# Properties

 Label 546.2.a.g.1.1 Level $546$ Weight $2$ Character 546.1 Self dual yes Analytic conductor $4.360$ 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] = [546,2,Mod(1,546)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, 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("546.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.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: $$4.35983195036$$ 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 $$\chi$$ $$=$$ 546.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{20} +1.00000 q^{21} -4.00000 q^{22} +1.00000 q^{24} -1.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} +2.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} +6.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} +10.0000 q^{37} -4.00000 q^{38} -1.00000 q^{39} +2.00000 q^{40} -6.00000 q^{41} +1.00000 q^{42} +4.00000 q^{43} -4.00000 q^{44} +2.00000 q^{45} +4.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +6.00000 q^{51} -1.00000 q^{52} +10.0000 q^{53} +1.00000 q^{54} -8.00000 q^{55} +1.00000 q^{56} -4.00000 q^{57} -6.00000 q^{58} +4.00000 q^{59} +2.00000 q^{60} -6.00000 q^{61} -8.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{66} -8.00000 q^{67} +6.00000 q^{68} +2.00000 q^{70} +1.00000 q^{72} -10.0000 q^{73} +10.0000 q^{74} -1.00000 q^{75} -4.00000 q^{76} -4.00000 q^{77} -1.00000 q^{78} -8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +4.00000 q^{83} +1.00000 q^{84} +12.0000 q^{85} +4.00000 q^{86} -6.00000 q^{87} -4.00000 q^{88} -6.00000 q^{89} +2.00000 q^{90} -1.00000 q^{91} -8.00000 q^{93} +4.00000 q^{94} -8.00000 q^{95} +1.00000 q^{96} -2.00000 q^{97} +1.00000 q^{98} -4.00000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ 1.00000 0.333333
$$10$$ 2.00000 0.632456
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 1.00000 0.288675
$$13$$ −1.00000 −0.277350
$$14$$ 1.00000 0.267261
$$15$$ 2.00000 0.516398
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 2.00000 0.447214
$$21$$ 1.00000 0.218218
$$22$$ −4.00000 −0.852803
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −1.00000 −0.200000
$$26$$ −1.00000 −0.196116
$$27$$ 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 $$0.688081\pi$$
$$30$$ 2.00000 0.365148
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −4.00000 −0.696311
$$34$$ 6.00000 1.02899
$$35$$ 2.00000 0.338062
$$36$$ 1.00000 0.166667
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ −1.00000 −0.160128
$$40$$ 2.00000 0.316228
$$41$$ −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 $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 2.00000 0.298142
$$46$$ 0 0
$$47$$ 4.00000 0.583460 0.291730 0.956501i $$-0.405769\pi$$
0.291730 + 0.956501i $$0.405769\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 1.00000 0.142857
$$50$$ −1.00000 −0.141421
$$51$$ 6.00000 0.840168
$$52$$ −1.00000 −0.138675
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ 1.00000 0.136083
$$55$$ −8.00000 −1.07872
$$56$$ 1.00000 0.133631
$$57$$ −4.00000 −0.529813
$$58$$ −6.00000 −0.787839
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 2.00000 0.258199
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ −8.00000 −1.01600
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ −2.00000 −0.248069
$$66$$ −4.00000 −0.492366
$$67$$ −8.00000 −0.977356 −0.488678 0.872464i $$-0.662521\pi$$
−0.488678 + 0.872464i $$0.662521\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 0 0
$$70$$ 2.00000 0.239046
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ 10.0000 1.16248
$$75$$ −1.00000 −0.115470
$$76$$ −4.00000 −0.458831
$$77$$ −4.00000 −0.455842
$$78$$ −1.00000 −0.113228
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 1.00000 0.109109
$$85$$ 12.0000 1.30158
$$86$$ 4.00000 0.431331
$$87$$ −6.00000 −0.643268
$$88$$ −4.00000 −0.426401
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 2.00000 0.210819
$$91$$ −1.00000 −0.104828
$$92$$ 0 0
$$93$$ −8.00000 −0.829561
$$94$$ 4.00000 0.412568
$$95$$ −8.00000 −0.820783
$$96$$ 1.00000 0.102062
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 1.00000 0.101015
$$99$$ −4.00000 −0.402015
$$100$$ −1.00000 −0.100000
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ 6.00000 0.594089
$$103$$ −12.0000 −1.18240 −0.591198 0.806527i $$-0.701345\pi$$
−0.591198 + 0.806527i $$0.701345\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 2.00000 0.195180
$$106$$ 10.0000 0.971286
$$107$$ 16.0000 1.54678 0.773389 0.633932i $$-0.218560\pi$$
0.773389 + 0.633932i $$0.218560\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ −8.00000 −0.762770
$$111$$ 10.0000 0.949158
$$112$$ 1.00000 0.0944911
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ −6.00000 −0.557086
$$117$$ −1.00000 −0.0924500
$$118$$ 4.00000 0.368230
$$119$$ 6.00000 0.550019
$$120$$ 2.00000 0.182574
$$121$$ 5.00000 0.454545
$$122$$ −6.00000 −0.543214
$$123$$ −6.00000 −0.541002
$$124$$ −8.00000 −0.718421
$$125$$ −12.0000 −1.07331
$$126$$ 1.00000 0.0890871
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 4.00000 0.352180
$$130$$ −2.00000 −0.175412
$$131$$ −20.0000 −1.74741 −0.873704 0.486458i $$-0.838289\pi$$
−0.873704 + 0.486458i $$0.838289\pi$$
$$132$$ −4.00000 −0.348155
$$133$$ −4.00000 −0.346844
$$134$$ −8.00000 −0.691095
$$135$$ 2.00000 0.172133
$$136$$ 6.00000 0.514496
$$137$$ 10.0000 0.854358 0.427179 0.904167i $$-0.359507\pi$$
0.427179 + 0.904167i $$0.359507\pi$$
$$138$$ 0 0
$$139$$ −12.0000 −1.01783 −0.508913 0.860818i $$-0.669953\pi$$
−0.508913 + 0.860818i $$0.669953\pi$$
$$140$$ 2.00000 0.169031
$$141$$ 4.00000 0.336861
$$142$$ 0 0
$$143$$ 4.00000 0.334497
$$144$$ 1.00000 0.0833333
$$145$$ −12.0000 −0.996546
$$146$$ −10.0000 −0.827606
$$147$$ 1.00000 0.0824786
$$148$$ 10.0000 0.821995
$$149$$ 14.0000 1.14692 0.573462 0.819232i $$-0.305600\pi$$
0.573462 + 0.819232i $$0.305600\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ 6.00000 0.485071
$$154$$ −4.00000 −0.322329
$$155$$ −16.0000 −1.28515
$$156$$ −1.00000 −0.0800641
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 10.0000 0.793052
$$160$$ 2.00000 0.158114
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ 16.0000 1.25322 0.626608 0.779334i $$-0.284443\pi$$
0.626608 + 0.779334i $$0.284443\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ −8.00000 −0.622799
$$166$$ 4.00000 0.310460
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 1.00000 0.0771517
$$169$$ 1.00000 0.0769231
$$170$$ 12.0000 0.920358
$$171$$ −4.00000 −0.305888
$$172$$ 4.00000 0.304997
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ −6.00000 −0.454859
$$175$$ −1.00000 −0.0755929
$$176$$ −4.00000 −0.301511
$$177$$ 4.00000 0.300658
$$178$$ −6.00000 −0.449719
$$179$$ 24.0000 1.79384 0.896922 0.442189i $$-0.145798\pi$$
0.896922 + 0.442189i $$0.145798\pi$$
$$180$$ 2.00000 0.149071
$$181$$ −22.0000 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$182$$ −1.00000 −0.0741249
$$183$$ −6.00000 −0.443533
$$184$$ 0 0
$$185$$ 20.0000 1.47043
$$186$$ −8.00000 −0.586588
$$187$$ −24.0000 −1.75505
$$188$$ 4.00000 0.291730
$$189$$ 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
$$193$$ −6.00000 −0.431889 −0.215945 0.976406i $$-0.569283\pi$$
−0.215945 + 0.976406i $$0.569283\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ −2.00000 −0.143223
$$196$$ 1.00000 0.0714286
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ −4.00000 −0.284268
$$199$$ 20.0000 1.41776 0.708881 0.705328i $$-0.249200\pi$$
0.708881 + 0.705328i $$0.249200\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ −8.00000 −0.564276
$$202$$ 10.0000 0.703598
$$203$$ −6.00000 −0.421117
$$204$$ 6.00000 0.420084
$$205$$ −12.0000 −0.838116
$$206$$ −12.0000 −0.836080
$$207$$ 0 0
$$208$$ −1.00000 −0.0693375
$$209$$ 16.0000 1.10674
$$210$$ 2.00000 0.138013
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 10.0000 0.686803
$$213$$ 0 0
$$214$$ 16.0000 1.09374
$$215$$ 8.00000 0.545595
$$216$$ 1.00000 0.0680414
$$217$$ −8.00000 −0.543075
$$218$$ −6.00000 −0.406371
$$219$$ −10.0000 −0.675737
$$220$$ −8.00000 −0.539360
$$221$$ −6.00000 −0.403604
$$222$$ 10.0000 0.671156
$$223$$ −8.00000 −0.535720 −0.267860 0.963458i $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −1.00000 −0.0666667
$$226$$ 2.00000 0.133038
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ −4.00000 −0.264906
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ −6.00000 −0.393919
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ −1.00000 −0.0653720
$$235$$ 8.00000 0.521862
$$236$$ 4.00000 0.260378
$$237$$ −8.00000 −0.519656
$$238$$ 6.00000 0.388922
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 2.00000 0.129099
$$241$$ 30.0000 1.93247 0.966235 0.257663i $$-0.0829523\pi$$
0.966235 + 0.257663i $$0.0829523\pi$$
$$242$$ 5.00000 0.321412
$$243$$ 1.00000 0.0641500
$$244$$ −6.00000 −0.384111
$$245$$ 2.00000 0.127775
$$246$$ −6.00000 −0.382546
$$247$$ 4.00000 0.254514
$$248$$ −8.00000 −0.508001
$$249$$ 4.00000 0.253490
$$250$$ −12.0000 −0.758947
$$251$$ −4.00000 −0.252478 −0.126239 0.992000i $$-0.540291\pi$$
−0.126239 + 0.992000i $$0.540291\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ 0 0
$$254$$ −16.0000 −1.00393
$$255$$ 12.0000 0.751469
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ 4.00000 0.249029
$$259$$ 10.0000 0.621370
$$260$$ −2.00000 −0.124035
$$261$$ −6.00000 −0.371391
$$262$$ −20.0000 −1.23560
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ 20.0000 1.22859
$$266$$ −4.00000 −0.245256
$$267$$ −6.00000 −0.367194
$$268$$ −8.00000 −0.488678
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 2.00000 0.121716
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 6.00000 0.363803
$$273$$ −1.00000 −0.0605228
$$274$$ 10.0000 0.604122
$$275$$ 4.00000 0.241209
$$276$$ 0 0
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ −12.0000 −0.719712
$$279$$ −8.00000 −0.478947
$$280$$ 2.00000 0.119523
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 4.00000 0.238197
$$283$$ 28.0000 1.66443 0.832214 0.554455i $$-0.187073\pi$$
0.832214 + 0.554455i $$0.187073\pi$$
$$284$$ 0 0
$$285$$ −8.00000 −0.473879
$$286$$ 4.00000 0.236525
$$287$$ −6.00000 −0.354169
$$288$$ 1.00000 0.0589256
$$289$$ 19.0000 1.11765
$$290$$ −12.0000 −0.704664
$$291$$ −2.00000 −0.117242
$$292$$ −10.0000 −0.585206
$$293$$ 2.00000 0.116841 0.0584206 0.998292i $$-0.481394\pi$$
0.0584206 + 0.998292i $$0.481394\pi$$
$$294$$ 1.00000 0.0583212
$$295$$ 8.00000 0.465778
$$296$$ 10.0000 0.581238
$$297$$ −4.00000 −0.232104
$$298$$ 14.0000 0.810998
$$299$$ 0 0
$$300$$ −1.00000 −0.0577350
$$301$$ 4.00000 0.230556
$$302$$ 16.0000 0.920697
$$303$$ 10.0000 0.574485
$$304$$ −4.00000 −0.229416
$$305$$ −12.0000 −0.687118
$$306$$ 6.00000 0.342997
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ −4.00000 −0.227921
$$309$$ −12.0000 −0.682656
$$310$$ −16.0000 −0.908739
$$311$$ −16.0000 −0.907277 −0.453638 0.891186i $$-0.649874\pi$$
−0.453638 + 0.891186i $$0.649874\pi$$
$$312$$ −1.00000 −0.0566139
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 18.0000 1.01580
$$315$$ 2.00000 0.112687
$$316$$ −8.00000 −0.450035
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 10.0000 0.560772
$$319$$ 24.0000 1.34374
$$320$$ 2.00000 0.111803
$$321$$ 16.0000 0.893033
$$322$$ 0 0
$$323$$ −24.0000 −1.33540
$$324$$ 1.00000 0.0555556
$$325$$ 1.00000 0.0554700
$$326$$ 16.0000 0.886158
$$327$$ −6.00000 −0.331801
$$328$$ −6.00000 −0.331295
$$329$$ 4.00000 0.220527
$$330$$ −8.00000 −0.440386
$$331$$ −8.00000 −0.439720 −0.219860 0.975531i $$-0.570560\pi$$
−0.219860 + 0.975531i $$0.570560\pi$$
$$332$$ 4.00000 0.219529
$$333$$ 10.0000 0.547997
$$334$$ −12.0000 −0.656611
$$335$$ −16.0000 −0.874173
$$336$$ 1.00000 0.0545545
$$337$$ 34.0000 1.85210 0.926049 0.377403i $$-0.123183\pi$$
0.926049 + 0.377403i $$0.123183\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ 2.00000 0.108625
$$340$$ 12.0000 0.650791
$$341$$ 32.0000 1.73290
$$342$$ −4.00000 −0.216295
$$343$$ 1.00000 0.0539949
$$344$$ 4.00000 0.215666
$$345$$ 0 0
$$346$$ 18.0000 0.967686
$$347$$ 8.00000 0.429463 0.214731 0.976673i $$-0.431112\pi$$
0.214731 + 0.976673i $$0.431112\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ −1.00000 −0.0534522
$$351$$ −1.00000 −0.0533761
$$352$$ −4.00000 −0.213201
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 4.00000 0.212598
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 6.00000 0.317554
$$358$$ 24.0000 1.26844
$$359$$ −16.0000 −0.844448 −0.422224 0.906492i $$-0.638750\pi$$
−0.422224 + 0.906492i $$0.638750\pi$$
$$360$$ 2.00000 0.105409
$$361$$ −3.00000 −0.157895
$$362$$ −22.0000 −1.15629
$$363$$ 5.00000 0.262432
$$364$$ −1.00000 −0.0524142
$$365$$ −20.0000 −1.04685
$$366$$ −6.00000 −0.313625
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 20.0000 1.03975
$$371$$ 10.0000 0.519174
$$372$$ −8.00000 −0.414781
$$373$$ 14.0000 0.724893 0.362446 0.932005i $$-0.381942\pi$$
0.362446 + 0.932005i $$0.381942\pi$$
$$374$$ −24.0000 −1.24101
$$375$$ −12.0000 −0.619677
$$376$$ 4.00000 0.206284
$$377$$ 6.00000 0.309016
$$378$$ 1.00000 0.0514344
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ −8.00000 −0.410391
$$381$$ −16.0000 −0.819705
$$382$$ 0 0
$$383$$ 4.00000 0.204390 0.102195 0.994764i $$-0.467413\pi$$
0.102195 + 0.994764i $$0.467413\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ −8.00000 −0.407718
$$386$$ −6.00000 −0.305392
$$387$$ 4.00000 0.203331
$$388$$ −2.00000 −0.101535
$$389$$ 34.0000 1.72387 0.861934 0.507020i $$-0.169253\pi$$
0.861934 + 0.507020i $$0.169253\pi$$
$$390$$ −2.00000 −0.101274
$$391$$ 0 0
$$392$$ 1.00000 0.0505076
$$393$$ −20.0000 −1.00887
$$394$$ 6.00000 0.302276
$$395$$ −16.0000 −0.805047
$$396$$ −4.00000 −0.201008
$$397$$ 18.0000 0.903394 0.451697 0.892171i $$-0.350819\pi$$
0.451697 + 0.892171i $$0.350819\pi$$
$$398$$ 20.0000 1.00251
$$399$$ −4.00000 −0.200250
$$400$$ −1.00000 −0.0500000
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ −8.00000 −0.399004
$$403$$ 8.00000 0.398508
$$404$$ 10.0000 0.497519
$$405$$ 2.00000 0.0993808
$$406$$ −6.00000 −0.297775
$$407$$ −40.0000 −1.98273
$$408$$ 6.00000 0.297044
$$409$$ 22.0000 1.08783 0.543915 0.839140i $$-0.316941\pi$$
0.543915 + 0.839140i $$0.316941\pi$$
$$410$$ −12.0000 −0.592638
$$411$$ 10.0000 0.493264
$$412$$ −12.0000 −0.591198
$$413$$ 4.00000 0.196827
$$414$$ 0 0
$$415$$ 8.00000 0.392705
$$416$$ −1.00000 −0.0490290
$$417$$ −12.0000 −0.587643
$$418$$ 16.0000 0.782586
$$419$$ 28.0000 1.36789 0.683945 0.729534i $$-0.260263\pi$$
0.683945 + 0.729534i $$0.260263\pi$$
$$420$$ 2.00000 0.0975900
$$421$$ −30.0000 −1.46211 −0.731055 0.682318i $$-0.760972\pi$$
−0.731055 + 0.682318i $$0.760972\pi$$
$$422$$ 12.0000 0.584151
$$423$$ 4.00000 0.194487
$$424$$ 10.0000 0.485643
$$425$$ −6.00000 −0.291043
$$426$$ 0 0
$$427$$ −6.00000 −0.290360
$$428$$ 16.0000 0.773389
$$429$$ 4.00000 0.193122
$$430$$ 8.00000 0.385794
$$431$$ −32.0000 −1.54139 −0.770693 0.637207i $$-0.780090\pi$$
−0.770693 + 0.637207i $$0.780090\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ 34.0000 1.63394 0.816968 0.576683i $$-0.195653\pi$$
0.816968 + 0.576683i $$0.195653\pi$$
$$434$$ −8.00000 −0.384012
$$435$$ −12.0000 −0.575356
$$436$$ −6.00000 −0.287348
$$437$$ 0 0
$$438$$ −10.0000 −0.477818
$$439$$ −20.0000 −0.954548 −0.477274 0.878755i $$-0.658375\pi$$
−0.477274 + 0.878755i $$0.658375\pi$$
$$440$$ −8.00000 −0.381385
$$441$$ 1.00000 0.0476190
$$442$$ −6.00000 −0.285391
$$443$$ 24.0000 1.14027 0.570137 0.821549i $$-0.306890\pi$$
0.570137 + 0.821549i $$0.306890\pi$$
$$444$$ 10.0000 0.474579
$$445$$ −12.0000 −0.568855
$$446$$ −8.00000 −0.378811
$$447$$ 14.0000 0.662177
$$448$$ 1.00000 0.0472456
$$449$$ 2.00000 0.0943858 0.0471929 0.998886i $$-0.484972\pi$$
0.0471929 + 0.998886i $$0.484972\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 24.0000 1.13012
$$452$$ 2.00000 0.0940721
$$453$$ 16.0000 0.751746
$$454$$ −12.0000 −0.563188
$$455$$ −2.00000 −0.0937614
$$456$$ −4.00000 −0.187317
$$457$$ −14.0000 −0.654892 −0.327446 0.944870i $$-0.606188\pi$$
−0.327446 + 0.944870i $$0.606188\pi$$
$$458$$ −6.00000 −0.280362
$$459$$ 6.00000 0.280056
$$460$$ 0 0
$$461$$ −14.0000 −0.652045 −0.326023 0.945362i $$-0.605709\pi$$
−0.326023 + 0.945362i $$0.605709\pi$$
$$462$$ −4.00000 −0.186097
$$463$$ −24.0000 −1.11537 −0.557687 0.830051i $$-0.688311\pi$$
−0.557687 + 0.830051i $$0.688311\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ −16.0000 −0.741982
$$466$$ 18.0000 0.833834
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ −1.00000 −0.0462250
$$469$$ −8.00000 −0.369406
$$470$$ 8.00000 0.369012
$$471$$ 18.0000 0.829396
$$472$$ 4.00000 0.184115
$$473$$ −16.0000 −0.735681
$$474$$ −8.00000 −0.367452
$$475$$ 4.00000 0.183533
$$476$$ 6.00000 0.275010
$$477$$ 10.0000 0.457869
$$478$$ −24.0000 −1.09773
$$479$$ −4.00000 −0.182765 −0.0913823 0.995816i $$-0.529129\pi$$
−0.0913823 + 0.995816i $$0.529129\pi$$
$$480$$ 2.00000 0.0912871
$$481$$ −10.0000 −0.455961
$$482$$ 30.0000 1.36646
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ −4.00000 −0.181631
$$486$$ 1.00000 0.0453609
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ −6.00000 −0.271607
$$489$$ 16.0000 0.723545
$$490$$ 2.00000 0.0903508
$$491$$ −32.0000 −1.44414 −0.722070 0.691820i $$-0.756809\pi$$
−0.722070 + 0.691820i $$0.756809\pi$$
$$492$$ −6.00000 −0.270501
$$493$$ −36.0000 −1.62136
$$494$$ 4.00000 0.179969
$$495$$ −8.00000 −0.359573
$$496$$ −8.00000 −0.359211
$$497$$ 0 0
$$498$$ 4.00000 0.179244
$$499$$ 32.0000 1.43252 0.716258 0.697835i $$-0.245853\pi$$
0.716258 + 0.697835i $$0.245853\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ −12.0000 −0.536120
$$502$$ −4.00000 −0.178529
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ 1.00000 0.0445435
$$505$$ 20.0000 0.889988
$$506$$ 0 0
$$507$$ 1.00000 0.0444116
$$508$$ −16.0000 −0.709885
$$509$$ −14.0000 −0.620539 −0.310270 0.950649i $$-0.600419\pi$$
−0.310270 + 0.950649i $$0.600419\pi$$
$$510$$ 12.0000 0.531369
$$511$$ −10.0000 −0.442374
$$512$$ 1.00000 0.0441942
$$513$$ −4.00000 −0.176604
$$514$$ 6.00000 0.264649
$$515$$ −24.0000 −1.05757
$$516$$ 4.00000 0.176090
$$517$$ −16.0000 −0.703679
$$518$$ 10.0000 0.439375
$$519$$ 18.0000 0.790112
$$520$$ −2.00000 −0.0877058
$$521$$ −26.0000 −1.13908 −0.569540 0.821963i $$-0.692879\pi$$
−0.569540 + 0.821963i $$0.692879\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ 4.00000 0.174908 0.0874539 0.996169i $$-0.472127\pi$$
0.0874539 + 0.996169i $$0.472127\pi$$
$$524$$ −20.0000 −0.873704
$$525$$ −1.00000 −0.0436436
$$526$$ 24.0000 1.04645
$$527$$ −48.0000 −2.09091
$$528$$ −4.00000 −0.174078
$$529$$ −23.0000 −1.00000
$$530$$ 20.0000 0.868744
$$531$$ 4.00000 0.173585
$$532$$ −4.00000 −0.173422
$$533$$ 6.00000 0.259889
$$534$$ −6.00000 −0.259645
$$535$$ 32.0000 1.38348
$$536$$ −8.00000 −0.345547
$$537$$ 24.0000 1.03568
$$538$$ 18.0000 0.776035
$$539$$ −4.00000 −0.172292
$$540$$ 2.00000 0.0860663
$$541$$ −46.0000 −1.97769 −0.988847 0.148933i $$-0.952416\pi$$
−0.988847 + 0.148933i $$0.952416\pi$$
$$542$$ 8.00000 0.343629
$$543$$ −22.0000 −0.944110
$$544$$ 6.00000 0.257248
$$545$$ −12.0000 −0.514024
$$546$$ −1.00000 −0.0427960
$$547$$ 12.0000 0.513083 0.256541 0.966533i $$-0.417417\pi$$
0.256541 + 0.966533i $$0.417417\pi$$
$$548$$ 10.0000 0.427179
$$549$$ −6.00000 −0.256074
$$550$$ 4.00000 0.170561
$$551$$ 24.0000 1.02243
$$552$$ 0 0
$$553$$ −8.00000 −0.340195
$$554$$ −10.0000 −0.424859
$$555$$ 20.0000 0.848953
$$556$$ −12.0000 −0.508913
$$557$$ 22.0000 0.932170 0.466085 0.884740i $$-0.345664\pi$$
0.466085 + 0.884740i $$0.345664\pi$$
$$558$$ −8.00000 −0.338667
$$559$$ −4.00000 −0.169182
$$560$$ 2.00000 0.0845154
$$561$$ −24.0000 −1.01328
$$562$$ 10.0000 0.421825
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 4.00000 0.168430
$$565$$ 4.00000 0.168281
$$566$$ 28.0000 1.17693
$$567$$ 1.00000 0.0419961
$$568$$ 0 0
$$569$$ −22.0000 −0.922288 −0.461144 0.887325i $$-0.652561\pi$$
−0.461144 + 0.887325i $$0.652561\pi$$
$$570$$ −8.00000 −0.335083
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 4.00000 0.167248
$$573$$ 0 0
$$574$$ −6.00000 −0.250435
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 22.0000 0.915872 0.457936 0.888985i $$-0.348589\pi$$
0.457936 + 0.888985i $$0.348589\pi$$
$$578$$ 19.0000 0.790296
$$579$$ −6.00000 −0.249351
$$580$$ −12.0000 −0.498273
$$581$$ 4.00000 0.165948
$$582$$ −2.00000 −0.0829027
$$583$$ −40.0000 −1.65663
$$584$$ −10.0000 −0.413803
$$585$$ −2.00000 −0.0826898
$$586$$ 2.00000 0.0826192
$$587$$ −28.0000 −1.15568 −0.577842 0.816149i $$-0.696105\pi$$
−0.577842 + 0.816149i $$0.696105\pi$$
$$588$$ 1.00000 0.0412393
$$589$$ 32.0000 1.31854
$$590$$ 8.00000 0.329355
$$591$$ 6.00000 0.246807
$$592$$ 10.0000 0.410997
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 12.0000 0.491952
$$596$$ 14.0000 0.573462
$$597$$ 20.0000 0.818546
$$598$$ 0 0
$$599$$ 8.00000 0.326871 0.163436 0.986554i $$-0.447742\pi$$
0.163436 + 0.986554i $$0.447742\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ 2.00000 0.0815817 0.0407909 0.999168i $$-0.487012\pi$$
0.0407909 + 0.999168i $$0.487012\pi$$
$$602$$ 4.00000 0.163028
$$603$$ −8.00000 −0.325785
$$604$$ 16.0000 0.651031
$$605$$ 10.0000 0.406558
$$606$$ 10.0000 0.406222
$$607$$ −28.0000 −1.13648 −0.568242 0.822861i $$-0.692376\pi$$
−0.568242 + 0.822861i $$0.692376\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ −6.00000 −0.243132
$$610$$ −12.0000 −0.485866
$$611$$ −4.00000 −0.161823
$$612$$ 6.00000 0.242536
$$613$$ −30.0000 −1.21169 −0.605844 0.795583i $$-0.707165\pi$$
−0.605844 + 0.795583i $$0.707165\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ −12.0000 −0.483887
$$616$$ −4.00000 −0.161165
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ −12.0000 −0.482711
$$619$$ 28.0000 1.12542 0.562708 0.826656i $$-0.309760\pi$$
0.562708 + 0.826656i $$0.309760\pi$$
$$620$$ −16.0000 −0.642575
$$621$$ 0 0
$$622$$ −16.0000 −0.641542
$$623$$ −6.00000 −0.240385
$$624$$ −1.00000 −0.0400320
$$625$$ −19.0000 −0.760000
$$626$$ 10.0000 0.399680
$$627$$ 16.0000 0.638978
$$628$$ 18.0000 0.718278
$$629$$ 60.0000 2.39236
$$630$$ 2.00000 0.0796819
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ 12.0000 0.476957
$$634$$ −18.0000 −0.714871
$$635$$ −32.0000 −1.26988
$$636$$ 10.0000 0.396526
$$637$$ −1.00000 −0.0396214
$$638$$ 24.0000 0.950169
$$639$$ 0 0
$$640$$ 2.00000 0.0790569
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 16.0000 0.631470
$$643$$ −28.0000 −1.10421 −0.552106 0.833774i $$-0.686176\pi$$
−0.552106 + 0.833774i $$0.686176\pi$$
$$644$$ 0 0
$$645$$ 8.00000 0.315000
$$646$$ −24.0000 −0.944267
$$647$$ 8.00000 0.314512 0.157256 0.987558i $$-0.449735\pi$$
0.157256 + 0.987558i $$0.449735\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ −16.0000 −0.628055
$$650$$ 1.00000 0.0392232
$$651$$ −8.00000 −0.313545
$$652$$ 16.0000 0.626608
$$653$$ −14.0000 −0.547862 −0.273931 0.961749i $$-0.588324\pi$$
−0.273931 + 0.961749i $$0.588324\pi$$
$$654$$ −6.00000 −0.234619
$$655$$ −40.0000 −1.56293
$$656$$ −6.00000 −0.234261
$$657$$ −10.0000 −0.390137
$$658$$ 4.00000 0.155936
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ −8.00000 −0.311400
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ −8.00000 −0.310929
$$663$$ −6.00000 −0.233021
$$664$$ 4.00000 0.155230
$$665$$ −8.00000 −0.310227
$$666$$ 10.0000 0.387492
$$667$$ 0 0
$$668$$ −12.0000 −0.464294
$$669$$ −8.00000 −0.309298
$$670$$ −16.0000 −0.618134
$$671$$ 24.0000 0.926510
$$672$$ 1.00000 0.0385758
$$673$$ 18.0000 0.693849 0.346925 0.937893i $$-0.387226\pi$$
0.346925 + 0.937893i $$0.387226\pi$$
$$674$$ 34.0000 1.30963
$$675$$ −1.00000 −0.0384900
$$676$$ 1.00000 0.0384615
$$677$$ −38.0000 −1.46046 −0.730229 0.683202i $$-0.760587\pi$$
−0.730229 + 0.683202i $$0.760587\pi$$
$$678$$ 2.00000 0.0768095
$$679$$ −2.00000 −0.0767530
$$680$$ 12.0000 0.460179
$$681$$ −12.0000 −0.459841
$$682$$ 32.0000 1.22534
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 20.0000 0.764161
$$686$$ 1.00000 0.0381802
$$687$$ −6.00000 −0.228914
$$688$$ 4.00000 0.152499
$$689$$ −10.0000 −0.380970
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 18.0000 0.684257
$$693$$ −4.00000 −0.151947
$$694$$ 8.00000 0.303676
$$695$$ −24.0000 −0.910372
$$696$$ −6.00000 −0.227429
$$697$$ −36.0000 −1.36360
$$698$$ 2.00000 0.0757011
$$699$$ 18.0000 0.680823
$$700$$ −1.00000 −0.0377964
$$701$$ 50.0000 1.88847 0.944237 0.329267i $$-0.106802\pi$$
0.944237 + 0.329267i $$0.106802\pi$$
$$702$$ −1.00000 −0.0377426
$$703$$ −40.0000 −1.50863
$$704$$ −4.00000 −0.150756
$$705$$ 8.00000 0.301297
$$706$$ −6.00000 −0.225813
$$707$$ 10.0000 0.376089
$$708$$ 4.00000 0.150329
$$709$$ −6.00000 −0.225335 −0.112667 0.993633i $$-0.535939\pi$$
−0.112667 + 0.993633i $$0.535939\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ −6.00000 −0.224860
$$713$$ 0 0
$$714$$ 6.00000 0.224544
$$715$$ 8.00000 0.299183
$$716$$ 24.0000 0.896922
$$717$$ −24.0000 −0.896296
$$718$$ −16.0000 −0.597115
$$719$$ 16.0000 0.596699 0.298350 0.954457i $$-0.403564\pi$$
0.298350 + 0.954457i $$0.403564\pi$$
$$720$$ 2.00000 0.0745356
$$721$$ −12.0000 −0.446903
$$722$$ −3.00000 −0.111648
$$723$$ 30.0000 1.11571
$$724$$ −22.0000 −0.817624
$$725$$ 6.00000 0.222834
$$726$$ 5.00000 0.185567
$$727$$ 28.0000 1.03846 0.519231 0.854634i $$-0.326218\pi$$
0.519231 + 0.854634i $$0.326218\pi$$
$$728$$ −1.00000 −0.0370625
$$729$$ 1.00000 0.0370370
$$730$$ −20.0000 −0.740233
$$731$$ 24.0000 0.887672
$$732$$ −6.00000 −0.221766
$$733$$ 2.00000 0.0738717 0.0369358 0.999318i $$-0.488240\pi$$
0.0369358 + 0.999318i $$0.488240\pi$$
$$734$$ −28.0000 −1.03350
$$735$$ 2.00000 0.0737711
$$736$$ 0 0
$$737$$ 32.0000 1.17874
$$738$$ −6.00000 −0.220863
$$739$$ 40.0000 1.47142 0.735712 0.677295i $$-0.236848\pi$$
0.735712 + 0.677295i $$0.236848\pi$$
$$740$$ 20.0000 0.735215
$$741$$ 4.00000 0.146944
$$742$$ 10.0000 0.367112
$$743$$ −8.00000 −0.293492 −0.146746 0.989174i $$-0.546880\pi$$
−0.146746 + 0.989174i $$0.546880\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ 28.0000 1.02584
$$746$$ 14.0000 0.512576
$$747$$ 4.00000 0.146352
$$748$$ −24.0000 −0.877527
$$749$$ 16.0000 0.584627
$$750$$ −12.0000 −0.438178
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ 4.00000 0.145865
$$753$$ −4.00000 −0.145768
$$754$$ 6.00000 0.218507
$$755$$ 32.0000 1.16460
$$756$$ 1.00000 0.0363696
$$757$$ −26.0000 −0.944986 −0.472493 0.881334i $$-0.656646\pi$$
−0.472493 + 0.881334i $$0.656646\pi$$
$$758$$ 16.0000 0.581146
$$759$$ 0 0
$$760$$ −8.00000 −0.290191
$$761$$ 26.0000 0.942499 0.471250 0.882000i $$-0.343803\pi$$
0.471250 + 0.882000i $$0.343803\pi$$
$$762$$ −16.0000 −0.579619
$$763$$ −6.00000 −0.217215
$$764$$ 0 0
$$765$$ 12.0000 0.433861
$$766$$ 4.00000 0.144526
$$767$$ −4.00000 −0.144432
$$768$$ 1.00000 0.0360844
$$769$$ −50.0000 −1.80305 −0.901523 0.432731i $$-0.857550\pi$$
−0.901523 + 0.432731i $$0.857550\pi$$
$$770$$ −8.00000 −0.288300
$$771$$ 6.00000 0.216085
$$772$$ −6.00000 −0.215945
$$773$$ 18.0000 0.647415 0.323708 0.946157i $$-0.395071\pi$$
0.323708 + 0.946157i $$0.395071\pi$$
$$774$$ 4.00000 0.143777
$$775$$ 8.00000 0.287368
$$776$$ −2.00000 −0.0717958
$$777$$ 10.0000 0.358748
$$778$$ 34.0000 1.21896
$$779$$ 24.0000 0.859889
$$780$$ −2.00000 −0.0716115
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −6.00000 −0.214423
$$784$$ 1.00000 0.0357143
$$785$$ 36.0000 1.28490
$$786$$ −20.0000 −0.713376
$$787$$ −12.0000 −0.427754 −0.213877 0.976861i $$-0.568609\pi$$
−0.213877 + 0.976861i $$0.568609\pi$$
$$788$$ 6.00000 0.213741
$$789$$ 24.0000 0.854423
$$790$$ −16.0000 −0.569254
$$791$$ 2.00000 0.0711118
$$792$$ −4.00000 −0.142134
$$793$$ 6.00000 0.213066
$$794$$ 18.0000 0.638796
$$795$$ 20.0000 0.709327
$$796$$ 20.0000 0.708881
$$797$$ −30.0000 −1.06265 −0.531327 0.847167i $$-0.678307\pi$$
−0.531327 + 0.847167i $$0.678307\pi$$
$$798$$ −4.00000 −0.141598
$$799$$ 24.0000 0.849059
$$800$$ −1.00000 −0.0353553
$$801$$ −6.00000 −0.212000
$$802$$ −30.0000 −1.05934
$$803$$ 40.0000 1.41157
$$804$$ −8.00000 −0.282138
$$805$$ 0 0
$$806$$ 8.00000 0.281788
$$807$$ 18.0000 0.633630
$$808$$ 10.0000 0.351799
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 2.00000 0.0702728
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ −6.00000 −0.210559
$$813$$ 8.00000 0.280572
$$814$$ −40.0000 −1.40200
$$815$$ 32.0000 1.12091
$$816$$ 6.00000 0.210042
$$817$$ −16.0000 −0.559769
$$818$$ 22.0000 0.769212
$$819$$ −1.00000 −0.0349428
$$820$$ −12.0000 −0.419058
$$821$$ −2.00000 −0.0698005 −0.0349002 0.999391i $$-0.511111\pi$$
−0.0349002 + 0.999391i $$0.511111\pi$$
$$822$$ 10.0000 0.348790
$$823$$ 32.0000 1.11545 0.557725 0.830026i $$-0.311674\pi$$
0.557725 + 0.830026i $$0.311674\pi$$
$$824$$ −12.0000 −0.418040
$$825$$ 4.00000 0.139262
$$826$$ 4.00000 0.139178
$$827$$ 20.0000 0.695468 0.347734 0.937593i $$-0.386951\pi$$
0.347734 + 0.937593i $$0.386951\pi$$
$$828$$ 0 0
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ 8.00000 0.277684
$$831$$ −10.0000 −0.346896
$$832$$ −1.00000 −0.0346688
$$833$$ 6.00000 0.207888
$$834$$ −12.0000 −0.415526
$$835$$ −24.0000 −0.830554
$$836$$ 16.0000 0.553372
$$837$$ −8.00000 −0.276520
$$838$$ 28.0000 0.967244
$$839$$ −4.00000 −0.138095 −0.0690477 0.997613i $$-0.521996\pi$$
−0.0690477 + 0.997613i $$0.521996\pi$$
$$840$$ 2.00000 0.0690066
$$841$$ 7.00000 0.241379
$$842$$ −30.0000 −1.03387
$$843$$ 10.0000 0.344418
$$844$$ 12.0000 0.413057
$$845$$ 2.00000 0.0688021
$$846$$ 4.00000 0.137523
$$847$$ 5.00000 0.171802
$$848$$ 10.0000 0.343401
$$849$$ 28.0000 0.960958
$$850$$ −6.00000 −0.205798
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −22.0000 −0.753266 −0.376633 0.926363i $$-0.622918\pi$$
−0.376633 + 0.926363i $$0.622918\pi$$
$$854$$ −6.00000 −0.205316
$$855$$ −8.00000 −0.273594
$$856$$ 16.0000 0.546869
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 4.00000 0.136558
$$859$$ 28.0000 0.955348 0.477674 0.878537i $$-0.341480\pi$$
0.477674 + 0.878537i $$0.341480\pi$$
$$860$$ 8.00000 0.272798
$$861$$ −6.00000 −0.204479
$$862$$ −32.0000 −1.08992
$$863$$ 56.0000 1.90626 0.953131 0.302558i $$-0.0978405\pi$$
0.953131 + 0.302558i $$0.0978405\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 36.0000 1.22404
$$866$$ 34.0000 1.15537
$$867$$ 19.0000 0.645274
$$868$$ −8.00000 −0.271538
$$869$$ 32.0000 1.08553
$$870$$ −12.0000 −0.406838
$$871$$ 8.00000 0.271070
$$872$$ −6.00000 −0.203186
$$873$$ −2.00000 −0.0676897
$$874$$ 0 0
$$875$$ −12.0000 −0.405674
$$876$$ −10.0000 −0.337869
$$877$$ −6.00000 −0.202606 −0.101303 0.994856i $$-0.532301\pi$$
−0.101303 + 0.994856i $$0.532301\pi$$
$$878$$ −20.0000 −0.674967
$$879$$ 2.00000 0.0674583
$$880$$ −8.00000 −0.269680
$$881$$ −18.0000 −0.606435 −0.303218 0.952921i $$-0.598061\pi$$
−0.303218 + 0.952921i $$0.598061\pi$$
$$882$$ 1.00000 0.0336718
$$883$$ 12.0000 0.403832 0.201916 0.979403i $$-0.435283\pi$$
0.201916 + 0.979403i $$0.435283\pi$$
$$884$$ −6.00000 −0.201802
$$885$$ 8.00000 0.268917
$$886$$ 24.0000 0.806296
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 10.0000 0.335578
$$889$$ −16.0000 −0.536623
$$890$$ −12.0000 −0.402241
$$891$$ −4.00000 −0.134005
$$892$$ −8.00000 −0.267860
$$893$$ −16.0000 −0.535420
$$894$$ 14.0000 0.468230
$$895$$ 48.0000 1.60446
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ 2.00000 0.0667409
$$899$$ 48.0000 1.60089
$$900$$ −1.00000 −0.0333333
$$901$$ 60.0000 1.99889
$$902$$ 24.0000 0.799113
$$903$$ 4.00000 0.133112
$$904$$ 2.00000 0.0665190
$$905$$ −44.0000 −1.46261
$$906$$ 16.0000 0.531564
$$907$$ 12.0000 0.398453 0.199227 0.979953i $$-0.436157\pi$$
0.199227 + 0.979953i $$0.436157\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 10.0000 0.331679
$$910$$ −2.00000 −0.0662994
$$911$$ −40.0000 −1.32526 −0.662630 0.748947i $$-0.730560\pi$$
−0.662630 + 0.748947i $$0.730560\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ −16.0000 −0.529523
$$914$$ −14.0000 −0.463079
$$915$$ −12.0000 −0.396708
$$916$$ −6.00000 −0.198246
$$917$$ −20.0000 −0.660458
$$918$$ 6.00000 0.198030
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ −12.0000 −0.395413
$$922$$ −14.0000 −0.461065
$$923$$ 0 0
$$924$$ −4.00000 −0.131590
$$925$$ −10.0000 −0.328798
$$926$$ −24.0000 −0.788689
$$927$$ −12.0000 −0.394132
$$928$$ −6.00000 −0.196960
$$929$$ 34.0000 1.11550 0.557752 0.830008i $$-0.311664\pi$$
0.557752 + 0.830008i $$0.311664\pi$$
$$930$$ −16.0000 −0.524661
$$931$$ −4.00000 −0.131095
$$932$$ 18.0000 0.589610
$$933$$ −16.0000 −0.523816
$$934$$ 12.0000 0.392652
$$935$$ −48.0000 −1.56977
$$936$$ −1.00000 −0.0326860
$$937$$ −22.0000 −0.718709 −0.359354 0.933201i $$-0.617003\pi$$
−0.359354 + 0.933201i $$0.617003\pi$$
$$938$$ −8.00000 −0.261209
$$939$$ 10.0000 0.326338
$$940$$ 8.00000 0.260931
$$941$$ −54.0000 −1.76035 −0.880175 0.474650i $$-0.842575\pi$$
−0.880175 + 0.474650i $$0.842575\pi$$
$$942$$ 18.0000 0.586472
$$943$$ 0 0
$$944$$ 4.00000 0.130189
$$945$$ 2.00000 0.0650600
$$946$$ −16.0000 −0.520205
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ −8.00000 −0.259828
$$949$$ 10.0000 0.324614
$$950$$ 4.00000 0.129777
$$951$$ −18.0000 −0.583690
$$952$$ 6.00000 0.194461
$$953$$ −14.0000 −0.453504 −0.226752 0.973952i $$-0.572811\pi$$
−0.226752 + 0.973952i $$0.572811\pi$$
$$954$$ 10.0000 0.323762
$$955$$ 0 0
$$956$$ −24.0000 −0.776215
$$957$$ 24.0000 0.775810
$$958$$ −4.00000 −0.129234
$$959$$ 10.0000 0.322917
$$960$$ 2.00000 0.0645497
$$961$$ 33.0000 1.06452
$$962$$ −10.0000 −0.322413
$$963$$ 16.0000 0.515593
$$964$$ 30.0000 0.966235
$$965$$ −12.0000 −0.386294
$$966$$ 0 0
$$967$$ 16.0000 0.514525 0.257263 0.966342i $$-0.417179\pi$$
0.257263 + 0.966342i $$0.417179\pi$$
$$968$$ 5.00000 0.160706
$$969$$ −24.0000 −0.770991
$$970$$ −4.00000 −0.128432
$$971$$ 4.00000 0.128366 0.0641831 0.997938i $$-0.479556\pi$$
0.0641831 + 0.997938i $$0.479556\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ −12.0000 −0.384702
$$974$$ 8.00000 0.256337
$$975$$ 1.00000 0.0320256
$$976$$ −6.00000 −0.192055
$$977$$ −30.0000 −0.959785 −0.479893 0.877327i $$-0.659324\pi$$
−0.479893 + 0.877327i $$0.659324\pi$$
$$978$$ 16.0000 0.511624
$$979$$ 24.0000 0.767043
$$980$$ 2.00000 0.0638877
$$981$$ −6.00000 −0.191565
$$982$$ −32.0000 −1.02116
$$983$$ 4.00000 0.127580 0.0637901 0.997963i $$-0.479681\pi$$
0.0637901 + 0.997963i $$0.479681\pi$$
$$984$$ −6.00000 −0.191273
$$985$$ 12.0000 0.382352
$$986$$ −36.0000 −1.14647
$$987$$ 4.00000 0.127321
$$988$$ 4.00000 0.127257
$$989$$ 0 0
$$990$$ −8.00000 −0.254257
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ −8.00000 −0.254000
$$993$$ −8.00000 −0.253872
$$994$$ 0 0
$$995$$ 40.0000 1.26809
$$996$$ 4.00000 0.126745
$$997$$ 2.00000 0.0633406 0.0316703 0.999498i $$-0.489917\pi$$
0.0316703 + 0.999498i $$0.489917\pi$$
$$998$$ 32.0000 1.01294
$$999$$ 10.0000 0.316386
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 546.2.a.g.1.1 1
3.2 odd 2 1638.2.a.d.1.1 1
4.3 odd 2 4368.2.a.k.1.1 1
7.6 odd 2 3822.2.a.t.1.1 1
13.12 even 2 7098.2.a.j.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.g.1.1 1 1.1 even 1 trivial
1638.2.a.d.1.1 1 3.2 odd 2
3822.2.a.t.1.1 1 7.6 odd 2
4368.2.a.k.1.1 1 4.3 odd 2
7098.2.a.j.1.1 1 13.12 even 2