# Properties

 Label 222.2.a.e.1.1 Level $222$ Weight $2$ Character 222.1 Self dual yes Analytic conductor $1.773$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [222,2,Mod(1,222)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(222, 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("222.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$222 = 2 \cdot 3 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 222.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: $$1.77267892487$$ 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 $$\chi$$ $$=$$ 222.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +1.00000 q^{18} -7.00000 q^{19} -1.00000 q^{21} +3.00000 q^{22} +3.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +2.00000 q^{31} +1.00000 q^{32} +3.00000 q^{33} -3.00000 q^{34} +1.00000 q^{36} +1.00000 q^{37} -7.00000 q^{38} -1.00000 q^{39} -6.00000 q^{41} -1.00000 q^{42} -4.00000 q^{43} +3.00000 q^{44} +3.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} -5.00000 q^{50} -3.00000 q^{51} -1.00000 q^{52} +9.00000 q^{53} +1.00000 q^{54} -1.00000 q^{56} -7.00000 q^{57} -10.0000 q^{61} +2.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} +2.00000 q^{67} -3.00000 q^{68} +3.00000 q^{69} +12.0000 q^{71} +1.00000 q^{72} +5.00000 q^{73} +1.00000 q^{74} -5.00000 q^{75} -7.00000 q^{76} -3.00000 q^{77} -1.00000 q^{78} +2.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +3.00000 q^{83} -1.00000 q^{84} -4.00000 q^{86} +3.00000 q^{88} -3.00000 q^{89} +1.00000 q^{91} +3.00000 q^{92} +2.00000 q^{93} +6.00000 q^{94} +1.00000 q^{96} +2.00000 q^{97} -6.00000 q^{98} +3.00000 q^{99} +O(q^{100})$$

## 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
$$7$$ −1.00000 −0.377964 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 3.00000 0.904534 0.452267 0.891883i $$-0.350615\pi$$
0.452267 + 0.891883i $$0.350615\pi$$
$$12$$ 1.00000 0.288675
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −7.00000 −1.60591 −0.802955 0.596040i $$-0.796740\pi$$
−0.802955 + 0.596040i $$0.796740\pi$$
$$20$$ 0 0
$$21$$ −1.00000 −0.218218
$$22$$ 3.00000 0.639602
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −5.00000 −1.00000
$$26$$ −1.00000 −0.196116
$$27$$ 1.00000 0.192450
$$28$$ −1.00000 −0.188982
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 3.00000 0.522233
$$34$$ −3.00000 −0.514496
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 1.00000 0.164399
$$38$$ −7.00000 −1.13555
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ −1.00000 −0.154303
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 3.00000 0.452267
$$45$$ 0 0
$$46$$ 3.00000 0.442326
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −6.00000 −0.857143
$$50$$ −5.00000 −0.707107
$$51$$ −3.00000 −0.420084
$$52$$ −1.00000 −0.138675
$$53$$ 9.00000 1.23625 0.618123 0.786082i $$-0.287894\pi$$
0.618123 + 0.786082i $$0.287894\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ −7.00000 −0.927173
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 2.00000 0.254000
$$63$$ −1.00000 −0.125988
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 3.00000 0.369274
$$67$$ 2.00000 0.244339 0.122169 0.992509i $$-0.461015\pi$$
0.122169 + 0.992509i $$0.461015\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ 3.00000 0.361158
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 5.00000 0.585206 0.292603 0.956234i $$-0.405479\pi$$
0.292603 + 0.956234i $$0.405479\pi$$
$$74$$ 1.00000 0.116248
$$75$$ −5.00000 −0.577350
$$76$$ −7.00000 −0.802955
$$77$$ −3.00000 −0.341882
$$78$$ −1.00000 −0.113228
$$79$$ 2.00000 0.225018 0.112509 0.993651i $$-0.464111\pi$$
0.112509 + 0.993651i $$0.464111\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ 3.00000 0.329293 0.164646 0.986353i $$-0.447352\pi$$
0.164646 + 0.986353i $$0.447352\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 3.00000 0.319801
$$89$$ −3.00000 −0.317999 −0.159000 0.987279i $$-0.550827\pi$$
−0.159000 + 0.987279i $$0.550827\pi$$
$$90$$ 0 0
$$91$$ 1.00000 0.104828
$$92$$ 3.00000 0.312772
$$93$$ 2.00000 0.207390
$$94$$ 6.00000 0.618853
$$95$$ 0 0
$$96$$ 1.00000 0.102062
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ −6.00000 −0.606092
$$99$$ 3.00000 0.301511
$$100$$ −5.00000 −0.500000
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ −3.00000 −0.297044
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ 9.00000 0.874157
$$107$$ 9.00000 0.870063 0.435031 0.900415i $$-0.356737\pi$$
0.435031 + 0.900415i $$0.356737\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 5.00000 0.478913 0.239457 0.970907i $$-0.423031\pi$$
0.239457 + 0.970907i $$0.423031\pi$$
$$110$$ 0 0
$$111$$ 1.00000 0.0949158
$$112$$ −1.00000 −0.0944911
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ −7.00000 −0.655610
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −1.00000 −0.0924500
$$118$$ 0 0
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ −10.0000 −0.905357
$$123$$ −6.00000 −0.541002
$$124$$ 2.00000 0.179605
$$125$$ 0 0
$$126$$ −1.00000 −0.0890871
$$127$$ 5.00000 0.443678 0.221839 0.975083i $$-0.428794\pi$$
0.221839 + 0.975083i $$0.428794\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 18.0000 1.57267 0.786334 0.617802i $$-0.211977\pi$$
0.786334 + 0.617802i $$0.211977\pi$$
$$132$$ 3.00000 0.261116
$$133$$ 7.00000 0.606977
$$134$$ 2.00000 0.172774
$$135$$ 0 0
$$136$$ −3.00000 −0.257248
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ 3.00000 0.255377
$$139$$ 14.0000 1.18746 0.593732 0.804663i $$-0.297654\pi$$
0.593732 + 0.804663i $$0.297654\pi$$
$$140$$ 0 0
$$141$$ 6.00000 0.505291
$$142$$ 12.0000 1.00702
$$143$$ −3.00000 −0.250873
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 5.00000 0.413803
$$147$$ −6.00000 −0.494872
$$148$$ 1.00000 0.0821995
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ −5.00000 −0.408248
$$151$$ 5.00000 0.406894 0.203447 0.979086i $$-0.434786\pi$$
0.203447 + 0.979086i $$0.434786\pi$$
$$152$$ −7.00000 −0.567775
$$153$$ −3.00000 −0.242536
$$154$$ −3.00000 −0.241747
$$155$$ 0 0
$$156$$ −1.00000 −0.0800641
$$157$$ 8.00000 0.638470 0.319235 0.947676i $$-0.396574\pi$$
0.319235 + 0.947676i $$0.396574\pi$$
$$158$$ 2.00000 0.159111
$$159$$ 9.00000 0.713746
$$160$$ 0 0
$$161$$ −3.00000 −0.236433
$$162$$ 1.00000 0.0785674
$$163$$ 11.0000 0.861586 0.430793 0.902451i $$-0.358234\pi$$
0.430793 + 0.902451i $$0.358234\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 3.00000 0.232845
$$167$$ −15.0000 −1.16073 −0.580367 0.814355i $$-0.697091\pi$$
−0.580367 + 0.814355i $$0.697091\pi$$
$$168$$ −1.00000 −0.0771517
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ −7.00000 −0.535303
$$172$$ −4.00000 −0.304997
$$173$$ −9.00000 −0.684257 −0.342129 0.939653i $$-0.611148\pi$$
−0.342129 + 0.939653i $$0.611148\pi$$
$$174$$ 0 0
$$175$$ 5.00000 0.377964
$$176$$ 3.00000 0.226134
$$177$$ 0 0
$$178$$ −3.00000 −0.224860
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ 20.0000 1.48659 0.743294 0.668965i $$-0.233262\pi$$
0.743294 + 0.668965i $$0.233262\pi$$
$$182$$ 1.00000 0.0741249
$$183$$ −10.0000 −0.739221
$$184$$ 3.00000 0.221163
$$185$$ 0 0
$$186$$ 2.00000 0.146647
$$187$$ −9.00000 −0.658145
$$188$$ 6.00000 0.437595
$$189$$ −1.00000 −0.0727393
$$190$$ 0 0
$$191$$ −9.00000 −0.651217 −0.325609 0.945505i $$-0.605569\pi$$
−0.325609 + 0.945505i $$0.605569\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 26.0000 1.87152 0.935760 0.352636i $$-0.114715\pi$$
0.935760 + 0.352636i $$0.114715\pi$$
$$194$$ 2.00000 0.143592
$$195$$ 0 0
$$196$$ −6.00000 −0.428571
$$197$$ −21.0000 −1.49619 −0.748094 0.663593i $$-0.769031\pi$$
−0.748094 + 0.663593i $$0.769031\pi$$
$$198$$ 3.00000 0.213201
$$199$$ −28.0000 −1.98487 −0.992434 0.122782i $$-0.960818\pi$$
−0.992434 + 0.122782i $$0.960818\pi$$
$$200$$ −5.00000 −0.353553
$$201$$ 2.00000 0.141069
$$202$$ 6.00000 0.422159
$$203$$ 0 0
$$204$$ −3.00000 −0.210042
$$205$$ 0 0
$$206$$ 14.0000 0.975426
$$207$$ 3.00000 0.208514
$$208$$ −1.00000 −0.0693375
$$209$$ −21.0000 −1.45260
$$210$$ 0 0
$$211$$ −28.0000 −1.92760 −0.963800 0.266627i $$-0.914091\pi$$
−0.963800 + 0.266627i $$0.914091\pi$$
$$212$$ 9.00000 0.618123
$$213$$ 12.0000 0.822226
$$214$$ 9.00000 0.615227
$$215$$ 0 0
$$216$$ 1.00000 0.0680414
$$217$$ −2.00000 −0.135769
$$218$$ 5.00000 0.338643
$$219$$ 5.00000 0.337869
$$220$$ 0 0
$$221$$ 3.00000 0.201802
$$222$$ 1.00000 0.0671156
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ −5.00000 −0.333333
$$226$$ −6.00000 −0.399114
$$227$$ −24.0000 −1.59294 −0.796468 0.604681i $$-0.793301\pi$$
−0.796468 + 0.604681i $$0.793301\pi$$
$$228$$ −7.00000 −0.463586
$$229$$ −16.0000 −1.05731 −0.528655 0.848837i $$-0.677303\pi$$
−0.528655 + 0.848837i $$0.677303\pi$$
$$230$$ 0 0
$$231$$ −3.00000 −0.197386
$$232$$ 0 0
$$233$$ 30.0000 1.96537 0.982683 0.185296i $$-0.0593245\pi$$
0.982683 + 0.185296i $$0.0593245\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 2.00000 0.129914
$$238$$ 3.00000 0.194461
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ −28.0000 −1.80364 −0.901819 0.432113i $$-0.857768\pi$$
−0.901819 + 0.432113i $$0.857768\pi$$
$$242$$ −2.00000 −0.128565
$$243$$ 1.00000 0.0641500
$$244$$ −10.0000 −0.640184
$$245$$ 0 0
$$246$$ −6.00000 −0.382546
$$247$$ 7.00000 0.445399
$$248$$ 2.00000 0.127000
$$249$$ 3.00000 0.190117
$$250$$ 0 0
$$251$$ 18.0000 1.13615 0.568075 0.822977i $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ −1.00000 −0.0629941
$$253$$ 9.00000 0.565825
$$254$$ 5.00000 0.313728
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −27.0000 −1.68421 −0.842107 0.539311i $$-0.818685\pi$$
−0.842107 + 0.539311i $$0.818685\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ −1.00000 −0.0621370
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 18.0000 1.11204
$$263$$ 6.00000 0.369976 0.184988 0.982741i $$-0.440775\pi$$
0.184988 + 0.982741i $$0.440775\pi$$
$$264$$ 3.00000 0.184637
$$265$$ 0 0
$$266$$ 7.00000 0.429198
$$267$$ −3.00000 −0.183597
$$268$$ 2.00000 0.122169
$$269$$ 15.0000 0.914566 0.457283 0.889321i $$-0.348823\pi$$
0.457283 + 0.889321i $$0.348823\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ −3.00000 −0.181902
$$273$$ 1.00000 0.0605228
$$274$$ −12.0000 −0.724947
$$275$$ −15.0000 −0.904534
$$276$$ 3.00000 0.180579
$$277$$ 17.0000 1.02143 0.510716 0.859750i $$-0.329381\pi$$
0.510716 + 0.859750i $$0.329381\pi$$
$$278$$ 14.0000 0.839664
$$279$$ 2.00000 0.119737
$$280$$ 0 0
$$281$$ 9.00000 0.536895 0.268447 0.963294i $$-0.413489\pi$$
0.268447 + 0.963294i $$0.413489\pi$$
$$282$$ 6.00000 0.357295
$$283$$ −1.00000 −0.0594438 −0.0297219 0.999558i $$-0.509462\pi$$
−0.0297219 + 0.999558i $$0.509462\pi$$
$$284$$ 12.0000 0.712069
$$285$$ 0 0
$$286$$ −3.00000 −0.177394
$$287$$ 6.00000 0.354169
$$288$$ 1.00000 0.0589256
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ 5.00000 0.292603
$$293$$ 9.00000 0.525786 0.262893 0.964825i $$-0.415323\pi$$
0.262893 + 0.964825i $$0.415323\pi$$
$$294$$ −6.00000 −0.349927
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ 3.00000 0.174078
$$298$$ −18.0000 −1.04271
$$299$$ −3.00000 −0.173494
$$300$$ −5.00000 −0.288675
$$301$$ 4.00000 0.230556
$$302$$ 5.00000 0.287718
$$303$$ 6.00000 0.344691
$$304$$ −7.00000 −0.401478
$$305$$ 0 0
$$306$$ −3.00000 −0.171499
$$307$$ −16.0000 −0.913168 −0.456584 0.889680i $$-0.650927\pi$$
−0.456584 + 0.889680i $$0.650927\pi$$
$$308$$ −3.00000 −0.170941
$$309$$ 14.0000 0.796432
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ −1.00000 −0.0566139
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ 8.00000 0.451466
$$315$$ 0 0
$$316$$ 2.00000 0.112509
$$317$$ 30.0000 1.68497 0.842484 0.538721i $$-0.181092\pi$$
0.842484 + 0.538721i $$0.181092\pi$$
$$318$$ 9.00000 0.504695
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 9.00000 0.502331
$$322$$ −3.00000 −0.167183
$$323$$ 21.0000 1.16847
$$324$$ 1.00000 0.0555556
$$325$$ 5.00000 0.277350
$$326$$ 11.0000 0.609234
$$327$$ 5.00000 0.276501
$$328$$ −6.00000 −0.331295
$$329$$ −6.00000 −0.330791
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 3.00000 0.164646
$$333$$ 1.00000 0.0547997
$$334$$ −15.0000 −0.820763
$$335$$ 0 0
$$336$$ −1.00000 −0.0545545
$$337$$ 5.00000 0.272367 0.136184 0.990684i $$-0.456516\pi$$
0.136184 + 0.990684i $$0.456516\pi$$
$$338$$ −12.0000 −0.652714
$$339$$ −6.00000 −0.325875
$$340$$ 0 0
$$341$$ 6.00000 0.324918
$$342$$ −7.00000 −0.378517
$$343$$ 13.0000 0.701934
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ −9.00000 −0.483843
$$347$$ 6.00000 0.322097 0.161048 0.986947i $$-0.448512\pi$$
0.161048 + 0.986947i $$0.448512\pi$$
$$348$$ 0 0
$$349$$ 26.0000 1.39175 0.695874 0.718164i $$-0.255017\pi$$
0.695874 + 0.718164i $$0.255017\pi$$
$$350$$ 5.00000 0.267261
$$351$$ −1.00000 −0.0533761
$$352$$ 3.00000 0.159901
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −3.00000 −0.159000
$$357$$ 3.00000 0.158777
$$358$$ 12.0000 0.634220
$$359$$ 30.0000 1.58334 0.791670 0.610949i $$-0.209212\pi$$
0.791670 + 0.610949i $$0.209212\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ 20.0000 1.05118
$$363$$ −2.00000 −0.104973
$$364$$ 1.00000 0.0524142
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ −25.0000 −1.30499 −0.652495 0.757793i $$-0.726278\pi$$
−0.652495 + 0.757793i $$0.726278\pi$$
$$368$$ 3.00000 0.156386
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −9.00000 −0.467257
$$372$$ 2.00000 0.103695
$$373$$ 20.0000 1.03556 0.517780 0.855514i $$-0.326758\pi$$
0.517780 + 0.855514i $$0.326758\pi$$
$$374$$ −9.00000 −0.465379
$$375$$ 0 0
$$376$$ 6.00000 0.309426
$$377$$ 0 0
$$378$$ −1.00000 −0.0514344
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ 0 0
$$381$$ 5.00000 0.256158
$$382$$ −9.00000 −0.460480
$$383$$ 21.0000 1.07305 0.536525 0.843884i $$-0.319737\pi$$
0.536525 + 0.843884i $$0.319737\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 26.0000 1.32337
$$387$$ −4.00000 −0.203331
$$388$$ 2.00000 0.101535
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ −9.00000 −0.455150
$$392$$ −6.00000 −0.303046
$$393$$ 18.0000 0.907980
$$394$$ −21.0000 −1.05796
$$395$$ 0 0
$$396$$ 3.00000 0.150756
$$397$$ 8.00000 0.401508 0.200754 0.979642i $$-0.435661\pi$$
0.200754 + 0.979642i $$0.435661\pi$$
$$398$$ −28.0000 −1.40351
$$399$$ 7.00000 0.350438
$$400$$ −5.00000 −0.250000
$$401$$ −3.00000 −0.149813 −0.0749064 0.997191i $$-0.523866\pi$$
−0.0749064 + 0.997191i $$0.523866\pi$$
$$402$$ 2.00000 0.0997509
$$403$$ −2.00000 −0.0996271
$$404$$ 6.00000 0.298511
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 3.00000 0.148704
$$408$$ −3.00000 −0.148522
$$409$$ −22.0000 −1.08783 −0.543915 0.839140i $$-0.683059\pi$$
−0.543915 + 0.839140i $$0.683059\pi$$
$$410$$ 0 0
$$411$$ −12.0000 −0.591916
$$412$$ 14.0000 0.689730
$$413$$ 0 0
$$414$$ 3.00000 0.147442
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 14.0000 0.685583
$$418$$ −21.0000 −1.02714
$$419$$ −15.0000 −0.732798 −0.366399 0.930458i $$-0.619409\pi$$
−0.366399 + 0.930458i $$0.619409\pi$$
$$420$$ 0 0
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ −28.0000 −1.36302
$$423$$ 6.00000 0.291730
$$424$$ 9.00000 0.437079
$$425$$ 15.0000 0.727607
$$426$$ 12.0000 0.581402
$$427$$ 10.0000 0.483934
$$428$$ 9.00000 0.435031
$$429$$ −3.00000 −0.144841
$$430$$ 0 0
$$431$$ −39.0000 −1.87856 −0.939282 0.343146i $$-0.888507\pi$$
−0.939282 + 0.343146i $$0.888507\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −25.0000 −1.20142 −0.600712 0.799466i $$-0.705116\pi$$
−0.600712 + 0.799466i $$0.705116\pi$$
$$434$$ −2.00000 −0.0960031
$$435$$ 0 0
$$436$$ 5.00000 0.239457
$$437$$ −21.0000 −1.00457
$$438$$ 5.00000 0.238909
$$439$$ −10.0000 −0.477274 −0.238637 0.971109i $$-0.576701\pi$$
−0.238637 + 0.971109i $$0.576701\pi$$
$$440$$ 0 0
$$441$$ −6.00000 −0.285714
$$442$$ 3.00000 0.142695
$$443$$ 36.0000 1.71041 0.855206 0.518289i $$-0.173431\pi$$
0.855206 + 0.518289i $$0.173431\pi$$
$$444$$ 1.00000 0.0474579
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ −18.0000 −0.851371
$$448$$ −1.00000 −0.0472456
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ −5.00000 −0.235702
$$451$$ −18.0000 −0.847587
$$452$$ −6.00000 −0.282216
$$453$$ 5.00000 0.234920
$$454$$ −24.0000 −1.12638
$$455$$ 0 0
$$456$$ −7.00000 −0.327805
$$457$$ 8.00000 0.374224 0.187112 0.982339i $$-0.440087\pi$$
0.187112 + 0.982339i $$0.440087\pi$$
$$458$$ −16.0000 −0.747631
$$459$$ −3.00000 −0.140028
$$460$$ 0 0
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ −3.00000 −0.139573
$$463$$ −34.0000 −1.58011 −0.790057 0.613033i $$-0.789949\pi$$
−0.790057 + 0.613033i $$0.789949\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 30.0000 1.38972
$$467$$ 6.00000 0.277647 0.138823 0.990317i $$-0.455668\pi$$
0.138823 + 0.990317i $$0.455668\pi$$
$$468$$ −1.00000 −0.0462250
$$469$$ −2.00000 −0.0923514
$$470$$ 0 0
$$471$$ 8.00000 0.368621
$$472$$ 0 0
$$473$$ −12.0000 −0.551761
$$474$$ 2.00000 0.0918630
$$475$$ 35.0000 1.60591
$$476$$ 3.00000 0.137505
$$477$$ 9.00000 0.412082
$$478$$ −24.0000 −1.09773
$$479$$ −39.0000 −1.78196 −0.890978 0.454047i $$-0.849980\pi$$
−0.890978 + 0.454047i $$0.849980\pi$$
$$480$$ 0 0
$$481$$ −1.00000 −0.0455961
$$482$$ −28.0000 −1.27537
$$483$$ −3.00000 −0.136505
$$484$$ −2.00000 −0.0909091
$$485$$ 0 0
$$486$$ 1.00000 0.0453609
$$487$$ 38.0000 1.72194 0.860972 0.508652i $$-0.169856\pi$$
0.860972 + 0.508652i $$0.169856\pi$$
$$488$$ −10.0000 −0.452679
$$489$$ 11.0000 0.497437
$$490$$ 0 0
$$491$$ 15.0000 0.676941 0.338470 0.940977i $$-0.390091\pi$$
0.338470 + 0.940977i $$0.390091\pi$$
$$492$$ −6.00000 −0.270501
$$493$$ 0 0
$$494$$ 7.00000 0.314945
$$495$$ 0 0
$$496$$ 2.00000 0.0898027
$$497$$ −12.0000 −0.538274
$$498$$ 3.00000 0.134433
$$499$$ −13.0000 −0.581960 −0.290980 0.956729i $$-0.593981\pi$$
−0.290980 + 0.956729i $$0.593981\pi$$
$$500$$ 0 0
$$501$$ −15.0000 −0.670151
$$502$$ 18.0000 0.803379
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ 0 0
$$506$$ 9.00000 0.400099
$$507$$ −12.0000 −0.532939
$$508$$ 5.00000 0.221839
$$509$$ 21.0000 0.930809 0.465404 0.885098i $$-0.345909\pi$$
0.465404 + 0.885098i $$0.345909\pi$$
$$510$$ 0 0
$$511$$ −5.00000 −0.221187
$$512$$ 1.00000 0.0441942
$$513$$ −7.00000 −0.309058
$$514$$ −27.0000 −1.19092
$$515$$ 0 0
$$516$$ −4.00000 −0.176090
$$517$$ 18.0000 0.791639
$$518$$ −1.00000 −0.0439375
$$519$$ −9.00000 −0.395056
$$520$$ 0 0
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ 0 0
$$523$$ 8.00000 0.349816 0.174908 0.984585i $$-0.444037\pi$$
0.174908 + 0.984585i $$0.444037\pi$$
$$524$$ 18.0000 0.786334
$$525$$ 5.00000 0.218218
$$526$$ 6.00000 0.261612
$$527$$ −6.00000 −0.261364
$$528$$ 3.00000 0.130558
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 7.00000 0.303488
$$533$$ 6.00000 0.259889
$$534$$ −3.00000 −0.129823
$$535$$ 0 0
$$536$$ 2.00000 0.0863868
$$537$$ 12.0000 0.517838
$$538$$ 15.0000 0.646696
$$539$$ −18.0000 −0.775315
$$540$$ 0 0
$$541$$ 11.0000 0.472927 0.236463 0.971640i $$-0.424012\pi$$
0.236463 + 0.971640i $$0.424012\pi$$
$$542$$ 8.00000 0.343629
$$543$$ 20.0000 0.858282
$$544$$ −3.00000 −0.128624
$$545$$ 0 0
$$546$$ 1.00000 0.0427960
$$547$$ −13.0000 −0.555840 −0.277920 0.960604i $$-0.589645\pi$$
−0.277920 + 0.960604i $$0.589645\pi$$
$$548$$ −12.0000 −0.512615
$$549$$ −10.0000 −0.426790
$$550$$ −15.0000 −0.639602
$$551$$ 0 0
$$552$$ 3.00000 0.127688
$$553$$ −2.00000 −0.0850487
$$554$$ 17.0000 0.722261
$$555$$ 0 0
$$556$$ 14.0000 0.593732
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ 2.00000 0.0846668
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ −9.00000 −0.379980
$$562$$ 9.00000 0.379642
$$563$$ 36.0000 1.51722 0.758610 0.651546i $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ 6.00000 0.252646
$$565$$ 0 0
$$566$$ −1.00000 −0.0420331
$$567$$ −1.00000 −0.0419961
$$568$$ 12.0000 0.503509
$$569$$ −21.0000 −0.880366 −0.440183 0.897908i $$-0.645086\pi$$
−0.440183 + 0.897908i $$0.645086\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ −3.00000 −0.125436
$$573$$ −9.00000 −0.375980
$$574$$ 6.00000 0.250435
$$575$$ −15.0000 −0.625543
$$576$$ 1.00000 0.0416667
$$577$$ 38.0000 1.58196 0.790980 0.611842i $$-0.209571\pi$$
0.790980 + 0.611842i $$0.209571\pi$$
$$578$$ −8.00000 −0.332756
$$579$$ 26.0000 1.08052
$$580$$ 0 0
$$581$$ −3.00000 −0.124461
$$582$$ 2.00000 0.0829027
$$583$$ 27.0000 1.11823
$$584$$ 5.00000 0.206901
$$585$$ 0 0
$$586$$ 9.00000 0.371787
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ −6.00000 −0.247436
$$589$$ −14.0000 −0.576860
$$590$$ 0 0
$$591$$ −21.0000 −0.863825
$$592$$ 1.00000 0.0410997
$$593$$ 24.0000 0.985562 0.492781 0.870153i $$-0.335980\pi$$
0.492781 + 0.870153i $$0.335980\pi$$
$$594$$ 3.00000 0.123091
$$595$$ 0 0
$$596$$ −18.0000 −0.737309
$$597$$ −28.0000 −1.14596
$$598$$ −3.00000 −0.122679
$$599$$ −6.00000 −0.245153 −0.122577 0.992459i $$-0.539116\pi$$
−0.122577 + 0.992459i $$0.539116\pi$$
$$600$$ −5.00000 −0.204124
$$601$$ −25.0000 −1.01977 −0.509886 0.860242i $$-0.670312\pi$$
−0.509886 + 0.860242i $$0.670312\pi$$
$$602$$ 4.00000 0.163028
$$603$$ 2.00000 0.0814463
$$604$$ 5.00000 0.203447
$$605$$ 0 0
$$606$$ 6.00000 0.243733
$$607$$ 14.0000 0.568242 0.284121 0.958788i $$-0.408298\pi$$
0.284121 + 0.958788i $$0.408298\pi$$
$$608$$ −7.00000 −0.283887
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −6.00000 −0.242734
$$612$$ −3.00000 −0.121268
$$613$$ 2.00000 0.0807792 0.0403896 0.999184i $$-0.487140\pi$$
0.0403896 + 0.999184i $$0.487140\pi$$
$$614$$ −16.0000 −0.645707
$$615$$ 0 0
$$616$$ −3.00000 −0.120873
$$617$$ −24.0000 −0.966204 −0.483102 0.875564i $$-0.660490\pi$$
−0.483102 + 0.875564i $$0.660490\pi$$
$$618$$ 14.0000 0.563163
$$619$$ 38.0000 1.52735 0.763674 0.645601i $$-0.223393\pi$$
0.763674 + 0.645601i $$0.223393\pi$$
$$620$$ 0 0
$$621$$ 3.00000 0.120386
$$622$$ 0 0
$$623$$ 3.00000 0.120192
$$624$$ −1.00000 −0.0400320
$$625$$ 25.0000 1.00000
$$626$$ −10.0000 −0.399680
$$627$$ −21.0000 −0.838659
$$628$$ 8.00000 0.319235
$$629$$ −3.00000 −0.119618
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ 2.00000 0.0795557
$$633$$ −28.0000 −1.11290
$$634$$ 30.0000 1.19145
$$635$$ 0 0
$$636$$ 9.00000 0.356873
$$637$$ 6.00000 0.237729
$$638$$ 0 0
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ −12.0000 −0.473972 −0.236986 0.971513i $$-0.576159\pi$$
−0.236986 + 0.971513i $$0.576159\pi$$
$$642$$ 9.00000 0.355202
$$643$$ 5.00000 0.197181 0.0985904 0.995128i $$-0.468567\pi$$
0.0985904 + 0.995128i $$0.468567\pi$$
$$644$$ −3.00000 −0.118217
$$645$$ 0 0
$$646$$ 21.0000 0.826234
$$647$$ 3.00000 0.117942 0.0589711 0.998260i $$-0.481218\pi$$
0.0589711 + 0.998260i $$0.481218\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0 0
$$650$$ 5.00000 0.196116
$$651$$ −2.00000 −0.0783862
$$652$$ 11.0000 0.430793
$$653$$ 24.0000 0.939193 0.469596 0.882881i $$-0.344399\pi$$
0.469596 + 0.882881i $$0.344399\pi$$
$$654$$ 5.00000 0.195515
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 5.00000 0.195069
$$658$$ −6.00000 −0.233904
$$659$$ 24.0000 0.934907 0.467454 0.884018i $$-0.345171\pi$$
0.467454 + 0.884018i $$0.345171\pi$$
$$660$$ 0 0
$$661$$ −25.0000 −0.972387 −0.486194 0.873851i $$-0.661615\pi$$
−0.486194 + 0.873851i $$0.661615\pi$$
$$662$$ −4.00000 −0.155464
$$663$$ 3.00000 0.116510
$$664$$ 3.00000 0.116423
$$665$$ 0 0
$$666$$ 1.00000 0.0387492
$$667$$ 0 0
$$668$$ −15.0000 −0.580367
$$669$$ −16.0000 −0.618596
$$670$$ 0 0
$$671$$ −30.0000 −1.15814
$$672$$ −1.00000 −0.0385758
$$673$$ −19.0000 −0.732396 −0.366198 0.930537i $$-0.619341\pi$$
−0.366198 + 0.930537i $$0.619341\pi$$
$$674$$ 5.00000 0.192593
$$675$$ −5.00000 −0.192450
$$676$$ −12.0000 −0.461538
$$677$$ 27.0000 1.03769 0.518847 0.854867i $$-0.326361\pi$$
0.518847 + 0.854867i $$0.326361\pi$$
$$678$$ −6.00000 −0.230429
$$679$$ −2.00000 −0.0767530
$$680$$ 0 0
$$681$$ −24.0000 −0.919682
$$682$$ 6.00000 0.229752
$$683$$ 18.0000 0.688751 0.344375 0.938832i $$-0.388091\pi$$
0.344375 + 0.938832i $$0.388091\pi$$
$$684$$ −7.00000 −0.267652
$$685$$ 0 0
$$686$$ 13.0000 0.496342
$$687$$ −16.0000 −0.610438
$$688$$ −4.00000 −0.152499
$$689$$ −9.00000 −0.342873
$$690$$ 0 0
$$691$$ −40.0000 −1.52167 −0.760836 0.648944i $$-0.775211\pi$$
−0.760836 + 0.648944i $$0.775211\pi$$
$$692$$ −9.00000 −0.342129
$$693$$ −3.00000 −0.113961
$$694$$ 6.00000 0.227757
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 18.0000 0.681799
$$698$$ 26.0000 0.984115
$$699$$ 30.0000 1.13470
$$700$$ 5.00000 0.188982
$$701$$ −24.0000 −0.906467 −0.453234 0.891392i $$-0.649730\pi$$
−0.453234 + 0.891392i $$0.649730\pi$$
$$702$$ −1.00000 −0.0377426
$$703$$ −7.00000 −0.264010
$$704$$ 3.00000 0.113067
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ −6.00000 −0.225653
$$708$$ 0 0
$$709$$ −25.0000 −0.938895 −0.469447 0.882960i $$-0.655547\pi$$
−0.469447 + 0.882960i $$0.655547\pi$$
$$710$$ 0 0
$$711$$ 2.00000 0.0750059
$$712$$ −3.00000 −0.112430
$$713$$ 6.00000 0.224702
$$714$$ 3.00000 0.112272
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ −24.0000 −0.896296
$$718$$ 30.0000 1.11959
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 0 0
$$721$$ −14.0000 −0.521387
$$722$$ 30.0000 1.11648
$$723$$ −28.0000 −1.04133
$$724$$ 20.0000 0.743294
$$725$$ 0 0
$$726$$ −2.00000 −0.0742270
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ 1.00000 0.0370625
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 12.0000 0.443836
$$732$$ −10.0000 −0.369611
$$733$$ −34.0000 −1.25582 −0.627909 0.778287i $$-0.716089\pi$$
−0.627909 + 0.778287i $$0.716089\pi$$
$$734$$ −25.0000 −0.922767
$$735$$ 0 0
$$736$$ 3.00000 0.110581
$$737$$ 6.00000 0.221013
$$738$$ −6.00000 −0.220863
$$739$$ 38.0000 1.39785 0.698926 0.715194i $$-0.253662\pi$$
0.698926 + 0.715194i $$0.253662\pi$$
$$740$$ 0 0
$$741$$ 7.00000 0.257151
$$742$$ −9.00000 −0.330400
$$743$$ −36.0000 −1.32071 −0.660356 0.750953i $$-0.729595\pi$$
−0.660356 + 0.750953i $$0.729595\pi$$
$$744$$ 2.00000 0.0733236
$$745$$ 0 0
$$746$$ 20.0000 0.732252
$$747$$ 3.00000 0.109764
$$748$$ −9.00000 −0.329073
$$749$$ −9.00000 −0.328853
$$750$$ 0 0
$$751$$ 32.0000 1.16770 0.583848 0.811863i $$-0.301546\pi$$
0.583848 + 0.811863i $$0.301546\pi$$
$$752$$ 6.00000 0.218797
$$753$$ 18.0000 0.655956
$$754$$ 0 0
$$755$$ 0 0
$$756$$ −1.00000 −0.0363696
$$757$$ 5.00000 0.181728 0.0908640 0.995863i $$-0.471037\pi$$
0.0908640 + 0.995863i $$0.471037\pi$$
$$758$$ −16.0000 −0.581146
$$759$$ 9.00000 0.326679
$$760$$ 0 0
$$761$$ 54.0000 1.95750 0.978749 0.205061i $$-0.0657392\pi$$
0.978749 + 0.205061i $$0.0657392\pi$$
$$762$$ 5.00000 0.181131
$$763$$ −5.00000 −0.181012
$$764$$ −9.00000 −0.325609
$$765$$ 0 0
$$766$$ 21.0000 0.758761
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ −16.0000 −0.576975 −0.288487 0.957484i $$-0.593152\pi$$
−0.288487 + 0.957484i $$0.593152\pi$$
$$770$$ 0 0
$$771$$ −27.0000 −0.972381
$$772$$ 26.0000 0.935760
$$773$$ −21.0000 −0.755318 −0.377659 0.925945i $$-0.623271\pi$$
−0.377659 + 0.925945i $$0.623271\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ −10.0000 −0.359211
$$776$$ 2.00000 0.0717958
$$777$$ −1.00000 −0.0358748
$$778$$ −6.00000 −0.215110
$$779$$ 42.0000 1.50481
$$780$$ 0 0
$$781$$ 36.0000 1.28818
$$782$$ −9.00000 −0.321839
$$783$$ 0 0
$$784$$ −6.00000 −0.214286
$$785$$ 0 0
$$786$$ 18.0000 0.642039
$$787$$ −34.0000 −1.21197 −0.605985 0.795476i $$-0.707221\pi$$
−0.605985 + 0.795476i $$0.707221\pi$$
$$788$$ −21.0000 −0.748094
$$789$$ 6.00000 0.213606
$$790$$ 0 0
$$791$$ 6.00000 0.213335
$$792$$ 3.00000 0.106600
$$793$$ 10.0000 0.355110
$$794$$ 8.00000 0.283909
$$795$$ 0 0
$$796$$ −28.0000 −0.992434
$$797$$ 12.0000 0.425062 0.212531 0.977154i $$-0.431829\pi$$
0.212531 + 0.977154i $$0.431829\pi$$
$$798$$ 7.00000 0.247797
$$799$$ −18.0000 −0.636794
$$800$$ −5.00000 −0.176777
$$801$$ −3.00000 −0.106000
$$802$$ −3.00000 −0.105934
$$803$$ 15.0000 0.529339
$$804$$ 2.00000 0.0705346
$$805$$ 0 0
$$806$$ −2.00000 −0.0704470
$$807$$ 15.0000 0.528025
$$808$$ 6.00000 0.211079
$$809$$ −27.0000 −0.949269 −0.474635 0.880183i $$-0.657420\pi$$
−0.474635 + 0.880183i $$0.657420\pi$$
$$810$$ 0 0
$$811$$ −52.0000 −1.82597 −0.912983 0.407997i $$-0.866228\pi$$
−0.912983 + 0.407997i $$0.866228\pi$$
$$812$$ 0 0
$$813$$ 8.00000 0.280572
$$814$$ 3.00000 0.105150
$$815$$ 0 0
$$816$$ −3.00000 −0.105021
$$817$$ 28.0000 0.979596
$$818$$ −22.0000 −0.769212
$$819$$ 1.00000 0.0349428
$$820$$ 0 0
$$821$$ 45.0000 1.57051 0.785255 0.619172i $$-0.212532\pi$$
0.785255 + 0.619172i $$0.212532\pi$$
$$822$$ −12.0000 −0.418548
$$823$$ 17.0000 0.592583 0.296291 0.955098i $$-0.404250\pi$$
0.296291 + 0.955098i $$0.404250\pi$$
$$824$$ 14.0000 0.487713
$$825$$ −15.0000 −0.522233
$$826$$ 0 0
$$827$$ −18.0000 −0.625921 −0.312961 0.949766i $$-0.601321\pi$$
−0.312961 + 0.949766i $$0.601321\pi$$
$$828$$ 3.00000 0.104257
$$829$$ 17.0000 0.590434 0.295217 0.955430i $$-0.404608\pi$$
0.295217 + 0.955430i $$0.404608\pi$$
$$830$$ 0 0
$$831$$ 17.0000 0.589723
$$832$$ −1.00000 −0.0346688
$$833$$ 18.0000 0.623663
$$834$$ 14.0000 0.484780
$$835$$ 0 0
$$836$$ −21.0000 −0.726300
$$837$$ 2.00000 0.0691301
$$838$$ −15.0000 −0.518166
$$839$$ −12.0000 −0.414286 −0.207143 0.978311i $$-0.566417\pi$$
−0.207143 + 0.978311i $$0.566417\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ −10.0000 −0.344623
$$843$$ 9.00000 0.309976
$$844$$ −28.0000 −0.963800
$$845$$ 0 0
$$846$$ 6.00000 0.206284
$$847$$ 2.00000 0.0687208
$$848$$ 9.00000 0.309061
$$849$$ −1.00000 −0.0343199
$$850$$ 15.0000 0.514496
$$851$$ 3.00000 0.102839
$$852$$ 12.0000 0.411113
$$853$$ −37.0000 −1.26686 −0.633428 0.773802i $$-0.718353\pi$$
−0.633428 + 0.773802i $$0.718353\pi$$
$$854$$ 10.0000 0.342193
$$855$$ 0 0
$$856$$ 9.00000 0.307614
$$857$$ −21.0000 −0.717346 −0.358673 0.933463i $$-0.616771\pi$$
−0.358673 + 0.933463i $$0.616771\pi$$
$$858$$ −3.00000 −0.102418
$$859$$ −13.0000 −0.443554 −0.221777 0.975097i $$-0.571186\pi$$
−0.221777 + 0.975097i $$0.571186\pi$$
$$860$$ 0 0
$$861$$ 6.00000 0.204479
$$862$$ −39.0000 −1.32835
$$863$$ −18.0000 −0.612727 −0.306364 0.951915i $$-0.599112\pi$$
−0.306364 + 0.951915i $$0.599112\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 0 0
$$866$$ −25.0000 −0.849535
$$867$$ −8.00000 −0.271694
$$868$$ −2.00000 −0.0678844
$$869$$ 6.00000 0.203536
$$870$$ 0 0
$$871$$ −2.00000 −0.0677674
$$872$$ 5.00000 0.169321
$$873$$ 2.00000 0.0676897
$$874$$ −21.0000 −0.710336
$$875$$ 0 0
$$876$$ 5.00000 0.168934
$$877$$ −58.0000 −1.95852 −0.979260 0.202606i $$-0.935059\pi$$
−0.979260 + 0.202606i $$0.935059\pi$$
$$878$$ −10.0000 −0.337484
$$879$$ 9.00000 0.303562
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ −6.00000 −0.202031
$$883$$ −25.0000 −0.841317 −0.420658 0.907219i $$-0.638201\pi$$
−0.420658 + 0.907219i $$0.638201\pi$$
$$884$$ 3.00000 0.100901
$$885$$ 0 0
$$886$$ 36.0000 1.20944
$$887$$ 18.0000 0.604381 0.302190 0.953248i $$-0.402282\pi$$
0.302190 + 0.953248i $$0.402282\pi$$
$$888$$ 1.00000 0.0335578
$$889$$ −5.00000 −0.167695
$$890$$ 0 0
$$891$$ 3.00000 0.100504
$$892$$ −16.0000 −0.535720
$$893$$ −42.0000 −1.40548
$$894$$ −18.0000 −0.602010
$$895$$ 0 0
$$896$$ −1.00000 −0.0334077
$$897$$ −3.00000 −0.100167
$$898$$ −18.0000 −0.600668
$$899$$ 0 0
$$900$$ −5.00000 −0.166667
$$901$$ −27.0000 −0.899500
$$902$$ −18.0000 −0.599334
$$903$$ 4.00000 0.133112
$$904$$ −6.00000 −0.199557
$$905$$ 0 0
$$906$$ 5.00000 0.166114
$$907$$ 23.0000 0.763702 0.381851 0.924224i $$-0.375287\pi$$
0.381851 + 0.924224i $$0.375287\pi$$
$$908$$ −24.0000 −0.796468
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ −7.00000 −0.231793
$$913$$ 9.00000 0.297857
$$914$$ 8.00000 0.264616
$$915$$ 0 0
$$916$$ −16.0000 −0.528655
$$917$$ −18.0000 −0.594412
$$918$$ −3.00000 −0.0990148
$$919$$ 38.0000 1.25350 0.626752 0.779219i $$-0.284384\pi$$
0.626752 + 0.779219i $$0.284384\pi$$
$$920$$ 0 0
$$921$$ −16.0000 −0.527218
$$922$$ −6.00000 −0.197599
$$923$$ −12.0000 −0.394985
$$924$$ −3.00000 −0.0986928
$$925$$ −5.00000 −0.164399
$$926$$ −34.0000 −1.11731
$$927$$ 14.0000 0.459820
$$928$$ 0 0
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ 42.0000 1.37649
$$932$$ 30.0000 0.982683
$$933$$ 0 0
$$934$$ 6.00000 0.196326
$$935$$ 0 0
$$936$$ −1.00000 −0.0326860
$$937$$ 38.0000 1.24141 0.620703 0.784046i $$-0.286847\pi$$
0.620703 + 0.784046i $$0.286847\pi$$
$$938$$ −2.00000 −0.0653023
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ −6.00000 −0.195594 −0.0977972 0.995206i $$-0.531180\pi$$
−0.0977972 + 0.995206i $$0.531180\pi$$
$$942$$ 8.00000 0.260654
$$943$$ −18.0000 −0.586161
$$944$$ 0 0
$$945$$ 0 0
$$946$$ −12.0000 −0.390154
$$947$$ 6.00000 0.194974 0.0974869 0.995237i $$-0.468920\pi$$
0.0974869 + 0.995237i $$0.468920\pi$$
$$948$$ 2.00000 0.0649570
$$949$$ −5.00000 −0.162307
$$950$$ 35.0000 1.13555
$$951$$ 30.0000 0.972817
$$952$$ 3.00000 0.0972306
$$953$$ −24.0000 −0.777436 −0.388718 0.921357i $$-0.627082\pi$$
−0.388718 + 0.921357i $$0.627082\pi$$
$$954$$ 9.00000 0.291386
$$955$$ 0 0
$$956$$ −24.0000 −0.776215
$$957$$ 0 0
$$958$$ −39.0000 −1.26003
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ −1.00000 −0.0322413
$$963$$ 9.00000 0.290021
$$964$$ −28.0000 −0.901819
$$965$$ 0 0
$$966$$ −3.00000 −0.0965234
$$967$$ 44.0000 1.41494 0.707472 0.706741i $$-0.249835\pi$$
0.707472 + 0.706741i $$0.249835\pi$$
$$968$$ −2.00000 −0.0642824
$$969$$ 21.0000 0.674617
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −14.0000 −0.448819
$$974$$ 38.0000 1.21760
$$975$$ 5.00000 0.160128
$$976$$ −10.0000 −0.320092
$$977$$ −3.00000 −0.0959785 −0.0479893 0.998848i $$-0.515281\pi$$
−0.0479893 + 0.998848i $$0.515281\pi$$
$$978$$ 11.0000 0.351741
$$979$$ −9.00000 −0.287641
$$980$$ 0 0
$$981$$ 5.00000 0.159638
$$982$$ 15.0000 0.478669
$$983$$ −18.0000 −0.574111 −0.287055 0.957914i $$-0.592676\pi$$
−0.287055 + 0.957914i $$0.592676\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −6.00000 −0.190982
$$988$$ 7.00000 0.222700
$$989$$ −12.0000 −0.381578
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 2.00000 0.0635001
$$993$$ −4.00000 −0.126936
$$994$$ −12.0000 −0.380617
$$995$$ 0 0
$$996$$ 3.00000 0.0950586
$$997$$ 53.0000 1.67853 0.839263 0.543725i $$-0.182987\pi$$
0.839263 + 0.543725i $$0.182987\pi$$
$$998$$ −13.0000 −0.411508
$$999$$ 1.00000 0.0316386
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 222.2.a.e.1.1 1
3.2 odd 2 666.2.a.a.1.1 1
4.3 odd 2 1776.2.a.c.1.1 1
5.4 even 2 5550.2.a.h.1.1 1
8.3 odd 2 7104.2.a.u.1.1 1
8.5 even 2 7104.2.a.g.1.1 1
12.11 even 2 5328.2.a.l.1.1 1
37.36 even 2 8214.2.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
222.2.a.e.1.1 1 1.1 even 1 trivial
666.2.a.a.1.1 1 3.2 odd 2
1776.2.a.c.1.1 1 4.3 odd 2
5328.2.a.l.1.1 1 12.11 even 2
5550.2.a.h.1.1 1 5.4 even 2
7104.2.a.g.1.1 1 8.5 even 2
7104.2.a.u.1.1 1 8.3 odd 2
8214.2.a.d.1.1 1 37.36 even 2