# Properties

 Label 63.2.a.a.1.1 Level $63$ Weight $2$ Character 63.1 Self dual yes Analytic conductor $0.503$ 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] = [63,2,Mod(1,63)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, 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("63.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 63.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: $$0.503057532734$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 63.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{7} -3.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{7} -3.00000 q^{8} +2.00000 q^{10} -4.00000 q^{11} -2.00000 q^{13} -1.00000 q^{14} -1.00000 q^{16} +6.00000 q^{17} +4.00000 q^{19} -2.00000 q^{20} -4.00000 q^{22} -1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{28} +2.00000 q^{29} +5.00000 q^{32} +6.00000 q^{34} -2.00000 q^{35} +6.00000 q^{37} +4.00000 q^{38} -6.00000 q^{40} -2.00000 q^{41} -4.00000 q^{43} +4.00000 q^{44} +1.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} -6.00000 q^{53} -8.00000 q^{55} +3.00000 q^{56} +2.00000 q^{58} -12.0000 q^{59} -2.00000 q^{61} +7.00000 q^{64} -4.00000 q^{65} +4.00000 q^{67} -6.00000 q^{68} -2.00000 q^{70} -6.00000 q^{73} +6.00000 q^{74} -4.00000 q^{76} +4.00000 q^{77} -16.0000 q^{79} -2.00000 q^{80} -2.00000 q^{82} +12.0000 q^{83} +12.0000 q^{85} -4.00000 q^{86} +12.0000 q^{88} +14.0000 q^{89} +2.00000 q^{91} +8.00000 q^{95} +18.0000 q^{97} +1.00000 q^{98} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ −3.00000 −1.06066
$$9$$ 0 0
$$10$$ 2.00000 0.632456
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ −2.00000 −0.447214
$$21$$ 0 0
$$22$$ −4.00000 −0.852803
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ 1.00000 0.188982
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 5.00000 0.883883
$$33$$ 0 0
$$34$$ 6.00000 1.02899
$$35$$ −2.00000 −0.338062
$$36$$ 0 0
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 0 0
$$40$$ −6.00000 −0.948683
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ 2.00000 0.277350
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ −8.00000 −1.07872
$$56$$ 3.00000 0.400892
$$57$$ 0 0
$$58$$ 2.00000 0.262613
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ −4.00000 −0.496139
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ 0 0
$$70$$ −2.00000 −0.239046
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −6.00000 −0.702247 −0.351123 0.936329i $$-0.614200\pi$$
−0.351123 + 0.936329i $$0.614200\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 4.00000 0.455842
$$78$$ 0 0
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ −2.00000 −0.223607
$$81$$ 0 0
$$82$$ −2.00000 −0.220863
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ 12.0000 1.30158
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 12.0000 1.27920
$$89$$ 14.0000 1.48400 0.741999 0.670402i $$-0.233878\pi$$
0.741999 + 0.670402i $$0.233878\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 8.00000 0.820783
$$96$$ 0 0
$$97$$ 18.0000 1.82762 0.913812 0.406138i $$-0.133125\pi$$
0.913812 + 0.406138i $$0.133125\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 0 0
$$109$$ −18.0000 −1.72409 −0.862044 0.506834i $$-0.830816\pi$$
−0.862044 + 0.506834i $$0.830816\pi$$
$$110$$ −8.00000 −0.762770
$$111$$ 0 0
$$112$$ 1.00000 0.0944911
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −2.00000 −0.185695
$$117$$ 0 0
$$118$$ −12.0000 −1.10469
$$119$$ −6.00000 −0.550019
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ −2.00000 −0.181071
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 0 0
$$130$$ −4.00000 −0.350823
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 0 0
$$133$$ −4.00000 −0.346844
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ −18.0000 −1.54349
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 2.00000 0.169031
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 8.00000 0.668994
$$144$$ 0 0
$$145$$ 4.00000 0.332182
$$146$$ −6.00000 −0.496564
$$147$$ 0 0
$$148$$ −6.00000 −0.493197
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ −12.0000 −0.973329
$$153$$ 0 0
$$154$$ 4.00000 0.322329
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ −16.0000 −1.27289
$$159$$ 0 0
$$160$$ 10.0000 0.790569
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 12.0000 0.920358
$$171$$ 0 0
$$172$$ 4.00000 0.304997
$$173$$ 10.0000 0.760286 0.380143 0.924928i $$-0.375875\pi$$
0.380143 + 0.924928i $$0.375875\pi$$
$$174$$ 0 0
$$175$$ 1.00000 0.0755929
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ 14.0000 1.04934
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ −26.0000 −1.93256 −0.966282 0.257485i $$-0.917106\pi$$
−0.966282 + 0.257485i $$0.917106\pi$$
$$182$$ 2.00000 0.148250
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 12.0000 0.882258
$$186$$ 0 0
$$187$$ −24.0000 −1.75505
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 8.00000 0.580381
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 18.0000 1.29232
$$195$$ 0 0
$$196$$ −1.00000 −0.0714286
$$197$$ −22.0000 −1.56744 −0.783718 0.621117i $$-0.786679\pi$$
−0.783718 + 0.621117i $$0.786679\pi$$
$$198$$ 0 0
$$199$$ 24.0000 1.70131 0.850657 0.525720i $$-0.176204\pi$$
0.850657 + 0.525720i $$0.176204\pi$$
$$200$$ 3.00000 0.212132
$$201$$ 0 0
$$202$$ −14.0000 −0.985037
$$203$$ −2.00000 −0.140372
$$204$$ 0 0
$$205$$ −4.00000 −0.279372
$$206$$ 8.00000 0.557386
$$207$$ 0 0
$$208$$ 2.00000 0.138675
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ −4.00000 −0.273434
$$215$$ −8.00000 −0.545595
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −18.0000 −1.21911
$$219$$ 0 0
$$220$$ 8.00000 0.539360
$$221$$ −12.0000 −0.807207
$$222$$ 0 0
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ −5.00000 −0.334077
$$225$$ 0 0
$$226$$ 14.0000 0.931266
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 12.0000 0.781133
$$237$$ 0 0
$$238$$ −6.00000 −0.388922
$$239$$ −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 $$0.479482\pi$$
$$242$$ 5.00000 0.321412
$$243$$ 0 0
$$244$$ 2.00000 0.128037
$$245$$ 2.00000 0.127775
$$246$$ 0 0
$$247$$ −8.00000 −0.509028
$$248$$ 0 0
$$249$$ 0 0
$$250$$ −12.0000 −0.758947
$$251$$ 20.0000 1.26239 0.631194 0.775625i $$-0.282565\pi$$
0.631194 + 0.775625i $$0.282565\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ −26.0000 −1.62184 −0.810918 0.585160i $$-0.801032\pi$$
−0.810918 + 0.585160i $$0.801032\pi$$
$$258$$ 0 0
$$259$$ −6.00000 −0.372822
$$260$$ 4.00000 0.248069
$$261$$ 0 0
$$262$$ −4.00000 −0.247121
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ 0 0
$$265$$ −12.0000 −0.737154
$$266$$ −4.00000 −0.245256
$$267$$ 0 0
$$268$$ −4.00000 −0.244339
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 4.00000 0.241209
$$276$$ 0 0
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ 12.0000 0.719712
$$279$$ 0 0
$$280$$ 6.00000 0.358569
$$281$$ 22.0000 1.31241 0.656205 0.754583i $$-0.272161\pi$$
0.656205 + 0.754583i $$0.272161\pi$$
$$282$$ 0 0
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ 2.00000 0.118056
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 4.00000 0.234888
$$291$$ 0 0
$$292$$ 6.00000 0.351123
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ 0 0
$$295$$ −24.0000 −1.39733
$$296$$ −18.0000 −1.04623
$$297$$ 0 0
$$298$$ −6.00000 −0.347571
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 8.00000 0.460348
$$303$$ 0 0
$$304$$ −4.00000 −0.229416
$$305$$ −4.00000 −0.229039
$$306$$ 0 0
$$307$$ 4.00000 0.228292 0.114146 0.993464i $$-0.463587\pi$$
0.114146 + 0.993464i $$0.463587\pi$$
$$308$$ −4.00000 −0.227921
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ 26.0000 1.46961 0.734803 0.678280i $$-0.237274\pi$$
0.734803 + 0.678280i $$0.237274\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ 16.0000 0.900070
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 0 0
$$319$$ −8.00000 −0.447914
$$320$$ 14.0000 0.782624
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 24.0000 1.33540
$$324$$ 0 0
$$325$$ 2.00000 0.110940
$$326$$ 4.00000 0.221540
$$327$$ 0 0
$$328$$ 6.00000 0.331295
$$329$$ 0 0
$$330$$ 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
$$334$$ 8.00000 0.437741
$$335$$ 8.00000 0.437087
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 0 0
$$340$$ −12.0000 −0.650791
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ 10.0000 0.537603
$$347$$ 28.0000 1.50312 0.751559 0.659665i $$-0.229302\pi$$
0.751559 + 0.659665i $$0.229302\pi$$
$$348$$ 0 0
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ 0 0
$$352$$ −20.0000 −1.06600
$$353$$ −10.0000 −0.532246 −0.266123 0.963939i $$-0.585743\pi$$
−0.266123 + 0.963939i $$0.585743\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −14.0000 −0.741999
$$357$$ 0 0
$$358$$ 4.00000 0.211407
$$359$$ −32.0000 −1.68890 −0.844448 0.535638i $$-0.820071\pi$$
−0.844448 + 0.535638i $$0.820071\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ −26.0000 −1.36653
$$363$$ 0 0
$$364$$ −2.00000 −0.104828
$$365$$ −12.0000 −0.628109
$$366$$ 0 0
$$367$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 12.0000 0.623850
$$371$$ 6.00000 0.311504
$$372$$ 0 0
$$373$$ −10.0000 −0.517780 −0.258890 0.965907i $$-0.583357\pi$$
−0.258890 + 0.965907i $$0.583357\pi$$
$$374$$ −24.0000 −1.24101
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ 12.0000 0.616399 0.308199 0.951322i $$-0.400274\pi$$
0.308199 + 0.951322i $$0.400274\pi$$
$$380$$ −8.00000 −0.410391
$$381$$ 0 0
$$382$$ 8.00000 0.409316
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 8.00000 0.407718
$$386$$ 2.00000 0.101797
$$387$$ 0 0
$$388$$ −18.0000 −0.913812
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −3.00000 −0.151523
$$393$$ 0 0
$$394$$ −22.0000 −1.10834
$$395$$ −32.0000 −1.61009
$$396$$ 0 0
$$397$$ −18.0000 −0.903394 −0.451697 0.892171i $$-0.649181\pi$$
−0.451697 + 0.892171i $$0.649181\pi$$
$$398$$ 24.0000 1.20301
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 14.0000 0.696526
$$405$$ 0 0
$$406$$ −2.00000 −0.0992583
$$407$$ −24.0000 −1.18964
$$408$$ 0 0
$$409$$ −22.0000 −1.08783 −0.543915 0.839140i $$-0.683059\pi$$
−0.543915 + 0.839140i $$0.683059\pi$$
$$410$$ −4.00000 −0.197546
$$411$$ 0 0
$$412$$ −8.00000 −0.394132
$$413$$ 12.0000 0.590481
$$414$$ 0 0
$$415$$ 24.0000 1.17811
$$416$$ −10.0000 −0.490290
$$417$$ 0 0
$$418$$ −16.0000 −0.782586
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ 38.0000 1.85201 0.926003 0.377515i $$-0.123221\pi$$
0.926003 + 0.377515i $$0.123221\pi$$
$$422$$ 4.00000 0.194717
$$423$$ 0 0
$$424$$ 18.0000 0.874157
$$425$$ −6.00000 −0.291043
$$426$$ 0 0
$$427$$ 2.00000 0.0967868
$$428$$ 4.00000 0.193347
$$429$$ 0 0
$$430$$ −8.00000 −0.385794
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 0 0
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 18.0000 0.862044
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −24.0000 −1.14546 −0.572729 0.819745i $$-0.694115\pi$$
−0.572729 + 0.819745i $$0.694115\pi$$
$$440$$ 24.0000 1.14416
$$441$$ 0 0
$$442$$ −12.0000 −0.570782
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ 0 0
$$445$$ 28.0000 1.32733
$$446$$ 16.0000 0.757622
$$447$$ 0 0
$$448$$ −7.00000 −0.330719
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ 8.00000 0.376705
$$452$$ −14.0000 −0.658505
$$453$$ 0 0
$$454$$ 12.0000 0.563188
$$455$$ 4.00000 0.187523
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 10.0000 0.465746 0.232873 0.972507i $$-0.425187\pi$$
0.232873 + 0.972507i $$0.425187\pi$$
$$462$$ 0 0
$$463$$ 16.0000 0.743583 0.371792 0.928316i $$-0.378744\pi$$
0.371792 + 0.928316i $$0.378744\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ −36.0000 −1.66588 −0.832941 0.553362i $$-0.813345\pi$$
−0.832941 + 0.553362i $$0.813345\pi$$
$$468$$ 0 0
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 36.0000 1.65703
$$473$$ 16.0000 0.735681
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 6.00000 0.275010
$$477$$ 0 0
$$478$$ −24.0000 −1.09773
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ −12.0000 −0.547153
$$482$$ 2.00000 0.0910975
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 36.0000 1.63468
$$486$$ 0 0
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ 6.00000 0.271607
$$489$$ 0 0
$$490$$ 2.00000 0.0903508
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ 0 0
$$493$$ 12.0000 0.540453
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 12.0000 0.536656
$$501$$ 0 0
$$502$$ 20.0000 0.892644
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ 0 0
$$505$$ −28.0000 −1.24598
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ 0 0
$$511$$ 6.00000 0.265424
$$512$$ −11.0000 −0.486136
$$513$$ 0 0
$$514$$ −26.0000 −1.14681
$$515$$ 16.0000 0.705044
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −6.00000 −0.263625
$$519$$ 0 0
$$520$$ 12.0000 0.526235
$$521$$ −18.0000 −0.788594 −0.394297 0.918983i $$-0.629012\pi$$
−0.394297 + 0.918983i $$0.629012\pi$$
$$522$$ 0 0
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ 4.00000 0.174741
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ −12.0000 −0.521247
$$531$$ 0 0
$$532$$ 4.00000 0.173422
$$533$$ 4.00000 0.173259
$$534$$ 0 0
$$535$$ −8.00000 −0.345870
$$536$$ −12.0000 −0.518321
$$537$$ 0 0
$$538$$ −6.00000 −0.258678
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 0 0
$$544$$ 30.0000 1.28624
$$545$$ −36.0000 −1.54207
$$546$$ 0 0
$$547$$ 4.00000 0.171028 0.0855138 0.996337i $$-0.472747\pi$$
0.0855138 + 0.996337i $$0.472747\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ 0 0
$$550$$ 4.00000 0.170561
$$551$$ 8.00000 0.340811
$$552$$ 0 0
$$553$$ 16.0000 0.680389
$$554$$ 22.0000 0.934690
$$555$$ 0 0
$$556$$ −12.0000 −0.508913
$$557$$ 2.00000 0.0847427 0.0423714 0.999102i $$-0.486509\pi$$
0.0423714 + 0.999102i $$0.486509\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ 2.00000 0.0845154
$$561$$ 0 0
$$562$$ 22.0000 0.928014
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ 0 0
$$565$$ 28.0000 1.17797
$$566$$ −20.0000 −0.840663
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −10.0000 −0.419222 −0.209611 0.977785i $$-0.567220\pi$$
−0.209611 + 0.977785i $$0.567220\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ −8.00000 −0.334497
$$573$$ 0 0
$$574$$ 2.00000 0.0834784
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 34.0000 1.41544 0.707719 0.706494i $$-0.249724\pi$$
0.707719 + 0.706494i $$0.249724\pi$$
$$578$$ 19.0000 0.790296
$$579$$ 0 0
$$580$$ −4.00000 −0.166091
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ 24.0000 0.993978
$$584$$ 18.0000 0.744845
$$585$$ 0 0
$$586$$ −14.0000 −0.578335
$$587$$ −28.0000 −1.15568 −0.577842 0.816149i $$-0.696105\pi$$
−0.577842 + 0.816149i $$0.696105\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ −24.0000 −0.988064
$$591$$ 0 0
$$592$$ −6.00000 −0.246598
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 0 0
$$595$$ −12.0000 −0.491952
$$596$$ 6.00000 0.245770
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −48.0000 −1.96123 −0.980613 0.195952i $$-0.937220\pi$$
−0.980613 + 0.195952i $$0.937220\pi$$
$$600$$ 0 0
$$601$$ −6.00000 −0.244745 −0.122373 0.992484i $$-0.539050\pi$$
−0.122373 + 0.992484i $$0.539050\pi$$
$$602$$ 4.00000 0.163028
$$603$$ 0 0
$$604$$ −8.00000 −0.325515
$$605$$ 10.0000 0.406558
$$606$$ 0 0
$$607$$ −16.0000 −0.649420 −0.324710 0.945814i $$-0.605267\pi$$
−0.324710 + 0.945814i $$0.605267\pi$$
$$608$$ 20.0000 0.811107
$$609$$ 0 0
$$610$$ −4.00000 −0.161955
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −26.0000 −1.05013 −0.525065 0.851062i $$-0.675959\pi$$
−0.525065 + 0.851062i $$0.675959\pi$$
$$614$$ 4.00000 0.161427
$$615$$ 0 0
$$616$$ −12.0000 −0.483494
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ 0 0
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 24.0000 0.962312
$$623$$ −14.0000 −0.560898
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 26.0000 1.03917
$$627$$ 0 0
$$628$$ 2.00000 0.0798087
$$629$$ 36.0000 1.43541
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ 48.0000 1.90934
$$633$$ 0 0
$$634$$ 18.0000 0.714871
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −2.00000 −0.0792429
$$638$$ −8.00000 −0.316723
$$639$$ 0 0
$$640$$ −6.00000 −0.237171
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 0 0
$$643$$ 20.0000 0.788723 0.394362 0.918955i $$-0.370966\pi$$
0.394362 + 0.918955i $$0.370966\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 24.0000 0.944267
$$647$$ 40.0000 1.57256 0.786281 0.617869i $$-0.212004\pi$$
0.786281 + 0.617869i $$0.212004\pi$$
$$648$$ 0 0
$$649$$ 48.0000 1.88416
$$650$$ 2.00000 0.0784465
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 0 0
$$655$$ −8.00000 −0.312586
$$656$$ 2.00000 0.0780869
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ −4.00000 −0.155464
$$663$$ 0 0
$$664$$ −36.0000 −1.39707
$$665$$ −8.00000 −0.310227
$$666$$ 0 0
$$667$$ 0 0
$$668$$ −8.00000 −0.309529
$$669$$ 0 0
$$670$$ 8.00000 0.309067
$$671$$ 8.00000 0.308837
$$672$$ 0 0
$$673$$ 34.0000 1.31060 0.655302 0.755367i $$-0.272541\pi$$
0.655302 + 0.755367i $$0.272541\pi$$
$$674$$ −14.0000 −0.539260
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ 0 0
$$679$$ −18.0000 −0.690777
$$680$$ −36.0000 −1.38054
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 0 0
$$685$$ 12.0000 0.458496
$$686$$ −1.00000 −0.0381802
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ −10.0000 −0.380143
$$693$$ 0 0
$$694$$ 28.0000 1.06287
$$695$$ 24.0000 0.910372
$$696$$ 0 0
$$697$$ −12.0000 −0.454532
$$698$$ −2.00000 −0.0757011
$$699$$ 0 0
$$700$$ −1.00000 −0.0377964
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ 0 0
$$703$$ 24.0000 0.905177
$$704$$ −28.0000 −1.05529
$$705$$ 0 0
$$706$$ −10.0000 −0.376355
$$707$$ 14.0000 0.526524
$$708$$ 0 0
$$709$$ 6.00000 0.225335 0.112667 0.993633i $$-0.464061\pi$$
0.112667 + 0.993633i $$0.464061\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −42.0000 −1.57402
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 16.0000 0.598366
$$716$$ −4.00000 −0.149487
$$717$$ 0 0
$$718$$ −32.0000 −1.19423
$$719$$ 48.0000 1.79010 0.895049 0.445968i $$-0.147140\pi$$
0.895049 + 0.445968i $$0.147140\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ −3.00000 −0.111648
$$723$$ 0 0
$$724$$ 26.0000 0.966282
$$725$$ −2.00000 −0.0742781
$$726$$ 0 0
$$727$$ −40.0000 −1.48352 −0.741759 0.670667i $$-0.766008\pi$$
−0.741759 + 0.670667i $$0.766008\pi$$
$$728$$ −6.00000 −0.222375
$$729$$ 0 0
$$730$$ −12.0000 −0.444140
$$731$$ −24.0000 −0.887672
$$732$$ 0 0
$$733$$ −18.0000 −0.664845 −0.332423 0.943131i $$-0.607866\pi$$
−0.332423 + 0.943131i $$0.607866\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −16.0000 −0.589368
$$738$$ 0 0
$$739$$ 36.0000 1.32428 0.662141 0.749380i $$-0.269648\pi$$
0.662141 + 0.749380i $$0.269648\pi$$
$$740$$ −12.0000 −0.441129
$$741$$ 0 0
$$742$$ 6.00000 0.220267
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ −12.0000 −0.439646
$$746$$ −10.0000 −0.366126
$$747$$ 0 0
$$748$$ 24.0000 0.877527
$$749$$ 4.00000 0.146157
$$750$$ 0 0
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ −4.00000 −0.145671
$$755$$ 16.0000 0.582300
$$756$$ 0 0
$$757$$ −10.0000 −0.363456 −0.181728 0.983349i $$-0.558169\pi$$
−0.181728 + 0.983349i $$0.558169\pi$$
$$758$$ 12.0000 0.435860
$$759$$ 0 0
$$760$$ −24.0000 −0.870572
$$761$$ −18.0000 −0.652499 −0.326250 0.945284i $$-0.605785\pi$$
−0.326250 + 0.945284i $$0.605785\pi$$
$$762$$ 0 0
$$763$$ 18.0000 0.651644
$$764$$ −8.00000 −0.289430
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 24.0000 0.866590
$$768$$ 0 0
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 8.00000 0.288300
$$771$$ 0 0
$$772$$ −2.00000 −0.0719816
$$773$$ −14.0000 −0.503545 −0.251773 0.967786i $$-0.581013\pi$$
−0.251773 + 0.967786i $$0.581013\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −54.0000 −1.93849
$$777$$ 0 0
$$778$$ −6.00000 −0.215110
$$779$$ −8.00000 −0.286630
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −1.00000 −0.0357143
$$785$$ −4.00000 −0.142766
$$786$$ 0 0
$$787$$ −44.0000 −1.56843 −0.784215 0.620489i $$-0.786934\pi$$
−0.784215 + 0.620489i $$0.786934\pi$$
$$788$$ 22.0000 0.783718
$$789$$ 0 0
$$790$$ −32.0000 −1.13851
$$791$$ −14.0000 −0.497783
$$792$$ 0 0
$$793$$ 4.00000 0.142044
$$794$$ −18.0000 −0.638796
$$795$$ 0 0
$$796$$ −24.0000 −0.850657
$$797$$ 26.0000 0.920967 0.460484 0.887668i $$-0.347676\pi$$
0.460484 + 0.887668i $$0.347676\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −5.00000 −0.176777
$$801$$ 0 0
$$802$$ 30.0000 1.05934
$$803$$ 24.0000 0.846942
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 42.0000 1.47755
$$809$$ −42.0000 −1.47664 −0.738321 0.674450i $$-0.764381\pi$$
−0.738321 + 0.674450i $$0.764381\pi$$
$$810$$ 0 0
$$811$$ 44.0000 1.54505 0.772524 0.634985i $$-0.218994\pi$$
0.772524 + 0.634985i $$0.218994\pi$$
$$812$$ 2.00000 0.0701862
$$813$$ 0 0
$$814$$ −24.0000 −0.841200
$$815$$ 8.00000 0.280228
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ −22.0000 −0.769212
$$819$$ 0 0
$$820$$ 4.00000 0.139686
$$821$$ −38.0000 −1.32621 −0.663105 0.748527i $$-0.730762\pi$$
−0.663105 + 0.748527i $$0.730762\pi$$
$$822$$ 0 0
$$823$$ 24.0000 0.836587 0.418294 0.908312i $$-0.362628\pi$$
0.418294 + 0.908312i $$0.362628\pi$$
$$824$$ −24.0000 −0.836080
$$825$$ 0 0
$$826$$ 12.0000 0.417533
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 0 0
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ 24.0000 0.833052
$$831$$ 0 0
$$832$$ −14.0000 −0.485363
$$833$$ 6.00000 0.207888
$$834$$ 0 0
$$835$$ 16.0000 0.553703
$$836$$ 16.0000 0.553372
$$837$$ 0 0
$$838$$ 12.0000 0.414533
$$839$$ 8.00000 0.276191 0.138095 0.990419i $$-0.455902\pi$$
0.138095 + 0.990419i $$0.455902\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 38.0000 1.30957
$$843$$ 0 0
$$844$$ −4.00000 −0.137686
$$845$$ −18.0000 −0.619219
$$846$$ 0 0
$$847$$ −5.00000 −0.171802
$$848$$ 6.00000 0.206041
$$849$$ 0 0
$$850$$ −6.00000 −0.205798
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −10.0000 −0.342393 −0.171197 0.985237i $$-0.554763\pi$$
−0.171197 + 0.985237i $$0.554763\pi$$
$$854$$ 2.00000 0.0684386
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ 14.0000 0.478231 0.239115 0.970991i $$-0.423143\pi$$
0.239115 + 0.970991i $$0.423143\pi$$
$$858$$ 0 0
$$859$$ 44.0000 1.50126 0.750630 0.660722i $$-0.229750\pi$$
0.750630 + 0.660722i $$0.229750\pi$$
$$860$$ 8.00000 0.272798
$$861$$ 0 0
$$862$$ 24.0000 0.817443
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 0 0
$$865$$ 20.0000 0.680020
$$866$$ −14.0000 −0.475739
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 64.0000 2.17105
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ 54.0000 1.82867
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 12.0000 0.405674
$$876$$ 0 0
$$877$$ 46.0000 1.55331 0.776655 0.629926i $$-0.216915\pi$$
0.776655 + 0.629926i $$0.216915\pi$$
$$878$$ −24.0000 −0.809961
$$879$$ 0 0
$$880$$ 8.00000 0.269680
$$881$$ 6.00000 0.202145 0.101073 0.994879i $$-0.467773\pi$$
0.101073 + 0.994879i $$0.467773\pi$$
$$882$$ 0 0
$$883$$ −28.0000 −0.942275 −0.471138 0.882060i $$-0.656156\pi$$
−0.471138 + 0.882060i $$0.656156\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ −36.0000 −1.20944
$$887$$ −8.00000 −0.268614 −0.134307 0.990940i $$-0.542881\pi$$
−0.134307 + 0.990940i $$0.542881\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 28.0000 0.938562
$$891$$ 0 0
$$892$$ −16.0000 −0.535720
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 8.00000 0.267411
$$896$$ 3.00000 0.100223
$$897$$ 0 0
$$898$$ 30.0000 1.00111
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −36.0000 −1.19933
$$902$$ 8.00000 0.266371
$$903$$ 0 0
$$904$$ −42.0000 −1.39690
$$905$$ −52.0000 −1.72854
$$906$$ 0 0
$$907$$ −4.00000 −0.132818 −0.0664089 0.997792i $$-0.521154\pi$$
−0.0664089 + 0.997792i $$0.521154\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 0 0
$$910$$ 4.00000 0.132599
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 0 0
$$913$$ −48.0000 −1.58857
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ 4.00000 0.132092
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 10.0000 0.329332
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −6.00000 −0.197279
$$926$$ 16.0000 0.525793
$$927$$ 0 0
$$928$$ 10.0000 0.328266
$$929$$ −26.0000 −0.853032 −0.426516 0.904480i $$-0.640259\pi$$
−0.426516 + 0.904480i $$0.640259\pi$$
$$930$$ 0 0
$$931$$ 4.00000 0.131095
$$932$$ −6.00000 −0.196537
$$933$$ 0 0
$$934$$ −36.0000 −1.17796
$$935$$ −48.0000 −1.56977
$$936$$ 0 0
$$937$$ 42.0000 1.37208 0.686040 0.727564i $$-0.259347\pi$$
0.686040 + 0.727564i $$0.259347\pi$$
$$938$$ −4.00000 −0.130605
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −38.0000 −1.23876 −0.619382 0.785090i $$-0.712617\pi$$
−0.619382 + 0.785090i $$0.712617\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ −44.0000 −1.42981 −0.714904 0.699223i $$-0.753530\pi$$
−0.714904 + 0.699223i $$0.753530\pi$$
$$948$$ 0 0
$$949$$ 12.0000 0.389536
$$950$$ −4.00000 −0.129777
$$951$$ 0 0
$$952$$ 18.0000 0.583383
$$953$$ −26.0000 −0.842223 −0.421111 0.907009i $$-0.638360\pi$$
−0.421111 + 0.907009i $$0.638360\pi$$
$$954$$ 0 0
$$955$$ 16.0000 0.517748
$$956$$ 24.0000 0.776215
$$957$$ 0 0
$$958$$ 16.0000 0.516937
$$959$$ −6.00000 −0.193750
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ −12.0000 −0.386896
$$963$$ 0 0
$$964$$ −2.00000 −0.0644157
$$965$$ 4.00000 0.128765
$$966$$ 0 0
$$967$$ 40.0000 1.28631 0.643157 0.765735i $$-0.277624\pi$$
0.643157 + 0.765735i $$0.277624\pi$$
$$968$$ −15.0000 −0.482118
$$969$$ 0 0
$$970$$ 36.0000 1.15589
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 0 0
$$973$$ −12.0000 −0.384702
$$974$$ −8.00000 −0.256337
$$975$$ 0 0
$$976$$ 2.00000 0.0640184
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 0 0
$$979$$ −56.0000 −1.78977
$$980$$ −2.00000 −0.0638877
$$981$$ 0 0
$$982$$ −20.0000 −0.638226
$$983$$ −24.0000 −0.765481 −0.382741 0.923856i $$-0.625020\pi$$
−0.382741 + 0.923856i $$0.625020\pi$$
$$984$$ 0 0
$$985$$ −44.0000 −1.40196
$$986$$ 12.0000 0.382158
$$987$$ 0 0
$$988$$ 8.00000 0.254514
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 48.0000 1.52170
$$996$$ 0 0
$$997$$ −26.0000 −0.823428 −0.411714 0.911313i $$-0.635070\pi$$
−0.411714 + 0.911313i $$0.635070\pi$$
$$998$$ 4.00000 0.126618
$$999$$ 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 63.2.a.a.1.1 1
3.2 odd 2 21.2.a.a.1.1 1
4.3 odd 2 1008.2.a.l.1.1 1
5.2 odd 4 1575.2.d.a.1324.2 2
5.3 odd 4 1575.2.d.a.1324.1 2
5.4 even 2 1575.2.a.c.1.1 1
7.2 even 3 441.2.e.a.361.1 2
7.3 odd 6 441.2.e.b.226.1 2
7.4 even 3 441.2.e.a.226.1 2
7.5 odd 6 441.2.e.b.361.1 2
7.6 odd 2 441.2.a.f.1.1 1
8.3 odd 2 4032.2.a.k.1.1 1
8.5 even 2 4032.2.a.h.1.1 1
9.2 odd 6 567.2.f.g.190.1 2
9.4 even 3 567.2.f.b.379.1 2
9.5 odd 6 567.2.f.g.379.1 2
9.7 even 3 567.2.f.b.190.1 2
11.10 odd 2 7623.2.a.g.1.1 1
12.11 even 2 336.2.a.a.1.1 1
15.2 even 4 525.2.d.a.274.1 2
15.8 even 4 525.2.d.a.274.2 2
15.14 odd 2 525.2.a.d.1.1 1
21.2 odd 6 147.2.e.b.67.1 2
21.5 even 6 147.2.e.c.67.1 2
21.11 odd 6 147.2.e.b.79.1 2
21.17 even 6 147.2.e.c.79.1 2
21.20 even 2 147.2.a.a.1.1 1
24.5 odd 2 1344.2.a.g.1.1 1
24.11 even 2 1344.2.a.s.1.1 1
28.27 even 2 7056.2.a.p.1.1 1
33.32 even 2 2541.2.a.j.1.1 1
39.38 odd 2 3549.2.a.c.1.1 1
48.5 odd 4 5376.2.c.r.2689.1 2
48.11 even 4 5376.2.c.l.2689.2 2
48.29 odd 4 5376.2.c.r.2689.2 2
48.35 even 4 5376.2.c.l.2689.1 2
51.50 odd 2 6069.2.a.b.1.1 1
57.56 even 2 7581.2.a.d.1.1 1
60.59 even 2 8400.2.a.bn.1.1 1
84.11 even 6 2352.2.q.x.961.1 2
84.23 even 6 2352.2.q.x.1537.1 2
84.47 odd 6 2352.2.q.e.1537.1 2
84.59 odd 6 2352.2.q.e.961.1 2
84.83 odd 2 2352.2.a.v.1.1 1
105.104 even 2 3675.2.a.n.1.1 1
168.83 odd 2 9408.2.a.m.1.1 1
168.125 even 2 9408.2.a.bv.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.2.a.a.1.1 1 3.2 odd 2
63.2.a.a.1.1 1 1.1 even 1 trivial
147.2.a.a.1.1 1 21.20 even 2
147.2.e.b.67.1 2 21.2 odd 6
147.2.e.b.79.1 2 21.11 odd 6
147.2.e.c.67.1 2 21.5 even 6
147.2.e.c.79.1 2 21.17 even 6
336.2.a.a.1.1 1 12.11 even 2
441.2.a.f.1.1 1 7.6 odd 2
441.2.e.a.226.1 2 7.4 even 3
441.2.e.a.361.1 2 7.2 even 3
441.2.e.b.226.1 2 7.3 odd 6
441.2.e.b.361.1 2 7.5 odd 6
525.2.a.d.1.1 1 15.14 odd 2
525.2.d.a.274.1 2 15.2 even 4
525.2.d.a.274.2 2 15.8 even 4
567.2.f.b.190.1 2 9.7 even 3
567.2.f.b.379.1 2 9.4 even 3
567.2.f.g.190.1 2 9.2 odd 6
567.2.f.g.379.1 2 9.5 odd 6
1008.2.a.l.1.1 1 4.3 odd 2
1344.2.a.g.1.1 1 24.5 odd 2
1344.2.a.s.1.1 1 24.11 even 2
1575.2.a.c.1.1 1 5.4 even 2
1575.2.d.a.1324.1 2 5.3 odd 4
1575.2.d.a.1324.2 2 5.2 odd 4
2352.2.a.v.1.1 1 84.83 odd 2
2352.2.q.e.961.1 2 84.59 odd 6
2352.2.q.e.1537.1 2 84.47 odd 6
2352.2.q.x.961.1 2 84.11 even 6
2352.2.q.x.1537.1 2 84.23 even 6
2541.2.a.j.1.1 1 33.32 even 2
3549.2.a.c.1.1 1 39.38 odd 2
3675.2.a.n.1.1 1 105.104 even 2
4032.2.a.h.1.1 1 8.5 even 2
4032.2.a.k.1.1 1 8.3 odd 2
5376.2.c.l.2689.1 2 48.35 even 4
5376.2.c.l.2689.2 2 48.11 even 4
5376.2.c.r.2689.1 2 48.5 odd 4
5376.2.c.r.2689.2 2 48.29 odd 4
6069.2.a.b.1.1 1 51.50 odd 2
7056.2.a.p.1.1 1 28.27 even 2
7581.2.a.d.1.1 1 57.56 even 2
7623.2.a.g.1.1 1 11.10 odd 2
8400.2.a.bn.1.1 1 60.59 even 2
9408.2.a.m.1.1 1 168.83 odd 2
9408.2.a.bv.1.1 1 168.125 even 2