Properties

 Label 114.2.a.a.1.1 Level $114$ Weight $2$ Character 114.1 Self dual yes Analytic conductor $0.910$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [114,2,Mod(1,114)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(114, 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("114.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$114 = 2 \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 114.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: $$0.910294583043$$ 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 $$\chi$$ $$=$$ 114.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} -1.00000 q^{12} -4.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} -4.00000 q^{21} -4.00000 q^{22} -2.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} -1.00000 q^{27} +4.00000 q^{28} -6.00000 q^{29} +6.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -8.00000 q^{37} -1.00000 q^{38} +10.0000 q^{41} +4.00000 q^{42} -12.0000 q^{43} +4.00000 q^{44} +2.00000 q^{46} +10.0000 q^{47} -1.00000 q^{48} +9.00000 q^{49} +5.00000 q^{50} +2.00000 q^{51} +2.00000 q^{53} +1.00000 q^{54} -4.00000 q^{56} -1.00000 q^{57} +6.00000 q^{58} +4.00000 q^{59} -10.0000 q^{61} -6.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} -2.00000 q^{68} +2.00000 q^{69} -16.0000 q^{71} -1.00000 q^{72} -2.00000 q^{73} +8.00000 q^{74} +5.00000 q^{75} +1.00000 q^{76} +16.0000 q^{77} +10.0000 q^{79} +1.00000 q^{81} -10.0000 q^{82} -16.0000 q^{83} -4.00000 q^{84} +12.0000 q^{86} +6.00000 q^{87} -4.00000 q^{88} -2.00000 q^{89} -2.00000 q^{92} -6.00000 q^{93} -10.0000 q^{94} +1.00000 q^{96} -10.0000 q^{97} -9.00000 q^{98} +4.00000 q^{99} +O(q^{100})$$

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
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 4.00000 1.51186 0.755929 0.654654i $$-0.227186\pi$$
0.755929 + 0.654654i $$0.227186\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ −4.00000 −1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ −4.00000 −0.852803
$$23$$ −2.00000 −0.417029 −0.208514 0.978019i $$-0.566863\pi$$
−0.208514 + 0.978019i $$0.566863\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −5.00000 −1.00000
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 4.00000 0.755929
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −4.00000 −0.696311
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −8.00000 −1.31519 −0.657596 0.753371i $$-0.728427\pi$$
−0.657596 + 0.753371i $$0.728427\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 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
$$45$$ 0 0
$$46$$ 2.00000 0.294884
$$47$$ 10.0000 1.45865 0.729325 0.684167i $$-0.239834\pi$$
0.729325 + 0.684167i $$0.239834\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 9.00000 1.28571
$$50$$ 5.00000 0.707107
$$51$$ 2.00000 0.280056
$$52$$ 0 0
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −4.00000 −0.534522
$$57$$ −1.00000 −0.132453
$$58$$ 6.00000 0.787839
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ −6.00000 −0.762001
$$63$$ 4.00000 0.503953
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 4.00000 0.492366
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ 2.00000 0.240772
$$70$$ 0 0
$$71$$ −16.0000 −1.89885 −0.949425 0.313993i $$-0.898333\pi$$
−0.949425 + 0.313993i $$0.898333\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 5.00000 0.577350
$$76$$ 1.00000 0.114708
$$77$$ 16.0000 1.82337
$$78$$ 0 0
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −10.0000 −1.10432
$$83$$ −16.0000 −1.75623 −0.878114 0.478451i $$-0.841198\pi$$
−0.878114 + 0.478451i $$0.841198\pi$$
$$84$$ −4.00000 −0.436436
$$85$$ 0 0
$$86$$ 12.0000 1.29399
$$87$$ 6.00000 0.643268
$$88$$ −4.00000 −0.426401
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −2.00000 −0.208514
$$93$$ −6.00000 −0.622171
$$94$$ −10.0000 −1.03142
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ −9.00000 −0.909137
$$99$$ 4.00000 0.402015
$$100$$ −5.00000 −0.500000
$$101$$ 8.00000 0.796030 0.398015 0.917379i $$-0.369699\pi$$
0.398015 + 0.917379i $$0.369699\pi$$
$$102$$ −2.00000 −0.198030
$$103$$ −6.00000 −0.591198 −0.295599 0.955312i $$-0.595519\pi$$
−0.295599 + 0.955312i $$0.595519\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −2.00000 −0.194257
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 0 0
$$111$$ 8.00000 0.759326
$$112$$ 4.00000 0.377964
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 1.00000 0.0936586
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ 0 0
$$118$$ −4.00000 −0.368230
$$119$$ −8.00000 −0.733359
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 10.0000 0.905357
$$123$$ −10.0000 −0.901670
$$124$$ 6.00000 0.538816
$$125$$ 0 0
$$126$$ −4.00000 −0.356348
$$127$$ 22.0000 1.95218 0.976092 0.217357i $$-0.0697436\pi$$
0.976092 + 0.217357i $$0.0697436\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 12.0000 1.05654
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ −4.00000 −0.348155
$$133$$ 4.00000 0.346844
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ −2.00000 −0.170251
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ −10.0000 −0.842152
$$142$$ 16.0000 1.34269
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 2.00000 0.165521
$$147$$ −9.00000 −0.742307
$$148$$ −8.00000 −0.657596
$$149$$ 20.0000 1.63846 0.819232 0.573462i $$-0.194400\pi$$
0.819232 + 0.573462i $$0.194400\pi$$
$$150$$ −5.00000 −0.408248
$$151$$ 10.0000 0.813788 0.406894 0.913475i $$-0.366612\pi$$
0.406894 + 0.913475i $$0.366612\pi$$
$$152$$ −1.00000 −0.0811107
$$153$$ −2.00000 −0.161690
$$154$$ −16.0000 −1.28932
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ −10.0000 −0.795557
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ −1.00000 −0.0785674
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 4.00000 0.308607
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 1.00000 0.0764719
$$172$$ −12.0000 −0.914991
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ −20.0000 −1.51186
$$176$$ 4.00000 0.301511
$$177$$ −4.00000 −0.300658
$$178$$ 2.00000 0.149906
$$179$$ −20.0000 −1.49487 −0.747435 0.664335i $$-0.768715\pi$$
−0.747435 + 0.664335i $$0.768715\pi$$
$$180$$ 0 0
$$181$$ 12.0000 0.891953 0.445976 0.895045i $$-0.352856\pi$$
0.445976 + 0.895045i $$0.352856\pi$$
$$182$$ 0 0
$$183$$ 10.0000 0.739221
$$184$$ 2.00000 0.147442
$$185$$ 0 0
$$186$$ 6.00000 0.439941
$$187$$ −8.00000 −0.585018
$$188$$ 10.0000 0.729325
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ −2.00000 −0.144715 −0.0723575 0.997379i $$-0.523052\pi$$
−0.0723575 + 0.997379i $$0.523052\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ −20.0000 −1.42494 −0.712470 0.701702i $$-0.752424\pi$$
−0.712470 + 0.701702i $$0.752424\pi$$
$$198$$ −4.00000 −0.284268
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 5.00000 0.353553
$$201$$ 0 0
$$202$$ −8.00000 −0.562878
$$203$$ −24.0000 −1.68447
$$204$$ 2.00000 0.140028
$$205$$ 0 0
$$206$$ 6.00000 0.418040
$$207$$ −2.00000 −0.139010
$$208$$ 0 0
$$209$$ 4.00000 0.276686
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 2.00000 0.137361
$$213$$ 16.0000 1.09630
$$214$$ 4.00000 0.273434
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ 24.0000 1.62923
$$218$$ 4.00000 0.270914
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 0 0
$$222$$ −8.00000 −0.536925
$$223$$ 6.00000 0.401790 0.200895 0.979613i $$-0.435615\pi$$
0.200895 + 0.979613i $$0.435615\pi$$
$$224$$ −4.00000 −0.267261
$$225$$ −5.00000 −0.333333
$$226$$ 14.0000 0.931266
$$227$$ 20.0000 1.32745 0.663723 0.747978i $$-0.268975\pi$$
0.663723 + 0.747978i $$0.268975\pi$$
$$228$$ −1.00000 −0.0662266
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 0 0
$$231$$ −16.0000 −1.05272
$$232$$ 6.00000 0.393919
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 4.00000 0.260378
$$237$$ −10.0000 −0.649570
$$238$$ 8.00000 0.518563
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ −5.00000 −0.321412
$$243$$ −1.00000 −0.0641500
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ 10.0000 0.637577
$$247$$ 0 0
$$248$$ −6.00000 −0.381000
$$249$$ 16.0000 1.01396
$$250$$ 0 0
$$251$$ −24.0000 −1.51487 −0.757433 0.652913i $$-0.773547\pi$$
−0.757433 + 0.652913i $$0.773547\pi$$
$$252$$ 4.00000 0.251976
$$253$$ −8.00000 −0.502956
$$254$$ −22.0000 −1.38040
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ −12.0000 −0.747087
$$259$$ −32.0000 −1.98838
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ −6.00000 −0.369976 −0.184988 0.982741i $$-0.559225\pi$$
−0.184988 + 0.982741i $$0.559225\pi$$
$$264$$ 4.00000 0.246183
$$265$$ 0 0
$$266$$ −4.00000 −0.245256
$$267$$ 2.00000 0.122398
$$268$$ 0 0
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ 0 0
$$271$$ −4.00000 −0.242983 −0.121491 0.992592i $$-0.538768\pi$$
−0.121491 + 0.992592i $$0.538768\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ −20.0000 −1.20605
$$276$$ 2.00000 0.120386
$$277$$ −2.00000 −0.120168 −0.0600842 0.998193i $$-0.519137\pi$$
−0.0600842 + 0.998193i $$0.519137\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 6.00000 0.359211
$$280$$ 0 0
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 10.0000 0.595491
$$283$$ 28.0000 1.66443 0.832214 0.554455i $$-0.187073\pi$$
0.832214 + 0.554455i $$0.187073\pi$$
$$284$$ −16.0000 −0.949425
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 40.0000 2.36113
$$288$$ −1.00000 −0.0589256
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 10.0000 0.586210
$$292$$ −2.00000 −0.117041
$$293$$ 14.0000 0.817889 0.408944 0.912559i $$-0.365897\pi$$
0.408944 + 0.912559i $$0.365897\pi$$
$$294$$ 9.00000 0.524891
$$295$$ 0 0
$$296$$ 8.00000 0.464991
$$297$$ −4.00000 −0.232104
$$298$$ −20.0000 −1.15857
$$299$$ 0 0
$$300$$ 5.00000 0.288675
$$301$$ −48.0000 −2.76667
$$302$$ −10.0000 −0.575435
$$303$$ −8.00000 −0.459588
$$304$$ 1.00000 0.0573539
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 16.0000 0.911685
$$309$$ 6.00000 0.341328
$$310$$ 0 0
$$311$$ 34.0000 1.92796 0.963982 0.265969i $$-0.0856919\pi$$
0.963982 + 0.265969i $$0.0856919\pi$$
$$312$$ 0 0
$$313$$ 6.00000 0.339140 0.169570 0.985518i $$-0.445762\pi$$
0.169570 + 0.985518i $$0.445762\pi$$
$$314$$ −18.0000 −1.01580
$$315$$ 0 0
$$316$$ 10.0000 0.562544
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ 2.00000 0.112154
$$319$$ −24.0000 −1.34374
$$320$$ 0 0
$$321$$ 4.00000 0.223258
$$322$$ 8.00000 0.445823
$$323$$ −2.00000 −0.111283
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −20.0000 −1.10770
$$327$$ 4.00000 0.221201
$$328$$ −10.0000 −0.552158
$$329$$ 40.0000 2.20527
$$330$$ 0 0
$$331$$ 24.0000 1.31916 0.659580 0.751635i $$-0.270734\pi$$
0.659580 + 0.751635i $$0.270734\pi$$
$$332$$ −16.0000 −0.878114
$$333$$ −8.00000 −0.438397
$$334$$ −12.0000 −0.656611
$$335$$ 0 0
$$336$$ −4.00000 −0.218218
$$337$$ 26.0000 1.41631 0.708155 0.706057i $$-0.249528\pi$$
0.708155 + 0.706057i $$0.249528\pi$$
$$338$$ 13.0000 0.707107
$$339$$ 14.0000 0.760376
$$340$$ 0 0
$$341$$ 24.0000 1.29967
$$342$$ −1.00000 −0.0540738
$$343$$ 8.00000 0.431959
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ −6.00000 −0.322562
$$347$$ 28.0000 1.50312 0.751559 0.659665i $$-0.229302\pi$$
0.751559 + 0.659665i $$0.229302\pi$$
$$348$$ 6.00000 0.321634
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 20.0000 1.06904
$$351$$ 0 0
$$352$$ −4.00000 −0.213201
$$353$$ −30.0000 −1.59674 −0.798369 0.602168i $$-0.794304\pi$$
−0.798369 + 0.602168i $$0.794304\pi$$
$$354$$ 4.00000 0.212598
$$355$$ 0 0
$$356$$ −2.00000 −0.106000
$$357$$ 8.00000 0.423405
$$358$$ 20.0000 1.05703
$$359$$ 6.00000 0.316668 0.158334 0.987386i $$-0.449388\pi$$
0.158334 + 0.987386i $$0.449388\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ −12.0000 −0.630706
$$363$$ −5.00000 −0.262432
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ −2.00000 −0.104257
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ 8.00000 0.415339
$$372$$ −6.00000 −0.311086
$$373$$ 8.00000 0.414224 0.207112 0.978317i $$-0.433593\pi$$
0.207112 + 0.978317i $$0.433593\pi$$
$$374$$ 8.00000 0.413670
$$375$$ 0 0
$$376$$ −10.0000 −0.515711
$$377$$ 0 0
$$378$$ 4.00000 0.205738
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ 0 0
$$381$$ −22.0000 −1.12709
$$382$$ 2.00000 0.102329
$$383$$ 8.00000 0.408781 0.204390 0.978889i $$-0.434479\pi$$
0.204390 + 0.978889i $$0.434479\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ −12.0000 −0.609994
$$388$$ −10.0000 −0.507673
$$389$$ 8.00000 0.405616 0.202808 0.979219i $$-0.434993\pi$$
0.202808 + 0.979219i $$0.434993\pi$$
$$390$$ 0 0
$$391$$ 4.00000 0.202289
$$392$$ −9.00000 −0.454569
$$393$$ 0 0
$$394$$ 20.0000 1.00759
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ −30.0000 −1.50566 −0.752828 0.658217i $$-0.771311\pi$$
−0.752828 + 0.658217i $$0.771311\pi$$
$$398$$ 4.00000 0.200502
$$399$$ −4.00000 −0.200250
$$400$$ −5.00000 −0.250000
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 8.00000 0.398015
$$405$$ 0 0
$$406$$ 24.0000 1.19110
$$407$$ −32.0000 −1.58618
$$408$$ −2.00000 −0.0990148
$$409$$ −18.0000 −0.890043 −0.445021 0.895520i $$-0.646804\pi$$
−0.445021 + 0.895520i $$0.646804\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ −6.00000 −0.295599
$$413$$ 16.0000 0.787309
$$414$$ 2.00000 0.0982946
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 4.00000 0.195881
$$418$$ −4.00000 −0.195646
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −8.00000 −0.389896 −0.194948 0.980814i $$-0.562454\pi$$
−0.194948 + 0.980814i $$0.562454\pi$$
$$422$$ 20.0000 0.973585
$$423$$ 10.0000 0.486217
$$424$$ −2.00000 −0.0971286
$$425$$ 10.0000 0.485071
$$426$$ −16.0000 −0.775203
$$427$$ −40.0000 −1.93574
$$428$$ −4.00000 −0.193347
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −4.00000 −0.192673 −0.0963366 0.995349i $$-0.530713\pi$$
−0.0963366 + 0.995349i $$0.530713\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −18.0000 −0.865025 −0.432512 0.901628i $$-0.642373\pi$$
−0.432512 + 0.901628i $$0.642373\pi$$
$$434$$ −24.0000 −1.15204
$$435$$ 0 0
$$436$$ −4.00000 −0.191565
$$437$$ −2.00000 −0.0956730
$$438$$ −2.00000 −0.0955637
$$439$$ 22.0000 1.05000 0.525001 0.851101i $$-0.324065\pi$$
0.525001 + 0.851101i $$0.324065\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 0 0
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 8.00000 0.379663
$$445$$ 0 0
$$446$$ −6.00000 −0.284108
$$447$$ −20.0000 −0.945968
$$448$$ 4.00000 0.188982
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 5.00000 0.235702
$$451$$ 40.0000 1.88353
$$452$$ −14.0000 −0.658505
$$453$$ −10.0000 −0.469841
$$454$$ −20.0000 −0.938647
$$455$$ 0 0
$$456$$ 1.00000 0.0468293
$$457$$ 2.00000 0.0935561 0.0467780 0.998905i $$-0.485105\pi$$
0.0467780 + 0.998905i $$0.485105\pi$$
$$458$$ 2.00000 0.0934539
$$459$$ 2.00000 0.0933520
$$460$$ 0 0
$$461$$ −12.0000 −0.558896 −0.279448 0.960161i $$-0.590151\pi$$
−0.279448 + 0.960161i $$0.590151\pi$$
$$462$$ 16.0000 0.744387
$$463$$ 36.0000 1.67306 0.836531 0.547920i $$-0.184580\pi$$
0.836531 + 0.547920i $$0.184580\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ 8.00000 0.370196 0.185098 0.982720i $$-0.440740\pi$$
0.185098 + 0.982720i $$0.440740\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −18.0000 −0.829396
$$472$$ −4.00000 −0.184115
$$473$$ −48.0000 −2.20704
$$474$$ 10.0000 0.459315
$$475$$ −5.00000 −0.229416
$$476$$ −8.00000 −0.366679
$$477$$ 2.00000 0.0915737
$$478$$ 6.00000 0.274434
$$479$$ 6.00000 0.274147 0.137073 0.990561i $$-0.456230\pi$$
0.137073 + 0.990561i $$0.456230\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 2.00000 0.0910975
$$483$$ 8.00000 0.364013
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 2.00000 0.0906287 0.0453143 0.998973i $$-0.485571\pi$$
0.0453143 + 0.998973i $$0.485571\pi$$
$$488$$ 10.0000 0.452679
$$489$$ −20.0000 −0.904431
$$490$$ 0 0
$$491$$ 24.0000 1.08310 0.541552 0.840667i $$-0.317837\pi$$
0.541552 + 0.840667i $$0.317837\pi$$
$$492$$ −10.0000 −0.450835
$$493$$ 12.0000 0.540453
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 6.00000 0.269408
$$497$$ −64.0000 −2.87079
$$498$$ −16.0000 −0.716977
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ −12.0000 −0.536120
$$502$$ 24.0000 1.07117
$$503$$ 34.0000 1.51599 0.757993 0.652263i $$-0.226180\pi$$
0.757993 + 0.652263i $$0.226180\pi$$
$$504$$ −4.00000 −0.178174
$$505$$ 0 0
$$506$$ 8.00000 0.355643
$$507$$ 13.0000 0.577350
$$508$$ 22.0000 0.976092
$$509$$ −10.0000 −0.443242 −0.221621 0.975133i $$-0.571135\pi$$
−0.221621 + 0.975133i $$0.571135\pi$$
$$510$$ 0 0
$$511$$ −8.00000 −0.353899
$$512$$ −1.00000 −0.0441942
$$513$$ −1.00000 −0.0441511
$$514$$ 18.0000 0.793946
$$515$$ 0 0
$$516$$ 12.0000 0.528271
$$517$$ 40.0000 1.75920
$$518$$ 32.0000 1.40600
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −26.0000 −1.13908 −0.569540 0.821963i $$-0.692879\pi$$
−0.569540 + 0.821963i $$0.692879\pi$$
$$522$$ 6.00000 0.262613
$$523$$ 16.0000 0.699631 0.349816 0.936819i $$-0.386244\pi$$
0.349816 + 0.936819i $$0.386244\pi$$
$$524$$ 0 0
$$525$$ 20.0000 0.872872
$$526$$ 6.00000 0.261612
$$527$$ −12.0000 −0.522728
$$528$$ −4.00000 −0.174078
$$529$$ −19.0000 −0.826087
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ 4.00000 0.173422
$$533$$ 0 0
$$534$$ −2.00000 −0.0865485
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 20.0000 0.863064
$$538$$ −14.0000 −0.603583
$$539$$ 36.0000 1.55063
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 4.00000 0.171815
$$543$$ −12.0000 −0.514969
$$544$$ 2.00000 0.0857493
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −16.0000 −0.684111 −0.342055 0.939680i $$-0.611123\pi$$
−0.342055 + 0.939680i $$0.611123\pi$$
$$548$$ 6.00000 0.256307
$$549$$ −10.0000 −0.426790
$$550$$ 20.0000 0.852803
$$551$$ −6.00000 −0.255609
$$552$$ −2.00000 −0.0851257
$$553$$ 40.0000 1.70097
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 4.00000 0.169485 0.0847427 0.996403i $$-0.472993\pi$$
0.0847427 + 0.996403i $$0.472993\pi$$
$$558$$ −6.00000 −0.254000
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 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 $$0.581374\pi$$
$$564$$ −10.0000 −0.421076
$$565$$ 0 0
$$566$$ −28.0000 −1.17693
$$567$$ 4.00000 0.167984
$$568$$ 16.0000 0.671345
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 0 0
$$573$$ 2.00000 0.0835512
$$574$$ −40.0000 −1.66957
$$575$$ 10.0000 0.417029
$$576$$ 1.00000 0.0416667
$$577$$ −14.0000 −0.582828 −0.291414 0.956597i $$-0.594126\pi$$
−0.291414 + 0.956597i $$0.594126\pi$$
$$578$$ 13.0000 0.540729
$$579$$ −14.0000 −0.581820
$$580$$ 0 0
$$581$$ −64.0000 −2.65517
$$582$$ −10.0000 −0.414513
$$583$$ 8.00000 0.331326
$$584$$ 2.00000 0.0827606
$$585$$ 0 0
$$586$$ −14.0000 −0.578335
$$587$$ 28.0000 1.15568 0.577842 0.816149i $$-0.303895\pi$$
0.577842 + 0.816149i $$0.303895\pi$$
$$588$$ −9.00000 −0.371154
$$589$$ 6.00000 0.247226
$$590$$ 0 0
$$591$$ 20.0000 0.822690
$$592$$ −8.00000 −0.328798
$$593$$ 10.0000 0.410651 0.205325 0.978694i $$-0.434175\pi$$
0.205325 + 0.978694i $$0.434175\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ 20.0000 0.819232
$$597$$ 4.00000 0.163709
$$598$$ 0 0
$$599$$ −4.00000 −0.163436 −0.0817178 0.996656i $$-0.526041\pi$$
−0.0817178 + 0.996656i $$0.526041\pi$$
$$600$$ −5.00000 −0.204124
$$601$$ 42.0000 1.71322 0.856608 0.515968i $$-0.172568\pi$$
0.856608 + 0.515968i $$0.172568\pi$$
$$602$$ 48.0000 1.95633
$$603$$ 0 0
$$604$$ 10.0000 0.406894
$$605$$ 0 0
$$606$$ 8.00000 0.324978
$$607$$ −22.0000 −0.892952 −0.446476 0.894795i $$-0.647321\pi$$
−0.446476 + 0.894795i $$0.647321\pi$$
$$608$$ −1.00000 −0.0405554
$$609$$ 24.0000 0.972529
$$610$$ 0 0
$$611$$ 0 0
$$612$$ −2.00000 −0.0808452
$$613$$ 2.00000 0.0807792 0.0403896 0.999184i $$-0.487140\pi$$
0.0403896 + 0.999184i $$0.487140\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ −16.0000 −0.644658
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ −6.00000 −0.241355
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 2.00000 0.0802572
$$622$$ −34.0000 −1.36328
$$623$$ −8.00000 −0.320513
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ −6.00000 −0.239808
$$627$$ −4.00000 −0.159745
$$628$$ 18.0000 0.718278
$$629$$ 16.0000 0.637962
$$630$$ 0 0
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ −10.0000 −0.397779
$$633$$ 20.0000 0.794929
$$634$$ −6.00000 −0.238290
$$635$$ 0 0
$$636$$ −2.00000 −0.0793052
$$637$$ 0 0
$$638$$ 24.0000 0.950169
$$639$$ −16.0000 −0.632950
$$640$$ 0 0
$$641$$ 46.0000 1.81689 0.908445 0.418004i $$-0.137270\pi$$
0.908445 + 0.418004i $$0.137270\pi$$
$$642$$ −4.00000 −0.157867
$$643$$ 28.0000 1.10421 0.552106 0.833774i $$-0.313824\pi$$
0.552106 + 0.833774i $$0.313824\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ 0 0
$$646$$ 2.00000 0.0786889
$$647$$ −18.0000 −0.707653 −0.353827 0.935311i $$-0.615120\pi$$
−0.353827 + 0.935311i $$0.615120\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 16.0000 0.628055
$$650$$ 0 0
$$651$$ −24.0000 −0.940634
$$652$$ 20.0000 0.783260
$$653$$ −36.0000 −1.40879 −0.704394 0.709809i $$-0.748781\pi$$
−0.704394 + 0.709809i $$0.748781\pi$$
$$654$$ −4.00000 −0.156412
$$655$$ 0 0
$$656$$ 10.0000 0.390434
$$657$$ −2.00000 −0.0780274
$$658$$ −40.0000 −1.55936
$$659$$ −20.0000 −0.779089 −0.389545 0.921008i $$-0.627368\pi$$
−0.389545 + 0.921008i $$0.627368\pi$$
$$660$$ 0 0
$$661$$ −8.00000 −0.311164 −0.155582 0.987823i $$-0.549725\pi$$
−0.155582 + 0.987823i $$0.549725\pi$$
$$662$$ −24.0000 −0.932786
$$663$$ 0 0
$$664$$ 16.0000 0.620920
$$665$$ 0 0
$$666$$ 8.00000 0.309994
$$667$$ 12.0000 0.464642
$$668$$ 12.0000 0.464294
$$669$$ −6.00000 −0.231973
$$670$$ 0 0
$$671$$ −40.0000 −1.54418
$$672$$ 4.00000 0.154303
$$673$$ 14.0000 0.539660 0.269830 0.962908i $$-0.413032\pi$$
0.269830 + 0.962908i $$0.413032\pi$$
$$674$$ −26.0000 −1.00148
$$675$$ 5.00000 0.192450
$$676$$ −13.0000 −0.500000
$$677$$ 42.0000 1.61419 0.807096 0.590421i $$-0.201038\pi$$
0.807096 + 0.590421i $$0.201038\pi$$
$$678$$ −14.0000 −0.537667
$$679$$ −40.0000 −1.53506
$$680$$ 0 0
$$681$$ −20.0000 −0.766402
$$682$$ −24.0000 −0.919007
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ 1.00000 0.0382360
$$685$$ 0 0
$$686$$ −8.00000 −0.305441
$$687$$ 2.00000 0.0763048
$$688$$ −12.0000 −0.457496
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −20.0000 −0.760836 −0.380418 0.924815i $$-0.624220\pi$$
−0.380418 + 0.924815i $$0.624220\pi$$
$$692$$ 6.00000 0.228086
$$693$$ 16.0000 0.607790
$$694$$ −28.0000 −1.06287
$$695$$ 0 0
$$696$$ −6.00000 −0.227429
$$697$$ −20.0000 −0.757554
$$698$$ −10.0000 −0.378506
$$699$$ 6.00000 0.226941
$$700$$ −20.0000 −0.755929
$$701$$ −8.00000 −0.302156 −0.151078 0.988522i $$-0.548274\pi$$
−0.151078 + 0.988522i $$0.548274\pi$$
$$702$$ 0 0
$$703$$ −8.00000 −0.301726
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ 30.0000 1.12906
$$707$$ 32.0000 1.20348
$$708$$ −4.00000 −0.150329
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 0 0
$$711$$ 10.0000 0.375029
$$712$$ 2.00000 0.0749532
$$713$$ −12.0000 −0.449404
$$714$$ −8.00000 −0.299392
$$715$$ 0 0
$$716$$ −20.0000 −0.747435
$$717$$ 6.00000 0.224074
$$718$$ −6.00000 −0.223918
$$719$$ 34.0000 1.26799 0.633993 0.773339i $$-0.281415\pi$$
0.633993 + 0.773339i $$0.281415\pi$$
$$720$$ 0 0
$$721$$ −24.0000 −0.893807
$$722$$ −1.00000 −0.0372161
$$723$$ 2.00000 0.0743808
$$724$$ 12.0000 0.445976
$$725$$ 30.0000 1.11417
$$726$$ 5.00000 0.185567
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 10.0000 0.369611
$$733$$ −38.0000 −1.40356 −0.701781 0.712393i $$-0.747612\pi$$
−0.701781 + 0.712393i $$0.747612\pi$$
$$734$$ 28.0000 1.03350
$$735$$ 0 0
$$736$$ 2.00000 0.0737210
$$737$$ 0 0
$$738$$ −10.0000 −0.368105
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −8.00000 −0.293689
$$743$$ −36.0000 −1.32071 −0.660356 0.750953i $$-0.729595\pi$$
−0.660356 + 0.750953i $$0.729595\pi$$
$$744$$ 6.00000 0.219971
$$745$$ 0 0
$$746$$ −8.00000 −0.292901
$$747$$ −16.0000 −0.585409
$$748$$ −8.00000 −0.292509
$$749$$ −16.0000 −0.584627
$$750$$ 0 0
$$751$$ 10.0000 0.364905 0.182453 0.983215i $$-0.441596\pi$$
0.182453 + 0.983215i $$0.441596\pi$$
$$752$$ 10.0000 0.364662
$$753$$ 24.0000 0.874609
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −4.00000 −0.145479
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ −8.00000 −0.290573
$$759$$ 8.00000 0.290382
$$760$$ 0 0
$$761$$ 14.0000 0.507500 0.253750 0.967270i $$-0.418336\pi$$
0.253750 + 0.967270i $$0.418336\pi$$
$$762$$ 22.0000 0.796976
$$763$$ −16.0000 −0.579239
$$764$$ −2.00000 −0.0723575
$$765$$ 0 0
$$766$$ −8.00000 −0.289052
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ 30.0000 1.08183 0.540914 0.841078i $$-0.318079\pi$$
0.540914 + 0.841078i $$0.318079\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ 14.0000 0.503871
$$773$$ 6.00000 0.215805 0.107903 0.994161i $$-0.465587\pi$$
0.107903 + 0.994161i $$0.465587\pi$$
$$774$$ 12.0000 0.431331
$$775$$ −30.0000 −1.07763
$$776$$ 10.0000 0.358979
$$777$$ 32.0000 1.14799
$$778$$ −8.00000 −0.286814
$$779$$ 10.0000 0.358287
$$780$$ 0 0
$$781$$ −64.0000 −2.29010
$$782$$ −4.00000 −0.143040
$$783$$ 6.00000 0.214423
$$784$$ 9.00000 0.321429
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 4.00000 0.142585 0.0712923 0.997455i $$-0.477288\pi$$
0.0712923 + 0.997455i $$0.477288\pi$$
$$788$$ −20.0000 −0.712470
$$789$$ 6.00000 0.213606
$$790$$ 0 0
$$791$$ −56.0000 −1.99113
$$792$$ −4.00000 −0.142134
$$793$$ 0 0
$$794$$ 30.0000 1.06466
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ −34.0000 −1.20434 −0.602171 0.798367i $$-0.705697\pi$$
−0.602171 + 0.798367i $$0.705697\pi$$
$$798$$ 4.00000 0.141598
$$799$$ −20.0000 −0.707549
$$800$$ 5.00000 0.176777
$$801$$ −2.00000 −0.0706665
$$802$$ 30.0000 1.05934
$$803$$ −8.00000 −0.282314
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −14.0000 −0.492823
$$808$$ −8.00000 −0.281439
$$809$$ 54.0000 1.89854 0.949269 0.314464i $$-0.101825\pi$$
0.949269 + 0.314464i $$0.101825\pi$$
$$810$$ 0 0
$$811$$ 16.0000 0.561836 0.280918 0.959732i $$-0.409361\pi$$
0.280918 + 0.959732i $$0.409361\pi$$
$$812$$ −24.0000 −0.842235
$$813$$ 4.00000 0.140286
$$814$$ 32.0000 1.12160
$$815$$ 0 0
$$816$$ 2.00000 0.0700140
$$817$$ −12.0000 −0.419827
$$818$$ 18.0000 0.629355
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$822$$ 6.00000 0.209274
$$823$$ 16.0000 0.557725 0.278862 0.960331i $$-0.410043\pi$$
0.278862 + 0.960331i $$0.410043\pi$$
$$824$$ 6.00000 0.209020
$$825$$ 20.0000 0.696311
$$826$$ −16.0000 −0.556711
$$827$$ −12.0000 −0.417281 −0.208640 0.977992i $$-0.566904\pi$$
−0.208640 + 0.977992i $$0.566904\pi$$
$$828$$ −2.00000 −0.0695048
$$829$$ −4.00000 −0.138926 −0.0694629 0.997585i $$-0.522129\pi$$
−0.0694629 + 0.997585i $$0.522129\pi$$
$$830$$ 0 0
$$831$$ 2.00000 0.0693792
$$832$$ 0 0
$$833$$ −18.0000 −0.623663
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ 4.00000 0.138343
$$837$$ −6.00000 −0.207390
$$838$$ 12.0000 0.414533
$$839$$ 12.0000 0.414286 0.207143 0.978311i $$-0.433583\pi$$
0.207143 + 0.978311i $$0.433583\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 8.00000 0.275698
$$843$$ −10.0000 −0.344418
$$844$$ −20.0000 −0.688428
$$845$$ 0 0
$$846$$ −10.0000 −0.343807
$$847$$ 20.0000 0.687208
$$848$$ 2.00000 0.0686803
$$849$$ −28.0000 −0.960958
$$850$$ −10.0000 −0.342997
$$851$$ 16.0000 0.548473
$$852$$ 16.0000 0.548151
$$853$$ 54.0000 1.84892 0.924462 0.381273i $$-0.124514\pi$$
0.924462 + 0.381273i $$0.124514\pi$$
$$854$$ 40.0000 1.36877
$$855$$ 0 0
$$856$$ 4.00000 0.136717
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ 0 0
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ −40.0000 −1.36320
$$862$$ 4.00000 0.136241
$$863$$ −28.0000 −0.953131 −0.476566 0.879139i $$-0.658119\pi$$
−0.476566 + 0.879139i $$0.658119\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ 18.0000 0.611665
$$867$$ 13.0000 0.441503
$$868$$ 24.0000 0.814613
$$869$$ 40.0000 1.35691
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 4.00000 0.135457
$$873$$ −10.0000 −0.338449
$$874$$ 2.00000 0.0676510
$$875$$ 0 0
$$876$$ 2.00000 0.0675737
$$877$$ 8.00000 0.270141 0.135070 0.990836i $$-0.456874\pi$$
0.135070 + 0.990836i $$0.456874\pi$$
$$878$$ −22.0000 −0.742464
$$879$$ −14.0000 −0.472208
$$880$$ 0 0
$$881$$ 42.0000 1.41502 0.707508 0.706705i $$-0.249819\pi$$
0.707508 + 0.706705i $$0.249819\pi$$
$$882$$ −9.00000 −0.303046
$$883$$ 36.0000 1.21150 0.605748 0.795656i $$-0.292874\pi$$
0.605748 + 0.795656i $$0.292874\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ −12.0000 −0.402921 −0.201460 0.979497i $$-0.564569\pi$$
−0.201460 + 0.979497i $$0.564569\pi$$
$$888$$ −8.00000 −0.268462
$$889$$ 88.0000 2.95143
$$890$$ 0 0
$$891$$ 4.00000 0.134005
$$892$$ 6.00000 0.200895
$$893$$ 10.0000 0.334637
$$894$$ 20.0000 0.668900
$$895$$ 0 0
$$896$$ −4.00000 −0.133631
$$897$$ 0 0
$$898$$ 18.0000 0.600668
$$899$$ −36.0000 −1.20067
$$900$$ −5.00000 −0.166667
$$901$$ −4.00000 −0.133259
$$902$$ −40.0000 −1.33185
$$903$$ 48.0000 1.59734
$$904$$ 14.0000 0.465633
$$905$$ 0 0
$$906$$ 10.0000 0.332228
$$907$$ −32.0000 −1.06254 −0.531271 0.847202i $$-0.678286\pi$$
−0.531271 + 0.847202i $$0.678286\pi$$
$$908$$ 20.0000 0.663723
$$909$$ 8.00000 0.265343
$$910$$ 0 0
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ −1.00000 −0.0331133
$$913$$ −64.0000 −2.11809
$$914$$ −2.00000 −0.0661541
$$915$$ 0 0
$$916$$ −2.00000 −0.0660819
$$917$$ 0 0
$$918$$ −2.00000 −0.0660098
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 12.0000 0.395199
$$923$$ 0 0
$$924$$ −16.0000 −0.526361
$$925$$ 40.0000 1.31519
$$926$$ −36.0000 −1.18303
$$927$$ −6.00000 −0.197066
$$928$$ 6.00000 0.196960
$$929$$ −42.0000 −1.37798 −0.688988 0.724773i $$-0.741945\pi$$
−0.688988 + 0.724773i $$0.741945\pi$$
$$930$$ 0 0
$$931$$ 9.00000 0.294963
$$932$$ −6.00000 −0.196537
$$933$$ −34.0000 −1.11311
$$934$$ −8.00000 −0.261768
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 38.0000 1.24141 0.620703 0.784046i $$-0.286847\pi$$
0.620703 + 0.784046i $$0.286847\pi$$
$$938$$ 0 0
$$939$$ −6.00000 −0.195803
$$940$$ 0 0
$$941$$ 46.0000 1.49956 0.749779 0.661689i $$-0.230160\pi$$
0.749779 + 0.661689i $$0.230160\pi$$
$$942$$ 18.0000 0.586472
$$943$$ −20.0000 −0.651290
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ 48.0000 1.56061
$$947$$ −8.00000 −0.259965 −0.129983 0.991516i $$-0.541492\pi$$
−0.129983 + 0.991516i $$0.541492\pi$$
$$948$$ −10.0000 −0.324785
$$949$$ 0 0
$$950$$ 5.00000 0.162221
$$951$$ −6.00000 −0.194563
$$952$$ 8.00000 0.259281
$$953$$ −30.0000 −0.971795 −0.485898 0.874016i $$-0.661507\pi$$
−0.485898 + 0.874016i $$0.661507\pi$$
$$954$$ −2.00000 −0.0647524
$$955$$ 0 0
$$956$$ −6.00000 −0.194054
$$957$$ 24.0000 0.775810
$$958$$ −6.00000 −0.193851
$$959$$ 24.0000 0.775000
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ −4.00000 −0.128898
$$964$$ −2.00000 −0.0644157
$$965$$ 0 0
$$966$$ −8.00000 −0.257396
$$967$$ 12.0000 0.385894 0.192947 0.981209i $$-0.438195\pi$$
0.192947 + 0.981209i $$0.438195\pi$$
$$968$$ −5.00000 −0.160706
$$969$$ 2.00000 0.0642493
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ −16.0000 −0.512936
$$974$$ −2.00000 −0.0640841
$$975$$ 0 0
$$976$$ −10.0000 −0.320092
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 20.0000 0.639529
$$979$$ −8.00000 −0.255681
$$980$$ 0 0
$$981$$ −4.00000 −0.127710
$$982$$ −24.0000 −0.765871
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 10.0000 0.318788
$$985$$ 0 0
$$986$$ −12.0000 −0.382158
$$987$$ −40.0000 −1.27321
$$988$$ 0 0
$$989$$ 24.0000 0.763156
$$990$$ 0 0
$$991$$ −50.0000 −1.58830 −0.794151 0.607720i $$-0.792084\pi$$
−0.794151 + 0.607720i $$0.792084\pi$$
$$992$$ −6.00000 −0.190500
$$993$$ −24.0000 −0.761617
$$994$$ 64.0000 2.02996
$$995$$ 0 0
$$996$$ 16.0000 0.506979
$$997$$ −22.0000 −0.696747 −0.348373 0.937356i $$-0.613266\pi$$
−0.348373 + 0.937356i $$0.613266\pi$$
$$998$$ 4.00000 0.126618
$$999$$ 8.00000 0.253109
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 114.2.a.a.1.1 1
3.2 odd 2 342.2.a.f.1.1 1
4.3 odd 2 912.2.a.h.1.1 1
5.2 odd 4 2850.2.d.s.799.1 2
5.3 odd 4 2850.2.d.s.799.2 2
5.4 even 2 2850.2.a.x.1.1 1
7.6 odd 2 5586.2.a.p.1.1 1
8.3 odd 2 3648.2.a.j.1.1 1
8.5 even 2 3648.2.a.bb.1.1 1
12.11 even 2 2736.2.a.j.1.1 1
15.14 odd 2 8550.2.a.a.1.1 1
19.18 odd 2 2166.2.a.i.1.1 1
57.56 even 2 6498.2.a.h.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.a.a.1.1 1 1.1 even 1 trivial
342.2.a.f.1.1 1 3.2 odd 2
912.2.a.h.1.1 1 4.3 odd 2
2166.2.a.i.1.1 1 19.18 odd 2
2736.2.a.j.1.1 1 12.11 even 2
2850.2.a.x.1.1 1 5.4 even 2
2850.2.d.s.799.1 2 5.2 odd 4
2850.2.d.s.799.2 2 5.3 odd 4
3648.2.a.j.1.1 1 8.3 odd 2
3648.2.a.bb.1.1 1 8.5 even 2
5586.2.a.p.1.1 1 7.6 odd 2
6498.2.a.h.1.1 1 57.56 even 2
8550.2.a.a.1.1 1 15.14 odd 2