# Properties

 Label 57.2.a.c.1.1 Level $57$ Weight $2$ Character 57.1 Self dual yes Analytic conductor $0.455$ 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] = [57,2,Mod(1,57)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(57, 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("57.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$57 = 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 57.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$0.455147291521$$ 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 $$\chi$$ $$=$$ 57.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{12} +6.00000 q^{13} -2.00000 q^{15} -1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} +2.00000 q^{20} +4.00000 q^{23} -3.00000 q^{24} -1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} +2.00000 q^{29} -2.00000 q^{30} +8.00000 q^{31} +5.00000 q^{32} -6.00000 q^{34} -1.00000 q^{36} -10.0000 q^{37} -1.00000 q^{38} +6.00000 q^{39} +6.00000 q^{40} -2.00000 q^{41} -4.00000 q^{43} -2.00000 q^{45} +4.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} -6.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -1.00000 q^{57} +2.00000 q^{58} -12.0000 q^{59} +2.00000 q^{60} -2.00000 q^{61} +8.00000 q^{62} +7.00000 q^{64} -12.0000 q^{65} -4.00000 q^{67} +6.00000 q^{68} +4.00000 q^{69} -3.00000 q^{72} +10.0000 q^{73} -10.0000 q^{74} -1.00000 q^{75} +1.00000 q^{76} +6.00000 q^{78} +2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +16.0000 q^{83} +12.0000 q^{85} -4.00000 q^{86} +2.00000 q^{87} -2.00000 q^{89} -2.00000 q^{90} -4.00000 q^{92} +8.00000 q^{93} +12.0000 q^{94} +2.00000 q^{95} +5.00000 q^{96} +10.0000 q^{97} -7.00000 q^{98} +O(q^{100})$$

## 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 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ 1.00000 0.577350
$$4$$ −1.00000 −0.500000
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 1.00000 0.333333
$$10$$ −2.00000 −0.632456
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 6.00000 1.66410 0.832050 0.554700i $$-0.187167\pi$$
0.832050 + 0.554700i $$0.187167\pi$$
$$14$$ 0 0
$$15$$ −2.00000 −0.516398
$$16$$ −1.00000 −0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −1.00000 −0.229416
$$20$$ 2.00000 0.447214
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ −3.00000 −0.612372
$$25$$ −1.00000 −0.200000
$$26$$ 6.00000 1.17670
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ −2.00000 −0.365148
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 5.00000 0.883883
$$33$$ 0 0
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ −1.00000 −0.166667
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ −1.00000 −0.162221
$$39$$ 6.00000 0.960769
$$40$$ 6.00000 0.948683
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ −2.00000 −0.298142
$$46$$ 4.00000 0.589768
$$47$$ 12.0000 1.75038 0.875190 0.483779i $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −7.00000 −1.00000
$$50$$ −1.00000 −0.141421
$$51$$ −6.00000 −0.840168
$$52$$ −6.00000 −0.832050
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 1.00000 0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −1.00000 −0.132453
$$58$$ 2.00000 0.262613
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 2.00000 0.258199
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 8.00000 1.01600
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ −12.0000 −1.48842
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 4.00000 0.481543
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ −3.00000 −0.353553
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ −1.00000 −0.115470
$$76$$ 1.00000 0.114708
$$77$$ 0 0
$$78$$ 6.00000 0.679366
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 1.00000 0.111111
$$82$$ −2.00000 −0.220863
$$83$$ 16.0000 1.75623 0.878114 0.478451i $$-0.158802\pi$$
0.878114 + 0.478451i $$0.158802\pi$$
$$84$$ 0 0
$$85$$ 12.0000 1.30158
$$86$$ −4.00000 −0.431331
$$87$$ 2.00000 0.214423
$$88$$ 0 0
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ −2.00000 −0.210819
$$91$$ 0 0
$$92$$ −4.00000 −0.417029
$$93$$ 8.00000 0.829561
$$94$$ 12.0000 1.23771
$$95$$ 2.00000 0.205196
$$96$$ 5.00000 0.510310
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ −7.00000 −0.707107
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ −6.00000 −0.594089
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ −18.0000 −1.76505
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ −10.0000 −0.949158
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ −1.00000 −0.0936586
$$115$$ −8.00000 −0.746004
$$116$$ −2.00000 −0.185695
$$117$$ 6.00000 0.554700
$$118$$ −12.0000 −1.10469
$$119$$ 0 0
$$120$$ 6.00000 0.547723
$$121$$ −11.0000 −1.00000
$$122$$ −2.00000 −0.181071
$$123$$ −2.00000 −0.180334
$$124$$ −8.00000 −0.718421
$$125$$ 12.0000 1.07331
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ −4.00000 −0.352180
$$130$$ −12.0000 −1.05247
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −4.00000 −0.345547
$$135$$ −2.00000 −0.172133
$$136$$ 18.0000 1.54349
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 4.00000 0.340503
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 12.0000 1.01058
$$142$$ 0 0
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ −4.00000 −0.332182
$$146$$ 10.0000 0.827606
$$147$$ −7.00000 −0.577350
$$148$$ 10.0000 0.821995
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 3.00000 0.243332
$$153$$ −6.00000 −0.485071
$$154$$ 0 0
$$155$$ −16.0000 −1.28515
$$156$$ −6.00000 −0.480384
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ −10.0000 −0.790569
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 2.00000 0.156174
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ 24.0000 1.85718 0.928588 0.371113i $$-0.121024\pi$$
0.928588 + 0.371113i $$0.121024\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 12.0000 0.920358
$$171$$ −1.00000 −0.0764719
$$172$$ 4.00000 0.304997
$$173$$ −22.0000 −1.67263 −0.836315 0.548250i $$-0.815294\pi$$
−0.836315 + 0.548250i $$0.815294\pi$$
$$174$$ 2.00000 0.151620
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −12.0000 −0.901975
$$178$$ −2.00000 −0.149906
$$179$$ −4.00000 −0.298974 −0.149487 0.988764i $$-0.547762\pi$$
−0.149487 + 0.988764i $$0.547762\pi$$
$$180$$ 2.00000 0.149071
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 0 0
$$183$$ −2.00000 −0.147844
$$184$$ −12.0000 −0.884652
$$185$$ 20.0000 1.47043
$$186$$ 8.00000 0.586588
$$187$$ 0 0
$$188$$ −12.0000 −0.875190
$$189$$ 0 0
$$190$$ 2.00000 0.145095
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 7.00000 0.505181
$$193$$ −14.0000 −1.00774 −0.503871 0.863779i $$-0.668091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 10.0000 0.717958
$$195$$ −12.0000 −0.859338
$$196$$ 7.00000 0.500000
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 0 0
$$199$$ −8.00000 −0.567105 −0.283552 0.958957i $$-0.591513\pi$$
−0.283552 + 0.958957i $$0.591513\pi$$
$$200$$ 3.00000 0.212132
$$201$$ −4.00000 −0.282138
$$202$$ −10.0000 −0.703598
$$203$$ 0 0
$$204$$ 6.00000 0.420084
$$205$$ 4.00000 0.279372
$$206$$ 8.00000 0.557386
$$207$$ 4.00000 0.278019
$$208$$ −6.00000 −0.416025
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ 4.00000 0.273434
$$215$$ 8.00000 0.545595
$$216$$ −3.00000 −0.204124
$$217$$ 0 0
$$218$$ −10.0000 −0.677285
$$219$$ 10.0000 0.675737
$$220$$ 0 0
$$221$$ −36.0000 −2.42162
$$222$$ −10.0000 −0.671156
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 6.00000 0.399114
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 1.00000 0.0662266
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ −8.00000 −0.527504
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ 10.0000 0.655122 0.327561 0.944830i $$-0.393773\pi$$
0.327561 + 0.944830i $$0.393773\pi$$
$$234$$ 6.00000 0.392232
$$235$$ −24.0000 −1.56559
$$236$$ 12.0000 0.781133
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 2.00000 0.129099
$$241$$ −6.00000 −0.386494 −0.193247 0.981150i $$-0.561902\pi$$
−0.193247 + 0.981150i $$0.561902\pi$$
$$242$$ −11.0000 −0.707107
$$243$$ 1.00000 0.0641500
$$244$$ 2.00000 0.128037
$$245$$ 14.0000 0.894427
$$246$$ −2.00000 −0.127515
$$247$$ −6.00000 −0.381771
$$248$$ −24.0000 −1.52400
$$249$$ 16.0000 1.01396
$$250$$ 12.0000 0.758947
$$251$$ −24.0000 −1.51487 −0.757433 0.652913i $$-0.773547\pi$$
−0.757433 + 0.652913i $$0.773547\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −8.00000 −0.501965
$$255$$ 12.0000 0.751469
$$256$$ −17.0000 −1.06250
$$257$$ 14.0000 0.873296 0.436648 0.899632i $$-0.356166\pi$$
0.436648 + 0.899632i $$0.356166\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ 0 0
$$260$$ 12.0000 0.744208
$$261$$ 2.00000 0.123797
$$262$$ 8.00000 0.494242
$$263$$ 12.0000 0.739952 0.369976 0.929041i $$-0.379366\pi$$
0.369976 + 0.929041i $$0.379366\pi$$
$$264$$ 0 0
$$265$$ 12.0000 0.737154
$$266$$ 0 0
$$267$$ −2.00000 −0.122398
$$268$$ 4.00000 0.244339
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ −2.00000 −0.121716
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 0 0
$$274$$ 18.0000 1.08742
$$275$$ 0 0
$$276$$ −4.00000 −0.240772
$$277$$ 22.0000 1.32185 0.660926 0.750451i $$-0.270164\pi$$
0.660926 + 0.750451i $$0.270164\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 12.0000 0.714590
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ 0 0
$$285$$ 2.00000 0.118470
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 5.00000 0.294628
$$289$$ 19.0000 1.11765
$$290$$ −4.00000 −0.234888
$$291$$ 10.0000 0.586210
$$292$$ −10.0000 −0.585206
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ −7.00000 −0.408248
$$295$$ 24.0000 1.39733
$$296$$ 30.0000 1.74371
$$297$$ 0 0
$$298$$ 6.00000 0.347571
$$299$$ 24.0000 1.38796
$$300$$ 1.00000 0.0577350
$$301$$ 0 0
$$302$$ −8.00000 −0.460348
$$303$$ −10.0000 −0.574485
$$304$$ 1.00000 0.0573539
$$305$$ 4.00000 0.229039
$$306$$ −6.00000 −0.342997
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ −16.0000 −0.908739
$$311$$ 4.00000 0.226819 0.113410 0.993548i $$-0.463823\pi$$
0.113410 + 0.993548i $$0.463823\pi$$
$$312$$ −18.0000 −1.01905
$$313$$ −22.0000 −1.24351 −0.621757 0.783210i $$-0.713581\pi$$
−0.621757 + 0.783210i $$0.713581\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ −6.00000 −0.336463
$$319$$ 0 0
$$320$$ −14.0000 −0.782624
$$321$$ 4.00000 0.223258
$$322$$ 0 0
$$323$$ 6.00000 0.333849
$$324$$ −1.00000 −0.0555556
$$325$$ −6.00000 −0.332820
$$326$$ −4.00000 −0.221540
$$327$$ −10.0000 −0.553001
$$328$$ 6.00000 0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ −16.0000 −0.878114
$$333$$ −10.0000 −0.547997
$$334$$ 24.0000 1.31322
$$335$$ 8.00000 0.437087
$$336$$ 0 0
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ 23.0000 1.25104
$$339$$ 6.00000 0.325875
$$340$$ −12.0000 −0.650791
$$341$$ 0 0
$$342$$ −1.00000 −0.0540738
$$343$$ 0 0
$$344$$ 12.0000 0.646997
$$345$$ −8.00000 −0.430706
$$346$$ −22.0000 −1.18273
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ −2.00000 −0.107211
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ 6.00000 0.320256
$$352$$ 0 0
$$353$$ −22.0000 −1.17094 −0.585471 0.810693i $$-0.699090\pi$$
−0.585471 + 0.810693i $$0.699090\pi$$
$$354$$ −12.0000 −0.637793
$$355$$ 0 0
$$356$$ 2.00000 0.106000
$$357$$ 0 0
$$358$$ −4.00000 −0.211407
$$359$$ −20.0000 −1.05556 −0.527780 0.849381i $$-0.676975\pi$$
−0.527780 + 0.849381i $$0.676975\pi$$
$$360$$ 6.00000 0.316228
$$361$$ 1.00000 0.0526316
$$362$$ 14.0000 0.735824
$$363$$ −11.0000 −0.577350
$$364$$ 0 0
$$365$$ −20.0000 −1.04685
$$366$$ −2.00000 −0.104542
$$367$$ 32.0000 1.67039 0.835193 0.549957i $$-0.185356\pi$$
0.835193 + 0.549957i $$0.185356\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ −2.00000 −0.104116
$$370$$ 20.0000 1.03975
$$371$$ 0 0
$$372$$ −8.00000 −0.414781
$$373$$ −10.0000 −0.517780 −0.258890 0.965907i $$-0.583357\pi$$
−0.258890 + 0.965907i $$0.583357\pi$$
$$374$$ 0 0
$$375$$ 12.0000 0.619677
$$376$$ −36.0000 −1.85656
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ 12.0000 0.616399 0.308199 0.951322i $$-0.400274\pi$$
0.308199 + 0.951322i $$0.400274\pi$$
$$380$$ −2.00000 −0.102598
$$381$$ −8.00000 −0.409852
$$382$$ −12.0000 −0.613973
$$383$$ 8.00000 0.408781 0.204390 0.978889i $$-0.434479\pi$$
0.204390 + 0.978889i $$0.434479\pi$$
$$384$$ −3.00000 −0.153093
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ −4.00000 −0.203331
$$388$$ −10.0000 −0.507673
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ −12.0000 −0.607644
$$391$$ −24.0000 −1.21373
$$392$$ 21.0000 1.06066
$$393$$ 8.00000 0.403547
$$394$$ −2.00000 −0.100759
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 14.0000 0.702640 0.351320 0.936255i $$-0.385733\pi$$
0.351320 + 0.936255i $$0.385733\pi$$
$$398$$ −8.00000 −0.401004
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 38.0000 1.89763 0.948815 0.315833i $$-0.102284\pi$$
0.948815 + 0.315833i $$0.102284\pi$$
$$402$$ −4.00000 −0.199502
$$403$$ 48.0000 2.39105
$$404$$ 10.0000 0.497519
$$405$$ −2.00000 −0.0993808
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 18.0000 0.891133
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 4.00000 0.197546
$$411$$ 18.0000 0.887875
$$412$$ −8.00000 −0.394132
$$413$$ 0 0
$$414$$ 4.00000 0.196589
$$415$$ −32.0000 −1.57082
$$416$$ 30.0000 1.47087
$$417$$ 4.00000 0.195881
$$418$$ 0 0
$$419$$ 8.00000 0.390826 0.195413 0.980721i $$-0.437395\pi$$
0.195413 + 0.980721i $$0.437395\pi$$
$$420$$ 0 0
$$421$$ 14.0000 0.682318 0.341159 0.940006i $$-0.389181\pi$$
0.341159 + 0.940006i $$0.389181\pi$$
$$422$$ −4.00000 −0.194717
$$423$$ 12.0000 0.583460
$$424$$ 18.0000 0.874157
$$425$$ 6.00000 0.291043
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −4.00000 −0.193347
$$429$$ 0 0
$$430$$ 8.00000 0.385794
$$431$$ −24.0000 −1.15604 −0.578020 0.816023i $$-0.696174\pi$$
−0.578020 + 0.816023i $$0.696174\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 0 0
$$435$$ −4.00000 −0.191785
$$436$$ 10.0000 0.478913
$$437$$ −4.00000 −0.191346
$$438$$ 10.0000 0.477818
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ −36.0000 −1.71235
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 10.0000 0.474579
$$445$$ 4.00000 0.189618
$$446$$ 16.0000 0.757622
$$447$$ 6.00000 0.283790
$$448$$ 0 0
$$449$$ −2.00000 −0.0943858 −0.0471929 0.998886i $$-0.515028\pi$$
−0.0471929 + 0.998886i $$0.515028\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ 0 0
$$452$$ −6.00000 −0.282216
$$453$$ −8.00000 −0.375873
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 3.00000 0.140488
$$457$$ −6.00000 −0.280668 −0.140334 0.990104i $$-0.544818\pi$$
−0.140334 + 0.990104i $$0.544818\pi$$
$$458$$ 6.00000 0.280362
$$459$$ −6.00000 −0.280056
$$460$$ 8.00000 0.373002
$$461$$ −18.0000 −0.838344 −0.419172 0.907907i $$-0.637680\pi$$
−0.419172 + 0.907907i $$0.637680\pi$$
$$462$$ 0 0
$$463$$ 32.0000 1.48717 0.743583 0.668644i $$-0.233125\pi$$
0.743583 + 0.668644i $$0.233125\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ −16.0000 −0.741982
$$466$$ 10.0000 0.463241
$$467$$ −32.0000 −1.48078 −0.740392 0.672176i $$-0.765360\pi$$
−0.740392 + 0.672176i $$0.765360\pi$$
$$468$$ −6.00000 −0.277350
$$469$$ 0 0
$$470$$ −24.0000 −1.10704
$$471$$ −2.00000 −0.0921551
$$472$$ 36.0000 1.65703
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ −6.00000 −0.274721
$$478$$ −12.0000 −0.548867
$$479$$ 20.0000 0.913823 0.456912 0.889512i $$-0.348956\pi$$
0.456912 + 0.889512i $$0.348956\pi$$
$$480$$ −10.0000 −0.456435
$$481$$ −60.0000 −2.73576
$$482$$ −6.00000 −0.273293
$$483$$ 0 0
$$484$$ 11.0000 0.500000
$$485$$ −20.0000 −0.908153
$$486$$ 1.00000 0.0453609
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ 6.00000 0.271607
$$489$$ −4.00000 −0.180886
$$490$$ 14.0000 0.632456
$$491$$ −32.0000 −1.44414 −0.722070 0.691820i $$-0.756809\pi$$
−0.722070 + 0.691820i $$0.756809\pi$$
$$492$$ 2.00000 0.0901670
$$493$$ −12.0000 −0.540453
$$494$$ −6.00000 −0.269953
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 0 0
$$498$$ 16.0000 0.716977
$$499$$ 28.0000 1.25345 0.626726 0.779240i $$-0.284395\pi$$
0.626726 + 0.779240i $$0.284395\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ 24.0000 1.07224
$$502$$ −24.0000 −1.07117
$$503$$ 12.0000 0.535054 0.267527 0.963550i $$-0.413794\pi$$
0.267527 + 0.963550i $$0.413794\pi$$
$$504$$ 0 0
$$505$$ 20.0000 0.889988
$$506$$ 0 0
$$507$$ 23.0000 1.02147
$$508$$ 8.00000 0.354943
$$509$$ −22.0000 −0.975133 −0.487566 0.873086i $$-0.662115\pi$$
−0.487566 + 0.873086i $$0.662115\pi$$
$$510$$ 12.0000 0.531369
$$511$$ 0 0
$$512$$ −11.0000 −0.486136
$$513$$ −1.00000 −0.0441511
$$514$$ 14.0000 0.617514
$$515$$ −16.0000 −0.705044
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −22.0000 −0.965693
$$520$$ 36.0000 1.57870
$$521$$ 14.0000 0.613351 0.306676 0.951814i $$-0.400783\pi$$
0.306676 + 0.951814i $$0.400783\pi$$
$$522$$ 2.00000 0.0875376
$$523$$ −28.0000 −1.22435 −0.612177 0.790721i $$-0.709706\pi$$
−0.612177 + 0.790721i $$0.709706\pi$$
$$524$$ −8.00000 −0.349482
$$525$$ 0 0
$$526$$ 12.0000 0.523225
$$527$$ −48.0000 −2.09091
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 12.0000 0.521247
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ −12.0000 −0.519778
$$534$$ −2.00000 −0.0865485
$$535$$ −8.00000 −0.345870
$$536$$ 12.0000 0.518321
$$537$$ −4.00000 −0.172613
$$538$$ −6.00000 −0.258678
$$539$$ 0 0
$$540$$ 2.00000 0.0860663
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ 0 0
$$543$$ 14.0000 0.600798
$$544$$ −30.0000 −1.28624
$$545$$ 20.0000 0.856706
$$546$$ 0 0
$$547$$ 4.00000 0.171028 0.0855138 0.996337i $$-0.472747\pi$$
0.0855138 + 0.996337i $$0.472747\pi$$
$$548$$ −18.0000 −0.768922
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ −2.00000 −0.0852029
$$552$$ −12.0000 −0.510754
$$553$$ 0 0
$$554$$ 22.0000 0.934690
$$555$$ 20.0000 0.848953
$$556$$ −4.00000 −0.169638
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ 8.00000 0.338667
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −10.0000 −0.421825
$$563$$ 20.0000 0.842900 0.421450 0.906852i $$-0.361521\pi$$
0.421450 + 0.906852i $$0.361521\pi$$
$$564$$ −12.0000 −0.505291
$$565$$ −12.0000 −0.504844
$$566$$ −20.0000 −0.840663
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ 2.00000 0.0837708
$$571$$ 44.0000 1.84134 0.920671 0.390339i $$-0.127642\pi$$
0.920671 + 0.390339i $$0.127642\pi$$
$$572$$ 0 0
$$573$$ −12.0000 −0.501307
$$574$$ 0 0
$$575$$ −4.00000 −0.166812
$$576$$ 7.00000 0.291667
$$577$$ 18.0000 0.749350 0.374675 0.927156i $$-0.377754\pi$$
0.374675 + 0.927156i $$0.377754\pi$$
$$578$$ 19.0000 0.790296
$$579$$ −14.0000 −0.581820
$$580$$ 4.00000 0.166091
$$581$$ 0 0
$$582$$ 10.0000 0.414513
$$583$$ 0 0
$$584$$ −30.0000 −1.24141
$$585$$ −12.0000 −0.496139
$$586$$ −14.0000 −0.578335
$$587$$ 8.00000 0.330195 0.165098 0.986277i $$-0.447206\pi$$
0.165098 + 0.986277i $$0.447206\pi$$
$$588$$ 7.00000 0.288675
$$589$$ −8.00000 −0.329634
$$590$$ 24.0000 0.988064
$$591$$ −2.00000 −0.0822690
$$592$$ 10.0000 0.410997
$$593$$ −30.0000 −1.23195 −0.615976 0.787765i $$-0.711238\pi$$
−0.615976 + 0.787765i $$0.711238\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −6.00000 −0.245770
$$597$$ −8.00000 −0.327418
$$598$$ 24.0000 0.981433
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 3.00000 0.122474
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ −4.00000 −0.162893
$$604$$ 8.00000 0.325515
$$605$$ 22.0000 0.894427
$$606$$ −10.0000 −0.406222
$$607$$ 24.0000 0.974130 0.487065 0.873366i $$-0.338067\pi$$
0.487065 + 0.873366i $$0.338067\pi$$
$$608$$ −5.00000 −0.202777
$$609$$ 0 0
$$610$$ 4.00000 0.161955
$$611$$ 72.0000 2.91281
$$612$$ 6.00000 0.242536
$$613$$ 6.00000 0.242338 0.121169 0.992632i $$-0.461336\pi$$
0.121169 + 0.992632i $$0.461336\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 4.00000 0.161296
$$616$$ 0 0
$$617$$ 2.00000 0.0805170 0.0402585 0.999189i $$-0.487182\pi$$
0.0402585 + 0.999189i $$0.487182\pi$$
$$618$$ 8.00000 0.321807
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 16.0000 0.642575
$$621$$ 4.00000 0.160514
$$622$$ 4.00000 0.160385
$$623$$ 0 0
$$624$$ −6.00000 −0.240192
$$625$$ −19.0000 −0.760000
$$626$$ −22.0000 −0.879297
$$627$$ 0 0
$$628$$ 2.00000 0.0798087
$$629$$ 60.0000 2.39236
$$630$$ 0 0
$$631$$ −32.0000 −1.27390 −0.636950 0.770905i $$-0.719804\pi$$
−0.636950 + 0.770905i $$0.719804\pi$$
$$632$$ 0 0
$$633$$ −4.00000 −0.158986
$$634$$ −6.00000 −0.238290
$$635$$ 16.0000 0.634941
$$636$$ 6.00000 0.237915
$$637$$ −42.0000 −1.66410
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 6.00000 0.237171
$$641$$ 38.0000 1.50091 0.750455 0.660922i $$-0.229834\pi$$
0.750455 + 0.660922i $$0.229834\pi$$
$$642$$ 4.00000 0.157867
$$643$$ 20.0000 0.788723 0.394362 0.918955i $$-0.370966\pi$$
0.394362 + 0.918955i $$0.370966\pi$$
$$644$$ 0 0
$$645$$ 8.00000 0.315000
$$646$$ 6.00000 0.236067
$$647$$ 36.0000 1.41531 0.707653 0.706560i $$-0.249754\pi$$
0.707653 + 0.706560i $$0.249754\pi$$
$$648$$ −3.00000 −0.117851
$$649$$ 0 0
$$650$$ −6.00000 −0.235339
$$651$$ 0 0
$$652$$ 4.00000 0.156652
$$653$$ 14.0000 0.547862 0.273931 0.961749i $$-0.411676\pi$$
0.273931 + 0.961749i $$0.411676\pi$$
$$654$$ −10.0000 −0.391031
$$655$$ −16.0000 −0.625172
$$656$$ 2.00000 0.0780869
$$657$$ 10.0000 0.390137
$$658$$ 0 0
$$659$$ −44.0000 −1.71400 −0.856998 0.515319i $$-0.827673\pi$$
−0.856998 + 0.515319i $$0.827673\pi$$
$$660$$ 0 0
$$661$$ 6.00000 0.233373 0.116686 0.993169i $$-0.462773\pi$$
0.116686 + 0.993169i $$0.462773\pi$$
$$662$$ 12.0000 0.466393
$$663$$ −36.0000 −1.39812
$$664$$ −48.0000 −1.86276
$$665$$ 0 0
$$666$$ −10.0000 −0.387492
$$667$$ 8.00000 0.309761
$$668$$ −24.0000 −0.928588
$$669$$ 16.0000 0.618596
$$670$$ 8.00000 0.309067
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −46.0000 −1.77317 −0.886585 0.462566i $$-0.846929\pi$$
−0.886585 + 0.462566i $$0.846929\pi$$
$$674$$ −22.0000 −0.847408
$$675$$ −1.00000 −0.0384900
$$676$$ −23.0000 −0.884615
$$677$$ −22.0000 −0.845529 −0.422764 0.906240i $$-0.638940\pi$$
−0.422764 + 0.906240i $$0.638940\pi$$
$$678$$ 6.00000 0.230429
$$679$$ 0 0
$$680$$ −36.0000 −1.38054
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ 1.00000 0.0382360
$$685$$ −36.0000 −1.37549
$$686$$ 0 0
$$687$$ 6.00000 0.228914
$$688$$ 4.00000 0.152499
$$689$$ −36.0000 −1.37149
$$690$$ −8.00000 −0.304555
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 22.0000 0.836315
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −8.00000 −0.303457
$$696$$ −6.00000 −0.227429
$$697$$ 12.0000 0.454532
$$698$$ −2.00000 −0.0757011
$$699$$ 10.0000 0.378235
$$700$$ 0 0
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 6.00000 0.226455
$$703$$ 10.0000 0.377157
$$704$$ 0 0
$$705$$ −24.0000 −0.903892
$$706$$ −22.0000 −0.827981
$$707$$ 0 0
$$708$$ 12.0000 0.450988
$$709$$ 38.0000 1.42712 0.713560 0.700594i $$-0.247082\pi$$
0.713560 + 0.700594i $$0.247082\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 6.00000 0.224860
$$713$$ 32.0000 1.19841
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 4.00000 0.149487
$$717$$ −12.0000 −0.448148
$$718$$ −20.0000 −0.746393
$$719$$ −20.0000 −0.745874 −0.372937 0.927857i $$-0.621649\pi$$
−0.372937 + 0.927857i $$0.621649\pi$$
$$720$$ 2.00000 0.0745356
$$721$$ 0 0
$$722$$ 1.00000 0.0372161
$$723$$ −6.00000 −0.223142
$$724$$ −14.0000 −0.520306
$$725$$ −2.00000 −0.0742781
$$726$$ −11.0000 −0.408248
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ −20.0000 −0.740233
$$731$$ 24.0000 0.887672
$$732$$ 2.00000 0.0739221
$$733$$ 46.0000 1.69905 0.849524 0.527549i $$-0.176889\pi$$
0.849524 + 0.527549i $$0.176889\pi$$
$$734$$ 32.0000 1.18114
$$735$$ 14.0000 0.516398
$$736$$ 20.0000 0.737210
$$737$$ 0 0
$$738$$ −2.00000 −0.0736210
$$739$$ −36.0000 −1.32428 −0.662141 0.749380i $$-0.730352\pi$$
−0.662141 + 0.749380i $$0.730352\pi$$
$$740$$ −20.0000 −0.735215
$$741$$ −6.00000 −0.220416
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ −24.0000 −0.879883
$$745$$ −12.0000 −0.439646
$$746$$ −10.0000 −0.366126
$$747$$ 16.0000 0.585409
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 12.0000 0.438178
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ −12.0000 −0.437595
$$753$$ −24.0000 −0.874609
$$754$$ 12.0000 0.437014
$$755$$ 16.0000 0.582300
$$756$$ 0 0
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ 12.0000 0.435860
$$759$$ 0 0
$$760$$ −6.00000 −0.217643
$$761$$ 50.0000 1.81250 0.906249 0.422744i $$-0.138933\pi$$
0.906249 + 0.422744i $$0.138933\pi$$
$$762$$ −8.00000 −0.289809
$$763$$ 0 0
$$764$$ 12.0000 0.434145
$$765$$ 12.0000 0.433861
$$766$$ 8.00000 0.289052
$$767$$ −72.0000 −2.59977
$$768$$ −17.0000 −0.613435
$$769$$ 18.0000 0.649097 0.324548 0.945869i $$-0.394788\pi$$
0.324548 + 0.945869i $$0.394788\pi$$
$$770$$ 0 0
$$771$$ 14.0000 0.504198
$$772$$ 14.0000 0.503871
$$773$$ 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
$$776$$ −30.0000 −1.07694
$$777$$ 0 0
$$778$$ 30.0000 1.07555
$$779$$ 2.00000 0.0716574
$$780$$ 12.0000 0.429669
$$781$$ 0 0
$$782$$ −24.0000 −0.858238
$$783$$ 2.00000 0.0714742
$$784$$ 7.00000 0.250000
$$785$$ 4.00000 0.142766
$$786$$ 8.00000 0.285351
$$787$$ 44.0000 1.56843 0.784215 0.620489i $$-0.213066\pi$$
0.784215 + 0.620489i $$0.213066\pi$$
$$788$$ 2.00000 0.0712470
$$789$$ 12.0000 0.427211
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −12.0000 −0.426132
$$794$$ 14.0000 0.496841
$$795$$ 12.0000 0.425596
$$796$$ 8.00000 0.283552
$$797$$ −6.00000 −0.212531 −0.106265 0.994338i $$-0.533889\pi$$
−0.106265 + 0.994338i $$0.533889\pi$$
$$798$$ 0 0
$$799$$ −72.0000 −2.54718
$$800$$ −5.00000 −0.176777
$$801$$ −2.00000 −0.0706665
$$802$$ 38.0000 1.34183
$$803$$ 0 0
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ 48.0000 1.69073
$$807$$ −6.00000 −0.211210
$$808$$ 30.0000 1.05540
$$809$$ 26.0000 0.914111 0.457056 0.889438i $$-0.348904\pi$$
0.457056 + 0.889438i $$0.348904\pi$$
$$810$$ −2.00000 −0.0702728
$$811$$ 44.0000 1.54505 0.772524 0.634985i $$-0.218994\pi$$
0.772524 + 0.634985i $$0.218994\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 8.00000 0.280228
$$816$$ 6.00000 0.210042
$$817$$ 4.00000 0.139942
$$818$$ −14.0000 −0.489499
$$819$$ 0 0
$$820$$ −4.00000 −0.139686
$$821$$ −42.0000 −1.46581 −0.732905 0.680331i $$-0.761836\pi$$
−0.732905 + 0.680331i $$0.761836\pi$$
$$822$$ 18.0000 0.627822
$$823$$ −32.0000 −1.11545 −0.557725 0.830026i $$-0.688326\pi$$
−0.557725 + 0.830026i $$0.688326\pi$$
$$824$$ −24.0000 −0.836080
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −28.0000 −0.973655 −0.486828 0.873498i $$-0.661846\pi$$
−0.486828 + 0.873498i $$0.661846\pi$$
$$828$$ −4.00000 −0.139010
$$829$$ −10.0000 −0.347314 −0.173657 0.984806i $$-0.555558\pi$$
−0.173657 + 0.984806i $$0.555558\pi$$
$$830$$ −32.0000 −1.11074
$$831$$ 22.0000 0.763172
$$832$$ 42.0000 1.45609
$$833$$ 42.0000 1.45521
$$834$$ 4.00000 0.138509
$$835$$ −48.0000 −1.66111
$$836$$ 0 0
$$837$$ 8.00000 0.276520
$$838$$ 8.00000 0.276355
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 14.0000 0.482472
$$843$$ −10.0000 −0.344418
$$844$$ 4.00000 0.137686
$$845$$ −46.0000 −1.58245
$$846$$ 12.0000 0.412568
$$847$$ 0 0
$$848$$ 6.00000 0.206041
$$849$$ −20.0000 −0.686398
$$850$$ 6.00000 0.205798
$$851$$ −40.0000 −1.37118
$$852$$ 0 0
$$853$$ 22.0000 0.753266 0.376633 0.926363i $$-0.377082\pi$$
0.376633 + 0.926363i $$0.377082\pi$$
$$854$$ 0 0
$$855$$ 2.00000 0.0683986
$$856$$ −12.0000 −0.410152
$$857$$ 30.0000 1.02478 0.512390 0.858753i $$-0.328760\pi$$
0.512390 + 0.858753i $$0.328760\pi$$
$$858$$ 0 0
$$859$$ −28.0000 −0.955348 −0.477674 0.878537i $$-0.658520\pi$$
−0.477674 + 0.878537i $$0.658520\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ 0 0
$$862$$ −24.0000 −0.817443
$$863$$ −40.0000 −1.36162 −0.680808 0.732462i $$-0.738371\pi$$
−0.680808 + 0.732462i $$0.738371\pi$$
$$864$$ 5.00000 0.170103
$$865$$ 44.0000 1.49604
$$866$$ −14.0000 −0.475739
$$867$$ 19.0000 0.645274
$$868$$ 0 0
$$869$$ 0 0
$$870$$ −4.00000 −0.135613
$$871$$ −24.0000 −0.813209
$$872$$ 30.0000 1.01593
$$873$$ 10.0000 0.338449
$$874$$ −4.00000 −0.135302
$$875$$ 0 0
$$876$$ −10.0000 −0.337869
$$877$$ −34.0000 −1.14810 −0.574049 0.818821i $$-0.694628\pi$$
−0.574049 + 0.818821i $$0.694628\pi$$
$$878$$ −8.00000 −0.269987
$$879$$ −14.0000 −0.472208
$$880$$ 0 0
$$881$$ −38.0000 −1.28025 −0.640126 0.768270i $$-0.721118\pi$$
−0.640126 + 0.768270i $$0.721118\pi$$
$$882$$ −7.00000 −0.235702
$$883$$ −36.0000 −1.21150 −0.605748 0.795656i $$-0.707126\pi$$
−0.605748 + 0.795656i $$0.707126\pi$$
$$884$$ 36.0000 1.21081
$$885$$ 24.0000 0.806751
$$886$$ 0 0
$$887$$ −40.0000 −1.34307 −0.671534 0.740973i $$-0.734364\pi$$
−0.671534 + 0.740973i $$0.734364\pi$$
$$888$$ 30.0000 1.00673
$$889$$ 0 0
$$890$$ 4.00000 0.134080
$$891$$ 0 0
$$892$$ −16.0000 −0.535720
$$893$$ −12.0000 −0.401565
$$894$$ 6.00000 0.200670
$$895$$ 8.00000 0.267411
$$896$$ 0 0
$$897$$ 24.0000 0.801337
$$898$$ −2.00000 −0.0667409
$$899$$ 16.0000 0.533630
$$900$$ 1.00000 0.0333333
$$901$$ 36.0000 1.19933
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −18.0000 −0.598671
$$905$$ −28.0000 −0.930751
$$906$$ −8.00000 −0.265782
$$907$$ 4.00000 0.132818 0.0664089 0.997792i $$-0.478846\pi$$
0.0664089 + 0.997792i $$0.478846\pi$$
$$908$$ 12.0000 0.398234
$$909$$ −10.0000 −0.331679
$$910$$ 0 0
$$911$$ −16.0000 −0.530104 −0.265052 0.964234i $$-0.585389\pi$$
−0.265052 + 0.964234i $$0.585389\pi$$
$$912$$ 1.00000 0.0331133
$$913$$ 0 0
$$914$$ −6.00000 −0.198462
$$915$$ 4.00000 0.132236
$$916$$ −6.00000 −0.198246
$$917$$ 0 0
$$918$$ −6.00000 −0.198030
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 24.0000 0.791257
$$921$$ −12.0000 −0.395413
$$922$$ −18.0000 −0.592798
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 10.0000 0.328798
$$926$$ 32.0000 1.05159
$$927$$ 8.00000 0.262754
$$928$$ 10.0000 0.328266
$$929$$ 34.0000 1.11550 0.557752 0.830008i $$-0.311664\pi$$
0.557752 + 0.830008i $$0.311664\pi$$
$$930$$ −16.0000 −0.524661
$$931$$ 7.00000 0.229416
$$932$$ −10.0000 −0.327561
$$933$$ 4.00000 0.130954
$$934$$ −32.0000 −1.04707
$$935$$ 0 0
$$936$$ −18.0000 −0.588348
$$937$$ −22.0000 −0.718709 −0.359354 0.933201i $$-0.617003\pi$$
−0.359354 + 0.933201i $$0.617003\pi$$
$$938$$ 0 0
$$939$$ −22.0000 −0.717943
$$940$$ 24.0000 0.782794
$$941$$ −22.0000 −0.717180 −0.358590 0.933495i $$-0.616742\pi$$
−0.358590 + 0.933495i $$0.616742\pi$$
$$942$$ −2.00000 −0.0651635
$$943$$ −8.00000 −0.260516
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −8.00000 −0.259965 −0.129983 0.991516i $$-0.541492\pi$$
−0.129983 + 0.991516i $$0.541492\pi$$
$$948$$ 0 0
$$949$$ 60.0000 1.94768
$$950$$ 1.00000 0.0324443
$$951$$ −6.00000 −0.194563
$$952$$ 0 0
$$953$$ 38.0000 1.23094 0.615470 0.788160i $$-0.288966\pi$$
0.615470 + 0.788160i $$0.288966\pi$$
$$954$$ −6.00000 −0.194257
$$955$$ 24.0000 0.776622
$$956$$ 12.0000 0.388108
$$957$$ 0 0
$$958$$ 20.0000 0.646171
$$959$$ 0 0
$$960$$ −14.0000 −0.451848
$$961$$ 33.0000 1.06452
$$962$$ −60.0000 −1.93448
$$963$$ 4.00000 0.128898
$$964$$ 6.00000 0.193247
$$965$$ 28.0000 0.901352
$$966$$ 0 0
$$967$$ −32.0000 −1.02905 −0.514525 0.857475i $$-0.672032\pi$$
−0.514525 + 0.857475i $$0.672032\pi$$
$$968$$ 33.0000 1.06066
$$969$$ 6.00000 0.192748
$$970$$ −20.0000 −0.642161
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ 32.0000 1.02535
$$975$$ −6.00000 −0.192154
$$976$$ 2.00000 0.0640184
$$977$$ −42.0000 −1.34370 −0.671850 0.740688i $$-0.734500\pi$$
−0.671850 + 0.740688i $$0.734500\pi$$
$$978$$ −4.00000 −0.127906
$$979$$ 0 0
$$980$$ −14.0000 −0.447214
$$981$$ −10.0000 −0.319275
$$982$$ −32.0000 −1.02116
$$983$$ −8.00000 −0.255160 −0.127580 0.991828i $$-0.540721\pi$$
−0.127580 + 0.991828i $$0.540721\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 4.00000 0.127451
$$986$$ −12.0000 −0.382158
$$987$$ 0 0
$$988$$ 6.00000 0.190885
$$989$$ −16.0000 −0.508770
$$990$$ 0 0
$$991$$ 40.0000 1.27064 0.635321 0.772248i $$-0.280868\pi$$
0.635321 + 0.772248i $$0.280868\pi$$
$$992$$ 40.0000 1.27000
$$993$$ 12.0000 0.380808
$$994$$ 0 0
$$995$$ 16.0000 0.507234
$$996$$ −16.0000 −0.506979
$$997$$ −58.0000 −1.83688 −0.918439 0.395562i $$-0.870550\pi$$
−0.918439 + 0.395562i $$0.870550\pi$$
$$998$$ 28.0000 0.886325
$$999$$ −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 57.2.a.c.1.1 1
3.2 odd 2 171.2.a.a.1.1 1
4.3 odd 2 912.2.a.b.1.1 1
5.2 odd 4 1425.2.c.g.799.2 2
5.3 odd 4 1425.2.c.g.799.1 2
5.4 even 2 1425.2.a.a.1.1 1
7.6 odd 2 2793.2.a.i.1.1 1
8.3 odd 2 3648.2.a.bf.1.1 1
8.5 even 2 3648.2.a.o.1.1 1
11.10 odd 2 6897.2.a.a.1.1 1
12.11 even 2 2736.2.a.s.1.1 1
13.12 even 2 9633.2.a.h.1.1 1
15.14 odd 2 4275.2.a.m.1.1 1
19.18 odd 2 1083.2.a.a.1.1 1
21.20 even 2 8379.2.a.e.1.1 1
57.56 even 2 3249.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.c.1.1 1 1.1 even 1 trivial
171.2.a.a.1.1 1 3.2 odd 2
912.2.a.b.1.1 1 4.3 odd 2
1083.2.a.a.1.1 1 19.18 odd 2
1425.2.a.a.1.1 1 5.4 even 2
1425.2.c.g.799.1 2 5.3 odd 4
1425.2.c.g.799.2 2 5.2 odd 4
2736.2.a.s.1.1 1 12.11 even 2
2793.2.a.i.1.1 1 7.6 odd 2
3249.2.a.g.1.1 1 57.56 even 2
3648.2.a.o.1.1 1 8.5 even 2
3648.2.a.bf.1.1 1 8.3 odd 2
4275.2.a.m.1.1 1 15.14 odd 2
6897.2.a.a.1.1 1 11.10 odd 2
8379.2.a.e.1.1 1 21.20 even 2
9633.2.a.h.1.1 1 13.12 even 2