# Properties

 Label 120.2.a.a.1.1 Level $120$ Weight $2$ Character 120.1 Self dual yes Analytic conductor $0.958$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [120,2,Mod(1,120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$120 = 2^{3} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 120.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: $$0.958204824255$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 120.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} -6.00000 q^{13} -1.00000 q^{15} -2.00000 q^{17} +4.00000 q^{19} +4.00000 q^{21} -8.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.00000 q^{29} -4.00000 q^{35} -6.00000 q^{37} -6.00000 q^{39} +10.0000 q^{41} -4.00000 q^{43} -1.00000 q^{45} +8.00000 q^{47} +9.00000 q^{49} -2.00000 q^{51} +10.0000 q^{53} +4.00000 q^{57} +6.00000 q^{61} +4.00000 q^{63} +6.00000 q^{65} -4.00000 q^{67} -8.00000 q^{69} -14.0000 q^{73} +1.00000 q^{75} +16.0000 q^{79} +1.00000 q^{81} +12.0000 q^{83} +2.00000 q^{85} -6.00000 q^{87} +2.00000 q^{89} -24.0000 q^{91} -4.00000 q^{95} +2.00000 q^{97} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 4.00000 1.51186 0.755929 0.654654i $$-0.227186\pi$$
0.755929 + 0.654654i $$0.227186\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 4.00000 0.872872
$$22$$ 0 0
$$23$$ −8.00000 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −4.00000 −0.676123
$$36$$ 0 0
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 0 0
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ −1.00000 −0.149071
$$46$$ 0 0
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ 0 0
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 4.00000 0.529813
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 0 0
$$63$$ 4.00000 0.503953
$$64$$ 0 0
$$65$$ 6.00000 0.744208
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 0 0
$$69$$ −8.00000 −0.963087
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −14.0000 −1.63858 −0.819288 0.573382i $$-0.805631\pi$$
−0.819288 + 0.573382i $$0.805631\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 16.0000 1.80014 0.900070 0.435745i $$-0.143515\pi$$
0.900070 + 0.435745i $$0.143515\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ 2.00000 0.216930
$$86$$ 0 0
$$87$$ −6.00000 −0.643268
$$88$$ 0 0
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ −24.0000 −2.51588
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −4.00000 −0.410391
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ −4.00000 −0.390360
$$106$$ 0 0
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ 0 0
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ −6.00000 −0.569495
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 8.00000 0.746004
$$116$$ 0 0
$$117$$ −6.00000 −0.554700
$$118$$ 0 0
$$119$$ −8.00000 −0.733359
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 10.0000 0.901670
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 16.0000 1.39793 0.698963 0.715158i $$-0.253645\pi$$
0.698963 + 0.715158i $$0.253645\pi$$
$$132$$ 0 0
$$133$$ 16.0000 1.38738
$$134$$ 0 0
$$135$$ −1.00000 −0.0860663
$$136$$ 0 0
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 0 0
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ 8.00000 0.673722
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 6.00000 0.498273
$$146$$ 0 0
$$147$$ 9.00000 0.742307
$$148$$ 0 0
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ 0 0
$$159$$ 10.0000 0.793052
$$160$$ 0 0
$$161$$ −32.0000 −2.52195
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 0 0
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ 0 0
$$175$$ 4.00000 0.302372
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 8.00000 0.597948 0.298974 0.954261i $$-0.403356\pi$$
0.298974 + 0.954261i $$0.403356\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 0 0
$$183$$ 6.00000 0.443533
$$184$$ 0 0
$$185$$ 6.00000 0.441129
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 0 0
$$193$$ −6.00000 −0.431889 −0.215945 0.976406i $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ 0 0
$$195$$ 6.00000 0.429669
$$196$$ 0 0
$$197$$ 10.0000 0.712470 0.356235 0.934396i $$-0.384060\pi$$
0.356235 + 0.934396i $$0.384060\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 0 0
$$203$$ −24.0000 −1.68447
$$204$$ 0 0
$$205$$ −10.0000 −0.698430
$$206$$ 0 0
$$207$$ −8.00000 −0.556038
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 4.00000 0.272798
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −14.0000 −0.946032
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ 0 0
$$223$$ 12.0000 0.803579 0.401790 0.915732i $$-0.368388\pi$$
0.401790 + 0.915732i $$0.368388\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ 0 0
$$235$$ −8.00000 −0.521862
$$236$$ 0 0
$$237$$ 16.0000 1.03931
$$238$$ 0 0
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −9.00000 −0.574989
$$246$$ 0 0
$$247$$ −24.0000 −1.52708
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ −8.00000 −0.504956 −0.252478 0.967603i $$-0.581245\pi$$
−0.252478 + 0.967603i $$0.581245\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 2.00000 0.125245
$$256$$ 0 0
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 0 0
$$259$$ −24.0000 −1.49129
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ −24.0000 −1.47990 −0.739952 0.672660i $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 0 0
$$265$$ −10.0000 −0.614295
$$266$$ 0 0
$$267$$ 2.00000 0.122398
$$268$$ 0 0
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ −24.0000 −1.45255
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 2.00000 0.119310 0.0596550 0.998219i $$-0.481000\pi$$
0.0596550 + 0.998219i $$0.481000\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 0 0
$$285$$ −4.00000 −0.236940
$$286$$ 0 0
$$287$$ 40.0000 2.36113
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ 0 0
$$293$$ 10.0000 0.584206 0.292103 0.956387i $$-0.405645\pi$$
0.292103 + 0.956387i $$0.405645\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 48.0000 2.77591
$$300$$ 0 0
$$301$$ −16.0000 −0.922225
$$302$$ 0 0
$$303$$ −14.0000 −0.804279
$$304$$ 0 0
$$305$$ −6.00000 −0.343559
$$306$$ 0 0
$$307$$ 4.00000 0.228292 0.114146 0.993464i $$-0.463587\pi$$
0.114146 + 0.993464i $$0.463587\pi$$
$$308$$ 0 0
$$309$$ 4.00000 0.227552
$$310$$ 0 0
$$311$$ 16.0000 0.907277 0.453638 0.891186i $$-0.350126\pi$$
0.453638 + 0.891186i $$0.350126\pi$$
$$312$$ 0 0
$$313$$ −14.0000 −0.791327 −0.395663 0.918396i $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ 0 0
$$315$$ −4.00000 −0.225374
$$316$$ 0 0
$$317$$ 2.00000 0.112331 0.0561656 0.998421i $$-0.482113\pi$$
0.0561656 + 0.998421i $$0.482113\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 4.00000 0.223258
$$322$$ 0 0
$$323$$ −8.00000 −0.445132
$$324$$ 0 0
$$325$$ −6.00000 −0.332820
$$326$$ 0 0
$$327$$ −10.0000 −0.553001
$$328$$ 0 0
$$329$$ 32.0000 1.76422
$$330$$ 0 0
$$331$$ 28.0000 1.53902 0.769510 0.638635i $$-0.220501\pi$$
0.769510 + 0.638635i $$0.220501\pi$$
$$332$$ 0 0
$$333$$ −6.00000 −0.328798
$$334$$ 0 0
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ 10.0000 0.544735 0.272367 0.962193i $$-0.412193\pi$$
0.272367 + 0.962193i $$0.412193\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 8.00000 0.431959
$$344$$ 0 0
$$345$$ 8.00000 0.430706
$$346$$ 0 0
$$347$$ −36.0000 −1.93258 −0.966291 0.257454i $$-0.917117\pi$$
−0.966291 + 0.257454i $$0.917117\pi$$
$$348$$ 0 0
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ −6.00000 −0.320256
$$352$$ 0 0
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −8.00000 −0.423405
$$358$$ 0 0
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ −11.0000 −0.577350
$$364$$ 0 0
$$365$$ 14.0000 0.732793
$$366$$ 0 0
$$367$$ 28.0000 1.46159 0.730794 0.682598i $$-0.239150\pi$$
0.730794 + 0.682598i $$0.239150\pi$$
$$368$$ 0 0
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ 40.0000 2.07670
$$372$$ 0 0
$$373$$ 2.00000 0.103556 0.0517780 0.998659i $$-0.483511\pi$$
0.0517780 + 0.998659i $$0.483511\pi$$
$$374$$ 0 0
$$375$$ −1.00000 −0.0516398
$$376$$ 0 0
$$377$$ 36.0000 1.85409
$$378$$ 0 0
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ 0 0
$$383$$ 8.00000 0.408781 0.204390 0.978889i $$-0.434479\pi$$
0.204390 + 0.978889i $$0.434479\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −4.00000 −0.203331
$$388$$ 0 0
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ 0 0
$$393$$ 16.0000 0.807093
$$394$$ 0 0
$$395$$ −16.0000 −0.805047
$$396$$ 0 0
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ 0 0
$$399$$ 16.0000 0.801002
$$400$$ 0 0
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −38.0000 −1.87898 −0.939490 0.342578i $$-0.888700\pi$$
−0.939490 + 0.342578i $$0.888700\pi$$
$$410$$ 0 0
$$411$$ −18.0000 −0.887875
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −12.0000 −0.589057
$$416$$ 0 0
$$417$$ −12.0000 −0.587643
$$418$$ 0 0
$$419$$ −16.0000 −0.781651 −0.390826 0.920465i $$-0.627810\pi$$
−0.390826 + 0.920465i $$0.627810\pi$$
$$420$$ 0 0
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ 0 0
$$423$$ 8.00000 0.388973
$$424$$ 0 0
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ 24.0000 1.16144
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −32.0000 −1.54139 −0.770693 0.637207i $$-0.780090\pi$$
−0.770693 + 0.637207i $$0.780090\pi$$
$$432$$ 0 0
$$433$$ 10.0000 0.480569 0.240285 0.970702i $$-0.422759\pi$$
0.240285 + 0.970702i $$0.422759\pi$$
$$434$$ 0 0
$$435$$ 6.00000 0.287678
$$436$$ 0 0
$$437$$ −32.0000 −1.53077
$$438$$ 0 0
$$439$$ −32.0000 −1.52728 −0.763638 0.645644i $$-0.776589\pi$$
−0.763638 + 0.645644i $$0.776589\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 0 0
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 0 0
$$445$$ −2.00000 −0.0948091
$$446$$ 0 0
$$447$$ −6.00000 −0.283790
$$448$$ 0 0
$$449$$ 26.0000 1.22702 0.613508 0.789689i $$-0.289758\pi$$
0.613508 + 0.789689i $$0.289758\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 24.0000 1.12514
$$456$$ 0 0
$$457$$ −38.0000 −1.77757 −0.888783 0.458329i $$-0.848448\pi$$
−0.888783 + 0.458329i $$0.848448\pi$$
$$458$$ 0 0
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ 2.00000 0.0931493 0.0465746 0.998915i $$-0.485169\pi$$
0.0465746 + 0.998915i $$0.485169\pi$$
$$462$$ 0 0
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −20.0000 −0.925490 −0.462745 0.886492i $$-0.653135\pi$$
−0.462745 + 0.886492i $$0.653135\pi$$
$$468$$ 0 0
$$469$$ −16.0000 −0.738811
$$470$$ 0 0
$$471$$ 10.0000 0.460776
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ 10.0000 0.457869
$$478$$ 0 0
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ 36.0000 1.64146
$$482$$ 0 0
$$483$$ −32.0000 −1.45605
$$484$$ 0 0
$$485$$ −2.00000 −0.0908153
$$486$$ 0 0
$$487$$ −36.0000 −1.63132 −0.815658 0.578535i $$-0.803625\pi$$
−0.815658 + 0.578535i $$0.803625\pi$$
$$488$$ 0 0
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ −40.0000 −1.80517 −0.902587 0.430507i $$-0.858335\pi$$
−0.902587 + 0.430507i $$0.858335\pi$$
$$492$$ 0 0
$$493$$ 12.0000 0.540453
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 20.0000 0.895323 0.447661 0.894203i $$-0.352257\pi$$
0.447661 + 0.894203i $$0.352257\pi$$
$$500$$ 0 0
$$501$$ −8.00000 −0.357414
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 14.0000 0.622992
$$506$$ 0 0
$$507$$ 23.0000 1.02147
$$508$$ 0 0
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ −56.0000 −2.47729
$$512$$ 0 0
$$513$$ 4.00000 0.176604
$$514$$ 0 0
$$515$$ −4.00000 −0.176261
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ −14.0000 −0.613351 −0.306676 0.951814i $$-0.599217\pi$$
−0.306676 + 0.951814i $$0.599217\pi$$
$$522$$ 0 0
$$523$$ 36.0000 1.57417 0.787085 0.616844i $$-0.211589\pi$$
0.787085 + 0.616844i $$0.211589\pi$$
$$524$$ 0 0
$$525$$ 4.00000 0.174574
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −60.0000 −2.59889
$$534$$ 0 0
$$535$$ −4.00000 −0.172935
$$536$$ 0 0
$$537$$ 8.00000 0.345225
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −10.0000 −0.429934 −0.214967 0.976621i $$-0.568964\pi$$
−0.214967 + 0.976621i $$0.568964\pi$$
$$542$$ 0 0
$$543$$ 14.0000 0.600798
$$544$$ 0 0
$$545$$ 10.0000 0.428353
$$546$$ 0 0
$$547$$ 4.00000 0.171028 0.0855138 0.996337i $$-0.472747\pi$$
0.0855138 + 0.996337i $$0.472747\pi$$
$$548$$ 0 0
$$549$$ 6.00000 0.256074
$$550$$ 0 0
$$551$$ −24.0000 −1.02243
$$552$$ 0 0
$$553$$ 64.0000 2.72156
$$554$$ 0 0
$$555$$ 6.00000 0.254686
$$556$$ 0 0
$$557$$ −14.0000 −0.593199 −0.296600 0.955002i $$-0.595853\pi$$
−0.296600 + 0.955002i $$0.595853\pi$$
$$558$$ 0 0
$$559$$ 24.0000 1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 36.0000 1.51722 0.758610 0.651546i $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ 0 0
$$567$$ 4.00000 0.167984
$$568$$ 0 0
$$569$$ −14.0000 −0.586911 −0.293455 0.955973i $$-0.594805\pi$$
−0.293455 + 0.955973i $$0.594805\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 0 0
$$573$$ 8.00000 0.334205
$$574$$ 0 0
$$575$$ −8.00000 −0.333623
$$576$$ 0 0
$$577$$ 34.0000 1.41544 0.707719 0.706494i $$-0.249724\pi$$
0.707719 + 0.706494i $$0.249724\pi$$
$$578$$ 0 0
$$579$$ −6.00000 −0.249351
$$580$$ 0 0
$$581$$ 48.0000 1.99138
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 6.00000 0.248069
$$586$$ 0 0
$$587$$ 36.0000 1.48588 0.742940 0.669359i $$-0.233431\pi$$
0.742940 + 0.669359i $$0.233431\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 10.0000 0.411345
$$592$$ 0 0
$$593$$ 22.0000 0.903432 0.451716 0.892162i $$-0.350812\pi$$
0.451716 + 0.892162i $$0.350812\pi$$
$$594$$ 0 0
$$595$$ 8.00000 0.327968
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 8.00000 0.326871 0.163436 0.986554i $$-0.447742\pi$$
0.163436 + 0.986554i $$0.447742\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 0 0
$$603$$ −4.00000 −0.162893
$$604$$ 0 0
$$605$$ 11.0000 0.447214
$$606$$ 0 0
$$607$$ −12.0000 −0.487065 −0.243532 0.969893i $$-0.578306\pi$$
−0.243532 + 0.969893i $$0.578306\pi$$
$$608$$ 0 0
$$609$$ −24.0000 −0.972529
$$610$$ 0 0
$$611$$ −48.0000 −1.94187
$$612$$ 0 0
$$613$$ 26.0000 1.05013 0.525065 0.851062i $$-0.324041\pi$$
0.525065 + 0.851062i $$0.324041\pi$$
$$614$$ 0 0
$$615$$ −10.0000 −0.403239
$$616$$ 0 0
$$617$$ −26.0000 −1.04672 −0.523360 0.852111i $$-0.675322\pi$$
−0.523360 + 0.852111i $$0.675322\pi$$
$$618$$ 0 0
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 0 0
$$621$$ −8.00000 −0.321029
$$622$$ 0 0
$$623$$ 8.00000 0.320513
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 12.0000 0.478471
$$630$$ 0 0
$$631$$ 40.0000 1.59237 0.796187 0.605050i $$-0.206847\pi$$
0.796187 + 0.605050i $$0.206847\pi$$
$$632$$ 0 0
$$633$$ 12.0000 0.476957
$$634$$ 0 0
$$635$$ 4.00000 0.158735
$$636$$ 0 0
$$637$$ −54.0000 −2.13956
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 0 0
$$643$$ −36.0000 −1.41970 −0.709851 0.704352i $$-0.751238\pi$$
−0.709851 + 0.704352i $$0.751238\pi$$
$$644$$ 0 0
$$645$$ 4.00000 0.157500
$$646$$ 0 0
$$647$$ 48.0000 1.88707 0.943537 0.331266i $$-0.107476\pi$$
0.943537 + 0.331266i $$0.107476\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 0 0
$$655$$ −16.0000 −0.625172
$$656$$ 0 0
$$657$$ −14.0000 −0.546192
$$658$$ 0 0
$$659$$ −16.0000 −0.623272 −0.311636 0.950202i $$-0.600877\pi$$
−0.311636 + 0.950202i $$0.600877\pi$$
$$660$$ 0 0
$$661$$ −2.00000 −0.0777910 −0.0388955 0.999243i $$-0.512384\pi$$
−0.0388955 + 0.999243i $$0.512384\pi$$
$$662$$ 0 0
$$663$$ 12.0000 0.466041
$$664$$ 0 0
$$665$$ −16.0000 −0.620453
$$666$$ 0 0
$$667$$ 48.0000 1.85857
$$668$$ 0 0
$$669$$ 12.0000 0.463947
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −22.0000 −0.848038 −0.424019 0.905653i $$-0.639381\pi$$
−0.424019 + 0.905653i $$0.639381\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ 0 0
$$679$$ 8.00000 0.307012
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ 0 0
$$685$$ 18.0000 0.687745
$$686$$ 0 0
$$687$$ 6.00000 0.228914
$$688$$ 0 0
$$689$$ −60.0000 −2.28582
$$690$$ 0 0
$$691$$ 4.00000 0.152167 0.0760836 0.997101i $$-0.475758\pi$$
0.0760836 + 0.997101i $$0.475758\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 12.0000 0.455186
$$696$$ 0 0
$$697$$ −20.0000 −0.757554
$$698$$ 0 0
$$699$$ −10.0000 −0.378235
$$700$$ 0 0
$$701$$ −38.0000 −1.43524 −0.717620 0.696435i $$-0.754769\pi$$
−0.717620 + 0.696435i $$0.754769\pi$$
$$702$$ 0 0
$$703$$ −24.0000 −0.905177
$$704$$ 0 0
$$705$$ −8.00000 −0.301297
$$706$$ 0 0
$$707$$ −56.0000 −2.10610
$$708$$ 0 0
$$709$$ 22.0000 0.826227 0.413114 0.910679i $$-0.364441\pi$$
0.413114 + 0.910679i $$0.364441\pi$$
$$710$$ 0 0
$$711$$ 16.0000 0.600047
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −24.0000 −0.896296
$$718$$ 0 0
$$719$$ 48.0000 1.79010 0.895049 0.445968i $$-0.147140\pi$$
0.895049 + 0.445968i $$0.147140\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ 0 0
$$723$$ 2.00000 0.0743808
$$724$$ 0 0
$$725$$ −6.00000 −0.222834
$$726$$ 0 0
$$727$$ −4.00000 −0.148352 −0.0741759 0.997245i $$-0.523633\pi$$
−0.0741759 + 0.997245i $$0.523633\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 8.00000 0.295891
$$732$$ 0 0
$$733$$ −46.0000 −1.69905 −0.849524 0.527549i $$-0.823111\pi$$
−0.849524 + 0.527549i $$0.823111\pi$$
$$734$$ 0 0
$$735$$ −9.00000 −0.331970
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ 0 0
$$741$$ −24.0000 −0.881662
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ 6.00000 0.219823
$$746$$ 0 0
$$747$$ 12.0000 0.439057
$$748$$ 0 0
$$749$$ 16.0000 0.584627
$$750$$ 0 0
$$751$$ −48.0000 −1.75154 −0.875772 0.482724i $$-0.839647\pi$$
−0.875772 + 0.482724i $$0.839647\pi$$
$$752$$ 0 0
$$753$$ −8.00000 −0.291536
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 42.0000 1.52652 0.763258 0.646094i $$-0.223599\pi$$
0.763258 + 0.646094i $$0.223599\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 50.0000 1.81250 0.906249 0.422744i $$-0.138933\pi$$
0.906249 + 0.422744i $$0.138933\pi$$
$$762$$ 0 0
$$763$$ −40.0000 −1.44810
$$764$$ 0 0
$$765$$ 2.00000 0.0723102
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −30.0000 −1.08183 −0.540914 0.841078i $$-0.681921\pi$$
−0.540914 + 0.841078i $$0.681921\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ 0 0
$$773$$ −38.0000 −1.36677 −0.683383 0.730061i $$-0.739492\pi$$
−0.683383 + 0.730061i $$0.739492\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −24.0000 −0.860995
$$778$$ 0 0
$$779$$ 40.0000 1.43315
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −6.00000 −0.214423
$$784$$ 0 0
$$785$$ −10.0000 −0.356915
$$786$$ 0 0
$$787$$ 12.0000 0.427754 0.213877 0.976861i $$-0.431391\pi$$
0.213877 + 0.976861i $$0.431391\pi$$
$$788$$ 0 0
$$789$$ −24.0000 −0.854423
$$790$$ 0 0
$$791$$ 24.0000 0.853342
$$792$$ 0 0
$$793$$ −36.0000 −1.27840
$$794$$ 0 0
$$795$$ −10.0000 −0.354663
$$796$$ 0 0
$$797$$ 34.0000 1.20434 0.602171 0.798367i $$-0.294303\pi$$
0.602171 + 0.798367i $$0.294303\pi$$
$$798$$ 0 0
$$799$$ −16.0000 −0.566039
$$800$$ 0 0
$$801$$ 2.00000 0.0706665
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 32.0000 1.12785
$$806$$ 0 0
$$807$$ −6.00000 −0.211210
$$808$$ 0 0
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ −12.0000 −0.421377 −0.210688 0.977553i $$-0.567571\pi$$
−0.210688 + 0.977553i $$0.567571\pi$$
$$812$$ 0 0
$$813$$ 8.00000 0.280572
$$814$$ 0 0
$$815$$ 4.00000 0.140114
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ 0 0
$$819$$ −24.0000 −0.838628
$$820$$ 0 0
$$821$$ 50.0000 1.74501 0.872506 0.488603i $$-0.162493\pi$$
0.872506 + 0.488603i $$0.162493\pi$$
$$822$$ 0 0
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −20.0000 −0.695468 −0.347734 0.937593i $$-0.613049\pi$$
−0.347734 + 0.937593i $$0.613049\pi$$
$$828$$ 0 0
$$829$$ 38.0000 1.31979 0.659897 0.751356i $$-0.270600\pi$$
0.659897 + 0.751356i $$0.270600\pi$$
$$830$$ 0 0
$$831$$ 10.0000 0.346896
$$832$$ 0 0
$$833$$ −18.0000 −0.623663
$$834$$ 0 0
$$835$$ 8.00000 0.276851
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −40.0000 −1.38095 −0.690477 0.723355i $$-0.742599\pi$$
−0.690477 + 0.723355i $$0.742599\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ 2.00000 0.0688837
$$844$$ 0 0
$$845$$ −23.0000 −0.791224
$$846$$ 0 0
$$847$$ −44.0000 −1.51186
$$848$$ 0 0
$$849$$ 4.00000 0.137280
$$850$$ 0 0
$$851$$ 48.0000 1.64542
$$852$$ 0 0
$$853$$ 10.0000 0.342393 0.171197 0.985237i $$-0.445237\pi$$
0.171197 + 0.985237i $$0.445237\pi$$
$$854$$ 0 0
$$855$$ −4.00000 −0.136797
$$856$$ 0 0
$$857$$ 38.0000 1.29806 0.649028 0.760765i $$-0.275176\pi$$
0.649028 + 0.760765i $$0.275176\pi$$
$$858$$ 0 0
$$859$$ 36.0000 1.22830 0.614152 0.789188i $$-0.289498\pi$$
0.614152 + 0.789188i $$0.289498\pi$$
$$860$$ 0 0
$$861$$ 40.0000 1.36320
$$862$$ 0 0
$$863$$ 16.0000 0.544646 0.272323 0.962206i $$-0.412208\pi$$
0.272323 + 0.962206i $$0.412208\pi$$
$$864$$ 0 0
$$865$$ −18.0000 −0.612018
$$866$$ 0 0
$$867$$ −13.0000 −0.441503
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ 0 0
$$873$$ 2.00000 0.0676897
$$874$$ 0 0
$$875$$ −4.00000 −0.135225
$$876$$ 0 0
$$877$$ 10.0000 0.337676 0.168838 0.985644i $$-0.445999\pi$$
0.168838 + 0.985644i $$0.445999\pi$$
$$878$$ 0 0
$$879$$ 10.0000 0.337292
$$880$$ 0 0
$$881$$ 26.0000 0.875962 0.437981 0.898984i $$-0.355694\pi$$
0.437981 + 0.898984i $$0.355694\pi$$
$$882$$ 0 0
$$883$$ 28.0000 0.942275 0.471138 0.882060i $$-0.343844\pi$$
0.471138 + 0.882060i $$0.343844\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ 0 0
$$889$$ −16.0000 −0.536623
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 32.0000 1.07084
$$894$$ 0 0
$$895$$ −8.00000 −0.267411
$$896$$ 0 0
$$897$$ 48.0000 1.60267
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −20.0000 −0.666297
$$902$$ 0 0
$$903$$ −16.0000 −0.532447
$$904$$ 0 0
$$905$$ −14.0000 −0.465376
$$906$$ 0 0
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ 0 0
$$909$$ −14.0000 −0.464351
$$910$$ 0 0
$$911$$ −32.0000 −1.06021 −0.530104 0.847933i $$-0.677847\pi$$
−0.530104 + 0.847933i $$0.677847\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −6.00000 −0.198354
$$916$$ 0 0
$$917$$ 64.0000 2.11347
$$918$$ 0 0
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 0 0
$$921$$ 4.00000 0.131804
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −6.00000 −0.197279
$$926$$ 0 0
$$927$$ 4.00000 0.131377
$$928$$ 0 0
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ 0 0
$$931$$ 36.0000 1.17985
$$932$$ 0 0
$$933$$ 16.0000 0.523816
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 26.0000 0.849383 0.424691 0.905338i $$-0.360383\pi$$
0.424691 + 0.905338i $$0.360383\pi$$
$$938$$ 0 0
$$939$$ −14.0000 −0.456873
$$940$$ 0 0
$$941$$ −14.0000 −0.456387 −0.228193 0.973616i $$-0.573282\pi$$
−0.228193 + 0.973616i $$0.573282\pi$$
$$942$$ 0 0
$$943$$ −80.0000 −2.60516
$$944$$ 0 0
$$945$$ −4.00000 −0.130120
$$946$$ 0 0
$$947$$ −44.0000 −1.42981 −0.714904 0.699223i $$-0.753530\pi$$
−0.714904 + 0.699223i $$0.753530\pi$$
$$948$$ 0 0
$$949$$ 84.0000 2.72676
$$950$$ 0 0
$$951$$ 2.00000 0.0648544
$$952$$ 0 0
$$953$$ −18.0000 −0.583077 −0.291539 0.956559i $$-0.594167\pi$$
−0.291539 + 0.956559i $$0.594167\pi$$
$$954$$ 0 0
$$955$$ −8.00000 −0.258874
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −72.0000 −2.32500
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 4.00000 0.128898
$$964$$ 0 0
$$965$$ 6.00000 0.193147
$$966$$ 0 0
$$967$$ 20.0000 0.643157 0.321578 0.946883i $$-0.395787\pi$$
0.321578 + 0.946883i $$0.395787\pi$$
$$968$$ 0 0
$$969$$ −8.00000 −0.256997
$$970$$ 0 0
$$971$$ −56.0000 −1.79713 −0.898563 0.438845i $$-0.855388\pi$$
−0.898563 + 0.438845i $$0.855388\pi$$
$$972$$ 0 0
$$973$$ −48.0000 −1.53881
$$974$$ 0 0
$$975$$ −6.00000 −0.192154
$$976$$ 0 0
$$977$$ −50.0000 −1.59964 −0.799821 0.600239i $$-0.795072\pi$$
−0.799821 + 0.600239i $$0.795072\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −10.0000 −0.319275
$$982$$ 0 0
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ −10.0000 −0.318626
$$986$$ 0 0
$$987$$ 32.0000 1.01857
$$988$$ 0 0
$$989$$ 32.0000 1.01754
$$990$$ 0 0
$$991$$ 56.0000 1.77890 0.889449 0.457034i $$-0.151088\pi$$
0.889449 + 0.457034i $$0.151088\pi$$
$$992$$ 0 0
$$993$$ 28.0000 0.888553
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −30.0000 −0.950110 −0.475055 0.879956i $$-0.657572\pi$$
−0.475055 + 0.879956i $$0.657572\pi$$
$$998$$ 0 0
$$999$$ −6.00000 −0.189832
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 120.2.a.a.1.1 1
3.2 odd 2 360.2.a.e.1.1 1
4.3 odd 2 240.2.a.a.1.1 1
5.2 odd 4 600.2.f.c.49.1 2
5.3 odd 4 600.2.f.c.49.2 2
5.4 even 2 600.2.a.a.1.1 1
7.6 odd 2 5880.2.a.p.1.1 1
8.3 odd 2 960.2.a.n.1.1 1
8.5 even 2 960.2.a.g.1.1 1
9.2 odd 6 3240.2.q.a.1081.1 2
9.4 even 3 3240.2.q.m.2161.1 2
9.5 odd 6 3240.2.q.a.2161.1 2
9.7 even 3 3240.2.q.m.1081.1 2
12.11 even 2 720.2.a.f.1.1 1
15.2 even 4 1800.2.f.g.649.2 2
15.8 even 4 1800.2.f.g.649.1 2
15.14 odd 2 1800.2.a.c.1.1 1
16.3 odd 4 3840.2.k.z.1921.1 2
16.5 even 4 3840.2.k.a.1921.1 2
16.11 odd 4 3840.2.k.z.1921.2 2
16.13 even 4 3840.2.k.a.1921.2 2
20.3 even 4 1200.2.f.f.49.1 2
20.7 even 4 1200.2.f.f.49.2 2
20.19 odd 2 1200.2.a.r.1.1 1
24.5 odd 2 2880.2.a.r.1.1 1
24.11 even 2 2880.2.a.b.1.1 1
40.3 even 4 4800.2.f.n.3649.2 2
40.13 odd 4 4800.2.f.u.3649.1 2
40.19 odd 2 4800.2.a.bh.1.1 1
40.27 even 4 4800.2.f.n.3649.1 2
40.29 even 2 4800.2.a.bl.1.1 1
40.37 odd 4 4800.2.f.u.3649.2 2
60.23 odd 4 3600.2.f.l.2449.2 2
60.47 odd 4 3600.2.f.l.2449.1 2
60.59 even 2 3600.2.a.bo.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.a.a.1.1 1 1.1 even 1 trivial
240.2.a.a.1.1 1 4.3 odd 2
360.2.a.e.1.1 1 3.2 odd 2
600.2.a.a.1.1 1 5.4 even 2
600.2.f.c.49.1 2 5.2 odd 4
600.2.f.c.49.2 2 5.3 odd 4
720.2.a.f.1.1 1 12.11 even 2
960.2.a.g.1.1 1 8.5 even 2
960.2.a.n.1.1 1 8.3 odd 2
1200.2.a.r.1.1 1 20.19 odd 2
1200.2.f.f.49.1 2 20.3 even 4
1200.2.f.f.49.2 2 20.7 even 4
1800.2.a.c.1.1 1 15.14 odd 2
1800.2.f.g.649.1 2 15.8 even 4
1800.2.f.g.649.2 2 15.2 even 4
2880.2.a.b.1.1 1 24.11 even 2
2880.2.a.r.1.1 1 24.5 odd 2
3240.2.q.a.1081.1 2 9.2 odd 6
3240.2.q.a.2161.1 2 9.5 odd 6
3240.2.q.m.1081.1 2 9.7 even 3
3240.2.q.m.2161.1 2 9.4 even 3
3600.2.a.bo.1.1 1 60.59 even 2
3600.2.f.l.2449.1 2 60.47 odd 4
3600.2.f.l.2449.2 2 60.23 odd 4
3840.2.k.a.1921.1 2 16.5 even 4
3840.2.k.a.1921.2 2 16.13 even 4
3840.2.k.z.1921.1 2 16.3 odd 4
3840.2.k.z.1921.2 2 16.11 odd 4
4800.2.a.bh.1.1 1 40.19 odd 2
4800.2.a.bl.1.1 1 40.29 even 2
4800.2.f.n.3649.1 2 40.27 even 4
4800.2.f.n.3649.2 2 40.3 even 4
4800.2.f.u.3649.1 2 40.13 odd 4
4800.2.f.u.3649.2 2 40.37 odd 4
5880.2.a.p.1.1 1 7.6 odd 2