# Properties

 Label 462.2.a.e.1.1 Level $462$ Weight $2$ Character 462.1 Self dual yes Analytic conductor $3.689$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [462,2,Mod(1,462)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(462, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("462.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$462 = 2 \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 462.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: $$3.68908857338$$ 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 $$\chi$$ $$=$$ 462.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -6.00000 q^{13} +1.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} -4.00000 q^{20} -1.00000 q^{21} -1.00000 q^{22} -8.00000 q^{23} -1.00000 q^{24} +11.0000 q^{25} -6.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} +4.00000 q^{30} +6.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} -4.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -2.00000 q^{38} +6.00000 q^{39} -4.00000 q^{40} +12.0000 q^{41} -1.00000 q^{42} +4.00000 q^{43} -1.00000 q^{44} -4.00000 q^{45} -8.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +11.0000 q^{50} +4.00000 q^{51} -6.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} +4.00000 q^{55} +1.00000 q^{56} +2.00000 q^{57} -6.00000 q^{58} +4.00000 q^{60} +10.0000 q^{61} +6.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +24.0000 q^{65} +1.00000 q^{66} +4.00000 q^{67} -4.00000 q^{68} +8.00000 q^{69} -4.00000 q^{70} -12.0000 q^{71} +1.00000 q^{72} -6.00000 q^{74} -11.0000 q^{75} -2.00000 q^{76} -1.00000 q^{77} +6.00000 q^{78} -16.0000 q^{79} -4.00000 q^{80} +1.00000 q^{81} +12.0000 q^{82} -14.0000 q^{83} -1.00000 q^{84} +16.0000 q^{85} +4.00000 q^{86} +6.00000 q^{87} -1.00000 q^{88} -14.0000 q^{89} -4.00000 q^{90} -6.00000 q^{91} -8.00000 q^{92} -6.00000 q^{93} +6.00000 q^{94} +8.00000 q^{95} -1.00000 q^{96} -14.0000 q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ −4.00000 −1.78885 −0.894427 0.447214i $$-0.852416\pi$$
−0.894427 + 0.447214i $$0.852416\pi$$
$$6$$ −1.00000 −0.408248
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ −4.00000 −1.26491
$$11$$ −1.00000 −0.301511
$$12$$ −1.00000 −0.288675
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 4.00000 1.03280
$$16$$ 1.00000 0.250000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ −4.00000 −0.894427
$$21$$ −1.00000 −0.218218
$$22$$ −1.00000 −0.213201
$$23$$ −8.00000 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 11.0000 2.20000
$$26$$ −6.00000 −1.17670
$$27$$ −1.00000 −0.192450
$$28$$ 1.00000 0.188982
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 4.00000 0.730297
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 1.00000 0.174078
$$34$$ −4.00000 −0.685994
$$35$$ −4.00000 −0.676123
$$36$$ 1.00000 0.166667
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ 6.00000 0.960769
$$40$$ −4.00000 −0.632456
$$41$$ 12.0000 1.87409 0.937043 0.349215i $$-0.113552\pi$$
0.937043 + 0.349215i $$0.113552\pi$$
$$42$$ −1.00000 −0.154303
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ −1.00000 −0.150756
$$45$$ −4.00000 −0.596285
$$46$$ −8.00000 −1.17954
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ 1.00000 0.142857
$$50$$ 11.0000 1.55563
$$51$$ 4.00000 0.560112
$$52$$ −6.00000 −0.832050
$$53$$ 2.00000 0.274721 0.137361 0.990521i $$-0.456138\pi$$
0.137361 + 0.990521i $$0.456138\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 4.00000 0.539360
$$56$$ 1.00000 0.133631
$$57$$ 2.00000 0.264906
$$58$$ −6.00000 −0.787839
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 4.00000 0.516398
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 6.00000 0.762001
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ 24.0000 2.97683
$$66$$ 1.00000 0.123091
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ −4.00000 −0.485071
$$69$$ 8.00000 0.963087
$$70$$ −4.00000 −0.478091
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 1.00000 0.117851
$$73$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ −11.0000 −1.27017
$$76$$ −2.00000 −0.229416
$$77$$ −1.00000 −0.113961
$$78$$ 6.00000 0.679366
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ −4.00000 −0.447214
$$81$$ 1.00000 0.111111
$$82$$ 12.0000 1.32518
$$83$$ −14.0000 −1.53670 −0.768350 0.640030i $$-0.778922\pi$$
−0.768350 + 0.640030i $$0.778922\pi$$
$$84$$ −1.00000 −0.109109
$$85$$ 16.0000 1.73544
$$86$$ 4.00000 0.431331
$$87$$ 6.00000 0.643268
$$88$$ −1.00000 −0.106600
$$89$$ −14.0000 −1.48400 −0.741999 0.670402i $$-0.766122\pi$$
−0.741999 + 0.670402i $$0.766122\pi$$
$$90$$ −4.00000 −0.421637
$$91$$ −6.00000 −0.628971
$$92$$ −8.00000 −0.834058
$$93$$ −6.00000 −0.622171
$$94$$ 6.00000 0.618853
$$95$$ 8.00000 0.820783
$$96$$ −1.00000 −0.102062
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 1.00000 0.101015
$$99$$ −1.00000 −0.100504
$$100$$ 11.0000 1.10000
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 4.00000 0.396059
$$103$$ −6.00000 −0.591198 −0.295599 0.955312i $$-0.595519\pi$$
−0.295599 + 0.955312i $$0.595519\pi$$
$$104$$ −6.00000 −0.588348
$$105$$ 4.00000 0.390360
$$106$$ 2.00000 0.194257
$$107$$ −8.00000 −0.773389 −0.386695 0.922208i $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 6.00000 0.574696 0.287348 0.957826i $$-0.407226\pi$$
0.287348 + 0.957826i $$0.407226\pi$$
$$110$$ 4.00000 0.381385
$$111$$ 6.00000 0.569495
$$112$$ 1.00000 0.0944911
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 2.00000 0.187317
$$115$$ 32.0000 2.98402
$$116$$ −6.00000 −0.557086
$$117$$ −6.00000 −0.554700
$$118$$ 0 0
$$119$$ −4.00000 −0.366679
$$120$$ 4.00000 0.365148
$$121$$ 1.00000 0.0909091
$$122$$ 10.0000 0.905357
$$123$$ −12.0000 −1.08200
$$124$$ 6.00000 0.538816
$$125$$ −24.0000 −2.14663
$$126$$ 1.00000 0.0890871
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −4.00000 −0.352180
$$130$$ 24.0000 2.10494
$$131$$ −2.00000 −0.174741 −0.0873704 0.996176i $$-0.527846\pi$$
−0.0873704 + 0.996176i $$0.527846\pi$$
$$132$$ 1.00000 0.0870388
$$133$$ −2.00000 −0.173422
$$134$$ 4.00000 0.345547
$$135$$ 4.00000 0.344265
$$136$$ −4.00000 −0.342997
$$137$$ 14.0000 1.19610 0.598050 0.801459i $$-0.295942\pi$$
0.598050 + 0.801459i $$0.295942\pi$$
$$138$$ 8.00000 0.681005
$$139$$ 2.00000 0.169638 0.0848189 0.996396i $$-0.472969\pi$$
0.0848189 + 0.996396i $$0.472969\pi$$
$$140$$ −4.00000 −0.338062
$$141$$ −6.00000 −0.505291
$$142$$ −12.0000 −1.00702
$$143$$ 6.00000 0.501745
$$144$$ 1.00000 0.0833333
$$145$$ 24.0000 1.99309
$$146$$ 0 0
$$147$$ −1.00000 −0.0824786
$$148$$ −6.00000 −0.493197
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ −11.0000 −0.898146
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ −2.00000 −0.162221
$$153$$ −4.00000 −0.323381
$$154$$ −1.00000 −0.0805823
$$155$$ −24.0000 −1.92773
$$156$$ 6.00000 0.480384
$$157$$ −4.00000 −0.319235 −0.159617 0.987179i $$-0.551026\pi$$
−0.159617 + 0.987179i $$0.551026\pi$$
$$158$$ −16.0000 −1.27289
$$159$$ −2.00000 −0.158610
$$160$$ −4.00000 −0.316228
$$161$$ −8.00000 −0.630488
$$162$$ 1.00000 0.0785674
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 12.0000 0.937043
$$165$$ −4.00000 −0.311400
$$166$$ −14.0000 −1.08661
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ −1.00000 −0.0771517
$$169$$ 23.0000 1.76923
$$170$$ 16.0000 1.22714
$$171$$ −2.00000 −0.152944
$$172$$ 4.00000 0.304997
$$173$$ 2.00000 0.152057 0.0760286 0.997106i $$-0.475776\pi$$
0.0760286 + 0.997106i $$0.475776\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 11.0000 0.831522
$$176$$ −1.00000 −0.0753778
$$177$$ 0 0
$$178$$ −14.0000 −1.04934
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ −4.00000 −0.298142
$$181$$ −8.00000 −0.594635 −0.297318 0.954779i $$-0.596092\pi$$
−0.297318 + 0.954779i $$0.596092\pi$$
$$182$$ −6.00000 −0.444750
$$183$$ −10.0000 −0.739221
$$184$$ −8.00000 −0.589768
$$185$$ 24.0000 1.76452
$$186$$ −6.00000 −0.439941
$$187$$ 4.00000 0.292509
$$188$$ 6.00000 0.437595
$$189$$ −1.00000 −0.0727393
$$190$$ 8.00000 0.580381
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ −24.0000 −1.71868
$$196$$ 1.00000 0.0714286
$$197$$ 10.0000 0.712470 0.356235 0.934396i $$-0.384060\pi$$
0.356235 + 0.934396i $$0.384060\pi$$
$$198$$ −1.00000 −0.0710669
$$199$$ 2.00000 0.141776 0.0708881 0.997484i $$-0.477417\pi$$
0.0708881 + 0.997484i $$0.477417\pi$$
$$200$$ 11.0000 0.777817
$$201$$ −4.00000 −0.282138
$$202$$ 14.0000 0.985037
$$203$$ −6.00000 −0.421117
$$204$$ 4.00000 0.280056
$$205$$ −48.0000 −3.35247
$$206$$ −6.00000 −0.418040
$$207$$ −8.00000 −0.556038
$$208$$ −6.00000 −0.416025
$$209$$ 2.00000 0.138343
$$210$$ 4.00000 0.276026
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 2.00000 0.137361
$$213$$ 12.0000 0.822226
$$214$$ −8.00000 −0.546869
$$215$$ −16.0000 −1.09119
$$216$$ −1.00000 −0.0680414
$$217$$ 6.00000 0.407307
$$218$$ 6.00000 0.406371
$$219$$ 0 0
$$220$$ 4.00000 0.269680
$$221$$ 24.0000 1.61441
$$222$$ 6.00000 0.402694
$$223$$ 6.00000 0.401790 0.200895 0.979613i $$-0.435615\pi$$
0.200895 + 0.979613i $$0.435615\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 11.0000 0.733333
$$226$$ 14.0000 0.931266
$$227$$ 6.00000 0.398234 0.199117 0.979976i $$-0.436193\pi$$
0.199117 + 0.979976i $$0.436193\pi$$
$$228$$ 2.00000 0.132453
$$229$$ 8.00000 0.528655 0.264327 0.964433i $$-0.414850\pi$$
0.264327 + 0.964433i $$0.414850\pi$$
$$230$$ 32.0000 2.11002
$$231$$ 1.00000 0.0657952
$$232$$ −6.00000 −0.393919
$$233$$ −14.0000 −0.917170 −0.458585 0.888650i $$-0.651644\pi$$
−0.458585 + 0.888650i $$0.651644\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ −24.0000 −1.56559
$$236$$ 0 0
$$237$$ 16.0000 1.03931
$$238$$ −4.00000 −0.259281
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 4.00000 0.258199
$$241$$ −8.00000 −0.515325 −0.257663 0.966235i $$-0.582952\pi$$
−0.257663 + 0.966235i $$0.582952\pi$$
$$242$$ 1.00000 0.0642824
$$243$$ −1.00000 −0.0641500
$$244$$ 10.0000 0.640184
$$245$$ −4.00000 −0.255551
$$246$$ −12.0000 −0.765092
$$247$$ 12.0000 0.763542
$$248$$ 6.00000 0.381000
$$249$$ 14.0000 0.887214
$$250$$ −24.0000 −1.51789
$$251$$ −16.0000 −1.00991 −0.504956 0.863145i $$-0.668491\pi$$
−0.504956 + 0.863145i $$0.668491\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ 8.00000 0.502956
$$254$$ 0 0
$$255$$ −16.0000 −1.00196
$$256$$ 1.00000 0.0625000
$$257$$ 26.0000 1.62184 0.810918 0.585160i $$-0.198968\pi$$
0.810918 + 0.585160i $$0.198968\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ −6.00000 −0.372822
$$260$$ 24.0000 1.48842
$$261$$ −6.00000 −0.371391
$$262$$ −2.00000 −0.123560
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 1.00000 0.0615457
$$265$$ −8.00000 −0.491436
$$266$$ −2.00000 −0.122628
$$267$$ 14.0000 0.856786
$$268$$ 4.00000 0.244339
$$269$$ −4.00000 −0.243884 −0.121942 0.992537i $$-0.538912\pi$$
−0.121942 + 0.992537i $$0.538912\pi$$
$$270$$ 4.00000 0.243432
$$271$$ −4.00000 −0.242983 −0.121491 0.992592i $$-0.538768\pi$$
−0.121491 + 0.992592i $$0.538768\pi$$
$$272$$ −4.00000 −0.242536
$$273$$ 6.00000 0.363137
$$274$$ 14.0000 0.845771
$$275$$ −11.0000 −0.663325
$$276$$ 8.00000 0.481543
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 2.00000 0.119952
$$279$$ 6.00000 0.359211
$$280$$ −4.00000 −0.239046
$$281$$ 26.0000 1.55103 0.775515 0.631329i $$-0.217490\pi$$
0.775515 + 0.631329i $$0.217490\pi$$
$$282$$ −6.00000 −0.357295
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ −8.00000 −0.473879
$$286$$ 6.00000 0.354787
$$287$$ 12.0000 0.708338
$$288$$ 1.00000 0.0589256
$$289$$ −1.00000 −0.0588235
$$290$$ 24.0000 1.40933
$$291$$ 14.0000 0.820695
$$292$$ 0 0
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ −1.00000 −0.0583212
$$295$$ 0 0
$$296$$ −6.00000 −0.348743
$$297$$ 1.00000 0.0580259
$$298$$ −10.0000 −0.579284
$$299$$ 48.0000 2.77591
$$300$$ −11.0000 −0.635085
$$301$$ 4.00000 0.230556
$$302$$ −8.00000 −0.460348
$$303$$ −14.0000 −0.804279
$$304$$ −2.00000 −0.114708
$$305$$ −40.0000 −2.29039
$$306$$ −4.00000 −0.228665
$$307$$ −14.0000 −0.799022 −0.399511 0.916728i $$-0.630820\pi$$
−0.399511 + 0.916728i $$0.630820\pi$$
$$308$$ −1.00000 −0.0569803
$$309$$ 6.00000 0.341328
$$310$$ −24.0000 −1.36311
$$311$$ 10.0000 0.567048 0.283524 0.958965i $$-0.408496\pi$$
0.283524 + 0.958965i $$0.408496\pi$$
$$312$$ 6.00000 0.339683
$$313$$ −34.0000 −1.92179 −0.960897 0.276907i $$-0.910691\pi$$
−0.960897 + 0.276907i $$0.910691\pi$$
$$314$$ −4.00000 −0.225733
$$315$$ −4.00000 −0.225374
$$316$$ −16.0000 −0.900070
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ −2.00000 −0.112154
$$319$$ 6.00000 0.335936
$$320$$ −4.00000 −0.223607
$$321$$ 8.00000 0.446516
$$322$$ −8.00000 −0.445823
$$323$$ 8.00000 0.445132
$$324$$ 1.00000 0.0555556
$$325$$ −66.0000 −3.66102
$$326$$ 4.00000 0.221540
$$327$$ −6.00000 −0.331801
$$328$$ 12.0000 0.662589
$$329$$ 6.00000 0.330791
$$330$$ −4.00000 −0.220193
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ −14.0000 −0.768350
$$333$$ −6.00000 −0.328798
$$334$$ −12.0000 −0.656611
$$335$$ −16.0000 −0.874173
$$336$$ −1.00000 −0.0545545
$$337$$ 6.00000 0.326841 0.163420 0.986557i $$-0.447747\pi$$
0.163420 + 0.986557i $$0.447747\pi$$
$$338$$ 23.0000 1.25104
$$339$$ −14.0000 −0.760376
$$340$$ 16.0000 0.867722
$$341$$ −6.00000 −0.324918
$$342$$ −2.00000 −0.108148
$$343$$ 1.00000 0.0539949
$$344$$ 4.00000 0.215666
$$345$$ −32.0000 −1.72282
$$346$$ 2.00000 0.107521
$$347$$ 8.00000 0.429463 0.214731 0.976673i $$-0.431112\pi$$
0.214731 + 0.976673i $$0.431112\pi$$
$$348$$ 6.00000 0.321634
$$349$$ 6.00000 0.321173 0.160586 0.987022i $$-0.448662\pi$$
0.160586 + 0.987022i $$0.448662\pi$$
$$350$$ 11.0000 0.587975
$$351$$ 6.00000 0.320256
$$352$$ −1.00000 −0.0533002
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ 48.0000 2.54758
$$356$$ −14.0000 −0.741999
$$357$$ 4.00000 0.211702
$$358$$ −4.00000 −0.211407
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ −4.00000 −0.210819
$$361$$ −15.0000 −0.789474
$$362$$ −8.00000 −0.420471
$$363$$ −1.00000 −0.0524864
$$364$$ −6.00000 −0.314485
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ −34.0000 −1.77479 −0.887393 0.461014i $$-0.847486\pi$$
−0.887393 + 0.461014i $$0.847486\pi$$
$$368$$ −8.00000 −0.417029
$$369$$ 12.0000 0.624695
$$370$$ 24.0000 1.24770
$$371$$ 2.00000 0.103835
$$372$$ −6.00000 −0.311086
$$373$$ −14.0000 −0.724893 −0.362446 0.932005i $$-0.618058\pi$$
−0.362446 + 0.932005i $$0.618058\pi$$
$$374$$ 4.00000 0.206835
$$375$$ 24.0000 1.23935
$$376$$ 6.00000 0.309426
$$377$$ 36.0000 1.85409
$$378$$ −1.00000 −0.0514344
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 8.00000 0.410391
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −22.0000 −1.12415 −0.562074 0.827087i $$-0.689996\pi$$
−0.562074 + 0.827087i $$0.689996\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 4.00000 0.203859
$$386$$ −14.0000 −0.712581
$$387$$ 4.00000 0.203331
$$388$$ −14.0000 −0.710742
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ −24.0000 −1.21529
$$391$$ 32.0000 1.61831
$$392$$ 1.00000 0.0505076
$$393$$ 2.00000 0.100887
$$394$$ 10.0000 0.503793
$$395$$ 64.0000 3.22019
$$396$$ −1.00000 −0.0502519
$$397$$ −20.0000 −1.00377 −0.501886 0.864934i $$-0.667360\pi$$
−0.501886 + 0.864934i $$0.667360\pi$$
$$398$$ 2.00000 0.100251
$$399$$ 2.00000 0.100125
$$400$$ 11.0000 0.550000
$$401$$ −26.0000 −1.29838 −0.649189 0.760627i $$-0.724892\pi$$
−0.649189 + 0.760627i $$0.724892\pi$$
$$402$$ −4.00000 −0.199502
$$403$$ −36.0000 −1.79329
$$404$$ 14.0000 0.696526
$$405$$ −4.00000 −0.198762
$$406$$ −6.00000 −0.297775
$$407$$ 6.00000 0.297409
$$408$$ 4.00000 0.198030
$$409$$ 4.00000 0.197787 0.0988936 0.995098i $$-0.468470\pi$$
0.0988936 + 0.995098i $$0.468470\pi$$
$$410$$ −48.0000 −2.37055
$$411$$ −14.0000 −0.690569
$$412$$ −6.00000 −0.295599
$$413$$ 0 0
$$414$$ −8.00000 −0.393179
$$415$$ 56.0000 2.74893
$$416$$ −6.00000 −0.294174
$$417$$ −2.00000 −0.0979404
$$418$$ 2.00000 0.0978232
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 4.00000 0.195180
$$421$$ 38.0000 1.85201 0.926003 0.377515i $$-0.123221\pi$$
0.926003 + 0.377515i $$0.123221\pi$$
$$422$$ −4.00000 −0.194717
$$423$$ 6.00000 0.291730
$$424$$ 2.00000 0.0971286
$$425$$ −44.0000 −2.13431
$$426$$ 12.0000 0.581402
$$427$$ 10.0000 0.483934
$$428$$ −8.00000 −0.386695
$$429$$ −6.00000 −0.289683
$$430$$ −16.0000 −0.771589
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ 6.00000 0.288009
$$435$$ −24.0000 −1.15071
$$436$$ 6.00000 0.287348
$$437$$ 16.0000 0.765384
$$438$$ 0 0
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ 4.00000 0.190693
$$441$$ 1.00000 0.0476190
$$442$$ 24.0000 1.14156
$$443$$ 28.0000 1.33032 0.665160 0.746701i $$-0.268363\pi$$
0.665160 + 0.746701i $$0.268363\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 56.0000 2.65465
$$446$$ 6.00000 0.284108
$$447$$ 10.0000 0.472984
$$448$$ 1.00000 0.0472456
$$449$$ 14.0000 0.660701 0.330350 0.943858i $$-0.392833\pi$$
0.330350 + 0.943858i $$0.392833\pi$$
$$450$$ 11.0000 0.518545
$$451$$ −12.0000 −0.565058
$$452$$ 14.0000 0.658505
$$453$$ 8.00000 0.375873
$$454$$ 6.00000 0.281594
$$455$$ 24.0000 1.12514
$$456$$ 2.00000 0.0936586
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 8.00000 0.373815
$$459$$ 4.00000 0.186704
$$460$$ 32.0000 1.49201
$$461$$ 14.0000 0.652045 0.326023 0.945362i $$-0.394291\pi$$
0.326023 + 0.945362i $$0.394291\pi$$
$$462$$ 1.00000 0.0465242
$$463$$ −24.0000 −1.11537 −0.557687 0.830051i $$-0.688311\pi$$
−0.557687 + 0.830051i $$0.688311\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 24.0000 1.11297
$$466$$ −14.0000 −0.648537
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ −6.00000 −0.277350
$$469$$ 4.00000 0.184703
$$470$$ −24.0000 −1.10704
$$471$$ 4.00000 0.184310
$$472$$ 0 0
$$473$$ −4.00000 −0.183920
$$474$$ 16.0000 0.734904
$$475$$ −22.0000 −1.00943
$$476$$ −4.00000 −0.183340
$$477$$ 2.00000 0.0915737
$$478$$ −16.0000 −0.731823
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 4.00000 0.182574
$$481$$ 36.0000 1.64146
$$482$$ −8.00000 −0.364390
$$483$$ 8.00000 0.364013
$$484$$ 1.00000 0.0454545
$$485$$ 56.0000 2.54283
$$486$$ −1.00000 −0.0453609
$$487$$ 20.0000 0.906287 0.453143 0.891438i $$-0.350303\pi$$
0.453143 + 0.891438i $$0.350303\pi$$
$$488$$ 10.0000 0.452679
$$489$$ −4.00000 −0.180886
$$490$$ −4.00000 −0.180702
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ −12.0000 −0.541002
$$493$$ 24.0000 1.08091
$$494$$ 12.0000 0.539906
$$495$$ 4.00000 0.179787
$$496$$ 6.00000 0.269408
$$497$$ −12.0000 −0.538274
$$498$$ 14.0000 0.627355
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ −24.0000 −1.07331
$$501$$ 12.0000 0.536120
$$502$$ −16.0000 −0.714115
$$503$$ 20.0000 0.891756 0.445878 0.895094i $$-0.352892\pi$$
0.445878 + 0.895094i $$0.352892\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ −56.0000 −2.49197
$$506$$ 8.00000 0.355643
$$507$$ −23.0000 −1.02147
$$508$$ 0 0
$$509$$ 20.0000 0.886484 0.443242 0.896402i $$-0.353828\pi$$
0.443242 + 0.896402i $$0.353828\pi$$
$$510$$ −16.0000 −0.708492
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 2.00000 0.0883022
$$514$$ 26.0000 1.14681
$$515$$ 24.0000 1.05757
$$516$$ −4.00000 −0.176090
$$517$$ −6.00000 −0.263880
$$518$$ −6.00000 −0.263625
$$519$$ −2.00000 −0.0877903
$$520$$ 24.0000 1.05247
$$521$$ 18.0000 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ −22.0000 −0.961993 −0.480996 0.876723i $$-0.659725\pi$$
−0.480996 + 0.876723i $$0.659725\pi$$
$$524$$ −2.00000 −0.0873704
$$525$$ −11.0000 −0.480079
$$526$$ 0 0
$$527$$ −24.0000 −1.04546
$$528$$ 1.00000 0.0435194
$$529$$ 41.0000 1.78261
$$530$$ −8.00000 −0.347498
$$531$$ 0 0
$$532$$ −2.00000 −0.0867110
$$533$$ −72.0000 −3.11867
$$534$$ 14.0000 0.605839
$$535$$ 32.0000 1.38348
$$536$$ 4.00000 0.172774
$$537$$ 4.00000 0.172613
$$538$$ −4.00000 −0.172452
$$539$$ −1.00000 −0.0430730
$$540$$ 4.00000 0.172133
$$541$$ −14.0000 −0.601907 −0.300954 0.953639i $$-0.597305\pi$$
−0.300954 + 0.953639i $$0.597305\pi$$
$$542$$ −4.00000 −0.171815
$$543$$ 8.00000 0.343313
$$544$$ −4.00000 −0.171499
$$545$$ −24.0000 −1.02805
$$546$$ 6.00000 0.256776
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ 14.0000 0.598050
$$549$$ 10.0000 0.426790
$$550$$ −11.0000 −0.469042
$$551$$ 12.0000 0.511217
$$552$$ 8.00000 0.340503
$$553$$ −16.0000 −0.680389
$$554$$ −10.0000 −0.424859
$$555$$ −24.0000 −1.01874
$$556$$ 2.00000 0.0848189
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ 6.00000 0.254000
$$559$$ −24.0000 −1.01509
$$560$$ −4.00000 −0.169031
$$561$$ −4.00000 −0.168880
$$562$$ 26.0000 1.09674
$$563$$ −26.0000 −1.09577 −0.547885 0.836554i $$-0.684567\pi$$
−0.547885 + 0.836554i $$0.684567\pi$$
$$564$$ −6.00000 −0.252646
$$565$$ −56.0000 −2.35594
$$566$$ −14.0000 −0.588464
$$567$$ 1.00000 0.0419961
$$568$$ −12.0000 −0.503509
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ −8.00000 −0.335083
$$571$$ −32.0000 −1.33916 −0.669579 0.742741i $$-0.733526\pi$$
−0.669579 + 0.742741i $$0.733526\pi$$
$$572$$ 6.00000 0.250873
$$573$$ 0 0
$$574$$ 12.0000 0.500870
$$575$$ −88.0000 −3.66985
$$576$$ 1.00000 0.0416667
$$577$$ −34.0000 −1.41544 −0.707719 0.706494i $$-0.750276\pi$$
−0.707719 + 0.706494i $$0.750276\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ 14.0000 0.581820
$$580$$ 24.0000 0.996546
$$581$$ −14.0000 −0.580818
$$582$$ 14.0000 0.580319
$$583$$ −2.00000 −0.0828315
$$584$$ 0 0
$$585$$ 24.0000 0.992278
$$586$$ −14.0000 −0.578335
$$587$$ 24.0000 0.990586 0.495293 0.868726i $$-0.335061\pi$$
0.495293 + 0.868726i $$0.335061\pi$$
$$588$$ −1.00000 −0.0412393
$$589$$ −12.0000 −0.494451
$$590$$ 0 0
$$591$$ −10.0000 −0.411345
$$592$$ −6.00000 −0.246598
$$593$$ −12.0000 −0.492781 −0.246390 0.969171i $$-0.579245\pi$$
−0.246390 + 0.969171i $$0.579245\pi$$
$$594$$ 1.00000 0.0410305
$$595$$ 16.0000 0.655936
$$596$$ −10.0000 −0.409616
$$597$$ −2.00000 −0.0818546
$$598$$ 48.0000 1.96287
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ −11.0000 −0.449073
$$601$$ 4.00000 0.163163 0.0815817 0.996667i $$-0.474003\pi$$
0.0815817 + 0.996667i $$0.474003\pi$$
$$602$$ 4.00000 0.163028
$$603$$ 4.00000 0.162893
$$604$$ −8.00000 −0.325515
$$605$$ −4.00000 −0.162623
$$606$$ −14.0000 −0.568711
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ −2.00000 −0.0811107
$$609$$ 6.00000 0.243132
$$610$$ −40.0000 −1.61955
$$611$$ −36.0000 −1.45640
$$612$$ −4.00000 −0.161690
$$613$$ −6.00000 −0.242338 −0.121169 0.992632i $$-0.538664\pi$$
−0.121169 + 0.992632i $$0.538664\pi$$
$$614$$ −14.0000 −0.564994
$$615$$ 48.0000 1.93555
$$616$$ −1.00000 −0.0402911
$$617$$ 26.0000 1.04672 0.523360 0.852111i $$-0.324678\pi$$
0.523360 + 0.852111i $$0.324678\pi$$
$$618$$ 6.00000 0.241355
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ −24.0000 −0.963863
$$621$$ 8.00000 0.321029
$$622$$ 10.0000 0.400963
$$623$$ −14.0000 −0.560898
$$624$$ 6.00000 0.240192
$$625$$ 41.0000 1.64000
$$626$$ −34.0000 −1.35891
$$627$$ −2.00000 −0.0798723
$$628$$ −4.00000 −0.159617
$$629$$ 24.0000 0.956943
$$630$$ −4.00000 −0.159364
$$631$$ 28.0000 1.11466 0.557331 0.830290i $$-0.311825\pi$$
0.557331 + 0.830290i $$0.311825\pi$$
$$632$$ −16.0000 −0.636446
$$633$$ 4.00000 0.158986
$$634$$ −6.00000 −0.238290
$$635$$ 0 0
$$636$$ −2.00000 −0.0793052
$$637$$ −6.00000 −0.237729
$$638$$ 6.00000 0.237542
$$639$$ −12.0000 −0.474713
$$640$$ −4.00000 −0.158114
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 8.00000 0.315735
$$643$$ 32.0000 1.26196 0.630978 0.775800i $$-0.282654\pi$$
0.630978 + 0.775800i $$0.282654\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ 16.0000 0.629999
$$646$$ 8.00000 0.314756
$$647$$ 26.0000 1.02217 0.511083 0.859532i $$-0.329245\pi$$
0.511083 + 0.859532i $$0.329245\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0 0
$$650$$ −66.0000 −2.58873
$$651$$ −6.00000 −0.235159
$$652$$ 4.00000 0.156652
$$653$$ 30.0000 1.17399 0.586995 0.809590i $$-0.300311\pi$$
0.586995 + 0.809590i $$0.300311\pi$$
$$654$$ −6.00000 −0.234619
$$655$$ 8.00000 0.312586
$$656$$ 12.0000 0.468521
$$657$$ 0 0
$$658$$ 6.00000 0.233904
$$659$$ −24.0000 −0.934907 −0.467454 0.884018i $$-0.654829\pi$$
−0.467454 + 0.884018i $$0.654829\pi$$
$$660$$ −4.00000 −0.155700
$$661$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$662$$ 20.0000 0.777322
$$663$$ −24.0000 −0.932083
$$664$$ −14.0000 −0.543305
$$665$$ 8.00000 0.310227
$$666$$ −6.00000 −0.232495
$$667$$ 48.0000 1.85857
$$668$$ −12.0000 −0.464294
$$669$$ −6.00000 −0.231973
$$670$$ −16.0000 −0.618134
$$671$$ −10.0000 −0.386046
$$672$$ −1.00000 −0.0385758
$$673$$ −10.0000 −0.385472 −0.192736 0.981251i $$-0.561736\pi$$
−0.192736 + 0.981251i $$0.561736\pi$$
$$674$$ 6.00000 0.231111
$$675$$ −11.0000 −0.423390
$$676$$ 23.0000 0.884615
$$677$$ 34.0000 1.30673 0.653363 0.757045i $$-0.273358\pi$$
0.653363 + 0.757045i $$0.273358\pi$$
$$678$$ −14.0000 −0.537667
$$679$$ −14.0000 −0.537271
$$680$$ 16.0000 0.613572
$$681$$ −6.00000 −0.229920
$$682$$ −6.00000 −0.229752
$$683$$ −36.0000 −1.37750 −0.688751 0.724998i $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ −56.0000 −2.13965
$$686$$ 1.00000 0.0381802
$$687$$ −8.00000 −0.305219
$$688$$ 4.00000 0.152499
$$689$$ −12.0000 −0.457164
$$690$$ −32.0000 −1.21822
$$691$$ 24.0000 0.913003 0.456502 0.889723i $$-0.349102\pi$$
0.456502 + 0.889723i $$0.349102\pi$$
$$692$$ 2.00000 0.0760286
$$693$$ −1.00000 −0.0379869
$$694$$ 8.00000 0.303676
$$695$$ −8.00000 −0.303457
$$696$$ 6.00000 0.227429
$$697$$ −48.0000 −1.81813
$$698$$ 6.00000 0.227103
$$699$$ 14.0000 0.529529
$$700$$ 11.0000 0.415761
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 6.00000 0.226455
$$703$$ 12.0000 0.452589
$$704$$ −1.00000 −0.0376889
$$705$$ 24.0000 0.903892
$$706$$ −6.00000 −0.225813
$$707$$ 14.0000 0.526524
$$708$$ 0 0
$$709$$ −30.0000 −1.12667 −0.563337 0.826227i $$-0.690483\pi$$
−0.563337 + 0.826227i $$0.690483\pi$$
$$710$$ 48.0000 1.80141
$$711$$ −16.0000 −0.600047
$$712$$ −14.0000 −0.524672
$$713$$ −48.0000 −1.79761
$$714$$ 4.00000 0.149696
$$715$$ −24.0000 −0.897549
$$716$$ −4.00000 −0.149487
$$717$$ 16.0000 0.597531
$$718$$ 24.0000 0.895672
$$719$$ 18.0000 0.671287 0.335643 0.941989i $$-0.391046\pi$$
0.335643 + 0.941989i $$0.391046\pi$$
$$720$$ −4.00000 −0.149071
$$721$$ −6.00000 −0.223452
$$722$$ −15.0000 −0.558242
$$723$$ 8.00000 0.297523
$$724$$ −8.00000 −0.297318
$$725$$ −66.0000 −2.45118
$$726$$ −1.00000 −0.0371135
$$727$$ −42.0000 −1.55769 −0.778847 0.627214i $$-0.784195\pi$$
−0.778847 + 0.627214i $$0.784195\pi$$
$$728$$ −6.00000 −0.222375
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −16.0000 −0.591781
$$732$$ −10.0000 −0.369611
$$733$$ 50.0000 1.84679 0.923396 0.383849i $$-0.125402\pi$$
0.923396 + 0.383849i $$0.125402\pi$$
$$734$$ −34.0000 −1.25496
$$735$$ 4.00000 0.147542
$$736$$ −8.00000 −0.294884
$$737$$ −4.00000 −0.147342
$$738$$ 12.0000 0.441726
$$739$$ −16.0000 −0.588570 −0.294285 0.955718i $$-0.595081\pi$$
−0.294285 + 0.955718i $$0.595081\pi$$
$$740$$ 24.0000 0.882258
$$741$$ −12.0000 −0.440831
$$742$$ 2.00000 0.0734223
$$743$$ 16.0000 0.586983 0.293492 0.955962i $$-0.405183\pi$$
0.293492 + 0.955962i $$0.405183\pi$$
$$744$$ −6.00000 −0.219971
$$745$$ 40.0000 1.46549
$$746$$ −14.0000 −0.512576
$$747$$ −14.0000 −0.512233
$$748$$ 4.00000 0.146254
$$749$$ −8.00000 −0.292314
$$750$$ 24.0000 0.876356
$$751$$ −28.0000 −1.02173 −0.510867 0.859660i $$-0.670676\pi$$
−0.510867 + 0.859660i $$0.670676\pi$$
$$752$$ 6.00000 0.218797
$$753$$ 16.0000 0.583072
$$754$$ 36.0000 1.31104
$$755$$ 32.0000 1.16460
$$756$$ −1.00000 −0.0363696
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 20.0000 0.726433
$$759$$ −8.00000 −0.290382
$$760$$ 8.00000 0.290191
$$761$$ 24.0000 0.869999 0.435000 0.900431i $$-0.356748\pi$$
0.435000 + 0.900431i $$0.356748\pi$$
$$762$$ 0 0
$$763$$ 6.00000 0.217215
$$764$$ 0 0
$$765$$ 16.0000 0.578481
$$766$$ −22.0000 −0.794892
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ 28.0000 1.00971 0.504853 0.863205i $$-0.331547\pi$$
0.504853 + 0.863205i $$0.331547\pi$$
$$770$$ 4.00000 0.144150
$$771$$ −26.0000 −0.936367
$$772$$ −14.0000 −0.503871
$$773$$ −24.0000 −0.863220 −0.431610 0.902060i $$-0.642054\pi$$
−0.431610 + 0.902060i $$0.642054\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 66.0000 2.37079
$$776$$ −14.0000 −0.502571
$$777$$ 6.00000 0.215249
$$778$$ 6.00000 0.215110
$$779$$ −24.0000 −0.859889
$$780$$ −24.0000 −0.859338
$$781$$ 12.0000 0.429394
$$782$$ 32.0000 1.14432
$$783$$ 6.00000 0.214423
$$784$$ 1.00000 0.0357143
$$785$$ 16.0000 0.571064
$$786$$ 2.00000 0.0713376
$$787$$ −50.0000 −1.78231 −0.891154 0.453701i $$-0.850103\pi$$
−0.891154 + 0.453701i $$0.850103\pi$$
$$788$$ 10.0000 0.356235
$$789$$ 0 0
$$790$$ 64.0000 2.27702
$$791$$ 14.0000 0.497783
$$792$$ −1.00000 −0.0355335
$$793$$ −60.0000 −2.13066
$$794$$ −20.0000 −0.709773
$$795$$ 8.00000 0.283731
$$796$$ 2.00000 0.0708881
$$797$$ 24.0000 0.850124 0.425062 0.905164i $$-0.360252\pi$$
0.425062 + 0.905164i $$0.360252\pi$$
$$798$$ 2.00000 0.0707992
$$799$$ −24.0000 −0.849059
$$800$$ 11.0000 0.388909
$$801$$ −14.0000 −0.494666
$$802$$ −26.0000 −0.918092
$$803$$ 0 0
$$804$$ −4.00000 −0.141069
$$805$$ 32.0000 1.12785
$$806$$ −36.0000 −1.26805
$$807$$ 4.00000 0.140807
$$808$$ 14.0000 0.492518
$$809$$ −2.00000 −0.0703163 −0.0351581 0.999382i $$-0.511193\pi$$
−0.0351581 + 0.999382i $$0.511193\pi$$
$$810$$ −4.00000 −0.140546
$$811$$ 26.0000 0.912983 0.456492 0.889728i $$-0.349106\pi$$
0.456492 + 0.889728i $$0.349106\pi$$
$$812$$ −6.00000 −0.210559
$$813$$ 4.00000 0.140286
$$814$$ 6.00000 0.210300
$$815$$ −16.0000 −0.560456
$$816$$ 4.00000 0.140028
$$817$$ −8.00000 −0.279885
$$818$$ 4.00000 0.139857
$$819$$ −6.00000 −0.209657
$$820$$ −48.0000 −1.67623
$$821$$ −26.0000 −0.907406 −0.453703 0.891153i $$-0.649897\pi$$
−0.453703 + 0.891153i $$0.649897\pi$$
$$822$$ −14.0000 −0.488306
$$823$$ 44.0000 1.53374 0.766872 0.641800i $$-0.221812\pi$$
0.766872 + 0.641800i $$0.221812\pi$$
$$824$$ −6.00000 −0.209020
$$825$$ 11.0000 0.382971
$$826$$ 0 0
$$827$$ −28.0000 −0.973655 −0.486828 0.873498i $$-0.661846\pi$$
−0.486828 + 0.873498i $$0.661846\pi$$
$$828$$ −8.00000 −0.278019
$$829$$ 16.0000 0.555703 0.277851 0.960624i $$-0.410378\pi$$
0.277851 + 0.960624i $$0.410378\pi$$
$$830$$ 56.0000 1.94379
$$831$$ 10.0000 0.346896
$$832$$ −6.00000 −0.208013
$$833$$ −4.00000 −0.138592
$$834$$ −2.00000 −0.0692543
$$835$$ 48.0000 1.66111
$$836$$ 2.00000 0.0691714
$$837$$ −6.00000 −0.207390
$$838$$ 12.0000 0.414533
$$839$$ −54.0000 −1.86429 −0.932144 0.362089i $$-0.882064\pi$$
−0.932144 + 0.362089i $$0.882064\pi$$
$$840$$ 4.00000 0.138013
$$841$$ 7.00000 0.241379
$$842$$ 38.0000 1.30957
$$843$$ −26.0000 −0.895488
$$844$$ −4.00000 −0.137686
$$845$$ −92.0000 −3.16490
$$846$$ 6.00000 0.206284
$$847$$ 1.00000 0.0343604
$$848$$ 2.00000 0.0686803
$$849$$ 14.0000 0.480479
$$850$$ −44.0000 −1.50919
$$851$$ 48.0000 1.64542
$$852$$ 12.0000 0.411113
$$853$$ −6.00000 −0.205436 −0.102718 0.994711i $$-0.532754\pi$$
−0.102718 + 0.994711i $$0.532754\pi$$
$$854$$ 10.0000 0.342193
$$855$$ 8.00000 0.273594
$$856$$ −8.00000 −0.273434
$$857$$ −40.0000 −1.36637 −0.683187 0.730243i $$-0.739407\pi$$
−0.683187 + 0.730243i $$0.739407\pi$$
$$858$$ −6.00000 −0.204837
$$859$$ 40.0000 1.36478 0.682391 0.730987i $$-0.260940\pi$$
0.682391 + 0.730987i $$0.260940\pi$$
$$860$$ −16.0000 −0.545595
$$861$$ −12.0000 −0.408959
$$862$$ −24.0000 −0.817443
$$863$$ −16.0000 −0.544646 −0.272323 0.962206i $$-0.587792\pi$$
−0.272323 + 0.962206i $$0.587792\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −8.00000 −0.272008
$$866$$ 26.0000 0.883516
$$867$$ 1.00000 0.0339618
$$868$$ 6.00000 0.203653
$$869$$ 16.0000 0.542763
$$870$$ −24.0000 −0.813676
$$871$$ −24.0000 −0.813209
$$872$$ 6.00000 0.203186
$$873$$ −14.0000 −0.473828
$$874$$ 16.0000 0.541208
$$875$$ −24.0000 −0.811348
$$876$$ 0 0
$$877$$ −18.0000 −0.607817 −0.303908 0.952701i $$-0.598292\pi$$
−0.303908 + 0.952701i $$0.598292\pi$$
$$878$$ −20.0000 −0.674967
$$879$$ 14.0000 0.472208
$$880$$ 4.00000 0.134840
$$881$$ −50.0000 −1.68454 −0.842271 0.539054i $$-0.818782\pi$$
−0.842271 + 0.539054i $$0.818782\pi$$
$$882$$ 1.00000 0.0336718
$$883$$ 28.0000 0.942275 0.471138 0.882060i $$-0.343844\pi$$
0.471138 + 0.882060i $$0.343844\pi$$
$$884$$ 24.0000 0.807207
$$885$$ 0 0
$$886$$ 28.0000 0.940678
$$887$$ −24.0000 −0.805841 −0.402921 0.915235i $$-0.632005\pi$$
−0.402921 + 0.915235i $$0.632005\pi$$
$$888$$ 6.00000 0.201347
$$889$$ 0 0
$$890$$ 56.0000 1.87712
$$891$$ −1.00000 −0.0335013
$$892$$ 6.00000 0.200895
$$893$$ −12.0000 −0.401565
$$894$$ 10.0000 0.334450
$$895$$ 16.0000 0.534821
$$896$$ 1.00000 0.0334077
$$897$$ −48.0000 −1.60267
$$898$$ 14.0000 0.467186
$$899$$ −36.0000 −1.20067
$$900$$ 11.0000 0.366667
$$901$$ −8.00000 −0.266519
$$902$$ −12.0000 −0.399556
$$903$$ −4.00000 −0.133112
$$904$$ 14.0000 0.465633
$$905$$ 32.0000 1.06372
$$906$$ 8.00000 0.265782
$$907$$ 52.0000 1.72663 0.863316 0.504664i $$-0.168384\pi$$
0.863316 + 0.504664i $$0.168384\pi$$
$$908$$ 6.00000 0.199117
$$909$$ 14.0000 0.464351
$$910$$ 24.0000 0.795592
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 2.00000 0.0662266
$$913$$ 14.0000 0.463332
$$914$$ 10.0000 0.330771
$$915$$ 40.0000 1.32236
$$916$$ 8.00000 0.264327
$$917$$ −2.00000 −0.0660458
$$918$$ 4.00000 0.132020
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 32.0000 1.05501
$$921$$ 14.0000 0.461316
$$922$$ 14.0000 0.461065
$$923$$ 72.0000 2.36991
$$924$$ 1.00000 0.0328976
$$925$$ −66.0000 −2.17007
$$926$$ −24.0000 −0.788689
$$927$$ −6.00000 −0.197066
$$928$$ −6.00000 −0.196960
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 24.0000 0.786991
$$931$$ −2.00000 −0.0655474
$$932$$ −14.0000 −0.458585
$$933$$ −10.0000 −0.327385
$$934$$ 36.0000 1.17796
$$935$$ −16.0000 −0.523256
$$936$$ −6.00000 −0.196116
$$937$$ −32.0000 −1.04539 −0.522697 0.852518i $$-0.675074\pi$$
−0.522697 + 0.852518i $$0.675074\pi$$
$$938$$ 4.00000 0.130605
$$939$$ 34.0000 1.10955
$$940$$ −24.0000 −0.782794
$$941$$ −30.0000 −0.977972 −0.488986 0.872292i $$-0.662633\pi$$
−0.488986 + 0.872292i $$0.662633\pi$$
$$942$$ 4.00000 0.130327
$$943$$ −96.0000 −3.12619
$$944$$ 0 0
$$945$$ 4.00000 0.130120
$$946$$ −4.00000 −0.130051
$$947$$ 36.0000 1.16984 0.584921 0.811090i $$-0.301125\pi$$
0.584921 + 0.811090i $$0.301125\pi$$
$$948$$ 16.0000 0.519656
$$949$$ 0 0
$$950$$ −22.0000 −0.713774
$$951$$ 6.00000 0.194563
$$952$$ −4.00000 −0.129641
$$953$$ 30.0000 0.971795 0.485898 0.874016i $$-0.338493\pi$$
0.485898 + 0.874016i $$0.338493\pi$$
$$954$$ 2.00000 0.0647524
$$955$$ 0 0
$$956$$ −16.0000 −0.517477
$$957$$ −6.00000 −0.193952
$$958$$ 24.0000 0.775405
$$959$$ 14.0000 0.452084
$$960$$ 4.00000 0.129099
$$961$$ 5.00000 0.161290
$$962$$ 36.0000 1.16069
$$963$$ −8.00000 −0.257796
$$964$$ −8.00000 −0.257663
$$965$$ 56.0000 1.80270
$$966$$ 8.00000 0.257396
$$967$$ −32.0000 −1.02905 −0.514525 0.857475i $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ 1.00000 0.0321412
$$969$$ −8.00000 −0.256997
$$970$$ 56.0000 1.79805
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 2.00000 0.0641171
$$974$$ 20.0000 0.640841
$$975$$ 66.0000 2.11369
$$976$$ 10.0000 0.320092
$$977$$ 2.00000 0.0639857 0.0319928 0.999488i $$-0.489815\pi$$
0.0319928 + 0.999488i $$0.489815\pi$$
$$978$$ −4.00000 −0.127906
$$979$$ 14.0000 0.447442
$$980$$ −4.00000 −0.127775
$$981$$ 6.00000 0.191565
$$982$$ −28.0000 −0.893516
$$983$$ −42.0000 −1.33959 −0.669796 0.742545i $$-0.733618\pi$$
−0.669796 + 0.742545i $$0.733618\pi$$
$$984$$ −12.0000 −0.382546
$$985$$ −40.0000 −1.27451
$$986$$ 24.0000 0.764316
$$987$$ −6.00000 −0.190982
$$988$$ 12.0000 0.381771
$$989$$ −32.0000 −1.01754
$$990$$ 4.00000 0.127128
$$991$$ 8.00000 0.254128 0.127064 0.991894i $$-0.459445\pi$$
0.127064 + 0.991894i $$0.459445\pi$$
$$992$$ 6.00000 0.190500
$$993$$ −20.0000 −0.634681
$$994$$ −12.0000 −0.380617
$$995$$ −8.00000 −0.253617
$$996$$ 14.0000 0.443607
$$997$$ −26.0000 −0.823428 −0.411714 0.911313i $$-0.635070\pi$$
−0.411714 + 0.911313i $$0.635070\pi$$
$$998$$ −28.0000 −0.886325
$$999$$ 6.00000 0.189832
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 462.2.a.e.1.1 1
3.2 odd 2 1386.2.a.e.1.1 1
4.3 odd 2 3696.2.a.p.1.1 1
7.6 odd 2 3234.2.a.v.1.1 1
11.10 odd 2 5082.2.a.a.1.1 1
21.20 even 2 9702.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.e.1.1 1 1.1 even 1 trivial
1386.2.a.e.1.1 1 3.2 odd 2
3234.2.a.v.1.1 1 7.6 odd 2
3696.2.a.p.1.1 1 4.3 odd 2
5082.2.a.a.1.1 1 11.10 odd 2
9702.2.a.b.1.1 1 21.20 even 2