# Properties

 Label 1155.2.a.c.1.1 Level $1155$ Weight $2$ Character 1155.1 Self dual yes Analytic conductor $9.223$ 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] = [1155,2,Mod(1,1155)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1155, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1155.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1155 = 3 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1155.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: $$9.22272143346$$ 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 $$\chi$$ $$=$$ 1155.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} +2.00000 q^{12} -6.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} -4.00000 q^{16} -7.00000 q^{17} -2.00000 q^{18} -5.00000 q^{19} +2.00000 q^{20} +1.00000 q^{21} -2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{25} +12.0000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -5.00000 q^{29} -2.00000 q^{30} -8.00000 q^{31} +8.00000 q^{32} +1.00000 q^{33} +14.0000 q^{34} +1.00000 q^{35} +2.00000 q^{36} -2.00000 q^{37} +10.0000 q^{38} -6.00000 q^{39} +12.0000 q^{41} -2.00000 q^{42} -11.0000 q^{43} +2.00000 q^{44} +1.00000 q^{45} +2.00000 q^{46} +8.00000 q^{47} -4.00000 q^{48} +1.00000 q^{49} -2.00000 q^{50} -7.00000 q^{51} -12.0000 q^{52} -11.0000 q^{53} -2.00000 q^{54} +1.00000 q^{55} -5.00000 q^{57} +10.0000 q^{58} -5.00000 q^{59} +2.00000 q^{60} +7.00000 q^{61} +16.0000 q^{62} +1.00000 q^{63} -8.00000 q^{64} -6.00000 q^{65} -2.00000 q^{66} -2.00000 q^{67} -14.0000 q^{68} -1.00000 q^{69} -2.00000 q^{70} +12.0000 q^{71} +4.00000 q^{73} +4.00000 q^{74} +1.00000 q^{75} -10.0000 q^{76} +1.00000 q^{77} +12.0000 q^{78} -10.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -24.0000 q^{82} -1.00000 q^{83} +2.00000 q^{84} -7.00000 q^{85} +22.0000 q^{86} -5.00000 q^{87} +15.0000 q^{89} -2.00000 q^{90} -6.00000 q^{91} -2.00000 q^{92} -8.00000 q^{93} -16.0000 q^{94} -5.00000 q^{95} +8.00000 q^{96} +3.00000 q^{97} -2.00000 q^{98} +1.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −1.41421 −0.707107 0.707107i $$-0.750000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ 1.00000 0.577350
$$4$$ 2.00000 1.00000
$$5$$ 1.00000 0.447214
$$6$$ −2.00000 −0.816497
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ −2.00000 −0.632456
$$11$$ 1.00000 0.301511
$$12$$ 2.00000 0.577350
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ −2.00000 −0.534522
$$15$$ 1.00000 0.258199
$$16$$ −4.00000 −1.00000
$$17$$ −7.00000 −1.69775 −0.848875 0.528594i $$-0.822719\pi$$
−0.848875 + 0.528594i $$0.822719\pi$$
$$18$$ −2.00000 −0.471405
$$19$$ −5.00000 −1.14708 −0.573539 0.819178i $$-0.694430\pi$$
−0.573539 + 0.819178i $$0.694430\pi$$
$$20$$ 2.00000 0.447214
$$21$$ 1.00000 0.218218
$$22$$ −2.00000 −0.426401
$$23$$ −1.00000 −0.208514 −0.104257 0.994550i $$-0.533247\pi$$
−0.104257 + 0.994550i $$0.533247\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 12.0000 2.35339
$$27$$ 1.00000 0.192450
$$28$$ 2.00000 0.377964
$$29$$ −5.00000 −0.928477 −0.464238 0.885710i $$-0.653672\pi$$
−0.464238 + 0.885710i $$0.653672\pi$$
$$30$$ −2.00000 −0.365148
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 8.00000 1.41421
$$33$$ 1.00000 0.174078
$$34$$ 14.0000 2.40098
$$35$$ 1.00000 0.169031
$$36$$ 2.00000 0.333333
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 10.0000 1.62221
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ 12.0000 1.87409 0.937043 0.349215i $$-0.113552\pi$$
0.937043 + 0.349215i $$0.113552\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ −11.0000 −1.67748 −0.838742 0.544529i $$-0.816708\pi$$
−0.838742 + 0.544529i $$0.816708\pi$$
$$44$$ 2.00000 0.301511
$$45$$ 1.00000 0.149071
$$46$$ 2.00000 0.294884
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ −4.00000 −0.577350
$$49$$ 1.00000 0.142857
$$50$$ −2.00000 −0.282843
$$51$$ −7.00000 −0.980196
$$52$$ −12.0000 −1.66410
$$53$$ −11.0000 −1.51097 −0.755483 0.655168i $$-0.772598\pi$$
−0.755483 + 0.655168i $$0.772598\pi$$
$$54$$ −2.00000 −0.272166
$$55$$ 1.00000 0.134840
$$56$$ 0 0
$$57$$ −5.00000 −0.662266
$$58$$ 10.0000 1.31306
$$59$$ −5.00000 −0.650945 −0.325472 0.945552i $$-0.605523\pi$$
−0.325472 + 0.945552i $$0.605523\pi$$
$$60$$ 2.00000 0.258199
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ 16.0000 2.03200
$$63$$ 1.00000 0.125988
$$64$$ −8.00000 −1.00000
$$65$$ −6.00000 −0.744208
$$66$$ −2.00000 −0.246183
$$67$$ −2.00000 −0.244339 −0.122169 0.992509i $$-0.538985\pi$$
−0.122169 + 0.992509i $$0.538985\pi$$
$$68$$ −14.0000 −1.69775
$$69$$ −1.00000 −0.120386
$$70$$ −2.00000 −0.239046
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ 4.00000 0.468165 0.234082 0.972217i $$-0.424791\pi$$
0.234082 + 0.972217i $$0.424791\pi$$
$$74$$ 4.00000 0.464991
$$75$$ 1.00000 0.115470
$$76$$ −10.0000 −1.14708
$$77$$ 1.00000 0.113961
$$78$$ 12.0000 1.35873
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ −4.00000 −0.447214
$$81$$ 1.00000 0.111111
$$82$$ −24.0000 −2.65036
$$83$$ −1.00000 −0.109764 −0.0548821 0.998493i $$-0.517478\pi$$
−0.0548821 + 0.998493i $$0.517478\pi$$
$$84$$ 2.00000 0.218218
$$85$$ −7.00000 −0.759257
$$86$$ 22.0000 2.37232
$$87$$ −5.00000 −0.536056
$$88$$ 0 0
$$89$$ 15.0000 1.59000 0.794998 0.606612i $$-0.207472\pi$$
0.794998 + 0.606612i $$0.207472\pi$$
$$90$$ −2.00000 −0.210819
$$91$$ −6.00000 −0.628971
$$92$$ −2.00000 −0.208514
$$93$$ −8.00000 −0.829561
$$94$$ −16.0000 −1.65027
$$95$$ −5.00000 −0.512989
$$96$$ 8.00000 0.816497
$$97$$ 3.00000 0.304604 0.152302 0.988334i $$-0.451331\pi$$
0.152302 + 0.988334i $$0.451331\pi$$
$$98$$ −2.00000 −0.202031
$$99$$ 1.00000 0.100504
$$100$$ 2.00000 0.200000
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 14.0000 1.38621
$$103$$ −1.00000 −0.0985329 −0.0492665 0.998786i $$-0.515688\pi$$
−0.0492665 + 0.998786i $$0.515688\pi$$
$$104$$ 0 0
$$105$$ 1.00000 0.0975900
$$106$$ 22.0000 2.13683
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 2.00000 0.192450
$$109$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$110$$ −2.00000 −0.190693
$$111$$ −2.00000 −0.189832
$$112$$ −4.00000 −0.377964
$$113$$ 9.00000 0.846649 0.423324 0.905978i $$-0.360863\pi$$
0.423324 + 0.905978i $$0.360863\pi$$
$$114$$ 10.0000 0.936586
$$115$$ −1.00000 −0.0932505
$$116$$ −10.0000 −0.928477
$$117$$ −6.00000 −0.554700
$$118$$ 10.0000 0.920575
$$119$$ −7.00000 −0.641689
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −14.0000 −1.26750
$$123$$ 12.0000 1.08200
$$124$$ −16.0000 −1.43684
$$125$$ 1.00000 0.0894427
$$126$$ −2.00000 −0.178174
$$127$$ −7.00000 −0.621150 −0.310575 0.950549i $$-0.600522\pi$$
−0.310575 + 0.950549i $$0.600522\pi$$
$$128$$ 0 0
$$129$$ −11.0000 −0.968496
$$130$$ 12.0000 1.05247
$$131$$ 2.00000 0.174741 0.0873704 0.996176i $$-0.472154\pi$$
0.0873704 + 0.996176i $$0.472154\pi$$
$$132$$ 2.00000 0.174078
$$133$$ −5.00000 −0.433555
$$134$$ 4.00000 0.345547
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ −22.0000 −1.87959 −0.939793 0.341743i $$-0.888983\pi$$
−0.939793 + 0.341743i $$0.888983\pi$$
$$138$$ 2.00000 0.170251
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 2.00000 0.169031
$$141$$ 8.00000 0.673722
$$142$$ −24.0000 −2.01404
$$143$$ −6.00000 −0.501745
$$144$$ −4.00000 −0.333333
$$145$$ −5.00000 −0.415227
$$146$$ −8.00000 −0.662085
$$147$$ 1.00000 0.0824786
$$148$$ −4.00000 −0.328798
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ −2.00000 −0.163299
$$151$$ 2.00000 0.162758 0.0813788 0.996683i $$-0.474068\pi$$
0.0813788 + 0.996683i $$0.474068\pi$$
$$152$$ 0 0
$$153$$ −7.00000 −0.565916
$$154$$ −2.00000 −0.161165
$$155$$ −8.00000 −0.642575
$$156$$ −12.0000 −0.960769
$$157$$ −17.0000 −1.35675 −0.678374 0.734717i $$-0.737315\pi$$
−0.678374 + 0.734717i $$0.737315\pi$$
$$158$$ 20.0000 1.59111
$$159$$ −11.0000 −0.872357
$$160$$ 8.00000 0.632456
$$161$$ −1.00000 −0.0788110
$$162$$ −2.00000 −0.157135
$$163$$ −6.00000 −0.469956 −0.234978 0.972001i $$-0.575502\pi$$
−0.234978 + 0.972001i $$0.575502\pi$$
$$164$$ 24.0000 1.87409
$$165$$ 1.00000 0.0778499
$$166$$ 2.00000 0.155230
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 14.0000 1.07375
$$171$$ −5.00000 −0.382360
$$172$$ −22.0000 −1.67748
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 10.0000 0.758098
$$175$$ 1.00000 0.0755929
$$176$$ −4.00000 −0.301511
$$177$$ −5.00000 −0.375823
$$178$$ −30.0000 −2.24860
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 2.00000 0.149071
$$181$$ 22.0000 1.63525 0.817624 0.575753i $$-0.195291\pi$$
0.817624 + 0.575753i $$0.195291\pi$$
$$182$$ 12.0000 0.889499
$$183$$ 7.00000 0.517455
$$184$$ 0 0
$$185$$ −2.00000 −0.147043
$$186$$ 16.0000 1.17318
$$187$$ −7.00000 −0.511891
$$188$$ 16.0000 1.16692
$$189$$ 1.00000 0.0727393
$$190$$ 10.0000 0.725476
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ −8.00000 −0.577350
$$193$$ −6.00000 −0.431889 −0.215945 0.976406i $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ −6.00000 −0.430775
$$195$$ −6.00000 −0.429669
$$196$$ 2.00000 0.142857
$$197$$ −12.0000 −0.854965 −0.427482 0.904024i $$-0.640599\pi$$
−0.427482 + 0.904024i $$0.640599\pi$$
$$198$$ −2.00000 −0.142134
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ 0 0
$$201$$ −2.00000 −0.141069
$$202$$ −4.00000 −0.281439
$$203$$ −5.00000 −0.350931
$$204$$ −14.0000 −0.980196
$$205$$ 12.0000 0.838116
$$206$$ 2.00000 0.139347
$$207$$ −1.00000 −0.0695048
$$208$$ 24.0000 1.66410
$$209$$ −5.00000 −0.345857
$$210$$ −2.00000 −0.138013
$$211$$ 22.0000 1.51454 0.757271 0.653101i $$-0.226532\pi$$
0.757271 + 0.653101i $$0.226532\pi$$
$$212$$ −22.0000 −1.51097
$$213$$ 12.0000 0.822226
$$214$$ 24.0000 1.64061
$$215$$ −11.0000 −0.750194
$$216$$ 0 0
$$217$$ −8.00000 −0.543075
$$218$$ 0 0
$$219$$ 4.00000 0.270295
$$220$$ 2.00000 0.134840
$$221$$ 42.0000 2.82523
$$222$$ 4.00000 0.268462
$$223$$ 9.00000 0.602685 0.301342 0.953516i $$-0.402565\pi$$
0.301342 + 0.953516i $$0.402565\pi$$
$$224$$ 8.00000 0.534522
$$225$$ 1.00000 0.0666667
$$226$$ −18.0000 −1.19734
$$227$$ 23.0000 1.52656 0.763282 0.646066i $$-0.223587\pi$$
0.763282 + 0.646066i $$0.223587\pi$$
$$228$$ −10.0000 −0.662266
$$229$$ −20.0000 −1.32164 −0.660819 0.750546i $$-0.729791\pi$$
−0.660819 + 0.750546i $$0.729791\pi$$
$$230$$ 2.00000 0.131876
$$231$$ 1.00000 0.0657952
$$232$$ 0 0
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 12.0000 0.784465
$$235$$ 8.00000 0.521862
$$236$$ −10.0000 −0.650945
$$237$$ −10.0000 −0.649570
$$238$$ 14.0000 0.907485
$$239$$ 5.00000 0.323423 0.161712 0.986838i $$-0.448299\pi$$
0.161712 + 0.986838i $$0.448299\pi$$
$$240$$ −4.00000 −0.258199
$$241$$ −18.0000 −1.15948 −0.579741 0.814801i $$-0.696846\pi$$
−0.579741 + 0.814801i $$0.696846\pi$$
$$242$$ −2.00000 −0.128565
$$243$$ 1.00000 0.0641500
$$244$$ 14.0000 0.896258
$$245$$ 1.00000 0.0638877
$$246$$ −24.0000 −1.53018
$$247$$ 30.0000 1.90885
$$248$$ 0 0
$$249$$ −1.00000 −0.0633724
$$250$$ −2.00000 −0.126491
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 2.00000 0.125988
$$253$$ −1.00000 −0.0628695
$$254$$ 14.0000 0.878438
$$255$$ −7.00000 −0.438357
$$256$$ 16.0000 1.00000
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 22.0000 1.36966
$$259$$ −2.00000 −0.124274
$$260$$ −12.0000 −0.744208
$$261$$ −5.00000 −0.309492
$$262$$ −4.00000 −0.247121
$$263$$ 4.00000 0.246651 0.123325 0.992366i $$-0.460644\pi$$
0.123325 + 0.992366i $$0.460644\pi$$
$$264$$ 0 0
$$265$$ −11.0000 −0.675725
$$266$$ 10.0000 0.613139
$$267$$ 15.0000 0.917985
$$268$$ −4.00000 −0.244339
$$269$$ −25.0000 −1.52428 −0.762138 0.647414i $$-0.775850\pi$$
−0.762138 + 0.647414i $$0.775850\pi$$
$$270$$ −2.00000 −0.121716
$$271$$ −3.00000 −0.182237 −0.0911185 0.995840i $$-0.529044\pi$$
−0.0911185 + 0.995840i $$0.529044\pi$$
$$272$$ 28.0000 1.69775
$$273$$ −6.00000 −0.363137
$$274$$ 44.0000 2.65814
$$275$$ 1.00000 0.0603023
$$276$$ −2.00000 −0.120386
$$277$$ −22.0000 −1.32185 −0.660926 0.750451i $$-0.729836\pi$$
−0.660926 + 0.750451i $$0.729836\pi$$
$$278$$ 0 0
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ −16.0000 −0.952786
$$283$$ 14.0000 0.832214 0.416107 0.909316i $$-0.363394\pi$$
0.416107 + 0.909316i $$0.363394\pi$$
$$284$$ 24.0000 1.42414
$$285$$ −5.00000 −0.296174
$$286$$ 12.0000 0.709575
$$287$$ 12.0000 0.708338
$$288$$ 8.00000 0.471405
$$289$$ 32.0000 1.88235
$$290$$ 10.0000 0.587220
$$291$$ 3.00000 0.175863
$$292$$ 8.00000 0.468165
$$293$$ 29.0000 1.69420 0.847099 0.531435i $$-0.178347\pi$$
0.847099 + 0.531435i $$0.178347\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ −5.00000 −0.291111
$$296$$ 0 0
$$297$$ 1.00000 0.0580259
$$298$$ 20.0000 1.15857
$$299$$ 6.00000 0.346989
$$300$$ 2.00000 0.115470
$$301$$ −11.0000 −0.634029
$$302$$ −4.00000 −0.230174
$$303$$ 2.00000 0.114897
$$304$$ 20.0000 1.14708
$$305$$ 7.00000 0.400819
$$306$$ 14.0000 0.800327
$$307$$ 18.0000 1.02731 0.513657 0.857996i $$-0.328290\pi$$
0.513657 + 0.857996i $$0.328290\pi$$
$$308$$ 2.00000 0.113961
$$309$$ −1.00000 −0.0568880
$$310$$ 16.0000 0.908739
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ −31.0000 −1.75222 −0.876112 0.482108i $$-0.839871\pi$$
−0.876112 + 0.482108i $$0.839871\pi$$
$$314$$ 34.0000 1.91873
$$315$$ 1.00000 0.0563436
$$316$$ −20.0000 −1.12509
$$317$$ −22.0000 −1.23564 −0.617822 0.786318i $$-0.711985\pi$$
−0.617822 + 0.786318i $$0.711985\pi$$
$$318$$ 22.0000 1.23370
$$319$$ −5.00000 −0.279946
$$320$$ −8.00000 −0.447214
$$321$$ −12.0000 −0.669775
$$322$$ 2.00000 0.111456
$$323$$ 35.0000 1.94745
$$324$$ 2.00000 0.111111
$$325$$ −6.00000 −0.332820
$$326$$ 12.0000 0.664619
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 8.00000 0.441054
$$330$$ −2.00000 −0.110096
$$331$$ −3.00000 −0.164895 −0.0824475 0.996595i $$-0.526274\pi$$
−0.0824475 + 0.996595i $$0.526274\pi$$
$$332$$ −2.00000 −0.109764
$$333$$ −2.00000 −0.109599
$$334$$ −16.0000 −0.875481
$$335$$ −2.00000 −0.109272
$$336$$ −4.00000 −0.218218
$$337$$ 13.0000 0.708155 0.354078 0.935216i $$-0.384795\pi$$
0.354078 + 0.935216i $$0.384795\pi$$
$$338$$ −46.0000 −2.50207
$$339$$ 9.00000 0.488813
$$340$$ −14.0000 −0.759257
$$341$$ −8.00000 −0.433224
$$342$$ 10.0000 0.540738
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ −1.00000 −0.0538382
$$346$$ 12.0000 0.645124
$$347$$ −32.0000 −1.71785 −0.858925 0.512101i $$-0.828867\pi$$
−0.858925 + 0.512101i $$0.828867\pi$$
$$348$$ −10.0000 −0.536056
$$349$$ −25.0000 −1.33822 −0.669110 0.743164i $$-0.733324\pi$$
−0.669110 + 0.743164i $$0.733324\pi$$
$$350$$ −2.00000 −0.106904
$$351$$ −6.00000 −0.320256
$$352$$ 8.00000 0.426401
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 10.0000 0.531494
$$355$$ 12.0000 0.636894
$$356$$ 30.0000 1.59000
$$357$$ −7.00000 −0.370479
$$358$$ 0 0
$$359$$ −15.0000 −0.791670 −0.395835 0.918322i $$-0.629545\pi$$
−0.395835 + 0.918322i $$0.629545\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ −44.0000 −2.31259
$$363$$ 1.00000 0.0524864
$$364$$ −12.0000 −0.628971
$$365$$ 4.00000 0.209370
$$366$$ −14.0000 −0.731792
$$367$$ −17.0000 −0.887393 −0.443696 0.896177i $$-0.646333\pi$$
−0.443696 + 0.896177i $$0.646333\pi$$
$$368$$ 4.00000 0.208514
$$369$$ 12.0000 0.624695
$$370$$ 4.00000 0.207950
$$371$$ −11.0000 −0.571092
$$372$$ −16.0000 −0.829561
$$373$$ 29.0000 1.50156 0.750782 0.660551i $$-0.229677\pi$$
0.750782 + 0.660551i $$0.229677\pi$$
$$374$$ 14.0000 0.723923
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ 30.0000 1.54508
$$378$$ −2.00000 −0.102869
$$379$$ 5.00000 0.256833 0.128416 0.991720i $$-0.459011\pi$$
0.128416 + 0.991720i $$0.459011\pi$$
$$380$$ −10.0000 −0.512989
$$381$$ −7.00000 −0.358621
$$382$$ 16.0000 0.818631
$$383$$ 34.0000 1.73732 0.868659 0.495410i $$-0.164982\pi$$
0.868659 + 0.495410i $$0.164982\pi$$
$$384$$ 0 0
$$385$$ 1.00000 0.0509647
$$386$$ 12.0000 0.610784
$$387$$ −11.0000 −0.559161
$$388$$ 6.00000 0.304604
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 12.0000 0.607644
$$391$$ 7.00000 0.354005
$$392$$ 0 0
$$393$$ 2.00000 0.100887
$$394$$ 24.0000 1.20910
$$395$$ −10.0000 −0.503155
$$396$$ 2.00000 0.100504
$$397$$ 38.0000 1.90717 0.953583 0.301131i $$-0.0973643\pi$$
0.953583 + 0.301131i $$0.0973643\pi$$
$$398$$ −40.0000 −2.00502
$$399$$ −5.00000 −0.250313
$$400$$ −4.00000 −0.200000
$$401$$ 12.0000 0.599251 0.299626 0.954057i $$-0.403138\pi$$
0.299626 + 0.954057i $$0.403138\pi$$
$$402$$ 4.00000 0.199502
$$403$$ 48.0000 2.39105
$$404$$ 4.00000 0.199007
$$405$$ 1.00000 0.0496904
$$406$$ 10.0000 0.496292
$$407$$ −2.00000 −0.0991363
$$408$$ 0 0
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ −24.0000 −1.18528
$$411$$ −22.0000 −1.08518
$$412$$ −2.00000 −0.0985329
$$413$$ −5.00000 −0.246034
$$414$$ 2.00000 0.0982946
$$415$$ −1.00000 −0.0490881
$$416$$ −48.0000 −2.35339
$$417$$ 0 0
$$418$$ 10.0000 0.489116
$$419$$ −15.0000 −0.732798 −0.366399 0.930458i $$-0.619409\pi$$
−0.366399 + 0.930458i $$0.619409\pi$$
$$420$$ 2.00000 0.0975900
$$421$$ −23.0000 −1.12095 −0.560476 0.828171i $$-0.689382\pi$$
−0.560476 + 0.828171i $$0.689382\pi$$
$$422$$ −44.0000 −2.14189
$$423$$ 8.00000 0.388973
$$424$$ 0 0
$$425$$ −7.00000 −0.339550
$$426$$ −24.0000 −1.16280
$$427$$ 7.00000 0.338754
$$428$$ −24.0000 −1.16008
$$429$$ −6.00000 −0.289683
$$430$$ 22.0000 1.06093
$$431$$ −8.00000 −0.385346 −0.192673 0.981263i $$-0.561716\pi$$
−0.192673 + 0.981263i $$0.561716\pi$$
$$432$$ −4.00000 −0.192450
$$433$$ −26.0000 −1.24948 −0.624740 0.780833i $$-0.714795\pi$$
−0.624740 + 0.780833i $$0.714795\pi$$
$$434$$ 16.0000 0.768025
$$435$$ −5.00000 −0.239732
$$436$$ 0 0
$$437$$ 5.00000 0.239182
$$438$$ −8.00000 −0.382255
$$439$$ −5.00000 −0.238637 −0.119318 0.992856i $$-0.538071\pi$$
−0.119318 + 0.992856i $$0.538071\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ −84.0000 −3.99547
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ −4.00000 −0.189832
$$445$$ 15.0000 0.711068
$$446$$ −18.0000 −0.852325
$$447$$ −10.0000 −0.472984
$$448$$ −8.00000 −0.377964
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ −2.00000 −0.0942809
$$451$$ 12.0000 0.565058
$$452$$ 18.0000 0.846649
$$453$$ 2.00000 0.0939682
$$454$$ −46.0000 −2.15889
$$455$$ −6.00000 −0.281284
$$456$$ 0 0
$$457$$ 33.0000 1.54367 0.771837 0.635820i $$-0.219338\pi$$
0.771837 + 0.635820i $$0.219338\pi$$
$$458$$ 40.0000 1.86908
$$459$$ −7.00000 −0.326732
$$460$$ −2.00000 −0.0932505
$$461$$ −38.0000 −1.76984 −0.884918 0.465746i $$-0.845786\pi$$
−0.884918 + 0.465746i $$0.845786\pi$$
$$462$$ −2.00000 −0.0930484
$$463$$ −6.00000 −0.278844 −0.139422 0.990233i $$-0.544524\pi$$
−0.139422 + 0.990233i $$0.544524\pi$$
$$464$$ 20.0000 0.928477
$$465$$ −8.00000 −0.370991
$$466$$ 12.0000 0.555889
$$467$$ 28.0000 1.29569 0.647843 0.761774i $$-0.275671\pi$$
0.647843 + 0.761774i $$0.275671\pi$$
$$468$$ −12.0000 −0.554700
$$469$$ −2.00000 −0.0923514
$$470$$ −16.0000 −0.738025
$$471$$ −17.0000 −0.783319
$$472$$ 0 0
$$473$$ −11.0000 −0.505781
$$474$$ 20.0000 0.918630
$$475$$ −5.00000 −0.229416
$$476$$ −14.0000 −0.641689
$$477$$ −11.0000 −0.503655
$$478$$ −10.0000 −0.457389
$$479$$ −10.0000 −0.456912 −0.228456 0.973554i $$-0.573368\pi$$
−0.228456 + 0.973554i $$0.573368\pi$$
$$480$$ 8.00000 0.365148
$$481$$ 12.0000 0.547153
$$482$$ 36.0000 1.63976
$$483$$ −1.00000 −0.0455016
$$484$$ 2.00000 0.0909091
$$485$$ 3.00000 0.136223
$$486$$ −2.00000 −0.0907218
$$487$$ −22.0000 −0.996915 −0.498458 0.866914i $$-0.666100\pi$$
−0.498458 + 0.866914i $$0.666100\pi$$
$$488$$ 0 0
$$489$$ −6.00000 −0.271329
$$490$$ −2.00000 −0.0903508
$$491$$ 27.0000 1.21849 0.609246 0.792981i $$-0.291472\pi$$
0.609246 + 0.792981i $$0.291472\pi$$
$$492$$ 24.0000 1.08200
$$493$$ 35.0000 1.57632
$$494$$ −60.0000 −2.69953
$$495$$ 1.00000 0.0449467
$$496$$ 32.0000 1.43684
$$497$$ 12.0000 0.538274
$$498$$ 2.00000 0.0896221
$$499$$ −25.0000 −1.11915 −0.559577 0.828778i $$-0.689036\pi$$
−0.559577 + 0.828778i $$0.689036\pi$$
$$500$$ 2.00000 0.0894427
$$501$$ 8.00000 0.357414
$$502$$ −24.0000 −1.07117
$$503$$ −11.0000 −0.490466 −0.245233 0.969464i $$-0.578864\pi$$
−0.245233 + 0.969464i $$0.578864\pi$$
$$504$$ 0 0
$$505$$ 2.00000 0.0889988
$$506$$ 2.00000 0.0889108
$$507$$ 23.0000 1.02147
$$508$$ −14.0000 −0.621150
$$509$$ −35.0000 −1.55135 −0.775674 0.631134i $$-0.782590\pi$$
−0.775674 + 0.631134i $$0.782590\pi$$
$$510$$ 14.0000 0.619930
$$511$$ 4.00000 0.176950
$$512$$ −32.0000 −1.41421
$$513$$ −5.00000 −0.220755
$$514$$ −36.0000 −1.58789
$$515$$ −1.00000 −0.0440653
$$516$$ −22.0000 −0.968496
$$517$$ 8.00000 0.351840
$$518$$ 4.00000 0.175750
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −23.0000 −1.00765 −0.503824 0.863806i $$-0.668074\pi$$
−0.503824 + 0.863806i $$0.668074\pi$$
$$522$$ 10.0000 0.437688
$$523$$ −16.0000 −0.699631 −0.349816 0.936819i $$-0.613756\pi$$
−0.349816 + 0.936819i $$0.613756\pi$$
$$524$$ 4.00000 0.174741
$$525$$ 1.00000 0.0436436
$$526$$ −8.00000 −0.348817
$$527$$ 56.0000 2.43940
$$528$$ −4.00000 −0.174078
$$529$$ −22.0000 −0.956522
$$530$$ 22.0000 0.955619
$$531$$ −5.00000 −0.216982
$$532$$ −10.0000 −0.433555
$$533$$ −72.0000 −3.11867
$$534$$ −30.0000 −1.29823
$$535$$ −12.0000 −0.518805
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 50.0000 2.15565
$$539$$ 1.00000 0.0430730
$$540$$ 2.00000 0.0860663
$$541$$ 12.0000 0.515920 0.257960 0.966156i $$-0.416950\pi$$
0.257960 + 0.966156i $$0.416950\pi$$
$$542$$ 6.00000 0.257722
$$543$$ 22.0000 0.944110
$$544$$ −56.0000 −2.40098
$$545$$ 0 0
$$546$$ 12.0000 0.513553
$$547$$ 3.00000 0.128271 0.0641354 0.997941i $$-0.479571\pi$$
0.0641354 + 0.997941i $$0.479571\pi$$
$$548$$ −44.0000 −1.87959
$$549$$ 7.00000 0.298753
$$550$$ −2.00000 −0.0852803
$$551$$ 25.0000 1.06504
$$552$$ 0 0
$$553$$ −10.0000 −0.425243
$$554$$ 44.0000 1.86938
$$555$$ −2.00000 −0.0848953
$$556$$ 0 0
$$557$$ 38.0000 1.61011 0.805056 0.593199i $$-0.202135\pi$$
0.805056 + 0.593199i $$0.202135\pi$$
$$558$$ 16.0000 0.677334
$$559$$ 66.0000 2.79150
$$560$$ −4.00000 −0.169031
$$561$$ −7.00000 −0.295540
$$562$$ 36.0000 1.51857
$$563$$ 4.00000 0.168580 0.0842900 0.996441i $$-0.473138\pi$$
0.0842900 + 0.996441i $$0.473138\pi$$
$$564$$ 16.0000 0.673722
$$565$$ 9.00000 0.378633
$$566$$ −28.0000 −1.17693
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ 15.0000 0.628833 0.314416 0.949285i $$-0.398191\pi$$
0.314416 + 0.949285i $$0.398191\pi$$
$$570$$ 10.0000 0.418854
$$571$$ −18.0000 −0.753277 −0.376638 0.926360i $$-0.622920\pi$$
−0.376638 + 0.926360i $$0.622920\pi$$
$$572$$ −12.0000 −0.501745
$$573$$ −8.00000 −0.334205
$$574$$ −24.0000 −1.00174
$$575$$ −1.00000 −0.0417029
$$576$$ −8.00000 −0.333333
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ −64.0000 −2.66205
$$579$$ −6.00000 −0.249351
$$580$$ −10.0000 −0.415227
$$581$$ −1.00000 −0.0414870
$$582$$ −6.00000 −0.248708
$$583$$ −11.0000 −0.455573
$$584$$ 0 0
$$585$$ −6.00000 −0.248069
$$586$$ −58.0000 −2.39596
$$587$$ −42.0000 −1.73353 −0.866763 0.498721i $$-0.833803\pi$$
−0.866763 + 0.498721i $$0.833803\pi$$
$$588$$ 2.00000 0.0824786
$$589$$ 40.0000 1.64817
$$590$$ 10.0000 0.411693
$$591$$ −12.0000 −0.493614
$$592$$ 8.00000 0.328798
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ −2.00000 −0.0820610
$$595$$ −7.00000 −0.286972
$$596$$ −20.0000 −0.819232
$$597$$ 20.0000 0.818546
$$598$$ −12.0000 −0.490716
$$599$$ −10.0000 −0.408589 −0.204294 0.978909i $$-0.565490\pi$$
−0.204294 + 0.978909i $$0.565490\pi$$
$$600$$ 0 0
$$601$$ −3.00000 −0.122373 −0.0611863 0.998126i $$-0.519488\pi$$
−0.0611863 + 0.998126i $$0.519488\pi$$
$$602$$ 22.0000 0.896653
$$603$$ −2.00000 −0.0814463
$$604$$ 4.00000 0.162758
$$605$$ 1.00000 0.0406558
$$606$$ −4.00000 −0.162489
$$607$$ −22.0000 −0.892952 −0.446476 0.894795i $$-0.647321\pi$$
−0.446476 + 0.894795i $$0.647321\pi$$
$$608$$ −40.0000 −1.62221
$$609$$ −5.00000 −0.202610
$$610$$ −14.0000 −0.566843
$$611$$ −48.0000 −1.94187
$$612$$ −14.0000 −0.565916
$$613$$ −6.00000 −0.242338 −0.121169 0.992632i $$-0.538664\pi$$
−0.121169 + 0.992632i $$0.538664\pi$$
$$614$$ −36.0000 −1.45284
$$615$$ 12.0000 0.483887
$$616$$ 0 0
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ 2.00000 0.0804518
$$619$$ 30.0000 1.20580 0.602901 0.797816i $$-0.294011\pi$$
0.602901 + 0.797816i $$0.294011\pi$$
$$620$$ −16.0000 −0.642575
$$621$$ −1.00000 −0.0401286
$$622$$ −24.0000 −0.962312
$$623$$ 15.0000 0.600962
$$624$$ 24.0000 0.960769
$$625$$ 1.00000 0.0400000
$$626$$ 62.0000 2.47802
$$627$$ −5.00000 −0.199681
$$628$$ −34.0000 −1.35675
$$629$$ 14.0000 0.558217
$$630$$ −2.00000 −0.0796819
$$631$$ −3.00000 −0.119428 −0.0597141 0.998216i $$-0.519019\pi$$
−0.0597141 + 0.998216i $$0.519019\pi$$
$$632$$ 0 0
$$633$$ 22.0000 0.874421
$$634$$ 44.0000 1.74746
$$635$$ −7.00000 −0.277787
$$636$$ −22.0000 −0.872357
$$637$$ −6.00000 −0.237729
$$638$$ 10.0000 0.395904
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 24.0000 0.947204
$$643$$ 19.0000 0.749287 0.374643 0.927169i $$-0.377765\pi$$
0.374643 + 0.927169i $$0.377765\pi$$
$$644$$ −2.00000 −0.0788110
$$645$$ −11.0000 −0.433125
$$646$$ −70.0000 −2.75411
$$647$$ 48.0000 1.88707 0.943537 0.331266i $$-0.107476\pi$$
0.943537 + 0.331266i $$0.107476\pi$$
$$648$$ 0 0
$$649$$ −5.00000 −0.196267
$$650$$ 12.0000 0.470679
$$651$$ −8.00000 −0.313545
$$652$$ −12.0000 −0.469956
$$653$$ −1.00000 −0.0391330 −0.0195665 0.999809i $$-0.506229\pi$$
−0.0195665 + 0.999809i $$0.506229\pi$$
$$654$$ 0 0
$$655$$ 2.00000 0.0781465
$$656$$ −48.0000 −1.87409
$$657$$ 4.00000 0.156055
$$658$$ −16.0000 −0.623745
$$659$$ 15.0000 0.584317 0.292159 0.956370i $$-0.405627\pi$$
0.292159 + 0.956370i $$0.405627\pi$$
$$660$$ 2.00000 0.0778499
$$661$$ −28.0000 −1.08907 −0.544537 0.838737i $$-0.683295\pi$$
−0.544537 + 0.838737i $$0.683295\pi$$
$$662$$ 6.00000 0.233197
$$663$$ 42.0000 1.63114
$$664$$ 0 0
$$665$$ −5.00000 −0.193892
$$666$$ 4.00000 0.154997
$$667$$ 5.00000 0.193601
$$668$$ 16.0000 0.619059
$$669$$ 9.00000 0.347960
$$670$$ 4.00000 0.154533
$$671$$ 7.00000 0.270232
$$672$$ 8.00000 0.308607
$$673$$ −1.00000 −0.0385472 −0.0192736 0.999814i $$-0.506135\pi$$
−0.0192736 + 0.999814i $$0.506135\pi$$
$$674$$ −26.0000 −1.00148
$$675$$ 1.00000 0.0384900
$$676$$ 46.0000 1.76923
$$677$$ 13.0000 0.499631 0.249815 0.968294i $$-0.419630\pi$$
0.249815 + 0.968294i $$0.419630\pi$$
$$678$$ −18.0000 −0.691286
$$679$$ 3.00000 0.115129
$$680$$ 0 0
$$681$$ 23.0000 0.881362
$$682$$ 16.0000 0.612672
$$683$$ −36.0000 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ −10.0000 −0.382360
$$685$$ −22.0000 −0.840577
$$686$$ −2.00000 −0.0763604
$$687$$ −20.0000 −0.763048
$$688$$ 44.0000 1.67748
$$689$$ 66.0000 2.51440
$$690$$ 2.00000 0.0761387
$$691$$ 42.0000 1.59776 0.798878 0.601494i $$-0.205427\pi$$
0.798878 + 0.601494i $$0.205427\pi$$
$$692$$ −12.0000 −0.456172
$$693$$ 1.00000 0.0379869
$$694$$ 64.0000 2.42941
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −84.0000 −3.18173
$$698$$ 50.0000 1.89253
$$699$$ −6.00000 −0.226941
$$700$$ 2.00000 0.0755929
$$701$$ −23.0000 −0.868698 −0.434349 0.900745i $$-0.643022\pi$$
−0.434349 + 0.900745i $$0.643022\pi$$
$$702$$ 12.0000 0.452911
$$703$$ 10.0000 0.377157
$$704$$ −8.00000 −0.301511
$$705$$ 8.00000 0.301297
$$706$$ 12.0000 0.451626
$$707$$ 2.00000 0.0752177
$$708$$ −10.0000 −0.375823
$$709$$ −35.0000 −1.31445 −0.657226 0.753693i $$-0.728270\pi$$
−0.657226 + 0.753693i $$0.728270\pi$$
$$710$$ −24.0000 −0.900704
$$711$$ −10.0000 −0.375029
$$712$$ 0 0
$$713$$ 8.00000 0.299602
$$714$$ 14.0000 0.523937
$$715$$ −6.00000 −0.224387
$$716$$ 0 0
$$717$$ 5.00000 0.186728
$$718$$ 30.0000 1.11959
$$719$$ −25.0000 −0.932343 −0.466171 0.884694i $$-0.654367\pi$$
−0.466171 + 0.884694i $$0.654367\pi$$
$$720$$ −4.00000 −0.149071
$$721$$ −1.00000 −0.0372419
$$722$$ −12.0000 −0.446594
$$723$$ −18.0000 −0.669427
$$724$$ 44.0000 1.63525
$$725$$ −5.00000 −0.185695
$$726$$ −2.00000 −0.0742270
$$727$$ −47.0000 −1.74313 −0.871567 0.490277i $$-0.836896\pi$$
−0.871567 + 0.490277i $$0.836896\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −8.00000 −0.296093
$$731$$ 77.0000 2.84795
$$732$$ 14.0000 0.517455
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ 34.0000 1.25496
$$735$$ 1.00000 0.0368856
$$736$$ −8.00000 −0.294884
$$737$$ −2.00000 −0.0736709
$$738$$ −24.0000 −0.883452
$$739$$ −10.0000 −0.367856 −0.183928 0.982940i $$-0.558881\pi$$
−0.183928 + 0.982940i $$0.558881\pi$$
$$740$$ −4.00000 −0.147043
$$741$$ 30.0000 1.10208
$$742$$ 22.0000 0.807645
$$743$$ −6.00000 −0.220119 −0.110059 0.993925i $$-0.535104\pi$$
−0.110059 + 0.993925i $$0.535104\pi$$
$$744$$ 0 0
$$745$$ −10.0000 −0.366372
$$746$$ −58.0000 −2.12353
$$747$$ −1.00000 −0.0365881
$$748$$ −14.0000 −0.511891
$$749$$ −12.0000 −0.438470
$$750$$ −2.00000 −0.0730297
$$751$$ 17.0000 0.620339 0.310169 0.950681i $$-0.399614\pi$$
0.310169 + 0.950681i $$0.399614\pi$$
$$752$$ −32.0000 −1.16692
$$753$$ 12.0000 0.437304
$$754$$ −60.0000 −2.18507
$$755$$ 2.00000 0.0727875
$$756$$ 2.00000 0.0727393
$$757$$ −22.0000 −0.799604 −0.399802 0.916602i $$-0.630921\pi$$
−0.399802 + 0.916602i $$0.630921\pi$$
$$758$$ −10.0000 −0.363216
$$759$$ −1.00000 −0.0362977
$$760$$ 0 0
$$761$$ 42.0000 1.52250 0.761249 0.648459i $$-0.224586\pi$$
0.761249 + 0.648459i $$0.224586\pi$$
$$762$$ 14.0000 0.507166
$$763$$ 0 0
$$764$$ −16.0000 −0.578860
$$765$$ −7.00000 −0.253086
$$766$$ −68.0000 −2.45694
$$767$$ 30.0000 1.08324
$$768$$ 16.0000 0.577350
$$769$$ 35.0000 1.26213 0.631066 0.775729i $$-0.282618\pi$$
0.631066 + 0.775729i $$0.282618\pi$$
$$770$$ −2.00000 −0.0720750
$$771$$ 18.0000 0.648254
$$772$$ −12.0000 −0.431889
$$773$$ −6.00000 −0.215805 −0.107903 0.994161i $$-0.534413\pi$$
−0.107903 + 0.994161i $$0.534413\pi$$
$$774$$ 22.0000 0.790774
$$775$$ −8.00000 −0.287368
$$776$$ 0 0
$$777$$ −2.00000 −0.0717496
$$778$$ −60.0000 −2.15110
$$779$$ −60.0000 −2.14972
$$780$$ −12.0000 −0.429669
$$781$$ 12.0000 0.429394
$$782$$ −14.0000 −0.500639
$$783$$ −5.00000 −0.178685
$$784$$ −4.00000 −0.142857
$$785$$ −17.0000 −0.606756
$$786$$ −4.00000 −0.142675
$$787$$ 28.0000 0.998092 0.499046 0.866575i $$-0.333684\pi$$
0.499046 + 0.866575i $$0.333684\pi$$
$$788$$ −24.0000 −0.854965
$$789$$ 4.00000 0.142404
$$790$$ 20.0000 0.711568
$$791$$ 9.00000 0.320003
$$792$$ 0 0
$$793$$ −42.0000 −1.49146
$$794$$ −76.0000 −2.69714
$$795$$ −11.0000 −0.390130
$$796$$ 40.0000 1.41776
$$797$$ 18.0000 0.637593 0.318796 0.947823i $$-0.396721\pi$$
0.318796 + 0.947823i $$0.396721\pi$$
$$798$$ 10.0000 0.353996
$$799$$ −56.0000 −1.98114
$$800$$ 8.00000 0.282843
$$801$$ 15.0000 0.529999
$$802$$ −24.0000 −0.847469
$$803$$ 4.00000 0.141157
$$804$$ −4.00000 −0.141069
$$805$$ −1.00000 −0.0352454
$$806$$ −96.0000 −3.38146
$$807$$ −25.0000 −0.880042
$$808$$ 0 0
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ −2.00000 −0.0702728
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ −10.0000 −0.350931
$$813$$ −3.00000 −0.105215
$$814$$ 4.00000 0.140200
$$815$$ −6.00000 −0.210171
$$816$$ 28.0000 0.980196
$$817$$ 55.0000 1.92421
$$818$$ 20.0000 0.699284
$$819$$ −6.00000 −0.209657
$$820$$ 24.0000 0.838116
$$821$$ −3.00000 −0.104701 −0.0523504 0.998629i $$-0.516671\pi$$
−0.0523504 + 0.998629i $$0.516671\pi$$
$$822$$ 44.0000 1.53468
$$823$$ 14.0000 0.488009 0.244005 0.969774i $$-0.421539\pi$$
0.244005 + 0.969774i $$0.421539\pi$$
$$824$$ 0 0
$$825$$ 1.00000 0.0348155
$$826$$ 10.0000 0.347945
$$827$$ 48.0000 1.66912 0.834562 0.550914i $$-0.185721\pi$$
0.834562 + 0.550914i $$0.185721\pi$$
$$828$$ −2.00000 −0.0695048
$$829$$ −30.0000 −1.04194 −0.520972 0.853574i $$-0.674430\pi$$
−0.520972 + 0.853574i $$0.674430\pi$$
$$830$$ 2.00000 0.0694210
$$831$$ −22.0000 −0.763172
$$832$$ 48.0000 1.66410
$$833$$ −7.00000 −0.242536
$$834$$ 0 0
$$835$$ 8.00000 0.276851
$$836$$ −10.0000 −0.345857
$$837$$ −8.00000 −0.276520
$$838$$ 30.0000 1.03633
$$839$$ 15.0000 0.517858 0.258929 0.965896i $$-0.416631\pi$$
0.258929 + 0.965896i $$0.416631\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 46.0000 1.58526
$$843$$ −18.0000 −0.619953
$$844$$ 44.0000 1.51454
$$845$$ 23.0000 0.791224
$$846$$ −16.0000 −0.550091
$$847$$ 1.00000 0.0343604
$$848$$ 44.0000 1.51097
$$849$$ 14.0000 0.480479
$$850$$ 14.0000 0.480196
$$851$$ 2.00000 0.0685591
$$852$$ 24.0000 0.822226
$$853$$ −36.0000 −1.23262 −0.616308 0.787505i $$-0.711372\pi$$
−0.616308 + 0.787505i $$0.711372\pi$$
$$854$$ −14.0000 −0.479070
$$855$$ −5.00000 −0.170996
$$856$$ 0 0
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ 12.0000 0.409673
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ −22.0000 −0.750194
$$861$$ 12.0000 0.408959
$$862$$ 16.0000 0.544962
$$863$$ 39.0000 1.32758 0.663788 0.747921i $$-0.268948\pi$$
0.663788 + 0.747921i $$0.268948\pi$$
$$864$$ 8.00000 0.272166
$$865$$ −6.00000 −0.204006
$$866$$ 52.0000 1.76703
$$867$$ 32.0000 1.08678
$$868$$ −16.0000 −0.543075
$$869$$ −10.0000 −0.339227
$$870$$ 10.0000 0.339032
$$871$$ 12.0000 0.406604
$$872$$ 0 0
$$873$$ 3.00000 0.101535
$$874$$ −10.0000 −0.338255
$$875$$ 1.00000 0.0338062
$$876$$ 8.00000 0.270295
$$877$$ 23.0000 0.776655 0.388327 0.921521i $$-0.373053\pi$$
0.388327 + 0.921521i $$0.373053\pi$$
$$878$$ 10.0000 0.337484
$$879$$ 29.0000 0.978146
$$880$$ −4.00000 −0.134840
$$881$$ −33.0000 −1.11180 −0.555899 0.831250i $$-0.687626\pi$$
−0.555899 + 0.831250i $$0.687626\pi$$
$$882$$ −2.00000 −0.0673435
$$883$$ 4.00000 0.134611 0.0673054 0.997732i $$-0.478560\pi$$
0.0673054 + 0.997732i $$0.478560\pi$$
$$884$$ 84.0000 2.82523
$$885$$ −5.00000 −0.168073
$$886$$ 72.0000 2.41889
$$887$$ 43.0000 1.44380 0.721899 0.691998i $$-0.243269\pi$$
0.721899 + 0.691998i $$0.243269\pi$$
$$888$$ 0 0
$$889$$ −7.00000 −0.234772
$$890$$ −30.0000 −1.00560
$$891$$ 1.00000 0.0335013
$$892$$ 18.0000 0.602685
$$893$$ −40.0000 −1.33855
$$894$$ 20.0000 0.668900
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 6.00000 0.200334
$$898$$ 0 0
$$899$$ 40.0000 1.33407
$$900$$ 2.00000 0.0666667
$$901$$ 77.0000 2.56524
$$902$$ −24.0000 −0.799113
$$903$$ −11.0000 −0.366057
$$904$$ 0 0
$$905$$ 22.0000 0.731305
$$906$$ −4.00000 −0.132891
$$907$$ −2.00000 −0.0664089 −0.0332045 0.999449i $$-0.510571\pi$$
−0.0332045 + 0.999449i $$0.510571\pi$$
$$908$$ 46.0000 1.52656
$$909$$ 2.00000 0.0663358
$$910$$ 12.0000 0.397796
$$911$$ 12.0000 0.397578 0.198789 0.980042i $$-0.436299\pi$$
0.198789 + 0.980042i $$0.436299\pi$$
$$912$$ 20.0000 0.662266
$$913$$ −1.00000 −0.0330952
$$914$$ −66.0000 −2.18309
$$915$$ 7.00000 0.231413
$$916$$ −40.0000 −1.32164
$$917$$ 2.00000 0.0660458
$$918$$ 14.0000 0.462069
$$919$$ 30.0000 0.989609 0.494804 0.869004i $$-0.335240\pi$$
0.494804 + 0.869004i $$0.335240\pi$$
$$920$$ 0 0
$$921$$ 18.0000 0.593120
$$922$$ 76.0000 2.50293
$$923$$ −72.0000 −2.36991
$$924$$ 2.00000 0.0657952
$$925$$ −2.00000 −0.0657596
$$926$$ 12.0000 0.394344
$$927$$ −1.00000 −0.0328443
$$928$$ −40.0000 −1.31306
$$929$$ −50.0000 −1.64045 −0.820223 0.572043i $$-0.806151\pi$$
−0.820223 + 0.572043i $$0.806151\pi$$
$$930$$ 16.0000 0.524661
$$931$$ −5.00000 −0.163868
$$932$$ −12.0000 −0.393073
$$933$$ 12.0000 0.392862
$$934$$ −56.0000 −1.83238
$$935$$ −7.00000 −0.228924
$$936$$ 0 0
$$937$$ 28.0000 0.914720 0.457360 0.889282i $$-0.348795\pi$$
0.457360 + 0.889282i $$0.348795\pi$$
$$938$$ 4.00000 0.130605
$$939$$ −31.0000 −1.01165
$$940$$ 16.0000 0.521862
$$941$$ 42.0000 1.36916 0.684580 0.728937i $$-0.259985\pi$$
0.684580 + 0.728937i $$0.259985\pi$$
$$942$$ 34.0000 1.10778
$$943$$ −12.0000 −0.390774
$$944$$ 20.0000 0.650945
$$945$$ 1.00000 0.0325300
$$946$$ 22.0000 0.715282
$$947$$ 23.0000 0.747400 0.373700 0.927550i $$-0.378089\pi$$
0.373700 + 0.927550i $$0.378089\pi$$
$$948$$ −20.0000 −0.649570
$$949$$ −24.0000 −0.779073
$$950$$ 10.0000 0.324443
$$951$$ −22.0000 −0.713399
$$952$$ 0 0
$$953$$ 34.0000 1.10137 0.550684 0.834714i $$-0.314367\pi$$
0.550684 + 0.834714i $$0.314367\pi$$
$$954$$ 22.0000 0.712276
$$955$$ −8.00000 −0.258874
$$956$$ 10.0000 0.323423
$$957$$ −5.00000 −0.161627
$$958$$ 20.0000 0.646171
$$959$$ −22.0000 −0.710417
$$960$$ −8.00000 −0.258199
$$961$$ 33.0000 1.06452
$$962$$ −24.0000 −0.773791
$$963$$ −12.0000 −0.386695
$$964$$ −36.0000 −1.15948
$$965$$ −6.00000 −0.193147
$$966$$ 2.00000 0.0643489
$$967$$ −47.0000 −1.51142 −0.755709 0.654907i $$-0.772708\pi$$
−0.755709 + 0.654907i $$0.772708\pi$$
$$968$$ 0 0
$$969$$ 35.0000 1.12436
$$970$$ −6.00000 −0.192648
$$971$$ −13.0000 −0.417190 −0.208595 0.978002i $$-0.566889\pi$$
−0.208595 + 0.978002i $$0.566889\pi$$
$$972$$ 2.00000 0.0641500
$$973$$ 0 0
$$974$$ 44.0000 1.40985
$$975$$ −6.00000 −0.192154
$$976$$ −28.0000 −0.896258
$$977$$ −7.00000 −0.223950 −0.111975 0.993711i $$-0.535718\pi$$
−0.111975 + 0.993711i $$0.535718\pi$$
$$978$$ 12.0000 0.383718
$$979$$ 15.0000 0.479402
$$980$$ 2.00000 0.0638877
$$981$$ 0 0
$$982$$ −54.0000 −1.72321
$$983$$ 14.0000 0.446531 0.223265 0.974758i $$-0.428328\pi$$
0.223265 + 0.974758i $$0.428328\pi$$
$$984$$ 0 0
$$985$$ −12.0000 −0.382352
$$986$$ −70.0000 −2.22925
$$987$$ 8.00000 0.254643
$$988$$ 60.0000 1.90885
$$989$$ 11.0000 0.349780
$$990$$ −2.00000 −0.0635642
$$991$$ −33.0000 −1.04828 −0.524140 0.851632i $$-0.675613\pi$$
−0.524140 + 0.851632i $$0.675613\pi$$
$$992$$ −64.0000 −2.03200
$$993$$ −3.00000 −0.0952021
$$994$$ −24.0000 −0.761234
$$995$$ 20.0000 0.634043
$$996$$ −2.00000 −0.0633724
$$997$$ −22.0000 −0.696747 −0.348373 0.937356i $$-0.613266\pi$$
−0.348373 + 0.937356i $$0.613266\pi$$
$$998$$ 50.0000 1.58272
$$999$$ −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 1155.2.a.c.1.1 1
3.2 odd 2 3465.2.a.r.1.1 1
5.4 even 2 5775.2.a.y.1.1 1
7.6 odd 2 8085.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.c.1.1 1 1.1 even 1 trivial
3465.2.a.r.1.1 1 3.2 odd 2
5775.2.a.y.1.1 1 5.4 even 2
8085.2.a.b.1.1 1 7.6 odd 2