# Properties

 Label 8550.2.a.bm.1.1 Level $8550$ Weight $2$ Character 8550.1 Self dual yes Analytic conductor $68.272$ 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] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, 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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8550.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: $$68.2720937282$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 8550.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +5.00000 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +5.00000 q^{7} +1.00000 q^{8} +4.00000 q^{11} +1.00000 q^{13} +5.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{19} +4.00000 q^{22} +7.00000 q^{23} +1.00000 q^{26} +5.00000 q^{28} +3.00000 q^{29} -2.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} +2.00000 q^{37} +1.00000 q^{38} +6.00000 q^{41} -6.00000 q^{43} +4.00000 q^{44} +7.00000 q^{46} +18.0000 q^{49} +1.00000 q^{52} -13.0000 q^{53} +5.00000 q^{56} +3.00000 q^{58} +9.00000 q^{59} -12.0000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +3.00000 q^{67} -3.00000 q^{68} -11.0000 q^{73} +2.00000 q^{74} +1.00000 q^{76} +20.0000 q^{77} -2.00000 q^{79} +6.00000 q^{82} -10.0000 q^{83} -6.00000 q^{86} +4.00000 q^{88} -2.00000 q^{89} +5.00000 q^{91} +7.00000 q^{92} +2.00000 q^{97} +18.0000 q^{98} +O(q^{100})$$

## 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
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 5.00000 1.88982 0.944911 0.327327i $$-0.106148\pi$$
0.944911 + 0.327327i $$0.106148\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ 5.00000 1.33631
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ 7.00000 1.45960 0.729800 0.683660i $$-0.239613\pi$$
0.729800 + 0.683660i $$0.239613\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 1.00000 0.196116
$$27$$ 0 0
$$28$$ 5.00000 0.944911
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −3.00000 −0.514496
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ −6.00000 −0.914991 −0.457496 0.889212i $$-0.651253\pi$$
−0.457496 + 0.889212i $$0.651253\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 7.00000 1.03209
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 18.0000 2.57143
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 1.00000 0.138675
$$53$$ −13.0000 −1.78569 −0.892844 0.450367i $$-0.851293\pi$$
−0.892844 + 0.450367i $$0.851293\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 5.00000 0.668153
$$57$$ 0 0
$$58$$ 3.00000 0.393919
$$59$$ 9.00000 1.17170 0.585850 0.810419i $$-0.300761\pi$$
0.585850 + 0.810419i $$0.300761\pi$$
$$60$$ 0 0
$$61$$ −12.0000 −1.53644 −0.768221 0.640184i $$-0.778858\pi$$
−0.768221 + 0.640184i $$0.778858\pi$$
$$62$$ −2.00000 −0.254000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 3.00000 0.366508 0.183254 0.983066i $$-0.441337\pi$$
0.183254 + 0.983066i $$0.441337\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −11.0000 −1.28745 −0.643726 0.765256i $$-0.722612\pi$$
−0.643726 + 0.765256i $$0.722612\pi$$
$$74$$ 2.00000 0.232495
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 20.0000 2.27921
$$78$$ 0 0
$$79$$ −2.00000 −0.225018 −0.112509 0.993651i $$-0.535889\pi$$
−0.112509 + 0.993651i $$0.535889\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 6.00000 0.662589
$$83$$ −10.0000 −1.09764 −0.548821 0.835940i $$-0.684923\pi$$
−0.548821 + 0.835940i $$0.684923\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −6.00000 −0.646997
$$87$$ 0 0
$$88$$ 4.00000 0.426401
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ 5.00000 0.524142
$$92$$ 7.00000 0.729800
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 18.0000 1.81827
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 8.00000 0.796030 0.398015 0.917379i $$-0.369699\pi$$
0.398015 + 0.917379i $$0.369699\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ −13.0000 −1.26267
$$107$$ −13.0000 −1.25676 −0.628379 0.777908i $$-0.716281\pi$$
−0.628379 + 0.777908i $$0.716281\pi$$
$$108$$ 0 0
$$109$$ 19.0000 1.81987 0.909935 0.414751i $$-0.136131\pi$$
0.909935 + 0.414751i $$0.136131\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 5.00000 0.472456
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 3.00000 0.278543
$$117$$ 0 0
$$118$$ 9.00000 0.828517
$$119$$ −15.0000 −1.37505
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ −12.0000 −1.08643
$$123$$ 0 0
$$124$$ −2.00000 −0.179605
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 6.00000 0.532414 0.266207 0.963916i $$-0.414230\pi$$
0.266207 + 0.963916i $$0.414230\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −16.0000 −1.39793 −0.698963 0.715158i $$-0.746355\pi$$
−0.698963 + 0.715158i $$0.746355\pi$$
$$132$$ 0 0
$$133$$ 5.00000 0.433555
$$134$$ 3.00000 0.259161
$$135$$ 0 0
$$136$$ −3.00000 −0.257248
$$137$$ 9.00000 0.768922 0.384461 0.923141i $$-0.374387\pi$$
0.384461 + 0.923141i $$0.374387\pi$$
$$138$$ 0 0
$$139$$ 16.0000 1.35710 0.678551 0.734553i $$-0.262608\pi$$
0.678551 + 0.734553i $$0.262608\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 4.00000 0.334497
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −11.0000 −0.910366
$$147$$ 0 0
$$148$$ 2.00000 0.164399
$$149$$ 4.00000 0.327693 0.163846 0.986486i $$-0.447610\pi$$
0.163846 + 0.986486i $$0.447610\pi$$
$$150$$ 0 0
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ 0 0
$$154$$ 20.0000 1.61165
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −6.00000 −0.478852 −0.239426 0.970915i $$-0.576959\pi$$
−0.239426 + 0.970915i $$0.576959\pi$$
$$158$$ −2.00000 −0.159111
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 35.0000 2.75839
$$162$$ 0 0
$$163$$ −22.0000 −1.72317 −0.861586 0.507611i $$-0.830529\pi$$
−0.861586 + 0.507611i $$0.830529\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ −10.0000 −0.776151
$$167$$ −2.00000 −0.154765 −0.0773823 0.997001i $$-0.524656\pi$$
−0.0773823 + 0.997001i $$0.524656\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −6.00000 −0.457496
$$173$$ −14.0000 −1.06440 −0.532200 0.846619i $$-0.678635\pi$$
−0.532200 + 0.846619i $$0.678635\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ −2.00000 −0.149906
$$179$$ 8.00000 0.597948 0.298974 0.954261i $$-0.403356\pi$$
0.298974 + 0.954261i $$0.403356\pi$$
$$180$$ 0 0
$$181$$ 26.0000 1.93256 0.966282 0.257485i $$-0.0828937\pi$$
0.966282 + 0.257485i $$0.0828937\pi$$
$$182$$ 5.00000 0.370625
$$183$$ 0 0
$$184$$ 7.00000 0.516047
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −12.0000 −0.877527
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −9.00000 −0.651217 −0.325609 0.945505i $$-0.605569\pi$$
−0.325609 + 0.945505i $$0.605569\pi$$
$$192$$ 0 0
$$193$$ −10.0000 −0.719816 −0.359908 0.932988i $$-0.617192\pi$$
−0.359908 + 0.932988i $$0.617192\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ 18.0000 1.28571
$$197$$ −22.0000 −1.56744 −0.783718 0.621117i $$-0.786679\pi$$
−0.783718 + 0.621117i $$0.786679\pi$$
$$198$$ 0 0
$$199$$ −15.0000 −1.06332 −0.531661 0.846957i $$-0.678432\pi$$
−0.531661 + 0.846957i $$0.678432\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 8.00000 0.562878
$$203$$ 15.0000 1.05279
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −4.00000 −0.278693
$$207$$ 0 0
$$208$$ 1.00000 0.0693375
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ −5.00000 −0.344214 −0.172107 0.985078i $$-0.555058\pi$$
−0.172107 + 0.985078i $$0.555058\pi$$
$$212$$ −13.0000 −0.892844
$$213$$ 0 0
$$214$$ −13.0000 −0.888662
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −10.0000 −0.678844
$$218$$ 19.0000 1.28684
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −3.00000 −0.201802
$$222$$ 0 0
$$223$$ 2.00000 0.133930 0.0669650 0.997755i $$-0.478668\pi$$
0.0669650 + 0.997755i $$0.478668\pi$$
$$224$$ 5.00000 0.334077
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 5.00000 0.331862 0.165931 0.986137i $$-0.446937\pi$$
0.165931 + 0.986137i $$0.446937\pi$$
$$228$$ 0 0
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 3.00000 0.196960
$$233$$ 10.0000 0.655122 0.327561 0.944830i $$-0.393773\pi$$
0.327561 + 0.944830i $$0.393773\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 9.00000 0.585850
$$237$$ 0 0
$$238$$ −15.0000 −0.972306
$$239$$ 11.0000 0.711531 0.355765 0.934575i $$-0.384220\pi$$
0.355765 + 0.934575i $$0.384220\pi$$
$$240$$ 0 0
$$241$$ −12.0000 −0.772988 −0.386494 0.922292i $$-0.626314\pi$$
−0.386494 + 0.922292i $$0.626314\pi$$
$$242$$ 5.00000 0.321412
$$243$$ 0 0
$$244$$ −12.0000 −0.768221
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.00000 0.0636285
$$248$$ −2.00000 −0.127000
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ 28.0000 1.76034
$$254$$ 6.00000 0.376473
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 22.0000 1.37232 0.686161 0.727450i $$-0.259294\pi$$
0.686161 + 0.727450i $$0.259294\pi$$
$$258$$ 0 0
$$259$$ 10.0000 0.621370
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −16.0000 −0.988483
$$263$$ 8.00000 0.493301 0.246651 0.969104i $$-0.420670\pi$$
0.246651 + 0.969104i $$0.420670\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 5.00000 0.306570
$$267$$ 0 0
$$268$$ 3.00000 0.183254
$$269$$ −2.00000 −0.121942 −0.0609711 0.998140i $$-0.519420\pi$$
−0.0609711 + 0.998140i $$0.519420\pi$$
$$270$$ 0 0
$$271$$ −27.0000 −1.64013 −0.820067 0.572268i $$-0.806064\pi$$
−0.820067 + 0.572268i $$0.806064\pi$$
$$272$$ −3.00000 −0.181902
$$273$$ 0 0
$$274$$ 9.00000 0.543710
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 8.00000 0.480673 0.240337 0.970690i $$-0.422742\pi$$
0.240337 + 0.970690i $$0.422742\pi$$
$$278$$ 16.0000 0.959616
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 0 0
$$283$$ 2.00000 0.118888 0.0594438 0.998232i $$-0.481067\pi$$
0.0594438 + 0.998232i $$0.481067\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ 30.0000 1.77084
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −11.0000 −0.643726
$$293$$ −27.0000 −1.57736 −0.788678 0.614806i $$-0.789234\pi$$
−0.788678 + 0.614806i $$0.789234\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 2.00000 0.116248
$$297$$ 0 0
$$298$$ 4.00000 0.231714
$$299$$ 7.00000 0.404820
$$300$$ 0 0
$$301$$ −30.0000 −1.72917
$$302$$ −10.0000 −0.575435
$$303$$ 0 0
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −4.00000 −0.228292 −0.114146 0.993464i $$-0.536413\pi$$
−0.114146 + 0.993464i $$0.536413\pi$$
$$308$$ 20.0000 1.13961
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −25.0000 −1.41762 −0.708810 0.705399i $$-0.750768\pi$$
−0.708810 + 0.705399i $$0.750768\pi$$
$$312$$ 0 0
$$313$$ 1.00000 0.0565233 0.0282617 0.999601i $$-0.491003\pi$$
0.0282617 + 0.999601i $$0.491003\pi$$
$$314$$ −6.00000 −0.338600
$$315$$ 0 0
$$316$$ −2.00000 −0.112509
$$317$$ 9.00000 0.505490 0.252745 0.967533i $$-0.418667\pi$$
0.252745 + 0.967533i $$0.418667\pi$$
$$318$$ 0 0
$$319$$ 12.0000 0.671871
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 35.0000 1.95047
$$323$$ −3.00000 −0.166924
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −22.0000 −1.21847
$$327$$ 0 0
$$328$$ 6.00000 0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −7.00000 −0.384755 −0.192377 0.981321i $$-0.561620\pi$$
−0.192377 + 0.981321i $$0.561620\pi$$
$$332$$ −10.0000 −0.548821
$$333$$ 0 0
$$334$$ −2.00000 −0.109435
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −6.00000 −0.326841 −0.163420 0.986557i $$-0.552253\pi$$
−0.163420 + 0.986557i $$0.552253\pi$$
$$338$$ −12.0000 −0.652714
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −8.00000 −0.433224
$$342$$ 0 0
$$343$$ 55.0000 2.96972
$$344$$ −6.00000 −0.323498
$$345$$ 0 0
$$346$$ −14.0000 −0.752645
$$347$$ −6.00000 −0.322097 −0.161048 0.986947i $$-0.551488\pi$$
−0.161048 + 0.986947i $$0.551488\pi$$
$$348$$ 0 0
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 4.00000 0.213201
$$353$$ 7.00000 0.372572 0.186286 0.982496i $$-0.440355\pi$$
0.186286 + 0.982496i $$0.440355\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −2.00000 −0.106000
$$357$$ 0 0
$$358$$ 8.00000 0.422813
$$359$$ 5.00000 0.263890 0.131945 0.991257i $$-0.457878\pi$$
0.131945 + 0.991257i $$0.457878\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 26.0000 1.36653
$$363$$ 0 0
$$364$$ 5.00000 0.262071
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 7.00000 0.364900
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −65.0000 −3.37463
$$372$$ 0 0
$$373$$ 23.0000 1.19089 0.595447 0.803394i $$-0.296975\pi$$
0.595447 + 0.803394i $$0.296975\pi$$
$$374$$ −12.0000 −0.620505
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 3.00000 0.154508
$$378$$ 0 0
$$379$$ −33.0000 −1.69510 −0.847548 0.530719i $$-0.821922\pi$$
−0.847548 + 0.530719i $$0.821922\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −9.00000 −0.460480
$$383$$ −4.00000 −0.204390 −0.102195 0.994764i $$-0.532587\pi$$
−0.102195 + 0.994764i $$0.532587\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −10.0000 −0.508987
$$387$$ 0 0
$$388$$ 2.00000 0.101535
$$389$$ 4.00000 0.202808 0.101404 0.994845i $$-0.467667\pi$$
0.101404 + 0.994845i $$0.467667\pi$$
$$390$$ 0 0
$$391$$ −21.0000 −1.06202
$$392$$ 18.0000 0.909137
$$393$$ 0 0
$$394$$ −22.0000 −1.10834
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 16.0000 0.803017 0.401508 0.915855i $$-0.368486\pi$$
0.401508 + 0.915855i $$0.368486\pi$$
$$398$$ −15.0000 −0.751882
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 6.00000 0.299626 0.149813 0.988714i $$-0.452133\pi$$
0.149813 + 0.988714i $$0.452133\pi$$
$$402$$ 0 0
$$403$$ −2.00000 −0.0996271
$$404$$ 8.00000 0.398015
$$405$$ 0 0
$$406$$ 15.0000 0.744438
$$407$$ 8.00000 0.396545
$$408$$ 0 0
$$409$$ 22.0000 1.08783 0.543915 0.839140i $$-0.316941\pi$$
0.543915 + 0.839140i $$0.316941\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −4.00000 −0.197066
$$413$$ 45.0000 2.21431
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 1.00000 0.0490290
$$417$$ 0 0
$$418$$ 4.00000 0.195646
$$419$$ −14.0000 −0.683945 −0.341972 0.939710i $$-0.611095\pi$$
−0.341972 + 0.939710i $$0.611095\pi$$
$$420$$ 0 0
$$421$$ 1.00000 0.0487370 0.0243685 0.999703i $$-0.492242\pi$$
0.0243685 + 0.999703i $$0.492242\pi$$
$$422$$ −5.00000 −0.243396
$$423$$ 0 0
$$424$$ −13.0000 −0.631336
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −60.0000 −2.90360
$$428$$ −13.0000 −0.628379
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 36.0000 1.73406 0.867029 0.498257i $$-0.166026\pi$$
0.867029 + 0.498257i $$0.166026\pi$$
$$432$$ 0 0
$$433$$ 16.0000 0.768911 0.384455 0.923144i $$-0.374389\pi$$
0.384455 + 0.923144i $$0.374389\pi$$
$$434$$ −10.0000 −0.480015
$$435$$ 0 0
$$436$$ 19.0000 0.909935
$$437$$ 7.00000 0.334855
$$438$$ 0 0
$$439$$ 26.0000 1.24091 0.620456 0.784241i $$-0.286947\pi$$
0.620456 + 0.784241i $$0.286947\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −3.00000 −0.142695
$$443$$ 36.0000 1.71041 0.855206 0.518289i $$-0.173431\pi$$
0.855206 + 0.518289i $$0.173431\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 2.00000 0.0947027
$$447$$ 0 0
$$448$$ 5.00000 0.236228
$$449$$ 22.0000 1.03824 0.519122 0.854700i $$-0.326259\pi$$
0.519122 + 0.854700i $$0.326259\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 5.00000 0.234662
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 29.0000 1.35656 0.678281 0.734802i $$-0.262725\pi$$
0.678281 + 0.734802i $$0.262725\pi$$
$$458$$ −6.00000 −0.280362
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −18.0000 −0.838344 −0.419172 0.907907i $$-0.637680\pi$$
−0.419172 + 0.907907i $$0.637680\pi$$
$$462$$ 0 0
$$463$$ 8.00000 0.371792 0.185896 0.982569i $$-0.440481\pi$$
0.185896 + 0.982569i $$0.440481\pi$$
$$464$$ 3.00000 0.139272
$$465$$ 0 0
$$466$$ 10.0000 0.463241
$$467$$ −8.00000 −0.370196 −0.185098 0.982720i $$-0.559260\pi$$
−0.185098 + 0.982720i $$0.559260\pi$$
$$468$$ 0 0
$$469$$ 15.0000 0.692636
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 9.00000 0.414259
$$473$$ −24.0000 −1.10352
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −15.0000 −0.687524
$$477$$ 0 0
$$478$$ 11.0000 0.503128
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ −12.0000 −0.546585
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 38.0000 1.72194 0.860972 0.508652i $$-0.169856\pi$$
0.860972 + 0.508652i $$0.169856\pi$$
$$488$$ −12.0000 −0.543214
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −18.0000 −0.812329 −0.406164 0.913800i $$-0.633134\pi$$
−0.406164 + 0.913800i $$0.633134\pi$$
$$492$$ 0 0
$$493$$ −9.00000 −0.405340
$$494$$ 1.00000 0.0449921
$$495$$ 0 0
$$496$$ −2.00000 −0.0898027
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 42.0000 1.88018 0.940089 0.340929i $$-0.110742\pi$$
0.940089 + 0.340929i $$0.110742\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 12.0000 0.535586
$$503$$ −21.0000 −0.936344 −0.468172 0.883637i $$-0.655087\pi$$
−0.468172 + 0.883637i $$0.655087\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 28.0000 1.24475
$$507$$ 0 0
$$508$$ 6.00000 0.266207
$$509$$ −2.00000 −0.0886484 −0.0443242 0.999017i $$-0.514113\pi$$
−0.0443242 + 0.999017i $$0.514113\pi$$
$$510$$ 0 0
$$511$$ −55.0000 −2.43306
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 22.0000 0.970378
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 10.0000 0.439375
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −24.0000 −1.05146 −0.525730 0.850652i $$-0.676208\pi$$
−0.525730 + 0.850652i $$0.676208\pi$$
$$522$$ 0 0
$$523$$ 9.00000 0.393543 0.196771 0.980449i $$-0.436954\pi$$
0.196771 + 0.980449i $$0.436954\pi$$
$$524$$ −16.0000 −0.698963
$$525$$ 0 0
$$526$$ 8.00000 0.348817
$$527$$ 6.00000 0.261364
$$528$$ 0 0
$$529$$ 26.0000 1.13043
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 5.00000 0.216777
$$533$$ 6.00000 0.259889
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 3.00000 0.129580
$$537$$ 0 0
$$538$$ −2.00000 −0.0862261
$$539$$ 72.0000 3.10126
$$540$$ 0 0
$$541$$ 16.0000 0.687894 0.343947 0.938989i $$-0.388236\pi$$
0.343947 + 0.938989i $$0.388236\pi$$
$$542$$ −27.0000 −1.15975
$$543$$ 0 0
$$544$$ −3.00000 −0.128624
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ 9.00000 0.384461
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 3.00000 0.127804
$$552$$ 0 0
$$553$$ −10.0000 −0.425243
$$554$$ 8.00000 0.339887
$$555$$ 0 0
$$556$$ 16.0000 0.678551
$$557$$ 12.0000 0.508456 0.254228 0.967144i $$-0.418179\pi$$
0.254228 + 0.967144i $$0.418179\pi$$
$$558$$ 0 0
$$559$$ −6.00000 −0.253773
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 18.0000 0.759284
$$563$$ 20.0000 0.842900 0.421450 0.906852i $$-0.361521\pi$$
0.421450 + 0.906852i $$0.361521\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 2.00000 0.0840663
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −24.0000 −1.00613 −0.503066 0.864248i $$-0.667795\pi$$
−0.503066 + 0.864248i $$0.667795\pi$$
$$570$$ 0 0
$$571$$ 6.00000 0.251092 0.125546 0.992088i $$-0.459932\pi$$
0.125546 + 0.992088i $$0.459932\pi$$
$$572$$ 4.00000 0.167248
$$573$$ 0 0
$$574$$ 30.0000 1.25218
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 7.00000 0.291414 0.145707 0.989328i $$-0.453454\pi$$
0.145707 + 0.989328i $$0.453454\pi$$
$$578$$ −8.00000 −0.332756
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −50.0000 −2.07435
$$582$$ 0 0
$$583$$ −52.0000 −2.15362
$$584$$ −11.0000 −0.455183
$$585$$ 0 0
$$586$$ −27.0000 −1.11536
$$587$$ 18.0000 0.742940 0.371470 0.928445i $$-0.378854\pi$$
0.371470 + 0.928445i $$0.378854\pi$$
$$588$$ 0 0
$$589$$ −2.00000 −0.0824086
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 2.00000 0.0821995
$$593$$ 30.0000 1.23195 0.615976 0.787765i $$-0.288762\pi$$
0.615976 + 0.787765i $$0.288762\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 4.00000 0.163846
$$597$$ 0 0
$$598$$ 7.00000 0.286251
$$599$$ 26.0000 1.06233 0.531166 0.847268i $$-0.321754\pi$$
0.531166 + 0.847268i $$0.321754\pi$$
$$600$$ 0 0
$$601$$ 42.0000 1.71322 0.856608 0.515968i $$-0.172568\pi$$
0.856608 + 0.515968i $$0.172568\pi$$
$$602$$ −30.0000 −1.22271
$$603$$ 0 0
$$604$$ −10.0000 −0.406894
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 26.0000 1.05531 0.527654 0.849460i $$-0.323072\pi$$
0.527654 + 0.849460i $$0.323072\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 20.0000 0.807792 0.403896 0.914805i $$-0.367656\pi$$
0.403896 + 0.914805i $$0.367656\pi$$
$$614$$ −4.00000 −0.161427
$$615$$ 0 0
$$616$$ 20.0000 0.805823
$$617$$ 14.0000 0.563619 0.281809 0.959470i $$-0.409065\pi$$
0.281809 + 0.959470i $$0.409065\pi$$
$$618$$ 0 0
$$619$$ −24.0000 −0.964641 −0.482321 0.875995i $$-0.660206\pi$$
−0.482321 + 0.875995i $$0.660206\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −25.0000 −1.00241
$$623$$ −10.0000 −0.400642
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 1.00000 0.0399680
$$627$$ 0 0
$$628$$ −6.00000 −0.239426
$$629$$ −6.00000 −0.239236
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ −2.00000 −0.0795557
$$633$$ 0 0
$$634$$ 9.00000 0.357436
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 18.0000 0.713186
$$638$$ 12.0000 0.475085
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −8.00000 −0.315981 −0.157991 0.987441i $$-0.550502\pi$$
−0.157991 + 0.987441i $$0.550502\pi$$
$$642$$ 0 0
$$643$$ −26.0000 −1.02534 −0.512670 0.858586i $$-0.671344\pi$$
−0.512670 + 0.858586i $$0.671344\pi$$
$$644$$ 35.0000 1.37919
$$645$$ 0 0
$$646$$ −3.00000 −0.118033
$$647$$ −21.0000 −0.825595 −0.412798 0.910823i $$-0.635448\pi$$
−0.412798 + 0.910823i $$0.635448\pi$$
$$648$$ 0 0
$$649$$ 36.0000 1.41312
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −22.0000 −0.861586
$$653$$ −16.0000 −0.626128 −0.313064 0.949732i $$-0.601356\pi$$
−0.313064 + 0.949732i $$0.601356\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 6.00000 0.234261
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −33.0000 −1.28550 −0.642749 0.766077i $$-0.722206\pi$$
−0.642749 + 0.766077i $$0.722206\pi$$
$$660$$ 0 0
$$661$$ 15.0000 0.583432 0.291716 0.956505i $$-0.405774\pi$$
0.291716 + 0.956505i $$0.405774\pi$$
$$662$$ −7.00000 −0.272063
$$663$$ 0 0
$$664$$ −10.0000 −0.388075
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 21.0000 0.813123
$$668$$ −2.00000 −0.0773823
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −48.0000 −1.85302
$$672$$ 0 0
$$673$$ 44.0000 1.69608 0.848038 0.529936i $$-0.177784\pi$$
0.848038 + 0.529936i $$0.177784\pi$$
$$674$$ −6.00000 −0.231111
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ −39.0000 −1.49889 −0.749446 0.662066i $$-0.769680\pi$$
−0.749446 + 0.662066i $$0.769680\pi$$
$$678$$ 0 0
$$679$$ 10.0000 0.383765
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −8.00000 −0.306336
$$683$$ −44.0000 −1.68361 −0.841807 0.539779i $$-0.818508\pi$$
−0.841807 + 0.539779i $$0.818508\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 55.0000 2.09991
$$687$$ 0 0
$$688$$ −6.00000 −0.228748
$$689$$ −13.0000 −0.495261
$$690$$ 0 0
$$691$$ −42.0000 −1.59776 −0.798878 0.601494i $$-0.794573\pi$$
−0.798878 + 0.601494i $$0.794573\pi$$
$$692$$ −14.0000 −0.532200
$$693$$ 0 0
$$694$$ −6.00000 −0.227757
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −18.0000 −0.681799
$$698$$ 14.0000 0.529908
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −24.0000 −0.906467 −0.453234 0.891392i $$-0.649730\pi$$
−0.453234 + 0.891392i $$0.649730\pi$$
$$702$$ 0 0
$$703$$ 2.00000 0.0754314
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ 7.00000 0.263448
$$707$$ 40.0000 1.50435
$$708$$ 0 0
$$709$$ 2.00000 0.0751116 0.0375558 0.999295i $$-0.488043\pi$$
0.0375558 + 0.999295i $$0.488043\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −2.00000 −0.0749532
$$713$$ −14.0000 −0.524304
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 8.00000 0.298974
$$717$$ 0 0
$$718$$ 5.00000 0.186598
$$719$$ 27.0000 1.00693 0.503465 0.864016i $$-0.332058\pi$$
0.503465 + 0.864016i $$0.332058\pi$$
$$720$$ 0 0
$$721$$ −20.0000 −0.744839
$$722$$ 1.00000 0.0372161
$$723$$ 0 0
$$724$$ 26.0000 0.966282
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −23.0000 −0.853023 −0.426511 0.904482i $$-0.640258\pi$$
−0.426511 + 0.904482i $$0.640258\pi$$
$$728$$ 5.00000 0.185312
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 18.0000 0.665754
$$732$$ 0 0
$$733$$ −36.0000 −1.32969 −0.664845 0.746981i $$-0.731502\pi$$
−0.664845 + 0.746981i $$0.731502\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 0 0
$$736$$ 7.00000 0.258023
$$737$$ 12.0000 0.442026
$$738$$ 0 0
$$739$$ −10.0000 −0.367856 −0.183928 0.982940i $$-0.558881\pi$$
−0.183928 + 0.982940i $$0.558881\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −65.0000 −2.38623
$$743$$ −18.0000 −0.660356 −0.330178 0.943919i $$-0.607109\pi$$
−0.330178 + 0.943919i $$0.607109\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 23.0000 0.842090
$$747$$ 0 0
$$748$$ −12.0000 −0.438763
$$749$$ −65.0000 −2.37505
$$750$$ 0 0
$$751$$ −26.0000 −0.948753 −0.474377 0.880322i $$-0.657327\pi$$
−0.474377 + 0.880322i $$0.657327\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 3.00000 0.109254
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 6.00000 0.218074 0.109037 0.994038i $$-0.465223\pi$$
0.109037 + 0.994038i $$0.465223\pi$$
$$758$$ −33.0000 −1.19861
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −11.0000 −0.398750 −0.199375 0.979923i $$-0.563891\pi$$
−0.199375 + 0.979923i $$0.563891\pi$$
$$762$$ 0 0
$$763$$ 95.0000 3.43923
$$764$$ −9.00000 −0.325609
$$765$$ 0 0
$$766$$ −4.00000 −0.144526
$$767$$ 9.00000 0.324971
$$768$$ 0 0
$$769$$ −47.0000 −1.69486 −0.847432 0.530904i $$-0.821852\pi$$
−0.847432 + 0.530904i $$0.821852\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −10.0000 −0.359908
$$773$$ −51.0000 −1.83434 −0.917171 0.398493i $$-0.869533\pi$$
−0.917171 + 0.398493i $$0.869533\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 2.00000 0.0717958
$$777$$ 0 0
$$778$$ 4.00000 0.143407
$$779$$ 6.00000 0.214972
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −21.0000 −0.750958
$$783$$ 0 0
$$784$$ 18.0000 0.642857
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 39.0000 1.39020 0.695100 0.718913i $$-0.255360\pi$$
0.695100 + 0.718913i $$0.255360\pi$$
$$788$$ −22.0000 −0.783718
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −12.0000 −0.426132
$$794$$ 16.0000 0.567819
$$795$$ 0 0
$$796$$ −15.0000 −0.531661
$$797$$ 31.0000 1.09808 0.549038 0.835797i $$-0.314994\pi$$
0.549038 + 0.835797i $$0.314994\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 6.00000 0.211867
$$803$$ −44.0000 −1.55273
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −2.00000 −0.0704470
$$807$$ 0 0
$$808$$ 8.00000 0.281439
$$809$$ −25.0000 −0.878953 −0.439477 0.898254i $$-0.644836\pi$$
−0.439477 + 0.898254i $$0.644836\pi$$
$$810$$ 0 0
$$811$$ 37.0000 1.29925 0.649623 0.760257i $$-0.274927\pi$$
0.649623 + 0.760257i $$0.274927\pi$$
$$812$$ 15.0000 0.526397
$$813$$ 0 0
$$814$$ 8.00000 0.280400
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −6.00000 −0.209913
$$818$$ 22.0000 0.769212
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 52.0000 1.81481 0.907406 0.420255i $$-0.138059\pi$$
0.907406 + 0.420255i $$0.138059\pi$$
$$822$$ 0 0
$$823$$ 43.0000 1.49889 0.749443 0.662069i $$-0.230321\pi$$
0.749443 + 0.662069i $$0.230321\pi$$
$$824$$ −4.00000 −0.139347
$$825$$ 0 0
$$826$$ 45.0000 1.56575
$$827$$ −3.00000 −0.104320 −0.0521601 0.998639i $$-0.516611\pi$$
−0.0521601 + 0.998639i $$0.516611\pi$$
$$828$$ 0 0
$$829$$ 35.0000 1.21560 0.607800 0.794090i $$-0.292052\pi$$
0.607800 + 0.794090i $$0.292052\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 1.00000 0.0346688
$$833$$ −54.0000 −1.87099
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 4.00000 0.138343
$$837$$ 0 0
$$838$$ −14.0000 −0.483622
$$839$$ 4.00000 0.138095 0.0690477 0.997613i $$-0.478004\pi$$
0.0690477 + 0.997613i $$0.478004\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 1.00000 0.0344623
$$843$$ 0 0
$$844$$ −5.00000 −0.172107
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 25.0000 0.859010
$$848$$ −13.0000 −0.446422
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 14.0000 0.479914
$$852$$ 0 0
$$853$$ −42.0000 −1.43805 −0.719026 0.694983i $$-0.755412\pi$$
−0.719026 + 0.694983i $$0.755412\pi$$
$$854$$ −60.0000 −2.05316
$$855$$ 0 0
$$856$$ −13.0000 −0.444331
$$857$$ −40.0000 −1.36637 −0.683187 0.730243i $$-0.739407\pi$$
−0.683187 + 0.730243i $$0.739407\pi$$
$$858$$ 0 0
$$859$$ −28.0000 −0.955348 −0.477674 0.878537i $$-0.658520\pi$$
−0.477674 + 0.878537i $$0.658520\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 36.0000 1.22616
$$863$$ 56.0000 1.90626 0.953131 0.302558i $$-0.0978405\pi$$
0.953131 + 0.302558i $$0.0978405\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 16.0000 0.543702
$$867$$ 0 0
$$868$$ −10.0000 −0.339422
$$869$$ −8.00000 −0.271381
$$870$$ 0 0
$$871$$ 3.00000 0.101651
$$872$$ 19.0000 0.643421
$$873$$ 0 0
$$874$$ 7.00000 0.236779
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −33.0000 −1.11433 −0.557165 0.830402i $$-0.688111\pi$$
−0.557165 + 0.830402i $$0.688111\pi$$
$$878$$ 26.0000 0.877457
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −10.0000 −0.336909 −0.168454 0.985709i $$-0.553878\pi$$
−0.168454 + 0.985709i $$0.553878\pi$$
$$882$$ 0 0
$$883$$ −30.0000 −1.00958 −0.504790 0.863242i $$-0.668430\pi$$
−0.504790 + 0.863242i $$0.668430\pi$$
$$884$$ −3.00000 −0.100901
$$885$$ 0 0
$$886$$ 36.0000 1.20944
$$887$$ −28.0000 −0.940148 −0.470074 0.882627i $$-0.655773\pi$$
−0.470074 + 0.882627i $$0.655773\pi$$
$$888$$ 0 0
$$889$$ 30.0000 1.00617
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 2.00000 0.0669650
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 5.00000 0.167038
$$897$$ 0 0
$$898$$ 22.0000 0.734150
$$899$$ −6.00000 −0.200111
$$900$$ 0 0
$$901$$ 39.0000 1.29928
$$902$$ 24.0000 0.799113
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 1.00000 0.0332045 0.0166022 0.999862i $$-0.494715\pi$$
0.0166022 + 0.999862i $$0.494715\pi$$
$$908$$ 5.00000 0.165931
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 0 0
$$913$$ −40.0000 −1.32381
$$914$$ 29.0000 0.959235
$$915$$ 0 0
$$916$$ −6.00000 −0.198246
$$917$$ −80.0000 −2.64183
$$918$$ 0 0
$$919$$ −5.00000 −0.164935 −0.0824674 0.996594i $$-0.526280\pi$$
−0.0824674 + 0.996594i $$0.526280\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −18.0000 −0.592798
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 8.00000 0.262896
$$927$$ 0 0
$$928$$ 3.00000 0.0984798
$$929$$ 3.00000 0.0984268 0.0492134 0.998788i $$-0.484329\pi$$
0.0492134 + 0.998788i $$0.484329\pi$$
$$930$$ 0 0
$$931$$ 18.0000 0.589926
$$932$$ 10.0000 0.327561
$$933$$ 0 0
$$934$$ −8.00000 −0.261768
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −47.0000 −1.53542 −0.767712 0.640796i $$-0.778605\pi$$
−0.767712 + 0.640796i $$0.778605\pi$$
$$938$$ 15.0000 0.489767
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 51.0000 1.66255 0.831276 0.555860i $$-0.187611\pi$$
0.831276 + 0.555860i $$0.187611\pi$$
$$942$$ 0 0
$$943$$ 42.0000 1.36771
$$944$$ 9.00000 0.292925
$$945$$ 0 0
$$946$$ −24.0000 −0.780307
$$947$$ −24.0000 −0.779895 −0.389948 0.920837i $$-0.627507\pi$$
−0.389948 + 0.920837i $$0.627507\pi$$
$$948$$ 0 0
$$949$$ −11.0000 −0.357075
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −15.0000 −0.486153
$$953$$ −24.0000 −0.777436 −0.388718 0.921357i $$-0.627082\pi$$
−0.388718 + 0.921357i $$0.627082\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 11.0000 0.355765
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 45.0000 1.45313
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 2.00000 0.0644826
$$963$$ 0 0
$$964$$ −12.0000 −0.386494
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 44.0000 1.41494 0.707472 0.706741i $$-0.249835\pi$$
0.707472 + 0.706741i $$0.249835\pi$$
$$968$$ 5.00000 0.160706
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −28.0000 −0.898563 −0.449281 0.893390i $$-0.648320\pi$$
−0.449281 + 0.893390i $$0.648320\pi$$
$$972$$ 0 0
$$973$$ 80.0000 2.56468
$$974$$ 38.0000 1.21760
$$975$$ 0 0
$$976$$ −12.0000 −0.384111
$$977$$ −62.0000 −1.98356 −0.991778 0.127971i $$-0.959153\pi$$
−0.991778 + 0.127971i $$0.959153\pi$$
$$978$$ 0 0
$$979$$ −8.00000 −0.255681
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −18.0000 −0.574403
$$983$$ −42.0000 −1.33959 −0.669796 0.742545i $$-0.733618\pi$$
−0.669796 + 0.742545i $$0.733618\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −9.00000 −0.286618
$$987$$ 0 0
$$988$$ 1.00000 0.0318142
$$989$$ −42.0000 −1.33552
$$990$$ 0 0
$$991$$ 30.0000 0.952981 0.476491 0.879180i $$-0.341909\pi$$
0.476491 + 0.879180i $$0.341909\pi$$
$$992$$ −2.00000 −0.0635001
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 50.0000 1.58352 0.791758 0.610835i $$-0.209166\pi$$
0.791758 + 0.610835i $$0.209166\pi$$
$$998$$ 42.0000 1.32949
$$999$$ 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 8550.2.a.bm.1.1 1
3.2 odd 2 950.2.a.c.1.1 1
5.4 even 2 1710.2.a.g.1.1 1
12.11 even 2 7600.2.a.a.1.1 1
15.2 even 4 950.2.b.a.799.1 2
15.8 even 4 950.2.b.a.799.2 2
15.14 odd 2 190.2.a.b.1.1 1
60.59 even 2 1520.2.a.j.1.1 1
105.104 even 2 9310.2.a.u.1.1 1
120.29 odd 2 6080.2.a.x.1.1 1
120.59 even 2 6080.2.a.b.1.1 1
285.284 even 2 3610.2.a.e.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.a.b.1.1 1 15.14 odd 2
950.2.a.c.1.1 1 3.2 odd 2
950.2.b.a.799.1 2 15.2 even 4
950.2.b.a.799.2 2 15.8 even 4
1520.2.a.j.1.1 1 60.59 even 2
1710.2.a.g.1.1 1 5.4 even 2
3610.2.a.e.1.1 1 285.284 even 2
6080.2.a.b.1.1 1 120.59 even 2
6080.2.a.x.1.1 1 120.29 odd 2
7600.2.a.a.1.1 1 12.11 even 2
8550.2.a.bm.1.1 1 1.1 even 1 trivial
9310.2.a.u.1.1 1 105.104 even 2