# Properties

 Label 5610.2.a.m.1.1 Level $5610$ Weight $2$ Character 5610.1 Self dual yes Analytic conductor $44.796$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5610,2,Mod(1,5610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5610, 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("5610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5610.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: $$44.7960755339$$ 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 $$\chi$$ $$=$$ 5610.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +4.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -8.00000 q^{19} -1.00000 q^{20} +1.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} +1.00000 q^{30} -6.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} -8.00000 q^{37} +8.00000 q^{38} +4.00000 q^{39} +1.00000 q^{40} -2.00000 q^{41} +4.00000 q^{43} -1.00000 q^{44} -1.00000 q^{45} -4.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} +1.00000 q^{51} +4.00000 q^{52} +8.00000 q^{53} -1.00000 q^{54} +1.00000 q^{55} -8.00000 q^{57} +4.00000 q^{59} -1.00000 q^{60} -10.0000 q^{61} +6.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} +1.00000 q^{66} -4.00000 q^{67} +1.00000 q^{68} +4.00000 q^{69} +8.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +8.00000 q^{74} +1.00000 q^{75} -8.00000 q^{76} -4.00000 q^{78} -8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -12.0000 q^{83} -1.00000 q^{85} -4.00000 q^{86} +1.00000 q^{88} -6.00000 q^{89} +1.00000 q^{90} +4.00000 q^{92} -6.00000 q^{93} +6.00000 q^{94} +8.00000 q^{95} -1.00000 q^{96} +10.0000 q^{97} +7.00000 q^{98} -1.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ −1.00000 −0.447214
$$6$$ −1.00000 −0.408248
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ −1.00000 −0.301511
$$12$$ 1.00000 0.288675
$$13$$ 4.00000 1.10940 0.554700 0.832050i $$-0.312833\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ 1.00000 0.242536
$$18$$ −1.00000 −0.235702
$$19$$ −8.00000 −1.83533 −0.917663 0.397360i $$-0.869927\pi$$
−0.917663 + 0.397360i $$0.869927\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ 1.00000 0.213201
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ −4.00000 −0.784465
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 1.00000 0.182574
$$31$$ −6.00000 −1.07763 −0.538816 0.842424i $$-0.681128\pi$$
−0.538816 + 0.842424i $$0.681128\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −1.00000 −0.174078
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −8.00000 −1.31519 −0.657596 0.753371i $$-0.728427\pi$$
−0.657596 + 0.753371i $$0.728427\pi$$
$$38$$ 8.00000 1.29777
$$39$$ 4.00000 0.640513
$$40$$ 1.00000 0.158114
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ −1.00000 −0.149071
$$46$$ −4.00000 −0.589768
$$47$$ −6.00000 −0.875190 −0.437595 0.899172i $$-0.644170\pi$$
−0.437595 + 0.899172i $$0.644170\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −7.00000 −1.00000
$$50$$ −1.00000 −0.141421
$$51$$ 1.00000 0.140028
$$52$$ 4.00000 0.554700
$$53$$ 8.00000 1.09888 0.549442 0.835532i $$-0.314840\pi$$
0.549442 + 0.835532i $$0.314840\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 1.00000 0.134840
$$56$$ 0 0
$$57$$ −8.00000 −1.05963
$$58$$ 0 0
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 6.00000 0.762001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −4.00000 −0.496139
$$66$$ 1.00000 0.123091
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 1.00000 0.121268
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 1.00000 0.115470
$$76$$ −8.00000 −0.917663
$$77$$ 0 0
$$78$$ −4.00000 −0.452911
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ 2.00000 0.220863
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ −1.00000 −0.108465
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 1.00000 0.106600
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ 4.00000 0.417029
$$93$$ −6.00000 −0.622171
$$94$$ 6.00000 0.618853
$$95$$ 8.00000 0.820783
$$96$$ −1.00000 −0.102062
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ 7.00000 0.707107
$$99$$ −1.00000 −0.100504
$$100$$ 1.00000 0.100000
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ −1.00000 −0.0990148
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ −4.00000 −0.392232
$$105$$ 0 0
$$106$$ −8.00000 −0.777029
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ −1.00000 −0.0953463
$$111$$ −8.00000 −0.759326
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 8.00000 0.749269
$$115$$ −4.00000 −0.373002
$$116$$ 0 0
$$117$$ 4.00000 0.369800
$$118$$ −4.00000 −0.368230
$$119$$ 0 0
$$120$$ 1.00000 0.0912871
$$121$$ 1.00000 0.0909091
$$122$$ 10.0000 0.905357
$$123$$ −2.00000 −0.180334
$$124$$ −6.00000 −0.538816
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −14.0000 −1.24230 −0.621150 0.783692i $$-0.713334\pi$$
−0.621150 + 0.783692i $$0.713334\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 4.00000 0.352180
$$130$$ 4.00000 0.350823
$$131$$ 16.0000 1.39793 0.698963 0.715158i $$-0.253645\pi$$
0.698963 + 0.715158i $$0.253645\pi$$
$$132$$ −1.00000 −0.0870388
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ −1.00000 −0.0860663
$$136$$ −1.00000 −0.0857493
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ −4.00000 −0.340503
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ −8.00000 −0.671345
$$143$$ −4.00000 −0.334497
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −6.00000 −0.496564
$$147$$ −7.00000 −0.577350
$$148$$ −8.00000 −0.657596
$$149$$ 2.00000 0.163846 0.0819232 0.996639i $$-0.473894\pi$$
0.0819232 + 0.996639i $$0.473894\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ 8.00000 0.648886
$$153$$ 1.00000 0.0808452
$$154$$ 0 0
$$155$$ 6.00000 0.481932
$$156$$ 4.00000 0.320256
$$157$$ −14.0000 −1.11732 −0.558661 0.829396i $$-0.688685\pi$$
−0.558661 + 0.829396i $$0.688685\pi$$
$$158$$ 8.00000 0.636446
$$159$$ 8.00000 0.634441
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ 12.0000 0.939913 0.469956 0.882690i $$-0.344270\pi$$
0.469956 + 0.882690i $$0.344270\pi$$
$$164$$ −2.00000 −0.156174
$$165$$ 1.00000 0.0778499
$$166$$ 12.0000 0.931381
$$167$$ −10.0000 −0.773823 −0.386912 0.922117i $$-0.626458\pi$$
−0.386912 + 0.922117i $$0.626458\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 1.00000 0.0766965
$$171$$ −8.00000 −0.611775
$$172$$ 4.00000 0.304997
$$173$$ −20.0000 −1.52057 −0.760286 0.649589i $$-0.774941\pi$$
−0.760286 + 0.649589i $$0.774941\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −1.00000 −0.0753778
$$177$$ 4.00000 0.300658
$$178$$ 6.00000 0.449719
$$179$$ −24.0000 −1.79384 −0.896922 0.442189i $$-0.854202\pi$$
−0.896922 + 0.442189i $$0.854202\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ 16.0000 1.18927 0.594635 0.803996i $$-0.297296\pi$$
0.594635 + 0.803996i $$0.297296\pi$$
$$182$$ 0 0
$$183$$ −10.0000 −0.739221
$$184$$ −4.00000 −0.294884
$$185$$ 8.00000 0.588172
$$186$$ 6.00000 0.439941
$$187$$ −1.00000 −0.0731272
$$188$$ −6.00000 −0.437595
$$189$$ 0 0
$$190$$ −8.00000 −0.580381
$$191$$ 6.00000 0.434145 0.217072 0.976156i $$-0.430349\pi$$
0.217072 + 0.976156i $$0.430349\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ −4.00000 −0.286446
$$196$$ −7.00000 −0.500000
$$197$$ −16.0000 −1.13995 −0.569976 0.821661i $$-0.693048\pi$$
−0.569976 + 0.821661i $$0.693048\pi$$
$$198$$ 1.00000 0.0710669
$$199$$ 18.0000 1.27599 0.637993 0.770042i $$-0.279765\pi$$
0.637993 + 0.770042i $$0.279765\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −4.00000 −0.282138
$$202$$ −18.0000 −1.26648
$$203$$ 0 0
$$204$$ 1.00000 0.0700140
$$205$$ 2.00000 0.139686
$$206$$ 8.00000 0.557386
$$207$$ 4.00000 0.278019
$$208$$ 4.00000 0.277350
$$209$$ 8.00000 0.553372
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 8.00000 0.549442
$$213$$ 8.00000 0.548151
$$214$$ 0 0
$$215$$ −4.00000 −0.272798
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ 2.00000 0.135457
$$219$$ 6.00000 0.405442
$$220$$ 1.00000 0.0674200
$$221$$ 4.00000 0.269069
$$222$$ 8.00000 0.536925
$$223$$ −28.0000 −1.87502 −0.937509 0.347960i $$-0.886874\pi$$
−0.937509 + 0.347960i $$0.886874\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 6.00000 0.399114
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ −8.00000 −0.529813
$$229$$ 18.0000 1.18947 0.594737 0.803921i $$-0.297256\pi$$
0.594737 + 0.803921i $$0.297256\pi$$
$$230$$ 4.00000 0.263752
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 14.0000 0.917170 0.458585 0.888650i $$-0.348356\pi$$
0.458585 + 0.888650i $$0.348356\pi$$
$$234$$ −4.00000 −0.261488
$$235$$ 6.00000 0.391397
$$236$$ 4.00000 0.260378
$$237$$ −8.00000 −0.519656
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ −1.00000 −0.0642824
$$243$$ 1.00000 0.0641500
$$244$$ −10.0000 −0.640184
$$245$$ 7.00000 0.447214
$$246$$ 2.00000 0.127515
$$247$$ −32.0000 −2.03611
$$248$$ 6.00000 0.381000
$$249$$ −12.0000 −0.760469
$$250$$ 1.00000 0.0632456
$$251$$ −16.0000 −1.00991 −0.504956 0.863145i $$-0.668491\pi$$
−0.504956 + 0.863145i $$0.668491\pi$$
$$252$$ 0 0
$$253$$ −4.00000 −0.251478
$$254$$ 14.0000 0.878438
$$255$$ −1.00000 −0.0626224
$$256$$ 1.00000 0.0625000
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ 0 0
$$260$$ −4.00000 −0.248069
$$261$$ 0 0
$$262$$ −16.0000 −0.988483
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ −8.00000 −0.491436
$$266$$ 0 0
$$267$$ −6.00000 −0.367194
$$268$$ −4.00000 −0.244339
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ −14.0000 −0.850439 −0.425220 0.905090i $$-0.639803\pi$$
−0.425220 + 0.905090i $$0.639803\pi$$
$$272$$ 1.00000 0.0606339
$$273$$ 0 0
$$274$$ −18.0000 −1.08742
$$275$$ −1.00000 −0.0603023
$$276$$ 4.00000 0.240772
$$277$$ −22.0000 −1.32185 −0.660926 0.750451i $$-0.729836\pi$$
−0.660926 + 0.750451i $$0.729836\pi$$
$$278$$ 12.0000 0.719712
$$279$$ −6.00000 −0.359211
$$280$$ 0 0
$$281$$ −22.0000 −1.31241 −0.656205 0.754583i $$-0.727839\pi$$
−0.656205 + 0.754583i $$0.727839\pi$$
$$282$$ 6.00000 0.357295
$$283$$ 28.0000 1.66443 0.832214 0.554455i $$-0.187073\pi$$
0.832214 + 0.554455i $$0.187073\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 8.00000 0.473879
$$286$$ 4.00000 0.236525
$$287$$ 0 0
$$288$$ −1.00000 −0.0589256
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 10.0000 0.586210
$$292$$ 6.00000 0.351123
$$293$$ 18.0000 1.05157 0.525786 0.850617i $$-0.323771\pi$$
0.525786 + 0.850617i $$0.323771\pi$$
$$294$$ 7.00000 0.408248
$$295$$ −4.00000 −0.232889
$$296$$ 8.00000 0.464991
$$297$$ −1.00000 −0.0580259
$$298$$ −2.00000 −0.115857
$$299$$ 16.0000 0.925304
$$300$$ 1.00000 0.0577350
$$301$$ 0 0
$$302$$ 10.0000 0.575435
$$303$$ 18.0000 1.03407
$$304$$ −8.00000 −0.458831
$$305$$ 10.0000 0.572598
$$306$$ −1.00000 −0.0571662
$$307$$ −8.00000 −0.456584 −0.228292 0.973593i $$-0.573314\pi$$
−0.228292 + 0.973593i $$0.573314\pi$$
$$308$$ 0 0
$$309$$ −8.00000 −0.455104
$$310$$ −6.00000 −0.340777
$$311$$ 4.00000 0.226819 0.113410 0.993548i $$-0.463823\pi$$
0.113410 + 0.993548i $$0.463823\pi$$
$$312$$ −4.00000 −0.226455
$$313$$ 18.0000 1.01742 0.508710 0.860938i $$-0.330123\pi$$
0.508710 + 0.860938i $$0.330123\pi$$
$$314$$ 14.0000 0.790066
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ −8.00000 −0.448618
$$319$$ 0 0
$$320$$ −1.00000 −0.0559017
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −8.00000 −0.445132
$$324$$ 1.00000 0.0555556
$$325$$ 4.00000 0.221880
$$326$$ −12.0000 −0.664619
$$327$$ −2.00000 −0.110600
$$328$$ 2.00000 0.110432
$$329$$ 0 0
$$330$$ −1.00000 −0.0550482
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ −8.00000 −0.438397
$$334$$ 10.0000 0.547176
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ −3.00000 −0.163178
$$339$$ −6.00000 −0.325875
$$340$$ −1.00000 −0.0542326
$$341$$ 6.00000 0.324918
$$342$$ 8.00000 0.432590
$$343$$ 0 0
$$344$$ −4.00000 −0.215666
$$345$$ −4.00000 −0.215353
$$346$$ 20.0000 1.07521
$$347$$ −8.00000 −0.429463 −0.214731 0.976673i $$-0.568888\pi$$
−0.214731 + 0.976673i $$0.568888\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ 4.00000 0.213504
$$352$$ 1.00000 0.0533002
$$353$$ −26.0000 −1.38384 −0.691920 0.721974i $$-0.743235\pi$$
−0.691920 + 0.721974i $$0.743235\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ −8.00000 −0.424596
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ 24.0000 1.26844
$$359$$ 28.0000 1.47778 0.738892 0.673824i $$-0.235349\pi$$
0.738892 + 0.673824i $$0.235349\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ 45.0000 2.36842
$$362$$ −16.0000 −0.840941
$$363$$ 1.00000 0.0524864
$$364$$ 0 0
$$365$$ −6.00000 −0.314054
$$366$$ 10.0000 0.522708
$$367$$ 14.0000 0.730794 0.365397 0.930852i $$-0.380933\pi$$
0.365397 + 0.930852i $$0.380933\pi$$
$$368$$ 4.00000 0.208514
$$369$$ −2.00000 −0.104116
$$370$$ −8.00000 −0.415900
$$371$$ 0 0
$$372$$ −6.00000 −0.311086
$$373$$ −12.0000 −0.621336 −0.310668 0.950518i $$-0.600553\pi$$
−0.310668 + 0.950518i $$0.600553\pi$$
$$374$$ 1.00000 0.0517088
$$375$$ −1.00000 −0.0516398
$$376$$ 6.00000 0.309426
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 24.0000 1.23280 0.616399 0.787434i $$-0.288591\pi$$
0.616399 + 0.787434i $$0.288591\pi$$
$$380$$ 8.00000 0.410391
$$381$$ −14.0000 −0.717242
$$382$$ −6.00000 −0.306987
$$383$$ −10.0000 −0.510976 −0.255488 0.966812i $$-0.582236\pi$$
−0.255488 + 0.966812i $$0.582236\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 14.0000 0.712581
$$387$$ 4.00000 0.203331
$$388$$ 10.0000 0.507673
$$389$$ −8.00000 −0.405616 −0.202808 0.979219i $$-0.565007\pi$$
−0.202808 + 0.979219i $$0.565007\pi$$
$$390$$ 4.00000 0.202548
$$391$$ 4.00000 0.202289
$$392$$ 7.00000 0.353553
$$393$$ 16.0000 0.807093
$$394$$ 16.0000 0.806068
$$395$$ 8.00000 0.402524
$$396$$ −1.00000 −0.0502519
$$397$$ −20.0000 −1.00377 −0.501886 0.864934i $$-0.667360\pi$$
−0.501886 + 0.864934i $$0.667360\pi$$
$$398$$ −18.0000 −0.902258
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −18.0000 −0.898877 −0.449439 0.893311i $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 4.00000 0.199502
$$403$$ −24.0000 −1.19553
$$404$$ 18.0000 0.895533
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ 8.00000 0.396545
$$408$$ −1.00000 −0.0495074
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ −2.00000 −0.0987730
$$411$$ 18.0000 0.887875
$$412$$ −8.00000 −0.394132
$$413$$ 0 0
$$414$$ −4.00000 −0.196589
$$415$$ 12.0000 0.589057
$$416$$ −4.00000 −0.196116
$$417$$ −12.0000 −0.587643
$$418$$ −8.00000 −0.391293
$$419$$ −20.0000 −0.977064 −0.488532 0.872546i $$-0.662467\pi$$
−0.488532 + 0.872546i $$0.662467\pi$$
$$420$$ 0 0
$$421$$ −14.0000 −0.682318 −0.341159 0.940006i $$-0.610819\pi$$
−0.341159 + 0.940006i $$0.610819\pi$$
$$422$$ 20.0000 0.973585
$$423$$ −6.00000 −0.291730
$$424$$ −8.00000 −0.388514
$$425$$ 1.00000 0.0485071
$$426$$ −8.00000 −0.387601
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −4.00000 −0.193122
$$430$$ 4.00000 0.192897
$$431$$ −10.0000 −0.481683 −0.240842 0.970564i $$-0.577423\pi$$
−0.240842 + 0.970564i $$0.577423\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −2.00000 −0.0961139 −0.0480569 0.998845i $$-0.515303\pi$$
−0.0480569 + 0.998845i $$0.515303\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −2.00000 −0.0957826
$$437$$ −32.0000 −1.53077
$$438$$ −6.00000 −0.286691
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ −1.00000 −0.0476731
$$441$$ −7.00000 −0.333333
$$442$$ −4.00000 −0.190261
$$443$$ 36.0000 1.71041 0.855206 0.518289i $$-0.173431\pi$$
0.855206 + 0.518289i $$0.173431\pi$$
$$444$$ −8.00000 −0.379663
$$445$$ 6.00000 0.284427
$$446$$ 28.0000 1.32584
$$447$$ 2.00000 0.0945968
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 2.00000 0.0941763
$$452$$ −6.00000 −0.282216
$$453$$ −10.0000 −0.469841
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ 8.00000 0.374634
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ −18.0000 −0.841085
$$459$$ 1.00000 0.0466760
$$460$$ −4.00000 −0.186501
$$461$$ 22.0000 1.02464 0.512321 0.858794i $$-0.328786\pi$$
0.512321 + 0.858794i $$0.328786\pi$$
$$462$$ 0 0
$$463$$ 40.0000 1.85896 0.929479 0.368875i $$-0.120257\pi$$
0.929479 + 0.368875i $$0.120257\pi$$
$$464$$ 0 0
$$465$$ 6.00000 0.278243
$$466$$ −14.0000 −0.648537
$$467$$ 40.0000 1.85098 0.925490 0.378773i $$-0.123654\pi$$
0.925490 + 0.378773i $$0.123654\pi$$
$$468$$ 4.00000 0.184900
$$469$$ 0 0
$$470$$ −6.00000 −0.276759
$$471$$ −14.0000 −0.645086
$$472$$ −4.00000 −0.184115
$$473$$ −4.00000 −0.183920
$$474$$ 8.00000 0.367452
$$475$$ −8.00000 −0.367065
$$476$$ 0 0
$$477$$ 8.00000 0.366295
$$478$$ 0 0
$$479$$ 34.0000 1.55350 0.776750 0.629809i $$-0.216867\pi$$
0.776750 + 0.629809i $$0.216867\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ −32.0000 −1.45907
$$482$$ −10.0000 −0.455488
$$483$$ 0 0
$$484$$ 1.00000 0.0454545
$$485$$ −10.0000 −0.454077
$$486$$ −1.00000 −0.0453609
$$487$$ 18.0000 0.815658 0.407829 0.913058i $$-0.366286\pi$$
0.407829 + 0.913058i $$0.366286\pi$$
$$488$$ 10.0000 0.452679
$$489$$ 12.0000 0.542659
$$490$$ −7.00000 −0.316228
$$491$$ 4.00000 0.180517 0.0902587 0.995918i $$-0.471231\pi$$
0.0902587 + 0.995918i $$0.471231\pi$$
$$492$$ −2.00000 −0.0901670
$$493$$ 0 0
$$494$$ 32.0000 1.43975
$$495$$ 1.00000 0.0449467
$$496$$ −6.00000 −0.269408
$$497$$ 0 0
$$498$$ 12.0000 0.537733
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −10.0000 −0.446767
$$502$$ 16.0000 0.714115
$$503$$ −2.00000 −0.0891756 −0.0445878 0.999005i $$-0.514197\pi$$
−0.0445878 + 0.999005i $$0.514197\pi$$
$$504$$ 0 0
$$505$$ −18.0000 −0.800989
$$506$$ 4.00000 0.177822
$$507$$ 3.00000 0.133235
$$508$$ −14.0000 −0.621150
$$509$$ 40.0000 1.77297 0.886484 0.462758i $$-0.153140\pi$$
0.886484 + 0.462758i $$0.153140\pi$$
$$510$$ 1.00000 0.0442807
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ −8.00000 −0.353209
$$514$$ 18.0000 0.793946
$$515$$ 8.00000 0.352522
$$516$$ 4.00000 0.176090
$$517$$ 6.00000 0.263880
$$518$$ 0 0
$$519$$ −20.0000 −0.877903
$$520$$ 4.00000 0.175412
$$521$$ 38.0000 1.66481 0.832405 0.554168i $$-0.186963\pi$$
0.832405 + 0.554168i $$0.186963\pi$$
$$522$$ 0 0
$$523$$ −16.0000 −0.699631 −0.349816 0.936819i $$-0.613756\pi$$
−0.349816 + 0.936819i $$0.613756\pi$$
$$524$$ 16.0000 0.698963
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ −6.00000 −0.261364
$$528$$ −1.00000 −0.0435194
$$529$$ −7.00000 −0.304348
$$530$$ 8.00000 0.347498
$$531$$ 4.00000 0.173585
$$532$$ 0 0
$$533$$ −8.00000 −0.346518
$$534$$ 6.00000 0.259645
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ −24.0000 −1.03568
$$538$$ 6.00000 0.258678
$$539$$ 7.00000 0.301511
$$540$$ −1.00000 −0.0430331
$$541$$ −6.00000 −0.257960 −0.128980 0.991647i $$-0.541170\pi$$
−0.128980 + 0.991647i $$0.541170\pi$$
$$542$$ 14.0000 0.601351
$$543$$ 16.0000 0.686626
$$544$$ −1.00000 −0.0428746
$$545$$ 2.00000 0.0856706
$$546$$ 0 0
$$547$$ −36.0000 −1.53925 −0.769624 0.638497i $$-0.779557\pi$$
−0.769624 + 0.638497i $$0.779557\pi$$
$$548$$ 18.0000 0.768922
$$549$$ −10.0000 −0.426790
$$550$$ 1.00000 0.0426401
$$551$$ 0 0
$$552$$ −4.00000 −0.170251
$$553$$ 0 0
$$554$$ 22.0000 0.934690
$$555$$ 8.00000 0.339581
$$556$$ −12.0000 −0.508913
$$557$$ −42.0000 −1.77960 −0.889799 0.456354i $$-0.849155\pi$$
−0.889799 + 0.456354i $$0.849155\pi$$
$$558$$ 6.00000 0.254000
$$559$$ 16.0000 0.676728
$$560$$ 0 0
$$561$$ −1.00000 −0.0422200
$$562$$ 22.0000 0.928014
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ −6.00000 −0.252646
$$565$$ 6.00000 0.252422
$$566$$ −28.0000 −1.17693
$$567$$ 0 0
$$568$$ −8.00000 −0.335673
$$569$$ −42.0000 −1.76073 −0.880366 0.474295i $$-0.842703\pi$$
−0.880366 + 0.474295i $$0.842703\pi$$
$$570$$ −8.00000 −0.335083
$$571$$ 28.0000 1.17176 0.585882 0.810397i $$-0.300748\pi$$
0.585882 + 0.810397i $$0.300748\pi$$
$$572$$ −4.00000 −0.167248
$$573$$ 6.00000 0.250654
$$574$$ 0 0
$$575$$ 4.00000 0.166812
$$576$$ 1.00000 0.0416667
$$577$$ 46.0000 1.91501 0.957503 0.288425i $$-0.0931316\pi$$
0.957503 + 0.288425i $$0.0931316\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ −14.0000 −0.581820
$$580$$ 0 0
$$581$$ 0 0
$$582$$ −10.0000 −0.414513
$$583$$ −8.00000 −0.331326
$$584$$ −6.00000 −0.248282
$$585$$ −4.00000 −0.165380
$$586$$ −18.0000 −0.743573
$$587$$ −28.0000 −1.15568 −0.577842 0.816149i $$-0.696105\pi$$
−0.577842 + 0.816149i $$0.696105\pi$$
$$588$$ −7.00000 −0.288675
$$589$$ 48.0000 1.97781
$$590$$ 4.00000 0.164677
$$591$$ −16.0000 −0.658152
$$592$$ −8.00000 −0.328798
$$593$$ −34.0000 −1.39621 −0.698106 0.715994i $$-0.745974\pi$$
−0.698106 + 0.715994i $$0.745974\pi$$
$$594$$ 1.00000 0.0410305
$$595$$ 0 0
$$596$$ 2.00000 0.0819232
$$597$$ 18.0000 0.736691
$$598$$ −16.0000 −0.654289
$$599$$ −10.0000 −0.408589 −0.204294 0.978909i $$-0.565490\pi$$
−0.204294 + 0.978909i $$0.565490\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ 38.0000 1.55005 0.775026 0.631929i $$-0.217737\pi$$
0.775026 + 0.631929i $$0.217737\pi$$
$$602$$ 0 0
$$603$$ −4.00000 −0.162893
$$604$$ −10.0000 −0.406894
$$605$$ −1.00000 −0.0406558
$$606$$ −18.0000 −0.731200
$$607$$ −32.0000 −1.29884 −0.649420 0.760430i $$-0.724988\pi$$
−0.649420 + 0.760430i $$0.724988\pi$$
$$608$$ 8.00000 0.324443
$$609$$ 0 0
$$610$$ −10.0000 −0.404888
$$611$$ −24.0000 −0.970936
$$612$$ 1.00000 0.0404226
$$613$$ −44.0000 −1.77714 −0.888572 0.458738i $$-0.848302\pi$$
−0.888572 + 0.458738i $$0.848302\pi$$
$$614$$ 8.00000 0.322854
$$615$$ 2.00000 0.0806478
$$616$$ 0 0
$$617$$ −42.0000 −1.69086 −0.845428 0.534089i $$-0.820655\pi$$
−0.845428 + 0.534089i $$0.820655\pi$$
$$618$$ 8.00000 0.321807
$$619$$ −44.0000 −1.76851 −0.884255 0.467005i $$-0.845333\pi$$
−0.884255 + 0.467005i $$0.845333\pi$$
$$620$$ 6.00000 0.240966
$$621$$ 4.00000 0.160514
$$622$$ −4.00000 −0.160385
$$623$$ 0 0
$$624$$ 4.00000 0.160128
$$625$$ 1.00000 0.0400000
$$626$$ −18.0000 −0.719425
$$627$$ 8.00000 0.319489
$$628$$ −14.0000 −0.558661
$$629$$ −8.00000 −0.318981
$$630$$ 0 0
$$631$$ −4.00000 −0.159237 −0.0796187 0.996825i $$-0.525370\pi$$
−0.0796187 + 0.996825i $$0.525370\pi$$
$$632$$ 8.00000 0.318223
$$633$$ −20.0000 −0.794929
$$634$$ 6.00000 0.238290
$$635$$ 14.0000 0.555573
$$636$$ 8.00000 0.317221
$$637$$ −28.0000 −1.10940
$$638$$ 0 0
$$639$$ 8.00000 0.316475
$$640$$ 1.00000 0.0395285
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 0 0
$$643$$ −40.0000 −1.57745 −0.788723 0.614749i $$-0.789257\pi$$
−0.788723 + 0.614749i $$0.789257\pi$$
$$644$$ 0 0
$$645$$ −4.00000 −0.157500
$$646$$ 8.00000 0.314756
$$647$$ 46.0000 1.80845 0.904223 0.427060i $$-0.140451\pi$$
0.904223 + 0.427060i $$0.140451\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −4.00000 −0.157014
$$650$$ −4.00000 −0.156893
$$651$$ 0 0
$$652$$ 12.0000 0.469956
$$653$$ 30.0000 1.17399 0.586995 0.809590i $$-0.300311\pi$$
0.586995 + 0.809590i $$0.300311\pi$$
$$654$$ 2.00000 0.0782062
$$655$$ −16.0000 −0.625172
$$656$$ −2.00000 −0.0780869
$$657$$ 6.00000 0.234082
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 1.00000 0.0389249
$$661$$ −50.0000 −1.94477 −0.972387 0.233373i $$-0.925024\pi$$
−0.972387 + 0.233373i $$0.925024\pi$$
$$662$$ 20.0000 0.777322
$$663$$ 4.00000 0.155347
$$664$$ 12.0000 0.465690
$$665$$ 0 0
$$666$$ 8.00000 0.309994
$$667$$ 0 0
$$668$$ −10.0000 −0.386912
$$669$$ −28.0000 −1.08254
$$670$$ −4.00000 −0.154533
$$671$$ 10.0000 0.386046
$$672$$ 0 0
$$673$$ 34.0000 1.31060 0.655302 0.755367i $$-0.272541\pi$$
0.655302 + 0.755367i $$0.272541\pi$$
$$674$$ 22.0000 0.847408
$$675$$ 1.00000 0.0384900
$$676$$ 3.00000 0.115385
$$677$$ 24.0000 0.922395 0.461197 0.887298i $$-0.347420\pi$$
0.461197 + 0.887298i $$0.347420\pi$$
$$678$$ 6.00000 0.230429
$$679$$ 0 0
$$680$$ 1.00000 0.0383482
$$681$$ −12.0000 −0.459841
$$682$$ −6.00000 −0.229752
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ −8.00000 −0.305888
$$685$$ −18.0000 −0.687745
$$686$$ 0 0
$$687$$ 18.0000 0.686743
$$688$$ 4.00000 0.152499
$$689$$ 32.0000 1.21910
$$690$$ 4.00000 0.152277
$$691$$ −40.0000 −1.52167 −0.760836 0.648944i $$-0.775211\pi$$
−0.760836 + 0.648944i $$0.775211\pi$$
$$692$$ −20.0000 −0.760286
$$693$$ 0 0
$$694$$ 8.00000 0.303676
$$695$$ 12.0000 0.455186
$$696$$ 0 0
$$697$$ −2.00000 −0.0757554
$$698$$ 0 0
$$699$$ 14.0000 0.529529
$$700$$ 0 0
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ −4.00000 −0.150970
$$703$$ 64.0000 2.41381
$$704$$ −1.00000 −0.0376889
$$705$$ 6.00000 0.225973
$$706$$ 26.0000 0.978523
$$707$$ 0 0
$$708$$ 4.00000 0.150329
$$709$$ 28.0000 1.05156 0.525781 0.850620i $$-0.323773\pi$$
0.525781 + 0.850620i $$0.323773\pi$$
$$710$$ 8.00000 0.300235
$$711$$ −8.00000 −0.300023
$$712$$ 6.00000 0.224860
$$713$$ −24.0000 −0.898807
$$714$$ 0 0
$$715$$ 4.00000 0.149592
$$716$$ −24.0000 −0.896922
$$717$$ 0 0
$$718$$ −28.0000 −1.04495
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 0 0
$$722$$ −45.0000 −1.67473
$$723$$ 10.0000 0.371904
$$724$$ 16.0000 0.594635
$$725$$ 0 0
$$726$$ −1.00000 −0.0371135
$$727$$ −20.0000 −0.741759 −0.370879 0.928681i $$-0.620944\pi$$
−0.370879 + 0.928681i $$0.620944\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 6.00000 0.222070
$$731$$ 4.00000 0.147945
$$732$$ −10.0000 −0.369611
$$733$$ −44.0000 −1.62518 −0.812589 0.582838i $$-0.801942\pi$$
−0.812589 + 0.582838i $$0.801942\pi$$
$$734$$ −14.0000 −0.516749
$$735$$ 7.00000 0.258199
$$736$$ −4.00000 −0.147442
$$737$$ 4.00000 0.147342
$$738$$ 2.00000 0.0736210
$$739$$ −36.0000 −1.32428 −0.662141 0.749380i $$-0.730352\pi$$
−0.662141 + 0.749380i $$0.730352\pi$$
$$740$$ 8.00000 0.294086
$$741$$ −32.0000 −1.17555
$$742$$ 0 0
$$743$$ 6.00000 0.220119 0.110059 0.993925i $$-0.464896\pi$$
0.110059 + 0.993925i $$0.464896\pi$$
$$744$$ 6.00000 0.219971
$$745$$ −2.00000 −0.0732743
$$746$$ 12.0000 0.439351
$$747$$ −12.0000 −0.439057
$$748$$ −1.00000 −0.0365636
$$749$$ 0 0
$$750$$ 1.00000 0.0365148
$$751$$ −42.0000 −1.53260 −0.766301 0.642482i $$-0.777905\pi$$
−0.766301 + 0.642482i $$0.777905\pi$$
$$752$$ −6.00000 −0.218797
$$753$$ −16.0000 −0.583072
$$754$$ 0 0
$$755$$ 10.0000 0.363937
$$756$$ 0 0
$$757$$ 22.0000 0.799604 0.399802 0.916602i $$-0.369079\pi$$
0.399802 + 0.916602i $$0.369079\pi$$
$$758$$ −24.0000 −0.871719
$$759$$ −4.00000 −0.145191
$$760$$ −8.00000 −0.290191
$$761$$ −10.0000 −0.362500 −0.181250 0.983437i $$-0.558014\pi$$
−0.181250 + 0.983437i $$0.558014\pi$$
$$762$$ 14.0000 0.507166
$$763$$ 0 0
$$764$$ 6.00000 0.217072
$$765$$ −1.00000 −0.0361551
$$766$$ 10.0000 0.361315
$$767$$ 16.0000 0.577727
$$768$$ 1.00000 0.0360844
$$769$$ 22.0000 0.793340 0.396670 0.917961i $$-0.370166\pi$$
0.396670 + 0.917961i $$0.370166\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ −14.0000 −0.503871
$$773$$ −4.00000 −0.143870 −0.0719350 0.997409i $$-0.522917\pi$$
−0.0719350 + 0.997409i $$0.522917\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ −6.00000 −0.215526
$$776$$ −10.0000 −0.358979
$$777$$ 0 0
$$778$$ 8.00000 0.286814
$$779$$ 16.0000 0.573259
$$780$$ −4.00000 −0.143223
$$781$$ −8.00000 −0.286263
$$782$$ −4.00000 −0.143040
$$783$$ 0 0
$$784$$ −7.00000 −0.250000
$$785$$ 14.0000 0.499681
$$786$$ −16.0000 −0.570701
$$787$$ 28.0000 0.998092 0.499046 0.866575i $$-0.333684\pi$$
0.499046 + 0.866575i $$0.333684\pi$$
$$788$$ −16.0000 −0.569976
$$789$$ −12.0000 −0.427211
$$790$$ −8.00000 −0.284627
$$791$$ 0 0
$$792$$ 1.00000 0.0355335
$$793$$ −40.0000 −1.42044
$$794$$ 20.0000 0.709773
$$795$$ −8.00000 −0.283731
$$796$$ 18.0000 0.637993
$$797$$ 16.0000 0.566749 0.283375 0.959009i $$-0.408546\pi$$
0.283375 + 0.959009i $$0.408546\pi$$
$$798$$ 0 0
$$799$$ −6.00000 −0.212265
$$800$$ −1.00000 −0.0353553
$$801$$ −6.00000 −0.212000
$$802$$ 18.0000 0.635602
$$803$$ −6.00000 −0.211735
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ 24.0000 0.845364
$$807$$ −6.00000 −0.211210
$$808$$ −18.0000 −0.633238
$$809$$ 14.0000 0.492214 0.246107 0.969243i $$-0.420849\pi$$
0.246107 + 0.969243i $$0.420849\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ −20.0000 −0.702295 −0.351147 0.936320i $$-0.614208\pi$$
−0.351147 + 0.936320i $$0.614208\pi$$
$$812$$ 0 0
$$813$$ −14.0000 −0.491001
$$814$$ −8.00000 −0.280400
$$815$$ −12.0000 −0.420342
$$816$$ 1.00000 0.0350070
$$817$$ −32.0000 −1.11954
$$818$$ −14.0000 −0.489499
$$819$$ 0 0
$$820$$ 2.00000 0.0698430
$$821$$ −44.0000 −1.53561 −0.767805 0.640683i $$-0.778651\pi$$
−0.767805 + 0.640683i $$0.778651\pi$$
$$822$$ −18.0000 −0.627822
$$823$$ 38.0000 1.32460 0.662298 0.749240i $$-0.269581\pi$$
0.662298 + 0.749240i $$0.269581\pi$$
$$824$$ 8.00000 0.278693
$$825$$ −1.00000 −0.0348155
$$826$$ 0 0
$$827$$ 4.00000 0.139094 0.0695468 0.997579i $$-0.477845\pi$$
0.0695468 + 0.997579i $$0.477845\pi$$
$$828$$ 4.00000 0.139010
$$829$$ 38.0000 1.31979 0.659897 0.751356i $$-0.270600\pi$$
0.659897 + 0.751356i $$0.270600\pi$$
$$830$$ −12.0000 −0.416526
$$831$$ −22.0000 −0.763172
$$832$$ 4.00000 0.138675
$$833$$ −7.00000 −0.242536
$$834$$ 12.0000 0.415526
$$835$$ 10.0000 0.346064
$$836$$ 8.00000 0.276686
$$837$$ −6.00000 −0.207390
$$838$$ 20.0000 0.690889
$$839$$ 36.0000 1.24286 0.621429 0.783470i $$-0.286552\pi$$
0.621429 + 0.783470i $$0.286552\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 14.0000 0.482472
$$843$$ −22.0000 −0.757720
$$844$$ −20.0000 −0.688428
$$845$$ −3.00000 −0.103203
$$846$$ 6.00000 0.206284
$$847$$ 0 0
$$848$$ 8.00000 0.274721
$$849$$ 28.0000 0.960958
$$850$$ −1.00000 −0.0342997
$$851$$ −32.0000 −1.09695
$$852$$ 8.00000 0.274075
$$853$$ 42.0000 1.43805 0.719026 0.694983i $$-0.244588\pi$$
0.719026 + 0.694983i $$0.244588\pi$$
$$854$$ 0 0
$$855$$ 8.00000 0.273594
$$856$$ 0 0
$$857$$ −6.00000 −0.204956 −0.102478 0.994735i $$-0.532677\pi$$
−0.102478 + 0.994735i $$0.532677\pi$$
$$858$$ 4.00000 0.136558
$$859$$ 44.0000 1.50126 0.750630 0.660722i $$-0.229750\pi$$
0.750630 + 0.660722i $$0.229750\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ 0 0
$$862$$ 10.0000 0.340601
$$863$$ −18.0000 −0.612727 −0.306364 0.951915i $$-0.599112\pi$$
−0.306364 + 0.951915i $$0.599112\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 20.0000 0.680020
$$866$$ 2.00000 0.0679628
$$867$$ 1.00000 0.0339618
$$868$$ 0 0
$$869$$ 8.00000 0.271381
$$870$$ 0 0
$$871$$ −16.0000 −0.542139
$$872$$ 2.00000 0.0677285
$$873$$ 10.0000 0.338449
$$874$$ 32.0000 1.08242
$$875$$ 0 0
$$876$$ 6.00000 0.202721
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ 20.0000 0.674967
$$879$$ 18.0000 0.607125
$$880$$ 1.00000 0.0337100
$$881$$ 54.0000 1.81931 0.909653 0.415369i $$-0.136347\pi$$
0.909653 + 0.415369i $$0.136347\pi$$
$$882$$ 7.00000 0.235702
$$883$$ −12.0000 −0.403832 −0.201916 0.979403i $$-0.564717\pi$$
−0.201916 + 0.979403i $$0.564717\pi$$
$$884$$ 4.00000 0.134535
$$885$$ −4.00000 −0.134459
$$886$$ −36.0000 −1.20944
$$887$$ 26.0000 0.872995 0.436497 0.899706i $$-0.356219\pi$$
0.436497 + 0.899706i $$0.356219\pi$$
$$888$$ 8.00000 0.268462
$$889$$ 0 0
$$890$$ −6.00000 −0.201120
$$891$$ −1.00000 −0.0335013
$$892$$ −28.0000 −0.937509
$$893$$ 48.0000 1.60626
$$894$$ −2.00000 −0.0668900
$$895$$ 24.0000 0.802232
$$896$$ 0 0
$$897$$ 16.0000 0.534224
$$898$$ 18.0000 0.600668
$$899$$ 0 0
$$900$$ 1.00000 0.0333333
$$901$$ 8.00000 0.266519
$$902$$ −2.00000 −0.0665927
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ −16.0000 −0.531858
$$906$$ 10.0000 0.332228
$$907$$ −4.00000 −0.132818 −0.0664089 0.997792i $$-0.521154\pi$$
−0.0664089 + 0.997792i $$0.521154\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 18.0000 0.597022
$$910$$ 0 0
$$911$$ 8.00000 0.265052 0.132526 0.991180i $$-0.457691\pi$$
0.132526 + 0.991180i $$0.457691\pi$$
$$912$$ −8.00000 −0.264906
$$913$$ 12.0000 0.397142
$$914$$ −10.0000 −0.330771
$$915$$ 10.0000 0.330590
$$916$$ 18.0000 0.594737
$$917$$ 0 0
$$918$$ −1.00000 −0.0330049
$$919$$ −6.00000 −0.197922 −0.0989609 0.995091i $$-0.531552\pi$$
−0.0989609 + 0.995091i $$0.531552\pi$$
$$920$$ 4.00000 0.131876
$$921$$ −8.00000 −0.263609
$$922$$ −22.0000 −0.724531
$$923$$ 32.0000 1.05329
$$924$$ 0 0
$$925$$ −8.00000 −0.263038
$$926$$ −40.0000 −1.31448
$$927$$ −8.00000 −0.262754
$$928$$ 0 0
$$929$$ 42.0000 1.37798 0.688988 0.724773i $$-0.258055\pi$$
0.688988 + 0.724773i $$0.258055\pi$$
$$930$$ −6.00000 −0.196748
$$931$$ 56.0000 1.83533
$$932$$ 14.0000 0.458585
$$933$$ 4.00000 0.130954
$$934$$ −40.0000 −1.30884
$$935$$ 1.00000 0.0327035
$$936$$ −4.00000 −0.130744
$$937$$ −10.0000 −0.326686 −0.163343 0.986569i $$-0.552228\pi$$
−0.163343 + 0.986569i $$0.552228\pi$$
$$938$$ 0 0
$$939$$ 18.0000 0.587408
$$940$$ 6.00000 0.195698
$$941$$ −4.00000 −0.130396 −0.0651981 0.997872i $$-0.520768\pi$$
−0.0651981 + 0.997872i $$0.520768\pi$$
$$942$$ 14.0000 0.456145
$$943$$ −8.00000 −0.260516
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ 4.00000 0.130051
$$947$$ 36.0000 1.16984 0.584921 0.811090i $$-0.301125\pi$$
0.584921 + 0.811090i $$0.301125\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ 24.0000 0.779073
$$950$$ 8.00000 0.259554
$$951$$ −6.00000 −0.194563
$$952$$ 0 0
$$953$$ 30.0000 0.971795 0.485898 0.874016i $$-0.338493\pi$$
0.485898 + 0.874016i $$0.338493\pi$$
$$954$$ −8.00000 −0.259010
$$955$$ −6.00000 −0.194155
$$956$$ 0 0
$$957$$ 0 0
$$958$$ −34.0000 −1.09849
$$959$$ 0 0
$$960$$ −1.00000 −0.0322749
$$961$$ 5.00000 0.161290
$$962$$ 32.0000 1.03172
$$963$$ 0 0
$$964$$ 10.0000 0.322078
$$965$$ 14.0000 0.450676
$$966$$ 0 0
$$967$$ −10.0000 −0.321578 −0.160789 0.986989i $$-0.551404\pi$$
−0.160789 + 0.986989i $$0.551404\pi$$
$$968$$ −1.00000 −0.0321412
$$969$$ −8.00000 −0.256997
$$970$$ 10.0000 0.321081
$$971$$ 20.0000 0.641831 0.320915 0.947108i $$-0.396010\pi$$
0.320915 + 0.947108i $$0.396010\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ −18.0000 −0.576757
$$975$$ 4.00000 0.128103
$$976$$ −10.0000 −0.320092
$$977$$ 34.0000 1.08776 0.543878 0.839164i $$-0.316955\pi$$
0.543878 + 0.839164i $$0.316955\pi$$
$$978$$ −12.0000 −0.383718
$$979$$ 6.00000 0.191761
$$980$$ 7.00000 0.223607
$$981$$ −2.00000 −0.0638551
$$982$$ −4.00000 −0.127645
$$983$$ 16.0000 0.510321 0.255160 0.966899i $$-0.417872\pi$$
0.255160 + 0.966899i $$0.417872\pi$$
$$984$$ 2.00000 0.0637577
$$985$$ 16.0000 0.509802
$$986$$ 0 0
$$987$$ 0 0
$$988$$ −32.0000 −1.01806
$$989$$ 16.0000 0.508770
$$990$$ −1.00000 −0.0317821
$$991$$ −58.0000 −1.84243 −0.921215 0.389053i $$-0.872802\pi$$
−0.921215 + 0.389053i $$0.872802\pi$$
$$992$$ 6.00000 0.190500
$$993$$ −20.0000 −0.634681
$$994$$ 0 0
$$995$$ −18.0000 −0.570638
$$996$$ −12.0000 −0.380235
$$997$$ 18.0000 0.570066 0.285033 0.958518i $$-0.407995\pi$$
0.285033 + 0.958518i $$0.407995\pi$$
$$998$$ 20.0000 0.633089
$$999$$ −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 5610.2.a.m.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
5610.2.a.m.1.1 1 1.1 even 1 trivial