# Properties

 Label 285.2.a.e.1.2 Level $285$ Weight $2$ Character 285.1 Self dual yes Analytic conductor $2.276$ Analytic rank $0$ 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] = [285,2,Mod(1,285)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(285, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("285.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$285 = 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 285.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: $$2.27573645761$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 285.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.73205 q^{6} +0.732051 q^{7} -1.73205 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.73205 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.73205 q^{6} +0.732051 q^{7} -1.73205 q^{8} +1.00000 q^{9} +1.73205 q^{10} +1.26795 q^{11} +1.00000 q^{12} -2.73205 q^{13} +1.26795 q^{14} +1.00000 q^{15} -5.00000 q^{16} +1.73205 q^{18} +1.00000 q^{19} +1.00000 q^{20} +0.732051 q^{21} +2.19615 q^{22} -3.46410 q^{23} -1.73205 q^{24} +1.00000 q^{25} -4.73205 q^{26} +1.00000 q^{27} +0.732051 q^{28} -2.19615 q^{29} +1.73205 q^{30} -4.92820 q^{31} -5.19615 q^{32} +1.26795 q^{33} +0.732051 q^{35} +1.00000 q^{36} +4.19615 q^{37} +1.73205 q^{38} -2.73205 q^{39} -1.73205 q^{40} +4.73205 q^{41} +1.26795 q^{42} -6.19615 q^{43} +1.26795 q^{44} +1.00000 q^{45} -6.00000 q^{46} +3.46410 q^{47} -5.00000 q^{48} -6.46410 q^{49} +1.73205 q^{50} -2.73205 q^{52} -2.53590 q^{53} +1.73205 q^{54} +1.26795 q^{55} -1.26795 q^{56} +1.00000 q^{57} -3.80385 q^{58} +9.46410 q^{59} +1.00000 q^{60} -13.4641 q^{61} -8.53590 q^{62} +0.732051 q^{63} +1.00000 q^{64} -2.73205 q^{65} +2.19615 q^{66} +8.00000 q^{67} -3.46410 q^{69} +1.26795 q^{70} +16.3923 q^{71} -1.73205 q^{72} -3.07180 q^{73} +7.26795 q^{74} +1.00000 q^{75} +1.00000 q^{76} +0.928203 q^{77} -4.73205 q^{78} +2.92820 q^{79} -5.00000 q^{80} +1.00000 q^{81} +8.19615 q^{82} +0.928203 q^{83} +0.732051 q^{84} -10.7321 q^{86} -2.19615 q^{87} -2.19615 q^{88} +7.26795 q^{89} +1.73205 q^{90} -2.00000 q^{91} -3.46410 q^{92} -4.92820 q^{93} +6.00000 q^{94} +1.00000 q^{95} -5.19615 q^{96} +4.19615 q^{97} -11.1962 q^{98} +1.26795 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 6 q^{11} + 2 q^{12} - 2 q^{13} + 6 q^{14} + 2 q^{15} - 10 q^{16} + 2 q^{19} + 2 q^{20} - 2 q^{21} - 6 q^{22} + 2 q^{25} - 6 q^{26} + 2 q^{27} - 2 q^{28} + 6 q^{29} + 4 q^{31} + 6 q^{33} - 2 q^{35} + 2 q^{36} - 2 q^{37} - 2 q^{39} + 6 q^{41} + 6 q^{42} - 2 q^{43} + 6 q^{44} + 2 q^{45} - 12 q^{46} - 10 q^{48} - 6 q^{49} - 2 q^{52} - 12 q^{53} + 6 q^{55} - 6 q^{56} + 2 q^{57} - 18 q^{58} + 12 q^{59} + 2 q^{60} - 20 q^{61} - 24 q^{62} - 2 q^{63} + 2 q^{64} - 2 q^{65} - 6 q^{66} + 16 q^{67} + 6 q^{70} + 12 q^{71} - 20 q^{73} + 18 q^{74} + 2 q^{75} + 2 q^{76} - 12 q^{77} - 6 q^{78} - 8 q^{79} - 10 q^{80} + 2 q^{81} + 6 q^{82} - 12 q^{83} - 2 q^{84} - 18 q^{86} + 6 q^{87} + 6 q^{88} + 18 q^{89} - 4 q^{91} + 4 q^{93} + 12 q^{94} + 2 q^{95} - 2 q^{97} - 12 q^{98} + 6 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 6 * q^11 + 2 * q^12 - 2 * q^13 + 6 * q^14 + 2 * q^15 - 10 * q^16 + 2 * q^19 + 2 * q^20 - 2 * q^21 - 6 * q^22 + 2 * q^25 - 6 * q^26 + 2 * q^27 - 2 * q^28 + 6 * q^29 + 4 * q^31 + 6 * q^33 - 2 * q^35 + 2 * q^36 - 2 * q^37 - 2 * q^39 + 6 * q^41 + 6 * q^42 - 2 * q^43 + 6 * q^44 + 2 * q^45 - 12 * q^46 - 10 * q^48 - 6 * q^49 - 2 * q^52 - 12 * q^53 + 6 * q^55 - 6 * q^56 + 2 * q^57 - 18 * q^58 + 12 * q^59 + 2 * q^60 - 20 * q^61 - 24 * q^62 - 2 * q^63 + 2 * q^64 - 2 * q^65 - 6 * q^66 + 16 * q^67 + 6 * q^70 + 12 * q^71 - 20 * q^73 + 18 * q^74 + 2 * q^75 + 2 * q^76 - 12 * q^77 - 6 * q^78 - 8 * q^79 - 10 * q^80 + 2 * q^81 + 6 * q^82 - 12 * q^83 - 2 * q^84 - 18 * q^86 + 6 * q^87 + 6 * q^88 + 18 * q^89 - 4 * q^91 + 4 * q^93 + 12 * q^94 + 2 * q^95 - 2 * q^97 - 12 * q^98 + 6 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.73205 1.22474 0.612372 0.790569i $$-0.290215\pi$$
0.612372 + 0.790569i $$0.290215\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 1.73205 0.707107
$$7$$ 0.732051 0.276689 0.138345 0.990384i $$-0.455822\pi$$
0.138345 + 0.990384i $$0.455822\pi$$
$$8$$ −1.73205 −0.612372
$$9$$ 1.00000 0.333333
$$10$$ 1.73205 0.547723
$$11$$ 1.26795 0.382301 0.191151 0.981561i $$-0.438778\pi$$
0.191151 + 0.981561i $$0.438778\pi$$
$$12$$ 1.00000 0.288675
$$13$$ −2.73205 −0.757735 −0.378867 0.925451i $$-0.623686\pi$$
−0.378867 + 0.925451i $$0.623686\pi$$
$$14$$ 1.26795 0.338874
$$15$$ 1.00000 0.258199
$$16$$ −5.00000 −1.25000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 1.73205 0.408248
$$19$$ 1.00000 0.229416
$$20$$ 1.00000 0.223607
$$21$$ 0.732051 0.159747
$$22$$ 2.19615 0.468221
$$23$$ −3.46410 −0.722315 −0.361158 0.932505i $$-0.617618\pi$$
−0.361158 + 0.932505i $$0.617618\pi$$
$$24$$ −1.73205 −0.353553
$$25$$ 1.00000 0.200000
$$26$$ −4.73205 −0.928032
$$27$$ 1.00000 0.192450
$$28$$ 0.732051 0.138345
$$29$$ −2.19615 −0.407815 −0.203908 0.978990i $$-0.565364\pi$$
−0.203908 + 0.978990i $$0.565364\pi$$
$$30$$ 1.73205 0.316228
$$31$$ −4.92820 −0.885131 −0.442566 0.896736i $$-0.645932\pi$$
−0.442566 + 0.896736i $$0.645932\pi$$
$$32$$ −5.19615 −0.918559
$$33$$ 1.26795 0.220722
$$34$$ 0 0
$$35$$ 0.732051 0.123739
$$36$$ 1.00000 0.166667
$$37$$ 4.19615 0.689843 0.344922 0.938631i $$-0.387905\pi$$
0.344922 + 0.938631i $$0.387905\pi$$
$$38$$ 1.73205 0.280976
$$39$$ −2.73205 −0.437478
$$40$$ −1.73205 −0.273861
$$41$$ 4.73205 0.739022 0.369511 0.929226i $$-0.379525\pi$$
0.369511 + 0.929226i $$0.379525\pi$$
$$42$$ 1.26795 0.195649
$$43$$ −6.19615 −0.944904 −0.472452 0.881356i $$-0.656631\pi$$
−0.472452 + 0.881356i $$0.656631\pi$$
$$44$$ 1.26795 0.191151
$$45$$ 1.00000 0.149071
$$46$$ −6.00000 −0.884652
$$47$$ 3.46410 0.505291 0.252646 0.967559i $$-0.418699\pi$$
0.252646 + 0.967559i $$0.418699\pi$$
$$48$$ −5.00000 −0.721688
$$49$$ −6.46410 −0.923443
$$50$$ 1.73205 0.244949
$$51$$ 0 0
$$52$$ −2.73205 −0.378867
$$53$$ −2.53590 −0.348332 −0.174166 0.984716i $$-0.555723\pi$$
−0.174166 + 0.984716i $$0.555723\pi$$
$$54$$ 1.73205 0.235702
$$55$$ 1.26795 0.170970
$$56$$ −1.26795 −0.169437
$$57$$ 1.00000 0.132453
$$58$$ −3.80385 −0.499470
$$59$$ 9.46410 1.23212 0.616061 0.787699i $$-0.288728\pi$$
0.616061 + 0.787699i $$0.288728\pi$$
$$60$$ 1.00000 0.129099
$$61$$ −13.4641 −1.72390 −0.861951 0.506992i $$-0.830757\pi$$
−0.861951 + 0.506992i $$0.830757\pi$$
$$62$$ −8.53590 −1.08406
$$63$$ 0.732051 0.0922297
$$64$$ 1.00000 0.125000
$$65$$ −2.73205 −0.338869
$$66$$ 2.19615 0.270328
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 0 0
$$69$$ −3.46410 −0.417029
$$70$$ 1.26795 0.151549
$$71$$ 16.3923 1.94541 0.972704 0.232048i $$-0.0745426\pi$$
0.972704 + 0.232048i $$0.0745426\pi$$
$$72$$ −1.73205 −0.204124
$$73$$ −3.07180 −0.359527 −0.179763 0.983710i $$-0.557533\pi$$
−0.179763 + 0.983710i $$0.557533\pi$$
$$74$$ 7.26795 0.844882
$$75$$ 1.00000 0.115470
$$76$$ 1.00000 0.114708
$$77$$ 0.928203 0.105779
$$78$$ −4.73205 −0.535799
$$79$$ 2.92820 0.329449 0.164724 0.986340i $$-0.447327\pi$$
0.164724 + 0.986340i $$0.447327\pi$$
$$80$$ −5.00000 −0.559017
$$81$$ 1.00000 0.111111
$$82$$ 8.19615 0.905114
$$83$$ 0.928203 0.101884 0.0509418 0.998702i $$-0.483778\pi$$
0.0509418 + 0.998702i $$0.483778\pi$$
$$84$$ 0.732051 0.0798733
$$85$$ 0 0
$$86$$ −10.7321 −1.15727
$$87$$ −2.19615 −0.235452
$$88$$ −2.19615 −0.234111
$$89$$ 7.26795 0.770401 0.385201 0.922833i $$-0.374132\pi$$
0.385201 + 0.922833i $$0.374132\pi$$
$$90$$ 1.73205 0.182574
$$91$$ −2.00000 −0.209657
$$92$$ −3.46410 −0.361158
$$93$$ −4.92820 −0.511031
$$94$$ 6.00000 0.618853
$$95$$ 1.00000 0.102598
$$96$$ −5.19615 −0.530330
$$97$$ 4.19615 0.426055 0.213027 0.977046i $$-0.431668\pi$$
0.213027 + 0.977046i $$0.431668\pi$$
$$98$$ −11.1962 −1.13098
$$99$$ 1.26795 0.127434
$$100$$ 1.00000 0.100000
$$101$$ 10.3923 1.03407 0.517036 0.855963i $$-0.327035\pi$$
0.517036 + 0.855963i $$0.327035\pi$$
$$102$$ 0 0
$$103$$ −17.8564 −1.75944 −0.879722 0.475488i $$-0.842271\pi$$
−0.879722 + 0.475488i $$0.842271\pi$$
$$104$$ 4.73205 0.464016
$$105$$ 0.732051 0.0714408
$$106$$ −4.39230 −0.426618
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 6.39230 0.612272 0.306136 0.951988i $$-0.400964\pi$$
0.306136 + 0.951988i $$0.400964\pi$$
$$110$$ 2.19615 0.209395
$$111$$ 4.19615 0.398281
$$112$$ −3.66025 −0.345861
$$113$$ 5.07180 0.477115 0.238557 0.971128i $$-0.423326\pi$$
0.238557 + 0.971128i $$0.423326\pi$$
$$114$$ 1.73205 0.162221
$$115$$ −3.46410 −0.323029
$$116$$ −2.19615 −0.203908
$$117$$ −2.73205 −0.252578
$$118$$ 16.3923 1.50903
$$119$$ 0 0
$$120$$ −1.73205 −0.158114
$$121$$ −9.39230 −0.853846
$$122$$ −23.3205 −2.11134
$$123$$ 4.73205 0.426675
$$124$$ −4.92820 −0.442566
$$125$$ 1.00000 0.0894427
$$126$$ 1.26795 0.112958
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ 12.1244 1.07165
$$129$$ −6.19615 −0.545541
$$130$$ −4.73205 −0.415028
$$131$$ 15.1244 1.32142 0.660711 0.750641i $$-0.270255\pi$$
0.660711 + 0.750641i $$0.270255\pi$$
$$132$$ 1.26795 0.110361
$$133$$ 0.732051 0.0634769
$$134$$ 13.8564 1.19701
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ −7.85641 −0.671218 −0.335609 0.942001i $$-0.608942\pi$$
−0.335609 + 0.942001i $$0.608942\pi$$
$$138$$ −6.00000 −0.510754
$$139$$ 12.3923 1.05110 0.525551 0.850762i $$-0.323859\pi$$
0.525551 + 0.850762i $$0.323859\pi$$
$$140$$ 0.732051 0.0618696
$$141$$ 3.46410 0.291730
$$142$$ 28.3923 2.38263
$$143$$ −3.46410 −0.289683
$$144$$ −5.00000 −0.416667
$$145$$ −2.19615 −0.182381
$$146$$ −5.32051 −0.440328
$$147$$ −6.46410 −0.533150
$$148$$ 4.19615 0.344922
$$149$$ 7.85641 0.643622 0.321811 0.946804i $$-0.395708\pi$$
0.321811 + 0.946804i $$0.395708\pi$$
$$150$$ 1.73205 0.141421
$$151$$ 14.0000 1.13930 0.569652 0.821886i $$-0.307078\pi$$
0.569652 + 0.821886i $$0.307078\pi$$
$$152$$ −1.73205 −0.140488
$$153$$ 0 0
$$154$$ 1.60770 0.129552
$$155$$ −4.92820 −0.395843
$$156$$ −2.73205 −0.218739
$$157$$ −14.3923 −1.14863 −0.574315 0.818634i $$-0.694732\pi$$
−0.574315 + 0.818634i $$0.694732\pi$$
$$158$$ 5.07180 0.403490
$$159$$ −2.53590 −0.201110
$$160$$ −5.19615 −0.410792
$$161$$ −2.53590 −0.199857
$$162$$ 1.73205 0.136083
$$163$$ 12.7321 0.997251 0.498626 0.866817i $$-0.333838\pi$$
0.498626 + 0.866817i $$0.333838\pi$$
$$164$$ 4.73205 0.369511
$$165$$ 1.26795 0.0987097
$$166$$ 1.60770 0.124781
$$167$$ −3.46410 −0.268060 −0.134030 0.990977i $$-0.542792\pi$$
−0.134030 + 0.990977i $$0.542792\pi$$
$$168$$ −1.26795 −0.0978244
$$169$$ −5.53590 −0.425838
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ −6.19615 −0.472452
$$173$$ −6.92820 −0.526742 −0.263371 0.964695i $$-0.584834\pi$$
−0.263371 + 0.964695i $$0.584834\pi$$
$$174$$ −3.80385 −0.288369
$$175$$ 0.732051 0.0553378
$$176$$ −6.33975 −0.477876
$$177$$ 9.46410 0.711365
$$178$$ 12.5885 0.943545
$$179$$ −11.3205 −0.846135 −0.423067 0.906098i $$-0.639047\pi$$
−0.423067 + 0.906098i $$0.639047\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ 18.3923 1.36709 0.683545 0.729909i $$-0.260437\pi$$
0.683545 + 0.729909i $$0.260437\pi$$
$$182$$ −3.46410 −0.256776
$$183$$ −13.4641 −0.995295
$$184$$ 6.00000 0.442326
$$185$$ 4.19615 0.308507
$$186$$ −8.53590 −0.625882
$$187$$ 0 0
$$188$$ 3.46410 0.252646
$$189$$ 0.732051 0.0532489
$$190$$ 1.73205 0.125656
$$191$$ 17.6603 1.27785 0.638926 0.769269i $$-0.279379\pi$$
0.638926 + 0.769269i $$0.279379\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −7.12436 −0.512822 −0.256411 0.966568i $$-0.582540\pi$$
−0.256411 + 0.966568i $$0.582540\pi$$
$$194$$ 7.26795 0.521808
$$195$$ −2.73205 −0.195646
$$196$$ −6.46410 −0.461722
$$197$$ −24.0000 −1.70993 −0.854965 0.518686i $$-0.826421\pi$$
−0.854965 + 0.518686i $$0.826421\pi$$
$$198$$ 2.19615 0.156074
$$199$$ 19.3205 1.36959 0.684797 0.728734i $$-0.259891\pi$$
0.684797 + 0.728734i $$0.259891\pi$$
$$200$$ −1.73205 −0.122474
$$201$$ 8.00000 0.564276
$$202$$ 18.0000 1.26648
$$203$$ −1.60770 −0.112838
$$204$$ 0 0
$$205$$ 4.73205 0.330501
$$206$$ −30.9282 −2.15487
$$207$$ −3.46410 −0.240772
$$208$$ 13.6603 0.947168
$$209$$ 1.26795 0.0877059
$$210$$ 1.26795 0.0874968
$$211$$ 14.9282 1.02770 0.513850 0.857880i $$-0.328219\pi$$
0.513850 + 0.857880i $$0.328219\pi$$
$$212$$ −2.53590 −0.174166
$$213$$ 16.3923 1.12318
$$214$$ 0 0
$$215$$ −6.19615 −0.422574
$$216$$ −1.73205 −0.117851
$$217$$ −3.60770 −0.244906
$$218$$ 11.0718 0.749877
$$219$$ −3.07180 −0.207573
$$220$$ 1.26795 0.0854851
$$221$$ 0 0
$$222$$ 7.26795 0.487793
$$223$$ 9.85641 0.660034 0.330017 0.943975i $$-0.392946\pi$$
0.330017 + 0.943975i $$0.392946\pi$$
$$224$$ −3.80385 −0.254155
$$225$$ 1.00000 0.0666667
$$226$$ 8.78461 0.584344
$$227$$ −10.3923 −0.689761 −0.344881 0.938647i $$-0.612081\pi$$
−0.344881 + 0.938647i $$0.612081\pi$$
$$228$$ 1.00000 0.0662266
$$229$$ −25.4641 −1.68272 −0.841358 0.540479i $$-0.818243\pi$$
−0.841358 + 0.540479i $$0.818243\pi$$
$$230$$ −6.00000 −0.395628
$$231$$ 0.928203 0.0610713
$$232$$ 3.80385 0.249735
$$233$$ 19.8564 1.30084 0.650418 0.759576i $$-0.274594\pi$$
0.650418 + 0.759576i $$0.274594\pi$$
$$234$$ −4.73205 −0.309344
$$235$$ 3.46410 0.225973
$$236$$ 9.46410 0.616061
$$237$$ 2.92820 0.190207
$$238$$ 0 0
$$239$$ −20.1962 −1.30638 −0.653190 0.757194i $$-0.726570\pi$$
−0.653190 + 0.757194i $$0.726570\pi$$
$$240$$ −5.00000 −0.322749
$$241$$ −16.9282 −1.09044 −0.545221 0.838293i $$-0.683554\pi$$
−0.545221 + 0.838293i $$0.683554\pi$$
$$242$$ −16.2679 −1.04574
$$243$$ 1.00000 0.0641500
$$244$$ −13.4641 −0.861951
$$245$$ −6.46410 −0.412976
$$246$$ 8.19615 0.522568
$$247$$ −2.73205 −0.173836
$$248$$ 8.53590 0.542030
$$249$$ 0.928203 0.0588225
$$250$$ 1.73205 0.109545
$$251$$ −10.0526 −0.634512 −0.317256 0.948340i $$-0.602761\pi$$
−0.317256 + 0.948340i $$0.602761\pi$$
$$252$$ 0.732051 0.0461149
$$253$$ −4.39230 −0.276142
$$254$$ −6.92820 −0.434714
$$255$$ 0 0
$$256$$ 19.0000 1.18750
$$257$$ −24.0000 −1.49708 −0.748539 0.663090i $$-0.769245\pi$$
−0.748539 + 0.663090i $$0.769245\pi$$
$$258$$ −10.7321 −0.668148
$$259$$ 3.07180 0.190872
$$260$$ −2.73205 −0.169435
$$261$$ −2.19615 −0.135938
$$262$$ 26.1962 1.61840
$$263$$ 6.00000 0.369976 0.184988 0.982741i $$-0.440775\pi$$
0.184988 + 0.982741i $$0.440775\pi$$
$$264$$ −2.19615 −0.135164
$$265$$ −2.53590 −0.155779
$$266$$ 1.26795 0.0777430
$$267$$ 7.26795 0.444791
$$268$$ 8.00000 0.488678
$$269$$ −30.5885 −1.86501 −0.932506 0.361156i $$-0.882382\pi$$
−0.932506 + 0.361156i $$0.882382\pi$$
$$270$$ 1.73205 0.105409
$$271$$ −20.3923 −1.23874 −0.619372 0.785098i $$-0.712613\pi$$
−0.619372 + 0.785098i $$0.712613\pi$$
$$272$$ 0 0
$$273$$ −2.00000 −0.121046
$$274$$ −13.6077 −0.822071
$$275$$ 1.26795 0.0764602
$$276$$ −3.46410 −0.208514
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ 21.4641 1.28733
$$279$$ −4.92820 −0.295044
$$280$$ −1.26795 −0.0757745
$$281$$ 4.73205 0.282290 0.141145 0.989989i $$-0.454922\pi$$
0.141145 + 0.989989i $$0.454922\pi$$
$$282$$ 6.00000 0.357295
$$283$$ −26.9808 −1.60384 −0.801920 0.597432i $$-0.796188\pi$$
−0.801920 + 0.597432i $$0.796188\pi$$
$$284$$ 16.3923 0.972704
$$285$$ 1.00000 0.0592349
$$286$$ −6.00000 −0.354787
$$287$$ 3.46410 0.204479
$$288$$ −5.19615 −0.306186
$$289$$ −17.0000 −1.00000
$$290$$ −3.80385 −0.223370
$$291$$ 4.19615 0.245983
$$292$$ −3.07180 −0.179763
$$293$$ −27.7128 −1.61900 −0.809500 0.587120i $$-0.800262\pi$$
−0.809500 + 0.587120i $$0.800262\pi$$
$$294$$ −11.1962 −0.652973
$$295$$ 9.46410 0.551021
$$296$$ −7.26795 −0.422441
$$297$$ 1.26795 0.0735739
$$298$$ 13.6077 0.788273
$$299$$ 9.46410 0.547323
$$300$$ 1.00000 0.0577350
$$301$$ −4.53590 −0.261445
$$302$$ 24.2487 1.39536
$$303$$ 10.3923 0.597022
$$304$$ −5.00000 −0.286770
$$305$$ −13.4641 −0.770952
$$306$$ 0 0
$$307$$ −11.6077 −0.662486 −0.331243 0.943545i $$-0.607468\pi$$
−0.331243 + 0.943545i $$0.607468\pi$$
$$308$$ 0.928203 0.0528893
$$309$$ −17.8564 −1.01582
$$310$$ −8.53590 −0.484806
$$311$$ −26.4449 −1.49955 −0.749775 0.661692i $$-0.769838\pi$$
−0.749775 + 0.661692i $$0.769838\pi$$
$$312$$ 4.73205 0.267900
$$313$$ −14.3923 −0.813501 −0.406751 0.913539i $$-0.633338\pi$$
−0.406751 + 0.913539i $$0.633338\pi$$
$$314$$ −24.9282 −1.40678
$$315$$ 0.732051 0.0412464
$$316$$ 2.92820 0.164724
$$317$$ 23.3205 1.30981 0.654905 0.755711i $$-0.272709\pi$$
0.654905 + 0.755711i $$0.272709\pi$$
$$318$$ −4.39230 −0.246308
$$319$$ −2.78461 −0.155908
$$320$$ 1.00000 0.0559017
$$321$$ 0 0
$$322$$ −4.39230 −0.244774
$$323$$ 0 0
$$324$$ 1.00000 0.0555556
$$325$$ −2.73205 −0.151547
$$326$$ 22.0526 1.22138
$$327$$ 6.39230 0.353495
$$328$$ −8.19615 −0.452557
$$329$$ 2.53590 0.139809
$$330$$ 2.19615 0.120894
$$331$$ 29.7128 1.63316 0.816582 0.577230i $$-0.195866\pi$$
0.816582 + 0.577230i $$0.195866\pi$$
$$332$$ 0.928203 0.0509418
$$333$$ 4.19615 0.229948
$$334$$ −6.00000 −0.328305
$$335$$ 8.00000 0.437087
$$336$$ −3.66025 −0.199683
$$337$$ −19.1244 −1.04177 −0.520885 0.853627i $$-0.674398\pi$$
−0.520885 + 0.853627i $$0.674398\pi$$
$$338$$ −9.58846 −0.521543
$$339$$ 5.07180 0.275462
$$340$$ 0 0
$$341$$ −6.24871 −0.338387
$$342$$ 1.73205 0.0936586
$$343$$ −9.85641 −0.532196
$$344$$ 10.7321 0.578633
$$345$$ −3.46410 −0.186501
$$346$$ −12.0000 −0.645124
$$347$$ 12.9282 0.694022 0.347011 0.937861i $$-0.387197\pi$$
0.347011 + 0.937861i $$0.387197\pi$$
$$348$$ −2.19615 −0.117726
$$349$$ −22.0000 −1.17763 −0.588817 0.808267i $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ 1.26795 0.0677747
$$351$$ −2.73205 −0.145826
$$352$$ −6.58846 −0.351166
$$353$$ 26.7846 1.42560 0.712800 0.701367i $$-0.247427\pi$$
0.712800 + 0.701367i $$0.247427\pi$$
$$354$$ 16.3923 0.871241
$$355$$ 16.3923 0.870013
$$356$$ 7.26795 0.385201
$$357$$ 0 0
$$358$$ −19.6077 −1.03630
$$359$$ 17.6603 0.932073 0.466036 0.884766i $$-0.345682\pi$$
0.466036 + 0.884766i $$0.345682\pi$$
$$360$$ −1.73205 −0.0912871
$$361$$ 1.00000 0.0526316
$$362$$ 31.8564 1.67434
$$363$$ −9.39230 −0.492968
$$364$$ −2.00000 −0.104828
$$365$$ −3.07180 −0.160785
$$366$$ −23.3205 −1.21898
$$367$$ 5.80385 0.302958 0.151479 0.988460i $$-0.451596\pi$$
0.151479 + 0.988460i $$0.451596\pi$$
$$368$$ 17.3205 0.902894
$$369$$ 4.73205 0.246341
$$370$$ 7.26795 0.377843
$$371$$ −1.85641 −0.0963798
$$372$$ −4.92820 −0.255515
$$373$$ 4.19615 0.217269 0.108634 0.994082i $$-0.465352\pi$$
0.108634 + 0.994082i $$0.465352\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ −6.00000 −0.309426
$$377$$ 6.00000 0.309016
$$378$$ 1.26795 0.0652163
$$379$$ 7.07180 0.363254 0.181627 0.983368i $$-0.441864\pi$$
0.181627 + 0.983368i $$0.441864\pi$$
$$380$$ 1.00000 0.0512989
$$381$$ −4.00000 −0.204926
$$382$$ 30.5885 1.56504
$$383$$ −30.9282 −1.58036 −0.790179 0.612877i $$-0.790012\pi$$
−0.790179 + 0.612877i $$0.790012\pi$$
$$384$$ 12.1244 0.618718
$$385$$ 0.928203 0.0473056
$$386$$ −12.3397 −0.628077
$$387$$ −6.19615 −0.314968
$$388$$ 4.19615 0.213027
$$389$$ −19.8564 −1.00676 −0.503380 0.864065i $$-0.667910\pi$$
−0.503380 + 0.864065i $$0.667910\pi$$
$$390$$ −4.73205 −0.239617
$$391$$ 0 0
$$392$$ 11.1962 0.565491
$$393$$ 15.1244 0.762923
$$394$$ −41.5692 −2.09423
$$395$$ 2.92820 0.147334
$$396$$ 1.26795 0.0637168
$$397$$ −4.92820 −0.247339 −0.123670 0.992323i $$-0.539466\pi$$
−0.123670 + 0.992323i $$0.539466\pi$$
$$398$$ 33.4641 1.67740
$$399$$ 0.732051 0.0366484
$$400$$ −5.00000 −0.250000
$$401$$ 4.05256 0.202375 0.101188 0.994867i $$-0.467736\pi$$
0.101188 + 0.994867i $$0.467736\pi$$
$$402$$ 13.8564 0.691095
$$403$$ 13.4641 0.670695
$$404$$ 10.3923 0.517036
$$405$$ 1.00000 0.0496904
$$406$$ −2.78461 −0.138198
$$407$$ 5.32051 0.263728
$$408$$ 0 0
$$409$$ −5.60770 −0.277283 −0.138641 0.990343i $$-0.544274\pi$$
−0.138641 + 0.990343i $$0.544274\pi$$
$$410$$ 8.19615 0.404779
$$411$$ −7.85641 −0.387528
$$412$$ −17.8564 −0.879722
$$413$$ 6.92820 0.340915
$$414$$ −6.00000 −0.294884
$$415$$ 0.928203 0.0455637
$$416$$ 14.1962 0.696024
$$417$$ 12.3923 0.606854
$$418$$ 2.19615 0.107417
$$419$$ 10.0526 0.491100 0.245550 0.969384i $$-0.421032\pi$$
0.245550 + 0.969384i $$0.421032\pi$$
$$420$$ 0.732051 0.0357204
$$421$$ 22.7846 1.11045 0.555227 0.831699i $$-0.312631\pi$$
0.555227 + 0.831699i $$0.312631\pi$$
$$422$$ 25.8564 1.25867
$$423$$ 3.46410 0.168430
$$424$$ 4.39230 0.213309
$$425$$ 0 0
$$426$$ 28.3923 1.37561
$$427$$ −9.85641 −0.476985
$$428$$ 0 0
$$429$$ −3.46410 −0.167248
$$430$$ −10.7321 −0.517545
$$431$$ 23.3205 1.12331 0.561655 0.827372i $$-0.310165\pi$$
0.561655 + 0.827372i $$0.310165\pi$$
$$432$$ −5.00000 −0.240563
$$433$$ 20.5885 0.989418 0.494709 0.869059i $$-0.335275\pi$$
0.494709 + 0.869059i $$0.335275\pi$$
$$434$$ −6.24871 −0.299948
$$435$$ −2.19615 −0.105297
$$436$$ 6.39230 0.306136
$$437$$ −3.46410 −0.165710
$$438$$ −5.32051 −0.254224
$$439$$ 13.0718 0.623883 0.311941 0.950101i $$-0.399021\pi$$
0.311941 + 0.950101i $$0.399021\pi$$
$$440$$ −2.19615 −0.104697
$$441$$ −6.46410 −0.307814
$$442$$ 0 0
$$443$$ 29.3205 1.39306 0.696530 0.717528i $$-0.254726\pi$$
0.696530 + 0.717528i $$0.254726\pi$$
$$444$$ 4.19615 0.199141
$$445$$ 7.26795 0.344534
$$446$$ 17.0718 0.808373
$$447$$ 7.85641 0.371595
$$448$$ 0.732051 0.0345861
$$449$$ 11.6603 0.550281 0.275141 0.961404i $$-0.411276\pi$$
0.275141 + 0.961404i $$0.411276\pi$$
$$450$$ 1.73205 0.0816497
$$451$$ 6.00000 0.282529
$$452$$ 5.07180 0.238557
$$453$$ 14.0000 0.657777
$$454$$ −18.0000 −0.844782
$$455$$ −2.00000 −0.0937614
$$456$$ −1.73205 −0.0811107
$$457$$ 11.4641 0.536268 0.268134 0.963382i $$-0.413593\pi$$
0.268134 + 0.963382i $$0.413593\pi$$
$$458$$ −44.1051 −2.06090
$$459$$ 0 0
$$460$$ −3.46410 −0.161515
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 1.60770 0.0747967
$$463$$ 9.51666 0.442277 0.221138 0.975242i $$-0.429023\pi$$
0.221138 + 0.975242i $$0.429023\pi$$
$$464$$ 10.9808 0.509769
$$465$$ −4.92820 −0.228540
$$466$$ 34.3923 1.59319
$$467$$ 27.4641 1.27089 0.635444 0.772147i $$-0.280817\pi$$
0.635444 + 0.772147i $$0.280817\pi$$
$$468$$ −2.73205 −0.126289
$$469$$ 5.85641 0.270424
$$470$$ 6.00000 0.276759
$$471$$ −14.3923 −0.663162
$$472$$ −16.3923 −0.754517
$$473$$ −7.85641 −0.361238
$$474$$ 5.07180 0.232955
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ −2.53590 −0.116111
$$478$$ −34.9808 −1.59998
$$479$$ −19.5167 −0.891739 −0.445869 0.895098i $$-0.647105\pi$$
−0.445869 + 0.895098i $$0.647105\pi$$
$$480$$ −5.19615 −0.237171
$$481$$ −11.4641 −0.522718
$$482$$ −29.3205 −1.33551
$$483$$ −2.53590 −0.115387
$$484$$ −9.39230 −0.426923
$$485$$ 4.19615 0.190537
$$486$$ 1.73205 0.0785674
$$487$$ −11.6077 −0.525995 −0.262997 0.964797i $$-0.584711\pi$$
−0.262997 + 0.964797i $$0.584711\pi$$
$$488$$ 23.3205 1.05567
$$489$$ 12.7321 0.575763
$$490$$ −11.1962 −0.505791
$$491$$ −22.0526 −0.995218 −0.497609 0.867401i $$-0.665789\pi$$
−0.497609 + 0.867401i $$0.665789\pi$$
$$492$$ 4.73205 0.213337
$$493$$ 0 0
$$494$$ −4.73205 −0.212905
$$495$$ 1.26795 0.0569901
$$496$$ 24.6410 1.10641
$$497$$ 12.0000 0.538274
$$498$$ 1.60770 0.0720425
$$499$$ 17.4641 0.781801 0.390900 0.920433i $$-0.372164\pi$$
0.390900 + 0.920433i $$0.372164\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ −3.46410 −0.154765
$$502$$ −17.4115 −0.777115
$$503$$ −36.9282 −1.64655 −0.823274 0.567645i $$-0.807855\pi$$
−0.823274 + 0.567645i $$0.807855\pi$$
$$504$$ −1.26795 −0.0564789
$$505$$ 10.3923 0.462451
$$506$$ −7.60770 −0.338203
$$507$$ −5.53590 −0.245858
$$508$$ −4.00000 −0.177471
$$509$$ 28.0526 1.24341 0.621704 0.783252i $$-0.286441\pi$$
0.621704 + 0.783252i $$0.286441\pi$$
$$510$$ 0 0
$$511$$ −2.24871 −0.0994771
$$512$$ 8.66025 0.382733
$$513$$ 1.00000 0.0441511
$$514$$ −41.5692 −1.83354
$$515$$ −17.8564 −0.786847
$$516$$ −6.19615 −0.272770
$$517$$ 4.39230 0.193173
$$518$$ 5.32051 0.233770
$$519$$ −6.92820 −0.304114
$$520$$ 4.73205 0.207514
$$521$$ 40.7321 1.78450 0.892252 0.451538i $$-0.149125\pi$$
0.892252 + 0.451538i $$0.149125\pi$$
$$522$$ −3.80385 −0.166490
$$523$$ 43.3205 1.89427 0.947137 0.320830i $$-0.103962\pi$$
0.947137 + 0.320830i $$0.103962\pi$$
$$524$$ 15.1244 0.660711
$$525$$ 0.732051 0.0319493
$$526$$ 10.3923 0.453126
$$527$$ 0 0
$$528$$ −6.33975 −0.275902
$$529$$ −11.0000 −0.478261
$$530$$ −4.39230 −0.190790
$$531$$ 9.46410 0.410707
$$532$$ 0.732051 0.0317384
$$533$$ −12.9282 −0.559983
$$534$$ 12.5885 0.544756
$$535$$ 0 0
$$536$$ −13.8564 −0.598506
$$537$$ −11.3205 −0.488516
$$538$$ −52.9808 −2.28416
$$539$$ −8.19615 −0.353033
$$540$$ 1.00000 0.0430331
$$541$$ −13.7128 −0.589560 −0.294780 0.955565i $$-0.595246\pi$$
−0.294780 + 0.955565i $$0.595246\pi$$
$$542$$ −35.3205 −1.51715
$$543$$ 18.3923 0.789289
$$544$$ 0 0
$$545$$ 6.39230 0.273816
$$546$$ −3.46410 −0.148250
$$547$$ 8.67949 0.371108 0.185554 0.982634i $$-0.440592\pi$$
0.185554 + 0.982634i $$0.440592\pi$$
$$548$$ −7.85641 −0.335609
$$549$$ −13.4641 −0.574634
$$550$$ 2.19615 0.0936443
$$551$$ −2.19615 −0.0935592
$$552$$ 6.00000 0.255377
$$553$$ 2.14359 0.0911549
$$554$$ 3.46410 0.147176
$$555$$ 4.19615 0.178117
$$556$$ 12.3923 0.525551
$$557$$ −12.9282 −0.547786 −0.273893 0.961760i $$-0.588311\pi$$
−0.273893 + 0.961760i $$0.588311\pi$$
$$558$$ −8.53590 −0.361353
$$559$$ 16.9282 0.715987
$$560$$ −3.66025 −0.154674
$$561$$ 0 0
$$562$$ 8.19615 0.345734
$$563$$ −20.5359 −0.865485 −0.432742 0.901518i $$-0.642454\pi$$
−0.432742 + 0.901518i $$0.642454\pi$$
$$564$$ 3.46410 0.145865
$$565$$ 5.07180 0.213372
$$566$$ −46.7321 −1.96429
$$567$$ 0.732051 0.0307432
$$568$$ −28.3923 −1.19131
$$569$$ −16.0526 −0.672958 −0.336479 0.941691i $$-0.609236\pi$$
−0.336479 + 0.941691i $$0.609236\pi$$
$$570$$ 1.73205 0.0725476
$$571$$ 14.2487 0.596290 0.298145 0.954521i $$-0.403632\pi$$
0.298145 + 0.954521i $$0.403632\pi$$
$$572$$ −3.46410 −0.144841
$$573$$ 17.6603 0.737768
$$574$$ 6.00000 0.250435
$$575$$ −3.46410 −0.144463
$$576$$ 1.00000 0.0416667
$$577$$ −47.1769 −1.96400 −0.982000 0.188879i $$-0.939515\pi$$
−0.982000 + 0.188879i $$0.939515\pi$$
$$578$$ −29.4449 −1.22474
$$579$$ −7.12436 −0.296078
$$580$$ −2.19615 −0.0911903
$$581$$ 0.679492 0.0281901
$$582$$ 7.26795 0.301266
$$583$$ −3.21539 −0.133168
$$584$$ 5.32051 0.220164
$$585$$ −2.73205 −0.112956
$$586$$ −48.0000 −1.98286
$$587$$ 3.46410 0.142979 0.0714894 0.997441i $$-0.477225\pi$$
0.0714894 + 0.997441i $$0.477225\pi$$
$$588$$ −6.46410 −0.266575
$$589$$ −4.92820 −0.203063
$$590$$ 16.3923 0.674861
$$591$$ −24.0000 −0.987228
$$592$$ −20.9808 −0.862304
$$593$$ 2.78461 0.114350 0.0571751 0.998364i $$-0.481791\pi$$
0.0571751 + 0.998364i $$0.481791\pi$$
$$594$$ 2.19615 0.0901092
$$595$$ 0 0
$$596$$ 7.85641 0.321811
$$597$$ 19.3205 0.790736
$$598$$ 16.3923 0.670331
$$599$$ 13.8564 0.566157 0.283079 0.959097i $$-0.408644\pi$$
0.283079 + 0.959097i $$0.408644\pi$$
$$600$$ −1.73205 −0.0707107
$$601$$ 15.1769 0.619079 0.309540 0.950887i $$-0.399825\pi$$
0.309540 + 0.950887i $$0.399825\pi$$
$$602$$ −7.85641 −0.320203
$$603$$ 8.00000 0.325785
$$604$$ 14.0000 0.569652
$$605$$ −9.39230 −0.381851
$$606$$ 18.0000 0.731200
$$607$$ −32.3923 −1.31476 −0.657382 0.753558i $$-0.728336\pi$$
−0.657382 + 0.753558i $$0.728336\pi$$
$$608$$ −5.19615 −0.210732
$$609$$ −1.60770 −0.0651471
$$610$$ −23.3205 −0.944220
$$611$$ −9.46410 −0.382877
$$612$$ 0 0
$$613$$ 21.6077 0.872727 0.436363 0.899771i $$-0.356266\pi$$
0.436363 + 0.899771i $$0.356266\pi$$
$$614$$ −20.1051 −0.811377
$$615$$ 4.73205 0.190815
$$616$$ −1.60770 −0.0647759
$$617$$ −27.7128 −1.11568 −0.557838 0.829950i $$-0.688369\pi$$
−0.557838 + 0.829950i $$0.688369\pi$$
$$618$$ −30.9282 −1.24411
$$619$$ 19.3205 0.776557 0.388278 0.921542i $$-0.373070\pi$$
0.388278 + 0.921542i $$0.373070\pi$$
$$620$$ −4.92820 −0.197921
$$621$$ −3.46410 −0.139010
$$622$$ −45.8038 −1.83657
$$623$$ 5.32051 0.213162
$$624$$ 13.6603 0.546848
$$625$$ 1.00000 0.0400000
$$626$$ −24.9282 −0.996331
$$627$$ 1.26795 0.0506370
$$628$$ −14.3923 −0.574315
$$629$$ 0 0
$$630$$ 1.26795 0.0505163
$$631$$ −21.0718 −0.838855 −0.419427 0.907789i $$-0.637769\pi$$
−0.419427 + 0.907789i $$0.637769\pi$$
$$632$$ −5.07180 −0.201745
$$633$$ 14.9282 0.593343
$$634$$ 40.3923 1.60418
$$635$$ −4.00000 −0.158735
$$636$$ −2.53590 −0.100555
$$637$$ 17.6603 0.699725
$$638$$ −4.82309 −0.190948
$$639$$ 16.3923 0.648470
$$640$$ 12.1244 0.479257
$$641$$ 17.4115 0.687715 0.343857 0.939022i $$-0.388266\pi$$
0.343857 + 0.939022i $$0.388266\pi$$
$$642$$ 0 0
$$643$$ −1.80385 −0.0711368 −0.0355684 0.999367i $$-0.511324\pi$$
−0.0355684 + 0.999367i $$0.511324\pi$$
$$644$$ −2.53590 −0.0999284
$$645$$ −6.19615 −0.243973
$$646$$ 0 0
$$647$$ −31.8564 −1.25240 −0.626202 0.779661i $$-0.715392\pi$$
−0.626202 + 0.779661i $$0.715392\pi$$
$$648$$ −1.73205 −0.0680414
$$649$$ 12.0000 0.471041
$$650$$ −4.73205 −0.185606
$$651$$ −3.60770 −0.141397
$$652$$ 12.7321 0.498626
$$653$$ 30.9282 1.21031 0.605157 0.796106i $$-0.293110\pi$$
0.605157 + 0.796106i $$0.293110\pi$$
$$654$$ 11.0718 0.432942
$$655$$ 15.1244 0.590957
$$656$$ −23.6603 −0.923778
$$657$$ −3.07180 −0.119842
$$658$$ 4.39230 0.171230
$$659$$ 18.9282 0.737338 0.368669 0.929561i $$-0.379814\pi$$
0.368669 + 0.929561i $$0.379814\pi$$
$$660$$ 1.26795 0.0493549
$$661$$ −23.1769 −0.901477 −0.450739 0.892656i $$-0.648839\pi$$
−0.450739 + 0.892656i $$0.648839\pi$$
$$662$$ 51.4641 2.00021
$$663$$ 0 0
$$664$$ −1.60770 −0.0623907
$$665$$ 0.732051 0.0283877
$$666$$ 7.26795 0.281627
$$667$$ 7.60770 0.294571
$$668$$ −3.46410 −0.134030
$$669$$ 9.85641 0.381071
$$670$$ 13.8564 0.535320
$$671$$ −17.0718 −0.659049
$$672$$ −3.80385 −0.146737
$$673$$ −7.12436 −0.274624 −0.137312 0.990528i $$-0.543846\pi$$
−0.137312 + 0.990528i $$0.543846\pi$$
$$674$$ −33.1244 −1.27590
$$675$$ 1.00000 0.0384900
$$676$$ −5.53590 −0.212919
$$677$$ 35.3205 1.35748 0.678739 0.734380i $$-0.262527\pi$$
0.678739 + 0.734380i $$0.262527\pi$$
$$678$$ 8.78461 0.337371
$$679$$ 3.07180 0.117885
$$680$$ 0 0
$$681$$ −10.3923 −0.398234
$$682$$ −10.8231 −0.414437
$$683$$ −18.9282 −0.724268 −0.362134 0.932126i $$-0.617952\pi$$
−0.362134 + 0.932126i $$0.617952\pi$$
$$684$$ 1.00000 0.0382360
$$685$$ −7.85641 −0.300178
$$686$$ −17.0718 −0.651804
$$687$$ −25.4641 −0.971516
$$688$$ 30.9808 1.18113
$$689$$ 6.92820 0.263944
$$690$$ −6.00000 −0.228416
$$691$$ −8.39230 −0.319258 −0.159629 0.987177i $$-0.551030\pi$$
−0.159629 + 0.987177i $$0.551030\pi$$
$$692$$ −6.92820 −0.263371
$$693$$ 0.928203 0.0352595
$$694$$ 22.3923 0.850000
$$695$$ 12.3923 0.470067
$$696$$ 3.80385 0.144184
$$697$$ 0 0
$$698$$ −38.1051 −1.44230
$$699$$ 19.8564 0.751038
$$700$$ 0.732051 0.0276689
$$701$$ 21.7128 0.820082 0.410041 0.912067i $$-0.365514\pi$$
0.410041 + 0.912067i $$0.365514\pi$$
$$702$$ −4.73205 −0.178600
$$703$$ 4.19615 0.158261
$$704$$ 1.26795 0.0477876
$$705$$ 3.46410 0.130466
$$706$$ 46.3923 1.74600
$$707$$ 7.60770 0.286117
$$708$$ 9.46410 0.355683
$$709$$ 33.1769 1.24599 0.622993 0.782228i $$-0.285917\pi$$
0.622993 + 0.782228i $$0.285917\pi$$
$$710$$ 28.3923 1.06554
$$711$$ 2.92820 0.109816
$$712$$ −12.5885 −0.471772
$$713$$ 17.0718 0.639344
$$714$$ 0 0
$$715$$ −3.46410 −0.129550
$$716$$ −11.3205 −0.423067
$$717$$ −20.1962 −0.754239
$$718$$ 30.5885 1.14155
$$719$$ 5.66025 0.211092 0.105546 0.994414i $$-0.466341\pi$$
0.105546 + 0.994414i $$0.466341\pi$$
$$720$$ −5.00000 −0.186339
$$721$$ −13.0718 −0.486819
$$722$$ 1.73205 0.0644603
$$723$$ −16.9282 −0.629567
$$724$$ 18.3923 0.683545
$$725$$ −2.19615 −0.0815631
$$726$$ −16.2679 −0.603760
$$727$$ 8.33975 0.309304 0.154652 0.987969i $$-0.450574\pi$$
0.154652 + 0.987969i $$0.450574\pi$$
$$728$$ 3.46410 0.128388
$$729$$ 1.00000 0.0370370
$$730$$ −5.32051 −0.196921
$$731$$ 0 0
$$732$$ −13.4641 −0.497648
$$733$$ 22.7846 0.841569 0.420784 0.907161i $$-0.361755\pi$$
0.420784 + 0.907161i $$0.361755\pi$$
$$734$$ 10.0526 0.371047
$$735$$ −6.46410 −0.238432
$$736$$ 18.0000 0.663489
$$737$$ 10.1436 0.373644
$$738$$ 8.19615 0.301705
$$739$$ 33.8564 1.24543 0.622714 0.782450i $$-0.286030\pi$$
0.622714 + 0.782450i $$0.286030\pi$$
$$740$$ 4.19615 0.154254
$$741$$ −2.73205 −0.100364
$$742$$ −3.21539 −0.118041
$$743$$ −44.7846 −1.64299 −0.821494 0.570217i $$-0.806859\pi$$
−0.821494 + 0.570217i $$0.806859\pi$$
$$744$$ 8.53590 0.312941
$$745$$ 7.85641 0.287836
$$746$$ 7.26795 0.266099
$$747$$ 0.928203 0.0339612
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 1.73205 0.0632456
$$751$$ 26.0000 0.948753 0.474377 0.880322i $$-0.342673\pi$$
0.474377 + 0.880322i $$0.342673\pi$$
$$752$$ −17.3205 −0.631614
$$753$$ −10.0526 −0.366336
$$754$$ 10.3923 0.378465
$$755$$ 14.0000 0.509512
$$756$$ 0.732051 0.0266244
$$757$$ −16.2487 −0.590569 −0.295285 0.955409i $$-0.595415\pi$$
−0.295285 + 0.955409i $$0.595415\pi$$
$$758$$ 12.2487 0.444893
$$759$$ −4.39230 −0.159431
$$760$$ −1.73205 −0.0628281
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ −6.92820 −0.250982
$$763$$ 4.67949 0.169409
$$764$$ 17.6603 0.638926
$$765$$ 0 0
$$766$$ −53.5692 −1.93553
$$767$$ −25.8564 −0.933621
$$768$$ 19.0000 0.685603
$$769$$ 48.6410 1.75404 0.877020 0.480454i $$-0.159528\pi$$
0.877020 + 0.480454i $$0.159528\pi$$
$$770$$ 1.60770 0.0579373
$$771$$ −24.0000 −0.864339
$$772$$ −7.12436 −0.256411
$$773$$ 37.1769 1.33716 0.668580 0.743640i $$-0.266902\pi$$
0.668580 + 0.743640i $$0.266902\pi$$
$$774$$ −10.7321 −0.385756
$$775$$ −4.92820 −0.177026
$$776$$ −7.26795 −0.260904
$$777$$ 3.07180 0.110200
$$778$$ −34.3923 −1.23302
$$779$$ 4.73205 0.169543
$$780$$ −2.73205 −0.0978231
$$781$$ 20.7846 0.743732
$$782$$ 0 0
$$783$$ −2.19615 −0.0784841
$$784$$ 32.3205 1.15430
$$785$$ −14.3923 −0.513683
$$786$$ 26.1962 0.934386
$$787$$ 43.3205 1.54421 0.772105 0.635495i $$-0.219204\pi$$
0.772105 + 0.635495i $$0.219204\pi$$
$$788$$ −24.0000 −0.854965
$$789$$ 6.00000 0.213606
$$790$$ 5.07180 0.180446
$$791$$ 3.71281 0.132012
$$792$$ −2.19615 −0.0780369
$$793$$ 36.7846 1.30626
$$794$$ −8.53590 −0.302928
$$795$$ −2.53590 −0.0899390
$$796$$ 19.3205 0.684797
$$797$$ 3.21539 0.113895 0.0569475 0.998377i $$-0.481863\pi$$
0.0569475 + 0.998377i $$0.481863\pi$$
$$798$$ 1.26795 0.0448849
$$799$$ 0 0
$$800$$ −5.19615 −0.183712
$$801$$ 7.26795 0.256800
$$802$$ 7.01924 0.247858
$$803$$ −3.89488 −0.137447
$$804$$ 8.00000 0.282138
$$805$$ −2.53590 −0.0893787
$$806$$ 23.3205 0.821430
$$807$$ −30.5885 −1.07676
$$808$$ −18.0000 −0.633238
$$809$$ −26.7846 −0.941697 −0.470848 0.882214i $$-0.656052\pi$$
−0.470848 + 0.882214i $$0.656052\pi$$
$$810$$ 1.73205 0.0608581
$$811$$ −45.5692 −1.60015 −0.800076 0.599899i $$-0.795207\pi$$
−0.800076 + 0.599899i $$0.795207\pi$$
$$812$$ −1.60770 −0.0564190
$$813$$ −20.3923 −0.715189
$$814$$ 9.21539 0.322999
$$815$$ 12.7321 0.445984
$$816$$ 0 0
$$817$$ −6.19615 −0.216776
$$818$$ −9.71281 −0.339601
$$819$$ −2.00000 −0.0698857
$$820$$ 4.73205 0.165250
$$821$$ −39.4641 −1.37731 −0.688653 0.725091i $$-0.741798\pi$$
−0.688653 + 0.725091i $$0.741798\pi$$
$$822$$ −13.6077 −0.474623
$$823$$ −38.9808 −1.35878 −0.679392 0.733776i $$-0.737756\pi$$
−0.679392 + 0.733776i $$0.737756\pi$$
$$824$$ 30.9282 1.07744
$$825$$ 1.26795 0.0441443
$$826$$ 12.0000 0.417533
$$827$$ 29.3205 1.01957 0.509787 0.860301i $$-0.329724\pi$$
0.509787 + 0.860301i $$0.329724\pi$$
$$828$$ −3.46410 −0.120386
$$829$$ −42.1051 −1.46237 −0.731186 0.682179i $$-0.761033\pi$$
−0.731186 + 0.682179i $$0.761033\pi$$
$$830$$ 1.60770 0.0558039
$$831$$ 2.00000 0.0693792
$$832$$ −2.73205 −0.0947168
$$833$$ 0 0
$$834$$ 21.4641 0.743241
$$835$$ −3.46410 −0.119880
$$836$$ 1.26795 0.0438529
$$837$$ −4.92820 −0.170344
$$838$$ 17.4115 0.601472
$$839$$ −40.3923 −1.39450 −0.697249 0.716829i $$-0.745593\pi$$
−0.697249 + 0.716829i $$0.745593\pi$$
$$840$$ −1.26795 −0.0437484
$$841$$ −24.1769 −0.833687
$$842$$ 39.4641 1.36002
$$843$$ 4.73205 0.162980
$$844$$ 14.9282 0.513850
$$845$$ −5.53590 −0.190441
$$846$$ 6.00000 0.206284
$$847$$ −6.87564 −0.236250
$$848$$ 12.6795 0.435416
$$849$$ −26.9808 −0.925977
$$850$$ 0 0
$$851$$ −14.5359 −0.498284
$$852$$ 16.3923 0.561591
$$853$$ −35.1769 −1.20443 −0.602217 0.798332i $$-0.705716\pi$$
−0.602217 + 0.798332i $$0.705716\pi$$
$$854$$ −17.0718 −0.584185
$$855$$ 1.00000 0.0341993
$$856$$ 0 0
$$857$$ 6.24871 0.213452 0.106726 0.994288i $$-0.465963\pi$$
0.106726 + 0.994288i $$0.465963\pi$$
$$858$$ −6.00000 −0.204837
$$859$$ 32.0000 1.09183 0.545913 0.837842i $$-0.316183\pi$$
0.545913 + 0.837842i $$0.316183\pi$$
$$860$$ −6.19615 −0.211287
$$861$$ 3.46410 0.118056
$$862$$ 40.3923 1.37577
$$863$$ −12.0000 −0.408485 −0.204242 0.978920i $$-0.565473\pi$$
−0.204242 + 0.978920i $$0.565473\pi$$
$$864$$ −5.19615 −0.176777
$$865$$ −6.92820 −0.235566
$$866$$ 35.6603 1.21178
$$867$$ −17.0000 −0.577350
$$868$$ −3.60770 −0.122453
$$869$$ 3.71281 0.125949
$$870$$ −3.80385 −0.128963
$$871$$ −21.8564 −0.740576
$$872$$ −11.0718 −0.374938
$$873$$ 4.19615 0.142018
$$874$$ −6.00000 −0.202953
$$875$$ 0.732051 0.0247478
$$876$$ −3.07180 −0.103786
$$877$$ 28.8756 0.975061 0.487531 0.873106i $$-0.337898\pi$$
0.487531 + 0.873106i $$0.337898\pi$$
$$878$$ 22.6410 0.764097
$$879$$ −27.7128 −0.934730
$$880$$ −6.33975 −0.213713
$$881$$ 15.4641 0.520999 0.260499 0.965474i $$-0.416113\pi$$
0.260499 + 0.965474i $$0.416113\pi$$
$$882$$ −11.1962 −0.376994
$$883$$ −14.9808 −0.504143 −0.252071 0.967709i $$-0.581112\pi$$
−0.252071 + 0.967709i $$0.581112\pi$$
$$884$$ 0 0
$$885$$ 9.46410 0.318132
$$886$$ 50.7846 1.70614
$$887$$ −24.0000 −0.805841 −0.402921 0.915235i $$-0.632005\pi$$
−0.402921 + 0.915235i $$0.632005\pi$$
$$888$$ −7.26795 −0.243896
$$889$$ −2.92820 −0.0982088
$$890$$ 12.5885 0.421966
$$891$$ 1.26795 0.0424779
$$892$$ 9.85641 0.330017
$$893$$ 3.46410 0.115922
$$894$$ 13.6077 0.455109
$$895$$ −11.3205 −0.378403
$$896$$ 8.87564 0.296514
$$897$$ 9.46410 0.315997
$$898$$ 20.1962 0.673954
$$899$$ 10.8231 0.360970
$$900$$ 1.00000 0.0333333
$$901$$ 0 0
$$902$$ 10.3923 0.346026
$$903$$ −4.53590 −0.150945
$$904$$ −8.78461 −0.292172
$$905$$ 18.3923 0.611381
$$906$$ 24.2487 0.805609
$$907$$ −11.6077 −0.385427 −0.192714 0.981255i $$-0.561729\pi$$
−0.192714 + 0.981255i $$0.561729\pi$$
$$908$$ −10.3923 −0.344881
$$909$$ 10.3923 0.344691
$$910$$ −3.46410 −0.114834
$$911$$ −41.0718 −1.36077 −0.680385 0.732855i $$-0.738187\pi$$
−0.680385 + 0.732855i $$0.738187\pi$$
$$912$$ −5.00000 −0.165567
$$913$$ 1.17691 0.0389502
$$914$$ 19.8564 0.656792
$$915$$ −13.4641 −0.445109
$$916$$ −25.4641 −0.841358
$$917$$ 11.0718 0.365623
$$918$$ 0 0
$$919$$ −59.4256 −1.96027 −0.980135 0.198330i $$-0.936448\pi$$
−0.980135 + 0.198330i $$0.936448\pi$$
$$920$$ 6.00000 0.197814
$$921$$ −11.6077 −0.382487
$$922$$ 10.3923 0.342252
$$923$$ −44.7846 −1.47410
$$924$$ 0.928203 0.0305356
$$925$$ 4.19615 0.137969
$$926$$ 16.4833 0.541676
$$927$$ −17.8564 −0.586481
$$928$$ 11.4115 0.374602
$$929$$ −22.3923 −0.734668 −0.367334 0.930089i $$-0.619729\pi$$
−0.367334 + 0.930089i $$0.619729\pi$$
$$930$$ −8.53590 −0.279903
$$931$$ −6.46410 −0.211852
$$932$$ 19.8564 0.650418
$$933$$ −26.4449 −0.865766
$$934$$ 47.5692 1.55651
$$935$$ 0 0
$$936$$ 4.73205 0.154672
$$937$$ 32.2487 1.05352 0.526760 0.850014i $$-0.323407\pi$$
0.526760 + 0.850014i $$0.323407\pi$$
$$938$$ 10.1436 0.331200
$$939$$ −14.3923 −0.469675
$$940$$ 3.46410 0.112987
$$941$$ −30.5885 −0.997155 −0.498578 0.866845i $$-0.666144\pi$$
−0.498578 + 0.866845i $$0.666144\pi$$
$$942$$ −24.9282 −0.812205
$$943$$ −16.3923 −0.533807
$$944$$ −47.3205 −1.54015
$$945$$ 0.732051 0.0238136
$$946$$ −13.6077 −0.442424
$$947$$ 55.8564 1.81509 0.907545 0.419956i $$-0.137954\pi$$
0.907545 + 0.419956i $$0.137954\pi$$
$$948$$ 2.92820 0.0951036
$$949$$ 8.39230 0.272426
$$950$$ 1.73205 0.0561951
$$951$$ 23.3205 0.756219
$$952$$ 0 0
$$953$$ 10.1436 0.328583 0.164292 0.986412i $$-0.447466\pi$$
0.164292 + 0.986412i $$0.447466\pi$$
$$954$$ −4.39230 −0.142206
$$955$$ 17.6603 0.571472
$$956$$ −20.1962 −0.653190
$$957$$ −2.78461 −0.0900136
$$958$$ −33.8038 −1.09215
$$959$$ −5.75129 −0.185719
$$960$$ 1.00000 0.0322749
$$961$$ −6.71281 −0.216542
$$962$$ −19.8564 −0.640196
$$963$$ 0 0
$$964$$ −16.9282 −0.545221
$$965$$ −7.12436 −0.229341
$$966$$ −4.39230 −0.141320
$$967$$ 29.1244 0.936576 0.468288 0.883576i $$-0.344871\pi$$
0.468288 + 0.883576i $$0.344871\pi$$
$$968$$ 16.2679 0.522872
$$969$$ 0 0
$$970$$ 7.26795 0.233360
$$971$$ 27.7128 0.889346 0.444673 0.895693i $$-0.353320\pi$$
0.444673 + 0.895693i $$0.353320\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 9.07180 0.290828
$$974$$ −20.1051 −0.644210
$$975$$ −2.73205 −0.0874957
$$976$$ 67.3205 2.15488
$$977$$ 51.0333 1.63270 0.816350 0.577557i $$-0.195994\pi$$
0.816350 + 0.577557i $$0.195994\pi$$
$$978$$ 22.0526 0.705163
$$979$$ 9.21539 0.294525
$$980$$ −6.46410 −0.206488
$$981$$ 6.39230 0.204091
$$982$$ −38.1962 −1.21889
$$983$$ −6.67949 −0.213043 −0.106521 0.994310i $$-0.533971\pi$$
−0.106521 + 0.994310i $$0.533971\pi$$
$$984$$ −8.19615 −0.261284
$$985$$ −24.0000 −0.764704
$$986$$ 0 0
$$987$$ 2.53590 0.0807185
$$988$$ −2.73205 −0.0869181
$$989$$ 21.4641 0.682519
$$990$$ 2.19615 0.0697983
$$991$$ 26.9282 0.855403 0.427701 0.903920i $$-0.359324\pi$$
0.427701 + 0.903920i $$0.359324\pi$$
$$992$$ 25.6077 0.813045
$$993$$ 29.7128 0.942908
$$994$$ 20.7846 0.659248
$$995$$ 19.3205 0.612501
$$996$$ 0.928203 0.0294112
$$997$$ −38.3923 −1.21590 −0.607948 0.793977i $$-0.708007\pi$$
−0.607948 + 0.793977i $$0.708007\pi$$
$$998$$ 30.2487 0.957506
$$999$$ 4.19615 0.132760
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 285.2.a.e.1.2 2
3.2 odd 2 855.2.a.f.1.1 2
4.3 odd 2 4560.2.a.bh.1.1 2
5.2 odd 4 1425.2.c.k.799.3 4
5.3 odd 4 1425.2.c.k.799.2 4
5.4 even 2 1425.2.a.o.1.1 2
15.14 odd 2 4275.2.a.t.1.2 2
19.18 odd 2 5415.2.a.r.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.e.1.2 2 1.1 even 1 trivial
855.2.a.f.1.1 2 3.2 odd 2
1425.2.a.o.1.1 2 5.4 even 2
1425.2.c.k.799.2 4 5.3 odd 4
1425.2.c.k.799.3 4 5.2 odd 4
4275.2.a.t.1.2 2 15.14 odd 2
4560.2.a.bh.1.1 2 4.3 odd 2
5415.2.a.r.1.1 2 19.18 odd 2