# Properties

 Label 128.2.a.a.1.1 Level $128$ Weight $2$ Character 128.1 Self dual yes Analytic conductor $1.022$ 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] = [128,2,Mod(1,128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(128, 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("128.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$128 = 2^{7}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 128.a (trivial)

## Newform invariants

comment: select newform

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

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

 Self dual: yes Analytic conductor: $$1.02208514587$$ 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 $$\chi$$ $$=$$ 128.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{3} -2.00000 q^{5} -4.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{3} -2.00000 q^{5} -4.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} -2.00000 q^{13} +4.00000 q^{15} -2.00000 q^{17} -2.00000 q^{19} +8.00000 q^{21} +4.00000 q^{23} -1.00000 q^{25} +4.00000 q^{27} +6.00000 q^{29} -4.00000 q^{33} +8.00000 q^{35} -10.0000 q^{37} +4.00000 q^{39} -6.00000 q^{41} -6.00000 q^{43} -2.00000 q^{45} -8.00000 q^{47} +9.00000 q^{49} +4.00000 q^{51} +6.00000 q^{53} -4.00000 q^{55} +4.00000 q^{57} -14.0000 q^{59} -2.00000 q^{61} -4.00000 q^{63} +4.00000 q^{65} -10.0000 q^{67} -8.00000 q^{69} +12.0000 q^{71} +14.0000 q^{73} +2.00000 q^{75} -8.00000 q^{77} -8.00000 q^{79} -11.0000 q^{81} +6.00000 q^{83} +4.00000 q^{85} -12.0000 q^{87} -2.00000 q^{89} +8.00000 q^{91} +4.00000 q^{95} -2.00000 q^{97} +2.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 0 0
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 0 0
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ 4.00000 1.03280
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ −2.00000 −0.458831 −0.229416 0.973329i $$-0.573682\pi$$
−0.229416 + 0.973329i $$0.573682\pi$$
$$20$$ 0 0
$$21$$ 8.00000 1.74574
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 4.00000 0.769800
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ −4.00000 −0.696311
$$34$$ 0 0
$$35$$ 8.00000 1.35225
$$36$$ 0 0
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ 0 0
$$39$$ 4.00000 0.640513
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ −6.00000 −0.914991 −0.457496 0.889212i $$-0.651253\pi$$
−0.457496 + 0.889212i $$0.651253\pi$$
$$44$$ 0 0
$$45$$ −2.00000 −0.298142
$$46$$ 0 0
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ 0 0
$$51$$ 4.00000 0.560112
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ −4.00000 −0.539360
$$56$$ 0 0
$$57$$ 4.00000 0.529813
$$58$$ 0 0
$$59$$ −14.0000 −1.82264 −0.911322 0.411693i $$-0.864937\pi$$
−0.911322 + 0.411693i $$0.864937\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 0 0
$$63$$ −4.00000 −0.503953
$$64$$ 0 0
$$65$$ 4.00000 0.496139
$$66$$ 0 0
$$67$$ −10.0000 −1.22169 −0.610847 0.791748i $$-0.709171\pi$$
−0.610847 + 0.791748i $$0.709171\pi$$
$$68$$ 0 0
$$69$$ −8.00000 −0.963087
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ 14.0000 1.63858 0.819288 0.573382i $$-0.194369\pi$$
0.819288 + 0.573382i $$0.194369\pi$$
$$74$$ 0 0
$$75$$ 2.00000 0.230940
$$76$$ 0 0
$$77$$ −8.00000 −0.911685
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 6.00000 0.658586 0.329293 0.944228i $$-0.393190\pi$$
0.329293 + 0.944228i $$0.393190\pi$$
$$84$$ 0 0
$$85$$ 4.00000 0.433861
$$86$$ 0 0
$$87$$ −12.0000 −1.28654
$$88$$ 0 0
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ 8.00000 0.838628
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 4.00000 0.410391
$$96$$ 0 0
$$97$$ −2.00000 −0.203069 −0.101535 0.994832i $$-0.532375\pi$$
−0.101535 + 0.994832i $$0.532375\pi$$
$$98$$ 0 0
$$99$$ 2.00000 0.201008
$$100$$ 0 0
$$101$$ 6.00000 0.597022 0.298511 0.954406i $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 0 0
$$105$$ −16.0000 −1.56144
$$106$$ 0 0
$$107$$ 2.00000 0.193347 0.0966736 0.995316i $$-0.469180\pi$$
0.0966736 + 0.995316i $$0.469180\pi$$
$$108$$ 0 0
$$109$$ 6.00000 0.574696 0.287348 0.957826i $$-0.407226\pi$$
0.287348 + 0.957826i $$0.407226\pi$$
$$110$$ 0 0
$$111$$ 20.0000 1.89832
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ −8.00000 −0.746004
$$116$$ 0 0
$$117$$ −2.00000 −0.184900
$$118$$ 0 0
$$119$$ 8.00000 0.733359
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 12.0000 1.08200
$$124$$ 0 0
$$125$$ 12.0000 1.07331
$$126$$ 0 0
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ 0 0
$$129$$ 12.0000 1.05654
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ 8.00000 0.693688
$$134$$ 0 0
$$135$$ −8.00000 −0.688530
$$136$$ 0 0
$$137$$ 10.0000 0.854358 0.427179 0.904167i $$-0.359507\pi$$
0.427179 + 0.904167i $$0.359507\pi$$
$$138$$ 0 0
$$139$$ 10.0000 0.848189 0.424094 0.905618i $$-0.360592\pi$$
0.424094 + 0.905618i $$0.360592\pi$$
$$140$$ 0 0
$$141$$ 16.0000 1.34744
$$142$$ 0 0
$$143$$ −4.00000 −0.334497
$$144$$ 0 0
$$145$$ −12.0000 −0.996546
$$146$$ 0 0
$$147$$ −18.0000 −1.48461
$$148$$ 0 0
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ 4.00000 0.325515 0.162758 0.986666i $$-0.447961\pi$$
0.162758 + 0.986666i $$0.447961\pi$$
$$152$$ 0 0
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −18.0000 −1.43656 −0.718278 0.695756i $$-0.755069\pi$$
−0.718278 + 0.695756i $$0.755069\pi$$
$$158$$ 0 0
$$159$$ −12.0000 −0.951662
$$160$$ 0 0
$$161$$ −16.0000 −1.26098
$$162$$ 0 0
$$163$$ −2.00000 −0.156652 −0.0783260 0.996928i $$-0.524958\pi$$
−0.0783260 + 0.996928i $$0.524958\pi$$
$$164$$ 0 0
$$165$$ 8.00000 0.622799
$$166$$ 0 0
$$167$$ −20.0000 −1.54765 −0.773823 0.633402i $$-0.781658\pi$$
−0.773823 + 0.633402i $$0.781658\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ −2.00000 −0.152944
$$172$$ 0 0
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ 0 0
$$175$$ 4.00000 0.302372
$$176$$ 0 0
$$177$$ 28.0000 2.10461
$$178$$ 0 0
$$179$$ 6.00000 0.448461 0.224231 0.974536i $$-0.428013\pi$$
0.224231 + 0.974536i $$0.428013\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ 4.00000 0.295689
$$184$$ 0 0
$$185$$ 20.0000 1.47043
$$186$$ 0 0
$$187$$ −4.00000 −0.292509
$$188$$ 0 0
$$189$$ −16.0000 −1.16383
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 0 0
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ 0 0
$$195$$ −8.00000 −0.572892
$$196$$ 0 0
$$197$$ 14.0000 0.997459 0.498729 0.866758i $$-0.333800\pi$$
0.498729 + 0.866758i $$0.333800\pi$$
$$198$$ 0 0
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ 20.0000 1.41069
$$202$$ 0 0
$$203$$ −24.0000 −1.68447
$$204$$ 0 0
$$205$$ 12.0000 0.838116
$$206$$ 0 0
$$207$$ 4.00000 0.278019
$$208$$ 0 0
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ 22.0000 1.51454 0.757271 0.653101i $$-0.226532\pi$$
0.757271 + 0.653101i $$0.226532\pi$$
$$212$$ 0 0
$$213$$ −24.0000 −1.64445
$$214$$ 0 0
$$215$$ 12.0000 0.818393
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −28.0000 −1.89206
$$220$$ 0 0
$$221$$ 4.00000 0.269069
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 0 0
$$227$$ −18.0000 −1.19470 −0.597351 0.801980i $$-0.703780\pi$$
−0.597351 + 0.801980i $$0.703780\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 16.0000 1.05272
$$232$$ 0 0
$$233$$ −18.0000 −1.17922 −0.589610 0.807688i $$-0.700718\pi$$
−0.589610 + 0.807688i $$0.700718\pi$$
$$234$$ 0 0
$$235$$ 16.0000 1.04372
$$236$$ 0 0
$$237$$ 16.0000 1.03931
$$238$$ 0 0
$$239$$ −24.0000 −1.55243 −0.776215 0.630468i $$-0.782863\pi$$
−0.776215 + 0.630468i $$0.782863\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 0 0
$$243$$ 10.0000 0.641500
$$244$$ 0 0
$$245$$ −18.0000 −1.14998
$$246$$ 0 0
$$247$$ 4.00000 0.254514
$$248$$ 0 0
$$249$$ −12.0000 −0.760469
$$250$$ 0 0
$$251$$ 18.0000 1.13615 0.568075 0.822977i $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ 0 0
$$253$$ 8.00000 0.502956
$$254$$ 0 0
$$255$$ −8.00000 −0.500979
$$256$$ 0 0
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ 40.0000 2.48548
$$260$$ 0 0
$$261$$ 6.00000 0.371391
$$262$$ 0 0
$$263$$ 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
$$267$$ 4.00000 0.244796
$$268$$ 0 0
$$269$$ −10.0000 −0.609711 −0.304855 0.952399i $$-0.598608\pi$$
−0.304855 + 0.952399i $$0.598608\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ −16.0000 −0.968364
$$274$$ 0 0
$$275$$ −2.00000 −0.120605
$$276$$ 0 0
$$277$$ 6.00000 0.360505 0.180253 0.983620i $$-0.442309\pi$$
0.180253 + 0.983620i $$0.442309\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 0 0
$$283$$ −6.00000 −0.356663 −0.178331 0.983970i $$-0.557070\pi$$
−0.178331 + 0.983970i $$0.557070\pi$$
$$284$$ 0 0
$$285$$ −8.00000 −0.473879
$$286$$ 0 0
$$287$$ 24.0000 1.41668
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 4.00000 0.234484
$$292$$ 0 0
$$293$$ 14.0000 0.817889 0.408944 0.912559i $$-0.365897\pi$$
0.408944 + 0.912559i $$0.365897\pi$$
$$294$$ 0 0
$$295$$ 28.0000 1.63022
$$296$$ 0 0
$$297$$ 8.00000 0.464207
$$298$$ 0 0
$$299$$ −8.00000 −0.462652
$$300$$ 0 0
$$301$$ 24.0000 1.38334
$$302$$ 0 0
$$303$$ −12.0000 −0.689382
$$304$$ 0 0
$$305$$ 4.00000 0.229039
$$306$$ 0 0
$$307$$ −18.0000 −1.02731 −0.513657 0.857996i $$-0.671710\pi$$
−0.513657 + 0.857996i $$0.671710\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ −28.0000 −1.58773 −0.793867 0.608091i $$-0.791935\pi$$
−0.793867 + 0.608091i $$0.791935\pi$$
$$312$$ 0 0
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 0 0
$$315$$ 8.00000 0.450749
$$316$$ 0 0
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ 0 0
$$319$$ 12.0000 0.671871
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ 0 0
$$323$$ 4.00000 0.222566
$$324$$ 0 0
$$325$$ 2.00000 0.110940
$$326$$ 0 0
$$327$$ −12.0000 −0.663602
$$328$$ 0 0
$$329$$ 32.0000 1.76422
$$330$$ 0 0
$$331$$ −14.0000 −0.769510 −0.384755 0.923019i $$-0.625714\pi$$
−0.384755 + 0.923019i $$0.625714\pi$$
$$332$$ 0 0
$$333$$ −10.0000 −0.547997
$$334$$ 0 0
$$335$$ 20.0000 1.09272
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ 0 0
$$339$$ −4.00000 −0.217250
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −8.00000 −0.431959
$$344$$ 0 0
$$345$$ 16.0000 0.861411
$$346$$ 0 0
$$347$$ 18.0000 0.966291 0.483145 0.875540i $$-0.339494\pi$$
0.483145 + 0.875540i $$0.339494\pi$$
$$348$$ 0 0
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ −8.00000 −0.427008
$$352$$ 0 0
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ −24.0000 −1.27379
$$356$$ 0 0
$$357$$ −16.0000 −0.846810
$$358$$ 0 0
$$359$$ −4.00000 −0.211112 −0.105556 0.994413i $$-0.533662\pi$$
−0.105556 + 0.994413i $$0.533662\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ 14.0000 0.734809
$$364$$ 0 0
$$365$$ −28.0000 −1.46559
$$366$$ 0 0
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −24.0000 −1.24602
$$372$$ 0 0
$$373$$ −10.0000 −0.517780 −0.258890 0.965907i $$-0.583357\pi$$
−0.258890 + 0.965907i $$0.583357\pi$$
$$374$$ 0 0
$$375$$ −24.0000 −1.23935
$$376$$ 0 0
$$377$$ −12.0000 −0.618031
$$378$$ 0 0
$$379$$ 2.00000 0.102733 0.0513665 0.998680i $$-0.483642\pi$$
0.0513665 + 0.998680i $$0.483642\pi$$
$$380$$ 0 0
$$381$$ −32.0000 −1.63941
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 16.0000 0.815436
$$386$$ 0 0
$$387$$ −6.00000 −0.304997
$$388$$ 0 0
$$389$$ −10.0000 −0.507020 −0.253510 0.967333i $$-0.581585\pi$$
−0.253510 + 0.967333i $$0.581585\pi$$
$$390$$ 0 0
$$391$$ −8.00000 −0.404577
$$392$$ 0 0
$$393$$ −12.0000 −0.605320
$$394$$ 0 0
$$395$$ 16.0000 0.805047
$$396$$ 0 0
$$397$$ 6.00000 0.301131 0.150566 0.988600i $$-0.451890\pi$$
0.150566 + 0.988600i $$0.451890\pi$$
$$398$$ 0 0
$$399$$ −16.0000 −0.801002
$$400$$ 0 0
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 22.0000 1.09319
$$406$$ 0 0
$$407$$ −20.0000 −0.991363
$$408$$ 0 0
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 0 0
$$411$$ −20.0000 −0.986527
$$412$$ 0 0
$$413$$ 56.0000 2.75558
$$414$$ 0 0
$$415$$ −12.0000 −0.589057
$$416$$ 0 0
$$417$$ −20.0000 −0.979404
$$418$$ 0 0
$$419$$ −26.0000 −1.27018 −0.635092 0.772437i $$-0.719038\pi$$
−0.635092 + 0.772437i $$0.719038\pi$$
$$420$$ 0 0
$$421$$ −34.0000 −1.65706 −0.828529 0.559946i $$-0.810822\pi$$
−0.828529 + 0.559946i $$0.810822\pi$$
$$422$$ 0 0
$$423$$ −8.00000 −0.388973
$$424$$ 0 0
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ 8.00000 0.387147
$$428$$ 0 0
$$429$$ 8.00000 0.386244
$$430$$ 0 0
$$431$$ 40.0000 1.92673 0.963366 0.268190i $$-0.0864254\pi$$
0.963366 + 0.268190i $$0.0864254\pi$$
$$432$$ 0 0
$$433$$ 30.0000 1.44171 0.720854 0.693087i $$-0.243750\pi$$
0.720854 + 0.693087i $$0.243750\pi$$
$$434$$ 0 0
$$435$$ 24.0000 1.15071
$$436$$ 0 0
$$437$$ −8.00000 −0.382692
$$438$$ 0 0
$$439$$ 36.0000 1.71819 0.859093 0.511819i $$-0.171028\pi$$
0.859093 + 0.511819i $$0.171028\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 0 0
$$443$$ −6.00000 −0.285069 −0.142534 0.989790i $$-0.545525\pi$$
−0.142534 + 0.989790i $$0.545525\pi$$
$$444$$ 0 0
$$445$$ 4.00000 0.189618
$$446$$ 0 0
$$447$$ 36.0000 1.70274
$$448$$ 0 0
$$449$$ −34.0000 −1.60456 −0.802280 0.596948i $$-0.796380\pi$$
−0.802280 + 0.596948i $$0.796380\pi$$
$$450$$ 0 0
$$451$$ −12.0000 −0.565058
$$452$$ 0 0
$$453$$ −8.00000 −0.375873
$$454$$ 0 0
$$455$$ −16.0000 −0.750092
$$456$$ 0 0
$$457$$ −6.00000 −0.280668 −0.140334 0.990104i $$-0.544818\pi$$
−0.140334 + 0.990104i $$0.544818\pi$$
$$458$$ 0 0
$$459$$ −8.00000 −0.373408
$$460$$ 0 0
$$461$$ −10.0000 −0.465746 −0.232873 0.972507i $$-0.574813\pi$$
−0.232873 + 0.972507i $$0.574813\pi$$
$$462$$ 0 0
$$463$$ −8.00000 −0.371792 −0.185896 0.982569i $$-0.559519\pi$$
−0.185896 + 0.982569i $$0.559519\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 14.0000 0.647843 0.323921 0.946084i $$-0.394999\pi$$
0.323921 + 0.946084i $$0.394999\pi$$
$$468$$ 0 0
$$469$$ 40.0000 1.84703
$$470$$ 0 0
$$471$$ 36.0000 1.65879
$$472$$ 0 0
$$473$$ −12.0000 −0.551761
$$474$$ 0 0
$$475$$ 2.00000 0.0917663
$$476$$ 0 0
$$477$$ 6.00000 0.274721
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ 0 0
$$483$$ 32.0000 1.45605
$$484$$ 0 0
$$485$$ 4.00000 0.181631
$$486$$ 0 0
$$487$$ −20.0000 −0.906287 −0.453143 0.891438i $$-0.649697\pi$$
−0.453143 + 0.891438i $$0.649697\pi$$
$$488$$ 0 0
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ 10.0000 0.451294 0.225647 0.974209i $$-0.427550\pi$$
0.225647 + 0.974209i $$0.427550\pi$$
$$492$$ 0 0
$$493$$ −12.0000 −0.540453
$$494$$ 0 0
$$495$$ −4.00000 −0.179787
$$496$$ 0 0
$$497$$ −48.0000 −2.15309
$$498$$ 0 0
$$499$$ 22.0000 0.984855 0.492428 0.870353i $$-0.336110\pi$$
0.492428 + 0.870353i $$0.336110\pi$$
$$500$$ 0 0
$$501$$ 40.0000 1.78707
$$502$$ 0 0
$$503$$ 20.0000 0.891756 0.445878 0.895094i $$-0.352892\pi$$
0.445878 + 0.895094i $$0.352892\pi$$
$$504$$ 0 0
$$505$$ −12.0000 −0.533993
$$506$$ 0 0
$$507$$ 18.0000 0.799408
$$508$$ 0 0
$$509$$ 14.0000 0.620539 0.310270 0.950649i $$-0.399581\pi$$
0.310270 + 0.950649i $$0.399581\pi$$
$$510$$ 0 0
$$511$$ −56.0000 −2.47729
$$512$$ 0 0
$$513$$ −8.00000 −0.353209
$$514$$ 0 0
$$515$$ 8.00000 0.352522
$$516$$ 0 0
$$517$$ −16.0000 −0.703679
$$518$$ 0 0
$$519$$ 36.0000 1.58022
$$520$$ 0 0
$$521$$ −22.0000 −0.963837 −0.481919 0.876216i $$-0.660060\pi$$
−0.481919 + 0.876216i $$0.660060\pi$$
$$522$$ 0 0
$$523$$ −14.0000 −0.612177 −0.306089 0.952003i $$-0.599020\pi$$
−0.306089 + 0.952003i $$0.599020\pi$$
$$524$$ 0 0
$$525$$ −8.00000 −0.349149
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 0 0
$$531$$ −14.0000 −0.607548
$$532$$ 0 0
$$533$$ 12.0000 0.519778
$$534$$ 0 0
$$535$$ −4.00000 −0.172935
$$536$$ 0 0
$$537$$ −12.0000 −0.517838
$$538$$ 0 0
$$539$$ 18.0000 0.775315
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 0 0
$$543$$ 4.00000 0.171656
$$544$$ 0 0
$$545$$ −12.0000 −0.514024
$$546$$ 0 0
$$547$$ 38.0000 1.62476 0.812381 0.583127i $$-0.198171\pi$$
0.812381 + 0.583127i $$0.198171\pi$$
$$548$$ 0 0
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ −12.0000 −0.511217
$$552$$ 0 0
$$553$$ 32.0000 1.36078
$$554$$ 0 0
$$555$$ −40.0000 −1.69791
$$556$$ 0 0
$$557$$ −2.00000 −0.0847427 −0.0423714 0.999102i $$-0.513491\pi$$
−0.0423714 + 0.999102i $$0.513491\pi$$
$$558$$ 0 0
$$559$$ 12.0000 0.507546
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 0 0
$$563$$ −18.0000 −0.758610 −0.379305 0.925272i $$-0.623837\pi$$
−0.379305 + 0.925272i $$0.623837\pi$$
$$564$$ 0 0
$$565$$ −4.00000 −0.168281
$$566$$ 0 0
$$567$$ 44.0000 1.84783
$$568$$ 0 0
$$569$$ 26.0000 1.08998 0.544988 0.838444i $$-0.316534\pi$$
0.544988 + 0.838444i $$0.316534\pi$$
$$570$$ 0 0
$$571$$ −38.0000 −1.59025 −0.795125 0.606445i $$-0.792595\pi$$
−0.795125 + 0.606445i $$0.792595\pi$$
$$572$$ 0 0
$$573$$ 32.0000 1.33682
$$574$$ 0 0
$$575$$ −4.00000 −0.166812
$$576$$ 0 0
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 0 0
$$579$$ 4.00000 0.166234
$$580$$ 0 0
$$581$$ −24.0000 −0.995688
$$582$$ 0 0
$$583$$ 12.0000 0.496989
$$584$$ 0 0
$$585$$ 4.00000 0.165380
$$586$$ 0 0
$$587$$ 34.0000 1.40333 0.701665 0.712507i $$-0.252440\pi$$
0.701665 + 0.712507i $$0.252440\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −28.0000 −1.15177
$$592$$ 0 0
$$593$$ 18.0000 0.739171 0.369586 0.929197i $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 0 0
$$595$$ −16.0000 −0.655936
$$596$$ 0 0
$$597$$ 8.00000 0.327418
$$598$$ 0 0
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ 30.0000 1.22373 0.611863 0.790964i $$-0.290420\pi$$
0.611863 + 0.790964i $$0.290420\pi$$
$$602$$ 0 0
$$603$$ −10.0000 −0.407231
$$604$$ 0 0
$$605$$ 14.0000 0.569181
$$606$$ 0 0
$$607$$ −16.0000 −0.649420 −0.324710 0.945814i $$-0.605267\pi$$
−0.324710 + 0.945814i $$0.605267\pi$$
$$608$$ 0 0
$$609$$ 48.0000 1.94506
$$610$$ 0 0
$$611$$ 16.0000 0.647291
$$612$$ 0 0
$$613$$ −34.0000 −1.37325 −0.686624 0.727013i $$-0.740908\pi$$
−0.686624 + 0.727013i $$0.740908\pi$$
$$614$$ 0 0
$$615$$ −24.0000 −0.967773
$$616$$ 0 0
$$617$$ −2.00000 −0.0805170 −0.0402585 0.999189i $$-0.512818\pi$$
−0.0402585 + 0.999189i $$0.512818\pi$$
$$618$$ 0 0
$$619$$ −46.0000 −1.84890 −0.924448 0.381308i $$-0.875474\pi$$
−0.924448 + 0.381308i $$0.875474\pi$$
$$620$$ 0 0
$$621$$ 16.0000 0.642058
$$622$$ 0 0
$$623$$ 8.00000 0.320513
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ 8.00000 0.319489
$$628$$ 0 0
$$629$$ 20.0000 0.797452
$$630$$ 0 0
$$631$$ −44.0000 −1.75161 −0.875806 0.482663i $$-0.839670\pi$$
−0.875806 + 0.482663i $$0.839670\pi$$
$$632$$ 0 0
$$633$$ −44.0000 −1.74884
$$634$$ 0 0
$$635$$ −32.0000 −1.26988
$$636$$ 0 0
$$637$$ −18.0000 −0.713186
$$638$$ 0 0
$$639$$ 12.0000 0.474713
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 0 0
$$643$$ −42.0000 −1.65632 −0.828159 0.560493i $$-0.810612\pi$$
−0.828159 + 0.560493i $$0.810612\pi$$
$$644$$ 0 0
$$645$$ −24.0000 −0.944999
$$646$$ 0 0
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ 0 0
$$649$$ −28.0000 −1.09910
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −42.0000 −1.64359 −0.821794 0.569785i $$-0.807026\pi$$
−0.821794 + 0.569785i $$0.807026\pi$$
$$654$$ 0 0
$$655$$ −12.0000 −0.468879
$$656$$ 0 0
$$657$$ 14.0000 0.546192
$$658$$ 0 0
$$659$$ 6.00000 0.233727 0.116863 0.993148i $$-0.462716\pi$$
0.116863 + 0.993148i $$0.462716\pi$$
$$660$$ 0 0
$$661$$ −34.0000 −1.32245 −0.661223 0.750189i $$-0.729962\pi$$
−0.661223 + 0.750189i $$0.729962\pi$$
$$662$$ 0 0
$$663$$ −8.00000 −0.310694
$$664$$ 0 0
$$665$$ −16.0000 −0.620453
$$666$$ 0 0
$$667$$ 24.0000 0.929284
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −4.00000 −0.154418
$$672$$ 0 0
$$673$$ −2.00000 −0.0770943 −0.0385472 0.999257i $$-0.512273\pi$$
−0.0385472 + 0.999257i $$0.512273\pi$$
$$674$$ 0 0
$$675$$ −4.00000 −0.153960
$$676$$ 0 0
$$677$$ 22.0000 0.845529 0.422764 0.906240i $$-0.361060\pi$$
0.422764 + 0.906240i $$0.361060\pi$$
$$678$$ 0 0
$$679$$ 8.00000 0.307012
$$680$$ 0 0
$$681$$ 36.0000 1.37952
$$682$$ 0 0
$$683$$ 42.0000 1.60709 0.803543 0.595247i $$-0.202946\pi$$
0.803543 + 0.595247i $$0.202946\pi$$
$$684$$ 0 0
$$685$$ −20.0000 −0.764161
$$686$$ 0 0
$$687$$ −28.0000 −1.06827
$$688$$ 0 0
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 6.00000 0.228251 0.114125 0.993466i $$-0.463593\pi$$
0.114125 + 0.993466i $$0.463593\pi$$
$$692$$ 0 0
$$693$$ −8.00000 −0.303895
$$694$$ 0 0
$$695$$ −20.0000 −0.758643
$$696$$ 0 0
$$697$$ 12.0000 0.454532
$$698$$ 0 0
$$699$$ 36.0000 1.36165
$$700$$ 0 0
$$701$$ −2.00000 −0.0755390 −0.0377695 0.999286i $$-0.512025\pi$$
−0.0377695 + 0.999286i $$0.512025\pi$$
$$702$$ 0 0
$$703$$ 20.0000 0.754314
$$704$$ 0 0
$$705$$ −32.0000 −1.20519
$$706$$ 0 0
$$707$$ −24.0000 −0.902613
$$708$$ 0 0
$$709$$ 22.0000 0.826227 0.413114 0.910679i $$-0.364441\pi$$
0.413114 + 0.910679i $$0.364441\pi$$
$$710$$ 0 0
$$711$$ −8.00000 −0.300023
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 8.00000 0.299183
$$716$$ 0 0
$$717$$ 48.0000 1.79259
$$718$$ 0 0
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ 0 0
$$723$$ 4.00000 0.148762
$$724$$ 0 0
$$725$$ −6.00000 −0.222834
$$726$$ 0 0
$$727$$ −12.0000 −0.445055 −0.222528 0.974926i $$-0.571431\pi$$
−0.222528 + 0.974926i $$0.571431\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 12.0000 0.443836
$$732$$ 0 0
$$733$$ 6.00000 0.221615 0.110808 0.993842i $$-0.464656\pi$$
0.110808 + 0.993842i $$0.464656\pi$$
$$734$$ 0 0
$$735$$ 36.0000 1.32788
$$736$$ 0 0
$$737$$ −20.0000 −0.736709
$$738$$ 0 0
$$739$$ −18.0000 −0.662141 −0.331070 0.943606i $$-0.607410\pi$$
−0.331070 + 0.943606i $$0.607410\pi$$
$$740$$ 0 0
$$741$$ −8.00000 −0.293887
$$742$$ 0 0
$$743$$ 44.0000 1.61420 0.807102 0.590412i $$-0.201035\pi$$
0.807102 + 0.590412i $$0.201035\pi$$
$$744$$ 0 0
$$745$$ 36.0000 1.31894
$$746$$ 0 0
$$747$$ 6.00000 0.219529
$$748$$ 0 0
$$749$$ −8.00000 −0.292314
$$750$$ 0 0
$$751$$ 8.00000 0.291924 0.145962 0.989290i $$-0.453372\pi$$
0.145962 + 0.989290i $$0.453372\pi$$
$$752$$ 0 0
$$753$$ −36.0000 −1.31191
$$754$$ 0 0
$$755$$ −8.00000 −0.291150
$$756$$ 0 0
$$757$$ 46.0000 1.67190 0.835949 0.548807i $$-0.184918\pi$$
0.835949 + 0.548807i $$0.184918\pi$$
$$758$$ 0 0
$$759$$ −16.0000 −0.580763
$$760$$ 0 0
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ 0 0
$$763$$ −24.0000 −0.868858
$$764$$ 0 0
$$765$$ 4.00000 0.144620
$$766$$ 0 0
$$767$$ 28.0000 1.01102
$$768$$ 0 0
$$769$$ −34.0000 −1.22607 −0.613036 0.790055i $$-0.710052\pi$$
−0.613036 + 0.790055i $$0.710052\pi$$
$$770$$ 0 0
$$771$$ −36.0000 −1.29651
$$772$$ 0 0
$$773$$ 54.0000 1.94225 0.971123 0.238581i $$-0.0766824\pi$$
0.971123 + 0.238581i $$0.0766824\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −80.0000 −2.86998
$$778$$ 0 0
$$779$$ 12.0000 0.429945
$$780$$ 0 0
$$781$$ 24.0000 0.858788
$$782$$ 0 0
$$783$$ 24.0000 0.857690
$$784$$ 0 0
$$785$$ 36.0000 1.28490
$$786$$ 0 0
$$787$$ 22.0000 0.784215 0.392108 0.919919i $$-0.371746\pi$$
0.392108 + 0.919919i $$0.371746\pi$$
$$788$$ 0 0
$$789$$ −24.0000 −0.854423
$$790$$ 0 0
$$791$$ −8.00000 −0.284447
$$792$$ 0 0
$$793$$ 4.00000 0.142044
$$794$$ 0 0
$$795$$ 24.0000 0.851192
$$796$$ 0 0
$$797$$ −18.0000 −0.637593 −0.318796 0.947823i $$-0.603279\pi$$
−0.318796 + 0.947823i $$0.603279\pi$$
$$798$$ 0 0
$$799$$ 16.0000 0.566039
$$800$$ 0 0
$$801$$ −2.00000 −0.0706665
$$802$$ 0 0
$$803$$ 28.0000 0.988099
$$804$$ 0 0
$$805$$ 32.0000 1.12785
$$806$$ 0 0
$$807$$ 20.0000 0.704033
$$808$$ 0 0
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ 18.0000 0.632065 0.316033 0.948748i $$-0.397649\pi$$
0.316033 + 0.948748i $$0.397649\pi$$
$$812$$ 0 0
$$813$$ −16.0000 −0.561144
$$814$$ 0 0
$$815$$ 4.00000 0.140114
$$816$$ 0 0
$$817$$ 12.0000 0.419827
$$818$$ 0 0
$$819$$ 8.00000 0.279543
$$820$$ 0 0
$$821$$ −10.0000 −0.349002 −0.174501 0.984657i $$-0.555831\pi$$
−0.174501 + 0.984657i $$0.555831\pi$$
$$822$$ 0 0
$$823$$ −28.0000 −0.976019 −0.488009 0.872838i $$-0.662277\pi$$
−0.488009 + 0.872838i $$0.662277\pi$$
$$824$$ 0 0
$$825$$ 4.00000 0.139262
$$826$$ 0 0
$$827$$ −22.0000 −0.765015 −0.382507 0.923952i $$-0.624939\pi$$
−0.382507 + 0.923952i $$0.624939\pi$$
$$828$$ 0 0
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ 0 0
$$831$$ −12.0000 −0.416275
$$832$$ 0 0
$$833$$ −18.0000 −0.623663
$$834$$ 0 0
$$835$$ 40.0000 1.38426
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −36.0000 −1.24286 −0.621429 0.783470i $$-0.713448\pi$$
−0.621429 + 0.783470i $$0.713448\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ 36.0000 1.23991
$$844$$ 0 0
$$845$$ 18.0000 0.619219
$$846$$ 0 0
$$847$$ 28.0000 0.962091
$$848$$ 0 0
$$849$$ 12.0000 0.411839
$$850$$ 0 0
$$851$$ −40.0000 −1.37118
$$852$$ 0 0
$$853$$ −26.0000 −0.890223 −0.445112 0.895475i $$-0.646836\pi$$
−0.445112 + 0.895475i $$0.646836\pi$$
$$854$$ 0 0
$$855$$ 4.00000 0.136797
$$856$$ 0 0
$$857$$ −6.00000 −0.204956 −0.102478 0.994735i $$-0.532677\pi$$
−0.102478 + 0.994735i $$0.532677\pi$$
$$858$$ 0 0
$$859$$ 50.0000 1.70598 0.852989 0.521929i $$-0.174787\pi$$
0.852989 + 0.521929i $$0.174787\pi$$
$$860$$ 0 0
$$861$$ −48.0000 −1.63584
$$862$$ 0 0
$$863$$ −32.0000 −1.08929 −0.544646 0.838666i $$-0.683336\pi$$
−0.544646 + 0.838666i $$0.683336\pi$$
$$864$$ 0 0
$$865$$ 36.0000 1.22404
$$866$$ 0 0
$$867$$ 26.0000 0.883006
$$868$$ 0 0
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ 20.0000 0.677674
$$872$$ 0 0
$$873$$ −2.00000 −0.0676897
$$874$$ 0 0
$$875$$ −48.0000 −1.62270
$$876$$ 0 0
$$877$$ 22.0000 0.742887 0.371444 0.928456i $$-0.378863\pi$$
0.371444 + 0.928456i $$0.378863\pi$$
$$878$$ 0 0
$$879$$ −28.0000 −0.944417
$$880$$ 0 0
$$881$$ −46.0000 −1.54978 −0.774890 0.632096i $$-0.782195\pi$$
−0.774890 + 0.632096i $$0.782195\pi$$
$$882$$ 0 0
$$883$$ −34.0000 −1.14419 −0.572096 0.820187i $$-0.693869\pi$$
−0.572096 + 0.820187i $$0.693869\pi$$
$$884$$ 0 0
$$885$$ −56.0000 −1.88242
$$886$$ 0 0
$$887$$ 36.0000 1.20876 0.604381 0.796696i $$-0.293421\pi$$
0.604381 + 0.796696i $$0.293421\pi$$
$$888$$ 0 0
$$889$$ −64.0000 −2.14649
$$890$$ 0 0
$$891$$ −22.0000 −0.737028
$$892$$ 0 0
$$893$$ 16.0000 0.535420
$$894$$ 0 0
$$895$$ −12.0000 −0.401116
$$896$$ 0 0
$$897$$ 16.0000 0.534224
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −12.0000 −0.399778
$$902$$ 0 0
$$903$$ −48.0000 −1.59734
$$904$$ 0 0
$$905$$ 4.00000 0.132964
$$906$$ 0 0
$$907$$ −38.0000 −1.26177 −0.630885 0.775877i $$-0.717308\pi$$
−0.630885 + 0.775877i $$0.717308\pi$$
$$908$$ 0 0
$$909$$ 6.00000 0.199007
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ 0 0
$$913$$ 12.0000 0.397142
$$914$$ 0 0
$$915$$ −8.00000 −0.264472
$$916$$ 0 0
$$917$$ −24.0000 −0.792550
$$918$$ 0 0
$$919$$ 36.0000 1.18753 0.593765 0.804638i $$-0.297641\pi$$
0.593765 + 0.804638i $$0.297641\pi$$
$$920$$ 0 0
$$921$$ 36.0000 1.18624
$$922$$ 0 0
$$923$$ −24.0000 −0.789970
$$924$$ 0 0
$$925$$ 10.0000 0.328798
$$926$$ 0 0
$$927$$ −4.00000 −0.131377
$$928$$ 0 0
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ −18.0000 −0.589926
$$932$$ 0 0
$$933$$ 56.0000 1.83336
$$934$$ 0 0
$$935$$ 8.00000 0.261628
$$936$$ 0 0
$$937$$ 46.0000 1.50275 0.751377 0.659873i $$-0.229390\pi$$
0.751377 + 0.659873i $$0.229390\pi$$
$$938$$ 0 0
$$939$$ −20.0000 −0.652675
$$940$$ 0 0
$$941$$ 38.0000 1.23876 0.619382 0.785090i $$-0.287383\pi$$
0.619382 + 0.785090i $$0.287383\pi$$
$$942$$ 0 0
$$943$$ −24.0000 −0.781548
$$944$$ 0 0
$$945$$ 32.0000 1.04096
$$946$$ 0 0
$$947$$ 14.0000 0.454939 0.227469 0.973785i $$-0.426955\pi$$
0.227469 + 0.973785i $$0.426955\pi$$
$$948$$ 0 0
$$949$$ −28.0000 −0.908918
$$950$$ 0 0
$$951$$ −12.0000 −0.389127
$$952$$ 0 0
$$953$$ 58.0000 1.87880 0.939402 0.342817i $$-0.111381\pi$$
0.939402 + 0.342817i $$0.111381\pi$$
$$954$$ 0 0
$$955$$ 32.0000 1.03550
$$956$$ 0 0
$$957$$ −24.0000 −0.775810
$$958$$ 0 0
$$959$$ −40.0000 −1.29167
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 2.00000 0.0644491
$$964$$ 0 0
$$965$$ 4.00000 0.128765
$$966$$ 0 0
$$967$$ 28.0000 0.900419 0.450210 0.892923i $$-0.351349\pi$$
0.450210 + 0.892923i $$0.351349\pi$$
$$968$$ 0 0
$$969$$ −8.00000 −0.256997
$$970$$ 0 0
$$971$$ −38.0000 −1.21948 −0.609739 0.792602i $$-0.708726\pi$$
−0.609739 + 0.792602i $$0.708726\pi$$
$$972$$ 0 0
$$973$$ −40.0000 −1.28234
$$974$$ 0 0
$$975$$ −4.00000 −0.128103
$$976$$ 0 0
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 0 0
$$979$$ −4.00000 −0.127841
$$980$$ 0 0
$$981$$ 6.00000 0.191565
$$982$$ 0 0
$$983$$ 20.0000 0.637901 0.318950 0.947771i $$-0.396670\pi$$
0.318950 + 0.947771i $$0.396670\pi$$
$$984$$ 0 0
$$985$$ −28.0000 −0.892154
$$986$$ 0 0
$$987$$ −64.0000 −2.03714
$$988$$ 0 0
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 0 0
$$993$$ 28.0000 0.888553
$$994$$ 0 0
$$995$$ 8.00000 0.253617
$$996$$ 0 0
$$997$$ 54.0000 1.71020 0.855099 0.518465i $$-0.173497\pi$$
0.855099 + 0.518465i $$0.173497\pi$$
$$998$$ 0 0
$$999$$ −40.0000 −1.26554
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 128.2.a.a.1.1 1
3.2 odd 2 1152.2.a.m.1.1 1
4.3 odd 2 128.2.a.c.1.1 yes 1
5.2 odd 4 3200.2.c.l.2049.2 2
5.3 odd 4 3200.2.c.l.2049.1 2
5.4 even 2 3200.2.a.x.1.1 1
7.6 odd 2 6272.2.a.h.1.1 1
8.3 odd 2 128.2.a.b.1.1 yes 1
8.5 even 2 128.2.a.d.1.1 yes 1
12.11 even 2 1152.2.a.r.1.1 1
16.3 odd 4 256.2.b.a.129.2 2
16.5 even 4 256.2.b.c.129.2 2
16.11 odd 4 256.2.b.a.129.1 2
16.13 even 4 256.2.b.c.129.1 2
20.3 even 4 3200.2.c.f.2049.2 2
20.7 even 4 3200.2.c.f.2049.1 2
20.19 odd 2 3200.2.a.e.1.1 1
24.5 odd 2 1152.2.a.c.1.1 1
24.11 even 2 1152.2.a.h.1.1 1
28.27 even 2 6272.2.a.b.1.1 1
32.3 odd 8 1024.2.e.m.257.2 4
32.5 even 8 1024.2.e.i.769.2 4
32.11 odd 8 1024.2.e.m.769.2 4
32.13 even 8 1024.2.e.i.257.2 4
32.19 odd 8 1024.2.e.m.257.1 4
32.21 even 8 1024.2.e.i.769.1 4
32.27 odd 8 1024.2.e.m.769.1 4
32.29 even 8 1024.2.e.i.257.1 4
40.3 even 4 3200.2.c.k.2049.1 2
40.13 odd 4 3200.2.c.e.2049.2 2
40.19 odd 2 3200.2.a.u.1.1 1
40.27 even 4 3200.2.c.k.2049.2 2
40.29 even 2 3200.2.a.h.1.1 1
40.37 odd 4 3200.2.c.e.2049.1 2
48.5 odd 4 2304.2.d.r.1153.2 2
48.11 even 4 2304.2.d.b.1153.2 2
48.29 odd 4 2304.2.d.r.1153.1 2
48.35 even 4 2304.2.d.b.1153.1 2
56.13 odd 2 6272.2.a.a.1.1 1
56.27 even 2 6272.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.a.a.1.1 1 1.1 even 1 trivial
128.2.a.b.1.1 yes 1 8.3 odd 2
128.2.a.c.1.1 yes 1 4.3 odd 2
128.2.a.d.1.1 yes 1 8.5 even 2
256.2.b.a.129.1 2 16.11 odd 4
256.2.b.a.129.2 2 16.3 odd 4
256.2.b.c.129.1 2 16.13 even 4
256.2.b.c.129.2 2 16.5 even 4
1024.2.e.i.257.1 4 32.29 even 8
1024.2.e.i.257.2 4 32.13 even 8
1024.2.e.i.769.1 4 32.21 even 8
1024.2.e.i.769.2 4 32.5 even 8
1024.2.e.m.257.1 4 32.19 odd 8
1024.2.e.m.257.2 4 32.3 odd 8
1024.2.e.m.769.1 4 32.27 odd 8
1024.2.e.m.769.2 4 32.11 odd 8
1152.2.a.c.1.1 1 24.5 odd 2
1152.2.a.h.1.1 1 24.11 even 2
1152.2.a.m.1.1 1 3.2 odd 2
1152.2.a.r.1.1 1 12.11 even 2
2304.2.d.b.1153.1 2 48.35 even 4
2304.2.d.b.1153.2 2 48.11 even 4
2304.2.d.r.1153.1 2 48.29 odd 4
2304.2.d.r.1153.2 2 48.5 odd 4
3200.2.a.e.1.1 1 20.19 odd 2
3200.2.a.h.1.1 1 40.29 even 2
3200.2.a.u.1.1 1 40.19 odd 2
3200.2.a.x.1.1 1 5.4 even 2
3200.2.c.e.2049.1 2 40.37 odd 4
3200.2.c.e.2049.2 2 40.13 odd 4
3200.2.c.f.2049.1 2 20.7 even 4
3200.2.c.f.2049.2 2 20.3 even 4
3200.2.c.k.2049.1 2 40.3 even 4
3200.2.c.k.2049.2 2 40.27 even 4
3200.2.c.l.2049.1 2 5.3 odd 4
3200.2.c.l.2049.2 2 5.2 odd 4
6272.2.a.a.1.1 1 56.13 odd 2
6272.2.a.b.1.1 1 28.27 even 2
6272.2.a.g.1.1 1 56.27 even 2
6272.2.a.h.1.1 1 7.6 odd 2