# Properties

 Label 285.2.a.b.1.1 Level $285$ Weight $2$ Character 285.1 Self dual yes Analytic conductor $2.276$ 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] = [285,2,Mod(1,285)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(285, 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("285.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$285 = 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 285.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: $$2.27573645761$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 285.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} -2.00000 q^{22} -4.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} -4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} +4.00000 q^{29} +1.00000 q^{30} +5.00000 q^{32} +2.00000 q^{33} +2.00000 q^{34} +2.00000 q^{35} -1.00000 q^{36} -1.00000 q^{38} +4.00000 q^{39} +3.00000 q^{40} +2.00000 q^{42} -10.0000 q^{43} +2.00000 q^{44} -1.00000 q^{45} -4.00000 q^{46} +12.0000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +4.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} +2.00000 q^{55} +6.00000 q^{56} +1.00000 q^{57} +4.00000 q^{58} +4.00000 q^{59} -1.00000 q^{60} +2.00000 q^{61} -2.00000 q^{63} +7.00000 q^{64} +4.00000 q^{65} +2.00000 q^{66} -16.0000 q^{67} -2.00000 q^{68} +4.00000 q^{69} +2.00000 q^{70} -3.00000 q^{72} -2.00000 q^{73} -1.00000 q^{75} +1.00000 q^{76} +4.00000 q^{77} +4.00000 q^{78} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -12.0000 q^{83} -2.00000 q^{84} -2.00000 q^{85} -10.0000 q^{86} -4.00000 q^{87} +6.00000 q^{88} -1.00000 q^{90} +8.00000 q^{91} +4.00000 q^{92} +12.0000 q^{94} +1.00000 q^{95} -5.00000 q^{96} -16.0000 q^{97} -3.00000 q^{98} -2.00000 q^{99} +O(q^{100})$$

## 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 $$0.384973\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ −1.00000 −0.500000
$$5$$ −1.00000 −0.447214
$$6$$ −1.00000 −0.408248
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 1.00000 0.333333
$$10$$ −1.00000 −0.316228
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 1.00000 0.288675
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 1.00000 0.258199
$$16$$ −1.00000 −0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −1.00000 −0.229416
$$20$$ 1.00000 0.223607
$$21$$ 2.00000 0.436436
$$22$$ −2.00000 −0.426401
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 3.00000 0.612372
$$25$$ 1.00000 0.200000
$$26$$ −4.00000 −0.784465
$$27$$ −1.00000 −0.192450
$$28$$ 2.00000 0.377964
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ 1.00000 0.182574
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 5.00000 0.883883
$$33$$ 2.00000 0.348155
$$34$$ 2.00000 0.342997
$$35$$ 2.00000 0.338062
$$36$$ −1.00000 −0.166667
$$37$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 4.00000 0.640513
$$40$$ 3.00000 0.474342
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 2.00000 0.308607
$$43$$ −10.0000 −1.52499 −0.762493 0.646997i $$-0.776025\pi$$
−0.762493 + 0.646997i $$0.776025\pi$$
$$44$$ 2.00000 0.301511
$$45$$ −1.00000 −0.149071
$$46$$ −4.00000 −0.589768
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −3.00000 −0.428571
$$50$$ 1.00000 0.141421
$$51$$ −2.00000 −0.280056
$$52$$ 4.00000 0.554700
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 2.00000 0.269680
$$56$$ 6.00000 0.801784
$$57$$ 1.00000 0.132453
$$58$$ 4.00000 0.525226
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ −2.00000 −0.251976
$$64$$ 7.00000 0.875000
$$65$$ 4.00000 0.496139
$$66$$ 2.00000 0.246183
$$67$$ −16.0000 −1.95471 −0.977356 0.211604i $$-0.932131\pi$$
−0.977356 + 0.211604i $$0.932131\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ 4.00000 0.481543
$$70$$ 2.00000 0.239046
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ −3.00000 −0.353553
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 0 0
$$75$$ −1.00000 −0.115470
$$76$$ 1.00000 0.114708
$$77$$ 4.00000 0.455842
$$78$$ 4.00000 0.452911
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ −2.00000 −0.216930
$$86$$ −10.0000 −1.07833
$$87$$ −4.00000 −0.428845
$$88$$ 6.00000 0.639602
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ −1.00000 −0.105409
$$91$$ 8.00000 0.838628
$$92$$ 4.00000 0.417029
$$93$$ 0 0
$$94$$ 12.0000 1.23771
$$95$$ 1.00000 0.102598
$$96$$ −5.00000 −0.510310
$$97$$ −16.0000 −1.62455 −0.812277 0.583272i $$-0.801772\pi$$
−0.812277 + 0.583272i $$0.801772\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ −2.00000 −0.201008
$$100$$ −1.00000 −0.100000
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ −2.00000 −0.198030
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 12.0000 1.17670
$$105$$ −2.00000 −0.195180
$$106$$ −2.00000 −0.194257
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 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 $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ 4.00000 0.373002
$$116$$ −4.00000 −0.371391
$$117$$ −4.00000 −0.369800
$$118$$ 4.00000 0.368230
$$119$$ −4.00000 −0.366679
$$120$$ −3.00000 −0.273861
$$121$$ −7.00000 −0.636364
$$122$$ 2.00000 0.181071
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ −2.00000 −0.178174
$$127$$ 4.00000 0.354943 0.177471 0.984126i $$-0.443208\pi$$
0.177471 + 0.984126i $$0.443208\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 10.0000 0.880451
$$130$$ 4.00000 0.350823
$$131$$ −14.0000 −1.22319 −0.611593 0.791173i $$-0.709471\pi$$
−0.611593 + 0.791173i $$0.709471\pi$$
$$132$$ −2.00000 −0.174078
$$133$$ 2.00000 0.173422
$$134$$ −16.0000 −1.38219
$$135$$ 1.00000 0.0860663
$$136$$ −6.00000 −0.514496
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 4.00000 0.340503
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ −12.0000 −1.01058
$$142$$ 0 0
$$143$$ 8.00000 0.668994
$$144$$ −1.00000 −0.0833333
$$145$$ −4.00000 −0.332182
$$146$$ −2.00000 −0.165521
$$147$$ 3.00000 0.247436
$$148$$ 0 0
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 24.0000 1.95309 0.976546 0.215308i $$-0.0690756\pi$$
0.976546 + 0.215308i $$0.0690756\pi$$
$$152$$ 3.00000 0.243332
$$153$$ 2.00000 0.161690
$$154$$ 4.00000 0.322329
$$155$$ 0 0
$$156$$ −4.00000 −0.320256
$$157$$ −18.0000 −1.43656 −0.718278 0.695756i $$-0.755069\pi$$
−0.718278 + 0.695756i $$0.755069\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 2.00000 0.158610
$$160$$ −5.00000 −0.395285
$$161$$ 8.00000 0.630488
$$162$$ 1.00000 0.0785674
$$163$$ −6.00000 −0.469956 −0.234978 0.972001i $$-0.575502\pi$$
−0.234978 + 0.972001i $$0.575502\pi$$
$$164$$ 0 0
$$165$$ −2.00000 −0.155700
$$166$$ −12.0000 −0.931381
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ −6.00000 −0.462910
$$169$$ 3.00000 0.230769
$$170$$ −2.00000 −0.153393
$$171$$ −1.00000 −0.0764719
$$172$$ 10.0000 0.762493
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ −4.00000 −0.303239
$$175$$ −2.00000 −0.151186
$$176$$ 2.00000 0.150756
$$177$$ −4.00000 −0.300658
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 1.00000 0.0745356
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 8.00000 0.592999
$$183$$ −2.00000 −0.147844
$$184$$ 12.0000 0.884652
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −4.00000 −0.292509
$$188$$ −12.0000 −0.875190
$$189$$ 2.00000 0.145479
$$190$$ 1.00000 0.0725476
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ −7.00000 −0.505181
$$193$$ 4.00000 0.287926 0.143963 0.989583i $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ −16.0000 −1.14873
$$195$$ −4.00000 −0.286446
$$196$$ 3.00000 0.214286
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ −2.00000 −0.142134
$$199$$ 12.0000 0.850657 0.425329 0.905039i $$-0.360158\pi$$
0.425329 + 0.905039i $$0.360158\pi$$
$$200$$ −3.00000 −0.212132
$$201$$ 16.0000 1.12855
$$202$$ 14.0000 0.985037
$$203$$ −8.00000 −0.561490
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ −4.00000 −0.278019
$$208$$ 4.00000 0.277350
$$209$$ 2.00000 0.138343
$$210$$ −2.00000 −0.138013
$$211$$ −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
$$216$$ 3.00000 0.204124
$$217$$ 0 0
$$218$$ 10.0000 0.677285
$$219$$ 2.00000 0.135147
$$220$$ −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
$$225$$ 1.00000 0.0666667
$$226$$ 14.0000 0.931266
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ −1.00000 −0.0662266
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 4.00000 0.263752
$$231$$ −4.00000 −0.263181
$$232$$ −12.0000 −0.787839
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ −12.0000 −0.782794
$$236$$ −4.00000 −0.260378
$$237$$ 8.00000 0.519656
$$238$$ −4.00000 −0.259281
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ 26.0000 1.67481 0.837404 0.546585i $$-0.184072\pi$$
0.837404 + 0.546585i $$0.184072\pi$$
$$242$$ −7.00000 −0.449977
$$243$$ −1.00000 −0.0641500
$$244$$ −2.00000 −0.128037
$$245$$ 3.00000 0.191663
$$246$$ 0 0
$$247$$ 4.00000 0.254514
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ −1.00000 −0.0632456
$$251$$ 22.0000 1.38863 0.694314 0.719672i $$-0.255708\pi$$
0.694314 + 0.719672i $$0.255708\pi$$
$$252$$ 2.00000 0.125988
$$253$$ 8.00000 0.502956
$$254$$ 4.00000 0.250982
$$255$$ 2.00000 0.125245
$$256$$ −17.0000 −1.06250
$$257$$ 22.0000 1.37232 0.686161 0.727450i $$-0.259294\pi$$
0.686161 + 0.727450i $$0.259294\pi$$
$$258$$ 10.0000 0.622573
$$259$$ 0 0
$$260$$ −4.00000 −0.248069
$$261$$ 4.00000 0.247594
$$262$$ −14.0000 −0.864923
$$263$$ −16.0000 −0.986602 −0.493301 0.869859i $$-0.664210\pi$$
−0.493301 + 0.869859i $$0.664210\pi$$
$$264$$ −6.00000 −0.369274
$$265$$ 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 $$-0.461088\pi$$
0.121942 + 0.992537i $$0.461088\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ 12.0000 0.728948 0.364474 0.931214i $$-0.381249\pi$$
0.364474 + 0.931214i $$0.381249\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ −8.00000 −0.484182
$$274$$ 6.00000 0.362473
$$275$$ −2.00000 −0.120605
$$276$$ −4.00000 −0.240772
$$277$$ 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 $$-0.185369\pi$$
0.835170 + 0.549992i $$0.185369\pi$$
$$282$$ −12.0000 −0.714590
$$283$$ 26.0000 1.54554 0.772770 0.634686i $$-0.218871\pi$$
0.772770 + 0.634686i $$0.218871\pi$$
$$284$$ 0 0
$$285$$ −1.00000 −0.0592349
$$286$$ 8.00000 0.473050
$$287$$ 0 0
$$288$$ 5.00000 0.294628
$$289$$ −13.0000 −0.764706
$$290$$ −4.00000 −0.234888
$$291$$ 16.0000 0.937937
$$292$$ 2.00000 0.117041
$$293$$ 26.0000 1.51894 0.759468 0.650545i $$-0.225459\pi$$
0.759468 + 0.650545i $$0.225459\pi$$
$$294$$ 3.00000 0.174964
$$295$$ −4.00000 −0.232889
$$296$$ 0 0
$$297$$ 2.00000 0.116052
$$298$$ −18.0000 −1.04271
$$299$$ 16.0000 0.925304
$$300$$ 1.00000 0.0577350
$$301$$ 20.0000 1.15278
$$302$$ 24.0000 1.38104
$$303$$ −14.0000 −0.804279
$$304$$ 1.00000 0.0573539
$$305$$ −2.00000 −0.114520
$$306$$ 2.00000 0.114332
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ −4.00000 −0.227921
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ −12.0000 −0.679366
$$313$$ −14.0000 −0.791327 −0.395663 0.918396i $$-0.629485\pi$$
−0.395663 + 0.918396i $$0.629485\pi$$
$$314$$ −18.0000 −1.01580
$$315$$ 2.00000 0.112687
$$316$$ 8.00000 0.450035
$$317$$ 14.0000 0.786318 0.393159 0.919470i $$-0.371382\pi$$
0.393159 + 0.919470i $$0.371382\pi$$
$$318$$ 2.00000 0.112154
$$319$$ −8.00000 −0.447914
$$320$$ −7.00000 −0.391312
$$321$$ 12.0000 0.669775
$$322$$ 8.00000 0.445823
$$323$$ −2.00000 −0.111283
$$324$$ −1.00000 −0.0555556
$$325$$ −4.00000 −0.221880
$$326$$ −6.00000 −0.332309
$$327$$ −10.0000 −0.553001
$$328$$ 0 0
$$329$$ −24.0000 −1.32316
$$330$$ −2.00000 −0.110096
$$331$$ −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
$$336$$ −2.00000 −0.109109
$$337$$ 16.0000 0.871576 0.435788 0.900049i $$-0.356470\pi$$
0.435788 + 0.900049i $$0.356470\pi$$
$$338$$ 3.00000 0.163178
$$339$$ −14.0000 −0.760376
$$340$$ 2.00000 0.108465
$$341$$ 0 0
$$342$$ −1.00000 −0.0540738
$$343$$ 20.0000 1.07990
$$344$$ 30.0000 1.61749
$$345$$ −4.00000 −0.215353
$$346$$ 18.0000 0.967686
$$347$$ −28.0000 −1.50312 −0.751559 0.659665i $$-0.770698\pi$$
−0.751559 + 0.659665i $$0.770698\pi$$
$$348$$ 4.00000 0.214423
$$349$$ −18.0000 −0.963518 −0.481759 0.876304i $$-0.660002\pi$$
−0.481759 + 0.876304i $$0.660002\pi$$
$$350$$ −2.00000 −0.106904
$$351$$ 4.00000 0.213504
$$352$$ −10.0000 −0.533002
$$353$$ −2.00000 −0.106449 −0.0532246 0.998583i $$-0.516950\pi$$
−0.0532246 + 0.998583i $$0.516950\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 4.00000 0.211702
$$358$$ −12.0000 −0.634220
$$359$$ −2.00000 −0.105556 −0.0527780 0.998606i $$-0.516808\pi$$
−0.0527780 + 0.998606i $$0.516808\pi$$
$$360$$ 3.00000 0.158114
$$361$$ 1.00000 0.0526316
$$362$$ 10.0000 0.525588
$$363$$ 7.00000 0.367405
$$364$$ −8.00000 −0.419314
$$365$$ 2.00000 0.104685
$$366$$ −2.00000 −0.104542
$$367$$ 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
$$375$$ 1.00000 0.0516398
$$376$$ −36.0000 −1.85656
$$377$$ −16.0000 −0.824042
$$378$$ 2.00000 0.102869
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ −1.00000 −0.0512989
$$381$$ −4.00000 −0.204926
$$382$$ −18.0000 −0.920960
$$383$$ 8.00000 0.408781 0.204390 0.978889i $$-0.434479\pi$$
0.204390 + 0.978889i $$0.434479\pi$$
$$384$$ 3.00000 0.153093
$$385$$ −4.00000 −0.203859
$$386$$ 4.00000 0.203595
$$387$$ −10.0000 −0.508329
$$388$$ 16.0000 0.812277
$$389$$ −30.0000 −1.52106 −0.760530 0.649303i $$-0.775061\pi$$
−0.760530 + 0.649303i $$0.775061\pi$$
$$390$$ −4.00000 −0.202548
$$391$$ −8.00000 −0.404577
$$392$$ 9.00000 0.454569
$$393$$ 14.0000 0.706207
$$394$$ −2.00000 −0.100759
$$395$$ 8.00000 0.402524
$$396$$ 2.00000 0.100504
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ 12.0000 0.601506
$$399$$ −2.00000 −0.100125
$$400$$ −1.00000 −0.0500000
$$401$$ 12.0000 0.599251 0.299626 0.954057i $$-0.403138\pi$$
0.299626 + 0.954057i $$0.403138\pi$$
$$402$$ 16.0000 0.798007
$$403$$ 0 0
$$404$$ −14.0000 −0.696526
$$405$$ −1.00000 −0.0496904
$$406$$ −8.00000 −0.397033
$$407$$ 0 0
$$408$$ 6.00000 0.297044
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ 8.00000 0.394132
$$413$$ −8.00000 −0.393654
$$414$$ −4.00000 −0.196589
$$415$$ 12.0000 0.589057
$$416$$ −20.0000 −0.980581
$$417$$ −8.00000 −0.391762
$$418$$ 2.00000 0.0978232
$$419$$ −26.0000 −1.27018 −0.635092 0.772437i $$-0.719038\pi$$
−0.635092 + 0.772437i $$0.719038\pi$$
$$420$$ 2.00000 0.0975900
$$421$$ −34.0000 −1.65706 −0.828529 0.559946i $$-0.810822\pi$$
−0.828529 + 0.559946i $$0.810822\pi$$
$$422$$ −12.0000 −0.584151
$$423$$ 12.0000 0.583460
$$424$$ 6.00000 0.291386
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ −4.00000 −0.193574
$$428$$ 12.0000 0.580042
$$429$$ −8.00000 −0.386244
$$430$$ 10.0000 0.482243
$$431$$ −4.00000 −0.192673 −0.0963366 0.995349i $$-0.530713\pi$$
−0.0963366 + 0.995349i $$0.530713\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −36.0000 −1.73005 −0.865025 0.501729i $$-0.832697\pi$$
−0.865025 + 0.501729i $$0.832697\pi$$
$$434$$ 0 0
$$435$$ 4.00000 0.191785
$$436$$ −10.0000 −0.478913
$$437$$ 4.00000 0.191346
$$438$$ 2.00000 0.0955637
$$439$$ −40.0000 −1.90910 −0.954548 0.298057i $$-0.903661\pi$$
−0.954548 + 0.298057i $$0.903661\pi$$
$$440$$ −6.00000 −0.286039
$$441$$ −3.00000 −0.142857
$$442$$ −8.00000 −0.380521
$$443$$ 16.0000 0.760183 0.380091 0.924949i $$-0.375893\pi$$
0.380091 + 0.924949i $$0.375893\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 24.0000 1.13643
$$447$$ 18.0000 0.851371
$$448$$ −14.0000 −0.661438
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 0 0
$$452$$ −14.0000 −0.658505
$$453$$ −24.0000 −1.12762
$$454$$ 12.0000 0.563188
$$455$$ −8.00000 −0.375046
$$456$$ −3.00000 −0.140488
$$457$$ −14.0000 −0.654892 −0.327446 0.944870i $$-0.606188\pi$$
−0.327446 + 0.944870i $$0.606188\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ −2.00000 −0.0933520
$$460$$ −4.00000 −0.186501
$$461$$ 6.00000 0.279448 0.139724 0.990190i $$-0.455378\pi$$
0.139724 + 0.990190i $$0.455378\pi$$
$$462$$ −4.00000 −0.186097
$$463$$ −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 $$-0.379291\pi$$
0.370196 + 0.928954i $$0.379291\pi$$
$$468$$ 4.00000 0.184900
$$469$$ 32.0000 1.47762
$$470$$ −12.0000 −0.553519
$$471$$ 18.0000 0.829396
$$472$$ −12.0000 −0.552345
$$473$$ 20.0000 0.919601
$$474$$ 8.00000 0.367452
$$475$$ −1.00000 −0.0458831
$$476$$ 4.00000 0.183340
$$477$$ −2.00000 −0.0915737
$$478$$ −6.00000 −0.274434
$$479$$ 38.0000 1.73626 0.868132 0.496333i $$-0.165321\pi$$
0.868132 + 0.496333i $$0.165321\pi$$
$$480$$ 5.00000 0.228218
$$481$$ 0 0
$$482$$ 26.0000 1.18427
$$483$$ −8.00000 −0.364013
$$484$$ 7.00000 0.318182
$$485$$ 16.0000 0.726523
$$486$$ −1.00000 −0.0453609
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ −6.00000 −0.271607
$$489$$ 6.00000 0.271329
$$490$$ 3.00000 0.135526
$$491$$ −6.00000 −0.270776 −0.135388 0.990793i $$-0.543228\pi$$
−0.135388 + 0.990793i $$0.543228\pi$$
$$492$$ 0 0
$$493$$ 8.00000 0.360302
$$494$$ 4.00000 0.179969
$$495$$ 2.00000 0.0898933
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 12.0000 0.537733
$$499$$ 32.0000 1.43252 0.716258 0.697835i $$-0.245853\pi$$
0.716258 + 0.697835i $$0.245853\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 8.00000 0.357414
$$502$$ 22.0000 0.981908
$$503$$ −8.00000 −0.356702 −0.178351 0.983967i $$-0.557076\pi$$
−0.178351 + 0.983967i $$0.557076\pi$$
$$504$$ 6.00000 0.267261
$$505$$ −14.0000 −0.622992
$$506$$ 8.00000 0.355643
$$507$$ −3.00000 −0.133235
$$508$$ −4.00000 −0.177471
$$509$$ −24.0000 −1.06378 −0.531891 0.846813i $$-0.678518\pi$$
−0.531891 + 0.846813i $$0.678518\pi$$
$$510$$ 2.00000 0.0885615
$$511$$ 4.00000 0.176950
$$512$$ −11.0000 −0.486136
$$513$$ 1.00000 0.0441511
$$514$$ 22.0000 0.970378
$$515$$ 8.00000 0.352522
$$516$$ −10.0000 −0.440225
$$517$$ −24.0000 −1.05552
$$518$$ 0 0
$$519$$ −18.0000 −0.790112
$$520$$ −12.0000 −0.526235
$$521$$ −12.0000 −0.525730 −0.262865 0.964833i $$-0.584667\pi$$
−0.262865 + 0.964833i $$0.584667\pi$$
$$522$$ 4.00000 0.175075
$$523$$ −28.0000 −1.22435 −0.612177 0.790721i $$-0.709706\pi$$
−0.612177 + 0.790721i $$0.709706\pi$$
$$524$$ 14.0000 0.611593
$$525$$ 2.00000 0.0872872
$$526$$ −16.0000 −0.697633
$$527$$ 0 0
$$528$$ −2.00000 −0.0870388
$$529$$ −7.00000 −0.304348
$$530$$ 2.00000 0.0868744
$$531$$ 4.00000 0.173585
$$532$$ −2.00000 −0.0867110
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 12.0000 0.518805
$$536$$ 48.0000 2.07328
$$537$$ 12.0000 0.517838
$$538$$ 4.00000 0.172452
$$539$$ 6.00000 0.258438
$$540$$ −1.00000 −0.0430331
$$541$$ −42.0000 −1.80572 −0.902861 0.429934i $$-0.858537\pi$$
−0.902861 + 0.429934i $$0.858537\pi$$
$$542$$ 12.0000 0.515444
$$543$$ −10.0000 −0.429141
$$544$$ 10.0000 0.428746
$$545$$ −10.0000 −0.428353
$$546$$ −8.00000 −0.342368
$$547$$ 16.0000 0.684111 0.342055 0.939680i $$-0.388877\pi$$
0.342055 + 0.939680i $$0.388877\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ 2.00000 0.0853579
$$550$$ −2.00000 −0.0852803
$$551$$ −4.00000 −0.170406
$$552$$ −12.0000 −0.510754
$$553$$ 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 $$-0.797865\pi$$
−0.805056 + 0.593199i $$0.797865\pi$$
$$558$$ 0 0
$$559$$ 40.0000 1.69182
$$560$$ −2.00000 −0.0845154
$$561$$ 4.00000 0.168880
$$562$$ 28.0000 1.18111
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ 12.0000 0.505291
$$565$$ −14.0000 −0.588984
$$566$$ 26.0000 1.09286
$$567$$ −2.00000 −0.0839921
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ −1.00000 −0.0418854
$$571$$ −24.0000 −1.00437 −0.502184 0.864761i $$-0.667470\pi$$
−0.502184 + 0.864761i $$0.667470\pi$$
$$572$$ −8.00000 −0.334497
$$573$$ 18.0000 0.751961
$$574$$ 0 0
$$575$$ −4.00000 −0.166812
$$576$$ 7.00000 0.291667
$$577$$ 42.0000 1.74848 0.874241 0.485491i $$-0.161359\pi$$
0.874241 + 0.485491i $$0.161359\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ −4.00000 −0.166234
$$580$$ 4.00000 0.166091
$$581$$ 24.0000 0.995688
$$582$$ 16.0000 0.663221
$$583$$ 4.00000 0.165663
$$584$$ 6.00000 0.248282
$$585$$ 4.00000 0.165380
$$586$$ 26.0000 1.07405
$$587$$ −32.0000 −1.32078 −0.660391 0.750922i $$-0.729609\pi$$
−0.660391 + 0.750922i $$0.729609\pi$$
$$588$$ −3.00000 −0.123718
$$589$$ 0 0
$$590$$ −4.00000 −0.164677
$$591$$ 2.00000 0.0822690
$$592$$ 0 0
$$593$$ −18.0000 −0.739171 −0.369586 0.929197i $$-0.620500\pi$$
−0.369586 + 0.929197i $$0.620500\pi$$
$$594$$ 2.00000 0.0820610
$$595$$ 4.00000 0.163984
$$596$$ 18.0000 0.737309
$$597$$ −12.0000 −0.491127
$$598$$ 16.0000 0.654289
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 3.00000 0.122474
$$601$$ 30.0000 1.22373 0.611863 0.790964i $$-0.290420\pi$$
0.611863 + 0.790964i $$0.290420\pi$$
$$602$$ 20.0000 0.815139
$$603$$ −16.0000 −0.651570
$$604$$ −24.0000 −0.976546
$$605$$ 7.00000 0.284590
$$606$$ −14.0000 −0.568711
$$607$$ −16.0000 −0.649420 −0.324710 0.945814i $$-0.605267\pi$$
−0.324710 + 0.945814i $$0.605267\pi$$
$$608$$ −5.00000 −0.202777
$$609$$ 8.00000 0.324176
$$610$$ −2.00000 −0.0809776
$$611$$ −48.0000 −1.94187
$$612$$ −2.00000 −0.0808452
$$613$$ −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 $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 8.00000 0.321807
$$619$$ 16.0000 0.643094 0.321547 0.946894i $$-0.395797\pi$$
0.321547 + 0.946894i $$0.395797\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ −18.0000 −0.721734
$$623$$ 0 0
$$624$$ −4.00000 −0.160128
$$625$$ 1.00000 0.0400000
$$626$$ −14.0000 −0.559553
$$627$$ −2.00000 −0.0798723
$$628$$ 18.0000 0.718278
$$629$$ 0 0
$$630$$ 2.00000 0.0796819
$$631$$ 48.0000 1.91085 0.955425 0.295234i $$-0.0953977\pi$$
0.955425 + 0.295234i $$0.0953977\pi$$
$$632$$ 24.0000 0.954669
$$633$$ 12.0000 0.476957
$$634$$ 14.0000 0.556011
$$635$$ −4.00000 −0.158735
$$636$$ −2.00000 −0.0793052
$$637$$ 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 $$-0.657180\pi$$
−0.473972 + 0.880540i $$0.657180\pi$$
$$642$$ 12.0000 0.473602
$$643$$ 26.0000 1.02534 0.512670 0.858586i $$-0.328656\pi$$
0.512670 + 0.858586i $$0.328656\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ −10.0000 −0.393750
$$646$$ −2.00000 −0.0786889
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ −3.00000 −0.117851
$$649$$ −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 $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ −10.0000 −0.391031
$$655$$ 14.0000 0.547025
$$656$$ 0 0
$$657$$ −2.00000 −0.0780274
$$658$$ −24.0000 −0.935617
$$659$$ 16.0000 0.623272 0.311636 0.950202i $$-0.399123\pi$$
0.311636 + 0.950202i $$0.399123\pi$$
$$660$$ 2.00000 0.0778499
$$661$$ 18.0000 0.700119 0.350059 0.936727i $$-0.386161\pi$$
0.350059 + 0.936727i $$0.386161\pi$$
$$662$$ −28.0000 −1.08825
$$663$$ 8.00000 0.310694
$$664$$ 36.0000 1.39707
$$665$$ −2.00000 −0.0775567
$$666$$ 0 0
$$667$$ −16.0000 −0.619522
$$668$$ 8.00000 0.309529
$$669$$ −24.0000 −0.927894
$$670$$ 16.0000 0.618134
$$671$$ −4.00000 −0.154418
$$672$$ 10.0000 0.385758
$$673$$ 24.0000 0.925132 0.462566 0.886585i $$-0.346929\pi$$
0.462566 + 0.886585i $$0.346929\pi$$
$$674$$ 16.0000 0.616297
$$675$$ −1.00000 −0.0384900
$$676$$ −3.00000 −0.115385
$$677$$ −50.0000 −1.92166 −0.960828 0.277145i $$-0.910612\pi$$
−0.960828 + 0.277145i $$0.910612\pi$$
$$678$$ −14.0000 −0.537667
$$679$$ 32.0000 1.22805
$$680$$ 6.00000 0.230089
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 1.00000 0.0382360
$$685$$ −6.00000 −0.229248
$$686$$ 20.0000 0.763604
$$687$$ 14.0000 0.534133
$$688$$ 10.0000 0.381246
$$689$$ 8.00000 0.304776
$$690$$ −4.00000 −0.152277
$$691$$ −16.0000 −0.608669 −0.304334 0.952565i $$-0.598434\pi$$
−0.304334 + 0.952565i $$0.598434\pi$$
$$692$$ −18.0000 −0.684257
$$693$$ 4.00000 0.151947
$$694$$ −28.0000 −1.06287
$$695$$ −8.00000 −0.303457
$$696$$ 12.0000 0.454859
$$697$$ 0 0
$$698$$ −18.0000 −0.681310
$$699$$ 18.0000 0.680823
$$700$$ 2.00000 0.0755929
$$701$$ −34.0000 −1.28416 −0.642081 0.766637i $$-0.721929\pi$$
−0.642081 + 0.766637i $$0.721929\pi$$
$$702$$ 4.00000 0.150970
$$703$$ 0 0
$$704$$ −14.0000 −0.527645
$$705$$ 12.0000 0.451946
$$706$$ −2.00000 −0.0752710
$$707$$ −28.0000 −1.05305
$$708$$ 4.00000 0.150329
$$709$$ 18.0000 0.676004 0.338002 0.941145i $$-0.390249\pi$$
0.338002 + 0.941145i $$0.390249\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 4.00000 0.149696
$$715$$ −8.00000 −0.299183
$$716$$ 12.0000 0.448461
$$717$$ 6.00000 0.224074
$$718$$ −2.00000 −0.0746393
$$719$$ 42.0000 1.56634 0.783168 0.621810i $$-0.213603\pi$$
0.783168 + 0.621810i $$0.213603\pi$$
$$720$$ 1.00000 0.0372678
$$721$$ 16.0000 0.595871
$$722$$ 1.00000 0.0372161
$$723$$ −26.0000 −0.966950
$$724$$ −10.0000 −0.371647
$$725$$ 4.00000 0.148556
$$726$$ 7.00000 0.259794
$$727$$ −14.0000 −0.519231 −0.259616 0.965712i $$-0.583596\pi$$
−0.259616 + 0.965712i $$0.583596\pi$$
$$728$$ −24.0000 −0.889499
$$729$$ 1.00000 0.0370370
$$730$$ 2.00000 0.0740233
$$731$$ −20.0000 −0.739727
$$732$$ 2.00000 0.0739221
$$733$$ 34.0000 1.25582 0.627909 0.778287i $$-0.283911\pi$$
0.627909 + 0.778287i $$0.283911\pi$$
$$734$$ 10.0000 0.369107
$$735$$ −3.00000 −0.110657
$$736$$ −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
$$741$$ −4.00000 −0.146944
$$742$$ 4.00000 0.146845
$$743$$ −8.00000 −0.293492 −0.146746 0.989174i $$-0.546880\pi$$
−0.146746 + 0.989174i $$0.546880\pi$$
$$744$$ 0 0
$$745$$ 18.0000 0.659469
$$746$$ 4.00000 0.146450
$$747$$ −12.0000 −0.439057
$$748$$ 4.00000 0.146254
$$749$$ 24.0000 0.876941
$$750$$ 1.00000 0.0365148
$$751$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$752$$ −12.0000 −0.437595
$$753$$ −22.0000 −0.801725
$$754$$ −16.0000 −0.582686
$$755$$ −24.0000 −0.873449
$$756$$ −2.00000 −0.0727393
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ −20.0000 −0.726433
$$759$$ −8.00000 −0.290382
$$760$$ −3.00000 −0.108821
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ −4.00000 −0.144905
$$763$$ −20.0000 −0.724049
$$764$$ 18.0000 0.651217
$$765$$ −2.00000 −0.0723102
$$766$$ 8.00000 0.289052
$$767$$ −16.0000 −0.577727
$$768$$ 17.0000 0.613435
$$769$$ −46.0000 −1.65880 −0.829401 0.558653i $$-0.811318\pi$$
−0.829401 + 0.558653i $$0.811318\pi$$
$$770$$ −4.00000 −0.144150
$$771$$ −22.0000 −0.792311
$$772$$ −4.00000 −0.143963
$$773$$ 14.0000 0.503545 0.251773 0.967786i $$-0.418987\pi$$
0.251773 + 0.967786i $$0.418987\pi$$
$$774$$ −10.0000 −0.359443
$$775$$ 0 0
$$776$$ 48.0000 1.72310
$$777$$ 0 0
$$778$$ −30.0000 −1.07555
$$779$$ 0 0
$$780$$ 4.00000 0.143223
$$781$$ 0 0
$$782$$ −8.00000 −0.286079
$$783$$ −4.00000 −0.142948
$$784$$ 3.00000 0.107143
$$785$$ 18.0000 0.642448
$$786$$ 14.0000 0.499363
$$787$$ 28.0000 0.998092 0.499046 0.866575i $$-0.333684\pi$$
0.499046 + 0.866575i $$0.333684\pi$$
$$788$$ 2.00000 0.0712470
$$789$$ 16.0000 0.569615
$$790$$ 8.00000 0.284627
$$791$$ −28.0000 −0.995565
$$792$$ 6.00000 0.213201
$$793$$ −8.00000 −0.284088
$$794$$ −2.00000 −0.0709773
$$795$$ −2.00000 −0.0709327
$$796$$ −12.0000 −0.425329
$$797$$ −14.0000 −0.495905 −0.247953 0.968772i $$-0.579758\pi$$
−0.247953 + 0.968772i $$0.579758\pi$$
$$798$$ −2.00000 −0.0707992
$$799$$ 24.0000 0.849059
$$800$$ 5.00000 0.176777
$$801$$ 0 0
$$802$$ 12.0000 0.423735
$$803$$ 4.00000 0.141157
$$804$$ −16.0000 −0.564276
$$805$$ −8.00000 −0.281963
$$806$$ 0 0
$$807$$ −4.00000 −0.140807
$$808$$ −42.0000 −1.47755
$$809$$ 2.00000 0.0703163 0.0351581 0.999382i $$-0.488807\pi$$
0.0351581 + 0.999382i $$0.488807\pi$$
$$810$$ −1.00000 −0.0351364
$$811$$ 4.00000 0.140459 0.0702295 0.997531i $$-0.477627\pi$$
0.0702295 + 0.997531i $$0.477627\pi$$
$$812$$ 8.00000 0.280745
$$813$$ −12.0000 −0.420858
$$814$$ 0 0
$$815$$ 6.00000 0.210171
$$816$$ 2.00000 0.0700140
$$817$$ 10.0000 0.349856
$$818$$ −26.0000 −0.909069
$$819$$ 8.00000 0.279543
$$820$$ 0 0
$$821$$ 6.00000 0.209401 0.104701 0.994504i $$-0.466612\pi$$
0.104701 + 0.994504i $$0.466612\pi$$
$$822$$ −6.00000 −0.209274
$$823$$ −14.0000 −0.488009 −0.244005 0.969774i $$-0.578461\pi$$
−0.244005 + 0.969774i $$0.578461\pi$$
$$824$$ 24.0000 0.836080
$$825$$ 2.00000 0.0696311
$$826$$ −8.00000 −0.278356
$$827$$ 36.0000 1.25184 0.625921 0.779886i $$-0.284723\pi$$
0.625921 + 0.779886i $$0.284723\pi$$
$$828$$ 4.00000 0.139010
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ 12.0000 0.416526
$$831$$ −22.0000 −0.763172
$$832$$ −28.0000 −0.970725
$$833$$ −6.00000 −0.207888
$$834$$ −8.00000 −0.277017
$$835$$ 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 $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 6.00000 0.207020
$$841$$ −13.0000 −0.448276
$$842$$ −34.0000 −1.17172
$$843$$ −28.0000 −0.964371
$$844$$ 12.0000 0.413057
$$845$$ −3.00000 −0.103203
$$846$$ 12.0000 0.412568
$$847$$ 14.0000 0.481046
$$848$$ 2.00000 0.0686803
$$849$$ −26.0000 −0.892318
$$850$$ 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
$$855$$ 1.00000 0.0341993
$$856$$ 36.0000 1.23045
$$857$$ 10.0000 0.341593 0.170797 0.985306i $$-0.445366\pi$$
0.170797 + 0.985306i $$0.445366\pi$$
$$858$$ −8.00000 −0.273115
$$859$$ −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 $$-0.261629\pi$$
0.680808 + 0.732462i $$0.261629\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ −18.0000 −0.612018
$$866$$ −36.0000 −1.22333
$$867$$ 13.0000 0.441503
$$868$$ 0 0
$$869$$ 16.0000 0.542763
$$870$$ 4.00000 0.135613
$$871$$ 64.0000 2.16856
$$872$$ −30.0000 −1.01593
$$873$$ −16.0000 −0.541518
$$874$$ 4.00000 0.135302
$$875$$ 2.00000 0.0676123
$$876$$ −2.00000 −0.0675737
$$877$$ 12.0000 0.405211 0.202606 0.979260i $$-0.435059\pi$$
0.202606 + 0.979260i $$0.435059\pi$$
$$878$$ −40.0000 −1.34993
$$879$$ −26.0000 −0.876958
$$880$$ −2.00000 −0.0674200
$$881$$ −10.0000 −0.336909 −0.168454 0.985709i $$-0.553878\pi$$
−0.168454 + 0.985709i $$0.553878\pi$$
$$882$$ −3.00000 −0.101015
$$883$$ −46.0000 −1.54802 −0.774012 0.633171i $$-0.781753\pi$$
−0.774012 + 0.633171i $$0.781753\pi$$
$$884$$ 8.00000 0.269069
$$885$$ 4.00000 0.134459
$$886$$ 16.0000 0.537531
$$887$$ −24.0000 −0.805841 −0.402921 0.915235i $$-0.632005\pi$$
−0.402921 + 0.915235i $$0.632005\pi$$
$$888$$ 0 0
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ −2.00000 −0.0670025
$$892$$ −24.0000 −0.803579
$$893$$ −12.0000 −0.401565
$$894$$ 18.0000 0.602010
$$895$$ 12.0000 0.401116
$$896$$ 6.00000 0.200446
$$897$$ −16.0000 −0.534224
$$898$$ 0 0
$$899$$ 0 0
$$900$$ −1.00000 −0.0333333
$$901$$ −4.00000 −0.133259
$$902$$ 0 0
$$903$$ −20.0000 −0.665558
$$904$$ −42.0000 −1.39690
$$905$$ −10.0000 −0.332411
$$906$$ −24.0000 −0.797347
$$907$$ 52.0000 1.72663 0.863316 0.504664i $$-0.168384\pi$$
0.863316 + 0.504664i $$0.168384\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 14.0000 0.464351
$$910$$ −8.00000 −0.265197
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ −1.00000 −0.0331133
$$913$$ 24.0000 0.794284
$$914$$ −14.0000 −0.463079
$$915$$ 2.00000 0.0661180
$$916$$ 14.0000 0.462573
$$917$$ 28.0000 0.924641
$$918$$ −2.00000 −0.0660098
$$919$$ 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
$$924$$ 4.00000 0.131590
$$925$$ 0 0
$$926$$ −22.0000 −0.722965
$$927$$ −8.00000 −0.262754
$$928$$ 20.0000 0.656532
$$929$$ 10.0000 0.328089 0.164045 0.986453i $$-0.447546\pi$$
0.164045 + 0.986453i $$0.447546\pi$$
$$930$$ 0 0
$$931$$ 3.00000 0.0983210
$$932$$ 18.0000 0.589610
$$933$$ 18.0000 0.589294
$$934$$ 16.0000 0.523536
$$935$$ 4.00000 0.130814
$$936$$ 12.0000 0.392232
$$937$$ −42.0000 −1.37208 −0.686040 0.727564i $$-0.740653\pi$$
−0.686040 + 0.727564i $$0.740653\pi$$
$$938$$ 32.0000 1.04484
$$939$$ 14.0000 0.456873
$$940$$ 12.0000 0.391397
$$941$$ −60.0000 −1.95594 −0.977972 0.208736i $$-0.933065\pi$$
−0.977972 + 0.208736i $$0.933065\pi$$
$$942$$ 18.0000 0.586472
$$943$$ 0 0
$$944$$ −4.00000 −0.130189
$$945$$ −2.00000 −0.0650600
$$946$$ 20.0000 0.650256
$$947$$ −52.0000 −1.68977 −0.844886 0.534946i $$-0.820332\pi$$
−0.844886 + 0.534946i $$0.820332\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ 8.00000 0.259691
$$950$$ −1.00000 −0.0324443
$$951$$ −14.0000 −0.453981
$$952$$ 12.0000 0.388922
$$953$$ 46.0000 1.49009 0.745043 0.667016i $$-0.232429\pi$$
0.745043 + 0.667016i $$0.232429\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ 18.0000 0.582466
$$956$$ 6.00000 0.194054
$$957$$ 8.00000 0.258603
$$958$$ 38.0000 1.22772
$$959$$ −12.0000 −0.387500
$$960$$ 7.00000 0.225924
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ −12.0000 −0.386695
$$964$$ −26.0000 −0.837404
$$965$$ −4.00000 −0.128765
$$966$$ −8.00000 −0.257396
$$967$$ 34.0000 1.09337 0.546683 0.837340i $$-0.315890\pi$$
0.546683 + 0.837340i $$0.315890\pi$$
$$968$$ 21.0000 0.674966
$$969$$ 2.00000 0.0642493
$$970$$ 16.0000 0.513729
$$971$$ −24.0000 −0.770197 −0.385098 0.922876i $$-0.625832\pi$$
−0.385098 + 0.922876i $$0.625832\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −16.0000 −0.512936
$$974$$ 8.00000 0.256337
$$975$$ 4.00000 0.128103
$$976$$ −2.00000 −0.0640184
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 6.00000 0.191859
$$979$$ 0 0
$$980$$ −3.00000 −0.0958315
$$981$$ 10.0000 0.319275
$$982$$ −6.00000 −0.191468
$$983$$ 16.0000 0.510321 0.255160 0.966899i $$-0.417872\pi$$
0.255160 + 0.966899i $$0.417872\pi$$
$$984$$ 0 0
$$985$$ 2.00000 0.0637253
$$986$$ 8.00000 0.254772
$$987$$ 24.0000 0.763928
$$988$$ −4.00000 −0.127257
$$989$$ 40.0000 1.27193
$$990$$ 2.00000 0.0635642
$$991$$ 8.00000 0.254128 0.127064 0.991894i $$-0.459445\pi$$
0.127064 + 0.991894i $$0.459445\pi$$
$$992$$ 0 0
$$993$$ 28.0000 0.888553
$$994$$ 0 0
$$995$$ −12.0000 −0.380426
$$996$$ −12.0000 −0.380235
$$997$$ −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 285.2.a.b.1.1 1
3.2 odd 2 855.2.a.b.1.1 1
4.3 odd 2 4560.2.a.v.1.1 1
5.2 odd 4 1425.2.c.d.799.2 2
5.3 odd 4 1425.2.c.d.799.1 2
5.4 even 2 1425.2.a.d.1.1 1
15.14 odd 2 4275.2.a.o.1.1 1
19.18 odd 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 1.1 even 1 trivial
855.2.a.b.1.1 1 3.2 odd 2
1425.2.a.d.1.1 1 5.4 even 2
1425.2.c.d.799.1 2 5.3 odd 4
1425.2.c.d.799.2 2 5.2 odd 4
4275.2.a.o.1.1 1 15.14 odd 2
4560.2.a.v.1.1 1 4.3 odd 2
5415.2.a.c.1.1 1 19.18 odd 2