Properties

 Label 162.2.a.b.1.1 Level $162$ Weight $2$ Character 162.1 Self dual yes Analytic conductor $1.294$ 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] = [162,2,Mod(1,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("162.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 162.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: $$1.29357651274$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 18) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 162.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{7} -1.00000 q^{8} +3.00000 q^{11} +2.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{19} -3.00000 q^{22} +6.00000 q^{23} -5.00000 q^{25} -2.00000 q^{26} +2.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -3.00000 q^{34} -4.00000 q^{37} +1.00000 q^{38} -9.00000 q^{41} -1.00000 q^{43} +3.00000 q^{44} -6.00000 q^{46} +6.00000 q^{47} -3.00000 q^{49} +5.00000 q^{50} +2.00000 q^{52} -12.0000 q^{53} -2.00000 q^{56} +6.00000 q^{58} -3.00000 q^{59} +8.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +5.00000 q^{67} +3.00000 q^{68} +12.0000 q^{71} +11.0000 q^{73} +4.00000 q^{74} -1.00000 q^{76} +6.00000 q^{77} -4.00000 q^{79} +9.00000 q^{82} -12.0000 q^{83} +1.00000 q^{86} -3.00000 q^{88} -6.00000 q^{89} +4.00000 q^{91} +6.00000 q^{92} -6.00000 q^{94} +5.00000 q^{97} +3.00000 q^{98} +O(q^{100})$$

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
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 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
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −3.00000 −0.639602
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0 0
$$25$$ −5.00000 −1.00000
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ 2.00000 0.377964
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 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
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −9.00000 −1.40556 −0.702782 0.711405i $$-0.748059\pi$$
−0.702782 + 0.711405i $$0.748059\pi$$
$$42$$ 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
$$46$$ −6.00000 −0.884652
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 5.00000 0.707107
$$51$$ 0 0
$$52$$ 2.00000 0.277350
$$53$$ −12.0000 −1.64833 −0.824163 0.566352i $$-0.808354\pi$$
−0.824163 + 0.566352i $$0.808354\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −2.00000 −0.267261
$$57$$ 0 0
$$58$$ 6.00000 0.787839
$$59$$ −3.00000 −0.390567 −0.195283 0.980747i $$-0.562563\pi$$
−0.195283 + 0.980747i $$0.562563\pi$$
$$60$$ 0 0
$$61$$ 8.00000 1.02430 0.512148 0.858898i $$-0.328850\pi$$
0.512148 + 0.858898i $$0.328850\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 5.00000 0.610847 0.305424 0.952217i $$-0.401202\pi$$
0.305424 + 0.952217i $$0.401202\pi$$
$$68$$ 3.00000 0.363803
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ 11.0000 1.28745 0.643726 0.765256i $$-0.277388\pi$$
0.643726 + 0.765256i $$0.277388\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 0 0
$$76$$ −1.00000 −0.114708
$$77$$ 6.00000 0.683763
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 9.00000 0.993884
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 1.00000 0.107833
$$87$$ 0 0
$$88$$ −3.00000 −0.319801
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ 6.00000 0.625543
$$93$$ 0 0
$$94$$ −6.00000 −0.618853
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 5.00000 0.507673 0.253837 0.967247i $$-0.418307\pi$$
0.253837 + 0.967247i $$0.418307\pi$$
$$98$$ 3.00000 0.303046
$$99$$ 0 0
$$100$$ −5.00000 −0.500000
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ 12.0000 1.16554
$$107$$ −3.00000 −0.290021 −0.145010 0.989430i $$-0.546322\pi$$
−0.145010 + 0.989430i $$0.546322\pi$$
$$108$$ 0 0
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 2.00000 0.188982
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 0 0
$$118$$ 3.00000 0.276172
$$119$$ 6.00000 0.550019
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ −8.00000 −0.724286
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ 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
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ −2.00000 −0.173422
$$134$$ −5.00000 −0.431934
$$135$$ 0 0
$$136$$ −3.00000 −0.257248
$$137$$ 3.00000 0.256307 0.128154 0.991754i $$-0.459095\pi$$
0.128154 + 0.991754i $$0.459095\pi$$
$$138$$ 0 0
$$139$$ −19.0000 −1.61156 −0.805779 0.592216i $$-0.798253\pi$$
−0.805779 + 0.592216i $$0.798253\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −12.0000 −1.00702
$$143$$ 6.00000 0.501745
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −11.0000 −0.910366
$$147$$ 0 0
$$148$$ −4.00000 −0.328798
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 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
$$154$$ −6.00000 −0.483494
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −4.00000 −0.319235 −0.159617 0.987179i $$-0.551026\pi$$
−0.159617 + 0.987179i $$0.551026\pi$$
$$158$$ 4.00000 0.318223
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 12.0000 0.945732
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ −9.00000 −0.702782
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 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
$$175$$ −10.0000 −0.755929
$$176$$ 3.00000 0.226134
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ −4.00000 −0.296500
$$183$$ 0 0
$$184$$ −6.00000 −0.442326
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 9.00000 0.658145
$$188$$ 6.00000 0.437595
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ 0 0
$$193$$ 5.00000 0.359908 0.179954 0.983675i $$-0.442405\pi$$
0.179954 + 0.983675i $$0.442405\pi$$
$$194$$ −5.00000 −0.358979
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 12.0000 0.854965 0.427482 0.904024i $$-0.359401\pi$$
0.427482 + 0.904024i $$0.359401\pi$$
$$198$$ 0 0
$$199$$ −10.0000 −0.708881 −0.354441 0.935079i $$-0.615329\pi$$
−0.354441 + 0.935079i $$0.615329\pi$$
$$200$$ 5.00000 0.353553
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −12.0000 −0.842235
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −14.0000 −0.975426
$$207$$ 0 0
$$208$$ 2.00000 0.138675
$$209$$ −3.00000 −0.207514
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ −12.0000 −0.824163
$$213$$ 0 0
$$214$$ 3.00000 0.205076
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −8.00000 −0.543075
$$218$$ 16.0000 1.08366
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 6.00000 0.403604
$$222$$ 0 0
$$223$$ 26.0000 1.74109 0.870544 0.492090i $$-0.163767\pi$$
0.870544 + 0.492090i $$0.163767\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ −21.0000 −1.39382 −0.696909 0.717159i $$-0.745442\pi$$
−0.696909 + 0.717159i $$0.745442\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 6.00000 0.393919
$$233$$ −3.00000 −0.196537 −0.0982683 0.995160i $$-0.531330\pi$$
−0.0982683 + 0.995160i $$0.531330\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −3.00000 −0.195283
$$237$$ 0 0
$$238$$ −6.00000 −0.388922
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ −7.00000 −0.450910 −0.225455 0.974254i $$-0.572387\pi$$
−0.225455 + 0.974254i $$0.572387\pi$$
$$242$$ 2.00000 0.128565
$$243$$ 0 0
$$244$$ 8.00000 0.512148
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −2.00000 −0.127257
$$248$$ 4.00000 0.254000
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −21.0000 −1.32551 −0.662754 0.748837i $$-0.730613\pi$$
−0.662754 + 0.748837i $$0.730613\pi$$
$$252$$ 0 0
$$253$$ 18.0000 1.13165
$$254$$ −2.00000 −0.125491
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 21.0000 1.30994 0.654972 0.755653i $$-0.272680\pi$$
0.654972 + 0.755653i $$0.272680\pi$$
$$258$$ 0 0
$$259$$ −8.00000 −0.497096
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −18.0000 −1.10993 −0.554964 0.831875i $$-0.687268\pi$$
−0.554964 + 0.831875i $$0.687268\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 2.00000 0.122628
$$267$$ 0 0
$$268$$ 5.00000 0.305424
$$269$$ 24.0000 1.46331 0.731653 0.681677i $$-0.238749\pi$$
0.731653 + 0.681677i $$0.238749\pi$$
$$270$$ 0 0
$$271$$ 20.0000 1.21491 0.607457 0.794353i $$-0.292190\pi$$
0.607457 + 0.794353i $$0.292190\pi$$
$$272$$ 3.00000 0.181902
$$273$$ 0 0
$$274$$ −3.00000 −0.181237
$$275$$ −15.0000 −0.904534
$$276$$ 0 0
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 19.0000 1.13954
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 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
$$286$$ −6.00000 −0.354787
$$287$$ −18.0000 −1.06251
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 11.0000 0.643726
$$293$$ −30.0000 −1.75262 −0.876309 0.481749i $$-0.840002\pi$$
−0.876309 + 0.481749i $$0.840002\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 4.00000 0.232495
$$297$$ 0 0
$$298$$ −6.00000 −0.347571
$$299$$ 12.0000 0.693978
$$300$$ 0 0
$$301$$ −2.00000 −0.115278
$$302$$ 10.0000 0.575435
$$303$$ 0 0
$$304$$ −1.00000 −0.0573539
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −7.00000 −0.399511 −0.199756 0.979846i $$-0.564015\pi$$
−0.199756 + 0.979846i $$0.564015\pi$$
$$308$$ 6.00000 0.341882
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 0 0
$$313$$ 29.0000 1.63918 0.819588 0.572953i $$-0.194202\pi$$
0.819588 + 0.572953i $$0.194202\pi$$
$$314$$ 4.00000 0.225733
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 0 0
$$319$$ −18.0000 −1.00781
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −12.0000 −0.668734
$$323$$ −3.00000 −0.166924
$$324$$ 0 0
$$325$$ −10.0000 −0.554700
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ 9.00000 0.496942
$$329$$ 12.0000 0.661581
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ 0 0
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −1.00000 −0.0544735 −0.0272367 0.999629i $$-0.508671\pi$$
−0.0272367 + 0.999629i $$0.508671\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −12.0000 −0.649836
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 1.00000 0.0539164
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ −33.0000 −1.77153 −0.885766 0.464131i $$-0.846367\pi$$
−0.885766 + 0.464131i $$0.846367\pi$$
$$348$$ 0 0
$$349$$ −16.0000 −0.856460 −0.428230 0.903670i $$-0.640863\pi$$
−0.428230 + 0.903670i $$0.640863\pi$$
$$350$$ 10.0000 0.534522
$$351$$ 0 0
$$352$$ −3.00000 −0.159901
$$353$$ 21.0000 1.11772 0.558859 0.829263i $$-0.311239\pi$$
0.558859 + 0.829263i $$0.311239\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ 18.0000 0.950004 0.475002 0.879985i $$-0.342447\pi$$
0.475002 + 0.879985i $$0.342447\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ −14.0000 −0.735824
$$363$$ 0 0
$$364$$ 4.00000 0.209657
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ 6.00000 0.312772
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −24.0000 −1.24602
$$372$$ 0 0
$$373$$ −34.0000 −1.76045 −0.880227 0.474554i $$-0.842610\pi$$
−0.880227 + 0.474554i $$0.842610\pi$$
$$374$$ −9.00000 −0.465379
$$375$$ 0 0
$$376$$ −6.00000 −0.309426
$$377$$ −12.0000 −0.618031
$$378$$ 0 0
$$379$$ 23.0000 1.18143 0.590715 0.806880i $$-0.298846\pi$$
0.590715 + 0.806880i $$0.298846\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −18.0000 −0.920960
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −5.00000 −0.254493
$$387$$ 0 0
$$388$$ 5.00000 0.253837
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ 0 0
$$391$$ 18.0000 0.910299
$$392$$ 3.00000 0.151523
$$393$$ 0 0
$$394$$ −12.0000 −0.604551
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 20.0000 1.00377 0.501886 0.864934i $$-0.332640\pi$$
0.501886 + 0.864934i $$0.332640\pi$$
$$398$$ 10.0000 0.501255
$$399$$ 0 0
$$400$$ −5.00000 −0.250000
$$401$$ 27.0000 1.34832 0.674158 0.738587i $$-0.264507\pi$$
0.674158 + 0.738587i $$0.264507\pi$$
$$402$$ 0 0
$$403$$ −8.00000 −0.398508
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 12.0000 0.595550
$$407$$ −12.0000 −0.594818
$$408$$ 0 0
$$409$$ 17.0000 0.840596 0.420298 0.907386i $$-0.361926\pi$$
0.420298 + 0.907386i $$0.361926\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 14.0000 0.689730
$$413$$ −6.00000 −0.295241
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ 0 0
$$418$$ 3.00000 0.146735
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ 20.0000 0.974740 0.487370 0.873195i $$-0.337956\pi$$
0.487370 + 0.873195i $$0.337956\pi$$
$$422$$ −20.0000 −0.973585
$$423$$ 0 0
$$424$$ 12.0000 0.582772
$$425$$ −15.0000 −0.727607
$$426$$ 0 0
$$427$$ 16.0000 0.774294
$$428$$ −3.00000 −0.145010
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 30.0000 1.44505 0.722525 0.691345i $$-0.242982\pi$$
0.722525 + 0.691345i $$0.242982\pi$$
$$432$$ 0 0
$$433$$ −7.00000 −0.336399 −0.168199 0.985753i $$-0.553795\pi$$
−0.168199 + 0.985753i $$0.553795\pi$$
$$434$$ 8.00000 0.384012
$$435$$ 0 0
$$436$$ −16.0000 −0.766261
$$437$$ −6.00000 −0.287019
$$438$$ 0 0
$$439$$ 8.00000 0.381819 0.190910 0.981608i $$-0.438856\pi$$
0.190910 + 0.981608i $$0.438856\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −6.00000 −0.285391
$$443$$ −3.00000 −0.142534 −0.0712672 0.997457i $$-0.522704\pi$$
−0.0712672 + 0.997457i $$0.522704\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −26.0000 −1.23114
$$447$$ 0 0
$$448$$ 2.00000 0.0944911
$$449$$ −9.00000 −0.424736 −0.212368 0.977190i $$-0.568118\pi$$
−0.212368 + 0.977190i $$0.568118\pi$$
$$450$$ 0 0
$$451$$ −27.0000 −1.27138
$$452$$ −6.00000 −0.282216
$$453$$ 0 0
$$454$$ 21.0000 0.985579
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 17.0000 0.795226 0.397613 0.917553i $$-0.369839\pi$$
0.397613 + 0.917553i $$0.369839\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ 0 0
$$463$$ 20.0000 0.929479 0.464739 0.885448i $$-0.346148\pi$$
0.464739 + 0.885448i $$0.346148\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 3.00000 0.138972
$$467$$ 15.0000 0.694117 0.347059 0.937843i $$-0.387180\pi$$
0.347059 + 0.937843i $$0.387180\pi$$
$$468$$ 0 0
$$469$$ 10.0000 0.461757
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 3.00000 0.138086
$$473$$ −3.00000 −0.137940
$$474$$ 0 0
$$475$$ 5.00000 0.229416
$$476$$ 6.00000 0.275010
$$477$$ 0 0
$$478$$ 6.00000 0.274434
$$479$$ 42.0000 1.91903 0.959514 0.281659i $$-0.0908848\pi$$
0.959514 + 0.281659i $$0.0908848\pi$$
$$480$$ 0 0
$$481$$ −8.00000 −0.364769
$$482$$ 7.00000 0.318841
$$483$$ 0 0
$$484$$ −2.00000 −0.0909091
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 26.0000 1.17817 0.589086 0.808070i $$-0.299488\pi$$
0.589086 + 0.808070i $$0.299488\pi$$
$$488$$ −8.00000 −0.362143
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 15.0000 0.676941 0.338470 0.940977i $$-0.390091\pi$$
0.338470 + 0.940977i $$0.390091\pi$$
$$492$$ 0 0
$$493$$ −18.0000 −0.810679
$$494$$ 2.00000 0.0899843
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 24.0000 1.07655
$$498$$ 0 0
$$499$$ −13.0000 −0.581960 −0.290980 0.956729i $$-0.593981\pi$$
−0.290980 + 0.956729i $$0.593981\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 21.0000 0.937276
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −18.0000 −0.800198
$$507$$ 0 0
$$508$$ 2.00000 0.0887357
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 22.0000 0.973223
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −21.0000 −0.926270
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 18.0000 0.791639
$$518$$ 8.00000 0.351500
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 3.00000 0.131432 0.0657162 0.997838i $$-0.479067\pi$$
0.0657162 + 0.997838i $$0.479067\pi$$
$$522$$ 0 0
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 18.0000 0.784837
$$527$$ −12.0000 −0.522728
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −2.00000 −0.0867110
$$533$$ −18.0000 −0.779667
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −5.00000 −0.215967
$$537$$ 0 0
$$538$$ −24.0000 −1.03471
$$539$$ −9.00000 −0.387657
$$540$$ 0 0
$$541$$ −4.00000 −0.171973 −0.0859867 0.996296i $$-0.527404\pi$$
−0.0859867 + 0.996296i $$0.527404\pi$$
$$542$$ −20.0000 −0.859074
$$543$$ 0 0
$$544$$ −3.00000 −0.128624
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −1.00000 −0.0427569 −0.0213785 0.999771i $$-0.506805\pi$$
−0.0213785 + 0.999771i $$0.506805\pi$$
$$548$$ 3.00000 0.128154
$$549$$ 0 0
$$550$$ 15.0000 0.639602
$$551$$ 6.00000 0.255609
$$552$$ 0 0
$$553$$ −8.00000 −0.340195
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ −19.0000 −0.805779
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ 0 0
$$559$$ −2.00000 −0.0845910
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6.00000 0.253095
$$563$$ 39.0000 1.64365 0.821827 0.569737i $$-0.192955\pi$$
0.821827 + 0.569737i $$0.192955\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 4.00000 0.168133
$$567$$ 0 0
$$568$$ −12.0000 −0.503509
$$569$$ −45.0000 −1.88650 −0.943249 0.332086i $$-0.892248\pi$$
−0.943249 + 0.332086i $$0.892248\pi$$
$$570$$ 0 0
$$571$$ −37.0000 −1.54840 −0.774201 0.632940i $$-0.781848\pi$$
−0.774201 + 0.632940i $$0.781848\pi$$
$$572$$ 6.00000 0.250873
$$573$$ 0 0
$$574$$ 18.0000 0.751305
$$575$$ −30.0000 −1.25109
$$576$$ 0 0
$$577$$ 11.0000 0.457936 0.228968 0.973434i $$-0.426465\pi$$
0.228968 + 0.973434i $$0.426465\pi$$
$$578$$ 8.00000 0.332756
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −24.0000 −0.995688
$$582$$ 0 0
$$583$$ −36.0000 −1.49097
$$584$$ −11.0000 −0.455183
$$585$$ 0 0
$$586$$ 30.0000 1.23929
$$587$$ 9.00000 0.371470 0.185735 0.982600i $$-0.440533\pi$$
0.185735 + 0.982600i $$0.440533\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −4.00000 −0.164399
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 0 0
$$598$$ −12.0000 −0.490716
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ −37.0000 −1.50926 −0.754631 0.656150i $$-0.772184\pi$$
−0.754631 + 0.656150i $$0.772184\pi$$
$$602$$ 2.00000 0.0815139
$$603$$ 0 0
$$604$$ −10.0000 −0.406894
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −28.0000 −1.13648 −0.568242 0.822861i $$-0.692376\pi$$
−0.568242 + 0.822861i $$0.692376\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ 0 0
$$613$$ −16.0000 −0.646234 −0.323117 0.946359i $$-0.604731\pi$$
−0.323117 + 0.946359i $$0.604731\pi$$
$$614$$ 7.00000 0.282497
$$615$$ 0 0
$$616$$ −6.00000 −0.241747
$$617$$ −27.0000 −1.08698 −0.543490 0.839416i $$-0.682897\pi$$
−0.543490 + 0.839416i $$0.682897\pi$$
$$618$$ 0 0
$$619$$ 35.0000 1.40677 0.703384 0.710810i $$-0.251671\pi$$
0.703384 + 0.710810i $$0.251671\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −18.0000 −0.721734
$$623$$ −12.0000 −0.480770
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ −29.0000 −1.15907
$$627$$ 0 0
$$628$$ −4.00000 −0.159617
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ 4.00000 0.159111
$$633$$ 0 0
$$634$$ −18.0000 −0.714871
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −6.00000 −0.237729
$$638$$ 18.0000 0.712627
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 3.00000 0.118493 0.0592464 0.998243i $$-0.481130\pi$$
0.0592464 + 0.998243i $$0.481130\pi$$
$$642$$ 0 0
$$643$$ 23.0000 0.907031 0.453516 0.891248i $$-0.350170\pi$$
0.453516 + 0.891248i $$0.350170\pi$$
$$644$$ 12.0000 0.472866
$$645$$ 0 0
$$646$$ 3.00000 0.118033
$$647$$ −18.0000 −0.707653 −0.353827 0.935311i $$-0.615120\pi$$
−0.353827 + 0.935311i $$0.615120\pi$$
$$648$$ 0 0
$$649$$ −9.00000 −0.353281
$$650$$ 10.0000 0.392232
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −9.00000 −0.351391
$$657$$ 0 0
$$658$$ −12.0000 −0.467809
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 0 0
$$661$$ −4.00000 −0.155582 −0.0777910 0.996970i $$-0.524787\pi$$
−0.0777910 + 0.996970i $$0.524787\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −36.0000 −1.39393
$$668$$ 12.0000 0.464294
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 24.0000 0.926510
$$672$$ 0 0
$$673$$ −22.0000 −0.848038 −0.424019 0.905653i $$-0.639381\pi$$
−0.424019 + 0.905653i $$0.639381\pi$$
$$674$$ 1.00000 0.0385186
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ −36.0000 −1.38359 −0.691796 0.722093i $$-0.743180\pi$$
−0.691796 + 0.722093i $$0.743180\pi$$
$$678$$ 0 0
$$679$$ 10.0000 0.383765
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 12.0000 0.459504
$$683$$ 9.00000 0.344375 0.172188 0.985064i $$-0.444916\pi$$
0.172188 + 0.985064i $$0.444916\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 20.0000 0.763604
$$687$$ 0 0
$$688$$ −1.00000 −0.0381246
$$689$$ −24.0000 −0.914327
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 0 0
$$694$$ 33.0000 1.25266
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −27.0000 −1.02270
$$698$$ 16.0000 0.605609
$$699$$ 0 0
$$700$$ −10.0000 −0.377964
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ 0 0
$$703$$ 4.00000 0.150863
$$704$$ 3.00000 0.113067
$$705$$ 0 0
$$706$$ −21.0000 −0.790345
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −4.00000 −0.150223 −0.0751116 0.997175i $$-0.523931\pi$$
−0.0751116 + 0.997175i $$0.523931\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 6.00000 0.224860
$$713$$ −24.0000 −0.898807
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ −18.0000 −0.671754
$$719$$ 36.0000 1.34257 0.671287 0.741198i $$-0.265742\pi$$
0.671287 + 0.741198i $$0.265742\pi$$
$$720$$ 0 0
$$721$$ 28.0000 1.04277
$$722$$ 18.0000 0.669891
$$723$$ 0 0
$$724$$ 14.0000 0.520306
$$725$$ 30.0000 1.11417
$$726$$ 0 0
$$727$$ 26.0000 0.964287 0.482143 0.876092i $$-0.339858\pi$$
0.482143 + 0.876092i $$0.339858\pi$$
$$728$$ −4.00000 −0.148250
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −3.00000 −0.110959
$$732$$ 0 0
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ 28.0000 1.03350
$$735$$ 0 0
$$736$$ −6.00000 −0.221163
$$737$$ 15.0000 0.552532
$$738$$ 0 0
$$739$$ 47.0000 1.72892 0.864461 0.502699i $$-0.167660\pi$$
0.864461 + 0.502699i $$0.167660\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 24.0000 0.881068
$$743$$ −6.00000 −0.220119 −0.110059 0.993925i $$-0.535104\pi$$
−0.110059 + 0.993925i $$0.535104\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 34.0000 1.24483
$$747$$ 0 0
$$748$$ 9.00000 0.329073
$$749$$ −6.00000 −0.219235
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 6.00000 0.218797
$$753$$ 0 0
$$754$$ 12.0000 0.437014
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ −23.0000 −0.835398
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −42.0000 −1.52250 −0.761249 0.648459i $$-0.775414\pi$$
−0.761249 + 0.648459i $$0.775414\pi$$
$$762$$ 0 0
$$763$$ −32.0000 −1.15848
$$764$$ 18.0000 0.651217
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −6.00000 −0.216647
$$768$$ 0 0
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 5.00000 0.179954
$$773$$ −18.0000 −0.647415 −0.323708 0.946157i $$-0.604929\pi$$
−0.323708 + 0.946157i $$0.604929\pi$$
$$774$$ 0 0
$$775$$ 20.0000 0.718421
$$776$$ −5.00000 −0.179490
$$777$$ 0 0
$$778$$ 18.0000 0.645331
$$779$$ 9.00000 0.322458
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ −18.0000 −0.643679
$$783$$ 0 0
$$784$$ −3.00000 −0.107143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −4.00000 −0.142585 −0.0712923 0.997455i $$-0.522712\pi$$
−0.0712923 + 0.997455i $$0.522712\pi$$
$$788$$ 12.0000 0.427482
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ 16.0000 0.568177
$$794$$ −20.0000 −0.709773
$$795$$ 0 0
$$796$$ −10.0000 −0.354441
$$797$$ 12.0000 0.425062 0.212531 0.977154i $$-0.431829\pi$$
0.212531 + 0.977154i $$0.431829\pi$$
$$798$$ 0 0
$$799$$ 18.0000 0.636794
$$800$$ 5.00000 0.176777
$$801$$ 0 0
$$802$$ −27.0000 −0.953403
$$803$$ 33.0000 1.16454
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 8.00000 0.281788
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −33.0000 −1.16022 −0.580109 0.814539i $$-0.696990\pi$$
−0.580109 + 0.814539i $$0.696990\pi$$
$$810$$ 0 0
$$811$$ −7.00000 −0.245803 −0.122902 0.992419i $$-0.539220\pi$$
−0.122902 + 0.992419i $$0.539220\pi$$
$$812$$ −12.0000 −0.421117
$$813$$ 0 0
$$814$$ 12.0000 0.420600
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 1.00000 0.0349856
$$818$$ −17.0000 −0.594391
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 18.0000 0.628204 0.314102 0.949389i $$-0.398297\pi$$
0.314102 + 0.949389i $$0.398297\pi$$
$$822$$ 0 0
$$823$$ −16.0000 −0.557725 −0.278862 0.960331i $$-0.589957\pi$$
−0.278862 + 0.960331i $$0.589957\pi$$
$$824$$ −14.0000 −0.487713
$$825$$ 0 0
$$826$$ 6.00000 0.208767
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 0 0
$$829$$ −10.0000 −0.347314 −0.173657 0.984806i $$-0.555558\pi$$
−0.173657 + 0.984806i $$0.555558\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 2.00000 0.0693375
$$833$$ −9.00000 −0.311832
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −3.00000 −0.103757
$$837$$ 0 0
$$838$$ −12.0000 −0.414533
$$839$$ −12.0000 −0.414286 −0.207143 0.978311i $$-0.566417\pi$$
−0.207143 + 0.978311i $$0.566417\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −20.0000 −0.689246
$$843$$ 0 0
$$844$$ 20.0000 0.688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −4.00000 −0.137442
$$848$$ −12.0000 −0.412082
$$849$$ 0 0
$$850$$ 15.0000 0.514496
$$851$$ −24.0000 −0.822709
$$852$$ 0 0
$$853$$ 26.0000 0.890223 0.445112 0.895475i $$-0.353164\pi$$
0.445112 + 0.895475i $$0.353164\pi$$
$$854$$ −16.0000 −0.547509
$$855$$ 0 0
$$856$$ 3.00000 0.102538
$$857$$ 42.0000 1.43469 0.717346 0.696717i $$-0.245357\pi$$
0.717346 + 0.696717i $$0.245357\pi$$
$$858$$ 0 0
$$859$$ 35.0000 1.19418 0.597092 0.802173i $$-0.296323\pi$$
0.597092 + 0.802173i $$0.296323\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −30.0000 −1.02180
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 7.00000 0.237870
$$867$$ 0 0
$$868$$ −8.00000 −0.271538
$$869$$ −12.0000 −0.407072
$$870$$ 0 0
$$871$$ 10.0000 0.338837
$$872$$ 16.0000 0.541828
$$873$$ 0 0
$$874$$ 6.00000 0.202953
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 8.00000 0.270141 0.135070 0.990836i $$-0.456874\pi$$
0.135070 + 0.990836i $$0.456874\pi$$
$$878$$ −8.00000 −0.269987
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ 0 0
$$883$$ −19.0000 −0.639401 −0.319700 0.947519i $$-0.603582\pi$$
−0.319700 + 0.947519i $$0.603582\pi$$
$$884$$ 6.00000 0.201802
$$885$$ 0 0
$$886$$ 3.00000 0.100787
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 4.00000 0.134156
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 26.0000 0.870544
$$893$$ −6.00000 −0.200782
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −2.00000 −0.0668153
$$897$$ 0 0
$$898$$ 9.00000 0.300334
$$899$$ 24.0000 0.800445
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 27.0000 0.899002
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −31.0000 −1.02934 −0.514669 0.857389i $$-0.672085\pi$$
−0.514669 + 0.857389i $$0.672085\pi$$
$$908$$ −21.0000 −0.696909
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −48.0000 −1.59031 −0.795155 0.606406i $$-0.792611\pi$$
−0.795155 + 0.606406i $$0.792611\pi$$
$$912$$ 0 0
$$913$$ −36.0000 −1.19143
$$914$$ −17.0000 −0.562310
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 38.0000 1.25350 0.626752 0.779219i $$-0.284384\pi$$
0.626752 + 0.779219i $$0.284384\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −30.0000 −0.987997
$$923$$ 24.0000 0.789970
$$924$$ 0 0
$$925$$ 20.0000 0.657596
$$926$$ −20.0000 −0.657241
$$927$$ 0 0
$$928$$ 6.00000 0.196960
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 0 0
$$931$$ 3.00000 0.0983210
$$932$$ −3.00000 −0.0982683
$$933$$ 0 0
$$934$$ −15.0000 −0.490815
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 14.0000 0.457360 0.228680 0.973502i $$-0.426559\pi$$
0.228680 + 0.973502i $$0.426559\pi$$
$$938$$ −10.0000 −0.326512
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 60.0000 1.95594 0.977972 0.208736i $$-0.0669349\pi$$
0.977972 + 0.208736i $$0.0669349\pi$$
$$942$$ 0 0
$$943$$ −54.0000 −1.75848
$$944$$ −3.00000 −0.0976417
$$945$$ 0 0
$$946$$ 3.00000 0.0975384
$$947$$ −3.00000 −0.0974869 −0.0487435 0.998811i $$-0.515522\pi$$
−0.0487435 + 0.998811i $$0.515522\pi$$
$$948$$ 0 0
$$949$$ 22.0000 0.714150
$$950$$ −5.00000 −0.162221
$$951$$ 0 0
$$952$$ −6.00000 −0.194461
$$953$$ −9.00000 −0.291539 −0.145769 0.989319i $$-0.546566\pi$$
−0.145769 + 0.989319i $$0.546566\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −6.00000 −0.194054
$$957$$ 0 0
$$958$$ −42.0000 −1.35696
$$959$$ 6.00000 0.193750
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 8.00000 0.257930
$$963$$ 0 0
$$964$$ −7.00000 −0.225455
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −22.0000 −0.707472 −0.353736 0.935345i $$-0.615089\pi$$
−0.353736 + 0.935345i $$0.615089\pi$$
$$968$$ 2.00000 0.0642824
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 36.0000 1.15529 0.577647 0.816286i $$-0.303971\pi$$
0.577647 + 0.816286i $$0.303971\pi$$
$$972$$ 0 0
$$973$$ −38.0000 −1.21822
$$974$$ −26.0000 −0.833094
$$975$$ 0 0
$$976$$ 8.00000 0.256074
$$977$$ 51.0000 1.63163 0.815817 0.578310i $$-0.196287\pi$$
0.815817 + 0.578310i $$0.196287\pi$$
$$978$$ 0 0
$$979$$ −18.0000 −0.575282
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −15.0000 −0.478669
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 18.0000 0.573237
$$987$$ 0 0
$$988$$ −2.00000 −0.0636285
$$989$$ −6.00000 −0.190789
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 0 0
$$994$$ −24.0000 −0.761234
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −28.0000 −0.886769 −0.443384 0.896332i $$-0.646222\pi$$
−0.443384 + 0.896332i $$0.646222\pi$$
$$998$$ 13.0000 0.411508
$$999$$ 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 162.2.a.b.1.1 1
3.2 odd 2 162.2.a.c.1.1 1
4.3 odd 2 1296.2.a.f.1.1 1
5.2 odd 4 4050.2.c.r.649.1 2
5.3 odd 4 4050.2.c.r.649.2 2
5.4 even 2 4050.2.a.v.1.1 1
7.6 odd 2 7938.2.a.i.1.1 1
8.3 odd 2 5184.2.a.p.1.1 1
8.5 even 2 5184.2.a.q.1.1 1
9.2 odd 6 18.2.c.a.13.1 yes 2
9.4 even 3 54.2.c.a.19.1 2
9.5 odd 6 18.2.c.a.7.1 2
9.7 even 3 54.2.c.a.37.1 2
12.11 even 2 1296.2.a.g.1.1 1
15.2 even 4 4050.2.c.c.649.2 2
15.8 even 4 4050.2.c.c.649.1 2
15.14 odd 2 4050.2.a.c.1.1 1
21.20 even 2 7938.2.a.x.1.1 1
24.5 odd 2 5184.2.a.r.1.1 1
24.11 even 2 5184.2.a.o.1.1 1
36.7 odd 6 432.2.i.b.145.1 2
36.11 even 6 144.2.i.c.49.1 2
36.23 even 6 144.2.i.c.97.1 2
36.31 odd 6 432.2.i.b.289.1 2
45.2 even 12 450.2.j.e.49.2 4
45.4 even 6 1350.2.e.c.451.1 2
45.7 odd 12 1350.2.j.a.199.1 4
45.13 odd 12 1350.2.j.a.1099.1 4
45.14 odd 6 450.2.e.i.151.1 2
45.22 odd 12 1350.2.j.a.1099.2 4
45.23 even 12 450.2.j.e.349.2 4
45.29 odd 6 450.2.e.i.301.1 2
45.32 even 12 450.2.j.e.349.1 4
45.34 even 6 1350.2.e.c.901.1 2
45.38 even 12 450.2.j.e.49.1 4
45.43 odd 12 1350.2.j.a.199.2 4
63.2 odd 6 882.2.h.c.67.1 2
63.4 even 3 2646.2.h.h.667.1 2
63.5 even 6 882.2.e.g.655.1 2
63.11 odd 6 882.2.e.i.373.1 2
63.13 odd 6 2646.2.f.g.883.1 2
63.16 even 3 2646.2.h.h.361.1 2
63.20 even 6 882.2.f.d.589.1 2
63.23 odd 6 882.2.e.i.655.1 2
63.25 even 3 2646.2.e.b.1549.1 2
63.31 odd 6 2646.2.h.i.667.1 2
63.32 odd 6 882.2.h.c.79.1 2
63.34 odd 6 2646.2.f.g.1765.1 2
63.38 even 6 882.2.e.g.373.1 2
63.40 odd 6 2646.2.e.c.2125.1 2
63.41 even 6 882.2.f.d.295.1 2
63.47 even 6 882.2.h.b.67.1 2
63.52 odd 6 2646.2.e.c.1549.1 2
63.58 even 3 2646.2.e.b.2125.1 2
63.59 even 6 882.2.h.b.79.1 2
63.61 odd 6 2646.2.h.i.361.1 2
72.5 odd 6 576.2.i.g.385.1 2
72.11 even 6 576.2.i.a.193.1 2
72.13 even 6 1728.2.i.e.1153.1 2
72.29 odd 6 576.2.i.g.193.1 2
72.43 odd 6 1728.2.i.f.577.1 2
72.59 even 6 576.2.i.a.385.1 2
72.61 even 6 1728.2.i.e.577.1 2
72.67 odd 6 1728.2.i.f.1153.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
18.2.c.a.7.1 2 9.5 odd 6
18.2.c.a.13.1 yes 2 9.2 odd 6
54.2.c.a.19.1 2 9.4 even 3
54.2.c.a.37.1 2 9.7 even 3
144.2.i.c.49.1 2 36.11 even 6
144.2.i.c.97.1 2 36.23 even 6
162.2.a.b.1.1 1 1.1 even 1 trivial
162.2.a.c.1.1 1 3.2 odd 2
432.2.i.b.145.1 2 36.7 odd 6
432.2.i.b.289.1 2 36.31 odd 6
450.2.e.i.151.1 2 45.14 odd 6
450.2.e.i.301.1 2 45.29 odd 6
450.2.j.e.49.1 4 45.38 even 12
450.2.j.e.49.2 4 45.2 even 12
450.2.j.e.349.1 4 45.32 even 12
450.2.j.e.349.2 4 45.23 even 12
576.2.i.a.193.1 2 72.11 even 6
576.2.i.a.385.1 2 72.59 even 6
576.2.i.g.193.1 2 72.29 odd 6
576.2.i.g.385.1 2 72.5 odd 6
882.2.e.g.373.1 2 63.38 even 6
882.2.e.g.655.1 2 63.5 even 6
882.2.e.i.373.1 2 63.11 odd 6
882.2.e.i.655.1 2 63.23 odd 6
882.2.f.d.295.1 2 63.41 even 6
882.2.f.d.589.1 2 63.20 even 6
882.2.h.b.67.1 2 63.47 even 6
882.2.h.b.79.1 2 63.59 even 6
882.2.h.c.67.1 2 63.2 odd 6
882.2.h.c.79.1 2 63.32 odd 6
1296.2.a.f.1.1 1 4.3 odd 2
1296.2.a.g.1.1 1 12.11 even 2
1350.2.e.c.451.1 2 45.4 even 6
1350.2.e.c.901.1 2 45.34 even 6
1350.2.j.a.199.1 4 45.7 odd 12
1350.2.j.a.199.2 4 45.43 odd 12
1350.2.j.a.1099.1 4 45.13 odd 12
1350.2.j.a.1099.2 4 45.22 odd 12
1728.2.i.e.577.1 2 72.61 even 6
1728.2.i.e.1153.1 2 72.13 even 6
1728.2.i.f.577.1 2 72.43 odd 6
1728.2.i.f.1153.1 2 72.67 odd 6
2646.2.e.b.1549.1 2 63.25 even 3
2646.2.e.b.2125.1 2 63.58 even 3
2646.2.e.c.1549.1 2 63.52 odd 6
2646.2.e.c.2125.1 2 63.40 odd 6
2646.2.f.g.883.1 2 63.13 odd 6
2646.2.f.g.1765.1 2 63.34 odd 6
2646.2.h.h.361.1 2 63.16 even 3
2646.2.h.h.667.1 2 63.4 even 3
2646.2.h.i.361.1 2 63.61 odd 6
2646.2.h.i.667.1 2 63.31 odd 6
4050.2.a.c.1.1 1 15.14 odd 2
4050.2.a.v.1.1 1 5.4 even 2
4050.2.c.c.649.1 2 15.8 even 4
4050.2.c.c.649.2 2 15.2 even 4
4050.2.c.r.649.1 2 5.2 odd 4
4050.2.c.r.649.2 2 5.3 odd 4
5184.2.a.o.1.1 1 24.11 even 2
5184.2.a.p.1.1 1 8.3 odd 2
5184.2.a.q.1.1 1 8.5 even 2
5184.2.a.r.1.1 1 24.5 odd 2
7938.2.a.i.1.1 1 7.6 odd 2
7938.2.a.x.1.1 1 21.20 even 2