# Properties

 Label 1638.2.a.l.1.1 Level $1638$ Weight $2$ Character 1638.1 Self dual yes Analytic conductor $13.079$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(1,1638)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, 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("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1638.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: $$13.0794958511$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1638.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.00000 q^{10} -3.00000 q^{11} +1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -7.00000 q^{19} -3.00000 q^{20} -3.00000 q^{22} -9.00000 q^{23} +4.00000 q^{25} +1.00000 q^{26} +1.00000 q^{28} +9.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} +3.00000 q^{34} -3.00000 q^{35} -7.00000 q^{37} -7.00000 q^{38} -3.00000 q^{40} -12.0000 q^{41} -1.00000 q^{43} -3.00000 q^{44} -9.00000 q^{46} +1.00000 q^{49} +4.00000 q^{50} +1.00000 q^{52} +6.00000 q^{53} +9.00000 q^{55} +1.00000 q^{56} +9.00000 q^{58} -12.0000 q^{59} -1.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -3.00000 q^{65} +14.0000 q^{67} +3.00000 q^{68} -3.00000 q^{70} -12.0000 q^{71} -7.00000 q^{73} -7.00000 q^{74} -7.00000 q^{76} -3.00000 q^{77} -10.0000 q^{79} -3.00000 q^{80} -12.0000 q^{82} +6.00000 q^{83} -9.00000 q^{85} -1.00000 q^{86} -3.00000 q^{88} +6.00000 q^{89} +1.00000 q^{91} -9.00000 q^{92} +21.0000 q^{95} -10.0000 q^{97} +1.00000 q^{98} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −3.00000 −1.34164 −0.670820 0.741620i $$-0.734058\pi$$
−0.670820 + 0.741620i $$0.734058\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ −3.00000 −0.948683
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 0 0
$$19$$ −7.00000 −1.60591 −0.802955 0.596040i $$-0.796740\pi$$
−0.802955 + 0.596040i $$0.796740\pi$$
$$20$$ −3.00000 −0.670820
$$21$$ 0 0
$$22$$ −3.00000 −0.639602
$$23$$ −9.00000 −1.87663 −0.938315 0.345782i $$-0.887614\pi$$
−0.938315 + 0.345782i $$0.887614\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ 1.00000 0.196116
$$27$$ 0 0
$$28$$ 1.00000 0.188982
$$29$$ 9.00000 1.67126 0.835629 0.549294i $$-0.185103\pi$$
0.835629 + 0.549294i $$0.185103\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 3.00000 0.514496
$$35$$ −3.00000 −0.507093
$$36$$ 0 0
$$37$$ −7.00000 −1.15079 −0.575396 0.817875i $$-0.695152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ −7.00000 −1.13555
$$39$$ 0 0
$$40$$ −3.00000 −0.474342
$$41$$ −12.0000 −1.87409 −0.937043 0.349215i $$-0.886448\pi$$
−0.937043 + 0.349215i $$0.886448\pi$$
$$42$$ 0 0
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ −3.00000 −0.452267
$$45$$ 0 0
$$46$$ −9.00000 −1.32698
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 4.00000 0.565685
$$51$$ 0 0
$$52$$ 1.00000 0.138675
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ 9.00000 1.21356
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ 9.00000 1.18176
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ −1.00000 −0.128037 −0.0640184 0.997949i $$-0.520392\pi$$
−0.0640184 + 0.997949i $$0.520392\pi$$
$$62$$ −4.00000 −0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −3.00000 −0.372104
$$66$$ 0 0
$$67$$ 14.0000 1.71037 0.855186 0.518321i $$-0.173443\pi$$
0.855186 + 0.518321i $$0.173443\pi$$
$$68$$ 3.00000 0.363803
$$69$$ 0 0
$$70$$ −3.00000 −0.358569
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ −7.00000 −0.819288 −0.409644 0.912245i $$-0.634347\pi$$
−0.409644 + 0.912245i $$0.634347\pi$$
$$74$$ −7.00000 −0.813733
$$75$$ 0 0
$$76$$ −7.00000 −0.802955
$$77$$ −3.00000 −0.341882
$$78$$ 0 0
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ −3.00000 −0.335410
$$81$$ 0 0
$$82$$ −12.0000 −1.32518
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ −9.00000 −0.976187
$$86$$ −1.00000 −0.107833
$$87$$ 0 0
$$88$$ −3.00000 −0.319801
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 1.00000 0.104828
$$92$$ −9.00000 −0.938315
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 21.0000 2.15455
$$96$$ 0 0
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ 4.00000 0.400000
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ 0 0
$$103$$ −13.0000 −1.28093 −0.640464 0.767988i $$-0.721258\pi$$
−0.640464 + 0.767988i $$0.721258\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 6.00000 0.580042 0.290021 0.957020i $$-0.406338\pi$$
0.290021 + 0.957020i $$0.406338\pi$$
$$108$$ 0 0
$$109$$ 11.0000 1.05361 0.526804 0.849987i $$-0.323390\pi$$
0.526804 + 0.849987i $$0.323390\pi$$
$$110$$ 9.00000 0.858116
$$111$$ 0 0
$$112$$ 1.00000 0.0944911
$$113$$ 18.0000 1.69330 0.846649 0.532152i $$-0.178617\pi$$
0.846649 + 0.532152i $$0.178617\pi$$
$$114$$ 0 0
$$115$$ 27.0000 2.51776
$$116$$ 9.00000 0.835629
$$117$$ 0 0
$$118$$ −12.0000 −1.10469
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ −1.00000 −0.0905357
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ 3.00000 0.268328
$$126$$ 0 0
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ −3.00000 −0.263117
$$131$$ 15.0000 1.31056 0.655278 0.755388i $$-0.272551\pi$$
0.655278 + 0.755388i $$0.272551\pi$$
$$132$$ 0 0
$$133$$ −7.00000 −0.606977
$$134$$ 14.0000 1.20942
$$135$$ 0 0
$$136$$ 3.00000 0.257248
$$137$$ −9.00000 −0.768922 −0.384461 0.923141i $$-0.625613\pi$$
−0.384461 + 0.923141i $$0.625613\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ −3.00000 −0.253546
$$141$$ 0 0
$$142$$ −12.0000 −1.00702
$$143$$ −3.00000 −0.250873
$$144$$ 0 0
$$145$$ −27.0000 −2.24223
$$146$$ −7.00000 −0.579324
$$147$$ 0 0
$$148$$ −7.00000 −0.575396
$$149$$ 12.0000 0.983078 0.491539 0.870855i $$-0.336434\pi$$
0.491539 + 0.870855i $$0.336434\pi$$
$$150$$ 0 0
$$151$$ 17.0000 1.38344 0.691720 0.722166i $$-0.256853\pi$$
0.691720 + 0.722166i $$0.256853\pi$$
$$152$$ −7.00000 −0.567775
$$153$$ 0 0
$$154$$ −3.00000 −0.241747
$$155$$ 12.0000 0.963863
$$156$$ 0 0
$$157$$ −13.0000 −1.03751 −0.518756 0.854922i $$-0.673605\pi$$
−0.518756 + 0.854922i $$0.673605\pi$$
$$158$$ −10.0000 −0.795557
$$159$$ 0 0
$$160$$ −3.00000 −0.237171
$$161$$ −9.00000 −0.709299
$$162$$ 0 0
$$163$$ 2.00000 0.156652 0.0783260 0.996928i $$-0.475042\pi$$
0.0783260 + 0.996928i $$0.475042\pi$$
$$164$$ −12.0000 −0.937043
$$165$$ 0 0
$$166$$ 6.00000 0.465690
$$167$$ −3.00000 −0.232147 −0.116073 0.993241i $$-0.537031\pi$$
−0.116073 + 0.993241i $$0.537031\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ −9.00000 −0.690268
$$171$$ 0 0
$$172$$ −1.00000 −0.0762493
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ 4.00000 0.302372
$$176$$ −3.00000 −0.226134
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 1.00000 0.0741249
$$183$$ 0 0
$$184$$ −9.00000 −0.663489
$$185$$ 21.0000 1.54395
$$186$$ 0 0
$$187$$ −9.00000 −0.658145
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 21.0000 1.52350
$$191$$ 15.0000 1.08536 0.542681 0.839939i $$-0.317409\pi$$
0.542681 + 0.839939i $$0.317409\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 12.0000 0.854965 0.427482 0.904024i $$-0.359401\pi$$
0.427482 + 0.904024i $$0.359401\pi$$
$$198$$ 0 0
$$199$$ −7.00000 −0.496217 −0.248108 0.968732i $$-0.579809\pi$$
−0.248108 + 0.968732i $$0.579809\pi$$
$$200$$ 4.00000 0.282843
$$201$$ 0 0
$$202$$ −12.0000 −0.844317
$$203$$ 9.00000 0.631676
$$204$$ 0 0
$$205$$ 36.0000 2.51435
$$206$$ −13.0000 −0.905753
$$207$$ 0 0
$$208$$ 1.00000 0.0693375
$$209$$ 21.0000 1.45260
$$210$$ 0 0
$$211$$ 5.00000 0.344214 0.172107 0.985078i $$-0.444942\pi$$
0.172107 + 0.985078i $$0.444942\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ 6.00000 0.410152
$$215$$ 3.00000 0.204598
$$216$$ 0 0
$$217$$ −4.00000 −0.271538
$$218$$ 11.0000 0.745014
$$219$$ 0 0
$$220$$ 9.00000 0.606780
$$221$$ 3.00000 0.201802
$$222$$ 0 0
$$223$$ 26.0000 1.74109 0.870544 0.492090i $$-0.163767\pi$$
0.870544 + 0.492090i $$0.163767\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ 18.0000 1.19734
$$227$$ 18.0000 1.19470 0.597351 0.801980i $$-0.296220\pi$$
0.597351 + 0.801980i $$0.296220\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 27.0000 1.78033
$$231$$ 0 0
$$232$$ 9.00000 0.590879
$$233$$ −24.0000 −1.57229 −0.786146 0.618041i $$-0.787927\pi$$
−0.786146 + 0.618041i $$0.787927\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ 0 0
$$238$$ 3.00000 0.194461
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ −2.00000 −0.128565
$$243$$ 0 0
$$244$$ −1.00000 −0.0640184
$$245$$ −3.00000 −0.191663
$$246$$ 0 0
$$247$$ −7.00000 −0.445399
$$248$$ −4.00000 −0.254000
$$249$$ 0 0
$$250$$ 3.00000 0.189737
$$251$$ −21.0000 −1.32551 −0.662754 0.748837i $$-0.730613\pi$$
−0.662754 + 0.748837i $$0.730613\pi$$
$$252$$ 0 0
$$253$$ 27.0000 1.69748
$$254$$ 2.00000 0.125491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ −7.00000 −0.434959
$$260$$ −3.00000 −0.186052
$$261$$ 0 0
$$262$$ 15.0000 0.926703
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 0 0
$$265$$ −18.0000 −1.10573
$$266$$ −7.00000 −0.429198
$$267$$ 0 0
$$268$$ 14.0000 0.855186
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 2.00000 0.121491 0.0607457 0.998153i $$-0.480652\pi$$
0.0607457 + 0.998153i $$0.480652\pi$$
$$272$$ 3.00000 0.181902
$$273$$ 0 0
$$274$$ −9.00000 −0.543710
$$275$$ −12.0000 −0.723627
$$276$$ 0 0
$$277$$ 26.0000 1.56219 0.781094 0.624413i $$-0.214662\pi$$
0.781094 + 0.624413i $$0.214662\pi$$
$$278$$ −4.00000 −0.239904
$$279$$ 0 0
$$280$$ −3.00000 −0.179284
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 0 0
$$286$$ −3.00000 −0.177394
$$287$$ −12.0000 −0.708338
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ −27.0000 −1.58549
$$291$$ 0 0
$$292$$ −7.00000 −0.409644
$$293$$ 18.0000 1.05157 0.525786 0.850617i $$-0.323771\pi$$
0.525786 + 0.850617i $$0.323771\pi$$
$$294$$ 0 0
$$295$$ 36.0000 2.09600
$$296$$ −7.00000 −0.406867
$$297$$ 0 0
$$298$$ 12.0000 0.695141
$$299$$ −9.00000 −0.520483
$$300$$ 0 0
$$301$$ −1.00000 −0.0576390
$$302$$ 17.0000 0.978240
$$303$$ 0 0
$$304$$ −7.00000 −0.401478
$$305$$ 3.00000 0.171780
$$306$$ 0 0
$$307$$ −16.0000 −0.913168 −0.456584 0.889680i $$-0.650927\pi$$
−0.456584 + 0.889680i $$0.650927\pi$$
$$308$$ −3.00000 −0.170941
$$309$$ 0 0
$$310$$ 12.0000 0.681554
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ 0 0
$$313$$ 8.00000 0.452187 0.226093 0.974106i $$-0.427405\pi$$
0.226093 + 0.974106i $$0.427405\pi$$
$$314$$ −13.0000 −0.733632
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$318$$ 0 0
$$319$$ −27.0000 −1.51171
$$320$$ −3.00000 −0.167705
$$321$$ 0 0
$$322$$ −9.00000 −0.501550
$$323$$ −21.0000 −1.16847
$$324$$ 0 0
$$325$$ 4.00000 0.221880
$$326$$ 2.00000 0.110770
$$327$$ 0 0
$$328$$ −12.0000 −0.662589
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −10.0000 −0.549650 −0.274825 0.961494i $$-0.588620\pi$$
−0.274825 + 0.961494i $$0.588620\pi$$
$$332$$ 6.00000 0.329293
$$333$$ 0 0
$$334$$ −3.00000 −0.164153
$$335$$ −42.0000 −2.29471
$$336$$ 0 0
$$337$$ −13.0000 −0.708155 −0.354078 0.935216i $$-0.615205\pi$$
−0.354078 + 0.935216i $$0.615205\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ 0 0
$$340$$ −9.00000 −0.488094
$$341$$ 12.0000 0.649836
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ −1.00000 −0.0539164
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ 0 0
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 4.00000 0.213809
$$351$$ 0 0
$$352$$ −3.00000 −0.159901
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ 0 0
$$355$$ 36.0000 1.91068
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ 2.00000 0.105118
$$363$$ 0 0
$$364$$ 1.00000 0.0524142
$$365$$ 21.0000 1.09919
$$366$$ 0 0
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ −9.00000 −0.469157
$$369$$ 0 0
$$370$$ 21.0000 1.09174
$$371$$ 6.00000 0.311504
$$372$$ 0 0
$$373$$ −4.00000 −0.207112 −0.103556 0.994624i $$-0.533022\pi$$
−0.103556 + 0.994624i $$0.533022\pi$$
$$374$$ −9.00000 −0.465379
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 9.00000 0.463524
$$378$$ 0 0
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ 21.0000 1.07728
$$381$$ 0 0
$$382$$ 15.0000 0.767467
$$383$$ 21.0000 1.07305 0.536525 0.843884i $$-0.319737\pi$$
0.536525 + 0.843884i $$0.319737\pi$$
$$384$$ 0 0
$$385$$ 9.00000 0.458682
$$386$$ −4.00000 −0.203595
$$387$$ 0 0
$$388$$ −10.0000 −0.507673
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ −27.0000 −1.36545
$$392$$ 1.00000 0.0505076
$$393$$ 0 0
$$394$$ 12.0000 0.604551
$$395$$ 30.0000 1.50946
$$396$$ 0 0
$$397$$ −34.0000 −1.70641 −0.853206 0.521575i $$-0.825345\pi$$
−0.853206 + 0.521575i $$0.825345\pi$$
$$398$$ −7.00000 −0.350878
$$399$$ 0 0
$$400$$ 4.00000 0.200000
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 0 0
$$403$$ −4.00000 −0.199254
$$404$$ −12.0000 −0.597022
$$405$$ 0 0
$$406$$ 9.00000 0.446663
$$407$$ 21.0000 1.04093
$$408$$ 0 0
$$409$$ −13.0000 −0.642809 −0.321404 0.946942i $$-0.604155\pi$$
−0.321404 + 0.946942i $$0.604155\pi$$
$$410$$ 36.0000 1.77791
$$411$$ 0 0
$$412$$ −13.0000 −0.640464
$$413$$ −12.0000 −0.590481
$$414$$ 0 0
$$415$$ −18.0000 −0.883585
$$416$$ 1.00000 0.0490290
$$417$$ 0 0
$$418$$ 21.0000 1.02714
$$419$$ −3.00000 −0.146560 −0.0732798 0.997311i $$-0.523347\pi$$
−0.0732798 + 0.997311i $$0.523347\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 5.00000 0.243396
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ 12.0000 0.582086
$$426$$ 0 0
$$427$$ −1.00000 −0.0483934
$$428$$ 6.00000 0.290021
$$429$$ 0 0
$$430$$ 3.00000 0.144673
$$431$$ −18.0000 −0.867029 −0.433515 0.901146i $$-0.642727\pi$$
−0.433515 + 0.901146i $$0.642727\pi$$
$$432$$ 0 0
$$433$$ −16.0000 −0.768911 −0.384455 0.923144i $$-0.625611\pi$$
−0.384455 + 0.923144i $$0.625611\pi$$
$$434$$ −4.00000 −0.192006
$$435$$ 0 0
$$436$$ 11.0000 0.526804
$$437$$ 63.0000 3.01370
$$438$$ 0 0
$$439$$ −1.00000 −0.0477274 −0.0238637 0.999715i $$-0.507597\pi$$
−0.0238637 + 0.999715i $$0.507597\pi$$
$$440$$ 9.00000 0.429058
$$441$$ 0 0
$$442$$ 3.00000 0.142695
$$443$$ −18.0000 −0.855206 −0.427603 0.903967i $$-0.640642\pi$$
−0.427603 + 0.903967i $$0.640642\pi$$
$$444$$ 0 0
$$445$$ −18.0000 −0.853282
$$446$$ 26.0000 1.23114
$$447$$ 0 0
$$448$$ 1.00000 0.0472456
$$449$$ −21.0000 −0.991051 −0.495526 0.868593i $$-0.665025\pi$$
−0.495526 + 0.868593i $$0.665025\pi$$
$$450$$ 0 0
$$451$$ 36.0000 1.69517
$$452$$ 18.0000 0.846649
$$453$$ 0 0
$$454$$ 18.0000 0.844782
$$455$$ −3.00000 −0.140642
$$456$$ 0 0
$$457$$ −28.0000 −1.30978 −0.654892 0.755722i $$-0.727286\pi$$
−0.654892 + 0.755722i $$0.727286\pi$$
$$458$$ 14.0000 0.654177
$$459$$ 0 0
$$460$$ 27.0000 1.25888
$$461$$ −33.0000 −1.53696 −0.768482 0.639872i $$-0.778987\pi$$
−0.768482 + 0.639872i $$0.778987\pi$$
$$462$$ 0 0
$$463$$ 41.0000 1.90543 0.952716 0.303863i $$-0.0982765\pi$$
0.952716 + 0.303863i $$0.0982765\pi$$
$$464$$ 9.00000 0.417815
$$465$$ 0 0
$$466$$ −24.0000 −1.11178
$$467$$ 3.00000 0.138823 0.0694117 0.997588i $$-0.477888\pi$$
0.0694117 + 0.997588i $$0.477888\pi$$
$$468$$ 0 0
$$469$$ 14.0000 0.646460
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −12.0000 −0.552345
$$473$$ 3.00000 0.137940
$$474$$ 0 0
$$475$$ −28.0000 −1.28473
$$476$$ 3.00000 0.137505
$$477$$ 0 0
$$478$$ −24.0000 −1.09773
$$479$$ 15.0000 0.685367 0.342684 0.939451i $$-0.388664\pi$$
0.342684 + 0.939451i $$0.388664\pi$$
$$480$$ 0 0
$$481$$ −7.00000 −0.319173
$$482$$ −10.0000 −0.455488
$$483$$ 0 0
$$484$$ −2.00000 −0.0909091
$$485$$ 30.0000 1.36223
$$486$$ 0 0
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ −1.00000 −0.0452679
$$489$$ 0 0
$$490$$ −3.00000 −0.135526
$$491$$ −30.0000 −1.35388 −0.676941 0.736038i $$-0.736695\pi$$
−0.676941 + 0.736038i $$0.736695\pi$$
$$492$$ 0 0
$$493$$ 27.0000 1.21602
$$494$$ −7.00000 −0.314945
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ −12.0000 −0.538274
$$498$$ 0 0
$$499$$ 32.0000 1.43252 0.716258 0.697835i $$-0.245853\pi$$
0.716258 + 0.697835i $$0.245853\pi$$
$$500$$ 3.00000 0.134164
$$501$$ 0 0
$$502$$ −21.0000 −0.937276
$$503$$ 6.00000 0.267527 0.133763 0.991013i $$-0.457294\pi$$
0.133763 + 0.991013i $$0.457294\pi$$
$$504$$ 0 0
$$505$$ 36.0000 1.60198
$$506$$ 27.0000 1.20030
$$507$$ 0 0
$$508$$ 2.00000 0.0887357
$$509$$ −9.00000 −0.398918 −0.199459 0.979906i $$-0.563918\pi$$
−0.199459 + 0.979906i $$0.563918\pi$$
$$510$$ 0 0
$$511$$ −7.00000 −0.309662
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 18.0000 0.793946
$$515$$ 39.0000 1.71855
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −7.00000 −0.307562
$$519$$ 0 0
$$520$$ −3.00000 −0.131559
$$521$$ 33.0000 1.44576 0.722878 0.690976i $$-0.242819\pi$$
0.722878 + 0.690976i $$0.242819\pi$$
$$522$$ 0 0
$$523$$ −16.0000 −0.699631 −0.349816 0.936819i $$-0.613756\pi$$
−0.349816 + 0.936819i $$0.613756\pi$$
$$524$$ 15.0000 0.655278
$$525$$ 0 0
$$526$$ −12.0000 −0.523225
$$527$$ −12.0000 −0.522728
$$528$$ 0 0
$$529$$ 58.0000 2.52174
$$530$$ −18.0000 −0.781870
$$531$$ 0 0
$$532$$ −7.00000 −0.303488
$$533$$ −12.0000 −0.519778
$$534$$ 0 0
$$535$$ −18.0000 −0.778208
$$536$$ 14.0000 0.604708
$$537$$ 0 0
$$538$$ −6.00000 −0.258678
$$539$$ −3.00000 −0.129219
$$540$$ 0 0
$$541$$ 11.0000 0.472927 0.236463 0.971640i $$-0.424012\pi$$
0.236463 + 0.971640i $$0.424012\pi$$
$$542$$ 2.00000 0.0859074
$$543$$ 0 0
$$544$$ 3.00000 0.128624
$$545$$ −33.0000 −1.41356
$$546$$ 0 0
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ −9.00000 −0.384461
$$549$$ 0 0
$$550$$ −12.0000 −0.511682
$$551$$ −63.0000 −2.68389
$$552$$ 0 0
$$553$$ −10.0000 −0.425243
$$554$$ 26.0000 1.10463
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$558$$ 0 0
$$559$$ −1.00000 −0.0422955
$$560$$ −3.00000 −0.126773
$$561$$ 0 0
$$562$$ 6.00000 0.253095
$$563$$ 15.0000 0.632175 0.316087 0.948730i $$-0.397631\pi$$
0.316087 + 0.948730i $$0.397631\pi$$
$$564$$ 0 0
$$565$$ −54.0000 −2.27180
$$566$$ −4.00000 −0.168133
$$567$$ 0 0
$$568$$ −12.0000 −0.503509
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ −3.00000 −0.125436
$$573$$ 0 0
$$574$$ −12.0000 −0.500870
$$575$$ −36.0000 −1.50130
$$576$$ 0 0
$$577$$ 38.0000 1.58196 0.790980 0.611842i $$-0.209571\pi$$
0.790980 + 0.611842i $$0.209571\pi$$
$$578$$ −8.00000 −0.332756
$$579$$ 0 0
$$580$$ −27.0000 −1.12111
$$581$$ 6.00000 0.248922
$$582$$ 0 0
$$583$$ −18.0000 −0.745484
$$584$$ −7.00000 −0.289662
$$585$$ 0 0
$$586$$ 18.0000 0.743573
$$587$$ 18.0000 0.742940 0.371470 0.928445i $$-0.378854\pi$$
0.371470 + 0.928445i $$0.378854\pi$$
$$588$$ 0 0
$$589$$ 28.0000 1.15372
$$590$$ 36.0000 1.48210
$$591$$ 0 0
$$592$$ −7.00000 −0.287698
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ −9.00000 −0.368964
$$596$$ 12.0000 0.491539
$$597$$ 0 0
$$598$$ −9.00000 −0.368037
$$599$$ 9.00000 0.367730 0.183865 0.982952i $$-0.441139\pi$$
0.183865 + 0.982952i $$0.441139\pi$$
$$600$$ 0 0
$$601$$ 8.00000 0.326327 0.163163 0.986599i $$-0.447830\pi$$
0.163163 + 0.986599i $$0.447830\pi$$
$$602$$ −1.00000 −0.0407570
$$603$$ 0 0
$$604$$ 17.0000 0.691720
$$605$$ 6.00000 0.243935
$$606$$ 0 0
$$607$$ 23.0000 0.933541 0.466771 0.884378i $$-0.345417\pi$$
0.466771 + 0.884378i $$0.345417\pi$$
$$608$$ −7.00000 −0.283887
$$609$$ 0 0
$$610$$ 3.00000 0.121466
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 11.0000 0.444286 0.222143 0.975014i $$-0.428695\pi$$
0.222143 + 0.975014i $$0.428695\pi$$
$$614$$ −16.0000 −0.645707
$$615$$ 0 0
$$616$$ −3.00000 −0.120873
$$617$$ −27.0000 −1.08698 −0.543490 0.839416i $$-0.682897\pi$$
−0.543490 + 0.839416i $$0.682897\pi$$
$$618$$ 0 0
$$619$$ 17.0000 0.683288 0.341644 0.939829i $$-0.389016\pi$$
0.341644 + 0.939829i $$0.389016\pi$$
$$620$$ 12.0000 0.481932
$$621$$ 0 0
$$622$$ −18.0000 −0.721734
$$623$$ 6.00000 0.240385
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 8.00000 0.319744
$$627$$ 0 0
$$628$$ −13.0000 −0.518756
$$629$$ −21.0000 −0.837325
$$630$$ 0 0
$$631$$ 29.0000 1.15447 0.577236 0.816577i $$-0.304131\pi$$
0.577236 + 0.816577i $$0.304131\pi$$
$$632$$ −10.0000 −0.397779
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −6.00000 −0.238103
$$636$$ 0 0
$$637$$ 1.00000 0.0396214
$$638$$ −27.0000 −1.06894
$$639$$ 0 0
$$640$$ −3.00000 −0.118585
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 0 0
$$643$$ −49.0000 −1.93237 −0.966186 0.257847i $$-0.916987\pi$$
−0.966186 + 0.257847i $$0.916987\pi$$
$$644$$ −9.00000 −0.354650
$$645$$ 0 0
$$646$$ −21.0000 −0.826234
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ 36.0000 1.41312
$$650$$ 4.00000 0.156893
$$651$$ 0 0
$$652$$ 2.00000 0.0783260
$$653$$ −9.00000 −0.352197 −0.176099 0.984373i $$-0.556348\pi$$
−0.176099 + 0.984373i $$0.556348\pi$$
$$654$$ 0 0
$$655$$ −45.0000 −1.75830
$$656$$ −12.0000 −0.468521
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −6.00000 −0.233727 −0.116863 0.993148i $$-0.537284\pi$$
−0.116863 + 0.993148i $$0.537284\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ −10.0000 −0.388661
$$663$$ 0 0
$$664$$ 6.00000 0.232845
$$665$$ 21.0000 0.814345
$$666$$ 0 0
$$667$$ −81.0000 −3.13633
$$668$$ −3.00000 −0.116073
$$669$$ 0 0
$$670$$ −42.0000 −1.62260
$$671$$ 3.00000 0.115814
$$672$$ 0 0
$$673$$ −1.00000 −0.0385472 −0.0192736 0.999814i $$-0.506135\pi$$
−0.0192736 + 0.999814i $$0.506135\pi$$
$$674$$ −13.0000 −0.500741
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ −18.0000 −0.691796 −0.345898 0.938272i $$-0.612426\pi$$
−0.345898 + 0.938272i $$0.612426\pi$$
$$678$$ 0 0
$$679$$ −10.0000 −0.383765
$$680$$ −9.00000 −0.345134
$$681$$ 0 0
$$682$$ 12.0000 0.459504
$$683$$ −33.0000 −1.26271 −0.631355 0.775494i $$-0.717501\pi$$
−0.631355 + 0.775494i $$0.717501\pi$$
$$684$$ 0 0
$$685$$ 27.0000 1.03162
$$686$$ 1.00000 0.0381802
$$687$$ 0 0
$$688$$ −1.00000 −0.0381246
$$689$$ 6.00000 0.228582
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 0 0
$$694$$ −12.0000 −0.455514
$$695$$ 12.0000 0.455186
$$696$$ 0 0
$$697$$ −36.0000 −1.36360
$$698$$ 26.0000 0.984115
$$699$$ 0 0
$$700$$ 4.00000 0.151186
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 0 0
$$703$$ 49.0000 1.84807
$$704$$ −3.00000 −0.113067
$$705$$ 0 0
$$706$$ 6.00000 0.225813
$$707$$ −12.0000 −0.451306
$$708$$ 0 0
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 36.0000 1.35106
$$711$$ 0 0
$$712$$ 6.00000 0.224860
$$713$$ 36.0000 1.34821
$$714$$ 0 0
$$715$$ 9.00000 0.336581
$$716$$ 0 0
$$717$$ 0 0
$$718$$ −24.0000 −0.895672
$$719$$ −18.0000 −0.671287 −0.335643 0.941989i $$-0.608954\pi$$
−0.335643 + 0.941989i $$0.608954\pi$$
$$720$$ 0 0
$$721$$ −13.0000 −0.484145
$$722$$ 30.0000 1.11648
$$723$$ 0 0
$$724$$ 2.00000 0.0743294
$$725$$ 36.0000 1.33701
$$726$$ 0 0
$$727$$ −1.00000 −0.0370879 −0.0185440 0.999828i $$-0.505903\pi$$
−0.0185440 + 0.999828i $$0.505903\pi$$
$$728$$ 1.00000 0.0370625
$$729$$ 0 0
$$730$$ 21.0000 0.777245
$$731$$ −3.00000 −0.110959
$$732$$ 0 0
$$733$$ 32.0000 1.18195 0.590973 0.806691i $$-0.298744\pi$$
0.590973 + 0.806691i $$0.298744\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ −9.00000 −0.331744
$$737$$ −42.0000 −1.54709
$$738$$ 0 0
$$739$$ −34.0000 −1.25071 −0.625355 0.780340i $$-0.715046\pi$$
−0.625355 + 0.780340i $$0.715046\pi$$
$$740$$ 21.0000 0.771975
$$741$$ 0 0
$$742$$ 6.00000 0.220267
$$743$$ −30.0000 −1.10059 −0.550297 0.834969i $$-0.685485\pi$$
−0.550297 + 0.834969i $$0.685485\pi$$
$$744$$ 0 0
$$745$$ −36.0000 −1.31894
$$746$$ −4.00000 −0.146450
$$747$$ 0 0
$$748$$ −9.00000 −0.329073
$$749$$ 6.00000 0.219235
$$750$$ 0 0
$$751$$ −4.00000 −0.145962 −0.0729810 0.997333i $$-0.523251\pi$$
−0.0729810 + 0.997333i $$0.523251\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 9.00000 0.327761
$$755$$ −51.0000 −1.85608
$$756$$ 0 0
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ −16.0000 −0.581146
$$759$$ 0 0
$$760$$ 21.0000 0.761750
$$761$$ 12.0000 0.435000 0.217500 0.976060i $$-0.430210\pi$$
0.217500 + 0.976060i $$0.430210\pi$$
$$762$$ 0 0
$$763$$ 11.0000 0.398227
$$764$$ 15.0000 0.542681
$$765$$ 0 0
$$766$$ 21.0000 0.758761
$$767$$ −12.0000 −0.433295
$$768$$ 0 0
$$769$$ −13.0000 −0.468792 −0.234396 0.972141i $$-0.575311\pi$$
−0.234396 + 0.972141i $$0.575311\pi$$
$$770$$ 9.00000 0.324337
$$771$$ 0 0
$$772$$ −4.00000 −0.143963
$$773$$ −33.0000 −1.18693 −0.593464 0.804861i $$-0.702240\pi$$
−0.593464 + 0.804861i $$0.702240\pi$$
$$774$$ 0 0
$$775$$ −16.0000 −0.574737
$$776$$ −10.0000 −0.358979
$$777$$ 0 0
$$778$$ 30.0000 1.07555
$$779$$ 84.0000 3.00961
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ −27.0000 −0.965518
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ 39.0000 1.39197
$$786$$ 0 0
$$787$$ −31.0000 −1.10503 −0.552515 0.833503i $$-0.686332\pi$$
−0.552515 + 0.833503i $$0.686332\pi$$
$$788$$ 12.0000 0.427482
$$789$$ 0 0
$$790$$ 30.0000 1.06735
$$791$$ 18.0000 0.640006
$$792$$ 0 0
$$793$$ −1.00000 −0.0355110
$$794$$ −34.0000 −1.20661
$$795$$ 0 0
$$796$$ −7.00000 −0.248108
$$797$$ −36.0000 −1.27519 −0.637593 0.770374i $$-0.720070\pi$$
−0.637593 + 0.770374i $$0.720070\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 4.00000 0.141421
$$801$$ 0 0
$$802$$ −18.0000 −0.635602
$$803$$ 21.0000 0.741074
$$804$$ 0 0
$$805$$ 27.0000 0.951625
$$806$$ −4.00000 −0.140894
$$807$$ 0 0
$$808$$ −12.0000 −0.422159
$$809$$ −54.0000 −1.89854 −0.949269 0.314464i $$-0.898175\pi$$
−0.949269 + 0.314464i $$0.898175\pi$$
$$810$$ 0 0
$$811$$ −43.0000 −1.50993 −0.754967 0.655763i $$-0.772347\pi$$
−0.754967 + 0.655763i $$0.772347\pi$$
$$812$$ 9.00000 0.315838
$$813$$ 0 0
$$814$$ 21.0000 0.736050
$$815$$ −6.00000 −0.210171
$$816$$ 0 0
$$817$$ 7.00000 0.244899
$$818$$ −13.0000 −0.454534
$$819$$ 0 0
$$820$$ 36.0000 1.25717
$$821$$ −24.0000 −0.837606 −0.418803 0.908077i $$-0.637550\pi$$
−0.418803 + 0.908077i $$0.637550\pi$$
$$822$$ 0 0
$$823$$ −40.0000 −1.39431 −0.697156 0.716919i $$-0.745552\pi$$
−0.697156 + 0.716919i $$0.745552\pi$$
$$824$$ −13.0000 −0.452876
$$825$$ 0 0
$$826$$ −12.0000 −0.417533
$$827$$ −27.0000 −0.938882 −0.469441 0.882964i $$-0.655545\pi$$
−0.469441 + 0.882964i $$0.655545\pi$$
$$828$$ 0 0
$$829$$ −25.0000 −0.868286 −0.434143 0.900844i $$-0.642949\pi$$
−0.434143 + 0.900844i $$0.642949\pi$$
$$830$$ −18.0000 −0.624789
$$831$$ 0 0
$$832$$ 1.00000 0.0346688
$$833$$ 3.00000 0.103944
$$834$$ 0 0
$$835$$ 9.00000 0.311458
$$836$$ 21.0000 0.726300
$$837$$ 0 0
$$838$$ −3.00000 −0.103633
$$839$$ 12.0000 0.414286 0.207143 0.978311i $$-0.433583\pi$$
0.207143 + 0.978311i $$0.433583\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ −10.0000 −0.344623
$$843$$ 0 0
$$844$$ 5.00000 0.172107
$$845$$ −3.00000 −0.103203
$$846$$ 0 0
$$847$$ −2.00000 −0.0687208
$$848$$ 6.00000 0.206041
$$849$$ 0 0
$$850$$ 12.0000 0.411597
$$851$$ 63.0000 2.15961
$$852$$ 0 0
$$853$$ 44.0000 1.50653 0.753266 0.657716i $$-0.228477\pi$$
0.753266 + 0.657716i $$0.228477\pi$$
$$854$$ −1.00000 −0.0342193
$$855$$ 0 0
$$856$$ 6.00000 0.205076
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ 0 0
$$859$$ −4.00000 −0.136478 −0.0682391 0.997669i $$-0.521738\pi$$
−0.0682391 + 0.997669i $$0.521738\pi$$
$$860$$ 3.00000 0.102299
$$861$$ 0 0
$$862$$ −18.0000 −0.613082
$$863$$ −6.00000 −0.204242 −0.102121 0.994772i $$-0.532563\pi$$
−0.102121 + 0.994772i $$0.532563\pi$$
$$864$$ 0 0
$$865$$ −18.0000 −0.612018
$$866$$ −16.0000 −0.543702
$$867$$ 0 0
$$868$$ −4.00000 −0.135769
$$869$$ 30.0000 1.01768
$$870$$ 0 0
$$871$$ 14.0000 0.474372
$$872$$ 11.0000 0.372507
$$873$$ 0 0
$$874$$ 63.0000 2.13101
$$875$$ 3.00000 0.101419
$$876$$ 0 0
$$877$$ 14.0000 0.472746 0.236373 0.971662i $$-0.424041\pi$$
0.236373 + 0.971662i $$0.424041\pi$$
$$878$$ −1.00000 −0.0337484
$$879$$ 0 0
$$880$$ 9.00000 0.303390
$$881$$ 3.00000 0.101073 0.0505363 0.998722i $$-0.483907\pi$$
0.0505363 + 0.998722i $$0.483907\pi$$
$$882$$ 0 0
$$883$$ −25.0000 −0.841317 −0.420658 0.907219i $$-0.638201\pi$$
−0.420658 + 0.907219i $$0.638201\pi$$
$$884$$ 3.00000 0.100901
$$885$$ 0 0
$$886$$ −18.0000 −0.604722
$$887$$ −36.0000 −1.20876 −0.604381 0.796696i $$-0.706579\pi$$
−0.604381 + 0.796696i $$0.706579\pi$$
$$888$$ 0 0
$$889$$ 2.00000 0.0670778
$$890$$ −18.0000 −0.603361
$$891$$ 0 0
$$892$$ 26.0000 0.870544
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ −21.0000 −0.700779
$$899$$ −36.0000 −1.20067
$$900$$ 0 0
$$901$$ 18.0000 0.599667
$$902$$ 36.0000 1.19867
$$903$$ 0 0
$$904$$ 18.0000 0.598671
$$905$$ −6.00000 −0.199447
$$906$$ 0 0
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ 18.0000 0.597351
$$909$$ 0 0
$$910$$ −3.00000 −0.0994490
$$911$$ 33.0000 1.09334 0.546669 0.837349i $$-0.315895\pi$$
0.546669 + 0.837349i $$0.315895\pi$$
$$912$$ 0 0
$$913$$ −18.0000 −0.595713
$$914$$ −28.0000 −0.926158
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ 15.0000 0.495344
$$918$$ 0 0
$$919$$ −34.0000 −1.12156 −0.560778 0.827966i $$-0.689498\pi$$
−0.560778 + 0.827966i $$0.689498\pi$$
$$920$$ 27.0000 0.890164
$$921$$ 0 0
$$922$$ −33.0000 −1.08680
$$923$$ −12.0000 −0.394985
$$924$$ 0 0
$$925$$ −28.0000 −0.920634
$$926$$ 41.0000 1.34734
$$927$$ 0 0
$$928$$ 9.00000 0.295439
$$929$$ −12.0000 −0.393707 −0.196854 0.980433i $$-0.563072\pi$$
−0.196854 + 0.980433i $$0.563072\pi$$
$$930$$ 0 0
$$931$$ −7.00000 −0.229416
$$932$$ −24.0000 −0.786146
$$933$$ 0 0
$$934$$ 3.00000 0.0981630
$$935$$ 27.0000 0.882994
$$936$$ 0 0
$$937$$ −34.0000 −1.11073 −0.555366 0.831606i $$-0.687422\pi$$
−0.555366 + 0.831606i $$0.687422\pi$$
$$938$$ 14.0000 0.457116
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ 0 0
$$943$$ 108.000 3.51696
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ 3.00000 0.0975384
$$947$$ 57.0000 1.85225 0.926126 0.377215i $$-0.123118\pi$$
0.926126 + 0.377215i $$0.123118\pi$$
$$948$$ 0 0
$$949$$ −7.00000 −0.227230
$$950$$ −28.0000 −0.908440
$$951$$ 0 0
$$952$$ 3.00000 0.0972306
$$953$$ 24.0000 0.777436 0.388718 0.921357i $$-0.372918\pi$$
0.388718 + 0.921357i $$0.372918\pi$$
$$954$$ 0 0
$$955$$ −45.0000 −1.45617
$$956$$ −24.0000 −0.776215
$$957$$ 0 0
$$958$$ 15.0000 0.484628
$$959$$ −9.00000 −0.290625
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −7.00000 −0.225689
$$963$$ 0 0
$$964$$ −10.0000 −0.322078
$$965$$ 12.0000 0.386294
$$966$$ 0 0
$$967$$ −13.0000 −0.418052 −0.209026 0.977910i $$-0.567029\pi$$
−0.209026 + 0.977910i $$0.567029\pi$$
$$968$$ −2.00000 −0.0642824
$$969$$ 0 0
$$970$$ 30.0000 0.963242
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ −4.00000 −0.128234
$$974$$ −16.0000 −0.512673
$$975$$ 0 0
$$976$$ −1.00000 −0.0320092
$$977$$ −15.0000 −0.479893 −0.239946 0.970786i $$-0.577130\pi$$
−0.239946 + 0.970786i $$0.577130\pi$$
$$978$$ 0 0
$$979$$ −18.0000 −0.575282
$$980$$ −3.00000 −0.0958315
$$981$$ 0 0
$$982$$ −30.0000 −0.957338
$$983$$ 39.0000 1.24391 0.621953 0.783054i $$-0.286339\pi$$
0.621953 + 0.783054i $$0.286339\pi$$
$$984$$ 0 0
$$985$$ −36.0000 −1.14706
$$986$$ 27.0000 0.859855
$$987$$ 0 0
$$988$$ −7.00000 −0.222700
$$989$$ 9.00000 0.286183
$$990$$ 0 0
$$991$$ 38.0000 1.20711 0.603555 0.797321i $$-0.293750\pi$$
0.603555 + 0.797321i $$0.293750\pi$$
$$992$$ −4.00000 −0.127000
$$993$$ 0 0
$$994$$ −12.0000 −0.380617
$$995$$ 21.0000 0.665745
$$996$$ 0 0
$$997$$ −46.0000 −1.45683 −0.728417 0.685134i $$-0.759744\pi$$
−0.728417 + 0.685134i $$0.759744\pi$$
$$998$$ 32.0000 1.01294
$$999$$ 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 1638.2.a.l.1.1 1
3.2 odd 2 546.2.a.d.1.1 1
12.11 even 2 4368.2.a.l.1.1 1
21.20 even 2 3822.2.a.a.1.1 1
39.38 odd 2 7098.2.a.w.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.d.1.1 1 3.2 odd 2
1638.2.a.l.1.1 1 1.1 even 1 trivial
3822.2.a.a.1.1 1 21.20 even 2
4368.2.a.l.1.1 1 12.11 even 2
7098.2.a.w.1.1 1 39.38 odd 2