# Properties

 Label 39.2.a.a.1.1 Level $39$ Weight $2$ Character 39.1 Self dual yes Analytic conductor $0.311$ 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] = [39,2,Mod(1,39)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(39, 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("39.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 39.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: $$0.311416567883$$ 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 $$\chi$$ $$=$$ 39.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} -4.00000 q^{14} -2.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -2.00000 q^{20} +4.00000 q^{21} +4.00000 q^{22} +3.00000 q^{24} -1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} -10.0000 q^{29} -2.00000 q^{30} +4.00000 q^{31} +5.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} -8.00000 q^{35} -1.00000 q^{36} -2.00000 q^{37} -1.00000 q^{39} -6.00000 q^{40} +6.00000 q^{41} +4.00000 q^{42} -12.0000 q^{43} -4.00000 q^{44} +2.00000 q^{45} +1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} -1.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +8.00000 q^{55} +12.0000 q^{56} -10.0000 q^{58} +12.0000 q^{59} +2.00000 q^{60} -2.00000 q^{61} +4.00000 q^{62} -4.00000 q^{63} +7.00000 q^{64} +2.00000 q^{65} -4.00000 q^{66} -8.00000 q^{67} -2.00000 q^{68} -8.00000 q^{70} -3.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -16.0000 q^{77} -1.00000 q^{78} +8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +4.00000 q^{83} -4.00000 q^{84} +4.00000 q^{85} -12.0000 q^{86} +10.0000 q^{87} -12.0000 q^{88} -2.00000 q^{89} +2.00000 q^{90} -4.00000 q^{91} -4.00000 q^{93} -5.00000 q^{96} +10.0000 q^{97} +9.00000 q^{98} +4.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 1.00000 0.333333
$$10$$ 2.00000 0.632456
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 1.00000 0.277350
$$14$$ −4.00000 −1.06904
$$15$$ −2.00000 −0.516398
$$16$$ −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
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ −2.00000 −0.447214
$$21$$ 4.00000 0.872872
$$22$$ 4.00000 0.852803
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 3.00000 0.612372
$$25$$ −1.00000 −0.200000
$$26$$ 1.00000 0.196116
$$27$$ −1.00000 −0.192450
$$28$$ 4.00000 0.755929
$$29$$ −10.0000 −1.85695 −0.928477 0.371391i $$-0.878881\pi$$
−0.928477 + 0.371391i $$0.878881\pi$$
$$30$$ −2.00000 −0.365148
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 5.00000 0.883883
$$33$$ −4.00000 −0.696311
$$34$$ 2.00000 0.342997
$$35$$ −8.00000 −1.35225
$$36$$ −1.00000 −0.166667
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ −6.00000 −0.948683
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 4.00000 0.617213
$$43$$ −12.0000 −1.82998 −0.914991 0.403473i $$-0.867803\pi$$
−0.914991 + 0.403473i $$0.867803\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 2.00000 0.298142
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 9.00000 1.28571
$$50$$ −1.00000 −0.141421
$$51$$ −2.00000 −0.280056
$$52$$ −1.00000 −0.138675
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 8.00000 1.07872
$$56$$ 12.0000 1.60357
$$57$$ 0 0
$$58$$ −10.0000 −1.31306
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 2.00000 0.258199
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 4.00000 0.508001
$$63$$ −4.00000 −0.503953
$$64$$ 7.00000 0.875000
$$65$$ 2.00000 0.248069
$$66$$ −4.00000 −0.492366
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ 0 0
$$70$$ −8.00000 −0.956183
$$71$$ 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 $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −2.00000 −0.232495
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ −16.0000 −1.82337
$$78$$ −1.00000 −0.113228
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ −2.00000 −0.223607
$$81$$ 1.00000 0.111111
$$82$$ 6.00000 0.662589
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ −4.00000 −0.436436
$$85$$ 4.00000 0.433861
$$86$$ −12.0000 −1.29399
$$87$$ 10.0000 1.07211
$$88$$ −12.0000 −1.27920
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 2.00000 0.210819
$$91$$ −4.00000 −0.419314
$$92$$ 0 0
$$93$$ −4.00000 −0.414781
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −5.00000 −0.510310
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ 9.00000 0.909137
$$99$$ 4.00000 0.402015
$$100$$ 1.00000 0.100000
$$101$$ −18.0000 −1.79107 −0.895533 0.444994i $$-0.853206\pi$$
−0.895533 + 0.444994i $$0.853206\pi$$
$$102$$ −2.00000 −0.198030
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ −3.00000 −0.294174
$$105$$ 8.00000 0.780720
$$106$$ 6.00000 0.582772
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 8.00000 0.762770
$$111$$ 2.00000 0.189832
$$112$$ 4.00000 0.377964
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 10.0000 0.928477
$$117$$ 1.00000 0.0924500
$$118$$ 12.0000 1.10469
$$119$$ −8.00000 −0.733359
$$120$$ 6.00000 0.547723
$$121$$ 5.00000 0.454545
$$122$$ −2.00000 −0.181071
$$123$$ −6.00000 −0.541002
$$124$$ −4.00000 −0.359211
$$125$$ −12.0000 −1.07331
$$126$$ −4.00000 −0.356348
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 12.0000 1.05654
$$130$$ 2.00000 0.175412
$$131$$ 4.00000 0.349482 0.174741 0.984614i $$-0.444091\pi$$
0.174741 + 0.984614i $$0.444091\pi$$
$$132$$ 4.00000 0.348155
$$133$$ 0 0
$$134$$ −8.00000 −0.691095
$$135$$ −2.00000 −0.172133
$$136$$ −6.00000 −0.514496
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 8.00000 0.676123
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 4.00000 0.334497
$$144$$ −1.00000 −0.0833333
$$145$$ −20.0000 −1.66091
$$146$$ 2.00000 0.165521
$$147$$ −9.00000 −0.742307
$$148$$ 2.00000 0.164399
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ 0 0
$$153$$ 2.00000 0.161690
$$154$$ −16.0000 −1.28932
$$155$$ 8.00000 0.642575
$$156$$ 1.00000 0.0800641
$$157$$ −18.0000 −1.43656 −0.718278 0.695756i $$-0.755069\pi$$
−0.718278 + 0.695756i $$0.755069\pi$$
$$158$$ 8.00000 0.636446
$$159$$ −6.00000 −0.475831
$$160$$ 10.0000 0.790569
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ 8.00000 0.626608 0.313304 0.949653i $$-0.398564\pi$$
0.313304 + 0.949653i $$0.398564\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ −8.00000 −0.622799
$$166$$ 4.00000 0.310460
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ −12.0000 −0.925820
$$169$$ 1.00000 0.0769231
$$170$$ 4.00000 0.306786
$$171$$ 0 0
$$172$$ 12.0000 0.914991
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 10.0000 0.758098
$$175$$ 4.00000 0.302372
$$176$$ −4.00000 −0.301511
$$177$$ −12.0000 −0.901975
$$178$$ −2.00000 −0.149906
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ −2.00000 −0.149071
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ −4.00000 −0.296500
$$183$$ 2.00000 0.147844
$$184$$ 0 0
$$185$$ −4.00000 −0.294086
$$186$$ −4.00000 −0.293294
$$187$$ 8.00000 0.585018
$$188$$ 0 0
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ −7.00000 −0.505181
$$193$$ 18.0000 1.29567 0.647834 0.761781i $$-0.275675\pi$$
0.647834 + 0.761781i $$0.275675\pi$$
$$194$$ 10.0000 0.717958
$$195$$ −2.00000 −0.143223
$$196$$ −9.00000 −0.642857
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 4.00000 0.284268
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 3.00000 0.212132
$$201$$ 8.00000 0.564276
$$202$$ −18.0000 −1.26648
$$203$$ 40.0000 2.80745
$$204$$ 2.00000 0.140028
$$205$$ 12.0000 0.838116
$$206$$ 0 0
$$207$$ 0 0
$$208$$ −1.00000 −0.0693375
$$209$$ 0 0
$$210$$ 8.00000 0.552052
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ −24.0000 −1.63679
$$216$$ 3.00000 0.204124
$$217$$ −16.0000 −1.08615
$$218$$ −2.00000 −0.135457
$$219$$ −2.00000 −0.135147
$$220$$ −8.00000 −0.539360
$$221$$ 2.00000 0.134535
$$222$$ 2.00000 0.134231
$$223$$ 4.00000 0.267860 0.133930 0.990991i $$-0.457240\pi$$
0.133930 + 0.990991i $$0.457240\pi$$
$$224$$ −20.0000 −1.33631
$$225$$ −1.00000 −0.0666667
$$226$$ −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 $$0.607187\pi$$
$$230$$ 0 0
$$231$$ 16.0000 1.05272
$$232$$ 30.0000 1.96960
$$233$$ −14.0000 −0.917170 −0.458585 0.888650i $$-0.651644\pi$$
−0.458585 + 0.888650i $$0.651644\pi$$
$$234$$ 1.00000 0.0653720
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ −8.00000 −0.519656
$$238$$ −8.00000 −0.518563
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 2.00000 0.129099
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 5.00000 0.321412
$$243$$ −1.00000 −0.0641500
$$244$$ 2.00000 0.128037
$$245$$ 18.0000 1.14998
$$246$$ −6.00000 −0.382546
$$247$$ 0 0
$$248$$ −12.0000 −0.762001
$$249$$ −4.00000 −0.253490
$$250$$ −12.0000 −0.758947
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 4.00000 0.251976
$$253$$ 0 0
$$254$$ −16.0000 −1.00393
$$255$$ −4.00000 −0.250490
$$256$$ −17.0000 −1.06250
$$257$$ 26.0000 1.62184 0.810918 0.585160i $$-0.198968\pi$$
0.810918 + 0.585160i $$0.198968\pi$$
$$258$$ 12.0000 0.747087
$$259$$ 8.00000 0.497096
$$260$$ −2.00000 −0.124035
$$261$$ −10.0000 −0.618984
$$262$$ 4.00000 0.247121
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 12.0000 0.738549
$$265$$ 12.0000 0.737154
$$266$$ 0 0
$$267$$ 2.00000 0.122398
$$268$$ 8.00000 0.488678
$$269$$ 22.0000 1.34136 0.670682 0.741745i $$-0.266002\pi$$
0.670682 + 0.741745i $$0.266002\pi$$
$$270$$ −2.00000 −0.121716
$$271$$ −12.0000 −0.728948 −0.364474 0.931214i $$-0.618751\pi$$
−0.364474 + 0.931214i $$0.618751\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ 4.00000 0.242091
$$274$$ 6.00000 0.362473
$$275$$ −4.00000 −0.241209
$$276$$ 0 0
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 12.0000 0.719712
$$279$$ 4.00000 0.239474
$$280$$ 24.0000 1.43427
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ 12.0000 0.713326 0.356663 0.934233i $$-0.383914\pi$$
0.356663 + 0.934233i $$0.383914\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 4.00000 0.236525
$$287$$ −24.0000 −1.41668
$$288$$ 5.00000 0.294628
$$289$$ −13.0000 −0.764706
$$290$$ −20.0000 −1.17444
$$291$$ −10.0000 −0.586210
$$292$$ −2.00000 −0.117041
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ −9.00000 −0.524891
$$295$$ 24.0000 1.39733
$$296$$ 6.00000 0.348743
$$297$$ −4.00000 −0.232104
$$298$$ −6.00000 −0.347571
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ 48.0000 2.76667
$$302$$ 4.00000 0.230174
$$303$$ 18.0000 1.03407
$$304$$ 0 0
$$305$$ −4.00000 −0.229039
$$306$$ 2.00000 0.114332
$$307$$ −16.0000 −0.913168 −0.456584 0.889680i $$-0.650927\pi$$
−0.456584 + 0.889680i $$0.650927\pi$$
$$308$$ 16.0000 0.911685
$$309$$ 0 0
$$310$$ 8.00000 0.454369
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 3.00000 0.169842
$$313$$ −6.00000 −0.339140 −0.169570 0.985518i $$-0.554238\pi$$
−0.169570 + 0.985518i $$0.554238\pi$$
$$314$$ −18.0000 −1.01580
$$315$$ −8.00000 −0.450749
$$316$$ −8.00000 −0.450035
$$317$$ 26.0000 1.46031 0.730153 0.683284i $$-0.239449\pi$$
0.730153 + 0.683284i $$0.239449\pi$$
$$318$$ −6.00000 −0.336463
$$319$$ −40.0000 −2.23957
$$320$$ 14.0000 0.782624
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ 0 0
$$324$$ −1.00000 −0.0555556
$$325$$ −1.00000 −0.0554700
$$326$$ 8.00000 0.443079
$$327$$ 2.00000 0.110600
$$328$$ −18.0000 −0.993884
$$329$$ 0 0
$$330$$ −8.00000 −0.440386
$$331$$ −16.0000 −0.879440 −0.439720 0.898135i $$-0.644922\pi$$
−0.439720 + 0.898135i $$0.644922\pi$$
$$332$$ −4.00000 −0.219529
$$333$$ −2.00000 −0.109599
$$334$$ −8.00000 −0.437741
$$335$$ −16.0000 −0.874173
$$336$$ −4.00000 −0.218218
$$337$$ 18.0000 0.980522 0.490261 0.871576i $$-0.336901\pi$$
0.490261 + 0.871576i $$0.336901\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ 6.00000 0.325875
$$340$$ −4.00000 −0.216930
$$341$$ 16.0000 0.866449
$$342$$ 0 0
$$343$$ −8.00000 −0.431959
$$344$$ 36.0000 1.94099
$$345$$ 0 0
$$346$$ 6.00000 0.322562
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ −10.0000 −0.536056
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 4.00000 0.213809
$$351$$ −1.00000 −0.0533761
$$352$$ 20.0000 1.06600
$$353$$ −2.00000 −0.106449 −0.0532246 0.998583i $$-0.516950\pi$$
−0.0532246 + 0.998583i $$0.516950\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ 2.00000 0.106000
$$357$$ 8.00000 0.423405
$$358$$ 4.00000 0.211407
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ −6.00000 −0.316228
$$361$$ −19.0000 −1.00000
$$362$$ −10.0000 −0.525588
$$363$$ −5.00000 −0.262432
$$364$$ 4.00000 0.209657
$$365$$ 4.00000 0.209370
$$366$$ 2.00000 0.104542
$$367$$ 16.0000 0.835193 0.417597 0.908633i $$-0.362873\pi$$
0.417597 + 0.908633i $$0.362873\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ −4.00000 −0.207950
$$371$$ −24.0000 −1.24602
$$372$$ 4.00000 0.207390
$$373$$ −26.0000 −1.34623 −0.673114 0.739538i $$-0.735044\pi$$
−0.673114 + 0.739538i $$0.735044\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 12.0000 0.619677
$$376$$ 0 0
$$377$$ −10.0000 −0.515026
$$378$$ 4.00000 0.205738
$$379$$ −24.0000 −1.23280 −0.616399 0.787434i $$-0.711409\pi$$
−0.616399 + 0.787434i $$0.711409\pi$$
$$380$$ 0 0
$$381$$ 16.0000 0.819705
$$382$$ 8.00000 0.409316
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ 3.00000 0.153093
$$385$$ −32.0000 −1.63087
$$386$$ 18.0000 0.916176
$$387$$ −12.0000 −0.609994
$$388$$ −10.0000 −0.507673
$$389$$ 22.0000 1.11544 0.557722 0.830028i $$-0.311675\pi$$
0.557722 + 0.830028i $$0.311675\pi$$
$$390$$ −2.00000 −0.101274
$$391$$ 0 0
$$392$$ −27.0000 −1.36371
$$393$$ −4.00000 −0.201773
$$394$$ 18.0000 0.906827
$$395$$ 16.0000 0.805047
$$396$$ −4.00000 −0.201008
$$397$$ 38.0000 1.90717 0.953583 0.301131i $$-0.0973643\pi$$
0.953583 + 0.301131i $$0.0973643\pi$$
$$398$$ 8.00000 0.401004
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 22.0000 1.09863 0.549314 0.835616i $$-0.314889\pi$$
0.549314 + 0.835616i $$0.314889\pi$$
$$402$$ 8.00000 0.399004
$$403$$ 4.00000 0.199254
$$404$$ 18.0000 0.895533
$$405$$ 2.00000 0.0993808
$$406$$ 40.0000 1.98517
$$407$$ −8.00000 −0.396545
$$408$$ 6.00000 0.297044
$$409$$ 34.0000 1.68119 0.840596 0.541663i $$-0.182205\pi$$
0.840596 + 0.541663i $$0.182205\pi$$
$$410$$ 12.0000 0.592638
$$411$$ −6.00000 −0.295958
$$412$$ 0 0
$$413$$ −48.0000 −2.36193
$$414$$ 0 0
$$415$$ 8.00000 0.392705
$$416$$ 5.00000 0.245145
$$417$$ −12.0000 −0.587643
$$418$$ 0 0
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ −8.00000 −0.390360
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ −20.0000 −0.973585
$$423$$ 0 0
$$424$$ −18.0000 −0.874157
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ 8.00000 0.387147
$$428$$ −12.0000 −0.580042
$$429$$ −4.00000 −0.193122
$$430$$ −24.0000 −1.15738
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 34.0000 1.63394 0.816968 0.576683i $$-0.195653\pi$$
0.816968 + 0.576683i $$0.195653\pi$$
$$434$$ −16.0000 −0.768025
$$435$$ 20.0000 0.958927
$$436$$ 2.00000 0.0957826
$$437$$ 0 0
$$438$$ −2.00000 −0.0955637
$$439$$ 32.0000 1.52728 0.763638 0.645644i $$-0.223411\pi$$
0.763638 + 0.645644i $$0.223411\pi$$
$$440$$ −24.0000 −1.14416
$$441$$ 9.00000 0.428571
$$442$$ 2.00000 0.0951303
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ −2.00000 −0.0949158
$$445$$ −4.00000 −0.189618
$$446$$ 4.00000 0.189405
$$447$$ 6.00000 0.283790
$$448$$ −28.0000 −1.32288
$$449$$ 22.0000 1.03824 0.519122 0.854700i $$-0.326259\pi$$
0.519122 + 0.854700i $$0.326259\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 24.0000 1.13012
$$452$$ 6.00000 0.282216
$$453$$ −4.00000 −0.187936
$$454$$ −20.0000 −0.938647
$$455$$ −8.00000 −0.375046
$$456$$ 0 0
$$457$$ 2.00000 0.0935561 0.0467780 0.998905i $$-0.485105\pi$$
0.0467780 + 0.998905i $$0.485105\pi$$
$$458$$ −10.0000 −0.467269
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ −38.0000 −1.76984 −0.884918 0.465746i $$-0.845786\pi$$
−0.884918 + 0.465746i $$0.845786\pi$$
$$462$$ 16.0000 0.744387
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ 10.0000 0.464238
$$465$$ −8.00000 −0.370991
$$466$$ −14.0000 −0.648537
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ −1.00000 −0.0462250
$$469$$ 32.0000 1.47762
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ −36.0000 −1.65703
$$473$$ −48.0000 −2.20704
$$474$$ −8.00000 −0.367452
$$475$$ 0 0
$$476$$ 8.00000 0.366679
$$477$$ 6.00000 0.274721
$$478$$ −24.0000 −1.09773
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ −10.0000 −0.456435
$$481$$ −2.00000 −0.0911922
$$482$$ 10.0000 0.455488
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 20.0000 0.908153
$$486$$ −1.00000 −0.0453609
$$487$$ 12.0000 0.543772 0.271886 0.962329i $$-0.412353\pi$$
0.271886 + 0.962329i $$0.412353\pi$$
$$488$$ 6.00000 0.271607
$$489$$ −8.00000 −0.361773
$$490$$ 18.0000 0.813157
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 6.00000 0.270501
$$493$$ −20.0000 −0.900755
$$494$$ 0 0
$$495$$ 8.00000 0.359573
$$496$$ −4.00000 −0.179605
$$497$$ 0 0
$$498$$ −4.00000 −0.179244
$$499$$ −24.0000 −1.07439 −0.537194 0.843459i $$-0.680516\pi$$
−0.537194 + 0.843459i $$0.680516\pi$$
$$500$$ 12.0000 0.536656
$$501$$ 8.00000 0.357414
$$502$$ −12.0000 −0.535586
$$503$$ −8.00000 −0.356702 −0.178351 0.983967i $$-0.557076\pi$$
−0.178351 + 0.983967i $$0.557076\pi$$
$$504$$ 12.0000 0.534522
$$505$$ −36.0000 −1.60198
$$506$$ 0 0
$$507$$ −1.00000 −0.0444116
$$508$$ 16.0000 0.709885
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ −4.00000 −0.177123
$$511$$ −8.00000 −0.353899
$$512$$ −11.0000 −0.486136
$$513$$ 0 0
$$514$$ 26.0000 1.14681
$$515$$ 0 0
$$516$$ −12.0000 −0.528271
$$517$$ 0 0
$$518$$ 8.00000 0.351500
$$519$$ −6.00000 −0.263371
$$520$$ −6.00000 −0.263117
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ −10.0000 −0.437688
$$523$$ 44.0000 1.92399 0.961993 0.273075i $$-0.0880406\pi$$
0.961993 + 0.273075i $$0.0880406\pi$$
$$524$$ −4.00000 −0.174741
$$525$$ −4.00000 −0.174574
$$526$$ 24.0000 1.04645
$$527$$ 8.00000 0.348485
$$528$$ 4.00000 0.174078
$$529$$ −23.0000 −1.00000
$$530$$ 12.0000 0.521247
$$531$$ 12.0000 0.520756
$$532$$ 0 0
$$533$$ 6.00000 0.259889
$$534$$ 2.00000 0.0865485
$$535$$ 24.0000 1.03761
$$536$$ 24.0000 1.03664
$$537$$ −4.00000 −0.172613
$$538$$ 22.0000 0.948487
$$539$$ 36.0000 1.55063
$$540$$ 2.00000 0.0860663
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ −12.0000 −0.515444
$$543$$ 10.0000 0.429141
$$544$$ 10.0000 0.428746
$$545$$ −4.00000 −0.171341
$$546$$ 4.00000 0.171184
$$547$$ 4.00000 0.171028 0.0855138 0.996337i $$-0.472747\pi$$
0.0855138 + 0.996337i $$0.472747\pi$$
$$548$$ −6.00000 −0.256307
$$549$$ −2.00000 −0.0853579
$$550$$ −4.00000 −0.170561
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −32.0000 −1.36078
$$554$$ −10.0000 −0.424859
$$555$$ 4.00000 0.169791
$$556$$ −12.0000 −0.508913
$$557$$ 18.0000 0.762684 0.381342 0.924434i $$-0.375462\pi$$
0.381342 + 0.924434i $$0.375462\pi$$
$$558$$ 4.00000 0.169334
$$559$$ −12.0000 −0.507546
$$560$$ 8.00000 0.338062
$$561$$ −8.00000 −0.337760
$$562$$ −10.0000 −0.421825
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 0 0
$$565$$ −12.0000 −0.504844
$$566$$ 12.0000 0.504398
$$567$$ −4.00000 −0.167984
$$568$$ 0 0
$$569$$ 34.0000 1.42535 0.712677 0.701492i $$-0.247483\pi$$
0.712677 + 0.701492i $$0.247483\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ −4.00000 −0.167248
$$573$$ −8.00000 −0.334205
$$574$$ −24.0000 −1.00174
$$575$$ 0 0
$$576$$ 7.00000 0.291667
$$577$$ −46.0000 −1.91501 −0.957503 0.288425i $$-0.906868\pi$$
−0.957503 + 0.288425i $$0.906868\pi$$
$$578$$ −13.0000 −0.540729
$$579$$ −18.0000 −0.748054
$$580$$ 20.0000 0.830455
$$581$$ −16.0000 −0.663792
$$582$$ −10.0000 −0.414513
$$583$$ 24.0000 0.993978
$$584$$ −6.00000 −0.248282
$$585$$ 2.00000 0.0826898
$$586$$ −6.00000 −0.247858
$$587$$ 28.0000 1.15568 0.577842 0.816149i $$-0.303895\pi$$
0.577842 + 0.816149i $$0.303895\pi$$
$$588$$ 9.00000 0.371154
$$589$$ 0 0
$$590$$ 24.0000 0.988064
$$591$$ −18.0000 −0.740421
$$592$$ 2.00000 0.0821995
$$593$$ −26.0000 −1.06769 −0.533846 0.845582i $$-0.679254\pi$$
−0.533846 + 0.845582i $$0.679254\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ −16.0000 −0.655936
$$596$$ 6.00000 0.245770
$$597$$ −8.00000 −0.327418
$$598$$ 0 0
$$599$$ −40.0000 −1.63436 −0.817178 0.576386i $$-0.804463\pi$$
−0.817178 + 0.576386i $$0.804463\pi$$
$$600$$ −3.00000 −0.122474
$$601$$ −38.0000 −1.55005 −0.775026 0.631929i $$-0.782263\pi$$
−0.775026 + 0.631929i $$0.782263\pi$$
$$602$$ 48.0000 1.95633
$$603$$ −8.00000 −0.325785
$$604$$ −4.00000 −0.162758
$$605$$ 10.0000 0.406558
$$606$$ 18.0000 0.731200
$$607$$ −16.0000 −0.649420 −0.324710 0.945814i $$-0.605267\pi$$
−0.324710 + 0.945814i $$0.605267\pi$$
$$608$$ 0 0
$$609$$ −40.0000 −1.62088
$$610$$ −4.00000 −0.161955
$$611$$ 0 0
$$612$$ −2.00000 −0.0808452
$$613$$ −2.00000 −0.0807792 −0.0403896 0.999184i $$-0.512860\pi$$
−0.0403896 + 0.999184i $$0.512860\pi$$
$$614$$ −16.0000 −0.645707
$$615$$ −12.0000 −0.483887
$$616$$ 48.0000 1.93398
$$617$$ 22.0000 0.885687 0.442843 0.896599i $$-0.353970\pi$$
0.442843 + 0.896599i $$0.353970\pi$$
$$618$$ 0 0
$$619$$ 24.0000 0.964641 0.482321 0.875995i $$-0.339794\pi$$
0.482321 + 0.875995i $$0.339794\pi$$
$$620$$ −8.00000 −0.321288
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 8.00000 0.320513
$$624$$ 1.00000 0.0400320
$$625$$ −19.0000 −0.760000
$$626$$ −6.00000 −0.239808
$$627$$ 0 0
$$628$$ 18.0000 0.718278
$$629$$ −4.00000 −0.159490
$$630$$ −8.00000 −0.318728
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ −24.0000 −0.954669
$$633$$ 20.0000 0.794929
$$634$$ 26.0000 1.03259
$$635$$ −32.0000 −1.26988
$$636$$ 6.00000 0.237915
$$637$$ 9.00000 0.356593
$$638$$ −40.0000 −1.58362
$$639$$ 0 0
$$640$$ −6.00000 −0.237171
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ −12.0000 −0.473602
$$643$$ −40.0000 −1.57745 −0.788723 0.614749i $$-0.789257\pi$$
−0.788723 + 0.614749i $$0.789257\pi$$
$$644$$ 0 0
$$645$$ 24.0000 0.944999
$$646$$ 0 0
$$647$$ −8.00000 −0.314512 −0.157256 0.987558i $$-0.550265\pi$$
−0.157256 + 0.987558i $$0.550265\pi$$
$$648$$ −3.00000 −0.117851
$$649$$ 48.0000 1.88416
$$650$$ −1.00000 −0.0392232
$$651$$ 16.0000 0.627089
$$652$$ −8.00000 −0.313304
$$653$$ 6.00000 0.234798 0.117399 0.993085i $$-0.462544\pi$$
0.117399 + 0.993085i $$0.462544\pi$$
$$654$$ 2.00000 0.0782062
$$655$$ 8.00000 0.312586
$$656$$ −6.00000 −0.234261
$$657$$ 2.00000 0.0780274
$$658$$ 0 0
$$659$$ 28.0000 1.09073 0.545363 0.838200i $$-0.316392\pi$$
0.545363 + 0.838200i $$0.316392\pi$$
$$660$$ 8.00000 0.311400
$$661$$ 30.0000 1.16686 0.583432 0.812162i $$-0.301709\pi$$
0.583432 + 0.812162i $$0.301709\pi$$
$$662$$ −16.0000 −0.621858
$$663$$ −2.00000 −0.0776736
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ −2.00000 −0.0774984
$$667$$ 0 0
$$668$$ 8.00000 0.309529
$$669$$ −4.00000 −0.154649
$$670$$ −16.0000 −0.618134
$$671$$ −8.00000 −0.308837
$$672$$ 20.0000 0.771517
$$673$$ −14.0000 −0.539660 −0.269830 0.962908i $$-0.586968\pi$$
−0.269830 + 0.962908i $$0.586968\pi$$
$$674$$ 18.0000 0.693334
$$675$$ 1.00000 0.0384900
$$676$$ −1.00000 −0.0384615
$$677$$ 22.0000 0.845529 0.422764 0.906240i $$-0.361060\pi$$
0.422764 + 0.906240i $$0.361060\pi$$
$$678$$ 6.00000 0.230429
$$679$$ −40.0000 −1.53506
$$680$$ −12.0000 −0.460179
$$681$$ 20.0000 0.766402
$$682$$ 16.0000 0.612672
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ 12.0000 0.458496
$$686$$ −8.00000 −0.305441
$$687$$ 10.0000 0.381524
$$688$$ 12.0000 0.457496
$$689$$ 6.00000 0.228582
$$690$$ 0 0
$$691$$ −24.0000 −0.913003 −0.456502 0.889723i $$-0.650898\pi$$
−0.456502 + 0.889723i $$0.650898\pi$$
$$692$$ −6.00000 −0.228086
$$693$$ −16.0000 −0.607790
$$694$$ −12.0000 −0.455514
$$695$$ 24.0000 0.910372
$$696$$ −30.0000 −1.13715
$$697$$ 12.0000 0.454532
$$698$$ −26.0000 −0.984115
$$699$$ 14.0000 0.529529
$$700$$ −4.00000 −0.151186
$$701$$ −34.0000 −1.28416 −0.642081 0.766637i $$-0.721929\pi$$
−0.642081 + 0.766637i $$0.721929\pi$$
$$702$$ −1.00000 −0.0377426
$$703$$ 0 0
$$704$$ 28.0000 1.05529
$$705$$ 0 0
$$706$$ −2.00000 −0.0752710
$$707$$ 72.0000 2.70784
$$708$$ 12.0000 0.450988
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 6.00000 0.224860
$$713$$ 0 0
$$714$$ 8.00000 0.299392
$$715$$ 8.00000 0.299183
$$716$$ −4.00000 −0.149487
$$717$$ 24.0000 0.896296
$$718$$ 24.0000 0.895672
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ −2.00000 −0.0745356
$$721$$ 0 0
$$722$$ −19.0000 −0.707107
$$723$$ −10.0000 −0.371904
$$724$$ 10.0000 0.371647
$$725$$ 10.0000 0.371391
$$726$$ −5.00000 −0.185567
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ 12.0000 0.444750
$$729$$ 1.00000 0.0370370
$$730$$ 4.00000 0.148047
$$731$$ −24.0000 −0.887672
$$732$$ −2.00000 −0.0739221
$$733$$ 30.0000 1.10808 0.554038 0.832492i $$-0.313086\pi$$
0.554038 + 0.832492i $$0.313086\pi$$
$$734$$ 16.0000 0.590571
$$735$$ −18.0000 −0.663940
$$736$$ 0 0
$$737$$ −32.0000 −1.17874
$$738$$ 6.00000 0.220863
$$739$$ 32.0000 1.17714 0.588570 0.808447i $$-0.299691\pi$$
0.588570 + 0.808447i $$0.299691\pi$$
$$740$$ 4.00000 0.147043
$$741$$ 0 0
$$742$$ −24.0000 −0.881068
$$743$$ 48.0000 1.76095 0.880475 0.474093i $$-0.157224\pi$$
0.880475 + 0.474093i $$0.157224\pi$$
$$744$$ 12.0000 0.439941
$$745$$ −12.0000 −0.439646
$$746$$ −26.0000 −0.951928
$$747$$ 4.00000 0.146352
$$748$$ −8.00000 −0.292509
$$749$$ −48.0000 −1.75388
$$750$$ 12.0000 0.438178
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 0 0
$$753$$ 12.0000 0.437304
$$754$$ −10.0000 −0.364179
$$755$$ 8.00000 0.291150
$$756$$ −4.00000 −0.145479
$$757$$ 22.0000 0.799604 0.399802 0.916602i $$-0.369079\pi$$
0.399802 + 0.916602i $$0.369079\pi$$
$$758$$ −24.0000 −0.871719
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −10.0000 −0.362500 −0.181250 0.983437i $$-0.558014\pi$$
−0.181250 + 0.983437i $$0.558014\pi$$
$$762$$ 16.0000 0.579619
$$763$$ 8.00000 0.289619
$$764$$ −8.00000 −0.289430
$$765$$ 4.00000 0.144620
$$766$$ −16.0000 −0.578103
$$767$$ 12.0000 0.433295
$$768$$ 17.0000 0.613435
$$769$$ −30.0000 −1.08183 −0.540914 0.841078i $$-0.681921\pi$$
−0.540914 + 0.841078i $$0.681921\pi$$
$$770$$ −32.0000 −1.15320
$$771$$ −26.0000 −0.936367
$$772$$ −18.0000 −0.647834
$$773$$ 10.0000 0.359675 0.179838 0.983696i $$-0.442443\pi$$
0.179838 + 0.983696i $$0.442443\pi$$
$$774$$ −12.0000 −0.431331
$$775$$ −4.00000 −0.143684
$$776$$ −30.0000 −1.07694
$$777$$ −8.00000 −0.286998
$$778$$ 22.0000 0.788738
$$779$$ 0 0
$$780$$ 2.00000 0.0716115
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 10.0000 0.357371
$$784$$ −9.00000 −0.321429
$$785$$ −36.0000 −1.28490
$$786$$ −4.00000 −0.142675
$$787$$ −32.0000 −1.14068 −0.570338 0.821410i $$-0.693188\pi$$
−0.570338 + 0.821410i $$0.693188\pi$$
$$788$$ −18.0000 −0.641223
$$789$$ −24.0000 −0.854423
$$790$$ 16.0000 0.569254
$$791$$ 24.0000 0.853342
$$792$$ −12.0000 −0.426401
$$793$$ −2.00000 −0.0710221
$$794$$ 38.0000 1.34857
$$795$$ −12.0000 −0.425596
$$796$$ −8.00000 −0.283552
$$797$$ 46.0000 1.62940 0.814702 0.579880i $$-0.196901\pi$$
0.814702 + 0.579880i $$0.196901\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −5.00000 −0.176777
$$801$$ −2.00000 −0.0706665
$$802$$ 22.0000 0.776847
$$803$$ 8.00000 0.282314
$$804$$ −8.00000 −0.282138
$$805$$ 0 0
$$806$$ 4.00000 0.140894
$$807$$ −22.0000 −0.774437
$$808$$ 54.0000 1.89971
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 2.00000 0.0702728
$$811$$ 8.00000 0.280918 0.140459 0.990086i $$-0.455142\pi$$
0.140459 + 0.990086i $$0.455142\pi$$
$$812$$ −40.0000 −1.40372
$$813$$ 12.0000 0.420858
$$814$$ −8.00000 −0.280400
$$815$$ 16.0000 0.560456
$$816$$ 2.00000 0.0700140
$$817$$ 0 0
$$818$$ 34.0000 1.18878
$$819$$ −4.00000 −0.139771
$$820$$ −12.0000 −0.419058
$$821$$ −22.0000 −0.767805 −0.383903 0.923374i $$-0.625420\pi$$
−0.383903 + 0.923374i $$0.625420\pi$$
$$822$$ −6.00000 −0.209274
$$823$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$824$$ 0 0
$$825$$ 4.00000 0.139262
$$826$$ −48.0000 −1.67013
$$827$$ 4.00000 0.139094 0.0695468 0.997579i $$-0.477845\pi$$
0.0695468 + 0.997579i $$0.477845\pi$$
$$828$$ 0 0
$$829$$ −34.0000 −1.18087 −0.590434 0.807086i $$-0.701044\pi$$
−0.590434 + 0.807086i $$0.701044\pi$$
$$830$$ 8.00000 0.277684
$$831$$ 10.0000 0.346896
$$832$$ 7.00000 0.242681
$$833$$ 18.0000 0.623663
$$834$$ −12.0000 −0.415526
$$835$$ −16.0000 −0.553703
$$836$$ 0 0
$$837$$ −4.00000 −0.138260
$$838$$ 4.00000 0.138178
$$839$$ 48.0000 1.65714 0.828572 0.559883i $$-0.189154\pi$$
0.828572 + 0.559883i $$0.189154\pi$$
$$840$$ −24.0000 −0.828079
$$841$$ 71.0000 2.44828
$$842$$ −10.0000 −0.344623
$$843$$ 10.0000 0.344418
$$844$$ 20.0000 0.688428
$$845$$ 2.00000 0.0688021
$$846$$ 0 0
$$847$$ −20.0000 −0.687208
$$848$$ −6.00000 −0.206041
$$849$$ −12.0000 −0.411839
$$850$$ −2.00000 −0.0685994
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 30.0000 1.02718 0.513590 0.858036i $$-0.328315\pi$$
0.513590 + 0.858036i $$0.328315\pi$$
$$854$$ 8.00000 0.273754
$$855$$ 0 0
$$856$$ −36.0000 −1.23045
$$857$$ −46.0000 −1.57133 −0.785665 0.618652i $$-0.787679\pi$$
−0.785665 + 0.618652i $$0.787679\pi$$
$$858$$ −4.00000 −0.136558
$$859$$ −44.0000 −1.50126 −0.750630 0.660722i $$-0.770250\pi$$
−0.750630 + 0.660722i $$0.770250\pi$$
$$860$$ 24.0000 0.818393
$$861$$ 24.0000 0.817918
$$862$$ 0 0
$$863$$ 16.0000 0.544646 0.272323 0.962206i $$-0.412208\pi$$
0.272323 + 0.962206i $$0.412208\pi$$
$$864$$ −5.00000 −0.170103
$$865$$ 12.0000 0.408012
$$866$$ 34.0000 1.15537
$$867$$ 13.0000 0.441503
$$868$$ 16.0000 0.543075
$$869$$ 32.0000 1.08553
$$870$$ 20.0000 0.678064
$$871$$ −8.00000 −0.271070
$$872$$ 6.00000 0.203186
$$873$$ 10.0000 0.338449
$$874$$ 0 0
$$875$$ 48.0000 1.62270
$$876$$ 2.00000 0.0675737
$$877$$ −10.0000 −0.337676 −0.168838 0.985644i $$-0.554001\pi$$
−0.168838 + 0.985644i $$0.554001\pi$$
$$878$$ 32.0000 1.07995
$$879$$ 6.00000 0.202375
$$880$$ −8.00000 −0.269680
$$881$$ 58.0000 1.95407 0.977035 0.213080i $$-0.0683494\pi$$
0.977035 + 0.213080i $$0.0683494\pi$$
$$882$$ 9.00000 0.303046
$$883$$ −44.0000 −1.48072 −0.740359 0.672212i $$-0.765344\pi$$
−0.740359 + 0.672212i $$0.765344\pi$$
$$884$$ −2.00000 −0.0672673
$$885$$ −24.0000 −0.806751
$$886$$ −4.00000 −0.134383
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ −6.00000 −0.201347
$$889$$ 64.0000 2.14649
$$890$$ −4.00000 −0.134080
$$891$$ 4.00000 0.134005
$$892$$ −4.00000 −0.133930
$$893$$ 0 0
$$894$$ 6.00000 0.200670
$$895$$ 8.00000 0.267411
$$896$$ 12.0000 0.400892
$$897$$ 0 0
$$898$$ 22.0000 0.734150
$$899$$ −40.0000 −1.33407
$$900$$ 1.00000 0.0333333
$$901$$ 12.0000 0.399778
$$902$$ 24.0000 0.799113
$$903$$ −48.0000 −1.59734
$$904$$ 18.0000 0.598671
$$905$$ −20.0000 −0.664822
$$906$$ −4.00000 −0.132891
$$907$$ 12.0000 0.398453 0.199227 0.979953i $$-0.436157\pi$$
0.199227 + 0.979953i $$0.436157\pi$$
$$908$$ 20.0000 0.663723
$$909$$ −18.0000 −0.597022
$$910$$ −8.00000 −0.265197
$$911$$ −40.0000 −1.32526 −0.662630 0.748947i $$-0.730560\pi$$
−0.662630 + 0.748947i $$0.730560\pi$$
$$912$$ 0 0
$$913$$ 16.0000 0.529523
$$914$$ 2.00000 0.0661541
$$915$$ 4.00000 0.132236
$$916$$ 10.0000 0.330409
$$917$$ −16.0000 −0.528367
$$918$$ −2.00000 −0.0660098
$$919$$ −48.0000 −1.58337 −0.791687 0.610927i $$-0.790797\pi$$
−0.791687 + 0.610927i $$0.790797\pi$$
$$920$$ 0 0
$$921$$ 16.0000 0.527218
$$922$$ −38.0000 −1.25146
$$923$$ 0 0
$$924$$ −16.0000 −0.526361
$$925$$ 2.00000 0.0657596
$$926$$ 4.00000 0.131448
$$927$$ 0 0
$$928$$ −50.0000 −1.64133
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ −8.00000 −0.262330
$$931$$ 0 0
$$932$$ 14.0000 0.458585
$$933$$ 0 0
$$934$$ 12.0000 0.392652
$$935$$ 16.0000 0.523256
$$936$$ −3.00000 −0.0980581
$$937$$ 26.0000 0.849383 0.424691 0.905338i $$-0.360383\pi$$
0.424691 + 0.905338i $$0.360383\pi$$
$$938$$ 32.0000 1.04484
$$939$$ 6.00000 0.195803
$$940$$ 0 0
$$941$$ −14.0000 −0.456387 −0.228193 0.973616i $$-0.573282\pi$$
−0.228193 + 0.973616i $$0.573282\pi$$
$$942$$ 18.0000 0.586472
$$943$$ 0 0
$$944$$ −12.0000 −0.390567
$$945$$ 8.00000 0.260240
$$946$$ −48.0000 −1.56061
$$947$$ −60.0000 −1.94974 −0.974869 0.222779i $$-0.928487\pi$$
−0.974869 + 0.222779i $$0.928487\pi$$
$$948$$ 8.00000 0.259828
$$949$$ 2.00000 0.0649227
$$950$$ 0 0
$$951$$ −26.0000 −0.843108
$$952$$ 24.0000 0.777844
$$953$$ −30.0000 −0.971795 −0.485898 0.874016i $$-0.661507\pi$$
−0.485898 + 0.874016i $$0.661507\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 16.0000 0.517748
$$956$$ 24.0000 0.776215
$$957$$ 40.0000 1.29302
$$958$$ −24.0000 −0.775405
$$959$$ −24.0000 −0.775000
$$960$$ −14.0000 −0.451848
$$961$$ −15.0000 −0.483871
$$962$$ −2.00000 −0.0644826
$$963$$ 12.0000 0.386695
$$964$$ −10.0000 −0.322078
$$965$$ 36.0000 1.15888
$$966$$ 0 0
$$967$$ −52.0000 −1.67221 −0.836104 0.548572i $$-0.815172\pi$$
−0.836104 + 0.548572i $$0.815172\pi$$
$$968$$ −15.0000 −0.482118
$$969$$ 0 0
$$970$$ 20.0000 0.642161
$$971$$ −60.0000 −1.92549 −0.962746 0.270408i $$-0.912841\pi$$
−0.962746 + 0.270408i $$0.912841\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −48.0000 −1.53881
$$974$$ 12.0000 0.384505
$$975$$ 1.00000 0.0320256
$$976$$ 2.00000 0.0640184
$$977$$ −42.0000 −1.34370 −0.671850 0.740688i $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ −8.00000 −0.255812
$$979$$ −8.00000 −0.255681
$$980$$ −18.0000 −0.574989
$$981$$ −2.00000 −0.0638551
$$982$$ −12.0000 −0.382935
$$983$$ 16.0000 0.510321 0.255160 0.966899i $$-0.417872\pi$$
0.255160 + 0.966899i $$0.417872\pi$$
$$984$$ 18.0000 0.573819
$$985$$ 36.0000 1.14706
$$986$$ −20.0000 −0.636930
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 8.00000 0.254257
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 20.0000 0.635001
$$993$$ 16.0000 0.507745
$$994$$ 0 0
$$995$$ 16.0000 0.507234
$$996$$ 4.00000 0.126745
$$997$$ −26.0000 −0.823428 −0.411714 0.911313i $$-0.635070\pi$$
−0.411714 + 0.911313i $$0.635070\pi$$
$$998$$ −24.0000 −0.759707
$$999$$ 2.00000 0.0632772
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 39.2.a.a.1.1 1
3.2 odd 2 117.2.a.a.1.1 1
4.3 odd 2 624.2.a.i.1.1 1
5.2 odd 4 975.2.c.f.274.2 2
5.3 odd 4 975.2.c.f.274.1 2
5.4 even 2 975.2.a.f.1.1 1
7.6 odd 2 1911.2.a.f.1.1 1
8.3 odd 2 2496.2.a.e.1.1 1
8.5 even 2 2496.2.a.q.1.1 1
9.2 odd 6 1053.2.e.d.352.1 2
9.4 even 3 1053.2.e.b.703.1 2
9.5 odd 6 1053.2.e.d.703.1 2
9.7 even 3 1053.2.e.b.352.1 2
11.10 odd 2 4719.2.a.c.1.1 1
12.11 even 2 1872.2.a.h.1.1 1
13.2 odd 12 507.2.j.e.316.2 4
13.3 even 3 507.2.e.a.22.1 2
13.4 even 6 507.2.e.b.484.1 2
13.5 odd 4 507.2.b.a.337.1 2
13.6 odd 12 507.2.j.e.361.1 4
13.7 odd 12 507.2.j.e.361.2 4
13.8 odd 4 507.2.b.a.337.2 2
13.9 even 3 507.2.e.a.484.1 2
13.10 even 6 507.2.e.b.22.1 2
13.11 odd 12 507.2.j.e.316.1 4
13.12 even 2 507.2.a.a.1.1 1
15.2 even 4 2925.2.c.e.2224.1 2
15.8 even 4 2925.2.c.e.2224.2 2
15.14 odd 2 2925.2.a.p.1.1 1
21.20 even 2 5733.2.a.e.1.1 1
24.5 odd 2 7488.2.a.bl.1.1 1
24.11 even 2 7488.2.a.by.1.1 1
39.5 even 4 1521.2.b.b.1351.2 2
39.8 even 4 1521.2.b.b.1351.1 2
39.38 odd 2 1521.2.a.e.1.1 1
52.51 odd 2 8112.2.a.s.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.a.a.1.1 1 1.1 even 1 trivial
117.2.a.a.1.1 1 3.2 odd 2
507.2.a.a.1.1 1 13.12 even 2
507.2.b.a.337.1 2 13.5 odd 4
507.2.b.a.337.2 2 13.8 odd 4
507.2.e.a.22.1 2 13.3 even 3
507.2.e.a.484.1 2 13.9 even 3
507.2.e.b.22.1 2 13.10 even 6
507.2.e.b.484.1 2 13.4 even 6
507.2.j.e.316.1 4 13.11 odd 12
507.2.j.e.316.2 4 13.2 odd 12
507.2.j.e.361.1 4 13.6 odd 12
507.2.j.e.361.2 4 13.7 odd 12
624.2.a.i.1.1 1 4.3 odd 2
975.2.a.f.1.1 1 5.4 even 2
975.2.c.f.274.1 2 5.3 odd 4
975.2.c.f.274.2 2 5.2 odd 4
1053.2.e.b.352.1 2 9.7 even 3
1053.2.e.b.703.1 2 9.4 even 3
1053.2.e.d.352.1 2 9.2 odd 6
1053.2.e.d.703.1 2 9.5 odd 6
1521.2.a.e.1.1 1 39.38 odd 2
1521.2.b.b.1351.1 2 39.8 even 4
1521.2.b.b.1351.2 2 39.5 even 4
1872.2.a.h.1.1 1 12.11 even 2
1911.2.a.f.1.1 1 7.6 odd 2
2496.2.a.e.1.1 1 8.3 odd 2
2496.2.a.q.1.1 1 8.5 even 2
2925.2.a.p.1.1 1 15.14 odd 2
2925.2.c.e.2224.1 2 15.2 even 4
2925.2.c.e.2224.2 2 15.8 even 4
4719.2.a.c.1.1 1 11.10 odd 2
5733.2.a.e.1.1 1 21.20 even 2
7488.2.a.bl.1.1 1 24.5 odd 2
7488.2.a.by.1.1 1 24.11 even 2
8112.2.a.s.1.1 1 52.51 odd 2