# Properties

 Label 1638.2.a.k.1.1 Level $1638$ Weight $2$ Character 1638.1 Self dual yes Analytic conductor $13.079$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 182) 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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -4.00000 q^{10} +1.00000 q^{11} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} +2.00000 q^{19} -4.00000 q^{20} +1.00000 q^{22} +7.00000 q^{23} +11.0000 q^{25} +1.00000 q^{26} -1.00000 q^{28} +8.00000 q^{29} +3.00000 q^{31} +1.00000 q^{32} -4.00000 q^{34} +4.00000 q^{35} +7.00000 q^{37} +2.00000 q^{38} -4.00000 q^{40} +7.00000 q^{41} -8.00000 q^{43} +1.00000 q^{44} +7.00000 q^{46} -3.00000 q^{47} +1.00000 q^{49} +11.0000 q^{50} +1.00000 q^{52} -4.00000 q^{55} -1.00000 q^{56} +8.00000 q^{58} +6.00000 q^{59} -13.0000 q^{61} +3.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +7.00000 q^{67} -4.00000 q^{68} +4.00000 q^{70} -4.00000 q^{71} +9.00000 q^{73} +7.00000 q^{74} +2.00000 q^{76} -1.00000 q^{77} -13.0000 q^{79} -4.00000 q^{80} +7.00000 q^{82} +16.0000 q^{83} +16.0000 q^{85} -8.00000 q^{86} +1.00000 q^{88} +6.00000 q^{89} -1.00000 q^{91} +7.00000 q^{92} -3.00000 q^{94} -8.00000 q^{95} +11.0000 q^{97} +1.00000 q^{98} +O(q^{100})$$

## 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
$$5$$ −4.00000 −1.78885 −0.894427 0.447214i $$-0.852416\pi$$
−0.894427 + 0.447214i $$0.852416\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ −4.00000 −1.26491
$$11$$ 1.00000 0.301511 0.150756 0.988571i $$-0.451829\pi$$
0.150756 + 0.988571i $$0.451829\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ −4.00000 −0.894427
$$21$$ 0 0
$$22$$ 1.00000 0.213201
$$23$$ 7.00000 1.45960 0.729800 0.683660i $$-0.239613\pi$$
0.729800 + 0.683660i $$0.239613\pi$$
$$24$$ 0 0
$$25$$ 11.0000 2.20000
$$26$$ 1.00000 0.196116
$$27$$ 0 0
$$28$$ −1.00000 −0.188982
$$29$$ 8.00000 1.48556 0.742781 0.669534i $$-0.233506\pi$$
0.742781 + 0.669534i $$0.233506\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −4.00000 −0.685994
$$35$$ 4.00000 0.676123
$$36$$ 0 0
$$37$$ 7.00000 1.15079 0.575396 0.817875i $$-0.304848\pi$$
0.575396 + 0.817875i $$0.304848\pi$$
$$38$$ 2.00000 0.324443
$$39$$ 0 0
$$40$$ −4.00000 −0.632456
$$41$$ 7.00000 1.09322 0.546608 0.837389i $$-0.315919\pi$$
0.546608 + 0.837389i $$0.315919\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 1.00000 0.150756
$$45$$ 0 0
$$46$$ 7.00000 1.03209
$$47$$ −3.00000 −0.437595 −0.218797 0.975770i $$-0.570213\pi$$
−0.218797 + 0.975770i $$0.570213\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 11.0000 1.55563
$$51$$ 0 0
$$52$$ 1.00000 0.138675
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ −4.00000 −0.539360
$$56$$ −1.00000 −0.133631
$$57$$ 0 0
$$58$$ 8.00000 1.05045
$$59$$ 6.00000 0.781133 0.390567 0.920575i $$-0.372279\pi$$
0.390567 + 0.920575i $$0.372279\pi$$
$$60$$ 0 0
$$61$$ −13.0000 −1.66448 −0.832240 0.554416i $$-0.812942\pi$$
−0.832240 + 0.554416i $$0.812942\pi$$
$$62$$ 3.00000 0.381000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −4.00000 −0.496139
$$66$$ 0 0
$$67$$ 7.00000 0.855186 0.427593 0.903971i $$-0.359362\pi$$
0.427593 + 0.903971i $$0.359362\pi$$
$$68$$ −4.00000 −0.485071
$$69$$ 0 0
$$70$$ 4.00000 0.478091
$$71$$ −4.00000 −0.474713 −0.237356 0.971423i $$-0.576281\pi$$
−0.237356 + 0.971423i $$0.576281\pi$$
$$72$$ 0 0
$$73$$ 9.00000 1.05337 0.526685 0.850060i $$-0.323435\pi$$
0.526685 + 0.850060i $$0.323435\pi$$
$$74$$ 7.00000 0.813733
$$75$$ 0 0
$$76$$ 2.00000 0.229416
$$77$$ −1.00000 −0.113961
$$78$$ 0 0
$$79$$ −13.0000 −1.46261 −0.731307 0.682048i $$-0.761089\pi$$
−0.731307 + 0.682048i $$0.761089\pi$$
$$80$$ −4.00000 −0.447214
$$81$$ 0 0
$$82$$ 7.00000 0.773021
$$83$$ 16.0000 1.75623 0.878114 0.478451i $$-0.158802\pi$$
0.878114 + 0.478451i $$0.158802\pi$$
$$84$$ 0 0
$$85$$ 16.0000 1.73544
$$86$$ −8.00000 −0.862662
$$87$$ 0 0
$$88$$ 1.00000 0.106600
$$89$$ 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
$$92$$ 7.00000 0.729800
$$93$$ 0 0
$$94$$ −3.00000 −0.309426
$$95$$ −8.00000 −0.820783
$$96$$ 0 0
$$97$$ 11.0000 1.11688 0.558440 0.829545i $$-0.311400\pi$$
0.558440 + 0.829545i $$0.311400\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ 11.0000 1.10000
$$101$$ −9.00000 −0.895533 −0.447767 0.894150i $$-0.647781\pi$$
−0.447767 + 0.894150i $$0.647781\pi$$
$$102$$ 0 0
$$103$$ 10.0000 0.985329 0.492665 0.870219i $$-0.336023\pi$$
0.492665 + 0.870219i $$0.336023\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ −4.00000 −0.381385
$$111$$ 0 0
$$112$$ −1.00000 −0.0944911
$$113$$ −1.00000 −0.0940721 −0.0470360 0.998893i $$-0.514978\pi$$
−0.0470360 + 0.998893i $$0.514978\pi$$
$$114$$ 0 0
$$115$$ −28.0000 −2.61101
$$116$$ 8.00000 0.742781
$$117$$ 0 0
$$118$$ 6.00000 0.552345
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ −13.0000 −1.17696
$$123$$ 0 0
$$124$$ 3.00000 0.269408
$$125$$ −24.0000 −2.14663
$$126$$ 0 0
$$127$$ 13.0000 1.15356 0.576782 0.816898i $$-0.304308\pi$$
0.576782 + 0.816898i $$0.304308\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ −4.00000 −0.350823
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ −2.00000 −0.173422
$$134$$ 7.00000 0.604708
$$135$$ 0 0
$$136$$ −4.00000 −0.342997
$$137$$ 14.0000 1.19610 0.598050 0.801459i $$-0.295942\pi$$
0.598050 + 0.801459i $$0.295942\pi$$
$$138$$ 0 0
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ 4.00000 0.338062
$$141$$ 0 0
$$142$$ −4.00000 −0.335673
$$143$$ 1.00000 0.0836242
$$144$$ 0 0
$$145$$ −32.0000 −2.65746
$$146$$ 9.00000 0.744845
$$147$$ 0 0
$$148$$ 7.00000 0.575396
$$149$$ −15.0000 −1.22885 −0.614424 0.788976i $$-0.710612\pi$$
−0.614424 + 0.788976i $$0.710612\pi$$
$$150$$ 0 0
$$151$$ −4.00000 −0.325515 −0.162758 0.986666i $$-0.552039\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ 2.00000 0.162221
$$153$$ 0 0
$$154$$ −1.00000 −0.0805823
$$155$$ −12.0000 −0.963863
$$156$$ 0 0
$$157$$ 7.00000 0.558661 0.279330 0.960195i $$-0.409888\pi$$
0.279330 + 0.960195i $$0.409888\pi$$
$$158$$ −13.0000 −1.03422
$$159$$ 0 0
$$160$$ −4.00000 −0.316228
$$161$$ −7.00000 −0.551677
$$162$$ 0 0
$$163$$ 12.0000 0.939913 0.469956 0.882690i $$-0.344270\pi$$
0.469956 + 0.882690i $$0.344270\pi$$
$$164$$ 7.00000 0.546608
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 16.0000 1.22714
$$171$$ 0 0
$$172$$ −8.00000 −0.609994
$$173$$ −14.0000 −1.06440 −0.532200 0.846619i $$-0.678635\pi$$
−0.532200 + 0.846619i $$0.678635\pi$$
$$174$$ 0 0
$$175$$ −11.0000 −0.831522
$$176$$ 1.00000 0.0753778
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ 18.0000 1.34538 0.672692 0.739923i $$-0.265138\pi$$
0.672692 + 0.739923i $$0.265138\pi$$
$$180$$ 0 0
$$181$$ 13.0000 0.966282 0.483141 0.875542i $$-0.339496\pi$$
0.483141 + 0.875542i $$0.339496\pi$$
$$182$$ −1.00000 −0.0741249
$$183$$ 0 0
$$184$$ 7.00000 0.516047
$$185$$ −28.0000 −2.05860
$$186$$ 0 0
$$187$$ −4.00000 −0.292509
$$188$$ −3.00000 −0.218797
$$189$$ 0 0
$$190$$ −8.00000 −0.580381
$$191$$ −24.0000 −1.73658 −0.868290 0.496058i $$-0.834780\pi$$
−0.868290 + 0.496058i $$0.834780\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 11.0000 0.789754
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ −3.00000 −0.213741 −0.106871 0.994273i $$-0.534083\pi$$
−0.106871 + 0.994273i $$0.534083\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 11.0000 0.777817
$$201$$ 0 0
$$202$$ −9.00000 −0.633238
$$203$$ −8.00000 −0.561490
$$204$$ 0 0
$$205$$ −28.0000 −1.95560
$$206$$ 10.0000 0.696733
$$207$$ 0 0
$$208$$ 1.00000 0.0693375
$$209$$ 2.00000 0.138343
$$210$$ 0 0
$$211$$ 22.0000 1.51454 0.757271 0.653101i $$-0.226532\pi$$
0.757271 + 0.653101i $$0.226532\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ 32.0000 2.18238
$$216$$ 0 0
$$217$$ −3.00000 −0.203653
$$218$$ 14.0000 0.948200
$$219$$ 0 0
$$220$$ −4.00000 −0.269680
$$221$$ −4.00000 −0.269069
$$222$$ 0 0
$$223$$ 9.00000 0.602685 0.301342 0.953516i $$-0.402565\pi$$
0.301342 + 0.953516i $$0.402565\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ −1.00000 −0.0665190
$$227$$ −28.0000 −1.85843 −0.929213 0.369546i $$-0.879513\pi$$
−0.929213 + 0.369546i $$0.879513\pi$$
$$228$$ 0 0
$$229$$ −20.0000 −1.32164 −0.660819 0.750546i $$-0.729791\pi$$
−0.660819 + 0.750546i $$0.729791\pi$$
$$230$$ −28.0000 −1.84627
$$231$$ 0 0
$$232$$ 8.00000 0.525226
$$233$$ 13.0000 0.851658 0.425829 0.904804i $$-0.359982\pi$$
0.425829 + 0.904804i $$0.359982\pi$$
$$234$$ 0 0
$$235$$ 12.0000 0.782794
$$236$$ 6.00000 0.390567
$$237$$ 0 0
$$238$$ 4.00000 0.259281
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ −10.0000 −0.642824
$$243$$ 0 0
$$244$$ −13.0000 −0.832240
$$245$$ −4.00000 −0.255551
$$246$$ 0 0
$$247$$ 2.00000 0.127257
$$248$$ 3.00000 0.190500
$$249$$ 0 0
$$250$$ −24.0000 −1.51789
$$251$$ −17.0000 −1.07303 −0.536515 0.843891i $$-0.680260\pi$$
−0.536515 + 0.843891i $$0.680260\pi$$
$$252$$ 0 0
$$253$$ 7.00000 0.440086
$$254$$ 13.0000 0.815693
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 12.0000 0.748539 0.374270 0.927320i $$-0.377893\pi$$
0.374270 + 0.927320i $$0.377893\pi$$
$$258$$ 0 0
$$259$$ −7.00000 −0.434959
$$260$$ −4.00000 −0.248069
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −2.00000 −0.122628
$$267$$ 0 0
$$268$$ 7.00000 0.427593
$$269$$ −3.00000 −0.182913 −0.0914566 0.995809i $$-0.529152\pi$$
−0.0914566 + 0.995809i $$0.529152\pi$$
$$270$$ 0 0
$$271$$ −3.00000 −0.182237 −0.0911185 0.995840i $$-0.529044\pi$$
−0.0911185 + 0.995840i $$0.529044\pi$$
$$272$$ −4.00000 −0.242536
$$273$$ 0 0
$$274$$ 14.0000 0.845771
$$275$$ 11.0000 0.663325
$$276$$ 0 0
$$277$$ −14.0000 −0.841178 −0.420589 0.907251i $$-0.638177\pi$$
−0.420589 + 0.907251i $$0.638177\pi$$
$$278$$ −20.0000 −1.19952
$$279$$ 0 0
$$280$$ 4.00000 0.239046
$$281$$ 20.0000 1.19310 0.596550 0.802576i $$-0.296538\pi$$
0.596550 + 0.802576i $$0.296538\pi$$
$$282$$ 0 0
$$283$$ 17.0000 1.01055 0.505273 0.862960i $$-0.331392\pi$$
0.505273 + 0.862960i $$0.331392\pi$$
$$284$$ −4.00000 −0.237356
$$285$$ 0 0
$$286$$ 1.00000 0.0591312
$$287$$ −7.00000 −0.413197
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ −32.0000 −1.87910
$$291$$ 0 0
$$292$$ 9.00000 0.526685
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ 0 0
$$295$$ −24.0000 −1.39733
$$296$$ 7.00000 0.406867
$$297$$ 0 0
$$298$$ −15.0000 −0.868927
$$299$$ 7.00000 0.404820
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ −4.00000 −0.230174
$$303$$ 0 0
$$304$$ 2.00000 0.114708
$$305$$ 52.0000 2.97751
$$306$$ 0 0
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ −1.00000 −0.0569803
$$309$$ 0 0
$$310$$ −12.0000 −0.681554
$$311$$ 30.0000 1.70114 0.850572 0.525859i $$-0.176256\pi$$
0.850572 + 0.525859i $$0.176256\pi$$
$$312$$ 0 0
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ 7.00000 0.395033
$$315$$ 0 0
$$316$$ −13.0000 −0.731307
$$317$$ 13.0000 0.730153 0.365076 0.930978i $$-0.381043\pi$$
0.365076 + 0.930978i $$0.381043\pi$$
$$318$$ 0 0
$$319$$ 8.00000 0.447914
$$320$$ −4.00000 −0.223607
$$321$$ 0 0
$$322$$ −7.00000 −0.390095
$$323$$ −8.00000 −0.445132
$$324$$ 0 0
$$325$$ 11.0000 0.610170
$$326$$ 12.0000 0.664619
$$327$$ 0 0
$$328$$ 7.00000 0.386510
$$329$$ 3.00000 0.165395
$$330$$ 0 0
$$331$$ 15.0000 0.824475 0.412237 0.911077i $$-0.364747\pi$$
0.412237 + 0.911077i $$0.364747\pi$$
$$332$$ 16.0000 0.878114
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −28.0000 −1.52980
$$336$$ 0 0
$$337$$ 9.00000 0.490261 0.245131 0.969490i $$-0.421169\pi$$
0.245131 + 0.969490i $$0.421169\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ 0 0
$$340$$ 16.0000 0.867722
$$341$$ 3.00000 0.162459
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ −14.0000 −0.752645
$$347$$ 32.0000 1.71785 0.858925 0.512101i $$-0.171133\pi$$
0.858925 + 0.512101i $$0.171133\pi$$
$$348$$ 0 0
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ −11.0000 −0.587975
$$351$$ 0 0
$$352$$ 1.00000 0.0533002
$$353$$ 11.0000 0.585471 0.292735 0.956193i $$-0.405434\pi$$
0.292735 + 0.956193i $$0.405434\pi$$
$$354$$ 0 0
$$355$$ 16.0000 0.849192
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ 18.0000 0.951330
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 13.0000 0.683265
$$363$$ 0 0
$$364$$ −1.00000 −0.0524142
$$365$$ −36.0000 −1.88433
$$366$$ 0 0
$$367$$ 12.0000 0.626395 0.313197 0.949688i $$-0.398600\pi$$
0.313197 + 0.949688i $$0.398600\pi$$
$$368$$ 7.00000 0.364900
$$369$$ 0 0
$$370$$ −28.0000 −1.45565
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −32.0000 −1.65690 −0.828449 0.560065i $$-0.810776\pi$$
−0.828449 + 0.560065i $$0.810776\pi$$
$$374$$ −4.00000 −0.206835
$$375$$ 0 0
$$376$$ −3.00000 −0.154713
$$377$$ 8.00000 0.412021
$$378$$ 0 0
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ −8.00000 −0.410391
$$381$$ 0 0
$$382$$ −24.0000 −1.22795
$$383$$ −21.0000 −1.07305 −0.536525 0.843884i $$-0.680263\pi$$
−0.536525 + 0.843884i $$0.680263\pi$$
$$384$$ 0 0
$$385$$ 4.00000 0.203859
$$386$$ −4.00000 −0.203595
$$387$$ 0 0
$$388$$ 11.0000 0.558440
$$389$$ −8.00000 −0.405616 −0.202808 0.979219i $$-0.565007\pi$$
−0.202808 + 0.979219i $$0.565007\pi$$
$$390$$ 0 0
$$391$$ −28.0000 −1.41602
$$392$$ 1.00000 0.0505076
$$393$$ 0 0
$$394$$ −3.00000 −0.151138
$$395$$ 52.0000 2.61640
$$396$$ 0 0
$$397$$ 8.00000 0.401508 0.200754 0.979642i $$-0.435661\pi$$
0.200754 + 0.979642i $$0.435661\pi$$
$$398$$ 16.0000 0.802008
$$399$$ 0 0
$$400$$ 11.0000 0.550000
$$401$$ −8.00000 −0.399501 −0.199750 0.979847i $$-0.564013\pi$$
−0.199750 + 0.979847i $$0.564013\pi$$
$$402$$ 0 0
$$403$$ 3.00000 0.149441
$$404$$ −9.00000 −0.447767
$$405$$ 0 0
$$406$$ −8.00000 −0.397033
$$407$$ 7.00000 0.346977
$$408$$ 0 0
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ −28.0000 −1.38282
$$411$$ 0 0
$$412$$ 10.0000 0.492665
$$413$$ −6.00000 −0.295241
$$414$$ 0 0
$$415$$ −64.0000 −3.14164
$$416$$ 1.00000 0.0490290
$$417$$ 0 0
$$418$$ 2.00000 0.0978232
$$419$$ 19.0000 0.928211 0.464105 0.885780i $$-0.346376\pi$$
0.464105 + 0.885780i $$0.346376\pi$$
$$420$$ 0 0
$$421$$ −11.0000 −0.536107 −0.268054 0.963404i $$-0.586380\pi$$
−0.268054 + 0.963404i $$0.586380\pi$$
$$422$$ 22.0000 1.07094
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −44.0000 −2.13431
$$426$$ 0 0
$$427$$ 13.0000 0.629114
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ 32.0000 1.54318
$$431$$ −2.00000 −0.0963366 −0.0481683 0.998839i $$-0.515338\pi$$
−0.0481683 + 0.998839i $$0.515338\pi$$
$$432$$ 0 0
$$433$$ −18.0000 −0.865025 −0.432512 0.901628i $$-0.642373\pi$$
−0.432512 + 0.901628i $$0.642373\pi$$
$$434$$ −3.00000 −0.144005
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ 14.0000 0.669711
$$438$$ 0 0
$$439$$ −2.00000 −0.0954548 −0.0477274 0.998860i $$-0.515198\pi$$
−0.0477274 + 0.998860i $$0.515198\pi$$
$$440$$ −4.00000 −0.190693
$$441$$ 0 0
$$442$$ −4.00000 −0.190261
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ −24.0000 −1.13771
$$446$$ 9.00000 0.426162
$$447$$ 0 0
$$448$$ −1.00000 −0.0472456
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ 7.00000 0.329617
$$452$$ −1.00000 −0.0470360
$$453$$ 0 0
$$454$$ −28.0000 −1.31411
$$455$$ 4.00000 0.187523
$$456$$ 0 0
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ −20.0000 −0.934539
$$459$$ 0 0
$$460$$ −28.0000 −1.30551
$$461$$ −16.0000 −0.745194 −0.372597 0.927993i $$-0.621533\pi$$
−0.372597 + 0.927993i $$0.621533\pi$$
$$462$$ 0 0
$$463$$ −12.0000 −0.557687 −0.278844 0.960337i $$-0.589951\pi$$
−0.278844 + 0.960337i $$0.589951\pi$$
$$464$$ 8.00000 0.371391
$$465$$ 0 0
$$466$$ 13.0000 0.602213
$$467$$ 20.0000 0.925490 0.462745 0.886492i $$-0.346865\pi$$
0.462745 + 0.886492i $$0.346865\pi$$
$$468$$ 0 0
$$469$$ −7.00000 −0.323230
$$470$$ 12.0000 0.553519
$$471$$ 0 0
$$472$$ 6.00000 0.276172
$$473$$ −8.00000 −0.367840
$$474$$ 0 0
$$475$$ 22.0000 1.00943
$$476$$ 4.00000 0.183340
$$477$$ 0 0
$$478$$ 6.00000 0.274434
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 7.00000 0.319173
$$482$$ −10.0000 −0.455488
$$483$$ 0 0
$$484$$ −10.0000 −0.454545
$$485$$ −44.0000 −1.99794
$$486$$ 0 0
$$487$$ 26.0000 1.17817 0.589086 0.808070i $$-0.299488\pi$$
0.589086 + 0.808070i $$0.299488\pi$$
$$488$$ −13.0000 −0.588482
$$489$$ 0 0
$$490$$ −4.00000 −0.180702
$$491$$ −40.0000 −1.80517 −0.902587 0.430507i $$-0.858335\pi$$
−0.902587 + 0.430507i $$0.858335\pi$$
$$492$$ 0 0
$$493$$ −32.0000 −1.44121
$$494$$ 2.00000 0.0899843
$$495$$ 0 0
$$496$$ 3.00000 0.134704
$$497$$ 4.00000 0.179425
$$498$$ 0 0
$$499$$ 37.0000 1.65635 0.828174 0.560471i $$-0.189380\pi$$
0.828174 + 0.560471i $$0.189380\pi$$
$$500$$ −24.0000 −1.07331
$$501$$ 0 0
$$502$$ −17.0000 −0.758747
$$503$$ 16.0000 0.713405 0.356702 0.934218i $$-0.383901\pi$$
0.356702 + 0.934218i $$0.383901\pi$$
$$504$$ 0 0
$$505$$ 36.0000 1.60198
$$506$$ 7.00000 0.311188
$$507$$ 0 0
$$508$$ 13.0000 0.576782
$$509$$ 18.0000 0.797836 0.398918 0.916987i $$-0.369386\pi$$
0.398918 + 0.916987i $$0.369386\pi$$
$$510$$ 0 0
$$511$$ −9.00000 −0.398137
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 12.0000 0.529297
$$515$$ −40.0000 −1.76261
$$516$$ 0 0
$$517$$ −3.00000 −0.131940
$$518$$ −7.00000 −0.307562
$$519$$ 0 0
$$520$$ −4.00000 −0.175412
$$521$$ 30.0000 1.31432 0.657162 0.753749i $$-0.271757\pi$$
0.657162 + 0.753749i $$0.271757\pi$$
$$522$$ 0 0
$$523$$ −15.0000 −0.655904 −0.327952 0.944694i $$-0.606358\pi$$
−0.327952 + 0.944694i $$0.606358\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ −12.0000 −0.522728
$$528$$ 0 0
$$529$$ 26.0000 1.13043
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −2.00000 −0.0867110
$$533$$ 7.00000 0.303204
$$534$$ 0 0
$$535$$ 48.0000 2.07522
$$536$$ 7.00000 0.302354
$$537$$ 0 0
$$538$$ −3.00000 −0.129339
$$539$$ 1.00000 0.0430730
$$540$$ 0 0
$$541$$ −38.0000 −1.63375 −0.816874 0.576816i $$-0.804295\pi$$
−0.816874 + 0.576816i $$0.804295\pi$$
$$542$$ −3.00000 −0.128861
$$543$$ 0 0
$$544$$ −4.00000 −0.171499
$$545$$ −56.0000 −2.39878
$$546$$ 0 0
$$547$$ −4.00000 −0.171028 −0.0855138 0.996337i $$-0.527253\pi$$
−0.0855138 + 0.996337i $$0.527253\pi$$
$$548$$ 14.0000 0.598050
$$549$$ 0 0
$$550$$ 11.0000 0.469042
$$551$$ 16.0000 0.681623
$$552$$ 0 0
$$553$$ 13.0000 0.552816
$$554$$ −14.0000 −0.594803
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ −31.0000 −1.31351 −0.656756 0.754103i $$-0.728072\pi$$
−0.656756 + 0.754103i $$0.728072\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 4.00000 0.169031
$$561$$ 0 0
$$562$$ 20.0000 0.843649
$$563$$ 31.0000 1.30649 0.653247 0.757145i $$-0.273406\pi$$
0.653247 + 0.757145i $$0.273406\pi$$
$$564$$ 0 0
$$565$$ 4.00000 0.168281
$$566$$ 17.0000 0.714563
$$567$$ 0 0
$$568$$ −4.00000 −0.167836
$$569$$ −29.0000 −1.21574 −0.607872 0.794035i $$-0.707976\pi$$
−0.607872 + 0.794035i $$0.707976\pi$$
$$570$$ 0 0
$$571$$ 26.0000 1.08807 0.544033 0.839064i $$-0.316897\pi$$
0.544033 + 0.839064i $$0.316897\pi$$
$$572$$ 1.00000 0.0418121
$$573$$ 0 0
$$574$$ −7.00000 −0.292174
$$575$$ 77.0000 3.21112
$$576$$ 0 0
$$577$$ −18.0000 −0.749350 −0.374675 0.927156i $$-0.622246\pi$$
−0.374675 + 0.927156i $$0.622246\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 0 0
$$580$$ −32.0000 −1.32873
$$581$$ −16.0000 −0.663792
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 9.00000 0.372423
$$585$$ 0 0
$$586$$ −14.0000 −0.578335
$$587$$ 18.0000 0.742940 0.371470 0.928445i $$-0.378854\pi$$
0.371470 + 0.928445i $$0.378854\pi$$
$$588$$ 0 0
$$589$$ 6.00000 0.247226
$$590$$ −24.0000 −0.988064
$$591$$ 0 0
$$592$$ 7.00000 0.287698
$$593$$ 10.0000 0.410651 0.205325 0.978694i $$-0.434175\pi$$
0.205325 + 0.978694i $$0.434175\pi$$
$$594$$ 0 0
$$595$$ −16.0000 −0.655936
$$596$$ −15.0000 −0.614424
$$597$$ 0 0
$$598$$ 7.00000 0.286251
$$599$$ −27.0000 −1.10319 −0.551595 0.834112i $$-0.685981\pi$$
−0.551595 + 0.834112i $$0.685981\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 8.00000 0.326056
$$603$$ 0 0
$$604$$ −4.00000 −0.162758
$$605$$ 40.0000 1.62623
$$606$$ 0 0
$$607$$ 14.0000 0.568242 0.284121 0.958788i $$-0.408298\pi$$
0.284121 + 0.958788i $$0.408298\pi$$
$$608$$ 2.00000 0.0811107
$$609$$ 0 0
$$610$$ 52.0000 2.10542
$$611$$ −3.00000 −0.121367
$$612$$ 0 0
$$613$$ 17.0000 0.686624 0.343312 0.939222i $$-0.388451\pi$$
0.343312 + 0.939222i $$0.388451\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ −1.00000 −0.0402911
$$617$$ 38.0000 1.52982 0.764911 0.644136i $$-0.222783\pi$$
0.764911 + 0.644136i $$0.222783\pi$$
$$618$$ 0 0
$$619$$ 14.0000 0.562708 0.281354 0.959604i $$-0.409217\pi$$
0.281354 + 0.959604i $$0.409217\pi$$
$$620$$ −12.0000 −0.481932
$$621$$ 0 0
$$622$$ 30.0000 1.20289
$$623$$ −6.00000 −0.240385
$$624$$ 0 0
$$625$$ 41.0000 1.64000
$$626$$ 6.00000 0.239808
$$627$$ 0 0
$$628$$ 7.00000 0.279330
$$629$$ −28.0000 −1.11643
$$630$$ 0 0
$$631$$ 22.0000 0.875806 0.437903 0.899022i $$-0.355721\pi$$
0.437903 + 0.899022i $$0.355721\pi$$
$$632$$ −13.0000 −0.517112
$$633$$ 0 0
$$634$$ 13.0000 0.516296
$$635$$ −52.0000 −2.06356
$$636$$ 0 0
$$637$$ 1.00000 0.0396214
$$638$$ 8.00000 0.316723
$$639$$ 0 0
$$640$$ −4.00000 −0.158114
$$641$$ 25.0000 0.987441 0.493720 0.869621i $$-0.335637\pi$$
0.493720 + 0.869621i $$0.335637\pi$$
$$642$$ 0 0
$$643$$ −28.0000 −1.10421 −0.552106 0.833774i $$-0.686176\pi$$
−0.552106 + 0.833774i $$0.686176\pi$$
$$644$$ −7.00000 −0.275839
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ 48.0000 1.88707 0.943537 0.331266i $$-0.107476\pi$$
0.943537 + 0.331266i $$0.107476\pi$$
$$648$$ 0 0
$$649$$ 6.00000 0.235521
$$650$$ 11.0000 0.431455
$$651$$ 0 0
$$652$$ 12.0000 0.469956
$$653$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 7.00000 0.273304
$$657$$ 0 0
$$658$$ 3.00000 0.116952
$$659$$ −6.00000 −0.233727 −0.116863 0.993148i $$-0.537284\pi$$
−0.116863 + 0.993148i $$0.537284\pi$$
$$660$$ 0 0
$$661$$ −8.00000 −0.311164 −0.155582 0.987823i $$-0.549725\pi$$
−0.155582 + 0.987823i $$0.549725\pi$$
$$662$$ 15.0000 0.582992
$$663$$ 0 0
$$664$$ 16.0000 0.620920
$$665$$ 8.00000 0.310227
$$666$$ 0 0
$$667$$ 56.0000 2.16833
$$668$$ 0 0
$$669$$ 0 0
$$670$$ −28.0000 −1.08173
$$671$$ −13.0000 −0.501859
$$672$$ 0 0
$$673$$ −33.0000 −1.27206 −0.636028 0.771666i $$-0.719424\pi$$
−0.636028 + 0.771666i $$0.719424\pi$$
$$674$$ 9.00000 0.346667
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ −13.0000 −0.499631 −0.249815 0.968294i $$-0.580370\pi$$
−0.249815 + 0.968294i $$0.580370\pi$$
$$678$$ 0 0
$$679$$ −11.0000 −0.422141
$$680$$ 16.0000 0.613572
$$681$$ 0 0
$$682$$ 3.00000 0.114876
$$683$$ −31.0000 −1.18618 −0.593091 0.805135i $$-0.702093\pi$$
−0.593091 + 0.805135i $$0.702093\pi$$
$$684$$ 0 0
$$685$$ −56.0000 −2.13965
$$686$$ −1.00000 −0.0381802
$$687$$ 0 0
$$688$$ −8.00000 −0.304997
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ −14.0000 −0.532200
$$693$$ 0 0
$$694$$ 32.0000 1.21470
$$695$$ 80.0000 3.03457
$$696$$ 0 0
$$697$$ −28.0000 −1.06058
$$698$$ 2.00000 0.0757011
$$699$$ 0 0
$$700$$ −11.0000 −0.415761
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 14.0000 0.528020
$$704$$ 1.00000 0.0376889
$$705$$ 0 0
$$706$$ 11.0000 0.413990
$$707$$ 9.00000 0.338480
$$708$$ 0 0
$$709$$ −51.0000 −1.91535 −0.957673 0.287860i $$-0.907056\pi$$
−0.957673 + 0.287860i $$0.907056\pi$$
$$710$$ 16.0000 0.600469
$$711$$ 0 0
$$712$$ 6.00000 0.224860
$$713$$ 21.0000 0.786456
$$714$$ 0 0
$$715$$ −4.00000 −0.149592
$$716$$ 18.0000 0.672692
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 22.0000 0.820462 0.410231 0.911982i $$-0.365448\pi$$
0.410231 + 0.911982i $$0.365448\pi$$
$$720$$ 0 0
$$721$$ −10.0000 −0.372419
$$722$$ −15.0000 −0.558242
$$723$$ 0 0
$$724$$ 13.0000 0.483141
$$725$$ 88.0000 3.26824
$$726$$ 0 0
$$727$$ 18.0000 0.667583 0.333792 0.942647i $$-0.391672\pi$$
0.333792 + 0.942647i $$0.391672\pi$$
$$728$$ −1.00000 −0.0370625
$$729$$ 0 0
$$730$$ −36.0000 −1.33242
$$731$$ 32.0000 1.18356
$$732$$ 0 0
$$733$$ −12.0000 −0.443230 −0.221615 0.975134i $$-0.571133\pi$$
−0.221615 + 0.975134i $$0.571133\pi$$
$$734$$ 12.0000 0.442928
$$735$$ 0 0
$$736$$ 7.00000 0.258023
$$737$$ 7.00000 0.257848
$$738$$ 0 0
$$739$$ −8.00000 −0.294285 −0.147142 0.989115i $$-0.547008\pi$$
−0.147142 + 0.989115i $$0.547008\pi$$
$$740$$ −28.0000 −1.02930
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 44.0000 1.61420 0.807102 0.590412i $$-0.201035\pi$$
0.807102 + 0.590412i $$0.201035\pi$$
$$744$$ 0 0
$$745$$ 60.0000 2.19823
$$746$$ −32.0000 −1.17160
$$747$$ 0 0
$$748$$ −4.00000 −0.146254
$$749$$ 12.0000 0.438470
$$750$$ 0 0
$$751$$ 37.0000 1.35015 0.675075 0.737749i $$-0.264111\pi$$
0.675075 + 0.737749i $$0.264111\pi$$
$$752$$ −3.00000 −0.109399
$$753$$ 0 0
$$754$$ 8.00000 0.291343
$$755$$ 16.0000 0.582300
$$756$$ 0 0
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ −8.00000 −0.290573
$$759$$ 0 0
$$760$$ −8.00000 −0.290191
$$761$$ −1.00000 −0.0362500 −0.0181250 0.999836i $$-0.505770\pi$$
−0.0181250 + 0.999836i $$0.505770\pi$$
$$762$$ 0 0
$$763$$ −14.0000 −0.506834
$$764$$ −24.0000 −0.868290
$$765$$ 0 0
$$766$$ −21.0000 −0.758761
$$767$$ 6.00000 0.216647
$$768$$ 0 0
$$769$$ 25.0000 0.901523 0.450762 0.892644i $$-0.351152\pi$$
0.450762 + 0.892644i $$0.351152\pi$$
$$770$$ 4.00000 0.144150
$$771$$ 0 0
$$772$$ −4.00000 −0.143963
$$773$$ −38.0000 −1.36677 −0.683383 0.730061i $$-0.739492\pi$$
−0.683383 + 0.730061i $$0.739492\pi$$
$$774$$ 0 0
$$775$$ 33.0000 1.18539
$$776$$ 11.0000 0.394877
$$777$$ 0 0
$$778$$ −8.00000 −0.286814
$$779$$ 14.0000 0.501602
$$780$$ 0 0
$$781$$ −4.00000 −0.143131
$$782$$ −28.0000 −1.00128
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ −28.0000 −0.999363
$$786$$ 0 0
$$787$$ −40.0000 −1.42585 −0.712923 0.701242i $$-0.752629\pi$$
−0.712923 + 0.701242i $$0.752629\pi$$
$$788$$ −3.00000 −0.106871
$$789$$ 0 0
$$790$$ 52.0000 1.85008
$$791$$ 1.00000 0.0355559
$$792$$ 0 0
$$793$$ −13.0000 −0.461644
$$794$$ 8.00000 0.283909
$$795$$ 0 0
$$796$$ 16.0000 0.567105
$$797$$ −21.0000 −0.743858 −0.371929 0.928261i $$-0.621304\pi$$
−0.371929 + 0.928261i $$0.621304\pi$$
$$798$$ 0 0
$$799$$ 12.0000 0.424529
$$800$$ 11.0000 0.388909
$$801$$ 0 0
$$802$$ −8.00000 −0.282490
$$803$$ 9.00000 0.317603
$$804$$ 0 0
$$805$$ 28.0000 0.986870
$$806$$ 3.00000 0.105670
$$807$$ 0 0
$$808$$ −9.00000 −0.316619
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ −8.00000 −0.280745
$$813$$ 0 0
$$814$$ 7.00000 0.245350
$$815$$ −48.0000 −1.68137
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ −10.0000 −0.349642
$$819$$ 0 0
$$820$$ −28.0000 −0.977802
$$821$$ −46.0000 −1.60541 −0.802706 0.596376i $$-0.796607\pi$$
−0.802706 + 0.596376i $$0.796607\pi$$
$$822$$ 0 0
$$823$$ −35.0000 −1.22002 −0.610012 0.792392i $$-0.708835\pi$$
−0.610012 + 0.792392i $$0.708835\pi$$
$$824$$ 10.0000 0.348367
$$825$$ 0 0
$$826$$ −6.00000 −0.208767
$$827$$ −4.00000 −0.139094 −0.0695468 0.997579i $$-0.522155\pi$$
−0.0695468 + 0.997579i $$0.522155\pi$$
$$828$$ 0 0
$$829$$ −18.0000 −0.625166 −0.312583 0.949890i $$-0.601194\pi$$
−0.312583 + 0.949890i $$0.601194\pi$$
$$830$$ −64.0000 −2.22147
$$831$$ 0 0
$$832$$ 1.00000 0.0346688
$$833$$ −4.00000 −0.138592
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 2.00000 0.0691714
$$837$$ 0 0
$$838$$ 19.0000 0.656344
$$839$$ −21.0000 −0.725001 −0.362500 0.931984i $$-0.618077\pi$$
−0.362500 + 0.931984i $$0.618077\pi$$
$$840$$ 0 0
$$841$$ 35.0000 1.20690
$$842$$ −11.0000 −0.379085
$$843$$ 0 0
$$844$$ 22.0000 0.757271
$$845$$ −4.00000 −0.137604
$$846$$ 0 0
$$847$$ 10.0000 0.343604
$$848$$ 0 0
$$849$$ 0 0
$$850$$ −44.0000 −1.50919
$$851$$ 49.0000 1.67970
$$852$$ 0 0
$$853$$ −44.0000 −1.50653 −0.753266 0.657716i $$-0.771523\pi$$
−0.753266 + 0.657716i $$0.771523\pi$$
$$854$$ 13.0000 0.444851
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ 0 0
$$859$$ 31.0000 1.05771 0.528853 0.848713i $$-0.322622\pi$$
0.528853 + 0.848713i $$0.322622\pi$$
$$860$$ 32.0000 1.09119
$$861$$ 0 0
$$862$$ −2.00000 −0.0681203
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 0 0
$$865$$ 56.0000 1.90406
$$866$$ −18.0000 −0.611665
$$867$$ 0 0
$$868$$ −3.00000 −0.101827
$$869$$ −13.0000 −0.440995
$$870$$ 0 0
$$871$$ 7.00000 0.237186
$$872$$ 14.0000 0.474100
$$873$$ 0 0
$$874$$ 14.0000 0.473557
$$875$$ 24.0000 0.811348
$$876$$ 0 0
$$877$$ 29.0000 0.979260 0.489630 0.871930i $$-0.337132\pi$$
0.489630 + 0.871930i $$0.337132\pi$$
$$878$$ −2.00000 −0.0674967
$$879$$ 0 0
$$880$$ −4.00000 −0.134840
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ 0 0
$$883$$ −34.0000 −1.14419 −0.572096 0.820187i $$-0.693869\pi$$
−0.572096 + 0.820187i $$0.693869\pi$$
$$884$$ −4.00000 −0.134535
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −14.0000 −0.470074 −0.235037 0.971986i $$-0.575521\pi$$
−0.235037 + 0.971986i $$0.575521\pi$$
$$888$$ 0 0
$$889$$ −13.0000 −0.436006
$$890$$ −24.0000 −0.804482
$$891$$ 0 0
$$892$$ 9.00000 0.301342
$$893$$ −6.00000 −0.200782
$$894$$ 0 0
$$895$$ −72.0000 −2.40669
$$896$$ −1.00000 −0.0334077
$$897$$ 0 0
$$898$$ −6.00000 −0.200223
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 7.00000 0.233075
$$903$$ 0 0
$$904$$ −1.00000 −0.0332595
$$905$$ −52.0000 −1.72854
$$906$$ 0 0
$$907$$ 54.0000 1.79304 0.896520 0.443003i $$-0.146087\pi$$
0.896520 + 0.443003i $$0.146087\pi$$
$$908$$ −28.0000 −0.929213
$$909$$ 0 0
$$910$$ 4.00000 0.132599
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 0 0
$$913$$ 16.0000 0.529523
$$914$$ 18.0000 0.595387
$$915$$ 0 0
$$916$$ −20.0000 −0.660819
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −37.0000 −1.22052 −0.610259 0.792202i $$-0.708935\pi$$
−0.610259 + 0.792202i $$0.708935\pi$$
$$920$$ −28.0000 −0.923133
$$921$$ 0 0
$$922$$ −16.0000 −0.526932
$$923$$ −4.00000 −0.131662
$$924$$ 0 0
$$925$$ 77.0000 2.53174
$$926$$ −12.0000 −0.394344
$$927$$ 0 0
$$928$$ 8.00000 0.262613
$$929$$ 27.0000 0.885841 0.442921 0.896561i $$-0.353942\pi$$
0.442921 + 0.896561i $$0.353942\pi$$
$$930$$ 0 0
$$931$$ 2.00000 0.0655474
$$932$$ 13.0000 0.425829
$$933$$ 0 0
$$934$$ 20.0000 0.654420
$$935$$ 16.0000 0.523256
$$936$$ 0 0
$$937$$ −48.0000 −1.56809 −0.784046 0.620703i $$-0.786847\pi$$
−0.784046 + 0.620703i $$0.786847\pi$$
$$938$$ −7.00000 −0.228558
$$939$$ 0 0
$$940$$ 12.0000 0.391397
$$941$$ −14.0000 −0.456387 −0.228193 0.973616i $$-0.573282\pi$$
−0.228193 + 0.973616i $$0.573282\pi$$
$$942$$ 0 0
$$943$$ 49.0000 1.59566
$$944$$ 6.00000 0.195283
$$945$$ 0 0
$$946$$ −8.00000 −0.260102
$$947$$ 8.00000 0.259965 0.129983 0.991516i $$-0.458508\pi$$
0.129983 + 0.991516i $$0.458508\pi$$
$$948$$ 0 0
$$949$$ 9.00000 0.292152
$$950$$ 22.0000 0.713774
$$951$$ 0 0
$$952$$ 4.00000 0.129641
$$953$$ 34.0000 1.10137 0.550684 0.834714i $$-0.314367\pi$$
0.550684 + 0.834714i $$0.314367\pi$$
$$954$$ 0 0
$$955$$ 96.0000 3.10649
$$956$$ 6.00000 0.194054
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −14.0000 −0.452084
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 7.00000 0.225689
$$963$$ 0 0
$$964$$ −10.0000 −0.322078
$$965$$ 16.0000 0.515058
$$966$$ 0 0
$$967$$ 18.0000 0.578841 0.289420 0.957202i $$-0.406537\pi$$
0.289420 + 0.957202i $$0.406537\pi$$
$$968$$ −10.0000 −0.321412
$$969$$ 0 0
$$970$$ −44.0000 −1.41275
$$971$$ 45.0000 1.44412 0.722059 0.691831i $$-0.243196\pi$$
0.722059 + 0.691831i $$0.243196\pi$$
$$972$$ 0 0
$$973$$ 20.0000 0.641171
$$974$$ 26.0000 0.833094
$$975$$ 0 0
$$976$$ −13.0000 −0.416120
$$977$$ −30.0000 −0.959785 −0.479893 0.877327i $$-0.659324\pi$$
−0.479893 + 0.877327i $$0.659324\pi$$
$$978$$ 0 0
$$979$$ 6.00000 0.191761
$$980$$ −4.00000 −0.127775
$$981$$ 0 0
$$982$$ −40.0000 −1.27645
$$983$$ −32.0000 −1.02064 −0.510321 0.859984i $$-0.670473\pi$$
−0.510321 + 0.859984i $$0.670473\pi$$
$$984$$ 0 0
$$985$$ 12.0000 0.382352
$$986$$ −32.0000 −1.01909
$$987$$ 0 0
$$988$$ 2.00000 0.0636285
$$989$$ −56.0000 −1.78070
$$990$$ 0 0
$$991$$ −33.0000 −1.04828 −0.524140 0.851632i $$-0.675613\pi$$
−0.524140 + 0.851632i $$0.675613\pi$$
$$992$$ 3.00000 0.0952501
$$993$$ 0 0
$$994$$ 4.00000 0.126872
$$995$$ −64.0000 −2.02894
$$996$$ 0 0
$$997$$ 31.0000 0.981780 0.490890 0.871222i $$-0.336672\pi$$
0.490890 + 0.871222i $$0.336672\pi$$
$$998$$ 37.0000 1.17121
$$999$$ 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.k.1.1 1
3.2 odd 2 182.2.a.a.1.1 1
12.11 even 2 1456.2.a.e.1.1 1
15.14 odd 2 4550.2.a.t.1.1 1
21.2 odd 6 1274.2.f.n.1145.1 2
21.5 even 6 1274.2.f.t.1145.1 2
21.11 odd 6 1274.2.f.n.79.1 2
21.17 even 6 1274.2.f.t.79.1 2
21.20 even 2 1274.2.a.b.1.1 1
24.5 odd 2 5824.2.a.g.1.1 1
24.11 even 2 5824.2.a.w.1.1 1
39.5 even 4 2366.2.d.g.337.2 2
39.8 even 4 2366.2.d.g.337.1 2
39.38 odd 2 2366.2.a.m.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
182.2.a.a.1.1 1 3.2 odd 2
1274.2.a.b.1.1 1 21.20 even 2
1274.2.f.n.79.1 2 21.11 odd 6
1274.2.f.n.1145.1 2 21.2 odd 6
1274.2.f.t.79.1 2 21.17 even 6
1274.2.f.t.1145.1 2 21.5 even 6
1456.2.a.e.1.1 1 12.11 even 2
1638.2.a.k.1.1 1 1.1 even 1 trivial
2366.2.a.m.1.1 1 39.38 odd 2
2366.2.d.g.337.1 2 39.8 even 4
2366.2.d.g.337.2 2 39.5 even 4
4550.2.a.t.1.1 1 15.14 odd 2
5824.2.a.g.1.1 1 24.5 odd 2
5824.2.a.w.1.1 1 24.11 even 2