# Properties

 Label 855.2.a.b.1.1 Level $855$ Weight $2$ Character 855.1 Self dual yes Analytic conductor $6.827$ 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] = [855,2,Mod(1,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("855.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$855 = 3^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 855.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: $$6.82720937282$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 285) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 855.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} +3.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} +3.00000 q^{8} -1.00000 q^{10} +2.00000 q^{11} -4.00000 q^{13} +2.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{19} -1.00000 q^{20} -2.00000 q^{22} +4.00000 q^{23} +1.00000 q^{25} +4.00000 q^{26} +2.00000 q^{28} -4.00000 q^{29} -5.00000 q^{32} +2.00000 q^{34} -2.00000 q^{35} +1.00000 q^{38} +3.00000 q^{40} -10.0000 q^{43} -2.00000 q^{44} -4.00000 q^{46} -12.0000 q^{47} -3.00000 q^{49} -1.00000 q^{50} +4.00000 q^{52} +2.00000 q^{53} +2.00000 q^{55} -6.00000 q^{56} +4.00000 q^{58} -4.00000 q^{59} +2.00000 q^{61} +7.00000 q^{64} -4.00000 q^{65} -16.0000 q^{67} +2.00000 q^{68} +2.00000 q^{70} -2.00000 q^{73} +1.00000 q^{76} -4.00000 q^{77} -8.00000 q^{79} -1.00000 q^{80} +12.0000 q^{83} -2.00000 q^{85} +10.0000 q^{86} +6.00000 q^{88} +8.00000 q^{91} -4.00000 q^{92} +12.0000 q^{94} -1.00000 q^{95} -16.0000 q^{97} +3.00000 q^{98} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107 −0.353553 0.935414i $$-0.615027\pi$$
−0.353553 + 0.935414i $$0.615027\pi$$
$$3$$ 0 0
$$4$$ −1.00000 −0.500000
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 3.00000 1.06066
$$9$$ 0 0
$$10$$ −1.00000 −0.316228
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ −2.00000 −0.426401
$$23$$ 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
$$26$$ 4.00000 0.784465
$$27$$ 0 0
$$28$$ 2.00000 0.377964
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −5.00000 −0.883883
$$33$$ 0 0
$$34$$ 2.00000 0.342997
$$35$$ −2.00000 −0.338062
$$36$$ 0 0
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 0 0
$$40$$ 3.00000 0.474342
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ −10.0000 −1.52499 −0.762493 0.646997i $$-0.776025\pi$$
−0.762493 + 0.646997i $$0.776025\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ −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
$$49$$ −3.00000 −0.428571
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ 4.00000 0.554700
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 0 0
$$55$$ 2.00000 0.269680
$$56$$ −6.00000 −0.801784
$$57$$ 0 0
$$58$$ 4.00000 0.525226
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ −4.00000 −0.496139
$$66$$ 0 0
$$67$$ −16.0000 −1.95471 −0.977356 0.211604i $$-0.932131\pi$$
−0.977356 + 0.211604i $$0.932131\pi$$
$$68$$ 2.00000 0.242536
$$69$$ 0 0
$$70$$ 2.00000 0.239046
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ −4.00000 −0.455842
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ −2.00000 −0.216930
$$86$$ 10.0000 1.07833
$$87$$ 0 0
$$88$$ 6.00000 0.639602
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 8.00000 0.838628
$$92$$ −4.00000 −0.417029
$$93$$ 0 0
$$94$$ 12.0000 1.23771
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ −16.0000 −1.62455 −0.812277 0.583272i $$-0.801772\pi$$
−0.812277 + 0.583272i $$0.801772\pi$$
$$98$$ 3.00000 0.303046
$$99$$ 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 $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ −12.0000 −1.17670
$$105$$ 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
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ −2.00000 −0.190693
$$111$$ 0 0
$$112$$ 2.00000 0.188982
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ 4.00000 0.371391
$$117$$ 0 0
$$118$$ 4.00000 0.368230
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ −2.00000 −0.181071
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ 3.00000 0.265165
$$129$$ 0 0
$$130$$ 4.00000 0.350823
$$131$$ 14.0000 1.22319 0.611593 0.791173i $$-0.290529\pi$$
0.611593 + 0.791173i $$0.290529\pi$$
$$132$$ 0 0
$$133$$ 2.00000 0.173422
$$134$$ 16.0000 1.38219
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 2.00000 0.169031
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −8.00000 −0.668994
$$144$$ 0 0
$$145$$ −4.00000 −0.332182
$$146$$ 2.00000 0.165521
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ 24.0000 1.95309 0.976546 0.215308i $$-0.0690756\pi$$
0.976546 + 0.215308i $$0.0690756\pi$$
$$152$$ −3.00000 −0.243332
$$153$$ 0 0
$$154$$ 4.00000 0.322329
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −18.0000 −1.43656 −0.718278 0.695756i $$-0.755069\pi$$
−0.718278 + 0.695756i $$0.755069\pi$$
$$158$$ 8.00000 0.636446
$$159$$ 0 0
$$160$$ −5.00000 −0.395285
$$161$$ −8.00000 −0.630488
$$162$$ 0 0
$$163$$ −6.00000 −0.469956 −0.234978 0.972001i $$-0.575502\pi$$
−0.234978 + 0.972001i $$0.575502\pi$$
$$164$$ 0 0
$$165$$ 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
$$169$$ 3.00000 0.230769
$$170$$ 2.00000 0.153393
$$171$$ 0 0
$$172$$ 10.0000 0.762493
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ 0 0
$$175$$ −2.00000 −0.151186
$$176$$ −2.00000 −0.150756
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ −8.00000 −0.592999
$$183$$ 0 0
$$184$$ 12.0000 0.884652
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −4.00000 −0.292509
$$188$$ 12.0000 0.875190
$$189$$ 0 0
$$190$$ 1.00000 0.0725476
$$191$$ 18.0000 1.30243 0.651217 0.758891i $$-0.274259\pi$$
0.651217 + 0.758891i $$0.274259\pi$$
$$192$$ 0 0
$$193$$ 4.00000 0.287926 0.143963 0.989583i $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ 16.0000 1.14873
$$195$$ 0 0
$$196$$ 3.00000 0.214286
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 0 0
$$199$$ 12.0000 0.850657 0.425329 0.905039i $$-0.360158\pi$$
0.425329 + 0.905039i $$0.360158\pi$$
$$200$$ 3.00000 0.212132
$$201$$ 0 0
$$202$$ 14.0000 0.985037
$$203$$ 8.00000 0.561490
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ 0 0
$$208$$ 4.00000 0.277350
$$209$$ −2.00000 −0.138343
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ −2.00000 −0.137361
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ −10.0000 −0.681994
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −10.0000 −0.677285
$$219$$ 0 0
$$220$$ −2.00000 −0.134840
$$221$$ 8.00000 0.538138
$$222$$ 0 0
$$223$$ 24.0000 1.60716 0.803579 0.595198i $$-0.202926\pi$$
0.803579 + 0.595198i $$0.202926\pi$$
$$224$$ 10.0000 0.668153
$$225$$ 0 0
$$226$$ 14.0000 0.931266
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ −4.00000 −0.263752
$$231$$ 0 0
$$232$$ −12.0000 −0.787839
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 0 0
$$235$$ −12.0000 −0.782794
$$236$$ 4.00000 0.260378
$$237$$ 0 0
$$238$$ −4.00000 −0.259281
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 0 0
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ 7.00000 0.449977
$$243$$ 0 0
$$244$$ −2.00000 −0.128037
$$245$$ −3.00000 −0.191663
$$246$$ 0 0
$$247$$ 4.00000 0.254514
$$248$$ 0 0
$$249$$ 0 0
$$250$$ −1.00000 −0.0632456
$$251$$ −22.0000 −1.38863 −0.694314 0.719672i $$-0.744292\pi$$
−0.694314 + 0.719672i $$0.744292\pi$$
$$252$$ 0 0
$$253$$ 8.00000 0.502956
$$254$$ −4.00000 −0.250982
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ −22.0000 −1.37232 −0.686161 0.727450i $$-0.740706\pi$$
−0.686161 + 0.727450i $$0.740706\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 4.00000 0.248069
$$261$$ 0 0
$$262$$ −14.0000 −0.864923
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 0 0
$$265$$ 2.00000 0.122859
$$266$$ −2.00000 −0.122628
$$267$$ 0 0
$$268$$ 16.0000 0.977356
$$269$$ −4.00000 −0.243884 −0.121942 0.992537i $$-0.538912\pi$$
−0.121942 + 0.992537i $$0.538912\pi$$
$$270$$ 0 0
$$271$$ 12.0000 0.728948 0.364474 0.931214i $$-0.381249\pi$$
0.364474 + 0.931214i $$0.381249\pi$$
$$272$$ 2.00000 0.121268
$$273$$ 0 0
$$274$$ 6.00000 0.362473
$$275$$ 2.00000 0.120605
$$276$$ 0 0
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ −8.00000 −0.479808
$$279$$ 0 0
$$280$$ −6.00000 −0.358569
$$281$$ −28.0000 −1.67034 −0.835170 0.549992i $$-0.814631\pi$$
−0.835170 + 0.549992i $$0.814631\pi$$
$$282$$ 0 0
$$283$$ 26.0000 1.54554 0.772770 0.634686i $$-0.218871\pi$$
0.772770 + 0.634686i $$0.218871\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 8.00000 0.473050
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 4.00000 0.234888
$$291$$ 0 0
$$292$$ 2.00000 0.117041
$$293$$ −26.0000 −1.51894 −0.759468 0.650545i $$-0.774541\pi$$
−0.759468 + 0.650545i $$0.774541\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ 0 0
$$297$$ 0 0
$$298$$ −18.0000 −1.04271
$$299$$ −16.0000 −0.925304
$$300$$ 0 0
$$301$$ 20.0000 1.15278
$$302$$ −24.0000 −1.38104
$$303$$ 0 0
$$304$$ 1.00000 0.0573539
$$305$$ 2.00000 0.114520
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 4.00000 0.227921
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 18.0000 1.02069 0.510343 0.859971i $$-0.329518\pi$$
0.510343 + 0.859971i $$0.329518\pi$$
$$312$$ 0 0
$$313$$ −14.0000 −0.791327 −0.395663 0.918396i $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ 18.0000 1.01580
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ −14.0000 −0.786318 −0.393159 0.919470i $$-0.628618\pi$$
−0.393159 + 0.919470i $$0.628618\pi$$
$$318$$ 0 0
$$319$$ −8.00000 −0.447914
$$320$$ 7.00000 0.391312
$$321$$ 0 0
$$322$$ 8.00000 0.445823
$$323$$ 2.00000 0.111283
$$324$$ 0 0
$$325$$ −4.00000 −0.221880
$$326$$ 6.00000 0.332309
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 24.0000 1.32316
$$330$$ 0 0
$$331$$ −28.0000 −1.53902 −0.769510 0.638635i $$-0.779499\pi$$
−0.769510 + 0.638635i $$0.779499\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ 0 0
$$334$$ −8.00000 −0.437741
$$335$$ −16.0000 −0.874173
$$336$$ 0 0
$$337$$ 16.0000 0.871576 0.435788 0.900049i $$-0.356470\pi$$
0.435788 + 0.900049i $$0.356470\pi$$
$$338$$ −3.00000 −0.163178
$$339$$ 0 0
$$340$$ 2.00000 0.108465
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ −30.0000 −1.61749
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ 28.0000 1.50312 0.751559 0.659665i $$-0.229302\pi$$
0.751559 + 0.659665i $$0.229302\pi$$
$$348$$ 0 0
$$349$$ −18.0000 −0.963518 −0.481759 0.876304i $$-0.660002\pi$$
−0.481759 + 0.876304i $$0.660002\pi$$
$$350$$ 2.00000 0.106904
$$351$$ 0 0
$$352$$ −10.0000 −0.533002
$$353$$ 2.00000 0.106449 0.0532246 0.998583i $$-0.483050\pi$$
0.0532246 + 0.998583i $$0.483050\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ −12.0000 −0.634220
$$359$$ 2.00000 0.105556 0.0527780 0.998606i $$-0.483192\pi$$
0.0527780 + 0.998606i $$0.483192\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −10.0000 −0.525588
$$363$$ 0 0
$$364$$ −8.00000 −0.419314
$$365$$ −2.00000 −0.104685
$$366$$ 0 0
$$367$$ 10.0000 0.521996 0.260998 0.965339i $$-0.415948\pi$$
0.260998 + 0.965339i $$0.415948\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −4.00000 −0.207670
$$372$$ 0 0
$$373$$ 4.00000 0.207112 0.103556 0.994624i $$-0.466978\pi$$
0.103556 + 0.994624i $$0.466978\pi$$
$$374$$ 4.00000 0.206835
$$375$$ 0 0
$$376$$ −36.0000 −1.85656
$$377$$ 16.0000 0.824042
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 1.00000 0.0512989
$$381$$ 0 0
$$382$$ −18.0000 −0.920960
$$383$$ −8.00000 −0.408781 −0.204390 0.978889i $$-0.565521\pi$$
−0.204390 + 0.978889i $$0.565521\pi$$
$$384$$ 0 0
$$385$$ −4.00000 −0.203859
$$386$$ −4.00000 −0.203595
$$387$$ 0 0
$$388$$ 16.0000 0.812277
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ −9.00000 −0.454569
$$393$$ 0 0
$$394$$ −2.00000 −0.100759
$$395$$ −8.00000 −0.402524
$$396$$ 0 0
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ −12.0000 −0.601506
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ −12.0000 −0.599251 −0.299626 0.954057i $$-0.596862\pi$$
−0.299626 + 0.954057i $$0.596862\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 14.0000 0.696526
$$405$$ 0 0
$$406$$ −8.00000 −0.397033
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 8.00000 0.394132
$$413$$ 8.00000 0.393654
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ 20.0000 0.980581
$$417$$ 0 0
$$418$$ 2.00000 0.0978232
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ 0 0
$$421$$ −34.0000 −1.65706 −0.828529 0.559946i $$-0.810822\pi$$
−0.828529 + 0.559946i $$0.810822\pi$$
$$422$$ 12.0000 0.584151
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ −4.00000 −0.193574
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ 10.0000 0.482243
$$431$$ 4.00000 0.192673 0.0963366 0.995349i $$-0.469287\pi$$
0.0963366 + 0.995349i $$0.469287\pi$$
$$432$$ 0 0
$$433$$ −36.0000 −1.73005 −0.865025 0.501729i $$-0.832697\pi$$
−0.865025 + 0.501729i $$0.832697\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −10.0000 −0.478913
$$437$$ −4.00000 −0.191346
$$438$$ 0 0
$$439$$ −40.0000 −1.90910 −0.954548 0.298057i $$-0.903661\pi$$
−0.954548 + 0.298057i $$0.903661\pi$$
$$440$$ 6.00000 0.286039
$$441$$ 0 0
$$442$$ −8.00000 −0.380521
$$443$$ −16.0000 −0.760183 −0.380091 0.924949i $$-0.624107\pi$$
−0.380091 + 0.924949i $$0.624107\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −24.0000 −1.13643
$$447$$ 0 0
$$448$$ −14.0000 −0.661438
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 14.0000 0.658505
$$453$$ 0 0
$$454$$ 12.0000 0.563188
$$455$$ 8.00000 0.375046
$$456$$ 0 0
$$457$$ −14.0000 −0.654892 −0.327446 0.944870i $$-0.606188\pi$$
−0.327446 + 0.944870i $$0.606188\pi$$
$$458$$ 14.0000 0.654177
$$459$$ 0 0
$$460$$ −4.00000 −0.186501
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ 0 0
$$463$$ −22.0000 −1.02243 −0.511213 0.859454i $$-0.670804\pi$$
−0.511213 + 0.859454i $$0.670804\pi$$
$$464$$ 4.00000 0.185695
$$465$$ 0 0
$$466$$ −18.0000 −0.833834
$$467$$ −16.0000 −0.740392 −0.370196 0.928954i $$-0.620709\pi$$
−0.370196 + 0.928954i $$0.620709\pi$$
$$468$$ 0 0
$$469$$ 32.0000 1.47762
$$470$$ 12.0000 0.553519
$$471$$ 0 0
$$472$$ −12.0000 −0.552345
$$473$$ −20.0000 −0.919601
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ −4.00000 −0.183340
$$477$$ 0 0
$$478$$ −6.00000 −0.274434
$$479$$ −38.0000 −1.73626 −0.868132 0.496333i $$-0.834679\pi$$
−0.868132 + 0.496333i $$0.834679\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −26.0000 −1.18427
$$483$$ 0 0
$$484$$ 7.00000 0.318182
$$485$$ −16.0000 −0.726523
$$486$$ 0 0
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ 6.00000 0.271607
$$489$$ 0 0
$$490$$ 3.00000 0.135526
$$491$$ 6.00000 0.270776 0.135388 0.990793i $$-0.456772\pi$$
0.135388 + 0.990793i $$0.456772\pi$$
$$492$$ 0 0
$$493$$ 8.00000 0.360302
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 32.0000 1.43252 0.716258 0.697835i $$-0.245853\pi$$
0.716258 + 0.697835i $$0.245853\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ 0 0
$$502$$ 22.0000 0.981908
$$503$$ 8.00000 0.356702 0.178351 0.983967i $$-0.442924\pi$$
0.178351 + 0.983967i $$0.442924\pi$$
$$504$$ 0 0
$$505$$ −14.0000 −0.622992
$$506$$ −8.00000 −0.355643
$$507$$ 0 0
$$508$$ −4.00000 −0.177471
$$509$$ 24.0000 1.06378 0.531891 0.846813i $$-0.321482\pi$$
0.531891 + 0.846813i $$0.321482\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ 11.0000 0.486136
$$513$$ 0 0
$$514$$ 22.0000 0.970378
$$515$$ −8.00000 −0.352522
$$516$$ 0 0
$$517$$ −24.0000 −1.05552
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −12.0000 −0.526235
$$521$$ 12.0000 0.525730 0.262865 0.964833i $$-0.415333\pi$$
0.262865 + 0.964833i $$0.415333\pi$$
$$522$$ 0 0
$$523$$ −28.0000 −1.22435 −0.612177 0.790721i $$-0.709706\pi$$
−0.612177 + 0.790721i $$0.709706\pi$$
$$524$$ −14.0000 −0.611593
$$525$$ 0 0
$$526$$ −16.0000 −0.697633
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ −2.00000 −0.0868744
$$531$$ 0 0
$$532$$ −2.00000 −0.0867110
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 12.0000 0.518805
$$536$$ −48.0000 −2.07328
$$537$$ 0 0
$$538$$ 4.00000 0.172452
$$539$$ −6.00000 −0.258438
$$540$$ 0 0
$$541$$ −42.0000 −1.80572 −0.902861 0.429934i $$-0.858537\pi$$
−0.902861 + 0.429934i $$0.858537\pi$$
$$542$$ −12.0000 −0.515444
$$543$$ 0 0
$$544$$ 10.0000 0.428746
$$545$$ 10.0000 0.428353
$$546$$ 0 0
$$547$$ 16.0000 0.684111 0.342055 0.939680i $$-0.388877\pi$$
0.342055 + 0.939680i $$0.388877\pi$$
$$548$$ 6.00000 0.256307
$$549$$ 0 0
$$550$$ −2.00000 −0.0852803
$$551$$ 4.00000 0.170406
$$552$$ 0 0
$$553$$ 16.0000 0.680389
$$554$$ −22.0000 −0.934690
$$555$$ 0 0
$$556$$ −8.00000 −0.339276
$$557$$ 38.0000 1.61011 0.805056 0.593199i $$-0.202135\pi$$
0.805056 + 0.593199i $$0.202135\pi$$
$$558$$ 0 0
$$559$$ 40.0000 1.69182
$$560$$ 2.00000 0.0845154
$$561$$ 0 0
$$562$$ 28.0000 1.18111
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 0 0
$$565$$ −14.0000 −0.588984
$$566$$ −26.0000 −1.09286
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ −24.0000 −1.00437 −0.502184 0.864761i $$-0.667470\pi$$
−0.502184 + 0.864761i $$0.667470\pi$$
$$572$$ 8.00000 0.334497
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 4.00000 0.166812
$$576$$ 0 0
$$577$$ 42.0000 1.74848 0.874241 0.485491i $$-0.161359\pi$$
0.874241 + 0.485491i $$0.161359\pi$$
$$578$$ 13.0000 0.540729
$$579$$ 0 0
$$580$$ 4.00000 0.166091
$$581$$ −24.0000 −0.995688
$$582$$ 0 0
$$583$$ 4.00000 0.165663
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ 26.0000 1.07405
$$587$$ 32.0000 1.32078 0.660391 0.750922i $$-0.270391\pi$$
0.660391 + 0.750922i $$0.270391\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 4.00000 0.164677
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 0 0
$$595$$ 4.00000 0.163984
$$596$$ −18.0000 −0.737309
$$597$$ 0 0
$$598$$ 16.0000 0.654289
$$599$$ −36.0000 −1.47092 −0.735460 0.677568i $$-0.763034\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 0 0
$$601$$ 30.0000 1.22373 0.611863 0.790964i $$-0.290420\pi$$
0.611863 + 0.790964i $$0.290420\pi$$
$$602$$ −20.0000 −0.815139
$$603$$ 0 0
$$604$$ −24.0000 −0.976546
$$605$$ −7.00000 −0.284590
$$606$$ 0 0
$$607$$ −16.0000 −0.649420 −0.324710 0.945814i $$-0.605267\pi$$
−0.324710 + 0.945814i $$0.605267\pi$$
$$608$$ 5.00000 0.202777
$$609$$ 0 0
$$610$$ −2.00000 −0.0809776
$$611$$ 48.0000 1.94187
$$612$$ 0 0
$$613$$ −18.0000 −0.727013 −0.363507 0.931592i $$-0.618421\pi$$
−0.363507 + 0.931592i $$0.618421\pi$$
$$614$$ 0 0
$$615$$ 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
$$619$$ 16.0000 0.643094 0.321547 0.946894i $$-0.395797\pi$$
0.321547 + 0.946894i $$0.395797\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −18.0000 −0.721734
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 14.0000 0.559553
$$627$$ 0 0
$$628$$ 18.0000 0.718278
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 48.0000 1.91085 0.955425 0.295234i $$-0.0953977\pi$$
0.955425 + 0.295234i $$0.0953977\pi$$
$$632$$ −24.0000 −0.954669
$$633$$ 0 0
$$634$$ 14.0000 0.556011
$$635$$ 4.00000 0.158735
$$636$$ 0 0
$$637$$ 12.0000 0.475457
$$638$$ 8.00000 0.316723
$$639$$ 0 0
$$640$$ 3.00000 0.118585
$$641$$ 24.0000 0.947943 0.473972 0.880540i $$-0.342820\pi$$
0.473972 + 0.880540i $$0.342820\pi$$
$$642$$ 0 0
$$643$$ 26.0000 1.02534 0.512670 0.858586i $$-0.328656\pi$$
0.512670 + 0.858586i $$0.328656\pi$$
$$644$$ 8.00000 0.315244
$$645$$ 0 0
$$646$$ −2.00000 −0.0786889
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ 0 0
$$649$$ −8.00000 −0.314027
$$650$$ 4.00000 0.156893
$$651$$ 0 0
$$652$$ 6.00000 0.234978
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 0 0
$$655$$ 14.0000 0.547025
$$656$$ 0 0
$$657$$ 0 0
$$658$$ −24.0000 −0.935617
$$659$$ −16.0000 −0.623272 −0.311636 0.950202i $$-0.600877\pi$$
−0.311636 + 0.950202i $$0.600877\pi$$
$$660$$ 0 0
$$661$$ 18.0000 0.700119 0.350059 0.936727i $$-0.386161\pi$$
0.350059 + 0.936727i $$0.386161\pi$$
$$662$$ 28.0000 1.08825
$$663$$ 0 0
$$664$$ 36.0000 1.39707
$$665$$ 2.00000 0.0775567
$$666$$ 0 0
$$667$$ −16.0000 −0.619522
$$668$$ −8.00000 −0.309529
$$669$$ 0 0
$$670$$ 16.0000 0.618134
$$671$$ 4.00000 0.154418
$$672$$ 0 0
$$673$$ 24.0000 0.925132 0.462566 0.886585i $$-0.346929\pi$$
0.462566 + 0.886585i $$0.346929\pi$$
$$674$$ −16.0000 −0.616297
$$675$$ 0 0
$$676$$ −3.00000 −0.115385
$$677$$ 50.0000 1.92166 0.960828 0.277145i $$-0.0893883\pi$$
0.960828 + 0.277145i $$0.0893883\pi$$
$$678$$ 0 0
$$679$$ 32.0000 1.22805
$$680$$ −6.00000 −0.230089
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ −6.00000 −0.229248
$$686$$ −20.0000 −0.763604
$$687$$ 0 0
$$688$$ 10.0000 0.381246
$$689$$ −8.00000 −0.304776
$$690$$ 0 0
$$691$$ −16.0000 −0.608669 −0.304334 0.952565i $$-0.598434\pi$$
−0.304334 + 0.952565i $$0.598434\pi$$
$$692$$ 18.0000 0.684257
$$693$$ 0 0
$$694$$ −28.0000 −1.06287
$$695$$ 8.00000 0.303457
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 18.0000 0.681310
$$699$$ 0 0
$$700$$ 2.00000 0.0755929
$$701$$ 34.0000 1.28416 0.642081 0.766637i $$-0.278071\pi$$
0.642081 + 0.766637i $$0.278071\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 14.0000 0.527645
$$705$$ 0 0
$$706$$ −2.00000 −0.0752710
$$707$$ 28.0000 1.05305
$$708$$ 0 0
$$709$$ 18.0000 0.676004 0.338002 0.941145i $$-0.390249\pi$$
0.338002 + 0.941145i $$0.390249\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ −8.00000 −0.299183
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ −2.00000 −0.0746393
$$719$$ −42.0000 −1.56634 −0.783168 0.621810i $$-0.786397\pi$$
−0.783168 + 0.621810i $$0.786397\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ −1.00000 −0.0372161
$$723$$ 0 0
$$724$$ −10.0000 −0.371647
$$725$$ −4.00000 −0.148556
$$726$$ 0 0
$$727$$ −14.0000 −0.519231 −0.259616 0.965712i $$-0.583596\pi$$
−0.259616 + 0.965712i $$0.583596\pi$$
$$728$$ 24.0000 0.889499
$$729$$ 0 0
$$730$$ 2.00000 0.0740233
$$731$$ 20.0000 0.739727
$$732$$ 0 0
$$733$$ 34.0000 1.25582 0.627909 0.778287i $$-0.283911\pi$$
0.627909 + 0.778287i $$0.283911\pi$$
$$734$$ −10.0000 −0.369107
$$735$$ 0 0
$$736$$ −20.0000 −0.737210
$$737$$ −32.0000 −1.17874
$$738$$ 0 0
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 4.00000 0.146845
$$743$$ 8.00000 0.293492 0.146746 0.989174i $$-0.453120\pi$$
0.146746 + 0.989174i $$0.453120\pi$$
$$744$$ 0 0
$$745$$ 18.0000 0.659469
$$746$$ −4.00000 −0.146450
$$747$$ 0 0
$$748$$ 4.00000 0.146254
$$749$$ −24.0000 −0.876941
$$750$$ 0 0
$$751$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$752$$ 12.0000 0.437595
$$753$$ 0 0
$$754$$ −16.0000 −0.582686
$$755$$ 24.0000 0.873449
$$756$$ 0 0
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ 20.0000 0.726433
$$759$$ 0 0
$$760$$ −3.00000 −0.108821
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 0 0
$$763$$ −20.0000 −0.724049
$$764$$ −18.0000 −0.651217
$$765$$ 0 0
$$766$$ 8.00000 0.289052
$$767$$ 16.0000 0.577727
$$768$$ 0 0
$$769$$ −46.0000 −1.65880 −0.829401 0.558653i $$-0.811318\pi$$
−0.829401 + 0.558653i $$0.811318\pi$$
$$770$$ 4.00000 0.144150
$$771$$ 0 0
$$772$$ −4.00000 −0.143963
$$773$$ −14.0000 −0.503545 −0.251773 0.967786i $$-0.581013\pi$$
−0.251773 + 0.967786i $$0.581013\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −48.0000 −1.72310
$$777$$ 0 0
$$778$$ −30.0000 −1.07555
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 8.00000 0.286079
$$783$$ 0 0
$$784$$ 3.00000 0.107143
$$785$$ −18.0000 −0.642448
$$786$$ 0 0
$$787$$ 28.0000 0.998092 0.499046 0.866575i $$-0.333684\pi$$
0.499046 + 0.866575i $$0.333684\pi$$
$$788$$ −2.00000 −0.0712470
$$789$$ 0 0
$$790$$ 8.00000 0.284627
$$791$$ 28.0000 0.995565
$$792$$ 0 0
$$793$$ −8.00000 −0.284088
$$794$$ 2.00000 0.0709773
$$795$$ 0 0
$$796$$ −12.0000 −0.425329
$$797$$ 14.0000 0.495905 0.247953 0.968772i $$-0.420242\pi$$
0.247953 + 0.968772i $$0.420242\pi$$
$$798$$ 0 0
$$799$$ 24.0000 0.849059
$$800$$ −5.00000 −0.176777
$$801$$ 0 0
$$802$$ 12.0000 0.423735
$$803$$ −4.00000 −0.141157
$$804$$ 0 0
$$805$$ −8.00000 −0.281963
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −42.0000 −1.47755
$$809$$ −2.00000 −0.0703163 −0.0351581 0.999382i $$-0.511193\pi$$
−0.0351581 + 0.999382i $$0.511193\pi$$
$$810$$ 0 0
$$811$$ 4.00000 0.140459 0.0702295 0.997531i $$-0.477627\pi$$
0.0702295 + 0.997531i $$0.477627\pi$$
$$812$$ −8.00000 −0.280745
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −6.00000 −0.210171
$$816$$ 0 0
$$817$$ 10.0000 0.349856
$$818$$ 26.0000 0.909069
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −6.00000 −0.209401 −0.104701 0.994504i $$-0.533388\pi$$
−0.104701 + 0.994504i $$0.533388\pi$$
$$822$$ 0 0
$$823$$ −14.0000 −0.488009 −0.244005 0.969774i $$-0.578461\pi$$
−0.244005 + 0.969774i $$0.578461\pi$$
$$824$$ −24.0000 −0.836080
$$825$$ 0 0
$$826$$ −8.00000 −0.278356
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ 0 0
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ −12.0000 −0.416526
$$831$$ 0 0
$$832$$ −28.0000 −0.970725
$$833$$ 6.00000 0.207888
$$834$$ 0 0
$$835$$ 8.00000 0.276851
$$836$$ 2.00000 0.0691714
$$837$$ 0 0
$$838$$ −26.0000 −0.898155
$$839$$ −24.0000 −0.828572 −0.414286 0.910147i $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 34.0000 1.17172
$$843$$ 0 0
$$844$$ 12.0000 0.413057
$$845$$ 3.00000 0.103203
$$846$$ 0 0
$$847$$ 14.0000 0.481046
$$848$$ −2.00000 −0.0686803
$$849$$ 0 0
$$850$$ 2.00000 0.0685994
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 10.0000 0.342393 0.171197 0.985237i $$-0.445237\pi$$
0.171197 + 0.985237i $$0.445237\pi$$
$$854$$ 4.00000 0.136877
$$855$$ 0 0
$$856$$ 36.0000 1.23045
$$857$$ −10.0000 −0.341593 −0.170797 0.985306i $$-0.554634\pi$$
−0.170797 + 0.985306i $$0.554634\pi$$
$$858$$ 0 0
$$859$$ −12.0000 −0.409435 −0.204717 0.978821i $$-0.565628\pi$$
−0.204717 + 0.978821i $$0.565628\pi$$
$$860$$ 10.0000 0.340997
$$861$$ 0 0
$$862$$ −4.00000 −0.136241
$$863$$ −40.0000 −1.36162 −0.680808 0.732462i $$-0.738371\pi$$
−0.680808 + 0.732462i $$0.738371\pi$$
$$864$$ 0 0
$$865$$ −18.0000 −0.612018
$$866$$ 36.0000 1.22333
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ 64.0000 2.16856
$$872$$ 30.0000 1.01593
$$873$$ 0 0
$$874$$ 4.00000 0.135302
$$875$$ −2.00000 −0.0676123
$$876$$ 0 0
$$877$$ 12.0000 0.405211 0.202606 0.979260i $$-0.435059\pi$$
0.202606 + 0.979260i $$0.435059\pi$$
$$878$$ 40.0000 1.34993
$$879$$ 0 0
$$880$$ −2.00000 −0.0674200
$$881$$ 10.0000 0.336909 0.168454 0.985709i $$-0.446122\pi$$
0.168454 + 0.985709i $$0.446122\pi$$
$$882$$ 0 0
$$883$$ −46.0000 −1.54802 −0.774012 0.633171i $$-0.781753\pi$$
−0.774012 + 0.633171i $$0.781753\pi$$
$$884$$ −8.00000 −0.269069
$$885$$ 0 0
$$886$$ 16.0000 0.537531
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ 0 0
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −24.0000 −0.803579
$$893$$ 12.0000 0.401565
$$894$$ 0 0
$$895$$ 12.0000 0.401116
$$896$$ −6.00000 −0.200446
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −4.00000 −0.133259
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −42.0000 −1.39690
$$905$$ 10.0000 0.332411
$$906$$ 0 0
$$907$$ 52.0000 1.72663 0.863316 0.504664i $$-0.168384\pi$$
0.863316 + 0.504664i $$0.168384\pi$$
$$908$$ 12.0000 0.398234
$$909$$ 0 0
$$910$$ −8.00000 −0.265197
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ 24.0000 0.794284
$$914$$ 14.0000 0.463079
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ −28.0000 −0.924641
$$918$$ 0 0
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 12.0000 0.395628
$$921$$ 0 0
$$922$$ 6.00000 0.197599
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 22.0000 0.722965
$$927$$ 0 0
$$928$$ 20.0000 0.656532
$$929$$ −10.0000 −0.328089 −0.164045 0.986453i $$-0.552454\pi$$
−0.164045 + 0.986453i $$0.552454\pi$$
$$930$$ 0 0
$$931$$ 3.00000 0.0983210
$$932$$ −18.0000 −0.589610
$$933$$ 0 0
$$934$$ 16.0000 0.523536
$$935$$ −4.00000 −0.130814
$$936$$ 0 0
$$937$$ −42.0000 −1.37208 −0.686040 0.727564i $$-0.740653\pi$$
−0.686040 + 0.727564i $$0.740653\pi$$
$$938$$ −32.0000 −1.04484
$$939$$ 0 0
$$940$$ 12.0000 0.391397
$$941$$ 60.0000 1.95594 0.977972 0.208736i $$-0.0669349\pi$$
0.977972 + 0.208736i $$0.0669349\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ 20.0000 0.650256
$$947$$ 52.0000 1.68977 0.844886 0.534946i $$-0.179668\pi$$
0.844886 + 0.534946i $$0.179668\pi$$
$$948$$ 0 0
$$949$$ 8.00000 0.259691
$$950$$ 1.00000 0.0324443
$$951$$ 0 0
$$952$$ 12.0000 0.388922
$$953$$ −46.0000 −1.49009 −0.745043 0.667016i $$-0.767571\pi$$
−0.745043 + 0.667016i $$0.767571\pi$$
$$954$$ 0 0
$$955$$ 18.0000 0.582466
$$956$$ −6.00000 −0.194054
$$957$$ 0 0
$$958$$ 38.0000 1.22772
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −26.0000 −0.837404
$$965$$ 4.00000 0.128765
$$966$$ 0 0
$$967$$ 34.0000 1.09337 0.546683 0.837340i $$-0.315890\pi$$
0.546683 + 0.837340i $$0.315890\pi$$
$$968$$ −21.0000 −0.674966
$$969$$ 0 0
$$970$$ 16.0000 0.513729
$$971$$ 24.0000 0.770197 0.385098 0.922876i $$-0.374168\pi$$
0.385098 + 0.922876i $$0.374168\pi$$
$$972$$ 0 0
$$973$$ −16.0000 −0.512936
$$974$$ −8.00000 −0.256337
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 3.00000 0.0958315
$$981$$ 0 0
$$982$$ −6.00000 −0.191468
$$983$$ −16.0000 −0.510321 −0.255160 0.966899i $$-0.582128\pi$$
−0.255160 + 0.966899i $$0.582128\pi$$
$$984$$ 0 0
$$985$$ 2.00000 0.0637253
$$986$$ −8.00000 −0.254772
$$987$$ 0 0
$$988$$ −4.00000 −0.127257
$$989$$ −40.0000 −1.27193
$$990$$ 0 0
$$991$$ 8.00000 0.254128 0.127064 0.991894i $$-0.459445\pi$$
0.127064 + 0.991894i $$0.459445\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 12.0000 0.380426
$$996$$ 0 0
$$997$$ −2.00000 −0.0633406 −0.0316703 0.999498i $$-0.510083\pi$$
−0.0316703 + 0.999498i $$0.510083\pi$$
$$998$$ −32.0000 −1.01294
$$999$$ 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 855.2.a.b.1.1 1
3.2 odd 2 285.2.a.b.1.1 1
5.4 even 2 4275.2.a.o.1.1 1
12.11 even 2 4560.2.a.v.1.1 1
15.2 even 4 1425.2.c.d.799.2 2
15.8 even 4 1425.2.c.d.799.1 2
15.14 odd 2 1425.2.a.d.1.1 1
57.56 even 2 5415.2.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.b.1.1 1 3.2 odd 2
855.2.a.b.1.1 1 1.1 even 1 trivial
1425.2.a.d.1.1 1 15.14 odd 2
1425.2.c.d.799.1 2 15.8 even 4
1425.2.c.d.799.2 2 15.2 even 4
4275.2.a.o.1.1 1 5.4 even 2
4560.2.a.v.1.1 1 12.11 even 2
5415.2.a.c.1.1 1 57.56 even 2