# Properties

 Label 55.2.a.a.1.1 Level $55$ Weight $2$ Character 55.1 Self dual yes Analytic conductor $0.439$ 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] = [55,2,Mod(1,55)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(55, 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("55.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$55 = 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 55.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: $$0.439177211117$$ 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 $$\chi$$ $$=$$ 55.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{8} -3.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -3.00000 q^{8} -3.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +2.00000 q^{13} -1.00000 q^{16} +6.00000 q^{17} -3.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} -1.00000 q^{22} +4.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +6.00000 q^{29} -8.00000 q^{31} +5.00000 q^{32} +6.00000 q^{34} +3.00000 q^{36} -2.00000 q^{37} -4.00000 q^{38} -3.00000 q^{40} +2.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} -3.00000 q^{45} +4.00000 q^{46} -12.0000 q^{47} -7.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{55} +6.00000 q^{58} +4.00000 q^{59} -10.0000 q^{61} -8.00000 q^{62} +7.00000 q^{64} +2.00000 q^{65} -16.0000 q^{67} -6.00000 q^{68} +8.00000 q^{71} +9.00000 q^{72} +14.0000 q^{73} -2.00000 q^{74} +4.00000 q^{76} +8.00000 q^{79} -1.00000 q^{80} +9.00000 q^{81} +2.00000 q^{82} -4.00000 q^{83} +6.00000 q^{85} +4.00000 q^{86} +3.00000 q^{88} +10.0000 q^{89} -3.00000 q^{90} -4.00000 q^{92} -12.0000 q^{94} -4.00000 q^{95} +10.0000 q^{97} -7.00000 q^{98} +3.00000 q^{99} +O(q^{100})$$

## 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ −3.00000 −1.00000
$$10$$ 1.00000 0.316228
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ −3.00000 −0.707107
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ −1.00000 −0.213201
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 2.00000 0.392232
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 5.00000 0.883883
$$33$$ 0 0
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 3.00000 0.500000
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 0 0
$$40$$ −3.00000 −0.474342
$$41$$ 2.00000 0.312348 0.156174 0.987730i $$-0.450084\pi$$
0.156174 + 0.987730i $$0.450084\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 1.00000 0.150756
$$45$$ −3.00000 −0.447214
$$46$$ 4.00000 0.589768
$$47$$ −12.0000 −1.75038 −0.875190 0.483779i $$-0.839264\pi$$
−0.875190 + 0.483779i $$0.839264\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 0 0
$$55$$ −1.00000 −0.134840
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 6.00000 0.787839
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ −8.00000 −1.01600
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ 2.00000 0.248069
$$66$$ 0 0
$$67$$ −16.0000 −1.95471 −0.977356 0.211604i $$-0.932131\pi$$
−0.977356 + 0.211604i $$0.932131\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 9.00000 1.06066
$$73$$ 14.0000 1.63858 0.819288 0.573382i $$-0.194369\pi$$
0.819288 + 0.573382i $$0.194369\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 9.00000 1.00000
$$82$$ 2.00000 0.220863
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 0 0
$$85$$ 6.00000 0.650791
$$86$$ 4.00000 0.431331
$$87$$ 0 0
$$88$$ 3.00000 0.319801
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ −3.00000 −0.316228
$$91$$ 0 0
$$92$$ −4.00000 −0.417029
$$93$$ 0 0
$$94$$ −12.0000 −1.23771
$$95$$ −4.00000 −0.410391
$$96$$ 0 0
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ −7.00000 −0.707107
$$99$$ 3.00000 0.301511
$$100$$ −1.00000 −0.100000
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ −18.0000 −1.72409 −0.862044 0.506834i $$-0.830816\pi$$
−0.862044 + 0.506834i $$0.830816\pi$$
$$110$$ −1.00000 −0.0953463
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ −6.00000 −0.557086
$$117$$ −6.00000 −0.554700
$$118$$ 4.00000 0.368230
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −10.0000 −0.905357
$$123$$ 0 0
$$124$$ 8.00000 0.718421
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 0 0
$$130$$ 2.00000 0.175412
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −16.0000 −1.38219
$$135$$ 0 0
$$136$$ −18.0000 −1.54349
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 8.00000 0.671345
$$143$$ −2.00000 −0.167248
$$144$$ 3.00000 0.250000
$$145$$ 6.00000 0.498273
$$146$$ 14.0000 1.15865
$$147$$ 0 0
$$148$$ 2.00000 0.164399
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 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
$$153$$ −18.0000 −1.45521
$$154$$ 0 0
$$155$$ −8.00000 −0.642575
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 8.00000 0.636446
$$159$$ 0 0
$$160$$ 5.00000 0.395285
$$161$$ 0 0
$$162$$ 9.00000 0.707107
$$163$$ 16.0000 1.25322 0.626608 0.779334i $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 6.00000 0.460179
$$171$$ 12.0000 0.917663
$$172$$ −4.00000 −0.304997
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 1.00000 0.0753778
$$177$$ 0 0
$$178$$ 10.0000 0.749532
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 3.00000 0.223607
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −12.0000 −0.884652
$$185$$ −2.00000 −0.147043
$$186$$ 0 0
$$187$$ −6.00000 −0.438763
$$188$$ 12.0000 0.875190
$$189$$ 0 0
$$190$$ −4.00000 −0.290191
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 0 0
$$193$$ −26.0000 −1.87152 −0.935760 0.352636i $$-0.885285\pi$$
−0.935760 + 0.352636i $$0.885285\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 7.00000 0.500000
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 3.00000 0.213201
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ −3.00000 −0.212132
$$201$$ 0 0
$$202$$ −10.0000 −0.703598
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 2.00000 0.139686
$$206$$ −4.00000 −0.278693
$$207$$ −12.0000 −0.834058
$$208$$ −2.00000 −0.138675
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 2.00000 0.137361
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ 4.00000 0.272798
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −18.0000 −1.21911
$$219$$ 0 0
$$220$$ 1.00000 0.0674200
$$221$$ 12.0000 0.807207
$$222$$ 0 0
$$223$$ −4.00000 −0.267860 −0.133930 0.990991i $$-0.542760\pi$$
−0.133930 + 0.990991i $$0.542760\pi$$
$$224$$ 0 0
$$225$$ −3.00000 −0.200000
$$226$$ −6.00000 −0.399114
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ 0 0
$$229$$ −10.0000 −0.660819 −0.330409 0.943838i $$-0.607187\pi$$
−0.330409 + 0.943838i $$0.607187\pi$$
$$230$$ 4.00000 0.263752
$$231$$ 0 0
$$232$$ −18.0000 −1.18176
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ −12.0000 −0.782794
$$236$$ −4.00000 −0.260378
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 8.00000 0.517477 0.258738 0.965947i $$-0.416693\pi$$
0.258738 + 0.965947i $$0.416693\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ 0 0
$$244$$ 10.0000 0.640184
$$245$$ −7.00000 −0.447214
$$246$$ 0 0
$$247$$ −8.00000 −0.509028
$$248$$ 24.0000 1.52400
$$249$$ 0 0
$$250$$ 1.00000 0.0632456
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ −4.00000 −0.251478
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −2.00000 −0.124035
$$261$$ −18.0000 −1.11417
$$262$$ −12.0000 −0.741362
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 0 0
$$265$$ −2.00000 −0.122859
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 16.0000 0.977356
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ 0 0
$$274$$ 18.0000 1.08742
$$275$$ −1.00000 −0.0603023
$$276$$ 0 0
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ 12.0000 0.719712
$$279$$ 24.0000 1.43684
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ −8.00000 −0.474713
$$285$$ 0 0
$$286$$ −2.00000 −0.118262
$$287$$ 0 0
$$288$$ −15.0000 −0.883883
$$289$$ 19.0000 1.11765
$$290$$ 6.00000 0.352332
$$291$$ 0 0
$$292$$ −14.0000 −0.819288
$$293$$ 10.0000 0.584206 0.292103 0.956387i $$-0.405645\pi$$
0.292103 + 0.956387i $$0.405645\pi$$
$$294$$ 0 0
$$295$$ 4.00000 0.232889
$$296$$ 6.00000 0.348743
$$297$$ 0 0
$$298$$ −10.0000 −0.579284
$$299$$ 8.00000 0.462652
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 8.00000 0.460348
$$303$$ 0 0
$$304$$ 4.00000 0.229416
$$305$$ −10.0000 −0.572598
$$306$$ −18.0000 −1.02899
$$307$$ 20.0000 1.14146 0.570730 0.821138i $$-0.306660\pi$$
0.570730 + 0.821138i $$0.306660\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −8.00000 −0.454369
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ −22.0000 −1.24351 −0.621757 0.783210i $$-0.713581\pi$$
−0.621757 + 0.783210i $$0.713581\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 0 0
$$319$$ −6.00000 −0.335936
$$320$$ 7.00000 0.391312
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −24.0000 −1.33540
$$324$$ −9.00000 −0.500000
$$325$$ 2.00000 0.110940
$$326$$ 16.0000 0.886158
$$327$$ 0 0
$$328$$ −6.00000 −0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 4.00000 0.219529
$$333$$ 6.00000 0.328798
$$334$$ −8.00000 −0.437741
$$335$$ −16.0000 −0.874173
$$336$$ 0 0
$$337$$ 6.00000 0.326841 0.163420 0.986557i $$-0.447747\pi$$
0.163420 + 0.986557i $$0.447747\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 0 0
$$340$$ −6.00000 −0.325396
$$341$$ 8.00000 0.433224
$$342$$ 12.0000 0.648886
$$343$$ 0 0
$$344$$ −12.0000 −0.646997
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ −4.00000 −0.214731 −0.107366 0.994220i $$-0.534242\pi$$
−0.107366 + 0.994220i $$0.534242\pi$$
$$348$$ 0 0
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −5.00000 −0.266501
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ 8.00000 0.424596
$$356$$ −10.0000 −0.529999
$$357$$ 0 0
$$358$$ 4.00000 0.211407
$$359$$ −32.0000 −1.68890 −0.844448 0.535638i $$-0.820071\pi$$
−0.844448 + 0.535638i $$0.820071\pi$$
$$360$$ 9.00000 0.474342
$$361$$ −3.00000 −0.157895
$$362$$ −10.0000 −0.525588
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 14.0000 0.732793
$$366$$ 0 0
$$367$$ 4.00000 0.208798 0.104399 0.994535i $$-0.466708\pi$$
0.104399 + 0.994535i $$0.466708\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ −6.00000 −0.312348
$$370$$ −2.00000 −0.103975
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 18.0000 0.932005 0.466002 0.884783i $$-0.345694\pi$$
0.466002 + 0.884783i $$0.345694\pi$$
$$374$$ −6.00000 −0.310253
$$375$$ 0 0
$$376$$ 36.0000 1.85656
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 4.00000 0.205196
$$381$$ 0 0
$$382$$ 8.00000 0.409316
$$383$$ −12.0000 −0.613171 −0.306586 0.951843i $$-0.599187\pi$$
−0.306586 + 0.951843i $$0.599187\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −26.0000 −1.32337
$$387$$ −12.0000 −0.609994
$$388$$ −10.0000 −0.507673
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ 24.0000 1.21373
$$392$$ 21.0000 1.06066
$$393$$ 0 0
$$394$$ 2.00000 0.100759
$$395$$ 8.00000 0.402524
$$396$$ −3.00000 −0.150756
$$397$$ 30.0000 1.50566 0.752828 0.658217i $$-0.228689\pi$$
0.752828 + 0.658217i $$0.228689\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ 0 0
$$403$$ −16.0000 −0.797017
$$404$$ 10.0000 0.497519
$$405$$ 9.00000 0.447214
$$406$$ 0 0
$$407$$ 2.00000 0.0991363
$$408$$ 0 0
$$409$$ −6.00000 −0.296681 −0.148340 0.988936i $$-0.547393\pi$$
−0.148340 + 0.988936i $$0.547393\pi$$
$$410$$ 2.00000 0.0987730
$$411$$ 0 0
$$412$$ 4.00000 0.197066
$$413$$ 0 0
$$414$$ −12.0000 −0.589768
$$415$$ −4.00000 −0.196352
$$416$$ 10.0000 0.490290
$$417$$ 0 0
$$418$$ 4.00000 0.195646
$$419$$ −28.0000 −1.36789 −0.683945 0.729534i $$-0.739737\pi$$
−0.683945 + 0.729534i $$0.739737\pi$$
$$420$$ 0 0
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ 4.00000 0.194717
$$423$$ 36.0000 1.75038
$$424$$ 6.00000 0.291386
$$425$$ 6.00000 0.291043
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ 4.00000 0.192897
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ 0 0
$$433$$ −22.0000 −1.05725 −0.528626 0.848855i $$-0.677293\pi$$
−0.528626 + 0.848855i $$0.677293\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 18.0000 0.862044
$$437$$ −16.0000 −0.765384
$$438$$ 0 0
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 3.00000 0.143019
$$441$$ 21.0000 1.00000
$$442$$ 12.0000 0.570782
$$443$$ 8.00000 0.380091 0.190046 0.981775i $$-0.439136\pi$$
0.190046 + 0.981775i $$0.439136\pi$$
$$444$$ 0 0
$$445$$ 10.0000 0.474045
$$446$$ −4.00000 −0.189405
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 2.00000 0.0943858 0.0471929 0.998886i $$-0.484972\pi$$
0.0471929 + 0.998886i $$0.484972\pi$$
$$450$$ −3.00000 −0.141421
$$451$$ −2.00000 −0.0941763
$$452$$ 6.00000 0.282216
$$453$$ 0 0
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −26.0000 −1.21623 −0.608114 0.793849i $$-0.708074\pi$$
−0.608114 + 0.793849i $$0.708074\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ 0 0
$$460$$ −4.00000 −0.186501
$$461$$ −34.0000 −1.58354 −0.791769 0.610821i $$-0.790840\pi$$
−0.791769 + 0.610821i $$0.790840\pi$$
$$462$$ 0 0
$$463$$ −36.0000 −1.67306 −0.836531 0.547920i $$-0.815420\pi$$
−0.836531 + 0.547920i $$0.815420\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 6.00000 0.277350
$$469$$ 0 0
$$470$$ −12.0000 −0.553519
$$471$$ 0 0
$$472$$ −12.0000 −0.552345
$$473$$ −4.00000 −0.183920
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ 0 0
$$477$$ 6.00000 0.274721
$$478$$ 8.00000 0.365911
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ −4.00000 −0.182384
$$482$$ 10.0000 0.455488
$$483$$ 0 0
$$484$$ −1.00000 −0.0454545
$$485$$ 10.0000 0.454077
$$486$$ 0 0
$$487$$ 28.0000 1.26880 0.634401 0.773004i $$-0.281247\pi$$
0.634401 + 0.773004i $$0.281247\pi$$
$$488$$ 30.0000 1.35804
$$489$$ 0 0
$$490$$ −7.00000 −0.316228
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ 0 0
$$493$$ 36.0000 1.62136
$$494$$ −8.00000 −0.359937
$$495$$ 3.00000 0.134840
$$496$$ 8.00000 0.359211
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −36.0000 −1.61158 −0.805791 0.592200i $$-0.798259\pi$$
−0.805791 + 0.592200i $$0.798259\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ 12.0000 0.535586
$$503$$ −16.0000 −0.713405 −0.356702 0.934218i $$-0.616099\pi$$
−0.356702 + 0.934218i $$0.616099\pi$$
$$504$$ 0 0
$$505$$ −10.0000 −0.444994
$$506$$ −4.00000 −0.177822
$$507$$ 0 0
$$508$$ −16.0000 −0.709885
$$509$$ 14.0000 0.620539 0.310270 0.950649i $$-0.399581\pi$$
0.310270 + 0.950649i $$0.399581\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −11.0000 −0.486136
$$513$$ 0 0
$$514$$ 18.0000 0.793946
$$515$$ −4.00000 −0.176261
$$516$$ 0 0
$$517$$ 12.0000 0.527759
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −6.00000 −0.263117
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ −18.0000 −0.787839
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ 12.0000 0.524222
$$525$$ 0 0
$$526$$ 24.0000 1.04645
$$527$$ −48.0000 −2.09091
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ −2.00000 −0.0868744
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ 4.00000 0.173259
$$534$$ 0 0
$$535$$ 12.0000 0.518805
$$536$$ 48.0000 2.07328
$$537$$ 0 0
$$538$$ −18.0000 −0.776035
$$539$$ 7.00000 0.301511
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 30.0000 1.28624
$$545$$ −18.0000 −0.771035
$$546$$ 0 0
$$547$$ 12.0000 0.513083 0.256541 0.966533i $$-0.417417\pi$$
0.256541 + 0.966533i $$0.417417\pi$$
$$548$$ −18.0000 −0.768922
$$549$$ 30.0000 1.28037
$$550$$ −1.00000 −0.0426401
$$551$$ −24.0000 −1.02243
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ −12.0000 −0.508913
$$557$$ 10.0000 0.423714 0.211857 0.977301i $$-0.432049\pi$$
0.211857 + 0.977301i $$0.432049\pi$$
$$558$$ 24.0000 1.01600
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 18.0000 0.759284
$$563$$ 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
$$566$$ 4.00000 0.168133
$$567$$ 0 0
$$568$$ −24.0000 −1.00702
$$569$$ 26.0000 1.08998 0.544988 0.838444i $$-0.316534\pi$$
0.544988 + 0.838444i $$0.316534\pi$$
$$570$$ 0 0
$$571$$ 36.0000 1.50655 0.753277 0.657704i $$-0.228472\pi$$
0.753277 + 0.657704i $$0.228472\pi$$
$$572$$ 2.00000 0.0836242
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 4.00000 0.166812
$$576$$ −21.0000 −0.875000
$$577$$ −22.0000 −0.915872 −0.457936 0.888985i $$-0.651411\pi$$
−0.457936 + 0.888985i $$0.651411\pi$$
$$578$$ 19.0000 0.790296
$$579$$ 0 0
$$580$$ −6.00000 −0.249136
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 2.00000 0.0828315
$$584$$ −42.0000 −1.73797
$$585$$ −6.00000 −0.248069
$$586$$ 10.0000 0.413096
$$587$$ −24.0000 −0.990586 −0.495293 0.868726i $$-0.664939\pi$$
−0.495293 + 0.868726i $$0.664939\pi$$
$$588$$ 0 0
$$589$$ 32.0000 1.31854
$$590$$ 4.00000 0.164677
$$591$$ 0 0
$$592$$ 2.00000 0.0821995
$$593$$ 22.0000 0.903432 0.451716 0.892162i $$-0.350812\pi$$
0.451716 + 0.892162i $$0.350812\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 0 0
$$598$$ 8.00000 0.327144
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 0 0
$$603$$ 48.0000 1.95471
$$604$$ −8.00000 −0.325515
$$605$$ 1.00000 0.0406558
$$606$$ 0 0
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ −20.0000 −0.811107
$$609$$ 0 0
$$610$$ −10.0000 −0.404888
$$611$$ −24.0000 −0.970936
$$612$$ 18.0000 0.727607
$$613$$ 34.0000 1.37325 0.686624 0.727013i $$-0.259092\pi$$
0.686624 + 0.727013i $$0.259092\pi$$
$$614$$ 20.0000 0.807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 2.00000 0.0805170 0.0402585 0.999189i $$-0.487182\pi$$
0.0402585 + 0.999189i $$0.487182\pi$$
$$618$$ 0 0
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 8.00000 0.321288
$$621$$ 0 0
$$622$$ −24.0000 −0.962312
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −22.0000 −0.879297
$$627$$ 0 0
$$628$$ 2.00000 0.0798087
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ 40.0000 1.59237 0.796187 0.605050i $$-0.206847\pi$$
0.796187 + 0.605050i $$0.206847\pi$$
$$632$$ −24.0000 −0.954669
$$633$$ 0 0
$$634$$ −18.0000 −0.714871
$$635$$ 16.0000 0.634941
$$636$$ 0 0
$$637$$ −14.0000 −0.554700
$$638$$ −6.00000 −0.237542
$$639$$ −24.0000 −0.949425
$$640$$ −3.00000 −0.118585
$$641$$ 34.0000 1.34292 0.671460 0.741041i $$-0.265668\pi$$
0.671460 + 0.741041i $$0.265668\pi$$
$$642$$ 0 0
$$643$$ −16.0000 −0.630978 −0.315489 0.948929i $$-0.602169\pi$$
−0.315489 + 0.948929i $$0.602169\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −24.0000 −0.944267
$$647$$ 20.0000 0.786281 0.393141 0.919478i $$-0.371389\pi$$
0.393141 + 0.919478i $$0.371389\pi$$
$$648$$ −27.0000 −1.06066
$$649$$ −4.00000 −0.157014
$$650$$ 2.00000 0.0784465
$$651$$ 0 0
$$652$$ −16.0000 −0.626608
$$653$$ −10.0000 −0.391330 −0.195665 0.980671i $$-0.562687\pi$$
−0.195665 + 0.980671i $$0.562687\pi$$
$$654$$ 0 0
$$655$$ −12.0000 −0.468879
$$656$$ −2.00000 −0.0780869
$$657$$ −42.0000 −1.63858
$$658$$ 0 0
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ 24.0000 0.929284
$$668$$ 8.00000 0.309529
$$669$$ 0 0
$$670$$ −16.0000 −0.618134
$$671$$ 10.0000 0.386046
$$672$$ 0 0
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ 6.00000 0.231111
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ −38.0000 −1.46046 −0.730229 0.683202i $$-0.760587\pi$$
−0.730229 + 0.683202i $$0.760587\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −18.0000 −0.690268
$$681$$ 0 0
$$682$$ 8.00000 0.306336
$$683$$ −16.0000 −0.612223 −0.306111 0.951996i $$-0.599028\pi$$
−0.306111 + 0.951996i $$0.599028\pi$$
$$684$$ −12.0000 −0.458831
$$685$$ 18.0000 0.687745
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −4.00000 −0.152499
$$689$$ −4.00000 −0.152388
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 0 0
$$694$$ −4.00000 −0.151838
$$695$$ 12.0000 0.455186
$$696$$ 0 0
$$697$$ 12.0000 0.454532
$$698$$ −10.0000 −0.378506
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 22.0000 0.830929 0.415464 0.909610i $$-0.363619\pi$$
0.415464 + 0.909610i $$0.363619\pi$$
$$702$$ 0 0
$$703$$ 8.00000 0.301726
$$704$$ −7.00000 −0.263822
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 8.00000 0.300235
$$711$$ −24.0000 −0.900070
$$712$$ −30.0000 −1.12430
$$713$$ −32.0000 −1.19841
$$714$$ 0 0
$$715$$ −2.00000 −0.0747958
$$716$$ −4.00000 −0.149487
$$717$$ 0 0
$$718$$ −32.0000 −1.19423
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 3.00000 0.111803
$$721$$ 0 0
$$722$$ −3.00000 −0.111648
$$723$$ 0 0
$$724$$ 10.0000 0.371647
$$725$$ 6.00000 0.222834
$$726$$ 0 0
$$727$$ 52.0000 1.92857 0.964287 0.264861i $$-0.0853260\pi$$
0.964287 + 0.264861i $$0.0853260\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 14.0000 0.518163
$$731$$ 24.0000 0.887672
$$732$$ 0 0
$$733$$ 42.0000 1.55131 0.775653 0.631160i $$-0.217421\pi$$
0.775653 + 0.631160i $$0.217421\pi$$
$$734$$ 4.00000 0.147643
$$735$$ 0 0
$$736$$ 20.0000 0.737210
$$737$$ 16.0000 0.589368
$$738$$ −6.00000 −0.220863
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ 2.00000 0.0735215
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 40.0000 1.46746 0.733729 0.679442i $$-0.237778\pi$$
0.733729 + 0.679442i $$0.237778\pi$$
$$744$$ 0 0
$$745$$ −10.0000 −0.366372
$$746$$ 18.0000 0.659027
$$747$$ 12.0000 0.439057
$$748$$ 6.00000 0.219382
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ 12.0000 0.437595
$$753$$ 0 0
$$754$$ 12.0000 0.437014
$$755$$ 8.00000 0.291150
$$756$$ 0 0
$$757$$ 6.00000 0.218074 0.109037 0.994038i $$-0.465223\pi$$
0.109037 + 0.994038i $$0.465223\pi$$
$$758$$ 20.0000 0.726433
$$759$$ 0 0
$$760$$ 12.0000 0.435286
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −8.00000 −0.289430
$$765$$ −18.0000 −0.650791
$$766$$ −12.0000 −0.433578
$$767$$ 8.00000 0.288863
$$768$$ 0 0
$$769$$ −22.0000 −0.793340 −0.396670 0.917961i $$-0.629834\pi$$
−0.396670 + 0.917961i $$0.629834\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 26.0000 0.935760
$$773$$ 14.0000 0.503545 0.251773 0.967786i $$-0.418987\pi$$
0.251773 + 0.967786i $$0.418987\pi$$
$$774$$ −12.0000 −0.431331
$$775$$ −8.00000 −0.287368
$$776$$ −30.0000 −1.07694
$$777$$ 0 0
$$778$$ 6.00000 0.215110
$$779$$ −8.00000 −0.286630
$$780$$ 0 0
$$781$$ −8.00000 −0.286263
$$782$$ 24.0000 0.858238
$$783$$ 0 0
$$784$$ 7.00000 0.250000
$$785$$ −2.00000 −0.0713831
$$786$$ 0 0
$$787$$ 52.0000 1.85360 0.926800 0.375555i $$-0.122548\pi$$
0.926800 + 0.375555i $$0.122548\pi$$
$$788$$ −2.00000 −0.0712470
$$789$$ 0 0
$$790$$ 8.00000 0.284627
$$791$$ 0 0
$$792$$ −9.00000 −0.319801
$$793$$ −20.0000 −0.710221
$$794$$ 30.0000 1.06466
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −2.00000 −0.0708436 −0.0354218 0.999372i $$-0.511277\pi$$
−0.0354218 + 0.999372i $$0.511277\pi$$
$$798$$ 0 0
$$799$$ −72.0000 −2.54718
$$800$$ 5.00000 0.176777
$$801$$ −30.0000 −1.06000
$$802$$ 2.00000 0.0706225
$$803$$ −14.0000 −0.494049
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −16.0000 −0.563576
$$807$$ 0 0
$$808$$ 30.0000 1.05540
$$809$$ 18.0000 0.632846 0.316423 0.948618i $$-0.397518\pi$$
0.316423 + 0.948618i $$0.397518\pi$$
$$810$$ 9.00000 0.316228
$$811$$ −12.0000 −0.421377 −0.210688 0.977553i $$-0.567571\pi$$
−0.210688 + 0.977553i $$0.567571\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 2.00000 0.0701000
$$815$$ 16.0000 0.560456
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ −6.00000 −0.209785
$$819$$ 0 0
$$820$$ −2.00000 −0.0698430
$$821$$ −18.0000 −0.628204 −0.314102 0.949389i $$-0.601703\pi$$
−0.314102 + 0.949389i $$0.601703\pi$$
$$822$$ 0 0
$$823$$ −36.0000 −1.25488 −0.627441 0.778664i $$-0.715897\pi$$
−0.627441 + 0.778664i $$0.715897\pi$$
$$824$$ 12.0000 0.418040
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 12.0000 0.417029
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ −4.00000 −0.138842
$$831$$ 0 0
$$832$$ 14.0000 0.485363
$$833$$ −42.0000 −1.45521
$$834$$ 0 0
$$835$$ −8.00000 −0.276851
$$836$$ −4.00000 −0.138343
$$837$$ 0 0
$$838$$ −28.0000 −0.967244
$$839$$ −32.0000 −1.10476 −0.552381 0.833592i $$-0.686281\pi$$
−0.552381 + 0.833592i $$0.686281\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 6.00000 0.206774
$$843$$ 0 0
$$844$$ −4.00000 −0.137686
$$845$$ −9.00000 −0.309609
$$846$$ 36.0000 1.23771
$$847$$ 0 0
$$848$$ 2.00000 0.0686803
$$849$$ 0 0
$$850$$ 6.00000 0.205798
$$851$$ −8.00000 −0.274236
$$852$$ 0 0
$$853$$ 2.00000 0.0684787 0.0342393 0.999414i $$-0.489099\pi$$
0.0342393 + 0.999414i $$0.489099\pi$$
$$854$$ 0 0
$$855$$ 12.0000 0.410391
$$856$$ −36.0000 −1.23045
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 0 0
$$859$$ 28.0000 0.955348 0.477674 0.878537i $$-0.341480\pi$$
0.477674 + 0.878537i $$0.341480\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ 0 0
$$862$$ −24.0000 −0.817443
$$863$$ 4.00000 0.136162 0.0680808 0.997680i $$-0.478312\pi$$
0.0680808 + 0.997680i $$0.478312\pi$$
$$864$$ 0 0
$$865$$ −6.00000 −0.204006
$$866$$ −22.0000 −0.747590
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −8.00000 −0.271381
$$870$$ 0 0
$$871$$ −32.0000 −1.08428
$$872$$ 54.0000 1.82867
$$873$$ −30.0000 −1.01535
$$874$$ −16.0000 −0.541208
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −22.0000 −0.742887 −0.371444 0.928456i $$-0.621137\pi$$
−0.371444 + 0.928456i $$0.621137\pi$$
$$878$$ −8.00000 −0.269987
$$879$$ 0 0
$$880$$ 1.00000 0.0337100
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ 21.0000 0.707107
$$883$$ −16.0000 −0.538443 −0.269221 0.963078i $$-0.586766\pi$$
−0.269221 + 0.963078i $$0.586766\pi$$
$$884$$ −12.0000 −0.403604
$$885$$ 0 0
$$886$$ 8.00000 0.268765
$$887$$ 56.0000 1.88030 0.940148 0.340766i $$-0.110687\pi$$
0.940148 + 0.340766i $$0.110687\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 10.0000 0.335201
$$891$$ −9.00000 −0.301511
$$892$$ 4.00000 0.133930
$$893$$ 48.0000 1.60626
$$894$$ 0 0
$$895$$ 4.00000 0.133705
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 2.00000 0.0667409
$$899$$ −48.0000 −1.60089
$$900$$ 3.00000 0.100000
$$901$$ −12.0000 −0.399778
$$902$$ −2.00000 −0.0665927
$$903$$ 0 0
$$904$$ 18.0000 0.598671
$$905$$ −10.0000 −0.332411
$$906$$ 0 0
$$907$$ −40.0000 −1.32818 −0.664089 0.747653i $$-0.731180\pi$$
−0.664089 + 0.747653i $$0.731180\pi$$
$$908$$ 20.0000 0.663723
$$909$$ 30.0000 0.995037
$$910$$ 0 0
$$911$$ 8.00000 0.265052 0.132526 0.991180i $$-0.457691\pi$$
0.132526 + 0.991180i $$0.457691\pi$$
$$912$$ 0 0
$$913$$ 4.00000 0.132381
$$914$$ −26.0000 −0.860004
$$915$$ 0 0
$$916$$ 10.0000 0.330409
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ −12.0000 −0.395628
$$921$$ 0 0
$$922$$ −34.0000 −1.11973
$$923$$ 16.0000 0.526646
$$924$$ 0 0
$$925$$ −2.00000 −0.0657596
$$926$$ −36.0000 −1.18303
$$927$$ 12.0000 0.394132
$$928$$ 30.0000 0.984798
$$929$$ −14.0000 −0.459325 −0.229663 0.973270i $$-0.573762\pi$$
−0.229663 + 0.973270i $$0.573762\pi$$
$$930$$ 0 0
$$931$$ 28.0000 0.917663
$$932$$ −6.00000 −0.196537
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −6.00000 −0.196221
$$936$$ 18.0000 0.588348
$$937$$ 6.00000 0.196011 0.0980057 0.995186i $$-0.468754\pi$$
0.0980057 + 0.995186i $$0.468754\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 12.0000 0.391397
$$941$$ −2.00000 −0.0651981 −0.0325991 0.999469i $$-0.510378\pi$$
−0.0325991 + 0.999469i $$0.510378\pi$$
$$942$$ 0 0
$$943$$ 8.00000 0.260516
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ −4.00000 −0.130051
$$947$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$948$$ 0 0
$$949$$ 28.0000 0.908918
$$950$$ −4.00000 −0.129777
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −42.0000 −1.36051 −0.680257 0.732974i $$-0.738132\pi$$
−0.680257 + 0.732974i $$0.738132\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 8.00000 0.258874
$$956$$ −8.00000 −0.258738
$$957$$ 0 0
$$958$$ 24.0000 0.775405
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ −4.00000 −0.128965
$$963$$ −36.0000 −1.16008
$$964$$ −10.0000 −0.322078
$$965$$ −26.0000 −0.836970
$$966$$ 0 0
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ −3.00000 −0.0964237
$$969$$ 0 0
$$970$$ 10.0000 0.321081
$$971$$ 36.0000 1.15529 0.577647 0.816286i $$-0.303971\pi$$
0.577647 + 0.816286i $$0.303971\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 28.0000 0.897178
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ 10.0000 0.319928 0.159964 0.987123i $$-0.448862\pi$$
0.159964 + 0.987123i $$0.448862\pi$$
$$978$$ 0 0
$$979$$ −10.0000 −0.319601
$$980$$ 7.00000 0.223607
$$981$$ 54.0000 1.72409
$$982$$ −28.0000 −0.893516
$$983$$ −4.00000 −0.127580 −0.0637901 0.997963i $$-0.520319\pi$$
−0.0637901 + 0.997963i $$0.520319\pi$$
$$984$$ 0 0
$$985$$ 2.00000 0.0637253
$$986$$ 36.0000 1.14647
$$987$$ 0 0
$$988$$ 8.00000 0.254514
$$989$$ 16.0000 0.508770
$$990$$ 3.00000 0.0953463
$$991$$ 8.00000 0.254128 0.127064 0.991894i $$-0.459445\pi$$
0.127064 + 0.991894i $$0.459445\pi$$
$$992$$ −40.0000 −1.27000
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −46.0000 −1.45683 −0.728417 0.685134i $$-0.759744\pi$$
−0.728417 + 0.685134i $$0.759744\pi$$
$$998$$ −36.0000 −1.13956
$$999$$ 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 55.2.a.a.1.1 1
3.2 odd 2 495.2.a.a.1.1 1
4.3 odd 2 880.2.a.h.1.1 1
5.2 odd 4 275.2.b.b.199.2 2
5.3 odd 4 275.2.b.b.199.1 2
5.4 even 2 275.2.a.a.1.1 1
7.6 odd 2 2695.2.a.c.1.1 1
8.3 odd 2 3520.2.a.n.1.1 1
8.5 even 2 3520.2.a.p.1.1 1
11.2 odd 10 605.2.g.c.81.1 4
11.3 even 5 605.2.g.a.251.1 4
11.4 even 5 605.2.g.a.511.1 4
11.5 even 5 605.2.g.a.366.1 4
11.6 odd 10 605.2.g.c.366.1 4
11.7 odd 10 605.2.g.c.511.1 4
11.8 odd 10 605.2.g.c.251.1 4
11.9 even 5 605.2.g.a.81.1 4
11.10 odd 2 605.2.a.b.1.1 1
12.11 even 2 7920.2.a.i.1.1 1
13.12 even 2 9295.2.a.b.1.1 1
15.2 even 4 2475.2.c.f.199.1 2
15.8 even 4 2475.2.c.f.199.2 2
15.14 odd 2 2475.2.a.i.1.1 1
20.3 even 4 4400.2.b.n.4049.2 2
20.7 even 4 4400.2.b.n.4049.1 2
20.19 odd 2 4400.2.a.p.1.1 1
33.32 even 2 5445.2.a.i.1.1 1
44.43 even 2 9680.2.a.r.1.1 1
55.54 odd 2 3025.2.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.a.1.1 1 1.1 even 1 trivial
275.2.a.a.1.1 1 5.4 even 2
275.2.b.b.199.1 2 5.3 odd 4
275.2.b.b.199.2 2 5.2 odd 4
495.2.a.a.1.1 1 3.2 odd 2
605.2.a.b.1.1 1 11.10 odd 2
605.2.g.a.81.1 4 11.9 even 5
605.2.g.a.251.1 4 11.3 even 5
605.2.g.a.366.1 4 11.5 even 5
605.2.g.a.511.1 4 11.4 even 5
605.2.g.c.81.1 4 11.2 odd 10
605.2.g.c.251.1 4 11.8 odd 10
605.2.g.c.366.1 4 11.6 odd 10
605.2.g.c.511.1 4 11.7 odd 10
880.2.a.h.1.1 1 4.3 odd 2
2475.2.a.i.1.1 1 15.14 odd 2
2475.2.c.f.199.1 2 15.2 even 4
2475.2.c.f.199.2 2 15.8 even 4
2695.2.a.c.1.1 1 7.6 odd 2
3025.2.a.f.1.1 1 55.54 odd 2
3520.2.a.n.1.1 1 8.3 odd 2
3520.2.a.p.1.1 1 8.5 even 2
4400.2.a.p.1.1 1 20.19 odd 2
4400.2.b.n.4049.1 2 20.7 even 4
4400.2.b.n.4049.2 2 20.3 even 4
5445.2.a.i.1.1 1 33.32 even 2
7920.2.a.i.1.1 1 12.11 even 2
9295.2.a.b.1.1 1 13.12 even 2
9680.2.a.r.1.1 1 44.43 even 2