# Properties

 Label 3200.2.a.br.1.3 Level $3200$ Weight $2$ Character 3200.1 Self dual yes Analytic conductor $25.552$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3200,2,Mod(1,3200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3200, 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("3200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3200.a (trivial)

## Newform invariants

comment: select newform

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

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

 Self dual: yes Analytic conductor: $$25.5521286468$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 640) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$2.17009$$ of defining polynomial Character $$\chi$$ $$=$$ 3200.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.70928 q^{3} +2.63090 q^{7} -0.0783777 q^{9} +O(q^{10})$$ $$q+1.70928 q^{3} +2.63090 q^{7} -0.0783777 q^{9} +5.41855 q^{11} +6.34017 q^{13} -3.41855 q^{17} +3.26180 q^{19} +4.49693 q^{21} +1.36910 q^{23} -5.26180 q^{27} -2.00000 q^{29} +4.68035 q^{31} +9.26180 q^{33} -5.75872 q^{37} +10.8371 q^{39} -7.75872 q^{41} -4.44748 q^{43} +4.78765 q^{47} -0.0783777 q^{49} -5.84324 q^{51} +1.65983 q^{53} +5.57531 q^{57} -3.26180 q^{59} +2.49693 q^{61} -0.206204 q^{63} -7.86603 q^{67} +2.34017 q^{69} +6.15676 q^{71} +13.5753 q^{73} +14.2557 q^{77} -12.6803 q^{79} -8.75872 q^{81} -14.9711 q^{83} -3.41855 q^{87} -8.52359 q^{89} +16.6803 q^{91} +8.00000 q^{93} +4.58145 q^{97} -0.424694 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{3} + 4 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^3 + 4 * q^7 + 3 * q^9 $$3 q - 2 q^{3} + 4 q^{7} + 3 q^{9} + 2 q^{11} + 8 q^{13} + 4 q^{17} + 2 q^{19} - 4 q^{21} + 8 q^{23} - 8 q^{27} - 6 q^{29} - 8 q^{31} + 20 q^{33} + 8 q^{37} + 4 q^{39} + 2 q^{41} - 14 q^{43} + 4 q^{47} + 3 q^{49} - 24 q^{51} + 16 q^{53} - 4 q^{57} - 2 q^{59} - 10 q^{61} + 24 q^{63} - 10 q^{67} - 4 q^{69} + 12 q^{71} + 20 q^{73} - 16 q^{79} - q^{81} - 30 q^{83} + 4 q^{87} - 10 q^{89} + 28 q^{91} + 24 q^{93} + 28 q^{97} - 22 q^{99}+O(q^{100})$$ 3 * q - 2 * q^3 + 4 * q^7 + 3 * q^9 + 2 * q^11 + 8 * q^13 + 4 * q^17 + 2 * q^19 - 4 * q^21 + 8 * q^23 - 8 * q^27 - 6 * q^29 - 8 * q^31 + 20 * q^33 + 8 * q^37 + 4 * q^39 + 2 * q^41 - 14 * q^43 + 4 * q^47 + 3 * q^49 - 24 * q^51 + 16 * q^53 - 4 * q^57 - 2 * q^59 - 10 * q^61 + 24 * q^63 - 10 * q^67 - 4 * q^69 + 12 * q^71 + 20 * q^73 - 16 * q^79 - q^81 - 30 * q^83 + 4 * q^87 - 10 * q^89 + 28 * q^91 + 24 * q^93 + 28 * q^97 - 22 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.70928 0.986851 0.493425 0.869788i $$-0.335745\pi$$
0.493425 + 0.869788i $$0.335745\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.63090 0.994386 0.497193 0.867640i $$-0.334364\pi$$
0.497193 + 0.867640i $$0.334364\pi$$
$$8$$ 0 0
$$9$$ −0.0783777 −0.0261259
$$10$$ 0 0
$$11$$ 5.41855 1.63375 0.816877 0.576812i $$-0.195703\pi$$
0.816877 + 0.576812i $$0.195703\pi$$
$$12$$ 0 0
$$13$$ 6.34017 1.75845 0.879224 0.476409i $$-0.158062\pi$$
0.879224 + 0.476409i $$0.158062\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −3.41855 −0.829120 −0.414560 0.910022i $$-0.636065\pi$$
−0.414560 + 0.910022i $$0.636065\pi$$
$$18$$ 0 0
$$19$$ 3.26180 0.748307 0.374154 0.927367i $$-0.377933\pi$$
0.374154 + 0.927367i $$0.377933\pi$$
$$20$$ 0 0
$$21$$ 4.49693 0.981310
$$22$$ 0 0
$$23$$ 1.36910 0.285478 0.142739 0.989760i $$-0.454409\pi$$
0.142739 + 0.989760i $$0.454409\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −5.26180 −1.01263
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 4.68035 0.840615 0.420307 0.907382i $$-0.361922\pi$$
0.420307 + 0.907382i $$0.361922\pi$$
$$32$$ 0 0
$$33$$ 9.26180 1.61227
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −5.75872 −0.946728 −0.473364 0.880867i $$-0.656961\pi$$
−0.473364 + 0.880867i $$0.656961\pi$$
$$38$$ 0 0
$$39$$ 10.8371 1.73533
$$40$$ 0 0
$$41$$ −7.75872 −1.21171 −0.605855 0.795575i $$-0.707169\pi$$
−0.605855 + 0.795575i $$0.707169\pi$$
$$42$$ 0 0
$$43$$ −4.44748 −0.678234 −0.339117 0.940744i $$-0.610128\pi$$
−0.339117 + 0.940744i $$0.610128\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 4.78765 0.698351 0.349175 0.937057i $$-0.386462\pi$$
0.349175 + 0.937057i $$0.386462\pi$$
$$48$$ 0 0
$$49$$ −0.0783777 −0.0111968
$$50$$ 0 0
$$51$$ −5.84324 −0.818218
$$52$$ 0 0
$$53$$ 1.65983 0.227995 0.113997 0.993481i $$-0.463634\pi$$
0.113997 + 0.993481i $$0.463634\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 5.57531 0.738467
$$58$$ 0 0
$$59$$ −3.26180 −0.424650 −0.212325 0.977199i $$-0.568103\pi$$
−0.212325 + 0.977199i $$0.568103\pi$$
$$60$$ 0 0
$$61$$ 2.49693 0.319699 0.159849 0.987141i $$-0.448899\pi$$
0.159849 + 0.987141i $$0.448899\pi$$
$$62$$ 0 0
$$63$$ −0.206204 −0.0259792
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −7.86603 −0.960989 −0.480494 0.876998i $$-0.659543\pi$$
−0.480494 + 0.876998i $$0.659543\pi$$
$$68$$ 0 0
$$69$$ 2.34017 0.281724
$$70$$ 0 0
$$71$$ 6.15676 0.730672 0.365336 0.930876i $$-0.380954\pi$$
0.365336 + 0.930876i $$0.380954\pi$$
$$72$$ 0 0
$$73$$ 13.5753 1.58887 0.794435 0.607350i $$-0.207767\pi$$
0.794435 + 0.607350i $$0.207767\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 14.2557 1.62458
$$78$$ 0 0
$$79$$ −12.6803 −1.42665 −0.713325 0.700833i $$-0.752812\pi$$
−0.713325 + 0.700833i $$0.752812\pi$$
$$80$$ 0 0
$$81$$ −8.75872 −0.973192
$$82$$ 0 0
$$83$$ −14.9711 −1.64329 −0.821644 0.570001i $$-0.806943\pi$$
−0.821644 + 0.570001i $$0.806943\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −3.41855 −0.366507
$$88$$ 0 0
$$89$$ −8.52359 −0.903499 −0.451749 0.892145i $$-0.649200\pi$$
−0.451749 + 0.892145i $$0.649200\pi$$
$$90$$ 0 0
$$91$$ 16.6803 1.74858
$$92$$ 0 0
$$93$$ 8.00000 0.829561
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 4.58145 0.465176 0.232588 0.972575i $$-0.425281\pi$$
0.232588 + 0.972575i $$0.425281\pi$$
$$98$$ 0 0
$$99$$ −0.424694 −0.0426833
$$100$$ 0 0
$$101$$ −2.31351 −0.230203 −0.115101 0.993354i $$-0.536719\pi$$
−0.115101 + 0.993354i $$0.536719\pi$$
$$102$$ 0 0
$$103$$ 16.2062 1.59684 0.798422 0.602098i $$-0.205668\pi$$
0.798422 + 0.602098i $$0.205668\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 15.6514 1.51308 0.756540 0.653948i $$-0.226888\pi$$
0.756540 + 0.653948i $$0.226888\pi$$
$$108$$ 0 0
$$109$$ 12.3402 1.18197 0.590987 0.806681i $$-0.298738\pi$$
0.590987 + 0.806681i $$0.298738\pi$$
$$110$$ 0 0
$$111$$ −9.84324 −0.934279
$$112$$ 0 0
$$113$$ −9.36069 −0.880580 −0.440290 0.897856i $$-0.645124\pi$$
−0.440290 + 0.897856i $$0.645124\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −0.496928 −0.0459411
$$118$$ 0 0
$$119$$ −8.99386 −0.824466
$$120$$ 0 0
$$121$$ 18.3607 1.66915
$$122$$ 0 0
$$123$$ −13.2618 −1.19578
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 1.95055 0.173083 0.0865417 0.996248i $$-0.472418\pi$$
0.0865417 + 0.996248i $$0.472418\pi$$
$$128$$ 0 0
$$129$$ −7.60197 −0.669316
$$130$$ 0 0
$$131$$ −15.5753 −1.36082 −0.680410 0.732831i $$-0.738198\pi$$
−0.680410 + 0.732831i $$0.738198\pi$$
$$132$$ 0 0
$$133$$ 8.58145 0.744106
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 15.2039 1.29896 0.649480 0.760379i $$-0.274987\pi$$
0.649480 + 0.760379i $$0.274987\pi$$
$$138$$ 0 0
$$139$$ 2.58145 0.218956 0.109478 0.993989i $$-0.465082\pi$$
0.109478 + 0.993989i $$0.465082\pi$$
$$140$$ 0 0
$$141$$ 8.18342 0.689168
$$142$$ 0 0
$$143$$ 34.3545 2.87287
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −0.133969 −0.0110496
$$148$$ 0 0
$$149$$ 1.81658 0.148820 0.0744101 0.997228i $$-0.476293\pi$$
0.0744101 + 0.997228i $$0.476293\pi$$
$$150$$ 0 0
$$151$$ 5.16290 0.420151 0.210075 0.977685i $$-0.432629\pi$$
0.210075 + 0.977685i $$0.432629\pi$$
$$152$$ 0 0
$$153$$ 0.267938 0.0216615
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 13.7587 1.09807 0.549033 0.835801i $$-0.314996\pi$$
0.549033 + 0.835801i $$0.314996\pi$$
$$158$$ 0 0
$$159$$ 2.83710 0.224997
$$160$$ 0 0
$$161$$ 3.60197 0.283875
$$162$$ 0 0
$$163$$ −6.29072 −0.492728 −0.246364 0.969177i $$-0.579236\pi$$
−0.246364 + 0.969177i $$0.579236\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −3.89269 −0.301226 −0.150613 0.988593i $$-0.548125\pi$$
−0.150613 + 0.988593i $$0.548125\pi$$
$$168$$ 0 0
$$169$$ 27.1978 2.09214
$$170$$ 0 0
$$171$$ −0.255652 −0.0195502
$$172$$ 0 0
$$173$$ −1.44521 −0.109877 −0.0549387 0.998490i $$-0.517496\pi$$
−0.0549387 + 0.998490i $$0.517496\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −5.57531 −0.419066
$$178$$ 0 0
$$179$$ −11.9421 −0.892598 −0.446299 0.894884i $$-0.647258\pi$$
−0.446299 + 0.894884i $$0.647258\pi$$
$$180$$ 0 0
$$181$$ −15.3607 −1.14175 −0.570876 0.821037i $$-0.693396\pi$$
−0.570876 + 0.821037i $$0.693396\pi$$
$$182$$ 0 0
$$183$$ 4.26794 0.315495
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −18.5236 −1.35458
$$188$$ 0 0
$$189$$ −13.8432 −1.00695
$$190$$ 0 0
$$191$$ −25.3607 −1.83504 −0.917518 0.397695i $$-0.869810\pi$$
−0.917518 + 0.397695i $$0.869810\pi$$
$$192$$ 0 0
$$193$$ 4.58145 0.329780 0.164890 0.986312i $$-0.447273\pi$$
0.164890 + 0.986312i $$0.447273\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 16.8638 1.20149 0.600747 0.799439i $$-0.294870\pi$$
0.600747 + 0.799439i $$0.294870\pi$$
$$198$$ 0 0
$$199$$ 9.84324 0.697769 0.348885 0.937166i $$-0.386561\pi$$
0.348885 + 0.937166i $$0.386561\pi$$
$$200$$ 0 0
$$201$$ −13.4452 −0.948352
$$202$$ 0 0
$$203$$ −5.26180 −0.369306
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −0.107307 −0.00745836
$$208$$ 0 0
$$209$$ 17.6742 1.22255
$$210$$ 0 0
$$211$$ −15.5753 −1.07225 −0.536124 0.844139i $$-0.680112\pi$$
−0.536124 + 0.844139i $$0.680112\pi$$
$$212$$ 0 0
$$213$$ 10.5236 0.721065
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 12.3135 0.835896
$$218$$ 0 0
$$219$$ 23.2039 1.56798
$$220$$ 0 0
$$221$$ −21.6742 −1.45796
$$222$$ 0 0
$$223$$ −16.9854 −1.13743 −0.568715 0.822535i $$-0.692559\pi$$
−0.568715 + 0.822535i $$0.692559\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 19.9649 1.32512 0.662559 0.749009i $$-0.269470\pi$$
0.662559 + 0.749009i $$0.269470\pi$$
$$228$$ 0 0
$$229$$ −23.6742 −1.56444 −0.782218 0.623005i $$-0.785912\pi$$
−0.782218 + 0.623005i $$0.785912\pi$$
$$230$$ 0 0
$$231$$ 24.3668 1.60322
$$232$$ 0 0
$$233$$ 13.5753 0.889348 0.444674 0.895693i $$-0.353320\pi$$
0.444674 + 0.895693i $$0.353320\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −21.6742 −1.40789
$$238$$ 0 0
$$239$$ 8.36683 0.541206 0.270603 0.962691i $$-0.412777\pi$$
0.270603 + 0.962691i $$0.412777\pi$$
$$240$$ 0 0
$$241$$ 9.91548 0.638712 0.319356 0.947635i $$-0.396533\pi$$
0.319356 + 0.947635i $$0.396533\pi$$
$$242$$ 0 0
$$243$$ 0.814315 0.0522383
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 20.6803 1.31586
$$248$$ 0 0
$$249$$ −25.5897 −1.62168
$$250$$ 0 0
$$251$$ −17.6163 −1.11193 −0.555967 0.831204i $$-0.687652\pi$$
−0.555967 + 0.831204i $$0.687652\pi$$
$$252$$ 0 0
$$253$$ 7.41855 0.466400
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −3.68649 −0.229957 −0.114978 0.993368i $$-0.536680\pi$$
−0.114978 + 0.993368i $$0.536680\pi$$
$$258$$ 0 0
$$259$$ −15.1506 −0.941413
$$260$$ 0 0
$$261$$ 0.156755 0.00970292
$$262$$ 0 0
$$263$$ −0.107307 −0.00661684 −0.00330842 0.999995i $$-0.501053\pi$$
−0.00330842 + 0.999995i $$0.501053\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −14.5692 −0.891618
$$268$$ 0 0
$$269$$ −3.85762 −0.235203 −0.117602 0.993061i $$-0.537521\pi$$
−0.117602 + 0.993061i $$0.537521\pi$$
$$270$$ 0 0
$$271$$ −21.3074 −1.29433 −0.647165 0.762350i $$-0.724046\pi$$
−0.647165 + 0.762350i $$0.724046\pi$$
$$272$$ 0 0
$$273$$ 28.5113 1.72558
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −1.44521 −0.0868344 −0.0434172 0.999057i $$-0.513824\pi$$
−0.0434172 + 0.999057i $$0.513824\pi$$
$$278$$ 0 0
$$279$$ −0.366835 −0.0219618
$$280$$ 0 0
$$281$$ 12.4391 0.742053 0.371026 0.928622i $$-0.379006\pi$$
0.371026 + 0.928622i $$0.379006\pi$$
$$282$$ 0 0
$$283$$ −6.29072 −0.373945 −0.186972 0.982365i $$-0.559867\pi$$
−0.186972 + 0.982365i $$0.559867\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −20.4124 −1.20491
$$288$$ 0 0
$$289$$ −5.31351 −0.312559
$$290$$ 0 0
$$291$$ 7.83096 0.459059
$$292$$ 0 0
$$293$$ −1.07838 −0.0629995 −0.0314998 0.999504i $$-0.510028\pi$$
−0.0314998 + 0.999504i $$0.510028\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −28.5113 −1.65439
$$298$$ 0 0
$$299$$ 8.68035 0.501997
$$300$$ 0 0
$$301$$ −11.7009 −0.674427
$$302$$ 0 0
$$303$$ −3.95443 −0.227176
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −25.2267 −1.43977 −0.719883 0.694096i $$-0.755804\pi$$
−0.719883 + 0.694096i $$0.755804\pi$$
$$308$$ 0 0
$$309$$ 27.7009 1.57585
$$310$$ 0 0
$$311$$ 18.8371 1.06815 0.534077 0.845436i $$-0.320659\pi$$
0.534077 + 0.845436i $$0.320659\pi$$
$$312$$ 0 0
$$313$$ 15.2039 0.859377 0.429689 0.902977i $$-0.358623\pi$$
0.429689 + 0.902977i $$0.358623\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −16.8638 −0.947163 −0.473582 0.880750i $$-0.657039\pi$$
−0.473582 + 0.880750i $$0.657039\pi$$
$$318$$ 0 0
$$319$$ −10.8371 −0.606761
$$320$$ 0 0
$$321$$ 26.7526 1.49318
$$322$$ 0 0
$$323$$ −11.1506 −0.620437
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 21.0928 1.16643
$$328$$ 0 0
$$329$$ 12.5958 0.694430
$$330$$ 0 0
$$331$$ 23.5753 1.29582 0.647908 0.761719i $$-0.275644\pi$$
0.647908 + 0.761719i $$0.275644\pi$$
$$332$$ 0 0
$$333$$ 0.451356 0.0247341
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 14.8371 0.808228 0.404114 0.914709i $$-0.367580\pi$$
0.404114 + 0.914709i $$0.367580\pi$$
$$338$$ 0 0
$$339$$ −16.0000 −0.869001
$$340$$ 0 0
$$341$$ 25.3607 1.37336
$$342$$ 0 0
$$343$$ −18.6225 −1.00552
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0.133969 0.00719184 0.00359592 0.999994i $$-0.498855\pi$$
0.00359592 + 0.999994i $$0.498855\pi$$
$$348$$ 0 0
$$349$$ −22.3135 −1.19441 −0.597207 0.802087i $$-0.703723\pi$$
−0.597207 + 0.802087i $$0.703723\pi$$
$$350$$ 0 0
$$351$$ −33.3607 −1.78066
$$352$$ 0 0
$$353$$ −22.8371 −1.21550 −0.607748 0.794130i $$-0.707927\pi$$
−0.607748 + 0.794130i $$0.707927\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −15.3730 −0.813624
$$358$$ 0 0
$$359$$ −31.8843 −1.68279 −0.841394 0.540422i $$-0.818265\pi$$
−0.841394 + 0.540422i $$0.818265\pi$$
$$360$$ 0 0
$$361$$ −8.36069 −0.440036
$$362$$ 0 0
$$363$$ 31.3835 1.64721
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 30.4619 1.59010 0.795048 0.606547i $$-0.207446\pi$$
0.795048 + 0.606547i $$0.207446\pi$$
$$368$$ 0 0
$$369$$ 0.608111 0.0316570
$$370$$ 0 0
$$371$$ 4.36683 0.226715
$$372$$ 0 0
$$373$$ 10.0722 0.521521 0.260760 0.965404i $$-0.416027\pi$$
0.260760 + 0.965404i $$0.416027\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −12.6803 −0.653071
$$378$$ 0 0
$$379$$ −14.7792 −0.759159 −0.379579 0.925159i $$-0.623931\pi$$
−0.379579 + 0.925159i $$0.623931\pi$$
$$380$$ 0 0
$$381$$ 3.33403 0.170808
$$382$$ 0 0
$$383$$ −14.4619 −0.738966 −0.369483 0.929237i $$-0.620465\pi$$
−0.369483 + 0.929237i $$0.620465\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0.348583 0.0177195
$$388$$ 0 0
$$389$$ 3.17727 0.161094 0.0805471 0.996751i $$-0.474333\pi$$
0.0805471 + 0.996751i $$0.474333\pi$$
$$390$$ 0 0
$$391$$ −4.68035 −0.236695
$$392$$ 0 0
$$393$$ −26.6225 −1.34293
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 10.8227 0.543177 0.271589 0.962413i $$-0.412451\pi$$
0.271589 + 0.962413i $$0.412451\pi$$
$$398$$ 0 0
$$399$$ 14.6681 0.734321
$$400$$ 0 0
$$401$$ −12.5236 −0.625398 −0.312699 0.949852i $$-0.601233\pi$$
−0.312699 + 0.949852i $$0.601233\pi$$
$$402$$ 0 0
$$403$$ 29.6742 1.47818
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −31.2039 −1.54672
$$408$$ 0 0
$$409$$ −13.2885 −0.657072 −0.328536 0.944491i $$-0.606555\pi$$
−0.328536 + 0.944491i $$0.606555\pi$$
$$410$$ 0 0
$$411$$ 25.9877 1.28188
$$412$$ 0 0
$$413$$ −8.58145 −0.422266
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 4.41241 0.216077
$$418$$ 0 0
$$419$$ −0.255652 −0.0124894 −0.00624471 0.999981i $$-0.501988\pi$$
−0.00624471 + 0.999981i $$0.501988\pi$$
$$420$$ 0 0
$$421$$ 2.49693 0.121693 0.0608464 0.998147i $$-0.480620\pi$$
0.0608464 + 0.998147i $$0.480620\pi$$
$$422$$ 0 0
$$423$$ −0.375245 −0.0182451
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 6.56916 0.317904
$$428$$ 0 0
$$429$$ 58.7214 2.83510
$$430$$ 0 0
$$431$$ 0.993857 0.0478724 0.0239362 0.999713i $$-0.492380\pi$$
0.0239362 + 0.999713i $$0.492380\pi$$
$$432$$ 0 0
$$433$$ 17.6286 0.847178 0.423589 0.905855i $$-0.360770\pi$$
0.423589 + 0.905855i $$0.360770\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 4.46573 0.213625
$$438$$ 0 0
$$439$$ 40.5113 1.93350 0.966750 0.255725i $$-0.0823142\pi$$
0.966750 + 0.255725i $$0.0823142\pi$$
$$440$$ 0 0
$$441$$ 0.00614307 0.000292527 0
$$442$$ 0 0
$$443$$ 17.7093 0.841393 0.420697 0.907201i $$-0.361786\pi$$
0.420697 + 0.907201i $$0.361786\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 3.10504 0.146863
$$448$$ 0 0
$$449$$ −1.28846 −0.0608061 −0.0304030 0.999538i $$-0.509679\pi$$
−0.0304030 + 0.999538i $$0.509679\pi$$
$$450$$ 0 0
$$451$$ −42.0410 −1.97964
$$452$$ 0 0
$$453$$ 8.82482 0.414626
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −26.3545 −1.23281 −0.616407 0.787428i $$-0.711412\pi$$
−0.616407 + 0.787428i $$0.711412\pi$$
$$458$$ 0 0
$$459$$ 17.9877 0.839595
$$460$$ 0 0
$$461$$ 41.0349 1.91119 0.955593 0.294690i $$-0.0952165\pi$$
0.955593 + 0.294690i $$0.0952165\pi$$
$$462$$ 0 0
$$463$$ 28.7708 1.33709 0.668547 0.743670i $$-0.266917\pi$$
0.668547 + 0.743670i $$0.266917\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 5.12783 0.237287 0.118644 0.992937i $$-0.462145\pi$$
0.118644 + 0.992937i $$0.462145\pi$$
$$468$$ 0 0
$$469$$ −20.6947 −0.955593
$$470$$ 0 0
$$471$$ 23.5174 1.08363
$$472$$ 0 0
$$473$$ −24.0989 −1.10807
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −0.130094 −0.00595657
$$478$$ 0 0
$$479$$ −13.6742 −0.624790 −0.312395 0.949952i $$-0.601131\pi$$
−0.312395 + 0.949952i $$0.601131\pi$$
$$480$$ 0 0
$$481$$ −36.5113 −1.66477
$$482$$ 0 0
$$483$$ 6.15676 0.280142
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −8.51971 −0.386065 −0.193033 0.981192i $$-0.561832\pi$$
−0.193033 + 0.981192i $$0.561832\pi$$
$$488$$ 0 0
$$489$$ −10.7526 −0.486249
$$490$$ 0 0
$$491$$ −4.73820 −0.213832 −0.106916 0.994268i $$-0.534098\pi$$
−0.106916 + 0.994268i $$0.534098\pi$$
$$492$$ 0 0
$$493$$ 6.83710 0.307928
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 16.1978 0.726570
$$498$$ 0 0
$$499$$ 11.0928 0.496580 0.248290 0.968686i $$-0.420131\pi$$
0.248290 + 0.968686i $$0.420131\pi$$
$$500$$ 0 0
$$501$$ −6.65368 −0.297265
$$502$$ 0 0
$$503$$ −8.42082 −0.375466 −0.187733 0.982220i $$-0.560114\pi$$
−0.187733 + 0.982220i $$0.560114\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 46.4885 2.06463
$$508$$ 0 0
$$509$$ −26.0000 −1.15243 −0.576215 0.817298i $$-0.695471\pi$$
−0.576215 + 0.817298i $$0.695471\pi$$
$$510$$ 0 0
$$511$$ 35.7152 1.57995
$$512$$ 0 0
$$513$$ −17.1629 −0.757760
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 25.9421 1.14093
$$518$$ 0 0
$$519$$ −2.47027 −0.108433
$$520$$ 0 0
$$521$$ −10.3135 −0.451843 −0.225922 0.974145i $$-0.572539\pi$$
−0.225922 + 0.974145i $$0.572539\pi$$
$$522$$ 0 0
$$523$$ −16.2784 −0.711806 −0.355903 0.934523i $$-0.615827\pi$$
−0.355903 + 0.934523i $$0.615827\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −16.0000 −0.696971
$$528$$ 0 0
$$529$$ −21.1256 −0.918503
$$530$$ 0 0
$$531$$ 0.255652 0.0110944
$$532$$ 0 0
$$533$$ −49.1917 −2.13073
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −20.4124 −0.880860
$$538$$ 0 0
$$539$$ −0.424694 −0.0182929
$$540$$ 0 0
$$541$$ 3.36069 0.144487 0.0722437 0.997387i $$-0.476984\pi$$
0.0722437 + 0.997387i $$0.476984\pi$$
$$542$$ 0 0
$$543$$ −26.2557 −1.12674
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −1.51148 −0.0646263 −0.0323132 0.999478i $$-0.510287\pi$$
−0.0323132 + 0.999478i $$0.510287\pi$$
$$548$$ 0 0
$$549$$ −0.195704 −0.00835243
$$550$$ 0 0
$$551$$ −6.52359 −0.277914
$$552$$ 0 0
$$553$$ −33.3607 −1.41864
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −36.2290 −1.53507 −0.767536 0.641006i $$-0.778517\pi$$
−0.767536 + 0.641006i $$0.778517\pi$$
$$558$$ 0 0
$$559$$ −28.1978 −1.19264
$$560$$ 0 0
$$561$$ −31.6619 −1.33677
$$562$$ 0 0
$$563$$ −2.17501 −0.0916656 −0.0458328 0.998949i $$-0.514594\pi$$
−0.0458328 + 0.998949i $$0.514594\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −23.0433 −0.967728
$$568$$ 0 0
$$569$$ 33.1194 1.38844 0.694219 0.719764i $$-0.255750\pi$$
0.694219 + 0.719764i $$0.255750\pi$$
$$570$$ 0 0
$$571$$ 18.4657 0.772767 0.386383 0.922338i $$-0.373724\pi$$
0.386383 + 0.922338i $$0.373724\pi$$
$$572$$ 0 0
$$573$$ −43.3484 −1.81091
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 14.2101 0.591573 0.295787 0.955254i $$-0.404418\pi$$
0.295787 + 0.955254i $$0.404418\pi$$
$$578$$ 0 0
$$579$$ 7.83096 0.325444
$$580$$ 0 0
$$581$$ −39.3874 −1.63406
$$582$$ 0 0
$$583$$ 8.99386 0.372487
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −11.7503 −0.484987 −0.242494 0.970153i $$-0.577965\pi$$
−0.242494 + 0.970153i $$0.577965\pi$$
$$588$$ 0 0
$$589$$ 15.2663 0.629038
$$590$$ 0 0
$$591$$ 28.8248 1.18569
$$592$$ 0 0
$$593$$ −8.00000 −0.328521 −0.164260 0.986417i $$-0.552524\pi$$
−0.164260 + 0.986417i $$0.552524\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 16.8248 0.688594
$$598$$ 0 0
$$599$$ 20.5646 0.840248 0.420124 0.907467i $$-0.361987\pi$$
0.420124 + 0.907467i $$0.361987\pi$$
$$600$$ 0 0
$$601$$ −8.60811 −0.351132 −0.175566 0.984468i $$-0.556176\pi$$
−0.175566 + 0.984468i $$0.556176\pi$$
$$602$$ 0 0
$$603$$ 0.616522 0.0251067
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −0.259528 −0.0105339 −0.00526695 0.999986i $$-0.501677\pi$$
−0.00526695 + 0.999986i $$0.501677\pi$$
$$608$$ 0 0
$$609$$ −8.99386 −0.364449
$$610$$ 0 0
$$611$$ 30.3545 1.22801
$$612$$ 0 0
$$613$$ −47.1650 −1.90498 −0.952488 0.304576i $$-0.901485\pi$$
−0.952488 + 0.304576i $$0.901485\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 40.2967 1.62228 0.811142 0.584849i $$-0.198846\pi$$
0.811142 + 0.584849i $$0.198846\pi$$
$$618$$ 0 0
$$619$$ −30.6102 −1.23033 −0.615164 0.788399i $$-0.710910\pi$$
−0.615164 + 0.788399i $$0.710910\pi$$
$$620$$ 0 0
$$621$$ −7.20394 −0.289084
$$622$$ 0 0
$$623$$ −22.4247 −0.898426
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 30.2101 1.20647
$$628$$ 0 0
$$629$$ 19.6865 0.784952
$$630$$ 0 0
$$631$$ −2.21008 −0.0879819 −0.0439909 0.999032i $$-0.514007\pi$$
−0.0439909 + 0.999032i $$0.514007\pi$$
$$632$$ 0 0
$$633$$ −26.6225 −1.05815
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −0.496928 −0.0196890
$$638$$ 0 0
$$639$$ −0.482553 −0.0190895
$$640$$ 0 0
$$641$$ 7.92777 0.313128 0.156564 0.987668i $$-0.449958\pi$$
0.156564 + 0.987668i $$0.449958\pi$$
$$642$$ 0 0
$$643$$ −5.18115 −0.204325 −0.102162 0.994768i $$-0.532576\pi$$
−0.102162 + 0.994768i $$0.532576\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −12.2062 −0.479875 −0.239938 0.970788i $$-0.577127\pi$$
−0.239938 + 0.970788i $$0.577127\pi$$
$$648$$ 0 0
$$649$$ −17.6742 −0.693773
$$650$$ 0 0
$$651$$ 21.0472 0.824904
$$652$$ 0 0
$$653$$ −32.4969 −1.27170 −0.635852 0.771811i $$-0.719351\pi$$
−0.635852 + 0.771811i $$0.719351\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −1.06400 −0.0415107
$$658$$ 0 0
$$659$$ 29.4186 1.14598 0.572992 0.819561i $$-0.305783\pi$$
0.572992 + 0.819561i $$0.305783\pi$$
$$660$$ 0 0
$$661$$ 26.0677 1.01392 0.506958 0.861971i $$-0.330770\pi$$
0.506958 + 0.861971i $$0.330770\pi$$
$$662$$ 0 0
$$663$$ −37.0472 −1.43879
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −2.73820 −0.106024
$$668$$ 0 0
$$669$$ −29.0328 −1.12247
$$670$$ 0 0
$$671$$ 13.5297 0.522310
$$672$$ 0 0
$$673$$ −9.62863 −0.371156 −0.185578 0.982629i $$-0.559416\pi$$
−0.185578 + 0.982629i $$0.559416\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −19.0205 −0.731018 −0.365509 0.930808i $$-0.619105\pi$$
−0.365509 + 0.930808i $$0.619105\pi$$
$$678$$ 0 0
$$679$$ 12.0533 0.462564
$$680$$ 0 0
$$681$$ 34.1256 1.30769
$$682$$ 0 0
$$683$$ −17.4947 −0.669415 −0.334707 0.942322i $$-0.608637\pi$$
−0.334707 + 0.942322i $$0.608637\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −40.4657 −1.54386
$$688$$ 0 0
$$689$$ 10.5236 0.400917
$$690$$ 0 0
$$691$$ −16.3090 −0.620423 −0.310211 0.950668i $$-0.600400\pi$$
−0.310211 + 0.950668i $$0.600400\pi$$
$$692$$ 0 0
$$693$$ −1.11733 −0.0424437
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 26.5236 1.00465
$$698$$ 0 0
$$699$$ 23.2039 0.877653
$$700$$ 0 0
$$701$$ −12.9672 −0.489764 −0.244882 0.969553i $$-0.578749\pi$$
−0.244882 + 0.969553i $$0.578749\pi$$
$$702$$ 0 0
$$703$$ −18.7838 −0.708444
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −6.08661 −0.228911
$$708$$ 0 0
$$709$$ 7.36069 0.276437 0.138218 0.990402i $$-0.455862\pi$$
0.138218 + 0.990402i $$0.455862\pi$$
$$710$$ 0 0
$$711$$ 0.993857 0.0372725
$$712$$ 0 0
$$713$$ 6.40787 0.239977
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 14.3012 0.534089
$$718$$ 0 0
$$719$$ 19.3197 0.720502 0.360251 0.932856i $$-0.382691\pi$$
0.360251 + 0.932856i $$0.382691\pi$$
$$720$$ 0 0
$$721$$ 42.6369 1.58788
$$722$$ 0 0
$$723$$ 16.9483 0.630313
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 21.1545 0.784577 0.392288 0.919842i $$-0.371684\pi$$
0.392288 + 0.919842i $$0.371684\pi$$
$$728$$ 0 0
$$729$$ 27.6681 1.02474
$$730$$ 0 0
$$731$$ 15.2039 0.562338
$$732$$ 0 0
$$733$$ 22.7526 0.840386 0.420193 0.907435i $$-0.361962\pi$$
0.420193 + 0.907435i $$0.361962\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −42.6225 −1.57002
$$738$$ 0 0
$$739$$ 28.4534 1.04668 0.523338 0.852125i $$-0.324686\pi$$
0.523338 + 0.852125i $$0.324686\pi$$
$$740$$ 0 0
$$741$$ 35.3484 1.29856
$$742$$ 0 0
$$743$$ −3.79380 −0.139181 −0.0695904 0.997576i $$-0.522169\pi$$
−0.0695904 + 0.997576i $$0.522169\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 1.17340 0.0429324
$$748$$ 0 0
$$749$$ 41.1773 1.50458
$$750$$ 0 0
$$751$$ −17.3607 −0.633501 −0.316750 0.948509i $$-0.602592\pi$$
−0.316750 + 0.948509i $$0.602592\pi$$
$$752$$ 0 0
$$753$$ −30.1112 −1.09731
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 4.76487 0.173182 0.0865910 0.996244i $$-0.472403\pi$$
0.0865910 + 0.996244i $$0.472403\pi$$
$$758$$ 0 0
$$759$$ 12.6803 0.460267
$$760$$ 0 0
$$761$$ −14.1978 −0.514670 −0.257335 0.966322i $$-0.582844\pi$$
−0.257335 + 0.966322i $$0.582844\pi$$
$$762$$ 0 0
$$763$$ 32.4657 1.17534
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −20.6803 −0.746724
$$768$$ 0 0
$$769$$ 54.7091 1.97286 0.986430 0.164181i $$-0.0524981\pi$$
0.986430 + 0.164181i $$0.0524981\pi$$
$$770$$ 0 0
$$771$$ −6.30122 −0.226933
$$772$$ 0 0
$$773$$ 11.0205 0.396381 0.198190 0.980164i $$-0.436494\pi$$
0.198190 + 0.980164i $$0.436494\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −25.8966 −0.929034
$$778$$ 0 0
$$779$$ −25.3074 −0.906731
$$780$$ 0 0
$$781$$ 33.3607 1.19374
$$782$$ 0 0
$$783$$ 10.5236 0.376082
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 33.6925 1.20101 0.600503 0.799622i $$-0.294967\pi$$
0.600503 + 0.799622i $$0.294967\pi$$
$$788$$ 0 0
$$789$$ −0.183417 −0.00652984
$$790$$ 0 0
$$791$$ −24.6270 −0.875636
$$792$$ 0 0
$$793$$ 15.8310 0.562174
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −23.7009 −0.839528 −0.419764 0.907633i $$-0.637887\pi$$
−0.419764 + 0.907633i $$0.637887\pi$$
$$798$$ 0 0
$$799$$ −16.3668 −0.579017
$$800$$ 0 0
$$801$$ 0.668060 0.0236047
$$802$$ 0 0
$$803$$ 73.5585 2.59582
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −6.59374 −0.232110
$$808$$ 0 0
$$809$$ −33.0349 −1.16145 −0.580723 0.814102i $$-0.697230\pi$$
−0.580723 + 0.814102i $$0.697230\pi$$
$$810$$ 0 0
$$811$$ −6.26794 −0.220097 −0.110049 0.993926i $$-0.535101\pi$$
−0.110049 + 0.993926i $$0.535101\pi$$
$$812$$ 0 0
$$813$$ −36.4202 −1.27731
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −14.5068 −0.507528
$$818$$ 0 0
$$819$$ −1.30737 −0.0456831
$$820$$ 0 0
$$821$$ −15.0616 −0.525652 −0.262826 0.964843i $$-0.584655\pi$$
−0.262826 + 0.964843i $$0.584655\pi$$
$$822$$ 0 0
$$823$$ 33.5669 1.17007 0.585034 0.811009i $$-0.301081\pi$$
0.585034 + 0.811009i $$0.301081\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 10.6576 0.370600 0.185300 0.982682i $$-0.440674\pi$$
0.185300 + 0.982682i $$0.440674\pi$$
$$828$$ 0 0
$$829$$ −52.4846 −1.82287 −0.911433 0.411447i $$-0.865023\pi$$
−0.911433 + 0.411447i $$0.865023\pi$$
$$830$$ 0 0
$$831$$ −2.47027 −0.0856926
$$832$$ 0 0
$$833$$ 0.267938 0.00928351
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −24.6270 −0.851234
$$838$$ 0 0
$$839$$ −18.2101 −0.628682 −0.314341 0.949310i $$-0.601783\pi$$
−0.314341 + 0.949310i $$0.601783\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ 21.2618 0.732295
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 48.3051 1.65978
$$848$$ 0 0
$$849$$ −10.7526 −0.369028
$$850$$ 0 0
$$851$$ −7.88428 −0.270270
$$852$$ 0 0
$$853$$ 19.7542 0.676371 0.338185 0.941080i $$-0.390187\pi$$
0.338185 + 0.941080i $$0.390187\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −48.9939 −1.67360 −0.836799 0.547510i $$-0.815576\pi$$
−0.836799 + 0.547510i $$0.815576\pi$$
$$858$$ 0 0
$$859$$ −24.3090 −0.829412 −0.414706 0.909956i $$-0.636115\pi$$
−0.414706 + 0.909956i $$0.636115\pi$$
$$860$$ 0 0
$$861$$ −34.8904 −1.18906
$$862$$ 0 0
$$863$$ −44.3584 −1.50998 −0.754989 0.655737i $$-0.772358\pi$$
−0.754989 + 0.655737i $$0.772358\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −9.08225 −0.308449
$$868$$ 0 0
$$869$$ −68.7091 −2.33080
$$870$$ 0 0
$$871$$ −49.8720 −1.68985
$$872$$ 0 0
$$873$$ −0.359084 −0.0121531
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 37.9565 1.28170 0.640850 0.767666i $$-0.278582\pi$$
0.640850 + 0.767666i $$0.278582\pi$$
$$878$$ 0 0
$$879$$ −1.84324 −0.0621711
$$880$$ 0 0
$$881$$ 25.0661 0.844498 0.422249 0.906480i $$-0.361241\pi$$
0.422249 + 0.906480i $$0.361241\pi$$
$$882$$ 0 0
$$883$$ −21.1278 −0.711008 −0.355504 0.934675i $$-0.615691\pi$$
−0.355504 + 0.934675i $$0.615691\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 36.1894 1.21512 0.607560 0.794274i $$-0.292148\pi$$
0.607560 + 0.794274i $$0.292148\pi$$
$$888$$ 0 0
$$889$$ 5.13170 0.172112
$$890$$ 0 0
$$891$$ −47.4596 −1.58996
$$892$$ 0 0
$$893$$ 15.6163 0.522581
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 14.8371 0.495396
$$898$$ 0 0
$$899$$ −9.36069 −0.312197
$$900$$ 0 0
$$901$$ −5.67420 −0.189035
$$902$$ 0 0
$$903$$ −20.0000 −0.665558
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −35.2678 −1.17105 −0.585523 0.810656i $$-0.699111\pi$$
−0.585523 + 0.810656i $$0.699111\pi$$
$$908$$ 0 0
$$909$$ 0.181328 0.00601426
$$910$$ 0 0
$$911$$ 15.0061 0.497176 0.248588 0.968609i $$-0.420034\pi$$
0.248588 + 0.968609i $$0.420034\pi$$
$$912$$ 0 0
$$913$$ −81.1215 −2.68473
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −40.9770 −1.35318
$$918$$ 0 0
$$919$$ −32.8781 −1.08455 −0.542275 0.840201i $$-0.682437\pi$$
−0.542275 + 0.840201i $$0.682437\pi$$
$$920$$ 0 0
$$921$$ −43.1194 −1.42083
$$922$$ 0 0
$$923$$ 39.0349 1.28485
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −1.27021 −0.0417190
$$928$$ 0 0
$$929$$ −30.2290 −0.991781 −0.495890 0.868385i $$-0.665158\pi$$
−0.495890 + 0.868385i $$0.665158\pi$$
$$930$$ 0 0
$$931$$ −0.255652 −0.00837866
$$932$$ 0 0
$$933$$ 32.1978 1.05411
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 28.4124 0.928193 0.464096 0.885785i $$-0.346379\pi$$
0.464096 + 0.885785i $$0.346379\pi$$
$$938$$ 0 0
$$939$$ 25.9877 0.848077
$$940$$ 0 0
$$941$$ −42.0821 −1.37184 −0.685918 0.727679i $$-0.740599\pi$$
−0.685918 + 0.727679i $$0.740599\pi$$
$$942$$ 0 0
$$943$$ −10.6225 −0.345916
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −22.2739 −0.723805 −0.361902 0.932216i $$-0.617873\pi$$
−0.361902 + 0.932216i $$0.617873\pi$$
$$948$$ 0 0
$$949$$ 86.0698 2.79394
$$950$$ 0 0
$$951$$ −28.8248 −0.934709
$$952$$ 0 0
$$953$$ 14.6681 0.475145 0.237573 0.971370i $$-0.423648\pi$$
0.237573 + 0.971370i $$0.423648\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −18.5236 −0.598783
$$958$$ 0 0
$$959$$ 40.0000 1.29167
$$960$$ 0 0
$$961$$ −9.09436 −0.293367
$$962$$ 0 0
$$963$$ −1.22672 −0.0395306
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −0.403997 −0.0129917 −0.00649584 0.999979i $$-0.502068\pi$$
−0.00649584 + 0.999979i $$0.502068\pi$$
$$968$$ 0 0
$$969$$ −19.0595 −0.612278
$$970$$ 0 0
$$971$$ −46.8326 −1.50293 −0.751464 0.659774i $$-0.770652\pi$$
−0.751464 + 0.659774i $$0.770652\pi$$
$$972$$ 0 0
$$973$$ 6.79153 0.217726
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −53.6041 −1.71495 −0.857473 0.514529i $$-0.827967\pi$$
−0.857473 + 0.514529i $$0.827967\pi$$
$$978$$ 0 0
$$979$$ −46.1855 −1.47610
$$980$$ 0 0
$$981$$ −0.967195 −0.0308802
$$982$$ 0 0
$$983$$ −19.9916 −0.637633 −0.318816 0.947817i $$-0.603285\pi$$
−0.318816 + 0.947817i $$0.603285\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 21.5297 0.685299
$$988$$ 0 0
$$989$$ −6.08906 −0.193621
$$990$$ 0 0
$$991$$ 24.6270 0.782303 0.391152 0.920326i $$-0.372077\pi$$
0.391152 + 0.920326i $$0.372077\pi$$
$$992$$ 0 0
$$993$$ 40.2967 1.27878
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 20.9795 0.664427 0.332213 0.943204i $$-0.392205\pi$$
0.332213 + 0.943204i $$0.392205\pi$$
$$998$$ 0 0
$$999$$ 30.3012 0.958688
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 3200.2.a.br.1.3 3
4.3 odd 2 3200.2.a.bs.1.1 3
5.2 odd 4 640.2.c.b.129.2 yes 6
5.3 odd 4 640.2.c.b.129.5 yes 6
5.4 even 2 3200.2.a.bt.1.1 3
8.3 odd 2 3200.2.a.bp.1.3 3
8.5 even 2 3200.2.a.bu.1.1 3
20.3 even 4 640.2.c.a.129.2 6
20.7 even 4 640.2.c.a.129.5 yes 6
20.19 odd 2 3200.2.a.bq.1.3 3
40.3 even 4 640.2.c.d.129.5 yes 6
40.13 odd 4 640.2.c.c.129.2 yes 6
40.19 odd 2 3200.2.a.bv.1.1 3
40.27 even 4 640.2.c.d.129.2 yes 6
40.29 even 2 3200.2.a.bo.1.3 3
40.37 odd 4 640.2.c.c.129.5 yes 6
80.3 even 4 1280.2.f.i.129.6 6
80.13 odd 4 1280.2.f.k.129.2 6
80.27 even 4 1280.2.f.i.129.5 6
80.37 odd 4 1280.2.f.k.129.1 6
80.43 even 4 1280.2.f.l.129.1 6
80.53 odd 4 1280.2.f.j.129.5 6
80.67 even 4 1280.2.f.l.129.2 6
80.77 odd 4 1280.2.f.j.129.6 6

By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.c.a.129.2 6 20.3 even 4
640.2.c.a.129.5 yes 6 20.7 even 4
640.2.c.b.129.2 yes 6 5.2 odd 4
640.2.c.b.129.5 yes 6 5.3 odd 4
640.2.c.c.129.2 yes 6 40.13 odd 4
640.2.c.c.129.5 yes 6 40.37 odd 4
640.2.c.d.129.2 yes 6 40.27 even 4
640.2.c.d.129.5 yes 6 40.3 even 4
1280.2.f.i.129.5 6 80.27 even 4
1280.2.f.i.129.6 6 80.3 even 4
1280.2.f.j.129.5 6 80.53 odd 4
1280.2.f.j.129.6 6 80.77 odd 4
1280.2.f.k.129.1 6 80.37 odd 4
1280.2.f.k.129.2 6 80.13 odd 4
1280.2.f.l.129.1 6 80.43 even 4
1280.2.f.l.129.2 6 80.67 even 4
3200.2.a.bo.1.3 3 40.29 even 2
3200.2.a.bp.1.3 3 8.3 odd 2
3200.2.a.bq.1.3 3 20.19 odd 2
3200.2.a.br.1.3 3 1.1 even 1 trivial
3200.2.a.bs.1.1 3 4.3 odd 2
3200.2.a.bt.1.1 3 5.4 even 2
3200.2.a.bu.1.1 3 8.5 even 2
3200.2.a.bv.1.1 3 40.19 odd 2