$$ 0 0
$$874$$ −0.690503 −0.0233566
$$875$$ −44.8779 −1.51715
$$876$$ 0 0
$$877$$ −31.8677 −1.07610 −0.538048 0.842914i $$-0.680838\pi$$
−0.538048 + 0.842914i $$0.680838\pi$$
$$878$$ 5.20819 0.175768
$$879$$ 0 0
$$880$$ 35.5784 1.19935
$$881$$ −34.7864 −1.17198 −0.585991 0.810317i $$-0.699295\pi$$
−0.585991 + 0.810317i $$0.699295\pi$$
$$882$$ 0 0
$$883$$ −30.3764 −1.02225 −0.511124 0.859507i $$-0.670771\pi$$
−0.511124 + 0.859507i $$0.670771\pi$$
$$884$$ 6.96831 0.234370
$$885$$ 0 0
$$886$$ 2.36312 0.0793907
$$887$$ 50.0276 1.67976 0.839882 0.542769i $$-0.182624\pi$$
0.839882 + 0.542769i $$0.182624\pi$$
$$888$$ 0 0
$$889$$ 63.0455 2.11448
$$890$$ −4.87263 −0.163331
$$891$$ 0 0
$$892$$ −46.4515 −1.55531
$$893$$ −0.677688 −0.0226780
$$894$$ 0 0
$$895$$ 33.9604 1.13517
$$896$$ −16.1581 −0.539803
$$897$$ 0 0
$$898$$ −3.80529 −0.126984
$$899$$ −42.9111 −1.43117
$$900$$ 0 0
$$901$$ 40.8797 1.36190
$$902$$ −2.26718 −0.0754889
$$903$$ 0 0
$$904$$ −6.42226 −0.213601
$$905$$ 26.6446 0.885696
$$906$$ 0 0
$$907$$ −44.0643 −1.46313 −0.731566 0.681771i $$-0.761210\pi$$
−0.731566 + 0.681771i $$0.761210\pi$$
$$908$$ 20.6546 0.685446
$$909$$ 0 0
$$910$$ 1.49919 0.0496975
$$911$$ 37.1783 1.23177 0.615885 0.787836i $$-0.288798\pi$$
0.615885 + 0.787836i $$0.288798\pi$$
$$912$$ 0 0
$$913$$ −22.4653 −0.743493
$$914$$ −0.257531 −0.00851838
$$915$$ 0 0
$$916$$ 27.3495 0.903652
$$917$$ −1.98815 −0.0656544
$$918$$ 0 0
$$919$$ 12.3976 0.408958 0.204479 0.978871i $$-0.434450\pi$$
0.204479 + 0.978871i $$0.434450\pi$$
$$920$$ 16.7416 0.551953
$$921$$ 0 0
$$922$$ −1.23044 −0.0405223
$$923$$ 3.76407 0.123896
$$924$$ 0 0
$$925$$ 44.4482 1.46145
$$926$$ −4.59091 −0.150867
$$927$$ 0 0
$$928$$ −13.3275 −0.437498
$$929$$ −33.3882 −1.09543 −0.547716 0.836664i $$-0.684502\pi$$
−0.547716 + 0.836664i $$0.684502\pi$$
$$930$$ 0 0
$$931$$ −1.33993 −0.0439143
$$932$$ −14.9719 −0.490421
$$933$$ 0 0
$$934$$ 4.52366 0.148019
$$935$$ 43.0399 1.40755
$$936$$ 0 0
$$937$$ −24.8441 −0.811620 −0.405810 0.913957i $$-0.633011\pi$$
−0.405810 + 0.913957i $$0.633011\pi$$
$$938$$ −0.635721 −0.0207570
$$939$$ 0 0
$$940$$ 8.15352 0.265938
$$941$$ −25.4299 −0.828990 −0.414495 0.910051i $$-0.636042\pi$$
−0.414495 + 0.910051i $$0.636042\pi$$
$$942$$ 0 0
$$943$$ 34.3198 1.11761
$$944$$ −45.2900 −1.47406
$$945$$ 0 0
$$946$$ −2.40094 −0.0780612
$$947$$ −16.6301 −0.540407 −0.270204 0.962803i $$-0.587091\pi$$
−0.270204 + 0.962803i $$0.587091\pi$$
$$948$$ 0 0
$$949$$ −3.28968 −0.106787
$$950$$ −0.948476 −0.0307726
$$951$$ 0 0
$$952$$ −9.62378 −0.311908
$$953$$ −14.2671 −0.462158 −0.231079 0.972935i $$-0.574226\pi$$
−0.231079 + 0.972935i $$0.574226\pi$$
$$954$$ 0 0
$$955$$ −44.6730 −1.44558
$$956$$ −32.6219 −1.05507
$$957$$ 0 0
$$958$$ −1.80008 −0.0581580
$$959$$ 26.0333 0.840659
$$960$$ 0 0
$$961$$ 11.9081 0.384131
$$962$$ −0.656418 −0.0211638
$$963$$ 0 0
$$964$$ 28.6026 0.921227
$$965$$ 33.1569 1.06736
$$966$$ 0 0
$$967$$ −39.0848 −1.25688 −0.628440 0.777858i $$-0.716306\pi$$
−0.628440 + 0.777858i $$0.716306\pi$$
$$968$$ 3.29042 0.105758
$$969$$ 0 0
$$970$$ 0.612955 0.0196808
$$971$$ 4.40370 0.141321 0.0706607 0.997500i $$-0.477489\pi$$
0.0706607 + 0.997500i $$0.477489\pi$$
$$972$$ 0 0
$$973$$ 40.7619 1.30677
$$974$$ −3.19424 −0.102350
$$975$$ 0 0
$$976$$ 31.2856 1.00143
$$977$$ −42.1266 −1.34775 −0.673874 0.738846i $$-0.735371\pi$$
−0.673874 + 0.738846i $$0.735371\pi$$
$$978$$ 0 0
$$979$$ −18.7845 −0.600356
$$980$$ 16.1211 0.514971
$$981$$ 0 0
$$982$$ −2.93608 −0.0936939
$$983$$ 20.4261 0.651493 0.325746 0.945457i $$-0.394384\pi$$
0.325746 + 0.945457i $$0.394384\pi$$
$$984$$ 0 0
$$985$$ −27.6427 −0.880770
$$986$$ −5.23829 −0.166821
$$987$$ 0 0
$$988$$ −0.922284 −0.0293418
$$989$$ 36.3445 1.15569
$$990$$ 0 0
$$991$$ 0.0680712 0.00216235 0.00108118 0.999999i $$-0.499656\pi$$
0.00108118 + 0.999999i $$0.499656\pi$$
$$992$$ 13.3266 0.423120
$$993$$ 0 0
$$994$$ −2.57965 −0.0818215
$$995$$ 38.7995 1.23003
$$996$$ 0 0
$$997$$ 10.6461 0.337166 0.168583 0.985688i $$-0.446081\pi$$
0.168583 + 0.985688i $$0.446081\pi$$
$$998$$ 4.26319 0.134949
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 729.2.a.b.1.4 6
3.2 odd 2 729.2.a.e.1.3 yes 6
9.2 odd 6 729.2.c.a.244.4 12
9.4 even 3 729.2.c.d.487.3 12
9.5 odd 6 729.2.c.a.487.4 12
9.7 even 3 729.2.c.d.244.3 12
27.2 odd 18 729.2.e.k.568.2 12
27.4 even 9 729.2.e.j.406.2 12
27.5 odd 18 729.2.e.l.649.1 12
27.7 even 9 729.2.e.j.325.2 12
27.11 odd 18 729.2.e.l.82.1 12
27.13 even 9 729.2.e.t.163.1 12
27.14 odd 18 729.2.e.k.163.2 12
27.16 even 9 729.2.e.s.82.2 12
27.20 odd 18 729.2.e.u.325.1 12
27.22 even 9 729.2.e.s.649.2 12
27.23 odd 18 729.2.e.u.406.1 12
27.25 even 9 729.2.e.t.568.1 12

By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.b.1.4 6 1.1 even 1 trivial
729.2.a.e.1.3 yes 6 3.2 odd 2
729.2.c.a.244.4 12 9.2 odd 6
729.2.c.a.487.4 12 9.5 odd 6
729.2.c.d.244.3 12 9.7 even 3
729.2.c.d.487.3 12 9.4 even 3
729.2.e.j.325.2 12 27.7 even 9
729.2.e.j.406.2 12 27.4 even 9
729.2.e.k.163.2 12 27.14 odd 18
729.2.e.k.568.2 12 27.2 odd 18
729.2.e.l.82.1 12 27.11 odd 18
729.2.e.l.649.1 12 27.5 odd 18
729.2.e.s.82.2 12 27.16 even 9
729.2.e.s.649.2 12 27.22 even 9
729.2.e.t.163.1 12 27.13 even 9
729.2.e.t.568.1 12 27.25 even 9
729.2.e.u.325.1 12 27.20 odd 18
729.2.e.u.406.1 12 27.23 odd 18