# Properties

 Label 1089.2.a.k.1.1 Level $1089$ Weight $2$ Character 1089.1 Self dual yes Analytic conductor $8.696$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,2,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1089.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1089.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: $$8.69570878012$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 363) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1089.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} +2.00000 q^{4} -4.00000 q^{5} +1.00000 q^{7} +O(q^{10})$$ $$q+2.00000 q^{2} +2.00000 q^{4} -4.00000 q^{5} +1.00000 q^{7} -8.00000 q^{10} -2.00000 q^{13} +2.00000 q^{14} -4.00000 q^{16} -4.00000 q^{17} -3.00000 q^{19} -8.00000 q^{20} -2.00000 q^{23} +11.0000 q^{25} -4.00000 q^{26} +2.00000 q^{28} -6.00000 q^{29} -5.00000 q^{31} -8.00000 q^{32} -8.00000 q^{34} -4.00000 q^{35} +3.00000 q^{37} -6.00000 q^{38} +2.00000 q^{41} +12.0000 q^{43} -4.00000 q^{46} -2.00000 q^{47} -6.00000 q^{49} +22.0000 q^{50} -4.00000 q^{52} -6.00000 q^{53} -12.0000 q^{58} +10.0000 q^{59} +3.00000 q^{61} -10.0000 q^{62} -8.00000 q^{64} +8.00000 q^{65} -1.00000 q^{67} -8.00000 q^{68} -8.00000 q^{70} -11.0000 q^{73} +6.00000 q^{74} -6.00000 q^{76} +11.0000 q^{79} +16.0000 q^{80} +4.00000 q^{82} -6.00000 q^{83} +16.0000 q^{85} +24.0000 q^{86} -12.0000 q^{89} -2.00000 q^{91} -4.00000 q^{92} -4.00000 q^{94} +12.0000 q^{95} +5.00000 q^{97} -12.0000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 1.41421 0.707107 0.707107i $$-0.250000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$3$$ 0 0
$$4$$ 2.00000 1.00000
$$5$$ −4.00000 −1.78885 −0.894427 0.447214i $$-0.852416\pi$$
−0.894427 + 0.447214i $$0.852416\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964 0.188982 0.981981i $$-0.439481\pi$$
0.188982 + 0.981981i $$0.439481\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ −8.00000 −2.52982
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 0 0
$$19$$ −3.00000 −0.688247 −0.344124 0.938924i $$-0.611824\pi$$
−0.344124 + 0.938924i $$0.611824\pi$$
$$20$$ −8.00000 −1.78885
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −2.00000 −0.417029 −0.208514 0.978019i $$-0.566863\pi$$
−0.208514 + 0.978019i $$0.566863\pi$$
$$24$$ 0 0
$$25$$ 11.0000 2.20000
$$26$$ −4.00000 −0.784465
$$27$$ 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
$$31$$ −5.00000 −0.898027 −0.449013 0.893525i $$-0.648224\pi$$
−0.449013 + 0.893525i $$0.648224\pi$$
$$32$$ −8.00000 −1.41421
$$33$$ 0 0
$$34$$ −8.00000 −1.37199
$$35$$ −4.00000 −0.676123
$$36$$ 0 0
$$37$$ 3.00000 0.493197 0.246598 0.969118i $$-0.420687\pi$$
0.246598 + 0.969118i $$0.420687\pi$$
$$38$$ −6.00000 −0.973329
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ 12.0000 1.82998 0.914991 0.403473i $$-0.132197\pi$$
0.914991 + 0.403473i $$0.132197\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ −2.00000 −0.291730 −0.145865 0.989305i $$-0.546597\pi$$
−0.145865 + 0.989305i $$0.546597\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 22.0000 3.11127
$$51$$ 0 0
$$52$$ −4.00000 −0.554700
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −12.0000 −1.57568
$$59$$ 10.0000 1.30189 0.650945 0.759125i $$-0.274373\pi$$
0.650945 + 0.759125i $$0.274373\pi$$
$$60$$ 0 0
$$61$$ 3.00000 0.384111 0.192055 0.981384i $$-0.438485\pi$$
0.192055 + 0.981384i $$0.438485\pi$$
$$62$$ −10.0000 −1.27000
$$63$$ 0 0
$$64$$ −8.00000 −1.00000
$$65$$ 8.00000 0.992278
$$66$$ 0 0
$$67$$ −1.00000 −0.122169 −0.0610847 0.998133i $$-0.519456\pi$$
−0.0610847 + 0.998133i $$0.519456\pi$$
$$68$$ −8.00000 −0.970143
$$69$$ 0 0
$$70$$ −8.00000 −0.956183
$$71$$ 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 $$0.722612\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ −6.00000 −0.688247
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 11.0000 1.23760 0.618798 0.785550i $$-0.287620\pi$$
0.618798 + 0.785550i $$0.287620\pi$$
$$80$$ 16.0000 1.78885
$$81$$ 0 0
$$82$$ 4.00000 0.441726
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ 0 0
$$85$$ 16.0000 1.73544
$$86$$ 24.0000 2.58799
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −12.0000 −1.27200 −0.635999 0.771690i $$-0.719412\pi$$
−0.635999 + 0.771690i $$0.719412\pi$$
$$90$$ 0 0
$$91$$ −2.00000 −0.209657
$$92$$ −4.00000 −0.417029
$$93$$ 0 0
$$94$$ −4.00000 −0.412568
$$95$$ 12.0000 1.23117
$$96$$ 0 0
$$97$$ 5.00000 0.507673 0.253837 0.967247i $$-0.418307\pi$$
0.253837 + 0.967247i $$0.418307\pi$$
$$98$$ −12.0000 −1.21218
$$99$$ 0 0
$$100$$ 22.0000 2.20000
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ 0 0
$$103$$ −7.00000 −0.689730 −0.344865 0.938652i $$-0.612075\pi$$
−0.344865 + 0.938652i $$0.612075\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −12.0000 −1.16554
$$107$$ 18.0000 1.74013 0.870063 0.492941i $$-0.164078\pi$$
0.870063 + 0.492941i $$0.164078\pi$$
$$108$$ 0 0
$$109$$ 1.00000 0.0957826 0.0478913 0.998853i $$-0.484750\pi$$
0.0478913 + 0.998853i $$0.484750\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −4.00000 −0.377964
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 8.00000 0.746004
$$116$$ −12.0000 −1.11417
$$117$$ 0 0
$$118$$ 20.0000 1.84115
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 6.00000 0.543214
$$123$$ 0 0
$$124$$ −10.0000 −0.898027
$$125$$ −24.0000 −2.14663
$$126$$ 0 0
$$127$$ −13.0000 −1.15356 −0.576782 0.816898i $$-0.695692\pi$$
−0.576782 + 0.816898i $$0.695692\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 16.0000 1.40329
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ −3.00000 −0.260133
$$134$$ −2.00000 −0.172774
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −8.00000 −0.683486 −0.341743 0.939793i $$-0.611017\pi$$
−0.341743 + 0.939793i $$0.611017\pi$$
$$138$$ 0 0
$$139$$ 16.0000 1.35710 0.678551 0.734553i $$-0.262608\pi$$
0.678551 + 0.734553i $$0.262608\pi$$
$$140$$ −8.00000 −0.676123
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 24.0000 1.99309
$$146$$ −22.0000 −1.82073
$$147$$ 0 0
$$148$$ 6.00000 0.493197
$$149$$ 16.0000 1.31077 0.655386 0.755295i $$-0.272506\pi$$
0.655386 + 0.755295i $$0.272506\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 20.0000 1.60644
$$156$$ 0 0
$$157$$ −1.00000 −0.0798087 −0.0399043 0.999204i $$-0.512705\pi$$
−0.0399043 + 0.999204i $$0.512705\pi$$
$$158$$ 22.0000 1.75023
$$159$$ 0 0
$$160$$ 32.0000 2.52982
$$161$$ −2.00000 −0.157622
$$162$$ 0 0
$$163$$ 25.0000 1.95815 0.979076 0.203497i $$-0.0652307\pi$$
0.979076 + 0.203497i $$0.0652307\pi$$
$$164$$ 4.00000 0.312348
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 18.0000 1.39288 0.696441 0.717614i $$-0.254766\pi$$
0.696441 + 0.717614i $$0.254766\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 32.0000 2.45429
$$171$$ 0 0
$$172$$ 24.0000 1.82998
$$173$$ −24.0000 −1.82469 −0.912343 0.409426i $$-0.865729\pi$$
−0.912343 + 0.409426i $$0.865729\pi$$
$$174$$ 0 0
$$175$$ 11.0000 0.831522
$$176$$ 0 0
$$177$$ 0 0
$$178$$ −24.0000 −1.79888
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ 0 0
$$181$$ −23.0000 −1.70958 −0.854788 0.518977i $$-0.826313\pi$$
−0.854788 + 0.518977i $$0.826313\pi$$
$$182$$ −4.00000 −0.296500
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −12.0000 −0.882258
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −4.00000 −0.291730
$$189$$ 0 0
$$190$$ 24.0000 1.74114
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 0 0
$$193$$ −5.00000 −0.359908 −0.179954 0.983675i $$-0.557595\pi$$
−0.179954 + 0.983675i $$0.557595\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ −12.0000 −0.857143
$$197$$ 8.00000 0.569976 0.284988 0.958531i $$-0.408010\pi$$
0.284988 + 0.958531i $$0.408010\pi$$
$$198$$ 0 0
$$199$$ −21.0000 −1.48865 −0.744325 0.667817i $$-0.767229\pi$$
−0.744325 + 0.667817i $$0.767229\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 20.0000 1.40720
$$203$$ −6.00000 −0.421117
$$204$$ 0 0
$$205$$ −8.00000 −0.558744
$$206$$ −14.0000 −0.975426
$$207$$ 0 0
$$208$$ 8.00000 0.554700
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −21.0000 −1.44570 −0.722850 0.691005i $$-0.757168\pi$$
−0.722850 + 0.691005i $$0.757168\pi$$
$$212$$ −12.0000 −0.824163
$$213$$ 0 0
$$214$$ 36.0000 2.46091
$$215$$ −48.0000 −3.27357
$$216$$ 0 0
$$217$$ −5.00000 −0.339422
$$218$$ 2.00000 0.135457
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 8.00000 0.538138
$$222$$ 0 0
$$223$$ −17.0000 −1.13840 −0.569202 0.822198i $$-0.692748\pi$$
−0.569202 + 0.822198i $$0.692748\pi$$
$$224$$ −8.00000 −0.534522
$$225$$ 0 0
$$226$$ −12.0000 −0.798228
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ −18.0000 −1.18947 −0.594737 0.803921i $$-0.702744\pi$$
−0.594737 + 0.803921i $$0.702744\pi$$
$$230$$ 16.0000 1.05501
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ 0 0
$$235$$ 8.00000 0.521862
$$236$$ 20.0000 1.30189
$$237$$ 0 0
$$238$$ −8.00000 −0.518563
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 6.00000 0.384111
$$245$$ 24.0000 1.53330
$$246$$ 0 0
$$247$$ 6.00000 0.381771
$$248$$ 0 0
$$249$$ 0 0
$$250$$ −48.0000 −3.03579
$$251$$ 2.00000 0.126239 0.0631194 0.998006i $$-0.479895\pi$$
0.0631194 + 0.998006i $$0.479895\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −26.0000 −1.63139
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 14.0000 0.873296 0.436648 0.899632i $$-0.356166\pi$$
0.436648 + 0.899632i $$0.356166\pi$$
$$258$$ 0 0
$$259$$ 3.00000 0.186411
$$260$$ 16.0000 0.992278
$$261$$ 0 0
$$262$$ 12.0000 0.741362
$$263$$ 10.0000 0.616626 0.308313 0.951285i $$-0.400236\pi$$
0.308313 + 0.951285i $$0.400236\pi$$
$$264$$ 0 0
$$265$$ 24.0000 1.47431
$$266$$ −6.00000 −0.367884
$$267$$ 0 0
$$268$$ −2.00000 −0.122169
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 16.0000 0.970143
$$273$$ 0 0
$$274$$ −16.0000 −0.966595
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −11.0000 −0.660926 −0.330463 0.943819i $$-0.607205\pi$$
−0.330463 + 0.943819i $$0.607205\pi$$
$$278$$ 32.0000 1.91923
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 12.0000 0.715860 0.357930 0.933748i $$-0.383483\pi$$
0.357930 + 0.933748i $$0.383483\pi$$
$$282$$ 0 0
$$283$$ −11.0000 −0.653882 −0.326941 0.945045i $$-0.606018\pi$$
−0.326941 + 0.945045i $$0.606018\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 2.00000 0.118056
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 48.0000 2.81866
$$291$$ 0 0
$$292$$ −22.0000 −1.28745
$$293$$ −12.0000 −0.701047 −0.350524 0.936554i $$-0.613996\pi$$
−0.350524 + 0.936554i $$0.613996\pi$$
$$294$$ 0 0
$$295$$ −40.0000 −2.32889
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 32.0000 1.85371
$$299$$ 4.00000 0.231326
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ −32.0000 −1.84139
$$303$$ 0 0
$$304$$ 12.0000 0.688247
$$305$$ −12.0000 −0.687118
$$306$$ 0 0
$$307$$ −19.0000 −1.08439 −0.542194 0.840254i $$-0.682406\pi$$
−0.542194 + 0.840254i $$0.682406\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 40.0000 2.27185
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ 22.0000 1.23760
$$317$$ 20.0000 1.12331 0.561656 0.827371i $$-0.310164\pi$$
0.561656 + 0.827371i $$0.310164\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 32.0000 1.78885
$$321$$ 0 0
$$322$$ −4.00000 −0.222911
$$323$$ 12.0000 0.667698
$$324$$ 0 0
$$325$$ −22.0000 −1.22034
$$326$$ 50.0000 2.76924
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −2.00000 −0.110264
$$330$$ 0 0
$$331$$ −11.0000 −0.604615 −0.302307 0.953211i $$-0.597757\pi$$
−0.302307 + 0.953211i $$0.597757\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ 0 0
$$334$$ 36.0000 1.96983
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ 5.00000 0.272367 0.136184 0.990684i $$-0.456516\pi$$
0.136184 + 0.990684i $$0.456516\pi$$
$$338$$ −18.0000 −0.979071
$$339$$ 0 0
$$340$$ 32.0000 1.73544
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −13.0000 −0.701934
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −48.0000 −2.58050
$$347$$ 2.00000 0.107366 0.0536828 0.998558i $$-0.482904\pi$$
0.0536828 + 0.998558i $$0.482904\pi$$
$$348$$ 0 0
$$349$$ 15.0000 0.802932 0.401466 0.915874i $$-0.368501\pi$$
0.401466 + 0.915874i $$0.368501\pi$$
$$350$$ 22.0000 1.17595
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 12.0000 0.638696 0.319348 0.947638i $$-0.396536\pi$$
0.319348 + 0.947638i $$0.396536\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −24.0000 −1.27200
$$357$$ 0 0
$$358$$ −12.0000 −0.634220
$$359$$ −4.00000 −0.211112 −0.105556 0.994413i $$-0.533662\pi$$
−0.105556 + 0.994413i $$0.533662\pi$$
$$360$$ 0 0
$$361$$ −10.0000 −0.526316
$$362$$ −46.0000 −2.41771
$$363$$ 0 0
$$364$$ −4.00000 −0.209657
$$365$$ 44.0000 2.30307
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 8.00000 0.417029
$$369$$ 0 0
$$370$$ −24.0000 −1.24770
$$371$$ −6.00000 −0.311504
$$372$$ 0 0
$$373$$ 7.00000 0.362446 0.181223 0.983442i $$-0.441994\pi$$
0.181223 + 0.983442i $$0.441994\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 24.0000 1.23117
$$381$$ 0 0
$$382$$ −16.0000 −0.818631
$$383$$ −26.0000 −1.32854 −0.664269 0.747494i $$-0.731257\pi$$
−0.664269 + 0.747494i $$0.731257\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −10.0000 −0.508987
$$387$$ 0 0
$$388$$ 10.0000 0.507673
$$389$$ −18.0000 −0.912636 −0.456318 0.889817i $$-0.650832\pi$$
−0.456318 + 0.889817i $$0.650832\pi$$
$$390$$ 0 0
$$391$$ 8.00000 0.404577
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 16.0000 0.806068
$$395$$ −44.0000 −2.21388
$$396$$ 0 0
$$397$$ 31.0000 1.55585 0.777923 0.628360i $$-0.216273\pi$$
0.777923 + 0.628360i $$0.216273\pi$$
$$398$$ −42.0000 −2.10527
$$399$$ 0 0
$$400$$ −44.0000 −2.20000
$$401$$ 28.0000 1.39825 0.699127 0.714998i $$-0.253572\pi$$
0.699127 + 0.714998i $$0.253572\pi$$
$$402$$ 0 0
$$403$$ 10.0000 0.498135
$$404$$ 20.0000 0.995037
$$405$$ 0 0
$$406$$ −12.0000 −0.595550
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 21.0000 1.03838 0.519192 0.854658i $$-0.326233\pi$$
0.519192 + 0.854658i $$0.326233\pi$$
$$410$$ −16.0000 −0.790184
$$411$$ 0 0
$$412$$ −14.0000 −0.689730
$$413$$ 10.0000 0.492068
$$414$$ 0 0
$$415$$ 24.0000 1.17811
$$416$$ 16.0000 0.784465
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −26.0000 −1.27018 −0.635092 0.772437i $$-0.719038\pi$$
−0.635092 + 0.772437i $$0.719038\pi$$
$$420$$ 0 0
$$421$$ −2.00000 −0.0974740 −0.0487370 0.998812i $$-0.515520\pi$$
−0.0487370 + 0.998812i $$0.515520\pi$$
$$422$$ −42.0000 −2.04453
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −44.0000 −2.13431
$$426$$ 0 0
$$427$$ 3.00000 0.145180
$$428$$ 36.0000 1.74013
$$429$$ 0 0
$$430$$ −96.0000 −4.62953
$$431$$ −18.0000 −0.867029 −0.433515 0.901146i $$-0.642727\pi$$
−0.433515 + 0.901146i $$0.642727\pi$$
$$432$$ 0 0
$$433$$ −17.0000 −0.816968 −0.408484 0.912766i $$-0.633942\pi$$
−0.408484 + 0.912766i $$0.633942\pi$$
$$434$$ −10.0000 −0.480015
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ 6.00000 0.287019
$$438$$ 0 0
$$439$$ 37.0000 1.76591 0.882957 0.469454i $$-0.155549\pi$$
0.882957 + 0.469454i $$0.155549\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 16.0000 0.761042
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ 0 0
$$445$$ 48.0000 2.27542
$$446$$ −34.0000 −1.60995
$$447$$ 0 0
$$448$$ −8.00000 −0.377964
$$449$$ −20.0000 −0.943858 −0.471929 0.881636i $$-0.656442\pi$$
−0.471929 + 0.881636i $$0.656442\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −12.0000 −0.564433
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 8.00000 0.375046
$$456$$ 0 0
$$457$$ 18.0000 0.842004 0.421002 0.907060i $$-0.361678\pi$$
0.421002 + 0.907060i $$0.361678\pi$$
$$458$$ −36.0000 −1.68217
$$459$$ 0 0
$$460$$ 16.0000 0.746004
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ 24.0000 1.11417
$$465$$ 0 0
$$466$$ −36.0000 −1.66767
$$467$$ −24.0000 −1.11059 −0.555294 0.831654i $$-0.687394\pi$$
−0.555294 + 0.831654i $$0.687394\pi$$
$$468$$ 0 0
$$469$$ −1.00000 −0.0461757
$$470$$ 16.0000 0.738025
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −33.0000 −1.51414
$$476$$ −8.00000 −0.366679
$$477$$ 0 0
$$478$$ −12.0000 −0.548867
$$479$$ 22.0000 1.00521 0.502603 0.864517i $$-0.332376\pi$$
0.502603 + 0.864517i $$0.332376\pi$$
$$480$$ 0 0
$$481$$ −6.00000 −0.273576
$$482$$ −28.0000 −1.27537
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −20.0000 −0.908153
$$486$$ 0 0
$$487$$ −40.0000 −1.81257 −0.906287 0.422664i $$-0.861095\pi$$
−0.906287 + 0.422664i $$0.861095\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 48.0000 2.16842
$$491$$ 14.0000 0.631811 0.315906 0.948791i $$-0.397692\pi$$
0.315906 + 0.948791i $$0.397692\pi$$
$$492$$ 0 0
$$493$$ 24.0000 1.08091
$$494$$ 12.0000 0.539906
$$495$$ 0 0
$$496$$ 20.0000 0.898027
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 23.0000 1.02962 0.514811 0.857304i $$-0.327862\pi$$
0.514811 + 0.857304i $$0.327862\pi$$
$$500$$ −48.0000 −2.14663
$$501$$ 0 0
$$502$$ 4.00000 0.178529
$$503$$ 32.0000 1.42681 0.713405 0.700752i $$-0.247152\pi$$
0.713405 + 0.700752i $$0.247152\pi$$
$$504$$ 0 0
$$505$$ −40.0000 −1.77998
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −26.0000 −1.15356
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ −11.0000 −0.486611
$$512$$ 32.0000 1.41421
$$513$$ 0 0
$$514$$ 28.0000 1.23503
$$515$$ 28.0000 1.23383
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 6.00000 0.263625
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 0 0
$$523$$ 29.0000 1.26808 0.634041 0.773300i $$-0.281395\pi$$
0.634041 + 0.773300i $$0.281395\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ 20.0000 0.872041
$$527$$ 20.0000 0.871214
$$528$$ 0 0
$$529$$ −19.0000 −0.826087
$$530$$ 48.0000 2.08499
$$531$$ 0 0
$$532$$ −6.00000 −0.260133
$$533$$ −4.00000 −0.173259
$$534$$ 0 0
$$535$$ −72.0000 −3.11283
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 28.0000 1.20717
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 0 0
$$544$$ 32.0000 1.37199
$$545$$ −4.00000 −0.171341
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ −16.0000 −0.683486
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 18.0000 0.766826
$$552$$ 0 0
$$553$$ 11.0000 0.467768
$$554$$ −22.0000 −0.934690
$$555$$ 0 0
$$556$$ 32.0000 1.35710
$$557$$ 8.00000 0.338971 0.169485 0.985533i $$-0.445789\pi$$
0.169485 + 0.985533i $$0.445789\pi$$
$$558$$ 0 0
$$559$$ −24.0000 −1.01509
$$560$$ 16.0000 0.676123
$$561$$ 0 0
$$562$$ 24.0000 1.01238
$$563$$ −28.0000 −1.18006 −0.590030 0.807382i $$-0.700884\pi$$
−0.590030 + 0.807382i $$0.700884\pi$$
$$564$$ 0 0
$$565$$ 24.0000 1.00969
$$566$$ −22.0000 −0.924729
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −12.0000 −0.503066 −0.251533 0.967849i $$-0.580935\pi$$
−0.251533 + 0.967849i $$0.580935\pi$$
$$570$$ 0 0
$$571$$ −25.0000 −1.04622 −0.523109 0.852266i $$-0.675228\pi$$
−0.523109 + 0.852266i $$0.675228\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 4.00000 0.166957
$$575$$ −22.0000 −0.917463
$$576$$ 0 0
$$577$$ 15.0000 0.624458 0.312229 0.950007i $$-0.398924\pi$$
0.312229 + 0.950007i $$0.398924\pi$$
$$578$$ −2.00000 −0.0831890
$$579$$ 0 0
$$580$$ 48.0000 1.99309
$$581$$ −6.00000 −0.248922
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ −24.0000 −0.991431
$$587$$ −4.00000 −0.165098 −0.0825488 0.996587i $$-0.526306\pi$$
−0.0825488 + 0.996587i $$0.526306\pi$$
$$588$$ 0 0
$$589$$ 15.0000 0.618064
$$590$$ −80.0000 −3.29355
$$591$$ 0 0
$$592$$ −12.0000 −0.493197
$$593$$ 46.0000 1.88899 0.944497 0.328521i $$-0.106550\pi$$
0.944497 + 0.328521i $$0.106550\pi$$
$$594$$ 0 0
$$595$$ 16.0000 0.655936
$$596$$ 32.0000 1.31077
$$597$$ 0 0
$$598$$ 8.00000 0.327144
$$599$$ 8.00000 0.326871 0.163436 0.986554i $$-0.447742\pi$$
0.163436 + 0.986554i $$0.447742\pi$$
$$600$$ 0 0
$$601$$ −1.00000 −0.0407909 −0.0203954 0.999792i $$-0.506493\pi$$
−0.0203954 + 0.999792i $$0.506493\pi$$
$$602$$ 24.0000 0.978167
$$603$$ 0 0
$$604$$ −32.0000 −1.30206
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 8.00000 0.324710 0.162355 0.986732i $$-0.448091\pi$$
0.162355 + 0.986732i $$0.448091\pi$$
$$608$$ 24.0000 0.973329
$$609$$ 0 0
$$610$$ −24.0000 −0.971732
$$611$$ 4.00000 0.161823
$$612$$ 0 0
$$613$$ −13.0000 −0.525065 −0.262533 0.964923i $$-0.584558\pi$$
−0.262533 + 0.964923i $$0.584558\pi$$
$$614$$ −38.0000 −1.53356
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 24.0000 0.966204 0.483102 0.875564i $$-0.339510\pi$$
0.483102 + 0.875564i $$0.339510\pi$$
$$618$$ 0 0
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 40.0000 1.60644
$$621$$ 0 0
$$622$$ −48.0000 −1.92462
$$623$$ −12.0000 −0.480770
$$624$$ 0 0
$$625$$ 41.0000 1.64000
$$626$$ −20.0000 −0.799361
$$627$$ 0 0
$$628$$ −2.00000 −0.0798087
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ −32.0000 −1.27390 −0.636950 0.770905i $$-0.719804\pi$$
−0.636950 + 0.770905i $$0.719804\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 40.0000 1.58860
$$635$$ 52.0000 2.06356
$$636$$ 0 0
$$637$$ 12.0000 0.475457
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 24.0000 0.947943 0.473972 0.880540i $$-0.342820\pi$$
0.473972 + 0.880540i $$0.342820\pi$$
$$642$$ 0 0
$$643$$ −37.0000 −1.45914 −0.729569 0.683907i $$-0.760279\pi$$
−0.729569 + 0.683907i $$0.760279\pi$$
$$644$$ −4.00000 −0.157622
$$645$$ 0 0
$$646$$ 24.0000 0.944267
$$647$$ 4.00000 0.157256 0.0786281 0.996904i $$-0.474946\pi$$
0.0786281 + 0.996904i $$0.474946\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ −44.0000 −1.72582
$$651$$ 0 0
$$652$$ 50.0000 1.95815
$$653$$ −10.0000 −0.391330 −0.195665 0.980671i $$-0.562687\pi$$
−0.195665 + 0.980671i $$0.562687\pi$$
$$654$$ 0 0
$$655$$ −24.0000 −0.937758
$$656$$ −8.00000 −0.312348
$$657$$ 0 0
$$658$$ −4.00000 −0.155936
$$659$$ −46.0000 −1.79191 −0.895953 0.444149i $$-0.853506\pi$$
−0.895953 + 0.444149i $$0.853506\pi$$
$$660$$ 0 0
$$661$$ −5.00000 −0.194477 −0.0972387 0.995261i $$-0.531001\pi$$
−0.0972387 + 0.995261i $$0.531001\pi$$
$$662$$ −22.0000 −0.855054
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 12.0000 0.465340
$$666$$ 0 0
$$667$$ 12.0000 0.464642
$$668$$ 36.0000 1.39288
$$669$$ 0 0
$$670$$ 8.00000 0.309067
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −13.0000 −0.501113 −0.250557 0.968102i $$-0.580614\pi$$
−0.250557 + 0.968102i $$0.580614\pi$$
$$674$$ 10.0000 0.385186
$$675$$ 0 0
$$676$$ −18.0000 −0.692308
$$677$$ −12.0000 −0.461197 −0.230599 0.973049i $$-0.574068\pi$$
−0.230599 + 0.973049i $$0.574068\pi$$
$$678$$ 0 0
$$679$$ 5.00000 0.191882
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 34.0000 1.30097 0.650487 0.759517i $$-0.274565\pi$$
0.650487 + 0.759517i $$0.274565\pi$$
$$684$$ 0 0
$$685$$ 32.0000 1.22266
$$686$$ −26.0000 −0.992685
$$687$$ 0 0
$$688$$ −48.0000 −1.82998
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ 11.0000 0.418460 0.209230 0.977866i $$-0.432904\pi$$
0.209230 + 0.977866i $$0.432904\pi$$
$$692$$ −48.0000 −1.82469
$$693$$ 0 0
$$694$$ 4.00000 0.151838
$$695$$ −64.0000 −2.42766
$$696$$ 0 0
$$697$$ −8.00000 −0.303022
$$698$$ 30.0000 1.13552
$$699$$ 0 0
$$700$$ 22.0000 0.831522
$$701$$ −50.0000 −1.88847 −0.944237 0.329267i $$-0.893198\pi$$
−0.944237 + 0.329267i $$0.893198\pi$$
$$702$$ 0 0
$$703$$ −9.00000 −0.339441
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 24.0000 0.903252
$$707$$ 10.0000 0.376089
$$708$$ 0 0
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 10.0000 0.374503
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ −8.00000 −0.298557
$$719$$ 6.00000 0.223762 0.111881 0.993722i $$-0.464312\pi$$
0.111881 + 0.993722i $$0.464312\pi$$
$$720$$ 0 0
$$721$$ −7.00000 −0.260694
$$722$$ −20.0000 −0.744323
$$723$$ 0 0
$$724$$ −46.0000 −1.70958
$$725$$ −66.0000 −2.45118
$$726$$ 0 0
$$727$$ −12.0000 −0.445055 −0.222528 0.974926i $$-0.571431\pi$$
−0.222528 + 0.974926i $$0.571431\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 88.0000 3.25703
$$731$$ −48.0000 −1.77534
$$732$$ 0 0
$$733$$ −30.0000 −1.10808 −0.554038 0.832492i $$-0.686914\pi$$
−0.554038 + 0.832492i $$0.686914\pi$$
$$734$$ −16.0000 −0.590571
$$735$$ 0 0
$$736$$ 16.0000 0.589768
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 41.0000 1.50821 0.754105 0.656754i $$-0.228071\pi$$
0.754105 + 0.656754i $$0.228071\pi$$
$$740$$ −24.0000 −0.882258
$$741$$ 0 0
$$742$$ −12.0000 −0.440534
$$743$$ 20.0000 0.733729 0.366864 0.930274i $$-0.380431\pi$$
0.366864 + 0.930274i $$0.380431\pi$$
$$744$$ 0 0
$$745$$ −64.0000 −2.34478
$$746$$ 14.0000 0.512576
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 18.0000 0.657706
$$750$$ 0 0
$$751$$ 19.0000 0.693320 0.346660 0.937991i $$-0.387316\pi$$
0.346660 + 0.937991i $$0.387316\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 0 0
$$754$$ 24.0000 0.874028
$$755$$ 64.0000 2.32920
$$756$$ 0 0
$$757$$ 5.00000 0.181728 0.0908640 0.995863i $$-0.471037\pi$$
0.0908640 + 0.995863i $$0.471037\pi$$
$$758$$ 32.0000 1.16229
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 24.0000 0.869999 0.435000 0.900431i $$-0.356748\pi$$
0.435000 + 0.900431i $$0.356748\pi$$
$$762$$ 0 0
$$763$$ 1.00000 0.0362024
$$764$$ −16.0000 −0.578860
$$765$$ 0 0
$$766$$ −52.0000 −1.87884
$$767$$ −20.0000 −0.722158
$$768$$ 0 0
$$769$$ 11.0000 0.396670 0.198335 0.980134i $$-0.436447\pi$$
0.198335 + 0.980134i $$0.436447\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −10.0000 −0.359908
$$773$$ −36.0000 −1.29483 −0.647415 0.762138i $$-0.724150\pi$$
−0.647415 + 0.762138i $$0.724150\pi$$
$$774$$ 0 0
$$775$$ −55.0000 −1.97566
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −36.0000 −1.29066
$$779$$ −6.00000 −0.214972
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 16.0000 0.572159
$$783$$ 0 0
$$784$$ 24.0000 0.857143
$$785$$ 4.00000 0.142766
$$786$$ 0 0
$$787$$ 4.00000 0.142585 0.0712923 0.997455i $$-0.477288\pi$$
0.0712923 + 0.997455i $$0.477288\pi$$
$$788$$ 16.0000 0.569976
$$789$$ 0 0
$$790$$ −88.0000 −3.13090
$$791$$ −6.00000 −0.213335
$$792$$ 0 0
$$793$$ −6.00000 −0.213066
$$794$$ 62.0000 2.20030
$$795$$ 0 0
$$796$$ −42.0000 −1.48865
$$797$$ 10.0000 0.354218 0.177109 0.984191i $$-0.443325\pi$$
0.177109 + 0.984191i $$0.443325\pi$$
$$798$$ 0 0
$$799$$ 8.00000 0.283020
$$800$$ −88.0000 −3.11127
$$801$$ 0 0
$$802$$ 56.0000 1.97743
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 8.00000 0.281963
$$806$$ 20.0000 0.704470
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 48.0000 1.68759 0.843795 0.536666i $$-0.180316\pi$$
0.843795 + 0.536666i $$0.180316\pi$$
$$810$$ 0 0
$$811$$ −17.0000 −0.596951 −0.298475 0.954417i $$-0.596478\pi$$
−0.298475 + 0.954417i $$0.596478\pi$$
$$812$$ −12.0000 −0.421117
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −100.000 −3.50285
$$816$$ 0 0
$$817$$ −36.0000 −1.25948
$$818$$ 42.0000 1.46850
$$819$$ 0 0
$$820$$ −16.0000 −0.558744
$$821$$ 38.0000 1.32621 0.663105 0.748527i $$-0.269238\pi$$
0.663105 + 0.748527i $$0.269238\pi$$
$$822$$ 0 0
$$823$$ −27.0000 −0.941161 −0.470580 0.882357i $$-0.655955\pi$$
−0.470580 + 0.882357i $$0.655955\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 20.0000 0.695889
$$827$$ 10.0000 0.347734 0.173867 0.984769i $$-0.444374\pi$$
0.173867 + 0.984769i $$0.444374\pi$$
$$828$$ 0 0
$$829$$ −11.0000 −0.382046 −0.191023 0.981586i $$-0.561180\pi$$
−0.191023 + 0.981586i $$0.561180\pi$$
$$830$$ 48.0000 1.66610
$$831$$ 0 0
$$832$$ 16.0000 0.554700
$$833$$ 24.0000 0.831551
$$834$$ 0 0
$$835$$ −72.0000 −2.49166
$$836$$ 0 0
$$837$$ 0 0
$$838$$ −52.0000 −1.79631
$$839$$ −16.0000 −0.552381 −0.276191 0.961103i $$-0.589072\pi$$
−0.276191 + 0.961103i $$0.589072\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −4.00000 −0.137849
$$843$$ 0 0
$$844$$ −42.0000 −1.44570
$$845$$ 36.0000 1.23844
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 24.0000 0.824163
$$849$$ 0 0
$$850$$ −88.0000 −3.01838
$$851$$ −6.00000 −0.205677
$$852$$ 0 0
$$853$$ 11.0000 0.376633 0.188316 0.982108i $$-0.439697\pi$$
0.188316 + 0.982108i $$0.439697\pi$$
$$854$$ 6.00000 0.205316
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 4.00000 0.136637 0.0683187 0.997664i $$-0.478237\pi$$
0.0683187 + 0.997664i $$0.478237\pi$$
$$858$$ 0 0
$$859$$ −45.0000 −1.53538 −0.767690 0.640821i $$-0.778594\pi$$
−0.767690 + 0.640821i $$0.778594\pi$$
$$860$$ −96.0000 −3.27357
$$861$$ 0 0
$$862$$ −36.0000 −1.22616
$$863$$ −30.0000 −1.02121 −0.510606 0.859815i $$-0.670579\pi$$
−0.510606 + 0.859815i $$0.670579\pi$$
$$864$$ 0 0
$$865$$ 96.0000 3.26410
$$866$$ −34.0000 −1.15537
$$867$$ 0 0
$$868$$ −10.0000 −0.339422
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 2.00000 0.0677674
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 12.0000 0.405906
$$875$$ −24.0000 −0.811348
$$876$$ 0 0
$$877$$ −45.0000 −1.51954 −0.759771 0.650191i $$-0.774689\pi$$
−0.759771 + 0.650191i $$0.774689\pi$$
$$878$$ 74.0000 2.49738
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −2.00000 −0.0673817 −0.0336909 0.999432i $$-0.510726\pi$$
−0.0336909 + 0.999432i $$0.510726\pi$$
$$882$$ 0 0
$$883$$ 49.0000 1.64898 0.824491 0.565876i $$-0.191462\pi$$
0.824491 + 0.565876i $$0.191462\pi$$
$$884$$ 16.0000 0.538138
$$885$$ 0 0
$$886$$ −8.00000 −0.268765
$$887$$ 22.0000 0.738688 0.369344 0.929293i $$-0.379582\pi$$
0.369344 + 0.929293i $$0.379582\pi$$
$$888$$ 0 0
$$889$$ −13.0000 −0.436006
$$890$$ 96.0000 3.21793
$$891$$ 0 0
$$892$$ −34.0000 −1.13840
$$893$$ 6.00000 0.200782
$$894$$ 0 0
$$895$$ 24.0000 0.802232
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −40.0000 −1.33482
$$899$$ 30.0000 1.00056
$$900$$ 0 0
$$901$$ 24.0000 0.799556
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 92.0000 3.05818
$$906$$ 0 0
$$907$$ 33.0000 1.09575 0.547874 0.836561i $$-0.315438\pi$$
0.547874 + 0.836561i $$0.315438\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 16.0000 0.530395
$$911$$ −36.0000 −1.19273 −0.596367 0.802712i $$-0.703390\pi$$
−0.596367 + 0.802712i $$0.703390\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 36.0000 1.19077
$$915$$ 0 0
$$916$$ −36.0000 −1.18947
$$917$$ 6.00000 0.198137
$$918$$ 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
$$922$$ 12.0000 0.395199
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 33.0000 1.08503
$$926$$ 32.0000 1.05159
$$927$$ 0 0
$$928$$ 48.0000 1.57568
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ 18.0000 0.589926
$$932$$ −36.0000 −1.17922
$$933$$ 0 0
$$934$$ −48.0000 −1.57061
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 23.0000 0.751377 0.375689 0.926746i $$-0.377406\pi$$
0.375689 + 0.926746i $$0.377406\pi$$
$$938$$ −2.00000 −0.0653023
$$939$$ 0 0
$$940$$ 16.0000 0.521862
$$941$$ −42.0000 −1.36916 −0.684580 0.728937i $$-0.740015\pi$$
−0.684580 + 0.728937i $$0.740015\pi$$
$$942$$ 0 0
$$943$$ −4.00000 −0.130258
$$944$$ −40.0000 −1.30189
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −54.0000 −1.75476 −0.877382 0.479792i $$-0.840712\pi$$
−0.877382 + 0.479792i $$0.840712\pi$$
$$948$$ 0 0
$$949$$ 22.0000 0.714150
$$950$$ −66.0000 −2.14132
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −34.0000 −1.10137 −0.550684 0.834714i $$-0.685633\pi$$
−0.550684 + 0.834714i $$0.685633\pi$$
$$954$$ 0 0
$$955$$ 32.0000 1.03550
$$956$$ −12.0000 −0.388108
$$957$$ 0 0
$$958$$ 44.0000 1.42158
$$959$$ −8.00000 −0.258333
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ −12.0000 −0.386896
$$963$$ 0 0
$$964$$ −28.0000 −0.901819
$$965$$ 20.0000 0.643823
$$966$$ 0 0
$$967$$ 13.0000 0.418052 0.209026 0.977910i $$-0.432971\pi$$
0.209026 + 0.977910i $$0.432971\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ −40.0000 −1.28432
$$971$$ −2.00000 −0.0641831 −0.0320915 0.999485i $$-0.510217\pi$$
−0.0320915 + 0.999485i $$0.510217\pi$$
$$972$$ 0 0
$$973$$ 16.0000 0.512936
$$974$$ −80.0000 −2.56337
$$975$$ 0 0
$$976$$ −12.0000 −0.384111
$$977$$ 42.0000 1.34370 0.671850 0.740688i $$-0.265500\pi$$
0.671850 + 0.740688i $$0.265500\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 48.0000 1.53330
$$981$$ 0 0
$$982$$ 28.0000 0.893516
$$983$$ −6.00000 −0.191370 −0.0956851 0.995412i $$-0.530504\pi$$
−0.0956851 + 0.995412i $$0.530504\pi$$
$$984$$ 0 0
$$985$$ −32.0000 −1.01960
$$986$$ 48.0000 1.52863
$$987$$ 0 0
$$988$$ 12.0000 0.381771
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ 40.0000 1.27000
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 84.0000 2.66298
$$996$$ 0 0
$$997$$ −49.0000 −1.55185 −0.775923 0.630828i $$-0.782715\pi$$
−0.775923 + 0.630828i $$0.782715\pi$$
$$998$$ 46.0000 1.45610
$$999$$ 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 1089.2.a.k.1.1 1
3.2 odd 2 363.2.a.a.1.1 1
11.10 odd 2 1089.2.a.a.1.1 1
12.11 even 2 5808.2.a.bh.1.1 1
15.14 odd 2 9075.2.a.t.1.1 1
33.2 even 10 363.2.e.d.202.1 4
33.5 odd 10 363.2.e.i.124.1 4
33.8 even 10 363.2.e.d.130.1 4
33.14 odd 10 363.2.e.i.130.1 4
33.17 even 10 363.2.e.d.124.1 4
33.20 odd 10 363.2.e.i.202.1 4
33.26 odd 10 363.2.e.i.148.1 4
33.29 even 10 363.2.e.d.148.1 4
33.32 even 2 363.2.a.c.1.1 yes 1
132.131 odd 2 5808.2.a.bi.1.1 1
165.164 even 2 9075.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
363.2.a.a.1.1 1 3.2 odd 2
363.2.a.c.1.1 yes 1 33.32 even 2
363.2.e.d.124.1 4 33.17 even 10
363.2.e.d.130.1 4 33.8 even 10
363.2.e.d.148.1 4 33.29 even 10
363.2.e.d.202.1 4 33.2 even 10
363.2.e.i.124.1 4 33.5 odd 10
363.2.e.i.130.1 4 33.14 odd 10
363.2.e.i.148.1 4 33.26 odd 10
363.2.e.i.202.1 4 33.20 odd 10
1089.2.a.a.1.1 1 11.10 odd 2
1089.2.a.k.1.1 1 1.1 even 1 trivial
5808.2.a.bh.1.1 1 12.11 even 2
5808.2.a.bi.1.1 1 132.131 odd 2
9075.2.a.b.1.1 1 165.164 even 2
9075.2.a.t.1.1 1 15.14 odd 2